LMFDB - Embedded newform 8004.2.a.k.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 55 x^{16} + 288 x^{15} + 1222 x^{14} - 6888 x^{13} - 13745 x^{12} + 88434 x^{11} + \cdots - 115488 $ x^18 - 5x^17 - 55x^16 + 288x^15 + 1222x^14 - 6888x^13 - 13745x^12 + 88434x^11 + 76571x^10 - 655793x^9 - 114554x^8 + 2789438x^7 - 855636x^6 - 6184176x^5 + 4260960x^4 + 5001480x^3 - 5659296x^2 + 1509552*x - 115488
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.25234$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.25234 q^{5} +3.02254 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.25234 q^{5} +3.02254 q^{7} +1.00000 q^{9} -4.27402 q^{11} +1.71624 q^{13} -2.25234 q^{15} +2.42113 q^{17} +1.07152 q^{19} +3.02254 q^{21} -1.00000 q^{23} +0.0730502 q^{25} +1.00000 q^{27} +1.00000 q^{29} +9.45536 q^{31} -4.27402 q^{33} -6.80781 q^{35} +7.34241 q^{37} +1.71624 q^{39} -3.99046 q^{41} -6.28811 q^{43} -2.25234 q^{45} +0.0164377 q^{47} +2.13577 q^{49} +2.42113 q^{51} -0.890152 q^{53} +9.62656 q^{55} +1.07152 q^{57} -7.48405 q^{59} +13.8255 q^{61} +3.02254 q^{63} -3.86555 q^{65} -1.64242 q^{67} -1.00000 q^{69} +1.92525 q^{71} -11.8987 q^{73} +0.0730502 q^{75} -12.9184 q^{77} -6.09596 q^{79} +1.00000 q^{81} +10.8286 q^{83} -5.45322 q^{85} +1.00000 q^{87} +0.796084 q^{89} +5.18740 q^{91} +9.45536 q^{93} -2.41344 q^{95} +17.5044 q^{97} -4.27402 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * 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
$5$	−2.25234	−1.00728	−0.503639	−	0.863914i	$-0.668006\pi$
$5$	−2.25234	−1.00728	−0.503639	+	0.863914i	$0.668006\pi$
$6$	0	0
$7$	3.02254	1.14241	0.571207	−	0.820806i	$-0.306475\pi$
$7$	3.02254	1.14241	0.571207	+	0.820806i	$0.306475\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.27402	−1.28867	−0.644333	−	0.764745i	$-0.722865\pi$
$11$	−4.27402	−1.28867	−0.644333	+	0.764745i	$0.722865\pi$
$12$	0	0
$13$	1.71624	0.475998	0.237999	−	0.971265i	$-0.423508\pi$
$13$	1.71624	0.475998	0.237999	+	0.971265i	$0.423508\pi$
$14$	0	0
$15$	−2.25234	−0.581553
$16$	0	0
$17$	2.42113	0.587211	0.293605	−	0.955927i	$-0.405145\pi$
$17$	2.42113	0.587211	0.293605	+	0.955927i	$0.405145\pi$
$18$	0	0
$19$	1.07152	0.245824	0.122912	−	0.992418i	$-0.460777\pi$
$19$	1.07152	0.245824	0.122912	+	0.992418i	$0.460777\pi$
$20$	0	0
$21$	3.02254	0.659573
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0.0730502	0.0146100
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	9.45536	1.69823	0.849117	−	0.528205i	$-0.177135\pi$
$31$	9.45536	1.69823	0.849117	+	0.528205i	$0.177135\pi$
$32$	0	0
$33$	−4.27402	−0.744012
$34$	0	0
$35$	−6.80781	−1.15073
$36$	0	0
$37$	7.34241	1.20708	0.603542	−	0.797331i	$-0.293755\pi$
$37$	7.34241	1.20708	0.603542	+	0.797331i	$0.293755\pi$
$38$	0	0
$39$	1.71624	0.274818
$40$	0	0
$41$	−3.99046	−0.623205	−0.311603	−	0.950213i	$-0.600866\pi$
$41$	−3.99046	−0.623205	−0.311603	+	0.950213i	$0.600866\pi$
$42$	0	0
$43$	−6.28811	−0.958928	−0.479464	−	0.877562i	$-0.659169\pi$
$43$	−6.28811	−0.958928	−0.479464	+	0.877562i	$0.659169\pi$
$44$	0	0
$45$	−2.25234	−0.335760
$46$	0	0
$47$	0.0164377	0.00239769	0.00119885	−	0.999999i	$-0.499618\pi$
$47$	0.0164377	0.00239769	0.00119885	+	0.999999i	$0.499618\pi$
$48$	0	0
$49$	2.13577	0.305110
$50$	0	0
$51$	2.42113	0.339026
$52$	0	0
$53$	−0.890152	−0.122272	−0.0611359	−	0.998129i	$-0.519472\pi$
$53$	−0.890152	−0.122272	−0.0611359	+	0.998129i	$0.519472\pi$
$54$	0	0
$55$	9.62656	1.29805
$56$	0	0
$57$	1.07152	0.141927
$58$	0	0
$59$	−7.48405	−0.974341	−0.487170	−	0.873307i	$-0.661971\pi$
$59$	−7.48405	−0.974341	−0.487170	+	0.873307i	$0.661971\pi$
$60$	0	0
$61$	13.8255	1.77017	0.885085	−	0.465430i	$-0.154100\pi$
$61$	13.8255	1.77017	0.885085	+	0.465430i	$0.154100\pi$
$62$	0	0
$63$	3.02254	0.380805
$64$	0	0
$65$	−3.86555	−0.479463
$66$	0	0
$67$	−1.64242	−0.200653	−0.100326	−	0.994955i	$-0.531989\pi$
$67$	−1.64242	−0.200653	−0.100326	+	0.994955i	$0.531989\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	1.92525	0.228485	0.114243	−	0.993453i	$-0.463556\pi$
$71$	1.92525	0.228485	0.114243	+	0.993453i	$0.463556\pi$
$72$	0	0
$73$	−11.8987	−1.39264	−0.696318	−	0.717733i	$-0.745180\pi$
$73$	−11.8987	−1.39264	−0.696318	+	0.717733i	$0.745180\pi$
$74$	0	0
$75$	0.0730502	0.00843511
$76$	0	0
$77$	−12.9184	−1.47219
$78$	0	0
$79$	−6.09596	−0.685849	−0.342924	−	0.939363i	$-0.611417\pi$
$79$	−6.09596	−0.685849	−0.342924	+	0.939363i	$0.611417\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.8286	1.18859	0.594295	−	0.804247i	$-0.297431\pi$
$83$	10.8286	1.18859	0.594295	+	0.804247i	$0.297431\pi$
$84$	0	0
