LMFDB - Embedded newform 1450.2.a.j.1.2

Error: no document with id 216593768 found in table mf_hecke_traces.

Error: table True does not exist

Error: no document with id 215039696 found in table mf_hecke_traces.

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1450,2,Mod(1,1450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1450.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1450 = 2 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1450.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,2,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$11.5783082931$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	1450.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} -4.44949 q^{7} -1.00000 q^{8} +3.00000 q^{9} +2.00000 q^{11} +2.44949 q^{12} +2.44949 q^{13} +4.44949 q^{14} +1.00000 q^{16} +2.00000 q^{17} -3.00000 q^{18} +6.44949 q^{19} -10.8990 q^{21} -2.00000 q^{22} +2.44949 q^{23} -2.44949 q^{24} -2.44949 q^{26} -4.44949 q^{28} -1.00000 q^{29} -3.00000 q^{31} -1.00000 q^{32} +4.89898 q^{33} -2.00000 q^{34} +3.00000 q^{36} +3.44949 q^{37} -6.44949 q^{38} +6.00000 q^{39} +11.3485 q^{41} +10.8990 q^{42} +8.89898 q^{43} +2.00000 q^{44} -2.44949 q^{46} -5.89898 q^{47} +2.44949 q^{48} +12.7980 q^{49} +4.89898 q^{51} +2.44949 q^{52} -10.4495 q^{53} +4.44949 q^{56} +15.7980 q^{57} +1.00000 q^{58} +8.55051 q^{59} +3.44949 q^{61} +3.00000 q^{62} -13.3485 q^{63} +1.00000 q^{64} -4.89898 q^{66} +13.4495 q^{67} +2.00000 q^{68} +6.00000 q^{69} +11.3485 q^{71} -3.00000 q^{72} -13.3485 q^{73} -3.44949 q^{74} +6.44949 q^{76} -8.89898 q^{77} -6.00000 q^{78} -2.89898 q^{79} -9.00000 q^{81} -11.3485 q^{82} +6.00000 q^{83} -10.8990 q^{84} -8.89898 q^{86} -2.44949 q^{87} -2.00000 q^{88} +3.55051 q^{89} -10.8990 q^{91} +2.44949 q^{92} -7.34847 q^{93} +5.89898 q^{94} -2.44949 q^{96} -7.34847 q^{97} -12.7980 q^{98} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{7} - 2 q^{8} + 6 q^{9} + 4 q^{11} + 4 q^{14} + 2 q^{16} + 4 q^{17} - 6 q^{18} + 8 q^{19} - 12 q^{21} - 4 q^{22} - 4 q^{28} - 2 q^{29} - 6 q^{31} - 2 q^{32} - 4 q^{34} + 6 q^{36}+ \cdots + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^7 - 2 * q^8 + 6 * q^9 + 4 * q^11 + 4 * q^14 + 2 * q^16 + 4 * q^17 - 6 * q^18 + 8 * q^19 - 12 * q^21 - 4 * q^22 - 4 * q^28 - 2 * q^29 - 6 * q^31 - 2 * q^32 - 4 * q^34 + 6 * q^36 + 2 * q^37 - 8 * q^38 + 12 * q^39 + 8 * q^41 + 12 * q^42 + 8 * q^43 + 4 * q^44 - 2 * q^47 + 6 * q^49 - 16 * q^53 + 4 * q^56 + 12 * q^57 + 2 * q^58 + 22 * q^59 + 2 * q^61 + 6 * q^62 - 12 * q^63 + 2 * q^64 + 22 * q^67 + 4 * q^68 + 12 * q^69 + 8 * q^71 - 6 * q^72 - 12 * q^73 - 2 * q^74 + 8 * q^76 - 8 * q^77 - 12 * q^78 + 4 * q^79 - 18 * q^81 - 8 * q^82 + 12 * q^83 - 12 * q^84 - 8 * q^86 - 4 * q^88 + 12 * q^89 - 12 * q^91 + 2 * q^94 - 6 * q^98 + 12 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	2.44949	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$3$	2.44949	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.44949	−1.00000
$7$	−4.44949	−1.68175	−0.840875	−	0.541230i	$-0.817959\pi$
$7$	−4.44949	−1.68175	−0.840875	+	0.541230i	$0.817959\pi$
$8$	−1.00000	−0.353553
$9$	3.00000	1.00000
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	2.44949	0.707107
$13$	2.44949	0.679366	0.339683	−	0.940540i	$-0.389680\pi$
$13$	2.44949	0.679366	0.339683	+	0.940540i	$0.389680\pi$
$14$	4.44949	1.18918
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−3.00000	−0.707107
$19$	6.44949	1.47961	0.739807	−	0.672819i	$-0.234917\pi$
$19$	6.44949	1.47961	0.739807	+	0.672819i	$0.234917\pi$
$20$	0	0
$21$	−10.8990	−2.37835
$22$	−2.00000	−0.426401
$23$	2.44949	0.510754	0.255377	−	0.966842i	$-0.417800\pi$
$23$	2.44949	0.510754	0.255377	+	0.966842i	$0.417800\pi$
$24$	−2.44949	−0.500000
$25$	0	0
$26$	−2.44949	−0.480384
$27$	0	0
$28$	−4.44949	−0.840875
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	−1.00000	−0.176777
$33$	4.89898	0.852803
$34$	−2.00000	−0.342997
$35$	0	0
$36$	3.00000	0.500000
$37$	3.44949	0.567093	0.283546	−	0.958959i	$-0.408489\pi$
$37$	3.44949	0.567093	0.283546	+	0.958959i	$0.408489\pi$
$38$	−6.44949	−1.04625
$39$	6.00000	0.960769
$40$	0	0
$41$	11.3485	1.77233	0.886167	−	0.463367i	$-0.153359\pi$
$41$	11.3485	1.77233	0.886167	+	0.463367i	$0.153359\pi$
$42$	10.8990	1.68175
$43$	8.89898	1.35708	0.678541	−	0.734563i	$-0.262613\pi$
$43$	8.89898	1.35708	0.678541	+	0.734563i	$0.262613\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	−2.44949	−0.361158
$47$	−5.89898	−0.860455	−0.430227	−	0.902721i	$-0.641567\pi$
$47$	−5.89898	−0.860455	−0.430227	+	0.902721i	$0.641567\pi$
$48$	2.44949	0.353553
$49$	12.7980	1.82828
$50$	0	0
$51$	4.89898	0.685994
$52$	2.44949	0.339683
$53$	−10.4495	−1.43535	−0.717674	−	0.696379i	$-0.754793\pi$
$53$	−10.4495	−1.43535	−0.717674	+	0.696379i	$0.754793\pi$
$54$	0	0
$55$	0	0
$56$	4.44949	0.594588
$57$	15.7980	2.09249
$58$	1.00000	0.131306
$59$	8.55051	1.11318	0.556591	−	0.830787i	$-0.312109\pi$
$59$	8.55051	1.11318	0.556591	+	0.830787i	$0.312109\pi$
$60$	0	0
$61$	3.44949	0.441662	0.220831	−	0.975312i	$-0.429123\pi$
$61$	3.44949	0.441662	0.220831	+	0.975312i	$0.429123\pi$
$62$	3.00000	0.381000
$63$	−13.3485	−1.68175
$64$	1.00000	0.125000
$65$	0	0
