LMFDB - Embedded newform 6040.2.a.r.1.3

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:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	$q-2.47861 q^{3} -1.00000 q^{5} -4.84176 q^{7} +3.14349 q^{9} +O(q^{10})$	$q-2.47861 q^{3} -1.00000 q^{5} -4.84176 q^{7} +3.14349 q^{9} +4.48047 q^{11} +6.58531 q^{13} +2.47861 q^{15} +4.51823 q^{17} -4.00151 q^{19} +12.0008 q^{21} +3.32200 q^{23} +1.00000 q^{25} -0.355649 q^{27} +3.90068 q^{29} -8.58596 q^{31} -11.1053 q^{33} +4.84176 q^{35} +8.80187 q^{37} -16.3224 q^{39} -5.90662 q^{41} -3.29400 q^{43} -3.14349 q^{45} -1.38603 q^{47} +16.4426 q^{49} -11.1989 q^{51} -3.19836 q^{53} -4.48047 q^{55} +9.91817 q^{57} -6.63102 q^{59} -11.7189 q^{61} -15.2200 q^{63} -6.58531 q^{65} +15.3601 q^{67} -8.23394 q^{69} +0.133010 q^{71} -5.43465 q^{73} -2.47861 q^{75} -21.6933 q^{77} +13.9742 q^{79} -8.54895 q^{81} +8.43018 q^{83} -4.51823 q^{85} -9.66826 q^{87} -13.3455 q^{89} -31.8845 q^{91} +21.2812 q^{93} +4.00151 q^{95} +3.08259 q^{97} +14.0843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 2 q^{3} - 23 q^{5} + 3 q^{7} + 31 q^{9}+O(q^{10}) $ 23 * q + 2 * q^3 - 23 * q^5 + 3 * q^7 + 31 * q^9	$ 23 q + 2 q^{3} - 23 q^{5} + 3 q^{7} + 31 q^{9} + 5 q^{11} - 2 q^{15} + 16 q^{17} - 14 q^{19} + 3 q^{21} + 11 q^{23} + 23 q^{25} + 14 q^{27} + 15 q^{29} - 14 q^{31} + 23 q^{33} - 3 q^{35} + 10 q^{37} - 5 q^{39} + 28 q^{41} + q^{43} - 31 q^{45} + 24 q^{47} + 40 q^{49} - 6 q^{51} - 11 q^{53} - 5 q^{55} + 42 q^{57} - 9 q^{59} - 14 q^{61} + 33 q^{63} + 9 q^{67} - 7 q^{69} + 22 q^{71} + 29 q^{73} + 2 q^{75} - 10 q^{79} + 67 q^{81} + 46 q^{83} - 16 q^{85} + 22 q^{87} + 33 q^{89} - 32 q^{91} + q^{93} + 14 q^{95} + 57 q^{97} + 42 q^{99}+O(q^{100}) $ 23 * q + 2 * q^3 - 23 * q^5 + 3 * q^7 + 31 * q^9 + 5 * q^11 - 2 * q^15 + 16 * q^17 - 14 * q^19 + 3 * q^21 + 11 * q^23 + 23 * q^25 + 14 * q^27 + 15 * q^29 - 14 * q^31 + 23 * q^33 - 3 * q^35 + 10 * q^37 - 5 * q^39 + 28 * q^41 + q^43 - 31 * q^45 + 24 * q^47 + 40 * q^49 - 6 * q^51 - 11 * q^53 - 5 * q^55 + 42 * q^57 - 9 * q^59 - 14 * q^61 + 33 * q^63 + 9 * q^67 - 7 * q^69 + 22 * q^71 + 29 * q^73 + 2 * q^75 - 10 * q^79 + 67 * q^81 + 46 * q^83 - 16 * q^85 + 22 * q^87 + 33 * q^89 - 32 * q^91 + q^93 + 14 * q^95 + 57 * q^97 + 42 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.47861	−1.43102	−0.715512	−	0.698601i	$-0.753806\pi$
$3$	−2.47861	−1.43102	−0.715512	+	0.698601i	$0.753806\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.84176	−1.83001	−0.915007	−	0.403439i	$-0.867815\pi$
$7$	−4.84176	−1.83001	−0.915007	+	0.403439i	$0.867815\pi$
$8$	0	0
$9$	3.14349	1.04783
$10$	0	0
$11$	4.48047	1.35091	0.675456	−	0.737401i	$-0.263947\pi$
$11$	4.48047	1.35091	0.675456	+	0.737401i	$0.263947\pi$
$12$	0	0
$13$	6.58531	1.82644	0.913218	−	0.407471i	$-0.133589\pi$
$13$	6.58531	1.82644	0.913218	+	0.407471i	$0.133589\pi$
$14$	0	0
$15$	2.47861	0.639973
$16$	0	0
$17$	4.51823	1.09583	0.547915	−	0.836534i	$-0.315422\pi$
$17$	4.51823	1.09583	0.547915	+	0.836534i	$0.315422\pi$
$18$	0	0
$19$	−4.00151	−0.918010	−0.459005	−	0.888434i	$-0.651794\pi$
$19$	−4.00151	−0.918010	−0.459005	+	0.888434i	$0.651794\pi$
$20$	0	0
$21$	12.0008	2.61879
$22$	0	0
$23$	3.32200	0.692686	0.346343	−	0.938108i	$-0.387423\pi$
$23$	3.32200	0.692686	0.346343	+	0.938108i	$0.387423\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.355649	−0.0684446
$28$	0	0
$29$	3.90068	0.724339	0.362169	−	0.932112i	$-0.382036\pi$
$29$	3.90068	0.724339	0.362169	+	0.932112i	$0.382036\pi$
$30$	0	0
$31$	−8.58596	−1.54208	−0.771042	−	0.636784i	$-0.780264\pi$
$31$	−8.58596	−1.54208	−0.771042	+	0.636784i	$0.780264\pi$
$32$	0	0
$33$	−11.1053	−1.93319
$34$	0	0
$35$	4.84176	0.818407
$36$	0	0
$37$	8.80187	1.44702	0.723509	−	0.690315i	$-0.242528\pi$
$37$	8.80187	1.44702	0.723509	+	0.690315i	$0.242528\pi$
$38$	0	0
$39$	−16.3224	−2.61367
$40$	0	0
$41$	−5.90662	−0.922460	−0.461230	−	0.887281i	$-0.652592\pi$
$41$	−5.90662	−0.922460	−0.461230	+	0.887281i	$0.652592\pi$
$42$	0	0
$43$	−3.29400	−0.502330	−0.251165	−	0.967944i	$-0.580814\pi$
$43$	−3.29400	−0.502330	−0.251165	+	0.967944i	$0.580814\pi$
$44$	0	0
$45$	−3.14349	−0.468603
$46$	0	0
$47$	−1.38603	−0.202173	−0.101087	−	0.994878i	$-0.532232\pi$
$47$	−1.38603	−0.202173	−0.101087	+	0.994878i	$0.532232\pi$
$48$	0	0
$49$	16.4426	2.34895
$50$	0	0
$51$	−11.1989	−1.56816
$52$	0	0
$53$	−3.19836	−0.439329	−0.219664	−	0.975575i	$-0.570496\pi$
$53$	−3.19836	−0.439329	−0.219664	+	0.975575i	$0.570496\pi$
$54$	0	0
$55$	−4.48047	−0.604146
$56$	0	0
$57$	9.91817	1.31369
$58$	0	0
$59$	−6.63102	−0.863285	−0.431642	−	0.902045i	$-0.642066\pi$
$59$	−6.63102	−0.863285	−0.431642	+	0.902045i	$0.642066\pi$
$60$	0	0
