LMFDB - Embedded newform 8048.2.a.y.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8048,2,Mod(1,8048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8048.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8048 = 2^{4} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8048.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:	$64.2636035467$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	no (minimal twist has level 4024)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8048.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.79707 q^{3} +2.90936 q^{5} -4.51295 q^{7} +0.229458 q^{9} +O(q^{10})$	$q-1.79707 q^{3} +2.90936 q^{5} -4.51295 q^{7} +0.229458 q^{9} +3.50985 q^{11} +3.43213 q^{13} -5.22832 q^{15} +5.02842 q^{17} -0.0967015 q^{19} +8.11008 q^{21} -2.63312 q^{23} +3.46436 q^{25} +4.97886 q^{27} -1.78443 q^{29} -0.0155257 q^{31} -6.30745 q^{33} -13.1298 q^{35} -1.64661 q^{37} -6.16778 q^{39} +2.90892 q^{41} +5.59252 q^{43} +0.667576 q^{45} +12.4298 q^{47} +13.3667 q^{49} -9.03642 q^{51} -3.37967 q^{53} +10.2114 q^{55} +0.173779 q^{57} -11.3874 q^{59} -0.777289 q^{61} -1.03553 q^{63} +9.98530 q^{65} -6.11970 q^{67} +4.73189 q^{69} -3.72562 q^{71} -7.62058 q^{73} -6.22569 q^{75} -15.8398 q^{77} +14.3607 q^{79} -9.63572 q^{81} +0.694674 q^{83} +14.6295 q^{85} +3.20675 q^{87} +0.821907 q^{89} -15.4890 q^{91} +0.0279007 q^{93} -0.281339 q^{95} -0.292073 q^{97} +0.805365 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) $ 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9	$ 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9} - 22 q^{11} + 25 q^{13} + 4 q^{15} + 17 q^{17} - 6 q^{19} + 18 q^{21} - 16 q^{23} + 47 q^{25} + 20 q^{27} + 47 q^{29} + 7 q^{31} - 6 q^{33} - 19 q^{35} + 75 q^{37} - 21 q^{39} + 22 q^{41} + 5 q^{43} + 33 q^{45} - 10 q^{47} + 31 q^{49} - 9 q^{51} + 64 q^{53} + 3 q^{55} + 5 q^{57} - 28 q^{59} + 49 q^{61} + 10 q^{63} + 46 q^{65} + 14 q^{67} + 30 q^{69} - 35 q^{71} + 19 q^{73} + 33 q^{75} + 32 q^{77} + 12 q^{79} + 57 q^{81} + 82 q^{85} + 5 q^{87} + 42 q^{89} + 15 q^{91} + 55 q^{93} - 33 q^{95} + 4 q^{97} - 22 q^{99}+O(q^{100}) $ 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9 - 22 * q^11 + 25 * q^13 + 4 * q^15 + 17 * q^17 - 6 * q^19 + 18 * q^21 - 16 * q^23 + 47 * q^25 + 20 * q^27 + 47 * q^29 + 7 * q^31 - 6 * q^33 - 19 * q^35 + 75 * q^37 - 21 * q^39 + 22 * q^41 + 5 * q^43 + 33 * q^45 - 10 * q^47 + 31 * q^49 - 9 * q^51 + 64 * q^53 + 3 * q^55 + 5 * q^57 - 28 * q^59 + 49 * q^61 + 10 * q^63 + 46 * q^65 + 14 * q^67 + 30 * q^69 - 35 * q^71 + 19 * q^73 + 33 * q^75 + 32 * q^77 + 12 * q^79 + 57 * q^81 + 82 * q^85 + 5 * q^87 + 42 * q^89 + 15 * q^91 + 55 * q^93 - 33 * q^95 + 4 * q^97 - 22 * 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
$3$	−1.79707	−1.03754	−0.518769	−	0.854914i	$-0.673610\pi$
$3$	−1.79707	−1.03754	−0.518769	+	0.854914i	$0.673610\pi$
$4$	0	0
$5$	2.90936	1.30110	0.650552	−	0.759462i	$-0.274538\pi$
$5$	2.90936	1.30110	0.650552	+	0.759462i	$0.274538\pi$
$6$	0	0
$7$	−4.51295	−1.70573	−0.852867	−	0.522128i	$-0.825138\pi$
$7$	−4.51295	−1.70573	−0.852867	+	0.522128i	$0.825138\pi$
$8$	0	0
$9$	0.229458	0.0764861
$10$	0	0
$11$	3.50985	1.05826	0.529130	−	0.848541i	$-0.322518\pi$
$11$	3.50985	1.05826	0.529130	+	0.848541i	$0.322518\pi$
$12$	0	0
$13$	3.43213	0.951902	0.475951	−	0.879472i	$-0.342104\pi$
$13$	3.43213	0.951902	0.475951	+	0.879472i	$0.342104\pi$
$14$	0	0
$15$	−5.22832	−1.34995
$16$	0	0
$17$	5.02842	1.21957	0.609786	−	0.792566i	$-0.291256\pi$
$17$	5.02842	1.21957	0.609786	+	0.792566i	$0.291256\pi$
$18$	0	0
$19$	−0.0967015	−0.0221848	−0.0110924	−	0.999938i	$-0.503531\pi$
$19$	−0.0967015	−0.0221848	−0.0110924	+	0.999938i	$0.503531\pi$
$20$	0	0
$21$	8.11008	1.76976
$22$	0	0
$23$	−2.63312	−0.549043	−0.274521	−	0.961581i	$-0.588519\pi$
$23$	−2.63312	−0.549043	−0.274521	+	0.961581i	$0.588519\pi$
$24$	0	0
$25$	3.46436	0.692872
$26$	0	0
$27$	4.97886	0.958181
$28$	0	0
$29$	−1.78443	−0.331361	−0.165681	−	0.986179i	$-0.552982\pi$
$29$	−1.78443	−0.331361	−0.165681	+	0.986179i	$0.552982\pi$
$30$	0	0
$31$	−0.0155257	−0.00278849	−0.00139424	−	0.999999i	$-0.500444\pi$
$31$	−0.0155257	−0.00278849	−0.00139424	+	0.999999i	$0.500444\pi$
$32$	0	0
$33$	−6.30745	−1.09799
$34$	0	0
$35$	−13.1298	−2.21934
$36$	0	0
$37$	−1.64661	−0.270701	−0.135351	−	0.990798i	$-0.543216\pi$
$37$	−1.64661	−0.270701	−0.135351	+	0.990798i	$0.543216\pi$
$38$	0	0
$39$	−6.16778	−0.987635
$40$	0	0
$41$	2.90892	0.454297	0.227149	−	0.973860i	$-0.427060\pi$
$41$	2.90892	0.454297	0.227149	+	0.973860i	$0.427060\pi$
$42$	0	0
$43$	5.59252	0.852851	0.426425	−	0.904523i	$-0.359773\pi$
$43$	5.59252	0.852851	0.426425	+	0.904523i	$0.359773\pi$
$44$	0	0
$45$	0.667576	0.0995163
$46$	0	0
$47$	12.4298	1.81307	0.906536	−	0.422129i	$-0.138717\pi$
