LMFDB - Embedded newform 8016.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0080822603$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 16x^{7} + 45x^{6} + 67x^{5} - 166x^{4} - 83x^{3} + 152x^{2} + 51x - 10 $ x^9 - 3x^8 - 16x^7 + 45x^6 + 67x^5 - 166x^4 - 83x^3 + 152x^2 + 51x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.17025$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.17025 q^{5} +1.78655 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.17025 q^{5} +1.78655 q^{7} +1.00000 q^{9} -4.35038 q^{11} +2.54982 q^{13} +4.17025 q^{15} +0.0716370 q^{17} +7.16710 q^{19} -1.78655 q^{21} +2.75103 q^{23} +12.3910 q^{25} -1.00000 q^{27} +5.44140 q^{29} -3.34520 q^{31} +4.35038 q^{33} -7.45034 q^{35} -6.04022 q^{37} -2.54982 q^{39} +3.33471 q^{41} -3.47637 q^{43} -4.17025 q^{45} -1.84888 q^{47} -3.80826 q^{49} -0.0716370 q^{51} -10.4810 q^{53} +18.1422 q^{55} -7.16710 q^{57} -7.84697 q^{59} -6.59987 q^{61} +1.78655 q^{63} -10.6334 q^{65} -3.06798 q^{67} -2.75103 q^{69} +9.41046 q^{71} -8.85175 q^{73} -12.3910 q^{75} -7.77214 q^{77} -0.0583476 q^{79} +1.00000 q^{81} -0.584902 q^{83} -0.298744 q^{85} -5.44140 q^{87} -3.61449 q^{89} +4.55536 q^{91} +3.34520 q^{93} -29.8886 q^{95} +0.334815 q^{97} -4.35038 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} - 6 q^{5} + 11 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q - 9 * q^3 - 6 * q^5 + 11 * q^7 + 9 * q^9	$ 9 q - 9 q^{3} - 6 q^{5} + 11 q^{7} + 9 q^{9} - q^{11} - 4 q^{13} + 6 q^{15} - 9 q^{17} + 8 q^{19} - 11 q^{21} + 7 q^{23} - q^{25} - 9 q^{27} - 9 q^{29} + 25 q^{31} + q^{33} - 5 q^{35} - 6 q^{37} + 4 q^{39} - 4 q^{41} + 24 q^{43} - 6 q^{45} + 16 q^{47} + 4 q^{49} + 9 q^{51} - 26 q^{53} + 29 q^{55} - 8 q^{57} + 4 q^{59} - 20 q^{61} + 11 q^{63} - 8 q^{65} + 25 q^{67} - 7 q^{69} + 15 q^{71} - 10 q^{73} + q^{75} - 20 q^{77} + 34 q^{79} + 9 q^{81} + 4 q^{83} - 13 q^{85} + 9 q^{87} + 13 q^{89} + 21 q^{91} - 25 q^{93} + 7 q^{95} - 4 q^{97} - q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 - 6 * q^5 + 11 * q^7 + 9 * q^9 - q^11 - 4 * q^13 + 6 * q^15 - 9 * q^17 + 8 * q^19 - 11 * q^21 + 7 * q^23 - q^25 - 9 * q^27 - 9 * q^29 + 25 * q^31 + q^33 - 5 * q^35 - 6 * q^37 + 4 * q^39 - 4 * q^41 + 24 * q^43 - 6 * q^45 + 16 * q^47 + 4 * q^49 + 9 * q^51 - 26 * q^53 + 29 * q^55 - 8 * q^57 + 4 * q^59 - 20 * q^61 + 11 * q^63 - 8 * q^65 + 25 * q^67 - 7 * q^69 + 15 * q^71 - 10 * q^73 + q^75 - 20 * q^77 + 34 * q^79 + 9 * q^81 + 4 * q^83 - 13 * q^85 + 9 * q^87 + 13 * q^89 + 21 * q^91 - 25 * q^93 + 7 * q^95 - 4 * q^97 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−4.17025	−1.86499	−0.932496	−	0.361180i	$-0.882374\pi$
$5$	−4.17025	−1.86499	−0.932496	+	0.361180i	$0.882374\pi$
$6$	0	0
$7$	1.78655	0.675251	0.337625	−	0.941281i	$-0.390376\pi$
$7$	1.78655	0.675251	0.337625	+	0.941281i	$0.390376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.35038	−1.31169	−0.655844	−	0.754897i	$-0.727687\pi$
$11$	−4.35038	−1.31169	−0.655844	+	0.754897i	$0.727687\pi$
$12$	0	0
$13$	2.54982	0.707192	0.353596	−	0.935398i	$-0.384959\pi$
$13$	2.54982	0.707192	0.353596	+	0.935398i	$0.384959\pi$
$14$	0	0
$15$	4.17025	1.07675
$16$	0	0
$17$	0.0716370	0.0173745	0.00868726	−	0.999962i	$-0.497235\pi$
$17$	0.0716370	0.0173745	0.00868726	+	0.999962i	$0.497235\pi$
$18$	0	0
$19$	7.16710	1.64425	0.822123	−	0.569310i	$-0.192789\pi$
$19$	7.16710	1.64425	0.822123	+	0.569310i	$0.192789\pi$
$20$	0	0
$21$	−1.78655	−0.389856
$22$	0	0
$23$	2.75103	0.573630	0.286815	−	0.957986i	$-0.407404\pi$
$23$	2.75103	0.573630	0.286815	+	0.957986i	$0.407404\pi$
$24$	0	0
$25$	12.3910	2.47820
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.44140	1.01044	0.505222	−	0.862990i	$-0.331411\pi$
$29$	5.44140	1.01044	0.505222	+	0.862990i	$0.331411\pi$
$30$	0	0
$31$	−3.34520	−0.600816	−0.300408	−	0.953811i	$-0.597123\pi$
$31$	−3.34520	−0.600816	−0.300408	+	0.953811i	$0.597123\pi$
$32$	0	0
$33$	4.35038	0.757303
$34$	0	0
$35$	−7.45034	−1.25934
$36$	0	0
$37$	−6.04022	−0.993005	−0.496503	−	0.868035i	$-0.665383\pi$
$37$	−6.04022	−0.993005	−0.496503	+	0.868035i	$0.665383\pi$
$38$	0	0
$39$	−2.54982	−0.408298
$40$	0	0
$41$	3.33471	0.520794	0.260397	−	0.965502i	$-0.416147\pi$
$41$	3.33471	0.520794	0.260397	+	0.965502i	$0.416147\pi$
$42$	0	0
$43$	−3.47637	−0.530142	−0.265071	−	0.964229i	$-0.585395\pi$
$43$	−3.47637	−0.530142	−0.265071	+	0.964229i	$0.585395\pi$
$44$	0	0
$45$	−4.17025	−0.621664
$46$	0	0
$47$	−1.84888	−0.269687	−0.134843	−	0.990867i	$-0.543053\pi$
$47$	−1.84888	−0.269687	−0.134843	+	0.990867i	$0.543053\pi$
$48$	0	0
$49$	−3.80826	−0.544036
$50$	0	0
$51$	−0.0716370	−0.0100312
$52$	0	0
$53$	−10.4810	−1.43968	−0.719838	−	0.694142i	$-0.755784\pi$
$53$	−10.4810	−1.43968	−0.719838	+	0.694142i	$0.755784\pi$
$54$	0	0
