LMFDB - Embedded newform 8040.2.a.bc.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8040,2,Mod(1,8040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8040.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:	$64.1997232251$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 37x^{8} + 132x^{7} + 358x^{6} - 1708x^{5} - 92x^{4} + 5969x^{3} - 3864x^{2} - 4752x + 3524 $ x^10 - 3x^9 - 37x^8 + 132x^7 + 358x^6 - 1708x^5 - 92x^4 + 5969x^3 - 3864x^2 - 4752*x + 3524
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-4.09581$ of defining polynomial
Character	$\chi$	$=$	8040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +1.83837 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +1.83837 q^{7} +1.00000 q^{9} -3.10197 q^{11} -3.40915 q^{13} -1.00000 q^{15} +3.12810 q^{17} +4.00619 q^{19} -1.83837 q^{21} +3.75152 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.96191 q^{29} -6.69616 q^{31} +3.10197 q^{33} +1.83837 q^{35} +9.55270 q^{37} +3.40915 q^{39} +1.84212 q^{41} -3.08508 q^{43} +1.00000 q^{45} -8.19434 q^{47} -3.62038 q^{49} -3.12810 q^{51} -4.80369 q^{53} -3.10197 q^{55} -4.00619 q^{57} -0.696043 q^{59} +2.50296 q^{61} +1.83837 q^{63} -3.40915 q^{65} -1.00000 q^{67} -3.75152 q^{69} -1.50115 q^{71} +4.39100 q^{73} -1.00000 q^{75} -5.70258 q^{77} +15.9021 q^{79} +1.00000 q^{81} +9.99211 q^{83} +3.12810 q^{85} -4.96191 q^{87} -3.33509 q^{89} -6.26730 q^{91} +6.69616 q^{93} +4.00619 q^{95} -1.40962 q^{97} -3.10197 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9	$ 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9} + 7 q^{11} - q^{13} - 10 q^{15} + 3 q^{17} + 4 q^{19} - q^{21} - 13 q^{23} + 10 q^{25} - 10 q^{27} + 18 q^{29} - 9 q^{31} - 7 q^{33} + q^{35} - 3 q^{37} + q^{39} + 19 q^{41} + 5 q^{43} + 10 q^{45} - 8 q^{47} + 43 q^{49} - 3 q^{51} + 17 q^{53} + 7 q^{55} - 4 q^{57} + 24 q^{59} + 21 q^{61} + q^{63} - q^{65} - 10 q^{67} + 13 q^{69} + 2 q^{71} + 25 q^{73} - 10 q^{75} + 15 q^{77} - q^{79} + 10 q^{81} - 6 q^{83} + 3 q^{85} - 18 q^{87} + 23 q^{89} + 29 q^{91} + 9 q^{93} + 4 q^{95} + 21 q^{97} + 7 q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9 + 7 * q^11 - q^13 - 10 * q^15 + 3 * q^17 + 4 * q^19 - q^21 - 13 * q^23 + 10 * q^25 - 10 * q^27 + 18 * q^29 - 9 * q^31 - 7 * q^33 + q^35 - 3 * q^37 + q^39 + 19 * q^41 + 5 * q^43 + 10 * q^45 - 8 * q^47 + 43 * q^49 - 3 * q^51 + 17 * q^53 + 7 * q^55 - 4 * q^57 + 24 * q^59 + 21 * q^61 + q^63 - q^65 - 10 * q^67 + 13 * q^69 + 2 * q^71 + 25 * q^73 - 10 * q^75 + 15 * q^77 - q^79 + 10 * q^81 - 6 * q^83 + 3 * q^85 - 18 * q^87 + 23 * q^89 + 29 * q^91 + 9 * q^93 + 4 * q^95 + 21 * q^97 + 7 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.83837	0.694840	0.347420	−	0.937710i	$-0.387058\pi$
$7$	1.83837	0.694840	0.347420	+	0.937710i	$0.387058\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.10197	−0.935279	−0.467639	−	0.883919i	$-0.654895\pi$
$11$	−3.10197	−0.935279	−0.467639	+	0.883919i	$0.654895\pi$
$12$	0	0
$13$	−3.40915	−0.945529	−0.472765	−	0.881189i	$-0.656744\pi$
$13$	−3.40915	−0.945529	−0.472765	+	0.881189i	$0.656744\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	3.12810	0.758675	0.379338	−	0.925258i	$-0.376152\pi$
$17$	3.12810	0.758675	0.379338	+	0.925258i	$0.376152\pi$
$18$	0	0
$19$	4.00619	0.919083	0.459542	−	0.888156i	$-0.348014\pi$
$19$	4.00619	0.919083	0.459542	+	0.888156i	$0.348014\pi$
$20$	0	0
$21$	−1.83837	−0.401166
$22$	0	0
$23$	3.75152	0.782246	0.391123	−	0.920338i	$-0.372087\pi$
$23$	3.75152	0.782246	0.391123	+	0.920338i	$0.372087\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.96191	0.921403	0.460701	−	0.887555i	$-0.347598\pi$
$29$	4.96191	0.921403	0.460701	+	0.887555i	$0.347598\pi$
$30$	0	0
$31$	−6.69616	−1.20267	−0.601333	−	0.798998i	$-0.705364\pi$
$31$	−6.69616	−1.20267	−0.601333	+	0.798998i	$0.705364\pi$
$32$	0	0
$33$	3.10197	0.539983
$34$	0	0
$35$	1.83837	0.310742
$36$	0	0
$37$	9.55270	1.57046	0.785228	−	0.619207i	$-0.212546\pi$
$37$	9.55270	1.57046	0.785228	+	0.619207i	$0.212546\pi$
$38$	0	0
$39$	3.40915	0.545902
$40$	0	0
$41$	1.84212	0.287691	0.143846	−	0.989600i	$-0.454053\pi$
$41$	1.84212	0.287691	0.143846	+	0.989600i	$0.454053\pi$
$42$	0	0
$43$	−3.08508	−0.470471	−0.235235	−	0.971938i	$-0.575586\pi$
$43$	−3.08508	−0.470471	−0.235235	+	0.971938i	$0.575586\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−8.19434	−1.19527	−0.597633	−	0.801770i	$-0.703892\pi$
$47$	−8.19434	−1.19527	−0.597633	+	0.801770i	$0.703892\pi$
$48$	0	0
$49$	−3.62038	−0.517197
$50$	0	0
$51$	−3.12810	−0.438021
$52$	0	0
$53$	−4.80369	−0.659838	−0.329919	−	0.944009i	$-0.607021\pi$
$53$	−4.80369	−0.659838	−0.329919	+	0.944009i	$0.607021\pi$
$54$	0	0
$55$	−3.10197	−0.418269
$56$	0	0
$57$	−4.00619	−0.530633
