LMFDB - Embedded newform 2450.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2450,2,Mod(1,2450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2450, 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("2450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2450 = 2 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2450.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:	$19.5633484952$
Analytic rank:	$1$
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:	no (minimal twist has level 70)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	2450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} -1.00000 q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} -1.00000 q^{8} +3.00000 q^{9} -4.89898 q^{11} +2.44949 q^{12} -4.44949 q^{13} +1.00000 q^{16} -2.00000 q^{17} -3.00000 q^{18} -1.55051 q^{19} +4.89898 q^{22} +2.89898 q^{23} -2.44949 q^{24} +4.44949 q^{26} +6.89898 q^{29} -8.89898 q^{31} -1.00000 q^{32} -12.0000 q^{33} +2.00000 q^{34} +3.00000 q^{36} +2.00000 q^{37} +1.55051 q^{38} -10.8990 q^{39} +1.10102 q^{41} -0.898979 q^{43} -4.89898 q^{44} -2.89898 q^{46} -8.89898 q^{47} +2.44949 q^{48} -4.89898 q^{51} -4.44949 q^{52} -10.8990 q^{53} -3.79796 q^{57} -6.89898 q^{58} +1.55051 q^{59} -3.55051 q^{61} +8.89898 q^{62} +1.00000 q^{64} +12.0000 q^{66} -8.00000 q^{67} -2.00000 q^{68} +7.10102 q^{69} -1.10102 q^{71} -3.00000 q^{72} -2.89898 q^{73} -2.00000 q^{74} -1.55051 q^{76} +10.8990 q^{78} +6.89898 q^{79} -9.00000 q^{81} -1.10102 q^{82} +2.44949 q^{83} +0.898979 q^{86} +16.8990 q^{87} +4.89898 q^{88} +10.0000 q^{89} +2.89898 q^{92} -21.7980 q^{93} +8.89898 q^{94} -2.44949 q^{96} -15.7980 q^{97} -14.6969 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 6 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 6 q^{9} - 4 q^{13} + 2 q^{16} - 4 q^{17} - 6 q^{18} - 8 q^{19} - 4 q^{23} + 4 q^{26} + 4 q^{29} - 8 q^{31} - 2 q^{32} - 24 q^{33} + 4 q^{34} + 6 q^{36} + 4 q^{37} + 8 q^{38} - 12 q^{39} + 12 q^{41} + 8 q^{43} + 4 q^{46} - 8 q^{47} - 4 q^{52} - 12 q^{53} + 12 q^{57} - 4 q^{58} + 8 q^{59} - 12 q^{61} + 8 q^{62} + 2 q^{64} + 24 q^{66} - 16 q^{67} - 4 q^{68} + 24 q^{69} - 12 q^{71} - 6 q^{72} + 4 q^{73} - 4 q^{74} - 8 q^{76} + 12 q^{78} + 4 q^{79} - 18 q^{81} - 12 q^{82} - 8 q^{86} + 24 q^{87} + 20 q^{89} - 4 q^{92} - 24 q^{93} + 8 q^{94} - 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 6 * q^9 - 4 * q^13 + 2 * q^16 - 4 * q^17 - 6 * q^18 - 8 * q^19 - 4 * q^23 + 4 * q^26 + 4 * q^29 - 8 * q^31 - 2 * q^32 - 24 * q^33 + 4 * q^34 + 6 * q^36 + 4 * q^37 + 8 * q^38 - 12 * q^39 + 12 * q^41 + 8 * q^43 + 4 * q^46 - 8 * q^47 - 4 * q^52 - 12 * q^53 + 12 * q^57 - 4 * q^58 + 8 * q^59 - 12 * q^61 + 8 * q^62 + 2 * q^64 + 24 * q^66 - 16 * q^67 - 4 * q^68 + 24 * q^69 - 12 * q^71 - 6 * q^72 + 4 * q^73 - 4 * q^74 - 8 * q^76 + 12 * q^78 + 4 * q^79 - 18 * q^81 - 12 * q^82 - 8 * q^86 + 24 * q^87 + 20 * q^89 - 4 * q^92 - 24 * q^93 + 8 * q^94 - 12 * q^97

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	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	3.00000	1.00000
$10$	0	0
$11$	−4.89898	−1.47710	−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898	−1.47710	−0.738549	+	0.674200i	$0.764489\pi$
$12$	2.44949	0.707107
$13$	−4.44949	−1.23407	−0.617033	−	0.786937i	$-0.711666\pi$
$13$	−4.44949	−1.23407	−0.617033	+	0.786937i	$0.711666\pi$
$14$	0	0
$15$	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	$-0.556916\pi$
$19$	−1.55051	−0.355711	−0.177856	+	0.984057i	$0.556916\pi$
$20$	0	0
$21$	0	0
$22$	4.89898	1.04447
$23$	2.89898	0.604479	0.302240	−	0.953232i	$-0.402266\pi$
$23$	2.89898	0.604479	0.302240	+	0.953232i	$0.402266\pi$
$24$	−2.44949	−0.500000
$25$	0	0
$26$	4.44949	0.872617
$27$	0	0
$28$	0	0
$29$	6.89898	1.28111	0.640554	−	0.767913i	$-0.278705\pi$
$29$	6.89898	1.28111	0.640554	+	0.767913i	$0.278705\pi$
$30$	0	0
$31$	−8.89898	−1.59830	−0.799152	−	0.601129i	$-0.794718\pi$
$31$	−8.89898	−1.59830	−0.799152	+	0.601129i	$0.794718\pi$
$32$	−1.00000	−0.176777
$33$	−12.0000	−2.08893
$34$	2.00000	0.342997
$35$	0	0
$36$	3.00000	0.500000
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	1.55051	0.251526
$39$	−10.8990	−1.74523
$40$	0	0
$41$	1.10102	0.171951	0.0859753	−	0.996297i	$-0.472599\pi$
$41$	1.10102	0.171951	0.0859753	+	0.996297i	$0.472599\pi$
$42$	0	0
$43$	−0.898979	−0.137093	−0.0685465	−	0.997648i	$-0.521836\pi$
$43$	−0.898979	−0.137093	−0.0685465	+	0.997648i	$0.521836\pi$
$44$	−4.89898	−0.738549
$45$	0	0
$46$	−2.89898	−0.427431
$47$	−8.89898	−1.29805	−0.649025	−	0.760767i	$-0.724823\pi$
$47$	−8.89898	−1.29805	−0.649025	+	0.760767i	$0.724823\pi$
$48$	2.44949	0.353553
$49$	0	0
$50$	0	0
$51$	−4.89898	−0.685994
$52$	−4.44949	−0.617033
$53$	−10.8990	−1.49709	−0.748545	−	0.663084i	$-0.769247\pi$
$53$	−10.8990	−1.49709	−0.748545	+	0.663084i	$0.769247\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.79796	−0.503052
$58$	−6.89898	−0.905880
$59$	1.55051	0.201859	0.100930	−	0.994894i	$-0.467818\pi$
$59$	1.55051	0.201859	0.100930	+	0.994894i	$0.467818\pi$
$60$	0	0
$61$	−3.55051	−0.454596	−0.227298	−	0.973825i	$-0.572989\pi$
$61$	−3.55051	−0.454596	−0.227298	+	0.973825i	$0.572989\pi$
$62$	8.89898	1.13017