$61$	−11.7189	−1.50045	−0.750226	−	0.661182i	$-0.770055\pi$
$61$	−11.7189	−1.50045	−0.750226	+	0.661182i	$0.770055\pi$
$62$	0	0
$63$	−15.2200	−1.91754
$64$	0	0
$65$	−6.58531	−0.816807
$66$	0	0
$67$	15.3601	1.87654	0.938268	−	0.345908i	$-0.112429\pi$
$67$	15.3601	1.87654	0.938268	+	0.345908i	$0.112429\pi$
$68$	0	0
$69$	−8.23394	−0.991250
$70$	0	0
$71$	0.133010	0.0157853	0.00789266	−	0.999969i	$-0.497488\pi$
$71$	0.133010	0.0157853	0.00789266	+	0.999969i	$0.497488\pi$
$72$	0	0
$73$	−5.43465	−0.636077	−0.318039	−	0.948078i	$-0.603024\pi$
$73$	−5.43465	−0.636077	−0.318039	+	0.948078i	$0.603024\pi$
$74$	0	0
$75$	−2.47861	−0.286205
$76$	0	0
$77$	−21.6933	−2.47219
$78$	0	0
$79$	13.9742	1.57222	0.786108	−	0.618090i	$-0.212093\pi$
$79$	13.9742	1.57222	0.786108	+	0.618090i	$0.212093\pi$
$80$	0	0
$81$	−8.54895	−0.949883
$82$	0	0
$83$	8.43018	0.925332	0.462666	−	0.886533i	$-0.346893\pi$
$83$	8.43018	0.925332	0.462666	+	0.886533i	$0.346893\pi$
$84$	0	0
$85$	−4.51823	−0.490070
$86$	0	0
$87$	−9.66826	−1.03655
$88$	0	0
$89$	−13.3455	−1.41462	−0.707308	−	0.706906i	$-0.750090\pi$
$89$	−13.3455	−1.41462	−0.707308	+	0.706906i	$0.750090\pi$
$90$	0	0
$91$	−31.8845	−3.34240
$92$	0	0
$93$	21.2812	2.20676
$94$	0	0
$95$	4.00151	0.410547
$96$	0	0
$97$	3.08259	0.312990	0.156495	−	0.987679i	$-0.449980\pi$
$97$	3.08259	0.312990	0.156495	+	0.987679i	$0.449980\pi$
$98$	0	0
$99$	14.0843	1.41552
$100$	0	0
$101$	−5.61104	−0.558319	−0.279160	−	0.960245i	$-0.590056\pi$
$101$	−5.61104	−0.558319	−0.279160	+	0.960245i	$0.590056\pi$
$102$	0	0
$103$	1.22362	0.120567	0.0602833	−	0.998181i	$-0.480800\pi$
$103$	1.22362	0.120567	0.0602833	+	0.998181i	$0.480800\pi$
$104$	0	0
$105$	−12.0008	−1.17116
$106$	0	0
$107$	−10.9204	−1.05572	−0.527858	−	0.849333i	$-0.677005\pi$
$107$	−10.9204	−1.05572	−0.527858	+	0.849333i	$0.677005\pi$
$108$	0	0
$109$	5.33156	0.510671	0.255336	−	0.966853i	$-0.417814\pi$
$109$	5.33156	0.510671	0.255336	+	0.966853i	$0.417814\pi$
$110$	0	0
$111$	−21.8164	−2.07072
$112$	0	0
$113$	−9.40815	−0.885045	−0.442522	−	0.896757i	$-0.645916\pi$
$113$	−9.40815	−0.885045	−0.442522	+	0.896757i	$0.645916\pi$
$114$	0	0
$115$	−3.32200	−0.309778
$116$	0	0
$117$	20.7008	1.91379
$118$	0	0
$119$	−21.8762	−2.00538
$120$	0	0
$121$	9.07458	0.824962
$122$	0	0
$123$	14.6402	1.32006
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.43850	−0.305118	−0.152559	−	0.988294i	$-0.548751\pi$
$127$	−3.43850	−0.305118	−0.152559	+	0.988294i	$0.548751\pi$
$128$	0	0
$129$	8.16453	0.718847
$130$	0	0
$131$	12.9636	1.13264	0.566319	−	0.824186i	$-0.308367\pi$
$131$	12.9636	1.13264	0.566319	+	0.824186i	$0.308367\pi$
$132$	0	0
$133$	19.3744	1.67997
$134$	0	0
$135$	0.355649	0.0306094
$136$	0	0
$137$	10.3056	0.880465	0.440233	−	0.897884i	$-0.354896\pi$
$137$	10.3056	0.880465	0.440233	+	0.897884i	$0.354896\pi$
$138$	0	0
$139$	3.18116	0.269823	0.134911	−	0.990858i	$-0.456925\pi$
$139$	3.18116	0.269823	0.134911	+	0.990858i	$0.456925\pi$
$140$	0	0
$141$	3.43542	0.289314
$142$	0	0
$143$	29.5053	2.46735
$144$	0	0
$145$	−3.90068	−0.323934
$146$	0	0
$147$	−40.7548	−3.36140
$148$	0	0
$149$	10.9446	0.896620	0.448310	−	0.893878i	$-0.352026\pi$
$149$	10.9446	0.896620	0.448310	+	0.893878i	$0.352026\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	14.2030	1.14824
$154$	0	0
$155$	8.58596	0.689641
$156$	0	0
$157$	−3.77865	−0.301569	−0.150784	−	0.988567i	$-0.548180\pi$
$157$	−3.77865	−0.301569	−0.150784	+	0.988567i	$0.548180\pi$
$158$	0	0
$159$	7.92748	0.628690
$160$	0	0
$161$	−16.0843	−1.26762
$162$	0	0
$163$	−0.511149	−0.0400363	−0.0200181	−	0.999800i	$-0.506372\pi$
$163$	−0.511149	−0.0400363	−0.0200181	+	0.999800i	$0.506372\pi$
$164$	0	0
$165$	11.1053	0.864547
$166$	0	0
$167$	7.96416	0.616285	0.308143	−	0.951340i	$-0.400293\pi$
$167$	7.96416	0.616285	0.308143	+	0.951340i	$0.400293\pi$
$168$	0	0
$169$	30.3663	2.33587
$170$	0	0
$171$	−12.5787	−0.961918
$172$	0	0
$173$	2.51280	0.191045	0.0955224	−	0.995427i	$-0.469548\pi$
$173$	2.51280	0.191045	0.0955224	+	0.995427i	$0.469548\pi$
$174$	0	0
$175$	−4.84176	−0.366003
$176$	0	0
$177$	16.4357	1.23538
$178$	0	0
$179$	−11.4943	−0.859128	−0.429564	−	0.903037i	$-0.641333\pi$
$179$	−11.4943	−0.859128	−0.429564	+	0.903037i	$0.641333\pi$
$180$	0	0
$181$	10.5671	0.785450	0.392725	−	0.919656i	$-0.371533\pi$
$181$	10.5671	0.785450	0.392725	+	0.919656i	$0.371533\pi$
$182$	0	0
$183$	29.0465	2.14718
$184$	0	0
$185$	−8.80187	−0.647126
$186$	0	0
$187$	20.2438	1.48037
$188$	0	0
$189$	1.72196	0.125255
$190$	0	0
$191$	−15.8866	−1.14952	−0.574758	−	0.818324i	$-0.694904\pi$
$191$	−15.8866	−1.14952	−0.574758	+	0.818324i	$0.694904\pi$
$192$	0	0
$193$	9.64984	0.694610	0.347305	−	0.937752i	$-0.387097\pi$
