LMFDB - Embedded newform 8008.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 15x^{8} + 43x^{7} + 66x^{6} - 173x^{5} - 127x^{4} + 246x^{3} + 99x^{2} - 82x + 8 $ x^10 - 3x^9 - 15x^8 + 43x^7 + 66x^6 - 173x^5 - 127x^4 + 246x^3 + 99x^2 - 82*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.79551$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.79551 q^{3} +3.68086 q^{5} +1.00000 q^{7} +0.223871 q^{9} +O(q^{10})$	$q-1.79551 q^{3} +3.68086 q^{5} +1.00000 q^{7} +0.223871 q^{9} +1.00000 q^{11} -1.00000 q^{13} -6.60904 q^{15} -5.37766 q^{17} -1.25558 q^{19} -1.79551 q^{21} +1.60669 q^{23} +8.54876 q^{25} +4.98458 q^{27} -4.82719 q^{29} -2.42888 q^{31} -1.79551 q^{33} +3.68086 q^{35} +4.86075 q^{37} +1.79551 q^{39} -12.3064 q^{41} -2.12752 q^{43} +0.824038 q^{45} +10.1172 q^{47} +1.00000 q^{49} +9.65567 q^{51} -2.26908 q^{53} +3.68086 q^{55} +2.25441 q^{57} -4.19132 q^{59} -6.96593 q^{61} +0.223871 q^{63} -3.68086 q^{65} -0.165980 q^{67} -2.88483 q^{69} -4.36876 q^{71} -2.85988 q^{73} -15.3494 q^{75} +1.00000 q^{77} +11.2207 q^{79} -9.62149 q^{81} -7.18700 q^{83} -19.7945 q^{85} +8.66730 q^{87} -2.16817 q^{89} -1.00000 q^{91} +4.36108 q^{93} -4.62162 q^{95} -4.17424 q^{97} +0.223871 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9}+O(q^{10}) $ 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9	$ 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9} + 10 q^{11} - 10 q^{13} - 5 q^{15} - 11 q^{17} + 2 q^{19} - 3 q^{21} - 8 q^{23} + 2 q^{25} - 15 q^{27} - 8 q^{29} - 23 q^{31} - 3 q^{33} - 4 q^{35} + 10 q^{37} + 3 q^{39} - 18 q^{41} + 12 q^{43} - 10 q^{45} - 36 q^{47} + 10 q^{49} + 9 q^{51} - 21 q^{53} - 4 q^{55} - 30 q^{57} - 13 q^{59} - 2 q^{61} + 9 q^{63} + 4 q^{65} - 4 q^{67} - 26 q^{69} - 24 q^{71} - 23 q^{73} - 28 q^{75} + 10 q^{77} + 14 q^{79} + 30 q^{81} - 9 q^{83} - 17 q^{85} + 7 q^{87} - 18 q^{89} - 10 q^{91} + q^{93} - 4 q^{95} - 9 q^{97} + 9 q^{99}+O(q^{100}) $ 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9 + 10 * q^11 - 10 * q^13 - 5 * q^15 - 11 * q^17 + 2 * q^19 - 3 * q^21 - 8 * q^23 + 2 * q^25 - 15 * q^27 - 8 * q^29 - 23 * q^31 - 3 * q^33 - 4 * q^35 + 10 * q^37 + 3 * q^39 - 18 * q^41 + 12 * q^43 - 10 * q^45 - 36 * q^47 + 10 * q^49 + 9 * q^51 - 21 * q^53 - 4 * q^55 - 30 * q^57 - 13 * q^59 - 2 * q^61 + 9 * q^63 + 4 * q^65 - 4 * q^67 - 26 * q^69 - 24 * q^71 - 23 * q^73 - 28 * q^75 + 10 * q^77 + 14 * q^79 + 30 * q^81 - 9 * q^83 - 17 * q^85 + 7 * q^87 - 18 * q^89 - 10 * q^91 + q^93 - 4 * q^95 - 9 * q^97 + 9 * 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
$3$	−1.79551	−1.03664	−0.518320	−	0.855187i	$-0.673442\pi$
$3$	−1.79551	−1.03664	−0.518320	+	0.855187i	$0.673442\pi$
$4$	0	0
$5$	3.68086	1.64613	0.823066	−	0.567945i	$-0.192262\pi$
$5$	3.68086	1.64613	0.823066	+	0.567945i	$0.192262\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0.223871	0.0746236
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−6.60904	−1.70645
$16$	0	0
$17$	−5.37766	−1.30428	−0.652138	−	0.758101i	$-0.726128\pi$
$17$	−5.37766	−1.30428	−0.652138	+	0.758101i	$0.726128\pi$
$18$	0	0
$19$	−1.25558	−0.288050	−0.144025	−	0.989574i	$-0.546005\pi$
$19$	−1.25558	−0.288050	−0.144025	+	0.989574i	$0.546005\pi$
$20$	0	0
$21$	−1.79551	−0.391813
$22$	0	0
$23$	1.60669	0.335018	0.167509	−	0.985871i	$-0.446428\pi$
$23$	1.60669	0.335018	0.167509	+	0.985871i	$0.446428\pi$
$24$	0	0
$25$	8.54876	1.70975
$26$	0	0
$27$	4.98458	0.959283
$28$	0	0
$29$	−4.82719	−0.896387	−0.448194	−	0.893936i	$-0.647933\pi$
$29$	−4.82719	−0.896387	−0.448194	+	0.893936i	$0.647933\pi$
$30$	0	0
$31$	−2.42888	−0.436239	−0.218119	−	0.975922i	$-0.569992\pi$
$31$	−2.42888	−0.436239	−0.218119	+	0.975922i	$0.569992\pi$
$32$	0	0
$33$	−1.79551	−0.312559
$34$	0	0
$35$	3.68086	0.622180
$36$	0	0
$37$	4.86075	0.799103	0.399551	−	0.916711i	$-0.369166\pi$
$37$	4.86075	0.799103	0.399551	+	0.916711i	$0.369166\pi$
$38$	0	0
$39$	1.79551	0.287512
$40$	0	0
$41$	−12.3064	−1.92194	−0.960970	−	0.276653i	$-0.910775\pi$
$41$	−12.3064	−1.92194	−0.960970	+	0.276653i	$0.910775\pi$
$42$	0	0
$43$	−2.12752	−0.324444	−0.162222	−	0.986754i	$-0.551866\pi$
$43$	−2.12752	−0.324444	−0.162222	+	0.986754i	$0.551866\pi$
$44$	0	0
$45$	0.824038	0.122840
$46$	0	0
$47$	10.1172	1.47575	0.737875	−	0.674938i	$-0.235830\pi$
$47$	10.1172	1.47575	0.737875	+	0.674938i	$0.235830\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	9.65567	1.35206
$52$	0	0
$53$	−2.26908	−0.311682	−0.155841	−	0.987782i	$-0.549809\pi$
$53$	−2.26908	−0.311682	−0.155841	+	0.987782i	$0.549809\pi$
$54$	0	0
$55$	3.68086	0.496328
$56$	0	0
$57$	2.25441	0.298604
$58$	0	0
$59$	−4.19132	−0.545663	−0.272831	−	0.962062i	$-0.587960\pi$
$59$	−4.19132	−0.545663	−0.272831	+	0.962062i	$0.587960\pi$
$60$	0	0