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	12.0000	1.47710
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−2.00000	−0.242536
$69$	7.10102	0.854862
$70$	0	0
$71$	−1.10102	−0.130667	−0.0653335	−	0.997863i	$-0.520811\pi$
$71$	−1.10102	−0.130667	−0.0653335	+	0.997863i	$0.520811\pi$
$72$	−3.00000	−0.353553
$73$	−2.89898	−0.339300	−0.169650	−	0.985504i	$-0.554264\pi$
$73$	−2.89898	−0.339300	−0.169650	+	0.985504i	$0.554264\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	−1.55051	−0.177856
$77$	0	0
$78$	10.8990	1.23407
$79$	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
$82$	−1.10102	−0.121587
$83$	2.44949	0.268866	0.134433	−	0.990923i	$-0.457079\pi$
$83$	2.44949	0.268866	0.134433	+	0.990923i	$0.457079\pi$
$84$	0	0
$85$	0	0
$86$	0.898979	0.0969395
$87$	16.8990	1.81176
$88$	4.89898	0.522233
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	0	0
$92$	2.89898	0.302240
$93$	−21.7980	−2.26034
$94$	8.89898	0.917860
$95$	0	0
$96$	−2.44949	−0.250000
$97$	−15.7980	−1.60404	−0.802020	−	0.597297i	$-0.796241\pi$
$97$	−15.7980	−1.60404	−0.802020	+	0.597297i	$0.796241\pi$
$98$	0	0
$99$	−14.6969	−1.47710
$100$	0	0
$101$	−3.55051	−0.353289	−0.176644	−	0.984275i	$-0.556524\pi$
$101$	−3.55051	−0.353289	−0.176644	+	0.984275i	$0.556524\pi$
$102$	4.89898	0.485071
$103$	−12.8990	−1.27097	−0.635487	−	0.772111i	$-0.719201\pi$
$103$	−12.8990	−1.27097	−0.635487	+	0.772111i	$0.719201\pi$
$104$	4.44949	0.436308
$105$	0	0
$106$	10.8990	1.05860
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	6.89898	0.660802	0.330401	−	0.943841i	$-0.392816\pi$
$109$	6.89898	0.660802	0.330401	+	0.943841i	$0.392816\pi$
$110$	0	0
$111$	4.89898	0.464991
$112$	0	0
$113$	19.7980	1.86244	0.931218	−	0.364464i	$-0.118748\pi$
$113$	19.7980	1.86244	0.931218	+	0.364464i	$0.118748\pi$
$114$	3.79796	0.355711
$115$	0	0
$116$	6.89898	0.640554
$117$	−13.3485	−1.23407
$118$	−1.55051	−0.142736
$119$	0	0
$120$	0	0
$121$	13.0000	1.18182
$122$	3.55051	0.321448
$123$	2.69694	0.243175
$124$	−8.89898	−0.799152
$125$	0	0
$126$	0	0
$127$	−14.8990	−1.32207	−0.661035	−	0.750355i	$-0.729883\pi$
$127$	−14.8990	−1.32207	−0.661035	+	0.750355i	$0.729883\pi$
$128$	−1.00000	−0.0883883
$129$	−2.20204	−0.193879
$130$	0	0
$131$	6.44949	0.563495	0.281747	−	0.959489i	$-0.409086\pi$
$131$	6.44949	0.563495	0.281747	+	0.959489i	$0.409086\pi$
$132$	−12.0000	−1.04447
$133$	0	0
$134$	8.00000	0.691095
$135$	0	0
$136$	2.00000	0.171499
$137$	−1.79796	−0.153610	−0.0768050	−	0.997046i	$-0.524472\pi$
$137$	−1.79796	−0.153610	−0.0768050	+	0.997046i	$0.524472\pi$
$138$	−7.10102	−0.604479
$139$	−1.55051	−0.131513	−0.0657563	−	0.997836i	$-0.520946\pi$
$139$	−1.55051	−0.131513	−0.0657563	+	0.997836i	$0.520946\pi$
$140$	0	0
$141$	−21.7980	−1.83572
$142$	1.10102	0.0923956
$143$	21.7980	1.82284
$144$	3.00000	0.250000
$145$	0	0
$146$	2.89898	0.239921
$147$	0	0
$148$	2.00000	0.164399
$149$	−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
$151$	19.5959	1.59469	0.797347	−	0.603522i	$-0.206236\pi$
$151$	19.5959	1.59469	0.797347	+	0.603522i	$0.206236\pi$
$152$	1.55051	0.125763
$153$	−6.00000	−0.485071
$154$	0	0
$155$	0	0
$156$	−10.8990	−0.872617
$157$	−3.55051	−0.283362	−0.141681	−	0.989912i	$-0.545251\pi$
$157$	−3.55051	−0.283362	−0.141681	+	0.989912i	$0.545251\pi$
$158$	−6.89898	−0.548853
$159$	−26.6969	−2.11720
$160$	0	0
$161$	0	0
$162$	9.00000	0.707107
$163$	−7.10102	−0.556195	−0.278097	−	0.960553i	$-0.589704\pi$
$163$	−7.10102	−0.556195	−0.278097	+	0.960553i	$0.589704\pi$
$164$	1.10102	0.0859753
$165$	0	0
$166$	−2.44949	−0.190117
$167$	4.89898	0.379094	0.189547	−	0.981872i	$-0.439298\pi$
$167$	4.89898	0.379094	0.189547	+	0.981872i	$0.439298\pi$
$168$	0	0
$169$	6.79796	0.522920
$170$	0	0
$171$	−4.65153	−0.355711
$172$	−0.898979	−0.0685465
$173$	6.24745	0.474985	0.237492	−	0.971389i	$-0.423675\pi$
$173$	6.24745	0.474985	0.237492	+	0.971389i	$0.423675\pi$
$174$	−16.8990	−1.28111
$175$	0	0
$176$	−4.89898	−0.369274
$177$	3.79796	0.285472
$178$	−10.0000	−0.749532
$179$	13.7980	1.03131	0.515654	−	0.856797i	$-0.327549\pi$
$179$	13.7980	1.03131	0.515654	+	0.856797i	$0.327549\pi$
$180$	0	0
$181$	10.2474	0.761687	0.380843	−	0.924640i	$-0.375634\pi$
$181$	10.2474	0.761687	0.380843	+	0.924640i	$0.375634\pi$
$182$	0	0
$183$	−8.69694	−0.642896
$184$	−2.89898	−0.213716
$185$	0	0
$186$	21.7980	1.59830
$187$	9.79796	0.716498
$188$	−8.89898	−0.649025
$189$	0	0
$190$	0	0
$191$	12.6969	0.918718	0.459359	−	0.888251i	$-0.348079\pi$
$191$	12.6969	0.918718	0.459359	+	0.888251i	$0.348079\pi$
$192$	2.44949	0.176777
$193$	−21.5959	−1.55451	−0.777254	−	0.629187i	$-0.783388\pi$
$193$	−21.5959	−1.55451	−0.777254	+	0.629187i	$0.783388\pi$
$194$	15.7980	1.13423
$195$	0	0
$196$	0	0
$197$	18.8990	1.34650	0.673248	−	0.739417i	$-0.264899\pi$
$197$	18.8990	1.34650	0.673248	+	0.739417i	$0.264899\pi$
$198$	14.6969	1.04447
$199$	−16.8990	−1.19794	−0.598968	−	0.800773i	$-0.704423\pi$
$199$	−16.8990	−1.19794	−0.598968	+	0.800773i	$0.704423\pi$
$200$	0	0
$201$	−19.5959	−1.38219
$202$	3.55051	0.249813
$203$	0	0
$204$	−4.89898	−0.342997
$205$	0	0
$206$	12.8990	0.898714
$207$	8.69694	0.604479
$208$	−4.44949	−0.308517
$209$	7.59592	0.525421
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	−10.8990	−0.748545
