LMFDB - Embedded newform 8024.2.a.y.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.29710 q^{3} -1.23439 q^{5} -2.46133 q^{7} -1.31752 q^{9} +O(q^{10})$	$q-1.29710 q^{3} -1.23439 q^{5} -2.46133 q^{7} -1.31752 q^{9} -0.0568786 q^{11} +4.73593 q^{13} +1.60113 q^{15} -1.00000 q^{17} -0.278794 q^{19} +3.19260 q^{21} +1.06619 q^{23} -3.47628 q^{25} +5.60027 q^{27} -6.61540 q^{29} +4.85624 q^{31} +0.0737774 q^{33} +3.03824 q^{35} +0.645264 q^{37} -6.14300 q^{39} +3.28997 q^{41} -2.78716 q^{43} +1.62634 q^{45} -4.10740 q^{47} -0.941872 q^{49} +1.29710 q^{51} +9.36717 q^{53} +0.0702104 q^{55} +0.361625 q^{57} -1.00000 q^{59} +9.88658 q^{61} +3.24285 q^{63} -5.84599 q^{65} -3.10620 q^{67} -1.38296 q^{69} +0.938212 q^{71} +5.16381 q^{73} +4.50909 q^{75} +0.139997 q^{77} -1.50059 q^{79} -3.31157 q^{81} +0.594577 q^{83} +1.23439 q^{85} +8.58086 q^{87} +16.4643 q^{89} -11.6567 q^{91} -6.29905 q^{93} +0.344141 q^{95} -3.06896 q^{97} +0.0749388 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.29710	−0.748883	−0.374442	−	0.927251i	$-0.622166\pi$
$3$	−1.29710	−0.748883	−0.374442	+	0.927251i	$0.622166\pi$
$4$	0	0
$5$	−1.23439	−0.552036	−0.276018	−	0.961152i	$-0.589015\pi$
$5$	−1.23439	−0.552036	−0.276018	+	0.961152i	$0.589015\pi$
$6$	0	0
$7$	−2.46133	−0.930294	−0.465147	−	0.885233i	$-0.653999\pi$
$7$	−2.46133	−0.930294	−0.465147	+	0.885233i	$0.653999\pi$
$8$	0	0
$9$	−1.31752	−0.439174
$10$	0	0
$11$	−0.0568786	−0.0171495	−0.00857477	−	0.999963i	$-0.502729\pi$
$11$	−0.0568786	−0.0171495	−0.00857477	+	0.999963i	$0.502729\pi$
$12$	0	0
$13$	4.73593	1.31351	0.656756	−	0.754103i	$-0.271928\pi$
$13$	4.73593	1.31351	0.656756	+	0.754103i	$0.271928\pi$
$14$	0	0
$15$	1.60113	0.413411
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−0.278794	−0.0639597	−0.0319799	−	0.999489i	$-0.510181\pi$
$19$	−0.278794	−0.0639597	−0.0319799	+	0.999489i	$0.510181\pi$
$20$	0	0
$21$	3.19260	0.696681
$22$	0	0
$23$	1.06619	0.222317	0.111158	−	0.993803i	$-0.464544\pi$
$23$	1.06619	0.222317	0.111158	+	0.993803i	$0.464544\pi$
$24$	0	0
$25$	−3.47628	−0.695256
$26$	0	0
$27$	5.60027	1.07777
$28$	0	0
$29$	−6.61540	−1.22845	−0.614224	−	0.789131i	$-0.710531\pi$
$29$	−6.61540	−1.22845	−0.614224	+	0.789131i	$0.710531\pi$
$30$	0	0
$31$	4.85624	0.872206	0.436103	−	0.899897i	$-0.356358\pi$
$31$	4.85624	0.872206	0.436103	+	0.899897i	$0.356358\pi$
$32$	0	0
$33$	0.0737774	0.0128430
$34$	0	0
$35$	3.03824	0.513556
$36$	0	0
$37$	0.645264	0.106081	0.0530404	−	0.998592i	$-0.483109\pi$
$37$	0.645264	0.106081	0.0530404	+	0.998592i	$0.483109\pi$
$38$	0	0
$39$	−6.14300	−0.983667
$40$	0	0
$41$	3.28997	0.513806	0.256903	−	0.966437i	$-0.417298\pi$
$41$	3.28997	0.513806	0.256903	+	0.966437i	$0.417298\pi$
$42$	0	0
$43$	−2.78716	−0.425037	−0.212519	−	0.977157i	$-0.568167\pi$
$43$	−2.78716	−0.425037	−0.212519	+	0.977157i	$0.568167\pi$
$44$	0	0
$45$	1.62634	0.242440
$46$	0	0
$47$	−4.10740	−0.599127	−0.299563	−	0.954076i	$-0.596841\pi$
$47$	−4.10740	−0.599127	−0.299563	+	0.954076i	$0.596841\pi$
$48$	0	0
$49$	−0.941872	−0.134553
$50$	0	0
$51$	1.29710	0.181631
$52$	0	0
$53$	9.36717	1.28668	0.643340	−	0.765581i	$-0.277548\pi$
$53$	9.36717	1.28668	0.643340	+	0.765581i	$0.277548\pi$
$54$	0	0
$55$	0.0702104	0.00946717
$56$	0	0
$57$	0.361625	0.0478984
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	9.88658	1.26585	0.632924	−	0.774214i	$-0.281855\pi$
$61$	9.88658	1.26585	0.632924	+	0.774214i	$0.281855\pi$
$62$	0	0
$63$	3.24285	0.408561
$64$	0	0
$65$	−5.84599	−0.725106
$66$	0	0
$67$	−3.10620	−0.379483	−0.189742	−	0.981834i	$-0.560765\pi$
$67$	−3.10620	−0.379483	−0.189742	+	0.981834i	$0.560765\pi$
$68$	0	0
$69$	−1.38296	−0.166489
$70$	0	0
$71$	0.938212	0.111345	0.0556727	−	0.998449i	$-0.482270\pi$
$71$	0.938212	0.111345	0.0556727	+	0.998449i	$0.482270\pi$
$72$	0	0
$73$	5.16381	0.604378	0.302189	−	0.953248i	$-0.402283\pi$
$73$	5.16381	0.604378	0.302189	+	0.953248i	$0.402283\pi$
$74$	0	0
$75$	4.50909	0.520665
$76$	0	0
$77$	0.139997	0.0159541
$78$	0	0
$79$	−1.50059	−0.168829	−0.0844147	−	0.996431i	$-0.526902\pi$
$79$	−1.50059	−0.168829	−0.0844147	+	0.996431i	$0.526902\pi$
$80$	0	0
$81$	−3.31157	−0.367952
$82$	0	0
$83$	0.594577	0.0652633	0.0326316	−	0.999467i	$-0.489611\pi$
$83$	0.594577	0.0652633	0.0326316	+	0.999467i	$0.489611\pi$
$84$	0	0
$85$	1.23439	0.133888
$86$	0	0
$87$	8.58086	0.919965
$88$	0	0
$89$	16.4643	1.74521	0.872607	−	0.488424i	$-0.162428\pi$
