LMFDB - Embedded newform 2450.2.a.bk.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:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 490)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +1.41421 q^{12} +4.24264 q^{13} +1.00000 q^{16} -5.65685 q^{17} +1.00000 q^{18} -4.24264 q^{19} -6.00000 q^{23} -1.41421 q^{24} -4.24264 q^{26} -5.65685 q^{27} +6.00000 q^{29} -8.48528 q^{31} -1.00000 q^{32} +5.65685 q^{34} -1.00000 q^{36} -6.00000 q^{37} +4.24264 q^{38} +6.00000 q^{39} +8.48528 q^{41} -12.0000 q^{43} +6.00000 q^{46} +2.82843 q^{47} +1.41421 q^{48} -8.00000 q^{51} +4.24264 q^{52} +6.00000 q^{53} +5.65685 q^{54} -6.00000 q^{57} -6.00000 q^{58} +4.24264 q^{59} -4.24264 q^{61} +8.48528 q^{62} +1.00000 q^{64} -5.65685 q^{68} -8.48528 q^{69} +6.00000 q^{71} +1.00000 q^{72} -8.48528 q^{73} +6.00000 q^{74} -4.24264 q^{76} -6.00000 q^{78} -10.0000 q^{79} -5.00000 q^{81} -8.48528 q^{82} -15.5563 q^{83} +12.0000 q^{86} +8.48528 q^{87} -6.00000 q^{92} -12.0000 q^{93} -2.82843 q^{94} -1.41421 q^{96} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} + 2 q^{16} + 2 q^{18} - 12 q^{23} + 12 q^{29} - 2 q^{32} - 2 q^{36} - 12 q^{37} + 12 q^{39} - 24 q^{43} + 12 q^{46} - 16 q^{51} + 12 q^{53} - 12 q^{57} - 12 q^{58} + 2 q^{64} + 12 q^{71} + 2 q^{72} + 12 q^{74} - 12 q^{78} - 20 q^{79} - 10 q^{81} + 24 q^{86} - 12 q^{92} - 24 q^{93}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 + 2 * q^16 + 2 * q^18 - 12 * q^23 + 12 * q^29 - 2 * q^32 - 2 * q^36 - 12 * q^37 + 12 * q^39 - 24 * q^43 + 12 * q^46 - 16 * q^51 + 12 * q^53 - 12 * q^57 - 12 * q^58 + 2 * q^64 + 12 * q^71 + 2 * q^72 + 12 * q^74 - 12 * q^78 - 20 * q^79 - 10 * q^81 + 24 * q^86 - 12 * q^92 - 24 * q^93

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
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.41421	−0.577350
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.41421	0.408248
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.65685	−1.37199	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	−5.65685	−1.37199	−0.685994	+	0.727607i	$0.740633\pi$
$18$	1.00000	0.235702
$19$	−4.24264	−0.973329	−0.486664	−	0.873589i	$-0.661786\pi$
$19$	−4.24264	−0.973329	−0.486664	+	0.873589i	$0.661786\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−1.41421	−0.288675
$25$	0	0
$26$	−4.24264	−0.832050
$27$	−5.65685	−1.08866
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.65685	0.970143
$35$	0	0
$36$	−1.00000	−0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	4.24264	0.688247
$39$	6.00000	0.960769
$40$	0	0
$41$	8.48528	1.32518	0.662589	−	0.748983i	$-0.269458\pi$
$41$	8.48528	1.32518	0.662589	+	0.748983i	$0.269458\pi$
$42$	0	0
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	0	0
$45$	0	0
$46$	6.00000	0.884652
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	1.41421	0.204124
$49$	0	0
$50$	0	0
$51$	−8.00000	−1.12022
$52$	4.24264	0.588348
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	5.65685	0.769800
$55$	0	0
$56$	0	0
$57$	−6.00000	−0.794719
$58$	−6.00000	−0.787839
$59$	4.24264	0.552345	0.276172	−	0.961108i	$-0.410934\pi$
$59$	4.24264	0.552345	0.276172	+	0.961108i	$0.410934\pi$
$60$	0	0
$61$	−4.24264	−0.543214	−0.271607	−	0.962408i	$-0.587555\pi$
$61$	−4.24264	−0.543214	−0.271607	+	0.962408i	$0.587555\pi$
$62$	8.48528	1.07763
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−5.65685	−0.685994
$69$	−8.48528	−1.02151
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	1.00000	0.117851
$73$	−8.48528	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−4.24264	−0.486664
$77$	0	0
$78$	−6.00000	−0.679366
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	−8.48528	−0.937043
$83$	−15.5563	−1.70753	−0.853766	−	0.520658i	$-0.825687\pi$
$83$	−15.5563	−1.70753	−0.853766	+	0.520658i	$0.825687\pi$
$84$	0	0
$85$	0	0
$86$	12.0000	1.29399
$87$	8.48528	0.909718
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	−12.0000	−1.24434
$94$	−2.82843	−0.291730
$95$	0	0
$96$	−1.41421	−0.144338
