LMFDB - Embedded newform 6040.2.a.m.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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 18 x^{10} + 54 x^{9} + 110 x^{8} - 335 x^{7} - 258 x^{6} + 825 x^{5} + 168 x^{4} + \cdots - 16 $ x^12 - 3x^11 - 18x^10 + 54x^9 + 110x^8 - 335x^7 - 258x^6 + 825x^5 + 168x^4 - 669x^3 + 39x^2 + 106*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.983156$ 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)

$f(q)$	$=$	$q+0.983156 q^{3} -1.00000 q^{5} -2.43489 q^{7} -2.03340 q^{9} +O(q^{10})$	$q+0.983156 q^{3} -1.00000 q^{5} -2.43489 q^{7} -2.03340 q^{9} +4.69188 q^{11} +1.64591 q^{13} -0.983156 q^{15} +0.351105 q^{17} -1.05554 q^{19} -2.39388 q^{21} +0.237686 q^{23} +1.00000 q^{25} -4.94862 q^{27} +10.7154 q^{29} +2.17061 q^{31} +4.61285 q^{33} +2.43489 q^{35} -7.57809 q^{37} +1.61818 q^{39} -9.89699 q^{41} -0.895195 q^{43} +2.03340 q^{45} +3.18773 q^{47} -1.07132 q^{49} +0.345191 q^{51} +7.34717 q^{53} -4.69188 q^{55} -1.03776 q^{57} -9.06435 q^{59} -2.85567 q^{61} +4.95111 q^{63} -1.64591 q^{65} +10.6840 q^{67} +0.233682 q^{69} -11.9942 q^{71} +8.27468 q^{73} +0.983156 q^{75} -11.4242 q^{77} +7.87957 q^{79} +1.23494 q^{81} -4.21882 q^{83} -0.351105 q^{85} +10.5350 q^{87} +12.9005 q^{89} -4.00760 q^{91} +2.13404 q^{93} +1.05554 q^{95} +18.7121 q^{97} -9.54048 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) $ 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9	$ 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9} + 10 q^{11} + 11 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} - q^{21} + 18 q^{23} + 12 q^{25} + 9 q^{27} + 16 q^{29} - q^{31} + 8 q^{33} - 5 q^{35} + 2 q^{37} + 6 q^{39} + 4 q^{41} + 7 q^{43} - 9 q^{45} + 3 q^{49} - 4 q^{51} + 39 q^{53} - 10 q^{55} - 15 q^{57} - 4 q^{59} - 32 q^{61} + 3 q^{63} - 11 q^{65} + 4 q^{67} + 12 q^{69} + 24 q^{71} - 10 q^{73} + 3 q^{75} + 38 q^{77} + 32 q^{79} - 8 q^{81} + 9 q^{83} + 4 q^{85} + 3 q^{87} + 15 q^{89} + 18 q^{91} + 36 q^{93} - 5 q^{95} + 15 q^{97} + 28 q^{99}+O(q^{100}) $ 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9 + 10 * q^11 + 11 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 - q^21 + 18 * q^23 + 12 * q^25 + 9 * q^27 + 16 * q^29 - q^31 + 8 * q^33 - 5 * q^35 + 2 * q^37 + 6 * q^39 + 4 * q^41 + 7 * q^43 - 9 * q^45 + 3 * q^49 - 4 * q^51 + 39 * q^53 - 10 * q^55 - 15 * q^57 - 4 * q^59 - 32 * q^61 + 3 * q^63 - 11 * q^65 + 4 * q^67 + 12 * q^69 + 24 * q^71 - 10 * q^73 + 3 * q^75 + 38 * q^77 + 32 * q^79 - 8 * q^81 + 9 * q^83 + 4 * q^85 + 3 * q^87 + 15 * q^89 + 18 * q^91 + 36 * q^93 - 5 * q^95 + 15 * q^97 + 28 * 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
$3$	0.983156	0.567626	0.283813	−	0.958880i	$-0.408401\pi$
$3$	0.983156	0.567626	0.283813	+	0.958880i	$0.408401\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.43489	−0.920301	−0.460150	−	0.887841i	$-0.652205\pi$
$7$	−2.43489	−0.920301	−0.460150	+	0.887841i	$0.652205\pi$
$8$	0	0
$9$	−2.03340	−0.677801
$10$	0	0
$11$	4.69188	1.41465	0.707327	−	0.706886i	$-0.249901\pi$
$11$	4.69188	1.41465	0.707327	+	0.706886i	$0.249901\pi$
$12$	0	0
$13$	1.64591	0.456492	0.228246	−	0.973603i	$-0.426701\pi$
$13$	1.64591	0.456492	0.228246	+	0.973603i	$0.426701\pi$
$14$	0	0
$15$	−0.983156	−0.253850
$16$	0	0
$17$	0.351105	0.0851554	0.0425777	−	0.999093i	$-0.486443\pi$
$17$	0.351105	0.0851554	0.0425777	+	0.999093i	$0.486443\pi$
$18$	0	0
$19$	−1.05554	−0.242158	−0.121079	−	0.992643i	$-0.538635\pi$
$19$	−1.05554	−0.242158	−0.121079	+	0.992643i	$0.538635\pi$
$20$	0	0
$21$	−2.39388	−0.522386
$22$	0	0
$23$	0.237686	0.0495609	0.0247805	−	0.999693i	$-0.492111\pi$
$23$	0.237686	0.0495609	0.0247805	+	0.999693i	$0.492111\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.94862	−0.952363
$28$	0	0
$29$	10.7154	1.98981	0.994904	−	0.100824i	$-0.0321479\pi$
$29$	10.7154	1.98981	0.994904	+	0.100824i	$0.0321479\pi$
$30$	0	0
$31$	2.17061	0.389852	0.194926	−	0.980818i	$-0.437553\pi$
$31$	2.17061	0.389852	0.194926	+	0.980818i	$0.437553\pi$
$32$	0	0
$33$	4.61285	0.802994
$34$	0	0
$35$	2.43489	0.411571
$36$	0	0
$37$	−7.57809	−1.24583	−0.622915	−	0.782289i	$-0.714052\pi$
$37$	−7.57809	−1.24583	−0.622915	+	0.782289i	$0.714052\pi$
$38$	0	0
$39$	1.61818	0.259117
$40$	0	0
$41$	−9.89699	−1.54565	−0.772825	−	0.634619i	$-0.781157\pi$
$41$	−9.89699	−1.54565	−0.772825	+	0.634619i	$0.781157\pi$
$42$	0	0
$43$	−0.895195	−0.136516	−0.0682580	−	0.997668i	$-0.521744\pi$
$43$	−0.895195	−0.136516	−0.0682580	+	0.997668i	$0.521744\pi$
$44$	0	0
$45$	2.03340	0.303122
$46$	0	0
$47$	3.18773	0.464978	0.232489	−	0.972599i	$-0.425313\pi$
$47$	3.18773	0.464978	0.232489	+	0.972599i	$0.425313\pi$
$48$	0	0
$49$	−1.07132	−0.153046
$50$	0	0
$51$	0.345191	0.0483364
$52$	0	0
$53$	7.34717	1.00921	0.504605	−	0.863350i	$-0.331638\pi$
