LMFDB - Embedded newform 3042.2.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, 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("3042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.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:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1014)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	3042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.13706 q^{5} +0.0489173 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +2.13706 q^{5} +0.0489173 q^{7} +1.00000 q^{8} +2.13706 q^{10} +6.29590 q^{11} +0.0489173 q^{14} +1.00000 q^{16} +2.89008 q^{17} +7.20775 q^{19} +2.13706 q^{20} +6.29590 q^{22} -2.71379 q^{23} -0.432960 q^{25} +0.0489173 q^{28} -4.91185 q^{29} -9.00969 q^{31} +1.00000 q^{32} +2.89008 q^{34} +0.104539 q^{35} +0.176292 q^{37} +7.20775 q^{38} +2.13706 q^{40} -8.59179 q^{41} +6.71379 q^{43} +6.29590 q^{44} -2.71379 q^{46} +7.20775 q^{47} -6.99761 q^{49} -0.432960 q^{50} -9.34481 q^{53} +13.4547 q^{55} +0.0489173 q^{56} -4.91185 q^{58} -4.26875 q^{59} +7.10992 q^{61} -9.00969 q^{62} +1.00000 q^{64} -5.38404 q^{67} +2.89008 q^{68} +0.104539 q^{70} -8.71379 q^{71} +14.9487 q^{73} +0.176292 q^{74} +7.20775 q^{76} +0.307979 q^{77} +13.8291 q^{79} +2.13706 q^{80} -8.59179 q^{82} -11.1347 q^{83} +6.17629 q^{85} +6.71379 q^{86} +6.29590 q^{88} +3.92154 q^{89} -2.71379 q^{92} +7.20775 q^{94} +15.4034 q^{95} +2.47889 q^{97} -6.99761 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + q^{5} - 9 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + q^5 - 9 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + q^{5} - 9 q^{7} + 3 q^{8} + q^{10} + 5 q^{11} - 9 q^{14} + 3 q^{16} + 8 q^{17} + 4 q^{19} + q^{20} + 5 q^{22} + 18 q^{25} - 9 q^{28} - 11 q^{29} - 5 q^{31} + 3 q^{32} + 8 q^{34} + 4 q^{35} + 8 q^{37} + 4 q^{38} + q^{40} + 2 q^{41} + 12 q^{43} + 5 q^{44} + 4 q^{47} + 20 q^{49} + 18 q^{50} - 5 q^{53} + 18 q^{55} - 9 q^{56} - 11 q^{58} - 5 q^{59} + 22 q^{61} - 5 q^{62} + 3 q^{64} - 6 q^{67} + 8 q^{68} + 4 q^{70} - 18 q^{71} + 13 q^{73} + 8 q^{74} + 4 q^{76} + 6 q^{77} + 31 q^{79} + q^{80} + 2 q^{82} + 13 q^{83} + 26 q^{85} + 12 q^{86} + 5 q^{88} - 14 q^{89} + 4 q^{94} - 8 q^{95} + 23 q^{97} + 20 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + q^5 - 9 * q^7 + 3 * q^8 + q^10 + 5 * q^11 - 9 * q^14 + 3 * q^16 + 8 * q^17 + 4 * q^19 + q^20 + 5 * q^22 + 18 * q^25 - 9 * q^28 - 11 * q^29 - 5 * q^31 + 3 * q^32 + 8 * q^34 + 4 * q^35 + 8 * q^37 + 4 * q^38 + q^40 + 2 * q^41 + 12 * q^43 + 5 * q^44 + 4 * q^47 + 20 * q^49 + 18 * q^50 - 5 * q^53 + 18 * q^55 - 9 * q^56 - 11 * q^58 - 5 * q^59 + 22 * q^61 - 5 * q^62 + 3 * q^64 - 6 * q^67 + 8 * q^68 + 4 * q^70 - 18 * q^71 + 13 * q^73 + 8 * q^74 + 4 * q^76 + 6 * q^77 + 31 * q^79 + q^80 + 2 * q^82 + 13 * q^83 + 26 * q^85 + 12 * q^86 + 5 * q^88 - 14 * q^89 + 4 * q^94 - 8 * q^95 + 23 * q^97 + 20 * 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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.13706	0.955724	0.477862	−	0.878435i	$-0.341412\pi$
$5$	2.13706	0.955724	0.477862	+	0.878435i	$0.341412\pi$
$6$	0	0
$7$	0.0489173	0.0184890	0.00924451	−	0.999957i	$-0.497057\pi$
$7$	0.0489173	0.0184890	0.00924451	+	0.999957i	$0.497057\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.13706	0.675799
$11$	6.29590	1.89828	0.949142	−	0.314848i	$-0.101953\pi$
$11$	6.29590	1.89828	0.949142	+	0.314848i	$0.101953\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0.0489173	0.0130737
$15$	0	0
$16$	1.00000	0.250000
$17$	2.89008	0.700948	0.350474	−	0.936572i	$-0.386020\pi$
$17$	2.89008	0.700948	0.350474	+	0.936572i	$0.386020\pi$
$18$	0	0
$19$	7.20775	1.65357	0.826786	−	0.562517i	$-0.190167\pi$
$19$	7.20775	1.65357	0.826786	+	0.562517i	$0.190167\pi$
$20$	2.13706	0.477862
$21$	0	0
$22$	6.29590	1.34229
$23$	−2.71379	−0.565865	−0.282932	−	0.959140i	$-0.591307\pi$
$23$	−2.71379	−0.565865	−0.282932	+	0.959140i	$0.591307\pi$
$24$	0	0
$25$	−0.432960	−0.0865921
$26$	0	0
$27$	0	0
$28$	0.0489173	0.00924451
$29$	−4.91185	−0.912108	−0.456054	−	0.889952i	$-0.650738\pi$
$29$	−4.91185	−0.912108	−0.456054	+	0.889952i	$0.650738\pi$
$30$	0	0
$31$	−9.00969	−1.61819	−0.809094	−	0.587679i	$-0.800042\pi$
$31$	−9.00969	−1.61819	−0.809094	+	0.587679i	$0.800042\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.89008	0.495645
$35$	0.104539	0.0176704
$36$	0	0
$37$	0.176292	0.0289822	0.0144911	−	0.999895i	$-0.495387\pi$
$37$	0.176292	0.0289822	0.0144911	+	0.999895i	$0.495387\pi$
$38$	7.20775	1.16925
$39$	0	0
$40$	2.13706	0.337899
$41$	−8.59179	−1.34181	−0.670906	−	0.741542i	$-0.734095\pi$
$41$	−8.59179	−1.34181	−0.670906	+	0.741542i	$0.734095\pi$
$42$	0	0
$43$	6.71379	1.02384	0.511922	−	0.859032i	$-0.328934\pi$
$43$	6.71379	1.02384	0.511922	+	0.859032i	$0.328934\pi$
$44$	6.29590	0.949142
$45$	0	0
$46$	−2.71379	−0.400127
$47$	7.20775	1.05136	0.525679	−	0.850683i	$-0.323811\pi$
$47$	7.20775	1.05136	0.525679	+	0.850683i	$0.323811\pi$
$48$	0	0
$49$	−6.99761	−0.999658
$50$	−0.432960	−0.0612298
$51$	0	0
$52$	0	0
$53$	−9.34481	−1.28361	−0.641804	−	0.766868i	$-0.721814\pi$
$53$	−9.34481	−1.28361	−0.641804	+	0.766868i	$0.721814\pi$
$54$	0	0
$55$	13.4547	1.81424
