LMFDB - Embedded newform 8664.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8664,2,Mod(1,8664)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8664, 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("8664.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8664 = 2^{3} \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8664.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:	$69.1823883112$
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} - 18x^{7} + 45x^{6} + 117x^{5} - 207x^{4} - 315x^{3} + 288x^{2} + 288x + 3 $ x^9 - 3x^8 - 18x^7 + 45x^6 + 117x^5 - 207x^4 - 315x^3 + 288x^2 + 288x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 456)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.755953$ of defining polynomial
Character	$\chi$	$=$	8664.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.05194 q^{5} -3.66740 q^{7} +1.00000 q^{9} +0.911449 q^{11} +2.23162 q^{13} +4.05194 q^{15} +8.01071 q^{17} +3.66740 q^{21} -6.79369 q^{23} +11.4182 q^{25} -1.00000 q^{27} -5.34666 q^{29} -0.820942 q^{31} -0.911449 q^{33} +14.8601 q^{35} -5.54834 q^{37} -2.23162 q^{39} +4.43635 q^{41} -7.46512 q^{43} -4.05194 q^{45} -8.70147 q^{47} +6.44984 q^{49} -8.01071 q^{51} -10.2713 q^{53} -3.69314 q^{55} -7.22752 q^{59} +8.09414 q^{61} -3.66740 q^{63} -9.04239 q^{65} -6.27143 q^{67} +6.79369 q^{69} +13.2757 q^{71} +1.49470 q^{73} -11.4182 q^{75} -3.34265 q^{77} -6.40819 q^{79} +1.00000 q^{81} +0.166370 q^{83} -32.4590 q^{85} +5.34666 q^{87} -9.65938 q^{89} -8.18424 q^{91} +0.820942 q^{93} +16.9755 q^{97} +0.911449 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} - 3 q^{5} - 6 q^{7} + 9 q^{9} - 9 q^{11} + 9 q^{13} + 3 q^{15} + 3 q^{17} + 6 q^{21} - 30 q^{23} + 30 q^{25} - 9 q^{27} + 3 q^{29} - 9 q^{31} + 9 q^{33} - 6 q^{35} + 15 q^{37} - 9 q^{39} + 3 q^{41}+ \cdots - 9 q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 - 3 * q^5 - 6 * q^7 + 9 * q^9 - 9 * q^11 + 9 * q^13 + 3 * q^15 + 3 * q^17 + 6 * q^21 - 30 * q^23 + 30 * q^25 - 9 * q^27 + 3 * q^29 - 9 * q^31 + 9 * q^33 - 6 * q^35 + 15 * q^37 - 9 * q^39 + 3 * q^41 - 12 * q^43 - 3 * q^45 - 18 * q^47 + 9 * q^49 - 3 * q^51 + 15 * q^53 - 12 * q^55 + 9 * q^59 - 6 * q^61 - 6 * q^63 - 3 * q^65 + 18 * q^67 + 30 * q^69 + 18 * q^71 + 33 * q^73 - 30 * q^75 - 33 * q^77 + 24 * q^79 + 9 * q^81 - 36 * q^83 - 3 * q^85 - 3 * q^87 + 9 * q^89 + 3 * q^91 + 9 * q^93 + 9 * q^97 - 9 * q^99

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
$5$	−4.05194	−1.81208	−0.906042	−	0.423188i	$-0.860911\pi$
$5$	−4.05194	−1.81208	−0.906042	+	0.423188i	$0.860911\pi$
$6$	0	0
$7$	−3.66740	−1.38615	−0.693074	−	0.720867i	$-0.743744\pi$
$7$	−3.66740	−1.38615	−0.693074	+	0.720867i	$0.743744\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.911449	0.274812	0.137406	−	0.990515i	$-0.456123\pi$
$11$	0.911449	0.274812	0.137406	+	0.990515i	$0.456123\pi$
$12$	0	0
$13$	2.23162	0.618939	0.309470	−	0.950909i	$-0.399848\pi$
$13$	2.23162	0.618939	0.309470	+	0.950909i	$0.399848\pi$
$14$	0	0
$15$	4.05194	1.04621
$16$	0	0
$17$	8.01071	1.94288	0.971442	−	0.237278i	$-0.0762552\pi$
$17$	8.01071	1.94288	0.971442	+	0.237278i	$0.0762552\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	3.66740	0.800293
$22$	0	0
$23$	−6.79369	−1.41658	−0.708292	−	0.705920i	$-0.750534\pi$
$23$	−6.79369	−1.41658	−0.708292	+	0.705920i	$0.750534\pi$
$24$	0	0
$25$	11.4182	2.28365
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.34666	−0.992851	−0.496425	−	0.868079i	$-0.665354\pi$
$29$	−5.34666	−0.992851	−0.496425	+	0.868079i	$0.665354\pi$
$30$	0	0
$31$	−0.820942	−0.147446	−0.0737228	−	0.997279i	$-0.523488\pi$
$31$	−0.820942	−0.147446	−0.0737228	+	0.997279i	$0.523488\pi$
$32$	0	0
$33$	−0.911449	−0.158663
$34$	0	0
$35$	14.8601	2.51182
$36$	0	0
$37$	−5.54834	−0.912142	−0.456071	−	0.889943i	$-0.650744\pi$
$37$	−5.54834	−0.912142	−0.456071	+	0.889943i	$0.650744\pi$
$38$	0	0
$39$	−2.23162	−0.357345
$40$	0	0
$41$	4.43635	0.692841	0.346421	−	0.938079i	$-0.387397\pi$
$41$	4.43635	0.692841	0.346421	+	0.938079i	$0.387397\pi$
$42$	0	0
$43$	−7.46512	−1.13842	−0.569210	−	0.822192i	$-0.692751\pi$
$43$	−7.46512	−1.13842	−0.569210	+	0.822192i	$0.692751\pi$
$44$	0	0
$45$	−4.05194	−0.604028
$46$	0	0
$47$	−8.70147	−1.26924	−0.634620	−	0.772824i	$-0.718843\pi$
$47$	−8.70147	−1.26924	−0.634620	+	0.772824i	$0.718843\pi$
$48$	0	0
$49$	6.44984	0.921406
$50$	0	0
$51$	−8.01071	−1.12172
$52$	0	0
$53$	−10.2713	−1.41087	−0.705437	−	0.708773i	$-0.749249\pi$
$53$	−10.2713	−1.41087	−0.705437	+	0.708773i	$0.749249\pi$
$54$	0	0
$55$	−3.69314	−0.497983
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.22752	−0.940943	−0.470472	−	0.882415i	$-0.655916\pi$
$59$	−7.22752	−0.940943	−0.470472	+	0.882415i	$0.655916\pi$
