LMFDB - Embedded newform 6080.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.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:	$48.5490444289$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3040)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.17009 q^{3} -1.00000 q^{5} +0.630898 q^{7} +7.04945 q^{9} +O(q^{10})$	$q-3.17009 q^{3} -1.00000 q^{5} +0.630898 q^{7} +7.04945 q^{9} -5.70928 q^{11} -1.80098 q^{13} +3.17009 q^{15} +2.00000 q^{17} +1.00000 q^{19} -2.00000 q^{21} -0.630898 q^{23} +1.00000 q^{25} -12.8371 q^{27} +10.4969 q^{29} -7.75872 q^{31} +18.0989 q^{33} -0.630898 q^{35} -3.80098 q^{37} +5.70928 q^{39} -4.49693 q^{41} +9.80817 q^{43} -7.04945 q^{45} +0.630898 q^{47} -6.60197 q^{49} -6.34017 q^{51} +7.80098 q^{53} +5.70928 q^{55} -3.17009 q^{57} +1.41855 q^{59} -8.94441 q^{61} +4.44748 q^{63} +1.80098 q^{65} +9.66701 q^{67} +2.00000 q^{69} -7.75872 q^{71} -2.49693 q^{73} -3.17009 q^{75} -3.60197 q^{77} +11.5753 q^{79} +19.5464 q^{81} +8.23287 q^{83} -2.00000 q^{85} -33.2762 q^{87} +6.49693 q^{89} -1.13624 q^{91} +24.5958 q^{93} -1.00000 q^{95} +12.2979 q^{97} -40.2472 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 4 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9	$ 3 q - 4 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 10 q^{11} + 4 q^{13} + 4 q^{15} + 6 q^{17} + 3 q^{19} - 6 q^{21} + 2 q^{23} + 3 q^{25} - 10 q^{27} + 14 q^{29} + 2 q^{31} + 18 q^{33} + 2 q^{35} - 2 q^{37} + 10 q^{39} + 4 q^{41} - 14 q^{43} - 3 q^{45} - 2 q^{47} - q^{49} - 8 q^{51} + 14 q^{53} + 10 q^{55} - 4 q^{57} - 10 q^{59} - 10 q^{61} + 14 q^{63} - 4 q^{65} + 6 q^{67} + 6 q^{69} + 2 q^{71} + 10 q^{73} - 4 q^{75} + 8 q^{77} + 14 q^{79} + 23 q^{81} + 2 q^{83} - 6 q^{85} - 24 q^{87} + 2 q^{89} - 30 q^{91} + 20 q^{93} - 3 q^{95} + 10 q^{97} - 30 q^{99}+O(q^{100}) $ 3 * q - 4 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 10 * q^11 + 4 * q^13 + 4 * q^15 + 6 * q^17 + 3 * q^19 - 6 * q^21 + 2 * q^23 + 3 * q^25 - 10 * q^27 + 14 * q^29 + 2 * q^31 + 18 * q^33 + 2 * q^35 - 2 * q^37 + 10 * q^39 + 4 * q^41 - 14 * q^43 - 3 * q^45 - 2 * q^47 - q^49 - 8 * q^51 + 14 * q^53 + 10 * q^55 - 4 * q^57 - 10 * q^59 - 10 * q^61 + 14 * q^63 - 4 * q^65 + 6 * q^67 + 6 * q^69 + 2 * q^71 + 10 * q^73 - 4 * q^75 + 8 * q^77 + 14 * q^79 + 23 * q^81 + 2 * q^83 - 6 * q^85 - 24 * q^87 + 2 * q^89 - 30 * q^91 + 20 * q^93 - 3 * q^95 + 10 * q^97 - 30 * 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
$3$	−3.17009	−1.83025	−0.915125	−	0.403170i	$-0.867908\pi$
$3$	−3.17009	−1.83025	−0.915125	+	0.403170i	$0.867908\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.630898	0.238457	0.119228	−	0.992867i	$-0.461958\pi$
$7$	0.630898	0.238457	0.119228	+	0.992867i	$0.461958\pi$
$8$	0	0
$9$	7.04945	2.34982
$10$	0	0
$11$	−5.70928	−1.72141	−0.860706	−	0.509103i	$-0.829977\pi$
$11$	−5.70928	−1.72141	−0.860706	+	0.509103i	$0.829977\pi$
$12$	0	0
$13$	−1.80098	−0.499503	−0.249752	−	0.968310i	$-0.580349\pi$
$13$	−1.80098	−0.499503	−0.249752	+	0.968310i	$0.580349\pi$
$14$	0	0
$15$	3.17009	0.818513
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	−0.630898	−0.131551	−0.0657756	−	0.997834i	$-0.520952\pi$
$23$	−0.630898	−0.131551	−0.0657756	+	0.997834i	$0.520952\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−12.8371	−2.47050
$28$	0	0
$29$	10.4969	1.94923	0.974615	−	0.223886i	$-0.0718743\pi$
$29$	10.4969	1.94923	0.974615	+	0.223886i	$0.0718743\pi$
$30$	0	0
$31$	−7.75872	−1.39351	−0.696754	−	0.717310i	$-0.745373\pi$
$31$	−7.75872	−1.39351	−0.696754	+	0.717310i	$0.745373\pi$
$32$	0	0
$33$	18.0989	3.15061
$34$	0	0
$35$	−0.630898	−0.106641
$36$	0	0
$37$	−3.80098	−0.624878	−0.312439	−	0.949938i	$-0.601146\pi$
$37$	−3.80098	−0.624878	−0.312439	+	0.949938i	$0.601146\pi$
$38$	0	0
$39$	5.70928	0.914216
$40$	0	0
$41$	−4.49693	−0.702302	−0.351151	−	0.936319i	$-0.614210\pi$
$41$	−4.49693	−0.702302	−0.351151	+	0.936319i	$0.614210\pi$
$42$	0	0
$43$	9.80817	1.49573	0.747866	−	0.663850i	$-0.231079\pi$
$43$	9.80817	1.49573	0.747866	+	0.663850i	$0.231079\pi$
$44$	0	0
$45$	−7.04945	−1.05087
$46$	0	0
$47$	0.630898	0.0920259	0.0460129	−	0.998941i	$-0.485348\pi$
$47$	0.630898	0.0920259	0.0460129	+	0.998941i	$0.485348\pi$
$48$	0	0
$49$	−6.60197	−0.943138
$50$	0	0
$51$	−6.34017	−0.887802
$52$	0	0
$53$	7.80098	1.07155	0.535774	−	0.844362i	$-0.320020\pi$
$53$	7.80098	1.07155	0.535774	+	0.844362i	$0.320020\pi$
$54$	0	0
$55$	5.70928	0.769839
$56$	0	0
$57$	−3.17009	−0.419888
$58$	0	0
$59$	1.41855	0.184680	0.0923398	−	0.995728i	$-0.470565\pi$
$59$	1.41855	0.184680	0.0923398	+	0.995728i	$0.470565\pi$
$60$	0	0
$61$	−8.94441	−1.14521	−0.572607	−	0.819830i	$-0.694068\pi$
$61$	−8.94441	−1.14521	−0.572607	+	0.819830i	$0.694068\pi$
$62$	0	0
$63$	4.44748	0.560330
