LMFDB - Embedded newform 8040.2.a.bc.1.5

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			1.5
Root			$-1.02133$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -1.55723 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -1.55723 q^{7} +1.00000 q^{9} -2.34461 q^{11} +1.77215 q^{13} -1.00000 q^{15} +5.20667 q^{17} -6.59714 q^{19} +1.55723 q^{21} +0.997661 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.57466 q^{29} +1.24774 q^{31} +2.34461 q^{33} -1.55723 q^{35} +0.683077 q^{37} -1.77215 q^{39} +5.02743 q^{41} +8.52606 q^{43} +1.00000 q^{45} -10.2821 q^{47} -4.57503 q^{49} -5.20667 q^{51} -6.10371 q^{53} -2.34461 q^{55} +6.59714 q^{57} +7.58122 q^{59} +9.98187 q^{61} -1.55723 q^{63} +1.77215 q^{65} -1.00000 q^{67} -0.997661 q^{69} -5.55969 q^{71} +5.11240 q^{73} -1.00000 q^{75} +3.65109 q^{77} -2.63518 q^{79} +1.00000 q^{81} -6.26804 q^{83} +5.20667 q^{85} +2.57466 q^{87} -6.48443 q^{89} -2.75964 q^{91} -1.24774 q^{93} -6.59714 q^{95} +0.717020 q^{97} -2.34461 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−1.55723	−0.588578	−0.294289	−	0.955716i	$-0.595083\pi$
$7$	−1.55723	−0.588578	−0.294289	+	0.955716i	$0.595083\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.34461	−0.706925	−0.353463	−	0.935449i	$-0.614996\pi$
$11$	−2.34461	−0.706925	−0.353463	+	0.935449i	$0.614996\pi$
$12$	0	0
$13$	1.77215	0.491506	0.245753	−	0.969333i	$-0.420965\pi$
$13$	1.77215	0.491506	0.245753	+	0.969333i	$0.420965\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	5.20667	1.26280	0.631402	−	0.775456i	$-0.282480\pi$
$17$	5.20667	1.26280	0.631402	+	0.775456i	$0.282480\pi$
$18$	0	0
$19$	−6.59714	−1.51349	−0.756744	−	0.653711i	$-0.773211\pi$
$19$	−6.59714	−1.51349	−0.756744	+	0.653711i	$0.773211\pi$
$20$	0	0
$21$	1.55723	0.339816
$22$	0	0
$23$	0.997661	0.208027	0.104013	−	0.994576i	$-0.466832\pi$
$23$	0.997661	0.208027	0.104013	+	0.994576i	$0.466832\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.57466	−0.478103	−0.239052	−	0.971007i	$-0.576836\pi$
$29$	−2.57466	−0.478103	−0.239052	+	0.971007i	$0.576836\pi$
$30$	0	0
$31$	1.24774	0.224100	0.112050	−	0.993703i	$-0.464258\pi$
$31$	1.24774	0.224100	0.112050	+	0.993703i	$0.464258\pi$
$32$	0	0
$33$	2.34461	0.408144
$34$	0	0
$35$	−1.55723	−0.263220
$36$	0	0
$37$	0.683077	0.112297	0.0561485	−	0.998422i	$-0.482118\pi$
$37$	0.683077	0.112297	0.0561485	+	0.998422i	$0.482118\pi$
$38$	0	0
$39$	−1.77215	−0.283771
$40$	0	0
$41$	5.02743	0.785152	0.392576	−	0.919720i	$-0.371584\pi$
$41$	5.02743	0.785152	0.392576	+	0.919720i	$0.371584\pi$
$42$	0	0
$43$	8.52606	1.30021	0.650106	−	0.759844i	$-0.274725\pi$
$43$	8.52606	1.30021	0.650106	+	0.759844i	$0.274725\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−10.2821	−1.49980	−0.749902	−	0.661548i	$-0.769900\pi$
$47$	−10.2821	−1.49980	−0.749902	+	0.661548i	$0.769900\pi$
$48$	0	0
$49$	−4.57503	−0.653576
$50$	0	0
$51$	−5.20667	−0.729080
$52$	0	0
$53$	−6.10371	−0.838409	−0.419204	−	0.907892i	$-0.637691\pi$
$53$	−6.10371	−0.838409	−0.419204	+	0.907892i	$0.637691\pi$
$54$	0	0
$55$	−2.34461	−0.316147
$56$	0	0
$57$	6.59714	0.873813
$58$	0	0
$59$	7.58122	0.986991	0.493495	−	0.869748i	$-0.335719\pi$
$59$	7.58122	0.986991	0.493495	+	0.869748i	$0.335719\pi$
$60$	0	0
$61$	9.98187	1.27805	0.639024	−	0.769187i	$-0.279339\pi$
$61$	9.98187	1.27805	0.639024	+	0.769187i	$0.279339\pi$
$62$	0	0
$63$	−1.55723	−0.196193
$64$	0	0
$65$	1.77215	0.219808
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	−0.997661	−0.120104
$70$	0	0
$71$	−5.55969	−0.659814	−0.329907	−	0.944013i	$-0.607017\pi$
$71$	−5.55969	−0.659814	−0.329907	+	0.944013i	$0.607017\pi$
$72$	0	0
$73$	5.11240	0.598361	0.299180	−	0.954197i	$-0.403287\pi$
$73$	5.11240	0.598361	0.299180	+	0.954197i	$0.403287\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	3.65109	0.416081
$78$	0	0
$79$	−2.63518	−0.296481	−0.148241	−	0.988951i	$-0.547361\pi$
$79$	−2.63518	−0.296481	−0.148241	+	0.988951i	$0.547361\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.26804	−0.688007	−0.344004	−	0.938968i	$-0.611783\pi$
$83$	−6.26804	−0.688007	−0.344004	+	0.938968i	$0.611783\pi$
$84$	0	0
$85$	5.20667	0.564743
$86$	0	0
$87$	2.57466	0.276033
$88$	0	0
$89$	−6.48443	−0.687348	−0.343674	−	0.939089i	$-0.611672\pi$
$89$	−6.48443	−0.687348	−0.343674	+	0.939089i	$0.611672\pi$
$90$	0	0
$91$	−2.75964	−0.289289
$92$	0	0
$93$	−1.24774	−0.129384