$56$	0.0489173	0.00653685
$57$	0	0
$58$	−4.91185	−0.644958
$59$	−4.26875	−0.555744	−0.277872	−	0.960618i	$-0.589629\pi$
$59$	−4.26875	−0.555744	−0.277872	+	0.960618i	$0.589629\pi$
$60$	0	0
$61$	7.10992	0.910331	0.455166	−	0.890407i	$-0.349580\pi$
$61$	7.10992	0.910331	0.455166	+	0.890407i	$0.349580\pi$
$62$	−9.00969	−1.14423
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−5.38404	−0.657766	−0.328883	−	0.944371i	$-0.606672\pi$
$67$	−5.38404	−0.657766	−0.328883	+	0.944371i	$0.606672\pi$
$68$	2.89008	0.350474
$69$	0	0
$70$	0.104539	0.0124949
$71$	−8.71379	−1.03414	−0.517068	−	0.855944i	$-0.672977\pi$
$71$	−8.71379	−1.03414	−0.517068	+	0.855944i	$0.672977\pi$
$72$	0	0
$73$	14.9487	1.74961	0.874806	−	0.484474i	$-0.160989\pi$
$73$	14.9487	1.74961	0.874806	+	0.484474i	$0.160989\pi$
$74$	0.176292	0.0204935
$75$	0	0
$76$	7.20775	0.826786
$77$	0.307979	0.0350974
$78$	0	0
$79$	13.8291	1.55589	0.777947	−	0.628330i	$-0.216261\pi$
$79$	13.8291	1.55589	0.777947	+	0.628330i	$0.216261\pi$
$80$	2.13706	0.238931
$81$	0	0
$82$	−8.59179	−0.948805
$83$	−11.1347	−1.22219	−0.611094	−	0.791558i	$-0.709270\pi$
$83$	−11.1347	−1.22219	−0.611094	+	0.791558i	$0.709270\pi$
$84$	0	0
$85$	6.17629	0.669913
$86$	6.71379	0.723967
$87$	0	0
$88$	6.29590	0.671145
$89$	3.92154	0.415683	0.207841	−	0.978163i	$-0.433356\pi$
$89$	3.92154	0.415683	0.207841	+	0.978163i	$0.433356\pi$
$90$	0	0
$91$	0	0
$92$	−2.71379	−0.282932
$93$	0	0
$94$	7.20775	0.743423
$95$	15.4034	1.58036
$96$	0	0
$97$	2.47889	0.251694	0.125847	−	0.992050i	$-0.459835\pi$
$97$	2.47889	0.251694	0.125847	+	0.992050i	$0.459835\pi$
$98$	−6.99761	−0.706865
$99$	0	0
$100$	−0.432960	−0.0432960
$101$	−1.65279	−0.164459	−0.0822295	−	0.996613i	$-0.526204\pi$
$101$	−1.65279	−0.164459	−0.0822295	+	0.996613i	$0.526204\pi$
$102$	0	0
$103$	−8.23490	−0.811409	−0.405704	−	0.914004i	$-0.632974\pi$
$103$	−8.23490	−0.811409	−0.405704	+	0.914004i	$0.632974\pi$
$104$	0	0
$105$	0	0
$106$	−9.34481	−0.907649
$107$	8.36658	0.808828	0.404414	−	0.914576i	$-0.367475\pi$
$107$	8.36658	0.808828	0.404414	+	0.914576i	$0.367475\pi$
$108$	0	0
$109$	17.4276	1.66926	0.834630	−	0.550811i	$-0.185682\pi$
$109$	17.4276	1.66926	0.834630	+	0.550811i	$0.185682\pi$
$110$	13.4547	1.28286
$111$	0	0
$112$	0.0489173	0.00462225
$113$	13.9758	1.31474	0.657368	−	0.753570i	$-0.271670\pi$
$113$	13.9758	1.31474	0.657368	+	0.753570i	$0.271670\pi$
$114$	0	0
$115$	−5.79954	−0.540810
$116$	−4.91185	−0.456054
$117$	0	0
$118$	−4.26875	−0.392970
$119$	0.141375	0.0129598
$120$	0	0
$121$	28.6383	2.60348
$122$	7.10992	0.643702
$123$	0	0
$124$	−9.00969	−0.809094
$125$	−11.6106	−1.03848
$126$	0	0
$127$	−7.52111	−0.667390	−0.333695	−	0.942681i	$-0.608296\pi$
$127$	−7.52111	−0.667390	−0.333695	+	0.942681i	$0.608296\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−5.12498	−0.447772	−0.223886	−	0.974615i	$-0.571874\pi$
$131$	−5.12498	−0.447772	−0.223886	+	0.974615i	$0.571874\pi$
$132$	0	0
$133$	0.352584	0.0305729
$134$	−5.38404	−0.465110
$135$	0	0
$136$	2.89008	0.247823
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	8.68963	0.737045	0.368522	−	0.929619i	$-0.379864\pi$
$139$	8.68963	0.737045	0.368522	+	0.929619i	$0.379864\pi$
$140$	0.104539	0.00883520
$141$	0	0
$142$	−8.71379	−0.731245
$143$	0	0
$144$	0	0
$145$	−10.4969	−0.871724
$146$	14.9487	1.23716
$147$	0	0
$148$	0.176292	0.0144911
$149$	−4.86831	−0.398828	−0.199414	−	0.979915i	$-0.563904\pi$
$149$	−4.86831	−0.398828	−0.199414	+	0.979915i	$0.563904\pi$
$150$	0	0
$151$	−14.7463	−1.20004	−0.600019	−	0.799986i	$-0.704840\pi$
$151$	−14.7463	−1.20004	−0.600019	+	0.799986i	$0.704840\pi$
$152$	7.20775	0.584626
$153$	0	0
$154$	0.307979	0.0248176
$155$	−19.2543	−1.54654
$156$	0	0
$157$	−16.7138	−1.33391	−0.666953	−	0.745100i	$-0.732402\pi$
$157$	−16.7138	−1.33391	−0.666953	+	0.745100i	$0.732402\pi$
$158$	13.8291	1.10018
$159$	0	0
$160$	2.13706	0.168950
$161$	−0.132751	−0.0104623
$162$	0	0
$163$	−5.54958	−0.434677	−0.217338	−	0.976096i	$-0.569738\pi$
$163$	−5.54958	−0.434677	−0.217338	+	0.976096i	$0.569738\pi$
$164$	−8.59179	−0.670906
$165$	0	0
$166$	−11.1347	−0.864218
$167$	−3.92154	−0.303458	−0.151729	−	0.988422i	$-0.548484\pi$
$167$	−3.92154	−0.303458	−0.151729	+	0.988422i	$0.548484\pi$
$168$	0	0
$169$	0	0
$170$	6.17629	0.473700
$171$	0	0
$172$	6.71379	0.511922
$173$	3.48427	0.264904	0.132452	−	0.991189i	$-0.457715\pi$
$173$	3.48427	0.264904	0.132452	+	0.991189i	$0.457715\pi$
$174$	0	0
$175$	−0.0211793	−0.00160100
$176$	6.29590	0.474571
$177$	0	0
$178$	3.92154	0.293932
$179$	−3.58881	−0.268240	−0.134120	−	0.990965i	$-0.542821\pi$
$179$	−3.58881	−0.268240	−0.134120	+	0.990965i	$0.542821\pi$
$180$	0	0
$181$	5.50604	0.409261	0.204630	−	0.978839i	$-0.434401\pi$
$181$	5.50604	0.409261	0.204630	+	0.978839i	$0.434401\pi$
$182$	0	0
$183$	0	0
$184$	−2.71379	−0.200063
$185$	0.376747	0.0276990
$186$	0	0
$187$	18.1957	1.33060
$188$	7.20775	0.525679
$189$	0	0
$190$	15.4034	1.11748
$191$	5.65817	0.409411	0.204705	−	0.978824i	$-0.434376\pi$
$191$	5.65817	0.409411	0.204705	+	0.978824i	$0.434376\pi$
$192$	0	0
$193$	13.4034	0.964799	0.482400	−	0.875951i	$-0.339765\pi$
$193$	13.4034	0.964799	0.482400	+	0.875951i	$0.339765\pi$
$194$	2.47889	0.177974
