LMFDB - Embedded newform 1450.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1450,2,Mod(1,1450)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 1450 = 2 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1450.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:	$11.5783082931$
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)

$f(q)$	$=$	$q+1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -0.449490 q^{7} +1.00000 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -0.449490 q^{7} +1.00000 q^{8} +3.00000 q^{9} +2.00000 q^{11} +2.44949 q^{12} +2.44949 q^{13} -0.449490 q^{14} +1.00000 q^{16} -2.00000 q^{17} +3.00000 q^{18} +1.55051 q^{19} -1.10102 q^{21} +2.00000 q^{22} +2.44949 q^{23} +2.44949 q^{24} +2.44949 q^{26} -0.449490 q^{28} -1.00000 q^{29} -3.00000 q^{31} +1.00000 q^{32} +4.89898 q^{33} -2.00000 q^{34} +3.00000 q^{36} +1.44949 q^{37} +1.55051 q^{38} +6.00000 q^{39} -3.34847 q^{41} -1.10102 q^{42} +0.898979 q^{43} +2.00000 q^{44} +2.44949 q^{46} -3.89898 q^{47} +2.44949 q^{48} -6.79796 q^{49} -4.89898 q^{51} +2.44949 q^{52} +5.55051 q^{53} -0.449490 q^{56} +3.79796 q^{57} -1.00000 q^{58} +13.4495 q^{59} -1.44949 q^{61} -3.00000 q^{62} -1.34847 q^{63} +1.00000 q^{64} +4.89898 q^{66} -8.55051 q^{67} -2.00000 q^{68} +6.00000 q^{69} -3.34847 q^{71} +3.00000 q^{72} -1.34847 q^{73} +1.44949 q^{74} +1.55051 q^{76} -0.898979 q^{77} +6.00000 q^{78} +6.89898 q^{79} -9.00000 q^{81} -3.34847 q^{82} -6.00000 q^{83} -1.10102 q^{84} +0.898979 q^{86} -2.44949 q^{87} +2.00000 q^{88} +8.44949 q^{89} -1.10102 q^{91} +2.44949 q^{92} -7.34847 q^{93} -3.89898 q^{94} +2.44949 q^{96} -7.34847 q^{97} -6.79796 q^{98} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8 + 6 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	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
$7$	−0.449490	−0.169891	−0.0849456	−	0.996386i	$-0.527072\pi$
$7$	−0.449490	−0.169891	−0.0849456	+	0.996386i	$0.527072\pi$
$8$	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	$0.389680\pi$
$14$	−0.449490	−0.120131
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	3.00000	0.707107
$19$	1.55051	0.355711	0.177856	−	0.984057i	$-0.443084\pi$
$19$	1.55051	0.355711	0.177856	+	0.984057i	$0.443084\pi$
$20$	0	0
$21$	−1.10102	−0.240262
$22$	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
$28$	−0.449490	−0.0849456
$29$	−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
$37$	1.44949	0.238295	0.119147	−	0.992877i	$-0.461984\pi$
$37$	1.44949	0.238295	0.119147	+	0.992877i	$0.461984\pi$
$38$	1.55051	0.251526
$39$	6.00000	0.960769
$40$	0	0
$41$	−3.34847	−0.522943	−0.261472	−	0.965211i	$-0.584208\pi$
$41$	−3.34847	−0.522943	−0.261472	+	0.965211i	$0.584208\pi$
$42$	−1.10102	−0.169891
$43$	0.898979	0.137093	0.0685465	−	0.997648i	$-0.478164\pi$
$43$	0.898979	0.137093	0.0685465	+	0.997648i	$0.478164\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	2.44949	0.361158
$47$	−3.89898	−0.568725	−0.284362	−	0.958717i	$-0.591782\pi$
$47$	−3.89898	−0.568725	−0.284362	+	0.958717i	$0.591782\pi$
$48$	2.44949	0.353553
$49$	−6.79796	−0.971137
$50$	0	0
$51$	−4.89898	−0.685994
$52$	2.44949	0.339683
$53$	5.55051	0.762421	0.381211	−	0.924488i	$-0.375507\pi$
$53$	5.55051	0.762421	0.381211	+	0.924488i	$0.375507\pi$
$54$	0	0
$55$	0	0
$56$	−0.449490	−0.0600656
$57$	3.79796	0.503052
$58$	−1.00000	−0.131306
$59$	13.4495	1.75097	0.875487	−	0.483241i	$-0.160541\pi$
$59$	13.4495	1.75097	0.875487	+	0.483241i	$0.160541\pi$
$60$	0	0
$61$	−1.44949	−0.185588	−0.0927941	−	0.995685i	$-0.529580\pi$
$61$	−1.44949	−0.185588	−0.0927941	+	0.995685i	$0.529580\pi$
$62$	−3.00000	−0.381000
$63$	−1.34847	−0.169891
$64$	1.00000	0.125000
$65$	0	0
$66$	4.89898	0.603023
$67$	−8.55051	−1.04461	−0.522306	−	0.852758i	$-0.674928\pi$
$67$	−8.55051	−1.04461	−0.522306	+	0.852758i	$0.674928\pi$
$68$	−2.00000	−0.242536
$69$	6.00000	0.722315
$70$	0	0
$71$	−3.34847	−0.397390	−0.198695	−	0.980061i	$-0.563670\pi$
$71$	−3.34847	−0.397390	−0.198695	+	0.980061i	$0.563670\pi$
$72$	3.00000	0.353553
$73$	−1.34847	−0.157826	−0.0789132	−	0.996881i	$-0.525145\pi$
$73$	−1.34847	−0.157826	−0.0789132	+	0.996881i	$0.525145\pi$
$74$	1.44949	0.168500
$75$	0	0
$76$	1.55051	0.177856
$77$	−0.898979	−0.102448
$78$	6.00000	0.679366
$79$	6.89898	0.776196	0.388098	−	0.921618i	$-0.373132\pi$
$79$	6.89898	0.776196	0.388098	+	0.921618i	$0.373132\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	−3.34847	−0.369777
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	−1.10102	−0.120131
$85$	0	0
$86$	0.898979	0.0969395
$87$	−2.44949	−0.262613
$88$	2.00000	0.213201
$89$	8.44949	0.895644	0.447822	−	0.894123i	$-0.352200\pi$
$89$	8.44949	0.895644	0.447822	+	0.894123i	$0.352200\pi$
$90$	0	0
$91$	−1.10102	−0.115418
$92$	2.44949	0.255377
$93$	−7.34847	−0.762001
$94$	−3.89898	−0.402149
$95$	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	$0.621692\pi$
$98$	−6.79796	−0.686698
$99$	6.00000	0.603023
$100$	0	0
$101$	8.55051	0.850808	0.425404	−	0.905004i	$-0.360132\pi$
