LMFDB - Embedded newform 4014.2.a.y.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, 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("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 14x^{6} + 28x^{5} + 43x^{4} - 90x^{3} - 23x^{2} + 82x - 28 $ x^8 - 2x^7 - 14x^6 + 28x^5 + 43x^4 - 90x^3 - 23x^2 + 82*x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$3.17253$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.585733 q^{5} -1.49689 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +0.585733 q^{5} -1.49689 q^{7} +1.00000 q^{8} +0.585733 q^{10} +3.18036 q^{11} -6.04779 q^{13} -1.49689 q^{14} +1.00000 q^{16} -4.54325 q^{17} +4.74794 q^{19} +0.585733 q^{20} +3.18036 q^{22} -1.06929 q^{23} -4.65692 q^{25} -6.04779 q^{26} -1.49689 q^{28} -10.4462 q^{29} -6.42013 q^{31} +1.00000 q^{32} -4.54325 q^{34} -0.876775 q^{35} -9.98632 q^{37} +4.74794 q^{38} +0.585733 q^{40} -3.37739 q^{41} -1.53155 q^{43} +3.18036 q^{44} -1.06929 q^{46} +13.0787 q^{47} -4.75933 q^{49} -4.65692 q^{50} -6.04779 q^{52} +7.71455 q^{53} +1.86284 q^{55} -1.49689 q^{56} -10.4462 q^{58} +3.21453 q^{59} +5.85140 q^{61} -6.42013 q^{62} +1.00000 q^{64} -3.54239 q^{65} +11.3418 q^{67} -4.54325 q^{68} -0.876775 q^{70} -15.3383 q^{71} +14.9439 q^{73} -9.98632 q^{74} +4.74794 q^{76} -4.76063 q^{77} +4.50495 q^{79} +0.585733 q^{80} -3.37739 q^{82} -1.09046 q^{83} -2.66113 q^{85} -1.53155 q^{86} +3.18036 q^{88} -4.43131 q^{89} +9.05285 q^{91} -1.06929 q^{92} +13.0787 q^{94} +2.78102 q^{95} -7.60980 q^{97} -4.75933 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} - 6 q^{5} - 6 q^{7} + 8 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^4 - 6 * q^5 - 6 * q^7 + 8 * q^8	$ 8 q + 8 q^{2} + 8 q^{4} - 6 q^{5} - 6 q^{7} + 8 q^{8} - 6 q^{10} - 11 q^{11} - q^{13} - 6 q^{14} + 8 q^{16} - 16 q^{17} - q^{19} - 6 q^{20} - 11 q^{22} - 14 q^{23} + 10 q^{25} - q^{26} - 6 q^{28} - 21 q^{29} - 6 q^{31} + 8 q^{32} - 16 q^{34} - 8 q^{35} - 14 q^{37} - q^{38} - 6 q^{40} - 16 q^{41} - 29 q^{43} - 11 q^{44} - 14 q^{46} - 9 q^{47} - 2 q^{49} + 10 q^{50} - q^{52} - 11 q^{53} - 22 q^{55} - 6 q^{56} - 21 q^{58} - 21 q^{59} + 3 q^{61} - 6 q^{62} + 8 q^{64} - 24 q^{65} - 20 q^{67} - 16 q^{68} - 8 q^{70} - 32 q^{71} + 13 q^{73} - 14 q^{74} - q^{76} - 4 q^{77} + 21 q^{79} - 6 q^{80} - 16 q^{82} - 28 q^{83} - 14 q^{85} - 29 q^{86} - 11 q^{88} - 54 q^{89} - 36 q^{91} - 14 q^{92} - 9 q^{94} - 30 q^{95} + 10 q^{97} - 2 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 - 6 * q^5 - 6 * q^7 + 8 * q^8 - 6 * q^10 - 11 * q^11 - q^13 - 6 * q^14 + 8 * q^16 - 16 * q^17 - q^19 - 6 * q^20 - 11 * q^22 - 14 * q^23 + 10 * q^25 - q^26 - 6 * q^28 - 21 * q^29 - 6 * q^31 + 8 * q^32 - 16 * q^34 - 8 * q^35 - 14 * q^37 - q^38 - 6 * q^40 - 16 * q^41 - 29 * q^43 - 11 * q^44 - 14 * q^46 - 9 * q^47 - 2 * q^49 + 10 * q^50 - q^52 - 11 * q^53 - 22 * q^55 - 6 * q^56 - 21 * q^58 - 21 * q^59 + 3 * q^61 - 6 * q^62 + 8 * q^64 - 24 * q^65 - 20 * q^67 - 16 * q^68 - 8 * q^70 - 32 * q^71 + 13 * q^73 - 14 * q^74 - q^76 - 4 * q^77 + 21 * q^79 - 6 * q^80 - 16 * q^82 - 28 * q^83 - 14 * q^85 - 29 * q^86 - 11 * q^88 - 54 * q^89 - 36 * q^91 - 14 * q^92 - 9 * q^94 - 30 * q^95 + 10 * q^97 - 2 * 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
$5$	0.585733	0.261948	0.130974	−	0.991386i	$-0.458190\pi$
$5$	0.585733	0.261948	0.130974	+	0.991386i	$0.458190\pi$
$6$	0	0
$7$	−1.49689	−0.565770	−0.282885	−	0.959154i	$-0.591291\pi$
$7$	−1.49689	−0.565770	−0.282885	+	0.959154i	$0.591291\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0.585733	0.185225
$11$	3.18036	0.958914	0.479457	−	0.877565i	$-0.340834\pi$
$11$	3.18036	0.958914	0.479457	+	0.877565i	$0.340834\pi$
$12$	0	0
$13$	−6.04779	−1.67735	−0.838677	−	0.544629i	$-0.816671\pi$
$13$	−6.04779	−1.67735	−0.838677	+	0.544629i	$0.816671\pi$
$14$	−1.49689	−0.400060
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.54325	−1.10190	−0.550949	−	0.834539i	$-0.685734\pi$
$17$	−4.54325	−1.10190	−0.550949	+	0.834539i	$0.685734\pi$
$18$	0	0
$19$	4.74794	1.08925	0.544626	−	0.838679i	$-0.316672\pi$
$19$	4.74794	1.08925	0.544626	+	0.838679i	$0.316672\pi$
$20$	0.585733	0.130974
$21$	0	0
$22$	3.18036	0.678054
$23$	−1.06929	−0.222962	−0.111481	−	0.993767i	$-0.535559\pi$
$23$	−1.06929	−0.222962	−0.111481	+	0.993767i	$0.535559\pi$
$24$	0	0
$25$	−4.65692	−0.931383
$26$	−6.04779	−1.18607
$27$	0	0
$28$	−1.49689	−0.282885
$29$	−10.4462	−1.93980	−0.969901	−	0.243499i	$-0.921705\pi$
$29$	−10.4462	−1.93980	−0.969901	+	0.243499i	$0.921705\pi$
$30$	0	0
$31$	−6.42013	−1.15309	−0.576545	−	0.817066i	$-0.695599\pi$
$31$	−6.42013	−1.15309	−0.576545	+	0.817066i	$0.695599\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.54325	−0.779160
$35$	−0.876775	−0.148202
$36$	0	0
$37$	−9.98632	−1.64174	−0.820871	−	0.571114i	$-0.806511\pi$
$37$	−9.98632	−1.64174	−0.820871	+	0.571114i	$0.806511\pi$
$38$	4.74794	0.770217
$39$	0	0
$40$	0.585733	0.0926125
$41$	−3.37739	−0.527459	−0.263730	−	0.964597i	$-0.584953\pi$
$41$	−3.37739	−0.527459	−0.263730	+	0.964597i	$0.584953\pi$
$42$	0	0
$43$	−1.53155	−0.233559	−0.116779	−	0.993158i	$-0.537257\pi$
$43$	−1.53155	−0.233559	−0.116779	+	0.993158i	$0.537257\pi$
$44$	3.18036	0.479457
$45$	0	0
