LMFDB - Embedded newform 8048.2.a.l.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
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)

$f(q)$	$=$	$q-2.23607 q^{3} +1.23607 q^{5} +4.23607 q^{7} +2.00000 q^{9} +O(q^{10})$	$q-2.23607 q^{3} +1.23607 q^{5} +4.23607 q^{7} +2.00000 q^{9} +4.23607 q^{11} -5.47214 q^{13} -2.76393 q^{15} +2.00000 q^{17} +7.70820 q^{19} -9.47214 q^{21} -2.23607 q^{23} -3.47214 q^{25} +2.23607 q^{27} -2.76393 q^{29} -7.70820 q^{31} -9.47214 q^{33} +5.23607 q^{35} -4.47214 q^{37} +12.2361 q^{39} -5.23607 q^{41} -10.2361 q^{43} +2.47214 q^{45} -4.23607 q^{47} +10.9443 q^{49} -4.47214 q^{51} -0.472136 q^{53} +5.23607 q^{55} -17.2361 q^{57} -8.94427 q^{59} +7.47214 q^{61} +8.47214 q^{63} -6.76393 q^{65} -4.70820 q^{67} +5.00000 q^{69} -0.763932 q^{71} -7.52786 q^{73} +7.76393 q^{75} +17.9443 q^{77} -12.9443 q^{79} -11.0000 q^{81} -16.7082 q^{83} +2.47214 q^{85} +6.18034 q^{87} -14.4721 q^{89} -23.1803 q^{91} +17.2361 q^{93} +9.52786 q^{95} -10.0000 q^{97} +8.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 + 4 * q^7 + 4 * q^9	$ 2 q - 2 q^{5} + 4 q^{7} + 4 q^{9} + 4 q^{11} - 2 q^{13} - 10 q^{15} + 4 q^{17} + 2 q^{19} - 10 q^{21} + 2 q^{25} - 10 q^{29} - 2 q^{31} - 10 q^{33} + 6 q^{35} + 20 q^{39} - 6 q^{41} - 16 q^{43} - 4 q^{45} - 4 q^{47} + 4 q^{49} + 8 q^{53} + 6 q^{55} - 30 q^{57} + 6 q^{61} + 8 q^{63} - 18 q^{65} + 4 q^{67} + 10 q^{69} - 6 q^{71} - 24 q^{73} + 20 q^{75} + 18 q^{77} - 8 q^{79} - 22 q^{81} - 20 q^{83} - 4 q^{85} - 10 q^{87} - 20 q^{89} - 24 q^{91} + 30 q^{93} + 28 q^{95} - 20 q^{97} + 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 + 4 * q^7 + 4 * q^9 + 4 * q^11 - 2 * q^13 - 10 * q^15 + 4 * q^17 + 2 * q^19 - 10 * q^21 + 2 * q^25 - 10 * q^29 - 2 * q^31 - 10 * q^33 + 6 * q^35 + 20 * q^39 - 6 * q^41 - 16 * q^43 - 4 * q^45 - 4 * q^47 + 4 * q^49 + 8 * q^53 + 6 * q^55 - 30 * q^57 + 6 * q^61 + 8 * q^63 - 18 * q^65 + 4 * q^67 + 10 * q^69 - 6 * q^71 - 24 * q^73 + 20 * q^75 + 18 * q^77 - 8 * q^79 - 22 * q^81 - 20 * q^83 - 4 * q^85 - 10 * q^87 - 20 * q^89 - 24 * q^91 + 30 * q^93 + 28 * q^95 - 20 * q^97 + 8 * 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
$3$	−2.23607	−1.29099	−0.645497	−	0.763763i	$-0.723350\pi$
$3$	−2.23607	−1.29099	−0.645497	+	0.763763i	$0.723350\pi$
$4$	0	0
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	0	0
$7$	4.23607	1.60108	0.800542	−	0.599277i	$-0.204545\pi$
$7$	4.23607	1.60108	0.800542	+	0.599277i	$0.204545\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	4.23607	1.27722	0.638611	−	0.769529i	$-0.279509\pi$
$11$	4.23607	1.27722	0.638611	+	0.769529i	$0.279509\pi$
$12$	0	0
$13$	−5.47214	−1.51770	−0.758849	−	0.651267i	$-0.774238\pi$
$13$	−5.47214	−1.51770	−0.758849	+	0.651267i	$0.774238\pi$
$14$	0	0
$15$	−2.76393	−0.713644
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	7.70820	1.76838	0.884192	−	0.467124i	$-0.154710\pi$
$19$	7.70820	1.76838	0.884192	+	0.467124i	$0.154710\pi$
$20$	0	0
$21$	−9.47214	−2.06699
$22$	0	0
$23$	−2.23607	−0.466252	−0.233126	−	0.972446i	$-0.574896\pi$
$23$	−2.23607	−0.466252	−0.233126	+	0.972446i	$0.574896\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	−2.76393	−0.513249	−0.256625	−	0.966511i	$-0.582610\pi$
$29$	−2.76393	−0.513249	−0.256625	+	0.966511i	$0.582610\pi$
$30$	0	0
$31$	−7.70820	−1.38443	−0.692217	−	0.721689i	$-0.743366\pi$
$31$	−7.70820	−1.38443	−0.692217	+	0.721689i	$0.743366\pi$
$32$	0	0
$33$	−9.47214	−1.64889
$34$	0	0
$35$	5.23607	0.885057
$36$	0	0
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	0	0
$39$	12.2361	1.95934
$40$	0	0
$41$	−5.23607	−0.817736	−0.408868	−	0.912593i	$-0.634076\pi$
$41$	−5.23607	−0.817736	−0.408868	+	0.912593i	$0.634076\pi$
$42$	0	0
$43$	−10.2361	−1.56099	−0.780493	−	0.625165i	$-0.785032\pi$
$43$	−10.2361	−1.56099	−0.780493	+	0.625165i	$0.785032\pi$
$44$	0	0
$45$	2.47214	0.368524
$46$	0	0
$47$	−4.23607	−0.617894	−0.308947	−	0.951079i	$-0.599977\pi$
$47$	−4.23607	−0.617894	−0.308947	+	0.951079i	$0.599977\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	0	0
$51$	−4.47214	−0.626224
$52$	0	0
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	0	0
$55$	5.23607	0.706031
$56$	0	0
$57$	−17.2361	−2.28297
$58$	0	0
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	0	0
$61$	7.47214	0.956709	0.478354	−	0.878167i	$-0.341233\pi$
$61$	7.47214	0.956709	0.478354	+	0.878167i	$0.341233\pi$
$62$	0	0
$63$	8.47214	1.06739
$64$	0	0
$65$	−6.76393	−0.838963
$66$	0	0
$67$	−4.70820	−0.575199	−0.287599	−	0.957751i	$-0.592857\pi$
$67$	−4.70820	−0.575199	−0.287599	+	0.957751i	$0.592857\pi$
$68$	0	0
$69$	5.00000	0.601929