$47$	12.4298	1.81307	0.906536	+	0.422129i	$0.138717\pi$
$48$	0	0
$49$	13.3667	1.90953
$50$	0	0
$51$	−9.03642	−1.26535
$52$	0	0
$53$	−3.37967	−0.464233	−0.232116	−	0.972688i	$-0.574565\pi$
$53$	−3.37967	−0.464233	−0.232116	+	0.972688i	$0.574565\pi$
$54$	0	0
$55$	10.2114	1.37691
$56$	0	0
$57$	0.173779	0.0230176
$58$	0	0
$59$	−11.3874	−1.48251	−0.741254	−	0.671225i	$-0.765768\pi$
$59$	−11.3874	−1.48251	−0.741254	+	0.671225i	$0.765768\pi$
$60$	0	0
$61$	−0.777289	−0.0995217	−0.0497608	−	0.998761i	$-0.515846\pi$
$61$	−0.777289	−0.0995217	−0.0497608	+	0.998761i	$0.515846\pi$
$62$	0	0
$63$	−1.03553	−0.130465
$64$	0	0
$65$	9.98530	1.23852
$66$	0	0
$67$	−6.11970	−0.747640	−0.373820	−	0.927501i	$-0.621952\pi$
$67$	−6.11970	−0.747640	−0.373820	+	0.927501i	$0.621952\pi$
$68$	0	0
$69$	4.73189	0.569653
$70$	0	0
$71$	−3.72562	−0.442149	−0.221075	−	0.975257i	$-0.570956\pi$
$71$	−3.72562	−0.442149	−0.221075	+	0.975257i	$0.570956\pi$
$72$	0	0
$73$	−7.62058	−0.891921	−0.445960	−	0.895053i	$-0.647138\pi$
$73$	−7.62058	−0.891921	−0.445960	+	0.895053i	$0.647138\pi$
$74$	0	0
$75$	−6.22569	−0.718881
$76$	0	0
$77$	−15.8398	−1.80511
$78$	0	0
$79$	14.3607	1.61571	0.807854	−	0.589382i	$-0.200629\pi$
$79$	14.3607	1.61571	0.807854	+	0.589382i	$0.200629\pi$
$80$	0	0
$81$	−9.63572	−1.07064
$82$	0	0
$83$	0.694674	0.0762503	0.0381252	−	0.999273i	$-0.487861\pi$
$83$	0.694674	0.0762503	0.0381252	+	0.999273i	$0.487861\pi$
$84$	0	0
$85$	14.6295	1.58679
$86$	0	0
$87$	3.20675	0.343800
$88$	0	0
$89$	0.821907	0.0871220	0.0435610	−	0.999051i	$-0.486130\pi$
$89$	0.821907	0.0871220	0.0435610	+	0.999051i	$0.486130\pi$
$90$	0	0
$91$	−15.4890	−1.62369
$92$	0	0
$93$	0.0279007	0.00289317
$94$	0	0
$95$	−0.281339	−0.0288648
$96$	0	0
$97$	−0.292073	−0.0296555	−0.0148278	−	0.999890i	$-0.504720\pi$
$97$	−0.292073	−0.0296555	−0.0148278	+	0.999890i	$0.504720\pi$
$98$	0	0
$99$	0.805365	0.0809422
$100$	0	0
$101$	8.83266	0.878882	0.439441	−	0.898271i	$-0.355176\pi$
$101$	8.83266	0.878882	0.439441	+	0.898271i	$0.355176\pi$
$102$	0	0
$103$	−7.64880	−0.753659	−0.376829	−	0.926283i	$-0.622986\pi$
$103$	−7.64880	−0.753659	−0.376829	+	0.926283i	$0.622986\pi$
$104$	0	0
$105$	23.5951	2.30265
$106$	0	0
$107$	−7.43541	−0.718808	−0.359404	−	0.933182i	$-0.617020\pi$
$107$	−7.43541	−0.718808	−0.359404	+	0.933182i	$0.617020\pi$
$108$	0	0
$109$	−6.67062	−0.638929	−0.319465	−	0.947598i	$-0.603503\pi$
$109$	−6.67062	−0.638929	−0.319465	+	0.947598i	$0.603503\pi$
$110$	0	0
$111$	2.95908	0.280863
$112$	0	0
$113$	−0.174052	−0.0163734	−0.00818670	−	0.999966i	$-0.502606\pi$
$113$	−0.174052	−0.0163734	−0.00818670	+	0.999966i	$0.502606\pi$
$114$	0	0
$115$	−7.66068	−0.714362
$116$	0	0
$117$	0.787531	0.0728073
$118$	0	0
$119$	−22.6930	−2.08026
$120$	0	0
$121$	1.31907	0.119916
$122$	0	0
$123$	−5.22754	−0.471351
$124$	0	0
$125$	−4.46773	−0.399606
$126$	0	0
$127$	7.46015	0.661982	0.330991	−	0.943634i	$-0.392617\pi$
$127$	7.46015	0.661982	0.330991	+	0.943634i	$0.392617\pi$
$128$	0	0
$129$	−10.0501	−0.884866
$130$	0	0
$131$	6.00939	0.525043	0.262521	−	0.964926i	$-0.415446\pi$
$131$	6.00939	0.525043	0.262521	+	0.964926i	$0.415446\pi$
$132$	0	0
$133$	0.436409	0.0378414
$134$	0	0
$135$	14.4853	1.24669
$136$	0	0
$137$	2.75896	0.235714	0.117857	−	0.993031i	$-0.462398\pi$
$137$	2.75896	0.235714	0.117857	+	0.993031i	$0.462398\pi$
$138$	0	0
$139$	−5.53223	−0.469238	−0.234619	−	0.972087i	$-0.575384\pi$
$139$	−5.53223	−0.469238	−0.234619	+	0.972087i	$0.575384\pi$
$140$	0	0
$141$	−22.3372	−1.88113
$142$	0	0
$143$	12.0463	1.00736
$144$	0	0
$145$	−5.19156	−0.431135
$146$	0	0
$147$	−24.0209	−1.98121
$148$	0	0
$149$	14.2217	1.16508	0.582542	−	0.812800i	$-0.302058\pi$
$149$	14.2217	1.16508	0.582542	+	0.812800i	$0.302058\pi$
$150$	0	0
$151$	9.80743	0.798117	0.399059	−	0.916925i	$-0.369337\pi$
$151$	9.80743	0.798117	0.399059	+	0.916925i	$0.369337\pi$
$152$	0	0
$153$	1.15381	0.0932802
$154$	0	0
$155$	−0.0451697	−0.00362812
$156$	0	0
$157$	−7.24110	−0.577903	−0.288951	−	0.957344i	$-0.593307\pi$
$157$	−7.24110	−0.577903	−0.288951	+	0.957344i	$0.593307\pi$
$158$	0	0
$159$	6.07350	0.481660
$160$	0	0
$161$	11.8831	0.936521
$162$	0	0
$163$	11.9350	0.934818	0.467409	−	0.884041i	$-0.345188\pi$
$163$	11.9350	0.934818	0.467409	+	0.884041i	$0.345188\pi$
$164$	0	0
$165$	−18.3506	−1.42859
$166$	0	0
$167$	16.3875	1.26810	0.634051	−	0.773291i	$-0.281391\pi$
$167$	16.3875	1.26810	0.634051	+	0.773291i	$0.281391\pi$
$168$	0	0
$169$	−1.22046	−0.0938817
$170$	0	0
$171$	−0.0221889	−0.00169683
$172$	0	0
$173$	18.2594	1.38824	0.694118	−	0.719861i	$-0.255795\pi$
$173$	18.2594	1.38824	0.694118	+	0.719861i	$0.255795\pi$
$174$	0	0
$175$	−15.6345	−1.18185
$176$	0	0
$177$	20.4639	1.53816
$178$	0	0
$179$	−0.0229534	−0.00171562	−0.000857809	−	1.00000i	$-0.500273\pi$
$179$	−0.0229534	−0.00171562	−0.000857809		1.00000i	$0.500273\pi$