$55$	18.1422	2.44629
$56$	0	0
$57$	−7.16710	−0.949306
$58$	0	0
$59$	−7.84697	−1.02159	−0.510794	−	0.859703i	$-0.670649\pi$
$59$	−7.84697	−1.02159	−0.510794	+	0.859703i	$0.670649\pi$
$60$	0	0
$61$	−6.59987	−0.845026	−0.422513	−	0.906357i	$-0.638852\pi$
$61$	−6.59987	−0.845026	−0.422513	+	0.906357i	$0.638852\pi$
$62$	0	0
$63$	1.78655	0.225084
$64$	0	0
$65$	−10.6334	−1.31891
$66$	0	0
$67$	−3.06798	−0.374814	−0.187407	−	0.982282i	$-0.560008\pi$
$67$	−3.06798	−0.374814	−0.187407	+	0.982282i	$0.560008\pi$
$68$	0	0
$69$	−2.75103	−0.331185
$70$	0	0
$71$	9.41046	1.11682	0.558408	−	0.829566i	$-0.311412\pi$
$71$	9.41046	1.11682	0.558408	+	0.829566i	$0.311412\pi$
$72$	0	0
$73$	−8.85175	−1.03602	−0.518009	−	0.855375i	$-0.673327\pi$
$73$	−8.85175	−1.03602	−0.518009	+	0.855375i	$0.673327\pi$
$74$	0	0
$75$	−12.3910	−1.43079
$76$	0	0
$77$	−7.77214	−0.885718
$78$	0	0
$79$	−0.0583476	−0.00656462	−0.00328231	−	0.999995i	$-0.501045\pi$
$79$	−0.0583476	−0.00656462	−0.00328231	+	0.999995i	$0.501045\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.584902	−0.0642013	−0.0321007	−	0.999485i	$-0.510220\pi$
$83$	−0.584902	−0.0642013	−0.0321007	+	0.999485i	$0.510220\pi$
$84$	0	0
$85$	−0.298744	−0.0324033
$86$	0	0
$87$	−5.44140	−0.583380
$88$	0	0
$89$	−3.61449	−0.383135	−0.191567	−	0.981479i	$-0.561357\pi$
$89$	−3.61449	−0.383135	−0.191567	+	0.981479i	$0.561357\pi$
$90$	0	0
$91$	4.55536	0.477532
$92$	0	0
$93$	3.34520	0.346881
$94$	0	0
$95$	−29.8886	−3.06651
$96$	0	0
$97$	0.334815	0.0339953	0.0169976	−	0.999856i	$-0.494589\pi$
$97$	0.334815	0.0339953	0.0169976	+	0.999856i	$0.494589\pi$
$98$	0	0
$99$	−4.35038	−0.437229
$100$	0	0
$101$	4.71758	0.469416	0.234708	−	0.972066i	$-0.424587\pi$
$101$	4.71758	0.469416	0.234708	+	0.972066i	$0.424587\pi$
$102$	0	0
$103$	20.2182	1.99216	0.996081	−	0.0884418i	$-0.0281887\pi$
$103$	20.2182	1.99216	0.996081	+	0.0884418i	$0.0281887\pi$
$104$	0	0
$105$	7.45034	0.727079
$106$	0	0
$107$	−4.02095	−0.388719	−0.194360	−	0.980930i	$-0.562263\pi$
$107$	−4.02095	−0.388719	−0.194360	+	0.980930i	$0.562263\pi$
$108$	0	0
$109$	7.32526	0.701633	0.350816	−	0.936444i	$-0.385904\pi$
$109$	7.32526	0.701633	0.350816	+	0.936444i	$0.385904\pi$
$110$	0	0
$111$	6.04022	0.573312
$112$	0	0
$113$	−14.8947	−1.40118	−0.700590	−	0.713564i	$-0.747080\pi$
$113$	−14.8947	−1.40118	−0.700590	+	0.713564i	$0.747080\pi$
$114$	0	0
$115$	−11.4725	−1.06981
$116$	0	0
$117$	2.54982	0.235731
$118$	0	0
$119$	0.127983	0.0117322
$120$	0	0
$121$	7.92577	0.720525
$122$	0	0
$123$	−3.33471	−0.300681
$124$	0	0
$125$	−30.8222	−2.75683
$126$	0	0
$127$	18.8761	1.67498	0.837490	−	0.546453i	$-0.184022\pi$
$127$	18.8761	1.67498	0.837490	+	0.546453i	$0.184022\pi$
$128$	0	0
$129$	3.47637	0.306078
$130$	0	0
$131$	−0.342949	−0.0299636	−0.0149818	−	0.999888i	$-0.504769\pi$
$131$	−0.342949	−0.0299636	−0.0149818	+	0.999888i	$0.504769\pi$
$132$	0	0
$133$	12.8044	1.11028
$134$	0	0
$135$	4.17025	0.358918
$136$	0	0
$137$	11.5389	0.985833	0.492917	−	0.870077i	$-0.335931\pi$
$137$	11.5389	0.985833	0.492917	+	0.870077i	$0.335931\pi$
$138$	0	0
$139$	15.7651	1.33718	0.668591	−	0.743630i	$-0.266898\pi$
$139$	15.7651	1.33718	0.668591	+	0.743630i	$0.266898\pi$
$140$	0	0
$141$	1.84888	0.155704
$142$	0	0
$143$	−11.0927	−0.927615
$144$	0	0
$145$	−22.6920	−1.88447
$146$	0	0
$147$	3.80826	0.314100
$148$	0	0
$149$	0.883298	0.0723626	0.0361813	−	0.999345i	$-0.488481\pi$
$149$	0.883298	0.0723626	0.0361813	+	0.999345i	$0.488481\pi$
$150$	0	0
$151$	4.06988	0.331202	0.165601	−	0.986193i	$-0.447044\pi$
$151$	4.06988	0.331202	0.165601	+	0.986193i	$0.447044\pi$
$152$	0	0
$153$	0.0716370	0.00579151
$154$	0	0
$155$	13.9503	1.12052
$156$	0	0
$157$	−12.5877	−1.00461	−0.502304	−	0.864691i	$-0.667514\pi$
$157$	−12.5877	−1.00461	−0.502304	+	0.864691i	$0.667514\pi$
$158$	0	0
$159$	10.4810	0.831198
$160$	0	0
$161$	4.91484	0.387344
$162$	0	0
$163$	10.2085	0.799593	0.399797	−	0.916604i	$-0.369081\pi$
$163$	10.2085	0.799593	0.399797	+	0.916604i	$0.369081\pi$
$164$	0	0
$165$	−18.1422	−1.41236
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−6.49843	−0.499879
$170$	0	0
$171$	7.16710	0.548082
$172$	0	0
$173$	−24.5779	−1.86862	−0.934310	−	0.356461i	$-0.883983\pi$
$173$	−24.5779	−1.86862	−0.934310	+	0.356461i	$0.883983\pi$
$174$	0	0
$175$	22.1371	1.67340
$176$	0	0
$177$	7.84697	0.589814
$178$	0	0
$179$	14.2545	1.06543	0.532717	−	0.846293i	$-0.321171\pi$
$179$	14.2545	1.06543	0.532717	+	0.846293i	$0.321171\pi$
$180$	0	0
$181$	22.8429	1.69790	0.848948	−	0.528476i	$-0.177236\pi$
$181$	22.8429	1.69790	0.848948	+	0.528476i	$0.177236\pi$
$182$	0	0
$183$	6.59987	0.487876
$184$	0	0
$185$	25.1892	1.85195
$186$	0	0
$187$	−0.311648	−0.0227899
$188$	0	0
$189$	−1.78655	−0.129952
$190$	0	0
$191$	−1.42405	−0.103041	−0.0515203	−	0.998672i	$-0.516407\pi$