$70$	0	0
$71$	−0.763932	−0.0906621	−0.0453310	−	0.998972i	$-0.514434\pi$
$71$	−0.763932	−0.0906621	−0.0453310	+	0.998972i	$0.514434\pi$
$72$	0	0
$73$	−7.52786	−0.881070	−0.440535	−	0.897735i	$-0.645211\pi$
$73$	−7.52786	−0.881070	−0.440535	+	0.897735i	$0.645211\pi$
$74$	0	0
$75$	7.76393	0.896502
$76$	0	0
$77$	17.9443	2.04494
$78$	0	0
$79$	−12.9443	−1.45634	−0.728172	−	0.685394i	$-0.759630\pi$
$79$	−12.9443	−1.45634	−0.728172	+	0.685394i	$0.759630\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−16.7082	−1.83396	−0.916982	−	0.398929i	$-0.869382\pi$
$83$	−16.7082	−1.83396	−0.916982	+	0.398929i	$0.869382\pi$
$84$	0	0
$85$	2.47214	0.268141
$86$	0	0
$87$	6.18034	0.662602
$88$	0	0
$89$	−14.4721	−1.53404	−0.767022	−	0.641621i	$-0.778262\pi$
$89$	−14.4721	−1.53404	−0.767022	+	0.641621i	$0.778262\pi$
$90$	0	0
$91$	−23.1803	−2.42996
$92$	0	0
$93$	17.2361	1.78730
$94$	0	0
$95$	9.52786	0.977538
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	8.47214	0.851482
$100$	0	0
$101$	−9.70820	−0.966002	−0.483001	−	0.875620i	$-0.660453\pi$
$101$	−9.70820	−0.966002	−0.483001	+	0.875620i	$0.660453\pi$
$102$	0	0
$103$	0.944272	0.0930419	0.0465209	−	0.998917i	$-0.485187\pi$
$103$	0.944272	0.0930419	0.0465209	+	0.998917i	$0.485187\pi$
$104$	0	0
$105$	−11.7082	−1.14260
$106$	0	0
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	8.47214	0.811483	0.405742	−	0.913988i	$-0.367013\pi$
$109$	8.47214	0.811483	0.405742	+	0.913988i	$0.367013\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	−8.52786	−0.802234	−0.401117	−	0.916027i	$-0.631378\pi$
$113$	−8.52786	−0.802234	−0.401117	+	0.916027i	$0.631378\pi$
$114$	0	0
$115$	−2.76393	−0.257738
$116$	0	0
$117$	−10.9443	−1.01180
$118$	0	0
$119$	8.47214	0.776639
$120$	0	0
$121$	6.94427	0.631297
$122$	0	0
$123$	11.7082	1.05569
$124$	0	0
$125$	−10.4721	−0.936656
$126$	0	0
$127$	−2.29180	−0.203364	−0.101682	−	0.994817i	$-0.532422\pi$
$127$	−2.29180	−0.203364	−0.101682	+	0.994817i	$0.532422\pi$
$128$	0	0
$129$	22.8885	2.01522
$130$	0	0
$131$	21.1803	1.85053	0.925267	−	0.379315i	$-0.123840\pi$
$131$	21.1803	1.85053	0.925267	+	0.379315i	$0.123840\pi$
$132$	0	0
$133$	32.6525	2.83133
$134$	0	0
$135$	2.76393	0.237881
$136$	0	0
$137$	−7.23607	−0.618219	−0.309110	−	0.951026i	$-0.600031\pi$
$137$	−7.23607	−0.618219	−0.309110	+	0.951026i	$0.600031\pi$
$138$	0	0
$139$	10.1803	0.863485	0.431743	−	0.901997i	$-0.357899\pi$
$139$	10.1803	0.863485	0.431743	+	0.901997i	$0.357899\pi$
$140$	0	0
$141$	9.47214	0.797698
$142$	0	0
$143$	−23.1803	−1.93844
$144$	0	0
$145$	−3.41641	−0.283717
$146$	0	0
$147$	−24.4721	−2.01843
$148$	0	0
$149$	−12.7639	−1.04566	−0.522831	−	0.852436i	$-0.675124\pi$
$149$	−12.7639	−1.04566	−0.522831	+	0.852436i	$0.675124\pi$
$150$	0	0
$151$	18.4721	1.50324	0.751621	−	0.659596i	$-0.229272\pi$
$151$	18.4721	1.50324	0.751621	+	0.659596i	$0.229272\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	−9.52786	−0.765296
$156$	0	0
$157$	12.4721	0.995385	0.497692	−	0.867354i	$-0.334181\pi$
$157$	12.4721	0.995385	0.497692	+	0.867354i	$0.334181\pi$
$158$	0	0
$159$	1.05573	0.0837247
$160$	0	0
$161$	−9.47214	−0.746509
$162$	0	0
$163$	11.7082	0.917057	0.458529	−	0.888680i	$-0.348377\pi$
$163$	11.7082	0.917057	0.458529	+	0.888680i	$0.348377\pi$
$164$	0	0
$165$	−11.7082	−0.911482
$166$	0	0
$167$	7.70820	0.596479	0.298239	−	0.954491i	$-0.403601\pi$
$167$	7.70820	0.596479	0.298239	+	0.954491i	$0.403601\pi$
$168$	0	0
$169$	16.9443	1.30341
$170$	0	0
$171$	15.4164	1.17892
$172$	0	0
$173$	11.0000	0.836315	0.418157	−	0.908375i	$-0.362676\pi$
$173$	11.0000	0.836315	0.418157	+	0.908375i	$0.362676\pi$
$174$	0	0
$175$	−14.7082	−1.11184
$176$	0	0
$177$	20.0000	1.50329
$178$	0	0
$179$	−5.70820	−0.426651	−0.213326	−	0.976981i	$-0.568430\pi$
$179$	−5.70820	−0.426651	−0.213326	+	0.976981i	$0.568430\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0	0
$183$	−16.7082	−1.23511
$184$	0	0
$185$	−5.52786	−0.406417
$186$	0	0
$187$	8.47214	0.619544
$188$	0	0
$189$	9.47214	0.688997
$190$	0	0
$191$	18.4721	1.33660	0.668298	−	0.743893i	$-0.267023\pi$
$191$	18.4721	1.33660	0.668298	+	0.743893i	$0.267023\pi$
$192$	0	0
$193$	−6.76393	−0.486878	−0.243439	−	0.969916i	$-0.578276\pi$
$193$	−6.76393	−0.486878	−0.243439	+	0.969916i	$0.578276\pi$
$194$	0	0
$195$	15.1246	1.08310
$196$	0	0
$197$	−20.4164	−1.45461	−0.727304	−	0.686315i	$-0.759227\pi$
$197$	−20.4164	−1.45461	−0.727304	+	0.686315i	$0.759227\pi$
$198$	0	0
$199$	−2.47214	−0.175245	−0.0876225	−	0.996154i	$-0.527927\pi$
$199$	−2.47214	−0.175245	−0.0876225	+	0.996154i	$0.527927\pi$
$200$	0	0
$201$	10.5279	0.742578
$202$	0	0
$203$	−11.7082	−0.821755