$53$	7.34717	1.00921	0.504605	+	0.863350i	$0.331638\pi$
$54$	0	0
$55$	−4.69188	−0.632653
$56$	0	0
$57$	−1.03776	−0.137455
$58$	0	0
$59$	−9.06435	−1.18008	−0.590039	−	0.807375i	$-0.700888\pi$
$59$	−9.06435	−1.18008	−0.590039	+	0.807375i	$0.700888\pi$
$60$	0	0
$61$	−2.85567	−0.365631	−0.182815	−	0.983147i	$-0.558521\pi$
$61$	−2.85567	−0.365631	−0.182815	+	0.983147i	$0.558521\pi$
$62$	0	0
$63$	4.95111	0.623781
$64$	0	0
$65$	−1.64591	−0.204150
$66$	0	0
$67$	10.6840	1.30526	0.652631	−	0.757676i	$-0.273665\pi$
$67$	10.6840	1.30526	0.652631	+	0.757676i	$0.273665\pi$
$68$	0	0
$69$	0.233682	0.0281321
$70$	0	0
$71$	−11.9942	−1.42345	−0.711723	−	0.702461i	$-0.752085\pi$
$71$	−11.9942	−1.42345	−0.711723	+	0.702461i	$0.752085\pi$
$72$	0	0
$73$	8.27468	0.968478	0.484239	−	0.874936i	$-0.339097\pi$
$73$	8.27468	0.968478	0.484239	+	0.874936i	$0.339097\pi$
$74$	0	0
$75$	0.983156	0.113525
$76$	0	0
$77$	−11.4242	−1.30191
$78$	0	0
$79$	7.87957	0.886521	0.443261	−	0.896393i	$-0.353822\pi$
$79$	7.87957	0.886521	0.443261	+	0.896393i	$0.353822\pi$
$80$	0	0
$81$	1.23494	0.137216
$82$	0	0
$83$	−4.21882	−0.463076	−0.231538	−	0.972826i	$-0.574376\pi$
$83$	−4.21882	−0.463076	−0.231538	+	0.972826i	$0.574376\pi$
$84$	0	0
$85$	−0.351105	−0.0380827
$86$	0	0
$87$	10.5350	1.12947
$88$	0	0
$89$	12.9005	1.36745	0.683726	−	0.729739i	$-0.260358\pi$
$89$	12.9005	1.36745	0.683726	+	0.729739i	$0.260358\pi$
$90$	0	0
$91$	−4.00760	−0.420110
$92$	0	0
$93$	2.13404	0.221290
$94$	0	0
$95$	1.05554	0.108296
$96$	0	0
$97$	18.7121	1.89993	0.949964	−	0.312360i	$-0.101119\pi$
$97$	18.7121	1.89993	0.949964	+	0.312360i	$0.101119\pi$
$98$	0	0
$99$	−9.54048	−0.958855
$100$	0	0
$101$	−12.5709	−1.25086	−0.625428	−	0.780282i	$-0.715076\pi$
$101$	−12.5709	−1.25086	−0.625428	+	0.780282i	$0.715076\pi$
$102$	0	0
$103$	14.1663	1.39585	0.697926	−	0.716170i	$-0.254107\pi$
$103$	14.1663	1.39585	0.697926	+	0.716170i	$0.254107\pi$
$104$	0	0
$105$	2.39388	0.233618
$106$	0	0
$107$	5.66740	0.547888	0.273944	−	0.961746i	$-0.411672\pi$
$107$	5.66740	0.547888	0.273944	+	0.961746i	$0.411672\pi$
$108$	0	0
$109$	14.0158	1.34247	0.671236	−	0.741244i	$-0.265764\pi$
$109$	14.0158	1.34247	0.671236	+	0.741244i	$0.265764\pi$
$110$	0	0
$111$	−7.45045	−0.707165
$112$	0	0
$113$	−0.598905	−0.0563402	−0.0281701	−	0.999603i	$-0.508968\pi$
$113$	−0.598905	−0.0563402	−0.0281701	+	0.999603i	$0.508968\pi$
$114$	0	0
$115$	−0.237686	−0.0221643
$116$	0	0
$117$	−3.34679	−0.309411
$118$	0	0
$119$	−0.854901	−0.0783686
$120$	0	0
$121$	11.0137	1.00125
$122$	0	0
$123$	−9.73029	−0.877351
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.18758	0.726530	0.363265	−	0.931686i	$-0.381662\pi$
$127$	8.18758	0.726530	0.363265	+	0.931686i	$0.381662\pi$
$128$	0	0
$129$	−0.880117	−0.0774900
$130$	0	0
$131$	−9.45376	−0.825979	−0.412990	−	0.910736i	$-0.635515\pi$
$131$	−9.45376	−0.825979	−0.412990	+	0.910736i	$0.635515\pi$
$132$	0	0
$133$	2.57013	0.222858
$134$	0	0
$135$	4.94862	0.425910
$136$	0	0
$137$	3.17183	0.270988	0.135494	−	0.990778i	$-0.456738\pi$
$137$	3.17183	0.270988	0.135494	+	0.990778i	$0.456738\pi$
$138$	0	0
$139$	4.38423	0.371866	0.185933	−	0.982562i	$-0.440469\pi$
$139$	4.38423	0.371866	0.185933	+	0.982562i	$0.440469\pi$
$140$	0	0
$141$	3.13404	0.263933
$142$	0	0
$143$	7.72239	0.645779
$144$	0	0
$145$	−10.7154	−0.889869
$146$	0	0
$147$	−1.05328	−0.0868729
$148$	0	0
$149$	11.9971	0.982838	0.491419	−	0.870923i	$-0.336478\pi$
$149$	11.9971	0.982838	0.491419	+	0.870923i	$0.336478\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−0.713938	−0.0577185
$154$	0	0
$155$	−2.17061	−0.174347
$156$	0	0
$157$	−0.122257	−0.00975721	−0.00487860	−	0.999988i	$-0.501553\pi$
$157$	−0.122257	−0.00975721	−0.00487860	+	0.999988i	$0.501553\pi$
$158$	0	0
$159$	7.22341	0.572854
$160$	0	0
$161$	−0.578738	−0.0456110
$162$	0	0
$163$	10.1159	0.792338	0.396169	−	0.918178i	$-0.370339\pi$
$163$	10.1159	0.792338	0.396169	+	0.918178i	$0.370339\pi$
$164$	0	0
$165$	−4.61285	−0.359110
$166$	0	0
$167$	7.17879	0.555511	0.277756	−	0.960652i	$-0.410409\pi$
$167$	7.17879	0.555511	0.277756	+	0.960652i	$0.410409\pi$
$168$	0	0
$169$	−10.2910	−0.791615
$170$	0	0
$171$	2.14634	0.164135
$172$	0	0
$173$	−16.5794	−1.26051	−0.630256	−	0.776388i	$-0.717050\pi$
$173$	−16.5794	−1.26051	−0.630256	+	0.776388i	$0.717050\pi$
$174$	0	0
$175$	−2.43489	−0.184060
$176$	0	0
$177$	−8.91167	−0.669842
$178$	0	0
$179$	20.5611	1.53681	0.768403	−	0.639966i	$-0.221052\pi$
$179$	20.5611	1.53681	0.768403	+	0.639966i	$0.221052\pi$
$180$	0	0
$181$	22.2370	1.65287	0.826433	−	0.563034i	$-0.190366\pi$
$181$	22.2370	1.65287	0.826433	+	0.563034i	$0.190366\pi$
$182$	0	0
$183$	−2.80757	−0.207541
$184$	0	0
$185$	7.57809	0.557152
$186$	0	0
$187$	1.64734	0.120466
$188$	0	0
$189$	12.0493	0.876460
$190$	0	0