$213$	−2.69694	−0.184791
$214$	8.00000	0.546869
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−6.89898	−0.467258
$219$	−7.10102	−0.479842
$220$	0	0
$221$	8.89898	0.598610
$222$	−4.89898	−0.328798
$223$	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	0
$225$	0	0
$226$	−19.7980	−1.31694
$227$	−7.34847	−0.487735	−0.243868	−	0.969809i	$-0.578416\pi$
$227$	−7.34847	−0.487735	−0.243868	+	0.969809i	$0.578416\pi$
$228$	−3.79796	−0.251526
$229$	19.1464	1.26523	0.632616	−	0.774466i	$-0.281981\pi$
$229$	19.1464	1.26523	0.632616	+	0.774466i	$0.281981\pi$
$230$	0	0
$231$	0	0
$232$	−6.89898	−0.452940
$233$	29.7980	1.95213	0.976065	−	0.217481i	$-0.0697840\pi$
$233$	29.7980	1.95213	0.976065	+	0.217481i	$0.0697840\pi$
$234$	13.3485	0.872617
$235$	0	0
$236$	1.55051	0.100930
$237$	16.8990	1.09771
$238$	0	0
$239$	−6.20204	−0.401177	−0.200588	−	0.979676i	$-0.564285\pi$
$239$	−6.20204	−0.401177	−0.200588	+	0.979676i	$0.564285\pi$
$240$	0	0
$241$	8.69694	0.560219	0.280110	−	0.959968i	$-0.409629\pi$
$241$	8.69694	0.560219	0.280110	+	0.959968i	$0.409629\pi$
$242$	−13.0000	−0.835672
$243$	−22.0454	−1.41421
$244$	−3.55051	−0.227298
$245$	0	0
$246$	−2.69694	−0.171951
$247$	6.89898	0.438972
$248$	8.89898	0.565086
$249$	6.00000	0.380235
$250$	0	0
$251$	6.44949	0.407088	0.203544	−	0.979066i	$-0.434754\pi$
$251$	6.44949	0.407088	0.203544	+	0.979066i	$0.434754\pi$
$252$	0	0
$253$	−14.2020	−0.892875
$254$	14.8990	0.934845
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.69694	0.542500	0.271250	−	0.962509i	$-0.412563\pi$
$257$	8.69694	0.542500	0.271250	+	0.962509i	$0.412563\pi$
$258$	2.20204	0.137093
$259$	0	0
$260$	0	0
$261$	20.6969	1.28111
$262$	−6.44949	−0.398451
$263$	9.79796	0.604168	0.302084	−	0.953281i	$-0.402318\pi$
$263$	9.79796	0.604168	0.302084	+	0.953281i	$0.402318\pi$
$264$	12.0000	0.738549
$265$	0	0
$266$	0	0
$267$	24.4949	1.49906
$268$	−8.00000	−0.488678
$269$	−19.1464	−1.16738	−0.583689	−	0.811977i	$-0.698391\pi$
$269$	−19.1464	−1.16738	−0.583689	+	0.811977i	$0.698391\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	1.79796	0.108619
$275$	0	0
$276$	7.10102	0.427431
$277$	−14.8990	−0.895193	−0.447596	−	0.894236i	$-0.647720\pi$
$277$	−14.8990	−0.895193	−0.447596	+	0.894236i	$0.647720\pi$
$278$	1.55051	0.0929934
$279$	−26.6969	−1.59830
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	21.7980	1.29805
$283$	−3.75255	−0.223066	−0.111533	−	0.993761i	$-0.535576\pi$
$283$	−3.75255	−0.223066	−0.111533	+	0.993761i	$0.535576\pi$
$284$	−1.10102	−0.0653335
$285$	0	0
$286$	−21.7980	−1.28894
$287$	0	0
$288$	−3.00000	−0.176777
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−38.6969	−2.26845
$292$	−2.89898	−0.169650
$293$	−18.2474	−1.06603	−0.533014	−	0.846107i	$-0.678941\pi$
$293$	−18.2474	−1.06603	−0.533014	+	0.846107i	$0.678941\pi$
$294$	0	0
$295$	0	0
$296$	−2.00000	−0.116248
$297$	0	0
$298$	3.79796	0.220010
$299$	−12.8990	−0.745967
$300$	0	0
$301$	0	0
$302$	−19.5959	−1.12762
$303$	−8.69694	−0.499626
$304$	−1.55051	−0.0889279
$305$	0	0
$306$	6.00000	0.342997
$307$	20.2474	1.15558	0.577791	−	0.816184i	$-0.303915\pi$
$307$	20.2474	1.15558	0.577791	+	0.816184i	$0.303915\pi$
$308$	0	0
$309$	−31.5959	−1.79743
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	10.8990	0.617033
$313$	21.5959	1.22067	0.610337	−	0.792142i	$-0.291034\pi$
$313$	21.5959	1.22067	0.610337	+	0.792142i	$0.291034\pi$
$314$	3.55051	0.200367
$315$	0	0
$316$	6.89898	0.388098
$317$	−22.4949	−1.26344	−0.631720	−	0.775197i	$-0.717651\pi$
$317$	−22.4949	−1.26344	−0.631720	+	0.775197i	$0.717651\pi$
$318$	26.6969	1.49709
$319$	−33.7980	−1.89232
$320$	0	0
$321$	−19.5959	−1.09374
$322$	0	0
$323$	3.10102	0.172545
$324$	−9.00000	−0.500000
$325$	0	0
$326$	7.10102	0.393289
$327$	16.8990	0.934516
$328$	−1.10102	−0.0607937
$329$	0	0
$330$	0	0
$331$	−18.6969	−1.02768	−0.513838	−	0.857887i	$-0.671777\pi$
$331$	−18.6969	−1.02768	−0.513838	+	0.857887i	$0.671777\pi$
$332$	2.44949	0.134433
$333$	6.00000	0.328798
$334$	−4.89898	−0.268060
$335$	0	0
$336$	0	0
$337$	9.59592	0.522723	0.261361	−	0.965241i	$-0.415829\pi$
$337$	9.59592	0.522723	0.261361	+	0.965241i	$0.415829\pi$
$338$	−6.79796	−0.369760
$339$	48.4949	2.63388
$340$	0	0
$341$	43.5959	2.36085
$342$	4.65153	0.251526
$343$	0	0
$344$	0.898979	0.0484697
$345$	0	0
$346$	−6.24745	−0.335865
$347$	28.8990	1.55138	0.775689	−	0.631115i	$-0.217402\pi$
$347$	28.8990	1.55138	0.775689	+	0.631115i	$0.217402\pi$
$348$	16.8990	0.905880
$349$	8.44949	0.452291	0.226145	−	0.974094i	$-0.427388\pi$
$349$	8.44949	0.452291	0.226145	+	0.974094i	$0.427388\pi$
$350$	0	0
$351$	0	0
$352$	4.89898	0.261116
$353$	−22.8990	−1.21879	−0.609395	−	0.792867i	$-0.708588\pi$
$353$	−22.8990	−1.21879	−0.609395	+	0.792867i	$0.708588\pi$
$354$	−3.79796	−0.201859
$355$	0	0
$356$	10.0000	0.529999
$357$	0	0
$358$	−13.7980	−0.729245
$359$	−27.5959	−1.45646	−0.728228	−	0.685334i	$-0.759656\pi$
$359$	−27.5959	−1.45646	−0.728228	+	0.685334i	$0.759656\pi$
$360$	0	0
$361$	−16.5959	−0.873469
$362$	−10.2474	−0.538594
$363$	31.8434	1.67134
$364$	0	0
$365$	0	0
$366$	8.69694	0.454596
$367$	−32.0000	−1.67039	−0.835193	−	0.549957i	$-0.814644\pi$
$367$	−32.0000	−1.67039	−0.835193	+	0.549957i	$0.814644\pi$
