LMFDB - Embedded newform 425.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,2,Mod(1,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.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:	$3.39364208590$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.73205 q^{2} +0.732051 q^{3} +1.00000 q^{4} -1.26795 q^{6} -0.732051 q^{7} +1.73205 q^{8} -2.46410 q^{9} +1.26795 q^{11} +0.732051 q^{12} +4.00000 q^{13} +1.26795 q^{14} -5.00000 q^{16} +1.00000 q^{17} +4.26795 q^{18} +5.46410 q^{19} -0.535898 q^{21} -2.19615 q^{22} -2.19615 q^{23} +1.26795 q^{24} -6.92820 q^{26} -4.00000 q^{27} -0.732051 q^{28} +3.46410 q^{29} +6.73205 q^{31} +5.19615 q^{32} +0.928203 q^{33} -1.73205 q^{34} -2.46410 q^{36} +7.46410 q^{37} -9.46410 q^{38} +2.92820 q^{39} +3.46410 q^{41} +0.928203 q^{42} +7.46410 q^{43} +1.26795 q^{44} +3.80385 q^{46} +0.928203 q^{47} -3.66025 q^{48} -6.46410 q^{49} +0.732051 q^{51} +4.00000 q^{52} -6.00000 q^{53} +6.92820 q^{54} -1.26795 q^{56} +4.00000 q^{57} -6.00000 q^{58} +9.46410 q^{59} +8.92820 q^{61} -11.6603 q^{62} +1.80385 q^{63} +1.00000 q^{64} -1.60770 q^{66} +10.0000 q^{67} +1.00000 q^{68} -1.60770 q^{69} -5.66025 q^{71} -4.26795 q^{72} +14.3923 q^{73} -12.9282 q^{74} +5.46410 q^{76} -0.928203 q^{77} -5.07180 q^{78} -16.5885 q^{79} +4.46410 q^{81} -6.00000 q^{82} -15.4641 q^{83} -0.535898 q^{84} -12.9282 q^{86} +2.53590 q^{87} +2.19615 q^{88} -16.3923 q^{89} -2.92820 q^{91} -2.19615 q^{92} +4.92820 q^{93} -1.60770 q^{94} +3.80385 q^{96} -8.92820 q^{97} +11.1962 q^{98} -3.12436 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{4} - 6 q^{6} + 2 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{12} + 8 q^{13} + 6 q^{14} - 10 q^{16} + 2 q^{17} + 12 q^{18} + 4 q^{19} - 8 q^{21} + 6 q^{22} + 6 q^{23} + 6 q^{24} - 8 q^{27} + 2 q^{28}+ \cdots + 18 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^4 - 6 * q^6 + 2 * q^7 + 2 * q^9 + 6 * q^11 - 2 * q^12 + 8 * q^13 + 6 * q^14 - 10 * q^16 + 2 * q^17 + 12 * q^18 + 4 * q^19 - 8 * q^21 + 6 * q^22 + 6 * q^23 + 6 * q^24 - 8 * q^27 + 2 * q^28 + 10 * q^31 - 12 * q^33 + 2 * q^36 + 8 * q^37 - 12 * q^38 - 8 * q^39 - 12 * q^42 + 8 * q^43 + 6 * q^44 + 18 * q^46 - 12 * q^47 + 10 * q^48 - 6 * q^49 - 2 * q^51 + 8 * q^52 - 12 * q^53 - 6 * q^56 + 8 * q^57 - 12 * q^58 + 12 * q^59 + 4 * q^61 - 6 * q^62 + 14 * q^63 + 2 * q^64 - 24 * q^66 + 20 * q^67 + 2 * q^68 - 24 * q^69 + 6 * q^71 - 12 * q^72 + 8 * q^73 - 12 * q^74 + 4 * q^76 + 12 * q^77 - 24 * q^78 - 2 * q^79 + 2 * q^81 - 12 * q^82 - 24 * q^83 - 8 * q^84 - 12 * q^86 + 12 * q^87 - 6 * q^88 - 12 * q^89 + 8 * q^91 + 6 * q^92 - 4 * q^93 - 24 * q^94 + 18 * q^96 - 4 * q^97 + 12 * q^98 + 18 * q^99

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 $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0.732051	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$3$	0.732051	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.26795	−0.517638
$7$	−0.732051	−0.276689	−0.138345	−	0.990384i	$-0.544178\pi$
$7$	−0.732051	−0.276689	−0.138345	+	0.990384i	$0.544178\pi$
$8$	1.73205	0.612372
$9$	−2.46410	−0.821367
$10$	0	0
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	0.732051	0.211325
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	1.26795	0.338874
$15$	0	0
$16$	−5.00000	−1.25000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	4.26795	1.00597
$19$	5.46410	1.25355	0.626775	−	0.779200i	$-0.284374\pi$
$19$	5.46410	1.25355	0.626775	+	0.779200i	$0.284374\pi$
$20$	0	0
$21$	−0.535898	−0.116943
$22$	−2.19615	−0.468221
$23$	−2.19615	−0.457929	−0.228965	−	0.973435i	$-0.573534\pi$
$23$	−2.19615	−0.457929	−0.228965	+	0.973435i	$0.573534\pi$
$24$	1.26795	0.258819
$25$	0	0
$26$	−6.92820	−1.35873
$27$	−4.00000	−0.769800
$28$	−0.732051	−0.138345
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	6.73205	1.20911	0.604556	−	0.796563i	$-0.293351\pi$
$31$	6.73205	1.20911	0.604556	+	0.796563i	$0.293351\pi$
$32$	5.19615	0.918559
$33$	0.928203	0.161579
$34$	−1.73205	−0.297044
$35$	0	0
$36$	−2.46410	−0.410684
$37$	7.46410	1.22709	0.613545	−	0.789659i	$-0.289743\pi$
$37$	7.46410	1.22709	0.613545	+	0.789659i	$0.289743\pi$
$38$	−9.46410	−1.53528
$39$	2.92820	0.468888
$40$	0	0
$41$	3.46410	0.541002	0.270501	−	0.962720i	$-0.412811\pi$
$41$	3.46410	0.541002	0.270501	+	0.962720i	$0.412811\pi$
$42$	0.928203	0.143225
$43$	7.46410	1.13826	0.569132	−	0.822246i	$-0.307279\pi$
$43$	7.46410	1.13826	0.569132	+	0.822246i	$0.307279\pi$
$44$	1.26795	0.191151
$45$	0	0
$46$	3.80385	0.560847
$47$	0.928203	0.135392	0.0676962	−	0.997706i	$-0.478435\pi$
$47$	0.928203	0.135392	0.0676962	+	0.997706i	$0.478435\pi$
$48$	−3.66025	−0.528312
$49$	−6.46410	−0.923443
$50$	0	0
$51$	0.732051	0.102508
$52$	4.00000	0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	6.92820	0.942809
$55$	0	0
$56$	−1.26795	−0.169437
$57$	4.00000	0.529813
$58$	−6.00000	−0.787839
$59$	9.46410	1.23212	0.616061	−	0.787699i	$-0.288728\pi$
$59$	9.46410	1.23212	0.616061	+	0.787699i	$0.288728\pi$
$60$	0	0
$61$	8.92820	1.14314	0.571570	−	0.820554i	$-0.306335\pi$
$61$	8.92820	1.14314	0.571570	+	0.820554i	$0.306335\pi$
$62$	−11.6603	−1.48085
$63$	1.80385	0.227263
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.60770	−0.197894
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	1.00000	0.121268
$69$	−1.60770	−0.193544