$60$	0	0
$61$	8.09414	1.03635	0.518174	−	0.855275i	$-0.326612\pi$
$61$	8.09414	1.03635	0.518174	+	0.855275i	$0.326612\pi$
$62$	0	0
$63$	−3.66740	−0.462049
$64$	0	0
$65$	−9.04239	−1.12157
$66$	0	0
$67$	−6.27143	−0.766177	−0.383088	−	0.923712i	$-0.625140\pi$
$67$	−6.27143	−0.766177	−0.383088	+	0.923712i	$0.625140\pi$
$68$	0	0
$69$	6.79369	0.817865
$70$	0	0
$71$	13.2757	1.57554	0.787769	−	0.615971i	$-0.211236\pi$
$71$	13.2757	1.57554	0.787769	+	0.615971i	$0.211236\pi$
$72$	0	0
$73$	1.49470	0.174942	0.0874708	−	0.996167i	$-0.472122\pi$
$73$	1.49470	0.174942	0.0874708	+	0.996167i	$0.472122\pi$
$74$	0	0
$75$	−11.4182	−1.31846
$76$	0	0
$77$	−3.34265	−0.380930
$78$	0	0
$79$	−6.40819	−0.720978	−0.360489	−	0.932764i	$-0.617390\pi$
$79$	−6.40819	−0.720978	−0.360489	+	0.932764i	$0.617390\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.166370	0.0182615	0.00913075	−	0.999958i	$-0.497094\pi$
$83$	0.166370	0.0182615	0.00913075	+	0.999958i	$0.497094\pi$
$84$	0	0
$85$	−32.4590	−3.52067
$86$	0	0
$87$	5.34666	0.573223
$88$	0	0
$89$	−9.65938	−1.02389	−0.511946	−	0.859018i	$-0.671075\pi$
$89$	−9.65938	−1.02389	−0.511946	+	0.859018i	$0.671075\pi$
$90$	0	0
$91$	−8.18424	−0.857941
$92$	0	0
$93$	0.820942	0.0851278
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.9755	1.72360	0.861800	−	0.507248i	$-0.169337\pi$
$97$	16.9755	1.72360	0.861800	+	0.507248i	$0.169337\pi$
$98$	0	0
$99$	0.911449	0.0916041
$100$	0	0
$101$	−0.499841	−0.0497360	−0.0248680	−	0.999691i	$-0.507917\pi$
$101$	−0.499841	−0.0497360	−0.0248680	+	0.999691i	$0.507917\pi$
$102$	0	0
$103$	−8.12979	−0.801052	−0.400526	−	0.916285i	$-0.631173\pi$
$103$	−8.12979	−0.801052	−0.400526	+	0.916285i	$0.631173\pi$
$104$	0	0
$105$	−14.8601	−1.45020
$106$	0	0
$107$	3.62051	0.350008	0.175004	−	0.984568i	$-0.444006\pi$
$107$	3.62051	0.350008	0.175004	+	0.984568i	$0.444006\pi$
$108$	0	0
$109$	−4.76292	−0.456205	−0.228102	−	0.973637i	$-0.573252\pi$
$109$	−4.76292	−0.456205	−0.228102	+	0.973637i	$0.573252\pi$
$110$	0	0
$111$	5.54834	0.526625
$112$	0	0
$113$	−10.3764	−0.976128	−0.488064	−	0.872808i	$-0.662297\pi$
$113$	−10.3764	−0.976128	−0.488064	+	0.872808i	$0.662297\pi$
$114$	0	0
$115$	27.5277	2.56697
$116$	0	0
$117$	2.23162	0.206313
$118$	0	0
$119$	−29.3785	−2.69312
$120$	0	0
$121$	−10.1693	−0.924478
$122$	0	0
$123$	−4.43635	−0.400012
$124$	0	0
$125$	−26.0063	−2.32608
$126$	0	0
$127$	−1.05738	−0.0938271	−0.0469135	−	0.998899i	$-0.514939\pi$
$127$	−1.05738	−0.0938271	−0.0469135	+	0.998899i	$0.514939\pi$
$128$	0	0
$129$	7.46512	0.657267
$130$	0	0
$131$	−7.64460	−0.667912	−0.333956	−	0.942589i	$-0.608384\pi$
$131$	−7.64460	−0.667912	−0.333956	+	0.942589i	$0.608384\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.05194	0.348736
$136$	0	0
$137$	2.82346	0.241225	0.120612	−	0.992700i	$-0.461514\pi$
$137$	2.82346	0.241225	0.120612	+	0.992700i	$0.461514\pi$
$138$	0	0
$139$	−6.67368	−0.566054	−0.283027	−	0.959112i	$-0.591339\pi$
$139$	−6.67368	−0.566054	−0.283027	+	0.959112i	$0.591339\pi$
$140$	0	0
$141$	8.70147	0.732796
$142$	0	0
$143$	2.03401	0.170092
$144$	0	0
$145$	21.6644	1.79913
$146$	0	0
$147$	−6.44984	−0.531974
$148$	0	0
$149$	−5.38013	−0.440757	−0.220379	−	0.975414i	$-0.570729\pi$
$149$	−5.38013	−0.440757	−0.220379	+	0.975414i	$0.570729\pi$
$150$	0	0
$151$	8.36023	0.680346	0.340173	−	0.940363i	$-0.389514\pi$
$151$	8.36023	0.680346	0.340173	+	0.940363i	$0.389514\pi$
$152$	0	0
$153$	8.01071	0.647628
$154$	0	0
$155$	3.32641	0.267184
$156$	0	0
$157$	9.36507	0.747414	0.373707	−	0.927547i	$-0.378087\pi$
$157$	9.36507	0.747414	0.373707	+	0.927547i	$0.378087\pi$
$158$	0	0
$159$	10.2713	0.814568
$160$	0	0
$161$	24.9152	1.96359
$162$	0	0
$163$	0.503153	0.0394100	0.0197050	−	0.999806i	$-0.493727\pi$
$163$	0.503153	0.0394100	0.0197050	+	0.999806i	$0.493727\pi$
$164$	0	0
$165$	3.69314	0.287511
$166$	0	0
$167$	−8.69456	−0.672806	−0.336403	−	0.941718i	$-0.609210\pi$
$167$	−8.69456	−0.672806	−0.336403	+	0.941718i	$0.609210\pi$
$168$	0	0
$169$	−8.01988	−0.616914
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.00363	0.228362	0.114181	−	0.993460i	$-0.463576\pi$
$173$	3.00363	0.228362	0.114181	+	0.993460i	$0.463576\pi$
$174$	0	0
$175$	−41.8753	−3.16547
$176$	0	0
$177$	7.22752	0.543254
$178$	0	0
$179$	16.8305	1.25797	0.628986	−	0.777417i	$-0.283470\pi$
$179$	16.8305	1.25797	0.628986	+	0.777417i	$0.283470\pi$
$180$	0	0
$181$	4.92802	0.366297	0.183149	−	0.983085i	$-0.441371\pi$
$181$	4.92802	0.366297	0.183149	+	0.983085i	$0.441371\pi$
$182$	0	0
$183$	−8.09414	−0.598336
$184$	0	0
$185$	22.4816	1.65288
$186$	0	0
$187$	7.30136	0.533928
$188$	0	0
$189$	3.66740	0.266764
$190$	0	0
$191$	4.08333	0.295459	0.147730	−	0.989028i	$-0.452803\pi$