$66$	−4.89898	−0.603023
$67$	13.4495	1.64312	0.821558	−	0.570124i	$-0.193105\pi$
$67$	13.4495	1.64312	0.821558	+	0.570124i	$0.193105\pi$
$68$	2.00000	0.242536
$69$	6.00000	0.722315
$70$	0	0
$71$	11.3485	1.34682	0.673408	−	0.739271i	$-0.264830\pi$
$71$	11.3485	1.34682	0.673408	+	0.739271i	$0.264830\pi$
$72$	−3.00000	−0.353553
$73$	−13.3485	−1.56232	−0.781160	−	0.624331i	$-0.785372\pi$
$73$	−13.3485	−1.56232	−0.781160	+	0.624331i	$0.785372\pi$
$74$	−3.44949	−0.400995
$75$	0	0
$76$	6.44949	0.739807
$77$	−8.89898	−1.01413
$78$	−6.00000	−0.679366
$79$	−2.89898	−0.326161	−0.163080	−	0.986613i	$-0.552143\pi$
$79$	−2.89898	−0.326161	−0.163080	+	0.986613i	$0.552143\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	−11.3485	−1.25323
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	−10.8990	−1.18918
$85$	0	0
$86$	−8.89898	−0.959602
$87$	−2.44949	−0.262613
$88$	−2.00000	−0.213201
$89$	3.55051	0.376353	0.188177	−	0.982135i	$-0.439742\pi$
$89$	3.55051	0.376353	0.188177	+	0.982135i	$0.439742\pi$
$90$	0	0
$91$	−10.8990	−1.14252
$92$	2.44949	0.255377
$93$	−7.34847	−0.762001
$94$	5.89898	0.608433
$95$	0	0
$96$	−2.44949	−0.250000
$97$	−7.34847	−0.746124	−0.373062	−	0.927806i	$-0.621692\pi$
$97$	−7.34847	−0.746124	−0.373062	+	0.927806i	$0.621692\pi$
$98$	−12.7980	−1.29279
$99$	6.00000	0.603023
$100$	0	0
$101$	13.4495	1.33827	0.669137	−	0.743139i	$-0.266664\pi$
$101$	13.4495	1.33827	0.669137	+	0.743139i	$0.266664\pi$
$102$	−4.89898	−0.485071
$103$	2.44949	0.241355	0.120678	−	0.992692i	$-0.461493\pi$
$103$	2.44949	0.241355	0.120678	+	0.992692i	$0.461493\pi$
$104$	−2.44949	−0.240192
$105$	0	0
$106$	10.4495	1.01494
$107$	−5.24745	−0.507290	−0.253645	−	0.967297i	$-0.581630\pi$
$107$	−5.24745	−0.507290	−0.253645	+	0.967297i	$0.581630\pi$
$108$	0	0
$109$	−9.34847	−0.895421	−0.447710	−	0.894179i	$-0.647760\pi$
$109$	−9.34847	−0.895421	−0.447710	+	0.894179i	$0.647760\pi$
$110$	0	0
$111$	8.44949	0.801990
$112$	−4.44949	−0.420437
$113$	−6.24745	−0.587711	−0.293855	−	0.955850i	$-0.594938\pi$
$113$	−6.24745	−0.587711	−0.293855	+	0.955850i	$0.594938\pi$
$114$	−15.7980	−1.47961
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	7.34847	0.679366
$118$	−8.55051	−0.787138
$119$	−8.89898	−0.815768
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−3.44949	−0.312302
$123$	27.7980	2.50646
$124$	−3.00000	−0.269408
$125$	0	0
$126$	13.3485	1.18918
$127$	−8.79796	−0.780693	−0.390346	−	0.920668i	$-0.627645\pi$
$127$	−8.79796	−0.780693	−0.390346	+	0.920668i	$0.627645\pi$
$128$	−1.00000	−0.0883883
$129$	21.7980	1.91920
$130$	0	0
$131$	−7.34847	−0.642039	−0.321019	−	0.947073i	$-0.604025\pi$
$131$	−7.34847	−0.642039	−0.321019	+	0.947073i	$0.604025\pi$
$132$	4.89898	0.426401
$133$	−28.6969	−2.48834
$134$	−13.4495	−1.16186
$135$	0	0
$136$	−2.00000	−0.171499
$137$	−0.898979	−0.0768050	−0.0384025	−	0.999262i	$-0.512227\pi$
$137$	−0.898979	−0.0768050	−0.0384025	+	0.999262i	$0.512227\pi$
$138$	−6.00000	−0.510754
$139$	−11.4495	−0.971133	−0.485567	−	0.874200i	$-0.661387\pi$
$139$	−11.4495	−0.971133	−0.485567	+	0.874200i	$0.661387\pi$
$140$	0	0
$141$	−14.4495	−1.21687
$142$	−11.3485	−0.952342
$143$	4.89898	0.409673
$144$	3.00000	0.250000
$145$	0	0
$146$	13.3485	1.10473
$147$	31.3485	2.58558
$148$	3.44949	0.283546
$149$	15.7980	1.29422	0.647110	−	0.762397i	$-0.275978\pi$
$149$	15.7980	1.29422	0.647110	+	0.762397i	$0.275978\pi$
$150$	0	0
$151$	11.3485	0.923525	0.461763	−	0.887004i	$-0.347217\pi$
$151$	11.3485	0.923525	0.461763	+	0.887004i	$0.347217\pi$
$152$	−6.44949	−0.523123
$153$	6.00000	0.485071
$154$	8.89898	0.717100
$155$	0	0
$156$	6.00000	0.480384
$157$	−12.3485	−0.985515	−0.492758	−	0.870167i	$-0.664011\pi$
$157$	−12.3485	−0.985515	−0.492758	+	0.870167i	$0.664011\pi$
$158$	2.89898	0.230630
$159$	−25.5959	−2.02989
$160$	0	0
$161$	−10.8990	−0.858960
$162$	9.00000	0.707107
$163$	12.4495	0.975119	0.487560	−	0.873090i	$-0.337887\pi$
$163$	12.4495	0.975119	0.487560	+	0.873090i	$0.337887\pi$
$164$	11.3485	0.886167
$165$	0	0
$166$	−6.00000	−0.465690
$167$	17.7980	1.37725	0.688624	−	0.725119i	$-0.258215\pi$
$167$	17.7980	1.37725	0.688624	+	0.725119i	$0.258215\pi$
$168$	10.8990	0.840875
$169$	−7.00000	−0.538462
$170$	0	0
$171$	19.3485	1.47961
$172$	8.89898	0.678541
$173$	16.0000	1.21646	0.608229	−	0.793762i	$-0.291880\pi$
$173$	16.0000	1.21646	0.608229	+	0.793762i	$0.291880\pi$
$174$	2.44949	0.185695
$175$	0	0
$176$	2.00000	0.150756
$177$	20.9444	1.57428
$178$	−3.55051	−0.266122
$179$	−17.2474	−1.28913	−0.644567	−	0.764547i	$-0.722962\pi$
$179$	−17.2474	−1.28913	−0.644567	+	0.764547i	$0.722962\pi$
$180$	0	0
$181$	−26.0454	−1.93594	−0.967970	−	0.251066i	$-0.919219\pi$
$181$	−26.0454	−1.93594	−0.967970	+	0.251066i	$0.919219\pi$
$182$	10.8990	0.807886
$183$	8.44949	0.624604
$184$	−2.44949	−0.180579
$185$	0	0
$186$	7.34847	0.538816
$187$	4.00000	0.292509
$188$	−5.89898	−0.430227
$189$	0	0
$190$	0	0
$191$	−11.6969	−0.846361	−0.423180	−	0.906045i	$-0.639086\pi$
$191$	−11.6969	−0.846361	−0.423180	+	0.906045i	$0.639086\pi$
$192$	2.44949	0.176777
$193$	5.34847	0.384991	0.192496	−	0.981298i	$-0.438342\pi$
$193$	5.34847	0.384991	0.192496	+	0.981298i	$0.438342\pi$
$194$	7.34847	0.527589
$195$	0	0