$89$	16.4643	1.74521	0.872607	+	0.488424i	$0.162428\pi$
$90$	0	0
$91$	−11.6567	−1.22195
$92$	0	0
$93$	−6.29905	−0.653181
$94$	0	0
$95$	0.344141	0.0353081
$96$	0	0
$97$	−3.06896	−0.311605	−0.155803	−	0.987788i	$-0.549796\pi$
$97$	−3.06896	−0.311605	−0.155803	+	0.987788i	$0.549796\pi$
$98$	0	0
$99$	0.0749388	0.00753163
$100$	0	0
$101$	9.78384	0.973529	0.486764	−	0.873533i	$-0.338177\pi$
$101$	9.78384	0.973529	0.486764	+	0.873533i	$0.338177\pi$
$102$	0	0
$103$	4.25123	0.418886	0.209443	−	0.977821i	$-0.432835\pi$
$103$	4.25123	0.418886	0.209443	+	0.977821i	$0.432835\pi$
$104$	0	0
$105$	−3.94091	−0.384594
$106$	0	0
$107$	−1.22464	−0.118390	−0.0591952	−	0.998246i	$-0.518853\pi$
$107$	−1.22464	−0.118390	−0.0591952	+	0.998246i	$0.518853\pi$
$108$	0	0
$109$	9.14796	0.876216	0.438108	−	0.898922i	$-0.355649\pi$
$109$	9.14796	0.876216	0.438108	+	0.898922i	$0.355649\pi$
$110$	0	0
$111$	−0.836975	−0.0794421
$112$	0	0
$113$	4.34210	0.408471	0.204235	−	0.978922i	$-0.434529\pi$
$113$	4.34210	0.408471	0.204235	+	0.978922i	$0.434529\pi$
$114$	0	0
$115$	−1.31610	−0.122727
$116$	0	0
$117$	−6.23970	−0.576860
$118$	0	0
$119$	2.46133	0.225629
$120$	0	0
$121$	−10.9968	−0.999706
$122$	0	0
$123$	−4.26743	−0.384781
$124$	0	0
$125$	10.4630	0.935843
$126$	0	0
$127$	−11.1112	−0.985963	−0.492982	−	0.870040i	$-0.664093\pi$
$127$	−11.1112	−0.985963	−0.492982	+	0.870040i	$0.664093\pi$
$128$	0	0
$129$	3.61523	0.318303
$130$	0	0
$131$	−1.43243	−0.125152	−0.0625759	−	0.998040i	$-0.519932\pi$
$131$	−1.43243	−0.125152	−0.0625759	+	0.998040i	$0.519932\pi$
$132$	0	0
$133$	0.686203	0.0595014
$134$	0	0
$135$	−6.91293	−0.594970
$136$	0	0
$137$	9.56972	0.817596	0.408798	−	0.912625i	$-0.365948\pi$
$137$	9.56972	0.817596	0.408798	+	0.912625i	$0.365948\pi$
$138$	0	0
$139$	18.7075	1.58675	0.793374	−	0.608735i	$-0.208323\pi$
$139$	18.7075	1.58675	0.793374	+	0.608735i	$0.208323\pi$
$140$	0	0
$141$	5.32773	0.448676
$142$	0	0
$143$	−0.269373	−0.0225261
$144$	0	0
$145$	8.16599	0.678148
$146$	0	0
$147$	1.22171	0.100765
$148$	0	0
$149$	12.8792	1.05511	0.527553	−	0.849522i	$-0.323110\pi$
$149$	12.8792	1.05511	0.527553	+	0.849522i	$0.323110\pi$
$150$	0	0
$151$	−13.6159	−1.10805	−0.554023	−	0.832502i	$-0.686908\pi$
$151$	−13.6159	−1.10805	−0.554023	+	0.832502i	$0.686908\pi$
$152$	0	0
$153$	1.31752	0.106515
$154$	0	0
$155$	−5.99450	−0.481490
$156$	0	0
$157$	−10.6662	−0.851257	−0.425629	−	0.904898i	$-0.639947\pi$
$157$	−10.6662	−0.851257	−0.425629	+	0.904898i	$0.639947\pi$
$158$	0	0
$159$	−12.1502	−0.963573
$160$	0	0
$161$	−2.62425	−0.206820
$162$	0	0
$163$	−17.3746	−1.36089	−0.680443	−	0.732801i	$-0.738212\pi$
$163$	−17.3746	−1.36089	−0.680443	+	0.732801i	$0.738212\pi$
$164$	0	0
$165$	−0.0910702	−0.00708981
$166$	0	0
$167$	−14.3787	−1.11266	−0.556330	−	0.830961i	$-0.687791\pi$
$167$	−14.3787	−1.11266	−0.556330	+	0.830961i	$0.687791\pi$
$168$	0	0
$169$	9.42907	0.725313
$170$	0	0
$171$	0.367317	0.0280895
$172$	0	0
$173$	21.9374	1.66787	0.833936	−	0.551861i	$-0.186082\pi$
$173$	21.9374	1.66787	0.833936	+	0.551861i	$0.186082\pi$
$174$	0	0
$175$	8.55626	0.646792
$176$	0	0
$177$	1.29710	0.0974963
$178$	0	0
$179$	−23.8613	−1.78348	−0.891740	−	0.452549i	$-0.850515\pi$
$179$	−23.8613	−1.78348	−0.891740	+	0.452549i	$0.850515\pi$
$180$	0	0
$181$	−12.3488	−0.917877	−0.458939	−	0.888468i	$-0.651770\pi$
$181$	−12.3488	−0.917877	−0.458939	+	0.888468i	$0.651770\pi$
$182$	0	0
$183$	−12.8239	−0.947972
$184$	0	0
$185$	−0.796509	−0.0585605
$186$	0	0
$187$	0.0568786	0.00415937
$188$	0	0
$189$	−13.7841	−1.00265
$190$	0	0
$191$	−16.5135	−1.19488	−0.597439	−	0.801914i	$-0.703815\pi$
$191$	−16.5135	−1.19488	−0.597439	+	0.801914i	$0.703815\pi$
$192$	0	0
$193$	−13.3093	−0.958026	−0.479013	−	0.877808i	$-0.659005\pi$
$193$	−13.3093	−0.958026	−0.479013	+	0.877808i	$0.659005\pi$
$194$	0	0
$195$	7.58286	0.543020
$196$	0	0
$197$	−0.348682	−0.0248426	−0.0124213	−	0.999923i	$-0.503954\pi$
$197$	−0.348682	−0.0248426	−0.0124213	+	0.999923i	$0.503954\pi$
$198$	0	0
$199$	−8.45657	−0.599470	−0.299735	−	0.954022i	$-0.596898\pi$
$199$	−8.45657	−0.599470	−0.299735	+	0.954022i	$0.596898\pi$
$200$	0	0
$201$	4.02907	0.284188
$202$	0	0
$203$	16.2827	1.14282
$204$	0	0
$205$	−4.06110	−0.283640
$206$	0	0
$207$	−1.40473	−0.0976358
$208$	0	0
$209$	0.0158574	0.00109688
$210$	0	0
$211$	24.9053	1.71455	0.857276	−	0.514858i	$-0.172155\pi$
$211$	24.9053	1.71455	0.857276	+	0.514858i	$0.172155\pi$
$212$	0	0
$213$	−1.21696	−0.0833846
$214$	0	0