$191$	4.08333	0.295459	0.147730	+	0.989028i	$0.452803\pi$
$192$	0	0
$193$	1.66621	0.119936	0.0599682	−	0.998200i	$-0.480900\pi$
$193$	1.66621	0.119936	0.0599682	+	0.998200i	$0.480900\pi$
$194$	0	0
$195$	9.04239	0.647539
$196$	0	0
$197$	−18.4354	−1.31347	−0.656734	−	0.754122i	$-0.728063\pi$
$197$	−18.4354	−1.31347	−0.656734	+	0.754122i	$0.728063\pi$
$198$	0	0
$199$	0.270909	0.0192043	0.00960213	−	0.999954i	$-0.496943\pi$
$199$	0.270909	0.0192043	0.00960213	+	0.999954i	$0.496943\pi$
$200$	0	0
$201$	6.27143	0.442352
$202$	0	0
$203$	19.6084	1.37624
$204$	0	0
$205$	−17.9758	−1.25549
$206$	0	0
$207$	−6.79369	−0.472194
$208$	0	0
$209$	0	0
$210$	0	0
$211$	22.5647	1.55341	0.776707	−	0.629862i	$-0.216888\pi$
$211$	22.5647	1.55341	0.776707	+	0.629862i	$0.216888\pi$
$212$	0	0
$213$	−13.2757	−0.909637
$214$	0	0
$215$	30.2482	2.06291
$216$	0	0
$217$	3.01073	0.204381
$218$	0	0
$219$	−1.49470	−0.101003
$220$	0	0
$221$	17.8769	1.20253
$222$	0	0
$223$	−2.99897	−0.200826	−0.100413	−	0.994946i	$-0.532016\pi$
$223$	−2.99897	−0.200826	−0.100413	+	0.994946i	$0.532016\pi$
$224$	0	0
$225$	11.4182	0.761216
$226$	0	0
$227$	9.03122	0.599423	0.299712	−	0.954030i	$-0.403110\pi$
$227$	9.03122	0.599423	0.299712	+	0.954030i	$0.403110\pi$
$228$	0	0
$229$	−28.4563	−1.88045	−0.940224	−	0.340556i	$-0.889385\pi$
$229$	−28.4563	−1.88045	−0.940224	+	0.340556i	$0.889385\pi$
$230$	0	0
$231$	3.34265	0.219930
$232$	0	0
$233$	−14.0458	−0.920172	−0.460086	−	0.887874i	$-0.652181\pi$
$233$	−14.0458	−0.920172	−0.460086	+	0.887874i	$0.652181\pi$
$234$	0	0
$235$	35.2579	2.29997
$236$	0	0
$237$	6.40819	0.416257
$238$	0	0
$239$	6.80492	0.440174	0.220087	−	0.975480i	$-0.429366\pi$
$239$	6.80492	0.440174	0.220087	+	0.975480i	$0.429366\pi$
$240$	0	0
$241$	6.29139	0.405264	0.202632	−	0.979255i	$-0.435050\pi$
$241$	6.29139	0.405264	0.202632	+	0.979255i	$0.435050\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−26.1344	−1.66966
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.166370	−0.0105433
$250$	0	0
$251$	−14.1772	−0.894860	−0.447430	−	0.894319i	$-0.647661\pi$
$251$	−14.1772	−0.894860	−0.447430	+	0.894319i	$0.647661\pi$
$252$	0	0
$253$	−6.19211	−0.389294
$254$	0	0
$255$	32.4590	2.03266
$256$	0	0
$257$	−22.2464	−1.38769	−0.693846	−	0.720124i	$-0.744085\pi$
$257$	−22.2464	−1.38769	−0.693846	+	0.720124i	$0.744085\pi$
$258$	0	0
$259$	20.3480	1.26436
$260$	0	0
$261$	−5.34666	−0.330950
$262$	0	0
$263$	−2.20425	−0.135920	−0.0679600	−	0.997688i	$-0.521649\pi$
$263$	−2.20425	−0.135920	−0.0679600	+	0.997688i	$0.521649\pi$
$264$	0	0
$265$	41.6188	2.55662
$266$	0	0
$267$	9.65938	0.591145
$268$	0	0
$269$	26.0828	1.59029	0.795147	−	0.606416i	$-0.207394\pi$
$269$	26.0828	1.59029	0.795147	+	0.606416i	$0.207394\pi$
$270$	0	0
$271$	15.2423	0.925904	0.462952	−	0.886383i	$-0.346790\pi$
$271$	15.2423	0.925904	0.462952	+	0.886383i	$0.346790\pi$
$272$	0	0
$273$	8.18424	0.495333
$274$	0	0
$275$	10.4071	0.627574
$276$	0	0
$277$	0.444117	0.0266844	0.0133422	−	0.999911i	$-0.495753\pi$
$277$	0.444117	0.0266844	0.0133422	+	0.999911i	$0.495753\pi$
$278$	0	0
$279$	−0.820942	−0.0491485
$280$	0	0
$281$	−13.7556	−0.820591	−0.410295	−	0.911953i	$-0.634574\pi$
$281$	−13.7556	−0.820591	−0.410295	+	0.911953i	$0.634574\pi$
$282$	0	0
$283$	−2.08267	−0.123802	−0.0619010	−	0.998082i	$-0.519716\pi$
$283$	−2.08267	−0.123802	−0.0619010	+	0.998082i	$0.519716\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−16.2699	−0.960381
$288$	0	0
$289$	47.1715	2.77480
$290$	0	0
$291$	−16.9755	−0.995121
$292$	0	0
$293$	31.2678	1.82668	0.913341	−	0.407196i	$-0.133493\pi$
$293$	31.2678	1.82668	0.913341	+	0.407196i	$0.133493\pi$
$294$	0	0
$295$	29.2855	1.70507
$296$	0	0
$297$	−0.911449	−0.0528876
$298$	0	0
$299$	−15.1609	−0.876779
$300$	0	0
$301$	27.3776	1.57802
$302$	0	0
$303$	0.499841	0.0287151
$304$	0	0
$305$	−32.7970	−1.87795
$306$	0	0
$307$	−26.2126	−1.49603	−0.748016	−	0.663681i	$-0.768993\pi$
$307$	−26.2126	−1.49603	−0.748016	+	0.663681i	$0.768993\pi$
$308$	0	0
$309$	8.12979	0.462488
$310$	0	0
$311$	−18.8902	−1.07117	−0.535583	−	0.844483i	$-0.679908\pi$
$311$	−18.8902	−1.07117	−0.535583	+	0.844483i	$0.679908\pi$
$312$	0	0
$313$	15.3430	0.867236	0.433618	−	0.901097i	$-0.357237\pi$
$313$	15.3430	0.867236	0.433618	+	0.901097i	$0.357237\pi$
$314$	0	0
$315$	14.8601	0.837272
$316$	0	0
$317$	20.0041	1.12354	0.561772	−	0.827292i	$-0.310120\pi$
$317$	20.0041	1.12354	0.561772	+	0.827292i	$0.310120\pi$
$318$	0	0
$319$	−4.87321	−0.272848
$320$	0	0
$321$	−3.62051	−0.202077
$322$	0	0
$323$	0	0
$324$	0	0
$325$	25.4811	1.41344
$326$	0	0
$327$	4.76292	0.263390
$328$	0	0
$329$	31.9118	1.75935
$330$	0	0
$331$	20.3645	1.11933	0.559667	−	0.828718i	$-0.310929\pi$