$58$	0	0
$59$	−0.696043	−0.0906171	−0.0453085	−	0.998973i	$-0.514427\pi$
$59$	−0.696043	−0.0906171	−0.0453085	+	0.998973i	$0.514427\pi$
$60$	0	0
$61$	2.50296	0.320471	0.160236	−	0.987079i	$-0.448775\pi$
$61$	2.50296	0.320471	0.160236	+	0.987079i	$0.448775\pi$
$62$	0	0
$63$	1.83837	0.231613
$64$	0	0
$65$	−3.40915	−0.422854
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	−3.75152	−0.451630
$70$	0	0
$71$	−1.50115	−0.178154	−0.0890768	−	0.996025i	$-0.528392\pi$
$71$	−1.50115	−0.178154	−0.0890768	+	0.996025i	$0.528392\pi$
$72$	0	0
$73$	4.39100	0.513928	0.256964	−	0.966421i	$-0.417278\pi$
$73$	4.39100	0.513928	0.256964	+	0.966421i	$0.417278\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−5.70258	−0.649869
$78$	0	0
$79$	15.9021	1.78912	0.894562	−	0.446944i	$-0.147488\pi$
$79$	15.9021	1.78912	0.894562	+	0.446944i	$0.147488\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.99211	1.09678	0.548388	−	0.836224i	$-0.315242\pi$
$83$	9.99211	1.09678	0.548388	+	0.836224i	$0.315242\pi$
$84$	0	0
$85$	3.12810	0.339290
$86$	0	0
$87$	−4.96191	−0.531972
$88$	0	0
$89$	−3.33509	−0.353519	−0.176759	−	0.984254i	$-0.556561\pi$
$89$	−3.33509	−0.353519	−0.176759	+	0.984254i	$0.556561\pi$
$90$	0	0
$91$	−6.26730	−0.656992
$92$	0	0
$93$	6.69616	0.694360
$94$	0	0
$95$	4.00619	0.411027
$96$	0	0
$97$	−1.40962	−0.143126	−0.0715628	−	0.997436i	$-0.522799\pi$
$97$	−1.40962	−0.143126	−0.0715628	+	0.997436i	$0.522799\pi$
$98$	0	0
$99$	−3.10197	−0.311760
$100$	0	0
$101$	−5.08178	−0.505656	−0.252828	−	0.967511i	$-0.581361\pi$
$101$	−5.08178	−0.505656	−0.252828	+	0.967511i	$0.581361\pi$
$102$	0	0
$103$	2.01121	0.198171	0.0990854	−	0.995079i	$-0.468408\pi$
$103$	2.01121	0.198171	0.0990854	+	0.995079i	$0.468408\pi$
$104$	0	0
$105$	−1.83837	−0.179407
$106$	0	0
$107$	16.8729	1.63116	0.815580	−	0.578644i	$-0.196418\pi$
$107$	16.8729	1.63116	0.815580	+	0.578644i	$0.196418\pi$
$108$	0	0
$109$	3.75195	0.359372	0.179686	−	0.983724i	$-0.442492\pi$
$109$	3.75195	0.359372	0.179686	+	0.983724i	$0.442492\pi$
$110$	0	0
$111$	−9.55270	−0.906703
$112$	0	0
$113$	−5.67326	−0.533695	−0.266848	−	0.963739i	$-0.585982\pi$
$113$	−5.67326	−0.533695	−0.266848	+	0.963739i	$0.585982\pi$
$114$	0	0
$115$	3.75152	0.349831
$116$	0	0
$117$	−3.40915	−0.315176
$118$	0	0
$119$	5.75061	0.527158
$120$	0	0
$121$	−1.37779	−0.125254
$122$	0	0
$123$	−1.84212	−0.166099
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.9942	−1.06432	−0.532158	−	0.846645i	$-0.678619\pi$
$127$	−11.9942	−1.06432	−0.532158	+	0.846645i	$0.678619\pi$
$128$	0	0
$129$	3.08508	0.271626
$130$	0	0
$131$	11.0566	0.966020	0.483010	−	0.875615i	$-0.339544\pi$
$131$	11.0566	0.966020	0.483010	+	0.875615i	$0.339544\pi$
$132$	0	0
$133$	7.36488	0.638616
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	11.2802	0.963736	0.481868	−	0.876244i	$-0.339958\pi$
$137$	11.2802	0.963736	0.481868	+	0.876244i	$0.339958\pi$
$138$	0	0
$139$	21.0811	1.78807	0.894037	−	0.447993i	$-0.147861\pi$
$139$	21.0811	1.78807	0.894037	+	0.447993i	$0.147861\pi$
$140$	0	0
$141$	8.19434	0.690088
$142$	0	0
$143$	10.5751	0.884333
$144$	0	0
$145$	4.96191	0.412064
$146$	0	0
$147$	3.62038	0.298604
$148$	0	0
$149$	22.9065	1.87657	0.938285	−	0.345863i	$-0.112414\pi$
$149$	22.9065	1.87657	0.938285	+	0.345863i	$0.112414\pi$
$150$	0	0
$151$	−19.9491	−1.62344	−0.811719	−	0.584048i	$-0.801468\pi$
$151$	−19.9491	−1.62344	−0.811719	+	0.584048i	$0.801468\pi$
$152$	0	0
$153$	3.12810	0.252892
$154$	0	0
$155$	−6.69616	−0.537849
$156$	0	0
$157$	17.4419	1.39202	0.696009	−	0.718033i	$-0.254957\pi$
$157$	17.4419	1.39202	0.696009	+	0.718033i	$0.254957\pi$
$158$	0	0
$159$	4.80369	0.380958
$160$	0	0
$161$	6.89670	0.543536
$162$	0	0
$163$	−17.7794	−1.39259	−0.696294	−	0.717756i	$-0.745169\pi$
$163$	−17.7794	−1.39259	−0.696294	+	0.717756i	$0.745169\pi$
$164$	0	0
$165$	3.10197	0.241488
$166$	0	0
$167$	−14.7180	−1.13891	−0.569457	−	0.822021i	$-0.692846\pi$
$167$	−14.7180	−1.13891	−0.569457	+	0.822021i	$0.692846\pi$
$168$	0	0
$169$	−1.37767	−0.105974
$170$	0	0
$171$	4.00619	0.306361
$172$	0	0
$173$	22.4514	1.70695	0.853475	−	0.521134i	$-0.174491\pi$
$173$	22.4514	1.70695	0.853475	+	0.521134i	$0.174491\pi$
$174$	0	0
$175$	1.83837	0.138968
$176$	0	0
$177$	0.696043	0.0523178
$178$	0	0
$179$	−0.427659	−0.0319647	−0.0159824	−	0.999872i	$-0.505088\pi$
$179$	−0.427659	−0.0319647	−0.0159824	+	0.999872i	$0.505088\pi$
$180$	0	0
$181$	−10.2598	−0.762605	−0.381303	−	0.924450i	$-0.624524\pi$
$181$	−10.2598	−0.762605	−0.381303	+	0.924450i	$0.624524\pi$
$182$	0	0
$183$	−2.50296	−0.185024
$184$	0	0
$185$	9.55270	0.702329
$186$	0	0
$187$	−9.70326	−0.709573
$188$	0	0
$189$	−1.83837	−0.133722
$190$	0	0
$191$	12.6157	0.912843	0.456422	−	0.889764i	$-0.349131\pi$