$94$	0	0
$95$	−6.59714	−0.676853
$96$	0	0
$97$	0.717020	0.0728024	0.0364012	−	0.999337i	$-0.488411\pi$
$97$	0.717020	0.0728024	0.0364012	+	0.999337i	$0.488411\pi$
$98$	0	0
$99$	−2.34461	−0.235642
$100$	0	0
$101$	6.86852	0.683443	0.341722	−	0.939801i	$-0.388990\pi$
$101$	6.86852	0.683443	0.341722	+	0.939801i	$0.388990\pi$
$102$	0	0
$103$	−8.82087	−0.869146	−0.434573	−	0.900637i	$-0.643101\pi$
$103$	−8.82087	−0.869146	−0.434573	+	0.900637i	$0.643101\pi$
$104$	0	0
$105$	1.55723	0.151970
$106$	0	0
$107$	16.9068	1.63444	0.817221	−	0.576324i	$-0.195513\pi$
$107$	16.9068	1.63444	0.817221	+	0.576324i	$0.195513\pi$
$108$	0	0
$109$	13.9721	1.33828	0.669140	−	0.743136i	$-0.266663\pi$
$109$	13.9721	1.33828	0.669140	+	0.743136i	$0.266663\pi$
$110$	0	0
$111$	−0.683077	−0.0648348
$112$	0	0
$113$	17.8078	1.67521	0.837607	−	0.546273i	$-0.183954\pi$
$113$	17.8078	1.67521	0.837607	+	0.546273i	$0.183954\pi$
$114$	0	0
$115$	0.997661	0.0930323
$116$	0	0
$117$	1.77215	0.163835
$118$	0	0
$119$	−8.10799	−0.743258
$120$	0	0
$121$	−5.50282	−0.500256
$122$	0	0
$123$	−5.02743	−0.453308
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.54455	−0.846942	−0.423471	−	0.905910i	$-0.639188\pi$
$127$	−9.54455	−0.846942	−0.423471	+	0.905910i	$0.639188\pi$
$128$	0	0
$129$	−8.52606	−0.750677
$130$	0	0
$131$	−6.67071	−0.582822	−0.291411	−	0.956598i	$-0.594125\pi$
$131$	−6.67071	−0.582822	−0.291411	+	0.956598i	$0.594125\pi$
$132$	0	0
$133$	10.2733	0.890806
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−11.6984	−0.999464	−0.499732	−	0.866180i	$-0.666568\pi$
$137$	−11.6984	−0.999464	−0.499732	+	0.866180i	$0.666568\pi$
$138$	0	0
$139$	10.7319	0.910267	0.455134	−	0.890423i	$-0.349591\pi$
$139$	10.7319	0.910267	0.455134	+	0.890423i	$0.349591\pi$
$140$	0	0
$141$	10.2821	0.865913
$142$	0	0
$143$	−4.15499	−0.347458
$144$	0	0
$145$	−2.57466	−0.213814
$146$	0	0
$147$	4.57503	0.377342
$148$	0	0
$149$	−9.65450	−0.790928	−0.395464	−	0.918482i	$-0.629416\pi$
$149$	−9.65450	−0.790928	−0.395464	+	0.918482i	$0.629416\pi$
$150$	0	0
$151$	17.6848	1.43917	0.719584	−	0.694406i	$-0.244333\pi$
$151$	17.6848	1.43917	0.719584	+	0.694406i	$0.244333\pi$
$152$	0	0
$153$	5.20667	0.420934
$154$	0	0
$155$	1.24774	0.100221
$156$	0	0
$157$	−7.05673	−0.563188	−0.281594	−	0.959534i	$-0.590863\pi$
$157$	−7.05673	−0.563188	−0.281594	+	0.959534i	$0.590863\pi$
$158$	0	0
$159$	6.10371	0.484056
$160$	0	0
$161$	−1.55359	−0.122440
$162$	0	0
$163$	21.4235	1.67802	0.839008	−	0.544119i	$-0.183136\pi$
$163$	21.4235	1.67802	0.839008	+	0.544119i	$0.183136\pi$
$164$	0	0
$165$	2.34461	0.182527
$166$	0	0
$167$	17.1477	1.32693	0.663465	−	0.748207i	$-0.269085\pi$
$167$	17.1477	1.32693	0.663465	+	0.748207i	$0.269085\pi$
$168$	0	0
$169$	−9.85949	−0.758422
$170$	0	0
$171$	−6.59714	−0.504496
$172$	0	0
$173$	−3.86767	−0.294054	−0.147027	−	0.989132i	$-0.546970\pi$
$173$	−3.86767	−0.294054	−0.147027	+	0.989132i	$0.546970\pi$
$174$	0	0
$175$	−1.55723	−0.117716
$176$	0	0
$177$	−7.58122	−0.569839
$178$	0	0
$179$	7.93505	0.593093	0.296547	−	0.955018i	$-0.404165\pi$
$179$	7.93505	0.593093	0.296547	+	0.955018i	$0.404165\pi$
$180$	0	0
$181$	7.00286	0.520518	0.260259	−	0.965539i	$-0.416192\pi$
$181$	7.00286	0.520518	0.260259	+	0.965539i	$0.416192\pi$
$182$	0	0
$183$	−9.98187	−0.737881
$184$	0	0
$185$	0.683077	0.0502208
$186$	0	0
$187$	−12.2076	−0.892708
$188$	0	0
$189$	1.55723	0.113272
$190$	0	0
$191$	−10.4435	−0.755668	−0.377834	−	0.925873i	$-0.623331\pi$
$191$	−10.4435	−0.755668	−0.377834	+	0.925873i	$0.623331\pi$
$192$	0	0
$193$	−9.21075	−0.663004	−0.331502	−	0.943454i	$-0.607555\pi$
$193$	−9.21075	−0.663004	−0.331502	+	0.943454i	$0.607555\pi$
$194$	0	0
$195$	−1.77215	−0.126906
$196$	0	0
$197$	24.5128	1.74647	0.873233	−	0.487304i	$-0.162020\pi$
$197$	24.5128	1.74647	0.873233	+	0.487304i	$0.162020\pi$
$198$	0	0
$199$	11.5716	0.820290	0.410145	−	0.912020i	$-0.365478\pi$
$199$	11.5716	0.820290	0.410145	+	0.912020i	$0.365478\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	4.00935	0.281401
$204$	0	0
$205$	5.02743	0.351131
$206$	0	0
$207$	0.997661	0.0693422
$208$	0	0
$209$	15.4677	1.06992
$210$	0	0
$211$	−16.5518	−1.13947	−0.569736	−	0.821828i	$-0.692955\pi$
$211$	−16.5518	−1.13947	−0.569736	+	0.821828i	$0.692955\pi$
$212$	0	0
$213$	5.55969	0.380944
$214$	0	0
$215$	8.52606	0.581472
$216$	0	0
$217$	−1.94301	−0.131900
$218$	0	0
$219$	−5.11240	−0.345464