$368$	2.89898	0.151120
$369$	3.30306	0.171951
$370$	0	0
$371$	0	0
$372$	−21.7980	−1.13017
$373$	−4.69694	−0.243198	−0.121599	−	0.992579i	$-0.538802\pi$
$373$	−4.69694	−0.243198	−0.121599	+	0.992579i	$0.538802\pi$
$374$	−9.79796	−0.506640
$375$	0	0
$376$	8.89898	0.458930
$377$	−30.6969	−1.58097
$378$	0	0
$379$	30.6969	1.57680	0.788398	−	0.615166i	$-0.210911\pi$
$379$	30.6969	1.57680	0.788398	+	0.615166i	$0.210911\pi$
$380$	0	0
$381$	−36.4949	−1.86969
$382$	−12.6969	−0.649632
$383$	7.10102	0.362845	0.181423	−	0.983405i	$-0.441930\pi$
$383$	7.10102	0.362845	0.181423	+	0.983405i	$0.441930\pi$
$384$	−2.44949	−0.125000
$385$	0	0
$386$	21.5959	1.09920
$387$	−2.69694	−0.137093
$388$	−15.7980	−0.802020
$389$	13.1010	0.664248	0.332124	−	0.943236i	$-0.392235\pi$
$389$	13.1010	0.664248	0.332124	+	0.943236i	$0.392235\pi$
$390$	0	0
$391$	−5.79796	−0.293215
$392$	0	0
$393$	15.7980	0.796902
$394$	−18.8990	−0.952117
$395$	0	0
$396$	−14.6969	−0.738549
$397$	2.65153	0.133077	0.0665383	−	0.997784i	$-0.478805\pi$
$397$	2.65153	0.133077	0.0665383	+	0.997784i	$0.478805\pi$
$398$	16.8990	0.847069
$399$	0	0
$400$	0	0
$401$	−29.3939	−1.46786	−0.733930	−	0.679225i	$-0.762316\pi$
$401$	−29.3939	−1.46786	−0.733930	+	0.679225i	$0.762316\pi$
$402$	19.5959	0.977356
$403$	39.5959	1.97241
$404$	−3.55051	−0.176644
$405$	0	0
$406$	0	0
$407$	−9.79796	−0.485667
$408$	4.89898	0.242536
$409$	−34.4949	−1.70566	−0.852831	−	0.522186i	$-0.825117\pi$
$409$	−34.4949	−1.70566	−0.852831	+	0.522186i	$0.825117\pi$
$410$	0	0
$411$	−4.40408	−0.217237
$412$	−12.8990	−0.635487
$413$	0	0
$414$	−8.69694	−0.427431
$415$	0	0
$416$	4.44949	0.218154
$417$	−3.79796	−0.185987
$418$	−7.59592	−0.371528
$419$	1.55051	0.0757474	0.0378737	−	0.999283i	$-0.487942\pi$
$419$	1.55051	0.0757474	0.0378737	+	0.999283i	$0.487942\pi$
$420$	0	0
$421$	−4.20204	−0.204795	−0.102397	−	0.994744i	$-0.532651\pi$
$421$	−4.20204	−0.204795	−0.102397	+	0.994744i	$0.532651\pi$
$422$	−12.0000	−0.584151
$423$	−26.6969	−1.29805
$424$	10.8990	0.529301
$425$	0	0
$426$	2.69694	0.130667
$427$	0	0
$428$	−8.00000	−0.386695
$429$	53.3939	2.57788
$430$	0	0
$431$	−1.79796	−0.0866046	−0.0433023	−	0.999062i	$-0.513788\pi$
$431$	−1.79796	−0.0866046	−0.0433023	+	0.999062i	$0.513788\pi$
$432$	0	0
$433$	0.202041	0.00970947	0.00485474	−	0.999988i	$-0.498455\pi$
$433$	0.202041	0.00970947	0.00485474	+	0.999988i	$0.498455\pi$
$434$	0	0
$435$	0	0
$436$	6.89898	0.330401
$437$	−4.49490	−0.215020
$438$	7.10102	0.339300
$439$	21.3939	1.02107	0.510537	−	0.859856i	$-0.329447\pi$
$439$	21.3939	1.02107	0.510537	+	0.859856i	$0.329447\pi$
$440$	0	0
$441$	0	0
$442$	−8.89898	−0.423281
$443$	9.79796	0.465515	0.232758	−	0.972535i	$-0.425225\pi$
$443$	9.79796	0.465515	0.232758	+	0.972535i	$0.425225\pi$
$444$	4.89898	0.232495
$445$	0	0
$446$	−4.00000	−0.189405
$447$	−9.30306	−0.440020
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−5.39388	−0.253988
$452$	19.7980	0.931218
$453$	48.0000	2.25524
$454$	7.34847	0.344881
$455$	0	0
$456$	3.79796	0.177856
$457$	29.5959	1.38444	0.692219	−	0.721687i	$-0.256633\pi$
$457$	29.5959	1.38444	0.692219	+	0.721687i	$0.256633\pi$
$458$	−19.1464	−0.894654
$459$	0	0
$460$	0	0
$461$	−17.3485	−0.807999	−0.403999	−	0.914759i	$-0.632380\pi$
$461$	−17.3485	−0.807999	−0.403999	+	0.914759i	$0.632380\pi$
$462$	0	0
$463$	3.59592	0.167116	0.0835582	−	0.996503i	$-0.473372\pi$
$463$	3.59592	0.167116	0.0835582	+	0.996503i	$0.473372\pi$
$464$	6.89898	0.320277
$465$	0	0
$466$	−29.7980	−1.38036
$467$	−10.4495	−0.483545	−0.241772	−	0.970333i	$-0.577729\pi$
$467$	−10.4495	−0.483545	−0.241772	+	0.970333i	$0.577729\pi$
$468$	−13.3485	−0.617033
$469$	0	0
$470$	0	0
$471$	−8.69694	−0.400734
$472$	−1.55051	−0.0713680
$473$	4.40408	0.202500
$474$	−16.8990	−0.776196
$475$	0	0
$476$	0	0
$477$	−32.6969	−1.49709
$478$	6.20204	0.283675
$479$	−9.30306	−0.425068	−0.212534	−	0.977154i	$-0.568172\pi$
$479$	−9.30306	−0.425068	−0.212534	+	0.977154i	$0.568172\pi$
$480$	0	0
$481$	−8.89898	−0.405759
$482$	−8.69694	−0.396135
$483$	0	0
$484$	13.0000	0.590909
$485$	0	0
$486$	22.0454	1.00000
$487$	−7.30306	−0.330933	−0.165467	−	0.986215i	$-0.552913\pi$
$487$	−7.30306	−0.330933	−0.165467	+	0.986215i	$0.552913\pi$
$488$	3.55051	0.160724
$489$	−17.3939	−0.786578
$490$	0	0
$491$	19.5959	0.884351	0.442176	−	0.896928i	$-0.354207\pi$
$491$	19.5959	0.884351	0.442176	+	0.896928i	$0.354207\pi$
$492$	2.69694	0.121587
$493$	−13.7980	−0.621429
$494$	−6.89898	−0.310400
$495$	0	0
$496$	−8.89898	−0.399576
$497$	0	0
$498$	−6.00000	−0.268866
$499$	6.20204	0.277641	0.138821	−	0.990318i	$-0.455669\pi$
$499$	6.20204	0.277641	0.138821	+	0.990318i	$0.455669\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	−6.44949	−0.287855
$503$	4.00000	0.178351	0.0891756	−	0.996016i	$-0.471577\pi$
$503$	4.00000	0.178351	0.0891756	+	0.996016i	$0.471577\pi$
$504$	0	0
$505$	0	0
$506$	14.2020	0.631358
$507$	16.6515	0.739520
$508$	−14.8990	−0.661035
$509$	31.5505	1.39845	0.699226	−	0.714901i	$-0.253528\pi$
$509$	31.5505	1.39845	0.699226	+	0.714901i	$0.253528\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−8.69694	−0.383606
$515$	0	0
$516$	−2.20204	−0.0969395
$517$	43.5959	1.91735
$518$	0	0
$519$	15.3031	0.671730