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.7279	1.26648	0.633238	−	0.773957i	$-0.281726\pi$
$101$	12.7279	1.26648	0.633238	+	0.773957i	$0.281726\pi$
$102$	8.00000	0.792118
$103$	8.48528	0.836080	0.418040	−	0.908429i	$-0.362717\pi$
$103$	8.48528	0.836080	0.418040	+	0.908429i	$0.362717\pi$
$104$	−4.24264	−0.416025
$105$	0	0
$106$	−6.00000	−0.582772
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−5.65685	−0.544331
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−8.48528	−0.805387
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	6.00000	0.561951
$115$	0	0
$116$	6.00000	0.557086
$117$	−4.24264	−0.392232
$118$	−4.24264	−0.390567
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	4.24264	0.384111
$123$	12.0000	1.08200
$124$	−8.48528	−0.762001
$125$	0	0
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	−1.00000	−0.0883883
$129$	−16.9706	−1.49417
$130$	0	0
$131$	12.7279	1.11204	0.556022	−	0.831168i	$-0.312327\pi$
$131$	12.7279	1.11204	0.556022	+	0.831168i	$0.312327\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	5.65685	0.485071
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	8.48528	0.722315
$139$	12.7279	1.07957	0.539784	−	0.841803i	$-0.318506\pi$
$139$	12.7279	1.07957	0.539784	+	0.841803i	$0.318506\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	−6.00000	−0.503509
$143$	0	0
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	8.48528	0.702247
$147$	0	0
$148$	−6.00000	−0.493197
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	4.24264	0.344124
$153$	5.65685	0.457330
$154$	0	0
$155$	0	0
$156$	6.00000	0.480384
$157$	−4.24264	−0.338600	−0.169300	−	0.985565i	$-0.554151\pi$
$157$	−4.24264	−0.338600	−0.169300	+	0.985565i	$0.554151\pi$
$158$	10.0000	0.795557
$159$	8.48528	0.672927
$160$	0	0
$161$	0	0
$162$	5.00000	0.392837
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	8.48528	0.662589
$165$	0	0
$166$	15.5563	1.20741
$167$	14.1421	1.09435	0.547176	−	0.837018i	$-0.315703\pi$
$167$	14.1421	1.09435	0.547176	+	0.837018i	$0.315703\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	4.24264	0.324443
$172$	−12.0000	−0.914991
$173$	−15.5563	−1.18273	−0.591364	−	0.806405i	$-0.701410\pi$
$173$	−15.5563	−1.18273	−0.591364	+	0.806405i	$0.701410\pi$
$174$	−8.48528	−0.643268
$175$	0	0
$176$	0	0
$177$	6.00000	0.450988
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−4.24264	−0.315353	−0.157676	−	0.987491i	$-0.550400\pi$
$181$	−4.24264	−0.315353	−0.157676	+	0.987491i	$0.550400\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	6.00000	0.442326
$185$	0	0
$186$	12.0000	0.879883
$187$	0	0
$188$	2.82843	0.206284
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	1.41421	0.102062
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	25.4558	1.80452	0.902258	−	0.431196i	$-0.141908\pi$
$199$	25.4558	1.80452	0.902258	+	0.431196i	$0.141908\pi$
$200$	0	0
$201$	0	0
$202$	−12.7279	−0.895533
$203$	0	0
$204$	−8.00000	−0.560112
$205$	0	0
$206$	−8.48528	−0.591198
$207$	6.00000	0.417029
$208$	4.24264	0.294174
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000	0.412082
$213$	8.48528	0.581402
$214$	0	0
$215$	0	0
$216$	5.65685	0.384900
$217$	0	0
$218$	2.00000	0.135457
$219$	−12.0000	−0.810885
$220$	0	0
$221$	−24.0000	−1.61441
$222$	8.48528	0.569495
$223$	16.9706	1.13643	0.568216	−	0.822879i	$-0.307634\pi$
$223$	16.9706	1.13643	0.568216	+	0.822879i	$0.307634\pi$
$224$	0	0
$225$	0	0
$226$	18.0000	1.19734
$227$	−15.5563	−1.03251	−0.516256	−	0.856435i	$-0.672675\pi$
$227$	−15.5563	−1.03251	−0.516256	+	0.856435i	$0.672675\pi$
$228$	−6.00000	−0.397360
$229$	21.2132	1.40181	0.700904	−	0.713256i	$-0.252780\pi$
$229$	21.2132	1.40181	0.700904	+	0.713256i	$0.252780\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	4.24264	0.277350
$235$	0	0
$236$	4.24264	0.276172
$237$	−14.1421	−0.918630
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	8.48528	0.546585	0.273293	−	0.961931i	$-0.411887\pi$
$241$	8.48528	0.546585	0.273293	+	0.961931i	$0.411887\pi$
$242$	11.0000	0.707107
$243$	9.89949	0.635053
$244$	−4.24264	−0.271607
$245$	0	0