$85$	−5.45322	−0.591485
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	0.796084	0.0843847	0.0421923	−	0.999110i	$-0.486566\pi$
$89$	0.796084	0.0843847	0.0421923	+	0.999110i	$0.486566\pi$
$90$	0	0
$91$	5.18740	0.543787
$92$	0	0
$93$	9.45536	0.980476
$94$	0	0
$95$	−2.41344	−0.247614
$96$	0	0
$97$	17.5044	1.77730	0.888650	−	0.458586i	$-0.151644\pi$
$97$	17.5044	1.77730	0.888650	+	0.458586i	$0.151644\pi$
$98$	0	0
$99$	−4.27402	−0.429555
$100$	0	0
$101$	19.8394	1.97410	0.987048	−	0.160423i	$-0.0512858\pi$
$101$	19.8394	1.97410	0.987048	+	0.160423i	$0.0512858\pi$
$102$	0	0
$103$	−3.01200	−0.296782	−0.148391	−	0.988929i	$-0.547409\pi$
$103$	−3.01200	−0.296782	−0.148391	+	0.988929i	$0.547409\pi$
$104$	0	0
$105$	−6.80781	−0.664374
$106$	0	0
$107$	−8.17067	−0.789888	−0.394944	−	0.918705i	$-0.629236\pi$
$107$	−8.17067	−0.789888	−0.394944	+	0.918705i	$0.629236\pi$
$108$	0	0
$109$	9.09121	0.870780	0.435390	−	0.900242i	$-0.356610\pi$
$109$	9.09121	0.870780	0.435390	+	0.900242i	$0.356610\pi$
$110$	0	0
$111$	7.34241	0.696911
$112$	0	0
$113$	−11.3428	−1.06704	−0.533520	−	0.845788i	$-0.679131\pi$
$113$	−11.3428	−1.06704	−0.533520	+	0.845788i	$0.679131\pi$
$114$	0	0
$115$	2.25234	0.210032
$116$	0	0
$117$	1.71624	0.158666
$118$	0	0
$119$	7.31798	0.670838
$120$	0	0
$121$	7.26725	0.660659
$122$	0	0
$123$	−3.99046	−0.359808
$124$	0	0
$125$	11.0972	0.992562
$126$	0	0
$127$	−13.3613	−1.18563	−0.592813	−	0.805340i	$-0.701983\pi$
$127$	−13.3613	−1.18563	−0.592813	+	0.805340i	$0.701983\pi$
$128$	0	0
$129$	−6.28811	−0.553637
$130$	0	0
$131$	6.61045	0.577558	0.288779	−	0.957396i	$-0.406751\pi$
$131$	6.61045	0.577558	0.288779	+	0.957396i	$0.406751\pi$
$132$	0	0
$133$	3.23873	0.280833
$134$	0	0
$135$	−2.25234	−0.193851
$136$	0	0
$137$	9.98315	0.852918	0.426459	−	0.904507i	$-0.359761\pi$
$137$	9.98315	0.852918	0.426459	+	0.904507i	$0.359761\pi$
$138$	0	0
$139$	4.89624	0.415294	0.207647	−	0.978204i	$-0.433420\pi$
$139$	4.89624	0.415294	0.207647	+	0.978204i	$0.433420\pi$
$140$	0	0
$141$	0.0164377	0.00138431
$142$	0	0
$143$	−7.33523	−0.613403
$144$	0	0
$145$	−2.25234	−0.187047
$146$	0	0
$147$	2.13577	0.176156
$148$	0	0
$149$	−5.92098	−0.485066	−0.242533	−	0.970143i	$-0.577978\pi$
$149$	−5.92098	−0.485066	−0.242533	+	0.970143i	$0.577978\pi$
$150$	0	0
$151$	11.7122	0.953129	0.476564	−	0.879140i	$-0.341882\pi$
$151$	11.7122	0.953129	0.476564	+	0.879140i	$0.341882\pi$
$152$	0	0
$153$	2.42113	0.195737
$154$	0	0
$155$	−21.2967	−1.71059
$156$	0	0
$157$	11.7712	0.939440	0.469720	−	0.882815i	$-0.344355\pi$
$157$	11.7712	0.939440	0.469720	+	0.882815i	$0.344355\pi$
$158$	0	0
$159$	−0.890152	−0.0705936
$160$	0	0
$161$	−3.02254	−0.238210
$162$	0	0
$163$	16.5214	1.29405	0.647027	−	0.762467i	$-0.276012\pi$
$163$	16.5214	1.29405	0.647027	+	0.762467i	$0.276012\pi$
$164$	0	0
$165$	9.62656	0.749427
$166$	0	0
$167$	14.9209	1.15461	0.577307	−	0.816527i	$-0.304104\pi$
$167$	14.9209	1.15461	0.577307	+	0.816527i	$0.304104\pi$
$168$	0	0
$169$	−10.0545	−0.773425
$170$	0	0
$171$	1.07152	0.0819415
$172$	0	0
$173$	−13.2895	−1.01038	−0.505192	−	0.863007i	$-0.668578\pi$
$173$	−13.2895	−1.01038	−0.505192	+	0.863007i	$0.668578\pi$
$174$	0	0
$175$	0.220797	0.0166907
$176$	0	0
$177$	−7.48405	−0.562536
$178$	0	0
$179$	23.8836	1.78514	0.892571	−	0.450907i	$-0.148899\pi$
$179$	23.8836	1.78514	0.892571	+	0.450907i	$0.148899\pi$
$180$	0	0
$181$	7.98704	0.593672	0.296836	−	0.954929i	$-0.404069\pi$
$181$	7.98704	0.593672	0.296836	+	0.954929i	$0.404069\pi$
$182$	0	0
$183$	13.8255	1.02201
$184$	0	0
$185$	−16.5376	−1.21587
$186$	0	0
$187$	−10.3480	−0.756718
$188$	0	0
$189$	3.02254	0.219858
$190$	0	0
$191$	−4.83917	−0.350150	−0.175075	−	0.984555i	$-0.556017\pi$
$191$	−4.83917	−0.350150	−0.175075	+	0.984555i	$0.556017\pi$
$192$	0	0
$193$	7.29224	0.524907	0.262453	−	0.964945i	$-0.415468\pi$
$193$	7.29224	0.524907	0.262453	+	0.964945i	$0.415468\pi$
$194$	0	0
$195$	−3.86555	−0.276818
$196$	0	0
$197$	11.5356	0.821879	0.410940	−	0.911663i	$-0.365201\pi$
$197$	11.5356	0.821879	0.410940	+	0.911663i	$0.365201\pi$
$198$	0	0
$199$	−14.1166	−1.00070	−0.500350	−	0.865823i	$-0.666795\pi$
$199$	−14.1166	−1.00070	−0.500350	+	0.865823i	$0.666795\pi$
$200$	0	0
$201$	−1.64242	−0.115847
$202$	0	0
$203$	3.02254	0.212141
$204$	0	0
$205$	8.98789	0.627741
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−4.57972	−0.316786
$210$	0	0
$211$	14.9509	1.02926	0.514632	−	0.857411i	$-0.327928\pi$
$211$	14.9509	1.02926	0.514632	+	0.857411i	$0.327928\pi$
$212$	0	0
$213$	1.92525	0.131916
$214$	0	0
$215$	14.1630	0.965907
$216$	0	0
$217$	28.5793	1.94009
