LMFDB - Embedded newform 8450.2.a.cj.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8450,2,Mod(1,8450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8450, 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("8450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8450 = 2 \cdot 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8450.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:	$67.4735897080$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 4 $ x^4 - 7*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.52434$ of defining polynomial
Character	$\chi$	$=$	8450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +2.52434 q^{3} +1.00000 q^{4} -2.52434 q^{6} +2.37228 q^{7} -1.00000 q^{8} +3.37228 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.52434 q^{3} +1.00000 q^{4} -2.52434 q^{6} +2.37228 q^{7} -1.00000 q^{8} +3.37228 q^{9} -1.58457 q^{11} +2.52434 q^{12} -2.37228 q^{14} +1.00000 q^{16} -5.98844 q^{17} -3.37228 q^{18} -3.46410 q^{19} +5.98844 q^{21} +1.58457 q^{22} +6.63325 q^{23} -2.52434 q^{24} +0.939764 q^{27} +2.37228 q^{28} -2.74456 q^{29} -3.46410 q^{31} -1.00000 q^{32} -4.00000 q^{33} +5.98844 q^{34} +3.37228 q^{36} -9.11684 q^{37} +3.46410 q^{38} -10.0974 q^{41} -5.98844 q^{42} -0.644810 q^{43} -1.58457 q^{44} -6.63325 q^{46} -10.3723 q^{47} +2.52434 q^{48} -1.37228 q^{49} -15.1168 q^{51} +5.04868 q^{53} -0.939764 q^{54} -2.37228 q^{56} -8.74456 q^{57} +2.74456 q^{58} -6.63325 q^{59} -6.74456 q^{61} +3.46410 q^{62} +8.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} -4.00000 q^{67} -5.98844 q^{68} +16.7446 q^{69} +12.6217 q^{71} -3.37228 q^{72} -10.0000 q^{73} +9.11684 q^{74} -3.46410 q^{76} -3.75906 q^{77} -4.74456 q^{79} -7.74456 q^{81} +10.0974 q^{82} +8.74456 q^{83} +5.98844 q^{84} +0.644810 q^{86} -6.92820 q^{87} +1.58457 q^{88} -1.87953 q^{89} +6.63325 q^{92} -8.74456 q^{93} +10.3723 q^{94} -2.52434 q^{96} -6.74456 q^{97} +1.37228 q^{98} -5.34363 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 2 q^{7} - 4 q^{8} + 2 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 - 2 * q^7 - 4 * q^8 + 2 * q^9	$ 4 q - 4 q^{2} + 4 q^{4} - 2 q^{7} - 4 q^{8} + 2 q^{9} + 2 q^{14} + 4 q^{16} - 2 q^{18} - 2 q^{28} + 12 q^{29} - 4 q^{32} - 16 q^{33} + 2 q^{36} - 2 q^{37} - 30 q^{47} + 6 q^{49} - 26 q^{51} + 2 q^{56} - 12 q^{57} - 12 q^{58} - 4 q^{61} + 32 q^{63} + 4 q^{64} + 16 q^{66} - 16 q^{67} + 44 q^{69} - 2 q^{72} - 40 q^{73} + 2 q^{74} + 4 q^{79} - 8 q^{81} + 12 q^{83} - 12 q^{93} + 30 q^{94} - 4 q^{97} - 6 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 2 * q^7 - 4 * q^8 + 2 * q^9 + 2 * q^14 + 4 * q^16 - 2 * q^18 - 2 * q^28 + 12 * q^29 - 4 * q^32 - 16 * q^33 + 2 * q^36 - 2 * q^37 - 30 * q^47 + 6 * q^49 - 26 * q^51 + 2 * q^56 - 12 * q^57 - 12 * q^58 - 4 * q^61 + 32 * q^63 + 4 * q^64 + 16 * q^66 - 16 * q^67 + 44 * q^69 - 2 * q^72 - 40 * q^73 + 2 * q^74 + 4 * q^79 - 8 * q^81 + 12 * q^83 - 12 * q^93 + 30 * q^94 - 4 * q^97 - 6 * q^98

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
$3$	2.52434	1.45743	0.728714	−	0.684819i	$-0.240119\pi$
$3$	2.52434	1.45743	0.728714	+	0.684819i	$0.240119\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−2.52434	−1.03056
$7$	2.37228	0.896638	0.448319	−	0.893874i	$-0.352023\pi$
$7$	2.37228	0.896638	0.448319	+	0.893874i	$0.352023\pi$
$8$	−1.00000	−0.353553
$9$	3.37228	1.12409
$10$	0	0
$11$	−1.58457	−0.477767	−0.238884	−	0.971048i	$-0.576781\pi$
$11$	−1.58457	−0.477767	−0.238884	+	0.971048i	$0.576781\pi$
$12$	2.52434	0.728714
$13$	0	0
$13$	0	0
$14$	−2.37228	−0.634019
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.98844	−1.45241	−0.726205	−	0.687478i	$-0.758718\pi$
$17$	−5.98844	−1.45241	−0.726205	+	0.687478i	$0.758718\pi$
$18$	−3.37228	−0.794854
$19$	−3.46410	−0.794719	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410	−0.794719	−0.397360	+	0.917663i	$0.630073\pi$
$20$	0	0
$21$	5.98844	1.30678
$22$	1.58457	0.337832
$23$	6.63325	1.38313	0.691564	−	0.722315i	$-0.256922\pi$
$23$	6.63325	1.38313	0.691564	+	0.722315i	$0.256922\pi$
$24$	−2.52434	−0.515278
$25$	0	0
$26$	0	0
$27$	0.939764	0.180858
$28$	2.37228	0.448319
$29$	−2.74456	−0.509652	−0.254826	−	0.966987i	$-0.582018\pi$
$29$	−2.74456	−0.509652	−0.254826	+	0.966987i	$0.582018\pi$
$30$	0	0
$31$	−3.46410	−0.622171	−0.311086	−	0.950382i	$-0.600693\pi$
$31$	−3.46410	−0.622171	−0.311086	+	0.950382i	$0.600693\pi$
$32$	−1.00000	−0.176777
$33$	−4.00000	−0.696311
$34$	5.98844	1.02701
$35$	0	0
$36$	3.37228	0.562047
$37$	−9.11684	−1.49880	−0.749400	−	0.662118i	$-0.769658\pi$
$37$	−9.11684	−1.49880	−0.749400	+	0.662118i	$0.769658\pi$
$38$	3.46410	0.561951
$39$	0	0
$40$	0	0
$41$	−10.0974	−1.57694	−0.788471	−	0.615072i	$-0.789127\pi$
$41$	−10.0974	−1.57694	−0.788471	+	0.615072i	$0.789127\pi$
$42$	−5.98844	−0.924036
$43$	−0.644810	−0.0983326	−0.0491663	−	0.998791i	$-0.515656\pi$
$43$	−0.644810	−0.0983326	−0.0491663	+	0.998791i	$0.515656\pi$
$44$	−1.58457	−0.238884
$45$	0	0
$46$	−6.63325	−0.978019
$47$	−10.3723	−1.51295	−0.756476	−	0.654021i	$-0.773081\pi$
$47$	−10.3723	−1.51295	−0.756476	+	0.654021i	$0.773081\pi$
$48$	2.52434	0.364357
$49$	−1.37228	−0.196040
$50$	0	0
$51$	−15.1168	−2.11678
$52$	0	0
$53$	5.04868	0.693489	0.346744	−	0.937960i	$-0.387287\pi$
$53$	5.04868	0.693489	0.346744	+	0.937960i	$0.387287\pi$
$54$	−0.939764	−0.127886
$55$	0	0
$56$	−2.37228	−0.317009