$61$	−6.96593	−0.891895	−0.445948	−	0.895059i	$-0.647133\pi$
$61$	−6.96593	−0.891895	−0.445948	+	0.895059i	$0.647133\pi$
$62$	0	0
$63$	0.223871	0.0282051
$64$	0	0
$65$	−3.68086	−0.456555
$66$	0	0
$67$	−0.165980	−0.0202777	−0.0101388	−	0.999949i	$-0.503227\pi$
$67$	−0.165980	−0.0202777	−0.0101388	+	0.999949i	$0.503227\pi$
$68$	0	0
$69$	−2.88483	−0.347293
$70$	0	0
$71$	−4.36876	−0.518476	−0.259238	−	0.965813i	$-0.583471\pi$
$71$	−4.36876	−0.518476	−0.259238	+	0.965813i	$0.583471\pi$
$72$	0	0
$73$	−2.85988	−0.334724	−0.167362	−	0.985896i	$-0.553525\pi$
$73$	−2.85988	−0.334724	−0.167362	+	0.985896i	$0.553525\pi$
$74$	0	0
$75$	−15.3494	−1.77240
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	11.2207	1.26243	0.631215	−	0.775608i	$-0.282556\pi$
$79$	11.2207	1.26243	0.631215	+	0.775608i	$0.282556\pi$
$80$	0	0
$81$	−9.62149	−1.06905
$82$	0	0
$83$	−7.18700	−0.788876	−0.394438	−	0.918923i	$-0.629061\pi$
$83$	−7.18700	−0.788876	−0.394438	+	0.918923i	$0.629061\pi$
$84$	0	0
$85$	−19.7945	−2.14701
$86$	0	0
$87$	8.66730	0.929232
$88$	0	0
$89$	−2.16817	−0.229826	−0.114913	−	0.993376i	$-0.536659\pi$
$89$	−2.16817	−0.229826	−0.114913	+	0.993376i	$0.536659\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	4.36108	0.452223
$94$	0	0
$95$	−4.62162	−0.474168
$96$	0	0
$97$	−4.17424	−0.423830	−0.211915	−	0.977288i	$-0.567970\pi$
$97$	−4.17424	−0.423830	−0.211915	+	0.977288i	$0.567970\pi$
$98$	0	0
$99$	0.223871	0.0224999
$100$	0	0
$101$	−10.8930	−1.08389	−0.541945	−	0.840414i	$-0.682312\pi$
$101$	−10.8930	−1.08389	−0.541945	+	0.840414i	$0.682312\pi$
$102$	0	0
$103$	−4.82674	−0.475593	−0.237796	−	0.971315i	$-0.576425\pi$
$103$	−4.82674	−0.475593	−0.237796	+	0.971315i	$0.576425\pi$
$104$	0	0
$105$	−6.60904	−0.644977
$106$	0	0
$107$	2.30369	0.222706	0.111353	−	0.993781i	$-0.464482\pi$
$107$	2.30369	0.222706	0.111353	+	0.993781i	$0.464482\pi$
$108$	0	0
$109$	−18.6646	−1.78774	−0.893871	−	0.448325i	$-0.852021\pi$
$109$	−18.6646	−1.78774	−0.893871	+	0.448325i	$0.852021\pi$
$110$	0	0
$111$	−8.72755	−0.828382
$112$	0	0
$113$	6.69225	0.629554	0.314777	−	0.949166i	$-0.398070\pi$
$113$	6.69225	0.629554	0.314777	+	0.949166i	$0.398070\pi$
$114$	0	0
$115$	5.91400	0.551483
$116$	0	0
$117$	−0.223871	−0.0206969
$118$	0	0
$119$	−5.37766	−0.492970
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	22.0963	1.99236
$124$	0	0
$125$	13.0625	1.16835
$126$	0	0
$127$	−7.43195	−0.659479	−0.329739	−	0.944072i	$-0.606961\pi$
$127$	−7.43195	−0.659479	−0.329739	+	0.944072i	$0.606961\pi$
$128$	0	0
$129$	3.81999	0.336332
$130$	0	0
$131$	−14.9745	−1.30833	−0.654163	−	0.756353i	$-0.726979\pi$
$131$	−14.9745	−1.30833	−0.654163	+	0.756353i	$0.726979\pi$
$132$	0	0
$133$	−1.25558	−0.108873
$134$	0	0
$135$	18.3476	1.57911
$136$	0	0
$137$	−13.2468	−1.13175	−0.565874	−	0.824492i	$-0.691461\pi$
$137$	−13.2468	−1.13175	−0.565874	+	0.824492i	$0.691461\pi$
$138$	0	0
$139$	13.8200	1.17220	0.586098	−	0.810240i	$-0.300663\pi$
$139$	13.8200	1.17220	0.586098	+	0.810240i	$0.300663\pi$
$140$	0	0
$141$	−18.1656	−1.52982
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−17.7682	−1.47557
$146$	0	0
$147$	−1.79551	−0.148092
$148$	0	0
$149$	−0.730080	−0.0598105	−0.0299052	−	0.999553i	$-0.509521\pi$
$149$	−0.730080	−0.0598105	−0.0299052	+	0.999553i	$0.509521\pi$
$150$	0	0
$151$	0.338715	0.0275642	0.0137821	−	0.999905i	$-0.495613\pi$
$151$	0.338715	0.0275642	0.0137821	+	0.999905i	$0.495613\pi$
$152$	0	0
$153$	−1.20390	−0.0973297
$154$	0	0
$155$	−8.94036	−0.718107
$156$	0	0
$157$	4.69390	0.374614	0.187307	−	0.982301i	$-0.440024\pi$
$157$	4.69390	0.374614	0.187307	+	0.982301i	$0.440024\pi$
$158$	0	0
$159$	4.07416	0.323102
$160$	0	0
$161$	1.60669	0.126625
$162$	0	0
$163$	13.8435	1.08431	0.542155	−	0.840279i	$-0.317609\pi$
$163$	13.8435	1.08431	0.542155	+	0.840279i	$0.317609\pi$
$164$	0	0
$165$	−6.60904	−0.514513
$166$	0	0
$167$	−10.7353	−0.830724	−0.415362	−	0.909656i	$-0.636345\pi$
$167$	−10.7353	−0.830724	−0.415362	+	0.909656i	$0.636345\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.281088	−0.0214953
$172$	0	0
$173$	11.3448	0.862529	0.431265	−	0.902225i	$-0.358068\pi$
$173$	11.3448	0.862529	0.431265	+	0.902225i	$0.358068\pi$
$174$	0	0
$175$	8.54876	0.646226
$176$	0	0
$177$	7.52557	0.565656
$178$	0	0
$179$	−16.5162	−1.23448	−0.617239	−	0.786776i	$-0.711749\pi$
$179$	−16.5162	−1.23448	−0.617239	+	0.786776i	$0.711749\pi$
$180$	0	0
$181$	13.6556	1.01502	0.507508	−	0.861647i	$-0.330567\pi$
$181$	13.6556	1.01502	0.507508	+	0.861647i	$0.330567\pi$
$182$	0	0
$183$	12.5074	0.924575
$184$	0	0
$185$	17.8918	1.31543
$186$	0	0
$187$	−5.37766	−0.393254
$188$	0	0
$189$	4.98458	0.362575
$190$	0	0
$191$	8.83130	0.639011	0.319505	−	0.947585i	$-0.396483\pi$
$191$	8.83130	0.639011	0.319505	+	0.947585i	$0.396483\pi$
$192$	0	0
