LMFDB - Embedded newform 6040.2.a.o.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 19 x^{13} + 119 x^{12} + 106 x^{11} - 1063 x^{10} - 48 x^{9} + 4510 x^{8} + \cdots + 272 $ x^15 - 5x^14 - 19x^13 + 119x^12 + 106x^11 - 1063x^10 - 48x^9 + 4510x^8 - 1130x^7 - 9512x^6 + 2839x^5 + 9383x^4 - 1781x^3 - 3692x^2 - 92x + 272
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.266097$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.266097 q^{3} +1.00000 q^{5} +3.77999 q^{7} -2.92919 q^{9} +O(q^{10})$	$q+0.266097 q^{3} +1.00000 q^{5} +3.77999 q^{7} -2.92919 q^{9} -1.90689 q^{11} +1.63336 q^{13} +0.266097 q^{15} -3.61052 q^{17} -4.01317 q^{19} +1.00584 q^{21} +2.09095 q^{23} +1.00000 q^{25} -1.57774 q^{27} +1.32071 q^{29} -3.64219 q^{31} -0.507417 q^{33} +3.77999 q^{35} -3.78278 q^{37} +0.434632 q^{39} +8.68857 q^{41} +2.05249 q^{43} -2.92919 q^{45} +12.6216 q^{47} +7.28831 q^{49} -0.960746 q^{51} +9.38276 q^{53} -1.90689 q^{55} -1.06789 q^{57} -2.81380 q^{59} +13.6738 q^{61} -11.0723 q^{63} +1.63336 q^{65} +15.5162 q^{67} +0.556394 q^{69} -9.50694 q^{71} +1.72533 q^{73} +0.266097 q^{75} -7.20803 q^{77} +0.651611 q^{79} +8.36775 q^{81} +17.9505 q^{83} -3.61052 q^{85} +0.351437 q^{87} -0.0643244 q^{89} +6.17408 q^{91} -0.969174 q^{93} -4.01317 q^{95} +10.9094 q^{97} +5.58565 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	$ 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) $ 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * 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
$3$	0.266097	0.153631	0.0768155	−	0.997045i	$-0.475525\pi$
$3$	0.266097	0.153631	0.0768155	+	0.997045i	$0.475525\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.77999	1.42870	0.714351	−	0.699788i	$-0.246722\pi$
$7$	3.77999	1.42870	0.714351	+	0.699788i	$0.246722\pi$
$8$	0	0
$9$	−2.92919	−0.976398
$10$	0	0
$11$	−1.90689	−0.574949	−0.287475	−	0.957788i	$-0.592816\pi$
$11$	−1.90689	−0.574949	−0.287475	+	0.957788i	$0.592816\pi$
$12$	0	0
$13$	1.63336	0.453013	0.226506	−	0.974010i	$-0.427270\pi$
$13$	1.63336	0.453013	0.226506	+	0.974010i	$0.427270\pi$
$14$	0	0
$15$	0.266097	0.0687058
$16$	0	0
$17$	−3.61052	−0.875679	−0.437839	−	0.899053i	$-0.644256\pi$
$17$	−3.61052	−0.875679	−0.437839	+	0.899053i	$0.644256\pi$
$18$	0	0
$19$	−4.01317	−0.920685	−0.460342	−	0.887742i	$-0.652273\pi$
$19$	−4.01317	−0.920685	−0.460342	+	0.887742i	$0.652273\pi$
$20$	0	0
$21$	1.00584	0.219493
$22$	0	0
$23$	2.09095	0.435993	0.217996	−	0.975950i	$-0.430048\pi$
$23$	2.09095	0.435993	0.217996	+	0.975950i	$0.430048\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.57774	−0.303636
$28$	0	0
$29$	1.32071	0.245250	0.122625	−	0.992453i	$-0.460869\pi$
$29$	1.32071	0.245250	0.122625	+	0.992453i	$0.460869\pi$
$30$	0	0
$31$	−3.64219	−0.654156	−0.327078	−	0.944997i	$-0.606064\pi$
$31$	−3.64219	−0.654156	−0.327078	+	0.944997i	$0.606064\pi$
$32$	0	0
$33$	−0.507417	−0.0883300
$34$	0	0
$35$	3.77999	0.638935
$36$	0	0
$37$	−3.78278	−0.621885	−0.310942	−	0.950429i	$-0.600645\pi$
$37$	−3.78278	−0.621885	−0.310942	+	0.950429i	$0.600645\pi$
$38$	0	0
$39$	0.434632	0.0695968
$40$	0	0
$41$	8.68857	1.35693	0.678463	−	0.734634i	$-0.262646\pi$
$41$	8.68857	1.35693	0.678463	+	0.734634i	$0.262646\pi$
$42$	0	0
$43$	2.05249	0.313001	0.156501	−	0.987678i	$-0.449979\pi$
$43$	2.05249	0.313001	0.156501	+	0.987678i	$0.449979\pi$
$44$	0	0
$45$	−2.92919	−0.436658
$46$	0	0
$47$	12.6216	1.84105	0.920527	−	0.390678i	$-0.127760\pi$
$47$	12.6216	1.84105	0.920527	+	0.390678i	$0.127760\pi$
$48$	0	0
$49$	7.28831	1.04119
$50$	0	0
$51$	−0.960746	−0.134531
$52$	0	0
$53$	9.38276	1.28882	0.644411	−	0.764680i	$-0.277103\pi$
$53$	9.38276	1.28882	0.644411	+	0.764680i	$0.277103\pi$
$54$	0	0
$55$	−1.90689	−0.257125
$56$	0	0
$57$	−1.06789	−0.141446
$58$	0	0
$59$	−2.81380	−0.366325	−0.183162	−	0.983083i	$-0.558633\pi$
$59$	−2.81380	−0.366325	−0.183162	+	0.983083i	$0.558633\pi$
$60$	0	0
$61$	13.6738	1.75075	0.875374	−	0.483447i	$-0.160615\pi$
$61$	13.6738	1.75075	0.875374	+	0.483447i	$0.160615\pi$
$62$	0	0
$63$	−11.0723	−1.39498
$64$	0	0
$65$	1.63336	0.202593
$66$	0	0
$67$	15.5162	1.89560	0.947800	−	0.318865i	$-0.103302\pi$
$67$	15.5162	1.89560	0.947800	+	0.318865i	$0.103302\pi$
$68$	0	0
$69$	0.556394	0.0669820
$70$	0	0
$71$	−9.50694	−1.12827	−0.564133	−	0.825684i	$-0.690790\pi$
$71$	−9.50694	−1.12827	−0.564133	+	0.825684i	$0.690790\pi$
$72$	0	0
$73$	1.72533	0.201934	0.100967	−	0.994890i	$-0.467806\pi$
$73$	1.72533	0.201934	0.100967	+	0.994890i	$0.467806\pi$
$74$	0	0
$75$	0.266097	0.0307262
$76$	0	0
$77$	−7.20803	−0.821431
$78$	0	0
$79$	0.651611	0.0733120	0.0366560	−	0.999328i	$-0.488329\pi$