$101$	8.55051	0.850808	0.425404	+	0.905004i	$0.360132\pi$
$102$	−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
$106$	5.55051	0.539113
$107$	−19.2474	−1.86072	−0.930361	−	0.366646i	$-0.880506\pi$
$107$	−19.2474	−1.86072	−0.930361	+	0.366646i	$0.880506\pi$
$108$	0	0
$109$	5.34847	0.512290	0.256145	−	0.966638i	$-0.417547\pi$
$109$	5.34847	0.512290	0.256145	+	0.966638i	$0.417547\pi$
$110$	0	0
$111$	3.55051	0.337000
$112$	−0.449490	−0.0424728
$113$	−18.2474	−1.71658	−0.858288	−	0.513169i	$-0.828472\pi$
$113$	−18.2474	−1.71658	−0.858288	+	0.513169i	$0.828472\pi$
$114$	3.79796	0.355711
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	7.34847	0.679366
$118$	13.4495	1.23813
$119$	0.898979	0.0824093
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−1.44949	−0.131231
$123$	−8.20204	−0.739553
$124$	−3.00000	−0.269408
$125$	0	0
$126$	−1.34847	−0.120131
$127$	−10.7980	−0.958164	−0.479082	−	0.877770i	$-0.659030\pi$
$127$	−10.7980	−0.958164	−0.479082	+	0.877770i	$0.659030\pi$
$128$	1.00000	0.0883883
$129$	2.20204	0.193879
$130$	0	0
$131$	7.34847	0.642039	0.321019	−	0.947073i	$-0.395975\pi$
$131$	7.34847	0.642039	0.321019	+	0.947073i	$0.395975\pi$
$132$	4.89898	0.426401
$133$	−0.696938	−0.0604322
$134$	−8.55051	−0.738652
$135$	0	0
$136$	−2.00000	−0.171499
$137$	−8.89898	−0.760291	−0.380146	−	0.924927i	$-0.624126\pi$
$137$	−8.89898	−0.760291	−0.380146	+	0.924927i	$0.624126\pi$
$138$	6.00000	0.510754
$139$	−6.55051	−0.555607	−0.277804	−	0.960638i	$-0.589606\pi$
$139$	−6.55051	−0.555607	−0.277804	+	0.960638i	$0.589606\pi$
$140$	0	0
$141$	−9.55051	−0.804298
$142$	−3.34847	−0.280997
$143$	4.89898	0.409673
$144$	3.00000	0.250000
$145$	0	0
$146$	−1.34847	−0.111600
$147$	−16.6515	−1.37340
$148$	1.44949	0.119147
$149$	−3.79796	−0.311141	−0.155570	−	0.987825i	$-0.549722\pi$
$149$	−3.79796	−0.311141	−0.155570	+	0.987825i	$0.549722\pi$
$150$	0	0
$151$	−3.34847	−0.272495	−0.136247	−	0.990675i	$-0.543504\pi$
$151$	−3.34847	−0.272495	−0.136247	+	0.990675i	$0.543504\pi$
$152$	1.55051	0.125763
$153$	−6.00000	−0.485071
$154$	−0.898979	−0.0724418
$155$	0	0
$156$	6.00000	0.480384
$157$	−2.34847	−0.187428	−0.0937141	−	0.995599i	$-0.529874\pi$
$157$	−2.34847	−0.187428	−0.0937141	+	0.995599i	$0.529874\pi$
$158$	6.89898	0.548853
$159$	13.5959	1.07823
$160$	0	0
$161$	−1.10102	−0.0867726
$162$	−9.00000	−0.707107
$163$	−7.55051	−0.591402	−0.295701	−	0.955281i	$-0.595553\pi$
$163$	−7.55051	−0.591402	−0.295701	+	0.955281i	$0.595553\pi$
$164$	−3.34847	−0.261472
$165$	0	0
$166$	−6.00000	−0.465690
$167$	1.79796	0.139130	0.0695651	−	0.997577i	$-0.477839\pi$
$167$	1.79796	0.139130	0.0695651	+	0.997577i	$0.477839\pi$
$168$	−1.10102	−0.0849456
$169$	−7.00000	−0.538462
$170$	0	0
$171$	4.65153	0.355711
$172$	0.898979	0.0685465
$173$	−16.0000	−1.21646	−0.608229	−	0.793762i	$-0.708120\pi$
$173$	−16.0000	−1.21646	−0.608229	+	0.793762i	$0.708120\pi$
$174$	−2.44949	−0.185695
$175$	0	0
$176$	2.00000	0.150756
$177$	32.9444	2.47625
$178$	8.44949	0.633316
$179$	7.24745	0.541700	0.270850	−	0.962622i	$-0.412695\pi$
$179$	7.24745	0.541700	0.270850	+	0.962622i	$0.412695\pi$
$180$	0	0
$181$	18.0454	1.34130	0.670652	−	0.741772i	$-0.266014\pi$
$181$	18.0454	1.34130	0.670652	+	0.741772i	$0.266014\pi$
$182$	−1.10102	−0.0816131
$183$	−3.55051	−0.262461
$184$	2.44949	0.180579
$185$	0	0
$186$	−7.34847	−0.538816
$187$	−4.00000	−0.292509
$188$	−3.89898	−0.284362
$189$	0	0
$190$	0	0
$191$	17.6969	1.28051	0.640253	−	0.768164i	$-0.278830\pi$
$191$	17.6969	1.28051	0.640253	+	0.768164i	$0.278830\pi$
$192$	2.44949	0.176777
$193$	9.34847	0.672918	0.336459	−	0.941698i	$-0.390771\pi$
$193$	9.34847	0.672918	0.336459	+	0.941698i	$0.390771\pi$
$194$	−7.34847	−0.527589
$195$	0	0
$196$	−6.79796	−0.485568
$197$	14.8990	1.06151	0.530754	−	0.847526i	$-0.321909\pi$
$197$	14.8990	1.06151	0.530754	+	0.847526i	$0.321909\pi$
$198$	6.00000	0.426401
$199$	−15.3485	−1.08802	−0.544012	−	0.839077i	$-0.683095\pi$
$199$	−15.3485	−1.08802	−0.544012	+	0.839077i	$0.683095\pi$
$200$	0	0
$201$	−20.9444	−1.47730
$202$	8.55051	0.601612
$203$	0.449490	0.0315480
$204$	−4.89898	−0.342997
$205$	0	0
$206$	2.44949	0.170664
$207$	7.34847	0.510754
$208$	2.44949	0.169842
$209$	3.10102	0.214502
$210$	0	0
$211$	26.4949	1.82398	0.911992	−	0.410208i	$-0.134544\pi$
$211$	26.4949	1.82398	0.911992	+	0.410208i	$0.134544\pi$
$212$	5.55051	0.381211
$213$	−8.20204	−0.561995
$214$	−19.2474	−1.31573
$215$	0	0
$216$	0	0
$217$	1.34847	0.0915401
$218$	5.34847	0.362244
$219$	−3.30306	−0.223200
$220$	0	0
$221$	−4.89898	−0.329541
$222$	3.55051	0.238295
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	−0.449490	−0.0300328
$225$	0	0
$226$	−18.2474	−1.21380
$227$	−9.24745	−0.613775	−0.306887	−	0.951746i	$-0.599287\pi$
$227$	−9.24745	−0.613775	−0.306887	+	0.951746i	$0.599287\pi$
$228$	3.79796	0.251526
$229$	−26.8990	−1.77753	−0.888767	−	0.458359i	$-0.848438\pi$
$229$	−26.8990	−1.77753	−0.888767	+	0.458359i	$0.848438\pi$
$230$	0	0
$231$	−2.20204	−0.144884