$191$	−1.42405	−0.103041	−0.0515203	+	0.998672i	$0.516407\pi$
$192$	0	0
$193$	−13.9804	−1.00633	−0.503166	−	0.864190i	$-0.667831\pi$
$193$	−13.9804	−1.00633	−0.503166	+	0.864190i	$0.667831\pi$
$194$	0	0
$195$	10.6334	0.761472
$196$	0	0
$197$	14.2442	1.01486	0.507428	−	0.861694i	$-0.330596\pi$
$197$	14.2442	1.01486	0.507428	+	0.861694i	$0.330596\pi$
$198$	0	0
$199$	16.6866	1.18288	0.591441	−	0.806348i	$-0.298559\pi$
$199$	16.6866	1.18288	0.591441	+	0.806348i	$0.298559\pi$
$200$	0	0
$201$	3.06798	0.216399
$202$	0	0
$203$	9.72131	0.682302
$204$	0	0
$205$	−13.9066	−0.971277
$206$	0	0
$207$	2.75103	0.191210
$208$	0	0
$209$	−31.1796	−2.15674
$210$	0	0
$211$	8.11871	0.558915	0.279458	−	0.960158i	$-0.409845\pi$
$211$	8.11871	0.558915	0.279458	+	0.960158i	$0.409845\pi$
$212$	0	0
$213$	−9.41046	−0.644794
$214$	0	0
$215$	14.4973	0.988711
$216$	0	0
$217$	−5.97636	−0.405701
$218$	0	0
$219$	8.85175	0.598146
$220$	0	0
$221$	0.182661	0.0122871
$222$	0	0
$223$	10.7903	0.722574	0.361287	−	0.932455i	$-0.382338\pi$
$223$	10.7903	0.722574	0.361287	+	0.932455i	$0.382338\pi$
$224$	0	0
$225$	12.3910	0.826066
$226$	0	0
$227$	0.189331	0.0125664	0.00628318	−	0.999980i	$-0.498000\pi$
$227$	0.189331	0.0125664	0.00628318	+	0.999980i	$0.498000\pi$
$228$	0	0
$229$	−14.4062	−0.951987	−0.475993	−	0.879449i	$-0.657911\pi$
$229$	−14.4062	−0.951987	−0.475993	+	0.879449i	$0.657911\pi$
$230$	0	0
$231$	7.77214	0.511370
$232$	0	0
$233$	29.7817	1.95107	0.975533	−	0.219854i	$-0.0705582\pi$
$233$	29.7817	1.95107	0.975533	+	0.219854i	$0.0705582\pi$
$234$	0	0
$235$	7.71029	0.502964
$236$	0	0
$237$	0.0583476	0.00379008
$238$	0	0
$239$	−7.50122	−0.485213	−0.242607	−	0.970125i	$-0.578002\pi$
$239$	−7.50122	−0.485213	−0.242607	+	0.970125i	$0.578002\pi$
$240$	0	0
$241$	15.6609	1.00881	0.504404	−	0.863468i	$-0.331712\pi$
$241$	15.6609	1.00881	0.504404	+	0.863468i	$0.331712\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	15.8814	1.01462
$246$	0	0
$247$	18.2748	1.16280
$248$	0	0
$249$	0.584902	0.0370667
$250$	0	0
$251$	7.02486	0.443405	0.221703	−	0.975114i	$-0.428839\pi$
$251$	7.02486	0.443405	0.221703	+	0.975114i	$0.428839\pi$
$252$	0	0
$253$	−11.9680	−0.752423
$254$	0	0
$255$	0.298744	0.0187081
$256$	0	0
$257$	8.17803	0.510132	0.255066	−	0.966924i	$-0.417903\pi$
$257$	8.17803	0.510132	0.255066	+	0.966924i	$0.417903\pi$
$258$	0	0
$259$	−10.7911	−0.670528
$260$	0	0
$261$	5.44140	0.336814
$262$	0	0
$263$	−22.1033	−1.36295	−0.681473	−	0.731843i	$-0.738660\pi$
$263$	−22.1033	−1.36295	−0.681473	+	0.731843i	$0.738660\pi$
$264$	0	0
$265$	43.7084	2.68499
$266$	0	0
$267$	3.61449	0.221203
$268$	0	0
$269$	−14.4088	−0.878519	−0.439259	−	0.898360i	$-0.644759\pi$
$269$	−14.4088	−0.878519	−0.439259	+	0.898360i	$0.644759\pi$
$270$	0	0
$271$	4.72029	0.286737	0.143369	−	0.989669i	$-0.454207\pi$
$271$	4.72029	0.286737	0.143369	+	0.989669i	$0.454207\pi$
$272$	0	0
$273$	−4.55536	−0.275703
$274$	0	0
$275$	−53.9054	−3.25062
$276$	0	0
$277$	−27.9962	−1.68213	−0.841063	−	0.540937i	$-0.818070\pi$
$277$	−27.9962	−1.68213	−0.841063	+	0.540937i	$0.818070\pi$
$278$	0	0
$279$	−3.34520	−0.200272
$280$	0	0
$281$	−21.2473	−1.26751	−0.633755	−	0.773534i	$-0.718487\pi$
$281$	−21.2473	−1.26751	−0.633755	+	0.773534i	$0.718487\pi$
$282$	0	0
$283$	6.07839	0.361322	0.180661	−	0.983545i	$-0.442176\pi$
$283$	6.07839	0.361322	0.180661	+	0.983545i	$0.442176\pi$
$284$	0	0
$285$	29.8886	1.77045
$286$	0	0
$287$	5.95761	0.351666
$288$	0	0
$289$	−16.9949	−0.999698
$290$	0	0
$291$	−0.334815	−0.0196272
$292$	0	0
$293$	15.5348	0.907553	0.453777	−	0.891115i	$-0.350076\pi$
$293$	15.5348	0.907553	0.453777	+	0.891115i	$0.350076\pi$
$294$	0	0
$295$	32.7238	1.90525
$296$	0	0
$297$	4.35038	0.252434
$298$	0	0
$299$	7.01463	0.405666
$300$	0	0
$301$	−6.21070	−0.357979
$302$	0	0
$303$	−4.71758	−0.271018
$304$	0	0
$305$	27.5231	1.57597
$306$	0	0
$307$	7.05233	0.402498	0.201249	−	0.979540i	$-0.435500\pi$
$307$	7.05233	0.402498	0.201249	+	0.979540i	$0.435500\pi$
$308$	0	0
$309$	−20.2182	−1.15018
$310$	0	0
$311$	−26.0059	−1.47466	−0.737330	−	0.675532i	$-0.763914\pi$
$311$	−26.0059	−1.47466	−0.737330	+	0.675532i	$0.763914\pi$
$312$	0	0
$313$	17.5377	0.991287	0.495644	−	0.868526i	$-0.334932\pi$
$313$	17.5377	0.991287	0.495644	+	0.868526i	$0.334932\pi$
$314$	0	0
$315$	−7.45034	−0.419779
$316$	0	0
$317$	20.8460	1.17083	0.585413	−	0.810735i	$-0.300932\pi$
$317$	20.8460	1.17083	0.585413	+	0.810735i	$0.300932\pi$
$318$	0	0
$319$	−23.6721	−1.32539
$320$	0	0
$321$	4.02095	0.224427
$322$	0	0
$323$	0.513430	0.0285680
$324$	0	0
$325$	31.5947	1.75256
$326$	0	0
$327$	−7.32526	−0.405088
$328$	0	0
$329$	−3.30311	−0.182106
$330$	0	0
$331$	−27.8470	−1.53061	−0.765305	−	0.643667i	$-0.777412\pi$