$64$	0	0
$65$	1.80098	0.223385
$66$	0	0
$67$	9.66701	1.18101	0.590507	−	0.807033i	$-0.298928\pi$
$67$	9.66701	1.18101	0.590507	+	0.807033i	$0.298928\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	−7.75872	−0.920791	−0.460396	−	0.887714i	$-0.652292\pi$
$71$	−7.75872	−0.920791	−0.460396	+	0.887714i	$0.652292\pi$
$72$	0	0
$73$	−2.49693	−0.292243	−0.146122	−	0.989267i	$-0.546679\pi$
$73$	−2.49693	−0.292243	−0.146122	+	0.989267i	$0.546679\pi$
$74$	0	0
$75$	−3.17009	−0.366050
$76$	0	0
$77$	−3.60197	−0.410482
$78$	0	0
$79$	11.5753	1.30232	0.651162	−	0.758939i	$-0.274282\pi$
$79$	11.5753	1.30232	0.651162	+	0.758939i	$0.274282\pi$
$80$	0	0
$81$	19.5464	2.17182
$82$	0	0
$83$	8.23287	0.903674	0.451837	−	0.892100i	$-0.350769\pi$
$83$	8.23287	0.903674	0.451837	+	0.892100i	$0.350769\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−33.2762	−3.56758
$88$	0	0
$89$	6.49693	0.688673	0.344337	−	0.938846i	$-0.388104\pi$
$89$	6.49693	0.688673	0.344337	+	0.938846i	$0.388104\pi$
$90$	0	0
$91$	−1.13624	−0.119110
$92$	0	0
$93$	24.5958	2.55047
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	12.2979	1.24866	0.624332	−	0.781159i	$-0.285371\pi$
$97$	12.2979	1.24866	0.624332	+	0.781159i	$0.285371\pi$
$98$	0	0
$99$	−40.2472	−4.04500
$100$	0	0
$101$	11.6514	1.15936	0.579680	−	0.814844i	$-0.303178\pi$
$101$	11.6514	1.15936	0.579680	+	0.814844i	$0.303178\pi$
$102$	0	0
$103$	−7.14342	−0.703863	−0.351931	−	0.936026i	$-0.614475\pi$
$103$	−7.14342	−0.703863	−0.351931	+	0.936026i	$0.614475\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	−9.35350	−0.904237	−0.452119	−	0.891958i	$-0.649332\pi$
$107$	−9.35350	−0.904237	−0.452119	+	0.891958i	$0.649332\pi$
$108$	0	0
$109$	12.0989	1.15886	0.579432	−	0.815021i	$-0.303274\pi$
$109$	12.0989	1.15886	0.579432	+	0.815021i	$0.303274\pi$
$110$	0	0
$111$	12.0494	1.14368
$112$	0	0
$113$	−10.6959	−1.00619	−0.503095	−	0.864231i	$-0.667805\pi$
$113$	−10.6959	−1.00619	−0.503095	+	0.864231i	$0.667805\pi$
$114$	0	0
$115$	0.630898	0.0588315
$116$	0	0
$117$	−12.6959	−1.17374
$118$	0	0
$119$	1.26180	0.115669
$120$	0	0
$121$	21.5958	1.96326
$122$	0	0
$123$	14.2557	1.28539
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	17.4257	1.54628	0.773142	−	0.634233i	$-0.218684\pi$
$127$	17.4257	1.54628	0.773142	+	0.634233i	$0.218684\pi$
$128$	0	0
$129$	−31.0928	−2.73756
$130$	0	0
$131$	−21.5441	−1.88232	−0.941159	−	0.337965i	$-0.890262\pi$
$131$	−21.5441	−1.88232	−0.941159	+	0.337965i	$0.890262\pi$
$132$	0	0
$133$	0.630898	0.0547058
$134$	0	0
$135$	12.8371	1.10484
$136$	0	0
$137$	18.0989	1.54629	0.773147	−	0.634227i	$-0.218682\pi$
$137$	18.0989	1.54629	0.773147	+	0.634227i	$0.218682\pi$
$138$	0	0
$139$	−17.1278	−1.45276	−0.726382	−	0.687292i	$-0.758799\pi$
$139$	−17.1278	−1.45276	−0.726382	+	0.687292i	$0.758799\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	10.2823	0.859850
$144$	0	0
$145$	−10.4969	−0.871722
$146$	0	0
$147$	20.9288	1.72618
$148$	0	0
$149$	−0.944409	−0.0773690	−0.0386845	−	0.999251i	$-0.512317\pi$
$149$	−0.944409	−0.0773690	−0.0386845	+	0.999251i	$0.512317\pi$
$150$	0	0
$151$	5.07838	0.413273	0.206636	−	0.978418i	$-0.433748\pi$
$151$	5.07838	0.413273	0.206636	+	0.978418i	$0.433748\pi$
$152$	0	0
$153$	14.0989	1.13983
$154$	0	0
$155$	7.75872	0.623196
$156$	0	0
$157$	2.09890	0.167510	0.0837551	−	0.996486i	$-0.473309\pi$
$157$	2.09890	0.167510	0.0837551	+	0.996486i	$0.473309\pi$
$158$	0	0
$159$	−24.7298	−1.96120
$160$	0	0
$161$	−0.398032	−0.0313693
$162$	0	0
$163$	−4.44748	−0.348354	−0.174177	−	0.984714i	$-0.555726\pi$
$163$	−4.44748	−0.348354	−0.174177	+	0.984714i	$0.555726\pi$
$164$	0	0
$165$	−18.0989	−1.40900
$166$	0	0
$167$	21.2423	1.64378	0.821890	−	0.569646i	$-0.192920\pi$
$167$	21.2423	1.64378	0.821890	+	0.569646i	$0.192920\pi$
$168$	0	0
$169$	−9.75646	−0.750497
$170$	0	0
$171$	7.04945	0.539085
$172$	0	0
$173$	1.70209	0.129407	0.0647037	−	0.997905i	$-0.479390\pi$
$173$	1.70209	0.129407	0.0647037	+	0.997905i	$0.479390\pi$
$174$	0	0
$175$	0.630898	0.0476914
$176$	0	0
$177$	−4.49693	−0.338010
$178$	0	0
$179$	1.41855	0.106027	0.0530137	−	0.998594i	$-0.483117\pi$
$179$	1.41855	0.106027	0.0530137	+	0.998594i	$0.483117\pi$
$180$	0	0
$181$	21.2039	1.57608	0.788038	−	0.615626i	$-0.211097\pi$
$181$	21.2039	1.57608	0.788038	+	0.615626i	$0.211097\pi$
$182$	0	0
$183$	28.3545	2.09603
$184$	0	0
$185$	3.80098	0.279454
$186$	0	0
$187$	−11.4186	−0.835007
$188$	0	0
$189$	−8.09890	−0.589108
$190$	0	0
$191$	−1.57531	−0.113985	−0.0569926	−	0.998375i	$-0.518151\pi$
$191$	−1.57531	−0.113985	−0.0569926	+	0.998375i	$0.518151\pi$
$192$	0	0
$193$	−6.90602	−0.497106	−0.248553	−	0.968618i	$-0.579955\pi$
$193$	−6.90602	−0.497106	−0.248553	+	0.968618i	$0.579955\pi$
$194$	0	0
$195$	−5.70928	−0.408850