$193$	9.64984	0.694610	0.347305	+	0.937752i	$0.387097\pi$
$194$	0	0
$195$	16.3224	1.16887
$196$	0	0
$197$	18.7721	1.33745	0.668727	−	0.743508i	$-0.266840\pi$
$197$	18.7721	1.33745	0.668727	+	0.743508i	$0.266840\pi$
$198$	0	0
$199$	11.5589	0.819388	0.409694	−	0.912223i	$-0.365635\pi$
$199$	11.5589	0.819388	0.409694	+	0.912223i	$0.365635\pi$
$200$	0	0
$201$	−38.0717	−2.68537
$202$	0	0
$203$	−18.8862	−1.32555
$204$	0	0
$205$	5.90662	0.412537
$206$	0	0
$207$	10.4427	0.725816
$208$	0	0
$209$	−17.9286	−1.24015
$210$	0	0
$211$	20.3724	1.40249	0.701246	−	0.712920i	$-0.252628\pi$
$211$	20.3724	1.40249	0.701246	+	0.712920i	$0.252628\pi$
$212$	0	0
$213$	−0.329678	−0.0225892
$214$	0	0
$215$	3.29400	0.224649
$216$	0	0
$217$	41.5712	2.82203
$218$	0	0
$219$	13.4703	0.910242
$220$	0	0
$221$	29.7539	2.00146
$222$	0	0
$223$	−9.80452	−0.656559	−0.328280	−	0.944581i	$-0.606469\pi$
$223$	−9.80452	−0.656559	−0.328280	+	0.944581i	$0.606469\pi$
$224$	0	0
$225$	3.14349	0.209566
$226$	0	0
$227$	11.9673	0.794296	0.397148	−	0.917755i	$-0.370000\pi$
$227$	11.9673	0.794296	0.397148	+	0.917755i	$0.370000\pi$
$228$	0	0
$229$	6.30719	0.416791	0.208395	−	0.978045i	$-0.433176\pi$
$229$	6.30719	0.416791	0.208395	+	0.978045i	$0.433176\pi$
$230$	0	0
$231$	53.7692	3.53776
$232$	0	0
$233$	−7.53563	−0.493676	−0.246838	−	0.969057i	$-0.579391\pi$
$233$	−7.53563	−0.493676	−0.246838	+	0.969057i	$0.579391\pi$
$234$	0	0
$235$	1.38603	0.0904145
$236$	0	0
$237$	−34.6364	−2.24988
$238$	0	0
$239$	−15.8694	−1.02651	−0.513255	−	0.858236i	$-0.671560\pi$
$239$	−15.8694	−1.02651	−0.513255	+	0.858236i	$0.671560\pi$
$240$	0	0
$241$	−15.5564	−1.00208	−0.501039	−	0.865425i	$-0.667049\pi$
$241$	−15.5564	−1.00208	−0.501039	+	0.865425i	$0.667049\pi$
$242$	0	0
$243$	22.2564	1.42775
$244$	0	0
$245$	−16.4426	−1.05048
$246$	0	0
$247$	−26.3512	−1.67669
$248$	0	0
$249$	−20.8951	−1.32417
$250$	0	0
$251$	14.1794	0.894997	0.447499	−	0.894285i	$-0.352315\pi$
$251$	14.1794	0.894997	0.447499	+	0.894285i	$0.352315\pi$
$252$	0	0
$253$	14.8841	0.935757
$254$	0	0
$255$	11.1989	0.701302
$256$	0	0
$257$	31.0138	1.93459	0.967293	−	0.253660i	$-0.0816345\pi$
$257$	31.0138	1.93459	0.967293	+	0.253660i	$0.0816345\pi$
$258$	0	0
$259$	−42.6165	−2.64806
$260$	0	0
$261$	12.2618	0.758983
$262$	0	0
$263$	−18.6416	−1.14949	−0.574744	−	0.818333i	$-0.694898\pi$
$263$	−18.6416	−1.14949	−0.574744	+	0.818333i	$0.694898\pi$
$264$	0	0
$265$	3.19836	0.196474
$266$	0	0
$267$	33.0781	2.02435
$268$	0	0
$269$	−27.1807	−1.65723	−0.828617	−	0.559815i	$-0.810872\pi$
$269$	−27.1807	−1.65723	−0.828617	+	0.559815i	$0.810872\pi$
$270$	0	0
$271$	6.28335	0.381686	0.190843	−	0.981621i	$-0.438878\pi$
$271$	6.28335	0.381686	0.190843	+	0.981621i	$0.438878\pi$
$272$	0	0
$273$	79.0291	4.78306
$274$	0	0
$275$	4.48047	0.270182
$276$	0	0
$277$	−22.5332	−1.35389	−0.676943	−	0.736036i	$-0.736696\pi$
$277$	−22.5332	−1.35389	−0.676943	+	0.736036i	$0.736696\pi$
$278$	0	0
$279$	−26.9899	−1.61584
$280$	0	0
$281$	−2.70652	−0.161457	−0.0807287	−	0.996736i	$-0.525725\pi$
$281$	−2.70652	−0.161457	−0.0807287	+	0.996736i	$0.525725\pi$
$282$	0	0
$283$	−14.2169	−0.845108	−0.422554	−	0.906338i	$-0.638866\pi$
$283$	−14.2169	−0.845108	−0.422554	+	0.906338i	$0.638866\pi$
$284$	0	0
$285$	−9.91817	−0.587502
$286$	0	0
$287$	28.5985	1.68811
$288$	0	0
$289$	3.41436	0.200845
$290$	0	0
$291$	−7.64053	−0.447896
$292$	0	0
$293$	−22.2614	−1.30052	−0.650261	−	0.759711i	$-0.725341\pi$
$293$	−22.2614	−1.30052	−0.650261	+	0.759711i	$0.725341\pi$
$294$	0	0
$295$	6.63102	0.386073
$296$	0	0
$297$	−1.59347	−0.0924626
$298$	0	0
$299$	21.8764	1.26515
$300$	0	0
$301$	15.9488	0.919271
$302$	0	0
$303$	13.9076	0.798968
$304$	0	0
$305$	11.7189	0.671022
$306$	0	0
$307$	−7.82361	−0.446517	−0.223258	−	0.974759i	$-0.571669\pi$
$307$	−7.82361	−0.446517	−0.223258	+	0.974759i	$0.571669\pi$
$308$	0	0
$309$	−3.03287	−0.172534
$310$	0	0
$311$	−20.6442	−1.17063	−0.585313	−	0.810808i	$-0.699028\pi$
$311$	−20.6442	−1.17063	−0.585313	+	0.810808i	$0.699028\pi$
$312$	0	0
$313$	17.8526	1.00909	0.504545	−	0.863385i	$-0.331660\pi$
$313$	17.8526	1.00909	0.504545	+	0.863385i	$0.331660\pi$
$314$	0	0
$315$	15.2200	0.857550
$316$	0	0
$317$	15.9827	0.897678	0.448839	−	0.893613i	$-0.351838\pi$
$317$	15.9827	0.897678	0.448839	+	0.893613i	$0.351838\pi$
$318$	0	0
$319$	17.4769	0.978518
$320$	0	0
$321$	27.0674	1.51076
$322$	0	0
$323$	−18.0797	−1.00598
$324$	0	0
$325$	6.58531	0.365287
$326$	0	0
$327$	−13.2148	−0.730782
$328$	0	0
$329$	6.71082	0.369979
$330$	0	0
$331$	−25.6083	−1.40756	−0.703781	−	0.710417i	$-0.748506\pi$
$331$	−25.6083	−1.40756	−0.703781	+	0.710417i	$0.748506\pi$
$332$	0	0
$333$	27.6686	1.51623
$334$	0	0
$335$	−15.3601	−0.839213