$191$	3.75159	0.271456	0.135728	−	0.990746i	$-0.456663\pi$
$191$	3.75159	0.271456	0.135728	+	0.990746i	$0.456663\pi$
$192$	0	0
$193$	1.27690	0.0919136	0.0459568	−	0.998943i	$-0.485366\pi$
$193$	1.27690	0.0919136	0.0459568	+	0.998943i	$0.485366\pi$
$194$	0	0
$195$	−1.61818	−0.115881
$196$	0	0
$197$	−2.96400	−0.211176	−0.105588	−	0.994410i	$-0.533673\pi$
$197$	−2.96400	−0.211176	−0.105588	+	0.994410i	$0.533673\pi$
$198$	0	0
$199$	21.4111	1.51779	0.758895	−	0.651214i	$-0.225740\pi$
$199$	21.4111	1.51779	0.758895	+	0.651214i	$0.225740\pi$
$200$	0	0
$201$	10.5041	0.740900
$202$	0	0
$203$	−26.0909	−1.83122
$204$	0	0
$205$	9.89699	0.691236
$206$	0	0
$207$	−0.483311	−0.0335925
$208$	0	0
$209$	−4.95248	−0.342570
$210$	0	0
$211$	−2.98340	−0.205386	−0.102693	−	0.994713i	$-0.532746\pi$
$211$	−2.98340	−0.205386	−0.102693	+	0.994713i	$0.532746\pi$
$212$	0	0
$213$	−11.7921	−0.807984
$214$	0	0
$215$	0.895195	0.0610518
$216$	0	0
$217$	−5.28518	−0.358781
$218$	0	0
$219$	8.13530	0.549733
$220$	0	0
$221$	0.577886	0.0388728
$222$	0	0
$223$	14.1986	0.950809	0.475405	−	0.879767i	$-0.342302\pi$
$223$	14.1986	0.950809	0.475405	+	0.879767i	$0.342302\pi$
$224$	0	0
$225$	−2.03340	−0.135560
$226$	0	0
$227$	27.0313	1.79413	0.897065	−	0.441900i	$-0.145695\pi$
$227$	27.0313	1.79413	0.897065	+	0.441900i	$0.145695\pi$
$228$	0	0
$229$	−3.72607	−0.246225	−0.123113	−	0.992393i	$-0.539288\pi$
$229$	−3.72607	−0.246225	−0.123113	+	0.992393i	$0.539288\pi$
$230$	0	0
$231$	−11.2318	−0.738996
$232$	0	0
$233$	25.0509	1.64114	0.820570	−	0.571546i	$-0.193656\pi$
$233$	25.0509	1.64114	0.820570	+	0.571546i	$0.193656\pi$
$234$	0	0
$235$	−3.18773	−0.207945
$236$	0	0
$237$	7.74685	0.503212
$238$	0	0
$239$	12.4779	0.807126	0.403563	−	0.914952i	$-0.367772\pi$
$239$	12.4779	0.807126	0.403563	+	0.914952i	$0.367772\pi$
$240$	0	0
$241$	−12.6966	−0.817857	−0.408928	−	0.912566i	$-0.634097\pi$
$241$	−12.6966	−0.817857	−0.408928	+	0.912566i	$0.634097\pi$
$242$	0	0
$243$	16.0600	1.03025
$244$	0	0
$245$	1.07132	0.0684443
$246$	0	0
$247$	−1.73732	−0.110543
$248$	0	0
$249$	−4.14776	−0.262854
$250$	0	0
$251$	−26.9115	−1.69864	−0.849319	−	0.527880i	$-0.822987\pi$
$251$	−26.9115	−1.69864	−0.849319	+	0.527880i	$0.822987\pi$
$252$	0	0
$253$	1.11519	0.0701116
$254$	0	0
$255$	−0.345191	−0.0216167
$256$	0	0
$257$	−2.64921	−0.165253	−0.0826264	−	0.996581i	$-0.526331\pi$
$257$	−2.64921	−0.165253	−0.0826264	+	0.996581i	$0.526331\pi$
$258$	0	0
$259$	18.4518	1.14654
$260$	0	0
$261$	−21.7888	−1.34869
$262$	0	0
$263$	30.3915	1.87402	0.937010	−	0.349303i	$-0.113582\pi$
$263$	30.3915	1.87402	0.937010	+	0.349303i	$0.113582\pi$
$264$	0	0
$265$	−7.34717	−0.451333
$266$	0	0
$267$	12.6832	0.776200
$268$	0	0
$269$	25.0076	1.52474	0.762370	−	0.647141i	$-0.224036\pi$
$269$	25.0076	1.52474	0.762370	+	0.647141i	$0.224036\pi$
$270$	0	0
$271$	−1.52045	−0.0923609	−0.0461804	−	0.998933i	$-0.514705\pi$
$271$	−1.52045	−0.0923609	−0.0461804	+	0.998933i	$0.514705\pi$
$272$	0	0
$273$	−3.94009	−0.238465
$274$	0	0
$275$	4.69188	0.282931
$276$	0	0
$277$	16.5379	0.993664	0.496832	−	0.867847i	$-0.334497\pi$
$277$	16.5379	0.993664	0.496832	+	0.867847i	$0.334497\pi$
$278$	0	0
$279$	−4.41372	−0.264242
$280$	0	0
$281$	5.78231	0.344944	0.172472	−	0.985014i	$-0.444825\pi$
$281$	5.78231	0.344944	0.172472	+	0.985014i	$0.444825\pi$
$282$	0	0
$283$	−12.4580	−0.740549	−0.370275	−	0.928922i	$-0.620736\pi$
$283$	−12.4580	−0.740549	−0.370275	+	0.928922i	$0.620736\pi$
$284$	0	0
$285$	1.03776	0.0614718
$286$	0	0
$287$	24.0981	1.42246
$288$	0	0
$289$	−16.8767	−0.992749
$290$	0	0
$291$	18.3969	1.07845
$292$	0	0
$293$	16.9527	0.990387	0.495193	−	0.868783i	$-0.335097\pi$
$293$	16.9527	0.990387	0.495193	+	0.868783i	$0.335097\pi$
$294$	0	0
$295$	9.06435	0.527747
$296$	0	0
$297$	−23.2183	−1.34726
$298$	0	0
$299$	0.391209	0.0226242
$300$	0	0
$301$	2.17970	0.125636
$302$	0	0
$303$	−12.3592	−0.710018
$304$	0	0
$305$	2.85567	0.163515
$306$	0	0
$307$	0.496534	0.0283387	0.0141694	−	0.999900i	$-0.495490\pi$
$307$	0.496534	0.0283387	0.0141694	+	0.999900i	$0.495490\pi$
$308$	0	0
$309$	13.9277	0.792321
$310$	0	0
$311$	20.0063	1.13445	0.567225	−	0.823563i	$-0.308017\pi$
$311$	20.0063	1.13445	0.567225	+	0.823563i	$0.308017\pi$
$312$	0	0
$313$	−26.4431	−1.49466	−0.747328	−	0.664456i	$-0.768663\pi$
$313$	−26.4431	−1.49466	−0.747328	+	0.664456i	$0.768663\pi$
$314$	0	0
$315$	−4.95111	−0.278963
$316$	0	0
$317$	3.71902	0.208881	0.104440	−	0.994531i	$-0.466695\pi$
$317$	3.71902	0.208881	0.104440	+	0.994531i	$0.466695\pi$
$318$	0	0
$319$	50.2756	2.81489
$320$	0	0
$321$	5.57194	0.310995
$322$	0	0
$323$	−0.370606	−0.0206211
$324$	0	0
$325$	1.64591	0.0912985
$326$	0	0
$327$	13.7797	0.762021
$328$	0	0
$329$	−7.76176	−0.427920
$330$	0	0
$331$	−1.17440	−0.0645506	−0.0322753	−	0.999479i	$-0.510275\pi$