$215$	3.44044	0.234636
$216$	0	0
$217$	−11.9528	−0.811408
$218$	0	0
$219$	−6.69800	−0.452609
$220$	0	0
$221$	−4.73593	−0.318573
$222$	0	0
$223$	12.8729	0.862035	0.431018	−	0.902343i	$-0.358155\pi$
$223$	12.8729	0.862035	0.431018	+	0.902343i	$0.358155\pi$
$224$	0	0
$225$	4.58007	0.305338
$226$	0	0
$227$	−10.6341	−0.705811	−0.352905	−	0.935659i	$-0.614806\pi$
$227$	−10.6341	−0.705811	−0.352905	+	0.935659i	$0.614806\pi$
$228$	0	0
$229$	−25.3482	−1.67506	−0.837529	−	0.546393i	$-0.816000\pi$
$229$	−25.3482	−1.67506	−0.837529	+	0.546393i	$0.816000\pi$
$230$	0	0
$231$	−0.181590	−0.0119478
$232$	0	0
$233$	−0.903128	−0.0591659	−0.0295829	−	0.999562i	$-0.509418\pi$
$233$	−0.903128	−0.0591659	−0.0295829	+	0.999562i	$0.509418\pi$
$234$	0	0
$235$	5.07014	0.330740
$236$	0	0
$237$	1.94642	0.126433
$238$	0	0
$239$	7.41826	0.479847	0.239924	−	0.970792i	$-0.422878\pi$
$239$	7.41826	0.479847	0.239924	+	0.970792i	$0.422878\pi$
$240$	0	0
$241$	−15.4288	−0.993854	−0.496927	−	0.867792i	$-0.665538\pi$
$241$	−15.4288	−0.993854	−0.496927	+	0.867792i	$0.665538\pi$
$242$	0	0
$243$	−12.5054	−0.802220
$244$	0	0
$245$	1.16264	0.0742783
$246$	0	0
$247$	−1.32035	−0.0840119
$248$	0	0
$249$	−0.771227	−0.0488746
$250$	0	0
$251$	−15.4448	−0.974867	−0.487434	−	0.873160i	$-0.662067\pi$
$251$	−15.4448	−0.974867	−0.487434	+	0.873160i	$0.662067\pi$
$252$	0	0
$253$	−0.0606436	−0.00381263
$254$	0	0
$255$	−1.60113	−0.100267
$256$	0	0
$257$	−2.64880	−0.165228	−0.0826138	−	0.996582i	$-0.526327\pi$
$257$	−2.64880	−0.165228	−0.0826138	+	0.996582i	$0.526327\pi$
$258$	0	0
$259$	−1.58821	−0.0986863
$260$	0	0
$261$	8.71594	0.539503
$262$	0	0
$263$	−8.40475	−0.518259	−0.259129	−	0.965843i	$-0.583436\pi$
$263$	−8.40475	−0.518259	−0.259129	+	0.965843i	$0.583436\pi$
$264$	0	0
$265$	−11.5628	−0.710294
$266$	0	0
$267$	−21.3559	−1.30696
$268$	0	0
$269$	−10.8550	−0.661840	−0.330920	−	0.943659i	$-0.607359\pi$
$269$	−10.8550	−0.661840	−0.330920	+	0.943659i	$0.607359\pi$
$270$	0	0
$271$	−3.96028	−0.240570	−0.120285	−	0.992739i	$-0.538381\pi$
$271$	−3.96028	−0.240570	−0.120285	+	0.992739i	$0.538381\pi$
$272$	0	0
$273$	15.1199	0.915099
$274$	0	0
$275$	0.197726	0.0119233
$276$	0	0
$277$	−6.36631	−0.382515	−0.191257	−	0.981540i	$-0.561256\pi$
$277$	−6.36631	−0.382515	−0.191257	+	0.981540i	$0.561256\pi$
$278$	0	0
$279$	−6.39820	−0.383050
$280$	0	0
$281$	−28.4860	−1.69933	−0.849667	−	0.527319i	$-0.823197\pi$
$281$	−28.4860	−1.69933	−0.849667	+	0.527319i	$0.823197\pi$
$282$	0	0
$283$	2.61282	0.155316	0.0776579	−	0.996980i	$-0.475256\pi$
$283$	2.61282	0.155316	0.0776579	+	0.996980i	$0.475256\pi$
$284$	0	0
$285$	−0.446386	−0.0264416
$286$	0	0
$287$	−8.09768	−0.477991
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	3.98075	0.233356
$292$	0	0
$293$	12.1151	0.707770	0.353885	−	0.935289i	$-0.384860\pi$
$293$	12.1151	0.707770	0.353885	+	0.935289i	$0.384860\pi$
$294$	0	0
$295$	1.23439	0.0718690
$296$	0	0
$297$	−0.318536	−0.0184833
$298$	0	0
$299$	5.04942	0.292016
$300$	0	0
$301$	6.86010	0.395410
$302$	0	0
$303$	−12.6907	−0.729059
$304$	0	0
$305$	−12.2039	−0.698794
$306$	0	0
$307$	−30.3898	−1.73444	−0.867220	−	0.497926i	$-0.834095\pi$
$307$	−30.3898	−1.73444	−0.867220	+	0.497926i	$0.834095\pi$
$308$	0	0
$309$	−5.51429	−0.313697
$310$	0	0
$311$	25.6621	1.45516	0.727582	−	0.686021i	$-0.240644\pi$
$311$	25.6621	1.45516	0.727582	+	0.686021i	$0.240644\pi$
$312$	0	0
$313$	13.3777	0.756152	0.378076	−	0.925775i	$-0.376586\pi$
$313$	13.3777	0.756152	0.378076	+	0.925775i	$0.376586\pi$
$314$	0	0
$315$	−4.00295	−0.225541
$316$	0	0
$317$	−26.4148	−1.48360	−0.741801	−	0.670620i	$-0.766028\pi$
$317$	−26.4148	−1.48360	−0.741801	+	0.670620i	$0.766028\pi$
$318$	0	0
$319$	0.376275	0.0210673
$320$	0	0
$321$	1.58849	0.0886606
$322$	0	0
$323$	0.278794	0.0155125
$324$	0	0
$325$	−16.4634	−0.913227
$326$	0	0
$327$	−11.8659	−0.656183
$328$	0	0
$329$	10.1097	0.557364
$330$	0	0
$331$	2.69457	0.148107	0.0740534	−	0.997254i	$-0.476406\pi$
$331$	2.69457	0.148107	0.0740534	+	0.997254i	$0.476406\pi$
$332$	0	0
$333$	−0.850150	−0.0465879
$334$	0	0
$335$	3.83427	0.209488
$336$	0	0
$337$	−12.4081	−0.675911	−0.337955	−	0.941162i	$-0.609735\pi$
$337$	−12.4081	−0.675911	−0.337955	+	0.941162i	$0.609735\pi$
$338$	0	0
$339$	−5.63216	−0.305897
$340$	0	0
$341$	−0.276216	−0.0149579
$342$	0	0
$343$	19.5475	1.05547
$344$	0	0
$345$	1.70712	0.0919082
$346$	0	0
$347$	−1.95860	−0.105143	−0.0525715	−	0.998617i	$-0.516742\pi$