$232$	−1.00000	−0.0656532
$233$	−11.0000	−0.720634	−0.360317	−	0.932830i	$-0.617331\pi$
$233$	−11.0000	−0.720634	−0.360317	+	0.932830i	$0.617331\pi$
$234$	7.34847	0.480384
$235$	0	0
$236$	13.4495	0.875487
$237$	16.8990	1.09771
$238$	0.898979	0.0582722
$239$	−17.5959	−1.13819	−0.569093	−	0.822273i	$-0.692705\pi$
$239$	−17.5959	−1.13819	−0.569093	+	0.822273i	$0.692705\pi$
$240$	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
$244$	−1.44949	−0.0927941
$245$	0	0
$246$	−8.20204	−0.522943
$247$	3.79796	0.241658
$248$	−3.00000	−0.190500
$249$	−14.6969	−0.931381
$250$	0	0
$251$	31.1464	1.96595	0.982973	−	0.183752i	$-0.0588245\pi$
$251$	31.1464	1.96595	0.982973	+	0.183752i	$0.0588245\pi$
$252$	−1.34847	−0.0849456
$253$	4.89898	0.307996
$254$	−10.7980	−0.677524
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.69694	−0.168230	−0.0841152	−	0.996456i	$-0.526806\pi$
$257$	−2.69694	−0.168230	−0.0841152	+	0.996456i	$0.526806\pi$
$258$	2.20204	0.137093
$259$	−0.651531	−0.0404842
$260$	0	0
$261$	−3.00000	−0.185695
$262$	7.34847	0.453990
$263$	2.79796	0.172530	0.0862648	−	0.996272i	$-0.472507\pi$
$263$	2.79796	0.172530	0.0862648	+	0.996272i	$0.472507\pi$
$264$	4.89898	0.301511
$265$	0	0
$266$	−0.696938	−0.0427320
$267$	20.6969	1.26663
$268$	−8.55051	−0.522306
$269$	9.65153	0.588464	0.294232	−	0.955734i	$-0.404936\pi$
$269$	9.65153	0.588464	0.294232	+	0.955734i	$0.404936\pi$
$270$	0	0
$271$	−13.6969	−0.832030	−0.416015	−	0.909358i	$-0.636574\pi$
$271$	−13.6969	−0.832030	−0.416015	+	0.909358i	$0.636574\pi$
$272$	−2.00000	−0.121268
$273$	−2.69694	−0.163226
$274$	−8.89898	−0.537607
$275$	0	0
$276$	6.00000	0.361158
$277$	30.9444	1.85927	0.929634	−	0.368484i	$-0.120123\pi$
$277$	30.9444	1.85927	0.929634	+	0.368484i	$0.120123\pi$
$278$	−6.55051	−0.392873
$279$	−9.00000	−0.538816
$280$	0	0
$281$	−9.20204	−0.548948	−0.274474	−	0.961595i	$-0.588504\pi$
$281$	−9.20204	−0.548948	−0.274474	+	0.961595i	$0.588504\pi$
$282$	−9.55051	−0.568725
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	−3.34847	−0.198695
$285$	0	0
$286$	4.89898	0.289683
$287$	1.50510	0.0888434
$288$	3.00000	0.176777
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−18.0000	−1.05518
$292$	−1.34847	−0.0789132
$293$	−2.55051	−0.149002	−0.0745012	−	0.997221i	$-0.523736\pi$
$293$	−2.55051	−0.149002	−0.0745012	+	0.997221i	$0.523736\pi$
$294$	−16.6515	−0.971137
$295$	0	0
$296$	1.44949	0.0842499
$297$	0	0
$298$	−3.79796	−0.220010
$299$	6.00000	0.346989
$300$	0	0
$301$	−0.404082	−0.0232909
$302$	−3.34847	−0.192683
$303$	20.9444	1.20322
$304$	1.55051	0.0889279
$305$	0	0
$306$	−6.00000	−0.342997
$307$	−33.3939	−1.90589	−0.952945	−	0.303144i	$-0.901964\pi$
$307$	−33.3939	−1.90589	−0.952945	+	0.303144i	$0.901964\pi$
$308$	−0.898979	−0.0512241
$309$	6.00000	0.341328
$310$	0	0
$311$	19.5959	1.11118	0.555591	−	0.831456i	$-0.312492\pi$
$311$	19.5959	1.11118	0.555591	+	0.831456i	$0.312492\pi$
$312$	6.00000	0.339683
$313$	6.59592	0.372823	0.186412	−	0.982472i	$-0.440314\pi$
$313$	6.59592	0.372823	0.186412	+	0.982472i	$0.440314\pi$
$314$	−2.34847	−0.132532
$315$	0	0
$316$	6.89898	0.388098
$317$	28.3485	1.59221	0.796104	−	0.605159i	$-0.206891\pi$
$317$	28.3485	1.59221	0.796104	+	0.605159i	$0.206891\pi$
$318$	13.5959	0.762421
$319$	−2.00000	−0.111979
$320$	0	0
$321$	−47.1464	−2.63146
$322$	−1.10102	−0.0613575
$323$	−3.10102	−0.172545
$324$	−9.00000	−0.500000
$325$	0	0
$326$	−7.55051	−0.418184
$327$	13.1010	0.724488
$328$	−3.34847	−0.184888
$329$	1.75255	0.0966213
$330$	0	0
$331$	−5.75255	−0.316189	−0.158094	−	0.987424i	$-0.550535\pi$
$331$	−5.75255	−0.316189	−0.158094	+	0.987424i	$0.550535\pi$
$332$	−6.00000	−0.329293
$333$	4.34847	0.238295
$334$	1.79796	0.0983799
$335$	0	0
$336$	−1.10102	−0.0600656
$337$	4.20204	0.228900	0.114450	−	0.993429i	$-0.463489\pi$
$337$	4.20204	0.228900	0.114450	+	0.993429i	$0.463489\pi$
$338$	−7.00000	−0.380750
$339$	−44.6969	−2.42760
$340$	0	0
$341$	−6.00000	−0.324918
$342$	4.65153	0.251526
$343$	6.20204	0.334879
$344$	0.898979	0.0484697
$345$	0	0
$346$	−16.0000	−0.860165
$347$	19.0454	1.02241	0.511206	−	0.859458i	$-0.329199\pi$
$347$	19.0454	1.02241	0.511206	+	0.859458i	$0.329199\pi$
$348$	−2.44949	−0.131306
$349$	22.9444	1.22818	0.614092	−	0.789234i	$-0.289522\pi$
$349$	22.9444	1.22818	0.614092	+	0.789234i	$0.289522\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	0.106600
$353$	2.79796	0.148920	0.0744602	−	0.997224i	$-0.476277\pi$
$353$	2.79796	0.148920	0.0744602	+	0.997224i	$0.476277\pi$
$354$	32.9444	1.75097
$355$	0	0
$356$	8.44949	0.447822
$357$	2.20204	0.116544
$358$	7.24745	0.383040
$359$	22.5959	1.19257	0.596283	−	0.802774i	$-0.296643\pi$
$359$	22.5959	1.19257	0.596283	+	0.802774i	$0.296643\pi$
$360$	0	0
$361$	−16.5959	−0.873469
$362$	18.0454	0.948446
$363$	−17.1464	−0.899954
$364$	−1.10102	−0.0577092
$365$	0	0
$366$	−3.55051	−0.185588
$367$	−20.7980	−1.08564	−0.542822	−	0.839848i	$-0.682644\pi$
$367$	−20.7980	−1.08564	−0.542822	+	0.839848i	$0.682644\pi$
$368$	2.44949	0.127688