$196$	12.7980	0.914140
$197$	−5.10102	−0.363433	−0.181716	−	0.983351i	$-0.558165\pi$
$197$	−5.10102	−0.363433	−0.181716	+	0.983351i	$0.558165\pi$
$198$	−6.00000	−0.426401
$199$	−0.651531	−0.0461858	−0.0230929	−	0.999733i	$-0.507351\pi$
$199$	−0.651531	−0.0461858	−0.0230929	+	0.999733i	$0.507351\pi$
$200$	0	0
$201$	32.9444	2.32372
$202$	−13.4495	−0.946303
$203$	4.44949	0.312293
$204$	4.89898	0.342997
$205$	0	0
$206$	−2.44949	−0.170664
$207$	7.34847	0.510754
$208$	2.44949	0.169842
$209$	12.8990	0.892241
$210$	0	0
$211$	−22.4949	−1.54861	−0.774306	−	0.632811i	$-0.781901\pi$
$211$	−22.4949	−1.54861	−0.774306	+	0.632811i	$0.781901\pi$
$212$	−10.4495	−0.717674
$213$	27.7980	1.90468
$214$	5.24745	0.358708
$215$	0	0
$216$	0	0
$217$	13.3485	0.906153
$218$	9.34847	0.633158
$219$	−32.6969	−2.20945
$220$	0	0
$221$	4.89898	0.329541
$222$	−8.44949	−0.567093
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	4.44949	0.297294
$225$	0	0
$226$	6.24745	0.415574
$227$	−15.2474	−1.01201	−0.506004	−	0.862531i	$-0.668878\pi$
$227$	−15.2474	−1.01201	−0.506004	+	0.862531i	$0.668878\pi$
$228$	15.7980	1.04625
$229$	−17.1010	−1.13007	−0.565034	−	0.825068i	$-0.691137\pi$
$229$	−17.1010	−1.13007	−0.565034	+	0.825068i	$0.691137\pi$
$230$	0	0
$231$	−21.7980	−1.43420
$232$	1.00000	0.0656532
$233$	11.0000	0.720634	0.360317	−	0.932830i	$-0.382669\pi$
$233$	11.0000	0.720634	0.360317	+	0.932830i	$0.382669\pi$
$234$	−7.34847	−0.480384
$235$	0	0
$236$	8.55051	0.556591
$237$	−7.10102	−0.461261
$238$	8.89898	0.576835
$239$	21.5959	1.39692	0.698462	−	0.715647i	$-0.253868\pi$
$239$	21.5959	1.39692	0.698462	+	0.715647i	$0.253868\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	7.00000	0.449977
$243$	−22.0454	−1.41421
$244$	3.44949	0.220831
$245$	0	0
$246$	−27.7980	−1.77233
$247$	15.7980	1.00520
$248$	3.00000	0.190500
$249$	14.6969	0.931381
$250$	0	0
$251$	−3.14643	−0.198601	−0.0993004	−	0.995058i	$-0.531660\pi$
$251$	−3.14643	−0.198601	−0.0993004	+	0.995058i	$0.531660\pi$
$252$	−13.3485	−0.840875
$253$	4.89898	0.307996
$254$	8.79796	0.552033
$255$	0	0
$256$	1.00000	0.0625000
$257$	−26.6969	−1.66531	−0.832655	−	0.553793i	$-0.813180\pi$
$257$	−26.6969	−1.66531	−0.832655	+	0.553793i	$0.813180\pi$
$258$	−21.7980	−1.35708
$259$	−15.3485	−0.953707
$260$	0	0
$261$	−3.00000	−0.185695
$262$	7.34847	0.453990
$263$	16.7980	1.03581	0.517903	−	0.855439i	$-0.326713\pi$
$263$	16.7980	1.03581	0.517903	+	0.855439i	$0.326713\pi$
$264$	−4.89898	−0.301511
$265$	0	0
$266$	28.6969	1.75952
$267$	8.69694	0.532244
$268$	13.4495	0.821558
$269$	24.3485	1.48455	0.742276	−	0.670094i	$-0.233746\pi$
$269$	24.3485	1.48455	0.742276	+	0.670094i	$0.233746\pi$
$270$	0	0
$271$	15.6969	0.953521	0.476761	−	0.879033i	$-0.341811\pi$
$271$	15.6969	0.953521	0.476761	+	0.879033i	$0.341811\pi$
$272$	2.00000	0.121268
$273$	−26.6969	−1.61577
$274$	0.898979	0.0543093
$275$	0	0
$276$	6.00000	0.361158
$277$	22.9444	1.37859	0.689297	−	0.724479i	$-0.257919\pi$
$277$	22.9444	1.37859	0.689297	+	0.724479i	$0.257919\pi$
$278$	11.4495	0.686695
$279$	−9.00000	−0.538816
$280$	0	0
$281$	−28.7980	−1.71794	−0.858971	−	0.512024i	$-0.828896\pi$
$281$	−28.7980	−1.71794	−0.858971	+	0.512024i	$0.828896\pi$
$282$	14.4495	0.860455
$283$	−24.0000	−1.42665	−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000	−1.42665	−0.713326	+	0.700832i	$0.752812\pi$
$284$	11.3485	0.673408
$285$	0	0
$286$	−4.89898	−0.289683
$287$	−50.4949	−2.98062
$288$	−3.00000	−0.176777
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−18.0000	−1.05518
$292$	−13.3485	−0.781160
$293$	7.44949	0.435204	0.217602	−	0.976038i	$-0.430177\pi$
$293$	7.44949	0.435204	0.217602	+	0.976038i	$0.430177\pi$
$294$	−31.3485	−1.82828
$295$	0	0
$296$	−3.44949	−0.200498
$297$	0	0
$298$	−15.7980	−0.915151
$299$	6.00000	0.346989
$300$	0	0
$301$	−39.5959	−2.28227
$302$	−11.3485	−0.653031
$303$	32.9444	1.89261
$304$	6.44949	0.369904
$305$	0	0
$306$	−6.00000	−0.342997
$307$	−25.3939	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$307$	−25.3939	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$308$	−8.89898	−0.507066
$309$	6.00000	0.341328
$310$	0	0
$311$	−19.5959	−1.11118	−0.555591	−	0.831456i	$-0.687508\pi$
$311$	−19.5959	−1.11118	−0.555591	+	0.831456i	$0.687508\pi$
$312$	−6.00000	−0.339683
$313$	32.5959	1.84243	0.921215	−	0.389054i	$-0.127198\pi$
$313$	32.5959	1.84243	0.921215	+	0.389054i	$0.127198\pi$
$314$	12.3485	0.696864
$315$	0	0
$316$	−2.89898	−0.163080
$317$	−13.6515	−0.766746	−0.383373	−	0.923594i	$-0.625238\pi$
$317$	−13.6515	−0.766746	−0.383373	+	0.923594i	$0.625238\pi$
$318$	25.5959	1.43535
$319$	−2.00000	−0.111979
$320$	0	0
$321$	−12.8536	−0.717416
$322$	10.8990	0.607376
$323$	12.8990	0.717718
$324$	−9.00000	−0.500000
$325$	0	0
$326$	−12.4495	−0.689513
$327$	−22.8990	−1.26632
$328$	−11.3485	−0.626614
$329$	26.2474	1.44707
$330$	0	0
$331$	−30.2474	−1.66255	−0.831275	−	0.555861i	$-0.812389\pi$
$331$	−30.2474	−1.66255	−0.831275	+	0.555861i	$0.812389\pi$
$332$	6.00000	0.329293
$333$	10.3485	0.567093
$334$	−17.7980	−0.973861
$335$	0	0
$336$	−10.8990	−0.594588
$337$	−23.7980	−1.29636	−0.648179	−	0.761488i	$-0.724469\pi$
$337$	−23.7980	−1.29636	−0.648179	+	0.761488i	$0.724469\pi$
$338$	7.00000	0.380750