$46$	−1.06929	−0.157658
$47$	13.0787	1.90772	0.953862	−	0.300246i	$-0.0970689\pi$
$47$	13.0787	1.90772	0.953862	+	0.300246i	$0.0970689\pi$
$48$	0	0
$49$	−4.75933	−0.679904
$50$	−4.65692	−0.658588
$51$	0	0
$52$	−6.04779	−0.838677
$53$	7.71455	1.05967	0.529837	−	0.848099i	$-0.322253\pi$
$53$	7.71455	1.05967	0.529837	+	0.848099i	$0.322253\pi$
$54$	0	0
$55$	1.86284	0.251185
$56$	−1.49689	−0.200030
$57$	0	0
$58$	−10.4462	−1.37165
$59$	3.21453	0.418496	0.209248	−	0.977863i	$-0.432898\pi$
$59$	3.21453	0.418496	0.209248	+	0.977863i	$0.432898\pi$
$60$	0	0
$61$	5.85140	0.749195	0.374597	−	0.927188i	$-0.377781\pi$
$61$	5.85140	0.749195	0.374597	+	0.927188i	$0.377781\pi$
$62$	−6.42013	−0.815357
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.54239	−0.439379
$66$	0	0
$67$	11.3418	1.38562	0.692812	−	0.721119i	$-0.256372\pi$
$67$	11.3418	1.38562	0.692812	+	0.721119i	$0.256372\pi$
$68$	−4.54325	−0.550949
$69$	0	0
$70$	−0.876775	−0.104795
$71$	−15.3383	−1.82032	−0.910161	−	0.414255i	$-0.864042\pi$
$71$	−15.3383	−1.82032	−0.910161	+	0.414255i	$0.864042\pi$
$72$	0	0
$73$	14.9439	1.74905	0.874523	−	0.484983i	$-0.161174\pi$
$73$	14.9439	1.74905	0.874523	+	0.484983i	$0.161174\pi$
$74$	−9.98632	−1.16089
$75$	0	0
$76$	4.74794	0.544626
$77$	−4.76063	−0.542525
$78$	0	0
$79$	4.50495	0.506846	0.253423	−	0.967356i	$-0.418444\pi$
$79$	4.50495	0.506846	0.253423	+	0.967356i	$0.418444\pi$
$80$	0.585733	0.0654869
$81$	0	0
$82$	−3.37739	−0.372970
$83$	−1.09046	−0.119694	−0.0598469	−	0.998208i	$-0.519061\pi$
$83$	−1.09046	−0.119694	−0.0598469	+	0.998208i	$0.519061\pi$
$84$	0	0
$85$	−2.66113	−0.288640
$86$	−1.53155	−0.165151
$87$	0	0
$88$	3.18036	0.339027
$89$	−4.43131	−0.469718	−0.234859	−	0.972029i	$-0.575463\pi$
$89$	−4.43131	−0.469718	−0.234859	+	0.972029i	$0.575463\pi$
$90$	0	0
$91$	9.05285	0.948997
$92$	−1.06929	−0.111481
$93$	0	0
$94$	13.0787	1.34896
$95$	2.78102	0.285327
$96$	0	0
$97$	−7.60980	−0.772658	−0.386329	−	0.922361i	$-0.626257\pi$
$97$	−7.60980	−0.772658	−0.386329	+	0.922361i	$0.626257\pi$
$98$	−4.75933	−0.480765
$99$	0	0
$100$	−4.65692	−0.465692
$101$	−11.3248	−1.12686	−0.563431	−	0.826163i	$-0.690519\pi$
$101$	−11.3248	−1.12686	−0.563431	+	0.826163i	$0.690519\pi$
$102$	0	0
$103$	−15.8368	−1.56045	−0.780224	−	0.625501i	$-0.784895\pi$
$103$	−15.8368	−1.56045	−0.780224	+	0.625501i	$0.784895\pi$
$104$	−6.04779	−0.593034
$105$	0	0
$106$	7.71455	0.749303
$107$	−8.73169	−0.844125	−0.422062	−	0.906567i	$-0.638694\pi$
$107$	−8.73169	−0.844125	−0.422062	+	0.906567i	$0.638694\pi$
$108$	0	0
$109$	−1.93898	−0.185720	−0.0928601	−	0.995679i	$-0.529601\pi$
$109$	−1.93898	−0.185720	−0.0928601	+	0.995679i	$0.529601\pi$
$110$	1.86284	0.177615
$111$	0	0
$112$	−1.49689	−0.141442
$113$	−11.9224	−1.12156	−0.560781	−	0.827964i	$-0.689499\pi$
$113$	−11.9224	−1.12156	−0.560781	+	0.827964i	$0.689499\pi$
$114$	0	0
$115$	−0.626318	−0.0584044
$116$	−10.4462	−0.969901
$117$	0	0
$118$	3.21453	0.295921
$119$	6.80072	0.623421
$120$	0	0
$121$	−0.885329	−0.0804845
$122$	5.85140	0.529761
$123$	0	0
$124$	−6.42013	−0.576545
$125$	−5.65637	−0.505921
$126$	0	0
$127$	2.46517	0.218748	0.109374	−	0.994001i	$-0.465115\pi$
$127$	2.46517	0.218748	0.109374	+	0.994001i	$0.465115\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−3.54239	−0.310688
$131$	−1.07564	−0.0939791	−0.0469895	−	0.998895i	$-0.514963\pi$
$131$	−1.07564	−0.0939791	−0.0469895	+	0.998895i	$0.514963\pi$
$132$	0	0
$133$	−7.10712	−0.616266
$134$	11.3418	0.979784
$135$	0	0
$136$	−4.54325	−0.389580
$137$	7.49845	0.640636	0.320318	−	0.947310i	$-0.396210\pi$
$137$	7.49845	0.640636	0.320318	+	0.947310i	$0.396210\pi$
$138$	0	0
$139$	−6.50751	−0.551960	−0.275980	−	0.961163i	$-0.589002\pi$
$139$	−6.50751	−0.551960	−0.275980	+	0.961163i	$0.589002\pi$
$140$	−0.876775	−0.0741010
$141$	0	0
$142$	−15.3383	−1.28716
$143$	−19.2341	−1.60844
$144$	0	0
$145$	−6.11865	−0.508127
$146$	14.9439	1.23676
$147$	0	0
$148$	−9.98632	−0.820871
$149$	−13.2807	−1.08800	−0.543999	−	0.839086i	$-0.683091\pi$
$149$	−13.2807	−1.08800	−0.543999	+	0.839086i	$0.683091\pi$
$150$	0	0
$151$	2.77304	0.225667	0.112833	−	0.993614i	$-0.464007\pi$
$151$	2.77304	0.225667	0.112833	+	0.993614i	$0.464007\pi$
$152$	4.74794	0.385109
$153$	0	0
$154$	−4.76063	−0.383623
$155$	−3.76048	−0.302049
$156$	0	0
$157$	3.97992	0.317633	0.158816	−	0.987308i	$-0.449232\pi$
$157$	3.97992	0.317633	0.158816	+	0.987308i	$0.449232\pi$
$158$	4.50495	0.358394
$159$	0	0
$160$	0.585733	0.0463062
$161$	1.60060	0.126145
$162$	0	0
$163$	17.0742	1.33735	0.668677	−	0.743553i	$-0.266861\pi$
$163$	17.0742	1.33735	0.668677	+	0.743553i	$0.266861\pi$
$164$	−3.37739	−0.263730
$165$	0	0
$166$	−1.09046	−0.0846362
$167$	−3.88787	−0.300853	−0.150426	−	0.988621i	$-0.548065\pi$
$167$	−3.88787	−0.300853	−0.150426	+	0.988621i	$0.548065\pi$
$168$	0	0
$169$	23.5757	1.81352
$170$	−2.66113	−0.204099
$171$	0	0
$172$	−1.53155	−0.116779
$173$	−6.81516	−0.518147	−0.259074	−	0.965858i	$-0.583417\pi$
$173$	−6.81516	−0.518147	−0.259074	+	0.965858i	$0.583417\pi$
$174$	0	0
$175$	6.97088	0.526949
$176$	3.18036	0.239728
$177$	0	0
$178$	−4.43131	−0.332141
$179$	1.64779	0.123162	0.0615808	−	0.998102i	$-0.480386\pi$
$179$	1.64779	0.123162	0.0615808	+	0.998102i	$0.480386\pi$