$369$	−10.0454	−0.522943
$370$	0	0
$371$	−2.49490	−0.129529
$372$	−7.34847	−0.381000
$373$	−20.4949	−1.06119	−0.530593	−	0.847627i	$-0.678031\pi$
$373$	−20.4949	−1.06119	−0.530593	+	0.847627i	$0.678031\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	−3.89898	−0.201075
$377$	−2.44949	−0.126155
$378$	0	0
$379$	−9.14643	−0.469820	−0.234910	−	0.972017i	$-0.575480\pi$
$379$	−9.14643	−0.469820	−0.234910	+	0.972017i	$0.575480\pi$
$380$	0	0
$381$	−26.4495	−1.35505
$382$	17.6969	0.905454
$383$	7.79796	0.398457	0.199229	−	0.979953i	$-0.436156\pi$
$383$	7.79796	0.398457	0.199229	+	0.979953i	$0.436156\pi$
$384$	2.44949	0.125000
$385$	0	0
$386$	9.34847	0.475825
$387$	2.69694	0.137093
$388$	−7.34847	−0.373062
$389$	−9.65153	−0.489352	−0.244676	−	0.969605i	$-0.578682\pi$
$389$	−9.65153	−0.489352	−0.244676	+	0.969605i	$0.578682\pi$
$390$	0	0
$391$	−4.89898	−0.247752
$392$	−6.79796	−0.343349
$393$	18.0000	0.907980
$394$	14.8990	0.750600
$395$	0	0
$396$	6.00000	0.301511
$397$	−21.1464	−1.06131	−0.530654	−	0.847588i	$-0.678054\pi$
$397$	−21.1464	−1.06131	−0.530654	+	0.847588i	$0.678054\pi$
$398$	−15.3485	−0.769349
$399$	−1.70714	−0.0854641
$400$	0	0
$401$	−0.595918	−0.0297587	−0.0148794	−	0.999889i	$-0.504736\pi$
$401$	−0.595918	−0.0297587	−0.0148794	+	0.999889i	$0.504736\pi$
$402$	−20.9444	−1.04461
$403$	−7.34847	−0.366053
$404$	8.55051	0.425404
$405$	0	0
$406$	0.449490	0.0223078
$407$	2.89898	0.143697
$408$	−4.89898	−0.242536
$409$	13.7980	0.682265	0.341133	−	0.940015i	$-0.389189\pi$
$409$	13.7980	0.682265	0.341133	+	0.940015i	$0.389189\pi$
$410$	0	0
$411$	−21.7980	−1.07521
$412$	2.44949	0.120678
$413$	−6.04541	−0.297475
$414$	7.34847	0.361158
$415$	0	0
$416$	2.44949	0.120096
$417$	−16.0454	−0.785747
$418$	3.10102	0.151676
$419$	24.1464	1.17963	0.589815	−	0.807538i	$-0.299201\pi$
$419$	24.1464	1.17963	0.589815	+	0.807538i	$0.299201\pi$
$420$	0	0
$421$	8.55051	0.416726	0.208363	−	0.978052i	$-0.433186\pi$
$421$	8.55051	0.416726	0.208363	+	0.978052i	$0.433186\pi$
$422$	26.4949	1.28975
$423$	−11.6969	−0.568725
$424$	5.55051	0.269557
$425$	0	0
$426$	−8.20204	−0.397390
$427$	0.651531	0.0315298
$428$	−19.2474	−0.930361
$429$	12.0000	0.579365
$430$	0	0
$431$	13.5505	0.652705	0.326353	−	0.945248i	$-0.394180\pi$
$431$	13.5505	0.652705	0.326353	+	0.945248i	$0.394180\pi$
$432$	0	0
$433$	25.5505	1.22788	0.613940	−	0.789353i	$-0.289584\pi$
$433$	25.5505	1.22788	0.613940	+	0.789353i	$0.289584\pi$
$434$	1.34847	0.0647286
$435$	0	0
$436$	5.34847	0.256145
$437$	3.79796	0.181681
$438$	−3.30306	−0.157826
$439$	−10.6969	−0.510537	−0.255269	−	0.966870i	$-0.582164\pi$
$439$	−10.6969	−0.510537	−0.255269	+	0.966870i	$0.582164\pi$
$440$	0	0
$441$	−20.3939	−0.971137
$442$	−4.89898	−0.233021
$443$	−22.8990	−1.08796	−0.543982	−	0.839097i	$-0.683084\pi$
$443$	−22.8990	−1.08796	−0.543982	+	0.839097i	$0.683084\pi$
$444$	3.55051	0.168500
$445$	0	0
$446$	4.00000	0.189405
$447$	−9.30306	−0.440020
$448$	−0.449490	−0.0212364
$449$	6.20204	0.292692	0.146346	−	0.989233i	$-0.453249\pi$
$449$	6.20204	0.292692	0.146346	+	0.989233i	$0.453249\pi$
$450$	0	0
$451$	−6.69694	−0.315347
$452$	−18.2474	−0.858288
$453$	−8.20204	−0.385366
$454$	−9.24745	−0.434004
$455$	0	0
$456$	3.79796	0.177856
$457$	1.79796	0.0841050	0.0420525	−	0.999115i	$-0.486610\pi$
$457$	1.79796	0.0841050	0.0420525	+	0.999115i	$0.486610\pi$
$458$	−26.8990	−1.25691
$459$	0	0
$460$	0	0
$461$	40.2929	1.87663	0.938313	−	0.345788i	$-0.112388\pi$
$461$	40.2929	1.87663	0.938313	+	0.345788i	$0.112388\pi$
$462$	−2.20204	−0.102448
$463$	−12.8990	−0.599466	−0.299733	−	0.954023i	$-0.596898\pi$
$463$	−12.8990	−0.599466	−0.299733	+	0.954023i	$0.596898\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	−11.0000	−0.509565
$467$	30.9444	1.43194	0.715968	−	0.698133i	$-0.245986\pi$
$467$	30.9444	1.43194	0.715968	+	0.698133i	$0.245986\pi$
$468$	7.34847	0.339683
$469$	3.84337	0.177470
$470$	0	0
$471$	−5.75255	−0.265064
$472$	13.4495	0.619063
$473$	1.79796	0.0826702
$474$	16.8990	0.776196
$475$	0	0
$476$	0.898979	0.0412047
$477$	16.6515	0.762421
$478$	−17.5959	−0.804819
$479$	−23.7980	−1.08736	−0.543678	−	0.839294i	$-0.682969\pi$
$479$	−23.7980	−1.08736	−0.543678	+	0.839294i	$0.682969\pi$
$480$	0	0
$481$	3.55051	0.161889
$482$	17.0000	0.774329
$483$	−2.69694	−0.122715
$484$	−7.00000	−0.318182
$485$	0	0
$486$	−22.0454	−1.00000
$487$	−18.8990	−0.856395	−0.428197	−	0.903685i	$-0.640851\pi$
$487$	−18.8990	−0.856395	−0.428197	+	0.903685i	$0.640851\pi$
$488$	−1.44949	−0.0656153
$489$	−18.4949	−0.836368
$490$	0	0
$491$	−22.4949	−1.01518	−0.507590	−	0.861599i	$-0.669464\pi$
$491$	−22.4949	−1.01518	−0.507590	+	0.861599i	$0.669464\pi$
$492$	−8.20204	−0.369777
$493$	2.00000	0.0900755
$494$	3.79796	0.170878
$495$	0	0
$496$	−3.00000	−0.134704
$497$	1.50510	0.0675131
$498$	−14.6969	−0.658586
$499$	−9.65153	−0.432062	−0.216031	−	0.976387i	$-0.569311\pi$
$499$	−9.65153	−0.432062	−0.216031	+	0.976387i	$0.569311\pi$
$500$	0	0
$501$	4.40408	0.196760
$502$	31.1464	1.39013