$70$	0	0
$71$	−5.66025	−0.671749	−0.335874	−	0.941907i	$-0.609032\pi$
$71$	−5.66025	−0.671749	−0.335874	+	0.941907i	$0.609032\pi$
$72$	−4.26795	−0.502983
$73$	14.3923	1.68449	0.842246	−	0.539093i	$-0.181233\pi$
$73$	14.3923	1.68449	0.842246	+	0.539093i	$0.181233\pi$
$74$	−12.9282	−1.50287
$75$	0	0
$76$	5.46410	0.626775
$77$	−0.928203	−0.105779
$78$	−5.07180	−0.574268
$79$	−16.5885	−1.86635	−0.933174	−	0.359426i	$-0.882973\pi$
$79$	−16.5885	−1.86635	−0.933174	+	0.359426i	$0.882973\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	−6.00000	−0.662589
$83$	−15.4641	−1.69741	−0.848703	−	0.528870i	$-0.822616\pi$
$83$	−15.4641	−1.69741	−0.848703	+	0.528870i	$0.822616\pi$
$84$	−0.535898	−0.0584713
$85$	0	0
$86$	−12.9282	−1.39408
$87$	2.53590	0.271877
$88$	2.19615	0.234111
$89$	−16.3923	−1.73758	−0.868790	−	0.495180i	$-0.835102\pi$
$89$	−16.3923	−1.73758	−0.868790	+	0.495180i	$0.835102\pi$
$90$	0	0
$91$	−2.92820	−0.306959
$92$	−2.19615	−0.228965
$93$	4.92820	0.511031
$94$	−1.60770	−0.165821
$95$	0	0
$96$	3.80385	0.388229
$97$	−8.92820	−0.906522	−0.453261	−	0.891378i	$-0.649739\pi$
$97$	−8.92820	−0.906522	−0.453261	+	0.891378i	$0.649739\pi$
$98$	11.1962	1.13098
$99$	−3.12436	−0.314010
$100$	0	0
$101$	−2.53590	−0.252331	−0.126166	−	0.992009i	$-0.540267\pi$
$101$	−2.53590	−0.252331	−0.126166	+	0.992009i	$0.540267\pi$
$102$	−1.26795	−0.125546
$103$	4.92820	0.485590	0.242795	−	0.970078i	$-0.421936\pi$
$103$	4.92820	0.485590	0.242795	+	0.970078i	$0.421936\pi$
$104$	6.92820	0.679366
$105$	0	0
$106$	10.3923	1.00939
$107$	−0.339746	−0.0328445	−0.0164222	−	0.999865i	$-0.505228\pi$
$107$	−0.339746	−0.0328445	−0.0164222	+	0.999865i	$0.505228\pi$
$108$	−4.00000	−0.384900
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	5.46410	0.518630
$112$	3.66025	0.345861
$113$	−17.3205	−1.62938	−0.814688	−	0.579899i	$-0.803092\pi$
$113$	−17.3205	−1.62938	−0.814688	+	0.579899i	$0.803092\pi$
$114$	−6.92820	−0.648886
$115$	0	0
$116$	3.46410	0.321634
$117$	−9.85641	−0.911225
$118$	−16.3923	−1.50903
$119$	−0.732051	−0.0671070
$120$	0	0
$121$	−9.39230	−0.853846
$122$	−15.4641	−1.40005
$123$	2.53590	0.228654
$124$	6.73205	0.604556
$125$	0	0
$126$	−3.12436	−0.278340
$127$	−6.39230	−0.567225	−0.283613	−	0.958939i	$-0.591533\pi$
$127$	−6.39230	−0.567225	−0.283613	+	0.958939i	$0.591533\pi$
$128$	−12.1244	−1.07165
$129$	5.46410	0.481087
$130$	0	0
$131$	8.19615	0.716101	0.358051	−	0.933702i	$-0.383442\pi$
$131$	8.19615	0.716101	0.358051	+	0.933702i	$0.383442\pi$
$132$	0.928203	0.0807897
$133$	−4.00000	−0.346844
$134$	−17.3205	−1.49626
$135$	0	0
$136$	1.73205	0.148522
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	2.78461	0.237042
$139$	13.6603	1.15865	0.579324	−	0.815097i	$-0.303317\pi$
$139$	13.6603	1.15865	0.579324	+	0.815097i	$0.303317\pi$
$140$	0	0
$141$	0.679492	0.0572235
$142$	9.80385	0.822721
$143$	5.07180	0.424125
$144$	12.3205	1.02671
$145$	0	0
$146$	−24.9282	−2.06307
$147$	−4.73205	−0.390293
$148$	7.46410	0.613545
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	5.46410	0.444662	0.222331	−	0.974971i	$-0.428633\pi$
$151$	5.46410	0.444662	0.222331	+	0.974971i	$0.428633\pi$
$152$	9.46410	0.767640
$153$	−2.46410	−0.199211
$154$	1.60770	0.129552
$155$	0	0
$156$	2.92820	0.234444
$157$	4.92820	0.393313	0.196657	−	0.980472i	$-0.436992\pi$
$157$	4.92820	0.393313	0.196657	+	0.980472i	$0.436992\pi$
$158$	28.7321	2.28580
$159$	−4.39230	−0.348332
$160$	0	0
$161$	1.60770	0.126704
$162$	−7.73205	−0.607487
$163$	−10.1962	−0.798624	−0.399312	−	0.916815i	$-0.630751\pi$
$163$	−10.1962	−0.798624	−0.399312	+	0.916815i	$0.630751\pi$
$164$	3.46410	0.270501
$165$	0	0
$166$	26.7846	2.07889
$167$	18.5885	1.43842	0.719209	−	0.694794i	$-0.244504\pi$
$167$	18.5885	1.43842	0.719209	+	0.694794i	$0.244504\pi$
$168$	−0.928203	−0.0716124
$169$	3.00000	0.230769
$170$	0	0
$171$	−13.4641	−1.02963
$172$	7.46410	0.569132
$173$	3.46410	0.263371	0.131685	−	0.991292i	$-0.457961\pi$
$173$	3.46410	0.263371	0.131685	+	0.991292i	$0.457961\pi$
$174$	−4.39230	−0.332980
$175$	0	0
$176$	−6.33975	−0.477876
$177$	6.92820	0.520756
$178$	28.3923	2.12809
$179$	23.3205	1.74306	0.871528	−	0.490345i	$-0.163129\pi$
$179$	23.3205	1.74306	0.871528	+	0.490345i	$0.163129\pi$
$180$	0	0
$181$	18.3923	1.36709	0.683545	−	0.729909i	$-0.260437\pi$
$181$	18.3923	1.36709	0.683545	+	0.729909i	$0.260437\pi$
$182$	5.07180	0.375947
$183$	6.53590	0.483148
$184$	−3.80385	−0.280423
$185$	0	0
$186$	−8.53590	−0.625882
$187$	1.26795	0.0927216
$188$	0.928203	0.0676962
$189$	2.92820	0.212995
$190$	0	0
$191$	−25.8564	−1.87090	−0.935452	−	0.353454i	$-0.885007\pi$
$191$	−25.8564	−1.87090	−0.935452	+	0.353454i	$0.885007\pi$
$192$	0.732051	0.0528312
$193$	−23.4641	−1.68898	−0.844491	−	0.535569i	$-0.820097\pi$
$193$	−23.4641	−1.68898	−0.844491	+	0.535569i	$0.820097\pi$
$194$	15.4641	1.11026
$195$	0	0
$196$	−6.46410	−0.461722
$197$	17.3205	1.23404	0.617018	−	0.786949i	$-0.288341\pi$
$197$	17.3205	1.23404	0.617018	+	0.786949i	$0.288341\pi$
$198$	5.41154	0.384582
$199$	−0.196152	−0.0139049	−0.00695244	−	0.999976i	$-0.502213\pi$
$199$	−0.196152	−0.0139049	−0.00695244	+	0.999976i	$0.502213\pi$
$200$	0	0
$201$	7.32051	0.516349
$202$	4.39230	0.309041
$203$	−2.53590	−0.177985
$204$	0.732051	0.0512538
$205$	0	0
$206$	−8.53590	−0.594724