$79$	0.651611	0.0733120	0.0366560	+	0.999328i	$0.488329\pi$
$80$	0	0
$81$	8.36775	0.929750
$82$	0	0
$83$	17.9505	1.97033	0.985163	−	0.171619i	$-0.0548998\pi$
$83$	17.9505	1.97033	0.985163	+	0.171619i	$0.0548998\pi$
$84$	0	0
$85$	−3.61052	−0.391616
$86$	0	0
$87$	0.351437	0.0376780
$88$	0	0
$89$	−0.0643244	−0.00681837	−0.00340918	−	0.999994i	$-0.501085\pi$
$89$	−0.0643244	−0.00681837	−0.00340918	+	0.999994i	$0.501085\pi$
$90$	0	0
$91$	6.17408	0.647220
$92$	0	0
$93$	−0.969174	−0.100499
$94$	0	0
$95$	−4.01317	−0.411743
$96$	0	0
$97$	10.9094	1.10768	0.553842	−	0.832622i	$-0.313161\pi$
$97$	10.9094	1.10768	0.553842	+	0.832622i	$0.313161\pi$
$98$	0	0
$99$	5.58565	0.561379
$100$	0	0
$101$	−3.45876	−0.344159	−0.172080	−	0.985083i	$-0.555049\pi$
$101$	−3.45876	−0.344159	−0.172080	+	0.985083i	$0.555049\pi$
$102$	0	0
$103$	7.05884	0.695528	0.347764	−	0.937582i	$-0.386941\pi$
$103$	7.05884	0.695528	0.347764	+	0.937582i	$0.386941\pi$
$104$	0	0
$105$	1.00584	0.0981601
$106$	0	0
$107$	5.98936	0.579013	0.289507	−	0.957176i	$-0.406509\pi$
$107$	5.98936	0.579013	0.289507	+	0.957176i	$0.406509\pi$
$108$	0	0
$109$	−6.10882	−0.585119	−0.292559	−	0.956247i	$-0.594507\pi$
$109$	−6.10882	−0.585119	−0.292559	+	0.956247i	$0.594507\pi$
$110$	0	0
$111$	−1.00658	−0.0955407
$112$	0	0
$113$	12.2686	1.15413	0.577064	−	0.816699i	$-0.304198\pi$
$113$	12.2686	1.15413	0.577064	+	0.816699i	$0.304198\pi$
$114$	0	0
$115$	2.09095	0.194982
$116$	0	0
$117$	−4.78443	−0.442320
$118$	0	0
$119$	−13.6477	−1.25108
$120$	0	0
$121$	−7.36377	−0.669433
$122$	0	0
$123$	2.31200	0.208466
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.23456	−0.109550	−0.0547748	−	0.998499i	$-0.517444\pi$
$127$	−1.23456	−0.109550	−0.0547748	+	0.998499i	$0.517444\pi$
$128$	0	0
$129$	0.546160	0.0480867
$130$	0	0
$131$	−0.825718	−0.0721433	−0.0360716	−	0.999349i	$-0.511484\pi$
$131$	−0.825718	−0.0721433	−0.0360716	+	0.999349i	$0.511484\pi$
$132$	0	0
$133$	−15.1697	−1.31538
$134$	0	0
$135$	−1.57774	−0.135790
$136$	0	0
$137$	−0.861326	−0.0735881	−0.0367940	−	0.999323i	$-0.511715\pi$
$137$	−0.861326	−0.0735881	−0.0367940	+	0.999323i	$0.511715\pi$
$138$	0	0
$139$	−3.81435	−0.323529	−0.161764	−	0.986829i	$-0.551718\pi$
$139$	−3.81435	−0.323529	−0.161764	+	0.986829i	$0.551718\pi$
$140$	0	0
$141$	3.35857	0.282843
$142$	0	0
$143$	−3.11464	−0.260459
$144$	0	0
$145$	1.32071	0.109679
$146$	0	0
$147$	1.93939	0.159959
$148$	0	0
$149$	−6.21838	−0.509430	−0.254715	−	0.967016i	$-0.581982\pi$
$149$	−6.21838	−0.509430	−0.254715	+	0.967016i	$0.581982\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	10.5759	0.855011
$154$	0	0
$155$	−3.64219	−0.292548
$156$	0	0
$157$	1.29223	0.103131	0.0515657	−	0.998670i	$-0.483579\pi$
$157$	1.29223	0.103131	0.0515657	+	0.998670i	$0.483579\pi$
$158$	0	0
$159$	2.49672	0.198003
$160$	0	0
$161$	7.90376	0.622904
$162$	0	0
$163$	4.62730	0.362438	0.181219	−	0.983443i	$-0.441996\pi$
$163$	4.62730	0.362438	0.181219	+	0.983443i	$0.441996\pi$
$164$	0	0
$165$	−0.507417	−0.0395024
$166$	0	0
$167$	3.50294	0.271066	0.135533	−	0.990773i	$-0.456725\pi$
$167$	3.50294	0.271066	0.135533	+	0.990773i	$0.456725\pi$
$168$	0	0
$169$	−10.3321	−0.794780
$170$	0	0
$171$	11.7554	0.898954
$172$	0	0
$173$	17.0791	1.29850	0.649250	−	0.760575i	$-0.275083\pi$
$173$	17.0791	1.29850	0.649250	+	0.760575i	$0.275083\pi$
$174$	0	0
$175$	3.77999	0.285740
$176$	0	0
$177$	−0.748741	−0.0562788
$178$	0	0
$179$	12.4551	0.930934	0.465467	−	0.885065i	$-0.345886\pi$
$179$	12.4551	0.930934	0.465467	+	0.885065i	$0.345886\pi$
$180$	0	0
$181$	−8.01198	−0.595526	−0.297763	−	0.954640i	$-0.596241\pi$
$181$	−8.01198	−0.595526	−0.297763	+	0.954640i	$0.596241\pi$
$182$	0	0
$183$	3.63854	0.268969
$184$	0	0
$185$	−3.78278	−0.278115
$186$	0	0
$187$	6.88486	0.503471
$188$	0	0
$189$	−5.96383	−0.433805
$190$	0	0
$191$	−13.0659	−0.945417	−0.472709	−	0.881219i	$-0.656724\pi$
$191$	−13.0659	−0.945417	−0.472709	+	0.881219i	$0.656724\pi$
$192$	0	0
$193$	9.24174	0.665235	0.332617	−	0.943062i	$-0.392068\pi$
$193$	9.24174	0.665235	0.332617	+	0.943062i	$0.392068\pi$
$194$	0	0
$195$	0.434632	0.0311246
$196$	0	0
$197$	11.0959	0.790551	0.395275	−	0.918563i	$-0.370649\pi$
$197$	11.0959	0.790551	0.395275	+	0.918563i	$0.370649\pi$
$198$	0	0
$199$	−3.44711	−0.244359	−0.122179	−	0.992508i	$-0.538988\pi$
$199$	−3.44711	−0.244359	−0.122179	+	0.992508i	$0.538988\pi$
$200$	0	0
$201$	4.12880	0.291223
$202$	0	0
$203$	4.99227	0.350389
$204$	0	0
$205$	8.68857	0.606836
$206$	0	0
$207$	−6.12479	−0.425702
$208$	0	0
$209$	7.65268	0.529347
$210$	0	0
$211$	0.425567	0.0292973	0.0146486	−	0.999893i	$-0.495337\pi$