$195$	0	0
$196$	−6.99761	−0.499829
$197$	−19.9812	−1.42360	−0.711801	−	0.702381i	$-0.752120\pi$
$197$	−19.9812	−1.42360	−0.711801	+	0.702381i	$0.752120\pi$
$198$	0	0
$199$	−6.24160	−0.442455	−0.221228	−	0.975222i	$-0.571006\pi$
$199$	−6.24160	−0.442455	−0.221228	+	0.975222i	$0.571006\pi$
$200$	−0.432960	−0.0306149
$201$	0	0
$202$	−1.65279	−0.116290
$203$	−0.240275	−0.0168640
$204$	0	0
$205$	−18.3612	−1.28240
$206$	−8.23490	−0.573753
$207$	0	0
$208$	0	0
$209$	45.3793	3.13895
$210$	0	0
$211$	−5.08575	−0.350118	−0.175059	−	0.984558i	$-0.556012\pi$
$211$	−5.08575	−0.350118	−0.175059	+	0.984558i	$0.556012\pi$
$212$	−9.34481	−0.641804
$213$	0	0
$214$	8.36658	0.571928
$215$	14.3478	0.978512
$216$	0	0
$217$	−0.440730	−0.0299187
$218$	17.4276	1.18034
$219$	0	0
$220$	13.4547	0.907118
$221$	0	0
$222$	0	0
$223$	−20.5483	−1.37601	−0.688006	−	0.725705i	$-0.741514\pi$
$223$	−20.5483	−1.37601	−0.688006	+	0.725705i	$0.741514\pi$
$224$	0.0489173	0.00326843
$225$	0	0
$226$	13.9758	0.929659
$227$	−4.41119	−0.292781	−0.146390	−	0.989227i	$-0.546766\pi$
$227$	−4.41119	−0.292781	−0.146390	+	0.989227i	$0.546766\pi$
$228$	0	0
$229$	0.230586	0.0152376	0.00761878	−	0.999971i	$-0.497575\pi$
$229$	0.230586	0.0152376	0.00761878	+	0.999971i	$0.497575\pi$
$230$	−5.79954	−0.382411
$231$	0	0
$232$	−4.91185	−0.322479
$233$	7.82371	0.512548	0.256274	−	0.966604i	$-0.417505\pi$
$233$	7.82371	0.512548	0.256274	+	0.966604i	$0.417505\pi$
$234$	0	0
$235$	15.4034	1.00481
$236$	−4.26875	−0.277872
$237$	0	0
$238$	0.141375	0.00916399
$239$	−11.1535	−0.721457	−0.360729	−	0.932671i	$-0.617472\pi$
$239$	−11.1535	−0.721457	−0.360729	+	0.932671i	$0.617472\pi$
$240$	0	0
$241$	−3.54527	−0.228371	−0.114185	−	0.993459i	$-0.536426\pi$
$241$	−3.54527	−0.228371	−0.114185	+	0.993459i	$0.536426\pi$
$242$	28.6383	1.84094
$243$	0	0
$244$	7.10992	0.455166
$245$	−14.9543	−0.955397
$246$	0	0
$247$	0	0
$248$	−9.00969	−0.572116
$249$	0	0
$250$	−11.6106	−0.734318
$251$	4.17092	0.263266	0.131633	−	0.991299i	$-0.457978\pi$
$251$	4.17092	0.263266	0.131633	+	0.991299i	$0.457978\pi$
$252$	0	0
$253$	−17.0858	−1.07417
$254$	−7.52111	−0.471916
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.8901	0.679305	0.339652	−	0.940551i	$-0.389691\pi$
$257$	10.8901	0.679305	0.339652	+	0.940551i	$0.389691\pi$
$258$	0	0
$259$	0.00862374	0.000535853 0
$260$	0	0
$261$	0	0
$262$	−5.12498	−0.316622
$263$	31.2271	1.92555	0.962774	−	0.270309i	$-0.0871259\pi$
$263$	31.2271	1.92555	0.962774	+	0.270309i	$0.0871259\pi$
$264$	0	0
$265$	−19.9705	−1.22678
$266$	0.352584	0.0216183
$267$	0	0
$268$	−5.38404	−0.328883
$269$	−15.9172	−0.970491	−0.485245	−	0.874378i	$-0.661270\pi$
$269$	−15.9172	−0.970491	−0.485245	+	0.874378i	$0.661270\pi$
$270$	0	0
$271$	3.52111	0.213892	0.106946	−	0.994265i	$-0.465893\pi$
$271$	3.52111	0.213892	0.106946	+	0.994265i	$0.465893\pi$
$272$	2.89008	0.175237
$273$	0	0
$274$	−4.00000	−0.241649
$275$	−2.72587	−0.164376
$276$	0	0
$277$	−8.58104	−0.515585	−0.257792	−	0.966200i	$-0.582995\pi$
$277$	−8.58104	−0.515585	−0.257792	+	0.966200i	$0.582995\pi$
$278$	8.68963	0.521169
$279$	0	0
$280$	0.104539	0.00624743
$281$	8.07846	0.481920	0.240960	−	0.970535i	$-0.422538\pi$
$281$	8.07846	0.481920	0.240960	+	0.970535i	$0.422538\pi$
$282$	0	0
$283$	−17.4034	−1.03453	−0.517263	−	0.855827i	$-0.673049\pi$
$283$	−17.4034	−1.03453	−0.517263	+	0.855827i	$0.673049\pi$
$284$	−8.71379	−0.517068
$285$	0	0
$286$	0	0
$287$	−0.420288	−0.0248088
$288$	0	0
$289$	−8.64742	−0.508672
$290$	−10.4969	−0.616402
$291$	0	0
$292$	14.9487	0.874806
$293$	19.3709	1.13166	0.565830	−	0.824522i	$-0.308556\pi$
$293$	19.3709	1.13166	0.565830	+	0.824522i	$0.308556\pi$
$294$	0	0
$295$	−9.12259	−0.531138
$296$	0.176292	0.0102468
$297$	0	0
$298$	−4.86831	−0.282014
$299$	0	0
$300$	0	0
$301$	0.328421	0.0189299
$302$	−14.7463	−0.848555
$303$	0	0
$304$	7.20775	0.413393
$305$	15.1943	0.870025
$306$	0	0
$307$	−12.4263	−0.709204	−0.354602	−	0.935017i	$-0.615384\pi$
$307$	−12.4263	−0.709204	−0.354602	+	0.935017i	$0.615384\pi$
$308$	0.307979	0.0175487
$309$	0	0
$310$	−19.2543	−1.09357
$311$	−2.71379	−0.153885	−0.0769425	−	0.997036i	$-0.524516\pi$
$311$	−2.71379	−0.153885	−0.0769425	+	0.997036i	$0.524516\pi$
$312$	0	0
$313$	15.3884	0.869801	0.434901	−	0.900478i	$-0.356783\pi$
$313$	15.3884	0.869801	0.434901	+	0.900478i	$0.356783\pi$
$314$	−16.7138	−0.943214
$315$	0	0
$316$	13.8291	0.777947
$317$	−25.6528	−1.44080	−0.720402	−	0.693557i	$-0.756043\pi$
$317$	−25.6528	−1.44080	−0.720402	+	0.693557i	$0.756043\pi$
$318$	0	0
$319$	−30.9245	−1.73144
$320$	2.13706	0.119465
$321$	0	0
$322$	−0.132751	−0.00739795
$323$	20.8310	1.15907
$324$	0	0
$325$	0	0
$326$	−5.54958	−0.307363
$327$	0	0
$328$	−8.59179	−0.474402
$329$	0.352584	0.0194386
$330$	0	0
$331$	−3.82371	−0.210170	−0.105085	−	0.994463i	$-0.533511\pi$
$331$	−3.82371	−0.210170	−0.105085	+	0.994463i	$0.533511\pi$
$332$	−11.1347	−0.611094
$333$	0	0
$334$	−3.92154	−0.214577
$335$	−11.5060	−0.628642
$336$	0	0
$337$	20.1304	1.09657	0.548285	−	0.836291i	$-0.315281\pi$
$337$	20.1304	1.09657	0.548285	+	0.836291i	$0.315281\pi$
$338$	0	0
$339$	0	0
$340$	6.17629	0.334956
$341$	−56.7241	−3.07178
$342$	0	0
$343$	−0.684726	−0.0369717