$204$	0	0
$205$	−6.47214	−0.452034
$206$	0	0
$207$	−4.47214	−0.310835
$208$	0	0
$209$	32.6525	2.25862
$210$	0	0
$211$	3.70820	0.255283	0.127642	−	0.991820i	$-0.459259\pi$
$211$	3.70820	0.255283	0.127642	+	0.991820i	$0.459259\pi$
$212$	0	0
$213$	1.70820	0.117044
$214$	0	0
$215$	−12.6525	−0.862892
$216$	0	0
$217$	−32.6525	−2.21659
$218$	0	0
$219$	16.8328	1.13746
$220$	0	0
$221$	−10.9443	−0.736191
$222$	0	0
$223$	−3.76393	−0.252052	−0.126026	−	0.992027i	$-0.540222\pi$
$223$	−3.76393	−0.252052	−0.126026	+	0.992027i	$0.540222\pi$
$224$	0	0
$225$	−6.94427	−0.462951
$226$	0	0
$227$	−11.7082	−0.777101	−0.388550	−	0.921427i	$-0.627024\pi$
$227$	−11.7082	−0.777101	−0.388550	+	0.921427i	$0.627024\pi$
$228$	0	0
$229$	18.4164	1.21699	0.608495	−	0.793558i	$-0.291773\pi$
$229$	18.4164	1.21699	0.608495	+	0.793558i	$0.291773\pi$
$230$	0	0
$231$	−40.1246	−2.64001
$232$	0	0
$233$	22.4164	1.46855	0.734274	−	0.678853i	$-0.237523\pi$
$233$	22.4164	1.46855	0.734274	+	0.678853i	$0.237523\pi$
$234$	0	0
$235$	−5.23607	−0.341563
$236$	0	0
$237$	28.9443	1.88013
$238$	0	0
$239$	25.4164	1.64405	0.822025	−	0.569451i	$-0.192844\pi$
$239$	25.4164	1.64405	0.822025	+	0.569451i	$0.192844\pi$
$240$	0	0
$241$	−4.94427	−0.318489	−0.159244	−	0.987239i	$-0.550906\pi$
$241$	−4.94427	−0.318489	−0.159244	+	0.987239i	$0.550906\pi$
$242$	0	0
$243$	17.8885	1.14755
$244$	0	0
$245$	13.5279	0.864264
$246$	0	0
$247$	−42.1803	−2.68387
$248$	0	0
$249$	37.3607	2.36764
$250$	0	0
$251$	4.18034	0.263861	0.131930	−	0.991259i	$-0.457882\pi$
$251$	4.18034	0.263861	0.131930	+	0.991259i	$0.457882\pi$
$252$	0	0
$253$	−9.47214	−0.595508
$254$	0	0
$255$	−5.52786	−0.346168
$256$	0	0
$257$	10.0557	0.627259	0.313630	−	0.949545i	$-0.398455\pi$
$257$	10.0557	0.627259	0.313630	+	0.949545i	$0.398455\pi$
$258$	0	0
$259$	−18.9443	−1.17714
$260$	0	0
$261$	−5.52786	−0.342166
$262$	0	0
$263$	−16.1246	−0.994286	−0.497143	−	0.867669i	$-0.665618\pi$
$263$	−16.1246	−0.994286	−0.497143	+	0.867669i	$0.665618\pi$
$264$	0	0
$265$	−0.583592	−0.0358498
$266$	0	0
$267$	32.3607	1.98044
$268$	0	0
$269$	16.9443	1.03311	0.516555	−	0.856254i	$-0.327214\pi$
$269$	16.9443	1.03311	0.516555	+	0.856254i	$0.327214\pi$
$270$	0	0
$271$	−26.7082	−1.62241	−0.811204	−	0.584763i	$-0.801187\pi$
$271$	−26.7082	−1.62241	−0.811204	+	0.584763i	$0.801187\pi$
$272$	0	0
$273$	51.8328	3.13706
$274$	0	0
$275$	−14.7082	−0.886938
$276$	0	0
$277$	26.1803	1.57302	0.786512	−	0.617575i	$-0.211885\pi$
$277$	26.1803	1.57302	0.786512	+	0.617575i	$0.211885\pi$
$278$	0	0
$279$	−15.4164	−0.922956
$280$	0	0
$281$	−28.4164	−1.69518	−0.847590	−	0.530651i	$-0.821947\pi$
$281$	−28.4164	−1.69518	−0.847590	+	0.530651i	$0.821947\pi$
$282$	0	0
$283$	18.4721	1.09805	0.549027	−	0.835804i	$-0.314998\pi$
$283$	18.4721	1.09805	0.549027	+	0.835804i	$0.314998\pi$
$284$	0	0
$285$	−21.3050	−1.26200
$286$	0	0
$287$	−22.1803	−1.30926
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	22.3607	1.31081
$292$	0	0
$293$	5.00000	0.292103	0.146052	−	0.989277i	$-0.453343\pi$
$293$	5.00000	0.292103	0.146052	+	0.989277i	$0.453343\pi$
$294$	0	0
$295$	−11.0557	−0.643689
$296$	0	0
$297$	9.47214	0.549629
$298$	0	0
$299$	12.2361	0.707630
$300$	0	0
$301$	−43.3607	−2.49927
$302$	0	0
$303$	21.7082	1.24710
$304$	0	0
$305$	9.23607	0.528856
$306$	0	0
$307$	−9.41641	−0.537423	−0.268711	−	0.963221i	$-0.586598\pi$
$307$	−9.41641	−0.537423	−0.268711	+	0.963221i	$0.586598\pi$
$308$	0	0
$309$	−2.11146	−0.120117
$310$	0	0
$311$	5.81966	0.330003	0.165001	−	0.986293i	$-0.447237\pi$
$311$	5.81966	0.330003	0.165001	+	0.986293i	$0.447237\pi$
$312$	0	0
$313$	−30.9443	−1.74907	−0.874537	−	0.484959i	$-0.838834\pi$
$313$	−30.9443	−1.74907	−0.874537	+	0.484959i	$0.838834\pi$
$314$	0	0
$315$	10.4721	0.590038
$316$	0	0
$317$	−8.05573	−0.452455	−0.226227	−	0.974075i	$-0.572639\pi$
$317$	−8.05573	−0.452455	−0.226227	+	0.974075i	$0.572639\pi$
$318$	0	0
$319$	−11.7082	−0.655534
$320$	0	0
$321$	−17.8885	−0.998441
$322$	0	0
$323$	15.4164	0.857792
$324$	0	0
$325$	19.0000	1.05393
$326$	0	0
$327$	−18.9443	−1.04762
$328$	0	0
$329$	−17.9443	−0.989300
$330$	0	0
$331$	−10.9443	−0.601552	−0.300776	−	0.953695i	$-0.597246\pi$
$331$	−10.9443	−0.601552	−0.300776	+	0.953695i	$0.597246\pi$
$332$	0	0
$333$	−8.94427	−0.490143
$334$	0	0
$335$	−5.81966	−0.317962
$336$	0	0
$337$	−30.9443	−1.68564	−0.842821	−	0.538194i	$-0.819107\pi$
$337$	−30.9443	−1.68564	−0.842821	+	0.538194i	$0.819107\pi$
$338$	0	0
$339$	19.0689	1.03568
$340$	0	0
$341$	−32.6525	−1.76823
$342$	0	0
$343$	16.7082	0.902158
$344$	0	0
$345$	6.18034	0.332738
$346$	0	0
$347$	8.00000	0.429463	0.214731	−	0.976673i	$-0.431112\pi$