$520$	0	0
$521$	−32.6969	−1.43248	−0.716239	−	0.697855i	$-0.754138\pi$
$521$	−32.6969	−1.43248	−0.716239	+	0.697855i	$0.754138\pi$
$522$	−20.6969	−0.905880
$523$	33.1464	1.44939	0.724696	−	0.689069i	$-0.241980\pi$
$523$	33.1464	1.44939	0.724696	+	0.689069i	$0.241980\pi$
$524$	6.44949	0.281747
$525$	0	0
$526$	−9.79796	−0.427211
$527$	17.7980	0.775291
$528$	−12.0000	−0.522233
$529$	−14.5959	−0.634605
$530$	0	0
$531$	4.65153	0.201859
$532$	0	0
$533$	−4.89898	−0.212198
$534$	−24.4949	−1.06000
$535$	0	0
$536$	8.00000	0.345547
$537$	33.7980	1.45849
$538$	19.1464	0.825461
$539$	0	0
$540$	0	0
$541$	9.59592	0.412561	0.206280	−	0.978493i	$-0.433864\pi$
$541$	9.59592	0.412561	0.206280	+	0.978493i	$0.433864\pi$
$542$	12.0000	0.515444
$543$	25.1010	1.07719
$544$	2.00000	0.0857493
$545$	0	0
$546$	0	0
$547$	−18.6969	−0.799423	−0.399712	−	0.916641i	$-0.630890\pi$
$547$	−18.6969	−0.799423	−0.399712	+	0.916641i	$0.630890\pi$
$548$	−1.79796	−0.0768050
$549$	−10.6515	−0.454596
$550$	0	0
$551$	−10.6969	−0.455705
$552$	−7.10102	−0.302240
$553$	0	0
$554$	14.8990	0.632997
$555$	0	0
$556$	−1.55051	−0.0657563
$557$	12.6969	0.537987	0.268993	−	0.963142i	$-0.413309\pi$
$557$	12.6969	0.537987	0.268993	+	0.963142i	$0.413309\pi$
$558$	26.6969	1.13017
$559$	4.00000	0.169182
$560$	0	0
$561$	24.0000	1.01328
$562$	18.0000	0.759284
$563$	30.0454	1.26626	0.633131	−	0.774044i	$-0.281769\pi$
$563$	30.0454	1.26626	0.633131	+	0.774044i	$0.281769\pi$
$564$	−21.7980	−0.917860
$565$	0	0
$566$	3.75255	0.157731
$567$	0	0
$568$	1.10102	0.0461978
$569$	33.7980	1.41688	0.708442	−	0.705769i	$-0.249398\pi$
$569$	33.7980	1.41688	0.708442	+	0.705769i	$0.249398\pi$
$570$	0	0
$571$	−11.1010	−0.464563	−0.232282	−	0.972649i	$-0.574619\pi$
$571$	−11.1010	−0.464563	−0.232282	+	0.972649i	$0.574619\pi$
$572$	21.7980	0.911418
$573$	31.1010	1.29926
$574$	0	0
$575$	0	0
$576$	3.00000	0.125000
$577$	2.49490	0.103864	0.0519320	−	0.998651i	$-0.483462\pi$
$577$	2.49490	0.103864	0.0519320	+	0.998651i	$0.483462\pi$
$578$	13.0000	0.540729
$579$	−52.8990	−2.19841
$580$	0	0
$581$	0	0
$582$	38.6969	1.60404
$583$	53.3939	2.21135
$584$	2.89898	0.119961
$585$	0	0
$586$	18.2474	0.753795
$587$	−1.14643	−0.0473182	−0.0236591	−	0.999720i	$-0.507532\pi$
$587$	−1.14643	−0.0473182	−0.0236591	+	0.999720i	$0.507532\pi$
$588$	0	0
$589$	13.7980	0.568535
$590$	0	0
$591$	46.2929	1.90423
$592$	2.00000	0.0821995
$593$	10.8990	0.447567	0.223784	−	0.974639i	$-0.428159\pi$
$593$	10.8990	0.447567	0.223784	+	0.974639i	$0.428159\pi$
$594$	0	0
$595$	0	0
$596$	−3.79796	−0.155570
$597$	−41.3939	−1.69414
$598$	12.8990	0.527478
$599$	−13.1010	−0.535293	−0.267647	−	0.963517i	$-0.586246\pi$
$599$	−13.1010	−0.535293	−0.267647	+	0.963517i	$0.586246\pi$
$600$	0	0
$601$	39.3939	1.60691	0.803455	−	0.595366i	$-0.202993\pi$
$601$	39.3939	1.60691	0.803455	+	0.595366i	$0.202993\pi$
$602$	0	0
$603$	−24.0000	−0.977356
$604$	19.5959	0.797347
$605$	0	0
$606$	8.69694	0.353289
$607$	−33.3939	−1.35542	−0.677708	−	0.735331i	$-0.737027\pi$
$607$	−33.3939	−1.35542	−0.677708	+	0.735331i	$0.737027\pi$
$608$	1.55051	0.0628815
$609$	0	0
$610$	0	0
$611$	39.5959	1.60188
$612$	−6.00000	−0.242536
$613$	−27.7980	−1.12275	−0.561374	−	0.827562i	$-0.689727\pi$
$613$	−27.7980	−1.12275	−0.561374	+	0.827562i	$0.689727\pi$
$614$	−20.2474	−0.817121
$615$	0	0
$616$	0	0
$617$	29.5959	1.19149	0.595743	−	0.803175i	$-0.296858\pi$
$617$	29.5959	1.19149	0.595743	+	0.803175i	$0.296858\pi$
$618$	31.5959	1.27097
$619$	41.5505	1.67006	0.835028	−	0.550207i	$-0.185451\pi$
$619$	41.5505	1.67006	0.835028	+	0.550207i	$0.185451\pi$
$620$	0	0
$621$	0	0
$622$	12.0000	0.481156
$623$	0	0
$624$	−10.8990	−0.436308
$625$	0	0
$626$	−21.5959	−0.863146
$627$	18.6061	0.743057
$628$	−3.55051	−0.141681
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−42.4949	−1.69170	−0.845848	−	0.533425i	$-0.820905\pi$
$631$	−42.4949	−1.69170	−0.845848	+	0.533425i	$0.820905\pi$
$632$	−6.89898	−0.274427
$633$	29.3939	1.16830
$634$	22.4949	0.893387
$635$	0	0
$636$	−26.6969	−1.05860
$637$	0	0
$638$	33.7980	1.33807
$639$	−3.30306	−0.130667
$640$	0	0
$641$	25.7980	1.01896	0.509479	−	0.860483i	$-0.329838\pi$
$641$	25.7980	1.01896	0.509479	+	0.860483i	$0.329838\pi$
$642$	19.5959	0.773389
$643$	−25.1464	−0.991678	−0.495839	−	0.868414i	$-0.665139\pi$
$643$	−25.1464	−0.991678	−0.495839	+	0.868414i	$0.665139\pi$
$644$	0	0
$645$	0	0
$646$	−3.10102	−0.122008
$647$	46.2929	1.81996	0.909980	−	0.414652i	$-0.136097\pi$
$647$	46.2929	1.81996	0.909980	+	0.414652i	$0.136097\pi$
$648$	9.00000	0.353553
$649$	−7.59592	−0.298166
$650$	0	0
$651$	0	0
$652$	−7.10102	−0.278097
$653$	−20.2020	−0.790567	−0.395283	−	0.918559i	$-0.629354\pi$
$653$	−20.2020	−0.790567	−0.395283	+	0.918559i	$0.629354\pi$
$654$	−16.8990	−0.660802
$655$	0	0
$656$	1.10102	0.0429876
$657$	−8.69694	−0.339300
$658$	0	0
$659$	16.8990	0.658291	0.329145	−	0.944279i	$-0.393239\pi$
$659$	16.8990	0.658291	0.329145	+	0.944279i	$0.393239\pi$
$660$	0	0
$661$	40.9444	1.59255	0.796276	−	0.604933i	$-0.206800\pi$
$661$	40.9444	1.59255	0.796276	+	0.604933i	$0.206800\pi$
$662$	18.6969	0.726677
$663$	21.7980	0.846563
$664$	−2.44949	−0.0950586
$665$	0	0
$666$	−6.00000	−0.232495
$667$	20.0000	0.774403