$344$	6.71379	0.361983
$345$	0	0
$346$	3.48427	0.187316
$347$	−2.93900	−0.157774	−0.0788869	−	0.996884i	$-0.525137\pi$
$347$	−2.93900	−0.157774	−0.0788869	+	0.996884i	$0.525137\pi$
$348$	0	0
$349$	7.37329	0.394683	0.197342	−	0.980335i	$-0.436769\pi$
$349$	7.37329	0.394683	0.197342	+	0.980335i	$0.436769\pi$
$350$	−0.0211793	−0.00113208
$351$	0	0
$352$	6.29590	0.335572
$353$	2.01075	0.107022	0.0535108	−	0.998567i	$-0.482959\pi$
$353$	2.01075	0.107022	0.0535108	+	0.998567i	$0.482959\pi$
$354$	0	0
$355$	−18.6219	−0.988349
$356$	3.92154	0.207841
$357$	0	0
$358$	−3.58881	−0.189675
$359$	−31.4577	−1.66027	−0.830137	−	0.557559i	$-0.811738\pi$
$359$	−31.4577	−1.66027	−0.830137	+	0.557559i	$0.811738\pi$
$360$	0	0
$361$	32.9517	1.73430
$362$	5.50604	0.289391
$363$	0	0
$364$	0	0
$365$	31.9463	1.67215
$366$	0	0
$367$	−4.49934	−0.234863	−0.117432	−	0.993081i	$-0.537466\pi$
$367$	−4.49934	−0.234863	−0.117432	+	0.993081i	$0.537466\pi$
$368$	−2.71379	−0.141466
$369$	0	0
$370$	0.376747	0.0195862
$371$	−0.457123	−0.0237327
$372$	0	0
$373$	37.8297	1.95875	0.979373	−	0.202060i	$-0.0647635\pi$
$373$	37.8297	1.95875	0.979373	+	0.202060i	$0.0647635\pi$
$374$	18.1957	0.940876
$375$	0	0
$376$	7.20775	0.371711
$377$	0	0
$378$	0	0
$379$	−8.43967	−0.433516	−0.216758	−	0.976225i	$-0.569548\pi$
$379$	−8.43967	−0.433516	−0.216758	+	0.976225i	$0.569548\pi$
$380$	15.4034	0.790179
$381$	0	0
$382$	5.65817	0.289497
$383$	−1.28621	−0.0657222	−0.0328611	−	0.999460i	$-0.510462\pi$
$383$	−1.28621	−0.0657222	−0.0328611	+	0.999460i	$0.510462\pi$
$384$	0	0
$385$	0.658170	0.0335434
$386$	13.4034	0.682216
$387$	0	0
$388$	2.47889	0.125847
$389$	−23.3924	−1.18604	−0.593021	−	0.805187i	$-0.702065\pi$
$389$	−23.3924	−1.18604	−0.593021	+	0.805187i	$0.702065\pi$
$390$	0	0
$391$	−7.84309	−0.396642
$392$	−6.99761	−0.353433
$393$	0	0
$394$	−19.9812	−1.00664
$395$	29.5536	1.48700
$396$	0	0
$397$	37.0858	1.86128	0.930640	−	0.365935i	$-0.119251\pi$
$397$	37.0858	1.86128	0.930640	+	0.365935i	$0.119251\pi$
$398$	−6.24160	−0.312863
$399$	0	0
$400$	−0.432960	−0.0216480
$401$	−4.86592	−0.242992	−0.121496	−	0.992592i	$-0.538769\pi$
$401$	−4.86592	−0.242992	−0.121496	+	0.992592i	$0.538769\pi$
$402$	0	0
$403$	0	0
$404$	−1.65279	−0.0822295
$405$	0	0
$406$	−0.240275	−0.0119246
$407$	1.10992	0.0550165
$408$	0	0
$409$	−0.445042	−0.0220059	−0.0110030	−	0.999939i	$-0.503502\pi$
$409$	−0.445042	−0.0220059	−0.0110030	+	0.999939i	$0.503502\pi$
$410$	−18.3612	−0.906795
$411$	0	0
$412$	−8.23490	−0.405704
$413$	−0.208816	−0.0102752
$414$	0	0
$415$	−23.7955	−1.16807
$416$	0	0
$417$	0	0
$418$	45.3793	2.21957
$419$	−17.9869	−0.878715	−0.439358	−	0.898312i	$-0.644794\pi$
$419$	−17.9869	−0.878715	−0.439358	+	0.898312i	$0.644794\pi$
$420$	0	0
$421$	−21.2814	−1.03719	−0.518597	−	0.855019i	$-0.673545\pi$
$421$	−21.2814	−1.03719	−0.518597	+	0.855019i	$0.673545\pi$
$422$	−5.08575	−0.247571
$423$	0	0
$424$	−9.34481	−0.453824
$425$	−1.25129	−0.0606966
$426$	0	0
$427$	0.347798	0.0168311
$428$	8.36658	0.404414
$429$	0	0
$430$	14.3478	0.691912
$431$	24.7138	1.19042	0.595211	−	0.803570i	$-0.297069\pi$
$431$	24.7138	1.19042	0.595211	+	0.803570i	$0.297069\pi$
$432$	0	0
$433$	32.2174	1.54827	0.774136	−	0.633020i	$-0.218185\pi$
$433$	32.2174	1.54827	0.774136	+	0.633020i	$0.218185\pi$
$434$	−0.440730	−0.0211557
$435$	0	0
$436$	17.4276	0.834630
$437$	−19.5603	−0.935698
$438$	0	0
$439$	32.0877	1.53146	0.765731	−	0.643162i	$-0.222378\pi$
$439$	32.0877	1.53146	0.765731	+	0.643162i	$0.222378\pi$
$440$	13.4547	0.641429
$441$	0	0
$442$	0	0
$443$	−20.5109	−0.974504	−0.487252	−	0.873261i	$-0.662001\pi$
$443$	−20.5109	−0.974504	−0.487252	+	0.873261i	$0.662001\pi$
$444$	0	0
$445$	8.38059	0.397278
$446$	−20.5483	−0.972988
$447$	0	0
$448$	0.0489173	0.00231113
$449$	−15.3163	−0.722823	−0.361411	−	0.932406i	$-0.617705\pi$
$449$	−15.3163	−0.722823	−0.361411	+	0.932406i	$0.617705\pi$
$450$	0	0
$451$	−54.0930	−2.54714
$452$	13.9758	0.657368
$453$	0	0
$454$	−4.41119	−0.207027
$455$	0	0
$456$	0	0
$457$	19.1860	0.897482	0.448741	−	0.893662i	$-0.351873\pi$
$457$	19.1860	0.897482	0.448741	+	0.893662i	$0.351873\pi$
$458$	0.230586	0.0107746
$459$	0	0
$460$	−5.79954	−0.270405
$461$	8.31229	0.387142	0.193571	−	0.981086i	$-0.437993\pi$
$461$	8.31229	0.387142	0.193571	+	0.981086i	$0.437993\pi$
$462$	0	0
$463$	6.32842	0.294107	0.147053	−	0.989129i	$-0.453021\pi$
$463$	6.32842	0.294107	0.147053	+	0.989129i	$0.453021\pi$
$464$	−4.91185	−0.228027
$465$	0	0
$466$	7.82371	0.362426
$467$	31.1879	1.44320	0.721602	−	0.692308i	$-0.243406\pi$
$467$	31.1879	1.44320	0.721602	+	0.692308i	$0.243406\pi$
$468$	0	0
$469$	−0.263373	−0.0121614
$470$	15.4034	0.710507
$471$	0	0
$472$	−4.26875	−0.196485
$473$	42.2693	1.94355
$474$	0	0
$475$	−3.12067	−0.143186
$476$	0.141375	0.00647992
$477$	0	0
$478$	−11.1535	−0.510147
$479$	−17.9323	−0.819348	−0.409674	−	0.912232i	$-0.634357\pi$
$479$	−17.9323	−0.819348	−0.409674	+	0.912232i	$0.634357\pi$
$480$	0	0
$481$	0	0
$482$	−3.54527	−0.161483
$483$	0	0
$484$	28.6383	1.30174
$485$	5.29755	0.240549
$486$	0	0
$487$	−30.3279	−1.37429	−0.687145	−	0.726520i	$-0.741136\pi$
$487$	−30.3279	−1.37429	−0.687145	+	0.726520i	$0.741136\pi$
$488$	7.10992	0.321851
$489$	0	0