$246$	−12.0000	−0.765092
$247$	−18.0000	−1.14531
$248$	8.48528	0.538816
$249$	−22.0000	−1.39419
$250$	0	0
$251$	−21.2132	−1.33897	−0.669483	−	0.742828i	$-0.733484\pi$
$251$	−21.2132	−1.33897	−0.669483	+	0.742828i	$0.733484\pi$
$252$	0	0
$253$	0	0
$254$	6.00000	0.376473
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.1421	0.882162	0.441081	−	0.897467i	$-0.354595\pi$
$257$	14.1421	0.882162	0.441081	+	0.897467i	$0.354595\pi$
$258$	16.9706	1.05654
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	−12.7279	−0.786334
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4.24264	−0.258678	−0.129339	−	0.991600i	$-0.541286\pi$
$269$	−4.24264	−0.258678	−0.129339	+	0.991600i	$0.541286\pi$
$270$	0	0
$271$	−16.9706	−1.03089	−0.515444	−	0.856923i	$-0.672373\pi$
$271$	−16.9706	−1.03089	−0.515444	+	0.856923i	$0.672373\pi$
$272$	−5.65685	−0.342997
$273$	0	0
$274$	−12.0000	−0.724947
$275$	0	0
$276$	−8.48528	−0.510754
$277$	−30.0000	−1.80253	−0.901263	−	0.433273i	$-0.857359\pi$
$277$	−30.0000	−1.80253	−0.901263	+	0.433273i	$0.857359\pi$
$278$	−12.7279	−0.763370
$279$	8.48528	0.508001
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	−4.00000	−0.238197
$283$	12.7279	0.756596	0.378298	−	0.925684i	$-0.376509\pi$
$283$	12.7279	0.756596	0.378298	+	0.925684i	$0.376509\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	15.0000	0.882353
$290$	0	0
$291$	0	0
$292$	−8.48528	−0.496564
$293$	−1.41421	−0.0826192	−0.0413096	−	0.999146i	$-0.513153\pi$
$293$	−1.41421	−0.0826192	−0.0413096	+	0.999146i	$0.513153\pi$
$294$	0	0
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−25.4558	−1.47215
$300$	0	0
$301$	0	0
$302$	8.00000	0.460348
$303$	18.0000	1.03407
$304$	−4.24264	−0.243332
$305$	0	0
$306$	−5.65685	−0.323381
$307$	−4.24264	−0.242140	−0.121070	−	0.992644i	$-0.538633\pi$
$307$	−4.24264	−0.242140	−0.121070	+	0.992644i	$0.538633\pi$
$308$	0	0
$309$	12.0000	0.682656
$310$	0	0
$311$	−16.9706	−0.962312	−0.481156	−	0.876635i	$-0.659783\pi$
$311$	−16.9706	−0.962312	−0.481156	+	0.876635i	$0.659783\pi$
$312$	−6.00000	−0.339683
$313$	−16.9706	−0.959233	−0.479616	−	0.877478i	$-0.659224\pi$
$313$	−16.9706	−0.959233	−0.479616	+	0.877478i	$0.659224\pi$
$314$	4.24264	0.239426
$315$	0	0
$316$	−10.0000	−0.562544
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	−8.48528	−0.475831
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	24.0000	1.33540
$324$	−5.00000	−0.277778
$325$	0	0
$326$	12.0000	0.664619
$327$	−2.82843	−0.156412
$328$	−8.48528	−0.468521
$329$	0	0
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−15.5563	−0.853766
$333$	6.00000	0.328798
$334$	−14.1421	−0.773823
$335$	0	0
$336$	0	0
$337$	30.0000	1.63420	0.817102	−	0.576493i	$-0.195579\pi$
$337$	30.0000	1.63420	0.817102	+	0.576493i	$0.195579\pi$
$338$	−5.00000	−0.271964
$339$	−25.4558	−1.38257
$340$	0	0
$341$	0	0
$342$	−4.24264	−0.229416
$343$	0	0
$344$	12.0000	0.646997
$345$	0	0
$346$	15.5563	0.836315
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	8.48528	0.454859
$349$	−4.24264	−0.227103	−0.113552	−	0.993532i	$-0.536223\pi$
$349$	−4.24264	−0.227103	−0.113552	+	0.993532i	$0.536223\pi$
$350$	0	0
$351$	−24.0000	−1.28103
$352$	0	0
$353$	2.82843	0.150542	0.0752710	−	0.997163i	$-0.476018\pi$
$353$	2.82843	0.150542	0.0752710	+	0.997163i	$0.476018\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	0	0
$357$	0	0
$358$	12.0000	0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	4.24264	0.222988
$363$	−15.5563	−0.816497
$364$	0	0
$365$	0	0
$366$	6.00000	0.313625
$367$	33.9411	1.77171	0.885856	−	0.463960i	$-0.153572\pi$
$367$	33.9411	1.77171	0.885856	+	0.463960i	$0.153572\pi$
$368$	−6.00000	−0.312772
$369$	−8.48528	−0.441726
$370$	0	0
$371$	0	0
$372$	−12.0000	−0.622171
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	0	0
$375$	0	0
$376$	−2.82843	−0.145865
$377$	25.4558	1.31104
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	−8.48528	−0.434714
$382$	6.00000	0.306987
$383$	14.1421	0.722629	0.361315	−	0.932444i	$-0.382328\pi$