$207$	5.41154	0.376128
$208$	−20.0000	−1.38675
$209$	6.92820	0.479234
$210$	0	0
$211$	−0.196152	−0.0135037	−0.00675184	−	0.999977i	$-0.502149\pi$
$211$	−0.196152	−0.0135037	−0.00675184	+	0.999977i	$0.502149\pi$
$212$	−6.00000	−0.412082
$213$	−4.14359	−0.283914
$214$	0.588457	0.0402261
$215$	0	0
$216$	−6.92820	−0.471405
$217$	−4.92820	−0.334548
$218$	17.3205	1.17309
$219$	10.5359	0.711950
$220$	0	0
$221$	4.00000	0.269069
$222$	−9.46410	−0.635189
$223$	5.60770	0.375519	0.187760	−	0.982215i	$-0.439877\pi$
$223$	5.60770	0.375519	0.187760	+	0.982215i	$0.439877\pi$
$224$	−3.80385	−0.254155
$225$	0	0
$226$	30.0000	1.99557
$227$	−19.2679	−1.27886	−0.639429	−	0.768850i	$-0.720829\pi$
$227$	−19.2679	−1.27886	−0.639429	+	0.768850i	$0.720829\pi$
$228$	4.00000	0.264906
$229$	12.3923	0.818907	0.409453	−	0.912331i	$-0.365719\pi$
$229$	12.3923	0.818907	0.409453	+	0.912331i	$0.365719\pi$
$230$	0	0
$231$	−0.679492	−0.0447073
$232$	6.00000	0.393919
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	17.0718	1.11602
$235$	0	0
$236$	9.46410	0.616061
$237$	−12.1436	−0.788811
$238$	1.26795	0.0821889
$239$	20.7846	1.34444	0.672222	−	0.740349i	$-0.265340\pi$
$239$	20.7846	1.34444	0.672222	+	0.740349i	$0.265340\pi$
$240$	0	0
$241$	−26.3923	−1.70008	−0.850039	−	0.526720i	$-0.823422\pi$
$241$	−26.3923	−1.70008	−0.850039	+	0.526720i	$0.823422\pi$
$242$	16.2679	1.04574
$243$	15.2679	0.979439
$244$	8.92820	0.571570
$245$	0	0
$246$	−4.39230	−0.280043
$247$	21.8564	1.39069
$248$	11.6603	0.740427
$249$	−11.3205	−0.717408
$250$	0	0
$251$	−6.92820	−0.437304	−0.218652	−	0.975803i	$-0.570166\pi$
$251$	−6.92820	−0.437304	−0.218652	+	0.975803i	$0.570166\pi$
$252$	1.80385	0.113632
$253$	−2.78461	−0.175067
$254$	11.0718	0.694706
$255$	0	0
$256$	19.0000	1.18750
$257$	−6.92820	−0.432169	−0.216085	−	0.976375i	$-0.569329\pi$
$257$	−6.92820	−0.432169	−0.216085	+	0.976375i	$0.569329\pi$
$258$	−9.46410	−0.589209
$259$	−5.46410	−0.339523
$260$	0	0
$261$	−8.53590	−0.528359
$262$	−14.1962	−0.877041
$263$	22.3923	1.38077	0.690384	−	0.723443i	$-0.257441\pi$
$263$	22.3923	1.38077	0.690384	+	0.723443i	$0.257441\pi$
$264$	1.60770	0.0989468
$265$	0	0
$266$	6.92820	0.424795
$267$	−12.0000	−0.734388
$268$	10.0000	0.610847
$269$	−12.9282	−0.788246	−0.394123	−	0.919058i	$-0.628952\pi$
$269$	−12.9282	−0.788246	−0.394123	+	0.919058i	$0.628952\pi$
$270$	0	0
$271$	−10.9282	−0.663841	−0.331921	−	0.943307i	$-0.607697\pi$
$271$	−10.9282	−0.663841	−0.331921	+	0.943307i	$0.607697\pi$
$272$	−5.00000	−0.303170
$273$	−2.14359	−0.129736
$274$	0	0
$275$	0	0
$276$	−1.60770	−0.0967719
$277$	−7.07180	−0.424903	−0.212452	−	0.977172i	$-0.568145\pi$
$277$	−7.07180	−0.424903	−0.212452	+	0.977172i	$0.568145\pi$
$278$	−23.6603	−1.41905
$279$	−16.5885	−0.993125
$280$	0	0
$281$	0.928203	0.0553720	0.0276860	−	0.999617i	$-0.491186\pi$
$281$	0.928203	0.0553720	0.0276860	+	0.999617i	$0.491186\pi$
$282$	−1.17691	−0.0700842
$283$	8.73205	0.519067	0.259533	−	0.965734i	$-0.416431\pi$
$283$	8.73205	0.519067	0.259533	+	0.965734i	$0.416431\pi$
$284$	−5.66025	−0.335874
$285$	0	0
$286$	−8.78461	−0.519445
$287$	−2.53590	−0.149689
$288$	−12.8038	−0.754474
$289$	1.00000	0.0588235
$290$	0	0
$291$	−6.53590	−0.383141
$292$	14.3923	0.842246
$293$	−12.9282	−0.755274	−0.377637	−	0.925954i	$-0.623263\pi$
$293$	−12.9282	−0.755274	−0.377637	+	0.925954i	$0.623263\pi$
$294$	8.19615	0.478009
$295$	0	0
$296$	12.9282	0.751437
$297$	−5.07180	−0.294295
$298$	10.3923	0.602010
$299$	−8.78461	−0.508027
$300$	0	0
$301$	−5.46410	−0.314946
$302$	−9.46410	−0.544598
$303$	−1.85641	−0.106648
$304$	−27.3205	−1.56694
$305$	0	0
$306$	4.26795	0.243982
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	−0.928203	−0.0528893
$309$	3.60770	0.205235
$310$	0	0
$311$	22.0526	1.25049	0.625243	−	0.780430i	$-0.285000\pi$
$311$	22.0526	1.25049	0.625243	+	0.780430i	$0.285000\pi$
$312$	5.07180	0.287134
$313$	5.60770	0.316966	0.158483	−	0.987362i	$-0.449340\pi$
$313$	5.60770	0.316966	0.158483	+	0.987362i	$0.449340\pi$
$314$	−8.53590	−0.481709
$315$	0	0
$316$	−16.5885	−0.933174
$317$	11.0718	0.621854	0.310927	−	0.950434i	$-0.399361\pi$
$317$	11.0718	0.621854	0.310927	+	0.950434i	$0.399361\pi$
$318$	7.60770	0.426618
$319$	4.39230	0.245922
$320$	0	0
$321$	−0.248711	−0.0138817
$322$	−2.78461	−0.155180
$323$	5.46410	0.304031
$324$	4.46410	0.248006
$325$	0	0
$326$	17.6603	0.978111
$327$	−7.32051	−0.404825
$328$	6.00000	0.331295
$329$	−0.679492	−0.0374616
$330$	0	0
$331$	−13.4641	−0.740054	−0.370027	−	0.929021i	$-0.620652\pi$
$331$	−13.4641	−0.740054	−0.370027	+	0.929021i	$0.620652\pi$
$332$	−15.4641	−0.848703
$333$	−18.3923	−1.00789
$334$	−32.1962	−1.76170
$335$	0	0
$336$	2.67949	0.146178
$337$	−34.7846	−1.89484	−0.947419	−	0.319995i	$-0.896319\pi$
$337$	−34.7846	−1.89484	−0.947419	+	0.319995i	$0.896319\pi$
$338$	−5.19615	−0.282633
$339$	−12.6795	−0.688655
$340$	0	0
$341$	8.53590	0.462245
$342$	23.3205	1.26103
$343$	9.85641	0.532196
$344$	12.9282	0.697042
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−14.1962	−0.762089	−0.381045	−	0.924557i	$-0.624436\pi$
$347$	−14.1962	−0.762089	−0.381045	+	0.924557i	$0.624436\pi$
$348$	2.53590	0.135938
$349$	−30.7846	−1.64786	−0.823931	−	0.566690i	$-0.808224\pi$
$349$	−30.7846	−1.64786	−0.823931	+	0.566690i	$0.808224\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	6.58846	0.351166
