LMFDB - Embedded newform 1764.3.d.g.685.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(685,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1764.685");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.d (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3288334336.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 12x^{6} + 30x^{4} + 12x^{2} + 1 $ x^8 + 12x^6 + 30x^4 + 12*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 7^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.2
Root			$0.595859i$ of defining polynomial
Character	$\chi$	$=$	1764.685
Dual form			1764.3.d.g.685.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.15626i q^{5} +O(q^{10})$	$q-6.15626i q^{5} +1.95169 q^{11} -0.317025i q^{13} +18.4688i q^{17} +6.94269i q^{19} -36.8859 q^{23} -12.8995 q^{25} -40.7893 q^{29} +30.2751i q^{31} -13.6985 q^{37} -30.1625i q^{41} +41.7990 q^{43} -48.6313i q^{47} -3.90338 q^{53} -12.0151i q^{55} +24.0063i q^{59} +79.8251i q^{61} -1.95169 q^{65} -91.3970 q^{67} +75.7236 q^{71} +132.133i q^{73} -25.1960 q^{79} +84.9501i q^{83} +113.698 q^{85} +128.044i q^{89} +42.7410 q^{95} +4.05760i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 24 q^{25} + 128 q^{37} + 176 q^{43} - 256 q^{67} + 432 q^{79} + 672 q^{85}+O(q^{100}) $ 8 * q - 24 * q^25 + 128 * q^37 + 176 * q^43 - 256 * q^67 + 432 * q^79 + 672 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 6.15626i	− 1.23125i	−0.788039	−	0.615626i	$-0.788903\pi$
$5$	− 6.15626i	− 1.23125i	0.788039	−	0.615626i	$-0.211097\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.95169	0.177426	0.0887132	−	0.996057i	$-0.471725\pi$
$11$	1.95169	0.177426	0.0887132	+	0.996057i	$0.471725\pi$
$12$	0	0
$13$	− 0.317025i	− 0.0243866i	−0.999926	−	0.0121933i	$-0.996119\pi$
$13$	− 0.317025i	− 0.0243866i	0.999926	−	0.0121933i	$-0.00388134\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	18.4688i	1.08640i	0.839604	+	0.543199i	$0.182787\pi$
$17$	18.4688i	1.08640i	−0.839604	+	0.543199i	$0.817213\pi$
$18$	0	0
$19$	6.94269i	0.365405i	0.983168	+	0.182702i	$0.0584845\pi$
$19$	6.94269i	0.365405i	−0.983168	+	0.182702i	$0.941515\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−36.8859	−1.60374	−0.801868	−	0.597501i	$-0.796160\pi$
$23$	−36.8859	−1.60374	−0.801868	+	0.597501i	$0.796160\pi$
$24$	0	0
$25$	−12.8995	−0.515980
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−40.7893	−1.40653	−0.703264	−	0.710929i	$-0.748275\pi$
$29$	−40.7893	−1.40653	−0.703264	+	0.710929i	$0.748275\pi$
$30$	0	0
$31$	30.2751i	0.976617i	0.872671	+	0.488308i	$0.162386\pi$
$31$	30.2751i	0.976617i	−0.872671	+	0.488308i	$0.837614\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−13.6985	−0.370229	−0.185115	−	0.982717i	$-0.559266\pi$
$37$	−13.6985	−0.370229	−0.185115	+	0.982717i	$0.559266\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 30.1625i	− 0.735672i	−0.929891	−	0.367836i	$-0.880099\pi$
$41$	− 30.1625i	− 0.735672i	0.929891	−	0.367836i	$-0.119901\pi$
$42$	0	0
$43$	41.7990	0.972070	0.486035	−	0.873939i	$-0.338443\pi$
$43$	41.7990	0.972070	0.486035	+	0.873939i	$0.338443\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 48.6313i	− 1.03471i	−0.855771	−	0.517354i	$-0.826917\pi$
$47$	− 48.6313i	− 1.03471i	0.855771	−	0.517354i	$-0.173083\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−3.90338	−0.0736487	−0.0368243	−	0.999322i	$-0.511724\pi$
$53$	−3.90338	−0.0736487	−0.0368243	+	0.999322i	$0.511724\pi$
$54$	0	0
$55$	− 12.0151i	− 0.218456i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	24.0063i	0.406886i	0.979087	+	0.203443i	$0.0652132\pi$
$59$	24.0063i	0.406886i	−0.979087	+	0.203443i	$0.934787\pi$
$60$	0	0
$61$	79.8251i	1.30861i	0.756232	+	0.654304i	$0.227038\pi$
$61$	79.8251i	1.30861i	−0.756232	+	0.654304i	$0.772962\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.95169	−0.0300260
$66$	0	0
$67$	−91.3970	−1.36413	−0.682067	−	0.731290i	$-0.738919\pi$
$67$	−91.3970	−1.36413	−0.682067	+	0.731290i	$0.738919\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	75.7236	1.06653	0.533265	−	0.845948i	$-0.320965\pi$
$71$	75.7236	1.06653	0.533265	+	0.845948i	$0.320965\pi$
$72$	0	0
$73$	132.133i	1.81004i	0.425373	+	0.905018i	$0.360143\pi$
$73$	132.133i	1.81004i	−0.425373	+	0.905018i	$0.639857\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−25.1960	−0.318936	−0.159468	−	0.987203i	$-0.550978\pi$