$193$	0.193276	0.0139123	0.00695614	−	0.999976i	$-0.497786\pi$
$193$	0.193276	0.0139123	0.00695614	+	0.999976i	$0.497786\pi$
$194$	0	0
$195$	6.60904	0.473283
$196$	0	0
$197$	−17.5627	−1.25129	−0.625646	−	0.780107i	$-0.715165\pi$
$197$	−17.5627	−1.25129	−0.625646	+	0.780107i	$0.715165\pi$
$198$	0	0
$199$	−0.152648	−0.0108210	−0.00541048	−	0.999985i	$-0.501722\pi$
$199$	−0.152648	−0.0108210	−0.00541048	+	0.999985i	$0.501722\pi$
$200$	0	0
$201$	0.298019	0.0210206
$202$	0	0
$203$	−4.82719	−0.338803
$204$	0	0
$205$	−45.2983	−3.16377
$206$	0	0
$207$	0.359691	0.0250002
$208$	0	0
$209$	−1.25558	−0.0868503
$210$	0	0
$211$	−6.35393	−0.437422	−0.218711	−	0.975790i	$-0.570185\pi$
$211$	−6.35393	−0.437422	−0.218711	+	0.975790i	$0.570185\pi$
$212$	0	0
$213$	7.84417	0.537474
$214$	0	0
$215$	−7.83111	−0.534078
$216$	0	0
$217$	−2.42888	−0.164883
$218$	0	0
$219$	5.13496	0.346989
$220$	0	0
$221$	5.37766	0.361741
$222$	0	0
$223$	17.8554	1.19569	0.597844	−	0.801612i	$-0.296024\pi$
$223$	17.8554	1.19569	0.597844	+	0.801612i	$0.296024\pi$
$224$	0	0
$225$	1.91382	0.127588
$226$	0	0
$227$	−17.4861	−1.16060	−0.580298	−	0.814404i	$-0.697064\pi$
$227$	−17.4861	−1.16060	−0.580298	+	0.814404i	$0.697064\pi$
$228$	0	0
$229$	9.16981	0.605958	0.302979	−	0.952997i	$-0.402019\pi$
$229$	9.16981	0.605958	0.302979	+	0.952997i	$0.402019\pi$
$230$	0	0
$231$	−1.79551	−0.118136
$232$	0	0
$233$	2.21970	0.145418	0.0727088	−	0.997353i	$-0.476836\pi$
$233$	2.21970	0.145418	0.0727088	+	0.997353i	$0.476836\pi$
$234$	0	0
$235$	37.2401	2.42928
$236$	0	0
$237$	−20.1470	−1.30869
$238$	0	0
$239$	−12.1509	−0.785976	−0.392988	−	0.919544i	$-0.628559\pi$
$239$	−12.1509	−0.785976	−0.392988	+	0.919544i	$0.628559\pi$
$240$	0	0
$241$	−15.2613	−0.983067	−0.491534	−	0.870859i	$-0.663564\pi$
$241$	−15.2613	−0.983067	−0.491534	+	0.870859i	$0.663564\pi$
$242$	0	0
$243$	2.32179	0.148943
$244$	0	0
$245$	3.68086	0.235162
$246$	0	0
$247$	1.25558	0.0798906
$248$	0	0
$249$	12.9044	0.817781
$250$	0	0
$251$	−1.20037	−0.0757664	−0.0378832	−	0.999282i	$-0.512061\pi$
$251$	−1.20037	−0.0757664	−0.0378832	+	0.999282i	$0.512061\pi$
$252$	0	0
$253$	1.60669	0.101012
$254$	0	0
$255$	35.5412	2.22568
$256$	0	0
$257$	10.9096	0.680523	0.340262	−	0.940331i	$-0.389484\pi$
$257$	10.9096	0.680523	0.340262	+	0.940331i	$0.389484\pi$
$258$	0	0
$259$	4.86075	0.302032
$260$	0	0
$261$	−1.08067	−0.0668917
$262$	0	0
$263$	25.1651	1.55175	0.775874	−	0.630888i	$-0.217309\pi$
$263$	25.1651	1.55175	0.775874	+	0.630888i	$0.217309\pi$
$264$	0	0
$265$	−8.35217	−0.513070
$266$	0	0
$267$	3.89299	0.238247
$268$	0	0
$269$	11.3114	0.689670	0.344835	−	0.938663i	$-0.387935\pi$
$269$	11.3114	0.689670	0.344835	+	0.938663i	$0.387935\pi$
$270$	0	0
$271$	−15.0755	−0.915769	−0.457884	−	0.889012i	$-0.651393\pi$
$271$	−15.0755	−0.915769	−0.457884	+	0.889012i	$0.651393\pi$
$272$	0	0
$273$	1.79551	0.108669
$274$	0	0
$275$	8.54876	0.515510
$276$	0	0
$277$	−13.1611	−0.790774	−0.395387	−	0.918515i	$-0.629390\pi$
$277$	−13.1611	−0.790774	−0.395387	+	0.918515i	$0.629390\pi$
$278$	0	0
$279$	−0.543754	−0.0325537
$280$	0	0
$281$	28.8572	1.72147	0.860737	−	0.509050i	$-0.170003\pi$
$281$	28.8572	1.72147	0.860737	+	0.509050i	$0.170003\pi$
$282$	0	0
$283$	10.3305	0.614085	0.307042	−	0.951696i	$-0.400661\pi$
$283$	10.3305	0.614085	0.307042	+	0.951696i	$0.400661\pi$
$284$	0	0
$285$	8.29818	0.491542
$286$	0	0
$287$	−12.3064	−0.726425
$288$	0	0
$289$	11.9193	0.701134
$290$	0	0
$291$	7.49491	0.439360
$292$	0	0
$293$	−16.8214	−0.982716	−0.491358	−	0.870958i	$-0.663499\pi$
$293$	−16.8214	−0.982716	−0.491358	+	0.870958i	$0.663499\pi$
$294$	0	0
$295$	−15.4277	−0.898234
$296$	0	0
$297$	4.98458	0.289235
$298$	0	0
$299$	−1.60669	−0.0929172
$300$	0	0
$301$	−2.12752	−0.122628
$302$	0	0
$303$	19.5584	1.12360
$304$	0	0
$305$	−25.6406	−1.46818
$306$	0	0
$307$	16.1499	0.921722	0.460861	−	0.887472i	$-0.347541\pi$
$307$	16.1499	0.921722	0.460861	+	0.887472i	$0.347541\pi$
$308$	0	0
$309$	8.66648	0.493019
$310$	0	0
$311$	28.0921	1.59296	0.796479	−	0.604666i	$-0.206693\pi$
$311$	28.0921	1.59296	0.796479	+	0.604666i	$0.206693\pi$
$312$	0	0
$313$	−22.2525	−1.25779	−0.628894	−	0.777491i	$-0.716492\pi$
$313$	−22.2525	−1.25779	−0.628894	+	0.777491i	$0.716492\pi$
$314$	0	0
$315$	0.824038	0.0464293
$316$	0	0
$317$	1.37871	0.0774363	0.0387182	−	0.999250i	$-0.487673\pi$
$317$	1.37871	0.0774363	0.0387182	+	0.999250i	$0.487673\pi$
$318$	0	0
$319$	−4.82719	−0.270271
$320$	0	0
$321$	−4.13631	−0.230866
$322$	0	0
$323$	6.75209	0.375696
$324$	0	0
$325$	−8.54876	−0.474200
$326$	0	0
$327$	33.5125	1.85325
$328$	0	0
$329$	10.1172	0.557781
$330$	0	0
$331$	2.66922	0.146713	0.0733567	−	0.997306i	$-0.476629\pi$
$331$	2.66922	0.146713	0.0733567	+	0.997306i	$0.476629\pi$
$332$	0	0
$333$	1.08818	0.0596319