$180$	0	0
$181$	−15.0095	−1.11565	−0.557824	−	0.829959i	$-0.688364\pi$
$181$	−15.0095	−1.11565	−0.557824	+	0.829959i	$0.688364\pi$
$182$	0	0
$183$	1.39684	0.103258
$184$	0	0
$185$	−4.79058	−0.352211
$186$	0	0
$187$	17.6490	1.29062
$188$	0	0
$189$	−22.4693	−1.63440
$190$	0	0
$191$	−5.34598	−0.386821	−0.193411	−	0.981118i	$-0.561955\pi$
$191$	−5.34598	−0.386821	−0.193411	+	0.981118i	$0.561955\pi$
$192$	0	0
$193$	−7.32387	−0.527184	−0.263592	−	0.964634i	$-0.584907\pi$
$193$	−7.32387	−0.527184	−0.263592	+	0.964634i	$0.584907\pi$
$194$	0	0
$195$	−17.9443	−1.28502
$196$	0	0
$197$	18.5986	1.32509	0.662547	−	0.749020i	$-0.269476\pi$
$197$	18.5986	1.32509	0.662547	+	0.749020i	$0.269476\pi$
$198$	0	0
$199$	18.5254	1.31323	0.656614	−	0.754227i	$-0.271988\pi$
$199$	18.5254	1.31323	0.656614	+	0.754227i	$0.271988\pi$
$200$	0	0
$201$	10.9975	0.775705
$202$	0	0
$203$	8.05306	0.565214
$204$	0	0
$205$	8.46309	0.591088
$206$	0	0
$207$	−0.604190	−0.0419941
$208$	0	0
$209$	−0.339408	−0.0234773
$210$	0	0
$211$	−18.6724	−1.28546	−0.642732	−	0.766091i	$-0.722199\pi$
$211$	−18.6724	−1.28546	−0.642732	+	0.766091i	$0.722199\pi$
$212$	0	0
$213$	6.69519	0.458747
$214$	0	0
$215$	16.2706	1.10965
$216$	0	0
$217$	0.0700665	0.00475642
$218$	0	0
$219$	13.6947	0.925402
$220$	0	0
$221$	17.2582	1.16091
$222$	0	0
$223$	0.887097	0.0594044	0.0297022	−	0.999559i	$-0.490544\pi$
$223$	0.887097	0.0594044	0.0297022	+	0.999559i	$0.490544\pi$
$224$	0	0
$225$	0.794926	0.0529950
$226$	0	0
$227$	19.5672	1.29872	0.649361	−	0.760480i	$-0.275036\pi$
$227$	19.5672	1.29872	0.649361	+	0.760480i	$0.275036\pi$
$228$	0	0
$229$	−10.7333	−0.709277	−0.354638	−	0.935004i	$-0.615396\pi$
$229$	−10.7333	−0.709277	−0.354638	+	0.935004i	$0.615396\pi$
$230$	0	0
$231$	28.4652	1.87287
$232$	0	0
$233$	−21.9280	−1.43655	−0.718274	−	0.695760i	$-0.755068\pi$
$233$	−21.9280	−1.43655	−0.718274	+	0.695760i	$0.755068\pi$
$234$	0	0
$235$	36.1627	2.35899
$236$	0	0
$237$	−25.8072	−1.67636
$238$	0	0
$239$	15.2718	0.987848	0.493924	−	0.869505i	$-0.335562\pi$
$239$	15.2718	0.987848	0.493924	+	0.869505i	$0.335562\pi$
$240$	0	0
$241$	16.5466	1.06586	0.532930	−	0.846159i	$-0.321091\pi$
$241$	16.5466	1.06586	0.532930	+	0.846159i	$0.321091\pi$
$242$	0	0
$243$	2.37950	0.152645
$244$	0	0
$245$	38.8885	2.48449
$246$	0	0
$247$	−0.331892	−0.0211178
$248$	0	0
$249$	−1.24838	−0.0791127
$250$	0	0
$251$	25.5407	1.61211	0.806057	−	0.591838i	$-0.201598\pi$
$251$	25.5407	1.61211	0.806057	+	0.591838i	$0.201598\pi$
$252$	0	0
$253$	−9.24186	−0.581030
$254$	0	0
$255$	−26.2902	−1.64635
$256$	0	0
$257$	27.2540	1.70006	0.850028	−	0.526737i	$-0.176585\pi$
$257$	27.2540	1.70006	0.850028	+	0.526737i	$0.176585\pi$
$258$	0	0
$259$	7.43108	0.461745
$260$	0	0
$261$	−0.409453	−0.0253445
$262$	0	0
$263$	−17.5800	−1.08403	−0.542015	−	0.840369i	$-0.682339\pi$
$263$	−17.5800	−1.08403	−0.542015	+	0.840369i	$0.682339\pi$
$264$	0	0
$265$	−9.83266	−0.604015
$266$	0	0
$267$	−1.47702	−0.0903924
$268$	0	0
$269$	23.7534	1.44827	0.724136	−	0.689658i	$-0.242239\pi$
$269$	23.7534	1.44827	0.724136	+	0.689658i	$0.242239\pi$
$270$	0	0
$271$	−4.21081	−0.255789	−0.127894	−	0.991788i	$-0.540822\pi$
$271$	−4.21081	−0.255789	−0.127894	+	0.991788i	$0.540822\pi$
$272$	0	0
$273$	27.8349	1.68464
$274$	0	0
$275$	12.1594	0.733239
$276$	0	0
$277$	7.69852	0.462559	0.231280	−	0.972887i	$-0.425709\pi$
$277$	7.69852	0.462559	0.231280	+	0.972887i	$0.425709\pi$
$278$	0	0
$279$	−0.00356249	−0.000213281 0
$280$	0	0
$281$	−21.9145	−1.30731	−0.653654	−	0.756794i	$-0.726765\pi$
$281$	−21.9145	−1.30731	−0.653654	+	0.756794i	$0.726765\pi$
$282$	0	0
$283$	2.41255	0.143411	0.0717057	−	0.997426i	$-0.477156\pi$
$283$	2.41255	0.143411	0.0717057	+	0.997426i	$0.477156\pi$
$284$	0	0
$285$	0.505586	0.0299483
$286$	0	0
$287$	−13.1278	−0.774910
$288$	0	0
$289$	8.28501	0.487354
$290$	0	0
$291$	0.524875	0.0307687
$292$	0	0
$293$	9.09890	0.531563	0.265782	−	0.964033i	$-0.414370\pi$
$293$	9.09890	0.531563	0.265782	+	0.964033i	$0.414370\pi$
$294$	0	0
$295$	−33.1299	−1.92890
$296$	0	0
$297$	17.4751	1.01401
$298$	0	0
$299$	−9.03721	−0.522635
$300$	0	0
$301$	−25.2387	−1.45474
$302$	0	0
$303$	−15.8729	−0.911874
$304$	0	0
$305$	−2.26141	−0.129488
$306$	0	0
$307$	3.92019	0.223737	0.111869	−	0.993723i	$-0.464316\pi$
$307$	3.92019	0.223737	0.111869	+	0.993723i	$0.464316\pi$
$308$	0	0
$309$	13.7454	0.781950
$310$	0	0
$311$	−23.0737	−1.30839	−0.654193	−	0.756327i	$-0.726992\pi$
$311$	−23.0737	−1.30839	−0.654193	+	0.756327i	$0.726992\pi$
$312$	0	0
$313$	−12.0480	−0.680995	−0.340497	−	0.940245i	$-0.610595\pi$
$313$	−12.0480	−0.680995	−0.340497	+	0.940245i	$0.610595\pi$
$314$	0	0
$315$	−3.01274	−0.169748
$316$	0	0
$317$	27.8367	1.56346	0.781732	−	0.623615i	$-0.214337\pi$
$317$	27.8367	1.56346	0.781732	+	0.623615i	$0.214337\pi$
$318$	0	0