$347$	8.00000	0.429463	0.214731	+	0.976673i	$0.431112\pi$
$348$	0	0
$349$	12.0000	0.642345	0.321173	−	0.947021i	$-0.395923\pi$
$349$	12.0000	0.642345	0.321173	+	0.947021i	$0.395923\pi$
$350$	0	0
$351$	−12.2361	−0.653113
$352$	0	0
$353$	−24.7639	−1.31805	−0.659026	−	0.752121i	$-0.729031\pi$
$353$	−24.7639	−1.31805	−0.659026	+	0.752121i	$0.729031\pi$
$354$	0	0
$355$	−0.944272	−0.0501167
$356$	0	0
$357$	−18.9443	−1.00264
$358$	0	0
$359$	9.70820	0.512379	0.256190	−	0.966627i	$-0.417533\pi$
$359$	9.70820	0.512379	0.256190	+	0.966627i	$0.417533\pi$
$360$	0	0
$361$	40.4164	2.12718
$362$	0	0
$363$	−15.5279	−0.815001
$364$	0	0
$365$	−9.30495	−0.487043
$366$	0	0
$367$	2.23607	0.116722	0.0583609	−	0.998296i	$-0.481413\pi$
$367$	2.23607	0.116722	0.0583609	+	0.998296i	$0.481413\pi$
$368$	0	0
$369$	−10.4721	−0.545158
$370$	0	0
$371$	−2.00000	−0.103835
$372$	0	0
$373$	26.3050	1.36202	0.681009	−	0.732275i	$-0.261541\pi$
$373$	26.3050	1.36202	0.681009	+	0.732275i	$0.261541\pi$
$374$	0	0
$375$	23.4164	1.20922
$376$	0	0
$377$	15.1246	0.778957
$378$	0	0
$379$	6.70820	0.344577	0.172289	−	0.985047i	$-0.444884\pi$
$379$	6.70820	0.344577	0.172289	+	0.985047i	$0.444884\pi$
$380$	0	0
$381$	5.12461	0.262542
$382$	0	0
$383$	12.9443	0.661421	0.330711	−	0.943732i	$-0.392712\pi$
$383$	12.9443	0.661421	0.330711	+	0.943732i	$0.392712\pi$
$384$	0	0
$385$	22.1803	1.13041
$386$	0	0
$387$	−20.4721	−1.04066
$388$	0	0
$389$	16.8328	0.853458	0.426729	−	0.904380i	$-0.359666\pi$
$389$	16.8328	0.853458	0.426729	+	0.904380i	$0.359666\pi$
$390$	0	0
$391$	−4.47214	−0.226166
$392$	0	0
$393$	−47.3607	−2.38903
$394$	0	0
$395$	−16.0000	−0.805047
$396$	0	0
$397$	37.3607	1.87508	0.937539	−	0.347879i	$-0.113098\pi$
$397$	37.3607	1.87508	0.937539	+	0.347879i	$0.113098\pi$
$398$	0	0
$399$	−73.0132	−3.65523
$400$	0	0
$401$	12.4164	0.620046	0.310023	−	0.950729i	$-0.399663\pi$
$401$	12.4164	0.620046	0.310023	+	0.950729i	$0.399663\pi$
$402$	0	0
$403$	42.1803	2.10115
$404$	0	0
$405$	−13.5967	−0.675628
$406$	0	0
$407$	−18.9443	−0.939033
$408$	0	0
$409$	1.70820	0.0844652	0.0422326	−	0.999108i	$-0.486553\pi$
$409$	1.70820	0.0844652	0.0422326	+	0.999108i	$0.486553\pi$
$410$	0	0
$411$	16.1803	0.798117
$412$	0	0
$413$	−37.8885	−1.86437
$414$	0	0
$415$	−20.6525	−1.01379
$416$	0	0
$417$	−22.7639	−1.11475
$418$	0	0
$419$	8.29180	0.405081	0.202540	−	0.979274i	$-0.435080\pi$
$419$	8.29180	0.405081	0.202540	+	0.979274i	$0.435080\pi$
$420$	0	0
$421$	−5.05573	−0.246401	−0.123201	−	0.992382i	$-0.539316\pi$
$421$	−5.05573	−0.246401	−0.123201	+	0.992382i	$0.539316\pi$
$422$	0	0
$423$	−8.47214	−0.411929
$424$	0	0
$425$	−6.94427	−0.336847
$426$	0	0
$427$	31.6525	1.53177
$428$	0	0
$429$	51.8328	2.50251
$430$	0	0
$431$	−1.34752	−0.0649080	−0.0324540	−	0.999473i	$-0.510332\pi$
$431$	−1.34752	−0.0649080	−0.0324540	+	0.999473i	$0.510332\pi$
$432$	0	0
$433$	20.4721	0.983828	0.491914	−	0.870644i	$-0.336297\pi$
$433$	20.4721	0.983828	0.491914	+	0.870644i	$0.336297\pi$
$434$	0	0
$435$	7.63932	0.366277
$436$	0	0
$437$	−17.2361	−0.824513
$438$	0	0
$439$	17.8885	0.853774	0.426887	−	0.904305i	$-0.359610\pi$
$439$	17.8885	0.853774	0.426887	+	0.904305i	$0.359610\pi$
$440$	0	0
$441$	21.8885	1.04231
$442$	0	0
$443$	−23.1803	−1.10133	−0.550666	−	0.834726i	$-0.685626\pi$
$443$	−23.1803	−1.10133	−0.550666	+	0.834726i	$0.685626\pi$
$444$	0	0
$445$	−17.8885	−0.847998
$446$	0	0
$447$	28.5410	1.34994
$448$	0	0
$449$	19.5967	0.924828	0.462414	−	0.886664i	$-0.346983\pi$
$449$	19.5967	0.924828	0.462414	+	0.886664i	$0.346983\pi$
$450$	0	0
$451$	−22.1803	−1.04443
$452$	0	0
$453$	−41.3050	−1.94068
$454$	0	0
$455$	−28.6525	−1.34325
$456$	0	0
$457$	−22.3607	−1.04599	−0.522994	−	0.852336i	$-0.675185\pi$
$457$	−22.3607	−1.04599	−0.522994	+	0.852336i	$0.675185\pi$
$458$	0	0
$459$	4.47214	0.208741
$460$	0	0
$461$	−16.0689	−0.748403	−0.374201	−	0.927348i	$-0.622083\pi$
$461$	−16.0689	−0.748403	−0.374201	+	0.927348i	$0.622083\pi$
$462$	0	0
$463$	−16.5967	−0.771316	−0.385658	−	0.922642i	$-0.626026\pi$
$463$	−16.5967	−0.771316	−0.385658	+	0.922642i	$0.626026\pi$
$464$	0	0
$465$	21.3050	0.987993
$466$	0	0
$467$	8.29180	0.383699	0.191849	−	0.981424i	$-0.438552\pi$
$467$	8.29180	0.383699	0.191849	+	0.981424i	$0.438552\pi$
$468$	0	0
$469$	−19.9443	−0.920941
$470$	0	0
$471$	−27.8885	−1.28504
$472$	0	0
$473$	−43.3607	−1.99373
$474$	0	0
$475$	−26.7639	−1.22801
$476$	0	0
$477$	−0.944272	−0.0432352
$478$	0	0
$479$	12.9443	0.591439	0.295719	−	0.955275i	$-0.404441\pi$
$479$	12.9443	0.591439	0.295719	+	0.955275i	$0.404441\pi$
$480$	0	0
$481$	24.4721	1.11583
$482$	0	0