$334$	0	0
$335$	−0.610949	−0.0333797
$336$	0	0
$337$	−12.0529	−0.656562	−0.328281	−	0.944580i	$-0.606469\pi$
$337$	−12.0529	−0.656562	−0.328281	+	0.944580i	$0.606469\pi$
$338$	0	0
$339$	−12.0160	−0.652621
$340$	0	0
$341$	−2.42888	−0.131531
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−10.6187	−0.571690
$346$	0	0
$347$	−18.6108	−0.999082	−0.499541	−	0.866290i	$-0.666498\pi$
$347$	−18.6108	−0.999082	−0.499541	+	0.866290i	$0.666498\pi$
$348$	0	0
$349$	14.3743	0.769438	0.384719	−	0.923034i	$-0.374298\pi$
$349$	14.3743	0.769438	0.384719	+	0.923034i	$0.374298\pi$
$350$	0	0
$351$	−4.98458	−0.266057
$352$	0	0
$353$	−16.1535	−0.859765	−0.429882	−	0.902885i	$-0.641445\pi$
$353$	−16.1535	−0.859765	−0.429882	+	0.902885i	$0.641445\pi$
$354$	0	0
$355$	−16.0808	−0.853481
$356$	0	0
$357$	9.65567	0.511032
$358$	0	0
$359$	7.33841	0.387306	0.193653	−	0.981070i	$-0.437966\pi$
$359$	7.33841	0.387306	0.193653	+	0.981070i	$0.437966\pi$
$360$	0	0
$361$	−17.4235	−0.917027
$362$	0	0
$363$	−1.79551	−0.0942400
$364$	0	0
$365$	−10.5268	−0.551000
$366$	0	0
$367$	4.70394	0.245544	0.122772	−	0.992435i	$-0.460822\pi$
$367$	4.70394	0.245544	0.122772	+	0.992435i	$0.460822\pi$
$368$	0	0
$369$	−2.75505	−0.143422
$370$	0	0
$371$	−2.26908	−0.117805
$372$	0	0
$373$	8.50129	0.440180	0.220090	−	0.975480i	$-0.429365\pi$
$373$	8.50129	0.440180	0.220090	+	0.975480i	$0.429365\pi$
$374$	0	0
$375$	−23.4539	−1.21115
$376$	0	0
$377$	4.82719	0.248613
$378$	0	0
$379$	9.81769	0.504301	0.252150	−	0.967688i	$-0.418862\pi$
$379$	9.81769	0.504301	0.252150	+	0.967688i	$0.418862\pi$
$380$	0	0
$381$	13.3442	0.683642
$382$	0	0
$383$	−1.53230	−0.0782970	−0.0391485	−	0.999233i	$-0.512465\pi$
$383$	−1.53230	−0.0782970	−0.0391485	+	0.999233i	$0.512465\pi$
$384$	0	0
$385$	3.68086	0.187594
$386$	0	0
$387$	−0.476290	−0.0242112
$388$	0	0
$389$	33.4035	1.69362	0.846812	−	0.531892i	$-0.178519\pi$
$389$	33.4035	1.69362	0.846812	+	0.531892i	$0.178519\pi$
$390$	0	0
$391$	−8.64023	−0.436955
$392$	0	0
$393$	26.8869	1.35626
$394$	0	0
$395$	41.3020	2.07813
$396$	0	0
$397$	−14.5053	−0.728002	−0.364001	−	0.931398i	$-0.618590\pi$
$397$	−14.5053	−0.728002	−0.364001	+	0.931398i	$0.618590\pi$
$398$	0	0
$399$	2.25441	0.112862
$400$	0	0
$401$	2.56192	0.127936	0.0639680	−	0.997952i	$-0.479624\pi$
$401$	2.56192	0.127936	0.0639680	+	0.997952i	$0.479624\pi$
$402$	0	0
$403$	2.42888	0.120991
$404$	0	0
$405$	−35.4154	−1.75981
$406$	0	0
$407$	4.86075	0.240939
$408$	0	0
$409$	9.10532	0.450229	0.225115	−	0.974332i	$-0.427724\pi$
$409$	9.10532	0.450229	0.225115	+	0.974332i	$0.427724\pi$
$410$	0	0
$411$	23.7848	1.17322
$412$	0	0
$413$	−4.19132	−0.206241
$414$	0	0
$415$	−26.4544	−1.29859
$416$	0	0
$417$	−24.8140	−1.21515
$418$	0	0
$419$	−12.4197	−0.606740	−0.303370	−	0.952873i	$-0.598112\pi$
$419$	−12.4197	−0.606740	−0.303370	+	0.952873i	$0.598112\pi$
$420$	0	0
$421$	−30.1632	−1.47006	−0.735032	−	0.678033i	$-0.762833\pi$
$421$	−30.1632	−1.47006	−0.735032	+	0.678033i	$0.762833\pi$
$422$	0	0
$423$	2.26495	0.110126
$424$	0	0
$425$	−45.9724	−2.22999
$426$	0	0
$427$	−6.96593	−0.337105
$428$	0	0
$429$	1.79551	0.0866882
$430$	0	0
$431$	19.1657	0.923178	0.461589	−	0.887094i	$-0.347279\pi$
$431$	19.1657	0.923178	0.461589	+	0.887094i	$0.347279\pi$
$432$	0	0
$433$	−37.5993	−1.80691	−0.903454	−	0.428686i	$-0.858977\pi$
$433$	−37.5993	−1.80691	−0.903454	+	0.428686i	$0.858977\pi$
$434$	0	0
$435$	31.9031	1.52964
$436$	0	0
$437$	−2.01733	−0.0965017
$438$	0	0
$439$	−19.2124	−0.916958	−0.458479	−	0.888705i	$-0.651606\pi$
$439$	−19.2124	−0.916958	−0.458479	+	0.888705i	$0.651606\pi$
$440$	0	0
$441$	0.223871	0.0106605
$442$	0	0
$443$	−32.1326	−1.52666	−0.763332	−	0.646006i	$-0.776438\pi$
$443$	−32.1326	−1.52666	−0.763332	+	0.646006i	$0.776438\pi$
$444$	0	0
$445$	−7.98075	−0.378324
$446$	0	0
$447$	1.31087	0.0620019
$448$	0	0
$449$	−39.4653	−1.86248	−0.931241	−	0.364403i	$-0.881273\pi$
$449$	−39.4653	−1.86248	−0.931241	+	0.364403i	$0.881273\pi$
$450$	0	0
$451$	−12.3064	−0.579487
$452$	0	0
$453$	−0.608167	−0.0285742
$454$	0	0
$455$	−3.68086	−0.172562
$456$	0	0
$457$	−37.1205	−1.73643	−0.868213	−	0.496192i	$-0.834731\pi$
$457$	−37.1205	−1.73643	−0.868213	+	0.496192i	$0.834731\pi$
$458$	0	0
$459$	−26.8054	−1.25117
$460$	0	0
$461$	−22.3117	−1.03916	−0.519580	−	0.854422i	$-0.673912\pi$
$461$	−22.3117	−1.03916	−0.519580	+	0.854422i	$0.673912\pi$
$462$	0	0
$463$	−26.0897	−1.21249	−0.606246	−	0.795277i	$-0.707325\pi$
$463$	−26.0897	−1.21249	−0.606246	+	0.795277i	$0.707325\pi$
$464$	0	0
$465$	16.0525	0.744419
$466$	0	0
$467$	−21.4441	−0.992315	−0.496158	−	0.868232i	$-0.665256\pi$
$467$	−21.4441	−0.992315	−0.496158	+	0.868232i	$0.665256\pi$
$468$	0	0
$469$	−0.165980	−0.00766423
$470$	0	0
$471$	−8.42796	−0.388340
$472$	0	0