$211$	0.425567	0.0292973	0.0146486	+	0.999893i	$0.495337\pi$
$212$	0	0
$213$	−2.52976	−0.173337
$214$	0	0
$215$	2.05249	0.139979
$216$	0	0
$217$	−13.7674	−0.934594
$218$	0	0
$219$	0.459104	0.0310233
$220$	0	0
$221$	−5.89728	−0.396694
$222$	0	0
$223$	−13.1234	−0.878808	−0.439404	−	0.898290i	$-0.644810\pi$
$223$	−13.1234	−0.878808	−0.439404	+	0.898290i	$0.644810\pi$
$224$	0	0
$225$	−2.92919	−0.195280
$226$	0	0
$227$	−16.0694	−1.06657	−0.533283	−	0.845937i	$-0.679042\pi$
$227$	−16.0694	−1.06657	−0.533283	+	0.845937i	$0.679042\pi$
$228$	0	0
$229$	4.78343	0.316098	0.158049	−	0.987431i	$-0.449480\pi$
$229$	4.78343	0.316098	0.158049	+	0.987431i	$0.449480\pi$
$230$	0	0
$231$	−1.91803	−0.126197
$232$	0	0
$233$	3.88853	0.254746	0.127373	−	0.991855i	$-0.459345\pi$
$233$	3.88853	0.254746	0.127373	+	0.991855i	$0.459345\pi$
$234$	0	0
$235$	12.6216	0.823345
$236$	0	0
$237$	0.173392	0.0112630
$238$	0	0
$239$	0.990378	0.0640622	0.0320311	−	0.999487i	$-0.489802\pi$
$239$	0.990378	0.0640622	0.0320311	+	0.999487i	$0.489802\pi$
$240$	0	0
$241$	−24.8572	−1.60120	−0.800598	−	0.599202i	$-0.795485\pi$
$241$	−24.8572	−1.60120	−0.800598	+	0.599202i	$0.795485\pi$
$242$	0	0
$243$	6.95984	0.446474
$244$	0	0
$245$	7.28831	0.465633
$246$	0	0
$247$	−6.55495	−0.417082
$248$	0	0
$249$	4.77657	0.302703
$250$	0	0
$251$	19.9719	1.26062	0.630308	−	0.776346i	$-0.282929\pi$
$251$	19.9719	1.26062	0.630308	+	0.776346i	$0.282929\pi$
$252$	0	0
$253$	−3.98721	−0.250674
$254$	0	0
$255$	−0.960746	−0.0601643
$256$	0	0
$257$	−19.5249	−1.21793	−0.608965	−	0.793197i	$-0.708415\pi$
$257$	−19.5249	−1.21793	−0.608965	+	0.793197i	$0.708415\pi$
$258$	0	0
$259$	−14.2989	−0.888487
$260$	0	0
$261$	−3.86862	−0.239461
$262$	0	0
$263$	13.3523	0.823337	0.411668	−	0.911334i	$-0.364946\pi$
$263$	13.3523	0.823337	0.411668	+	0.911334i	$0.364946\pi$
$264$	0	0
$265$	9.38276	0.576378
$266$	0	0
$267$	−0.0171165	−0.00104751
$268$	0	0
$269$	−29.9153	−1.82397	−0.911984	−	0.410225i	$-0.865450\pi$
$269$	−29.9153	−1.82397	−0.911984	+	0.410225i	$0.865450\pi$
$270$	0	0
$271$	14.7765	0.897607	0.448803	−	0.893631i	$-0.351850\pi$
$271$	14.7765	0.897607	0.448803	+	0.893631i	$0.351850\pi$
$272$	0	0
$273$	1.64290	0.0994330
$274$	0	0
$275$	−1.90689	−0.114990
$276$	0	0
$277$	−30.8073	−1.85103	−0.925514	−	0.378712i	$-0.876367\pi$
$277$	−30.8073	−1.85103	−0.925514	+	0.378712i	$0.876367\pi$
$278$	0	0
$279$	10.6687	0.638717
$280$	0	0
$281$	−19.5462	−1.16603	−0.583015	−	0.812461i	$-0.698127\pi$
$281$	−19.5462	−1.16603	−0.583015	+	0.812461i	$0.698127\pi$
$282$	0	0
$283$	28.3931	1.68779	0.843897	−	0.536505i	$-0.180256\pi$
$283$	28.3931	1.68779	0.843897	+	0.536505i	$0.180256\pi$
$284$	0	0
$285$	−1.06789	−0.0632564
$286$	0	0
$287$	32.8427	1.93864
$288$	0	0
$289$	−3.96417	−0.233186
$290$	0	0
$291$	2.90296	0.170175
$292$	0	0
$293$	−1.76941	−0.103370	−0.0516851	−	0.998663i	$-0.516459\pi$
$293$	−1.76941	−0.103370	−0.0516851	+	0.998663i	$0.516459\pi$
$294$	0	0
$295$	−2.81380	−0.163826
$296$	0	0
$297$	3.00857	0.174575
$298$	0	0
$299$	3.41527	0.197510
$300$	0	0
$301$	7.75838	0.447186
$302$	0	0
$303$	−0.920364	−0.0528735
$304$	0	0
$305$	13.6738	0.782958
$306$	0	0
$307$	−14.9482	−0.853141	−0.426570	−	0.904454i	$-0.640278\pi$
$307$	−14.9482	−0.853141	−0.426570	+	0.904454i	$0.640278\pi$
$308$	0	0
$309$	1.87833	0.106855
$310$	0	0
$311$	14.4295	0.818224	0.409112	−	0.912484i	$-0.365838\pi$
$311$	14.4295	0.818224	0.409112	+	0.912484i	$0.365838\pi$
$312$	0	0
$313$	−31.2138	−1.76431	−0.882155	−	0.470959i	$-0.843908\pi$
$313$	−31.2138	−1.76431	−0.882155	+	0.470959i	$0.843908\pi$
$314$	0	0
$315$	−11.0723	−0.623854
$316$	0	0
$317$	−18.4510	−1.03631	−0.518156	−	0.855286i	$-0.673381\pi$
$317$	−18.4510	−1.03631	−0.518156	+	0.855286i	$0.673381\pi$
$318$	0	0
$319$	−2.51845	−0.141006
$320$	0	0
$321$	1.59375	0.0889543
$322$	0	0
$323$	14.4896	0.806224
$324$	0	0
$325$	1.63336	0.0906025
$326$	0	0
$327$	−1.62554	−0.0898923
$328$	0	0
$329$	47.7096	2.63032
$330$	0	0
$331$	−19.6346	−1.07922	−0.539608	−	0.841916i	$-0.681428\pi$
$331$	−19.6346	−1.07922	−0.539608	+	0.841916i	$0.681428\pi$
$332$	0	0
$333$	11.0805	0.607207
$334$	0	0
$335$	15.5162	0.847738
$336$	0	0
$337$	1.01618	0.0553546	0.0276773	−	0.999617i	$-0.491189\pi$
$337$	1.01618	0.0553546	0.0276773	+	0.999617i	$0.491189\pi$
$338$	0	0
$339$	3.26462	0.177310
$340$	0	0
$341$	6.94526	0.376107
$342$	0	0
$343$	1.08980	0.0588438
$344$	0	0
$345$	0.556394	0.0299553
$346$	0	0
$347$	2.78213	0.149353	0.0746763	−	0.997208i	$-0.476208\pi$
$347$	2.78213	0.149353	0.0746763	+	0.997208i	$0.476208\pi$
$348$	0	0
$349$	−23.6176	−1.26422	−0.632109	−	0.774879i	$-0.717811\pi$
$349$	−23.6176	−1.26422	−0.632109	+	0.774879i	$0.717811\pi$
