LMFDB - Embedded newform 1425.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(1,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, 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("1425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.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:	$11.3786822880$
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 285)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	1425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{6} +2.73205 q^{7} -1.73205 q^{8} +1.00000 q^{9} +4.73205 q^{11} -1.00000 q^{12} -0.732051 q^{13} +4.73205 q^{14} -5.00000 q^{16} +1.73205 q^{18} +1.00000 q^{19} -2.73205 q^{21} +8.19615 q^{22} -3.46410 q^{23} +1.73205 q^{24} -1.26795 q^{26} -1.00000 q^{27} +2.73205 q^{28} +8.19615 q^{29} +8.92820 q^{31} -5.19615 q^{32} -4.73205 q^{33} +1.00000 q^{36} +6.19615 q^{37} +1.73205 q^{38} +0.732051 q^{39} +1.26795 q^{41} -4.73205 q^{42} -4.19615 q^{43} +4.73205 q^{44} -6.00000 q^{46} +3.46410 q^{47} +5.00000 q^{48} +0.464102 q^{49} -0.732051 q^{52} +9.46410 q^{53} -1.73205 q^{54} -4.73205 q^{56} -1.00000 q^{57} +14.1962 q^{58} +2.53590 q^{59} -6.53590 q^{61} +15.4641 q^{62} +2.73205 q^{63} +1.00000 q^{64} -8.19615 q^{66} -8.00000 q^{67} +3.46410 q^{69} -4.39230 q^{71} -1.73205 q^{72} +16.9282 q^{73} +10.7321 q^{74} +1.00000 q^{76} +12.9282 q^{77} +1.26795 q^{78} -10.9282 q^{79} +1.00000 q^{81} +2.19615 q^{82} +12.9282 q^{83} -2.73205 q^{84} -7.26795 q^{86} -8.19615 q^{87} -8.19615 q^{88} +10.7321 q^{89} -2.00000 q^{91} -3.46410 q^{92} -8.92820 q^{93} +6.00000 q^{94} +5.19615 q^{96} +6.19615 q^{97} +0.803848 q^{98} +4.73205 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{4} + 2 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{12} + 2 q^{13} + 6 q^{14} - 10 q^{16} + 2 q^{19} - 2 q^{21} + 6 q^{22} - 6 q^{26} - 2 q^{27} + 2 q^{28} + 6 q^{29} + 4 q^{31} - 6 q^{33} + 2 q^{36}+ \cdots + 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^7 + 2 * q^9 + 6 * q^11 - 2 * q^12 + 2 * q^13 + 6 * q^14 - 10 * q^16 + 2 * q^19 - 2 * q^21 + 6 * q^22 - 6 * q^26 - 2 * q^27 + 2 * q^28 + 6 * q^29 + 4 * q^31 - 6 * q^33 + 2 * q^36 + 2 * q^37 - 2 * q^39 + 6 * q^41 - 6 * q^42 + 2 * q^43 + 6 * q^44 - 12 * q^46 + 10 * q^48 - 6 * q^49 + 2 * q^52 + 12 * q^53 - 6 * q^56 - 2 * q^57 + 18 * q^58 + 12 * q^59 - 20 * q^61 + 24 * q^62 + 2 * q^63 + 2 * q^64 - 6 * q^66 - 16 * q^67 + 12 * q^71 + 20 * q^73 + 18 * q^74 + 2 * q^76 + 12 * q^77 + 6 * q^78 - 8 * q^79 + 2 * q^81 - 6 * q^82 + 12 * q^83 - 2 * q^84 - 18 * q^86 - 6 * q^87 - 6 * q^88 + 18 * q^89 - 4 * q^91 - 4 * q^93 + 12 * q^94 + 2 * q^97 + 12 * q^98 + 6 * 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	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.73205	−0.707107
$7$	2.73205	1.03262	0.516309	−	0.856402i	$-0.327306\pi$
$7$	2.73205	1.03262	0.516309	+	0.856402i	$0.327306\pi$
$8$	−1.73205	−0.612372
$9$	1.00000	0.333333
$10$	0	0
$11$	4.73205	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$12$	−1.00000	−0.288675
$13$	−0.732051	−0.203034	−0.101517	−	0.994834i	$-0.532370\pi$
$13$	−0.732051	−0.203034	−0.101517	+	0.994834i	$0.532370\pi$
$14$	4.73205	1.26469
$15$	0	0
$16$	−5.00000	−1.25000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.73205	0.408248
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−2.73205	−0.596182
$22$	8.19615	1.74743
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	1.73205	0.353553
$25$	0	0
$26$	−1.26795	−0.248665
$27$	−1.00000	−0.192450
$28$	2.73205	0.516309
$29$	8.19615	1.52199	0.760994	−	0.648759i	$-0.224712\pi$
$29$	8.19615	1.52199	0.760994	+	0.648759i	$0.224712\pi$
$30$	0	0
$31$	8.92820	1.60355	0.801776	−	0.597624i	$-0.203889\pi$
$31$	8.92820	1.60355	0.801776	+	0.597624i	$0.203889\pi$
$32$	−5.19615	−0.918559
$33$	−4.73205	−0.823744
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	6.19615	1.01864	0.509321	−	0.860577i	$-0.329897\pi$
$37$	6.19615	1.01864	0.509321	+	0.860577i	$0.329897\pi$
$38$	1.73205	0.280976
$39$	0.732051	0.117222
$40$	0	0
$41$	1.26795	0.198020	0.0990102	−	0.995086i	$-0.468432\pi$
$41$	1.26795	0.198020	0.0990102	+	0.995086i	$0.468432\pi$
$42$	−4.73205	−0.730171
$43$	−4.19615	−0.639907	−0.319954	−	0.947433i	$-0.603667\pi$
$43$	−4.19615	−0.639907	−0.319954	+	0.947433i	$0.603667\pi$
$44$	4.73205	0.713384
$45$	0	0
$46$	−6.00000	−0.884652
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	5.00000	0.721688
$49$	0.464102	0.0663002
$50$	0	0
$51$	0	0
$52$	−0.732051	−0.101517
$53$	9.46410	1.29999	0.649997	−	0.759937i	$-0.274770\pi$
$53$	9.46410	1.29999	0.649997	+	0.759937i	$0.274770\pi$
$54$	−1.73205	−0.235702
$55$	0	0
$56$	−4.73205	−0.632347
$57$	−1.00000	−0.132453
$58$	14.1962	1.86405
$59$	2.53590	0.330146	0.165073	−	0.986281i	$-0.447214\pi$
$59$	2.53590	0.330146	0.165073	+	0.986281i	$0.447214\pi$
$60$	0	0
$61$	−6.53590	−0.836836	−0.418418	−	0.908255i	$-0.637415\pi$
$61$	−6.53590	−0.836836	−0.418418	+	0.908255i	$0.637415\pi$
$62$	15.4641	1.96394
$63$	2.73205	0.344206
$64$	1.00000	0.125000
$65$	0	0
$66$	−8.19615	−1.00888
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	3.46410	0.417029
$70$	0	0