$180$	0	0
$181$	16.1341	1.19924	0.599621	−	0.800284i	$-0.295318\pi$
$181$	16.1341	1.19924	0.599621	+	0.800284i	$0.295318\pi$
$182$	9.05285	0.671042
$183$	0	0
$184$	−1.06929	−0.0788291
$185$	−5.84932	−0.430050
$186$	0	0
$187$	−14.4491	−1.05663
$188$	13.0787	0.953862
$189$	0	0
$190$	2.78102	0.201757
$191$	0.988248	0.0715071	0.0357536	−	0.999361i	$-0.488617\pi$
$191$	0.988248	0.0715071	0.0357536	+	0.999361i	$0.488617\pi$
$192$	0	0
$193$	16.2266	1.16801	0.584006	−	0.811749i	$-0.301484\pi$
$193$	16.2266	1.16801	0.584006	+	0.811749i	$0.301484\pi$
$194$	−7.60980	−0.546351
$195$	0	0
$196$	−4.75933	−0.339952
$197$	21.1776	1.50884	0.754422	−	0.656389i	$-0.227917\pi$
$197$	21.1776	1.50884	0.754422	+	0.656389i	$0.227917\pi$
$198$	0	0
$199$	−5.43075	−0.384975	−0.192488	−	0.981299i	$-0.561656\pi$
$199$	−5.43075	−0.384975	−0.192488	+	0.981299i	$0.561656\pi$
$200$	−4.65692	−0.329294
$201$	0	0
$202$	−11.3248	−0.796812
$203$	15.6367	1.09748
$204$	0	0
$205$	−1.97825	−0.138167
$206$	−15.8368	−1.10340
$207$	0	0
$208$	−6.04779	−0.419339
$209$	15.1001	1.04450
$210$	0	0
$211$	8.70251	0.599105	0.299553	−	0.954080i	$-0.403163\pi$
$211$	8.70251	0.599105	0.299553	+	0.954080i	$0.403163\pi$
$212$	7.71455	0.529837
$213$	0	0
$214$	−8.73169	−0.596886
$215$	−0.897078	−0.0611802
$216$	0	0
$217$	9.61020	0.652383
$218$	−1.93898	−0.131324
$219$	0	0
$220$	1.86284	0.125593
$221$	27.4766	1.84827
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	−1.49689	−0.100015
$225$	0	0
$226$	−11.9224	−0.793064
$227$	−5.33837	−0.354320	−0.177160	−	0.984182i	$-0.556691\pi$
$227$	−5.33837	−0.354320	−0.177160	+	0.984182i	$0.556691\pi$
$228$	0	0
$229$	−16.9242	−1.11838	−0.559191	−	0.829039i	$-0.688888\pi$
$229$	−16.9242	−1.11838	−0.559191	+	0.829039i	$0.688888\pi$
$230$	−0.626318	−0.0412982
$231$	0	0
$232$	−10.4462	−0.685824
$233$	−16.5149	−1.08193	−0.540963	−	0.841046i	$-0.681940\pi$
$233$	−16.5149	−1.08193	−0.540963	+	0.841046i	$0.681940\pi$
$234$	0	0
$235$	7.66062	0.499724
$236$	3.21453	0.209248
$237$	0	0
$238$	6.80072	0.440825
$239$	−14.8861	−0.962903	−0.481451	−	0.876473i	$-0.659890\pi$
$239$	−14.8861	−0.962903	−0.481451	+	0.876473i	$0.659890\pi$
$240$	0	0
$241$	1.70313	0.109708	0.0548541	−	0.998494i	$-0.482531\pi$
$241$	1.70313	0.109708	0.0548541	+	0.998494i	$0.482531\pi$
$242$	−0.885329	−0.0569111
$243$	0	0
$244$	5.85140	0.374597
$245$	−2.78770	−0.178099
$246$	0	0
$247$	−28.7145	−1.82706
$248$	−6.42013	−0.407679
$249$	0	0
$250$	−5.65637	−0.357740
$251$	−28.4403	−1.79514	−0.897569	−	0.440873i	$-0.854669\pi$
$251$	−28.4403	−1.79514	−0.897569	+	0.440873i	$0.854669\pi$
$252$	0	0
$253$	−3.40072	−0.213802
$254$	2.46517	0.154678
$255$	0	0
$256$	1.00000	0.0625000
$257$	−17.2851	−1.07821	−0.539106	−	0.842238i	$-0.681238\pi$
$257$	−17.2851	−1.07821	−0.539106	+	0.842238i	$0.681238\pi$
$258$	0	0
$259$	14.9484	0.928848
$260$	−3.54239	−0.219690
$261$	0	0
$262$	−1.07564	−0.0664532
$263$	3.84344	0.236996	0.118498	−	0.992954i	$-0.462192\pi$
$263$	3.84344	0.236996	0.118498	+	0.992954i	$0.462192\pi$
$264$	0	0
$265$	4.51866	0.277579
$266$	−7.10712	−0.435766
$267$	0	0
$268$	11.3418	0.692812
$269$	1.43863	0.0877150	0.0438575	−	0.999038i	$-0.486035\pi$
$269$	1.43863	0.0877150	0.0438575	+	0.999038i	$0.486035\pi$
$270$	0	0
$271$	−8.44794	−0.513176	−0.256588	−	0.966521i	$-0.582598\pi$
$271$	−8.44794	−0.513176	−0.256588	+	0.966521i	$0.582598\pi$
$272$	−4.54325	−0.275475
$273$	0	0
$274$	7.49845	0.452998
$275$	−14.8107	−0.893116
$276$	0	0
$277$	20.1492	1.21065	0.605324	−	0.795979i	$-0.293043\pi$
$277$	20.1492	1.21065	0.605324	+	0.795979i	$0.293043\pi$
$278$	−6.50751	−0.390294
$279$	0	0
$280$	−0.876775	−0.0523973
$281$	−0.863739	−0.0515264	−0.0257632	−	0.999668i	$-0.508202\pi$
$281$	−0.863739	−0.0515264	−0.0257632	+	0.999668i	$0.508202\pi$
$282$	0	0
$283$	14.7643	0.877646	0.438823	−	0.898573i	$-0.355395\pi$
$283$	14.7643	0.877646	0.438823	+	0.898573i	$0.355395\pi$
$284$	−15.3383	−0.910161
$285$	0	0
$286$	−19.2341	−1.13734
$287$	5.05556	0.298421
$288$	0	0
$289$	3.64108	0.214181
$290$	−6.11865	−0.359300
$291$	0	0
$292$	14.9439	0.874523
$293$	27.7810	1.62298	0.811491	−	0.584364i	$-0.198656\pi$
$293$	27.7810	1.62298	0.811491	+	0.584364i	$0.198656\pi$
$294$	0	0
$295$	1.88285	0.109624
$296$	−9.98632	−0.580443
$297$	0	0
$298$	−13.2807	−0.769331
$299$	6.46684	0.373987
$300$	0	0
$301$	2.29255	0.132141
$302$	2.77304	0.159570
$303$	0	0
$304$	4.74794	0.272313
$305$	3.42735	0.196250
$306$	0	0
$307$	−24.5648	−1.40199	−0.700994	−	0.713167i	$-0.747260\pi$
$307$	−24.5648	−1.40199	−0.700994	+	0.713167i	$0.747260\pi$
$308$	−4.76063	−0.271262
$309$	0	0
$310$	−3.76048	−0.213581
$311$	−23.3167	−1.32217	−0.661083	−	0.750313i	$-0.729903\pi$
$311$	−23.3167	−1.32217	−0.661083	+	0.750313i	$0.729903\pi$
$312$	0	0
$313$	33.8040	1.91071	0.955357	−	0.295453i	$-0.0954704\pi$
$313$	33.8040	1.91071	0.955357	+	0.295453i	$0.0954704\pi$
$314$	3.97992	0.224600
$315$	0	0
$316$	4.50495	0.253423
$317$	−28.4355	−1.59709	−0.798547	−	0.601932i	$-0.794398\pi$
$317$	−28.4355	−1.59709	−0.798547	+	0.601932i	$0.794398\pi$
$318$	0	0
$319$	−33.2225	−1.86010
$320$	0.585733	0.0327435
$321$	0	0
$322$	1.60060	0.0891982
$323$	−21.5710	−1.20024
$324$	0	0
$325$	28.1640	1.56226