$353$	14.7846	0.786905	0.393453	−	0.919345i	$-0.371281\pi$
$353$	14.7846	0.786905	0.393453	+	0.919345i	$0.371281\pi$
$354$	−12.0000	−0.637793
$355$	0	0
$356$	−16.3923	−0.868790
$357$	−0.535898	−0.0283628
$358$	−40.3923	−2.13480
$359$	−14.5359	−0.767175	−0.383588	−	0.923504i	$-0.625312\pi$
$359$	−14.5359	−0.767175	−0.383588	+	0.923504i	$0.625312\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	−31.8564	−1.67434
$363$	−6.87564	−0.360878
$364$	−2.92820	−0.153480
$365$	0	0
$366$	−11.3205	−0.591732
$367$	18.1962	0.949831	0.474916	−	0.880031i	$-0.342479\pi$
$367$	18.1962	0.949831	0.474916	+	0.880031i	$0.342479\pi$
$368$	10.9808	0.572412
$369$	−8.53590	−0.444361
$370$	0	0
$371$	4.39230	0.228037
$372$	4.92820	0.255515
$373$	−20.0000	−1.03556	−0.517780	−	0.855514i	$-0.673242\pi$
$373$	−20.0000	−1.03556	−0.517780	+	0.855514i	$0.673242\pi$
$374$	−2.19615	−0.113560
$375$	0	0
$376$	1.60770	0.0829105
$377$	13.8564	0.713641
$378$	−5.07180	−0.260865
$379$	28.1962	1.44834	0.724170	−	0.689622i	$-0.242223\pi$
$379$	28.1962	1.44834	0.724170	+	0.689622i	$0.242223\pi$
$380$	0	0
$381$	−4.67949	−0.239738
$382$	44.7846	2.29138
$383$	−8.53590	−0.436164	−0.218082	−	0.975930i	$-0.569980\pi$
$383$	−8.53590	−0.436164	−0.218082	+	0.975930i	$0.569980\pi$
$384$	−8.87564	−0.452933
$385$	0	0
$386$	40.6410	2.06857
$387$	−18.3923	−0.934933
$388$	−8.92820	−0.453261
$389$	16.3923	0.831123	0.415561	−	0.909565i	$-0.363585\pi$
$389$	16.3923	0.831123	0.415561	+	0.909565i	$0.363585\pi$
$390$	0	0
$391$	−2.19615	−0.111064
$392$	−11.1962	−0.565491
$393$	6.00000	0.302660
$394$	−30.0000	−1.51138
$395$	0	0
$396$	−3.12436	−0.157005
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0.339746	0.0170299
$399$	−2.92820	−0.146594
$400$	0	0
$401$	−0.928203	−0.0463523	−0.0231761	−	0.999731i	$-0.507378\pi$
$401$	−0.928203	−0.0463523	−0.0231761	+	0.999731i	$0.507378\pi$
$402$	−12.6795	−0.632396
$403$	26.9282	1.34139
$404$	−2.53590	−0.126166
$405$	0	0
$406$	4.39230	0.217986
$407$	9.46410	0.469118
$408$	1.26795	0.0627728
$409$	26.0000	1.28562	0.642809	−	0.766027i	$-0.277769\pi$
$409$	26.0000	1.28562	0.642809	+	0.766027i	$0.277769\pi$
$410$	0	0
$411$	0	0
$412$	4.92820	0.242795
$413$	−6.92820	−0.340915
$414$	−9.37307	−0.460661
$415$	0	0
$416$	20.7846	1.01905
$417$	10.0000	0.489702
$418$	−12.0000	−0.586939
$419$	27.8038	1.35831	0.679153	−	0.733996i	$-0.262347\pi$
$419$	27.8038	1.35831	0.679153	+	0.733996i	$0.262347\pi$
$420$	0	0
$421$	−1.46410	−0.0713559	−0.0356780	−	0.999363i	$-0.511359\pi$
$421$	−1.46410	−0.0713559	−0.0356780	+	0.999363i	$0.511359\pi$
$422$	0.339746	0.0165386
$423$	−2.28719	−0.111207
$424$	−10.3923	−0.504695
$425$	0	0
$426$	7.17691	0.347723
$427$	−6.53590	−0.316294
$428$	−0.339746	−0.0164222
$429$	3.71281	0.179256
$430$	0	0
$431$	20.1962	0.972814	0.486407	−	0.873732i	$-0.338307\pi$
$431$	20.1962	0.972814	0.486407	+	0.873732i	$0.338307\pi$
$432$	20.0000	0.962250
$433$	−9.85641	−0.473669	−0.236834	−	0.971550i	$-0.576110\pi$
$433$	−9.85641	−0.473669	−0.236834	+	0.971550i	$0.576110\pi$
$434$	8.53590	0.409736
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−12.0000	−0.574038
$438$	−18.2487	−0.871957
$439$	18.0526	0.861602	0.430801	−	0.902447i	$-0.358231\pi$
$439$	18.0526	0.861602	0.430801	+	0.902447i	$0.358231\pi$
$440$	0	0
$441$	15.9282	0.758486
$442$	−6.92820	−0.329541
$443$	−0.928203	−0.0441003	−0.0220501	−	0.999757i	$-0.507019\pi$
$443$	−0.928203	−0.0441003	−0.0220501	+	0.999757i	$0.507019\pi$
$444$	5.46410	0.259315
$445$	0	0
$446$	−9.71281	−0.459915
$447$	−4.39230	−0.207749
$448$	−0.732051	−0.0345861
$449$	−13.6077	−0.642187	−0.321093	−	0.947048i	$-0.604050\pi$
$449$	−13.6077	−0.642187	−0.321093	+	0.947048i	$0.604050\pi$
$450$	0	0
$451$	4.39230	0.206826
$452$	−17.3205	−0.814688
$453$	4.00000	0.187936
$454$	33.3731	1.56628
$455$	0	0
$456$	6.92820	0.324443
$457$	−4.78461	−0.223815	−0.111907	−	0.993719i	$-0.535696\pi$
$457$	−4.78461	−0.223815	−0.111907	+	0.993719i	$0.535696\pi$
$458$	−21.4641	−1.00295
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−11.0718	−0.515665	−0.257832	−	0.966190i	$-0.583008\pi$
$461$	−11.0718	−0.515665	−0.257832	+	0.966190i	$0.583008\pi$
$462$	1.17691	0.0547550
$463$	−3.85641	−0.179222	−0.0896112	−	0.995977i	$-0.528562\pi$
$463$	−3.85641	−0.179222	−0.0896112	+	0.995977i	$0.528562\pi$
$464$	−17.3205	−0.804084
$465$	0	0
$466$	10.3923	0.481414
$467$	−22.3923	−1.03619	−0.518096	−	0.855322i	$-0.673359\pi$
$467$	−22.3923	−1.03619	−0.518096	+	0.855322i	$0.673359\pi$
$468$	−9.85641	−0.455613
$469$	−7.32051	−0.338030
$470$	0	0
$471$	3.60770	0.166234
$472$	16.3923	0.754517
$473$	9.46410	0.435160
$474$	21.0333	0.966092
$475$	0	0
$476$	−0.732051	−0.0335535
$477$	14.7846	0.676941
$478$	−36.0000	−1.64660
$479$	5.66025	0.258624	0.129312	−	0.991604i	$-0.458723\pi$
$479$	5.66025	0.258624	0.129312	+	0.991604i	$0.458723\pi$
$480$	0	0
$481$	29.8564	1.36133
$482$	45.7128	2.08216
$483$	1.17691	0.0535515
$484$	−9.39230	−0.426923
$485$	0	0
$486$	−26.4449	−1.19956
$487$	26.9808	1.22262	0.611308	−	0.791393i	$-0.290644\pi$
$487$	26.9808	1.22262	0.611308	+	0.791393i	$0.290644\pi$
$488$	15.4641	0.700027
$489$	−7.46410	−0.337538
$490$	0	0
$491$	−40.3923	−1.82288	−0.911440	−	0.411434i	$-0.865028\pi$
$491$	−40.3923	−1.82288	−0.911440	+	0.411434i	$0.865028\pi$