$319$	−6.26311	−0.350667
$320$	0	0
$321$	13.3619	0.745791
$322$	0	0
$323$	−0.486256	−0.0270560
$324$	0	0
$325$	11.8901	0.659546
$326$	0	0
$327$	11.9876	0.662914
$328$	0	0
$329$	−56.0950	−3.09262
$330$	0	0
$331$	−28.4433	−1.56339	−0.781694	−	0.623663i	$-0.785644\pi$
$331$	−28.4433	−1.56339	−0.781694	+	0.623663i	$0.785644\pi$
$332$	0	0
$333$	−0.377829	−0.0207049
$334$	0	0
$335$	−17.8044	−0.972758
$336$	0	0
$337$	−0.547783	−0.0298397	−0.0149198	−	0.999889i	$-0.504749\pi$
$337$	−0.547783	−0.0298397	−0.0149198	+	0.999889i	$0.504749\pi$
$338$	0	0
$339$	0.312783	0.0169880
$340$	0	0
$341$	−0.0544928	−0.00295095
$342$	0	0
$343$	−28.7326	−1.55141
$344$	0	0
$345$	13.7668	0.741178
$346$	0	0
$347$	−22.9120	−1.22998	−0.614991	−	0.788534i	$-0.710840\pi$
$347$	−22.9120	−1.22998	−0.614991	+	0.788534i	$0.710840\pi$
$348$	0	0
$349$	1.92123	0.102841	0.0514206	−	0.998677i	$-0.483625\pi$
$349$	1.92123	0.102841	0.0514206	+	0.998677i	$0.483625\pi$
$350$	0	0
$351$	17.0881	0.912095
$352$	0	0
$353$	32.9537	1.75395	0.876975	−	0.480536i	$-0.159558\pi$
$353$	32.9537	1.75395	0.876975	+	0.480536i	$0.159558\pi$
$354$	0	0
$355$	−10.8392	−0.575282
$356$	0	0
$357$	40.7809	2.15835
$358$	0	0
$359$	16.7967	0.886495	0.443247	−	0.896399i	$-0.353826\pi$
$359$	16.7967	0.886495	0.443247	+	0.896399i	$0.353826\pi$
$360$	0	0
$361$	−18.9906	−0.999508
$362$	0	0
$363$	−2.37047	−0.124417
$364$	0	0
$365$	−22.1710	−1.16048
$366$	0	0
$367$	−4.43893	−0.231710	−0.115855	−	0.993266i	$-0.536961\pi$
$367$	−4.43893	−0.231710	−0.115855	+	0.993266i	$0.536961\pi$
$368$	0	0
$369$	0.667476	0.0347474
$370$	0	0
$371$	15.2523	0.791858
$372$	0	0
$373$	9.13372	0.472926	0.236463	−	0.971641i	$-0.424012\pi$
$373$	9.13372	0.472926	0.236463	+	0.971641i	$0.424012\pi$
$374$	0	0
$375$	8.02882	0.414606
$376$	0	0
$377$	−6.12442	−0.315424
$378$	0	0
$379$	−0.00960407	−0.000493328 0	−0.000246664	−	1.00000i	$-0.500079\pi$
$379$	−0.00960407	−0.000493328 0	−0.000246664		1.00000i	$0.500079\pi$
$380$	0	0
$381$	−13.4064	−0.686831
$382$	0	0
$383$	−9.24694	−0.472497	−0.236248	−	0.971693i	$-0.575918\pi$
$383$	−9.24694	−0.472497	−0.236248	+	0.971693i	$0.575918\pi$
$384$	0	0
$385$	−46.0836	−2.34864
$386$	0	0
$387$	1.28325	0.0652312
$388$	0	0
$389$	−3.38044	−0.171395	−0.0856977	−	0.996321i	$-0.527312\pi$
$389$	−3.38044	−0.171395	−0.0856977	+	0.996321i	$0.527312\pi$
$390$	0	0
$391$	−13.2404	−0.669597
$392$	0	0
$393$	−10.7993	−0.544752
$394$	0	0
$395$	41.7805	2.10220
$396$	0	0
$397$	9.77660	0.490674	0.245337	−	0.969438i	$-0.421101\pi$
$397$	9.77660	0.490674	0.245337	+	0.969438i	$0.421101\pi$
$398$	0	0
$399$	−0.784257	−0.0392619
$400$	0	0
$401$	9.97083	0.497919	0.248960	−	0.968514i	$-0.419911\pi$
$401$	9.97083	0.497919	0.248960	+	0.968514i	$0.419911\pi$
$402$	0	0
$403$	−0.0532861	−0.00265437
$404$	0	0
$405$	−28.0338	−1.39301
$406$	0	0
$407$	−5.77937	−0.286473
$408$	0	0
$409$	5.80787	0.287181	0.143590	−	0.989637i	$-0.454135\pi$
$409$	5.80787	0.287181	0.143590	+	0.989637i	$0.454135\pi$
$410$	0	0
$411$	−4.95804	−0.244562
$412$	0	0
$413$	51.3905	2.52876
$414$	0	0
$415$	2.02105	0.0992096
$416$	0	0
$417$	9.94180	0.486852
$418$	0	0
$419$	27.0778	1.32284	0.661419	−	0.750017i	$-0.269955\pi$
$419$	27.0778	1.32284	0.661419	+	0.750017i	$0.269955\pi$
$420$	0	0
$421$	5.56971	0.271451	0.135726	−	0.990746i	$-0.456663\pi$
$421$	5.56971	0.271451	0.135726	+	0.990746i	$0.456663\pi$
$422$	0	0
$423$	2.85212	0.138675
$424$	0	0
$425$	17.4203	0.845006
$426$	0	0
$427$	3.50787	0.169758
$428$	0	0
$429$	−21.6480	−1.04518
$430$	0	0
$431$	17.8123	0.857987	0.428993	−	0.903308i	$-0.358868\pi$
$431$	17.8123	0.857987	0.428993	+	0.903308i	$0.358868\pi$
$432$	0	0
$433$	12.9691	0.623254	0.311627	−	0.950204i	$-0.399126\pi$
$433$	12.9691	0.623254	0.311627	+	0.950204i	$0.399126\pi$
$434$	0	0
$435$	9.32959	0.447320
$436$	0	0
$437$	0.254626	0.0121804
$438$	0	0
$439$	14.5928	0.696476	0.348238	−	0.937406i	$-0.386780\pi$
$439$	14.5928	0.696476	0.348238	+	0.937406i	$0.386780\pi$
$440$	0	0
$441$	3.06710	0.146052
$442$	0	0
$443$	−31.5304	−1.49805	−0.749026	−	0.662540i	$-0.769478\pi$
$443$	−31.5304	−1.49805	−0.749026	+	0.662540i	$0.769478\pi$
$444$	0	0
$445$	2.39122	0.113355
$446$	0	0
$447$	−25.5573	−1.20882
$448$	0	0
$449$	−6.51190	−0.307316	−0.153658	−	0.988124i	$-0.549105\pi$
$449$	−6.51190	−0.307316	−0.153658	+	0.988124i	$0.549105\pi$
$450$	0	0
$451$	10.2099	0.480765
$452$	0	0
$453$	−17.6246	−0.828077
$454$	0	0
$455$	−45.0631	−2.11259
$456$	0	0
$457$	28.9663	1.35499	0.677494	−	0.735529i	$-0.263066\pi$
$457$	28.9663	1.35499	0.677494	+	0.735529i	$0.263066\pi$
$458$	0	0
$459$	25.0358	1.16857
$460$	0	0
$461$	−12.2085	−0.568607	−0.284304	−	0.958734i	$-0.591762\pi$
$461$	−12.2085	−0.568607	−0.284304	+	0.958734i	$0.591762\pi$
$462$	0	0
$463$	9.59300	0.445825	0.222912	−	0.974839i	$-0.428444\pi$