$326$	17.0742	0.945652
$327$	0	0
$328$	−3.37739	−0.186485
$329$	−19.5773	−1.07933
$330$	0	0
$331$	−33.1657	−1.82295	−0.911475	−	0.411356i	$-0.865055\pi$
$331$	−33.1657	−1.82295	−0.911475	+	0.411356i	$0.865055\pi$
$332$	−1.09046	−0.0598469
$333$	0	0
$334$	−3.88787	−0.212735
$335$	6.64327	0.362961
$336$	0	0
$337$	6.40874	0.349107	0.174553	−	0.984648i	$-0.444152\pi$
$337$	6.40874	0.349107	0.174553	+	0.984648i	$0.444152\pi$
$338$	23.5757	1.28235
$339$	0	0
$340$	−2.66113	−0.144320
$341$	−20.4183	−1.10571
$342$	0	0
$343$	17.6024	0.950439
$344$	−1.53155	−0.0825756
$345$	0	0
$346$	−6.81516	−0.366385
$347$	30.3873	1.63128	0.815639	−	0.578562i	$-0.196386\pi$
$347$	30.3873	1.63128	0.815639	+	0.578562i	$0.196386\pi$
$348$	0	0
$349$	13.1951	0.706319	0.353159	−	0.935563i	$-0.385107\pi$
$349$	13.1951	0.706319	0.353159	+	0.935563i	$0.385107\pi$
$350$	6.97088	0.372609
$351$	0	0
$352$	3.18036	0.169514
$353$	−16.0690	−0.855265	−0.427633	−	0.903953i	$-0.640652\pi$
$353$	−16.0690	−0.855265	−0.427633	+	0.903953i	$0.640652\pi$
$354$	0	0
$355$	−8.98414	−0.476829
$356$	−4.43131	−0.234859
$357$	0	0
$358$	1.64779	0.0870884
$359$	−10.4685	−0.552507	−0.276253	−	0.961085i	$-0.589093\pi$
$359$	−10.4685	−0.552507	−0.276253	+	0.961085i	$0.589093\pi$
$360$	0	0
$361$	3.54291	0.186469
$362$	16.1341	0.847992
$363$	0	0
$364$	9.05285	0.474498
$365$	8.75311	0.458159
$366$	0	0
$367$	26.1262	1.36378	0.681888	−	0.731457i	$-0.261159\pi$
$367$	26.1262	1.36378	0.681888	+	0.731457i	$0.261159\pi$
$368$	−1.06929	−0.0557406
$369$	0	0
$370$	−5.84932	−0.304091
$371$	−11.5478	−0.599532
$372$	0	0
$373$	−26.2059	−1.35689	−0.678446	−	0.734650i	$-0.737346\pi$
$373$	−26.2059	−1.35689	−0.678446	+	0.734650i	$0.737346\pi$
$374$	−14.4491	−0.747147
$375$	0	0
$376$	13.0787	0.674482
$377$	63.1761	3.25374
$378$	0	0
$379$	32.9269	1.69134	0.845670	−	0.533706i	$-0.179201\pi$
$379$	32.9269	1.69134	0.845670	+	0.533706i	$0.179201\pi$
$380$	2.78102	0.142663
$381$	0	0
$382$	0.988248	0.0505632
$383$	20.5468	1.04989	0.524946	−	0.851136i	$-0.324086\pi$
$383$	20.5468	1.04989	0.524946	+	0.851136i	$0.324086\pi$
$384$	0	0
$385$	−2.78846	−0.142113
$386$	16.2266	0.825910
$387$	0	0
$388$	−7.60980	−0.386329
$389$	22.5931	1.14551	0.572757	−	0.819725i	$-0.305874\pi$
$389$	22.5931	1.14551	0.572757	+	0.819725i	$0.305874\pi$
$390$	0	0
$391$	4.85804	0.245682
$392$	−4.75933	−0.240383
$393$	0	0
$394$	21.1776	1.06691
$395$	2.63870	0.132767
$396$	0	0
$397$	7.21751	0.362236	0.181118	−	0.983461i	$-0.442028\pi$
$397$	7.21751	0.362236	0.181118	+	0.983461i	$0.442028\pi$
$398$	−5.43075	−0.272219
$399$	0	0
$400$	−4.65692	−0.232846
$401$	−11.3770	−0.568143	−0.284071	−	0.958803i	$-0.591685\pi$
$401$	−11.3770	−0.568143	−0.284071	+	0.958803i	$0.591685\pi$
$402$	0	0
$403$	38.8276	1.93414
$404$	−11.3248	−0.563431
$405$	0	0
$406$	15.6367	0.776037
$407$	−31.7601	−1.57429
$408$	0	0
$409$	2.13979	0.105806	0.0529030	−	0.998600i	$-0.483153\pi$
$409$	2.13979	0.105806	0.0529030	+	0.998600i	$0.483153\pi$
$410$	−1.97825	−0.0976986
$411$	0	0
$412$	−15.8368	−0.780224
$413$	−4.81178	−0.236772
$414$	0	0
$415$	−0.638719	−0.0313535
$416$	−6.04779	−0.296517
$417$	0	0
$418$	15.1001	0.738572
$419$	33.3352	1.62853	0.814266	−	0.580493i	$-0.197140\pi$
$419$	33.3352	1.62853	0.814266	+	0.580493i	$0.197140\pi$
$420$	0	0
$421$	−25.1838	−1.22738	−0.613691	−	0.789546i	$-0.710316\pi$
$421$	−25.1838	−1.22738	−0.613691	+	0.789546i	$0.710316\pi$
$422$	8.70251	0.423631
$423$	0	0
$424$	7.71455	0.374652
$425$	21.1575	1.02629
$426$	0	0
$427$	−8.75888	−0.423872
$428$	−8.73169	−0.422062
$429$	0	0
$430$	−0.897078	−0.0432609
$431$	−21.3427	−1.02804	−0.514020	−	0.857778i	$-0.671844\pi$
$431$	−21.3427	−1.02804	−0.514020	+	0.857778i	$0.671844\pi$
$432$	0	0
$433$	31.4723	1.51246	0.756231	−	0.654305i	$-0.227039\pi$
$433$	31.4723	1.51246	0.756231	+	0.654305i	$0.227039\pi$
$434$	9.61020	0.461305
$435$	0	0
$436$	−1.93898	−0.0928601
$437$	−5.07692	−0.242862
$438$	0	0
$439$	8.30672	0.396458	0.198229	−	0.980156i	$-0.436481\pi$
$439$	8.30672	0.396458	0.198229	+	0.980156i	$0.436481\pi$
$440$	1.86284	0.0888074
$441$	0	0
$442$	27.4766	1.30693
$443$	5.18961	0.246566	0.123283	−	0.992372i	$-0.460658\pi$
$443$	5.18961	0.246566	0.123283	+	0.992372i	$0.460658\pi$
$444$	0	0
$445$	−2.59556	−0.123041
$446$	1.00000	0.0473514
$447$	0	0
$448$	−1.49689	−0.0707212
$449$	−12.2820	−0.579623	−0.289811	−	0.957084i	$-0.593593\pi$
$449$	−12.2820	−0.579623	−0.289811	+	0.957084i	$0.593593\pi$
$450$	0	0
$451$	−10.7413	−0.505788
$452$	−11.9224	−0.560781
$453$	0	0
$454$	−5.33837	−0.250542
$455$	5.30255	0.248587
$456$	0	0
$457$	36.6297	1.71347	0.856733	−	0.515760i	$-0.172490\pi$
$457$	36.6297	1.71347	0.856733	+	0.515760i	$0.172490\pi$
$458$	−16.9242	−0.790815
$459$	0	0
$460$	−0.626318	−0.0292022
$461$	−23.3611	−1.08804	−0.544018	−	0.839074i	$-0.683098\pi$
$461$	−23.3611	−1.08804	−0.544018	+	0.839074i	$0.683098\pi$
$462$	0	0
$463$	−31.6070	−1.46890	−0.734451	−	0.678661i	$-0.762560\pi$
$463$	−31.6070	−1.46890	−0.734451	+	0.678661i	$0.762560\pi$
$464$	−10.4462	−0.484951
$465$	0	0
$466$	−16.5149	−0.765037
$467$	41.2962	1.91096	0.955479	−	0.295058i	$-0.0953391\pi$
$467$	41.2962	1.91096	0.955479	+	0.295058i	$0.0953391\pi$
$468$	0	0
$469$	−16.9774	−0.783944