$492$	2.53590	0.114327
$493$	3.46410	0.156015
$494$	−37.8564	−1.70324
$495$	0	0
$496$	−33.6603	−1.51139
$497$	4.14359	0.185866
$498$	19.6077	0.878642
$499$	1.66025	0.0743232	0.0371616	−	0.999309i	$-0.488168\pi$
$499$	1.66025	0.0743232	0.0371616	+	0.999309i	$0.488168\pi$
$500$	0	0
$501$	13.6077	0.607947
$502$	12.0000	0.535586
$503$	9.12436	0.406835	0.203417	−	0.979092i	$-0.434795\pi$
$503$	9.12436	0.406835	0.203417	+	0.979092i	$0.434795\pi$
$504$	3.12436	0.139170
$505$	0	0
$506$	4.82309	0.214412
$507$	2.19615	0.0975346
$508$	−6.39230	−0.283613
$509$	−7.85641	−0.348229	−0.174115	−	0.984725i	$-0.555706\pi$
$509$	−7.85641	−0.348229	−0.174115	+	0.984725i	$0.555706\pi$
$510$	0	0
$511$	−10.5359	−0.466081
$512$	−8.66025	−0.382733
$513$	−21.8564	−0.964984
$514$	12.0000	0.529297
$515$	0	0
$516$	5.46410	0.240544
$517$	1.17691	0.0517606
$518$	9.46410	0.415829
$519$	2.53590	0.111314
$520$	0	0
$521$	31.8564	1.39565	0.697827	−	0.716266i	$-0.254150\pi$
$521$	31.8564	1.39565	0.697827	+	0.716266i	$0.254150\pi$
$522$	14.7846	0.647105
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	8.19615	0.358051
$525$	0	0
$526$	−38.7846	−1.69109
$527$	6.73205	0.293253
$528$	−4.64102	−0.201974
$529$	−18.1769	−0.790301
$530$	0	0
$531$	−23.3205	−1.01202
$532$	−4.00000	−0.173422
$533$	13.8564	0.600188
$534$	20.7846	0.899438
$535$	0	0
$536$	17.3205	0.748132
$537$	17.0718	0.736702
$538$	22.3923	0.965401
$539$	−8.19615	−0.353033
$540$	0	0
$541$	−23.1769	−0.996453	−0.498227	−	0.867047i	$-0.666015\pi$
$541$	−23.1769	−0.996453	−0.498227	+	0.867047i	$0.666015\pi$
$542$	18.9282	0.813036
$543$	13.4641	0.577800
$544$	5.19615	0.222783
$545$	0	0
$546$	3.71281	0.158894
$547$	−25.9090	−1.10779	−0.553894	−	0.832587i	$-0.686859\pi$
$547$	−25.9090	−1.10779	−0.553894	+	0.832587i	$0.686859\pi$
$548$	0	0
$549$	−22.0000	−0.938937
$550$	0	0
$551$	18.9282	0.806369
$552$	−2.78461	−0.118521
$553$	12.1436	0.516398
$554$	12.2487	0.520398
$555$	0	0
$556$	13.6603	0.579324
$557$	−6.92820	−0.293557	−0.146779	−	0.989169i	$-0.546891\pi$
$557$	−6.92820	−0.293557	−0.146779	+	0.989169i	$0.546891\pi$
$558$	28.7321	1.21632
$559$	29.8564	1.26279
$560$	0	0
$561$	0.928203	0.0391888
$562$	−1.60770	−0.0678165
$563$	−20.5359	−0.865485	−0.432742	−	0.901518i	$-0.642454\pi$
$563$	−20.5359	−0.865485	−0.432742	+	0.901518i	$0.642454\pi$
$564$	0.679492	0.0286118
$565$	0	0
$566$	−15.1244	−0.635724
$567$	−3.26795	−0.137241
$568$	−9.80385	−0.411360
$569$	−28.6410	−1.20069	−0.600347	−	0.799740i	$-0.704971\pi$
$569$	−28.6410	−1.20069	−0.600347	+	0.799740i	$0.704971\pi$
$570$	0	0
$571$	22.4449	0.939288	0.469644	−	0.882856i	$-0.344382\pi$
$571$	22.4449	0.939288	0.469644	+	0.882856i	$0.344382\pi$
$572$	5.07180	0.212062
$573$	−18.9282	−0.790737
$574$	4.39230	0.183331
$575$	0	0
$576$	−2.46410	−0.102671
$577$	−30.6410	−1.27560	−0.637801	−	0.770201i	$-0.720156\pi$
$577$	−30.6410	−1.27560	−0.637801	+	0.770201i	$0.720156\pi$
$578$	−1.73205	−0.0720438
$579$	−17.1769	−0.713848
$580$	0	0
$581$	11.3205	0.469654
$582$	11.3205	0.469250
$583$	−7.60770	−0.315079
$584$	24.9282	1.03154
$585$	0	0
$586$	22.3923	0.925018
$587$	−25.6077	−1.05694	−0.528471	−	0.848951i	$-0.677235\pi$
$587$	−25.6077	−1.05694	−0.528471	+	0.848951i	$0.677235\pi$
$588$	−4.73205	−0.195146
$589$	36.7846	1.51568
$590$	0	0
$591$	12.6795	0.521565
$592$	−37.3205	−1.53386
$593$	7.85641	0.322624	0.161312	−	0.986903i	$-0.448427\pi$
$593$	7.85641	0.322624	0.161312	+	0.986903i	$0.448427\pi$
$594$	8.78461	0.360437
$595$	0	0
$596$	−6.00000	−0.245770
$597$	−0.143594	−0.00587689
$598$	15.2154	0.622204
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	−40.2487	−1.64178	−0.820890	−	0.571087i	$-0.806522\pi$
$601$	−40.2487	−1.64178	−0.820890	+	0.571087i	$0.806522\pi$
$602$	9.46410	0.385728
$603$	−24.6410	−1.00346
$604$	5.46410	0.222331
$605$	0	0
$606$	3.21539	0.130616
$607$	4.33975	0.176145	0.0880724	−	0.996114i	$-0.471929\pi$
$607$	4.33975	0.176145	0.0880724	+	0.996114i	$0.471929\pi$
$608$	28.3923	1.15146
$609$	−1.85641	−0.0752254
$610$	0	0
$611$	3.71281	0.150204
$612$	−2.46410	−0.0996054
$613$	11.8564	0.478876	0.239438	−	0.970912i	$-0.423037\pi$
$613$	11.8564	0.478876	0.239438	+	0.970912i	$0.423037\pi$
$614$	−17.3205	−0.698999
$615$	0	0
$616$	−1.60770	−0.0647759
$617$	20.5359	0.826744	0.413372	−	0.910562i	$-0.364351\pi$
$617$	20.5359	0.826744	0.413372	+	0.910562i	$0.364351\pi$
$618$	−6.24871	−0.251360
$619$	7.41154	0.297895	0.148948	−	0.988845i	$-0.452411\pi$
$619$	7.41154	0.297895	0.148948	+	0.988845i	$0.452411\pi$
$620$	0	0
$621$	8.78461	0.352514
$622$	−38.1962	−1.53153
$623$	12.0000	0.480770
$624$	−14.6410	−0.586110
$625$	0	0
$626$	−9.71281	−0.388202
$627$	5.07180	0.202548
$628$	4.92820	0.196657
$629$	7.46410	0.297613
$630$	0	0
$631$	−11.6077	−0.462095	−0.231048	−	0.972942i	$-0.574215\pi$
$631$	−11.6077	−0.462095	−0.231048	+	0.972942i	$0.574215\pi$
$632$	−28.7321	−1.14290
$633$	−0.143594	−0.00570733
$634$	−19.1769	−0.761613
$635$	0	0
$636$	−4.39230	−0.174166
$637$	−25.8564	−1.02447
$638$	−7.60770	−0.301192
$639$	13.9474	0.551752
$640$	0	0
$641$	−31.1769	−1.23141	−0.615707	−	0.787975i	$-0.711130\pi$
$641$	−31.1769	−1.23141	−0.615707	+	0.787975i	$0.711130\pi$
$642$	0.430781	0.0170016
$643$	13.8038	0.544371	0.272185	−	0.962245i	$-0.412254\pi$
$643$	13.8038	0.544371	0.272185	+	0.962245i	$0.412254\pi$