$71$	−4.39230	−0.521271	−0.260635	−	0.965437i	$-0.583932\pi$
$71$	−4.39230	−0.521271	−0.260635	+	0.965437i	$0.583932\pi$
$72$	−1.73205	−0.204124
$73$	16.9282	1.98130	0.990648	−	0.136441i	$-0.0435665\pi$
$73$	16.9282	1.98130	0.990648	+	0.136441i	$0.0435665\pi$
$74$	10.7321	1.24758
$75$	0	0
$76$	1.00000	0.114708
$77$	12.9282	1.47331
$78$	1.26795	0.143567
$79$	−10.9282	−1.22952	−0.614759	−	0.788715i	$-0.710747\pi$
$79$	−10.9282	−1.22952	−0.614759	+	0.788715i	$0.710747\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.19615	0.242524
$83$	12.9282	1.41905	0.709527	−	0.704678i	$-0.248908\pi$
$83$	12.9282	1.41905	0.709527	+	0.704678i	$0.248908\pi$
$84$	−2.73205	−0.298091
$85$	0	0
$86$	−7.26795	−0.783723
$87$	−8.19615	−0.878720
$88$	−8.19615	−0.873713
$89$	10.7321	1.13760	0.568798	−	0.822478i	$-0.307409\pi$
$89$	10.7321	1.13760	0.568798	+	0.822478i	$0.307409\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	−3.46410	−0.361158
$93$	−8.92820	−0.925812
$94$	6.00000	0.618853
$95$	0	0
$96$	5.19615	0.530330
$97$	6.19615	0.629124	0.314562	−	0.949237i	$-0.398142\pi$
$97$	6.19615	0.629124	0.314562	+	0.949237i	$0.398142\pi$
$98$	0.803848	0.0812009
$99$	4.73205	0.475589
$100$	0	0
$101$	−10.3923	−1.03407	−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923	−1.03407	−0.517036	+	0.855963i	$0.672965\pi$
$102$	0	0
$103$	−9.85641	−0.971181	−0.485590	−	0.874187i	$-0.661395\pi$
$103$	−9.85641	−0.971181	−0.485590	+	0.874187i	$0.661395\pi$
$104$	1.26795	0.124333
$105$	0	0
$106$	16.3923	1.59216
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−1.00000	−0.0962250
$109$	−14.3923	−1.37853	−0.689266	−	0.724508i	$-0.742067\pi$
$109$	−14.3923	−1.37853	−0.689266	+	0.724508i	$0.742067\pi$
$110$	0	0
$111$	−6.19615	−0.588113
$112$	−13.6603	−1.29077
$113$	−18.9282	−1.78062	−0.890308	−	0.455359i	$-0.849511\pi$
$113$	−18.9282	−1.78062	−0.890308	+	0.455359i	$0.849511\pi$
$114$	−1.73205	−0.162221
$115$	0	0
$116$	8.19615	0.760994
$117$	−0.732051	−0.0676781
$118$	4.39230	0.404344
$119$	0	0
$120$	0	0
$121$	11.3923	1.03566
$122$	−11.3205	−1.02491
$123$	−1.26795	−0.114327
$124$	8.92820	0.801776
$125$	0	0
$126$	4.73205	0.421565
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	12.1244	1.07165
$129$	4.19615	0.369451
$130$	0	0
$131$	−9.12436	−0.797199	−0.398599	−	0.917125i	$-0.630504\pi$
$131$	−9.12436	−0.797199	−0.398599	+	0.917125i	$0.630504\pi$
$132$	−4.73205	−0.411872
$133$	2.73205	0.236899
$134$	−13.8564	−1.19701
$135$	0	0
$136$	0	0
$137$	−19.8564	−1.69645	−0.848224	−	0.529638i	$-0.822328\pi$
$137$	−19.8564	−1.69645	−0.848224	+	0.529638i	$0.822328\pi$
$138$	6.00000	0.510754
$139$	−8.39230	−0.711826	−0.355913	−	0.934519i	$-0.615830\pi$
$139$	−8.39230	−0.711826	−0.355913	+	0.934519i	$0.615830\pi$
$140$	0	0
$141$	−3.46410	−0.291730
$142$	−7.60770	−0.638424
$143$	−3.46410	−0.289683
$144$	−5.00000	−0.416667
$145$	0	0
$146$	29.3205	2.42658
$147$	−0.464102	−0.0382785
$148$	6.19615	0.509321
$149$	−19.8564	−1.62670	−0.813350	−	0.581775i	$-0.802359\pi$
$149$	−19.8564	−1.62670	−0.813350	+	0.581775i	$0.802359\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	−1.73205	−0.140488
$153$	0	0
$154$	22.3923	1.80442
$155$	0	0
$156$	0.732051	0.0586110
$157$	−6.39230	−0.510161	−0.255081	−	0.966920i	$-0.582102\pi$
$157$	−6.39230	−0.510161	−0.255081	+	0.966920i	$0.582102\pi$
$158$	−18.9282	−1.50585
$159$	−9.46410	−0.750552
$160$	0	0
$161$	−9.46410	−0.745876
$162$	1.73205	0.136083
$163$	−9.26795	−0.725922	−0.362961	−	0.931804i	$-0.618234\pi$
$163$	−9.26795	−0.725922	−0.362961	+	0.931804i	$0.618234\pi$
$164$	1.26795	0.0990102
$165$	0	0
$166$	22.3923	1.73798
$167$	−3.46410	−0.268060	−0.134030	−	0.990977i	$-0.542792\pi$
$167$	−3.46410	−0.268060	−0.134030	+	0.990977i	$0.542792\pi$
$168$	4.73205	0.365086
$169$	−12.4641	−0.958777
$170$	0	0
$171$	1.00000	0.0764719
$172$	−4.19615	−0.319954
$173$	−6.92820	−0.526742	−0.263371	−	0.964695i	$-0.584834\pi$
$173$	−6.92820	−0.526742	−0.263371	+	0.964695i	$0.584834\pi$
$174$	−14.1962	−1.07621
$175$	0	0
$176$	−23.6603	−1.78346
$177$	−2.53590	−0.190610
$178$	18.5885	1.39326
$179$	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
$181$	−2.39230	−0.177819	−0.0889093	−	0.996040i	$-0.528338\pi$
$181$	−2.39230	−0.177819	−0.0889093	+	0.996040i	$0.528338\pi$
$182$	−3.46410	−0.256776
$183$	6.53590	0.483148
$184$	6.00000	0.442326
$185$	0	0
$186$	−15.4641	−1.13388
$187$	0	0
$188$	3.46410	0.252646
$189$	−2.73205	−0.198727
$190$	0	0
$191$	0.339746	0.0245832	0.0122916	−	0.999924i	$-0.496087\pi$
$191$	0.339746	0.0245832	0.0122916	+	0.999924i	$0.496087\pi$
$192$	−1.00000	−0.0721688
$193$	−17.1244	−1.23264	−0.616319	−	0.787497i	$-0.711377\pi$
$193$	−17.1244	−1.23264	−0.616319	+	0.787497i	$0.711377\pi$
$194$	10.7321	0.770516
$195$	0	0
$196$	0.464102	0.0331501
$197$	24.0000	1.70993	0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000	1.70993	0.854965	+	0.518686i	$0.173579\pi$
$198$	8.19615	0.582475
$199$	−15.3205	−1.08604	−0.543021	−	0.839719i	$-0.682720\pi$
$199$	−15.3205	−1.08604	−0.543021	+	0.839719i	$0.682720\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	−18.0000	−1.26648