$503$	−14.7980	−0.659808	−0.329904	−	0.944014i	$-0.607016\pi$
$503$	−14.7980	−0.659808	−0.329904	+	0.944014i	$0.607016\pi$
$504$	−1.34847	−0.0600656
$505$	0	0
$506$	4.89898	0.217786
$507$	−17.1464	−0.761500
$508$	−10.7980	−0.479082
$509$	41.3939	1.83475	0.917376	−	0.398022i	$-0.130303\pi$
$509$	41.3939	1.83475	0.917376	+	0.398022i	$0.130303\pi$
$510$	0	0
$511$	0.606123	0.0268133
$512$	1.00000	0.0441942
$513$	0	0
$514$	−2.69694	−0.118957
$515$	0	0
$516$	2.20204	0.0969395
$517$	−7.79796	−0.342954
$518$	−0.651531	−0.0286266
$519$	−39.1918	−1.72033
$520$	0	0
$521$	−19.8990	−0.871790	−0.435895	−	0.899997i	$-0.643568\pi$
$521$	−19.8990	−0.871790	−0.435895	+	0.899997i	$0.643568\pi$
$522$	−3.00000	−0.131306
$523$	−22.5505	−0.986065	−0.493032	−	0.870011i	$-0.664112\pi$
$523$	−22.5505	−0.986065	−0.493032	+	0.870011i	$0.664112\pi$
$524$	7.34847	0.321019
$525$	0	0
$526$	2.79796	0.121997
$527$	6.00000	0.261364
$528$	4.89898	0.213201
$529$	−17.0000	−0.739130
$530$	0	0
$531$	40.3485	1.75097
$532$	−0.696938	−0.0302161
$533$	−8.20204	−0.355270
$534$	20.6969	0.895644
$535$	0	0
$536$	−8.55051	−0.369326
$537$	17.7526	0.766079
$538$	9.65153	0.416107
$539$	−13.5959	−0.585618
$540$	0	0
$541$	23.0454	0.990799	0.495400	−	0.868665i	$-0.335022\pi$
$541$	23.0454	0.990799	0.495400	+	0.868665i	$0.335022\pi$
$542$	−13.6969	−0.588334
$543$	44.2020	1.89689
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	−2.69694	−0.115418
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	−8.89898	−0.380146
$549$	−4.34847	−0.185588
$550$	0	0
$551$	−1.55051	−0.0660540
$552$	6.00000	0.255377
$553$	−3.10102	−0.131869
$554$	30.9444	1.31470
$555$	0	0
$556$	−6.55051	−0.277804
$557$	42.4949	1.80057	0.900283	−	0.435304i	$-0.143359\pi$
$557$	42.4949	1.80057	0.900283	+	0.435304i	$0.143359\pi$
$558$	−9.00000	−0.381000
$559$	2.20204	0.0931364
$560$	0	0
$561$	−9.79796	−0.413670
$562$	−9.20204	−0.388165
$563$	−2.20204	−0.0928050	−0.0464025	−	0.998923i	$-0.514776\pi$
$563$	−2.20204	−0.0928050	−0.0464025	+	0.998923i	$0.514776\pi$
$564$	−9.55051	−0.402149
$565$	0	0
$566$	24.0000	1.00880
$567$	4.04541	0.169891
$568$	−3.34847	−0.140499
$569$	20.6969	0.867661	0.433830	−	0.900995i	$-0.357162\pi$
$569$	20.6969	0.867661	0.433830	+	0.900995i	$0.357162\pi$
$570$	0	0
$571$	−19.0454	−0.797026	−0.398513	−	0.917163i	$-0.630474\pi$
$571$	−19.0454	−0.797026	−0.398513	+	0.917163i	$0.630474\pi$
$572$	4.89898	0.204837
$573$	43.3485	1.81091
$574$	1.50510	0.0628218
$575$	0	0
$576$	3.00000	0.125000
$577$	−24.9444	−1.03845	−0.519224	−	0.854638i	$-0.673779\pi$
$577$	−24.9444	−1.03845	−0.519224	+	0.854638i	$0.673779\pi$
$578$	−13.0000	−0.540729
$579$	22.8990	0.951649
$580$	0	0
$581$	2.69694	0.111888
$582$	−18.0000	−0.746124
$583$	11.1010	0.459757
$584$	−1.34847	−0.0558001
$585$	0	0
$586$	−2.55051	−0.105361
$587$	−39.2474	−1.61992	−0.809958	−	0.586488i	$-0.800510\pi$
$587$	−39.2474	−1.61992	−0.809958	+	0.586488i	$0.800510\pi$
$588$	−16.6515	−0.686698
$589$	−4.65153	−0.191663
$590$	0	0
$591$	36.4949	1.50120
$592$	1.44949	0.0595737
$593$	40.3939	1.65878	0.829389	−	0.558672i	$-0.188689\pi$
$593$	40.3939	1.65878	0.829389	+	0.558672i	$0.188689\pi$
$594$	0	0
$595$	0	0
$596$	−3.79796	−0.155570
$597$	−37.5959	−1.53870
$598$	6.00000	0.245358
$599$	−21.8990	−0.894768	−0.447384	−	0.894342i	$-0.647644\pi$
$599$	−21.8990	−0.894768	−0.447384	+	0.894342i	$0.647644\pi$
$600$	0	0
$601$	22.6969	0.925827	0.462914	−	0.886403i	$-0.346804\pi$
$601$	22.6969	0.925827	0.462914	+	0.886403i	$0.346804\pi$
$602$	−0.404082	−0.0164692
$603$	−25.6515	−1.04461
$604$	−3.34847	−0.136247
$605$	0	0
$606$	20.9444	0.850808
$607$	24.3939	0.990117	0.495058	−	0.868860i	$-0.335147\pi$
$607$	24.3939	0.990117	0.495058	+	0.868860i	$0.335147\pi$
$608$	1.55051	0.0628815
$609$	1.10102	0.0446156
$610$	0	0
$611$	−9.55051	−0.386372
$612$	−6.00000	−0.242536
$613$	37.1010	1.49850	0.749248	−	0.662289i	$-0.230415\pi$
$613$	37.1010	1.49850	0.749248	+	0.662289i	$0.230415\pi$
$614$	−33.3939	−1.34767
$615$	0	0
$616$	−0.898979	−0.0362209
$617$	9.55051	0.384489	0.192245	−	0.981347i	$-0.438423\pi$
$617$	9.55051	0.384489	0.192245	+	0.981347i	$0.438423\pi$
$618$	6.00000	0.241355
$619$	−6.89898	−0.277293	−0.138647	−	0.990342i	$-0.544275\pi$
$619$	−6.89898	−0.277293	−0.138647	+	0.990342i	$0.544275\pi$
$620$	0	0
$621$	0	0
$622$	19.5959	0.785725
$623$	−3.79796	−0.152162
$624$	6.00000	0.240192
$625$	0	0
$626$	6.59592	0.263626
$627$	7.59592	0.303352
$628$	−2.34847	−0.0937141
$629$	−2.89898	−0.115590
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	6.89898	0.274427
$633$	64.8990	2.57950
$634$	28.3485	1.12586
$635$	0	0
$636$	13.5959	0.539113
$637$	−16.6515	−0.659758
$638$	−2.00000	−0.0791808
$639$	−10.0454	−0.397390
$640$	0	0
$641$	−0.247449	−0.00977364	−0.00488682	−	0.999988i	$-0.501556\pi$
$641$	−0.247449	−0.00977364	−0.00488682	+	0.999988i	$0.501556\pi$
$642$	−47.1464	−1.86072
$643$	38.8434	1.53183	0.765916	−	0.642940i	$-0.222286\pi$
$643$	38.8434	1.53183	0.765916	+	0.642940i	$0.222286\pi$