$339$	−15.3031	−0.831148
$340$	0	0
$341$	−6.00000	−0.324918
$342$	−19.3485	−1.04625
$343$	−25.7980	−1.39296
$344$	−8.89898	−0.479801
$345$	0	0
$346$	−16.0000	−0.860165
$347$	25.0454	1.34451	0.672254	−	0.740321i	$-0.265326\pi$
$347$	25.0454	1.34451	0.672254	+	0.740321i	$0.265326\pi$
$348$	−2.44949	−0.131306
$349$	−30.9444	−1.65642	−0.828208	−	0.560422i	$-0.810639\pi$
$349$	−30.9444	−1.65642	−0.828208	+	0.560422i	$0.810639\pi$
$350$	0	0
$351$	0	0
$352$	−2.00000	−0.106600
$353$	16.7980	0.894065	0.447033	−	0.894518i	$-0.352481\pi$
$353$	16.7980	0.894065	0.447033	+	0.894518i	$0.352481\pi$
$354$	−20.9444	−1.11318
$355$	0	0
$356$	3.55051	0.188177
$357$	−21.7980	−1.15367
$358$	17.2474	0.911556
$359$	−16.5959	−0.875899	−0.437950	−	0.899000i	$-0.644295\pi$
$359$	−16.5959	−0.875899	−0.437950	+	0.899000i	$0.644295\pi$
$360$	0	0
$361$	22.5959	1.18926
$362$	26.0454	1.36892
$363$	−17.1464	−0.899954
$364$	−10.8990	−0.571262
$365$	0	0
$366$	−8.44949	−0.441662
$367$	1.20204	0.0627460	0.0313730	−	0.999508i	$-0.490012\pi$
$367$	1.20204	0.0627460	0.0313730	+	0.999508i	$0.490012\pi$
$368$	2.44949	0.127688
$369$	34.0454	1.77233
$370$	0	0
$371$	46.4949	2.41389
$372$	−7.34847	−0.381000
$373$	−28.4949	−1.47541	−0.737705	−	0.675123i	$-0.764090\pi$
$373$	−28.4949	−1.47541	−0.737705	+	0.675123i	$0.764090\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	5.89898	0.304217
$377$	−2.44949	−0.126155
$378$	0	0
$379$	25.1464	1.29169	0.645843	−	0.763471i	$-0.276506\pi$
$379$	25.1464	1.29169	0.645843	+	0.763471i	$0.276506\pi$
$380$	0	0
$381$	−21.5505	−1.10407
$382$	11.6969	0.598467
$383$	11.7980	0.602848	0.301424	−	0.953490i	$-0.402538\pi$
$383$	11.7980	0.602848	0.301424	+	0.953490i	$0.402538\pi$
$384$	−2.44949	−0.125000
$385$	0	0
$386$	−5.34847	−0.272230
$387$	26.6969	1.35708
$388$	−7.34847	−0.373062
$389$	−24.3485	−1.23452	−0.617258	−	0.786761i	$-0.711757\pi$
$389$	−24.3485	−1.23452	−0.617258	+	0.786761i	$0.711757\pi$
$390$	0	0
$391$	4.89898	0.247752
$392$	−12.7980	−0.646395
$393$	−18.0000	−0.907980
$394$	5.10102	0.256986
$395$	0	0
$396$	6.00000	0.301511
$397$	−13.1464	−0.659800	−0.329900	−	0.944016i	$-0.607015\pi$
$397$	−13.1464	−0.659800	−0.329900	+	0.944016i	$0.607015\pi$
$398$	0.651531	0.0326583
$399$	−70.2929	−3.51904
$400$	0	0
$401$	38.5959	1.92739	0.963694	−	0.267009i	$-0.0860353\pi$
$401$	38.5959	1.92739	0.963694	+	0.267009i	$0.0860353\pi$
$402$	−32.9444	−1.64312
$403$	−7.34847	−0.366053
$404$	13.4495	0.669137
$405$	0	0
$406$	−4.44949	−0.220824
$407$	6.89898	0.341970
$408$	−4.89898	−0.242536
$409$	−5.79796	−0.286691	−0.143345	−	0.989673i	$-0.545786\pi$
$409$	−5.79796	−0.286691	−0.143345	+	0.989673i	$0.545786\pi$
$410$	0	0
$411$	−2.20204	−0.108619
$412$	2.44949	0.120678
$413$	−38.0454	−1.87209
$414$	−7.34847	−0.361158
$415$	0	0
$416$	−2.44949	−0.120096
$417$	−28.0454	−1.37339
$418$	−12.8990	−0.630910
$419$	−10.1464	−0.495685	−0.247843	−	0.968800i	$-0.579722\pi$
$419$	−10.1464	−0.495685	−0.247843	+	0.968800i	$0.579722\pi$
$420$	0	0
$421$	13.4495	0.655488	0.327744	−	0.944767i	$-0.393712\pi$
$421$	13.4495	0.655488	0.327744	+	0.944767i	$0.393712\pi$
$422$	22.4949	1.09503
$423$	−17.6969	−0.860455
$424$	10.4495	0.507472
$425$	0	0
$426$	−27.7980	−1.34682
$427$	−15.3485	−0.742764
$428$	−5.24745	−0.253645
$429$	12.0000	0.579365
$430$	0	0
$431$	18.4495	0.888681	0.444340	−	0.895858i	$-0.353438\pi$
$431$	18.4495	0.888681	0.444340	+	0.895858i	$0.353438\pi$
$432$	0	0
$433$	−30.4495	−1.46331	−0.731655	−	0.681676i	$-0.761252\pi$
$433$	−30.4495	−1.46331	−0.731655	+	0.681676i	$0.761252\pi$
$434$	−13.3485	−0.640747
$435$	0	0
$436$	−9.34847	−0.447710
$437$	15.7980	0.755719
$438$	32.6969	1.56232
$439$	18.6969	0.892356	0.446178	−	0.894944i	$-0.352785\pi$
$439$	18.6969	0.892356	0.446178	+	0.894944i	$0.352785\pi$
$440$	0	0
$441$	38.3939	1.82828
$442$	−4.89898	−0.233021
$443$	13.1010	0.622448	0.311224	−	0.950337i	$-0.399261\pi$
$443$	13.1010	0.622448	0.311224	+	0.950337i	$0.399261\pi$
$444$	8.44949	0.400995
$445$	0	0
$446$	4.00000	0.189405
$447$	38.6969	1.83030
$448$	−4.44949	−0.210219
$449$	25.7980	1.21748	0.608740	−	0.793369i	$-0.291675\pi$
$449$	25.7980	1.21748	0.608740	+	0.793369i	$0.291675\pi$
$450$	0	0
$451$	22.6969	1.06876
$452$	−6.24745	−0.293855
$453$	27.7980	1.30606
$454$	15.2474	0.715598
$455$	0	0
$456$	−15.7980	−0.739807
$457$	17.7980	0.832553	0.416277	−	0.909238i	$-0.363335\pi$
$457$	17.7980	0.832553	0.416277	+	0.909238i	$0.363335\pi$
$458$	17.1010	0.799078
$459$	0	0
$460$	0	0
$461$	−28.2929	−1.31773	−0.658865	−	0.752261i	$-0.728963\pi$
$461$	−28.2929	−1.31773	−0.658865	+	0.752261i	$0.728963\pi$
$462$	21.7980	1.01413
$463$	3.10102	0.144117	0.0720583	−	0.997400i	$-0.477043\pi$
$463$	3.10102	0.144117	0.0720583	+	0.997400i	$0.477043\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	−11.0000	−0.509565
$467$	22.9444	1.06174	0.530870	−	0.847453i	$-0.321865\pi$
$467$	22.9444	1.06174	0.530870	+	0.847453i	$0.321865\pi$
$468$	7.34847	0.339683
$469$	−59.8434	−2.76331
$470$	0	0
$471$	−30.2474	−1.39373
$472$	−8.55051	−0.393569
$473$	17.7980	0.818351
$474$	7.10102	0.326161
$475$	0	0
$476$	−8.89898	−0.407884
$477$	−31.3485	−1.43535
$478$	−21.5959	−0.987774