$336$	0	0
$337$	33.2157	1.80937	0.904687	−	0.426077i	$-0.140105\pi$
$337$	33.2157	1.80937	0.904687	+	0.426077i	$0.140105\pi$
$338$	0	0
$339$	23.3191	1.26652
$340$	0	0
$341$	−38.4691	−2.08322
$342$	0	0
$343$	−45.7190	−2.46859
$344$	0	0
$345$	8.23394	0.443300
$346$	0	0
$347$	−24.8003	−1.33135	−0.665675	−	0.746242i	$-0.731856\pi$
$347$	−24.8003	−1.33135	−0.665675	+	0.746242i	$0.731856\pi$
$348$	0	0
$349$	23.6512	1.26602	0.633009	−	0.774145i	$-0.281820\pi$
$349$	23.6512	1.26602	0.633009	+	0.774145i	$0.281820\pi$
$350$	0	0
$351$	−2.34206	−0.125010
$352$	0	0
$353$	36.2866	1.93134	0.965670	−	0.259772i	$-0.0836475\pi$
$353$	36.2866	1.93134	0.965670	+	0.259772i	$0.0836475\pi$
$354$	0	0
$355$	−0.133010	−0.00705941
$356$	0	0
$357$	54.2224	2.86975
$358$	0	0
$359$	33.0234	1.74291	0.871454	−	0.490478i	$-0.163178\pi$
$359$	33.0234	1.74291	0.871454	+	0.490478i	$0.163178\pi$
$360$	0	0
$361$	−2.98789	−0.157257
$362$	0	0
$363$	−22.4923	−1.18054
$364$	0	0
$365$	5.43465	0.284462
$366$	0	0
$367$	28.7291	1.49964	0.749822	−	0.661639i	$-0.230139\pi$
$367$	28.7291	1.49964	0.749822	+	0.661639i	$0.230139\pi$
$368$	0	0
$369$	−18.5674	−0.966580
$370$	0	0
$371$	15.4857	0.803977
$372$	0	0
$373$	−6.25944	−0.324102	−0.162051	−	0.986782i	$-0.551811\pi$
$373$	−6.25944	−0.324102	−0.162051	+	0.986782i	$0.551811\pi$
$374$	0	0
$375$	2.47861	0.127995
$376$	0	0
$377$	25.6872	1.32296
$378$	0	0
$379$	33.4315	1.71726	0.858630	−	0.512596i	$-0.171316\pi$
$379$	33.4315	1.71726	0.858630	+	0.512596i	$0.171316\pi$
$380$	0	0
$381$	8.52269	0.436630
$382$	0	0
$383$	9.72044	0.496691	0.248346	−	0.968671i	$-0.420113\pi$
$383$	9.72044	0.496691	0.248346	+	0.968671i	$0.420113\pi$
$384$	0	0
$385$	21.6933	1.10559
$386$	0	0
$387$	−10.3546	−0.526356
$388$	0	0
$389$	−26.7591	−1.35674	−0.678371	−	0.734719i	$-0.737314\pi$
$389$	−26.7591	−1.35674	−0.678371	+	0.734719i	$0.737314\pi$
$390$	0	0
$391$	15.0096	0.759066
$392$	0	0
$393$	−32.1318	−1.62083
$394$	0	0
$395$	−13.9742	−0.703116
$396$	0	0
$397$	−31.3595	−1.57389	−0.786944	−	0.617025i	$-0.788338\pi$
$397$	−31.3595	−1.57389	−0.786944	+	0.617025i	$0.788338\pi$
$398$	0	0
$399$	−48.0214	−2.40408
$400$	0	0
$401$	36.8310	1.83925	0.919626	−	0.392795i	$-0.128492\pi$
$401$	36.8310	1.83925	0.919626	+	0.392795i	$0.128492\pi$
$402$	0	0
$403$	−56.5412	−2.81652
$404$	0	0
$405$	8.54895	0.424801
$406$	0	0
$407$	39.4365	1.95479
$408$	0	0
$409$	−18.1266	−0.896303	−0.448152	−	0.893958i	$-0.647918\pi$
$409$	−18.1266	−0.896303	−0.448152	+	0.893958i	$0.647918\pi$
$410$	0	0
$411$	−25.5435	−1.25997
$412$	0	0
$413$	32.1058	1.57982
$414$	0	0
$415$	−8.43018	−0.413821
$416$	0	0
$417$	−7.88485	−0.386123
$418$	0	0
$419$	−10.1106	−0.493934	−0.246967	−	0.969024i	$-0.579434\pi$
$419$	−10.1106	−0.493934	−0.246967	+	0.969024i	$0.579434\pi$
$420$	0	0
$421$	−30.6158	−1.49212	−0.746062	−	0.665876i	$-0.768058\pi$
$421$	−30.6158	−1.49212	−0.746062	+	0.665876i	$0.768058\pi$
$422$	0	0
$423$	−4.35696	−0.211843
$424$	0	0
$425$	4.51823	0.219166
$426$	0	0
$427$	56.7401	2.74585
$428$	0	0
$429$	−73.1319	−3.53084
$430$	0	0
$431$	34.8316	1.67778	0.838890	−	0.544301i	$-0.183205\pi$
$431$	34.8316	1.67778	0.838890	+	0.544301i	$0.183205\pi$
$432$	0	0
$433$	−8.08380	−0.388483	−0.194241	−	0.980954i	$-0.562224\pi$
$433$	−8.08380	−0.388483	−0.194241	+	0.980954i	$0.562224\pi$
$434$	0	0
$435$	9.66826	0.463558
$436$	0	0
$437$	−13.2930	−0.635892
$438$	0	0
$439$	−12.4252	−0.593021	−0.296511	−	0.955030i	$-0.595823\pi$
$439$	−12.4252	−0.593021	−0.296511	+	0.955030i	$0.595823\pi$
$440$	0	0
$441$	51.6872	2.46130
$442$	0	0
$443$	18.4082	0.874598	0.437299	−	0.899316i	$-0.355935\pi$
$443$	18.4082	0.874598	0.437299	+	0.899316i	$0.355935\pi$
$444$	0	0
$445$	13.3455	0.632635
$446$	0	0
$447$	−27.1275	−1.28308
$448$	0	0
$449$	26.5194	1.25153	0.625763	−	0.780013i	$-0.284788\pi$
$449$	26.5194	1.25153	0.625763	+	0.780013i	$0.284788\pi$
$450$	0	0
$451$	−26.4644	−1.24616
$452$	0	0
$453$	2.47861	0.116455
$454$	0	0
$455$	31.8845	1.49477
$456$	0	0
$457$	21.2283	0.993017	0.496509	−	0.868032i	$-0.334615\pi$
$457$	21.2283	0.993017	0.496509	+	0.868032i	$0.334615\pi$
$458$	0	0
$459$	−1.60690	−0.0750037
$460$	0	0
$461$	−41.2228	−1.91994	−0.959969	−	0.280105i	$-0.909631\pi$
$461$	−41.2228	−1.91994	−0.959969	+	0.280105i	$0.909631\pi$
$462$	0	0
$463$	−14.9598	−0.695240	−0.347620	−	0.937636i	$-0.613010\pi$
$463$	−14.9598	−0.695240	−0.347620	+	0.937636i	$0.613010\pi$
$464$	0	0
$465$	−21.2812	−0.986893
$466$	0	0
$467$	26.6174	1.23171	0.615854	−	0.787860i	$-0.288811\pi$
$467$	26.6174	1.23171	0.615854	+	0.787860i	$0.288811\pi$
$468$	0	0
$469$	−74.3700	−3.43409
$470$	0	0
$471$	9.36578	0.431552
$472$	0	0
$473$	−14.7587	−0.678604
$474$	0	0
$475$	−4.00151	−0.183602