$644$	−1.10102	−0.0433863
$645$	0	0
$646$	−3.10102	−0.122008
$647$	0.404082	0.0158861	0.00794305	−	0.999968i	$-0.497472\pi$
$647$	0.404082	0.0158861	0.00794305	+	0.999968i	$0.497472\pi$
$648$	−9.00000	−0.353553
$649$	26.8990	1.05588
$650$	0	0
$651$	3.30306	0.129457
$652$	−7.55051	−0.295701
$653$	−13.2474	−0.518413	−0.259206	−	0.965822i	$-0.583461\pi$
$653$	−13.2474	−0.518413	−0.259206	+	0.965822i	$0.583461\pi$
$654$	13.1010	0.512290
$655$	0	0
$656$	−3.34847	−0.130736
$657$	−4.04541	−0.157826
$658$	1.75255	0.0683216
$659$	−9.30306	−0.362396	−0.181198	−	0.983447i	$-0.557997\pi$
$659$	−9.30306	−0.362396	−0.181198	+	0.983447i	$0.557997\pi$
$660$	0	0
$661$	34.2474	1.33207	0.666036	−	0.745920i	$-0.267990\pi$
$661$	34.2474	1.33207	0.666036	+	0.745920i	$0.267990\pi$
$662$	−5.75255	−0.223579
$663$	−12.0000	−0.466041
$664$	−6.00000	−0.232845
$665$	0	0
$666$	4.34847	0.168500
$667$	−2.44949	−0.0948446
$668$	1.79796	0.0695651
$669$	9.79796	0.378811
$670$	0	0
$671$	−2.89898	−0.111914
$672$	−1.10102	−0.0424728
$673$	2.79796	0.107853	0.0539267	−	0.998545i	$-0.482826\pi$
$673$	2.79796	0.107853	0.0539267	+	0.998545i	$0.482826\pi$
$674$	4.20204	0.161857
$675$	0	0
$676$	−7.00000	−0.269231
$677$	19.0454	0.731974	0.365987	−	0.930620i	$-0.380731\pi$
$677$	19.0454	0.731974	0.365987	+	0.930620i	$0.380731\pi$
$678$	−44.6969	−1.71658
$679$	3.30306	0.126760
$680$	0	0
$681$	−22.6515	−0.868009
$682$	−6.00000	−0.229752
$683$	20.8990	0.799677	0.399839	−	0.916586i	$-0.369066\pi$
$683$	20.8990	0.799677	0.399839	+	0.916586i	$0.369066\pi$
$684$	4.65153	0.177856
$685$	0	0
$686$	6.20204	0.236795
$687$	−65.8888	−2.51381
$688$	0.898979	0.0342733
$689$	13.5959	0.517963
$690$	0	0
$691$	9.94439	0.378302	0.189151	−	0.981948i	$-0.439426\pi$
$691$	9.94439	0.378302	0.189151	+	0.981948i	$0.439426\pi$
$692$	−16.0000	−0.608229
$693$	−2.69694	−0.102448
$694$	19.0454	0.722954
$695$	0	0
$696$	−2.44949	−0.0928477
$697$	6.69694	0.253665
$698$	22.9444	0.868458
$699$	−26.9444	−1.01913
$700$	0	0
$701$	33.3939	1.26127	0.630635	−	0.776080i	$-0.282795\pi$
$701$	33.3939	1.26127	0.630635	+	0.776080i	$0.282795\pi$
$702$	0	0
$703$	2.24745	0.0847641
$704$	2.00000	0.0753778
$705$	0	0
$706$	2.79796	0.105303
$707$	−3.84337	−0.144545
$708$	32.9444	1.23813
$709$	31.3939	1.17902	0.589511	−	0.807760i	$-0.299321\pi$
$709$	31.3939	1.17902	0.589511	+	0.807760i	$0.299321\pi$
$710$	0	0
$711$	20.6969	0.776196
$712$	8.44949	0.316658
$713$	−7.34847	−0.275202
$714$	2.20204	0.0824093
$715$	0	0
$716$	7.24745	0.270850
$717$	−43.1010	−1.60964
$718$	22.5959	0.843272
$719$	36.7423	1.37026	0.685129	−	0.728422i	$-0.259746\pi$
$719$	36.7423	1.37026	0.685129	+	0.728422i	$0.259746\pi$
$720$	0	0
$721$	−1.10102	−0.0410041
$722$	−16.5959	−0.617636
$723$	41.6413	1.54866
$724$	18.0454	0.670652
$725$	0	0
$726$	−17.1464	−0.636364
$727$	5.59592	0.207541	0.103771	−	0.994601i	$-0.466909\pi$
$727$	5.59592	0.207541	0.103771	+	0.994601i	$0.466909\pi$
$728$	−1.10102	−0.0408065
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−1.79796	−0.0664999
$732$	−3.55051	−0.131231
$733$	10.8990	0.402563	0.201281	−	0.979533i	$-0.435489\pi$
$733$	10.8990	0.402563	0.201281	+	0.979533i	$0.435489\pi$
$734$	−20.7980	−0.767667
$735$	0	0
$736$	2.44949	0.0902894
$737$	−17.1010	−0.629924
$738$	−10.0454	−0.369777
$739$	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
$741$	9.30306	0.341757
$742$	−2.49490	−0.0915906
$743$	−16.6969	−0.612551	−0.306276	−	0.951943i	$-0.599083\pi$
$743$	−16.6969	−0.612551	−0.306276	+	0.951943i	$0.599083\pi$
$744$	−7.34847	−0.269408
$745$	0	0
$746$	−20.4949	−0.750372
$747$	−18.0000	−0.658586
$748$	−4.00000	−0.146254
$749$	8.65153	0.316120
$750$	0	0
$751$	33.8990	1.23699	0.618496	−	0.785788i	$-0.287742\pi$
$751$	33.8990	1.23699	0.618496	+	0.785788i	$0.287742\pi$
$752$	−3.89898	−0.142181
$753$	76.2929	2.78027
$754$	−2.44949	−0.0892052
$755$	0	0
$756$	0	0
$757$	39.3939	1.43179	0.715897	−	0.698205i	$-0.246018\pi$
$757$	39.3939	1.43179	0.715897	+	0.698205i	$0.246018\pi$
$758$	−9.14643	−0.332213
$759$	12.0000	0.435572
$760$	0	0
$761$	23.8990	0.866337	0.433169	−	0.901313i	$-0.357395\pi$
$761$	23.8990	0.866337	0.433169	+	0.901313i	$0.357395\pi$
$762$	−26.4495	−0.958164
$763$	−2.40408	−0.0870336
$764$	17.6969	0.640253
$765$	0	0
$766$	7.79796	0.281752
$767$	32.9444	1.18955
$768$	2.44949	0.0883883
$769$	−8.44949	−0.304696	−0.152348	−	0.988327i	$-0.548684\pi$
$769$	−8.44949	−0.304696	−0.152348	+	0.988327i	$0.548684\pi$
$770$	0	0
$771$	−6.60612	−0.237914
$772$	9.34847	0.336459
$773$	−12.5505	−0.451410	−0.225705	−	0.974196i	$-0.572469\pi$
$773$	−12.5505	−0.451410	−0.225705	+	0.974196i	$0.572469\pi$
$774$	2.69694	0.0969395
$775$	0	0
$776$	−7.34847	−0.263795
$777$	−1.59592	−0.0572532
$778$	−9.65153	−0.346024
$779$	−5.19184	−0.186017
$780$	0	0
$781$	−6.69694	−0.239635
$782$	−4.89898	−0.175187
$783$	0	0
$784$	−6.79796	−0.242784
$785$	0	0
$786$	18.0000	0.642039
$787$	−45.4495	−1.62010	−0.810050	−	0.586361i	$-0.800560\pi$
$787$	−45.4495	−1.62010	−0.810050	+	0.586361i	$0.800560\pi$