$191$	12.6157	0.912843	0.456422	+	0.889764i	$0.349131\pi$
$192$	0	0
$193$	10.5900	0.762285	0.381142	−	0.924516i	$-0.375531\pi$
$193$	10.5900	0.762285	0.381142	+	0.924516i	$0.375531\pi$
$194$	0	0
$195$	3.40915	0.244135
$196$	0	0
$197$	−12.4337	−0.885866	−0.442933	−	0.896555i	$-0.646062\pi$
$197$	−12.4337	−0.885866	−0.442933	+	0.896555i	$0.646062\pi$
$198$	0	0
$199$	−22.1662	−1.57132	−0.785661	−	0.618657i	$-0.787677\pi$
$199$	−22.1662	−1.57132	−0.785661	+	0.618657i	$0.787677\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	9.12184	0.640227
$204$	0	0
$205$	1.84212	0.128659
$206$	0	0
$207$	3.75152	0.260749
$208$	0	0
$209$	−12.4271	−0.859599
$210$	0	0
$211$	−17.2046	−1.18441	−0.592206	−	0.805787i	$-0.701743\pi$
$211$	−17.2046	−1.18441	−0.592206	+	0.805787i	$0.701743\pi$
$212$	0	0
$213$	1.50115	0.102857
$214$	0	0
$215$	−3.08508	−0.210401
$216$	0	0
$217$	−12.3100	−0.835661
$218$	0	0
$219$	−4.39100	−0.296716
$220$	0	0
$221$	−10.6642	−0.717350
$222$	0	0
$223$	5.30838	0.355476	0.177738	−	0.984078i	$-0.443122\pi$
$223$	5.30838	0.355476	0.177738	+	0.984078i	$0.443122\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	0.998660	0.0662834	0.0331417	−	0.999451i	$-0.489449\pi$
$227$	0.998660	0.0662834	0.0331417	+	0.999451i	$0.489449\pi$
$228$	0	0
$229$	15.2161	1.00551	0.502753	−	0.864430i	$-0.332320\pi$
$229$	15.2161	1.00551	0.502753	+	0.864430i	$0.332320\pi$
$230$	0	0
$231$	5.70258	0.375202
$232$	0	0
$233$	−9.88929	−0.647869	−0.323934	−	0.946080i	$-0.605006\pi$
$233$	−9.88929	−0.647869	−0.323934	+	0.946080i	$0.605006\pi$
$234$	0	0
$235$	−8.19434	−0.534540
$236$	0	0
$237$	−15.9021	−1.03295
$238$	0	0
$239$	3.26053	0.210906	0.105453	−	0.994424i	$-0.466371\pi$
$239$	3.26053	0.210906	0.105453	+	0.994424i	$0.466371\pi$
$240$	0	0
$241$	22.9369	1.47750	0.738748	−	0.673982i	$-0.235418\pi$
$241$	22.9369	1.47750	0.738748	+	0.673982i	$0.235418\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.62038	−0.231298
$246$	0	0
$247$	−13.6577	−0.869020
$248$	0	0
$249$	−9.99211	−0.633224
$250$	0	0
$251$	−5.57786	−0.352072	−0.176036	−	0.984384i	$-0.556327\pi$
$251$	−5.57786	−0.352072	−0.176036	+	0.984384i	$0.556327\pi$
$252$	0	0
$253$	−11.6371	−0.731618
$254$	0	0
$255$	−3.12810	−0.195889
$256$	0	0
$257$	17.7818	1.10920	0.554598	−	0.832118i	$-0.312872\pi$
$257$	17.7818	1.10920	0.554598	+	0.832118i	$0.312872\pi$
$258$	0	0
$259$	17.5614	1.09121
$260$	0	0
$261$	4.96191	0.307134
$262$	0	0
$263$	−28.0928	−1.73228	−0.866139	−	0.499803i	$-0.833406\pi$
$263$	−28.0928	−1.73228	−0.866139	+	0.499803i	$0.833406\pi$
$264$	0	0
$265$	−4.80369	−0.295088
$266$	0	0
$267$	3.33509	0.204104
$268$	0	0
$269$	17.7854	1.08439	0.542196	−	0.840252i	$-0.317593\pi$
$269$	17.7854	1.08439	0.542196	+	0.840252i	$0.317593\pi$
$270$	0	0
$271$	−7.99801	−0.485844	−0.242922	−	0.970046i	$-0.578106\pi$
$271$	−7.99801	−0.485844	−0.242922	+	0.970046i	$0.578106\pi$
$272$	0	0
$273$	6.26730	0.379314
$274$	0	0
$275$	−3.10197	−0.187056
$276$	0	0
$277$	27.5698	1.65651	0.828253	−	0.560354i	$-0.189335\pi$
$277$	27.5698	1.65651	0.828253	+	0.560354i	$0.189335\pi$
$278$	0	0
$279$	−6.69616	−0.400889
$280$	0	0
$281$	29.7744	1.77619	0.888097	−	0.459656i	$-0.152027\pi$
$281$	29.7744	1.77619	0.888097	+	0.459656i	$0.152027\pi$
$282$	0	0
$283$	4.37304	0.259950	0.129975	−	0.991517i	$-0.458510\pi$
$283$	4.37304	0.259950	0.129975	+	0.991517i	$0.458510\pi$
$284$	0	0
$285$	−4.00619	−0.237306
$286$	0	0
$287$	3.38651	0.199899
$288$	0	0
$289$	−7.21500	−0.424412
$290$	0	0
$291$	1.40962	0.0826336
$292$	0	0
$293$	18.4130	1.07570	0.537848	−	0.843042i	$-0.319237\pi$
$293$	18.4130	1.07570	0.537848	+	0.843042i	$0.319237\pi$
$294$	0	0
$295$	−0.696043	−0.0405252
$296$	0	0
$297$	3.10197	0.179994
$298$	0	0
$299$	−12.7895	−0.739637
$300$	0	0
$301$	−5.67153	−0.326902
$302$	0	0
$303$	5.08178	0.291940
$304$	0	0
$305$	2.50296	0.143319
$306$	0	0
$307$	−27.6825	−1.57992	−0.789962	−	0.613156i	$-0.789900\pi$
$307$	−27.6825	−1.57992	−0.789962	+	0.613156i	$0.789900\pi$
$308$	0	0
$309$	−2.01121	−0.114414
$310$	0	0
$311$	34.7004	1.96768	0.983839	−	0.179054i	$-0.0573036\pi$
$311$	34.7004	1.96768	0.983839	+	0.179054i	$0.0573036\pi$
$312$	0	0
$313$	20.8000	1.17568	0.587842	−	0.808976i	$-0.299978\pi$
$313$	20.8000	1.17568	0.587842	+	0.808976i	$0.299978\pi$
$314$	0	0
$315$	1.83837	0.103581
$316$	0	0
$317$	−4.89735	−0.275062	−0.137531	−	0.990497i	$-0.543917\pi$
$317$	−4.89735	−0.275062	−0.137531	+	0.990497i	$0.543917\pi$
$318$	0	0
$319$	−15.3917	−0.861768
$320$	0	0
$321$	−16.8729	−0.941751
$322$	0	0
$323$	12.5318	0.697286
$324$	0	0
$325$	−3.40915	−0.189106
$326$	0	0
$327$	−3.75195	−0.207484
$328$	0	0
$329$	−15.0643	−0.830519
$330$	0	0
$331$	33.5094	1.84184	0.920920	−	0.389751i	$-0.127439\pi$