$350$	0	0
$351$	−2.57701	−0.137551
$352$	0	0
$353$	18.1588	0.966498	0.483249	−	0.875483i	$-0.339457\pi$
$353$	18.1588	0.966498	0.483249	+	0.875483i	$0.339457\pi$
$354$	0	0
$355$	−9.50694	−0.504576
$356$	0	0
$357$	−3.63161	−0.192205
$358$	0	0
$359$	30.3794	1.60337	0.801683	−	0.597750i	$-0.203938\pi$
$359$	30.3794	1.60337	0.801683	+	0.597750i	$0.203938\pi$
$360$	0	0
$361$	−2.89446	−0.152340
$362$	0	0
$363$	−1.95947	−0.102846
$364$	0	0
$365$	1.72533	0.0903077
$366$	0	0
$367$	−7.87211	−0.410921	−0.205460	−	0.978665i	$-0.565869\pi$
$367$	−7.87211	−0.410921	−0.205460	+	0.978665i	$0.565869\pi$
$368$	0	0
$369$	−25.4505	−1.32490
$370$	0	0
$371$	35.4667	1.84134
$372$	0	0
$373$	−3.85062	−0.199378	−0.0996889	−	0.995019i	$-0.531785\pi$
$373$	−3.85062	−0.199378	−0.0996889	+	0.995019i	$0.531785\pi$
$374$	0	0
$375$	0.266097	0.0137412
$376$	0	0
$377$	2.15720	0.111101
$378$	0	0
$379$	9.34202	0.479868	0.239934	−	0.970789i	$-0.422874\pi$
$379$	9.34202	0.479868	0.239934	+	0.970789i	$0.422874\pi$
$380$	0	0
$381$	−0.328513	−0.0168302
$382$	0	0
$383$	37.0281	1.89205	0.946024	−	0.324098i	$-0.105061\pi$
$383$	37.0281	1.89205	0.946024	+	0.324098i	$0.105061\pi$
$384$	0	0
$385$	−7.20803	−0.367355
$386$	0	0
$387$	−6.01213	−0.305614
$388$	0	0
$389$	10.1957	0.516940	0.258470	−	0.966019i	$-0.416782\pi$
$389$	10.1957	0.516940	0.258470	+	0.966019i	$0.416782\pi$
$390$	0	0
$391$	−7.54941	−0.381790
$392$	0	0
$393$	−0.219721	−0.0110834
$394$	0	0
$395$	0.651611	0.0327861
$396$	0	0
$397$	−7.24768	−0.363751	−0.181875	−	0.983322i	$-0.558217\pi$
$397$	−7.24768	−0.363751	−0.181875	+	0.983322i	$0.558217\pi$
$398$	0	0
$399$	−4.03662	−0.202084
$400$	0	0
$401$	33.5252	1.67417	0.837084	−	0.547074i	$-0.184258\pi$
$401$	33.5252	1.67417	0.837084	+	0.547074i	$0.184258\pi$
$402$	0	0
$403$	−5.94901	−0.296341
$404$	0	0
$405$	8.36775	0.415797
$406$	0	0
$407$	7.21334	0.357552
$408$	0	0
$409$	−15.6954	−0.776088	−0.388044	−	0.921641i	$-0.626849\pi$
$409$	−15.6954	−0.776088	−0.388044	+	0.921641i	$0.626849\pi$
$410$	0	0
$411$	−0.229196	−0.0113054
$412$	0	0
$413$	−10.6361	−0.523369
$414$	0	0
$415$	17.9505	0.881157
$416$	0	0
$417$	−1.01498	−0.0497040
$418$	0	0
$419$	9.79302	0.478420	0.239210	−	0.970968i	$-0.423112\pi$
$419$	9.79302	0.478420	0.239210	+	0.970968i	$0.423112\pi$
$420$	0	0
$421$	24.9090	1.21399	0.606995	−	0.794705i	$-0.292375\pi$
$421$	24.9090	1.21399	0.606995	+	0.794705i	$0.292375\pi$
$422$	0	0
$423$	−36.9712	−1.79760
$424$	0	0
$425$	−3.61052	−0.175136
$426$	0	0
$427$	51.6867	2.50129
$428$	0	0
$429$	−0.828795	−0.0400146
$430$	0	0
$431$	−9.79873	−0.471988	−0.235994	−	0.971754i	$-0.575835\pi$
$431$	−9.79873	−0.471988	−0.235994	+	0.971754i	$0.575835\pi$
$432$	0	0
$433$	36.8112	1.76903	0.884516	−	0.466511i	$-0.154489\pi$
$433$	36.8112	1.76903	0.884516	+	0.466511i	$0.154489\pi$
$434$	0	0
$435$	0.351437	0.0168501
$436$	0	0
$437$	−8.39133	−0.401412
$438$	0	0
$439$	−0.171822	−0.00820061	−0.00410031	−	0.999992i	$-0.501305\pi$
$439$	−0.171822	−0.00820061	−0.00410031	+	0.999992i	$0.501305\pi$
$440$	0	0
$441$	−21.3489	−1.01661
$442$	0	0
$443$	−31.9507	−1.51802	−0.759012	−	0.651077i	$-0.774318\pi$
$443$	−31.9507	−1.51802	−0.759012	+	0.651077i	$0.774318\pi$
$444$	0	0
$445$	−0.0643244	−0.00304927
$446$	0	0
$447$	−1.65469	−0.0782641
$448$	0	0
$449$	−2.96068	−0.139723	−0.0698616	−	0.997557i	$-0.522256\pi$
$449$	−2.96068	−0.139723	−0.0698616	+	0.997557i	$0.522256\pi$
$450$	0	0
$451$	−16.5682	−0.780164
$452$	0	0
$453$	−0.266097	−0.0125023
$454$	0	0
$455$	6.17408	0.289445
$456$	0	0
$457$	15.4719	0.723743	0.361871	−	0.932228i	$-0.382138\pi$
$457$	15.4719	0.723743	0.361871	+	0.932228i	$0.382138\pi$
$458$	0	0
$459$	5.69645	0.265887
$460$	0	0
$461$	−6.89748	−0.321248	−0.160624	−	0.987016i	$-0.551351\pi$
$461$	−6.89748	−0.321248	−0.160624	+	0.987016i	$0.551351\pi$
$462$	0	0
$463$	−16.5233	−0.767903	−0.383951	−	0.923353i	$-0.625437\pi$
$463$	−16.5233	−0.767903	−0.383951	+	0.923353i	$0.625437\pi$
$464$	0	0
$465$	−0.969174	−0.0449444
$466$	0	0
$467$	10.3972	0.481127	0.240563	−	0.970633i	$-0.422668\pi$
$467$	10.3972	0.481127	0.240563	+	0.970633i	$0.422668\pi$
$468$	0	0
$469$	58.6509	2.70825
$470$	0	0
$471$	0.343859	0.0158442
$472$	0	0
$473$	−3.91387	−0.179960
$474$	0	0
$475$	−4.01317	−0.184137
$476$	0	0
$477$	−27.4839	−1.25840
$478$	0	0
$479$	0.178995	0.00817848	0.00408924	−	0.999992i	$-0.498698\pi$
$479$	0.178995	0.00817848	0.00408924	+	0.999992i	$0.498698\pi$
$480$	0	0
$481$	−6.17864	−0.281722
$482$	0	0
$483$	2.10316	0.0956973
$484$	0	0
$485$	10.9094	0.495372
$486$	0	0
$487$	11.6922	0.529822	0.264911	−	0.964273i	$-0.414657\pi$