$203$	22.3923	1.57163
$204$	0	0
$205$	0	0
$206$	−17.0718	−1.18945
$207$	−3.46410	−0.240772
$208$	3.66025	0.253793
$209$	4.73205	0.327323
$210$	0	0
$211$	1.07180	0.0737855	0.0368928	−	0.999319i	$-0.488254\pi$
$211$	1.07180	0.0737855	0.0368928	+	0.999319i	$0.488254\pi$
$212$	9.46410	0.649997
$213$	4.39230	0.300956
$214$	0	0
$215$	0	0
$216$	1.73205	0.117851
$217$	24.3923	1.65586
$218$	−24.9282	−1.68835
$219$	−16.9282	−1.14390
$220$	0	0
$221$	0	0
$222$	−10.7321	−0.720288
$223$	17.8564	1.19575	0.597877	−	0.801588i	$-0.296011\pi$
$223$	17.8564	1.19575	0.597877	+	0.801588i	$0.296011\pi$
$224$	−14.1962	−0.948520
$225$	0	0
$226$	−32.7846	−2.18080
$227$	−10.3923	−0.689761	−0.344881	−	0.938647i	$-0.612081\pi$
$227$	−10.3923	−0.689761	−0.344881	+	0.938647i	$0.612081\pi$
$228$	−1.00000	−0.0662266
$229$	−18.5359	−1.22489	−0.612443	−	0.790515i	$-0.709813\pi$
$229$	−18.5359	−1.22489	−0.612443	+	0.790515i	$0.709813\pi$
$230$	0	0
$231$	−12.9282	−0.850613
$232$	−14.1962	−0.932023
$233$	7.85641	0.514690	0.257345	−	0.966320i	$-0.417152\pi$
$233$	7.85641	0.514690	0.257345	+	0.966320i	$0.417152\pi$
$234$	−1.26795	−0.0828884
$235$	0	0
$236$	2.53590	0.165073
$237$	10.9282	0.709863
$238$	0	0
$239$	−9.80385	−0.634158	−0.317079	−	0.948399i	$-0.602702\pi$
$239$	−9.80385	−0.634158	−0.317079	+	0.948399i	$0.602702\pi$
$240$	0	0
$241$	−3.07180	−0.197872	−0.0989359	−	0.995094i	$-0.531544\pi$
$241$	−3.07180	−0.197872	−0.0989359	+	0.995094i	$0.531544\pi$
$242$	19.7321	1.26842
$243$	−1.00000	−0.0641500
$244$	−6.53590	−0.418418
$245$	0	0
$246$	−2.19615	−0.140022
$247$	−0.732051	−0.0465793
$248$	−15.4641	−0.981971
$249$	−12.9282	−0.819292
$250$	0	0
$251$	28.0526	1.77066	0.885331	−	0.464961i	$-0.153932\pi$
$251$	28.0526	1.77066	0.885331	+	0.464961i	$0.153932\pi$
$252$	2.73205	0.172103
$253$	−16.3923	−1.03058
$254$	6.92820	0.434714
$255$	0	0
$256$	19.0000	1.18750
$257$	24.0000	1.49708	0.748539	−	0.663090i	$-0.230755\pi$
$257$	24.0000	1.49708	0.748539	+	0.663090i	$0.230755\pi$
$258$	7.26795	0.452483
$259$	16.9282	1.05187
$260$	0	0
$261$	8.19615	0.507329
$262$	−15.8038	−0.976365
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	8.19615	0.504438
$265$	0	0
$266$	4.73205	0.290141
$267$	−10.7321	−0.656791
$268$	−8.00000	−0.488678
$269$	0.588457	0.0358789	0.0179394	−	0.999839i	$-0.494289\pi$
$269$	0.588457	0.0358789	0.0179394	+	0.999839i	$0.494289\pi$
$270$	0	0
$271$	0.392305	0.0238308	0.0119154	−	0.999929i	$-0.496207\pi$
$271$	0.392305	0.0238308	0.0119154	+	0.999929i	$0.496207\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	−34.3923	−2.07772
$275$	0	0
$276$	3.46410	0.208514
$277$	−2.00000	−0.120168	−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000	−0.120168	−0.0600842	+	0.998193i	$0.519137\pi$
$278$	−14.5359	−0.871805
$279$	8.92820	0.534518
$280$	0	0
$281$	1.26795	0.0756395	0.0378198	−	0.999285i	$-0.487959\pi$
$281$	1.26795	0.0756395	0.0378198	+	0.999285i	$0.487959\pi$
$282$	−6.00000	−0.357295
$283$	−24.9808	−1.48495	−0.742476	−	0.669873i	$-0.766349\pi$
$283$	−24.9808	−1.48495	−0.742476	+	0.669873i	$0.766349\pi$
$284$	−4.39230	−0.260635
$285$	0	0
$286$	−6.00000	−0.354787
$287$	3.46410	0.204479
$288$	−5.19615	−0.306186
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−6.19615	−0.363225
$292$	16.9282	0.990648
$293$	−27.7128	−1.61900	−0.809500	−	0.587120i	$-0.800262\pi$
$293$	−27.7128	−1.61900	−0.809500	+	0.587120i	$0.800262\pi$
$294$	−0.803848	−0.0468813
$295$	0	0
$296$	−10.7321	−0.623788
$297$	−4.73205	−0.274581
$298$	−34.3923	−1.99229
$299$	2.53590	0.146655
$300$	0	0
$301$	−11.4641	−0.660780
$302$	24.2487	1.39536
$303$	10.3923	0.597022
$304$	−5.00000	−0.286770
$305$	0	0
$306$	0	0
$307$	32.3923	1.84873	0.924363	−	0.381514i	$-0.124597\pi$
$307$	32.3923	1.84873	0.924363	+	0.381514i	$0.124597\pi$
$308$	12.9282	0.736653
$309$	9.85641	0.560711
$310$	0	0
$311$	32.4449	1.83978	0.919890	−	0.392177i	$-0.128278\pi$
$311$	32.4449	1.83978	0.919890	+	0.392177i	$0.128278\pi$
$312$	−1.26795	−0.0717835
$313$	−6.39230	−0.361314	−0.180657	−	0.983546i	$-0.557822\pi$
$313$	−6.39230	−0.361314	−0.180657	+	0.983546i	$0.557822\pi$
$314$	−11.0718	−0.624818
$315$	0	0
$316$	−10.9282	−0.614759
$317$	11.3205	0.635823	0.317912	−	0.948120i	$-0.397018\pi$
$317$	11.3205	0.635823	0.317912	+	0.948120i	$0.397018\pi$
$318$	−16.3923	−0.919235
$319$	38.7846	2.17152
$320$	0	0
$321$	0	0
$322$	−16.3923	−0.913507
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	−16.0526	−0.889069
$327$	14.3923	0.795896
$328$	−2.19615	−0.121262
$329$	9.46410	0.521773
$330$	0	0
$331$	−25.7128	−1.41330	−0.706652	−	0.707561i	$-0.749795\pi$
$331$	−25.7128	−1.41330	−0.706652	+	0.707561i	$0.749795\pi$
$332$	12.9282	0.709527
$333$	6.19615	0.339547
$334$	−6.00000	−0.328305
$335$	0	0
$336$	13.6603	0.745228
$337$	−5.12436	−0.279141	−0.139571	−	0.990212i	$-0.544572\pi$
$337$	−5.12436	−0.279141	−0.139571	+	0.990212i	$0.544572\pi$
$338$	−21.5885	−1.17426
$339$	18.9282	1.02804
$340$	0	0
$341$	42.2487	2.28790
$342$	1.73205	0.0936586
$343$	−17.8564	−0.964155
$344$	7.26795	0.391862
$345$	0	0
$346$	−12.0000	−0.645124
$347$	0.928203	0.0498286	0.0249143	−	0.999690i	$-0.492069\pi$