$79$	−25.1960	−0.318936	−0.159468	+	0.987203i	$0.550978\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	84.9501i	1.02350i	0.859136	+	0.511748i	$0.171002\pi$
$83$	84.9501i	1.02350i	−0.859136	+	0.511748i	$0.828998\pi$
$84$	0	0
$85$	113.698	1.33763
$86$	0	0
$87$	0	0
$88$	0	0
$89$	128.044i	1.43870i	0.694650	+	0.719348i	$0.255559\pi$
$89$	128.044i	1.43870i	−0.694650	+	0.719348i	$0.744441\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	42.7410	0.449905
$96$	0	0
$97$	4.05760i	0.0418310i	0.999781	+	0.0209155i	$0.00665809\pi$
$97$	4.05760i	0.0418310i	−0.999781	+	0.0209155i	$0.993342\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 152.050i	− 1.50545i	−0.658336	−	0.752724i	$-0.728740\pi$
$101$	− 152.050i	− 1.50545i	0.658336	−	0.752724i	$-0.271260\pi$
$102$	0	0
$103$	94.6934i	0.919353i	0.888086	+	0.459677i	$0.152035\pi$
$103$	94.6934i	0.919353i	−0.888086	+	0.459677i	$0.847965\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	188.333	1.76012	0.880061	−	0.474860i	$-0.157501\pi$
$107$	188.333	1.76012	0.880061	+	0.474860i	$0.157501\pi$
$108$	0	0
$109$	−118.894	−1.09077	−0.545387	−	0.838184i	$-0.683617\pi$
$109$	−118.894	−1.09077	−0.545387	+	0.838184i	$0.683617\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−19.3207	−0.170980	−0.0854900	−	0.996339i	$-0.527246\pi$
$113$	−19.3207	−0.170980	−0.0854900	+	0.996339i	$0.527246\pi$
$114$	0	0
$115$	227.079i	1.97460i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−117.191	−0.968520
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 74.4938i	− 0.595951i
$126$	0	0
$127$	174.995	1.37791	0.688956	−	0.724803i	$-0.258069\pi$
$127$	174.995	1.37791	0.688956	+	0.724803i	$0.258069\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	35.7001i	0.272520i	0.990673	+	0.136260i	$0.0435082\pi$
$131$	35.7001i	0.272520i	−0.990673	+	0.136260i	$0.956492\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−200.043	−1.46017	−0.730085	−	0.683357i	$-0.760519\pi$
$137$	−200.043	−1.46017	−0.730085	+	0.683357i	$0.760519\pi$
$138$	0	0
$139$	− 92.8231i	− 0.667792i	−0.942610	−	0.333896i	$-0.891637\pi$
$139$	− 92.8231i	− 0.667792i	0.942610	−	0.333896i	$-0.108363\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 0.618735i	− 0.00432682i
$144$	0	0
$145$	251.110i	1.73179i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	85.4820	0.573705	0.286852	−	0.957975i	$-0.407391\pi$
$149$	85.4820	0.573705	0.286852	+	0.957975i	$0.407391\pi$
$150$	0	0
$151$	−244.593	−1.61982	−0.809910	−	0.586554i	$-0.800484\pi$
$151$	−244.593	−1.61982	−0.809910	+	0.586554i	$0.800484\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	186.381	1.20246
$156$	0	0
$157$	226.699i	1.44394i	0.691924	+	0.721970i	$0.256763\pi$
$157$	226.699i	1.44394i	−0.691924	+	0.721970i	$0.743237\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	45.0051	0.276105	0.138052	−	0.990425i	$-0.455916\pi$
$163$	45.0051	0.276105	0.138052	+	0.990425i	$0.455916\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	86.8063i	0.519798i	0.965636	+	0.259899i	$0.0836893\pi$
$167$	86.8063i	0.519798i	−0.965636	+	0.259899i	$0.916311\pi$
$168$	0	0
$169$	168.899	0.999405
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3.68132i	− 0.0212793i	−0.999943	−	0.0106396i	$-0.996613\pi$
$173$	− 3.68132i	− 0.0212793i	0.999943	−	0.0106396i	$-0.00338677\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	297.039	1.65944	0.829719	−	0.558182i	$-0.188501\pi$
$179$	297.039	1.65944	0.829719	+	0.558182i	$0.188501\pi$
$180$	0	0
$181$	119.483i	0.660130i	0.943958	+	0.330065i	$0.107071\pi$
$181$	119.483i	0.660130i	−0.943958	+	0.330065i	$0.892929\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	84.3314i	0.455845i
$186$	0	0
$187$	36.0453i	0.192756i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−336.073	−1.75954	−0.879772	−	0.475395i	$-0.842305\pi$
$191$	−336.073	−1.75954	−0.879772	+	0.475395i	$0.842305\pi$
$192$	0	0
$193$	−49.5980	−0.256984	−0.128492	−	0.991711i	$-0.541014\pi$
$193$	−49.5980	−0.256984	−0.128492	+	0.991711i	$0.541014\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−217.608	−1.10461	−0.552306	−	0.833642i	$-0.686252\pi$
$197$	−217.608	−1.10461	−0.552306	+	0.833642i	$0.686252\pi$
$198$	0	0
$199$	290.832i	1.46147i	0.682664	+	0.730733i	$0.260821\pi$
$199$	290.832i	1.46147i	−0.682664	+	0.730733i	$0.739179\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−185.688	−0.905797
$206$	0	0
$207$	0	0
$208$	0	0