$644$	1.60770	0.0633521
$645$	0	0
$646$	−9.46410	−0.372360
$647$	−2.78461	−0.109474	−0.0547372	−	0.998501i	$-0.517432\pi$
$647$	−2.78461	−0.109474	−0.0547372	+	0.998501i	$0.517432\pi$
$648$	7.73205	0.303744
$649$	12.0000	0.471041
$650$	0	0
$651$	−3.60770	−0.141397
$652$	−10.1962	−0.399312
$653$	46.3923	1.81547	0.907736	−	0.419543i	$-0.137810\pi$
$653$	46.3923	1.81547	0.907736	+	0.419543i	$0.137810\pi$
$654$	12.6795	0.495807
$655$	0	0
$656$	−17.3205	−0.676252
$657$	−35.4641	−1.38359
$658$	1.17691	0.0458809
$659$	8.78461	0.342200	0.171100	−	0.985254i	$-0.445268\pi$
$659$	8.78461	0.342200	0.171100	+	0.985254i	$0.445268\pi$
$660$	0	0
$661$	−35.8564	−1.39465	−0.697326	−	0.716754i	$-0.745627\pi$
$661$	−35.8564	−1.39465	−0.697326	+	0.716754i	$0.745627\pi$
$662$	23.3205	0.906377
$663$	2.92820	0.113722
$664$	−26.7846	−1.03944
$665$	0	0
$666$	31.8564	1.23441
$667$	−7.60770	−0.294571
$668$	18.5885	0.719209
$669$	4.10512	0.158713
$670$	0	0
$671$	11.3205	0.437023
$672$	−2.78461	−0.107419
$673$	−16.5359	−0.637412	−0.318706	−	0.947854i	$-0.603248\pi$
$673$	−16.5359	−0.637412	−0.318706	+	0.947854i	$0.603248\pi$
$674$	60.2487	2.32069
$675$	0	0
$676$	3.00000	0.115385
$677$	38.7846	1.49061	0.745307	−	0.666722i	$-0.232303\pi$
$677$	38.7846	1.49061	0.745307	+	0.666722i	$0.232303\pi$
$678$	21.9615	0.843427
$679$	6.53590	0.250825
$680$	0	0
$681$	−14.1051	−0.540509
$682$	−14.7846	−0.566132
$683$	−48.8372	−1.86870	−0.934351	−	0.356354i	$-0.884020\pi$
$683$	−48.8372	−1.86870	−0.934351	+	0.356354i	$0.884020\pi$
$684$	−13.4641	−0.514813
$685$	0	0
$686$	−17.0718	−0.651804
$687$	9.07180	0.346111
$688$	−37.3205	−1.42283
$689$	−24.0000	−0.914327
$690$	0	0
$691$	12.9808	0.493811	0.246906	−	0.969040i	$-0.420586\pi$
$691$	12.9808	0.493811	0.246906	+	0.969040i	$0.420586\pi$
$692$	3.46410	0.131685
$693$	2.28719	0.0868831
$694$	24.5885	0.933365
$695$	0	0
$696$	4.39230	0.166490
$697$	3.46410	0.131212
$698$	53.3205	2.01821
$699$	−4.39230	−0.166132
$700$	0	0
$701$	−23.3205	−0.880803	−0.440402	−	0.897801i	$-0.645164\pi$
$701$	−23.3205	−0.880803	−0.440402	+	0.897801i	$0.645164\pi$
$702$	27.7128	1.04595
$703$	40.7846	1.53822
$704$	1.26795	0.0477876
$705$	0	0
$706$	−25.6077	−0.963758
$707$	1.85641	0.0698174
$708$	6.92820	0.260378
$709$	11.4641	0.430543	0.215272	−	0.976554i	$-0.430936\pi$
$709$	11.4641	0.430543	0.215272	+	0.976554i	$0.430936\pi$
$710$	0	0
$711$	40.8756	1.53296
$712$	−28.3923	−1.06405
$713$	−14.7846	−0.553688
$714$	0.928203	0.0347371
$715$	0	0
$716$	23.3205	0.871528
$717$	15.2154	0.568229
$718$	25.1769	0.939594
$719$	−36.5885	−1.36452	−0.682260	−	0.731110i	$-0.739003\pi$
$719$	−36.5885	−1.36452	−0.682260	+	0.731110i	$0.739003\pi$
$720$	0	0
$721$	−3.60770	−0.134358
$722$	−18.8038	−0.699807
$723$	−19.3205	−0.718537
$724$	18.3923	0.683545
$725$	0	0
$726$	11.9090	0.441983
$727$	−27.8564	−1.03314	−0.516568	−	0.856246i	$-0.672791\pi$
$727$	−27.8564	−1.03314	−0.516568	+	0.856246i	$0.672791\pi$
$728$	−5.07180	−0.187973
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	7.46410	0.276070
$732$	6.53590	0.241574
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	−31.5167	−1.16330
$735$	0	0
$736$	−11.4115	−0.420635
$737$	12.6795	0.467055
$738$	14.7846	0.544229
$739$	−32.3923	−1.19157	−0.595785	−	0.803144i	$-0.703159\pi$
$739$	−32.3923	−1.19157	−0.595785	+	0.803144i	$0.703159\pi$
$740$	0	0
$741$	16.0000	0.587775
$742$	−7.60770	−0.279287
$743$	19.2679	0.706872	0.353436	−	0.935459i	$-0.385013\pi$
$743$	19.2679	0.706872	0.353436	+	0.935459i	$0.385013\pi$
$744$	8.53590	0.312941
$745$	0	0
$746$	34.6410	1.26830
$747$	38.1051	1.39419
$748$	1.26795	0.0463608
$749$	0.248711	0.00908771
$750$	0	0
$751$	26.3397	0.961151	0.480575	−	0.876953i	$-0.340428\pi$
$751$	26.3397	0.961151	0.480575	+	0.876953i	$0.340428\pi$
$752$	−4.64102	−0.169240
$753$	−5.07180	−0.184827
$754$	−24.0000	−0.874028
$755$	0	0
$756$	2.92820	0.106498
$757$	38.6410	1.40443	0.702216	−	0.711964i	$-0.252194\pi$
$757$	38.6410	1.40443	0.702216	+	0.711964i	$0.252194\pi$
$758$	−48.8372	−1.77385
$759$	−2.03848	−0.0739920
$760$	0	0
$761$	−7.60770	−0.275779	−0.137889	−	0.990448i	$-0.544032\pi$
$761$	−7.60770	−0.275779	−0.137889	+	0.990448i	$0.544032\pi$
$762$	8.10512	0.293617
$763$	7.32051	0.265020
$764$	−25.8564	−0.935452
$765$	0	0
$766$	14.7846	0.534190
$767$	37.8564	1.36692
$768$	13.9090	0.501897
$769$	15.6077	0.562828	0.281414	−	0.959586i	$-0.409197\pi$
$769$	15.6077	0.562828	0.281414	+	0.959586i	$0.409197\pi$
$770$	0	0
$771$	−5.07180	−0.182656
$772$	−23.4641	−0.844491
$773$	−22.6410	−0.814341	−0.407170	−	0.913352i	$-0.633484\pi$
$773$	−22.6410	−0.814341	−0.407170	+	0.913352i	$0.633484\pi$
$774$	31.8564	1.14505
$775$	0	0
$776$	−15.4641	−0.555129
$777$	−4.00000	−0.143499
$778$	−28.3923	−1.01791
$779$	18.9282	0.678173
$780$	0	0
$781$	−7.17691	−0.256810
$782$	3.80385	0.136025
$783$	−13.8564	−0.495188
$784$	32.3205	1.15430
$785$	0	0
$786$	−10.3923	−0.370681
$787$	21.9090	0.780970	0.390485	−	0.920609i	$-0.372307\pi$
$787$	21.9090	0.780970	0.390485	+	0.920609i	$0.372307\pi$
$788$	17.3205	0.617018
$789$	16.3923	0.583582
$790$	0	0
$791$	12.6795	0.450831
$792$	−5.41154	−0.192291
$793$	35.7128	1.26820
$794$	24.2487	0.860555
$795$	0	0
$796$	−0.196152	−0.00695244
$797$	11.0718	0.392183	0.196092	−	0.980586i	$-0.437175\pi$