$331$	33.5094	1.84184	0.920920	+	0.389751i	$0.127439\pi$
$332$	0	0
$333$	9.55270	0.523485
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−8.67437	−0.472523	−0.236262	−	0.971689i	$-0.575922\pi$
$337$	−8.67437	−0.472523	−0.236262	+	0.971689i	$0.575922\pi$
$338$	0	0
$339$	5.67326	0.308129
$340$	0	0
$341$	20.7713	1.12483
$342$	0	0
$343$	−19.5242	−1.05421
$344$	0	0
$345$	−3.75152	−0.201975
$346$	0	0
$347$	−0.896079	−0.0481041	−0.0240520	−	0.999711i	$-0.507657\pi$
$347$	−0.896079	−0.0481041	−0.0240520	+	0.999711i	$0.507657\pi$
$348$	0	0
$349$	4.42612	0.236925	0.118462	−	0.992959i	$-0.462204\pi$
$349$	4.42612	0.236925	0.118462	+	0.992959i	$0.462204\pi$
$350$	0	0
$351$	3.40915	0.181967
$352$	0	0
$353$	−17.7163	−0.942945	−0.471472	−	0.881881i	$-0.656277\pi$
$353$	−17.7163	−0.942945	−0.471472	+	0.881881i	$0.656277\pi$
$354$	0	0
$355$	−1.50115	−0.0796727
$356$	0	0
$357$	−5.75061	−0.304355
$358$	0	0
$359$	−18.4043	−0.971340	−0.485670	−	0.874142i	$-0.661424\pi$
$359$	−18.4043	−0.971340	−0.485670	+	0.874142i	$0.661424\pi$
$360$	0	0
$361$	−2.95043	−0.155286
$362$	0	0
$363$	1.37779	0.0723153
$364$	0	0
$365$	4.39100	0.229835
$366$	0	0
$367$	3.81577	0.199182	0.0995909	−	0.995028i	$-0.468247\pi$
$367$	3.81577	0.199182	0.0995909	+	0.995028i	$0.468247\pi$
$368$	0	0
$369$	1.84212	0.0958970
$370$	0	0
$371$	−8.83098	−0.458482
$372$	0	0
$373$	−5.25423	−0.272054	−0.136027	−	0.990705i	$-0.543433\pi$
$373$	−5.25423	−0.272054	−0.136027	+	0.990705i	$0.543433\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−16.9159	−0.871213
$378$	0	0
$379$	22.2674	1.14380	0.571900	−	0.820323i	$-0.306206\pi$
$379$	22.2674	1.14380	0.571900	+	0.820323i	$0.306206\pi$
$380$	0	0
$381$	11.9942	0.614484
$382$	0	0
$383$	5.62996	0.287677	0.143839	−	0.989601i	$-0.454055\pi$
$383$	5.62996	0.287677	0.143839	+	0.989601i	$0.454055\pi$
$384$	0	0
$385$	−5.70258	−0.290630
$386$	0	0
$387$	−3.08508	−0.156824
$388$	0	0
$389$	−13.7983	−0.699599	−0.349800	−	0.936825i	$-0.613750\pi$
$389$	−13.7983	−0.699599	−0.349800	+	0.936825i	$0.613750\pi$
$390$	0	0
$391$	11.7351	0.593471
$392$	0	0
$393$	−11.0566	−0.557732
$394$	0	0
$395$	15.9021	0.800121
$396$	0	0
$397$	9.42406	0.472980	0.236490	−	0.971634i	$-0.424003\pi$
$397$	9.42406	0.472980	0.236490	+	0.971634i	$0.424003\pi$
$398$	0	0
$399$	−7.36488	−0.368705
$400$	0	0
$401$	−15.7502	−0.786527	−0.393264	−	0.919426i	$-0.628654\pi$
$401$	−15.7502	−0.786527	−0.393264	+	0.919426i	$0.628654\pi$
$402$	0	0
$403$	22.8283	1.13716
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−29.6322	−1.46881
$408$	0	0
$409$	21.7004	1.07301	0.536507	−	0.843896i	$-0.319743\pi$
$409$	21.7004	1.07301	0.536507	+	0.843896i	$0.319743\pi$
$410$	0	0
$411$	−11.2802	−0.556413
$412$	0	0
$413$	−1.27959	−0.0629644
$414$	0	0
$415$	9.99211	0.490493
$416$	0	0
$417$	−21.0811	−1.03235
$418$	0	0
$419$	27.8209	1.35914	0.679569	−	0.733612i	$-0.262167\pi$
$419$	27.8209	1.35914	0.679569	+	0.733612i	$0.262167\pi$
$420$	0	0
$421$	1.39909	0.0681876	0.0340938	−	0.999419i	$-0.489146\pi$
$421$	1.39909	0.0681876	0.0340938	+	0.999419i	$0.489146\pi$
$422$	0	0
$423$	−8.19434	−0.398422
$424$	0	0
$425$	3.12810	0.151735
$426$	0	0
$427$	4.60138	0.222676
$428$	0	0
$429$	−10.5751	−0.510570
$430$	0	0
$431$	40.9595	1.97295	0.986475	−	0.163913i	$-0.0524118\pi$
$431$	40.9595	1.97295	0.986475	+	0.163913i	$0.0524118\pi$
$432$	0	0
$433$	−3.41476	−0.164103	−0.0820514	−	0.996628i	$-0.526147\pi$
$433$	−3.41476	−0.164103	−0.0820514	+	0.996628i	$0.526147\pi$
$434$	0	0
$435$	−4.96191	−0.237905
$436$	0	0
$437$	15.0293	0.718949
$438$	0	0
$439$	−17.5676	−0.838455	−0.419228	−	0.907881i	$-0.637699\pi$
$439$	−17.5676	−0.838455	−0.419228	+	0.907881i	$0.637699\pi$
$440$	0	0
$441$	−3.62038	−0.172399
$442$	0	0
$443$	−6.24566	−0.296740	−0.148370	−	0.988932i	$-0.547403\pi$
$443$	−6.24566	−0.296740	−0.148370	+	0.988932i	$0.547403\pi$
$444$	0	0
$445$	−3.33509	−0.158098
$446$	0	0
$447$	−22.9065	−1.08344
$448$	0	0
$449$	9.79797	0.462395	0.231197	−	0.972907i	$-0.425736\pi$
$449$	9.79797	0.462395	0.231197	+	0.972907i	$0.425736\pi$
$450$	0	0
$451$	−5.71420	−0.269071
$452$	0	0
$453$	19.9491	0.937292
$454$	0	0
$455$	−6.26730	−0.293816
$456$	0	0
$457$	6.35281	0.297172	0.148586	−	0.988899i	$-0.452528\pi$
$457$	6.35281	0.297172	0.148586	+	0.988899i	$0.452528\pi$
$458$	0	0
$459$	−3.12810	−0.146007
$460$	0	0
$461$	29.8215	1.38893	0.694463	−	0.719528i	$-0.255642\pi$
$461$	29.8215	1.38893	0.694463	+	0.719528i	$0.255642\pi$
$462$	0	0
$463$	4.94399	0.229767	0.114883	−	0.993379i	$-0.463351\pi$
$463$	4.94399	0.229767	0.114883	+	0.993379i	$0.463351\pi$
$464$	0	0
$465$	6.69616	0.310527
$466$	0	0
$467$	−24.6621	−1.14123	−0.570613	−	0.821219i	$-0.693294\pi$
$467$	−24.6621	−1.14123	−0.570613	+	0.821219i	$0.693294\pi$