$331$	20.3645	1.11933	0.559667	+	0.828718i	$0.310929\pi$
$332$	0	0
$333$	−5.54834	−0.304047
$334$	0	0
$335$	25.4115	1.38838
$336$	0	0
$337$	1.23503	0.0672762	0.0336381	−	0.999434i	$-0.489291\pi$
$337$	1.23503	0.0672762	0.0336381	+	0.999434i	$0.489291\pi$
$338$	0	0
$339$	10.3764	0.563568
$340$	0	0
$341$	−0.748247	−0.0405199
$342$	0	0
$343$	2.01766	0.108943
$344$	0	0
$345$	−27.5277	−1.48204
$346$	0	0
$347$	8.57821	0.460502	0.230251	−	0.973131i	$-0.426045\pi$
$347$	8.57821	0.460502	0.230251	+	0.973131i	$0.426045\pi$
$348$	0	0
$349$	4.01831	0.215095	0.107548	−	0.994200i	$-0.465700\pi$
$349$	4.01831	0.215095	0.107548	+	0.994200i	$0.465700\pi$
$350$	0	0
$351$	−2.23162	−0.119115
$352$	0	0
$353$	−14.4922	−0.771342	−0.385671	−	0.922636i	$-0.626030\pi$
$353$	−14.4922	−0.771342	−0.385671	+	0.922636i	$0.626030\pi$
$354$	0	0
$355$	−53.7924	−2.85501
$356$	0	0
$357$	29.3785	1.55488
$358$	0	0
$359$	−15.2966	−0.807326	−0.403663	−	0.914908i	$-0.632263\pi$
$359$	−15.2966	−0.807326	−0.403663	+	0.914908i	$0.632263\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	10.1693	0.533748
$364$	0	0
$365$	−6.05645	−0.317009
$366$	0	0
$367$	35.7480	1.86603	0.933014	−	0.359840i	$-0.117169\pi$
$367$	35.7480	1.86603	0.933014	+	0.359840i	$0.117169\pi$
$368$	0	0
$369$	4.43635	0.230947
$370$	0	0
$371$	37.6690	1.95568
$372$	0	0
$373$	12.8935	0.667600	0.333800	−	0.942644i	$-0.391669\pi$
$373$	12.8935	0.667600	0.333800	+	0.942644i	$0.391669\pi$
$374$	0	0
$375$	26.0063	1.34296
$376$	0	0
$377$	−11.9317	−0.614514
$378$	0	0
$379$	24.3960	1.25314	0.626570	−	0.779366i	$-0.284458\pi$
$379$	24.3960	1.25314	0.626570	+	0.779366i	$0.284458\pi$
$380$	0	0
$381$	1.05738	0.0541711
$382$	0	0
$383$	30.4375	1.55528	0.777641	−	0.628709i	$-0.216416\pi$
$383$	30.4375	1.55528	0.777641	+	0.628709i	$0.216416\pi$
$384$	0	0
$385$	13.5442	0.690278
$386$	0	0
$387$	−7.46512	−0.379474
$388$	0	0
$389$	−13.6339	−0.691264	−0.345632	−	0.938370i	$-0.612335\pi$
$389$	−13.6339	−0.691264	−0.345632	+	0.938370i	$0.612335\pi$
$390$	0	0
$391$	−54.4223	−2.75226
$392$	0	0
$393$	7.64460	0.385619
$394$	0	0
$395$	25.9656	1.30647
$396$	0	0
$397$	3.54632	0.177985	0.0889924	−	0.996032i	$-0.471635\pi$
$397$	3.54632	0.177985	0.0889924	+	0.996032i	$0.471635\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	20.3860	1.01803	0.509015	−	0.860758i	$-0.330010\pi$
$401$	20.3860	1.01803	0.509015	+	0.860758i	$0.330010\pi$
$402$	0	0
$403$	−1.83203	−0.0912599
$404$	0	0
$405$	−4.05194	−0.201343
$406$	0	0
$407$	−5.05703	−0.250668
$408$	0	0
$409$	34.4883	1.70533	0.852667	−	0.522455i	$-0.174984\pi$
$409$	34.4883	1.70533	0.852667	+	0.522455i	$0.174984\pi$
$410$	0	0
$411$	−2.82346	−0.139271
$412$	0	0
$413$	26.5062	1.30429
$414$	0	0
$415$	−0.674123	−0.0330914
$416$	0	0
$417$	6.67368	0.326811
$418$	0	0
$419$	−25.4521	−1.24342	−0.621709	−	0.783248i	$-0.713561\pi$
$419$	−25.4521	−1.24342	−0.621709	+	0.783248i	$0.713561\pi$
$420$	0	0
$421$	20.0702	0.978159	0.489080	−	0.872239i	$-0.337333\pi$
$421$	20.0702	0.978159	0.489080	+	0.872239i	$0.337333\pi$
$422$	0	0
$423$	−8.70147	−0.423080
$424$	0	0
$425$	91.4682	4.43686
$426$	0	0
$427$	−29.6845	−1.43653
$428$	0	0
$429$	−2.03401	−0.0982028
$430$	0	0
$431$	19.6369	0.945875	0.472938	−	0.881096i	$-0.343194\pi$
$431$	19.6369	0.945875	0.472938	+	0.881096i	$0.343194\pi$
$432$	0	0
$433$	−8.28909	−0.398348	−0.199174	−	0.979964i	$-0.563826\pi$
$433$	−8.28909	−0.398348	−0.199174	+	0.979964i	$0.563826\pi$
$434$	0	0
$435$	−21.6644	−1.03873
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−4.47702	−0.213676	−0.106838	−	0.994276i	$-0.534073\pi$
$439$	−4.47702	−0.213676	−0.106838	+	0.994276i	$0.534073\pi$
$440$	0	0
$441$	6.44984	0.307135
$442$	0	0
$443$	−33.6495	−1.59874	−0.799368	−	0.600842i	$-0.794832\pi$
$443$	−33.6495	−1.59874	−0.799368	+	0.600842i	$0.794832\pi$
$444$	0	0
$445$	39.1393	1.85538
$446$	0	0
$447$	5.38013	0.254471
$448$	0	0
$449$	−0.190932	−0.00901066	−0.00450533	−	0.999990i	$-0.501434\pi$
$449$	−0.190932	−0.00901066	−0.00450533	+	0.999990i	$0.501434\pi$
$450$	0	0
$451$	4.04351	0.190401
$452$	0	0
$453$	−8.36023	−0.392798
$454$	0	0
$455$	33.1621	1.55466
$456$	0	0
$457$	41.3845	1.93589	0.967943	−	0.251168i	$-0.0808148\pi$
$457$	41.3845	1.93589	0.967943	+	0.251168i	$0.0808148\pi$
$458$	0	0
$459$	−8.01071	−0.373908
$460$	0	0
$461$	8.21163	0.382454	0.191227	−	0.981546i	$-0.438753\pi$
$461$	8.21163	0.382454	0.191227	+	0.981546i	$0.438753\pi$
$462$	0	0
$463$	10.1174	0.470195	0.235097	−	0.971972i	$-0.424459\pi$
$463$	10.1174	0.470195	0.235097	+	0.971972i	$0.424459\pi$
$464$	0	0
$465$	−3.32641	−0.154259
$466$	0	0
$467$	−19.1242	−0.884963	−0.442482	−	0.896778i	$-0.645902\pi$
$467$	−19.1242	−0.884963	−0.442482	+	0.896778i	$0.645902\pi$
$468$	0	0
$469$	22.9998	1.06203