$797$	11.0718	0.392183	0.196092	+	0.980586i	$0.437175\pi$
$798$	5.07180	0.179540
$799$	0.928203	0.0328375
$800$	0	0
$801$	40.3923	1.42719
$802$	1.60770	0.0567697
$803$	18.2487	0.643983
$804$	7.32051	0.258174
$805$	0	0
$806$	−46.6410	−1.64286
$807$	−9.46410	−0.333152
$808$	−4.39230	−0.154521
$809$	−28.1436	−0.989476	−0.494738	−	0.869042i	$-0.664736\pi$
$809$	−28.1436	−0.989476	−0.494738	+	0.869042i	$0.664736\pi$
$810$	0	0
$811$	−34.8372	−1.22330	−0.611649	−	0.791129i	$-0.709494\pi$
$811$	−34.8372	−1.22330	−0.611649	+	0.791129i	$0.709494\pi$
$812$	−2.53590	−0.0889926
$813$	−8.00000	−0.280572
$814$	−16.3923	−0.574550
$815$	0	0
$816$	−3.66025	−0.128135
$817$	40.7846	1.42687
$818$	−45.0333	−1.57455
$819$	7.21539	0.252126
$820$	0	0
$821$	−11.0718	−0.386408	−0.193204	−	0.981159i	$-0.561888\pi$
$821$	−11.0718	−0.386408	−0.193204	+	0.981159i	$0.561888\pi$
$822$	0	0
$823$	−42.9808	−1.49822	−0.749108	−	0.662448i	$-0.769517\pi$
$823$	−42.9808	−1.49822	−0.749108	+	0.662448i	$0.769517\pi$
$824$	8.53590	0.297362
$825$	0	0
$826$	12.0000	0.417533
$827$	26.1962	0.910929	0.455465	−	0.890254i	$-0.349473\pi$
$827$	26.1962	0.910929	0.455465	+	0.890254i	$0.349473\pi$
$828$	5.41154	0.188064
$829$	−37.7128	−1.30982	−0.654910	−	0.755707i	$-0.727294\pi$
$829$	−37.7128	−1.30982	−0.654910	+	0.755707i	$0.727294\pi$
$830$	0	0
$831$	−5.17691	−0.179585
$832$	4.00000	0.138675
$833$	−6.46410	−0.223968
$834$	−17.3205	−0.599760
$835$	0	0
$836$	6.92820	0.239617
$837$	−26.9282	−0.930775
$838$	−48.1577	−1.66358
$839$	22.7321	0.784798	0.392399	−	0.919795i	$-0.371645\pi$
$839$	22.7321	0.784798	0.392399	+	0.919795i	$0.371645\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	2.53590	0.0873928
$843$	0.679492	0.0234029
$844$	−0.196152	−0.00675184
$845$	0	0
$846$	3.96152	0.136200
$847$	6.87564	0.236250
$848$	30.0000	1.03020
$849$	6.39230	0.219383
$850$	0	0
$851$	−16.3923	−0.561921
$852$	−4.14359	−0.141957
$853$	−39.1769	−1.34139	−0.670696	−	0.741732i	$-0.734004\pi$
$853$	−39.1769	−1.34139	−0.670696	+	0.741732i	$0.734004\pi$
$854$	11.3205	0.387380
$855$	0	0
$856$	−0.588457	−0.0201131
$857$	31.1769	1.06498	0.532492	−	0.846435i	$-0.321256\pi$
$857$	31.1769	1.06498	0.532492	+	0.846435i	$0.321256\pi$
$858$	−6.43078	−0.219543
$859$	−18.5359	−0.632437	−0.316218	−	0.948686i	$-0.602413\pi$
$859$	−18.5359	−0.632437	−0.316218	+	0.948686i	$0.602413\pi$
$860$	0	0
$861$	−1.85641	−0.0632662
$862$	−34.9808	−1.19145
$863$	−36.9282	−1.25705	−0.628525	−	0.777789i	$-0.716341\pi$
$863$	−36.9282	−1.25705	−0.628525	+	0.777789i	$0.716341\pi$
$864$	−20.7846	−0.707107
$865$	0	0
$866$	17.0718	0.580123
$867$	0.732051	0.0248617
$868$	−4.92820	−0.167274
$869$	−21.0333	−0.713507
$870$	0	0
$871$	40.0000	1.35535
$872$	−17.3205	−0.586546
$873$	22.0000	0.744587
$874$	20.7846	0.703050
$875$	0	0
$876$	10.5359	0.355975
$877$	42.7846	1.44473	0.722367	−	0.691510i	$-0.243054\pi$
$877$	42.7846	1.44473	0.722367	+	0.691510i	$0.243054\pi$
$878$	−31.2679	−1.05524
$879$	−9.46410	−0.319216
$880$	0	0
$881$	−6.67949	−0.225038	−0.112519	−	0.993650i	$-0.535892\pi$
$881$	−6.67949	−0.225038	−0.112519	+	0.993650i	$0.535892\pi$
$882$	−27.5885	−0.928952
$883$	10.0000	0.336527	0.168263	−	0.985742i	$-0.446184\pi$
$883$	10.0000	0.336527	0.168263	+	0.985742i	$0.446184\pi$
$884$	4.00000	0.134535
$885$	0	0
$886$	1.60770	0.0540116
$887$	21.1244	0.709286	0.354643	−	0.935002i	$-0.384602\pi$
$887$	21.1244	0.709286	0.354643	+	0.935002i	$0.384602\pi$
$888$	9.46410	0.317594
$889$	4.67949	0.156945
$890$	0	0
$891$	5.66025	0.189626
$892$	5.60770	0.187760
$893$	5.07180	0.169721
$894$	7.60770	0.254439
$895$	0	0
$896$	8.87564	0.296514
$897$	−6.43078	−0.214718
$898$	23.5692	0.786515
$899$	23.3205	0.777782
$900$	0	0
$901$	−6.00000	−0.199889
$902$	−7.60770	−0.253309
$903$	−4.00000	−0.133112
$904$	−30.0000	−0.997785
$905$	0	0
$906$	−6.92820	−0.230174
$907$	−27.2679	−0.905417	−0.452709	−	0.891658i	$-0.649542\pi$
$907$	−27.2679	−0.905417	−0.452709	+	0.891658i	$0.649542\pi$
$908$	−19.2679	−0.639429
$909$	6.24871	0.207257
$910$	0	0
$911$	53.6603	1.77784	0.888922	−	0.458059i	$-0.151455\pi$
$911$	53.6603	1.77784	0.888922	+	0.458059i	$0.151455\pi$
$912$	−20.0000	−0.662266
$913$	−19.6077	−0.648920
$914$	8.28719	0.274116
$915$	0	0
$916$	12.3923	0.409453
$917$	−6.00000	−0.198137
$918$	6.92820	0.228665
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	7.32051	0.241219
$922$	19.1769	0.631558
$923$	−22.6410	−0.745238
$924$	−0.679492	−0.0223536
$925$	0	0
$926$	6.67949	0.219502
$927$	−12.1436	−0.398848
$928$	18.0000	0.590879
$929$	3.46410	0.113653	0.0568267	−	0.998384i	$-0.481902\pi$
$929$	3.46410	0.113653	0.0568267	+	0.998384i	$0.481902\pi$
$930$	0	0
$931$	−35.3205	−1.15758
$932$	−6.00000	−0.196537
$933$	16.1436	0.528518
$934$	38.7846	1.26907
$935$	0	0
$936$	−17.0718	−0.558009
$937$	−36.6410	−1.19701	−0.598505	−	0.801119i	$-0.704238\pi$
$937$	−36.6410	−1.19701	−0.598505	+	0.801119i	$0.704238\pi$
$938$	12.6795	0.414000
$939$	4.10512	0.133965
$940$	0	0
$941$	−0.248711	−0.00810776	−0.00405388	−	0.999992i	$-0.501290\pi$
$941$	−0.248711	−0.00810776	−0.00405388	+	0.999992i	$0.501290\pi$
$942$	−6.24871	−0.203594
$943$	−7.60770	−0.247741
$944$	−47.3205	−1.54015
$945$	0	0
$946$	−16.3923	−0.532960
$947$	2.19615	0.0713654	0.0356827	−	0.999363i	$-0.488639\pi$