$209$	13.5500i	0.0648325i
$210$	0	0
$211$	237.990	1.12791	0.563957	−	0.825804i	$-0.309278\pi$
$211$	237.990	1.12791	0.563957	+	0.825804i	$0.309278\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 257.325i	− 1.19686i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.85507	0.0264935
$222$	0	0
$223$	− 179.242i	− 0.803776i	−0.915689	−	0.401888i	$-0.868354\pi$
$223$	− 179.242i	− 0.803776i	0.915689	−	0.401888i	$-0.131646\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	86.8063i	0.382407i	0.981550	+	0.191203i	$0.0612390\pi$
$227$	86.8063i	0.382407i	−0.981550	+	0.191203i	$0.938761\pi$
$228$	0	0
$229$	316.415i	1.38173i	0.722986	+	0.690863i	$0.242769\pi$
$229$	316.415i	1.38173i	−0.722986	+	0.690863i	$0.757231\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−149.299	−0.640770	−0.320385	−	0.947287i	$-0.603812\pi$
$233$	−149.299	−0.640770	−0.320385	+	0.947287i	$0.603812\pi$
$234$	0	0
$235$	−299.387	−1.27399
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−196.140	−0.820669	−0.410334	−	0.911935i	$-0.634588\pi$
$239$	−196.140	−0.820669	−0.410334	+	0.911935i	$0.634588\pi$
$240$	0	0
$241$	127.662i	0.529720i	0.964287	+	0.264860i	$0.0853257\pi$
$241$	127.662i	0.529720i	−0.964287	+	0.264860i	$0.914674\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.20101	0.00891097
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 329.344i	− 1.31213i	−0.754705	−	0.656064i	$-0.772220\pi$
$251$	− 329.344i	− 1.31213i	0.754705	−	0.656064i	$-0.227780\pi$
$252$	0	0
$253$	−71.9899	−0.284545
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 92.9626i	− 0.361722i	−0.983509	−	0.180861i	$-0.942112\pi$
$257$	− 92.9626i	− 0.361722i	0.983509	−	0.180861i	$-0.0578884\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−494.935	−1.88188	−0.940940	−	0.338572i	$-0.890056\pi$
$263$	−494.935	−1.88188	−0.940940	+	0.338572i	$0.890056\pi$
$264$	0	0
$265$	24.0302i	0.0906800i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	387.225i	1.43950i	0.694233	+	0.719750i	$0.255744\pi$
$269$	387.225i	1.43950i	−0.694233	+	0.719750i	$0.744256\pi$
$270$	0	0
$271$	− 222.641i	− 0.821553i	−0.911736	−	0.410777i	$-0.865258\pi$
$271$	− 222.641i	− 0.821553i	0.911736	−	0.410777i	$-0.134742\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−25.1758	−0.0915484
$276$	0	0
$277$	139.397	0.503238	0.251619	−	0.967826i	$-0.419037\pi$
$277$	139.397	0.503238	0.251619	+	0.967826i	$0.419037\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−91.1409	−0.324345	−0.162172	−	0.986762i	$-0.551850\pi$
$281$	−91.1409	−0.324345	−0.162172	+	0.986762i	$0.551850\pi$
$282$	0	0
$283$	0.0318627i	0 0.000112589i	1.00000		5.62944e-5i	$1.79191e-5\pi$
$283$	0.0318627i	0 0.000112589i	−1.00000		5.62944e-5i	$0.999982\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−52.0955	−0.180261
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 32.6375i	− 0.111391i	−0.998448	−	0.0556954i	$-0.982262\pi$
$293$	− 32.6375i	− 0.111391i	0.998448	−	0.0556954i	$-0.0177376\pi$
$294$	0	0
$295$	147.789	0.500979
$296$	0	0
$297$	0	0
$298$	0	0
$299$	11.6938i	0.0391096i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	491.424	1.61122
$306$	0	0
$307$	93.2341i	0.303694i	0.988404	+	0.151847i	$0.0485221\pi$
$307$	93.2341i	0.303694i	−0.988404	+	0.151847i	$0.951478\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 181.594i	− 0.583904i	−0.956433	−	0.291952i	$-0.905695\pi$
$311$	− 181.594i	− 0.583904i	0.956433	−	0.291952i	$-0.0943047\pi$
$312$	0	0
$313$	297.362i	0.950038i	0.879976	+	0.475019i	$0.157559\pi$
$313$	297.362i	0.950038i	−0.879976	+	0.475019i	$0.842441\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−147.740	−0.466057	−0.233028	−	0.972470i	$-0.574863\pi$
$317$	−147.740	−0.466057	−0.233028	+	0.972470i	$0.574863\pi$
$318$	0	0
$319$	−79.6081	−0.249555
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−128.223	−0.396975
$324$	0	0
$325$	4.08947i	0.0125830i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	321.186	0.970350	0.485175	−	0.874417i	$-0.338756\pi$
$331$	321.186	0.970350	0.485175	+	0.874417i	$0.338756\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	562.663i	1.67959i
$336$	0	0
$337$	−44.8944	−0.133218	−0.0666090	−	0.997779i	$-0.521218\pi$
$337$	−44.8944	−0.133218	−0.0666090	+	0.997779i	$0.521218\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	59.0876i	0.173277i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	145.396	0.419008	0.209504	−	0.977808i	$-0.432815\pi$
