LMFDB - Embedded newform 1764.3.d.g.685.4

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			685.4
Root			$0.338766i$ of defining polynomial
Character	$\chi$	$=$	1764.685
Dual form			1764.3.d.g.685.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.25447i q^{5} +O(q^{10})$	$q-4.25447i q^{5} +18.9787 q^{11} -4.46088i q^{13} +12.7634i q^{17} +9.37011i q^{19} +17.0712 q^{23} +6.89949 q^{25} -20.8862 q^{29} -15.6018i q^{31} +45.6985 q^{37} +63.3894i q^{41} +2.20101 q^{43} +50.6260i q^{47} -37.9574 q^{53} -80.7443i q^{55} -67.6439i q^{59} +63.3716i q^{61} -18.9787 q^{65} +27.3970 q^{67} -15.1638 q^{71} -83.8151i q^{73} +133.196 q^{79} -109.761i q^{83} +54.3015 q^{85} -79.9797i q^{89} +39.8648 q^{95} -119.547i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 4.25447i	− 0.850894i	−0.904983	−	0.425447i	$-0.860117\pi$
$5$	− 4.25447i	− 0.850894i	0.904983	−	0.425447i	$-0.139883\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	18.9787	1.72534	0.862668	−	0.505771i	$-0.168792\pi$
$11$	18.9787	1.72534	0.862668	+	0.505771i	$0.168792\pi$
$12$	0	0
$13$	− 4.46088i	− 0.343145i	−0.985171	−	0.171572i	$-0.945115\pi$
$13$	− 4.46088i	− 0.343145i	0.985171	−	0.171572i	$-0.0548848\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	12.7634i	0.750789i	0.926865	+	0.375394i	$0.122493\pi$
$17$	12.7634i	0.750789i	−0.926865	+	0.375394i	$0.877507\pi$
$18$	0	0
$19$	9.37011i	0.493164i	0.969122	+	0.246582i	$0.0793074\pi$
$19$	9.37011i	0.493164i	−0.969122	+	0.246582i	$0.920693\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	17.0712	0.742228	0.371114	−	0.928587i	$-0.378976\pi$
$23$	17.0712	0.742228	0.371114	+	0.928587i	$0.378976\pi$
$24$	0	0
$25$	6.89949	0.275980
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−20.8862	−0.720212	−0.360106	−	0.932911i	$-0.617259\pi$
$29$	−20.8862	−0.720212	−0.360106	+	0.932911i	$0.617259\pi$
$30$	0	0
$31$	− 15.6018i	− 0.503285i	−0.967820	−	0.251642i	$-0.919029\pi$
$31$	− 15.6018i	− 0.503285i	0.967820	−	0.251642i	$-0.0809707\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	45.6985	1.23509	0.617547	−	0.786534i	$-0.288126\pi$
$37$	45.6985	1.23509	0.617547	+	0.786534i	$0.288126\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	63.3894i	1.54608i	0.634355	+	0.773042i	$0.281266\pi$
$41$	63.3894i	1.54608i	−0.634355	+	0.773042i	$0.718734\pi$
$42$	0	0
$43$	2.20101	0.0511863	0.0255931	−	0.999672i	$-0.491853\pi$
$43$	2.20101	0.0511863	0.0255931	+	0.999672i	$0.491853\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	50.6260i	1.07715i	0.842578	+	0.538575i	$0.181037\pi$
$47$	50.6260i	1.07715i	−0.842578	+	0.538575i	$0.818963\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−37.9574	−0.716177	−0.358089	−	0.933688i	$-0.616571\pi$
$53$	−37.9574	−0.716177	−0.358089	+	0.933688i	$0.616571\pi$
$54$	0	0
$55$	− 80.7443i	− 1.46808i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 67.6439i	− 1.14651i	−0.819378	−	0.573253i	$-0.805681\pi$
$59$	− 67.6439i	− 1.14651i	0.819378	−	0.573253i	$-0.194319\pi$
$60$	0	0
$61$	63.3716i	1.03888i	0.854507	+	0.519439i	$0.173859\pi$
$61$	63.3716i	1.03888i	−0.854507	+	0.519439i	$0.826141\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−18.9787	−0.291980
$66$	0	0
$67$	27.3970	0.408910	0.204455	−	0.978876i	$-0.434458\pi$
$67$	27.3970	0.408910	0.204455	+	0.978876i	$0.434458\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−15.1638	−0.213574	−0.106787	−	0.994282i	$-0.534056\pi$
$71$	−15.1638	−0.213574	−0.106787	+	0.994282i	$0.534056\pi$
$72$	0	0