$470$	0	0
$471$	−9.36507	−0.431520
$472$	0	0
$473$	−6.80408	−0.312852
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−10.2713	−0.470291
$478$	0	0
$479$	3.46439	0.158292	0.0791459	−	0.996863i	$-0.474781\pi$
$479$	3.46439	0.158292	0.0791459	+	0.996863i	$0.474781\pi$
$480$	0	0
$481$	−12.3818	−0.564560
$482$	0	0
$483$	−24.9152	−1.13368
$484$	0	0
$485$	−68.7837	−3.12331
$486$	0	0
$487$	−37.2125	−1.68626	−0.843130	−	0.537709i	$-0.819290\pi$
$487$	−37.2125	−1.68626	−0.843130	+	0.537709i	$0.819290\pi$
$488$	0	0
$489$	−0.503153	−0.0227534
$490$	0	0
$491$	−27.5605	−1.24379	−0.621894	−	0.783101i	$-0.713637\pi$
$491$	−27.5605	−1.24379	−0.621894	+	0.783101i	$0.713637\pi$
$492$	0	0
$493$	−42.8306	−1.92899
$494$	0	0
$495$	−3.69314	−0.165994
$496$	0	0
$497$	−48.6874	−2.18393
$498$	0	0
$499$	−10.9219	−0.488932	−0.244466	−	0.969658i	$-0.578613\pi$
$499$	−10.9219	−0.488932	−0.244466	+	0.969658i	$0.578613\pi$
$500$	0	0
$501$	8.69456	0.388444
$502$	0	0
$503$	6.45328	0.287738	0.143869	−	0.989597i	$-0.454046\pi$
$503$	6.45328	0.287738	0.143869	+	0.989597i	$0.454046\pi$
$504$	0	0
$505$	2.02533	0.0901259
$506$	0	0
$507$	8.01988	0.356175
$508$	0	0
$509$	−29.4907	−1.30715	−0.653575	−	0.756862i	$-0.726732\pi$
$509$	−29.4907	−1.30715	−0.653575	+	0.756862i	$0.726732\pi$
$510$	0	0
$511$	−5.48167	−0.242495
$512$	0	0
$513$	0	0
$514$	0	0
$515$	32.9414	1.45157
$516$	0	0
$517$	−7.93095	−0.348803
$518$	0	0
$519$	−3.00363	−0.131845
$520$	0	0
$521$	−2.63414	−0.115404	−0.0577019	−	0.998334i	$-0.518377\pi$
$521$	−2.63414	−0.115404	−0.0577019	+	0.998334i	$0.518377\pi$
$522$	0	0
$523$	−39.9597	−1.74731	−0.873657	−	0.486543i	$-0.838258\pi$
$523$	−39.9597	−1.74731	−0.873657	+	0.486543i	$0.838258\pi$
$524$	0	0
$525$	41.8753	1.82759
$526$	0	0
$527$	−6.57633	−0.286470
$528$	0	0
$529$	23.1543	1.00671
$530$	0	0
$531$	−7.22752	−0.313648
$532$	0	0
$533$	9.90024	0.428827
$534$	0	0
$535$	−14.6701	−0.634243
$536$	0	0
$537$	−16.8305	−0.726290
$538$	0	0
$539$	5.87870	0.253214
$540$	0	0
$541$	21.9277	0.942745	0.471372	−	0.881934i	$-0.343759\pi$
$541$	21.9277	0.942745	0.471372	+	0.881934i	$0.343759\pi$
$542$	0	0
$543$	−4.92802	−0.211482
$544$	0	0
$545$	19.2991	0.826681
$546$	0	0
$547$	−1.64309	−0.0702535	−0.0351267	−	0.999383i	$-0.511183\pi$
$547$	−1.64309	−0.0702535	−0.0351267	+	0.999383i	$0.511183\pi$
$548$	0	0
$549$	8.09414	0.345449
$550$	0	0
$551$	0	0
$552$	0	0
$553$	23.5014	0.999381
$554$	0	0
$555$	−22.4816	−0.954289
$556$	0	0
$557$	−22.7892	−0.965611	−0.482806	−	0.875728i	$-0.660382\pi$
$557$	−22.7892	−0.965611	−0.482806	+	0.875728i	$0.660382\pi$
$558$	0	0
$559$	−16.6593	−0.704613
$560$	0	0
$561$	−7.30136	−0.308264
$562$	0	0
$563$	−37.5667	−1.58325	−0.791624	−	0.611009i	$-0.790764\pi$
$563$	−37.5667	−1.58325	−0.791624	+	0.611009i	$0.790764\pi$
$564$	0	0
$565$	42.0445	1.76882
$566$	0	0
$567$	−3.66740	−0.154016
$568$	0	0
$569$	4.37557	0.183433	0.0917167	−	0.995785i	$-0.470765\pi$
$569$	4.37557	0.183433	0.0917167	+	0.995785i	$0.470765\pi$
$570$	0	0
$571$	10.3397	0.432703	0.216351	−	0.976316i	$-0.430584\pi$
$571$	10.3397	0.432703	0.216351	+	0.976316i	$0.430584\pi$
$572$	0	0
$573$	−4.08333	−0.170583
$574$	0	0
$575$	−77.5720	−3.23498
$576$	0	0
$577$	37.4709	1.55994	0.779968	−	0.625820i	$-0.215235\pi$
$577$	37.4709	1.55994	0.779968	+	0.625820i	$0.215235\pi$
$578$	0	0
$579$	−1.66621	−0.0692453
$580$	0	0
$581$	−0.610147	−0.0253131
$582$	0	0
$583$	−9.36178	−0.387725
$584$	0	0
$585$	−9.04239	−0.373857
$586$	0	0
$587$	−12.2553	−0.505830	−0.252915	−	0.967489i	$-0.581389\pi$
$587$	−12.2553	−0.505830	−0.252915	+	0.967489i	$0.581389\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	18.4354	0.758331
$592$	0	0
$593$	−11.3024	−0.464132	−0.232066	−	0.972700i	$-0.574549\pi$
$593$	−11.3024	−0.464132	−0.232066	+	0.972700i	$0.574549\pi$
$594$	0	0
$595$	119.040	4.88017
$596$	0	0
$597$	−0.270909	−0.0110876
$598$	0	0
$599$	22.0716	0.901822	0.450911	−	0.892569i	$-0.351099\pi$
$599$	22.0716	0.901822	0.450911	+	0.892569i	$0.351099\pi$
$600$	0	0
$601$	−15.5406	−0.633915	−0.316958	−	0.948440i	$-0.602661\pi$
$601$	−15.5406	−0.633915	−0.316958	+	0.948440i	$0.602661\pi$
$602$	0	0
$603$	−6.27143	−0.255392
$604$	0	0
$605$	41.2053	1.67523
$606$	0	0
$607$	1.74786	0.0709436	0.0354718	−	0.999371i	$-0.488707\pi$
$607$	1.74786	0.0709436	0.0354718	+	0.999371i	$0.488707\pi$
$608$	0	0
$609$	−19.6084	−0.794571
$610$	0	0
$611$	−19.4184	−0.785583
$612$	0	0
$613$	10.1271	0.409029	0.204515	−	0.978864i	$-0.434438\pi$
$613$	10.1271	0.409029	0.204515	+	0.978864i	$0.434438\pi$
$614$	0	0
$615$	17.9758	0.724856
$616$	0	0
$617$	−26.7840	−1.07828	−0.539142	−	0.842215i	$-0.681251\pi$
$617$	−26.7840	−1.07828	−0.539142	+	0.842215i	$0.681251\pi$
$618$	0	0