$220$	0	0
$221$	9.22700	0.620675
$222$	0	0
$223$	1.95974	0.131234	0.0656168	−	0.997845i	$-0.479099\pi$
$223$	1.95974	0.131234	0.0656168	+	0.997845i	$0.479099\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−6.63644	−0.440476	−0.220238	−	0.975446i	$-0.570683\pi$
$227$	−6.63644	−0.440476	−0.220238	+	0.975446i	$0.570683\pi$
$228$	0	0
$229$	5.90298	0.390080	0.195040	−	0.980795i	$-0.437516\pi$
$229$	5.90298	0.390080	0.195040	+	0.980795i	$0.437516\pi$
$230$	0	0
$231$	−3.65109	−0.240224
$232$	0	0
$233$	11.4130	0.747690	0.373845	−	0.927491i	$-0.378039\pi$
$233$	11.4130	0.747690	0.373845	+	0.927491i	$0.378039\pi$
$234$	0	0
$235$	−10.2821	−0.670733
$236$	0	0
$237$	2.63518	0.171173
$238$	0	0
$239$	12.9066	0.834859	0.417429	−	0.908709i	$-0.362931\pi$
$239$	12.9066	0.834859	0.417429	+	0.908709i	$0.362931\pi$
$240$	0	0
$241$	−7.82521	−0.504066	−0.252033	−	0.967719i	$-0.581099\pi$
$241$	−7.82521	−0.504066	−0.252033	+	0.967719i	$0.581099\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−4.57503	−0.292288
$246$	0	0
$247$	−11.6911	−0.743888
$248$	0	0
$249$	6.26804	0.397221
$250$	0	0
$251$	26.3125	1.66083	0.830414	−	0.557146i	$-0.188104\pi$
$251$	26.3125	1.66083	0.830414	+	0.557146i	$0.188104\pi$
$252$	0	0
$253$	−2.33912	−0.147059
$254$	0	0
$255$	−5.20667	−0.326054
$256$	0	0
$257$	−7.53304	−0.469898	−0.234949	−	0.972008i	$-0.575492\pi$
$257$	−7.53304	−0.469898	−0.234949	+	0.972008i	$0.575492\pi$
$258$	0	0
$259$	−1.06371	−0.0660956
$260$	0	0
$261$	−2.57466	−0.159368
$262$	0	0
$263$	5.65385	0.348631	0.174316	−	0.984690i	$-0.444229\pi$
$263$	5.65385	0.348631	0.174316	+	0.984690i	$0.444229\pi$
$264$	0	0
$265$	−6.10371	−0.374948
$266$	0	0
$267$	6.48443	0.396841
$268$	0	0
$269$	9.76777	0.595552	0.297776	−	0.954636i	$-0.403755\pi$
$269$	9.76777	0.595552	0.297776	+	0.954636i	$0.403755\pi$
$270$	0	0
$271$	6.69983	0.406986	0.203493	−	0.979076i	$-0.434771\pi$
$271$	6.69983	0.406986	0.203493	+	0.979076i	$0.434771\pi$
$272$	0	0
$273$	2.75964	0.167021
$274$	0	0
$275$	−2.34461	−0.141385
$276$	0	0
$277$	−26.0185	−1.56330	−0.781649	−	0.623719i	$-0.785621\pi$
$277$	−26.0185	−1.56330	−0.781649	+	0.623719i	$0.785621\pi$
$278$	0	0
$279$	1.24774	0.0747000
$280$	0	0
$281$	−8.13846	−0.485500	−0.242750	−	0.970089i	$-0.578049\pi$
$281$	−8.13846	−0.485500	−0.242750	+	0.970089i	$0.578049\pi$
$282$	0	0
$283$	17.9127	1.06480	0.532399	−	0.846494i	$-0.321291\pi$
$283$	17.9127	1.06480	0.532399	+	0.846494i	$0.321291\pi$
$284$	0	0
$285$	6.59714	0.390781
$286$	0	0
$287$	−7.82887	−0.462123
$288$	0	0
$289$	10.1094	0.594672
$290$	0	0
$291$	−0.717020	−0.0420325
$292$	0	0
$293$	−4.45410	−0.260211	−0.130106	−	0.991500i	$-0.541532\pi$
$293$	−4.45410	−0.260211	−0.130106	+	0.991500i	$0.541532\pi$
$294$	0	0
$295$	7.58122	0.441396
$296$	0	0
$297$	2.34461	0.136048
$298$	0	0
$299$	1.76800	0.102246
$300$	0	0
$301$	−13.2770	−0.765276
$302$	0	0
$303$	−6.86852	−0.394586
$304$	0	0
$305$	9.98187	0.571560
$306$	0	0
$307$	17.2906	0.986829	0.493414	−	0.869794i	$-0.335749\pi$
$307$	17.2906	0.986829	0.493414	+	0.869794i	$0.335749\pi$
$308$	0	0
$309$	8.82087	0.501802
$310$	0	0
$311$	13.1005	0.742860	0.371430	−	0.928461i	$-0.378868\pi$
$311$	13.1005	0.742860	0.371430	+	0.928461i	$0.378868\pi$
$312$	0	0
$313$	−23.1980	−1.31123	−0.655613	−	0.755097i	$-0.727590\pi$
$313$	−23.1980	−1.31123	−0.655613	+	0.755097i	$0.727590\pi$
$314$	0	0
$315$	−1.55723	−0.0877400
$316$	0	0
$317$	11.3050	0.634950	0.317475	−	0.948267i	$-0.397165\pi$
$317$	11.3050	0.634950	0.317475	+	0.948267i	$0.397165\pi$
$318$	0	0
$319$	6.03657	0.337983
$320$	0	0
$321$	−16.9068	−0.943646
$322$	0	0
$323$	−34.3492	−1.91124
$324$	0	0
$325$	1.77215	0.0983011
$326$	0	0
$327$	−13.9721	−0.772657
$328$	0	0
$329$	16.0117	0.882752
$330$	0	0
$331$	30.0587	1.65218	0.826088	−	0.563541i	$-0.190562\pi$
$331$	30.0587	1.65218	0.826088	+	0.563541i	$0.190562\pi$
$332$	0	0
$333$	0.683077	0.0374324
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	9.56119	0.520831	0.260416	−	0.965497i	$-0.416140\pi$
$337$	9.56119	0.520831	0.260416	+	0.965497i	$0.416140\pi$
$338$	0	0
$339$	−17.8078	−0.967186
$340$	0	0
$341$	−2.92545	−0.158422
$342$	0	0
$343$	18.0250	0.973258
$344$	0	0
$345$	−0.997661	−0.0537122
$346$	0	0
$347$	3.66723	0.196867	0.0984336	−	0.995144i	$-0.468617\pi$
$347$	3.66723	0.196867	0.0984336	+	0.995144i	$0.468617\pi$
$348$	0	0
$349$	24.2693	1.29911	0.649554	−	0.760316i	$-0.274956\pi$