$347$	−1.95860	−0.105143	−0.0525715	+	0.998617i	$0.516742\pi$
$348$	0	0
$349$	13.4538	0.720167	0.360083	−	0.932920i	$-0.382748\pi$
$349$	13.4538	0.720167	0.360083	+	0.932920i	$0.382748\pi$
$350$	0	0
$351$	26.5225	1.41567
$352$	0	0
$353$	4.13084	0.219862	0.109931	−	0.993939i	$-0.464937\pi$
$353$	4.13084	0.219862	0.109931	+	0.993939i	$0.464937\pi$
$354$	0	0
$355$	−1.15812	−0.0614667
$356$	0	0
$357$	−3.19260	−0.168970
$358$	0	0
$359$	15.9796	0.843372	0.421686	−	0.906742i	$-0.361438\pi$
$359$	15.9796	0.843372	0.421686	+	0.906742i	$0.361438\pi$
$360$	0	0
$361$	−18.9223	−0.995909
$362$	0	0
$363$	14.2639	0.748663
$364$	0	0
$365$	−6.37416	−0.333639
$366$	0	0
$367$	−4.20086	−0.219283	−0.109642	−	0.993971i	$-0.534970\pi$
$367$	−4.20086	−0.219283	−0.109642	+	0.993971i	$0.534970\pi$
$368$	0	0
$369$	−4.33460	−0.225650
$370$	0	0
$371$	−23.0557	−1.19699
$372$	0	0
$373$	−18.1911	−0.941898	−0.470949	−	0.882161i	$-0.656088\pi$
$373$	−18.1911	−0.941898	−0.470949	+	0.882161i	$0.656088\pi$
$374$	0	0
$375$	−13.5716	−0.700837
$376$	0	0
$377$	−31.3301	−1.61358
$378$	0	0
$379$	−36.3502	−1.86718	−0.933592	−	0.358339i	$-0.883343\pi$
$379$	−36.3502	−1.86718	−0.933592	+	0.358339i	$0.883343\pi$
$380$	0	0
$381$	14.4124	0.738371
$382$	0	0
$383$	−20.8142	−1.06356	−0.531779	−	0.846883i	$-0.678476\pi$
$383$	−20.8142	−1.06356	−0.531779	+	0.846883i	$0.678476\pi$
$384$	0	0
$385$	−0.172811	−0.00880725
$386$	0	0
$387$	3.67214	0.186665
$388$	0	0
$389$	−1.10874	−0.0562156	−0.0281078	−	0.999605i	$-0.508948\pi$
$389$	−1.10874	−0.0562156	−0.0281078	+	0.999605i	$0.508948\pi$
$390$	0	0
$391$	−1.06619	−0.0539197
$392$	0	0
$393$	1.85801	0.0937240
$394$	0	0
$395$	1.85231	0.0931999
$396$	0	0
$397$	6.78384	0.340471	0.170236	−	0.985403i	$-0.445547\pi$
$397$	6.78384	0.340471	0.170236	+	0.985403i	$0.445547\pi$
$398$	0	0
$399$	−0.890076	−0.0445596
$400$	0	0
$401$	−18.4040	−0.919052	−0.459526	−	0.888164i	$-0.651981\pi$
$401$	−18.4040	−0.919052	−0.459526	+	0.888164i	$0.651981\pi$
$402$	0	0
$403$	22.9988	1.14565
$404$	0	0
$405$	4.08777	0.203123
$406$	0	0
$407$	−0.0367017	−0.00181924
$408$	0	0
$409$	27.9777	1.38341	0.691704	−	0.722181i	$-0.256860\pi$
$409$	27.9777	1.38341	0.691704	+	0.722181i	$0.256860\pi$
$410$	0	0
$411$	−12.4129	−0.612284
$412$	0	0
$413$	2.46133	0.121114
$414$	0	0
$415$	−0.733940	−0.0360277
$416$	0	0
$417$	−24.2655	−1.18829
$418$	0	0
$419$	−9.68779	−0.473279	−0.236640	−	0.971597i	$-0.576046\pi$
$419$	−9.68779	−0.473279	−0.236640	+	0.971597i	$0.576046\pi$
$420$	0	0
$421$	−36.2543	−1.76692	−0.883462	−	0.468503i	$-0.844794\pi$
$421$	−36.2543	−1.76692	−0.883462	+	0.468503i	$0.844794\pi$
$422$	0	0
$423$	5.41160	0.263121
$424$	0	0
$425$	3.47628	0.168624
$426$	0	0
$427$	−24.3341	−1.17761
$428$	0	0
$429$	0.349405	0.0168694
$430$	0	0
$431$	−20.9131	−1.00735	−0.503674	−	0.863894i	$-0.668019\pi$
$431$	−20.9131	−1.00735	−0.503674	+	0.863894i	$0.668019\pi$
$432$	0	0
$433$	33.7397	1.62143	0.810713	−	0.585443i	$-0.199080\pi$
$433$	33.7397	1.62143	0.810713	+	0.585443i	$0.199080\pi$
$434$	0	0
$435$	−10.5921	−0.507854
$436$	0	0
$437$	−0.297248	−0.0142193
$438$	0	0
$439$	−21.2782	−1.01555	−0.507777	−	0.861488i	$-0.669533\pi$
$439$	−21.2782	−1.01555	−0.507777	+	0.861488i	$0.669533\pi$
$440$	0	0
$441$	1.24094	0.0590923
$442$	0	0
$443$	−1.49998	−0.0712660	−0.0356330	−	0.999365i	$-0.511345\pi$
$443$	−1.49998	−0.0712660	−0.0356330	+	0.999365i	$0.511345\pi$
$444$	0	0
$445$	−20.3234	−0.963421
$446$	0	0
$447$	−16.7057	−0.790151
$448$	0	0
$449$	28.5674	1.34818	0.674090	−	0.738649i	$-0.264536\pi$
$449$	28.5674	1.34818	0.674090	+	0.738649i	$0.264536\pi$
$450$	0	0
$451$	−0.187129	−0.00881155
$452$	0	0
$453$	17.6612	0.829796
$454$	0	0
$455$	14.3889	0.674562
$456$	0	0
$457$	28.1463	1.31663	0.658315	−	0.752743i	$-0.271270\pi$
$457$	28.1463	1.31663	0.658315	+	0.752743i	$0.271270\pi$
$458$	0	0
$459$	−5.60027	−0.261398
$460$	0	0
$461$	15.4758	0.720782	0.360391	−	0.932801i	$-0.382643\pi$
$461$	15.4758	0.720782	0.360391	+	0.932801i	$0.382643\pi$
$462$	0	0
$463$	−20.0767	−0.933042	−0.466521	−	0.884510i	$-0.654493\pi$
$463$	−20.0767	−0.933042	−0.466521	+	0.884510i	$0.654493\pi$
$464$	0	0
$465$	7.77548	0.360579
$466$	0	0
$467$	−27.8613	−1.28927	−0.644634	−	0.764491i	$-0.722990\pi$
$467$	−27.8613	−1.28927	−0.644634	+	0.764491i	$0.722990\pi$
$468$	0	0
$469$	7.64538	0.353031
$470$	0	0
$471$	13.8352	0.637492
$472$	0	0
$473$	0.158530	0.00728920
$474$	0	0
$475$	0.969166	0.0444684
$476$	0	0