$73$	− 83.8151i	− 1.14815i	−0.818802	−	0.574076i	$-0.805361\pi$
$73$	− 83.8151i	− 1.14815i	0.818802	−	0.574076i	$-0.194639\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	133.196	1.68602	0.843012	−	0.537894i	$-0.180780\pi$
$79$	133.196	1.68602	0.843012	+	0.537894i	$0.180780\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 109.761i	− 1.32242i	−0.750200	−	0.661211i	$-0.770043\pi$
$83$	− 109.761i	− 1.32242i	0.750200	−	0.661211i	$-0.229957\pi$
$84$	0	0
$85$	54.3015	0.638841
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 79.9797i	− 0.898649i	−0.893369	−	0.449324i	$-0.851665\pi$
$89$	− 79.9797i	− 0.898649i	0.893369	−	0.449324i	$-0.148335\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	39.8648	0.419630
$96$	0	0
$97$	− 119.547i	− 1.23245i	−0.787572	−	0.616223i	$-0.788662\pi$
$97$	− 119.547i	− 1.23245i	0.787572	−	0.616223i	$-0.211338\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	147.624i	1.46162i	0.682581	+	0.730810i	$0.260857\pi$
$101$	147.624i	1.46162i	−0.682581	+	0.730810i	$0.739143\pi$
$102$	0	0
$103$	184.264i	1.78897i	0.447099	+	0.894485i	$0.352457\pi$
$103$	184.264i	1.78897i	−0.447099	+	0.894485i	$0.647543\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−47.3988	−0.442980	−0.221490	−	0.975163i	$-0.571092\pi$
$107$	−47.3988	−0.442980	−0.221490	+	0.975163i	$0.571092\pi$
$108$	0	0
$109$	98.8944	0.907288	0.453644	−	0.891183i	$-0.350124\pi$
$109$	98.8944	0.907288	0.453644	+	0.891183i	$0.350124\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	187.880	1.66265	0.831325	−	0.555786i	$-0.187583\pi$
$113$	187.880	1.66265	0.831325	+	0.555786i	$0.187583\pi$
$114$	0	0
$115$	− 72.6291i	− 0.631557i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	239.191	1.97678
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 135.715i	− 1.08572i
$126$	0	0
$127$	−22.9949	−0.181063	−0.0905313	−	0.995894i	$-0.528857\pi$
$127$	−22.9949	−0.181063	−0.0905313	+	0.995894i	$0.528857\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 143.797i	− 1.09769i	−0.835926	−	0.548843i	$-0.815069\pi$
$131$	− 143.797i	− 1.09769i	0.835926	−	0.548843i	$-0.184931\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−66.4734	−0.485207	−0.242604	−	0.970126i	$-0.578001\pi$
$137$	−66.4734	−0.485207	−0.242604	+	0.970126i	$0.578001\pi$
$138$	0	0
$139$	− 246.268i	− 1.77171i	−0.463961	−	0.885856i	$-0.653572\pi$
$139$	− 246.268i	− 1.77171i	0.463961	−	0.885856i	$-0.346428\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 84.6618i	− 0.592040i
$144$	0	0
$145$	88.8595i	0.612824i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	79.7297	0.535099	0.267549	−	0.963544i	$-0.413786\pi$
$149$	79.7297	0.535099	0.267549	+	0.963544i	$0.413786\pi$
$150$	0	0
$151$	32.5929	0.215847	0.107924	−	0.994159i	$-0.465580\pi$
$151$	32.5929	0.215847	0.107924	+	0.994159i	$0.465580\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−66.3775	−0.428242
$156$	0	0
$157$	− 254.629i	− 1.62184i	−0.585158	−	0.810919i	$-0.698968\pi$
$157$	− 254.629i	− 1.62184i	0.585158	−	0.810919i	$-0.301032\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	242.995	1.49077	0.745383	−	0.666636i	$-0.232266\pi$
$163$	242.995	1.49077	0.745383	+	0.666636i	$0.232266\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	144.224i	0.863619i	0.901965	+	0.431809i	$0.142125\pi$
$167$	144.224i	0.863619i	−0.901965	+	0.431809i	$0.857875\pi$
$168$	0	0
$169$	149.101	0.882252
$170$	0	0
$171$	0	0
$172$	0	0
$173$	334.393i	1.93291i	0.256844	+	0.966453i	$0.417317\pi$