$470$	7.66062	0.353358
$471$	0	0
$472$	3.21453	0.147961
$473$	−4.87087	−0.223963
$474$	0	0
$475$	−22.1108	−1.01451
$476$	6.80072	0.311711
$477$	0	0
$478$	−14.8861	−0.680875
$479$	18.4305	0.842112	0.421056	−	0.907035i	$-0.361660\pi$
$479$	18.4305	0.842112	0.421056	+	0.907035i	$0.361660\pi$
$480$	0	0
$481$	60.3952	2.75378
$482$	1.70313	0.0775754
$483$	0	0
$484$	−0.885329	−0.0402422
$485$	−4.45731	−0.202396
$486$	0	0
$487$	−10.4137	−0.471891	−0.235945	−	0.971766i	$-0.575819\pi$
$487$	−10.4137	−0.471891	−0.235945	+	0.971766i	$0.575819\pi$
$488$	5.85140	0.264880
$489$	0	0
$490$	−2.78770	−0.125935
$491$	23.7037	1.06973	0.534867	−	0.844936i	$-0.320362\pi$
$491$	23.7037	1.06973	0.534867	+	0.844936i	$0.320362\pi$
$492$	0	0
$493$	47.4594	2.13747
$494$	−28.7145	−1.29193
$495$	0	0
$496$	−6.42013	−0.288272
$497$	22.9597	1.02988
$498$	0	0
$499$	−24.2061	−1.08361	−0.541807	−	0.840503i	$-0.682260\pi$
$499$	−24.2061	−1.08361	−0.541807	+	0.840503i	$0.682260\pi$
$500$	−5.65637	−0.252961
$501$	0	0
$502$	−28.4403	−1.26935
$503$	−3.47026	−0.154731	−0.0773657	−	0.997003i	$-0.524651\pi$
$503$	−3.47026	−0.154731	−0.0773657	+	0.997003i	$0.524651\pi$
$504$	0	0
$505$	−6.63332	−0.295179
$506$	−3.40072	−0.151181
$507$	0	0
$508$	2.46517	0.109374
$509$	−1.53550	−0.0680600	−0.0340300	−	0.999421i	$-0.510834\pi$
$509$	−1.53550	−0.0680600	−0.0340300	+	0.999421i	$0.510834\pi$
$510$	0	0
$511$	−22.3693	−0.989558
$512$	1.00000	0.0441942
$513$	0	0
$514$	−17.2851	−0.762411
$515$	−9.27614	−0.408755
$516$	0	0
$517$	41.5949	1.82934
$518$	14.9484	0.656795
$519$	0	0
$520$	−3.54239	−0.155344
$521$	16.9837	0.744068	0.372034	−	0.928219i	$-0.378660\pi$
$521$	16.9837	0.744068	0.372034	+	0.928219i	$0.378660\pi$
$522$	0	0
$523$	−2.10471	−0.0920326	−0.0460163	−	0.998941i	$-0.514653\pi$
$523$	−2.10471	−0.0920326	−0.0460163	+	0.998941i	$0.514653\pi$
$524$	−1.07564	−0.0469895
$525$	0	0
$526$	3.84344	0.167582
$527$	29.1682	1.27059
$528$	0	0
$529$	−21.8566	−0.950288
$530$	4.51866	0.196278
$531$	0	0
$532$	−7.10712	−0.308133
$533$	20.4257	0.884736
$534$	0	0
$535$	−5.11444	−0.221116
$536$	11.3418	0.489892
$537$	0	0
$538$	1.43863	0.0620239
$539$	−15.1364	−0.651970
$540$	0	0
$541$	13.3108	0.572277	0.286138	−	0.958188i	$-0.407628\pi$
$541$	13.3108	0.572277	0.286138	+	0.958188i	$0.407628\pi$
$542$	−8.44794	−0.362870
$543$	0	0
$544$	−4.54325	−0.194790
$545$	−1.13572	−0.0486490
$546$	0	0
$547$	−11.3342	−0.484617	−0.242308	−	0.970199i	$-0.577905\pi$
$547$	−11.3342	−0.484617	−0.242308	+	0.970199i	$0.577905\pi$
$548$	7.49845	0.320318
$549$	0	0
$550$	−14.8107	−0.631529
$551$	−49.5977	−2.11293
$552$	0	0
$553$	−6.74340	−0.286758
$554$	20.1492	0.856058
$555$	0	0
$556$	−6.50751	−0.275980
$557$	−37.7363	−1.59894	−0.799470	−	0.600706i	$-0.794886\pi$
$557$	−37.7363	−1.59894	−0.799470	+	0.600706i	$0.794886\pi$
$558$	0	0
$559$	9.26248	0.391761
$560$	−0.876775	−0.0370505
$561$	0	0
$562$	−0.863739	−0.0364346
$563$	18.2752	0.770209	0.385105	−	0.922873i	$-0.374165\pi$
$563$	18.2752	0.770209	0.385105	+	0.922873i	$0.374165\pi$
$564$	0	0
$565$	−6.98332	−0.293790
$566$	14.7643	0.620590
$567$	0	0
$568$	−15.3383	−0.643581
$569$	−12.2166	−0.512146	−0.256073	−	0.966657i	$-0.582429\pi$
$569$	−12.2166	−0.512146	−0.256073	+	0.966657i	$0.582429\pi$
$570$	0	0
$571$	0.810246	0.0339077	0.0169539	−	0.999856i	$-0.494603\pi$
$571$	0.810246	0.0339077	0.0169539	+	0.999856i	$0.494603\pi$
$572$	−19.2341	−0.804219
$573$	0	0
$574$	5.05556	0.211015
$575$	4.97959	0.207663
$576$	0	0
$577$	−19.9944	−0.832376	−0.416188	−	0.909279i	$-0.636634\pi$
$577$	−19.9944	−0.832376	−0.416188	+	0.909279i	$0.636634\pi$
$578$	3.64108	0.151449
$579$	0	0
$580$	−6.11865	−0.254063
$581$	1.63230	0.0677191
$582$	0	0
$583$	24.5350	1.01614
$584$	14.9439	0.618381
$585$	0	0
$586$	27.7810	1.14762
$587$	−15.1575	−0.625617	−0.312808	−	0.949816i	$-0.601270\pi$
$587$	−15.1575	−0.625617	−0.312808	+	0.949816i	$0.601270\pi$
$588$	0	0
$589$	−30.4824	−1.25600
$590$	1.88285	0.0775159
$591$	0	0
$592$	−9.98632	−0.410435
$593$	10.1080	0.415085	0.207542	−	0.978226i	$-0.433454\pi$
$593$	10.1080	0.415085	0.207542	+	0.978226i	$0.433454\pi$
$594$	0	0
$595$	3.98341	0.163304
$596$	−13.2807	−0.543999
$597$	0	0
$598$	6.46684	0.264449
$599$	−13.7754	−0.562849	−0.281424	−	0.959583i	$-0.590807\pi$
$599$	−13.7754	−0.562849	−0.281424	+	0.959583i	$0.590807\pi$
$600$	0	0
$601$	−8.92779	−0.364172	−0.182086	−	0.983283i	$-0.558285\pi$
$601$	−8.92779	−0.364172	−0.182086	+	0.983283i	$0.558285\pi$
$602$	2.29255	0.0934375
$603$	0	0
$604$	2.77304	0.112833
$605$	−0.518566	−0.0210827
$606$	0	0
$607$	12.3237	0.500204	0.250102	−	0.968219i	$-0.419536\pi$
$607$	12.3237	0.500204	0.250102	+	0.968219i	$0.419536\pi$
$608$	4.74794	0.192554
$609$	0	0
$610$	3.42735	0.138770
$611$	−79.0972	−3.19993
$612$	0	0
$613$	−26.9032	−1.08661	−0.543306	−	0.839535i	$-0.682828\pi$
$613$	−26.9032	−1.08661	−0.543306	+	0.839535i	$0.682828\pi$
$614$	−24.5648	−0.991355
$615$	0	0
$616$	−4.76063	−0.191811
$617$	−1.35067	−0.0543758	−0.0271879	−	0.999630i	$-0.508655\pi$
$617$	−1.35067	−0.0543758	−0.0271879	+	0.999630i	$0.508655\pi$
$618$	0	0
$619$	9.41979	0.378613	0.189307	−	0.981918i	$-0.439376\pi$
$619$	9.41979	0.378613	0.189307	+	0.981918i	$0.439376\pi$
$620$	−3.76048	−0.151024