$331$	−27.8470	−1.53061	−0.765305	+	0.643667i	$0.777412\pi$
$332$	0	0
$333$	−6.04022	−0.331002
$334$	0	0
$335$	12.7943	0.699025
$336$	0	0
$337$	7.22298	0.393461	0.196731	−	0.980458i	$-0.436968\pi$
$337$	7.22298	0.393461	0.196731	+	0.980458i	$0.436968\pi$
$338$	0	0
$339$	14.8947	0.808971
$340$	0	0
$341$	14.5529	0.788083
$342$	0	0
$343$	−19.3094	−1.04261
$344$	0	0
$345$	11.4725	0.617658
$346$	0	0
$347$	−14.7990	−0.794452	−0.397226	−	0.917721i	$-0.630027\pi$
$347$	−14.7990	−0.794452	−0.397226	+	0.917721i	$0.630027\pi$
$348$	0	0
$349$	−1.40557	−0.0752384	−0.0376192	−	0.999292i	$-0.511977\pi$
$349$	−1.40557	−0.0752384	−0.0376192	+	0.999292i	$0.511977\pi$
$350$	0	0
$351$	−2.54982	−0.136099
$352$	0	0
$353$	−11.8156	−0.628883	−0.314442	−	0.949277i	$-0.601817\pi$
$353$	−11.8156	−0.628883	−0.314442	+	0.949277i	$0.601817\pi$
$354$	0	0
$355$	−39.2440	−2.08285
$356$	0	0
$357$	−0.127983	−0.00677356
$358$	0	0
$359$	5.61217	0.296199	0.148099	−	0.988972i	$-0.452684\pi$
$359$	5.61217	0.296199	0.148099	+	0.988972i	$0.452684\pi$
$360$	0	0
$361$	32.3674	1.70355
$362$	0	0
$363$	−7.92577	−0.415995
$364$	0	0
$365$	36.9140	1.93217
$366$	0	0
$367$	31.0874	1.62275	0.811374	−	0.584527i	$-0.198720\pi$
$367$	31.0874	1.62275	0.811374	+	0.584527i	$0.198720\pi$
$368$	0	0
$369$	3.33471	0.173598
$370$	0	0
$371$	−18.7248	−0.972143
$372$	0	0
$373$	−1.78421	−0.0923828	−0.0461914	−	0.998933i	$-0.514708\pi$
$373$	−1.78421	−0.0923828	−0.0461914	+	0.998933i	$0.514708\pi$
$374$	0	0
$375$	30.8222	1.59165
$376$	0	0
$377$	13.8746	0.714577
$378$	0	0
$379$	−8.79107	−0.451567	−0.225783	−	0.974178i	$-0.572494\pi$
$379$	−8.79107	−0.451567	−0.225783	+	0.974178i	$0.572494\pi$
$380$	0	0
$381$	−18.8761	−0.967050
$382$	0	0
$383$	19.5651	0.999729	0.499865	−	0.866103i	$-0.333383\pi$
$383$	19.5651	0.999729	0.499865	+	0.866103i	$0.333383\pi$
$384$	0	0
$385$	32.4118	1.65186
$386$	0	0
$387$	−3.47637	−0.176714
$388$	0	0
$389$	−19.6033	−0.993927	−0.496963	−	0.867772i	$-0.665552\pi$
$389$	−19.6033	−0.993927	−0.496963	+	0.867772i	$0.665552\pi$
$390$	0	0
$391$	0.197075	0.00996654
$392$	0	0
$393$	0.342949	0.0172995
$394$	0	0
$395$	0.243324	0.0122430
$396$	0	0
$397$	−8.41521	−0.422347	−0.211174	−	0.977449i	$-0.567729\pi$
$397$	−8.41521	−0.422347	−0.211174	+	0.977449i	$0.567729\pi$
$398$	0	0
$399$	−12.8044	−0.641020
$400$	0	0
$401$	1.30360	0.0650989	0.0325494	−	0.999470i	$-0.489637\pi$
$401$	1.30360	0.0650989	0.0325494	+	0.999470i	$0.489637\pi$
$402$	0	0
$403$	−8.52965	−0.424892
$404$	0	0
$405$	−4.17025	−0.207221
$406$	0	0
$407$	26.2772	1.30251
$408$	0	0
$409$	10.4075	0.514616	0.257308	−	0.966329i	$-0.417165\pi$
$409$	10.4075	0.514616	0.257308	+	0.966329i	$0.417165\pi$
$410$	0	0
$411$	−11.5389	−0.569171
$412$	0	0
$413$	−14.0190	−0.689828
$414$	0	0
$415$	2.43919	0.119735
$416$	0	0
$417$	−15.7651	−0.772022
$418$	0	0
$419$	−8.00995	−0.391312	−0.195656	−	0.980673i	$-0.562684\pi$
$419$	−8.00995	−0.391312	−0.195656	+	0.980673i	$0.562684\pi$
$420$	0	0
$421$	23.1739	1.12943	0.564714	−	0.825287i	$-0.308986\pi$
$421$	23.1739	1.12943	0.564714	+	0.825287i	$0.308986\pi$
$422$	0	0
$423$	−1.84888	−0.0898956
$424$	0	0
$425$	0.887653	0.0430575
$426$	0	0
$427$	−11.7910	−0.570605
$428$	0	0
$429$	11.0927	0.535559
$430$	0	0
$431$	36.5254	1.75937	0.879684	−	0.475559i	$-0.157754\pi$
$431$	36.5254	1.75937	0.879684	+	0.475559i	$0.157754\pi$
$432$	0	0
$433$	−11.3450	−0.545207	−0.272603	−	0.962127i	$-0.587885\pi$
$433$	−11.3450	−0.545207	−0.272603	+	0.962127i	$0.587885\pi$
$434$	0	0
$435$	22.6920	1.08800
$436$	0	0
$437$	19.7169	0.943188
$438$	0	0
$439$	−1.47796	−0.0705391	−0.0352696	−	0.999378i	$-0.511229\pi$
$439$	−1.47796	−0.0705391	−0.0352696	+	0.999378i	$0.511229\pi$
$440$	0	0
$441$	−3.80826	−0.181345
$442$	0	0
$443$	30.7173	1.45943	0.729713	−	0.683754i	$-0.239654\pi$
$443$	30.7173	1.45943	0.729713	+	0.683754i	$0.239654\pi$
$444$	0	0
$445$	15.0733	0.714544
$446$	0	0
$447$	−0.883298	−0.0417786
$448$	0	0
$449$	13.5252	0.638292	0.319146	−	0.947705i	$-0.396604\pi$
$449$	13.5252	0.638292	0.319146	+	0.947705i	$0.396604\pi$
$450$	0	0
$451$	−14.5072	−0.683119
$452$	0	0
$453$	−4.06988	−0.191220
$454$	0	0
$455$	−18.9970	−0.890593
$456$	0	0
$457$	18.2676	0.854523	0.427262	−	0.904128i	$-0.359478\pi$
$457$	18.2676	0.854523	0.427262	+	0.904128i	$0.359478\pi$
$458$	0	0
$459$	−0.0716370	−0.00334373
$460$	0	0
$461$	21.1705	0.986007	0.493003	−	0.870027i	$-0.335899\pi$
$461$	21.1705	0.986007	0.493003	+	0.870027i	$0.335899\pi$
$462$	0	0
$463$	25.4679	1.18359	0.591796	−	0.806088i	$-0.298419\pi$
$463$	25.4679	1.18359	0.591796	+	0.806088i	$0.298419\pi$
$464$	0	0
$465$	−13.9503	−0.646931
$466$	0	0
$467$	18.1847	0.841488	0.420744	−	0.907179i	$-0.361769\pi$
$467$	18.1847	0.841488	0.420744	+	0.907179i	$0.361769\pi$
$468$	0	0
$469$	−5.48109	−0.253093
$470$	0	0