$173$	334.393i	1.93291i	−0.256844	+	0.966453i	$0.582683\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−117.591	−0.656934	−0.328467	−	0.944515i	$-0.606532\pi$
$179$	−117.591	−0.656934	−0.328467	+	0.944515i	$0.606532\pi$
$180$	0	0
$181$	− 173.481i	− 0.958459i	−0.877690	−	0.479230i	$-0.840916\pi$
$181$	− 173.481i	− 0.958459i	0.877690	−	0.479230i	$-0.159084\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 194.423i	− 1.05093i
$186$	0	0
$187$	242.233i	1.29536i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−261.983	−1.37164	−0.685819	−	0.727773i	$-0.740556\pi$
$191$	−261.983	−1.37164	−0.685819	+	0.727773i	$0.740556\pi$
$192$	0	0
$193$	29.5980	0.153357	0.0766787	−	0.997056i	$-0.475568\pi$
$193$	29.5980	0.153357	0.0766787	+	0.997056i	$0.475568\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−237.282	−1.20448	−0.602238	−	0.798317i	$-0.705724\pi$
$197$	−237.282	−1.20448	−0.602238	+	0.798317i	$0.705724\pi$
$198$	0	0
$199$	29.5455i	0.148470i	0.997241	+	0.0742349i	$0.0236515\pi$
$199$	29.5455i	0.148470i	−0.997241	+	0.0742349i	$0.976349\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	269.688	1.31555
$206$	0	0
$207$	0	0
$208$	0	0
$209$	177.832i	0.850873i
$210$	0	0
$211$	−157.990	−0.748767	−0.374384	−	0.927274i	$-0.622146\pi$
$211$	−157.990	−0.748767	−0.374384	+	0.927274i	$0.622146\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 9.36413i	− 0.0435541i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	56.9361	0.257629
$222$	0	0
$223$	− 225.779i	− 1.01246i	−0.862397	−	0.506232i	$-0.831038\pi$
$223$	− 225.779i	− 1.01246i	0.862397	−	0.506232i	$-0.168962\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	144.224i	0.635350i	0.948200	+	0.317675i	$0.102902\pi$
$227$	144.224i	0.635350i	−0.948200	+	0.317675i	$0.897098\pi$
$228$	0	0
$229$	124.569i	0.543970i	0.962302	+	0.271985i	$0.0876801\pi$
$229$	124.569i	0.543970i	−0.962302	+	0.271985i	$0.912320\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	426.973	1.83250	0.916251	−	0.400606i	$-0.131200\pi$
$233$	426.973	1.83250	0.916251	+	0.400606i	$0.131200\pi$
$234$	0	0
$235$	215.387	0.916540
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−28.5160	−0.119314	−0.0596568	−	0.998219i	$-0.519001\pi$
$239$	−28.5160	−0.119314	−0.0596568	+	0.998219i	$0.519001\pi$
$240$	0	0
$241$	− 235.037i	− 0.975257i	−0.873051	−	0.487628i	$-0.837862\pi$
$241$	− 235.037i	− 0.975257i	0.873051	−	0.487628i	$-0.162138\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	41.7990	0.169227
$248$	0	0
$249$	0	0
$250$	0	0
$251$	193.568i	0.771186i	0.922669	+	0.385593i	$0.126003\pi$
$251$	193.568i	0.771186i	−0.922669	+	0.385593i	$0.873997\pi$
$252$	0	0
$253$	323.990	1.28059
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 148.479i	− 0.577739i	−0.957369	−	0.288869i	$-0.906721\pi$
$257$	− 148.479i	− 0.577739i	0.957369	−	0.288869i	$-0.0932793\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	447.763	1.70252	0.851261	−	0.524743i	$-0.175839\pi$
$263$	447.763	1.70252	0.851261	+	0.524743i	$0.175839\pi$
$264$	0	0
$265$	161.489i	0.609391i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	183.370i	0.681672i	0.940123	+	0.340836i	$0.110710\pi$
$269$	183.370i	0.681672i	−0.940123	+	0.340836i	$0.889290\pi$
$270$	0	0
$271$	135.081i	0.498456i	0.968445	+	0.249228i	$0.0801768\pi$
$271$	135.081i	0.498456i	−0.968445	+	0.249228i	$0.919823\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	130.943	0.476158
$276$	0	0
$277$	20.6030	0.0743792	0.0371896	−	0.999308i	$-0.488159\pi$