$476$	0	0
$477$	−10.0540	−0.460342
$478$	0	0
$479$	34.9855	1.59853	0.799263	−	0.600981i	$-0.205223\pi$
$479$	34.9855	1.59853	0.799263	+	0.600981i	$0.205223\pi$
$480$	0	0
$481$	57.9631	2.64289
$482$	0	0
$483$	39.8667	1.81400
$484$	0	0
$485$	−3.08259	−0.139973
$486$	0	0
$487$	−26.7009	−1.20993	−0.604967	−	0.796250i	$-0.706814\pi$
$487$	−26.7009	−1.20993	−0.604967	+	0.796250i	$0.706814\pi$
$488$	0	0
$489$	1.26694	0.0572929
$490$	0	0
$491$	7.68050	0.346616	0.173308	−	0.984868i	$-0.444554\pi$
$491$	7.68050	0.346616	0.173308	+	0.984868i	$0.444554\pi$
$492$	0	0
$493$	17.6242	0.793753
$494$	0	0
$495$	−14.0843	−0.633042
$496$	0	0
$497$	−0.644000	−0.0288874
$498$	0	0
$499$	17.5861	0.787261	0.393631	−	0.919269i	$-0.371219\pi$
$499$	17.5861	0.787261	0.393631	+	0.919269i	$0.371219\pi$
$500$	0	0
$501$	−19.7400	−0.881919
$502$	0	0
$503$	−8.62597	−0.384613	−0.192307	−	0.981335i	$-0.561597\pi$
$503$	−8.62597	−0.384613	−0.192307	+	0.981335i	$0.561597\pi$
$504$	0	0
$505$	5.61104	0.249688
$506$	0	0
$507$	−75.2661	−3.34269
$508$	0	0
$509$	−9.92051	−0.439719	−0.219859	−	0.975532i	$-0.570560\pi$
$509$	−9.92051	−0.439719	−0.219859	+	0.975532i	$0.570560\pi$
$510$	0	0
$511$	26.3133	1.16403
$512$	0	0
$513$	1.42313	0.0628328
$514$	0	0
$515$	−1.22362	−0.0539190
$516$	0	0
$517$	−6.21005	−0.273118
$518$	0	0
$519$	−6.22825	−0.273390
$520$	0	0
$521$	12.9464	0.567190	0.283595	−	0.958944i	$-0.408473\pi$
$521$	12.9464	0.567190	0.283595	+	0.958944i	$0.408473\pi$
$522$	0	0
$523$	−26.5551	−1.16117	−0.580586	−	0.814199i	$-0.697177\pi$
$523$	−26.5551	−1.16117	−0.580586	+	0.814199i	$0.697177\pi$
$524$	0	0
$525$	12.0008	0.523758
$526$	0	0
$527$	−38.7933	−1.68986
$528$	0	0
$529$	−11.9643	−0.520187
$530$	0	0
$531$	−20.8445	−0.904575
$532$	0	0
$533$	−38.8970	−1.68481
$534$	0	0
$535$	10.9204	0.472131
$536$	0	0
$537$	28.4899	1.22943
$538$	0	0
$539$	73.6707	3.17322
$540$	0	0
$541$	−33.7600	−1.45146	−0.725728	−	0.687982i	$-0.758497\pi$
$541$	−33.7600	−1.45146	−0.725728	+	0.687982i	$0.758497\pi$
$542$	0	0
$543$	−26.1918	−1.12400
$544$	0	0
$545$	−5.33156	−0.228379
$546$	0	0
$547$	16.2412	0.694423	0.347212	−	0.937787i	$-0.387129\pi$
$547$	16.2412	0.694423	0.347212	+	0.937787i	$0.387129\pi$
$548$	0	0
$549$	−36.8382	−1.57222
$550$	0	0
$551$	−15.6086	−0.664950
$552$	0	0
$553$	−67.6595	−2.87717
$554$	0	0
$555$	21.8164	0.926053
$556$	0	0
$557$	−10.1854	−0.431570	−0.215785	−	0.976441i	$-0.569231\pi$
$557$	−10.1854	−0.431570	−0.215785	+	0.976441i	$0.569231\pi$
$558$	0	0
$559$	−21.6920	−0.917474
$560$	0	0
$561$	−50.1763	−2.11844
$562$	0	0
$563$	15.8719	0.668921	0.334461	−	0.942410i	$-0.391446\pi$
$563$	15.8719	0.668921	0.334461	+	0.942410i	$0.391446\pi$
$564$	0	0
$565$	9.40815	0.395804
$566$	0	0
$567$	41.3920	1.73830
$568$	0	0
$569$	5.33895	0.223820	0.111910	−	0.993718i	$-0.464303\pi$
$569$	5.33895	0.223820	0.111910	+	0.993718i	$0.464303\pi$
$570$	0	0
$571$	−6.54840	−0.274042	−0.137021	−	0.990568i	$-0.543753\pi$
$571$	−6.54840	−0.274042	−0.137021	+	0.990568i	$0.543753\pi$
$572$	0	0
$573$	39.3767	1.64498
$574$	0	0
$575$	3.32200	0.138537
$576$	0	0
$577$	46.1726	1.92219	0.961095	−	0.276218i	$-0.0890812\pi$
$577$	46.1726	1.92219	0.961095	+	0.276218i	$0.0890812\pi$
$578$	0	0
$579$	−23.9181	−0.994004
$580$	0	0
$581$	−40.8169	−1.69337
$582$	0	0
$583$	−14.3302	−0.593494
$584$	0	0
$585$	−20.7008	−0.855874
$586$	0	0
$587$	−27.9894	−1.15525	−0.577624	−	0.816303i	$-0.696020\pi$
$587$	−27.9894	−1.15525	−0.577624	+	0.816303i	$0.696020\pi$
$588$	0	0
$589$	34.3568	1.41565
$590$	0	0
$591$	−46.5285	−1.91393
$592$	0	0
$593$	14.3962	0.591180	0.295590	−	0.955315i	$-0.404484\pi$
$593$	14.3962	0.591180	0.295590	+	0.955315i	$0.404484\pi$
$594$	0	0
$595$	21.8762	0.896835
$596$	0	0
$597$	−28.6499	−1.17256
$598$	0	0
$599$	−13.3571	−0.545756	−0.272878	−	0.962049i	$-0.587976\pi$
$599$	−13.3571	−0.545756	−0.272878	+	0.962049i	$0.587976\pi$
$600$	0	0
$601$	5.28015	0.215382	0.107691	−	0.994184i	$-0.465654\pi$
$601$	5.28015	0.215382	0.107691	+	0.994184i	$0.465654\pi$
$602$	0	0
$603$	48.2843	1.96629
$604$	0	0
$605$	−9.07458	−0.368934
$606$	0	0
$607$	6.48378	0.263168	0.131584	−	0.991305i	$-0.457994\pi$
$607$	6.48378	0.263168	0.131584	+	0.991305i	$0.457994\pi$
$608$	0	0
$609$	46.8114	1.89689
$610$	0	0
$611$	−9.12743	−0.369256
$612$	0	0
$613$	26.1455	1.05601	0.528003	−	0.849243i	$-0.322941\pi$
$613$	26.1455	1.05601	0.528003	+	0.849243i	$0.322941\pi$
$614$	0	0
$615$	−14.6402	−0.590350
$616$	0	0
$617$	29.6034	1.19179	0.595894	−	0.803063i	$-0.296798\pi$
$617$	29.6034	1.19179	0.595894	+	0.803063i	$0.296798\pi$
$618$	0	0
$619$	−28.5809	−1.14876	−0.574381	−	0.818588i	$-0.694757\pi$
$619$	−28.5809	−1.14876	−0.574381	+	0.818588i	$0.694757\pi$