$331$	−1.17440	−0.0645506	−0.0322753	+	0.999479i	$0.510275\pi$
$332$	0	0
$333$	15.4093	0.844425
$334$	0	0
$335$	−10.6840	−0.583731
$336$	0	0
$337$	−5.72030	−0.311605	−0.155802	−	0.987788i	$-0.549796\pi$
$337$	−5.72030	−0.311605	−0.155802	+	0.987788i	$0.549796\pi$
$338$	0	0
$339$	−0.588817	−0.0319801
$340$	0	0
$341$	10.1842	0.551506
$342$	0	0
$343$	19.6528	1.06115
$344$	0	0
$345$	−0.233682	−0.0125810
$346$	0	0
$347$	−33.1210	−1.77803	−0.889014	−	0.457879i	$-0.848609\pi$
$347$	−33.1210	−1.77803	−0.889014	+	0.457879i	$0.848609\pi$
$348$	0	0
$349$	−5.18679	−0.277642	−0.138821	−	0.990317i	$-0.544331\pi$
$349$	−5.18679	−0.277642	−0.138821	+	0.990317i	$0.544331\pi$
$350$	0	0
$351$	−8.14497	−0.434746
$352$	0	0
$353$	−1.54366	−0.0821607	−0.0410803	−	0.999156i	$-0.513080\pi$
$353$	−1.54366	−0.0821607	−0.0410803	+	0.999156i	$0.513080\pi$
$354$	0	0
$355$	11.9942	0.636584
$356$	0	0
$357$	−0.840501	−0.0444840
$358$	0	0
$359$	22.3218	1.17810	0.589049	−	0.808097i	$-0.299502\pi$
$359$	22.3218	1.17810	0.589049	+	0.808097i	$0.299502\pi$
$360$	0	0
$361$	−17.8858	−0.941360
$362$	0	0
$363$	10.8282	0.568334
$364$	0	0
$365$	−8.27468	−0.433116
$366$	0	0
$367$	−28.7558	−1.50104	−0.750520	−	0.660848i	$-0.770197\pi$
$367$	−28.7558	−1.50104	−0.750520	+	0.660848i	$0.770197\pi$
$368$	0	0
$369$	20.1246	1.04764
$370$	0	0
$371$	−17.8895	−0.928778
$372$	0	0
$373$	−12.0620	−0.624548	−0.312274	−	0.949992i	$-0.601091\pi$
$373$	−12.0620	−0.624548	−0.312274	+	0.949992i	$0.601091\pi$
$374$	0	0
$375$	−0.983156	−0.0507700
$376$	0	0
$377$	17.6366	0.908332
$378$	0	0
$379$	−36.4441	−1.87201	−0.936005	−	0.351987i	$-0.885506\pi$
$379$	−36.4441	−1.87201	−0.936005	+	0.351987i	$0.885506\pi$
$380$	0	0
$381$	8.04967	0.412397
$382$	0	0
$383$	−4.90950	−0.250864	−0.125432	−	0.992102i	$-0.540032\pi$
$383$	−4.90950	−0.250864	−0.125432	+	0.992102i	$0.540032\pi$
$384$	0	0
$385$	11.4242	0.582231
$386$	0	0
$387$	1.82029	0.0925307
$388$	0	0
$389$	−15.6690	−0.794451	−0.397226	−	0.917721i	$-0.630027\pi$
$389$	−15.6690	−0.794451	−0.397226	+	0.917721i	$0.630027\pi$
$390$	0	0
$391$	0.0834527	0.00422038
$392$	0	0
$393$	−9.29453	−0.468847
$394$	0	0
$395$	−7.87957	−0.396464
$396$	0	0
$397$	−19.4041	−0.973862	−0.486931	−	0.873440i	$-0.661884\pi$
$397$	−19.4041	−0.973862	−0.486931	+	0.873440i	$0.661884\pi$
$398$	0	0
$399$	2.52684	0.126500
$400$	0	0
$401$	16.6271	0.830316	0.415158	−	0.909749i	$-0.363726\pi$
$401$	16.6271	0.830316	0.415158	+	0.909749i	$0.363726\pi$
$402$	0	0
$403$	3.57261	0.177965
$404$	0	0
$405$	−1.23494	−0.0613647
$406$	0	0
$407$	−35.5555	−1.76242
$408$	0	0
$409$	−33.7883	−1.67072	−0.835362	−	0.549700i	$-0.814742\pi$
$409$	−33.7883	−1.67072	−0.835362	+	0.549700i	$0.814742\pi$
$410$	0	0
$411$	3.11841	0.153820
$412$	0	0
$413$	22.0707	1.08603
$414$	0	0
$415$	4.21882	0.207094
$416$	0	0
$417$	4.31038	0.211080
$418$	0	0
$419$	−11.6173	−0.567542	−0.283771	−	0.958892i	$-0.591586\pi$
$419$	−11.6173	−0.567542	−0.283771	+	0.958892i	$0.591586\pi$
$420$	0	0
$421$	−28.5920	−1.39349	−0.696745	−	0.717319i	$-0.745369\pi$
$421$	−28.5920	−1.39349	−0.696745	+	0.717319i	$0.745369\pi$
$422$	0	0
$423$	−6.48194	−0.315163
$424$	0	0
$425$	0.351105	0.0170311
$426$	0	0
$427$	6.95323	0.336490
$428$	0	0
$429$	7.59232	0.366561
$430$	0	0
$431$	20.4690	0.985957	0.492979	−	0.870041i	$-0.335908\pi$
$431$	20.4690	0.985957	0.492979	+	0.870041i	$0.335908\pi$
$432$	0	0
$433$	−15.5603	−0.747781	−0.373891	−	0.927473i	$-0.621976\pi$
$433$	−15.5603	−0.747781	−0.373891	+	0.927473i	$0.621976\pi$
$434$	0	0
$435$	−10.5350	−0.505113
$436$	0	0
$437$	−0.250887	−0.0120016
$438$	0	0
$439$	−29.2067	−1.39396	−0.696981	−	0.717090i	$-0.745474\pi$
$439$	−29.2067	−1.39396	−0.696981	+	0.717090i	$0.745474\pi$
$440$	0	0
$441$	2.17843	0.103735
$442$	0	0
$443$	−0.690099	−0.0327876	−0.0163938	−	0.999866i	$-0.505219\pi$
$443$	−0.690099	−0.0327876	−0.0163938	+	0.999866i	$0.505219\pi$
$444$	0	0
$445$	−12.9005	−0.611543
$446$	0	0
$447$	11.7950	0.557884
$448$	0	0
$449$	−13.0757	−0.617083	−0.308541	−	0.951211i	$-0.599841\pi$
$449$	−13.0757	−0.617083	−0.308541	+	0.951211i	$0.599841\pi$
$450$	0	0
$451$	−46.4355	−2.18656
$452$	0	0
$453$	0.983156	0.0461927
$454$	0	0
$455$	4.00760	0.187879
$456$	0	0
$457$	29.7939	1.39370	0.696850	−	0.717217i	$-0.254584\pi$
$457$	29.7939	1.39370	0.696850	+	0.717217i	$0.254584\pi$
$458$	0	0
$459$	−1.73749	−0.0810989
$460$	0	0
$461$	2.25313	0.104939	0.0524693	−	0.998623i	$-0.483291\pi$
$461$	2.25313	0.104939	0.0524693	+	0.998623i	$0.483291\pi$
$462$	0	0
$463$	34.4685	1.60189	0.800944	−	0.598739i	$-0.204331\pi$
$463$	34.4685	1.60189	0.800944	+	0.598739i	$0.204331\pi$
$464$	0	0
$465$	−2.13404	−0.0989640
$466$	0	0
$467$	4.65536	0.215425	0.107712	−	0.994182i	$-0.465647\pi$
$467$	4.65536	0.215425	0.107712	+	0.994182i	$0.465647\pi$