$947$	2.19615	0.0713654	0.0356827	+	0.999363i	$0.488639\pi$
$948$	−12.1436	−0.394406
$949$	57.5692	1.86878
$950$	0	0
$951$	8.10512	0.262826
$952$	−1.26795	−0.0410945
$953$	−32.7846	−1.06200	−0.530999	−	0.847373i	$-0.678183\pi$
$953$	−32.7846	−1.06200	−0.530999	+	0.847373i	$0.678183\pi$
$954$	−25.6077	−0.829080
$955$	0	0
$956$	20.7846	0.672222
$957$	3.21539	0.103939
$958$	−9.80385	−0.316748
$959$	0	0
$960$	0	0
$961$	14.3205	0.461952
$962$	−51.7128	−1.66729
$963$	0.837169	0.0269774
$964$	−26.3923	−0.850039
$965$	0	0
$966$	−2.03848	−0.0655869
$967$	23.1769	0.745319	0.372660	−	0.927968i	$-0.378446\pi$
$967$	23.1769	0.745319	0.372660	+	0.927968i	$0.378446\pi$
$968$	−16.2679	−0.522872
$969$	4.00000	0.128499
$970$	0	0
$971$	5.75129	0.184568	0.0922838	−	0.995733i	$-0.470583\pi$
$971$	5.75129	0.184568	0.0922838	+	0.995733i	$0.470583\pi$
$972$	15.2679	0.489720
$973$	−10.0000	−0.320585
$974$	−46.7321	−1.49739
$975$	0	0
$976$	−44.6410	−1.42892
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	12.9282	0.413398
$979$	−20.7846	−0.664279
$980$	0	0
$981$	24.6410	0.786727
$982$	69.9615	2.23256
$983$	−55.2679	−1.76277	−0.881387	−	0.472395i	$-0.843390\pi$
$983$	−55.2679	−1.76277	−0.881387	+	0.472395i	$0.843390\pi$
$984$	4.39230	0.140022
$985$	0	0
$986$	−6.00000	−0.191079
$987$	−0.497423	−0.0158331
$988$	21.8564	0.695345
$989$	−16.3923	−0.521245
$990$	0	0
$991$	48.9808	1.55593	0.777963	−	0.628311i	$-0.216253\pi$
$991$	48.9808	1.55593	0.777963	+	0.628311i	$0.216253\pi$
$992$	34.9808	1.11064
$993$	−9.85641	−0.312784
$994$	−7.17691	−0.227638
$995$	0	0
$996$	−11.3205	−0.358704
$997$	−18.3923	−0.582490	−0.291245	−	0.956648i	$-0.594070\pi$
$997$	−18.3923	−0.582490	−0.291245	+	0.956648i	$0.594070\pi$
$998$	−2.87564	−0.0910269
$999$	−29.8564	−0.944615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.a.e.1.1		2
3.2	odd	2		3825.2.a.v.1.2		2
4.3	odd	2		6800.2.a.bg.1.1		2
5.2	odd	4		425.2.b.d.324.2		4
5.3	odd	4		425.2.b.d.324.3		4
5.4	even	2		85.2.a.c.1.2	✓	2
15.14	odd	2		765.2.a.g.1.1		2
17.16	even	2		7225.2.a.l.1.1		2
20.19	odd	2		1360.2.a.k.1.2		2
35.34	odd	2		4165.2.a.t.1.2		2
40.19	odd	2		5440.2.a.bl.1.1		2
40.29	even	2		5440.2.a.bb.1.2		2
85.4	even	4		1445.2.d.e.866.2		4
85.64	even	4		1445.2.d.e.866.1		4
85.84	even	2		1445.2.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.a.c.1.2	✓	2	5.4	even	2
425.2.a.e.1.1		2	1.1	even	1	trivial
425.2.b.d.324.2		4	5.2	odd	4
425.2.b.d.324.3		4	5.3	odd	4
765.2.a.g.1.1		2	15.14	odd	2
1360.2.a.k.1.2		2	20.19	odd	2
1445.2.a.g.1.2		2	85.84	even	2
1445.2.d.e.866.1		4	85.64	even	4
1445.2.d.e.866.2		4	85.4	even	4
3825.2.a.v.1.2		2	3.2	odd	2
4165.2.a.t.1.2		2	35.34	odd	2
5440.2.a.bb.1.2		2	40.29	even	2
5440.2.a.bl.1.1		2	40.19	odd	2
6800.2.a.bg.1.1		2	4.3	odd	2
7225.2.a.l.1.1		2	17.16	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:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +0.732051 q^{3} +1.00000 q^{4} -1.26795 q^{6} -0.732051 q^{7} +1.73205 q^{8} -2.46410 q^{9} +1.26795 q^{11} +0.732051 q^{12} +4.00000 q^{13} +1.26795 q^{14} -5.00000 q^{16} +1.00000 q^{17} +4.26795 q^{18} +5.46410 q^{19} -0.535898 q^{21} -2.19615 q^{22} -2.19615 q^{23} +1.26795 q^{24} -6.92820 q^{26} -4.00000 q^{27} -0.732051 q^{28} +3.46410 q^{29} +6.73205 q^{31} +5.19615 q^{32} +0.928203 q^{33} -1.73205 q^{34} -2.46410 q^{36} +7.46410 q^{37} -9.46410 q^{38} +2.92820 q^{39} +3.46410 q^{41} +0.928203 q^{42} +7.46410 q^{43} +1.26795 q^{44} +3.80385 q^{46} +0.928203 q^{47} -3.66025 q^{48} -6.46410 q^{49} +0.732051 q^{51} +4.00000 q^{52} -6.00000 q^{53} +6.92820 q^{54} -1.26795 q^{56} +4.00000 q^{57} -6.00000 q^{58} +9.46410 q^{59} +8.92820 q^{61} -11.6603 q^{62} +1.80385 q^{63} +1.00000 q^{64} -1.60770 q^{66} +10.0000 q^{67} +1.00000 q^{68} -1.60770 q^{69} -5.66025 q^{71} -4.26795 q^{72} +14.3923 q^{73} -12.9282 q^{74} +5.46410 q^{76} -0.928203 q^{77} -5.07180 q^{78} -16.5885 q^{79} +4.46410 q^{81} -6.00000 q^{82} -15.4641 q^{83} -0.535898 q^{84} -12.9282 q^{86} +2.53590 q^{87} +2.19615 q^{88} -16.3923 q^{89} -2.92820 q^{91} -2.19615 q^{92} +4.92820 q^{93} -1.60770 q^{94} +3.80385 q^{96} -8.92820 q^{97} +11.1962 q^{98} -3.12436 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{4} - 6 q^{6} + 2 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{12} + 8 q^{13} + 6 q^{14} - 10 q^{16} + 2 q^{17} + 12 q^{18} + 4 q^{19} - 8 q^{21} + 6 q^{22} + 6 q^{23} + 6 q^{24} - 8 q^{27} + 2 q^{28}+ \cdots + 18 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^4 - 6 * q^6 + 2 * q^7 + 2 * q^9 + 6 * q^11 - 2 * q^12 + 8 * q^13 + 6 * q^14 - 10 * q^16 + 2 * q^17 + 12 * q^18 + 4 * q^19 - 8 * q^21 + 6 * q^22 + 6 * q^23 + 6 * q^24 - 8 * q^27 + 2 * q^28 + 10 * q^31 - 12 * q^33 + 2 * q^36 + 8 * q^37 - 12 * q^38 - 8 * q^39 - 12 * q^42 + 8 * q^43 + 6 * q^44 + 18 * q^46 - 12 * q^47 + 10 * q^48 - 6 * q^49 - 2 * q^51 + 8 * q^52 - 12 * q^53 - 6 * q^56 + 8 * q^57 - 12 * q^58 + 12 * q^59 + 4 * q^61 - 6 * q^62 + 14 * q^63 + 2 * q^64 - 24 * q^66 + 20 * q^67 + 2 * q^68 - 24 * q^69 + 6 * q^71 - 12 * q^72 + 8 * q^73 - 12 * q^74 + 4 * q^76 + 12 * q^77 - 24 * q^78 - 2 * q^79 + 2 * q^81 - 12 * q^82 - 24 * q^83 - 8 * q^84 - 12 * q^86 + 12 * q^87 - 6 * q^88 - 12 * q^89 + 8 * q^91 + 6 * q^92 - 4 * q^93 - 24 * q^94 + 18 * q^96 - 4 * q^97 + 12 * q^98 + 18 * q^99

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