$479$	−4.20204	−0.191996	−0.0959981	−	0.995382i	$-0.530604\pi$
$479$	−4.20204	−0.191996	−0.0959981	+	0.995382i	$0.530604\pi$
$480$	0	0
$481$	8.44949	0.385264
$482$	−17.0000	−0.774329
$483$	−26.6969	−1.21475
$484$	−7.00000	−0.318182
$485$	0	0
$486$	22.0454	1.00000
$487$	9.10102	0.412407	0.206203	−	0.978509i	$-0.433889\pi$
$487$	9.10102	0.412407	0.206203	+	0.978509i	$0.433889\pi$
$488$	−3.44949	−0.156151
$489$	30.4949	1.37903
$490$	0	0
$491$	26.4949	1.19570	0.597849	−	0.801609i	$-0.296022\pi$
$491$	26.4949	1.19570	0.597849	+	0.801609i	$0.296022\pi$
$492$	27.7980	1.25323
$493$	−2.00000	−0.0900755
$494$	−15.7980	−0.710784
$495$	0	0
$496$	−3.00000	−0.134704
$497$	−50.4949	−2.26501
$498$	−14.6969	−0.658586
$499$	−24.3485	−1.08999	−0.544994	−	0.838440i	$-0.683468\pi$
$499$	−24.3485	−1.08999	−0.544994	+	0.838440i	$0.683468\pi$
$500$	0	0
$501$	43.5959	1.94772
$502$	3.14643	0.140432
$503$	−4.79796	−0.213930	−0.106965	−	0.994263i	$-0.534113\pi$
$503$	−4.79796	−0.213930	−0.106965	+	0.994263i	$0.534113\pi$
$504$	13.3485	0.594588
$505$	0	0
$506$	−4.89898	−0.217786
$507$	−17.1464	−0.761500
$508$	−8.79796	−0.390346
$509$	−17.3939	−0.770970	−0.385485	−	0.922714i	$-0.625966\pi$
$509$	−17.3939	−0.770970	−0.385485	+	0.922714i	$0.625966\pi$
$510$	0	0
$511$	59.3939	2.62743
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	26.6969	1.17755
$515$	0	0
$516$	21.7980	0.959602
$517$	−11.7980	−0.518874
$518$	15.3485	0.674373
$519$	39.1918	1.72033
$520$	0	0
$521$	−10.1010	−0.442534	−0.221267	−	0.975213i	$-0.571019\pi$
$521$	−10.1010	−0.442534	−0.221267	+	0.975213i	$0.571019\pi$
$522$	3.00000	0.131306
$523$	27.4495	1.20028	0.600141	−	0.799894i	$-0.295111\pi$
$523$	27.4495	1.20028	0.600141	+	0.799894i	$0.295111\pi$
$524$	−7.34847	−0.321019
$525$	0	0
$526$	−16.7980	−0.732426
$527$	−6.00000	−0.261364
$528$	4.89898	0.213201
$529$	−17.0000	−0.739130
$530$	0	0
$531$	25.6515	1.11318
$532$	−28.6969	−1.24417
$533$	27.7980	1.20406
$534$	−8.69694	−0.376353
$535$	0	0
$536$	−13.4495	−0.580929
$537$	−42.2474	−1.82311
$538$	−24.3485	−1.04974
$539$	25.5959	1.10249
$540$	0	0
$541$	−21.0454	−0.904813	−0.452406	−	0.891812i	$-0.649434\pi$
$541$	−21.0454	−0.904813	−0.452406	+	0.891812i	$0.649434\pi$
$542$	−15.6969	−0.674241
$543$	−63.7980	−2.73783
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	26.6969	1.14252
$547$	22.0000	0.940652	0.470326	−	0.882493i	$-0.344136\pi$
$547$	22.0000	0.940652	0.470326	+	0.882493i	$0.344136\pi$
$548$	−0.898979	−0.0384025
$549$	10.3485	0.441662
$550$	0	0
$551$	−6.44949	−0.274758
$552$	−6.00000	−0.255377
$553$	12.8990	0.548520
$554$	−22.9444	−0.974814
$555$	0	0
$556$	−11.4495	−0.485567
$557$	6.49490	0.275198	0.137599	−	0.990488i	$-0.456062\pi$
$557$	6.49490	0.275198	0.137599	+	0.990488i	$0.456062\pi$
$558$	9.00000	0.381000
$559$	21.7980	0.921955
$560$	0	0
$561$	9.79796	0.413670
$562$	28.7980	1.21477
$563$	21.7980	0.918674	0.459337	−	0.888262i	$-0.348087\pi$
$563$	21.7980	0.918674	0.459337	+	0.888262i	$0.348087\pi$
$564$	−14.4495	−0.608433
$565$	0	0
$566$	24.0000	1.00880
$567$	40.0454	1.68175
$568$	−11.3485	−0.476171
$569$	−8.69694	−0.364595	−0.182297	−	0.983243i	$-0.558353\pi$
$569$	−8.69694	−0.364595	−0.182297	+	0.983243i	$0.558353\pi$
$570$	0	0
$571$	25.0454	1.04812	0.524059	−	0.851682i	$-0.324417\pi$
$571$	25.0454	1.04812	0.524059	+	0.851682i	$0.324417\pi$
$572$	4.89898	0.204837
$573$	−28.6515	−1.19693
$574$	50.4949	2.10762
$575$	0	0
$576$	3.00000	0.125000
$577$	−28.9444	−1.20497	−0.602485	−	0.798130i	$-0.705823\pi$
$577$	−28.9444	−1.20497	−0.602485	+	0.798130i	$0.705823\pi$
$578$	13.0000	0.540729
$579$	13.1010	0.544460
$580$	0	0
$581$	−26.6969	−1.10758
$582$	18.0000	0.746124
$583$	−20.8990	−0.865547
$584$	13.3485	0.552364
$585$	0	0
$586$	−7.44949	−0.307736
$587$	14.7526	0.608903	0.304451	−	0.952528i	$-0.401527\pi$
$587$	14.7526	0.608903	0.304451	+	0.952528i	$0.401527\pi$
$588$	31.3485	1.29279
$589$	−19.3485	−0.797240
$590$	0	0
$591$	−12.4949	−0.513971
$592$	3.44949	0.141773
$593$	18.3939	0.755346	0.377673	−	0.925939i	$-0.376724\pi$
$593$	18.3939	0.755346	0.377673	+	0.925939i	$0.376724\pi$
$594$	0	0
$595$	0	0
$596$	15.7980	0.647110
$597$	−1.59592	−0.0653166
$598$	−6.00000	−0.245358
$599$	−12.1010	−0.494434	−0.247217	−	0.968960i	$-0.579516\pi$
$599$	−12.1010	−0.494434	−0.247217	+	0.968960i	$0.579516\pi$
$600$	0	0
$601$	−6.69694	−0.273174	−0.136587	−	0.990628i	$-0.543613\pi$
$601$	−6.69694	−0.273174	−0.136587	+	0.990628i	$0.543613\pi$
$602$	39.5959	1.61381
$603$	40.3485	1.64312
$604$	11.3485	0.461763
$605$	0	0
$606$	−32.9444	−1.33827
$607$	34.3939	1.39600	0.698002	−	0.716096i	$-0.254073\pi$
$607$	34.3939	1.39600	0.698002	+	0.716096i	$0.254073\pi$
$608$	−6.44949	−0.261561
$609$	10.8990	0.441649
$610$	0	0
$611$	−14.4495	−0.584564
$612$	6.00000	0.242536
$613$	−46.8990	−1.89423	−0.947116	−	0.320891i	$-0.896018\pi$
$613$	−46.8990	−1.89423	−0.947116	+	0.320891i	$0.896018\pi$
$614$	25.3939	1.02481
$615$	0	0
$616$	8.89898	0.358550
$617$	−14.4495	−0.581715	−0.290857	−	0.956766i	$-0.593940\pi$
$617$	−14.4495	−0.581715	−0.290857	+	0.956766i	$0.593940\pi$
$618$	−6.00000	−0.241355
$619$	2.89898	0.116520	0.0582599	−	0.998301i	$-0.481445\pi$
$619$	2.89898	0.116520	0.0582599	+	0.998301i	$0.481445\pi$