$483$	21.1803	0.963739
$484$	0	0
$485$	−12.3607	−0.561270
$486$	0	0
$487$	−35.5967	−1.61304	−0.806521	−	0.591205i	$-0.798652\pi$
$487$	−35.5967	−1.61304	−0.806521	+	0.591205i	$0.798652\pi$
$488$	0	0
$489$	−26.1803	−1.18392
$490$	0	0
$491$	1.12461	0.0507530	0.0253765	−	0.999678i	$-0.491922\pi$
$491$	1.12461	0.0507530	0.0253765	+	0.999678i	$0.491922\pi$
$492$	0	0
$493$	−5.52786	−0.248962
$494$	0	0
$495$	10.4721	0.470688
$496$	0	0
$497$	−3.23607	−0.145157
$498$	0	0
$499$	35.8885	1.60659	0.803296	−	0.595580i	$-0.203078\pi$
$499$	35.8885	1.60659	0.803296	+	0.595580i	$0.203078\pi$
$500$	0	0
$501$	−17.2361	−0.770051
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−12.0000	−0.533993
$506$	0	0
$507$	−37.8885	−1.68269
$508$	0	0
$509$	14.8885	0.659923	0.329962	−	0.943994i	$-0.392964\pi$
$509$	14.8885	0.659923	0.329962	+	0.943994i	$0.392964\pi$
$510$	0	0
$511$	−31.8885	−1.41067
$512$	0	0
$513$	17.2361	0.760991
$514$	0	0
$515$	1.16718	0.0514323
$516$	0	0
$517$	−17.9443	−0.789188
$518$	0	0
$519$	−24.5967	−1.07968
$520$	0	0
$521$	−41.0000	−1.79624	−0.898121	−	0.439748i	$-0.855068\pi$
$521$	−41.0000	−1.79624	−0.898121	+	0.439748i	$0.855068\pi$
$522$	0	0
$523$	−12.3607	−0.540495	−0.270247	−	0.962791i	$-0.587105\pi$
$523$	−12.3607	−0.540495	−0.270247	+	0.962791i	$0.587105\pi$
$524$	0	0
$525$	32.8885	1.43537
$526$	0	0
$527$	−15.4164	−0.671549
$528$	0	0
$529$	−18.0000	−0.782609
$530$	0	0
$531$	−17.8885	−0.776297
$532$	0	0
$533$	28.6525	1.24108
$534$	0	0
$535$	9.88854	0.427519
$536$	0	0
$537$	12.7639	0.550804
$538$	0	0
$539$	46.3607	1.99690
$540$	0	0
$541$	6.65248	0.286012	0.143006	−	0.989722i	$-0.454323\pi$
$541$	6.65248	0.286012	0.143006	+	0.989722i	$0.454323\pi$
$542$	0	0
$543$	44.7214	1.91918
$544$	0	0
$545$	10.4721	0.448577
$546$	0	0
$547$	−36.3607	−1.55467	−0.777335	−	0.629087i	$-0.783429\pi$
$547$	−36.3607	−1.55467	−0.777335	+	0.629087i	$0.783429\pi$
$548$	0	0
$549$	14.9443	0.637806
$550$	0	0
$551$	−21.3050	−0.907621
$552$	0	0
$553$	−54.8328	−2.33173
$554$	0	0
$555$	12.3607	0.524682
$556$	0	0
$557$	−37.2492	−1.57830	−0.789150	−	0.614200i	$-0.789479\pi$
$557$	−37.2492	−1.57830	−0.789150	+	0.614200i	$0.789479\pi$
$558$	0	0
$559$	56.0132	2.36910
$560$	0	0
$561$	−18.9443	−0.799828
$562$	0	0
$563$	−35.7082	−1.50492	−0.752461	−	0.658637i	$-0.771133\pi$
$563$	−35.7082	−1.50492	−0.752461	+	0.658637i	$0.771133\pi$
$564$	0	0
$565$	−10.5410	−0.443464
$566$	0	0
$567$	−46.5967	−1.95688
$568$	0	0
$569$	39.3050	1.64775	0.823875	−	0.566772i	$-0.191808\pi$
$569$	39.3050	1.64775	0.823875	+	0.566772i	$0.191808\pi$
$570$	0	0
$571$	−15.8885	−0.664915	−0.332457	−	0.943118i	$-0.607878\pi$
$571$	−15.8885	−0.664915	−0.332457	+	0.943118i	$0.607878\pi$
$572$	0	0
$573$	−41.3050	−1.72554
$574$	0	0
$575$	7.76393	0.323778
$576$	0	0
$577$	−22.3607	−0.930887	−0.465444	−	0.885078i	$-0.654105\pi$
$577$	−22.3607	−0.930887	−0.465444	+	0.885078i	$0.654105\pi$
$578$	0	0
$579$	15.1246	0.628557
$580$	0	0
$581$	−70.7771	−2.93633
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	−13.5279	−0.559308
$586$	0	0
$587$	14.8328	0.612216	0.306108	−	0.951997i	$-0.400973\pi$
$587$	14.8328	0.612216	0.306108	+	0.951997i	$0.400973\pi$
$588$	0	0
$589$	−59.4164	−2.44821
$590$	0	0
$591$	45.6525	1.87789
$592$	0	0
$593$	16.4721	0.676430	0.338215	−	0.941069i	$-0.390177\pi$
$593$	16.4721	0.676430	0.338215	+	0.941069i	$0.390177\pi$
$594$	0	0
$595$	10.4721	0.429316
$596$	0	0
$597$	5.52786	0.226240
$598$	0	0
$599$	−25.5279	−1.04304	−0.521520	−	0.853239i	$-0.674635\pi$
$599$	−25.5279	−1.04304	−0.521520	+	0.853239i	$0.674635\pi$
$600$	0	0
$601$	−9.47214	−0.386376	−0.193188	−	0.981162i	$-0.561883\pi$
$601$	−9.47214	−0.386376	−0.193188	+	0.981162i	$0.561883\pi$
$602$	0	0
$603$	−9.41641	−0.383466
$604$	0	0
$605$	8.58359	0.348973
$606$	0	0
$607$	−10.2361	−0.415469	−0.207735	−	0.978185i	$-0.566609\pi$
$607$	−10.2361	−0.415469	−0.207735	+	0.978185i	$0.566609\pi$
$608$	0	0
$609$	26.1803	1.06088
$610$	0	0
$611$	23.1803	0.937776
$612$	0	0
$613$	7.81966	0.315833	0.157917	−	0.987452i	$-0.449522\pi$
$613$	7.81966	0.315833	0.157917	+	0.987452i	$0.449522\pi$
$614$	0	0
$615$	14.4721	0.583573
$616$	0	0
$617$	−23.8885	−0.961717	−0.480858	−	0.876798i	$-0.659675\pi$
$617$	−23.8885	−0.961717	−0.480858	+	0.876798i	$0.659675\pi$
$618$	0	0
$619$	−31.7082	−1.27446	−0.637230	−	0.770674i	$-0.719920\pi$
$619$	−31.7082	−1.27446	−0.637230	+	0.770674i	$0.719920\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	0	0
$623$	−61.3050	−2.45613
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	−73.0132	−2.91586
$628$	0	0
$629$	−8.94427	−0.356631
$630$	0	0