$468$	0	0
$469$	−26.0144	−1.20123
$470$	0	0
$471$	−0.120198	−0.00553844
$472$	0	0
$473$	−4.20015	−0.193123
$474$	0	0
$475$	−1.05554	−0.0484316
$476$	0	0
$477$	−14.9398	−0.684044
$478$	0	0
$479$	−21.0261	−0.960709	−0.480355	−	0.877074i	$-0.659492\pi$
$479$	−21.0261	−0.960709	−0.480355	+	0.877074i	$0.659492\pi$
$480$	0	0
$481$	−12.4728	−0.568712
$482$	0	0
$483$	−0.568990	−0.0258900
$484$	0	0
$485$	−18.7121	−0.849674
$486$	0	0
$487$	−15.5723	−0.705648	−0.352824	−	0.935690i	$-0.614779\pi$
$487$	−15.5723	−0.705648	−0.352824	+	0.935690i	$0.614779\pi$
$488$	0	0
$489$	9.94551	0.449751
$490$	0	0
$491$	−29.6177	−1.33663	−0.668314	−	0.743879i	$-0.732984\pi$
$491$	−29.6177	−1.33663	−0.668314	+	0.743879i	$0.732984\pi$
$492$	0	0
$493$	3.76225	0.169443
$494$	0	0
$495$	9.54048	0.428813
$496$	0	0
$497$	29.2044	1.31000
$498$	0	0
$499$	20.3147	0.909411	0.454706	−	0.890642i	$-0.349745\pi$
$499$	20.3147	0.909411	0.454706	+	0.890642i	$0.349745\pi$
$500$	0	0
$501$	7.05787	0.315322
$502$	0	0
$503$	6.21712	0.277208	0.138604	−	0.990348i	$-0.455739\pi$
$503$	6.21712	0.277208	0.138604	+	0.990348i	$0.455739\pi$
$504$	0	0
$505$	12.5709	0.559400
$506$	0	0
$507$	−10.1177	−0.449341
$508$	0	0
$509$	−39.5195	−1.75167	−0.875836	−	0.482609i	$-0.839689\pi$
$509$	−39.5195	−1.75167	−0.875836	+	0.482609i	$0.839689\pi$
$510$	0	0
$511$	−20.1479	−0.891291
$512$	0	0
$513$	5.22348	0.230622
$514$	0	0
$515$	−14.1663	−0.624244
$516$	0	0
$517$	14.9564	0.657783
$518$	0	0
$519$	−16.3002	−0.715499
$520$	0	0
$521$	−4.60604	−0.201794	−0.100897	−	0.994897i	$-0.532171\pi$
$521$	−4.60604	−0.201794	−0.100897	+	0.994897i	$0.532171\pi$
$522$	0	0
$523$	−9.15922	−0.400505	−0.200252	−	0.979744i	$-0.564176\pi$
$523$	−9.15922	−0.400505	−0.200252	+	0.979744i	$0.564176\pi$
$524$	0	0
$525$	−2.39388	−0.104477
$526$	0	0
$527$	0.762110	0.0331980
$528$	0	0
$529$	−22.9435	−0.997544
$530$	0	0
$531$	18.4315	0.799858
$532$	0	0
$533$	−16.2895	−0.705578
$534$	0	0
$535$	−5.66740	−0.245023
$536$	0	0
$537$	20.2147	0.872330
$538$	0	0
$539$	−5.02652	−0.216507
$540$	0	0
$541$	30.5972	1.31547	0.657737	−	0.753248i	$-0.271514\pi$
$541$	30.5972	1.31547	0.657737	+	0.753248i	$0.271514\pi$
$542$	0	0
$543$	21.8625	0.938209
$544$	0	0
$545$	−14.0158	−0.600371
$546$	0	0
$547$	−20.7301	−0.886353	−0.443176	−	0.896434i	$-0.646149\pi$
$547$	−20.7301	−0.886353	−0.443176	+	0.896434i	$0.646149\pi$
$548$	0	0
$549$	5.80672	0.247825
$550$	0	0
$551$	−11.3106	−0.481848
$552$	0	0
$553$	−19.1859	−0.815866
$554$	0	0
$555$	7.45045	0.316254
$556$	0	0
$557$	−27.6139	−1.17004	−0.585019	−	0.811020i	$-0.698913\pi$
$557$	−27.6139	−1.17004	−0.585019	+	0.811020i	$0.698913\pi$
$558$	0	0
$559$	−1.47341	−0.0623185
$560$	0	0
$561$	1.61959	0.0683793
$562$	0	0
$563$	37.0241	1.56038	0.780189	−	0.625544i	$-0.215123\pi$
$563$	37.0241	1.56038	0.780189	+	0.625544i	$0.215123\pi$
$564$	0	0
$565$	0.598905	0.0251961
$566$	0	0
$567$	−3.00694	−0.126280
$568$	0	0
$569$	39.6535	1.66236	0.831180	−	0.556003i	$-0.187666\pi$
$569$	39.6535	1.66236	0.831180	+	0.556003i	$0.187666\pi$
$570$	0	0
$571$	46.3918	1.94144	0.970718	−	0.240223i	$-0.0772207\pi$
$571$	46.3918	1.94144	0.970718	+	0.240223i	$0.0772207\pi$
$572$	0	0
$573$	3.68840	0.154085
$574$	0	0
$575$	0.237686	0.00991219
$576$	0	0
$577$	−31.8817	−1.32725	−0.663626	−	0.748065i	$-0.730983\pi$
$577$	−31.8817	−1.32725	−0.663626	+	0.748065i	$0.730983\pi$
$578$	0	0
$579$	1.25540	0.0521725
$580$	0	0
$581$	10.2724	0.426169
$582$	0	0
$583$	34.4720	1.42768
$584$	0	0
$585$	3.34679	0.138373
$586$	0	0
$587$	−10.2175	−0.421721	−0.210861	−	0.977516i	$-0.567627\pi$
$587$	−10.2175	−0.421721	−0.210861	+	0.977516i	$0.567627\pi$
$588$	0	0
$589$	−2.29117	−0.0944058
$590$	0	0
$591$	−2.91408	−0.119869
$592$	0	0
$593$	38.2280	1.56983	0.784917	−	0.619600i	$-0.212705\pi$
$593$	38.2280	1.56983	0.784917	+	0.619600i	$0.212705\pi$
$594$	0	0
$595$	0.854901	0.0350475
$596$	0	0
$597$	21.0504	0.861536
$598$	0	0
$599$	23.4678	0.958868	0.479434	−	0.877578i	$-0.340842\pi$
$599$	23.4678	0.958868	0.479434	+	0.877578i	$0.340842\pi$
$600$	0	0
$601$	−32.8822	−1.34129	−0.670647	−	0.741776i	$-0.733984\pi$
$601$	−32.8822	−1.34129	−0.670647	+	0.741776i	$0.733984\pi$
$602$	0	0
$603$	−21.7249	−0.884708
$604$	0	0
$605$	−11.0137	−0.447772
$606$	0	0
$607$	37.7564	1.53248	0.766242	−	0.642552i	$-0.222124\pi$
$607$	37.7564	1.53248	0.766242	+	0.642552i	$0.222124\pi$
$608$	0	0
$609$	−25.6514	−1.03945
$610$	0	0
$611$	5.24670	0.212259
$612$	0	0
$613$	9.13196	0.368836	0.184418	−	0.982848i	$-0.440960\pi$
$613$	9.13196	0.368836	0.184418	+	0.982848i	$0.440960\pi$
$614$	0	0
$615$	9.73029	0.392363
$616$	0	0
$617$	−3.05500	−0.122990	−0.0614948	−	0.998107i	$-0.519587\pi$
$617$	−3.05500	−0.122990	−0.0614948	+	0.998107i	$0.519587\pi$
$618$	0	0
$619$	37.9598	1.52573	0.762866	−	0.646557i	$-0.223792\pi$