$621$	0	0
$622$	−23.3167	−0.934913
$623$	6.63317	0.265752
$624$	0	0
$625$	19.9715	0.798859
$626$	33.8040	1.35108
$627$	0	0
$628$	3.97992	0.158816
$629$	45.3703	1.80903
$630$	0	0
$631$	−32.1621	−1.28035	−0.640177	−	0.768228i	$-0.721139\pi$
$631$	−32.1621	−1.28035	−0.640177	+	0.768228i	$0.721139\pi$
$632$	4.50495	0.179197
$633$	0	0
$634$	−28.4355	−1.12932
$635$	1.44393	0.0573006
$636$	0	0
$637$	28.7834	1.14044
$638$	−33.2225	−1.31529
$639$	0	0
$640$	0.585733	0.0231531
$641$	−10.5434	−0.416441	−0.208220	−	0.978082i	$-0.566767\pi$
$641$	−10.5434	−0.416441	−0.208220	+	0.978082i	$0.566767\pi$
$642$	0	0
$643$	42.5228	1.67694	0.838468	−	0.544951i	$-0.183452\pi$
$643$	42.5228	1.67694	0.838468	+	0.544951i	$0.183452\pi$
$644$	1.60060	0.0630727
$645$	0	0
$646$	−21.5710	−0.848701
$647$	−5.84286	−0.229707	−0.114853	−	0.993382i	$-0.536640\pi$
$647$	−5.84286	−0.229707	−0.114853	+	0.993382i	$0.536640\pi$
$648$	0	0
$649$	10.2233	0.401301
$650$	28.1640	1.10468
$651$	0	0
$652$	17.0742	0.668677
$653$	−5.00429	−0.195833	−0.0979164	−	0.995195i	$-0.531218\pi$
$653$	−5.00429	−0.195833	−0.0979164	+	0.995195i	$0.531218\pi$
$654$	0	0
$655$	−0.630037	−0.0246176
$656$	−3.37739	−0.131865
$657$	0	0
$658$	−19.5773	−0.763203
$659$	−31.8454	−1.24052	−0.620261	−	0.784395i	$-0.712973\pi$
$659$	−31.8454	−1.24052	−0.620261	+	0.784395i	$0.712973\pi$
$660$	0	0
$661$	18.9484	0.737005	0.368503	−	0.929627i	$-0.379871\pi$
$661$	18.9484	0.737005	0.368503	+	0.929627i	$0.379871\pi$
$662$	−33.1657	−1.28902
$663$	0	0
$664$	−1.09046	−0.0423181
$665$	−4.16287	−0.161429
$666$	0	0
$667$	11.1700	0.432503
$668$	−3.88787	−0.150426
$669$	0	0
$670$	6.64327	0.256652
$671$	18.6095	0.718413
$672$	0	0
$673$	25.9693	1.00104	0.500521	−	0.865725i	$-0.333142\pi$
$673$	25.9693	1.00104	0.500521	+	0.865725i	$0.333142\pi$
$674$	6.40874	0.246856
$675$	0	0
$676$	23.5757	0.906759
$677$	42.6732	1.64006	0.820032	−	0.572317i	$-0.193955\pi$
$677$	42.6732	1.64006	0.820032	+	0.572317i	$0.193955\pi$
$678$	0	0
$679$	11.3910	0.437146
$680$	−2.66113	−0.102050
$681$	0	0
$682$	−20.4183	−0.781857
$683$	43.2949	1.65663	0.828317	−	0.560260i	$-0.189299\pi$
$683$	43.2949	1.65663	0.828317	+	0.560260i	$0.189299\pi$
$684$	0	0
$685$	4.39209	0.167813
$686$	17.6024	0.672062
$687$	0	0
$688$	−1.53155	−0.0583897
$689$	−46.6559	−1.77745
$690$	0	0
$691$	−19.8538	−0.755273	−0.377636	−	0.925954i	$-0.623263\pi$
$691$	−19.8538	−0.755273	−0.377636	+	0.925954i	$0.623263\pi$
$692$	−6.81516	−0.259074
$693$	0	0
$694$	30.3873	1.15349
$695$	−3.81166	−0.144584
$696$	0	0
$697$	15.3443	0.581207
$698$	13.1951	0.499443
$699$	0	0
$700$	6.97088	0.263474
$701$	−27.2877	−1.03064	−0.515321	−	0.856997i	$-0.672327\pi$
$701$	−27.2877	−1.03064	−0.515321	+	0.856997i	$0.672327\pi$
$702$	0	0
$703$	−47.4144	−1.78827
$704$	3.18036	0.119864
$705$	0	0
$706$	−16.0690	−0.604764
$707$	16.9520	0.637545
$708$	0	0
$709$	−24.4005	−0.916382	−0.458191	−	0.888854i	$-0.651502\pi$
$709$	−24.4005	−0.916382	−0.458191	+	0.888854i	$0.651502\pi$
$710$	−8.98414	−0.337169
$711$	0	0
$712$	−4.43131	−0.166070
$713$	6.86498	0.257095
$714$	0	0
$715$	−11.2661	−0.421327
$716$	1.64779	0.0615808
$717$	0	0
$718$	−10.4685	−0.390681
$719$	−5.73870	−0.214018	−0.107009	−	0.994258i	$-0.534127\pi$
$719$	−5.73870	−0.214018	−0.107009	+	0.994258i	$0.534127\pi$
$720$	0	0
$721$	23.7059	0.882854
$722$	3.54291	0.131853
$723$	0	0
$724$	16.1341	0.599621
$725$	48.6469	1.80670
$726$	0	0
$727$	3.25441	0.120699	0.0603497	−	0.998177i	$-0.480778\pi$
$727$	3.25441	0.120699	0.0603497	+	0.998177i	$0.480778\pi$
$728$	9.05285	0.335521
$729$	0	0
$730$	8.75311	0.323967
$731$	6.95820	0.257358
$732$	0	0
$733$	39.2895	1.45119	0.725595	−	0.688122i	$-0.241564\pi$
$733$	39.2895	1.45119	0.725595	+	0.688122i	$0.241564\pi$
$734$	26.1262	0.964335
$735$	0	0
$736$	−1.06929	−0.0394145
$737$	36.0710	1.32869
$738$	0	0
$739$	22.5580	0.829809	0.414904	−	0.909865i	$-0.363815\pi$
$739$	22.5580	0.829809	0.414904	+	0.909865i	$0.363815\pi$
$740$	−5.84932	−0.215025
$741$	0	0
$742$	−11.5478	−0.423933
$743$	−16.1152	−0.591210	−0.295605	−	0.955310i	$-0.595521\pi$
$743$	−16.1152	−0.591210	−0.295605	+	0.955310i	$0.595521\pi$
$744$	0	0
$745$	−7.77895	−0.284999
$746$	−26.2059	−0.959467
$747$	0	0
$748$	−14.4491	−0.528313
$749$	13.0704	0.477580
$750$	0	0
$751$	11.6657	0.425688	0.212844	−	0.977086i	$-0.431727\pi$
$751$	11.6657	0.425688	0.212844	+	0.977086i	$0.431727\pi$
$752$	13.0787	0.476931
$753$	0	0
$754$	63.1761	2.30074
$755$	1.62426	0.0591128
$756$	0	0
$757$	44.1726	1.60548	0.802741	−	0.596328i	$-0.203374\pi$
$757$	44.1726	1.60548	0.802741	+	0.596328i	$0.203374\pi$
$758$	32.9269	1.19596
$759$	0	0
$760$	2.78102	0.100878
$761$	4.55042	0.164953	0.0824763	−	0.996593i	$-0.473717\pi$
$761$	4.55042	0.164953	0.0824763	+	0.996593i	$0.473717\pi$
$762$	0	0
$763$	2.90243	0.105075
$764$	0.988248	0.0357536
$765$	0	0
$766$	20.5468	0.742386
$767$	−19.4408	−0.701966
$768$	0	0
$769$	12.5915	0.454063	0.227031	−	0.973887i	$-0.427098\pi$
$769$	12.5915	0.454063	0.227031	+	0.973887i	$0.427098\pi$
$770$	−2.78846	−0.100489
$771$	0	0
$772$	16.2266	0.584006
$773$	−37.3896	−1.34481	−0.672405	−	0.740183i	$-0.734739\pi$
$773$	−37.3896	−1.34481	−0.672405	+	0.740183i	$0.734739\pi$
$774$	0	0