$619$	28.8062	1.15782	0.578910	−	0.815391i	$-0.303478\pi$
$619$	28.8062	1.15782	0.578910	+	0.815391i	$0.303478\pi$
$620$	0	0
$621$	6.79369	0.272622
$622$	0	0
$623$	35.4248	1.41927
$624$	0	0
$625$	48.2849	1.93140
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−44.4462	−1.77219
$630$	0	0
$631$	41.5922	1.65576	0.827881	−	0.560904i	$-0.189546\pi$
$631$	41.5922	1.65576	0.827881	+	0.560904i	$0.189546\pi$
$632$	0	0
$633$	−22.5647	−0.896864
$634$	0	0
$635$	4.28443	0.170023
$636$	0	0
$637$	14.3936	0.570294
$638$	0	0
$639$	13.2757	0.525179
$640$	0	0
$641$	12.5914	0.497332	0.248666	−	0.968589i	$-0.420008\pi$
$641$	12.5914	0.497332	0.248666	+	0.968589i	$0.420008\pi$
$642$	0	0
$643$	−4.74820	−0.187251	−0.0936253	−	0.995608i	$-0.529846\pi$
$643$	−4.74820	−0.187251	−0.0936253	+	0.995608i	$0.529846\pi$
$644$	0	0
$645$	−30.2482	−1.19102
$646$	0	0
$647$	−7.67738	−0.301829	−0.150914	−	0.988547i	$-0.548222\pi$
$647$	−7.67738	−0.301829	−0.150914	+	0.988547i	$0.548222\pi$
$648$	0	0
$649$	−6.58752	−0.258583
$650$	0	0
$651$	−3.01073	−0.118000
$652$	0	0
$653$	−15.5384	−0.608064	−0.304032	−	0.952662i	$-0.598333\pi$
$653$	−15.5384	−0.608064	−0.304032	+	0.952662i	$0.598333\pi$
$654$	0	0
$655$	30.9755	1.21031
$656$	0	0
$657$	1.49470	0.0583139
$658$	0	0
$659$	−23.1638	−0.902333	−0.451166	−	0.892440i	$-0.648992\pi$
$659$	−23.1638	−0.902333	−0.451166	+	0.892440i	$0.648992\pi$
$660$	0	0
$661$	−1.06738	−0.0415164	−0.0207582	−	0.999785i	$-0.506608\pi$
$661$	−1.06738	−0.0415164	−0.0207582	+	0.999785i	$0.506608\pi$
$662$	0	0
$663$	−17.8769	−0.694279
$664$	0	0
$665$	0	0
$666$	0	0
$667$	36.3236	1.40646
$668$	0	0
$669$	2.99897	0.115947
$670$	0	0
$671$	7.37740	0.284801
$672$	0	0
$673$	−26.0316	−1.00345	−0.501723	−	0.865029i	$-0.667300\pi$
$673$	−26.0316	−1.00345	−0.501723	+	0.865029i	$0.667300\pi$
$674$	0	0
$675$	−11.4182	−0.439488
$676$	0	0
$677$	19.5757	0.752357	0.376178	−	0.926547i	$-0.377238\pi$
$677$	19.5757	0.752357	0.376178	+	0.926547i	$0.377238\pi$
$678$	0	0
$679$	−62.2560	−2.38917
$680$	0	0
$681$	−9.03122	−0.346077
$682$	0	0
$683$	−30.0261	−1.14892	−0.574459	−	0.818533i	$-0.694787\pi$
$683$	−30.0261	−1.14892	−0.574459	+	0.818533i	$0.694787\pi$
$684$	0	0
$685$	−11.4405	−0.437119
$686$	0	0
$687$	28.4563	1.08568
$688$	0	0
$689$	−22.9216	−0.873245
$690$	0	0
$691$	−27.9591	−1.06362	−0.531808	−	0.846865i	$-0.678487\pi$
$691$	−27.9591	−1.06362	−0.531808	+	0.846865i	$0.678487\pi$
$692$	0	0
$693$	−3.34265	−0.126977
$694$	0	0
$695$	27.0414	1.02574
$696$	0	0
$697$	35.5383	1.34611
$698$	0	0
$699$	14.0458	0.531262
$700$	0	0
$701$	−6.23189	−0.235375	−0.117688	−	0.993051i	$-0.537548\pi$
$701$	−6.23189	−0.235375	−0.117688	+	0.993051i	$0.537548\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−35.2579	−1.32789
$706$	0	0
$707$	1.83312	0.0689415
$708$	0	0
$709$	46.1303	1.73246	0.866230	−	0.499646i	$-0.166537\pi$
$709$	46.1303	1.73246	0.866230	+	0.499646i	$0.166537\pi$
$710$	0	0
$711$	−6.40819	−0.240326
$712$	0	0
$713$	5.57723	0.208869
$714$	0	0
$715$	−8.24168	−0.308221
$716$	0	0
$717$	−6.80492	−0.254135
$718$	0	0
$719$	35.3676	1.31899	0.659495	−	0.751709i	$-0.270770\pi$
$719$	35.3676	1.31899	0.659495	+	0.751709i	$0.270770\pi$
$720$	0	0
$721$	29.8152	1.11038
$722$	0	0
$723$	−6.29139	−0.233979
$724$	0	0
$725$	−61.0495	−2.26732
$726$	0	0
$727$	−5.80007	−0.215113	−0.107556	−	0.994199i	$-0.534303\pi$
$727$	−5.80007	−0.215113	−0.107556	+	0.994199i	$0.534303\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−59.8010	−2.21182
$732$	0	0
$733$	−11.0708	−0.408908	−0.204454	−	0.978876i	$-0.565542\pi$
$733$	−11.0708	−0.408908	−0.204454	+	0.978876i	$0.565542\pi$
$734$	0	0
$735$	26.1344	0.963981
$736$	0	0
$737$	−5.71609	−0.210555
$738$	0	0
$739$	29.9545	1.10189	0.550947	−	0.834540i	$-0.314267\pi$
$739$	29.9545	1.10189	0.550947	+	0.834540i	$0.314267\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	52.8735	1.93974	0.969870	−	0.243621i	$-0.0783354\pi$
$743$	52.8735	1.93974	0.969870	+	0.243621i	$0.0783354\pi$
$744$	0	0
$745$	21.8000	0.798689
$746$	0	0
$747$	0.166370	0.00608717
$748$	0	0
$749$	−13.2779	−0.485162
$750$	0	0
$751$	−30.9624	−1.12983	−0.564917	−	0.825148i	$-0.691092\pi$
$751$	−30.9624	−1.12983	−0.564917	+	0.825148i	$0.691092\pi$
$752$	0	0
$753$	14.1772	0.516648
$754$	0	0
$755$	−33.8752	−1.23284
$756$	0	0
$757$	39.9530	1.45212	0.726059	−	0.687633i	$-0.241350\pi$
$757$	39.9530	1.45212	0.726059	+	0.687633i	$0.241350\pi$
$758$	0	0
$759$	6.19211	0.224759
$760$	0	0
$761$	18.7341	0.679111	0.339555	−	0.940586i	$-0.389723\pi$
$761$	18.7341	0.679111	0.339555	+	0.940586i	$0.389723\pi$
$762$	0	0
$763$	17.4675	0.632367
$764$	0	0
$765$	−32.4590	−1.17356
$766$	0	0
$767$	−16.1291	−0.582387
$768$	0	0
$769$	−25.1054	−0.905324	−0.452662	−	0.891682i	$-0.649526\pi$