$349$	24.2693	1.29911	0.649554	+	0.760316i	$0.274956\pi$
$350$	0	0
$351$	−1.77215	−0.0945903
$352$	0	0
$353$	16.1332	0.858684	0.429342	−	0.903142i	$-0.358745\pi$
$353$	16.1332	0.858684	0.429342	+	0.903142i	$0.358745\pi$
$354$	0	0
$355$	−5.55969	−0.295078
$356$	0	0
$357$	8.10799	0.429120
$358$	0	0
$359$	31.7376	1.67505	0.837523	−	0.546403i	$-0.184003\pi$
$359$	31.7376	1.67505	0.837523	+	0.546403i	$0.184003\pi$
$360$	0	0
$361$	24.5223	1.29065
$362$	0	0
$363$	5.50282	0.288823
$364$	0	0
$365$	5.11240	0.267595
$366$	0	0
$367$	−10.7755	−0.562478	−0.281239	−	0.959638i	$-0.590745\pi$
$367$	−10.7755	−0.562478	−0.281239	+	0.959638i	$0.590745\pi$
$368$	0	0
$369$	5.02743	0.261717
$370$	0	0
$371$	9.50488	0.493469
$372$	0	0
$373$	8.76358	0.453761	0.226880	−	0.973923i	$-0.427147\pi$
$373$	8.76358	0.453761	0.226880	+	0.973923i	$0.427147\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−4.56269	−0.234990
$378$	0	0
$379$	−31.4971	−1.61790	−0.808949	−	0.587878i	$-0.799963\pi$
$379$	−31.4971	−1.61790	−0.808949	+	0.587878i	$0.799963\pi$
$380$	0	0
$381$	9.54455	0.488982
$382$	0	0
$383$	−10.4342	−0.533160	−0.266580	−	0.963813i	$-0.585894\pi$
$383$	−10.4342	−0.533160	−0.266580	+	0.963813i	$0.585894\pi$
$384$	0	0
$385$	3.65109	0.186077
$386$	0	0
$387$	8.52606	0.433404
$388$	0	0
$389$	31.8823	1.61650	0.808248	−	0.588843i	$-0.200416\pi$
$389$	31.8823	1.61650	0.808248	+	0.588843i	$0.200416\pi$
$390$	0	0
$391$	5.19449	0.262697
$392$	0	0
$393$	6.67071	0.336493
$394$	0	0
$395$	−2.63518	−0.132590
$396$	0	0
$397$	−1.38400	−0.0694610	−0.0347305	−	0.999397i	$-0.511057\pi$
$397$	−1.38400	−0.0694610	−0.0347305	+	0.999397i	$0.511057\pi$
$398$	0	0
$399$	−10.2733	−0.514307
$400$	0	0
$401$	28.8741	1.44190	0.720952	−	0.692985i	$-0.243705\pi$
$401$	28.8741	1.44190	0.720952	+	0.692985i	$0.243705\pi$
$402$	0	0
$403$	2.21117	0.110146
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−1.60155	−0.0793857
$408$	0	0
$409$	9.12021	0.450965	0.225483	−	0.974247i	$-0.427604\pi$
$409$	9.12021	0.450965	0.225483	+	0.974247i	$0.427604\pi$
$410$	0	0
$411$	11.6984	0.577041
$412$	0	0
$413$	−11.8057	−0.580921
$414$	0	0
$415$	−6.26804	−0.307686
$416$	0	0
$417$	−10.7319	−0.525543
$418$	0	0
$419$	14.1895	0.693202	0.346601	−	0.938013i	$-0.387336\pi$
$419$	14.1895	0.693202	0.346601	+	0.938013i	$0.387336\pi$
$420$	0	0
$421$	7.17624	0.349749	0.174874	−	0.984591i	$-0.444048\pi$
$421$	7.17624	0.349749	0.174874	+	0.984591i	$0.444048\pi$
$422$	0	0
$423$	−10.2821	−0.499935
$424$	0	0
$425$	5.20667	0.252561
$426$	0	0
$427$	−15.5441	−0.752231
$428$	0	0
$429$	4.15499	0.200605
$430$	0	0
$431$	9.02371	0.434657	0.217328	−	0.976099i	$-0.430266\pi$
$431$	9.02371	0.434657	0.217328	+	0.976099i	$0.430266\pi$
$432$	0	0
$433$	−6.29778	−0.302652	−0.151326	−	0.988484i	$-0.548354\pi$
$433$	−6.29778	−0.302652	−0.151326	+	0.988484i	$0.548354\pi$
$434$	0	0
$435$	2.57466	0.123446
$436$	0	0
$437$	−6.58171	−0.314846
$438$	0	0
$439$	−28.9143	−1.38001	−0.690003	−	0.723806i	$-0.742391\pi$
$439$	−28.9143	−1.38001	−0.690003	+	0.723806i	$0.742391\pi$
$440$	0	0
$441$	−4.57503	−0.217859
$442$	0	0
$443$	−35.3238	−1.67828	−0.839142	−	0.543912i	$-0.816942\pi$
$443$	−35.3238	−1.67828	−0.839142	+	0.543912i	$0.816942\pi$
$444$	0	0
$445$	−6.48443	−0.307391
$446$	0	0
$447$	9.65450	0.456642
$448$	0	0
$449$	−12.9857	−0.612831	−0.306416	−	0.951898i	$-0.599130\pi$
$449$	−12.9857	−0.612831	−0.306416	+	0.951898i	$0.599130\pi$
$450$	0	0
$451$	−11.7873	−0.555044
$452$	0	0
$453$	−17.6848	−0.830903
$454$	0	0
$455$	−2.75964	−0.129374
$456$	0	0
$457$	12.1440	0.568072	0.284036	−	0.958814i	$-0.408327\pi$
$457$	12.1440	0.568072	0.284036	+	0.958814i	$0.408327\pi$
$458$	0	0
$459$	−5.20667	−0.243027
$460$	0	0
$461$	−0.977404	−0.0455222	−0.0227611	−	0.999741i	$-0.507246\pi$
$461$	−0.977404	−0.0455222	−0.0227611	+	0.999741i	$0.507246\pi$
$462$	0	0
$463$	−11.2533	−0.522986	−0.261493	−	0.965205i	$-0.584215\pi$
$463$	−11.2533	−0.522986	−0.261493	+	0.965205i	$0.584215\pi$
$464$	0	0
$465$	−1.24774	−0.0578624
$466$	0	0
$467$	−7.07399	−0.327345	−0.163673	−	0.986515i	$-0.552334\pi$
$467$	−7.07399	−0.327345	−0.163673	+	0.986515i	$0.552334\pi$
$468$	0	0
$469$	1.55723	0.0719062
$470$	0	0
$471$	7.05673	0.325157
$472$	0	0
$473$	−19.9902	−0.919152
$474$	0	0
$475$	−6.59714	−0.302698
$476$	0	0
$477$	−6.10371	−0.279470
$478$	0	0
$479$	−7.89392	−0.360683	−0.180341	−	0.983604i	$-0.557720\pi$