$468$	0	0
$469$	−1.83837	−0.0848882
$470$	0	0
$471$	−17.4419	−0.803682
$472$	0	0
$473$	9.56983	0.440021
$474$	0	0
$475$	4.00619	0.183817
$476$	0	0
$477$	−4.80369	−0.219946
$478$	0	0
$479$	36.3794	1.66222	0.831108	−	0.556110i	$-0.187707\pi$
$479$	36.3794	1.66222	0.831108	+	0.556110i	$0.187707\pi$
$480$	0	0
$481$	−32.5666	−1.48491
$482$	0	0
$483$	−6.89670	−0.313811
$484$	0	0
$485$	−1.40962	−0.0640077
$486$	0	0
$487$	−40.0017	−1.81265	−0.906325	−	0.422582i	$-0.861124\pi$
$487$	−40.0017	−1.81265	−0.906325	+	0.422582i	$0.861124\pi$
$488$	0	0
$489$	17.7794	0.804011
$490$	0	0
$491$	16.1681	0.729655	0.364827	−	0.931075i	$-0.381128\pi$
$491$	16.1681	0.729655	0.364827	+	0.931075i	$0.381128\pi$
$492$	0	0
$493$	15.5213	0.699046
$494$	0	0
$495$	−3.10197	−0.139423
$496$	0	0
$497$	−2.75967	−0.123788
$498$	0	0
$499$	2.03642	0.0911628	0.0455814	−	0.998961i	$-0.485486\pi$
$499$	2.03642	0.0911628	0.0455814	+	0.998961i	$0.485486\pi$
$500$	0	0
$501$	14.7180	0.657552
$502$	0	0
$503$	27.2166	1.21353	0.606764	−	0.794882i	$-0.292467\pi$
$503$	27.2166	1.21353	0.606764	+	0.794882i	$0.292467\pi$
$504$	0	0
$505$	−5.08178	−0.226136
$506$	0	0
$507$	1.37767	0.0611843
$508$	0	0
$509$	10.1865	0.451509	0.225755	−	0.974184i	$-0.427515\pi$
$509$	10.1865	0.451509	0.225755	+	0.974184i	$0.427515\pi$
$510$	0	0
$511$	8.07230	0.357097
$512$	0	0
$513$	−4.00619	−0.176878
$514$	0	0
$515$	2.01121	0.0886247
$516$	0	0
$517$	25.4186	1.11791
$518$	0	0
$519$	−22.4514	−0.985508
$520$	0	0
$521$	−35.7780	−1.56746	−0.783731	−	0.621100i	$-0.786686\pi$
$521$	−35.7780	−1.56746	−0.783731	+	0.621100i	$0.786686\pi$
$522$	0	0
$523$	9.54830	0.417518	0.208759	−	0.977967i	$-0.433058\pi$
$523$	9.54830	0.417518	0.208759	+	0.977967i	$0.433058\pi$
$524$	0	0
$525$	−1.83837	−0.0802332
$526$	0	0
$527$	−20.9463	−0.912433
$528$	0	0
$529$	−8.92609	−0.388091
$530$	0	0
$531$	−0.696043	−0.0302057
$532$	0	0
$533$	−6.28008	−0.272020
$534$	0	0
$535$	16.8729	0.729477
$536$	0	0
$537$	0.427659	0.0184549
$538$	0	0
$539$	11.2303	0.483724
$540$	0	0
$541$	30.0687	1.29275	0.646377	−	0.763018i	$-0.276283\pi$
$541$	30.0687	1.29275	0.646377	+	0.763018i	$0.276283\pi$
$542$	0	0
$543$	10.2598	0.440290
$544$	0	0
$545$	3.75195	0.160716
$546$	0	0
$547$	17.8219	0.762010	0.381005	−	0.924573i	$-0.375578\pi$
$547$	17.8219	0.762010	0.381005	+	0.924573i	$0.375578\pi$
$548$	0	0
$549$	2.50296	0.106824
$550$	0	0
$551$	19.8783	0.846846
$552$	0	0
$553$	29.2340	1.24315
$554$	0	0
$555$	−9.55270	−0.405490
$556$	0	0
$557$	42.0269	1.78074	0.890368	−	0.455241i	$-0.150447\pi$
$557$	42.0269	1.78074	0.890368	+	0.455241i	$0.150447\pi$
$558$	0	0
$559$	10.5175	0.444844
$560$	0	0
$561$	9.70326	0.409672
$562$	0	0
$563$	−25.1924	−1.06173	−0.530866	−	0.847456i	$-0.678133\pi$
$563$	−25.1924	−1.06173	−0.530866	+	0.847456i	$0.678133\pi$
$564$	0	0
$565$	−5.67326	−0.238676
$566$	0	0
$567$	1.83837	0.0772044
$568$	0	0
$569$	30.4138	1.27501	0.637507	−	0.770444i	$-0.279966\pi$
$569$	30.4138	1.27501	0.637507	+	0.770444i	$0.279966\pi$
$570$	0	0
$571$	23.4357	0.980752	0.490376	−	0.871511i	$-0.336859\pi$
$571$	23.4357	0.980752	0.490376	+	0.871511i	$0.336859\pi$
$572$	0	0
$573$	−12.6157	−0.527030
$574$	0	0
$575$	3.75152	0.156449
$576$	0	0
$577$	−9.77422	−0.406906	−0.203453	−	0.979085i	$-0.565216\pi$
$577$	−9.77422	−0.406906	−0.203453	+	0.979085i	$0.565216\pi$
$578$	0	0
$579$	−10.5900	−0.440105
$580$	0	0
$581$	18.3692	0.762084
$582$	0	0
$583$	14.9009	0.617132
$584$	0	0
$585$	−3.40915	−0.140951
$586$	0	0
$587$	−2.75752	−0.113815	−0.0569076	−	0.998379i	$-0.518124\pi$
$587$	−2.75752	−0.113815	−0.0569076	+	0.998379i	$0.518124\pi$
$588$	0	0
$589$	−26.8261	−1.10535
$590$	0	0
$591$	12.4337	0.511455
$592$	0	0
$593$	11.3121	0.464532	0.232266	−	0.972652i	$-0.425386\pi$
$593$	11.3121	0.464532	0.232266	+	0.972652i	$0.425386\pi$
$594$	0	0
$595$	5.75061	0.235752
$596$	0	0
$597$	22.1662	0.907204
$598$	0	0
$599$	9.54217	0.389883	0.194941	−	0.980815i	$-0.437548\pi$
$599$	9.54217	0.389883	0.194941	+	0.980815i	$0.437548\pi$
$600$	0	0
$601$	39.2205	1.59984	0.799918	−	0.600109i	$-0.204876\pi$
$601$	39.2205	1.59984	0.799918	+	0.600109i	$0.204876\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−1.37779	−0.0560152
$606$	0	0
$607$	34.6772	1.40751	0.703753	−	0.710445i	$-0.251506\pi$
$607$	34.6772	1.40751	0.703753	+	0.710445i	$0.251506\pi$
$608$	0	0
$609$	−9.12184	−0.369635
$610$	0	0
$611$	27.9358	1.13016
$612$	0	0
$613$	27.2981	1.10256	0.551280	−	0.834320i	$-0.314139\pi$
$613$	27.2981	1.10256	0.551280	+	0.834320i	$0.314139\pi$
$614$	0	0
$615$	−1.84212	−0.0742815
$616$	0	0
$617$	28.5352	1.14878	0.574392	−	0.818580i	$-0.305239\pi$
$617$	28.5352	1.14878	0.574392	+	0.818580i	$0.305239\pi$
$618$	0	0
$619$	−40.0717	−1.61062	−0.805309	−	0.592855i	$-0.798001\pi$