$769$	−25.1054	−0.905324	−0.452662	+	0.891682i	$0.649526\pi$
$770$	0	0
$771$	22.2464	0.801184
$772$	0	0
$773$	−27.5463	−0.990773	−0.495386	−	0.868673i	$-0.664973\pi$
$773$	−27.5463	−0.990773	−0.495386	+	0.868673i	$0.664973\pi$
$774$	0	0
$775$	−9.37371	−0.336714
$776$	0	0
$777$	−20.3480	−0.729980
$778$	0	0
$779$	0	0
$780$	0	0
$781$	12.1001	0.432977
$782$	0	0
$783$	5.34666	0.191074
$784$	0	0
$785$	−37.9467	−1.35438
$786$	0	0
$787$	43.3134	1.54396	0.771978	−	0.635649i	$-0.219267\pi$
$787$	43.3134	1.54396	0.771978	+	0.635649i	$0.219267\pi$
$788$	0	0
$789$	2.20425	0.0784734
$790$	0	0
$791$	38.0544	1.35306
$792$	0	0
$793$	18.0630	0.641437
$794$	0	0
$795$	−41.6188	−1.47607
$796$	0	0
$797$	−28.2922	−1.00216	−0.501081	−	0.865401i	$-0.667064\pi$
$797$	−28.2922	−1.00216	−0.501081	+	0.865401i	$0.667064\pi$
$798$	0	0
$799$	−69.7050	−2.46599
$800$	0	0
$801$	−9.65938	−0.341297
$802$	0	0
$803$	1.36235	0.0480761
$804$	0	0
$805$	−100.955	−3.55820
$806$	0	0
$807$	−26.0828	−0.918157
$808$	0	0
$809$	48.3708	1.70063	0.850313	−	0.526278i	$-0.176413\pi$
$809$	48.3708	1.70063	0.850313	+	0.526278i	$0.176413\pi$
$810$	0	0
$811$	−29.3861	−1.03189	−0.515943	−	0.856623i	$-0.672558\pi$
$811$	−29.3861	−1.03189	−0.515943	+	0.856623i	$0.672558\pi$
$812$	0	0
$813$	−15.2423	−0.534571
$814$	0	0
$815$	−2.03875	−0.0714142
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−8.18424	−0.285980
$820$	0	0
$821$	12.3501	0.431021	0.215511	−	0.976502i	$-0.430858\pi$
$821$	12.3501	0.431021	0.215511	+	0.976502i	$0.430858\pi$
$822$	0	0
$823$	0.930735	0.0324434	0.0162217	−	0.999868i	$-0.494836\pi$
$823$	0.930735	0.0324434	0.0162217	+	0.999868i	$0.494836\pi$
$824$	0	0
$825$	−10.4071	−0.362330
$826$	0	0
$827$	13.3750	0.465094	0.232547	−	0.972585i	$-0.425294\pi$
$827$	13.3750	0.465094	0.232547	+	0.972585i	$0.425294\pi$
$828$	0	0
$829$	15.1595	0.526511	0.263256	−	0.964726i	$-0.415204\pi$
$829$	15.1595	0.526511	0.263256	+	0.964726i	$0.415204\pi$
$830$	0	0
$831$	−0.444117	−0.0154063
$832$	0	0
$833$	51.6678	1.79018
$834$	0	0
$835$	35.2299	1.21918
$836$	0	0
$837$	0.820942	0.0283759
$838$	0	0
$839$	−8.49605	−0.293316	−0.146658	−	0.989187i	$-0.546852\pi$
$839$	−8.49605	−0.293316	−0.146658	+	0.989187i	$0.546852\pi$
$840$	0	0
$841$	−0.413174	−0.0142474
$842$	0	0
$843$	13.7556	0.473768
$844$	0	0
$845$	32.4961	1.11790
$846$	0	0
$847$	37.2948	1.28146
$848$	0	0
$849$	2.08267	0.0714771
$850$	0	0
$851$	37.6937	1.29212
$852$	0	0
$853$	25.7442	0.881463	0.440732	−	0.897639i	$-0.354719\pi$
$853$	25.7442	0.881463	0.440732	+	0.897639i	$0.354719\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.7988	0.505517	0.252759	−	0.967529i	$-0.418662\pi$
$857$	14.7988	0.505517	0.252759	+	0.967529i	$0.418662\pi$
$858$	0	0
$859$	−47.2516	−1.61220	−0.806101	−	0.591778i	$-0.798426\pi$
$859$	−47.2516	−1.61220	−0.806101	+	0.591778i	$0.798426\pi$
$860$	0	0
$861$	16.2699	0.554476
$862$	0	0
$863$	10.7114	0.364619	0.182310	−	0.983241i	$-0.441643\pi$
$863$	10.7114	0.364619	0.182310	+	0.983241i	$0.441643\pi$
$864$	0	0
$865$	−12.1705	−0.413811
$866$	0	0
$867$	−47.1715	−1.60203
$868$	0	0
$869$	−5.84074	−0.198133
$870$	0	0
$871$	−13.9954	−0.474217
$872$	0	0
$873$	16.9755	0.574534
$874$	0	0
$875$	95.3757	3.22429
$876$	0	0
$877$	50.7594	1.71402	0.857012	−	0.515296i	$-0.172318\pi$
$877$	50.7594	1.71402	0.857012	+	0.515296i	$0.172318\pi$
$878$	0	0
$879$	−31.2678	−1.05464
$880$	0	0
$881$	51.6623	1.74055	0.870274	−	0.492568i	$-0.163942\pi$
$881$	51.6623	1.74055	0.870274	+	0.492568i	$0.163942\pi$
$882$	0	0
$883$	8.63712	0.290662	0.145331	−	0.989383i	$-0.453575\pi$
$883$	8.63712	0.290662	0.145331	+	0.989383i	$0.453575\pi$
$884$	0	0
$885$	−29.2855	−0.984421
$886$	0	0
$887$	−11.8955	−0.399411	−0.199705	−	0.979856i	$-0.563999\pi$
$887$	−11.8955	−0.399411	−0.199705	+	0.979856i	$0.563999\pi$
$888$	0	0
$889$	3.87783	0.130058
$890$	0	0
$891$	0.911449	0.0305347
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−68.1962	−2.27955
$896$	0	0
$897$	15.1609	0.506209
$898$	0	0
$899$	4.38930	0.146391
$900$	0	0
$901$	−82.2805	−2.74116
$902$	0	0
$903$	−27.3776	−0.911070
$904$	0	0
$905$	−19.9681	−0.663761
$906$	0	0
$907$	26.8976	0.893122	0.446561	−	0.894753i	$-0.352649\pi$
$907$	26.8976	0.893122	0.446561	+	0.894753i	$0.352649\pi$
$908$	0	0
$909$	−0.499841	−0.0165787
$910$	0	0
$911$	−26.1368	−0.865950	−0.432975	−	0.901406i	$-0.642536\pi$
$911$	−26.1368	−0.865950	−0.432975	+	0.901406i	$0.642536\pi$
$912$	0	0
$913$	0.151638	0.00501849
$914$	0	0
$915$	32.7970	1.08423
$916$	0	0
$917$	28.0358	0.925824
$918$	0	0
$919$	−53.4724	−1.76389	−0.881945	−	0.471352i	$-0.843766\pi$
$919$	−53.4724	−1.76389	−0.881945	+	0.471352i	$0.843766\pi$
$920$	0	0
$921$	26.2126	0.863735
$922$	0	0