$463$	9.59300	0.445825	0.222912	+	0.974839i	$0.428444\pi$
$464$	0	0
$465$	0.0811730	0.00376431
$466$	0	0
$467$	30.1704	1.39612	0.698060	−	0.716039i	$-0.254047\pi$
$467$	30.1704	1.39612	0.698060	+	0.716039i	$0.254047\pi$
$468$	0	0
$469$	27.6179	1.27528
$470$	0	0
$471$	13.0128	0.599596
$472$	0	0
$473$	19.6289	0.902539
$474$	0	0
$475$	−0.335009	−0.0153712
$476$	0	0
$477$	−0.775492	−0.0355074
$478$	0	0
$479$	24.9754	1.14116	0.570578	−	0.821243i	$-0.306719\pi$
$479$	24.9754	1.14116	0.570578	+	0.821243i	$0.306719\pi$
$480$	0	0
$481$	−5.65139	−0.257681
$482$	0	0
$483$	−21.3548	−0.971677
$484$	0	0
$485$	−0.849744	−0.0385849
$486$	0	0
$487$	21.7967	0.987705	0.493852	−	0.869546i	$-0.335588\pi$
$487$	21.7967	0.987705	0.493852	+	0.869546i	$0.335588\pi$
$488$	0	0
$489$	−21.4479	−0.969909
$490$	0	0
$491$	33.6881	1.52032	0.760162	−	0.649734i	$-0.225120\pi$
$491$	33.6881	1.52032	0.760162	+	0.649734i	$0.225120\pi$
$492$	0	0
$493$	−8.97289	−0.404119
$494$	0	0
$495$	2.34309	0.105314
$496$	0	0
$497$	16.8135	0.754189
$498$	0	0
$499$	−0.927617	−0.0415258	−0.0207629	−	0.999784i	$-0.506610\pi$
$499$	−0.927617	−0.0415258	−0.0207629	+	0.999784i	$0.506610\pi$
$500$	0	0
$501$	−29.4495	−1.31571
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	25.6974	1.14352
$506$	0	0
$507$	2.19326	0.0974059
$508$	0	0
$509$	30.1428	1.33606	0.668029	−	0.744135i	$-0.267138\pi$
$509$	30.1428	1.33606	0.668029	+	0.744135i	$0.267138\pi$
$510$	0	0
$511$	34.3913	1.52138
$512$	0	0
$513$	−0.481463	−0.0212571
$514$	0	0
$515$	−22.2531	−0.980588
$516$	0	0
$517$	43.6268	1.91870
$518$	0	0
$519$	−32.8134	−1.44035
$520$	0	0
$521$	16.2092	0.710139	0.355069	−	0.934840i	$-0.384457\pi$
$521$	16.2092	0.710139	0.355069	+	0.934840i	$0.384457\pi$
$522$	0	0
$523$	20.7447	0.907101	0.453551	−	0.891230i	$-0.350157\pi$
$523$	20.7447	0.907101	0.453551	+	0.891230i	$0.350157\pi$
$524$	0	0
$525$	28.0962	1.22622
$526$	0	0
$527$	−0.0780695	−0.00340076
$528$	0	0
$529$	−16.0667	−0.698552
$530$	0	0
$531$	−2.61292	−0.113391
$532$	0	0
$533$	9.98381	0.432447
$534$	0	0
$535$	−21.6322	−0.935244
$536$	0	0
$537$	0.0412489	0.00178002
$538$	0	0
$539$	46.9152	2.02078
$540$	0	0
$541$	−21.7457	−0.934922	−0.467461	−	0.884014i	$-0.654831\pi$
$541$	−21.7457	−0.934922	−0.467461	+	0.884014i	$0.654831\pi$
$542$	0	0
$543$	26.9731	1.15753
$544$	0	0
$545$	−19.4072	−0.831314
$546$	0	0
$547$	24.3993	1.04324	0.521620	−	0.853178i	$-0.325328\pi$
$547$	24.3993	1.04324	0.521620	+	0.853178i	$0.325328\pi$
$548$	0	0
$549$	−0.178355	−0.00761202
$550$	0	0
$551$	0.172557	0.00735120
$552$	0	0
$553$	−64.8092	−2.75597
$554$	0	0
$555$	8.60901	0.365432
$556$	0	0
$557$	−34.3012	−1.45339	−0.726694	−	0.686961i	$-0.758944\pi$
$557$	−34.3012	−1.45339	−0.726694	+	0.686961i	$0.758944\pi$
$558$	0	0
$559$	19.1943	0.811831
$560$	0	0
$561$	−31.7165	−1.33907
$562$	0	0
$563$	37.8285	1.59428	0.797140	−	0.603795i	$-0.206345\pi$
$563$	37.8285	1.59428	0.797140	+	0.603795i	$0.206345\pi$
$564$	0	0
$565$	−0.506378	−0.0213035
$566$	0	0
$567$	43.4855	1.82622
$568$	0	0
$569$	40.4172	1.69438	0.847189	−	0.531292i	$-0.178293\pi$
$569$	40.4172	1.69438	0.847189	+	0.531292i	$0.178293\pi$
$570$	0	0
$571$	−43.7573	−1.83119	−0.915593	−	0.402105i	$-0.868279\pi$
$571$	−43.7573	−1.83119	−0.915593	+	0.402105i	$0.868279\pi$
$572$	0	0
$573$	9.60710	0.401342
$574$	0	0
$575$	−9.12206	−0.380416
$576$	0	0
$577$	2.50872	0.104439	0.0522197	−	0.998636i	$-0.483370\pi$
$577$	2.50872	0.104439	0.0522197	+	0.998636i	$0.483370\pi$
$578$	0	0
$579$	13.1615	0.546973
$580$	0	0
$581$	−3.13503	−0.130063
$582$	0	0
$583$	−11.8621	−0.491279
$584$	0	0
$585$	2.29121	0.0947299
$586$	0	0
$587$	−39.1097	−1.61423	−0.807115	−	0.590394i	$-0.798972\pi$
$587$	−39.1097	−1.61423	−0.807115	+	0.590394i	$0.798972\pi$
$588$	0	0
$589$	0.00150135	6.18622e−5 0
$590$	0	0
$591$	−33.4229	−1.37484
$592$	0	0
$593$	22.0855	0.906944	0.453472	−	0.891271i	$-0.350185\pi$
$593$	22.0855	0.906944	0.453472	+	0.891271i	$0.350185\pi$
$594$	0	0
$595$	−66.0220	−2.70664
$596$	0	0
$597$	−33.2914	−1.36253
$598$	0	0
$599$	16.1771	0.660978	0.330489	−	0.943810i	$-0.392786\pi$
$599$	16.1771	0.660978	0.330489	+	0.943810i	$0.392786\pi$
$600$	0	0
$601$	21.2401	0.866402	0.433201	−	0.901297i	$-0.357384\pi$
$601$	21.2401	0.866402	0.433201	+	0.901297i	$0.357384\pi$
$602$	0	0
$603$	−1.40422	−0.0571841
$604$	0	0
$605$	3.83765	0.156023
$606$	0	0
$607$	19.2390	0.780887	0.390443	−	0.920627i	$-0.372322\pi$
$607$	19.2390	0.780887	0.390443	+	0.920627i	$0.372322\pi$
$608$	0	0
$609$	−14.4719	−0.586431
$610$	0	0
$611$	42.6607	1.72587
$612$	0	0
$613$	24.6007	0.993613	0.496806	−	0.867861i	$-0.334506\pi$
$613$	24.6007	0.993613	0.496806	+	0.867861i	$0.334506\pi$
$614$	0	0
$615$	−15.2088	−0.613277
$616$	0	0
$617$	40.2246	1.61938	0.809690	−	0.586857i	$-0.199635\pi$
$617$	40.2246	1.61938	0.809690	+	0.586857i	$0.199635\pi$