$487$	11.6922	0.529822	0.264911	+	0.964273i	$0.414657\pi$
$488$	0	0
$489$	1.23131	0.0556817
$490$	0	0
$491$	33.3098	1.50325	0.751624	−	0.659591i	$-0.229271\pi$
$491$	33.3098	1.50325	0.751624	+	0.659591i	$0.229271\pi$
$492$	0	0
$493$	−4.76845	−0.214760
$494$	0	0
$495$	5.58565	0.251056
$496$	0	0
$497$	−35.9361	−1.61196
$498$	0	0
$499$	−10.7113	−0.479502	−0.239751	−	0.970834i	$-0.577066\pi$
$499$	−10.7113	−0.479502	−0.239751	+	0.970834i	$0.577066\pi$
$500$	0	0
$501$	0.932120	0.0416440
$502$	0	0
$503$	−42.3150	−1.88673	−0.943367	−	0.331752i	$-0.892360\pi$
$503$	−42.3150	−1.88673	−0.943367	+	0.331752i	$0.892360\pi$
$504$	0	0
$505$	−3.45876	−0.153913
$506$	0	0
$507$	−2.74935	−0.122103
$508$	0	0
$509$	3.80484	0.168647	0.0843233	−	0.996438i	$-0.473127\pi$
$509$	3.80484	0.168647	0.0843233	+	0.996438i	$0.473127\pi$
$510$	0	0
$511$	6.52171	0.288504
$512$	0	0
$513$	6.33173	0.279553
$514$	0	0
$515$	7.05884	0.311049
$516$	0	0
$517$	−24.0681	−1.05851
$518$	0	0
$519$	4.54469	0.199490
$520$	0	0
$521$	14.5614	0.637946	0.318973	−	0.947764i	$-0.396662\pi$
$521$	14.5614	0.637946	0.318973	+	0.947764i	$0.396662\pi$
$522$	0	0
$523$	27.3218	1.19470	0.597348	−	0.801982i	$-0.296221\pi$
$523$	27.3218	1.19470	0.597348	+	0.801982i	$0.296221\pi$
$524$	0	0
$525$	1.00584	0.0438985
$526$	0	0
$527$	13.1502	0.572831
$528$	0	0
$529$	−18.6279	−0.809910
$530$	0	0
$531$	8.24215	0.357679
$532$	0	0
$533$	14.1916	0.614705
$534$	0	0
$535$	5.98936	0.258943
$536$	0	0
$537$	3.31425	0.143020
$538$	0	0
$539$	−13.8980	−0.598630
$540$	0	0
$541$	2.01130	0.0864724	0.0432362	−	0.999065i	$-0.486233\pi$
$541$	2.01130	0.0864724	0.0432362	+	0.999065i	$0.486233\pi$
$542$	0	0
$543$	−2.13196	−0.0914912
$544$	0	0
$545$	−6.10882	−0.261673
$546$	0	0
$547$	−9.92779	−0.424482	−0.212241	−	0.977217i	$-0.568076\pi$
$547$	−9.92779	−0.424482	−0.212241	+	0.977217i	$0.568076\pi$
$548$	0	0
$549$	−40.0531	−1.70943
$550$	0	0
$551$	−5.30024	−0.225798
$552$	0	0
$553$	2.46308	0.104741
$554$	0	0
$555$	−1.00658	−0.0427271
$556$	0	0
$557$	16.9294	0.717320	0.358660	−	0.933468i	$-0.383234\pi$
$557$	16.9294	0.717320	0.358660	+	0.933468i	$0.383234\pi$
$558$	0	0
$559$	3.35245	0.141794
$560$	0	0
$561$	1.83204	0.0773487
$562$	0	0
$563$	40.7329	1.71669	0.858344	−	0.513074i	$-0.171494\pi$
$563$	40.7329	1.71669	0.858344	+	0.513074i	$0.171494\pi$
$564$	0	0
$565$	12.2686	0.516142
$566$	0	0
$567$	31.6300	1.32833
$568$	0	0
$569$	−28.2648	−1.18492	−0.592460	−	0.805600i	$-0.701843\pi$
$569$	−28.2648	−1.18492	−0.592460	+	0.805600i	$0.701843\pi$
$570$	0	0
$571$	−3.14818	−0.131747	−0.0658737	−	0.997828i	$-0.520983\pi$
$571$	−3.14818	−0.131747	−0.0658737	+	0.997828i	$0.520983\pi$
$572$	0	0
$573$	−3.47680	−0.145245
$574$	0	0
$575$	2.09095	0.0871986
$576$	0	0
$577$	−31.6967	−1.31955	−0.659776	−	0.751462i	$-0.729349\pi$
$577$	−31.6967	−1.31955	−0.659776	+	0.751462i	$0.729349\pi$
$578$	0	0
$579$	2.45919	0.102201
$580$	0	0
$581$	67.8528	2.81501
$582$	0	0
$583$	−17.8919	−0.741007
$584$	0	0
$585$	−4.78443	−0.197812
$586$	0	0
$587$	32.3600	1.33564	0.667820	−	0.744323i	$-0.267228\pi$
$587$	32.3600	1.33564	0.667820	+	0.744323i	$0.267228\pi$
$588$	0	0
$589$	14.6167	0.602272
$590$	0	0
$591$	2.95258	0.121453
$592$	0	0
$593$	−16.0351	−0.658481	−0.329241	−	0.944246i	$-0.606793\pi$
$593$	−16.0351	−0.658481	−0.329241	+	0.944246i	$0.606793\pi$
$594$	0	0
$595$	−13.6477	−0.559502
$596$	0	0
$597$	−0.917263	−0.0375411
$598$	0	0
$599$	7.50350	0.306585	0.153292	−	0.988181i	$-0.451012\pi$
$599$	7.50350	0.306585	0.153292	+	0.988181i	$0.451012\pi$
$600$	0	0
$601$	16.8425	0.687020	0.343510	−	0.939149i	$-0.388384\pi$
$601$	16.8425	0.687020	0.343510	+	0.939149i	$0.388384\pi$
$602$	0	0
$603$	−45.4498	−1.85086
$604$	0	0
$605$	−7.36377	−0.299380
$606$	0	0
$607$	−1.83111	−0.0743225	−0.0371613	−	0.999309i	$-0.511832\pi$
$607$	−1.83111	−0.0743225	−0.0371613	+	0.999309i	$0.511832\pi$
$608$	0	0
$609$	1.32843	0.0538306
$610$	0	0
$611$	20.6157	0.834021
$612$	0	0
$613$	−44.4413	−1.79497	−0.897483	−	0.441050i	$-0.854606\pi$
$613$	−44.4413	−1.79497	−0.897483	+	0.441050i	$0.854606\pi$
$614$	0	0
$615$	2.31200	0.0932288
$616$	0	0
$617$	−31.6788	−1.27534	−0.637671	−	0.770309i	$-0.720102\pi$
$617$	−31.6788	−1.27534	−0.637671	+	0.770309i	$0.720102\pi$
$618$	0	0
$619$	5.54970	0.223061	0.111531	−	0.993761i	$-0.464425\pi$
$619$	5.54970	0.223061	0.111531	+	0.993761i	$0.464425\pi$
$620$	0	0
$621$	−3.29897	−0.132383
$622$	0	0
$623$	−0.243145	−0.00974141
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.03635	0.0813241
$628$	0	0
$629$	13.6578	0.544571
$630$	0	0
$631$	29.4363	1.17184	0.585920	−	0.810369i	$-0.300733\pi$
$631$	29.4363	1.17184	0.585920	+	0.810369i	$0.300733\pi$