$347$	145.396	0.419008	0.209504	+	0.977808i	$0.432815\pi$
$348$	0	0
$349$	187.199i	0.536388i	0.963365	+	0.268194i	$0.0864268\pi$
$349$	187.199i	0.536388i	−0.963365	+	0.268194i	$0.913573\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 462.338i	− 1.30974i	−0.755742	−	0.654870i	$-0.772724\pi$
$353$	− 462.338i	− 1.30974i	0.755742	−	0.654870i	$-0.227276\pi$
$354$	0	0
$355$	− 466.174i	− 1.31317i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	355.394	0.989955	0.494977	−	0.868906i	$-0.335176\pi$
$359$	355.394	0.989955	0.494977	+	0.868906i	$0.335176\pi$
$360$	0	0
$361$	312.799	0.866479
$362$	0	0
$363$	0	0
$364$	0	0
$365$	813.442	2.22861
$366$	0	0
$367$	− 329.824i	− 0.898704i	−0.893355	−	0.449352i	$-0.851655\pi$
$367$	− 329.824i	− 0.898704i	0.893355	−	0.449352i	$-0.148345\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−406.593	−1.09006	−0.545031	−	0.838416i	$-0.683482\pi$
$373$	−406.593	−1.09006	−0.545031	+	0.838416i	$0.683482\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	12.9312i	0.0343004i
$378$	0	0
$379$	−435.196	−1.14827	−0.574137	−	0.818759i	$-0.694662\pi$
$379$	−435.196	−1.14827	−0.574137	+	0.818759i	$0.694662\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	470.350i	1.22807i	0.789279	+	0.614035i	$0.210454\pi$
$383$	470.350i	1.22807i	−0.789279	+	0.614035i	$0.789546\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−235.174	−0.604559	−0.302280	−	0.953219i	$-0.597748\pi$
$389$	−235.174	−0.604559	−0.302280	+	0.953219i	$0.597748\pi$
$390$	0	0
$391$	− 681.238i	− 1.74230i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	155.113i	0.392691i
$396$	0	0
$397$	62.5145i	0.157467i	0.996896	+	0.0787336i	$0.0250877\pi$
$397$	62.5145i	0.157467i	−0.996896	+	0.0787336i	$0.974912\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	510.352	1.27270	0.636349	−	0.771401i	$-0.280444\pi$
$401$	510.352	1.27270	0.636349	+	0.771401i	$0.280444\pi$
$402$	0	0
$403$	9.59798	0.0238163
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−26.7352	−0.0656884
$408$	0	0
$409$	− 611.083i	− 1.49409i	−0.664773	−	0.747045i	$-0.731472\pi$
$409$	− 611.083i	− 1.49409i	0.664773	−	0.747045i	$-0.268528\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	522.975	1.26018
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 639.632i	− 1.52657i	−0.646063	−	0.763284i	$-0.723586\pi$
$419$	− 639.632i	− 1.52657i	0.646063	−	0.763284i	$-0.276414\pi$
$420$	0	0
$421$	−369.186	−0.876926	−0.438463	−	0.898749i	$-0.644477\pi$
$421$	−369.186	−0.876926	−0.438463	+	0.898749i	$0.644477\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 238.238i	− 0.560560i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−460.589	−1.06865	−0.534326	−	0.845279i	$-0.679434\pi$
$431$	−460.589	−1.06865	−0.534326	+	0.845279i	$0.679434\pi$
$432$	0	0
$433$	801.072i	1.85005i	0.379905	+	0.925025i	$0.375957\pi$
$433$	801.072i	1.85005i	−0.379905	+	0.925025i	$0.624043\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 256.088i	− 0.586013i
$438$	0	0
$439$	− 729.837i	− 1.66250i	−0.555900	−	0.831249i	$-0.687626\pi$
$439$	− 729.837i	− 1.66250i	0.555900	−	0.831249i	$-0.312374\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	382.914	0.864365	0.432182	−	0.901786i	$-0.357744\pi$
$443$	382.914	0.864365	0.432182	+	0.901786i	$0.357744\pi$
$444$	0	0
$445$	788.271	1.77140
$446$	0	0
$447$	0	0
$448$	0	0
$449$	271.863	0.605486	0.302743	−	0.953072i	$-0.402098\pi$
$449$	271.863	0.605486	0.302743	+	0.953072i	$0.402098\pi$
$450$	0	0
$451$	− 58.8679i	− 0.130528i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−197.809	−0.432843	−0.216421	−	0.976300i	$-0.569438\pi$
$457$	−197.809	−0.432843	−0.216421	+	0.976300i	$0.569438\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	486.344i	1.05498i	0.849562	+	0.527488i	$0.176866\pi$
$461$	486.344i	1.05498i	−0.849562	+	0.527488i	$0.823134\pi$
$462$	0	0
$463$	73.1859	0.158069	0.0790344	−	0.996872i	$-0.474816\pi$
$463$	73.1859	0.158069	0.0790344	+	0.996872i	$0.474816\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	781.195i	1.67279i	0.548124	+	0.836397i	$0.315342\pi$
$467$	781.195i	1.67279i	−0.548124	+	0.836397i	$0.684658\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	81.5786	0.172471
$474$	0	0
$475$	− 89.5573i	− 0.188542i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 708.557i	− 1.47924i	−0.673023	−	0.739621i	$-0.735005\pi$