$631$	−12.2361	−0.487110	−0.243555	−	0.969887i	$-0.578314\pi$
$631$	−12.2361	−0.487110	−0.243555	+	0.969887i	$0.578314\pi$
$632$	0	0
$633$	−8.29180	−0.329569
$634$	0	0
$635$	−2.83282	−0.112417
$636$	0	0
$637$	−59.8885	−2.37287
$638$	0	0
$639$	−1.52786	−0.0604414
$640$	0	0
$641$	−10.4164	−0.411423	−0.205712	−	0.978613i	$-0.565951\pi$
$641$	−10.4164	−0.411423	−0.205712	+	0.978613i	$0.565951\pi$
$642$	0	0
$643$	−21.0557	−0.830357	−0.415178	−	0.909740i	$-0.636281\pi$
$643$	−21.0557	−0.830357	−0.415178	+	0.909740i	$0.636281\pi$
$644$	0	0
$645$	28.2918	1.11399
$646$	0	0
$647$	44.9443	1.76694	0.883471	−	0.468486i	$-0.155200\pi$
$647$	44.9443	1.76694	0.883471	+	0.468486i	$0.155200\pi$
$648$	0	0
$649$	−37.8885	−1.48726
$650$	0	0
$651$	73.0132	2.86161
$652$	0	0
$653$	9.00000	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$653$	9.00000	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$654$	0	0
$655$	26.1803	1.02295
$656$	0	0
$657$	−15.0557	−0.587380
$658$	0	0
$659$	−30.0132	−1.16915	−0.584573	−	0.811341i	$-0.698738\pi$
$659$	−30.0132	−1.16915	−0.584573	+	0.811341i	$0.698738\pi$
$660$	0	0
$661$	−47.3607	−1.84212	−0.921058	−	0.389424i	$-0.872674\pi$
$661$	−47.3607	−1.84212	−0.921058	+	0.389424i	$0.872674\pi$
$662$	0	0
$663$	24.4721	0.950419
$664$	0	0
$665$	40.3607	1.56512
$666$	0	0
$667$	6.18034	0.239304
$668$	0	0
$669$	8.41641	0.325397
$670$	0	0
$671$	31.6525	1.22193
$672$	0	0
$673$	22.9443	0.884437	0.442218	−	0.896907i	$-0.354192\pi$
$673$	22.9443	0.884437	0.442218	+	0.896907i	$0.354192\pi$
$674$	0	0
$675$	−7.76393	−0.298834
$676$	0	0
$677$	27.5279	1.05798	0.528991	−	0.848628i	$-0.322571\pi$
$677$	27.5279	1.05798	0.528991	+	0.848628i	$0.322571\pi$
$678$	0	0
$679$	−42.3607	−1.62565
$680$	0	0
$681$	26.1803	1.00323
$682$	0	0
$683$	23.8197	0.911434	0.455717	−	0.890125i	$-0.349383\pi$
$683$	23.8197	0.911434	0.455717	+	0.890125i	$0.349383\pi$
$684$	0	0
$685$	−8.94427	−0.341743
$686$	0	0
$687$	−41.1803	−1.57113
$688$	0	0
$689$	2.58359	0.0984270
$690$	0	0
$691$	8.00000	0.304334	0.152167	−	0.988355i	$-0.451375\pi$
$691$	8.00000	0.304334	0.152167	+	0.988355i	$0.451375\pi$
$692$	0	0
$693$	35.8885	1.36329
$694$	0	0
$695$	12.5836	0.477323
$696$	0	0
$697$	−10.4721	−0.396660
$698$	0	0
$699$	−50.1246	−1.89589
$700$	0	0
$701$	9.00000	0.339925	0.169963	−	0.985451i	$-0.445635\pi$
$701$	9.00000	0.339925	0.169963	+	0.985451i	$0.445635\pi$
$702$	0	0
$703$	−34.4721	−1.30014
$704$	0	0
$705$	11.7082	0.440956
$706$	0	0
$707$	−41.1246	−1.54665
$708$	0	0
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	0	0
$711$	−25.8885	−0.970896
$712$	0	0
$713$	17.2361	0.645496
$714$	0	0
$715$	−28.6525	−1.07154
$716$	0	0
$717$	−56.8328	−2.12246
$718$	0	0
$719$	9.88854	0.368780	0.184390	−	0.982853i	$-0.440969\pi$
$719$	9.88854	0.368780	0.184390	+	0.982853i	$0.440969\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	0	0
$723$	11.0557	0.411167
$724$	0	0
$725$	9.59675	0.356414
$726$	0	0
$727$	−21.0689	−0.781402	−0.390701	−	0.920518i	$-0.627767\pi$
$727$	−21.0689	−0.781402	−0.390701	+	0.920518i	$0.627767\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	0	0
$731$	−20.4721	−0.757189
$732$	0	0
$733$	−29.2361	−1.07986	−0.539929	−	0.841710i	$-0.681549\pi$
$733$	−29.2361	−1.07986	−0.539929	+	0.841710i	$0.681549\pi$
$734$	0	0
$735$	−30.2492	−1.11576
$736$	0	0
$737$	−19.9443	−0.734657
$738$	0	0
$739$	30.2361	1.11225	0.556126	−	0.831098i	$-0.312287\pi$
$739$	30.2361	1.11225	0.556126	+	0.831098i	$0.312287\pi$
$740$	0	0
$741$	94.3181	3.46486
$742$	0	0
$743$	36.2918	1.33142	0.665708	−	0.746212i	$-0.268129\pi$
$743$	36.2918	1.33142	0.665708	+	0.746212i	$0.268129\pi$
$744$	0	0
$745$	−15.7771	−0.578028
$746$	0	0
$747$	−33.4164	−1.22264
$748$	0	0
$749$	33.8885	1.23826
$750$	0	0
$751$	40.3607	1.47278	0.736391	−	0.676556i	$-0.236528\pi$
$751$	40.3607	1.47278	0.736391	+	0.676556i	$0.236528\pi$
$752$	0	0
$753$	−9.34752	−0.340643
$754$	0	0
$755$	22.8328	0.830971
$756$	0	0
$757$	42.3607	1.53963	0.769813	−	0.638270i	$-0.220350\pi$
$757$	42.3607	1.53963	0.769813	+	0.638270i	$0.220350\pi$
$758$	0	0
$759$	21.1803	0.768798
$760$	0	0
$761$	7.00000	0.253750	0.126875	−	0.991919i	$-0.459505\pi$
$761$	7.00000	0.253750	0.126875	+	0.991919i	$0.459505\pi$
$762$	0	0
$763$	35.8885	1.29925
$764$	0	0
$765$	4.94427	0.178761
$766$	0	0
$767$	48.9443	1.76728
$768$	0	0
$769$	45.7771	1.65076	0.825382	−	0.564575i	$-0.190960\pi$
$769$	45.7771	1.65076	0.825382	+	0.564575i	$0.190960\pi$
$770$	0	0
$771$	−22.4853	−0.809788
$772$	0	0
$773$	33.5967	1.20839	0.604196	−	0.796836i	$-0.293495\pi$
$773$	33.5967	1.20839	0.604196	+	0.796836i	$0.293495\pi$
$774$	0	0
$775$	26.7639	0.961389