$471$	12.5877	0.580010
$472$	0	0
$473$	15.1235	0.695381
$474$	0	0
$475$	88.8075	4.07477
$476$	0	0
$477$	−10.4810	−0.479892
$478$	0	0
$479$	31.4363	1.43636	0.718182	−	0.695856i	$-0.244975\pi$
$479$	31.4363	1.43636	0.718182	+	0.695856i	$0.244975\pi$
$480$	0	0
$481$	−15.4014	−0.702246
$482$	0	0
$483$	−4.91484	−0.223633
$484$	0	0
$485$	−1.39626	−0.0634009
$486$	0	0
$487$	−7.63846	−0.346132	−0.173066	−	0.984910i	$-0.555367\pi$
$487$	−7.63846	−0.346132	−0.173066	+	0.984910i	$0.555367\pi$
$488$	0	0
$489$	−10.2085	−0.461645
$490$	0	0
$491$	−2.60219	−0.117435	−0.0587176	−	0.998275i	$-0.518701\pi$
$491$	−2.60219	−0.117435	−0.0587176	+	0.998275i	$0.518701\pi$
$492$	0	0
$493$	0.389806	0.0175560
$494$	0	0
$495$	18.1422	0.815429
$496$	0	0
$497$	16.8122	0.754131
$498$	0	0
$499$	11.7781	0.527261	0.263631	−	0.964624i	$-0.415080\pi$
$499$	11.7781	0.527261	0.263631	+	0.964624i	$0.415080\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−19.8128	−0.883410	−0.441705	−	0.897160i	$-0.645626\pi$
$503$	−19.8128	−0.883410	−0.441705	+	0.897160i	$0.645626\pi$
$504$	0	0
$505$	−19.6735	−0.875458
$506$	0	0
$507$	6.49843	0.288606
$508$	0	0
$509$	41.0164	1.81802	0.909010	−	0.416774i	$-0.136839\pi$
$509$	41.0164	1.81802	0.909010	+	0.416774i	$0.136839\pi$
$510$	0	0
$511$	−15.8140	−0.699572
$512$	0	0
$513$	−7.16710	−0.316435
$514$	0	0
$515$	−84.3151	−3.71537
$516$	0	0
$517$	8.04332	0.353745
$518$	0	0
$519$	24.5779	1.07885
$520$	0	0
$521$	−24.5419	−1.07520	−0.537599	−	0.843201i	$-0.680669\pi$
$521$	−24.5419	−1.07520	−0.537599	+	0.843201i	$0.680669\pi$
$522$	0	0
$523$	−42.2257	−1.84640	−0.923201	−	0.384318i	$-0.874437\pi$
$523$	−42.2257	−1.84640	−0.923201	+	0.384318i	$0.874437\pi$
$524$	0	0
$525$	−22.1371	−0.966140
$526$	0	0
$527$	−0.239640	−0.0104389
$528$	0	0
$529$	−15.4318	−0.670949
$530$	0	0
$531$	−7.84697	−0.340530
$532$	0	0
$533$	8.50290	0.368301
$534$	0	0
$535$	16.7683	0.724959
$536$	0	0
$537$	−14.2545	−0.615129
$538$	0	0
$539$	16.5673	0.713606
$540$	0	0
$541$	39.1790	1.68444	0.842219	−	0.539135i	$-0.181249\pi$
$541$	39.1790	1.68444	0.842219	+	0.539135i	$0.181249\pi$
$542$	0	0
$543$	−22.8429	−0.980281
$544$	0	0
$545$	−30.5482	−1.30854
$546$	0	0
$547$	−30.0775	−1.28602	−0.643010	−	0.765858i	$-0.722315\pi$
$547$	−30.0775	−1.28602	−0.643010	+	0.765858i	$0.722315\pi$
$548$	0	0
$549$	−6.59987	−0.281675
$550$	0	0
$551$	38.9991	1.66142
$552$	0	0
$553$	−0.104241	−0.00443276
$554$	0	0
$555$	−25.1892	−1.06922
$556$	0	0
$557$	37.0920	1.57164	0.785819	−	0.618456i	$-0.212242\pi$
$557$	37.0920	1.57164	0.785819	+	0.618456i	$0.212242\pi$
$558$	0	0
$559$	−8.86412	−0.374912
$560$	0	0
$561$	0.311648	0.0131578
$562$	0	0
$563$	41.1589	1.73464	0.867321	−	0.497749i	$-0.165840\pi$
$563$	41.1589	1.73464	0.867321	+	0.497749i	$0.165840\pi$
$564$	0	0
$565$	62.1148	2.61319
$566$	0	0
$567$	1.78655	0.0750279
$568$	0	0
$569$	14.7976	0.620347	0.310173	−	0.950680i	$-0.399613\pi$
$569$	14.7976	0.620347	0.310173	+	0.950680i	$0.399613\pi$
$570$	0	0
$571$	19.6533	0.822464	0.411232	−	0.911531i	$-0.365099\pi$
$571$	19.6533	0.822464	0.411232	+	0.911531i	$0.365099\pi$
$572$	0	0
$573$	1.42405	0.0594906
$574$	0	0
$575$	34.0880	1.42157
$576$	0	0
$577$	34.3832	1.43139	0.715696	−	0.698412i	$-0.246109\pi$
$577$	34.3832	1.43139	0.715696	+	0.698412i	$0.246109\pi$
$578$	0	0
$579$	13.9804	0.581006
$580$	0	0
$581$	−1.04495	−0.0433520
$582$	0	0
$583$	45.5963	1.88841
$584$	0	0
$585$	−10.6334	−0.439636
$586$	0	0
$587$	11.7539	0.485135	0.242567	−	0.970135i	$-0.422010\pi$
$587$	11.7539	0.485135	0.242567	+	0.970135i	$0.422010\pi$
$588$	0	0
$589$	−23.9754	−0.987889
$590$	0	0
$591$	−14.2442	−0.585928
$592$	0	0
$593$	−30.1475	−1.23801	−0.619005	−	0.785387i	$-0.712464\pi$
$593$	−30.1475	−1.23801	−0.619005	+	0.785387i	$0.712464\pi$
$594$	0	0
$595$	−0.533720	−0.0218804
$596$	0	0
$597$	−16.6866	−0.682938
$598$	0	0
$599$	−39.4902	−1.61353	−0.806764	−	0.590874i	$-0.798783\pi$
$599$	−39.4902	−1.61353	−0.806764	+	0.590874i	$0.798783\pi$
$600$	0	0
$601$	−8.04132	−0.328012	−0.164006	−	0.986459i	$-0.552442\pi$
$601$	−8.04132	−0.328012	−0.164006	+	0.986459i	$0.552442\pi$
$602$	0	0
$603$	−3.06798	−0.124938
$604$	0	0
$605$	−33.0524	−1.34377
$606$	0	0
$607$	−2.36619	−0.0960408	−0.0480204	−	0.998846i	$-0.515291\pi$
$607$	−2.36619	−0.0960408	−0.0480204	+	0.998846i	$0.515291\pi$
$608$	0	0
$609$	−9.72131	−0.393927
$610$	0	0
$611$	−4.71430	−0.190720
$612$	0	0
$613$	38.3389	1.54849	0.774247	−	0.632884i	$-0.218129\pi$
$613$	38.3389	1.54849	0.774247	+	0.632884i	$0.218129\pi$
$614$	0	0
$615$	13.9066	0.560767
$616$	0	0
$617$	23.7019	0.954201	0.477100	−	0.878849i	$-0.341688\pi$
$617$	23.7019	0.954201	0.477100	+	0.878849i	$0.341688\pi$
$618$	0	0
$619$	−21.6224	−0.869078	−0.434539	−	0.900653i	$-0.643089\pi$