$632$	0	0
$633$	0.113242	0.00450097
$634$	0	0
$635$	−1.23456	−0.0489921
$636$	0	0
$637$	11.9044	0.471671
$638$	0	0
$639$	27.8477	1.10164
$640$	0	0
$641$	11.1579	0.440711	0.220356	−	0.975420i	$-0.429278\pi$
$641$	11.1579	0.440711	0.220356	+	0.975420i	$0.429278\pi$
$642$	0	0
$643$	19.1586	0.755540	0.377770	−	0.925899i	$-0.376691\pi$
$643$	19.1586	0.755540	0.377770	+	0.925899i	$0.376691\pi$
$644$	0	0
$645$	0.546160	0.0215050
$646$	0	0
$647$	−21.1657	−0.832110	−0.416055	−	0.909340i	$-0.636588\pi$
$647$	−21.1657	−0.832110	−0.416055	+	0.909340i	$0.636588\pi$
$648$	0	0
$649$	5.36560	0.210618
$650$	0	0
$651$	−3.66346	−0.143583
$652$	0	0
$653$	−18.3004	−0.716150	−0.358075	−	0.933693i	$-0.616567\pi$
$653$	−18.3004	−0.716150	−0.358075	+	0.933693i	$0.616567\pi$
$654$	0	0
$655$	−0.825718	−0.0322635
$656$	0	0
$657$	−5.05381	−0.197168
$658$	0	0
$659$	19.7504	0.769367	0.384683	−	0.923049i	$-0.374311\pi$
$659$	19.7504	0.769367	0.384683	+	0.923049i	$0.374311\pi$
$660$	0	0
$661$	30.2579	1.17690	0.588448	−	0.808535i	$-0.299739\pi$
$661$	30.2579	1.17690	0.588448	+	0.808535i	$0.299739\pi$
$662$	0	0
$663$	−1.56924	−0.0609444
$664$	0	0
$665$	−15.1697	−0.588257
$666$	0	0
$667$	2.76154	0.106927
$668$	0	0
$669$	−3.49209	−0.135012
$670$	0	0
$671$	−26.0744	−1.00659
$672$	0	0
$673$	6.61931	0.255156	0.127578	−	0.991829i	$-0.459280\pi$
$673$	6.61931	0.255156	0.127578	+	0.991829i	$0.459280\pi$
$674$	0	0
$675$	−1.57774	−0.0607272
$676$	0	0
$677$	13.3242	0.512090	0.256045	−	0.966665i	$-0.417581\pi$
$677$	13.3242	0.512090	0.256045	+	0.966665i	$0.417581\pi$
$678$	0	0
$679$	41.2375	1.58255
$680$	0	0
$681$	−4.27602	−0.163857
$682$	0	0
$683$	−0.607531	−0.0232465	−0.0116233	−	0.999932i	$-0.503700\pi$
$683$	−0.607531	−0.0232465	−0.0116233	+	0.999932i	$0.503700\pi$
$684$	0	0
$685$	−0.861326	−0.0329096
$686$	0	0
$687$	1.27285	0.0485624
$688$	0	0
$689$	15.3254	0.583852
$690$	0	0
$691$	4.66452	0.177447	0.0887234	−	0.996056i	$-0.471721\pi$
$691$	4.66452	0.177447	0.0887234	+	0.996056i	$0.471721\pi$
$692$	0	0
$693$	21.1137	0.802043
$694$	0	0
$695$	−3.81435	−0.144686
$696$	0	0
$697$	−31.3702	−1.18823
$698$	0	0
$699$	1.03473	0.0391369
$700$	0	0
$701$	−7.14827	−0.269986	−0.134993	−	0.990847i	$-0.543101\pi$
$701$	−7.14827	−0.269986	−0.134993	+	0.990847i	$0.543101\pi$
$702$	0	0
$703$	15.1809	0.572560
$704$	0	0
$705$	3.35857	0.126491
$706$	0	0
$707$	−13.0741	−0.491701
$708$	0	0
$709$	40.8733	1.53503	0.767515	−	0.641031i	$-0.221493\pi$
$709$	40.8733	1.53503	0.767515	+	0.641031i	$0.221493\pi$
$710$	0	0
$711$	−1.90870	−0.0715817
$712$	0	0
$713$	−7.61563	−0.285208
$714$	0	0
$715$	−3.11464	−0.116481
$716$	0	0
$717$	0.263536	0.00984194
$718$	0	0
$719$	−37.6221	−1.40307	−0.701535	−	0.712635i	$-0.747501\pi$
$719$	−37.6221	−1.40307	−0.701535	+	0.712635i	$0.747501\pi$
$720$	0	0
$721$	26.6823	0.993701
$722$	0	0
$723$	−6.61443	−0.245993
$724$	0	0
$725$	1.32071	0.0490500
$726$	0	0
$727$	0.000527053 0	1.95473e−5 0	9.77366e−6	−	1.00000i	$-0.499997\pi$
$727$	0.000527053 0	1.95473e−5 0	9.77366e−6		1.00000i	$0.499997\pi$
$728$	0	0
$729$	−23.2513	−0.861157
$730$	0	0
$731$	−7.41054	−0.274089
$732$	0	0
$733$	29.1385	1.07625	0.538127	−	0.842864i	$-0.319132\pi$
$733$	29.1385	1.07625	0.538127	+	0.842864i	$0.319132\pi$
$734$	0	0
$735$	1.93939	0.0715356
$736$	0	0
$737$	−29.5876	−1.08987
$738$	0	0
$739$	−12.1517	−0.447008	−0.223504	−	0.974703i	$-0.571750\pi$
$739$	−12.1517	−0.447008	−0.223504	+	0.974703i	$0.571750\pi$
$740$	0	0
$741$	−1.74425	−0.0640767
$742$	0	0
$743$	−1.88268	−0.0690690	−0.0345345	−	0.999404i	$-0.510995\pi$
$743$	−1.88268	−0.0690690	−0.0345345	+	0.999404i	$0.510995\pi$
$744$	0	0
$745$	−6.21838	−0.227824
$746$	0	0
$747$	−52.5806	−1.92382
$748$	0	0
$749$	22.6397	0.827237
$750$	0	0
$751$	18.1147	0.661015	0.330508	−	0.943803i	$-0.392780\pi$
$751$	18.1147	0.661015	0.330508	+	0.943803i	$0.392780\pi$
$752$	0	0
$753$	5.31445	0.193669
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	38.2742	1.39110	0.695550	−	0.718478i	$-0.255161\pi$
$757$	38.2742	1.39110	0.695550	+	0.718478i	$0.255161\pi$
$758$	0	0
$759$	−1.06098	−0.0385113
$760$	0	0
$761$	−22.0654	−0.799869	−0.399935	−	0.916544i	$-0.630967\pi$
$761$	−22.0654	−0.799869	−0.399935	+	0.916544i	$0.630967\pi$
$762$	0	0
$763$	−23.0913	−0.835960
$764$	0	0
$765$	10.5759	0.382372
$766$	0	0
$767$	−4.59594	−0.165950
$768$	0	0
$769$	−26.9167	−0.970642	−0.485321	−	0.874336i	$-0.661297\pi$
$769$	−26.9167	−0.970642	−0.485321	+	0.874336i	$0.661297\pi$
$770$	0	0
$771$	−5.19551	−0.187112
$772$	0	0
$773$	−45.1737	−1.62479	−0.812393	−	0.583110i	$-0.801836\pi$
$773$	−45.1737	−1.62479	−0.812393	+	0.583110i	$0.801836\pi$