$618$	0	0
$619$	37.2253	1.49621	0.748106	−	0.663579i	$-0.230963\pi$
$619$	37.2253	1.49621	0.748106	+	0.663579i	$0.230963\pi$
$620$	0	0
$621$	−13.1099	−0.526083
$622$	0	0
$623$	−3.70922	−0.148607
$624$	0	0
$625$	−30.3200	−1.21280
$626$	0	0
$627$	0.609940	0.0243586
$628$	0	0
$629$	−8.27986	−0.330140
$630$	0	0
$631$	−40.6573	−1.61854	−0.809270	−	0.587437i	$-0.800137\pi$
$631$	−40.6573	−1.61854	−0.809270	+	0.587437i	$0.800137\pi$
$632$	0	0
$633$	33.5557	1.33372
$634$	0	0
$635$	21.7043	0.861307
$636$	0	0
$637$	45.8763	1.81768
$638$	0	0
$639$	−0.854874	−0.0338183
$640$	0	0
$641$	−26.1770	−1.03393	−0.516965	−	0.856006i	$-0.672938\pi$
$641$	−26.1770	−1.03393	−0.516965	+	0.856006i	$0.672938\pi$
$642$	0	0
$643$	−9.69249	−0.382234	−0.191117	−	0.981567i	$-0.561211\pi$
$643$	−9.69249	−0.382234	−0.191117	+	0.981567i	$0.561211\pi$
$644$	0	0
$645$	−29.2394	−1.15130
$646$	0	0
$647$	0.426109	0.0167521	0.00837603	−	0.999965i	$-0.497334\pi$
$647$	0.426109	0.0167521	0.00837603	+	0.999965i	$0.497334\pi$
$648$	0	0
$649$	−39.9680	−1.56888
$650$	0	0
$651$	−0.125914	−0.00493497
$652$	0	0
$653$	−43.2107	−1.69096	−0.845482	−	0.534004i	$-0.820687\pi$
$653$	−43.2107	−1.69096	−0.845482	+	0.534004i	$0.820687\pi$
$654$	0	0
$655$	17.4835	0.683135
$656$	0	0
$657$	−1.74860	−0.0682195
$658$	0	0
$659$	−1.65428	−0.0644416	−0.0322208	−	0.999481i	$-0.510258\pi$
$659$	−1.65428	−0.0644416	−0.0322208	+	0.999481i	$0.510258\pi$
$660$	0	0
$661$	−20.9766	−0.815897	−0.407948	−	0.913005i	$-0.633756\pi$
$661$	−20.9766	−0.815897	−0.407948	+	0.913005i	$0.633756\pi$
$662$	0	0
$663$	−31.0142	−1.20449
$664$	0	0
$665$	1.26967	0.0492356
$666$	0	0
$667$	4.69863	0.181932
$668$	0	0
$669$	−1.59418	−0.0616344
$670$	0	0
$671$	−2.72817	−0.105320
$672$	0	0
$673$	−21.1557	−0.815492	−0.407746	−	0.913095i	$-0.633685\pi$
$673$	−21.1557	−0.815492	−0.407746	+	0.913095i	$0.633685\pi$
$674$	0	0
$675$	17.2485	0.663897
$676$	0	0
$677$	−8.01532	−0.308054	−0.154027	−	0.988067i	$-0.549224\pi$
$677$	−8.01532	−0.308054	−0.154027	+	0.988067i	$0.549224\pi$
$678$	0	0
$679$	1.31811	0.0505844
$680$	0	0
$681$	−35.1637	−1.34747
$682$	0	0
$683$	45.5328	1.74226	0.871131	−	0.491050i	$-0.163387\pi$
$683$	45.5328	1.74226	0.871131	+	0.491050i	$0.163387\pi$
$684$	0	0
$685$	8.02680	0.306688
$686$	0	0
$687$	19.2885	0.735902
$688$	0	0
$689$	−11.5995	−0.441904
$690$	0	0
$691$	12.8409	0.488491	0.244245	−	0.969713i	$-0.421460\pi$
$691$	12.8409	0.488491	0.244245	+	0.969713i	$0.421460\pi$
$692$	0	0
$693$	−3.63457	−0.138066
$694$	0	0
$695$	−16.0952	−0.610527
$696$	0	0
$697$	14.6273	0.554048
$698$	0	0
$699$	39.4061	1.49047
$700$	0	0
$701$	−21.9784	−0.830113	−0.415056	−	0.909796i	$-0.636238\pi$
$701$	−21.9784	−0.830113	−0.415056	+	0.909796i	$0.636238\pi$
$702$	0	0
$703$	0.159230	0.00600547
$704$	0	0
$705$	−64.9869	−2.44755
$706$	0	0
$707$	−39.8613	−1.49914
$708$	0	0
$709$	−12.0946	−0.454224	−0.227112	−	0.973869i	$-0.572928\pi$
$709$	−12.0946	−0.454224	−0.227112	+	0.973869i	$0.572928\pi$
$710$	0	0
$711$	3.29519	0.123579
$712$	0	0
$713$	0.0408809	0.00153100
$714$	0	0
$715$	35.0469	1.31068
$716$	0	0
$717$	−27.4444	−1.02493
$718$	0	0
$719$	−30.6707	−1.14383	−0.571913	−	0.820314i	$-0.693798\pi$
$719$	−30.6707	−1.14383	−0.571913	+	0.820314i	$0.693798\pi$
$720$	0	0
$721$	34.5186	1.28554
$722$	0	0
$723$	−29.7354	−1.10587
$724$	0	0
$725$	−6.18192	−0.229591
$726$	0	0
$727$	−11.0624	−0.410281	−0.205141	−	0.978733i	$-0.565765\pi$
$727$	−11.0624	−0.410281	−0.205141	+	0.978733i	$0.565765\pi$
$728$	0	0
$729$	24.6311	0.912261
$730$	0	0
$731$	28.1215	1.04011
$732$	0	0
$733$	−3.04875	−0.112608	−0.0563040	−	0.998414i	$-0.517932\pi$
$733$	−3.04875	−0.112608	−0.0563040	+	0.998414i	$0.517932\pi$
$734$	0	0
$735$	−69.8853	−2.57776
$736$	0	0
$737$	−21.4792	−0.791198
$738$	0	0
$739$	−21.5194	−0.791605	−0.395803	−	0.918336i	$-0.629534\pi$
$739$	−21.5194	−0.791605	−0.395803	+	0.918336i	$0.629534\pi$
$740$	0	0
$741$	0.596433	0.0219105
$742$	0	0
$743$	22.8673	0.838919	0.419460	−	0.907774i	$-0.362220\pi$
$743$	22.8673	0.838919	0.419460	+	0.907774i	$0.362220\pi$
$744$	0	0
$745$	41.3759	1.51590
$746$	0	0
$747$	0.159399	0.00583209
$748$	0	0
$749$	33.5556	1.22609
$750$	0	0
$751$	7.44444	0.271651	0.135826	−	0.990733i	$-0.456631\pi$
$751$	7.44444	0.271651	0.135826	+	0.990733i	$0.456631\pi$
$752$	0	0
$753$	−45.8984	−1.67263
$754$	0	0
$755$	28.5333	1.03843
$756$	0	0
$757$	16.4773	0.598879	0.299439	−	0.954115i	$-0.403200\pi$
$757$	16.4773	0.598879	0.299439	+	0.954115i	$0.403200\pi$
$758$	0	0
$759$	16.6083	0.602841
$760$	0	0
$761$	−3.94946	−0.143168	−0.0715839	−	0.997435i	$-0.522805\pi$
$761$	−3.94946	−0.143168	−0.0715839	+	0.997435i	$0.522805\pi$
$762$	0	0
$763$	30.1042	1.08984
$764$	0	0
$765$	3.35685	0.121367
$766$	0	0
$767$	−39.0829	−1.41120
$768$	0	0
$769$	18.4542	0.665475	0.332738	−	0.943019i	$-0.392028\pi$