$490$	−14.9543	−0.675568
$491$	−30.0954	−1.35819	−0.679094	−	0.734051i	$-0.737627\pi$
$491$	−30.0954	−1.35819	−0.679094	+	0.734051i	$0.737627\pi$
$492$	0	0
$493$	−14.1957	−0.639341
$494$	0	0
$495$	0	0
$496$	−9.00969	−0.404547
$497$	−0.426256	−0.0191202
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	−11.6106	−0.519241
$501$	0	0
$502$	4.17092	0.186157
$503$	−33.7512	−1.50489	−0.752446	−	0.658654i	$-0.771126\pi$
$503$	−33.7512	−1.50489	−0.752446	+	0.658654i	$0.771126\pi$
$504$	0	0
$505$	−3.53212	−0.157177
$506$	−17.0858	−0.759554
$507$	0	0
$508$	−7.52111	−0.333695
$509$	−14.7439	−0.653513	−0.326756	−	0.945109i	$-0.605956\pi$
$509$	−14.7439	−0.653513	−0.326756	+	0.945109i	$0.605956\pi$
$510$	0	0
$511$	0.731250	0.0323486
$512$	1.00000	0.0441942
$513$	0	0
$514$	10.8901	0.480341
$515$	−17.5985	−0.775483
$516$	0	0
$517$	45.3793	1.99578
$518$	0.00862374	0.000378905 0
$519$	0	0
$520$	0	0
$521$	−22.1086	−0.968595	−0.484297	−	0.874903i	$-0.660925\pi$
$521$	−22.1086	−0.968595	−0.484297	+	0.874903i	$0.660925\pi$
$522$	0	0
$523$	−17.3599	−0.759095	−0.379547	−	0.925172i	$-0.623920\pi$
$523$	−17.3599	−0.759095	−0.379547	+	0.925172i	$0.623920\pi$
$524$	−5.12498	−0.223886
$525$	0	0
$526$	31.2271	1.36157
$527$	−26.0388	−1.13427
$528$	0	0
$529$	−15.6353	−0.679797
$530$	−19.9705	−0.867461
$531$	0	0
$532$	0.352584	0.0152865
$533$	0	0
$534$	0	0
$535$	17.8799	0.773016
$536$	−5.38404	−0.232555
$537$	0	0
$538$	−15.9172	−0.686241
$539$	−44.0562	−1.89764
$540$	0	0
$541$	8.83579	0.379880	0.189940	−	0.981796i	$-0.439171\pi$
$541$	8.83579	0.379880	0.189940	+	0.981796i	$0.439171\pi$
$542$	3.52111	0.151244
$543$	0	0
$544$	2.89008	0.123911
$545$	37.2438	1.59535
$546$	0	0
$547$	−8.10859	−0.346698	−0.173349	−	0.984860i	$-0.555459\pi$
$547$	−8.10859	−0.346698	−0.173349	+	0.984860i	$0.555459\pi$
$548$	−4.00000	−0.170872
$549$	0	0
$550$	−2.72587	−0.116232
$551$	−35.4034	−1.50824
$552$	0	0
$553$	0.676482	0.0287669
$554$	−8.58104	−0.364573
$555$	0	0
$556$	8.68963	0.368522
$557$	−9.56033	−0.405084	−0.202542	−	0.979274i	$-0.564920\pi$
$557$	−9.56033	−0.405084	−0.202542	+	0.979274i	$0.564920\pi$
$558$	0	0
$559$	0	0
$560$	0.104539	0.00441760
$561$	0	0
$562$	8.07846	0.340769
$563$	45.1637	1.90342	0.951712	−	0.306991i	$-0.0993223\pi$
$563$	45.1637	1.90342	0.951712	+	0.306991i	$0.0993223\pi$
$564$	0	0
$565$	29.8672	1.25652
$566$	−17.4034	−0.731520
$567$	0	0
$568$	−8.71379	−0.365623
$569$	3.20775	0.134476	0.0672380	−	0.997737i	$-0.478581\pi$
$569$	3.20775	0.134476	0.0672380	+	0.997737i	$0.478581\pi$
$570$	0	0
$571$	0.0241632	0.00101120	0.000505599	−	1.00000i	$-0.499839\pi$
$571$	0.0241632	0.00101120	0.000505599		1.00000i	$0.499839\pi$
$572$	0	0
$573$	0	0
$574$	−0.420288	−0.0175425
$575$	1.17496	0.0489994
$576$	0	0
$577$	−46.0200	−1.91584	−0.957918	−	0.287041i	$-0.907328\pi$
$577$	−46.0200	−1.91584	−0.957918	+	0.287041i	$0.907328\pi$
$578$	−8.64742	−0.359685
$579$	0	0
$580$	−10.4969	−0.435862
$581$	−0.544678	−0.0225971
$582$	0	0
$583$	−58.8340	−2.43665
$584$	14.9487	0.618581
$585$	0	0
$586$	19.3709	0.800204
$587$	25.9385	1.07060	0.535299	−	0.844663i	$-0.320199\pi$
$587$	25.9385	1.07060	0.535299	+	0.844663i	$0.320199\pi$
$588$	0	0
$589$	−64.9396	−2.67579
$590$	−9.12259	−0.375571
$591$	0	0
$592$	0.176292	0.00724556
$593$	8.30691	0.341124	0.170562	−	0.985347i	$-0.445442\pi$
$593$	8.30691	0.341124	0.170562	+	0.985347i	$0.445442\pi$
$594$	0	0
$595$	0.302128	0.0123860
$596$	−4.86831	−0.199414
$597$	0	0
$598$	0	0
$599$	−37.1702	−1.51873	−0.759366	−	0.650664i	$-0.774491\pi$
$599$	−37.1702	−1.51873	−0.759366	+	0.650664i	$0.774491\pi$
$600$	0	0
$601$	−12.5700	−0.512742	−0.256371	−	0.966578i	$-0.582527\pi$
$601$	−12.5700	−0.512742	−0.256371	+	0.966578i	$0.582527\pi$
$602$	0.328421	0.0133854
$603$	0	0
$604$	−14.7463	−0.600019
$605$	61.2019	2.48821
$606$	0	0
$607$	8.04892	0.326695	0.163348	−	0.986569i	$-0.447771\pi$
$607$	8.04892	0.326695	0.163348	+	0.986569i	$0.447771\pi$
$608$	7.20775	0.292313
$609$	0	0
$610$	15.1943	0.615201
$611$	0	0
$612$	0	0
$613$	30.4590	1.23023	0.615115	−	0.788438i	$-0.289110\pi$
$613$	30.4590	1.23023	0.615115	+	0.788438i	$0.289110\pi$
$614$	−12.4263	−0.501483
$615$	0	0
$616$	0.307979	0.0124088
$617$	−4.73317	−0.190550	−0.0952751	−	0.995451i	$-0.530373\pi$
$617$	−4.73317	−0.190550	−0.0952751	+	0.995451i	$0.530373\pi$
$618$	0	0
$619$	−24.9095	−1.00120	−0.500598	−	0.865680i	$-0.666886\pi$
$619$	−24.9095	−1.00120	−0.500598	+	0.865680i	$0.666886\pi$
$620$	−19.2543	−0.773270
$621$	0	0
$622$	−2.71379	−0.108813
$623$	0.191831	0.00768556
$624$	0	0
$625$	−22.6477	−0.905910
$626$	15.3884	0.615042
$627$	0	0
$628$	−16.7138	−0.666953
$629$	0.509499	0.0203150
$630$	0	0
$631$	24.4295	0.972523	0.486262	−	0.873813i	$-0.338360\pi$
$631$	24.4295	0.972523	0.486262	+	0.873813i	$0.338360\pi$
$632$	13.8291	0.550091
$633$	0	0
$634$	−25.6528	−1.01880
$635$	−16.0731	−0.637841
$636$	0	0
$637$	0	0
$638$	−30.9245	−1.22431
$639$	0	0
$640$	2.13706	0.0844748
$641$	−48.7982	−1.92741	−0.963707	−	0.266963i	$-0.913980\pi$
$641$	−48.7982	−1.92741	−0.963707	+	0.266963i	$0.913980\pi$
$642$	0	0
$643$	−28.2693	−1.11483	−0.557417	−	0.830233i	$-0.688208\pi$
$643$	−28.2693	−1.11483	−0.557417	+	0.830233i	$0.688208\pi$