$477$	−12.3415	−0.565077
$478$	0	0
$479$	12.4137	0.567198	0.283599	−	0.958943i	$-0.408472\pi$
$479$	12.4137	0.567198	0.283599	+	0.958943i	$0.408472\pi$
$480$	0	0
$481$	3.05593	0.139338
$482$	0	0
$483$	3.40393	0.154884
$484$	0	0
$485$	3.78829	0.172017
$486$	0	0
$487$	−8.51783	−0.385980	−0.192990	−	0.981201i	$-0.561818\pi$
$487$	−8.51783	−0.385980	−0.192990	+	0.981201i	$0.561818\pi$
$488$	0	0
$489$	22.5367	1.01914
$490$	0	0
$491$	−21.1020	−0.952318	−0.476159	−	0.879359i	$-0.657971\pi$
$491$	−21.1020	−0.952318	−0.476159	+	0.879359i	$0.657971\pi$
$492$	0	0
$493$	6.61540	0.297943
$494$	0	0
$495$	−0.0925038	−0.00415774
$496$	0	0
$497$	−2.30925	−0.103584
$498$	0	0
$499$	35.6039	1.59385	0.796925	−	0.604078i	$-0.206458\pi$
$499$	35.6039	1.59385	0.796925	+	0.604078i	$0.206458\pi$
$500$	0	0
$501$	18.6507	0.833253
$502$	0	0
$503$	−1.21833	−0.0543226	−0.0271613	−	0.999631i	$-0.508647\pi$
$503$	−1.21833	−0.0543226	−0.0271613	+	0.999631i	$0.508647\pi$
$504$	0	0
$505$	−12.0771	−0.537423
$506$	0	0
$507$	−12.2305	−0.543175
$508$	0	0
$509$	29.1474	1.29194	0.645969	−	0.763364i	$-0.276454\pi$
$509$	29.1474	1.29194	0.645969	+	0.763364i	$0.276454\pi$
$510$	0	0
$511$	−12.7098	−0.562250
$512$	0	0
$513$	−1.56132	−0.0689341
$514$	0	0
$515$	−5.24768	−0.231240
$516$	0	0
$517$	0.233623	0.0102747
$518$	0	0
$519$	−28.4551	−1.24904
$520$	0	0
$521$	27.2902	1.19561	0.597803	−	0.801643i	$-0.296041\pi$
$521$	27.2902	1.19561	0.597803	+	0.801643i	$0.296041\pi$
$522$	0	0
$523$	−29.9036	−1.30759	−0.653796	−	0.756671i	$-0.726825\pi$
$523$	−29.9036	−1.30759	−0.653796	+	0.756671i	$0.726825\pi$
$524$	0	0
$525$	−11.0984	−0.484372
$526$	0	0
$527$	−4.85624	−0.211541
$528$	0	0
$529$	−21.8632	−0.950575
$530$	0	0
$531$	1.31752	0.0571756
$532$	0	0
$533$	15.5811	0.674891
$534$	0	0
$535$	1.51168	0.0653558
$536$	0	0
$537$	30.9506	1.33562
$538$	0	0
$539$	0.0535724	0.00230753
$540$	0	0
$541$	−4.82213	−0.207319	−0.103660	−	0.994613i	$-0.533055\pi$
$541$	−4.82213	−0.207319	−0.103660	+	0.994613i	$0.533055\pi$
$542$	0	0
$543$	16.0176	0.687383
$544$	0	0
$545$	−11.2922	−0.483703
$546$	0	0
$547$	31.3028	1.33841	0.669206	−	0.743077i	$-0.266634\pi$
$547$	31.3028	1.33841	0.669206	+	0.743077i	$0.266634\pi$
$548$	0	0
$549$	−13.0258	−0.555927
$550$	0	0
$551$	1.84433	0.0785713
$552$	0	0
$553$	3.69344	0.157061
$554$	0	0
$555$	1.03315	0.0438549
$556$	0	0
$557$	0.301869	0.0127906	0.00639529	−	0.999980i	$-0.497964\pi$
$557$	0.301869	0.0127906	0.00639529	+	0.999980i	$0.497964\pi$
$558$	0	0
$559$	−13.1998	−0.558292
$560$	0	0
$561$	−0.0737774	−0.00311489
$562$	0	0
$563$	4.21394	0.177596	0.0887981	−	0.996050i	$-0.471697\pi$
$563$	4.21394	0.177596	0.0887981	+	0.996050i	$0.471697\pi$
$564$	0	0
$565$	−5.35985	−0.225491
$566$	0	0
$567$	8.15085	0.342304
$568$	0	0
$569$	39.7024	1.66441	0.832205	−	0.554468i	$-0.187078\pi$
$569$	39.7024	1.66441	0.832205	+	0.554468i	$0.187078\pi$
$570$	0	0
$571$	−35.0906	−1.46850	−0.734248	−	0.678881i	$-0.762465\pi$
$571$	−35.0906	−1.46850	−0.734248	+	0.678881i	$0.762465\pi$
$572$	0	0
$573$	21.4198	0.894824
$574$	0	0
$575$	−3.70639	−0.154567
$576$	0	0
$577$	−15.0787	−0.627733	−0.313867	−	0.949467i	$-0.601624\pi$
$577$	−15.0787	−0.627733	−0.313867	+	0.949467i	$0.601624\pi$
$578$	0	0
$579$	17.2636	0.717449
$580$	0	0
$581$	−1.46345	−0.0607140
$582$	0	0
$583$	−0.532792	−0.0220660
$584$	0	0
$585$	7.70223	0.318448
$586$	0	0
$587$	21.0474	0.868718	0.434359	−	0.900740i	$-0.356975\pi$
$587$	21.0474	0.868718	0.434359	+	0.900740i	$0.356975\pi$
$588$	0	0
$589$	−1.35389	−0.0557861
$590$	0	0
$591$	0.452277	0.0186042
$592$	0	0
$593$	−46.2794	−1.90047	−0.950234	−	0.311538i	$-0.899156\pi$
$593$	−46.2794	−1.90047	−0.950234	+	0.311538i	$0.899156\pi$
$594$	0	0
$595$	−3.03824	−0.124556
$596$	0	0
$597$	10.9690	0.448933
$598$	0	0
$599$	−5.37923	−0.219789	−0.109895	−	0.993943i	$-0.535051\pi$
$599$	−5.37923	−0.219789	−0.109895	+	0.993943i	$0.535051\pi$
$600$	0	0
$601$	24.2604	0.989603	0.494801	−	0.869006i	$-0.335241\pi$
$601$	24.2604	0.989603	0.494801	+	0.869006i	$0.335241\pi$
$602$	0	0
$603$	4.09249	0.166659
$604$	0	0
$605$	13.5743	0.551874
$606$	0	0
$607$	−5.00634	−0.203201	−0.101601	−	0.994825i	$-0.532396\pi$
$607$	−5.00634	−0.203201	−0.101601	+	0.994825i	$0.532396\pi$
$608$	0	0
$609$	−21.1203	−0.855837
$610$	0	0
$611$	−19.4524	−0.786960
$612$	0	0
$613$	14.1069	0.569771	0.284886	−	0.958562i	$-0.408044\pi$
$613$	14.1069	0.569771	0.284886	+	0.958562i	$0.408044\pi$
$614$	0	0