$196$	0	0
$197$	10.4969	0.747875	0.373938	−	0.927454i	$-0.378007\pi$
$197$	10.4969	0.747875	0.373938	+	0.927454i	$0.378007\pi$
$198$	0	0
$199$	−10.1568	−0.719993	−0.359997	−	0.932954i	$-0.617222\pi$
$199$	−10.1568	−0.719993	−0.359997	+	0.932954i	$0.617222\pi$
$200$	0	0
$201$	−30.6453	−2.16155
$202$	0	0
$203$	6.62249	0.464807
$204$	0	0
$205$	4.49693	0.314079
$206$	0	0
$207$	−4.44748	−0.309121
$208$	0	0
$209$	−5.70928	−0.394919
$210$	0	0
$211$	−25.0784	−1.72647	−0.863233	−	0.504805i	$-0.831564\pi$
$211$	−25.0784	−1.72647	−0.863233	+	0.504805i	$0.831564\pi$
$212$	0	0
$213$	24.5958	1.68528
$214$	0	0
$215$	−9.80817	−0.668912
$216$	0	0
$217$	−4.89496	−0.332292
$218$	0	0
$219$	7.91548	0.534879
$220$	0	0
$221$	−3.60197	−0.242295
$222$	0	0
$223$	2.19061	0.146694	0.0733469	−	0.997306i	$-0.476632\pi$
$223$	2.19061	0.146694	0.0733469	+	0.997306i	$0.476632\pi$
$224$	0	0
$225$	7.04945	0.469963
$226$	0	0
$227$	−5.88163	−0.390377	−0.195189	−	0.980766i	$-0.562532\pi$
$227$	−5.88163	−0.390377	−0.195189	+	0.980766i	$0.562532\pi$
$228$	0	0
$229$	−3.65142	−0.241292	−0.120646	−	0.992696i	$-0.538497\pi$
$229$	−3.65142	−0.241292	−0.120646	+	0.992696i	$0.538497\pi$
$230$	0	0
$231$	11.4186	0.751285
$232$	0	0
$233$	−6.49693	−0.425628	−0.212814	−	0.977093i	$-0.568263\pi$
$233$	−6.49693	−0.425628	−0.212814	+	0.977093i	$0.568263\pi$
$234$	0	0
$235$	−0.630898	−0.0411552
$236$	0	0
$237$	−36.6947	−2.38358
$238$	0	0
$239$	7.91548	0.512010	0.256005	−	0.966675i	$-0.417594\pi$
$239$	7.91548	0.512010	0.256005	+	0.966675i	$0.417594\pi$
$240$	0	0
$241$	−24.5958	−1.58436	−0.792178	−	0.610290i	$-0.791053\pi$
$241$	−24.5958	−1.58436	−0.792178	+	0.610290i	$0.791053\pi$
$242$	0	0
$243$	−23.4524	−1.50447
$244$	0	0
$245$	6.60197	0.421784
$246$	0	0
$247$	−1.80098	−0.114594
$248$	0	0
$249$	−26.0989	−1.65395
$250$	0	0
$251$	6.93600	0.437796	0.218898	−	0.975748i	$-0.429754\pi$
$251$	6.93600	0.437796	0.218898	+	0.975748i	$0.429754\pi$
$252$	0	0
$253$	3.60197	0.226454
$254$	0	0
$255$	6.34017	0.397037
$256$	0	0
$257$	−28.3968	−1.77134	−0.885672	−	0.464311i	$-0.846302\pi$
$257$	−28.3968	−1.77134	−0.885672	+	0.464311i	$0.846302\pi$
$258$	0	0
$259$	−2.39803	−0.149006
$260$	0	0
$261$	73.9976	4.58033
$262$	0	0
$263$	−1.89269	−0.116708	−0.0583542	−	0.998296i	$-0.518585\pi$
$263$	−1.89269	−0.116708	−0.0583542	+	0.998296i	$0.518585\pi$
$264$	0	0
$265$	−7.80098	−0.479211
$266$	0	0
$267$	−20.5958	−1.26044
$268$	0	0
$269$	−12.0000	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$270$	0	0
$271$	−4.13397	−0.251121	−0.125560	−	0.992086i	$-0.540073\pi$
$271$	−4.13397	−0.251121	−0.125560	+	0.992086i	$0.540073\pi$
$272$	0	0
$273$	3.60197	0.218001
$274$	0	0
$275$	−5.70928	−0.344282
$276$	0	0
$277$	−29.3028	−1.76064	−0.880318	−	0.474383i	$-0.842671\pi$
$277$	−29.3028	−1.76064	−0.880318	+	0.474383i	$0.842671\pi$
$278$	0	0
$279$	−54.6947	−3.27449
$280$	0	0
$281$	11.7009	0.698015	0.349008	−	0.937120i	$-0.386519\pi$
$281$	11.7009	0.698015	0.349008	+	0.937120i	$0.386519\pi$
$282$	0	0
$283$	−18.3896	−1.09315	−0.546575	−	0.837410i	$-0.684069\pi$
$283$	−18.3896	−1.09315	−0.546575	+	0.837410i	$0.684069\pi$
$284$	0	0
$285$	3.17009	0.187780
$286$	0	0
$287$	−2.83710	−0.167469
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−38.9854	−2.28537
$292$	0	0
$293$	27.8999	1.62993	0.814964	−	0.579511i	$-0.196757\pi$
$293$	27.8999	1.62993	0.814964	+	0.579511i	$0.196757\pi$
$294$	0	0
$295$	−1.41855	−0.0825912
$296$	0	0
$297$	73.2905	4.25275
$298$	0	0
$299$	1.13624	0.0657103
$300$	0	0
$301$	6.18795	0.356668
$302$	0	0
$303$	−36.9360	−2.12192
$304$	0	0
$305$	8.94441	0.512155
$306$	0	0
$307$	26.1639	1.49326	0.746628	−	0.665242i	$-0.231672\pi$
$307$	26.1639	1.49326	0.746628	+	0.665242i	$0.231672\pi$
$308$	0	0
$309$	22.6453	1.28824
$310$	0	0
$311$	0.348583	0.0197664	0.00988318	−	0.999951i	$-0.496854\pi$
$311$	0.348583	0.0197664	0.00988318	+	0.999951i	$0.496854\pi$
$312$	0	0
$313$	10.4969	0.593321	0.296661	−	0.954983i	$-0.404127\pi$
$313$	10.4969	0.593321	0.296661	+	0.954983i	$0.404127\pi$
$314$	0	0
$315$	−4.44748	−0.250587
$316$	0	0
$317$	−17.8999	−1.00536	−0.502679	−	0.864473i	$-0.667652\pi$
$317$	−17.8999	−1.00536	−0.502679	+	0.864473i	$0.667652\pi$
$318$	0	0
$319$	−59.9299	−3.35543
$320$	0	0
$321$	29.6514	1.65498
$322$	0	0
$323$	2.00000	0.111283
$324$	0	0
$325$	−1.80098	−0.0999006
$326$	0	0
$327$	−38.3545	−2.12101
$328$	0	0
$329$	0.398032	0.0219442
$330$	0	0
$331$	−26.3402	−1.44779	−0.723893	−	0.689912i	$-0.757649\pi$
$331$	−26.3402	−1.44779	−0.723893	+	0.689912i	$0.757649\pi$
$332$	0	0
$333$	−26.7948	−1.46835
$334$	0	0
$335$	−9.66701	−0.528165
$336$	0	0
$337$	−21.0049	−1.14421	−0.572105	−	0.820180i	$-0.693873\pi$
$337$	−21.0049	−1.14421	−0.572105	+	0.820180i	$0.693873\pi$
$338$	0	0