$788$	14.8990	0.530754
$789$	6.85357	0.243994
$790$	0	0
$791$	8.20204	0.291631
$792$	6.00000	0.213201
$793$	−3.55051	−0.126082
$794$	−21.1464	−0.750459
$795$	0	0
$796$	−15.3485	−0.544012
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	−1.70714	−0.0604322
$799$	7.79796	0.275872
$800$	0	0
$801$	25.3485	0.895644
$802$	−0.595918	−0.0210426
$803$	−2.69694	−0.0951729
$804$	−20.9444	−0.738652
$805$	0	0
$806$	−7.34847	−0.258839
$807$	23.6413	0.832214
$808$	8.55051	0.300806
$809$	−7.59592	−0.267058	−0.133529	−	0.991045i	$-0.542631\pi$
$809$	−7.59592	−0.267058	−0.133529	+	0.991045i	$0.542631\pi$
$810$	0	0
$811$	−24.5505	−0.862085	−0.431043	−	0.902332i	$-0.641854\pi$
$811$	−24.5505	−0.862085	−0.431043	+	0.902332i	$0.641854\pi$
$812$	0.449490	0.0157740
$813$	−33.5505	−1.17667
$814$	2.89898	0.101609
$815$	0	0
$816$	−4.89898	−0.171499
$817$	1.39388	0.0487656
$818$	13.7980	0.482434
$819$	−3.30306	−0.115418
$820$	0	0
$821$	44.2474	1.54425	0.772123	−	0.635473i	$-0.219195\pi$
$821$	44.2474	1.54425	0.772123	+	0.635473i	$0.219195\pi$
$822$	−21.7980	−0.760291
$823$	7.79796	0.271820	0.135910	−	0.990721i	$-0.456604\pi$
$823$	7.79796	0.271820	0.135910	+	0.990721i	$0.456604\pi$
$824$	2.44949	0.0853320
$825$	0	0
$826$	−6.04541	−0.210347
$827$	48.6969	1.69336	0.846679	−	0.532104i	$-0.178598\pi$
$827$	48.6969	1.69336	0.846679	+	0.532104i	$0.178598\pi$
$828$	7.34847	0.255377
$829$	−34.4949	−1.19806	−0.599029	−	0.800728i	$-0.704446\pi$
$829$	−34.4949	−1.19806	−0.599029	+	0.800728i	$0.704446\pi$
$830$	0	0
$831$	75.7980	2.62940
$832$	2.44949	0.0849208
$833$	13.5959	0.471071
$834$	−16.0454	−0.555607
$835$	0	0
$836$	3.10102	0.107251
$837$	0	0
$838$	24.1464	0.834125
$839$	−11.2020	−0.386737	−0.193369	−	0.981126i	$-0.561941\pi$
$839$	−11.2020	−0.386737	−0.193369	+	0.981126i	$0.561941\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	8.55051	0.294670
$843$	−22.5403	−0.776329
$844$	26.4949	0.911992
$845$	0	0
$846$	−11.6969	−0.402149
$847$	3.14643	0.108113
$848$	5.55051	0.190605
$849$	58.7878	2.01759
$850$	0	0
$851$	3.55051	0.121710
$852$	−8.20204	−0.280997
$853$	27.7980	0.951784	0.475892	−	0.879504i	$-0.342125\pi$
$853$	27.7980	0.951784	0.475892	+	0.879504i	$0.342125\pi$
$854$	0.651531	0.0222949
$855$	0	0
$856$	−19.2474	−0.657864
$857$	19.3939	0.662482	0.331241	−	0.943546i	$-0.392533\pi$
$857$	19.3939	0.662482	0.331241	+	0.943546i	$0.392533\pi$
$858$	12.0000	0.409673
$859$	13.1010	0.447001	0.223501	−	0.974704i	$-0.428252\pi$
$859$	13.1010	0.447001	0.223501	+	0.974704i	$0.428252\pi$
$860$	0	0
$861$	3.68673	0.125644
$862$	13.5505	0.461532
$863$	36.2474	1.23388	0.616939	−	0.787011i	$-0.288373\pi$
$863$	36.2474	1.23388	0.616939	+	0.787011i	$0.288373\pi$
$864$	0	0
$865$	0	0
$866$	25.5505	0.868242
$867$	−31.8434	−1.08146
$868$	1.34847	0.0457700
$869$	13.7980	0.468064
$870$	0	0
$871$	−20.9444	−0.709673
$872$	5.34847	0.181122
$873$	−22.0454	−0.746124
$874$	3.79796	0.128468
$875$	0	0
$876$	−3.30306	−0.111600
$877$	8.00000	0.270141	0.135070	−	0.990836i	$-0.456874\pi$
$877$	8.00000	0.270141	0.135070	+	0.990836i	$0.456874\pi$
$878$	−10.6969	−0.361004
$879$	−6.24745	−0.210721
$880$	0	0
$881$	12.0000	0.404290	0.202145	−	0.979356i	$-0.435209\pi$
$881$	12.0000	0.404290	0.202145	+	0.979356i	$0.435209\pi$
$882$	−20.3939	−0.686698
$883$	−23.9444	−0.805793	−0.402896	−	0.915246i	$-0.631997\pi$
$883$	−23.9444	−0.805793	−0.402896	+	0.915246i	$0.631997\pi$
$884$	−4.89898	−0.164771
$885$	0	0
$886$	−22.8990	−0.769306
$887$	−28.2020	−0.946932	−0.473466	−	0.880812i	$-0.656997\pi$
$887$	−28.2020	−0.946932	−0.473466	+	0.880812i	$0.656997\pi$
$888$	3.55051	0.119147
$889$	4.85357	0.162784
$890$	0	0
$891$	−18.0000	−0.603023
$892$	4.00000	0.133930
$893$	−6.04541	−0.202302
$894$	−9.30306	−0.311141
$895$	0	0
$896$	−0.449490	−0.0150164
$897$	14.6969	0.490716
$898$	6.20204	0.206965
$899$	3.00000	0.100056
$900$	0	0
$901$	−11.1010	−0.369829
$902$	−6.69694	−0.222984
$903$	−0.989795	−0.0329383
$904$	−18.2474	−0.606901
$905$	0	0
$906$	−8.20204	−0.272495
$907$	−6.65153	−0.220860	−0.110430	−	0.993884i	$-0.535223\pi$
$907$	−6.65153	−0.220860	−0.110430	+	0.993884i	$0.535223\pi$
$908$	−9.24745	−0.306887
$909$	25.6515	0.850808
$910$	0	0
$911$	2.50510	0.0829978	0.0414989	−	0.999139i	$-0.486787\pi$
$911$	2.50510	0.0829978	0.0414989	+	0.999139i	$0.486787\pi$
$912$	3.79796	0.125763
$913$	−12.0000	−0.397142
$914$	1.79796	0.0594712
$915$	0	0
$916$	−26.8990	−0.888767
$917$	−3.30306	−0.109077
$918$	0	0
$919$	−59.1464	−1.95106	−0.975530	−	0.219865i	$-0.929438\pi$
$919$	−59.1464	−1.95106	−0.975530	+	0.219865i	$0.929438\pi$
$920$	0	0
$921$	−81.7980	−2.69533
$922$	40.2929	1.32697
$923$	−8.20204	−0.269973
$924$	−2.20204	−0.0724418
$925$	0	0
$926$	−12.8990	−0.423887
$927$	7.34847	0.241355
$928$	−1.00000	−0.0328266
$929$	9.49490	0.311517	0.155759	−	0.987795i	$-0.450218\pi$
$929$	9.49490	0.311517	0.155759	+	0.987795i	$0.450218\pi$
$930$	0	0
$931$	−10.5403	−0.345445
$932$	−11.0000	−0.360317
$933$	48.0000	1.57145