$769$	18.4542	0.665475	0.332738	+	0.943019i	$0.392028\pi$
$770$	0	0
$771$	−48.9773	−1.76387
$772$	0	0
$773$	17.5271	0.630406	0.315203	−	0.949024i	$-0.397927\pi$
$773$	17.5271	0.630406	0.315203	+	0.949024i	$0.397927\pi$
$774$	0	0
$775$	−0.0537864	−0.00193207
$776$	0	0
$777$	−13.3542	−0.479078
$778$	0	0
$779$	−0.281297	−0.0100785
$780$	0	0
$781$	−13.0764	−0.467909
$782$	0	0
$783$	−8.88444	−0.317504
$784$	0	0
$785$	−21.0669	−0.751911
$786$	0	0
$787$	6.05698	0.215908	0.107954	−	0.994156i	$-0.465570\pi$
$787$	6.05698	0.215908	0.107954	+	0.994156i	$0.465570\pi$
$788$	0	0
$789$	31.5925	1.12472
$790$	0	0
$791$	0.785486	0.0279287
$792$	0	0
$793$	−2.66776	−0.0947349
$794$	0	0
$795$	17.6700	0.626689
$796$	0	0
$797$	36.4611	1.29152	0.645759	−	0.763541i	$-0.276541\pi$
$797$	36.4611	1.29152	0.645759	+	0.763541i	$0.276541\pi$
$798$	0	0
$799$	62.5022	2.21117
$800$	0	0
$801$	0.188593	0.00666362
$802$	0	0
$803$	−26.7471	−0.943885
$804$	0	0
$805$	34.5722	1.21851
$806$	0	0
$807$	−42.6865	−1.50264
$808$	0	0
$809$	−2.59186	−0.0911250	−0.0455625	−	0.998961i	$-0.514508\pi$
$809$	−2.59186	−0.0911250	−0.0455625	+	0.998961i	$0.514508\pi$
$810$	0	0
$811$	−42.4795	−1.49166	−0.745828	−	0.666139i	$-0.767946\pi$
$811$	−42.4795	−1.49166	−0.745828	+	0.666139i	$0.767946\pi$
$812$	0	0
$813$	7.56712	0.265391
$814$	0	0
$815$	34.7230	1.21630
$816$	0	0
$817$	−0.540805	−0.0189204
$818$	0	0
$819$	−3.55409	−0.124190
$820$	0	0
$821$	−5.09070	−0.177667	−0.0888333	−	0.996047i	$-0.528314\pi$
$821$	−5.09070	−0.177667	−0.0888333	+	0.996047i	$0.528314\pi$
$822$	0	0
$823$	−20.0353	−0.698387	−0.349193	−	0.937051i	$-0.613544\pi$
$823$	−20.0353	−0.698387	−0.349193	+	0.937051i	$0.613544\pi$
$824$	0	0
$825$	−21.8513	−0.760764
$826$	0	0
$827$	−40.8193	−1.41942	−0.709712	−	0.704492i	$-0.751175\pi$
$827$	−40.8193	−1.41942	−0.709712	+	0.704492i	$0.751175\pi$
$828$	0	0
$829$	27.6181	0.959217	0.479609	−	0.877482i	$-0.340779\pi$
$829$	27.6181	0.959217	0.479609	+	0.877482i	$0.340779\pi$
$830$	0	0
$831$	−13.8348	−0.479923
$832$	0	0
$833$	67.2134	2.32881
$834$	0	0
$835$	47.6771	1.64993
$836$	0	0
$837$	−0.0773000	−0.00267188
$838$	0	0
$839$	14.5580	0.502597	0.251299	−	0.967910i	$-0.419142\pi$
$839$	14.5580	0.502597	0.251299	+	0.967910i	$0.419142\pi$
$840$	0	0
$841$	−25.8158	−0.890200
$842$	0	0
$843$	39.3818	1.35638
$844$	0	0
$845$	−3.55076	−0.122150
$846$	0	0
$847$	−5.95291	−0.204544
$848$	0	0
$849$	−4.33552	−0.148795
$850$	0	0
$851$	4.33572	0.148627
$852$	0	0
$853$	38.4709	1.31722	0.658609	−	0.752486i	$-0.271145\pi$
$853$	38.4709	1.31722	0.658609	+	0.752486i	$0.271145\pi$
$854$	0	0
$855$	−0.0645556	−0.00220775
$856$	0	0
$857$	15.1860	0.518743	0.259371	−	0.965778i	$-0.416485\pi$
$857$	15.1860	0.518743	0.259371	+	0.965778i	$0.416485\pi$
$858$	0	0
$859$	−32.1712	−1.09767	−0.548834	−	0.835931i	$-0.684928\pi$
$859$	−32.1712	−1.09767	−0.548834	+	0.835931i	$0.684928\pi$
$860$	0	0
$861$	23.5916	0.803999
$862$	0	0
$863$	44.3008	1.50802	0.754008	−	0.656865i	$-0.228118\pi$
$863$	44.3008	1.50802	0.754008	+	0.656865i	$0.228118\pi$
$864$	0	0
$865$	53.1231	1.80624
$866$	0	0
$867$	−14.8887	−0.505648
$868$	0	0
$869$	50.4041	1.70984
$870$	0	0
$871$	−21.0036	−0.711681
$872$	0	0
$873$	−0.0670185	−0.00226823
$874$	0	0
$875$	20.1626	0.681621
$876$	0	0
$877$	19.5830	0.661272	0.330636	−	0.943758i	$-0.392737\pi$
$877$	19.5830	0.661272	0.330636	+	0.943758i	$0.392737\pi$
$878$	0	0
$879$	−16.3514	−0.551518
$880$	0	0
$881$	46.5584	1.56859	0.784295	−	0.620388i	$-0.213025\pi$
$881$	46.5584	1.56859	0.784295	+	0.620388i	$0.213025\pi$
$882$	0	0
$883$	13.8535	0.466208	0.233104	−	0.972452i	$-0.425112\pi$
$883$	13.8535	0.466208	0.233104	+	0.972452i	$0.425112\pi$
$884$	0	0
$885$	59.5367	2.00130
$886$	0	0
$887$	51.7642	1.73807	0.869035	−	0.494750i	$-0.164740\pi$
$887$	51.7642	1.73807	0.869035	+	0.494750i	$0.164740\pi$
$888$	0	0
$889$	−33.6673	−1.12916
$890$	0	0
$891$	−33.8200	−1.13301
$892$	0	0
$893$	−1.20198	−0.0402227
$894$	0	0
$895$	−0.0667797	−0.00223220
$896$	0	0
$897$	16.2405	0.542254
$898$	0	0
$899$	0.0277045	0.000923998 0
$900$	0	0
$901$	−16.9944	−0.566165
$902$	0	0
$903$	45.3558	1.50935
$904$	0	0
$905$	−43.6680	−1.45157
$906$	0	0
$907$	33.5420	1.11374	0.556871	−	0.830599i	$-0.312002\pi$
$907$	33.5420	1.11374	0.556871	+	0.830599i	$0.312002\pi$
$908$	0	0
$909$	2.02673	0.0672223
$910$	0	0
$911$	−18.6894	−0.619208	−0.309604	−	0.950866i	$-0.600196\pi$
$911$	−18.6894	−0.619208	−0.309604	+	0.950866i	$0.600196\pi$
$912$	0	0
$913$	2.43820	0.0806927
$914$	0	0
$915$	4.06391	0.134349
$916$	0	0
$917$	−27.1200	−0.895583
$918$	0	0
$919$	4.65205	0.153457	0.0767286	−	0.997052i	$-0.475553\pi$
$919$	4.65205	0.153457	0.0767286	+	0.997052i	$0.475553\pi$
$920$	0	0
$921$	−7.04486	−0.232136
$922$	0	0
$923$	−12.7868	−0.420883
$924$	0	0
$925$	−5.70446	−0.187561
$926$	0	0