$218$	0	0
$219$	−11.8987	−0.804039
$220$	0	0
$221$	4.15524	0.279511
$222$	0	0
$223$	17.9712	1.20344	0.601719	−	0.798708i	$-0.294483\pi$
$223$	17.9712	1.20344	0.601719	+	0.798708i	$0.294483\pi$
$224$	0	0
$225$	0.0730502	0.00487001
$226$	0	0
$227$	26.9039	1.78568	0.892838	−	0.450379i	$-0.148711\pi$
$227$	26.9039	1.78568	0.892838	+	0.450379i	$0.148711\pi$
$228$	0	0
$229$	20.7914	1.37393	0.686967	−	0.726688i	$-0.258942\pi$
$229$	20.7914	1.37393	0.686967	+	0.726688i	$0.258942\pi$
$230$	0	0
$231$	−12.9184	−0.849969
$232$	0	0
$233$	−14.5056	−0.950295	−0.475147	−	0.879906i	$-0.657605\pi$
$233$	−14.5056	−0.950295	−0.475147	+	0.879906i	$0.657605\pi$
$234$	0	0
$235$	−0.0370234	−0.00241514
$236$	0	0
$237$	−6.09596	−0.395975
$238$	0	0
$239$	−17.3103	−1.11971	−0.559855	−	0.828591i	$-0.689143\pi$
$239$	−17.3103	−1.11971	−0.559855	+	0.828591i	$0.689143\pi$
$240$	0	0
$241$	−11.2793	−0.726564	−0.363282	−	0.931679i	$-0.618344\pi$
$241$	−11.2793	−0.726564	−0.363282	+	0.931679i	$0.618344\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.81049	−0.307331
$246$	0	0
$247$	1.83899	0.117012
$248$	0	0
$249$	10.8286	0.686233
$250$	0	0
$251$	8.19977	0.517565	0.258782	−	0.965936i	$-0.416679\pi$
$251$	8.19977	0.517565	0.258782	+	0.965936i	$0.416679\pi$
$252$	0	0
$253$	4.27402	0.268705
$254$	0	0
$255$	−5.45322	−0.341494
$256$	0	0
$257$	−4.07372	−0.254112	−0.127056	−	0.991896i	$-0.540553\pi$
$257$	−4.07372	−0.254112	−0.127056	+	0.991896i	$0.540553\pi$
$258$	0	0
$259$	22.1928	1.37899
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−32.3840	−1.99688	−0.998440	−	0.0558352i	$-0.982218\pi$
$263$	−32.3840	−1.99688	−0.998440	+	0.0558352i	$0.982218\pi$
$264$	0	0
$265$	2.00493	0.123162
$266$	0	0
$267$	0.796084	0.0487195
$268$	0	0
$269$	3.36209	0.204990	0.102495	−	0.994734i	$-0.467317\pi$
$269$	3.36209	0.204990	0.102495	+	0.994734i	$0.467317\pi$
$270$	0	0
$271$	25.9818	1.57828	0.789140	−	0.614213i	$-0.210527\pi$
$271$	25.9818	1.57828	0.789140	+	0.614213i	$0.210527\pi$
$272$	0	0
$273$	5.18740	0.313956
$274$	0	0
$275$	−0.312218	−0.0188274
$276$	0	0
$277$	−22.8258	−1.37147	−0.685735	−	0.727851i	$-0.740519\pi$
$277$	−22.8258	−1.37147	−0.685735	+	0.727851i	$0.740519\pi$
$278$	0	0
$279$	9.45536	0.566078
$280$	0	0
$281$	2.25163	0.134321	0.0671605	−	0.997742i	$-0.478606\pi$
$281$	2.25163	0.134321	0.0671605	+	0.997742i	$0.478606\pi$
$282$	0	0
$283$	1.21651	0.0723140	0.0361570	−	0.999346i	$-0.488488\pi$
$283$	1.21651	0.0723140	0.0361570	+	0.999346i	$0.488488\pi$
$284$	0	0
$285$	−2.41344	−0.142960
$286$	0	0
$287$	−12.0613	−0.711959
$288$	0	0
$289$	−11.1381	−0.655184
$290$	0	0
$291$	17.5044	1.02612
$292$	0	0
$293$	−15.8302	−0.924812	−0.462406	−	0.886668i	$-0.653014\pi$
$293$	−15.8302	−0.924812	−0.462406	+	0.886668i	$0.653014\pi$
$294$	0	0
$295$	16.8567	0.981432
$296$	0	0
$297$	−4.27402	−0.248004
$298$	0	0
$299$	−1.71624	−0.0992525
$300$	0	0
$301$	−19.0061	−1.09549
$302$	0	0
$303$	19.8394	1.13975
$304$	0	0
$305$	−31.1397	−1.78305
$306$	0	0
$307$	−5.45684	−0.311439	−0.155719	−	0.987801i	$-0.549770\pi$
$307$	−5.45684	−0.311439	−0.155719	+	0.987801i	$0.549770\pi$
$308$	0	0
$309$	−3.01200	−0.171347
$310$	0	0
$311$	3.14591	0.178388	0.0891941	−	0.996014i	$-0.471571\pi$
$311$	3.14591	0.178388	0.0891941	+	0.996014i	$0.471571\pi$
$312$	0	0
$313$	15.0339	0.849766	0.424883	−	0.905248i	$-0.360315\pi$
$313$	15.0339	0.849766	0.424883	+	0.905248i	$0.360315\pi$
$314$	0	0
$315$	−6.80781	−0.383576
$316$	0	0
$317$	−10.7472	−0.603622	−0.301811	−	0.953368i	$-0.597591\pi$
$317$	−10.7472	−0.603622	−0.301811	+	0.953368i	$0.597591\pi$
$318$	0	0
$319$	−4.27402	−0.239299
$320$	0	0
$321$	−8.17067	−0.456042
$322$	0	0
$323$	2.59430	0.144351
$324$	0	0
$325$	0.125371	0.00695435
$326$	0	0
$327$	9.09121	0.502745
$328$	0	0
$329$	0.0496838	0.00273916
$330$	0	0
$331$	−4.21791	−0.231838	−0.115919	−	0.993259i	$-0.536981\pi$
$331$	−4.21791	−0.231838	−0.115919	+	0.993259i	$0.536981\pi$
$332$	0	0
$333$	7.34241	0.402361
$334$	0	0
$335$	3.69928	0.202113
$336$	0	0
$337$	−26.1316	−1.42348	−0.711741	−	0.702442i	$-0.752093\pi$
$337$	−26.1316	−1.42348	−0.711741	+	0.702442i	$0.752093\pi$
$338$	0	0
$339$	−11.3428	−0.616056
$340$	0	0
$341$	−40.4124	−2.18846
$342$	0	0
$343$	−14.7023	−0.793852
$344$	0	0
$345$	2.25234	0.121262
$346$	0	0
$347$	−1.35427	−0.0727010	−0.0363505	−	0.999339i	$-0.511573\pi$
$347$	−1.35427	−0.0727010	−0.0363505	+	0.999339i	$0.511573\pi$
$348$	0	0
$349$	8.47566	0.453692	0.226846	−	0.973931i	$-0.427159\pi$
$349$	8.47566	0.453692	0.226846	+	0.973931i	$0.427159\pi$
$350$	0	0
$351$	1.71624	0.0916059
$352$	0	0