$644$	−0.132751	−0.00523114
$645$	0	0
$646$	20.8310	0.819585
$647$	−34.8961	−1.37191	−0.685953	−	0.727646i	$-0.740614\pi$
$647$	−34.8961	−1.37191	−0.685953	+	0.727646i	$0.740614\pi$
$648$	0	0
$649$	−26.8756	−1.05496
$650$	0	0
$651$	0	0
$652$	−5.54958	−0.217338
$653$	11.7157	0.458471	0.229236	−	0.973371i	$-0.426377\pi$
$653$	11.7157	0.458471	0.229236	+	0.973371i	$0.426377\pi$
$654$	0	0
$655$	−10.9524	−0.427946
$656$	−8.59179	−0.335453
$657$	0	0
$658$	0.352584	0.0137452
$659$	7.13467	0.277927	0.138964	−	0.990297i	$-0.455623\pi$
$659$	7.13467	0.277927	0.138964	+	0.990297i	$0.455623\pi$
$660$	0	0
$661$	−8.52888	−0.331735	−0.165867	−	0.986148i	$-0.553042\pi$
$661$	−8.52888	−0.331735	−0.165867	+	0.986148i	$0.553042\pi$
$662$	−3.82371	−0.148613
$663$	0	0
$664$	−11.1347	−0.432109
$665$	0.753494	0.0292193
$666$	0	0
$667$	13.3297	0.516130
$668$	−3.92154	−0.151729
$669$	0	0
$670$	−11.5060	−0.444517
$671$	44.7633	1.72807
$672$	0	0
$673$	−31.5937	−1.21785	−0.608924	−	0.793229i	$-0.708399\pi$
$673$	−31.5937	−1.21785	−0.608924	+	0.793229i	$0.708399\pi$
$674$	20.1304	0.775392
$675$	0	0
$676$	0	0
$677$	17.3002	0.664901	0.332451	−	0.943121i	$-0.392125\pi$
$677$	17.3002	0.664901	0.332451	+	0.943121i	$0.392125\pi$
$678$	0	0
$679$	0.121261	0.00465357
$680$	6.17629	0.236850
$681$	0	0
$682$	−56.7241	−2.17208
$683$	−30.3957	−1.16306	−0.581529	−	0.813526i	$-0.697545\pi$
$683$	−30.3957	−1.16306	−0.581529	+	0.813526i	$0.697545\pi$
$684$	0	0
$685$	−8.54825	−0.326612
$686$	−0.684726	−0.0261429
$687$	0	0
$688$	6.71379	0.255961
$689$	0	0
$690$	0	0
$691$	35.7318	1.35930	0.679652	−	0.733535i	$-0.262131\pi$
$691$	35.7318	1.35930	0.679652	+	0.733535i	$0.262131\pi$
$692$	3.48427	0.132452
$693$	0	0
$694$	−2.93900	−0.111563
$695$	18.5703	0.704411
$696$	0	0
$697$	−24.8310	−0.940541
$698$	7.37329	0.279083
$699$	0	0
$700$	−0.0211793	−0.000800501 0
$701$	−20.1328	−0.760404	−0.380202	−	0.924904i	$-0.624145\pi$
$701$	−20.1328	−0.760404	−0.380202	+	0.924904i	$0.624145\pi$
$702$	0	0
$703$	1.27067	0.0479242
$704$	6.29590	0.237286
$705$	0	0
$706$	2.01075	0.0756757
$707$	−0.0808502	−0.00304069
$708$	0	0
$709$	16.6160	0.624025	0.312013	−	0.950078i	$-0.398997\pi$
$709$	16.6160	0.624025	0.312013	+	0.950078i	$0.398997\pi$
$710$	−18.6219	−0.698868
$711$	0	0
$712$	3.92154	0.146966
$713$	24.4504	0.915675
$714$	0	0
$715$	0	0
$716$	−3.58881	−0.134120
$717$	0	0
$718$	−31.4577	−1.17399
$719$	−2.04833	−0.0763897	−0.0381948	−	0.999270i	$-0.512161\pi$
$719$	−2.04833	−0.0763897	−0.0381948	+	0.999270i	$0.512161\pi$
$720$	0	0
$721$	−0.402829	−0.0150021
$722$	32.9517	1.22633
$723$	0	0
$724$	5.50604	0.204630
$725$	2.12664	0.0789813
$726$	0	0
$727$	29.9377	1.11033	0.555163	−	0.831741i	$-0.312656\pi$
$727$	29.9377	1.11033	0.555163	+	0.831741i	$0.312656\pi$
$728$	0	0
$729$	0	0
$730$	31.9463	1.18239
$731$	19.4034	0.717661
$732$	0	0
$733$	−1.46250	−0.0540187	−0.0270093	−	0.999635i	$-0.508598\pi$
$733$	−1.46250	−0.0540187	−0.0270093	+	0.999635i	$0.508598\pi$
$734$	−4.49934	−0.166074
$735$	0	0
$736$	−2.71379	−0.100032
$737$	−33.8974	−1.24863
$738$	0	0
$739$	40.3866	1.48564	0.742822	−	0.669489i	$-0.233487\pi$
$739$	40.3866	1.48564	0.742822	+	0.669489i	$0.233487\pi$
$740$	0.376747	0.0138495
$741$	0	0
$742$	−0.457123	−0.0167815
$743$	9.77538	0.358624	0.179312	−	0.983792i	$-0.442613\pi$
$743$	9.77538	0.358624	0.179312	+	0.983792i	$0.442613\pi$
$744$	0	0
$745$	−10.4039	−0.381169
$746$	37.8297	1.38504
$747$	0	0
$748$	18.1957	0.665300
$749$	0.409271	0.0149544
$750$	0	0
$751$	34.5459	1.26060	0.630298	−	0.776353i	$-0.282933\pi$
$751$	34.5459	1.26060	0.630298	+	0.776353i	$0.282933\pi$
$752$	7.20775	0.262840
$753$	0	0
$754$	0	0
$755$	−31.5138	−1.14690
$756$	0	0
$757$	−5.14483	−0.186992	−0.0934961	−	0.995620i	$-0.529804\pi$
$757$	−5.14483	−0.186992	−0.0934961	+	0.995620i	$0.529804\pi$
$758$	−8.43967	−0.306542
$759$	0	0
$760$	15.4034	0.558741
$761$	19.5120	0.707310	0.353655	−	0.935376i	$-0.384939\pi$
$761$	19.5120	0.707310	0.353655	+	0.935376i	$0.384939\pi$
$762$	0	0
$763$	0.852511	0.0308630
$764$	5.65817	0.204705
$765$	0	0
$766$	−1.28621	−0.0464726
$767$	0	0
$768$	0	0
$769$	5.75600	0.207567	0.103783	−	0.994600i	$-0.466905\pi$
$769$	5.75600	0.207567	0.103783	+	0.994600i	$0.466905\pi$
$770$	0.658170	0.0237188
$771$	0	0
$772$	13.4034	0.482400
$773$	4.12737	0.148451	0.0742257	−	0.997241i	$-0.476351\pi$
$773$	4.12737	0.148451	0.0742257	+	0.997241i	$0.476351\pi$
$774$	0	0
$775$	3.90084	0.140122
$776$	2.47889	0.0889871
$777$	0	0
$778$	−23.3924	−0.838658
$779$	−61.9275	−2.21878
$780$	0	0
$781$	−54.8611	−1.96309
$782$	−7.84309	−0.280468
$783$	0	0
$784$	−6.99761	−0.249915
$785$	−35.7184	−1.27485
$786$	0	0
$787$	−16.2392	−0.578865	−0.289433	−	0.957198i	$-0.593467\pi$
$787$	−16.2392	−0.578865	−0.289433	+	0.957198i	$0.593467\pi$
$788$	−19.9812	−0.711801
$789$	0	0
$790$	29.5536	1.05147
$791$	0.683661	0.0243082
$792$	0	0
$793$	0	0
$794$	37.0858	1.31612
$795$	0	0
$796$	−6.24160	−0.221228
$797$	−16.1148	−0.570816	−0.285408	−	0.958406i	$-0.592129\pi$
$797$	−16.1148	−0.570816	−0.285408	+	0.958406i	$0.592129\pi$
$798$	0	0
$799$	20.8310	0.736948
$800$	−0.432960	−0.0153075
$801$	0	0
$802$	−4.86592	−0.171822
$803$	94.1154	3.32126
$804$	0	0