$479$	− 708.557i	− 1.47924i	0.673023	−	0.739621i	$-0.264995\pi$
$480$	0	0
$481$	4.34277i	0.00902862i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24.9797	0.0515044
$486$	0	0
$487$	−46.2010	−0.0948686	−0.0474343	−	0.998874i	$-0.515104\pi$
$487$	−46.2010	−0.0948686	−0.0474343	+	0.998874i	$0.515104\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−627.453	−1.27791	−0.638955	−	0.769245i	$-0.720633\pi$
$491$	−627.453	−1.27791	−0.638955	+	0.769245i	$0.720633\pi$
$492$	0	0
$493$	− 753.329i	− 1.52805i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	411.568	0.824785	0.412392	−	0.911006i	$-0.364693\pi$
$499$	411.568	0.824785	0.412392	+	0.911006i	$0.364693\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 137.913i	− 0.274180i	−0.990559	−	0.137090i	$-0.956225\pi$
$503$	− 137.913i	− 0.274180i	0.990559	−	0.137090i	$-0.0437750\pi$
$504$	0	0
$505$	−936.060	−1.85358
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 265.338i	− 0.521292i	−0.965434	−	0.260646i	$-0.916064\pi$
$509$	− 265.338i	− 0.521292i	0.965434	−	0.260646i	$-0.0839356\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	582.957	1.13195
$516$	0	0
$517$	− 94.9132i	− 0.183585i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	363.838i	0.698345i	0.937058	+	0.349173i	$0.113537\pi$
$521$	363.838i	0.698345i	−0.937058	+	0.349173i	$0.886463\pi$
$522$	0	0
$523$	− 784.493i	− 1.49999i	−0.661446	−	0.749993i	$-0.730057\pi$
$523$	− 784.493i	− 1.49999i	0.661446	−	0.749993i	$-0.269943\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−559.144	−1.06099
$528$	0	0
$529$	831.573	1.57197
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9.56229	−0.0179405
$534$	0	0
$535$	− 1159.43i	− 2.16715i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	402.181	0.743403	0.371701	−	0.928352i	$-0.378775\pi$
$541$	402.181	0.743403	0.371701	+	0.928352i	$0.378775\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	731.945i	1.34302i
$546$	0	0
$547$	−411.206	−0.751748	−0.375874	−	0.926671i	$-0.622657\pi$
$547$	−411.206	−0.751748	−0.375874	+	0.926671i	$0.622657\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 283.188i	− 0.513952i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	206.291	0.370360	0.185180	−	0.982705i	$-0.440713\pi$
$557$	206.291	0.370360	0.185180	+	0.982705i	$0.440713\pi$
$558$	0	0
$559$	− 13.2513i	− 0.0237054i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	972.070i	1.72659i	0.504700	+	0.863295i	$0.331603\pi$
$563$	972.070i	1.72659i	−0.504700	+	0.863295i	$0.668397\pi$
$564$	0	0
$565$	118.943i	0.210519i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−568.903	−0.999829	−0.499914	−	0.866075i	$-0.666635\pi$
$569$	−568.903	−0.999829	−0.499914	+	0.866075i	$0.666635\pi$
$570$	0	0
$571$	43.3869	0.0759840	0.0379920	−	0.999278i	$-0.487904\pi$
$571$	43.3869	0.0759840	0.0379920	+	0.999278i	$0.487904\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	475.810	0.827496
$576$	0	0
$577$	− 248.699i	− 0.431021i	−0.976502	−	0.215511i	$-0.930858\pi$
$577$	− 248.699i	− 0.431021i	0.976502	−	0.215511i	$-0.0691416\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−7.61818	−0.0130672
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 804.582i	− 1.37067i	−0.728229	−	0.685334i	$-0.759656\pi$
$587$	− 804.582i	− 1.37067i	0.728229	−	0.685334i	$-0.240344\pi$
$588$	0	0
$589$	−210.191	−0.356861
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 581.132i	− 0.979986i	−0.871726	−	0.489993i	$-0.836999\pi$
$593$	− 581.132i	− 0.979986i	0.871726	−	0.489993i	$-0.163001\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−716.250	−1.19574	−0.597872	−	0.801592i	$-0.703987\pi$
$599$	−716.250	−1.19574	−0.597872	+	0.801592i	$0.703987\pi$
$600$	0	0
$601$	135.622i	0.225660i	0.993614	+	0.112830i	$0.0359915\pi$
$601$	135.622i	0.225660i	−0.993614	+	0.112830i	$0.964008\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	721.457i	1.19249i
$606$	0	0
$607$	− 999.681i	− 1.64692i	−0.567374	−	0.823461i	$-0.692040\pi$
$607$	− 999.681i	− 1.64692i	0.567374	−	0.823461i	$-0.307960\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.4174	−0.0252330
$612$	0	0
$613$	814.492	1.32870	0.664349	−	0.747422i	$-0.268709\pi$
$613$	814.492	1.32870	0.664349	+	0.747422i	$0.268709\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−417.848	−0.677225	−0.338612	−	0.940926i	$-0.609958\pi$
$617$	−417.848	−0.677225	−0.338612	+	0.940926i	$0.609958\pi$