$57$	−8.74456	−1.15825
$58$	2.74456	0.360379
$59$	−6.63325	−0.863576	−0.431788	−	0.901975i	$-0.642117\pi$
$59$	−6.63325	−0.863576	−0.431788	+	0.901975i	$0.642117\pi$
$60$	0	0
$61$	−6.74456	−0.863553	−0.431776	−	0.901981i	$-0.642113\pi$
$61$	−6.74456	−0.863553	−0.431776	+	0.901981i	$0.642113\pi$
$62$	3.46410	0.439941
$63$	8.00000	1.00791
$64$	1.00000	0.125000
$65$	0	0
$66$	4.00000	0.492366
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−5.98844	−0.726205
$69$	16.7446	2.01581
$70$	0	0
$71$	12.6217	1.49792	0.748959	−	0.662616i	$-0.230554\pi$
$71$	12.6217	1.49792	0.748959	+	0.662616i	$0.230554\pi$
$72$	−3.37228	−0.397427
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	9.11684	1.05981
$75$	0	0
$76$	−3.46410	−0.397360
$77$	−3.75906	−0.428384
$78$	0	0
$79$	−4.74456	−0.533805	−0.266903	−	0.963724i	$-0.586000\pi$
$79$	−4.74456	−0.533805	−0.266903	+	0.963724i	$0.586000\pi$
$80$	0	0
$81$	−7.74456	−0.860507
$82$	10.0974	1.11507
$83$	8.74456	0.959840	0.479920	−	0.877312i	$-0.340666\pi$
$83$	8.74456	0.959840	0.479920	+	0.877312i	$0.340666\pi$
$84$	5.98844	0.653392
$85$	0	0
$86$	0.644810	0.0695317
$87$	−6.92820	−0.742781
$88$	1.58457	0.168916
$89$	−1.87953	−0.199230	−0.0996148	−	0.995026i	$-0.531761\pi$
$89$	−1.87953	−0.199230	−0.0996148	+	0.995026i	$0.531761\pi$
$90$	0	0
$91$	0	0
$92$	6.63325	0.691564
$93$	−8.74456	−0.906769
$94$	10.3723	1.06982
$95$	0	0
$96$	−2.52434	−0.257639
$97$	−6.74456	−0.684807	−0.342403	−	0.939553i	$-0.611241\pi$
$97$	−6.74456	−0.684807	−0.342403	+	0.939553i	$0.611241\pi$
$98$	1.37228	0.138621
$99$	−5.34363	−0.537055
$100$	0	0
$101$	14.7446	1.46714	0.733569	−	0.679615i	$-0.237853\pi$
$101$	14.7446	1.46714	0.733569	+	0.679615i	$0.237853\pi$
$102$	15.1168	1.49679
$103$	10.3923	1.02398	0.511992	−	0.858990i	$-0.328908\pi$
$103$	10.3923	1.02398	0.511992	+	0.858990i	$0.328908\pi$
$104$	0	0
$105$	0	0
$106$	−5.04868	−0.490371
$107$	13.5615	1.31104	0.655518	−	0.755180i	$-0.272451\pi$
$107$	13.5615	1.31104	0.655518	+	0.755180i	$0.272451\pi$
$108$	0.939764	0.0904288
$109$	−2.81929	−0.270039	−0.135020	−	0.990843i	$-0.543110\pi$
$109$	−2.81929	−0.270039	−0.135020	+	0.990843i	$0.543110\pi$
$110$	0	0
$111$	−23.0140	−2.18439
$112$	2.37228	0.224160
$113$	3.75906	0.353622	0.176811	−	0.984245i	$-0.443422\pi$
$113$	3.75906	0.353622	0.176811	+	0.984245i	$0.443422\pi$
$114$	8.74456	0.819003
$115$	0	0
$116$	−2.74456	−0.254826
$117$	0	0
$118$	6.63325	0.610640
$119$	−14.2063	−1.30229
$120$	0	0
$121$	−8.48913	−0.771739
$122$	6.74456	0.610624
$123$	−25.4891	−2.29828
$124$	−3.46410	−0.311086
$125$	0	0
$126$	−8.00000	−0.712697
$127$	3.46410	0.307389	0.153695	−	0.988118i	$-0.450883\pi$
$127$	3.46410	0.307389	0.153695	+	0.988118i	$0.450883\pi$
$128$	−1.00000	−0.0883883
$129$	−1.62772	−0.143313
$130$	0	0
$131$	−1.62772	−0.142214	−0.0711072	−	0.997469i	$-0.522653\pi$
$131$	−1.62772	−0.142214	−0.0711072	+	0.997469i	$0.522653\pi$
$132$	−4.00000	−0.348155
$133$	−8.21782	−0.712576
$134$	4.00000	0.345547
$135$	0	0
$136$	5.98844	0.513504
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	−16.7446	−1.42539
$139$	−6.37228	−0.540490	−0.270245	−	0.962792i	$-0.587105\pi$
$139$	−6.37228	−0.540490	−0.270245	+	0.962792i	$0.587105\pi$
$140$	0	0
$141$	−26.1831	−2.20502
$142$	−12.6217	−1.05919
$143$	0	0
$144$	3.37228	0.281023
$145$	0	0
$146$	10.0000	0.827606
$147$	−3.46410	−0.285714
$148$	−9.11684	−0.749400
$149$	17.0256	1.39479	0.697394	−	0.716688i	$-0.254343\pi$
$149$	17.0256	1.39479	0.697394	+	0.716688i	$0.254343\pi$
$150$	0	0
$151$	7.57301	0.616283	0.308142	−	0.951341i	$-0.400293\pi$
$151$	7.57301	0.616283	0.308142	+	0.951341i	$0.400293\pi$
$152$	3.46410	0.280976
$153$	−20.1947	−1.63264
$154$	3.75906	0.302913
$155$	0	0
$156$	0	0
$157$	−15.1460	−1.20878	−0.604392	−	0.796687i	$-0.706584\pi$
$157$	−15.1460	−1.20878	−0.604392	+	0.796687i	$0.706584\pi$
$158$	4.74456	0.377457
$159$	12.7446	1.01071
$160$	0	0
$161$	15.7359	1.24017
$162$	7.74456	0.608470
$163$	−24.7446	−1.93814	−0.969072	−	0.246779i	$-0.920628\pi$
$163$	−24.7446	−1.93814	−0.969072	+	0.246779i	$0.920628\pi$
$164$	−10.0974	−0.788471
$165$	0	0
$166$	−8.74456	−0.678710
$167$	17.4891	1.35335	0.676675	−	0.736282i	$-0.263420\pi$
$167$	17.4891	1.35335	0.676675	+	0.736282i	$0.263420\pi$
$168$	−5.98844	−0.462018
$169$	0	0
$170$	0	0
$171$	−11.6819	−0.893339
$172$	−0.644810	−0.0491663
$173$	−18.9051	−1.43733	−0.718663	−	0.695358i	$-0.755246\pi$
$173$	−18.9051	−1.43733	−0.718663	+	0.695358i	$0.755246\pi$
$174$	6.92820	0.525226
$175$	0	0
$176$	−1.58457	−0.119442
$177$	−16.7446	−1.25860
$178$	1.87953	0.140877
$179$	4.88316	0.364984	0.182492	−	0.983207i	$-0.441584\pi$
$179$	4.88316	0.364984	0.182492	+	0.983207i	$0.441584\pi$
$180$	0	0
$181$	3.48913	0.259345	0.129672	−	0.991557i	$-0.458607\pi$
$181$	3.48913	0.259345	0.129672	+	0.991557i	$0.458607\pi$
$182$	0	0
$183$	−17.0256	−1.25857
$184$	−6.63325	−0.489010
$185$	0	0
$186$	8.74456	0.641182
$187$	9.48913	0.693914
$188$	−10.3723	−0.756476
$189$	2.22938	0.162164
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	2.52434	0.182178
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	6.74456	0.484231
$195$	0	0
$196$	−1.37228	−0.0980201
$197$	4.37228	0.311512	0.155756	−	0.987796i	$-0.450219\pi$
$197$	4.37228	0.311512	0.155756	+	0.987796i	$0.450219\pi$
$198$	5.34363	0.379755
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−10.0974	−0.712212