$668$	4.89898	0.189547
$669$	9.79796	0.378811
$670$	0	0
$671$	17.3939	0.671483
$672$	0	0
$673$	−17.7980	−0.686061	−0.343030	−	0.939324i	$-0.611453\pi$
$673$	−17.7980	−0.686061	−0.343030	+	0.939324i	$0.611453\pi$
$674$	−9.59592	−0.369621
$675$	0	0
$676$	6.79796	0.261460
$677$	36.4495	1.40087	0.700434	−	0.713717i	$-0.252990\pi$
$677$	36.4495	1.40087	0.700434	+	0.713717i	$0.252990\pi$
$678$	−48.4949	−1.86244
$679$	0	0
$680$	0	0
$681$	−18.0000	−0.689761
$682$	−43.5959	−1.66937
$683$	3.59592	0.137594	0.0687970	−	0.997631i	$-0.478084\pi$
$683$	3.59592	0.137594	0.0687970	+	0.997631i	$0.478084\pi$
$684$	−4.65153	−0.177856
$685$	0	0
$686$	0	0
$687$	46.8990	1.78931
$688$	−0.898979	−0.0342733
$689$	48.4949	1.84751
$690$	0	0
$691$	−21.1464	−0.804448	−0.402224	−	0.915541i	$-0.631763\pi$
$691$	−21.1464	−0.804448	−0.402224	+	0.915541i	$0.631763\pi$
$692$	6.24745	0.237492
$693$	0	0
$694$	−28.8990	−1.09699
$695$	0	0
$696$	−16.8990	−0.640554
$697$	−2.20204	−0.0834083
$698$	−8.44949	−0.319818
$699$	72.9898	2.76073
$700$	0	0
$701$	11.3031	0.426911	0.213455	−	0.976953i	$-0.431528\pi$
$701$	11.3031	0.426911	0.213455	+	0.976953i	$0.431528\pi$
$702$	0	0
$703$	−3.10102	−0.116957
$704$	−4.89898	−0.184637
$705$	0	0
$706$	22.8990	0.861814
$707$	0	0
$708$	3.79796	0.142736
$709$	−28.2929	−1.06256	−0.531280	−	0.847196i	$-0.678289\pi$
$709$	−28.2929	−1.06256	−0.531280	+	0.847196i	$0.678289\pi$
$710$	0	0
$711$	20.6969	0.776196
$712$	−10.0000	−0.374766
$713$	−25.7980	−0.966141
$714$	0	0
$715$	0	0
$716$	13.7980	0.515654
$717$	−15.1918	−0.567350
$718$	27.5959	1.02987
$719$	4.49490	0.167631	0.0838157	−	0.996481i	$-0.473289\pi$
$719$	4.49490	0.167631	0.0838157	+	0.996481i	$0.473289\pi$
$720$	0	0
$721$	0	0
$722$	16.5959	0.617636
$723$	21.3031	0.792269
$724$	10.2474	0.380843
$725$	0	0
$726$	−31.8434	−1.18182
$727$	−22.6969	−0.841783	−0.420891	−	0.907111i	$-0.638283\pi$
$727$	−22.6969	−0.841783	−0.420891	+	0.907111i	$0.638283\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	1.79796	0.0664999
$732$	−8.69694	−0.321448
$733$	−39.6413	−1.46419	−0.732093	−	0.681205i	$-0.761456\pi$
$733$	−39.6413	−1.46419	−0.732093	+	0.681205i	$0.761456\pi$
$734$	32.0000	1.18114
$735$	0	0
$736$	−2.89898	−0.106858
$737$	39.1918	1.44365
$738$	−3.30306	−0.121587
$739$	4.49490	0.165347	0.0826737	−	0.996577i	$-0.473654\pi$
$739$	4.49490	0.165347	0.0826737	+	0.996577i	$0.473654\pi$
$740$	0	0
$741$	16.8990	0.620800
$742$	0	0
$743$	−44.6969	−1.63977	−0.819886	−	0.572527i	$-0.805963\pi$
$743$	−44.6969	−1.63977	−0.819886	+	0.572527i	$0.805963\pi$
$744$	21.7980	0.799152
$745$	0	0
$746$	4.69694	0.171967
$747$	7.34847	0.268866
$748$	9.79796	0.358249
$749$	0	0
$750$	0	0
$751$	−41.7980	−1.52523	−0.762615	−	0.646853i	$-0.776085\pi$
$751$	−41.7980	−1.52523	−0.762615	+	0.646853i	$0.776085\pi$
$752$	−8.89898	−0.324512
$753$	15.7980	0.575710
$754$	30.6969	1.11792
$755$	0	0
$756$	0	0
$757$	−51.7980	−1.88263	−0.941314	−	0.337531i	$-0.890408\pi$
$757$	−51.7980	−1.88263	−0.941314	+	0.337531i	$0.890408\pi$
$758$	−30.6969	−1.11496
$759$	−34.7878	−1.26272
$760$	0	0
$761$	21.1010	0.764911	0.382456	−	0.923974i	$-0.375078\pi$
$761$	21.1010	0.764911	0.382456	+	0.923974i	$0.375078\pi$
$762$	36.4949	1.32207
$763$	0	0
$764$	12.6969	0.459359
$765$	0	0
$766$	−7.10102	−0.256570
$767$	−6.89898	−0.249108
$768$	2.44949	0.0883883
$769$	40.6969	1.46757	0.733785	−	0.679382i	$-0.237752\pi$
$769$	40.6969	1.46757	0.733785	+	0.679382i	$0.237752\pi$
$770$	0	0
$771$	21.3031	0.767211
$772$	−21.5959	−0.777254
$773$	−1.34847	−0.0485011	−0.0242505	−	0.999706i	$-0.507720\pi$
$773$	−1.34847	−0.0485011	−0.0242505	+	0.999706i	$0.507720\pi$
$774$	2.69694	0.0969395
$775$	0	0
$776$	15.7980	0.567114
$777$	0	0
$778$	−13.1010	−0.469694
$779$	−1.70714	−0.0611648
$780$	0	0
$781$	5.39388	0.193008
$782$	5.79796	0.207335
$783$	0	0
$784$	0	0
$785$	0	0
$786$	−15.7980	−0.563495
$787$	−50.4495	−1.79833	−0.899165	−	0.437610i	$-0.855825\pi$
$787$	−50.4495	−1.79833	−0.899165	+	0.437610i	$0.855825\pi$
$788$	18.8990	0.673248
$789$	24.0000	0.854423
$790$	0	0
$791$	0	0
$792$	14.6969	0.522233
$793$	15.7980	0.561002
$794$	−2.65153	−0.0940993
$795$	0	0
$796$	−16.8990	−0.598968
$797$	0.944387	0.0334519	0.0167260	−	0.999860i	$-0.494676\pi$
$797$	0.944387	0.0334519	0.0167260	+	0.999860i	$0.494676\pi$
$798$	0	0
$799$	17.7980	0.629647
$800$	0	0
$801$	30.0000	1.06000
$802$	29.3939	1.03793
$803$	14.2020	0.501179
$804$	−19.5959	−0.691095
$805$	0	0
$806$	−39.5959	−1.39471
$807$	−46.8990	−1.65092
$808$	3.55051	0.124907
$809$	47.5959	1.67338	0.836692	−	0.547674i	$-0.184487\pi$
$809$	47.5959	1.67338	0.836692	+	0.547674i	$0.184487\pi$
$810$	0	0
$811$	−14.9444	−0.524768	−0.262384	−	0.964963i	$-0.584509\pi$
$811$	−14.9444	−0.524768	−0.262384	+	0.964963i	$0.584509\pi$
$812$	0	0
$813$	−29.3939	−1.03089
$814$	9.79796	0.343418
$815$	0	0
$816$	−4.89898	−0.171499
$817$	1.39388	0.0487656
$818$	34.4949	1.20609
$819$	0	0
$820$	0	0
$821$	8.20204	0.286253	0.143127	−	0.989704i	$-0.454284\pi$
$821$	8.20204	0.286253	0.143127	+	0.989704i	$0.454284\pi$
$822$	4.40408	0.153610
$823$	−39.1918	−1.36614	−0.683071	−	0.730352i	$-0.739356\pi$
$823$	−39.1918	−1.36614	−0.683071	+	0.730352i	$0.739356\pi$
$824$	12.8990	0.449357