$619$	37.9598	1.52573	0.762866	+	0.646557i	$0.223792\pi$
$620$	0	0
$621$	−1.17622	−0.0472000
$622$	0	0
$623$	−31.4113	−1.25847
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.86906	−0.194451
$628$	0	0
$629$	−2.66070	−0.106089
$630$	0	0
$631$	39.6733	1.57937	0.789684	−	0.613514i	$-0.210244\pi$
$631$	39.6733	1.57937	0.789684	+	0.613514i	$0.210244\pi$
$632$	0	0
$633$	−2.93315	−0.116582
$634$	0	0
$635$	−8.18758	−0.324914
$636$	0	0
$637$	−1.76330	−0.0698644
$638$	0	0
$639$	24.3890	0.964813
$640$	0	0
$641$	−36.1593	−1.42821	−0.714104	−	0.700040i	$-0.753166\pi$
$641$	−36.1593	−1.42821	−0.714104	+	0.700040i	$0.753166\pi$
$642$	0	0
$643$	44.1024	1.73923	0.869614	−	0.493732i	$-0.164368\pi$
$643$	44.1024	1.73923	0.869614	+	0.493732i	$0.164368\pi$
$644$	0	0
$645$	0.880117	0.0346546
$646$	0	0
$647$	−15.7076	−0.617530	−0.308765	−	0.951138i	$-0.599916\pi$
$647$	−15.7076	−0.617530	−0.308765	+	0.951138i	$0.599916\pi$
$648$	0	0
$649$	−42.5288	−1.66940
$650$	0	0
$651$	−5.19616	−0.203654
$652$	0	0
$653$	−1.68614	−0.0659838	−0.0329919	−	0.999456i	$-0.510504\pi$
$653$	−1.68614	−0.0659838	−0.0329919	+	0.999456i	$0.510504\pi$
$654$	0	0
$655$	9.45376	0.369389
$656$	0	0
$657$	−16.8258	−0.656435
$658$	0	0
$659$	10.3060	0.401464	0.200732	−	0.979646i	$-0.435668\pi$
$659$	10.3060	0.401464	0.200732	+	0.979646i	$0.435668\pi$
$660$	0	0
$661$	30.9462	1.20367	0.601834	−	0.798621i	$-0.294437\pi$
$661$	30.9462	1.20367	0.601834	+	0.798621i	$0.294437\pi$
$662$	0	0
$663$	0.568152	0.0220652
$664$	0	0
$665$	−2.57013	−0.0996652
$666$	0	0
$667$	2.54691	0.0986168
$668$	0	0
$669$	13.9595	0.539704
$670$	0	0
$671$	−13.3984	−0.517241
$672$	0	0
$673$	−49.4425	−1.90587	−0.952935	−	0.303176i	$-0.901953\pi$
$673$	−49.4425	−1.90587	−0.952935	+	0.303176i	$0.901953\pi$
$674$	0	0
$675$	−4.94862	−0.190473
$676$	0	0
$677$	−29.5482	−1.13563	−0.567814	−	0.823157i	$-0.692211\pi$
$677$	−29.5482	−1.13563	−0.567814	+	0.823157i	$0.692211\pi$
$678$	0	0
$679$	−45.5619	−1.74851
$680$	0	0
$681$	26.5760	1.01839
$682$	0	0
$683$	18.0679	0.691349	0.345674	−	0.938354i	$-0.387650\pi$
$683$	18.0679	0.691349	0.345674	+	0.938354i	$0.387650\pi$
$684$	0	0
$685$	−3.17183	−0.121190
$686$	0	0
$687$	−3.66330	−0.139764
$688$	0	0
$689$	12.0927	0.460697
$690$	0	0
$691$	42.2767	1.60828	0.804141	−	0.594439i	$-0.202626\pi$
$691$	42.2767	1.60828	0.804141	+	0.594439i	$0.202626\pi$
$692$	0	0
$693$	23.2300	0.882435
$694$	0	0
$695$	−4.38423	−0.166303
$696$	0	0
$697$	−3.47488	−0.131621
$698$	0	0
$699$	24.6290	0.931553
$700$	0	0
$701$	4.32820	0.163474	0.0817370	−	0.996654i	$-0.473953\pi$
$701$	4.32820	0.163474	0.0817370	+	0.996654i	$0.473953\pi$
$702$	0	0
$703$	7.99899	0.301688
$704$	0	0
$705$	−3.13404	−0.118035
$706$	0	0
$707$	30.6088	1.15116
$708$	0	0
$709$	−3.15253	−0.118396	−0.0591979	−	0.998246i	$-0.518854\pi$
$709$	−3.15253	−0.118396	−0.0591979	+	0.998246i	$0.518854\pi$
$710$	0	0
$711$	−16.0223	−0.600885
$712$	0	0
$713$	0.515922	0.0193214
$714$	0	0
$715$	−7.72239	−0.288801
$716$	0	0
$717$	12.2677	0.458146
$718$	0	0
$719$	25.7683	0.960997	0.480499	−	0.876996i	$-0.340456\pi$
$719$	25.7683	0.960997	0.480499	+	0.876996i	$0.340456\pi$
$720$	0	0
$721$	−34.4935	−1.28460
$722$	0	0
$723$	−12.4827	−0.464236
$724$	0	0
$725$	10.7154	0.397962
$726$	0	0
$727$	42.7221	1.58447	0.792237	−	0.610213i	$-0.208916\pi$
$727$	42.7221	1.58447	0.792237	+	0.610213i	$0.208916\pi$
$728$	0	0
$729$	12.0847	0.447581
$730$	0	0
$731$	−0.314307	−0.0116251
$732$	0	0
$733$	−19.9094	−0.735372	−0.367686	−	0.929950i	$-0.619850\pi$
$733$	−19.9094	−0.735372	−0.367686	+	0.929950i	$0.619850\pi$
$734$	0	0
$735$	1.05328	0.0388507
$736$	0	0
$737$	50.1281	1.84649
$738$	0	0
$739$	2.47692	0.0911151	0.0455575	−	0.998962i	$-0.485494\pi$
$739$	2.47692	0.0911151	0.0455575	+	0.998962i	$0.485494\pi$
$740$	0	0
$741$	−1.70806	−0.0627472
$742$	0	0
$743$	−25.2975	−0.928076	−0.464038	−	0.885815i	$-0.653600\pi$
$743$	−25.2975	−0.928076	−0.464038	+	0.885815i	$0.653600\pi$
$744$	0	0
$745$	−11.9971	−0.439539
$746$	0	0
$747$	8.57857	0.313873
$748$	0	0
$749$	−13.7995	−0.504222
$750$	0	0
$751$	−30.8338	−1.12514	−0.562571	−	0.826749i	$-0.690188\pi$
$751$	−30.8338	−1.12514	−0.562571	+	0.826749i	$0.690188\pi$
$752$	0	0
$753$	−26.4582	−0.964190
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	34.2835	1.24605	0.623027	−	0.782200i	$-0.285903\pi$
$757$	34.2835	1.24605	0.623027	+	0.782200i	$0.285903\pi$
$758$	0	0
$759$	1.09641	0.0397971
$760$	0	0
$761$	−30.9065	−1.12036	−0.560180	−	0.828371i	$-0.689268\pi$
$761$	−30.9065	−1.12036	−0.560180	+	0.828371i	$0.689268\pi$
$762$	0	0
$763$	−34.1269	−1.23548
$764$	0	0
$765$	0.713938	0.0258125
$766$	0	0
$767$	−14.9191	−0.538696
$768$	0	0
$769$	−25.6188	−0.923836	−0.461918	−	0.886923i	$-0.652839\pi$
$769$	−25.6188	−0.923836	−0.461918	+	0.886923i	$0.652839\pi$