$619$	−40.0717	−1.61062	−0.805309	+	0.592855i	$0.798001\pi$
$620$	0	0
$621$	−3.75152	−0.150543
$622$	0	0
$623$	−6.13114	−0.245639
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	12.4271	0.496290
$628$	0	0
$629$	29.8818	1.19147
$630$	0	0
$631$	−12.2211	−0.486513	−0.243257	−	0.969962i	$-0.578216\pi$
$631$	−12.2211	−0.486513	−0.243257	+	0.969962i	$0.578216\pi$
$632$	0	0
$633$	17.2046	0.683821
$634$	0	0
$635$	−11.9942	−0.475977
$636$	0	0
$637$	12.3424	0.489025
$638$	0	0
$639$	−1.50115	−0.0593846
$640$	0	0
$641$	14.0583	0.555270	0.277635	−	0.960687i	$-0.410449\pi$
$641$	14.0583	0.555270	0.277635	+	0.960687i	$0.410449\pi$
$642$	0	0
$643$	42.5563	1.67826	0.839128	−	0.543934i	$-0.183066\pi$
$643$	42.5563	1.67826	0.839128	+	0.543934i	$0.183066\pi$
$644$	0	0
$645$	3.08508	0.121475
$646$	0	0
$647$	5.09616	0.200351	0.100175	−	0.994970i	$-0.468060\pi$
$647$	5.09616	0.200351	0.100175	+	0.994970i	$0.468060\pi$
$648$	0	0
$649$	2.15910	0.0847522
$650$	0	0
$651$	12.3100	0.482469
$652$	0	0
$653$	28.1394	1.10118	0.550591	−	0.834775i	$-0.314403\pi$
$653$	28.1394	1.10118	0.550591	+	0.834775i	$0.314403\pi$
$654$	0	0
$655$	11.0566	0.432017
$656$	0	0
$657$	4.39100	0.171309
$658$	0	0
$659$	−30.2651	−1.17896	−0.589481	−	0.807782i	$-0.700668\pi$
$659$	−30.2651	−1.17896	−0.589481	+	0.807782i	$0.700668\pi$
$660$	0	0
$661$	−45.8313	−1.78263	−0.891315	−	0.453385i	$-0.850216\pi$
$661$	−45.8313	−1.78263	−0.891315	+	0.453385i	$0.850216\pi$
$662$	0	0
$663$	10.6642	0.414162
$664$	0	0
$665$	7.36488	0.285598
$666$	0	0
$667$	18.6147	0.720764
$668$	0	0
$669$	−5.30838	−0.205234
$670$	0	0
$671$	−7.76411	−0.299730
$672$	0	0
$673$	−32.6004	−1.25665	−0.628326	−	0.777950i	$-0.716260\pi$
$673$	−32.6004	−1.25665	−0.628326	+	0.777950i	$0.716260\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	31.6513	1.21646	0.608229	−	0.793762i	$-0.291880\pi$
$677$	31.6513	1.21646	0.608229	+	0.793762i	$0.291880\pi$
$678$	0	0
$679$	−2.59141	−0.0994493
$680$	0	0
$681$	−0.998660	−0.0382687
$682$	0	0
$683$	−45.4176	−1.73786	−0.868929	−	0.494937i	$-0.835191\pi$
$683$	−45.4176	−1.73786	−0.868929	+	0.494937i	$0.835191\pi$
$684$	0	0
$685$	11.2802	0.430996
$686$	0	0
$687$	−15.2161	−0.580529
$688$	0	0
$689$	16.3765	0.623896
$690$	0	0
$691$	6.69109	0.254541	0.127271	−	0.991868i	$-0.459378\pi$
$691$	6.69109	0.254541	0.127271	+	0.991868i	$0.459378\pi$
$692$	0	0
$693$	−5.70258	−0.216623
$694$	0	0
$695$	21.0811	0.799651
$696$	0	0
$697$	5.76234	0.218264
$698$	0	0
$699$	9.88929	0.374047
$700$	0	0
$701$	8.78022	0.331624	0.165812	−	0.986157i	$-0.446975\pi$
$701$	8.78022	0.331624	0.165812	+	0.986157i	$0.446975\pi$
$702$	0	0
$703$	38.2700	1.44338
$704$	0	0
$705$	8.19434	0.308617
$706$	0	0
$707$	−9.34220	−0.351350
$708$	0	0
$709$	−22.1583	−0.832174	−0.416087	−	0.909325i	$-0.636599\pi$
$709$	−22.1583	−0.832174	−0.416087	+	0.909325i	$0.636599\pi$
$710$	0	0
$711$	15.9021	0.596375
$712$	0	0
$713$	−25.1208	−0.940781
$714$	0	0
$715$	10.5751	0.395486
$716$	0	0
$717$	−3.26053	−0.121767
$718$	0	0
$719$	−12.9998	−0.484812	−0.242406	−	0.970175i	$-0.577937\pi$
$719$	−12.9998	−0.484812	−0.242406	+	0.970175i	$0.577937\pi$
$720$	0	0
$721$	3.69736	0.137697
$722$	0	0
$723$	−22.9369	−0.853032
$724$	0	0
$725$	4.96191	0.184281
$726$	0	0
$727$	−48.9214	−1.81439	−0.907197	−	0.420706i	$-0.861782\pi$
$727$	−48.9214	−1.81439	−0.907197	+	0.420706i	$0.861782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.65044	−0.356935
$732$	0	0
$733$	−25.1206	−0.927849	−0.463925	−	0.885875i	$-0.653559\pi$
$733$	−25.1206	−0.927849	−0.463925	+	0.885875i	$0.653559\pi$
$734$	0	0
$735$	3.62038	0.133540
$736$	0	0
$737$	3.10197	0.114262
$738$	0	0
$739$	−0.553507	−0.0203611	−0.0101805	−	0.999948i	$-0.503241\pi$
$739$	−0.553507	−0.0203611	−0.0101805	+	0.999948i	$0.503241\pi$
$740$	0	0
$741$	13.6577	0.501729
$742$	0	0
$743$	1.81447	0.0665664	0.0332832	−	0.999446i	$-0.489404\pi$
$743$	1.81447	0.0665664	0.0332832	+	0.999446i	$0.489404\pi$
$744$	0	0
$745$	22.9065	0.839228
$746$	0	0
$747$	9.99211	0.365592
$748$	0	0
$749$	31.0186	1.13340
$750$	0	0
$751$	10.5949	0.386613	0.193306	−	0.981138i	$-0.438079\pi$
$751$	10.5949	0.386613	0.193306	+	0.981138i	$0.438079\pi$
$752$	0	0
$753$	5.57786	0.203269
$754$	0	0
$755$	−19.9491	−0.726024
$756$	0	0
$757$	−38.3933	−1.39543	−0.697714	−	0.716377i	$-0.745799\pi$
$757$	−38.3933	−1.39543	−0.697714	+	0.716377i	$0.745799\pi$
$758$	0	0
$759$	11.6371	0.422400
$760$	0	0
$761$	33.0472	1.19796	0.598981	−	0.800764i	$-0.295573\pi$
$761$	33.0472	1.19796	0.598981	+	0.800764i	$0.295573\pi$
$762$	0	0
$763$	6.89749	0.249706
$764$	0	0
$765$	3.12810	0.113097
$766$	0	0
$767$	2.37292	0.0856811
$768$	0	0
$769$	43.9743	1.58575	0.792877	−	0.609382i	$-0.208582\pi$