$339$	33.9071	1.84158
$340$	0	0
$341$	44.2967	2.39880
$342$	0	0
$343$	−8.58145	−0.463355
$344$	0	0
$345$	−2.00000	−0.107676
$346$	0	0
$347$	−2.20620	−0.118435	−0.0592176	−	0.998245i	$-0.518861\pi$
$347$	−2.20620	−0.118435	−0.0592176	+	0.998245i	$0.518861\pi$
$348$	0	0
$349$	1.60197	0.0857514	0.0428757	−	0.999080i	$-0.486348\pi$
$349$	1.60197	0.0857514	0.0428757	+	0.999080i	$0.486348\pi$
$350$	0	0
$351$	23.1194	1.23402
$352$	0	0
$353$	−5.20394	−0.276978	−0.138489	−	0.990364i	$-0.544224\pi$
$353$	−5.20394	−0.276978	−0.138489	+	0.990364i	$0.544224\pi$
$354$	0	0
$355$	7.75872	0.411790
$356$	0	0
$357$	−4.00000	−0.211702
$358$	0	0
$359$	25.0121	1.32009	0.660044	−	0.751227i	$-0.270537\pi$
$359$	25.0121	1.32009	0.660044	+	0.751227i	$0.270537\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−68.4606	−3.59325
$364$	0	0
$365$	2.49693	0.130695
$366$	0	0
$367$	−18.3896	−0.959930	−0.479965	−	0.877288i	$-0.659351\pi$
$367$	−18.3896	−0.959930	−0.479965	+	0.877288i	$0.659351\pi$
$368$	0	0
$369$	−31.7009	−1.65028
$370$	0	0
$371$	4.92162	0.255518
$372$	0	0
$373$	−10.0878	−0.522328	−0.261164	−	0.965294i	$-0.584106\pi$
$373$	−10.0878	−0.522328	−0.261164	+	0.965294i	$0.584106\pi$
$374$	0	0
$375$	3.17009	0.163703
$376$	0	0
$377$	−18.9048	−0.973647
$378$	0	0
$379$	18.3545	0.942810	0.471405	−	0.881917i	$-0.343747\pi$
$379$	18.3545	0.942810	0.471405	+	0.881917i	$0.343747\pi$
$380$	0	0
$381$	−55.2411	−2.83009
$382$	0	0
$383$	4.30632	0.220043	0.110021	−	0.993929i	$-0.464908\pi$
$383$	4.30632	0.220043	0.110021	+	0.993929i	$0.464908\pi$
$384$	0	0
$385$	3.60197	0.183573
$386$	0	0
$387$	69.1422	3.51470
$388$	0	0
$389$	−28.8059	−1.46052	−0.730259	−	0.683171i	$-0.760600\pi$
$389$	−28.8059	−1.46052	−0.730259	+	0.683171i	$0.760600\pi$
$390$	0	0
$391$	−1.26180	−0.0638117
$392$	0	0
$393$	68.2967	3.44511
$394$	0	0
$395$	−11.5753	−0.582417
$396$	0	0
$397$	−13.6020	−0.682663	−0.341332	−	0.939943i	$-0.610878\pi$
$397$	−13.6020	−0.682663	−0.341332	+	0.939943i	$0.610878\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	0	0
$401$	−9.10504	−0.454684	−0.227342	−	0.973815i	$-0.573004\pi$
$401$	−9.10504	−0.454684	−0.227342	+	0.973815i	$0.573004\pi$
$402$	0	0
$403$	13.9733	0.696062
$404$	0	0
$405$	−19.5464	−0.971267
$406$	0	0
$407$	21.7009	1.07567
$408$	0	0
$409$	−36.1978	−1.78987	−0.894933	−	0.446201i	$-0.852777\pi$
$409$	−36.1978	−1.78987	−0.894933	+	0.446201i	$0.852777\pi$
$410$	0	0
$411$	−57.3751	−2.83010
$412$	0	0
$413$	0.894960	0.0440381
$414$	0	0
$415$	−8.23287	−0.404135
$416$	0	0
$417$	54.2967	2.65892
$418$	0	0
$419$	0.282314	0.0137919	0.00689597	−	0.999976i	$-0.497805\pi$
$419$	0.282314	0.0137919	0.00689597	+	0.999976i	$0.497805\pi$
$420$	0	0
$421$	−23.7998	−1.15993	−0.579965	−	0.814642i	$-0.696934\pi$
$421$	−23.7998	−1.15993	−0.579965	+	0.814642i	$0.696934\pi$
$422$	0	0
$423$	4.44748	0.216244
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	−5.64301	−0.273084
$428$	0	0
$429$	−32.5958	−1.57374
$430$	0	0
$431$	−6.34017	−0.305395	−0.152698	−	0.988273i	$-0.548796\pi$
$431$	−6.34017	−0.305395	−0.152698	+	0.988273i	$0.548796\pi$
$432$	0	0
$433$	−37.1038	−1.78310	−0.891548	−	0.452927i	$-0.850380\pi$
$433$	−37.1038	−1.78310	−0.891548	+	0.452927i	$0.850380\pi$
$434$	0	0
$435$	33.2762	1.59547
$436$	0	0
$437$	−0.630898	−0.0301799
$438$	0	0
$439$	41.3484	1.97345	0.986726	−	0.162395i	$-0.0519218\pi$
$439$	41.3484	1.97345	0.986726	+	0.162395i	$0.0519218\pi$
$440$	0	0
$441$	−46.5402	−2.21620
$442$	0	0
$443$	−1.57918	−0.0750292	−0.0375146	−	0.999296i	$-0.511944\pi$
$443$	−1.57918	−0.0750292	−0.0375146	+	0.999296i	$0.511944\pi$
$444$	0	0
$445$	−6.49693	−0.307984
$446$	0	0
$447$	2.99386	0.141605
$448$	0	0
$449$	0.0988967	0.00466722	0.00233361	−	0.999997i	$-0.499257\pi$
$449$	0.0988967	0.00466722	0.00233361	+	0.999997i	$0.499257\pi$
$450$	0	0
$451$	25.6742	1.20895
$452$	0	0
$453$	−16.0989	−0.756392
$454$	0	0
$455$	1.13624	0.0532676
$456$	0	0
$457$	0.210079	0.00982710	0.00491355	−	0.999988i	$-0.498436\pi$
$457$	0.210079	0.00982710	0.00491355	+	0.999988i	$0.498436\pi$
$458$	0	0
$459$	−25.6742	−1.19837
$460$	0	0
$461$	−26.0000	−1.21094	−0.605470	−	0.795868i	$-0.707015\pi$
$461$	−26.0000	−1.21094	−0.605470	+	0.795868i	$0.707015\pi$
$462$	0	0
$463$	−5.11345	−0.237642	−0.118821	−	0.992916i	$-0.537911\pi$
$463$	−5.11345	−0.237642	−0.118821	+	0.992916i	$0.537911\pi$
$464$	0	0
$465$	−24.5958	−1.14060
$466$	0	0
$467$	−5.04331	−0.233376	−0.116688	−	0.993169i	$-0.537228\pi$
$467$	−5.04331	−0.233376	−0.116688	+	0.993169i	$0.537228\pi$
$468$	0	0
$469$	6.09890	0.281621
$470$	0	0
$471$	−6.65368	−0.306586
$472$	0	0
$473$	−55.9976	−2.57477
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	54.9926	2.51794
$478$	0	0
$479$	−20.5997	−0.941224	−0.470612	−	0.882340i	$-0.655967\pi$