$615$	5.26767	0.212413
$616$	0	0
$617$	33.7805	1.35995	0.679976	−	0.733234i	$-0.261990\pi$
$617$	33.7805	1.35995	0.679976	+	0.733234i	$0.261990\pi$
$618$	0	0
$619$	13.2756	0.533591	0.266796	−	0.963753i	$-0.414035\pi$
$619$	13.2756	0.533591	0.266796	+	0.963753i	$0.414035\pi$
$620$	0	0
$621$	5.97098	0.239607
$622$	0	0
$623$	−40.5240	−1.62356
$624$	0	0
$625$	4.46591	0.178636
$626$	0	0
$627$	−0.0205687	−0.000821435 0
$628$	0	0
$629$	−0.645264	−0.0257284
$630$	0	0
$631$	−20.8175	−0.828731	−0.414366	−	0.910110i	$-0.635997\pi$
$631$	−20.8175	−0.828731	−0.414366	+	0.910110i	$0.635997\pi$
$632$	0	0
$633$	−32.3048	−1.28400
$634$	0	0
$635$	13.7156	0.544288
$636$	0	0
$637$	−4.46065	−0.176737
$638$	0	0
$639$	−1.23612	−0.0489000
$640$	0	0
$641$	23.0983	0.912327	0.456164	−	0.889896i	$-0.349223\pi$
$641$	23.0983	0.912327	0.456164	+	0.889896i	$0.349223\pi$
$642$	0	0
$643$	10.1158	0.398927	0.199464	−	0.979905i	$-0.436080\pi$
$643$	10.1158	0.398927	0.199464	+	0.979905i	$0.436080\pi$
$644$	0	0
$645$	−4.46261	−0.175715
$646$	0	0
$647$	3.05842	0.120239	0.0601195	−	0.998191i	$-0.480852\pi$
$647$	3.05842	0.120239	0.0601195	+	0.998191i	$0.480852\pi$
$648$	0	0
$649$	0.0568786	0.00223268
$650$	0	0
$651$	15.5040	0.607650
$652$	0	0
$653$	−20.5635	−0.804713	−0.402356	−	0.915483i	$-0.631809\pi$
$653$	−20.5635	−0.804713	−0.402356	+	0.915483i	$0.631809\pi$
$654$	0	0
$655$	1.76818	0.0690883
$656$	0	0
$657$	−6.80344	−0.265427
$658$	0	0
$659$	−39.1876	−1.52653	−0.763266	−	0.646085i	$-0.776405\pi$
$659$	−39.1876	−1.52653	−0.763266	+	0.646085i	$0.776405\pi$
$660$	0	0
$661$	−13.8958	−0.540483	−0.270241	−	0.962793i	$-0.587103\pi$
$661$	−13.8958	−0.540483	−0.270241	+	0.962793i	$0.587103\pi$
$662$	0	0
$663$	6.14300	0.238574
$664$	0	0
$665$	−0.847043	−0.0328469
$666$	0	0
$667$	−7.05330	−0.273105
$668$	0	0
$669$	−16.6975	−0.645564
$670$	0	0
$671$	−0.562335	−0.0217087
$672$	0	0
$673$	28.1016	1.08324	0.541619	−	0.840624i	$-0.317812\pi$
$673$	28.1016	1.08324	0.541619	+	0.840624i	$0.317812\pi$
$674$	0	0
$675$	−19.4681	−0.749328
$676$	0	0
$677$	−17.3079	−0.665197	−0.332599	−	0.943069i	$-0.607925\pi$
$677$	−17.3079	−0.665197	−0.332599	+	0.943069i	$0.607925\pi$
$678$	0	0
$679$	7.55370	0.289885
$680$	0	0
$681$	13.7935	0.528570
$682$	0	0
$683$	4.64602	0.177775	0.0888875	−	0.996042i	$-0.471669\pi$
$683$	4.64602	0.177775	0.0888875	+	0.996042i	$0.471669\pi$
$684$	0	0
$685$	−11.8128	−0.451343
$686$	0	0
$687$	32.8793	1.25442
$688$	0	0
$689$	44.3623	1.69007
$690$	0	0
$691$	31.2186	1.18761	0.593806	−	0.804609i	$-0.297625\pi$
$691$	31.2186	1.18761	0.593806	+	0.804609i	$0.297625\pi$
$692$	0	0
$693$	−0.184449	−0.00700663
$694$	0	0
$695$	−23.0923	−0.875943
$696$	0	0
$697$	−3.28997	−0.124616
$698$	0	0
$699$	1.17145	0.0443083
$700$	0	0
$701$	47.6324	1.79905	0.899525	−	0.436870i	$-0.143913\pi$
$701$	47.6324	1.79905	0.899525	+	0.436870i	$0.143913\pi$
$702$	0	0
$703$	−0.179896	−0.00678490
$704$	0	0
$705$	−6.57650	−0.247685
$706$	0	0
$707$	−24.0812	−0.905668
$708$	0	0
$709$	−34.7195	−1.30392	−0.651958	−	0.758255i	$-0.726052\pi$
$709$	−34.7195	−1.30392	−0.651958	+	0.758255i	$0.726052\pi$
$710$	0	0
$711$	1.97706	0.0741455
$712$	0	0
$713$	5.17769	0.193906
$714$	0	0
$715$	0.332512	0.0124352
$716$	0	0
$717$	−9.62225	−0.359350
$718$	0	0
$719$	−37.8312	−1.41087	−0.705433	−	0.708777i	$-0.749247\pi$
$719$	−37.8312	−1.41087	−0.705433	+	0.708777i	$0.749247\pi$
$720$	0	0
$721$	−10.4637	−0.389687
$722$	0	0
$723$	20.0127	0.744280
$724$	0	0
$725$	22.9970	0.854086
$726$	0	0
$727$	24.6336	0.913610	0.456805	−	0.889567i	$-0.348994\pi$
$727$	24.6336	0.913610	0.456805	+	0.889567i	$0.348994\pi$
$728$	0	0
$729$	26.1555	0.968721
$730$	0	0
$731$	2.78716	0.103087
$732$	0	0
$733$	−30.3216	−1.11995	−0.559977	−	0.828508i	$-0.689190\pi$
$733$	−30.3216	−1.11995	−0.559977	+	0.828508i	$0.689190\pi$
$734$	0	0
$735$	−1.50806	−0.0556257
$736$	0	0
$737$	0.176676	0.00650796
$738$	0	0
$739$	−15.2402	−0.560618	−0.280309	−	0.959910i	$-0.590437\pi$
$739$	−15.2402	−0.560618	−0.280309	+	0.959910i	$0.590437\pi$
$740$	0	0
$741$	1.71263	0.0629151
$742$	0	0
$743$	2.24737	0.0824480	0.0412240	−	0.999150i	$-0.486874\pi$
$743$	2.24737	0.0824480	0.0412240	+	0.999150i	$0.486874\pi$
$744$	0	0
$745$	−15.8980	−0.582457
$746$	0	0
$747$	−0.783368	−0.0286619
$748$	0	0
$749$	3.01424	0.110138
$750$	0	0
$751$	12.1025	0.441625	0.220812	−	0.975316i	$-0.429129\pi$
$751$	12.1025	0.441625	0.220812	+	0.975316i	$0.429129\pi$
$752$	0	0