$769$	43.9743	1.58575	0.792877	+	0.609382i	$0.208582\pi$
$770$	0	0
$771$	−17.7818	−0.640395
$772$	0	0
$773$	4.62853	0.166477	0.0832384	−	0.996530i	$-0.473474\pi$
$773$	4.62853	0.166477	0.0832384	+	0.996530i	$0.473474\pi$
$774$	0	0
$775$	−6.69616	−0.240533
$776$	0	0
$777$	−17.5614	−0.630013
$778$	0	0
$779$	7.37989	0.264412
$780$	0	0
$781$	4.65652	0.166623
$782$	0	0
$783$	−4.96191	−0.177324
$784$	0	0
$785$	17.4419	0.622529
$786$	0	0
$787$	23.5337	0.838886	0.419443	−	0.907782i	$-0.362225\pi$
$787$	23.5337	0.838886	0.419443	+	0.907782i	$0.362225\pi$
$788$	0	0
$789$	28.0928	1.00013
$790$	0	0
$791$	−10.4296	−0.370833
$792$	0	0
$793$	−8.53298	−0.303015
$794$	0	0
$795$	4.80369	0.170369
$796$	0	0
$797$	14.4551	0.512025	0.256012	−	0.966673i	$-0.417591\pi$
$797$	14.4551	0.512025	0.256012	+	0.966673i	$0.417591\pi$
$798$	0	0
$799$	−25.6327	−0.906819
$800$	0	0
$801$	−3.33509	−0.117840
$802$	0	0
$803$	−13.6207	−0.480666
$804$	0	0
$805$	6.89670	0.243077
$806$	0	0
$807$	−17.7854	−0.626074
$808$	0	0
$809$	35.2868	1.24062	0.620309	−	0.784358i	$-0.287007\pi$
$809$	35.2868	1.24062	0.620309	+	0.784358i	$0.287007\pi$
$810$	0	0
$811$	−35.8015	−1.25716	−0.628580	−	0.777745i	$-0.716363\pi$
$811$	−35.8015	−1.25716	−0.628580	+	0.777745i	$0.716363\pi$
$812$	0	0
$813$	7.99801	0.280502
$814$	0	0
$815$	−17.7794	−0.622785
$816$	0	0
$817$	−12.3594	−0.432402
$818$	0	0
$819$	−6.26730	−0.218997
$820$	0	0
$821$	3.76511	0.131403	0.0657015	−	0.997839i	$-0.479071\pi$
$821$	3.76511	0.131403	0.0657015	+	0.997839i	$0.479071\pi$
$822$	0	0
$823$	−6.64876	−0.231761	−0.115881	−	0.993263i	$-0.536969\pi$
$823$	−6.64876	−0.231761	−0.115881	+	0.993263i	$0.536969\pi$
$824$	0	0
$825$	3.10197	0.107997
$826$	0	0
$827$	−8.33884	−0.289970	−0.144985	−	0.989434i	$-0.546313\pi$
$827$	−8.33884	−0.289970	−0.144985	+	0.989434i	$0.546313\pi$
$828$	0	0
$829$	−39.9859	−1.38877	−0.694384	−	0.719605i	$-0.744323\pi$
$829$	−39.9859	−1.38877	−0.694384	+	0.719605i	$0.744323\pi$
$830$	0	0
$831$	−27.5698	−0.956385
$832$	0	0
$833$	−11.3249	−0.392385
$834$	0	0
$835$	−14.7180	−0.509337
$836$	0	0
$837$	6.69616	0.231453
$838$	0	0
$839$	25.8319	0.891815	0.445907	−	0.895079i	$-0.352881\pi$
$839$	25.8319	0.891815	0.445907	+	0.895079i	$0.352881\pi$
$840$	0	0
$841$	−4.37949	−0.151017
$842$	0	0
$843$	−29.7744	−1.02549
$844$	0	0
$845$	−1.37767	−0.0473932
$846$	0	0
$847$	−2.53289	−0.0870313
$848$	0	0
$849$	−4.37304	−0.150082
$850$	0	0
$851$	35.8372	1.22848
$852$	0	0
$853$	−46.4114	−1.58910	−0.794549	−	0.607200i	$-0.792293\pi$
$853$	−46.4114	−1.58910	−0.794549	+	0.607200i	$0.792293\pi$
$854$	0	0
$855$	4.00619	0.137009
$856$	0	0
$857$	−53.5746	−1.83007	−0.915037	−	0.403370i	$-0.867839\pi$
$857$	−53.5746	−1.83007	−0.915037	+	0.403370i	$0.867839\pi$
$858$	0	0
$859$	7.94340	0.271025	0.135513	−	0.990776i	$-0.456732\pi$
$859$	7.94340	0.271025	0.135513	+	0.990776i	$0.456732\pi$
$860$	0	0
$861$	−3.38651	−0.115412
$862$	0	0
$863$	−14.1176	−0.480570	−0.240285	−	0.970702i	$-0.577241\pi$
$863$	−14.1176	−0.480570	−0.240285	+	0.970702i	$0.577241\pi$
$864$	0	0
$865$	22.4514	0.763371
$866$	0	0
$867$	7.21500	0.245034
$868$	0	0
$869$	−49.3278	−1.67333
$870$	0	0
$871$	3.40915	0.115515
$872$	0	0
$873$	−1.40962	−0.0477085
$874$	0	0
$875$	1.83837	0.0621484
$876$	0	0
$877$	−7.40335	−0.249993	−0.124997	−	0.992157i	$-0.539892\pi$
$877$	−7.40335	−0.249993	−0.124997	+	0.992157i	$0.539892\pi$
$878$	0	0
$879$	−18.4130	−0.621054
$880$	0	0
$881$	4.05059	0.136468	0.0682340	−	0.997669i	$-0.478264\pi$
$881$	4.05059	0.136468	0.0682340	+	0.997669i	$0.478264\pi$
$882$	0	0
$883$	−29.3852	−0.988890	−0.494445	−	0.869209i	$-0.664629\pi$
$883$	−29.3852	−0.988890	−0.494445	+	0.869209i	$0.664629\pi$
$884$	0	0
$885$	0.696043	0.0233972
$886$	0	0
$887$	3.19213	0.107181	0.0535906	−	0.998563i	$-0.482933\pi$
$887$	3.19213	0.107181	0.0535906	+	0.998563i	$0.482933\pi$
$888$	0	0
$889$	−22.0499	−0.739530
$890$	0	0
$891$	−3.10197	−0.103920
$892$	0	0
$893$	−32.8281	−1.09855
$894$	0	0
$895$	−0.427659	−0.0142951
$896$	0	0
$897$	12.7895	0.427029
$898$	0	0
$899$	−33.2257	−1.10814
$900$	0	0
$901$	−15.0264	−0.500603
$902$	0	0
$903$	5.67153	0.188737
$904$	0	0
$905$	−10.2598	−0.341048
$906$	0	0
$907$	−32.7390	−1.08708	−0.543540	−	0.839383i	$-0.682916\pi$
$907$	−32.7390	−1.08708	−0.543540	+	0.839383i	$0.682916\pi$
$908$	0	0
$909$	−5.08178	−0.168552
$910$	0	0
$911$	21.2768	0.704931	0.352465	−	0.935825i	$-0.385343\pi$
$911$	21.2768	0.704931	0.352465	+	0.935825i	$0.385343\pi$
$912$	0	0
$913$	−30.9952	−1.02579
$914$	0	0
$915$	−2.50296	−0.0827454
$916$	0	0
$917$	20.3262	0.671229
$918$	0	0
$919$	17.8484	0.588763	0.294382	−	0.955688i	$-0.404886\pi$
$919$	17.8484	0.588763	0.294382	+	0.955688i	$0.404886\pi$
$920$	0	0
$921$	27.6825	0.912169