$353$	26.3181	1.40077	0.700387	−	0.713764i	$-0.253011\pi$
$353$	26.3181	1.40077	0.700387	+	0.713764i	$0.253011\pi$
$354$	0	0
$355$	−4.33632	−0.230148
$356$	0	0
$357$	7.31798	0.387308
$358$	0	0
$359$	21.6943	1.14498	0.572489	−	0.819912i	$-0.305978\pi$
$359$	21.6943	1.14498	0.572489	+	0.819912i	$0.305978\pi$
$360$	0	0
$361$	−17.8518	−0.939570
$362$	0	0
$363$	7.26725	0.381432
$364$	0	0
$365$	26.7999	1.40277
$366$	0	0
$367$	−7.29335	−0.380710	−0.190355	−	0.981715i	$-0.560964\pi$
$367$	−7.29335	−0.380710	−0.190355	+	0.981715i	$0.560964\pi$
$368$	0	0
$369$	−3.99046	−0.207735
$370$	0	0
$371$	−2.69052	−0.139685
$372$	0	0
$373$	36.7193	1.90125	0.950627	−	0.310335i	$-0.100441\pi$
$373$	36.7193	1.90125	0.950627	+	0.310335i	$0.100441\pi$
$374$	0	0
$375$	11.0972	0.573056
$376$	0	0
$377$	1.71624	0.0883907
$378$	0	0
$379$	−4.92538	−0.253000	−0.126500	−	0.991967i	$-0.540374\pi$
$379$	−4.92538	−0.253000	−0.126500	+	0.991967i	$0.540374\pi$
$380$	0	0
$381$	−13.3613	−0.684522
$382$	0	0
$383$	25.7876	1.31769	0.658843	−	0.752281i	$-0.271046\pi$
$383$	25.7876	1.31769	0.658843	+	0.752281i	$0.271046\pi$
$384$	0	0
$385$	29.0967	1.48291
$386$	0	0
$387$	−6.28811	−0.319643
$388$	0	0
$389$	24.3495	1.23457	0.617285	−	0.786740i	$-0.288233\pi$
$389$	24.3495	1.23457	0.617285	+	0.786740i	$0.288233\pi$
$390$	0	0
$391$	−2.42113	−0.122442
$392$	0	0
$393$	6.61045	0.333453
$394$	0	0
$395$	13.7302	0.690841
$396$	0	0
$397$	28.5025	1.43050	0.715251	−	0.698868i	$-0.246313\pi$
$397$	28.5025	1.43050	0.715251	+	0.698868i	$0.246313\pi$
$398$	0	0
$399$	3.23873	0.162139
$400$	0	0
$401$	32.7116	1.63354	0.816769	−	0.576965i	$-0.195763\pi$
$401$	32.7116	1.63354	0.816769	+	0.576965i	$0.195763\pi$
$402$	0	0
$403$	16.2276	0.808357
$404$	0	0
$405$	−2.25234	−0.111920
$406$	0	0
$407$	−31.3816	−1.55553
$408$	0	0
$409$	−2.55660	−0.126416	−0.0632078	−	0.998000i	$-0.520133\pi$
$409$	−2.55660	−0.126416	−0.0632078	+	0.998000i	$0.520133\pi$
$410$	0	0
$411$	9.98315	0.492432
$412$	0	0
$413$	−22.6209	−1.11310
$414$	0	0
$415$	−24.3897	−1.19724
$416$	0	0
$417$	4.89624	0.239770
$418$	0	0
$419$	−30.3010	−1.48030	−0.740151	−	0.672441i	$-0.765246\pi$
$419$	−30.3010	−1.48030	−0.740151	+	0.672441i	$0.765246\pi$
$420$	0	0
$421$	18.0514	0.879774	0.439887	−	0.898053i	$-0.355019\pi$
$421$	18.0514	0.879774	0.439887	+	0.898053i	$0.355019\pi$
$422$	0	0
$423$	0.0164377	0.000799230 0
$424$	0	0
$425$	0.176864	0.00857917
$426$	0	0
$427$	41.7881	2.02227
$428$	0	0
$429$	−7.33523	−0.354148
$430$	0	0
$431$	−1.57432	−0.0758325	−0.0379162	−	0.999281i	$-0.512072\pi$
$431$	−1.57432	−0.0758325	−0.0379162	+	0.999281i	$0.512072\pi$
$432$	0	0
$433$	22.2111	1.06740	0.533698	−	0.845675i	$-0.320802\pi$
$433$	22.2111	1.06740	0.533698	+	0.845675i	$0.320802\pi$
$434$	0	0
$435$	−2.25234	−0.107992
$436$	0	0
$437$	−1.07152	−0.0512579
$438$	0	0
$439$	−18.1457	−0.866046	−0.433023	−	0.901383i	$-0.642553\pi$
$439$	−18.1457	−0.866046	−0.433023	+	0.901383i	$0.642553\pi$
$440$	0	0
$441$	2.13577	0.101703
$442$	0	0
$443$	−0.427656	−0.0203185	−0.0101593	−	0.999948i	$-0.503234\pi$
$443$	−0.427656	−0.0203185	−0.0101593	+	0.999948i	$0.503234\pi$
$444$	0	0
$445$	−1.79305	−0.0849989
$446$	0	0
$447$	−5.92098	−0.280053
$448$	0	0
$449$	−2.20151	−0.103896	−0.0519480	−	0.998650i	$-0.516543\pi$
$449$	−2.20151	−0.103896	−0.0519480	+	0.998650i	$0.516543\pi$
$450$	0	0
$451$	17.0553	0.803103
$452$	0	0
$453$	11.7122	0.550289
$454$	0	0
$455$	−11.6838	−0.547745
$456$	0	0
$457$	−27.9211	−1.30609	−0.653047	−	0.757318i	$-0.726510\pi$
$457$	−27.9211	−1.30609	−0.653047	+	0.757318i	$0.726510\pi$
$458$	0	0
$459$	2.42113	0.113009
$460$	0	0
$461$	8.19068	0.381478	0.190739	−	0.981641i	$-0.438912\pi$
$461$	8.19068	0.381478	0.190739	+	0.981641i	$0.438912\pi$
$462$	0	0
$463$	23.2385	1.07998	0.539991	−	0.841671i	$-0.318427\pi$
$463$	23.2385	1.07998	0.539991	+	0.841671i	$0.318427\pi$
$464$	0	0
$465$	−21.2967	−0.987612
$466$	0	0
$467$	33.9942	1.57306	0.786531	−	0.617551i	$-0.211875\pi$
$467$	33.9942	1.57306	0.786531	+	0.617551i	$0.211875\pi$
$468$	0	0
$469$	−4.96427	−0.229229
$470$	0	0
$471$	11.7712	0.542386
$472$	0	0
$473$	26.8755	1.23574
$474$	0	0
$475$	0.0782750	0.00359150
$476$	0	0
$477$	−0.890152	−0.0407573
$478$	0	0
$479$	39.4787	1.80383	0.901915	−	0.431914i	$-0.142162\pi$
$479$	39.4787	1.80383	0.901915	+	0.431914i	$0.142162\pi$
$480$	0	0
$481$	12.6013	0.574570
$482$	0	0
$483$	−3.02254	−0.137531
$484$	0	0
$485$	−39.4259	−1.79024
$486$	0	0
$487$	−5.67209	−0.257027	−0.128513	−	0.991708i	$-0.541021\pi$
$487$	−5.67209	−0.257027	−0.128513	+	0.991708i	$0.541021\pi$
$488$	0	0
$489$	16.5214	0.747122
$490$	0	0