$479$	−20.5997	−0.941224	−0.470612	+	0.882340i	$0.655967\pi$
$480$	0	0
$481$	6.84551	0.312128
$482$	0	0
$483$	1.26180	0.0574137
$484$	0	0
$485$	−12.2979	−0.558419
$486$	0	0
$487$	−30.9288	−1.40152	−0.700759	−	0.713398i	$-0.747155\pi$
$487$	−30.9288	−1.40152	−0.700759	+	0.713398i	$0.747155\pi$
$488$	0	0
$489$	14.0989	0.637574
$490$	0	0
$491$	−23.7854	−1.07342	−0.536710	−	0.843767i	$-0.680333\pi$
$491$	−23.7854	−1.07342	−0.536710	+	0.843767i	$0.680333\pi$
$492$	0	0
$493$	20.9939	0.945516
$494$	0	0
$495$	40.2472	1.80898
$496$	0	0
$497$	−4.89496	−0.219569
$498$	0	0
$499$	−21.5402	−0.964273	−0.482137	−	0.876096i	$-0.660139\pi$
$499$	−21.5402	−0.964273	−0.482137	+	0.876096i	$0.660139\pi$
$500$	0	0
$501$	−67.3400	−3.00853
$502$	0	0
$503$	21.8927	0.976147	0.488073	−	0.872803i	$-0.337700\pi$
$503$	21.8927	0.976147	0.488073	+	0.872803i	$0.337700\pi$
$504$	0	0
$505$	−11.6514	−0.518481
$506$	0	0
$507$	30.9288	1.37360
$508$	0	0
$509$	−34.5958	−1.53343	−0.766716	−	0.641986i	$-0.778111\pi$
$509$	−34.5958	−1.53343	−0.766716	+	0.641986i	$0.778111\pi$
$510$	0	0
$511$	−1.57531	−0.0696874
$512$	0	0
$513$	−12.8371	−0.566772
$514$	0	0
$515$	7.14342	0.314777
$516$	0	0
$517$	−3.60197	−0.158414
$518$	0	0
$519$	−5.39576	−0.236848
$520$	0	0
$521$	42.2967	1.85305	0.926526	−	0.376231i	$-0.122780\pi$
$521$	42.2967	1.85305	0.926526	+	0.376231i	$0.122780\pi$
$522$	0	0
$523$	−30.3884	−1.32879	−0.664396	−	0.747381i	$-0.731311\pi$
$523$	−30.3884	−1.32879	−0.664396	+	0.747381i	$0.731311\pi$
$524$	0	0
$525$	−2.00000	−0.0872872
$526$	0	0
$527$	−15.5174	−0.675951
$528$	0	0
$529$	−22.6020	−0.982694
$530$	0	0
$531$	10.0000	0.433963
$532$	0	0
$533$	8.09890	0.350802
$534$	0	0
$535$	9.35350	0.404387
$536$	0	0
$537$	−4.49693	−0.194057
$538$	0	0
$539$	37.6925	1.62353
$540$	0	0
$541$	−39.6514	−1.70475	−0.852374	−	0.522933i	$-0.824838\pi$
$541$	−39.6514	−1.70475	−0.852374	+	0.522933i	$0.824838\pi$
$542$	0	0
$543$	−67.2183	−2.88461
$544$	0	0
$545$	−12.0989	−0.518260
$546$	0	0
$547$	34.5574	1.47757	0.738785	−	0.673941i	$-0.235400\pi$
$547$	34.5574	1.47757	0.738785	+	0.673941i	$0.235400\pi$
$548$	0	0
$549$	−63.0531	−2.69104
$550$	0	0
$551$	10.4969	0.447184
$552$	0	0
$553$	7.30283	0.310548
$554$	0	0
$555$	−12.0494	−0.511471
$556$	0	0
$557$	−22.8950	−0.970091	−0.485045	−	0.874489i	$-0.661197\pi$
$557$	−22.8950	−0.970091	−0.485045	+	0.874489i	$0.661197\pi$
$558$	0	0
$559$	−17.6644	−0.747123
$560$	0	0
$561$	36.1978	1.52827
$562$	0	0
$563$	−24.7142	−1.04158	−0.520790	−	0.853685i	$-0.674362\pi$
$563$	−24.7142	−1.04158	−0.520790	+	0.853685i	$0.674362\pi$
$564$	0	0
$565$	10.6959	0.449982
$566$	0	0
$567$	12.3318	0.517885
$568$	0	0
$569$	16.9939	0.712420	0.356210	−	0.934406i	$-0.384069\pi$
$569$	16.9939	0.712420	0.356210	+	0.934406i	$0.384069\pi$
$570$	0	0
$571$	−0.348583	−0.0145878	−0.00729388	−	0.999973i	$-0.502322\pi$
$571$	−0.348583	−0.0145878	−0.00729388	+	0.999973i	$0.502322\pi$
$572$	0	0
$573$	4.99386	0.208621
$574$	0	0
$575$	−0.630898	−0.0263102
$576$	0	0
$577$	15.4908	0.644890	0.322445	−	0.946588i	$-0.395495\pi$
$577$	15.4908	0.644890	0.322445	+	0.946588i	$0.395495\pi$
$578$	0	0
$579$	21.8927	0.909829
$580$	0	0
$581$	5.19410	0.215487
$582$	0	0
$583$	−44.5380	−1.84457
$584$	0	0
$585$	12.6959	0.524913
$586$	0	0
$587$	1.92389	0.0794074	0.0397037	−	0.999211i	$-0.487359\pi$
$587$	1.92389	0.0794074	0.0397037	+	0.999211i	$0.487359\pi$
$588$	0	0
$589$	−7.75872	−0.319693
$590$	0	0
$591$	−33.2762	−1.36880
$592$	0	0
$593$	19.1917	0.788107	0.394053	−	0.919088i	$-0.371073\pi$
$593$	19.1917	0.788107	0.394053	+	0.919088i	$0.371073\pi$
$594$	0	0
$595$	−1.26180	−0.0517286
$596$	0	0
$597$	32.1978	1.31777
$598$	0	0
$599$	2.68035	0.109516	0.0547580	−	0.998500i	$-0.482561\pi$
$599$	2.68035	0.109516	0.0547580	+	0.998500i	$0.482561\pi$
$600$	0	0
$601$	−41.5897	−1.69648	−0.848239	−	0.529613i	$-0.822337\pi$
$601$	−41.5897	−1.69648	−0.848239	+	0.529613i	$0.822337\pi$
$602$	0	0
$603$	68.1471	2.77517
$604$	0	0
$605$	−21.5958	−0.877995
$606$	0	0
$607$	39.7224	1.61228	0.806142	−	0.591722i	$-0.201552\pi$
$607$	39.7224	1.61228	0.806142	+	0.591722i	$0.201552\pi$
$608$	0	0
$609$	−20.9939	−0.850714
$610$	0	0
$611$	−1.13624	−0.0459672
$612$	0	0
$613$	−2.29914	−0.0928612	−0.0464306	−	0.998922i	$-0.514785\pi$
$613$	−2.29914	−0.0928612	−0.0464306	+	0.998922i	$0.514785\pi$
$614$	0	0
$615$	−14.2557	−0.574843
$616$	0	0
$617$	21.7009	0.873644	0.436822	−	0.899548i	$-0.356104\pi$
$617$	21.7009	0.873644	0.436822	+	0.899548i	$0.356104\pi$
$618$	0	0
$619$	11.0700	0.444940	0.222470	−	0.974940i	$-0.428588\pi$
$619$	11.0700	0.444940	0.222470	+	0.974940i	$0.428588\pi$
$620$	0	0
$621$	8.09890	0.324998
$622$	0	0
$623$	4.09890	0.164219
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	18.0989	0.722800