$479$	−7.89392	−0.360683	−0.180341	+	0.983604i	$0.557720\pi$
$480$	0	0
$481$	1.21051	0.0551947
$482$	0	0
$483$	1.55359	0.0706907
$484$	0	0
$485$	0.717020	0.0325582
$486$	0	0
$487$	1.65950	0.0751991	0.0375995	−	0.999293i	$-0.488029\pi$
$487$	1.65950	0.0751991	0.0375995	+	0.999293i	$0.488029\pi$
$488$	0	0
$489$	−21.4235	−0.968803
$490$	0	0
$491$	40.5111	1.82824	0.914121	−	0.405442i	$-0.132882\pi$
$491$	40.5111	1.82824	0.914121	+	0.405442i	$0.132882\pi$
$492$	0	0
$493$	−13.4054	−0.603750
$494$	0	0
$495$	−2.34461	−0.105382
$496$	0	0
$497$	8.65773	0.388352
$498$	0	0
$499$	4.72806	0.211657	0.105829	−	0.994384i	$-0.466251\pi$
$499$	4.72806	0.211657	0.105829	+	0.994384i	$0.466251\pi$
$500$	0	0
$501$	−17.1477	−0.766103
$502$	0	0
$503$	−13.1551	−0.586555	−0.293278	−	0.956027i	$-0.594746\pi$
$503$	−13.1551	−0.586555	−0.293278	+	0.956027i	$0.594746\pi$
$504$	0	0
$505$	6.86852	0.305645
$506$	0	0
$507$	9.85949	0.437875
$508$	0	0
$509$	29.0023	1.28551	0.642753	−	0.766074i	$-0.277792\pi$
$509$	29.0023	1.28551	0.642753	+	0.766074i	$0.277792\pi$
$510$	0	0
$511$	−7.96118	−0.352182
$512$	0	0
$513$	6.59714	0.291271
$514$	0	0
$515$	−8.82087	−0.388694
$516$	0	0
$517$	24.1076	1.06025
$518$	0	0
$519$	3.86767	0.169772
$520$	0	0
$521$	20.2865	0.888768	0.444384	−	0.895836i	$-0.353423\pi$
$521$	20.2865	0.888768	0.444384	+	0.895836i	$0.353423\pi$
$522$	0	0
$523$	11.0838	0.484662	0.242331	−	0.970194i	$-0.422088\pi$
$523$	11.0838	0.484662	0.242331	+	0.970194i	$0.422088\pi$
$524$	0	0
$525$	1.55723	0.0679631
$526$	0	0
$527$	6.49655	0.282994
$528$	0	0
$529$	−22.0047	−0.956725
$530$	0	0
$531$	7.58122	0.328997
$532$	0	0
$533$	8.90935	0.385907
$534$	0	0
$535$	16.9068	0.730945
$536$	0	0
$537$	−7.93505	−0.342423
$538$	0	0
$539$	10.7266	0.462029
$540$	0	0
$541$	25.6457	1.10259	0.551297	−	0.834309i	$-0.314133\pi$
$541$	25.6457	1.10259	0.551297	+	0.834309i	$0.314133\pi$
$542$	0	0
$543$	−7.00286	−0.300521
$544$	0	0
$545$	13.9721	0.598497
$546$	0	0
$547$	−43.8169	−1.87347	−0.936737	−	0.350033i	$-0.886170\pi$
$547$	−43.8169	−1.87347	−0.936737	+	0.350033i	$0.886170\pi$
$548$	0	0
$549$	9.98187	0.426016
$550$	0	0
$551$	16.9854	0.723603
$552$	0	0
$553$	4.10359	0.174502
$554$	0	0
$555$	−0.683077	−0.0289950
$556$	0	0
$557$	−16.2757	−0.689622	−0.344811	−	0.938672i	$-0.612057\pi$
$557$	−16.2757	−0.689622	−0.344811	+	0.938672i	$0.612057\pi$
$558$	0	0
$559$	15.1094	0.639061
$560$	0	0
$561$	12.2076	0.515405
$562$	0	0
$563$	23.5724	0.993458	0.496729	−	0.867906i	$-0.334534\pi$
$563$	23.5724	0.993458	0.496729	+	0.867906i	$0.334534\pi$
$564$	0	0
$565$	17.8078	0.749179
$566$	0	0
$567$	−1.55723	−0.0653976
$568$	0	0
$569$	12.8828	0.540073	0.270037	−	0.962850i	$-0.412964\pi$
$569$	12.8828	0.540073	0.270037	+	0.962850i	$0.412964\pi$
$570$	0	0
$571$	−10.3040	−0.431207	−0.215604	−	0.976481i	$-0.569172\pi$
$571$	−10.3040	−0.431207	−0.215604	+	0.976481i	$0.569172\pi$
$572$	0	0
$573$	10.4435	0.436285
$574$	0	0
$575$	0.997661	0.0416053
$576$	0	0
$577$	28.1140	1.17040	0.585200	−	0.810889i	$-0.301016\pi$
$577$	28.1140	1.17040	0.585200	+	0.810889i	$0.301016\pi$
$578$	0	0
$579$	9.21075	0.382786
$580$	0	0
$581$	9.76079	0.404946
$582$	0	0
$583$	14.3108	0.592692
$584$	0	0
$585$	1.77215	0.0732693
$586$	0	0
$587$	−37.1720	−1.53425	−0.767126	−	0.641496i	$-0.778314\pi$
$587$	−37.1720	−1.53425	−0.767126	+	0.641496i	$0.778314\pi$
$588$	0	0
$589$	−8.23149	−0.339173
$590$	0	0
$591$	−24.5128	−1.00832
$592$	0	0
$593$	−31.0522	−1.27516	−0.637580	−	0.770384i	$-0.720064\pi$
$593$	−31.0522	−1.27516	−0.637580	+	0.770384i	$0.720064\pi$
$594$	0	0
$595$	−8.10799	−0.332395
$596$	0	0
$597$	−11.5716	−0.473595
$598$	0	0
$599$	5.99710	0.245035	0.122517	−	0.992466i	$-0.460903\pi$
$599$	5.99710	0.245035	0.122517	+	0.992466i	$0.460903\pi$
$600$	0	0
$601$	24.1559	0.985340	0.492670	−	0.870216i	$-0.336021\pi$
$601$	24.1559	0.985340	0.492670	+	0.870216i	$0.336021\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−5.50282	−0.223721
$606$	0	0
$607$	28.0587	1.13887	0.569434	−	0.822037i	$-0.307162\pi$
$607$	28.0587	1.13887	0.569434	+	0.822037i	$0.307162\pi$
$608$	0	0
$609$	−4.00935	−0.162467
$610$	0	0
$611$	−18.2215	−0.737162
$612$	0	0
$613$	0.726673	0.0293500	0.0146750	−	0.999892i	$-0.495329\pi$
$613$	0.726673	0.0293500	0.0146750	+	0.999892i	$0.495329\pi$
$614$	0	0
$615$	−5.02743	−0.202725
$616$	0	0