$347$	0.928203	0.0498286	0.0249143	+	0.999690i	$0.492069\pi$
$348$	−8.19615	−0.439360
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	0.732051	0.0390740
$352$	−24.5885	−1.31057
$353$	14.7846	0.786905	0.393453	−	0.919345i	$-0.371281\pi$
$353$	14.7846	0.786905	0.393453	+	0.919345i	$0.371281\pi$
$354$	−4.39230	−0.233448
$355$	0	0
$356$	10.7321	0.568798
$357$	0	0
$358$	40.3923	2.13480
$359$	0.339746	0.0179311	0.00896555	−	0.999960i	$-0.497146\pi$
$359$	0.339746	0.0179311	0.00896555	+	0.999960i	$0.497146\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−4.14359	−0.217782
$363$	−11.3923	−0.597941
$364$	−2.00000	−0.104828
$365$	0	0
$366$	11.3205	0.591732
$367$	−16.1962	−0.845432	−0.422716	−	0.906262i	$-0.638923\pi$
$367$	−16.1962	−0.845432	−0.422716	+	0.906262i	$0.638923\pi$
$368$	17.3205	0.902894
$369$	1.26795	0.0660068
$370$	0	0
$371$	25.8564	1.34240
$372$	−8.92820	−0.462906
$373$	6.19615	0.320825	0.160412	−	0.987050i	$-0.448718\pi$
$373$	6.19615	0.320825	0.160412	+	0.987050i	$0.448718\pi$
$374$	0	0
$375$	0	0
$376$	−6.00000	−0.309426
$377$	−6.00000	−0.309016
$378$	−4.73205	−0.243390
$379$	20.9282	1.07501	0.537505	−	0.843261i	$-0.319367\pi$
$379$	20.9282	1.07501	0.537505	+	0.843261i	$0.319367\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0.588457	0.0301081
$383$	17.0718	0.872328	0.436164	−	0.899867i	$-0.356337\pi$
$383$	17.0718	0.872328	0.436164	+	0.899867i	$0.356337\pi$
$384$	−12.1244	−0.618718
$385$	0	0
$386$	−29.6603	−1.50967
$387$	−4.19615	−0.213302
$388$	6.19615	0.314562
$389$	7.85641	0.398336	0.199168	−	0.979965i	$-0.436176\pi$
$389$	7.85641	0.398336	0.199168	+	0.979965i	$0.436176\pi$
$390$	0	0
$391$	0	0
$392$	−0.803848	−0.0406004
$393$	9.12436	0.460263
$394$	41.5692	2.09423
$395$	0	0
$396$	4.73205	0.237795
$397$	−8.92820	−0.448094	−0.224047	−	0.974578i	$-0.571927\pi$
$397$	−8.92820	−0.448094	−0.224047	+	0.974578i	$0.571927\pi$
$398$	−26.5359	−1.33012
$399$	−2.73205	−0.136774
$400$	0	0
$401$	−34.0526	−1.70050	−0.850252	−	0.526376i	$-0.823550\pi$
$401$	−34.0526	−1.70050	−0.850252	+	0.526376i	$0.823550\pi$
$402$	13.8564	0.691095
$403$	−6.53590	−0.325576
$404$	−10.3923	−0.517036
$405$	0	0
$406$	38.7846	1.92485
$407$	29.3205	1.45336
$408$	0	0
$409$	−26.3923	−1.30502	−0.652508	−	0.757782i	$-0.726283\pi$
$409$	−26.3923	−1.30502	−0.652508	+	0.757782i	$0.726283\pi$
$410$	0	0
$411$	19.8564	0.979444
$412$	−9.85641	−0.485590
$413$	6.92820	0.340915
$414$	−6.00000	−0.294884
$415$	0	0
$416$	3.80385	0.186499
$417$	8.39230	0.410973
$418$	8.19615	0.400887
$419$	−28.0526	−1.37046	−0.685229	−	0.728328i	$-0.740298\pi$
$419$	−28.0526	−1.37046	−0.685229	+	0.728328i	$0.740298\pi$
$420$	0	0
$421$	−18.7846	−0.915506	−0.457753	−	0.889079i	$-0.651346\pi$
$421$	−18.7846	−0.915506	−0.457753	+	0.889079i	$0.651346\pi$
$422$	1.85641	0.0903685
$423$	3.46410	0.168430
$424$	−16.3923	−0.796081
$425$	0	0
$426$	7.60770	0.368594
$427$	−17.8564	−0.864132
$428$	0	0
$429$	3.46410	0.167248
$430$	0	0
$431$	−11.3205	−0.545290	−0.272645	−	0.962115i	$-0.587898\pi$
$431$	−11.3205	−0.545290	−0.272645	+	0.962115i	$0.587898\pi$
$432$	5.00000	0.240563
$433$	10.5885	0.508849	0.254424	−	0.967093i	$-0.418114\pi$
$433$	10.5885	0.508849	0.254424	+	0.967093i	$0.418114\pi$
$434$	42.2487	2.02800
$435$	0	0
$436$	−14.3923	−0.689266
$437$	−3.46410	−0.165710
$438$	−29.3205	−1.40099
$439$	26.9282	1.28521	0.642607	−	0.766196i	$-0.277853\pi$
$439$	26.9282	1.28521	0.642607	+	0.766196i	$0.277853\pi$
$440$	0	0
$441$	0.464102	0.0221001
$442$	0	0
$443$	5.32051	0.252785	0.126392	−	0.991980i	$-0.459660\pi$
$443$	5.32051	0.252785	0.126392	+	0.991980i	$0.459660\pi$
$444$	−6.19615	−0.294056
$445$	0	0
$446$	30.9282	1.46449
$447$	19.8564	0.939176
$448$	2.73205	0.129077
$449$	−5.66025	−0.267124	−0.133562	−	0.991040i	$-0.542642\pi$
$449$	−5.66025	−0.267124	−0.133562	+	0.991040i	$0.542642\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	−18.9282	−0.890308
$453$	−14.0000	−0.657777
$454$	−18.0000	−0.844782
$455$	0	0
$456$	1.73205	0.0811107
$457$	−4.53590	−0.212180	−0.106090	−	0.994357i	$-0.533833\pi$
$457$	−4.53590	−0.212180	−0.106090	+	0.994357i	$0.533833\pi$
$458$	−32.1051	−1.50017
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	−22.3923	−1.04178
$463$	35.5167	1.65060	0.825300	−	0.564695i	$-0.191006\pi$
$463$	35.5167	1.65060	0.825300	+	0.564695i	$0.191006\pi$
$464$	−40.9808	−1.90248
$465$	0	0
$466$	13.6077	0.630364
$467$	−20.5359	−0.950288	−0.475144	−	0.879908i	$-0.657604\pi$
$467$	−20.5359	−0.950288	−0.475144	+	0.879908i	$0.657604\pi$
$468$	−0.732051	−0.0338391
$469$	−21.8564	−1.00924
$470$	0	0
$471$	6.39230	0.294542
$472$	−4.39230	−0.202172
$473$	−19.8564	−0.912999
$474$	18.9282	0.869401
$475$	0	0
$476$	0	0
$477$	9.46410	0.433331
$478$	−16.9808	−0.776682
$479$	25.5167	1.16589	0.582943	−	0.812513i	$-0.301901\pi$
$479$	25.5167	1.16589	0.582943	+	0.812513i	$0.301901\pi$
$480$	0	0
$481$	−4.53590	−0.206819
$482$	−5.32051	−0.242343
$483$	9.46410	0.430632
$484$	11.3923	0.517832
$485$	0	0
$486$	−1.73205	−0.0785674
$487$	32.3923	1.46784	0.733918	−	0.679238i	$-0.237690\pi$
$487$	32.3923	1.46784	0.733918	+	0.679238i	$0.237690\pi$