$628$	0	0
$629$	−7.60197	−0.303110
$630$	0	0
$631$	1.22672	0.0488351	0.0244175	−	0.999702i	$-0.492227\pi$
$631$	1.22672	0.0488351	0.0244175	+	0.999702i	$0.492227\pi$
$632$	0	0
$633$	79.5006	3.15987
$634$	0	0
$635$	−17.4257	−0.691519
$636$	0	0
$637$	11.8900	0.471101
$638$	0	0
$639$	−54.6947	−2.16369
$640$	0	0
$641$	−5.39189	−0.212967	−0.106483	−	0.994314i	$-0.533959\pi$
$641$	−5.39189	−0.212967	−0.106483	+	0.994314i	$0.533959\pi$
$642$	0	0
$643$	−26.6186	−1.04974	−0.524868	−	0.851184i	$-0.675885\pi$
$643$	−26.6186	−1.04974	−0.524868	+	0.851184i	$0.675885\pi$
$644$	0	0
$645$	31.0928	1.22428
$646$	0	0
$647$	34.8554	1.37031	0.685153	−	0.728399i	$-0.259735\pi$
$647$	34.8554	1.37031	0.685153	+	0.728399i	$0.259735\pi$
$648$	0	0
$649$	−8.09890	−0.317910
$650$	0	0
$651$	15.5174	0.608177
$652$	0	0
$653$	42.4969	1.66303	0.831517	−	0.555500i	$-0.187473\pi$
$653$	42.4969	1.66303	0.831517	+	0.555500i	$0.187473\pi$
$654$	0	0
$655$	21.5441	0.841798
$656$	0	0
$657$	−17.6020	−0.686718
$658$	0	0
$659$	10.9483	0.426485	0.213242	−	0.976999i	$-0.431598\pi$
$659$	10.9483	0.426485	0.213242	+	0.976999i	$0.431598\pi$
$660$	0	0
$661$	28.5958	1.11225	0.556124	−	0.831099i	$-0.312288\pi$
$661$	28.5958	1.11225	0.556124	+	0.831099i	$0.312288\pi$
$662$	0	0
$663$	11.4186	0.443460
$664$	0	0
$665$	−0.630898	−0.0244652
$666$	0	0
$667$	−6.62249	−0.256424
$668$	0	0
$669$	−6.94441	−0.268486
$670$	0	0
$671$	51.0661	1.97138
$672$	0	0
$673$	−7.19287	−0.277265	−0.138632	−	0.990344i	$-0.544271\pi$
$673$	−7.19287	−0.277265	−0.138632	+	0.990344i	$0.544271\pi$
$674$	0	0
$675$	−12.8371	−0.494100
$676$	0	0
$677$	24.6959	0.949142	0.474571	−	0.880217i	$-0.342603\pi$
$677$	24.6959	0.949142	0.474571	+	0.880217i	$0.342603\pi$
$678$	0	0
$679$	7.75872	0.297752
$680$	0	0
$681$	18.6453	0.714488
$682$	0	0
$683$	18.0605	0.691066	0.345533	−	0.938407i	$-0.387698\pi$
$683$	18.0605	0.691066	0.345533	+	0.938407i	$0.387698\pi$
$684$	0	0
$685$	−18.0989	−0.691523
$686$	0	0
$687$	11.5753	0.441625
$688$	0	0
$689$	−14.0494	−0.535241
$690$	0	0
$691$	39.6514	1.50841	0.754205	−	0.656638i	$-0.228022\pi$
$691$	39.6514	1.50841	0.754205	+	0.656638i	$0.228022\pi$
$692$	0	0
$693$	−25.3919	−0.964558
$694$	0	0
$695$	17.1278	0.649695
$696$	0	0
$697$	−8.99386	−0.340667
$698$	0	0
$699$	20.5958	0.779006
$700$	0	0
$701$	−4.84551	−0.183012	−0.0915062	−	0.995805i	$-0.529168\pi$
$701$	−4.84551	−0.183012	−0.0915062	+	0.995805i	$0.529168\pi$
$702$	0	0
$703$	−3.80098	−0.143357
$704$	0	0
$705$	2.00000	0.0753244
$706$	0	0
$707$	7.35085	0.276457
$708$	0	0
$709$	24.3857	0.915826	0.457913	−	0.888997i	$-0.348597\pi$
$709$	24.3857	0.915826	0.457913	+	0.888997i	$0.348597\pi$
$710$	0	0
$711$	81.5995	3.06022
$712$	0	0
$713$	4.89496	0.183318
$714$	0	0
$715$	−10.2823	−0.384537
$716$	0	0
$717$	−25.0928	−0.937106
$718$	0	0
$719$	−8.93003	−0.333034	−0.166517	−	0.986039i	$-0.553252\pi$
$719$	−8.93003	−0.333034	−0.166517	+	0.986039i	$0.553252\pi$
$720$	0	0
$721$	−4.50677	−0.167841
$722$	0	0
$723$	77.9709	2.89977
$724$	0	0
$725$	10.4969	0.389846
$726$	0	0
$727$	−36.4307	−1.35114	−0.675569	−	0.737297i	$-0.736102\pi$
$727$	−36.4307	−1.35114	−0.675569	+	0.737297i	$0.736102\pi$
$728$	0	0
$729$	15.7070	0.581741
$730$	0	0
$731$	19.6163	0.725537
$732$	0	0
$733$	16.8059	0.620740	0.310370	−	0.950616i	$-0.399547\pi$
$733$	16.8059	0.620740	0.310370	+	0.950616i	$0.399547\pi$
$734$	0	0
$735$	−20.9288	−0.771971
$736$	0	0
$737$	−55.1917	−2.03301
$738$	0	0
$739$	−4.73367	−0.174131	−0.0870654	−	0.996203i	$-0.527749\pi$
$739$	−4.73367	−0.174131	−0.0870654	+	0.996203i	$0.527749\pi$
$740$	0	0
$741$	5.70928	0.209735
$742$	0	0
$743$	31.7515	1.16485	0.582425	−	0.812884i	$-0.302104\pi$
$743$	31.7515	1.16485	0.582425	+	0.812884i	$0.302104\pi$
$744$	0	0
$745$	0.944409	0.0346005
$746$	0	0
$747$	58.0372	2.12347
$748$	0	0
$749$	−5.90110	−0.215622
$750$	0	0
$751$	−45.0082	−1.64237	−0.821187	−	0.570659i	$-0.806688\pi$
$751$	−45.0082	−1.64237	−0.821187	+	0.570659i	$0.806688\pi$
$752$	0	0
$753$	−21.9877	−0.801277
$754$	0	0
$755$	−5.07838	−0.184821
$756$	0	0
$757$	26.4969	0.963047	0.481524	−	0.876433i	$-0.340084\pi$
$757$	26.4969	0.963047	0.481524	+	0.876433i	$0.340084\pi$
$758$	0	0
$759$	−11.4186	−0.414467
$760$	0	0
$761$	−23.6514	−0.857363	−0.428682	−	0.903456i	$-0.641022\pi$
$761$	−23.6514	−0.857363	−0.428682	+	0.903456i	$0.641022\pi$
$762$	0	0
$763$	7.63317	0.276339
$764$	0	0
$765$	−14.0989	−0.509747
$766$	0	0
$767$	−2.55479	−0.0922480
$768$	0	0
$769$	−6.85535	−0.247210	−0.123605	−	0.992331i	$-0.539446\pi$
$769$	−6.85535	−0.247210	−0.123605	+	0.992331i	$0.539446\pi$
$770$	0	0
$771$	90.0203	3.24200
$772$	0	0
$773$	−3.49201	−0.125599	−0.0627994	−	0.998026i	$-0.520003\pi$
$773$	−3.49201	−0.125599	−0.0627994	+	0.998026i	$0.520003\pi$
$774$	0	0