$617$	34.8779	1.40413	0.702066	−	0.712112i	$-0.252261\pi$
$617$	34.8779	1.40413	0.702066	+	0.712112i	$0.252261\pi$
$618$	0	0
$619$	30.5773	1.22900	0.614502	−	0.788915i	$-0.289357\pi$
$619$	30.5773	1.22900	0.614502	+	0.788915i	$0.289357\pi$
$620$	0	0
$621$	−0.997661	−0.0400347
$622$	0	0
$623$	10.0978	0.404558
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−15.4677	−0.617721
$628$	0	0
$629$	3.55655	0.141809
$630$	0	0
$631$	−12.7166	−0.506240	−0.253120	−	0.967435i	$-0.581457\pi$
$631$	−12.7166	−0.506240	−0.253120	+	0.967435i	$0.581457\pi$
$632$	0	0
$633$	16.5518	0.657874
$634$	0	0
$635$	−9.54455	−0.378764
$636$	0	0
$637$	−8.10764	−0.321236
$638$	0	0
$639$	−5.55969	−0.219938
$640$	0	0
$641$	17.0359	0.672876	0.336438	−	0.941706i	$-0.390778\pi$
$641$	17.0359	0.672876	0.336438	+	0.941706i	$0.390778\pi$
$642$	0	0
$643$	42.2829	1.66748	0.833738	−	0.552160i	$-0.186196\pi$
$643$	42.2829	1.66748	0.833738	+	0.552160i	$0.186196\pi$
$644$	0	0
$645$	−8.52606	−0.335713
$646$	0	0
$647$	−35.6927	−1.40322	−0.701612	−	0.712559i	$-0.747536\pi$
$647$	−35.6927	−1.40322	−0.701612	+	0.712559i	$0.747536\pi$
$648$	0	0
$649$	−17.7750	−0.697729
$650$	0	0
$651$	1.94301	0.0761527
$652$	0	0
$653$	13.4161	0.525012	0.262506	−	0.964930i	$-0.415451\pi$
$653$	13.4161	0.525012	0.262506	+	0.964930i	$0.415451\pi$
$654$	0	0
$655$	−6.67071	−0.260646
$656$	0	0
$657$	5.11240	0.199454
$658$	0	0
$659$	5.21776	0.203255	0.101628	−	0.994823i	$-0.467595\pi$
$659$	5.21776	0.203255	0.101628	+	0.994823i	$0.467595\pi$
$660$	0	0
$661$	3.79498	0.147608	0.0738039	−	0.997273i	$-0.476486\pi$
$661$	3.79498	0.147608	0.0738039	+	0.997273i	$0.476486\pi$
$662$	0	0
$663$	−9.22700	−0.358347
$664$	0	0
$665$	10.2733	0.398381
$666$	0	0
$667$	−2.56864	−0.0994582
$668$	0	0
$669$	−1.95974	−0.0757678
$670$	0	0
$671$	−23.4036	−0.903484
$672$	0	0
$673$	38.0690	1.46745	0.733725	−	0.679446i	$-0.237780\pi$
$673$	38.0690	1.46745	0.733725	+	0.679446i	$0.237780\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−34.0886	−1.31013	−0.655065	−	0.755572i	$-0.727359\pi$
$677$	−34.0886	−1.31013	−0.655065	+	0.755572i	$0.727359\pi$
$678$	0	0
$679$	−1.11657	−0.0428499
$680$	0	0
$681$	6.63644	0.254309
$682$	0	0
$683$	9.90557	0.379026	0.189513	−	0.981878i	$-0.439309\pi$
$683$	9.90557	0.379026	0.189513	+	0.981878i	$0.439309\pi$
$684$	0	0
$685$	−11.6984	−0.446974
$686$	0	0
$687$	−5.90298	−0.225213
$688$	0	0
$689$	−10.8167	−0.412083
$690$	0	0
$691$	28.6438	1.08966	0.544831	−	0.838546i	$-0.316594\pi$
$691$	28.6438	1.08966	0.544831	+	0.838546i	$0.316594\pi$
$692$	0	0
$693$	3.65109	0.138694
$694$	0	0
$695$	10.7319	0.407084
$696$	0	0
$697$	26.1762	0.991493
$698$	0	0
$699$	−11.4130	−0.431679
$700$	0	0
$701$	20.7949	0.785411	0.392705	−	0.919664i	$-0.371539\pi$
$701$	20.7949	0.785411	0.392705	+	0.919664i	$0.371539\pi$
$702$	0	0
$703$	−4.50635	−0.169960
$704$	0	0
$705$	10.2821	0.387248
$706$	0	0
$707$	−10.6959	−0.402260
$708$	0	0
$709$	−12.6545	−0.475248	−0.237624	−	0.971357i	$-0.576369\pi$
$709$	−12.6545	−0.475248	−0.237624	+	0.971357i	$0.576369\pi$
$710$	0	0
$711$	−2.63518	−0.0988270
$712$	0	0
$713$	1.24482	0.0466188
$714$	0	0
$715$	−4.15499	−0.155388
$716$	0	0
$717$	−12.9066	−0.482006
$718$	0	0
$719$	32.3045	1.20476	0.602378	−	0.798211i	$-0.294220\pi$
$719$	32.3045	1.20476	0.602378	+	0.798211i	$0.294220\pi$
$720$	0	0
$721$	13.7361	0.511560
$722$	0	0
$723$	7.82521	0.291023
$724$	0	0
$725$	−2.57466	−0.0956206
$726$	0	0
$727$	−28.7538	−1.06642	−0.533209	−	0.845984i	$-0.679014\pi$
$727$	−28.7538	−1.06642	−0.533209	+	0.845984i	$0.679014\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	44.3924	1.64191
$732$	0	0
$733$	−31.9932	−1.18170	−0.590848	−	0.806783i	$-0.701207\pi$
$733$	−31.9932	−1.18170	−0.590848	+	0.806783i	$0.701207\pi$
$734$	0	0
$735$	4.57503	0.168753
$736$	0	0
$737$	2.34461	0.0863647
$738$	0	0
$739$	18.1166	0.666428	0.333214	−	0.942851i	$-0.391867\pi$
$739$	18.1166	0.666428	0.333214	+	0.942851i	$0.391867\pi$
$740$	0	0
$741$	11.6911	0.429484
$742$	0	0
$743$	43.0771	1.58034	0.790172	−	0.612885i	$-0.209991\pi$
$743$	43.0771	1.58034	0.790172	+	0.612885i	$0.209991\pi$
$744$	0	0
$745$	−9.65450	−0.353714
$746$	0	0
$747$	−6.26804	−0.229336
$748$	0	0
$749$	−26.3278	−0.961997
$750$	0	0
$751$	4.55259	0.166127	0.0830633	−	0.996544i	$-0.473530\pi$
$751$	4.55259	0.166127	0.0830633	+	0.996544i	$0.473530\pi$
$752$	0	0
$753$	−26.3125	−0.958880
$754$	0	0
$755$	17.6848	0.643615