$277$	20.6030	0.0743792	0.0371896	+	0.999308i	$0.488159\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	241.001	0.857654	0.428827	−	0.903387i	$-0.358927\pi$
$281$	241.001	0.857654	0.428827	+	0.903387i	$0.358927\pi$
$282$	0	0
$283$	88.7694i	0.313673i	0.987625	+	0.156836i	$0.0501295\pi$
$283$	88.7694i	0.313673i	−0.987625	+	0.156836i	$0.949870\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	126.095	0.436316
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 275.258i	− 0.939446i	−0.882814	−	0.469723i	$-0.844354\pi$
$293$	− 275.258i	− 0.939446i	0.882814	−	0.469723i	$-0.155646\pi$
$294$	0	0
$295$	−287.789	−0.975556
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 76.1528i	− 0.254692i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	269.613	0.883975
$306$	0	0
$307$	− 366.196i	− 1.19282i	−0.802680	−	0.596410i	$-0.796593\pi$
$307$	− 366.196i	− 1.19282i	0.802680	−	0.596410i	$-0.203407\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	295.675i	0.950723i	0.879791	+	0.475361i	$0.157683\pi$
$311$	295.675i	0.950723i	−0.879791	+	0.475361i	$0.842317\pi$
$312$	0	0
$313$	− 231.853i	− 0.740746i	−0.928883	−	0.370373i	$-0.879230\pi$
$313$	− 231.853i	− 0.740746i	0.928883	−	0.370373i	$-0.120770\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−309.382	−0.975967	−0.487983	−	0.872853i	$-0.662267\pi$
$317$	−309.382	−0.975967	−0.487983	+	0.872853i	$0.662267\pi$
$318$	0	0
$319$	−396.392	−1.24261
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−119.595	−0.370262
$324$	0	0
$325$	− 30.7779i	− 0.0947011i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−233.186	−0.704489	−0.352244	−	0.935908i	$-0.614581\pi$
$331$	−233.186	−0.704489	−0.352244	+	0.935908i	$0.614581\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 116.560i	− 0.347939i
$336$	0	0
$337$	172.894	0.513040	0.256520	−	0.966539i	$-0.417424\pi$
$337$	172.894	0.513040	0.256520	+	0.966539i	$0.417424\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 296.102i	− 0.868336i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−464.930	−1.33986	−0.669928	−	0.742426i	$-0.733675\pi$
$347$	−464.930	−1.33986	−0.669928	+	0.742426i	$0.733675\pi$
$348$	0	0
$349$	426.071i	1.22083i	0.792080	+	0.610417i	$0.208998\pi$
$349$	426.071i	1.22083i	−0.792080	+	0.610417i	$0.791002\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 403.747i	− 1.14376i	−0.820338	−	0.571880i	$-0.806214\pi$
$353$	− 403.747i	− 1.14376i	0.820338	−	0.571880i	$-0.193786\pi$
$354$	0	0
$355$	64.5139i	0.181729i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	74.1032	0.206416	0.103208	−	0.994660i	$-0.467089\pi$
$359$	74.1032	0.206416	0.103208	+	0.994660i	$0.467089\pi$
$360$	0	0
$361$	273.201	0.756790
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−356.589	−0.976955
$366$	0	0
$367$	304.998i	0.831058i	0.909580	+	0.415529i	$0.136404\pi$
$367$	304.998i	0.831058i	−0.909580	+	0.415529i	$0.863596\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−129.407	−0.346936	−0.173468	−	0.984840i	$-0.555497\pi$
$373$	−129.407	−0.346936	−0.173468	+	0.984840i	$0.555497\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	93.1707i	0.247137i
$378$	0	0
$379$	−276.804	−0.730354	−0.365177	−	0.930938i	$-0.618991\pi$
$379$	−276.804	−0.730354	−0.365177	+	0.930938i	$0.618991\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	661.987i	1.72842i	0.503127	+	0.864212i	$0.332183\pi$
$383$	661.987i	1.72842i	−0.503127	+	0.864212i	$0.667817\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−408.090	−1.04907	−0.524537	−	0.851388i	$-0.675762\pi$