$202$	−14.7446	−1.03742
$203$	−6.51087	−0.456974
$204$	−15.1168	−1.05839
$205$	0	0
$206$	−10.3923	−0.724066
$207$	22.3692	1.55477
$208$	0	0
$209$	5.48913	0.379691
$210$	0	0
$211$	3.11684	0.214572	0.107286	−	0.994228i	$-0.465784\pi$
$211$	3.11684	0.214572	0.107286	+	0.994228i	$0.465784\pi$
$212$	5.04868	0.346744
$213$	31.8614	2.18311
$214$	−13.5615	−0.927042
$215$	0	0
$216$	−0.939764	−0.0639428
$217$	−8.21782	−0.557862
$218$	2.81929	0.190947
$219$	−25.2434	−1.70579
$220$	0	0
$221$	0	0
$222$	23.0140	1.54460
$223$	−15.1168	−1.01230	−0.506149	−	0.862446i	$-0.668932\pi$
$223$	−15.1168	−1.01230	−0.506149	+	0.862446i	$0.668932\pi$
$224$	−2.37228	−0.158505
$225$	0	0
$226$	−3.75906	−0.250049
$227$	5.48913	0.364326	0.182163	−	0.983268i	$-0.441690\pi$
$227$	5.48913	0.364326	0.182163	+	0.983268i	$0.441690\pi$
$228$	−8.74456	−0.579123
$229$	24.8935	1.64501	0.822505	−	0.568758i	$-0.192576\pi$
$229$	24.8935	1.64501	0.822505	+	0.568758i	$0.192576\pi$
$230$	0	0
$231$	−9.48913	−0.624339
$232$	2.74456	0.180189
$233$	−7.86797	−0.515448	−0.257724	−	0.966219i	$-0.582972\pi$
$233$	−7.86797	−0.515448	−0.257724	+	0.966219i	$0.582972\pi$
$234$	0	0
$235$	0	0
$236$	−6.63325	−0.431788
$237$	−11.9769	−0.777982
$238$	14.2063	0.920855
$239$	9.45254	0.611434	0.305717	−	0.952122i	$-0.401104\pi$
$239$	9.45254	0.611434	0.305717	+	0.952122i	$0.401104\pi$
$240$	0	0
$241$	−8.21782	−0.529357	−0.264678	−	0.964337i	$-0.585266\pi$
$241$	−8.21782	−0.529357	−0.264678	+	0.964337i	$0.585266\pi$
$242$	8.48913	0.545702
$243$	−22.3692	−1.43498
$244$	−6.74456	−0.431776
$245$	0	0
$246$	25.4891	1.62513
$247$	0	0
$248$	3.46410	0.219971
$249$	22.0742	1.39890
$250$	0	0
$251$	22.9783	1.45037	0.725187	−	0.688552i	$-0.241753\pi$
$251$	22.9783	1.45037	0.725187	+	0.688552i	$0.241753\pi$
$252$	8.00000	0.503953
$253$	−10.5109	−0.660813
$254$	−3.46410	−0.217357
$255$	0	0
$256$	1.00000	0.0625000
$257$	−14.2063	−0.886162	−0.443081	−	0.896481i	$-0.646115\pi$
$257$	−14.2063	−0.886162	−0.443081	+	0.896481i	$0.646115\pi$
$258$	1.62772	0.101337
$259$	−21.6277	−1.34388
$260$	0	0
$261$	−9.25544	−0.572897
$262$	1.62772	0.100561
$263$	18.0202	1.11117	0.555587	−	0.831458i	$-0.312493\pi$
$263$	18.0202	1.11117	0.555587	+	0.831458i	$0.312493\pi$
$264$	4.00000	0.246183
$265$	0	0
$266$	8.21782	0.503867
$267$	−4.74456	−0.290363
$268$	−4.00000	−0.244339
$269$	−2.74456	−0.167339	−0.0836695	−	0.996494i	$-0.526664\pi$
$269$	−2.74456	−0.167339	−0.0836695	+	0.996494i	$0.526664\pi$
$270$	0	0
$271$	21.4294	1.30174	0.650872	−	0.759187i	$-0.274403\pi$
$271$	21.4294	1.30174	0.650872	+	0.759187i	$0.274403\pi$
$272$	−5.98844	−0.363102
$273$	0	0
$274$	6.00000	0.362473
$275$	0	0
$276$	16.7446	1.00790
$277$	−23.3639	−1.40380	−0.701899	−	0.712277i	$-0.747664\pi$
$277$	−23.3639	−1.40380	−0.701899	+	0.712277i	$0.747664\pi$
$278$	6.37228	0.382184
$279$	−11.6819	−0.699379
$280$	0	0
$281$	1.87953	0.112123	0.0560616	−	0.998427i	$-0.482146\pi$
$281$	1.87953	0.112123	0.0560616	+	0.998427i	$0.482146\pi$
$282$	26.1831	1.55918
$283$	17.3205	1.02960	0.514799	−	0.857311i	$-0.327867\pi$
$283$	17.3205	1.02960	0.514799	+	0.857311i	$0.327867\pi$
$284$	12.6217	0.748959
$285$	0	0
$286$	0	0
$287$	−23.9538	−1.41395
$288$	−3.37228	−0.198714
$289$	18.8614	1.10949
$290$	0	0
$291$	−17.0256	−0.998056
$292$	−10.0000	−0.585206
$293$	−2.13859	−0.124938	−0.0624690	−	0.998047i	$-0.519897\pi$
$293$	−2.13859	−0.124938	−0.0624690	+	0.998047i	$0.519897\pi$
$294$	3.46410	0.202031
$295$	0	0
$296$	9.11684	0.529906
$297$	−1.48913	−0.0864078
$298$	−17.0256	−0.986264
$299$	0	0
$300$	0	0
$301$	−1.52967	−0.0881688
$302$	−7.57301	−0.435778
$303$	37.2203	2.13825
$304$	−3.46410	−0.198680
$305$	0	0
$306$	20.1947	1.15445
$307$	18.2337	1.04065	0.520326	−	0.853968i	$-0.325811\pi$
$307$	18.2337	1.04065	0.520326	+	0.853968i	$0.325811\pi$
$308$	−3.75906	−0.214192
$309$	26.2337	1.49238
$310$	0	0
$311$	−3.25544	−0.184599	−0.0922995	−	0.995731i	$-0.529422\pi$
$311$	−3.25544	−0.184599	−0.0922995	+	0.995731i	$0.529422\pi$
$312$	0	0
$313$	−12.3267	−0.696748	−0.348374	−	0.937356i	$-0.613266\pi$
$313$	−12.3267	−0.696748	−0.348374	+	0.937356i	$0.613266\pi$
$314$	15.1460	0.854740
$315$	0	0
$316$	−4.74456	−0.266903
$317$	−16.9783	−0.953594	−0.476797	−	0.879014i	$-0.658202\pi$
$317$	−16.9783	−0.953594	−0.476797	+	0.879014i	$0.658202\pi$
$318$	−12.7446	−0.714680
$319$	4.34896	0.243495
$320$	0	0
$321$	34.2337	1.91074
$322$	−15.7359	−0.876929
$323$	20.7446	1.15426
$324$	−7.74456	−0.430253
$325$	0	0
$326$	24.7446	1.37047
$327$	−7.11684	−0.393562
$328$	10.0974	0.557533
$329$	−24.6060	−1.35657
$330$	0	0
$331$	−10.3923	−0.571213	−0.285606	−	0.958347i	$-0.592195\pi$
$331$	−10.3923	−0.571213	−0.285606	+	0.958347i	$0.592195\pi$
$332$	8.74456	0.479920
$333$	−30.7446	−1.68479
$334$	−17.4891	−0.956962
$335$	0	0
$336$	5.98844	0.326696
$337$	1.52967	0.0833265	0.0416632	−	0.999132i	$-0.486734\pi$
$337$	1.52967	0.0833265	0.0416632	+	0.999132i	$0.486734\pi$
$338$	0	0
$339$	9.48913	0.515379
$340$	0	0
$341$	5.48913	0.297253
$342$	11.6819	0.631686
$343$	−19.8614	−1.07242
$344$	0.644810	0.0347658
$345$	0	0
$346$	18.9051	1.01634
$347$	−20.8395	−1.11872	−0.559362	−	0.828924i	$-0.688954\pi$
$347$	−20.8395	−1.11872	−0.559362	+	0.828924i	$0.688954\pi$
$348$	−6.92820	−0.371391
$349$	27.4728	1.47058	0.735292	−	0.677751i	$-0.237045\pi$