$927$	−1.75508	−0.0576444
$928$	0	0
$929$	−25.1669	−0.825698	−0.412849	−	0.910799i	$-0.635466\pi$
$929$	−25.1669	−0.825698	−0.412849	+	0.910799i	$0.635466\pi$
$930$	0	0
$931$	−1.29258	−0.0423626
$932$	0	0
$933$	41.4650	1.35750
$934$	0	0
$935$	51.3473	1.67924
$936$	0	0
$937$	−43.8197	−1.43153	−0.715764	−	0.698342i	$-0.753921\pi$
$937$	−43.8197	−1.43153	−0.715764	+	0.698342i	$0.753921\pi$
$938$	0	0
$939$	21.6511	0.706558
$940$	0	0
$941$	29.4690	0.960663	0.480332	−	0.877087i	$-0.340516\pi$
$941$	29.4690	0.960663	0.480332	+	0.877087i	$0.340516\pi$
$942$	0	0
$943$	−7.65953	−0.249429
$944$	0	0
$945$	−65.3713	−2.12653
$946$	0	0
$947$	41.3093	1.34237	0.671186	−	0.741289i	$-0.265785\pi$
$947$	41.3093	1.34237	0.671186	+	0.741289i	$0.265785\pi$
$948$	0	0
$949$	−26.1548	−0.849022
$950$	0	0
$951$	−50.0244	−1.62215
$952$	0	0
$953$	−20.7688	−0.672767	−0.336383	−	0.941725i	$-0.609204\pi$
$953$	−20.7688	−0.672767	−0.336383	+	0.941725i	$0.609204\pi$
$954$	0	0
$955$	−15.5534	−0.503295
$956$	0	0
$957$	11.2552	0.363830
$958$	0	0
$959$	−12.4510	−0.402065
$960$	0	0
$961$	−30.9998	−0.999992
$962$	0	0
$963$	−1.70611	−0.0549788
$964$	0	0
$965$	−21.3078	−0.685921
$966$	0	0
$967$	58.0305	1.86613	0.933067	−	0.359702i	$-0.117122\pi$
$967$	58.0305	1.86613	0.933067	+	0.359702i	$0.117122\pi$
$968$	0	0
$969$	0.873835	0.0280716
$970$	0	0
$971$	49.3578	1.58397	0.791984	−	0.610542i	$-0.209048\pi$
$971$	49.3578	1.58397	0.791984	+	0.610542i	$0.209048\pi$
$972$	0	0
$973$	24.9667	0.800395
$974$	0	0
$975$	−21.3674	−0.684305
$976$	0	0
$977$	−60.7177	−1.94253	−0.971266	−	0.237996i	$-0.923510\pi$
$977$	−60.7177	−1.94253	−0.971266	+	0.237996i	$0.923510\pi$
$978$	0	0
$979$	2.88477	0.0921977
$980$	0	0
$981$	−1.53063	−0.0488692
$982$	0	0
$983$	−51.5209	−1.64326	−0.821631	−	0.570020i	$-0.806935\pi$
$983$	−51.5209	−1.64326	−0.821631	+	0.570020i	$0.806935\pi$
$984$	0	0
$985$	54.1099	1.72409
$986$	0	0
$987$	100.807	3.20871
$988$	0	0
$989$	−14.7258	−0.468252
$990$	0	0
$991$	46.2073	1.46782	0.733912	−	0.679244i	$-0.237692\pi$
$991$	46.2073	1.46782	0.733912	+	0.679244i	$0.237692\pi$
$992$	0	0
$993$	51.1147	1.62207
$994$	0	0
$995$	53.8969	1.70865
$996$	0	0
$997$	6.03128	0.191012	0.0955062	−	0.995429i	$-0.469553\pi$
$997$	6.03128	0.191012	0.0955062	+	0.995429i	$0.469553\pi$
$998$	0	0
$999$	−8.19825	−0.259381

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.y.1.8		33
4.3	odd	2		4024.2.a.f.1.26	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.26	✓	33	4.3	odd	2
8048.2.a.y.1.8		33	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:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79707 q^{3} +2.90936 q^{5} -4.51295 q^{7} +0.229458 q^{9} +O(q^{10})\)	\(q-1.79707 q^{3} +2.90936 q^{5} -4.51295 q^{7} +0.229458 q^{9} +3.50985 q^{11} +3.43213 q^{13} -5.22832 q^{15} +5.02842 q^{17} -0.0967015 q^{19} +8.11008 q^{21} -2.63312 q^{23} +3.46436 q^{25} +4.97886 q^{27} -1.78443 q^{29} -0.0155257 q^{31} -6.30745 q^{33} -13.1298 q^{35} -1.64661 q^{37} -6.16778 q^{39} +2.90892 q^{41} +5.59252 q^{43} +0.667576 q^{45} +12.4298 q^{47} +13.3667 q^{49} -9.03642 q^{51} -3.37967 q^{53} +10.2114 q^{55} +0.173779 q^{57} -11.3874 q^{59} -0.777289 q^{61} -1.03553 q^{63} +9.98530 q^{65} -6.11970 q^{67} +4.73189 q^{69} -3.72562 q^{71} -7.62058 q^{73} -6.22569 q^{75} -15.8398 q^{77} +14.3607 q^{79} -9.63572 q^{81} +0.694674 q^{83} +14.6295 q^{85} +3.20675 q^{87} +0.821907 q^{89} -15.4890 q^{91} +0.0279007 q^{93} -0.281339 q^{95} -0.292073 q^{97} +0.805365 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9}+O(q^{10}) \) 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9	\( 33 q + 2 q^{3} + 12 q^{5} - 4 q^{7} + 43 q^{9} - 22 q^{11} + 25 q^{13} + 4 q^{15} + 17 q^{17} - 6 q^{19} + 18 q^{21} - 16 q^{23} + 47 q^{25} + 20 q^{27} + 47 q^{29} + 7 q^{31} - 6 q^{33} - 19 q^{35} + 75 q^{37} - 21 q^{39} + 22 q^{41} + 5 q^{43} + 33 q^{45} - 10 q^{47} + 31 q^{49} - 9 q^{51} + 64 q^{53} + 3 q^{55} + 5 q^{57} - 28 q^{59} + 49 q^{61} + 10 q^{63} + 46 q^{65} + 14 q^{67} + 30 q^{69} - 35 q^{71} + 19 q^{73} + 33 q^{75} + 32 q^{77} + 12 q^{79} + 57 q^{81} + 82 q^{85} + 5 q^{87} + 42 q^{89} + 15 q^{91} + 55 q^{93} - 33 q^{95} + 4 q^{97} - 22 q^{99}+O(q^{100}) \) 33 * q + 2 * q^3 + 12 * q^5 - 4 * q^7 + 43 * q^9 - 22 * q^11 + 25 * q^13 + 4 * q^15 + 17 * q^17 - 6 * q^19 + 18 * q^21 - 16 * q^23 + 47 * q^25 + 20 * q^27 + 47 * q^29 + 7 * q^31 - 6 * q^33 - 19 * q^35 + 75 * q^37 - 21 * q^39 + 22 * q^41 + 5 * q^43 + 33 * q^45 - 10 * q^47 + 31 * q^49 - 9 * q^51 + 64 * q^53 + 3 * q^55 + 5 * q^57 - 28 * q^59 + 49 * q^61 + 10 * q^63 + 46 * q^65 + 14 * q^67 + 30 * q^69 - 35 * q^71 + 19 * q^73 + 33 * q^75 + 32 * q^77 + 12 * q^79 + 57 * q^81 + 82 * q^85 + 5 * q^87 + 42 * q^89 + 15 * q^91 + 55 * q^93 - 33 * q^95 + 4 * q^97 - 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more