$488$	11.3205	0.512455
$489$	9.26795	0.419111
$490$	0	0
$491$	16.0526	0.724442	0.362221	−	0.932092i	$-0.382019\pi$
$491$	16.0526	0.724442	0.362221	+	0.932092i	$0.382019\pi$
$492$	−1.26795	−0.0571636
$493$	0	0
$494$	−1.26795	−0.0570477
$495$	0	0
$496$	−44.6410	−2.00444
$497$	−12.0000	−0.538274
$498$	−22.3923	−1.00342
$499$	10.5359	0.471652	0.235826	−	0.971795i	$-0.424221\pi$
$499$	10.5359	0.471652	0.235826	+	0.971795i	$0.424221\pi$
$500$	0	0
$501$	3.46410	0.154765
$502$	48.5885	2.16861
$503$	23.0718	1.02872	0.514360	−	0.857574i	$-0.328029\pi$
$503$	23.0718	1.02872	0.514360	+	0.857574i	$0.328029\pi$
$504$	−4.73205	−0.210782
$505$	0	0
$506$	−28.3923	−1.26219
$507$	12.4641	0.553550
$508$	4.00000	0.177471
$509$	−10.0526	−0.445572	−0.222786	−	0.974867i	$-0.571515\pi$
$509$	−10.0526	−0.445572	−0.222786	+	0.974867i	$0.571515\pi$
$510$	0	0
$511$	46.2487	2.04592
$512$	8.66025	0.382733
$513$	−1.00000	−0.0441511
$514$	41.5692	1.83354
$515$	0	0
$516$	4.19615	0.184725
$517$	16.3923	0.720933
$518$	29.3205	1.28827
$519$	6.92820	0.304114
$520$	0	0
$521$	37.2679	1.63274	0.816369	−	0.577530i	$-0.195983\pi$
$521$	37.2679	1.63274	0.816369	+	0.577530i	$0.195983\pi$
$522$	14.1962	0.621349
$523$	−8.67949	−0.379528	−0.189764	−	0.981830i	$-0.560772\pi$
$523$	−8.67949	−0.379528	−0.189764	+	0.981830i	$0.560772\pi$
$524$	−9.12436	−0.398599
$525$	0	0
$526$	−10.3923	−0.453126
$527$	0	0
$528$	23.6603	1.02968
$529$	−11.0000	−0.478261
$530$	0	0
$531$	2.53590	0.110049
$532$	2.73205	0.118449
$533$	−0.928203	−0.0402049
$534$	−18.5885	−0.804401
$535$	0	0
$536$	13.8564	0.598506
$537$	−23.3205	−1.00635
$538$	1.01924	0.0439425
$539$	2.19615	0.0945950
$540$	0	0
$541$	41.7128	1.79337	0.896687	−	0.442665i	$-0.145967\pi$
$541$	41.7128	1.79337	0.896687	+	0.442665i	$0.145967\pi$
$542$	0.679492	0.0291867
$543$	2.39230	0.102664
$544$	0	0
$545$	0	0
$546$	3.46410	0.148250
$547$	−43.3205	−1.85225	−0.926126	−	0.377215i	$-0.876882\pi$
$547$	−43.3205	−1.85225	−0.926126	+	0.377215i	$0.876882\pi$
$548$	−19.8564	−0.848224
$549$	−6.53590	−0.278945
$550$	0	0
$551$	8.19615	0.349168
$552$	−6.00000	−0.255377
$553$	−29.8564	−1.26962
$554$	−3.46410	−0.147176
$555$	0	0
$556$	−8.39230	−0.355913
$557$	−0.928203	−0.0393292	−0.0196646	−	0.999807i	$-0.506260\pi$
$557$	−0.928203	−0.0393292	−0.0196646	+	0.999807i	$0.506260\pi$
$558$	15.4641	0.654648
$559$	3.07180	0.129923
$560$	0	0
$561$	0	0
$562$	2.19615	0.0926391
$563$	27.4641	1.15747	0.578737	−	0.815514i	$-0.303546\pi$
$563$	27.4641	1.15747	0.578737	+	0.815514i	$0.303546\pi$
$564$	−3.46410	−0.145865
$565$	0	0
$566$	−43.2679	−1.81869
$567$	2.73205	0.114735
$568$	7.60770	0.319212
$569$	22.0526	0.924491	0.462246	−	0.886752i	$-0.347044\pi$
$569$	22.0526	0.924491	0.462246	+	0.886752i	$0.347044\pi$
$570$	0	0
$571$	−34.2487	−1.43326	−0.716632	−	0.697452i	$-0.754317\pi$
$571$	−34.2487	−1.43326	−0.716632	+	0.697452i	$0.754317\pi$
$572$	−3.46410	−0.144841
$573$	−0.339746	−0.0141931
$574$	6.00000	0.250435
$575$	0	0
$576$	1.00000	0.0416667
$577$	−15.1769	−0.631823	−0.315912	−	0.948789i	$-0.602310\pi$
$577$	−15.1769	−0.631823	−0.315912	+	0.948789i	$0.602310\pi$
$578$	−29.4449	−1.22474
$579$	17.1244	0.711664
$580$	0	0
$581$	35.3205	1.46534
$582$	−10.7321	−0.444858
$583$	44.7846	1.85479
$584$	−29.3205	−1.21329
$585$	0	0
$586$	−48.0000	−1.98286
$587$	3.46410	0.142979	0.0714894	−	0.997441i	$-0.477225\pi$
$587$	3.46410	0.142979	0.0714894	+	0.997441i	$0.477225\pi$
$588$	−0.464102	−0.0191392
$589$	8.92820	0.367880
$590$	0	0
$591$	−24.0000	−0.987228
$592$	−30.9808	−1.27330
$593$	38.7846	1.59269	0.796347	−	0.604841i	$-0.206763\pi$
$593$	38.7846	1.59269	0.796347	+	0.604841i	$0.206763\pi$
$594$	−8.19615	−0.336292
$595$	0	0
$596$	−19.8564	−0.813350
$597$	15.3205	0.627027
$598$	4.39230	0.179615
$599$	−13.8564	−0.566157	−0.283079	−	0.959097i	$-0.591356\pi$
$599$	−13.8564	−0.566157	−0.283079	+	0.959097i	$0.591356\pi$
$600$	0	0
$601$	−47.1769	−1.92439	−0.962193	−	0.272368i	$-0.912193\pi$
$601$	−47.1769	−1.92439	−0.962193	+	0.272368i	$0.912193\pi$
$602$	−19.8564	−0.809287
$603$	−8.00000	−0.325785
$604$	14.0000	0.569652
$605$	0	0
$606$	18.0000	0.731200
$607$	11.6077	0.471142	0.235571	−	0.971857i	$-0.424304\pi$
$607$	11.6077	0.471142	0.235571	+	0.971857i	$0.424304\pi$
$608$	−5.19615	−0.210732
$609$	−22.3923	−0.907382
$610$	0	0
$611$	−2.53590	−0.102591
$612$	0	0
$613$	−42.3923	−1.71221	−0.856105	−	0.516803i	$-0.827122\pi$
$613$	−42.3923	−1.71221	−0.856105	+	0.516803i	$0.827122\pi$
$614$	56.1051	2.26422
$615$	0	0
$616$	−22.3923	−0.902212
$617$	−27.7128	−1.11568	−0.557838	−	0.829950i	$-0.688369\pi$
$617$	−27.7128	−1.11568	−0.557838	+	0.829950i	$0.688369\pi$
$618$	17.0718	0.686728
$619$	−15.3205	−0.615783	−0.307892	−	0.951421i	$-0.599623\pi$
$619$	−15.3205	−0.615783	−0.307892	+	0.951421i	$0.599623\pi$
$620$	0	0
$621$	3.46410	0.139010
$622$	56.1962	2.25326
$623$	29.3205	1.17470
$624$	−3.66025	−0.146527
$625$	0	0
$626$	−11.0718	−0.442518
$627$	−4.73205	−0.188980
$628$	−6.39230	−0.255081
$629$	0	0
$630$	0	0
$631$	−34.9282	−1.39047	−0.695235	−	0.718783i	$-0.744700\pi$
$631$	−34.9282	−1.39047	−0.695235	+	0.718783i	$0.744700\pi$