$775$	−7.75872	−0.278702
$776$	0	0
$777$	7.60197	0.272719
$778$	0	0
$779$	−4.49693	−0.161119
$780$	0	0
$781$	44.2967	1.58506
$782$	0	0
$783$	−134.750	−4.81558
$784$	0	0
$785$	−2.09890	−0.0749128
$786$	0	0
$787$	9.19675	0.327829	0.163914	−	0.986475i	$-0.447588\pi$
$787$	9.19675	0.327829	0.163914	+	0.986475i	$0.447588\pi$
$788$	0	0
$789$	6.00000	0.213606
$790$	0	0
$791$	−6.74805	−0.239933
$792$	0	0
$793$	16.1087	0.572038
$794$	0	0
$795$	24.7298	0.877075
$796$	0	0
$797$	−6.98279	−0.247343	−0.123672	−	0.992323i	$-0.539467\pi$
$797$	−6.98279	−0.247343	−0.123672	+	0.992323i	$0.539467\pi$
$798$	0	0
$799$	1.26180	0.0446391
$800$	0	0
$801$	45.7998	1.61826
$802$	0	0
$803$	14.2557	0.503071
$804$	0	0
$805$	0.398032	0.0140288
$806$	0	0
$807$	38.0410	1.33911
$808$	0	0
$809$	−36.3857	−1.27925	−0.639627	−	0.768685i	$-0.720911\pi$
$809$	−36.3857	−1.27925	−0.639627	+	0.768685i	$0.720911\pi$
$810$	0	0
$811$	2.64915	0.0930242	0.0465121	−	0.998918i	$-0.485189\pi$
$811$	2.64915	0.0930242	0.0465121	+	0.998918i	$0.485189\pi$
$812$	0	0
$813$	13.1050	0.459614
$814$	0	0
$815$	4.44748	0.155788
$816$	0	0
$817$	9.80817	0.343145
$818$	0	0
$819$	−8.00984	−0.279886
$820$	0	0
$821$	−2.39803	−0.0836919	−0.0418459	−	0.999124i	$-0.513324\pi$
$821$	−2.39803	−0.0836919	−0.0418459	+	0.999124i	$0.513324\pi$
$822$	0	0
$823$	−13.9772	−0.487215	−0.243608	−	0.969874i	$-0.578331\pi$
$823$	−13.9772	−0.487215	−0.243608	+	0.969874i	$0.578331\pi$
$824$	0	0
$825$	18.0989	0.630123
$826$	0	0
$827$	43.7347	1.52081	0.760403	−	0.649452i	$-0.225002\pi$
$827$	43.7347	1.52081	0.760403	+	0.649452i	$0.225002\pi$
$828$	0	0
$829$	−38.0866	−1.32280	−0.661401	−	0.750032i	$-0.730038\pi$
$829$	−38.0866	−1.32280	−0.661401	+	0.750032i	$0.730038\pi$
$830$	0	0
$831$	92.8925	3.22241
$832$	0	0
$833$	−13.2039	−0.457489
$834$	0	0
$835$	−21.2423	−0.735121
$836$	0	0
$837$	99.5995	3.44266
$838$	0	0
$839$	37.8843	1.30791	0.653955	−	0.756533i	$-0.273108\pi$
$839$	37.8843	1.30791	0.653955	+	0.756533i	$0.273108\pi$
$840$	0	0
$841$	81.1855	2.79950
$842$	0	0
$843$	−37.0928	−1.27754
$844$	0	0
$845$	9.75646	0.335632
$846$	0	0
$847$	13.6248	0.468152
$848$	0	0
$849$	58.2967	2.00074
$850$	0	0
$851$	2.39803	0.0822035
$852$	0	0
$853$	−47.2905	−1.61920	−0.809599	−	0.586984i	$-0.800315\pi$
$853$	−47.2905	−1.61920	−0.809599	+	0.586984i	$0.800315\pi$
$854$	0	0
$855$	−7.04945	−0.241086
$856$	0	0
$857$	49.1904	1.68031	0.840157	−	0.542344i	$-0.182463\pi$
$857$	49.1904	1.68031	0.840157	+	0.542344i	$0.182463\pi$
$858$	0	0
$859$	−15.2351	−0.519816	−0.259908	−	0.965633i	$-0.583692\pi$
$859$	−15.2351	−0.519816	−0.259908	+	0.965633i	$0.583692\pi$
$860$	0	0
$861$	8.99386	0.306510
$862$	0	0
$863$	33.6404	1.14513	0.572565	−	0.819859i	$-0.305948\pi$
$863$	33.6404	1.14513	0.572565	+	0.819859i	$0.305948\pi$
$864$	0	0
$865$	−1.70209	−0.0578727
$866$	0	0
$867$	41.2111	1.39960
$868$	0	0
$869$	−66.0866	−2.24183
$870$	0	0
$871$	−17.4101	−0.589920
$872$	0	0
$873$	86.6935	2.93413
$874$	0	0
$875$	−0.630898	−0.0213282
$876$	0	0
$877$	50.3078	1.69877	0.849386	−	0.527772i	$-0.176972\pi$
$877$	50.3078	1.69877	0.849386	+	0.527772i	$0.176972\pi$
$878$	0	0
$879$	−88.4450	−2.98318
$880$	0	0
$881$	4.04945	0.136429	0.0682147	−	0.997671i	$-0.478270\pi$
$881$	4.04945	0.136429	0.0682147	+	0.997671i	$0.478270\pi$
$882$	0	0
$883$	−27.5669	−0.927700	−0.463850	−	0.885914i	$-0.653532\pi$
$883$	−27.5669	−0.927700	−0.463850	+	0.885914i	$0.653532\pi$
$884$	0	0
$885$	4.49693	0.151163
$886$	0	0
$887$	−43.2567	−1.45242	−0.726209	−	0.687474i	$-0.758719\pi$
$887$	−43.2567	−1.45242	−0.726209	+	0.687474i	$0.758719\pi$
$888$	0	0
$889$	10.9939	0.368722
$890$	0	0
$891$	−111.596	−3.73859
$892$	0	0
$893$	0.630898	0.0211122
$894$	0	0
$895$	−1.41855	−0.0474169
$896$	0	0
$897$	−3.60197	−0.120266
$898$	0	0
$899$	−81.4428	−2.71627
$900$	0	0
$901$	15.6020	0.519777
$902$	0	0
$903$	−19.6163	−0.652791
$904$	0	0
$905$	−21.2039	−0.704843
$906$	0	0
$907$	12.5431	0.416486	0.208243	−	0.978077i	$-0.433226\pi$
$907$	12.5431	0.416486	0.208243	+	0.978077i	$0.433226\pi$
$908$	0	0
$909$	82.1361	2.72428
$910$	0	0
$911$	−4.44360	−0.147223	−0.0736116	−	0.997287i	$-0.523453\pi$
$911$	−4.44360	−0.147223	−0.0736116	+	0.997287i	$0.523453\pi$
$912$	0	0
$913$	−47.0037	−1.55560
$914$	0	0
$915$	−28.3545	−0.937372
$916$	0	0
$917$	−13.5921	−0.448852
$918$	0	0
$919$	26.0189	0.858285	0.429142	−	0.903237i	$-0.358816\pi$
$919$	26.0189	0.858285	0.429142	+	0.903237i	$0.358816\pi$
$920$	0	0
$921$	−82.9420	−2.73303
$922$	0	0
$923$	13.9733	0.459938
$924$	0	0
$925$	−3.80098	−0.124976
$926$	0	0
$927$	−50.3572	−1.65395
$928$	0	0
$929$	−48.1855	−1.58092	−0.790458	−	0.612517i	$-0.790157\pi$
$929$	−48.1855	−1.58092	−0.790458	+	0.612517i	$0.790157\pi$
$930$	0	0
$931$	−6.60197	−0.216371
$932$	0	0