$620$	0	0
$621$	0	0
$622$	19.5959	0.785725
$623$	−15.7980	−0.632932
$624$	6.00000	0.240192
$625$	0	0
$626$	−32.5959	−1.30279
$627$	31.5959	1.26182
$628$	−12.3485	−0.492758
$629$	6.89898	0.275080
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	2.89898	0.115315
$633$	−55.1010	−2.19007
$634$	13.6515	0.542172
$635$	0	0
$636$	−25.5959	−1.01494
$637$	31.3485	1.24207
$638$	2.00000	0.0791808
$639$	34.0454	1.34682
$640$	0	0
$641$	24.2474	0.957717	0.478858	−	0.877892i	$-0.341051\pi$
$641$	24.2474	0.957717	0.478858	+	0.877892i	$0.341051\pi$
$642$	12.8536	0.507290
$643$	24.8434	0.979727	0.489863	−	0.871799i	$-0.337047\pi$
$643$	24.8434	0.979727	0.489863	+	0.871799i	$0.337047\pi$
$644$	−10.8990	−0.429480
$645$	0	0
$646$	−12.8990	−0.507504
$647$	−39.5959	−1.55668	−0.778338	−	0.627845i	$-0.783937\pi$
$647$	−39.5959	−1.55668	−0.778338	+	0.627845i	$0.783937\pi$
$648$	9.00000	0.353553
$649$	17.1010	0.671274
$650$	0	0
$651$	32.6969	1.28149
$652$	12.4495	0.487560
$653$	−11.2474	−0.440147	−0.220073	−	0.975483i	$-0.570630\pi$
$653$	−11.2474	−0.440147	−0.220073	+	0.975483i	$0.570630\pi$
$654$	22.8990	0.895421
$655$	0	0
$656$	11.3485	0.443083
$657$	−40.0454	−1.56232
$658$	−26.2474	−1.02323
$659$	−38.6969	−1.50742	−0.753709	−	0.657208i	$-0.771737\pi$
$659$	−38.6969	−1.50742	−0.753709	+	0.657208i	$0.771737\pi$
$660$	0	0
$661$	9.75255	0.379330	0.189665	−	0.981849i	$-0.439260\pi$
$661$	9.75255	0.379330	0.189665	+	0.981849i	$0.439260\pi$
$662$	30.2474	1.17560
$663$	12.0000	0.466041
$664$	−6.00000	−0.232845
$665$	0	0
$666$	−10.3485	−0.400995
$667$	−2.44949	−0.0948446
$668$	17.7980	0.688624
$669$	−9.79796	−0.378811
$670$	0	0
$671$	6.89898	0.266332
$672$	10.8990	0.420437
$673$	16.7980	0.647514	0.323757	−	0.946140i	$-0.395054\pi$
$673$	16.7980	0.647514	0.323757	+	0.946140i	$0.395054\pi$
$674$	23.7980	0.916663
$675$	0	0
$676$	−7.00000	−0.269231
$677$	25.0454	0.962573	0.481287	−	0.876563i	$-0.340170\pi$
$677$	25.0454	0.962573	0.481287	+	0.876563i	$0.340170\pi$
$678$	15.3031	0.587711
$679$	32.6969	1.25479
$680$	0	0
$681$	−37.3485	−1.43120
$682$	6.00000	0.229752
$683$	−11.1010	−0.424769	−0.212384	−	0.977186i	$-0.568123\pi$
$683$	−11.1010	−0.424769	−0.212384	+	0.977186i	$0.568123\pi$
$684$	19.3485	0.739807
$685$	0	0
$686$	25.7980	0.984971
$687$	−41.8888	−1.59816
$688$	8.89898	0.339270
$689$	−25.5959	−0.975127
$690$	0	0
$691$	−43.9444	−1.67172	−0.835862	−	0.548940i	$-0.815031\pi$
$691$	−43.9444	−1.67172	−0.835862	+	0.548940i	$0.815031\pi$
$692$	16.0000	0.608229
$693$	−26.6969	−1.01413
$694$	−25.0454	−0.950711
$695$	0	0
$696$	2.44949	0.0928477
$697$	22.6969	0.859708
$698$	30.9444	1.17126
$699$	26.9444	1.01913
$700$	0	0
$701$	−25.3939	−0.959113	−0.479557	−	0.877511i	$-0.659203\pi$
$701$	−25.3939	−0.959113	−0.479557	+	0.877511i	$0.659203\pi$
$702$	0	0
$703$	22.2474	0.839078
$704$	2.00000	0.0753778
$705$	0	0
$706$	−16.7980	−0.632200
$707$	−59.8434	−2.25064
$708$	20.9444	0.787138
$709$	−27.3939	−1.02880	−0.514399	−	0.857551i	$-0.671985\pi$
$709$	−27.3939	−1.02880	−0.514399	+	0.857551i	$0.671985\pi$
$710$	0	0
$711$	−8.69694	−0.326161
$712$	−3.55051	−0.133061
$713$	−7.34847	−0.275202
$714$	21.7980	0.815768
$715$	0	0
$716$	−17.2474	−0.644567
$717$	52.8990	1.97555
$718$	16.5959	0.619354
$719$	−36.7423	−1.37026	−0.685129	−	0.728422i	$-0.740254\pi$
$719$	−36.7423	−1.37026	−0.685129	+	0.728422i	$0.740254\pi$
$720$	0	0
$721$	−10.8990	−0.405899
$722$	−22.5959	−0.840933
$723$	41.6413	1.54866
$724$	−26.0454	−0.967970
$725$	0	0
$726$	17.1464	0.636364
$727$	33.5959	1.24600	0.623002	−	0.782220i	$-0.285913\pi$
$727$	33.5959	1.24600	0.623002	+	0.782220i	$0.285913\pi$
$728$	10.8990	0.403943
$729$	−27.0000	−1.00000
$730$	0	0
$731$	17.7980	0.658281
$732$	8.44949	0.312302
$733$	−1.10102	−0.0406671	−0.0203336	−	0.999793i	$-0.506473\pi$
$733$	−1.10102	−0.0406671	−0.0203336	+	0.999793i	$0.506473\pi$
$734$	−1.20204	−0.0443681
$735$	0	0
$736$	−2.44949	−0.0902894
$737$	26.8990	0.990837
$738$	−34.0454	−1.25323
$739$	10.0000	0.367856	0.183928	−	0.982940i	$-0.441119\pi$
$739$	10.0000	0.367856	0.183928	+	0.982940i	$0.441119\pi$
$740$	0	0
$741$	38.6969	1.42157
$742$	−46.4949	−1.70688
$743$	−12.6969	−0.465805	−0.232903	−	0.972500i	$-0.574822\pi$
$743$	−12.6969	−0.465805	−0.232903	+	0.972500i	$0.574822\pi$
$744$	7.34847	0.269408
$745$	0	0
$746$	28.4949	1.04327
$747$	18.0000	0.658586
$748$	4.00000	0.146254
$749$	23.3485	0.853134
$750$	0	0
$751$	24.1010	0.879459	0.439729	−	0.898130i	$-0.355074\pi$
$751$	24.1010	0.879459	0.439729	+	0.898130i	$0.355074\pi$
$752$	−5.89898	−0.215114
$753$	−7.70714	−0.280864
$754$	2.44949	0.0892052
$755$	0	0
$756$	0	0
$757$	19.3939	0.704882	0.352441	−	0.935834i	$-0.385352\pi$
$757$	19.3939	0.704882	0.352441	+	0.935834i	$0.385352\pi$
$758$	−25.1464	−0.913359
$759$	12.0000	0.435572
$760$	0	0
$761$	14.1010	0.511162	0.255581	−	0.966788i	$-0.417733\pi$
$761$	14.1010	0.511162	0.255581	+	0.966788i	$0.417733\pi$
$762$	21.5505	0.780693
$763$	41.5959	1.50587
$764$	−11.6969	−0.423180
$765$	0	0
$766$	−11.7980	−0.426278
$767$	20.9444	0.756258
$768$	2.44949	0.0883883
$769$	−3.55051	−0.128035	−0.0640173	−	0.997949i	$-0.520391\pi$
$769$	−3.55051	−0.128035	−0.0640173	+	0.997949i	$0.520391\pi$
$770$	0	0