$825$	0	0
$826$	0	0
$827$	−15.5959	−0.542323	−0.271162	−	0.962534i	$-0.587408\pi$
$827$	−15.5959	−0.542323	−0.271162	+	0.962534i	$0.587408\pi$
$828$	8.69694	0.302240
$829$	43.6413	1.51573	0.757863	−	0.652414i	$-0.226244\pi$
$829$	43.6413	1.51573	0.757863	+	0.652414i	$0.226244\pi$
$830$	0	0
$831$	−36.4949	−1.26599
$832$	−4.44949	−0.154258
$833$	0	0
$834$	3.79796	0.131513
$835$	0	0
$836$	7.59592	0.262710
$837$	0	0
$838$	−1.55051	−0.0535615
$839$	−36.8990	−1.27389	−0.636947	−	0.770907i	$-0.719803\pi$
$839$	−36.8990	−1.27389	−0.636947	+	0.770907i	$0.719803\pi$
$840$	0	0
$841$	18.5959	0.641239
$842$	4.20204	0.144812
$843$	−44.0908	−1.51857
$844$	12.0000	0.413057
$845$	0	0
$846$	26.6969	0.917860
$847$	0	0
$848$	−10.8990	−0.374272
$849$	−9.19184	−0.315463
$850$	0	0
$851$	5.79796	0.198751
$852$	−2.69694	−0.0923956
$853$	33.8434	1.15877	0.579387	−	0.815052i	$-0.303292\pi$
$853$	33.8434	1.15877	0.579387	+	0.815052i	$0.303292\pi$
$854$	0	0
$855$	0	0
$856$	8.00000	0.273434
$857$	53.1918	1.81700	0.908499	−	0.417886i	$-0.137229\pi$
$857$	53.1918	1.81700	0.908499	+	0.417886i	$0.137229\pi$
$858$	−53.3939	−1.82284
$859$	−53.6413	−1.83022	−0.915109	−	0.403206i	$-0.867896\pi$
$859$	−53.6413	−1.83022	−0.915109	+	0.403206i	$0.867896\pi$
$860$	0	0
$861$	0	0
$862$	1.79796	0.0612387
$863$	−45.3939	−1.54523	−0.772613	−	0.634878i	$-0.781051\pi$
$863$	−45.3939	−1.54523	−0.772613	+	0.634878i	$0.781051\pi$
$864$	0	0
$865$	0	0
$866$	−0.202041	−0.00686563
$867$	−31.8434	−1.08146
$868$	0	0
$869$	−33.7980	−1.14652
$870$	0	0
$871$	35.5959	1.20612
$872$	−6.89898	−0.233629
$873$	−47.3939	−1.60404
$874$	4.49490	0.152042
$875$	0	0
$876$	−7.10102	−0.239921
$877$	−39.3939	−1.33024	−0.665118	−	0.746738i	$-0.731619\pi$
$877$	−39.3939	−1.33024	−0.665118	+	0.746738i	$0.731619\pi$
$878$	−21.3939	−0.722008
$879$	−44.6969	−1.50759
$880$	0	0
$881$	−8.20204	−0.276334	−0.138167	−	0.990409i	$-0.544121\pi$
$881$	−8.20204	−0.276334	−0.138167	+	0.990409i	$0.544121\pi$
$882$	0	0
$883$	22.2020	0.747158	0.373579	−	0.927598i	$-0.378130\pi$
$883$	22.2020	0.747158	0.373579	+	0.927598i	$0.378130\pi$
$884$	8.89898	0.299305
$885$	0	0
$886$	−9.79796	−0.329169
$887$	−2.69694	−0.0905543	−0.0452772	−	0.998974i	$-0.514417\pi$
$887$	−2.69694	−0.0905543	−0.0452772	+	0.998974i	$0.514417\pi$
$888$	−4.89898	−0.164399
$889$	0	0
$890$	0	0
$891$	44.0908	1.47710
$892$	4.00000	0.133930
$893$	13.7980	0.461731
$894$	9.30306	0.311141
$895$	0	0
$896$	0	0
$897$	−31.5959	−1.05496
$898$	10.0000	0.333704
$899$	−61.3939	−2.04760
$900$	0	0
$901$	21.7980	0.726195
$902$	5.39388	0.179596
$903$	0	0
$904$	−19.7980	−0.658470
$905$	0	0
$906$	−48.0000	−1.59469
$907$	−41.7980	−1.38788	−0.693939	−	0.720034i	$-0.744126\pi$
$907$	−41.7980	−1.38788	−0.693939	+	0.720034i	$0.744126\pi$
$908$	−7.34847	−0.243868
$909$	−10.6515	−0.353289
$910$	0	0
$911$	−35.5959	−1.17935	−0.589673	−	0.807642i	$-0.700743\pi$
$911$	−35.5959	−1.17935	−0.589673	+	0.807642i	$0.700743\pi$
$912$	−3.79796	−0.125763
$913$	−12.0000	−0.397142
$914$	−29.5959	−0.978946
$915$	0	0
$916$	19.1464	0.632616
$917$	0	0
$918$	0	0
$919$	−26.8990	−0.887315	−0.443658	−	0.896196i	$-0.646319\pi$
$919$	−26.8990	−0.887315	−0.443658	+	0.896196i	$0.646319\pi$
$920$	0	0
$921$	49.5959	1.63424
$922$	17.3485	0.571341
$923$	4.89898	0.161252
$924$	0	0
$925$	0	0
$926$	−3.59592	−0.118169
$927$	−38.6969	−1.27097
$928$	−6.89898	−0.226470
$929$	28.2929	0.928259	0.464129	−	0.885767i	$-0.346367\pi$
$929$	28.2929	0.928259	0.464129	+	0.885767i	$0.346367\pi$
$930$	0	0
$931$	0	0
$932$	29.7980	0.976065
$933$	−29.3939	−0.962312
$934$	10.4495	0.341918
$935$	0	0
$936$	13.3485	0.436308
$937$	41.1010	1.34271	0.671356	−	0.741135i	$-0.265712\pi$
$937$	41.1010	1.34271	0.671356	+	0.741135i	$0.265712\pi$
$938$	0	0
$939$	52.8990	1.72629
$940$	0	0
$941$	19.5505	0.637328	0.318664	−	0.947868i	$-0.396766\pi$
$941$	19.5505	0.637328	0.318664	+	0.947868i	$0.396766\pi$
$942$	8.69694	0.283362
$943$	3.19184	0.103940
$944$	1.55051	0.0504648
$945$	0	0
$946$	−4.40408	−0.143189
$947$	44.0908	1.43276	0.716379	−	0.697711i	$-0.245798\pi$
$947$	44.0908	1.43276	0.716379	+	0.697711i	$0.245798\pi$
$948$	16.8990	0.548853
$949$	12.8990	0.418719
$950$	0	0
$951$	−55.1010	−1.78677
$952$	0	0
$953$	2.20204	0.0713311	0.0356656	−	0.999364i	$-0.488645\pi$
$953$	2.20204	0.0713311	0.0356656	+	0.999364i	$0.488645\pi$
$954$	32.6969	1.05860
$955$	0	0
$956$	−6.20204	−0.200588
$957$	−82.7878	−2.67615
$958$	9.30306	0.300568
$959$	0	0
$960$	0	0
$961$	48.1918	1.55458
$962$	8.89898	0.286915
$963$	−24.0000	−0.773389
$964$	8.69694	0.280110
$965$	0	0
$966$	0	0
$967$	−36.2929	−1.16710	−0.583550	−	0.812077i	$-0.698337\pi$
$967$	−36.2929	−1.16710	−0.583550	+	0.812077i	$0.698337\pi$
$968$	−13.0000	−0.417836
$969$	7.59592	0.244016
$970$	0	0
$971$	9.55051	0.306490	0.153245	−	0.988188i	$-0.451028\pi$
$971$	9.55051	0.306490	0.153245	+	0.988188i	$0.451028\pi$
$972$	−22.0454	−0.707107
$973$	0	0
$974$	7.30306	0.234005
$975$	0	0
$976$	−3.55051	−0.113649
$977$	−29.3939	−0.940393	−0.470197	−	0.882562i	$-0.655817\pi$
$977$	−29.3939	−0.940393	−0.470197	+	0.882562i	$0.655817\pi$
$978$	17.3939	0.556195
$979$	−48.9898	−1.56572
$980$	0	0
$981$	20.6969	0.660802
$982$	−19.5959	−0.625331