$619$	−21.6224	−0.869078	−0.434539	+	0.900653i	$0.643089\pi$
$620$	0	0
$621$	−2.75103	−0.110395
$622$	0	0
$623$	−6.45745	−0.258712
$624$	0	0
$625$	66.5816	2.66326
$626$	0	0
$627$	31.1796	1.24519
$628$	0	0
$629$	−0.432703	−0.0172530
$630$	0	0
$631$	22.0461	0.877643	0.438822	−	0.898574i	$-0.355396\pi$
$631$	22.0461	0.877643	0.438822	+	0.898574i	$0.355396\pi$
$632$	0	0
$633$	−8.11871	−0.322690
$634$	0	0
$635$	−78.7179	−3.12383
$636$	0	0
$637$	−9.71036	−0.384738
$638$	0	0
$639$	9.41046	0.372272
$640$	0	0
$641$	−2.39872	−0.0947438	−0.0473719	−	0.998877i	$-0.515085\pi$
$641$	−2.39872	−0.0947438	−0.0473719	+	0.998877i	$0.515085\pi$
$642$	0	0
$643$	−4.16347	−0.164191	−0.0820957	−	0.996624i	$-0.526161\pi$
$643$	−4.16347	−0.164191	−0.0820957	+	0.996624i	$0.526161\pi$
$644$	0	0
$645$	−14.4973	−0.570832
$646$	0	0
$647$	18.0107	0.708072	0.354036	−	0.935232i	$-0.384809\pi$
$647$	18.0107	0.708072	0.354036	+	0.935232i	$0.384809\pi$
$648$	0	0
$649$	34.1373	1.34001
$650$	0	0
$651$	5.97636	0.234232
$652$	0	0
$653$	−12.0581	−0.471871	−0.235935	−	0.971769i	$-0.575815\pi$
$653$	−12.0581	−0.471871	−0.235935	+	0.971769i	$0.575815\pi$
$654$	0	0
$655$	1.43018	0.0558819
$656$	0	0
$657$	−8.85175	−0.345340
$658$	0	0
$659$	−3.11754	−0.121442	−0.0607211	−	0.998155i	$-0.519340\pi$
$659$	−3.11754	−0.121442	−0.0607211	+	0.998155i	$0.519340\pi$
$660$	0	0
$661$	6.44996	0.250874	0.125437	−	0.992102i	$-0.459967\pi$
$661$	6.44996	0.250874	0.125437	+	0.992102i	$0.459967\pi$
$662$	0	0
$663$	−0.182661	−0.00709397
$664$	0	0
$665$	−53.3974	−2.07066
$666$	0	0
$667$	14.9695	0.579620
$668$	0	0
$669$	−10.7903	−0.417178
$670$	0	0
$671$	28.7119	1.10841
$672$	0	0
$673$	−49.1326	−1.89392	−0.946960	−	0.321350i	$-0.895863\pi$
$673$	−49.1326	−1.89392	−0.946960	+	0.321350i	$0.895863\pi$
$674$	0	0
$675$	−12.3910	−0.476929
$676$	0	0
$677$	−12.1529	−0.467074	−0.233537	−	0.972348i	$-0.575030\pi$
$677$	−12.1529	−0.467074	−0.233537	+	0.972348i	$0.575030\pi$
$678$	0	0
$679$	0.598161	0.0229553
$680$	0	0
$681$	−0.189331	−0.00725519
$682$	0	0
$683$	−34.4616	−1.31863	−0.659317	−	0.751865i	$-0.729155\pi$
$683$	−34.4616	−1.31863	−0.659317	+	0.751865i	$0.729155\pi$
$684$	0	0
$685$	−48.1200	−1.83857
$686$	0	0
$687$	14.4062	0.549630
$688$	0	0
$689$	−26.7247	−1.01813
$690$	0	0
$691$	3.60365	0.137089	0.0685446	−	0.997648i	$-0.478164\pi$
$691$	3.60365	0.137089	0.0685446	+	0.997648i	$0.478164\pi$
$692$	0	0
$693$	−7.77214	−0.295239
$694$	0	0
$695$	−65.7446	−2.49383
$696$	0	0
$697$	0.238888	0.00904854
$698$	0	0
$699$	−29.7817	−1.12645
$700$	0	0
$701$	40.8273	1.54203	0.771013	−	0.636819i	$-0.219750\pi$
$701$	40.8273	1.54203	0.771013	+	0.636819i	$0.219750\pi$
$702$	0	0
$703$	−43.2909	−1.63275
$704$	0	0
$705$	−7.71029	−0.290386
$706$	0	0
$707$	8.42816	0.316974
$708$	0	0
$709$	29.4473	1.10592	0.552959	−	0.833209i	$-0.313499\pi$
$709$	29.4473	1.10592	0.552959	+	0.833209i	$0.313499\pi$
$710$	0	0
$711$	−0.0583476	−0.00218821
$712$	0	0
$713$	−9.20275	−0.344646
$714$	0	0
$715$	46.2592	1.73000
$716$	0	0
$717$	7.50122	0.280138
$718$	0	0
$719$	15.9152	0.593535	0.296768	−	0.954950i	$-0.404091\pi$
$719$	15.9152	0.593535	0.296768	+	0.954950i	$0.404091\pi$
$720$	0	0
$721$	36.1208	1.34521
$722$	0	0
$723$	−15.6609	−0.582435
$724$	0	0
$725$	67.4243	2.50408
$726$	0	0
$727$	52.3345	1.94098	0.970490	−	0.241142i	$-0.0775219\pi$
$727$	52.3345	1.94098	0.970490	+	0.241142i	$0.0775219\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.249037	−0.00921096
$732$	0	0
$733$	−25.8494	−0.954771	−0.477385	−	0.878694i	$-0.658415\pi$
$733$	−25.8494	−0.954771	−0.477385	+	0.878694i	$0.658415\pi$
$734$	0	0
$735$	−15.8814	−0.585793
$736$	0	0
$737$	13.3469	0.491639
$738$	0	0
$739$	15.9750	0.587650	0.293825	−	0.955859i	$-0.405072\pi$
$739$	15.9750	0.587650	0.293825	+	0.955859i	$0.405072\pi$
$740$	0	0
$741$	−18.2748	−0.671342
$742$	0	0
$743$	23.6409	0.867302	0.433651	−	0.901081i	$-0.357225\pi$
$743$	23.6409	0.867302	0.433651	+	0.901081i	$0.357225\pi$
$744$	0	0
$745$	−3.68357	−0.134956
$746$	0	0
$747$	−0.584902	−0.0214004
$748$	0	0
$749$	−7.18360	−0.262483
$750$	0	0
$751$	20.1018	0.733526	0.366763	−	0.930314i	$-0.380466\pi$
$751$	20.1018	0.733526	0.366763	+	0.930314i	$0.380466\pi$
$752$	0	0
$753$	−7.02486	−0.256000
$754$	0	0
$755$	−16.9724	−0.617689
$756$	0	0
$757$	4.43011	0.161015	0.0805075	−	0.996754i	$-0.474346\pi$
$757$	4.43011	0.161015	0.0805075	+	0.996754i	$0.474346\pi$
$758$	0	0
$759$	11.9680	0.434412
$760$	0	0
$761$	12.1245	0.439512	0.219756	−	0.975555i	$-0.429474\pi$
$761$	12.1245	0.439512	0.219756	+	0.975555i	$0.429474\pi$
$762$	0	0
$763$	13.0869	0.473778
$764$	0	0
$765$	−0.298744	−0.0108011
$766$	0	0
$767$	−20.0083	−0.722459
$768$	0	0
$769$	−27.8010	−1.00253	−0.501265	−	0.865294i	$-0.667132\pi$
$769$	−27.8010	−1.00253	−0.501265	+	0.865294i	$0.667132\pi$