$389$	−408.090	−1.04907	−0.524537	+	0.851388i	$0.675762\pi$
$390$	0	0
$391$	217.887i	0.557256i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 566.678i	− 1.43463i
$396$	0	0
$397$	− 710.132i	− 1.78875i	−0.447322	−	0.894373i	$-0.647622\pi$
$397$	− 710.132i	− 1.78875i	0.447322	−	0.894373i	$-0.352378\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−673.600	−1.67980	−0.839900	−	0.542741i	$-0.817387\pi$
$401$	−673.600	−1.67980	−0.839900	+	0.542741i	$0.817387\pi$
$402$	0	0
$403$	−69.5980	−0.172700
$404$	0	0
$405$	0	0
$406$	0	0
$407$	867.298	2.13095
$408$	0	0
$409$	− 296.415i	− 0.724730i	−0.932036	−	0.362365i	$-0.881969\pi$
$409$	− 296.415i	− 0.724730i	0.932036	−	0.362365i	$-0.118031\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−466.975	−1.12524
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 357.803i	− 0.853945i	−0.904265	−	0.426973i	$-0.859580\pi$
$419$	− 357.803i	− 0.853945i	0.904265	−	0.426973i	$-0.140420\pi$
$420$	0	0
$421$	185.186	0.439871	0.219936	−	0.975514i	$-0.429415\pi$
$421$	185.186	0.439871	0.219936	+	0.975514i	$0.429415\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	88.0611i	0.207202i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−721.286	−1.67352	−0.836759	−	0.547571i	$-0.815553\pi$
$431$	−721.286	−1.67352	−0.836759	+	0.547571i	$0.815553\pi$
$432$	0	0
$433$	585.095i	1.35126i	0.737242	+	0.675629i	$0.236128\pi$
$433$	585.095i	1.35126i	−0.737242	+	0.675629i	$0.763872\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	159.959i	0.366040i
$438$	0	0
$439$	152.296i	0.346917i	0.984841	+	0.173458i	$0.0554942\pi$
$439$	152.296i	0.346917i	−0.984841	+	0.173458i	$0.944506\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	717.471	1.61957	0.809787	−	0.586724i	$-0.199583\pi$
$443$	717.471	1.61957	0.809787	+	0.586724i	$0.199583\pi$
$444$	0	0
$445$	−340.271	−0.764654
$446$	0	0
$447$	0	0
$448$	0	0
$449$	13.3522	0.0297376	0.0148688	−	0.999889i	$-0.495267\pi$
$449$	13.3522	0.0297376	0.0148688	+	0.999889i	$0.495267\pi$
$450$	0	0
$451$	1203.05i	2.66751i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−554.191	−1.21267	−0.606336	−	0.795209i	$-0.707361\pi$
$457$	−554.191	−1.21267	−0.606336	+	0.795209i	$0.707361\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	336.103i	0.729074i	0.931189	+	0.364537i	$0.118773\pi$
$461$	336.103i	0.729074i	−0.931189	+	0.364537i	$0.881227\pi$
$462$	0	0
$463$	−481.186	−1.03928	−0.519639	−	0.854386i	$-0.673934\pi$
$463$	−481.186	−1.03928	−0.519639	+	0.854386i	$0.673934\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 386.708i	− 0.828068i	−0.910262	−	0.414034i	$-0.864120\pi$
$467$	− 386.708i	− 0.828068i	0.910262	−	0.414034i	$-0.135880\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	41.7723	0.0883135
$474$	0	0
$475$	64.6490i	0.136103i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	268.438i	0.560413i	0.959940	+	0.280206i	$0.0904029\pi$
$479$	268.438i	0.560413i	−0.959940	+	0.280206i	$0.909597\pi$
$480$	0	0
$481$	− 203.856i	− 0.423816i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−508.610	−1.04868
$486$	0	0
$487$	−85.7990	−0.176179	−0.0880893	−	0.996113i	$-0.528076\pi$
$487$	−85.7990	−0.176179	−0.0880893	+	0.996113i	$0.528076\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−465.122	−0.947295	−0.473647	−	0.880715i	$-0.657063\pi$
$491$	−465.122	−0.947295	−0.473647	+	0.880715i	$0.657063\pi$
$492$	0	0
$493$	− 266.578i	− 0.540727i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−855.568	−1.71456	−0.857282	−	0.514847i	$-0.827849\pi$