$620$	0	0
$621$	−1.18147	−0.0474106
$622$	0	0
$623$	64.6155	2.58876
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	44.4380	1.77468
$628$	0	0
$629$	39.7688	1.58569
$630$	0	0
$631$	41.2909	1.64376	0.821881	−	0.569659i	$-0.192925\pi$
$631$	41.2909	1.64376	0.821881	+	0.569659i	$0.192925\pi$
$632$	0	0
$633$	−50.4951	−2.00700
$634$	0	0
$635$	3.43850	0.136453
$636$	0	0
$637$	108.280	4.29020
$638$	0	0
$639$	0.418114	0.0165403
$640$	0	0
$641$	8.89283	0.351246	0.175623	−	0.984458i	$-0.443806\pi$
$641$	8.89283	0.351246	0.175623	+	0.984458i	$0.443806\pi$
$642$	0	0
$643$	27.8240	1.09727	0.548635	−	0.836062i	$-0.315148\pi$
$643$	27.8240	1.09727	0.548635	+	0.836062i	$0.315148\pi$
$644$	0	0
$645$	−8.16453	−0.321478
$646$	0	0
$647$	15.3016	0.601568	0.300784	−	0.953692i	$-0.402752\pi$
$647$	15.3016	0.601568	0.300784	+	0.953692i	$0.402752\pi$
$648$	0	0
$649$	−29.7100	−1.16622
$650$	0	0
$651$	−103.039	−4.03840
$652$	0	0
$653$	32.4527	1.26997	0.634986	−	0.772523i	$-0.281006\pi$
$653$	32.4527	1.26997	0.634986	+	0.772523i	$0.281006\pi$
$654$	0	0
$655$	−12.9636	−0.506531
$656$	0	0
$657$	−17.0837	−0.666500
$658$	0	0
$659$	28.1544	1.09674	0.548369	−	0.836236i	$-0.315249\pi$
$659$	28.1544	1.09674	0.548369	+	0.836236i	$0.315249\pi$
$660$	0	0
$661$	44.9831	1.74964	0.874819	−	0.484449i	$-0.160980\pi$
$661$	44.9831	1.74964	0.874819	+	0.484449i	$0.160980\pi$
$662$	0	0
$663$	−73.7482	−2.86414
$664$	0	0
$665$	−19.3744	−0.751306
$666$	0	0
$667$	12.9581	0.501739
$668$	0	0
$669$	24.3015	0.939552
$670$	0	0
$671$	−52.5061	−2.02698
$672$	0	0
$673$	21.9899	0.847650	0.423825	−	0.905744i	$-0.360687\pi$
$673$	21.9899	0.847650	0.423825	+	0.905744i	$0.360687\pi$
$674$	0	0
$675$	−0.355649	−0.0136889
$676$	0	0
$677$	9.45474	0.363375	0.181688	−	0.983356i	$-0.441844\pi$
$677$	9.45474	0.363375	0.181688	+	0.983356i	$0.441844\pi$
$678$	0	0
$679$	−14.9252	−0.572776
$680$	0	0
$681$	−29.6622	−1.13666
$682$	0	0
$683$	−29.5027	−1.12889	−0.564444	−	0.825471i	$-0.690910\pi$
$683$	−29.5027	−1.12889	−0.564444	+	0.825471i	$0.690910\pi$
$684$	0	0
$685$	−10.3056	−0.393756
$686$	0	0
$687$	−15.6330	−0.596437
$688$	0	0
$689$	−21.0622	−0.802406
$690$	0	0
$691$	12.1656	0.462799	0.231400	−	0.972859i	$-0.425669\pi$
$691$	12.1656	0.462799	0.231400	+	0.972859i	$0.425669\pi$
$692$	0	0
$693$	−68.1927	−2.59043
$694$	0	0
$695$	−3.18116	−0.120668
$696$	0	0
$697$	−26.6875	−1.01086
$698$	0	0
$699$	18.6779	0.706462
$700$	0	0
$701$	23.5272	0.888610	0.444305	−	0.895876i	$-0.353451\pi$
$701$	23.5272	0.888610	0.444305	+	0.895876i	$0.353451\pi$
$702$	0	0
$703$	−35.2208	−1.32838
$704$	0	0
$705$	−3.43542	−0.129385
$706$	0	0
$707$	27.1673	1.02173
$708$	0	0
$709$	−18.5759	−0.697631	−0.348816	−	0.937191i	$-0.613416\pi$
$709$	−18.5759	−0.697631	−0.348816	+	0.937191i	$0.613416\pi$
$710$	0	0
$711$	43.9276	1.64741
$712$	0	0
$713$	−28.5226	−1.06818
$714$	0	0
$715$	−29.5053	−1.10343
$716$	0	0
$717$	39.3341	1.46896
$718$	0	0
$719$	28.5913	1.06627	0.533137	−	0.846029i	$-0.321013\pi$
$719$	28.5913	1.06627	0.533137	+	0.846029i	$0.321013\pi$
$720$	0	0
$721$	−5.92446	−0.220639
$722$	0	0
$723$	38.5583	1.43400
$724$	0	0
$725$	3.90068	0.144868
$726$	0	0
$727$	19.6845	0.730058	0.365029	−	0.930996i	$-0.381059\pi$
$727$	19.6845	0.730058	0.365029	+	0.930996i	$0.381059\pi$
$728$	0	0
$729$	−29.5181	−1.09326
$730$	0	0
$731$	−14.8830	−0.550469
$732$	0	0
$733$	−34.2167	−1.26382	−0.631912	−	0.775040i	$-0.717729\pi$
$733$	−34.2167	−1.26382	−0.631912	+	0.775040i	$0.717729\pi$
$734$	0	0
$735$	40.7548	1.50326
$736$	0	0
$737$	68.8205	2.53503
$738$	0	0
$739$	6.87100	0.252754	0.126377	−	0.991982i	$-0.459665\pi$
$739$	6.87100	0.252754	0.126377	+	0.991982i	$0.459665\pi$
$740$	0	0
$741$	65.3143	2.39938
$742$	0	0
$743$	7.83969	0.287610	0.143805	−	0.989606i	$-0.454066\pi$
$743$	7.83969	0.287610	0.143805	+	0.989606i	$0.454066\pi$
$744$	0	0
$745$	−10.9446	−0.400981
$746$	0	0
$747$	26.5001	0.969590
$748$	0	0
$749$	52.8740	1.93197
$750$	0	0
$751$	12.2871	0.448364	0.224182	−	0.974547i	$-0.428029\pi$
$751$	12.2871	0.448364	0.224182	+	0.974547i	$0.428029\pi$
$752$	0	0
$753$	−35.1452	−1.28076
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	−16.6135	−0.603830	−0.301915	−	0.953335i	$-0.597626\pi$
$757$	−16.6135	−0.603830	−0.301915	+	0.953335i	$0.597626\pi$
$758$	0	0
$759$	−36.8919	−1.33909
$760$	0	0
$761$	−20.1665	−0.731035	−0.365517	−	0.930805i	$-0.619108\pi$
$761$	−20.1665	−0.731035	−0.365517	+	0.930805i	$0.619108\pi$
$762$	0	0
$763$	−25.8141	−0.934535
$764$	0	0
$765$	−14.2030	−0.513510
$766$	0	0
$767$	−43.6673	−1.57673
$768$	0	0
$769$	33.4982	1.20798	0.603989	−	0.796993i	$-0.293577\pi$
$769$	33.4982	1.20798	0.603989	+	0.796993i	$0.293577\pi$
$770$	0	0
$771$	−76.8710	−2.76844