$922$	0	0
$923$	5.11765	0.168450
$924$	0	0
$925$	9.55270	0.314091
$926$	0	0
$927$	2.01121	0.0660569
$928$	0	0
$929$	−39.6908	−1.30221	−0.651107	−	0.758986i	$-0.725695\pi$
$929$	−39.6908	−1.30221	−0.651107	+	0.758986i	$0.725695\pi$
$930$	0	0
$931$	−14.5039	−0.475348
$932$	0	0
$933$	−34.7004	−1.13604
$934$	0	0
$935$	−9.70326	−0.317331
$936$	0	0
$937$	−21.7600	−0.710867	−0.355433	−	0.934702i	$-0.615667\pi$
$937$	−21.7600	−0.710867	−0.355433	+	0.934702i	$0.615667\pi$
$938$	0	0
$939$	−20.8000	−0.678781
$940$	0	0
$941$	52.4892	1.71110	0.855550	−	0.517720i	$-0.173219\pi$
$941$	52.4892	1.71110	0.855550	+	0.517720i	$0.173219\pi$
$942$	0	0
$943$	6.91076	0.225045
$944$	0	0
$945$	−1.83837	−0.0598023
$946$	0	0
$947$	−24.1097	−0.783459	−0.391729	−	0.920080i	$-0.628123\pi$
$947$	−24.1097	−0.783459	−0.391729	+	0.920080i	$0.628123\pi$
$948$	0	0
$949$	−14.9696	−0.485934
$950$	0	0
$951$	4.89735	0.158807
$952$	0	0
$953$	−6.93750	−0.224728	−0.112364	−	0.993667i	$-0.535842\pi$
$953$	−6.93750	−0.224728	−0.112364	+	0.993667i	$0.535842\pi$
$954$	0	0
$955$	12.6157	0.408236
$956$	0	0
$957$	15.3917	0.497542
$958$	0	0
$959$	20.7373	0.669642
$960$	0	0
$961$	13.8386	0.446406
$962$	0	0
$963$	16.8729	0.543720
$964$	0	0
$965$	10.5900	0.340904
$966$	0	0
$967$	−25.1364	−0.808333	−0.404166	−	0.914685i	$-0.632438\pi$
$967$	−25.1364	−0.808333	−0.404166	+	0.914685i	$0.632438\pi$
$968$	0	0
$969$	−12.5318	−0.402578
$970$	0	0
$971$	32.1316	1.03115	0.515576	−	0.856844i	$-0.327578\pi$
$971$	32.1316	1.03115	0.515576	+	0.856844i	$0.327578\pi$
$972$	0	0
$973$	38.7549	1.24243
$974$	0	0
$975$	3.40915	0.109180
$976$	0	0
$977$	12.0083	0.384178	0.192089	−	0.981377i	$-0.438474\pi$
$977$	12.0083	0.384178	0.192089	+	0.981377i	$0.438474\pi$
$978$	0	0
$979$	10.3453	0.330639
$980$	0	0
$981$	3.75195	0.119791
$982$	0	0
$983$	19.7610	0.630278	0.315139	−	0.949046i	$-0.397949\pi$
$983$	19.7610	0.630278	0.315139	+	0.949046i	$0.397949\pi$
$984$	0	0
$985$	−12.4337	−0.396171
$986$	0	0
$987$	15.0643	0.479500
$988$	0	0
$989$	−11.5738	−0.368024
$990$	0	0
$991$	44.2763	1.40648	0.703242	−	0.710951i	$-0.251735\pi$
$991$	44.2763	1.40648	0.703242	+	0.710951i	$0.251735\pi$
$992$	0	0
$993$	−33.5094	−1.06339
$994$	0	0
$995$	−22.1662	−0.702717
$996$	0	0
$997$	25.2618	0.800050	0.400025	−	0.916504i	$-0.369001\pi$
$997$	25.2618	0.800050	0.400025	+	0.916504i	$0.369001\pi$
$998$	0	0
$999$	−9.55270	−0.302234

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8040.2.a.bc.1.6	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8040.2.a.bc.1.6	✓	10	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:	\( N \)	\(=\)	\( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +1.83837 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +1.83837 q^{7} +1.00000 q^{9} -3.10197 q^{11} -3.40915 q^{13} -1.00000 q^{15} +3.12810 q^{17} +4.00619 q^{19} -1.83837 q^{21} +3.75152 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.96191 q^{29} -6.69616 q^{31} +3.10197 q^{33} +1.83837 q^{35} +9.55270 q^{37} +3.40915 q^{39} +1.84212 q^{41} -3.08508 q^{43} +1.00000 q^{45} -8.19434 q^{47} -3.62038 q^{49} -3.12810 q^{51} -4.80369 q^{53} -3.10197 q^{55} -4.00619 q^{57} -0.696043 q^{59} +2.50296 q^{61} +1.83837 q^{63} -3.40915 q^{65} -1.00000 q^{67} -3.75152 q^{69} -1.50115 q^{71} +4.39100 q^{73} -1.00000 q^{75} -5.70258 q^{77} +15.9021 q^{79} +1.00000 q^{81} +9.99211 q^{83} +3.12810 q^{85} -4.96191 q^{87} -3.33509 q^{89} -6.26730 q^{91} +6.69616 q^{93} +4.00619 q^{95} -1.40962 q^{97} -3.10197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9	\( 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9} + 7 q^{11} - q^{13} - 10 q^{15} + 3 q^{17} + 4 q^{19} - q^{21} - 13 q^{23} + 10 q^{25} - 10 q^{27} + 18 q^{29} - 9 q^{31} - 7 q^{33} + q^{35} - 3 q^{37} + q^{39} + 19 q^{41} + 5 q^{43} + 10 q^{45} - 8 q^{47} + 43 q^{49} - 3 q^{51} + 17 q^{53} + 7 q^{55} - 4 q^{57} + 24 q^{59} + 21 q^{61} + q^{63} - q^{65} - 10 q^{67} + 13 q^{69} + 2 q^{71} + 25 q^{73} - 10 q^{75} + 15 q^{77} - q^{79} + 10 q^{81} - 6 q^{83} + 3 q^{85} - 18 q^{87} + 23 q^{89} + 29 q^{91} + 9 q^{93} + 4 q^{95} + 21 q^{97} + 7 q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 + 10 * q^5 + q^7 + 10 * q^9 + 7 * q^11 - q^13 - 10 * q^15 + 3 * q^17 + 4 * q^19 - q^21 - 13 * q^23 + 10 * q^25 - 10 * q^27 + 18 * q^29 - 9 * q^31 - 7 * q^33 + q^35 - 3 * q^37 + q^39 + 19 * q^41 + 5 * q^43 + 10 * q^45 - 8 * q^47 + 43 * q^49 - 3 * q^51 + 17 * q^53 + 7 * q^55 - 4 * q^57 + 24 * q^59 + 21 * q^61 + q^63 - q^65 - 10 * q^67 + 13 * q^69 + 2 * q^71 + 25 * q^73 - 10 * q^75 + 15 * q^77 - q^79 + 10 * q^81 - 6 * q^83 + 3 * q^85 - 18 * q^87 + 23 * q^89 + 29 * q^91 + 9 * q^93 + 4 * q^95 + 21 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more