$774$	0	0
$775$	−3.64219	−0.130831
$776$	0	0
$777$	−3.80488	−0.136499
$778$	0	0
$779$	−34.8687	−1.24930
$780$	0	0
$781$	18.1287	0.648696
$782$	0	0
$783$	−2.08374	−0.0744667
$784$	0	0
$785$	1.29223	0.0461217
$786$	0	0
$787$	47.5719	1.69576	0.847878	−	0.530192i	$-0.177880\pi$
$787$	47.5719	1.69576	0.847878	+	0.530192i	$0.177880\pi$
$788$	0	0
$789$	3.55299	0.126490
$790$	0	0
$791$	46.3750	1.64890
$792$	0	0
$793$	22.3342	0.793111
$794$	0	0
$795$	2.49672	0.0885496
$796$	0	0
$797$	37.4057	1.32498	0.662490	−	0.749071i	$-0.269500\pi$
$797$	37.4057	1.32498	0.662490	+	0.749071i	$0.269500\pi$
$798$	0	0
$799$	−45.5706	−1.61217
$800$	0	0
$801$	0.188418	0.00665744
$802$	0	0
$803$	−3.29001	−0.116102
$804$	0	0
$805$	7.90376	0.278571
$806$	0	0
$807$	−7.96036	−0.280218
$808$	0	0
$809$	−3.19399	−0.112295	−0.0561473	−	0.998422i	$-0.517882\pi$
$809$	−3.19399	−0.112295	−0.0561473	+	0.998422i	$0.517882\pi$
$810$	0	0
$811$	16.7239	0.587256	0.293628	−	0.955920i	$-0.405137\pi$
$811$	16.7239	0.587256	0.293628	+	0.955920i	$0.405137\pi$
$812$	0	0
$813$	3.93197	0.137900
$814$	0	0
$815$	4.62730	0.162087
$816$	0	0
$817$	−8.23698	−0.288176
$818$	0	0
$819$	−18.0851	−0.631944
$820$	0	0
$821$	−11.9762	−0.417974	−0.208987	−	0.977918i	$-0.567017\pi$
$821$	−11.9762	−0.417974	−0.208987	+	0.977918i	$0.567017\pi$
$822$	0	0
$823$	11.1237	0.387747	0.193874	−	0.981026i	$-0.437895\pi$
$823$	11.1237	0.387747	0.193874	+	0.981026i	$0.437895\pi$
$824$	0	0
$825$	−0.507417	−0.0176660
$826$	0	0
$827$	−3.39504	−0.118057	−0.0590285	−	0.998256i	$-0.518800\pi$
$827$	−3.39504	−0.118057	−0.0590285	+	0.998256i	$0.518800\pi$
$828$	0	0
$829$	−44.9685	−1.56182	−0.780911	−	0.624642i	$-0.785245\pi$
$829$	−44.9685	−1.56182	−0.780911	+	0.624642i	$0.785245\pi$
$830$	0	0
$831$	−8.19771	−0.284375
$832$	0	0
$833$	−26.3146	−0.911745
$834$	0	0
$835$	3.50294	0.121224
$836$	0	0
$837$	5.74642	0.198625
$838$	0	0
$839$	2.49041	0.0859784	0.0429892	−	0.999076i	$-0.486312\pi$
$839$	2.49041	0.0859784	0.0429892	+	0.999076i	$0.486312\pi$
$840$	0	0
$841$	−27.2557	−0.939852
$842$	0	0
$843$	−5.20118	−0.179138
$844$	0	0
$845$	−10.3321	−0.355436
$846$	0	0
$847$	−27.8349	−0.956420
$848$	0	0
$849$	7.55531	0.259297
$850$	0	0
$851$	−7.90959	−0.271137
$852$	0	0
$853$	10.1000	0.345816	0.172908	−	0.984938i	$-0.444684\pi$
$853$	10.1000	0.345816	0.172908	+	0.984938i	$0.444684\pi$
$854$	0	0
$855$	11.7554	0.402025
$856$	0	0
$857$	−53.0014	−1.81049	−0.905247	−	0.424886i	$-0.860314\pi$
$857$	−53.0014	−1.81049	−0.905247	+	0.424886i	$0.860314\pi$
$858$	0	0
$859$	−40.7923	−1.39181	−0.695907	−	0.718132i	$-0.744997\pi$
$859$	−40.7923	−1.39181	−0.695907	+	0.718132i	$0.744997\pi$
$860$	0	0
$861$	8.73933	0.297835
$862$	0	0
$863$	−11.4806	−0.390803	−0.195401	−	0.980723i	$-0.562601\pi$
$863$	−11.4806	−0.390803	−0.195401	+	0.980723i	$0.562601\pi$
$864$	0	0
$865$	17.0791	0.580707
$866$	0	0
$867$	−1.05485	−0.0358246
$868$	0	0
$869$	−1.24255	−0.0421507
$870$	0	0
$871$	25.3435	0.858731
$872$	0	0
$873$	−31.9558	−1.08154
$874$	0	0
$875$	3.77999	0.127787
$876$	0	0
$877$	30.3140	1.02363	0.511815	−	0.859096i	$-0.328973\pi$
$877$	30.3140	1.02363	0.511815	+	0.859096i	$0.328973\pi$
$878$	0	0
$879$	−0.470834	−0.0158808
$880$	0	0
$881$	22.3639	0.753460	0.376730	−	0.926323i	$-0.377049\pi$
$881$	22.3639	0.753460	0.376730	+	0.926323i	$0.377049\pi$
$882$	0	0
$883$	−50.2844	−1.69220	−0.846102	−	0.533021i	$-0.821057\pi$
$883$	−50.2844	−1.69220	−0.846102	+	0.533021i	$0.821057\pi$
$884$	0	0
$885$	−0.748741	−0.0251687
$886$	0	0
$887$	−42.1347	−1.41474	−0.707372	−	0.706842i	$-0.750119\pi$
$887$	−42.1347	−1.41474	−0.707372	+	0.706842i	$0.750119\pi$
$888$	0	0
$889$	−4.66663	−0.156514
$890$	0	0
$891$	−15.9564	−0.534559
$892$	0	0
$893$	−50.6528	−1.69503
$894$	0	0
$895$	12.4551	0.416326
$896$	0	0
$897$	0.908792	0.0303437
$898$	0	0
$899$	−4.81028	−0.160432
$900$	0	0
$901$	−33.8766	−1.12859
$902$	0	0
$903$	2.06448	0.0687015
$904$	0	0
$905$	−8.01198	−0.266327
$906$	0	0
$907$	−35.0912	−1.16519	−0.582593	−	0.812764i	$-0.697962\pi$
$907$	−35.0912	−1.16519	−0.582593	+	0.812764i	$0.697962\pi$
$908$	0	0
$909$	10.1314	0.336036
$910$	0	0
$911$	43.6691	1.44682	0.723411	−	0.690417i	$-0.242573\pi$
$911$	43.6691	1.44682	0.723411	+	0.690417i	$0.242573\pi$
$912$	0	0
$913$	−34.2297	−1.13284
$914$	0	0
$915$	3.63854	0.120287
$916$	0	0
$917$	−3.12120	−0.103071
$918$	0	0
$919$	−44.3867	−1.46418	−0.732092	−	0.681206i	$-0.761456\pi$
$919$	−44.3867	−1.46418	−0.732092	+	0.681206i	$0.761456\pi$
$920$	0	0
$921$	−3.97767	−0.131069
$922$	0	0
$923$	−15.5283	−0.511119
$924$	0	0
$925$	−3.78278	−0.124377
$926$	0	0
$927$	−20.6767	−0.679112
$928$	0	0