$776$	0	0
$777$	42.3607	1.51968
$778$	0	0
$779$	−40.3607	−1.44607
$780$	0	0
$781$	−3.23607	−0.115796
$782$	0	0
$783$	−6.18034	−0.220867
$784$	0	0
$785$	15.4164	0.550235
$786$	0	0
$787$	−52.0689	−1.85606	−0.928028	−	0.372511i	$-0.878497\pi$
$787$	−52.0689	−1.85606	−0.928028	+	0.372511i	$0.878497\pi$
$788$	0	0
$789$	36.0557	1.28362
$790$	0	0
$791$	−36.1246	−1.28444
$792$	0	0
$793$	−40.8885	−1.45199
$794$	0	0
$795$	1.30495	0.0462819
$796$	0	0
$797$	−4.83282	−0.171187	−0.0855936	−	0.996330i	$-0.527279\pi$
$797$	−4.83282	−0.171187	−0.0855936	+	0.996330i	$0.527279\pi$
$798$	0	0
$799$	−8.47214	−0.299723
$800$	0	0
$801$	−28.9443	−1.02270
$802$	0	0
$803$	−31.8885	−1.12532
$804$	0	0
$805$	−11.7082	−0.412660
$806$	0	0
$807$	−37.8885	−1.33374
$808$	0	0
$809$	−6.18034	−0.217289	−0.108645	−	0.994081i	$-0.534651\pi$
$809$	−6.18034	−0.217289	−0.108645	+	0.994081i	$0.534651\pi$
$810$	0	0
$811$	8.23607	0.289207	0.144604	−	0.989490i	$-0.453809\pi$
$811$	8.23607	0.289207	0.144604	+	0.989490i	$0.453809\pi$
$812$	0	0
$813$	59.7214	2.09452
$814$	0	0
$815$	14.4721	0.506937
$816$	0	0
$817$	−78.9017	−2.76042
$818$	0	0
$819$	−46.3607	−1.61997
$820$	0	0
$821$	40.9443	1.42896	0.714482	−	0.699653i	$-0.246662\pi$
$821$	40.9443	1.42896	0.714482	+	0.699653i	$0.246662\pi$
$822$	0	0
$823$	24.1803	0.842874	0.421437	−	0.906858i	$-0.361526\pi$
$823$	24.1803	0.842874	0.421437	+	0.906858i	$0.361526\pi$
$824$	0	0
$825$	32.8885	1.14503
$826$	0	0
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	0	0
$829$	−45.5967	−1.58364	−0.791820	−	0.610754i	$-0.790866\pi$
$829$	−45.5967	−1.58364	−0.791820	+	0.610754i	$0.790866\pi$
$830$	0	0
$831$	−58.5410	−2.03077
$832$	0	0
$833$	21.8885	0.758393
$834$	0	0
$835$	9.52786	0.329725
$836$	0	0
$837$	−17.2361	−0.595766
$838$	0	0
$839$	−34.5967	−1.19441	−0.597206	−	0.802088i	$-0.703723\pi$
$839$	−34.5967	−1.19441	−0.597206	+	0.802088i	$0.703723\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	0	0
$843$	63.5410	2.18847
$844$	0	0
$845$	20.9443	0.720505
$846$	0	0
$847$	29.4164	1.01076
$848$	0	0
$849$	−41.3050	−1.41758
$850$	0	0
$851$	10.0000	0.342796
$852$	0	0
$853$	15.8328	0.542105	0.271053	−	0.962565i	$-0.412628\pi$
$853$	15.8328	0.542105	0.271053	+	0.962565i	$0.412628\pi$
$854$	0	0
$855$	19.0557	0.651692
$856$	0	0
$857$	16.0557	0.548453	0.274227	−	0.961665i	$-0.411578\pi$
$857$	16.0557	0.548453	0.274227	+	0.961665i	$0.411578\pi$
$858$	0	0
$859$	14.2918	0.487630	0.243815	−	0.969822i	$-0.421601\pi$
$859$	14.2918	0.487630	0.243815	+	0.969822i	$0.421601\pi$
$860$	0	0
$861$	49.5967	1.69025
$862$	0	0
$863$	−7.23607	−0.246319	−0.123159	−	0.992387i	$-0.539303\pi$
$863$	−7.23607	−0.246319	−0.123159	+	0.992387i	$0.539303\pi$
$864$	0	0
$865$	13.5967	0.462303
$866$	0	0
$867$	29.0689	0.987231
$868$	0	0
$869$	−54.8328	−1.86008
$870$	0	0
$871$	25.7639	0.872978
$872$	0	0
$873$	−20.0000	−0.676897
$874$	0	0
$875$	−44.3607	−1.49966
$876$	0	0
$877$	−17.7771	−0.600290	−0.300145	−	0.953894i	$-0.597035\pi$
$877$	−17.7771	−0.600290	−0.300145	+	0.953894i	$0.597035\pi$
$878$	0	0
$879$	−11.1803	−0.377104
$880$	0	0
$881$	−11.8885	−0.400535	−0.200268	−	0.979741i	$-0.564181\pi$
$881$	−11.8885	−0.400535	−0.200268	+	0.979741i	$0.564181\pi$
$882$	0	0
$883$	−41.5279	−1.39752	−0.698762	−	0.715354i	$-0.746265\pi$
$883$	−41.5279	−1.39752	−0.698762	+	0.715354i	$0.746265\pi$
$884$	0	0
$885$	24.7214	0.830999
$886$	0	0
$887$	40.7214	1.36729	0.683645	−	0.729815i	$-0.260394\pi$
$887$	40.7214	1.36729	0.683645	+	0.729815i	$0.260394\pi$
$888$	0	0
$889$	−9.70820	−0.325603
$890$	0	0
$891$	−46.5967	−1.56105
$892$	0	0
$893$	−32.6525	−1.09267
$894$	0	0
$895$	−7.05573	−0.235847
$896$	0	0
$897$	−27.3607	−0.913547
$898$	0	0
$899$	21.3050	0.710560
$900$	0	0
$901$	−0.944272	−0.0314583
$902$	0	0
$903$	96.9574	3.22654
$904$	0	0
$905$	−24.7214	−0.821766
$906$	0	0
$907$	2.94427	0.0977629	0.0488815	−	0.998805i	$-0.484434\pi$
$907$	2.94427	0.0977629	0.0488815	+	0.998805i	$0.484434\pi$
$908$	0	0
$909$	−19.4164	−0.644002
$910$	0	0
$911$	−6.76393	−0.224099	−0.112050	−	0.993703i	$-0.535742\pi$
$911$	−6.76393	−0.224099	−0.112050	+	0.993703i	$0.535742\pi$
$912$	0	0
$913$	−70.7771	−2.34238
$914$	0	0
$915$	−20.6525	−0.682750
$916$	0	0
$917$	89.7214	2.96286
$918$	0	0
$919$	−4.94427	−0.163096	−0.0815482	−	0.996669i	$-0.525986\pi$
$919$	−4.94427	−0.163096	−0.0815482	+	0.996669i	$0.525986\pi$
$920$	0	0
$921$	21.0557	0.693810
$922$	0	0
$923$	4.18034	0.137598
$924$	0	0
$925$	15.5279	0.510553
$926$	0	0
$927$	1.88854	0.0620279
$928$	0	0
$929$	26.6525	0.874439	0.437220	−	0.899355i	$-0.355963\pi$
$929$	26.6525	0.874439	0.437220	+	0.899355i	$0.355963\pi$