$632$	18.9282	0.752923
$633$	−1.07180	−0.0426001
$634$	19.6077	0.778721
$635$	0	0
$636$	−9.46410	−0.375276
$637$	−0.339746	−0.0134612
$638$	67.1769	2.65956
$639$	−4.39230	−0.173757
$640$	0	0
$641$	48.5885	1.91913	0.959564	−	0.281489i	$-0.0908284\pi$
$641$	48.5885	1.91913	0.959564	+	0.281489i	$0.0908284\pi$
$642$	0	0
$643$	12.1962	0.480969	0.240485	−	0.970653i	$-0.422694\pi$
$643$	12.1962	0.480969	0.240485	+	0.970653i	$0.422694\pi$
$644$	−9.46410	−0.372938
$645$	0	0
$646$	0	0
$647$	4.14359	0.162901	0.0814507	−	0.996677i	$-0.474045\pi$
$647$	4.14359	0.162901	0.0814507	+	0.996677i	$0.474045\pi$
$648$	−1.73205	−0.0680414
$649$	12.0000	0.471041
$650$	0	0
$651$	−24.3923	−0.956010
$652$	−9.26795	−0.362961
$653$	−17.0718	−0.668071	−0.334036	−	0.942560i	$-0.608411\pi$
$653$	−17.0718	−0.668071	−0.334036	+	0.942560i	$0.608411\pi$
$654$	24.9282	0.974770
$655$	0	0
$656$	−6.33975	−0.247525
$657$	16.9282	0.660432
$658$	16.3923	0.639039
$659$	5.07180	0.197569	0.0987846	−	0.995109i	$-0.468505\pi$
$659$	5.07180	0.197569	0.0987846	+	0.995109i	$0.468505\pi$
$660$	0	0
$661$	39.1769	1.52381	0.761903	−	0.647692i	$-0.224265\pi$
$661$	39.1769	1.52381	0.761903	+	0.647692i	$0.224265\pi$
$662$	−44.5359	−1.73094
$663$	0	0
$664$	−22.3923	−0.868990
$665$	0	0
$666$	10.7321	0.415859
$667$	−28.3923	−1.09935
$668$	−3.46410	−0.134030
$669$	−17.8564	−0.690369
$670$	0	0
$671$	−30.9282	−1.19397
$672$	14.1962	0.547628
$673$	−17.1244	−0.660095	−0.330048	−	0.943964i	$-0.607065\pi$
$673$	−17.1244	−0.660095	−0.330048	+	0.943964i	$0.607065\pi$
$674$	−8.87564	−0.341877
$675$	0	0
$676$	−12.4641	−0.479389
$677$	−0.679492	−0.0261150	−0.0130575	−	0.999915i	$-0.504156\pi$
$677$	−0.679492	−0.0261150	−0.0130575	+	0.999915i	$0.504156\pi$
$678$	32.7846	1.25909
$679$	16.9282	0.649645
$680$	0	0
$681$	10.3923	0.398234
$682$	73.1769	2.80209
$683$	5.07180	0.194067	0.0970335	−	0.995281i	$-0.469065\pi$
$683$	5.07180	0.194067	0.0970335	+	0.995281i	$0.469065\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	−30.9282	−1.18084
$687$	18.5359	0.707189
$688$	20.9808	0.799884
$689$	−6.92820	−0.263944
$690$	0	0
$691$	12.3923	0.471425	0.235713	−	0.971823i	$-0.424258\pi$
$691$	12.3923	0.471425	0.235713	+	0.971823i	$0.424258\pi$
$692$	−6.92820	−0.263371
$693$	12.9282	0.491102
$694$	1.60770	0.0610273
$695$	0	0
$696$	14.1962	0.538104
$697$	0	0
$698$	−38.1051	−1.44230
$699$	−7.85641	−0.297157
$700$	0	0
$701$	−33.7128	−1.27332	−0.636658	−	0.771147i	$-0.719684\pi$
$701$	−33.7128	−1.27332	−0.636658	+	0.771147i	$0.719684\pi$
$702$	1.26795	0.0478557
$703$	6.19615	0.233692
$704$	4.73205	0.178346
$705$	0	0
$706$	25.6077	0.963758
$707$	−28.3923	−1.06780
$708$	−2.53590	−0.0953049
$709$	−29.1769	−1.09576	−0.547881	−	0.836556i	$-0.684565\pi$
$709$	−29.1769	−1.09576	−0.547881	+	0.836556i	$0.684565\pi$
$710$	0	0
$711$	−10.9282	−0.409840
$712$	−18.5885	−0.696632
$713$	−30.9282	−1.15827
$714$	0	0
$715$	0	0
$716$	23.3205	0.871528
$717$	9.80385	0.366131
$718$	0.588457	0.0219610
$719$	−11.6603	−0.434854	−0.217427	−	0.976077i	$-0.569766\pi$
$719$	−11.6603	−0.434854	−0.217427	+	0.976077i	$0.569766\pi$
$720$	0	0
$721$	−26.9282	−1.00286
$722$	1.73205	0.0644603
$723$	3.07180	0.114241
$724$	−2.39230	−0.0889093
$725$	0	0
$726$	−19.7321	−0.732325
$727$	−25.6603	−0.951686	−0.475843	−	0.879530i	$-0.657857\pi$
$727$	−25.6603	−0.951686	−0.475843	+	0.879530i	$0.657857\pi$
$728$	3.46410	0.128388
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	6.53590	0.241574
$733$	18.7846	0.693825	0.346913	−	0.937897i	$-0.387230\pi$
$733$	18.7846	0.693825	0.346913	+	0.937897i	$0.387230\pi$
$734$	−28.0526	−1.03544
$735$	0	0
$736$	18.0000	0.663489
$737$	−37.8564	−1.39446
$738$	2.19615	0.0808415
$739$	6.14359	0.225996	0.112998	−	0.993595i	$-0.463955\pi$
$739$	6.14359	0.225996	0.112998	+	0.993595i	$0.463955\pi$
$740$	0	0
$741$	0.732051	0.0268926
$742$	44.7846	1.64409
$743$	3.21539	0.117961	0.0589806	−	0.998259i	$-0.481215\pi$
$743$	3.21539	0.117961	0.0589806	+	0.998259i	$0.481215\pi$
$744$	15.4641	0.566941
$745$	0	0
$746$	10.7321	0.392928
$747$	12.9282	0.473018
$748$	0	0
$749$	0	0
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	−17.3205	−0.631614
$753$	−28.0526	−1.02229
$754$	−10.3923	−0.378465
$755$	0	0
$756$	−2.73205	−0.0993637
$757$	−32.2487	−1.17210	−0.586050	−	0.810275i	$-0.699318\pi$
$757$	−32.2487	−1.17210	−0.586050	+	0.810275i	$0.699318\pi$
$758$	36.2487	1.31661
$759$	16.3923	0.595003
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	−6.92820	−0.250982
$763$	−39.3205	−1.42350
$764$	0.339746	0.0122916
$765$	0	0
$766$	29.5692	1.06838
$767$	−1.85641	−0.0670310
$768$	−19.0000	−0.685603
$769$	−20.6410	−0.744334	−0.372167	−	0.928166i	$-0.621385\pi$
$769$	−20.6410	−0.744334	−0.372167	+	0.928166i	$0.621385\pi$
$770$	0	0
$771$	−24.0000	−0.864339
$772$	−17.1244	−0.616319
$773$	25.1769	0.905551	0.452775	−	0.891625i	$-0.350434\pi$
$773$	25.1769	0.905551	0.452775	+	0.891625i	$0.350434\pi$
$774$	−7.26795	−0.261241
$775$	0	0
$776$	−10.7321	−0.385258
$777$	−16.9282	−0.607296
$778$	13.6077	0.487860
$779$	1.26795	0.0454290
$780$	0	0
$781$	−20.7846	−0.743732
$782$	0	0