$770$	0	0
$771$	−8.17803	−0.294525
$772$	0	0
$773$	23.2674	0.836869	0.418435	−	0.908247i	$-0.362579\pi$
$773$	23.2674	0.836869	0.418435	+	0.908247i	$0.362579\pi$
$774$	0	0
$775$	−41.4503	−1.48894
$776$	0	0
$777$	10.7911	0.387129
$778$	0	0
$779$	23.9002	0.856313
$780$	0	0
$781$	−40.9390	−1.46491
$782$	0	0
$783$	−5.44140	−0.194460
$784$	0	0
$785$	52.4938	1.87358
$786$	0	0
$787$	51.5377	1.83712	0.918560	−	0.395282i	$-0.129353\pi$
$787$	51.5377	1.83712	0.918560	+	0.395282i	$0.129353\pi$
$788$	0	0
$789$	22.1033	0.786897
$790$	0	0
$791$	−26.6101	−0.946148
$792$	0	0
$793$	−16.8285	−0.597596
$794$	0	0
$795$	−43.7084	−1.55018
$796$	0	0
$797$	−5.11408	−0.181150	−0.0905750	−	0.995890i	$-0.528870\pi$
$797$	−5.11408	−0.181150	−0.0905750	+	0.995890i	$0.528870\pi$
$798$	0	0
$799$	−0.132448	−0.00468568
$800$	0	0
$801$	−3.61449	−0.127712
$802$	0	0
$803$	38.5084	1.35893
$804$	0	0
$805$	−20.4961	−0.722393
$806$	0	0
$807$	14.4088	0.507213
$808$	0	0
$809$	36.7775	1.29303	0.646514	−	0.762902i	$-0.276226\pi$
$809$	36.7775	1.29303	0.646514	+	0.762902i	$0.276226\pi$
$810$	0	0
$811$	−38.7923	−1.36218	−0.681090	−	0.732200i	$-0.738494\pi$
$811$	−38.7923	−1.36218	−0.681090	+	0.732200i	$0.738494\pi$
$812$	0	0
$813$	−4.72029	−0.165548
$814$	0	0
$815$	−42.5721	−1.49124
$816$	0	0
$817$	−24.9155	−0.871684
$818$	0	0
$819$	4.55536	0.159177
$820$	0	0
$821$	−26.0951	−0.910725	−0.455363	−	0.890306i	$-0.650490\pi$
$821$	−26.0951	−0.910725	−0.455363	+	0.890306i	$0.650490\pi$
$822$	0	0
$823$	−41.9826	−1.46342	−0.731710	−	0.681616i	$-0.761277\pi$
$823$	−41.9826	−1.46342	−0.731710	+	0.681616i	$0.761277\pi$
$824$	0	0
$825$	53.9054	1.87675
$826$	0	0
$827$	8.82167	0.306759	0.153380	−	0.988167i	$-0.450984\pi$
$827$	8.82167	0.306759	0.153380	+	0.988167i	$0.450984\pi$
$828$	0	0
$829$	−27.5764	−0.957768	−0.478884	−	0.877878i	$-0.658959\pi$
$829$	−27.5764	−0.957768	−0.478884	+	0.877878i	$0.658959\pi$
$830$	0	0
$831$	27.9962	0.971176
$832$	0	0
$833$	−0.272812	−0.00945237
$834$	0	0
$835$	−4.17025	−0.144317
$836$	0	0
$837$	3.34520	0.115627
$838$	0	0
$839$	35.2189	1.21589	0.607945	−	0.793979i	$-0.291994\pi$
$839$	35.2189	1.21589	0.607945	+	0.793979i	$0.291994\pi$
$840$	0	0
$841$	0.608865	0.0209953
$842$	0	0
$843$	21.2473	0.731797
$844$	0	0
$845$	27.1001	0.932271
$846$	0	0
$847$	14.1598	0.486535
$848$	0	0
$849$	−6.07839	−0.208610
$850$	0	0
$851$	−16.6168	−0.569617
$852$	0	0
$853$	−0.579339	−0.0198362	−0.00991810	−	0.999951i	$-0.503157\pi$
$853$	−0.579339	−0.0198362	−0.00991810	+	0.999951i	$0.503157\pi$
$854$	0	0
$855$	−29.8886	−1.02217
$856$	0	0
$857$	−11.0178	−0.376359	−0.188180	−	0.982135i	$-0.560259\pi$
$857$	−11.0178	−0.376359	−0.188180	+	0.982135i	$0.560259\pi$
$858$	0	0
$859$	18.7589	0.640045	0.320022	−	0.947410i	$-0.396310\pi$
$859$	18.7589	0.640045	0.320022	+	0.947410i	$0.396310\pi$
$860$	0	0
$861$	−5.95761	−0.203035
$862$	0	0
$863$	5.38140	0.183185	0.0915925	−	0.995797i	$-0.470804\pi$
$863$	5.38140	0.183185	0.0915925	+	0.995797i	$0.470804\pi$
$864$	0	0
$865$	102.496	3.48496
$866$	0	0
$867$	16.9949	0.577176
$868$	0	0
$869$	0.253834	0.00861073
$870$	0	0
$871$	−7.82280	−0.265065
$872$	0	0
$873$	0.334815	0.0113318
$874$	0	0
$875$	−55.0653	−1.86155
$876$	0	0
$877$	−7.93120	−0.267818	−0.133909	−	0.990994i	$-0.542753\pi$
$877$	−7.93120	−0.267818	−0.133909	+	0.990994i	$0.542753\pi$
$878$	0	0
$879$	−15.5348	−0.523976
$880$	0	0
$881$	−27.7441	−0.934721	−0.467361	−	0.884067i	$-0.654795\pi$
$881$	−27.7441	−0.934721	−0.467361	+	0.884067i	$0.654795\pi$
$882$	0	0
$883$	−10.6482	−0.358340	−0.179170	−	0.983818i	$-0.557341\pi$
$883$	−10.6482	−0.358340	−0.179170	+	0.983818i	$0.557341\pi$
$884$	0	0
$885$	−32.7238	−1.10000
$886$	0	0
$887$	41.5610	1.39548	0.697741	−	0.716350i	$-0.254189\pi$
$887$	41.5610	1.39548	0.697741	+	0.716350i	$0.254189\pi$
$888$	0	0
$889$	33.7230	1.13103
$890$	0	0
$891$	−4.35038	−0.145743
$892$	0	0
$893$	−13.2511	−0.443431
$894$	0	0
$895$	−59.4450	−1.98703
$896$	0	0
$897$	−7.01463	−0.234212
$898$	0	0
$899$	−18.2026	−0.607090
$900$	0	0
$901$	−0.750828	−0.0250137
$902$	0	0
$903$	6.21070	0.206679
$904$	0	0
$905$	−95.2604	−3.16656
$906$	0	0
$907$	−58.4184	−1.93975	−0.969876	−	0.243599i	$-0.921672\pi$
$907$	−58.4184	−1.93975	−0.969876	+	0.243599i	$0.921672\pi$
$908$	0	0
$909$	4.71758	0.156472
$910$	0	0
$911$	21.0024	0.695840	0.347920	−	0.937524i	$-0.386888\pi$
$911$	21.0024	0.695840	0.347920	+	0.937524i	$0.386888\pi$
$912$	0	0
$913$	2.54454	0.0842121
$914$	0	0
$915$	−27.5231	−0.909885
$916$	0	0
$917$	−0.612695	−0.0202330
$918$	0	0
$919$	10.2975	0.339683	0.169842	−	0.985471i	$-0.445674\pi$
$919$	10.2975	0.339683	0.169842	+	0.985471i	$0.445674\pi$
$920$	0	0
$921$	−7.05233	−0.232382
$922$	0	0
$923$	23.9950	0.789804
$924$	0	0
$925$	−74.8442	−2.46086
$926$	0	0