$756$	0	0
$757$	7.65852	0.278354	0.139177	−	0.990268i	$-0.455554\pi$
$757$	7.65852	0.278354	0.139177	+	0.990268i	$0.455554\pi$
$758$	0	0
$759$	2.33912	0.0849047
$760$	0	0
$761$	48.5851	1.76121	0.880605	−	0.473852i	$-0.157137\pi$
$761$	48.5851	1.76121	0.880605	+	0.473852i	$0.157137\pi$
$762$	0	0
$763$	−21.7577	−0.787683
$764$	0	0
$765$	5.20667	0.188248
$766$	0	0
$767$	13.4350	0.485112
$768$	0	0
$769$	13.2474	0.477715	0.238857	−	0.971055i	$-0.423227\pi$
$769$	13.2474	0.477715	0.238857	+	0.971055i	$0.423227\pi$
$770$	0	0
$771$	7.53304	0.271296
$772$	0	0
$773$	−22.9487	−0.825408	−0.412704	−	0.910865i	$-0.635416\pi$
$773$	−22.9487	−0.825408	−0.412704	+	0.910865i	$0.635416\pi$
$774$	0	0
$775$	1.24774	0.0448200
$776$	0	0
$777$	1.06371	0.0381603
$778$	0	0
$779$	−33.1667	−1.18832
$780$	0	0
$781$	13.0353	0.466439
$782$	0	0
$783$	2.57466	0.0920110
$784$	0	0
$785$	−7.05673	−0.251866
$786$	0	0
$787$	−7.99466	−0.284979	−0.142489	−	0.989796i	$-0.545511\pi$
$787$	−7.99466	−0.284979	−0.142489	+	0.989796i	$0.545511\pi$
$788$	0	0
$789$	−5.65385	−0.201282
$790$	0	0
$791$	−27.7308	−0.985995
$792$	0	0
$793$	17.6894	0.628168
$794$	0	0
$795$	6.10371	0.216476
$796$	0	0
$797$	−0.612757	−0.0217050	−0.0108525	−	0.999941i	$-0.503455\pi$
$797$	−0.612757	−0.0217050	−0.0108525	+	0.999941i	$0.503455\pi$
$798$	0	0
$799$	−53.5357	−1.89396
$800$	0	0
$801$	−6.48443	−0.229116
$802$	0	0
$803$	−11.9866	−0.422996
$804$	0	0
$805$	−1.55359	−0.0547568
$806$	0	0
$807$	−9.76777	−0.343842
$808$	0	0
$809$	5.38439	0.189305	0.0946525	−	0.995510i	$-0.469826\pi$
$809$	5.38439	0.189305	0.0946525	+	0.995510i	$0.469826\pi$
$810$	0	0
$811$	−5.60449	−0.196800	−0.0984001	−	0.995147i	$-0.531372\pi$
$811$	−5.60449	−0.196800	−0.0984001	+	0.995147i	$0.531372\pi$
$812$	0	0
$813$	−6.69983	−0.234973
$814$	0	0
$815$	21.4235	0.750432
$816$	0	0
$817$	−56.2476	−1.96785
$818$	0	0
$819$	−2.75964	−0.0964298
$820$	0	0
$821$	35.1571	1.22699	0.613495	−	0.789698i	$-0.289763\pi$
$821$	35.1571	1.22699	0.613495	+	0.789698i	$0.289763\pi$
$822$	0	0
$823$	−17.7892	−0.620092	−0.310046	−	0.950722i	$-0.600344\pi$
$823$	−17.7892	−0.620092	−0.310046	+	0.950722i	$0.600344\pi$
$824$	0	0
$825$	2.34461	0.0816287
$826$	0	0
$827$	14.2074	0.494041	0.247020	−	0.969010i	$-0.420549\pi$
$827$	14.2074	0.494041	0.247020	+	0.969010i	$0.420549\pi$
$828$	0	0
$829$	−51.0178	−1.77192	−0.885962	−	0.463759i	$-0.846500\pi$
$829$	−51.0178	−1.77192	−0.885962	+	0.463759i	$0.846500\pi$
$830$	0	0
$831$	26.0185	0.902570
$832$	0	0
$833$	−23.8207	−0.825338
$834$	0	0
$835$	17.1477	0.593421
$836$	0	0
$837$	−1.24774	−0.0431281
$838$	0	0
$839$	5.30145	0.183026	0.0915132	−	0.995804i	$-0.470830\pi$
$839$	5.30145	0.183026	0.0915132	+	0.995804i	$0.470830\pi$
$840$	0	0
$841$	−22.3711	−0.771417
$842$	0	0
$843$	8.13846	0.280303
$844$	0	0
$845$	−9.85949	−0.339177
$846$	0	0
$847$	8.56916	0.294440
$848$	0	0
$849$	−17.9127	−0.614761
$850$	0	0
$851$	0.681479	0.0233608
$852$	0	0
$853$	3.98369	0.136399	0.0681994	−	0.997672i	$-0.478275\pi$
$853$	3.98369	0.136399	0.0681994	+	0.997672i	$0.478275\pi$
$854$	0	0
$855$	−6.59714	−0.225618
$856$	0	0
$857$	28.4229	0.970906	0.485453	−	0.874263i	$-0.338655\pi$
$857$	28.4229	0.970906	0.485453	+	0.874263i	$0.338655\pi$
$858$	0	0
$859$	−12.2487	−0.417921	−0.208961	−	0.977924i	$-0.567008\pi$
$859$	−12.2487	−0.417921	−0.208961	+	0.977924i	$0.567008\pi$
$860$	0	0
$861$	7.82887	0.266807
$862$	0	0
$863$	10.7623	0.366354	0.183177	−	0.983080i	$-0.441362\pi$
$863$	10.7623	0.366354	0.183177	+	0.983080i	$0.441362\pi$
$864$	0	0
$865$	−3.86767	−0.131505
$866$	0	0
$867$	−10.1094	−0.343334
$868$	0	0
$869$	6.17846	0.209590
$870$	0	0
$871$	−1.77215	−0.0600470
$872$	0	0
$873$	0.717020	0.0242675
$874$	0	0
$875$	−1.55723	−0.0526440
$876$	0	0
$877$	−22.2444	−0.751140	−0.375570	−	0.926794i	$-0.622553\pi$
$877$	−22.2444	−0.751140	−0.375570	+	0.926794i	$0.622553\pi$
$878$	0	0
$879$	4.45410	0.150233
$880$	0	0
$881$	−10.3526	−0.348789	−0.174395	−	0.984676i	$-0.555797\pi$
$881$	−10.3526	−0.348789	−0.174395	+	0.984676i	$0.555797\pi$
$882$	0	0
$883$	46.4770	1.56407	0.782037	−	0.623232i	$-0.214181\pi$
$883$	46.4770	1.56407	0.782037	+	0.623232i	$0.214181\pi$
$884$	0	0
$885$	−7.58122	−0.254840
$886$	0	0
$887$	1.26882	0.0426028	0.0213014	−	0.999773i	$-0.493219\pi$
$887$	1.26882	0.0426028	0.0213014	+	0.999773i	$0.493219\pi$
$888$	0	0
$889$	14.8631	0.498491
$890$	0	0