$753$	20.0335	0.730062
$754$	0	0
$755$	16.8073	0.611681
$756$	0	0
$757$	−35.5942	−1.29369	−0.646846	−	0.762621i	$-0.723912\pi$
$757$	−35.5942	−1.29369	−0.646846	+	0.762621i	$0.723912\pi$
$758$	0	0
$759$	0.0786611	0.00285522
$760$	0	0
$761$	−53.5622	−1.94163	−0.970815	−	0.239832i	$-0.922908\pi$
$761$	−53.5622	−1.94163	−0.970815	+	0.239832i	$0.922908\pi$
$762$	0	0
$763$	−22.5161	−0.815138
$764$	0	0
$765$	−1.62634	−0.0588004
$766$	0	0
$767$	−4.73593	−0.171005
$768$	0	0
$769$	−44.1430	−1.59184	−0.795918	−	0.605404i	$-0.793011\pi$
$769$	−44.1430	−1.59184	−0.795918	+	0.605404i	$0.793011\pi$
$770$	0	0
$771$	3.43577	0.123736
$772$	0	0
$773$	−31.3459	−1.12743	−0.563717	−	0.825968i	$-0.690629\pi$
$773$	−31.3459	−1.12743	−0.563717	+	0.825968i	$0.690629\pi$
$774$	0	0
$775$	−16.8816	−0.606407
$776$	0	0
$777$	2.06007	0.0739045
$778$	0	0
$779$	−0.917223	−0.0328629
$780$	0	0
$781$	−0.0533642	−0.00190952
$782$	0	0
$783$	−37.0480	−1.32399
$784$	0	0
$785$	13.1663	0.469925
$786$	0	0
$787$	22.3412	0.796378	0.398189	−	0.917303i	$-0.369639\pi$
$787$	22.3412	0.796378	0.398189	+	0.917303i	$0.369639\pi$
$788$	0	0
$789$	10.9018	0.388115
$790$	0	0
$791$	−10.6873	−0.379998
$792$	0	0
$793$	46.8222	1.66271
$794$	0	0
$795$	14.9981	0.531927
$796$	0	0
$797$	20.3273	0.720030	0.360015	−	0.932946i	$-0.382771\pi$
$797$	20.3273	0.720030	0.360015	+	0.932946i	$0.382771\pi$
$798$	0	0
$799$	4.10740	0.145310
$800$	0	0
$801$	−21.6921	−0.766452
$802$	0	0
$803$	−0.293710	−0.0103648
$804$	0	0
$805$	3.23935	0.114172
$806$	0	0
$807$	14.0800	0.495641
$808$	0	0
$809$	−20.4906	−0.720410	−0.360205	−	0.932873i	$-0.617293\pi$
$809$	−20.4906	−0.720410	−0.360205	+	0.932873i	$0.617293\pi$
$810$	0	0
$811$	−30.7105	−1.07839	−0.539196	−	0.842180i	$-0.681272\pi$
$811$	−30.7105	−1.07839	−0.539196	+	0.842180i	$0.681272\pi$
$812$	0	0
$813$	5.13689	0.180159
$814$	0	0
$815$	21.4471	0.751258
$816$	0	0
$817$	0.777043	0.0271853
$818$	0	0
$819$	15.3579	0.536650
$820$	0	0
$821$	−23.1768	−0.808876	−0.404438	−	0.914566i	$-0.632533\pi$
$821$	−23.1768	−0.808876	−0.404438	+	0.914566i	$0.632533\pi$
$822$	0	0
$823$	−1.00721	−0.0351093	−0.0175546	−	0.999846i	$-0.505588\pi$
$823$	−1.00721	−0.0351093	−0.0175546	+	0.999846i	$0.505588\pi$
$824$	0	0
$825$	−0.256471	−0.00892917
$826$	0	0
$827$	−0.370447	−0.0128817	−0.00644085	−	0.999979i	$-0.502050\pi$
$827$	−0.370447	−0.0128817	−0.00644085	+	0.999979i	$0.502050\pi$
$828$	0	0
$829$	−8.66441	−0.300927	−0.150464	−	0.988616i	$-0.548077\pi$
$829$	−8.66441	−0.300927	−0.150464	+	0.988616i	$0.548077\pi$
$830$	0	0
$831$	8.25776	0.286459
$832$	0	0
$833$	0.941872	0.0326339
$834$	0	0
$835$	17.7490	0.614229
$836$	0	0
$837$	27.1963	0.940041
$838$	0	0
$839$	−27.9570	−0.965184	−0.482592	−	0.875845i	$-0.660305\pi$
$839$	−27.9570	−0.965184	−0.482592	+	0.875845i	$0.660305\pi$
$840$	0	0
$841$	14.7635	0.509086
$842$	0	0
$843$	36.9493	1.27260
$844$	0	0
$845$	−11.6392	−0.400399
$846$	0	0
$847$	27.0666	0.930020
$848$	0	0
$849$	−3.38909	−0.116313
$850$	0	0
$851$	0.687977	0.0235835
$852$	0	0
$853$	−16.9894	−0.581705	−0.290852	−	0.956768i	$-0.593939\pi$
$853$	−16.9894	−0.581705	−0.290852	+	0.956768i	$0.593939\pi$
$854$	0	0
$855$	−0.453413	−0.0155064
$856$	0	0
$857$	−2.37942	−0.0812794	−0.0406397	−	0.999174i	$-0.512940\pi$
$857$	−2.37942	−0.0812794	−0.0406397	+	0.999174i	$0.512940\pi$
$858$	0	0
$859$	−34.4718	−1.17616	−0.588082	−	0.808801i	$-0.700117\pi$
$859$	−34.4718	−1.17616	−0.588082	+	0.808801i	$0.700117\pi$
$860$	0	0
$861$	10.5035	0.357959
$862$	0	0
$863$	−1.14835	−0.0390903	−0.0195452	−	0.999809i	$-0.506222\pi$
$863$	−1.14835	−0.0390903	−0.0195452	+	0.999809i	$0.506222\pi$
$864$	0	0
$865$	−27.0794	−0.920726
$866$	0	0
$867$	−1.29710	−0.0440519
$868$	0	0
$869$	0.0853513	0.00289535
$870$	0	0
$871$	−14.7108	−0.498456
$872$	0	0
$873$	4.04342	0.136849
$874$	0	0
$875$	−25.7530	−0.870609
$876$	0	0
$877$	−23.1612	−0.782099	−0.391050	−	0.920370i	$-0.627888\pi$
$877$	−23.1612	−0.782099	−0.391050	+	0.920370i	$0.627888\pi$
$878$	0	0
$879$	−15.7145	−0.530037
$880$	0	0
$881$	33.9290	1.14310	0.571548	−	0.820569i	$-0.306343\pi$
$881$	33.9290	1.14310	0.571548	+	0.820569i	$0.306343\pi$
$882$	0	0
$883$	26.0128	0.875399	0.437699	−	0.899121i	$-0.355793\pi$
$883$	26.0128	0.875399	0.437699	+	0.899121i	$0.355793\pi$
$884$	0	0
$885$	−1.60113	−0.0538215
$886$	0	0
$887$	28.4546	0.955414	0.477707	−	0.878519i	$-0.341468\pi$
$887$	28.4546	0.955414	0.477707	+	0.878519i	$0.341468\pi$
$888$	0	0