$383$	14.1421	0.722629	0.361315	+	0.932444i	$0.382328\pi$
$384$	−1.41421	−0.0721688
$385$	0	0
$386$	6.00000	0.305392
$387$	12.0000	0.609994
$388$	0	0
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	33.9411	1.71648
$392$	0	0
$393$	18.0000	0.907980
$394$	6.00000	0.302276
$395$	0	0
$396$	0	0
$397$	−21.2132	−1.06466	−0.532330	−	0.846537i	$-0.678683\pi$
$397$	−21.2132	−1.06466	−0.532330	+	0.846537i	$0.678683\pi$
$398$	−25.4558	−1.27599
$399$	0	0
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	−36.0000	−1.79329
$404$	12.7279	0.633238
$405$	0	0
$406$	0	0
$407$	0	0
$408$	8.00000	0.396059
$409$	8.48528	0.419570	0.209785	−	0.977748i	$-0.432724\pi$
$409$	8.48528	0.419570	0.209785	+	0.977748i	$0.432724\pi$
$410$	0	0
$411$	16.9706	0.837096
$412$	8.48528	0.418040
$413$	0	0
$414$	−6.00000	−0.294884
$415$	0	0
$416$	−4.24264	−0.208013
$417$	18.0000	0.881464
$418$	0	0
$419$	4.24264	0.207267	0.103633	−	0.994616i	$-0.466953\pi$
$419$	4.24264	0.207267	0.103633	+	0.994616i	$0.466953\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	4.00000	0.194717
$423$	−2.82843	−0.137523
$424$	−6.00000	−0.291386
$425$	0	0
$426$	−8.48528	−0.411113
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	−5.65685	−0.272166
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	25.4558	1.21772
$438$	12.0000	0.573382
$439$	16.9706	0.809961	0.404980	−	0.914325i	$-0.367278\pi$
$439$	16.9706	0.809961	0.404980	+	0.914325i	$0.367278\pi$
$440$	0	0
$441$	0	0
$442$	24.0000	1.14156
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	−8.48528	−0.402694
$445$	0	0
$446$	−16.9706	−0.803579
$447$	8.48528	0.401340
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	−18.0000	−0.846649
$453$	−11.3137	−0.531564
$454$	15.5563	0.730096
$455$	0	0
$456$	6.00000	0.280976
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	−21.2132	−0.991228
$459$	32.0000	1.49363
$460$	0	0
$461$	29.6985	1.38320	0.691598	−	0.722282i	$-0.256907\pi$
$461$	29.6985	1.38320	0.691598	+	0.722282i	$0.256907\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	0	0
$467$	−7.07107	−0.327210	−0.163605	−	0.986526i	$-0.552312\pi$
$467$	−7.07107	−0.327210	−0.163605	+	0.986526i	$0.552312\pi$
$468$	−4.24264	−0.196116
$469$	0	0
$470$	0	0
$471$	−6.00000	−0.276465
$472$	−4.24264	−0.195283
$473$	0	0
$474$	14.1421	0.649570
$475$	0	0
$476$	0	0
$477$	−6.00000	−0.274721
$478$	24.0000	1.09773
$479$	8.48528	0.387702	0.193851	−	0.981031i	$-0.437902\pi$
$479$	8.48528	0.387702	0.193851	+	0.981031i	$0.437902\pi$
$480$	0	0
$481$	−25.4558	−1.16069
$482$	−8.48528	−0.386494
$483$	0	0
$484$	−11.0000	−0.500000
$485$	0	0
$486$	−9.89949	−0.449050
$487$	−18.0000	−0.815658	−0.407829	−	0.913058i	$-0.633714\pi$
$487$	−18.0000	−0.815658	−0.407829	+	0.913058i	$0.633714\pi$
$488$	4.24264	0.192055
$489$	−16.9706	−0.767435
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	12.0000	0.541002
$493$	−33.9411	−1.52863
$494$	18.0000	0.809858
$495$	0	0
$496$	−8.48528	−0.381000
$497$	0	0
$498$	22.0000	0.985844
$499$	−40.0000	−1.79065	−0.895323	−	0.445418i	$-0.853055\pi$
$499$	−40.0000	−1.79065	−0.895323	+	0.445418i	$0.853055\pi$
$500$	0	0
$501$	20.0000	0.893534
$502$	21.2132	0.946792
$503$	5.65685	0.252227	0.126113	−	0.992016i	$-0.459750\pi$
$503$	5.65685	0.252227	0.126113	+	0.992016i	$0.459750\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.07107	0.314037
$508$	−6.00000	−0.266207
$509$	−29.6985	−1.31636	−0.658181	−	0.752860i	$-0.728674\pi$
$509$	−29.6985	−1.31636	−0.658181	+	0.752860i	$0.728674\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	24.0000	1.05963
$514$	−14.1421	−0.623783
$515$	0	0
$516$	−16.9706	−0.747087
$517$	0	0
$518$	0	0
$519$	−22.0000	−0.965693
$520$	0	0
$521$	−8.48528	−0.371747	−0.185873	−	0.982574i	$-0.559511\pi$
$521$	−8.48528	−0.371747	−0.185873	+	0.982574i	$0.559511\pi$
$522$	6.00000	0.262613
$523$	21.2132	0.927589	0.463794	−	0.885943i	$-0.346488\pi$
$523$	21.2132	0.927589	0.463794	+	0.885943i	$0.346488\pi$
$524$	12.7279	0.556022
$525$	0	0
$526$	0	0
$527$	48.0000	2.09091
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−4.24264	−0.184115