$349$	27.4728	1.47058	0.735292	+	0.677751i	$0.237045\pi$
$350$	0	0
$351$	0	0
$352$	1.58457	0.0844581
$353$	11.4891	0.611504	0.305752	−	0.952111i	$-0.401092\pi$
$353$	11.4891	0.611504	0.305752	+	0.952111i	$0.401092\pi$
$354$	16.7446	0.889963
$355$	0	0
$356$	−1.87953	−0.0996148
$357$	−35.8614	−1.89799
$358$	−4.88316	−0.258083
$359$	2.87419	0.151694	0.0758471	−	0.997119i	$-0.475834\pi$
$359$	2.87419	0.151694	0.0758471	+	0.997119i	$0.475834\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	−3.48913	−0.183384
$363$	−21.4294	−1.12475
$364$	0	0
$365$	0	0
$366$	17.0256	0.889940
$367$	4.75372	0.248142	0.124071	−	0.992273i	$-0.460405\pi$
$367$	4.75372	0.248142	0.124071	+	0.992273i	$0.460405\pi$
$368$	6.63325	0.345782
$369$	−34.0511	−1.77263
$370$	0	0
$371$	11.9769	0.621809
$372$	−8.74456	−0.453384
$373$	9.50744	0.492277	0.246138	−	0.969235i	$-0.420838\pi$
$373$	9.50744	0.492277	0.246138	+	0.969235i	$0.420838\pi$
$374$	−9.48913	−0.490671
$375$	0	0
$376$	10.3723	0.534910
$377$	0	0
$378$	−2.22938	−0.114667
$379$	−17.3205	−0.889695	−0.444847	−	0.895606i	$-0.646742\pi$
$379$	−17.3205	−0.889695	−0.444847	+	0.895606i	$0.646742\pi$
$380$	0	0
$381$	8.74456	0.447998
$382$	0	0
$383$	−16.8832	−0.862689	−0.431344	−	0.902187i	$-0.641961\pi$
$383$	−16.8832	−0.862689	−0.431344	+	0.902187i	$0.641961\pi$
$384$	−2.52434	−0.128820
$385$	0	0
$386$	−14.0000	−0.712581
$387$	−2.17448	−0.110535
$388$	−6.74456	−0.342403
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−39.7228	−2.00887
$392$	1.37228	0.0693107
$393$	−4.10891	−0.207267
$394$	−4.37228	−0.220272
$395$	0	0
$396$	−5.34363	−0.268527
$397$	−8.51087	−0.427149	−0.213574	−	0.976927i	$-0.568511\pi$
$397$	−8.51087	−0.427149	−0.213574	+	0.976927i	$0.568511\pi$
$398$	−8.00000	−0.401004
$399$	−20.7446	−1.03853
$400$	0	0
$401$	32.1716	1.60657	0.803286	−	0.595593i	$-0.203083\pi$
$401$	32.1716	1.60657	0.803286	+	0.595593i	$0.203083\pi$
$402$	10.0974	0.503610
$403$	0	0
$404$	14.7446	0.733569
$405$	0	0
$406$	6.51087	0.323129
$407$	14.4463	0.716077
$408$	15.1168	0.748395
$409$	−5.63858	−0.278810	−0.139405	−	0.990235i	$-0.544519\pi$
$409$	−5.63858	−0.278810	−0.139405	+	0.990235i	$0.544519\pi$
$410$	0	0
$411$	−15.1460	−0.747098
$412$	10.3923	0.511992
$413$	−15.7359	−0.774315
$414$	−22.3692	−1.09939
$415$	0	0
$416$	0	0
$417$	−16.0858	−0.787725
$418$	−5.48913	−0.268482
$419$	−19.1168	−0.933919	−0.466959	−	0.884279i	$-0.654651\pi$
$419$	−19.1168	−0.933919	−0.466959	+	0.884279i	$0.654651\pi$
$420$	0	0
$421$	−11.0371	−0.537916	−0.268958	−	0.963152i	$-0.586679\pi$
$421$	−11.0371	−0.537916	−0.268958	+	0.963152i	$0.586679\pi$
$422$	−3.11684	−0.151726
$423$	−34.9783	−1.70070
$424$	−5.04868	−0.245185
$425$	0	0
$426$	−31.8614	−1.54369
$427$	−16.0000	−0.774294
$428$	13.5615	0.655518
$429$	0	0
$430$	0	0
$431$	26.4781	1.27540	0.637702	−	0.770283i	$-0.279885\pi$
$431$	26.4781	1.27540	0.637702	+	0.770283i	$0.279885\pi$
$432$	0.939764	0.0452144
$433$	−34.4010	−1.65320	−0.826602	−	0.562786i	$-0.809729\pi$
$433$	−34.4010	−1.65320	−0.826602	+	0.562786i	$0.809729\pi$
$434$	8.21782	0.394468
$435$	0	0
$436$	−2.81929	−0.135020
$437$	−22.9783	−1.09920
$438$	25.2434	1.20618
$439$	22.2337	1.06116	0.530578	−	0.847636i	$-0.321975\pi$
$439$	22.2337	1.06116	0.530578	+	0.847636i	$0.321975\pi$
$440$	0	0
$441$	−4.62772	−0.220368
$442$	0	0
$443$	4.40387	0.209234	0.104617	−	0.994513i	$-0.466638\pi$
$443$	4.40387	0.209234	0.104617	+	0.994513i	$0.466638\pi$
$444$	−23.0140	−1.09220
$445$	0	0
$446$	15.1168	0.715803
$447$	42.9783	2.03280
$448$	2.37228	0.112080
$449$	−7.51811	−0.354802	−0.177401	−	0.984139i	$-0.556769\pi$
$449$	−7.51811	−0.354802	−0.177401	+	0.984139i	$0.556769\pi$
$450$	0	0
$451$	16.0000	0.753411
$452$	3.75906	0.176811
$453$	19.1168	0.898188
$454$	−5.48913	−0.257617
$455$	0	0
$456$	8.74456	0.409502
$457$	24.9783	1.16843	0.584217	−	0.811598i	$-0.301402\pi$
$457$	24.9783	1.16843	0.584217	+	0.811598i	$0.301402\pi$
$458$	−24.8935	−1.16320
$459$	−5.62772	−0.262679
$460$	0	0
$461$	−1.63948	−0.0763580	−0.0381790	−	0.999271i	$-0.512156\pi$
$461$	−1.63948	−0.0763580	−0.0381790	+	0.999271i	$0.512156\pi$
$462$	9.48913	0.441474
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	−2.74456	−0.127413
$465$	0	0
$466$	7.86797	0.364477
$467$	−27.4179	−1.26875	−0.634374	−	0.773027i	$-0.718742\pi$
$467$	−27.4179	−1.26875	−0.634374	+	0.773027i	$0.718742\pi$
$468$	0	0
$469$	−9.48913	−0.438167
$470$	0	0
$471$	−38.2337	−1.76172
$472$	6.63325	0.305320
$473$	1.02175	0.0469801
$474$	11.9769	0.550116
$475$	0	0
$476$	−14.2063	−0.651143
$477$	17.0256	0.779547
$478$	−9.45254	−0.432349
$479$	18.2603	0.834333	0.417167	−	0.908830i	$-0.363023\pi$
$479$	18.2603	0.834333	0.417167	+	0.908830i	$0.363023\pi$
$480$	0	0
$481$	0	0
$482$	8.21782	0.374312
$483$	39.7228	1.80745
$484$	−8.48913	−0.385869
$485$	0	0
$486$	22.3692	1.01469
$487$	1.48913	0.0674787	0.0337394	−	0.999431i	$-0.489258\pi$
$487$	1.48913	0.0674787	0.0337394	+	0.999431i	$0.489258\pi$
$488$	6.74456	0.305312
$489$	−62.4636	−2.82470
$490$	0	0
$491$	25.6277	1.15656	0.578281	−	0.815837i	$-0.303724\pi$
$491$	25.6277	1.15656	0.578281	+	0.815837i	$0.303724\pi$
$492$	−25.4891	−1.14914
$493$	16.4356	0.740224
$494$	0	0
$495$	0	0
$496$	−3.46410	−0.155543
$497$	29.9422	1.34309
$498$	−22.0742	−0.989170
$499$	16.0309	0.717641	0.358821	−	0.933407i	$-0.383179\pi$
$499$	16.0309	0.717641	0.358821	+	0.933407i	$0.383179\pi$