$491$	20.3339	0.917656	0.458828	−	0.888525i	$-0.348269\pi$
$491$	20.3339	0.917656	0.458828	+	0.888525i	$0.348269\pi$
$492$	0	0
$493$	2.42113	0.109042
$494$	0	0
$495$	9.62656	0.432682
$496$	0	0
$497$	5.81915	0.261025
$498$	0	0
$499$	−34.7001	−1.55339	−0.776696	−	0.629876i	$-0.783106\pi$
$499$	−34.7001	−1.55339	−0.776696	+	0.629876i	$0.783106\pi$
$500$	0	0
$501$	14.9209	0.666617
$502$	0	0
$503$	−7.96438	−0.355114	−0.177557	−	0.984110i	$-0.556819\pi$
$503$	−7.96438	−0.355114	−0.177557	+	0.984110i	$0.556819\pi$
$504$	0	0
$505$	−44.6852	−1.98847
$506$	0	0
$507$	−10.0545	−0.446537
$508$	0	0
$509$	−27.2456	−1.20764	−0.603820	−	0.797121i	$-0.706355\pi$
$509$	−27.2456	−1.20764	−0.603820	+	0.797121i	$0.706355\pi$
$510$	0	0
$511$	−35.9643	−1.59097
$512$	0	0
$513$	1.07152	0.0473089
$514$	0	0
$515$	6.78407	0.298942
$516$	0	0
$517$	−0.0702553	−0.00308982
$518$	0	0
$519$	−13.2895	−0.583345
$520$	0	0
$521$	−4.38065	−0.191920	−0.0959598	−	0.995385i	$-0.530592\pi$
$521$	−4.38065	−0.191920	−0.0959598	+	0.995385i	$0.530592\pi$
$522$	0	0
$523$	34.1909	1.49507	0.747533	−	0.664225i	$-0.231238\pi$
$523$	34.1909	1.49507	0.747533	+	0.664225i	$0.231238\pi$
$524$	0	0
$525$	0.220797	0.00963638
$526$	0	0
$527$	22.8927	0.997221
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−7.48405	−0.324780
$532$	0	0
$533$	−6.84858	−0.296645
$534$	0	0
$535$	18.4031	0.795637
$536$	0	0
$537$	23.8836	1.03065
$538$	0	0
$539$	−9.12834	−0.393185
$540$	0	0
$541$	−36.7491	−1.57997	−0.789983	−	0.613128i	$-0.789911\pi$
$541$	−36.7491	−1.57997	−0.789983	+	0.613128i	$0.789911\pi$
$542$	0	0
$543$	7.98704	0.342757
$544$	0	0
$545$	−20.4765	−0.877118
$546$	0	0
$547$	5.19168	0.221980	0.110990	−	0.993822i	$-0.464598\pi$
$547$	5.19168	0.221980	0.110990	+	0.993822i	$0.464598\pi$
$548$	0	0
$549$	13.8255	0.590056
$550$	0	0
$551$	1.07152	0.0456485
$552$	0	0
$553$	−18.4253	−0.783524
$554$	0	0
$555$	−16.5376	−0.701983
$556$	0	0
$557$	−30.2879	−1.28334	−0.641669	−	0.766982i	$-0.721758\pi$
$557$	−30.2879	−1.28334	−0.641669	+	0.766982i	$0.721758\pi$
$558$	0	0
$559$	−10.7919	−0.456448
$560$	0	0
$561$	−10.3480	−0.436892
$562$	0	0
$563$	20.1410	0.848843	0.424421	−	0.905465i	$-0.360478\pi$
$563$	20.1410	0.848843	0.424421	+	0.905465i	$0.360478\pi$
$564$	0	0
$565$	25.5478	1.07481
$566$	0	0
$567$	3.02254	0.126935
$568$	0	0
$569$	13.8183	0.579295	0.289648	−	0.957133i	$-0.406462\pi$
$569$	13.8183	0.579295	0.289648	+	0.957133i	$0.406462\pi$
$570$	0	0
$571$	42.5633	1.78122	0.890610	−	0.454769i	$-0.150278\pi$
$571$	42.5633	1.78122	0.890610	+	0.454769i	$0.150278\pi$
$572$	0	0
$573$	−4.83917	−0.202159
$574$	0	0
$575$	−0.0730502	−0.00304640
$576$	0	0
$577$	−19.6148	−0.816575	−0.408287	−	0.912853i	$-0.633874\pi$
$577$	−19.6148	−0.816575	−0.408287	+	0.912853i	$0.633874\pi$
$578$	0	0
$579$	7.29224	0.303055
$580$	0	0
$581$	32.7298	1.35786
$582$	0	0
$583$	3.80453	0.157567
$584$	0	0
$585$	−3.86555	−0.159821
$586$	0	0
$587$	12.2590	0.505982	0.252991	−	0.967469i	$-0.418586\pi$
$587$	12.2590	0.505982	0.252991	+	0.967469i	$0.418586\pi$
$588$	0	0
$589$	10.1317	0.417467
$590$	0	0
$591$	11.5356	0.474512
$592$	0	0
$593$	−38.0414	−1.56217	−0.781086	−	0.624423i	$-0.785334\pi$
$593$	−38.0414	−1.56217	−0.781086	+	0.624423i	$0.785334\pi$
$594$	0	0
$595$	−16.4826	−0.675721
$596$	0	0
$597$	−14.1166	−0.577755
$598$	0	0
$599$	−39.1566	−1.59989	−0.799947	−	0.600071i	$-0.795139\pi$
$599$	−39.1566	−1.59989	−0.799947	+	0.600071i	$0.795139\pi$
$600$	0	0
$601$	−18.8521	−0.768992	−0.384496	−	0.923127i	$-0.625625\pi$
$601$	−18.8521	−0.768992	−0.384496	+	0.923127i	$0.625625\pi$
$602$	0	0
$603$	−1.64242	−0.0668843
$604$	0	0
$605$	−16.3684	−0.665468
$606$	0	0
$607$	−13.0919	−0.531385	−0.265693	−	0.964058i	$-0.585601\pi$
$607$	−13.0919	−0.531385	−0.265693	+	0.964058i	$0.585601\pi$
$608$	0	0
$609$	3.02254	0.122480
$610$	0	0
$611$	0.0282111	0.00114130
$612$	0	0
$613$	8.12461	0.328150	0.164075	−	0.986448i	$-0.447536\pi$
$613$	8.12461	0.328150	0.164075	+	0.986448i	$0.447536\pi$
$614$	0	0
$615$	8.98789	0.362427
$616$	0	0
$617$	28.3554	1.14155	0.570773	−	0.821108i	$-0.306644\pi$
$617$	28.3554	1.14155	0.570773	+	0.821108i	$0.306644\pi$
$618$	0	0
$619$	5.07978	0.204174	0.102087	−	0.994775i	$-0.467448\pi$
$619$	5.07978	0.204174	0.102087	+	0.994775i	$0.467448\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	2.40620	0.0964023
$624$	0	0
$625$	−25.3599	−1.01440
$626$	0	0
$627$	−4.57972	−0.182896
$628$	0	0
$629$	17.7769	0.708813
$630$	0	0
$631$	32.1184	1.27861	0.639307	−	0.768952i	$-0.279221\pi$
$631$	32.1184	1.27861	0.639307	+	0.768952i	$0.279221\pi$
$632$	0	0
$633$	14.9509	0.594246