$771$	−65.3939	−2.35510
$772$	5.34847	0.192496
$773$	17.4495	0.627615	0.313807	−	0.949487i	$-0.398395\pi$
$773$	17.4495	0.627615	0.313807	+	0.949487i	$0.398395\pi$
$774$	−26.6969	−0.959602
$775$	0	0
$776$	7.34847	0.263795
$777$	−37.5959	−1.34875
$778$	24.3485	0.872935
$779$	73.1918	2.62237
$780$	0	0
$781$	22.6969	0.812160
$782$	−4.89898	−0.175187
$783$	0	0
$784$	12.7980	0.457070
$785$	0	0
$786$	18.0000	0.642039
$787$	40.5505	1.44547	0.722735	−	0.691125i	$-0.242885\pi$
$787$	40.5505	1.44547	0.722735	+	0.691125i	$0.242885\pi$
$788$	−5.10102	−0.181716
$789$	41.1464	1.46485
$790$	0	0
$791$	27.7980	0.988382
$792$	−6.00000	−0.213201
$793$	8.44949	0.300050
$794$	13.1464	0.466549
$795$	0	0
$796$	−0.651531	−0.0230929
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	70.2929	2.48834
$799$	−11.7980	−0.417382
$800$	0	0
$801$	10.6515	0.376353
$802$	−38.5959	−1.36287
$803$	−26.6969	−0.942114
$804$	32.9444	1.16186
$805$	0	0
$806$	7.34847	0.258839
$807$	59.6413	2.09947
$808$	−13.4495	−0.473151
$809$	31.5959	1.11085	0.555427	−	0.831566i	$-0.312555\pi$
$809$	31.5959	1.11085	0.555427	+	0.831566i	$0.312555\pi$
$810$	0	0
$811$	−29.4495	−1.03411	−0.517056	−	0.855952i	$-0.672972\pi$
$811$	−29.4495	−1.03411	−0.517056	+	0.855952i	$0.672972\pi$
$812$	4.44949	0.156146
$813$	38.4495	1.34848
$814$	−6.89898	−0.241809
$815$	0	0
$816$	4.89898	0.171499
$817$	57.3939	2.00796
$818$	5.79796	0.202721
$819$	−32.6969	−1.14252
$820$	0	0
$821$	19.7526	0.689369	0.344684	−	0.938719i	$-0.387986\pi$
$821$	19.7526	0.689369	0.344684	+	0.938719i	$0.387986\pi$
$822$	2.20204	0.0768050
$823$	11.7980	0.411251	0.205625	−	0.978631i	$-0.434077\pi$
$823$	11.7980	0.411251	0.205625	+	0.978631i	$0.434077\pi$
$824$	−2.44949	−0.0853320
$825$	0	0
$826$	38.0454	1.32377
$827$	−19.3031	−0.671233	−0.335617	−	0.941999i	$-0.608945\pi$
$827$	−19.3031	−0.671233	−0.335617	+	0.941999i	$0.608945\pi$
$828$	7.34847	0.255377
$829$	14.4949	0.503429	0.251714	−	0.967802i	$-0.419006\pi$
$829$	14.4949	0.503429	0.251714	+	0.967802i	$0.419006\pi$
$830$	0	0
$831$	56.2020	1.94963
$832$	2.44949	0.0849208
$833$	25.5959	0.886846
$834$	28.0454	0.971133
$835$	0	0
$836$	12.8990	0.446121
$837$	0	0
$838$	10.1464	0.350503
$839$	−30.7980	−1.06326	−0.531632	−	0.846976i	$-0.678421\pi$
$839$	−30.7980	−1.06326	−0.531632	+	0.846976i	$0.678421\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−13.4495	−0.463500
$843$	−70.5403	−2.42954
$844$	−22.4949	−0.774306
$845$	0	0
$846$	17.6969	0.608433
$847$	31.1464	1.07020
$848$	−10.4495	−0.358837
$849$	−58.7878	−2.01759
$850$	0	0
$851$	8.44949	0.289645
$852$	27.7980	0.952342
$853$	−8.20204	−0.280833	−0.140416	−	0.990093i	$-0.544844\pi$
$853$	−8.20204	−0.280833	−0.140416	+	0.990093i	$0.544844\pi$
$854$	15.3485	0.525214
$855$	0	0
$856$	5.24745	0.179354
$857$	39.3939	1.34567	0.672835	−	0.739793i	$-0.265077\pi$
$857$	39.3939	1.34567	0.672835	+	0.739793i	$0.265077\pi$
$858$	−12.0000	−0.409673
$859$	22.8990	0.781303	0.390652	−	0.920539i	$-0.372250\pi$
$859$	22.8990	0.781303	0.390652	+	0.920539i	$0.372250\pi$
$860$	0	0
$861$	−123.687	−4.21523
$862$	−18.4495	−0.628392
$863$	−11.7526	−0.400061	−0.200031	−	0.979790i	$-0.564104\pi$
$863$	−11.7526	−0.400061	−0.200031	+	0.979790i	$0.564104\pi$
$864$	0	0
$865$	0	0
$866$	30.4495	1.03472
$867$	−31.8434	−1.08146
$868$	13.3485	0.453077
$869$	−5.79796	−0.196682
$870$	0	0
$871$	32.9444	1.11628
$872$	9.34847	0.316579
$873$	−22.0454	−0.746124
$874$	−15.7980	−0.534374
$875$	0	0
$876$	−32.6969	−1.10473
$877$	−8.00000	−0.270141	−0.135070	−	0.990836i	$-0.543126\pi$
$877$	−8.00000	−0.270141	−0.135070	+	0.990836i	$0.543126\pi$
$878$	−18.6969	−0.630991
$879$	18.2474	0.615471
$880$	0	0
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	−38.3939	−1.29279
$883$	−29.9444	−1.00771	−0.503854	−	0.863789i	$-0.668085\pi$
$883$	−29.9444	−1.00771	−0.503854	+	0.863789i	$0.668085\pi$
$884$	4.89898	0.164771
$885$	0	0
$886$	−13.1010	−0.440137
$887$	47.7980	1.60490	0.802449	−	0.596720i	$-0.203530\pi$
$887$	47.7980	1.60490	0.802449	+	0.596720i	$0.203530\pi$
$888$	−8.44949	−0.283546
$889$	39.1464	1.31293
$890$	0	0
$891$	−18.0000	−0.603023
$892$	−4.00000	−0.133930
$893$	−38.0454	−1.27314
$894$	−38.6969	−1.29422
$895$	0	0
$896$	4.44949	0.148647
$897$	14.6969	0.490716
$898$	−25.7980	−0.860889
$899$	3.00000	0.100056
$900$	0	0
$901$	−20.8990	−0.696246
$902$	−22.6969	−0.755725
$903$	−96.9898	−3.22762
$904$	6.24745	0.207787
$905$	0	0
$906$	−27.7980	−0.923525
$907$	21.3485	0.708864	0.354432	−	0.935082i	$-0.384674\pi$
$907$	21.3485	0.708864	0.354432	+	0.935082i	$0.384674\pi$
$908$	−15.2474	−0.506004
$909$	40.3485	1.33827
$910$	0	0
$911$	51.4949	1.70610	0.853051	−	0.521827i	$-0.174750\pi$
$911$	51.4949	1.70610	0.853051	+	0.521827i	$0.174750\pi$
$912$	15.7980	0.523123
$913$	12.0000	0.397142
$914$	−17.7980	−0.588704
$915$	0	0
$916$	−17.1010	−0.565034
$917$	32.6969	1.07975
$918$	0	0
$919$	−24.8536	−0.819844	−0.409922	−	0.912121i	$-0.634444\pi$
$919$	−24.8536	−0.819844	−0.409922	+	0.912121i	$0.634444\pi$
$920$	0	0
$921$	−62.2020	−2.04963
$922$	28.2929	0.931776
$923$	27.7980	0.914981
$924$	−21.7980	−0.717100
$925$	0	0
$926$	−3.10102	−0.101906
$927$	7.34847	0.241355
$928$	1.00000	0.0328266