$930$	0	0
$931$	84.3607	2.76481
$932$	0	0
$933$	−13.0132	−0.426032
$934$	0	0
$935$	10.4721	0.342475
$936$	0	0
$937$	−16.0689	−0.524948	−0.262474	−	0.964939i	$-0.584538\pi$
$937$	−16.0689	−0.524948	−0.262474	+	0.964939i	$0.584538\pi$
$938$	0	0
$939$	69.1935	2.25804
$940$	0	0
$941$	36.4164	1.18714	0.593570	−	0.804782i	$-0.297718\pi$
$941$	36.4164	1.18714	0.593570	+	0.804782i	$0.297718\pi$
$942$	0	0
$943$	11.7082	0.381272
$944$	0	0
$945$	11.7082	0.380868
$946$	0	0
$947$	29.2361	0.950045	0.475022	−	0.879974i	$-0.342440\pi$
$947$	29.2361	0.950045	0.475022	+	0.879974i	$0.342440\pi$
$948$	0	0
$949$	41.1935	1.33720
$950$	0	0
$951$	18.0132	0.584117
$952$	0	0
$953$	20.0557	0.649669	0.324834	−	0.945771i	$-0.394691\pi$
$953$	20.0557	0.649669	0.324834	+	0.945771i	$0.394691\pi$
$954$	0	0
$955$	22.8328	0.738853
$956$	0	0
$957$	26.1803	0.846290
$958$	0	0
$959$	−30.6525	−0.989820
$960$	0	0
$961$	28.4164	0.916658
$962$	0	0
$963$	16.0000	0.515593
$964$	0	0
$965$	−8.36068	−0.269140
$966$	0	0
$967$	−61.3050	−1.97143	−0.985717	−	0.168409i	$-0.946137\pi$
$967$	−61.3050	−1.97143	−0.985717	+	0.168409i	$0.946137\pi$
$968$	0	0
$969$	−34.4721	−1.10740
$970$	0	0
$971$	57.0689	1.83143	0.915714	−	0.401831i	$-0.131626\pi$
$971$	57.0689	1.83143	0.915714	+	0.401831i	$0.131626\pi$
$972$	0	0
$973$	43.1246	1.38251
$974$	0	0
$975$	−42.4853	−1.36062
$976$	0	0
$977$	39.8885	1.27615	0.638074	−	0.769975i	$-0.279731\pi$
$977$	39.8885	1.27615	0.638074	+	0.769975i	$0.279731\pi$
$978$	0	0
$979$	−61.3050	−1.95931
$980$	0	0
$981$	16.9443	0.540989
$982$	0	0
$983$	39.2361	1.25144	0.625718	−	0.780049i	$-0.284806\pi$
$983$	39.2361	1.25144	0.625718	+	0.780049i	$0.284806\pi$
$984$	0	0
$985$	−25.2361	−0.804088
$986$	0	0
$987$	40.1246	1.27718
$988$	0	0
$989$	22.8885	0.727813
$990$	0	0
$991$	36.9574	1.17399	0.586996	−	0.809590i	$-0.300311\pi$
$991$	36.9574	1.17399	0.586996	+	0.809590i	$0.300311\pi$
$992$	0	0
$993$	24.4721	0.776600
$994$	0	0
$995$	−3.05573	−0.0968731
$996$	0	0
$997$	−2.06888	−0.0655222	−0.0327611	−	0.999463i	$-0.510430\pi$
$997$	−2.06888	−0.0655222	−0.0327611	+	0.999463i	$0.510430\pi$
$998$	0	0
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8048.2.a.l.1.1		2
4.3	odd	2		1006.2.a.f.1.2	✓	2
12.11	even	2		9054.2.a.y.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.f.1.2	✓	2	4.3	odd	2
8048.2.a.l.1.1		2	1.1	even	1	trivial
9054.2.a.y.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8048 = 2^{4} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8048.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{3} +1.23607 q^{5} +4.23607 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q-2.23607 q^{3} +1.23607 q^{5} +4.23607 q^{7} +2.00000 q^{9} +4.23607 q^{11} -5.47214 q^{13} -2.76393 q^{15} +2.00000 q^{17} +7.70820 q^{19} -9.47214 q^{21} -2.23607 q^{23} -3.47214 q^{25} +2.23607 q^{27} -2.76393 q^{29} -7.70820 q^{31} -9.47214 q^{33} +5.23607 q^{35} -4.47214 q^{37} +12.2361 q^{39} -5.23607 q^{41} -10.2361 q^{43} +2.47214 q^{45} -4.23607 q^{47} +10.9443 q^{49} -4.47214 q^{51} -0.472136 q^{53} +5.23607 q^{55} -17.2361 q^{57} -8.94427 q^{59} +7.47214 q^{61} +8.47214 q^{63} -6.76393 q^{65} -4.70820 q^{67} +5.00000 q^{69} -0.763932 q^{71} -7.52786 q^{73} +7.76393 q^{75} +17.9443 q^{77} -12.9443 q^{79} -11.0000 q^{81} -16.7082 q^{83} +2.47214 q^{85} +6.18034 q^{87} -14.4721 q^{89} -23.1803 q^{91} +17.2361 q^{93} +9.52786 q^{95} -10.0000 q^{97} +8.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 + 4 * q^7 + 4 * q^9	\( 2 q - 2 q^{5} + 4 q^{7} + 4 q^{9} + 4 q^{11} - 2 q^{13} - 10 q^{15} + 4 q^{17} + 2 q^{19} - 10 q^{21} + 2 q^{25} - 10 q^{29} - 2 q^{31} - 10 q^{33} + 6 q^{35} + 20 q^{39} - 6 q^{41} - 16 q^{43} - 4 q^{45} - 4 q^{47} + 4 q^{49} + 8 q^{53} + 6 q^{55} - 30 q^{57} + 6 q^{61} + 8 q^{63} - 18 q^{65} + 4 q^{67} + 10 q^{69} - 6 q^{71} - 24 q^{73} + 20 q^{75} + 18 q^{77} - 8 q^{79} - 22 q^{81} - 20 q^{83} - 4 q^{85} - 10 q^{87} - 20 q^{89} - 24 q^{91} + 30 q^{93} + 28 q^{95} - 20 q^{97} + 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 + 4 * q^7 + 4 * q^9 + 4 * q^11 - 2 * q^13 - 10 * q^15 + 4 * q^17 + 2 * q^19 - 10 * q^21 + 2 * q^25 - 10 * q^29 - 2 * q^31 - 10 * q^33 + 6 * q^35 + 20 * q^39 - 6 * q^41 - 16 * q^43 - 4 * q^45 - 4 * q^47 + 4 * q^49 + 8 * q^53 + 6 * q^55 - 30 * q^57 + 6 * q^61 + 8 * q^63 - 18 * q^65 + 4 * q^67 + 10 * q^69 - 6 * q^71 - 24 * q^73 + 20 * q^75 + 18 * q^77 - 8 * q^79 - 22 * q^81 - 20 * q^83 - 4 * q^85 - 10 * q^87 - 20 * q^89 - 24 * q^91 + 30 * q^93 + 28 * q^95 - 20 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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