$500$	0	0
$501$	44.1485	1.97241
$502$	−22.9783	−1.02557
$503$	−6.63325	−0.295762	−0.147881	−	0.989005i	$-0.547245\pi$
$503$	−6.63325	−0.295762	−0.147881	+	0.989005i	$0.547245\pi$
$504$	−8.00000	−0.356348
$505$	0	0
$506$	10.5109	0.467265
$507$	0	0
$508$	3.46410	0.153695
$509$	3.16915	0.140470	0.0702350	−	0.997530i	$-0.477625\pi$
$509$	3.16915	0.140470	0.0702350	+	0.997530i	$0.477625\pi$
$510$	0	0
$511$	−23.7228	−1.04944
$512$	−1.00000	−0.0441942
$513$	−3.25544	−0.143731
$514$	14.2063	0.626611
$515$	0	0
$516$	−1.62772	−0.0716563
$517$	16.4356	0.722839
$518$	21.6277	0.950267
$519$	−47.7228	−2.09480
$520$	0	0
$521$	18.6060	0.815142	0.407571	−	0.913173i	$-0.366376\pi$
$521$	18.6060	0.815142	0.407571	+	0.913173i	$0.366376\pi$
$522$	9.25544	0.405099
$523$	10.3923	0.454424	0.227212	−	0.973845i	$-0.427039\pi$
$523$	10.3923	0.454424	0.227212	+	0.973845i	$0.427039\pi$
$524$	−1.62772	−0.0711072
$525$	0	0
$526$	−18.0202	−0.785719
$527$	20.7446	0.903647
$528$	−4.00000	−0.174078
$529$	21.0000	0.913043
$530$	0	0
$531$	−22.3692	−0.970740
$532$	−8.21782	−0.356288
$533$	0	0
$534$	4.74456	0.205317
$535$	0	0
$536$	4.00000	0.172774
$537$	12.3267	0.531938
$538$	2.74456	0.118326
$539$	2.17448	0.0936615
$540$	0	0
$541$	−30.5321	−1.31268	−0.656339	−	0.754466i	$-0.727896\pi$
$541$	−30.5321	−1.31268	−0.656339	+	0.754466i	$0.727896\pi$
$542$	−21.4294	−0.920472
$543$	8.80773	0.377976
$544$	5.98844	0.256752
$545$	0	0
$546$	0	0
$547$	21.4294	0.916256	0.458128	−	0.888886i	$-0.348520\pi$
$547$	21.4294	0.916256	0.458128	+	0.888886i	$0.348520\pi$
$548$	−6.00000	−0.256307
$549$	−22.7446	−0.970714
$550$	0	0
$551$	9.50744	0.405031
$552$	−16.7446	−0.712696
$553$	−11.2554	−0.478630
$554$	23.3639	0.992635
$555$	0	0
$556$	−6.37228	−0.270245
$557$	−1.11684	−0.0473222	−0.0236611	−	0.999720i	$-0.507532\pi$
$557$	−1.11684	−0.0473222	−0.0236611	+	0.999720i	$0.507532\pi$
$558$	11.6819	0.494535
$559$	0	0
$560$	0	0
$561$	23.9538	1.01133
$562$	−1.87953	−0.0792831
$563$	−1.82462	−0.0768988	−0.0384494	−	0.999261i	$-0.512242\pi$
$563$	−1.82462	−0.0768988	−0.0384494	+	0.999261i	$0.512242\pi$
$564$	−26.1831	−1.10251
$565$	0	0
$566$	−17.3205	−0.728035
$567$	−18.3723	−0.771563
$568$	−12.6217	−0.529594
$569$	−16.3723	−0.686362	−0.343181	−	0.939269i	$-0.611504\pi$
$569$	−16.3723	−0.686362	−0.343181	+	0.939269i	$0.611504\pi$
$570$	0	0
$571$	−12.8832	−0.539143	−0.269572	−	0.962980i	$-0.586882\pi$
$571$	−12.8832	−0.539143	−0.269572	+	0.962980i	$0.586882\pi$
$572$	0	0
$573$	0	0
$574$	23.9538	0.999811
$575$	0	0
$576$	3.37228	0.140512
$577$	−20.9783	−0.873336	−0.436668	−	0.899623i	$-0.643842\pi$
$577$	−20.9783	−0.873336	−0.436668	+	0.899623i	$0.643842\pi$
$578$	−18.8614	−0.784531
$579$	35.3407	1.46871
$580$	0	0
$581$	20.7446	0.860629
$582$	17.0256	0.705732
$583$	−8.00000	−0.331326
$584$	10.0000	0.413803
$585$	0	0
$586$	2.13859	0.0883445
$587$	−19.7228	−0.814048	−0.407024	−	0.913418i	$-0.633433\pi$
$587$	−19.7228	−0.814048	−0.407024	+	0.913418i	$0.633433\pi$
$588$	−3.46410	−0.142857
$589$	12.0000	0.494451
$590$	0	0
$591$	11.0371	0.454006
$592$	−9.11684	−0.374700
$593$	−32.2337	−1.32368	−0.661839	−	0.749646i	$-0.730224\pi$
$593$	−32.2337	−1.32368	−0.661839	+	0.749646i	$0.730224\pi$
$594$	1.48913	0.0610996
$595$	0	0
$596$	17.0256	0.697394
$597$	20.1947	0.826514
$598$	0	0
$599$	−31.7228	−1.29616	−0.648080	−	0.761573i	$-0.724428\pi$
$599$	−31.7228	−1.29616	−0.648080	+	0.761573i	$0.724428\pi$
$600$	0	0
$601$	−32.3723	−1.32049	−0.660246	−	0.751049i	$-0.729548\pi$
$601$	−32.3723	−1.32049	−0.660246	+	0.751049i	$0.729548\pi$
$602$	1.52967	0.0623447
$603$	−13.4891	−0.549320
$604$	7.57301	0.308142
$605$	0	0
$606$	−37.2203	−1.51197
$607$	−3.46410	−0.140604	−0.0703018	−	0.997526i	$-0.522396\pi$
$607$	−3.46410	−0.140604	−0.0703018	+	0.997526i	$0.522396\pi$
$608$	3.46410	0.140488
$609$	−16.4356	−0.666006
$610$	0	0
$611$	0	0
$612$	−20.1947	−0.816322
$613$	−19.4891	−0.787158	−0.393579	−	0.919291i	$-0.628763\pi$
$613$	−19.4891	−0.787158	−0.393579	+	0.919291i	$0.628763\pi$
$614$	−18.2337	−0.735852
$615$	0	0
$616$	3.75906	0.151457
$617$	−14.7446	−0.593594	−0.296797	−	0.954941i	$-0.595918\pi$
$617$	−14.7446	−0.593594	−0.296797	+	0.954941i	$0.595918\pi$
$618$	−26.2337	−1.05527
$619$	9.10268	0.365868	0.182934	−	0.983125i	$-0.441441\pi$
$619$	9.10268	0.365868	0.182934	+	0.983125i	$0.441441\pi$
$620$	0	0
$621$	6.23369	0.250149
$622$	3.25544	0.130531
$623$	−4.45877	−0.178637
$624$	0	0
$625$	0	0
$626$	12.3267	0.492675
$627$	13.8564	0.553372
$628$	−15.1460	−0.604392
$629$	54.5957	2.17687
$630$	0	0
$631$	40.4443	1.61006	0.805031	−	0.593232i	$-0.202148\pi$
$631$	40.4443	1.61006	0.805031	+	0.593232i	$0.202148\pi$
$632$	4.74456	0.188729
$633$	7.86797	0.312724
$634$	16.9783	0.674292
$635$	0	0
$636$	12.7446	0.505355
$637$	0	0
$638$	−4.34896	−0.172177
$639$	42.5639	1.68380
$640$	0	0
$641$	16.9783	0.670601	0.335300	−	0.942111i	$-0.391162\pi$
$641$	16.9783	0.670601	0.335300	+	0.942111i	$0.391162\pi$
$642$	−34.2337	−1.35110
$643$	−37.4891	−1.47843	−0.739213	−	0.673471i	$-0.764803\pi$
$643$	−37.4891	−1.47843	−0.739213	+	0.673471i	$0.764803\pi$
$644$	15.7359	0.620083
$645$	0	0
$646$	−20.7446	−0.816184
$647$	21.0796	0.828723	0.414362	−	0.910112i	$-0.364005\pi$
$647$	21.0796	0.828723	0.414362	+	0.910112i	$0.364005\pi$
$648$	7.74456	0.304235
$649$	10.5109	0.412588
$650$	0	0
$651$	−20.7446	−0.813044