$618$	0	0
$619$	− 287.977i	− 0.465229i	−0.972569	−	0.232615i	$-0.925272\pi$
$619$	− 287.977i	− 0.465229i	0.972569	−	0.232615i	$-0.0747281\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−781.090	−1.24974
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 252.994i	− 0.402216i
$630$	0	0
$631$	−474.412	−0.751842	−0.375921	−	0.926652i	$-0.622674\pi$
$631$	−474.412	−0.751842	−0.375921	+	0.926652i	$0.622674\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1077.31i	− 1.69656i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	71.6240	0.111738	0.0558690	−	0.998438i	$-0.482207\pi$
$641$	71.6240	0.111738	0.0558690	+	0.998438i	$0.482207\pi$
$642$	0	0
$643$	− 256.561i	− 0.399006i	−0.979897	−	0.199503i	$-0.936067\pi$
$643$	− 256.561i	− 0.399006i	0.979897	−	0.199503i	$-0.0639328\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 709.795i	− 1.09706i	−0.836132	−	0.548528i	$-0.815189\pi$
$647$	− 709.795i	− 1.09706i	0.836132	−	0.548528i	$-0.184811\pi$
$648$	0	0
$649$	46.8528i	0.0721923i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−417.063	−0.638688	−0.319344	−	0.947639i	$-0.603463\pi$
$653$	−417.063	−0.638688	−0.319344	+	0.947639i	$0.603463\pi$
$654$	0	0
$655$	219.779	0.335540
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−755.676	−1.14670	−0.573351	−	0.819310i	$-0.694357\pi$
$659$	−755.676	−1.14670	−0.573351	+	0.819310i	$0.694357\pi$
$660$	0	0
$661$	− 862.193i	− 1.30438i	−0.758057	−	0.652188i	$-0.773851\pi$
$661$	− 862.193i	− 1.30438i	0.758057	−	0.652188i	$-0.226149\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1504.55	2.25570
$668$	0	0
$669$	0	0
$670$	0	0
$671$	155.794i	0.232181i
$672$	0	0
$673$	−529.266	−0.786428	−0.393214	−	0.919447i	$-0.628637\pi$
$673$	−529.266	−0.786428	−0.393214	+	0.919447i	$0.628637\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	450.644i	0.665649i	0.942989	+	0.332824i	$0.108002\pi$
$677$	450.644i	0.665649i	−0.942989	+	0.332824i	$0.891998\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	121.779	0.178301	0.0891504	−	0.996018i	$-0.471585\pi$
$683$	121.779	0.178301	0.0891504	+	0.996018i	$0.471585\pi$
$684$	0	0
$685$	1231.52i	1.79784i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.23747i	0.00179604i
$690$	0	0
$691$	− 1077.48i	− 1.55930i	−0.626214	−	0.779651i	$-0.715396\pi$
$691$	− 1077.48i	− 1.55930i	0.626214	−	0.779651i	$-0.284604\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−571.443	−0.822220
$696$	0	0
$697$	557.065	0.799233
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−770.701	−1.09943	−0.549716	−	0.835352i	$-0.685264\pi$
$701$	−770.701	−1.09943	−0.549716	+	0.835352i	$0.685264\pi$
$702$	0	0
$703$	− 95.1044i	− 0.135284i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−248.503	−0.350497	−0.175249	−	0.984524i	$-0.556073\pi$
$709$	−248.503	−0.350497	−0.175249	+	0.984524i	$0.556073\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1116.73i	− 1.56624i
$714$	0	0
$715$	−3.80909	−0.00532740
$716$	0	0
$717$	0	0
$718$	0	0
$719$	301.688i	0.419593i	0.977745	+	0.209797i	$0.0672802\pi$
$719$	301.688i	0.419593i	−0.977745	+	0.209797i	$0.932720\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	526.162	0.725740
$726$	0	0
$727$	− 1082.52i	− 1.48902i	−0.667610	−	0.744511i	$-0.732683\pi$
$727$	− 1082.52i	− 1.48902i	0.667610	−	0.744511i	$-0.267317\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	771.976i	1.05605i
$732$	0	0
$733$	939.671i	1.28195i	0.767561	+	0.640976i	$0.221470\pi$
$733$	939.671i	1.28195i	−0.767561	+	0.640976i	$0.778530\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−178.378	−0.242033
$738$	0	0
$739$	−562.352	−0.760963	−0.380481	−	0.924789i	$-0.624242\pi$
$739$	−562.352	−0.760963	−0.380481	+	0.924789i	$0.624242\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−52.3033	−0.0703948	−0.0351974	−	0.999380i	$-0.511206\pi$
$743$	−52.3033	−0.0703948	−0.0351974	+	0.999380i	$0.511206\pi$
$744$	0	0
$745$	− 526.249i	− 0.706375i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−319.417	−0.425322	−0.212661	−	0.977126i	$-0.568213\pi$
$751$	−319.417	−0.425322	−0.212661	+	0.977126i	$0.568213\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1505.78i	1.99441i
$756$	0	0
$757$	−784.060	−1.03575	−0.517873	−	0.855457i	$-0.673276\pi$
$757$	−784.060	−1.03575	−0.517873	+	0.855457i	$0.673276\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	916.045i	1.20374i	0.798595	+	0.601869i	$0.205577\pi$