$783$	−8.19615	−0.292907
$784$	−2.32051	−0.0828753
$785$	0	0
$786$	15.8038	0.563705
$787$	−8.67949	−0.309390	−0.154695	−	0.987962i	$-0.549440\pi$
$787$	−8.67949	−0.309390	−0.154695	+	0.987962i	$0.549440\pi$
$788$	24.0000	0.854965
$789$	6.00000	0.213606
$790$	0	0
$791$	−51.7128	−1.83870
$792$	−8.19615	−0.291238
$793$	4.78461	0.169906
$794$	−15.4641	−0.548800
$795$	0	0
$796$	−15.3205	−0.543021
$797$	−44.7846	−1.58635	−0.793176	−	0.608992i	$-0.791574\pi$
$797$	−44.7846	−1.58635	−0.793176	+	0.608992i	$0.791574\pi$
$798$	−4.73205	−0.167513
$799$	0	0
$800$	0	0
$801$	10.7321	0.379198
$802$	−58.9808	−2.08268
$803$	80.1051	2.82685
$804$	8.00000	0.282138
$805$	0	0
$806$	−11.3205	−0.398748
$807$	−0.588457	−0.0207147
$808$	18.0000	0.633238
$809$	14.7846	0.519799	0.259900	−	0.965636i	$-0.416311\pi$
$809$	14.7846	0.519799	0.259900	+	0.965636i	$0.416311\pi$
$810$	0	0
$811$	37.5692	1.31923	0.659617	−	0.751602i	$-0.270719\pi$
$811$	37.5692	1.31923	0.659617	+	0.751602i	$0.270719\pi$
$812$	22.3923	0.785816
$813$	−0.392305	−0.0137587
$814$	50.7846	1.78000
$815$	0	0
$816$	0	0
$817$	−4.19615	−0.146805
$818$	−45.7128	−1.59831
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	−32.5359	−1.13551	−0.567755	−	0.823197i	$-0.692188\pi$
$821$	−32.5359	−1.13551	−0.567755	+	0.823197i	$0.692188\pi$
$822$	34.3923	1.19957
$823$	−12.9808	−0.452481	−0.226240	−	0.974071i	$-0.572644\pi$
$823$	−12.9808	−0.452481	−0.226240	+	0.974071i	$0.572644\pi$
$824$	17.0718	0.594724
$825$	0	0
$826$	12.0000	0.417533
$827$	5.32051	0.185012	0.0925061	−	0.995712i	$-0.470512\pi$
$827$	5.32051	0.185012	0.0925061	+	0.995712i	$0.470512\pi$
$828$	−3.46410	−0.120386
$829$	34.1051	1.18452	0.592260	−	0.805747i	$-0.298236\pi$
$829$	34.1051	1.18452	0.592260	+	0.805747i	$0.298236\pi$
$830$	0	0
$831$	2.00000	0.0693792
$832$	−0.732051	−0.0253793
$833$	0	0
$834$	14.5359	0.503337
$835$	0	0
$836$	4.73205	0.163661
$837$	−8.92820	−0.308604
$838$	−48.5885	−1.67846
$839$	−19.6077	−0.676933	−0.338466	−	0.940978i	$-0.609908\pi$
$839$	−19.6077	−0.676933	−0.338466	+	0.940978i	$0.609908\pi$
$840$	0	0
$841$	38.1769	1.31645
$842$	−32.5359	−1.12126
$843$	−1.26795	−0.0436705
$844$	1.07180	0.0368928
$845$	0	0
$846$	6.00000	0.206284
$847$	31.1244	1.06945
$848$	−47.3205	−1.62499
$849$	24.9808	0.857338
$850$	0	0
$851$	−21.4641	−0.735780
$852$	4.39230	0.150478
$853$	−27.1769	−0.930520	−0.465260	−	0.885174i	$-0.654039\pi$
$853$	−27.1769	−0.930520	−0.465260	+	0.885174i	$0.654039\pi$
$854$	−30.9282	−1.05834
$855$	0	0
$856$	0	0
$857$	42.2487	1.44319	0.721594	−	0.692316i	$-0.243410\pi$
$857$	42.2487	1.44319	0.721594	+	0.692316i	$0.243410\pi$
$858$	6.00000	0.204837
$859$	32.0000	1.09183	0.545913	−	0.837842i	$-0.316183\pi$
$859$	32.0000	1.09183	0.545913	+	0.837842i	$0.316183\pi$
$860$	0	0
$861$	−3.46410	−0.118056
$862$	−19.6077	−0.667841
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	5.19615	0.176777
$865$	0	0
$866$	18.3397	0.623210
$867$	17.0000	0.577350
$868$	24.3923	0.827929
$869$	−51.7128	−1.75424
$870$	0	0
$871$	5.85641	0.198437
$872$	24.9282	0.844175
$873$	6.19615	0.209708
$874$	−6.00000	−0.202953
$875$	0	0
$876$	−16.9282	−0.571951
$877$	−53.1244	−1.79388	−0.896941	−	0.442150i	$-0.854216\pi$
$877$	−53.1244	−1.79388	−0.896941	+	0.442150i	$0.854216\pi$
$878$	46.6410	1.57406
$879$	27.7128	0.934730
$880$	0	0
$881$	8.53590	0.287582	0.143791	−	0.989608i	$-0.454071\pi$
$881$	8.53590	0.287582	0.143791	+	0.989608i	$0.454071\pi$
$882$	0.803848	0.0270670
$883$	−36.9808	−1.24450	−0.622251	−	0.782818i	$-0.713782\pi$
$883$	−36.9808	−1.24450	−0.622251	+	0.782818i	$0.713782\pi$
$884$	0	0
$885$	0	0
$886$	9.21539	0.309597
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	10.7321	0.360144
$889$	10.9282	0.366520
$890$	0	0
$891$	4.73205	0.158530
$892$	17.8564	0.597877
$893$	3.46410	0.115922
$894$	34.3923	1.15025
$895$	0	0
$896$	33.1244	1.10661
$897$	−2.53590	−0.0846712
$898$	−9.80385	−0.327159
$899$	73.1769	2.44059
$900$	0	0
$901$	0	0
$902$	10.3923	0.346026
$903$	11.4641	0.381501
$904$	32.7846	1.09040
$905$	0	0
$906$	−24.2487	−0.805609
$907$	32.3923	1.07557	0.537784	−	0.843082i	$-0.319261\pi$
$907$	32.3923	1.07557	0.537784	+	0.843082i	$0.319261\pi$
$908$	−10.3923	−0.344881
$909$	−10.3923	−0.344691
$910$	0	0
$911$	−54.9282	−1.81985	−0.909926	−	0.414770i	$-0.863862\pi$
$911$	−54.9282	−1.81985	−0.909926	+	0.414770i	$0.863862\pi$
$912$	5.00000	0.165567
$913$	61.1769	2.02466
$914$	−7.85641	−0.259867
$915$	0	0
$916$	−18.5359	−0.612443
$917$	−24.9282	−0.823202
$918$	0	0
$919$	51.4256	1.69637	0.848187	−	0.529696i	$-0.177694\pi$
$919$	51.4256	1.69637	0.848187	+	0.529696i	$0.177694\pi$
$920$	0	0
$921$	−32.3923	−1.06736
$922$	10.3923	0.342252
$923$	3.21539	0.105836
$924$	−12.9282	−0.425307
$925$	0	0
$926$	61.5167	2.02156
$927$	−9.85641	−0.323727
$928$	−42.5885	−1.39803
$929$	−1.60770	−0.0527468	−0.0263734	−	0.999652i	$-0.508396\pi$
$929$	−1.60770	−0.0527468	−0.0263734	+	0.999652i	$0.508396\pi$
$930$	0	0
$931$	0.464102	0.0152103
$932$	7.85641	0.257345
$933$	−32.4449	−1.06220
$934$	−35.5692	−1.16386
$935$	0	0
$936$	1.26795	0.0414442
$937$	16.2487	0.530822	0.265411	−	0.964135i	$-0.414492\pi$