$532$	0	0
$533$	36.0000	1.55933
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−16.9706	−0.732334
$538$	4.24264	0.182913
$539$	0	0
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	16.9706	0.728948
$543$	−6.00000	−0.257485
$544$	5.65685	0.242536
$545$	0	0
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	12.0000	0.512615
$549$	4.24264	0.181071
$550$	0	0
$551$	−25.4558	−1.08446
$552$	8.48528	0.361158
$553$	0	0
$554$	30.0000	1.27458
$555$	0	0
$556$	12.7279	0.539784
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	−8.48528	−0.359211
$559$	−50.9117	−2.15333
$560$	0	0
$561$	0	0
$562$	−18.0000	−0.759284
$563$	1.41421	0.0596020	0.0298010	−	0.999556i	$-0.490513\pi$
$563$	1.41421	0.0596020	0.0298010	+	0.999556i	$0.490513\pi$
$564$	4.00000	0.168430
$565$	0	0
$566$	−12.7279	−0.534994
$567$	0	0
$568$	−6.00000	−0.251754
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	0	0
$573$	−8.48528	−0.354478
$574$	0	0
$575$	0	0
$576$	−1.00000	−0.0416667
$577$	−25.4558	−1.05974	−0.529870	−	0.848079i	$-0.677759\pi$
$577$	−25.4558	−1.05974	−0.529870	+	0.848079i	$0.677759\pi$
$578$	−15.0000	−0.623918
$579$	−8.48528	−0.352636
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	8.48528	0.351123
$585$	0	0
$586$	1.41421	0.0584206
$587$	−9.89949	−0.408596	−0.204298	−	0.978909i	$-0.565491\pi$
$587$	−9.89949	−0.408596	−0.204298	+	0.978909i	$0.565491\pi$
$588$	0	0
$589$	36.0000	1.48335
$590$	0	0
$591$	−8.48528	−0.349038
$592$	−6.00000	−0.246598
$593$	19.7990	0.813047	0.406524	−	0.913640i	$-0.366741\pi$
$593$	19.7990	0.813047	0.406524	+	0.913640i	$0.366741\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	36.0000	1.47338
$598$	25.4558	1.04097
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	−16.9706	−0.692244	−0.346122	−	0.938190i	$-0.612502\pi$
$601$	−16.9706	−0.692244	−0.346122	+	0.938190i	$0.612502\pi$
$602$	0	0
$603$	0	0
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−18.0000	−0.731200
$607$	33.9411	1.37763	0.688814	−	0.724938i	$-0.258132\pi$
$607$	33.9411	1.37763	0.688814	+	0.724938i	$0.258132\pi$
$608$	4.24264	0.172062
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	5.65685	0.228665
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	4.24264	0.171219
$615$	0	0
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	−12.0000	−0.482711
$619$	−12.7279	−0.511578	−0.255789	−	0.966733i	$-0.582335\pi$
$619$	−12.7279	−0.511578	−0.255789	+	0.966733i	$0.582335\pi$
$620$	0	0
$621$	33.9411	1.36201
$622$	16.9706	0.680458
$623$	0	0
$624$	6.00000	0.240192
$625$	0	0
$626$	16.9706	0.678280
$627$	0	0
$628$	−4.24264	−0.169300
$629$	33.9411	1.35332
$630$	0	0
$631$	−38.0000	−1.51276	−0.756378	−	0.654135i	$-0.773033\pi$
$631$	−38.0000	−1.51276	−0.756378	+	0.654135i	$0.773033\pi$
$632$	10.0000	0.397779
$633$	−5.65685	−0.224840
$634$	−6.00000	−0.238290
$635$	0	0
$636$	8.48528	0.336463
$637$	0	0
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	12.7279	0.501940	0.250970	−	0.967995i	$-0.419250\pi$
$643$	12.7279	0.501940	0.250970	+	0.967995i	$0.419250\pi$
$644$	0	0
$645$	0	0
$646$	−24.0000	−0.944267
$647$	−19.7990	−0.778379	−0.389189	−	0.921158i	$-0.627245\pi$
$647$	−19.7990	−0.778379	−0.389189	+	0.921158i	$0.627245\pi$
$648$	5.00000	0.196419
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−12.0000	−0.469956
$653$	42.0000	1.64359	0.821794	−	0.569785i	$-0.192974\pi$
$653$	42.0000	1.64359	0.821794	+	0.569785i	$0.192974\pi$
$654$	2.82843	0.110600
$655$	0	0
$656$	8.48528	0.331295
$657$	8.48528	0.331042
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−12.7279	−0.495059	−0.247529	−	0.968880i	$-0.579619\pi$
$661$	−12.7279	−0.495059	−0.247529	+	0.968880i	$0.579619\pi$
$662$	28.0000	1.08825
$663$	−33.9411	−1.31816
$664$	15.5563	0.603703
$665$	0	0
$666$	−6.00000	−0.232495
$667$	−36.0000	−1.39393
$668$	14.1421	0.547176
$669$	24.0000	0.927894
$670$	0	0
$671$	0	0
$672$	0	0
$673$	36.0000	1.38770	0.693849	−	0.720121i	$-0.255914\pi$
$673$	36.0000	1.38770	0.693849	+	0.720121i	$0.255914\pi$