$772$	0	0
$773$	−27.5951	−0.992525	−0.496263	−	0.868172i	$-0.665295\pi$
$773$	−27.5951	−0.992525	−0.496263	+	0.868172i	$0.665295\pi$
$774$	0	0
$775$	−8.58596	−0.308417
$776$	0	0
$777$	105.630	3.78944
$778$	0	0
$779$	23.6354	0.846827
$780$	0	0
$781$	0.595945	0.0213246
$782$	0	0
$783$	−1.38727	−0.0495771
$784$	0	0
$785$	3.77865	0.134866
$786$	0	0
$787$	−29.1757	−1.04000	−0.520000	−	0.854166i	$-0.674068\pi$
$787$	−29.1757	−1.04000	−0.520000	+	0.854166i	$0.674068\pi$
$788$	0	0
$789$	46.2051	1.64494
$790$	0	0
$791$	45.5520	1.61964
$792$	0	0
$793$	−77.1726	−2.74048
$794$	0	0
$795$	−7.92748	−0.281159
$796$	0	0
$797$	−13.7238	−0.486121	−0.243061	−	0.970011i	$-0.578151\pi$
$797$	−13.7238	−0.486121	−0.243061	+	0.970011i	$0.578151\pi$
$798$	0	0
$799$	−6.26239	−0.221547
$800$	0	0
$801$	−41.9513	−1.48228
$802$	0	0
$803$	−24.3498	−0.859284
$804$	0	0
$805$	16.0843	0.566899
$806$	0	0
$807$	67.3702	2.37154
$808$	0	0
$809$	16.2746	0.572184	0.286092	−	0.958202i	$-0.407644\pi$
$809$	16.2746	0.572184	0.286092	+	0.958202i	$0.407644\pi$
$810$	0	0
$811$	47.4362	1.66571	0.832856	−	0.553490i	$-0.186704\pi$
$811$	47.4362	1.66571	0.832856	+	0.553490i	$0.186704\pi$
$812$	0	0
$813$	−15.5740	−0.546202
$814$	0	0
$815$	0.511149	0.0179048
$816$	0	0
$817$	13.1810	0.461144
$818$	0	0
$819$	−100.228	−3.50227
$820$	0	0
$821$	44.5926	1.55629	0.778146	−	0.628084i	$-0.216160\pi$
$821$	44.5926	1.55629	0.778146	+	0.628084i	$0.216160\pi$
$822$	0	0
$823$	−39.8413	−1.38878	−0.694391	−	0.719598i	$-0.744326\pi$
$823$	−39.8413	−1.38878	−0.694391	+	0.719598i	$0.744326\pi$
$824$	0	0
$825$	−11.1053	−0.386637
$826$	0	0
$827$	5.70811	0.198490	0.0992452	−	0.995063i	$-0.468357\pi$
$827$	5.70811	0.198490	0.0992452	+	0.995063i	$0.468357\pi$
$828$	0	0
$829$	−11.8627	−0.412009	−0.206004	−	0.978551i	$-0.566046\pi$
$829$	−11.8627	−0.412009	−0.206004	+	0.978551i	$0.566046\pi$
$830$	0	0
$831$	55.8508	1.93744
$832$	0	0
$833$	74.2915	2.57405
$834$	0	0
$835$	−7.96416	−0.275611
$836$	0	0
$837$	3.05359	0.105547
$838$	0	0
$839$	22.1725	0.765478	0.382739	−	0.923856i	$-0.374981\pi$
$839$	22.1725	0.765478	0.382739	+	0.923856i	$0.374981\pi$
$840$	0	0
$841$	−13.7847	−0.475333
$842$	0	0
$843$	6.70840	0.231050
$844$	0	0
$845$	−30.3663	−1.04463
$846$	0	0
$847$	−43.9369	−1.50969
$848$	0	0
$849$	35.2381	1.20937
$850$	0	0
$851$	29.2399	1.00233
$852$	0	0
$853$	34.7796	1.19083	0.595416	−	0.803417i	$-0.296987\pi$
$853$	34.7796	1.19083	0.595416	+	0.803417i	$0.296987\pi$
$854$	0	0
$855$	12.5787	0.430183
$856$	0	0
$857$	3.95580	0.135127	0.0675637	−	0.997715i	$-0.478477\pi$
$857$	3.95580	0.135127	0.0675637	+	0.997715i	$0.478477\pi$
$858$	0	0
$859$	55.8398	1.90523	0.952614	−	0.304182i	$-0.0983830\pi$
$859$	55.8398	1.90523	0.952614	+	0.304182i	$0.0983830\pi$
$860$	0	0
$861$	−70.8843	−2.41573
$862$	0	0
$863$	5.76706	0.196313	0.0981565	−	0.995171i	$-0.468705\pi$
$863$	5.76706	0.196313	0.0981565	+	0.995171i	$0.468705\pi$
$864$	0	0
$865$	−2.51280	−0.0854378
$866$	0	0
$867$	−8.46285	−0.287413
$868$	0	0
$869$	62.6107	2.12392
$870$	0	0
$871$	101.151	3.42737
$872$	0	0
$873$	9.69009	0.327960
$874$	0	0
$875$	4.84176	0.163681
$876$	0	0
$877$	−23.2242	−0.784225	−0.392113	−	0.919917i	$-0.628256\pi$
$877$	−23.2242	−0.784225	−0.392113	+	0.919917i	$0.628256\pi$
$878$	0	0
$879$	55.1771	1.86108
$880$	0	0
$881$	−19.6956	−0.663563	−0.331782	−	0.943356i	$-0.607650\pi$
$881$	−19.6956	−0.663563	−0.331782	+	0.943356i	$0.607650\pi$
$882$	0	0
$883$	−0.245903	−0.00827529	−0.00413764	−	0.999991i	$-0.501317\pi$
$883$	−0.245903	−0.00827529	−0.00413764	+	0.999991i	$0.501317\pi$
$884$	0	0
$885$	−16.4357	−0.552479
$886$	0	0
$887$	40.8402	1.37128	0.685641	−	0.727940i	$-0.259522\pi$
$887$	40.8402	1.37128	0.685641	+	0.727940i	$0.259522\pi$
$888$	0	0
$889$	16.6484	0.558369
$890$	0	0
$891$	−38.3033	−1.28321
$892$	0	0
$893$	5.54621	0.185597
$894$	0	0
$895$	11.4943	0.384214
$896$	0	0
$897$	−54.2230	−1.81045
$898$	0	0
$899$	−33.4911	−1.11699
$900$	0	0
$901$	−14.4509	−0.481430
$902$	0	0
$903$	−39.5307	−1.31550
$904$	0	0
$905$	−10.5671	−0.351264
$906$	0	0
$907$	45.2863	1.50371	0.751853	−	0.659331i	$-0.229160\pi$
$907$	45.2863	1.50371	0.751853	+	0.659331i	$0.229160\pi$
$908$	0	0
$909$	−17.6382	−0.585023
$910$	0	0
$911$	30.3777	1.00646	0.503229	−	0.864153i	$-0.332145\pi$
$911$	30.3777	1.00646	0.503229	+	0.864153i	$0.332145\pi$
$912$	0	0
$913$	37.7711	1.25004
$914$	0	0
$915$	−29.0465	−0.960249
$916$	0	0
$917$	−62.7668	−2.07274
$918$	0	0
$919$	−13.6315	−0.449663	−0.224832	−	0.974398i	$-0.572183\pi$
$919$	−13.6315	−0.449663	−0.224832	+	0.974398i	$0.572183\pi$
$920$	0	0
$921$	19.3916	0.638976
$922$	0	0
$923$	0.875909	0.0288309
$924$	0	0
$925$	8.80187	0.289404
$926$	0	0