$473$	−2.12752	−0.0978235
$474$	0	0
$475$	−10.7337	−0.492494
$476$	0	0
$477$	−0.507981	−0.0232588
$478$	0	0
$479$	−22.9762	−1.04981	−0.524905	−	0.851161i	$-0.675899\pi$
$479$	−22.9762	−1.04981	−0.524905	+	0.851161i	$0.675899\pi$
$480$	0	0
$481$	−4.86075	−0.221631
$482$	0	0
$483$	−2.88483	−0.131264
$484$	0	0
$485$	−15.3648	−0.697681
$486$	0	0
$487$	0.754619	0.0341951	0.0170975	−	0.999854i	$-0.494557\pi$
$487$	0.754619	0.0341951	0.0170975	+	0.999854i	$0.494557\pi$
$488$	0	0
$489$	−24.8563	−1.12404
$490$	0	0
$491$	3.40184	0.153523	0.0767614	−	0.997049i	$-0.475542\pi$
$491$	3.40184	0.153523	0.0767614	+	0.997049i	$0.475542\pi$
$492$	0	0
$493$	25.9590	1.16914
$494$	0	0
$495$	0.824038	0.0370378
$496$	0	0
$497$	−4.36876	−0.195966
$498$	0	0
$499$	−16.0911	−0.720335	−0.360167	−	0.932888i	$-0.617280\pi$
$499$	−16.0911	−0.720335	−0.360167	+	0.932888i	$0.617280\pi$
$500$	0	0
$501$	19.2754	0.861162
$502$	0	0
$503$	0.891504	0.0397502	0.0198751	−	0.999802i	$-0.493673\pi$
$503$	0.891504	0.0397502	0.0198751	+	0.999802i	$0.493673\pi$
$504$	0	0
$505$	−40.0955	−1.78423
$506$	0	0
$507$	−1.79551	−0.0797416
$508$	0	0
$509$	22.7115	1.00667	0.503334	−	0.864092i	$-0.332106\pi$
$509$	22.7115	1.00667	0.503334	+	0.864092i	$0.332106\pi$
$510$	0	0
$511$	−2.85988	−0.126514
$512$	0	0
$513$	−6.25854	−0.276321
$514$	0	0
$515$	−17.7666	−0.782889
$516$	0	0
$517$	10.1172	0.444955
$518$	0	0
$519$	−20.3698	−0.894133
$520$	0	0
$521$	44.4678	1.94817	0.974084	−	0.226186i	$-0.0726259\pi$
$521$	44.4678	1.94817	0.974084	+	0.226186i	$0.0726259\pi$
$522$	0	0
$523$	−31.6890	−1.38566	−0.692832	−	0.721099i	$-0.743637\pi$
$523$	−31.6890	−1.38566	−0.692832	+	0.721099i	$0.743637\pi$
$524$	0	0
$525$	−15.3494	−0.669904
$526$	0	0
$527$	13.0617	0.568976
$528$	0	0
$529$	−20.4186	−0.887763
$530$	0	0
$531$	−0.938314	−0.0407193
$532$	0	0
$533$	12.3064	0.533050
$534$	0	0
$535$	8.47957	0.366604
$536$	0	0
$537$	29.6551	1.27971
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	37.4062	1.60822	0.804110	−	0.594481i	$-0.202643\pi$
$541$	37.4062	1.60822	0.804110	+	0.594481i	$0.202643\pi$
$542$	0	0
$543$	−24.5189	−1.05221
$544$	0	0
$545$	−68.7017	−2.94286
$546$	0	0
$547$	−7.09451	−0.303339	−0.151670	−	0.988431i	$-0.548465\pi$
$547$	−7.09451	−0.303339	−0.151670	+	0.988431i	$0.548465\pi$
$548$	0	0
$549$	−1.55947	−0.0665565
$550$	0	0
$551$	6.06093	0.258204
$552$	0	0
$553$	11.2207	0.477154
$554$	0	0
$555$	−32.1249	−1.36363
$556$	0	0
$557$	34.4001	1.45758	0.728789	−	0.684739i	$-0.240084\pi$
$557$	34.4001	1.45758	0.728789	+	0.684739i	$0.240084\pi$
$558$	0	0
$559$	2.12752	0.0899845
$560$	0	0
$561$	9.65567	0.407663
$562$	0	0
$563$	−8.87209	−0.373914	−0.186957	−	0.982368i	$-0.559862\pi$
$563$	−8.87209	−0.373914	−0.186957	+	0.982368i	$0.559862\pi$
$564$	0	0
$565$	24.6333	1.03633
$566$	0	0
$567$	−9.62149	−0.404065
$568$	0	0
$569$	11.9493	0.500941	0.250471	−	0.968124i	$-0.419415\pi$
$569$	11.9493	0.500941	0.250471	+	0.968124i	$0.419415\pi$
$570$	0	0
$571$	44.2558	1.85205	0.926024	−	0.377464i	$-0.123204\pi$
$571$	44.2558	1.85205	0.926024	+	0.377464i	$0.123204\pi$
$572$	0	0
$573$	−15.8567	−0.662424
$574$	0	0
$575$	13.7352	0.572797
$576$	0	0
$577$	−27.2826	−1.13579	−0.567894	−	0.823102i	$-0.692242\pi$
$577$	−27.2826	−1.13579	−0.567894	+	0.823102i	$0.692242\pi$
$578$	0	0
$579$	−0.347029	−0.0144220
$580$	0	0
$581$	−7.18700	−0.298167
$582$	0	0
$583$	−2.26908	−0.0939756
$584$	0	0
$585$	−0.824038	−0.0340698
$586$	0	0
$587$	−14.9960	−0.618950	−0.309475	−	0.950908i	$-0.600153\pi$
$587$	−14.9960	−0.618950	−0.309475	+	0.950908i	$0.600153\pi$
$588$	0	0
$589$	3.04965	0.125659
$590$	0	0
$591$	31.5341	1.29714
$592$	0	0
$593$	−17.9946	−0.738952	−0.369476	−	0.929240i	$-0.620463\pi$
$593$	−17.9946	−0.738952	−0.369476	+	0.929240i	$0.620463\pi$
$594$	0	0
$595$	−19.7945	−0.811493
$596$	0	0
$597$	0.274082	0.0112174
$598$	0	0
$599$	7.18381	0.293523	0.146761	−	0.989172i	$-0.453115\pi$
$599$	7.18381	0.293523	0.146761	+	0.989172i	$0.453115\pi$
$600$	0	0
$601$	−38.5416	−1.57215	−0.786073	−	0.618134i	$-0.787889\pi$
$601$	−38.5416	−1.57215	−0.786073	+	0.618134i	$0.787889\pi$
$602$	0	0
$603$	−0.0371580	−0.00151319
$604$	0	0
$605$	3.68086	0.149648
$606$	0	0
$607$	−27.5175	−1.11690	−0.558451	−	0.829537i	$-0.688604\pi$
$607$	−27.5175	−1.11690	−0.558451	+	0.829537i	$0.688604\pi$
$608$	0	0
$609$	8.66730	0.351217
$610$	0	0
$611$	−10.1172	−0.409299
$612$	0	0
$613$	21.2006	0.856286	0.428143	−	0.903711i	$-0.359168\pi$
$613$	21.2006	0.856286	0.428143	+	0.903711i	$0.359168\pi$
$614$	0	0
$615$	81.3337	3.27969
$616$	0	0
$617$	7.19317	0.289586	0.144793	−	0.989462i	$-0.453748\pi$
$617$	7.19317	0.289586	0.144793	+	0.989462i	$0.453748\pi$
$618$	0	0
$619$	2.34193	0.0941301	0.0470650	−	0.998892i	$-0.485013\pi$
$619$	2.34193	0.0941301	0.0470650	+	0.998892i	$0.485013\pi$