$674$	−30.0000	−1.15556
$675$	0	0
$676$	5.00000	0.192308
$677$	−15.5563	−0.597879	−0.298940	−	0.954272i	$-0.596633\pi$
$677$	−15.5563	−0.597879	−0.298940	+	0.954272i	$0.596633\pi$
$678$	25.4558	0.977626
$679$	0	0
$680$	0	0
$681$	−22.0000	−0.843042
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	4.24264	0.162221
$685$	0	0
$686$	0	0
$687$	30.0000	1.14457
$688$	−12.0000	−0.457496
$689$	25.4558	0.969790
$690$	0	0
$691$	46.6690	1.77537	0.887687	−	0.460447i	$-0.152311\pi$
$691$	46.6690	1.77537	0.887687	+	0.460447i	$0.152311\pi$
$692$	−15.5563	−0.591364
$693$	0	0
$694$	0	0
$695$	0	0
$696$	−8.48528	−0.321634
$697$	−48.0000	−1.81813
$698$	4.24264	0.160586
$699$	0	0
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	24.0000	0.905822
$703$	25.4558	0.960085
$704$	0	0
$705$	0	0
$706$	−2.82843	−0.106449
$707$	0	0
$708$	6.00000	0.225494
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	50.9117	1.90666
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	−33.9411	−1.26755
$718$	0	0
$719$	42.4264	1.58224	0.791119	−	0.611662i	$-0.209499\pi$
$719$	42.4264	1.58224	0.791119	+	0.611662i	$0.209499\pi$
$720$	0	0
$721$	0	0
$722$	1.00000	0.0372161
$723$	12.0000	0.446285
$724$	−4.24264	−0.157676
$725$	0	0
$726$	15.5563	0.577350
$727$	−42.4264	−1.57351	−0.786754	−	0.617266i	$-0.788240\pi$
$727$	−42.4264	−1.57351	−0.786754	+	0.617266i	$0.788240\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	67.8823	2.51072
$732$	−6.00000	−0.221766
$733$	4.24264	0.156706	0.0783528	−	0.996926i	$-0.475034\pi$
$733$	4.24264	0.156706	0.0783528	+	0.996926i	$0.475034\pi$
$734$	−33.9411	−1.25279
$735$	0	0
$736$	6.00000	0.221163
$737$	0	0
$738$	8.48528	0.312348
$739$	52.0000	1.91285	0.956425	−	0.291977i	$-0.0943129\pi$
$739$	52.0000	1.91285	0.956425	+	0.291977i	$0.0943129\pi$
$740$	0	0
$741$	−25.4558	−0.935144
$742$	0	0
$743$	−18.0000	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$744$	12.0000	0.439941
$745$	0	0
$746$	−6.00000	−0.219676
$747$	15.5563	0.569177
$748$	0	0
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	2.82843	0.103142
$753$	−30.0000	−1.09326
$754$	−25.4558	−0.927047
$755$	0	0
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	16.0000	0.581146
$759$	0	0
$760$	0	0
$761$	42.4264	1.53796	0.768978	−	0.639275i	$-0.220766\pi$
$761$	42.4264	1.53796	0.768978	+	0.639275i	$0.220766\pi$
$762$	8.48528	0.307389
$763$	0	0
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−14.1421	−0.510976
$767$	18.0000	0.649942
$768$	1.41421	0.0510310
$769$	42.4264	1.52994	0.764968	−	0.644069i	$-0.222755\pi$
$769$	42.4264	1.52994	0.764968	+	0.644069i	$0.222755\pi$
$770$	0	0
$771$	20.0000	0.720282
$772$	−6.00000	−0.215945
$773$	18.3848	0.661254	0.330627	−	0.943761i	$-0.392740\pi$
$773$	18.3848	0.661254	0.330627	+	0.943761i	$0.392740\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	0	0
$777$	0	0
$778$	18.0000	0.645331
$779$	−36.0000	−1.28983
$780$	0	0
$781$	0	0
$782$	−33.9411	−1.21373
$783$	−33.9411	−1.21296
$784$	0	0
$785$	0	0
$786$	−18.0000	−0.642039
$787$	−29.6985	−1.05864	−0.529318	−	0.848423i	$-0.677552\pi$
$787$	−29.6985	−1.05864	−0.529318	+	0.848423i	$0.677552\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−18.0000	−0.639199
$794$	21.2132	0.752828
$795$	0	0
$796$	25.4558	0.902258
$797$	−52.3259	−1.85348	−0.926739	−	0.375705i	$-0.877401\pi$
$797$	−52.3259	−1.85348	−0.926739	+	0.375705i	$0.877401\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	0	0
$802$	−24.0000	−0.847469
$803$	0	0
$804$	0	0
$805$	0	0
$806$	36.0000	1.26805
$807$	−6.00000	−0.211210
$808$	−12.7279	−0.447767
$809$	−24.0000	−0.843795	−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000	−0.843795	−0.421898	+	0.906644i	$0.638636\pi$
$810$	0	0
$811$	12.7279	0.446938	0.223469	−	0.974711i	$-0.428262\pi$
$811$	12.7279	0.446938	0.223469	+	0.974711i	$0.428262\pi$
$812$	0	0
$813$	−24.0000	−0.841717
$814$	0	0
$815$	0	0
$816$	−8.00000	−0.280056
$817$	50.9117	1.78117
$818$	−8.48528	−0.296681
$819$	0	0
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	−16.9706	−0.591916