$933$	−1.10504	−0.0361774
$934$	0	0
$935$	11.4186	0.373427
$936$	0	0
$937$	−32.4846	−1.06123	−0.530614	−	0.847614i	$-0.678038\pi$
$937$	−32.4846	−1.06123	−0.530614	+	0.847614i	$0.678038\pi$
$938$	0	0
$939$	−33.2762	−1.08593
$940$	0	0
$941$	−17.3028	−0.564056	−0.282028	−	0.959406i	$-0.591007\pi$
$941$	−17.3028	−0.564056	−0.282028	+	0.959406i	$0.591007\pi$
$942$	0	0
$943$	2.83710	0.0923887
$944$	0	0
$945$	8.09890	0.263457
$946$	0	0
$947$	32.0494	1.04147	0.520734	−	0.853719i	$-0.325658\pi$
$947$	32.0494	1.04147	0.520734	+	0.853719i	$0.325658\pi$
$948$	0	0
$949$	4.49693	0.145976
$950$	0	0
$951$	56.7442	1.84006
$952$	0	0
$953$	3.70209	0.119922	0.0599612	−	0.998201i	$-0.480902\pi$
$953$	3.70209	0.119922	0.0599612	+	0.998201i	$0.480902\pi$
$954$	0	0
$955$	1.57531	0.0509757
$956$	0	0
$957$	189.983	6.14127
$958$	0	0
$959$	11.4186	0.368724
$960$	0	0
$961$	29.1978	0.941864
$962$	0	0
$963$	−65.9370	−2.12479
$964$	0	0
$965$	6.90602	0.222313
$966$	0	0
$967$	33.3424	1.07222	0.536110	−	0.844148i	$-0.319893\pi$
$967$	33.3424	1.07222	0.536110	+	0.844148i	$0.319893\pi$
$968$	0	0
$969$	−6.34017	−0.203676
$970$	0	0
$971$	−41.7009	−1.33824	−0.669122	−	0.743152i	$-0.733330\pi$
$971$	−41.7009	−1.33824	−0.669122	+	0.743152i	$0.733330\pi$
$972$	0	0
$973$	−10.8059	−0.346421
$974$	0	0
$975$	5.70928	0.182843
$976$	0	0
$977$	−43.8108	−1.40163	−0.700816	−	0.713342i	$-0.747181\pi$
$977$	−43.8108	−1.40163	−0.700816	+	0.713342i	$0.747181\pi$
$978$	0	0
$979$	−37.0928	−1.18549
$980$	0	0
$981$	85.2905	2.72312
$982$	0	0
$983$	51.1799	1.63239	0.816193	−	0.577779i	$-0.196081\pi$
$983$	51.1799	1.63239	0.816193	+	0.577779i	$0.196081\pi$
$984$	0	0
$985$	−10.4969	−0.334460
$986$	0	0
$987$	−1.26180	−0.0401634
$988$	0	0
$989$	−6.18795	−0.196765
$990$	0	0
$991$	48.4801	1.54002	0.770011	−	0.638031i	$-0.220251\pi$
$991$	48.4801	1.54002	0.770011	+	0.638031i	$0.220251\pi$
$992$	0	0
$993$	83.5006	2.64981
$994$	0	0
$995$	10.1568	0.321991
$996$	0	0
$997$	−1.80221	−0.0570765	−0.0285382	−	0.999593i	$-0.509085\pi$
$997$	−1.80221	−0.0570765	−0.0285382	+	0.999593i	$0.509085\pi$
$998$	0	0
$999$	48.7936	1.54376

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.bm.1.1		3
4.3	odd	2		6080.2.a.ca.1.3		3
8.3	odd	2		3040.2.a.j.1.1	✓	3
8.5	even	2		3040.2.a.p.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.j.1.1	✓	3	8.3	odd	2
3040.2.a.p.1.3	yes	3	8.5	even	2
6080.2.a.bm.1.1		3	1.1	even	1	trivial
6080.2.a.ca.1.3		3	4.3	odd	2

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:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q-3.17009 q^{3} -1.00000 q^{5} +0.630898 q^{7} +7.04945 q^{9} +O(q^{10})\)	\(q-3.17009 q^{3} -1.00000 q^{5} +0.630898 q^{7} +7.04945 q^{9} -5.70928 q^{11} -1.80098 q^{13} +3.17009 q^{15} +2.00000 q^{17} +1.00000 q^{19} -2.00000 q^{21} -0.630898 q^{23} +1.00000 q^{25} -12.8371 q^{27} +10.4969 q^{29} -7.75872 q^{31} +18.0989 q^{33} -0.630898 q^{35} -3.80098 q^{37} +5.70928 q^{39} -4.49693 q^{41} +9.80817 q^{43} -7.04945 q^{45} +0.630898 q^{47} -6.60197 q^{49} -6.34017 q^{51} +7.80098 q^{53} +5.70928 q^{55} -3.17009 q^{57} +1.41855 q^{59} -8.94441 q^{61} +4.44748 q^{63} +1.80098 q^{65} +9.66701 q^{67} +2.00000 q^{69} -7.75872 q^{71} -2.49693 q^{73} -3.17009 q^{75} -3.60197 q^{77} +11.5753 q^{79} +19.5464 q^{81} +8.23287 q^{83} -2.00000 q^{85} -33.2762 q^{87} +6.49693 q^{89} -1.13624 q^{91} +24.5958 q^{93} -1.00000 q^{95} +12.2979 q^{97} -40.2472 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 4 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9	\( 3 q - 4 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 10 q^{11} + 4 q^{13} + 4 q^{15} + 6 q^{17} + 3 q^{19} - 6 q^{21} + 2 q^{23} + 3 q^{25} - 10 q^{27} + 14 q^{29} + 2 q^{31} + 18 q^{33} + 2 q^{35} - 2 q^{37} + 10 q^{39} + 4 q^{41} - 14 q^{43} - 3 q^{45} - 2 q^{47} - q^{49} - 8 q^{51} + 14 q^{53} + 10 q^{55} - 4 q^{57} - 10 q^{59} - 10 q^{61} + 14 q^{63} - 4 q^{65} + 6 q^{67} + 6 q^{69} + 2 q^{71} + 10 q^{73} - 4 q^{75} + 8 q^{77} + 14 q^{79} + 23 q^{81} + 2 q^{83} - 6 q^{85} - 24 q^{87} + 2 q^{89} - 30 q^{91} + 20 q^{93} - 3 q^{95} + 10 q^{97} - 30 q^{99}+O(q^{100}) \) 3 * q - 4 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 10 * q^11 + 4 * q^13 + 4 * q^15 + 6 * q^17 + 3 * q^19 - 6 * q^21 + 2 * q^23 + 3 * q^25 - 10 * q^27 + 14 * q^29 + 2 * q^31 + 18 * q^33 + 2 * q^35 - 2 * q^37 + 10 * q^39 + 4 * q^41 - 14 * q^43 - 3 * q^45 - 2 * q^47 - q^49 - 8 * q^51 + 14 * q^53 + 10 * q^55 - 4 * q^57 - 10 * q^59 - 10 * q^61 + 14 * q^63 - 4 * q^65 + 6 * q^67 + 6 * q^69 + 2 * q^71 + 10 * q^73 - 4 * q^75 + 8 * q^77 + 14 * q^79 + 23 * q^81 + 2 * q^83 - 6 * q^85 - 24 * q^87 + 2 * q^89 - 30 * q^91 + 20 * q^93 - 3 * q^95 + 10 * q^97 - 30 * q^99

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