$937$	16.2487	0.530822	0.265411	+	0.964135i	$0.414492\pi$
$938$	−37.8564	−1.23606
$939$	6.39230	0.208605
$940$	0	0
$941$	0.588457	0.0191832	0.00959158	−	0.999954i	$-0.496947\pi$
$941$	0.588457	0.0191832	0.00959158	+	0.999954i	$0.496947\pi$
$942$	11.0718	0.360739
$943$	−4.39230	−0.143033
$944$	−12.6795	−0.412682
$945$	0	0
$946$	−34.3923	−1.11819
$947$	−28.1436	−0.914544	−0.457272	−	0.889327i	$-0.651173\pi$
$947$	−28.1436	−0.914544	−0.457272	+	0.889327i	$0.651173\pi$
$948$	10.9282	0.354932
$949$	−12.3923	−0.402271
$950$	0	0
$951$	−11.3205	−0.367093
$952$	0	0
$953$	−37.8564	−1.22629	−0.613145	−	0.789971i	$-0.710096\pi$
$953$	−37.8564	−1.22629	−0.613145	+	0.789971i	$0.710096\pi$
$954$	16.3923	0.530720
$955$	0	0
$956$	−9.80385	−0.317079
$957$	−38.7846	−1.25373
$958$	44.1962	1.42791
$959$	−54.2487	−1.75178
$960$	0	0
$961$	48.7128	1.57138
$962$	−7.85641	−0.253301
$963$	0	0
$964$	−3.07180	−0.0989359
$965$	0	0
$966$	16.3923	0.527414
$967$	−4.87564	−0.156790	−0.0783951	−	0.996922i	$-0.524980\pi$
$967$	−4.87564	−0.156790	−0.0783951	+	0.996922i	$0.524980\pi$
$968$	−19.7321	−0.634212
$969$	0	0
$970$	0	0
$971$	−27.7128	−0.889346	−0.444673	−	0.895693i	$-0.646680\pi$
$971$	−27.7128	−0.889346	−0.444673	+	0.895693i	$0.646680\pi$
$972$	−1.00000	−0.0320750
$973$	−22.9282	−0.735044
$974$	56.1051	1.79772
$975$	0	0
$976$	32.6795	1.04605
$977$	39.0333	1.24879	0.624393	−	0.781110i	$-0.285346\pi$
$977$	39.0333	1.24879	0.624393	+	0.781110i	$0.285346\pi$
$978$	16.0526	0.513304
$979$	50.7846	1.62308
$980$	0	0
$981$	−14.3923	−0.459511
$982$	27.8038	0.887256
$983$	41.3205	1.31792	0.658960	−	0.752178i	$-0.270997\pi$
$983$	41.3205	1.31792	0.658960	+	0.752178i	$0.270997\pi$
$984$	2.19615	0.0700108
$985$	0	0
$986$	0	0
$987$	−9.46410	−0.301246
$988$	−0.732051	−0.0232896
$989$	14.5359	0.462215
$990$	0	0
$991$	13.0718	0.415239	0.207620	−	0.978210i	$-0.433428\pi$
$991$	13.0718	0.415239	0.207620	+	0.978210i	$0.433428\pi$
$992$	−46.3923	−1.47296
$993$	25.7128	0.815971
$994$	−20.7846	−0.659248
$995$	0	0
$996$	−12.9282	−0.409646
$997$	17.6077	0.557641	0.278821	−	0.960343i	$-0.410056\pi$
$997$	17.6077	0.557641	0.278821	+	0.960343i	$0.410056\pi$
$998$	18.2487	0.577653
$999$	−6.19615	−0.196038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.a.o.1.2		2
3.2	odd	2		4275.2.a.t.1.1		2
5.2	odd	4		1425.2.c.k.799.4		4
5.3	odd	4		1425.2.c.k.799.1		4
5.4	even	2		285.2.a.e.1.1	✓	2
15.14	odd	2		855.2.a.f.1.2		2
20.19	odd	2		4560.2.a.bh.1.2		2
95.94	odd	2		5415.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.e.1.1	✓	2	5.4	even	2
855.2.a.f.1.2		2	15.14	odd	2
1425.2.a.o.1.2		2	1.1	even	1	trivial
1425.2.c.k.799.1		4	5.3	odd	4
1425.2.c.k.799.4		4	5.2	odd	4
4275.2.a.t.1.1		2	3.2	odd	2
4560.2.a.bh.1.2		2	20.19	odd	2
5415.2.a.r.1.2		2	95.94	odd	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:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.73205 q^{6} +2.73205 q^{7} -1.73205 q^{8} +1.00000 q^{9} +4.73205 q^{11} -1.00000 q^{12} -0.732051 q^{13} +4.73205 q^{14} -5.00000 q^{16} +1.73205 q^{18} +1.00000 q^{19} -2.73205 q^{21} +8.19615 q^{22} -3.46410 q^{23} +1.73205 q^{24} -1.26795 q^{26} -1.00000 q^{27} +2.73205 q^{28} +8.19615 q^{29} +8.92820 q^{31} -5.19615 q^{32} -4.73205 q^{33} +1.00000 q^{36} +6.19615 q^{37} +1.73205 q^{38} +0.732051 q^{39} +1.26795 q^{41} -4.73205 q^{42} -4.19615 q^{43} +4.73205 q^{44} -6.00000 q^{46} +3.46410 q^{47} +5.00000 q^{48} +0.464102 q^{49} -0.732051 q^{52} +9.46410 q^{53} -1.73205 q^{54} -4.73205 q^{56} -1.00000 q^{57} +14.1962 q^{58} +2.53590 q^{59} -6.53590 q^{61} +15.4641 q^{62} +2.73205 q^{63} +1.00000 q^{64} -8.19615 q^{66} -8.00000 q^{67} +3.46410 q^{69} -4.39230 q^{71} -1.73205 q^{72} +16.9282 q^{73} +10.7321 q^{74} +1.00000 q^{76} +12.9282 q^{77} +1.26795 q^{78} -10.9282 q^{79} +1.00000 q^{81} +2.19615 q^{82} +12.9282 q^{83} -2.73205 q^{84} -7.26795 q^{86} -8.19615 q^{87} -8.19615 q^{88} +10.7321 q^{89} -2.00000 q^{91} -3.46410 q^{92} -8.92820 q^{93} +6.00000 q^{94} +5.19615 q^{96} +6.19615 q^{97} +0.803848 q^{98} +4.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{4} + 2 q^{7} + 2 q^{9} + 6 q^{11} - 2 q^{12} + 2 q^{13} + 6 q^{14} - 10 q^{16} + 2 q^{19} - 2 q^{21} + 6 q^{22} - 6 q^{26} - 2 q^{27} + 2 q^{28} + 6 q^{29} + 4 q^{31} - 6 q^{33} + 2 q^{36}+ \cdots + 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^4 + 2 * q^7 + 2 * q^9 + 6 * q^11 - 2 * q^12 + 2 * q^13 + 6 * q^14 - 10 * q^16 + 2 * q^19 - 2 * q^21 + 6 * q^22 - 6 * q^26 - 2 * q^27 + 2 * q^28 + 6 * q^29 + 4 * q^31 - 6 * q^33 + 2 * q^36 + 2 * q^37 - 2 * q^39 + 6 * q^41 - 6 * q^42 + 2 * q^43 + 6 * q^44 - 12 * q^46 + 10 * q^48 - 6 * q^49 + 2 * q^52 + 12 * q^53 - 6 * q^56 - 2 * q^57 + 18 * q^58 + 12 * q^59 - 20 * q^61 + 24 * q^62 + 2 * q^63 + 2 * q^64 - 6 * q^66 - 16 * q^67 + 12 * q^71 + 20 * q^73 + 18 * q^74 + 2 * q^76 + 12 * q^77 + 6 * q^78 - 8 * q^79 + 2 * q^81 - 6 * q^82 + 12 * q^83 - 2 * q^84 - 18 * q^86 - 6 * q^87 - 6 * q^88 + 18 * q^89 - 4 * q^91 - 4 * q^93 + 12 * q^94 + 2 * q^97 + 12 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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