$652$	−24.7446	−0.969072
$653$	−17.0256	−0.666261	−0.333131	−	0.942881i	$-0.608105\pi$
$653$	−17.0256	−0.666261	−0.333131	+	0.942881i	$0.608105\pi$
$654$	7.11684	0.278291
$655$	0	0
$656$	−10.0974	−0.394235
$657$	−33.7228	−1.31565
$658$	24.6060	0.959241
$659$	40.4674	1.57639	0.788193	−	0.615429i	$-0.211017\pi$
$659$	40.4674	1.57639	0.788193	+	0.615429i	$0.211017\pi$
$660$	0	0
$661$	9.50744	0.369797	0.184898	−	0.982758i	$-0.440804\pi$
$661$	9.50744	0.369797	0.184898	+	0.982758i	$0.440804\pi$
$662$	10.3923	0.403908
$663$	0	0
$664$	−8.74456	−0.339355
$665$	0	0
$666$	30.7446	1.19133
$667$	−18.2054	−0.704915
$668$	17.4891	0.676675
$669$	−38.1600	−1.47535
$670$	0	0
$671$	10.6873	0.412577
$672$	−5.98844	−0.231009
$673$	9.74749	0.375738	0.187869	−	0.982194i	$-0.439842\pi$
$673$	9.74749	0.375738	0.187869	+	0.982194i	$0.439842\pi$
$674$	−1.52967	−0.0589207
$675$	0	0
$676$	0	0
$677$	39.0998	1.50273	0.751363	−	0.659889i	$-0.229397\pi$
$677$	39.0998	1.50273	0.751363	+	0.659889i	$0.229397\pi$
$678$	−9.48913	−0.364428
$679$	−16.0000	−0.614024
$680$	0	0
$681$	13.8564	0.530979
$682$	−5.48913	−0.210189
$683$	8.74456	0.334601	0.167301	−	0.985906i	$-0.446495\pi$
$683$	8.74456	0.334601	0.167301	+	0.985906i	$0.446495\pi$
$684$	−11.6819	−0.446670
$685$	0	0
$686$	19.8614	0.758312
$687$	62.8397	2.39748
$688$	−0.644810	−0.0245832
$689$	0	0
$690$	0	0
$691$	39.3947	1.49865	0.749323	−	0.662204i	$-0.230379\pi$
$691$	39.3947	1.49865	0.749323	+	0.662204i	$0.230379\pi$
$692$	−18.9051	−0.718663
$693$	−12.6766	−0.481544
$694$	20.8395	0.791057
$695$	0	0
$696$	6.92820	0.262613
$697$	60.4674	2.29037
$698$	−27.4728	−1.03986
$699$	−19.8614	−0.751227
$700$	0	0
$701$	−16.9783	−0.641260	−0.320630	−	0.947205i	$-0.603895\pi$
$701$	−16.9783	−0.641260	−0.320630	+	0.947205i	$0.603895\pi$
$702$	0	0
$703$	31.5817	1.19113
$704$	−1.58457	−0.0597209
$705$	0	0
$706$	−11.4891	−0.432399
$707$	34.9783	1.31549
$708$	−16.7446	−0.629299
$709$	−20.7846	−0.780582	−0.390291	−	0.920691i	$-0.627626\pi$
$709$	−20.7846	−0.780582	−0.390291	+	0.920691i	$0.627626\pi$
$710$	0	0
$711$	−16.0000	−0.600047
$712$	1.87953	0.0704383
$713$	−22.9783	−0.860542
$714$	35.8614	1.34208
$715$	0	0
$716$	4.88316	0.182492
$717$	23.8614	0.891121
$718$	−2.87419	−0.107264
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	24.6535	0.918143
$722$	7.00000	0.260513
$723$	−20.7446	−0.771499
$724$	3.48913	0.129672
$725$	0	0
$726$	21.4294	0.795320
$727$	−2.17448	−0.0806470	−0.0403235	−	0.999187i	$-0.512839\pi$
$727$	−2.17448	−0.0806470	−0.0403235	+	0.999187i	$0.512839\pi$
$728$	0	0
$729$	−33.2337	−1.23088
$730$	0	0
$731$	3.86141	0.142819
$732$	−17.0256	−0.629283
$733$	16.0951	0.594486	0.297243	−	0.954802i	$-0.403933\pi$
$733$	16.0951	0.594486	0.297243	+	0.954802i	$0.403933\pi$
$734$	−4.75372	−0.175463
$735$	0	0
$736$	−6.63325	−0.244505
$737$	6.33830	0.233474
$738$	34.0511	1.25344
$739$	28.1176	1.03432	0.517161	−	0.855888i	$-0.326989\pi$
$739$	28.1176	1.03432	0.517161	+	0.855888i	$0.326989\pi$
$740$	0	0
$741$	0	0
$742$	−11.9769	−0.439685
$743$	−7.11684	−0.261092	−0.130546	−	0.991442i	$-0.541673\pi$
$743$	−7.11684	−0.261092	−0.130546	+	0.991442i	$0.541673\pi$
$744$	8.74456	0.320591
$745$	0	0
$746$	−9.50744	−0.348092
$747$	29.4891	1.07895
$748$	9.48913	0.346957
$749$	32.1716	1.17552
$750$	0	0
$751$	−25.4891	−0.930111	−0.465056	−	0.885281i	$-0.653966\pi$
$751$	−25.4891	−0.930111	−0.465056	+	0.885281i	$0.653966\pi$
$752$	−10.3723	−0.378238
$753$	58.0049	2.11381
$754$	0	0
$755$	0	0
$756$	2.22938	0.0810819
$757$	−45.4381	−1.65148	−0.825738	−	0.564055i	$-0.809241\pi$
$757$	−45.4381	−1.65148	−0.825738	+	0.564055i	$0.809241\pi$
$758$	17.3205	0.629109
$759$	−26.5330	−0.963087
$760$	0	0
$761$	1.87953	0.0681328	0.0340664	−	0.999420i	$-0.489154\pi$
$761$	1.87953	0.0681328	0.0340664	+	0.999420i	$0.489154\pi$
$762$	−8.74456	−0.316782
$763$	−6.68815	−0.242127
$764$	0	0
$765$	0	0
$766$	16.8832	0.610013
$767$	0	0
$768$	2.52434	0.0910892
$769$	−16.4356	−0.592685	−0.296342	−	0.955082i	$-0.595767\pi$
$769$	−16.4356	−0.592685	−0.296342	+	0.955082i	$0.595767\pi$
$770$	0	0
$771$	−35.8614	−1.29152
$772$	14.0000	0.503871
$773$	30.6060	1.10082	0.550410	−	0.834894i	$-0.314471\pi$
$773$	30.6060	1.10082	0.550410	+	0.834894i	$0.314471\pi$
$774$	2.17448	0.0781601
$775$	0	0
$776$	6.74456	0.242116
$777$	−54.5957	−1.95861
$778$	−6.00000	−0.215110
$779$	34.9783	1.25323
$780$	0	0
$781$	−20.0000	−0.715656
$782$	39.7228	1.42048
$783$	−2.57924	−0.0921745
$784$	−1.37228	−0.0490100
$785$	0	0
$786$	4.10891	0.146560
$787$	−44.0000	−1.56843	−0.784215	−	0.620489i	$-0.786934\pi$
$787$	−44.0000	−1.56843	−0.784215	+	0.620489i	$0.786934\pi$
$788$	4.37228	0.155756
$789$	45.4891	1.61946
$790$	0	0
$791$	8.91754	0.317071
$792$	5.34363	0.189878
$793$	0	0
$794$	8.51087	0.302040
$795$	0	0
$796$	8.00000	0.283552
$797$	−25.2434	−0.894166	−0.447083	−	0.894492i	$-0.647537\pi$
$797$	−25.2434	−0.894166	−0.447083	+	0.894492i	$0.647537\pi$
$798$	20.7446	0.734350
$799$	62.1138	2.19743
$800$	0	0
$801$	−6.33830	−0.223953
$802$	−32.1716	−1.13602
$803$	15.8457	0.559184
$804$	−10.0974	−0.356106
$805$	0	0
$806$	0	0
$807$	−6.92820	−0.243884
$808$	−14.7446	−0.518712
$809$	−39.3505	−1.38349	−0.691746	−	0.722141i	$-0.743158\pi$
$809$	−39.3505	−1.38349	−0.691746	+	0.722141i	$0.743158\pi$
$810$	0	0
$811$	41.9740	1.47391	0.736953	−	0.675944i	$-0.236264\pi$