$761$	916.045i	1.20374i	−0.798595	+	0.601869i	$0.794423\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.61060	0.00992256
$768$	0	0
$769$	− 452.414i	− 0.588315i	−0.955757	−	0.294157i	$-0.904961\pi$
$769$	− 452.414i	− 0.588315i	0.955757	−	0.294157i	$-0.0950390\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 828.620i	− 1.07195i	−0.844233	−	0.535977i	$-0.819944\pi$
$773$	− 828.620i	− 1.07195i	0.844233	−	0.535977i	$-0.180056\pi$
$774$	0	0
$775$	− 390.534i	− 0.503915i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	209.409	0.268818
$780$	0	0
$781$	147.789	0.189230
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1395.61	1.77785
$786$	0	0
$787$	− 246.448i	− 0.313149i	−0.987666	−	0.156574i	$-0.949955\pi$
$787$	− 246.448i	− 0.313149i	0.987666	−	0.156574i	$-0.0500451\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	25.3066	0.0319124
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 426.638i	− 0.535305i	−0.963516	−	0.267652i	$-0.913752\pi$
$797$	− 426.638i	− 0.535305i	0.963516	−	0.267652i	$-0.0862479\pi$
$798$	0	0
$799$	898.161	1.12411
$800$	0	0
$801$	0	0
$802$	0	0
$803$	257.882i	0.321148i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1106.58	−1.36783	−0.683917	−	0.729560i	$-0.739725\pi$
$809$	−1106.58	−1.36783	−0.683917	+	0.729560i	$0.739725\pi$
$810$	0	0
$811$	1002.47i	1.23609i	0.786143	+	0.618045i	$0.212075\pi$
$811$	1002.47i	1.23609i	−0.786143	+	0.618045i	$0.787925\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 277.063i	− 0.339954i
$816$	0	0
$817$	290.198i	0.355199i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1530.67	1.86440	0.932201	−	0.361942i	$-0.117886\pi$
$821$	1530.67	1.86440	0.932201	+	0.361942i	$0.117886\pi$
$822$	0	0
$823$	504.553	0.613065	0.306533	−	0.951860i	$-0.400831\pi$
$823$	504.553	0.613065	0.306533	+	0.951860i	$0.400831\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1105.22	1.33641	0.668207	−	0.743975i	$-0.267062\pi$
$827$	1105.22	1.33641	0.668207	+	0.743975i	$0.267062\pi$
$828$	0	0
$829$	828.683i	0.999618i	0.866136	+	0.499809i	$0.166596\pi$
$829$	828.683i	0.999618i	−0.866136	+	0.499809i	$0.833404\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	534.402	0.640002
$836$	0	0
$837$	0	0
$838$	0	0
$839$	594.032i	0.708024i	0.935241	+	0.354012i	$0.115183\pi$
$839$	594.032i	0.708024i	−0.935241	+	0.354012i	$0.884817\pi$
$840$	0	0
$841$	822.769	0.978322
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1039.79i	− 1.23052i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	505.282	0.593750
$852$	0	0
$853$	481.329i	0.564278i	0.959374	+	0.282139i	$0.0910439\pi$
$853$	481.329i	0.564278i	−0.959374	+	0.282139i	$0.908956\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	384.194i	0.448301i	0.974555	+	0.224150i	$0.0719607\pi$
$857$	384.194i	0.448301i	−0.974555	+	0.224150i	$0.928039\pi$
$858$	0	0
$859$	− 402.960i	− 0.469103i	−0.972104	−	0.234552i	$-0.924638\pi$
$859$	− 402.960i	− 0.469103i	0.972104	−	0.234552i	$-0.0753622\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	595.442	0.689967	0.344984	−	0.938609i	$-0.387884\pi$
$863$	595.442	0.689967	0.344984	+	0.938609i	$0.387884\pi$
$864$	0	0
$865$	−22.6631	−0.0262002
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−49.1747	−0.0565877
$870$	0	0
$871$	28.9752i	0.0332665i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−999.658	−1.13986	−0.569930	−	0.821693i	$-0.693030\pi$
$877$	−999.658	−1.13986	−0.569930	+	0.821693i	$0.693030\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 39.4436i	− 0.0447714i	−0.999749	−	0.0223857i	$-0.992874\pi$
$881$	− 39.4436i	− 0.0447714i	0.999749	−	0.0223857i	$-0.00712618\pi$
$882$	0	0
$883$	−1168.75	−1.32362	−0.661808	−	0.749673i	$-0.730211\pi$
$883$	−1168.75	−1.32362	−0.661808	+	0.749673i	$0.730211\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 424.194i	− 0.478235i	−0.970991	−	0.239117i	$-0.923142\pi$
$887$	− 424.194i	− 0.478235i	0.970991	−	0.239117i	$-0.0768580\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	337.632	0.378088
$894$	0	0
$895$	− 1828.65i	− 2.04318i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1234.90i	− 1.37364i
$900$	0	0
$901$	− 72.0906i	− 0.0800118i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	735.571	0.812786
$906$	0	0
$907$	−1258.16	−1.38717	−0.693584	−	0.720376i	$-0.743969\pi$