$620$	0	0
$621$	8.00866	0.321377
$622$	0	0
$623$	−2.16817	−0.0868660
$624$	0	0
$625$	5.33750	0.213500
$626$	0	0
$627$	2.25441	0.0900325
$628$	0	0
$629$	−26.1395	−1.04225
$630$	0	0
$631$	44.3461	1.76539	0.882695	−	0.469946i	$-0.155727\pi$
$631$	44.3461	1.76539	0.882695	+	0.469946i	$0.155727\pi$
$632$	0	0
$633$	11.4086	0.453450
$634$	0	0
$635$	−27.3560	−1.08559
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−0.978038	−0.0386906
$640$	0	0
$641$	−36.9353	−1.45886	−0.729428	−	0.684057i	$-0.760214\pi$
$641$	−36.9353	−1.45886	−0.729428	+	0.684057i	$0.760214\pi$
$642$	0	0
$643$	−18.1090	−0.714148	−0.357074	−	0.934076i	$-0.616226\pi$
$643$	−18.1090	−0.714148	−0.357074	+	0.934076i	$0.616226\pi$
$644$	0	0
$645$	14.0609	0.553646
$646$	0	0
$647$	−5.40656	−0.212554	−0.106277	−	0.994337i	$-0.533893\pi$
$647$	−5.40656	−0.212554	−0.106277	+	0.994337i	$0.533893\pi$
$648$	0	0
$649$	−4.19132	−0.164524
$650$	0	0
$651$	4.36108	0.170924
$652$	0	0
$653$	−5.52296	−0.216130	−0.108065	−	0.994144i	$-0.534465\pi$
$653$	−5.52296	−0.216130	−0.108065	+	0.994144i	$0.534465\pi$
$654$	0	0
$655$	−55.1190	−2.15368
$656$	0	0
$657$	−0.640245	−0.0249783
$658$	0	0
$659$	11.3753	0.443118	0.221559	−	0.975147i	$-0.428885\pi$
$659$	11.3753	0.443118	0.221559	+	0.975147i	$0.428885\pi$
$660$	0	0
$661$	−16.8463	−0.655245	−0.327623	−	0.944809i	$-0.606247\pi$
$661$	−16.8463	−0.655245	−0.327623	+	0.944809i	$0.606247\pi$
$662$	0	0
$663$	−9.65567	−0.374995
$664$	0	0
$665$	−4.62162	−0.179219
$666$	0	0
$667$	−7.75580	−0.300306
$668$	0	0
$669$	−32.0597	−1.23950
$670$	0	0
$671$	−6.96593	−0.268917
$672$	0	0
$673$	−5.99402	−0.231052	−0.115526	−	0.993304i	$-0.536855\pi$
$673$	−5.99402	−0.231052	−0.115526	+	0.993304i	$0.536855\pi$
$674$	0	0
$675$	42.6120	1.64014
$676$	0	0
$677$	6.43633	0.247368	0.123684	−	0.992322i	$-0.460529\pi$
$677$	6.43633	0.247368	0.123684	+	0.992322i	$0.460529\pi$
$678$	0	0
$679$	−4.17424	−0.160193
$680$	0	0
$681$	31.3966	1.20312
$682$	0	0
$683$	−10.3339	−0.395417	−0.197709	−	0.980261i	$-0.563350\pi$
$683$	−10.3339	−0.395417	−0.197709	+	0.980261i	$0.563350\pi$
$684$	0	0
$685$	−48.7596	−1.86301
$686$	0	0
$687$	−16.4645	−0.628161
$688$	0	0
$689$	2.26908	0.0864450
$690$	0	0
$691$	−20.7200	−0.788226	−0.394113	−	0.919062i	$-0.628948\pi$
$691$	−20.7200	−0.788226	−0.394113	+	0.919062i	$0.628948\pi$
$692$	0	0
$693$	0.223871	0.00850415
$694$	0	0
$695$	50.8695	1.92959
$696$	0	0
$697$	66.1798	2.50674
$698$	0	0
$699$	−3.98551	−0.150746
$700$	0	0
$701$	18.2639	0.689818	0.344909	−	0.938636i	$-0.387910\pi$
$701$	18.2639	0.689818	0.344909	+	0.938636i	$0.387910\pi$
$702$	0	0
$703$	−6.10306	−0.230181
$704$	0	0
$705$	−66.8652	−2.51829
$706$	0	0
$707$	−10.8930	−0.409672
$708$	0	0
$709$	4.59644	0.172623	0.0863114	−	0.996268i	$-0.472492\pi$
$709$	4.59644	0.172623	0.0863114	+	0.996268i	$0.472492\pi$
$710$	0	0
$711$	2.51199	0.0942072
$712$	0	0
$713$	−3.90245	−0.146148
$714$	0	0
$715$	−3.68086	−0.137657
$716$	0	0
$717$	21.8171	0.814775
$718$	0	0
$719$	24.7422	0.922730	0.461365	−	0.887210i	$-0.347360\pi$
$719$	24.7422	0.922730	0.461365	+	0.887210i	$0.347360\pi$
$720$	0	0
$721$	−4.82674	−0.179757
$722$	0	0
$723$	27.4019	1.01909
$724$	0	0
$725$	−41.2665	−1.53260
$726$	0	0
$727$	23.2628	0.862770	0.431385	−	0.902168i	$-0.358025\pi$
$727$	23.2628	0.862770	0.431385	+	0.902168i	$0.358025\pi$
$728$	0	0
$729$	24.6957	0.914655
$730$	0	0
$731$	11.4411	0.423164
$732$	0	0
$733$	45.2063	1.66973	0.834867	−	0.550451i	$-0.185544\pi$
$733$	45.2063	1.66973	0.834867	+	0.550451i	$0.185544\pi$
$734$	0	0
$735$	−6.60904	−0.243778
$736$	0	0
$737$	−0.165980	−0.00611394
$738$	0	0
$739$	3.30330	0.121514	0.0607570	−	0.998153i	$-0.480649\pi$
$739$	3.30330	0.121514	0.0607570	+	0.998153i	$0.480649\pi$
$740$	0	0
$741$	−2.25441	−0.0828179
$742$	0	0
$743$	16.1548	0.592661	0.296331	−	0.955085i	$-0.404237\pi$
$743$	16.1548	0.592661	0.296331	+	0.955085i	$0.404237\pi$
$744$	0	0
$745$	−2.68732	−0.0984559
$746$	0	0
$747$	−1.60896	−0.0588688
$748$	0	0
$749$	2.30369	0.0841750
$750$	0	0
$751$	−6.93288	−0.252985	−0.126492	−	0.991968i	$-0.540372\pi$
$751$	−6.93288	−0.252985	−0.126492	+	0.991968i	$0.540372\pi$
$752$	0	0
$753$	2.15527	0.0785425
$754$	0	0
$755$	1.24676	0.0453743
$756$	0	0
$757$	−4.44437	−0.161533	−0.0807666	−	0.996733i	$-0.525737\pi$
$757$	−4.44437	−0.161533	−0.0807666	+	0.996733i	$0.525737\pi$
$758$	0	0
$759$	−2.88483	−0.104713
$760$	0	0
$761$	−20.1160	−0.729203	−0.364602	−	0.931164i	$-0.618795\pi$
$761$	−20.1160	−0.729203	−0.364602	+	0.931164i	$0.618795\pi$
$762$	0	0
$763$	−18.6646	−0.675703
$764$	0	0
$765$	−4.43140	−0.160218
$766$	0	0
$767$	4.19132	0.151340
$768$	0	0
$769$	35.2030	1.26945	0.634726	−	0.772737i	$-0.281113\pi$
$769$	35.2030	1.26945	0.634726	+	0.772737i	$0.281113\pi$