$934$	30.9444	1.01253
$935$	0	0
$936$	7.34847	0.240192
$937$	−15.2929	−0.499596	−0.249798	−	0.968298i	$-0.580364\pi$
$937$	−15.2929	−0.499596	−0.249798	+	0.968298i	$0.580364\pi$
$938$	3.84337	0.125490
$939$	16.1566	0.527252
$940$	0	0
$941$	3.55051	0.115743	0.0578717	−	0.998324i	$-0.481569\pi$
$941$	3.55051	0.115743	0.0578717	+	0.998324i	$0.481569\pi$
$942$	−5.75255	−0.187428
$943$	−8.20204	−0.267095
$944$	13.4495	0.437744
$945$	0	0
$946$	1.79796	0.0584567
$947$	22.6515	0.736076	0.368038	−	0.929811i	$-0.380030\pi$
$947$	22.6515	0.736076	0.368038	+	0.929811i	$0.380030\pi$
$948$	16.8990	0.548853
$949$	−3.30306	−0.107222
$950$	0	0
$951$	69.4393	2.25172
$952$	0.898979	0.0291361
$953$	−34.2929	−1.11085	−0.555427	−	0.831565i	$-0.687445\pi$
$953$	−34.2929	−1.11085	−0.555427	+	0.831565i	$0.687445\pi$
$954$	16.6515	0.539113
$955$	0	0
$956$	−17.5959	−0.569093
$957$	−4.89898	−0.158362
$958$	−23.7980	−0.768877
$959$	4.00000	0.129167
$960$	0	0
$961$	−22.0000	−0.709677
$962$	3.55051	0.114473
$963$	−57.7423	−1.86072
$964$	17.0000	0.547533
$965$	0	0
$966$	−2.69694	−0.0867726
$967$	28.0000	0.900419	0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000	0.900419	0.450210	+	0.892923i	$0.351349\pi$
$968$	−7.00000	−0.224989
$969$	−7.59592	−0.244016
$970$	0	0
$971$	−30.9444	−0.993053	−0.496526	−	0.868022i	$-0.665391\pi$
$971$	−30.9444	−0.993053	−0.496526	+	0.868022i	$0.665391\pi$
$972$	−22.0454	−0.707107
$973$	2.94439	0.0943927
$974$	−18.8990	−0.605562
$975$	0	0
$976$	−1.44949	−0.0463970
$977$	9.89898	0.316696	0.158348	−	0.987383i	$-0.449383\pi$
$977$	9.89898	0.316696	0.158348	+	0.987383i	$0.449383\pi$
$978$	−18.4949	−0.591402
$979$	16.8990	0.540094
$980$	0	0
$981$	16.0454	0.512290
$982$	−22.4949	−0.717841
$983$	39.0000	1.24391	0.621953	−	0.783054i	$-0.286339\pi$
$983$	39.0000	1.24391	0.621953	+	0.783054i	$0.286339\pi$
$984$	−8.20204	−0.261472
$985$	0	0
$986$	2.00000	0.0636930
$987$	4.29286	0.136643
$988$	3.79796	0.120829
$989$	2.20204	0.0700208
$990$	0	0
$991$	−52.4949	−1.66756	−0.833778	−	0.552100i	$-0.813827\pi$
$991$	−52.4949	−1.66756	−0.833778	+	0.552100i	$0.813827\pi$
$992$	−3.00000	−0.0952501
$993$	−14.0908	−0.447159
$994$	1.50510	0.0477390
$995$	0	0
$996$	−14.6969	−0.465690
$997$	−43.7423	−1.38533	−0.692667	−	0.721258i	$-0.743564\pi$
$997$	−43.7423	−1.38533	−0.692667	+	0.721258i	$0.743564\pi$
$998$	−9.65153	−0.305514
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1450.2.a.o.1.2	yes	2
5.2	odd	4		1450.2.b.i.349.3		4
5.3	odd	4		1450.2.b.i.349.2		4
5.4	even	2		1450.2.a.j.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1450.2.a.j.1.1	✓	2	5.4	even	2
1450.2.a.o.1.2	yes	2	1.1	even	1	trivial
1450.2.b.i.349.2		4	5.3	odd	4
1450.2.b.i.349.3		4	5.2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -0.449490 q^{7} +1.00000 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} +2.44949 q^{6} -0.449490 q^{7} +1.00000 q^{8} +3.00000 q^{9} +2.00000 q^{11} +2.44949 q^{12} +2.44949 q^{13} -0.449490 q^{14} +1.00000 q^{16} -2.00000 q^{17} +3.00000 q^{18} +1.55051 q^{19} -1.10102 q^{21} +2.00000 q^{22} +2.44949 q^{23} +2.44949 q^{24} +2.44949 q^{26} -0.449490 q^{28} -1.00000 q^{29} -3.00000 q^{31} +1.00000 q^{32} +4.89898 q^{33} -2.00000 q^{34} +3.00000 q^{36} +1.44949 q^{37} +1.55051 q^{38} +6.00000 q^{39} -3.34847 q^{41} -1.10102 q^{42} +0.898979 q^{43} +2.00000 q^{44} +2.44949 q^{46} -3.89898 q^{47} +2.44949 q^{48} -6.79796 q^{49} -4.89898 q^{51} +2.44949 q^{52} +5.55051 q^{53} -0.449490 q^{56} +3.79796 q^{57} -1.00000 q^{58} +13.4495 q^{59} -1.44949 q^{61} -3.00000 q^{62} -1.34847 q^{63} +1.00000 q^{64} +4.89898 q^{66} -8.55051 q^{67} -2.00000 q^{68} +6.00000 q^{69} -3.34847 q^{71} +3.00000 q^{72} -1.34847 q^{73} +1.44949 q^{74} +1.55051 q^{76} -0.898979 q^{77} +6.00000 q^{78} +6.89898 q^{79} -9.00000 q^{81} -3.34847 q^{82} -6.00000 q^{83} -1.10102 q^{84} +0.898979 q^{86} -2.44949 q^{87} +2.00000 q^{88} +8.44949 q^{89} -1.10102 q^{91} +2.44949 q^{92} -7.34847 q^{93} -3.89898 q^{94} +2.44949 q^{96} -7.34847 q^{97} -6.79796 q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^7 + 2 * q^8 + 6 * q^9	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{7} + 2 q^{8} + 6 q^{9} + 4 q^{11} + 4 q^{14} + 2 q^{16} - 4 q^{17} + 6 q^{18} + 8 q^{19} - 12 q^{21} + 4 q^{22} + 4 q^{28} - 2 q^{29} - 6 q^{31} + 2 q^{32} - 4 q^{34} + 6 q^{36} - 2 q^{37} + 8 q^{38} + 12 q^{39} + 8 q^{41} - 12 q^{42} - 8 q^{43} + 4 q^{44} + 2 q^{47} + 6 q^{49} + 16 q^{53} + 4 q^{56} - 12 q^{57} - 2 q^{58} + 22 q^{59} + 2 q^{61} - 6 q^{62} + 12 q^{63} + 2 q^{64} - 22 q^{67} - 4 q^{68} + 12 q^{69} + 8 q^{71} + 6 q^{72} + 12 q^{73} - 2 q^{74} + 8 q^{76} + 8 q^{77} + 12 q^{78} + 4 q^{79} - 18 q^{81} + 8 q^{82} - 12 q^{83} - 12 q^{84} - 8 q^{86} + 4 q^{88} + 12 q^{89} - 12 q^{91} + 2 q^{94} + 6 q^{98} + 12 q^{99}+O(q^{100}) \) 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

Varieties

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Database

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