$770$	0	0
$771$	−2.60458	−0.0938018
$772$	0	0
$773$	33.7210	1.21286	0.606430	−	0.795137i	$-0.292601\pi$
$773$	33.7210	1.21286	0.606430	+	0.795137i	$0.292601\pi$
$774$	0	0
$775$	2.17061	0.0779705
$776$	0	0
$777$	18.1410	0.650805
$778$	0	0
$779$	10.4467	0.374292
$780$	0	0
$781$	−56.2751	−2.01368
$782$	0	0
$783$	−53.0267	−1.89502
$784$	0	0
$785$	0.122257	0.00436356
$786$	0	0
$787$	−5.86096	−0.208921	−0.104460	−	0.994529i	$-0.533312\pi$
$787$	−5.86096	−0.208921	−0.104460	+	0.994529i	$0.533312\pi$
$788$	0	0
$789$	29.8796	1.06374
$790$	0	0
$791$	1.45827	0.0518499
$792$	0	0
$793$	−4.70016	−0.166908
$794$	0	0
$795$	−7.22341	−0.256188
$796$	0	0
$797$	−25.8540	−0.915796	−0.457898	−	0.889005i	$-0.651397\pi$
$797$	−25.8540	−0.915796	−0.457898	+	0.889005i	$0.651397\pi$
$798$	0	0
$799$	1.11923	0.0395954
$800$	0	0
$801$	−26.2319	−0.926860
$802$	0	0
$803$	38.8238	1.37006
$804$	0	0
$805$	0.578738	0.0203978
$806$	0	0
$807$	24.5864	0.865481
$808$	0	0
$809$	−4.35898	−0.153254	−0.0766269	−	0.997060i	$-0.524415\pi$
$809$	−4.35898	−0.153254	−0.0766269	+	0.997060i	$0.524415\pi$
$810$	0	0
$811$	−33.4945	−1.17615	−0.588076	−	0.808805i	$-0.700115\pi$
$811$	−33.4945	−1.17615	−0.588076	+	0.808805i	$0.700115\pi$
$812$	0	0
$813$	−1.49484	−0.0524264
$814$	0	0
$815$	−10.1159	−0.354344
$816$	0	0
$817$	0.944916	0.0330584
$818$	0	0
$819$	8.14906	0.284751
$820$	0	0
$821$	10.4359	0.364216	0.182108	−	0.983279i	$-0.441708\pi$
$821$	10.4359	0.364216	0.182108	+	0.983279i	$0.441708\pi$
$822$	0	0
$823$	−50.1383	−1.74771	−0.873855	−	0.486187i	$-0.838388\pi$
$823$	−50.1383	−1.74771	−0.873855	+	0.486187i	$0.838388\pi$
$824$	0	0
$825$	4.61285	0.160599
$826$	0	0
$827$	15.7514	0.547730	0.273865	−	0.961768i	$-0.411698\pi$
$827$	15.7514	0.547730	0.273865	+	0.961768i	$0.411698\pi$
$828$	0	0
$829$	−1.12627	−0.0391171	−0.0195585	−	0.999809i	$-0.506226\pi$
$829$	−1.12627	−0.0391171	−0.0195585	+	0.999809i	$0.506226\pi$
$830$	0	0
$831$	16.2593	0.564029
$832$	0	0
$833$	−0.376147	−0.0130327
$834$	0	0
$835$	−7.17879	−0.248432
$836$	0	0
$837$	−10.7415	−0.371281
$838$	0	0
$839$	10.1775	0.351365	0.175683	−	0.984447i	$-0.443787\pi$
$839$	10.1775	0.351365	0.175683	+	0.984447i	$0.443787\pi$
$840$	0	0
$841$	85.8208	2.95934
$842$	0	0
$843$	5.68492	0.195799
$844$	0	0
$845$	10.2910	0.354021
$846$	0	0
$847$	−26.8172	−0.921449
$848$	0	0
$849$	−12.2481	−0.420355
$850$	0	0
$851$	−1.80121	−0.0617445
$852$	0	0
$853$	−6.35635	−0.217637	−0.108819	−	0.994062i	$-0.534707\pi$
$853$	−6.35635	−0.217637	−0.108819	+	0.994062i	$0.534707\pi$
$854$	0	0
$855$	−2.14634	−0.0734034
$856$	0	0
$857$	−33.3248	−1.13835	−0.569177	−	0.822215i	$-0.692738\pi$
$857$	−33.3248	−1.13835	−0.569177	+	0.822215i	$0.692738\pi$
$858$	0	0
$859$	27.4563	0.936797	0.468398	−	0.883517i	$-0.344831\pi$
$859$	27.4563	0.936797	0.468398	+	0.883517i	$0.344831\pi$
$860$	0	0
$861$	23.6922	0.807427
$862$	0	0
$863$	26.7938	0.912071	0.456035	−	0.889962i	$-0.349269\pi$
$863$	26.7938	0.912071	0.456035	+	0.889962i	$0.349269\pi$
$864$	0	0
$865$	16.5794	0.563718
$866$	0	0
$867$	−16.5925	−0.563509
$868$	0	0
$869$	36.9700	1.25412
$870$	0	0
$871$	17.5849	0.595842
$872$	0	0
$873$	−38.0493	−1.28777
$874$	0	0
$875$	2.43489	0.0823142
$876$	0	0
$877$	9.81653	0.331481	0.165740	−	0.986169i	$-0.446999\pi$
$877$	9.81653	0.331481	0.165740	+	0.986169i	$0.446999\pi$
$878$	0	0
$879$	16.6671	0.562169
$880$	0	0
$881$	49.9839	1.68400	0.842001	−	0.539476i	$-0.181378\pi$
$881$	49.9839	1.68400	0.842001	+	0.539476i	$0.181378\pi$
$882$	0	0
$883$	−7.71252	−0.259547	−0.129774	−	0.991544i	$-0.541425\pi$
$883$	−7.71252	−0.259547	−0.129774	+	0.991544i	$0.541425\pi$
$884$	0	0
$885$	8.91167	0.299562
$886$	0	0
$887$	3.53442	0.118674	0.0593371	−	0.998238i	$-0.481101\pi$
$887$	3.53442	0.118674	0.0593371	+	0.998238i	$0.481101\pi$
$888$	0	0
$889$	−19.9358	−0.668626
$890$	0	0
$891$	5.79419	0.194113
$892$	0	0
$893$	−3.36478	−0.112598
$894$	0	0
$895$	−20.5611	−0.687281
$896$	0	0
$897$	0.384619	0.0128421
$898$	0	0
$899$	23.2590	0.775731
$900$	0	0
$901$	2.57963	0.0859398
$902$	0	0
$903$	2.14299	0.0713141
$904$	0	0
$905$	−22.2370	−0.739185
$906$	0	0
$907$	18.1593	0.602969	0.301485	−	0.953471i	$-0.402518\pi$
$907$	18.1593	0.602969	0.301485	+	0.953471i	$0.402518\pi$
$908$	0	0
$909$	25.5618	0.847832
$910$	0	0
$911$	39.6112	1.31238	0.656189	−	0.754597i	$-0.272168\pi$
$911$	39.6112	1.31238	0.656189	+	0.754597i	$0.272168\pi$
$912$	0	0
$913$	−19.7942	−0.655092
$914$	0	0
$915$	2.80757	0.0928153
$916$	0	0
$917$	23.0188	0.760149
$918$	0	0
$919$	28.8318	0.951075	0.475537	−	0.879696i	$-0.342254\pi$
$919$	28.8318	0.951075	0.475537	+	0.879696i	$0.342254\pi$
$920$	0	0
$921$	0.488171	0.0160858
$922$	0	0
$923$	−19.7413	−0.649792
$924$	0	0
$925$	−7.57809	−0.249166
$926$	0	0
$927$	−28.8059	−0.946110