$775$	29.8980	1.07397
$776$	−7.60980	−0.273176
$777$	0	0
$778$	22.5931	0.810001
$779$	−16.0356	−0.574536
$780$	0	0
$781$	−48.7813	−1.74553
$782$	4.85804	0.173723
$783$	0	0
$784$	−4.75933	−0.169976
$785$	2.33117	0.0832031
$786$	0	0
$787$	−20.6138	−0.734803	−0.367402	−	0.930062i	$-0.619753\pi$
$787$	−20.6138	−0.734803	−0.367402	+	0.930062i	$0.619753\pi$
$788$	21.1776	0.754422
$789$	0	0
$790$	2.63870	0.0938806
$791$	17.8464	0.634546
$792$	0	0
$793$	−35.3880	−1.25667
$794$	7.21751	0.256140
$795$	0	0
$796$	−5.43075	−0.192488
$797$	23.8712	0.845560	0.422780	−	0.906232i	$-0.361054\pi$
$797$	23.8712	0.845560	0.422780	+	0.906232i	$0.361054\pi$
$798$	0	0
$799$	−59.4197	−2.10212
$800$	−4.65692	−0.164647
$801$	0	0
$802$	−11.3770	−0.401738
$803$	47.5268	1.67718
$804$	0	0
$805$	0.937527	0.0330435
$806$	38.8276	1.36764
$807$	0	0
$808$	−11.3248	−0.398406
$809$	−12.5712	−0.441978	−0.220989	−	0.975276i	$-0.570929\pi$
$809$	−12.5712	−0.441978	−0.220989	+	0.975276i	$0.570929\pi$
$810$	0	0
$811$	31.4170	1.10320	0.551600	−	0.834109i	$-0.314018\pi$
$811$	31.4170	1.10320	0.551600	+	0.834109i	$0.314018\pi$
$812$	15.6367	0.548741
$813$	0	0
$814$	−31.7601	−1.11319
$815$	10.0009	0.350316
$816$	0	0
$817$	−7.27170	−0.254404
$818$	2.13979	0.0748162
$819$	0	0
$820$	−1.97825	−0.0690833
$821$	34.9424	1.21950	0.609748	−	0.792595i	$-0.291270\pi$
$821$	34.9424	1.21950	0.609748	+	0.792595i	$0.291270\pi$
$822$	0	0
$823$	−16.4458	−0.573263	−0.286632	−	0.958041i	$-0.592536\pi$
$823$	−16.4458	−0.573263	−0.286632	+	0.958041i	$0.592536\pi$
$824$	−15.8368	−0.551701
$825$	0	0
$826$	−4.81178	−0.167423
$827$	5.00885	0.174175	0.0870874	−	0.996201i	$-0.472244\pi$
$827$	5.00885	0.174175	0.0870874	+	0.996201i	$0.472244\pi$
$828$	0	0
$829$	−12.1491	−0.421954	−0.210977	−	0.977491i	$-0.567665\pi$
$829$	−12.1491	−0.421954	−0.210977	+	0.977491i	$0.567665\pi$
$830$	−0.638719	−0.0221703
$831$	0	0
$832$	−6.04779	−0.209669
$833$	21.6228	0.749186
$834$	0	0
$835$	−2.27726	−0.0788077
$836$	15.1001	0.522249
$837$	0	0
$838$	33.3352	1.15155
$839$	28.3298	0.978053	0.489026	−	0.872269i	$-0.337352\pi$
$839$	28.3298	0.978053	0.489026	+	0.872269i	$0.337352\pi$
$840$	0	0
$841$	80.1222	2.76283
$842$	−25.1838	−0.867890
$843$	0	0
$844$	8.70251	0.299553
$845$	13.8091	0.475047
$846$	0	0
$847$	1.32524	0.0455357
$848$	7.71455	0.264919
$849$	0	0
$850$	21.1575	0.725697
$851$	10.6783	0.366046
$852$	0	0
$853$	−34.7850	−1.19101	−0.595507	−	0.803350i	$-0.703049\pi$
$853$	−34.7850	−1.19101	−0.595507	+	0.803350i	$0.703049\pi$
$854$	−8.75888	−0.299723
$855$	0	0
$856$	−8.73169	−0.298443
$857$	29.5698	1.01008	0.505042	−	0.863095i	$-0.331477\pi$
$857$	29.5698	1.01008	0.505042	+	0.863095i	$0.331477\pi$
$858$	0	0
$859$	13.1666	0.449240	0.224620	−	0.974446i	$-0.427886\pi$
$859$	13.1666	0.449240	0.224620	+	0.974446i	$0.427886\pi$
$860$	−0.897078	−0.0305901
$861$	0	0
$862$	−21.3427	−0.726934
$863$	26.5705	0.904469	0.452235	−	0.891899i	$-0.350627\pi$
$863$	26.5705	0.904469	0.452235	+	0.891899i	$0.350627\pi$
$864$	0	0
$865$	−3.99186	−0.135727
$866$	31.4723	1.06947
$867$	0	0
$868$	9.61020	0.326192
$869$	14.3273	0.486022
$870$	0	0
$871$	−68.5929	−2.32418
$872$	−1.93898	−0.0656620
$873$	0	0
$874$	−5.07692	−0.171729
$875$	8.46695	0.286235
$876$	0	0
$877$	−53.8177	−1.81730	−0.908648	−	0.417563i	$-0.862884\pi$
$877$	−53.8177	−1.81730	−0.908648	+	0.417563i	$0.862884\pi$
$878$	8.30672	0.280338
$879$	0	0
$880$	1.86284	0.0627963
$881$	−16.5945	−0.559082	−0.279541	−	0.960134i	$-0.590182\pi$
$881$	−16.5945	−0.559082	−0.279541	+	0.960134i	$0.590182\pi$
$882$	0	0
$883$	−51.7445	−1.74134	−0.870671	−	0.491866i	$-0.836315\pi$
$883$	−51.7445	−1.74134	−0.870671	+	0.491866i	$0.836315\pi$
$884$	27.4766	0.924137
$885$	0	0
$886$	5.18961	0.174348
$887$	−39.9001	−1.33972	−0.669858	−	0.742489i	$-0.733645\pi$
$887$	−39.9001	−1.33972	−0.669858	+	0.742489i	$0.733645\pi$
$888$	0	0
$889$	−3.69007	−0.123761
$890$	−2.59556	−0.0870034
$891$	0	0
$892$	1.00000	0.0334825
$893$	62.0968	2.07799
$894$	0	0
$895$	0.965164	0.0322619
$896$	−1.49689	−0.0500075
$897$	0	0
$898$	−12.2820	−0.409855
$899$	67.0657	2.23677
$900$	0	0
$901$	−35.0491	−1.16765
$902$	−10.7413	−0.357646
$903$	0	0
$904$	−11.9224	−0.396532
$905$	9.45030	0.314138
$906$	0	0
$907$	26.9406	0.894549	0.447275	−	0.894397i	$-0.352395\pi$
$907$	26.9406	0.894549	0.447275	+	0.894397i	$0.352395\pi$
$908$	−5.33837	−0.177160
$909$	0	0
$910$	5.30255	0.175778
$911$	19.2901	0.639111	0.319555	−	0.947568i	$-0.396466\pi$
$911$	19.2901	0.639111	0.319555	+	0.947568i	$0.396466\pi$
$912$	0	0
$913$	−3.46806	−0.114776
$914$	36.6297	1.21160
$915$	0	0
$916$	−16.9242	−0.559191
$917$	1.61011	0.0531705
$918$	0	0
$919$	46.0123	1.51781	0.758903	−	0.651204i	$-0.225736\pi$
$919$	46.0123	1.51781	0.758903	+	0.651204i	$0.225736\pi$
$920$	−0.626318	−0.0206491
$921$	0	0
$922$	−23.3611	−0.769358
$923$	92.7628	3.05332
$924$	0	0
$925$	46.5055	1.52909
$926$	−31.6070	−1.03867
$927$	0	0
$928$	−10.4462	−0.342912
$929$	−40.9071	−1.34212	−0.671059	−	0.741404i	$-0.734160\pi$
$929$	−40.9071	−1.34212	−0.671059	+	0.741404i	$0.734160\pi$
$930$	0	0
$931$	−22.5970	−0.740587
$932$	−16.5149	−0.540963
$933$	0	0
$934$	41.2962	1.35125
$935$	−8.46333	−0.276781
$936$	0	0