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	−8.48528	−0.295599
$825$	0	0
$826$	0	0
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	6.00000	0.208514
$829$	12.7279	0.442059	0.221030	−	0.975267i	$-0.429058\pi$
$829$	12.7279	0.442059	0.221030	+	0.975267i	$0.429058\pi$
$830$	0	0
$831$	−42.4264	−1.47176
$832$	4.24264	0.147087
$833$	0	0
$834$	−18.0000	−0.623289
$835$	0	0
$836$	0	0
$837$	48.0000	1.65912
$838$	−4.24264	−0.146560
$839$	8.48528	0.292944	0.146472	−	0.989215i	$-0.453208\pi$
$839$	8.48528	0.292944	0.146472	+	0.989215i	$0.453208\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−10.0000	−0.344623
$843$	25.4558	0.876746
$844$	−4.00000	−0.137686
$845$	0	0
$846$	2.82843	0.0972433
$847$	0	0
$848$	6.00000	0.206041
$849$	18.0000	0.617758
$850$	0	0
$851$	36.0000	1.23406
$852$	8.48528	0.290701
$853$	−38.1838	−1.30739	−0.653694	−	0.756759i	$-0.726781\pi$
$853$	−38.1838	−1.30739	−0.653694	+	0.756759i	$0.726781\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.2843	0.966172	0.483086	−	0.875573i	$-0.339516\pi$
$857$	28.2843	0.966172	0.483086	+	0.875573i	$0.339516\pi$
$858$	0	0
$859$	−29.6985	−1.01330	−0.506650	−	0.862152i	$-0.669116\pi$
$859$	−29.6985	−1.01330	−0.506650	+	0.862152i	$0.669116\pi$
$860$	0	0
$861$	0	0
$862$	24.0000	0.817443
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	5.65685	0.192450
$865$	0	0
$866$	0	0
$867$	21.2132	0.720438
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	2.00000	0.0677285
$873$	0	0
$874$	−25.4558	−0.861057
$875$	0	0
$876$	−12.0000	−0.405442
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	−16.9706	−0.572729
$879$	−2.00000	−0.0674583
$880$	0	0
$881$	−50.9117	−1.71526	−0.857629	−	0.514269i	$-0.828063\pi$
$881$	−50.9117	−1.71526	−0.857629	+	0.514269i	$0.828063\pi$
$882$	0	0
$883$	24.0000	0.807664	0.403832	−	0.914833i	$-0.367678\pi$
$883$	24.0000	0.807664	0.403832	+	0.914833i	$0.367678\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	−24.0000	−0.806296
$887$	−14.1421	−0.474846	−0.237423	−	0.971406i	$-0.576303\pi$
$887$	−14.1421	−0.474846	−0.237423	+	0.971406i	$0.576303\pi$
$888$	8.48528	0.284747
$889$	0	0
$890$	0	0
$891$	0	0
$892$	16.9706	0.568216
$893$	−12.0000	−0.401565
$894$	−8.48528	−0.283790
$895$	0	0
$896$	0	0
$897$	−36.0000	−1.20201
$898$	−30.0000	−1.00111
$899$	−50.9117	−1.69800
$900$	0	0
$901$	−33.9411	−1.13074
$902$	0	0
$903$	0	0
$904$	18.0000	0.598671
$905$	0	0
$906$	11.3137	0.375873
$907$	−24.0000	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$908$	−15.5563	−0.516256
$909$	−12.7279	−0.422159
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	−6.00000	−0.198680
$913$	0	0
$914$	18.0000	0.595387
$915$	0	0
$916$	21.2132	0.700904
$917$	0	0
$918$	−32.0000	−1.05616
$919$	−2.00000	−0.0659739	−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000	−0.0659739	−0.0329870	+	0.999456i	$0.510502\pi$
$920$	0	0
$921$	−6.00000	−0.197707
$922$	−29.6985	−0.978068
$923$	25.4558	0.837889
$924$	0	0
$925$	0	0
$926$	−24.0000	−0.788689
$927$	−8.48528	−0.278693
$928$	−6.00000	−0.196960
$929$	42.4264	1.39197	0.695983	−	0.718059i	$-0.254969\pi$
$929$	42.4264	1.39197	0.695983	+	0.718059i	$0.254969\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−24.0000	−0.785725
$934$	7.07107	0.231372
$935$	0	0
$936$	4.24264	0.138675
$937$	59.3970	1.94041	0.970207	−	0.242277i	$-0.0778942\pi$
$937$	59.3970	1.94041	0.970207	+	0.242277i	$0.0778942\pi$
$938$	0	0
$939$	−24.0000	−0.783210
$940$	0	0
$941$	−29.6985	−0.968143	−0.484071	−	0.875028i	$-0.660843\pi$
$941$	−29.6985	−0.968143	−0.484071	+	0.875028i	$0.660843\pi$
$942$	6.00000	0.195491
$943$	−50.9117	−1.65791
$944$	4.24264	0.138086
$945$	0	0
$946$	0	0
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	−14.1421	−0.459315
$949$	−36.0000	−1.16861
$950$	0	0
$951$	8.48528	0.275154
$952$	0	0
$953$	−36.0000	−1.16615	−0.583077	−	0.812417i	$-0.698151\pi$
$953$	−36.0000	−1.16615	−0.583077	+	0.812417i	$0.698151\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	−24.0000	−0.776215
$957$	0	0
$958$	−8.48528	−0.274147