$929$	30.3128	0.994532	0.497266	−	0.867598i	$-0.334337\pi$
$929$	30.3128	0.994532	0.497266	+	0.867598i	$0.334337\pi$
$930$	0	0
$931$	−29.2492	−0.958605
$932$	0	0
$933$	3.83965	0.125705
$934$	0	0
$935$	6.88486	0.225159
$936$	0	0
$937$	16.0419	0.524064	0.262032	−	0.965059i	$-0.415607\pi$
$937$	16.0419	0.524064	0.262032	+	0.965059i	$0.415607\pi$
$938$	0	0
$939$	−8.30589	−0.271053
$940$	0	0
$941$	27.9962	0.912650	0.456325	−	0.889813i	$-0.349166\pi$
$941$	27.9962	0.912650	0.456325	+	0.889813i	$0.349166\pi$
$942$	0	0
$943$	18.1674	0.591610
$944$	0	0
$945$	−5.96383	−0.194003
$946$	0	0
$947$	19.1829	0.623360	0.311680	−	0.950187i	$-0.399108\pi$
$947$	19.1829	0.623360	0.311680	+	0.950187i	$0.399108\pi$
$948$	0	0
$949$	2.81808	0.0914788
$950$	0	0
$951$	−4.90975	−0.159210
$952$	0	0
$953$	28.0295	0.907966	0.453983	−	0.891010i	$-0.350003\pi$
$953$	28.0295	0.907966	0.453983	+	0.891010i	$0.350003\pi$
$954$	0	0
$955$	−13.0659	−0.422803
$956$	0	0
$957$	−0.670152	−0.0216629
$958$	0	0
$959$	−3.25580	−0.105135
$960$	0	0
$961$	−17.7345	−0.572080
$962$	0	0
$963$	−17.5440	−0.565347
$964$	0	0
$965$	9.24174	0.297502
$966$	0	0
$967$	38.2089	1.22872	0.614358	−	0.789027i	$-0.289415\pi$
$967$	38.2089	1.22872	0.614358	+	0.789027i	$0.289415\pi$
$968$	0	0
$969$	3.85564	0.123861
$970$	0	0
$971$	29.7390	0.954370	0.477185	−	0.878803i	$-0.341657\pi$
$971$	29.7390	0.954370	0.477185	+	0.878803i	$0.341657\pi$
$972$	0	0
$973$	−14.4182	−0.462226
$974$	0	0
$975$	0.434632	0.0139194
$976$	0	0
$977$	−30.8114	−0.985743	−0.492872	−	0.870102i	$-0.664053\pi$
$977$	−30.8114	−0.985743	−0.492872	+	0.870102i	$0.664053\pi$
$978$	0	0
$979$	0.122660	0.00392022
$980$	0	0
$981$	17.8939	0.571309
$982$	0	0
$983$	−23.5219	−0.750231	−0.375116	−	0.926978i	$-0.622397\pi$
$983$	−23.5219	−0.750231	−0.375116	+	0.926978i	$0.622397\pi$
$984$	0	0
$985$	11.0959	0.353545
$986$	0	0
$987$	12.6954	0.404098
$988$	0	0
$989$	4.29165	0.136466
$990$	0	0
$991$	−19.4530	−0.617944	−0.308972	−	0.951071i	$-0.599985\pi$
$991$	−19.4530	−0.617944	−0.308972	+	0.951071i	$0.599985\pi$
$992$	0	0
$993$	−5.22471	−0.165801
$994$	0	0
$995$	−3.44711	−0.109281
$996$	0	0
$997$	5.34674	0.169333	0.0846665	−	0.996409i	$-0.473018\pi$
$997$	5.34674	0.169333	0.0846665	+	0.996409i	$0.473018\pi$
$998$	0	0
$999$	5.96823	0.188826

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.o.1.8	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.o.1.8	✓	15	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q+0.266097 q^{3} +1.00000 q^{5} +3.77999 q^{7} -2.92919 q^{9} +O(q^{10})\)	\(q+0.266097 q^{3} +1.00000 q^{5} +3.77999 q^{7} -2.92919 q^{9} -1.90689 q^{11} +1.63336 q^{13} +0.266097 q^{15} -3.61052 q^{17} -4.01317 q^{19} +1.00584 q^{21} +2.09095 q^{23} +1.00000 q^{25} -1.57774 q^{27} +1.32071 q^{29} -3.64219 q^{31} -0.507417 q^{33} +3.77999 q^{35} -3.78278 q^{37} +0.434632 q^{39} +8.68857 q^{41} +2.05249 q^{43} -2.92919 q^{45} +12.6216 q^{47} +7.28831 q^{49} -0.960746 q^{51} +9.38276 q^{53} -1.90689 q^{55} -1.06789 q^{57} -2.81380 q^{59} +13.6738 q^{61} -11.0723 q^{63} +1.63336 q^{65} +15.5162 q^{67} +0.556394 q^{69} -9.50694 q^{71} +1.72533 q^{73} +0.266097 q^{75} -7.20803 q^{77} +0.651611 q^{79} +8.36775 q^{81} +17.9505 q^{83} -3.61052 q^{85} +0.351437 q^{87} -0.0643244 q^{89} +6.17408 q^{91} -0.969174 q^{93} -4.01317 q^{95} +10.9094 q^{97} +5.58565 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9	\( 15 q + 5 q^{3} + 15 q^{5} + 7 q^{7} + 18 q^{9} + 7 q^{11} + 2 q^{13} + 5 q^{15} - 3 q^{17} + 8 q^{19} + 7 q^{21} + 15 q^{23} + 15 q^{25} + 23 q^{27} + 5 q^{29} + 27 q^{31} - 5 q^{33} + 7 q^{35} - 4 q^{37} + 11 q^{39} + 20 q^{41} + 25 q^{43} + 18 q^{45} + 35 q^{47} - 14 q^{49} + 25 q^{51} - 2 q^{53} + 7 q^{55} - 24 q^{57} + 39 q^{59} + 23 q^{61} + 39 q^{63} + 2 q^{65} + 32 q^{67} + 13 q^{69} + 30 q^{71} + 7 q^{73} + 5 q^{75} - 4 q^{77} + 38 q^{79} + 11 q^{81} + 29 q^{83} - 3 q^{85} + 4 q^{87} + 19 q^{89} + 16 q^{91} + 8 q^{93} + 8 q^{95} - 8 q^{97} + 16 q^{99}+O(q^{100}) \) 15 * q + 5 * q^3 + 15 * q^5 + 7 * q^7 + 18 * q^9 + 7 * q^11 + 2 * q^13 + 5 * q^15 - 3 * q^17 + 8 * q^19 + 7 * q^21 + 15 * q^23 + 15 * q^25 + 23 * q^27 + 5 * q^29 + 27 * q^31 - 5 * q^33 + 7 * q^35 - 4 * q^37 + 11 * q^39 + 20 * q^41 + 25 * q^43 + 18 * q^45 + 35 * q^47 - 14 * q^49 + 25 * q^51 - 2 * q^53 + 7 * q^55 - 24 * q^57 + 39 * q^59 + 23 * q^61 + 39 * q^63 + 2 * q^65 + 32 * q^67 + 13 * q^69 + 30 * q^71 + 7 * q^73 + 5 * q^75 - 4 * q^77 + 38 * q^79 + 11 * q^81 + 29 * q^83 - 3 * q^85 + 4 * q^87 + 19 * q^89 + 16 * q^91 + 8 * q^93 + 8 * q^95 - 8 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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