$805$	−0.283698	−0.00999905
$806$	0	0
$807$	0	0
$808$	−1.65279	−0.0581450
$809$	−40.1521	−1.41167	−0.705837	−	0.708374i	$-0.749429\pi$
$809$	−40.1521	−1.41167	−0.705837	+	0.708374i	$0.749429\pi$
$810$	0	0
$811$	39.8646	1.39984	0.699918	−	0.714224i	$-0.253220\pi$
$811$	39.8646	1.39984	0.699918	+	0.714224i	$0.253220\pi$
$812$	−0.240275	−0.00843199
$813$	0	0
$814$	1.10992	0.0389025
$815$	−11.8598	−0.415431
$816$	0	0
$817$	48.3913	1.69300
$818$	−0.445042	−0.0155605
$819$	0	0
$820$	−18.3612	−0.641201
$821$	6.25428	0.218276	0.109138	−	0.994027i	$-0.465191\pi$
$821$	6.25428	0.218276	0.109138	+	0.994027i	$0.465191\pi$
$822$	0	0
$823$	−4.74333	−0.165342	−0.0826711	−	0.996577i	$-0.526345\pi$
$823$	−4.74333	−0.165342	−0.0826711	+	0.996577i	$0.526345\pi$
$824$	−8.23490	−0.286876
$825$	0	0
$826$	−0.208816	−0.00726563
$827$	0.716185	0.0249042	0.0124521	−	0.999922i	$-0.496036\pi$
$827$	0.716185	0.0249042	0.0124521	+	0.999922i	$0.496036\pi$
$828$	0	0
$829$	−36.0060	−1.25054	−0.625269	−	0.780409i	$-0.715011\pi$
$829$	−36.0060	−1.25054	−0.625269	+	0.780409i	$0.715011\pi$
$830$	−23.7955	−0.825954
$831$	0	0
$832$	0	0
$833$	−20.2237	−0.700709
$834$	0	0
$835$	−8.38059	−0.290022
$836$	45.3793	1.56947
$837$	0	0
$838$	−17.9869	−0.621346
$839$	21.5555	0.744180	0.372090	−	0.928197i	$-0.378641\pi$
$839$	21.5555	0.744180	0.372090	+	0.928197i	$0.378641\pi$
$840$	0	0
$841$	−4.87369	−0.168058
$842$	−21.2814	−0.733406
$843$	0	0
$844$	−5.08575	−0.175059
$845$	0	0
$846$	0	0
$847$	1.40091	0.0481358
$848$	−9.34481	−0.320902
$849$	0	0
$850$	−1.25129	−0.0429189
$851$	−0.478420	−0.0164000
$852$	0	0
$853$	37.8237	1.29506	0.647530	−	0.762040i	$-0.275802\pi$
$853$	37.8237	1.29506	0.647530	+	0.762040i	$0.275802\pi$
$854$	0.347798	0.0119014
$855$	0	0
$856$	8.36658	0.285964
$857$	6.58317	0.224877	0.112438	−	0.993659i	$-0.464134\pi$
$857$	6.58317	0.224877	0.112438	+	0.993659i	$0.464134\pi$
$858$	0	0
$859$	−20.6246	−0.703702	−0.351851	−	0.936056i	$-0.614448\pi$
$859$	−20.6246	−0.703702	−0.351851	+	0.936056i	$0.614448\pi$
$860$	14.3478	0.489256
$861$	0	0
$862$	24.7138	0.841755
$863$	15.9081	0.541519	0.270760	−	0.962647i	$-0.412725\pi$
$863$	15.9081	0.541519	0.270760	+	0.962647i	$0.412725\pi$
$864$	0	0
$865$	7.44611	0.253175
$866$	32.2174	1.09479
$867$	0	0
$868$	−0.440730	−0.0149594
$869$	87.0665	2.95353
$870$	0	0
$871$	0	0
$872$	17.4276	0.590172
$873$	0	0
$874$	−19.5603	−0.661638
$875$	−0.567959	−0.0192005
$876$	0	0
$877$	15.6039	0.526905	0.263453	−	0.964672i	$-0.415139\pi$
$877$	15.6039	0.526905	0.263453	+	0.964672i	$0.415139\pi$
$878$	32.0877	1.08291
$879$	0	0
$880$	13.4547	0.453559
$881$	14.2547	0.480255	0.240127	−	0.970741i	$-0.422811\pi$
$881$	14.2547	0.480255	0.240127	+	0.970741i	$0.422811\pi$
$882$	0	0
$883$	1.65817	0.0558019	0.0279009	−	0.999611i	$-0.491118\pi$
$883$	1.65817	0.0558019	0.0279009	+	0.999611i	$0.491118\pi$
$884$	0	0
$885$	0	0
$886$	−20.5109	−0.689079
$887$	19.2030	0.644772	0.322386	−	0.946608i	$-0.395515\pi$
$887$	19.2030	0.644772	0.322386	+	0.946608i	$0.395515\pi$
$888$	0	0
$889$	−0.367913	−0.0123394
$890$	8.38059	0.280918
$891$	0	0
$892$	−20.5483	−0.688006
$893$	51.9517	1.73850
$894$	0	0
$895$	−7.66951	−0.256364
$896$	0.0489173	0.00163421
$897$	0	0
$898$	−15.3163	−0.511113
$899$	44.2543	1.47596
$900$	0	0
$901$	−27.0073	−0.899743
$902$	−54.0930	−1.80110
$903$	0	0
$904$	13.9758	0.464829
$905$	11.7668	0.391140
$906$	0	0
$907$	−32.3672	−1.07473	−0.537367	−	0.843348i	$-0.680581\pi$
$907$	−32.3672	−1.07473	−0.537367	+	0.843348i	$0.680581\pi$
$908$	−4.41119	−0.146390
$909$	0	0
$910$	0	0
$911$	33.9624	1.12523	0.562613	−	0.826721i	$-0.309796\pi$
$911$	33.9624	1.12523	0.562613	+	0.826721i	$0.309796\pi$
$912$	0	0
$913$	−70.1027	−2.32006
$914$	19.1860	0.634616
$915$	0	0
$916$	0.230586	0.00761878
$917$	−0.250700	−0.00827886
$918$	0	0
$919$	−20.0780	−0.662312	−0.331156	−	0.943576i	$-0.607439\pi$
$919$	−20.0780	−0.662312	−0.331156	+	0.943576i	$0.607439\pi$
$920$	−5.79954	−0.191205
$921$	0	0
$922$	8.31229	0.273751
$923$	0	0
$924$	0	0
$925$	−0.0763274	−0.00250963
$926$	6.32842	0.207965
$927$	0	0
$928$	−4.91185	−0.161240
$929$	56.1280	1.84150	0.920749	−	0.390154i	$-0.127578\pi$
$929$	56.1280	1.84150	0.920749	+	0.390154i	$0.127578\pi$
$930$	0	0
$931$	−50.4370	−1.65301
$932$	7.82371	0.256274
$933$	0	0
$934$	31.1879	1.02050
$935$	38.8853	1.27169
$936$	0	0
$937$	27.0291	0.883001	0.441501	−	0.897261i	$-0.354446\pi$
$937$	27.0291	0.883001	0.441501	+	0.897261i	$0.354446\pi$
$938$	−0.263373	−0.00859944
$939$	0	0
$940$	15.4034	0.502404
$941$	−24.0277	−0.783282	−0.391641	−	0.920118i	$-0.628092\pi$
$941$	−24.0277	−0.783282	−0.391641	+	0.920118i	$0.628092\pi$
$942$	0	0
$943$	23.3163	0.759284
$944$	−4.26875	−0.138936
$945$	0	0
$946$	42.2693	1.37429
$947$	−26.7966	−0.870771	−0.435386	−	0.900244i	$-0.643388\pi$
$947$	−26.7966	−0.870771	−0.435386	+	0.900244i	$0.643388\pi$
$948$	0	0
$949$	0	0
$950$	−3.12067	−0.101248
$951$	0	0
$952$	0.141375	0.00458200
$953$	−34.4215	−1.11502	−0.557510	−	0.830170i	$-0.688243\pi$
$953$	−34.4215	−1.11502	−0.557510	+	0.830170i	$0.688243\pi$
$954$	0	0
$955$	12.0919	0.391284
$956$	−11.1535	−0.360729
$957$	0	0
$958$	−17.9323	−0.579366
$959$	−0.195669	−0.00631849
$960$	0	0
$961$	50.1745	1.61853