$983$	13.3031	0.424302	0.212151	−	0.977237i	$-0.431953\pi$
$983$	13.3031	0.424302	0.212151	+	0.977237i	$0.431953\pi$
$984$	−2.69694	−0.0859753
$985$	0	0
$986$	13.7980	0.439417
$987$	0	0
$988$	6.89898	0.219486
$989$	−2.60612	−0.0828699
$990$	0	0
$991$	31.3031	0.994375	0.497187	−	0.867643i	$-0.334366\pi$
$991$	31.3031	0.994375	0.497187	+	0.867643i	$0.334366\pi$
$992$	8.89898	0.282543
$993$	−45.7980	−1.45335
$994$	0	0
$995$	0	0
$996$	6.00000	0.190117
$997$	−57.3485	−1.81624	−0.908122	−	0.418705i	$-0.862484\pi$
$997$	−57.3485	−1.81624	−0.908122	+	0.418705i	$0.862484\pi$
$998$	−6.20204	−0.196322
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.a.bl.1.2		2
5.2	odd	4		490.2.c.e.99.1		4
5.3	odd	4		490.2.c.e.99.4		4
5.4	even	2		2450.2.a.bq.1.1		2
7.6	odd	2		350.2.a.g.1.1		2
21.20	even	2		3150.2.a.bt.1.2		2
28.27	even	2		2800.2.a.bm.1.2		2
35.2	odd	12		490.2.i.f.459.2		8
35.3	even	12		490.2.i.c.79.1		8
35.12	even	12		490.2.i.c.459.1		8
35.13	even	4		70.2.c.a.29.3	yes	4
35.17	even	12		490.2.i.c.79.4		8
35.18	odd	12		490.2.i.f.79.2		8
35.23	odd	12		490.2.i.f.459.3		8
35.27	even	4		70.2.c.a.29.2	✓	4
35.32	odd	12		490.2.i.f.79.3		8
35.33	even	12		490.2.i.c.459.4		8
35.34	odd	2		350.2.a.h.1.2		2
105.62	odd	4		630.2.g.g.379.3		4
105.83	odd	4		630.2.g.g.379.1		4
105.104	even	2		3150.2.a.bs.1.2		2
140.27	odd	4		560.2.g.e.449.2		4
140.83	odd	4		560.2.g.e.449.4		4
140.139	even	2		2800.2.a.bl.1.1		2
280.13	even	4		2240.2.g.j.449.3		4
280.27	odd	4		2240.2.g.i.449.3		4
280.83	odd	4		2240.2.g.i.449.1		4
280.237	even	4		2240.2.g.j.449.1		4
420.83	even	4		5040.2.t.t.1009.1		4
420.167	even	4		5040.2.t.t.1009.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.2.c.a.29.2	✓	4	35.27	even	4
70.2.c.a.29.3	yes	4	35.13	even	4
350.2.a.g.1.1		2	7.6	odd	2
350.2.a.h.1.2		2	35.34	odd	2
490.2.c.e.99.1		4	5.2	odd	4
490.2.c.e.99.4		4	5.3	odd	4
490.2.i.c.79.1		8	35.3	even	12
490.2.i.c.79.4		8	35.17	even	12
490.2.i.c.459.1		8	35.12	even	12
490.2.i.c.459.4		8	35.33	even	12
490.2.i.f.79.2		8	35.18	odd	12
490.2.i.f.79.3		8	35.32	odd	12
490.2.i.f.459.2		8	35.2	odd	12
490.2.i.f.459.3		8	35.23	odd	12
560.2.g.e.449.2		4	140.27	odd	4
560.2.g.e.449.4		4	140.83	odd	4
630.2.g.g.379.1		4	105.83	odd	4
630.2.g.g.379.3		4	105.62	odd	4
2240.2.g.i.449.1		4	280.83	odd	4
2240.2.g.i.449.3		4	280.27	odd	4
2240.2.g.j.449.1		4	280.237	even	4
2240.2.g.j.449.3		4	280.13	even	4
2450.2.a.bl.1.2		2	1.1	even	1	trivial
2450.2.a.bq.1.1		2	5.4	even	2
2800.2.a.bl.1.1		2	140.139	even	2
2800.2.a.bm.1.2		2	28.27	even	2
3150.2.a.bs.1.2		2	105.104	even	2
3150.2.a.bt.1.2		2	21.20	even	2
5040.2.t.t.1009.1		4	420.83	even	4
5040.2.t.t.1009.2		4	420.167	even	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:	\( N \)	\(=\)	\( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} -1.00000 q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.44949 q^{3} +1.00000 q^{4} -2.44949 q^{6} -1.00000 q^{8} +3.00000 q^{9} -4.89898 q^{11} +2.44949 q^{12} -4.44949 q^{13} +1.00000 q^{16} -2.00000 q^{17} -3.00000 q^{18} -1.55051 q^{19} +4.89898 q^{22} +2.89898 q^{23} -2.44949 q^{24} +4.44949 q^{26} +6.89898 q^{29} -8.89898 q^{31} -1.00000 q^{32} -12.0000 q^{33} +2.00000 q^{34} +3.00000 q^{36} +2.00000 q^{37} +1.55051 q^{38} -10.8990 q^{39} +1.10102 q^{41} -0.898979 q^{43} -4.89898 q^{44} -2.89898 q^{46} -8.89898 q^{47} +2.44949 q^{48} -4.89898 q^{51} -4.44949 q^{52} -10.8990 q^{53} -3.79796 q^{57} -6.89898 q^{58} +1.55051 q^{59} -3.55051 q^{61} +8.89898 q^{62} +1.00000 q^{64} +12.0000 q^{66} -8.00000 q^{67} -2.00000 q^{68} +7.10102 q^{69} -1.10102 q^{71} -3.00000 q^{72} -2.89898 q^{73} -2.00000 q^{74} -1.55051 q^{76} +10.8990 q^{78} +6.89898 q^{79} -9.00000 q^{81} -1.10102 q^{82} +2.44949 q^{83} +0.898979 q^{86} +16.8990 q^{87} +4.89898 q^{88} +10.0000 q^{89} +2.89898 q^{92} -21.7980 q^{93} +8.89898 q^{94} -2.44949 q^{96} -15.7980 q^{97} -14.6969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 6 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} + 6 q^{9} - 4 q^{13} + 2 q^{16} - 4 q^{17} - 6 q^{18} - 8 q^{19} - 4 q^{23} + 4 q^{26} + 4 q^{29} - 8 q^{31} - 2 q^{32} - 24 q^{33} + 4 q^{34} + 6 q^{36} + 4 q^{37} + 8 q^{38} - 12 q^{39} + 12 q^{41} + 8 q^{43} + 4 q^{46} - 8 q^{47} - 4 q^{52} - 12 q^{53} + 12 q^{57} - 4 q^{58} + 8 q^{59} - 12 q^{61} + 8 q^{62} + 2 q^{64} + 24 q^{66} - 16 q^{67} - 4 q^{68} + 24 q^{69} - 12 q^{71} - 6 q^{72} + 4 q^{73} - 4 q^{74} - 8 q^{76} + 12 q^{78} + 4 q^{79} - 18 q^{81} - 12 q^{82} - 8 q^{86} + 24 q^{87} + 20 q^{89} - 4 q^{92} - 24 q^{93} + 8 q^{94} - 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 + 6 * q^9 - 4 * q^13 + 2 * q^16 - 4 * q^17 - 6 * q^18 - 8 * q^19 - 4 * q^23 + 4 * q^26 + 4 * q^29 - 8 * q^31 - 2 * q^32 - 24 * q^33 + 4 * q^34 + 6 * q^36 + 4 * q^37 + 8 * q^38 - 12 * q^39 + 12 * q^41 + 8 * q^43 + 4 * q^46 - 8 * q^47 - 4 * q^52 - 12 * q^53 + 12 * q^57 - 4 * q^58 + 8 * q^59 - 12 * q^61 + 8 * q^62 + 2 * q^64 + 24 * q^66 - 16 * q^67 - 4 * q^68 + 24 * q^69 - 12 * q^71 - 6 * q^72 + 4 * q^73 - 4 * q^74 - 8 * q^76 + 12 * q^78 + 4 * q^79 - 18 * q^81 - 12 * q^82 - 8 * q^86 + 24 * q^87 + 20 * q^89 - 4 * q^92 - 24 * q^93 + 8 * q^94 - 12 * q^97

Introduction

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