$811$	41.9740	1.47391	0.736953	+	0.675944i	$0.236264\pi$
$812$	−6.51087	−0.228487
$813$	54.0951	1.89720
$814$	−14.4463	−0.506343
$815$	0	0
$816$	−15.1168	−0.529195
$817$	2.23369	0.0781468
$818$	5.63858	0.197148
$819$	0	0
$820$	0	0
$821$	25.5932	0.893210	0.446605	−	0.894731i	$-0.352633\pi$
$821$	25.5932	0.893210	0.446605	+	0.894731i	$0.352633\pi$
$822$	15.1460	0.528278
$823$	54.5408	1.90117	0.950586	−	0.310462i	$-0.100484\pi$
$823$	54.5408	1.90117	0.950586	+	0.310462i	$0.100484\pi$
$824$	−10.3923	−0.362033
$825$	0	0
$826$	15.7359	0.547523
$827$	−15.2554	−0.530484	−0.265242	−	0.964182i	$-0.585452\pi$
$827$	−15.2554	−0.530484	−0.265242	+	0.964182i	$0.585452\pi$
$828$	22.3692	0.777383
$829$	−48.2337	−1.67523	−0.837613	−	0.546265i	$-0.816049\pi$
$829$	−48.2337	−1.67523	−0.837613	+	0.546265i	$0.816049\pi$
$830$	0	0
$831$	−58.9783	−2.04593
$832$	0	0
$833$	8.21782	0.284731
$834$	16.0858	0.557005
$835$	0	0
$836$	5.48913	0.189845
$837$	−3.25544	−0.112524
$838$	19.1168	0.660380
$839$	−13.5615	−0.468193	−0.234097	−	0.972213i	$-0.575213\pi$
$839$	−13.5615	−0.468193	−0.234097	+	0.972213i	$0.575213\pi$
$840$	0	0
$841$	−21.4674	−0.740254
$842$	11.0371	0.380364
$843$	4.74456	0.163411
$844$	3.11684	0.107286
$845$	0	0
$846$	34.9783	1.20258
$847$	−20.1386	−0.691970
$848$	5.04868	0.173372
$849$	43.7228	1.50056
$850$	0	0
$851$	−60.4743	−2.07303
$852$	31.8614	1.09155
$853$	−5.11684	−0.175197	−0.0875987	−	0.996156i	$-0.527919\pi$
$853$	−5.11684	−0.175197	−0.0875987	+	0.996156i	$0.527919\pi$
$854$	16.0000	0.547509
$855$	0	0
$856$	−13.5615	−0.463521
$857$	22.7739	0.777943	0.388972	−	0.921250i	$-0.372830\pi$
$857$	22.7739	0.777943	0.388972	+	0.921250i	$0.372830\pi$
$858$	0	0
$859$	32.4674	1.10777	0.553886	−	0.832592i	$-0.313144\pi$
$859$	32.4674	1.10777	0.553886	+	0.832592i	$0.313144\pi$
$860$	0	0
$861$	−60.4674	−2.06072
$862$	−26.4781	−0.901848
$863$	−3.86141	−0.131444	−0.0657219	−	0.997838i	$-0.520935\pi$
$863$	−3.86141	−0.131444	−0.0657219	+	0.997838i	$0.520935\pi$
$864$	−0.939764	−0.0319714
$865$	0	0
$866$	34.4010	1.16899
$867$	47.6126	1.61701
$868$	−8.21782	−0.278931
$869$	7.51811	0.255034
$870$	0	0
$871$	0	0
$872$	2.81929	0.0954733
$873$	−22.7446	−0.769787
$874$	22.9783	0.777251
$875$	0	0
$876$	−25.2434	−0.852895
$877$	47.3505	1.59891	0.799457	−	0.600723i	$-0.205121\pi$
$877$	47.3505	1.59891	0.799457	+	0.600723i	$0.205121\pi$
$878$	−22.2337	−0.750351
$879$	−5.39853	−0.182088
$880$	0	0
$881$	54.6060	1.83972	0.919861	−	0.392245i	$-0.128301\pi$
$881$	54.6060	1.83972	0.919861	+	0.392245i	$0.128301\pi$
$882$	4.62772	0.155823
$883$	−58.6497	−1.97372	−0.986859	−	0.161581i	$-0.948341\pi$
$883$	−58.6497	−1.97372	−0.986859	+	0.161581i	$0.948341\pi$
$884$	0	0
$885$	0	0
$886$	−4.40387	−0.147951
$887$	−45.7330	−1.53556	−0.767782	−	0.640711i	$-0.778640\pi$
$887$	−45.7330	−1.53556	−0.767782	+	0.640711i	$0.778640\pi$
$888$	23.0140	0.772299
$889$	8.21782	0.275617
$890$	0	0
$891$	12.2718	0.411122
$892$	−15.1168	−0.506149
$893$	35.9306	1.20237
$894$	−42.9783	−1.43741
$895$	0	0
$896$	−2.37228	−0.0792524
$897$	0	0
$898$	7.51811	0.250883
$899$	9.50744	0.317091
$900$	0	0
$901$	−30.2337	−1.00723
$902$	−16.0000	−0.532742
$903$	−3.86141	−0.128500
$904$	−3.75906	−0.125024
$905$	0	0
$906$	−19.1168	−0.635115
$907$	0.644810	0.0214106	0.0107053	−	0.999943i	$-0.496592\pi$
$907$	0.644810	0.0214106	0.0107053	+	0.999943i	$0.496592\pi$
$908$	5.48913	0.182163
$909$	49.7228	1.64920
$910$	0	0
$911$	34.9783	1.15888	0.579441	−	0.815014i	$-0.303271\pi$
$911$	34.9783	1.15888	0.579441	+	0.815014i	$0.303271\pi$
$912$	−8.74456	−0.289561
$913$	−13.8564	−0.458580
$914$	−24.9783	−0.826207
$915$	0	0
$916$	24.8935	0.822505
$917$	−3.86141	−0.127515
$918$	5.62772	0.185742
$919$	−42.9783	−1.41772	−0.708861	−	0.705348i	$-0.750791\pi$
$919$	−42.9783	−1.41772	−0.708861	+	0.705348i	$0.750791\pi$
$920$	0	0
$921$	46.0280	1.51667
$922$	1.63948	0.0539933
$923$	0	0
$924$	−9.48913	−0.312169
$925$	0	0
$926$	16.0000	0.525793
$927$	35.0458	1.15105
$928$	2.74456	0.0900947
$929$	−47.9075	−1.57179	−0.785897	−	0.618357i	$-0.787799\pi$
$929$	−47.9075	−1.57179	−0.785897	+	0.618357i	$0.787799\pi$
$930$	0	0
$931$	4.75372	0.155797
$932$	−7.86797	−0.257724
$933$	−8.21782	−0.269039
$934$	27.4179	0.897140
$935$	0	0
$936$	0	0
$937$	30.2921	0.989598	0.494799	−	0.869007i	$-0.335242\pi$
$937$	30.2921	0.989598	0.494799	+	0.869007i	$0.335242\pi$
$938$	9.48913	0.309831
$939$	−31.1168	−1.01546
$940$	0	0
$941$	−40.6295	−1.32448	−0.662241	−	0.749291i	$-0.730395\pi$
$941$	−40.6295	−1.32448	−0.662241	+	0.749291i	$0.730395\pi$
$942$	38.2337	1.24572
$943$	−66.9783	−2.18111
$944$	−6.63325	−0.215894
$945$	0	0
$946$	−1.02175	−0.0332199
$947$	29.4891	0.958268	0.479134	−	0.877742i	$-0.340951\pi$
$947$	29.4891	0.958268	0.479134	+	0.877742i	$0.340951\pi$
$948$	−11.9769	−0.388991
$949$	0	0
$950$	0	0
$951$	−42.8588	−1.38979
$952$	14.2063	0.460428
$953$	−16.0858	−0.521070	−0.260535	−	0.965464i	$-0.583899\pi$
$953$	−16.0858	−0.521070	−0.260535	+	0.965464i	$0.583899\pi$
$954$	−17.0256	−0.551223
$955$	0	0
$956$	9.45254	0.305717
$957$	10.9783	0.354876
$958$	−18.2603	−0.589963
$959$	−14.2337	−0.459630
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	45.7330	1.47373
$964$	−8.21782	−0.264678
$965$	0	0
$966$	−39.7228	−1.27806
$967$	5.35053	0.172062	0.0860308	−	0.996292i	$-0.472582\pi$
$967$	5.35053	0.172062	0.0860308	+	0.996292i	$0.472582\pi$