$927$	3.84643	0.126333
$928$	0	0
$929$	12.4329	0.407910	0.203955	−	0.978980i	$-0.434620\pi$
$929$	12.4329	0.407910	0.203955	+	0.978980i	$0.434620\pi$
$930$	0	0
$931$	−65.7954	−2.15636
$932$	0	0
$933$	51.1689	1.67519
$934$	0	0
$935$	−20.2438	−0.662042
$936$	0	0
$937$	−49.2035	−1.60741	−0.803704	−	0.595030i	$-0.797140\pi$
$937$	−49.2035	−1.60741	−0.803704	+	0.595030i	$0.797140\pi$
$938$	0	0
$939$	−44.2497	−1.44403
$940$	0	0
$941$	27.7894	0.905910	0.452955	−	0.891533i	$-0.350370\pi$
$941$	27.7894	0.905910	0.452955	+	0.891533i	$0.350370\pi$
$942$	0	0
$943$	−19.6218	−0.638975
$944$	0	0
$945$	−1.72196	−0.0560155
$946$	0	0
$947$	−36.0244	−1.17064	−0.585318	−	0.810804i	$-0.699030\pi$
$947$	−36.0244	−1.17064	−0.585318	+	0.810804i	$0.699030\pi$
$948$	0	0
$949$	−35.7888	−1.16175
$950$	0	0
$951$	−39.6148	−1.28460
$952$	0	0
$953$	18.5907	0.602213	0.301106	−	0.953591i	$-0.402644\pi$
$953$	18.5907	0.602213	0.301106	+	0.953591i	$0.402644\pi$
$954$	0	0
$955$	15.8866	0.514079
$956$	0	0
$957$	−43.3183	−1.40028
$958$	0	0
$959$	−49.8971	−1.61126
$960$	0	0
$961$	42.7188	1.37802
$962$	0	0
$963$	−34.3282	−1.10621
$964$	0	0
$965$	−9.64984	−0.310639
$966$	0	0
$967$	21.1017	0.678583	0.339292	−	0.940681i	$-0.389813\pi$
$967$	21.1017	0.678583	0.339292	+	0.940681i	$0.389813\pi$
$968$	0	0
$969$	44.8125	1.43959
$970$	0	0
$971$	−9.04765	−0.290353	−0.145176	−	0.989406i	$-0.546375\pi$
$971$	−9.04765	−0.290353	−0.145176	+	0.989406i	$0.546375\pi$
$972$	0	0
$973$	−15.4024	−0.493779
$974$	0	0
$975$	−16.3224	−0.522735
$976$	0	0
$977$	−46.2342	−1.47916	−0.739582	−	0.673067i	$-0.764977\pi$
$977$	−46.2342	−1.47916	−0.739582	+	0.673067i	$0.764977\pi$
$978$	0	0
$979$	−59.7939	−1.91102
$980$	0	0
$981$	16.7597	0.535096
$982$	0	0
$983$	−36.4494	−1.16255	−0.581277	−	0.813706i	$-0.697447\pi$
$983$	−36.4494	−1.16255	−0.581277	+	0.813706i	$0.697447\pi$
$984$	0	0
$985$	−18.7721	−0.598128
$986$	0	0
$987$	−16.6335	−0.529449
$988$	0	0
$989$	−10.9427	−0.347957
$990$	0	0
$991$	−45.2928	−1.43877	−0.719386	−	0.694611i	$-0.755577\pi$
$991$	−45.2928	−1.43877	−0.719386	+	0.694611i	$0.755577\pi$
$992$	0	0
$993$	63.4730	2.01425
$994$	0	0
$995$	−11.5589	−0.366442
$996$	0	0
$997$	49.2339	1.55925	0.779626	−	0.626246i	$-0.215409\pi$
$997$	49.2339	1.55925	0.779626	+	0.626246i	$0.215409\pi$
$998$	0	0
$999$	−3.13037	−0.0990406

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.r.1.3	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.r.1.3	✓	23	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)

\(f(q)\)	\(=\)	\(q-2.47861 q^{3} -1.00000 q^{5} -4.84176 q^{7} +3.14349 q^{9} +O(q^{10})\)	\(q-2.47861 q^{3} -1.00000 q^{5} -4.84176 q^{7} +3.14349 q^{9} +4.48047 q^{11} +6.58531 q^{13} +2.47861 q^{15} +4.51823 q^{17} -4.00151 q^{19} +12.0008 q^{21} +3.32200 q^{23} +1.00000 q^{25} -0.355649 q^{27} +3.90068 q^{29} -8.58596 q^{31} -11.1053 q^{33} +4.84176 q^{35} +8.80187 q^{37} -16.3224 q^{39} -5.90662 q^{41} -3.29400 q^{43} -3.14349 q^{45} -1.38603 q^{47} +16.4426 q^{49} -11.1989 q^{51} -3.19836 q^{53} -4.48047 q^{55} +9.91817 q^{57} -6.63102 q^{59} -11.7189 q^{61} -15.2200 q^{63} -6.58531 q^{65} +15.3601 q^{67} -8.23394 q^{69} +0.133010 q^{71} -5.43465 q^{73} -2.47861 q^{75} -21.6933 q^{77} +13.9742 q^{79} -8.54895 q^{81} +8.43018 q^{83} -4.51823 q^{85} -9.66826 q^{87} -13.3455 q^{89} -31.8845 q^{91} +21.2812 q^{93} +4.00151 q^{95} +3.08259 q^{97} +14.0843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 2 q^{3} - 23 q^{5} + 3 q^{7} + 31 q^{9}+O(q^{10}) \) 23 * q + 2 * q^3 - 23 * q^5 + 3 * q^7 + 31 * q^9	\( 23 q + 2 q^{3} - 23 q^{5} + 3 q^{7} + 31 q^{9} + 5 q^{11} - 2 q^{15} + 16 q^{17} - 14 q^{19} + 3 q^{21} + 11 q^{23} + 23 q^{25} + 14 q^{27} + 15 q^{29} - 14 q^{31} + 23 q^{33} - 3 q^{35} + 10 q^{37} - 5 q^{39} + 28 q^{41} + q^{43} - 31 q^{45} + 24 q^{47} + 40 q^{49} - 6 q^{51} - 11 q^{53} - 5 q^{55} + 42 q^{57} - 9 q^{59} - 14 q^{61} + 33 q^{63} + 9 q^{67} - 7 q^{69} + 22 q^{71} + 29 q^{73} + 2 q^{75} - 10 q^{79} + 67 q^{81} + 46 q^{83} - 16 q^{85} + 22 q^{87} + 33 q^{89} - 32 q^{91} + q^{93} + 14 q^{95} + 57 q^{97} + 42 q^{99}+O(q^{100}) \) 23 * q + 2 * q^3 - 23 * q^5 + 3 * q^7 + 31 * q^9 + 5 * q^11 - 2 * q^15 + 16 * q^17 - 14 * q^19 + 3 * q^21 + 11 * q^23 + 23 * q^25 + 14 * q^27 + 15 * q^29 - 14 * q^31 + 23 * q^33 - 3 * q^35 + 10 * q^37 - 5 * q^39 + 28 * q^41 + q^43 - 31 * q^45 + 24 * q^47 + 40 * q^49 - 6 * q^51 - 11 * q^53 - 5 * q^55 + 42 * q^57 - 9 * q^59 - 14 * q^61 + 33 * q^63 + 9 * q^67 - 7 * q^69 + 22 * q^71 + 29 * q^73 + 2 * q^75 - 10 * q^79 + 67 * q^81 + 46 * q^83 - 16 * q^85 + 22 * q^87 + 33 * q^89 - 32 * q^91 + q^93 + 14 * q^95 + 57 * q^97 + 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more