$923$	29.6263	0.975162
$924$	0	0
$925$	−63.3523	−2.08301
$926$	0	0
$927$	−8.12979	−0.267017
$928$	0	0
$929$	−19.4330	−0.637577	−0.318788	−	0.947826i	$-0.603276\pi$
$929$	−19.4330	−0.637577	−0.318788	+	0.947826i	$0.603276\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	18.8902	0.618438
$934$	0	0
$935$	−29.5847	−0.967523
$936$	0	0
$937$	12.4554	0.406899	0.203449	−	0.979085i	$-0.434785\pi$
$937$	12.4554	0.406899	0.203449	+	0.979085i	$0.434785\pi$
$938$	0	0
$939$	−15.3430	−0.500699
$940$	0	0
$941$	−29.2509	−0.953551	−0.476775	−	0.879025i	$-0.658194\pi$
$941$	−29.2509	−0.953551	−0.476775	+	0.879025i	$0.658194\pi$
$942$	0	0
$943$	−30.1392	−0.981467
$944$	0	0
$945$	−14.8601	−0.483399
$946$	0	0
$947$	−16.6902	−0.542359	−0.271180	−	0.962529i	$-0.587414\pi$
$947$	−16.6902	−0.542359	−0.271180	+	0.962529i	$0.587414\pi$
$948$	0	0
$949$	3.33560	0.108278
$950$	0	0
$951$	−20.0041	−0.648678
$952$	0	0
$953$	39.6511	1.28443	0.642213	−	0.766527i	$-0.278017\pi$
$953$	39.6511	1.28443	0.642213	+	0.766527i	$0.278017\pi$
$954$	0	0
$955$	−16.5454	−0.535397
$956$	0	0
$957$	4.87321	0.157529
$958$	0	0
$959$	−10.3548	−0.334373
$960$	0	0
$961$	−30.3261	−0.978260
$962$	0	0
$963$	3.62051	0.116669
$964$	0	0
$965$	−6.75139	−0.217335
$966$	0	0
$967$	38.2767	1.23090	0.615448	−	0.788178i	$-0.288975\pi$
$967$	38.2767	1.23090	0.615448	+	0.788178i	$0.288975\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.6125	0.404754	0.202377	−	0.979308i	$-0.435133\pi$
$971$	12.6125	0.404754	0.202377	+	0.979308i	$0.435133\pi$
$972$	0	0
$973$	24.4751	0.784634
$974$	0	0
$975$	−25.4811	−0.816050
$976$	0	0
$977$	−36.8021	−1.17740	−0.588702	−	0.808350i	$-0.700361\pi$
$977$	−36.8021	−1.17740	−0.588702	+	0.808350i	$0.700361\pi$
$978$	0	0
$979$	−8.80404	−0.281378
$980$	0	0
$981$	−4.76292	−0.152068
$982$	0	0
$983$	7.63548	0.243534	0.121767	−	0.992559i	$-0.461144\pi$
$983$	7.63548	0.243534	0.121767	+	0.992559i	$0.461144\pi$
$984$	0	0
$985$	74.6992	2.38011
$986$	0	0
$987$	−31.9118	−1.01576
$988$	0	0
$989$	50.7158	1.61267
$990$	0	0
$991$	46.5262	1.47795	0.738977	−	0.673731i	$-0.235309\pi$
$991$	46.5262	1.47795	0.738977	+	0.673731i	$0.235309\pi$
$992$	0	0
$993$	−20.3645	−0.646248
$994$	0	0
$995$	−1.09771	−0.0347997
$996$	0	0
$997$	5.14778	0.163032	0.0815160	−	0.996672i	$-0.474024\pi$
$997$	5.14778	0.163032	0.0815160	+	0.996672i	$0.474024\pi$
$998$	0	0
$999$	5.54834	0.175542

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8664.2.a.bn.1.2		9
19.9	even	9		456.2.bg.d.385.1	yes	18
19.17	even	9		456.2.bg.d.289.1	✓	18
19.18	odd	2		8664.2.a.bp.1.2		9
76.47	odd	18		912.2.bo.l.385.1		18
76.55	odd	18		912.2.bo.l.289.1		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.bg.d.289.1	✓	18	19.17	even	9
456.2.bg.d.385.1	yes	18	19.9	even	9
912.2.bo.l.289.1		18	76.55	odd	18
912.2.bo.l.385.1		18	76.47	odd	18
8664.2.a.bn.1.2		9	1.1	even	1	trivial
8664.2.a.bp.1.2		9	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8664 = 2^{3} \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8664.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.05194 q^{5} -3.66740 q^{7} +1.00000 q^{9} +0.911449 q^{11} +2.23162 q^{13} +4.05194 q^{15} +8.01071 q^{17} +3.66740 q^{21} -6.79369 q^{23} +11.4182 q^{25} -1.00000 q^{27} -5.34666 q^{29} -0.820942 q^{31} -0.911449 q^{33} +14.8601 q^{35} -5.54834 q^{37} -2.23162 q^{39} +4.43635 q^{41} -7.46512 q^{43} -4.05194 q^{45} -8.70147 q^{47} +6.44984 q^{49} -8.01071 q^{51} -10.2713 q^{53} -3.69314 q^{55} -7.22752 q^{59} +8.09414 q^{61} -3.66740 q^{63} -9.04239 q^{65} -6.27143 q^{67} +6.79369 q^{69} +13.2757 q^{71} +1.49470 q^{73} -11.4182 q^{75} -3.34265 q^{77} -6.40819 q^{79} +1.00000 q^{81} +0.166370 q^{83} -32.4590 q^{85} +5.34666 q^{87} -9.65938 q^{89} -8.18424 q^{91} +0.820942 q^{93} +16.9755 q^{97} +0.911449 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} - 3 q^{5} - 6 q^{7} + 9 q^{9} - 9 q^{11} + 9 q^{13} + 3 q^{15} + 3 q^{17} + 6 q^{21} - 30 q^{23} + 30 q^{25} - 9 q^{27} + 3 q^{29} - 9 q^{31} + 9 q^{33} - 6 q^{35} + 15 q^{37} - 9 q^{39} + 3 q^{41}+ \cdots - 9 q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 - 3 * q^5 - 6 * q^7 + 9 * q^9 - 9 * q^11 + 9 * q^13 + 3 * q^15 + 3 * q^17 + 6 * q^21 - 30 * q^23 + 30 * q^25 - 9 * q^27 + 3 * q^29 - 9 * q^31 + 9 * q^33 - 6 * q^35 + 15 * q^37 - 9 * q^39 + 3 * q^41 - 12 * q^43 - 3 * q^45 - 18 * q^47 + 9 * q^49 - 3 * q^51 + 15 * q^53 - 12 * q^55 + 9 * q^59 - 6 * q^61 - 6 * q^63 - 3 * q^65 + 18 * q^67 + 30 * q^69 + 18 * q^71 + 33 * q^73 - 30 * q^75 - 33 * q^77 + 24 * q^79 + 9 * q^81 - 36 * q^83 - 3 * q^85 - 3 * q^87 + 9 * q^89 + 3 * q^91 + 9 * q^93 + 9 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

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