$968$	8.48913	0.272851
$969$	52.3663	1.68225
$970$	0	0
$971$	−12.6060	−0.404545	−0.202272	−	0.979329i	$-0.564833\pi$
$971$	−12.6060	−0.404545	−0.202272	+	0.979329i	$0.564833\pi$
$972$	−22.3692	−0.717492
$973$	−15.1168	−0.484624
$974$	−1.48913	−0.0477147
$975$	0	0
$976$	−6.74456	−0.215888
$977$	−25.7228	−0.822946	−0.411473	−	0.911422i	$-0.634986\pi$
$977$	−25.7228	−0.822946	−0.411473	+	0.911422i	$0.634986\pi$
$978$	62.4636	1.99737
$979$	2.97825	0.0951853
$980$	0	0
$981$	−9.50744	−0.303549
$982$	−25.6277	−0.817813
$983$	42.0951	1.34263	0.671313	−	0.741174i	$-0.265731\pi$
$983$	42.0951	1.34263	0.671313	+	0.741174i	$0.265731\pi$
$984$	25.4891	0.812564
$985$	0	0
$986$	−16.4356	−0.523418
$987$	−62.1138	−1.97710
$988$	0	0
$989$	−4.27719	−0.136007
$990$	0	0
$991$	−1.76631	−0.0561088	−0.0280544	−	0.999606i	$-0.508931\pi$
$991$	−1.76631	−0.0561088	−0.0280544	+	0.999606i	$0.508931\pi$
$992$	3.46410	0.109985
$993$	−26.2337	−0.832501
$994$	−29.9422	−0.949709
$995$	0	0
$996$	22.0742	0.699449
$997$	−4.34896	−0.137733	−0.0688665	−	0.997626i	$-0.521938\pi$
$997$	−4.34896	−0.137733	−0.0688665	+	0.997626i	$0.521938\pi$
$998$	−16.0309	−0.507449
$999$	−8.56768	−0.271069

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8450.2.a.cj.1.4		4
5.2	odd	4		1690.2.b.d.339.1		8
5.3	odd	4		1690.2.b.d.339.8		8
5.4	even	2		8450.2.a.cn.1.1		4
13.5	odd	4		650.2.d.e.51.8		8
13.8	odd	4		650.2.d.e.51.4		8
13.12	even	2		8450.2.a.cn.1.4		4
65.8	even	4		130.2.c.a.129.4	yes	4
65.12	odd	4		1690.2.b.d.339.5		8
65.18	even	4		130.2.c.b.129.4	yes	4
65.34	odd	4		650.2.d.e.51.5		8
65.38	odd	4		1690.2.b.d.339.4		8
65.44	odd	4		650.2.d.e.51.1		8
65.47	even	4		130.2.c.b.129.1	yes	4
65.57	even	4		130.2.c.a.129.1	✓	4
65.64	even	2	inner	8450.2.a.cj.1.1		4
195.8	odd	4		1170.2.f.b.649.1		4
195.47	odd	4		1170.2.f.a.649.3		4
195.83	odd	4		1170.2.f.a.649.4		4
195.122	odd	4		1170.2.f.b.649.2		4
260.47	odd	4		1040.2.f.c.129.4		4
260.83	odd	4		1040.2.f.c.129.1		4
260.187	odd	4		1040.2.f.d.129.4		4
260.203	odd	4		1040.2.f.d.129.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.c.a.129.1	✓	4	65.57	even	4
130.2.c.a.129.4	yes	4	65.8	even	4
130.2.c.b.129.1	yes	4	65.47	even	4
130.2.c.b.129.4	yes	4	65.18	even	4
650.2.d.e.51.1		8	65.44	odd	4
650.2.d.e.51.4		8	13.8	odd	4
650.2.d.e.51.5		8	65.34	odd	4
650.2.d.e.51.8		8	13.5	odd	4
1040.2.f.c.129.1		4	260.83	odd	4
1040.2.f.c.129.4		4	260.47	odd	4
1040.2.f.d.129.1		4	260.203	odd	4
1040.2.f.d.129.4		4	260.187	odd	4
1170.2.f.a.649.3		4	195.47	odd	4
1170.2.f.a.649.4		4	195.83	odd	4
1170.2.f.b.649.1		4	195.8	odd	4
1170.2.f.b.649.2		4	195.122	odd	4
1690.2.b.d.339.1		8	5.2	odd	4
1690.2.b.d.339.4		8	65.38	odd	4
1690.2.b.d.339.5		8	65.12	odd	4
1690.2.b.d.339.8		8	5.3	odd	4
8450.2.a.cj.1.1		4	65.64	even	2	inner
8450.2.a.cj.1.4		4	1.1	even	1	trivial
8450.2.a.cn.1.1		4	5.4	even	2
8450.2.a.cn.1.4		4	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8450 = 2 \cdot 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.52434 q^{3} +1.00000 q^{4} -2.52434 q^{6} +2.37228 q^{7} -1.00000 q^{8} +3.37228 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.52434 q^{3} +1.00000 q^{4} -2.52434 q^{6} +2.37228 q^{7} -1.00000 q^{8} +3.37228 q^{9} -1.58457 q^{11} +2.52434 q^{12} -2.37228 q^{14} +1.00000 q^{16} -5.98844 q^{17} -3.37228 q^{18} -3.46410 q^{19} +5.98844 q^{21} +1.58457 q^{22} +6.63325 q^{23} -2.52434 q^{24} +0.939764 q^{27} +2.37228 q^{28} -2.74456 q^{29} -3.46410 q^{31} -1.00000 q^{32} -4.00000 q^{33} +5.98844 q^{34} +3.37228 q^{36} -9.11684 q^{37} +3.46410 q^{38} -10.0974 q^{41} -5.98844 q^{42} -0.644810 q^{43} -1.58457 q^{44} -6.63325 q^{46} -10.3723 q^{47} +2.52434 q^{48} -1.37228 q^{49} -15.1168 q^{51} +5.04868 q^{53} -0.939764 q^{54} -2.37228 q^{56} -8.74456 q^{57} +2.74456 q^{58} -6.63325 q^{59} -6.74456 q^{61} +3.46410 q^{62} +8.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} -4.00000 q^{67} -5.98844 q^{68} +16.7446 q^{69} +12.6217 q^{71} -3.37228 q^{72} -10.0000 q^{73} +9.11684 q^{74} -3.46410 q^{76} -3.75906 q^{77} -4.74456 q^{79} -7.74456 q^{81} +10.0974 q^{82} +8.74456 q^{83} +5.98844 q^{84} +0.644810 q^{86} -6.92820 q^{87} +1.58457 q^{88} -1.87953 q^{89} +6.63325 q^{92} -8.74456 q^{93} +10.3723 q^{94} -2.52434 q^{96} -6.74456 q^{97} +1.37228 q^{98} -5.34363 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 2 q^{7} - 4 q^{8} + 2 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 - 2 * q^7 - 4 * q^8 + 2 * q^9	\( 4 q - 4 q^{2} + 4 q^{4} - 2 q^{7} - 4 q^{8} + 2 q^{9} + 2 q^{14} + 4 q^{16} - 2 q^{18} - 2 q^{28} + 12 q^{29} - 4 q^{32} - 16 q^{33} + 2 q^{36} - 2 q^{37} - 30 q^{47} + 6 q^{49} - 26 q^{51} + 2 q^{56} - 12 q^{57} - 12 q^{58} - 4 q^{61} + 32 q^{63} + 4 q^{64} + 16 q^{66} - 16 q^{67} + 44 q^{69} - 2 q^{72} - 40 q^{73} + 2 q^{74} + 4 q^{79} - 8 q^{81} + 12 q^{83} - 12 q^{93} + 30 q^{94} - 4 q^{97} - 6 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 2 * q^7 - 4 * q^8 + 2 * q^9 + 2 * q^14 + 4 * q^16 - 2 * q^18 - 2 * q^28 + 12 * q^29 - 4 * q^32 - 16 * q^33 + 2 * q^36 - 2 * q^37 - 30 * q^47 + 6 * q^49 - 26 * q^51 + 2 * q^56 - 12 * q^57 - 12 * q^58 - 4 * q^61 + 32 * q^63 + 4 * q^64 + 16 * q^66 - 16 * q^67 + 44 * q^69 - 2 * q^72 - 40 * q^73 + 2 * q^74 + 4 * q^79 - 8 * q^81 + 12 * q^83 - 12 * q^93 + 30 * q^94 - 4 * q^97 - 6 * q^98

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