$891$	−2.34461	−0.0785473
$892$	0	0
$893$	67.8328	2.26994
$894$	0	0
$895$	7.93505	0.265239
$896$	0	0
$897$	−1.76800	−0.0590319
$898$	0	0
$899$	−3.21250	−0.107143
$900$	0	0
$901$	−31.7800	−1.05875
$902$	0	0
$903$	13.2770	0.441832
$904$	0	0
$905$	7.00286	0.232783
$906$	0	0
$907$	−50.6519	−1.68187	−0.840934	−	0.541138i	$-0.817994\pi$
$907$	−50.6519	−1.68187	−0.840934	+	0.541138i	$0.817994\pi$
$908$	0	0
$909$	6.86852	0.227814
$910$	0	0
$911$	−8.88879	−0.294499	−0.147249	−	0.989099i	$-0.547042\pi$
$911$	−8.88879	−0.294499	−0.147249	+	0.989099i	$0.547042\pi$
$912$	0	0
$913$	14.6961	0.486370
$914$	0	0
$915$	−9.98187	−0.329990
$916$	0	0
$917$	10.3878	0.343036
$918$	0	0
$919$	3.78193	0.124754	0.0623771	−	0.998053i	$-0.480132\pi$
$919$	3.78193	0.124754	0.0623771	+	0.998053i	$0.480132\pi$
$920$	0	0
$921$	−17.2906	−0.569746
$922$	0	0
$923$	−9.85260	−0.324302
$924$	0	0
$925$	0.683077	0.0224594
$926$	0	0
$927$	−8.82087	−0.289715
$928$	0	0
$929$	21.6695	0.710952	0.355476	−	0.934685i	$-0.384319\pi$
$929$	21.6695	0.710952	0.355476	+	0.934685i	$0.384319\pi$
$930$	0	0
$931$	30.1821	0.989180
$932$	0	0
$933$	−13.1005	−0.428891
$934$	0	0
$935$	−12.2076	−0.399231
$936$	0	0
$937$	−48.4753	−1.58362	−0.791810	−	0.610767i	$-0.790861\pi$
$937$	−48.4753	−1.58362	−0.791810	+	0.610767i	$0.790861\pi$
$938$	0	0
$939$	23.1980	0.757037
$940$	0	0
$941$	47.9342	1.56261	0.781304	−	0.624150i	$-0.214555\pi$
$941$	47.9342	1.56261	0.781304	+	0.624150i	$0.214555\pi$
$942$	0	0
$943$	5.01567	0.163333
$944$	0	0
$945$	1.55723	0.0506567
$946$	0	0
$947$	−10.8229	−0.351697	−0.175848	−	0.984417i	$-0.556267\pi$
$947$	−10.8229	−0.351697	−0.175848	+	0.984417i	$0.556267\pi$
$948$	0	0
$949$	9.05992	0.294098
$950$	0	0
$951$	−11.3050	−0.366589
$952$	0	0
$953$	4.09914	0.132784	0.0663920	−	0.997794i	$-0.478851\pi$
$953$	4.09914	0.132784	0.0663920	+	0.997794i	$0.478851\pi$
$954$	0	0
$955$	−10.4435	−0.337945
$956$	0	0
$957$	−6.03657	−0.195135
$958$	0	0
$959$	18.2172	0.588263
$960$	0	0
$961$	−29.4432	−0.949779
$962$	0	0
$963$	16.9068	0.544814
$964$	0	0
$965$	−9.21075	−0.296505
$966$	0	0
$967$	−24.7850	−0.797030	−0.398515	−	0.917162i	$-0.630474\pi$
$967$	−24.7850	−0.797030	−0.398515	+	0.917162i	$0.630474\pi$
$968$	0	0
$969$	34.3492	1.10345
$970$	0	0
$971$	−2.08887	−0.0670349	−0.0335175	−	0.999438i	$-0.510671\pi$
$971$	−2.08887	−0.0670349	−0.0335175	+	0.999438i	$0.510671\pi$
$972$	0	0
$973$	−16.7120	−0.535763
$974$	0	0
$975$	−1.77215	−0.0567542
$976$	0	0
$977$	29.7150	0.950668	0.475334	−	0.879806i	$-0.342327\pi$
$977$	29.7150	0.950668	0.475334	+	0.879806i	$0.342327\pi$
$978$	0	0
$979$	15.2034	0.485904
$980$	0	0
$981$	13.9721	0.446094
$982$	0	0
$983$	−5.94167	−0.189510	−0.0947550	−	0.995501i	$-0.530207\pi$
$983$	−5.94167	−0.189510	−0.0947550	+	0.995501i	$0.530207\pi$
$984$	0	0
$985$	24.5128	0.781043
$986$	0	0
$987$	−16.0117	−0.509657
$988$	0	0
$989$	8.50611	0.270479
$990$	0	0
$991$	44.2297	1.40500	0.702501	−	0.711682i	$-0.252066\pi$
$991$	44.2297	1.40500	0.702501	+	0.711682i	$0.252066\pi$
$992$	0	0
$993$	−30.0587	−0.953884
$994$	0	0
$995$	11.5716	0.366845
$996$	0	0
$997$	4.70874	0.149127	0.0745636	−	0.997216i	$-0.476244\pi$
$997$	4.70874	0.149127	0.0745636	+	0.997216i	$0.476244\pi$
$998$	0	0
$999$	−0.683077	−0.0216116

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8040.2.a.bc.1.5	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8040.2.a.bc.1.5	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.55723 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.55723 q^{7} +1.00000 q^{9} -2.34461 q^{11} +1.77215 q^{13} -1.00000 q^{15} +5.20667 q^{17} -6.59714 q^{19} +1.55723 q^{21} +0.997661 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.57466 q^{29} +1.24774 q^{31} +2.34461 q^{33} -1.55723 q^{35} +0.683077 q^{37} -1.77215 q^{39} +5.02743 q^{41} +8.52606 q^{43} +1.00000 q^{45} -10.2821 q^{47} -4.57503 q^{49} -5.20667 q^{51} -6.10371 q^{53} -2.34461 q^{55} +6.59714 q^{57} +7.58122 q^{59} +9.98187 q^{61} -1.55723 q^{63} +1.77215 q^{65} -1.00000 q^{67} -0.997661 q^{69} -5.55969 q^{71} +5.11240 q^{73} -1.00000 q^{75} +3.65109 q^{77} -2.63518 q^{79} +1.00000 q^{81} -6.26804 q^{83} +5.20667 q^{85} +2.57466 q^{87} -6.48443 q^{89} -2.75964 q^{91} -1.24774 q^{93} -6.59714 q^{95} +0.717020 q^{97} -2.34461 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) \) 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