$634$	0	0
$635$	30.0943	1.19426
$636$	0	0
$637$	3.66549	0.145232
$638$	0	0
$639$	1.92525	0.0761617
$640$	0	0
$641$	40.1716	1.58668	0.793342	−	0.608776i	$-0.208339\pi$
$641$	40.1716	1.58668	0.793342	+	0.608776i	$0.208339\pi$
$642$	0	0
$643$	30.3421	1.19658	0.598289	−	0.801281i	$-0.295848\pi$
$643$	30.3421	1.19658	0.598289	+	0.801281i	$0.295848\pi$
$644$	0	0
$645$	14.1630	0.557667
$646$	0	0
$647$	−8.06943	−0.317242	−0.158621	−	0.987340i	$-0.550705\pi$
$647$	−8.06943	−0.317242	−0.158621	+	0.987340i	$0.550705\pi$
$648$	0	0
$649$	31.9870	1.25560
$650$	0	0
$651$	28.5793	1.12011
$652$	0	0
$653$	30.4308	1.19085	0.595425	−	0.803411i	$-0.296984\pi$
$653$	30.4308	1.19085	0.595425	+	0.803411i	$0.296984\pi$
$654$	0	0
$655$	−14.8890	−0.581761
$656$	0	0
$657$	−11.8987	−0.464212
$658$	0	0
$659$	−5.66502	−0.220678	−0.110339	−	0.993894i	$-0.535194\pi$
$659$	−5.66502	−0.220678	−0.110339	+	0.993894i	$0.535194\pi$
$660$	0	0
$661$	−42.4591	−1.65147	−0.825734	−	0.564059i	$-0.809239\pi$
$661$	−42.4591	−1.65147	−0.825734	+	0.564059i	$0.809239\pi$
$662$	0	0
$663$	4.15524	0.161376
$664$	0	0
$665$	−7.29473	−0.282877
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	17.9712	0.694805
$670$	0	0
$671$	−59.0903	−2.28116
$672$	0	0
$673$	42.9919	1.65722	0.828609	−	0.559828i	$-0.189133\pi$
$673$	42.9919	1.65722	0.828609	+	0.559828i	$0.189133\pi$
$674$	0	0
$675$	0.0730502	0.00281170
$676$	0	0
$677$	34.6967	1.33350	0.666751	−	0.745280i	$-0.267684\pi$
$677$	34.6967	1.33350	0.666751	+	0.745280i	$0.267684\pi$
$678$	0	0
$679$	52.9078	2.03041
$680$	0	0
$681$	26.9039	1.03096
$682$	0	0
$683$	18.0806	0.691836	0.345918	−	0.938265i	$-0.387568\pi$
$683$	18.0806	0.691836	0.345918	+	0.938265i	$0.387568\pi$
$684$	0	0
$685$	−22.4855	−0.859126
$686$	0	0
$687$	20.7914	0.793241
$688$	0	0
$689$	−1.52771	−0.0582012
$690$	0	0
$691$	−35.2629	−1.34147	−0.670733	−	0.741699i	$-0.734020\pi$
$691$	−35.2629	−1.34147	−0.670733	+	0.741699i	$0.734020\pi$
$692$	0	0
$693$	−12.9184	−0.490730
$694$	0	0
$695$	−11.0280	−0.418316
$696$	0	0
$697$	−9.66143	−0.365953
$698$	0	0
$699$	−14.5056	−0.548653
$700$	0	0
$701$	14.8367	0.560373	0.280187	−	0.959946i	$-0.409604\pi$
$701$	14.8367	0.560373	0.280187	+	0.959946i	$0.409604\pi$
$702$	0	0
$703$	7.86757	0.296731
$704$	0	0
$705$	−0.0370234	−0.00139438
$706$	0	0
$707$	59.9655	2.25524
$708$	0	0
$709$	−24.5737	−0.922884	−0.461442	−	0.887170i	$-0.652668\pi$
$709$	−24.5737	−0.922884	−0.461442	+	0.887170i	$0.652668\pi$
$710$	0	0
$711$	−6.09596	−0.228616
$712$	0	0
$713$	−9.45536	−0.354106
$714$	0	0
$715$	16.5215	0.617868
$716$	0	0
$717$	−17.3103	−0.646464
$718$	0	0
$719$	−20.7626	−0.774316	−0.387158	−	0.922013i	$-0.626543\pi$
$719$	−20.7626	−0.774316	−0.387158	+	0.922013i	$0.626543\pi$
$720$	0	0
$721$	−9.10392	−0.339048
$722$	0	0
$723$	−11.2793	−0.419482
$724$	0	0
$725$	0.0730502	0.00271301
$726$	0	0
$727$	−20.8106	−0.771822	−0.385911	−	0.922536i	$-0.626113\pi$
$727$	−20.8106	−0.771822	−0.385911	+	0.922536i	$0.626113\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−15.2243	−0.563093
$732$	0	0
$733$	−0.414188	−0.0152984	−0.00764919	−	0.999971i	$-0.502435\pi$
$733$	−0.414188	−0.0152984	−0.00764919	+	0.999971i	$0.502435\pi$
$734$	0	0
$735$	−4.81049	−0.177438
$736$	0	0
$737$	7.01972	0.258575
$738$	0	0
$739$	−39.4762	−1.45216	−0.726078	−	0.687613i	$-0.758659\pi$
$739$	−39.4762	−1.45216	−0.726078	+	0.687613i	$0.758659\pi$
$740$	0	0
$741$	1.83899	0.0675570
$742$	0	0
$743$	−31.6106	−1.15968	−0.579841	−	0.814730i	$-0.696885\pi$
$743$	−31.6106	−1.15968	−0.579841	+	0.814730i	$0.696885\pi$
$744$	0	0
$745$	13.3361	0.488596
$746$	0	0
$747$	10.8286	0.396197
$748$	0	0
$749$	−24.6962	−0.902380
$750$	0	0
$751$	6.80121	0.248180	0.124090	−	0.992271i	$-0.460399\pi$
$751$	6.80121	0.248180	0.124090	+	0.992271i	$0.460399\pi$
$752$	0	0
$753$	8.19977	0.298816
$754$	0	0
$755$	−26.3800	−0.960066
$756$	0	0
$757$	3.87396	0.140802	0.0704008	−	0.997519i	$-0.477572\pi$
$757$	3.87396	0.140802	0.0704008	+	0.997519i	$0.477572\pi$
$758$	0	0
$759$	4.27402	0.155137
$760$	0	0
$761$	7.93276	0.287562	0.143781	−	0.989610i	$-0.454074\pi$
$761$	7.93276	0.287562	0.143781	+	0.989610i	$0.454074\pi$
$762$	0	0
$763$	27.4786	0.994792
$764$	0	0
$765$	−5.45322	−0.197162
$766$	0	0
$767$	−12.8444	−0.463785
$768$	0	0
$769$	−17.7170	−0.638892	−0.319446	−	0.947605i	$-0.603497\pi$
$769$	−17.7170	−0.638892	−0.319446	+	0.947605i	$0.603497\pi$
$770$	0	0
$771$	−4.07372	−0.146712
$772$	0	0
$773$	−39.3108	−1.41391	−0.706956	−	0.707257i	$-0.749932\pi$
$773$	−39.3108	−1.41391	−0.706956	+	0.707257i	$0.749932\pi$
$774$	0	0
$775$	0.690716	0.0248112