$928$	0	0
$929$	−32.0982	−1.05311	−0.526554	−	0.850142i	$-0.676516\pi$
$929$	−32.0982	−1.05311	−0.526554	+	0.850142i	$0.676516\pi$
$930$	0	0
$931$	1.13083	0.0370613
$932$	0	0
$933$	19.6693	0.643943
$934$	0	0
$935$	−1.64734	−0.0538738
$936$	0	0
$937$	−18.7523	−0.612610	−0.306305	−	0.951933i	$-0.599093\pi$
$937$	−18.7523	−0.612610	−0.306305	+	0.951933i	$0.599093\pi$
$938$	0	0
$939$	−25.9977	−0.848405
$940$	0	0
$941$	1.49219	0.0486440	0.0243220	−	0.999704i	$-0.492257\pi$
$941$	1.49219	0.0486440	0.0243220	+	0.999704i	$0.492257\pi$
$942$	0	0
$943$	−2.35238	−0.0766039
$944$	0	0
$945$	−12.0493	−0.391965
$946$	0	0
$947$	−8.73808	−0.283950	−0.141975	−	0.989870i	$-0.545345\pi$
$947$	−8.73808	−0.283950	−0.141975	+	0.989870i	$0.545345\pi$
$948$	0	0
$949$	13.6193	0.442102
$950$	0	0
$951$	3.65637	0.118566
$952$	0	0
$953$	27.1332	0.878932	0.439466	−	0.898259i	$-0.355168\pi$
$953$	27.1332	0.878932	0.439466	+	0.898259i	$0.355168\pi$
$954$	0	0
$955$	−3.75159	−0.121399
$956$	0	0
$957$	49.4288	1.59780
$958$	0	0
$959$	−7.72306	−0.249391
$960$	0	0
$961$	−26.2885	−0.848015
$962$	0	0
$963$	−11.5241	−0.371359
$964$	0	0
$965$	−1.27690	−0.0411050
$966$	0	0
$967$	−2.14586	−0.0690063	−0.0345031	−	0.999405i	$-0.510985\pi$
$967$	−2.14586	−0.0690063	−0.0345031	+	0.999405i	$0.510985\pi$
$968$	0	0
$969$	−0.364364	−0.0117050
$970$	0	0
$971$	−2.94010	−0.0943522	−0.0471761	−	0.998887i	$-0.515022\pi$
$971$	−2.94010	−0.0943522	−0.0471761	+	0.998887i	$0.515022\pi$
$972$	0	0
$973$	−10.6751	−0.342228
$974$	0	0
$975$	1.61818	0.0518233
$976$	0	0
$977$	30.1785	0.965495	0.482748	−	0.875759i	$-0.339639\pi$
$977$	30.1785	0.965495	0.482748	+	0.875759i	$0.339639\pi$
$978$	0	0
$979$	60.5276	1.93447
$980$	0	0
$981$	−28.4998	−0.909929
$982$	0	0
$983$	−2.04407	−0.0651956	−0.0325978	−	0.999469i	$-0.510378\pi$
$983$	−2.04407	−0.0651956	−0.0325978	+	0.999469i	$0.510378\pi$
$984$	0	0
$985$	2.96400	0.0944409
$986$	0	0
$987$	−7.63103	−0.242898
$988$	0	0
$989$	−0.212775	−0.00676586
$990$	0	0
$991$	25.0929	0.797102	0.398551	−	0.917146i	$-0.369513\pi$
$991$	25.0929	0.797102	0.398551	+	0.917146i	$0.369513\pi$
$992$	0	0
$993$	−1.15461	−0.0366406
$994$	0	0
$995$	−21.4111	−0.678776
$996$	0	0
$997$	14.4245	0.456828	0.228414	−	0.973564i	$-0.426646\pi$
$997$	14.4245	0.456828	0.228414	+	0.973564i	$0.426646\pi$
$998$	0	0
$999$	37.5011	1.18648

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.m.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.m.1.8	✓	12	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)

\(f(q)\)	\(=\)	\(q+0.983156 q^{3} -1.00000 q^{5} -2.43489 q^{7} -2.03340 q^{9} +O(q^{10})\)	\(q+0.983156 q^{3} -1.00000 q^{5} -2.43489 q^{7} -2.03340 q^{9} +4.69188 q^{11} +1.64591 q^{13} -0.983156 q^{15} +0.351105 q^{17} -1.05554 q^{19} -2.39388 q^{21} +0.237686 q^{23} +1.00000 q^{25} -4.94862 q^{27} +10.7154 q^{29} +2.17061 q^{31} +4.61285 q^{33} +2.43489 q^{35} -7.57809 q^{37} +1.61818 q^{39} -9.89699 q^{41} -0.895195 q^{43} +2.03340 q^{45} +3.18773 q^{47} -1.07132 q^{49} +0.345191 q^{51} +7.34717 q^{53} -4.69188 q^{55} -1.03776 q^{57} -9.06435 q^{59} -2.85567 q^{61} +4.95111 q^{63} -1.64591 q^{65} +10.6840 q^{67} +0.233682 q^{69} -11.9942 q^{71} +8.27468 q^{73} +0.983156 q^{75} -11.4242 q^{77} +7.87957 q^{79} +1.23494 q^{81} -4.21882 q^{83} -0.351105 q^{85} +10.5350 q^{87} +12.9005 q^{89} -4.00760 q^{91} +2.13404 q^{93} +1.05554 q^{95} +18.7121 q^{97} -9.54048 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) \) 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9	\( 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9} + 10 q^{11} + 11 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} - q^{21} + 18 q^{23} + 12 q^{25} + 9 q^{27} + 16 q^{29} - q^{31} + 8 q^{33} - 5 q^{35} + 2 q^{37} + 6 q^{39} + 4 q^{41} + 7 q^{43} - 9 q^{45} + 3 q^{49} - 4 q^{51} + 39 q^{53} - 10 q^{55} - 15 q^{57} - 4 q^{59} - 32 q^{61} + 3 q^{63} - 11 q^{65} + 4 q^{67} + 12 q^{69} + 24 q^{71} - 10 q^{73} + 3 q^{75} + 38 q^{77} + 32 q^{79} - 8 q^{81} + 9 q^{83} + 4 q^{85} + 3 q^{87} + 15 q^{89} + 18 q^{91} + 36 q^{93} - 5 q^{95} + 15 q^{97} + 28 q^{99}+O(q^{100}) \) 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9 + 10 * q^11 + 11 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 - q^21 + 18 * q^23 + 12 * q^25 + 9 * q^27 + 16 * q^29 - q^31 + 8 * q^33 - 5 * q^35 + 2 * q^37 + 6 * q^39 + 4 * q^41 + 7 * q^43 - 9 * q^45 + 3 * q^49 - 4 * q^51 + 39 * q^53 - 10 * q^55 - 15 * q^57 - 4 * q^59 - 32 * q^61 + 3 * q^63 - 11 * q^65 + 4 * q^67 + 12 * q^69 + 24 * q^71 - 10 * q^73 + 3 * q^75 + 38 * q^77 + 32 * q^79 - 8 * q^81 + 9 * q^83 + 4 * q^85 + 3 * q^87 + 15 * q^89 + 18 * q^91 + 36 * q^93 - 5 * q^95 + 15 * q^97 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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