$889$	27.3484	0.917236
$890$	0	0
$891$	0.188357	0.00631021
$892$	0	0
$893$	1.14512	0.0383200
$894$	0	0
$895$	29.4542	0.984546
$896$	0	0
$897$	−6.54963	−0.218686
$898$	0	0
$899$	−32.1260	−1.07146
$900$	0	0
$901$	−9.36717	−0.312066
$902$	0	0
$903$	−8.89826	−0.296116
$904$	0	0
$905$	15.2432	0.506702
$906$	0	0
$907$	−8.04234	−0.267041	−0.133521	−	0.991046i	$-0.542628\pi$
$907$	−8.04234	−0.267041	−0.133521	+	0.991046i	$0.542628\pi$
$908$	0	0
$909$	−12.8904	−0.427549
$910$	0	0
$911$	42.6491	1.41303	0.706513	−	0.707700i	$-0.250267\pi$
$911$	42.6491	1.41303	0.706513	+	0.707700i	$0.250267\pi$
$912$	0	0
$913$	−0.0338187	−0.00111924
$914$	0	0
$915$	15.8297	0.523315
$916$	0	0
$917$	3.52567	0.116428
$918$	0	0
$919$	−46.8344	−1.54492	−0.772462	−	0.635061i	$-0.780975\pi$
$919$	−46.8344	−1.54492	−0.772462	+	0.635061i	$0.780975\pi$
$920$	0	0
$921$	39.4188	1.29889
$922$	0	0
$923$	4.44331	0.146253
$924$	0	0
$925$	−2.24312	−0.0737533
$926$	0	0
$927$	−5.60109	−0.183964
$928$	0	0
$929$	12.7665	0.418855	0.209428	−	0.977824i	$-0.432840\pi$
$929$	12.7665	0.418855	0.209428	+	0.977824i	$0.432840\pi$
$930$	0	0
$931$	0.262588	0.00860599
$932$	0	0
$933$	−33.2864	−1.08975
$934$	0	0
$935$	−0.0702104	−0.00229613
$936$	0	0
$937$	42.0096	1.37239	0.686197	−	0.727415i	$-0.259279\pi$
$937$	42.0096	1.37239	0.686197	+	0.727415i	$0.259279\pi$
$938$	0	0
$939$	−17.3522	−0.566269
$940$	0	0
$941$	−7.12858	−0.232385	−0.116193	−	0.993227i	$-0.537069\pi$
$941$	−7.12858	−0.232385	−0.116193	+	0.993227i	$0.537069\pi$
$942$	0	0
$943$	3.50774	0.114228
$944$	0	0
$945$	17.0150	0.553497
$946$	0	0
$947$	31.2060	1.01406	0.507029	−	0.861929i	$-0.330744\pi$
$947$	31.2060	1.01406	0.507029	+	0.861929i	$0.330744\pi$
$948$	0	0
$949$	24.4555	0.793858
$950$	0	0
$951$	34.2627	1.11104
$952$	0	0
$953$	26.9057	0.871562	0.435781	−	0.900053i	$-0.356472\pi$
$953$	26.9057	0.871562	0.435781	+	0.900053i	$0.356472\pi$
$954$	0	0
$955$	20.3842	0.659616
$956$	0	0
$957$	−0.488067	−0.0157770
$958$	0	0
$959$	−23.5542	−0.760605
$960$	0	0
$961$	−7.41694	−0.239256
$962$	0	0
$963$	1.61349	0.0519940
$964$	0	0
$965$	16.4289	0.528865
$966$	0	0
$967$	55.0536	1.77040	0.885202	−	0.465208i	$-0.154020\pi$
$967$	55.0536	1.77040	0.885202	+	0.465208i	$0.154020\pi$
$968$	0	0
$969$	−0.361625	−0.0116171
$970$	0	0
$971$	−51.5144	−1.65317	−0.826587	−	0.562808i	$-0.809721\pi$
$971$	−51.5144	−1.65317	−0.826587	+	0.562808i	$0.809721\pi$
$972$	0	0
$973$	−46.0452	−1.47614
$974$	0	0
$975$	21.3548	0.683900
$976$	0	0
$977$	6.46003	0.206675	0.103337	−	0.994646i	$-0.467048\pi$
$977$	6.46003	0.206675	0.103337	+	0.994646i	$0.467048\pi$
$978$	0	0
$979$	−0.936467	−0.0299296
$980$	0	0
$981$	−12.0526	−0.384811
$982$	0	0
$983$	8.87906	0.283198	0.141599	−	0.989924i	$-0.454776\pi$
$983$	8.87906	0.283198	0.141599	+	0.989924i	$0.454776\pi$
$984$	0	0
$985$	0.430410	0.0137140
$986$	0	0
$987$	−13.1133	−0.417400
$988$	0	0
$989$	−2.97165	−0.0944930
$990$	0	0
$991$	−31.5625	−1.00261	−0.501307	−	0.865269i	$-0.667147\pi$
$991$	−31.5625	−1.00261	−0.501307	+	0.865269i	$0.667147\pi$
$992$	0	0
$993$	−3.49513	−0.110915
$994$	0	0
$995$	10.4387	0.330929
$996$	0	0
$997$	30.7798	0.974806	0.487403	−	0.873177i	$-0.337944\pi$
$997$	30.7798	0.974806	0.487403	+	0.873177i	$0.337944\pi$
$998$	0	0
$999$	3.61366	0.114331

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.y.1.10	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.y.1.10	✓	23	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.29710 q^{3} -1.23439 q^{5} -2.46133 q^{7} -1.31752 q^{9} +O(q^{10})\)	\(q-1.29710 q^{3} -1.23439 q^{5} -2.46133 q^{7} -1.31752 q^{9} -0.0568786 q^{11} +4.73593 q^{13} +1.60113 q^{15} -1.00000 q^{17} -0.278794 q^{19} +3.19260 q^{21} +1.06619 q^{23} -3.47628 q^{25} +5.60027 q^{27} -6.61540 q^{29} +4.85624 q^{31} +0.0737774 q^{33} +3.03824 q^{35} +0.645264 q^{37} -6.14300 q^{39} +3.28997 q^{41} -2.78716 q^{43} +1.62634 q^{45} -4.10740 q^{47} -0.941872 q^{49} +1.29710 q^{51} +9.36717 q^{53} +0.0702104 q^{55} +0.361625 q^{57} -1.00000 q^{59} +9.88658 q^{61} +3.24285 q^{63} -5.84599 q^{65} -3.10620 q^{67} -1.38296 q^{69} +0.938212 q^{71} +5.16381 q^{73} +4.50909 q^{75} +0.139997 q^{77} -1.50059 q^{79} -3.31157 q^{81} +0.594577 q^{83} +1.23439 q^{85} +8.58086 q^{87} +16.4643 q^{89} -11.6567 q^{91} -6.29905 q^{93} +0.344141 q^{95} -3.06896 q^{97} +0.0749388 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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