$937$	−34.4636	−1.12588	−0.562938	−	0.826499i	$-0.690329\pi$
$937$	−34.4636	−1.12588	−0.562938	+	0.826499i	$0.690329\pi$
$938$	−16.9774	−0.554332
$939$	0	0
$940$	7.66062	0.249862
$941$	12.9895	0.423445	0.211723	−	0.977330i	$-0.432093\pi$
$941$	12.9895	0.423445	0.211723	+	0.977330i	$0.432093\pi$
$942$	0	0
$943$	3.61140	0.117604
$944$	3.21453	0.104624
$945$	0	0
$946$	−4.87087	−0.158366
$947$	−1.02790	−0.0334022	−0.0167011	−	0.999861i	$-0.505316\pi$
$947$	−1.02790	−0.0334022	−0.0167011	+	0.999861i	$0.505316\pi$
$948$	0	0
$949$	−90.3773	−2.93377
$950$	−22.1108	−0.717367
$951$	0	0
$952$	6.80072	0.220413
$953$	−55.1750	−1.78729	−0.893647	−	0.448770i	$-0.851862\pi$
$953$	−55.1750	−1.78729	−0.893647	+	0.448770i	$0.851862\pi$
$954$	0	0
$955$	0.578849	0.0187311
$956$	−14.8861	−0.481451
$957$	0	0
$958$	18.4305	0.595463
$959$	−11.2243	−0.362452
$960$	0	0
$961$	10.2181	0.329615
$962$	60.3952	1.94722
$963$	0	0
$964$	1.70313	0.0548541
$965$	9.50442	0.305958
$966$	0	0
$967$	−55.2258	−1.77594	−0.887971	−	0.459899i	$-0.847886\pi$
$967$	−55.2258	−1.77594	−0.887971	+	0.459899i	$0.847886\pi$
$968$	−0.885329	−0.0284556
$969$	0	0
$970$	−4.45731	−0.143115
$971$	−52.7978	−1.69436	−0.847180	−	0.531305i	$-0.821702\pi$
$971$	−52.7978	−1.69436	−0.847180	+	0.531305i	$0.821702\pi$
$972$	0	0
$973$	9.74100	0.312282
$974$	−10.4137	−0.333677
$975$	0	0
$976$	5.85140	0.187299
$977$	−24.8695	−0.795647	−0.397824	−	0.917462i	$-0.630234\pi$
$977$	−24.8695	−0.795647	−0.397824	+	0.917462i	$0.630234\pi$
$978$	0	0
$979$	−14.0931	−0.450419
$980$	−2.78770	−0.0890497
$981$	0	0
$982$	23.7037	0.756417
$983$	29.7887	0.950111	0.475056	−	0.879956i	$-0.342428\pi$
$983$	29.7887	0.950111	0.475056	+	0.879956i	$0.342428\pi$
$984$	0	0
$985$	12.4044	0.395238
$986$	47.4594	1.51142
$987$	0	0
$988$	−28.7145	−0.913530
$989$	1.63767	0.0520748
$990$	0	0
$991$	15.9480	0.506604	0.253302	−	0.967387i	$-0.418483\pi$
$991$	15.9480	0.506604	0.253302	+	0.967387i	$0.418483\pi$
$992$	−6.42013	−0.203839
$993$	0	0
$994$	22.9597	0.728237
$995$	−3.18097	−0.100843
$996$	0	0
$997$	25.9344	0.821352	0.410676	−	0.911781i	$-0.365293\pi$
$997$	25.9344	0.821352	0.410676	+	0.911781i	$0.365293\pi$
$998$	−24.2061	−0.766231
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.y.1.6	yes	8
3.2	odd	2		4014.2.a.x.1.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.a.x.1.3	✓	8	3.2	odd	2
4014.2.a.y.1.6	yes	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.585733 q^{5} -1.49689 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.585733 q^{5} -1.49689 q^{7} +1.00000 q^{8} +0.585733 q^{10} +3.18036 q^{11} -6.04779 q^{13} -1.49689 q^{14} +1.00000 q^{16} -4.54325 q^{17} +4.74794 q^{19} +0.585733 q^{20} +3.18036 q^{22} -1.06929 q^{23} -4.65692 q^{25} -6.04779 q^{26} -1.49689 q^{28} -10.4462 q^{29} -6.42013 q^{31} +1.00000 q^{32} -4.54325 q^{34} -0.876775 q^{35} -9.98632 q^{37} +4.74794 q^{38} +0.585733 q^{40} -3.37739 q^{41} -1.53155 q^{43} +3.18036 q^{44} -1.06929 q^{46} +13.0787 q^{47} -4.75933 q^{49} -4.65692 q^{50} -6.04779 q^{52} +7.71455 q^{53} +1.86284 q^{55} -1.49689 q^{56} -10.4462 q^{58} +3.21453 q^{59} +5.85140 q^{61} -6.42013 q^{62} +1.00000 q^{64} -3.54239 q^{65} +11.3418 q^{67} -4.54325 q^{68} -0.876775 q^{70} -15.3383 q^{71} +14.9439 q^{73} -9.98632 q^{74} +4.74794 q^{76} -4.76063 q^{77} +4.50495 q^{79} +0.585733 q^{80} -3.37739 q^{82} -1.09046 q^{83} -2.66113 q^{85} -1.53155 q^{86} +3.18036 q^{88} -4.43131 q^{89} +9.05285 q^{91} -1.06929 q^{92} +13.0787 q^{94} +2.78102 q^{95} -7.60980 q^{97} -4.75933 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} - 6 q^{5} - 6 q^{7} + 8 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^4 - 6 * q^5 - 6 * q^7 + 8 * q^8	\( 8 q + 8 q^{2} + 8 q^{4} - 6 q^{5} - 6 q^{7} + 8 q^{8} - 6 q^{10} - 11 q^{11} - q^{13} - 6 q^{14} + 8 q^{16} - 16 q^{17} - q^{19} - 6 q^{20} - 11 q^{22} - 14 q^{23} + 10 q^{25} - q^{26} - 6 q^{28} - 21 q^{29} - 6 q^{31} + 8 q^{32} - 16 q^{34} - 8 q^{35} - 14 q^{37} - q^{38} - 6 q^{40} - 16 q^{41} - 29 q^{43} - 11 q^{44} - 14 q^{46} - 9 q^{47} - 2 q^{49} + 10 q^{50} - q^{52} - 11 q^{53} - 22 q^{55} - 6 q^{56} - 21 q^{58} - 21 q^{59} + 3 q^{61} - 6 q^{62} + 8 q^{64} - 24 q^{65} - 20 q^{67} - 16 q^{68} - 8 q^{70} - 32 q^{71} + 13 q^{73} - 14 q^{74} - q^{76} - 4 q^{77} + 21 q^{79} - 6 q^{80} - 16 q^{82} - 28 q^{83} - 14 q^{85} - 29 q^{86} - 11 q^{88} - 54 q^{89} - 36 q^{91} - 14 q^{92} - 9 q^{94} - 30 q^{95} + 10 q^{97} - 2 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 - 6 * q^5 - 6 * q^7 + 8 * q^8 - 6 * q^10 - 11 * q^11 - q^13 - 6 * q^14 + 8 * q^16 - 16 * q^17 - q^19 - 6 * q^20 - 11 * q^22 - 14 * q^23 + 10 * q^25 - q^26 - 6 * q^28 - 21 * q^29 - 6 * q^31 + 8 * q^32 - 16 * q^34 - 8 * q^35 - 14 * q^37 - q^38 - 6 * q^40 - 16 * q^41 - 29 * q^43 - 11 * q^44 - 14 * q^46 - 9 * q^47 - 2 * q^49 + 10 * q^50 - q^52 - 11 * q^53 - 22 * q^55 - 6 * q^56 - 21 * q^58 - 21 * q^59 + 3 * q^61 - 6 * q^62 + 8 * q^64 - 24 * q^65 - 20 * q^67 - 16 * q^68 - 8 * q^70 - 32 * q^71 + 13 * q^73 - 14 * q^74 - q^76 - 4 * q^77 + 21 * q^79 - 6 * q^80 - 16 * q^82 - 28 * q^83 - 14 * q^85 - 29 * q^86 - 11 * q^88 - 54 * q^89 - 36 * q^91 - 14 * q^92 - 9 * q^94 - 30 * q^95 + 10 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more