$776$	0	0
$777$	22.1928	0.796161
$778$	0	0
$779$	−4.27587	−0.153199
$780$	0	0
$781$	−8.22855	−0.294441
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−26.5127	−0.946278
$786$	0	0
$787$	27.1424	0.967522	0.483761	−	0.875200i	$-0.339270\pi$
$787$	27.1424	0.967522	0.483761	+	0.875200i	$0.339270\pi$
$788$	0	0
$789$	−32.3840	−1.15290
$790$	0	0
$791$	−34.2841	−1.21900
$792$	0	0
$793$	23.7278	0.842598
$794$	0	0
$795$	2.00493	0.0711075
$796$	0	0
$797$	12.5458	0.444395	0.222197	−	0.975002i	$-0.428677\pi$
$797$	12.5458	0.444395	0.222197	+	0.975002i	$0.428677\pi$
$798$	0	0
$799$	0.0397979	0.00140795
$800$	0	0
$801$	0.796084	0.0281282
$802$	0	0
$803$	50.8552	1.79464
$804$	0	0
$805$	6.80781	0.239944
$806$	0	0
$807$	3.36209	0.118351
$808$	0	0
$809$	51.9171	1.82531	0.912654	−	0.408734i	$-0.134030\pi$
$809$	51.9171	1.82531	0.912654	+	0.408734i	$0.134030\pi$
$810$	0	0
$811$	−52.1483	−1.83117	−0.915587	−	0.402120i	$-0.868273\pi$
$811$	−52.1483	−1.83117	−0.915587	+	0.402120i	$0.868273\pi$
$812$	0	0
$813$	25.9818	0.911220
$814$	0	0
$815$	−37.2118	−1.30347
$816$	0	0
$817$	−6.73786	−0.235728
$818$	0	0
$819$	5.18740	0.181262
$820$	0	0
$821$	−18.0772	−0.630898	−0.315449	−	0.948942i	$-0.602155\pi$
$821$	−18.0772	−0.630898	−0.315449	+	0.948942i	$0.602155\pi$
$822$	0	0
$823$	−20.3177	−0.708231	−0.354116	−	0.935202i	$-0.615218\pi$
$823$	−20.3177	−0.708231	−0.354116	+	0.935202i	$0.615218\pi$
$824$	0	0
$825$	−0.312218	−0.0108700
$826$	0	0
$827$	−14.4491	−0.502446	−0.251223	−	0.967929i	$-0.580833\pi$
$827$	−14.4491	−0.502446	−0.251223	+	0.967929i	$0.580833\pi$
$828$	0	0
$829$	10.2416	0.355706	0.177853	−	0.984057i	$-0.443085\pi$
$829$	10.2416	0.355706	0.177853	+	0.984057i	$0.443085\pi$
$830$	0	0
$831$	−22.8258	−0.791818
$832$	0	0
$833$	5.17099	0.179164
$834$	0	0
$835$	−33.6070	−1.16302
$836$	0	0
$837$	9.45536	0.326825
$838$	0	0
$839$	7.50380	0.259060	0.129530	−	0.991575i	$-0.458653\pi$
$839$	7.50380	0.259060	0.129530	+	0.991575i	$0.458653\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	2.25163	0.0775502
$844$	0	0
$845$	22.6463	0.779055
$846$	0	0
$847$	21.9656	0.754747
$848$	0	0
$849$	1.21651	0.0417505
$850$	0	0
$851$	−7.34241	−0.251695
$852$	0	0
$853$	−18.1350	−0.620931	−0.310466	−	0.950585i	$-0.600485\pi$
$853$	−18.1350	−0.620931	−0.310466	+	0.950585i	$0.600485\pi$
$854$	0	0
$855$	−2.41344	−0.0825379
$856$	0	0
$857$	36.1466	1.23475	0.617373	−	0.786671i	$-0.288197\pi$
$857$	36.1466	1.23475	0.617373	+	0.786671i	$0.288197\pi$
$858$	0	0
$859$	3.91777	0.133673	0.0668363	−	0.997764i	$-0.478709\pi$
$859$	3.91777	0.133673	0.0668363	+	0.997764i	$0.478709\pi$
$860$	0	0
$861$	−12.0613	−0.411050
$862$	0	0
$863$	−46.0890	−1.56889	−0.784444	−	0.620199i	$-0.787052\pi$
$863$	−46.0890	−1.56889	−0.784444	+	0.620199i	$0.787052\pi$
$864$	0	0
$865$	29.9326	1.01774
$866$	0	0
$867$	−11.1381	−0.378270
$868$	0	0
$869$	26.0543	0.883830
$870$	0	0
$871$	−2.81877	−0.0955105
$872$	0	0
$873$	17.5044	0.592433
$874$	0	0
$875$	33.5417	1.13392
$876$	0	0
$877$	−53.4301	−1.80421	−0.902104	−	0.431520i	$-0.857978\pi$
$877$	−53.4301	−1.80421	−0.902104	+	0.431520i	$0.857978\pi$
$878$	0	0
$879$	−15.8302	−0.533940
$880$	0	0
$881$	2.17882	0.0734063	0.0367032	−	0.999326i	$-0.488314\pi$
$881$	2.17882	0.0734063	0.0367032	+	0.999326i	$0.488314\pi$
$882$	0	0
$883$	36.7994	1.23840	0.619199	−	0.785234i	$-0.287457\pi$
$883$	36.7994	1.23840	0.619199	+	0.785234i	$0.287457\pi$
$884$	0	0
$885$	16.8567	0.566630
$886$	0	0
$887$	−22.1743	−0.744542	−0.372271	−	0.928124i	$-0.621421\pi$
$887$	−22.1743	−0.744542	−0.372271	+	0.928124i	$0.621421\pi$
$888$	0	0
$889$	−40.3852	−1.35448
$890$	0	0
$891$	−4.27402	−0.143185
$892$	0	0
$893$	0.0176134	0.000589411 0
$894$	0	0
$895$	−53.7940	−1.79814
$896$	0	0
$897$	−1.71624	−0.0573035
$898$	0	0
$899$	9.45536	0.315354
$900$	0	0
$901$	−2.15517	−0.0717993
$902$	0	0
$903$	−19.0061	−0.632483
$904$	0	0
$905$	−17.9895	−0.597993
$906$	0	0
$907$	−43.2546	−1.43625	−0.718123	−	0.695916i	$-0.754998\pi$
$907$	−43.2546	−1.43625	−0.718123	+	0.695916i	$0.754998\pi$
$908$	0	0
$909$	19.8394	0.658032
$910$	0	0
$911$	−50.0825	−1.65931	−0.829653	−	0.558279i	$-0.811462\pi$
$911$	−50.0825	−1.65931	−0.829653	+	0.558279i	$0.811462\pi$
$912$	0	0
$913$	−46.2815	−1.53170
$914$	0	0
$915$	−31.1397	−1.02945
$916$	0	0
$917$	19.9804	0.659810
$918$	0	0
$919$	−9.22352	−0.304256	−0.152128	−	0.988361i	$-0.548613\pi$
$919$	−9.22352	−0.304256	−0.152128	+	0.988361i	$0.548613\pi$
$920$	0	0
$921$	−5.45684	−0.179809
$922$	0	0
$923$	3.30418	0.108759
$924$	0	0
$925$	0.536364	0.0176355
$926$	0	0
$927$	−3.01200	−0.0989272
$928$	0	0