$929$	−39.4949	−1.29579	−0.647893	−	0.761732i	$-0.724350\pi$
$929$	−39.4949	−1.29579	−0.647893	+	0.761732i	$0.724350\pi$
$930$	0	0
$931$	82.5403	2.70515
$932$	11.0000	0.360317
$933$	−48.0000	−1.57145
$934$	−22.9444	−0.750763
$935$	0	0
$936$	−7.34847	−0.240192
$937$	−53.2929	−1.74100	−0.870501	−	0.492167i	$-0.836205\pi$
$937$	−53.2929	−1.74100	−0.870501	+	0.492167i	$0.836205\pi$
$938$	59.8434	1.95396
$939$	79.8434	2.60559
$940$	0	0
$941$	8.44949	0.275445	0.137723	−	0.990471i	$-0.456022\pi$
$941$	8.44949	0.275445	0.137723	+	0.990471i	$0.456022\pi$
$942$	30.2474	0.985515
$943$	27.7980	0.905226
$944$	8.55051	0.278295
$945$	0	0
$946$	−17.7980	−0.578662
$947$	−37.3485	−1.21366	−0.606831	−	0.794831i	$-0.707560\pi$
$947$	−37.3485	−1.21366	−0.606831	+	0.794831i	$0.707560\pi$
$948$	−7.10102	−0.230630
$949$	−32.6969	−1.06139
$950$	0	0
$951$	−33.4393	−1.08434
$952$	8.89898	0.288418
$953$	−34.2929	−1.11085	−0.555427	−	0.831565i	$-0.687445\pi$
$953$	−34.2929	−1.11085	−0.555427	+	0.831565i	$0.687445\pi$
$954$	31.3485	1.01494
$955$	0	0
$956$	21.5959	0.698462
$957$	−4.89898	−0.158362
$958$	4.20204	0.135762
$959$	4.00000	0.129167
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−8.44949	−0.272422
$963$	−15.7423	−0.507290
$964$	17.0000	0.547533
$965$	0	0
$966$	26.6969	0.858960
$967$	−28.0000	−0.900419	−0.450210	−	0.892923i	$-0.648651\pi$
$967$	−28.0000	−0.900419	−0.450210	+	0.892923i	$0.648651\pi$
$968$	7.00000	0.224989
$969$	31.5959	1.01501
$970$	0	0
$971$	22.9444	0.736320	0.368160	−	0.929762i	$-0.379988\pi$
$971$	22.9444	0.736320	0.368160	+	0.929762i	$0.379988\pi$
$972$	−22.0454	−0.707107
$973$	50.9444	1.63320
$974$	−9.10102	−0.291616
$975$	0	0
$976$	3.44949	0.110415
$977$	−0.101021	−0.00323193	−0.00161597	−	0.999999i	$-0.500514\pi$
$977$	−0.101021	−0.00323193	−0.00161597	+	0.999999i	$0.500514\pi$
$978$	−30.4949	−0.975119
$979$	7.10102	0.226950
$980$	0	0
$981$	−28.0454	−0.895421
$982$	−26.4949	−0.845486
$983$	−39.0000	−1.24391	−0.621953	−	0.783054i	$-0.713661\pi$
$983$	−39.0000	−1.24391	−0.621953	+	0.783054i	$0.713661\pi$
$984$	−27.7980	−0.886167
$985$	0	0
$986$	2.00000	0.0636930
$987$	64.2929	2.04646
$988$	15.7980	0.502600
$989$	21.7980	0.693135
$990$	0	0
$991$	−3.50510	−0.111343	−0.0556716	−	0.998449i	$-0.517730\pi$
$991$	−3.50510	−0.111343	−0.0556716	+	0.998449i	$0.517730\pi$
$992$	3.00000	0.0952501
$993$	−74.0908	−2.35120
$994$	50.4949	1.60160
$995$	0	0
$996$	14.6969	0.465690
$997$	−29.7423	−0.941950	−0.470975	−	0.882147i	$-0.656098\pi$
$997$	−29.7423	−0.941950	−0.470975	+	0.882147i	$0.656098\pi$
$998$	24.3485	0.770737
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1450.2.a.j.1.2	✓	2
5.2	odd	4		1450.2.b.i.349.1		4
5.3	odd	4		1450.2.b.i.349.4		4
5.4	even	2		1450.2.a.o.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1450.2.a.j.1.2	✓	2	1.1	even	1	trivial
1450.2.a.o.1.1	yes	2	5.4	even	2
1450.2.b.i.349.1		4	5.2	odd	4
1450.2.b.i.349.4		4	5.3	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} -4.44949 q^{7} -1.00000 q^{8} +3.00000 q^{9} +2.00000 q^{11} +2.44949 q^{12} +2.44949 q^{13} +4.44949 q^{14} +1.00000 q^{16} +2.00000 q^{17} -3.00000 q^{18} +6.44949 q^{19} -10.8990 q^{21} -2.00000 q^{22} +2.44949 q^{23} -2.44949 q^{24} -2.44949 q^{26} -4.44949 q^{28} -1.00000 q^{29} -3.00000 q^{31} -1.00000 q^{32} +4.89898 q^{33} -2.00000 q^{34} +3.00000 q^{36} +3.44949 q^{37} -6.44949 q^{38} +6.00000 q^{39} +11.3485 q^{41} +10.8990 q^{42} +8.89898 q^{43} +2.00000 q^{44} -2.44949 q^{46} -5.89898 q^{47} +2.44949 q^{48} +12.7980 q^{49} +4.89898 q^{51} +2.44949 q^{52} -10.4495 q^{53} +4.44949 q^{56} +15.7980 q^{57} +1.00000 q^{58} +8.55051 q^{59} +3.44949 q^{61} +3.00000 q^{62} -13.3485 q^{63} +1.00000 q^{64} -4.89898 q^{66} +13.4495 q^{67} +2.00000 q^{68} +6.00000 q^{69} +11.3485 q^{71} -3.00000 q^{72} -13.3485 q^{73} -3.44949 q^{74} +6.44949 q^{76} -8.89898 q^{77} -6.00000 q^{78} -2.89898 q^{79} -9.00000 q^{81} -11.3485 q^{82} +6.00000 q^{83} -10.8990 q^{84} -8.89898 q^{86} -2.44949 q^{87} -2.00000 q^{88} +3.55051 q^{89} -10.8990 q^{91} +2.44949 q^{92} -7.34847 q^{93} +5.89898 q^{94} -2.44949 q^{96} -7.34847 q^{97} -12.7980 q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{7} - 2 q^{8} + 6 q^{9} + 4 q^{11} + 4 q^{14} + 2 q^{16} + 4 q^{17} - 6 q^{18} + 8 q^{19} - 12 q^{21} - 4 q^{22} - 4 q^{28} - 2 q^{29} - 6 q^{31} - 2 q^{32} - 4 q^{34} + 6 q^{36}+ \cdots + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^7 - 2 * q^8 + 6 * q^9 + 4 * q^11 + 4 * q^14 + 2 * q^16 + 4 * q^17 - 6 * q^18 + 8 * q^19 - 12 * q^21 - 4 * q^22 - 4 * q^28 - 2 * q^29 - 6 * q^31 - 2 * q^32 - 4 * q^34 + 6 * q^36 + 2 * q^37 - 8 * q^38 + 12 * q^39 + 8 * q^41 + 12 * q^42 + 8 * q^43 + 4 * q^44 - 2 * q^47 + 6 * q^49 - 16 * q^53 + 4 * q^56 + 12 * q^57 + 2 * q^58 + 22 * q^59 + 2 * q^61 + 6 * q^62 - 12 * q^63 + 2 * q^64 + 22 * q^67 + 4 * q^68 + 12 * q^69 + 8 * q^71 - 6 * q^72 - 12 * q^73 - 2 * q^74 + 8 * q^76 - 8 * q^77 - 12 * q^78 + 4 * q^79 - 18 * q^81 - 8 * q^82 + 12 * q^83 - 12 * q^84 - 8 * q^86 - 4 * q^88 + 12 * q^89 - 12 * q^91 + 2 * q^94 - 6 * q^98 + 12 * q^99

Introduction

L-functions

Modular forms

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Database

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