$770$	0	0
$771$	−19.5884	−0.705458
$772$	0	0
$773$	−2.04245	−0.0734618	−0.0367309	−	0.999325i	$-0.511694\pi$
$773$	−2.04245	−0.0734618	−0.0367309	+	0.999325i	$0.511694\pi$
$774$	0	0
$775$	−20.7639	−0.745860
$776$	0	0
$777$	−8.72755	−0.313099
$778$	0	0
$779$	15.4517	0.553614
$780$	0	0
$781$	−4.36876	−0.156327
$782$	0	0
$783$	−24.0615	−0.859889
$784$	0	0
$785$	17.2776	0.616664
$786$	0	0
$787$	49.0789	1.74947	0.874737	−	0.484599i	$-0.161034\pi$
$787$	49.0789	1.74947	0.874737	+	0.484599i	$0.161034\pi$
$788$	0	0
$789$	−45.1843	−1.60860
$790$	0	0
$791$	6.69225	0.237949
$792$	0	0
$793$	6.96593	0.247367
$794$	0	0
$795$	14.9964	0.531869
$796$	0	0
$797$	−45.1862	−1.60058	−0.800289	−	0.599615i	$-0.795321\pi$
$797$	−45.1862	−1.60058	−0.800289	+	0.599615i	$0.795321\pi$
$798$	0	0
$799$	−54.4070	−1.92478
$800$	0	0
$801$	−0.485391	−0.0171504
$802$	0	0
$803$	−2.85988	−0.100923
$804$	0	0
$805$	5.91400	0.208441
$806$	0	0
$807$	−20.3098	−0.714939
$808$	0	0
$809$	15.1987	0.534356	0.267178	−	0.963647i	$-0.413909\pi$
$809$	15.1987	0.534356	0.267178	+	0.963647i	$0.413909\pi$
$810$	0	0
$811$	12.0007	0.421401	0.210700	−	0.977551i	$-0.432426\pi$
$811$	12.0007	0.421401	0.210700	+	0.977551i	$0.432426\pi$
$812$	0	0
$813$	27.0682	0.949323
$814$	0	0
$815$	50.9562	1.78492
$816$	0	0
$817$	2.67127	0.0934560
$818$	0	0
$819$	−0.223871	−0.00782268
$820$	0	0
$821$	2.36960	0.0826996	0.0413498	−	0.999145i	$-0.486834\pi$
$821$	2.36960	0.0826996	0.0413498	+	0.999145i	$0.486834\pi$
$822$	0	0
$823$	−26.7016	−0.930759	−0.465379	−	0.885111i	$-0.654082\pi$
$823$	−26.7016	−0.930759	−0.465379	+	0.885111i	$0.654082\pi$
$824$	0	0
$825$	−15.3494	−0.534398
$826$	0	0
$827$	−29.9384	−1.04106	−0.520530	−	0.853844i	$-0.674265\pi$
$827$	−29.9384	−1.04106	−0.520530	+	0.853844i	$0.674265\pi$
$828$	0	0
$829$	53.3611	1.85331	0.926654	−	0.375916i	$-0.122672\pi$
$829$	53.3611	1.85331	0.926654	+	0.375916i	$0.122672\pi$
$830$	0	0
$831$	23.6310	0.819749
$832$	0	0
$833$	−5.37766	−0.186325
$834$	0	0
$835$	−39.5153	−1.36748
$836$	0	0
$837$	−12.1069	−0.418476
$838$	0	0
$839$	−11.0376	−0.381060	−0.190530	−	0.981681i	$-0.561021\pi$
$839$	−11.0376	−0.381060	−0.190530	+	0.981681i	$0.561021\pi$
$840$	0	0
$841$	−5.69820	−0.196490
$842$	0	0
$843$	−51.8134	−1.78455
$844$	0	0
$845$	3.68086	0.126626
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−18.5486	−0.636585
$850$	0	0
$851$	7.80971	0.267713
$852$	0	0
$853$	5.29148	0.181177	0.0905884	−	0.995888i	$-0.471125\pi$
$853$	5.29148	0.181177	0.0905884	+	0.995888i	$0.471125\pi$
$854$	0	0
$855$	−1.03465	−0.0353841
$856$	0	0
$857$	3.05059	0.104206	0.0521031	−	0.998642i	$-0.483408\pi$
$857$	3.05059	0.104206	0.0521031	+	0.998642i	$0.483408\pi$
$858$	0	0
$859$	−32.8734	−1.12163	−0.560813	−	0.827943i	$-0.689511\pi$
$859$	−32.8734	−1.12163	−0.560813	+	0.827943i	$0.689511\pi$
$860$	0	0
$861$	22.0963	0.753042
$862$	0	0
$863$	−20.5846	−0.700707	−0.350354	−	0.936618i	$-0.613939\pi$
$863$	−20.5846	−0.700707	−0.350354	+	0.936618i	$0.613939\pi$
$864$	0	0
$865$	41.7587	1.41984
$866$	0	0
$867$	−21.4012	−0.726824
$868$	0	0
$869$	11.2207	0.380637
$870$	0	0
$871$	0.165980	0.00562401
$872$	0	0
$873$	−0.934492	−0.0316277
$874$	0	0
$875$	13.0625	0.441593
$876$	0	0
$877$	−18.1596	−0.613205	−0.306603	−	0.951838i	$-0.599192\pi$
$877$	−18.1596	−0.613205	−0.306603	+	0.951838i	$0.599192\pi$
$878$	0	0
$879$	30.2030	1.01872
$880$	0	0
$881$	27.1891	0.916024	0.458012	−	0.888946i	$-0.348562\pi$
$881$	27.1891	0.916024	0.458012	+	0.888946i	$0.348562\pi$
$882$	0	0
$883$	−7.04528	−0.237093	−0.118546	−	0.992949i	$-0.537823\pi$
$883$	−7.04528	−0.237093	−0.118546	+	0.992949i	$0.537823\pi$
$884$	0	0
$885$	27.7006	0.931145
$886$	0	0
$887$	38.0382	1.27720	0.638600	−	0.769539i	$-0.279514\pi$
$887$	38.0382	1.27720	0.638600	+	0.769539i	$0.279514\pi$
$888$	0	0
$889$	−7.43195	−0.249260
$890$	0	0
$891$	−9.62149	−0.322332
$892$	0	0
$893$	−12.7030	−0.425089
$894$	0	0
$895$	−60.7939	−2.03211
$896$	0	0
$897$	2.88483	0.0963217
$898$	0	0
$899$	11.7247	0.391039
$900$	0	0
$901$	12.2023	0.406519
$902$	0	0
$903$	3.81999	0.127121
$904$	0	0
$905$	50.2646	1.67085
$906$	0	0
$907$	−35.9806	−1.19472	−0.597359	−	0.801974i	$-0.703783\pi$
$907$	−35.9806	−1.19472	−0.597359	+	0.801974i	$0.703783\pi$
$908$	0	0
$909$	−2.43861	−0.0808837
$910$	0	0
$911$	−33.7315	−1.11757	−0.558787	−	0.829311i	$-0.688733\pi$
$911$	−33.7315	−1.11757	−0.558787	+	0.829311i	$0.688733\pi$
$912$	0	0
$913$	−7.18700	−0.237855
$914$	0	0
$915$	46.0381	1.52197
$916$	0	0
$917$	−14.9745	−0.494501
$918$	0	0
$919$	−11.5445	−0.380819	−0.190410	−	0.981705i	$-0.560982\pi$
$919$	−11.5445	−0.380819	−0.190410	+	0.981705i	$0.560982\pi$
$920$	0	0
$921$	−28.9973	−0.955495
$922$	0	0
$923$	4.36876	0.143800
$924$	0	0
$925$	41.5534	1.36627
$926$	0	0
$927$	−1.08057	−0.0354904