$959$	0	0
$960$	0	0
$961$	41.0000	1.32258
$962$	25.4558	0.820729
$963$	0	0
$964$	8.48528	0.273293
$965$	0	0
$966$	0	0
$967$	18.0000	0.578841	0.289420	−	0.957202i	$-0.406537\pi$
$967$	18.0000	0.578841	0.289420	+	0.957202i	$0.406537\pi$
$968$	11.0000	0.353553
$969$	33.9411	1.09035
$970$	0	0
$971$	−29.6985	−0.953070	−0.476535	−	0.879156i	$-0.658107\pi$
$971$	−29.6985	−0.953070	−0.476535	+	0.879156i	$0.658107\pi$
$972$	9.89949	0.317526
$973$	0	0
$974$	18.0000	0.576757
$975$	0	0
$976$	−4.24264	−0.135804
$977$	−36.0000	−1.15174	−0.575871	−	0.817541i	$-0.695337\pi$
$977$	−36.0000	−1.15174	−0.575871	+	0.817541i	$0.695337\pi$
$978$	16.9706	0.542659
$979$	0	0
$980$	0	0
$981$	2.00000	0.0638551
$982$	−24.0000	−0.765871
$983$	−48.0833	−1.53362	−0.766809	−	0.641875i	$-0.778157\pi$
$983$	−48.0833	−1.53362	−0.766809	+	0.641875i	$0.778157\pi$
$984$	−12.0000	−0.382546
$985$	0	0
$986$	33.9411	1.08091
$987$	0	0
$988$	−18.0000	−0.572656
$989$	72.0000	2.28947
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	8.48528	0.269408
$993$	−39.5980	−1.25660
$994$	0	0
$995$	0	0
$996$	−22.0000	−0.697097
$997$	46.6690	1.47802	0.739012	−	0.673693i	$-0.235293\pi$
$997$	46.6690	1.47802	0.739012	+	0.673693i	$0.235293\pi$
$998$	40.0000	1.26618
$999$	33.9411	1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2450.2.a.bk.1.2		2
5.2	odd	4		490.2.c.f.99.1	✓	4
5.3	odd	4		490.2.c.f.99.4	yes	4
5.4	even	2		2450.2.a.bp.1.1		2
7.6	odd	2	inner	2450.2.a.bk.1.1		2
35.2	odd	12		490.2.i.e.459.2		8
35.3	even	12		490.2.i.e.79.1		8
35.12	even	12		490.2.i.e.459.1		8
35.13	even	4		490.2.c.f.99.3	yes	4
35.17	even	12		490.2.i.e.79.4		8
35.18	odd	12		490.2.i.e.79.2		8
35.23	odd	12		490.2.i.e.459.3		8
35.27	even	4		490.2.c.f.99.2	yes	4
35.32	odd	12		490.2.i.e.79.3		8
35.33	even	12		490.2.i.e.459.4		8
35.34	odd	2		2450.2.a.bp.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
490.2.c.f.99.1	✓	4	5.2	odd	4
490.2.c.f.99.2	yes	4	35.27	even	4
490.2.c.f.99.3	yes	4	35.13	even	4
490.2.c.f.99.4	yes	4	5.3	odd	4
490.2.i.e.79.1		8	35.3	even	12
490.2.i.e.79.2		8	35.18	odd	12
490.2.i.e.79.3		8	35.32	odd	12
490.2.i.e.79.4		8	35.17	even	12
490.2.i.e.459.1		8	35.12	even	12
490.2.i.e.459.2		8	35.2	odd	12
490.2.i.e.459.3		8	35.23	odd	12
490.2.i.e.459.4		8	35.33	even	12
2450.2.a.bk.1.1		2	7.6	odd	2	inner
2450.2.a.bk.1.2		2	1.1	even	1	trivial
2450.2.a.bp.1.1		2	5.4	even	2
2450.2.a.bp.1.2		2	35.34	odd	2

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +1.41421 q^{12} +4.24264 q^{13} +1.00000 q^{16} -5.65685 q^{17} +1.00000 q^{18} -4.24264 q^{19} -6.00000 q^{23} -1.41421 q^{24} -4.24264 q^{26} -5.65685 q^{27} +6.00000 q^{29} -8.48528 q^{31} -1.00000 q^{32} +5.65685 q^{34} -1.00000 q^{36} -6.00000 q^{37} +4.24264 q^{38} +6.00000 q^{39} +8.48528 q^{41} -12.0000 q^{43} +6.00000 q^{46} +2.82843 q^{47} +1.41421 q^{48} -8.00000 q^{51} +4.24264 q^{52} +6.00000 q^{53} +5.65685 q^{54} -6.00000 q^{57} -6.00000 q^{58} +4.24264 q^{59} -4.24264 q^{61} +8.48528 q^{62} +1.00000 q^{64} -5.65685 q^{68} -8.48528 q^{69} +6.00000 q^{71} +1.00000 q^{72} -8.48528 q^{73} +6.00000 q^{74} -4.24264 q^{76} -6.00000 q^{78} -10.0000 q^{79} -5.00000 q^{81} -8.48528 q^{82} -15.5563 q^{83} +12.0000 q^{86} +8.48528 q^{87} -6.00000 q^{92} -12.0000 q^{93} -2.82843 q^{94} -1.41421 q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} + 2 q^{16} + 2 q^{18} - 12 q^{23} + 12 q^{29} - 2 q^{32} - 2 q^{36} - 12 q^{37} + 12 q^{39} - 24 q^{43} + 12 q^{46} - 16 q^{51} + 12 q^{53} - 12 q^{57} - 12 q^{58} + 2 q^{64} + 12 q^{71} + 2 q^{72} + 12 q^{74} - 12 q^{78} - 20 q^{79} - 10 q^{81} + 24 q^{86} - 12 q^{92} - 24 q^{93}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^8 - 2 * q^9 + 2 * q^16 + 2 * q^18 - 12 * q^23 + 12 * q^29 - 2 * q^32 - 2 * q^36 - 12 * q^37 + 12 * q^39 - 24 * q^43 + 12 * q^46 - 16 * q^51 + 12 * q^53 - 12 * q^57 - 12 * q^58 + 2 * q^64 + 12 * q^71 + 2 * q^72 + 12 * q^74 - 12 * q^78 - 20 * q^79 - 10 * q^81 + 24 * q^86 - 12 * q^92 - 24 * q^93

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