$907$	−1258.16	−1.38717	−0.693584	+	0.720376i	$0.743969\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	467.611	0.513294	0.256647	−	0.966505i	$-0.417382\pi$
$911$	467.611	0.513294	0.256647	+	0.966505i	$0.417382\pi$
$912$	0	0
$913$	165.796i	0.181595i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	222.603	0.242223	0.121112	−	0.992639i	$-0.461354\pi$
$919$	222.603	0.242223	0.121112	+	0.992639i	$0.461354\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 24.0063i	− 0.0260090i
$924$	0	0
$925$	176.704	0.191031
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1573.50i	1.69375i	0.531790	+	0.846876i	$0.321519\pi$
$929$	1573.50i	1.69375i	−0.531790	+	0.846876i	$0.678481\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	221.904	0.237331
$936$	0	0
$937$	1165.80i	1.24418i	0.782945	+	0.622090i	$0.213716\pi$
$937$	1165.80i	1.24418i	−0.782945	+	0.622090i	$0.786284\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1417.76i	− 1.50666i	−0.657644	−	0.753328i	$-0.728447\pi$
$941$	− 1417.76i	− 1.50666i	0.657644	−	0.753328i	$-0.271553\pi$
$942$	0	0
$943$	1112.57i	1.17982i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1260.57	1.33111	0.665557	−	0.746347i	$-0.268194\pi$
$947$	1260.57	1.33111	0.665557	+	0.746347i	$0.268194\pi$
$948$	0	0
$949$	41.8894	0.0441406
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1479.73	−1.55271	−0.776355	−	0.630295i	$-0.782934\pi$
$953$	−1479.73	−1.55271	−0.776355	+	0.630295i	$0.782934\pi$
$954$	0	0
$955$	2068.95i	2.16644i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	44.4172	0.0462197
$962$	0	0
$963$	0	0
$964$	0	0
$965$	305.338i	0.316412i
$966$	0	0
$967$	1725.53	1.78441	0.892206	−	0.451628i	$-0.149156\pi$
$967$	1725.53	1.78441	0.892206	+	0.451628i	$0.149156\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	902.526i	0.929481i	0.885447	+	0.464740i	$0.153852\pi$
$971$	902.526i	0.929481i	−0.885447	+	0.464740i	$0.846148\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1198.90	−1.22712	−0.613560	−	0.789648i	$-0.710263\pi$
$977$	−1198.90	−1.22712	−0.613560	+	0.789648i	$0.710263\pi$
$978$	0	0
$979$	249.902i	0.255262i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1082.26i	1.10098i	0.834842	+	0.550490i	$0.185559\pi$
$983$	1082.26i	1.10098i	−0.834842	+	0.550490i	$0.814441\pi$
$984$	0	0
$985$	1339.65i	1.36005i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1541.80	−1.55894
$990$	0	0
$991$	−1648.75	−1.66373	−0.831864	−	0.554980i	$-0.812726\pi$
$991$	−1648.75	−1.66373	−0.831864	+	0.554980i	$0.812726\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1790.43	1.79943
$996$	0	0
$997$	393.196i	0.394379i	0.980365	+	0.197189i	$0.0631814\pi$
$997$	393.196i	0.394379i	−0.980365	+	0.197189i	$0.936819\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.d.g.685.2	yes	8
3.2	odd	2	inner	1764.3.d.g.685.7	yes	8
7.2	even	3		1764.3.z.n.325.1		16
7.3	odd	6		1764.3.z.n.901.1		16
7.4	even	3		1764.3.z.n.901.7		16
7.5	odd	6		1764.3.z.n.325.7		16
7.6	odd	2	inner	1764.3.d.g.685.8	yes	8
21.2	odd	6		1764.3.z.n.325.8		16
21.5	even	6		1764.3.z.n.325.2		16
21.11	odd	6		1764.3.z.n.901.2		16
21.17	even	6		1764.3.z.n.901.8		16
21.20	even	2	inner	1764.3.d.g.685.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.3.d.g.685.1	✓	8	21.20	even	2	inner
1764.3.d.g.685.2	yes	8	1.1	even	1	trivial
1764.3.d.g.685.7	yes	8	3.2	odd	2	inner
1764.3.d.g.685.8	yes	8	7.6	odd	2	inner
1764.3.z.n.325.1		16	7.2	even	3
1764.3.z.n.325.2		16	21.5	even	6
1764.3.z.n.325.7		16	7.5	odd	6
1764.3.z.n.325.8		16	21.2	odd	6
1764.3.z.n.901.1		16	7.3	odd	6
1764.3.z.n.901.2		16	21.11	odd	6
1764.3.z.n.901.7		16	7.4	even	3
1764.3.z.n.901.8		16	21.17	even	6

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.15626i q^{5} +O(q^{10})\)	\(q-6.15626i q^{5} +1.95169 q^{11} -0.317025i q^{13} +18.4688i q^{17} +6.94269i q^{19} -36.8859 q^{23} -12.8995 q^{25} -40.7893 q^{29} +30.2751i q^{31} -13.6985 q^{37} -30.1625i q^{41} +41.7990 q^{43} -48.6313i q^{47} -3.90338 q^{53} -12.0151i q^{55} +24.0063i q^{59} +79.8251i q^{61} -1.95169 q^{65} -91.3970 q^{67} +75.7236 q^{71} +132.133i q^{73} -25.1960 q^{79} +84.9501i q^{83} +113.698 q^{85} +128.044i q^{89} +42.7410 q^{95} +4.05760i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 24 q^{25} + 128 q^{37} + 176 q^{43} - 256 q^{67} + 432 q^{79} + 672 q^{85}+O(q^{100}) \) 8 * q - 24 * q^25 + 128 * q^37 + 176 * q^43 - 256 * q^67 + 432 * q^79 + 672 * q^85

Introduction

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