$962$	0	0
$963$	0	0
$964$	−3.54527	−0.114185
$965$	28.6440	0.922082
$966$	0	0
$967$	−26.2631	−0.844565	−0.422282	−	0.906464i	$-0.638771\pi$
$967$	−26.2631	−0.844565	−0.422282	+	0.906464i	$0.638771\pi$
$968$	28.6383	0.920470
$969$	0	0
$970$	5.29755	0.170094
$971$	39.9842	1.28315	0.641577	−	0.767059i	$-0.278280\pi$
$971$	39.9842	1.28315	0.641577	+	0.767059i	$0.278280\pi$
$972$	0	0
$973$	0.425074	0.0136272
$974$	−30.3279	−0.971770
$975$	0	0
$976$	7.10992	0.227583
$977$	4.58317	0.146629	0.0733143	−	0.997309i	$-0.476642\pi$
$977$	4.58317	0.146629	0.0733143	+	0.997309i	$0.476642\pi$
$978$	0	0
$979$	24.6896	0.789084
$980$	−14.9543	−0.477699
$981$	0	0
$982$	−30.0954	−0.960384
$983$	−10.6848	−0.340794	−0.170397	−	0.985376i	$-0.554505\pi$
$983$	−10.6848	−0.340794	−0.170397	+	0.985376i	$0.554505\pi$
$984$	0	0
$985$	−42.7011	−1.36057
$986$	−14.1957	−0.452082
$987$	0	0
$988$	0	0
$989$	−18.2198	−0.579357
$990$	0	0
$991$	50.6021	1.60743	0.803714	−	0.595016i	$-0.202854\pi$
$991$	50.6021	1.60743	0.803714	+	0.595016i	$0.202854\pi$
$992$	−9.00969	−0.286058
$993$	0	0
$994$	−0.426256	−0.0135200
$995$	−13.3387	−0.422865
$996$	0	0
$997$	−35.9603	−1.13887	−0.569437	−	0.822035i	$-0.692839\pi$
$997$	−35.9603	−1.13887	−0.569437	+	0.822035i	$0.692839\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.bh.1.2		3
3.2	odd	2		1014.2.a.l.1.2	✓	3
12.11	even	2		8112.2.a.cj.1.2		3
13.5	odd	4		3042.2.b.o.1351.2		6
13.8	odd	4		3042.2.b.o.1351.5		6
13.12	even	2		3042.2.a.ba.1.2		3
39.2	even	12		1014.2.i.h.823.2		12
39.5	even	4		1014.2.b.f.337.5		6
39.8	even	4		1014.2.b.f.337.2		6
39.11	even	12		1014.2.i.h.823.5		12
39.17	odd	6		1014.2.e.l.991.2		6
39.20	even	12		1014.2.i.h.361.2		12
39.23	odd	6		1014.2.e.l.529.2		6
39.29	odd	6		1014.2.e.n.529.2		6
39.32	even	12		1014.2.i.h.361.5		12
39.35	odd	6		1014.2.e.n.991.2		6
39.38	odd	2		1014.2.a.n.1.2	yes	3
156.155	even	2		8112.2.a.cm.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1014.2.a.l.1.2	✓	3	3.2	odd	2
1014.2.a.n.1.2	yes	3	39.38	odd	2
1014.2.b.f.337.2		6	39.8	even	4
1014.2.b.f.337.5		6	39.5	even	4
1014.2.e.l.529.2		6	39.23	odd	6
1014.2.e.l.991.2		6	39.17	odd	6
1014.2.e.n.529.2		6	39.29	odd	6
1014.2.e.n.991.2		6	39.35	odd	6
1014.2.i.h.361.2		12	39.20	even	12
1014.2.i.h.361.5		12	39.32	even	12
1014.2.i.h.823.2		12	39.2	even	12
1014.2.i.h.823.5		12	39.11	even	12
3042.2.a.ba.1.2		3	13.12	even	2
3042.2.a.bh.1.2		3	1.1	even	1	trivial
3042.2.b.o.1351.2		6	13.5	odd	4
3042.2.b.o.1351.5		6	13.8	odd	4
8112.2.a.cj.1.2		3	12.11	even	2
8112.2.a.cm.1.2		3	156.155	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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.13706 q^{5} +0.0489173 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.13706 q^{5} +0.0489173 q^{7} +1.00000 q^{8} +2.13706 q^{10} +6.29590 q^{11} +0.0489173 q^{14} +1.00000 q^{16} +2.89008 q^{17} +7.20775 q^{19} +2.13706 q^{20} +6.29590 q^{22} -2.71379 q^{23} -0.432960 q^{25} +0.0489173 q^{28} -4.91185 q^{29} -9.00969 q^{31} +1.00000 q^{32} +2.89008 q^{34} +0.104539 q^{35} +0.176292 q^{37} +7.20775 q^{38} +2.13706 q^{40} -8.59179 q^{41} +6.71379 q^{43} +6.29590 q^{44} -2.71379 q^{46} +7.20775 q^{47} -6.99761 q^{49} -0.432960 q^{50} -9.34481 q^{53} +13.4547 q^{55} +0.0489173 q^{56} -4.91185 q^{58} -4.26875 q^{59} +7.10992 q^{61} -9.00969 q^{62} +1.00000 q^{64} -5.38404 q^{67} +2.89008 q^{68} +0.104539 q^{70} -8.71379 q^{71} +14.9487 q^{73} +0.176292 q^{74} +7.20775 q^{76} +0.307979 q^{77} +13.8291 q^{79} +2.13706 q^{80} -8.59179 q^{82} -11.1347 q^{83} +6.17629 q^{85} +6.71379 q^{86} +6.29590 q^{88} +3.92154 q^{89} -2.71379 q^{92} +7.20775 q^{94} +15.4034 q^{95} +2.47889 q^{97} -6.99761 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + q^{5} - 9 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + q^5 - 9 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + q^{5} - 9 q^{7} + 3 q^{8} + q^{10} + 5 q^{11} - 9 q^{14} + 3 q^{16} + 8 q^{17} + 4 q^{19} + q^{20} + 5 q^{22} + 18 q^{25} - 9 q^{28} - 11 q^{29} - 5 q^{31} + 3 q^{32} + 8 q^{34} + 4 q^{35} + 8 q^{37} + 4 q^{38} + q^{40} + 2 q^{41} + 12 q^{43} + 5 q^{44} + 4 q^{47} + 20 q^{49} + 18 q^{50} - 5 q^{53} + 18 q^{55} - 9 q^{56} - 11 q^{58} - 5 q^{59} + 22 q^{61} - 5 q^{62} + 3 q^{64} - 6 q^{67} + 8 q^{68} + 4 q^{70} - 18 q^{71} + 13 q^{73} + 8 q^{74} + 4 q^{76} + 6 q^{77} + 31 q^{79} + q^{80} + 2 q^{82} + 13 q^{83} + 26 q^{85} + 12 q^{86} + 5 q^{88} - 14 q^{89} + 4 q^{94} - 8 q^{95} + 23 q^{97} + 20 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + q^5 - 9 * q^7 + 3 * q^8 + q^10 + 5 * q^11 - 9 * q^14 + 3 * q^16 + 8 * q^17 + 4 * q^19 + q^20 + 5 * q^22 + 18 * q^25 - 9 * q^28 - 11 * q^29 - 5 * q^31 + 3 * q^32 + 8 * q^34 + 4 * q^35 + 8 * q^37 + 4 * q^38 + q^40 + 2 * q^41 + 12 * q^43 + 5 * q^44 + 4 * q^47 + 20 * q^49 + 18 * q^50 - 5 * q^53 + 18 * q^55 - 9 * q^56 - 11 * q^58 - 5 * q^59 + 22 * q^61 - 5 * q^62 + 3 * q^64 - 6 * q^67 + 8 * q^68 + 4 * q^70 - 18 * q^71 + 13 * q^73 + 8 * q^74 + 4 * q^76 + 6 * q^77 + 31 * q^79 + q^80 + 2 * q^82 + 13 * q^83 + 26 * q^85 + 12 * q^86 + 5 * q^88 - 14 * q^89 + 4 * q^94 - 8 * q^95 + 23 * q^97 + 20 * q^98

Introduction

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