$928$	0	0
$929$	−31.8130	−1.04375	−0.521875	−	0.853022i	$-0.674767\pi$
$929$	−31.8130	−1.04375	−0.521875	+	0.853022i	$0.674767\pi$
$930$	0	0
$931$	−1.25558	−0.0411500
$932$	0	0
$933$	−50.4398	−1.65133
$934$	0	0
$935$	−19.7945	−0.647348
$936$	0	0
$937$	−22.9158	−0.748626	−0.374313	−	0.927302i	$-0.622121\pi$
$937$	−22.9158	−0.748626	−0.374313	+	0.927302i	$0.622121\pi$
$938$	0	0
$939$	39.9547	1.30387
$940$	0	0
$941$	−37.4434	−1.22062	−0.610311	−	0.792162i	$-0.708955\pi$
$941$	−37.4434	−1.22062	−0.610311	+	0.792162i	$0.708955\pi$
$942$	0	0
$943$	−19.7726	−0.643884
$944$	0	0
$945$	18.3476	0.596846
$946$	0	0
$947$	3.00193	0.0975496	0.0487748	−	0.998810i	$-0.484468\pi$
$947$	3.00193	0.0975496	0.0487748	+	0.998810i	$0.484468\pi$
$948$	0	0
$949$	2.85988	0.0928358
$950$	0	0
$951$	−2.47550	−0.0802736
$952$	0	0
$953$	−32.7305	−1.06024	−0.530122	−	0.847921i	$-0.677854\pi$
$953$	−32.7305	−1.06024	−0.530122	+	0.847921i	$0.677854\pi$
$954$	0	0
$955$	32.5068	1.05190
$956$	0	0
$957$	8.66730	0.280174
$958$	0	0
$959$	−13.2468	−0.427761
$960$	0	0
$961$	−25.1006	−0.809696
$962$	0	0
$963$	0.515729	0.0166191
$964$	0	0
$965$	0.711421	0.0229015
$966$	0	0
$967$	53.6983	1.72682	0.863411	−	0.504501i	$-0.168324\pi$
$967$	53.6983	1.72682	0.863411	+	0.504501i	$0.168324\pi$
$968$	0	0
$969$	−12.1235	−0.389462
$970$	0	0
$971$	−49.1041	−1.57583	−0.787913	−	0.615786i	$-0.788839\pi$
$971$	−49.1041	−1.57583	−0.787913	+	0.615786i	$0.788839\pi$
$972$	0	0
$973$	13.8200	0.443048
$974$	0	0
$975$	15.3494	0.491575
$976$	0	0
$977$	13.7258	0.439127	0.219563	−	0.975598i	$-0.429537\pi$
$977$	13.7258	0.439127	0.219563	+	0.975598i	$0.429537\pi$
$978$	0	0
$979$	−2.16817	−0.0692951
$980$	0	0
$981$	−4.17845	−0.133408
$982$	0	0
$983$	−15.6448	−0.498991	−0.249496	−	0.968376i	$-0.580265\pi$
$983$	−15.6448	−0.498991	−0.249496	+	0.968376i	$0.580265\pi$
$984$	0	0
$985$	−64.6460	−2.05979
$986$	0	0
$987$	−18.1656	−0.578218
$988$	0	0
$989$	−3.41826	−0.108694
$990$	0	0
$991$	−11.6886	−0.371302	−0.185651	−	0.982616i	$-0.559439\pi$
$991$	−11.6886	−0.371302	−0.185651	+	0.982616i	$0.559439\pi$
$992$	0	0
$993$	−4.79262	−0.152089
$994$	0	0
$995$	−0.561878	−0.0178127
$996$	0	0
$997$	33.5000	1.06096	0.530478	−	0.847699i	$-0.322012\pi$
$997$	33.5000	1.06096	0.530478	+	0.847699i	$0.322012\pi$
$998$	0	0
$999$	24.2288	0.766565

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.s.1.3	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.s.1.3	✓	10	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79551 q^{3} +3.68086 q^{5} +1.00000 q^{7} +0.223871 q^{9} +O(q^{10})\)	\(q-1.79551 q^{3} +3.68086 q^{5} +1.00000 q^{7} +0.223871 q^{9} +1.00000 q^{11} -1.00000 q^{13} -6.60904 q^{15} -5.37766 q^{17} -1.25558 q^{19} -1.79551 q^{21} +1.60669 q^{23} +8.54876 q^{25} +4.98458 q^{27} -4.82719 q^{29} -2.42888 q^{31} -1.79551 q^{33} +3.68086 q^{35} +4.86075 q^{37} +1.79551 q^{39} -12.3064 q^{41} -2.12752 q^{43} +0.824038 q^{45} +10.1172 q^{47} +1.00000 q^{49} +9.65567 q^{51} -2.26908 q^{53} +3.68086 q^{55} +2.25441 q^{57} -4.19132 q^{59} -6.96593 q^{61} +0.223871 q^{63} -3.68086 q^{65} -0.165980 q^{67} -2.88483 q^{69} -4.36876 q^{71} -2.85988 q^{73} -15.3494 q^{75} +1.00000 q^{77} +11.2207 q^{79} -9.62149 q^{81} -7.18700 q^{83} -19.7945 q^{85} +8.66730 q^{87} -2.16817 q^{89} -1.00000 q^{91} +4.36108 q^{93} -4.62162 q^{95} -4.17424 q^{97} +0.223871 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9}+O(q^{10}) \) 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9	\( 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9} + 10 q^{11} - 10 q^{13} - 5 q^{15} - 11 q^{17} + 2 q^{19} - 3 q^{21} - 8 q^{23} + 2 q^{25} - 15 q^{27} - 8 q^{29} - 23 q^{31} - 3 q^{33} - 4 q^{35} + 10 q^{37} + 3 q^{39} - 18 q^{41} + 12 q^{43} - 10 q^{45} - 36 q^{47} + 10 q^{49} + 9 q^{51} - 21 q^{53} - 4 q^{55} - 30 q^{57} - 13 q^{59} - 2 q^{61} + 9 q^{63} + 4 q^{65} - 4 q^{67} - 26 q^{69} - 24 q^{71} - 23 q^{73} - 28 q^{75} + 10 q^{77} + 14 q^{79} + 30 q^{81} - 9 q^{83} - 17 q^{85} + 7 q^{87} - 18 q^{89} - 10 q^{91} + q^{93} - 4 q^{95} - 9 q^{97} + 9 q^{99}+O(q^{100}) \) 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9 + 10 * q^11 - 10 * q^13 - 5 * q^15 - 11 * q^17 + 2 * q^19 - 3 * q^21 - 8 * q^23 + 2 * q^25 - 15 * q^27 - 8 * q^29 - 23 * q^31 - 3 * q^33 - 4 * q^35 + 10 * q^37 + 3 * q^39 - 18 * q^41 + 12 * q^43 - 10 * q^45 - 36 * q^47 + 10 * q^49 + 9 * q^51 - 21 * q^53 - 4 * q^55 - 30 * q^57 - 13 * q^59 - 2 * q^61 + 9 * q^63 + 4 * q^65 - 4 * q^67 - 26 * q^69 - 24 * q^71 - 23 * q^73 - 28 * q^75 + 10 * q^77 + 14 * q^79 + 30 * q^81 - 9 * q^83 - 17 * q^85 + 7 * q^87 - 18 * q^89 - 10 * q^91 + q^93 - 4 * q^95 - 9 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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