$927$	20.2182	0.664054
$928$	0	0
$929$	−47.3424	−1.55325	−0.776627	−	0.629960i	$-0.783071\pi$
$929$	−47.3424	−1.55325	−0.776627	+	0.629960i	$0.783071\pi$
$930$	0	0
$931$	−27.2942	−0.894530
$932$	0	0
$933$	26.0059	0.851396
$934$	0	0
$935$	1.29965	0.0425031
$936$	0	0
$937$	33.2393	1.08588	0.542941	−	0.839771i	$-0.317311\pi$
$937$	33.2393	1.08588	0.542941	+	0.839771i	$0.317311\pi$
$938$	0	0
$939$	−17.5377	−0.572320
$940$	0	0
$941$	50.1401	1.63452	0.817260	−	0.576270i	$-0.195492\pi$
$941$	50.1401	1.63452	0.817260	+	0.576270i	$0.195492\pi$
$942$	0	0
$943$	9.17388	0.298743
$944$	0	0
$945$	7.45034	0.242360
$946$	0	0
$947$	−16.5748	−0.538609	−0.269304	−	0.963055i	$-0.586794\pi$
$947$	−16.5748	−0.538609	−0.269304	+	0.963055i	$0.586794\pi$
$948$	0	0
$949$	−22.5703	−0.732664
$950$	0	0
$951$	−20.8460	−0.675977
$952$	0	0
$953$	−26.0587	−0.844123	−0.422062	−	0.906567i	$-0.638693\pi$
$953$	−26.0587	−0.844123	−0.422062	+	0.906567i	$0.638693\pi$
$954$	0	0
$955$	5.93865	0.192170
$956$	0	0
$957$	23.6721	0.765212
$958$	0	0
$959$	20.6147	0.665685
$960$	0	0
$961$	−19.8096	−0.639020
$962$	0	0
$963$	−4.02095	−0.129573
$964$	0	0
$965$	58.3018	1.87680
$966$	0	0
$967$	12.6689	0.407403	0.203702	−	0.979033i	$-0.434703\pi$
$967$	12.6689	0.407403	0.203702	+	0.979033i	$0.434703\pi$
$968$	0	0
$969$	−0.513430	−0.0164937
$970$	0	0
$971$	49.9738	1.60373	0.801867	−	0.597502i	$-0.203840\pi$
$971$	49.9738	1.60373	0.801867	+	0.597502i	$0.203840\pi$
$972$	0	0
$973$	28.1651	0.902933
$974$	0	0
$975$	−31.5947	−1.01184
$976$	0	0
$977$	33.0974	1.05888	0.529440	−	0.848347i	$-0.322402\pi$
$977$	33.0974	1.05888	0.529440	+	0.848347i	$0.322402\pi$
$978$	0	0
$979$	15.7244	0.502553
$980$	0	0
$981$	7.32526	0.233878
$982$	0	0
$983$	39.4423	1.25801	0.629007	−	0.777400i	$-0.283462\pi$
$983$	39.4423	1.25801	0.629007	+	0.777400i	$0.283462\pi$
$984$	0	0
$985$	−59.4018	−1.89270
$986$	0	0
$987$	3.30311	0.105139
$988$	0	0
$989$	−9.56361	−0.304105
$990$	0	0
$991$	−53.1881	−1.68957	−0.844787	−	0.535102i	$-0.820273\pi$
$991$	−53.1881	−1.68957	−0.844787	+	0.535102i	$0.820273\pi$
$992$	0	0
$993$	27.8470	0.883698
$994$	0	0
$995$	−69.5874	−2.20607
$996$	0	0
$997$	−30.5032	−0.966046	−0.483023	−	0.875608i	$-0.660461\pi$
$997$	−30.5032	−0.966046	−0.483023	+	0.875608i	$0.660461\pi$
$998$	0	0
$999$	6.04022	0.191104

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.ba.1.1		9
4.3	odd	2		4008.2.a.i.1.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.i.1.1	✓	9	4.3	odd	2
8016.2.a.ba.1.1		9	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.17025 q^{5} +1.78655 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.17025 q^{5} +1.78655 q^{7} +1.00000 q^{9} -4.35038 q^{11} +2.54982 q^{13} +4.17025 q^{15} +0.0716370 q^{17} +7.16710 q^{19} -1.78655 q^{21} +2.75103 q^{23} +12.3910 q^{25} -1.00000 q^{27} +5.44140 q^{29} -3.34520 q^{31} +4.35038 q^{33} -7.45034 q^{35} -6.04022 q^{37} -2.54982 q^{39} +3.33471 q^{41} -3.47637 q^{43} -4.17025 q^{45} -1.84888 q^{47} -3.80826 q^{49} -0.0716370 q^{51} -10.4810 q^{53} +18.1422 q^{55} -7.16710 q^{57} -7.84697 q^{59} -6.59987 q^{61} +1.78655 q^{63} -10.6334 q^{65} -3.06798 q^{67} -2.75103 q^{69} +9.41046 q^{71} -8.85175 q^{73} -12.3910 q^{75} -7.77214 q^{77} -0.0583476 q^{79} +1.00000 q^{81} -0.584902 q^{83} -0.298744 q^{85} -5.44140 q^{87} -3.61449 q^{89} +4.55536 q^{91} +3.34520 q^{93} -29.8886 q^{95} +0.334815 q^{97} -4.35038 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} - 6 q^{5} + 11 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q - 9 * q^3 - 6 * q^5 + 11 * q^7 + 9 * q^9	\( 9 q - 9 q^{3} - 6 q^{5} + 11 q^{7} + 9 q^{9} - q^{11} - 4 q^{13} + 6 q^{15} - 9 q^{17} + 8 q^{19} - 11 q^{21} + 7 q^{23} - q^{25} - 9 q^{27} - 9 q^{29} + 25 q^{31} + q^{33} - 5 q^{35} - 6 q^{37} + 4 q^{39} - 4 q^{41} + 24 q^{43} - 6 q^{45} + 16 q^{47} + 4 q^{49} + 9 q^{51} - 26 q^{53} + 29 q^{55} - 8 q^{57} + 4 q^{59} - 20 q^{61} + 11 q^{63} - 8 q^{65} + 25 q^{67} - 7 q^{69} + 15 q^{71} - 10 q^{73} + q^{75} - 20 q^{77} + 34 q^{79} + 9 q^{81} + 4 q^{83} - 13 q^{85} + 9 q^{87} + 13 q^{89} + 21 q^{91} - 25 q^{93} + 7 q^{95} - 4 q^{97} - q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 - 6 * q^5 + 11 * q^7 + 9 * q^9 - q^11 - 4 * q^13 + 6 * q^15 - 9 * q^17 + 8 * q^19 - 11 * q^21 + 7 * q^23 - q^25 - 9 * q^27 - 9 * q^29 + 25 * q^31 + q^33 - 5 * q^35 - 6 * q^37 + 4 * q^39 - 4 * q^41 + 24 * q^43 - 6 * q^45 + 16 * q^47 + 4 * q^49 + 9 * q^51 - 26 * q^53 + 29 * q^55 - 8 * q^57 + 4 * q^59 - 20 * q^61 + 11 * q^63 - 8 * q^65 + 25 * q^67 - 7 * q^69 + 15 * q^71 - 10 * q^73 + q^75 - 20 * q^77 + 34 * q^79 + 9 * q^81 + 4 * q^83 - 13 * q^85 + 9 * q^87 + 13 * q^89 + 21 * q^91 - 25 * q^93 + 7 * q^95 - 4 * q^97 - q^99

Introduction

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