$929$	29.1835	0.957478	0.478739	−	0.877957i	$-0.341094\pi$
$929$	29.1835	0.957478	0.478739	+	0.877957i	$0.341094\pi$
$930$	0	0
$931$	2.28853	0.0750036
$932$	0	0
$933$	3.14591	0.102992
$934$	0	0
$935$	23.3072	0.762226
$936$	0	0
$937$	−17.8703	−0.583796	−0.291898	−	0.956449i	$-0.594287\pi$
$937$	−17.8703	−0.583796	−0.291898	+	0.956449i	$0.594287\pi$
$938$	0	0
$939$	15.0339	0.490613
$940$	0	0
$941$	36.9109	1.20326	0.601630	−	0.798775i	$-0.294518\pi$
$941$	36.9109	1.20326	0.601630	+	0.798775i	$0.294518\pi$
$942$	0	0
$943$	3.99046	0.129947
$944$	0	0
$945$	−6.80781	−0.221458
$946$	0	0
$947$	−57.1022	−1.85557	−0.927786	−	0.373112i	$-0.878291\pi$
$947$	−57.1022	−1.85557	−0.927786	+	0.373112i	$0.878291\pi$
$948$	0	0
$949$	−20.4210	−0.662893
$950$	0	0
$951$	−10.7472	−0.348501
$952$	0	0
$953$	3.46913	0.112376	0.0561881	−	0.998420i	$-0.482105\pi$
$953$	3.46913	0.112376	0.0561881	+	0.998420i	$0.482105\pi$
$954$	0	0
$955$	10.8995	0.352699
$956$	0	0
$957$	−4.27402	−0.138159
$958$	0	0
$959$	30.1745	0.974386
$960$	0	0
$961$	58.4039	1.88400
$962$	0	0
$963$	−8.17067	−0.263296
$964$	0	0
$965$	−16.4246	−0.528727
$966$	0	0
$967$	−39.5401	−1.27153	−0.635763	−	0.771885i	$-0.719314\pi$
$967$	−39.5401	−1.27153	−0.635763	+	0.771885i	$0.719314\pi$
$968$	0	0
$969$	2.59430	0.0833410
$970$	0	0
$971$	−18.8056	−0.603501	−0.301750	−	0.953387i	$-0.597571\pi$
$971$	−18.8056	−0.603501	−0.301750	+	0.953387i	$0.597571\pi$
$972$	0	0
$973$	14.7991	0.474437
$974$	0	0
$975$	0.125371	0.00401510
$976$	0	0
$977$	−29.4298	−0.941542	−0.470771	−	0.882255i	$-0.656024\pi$
$977$	−29.4298	−0.941542	−0.470771	+	0.882255i	$0.656024\pi$
$978$	0	0
$979$	−3.40248	−0.108744
$980$	0	0
$981$	9.09121	0.290260
$982$	0	0
$983$	−16.7947	−0.535667	−0.267833	−	0.963465i	$-0.586308\pi$
$983$	−16.7947	−0.535667	−0.267833	+	0.963465i	$0.586308\pi$
$984$	0	0
$985$	−25.9822	−0.827861
$986$	0	0
$987$	0.0496838	0.00158145
$988$	0	0
$989$	6.28811	0.199950
$990$	0	0
$991$	47.3114	1.50290	0.751448	−	0.659792i	$-0.229356\pi$
$991$	47.3114	1.50290	0.751448	+	0.659792i	$0.229356\pi$
$992$	0	0
$993$	−4.21791	−0.133851
$994$	0	0
$995$	31.7955	1.00798
$996$	0	0
$997$	−6.25401	−0.198066	−0.0990332	−	0.995084i	$-0.531575\pi$
$997$	−6.25401	−0.198066	−0.0990332	+	0.995084i	$0.531575\pi$
$998$	0	0
$999$	7.34241	0.232304

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.k.1.6	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.k.1.6	✓	18	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.25234 q^{5} +3.02254 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.25234 q^{5} +3.02254 q^{7} +1.00000 q^{9} -4.27402 q^{11} +1.71624 q^{13} -2.25234 q^{15} +2.42113 q^{17} +1.07152 q^{19} +3.02254 q^{21} -1.00000 q^{23} +0.0730502 q^{25} +1.00000 q^{27} +1.00000 q^{29} +9.45536 q^{31} -4.27402 q^{33} -6.80781 q^{35} +7.34241 q^{37} +1.71624 q^{39} -3.99046 q^{41} -6.28811 q^{43} -2.25234 q^{45} +0.0164377 q^{47} +2.13577 q^{49} +2.42113 q^{51} -0.890152 q^{53} +9.62656 q^{55} +1.07152 q^{57} -7.48405 q^{59} +13.8255 q^{61} +3.02254 q^{63} -3.86555 q^{65} -1.64242 q^{67} -1.00000 q^{69} +1.92525 q^{71} -11.8987 q^{73} +0.0730502 q^{75} -12.9184 q^{77} -6.09596 q^{79} +1.00000 q^{81} +10.8286 q^{83} -5.45322 q^{85} +1.00000 q^{87} +0.796084 q^{89} +5.18740 q^{91} +9.45536 q^{93} -2.41344 q^{95} +17.5044 q^{97} -4.27402 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + 5 q^{5} + 6 q^{7} + 18 q^{9} + 5 q^{11} + 6 q^{13} + 5 q^{15} + 7 q^{17} + 15 q^{19} + 6 q^{21} - 18 q^{23} + 45 q^{25} + 18 q^{27} + 18 q^{29} + 10 q^{31} + 5 q^{33} - 7 q^{35} + 22 q^{37} + 6 q^{39} + 17 q^{41} + 25 q^{43} + 5 q^{45} - 6 q^{47} + 64 q^{49} + 7 q^{51} + 21 q^{53} + 3 q^{55} + 15 q^{57} + 6 q^{59} + 7 q^{61} + 6 q^{63} + 44 q^{65} + 35 q^{67} - 18 q^{69} + q^{71} + 27 q^{73} + 45 q^{75} + 4 q^{77} + 18 q^{81} + 13 q^{83} + 24 q^{85} + 18 q^{87} + 30 q^{89} + 53 q^{91} + 10 q^{93} + 17 q^{95} + 27 q^{97} + 5 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + 5 * q^5 + 6 * q^7 + 18 * q^9 + 5 * q^11 + 6 * q^13 + 5 * q^15 + 7 * q^17 + 15 * q^19 + 6 * q^21 - 18 * q^23 + 45 * q^25 + 18 * q^27 + 18 * q^29 + 10 * q^31 + 5 * q^33 - 7 * q^35 + 22 * q^37 + 6 * q^39 + 17 * q^41 + 25 * q^43 + 5 * q^45 - 6 * q^47 + 64 * q^49 + 7 * q^51 + 21 * q^53 + 3 * q^55 + 15 * q^57 + 6 * q^59 + 7 * q^61 + 6 * q^63 + 44 * q^65 + 35 * q^67 - 18 * q^69 + q^71 + 27 * q^73 + 45 * q^75 + 4 * q^77 + 18 * q^81 + 13 * q^83 + 24 * q^85 + 18 * q^87 + 30 * q^89 + 53 * q^91 + 10 * q^93 + 17 * q^95 + 27 * q^97 + 5 * q^99

Introduction

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