$499$	−855.568	−1.71456	−0.857282	+	0.514847i	$0.827849\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 432.245i	− 0.859335i	−0.902987	−	0.429667i	$-0.858631\pi$
$503$	− 432.245i	− 0.859335i	0.902987	−	0.429667i	$-0.141369\pi$
$504$	0	0
$505$	628.060	1.24368
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 267.604i	− 0.525744i	−0.964831	−	0.262872i	$-0.915330\pi$
$509$	− 267.604i	− 0.525744i	0.964831	−	0.262872i	$-0.0846698\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	783.945	1.52222
$516$	0	0
$517$	960.816i	1.85844i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	335.675i	0.644291i	0.946690	+	0.322145i	$0.104404\pi$
$521$	335.675i	0.644291i	−0.946690	+	0.322145i	$0.895596\pi$
$522$	0	0
$523$	− 970.053i	− 1.85479i	−0.374088	−	0.927393i	$-0.622044\pi$
$523$	− 970.053i	− 1.85479i	0.374088	−	0.927393i	$-0.377956\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	199.133	0.377861
$528$	0	0
$529$	−237.573	−0.449098
$530$	0	0
$531$	0	0
$532$	0	0
$533$	282.773	0.530531
$534$	0	0
$535$	201.657i	0.376929i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−350.181	−0.647284	−0.323642	−	0.946180i	$-0.604907\pi$
$541$	−350.181	−0.647284	−0.323642	+	0.946180i	$0.604907\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 420.743i	− 0.772006i
$546$	0	0
$547$	−648.794	−1.18609	−0.593047	−	0.805167i	$-0.702075\pi$
$547$	−648.794	−1.18609	−0.593047	+	0.805167i	$0.702075\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 195.706i	− 0.355183i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	878.742	1.57763	0.788817	−	0.614628i	$-0.210694\pi$
$557$	878.742	1.57763	0.788817	+	0.614628i	$0.210694\pi$
$558$	0	0
$559$	− 9.81845i	− 0.0175643i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	587.544i	1.04360i	0.853069	+	0.521798i	$0.174738\pi$
$563$	587.544i	1.04360i	−0.853069	+	0.521798i	$0.825262\pi$
$564$	0	0
$565$	− 799.328i	− 1.41474i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	104.239	0.183197	0.0915985	−	0.995796i	$-0.470802\pi$
$569$	104.239	0.183197	0.0915985	+	0.995796i	$0.470802\pi$
$570$	0	0
$571$	−471.387	−0.825546	−0.412773	−	0.910834i	$-0.635440\pi$
$571$	−471.387	−0.825546	−0.412773	+	0.910834i	$0.635440\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	117.783	0.204840
$576$	0	0
$577$	474.983i	0.823194i	0.911366	+	0.411597i	$0.135029\pi$
$577$	474.983i	0.823194i	−0.911366	+	0.411597i	$0.864971\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−720.382	−1.23565
$584$	0	0
$585$	0	0
$586$	0	0
$587$	539.013i	0.918251i	0.888371	+	0.459125i	$0.151837\pi$
$587$	539.013i	0.918251i	−0.888371	+	0.459125i	$0.848163\pi$
$588$	0	0
$589$	146.191	0.248202
$590$	0	0
$591$	0	0
$592$	0	0
$593$	103.796i	0.175036i	0.996163	+	0.0875178i	$0.0278935\pi$
$593$	103.796i	0.175036i	−0.996163	+	0.0875178i	$0.972107\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	550.190	0.918515	0.459257	−	0.888303i	$-0.348116\pi$
$599$	550.190	0.918515	0.459257	+	0.888303i	$0.348116\pi$
$600$	0	0
$601$	848.487i	1.41179i	0.708315	+	0.705896i	$0.249456\pi$
$601$	848.487i	1.41179i	−0.708315	+	0.705896i	$0.750544\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 1017.63i	− 1.68203i
$606$	0	0
$607$	594.708i	0.979750i	0.871793	+	0.489875i	$0.162958\pi$
$607$	594.708i	0.979750i	−0.871793	+	0.489875i	$0.837042\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	225.837	0.369618
$612$	0	0
$613$	517.508	0.844221	0.422111	−	0.906544i	$-0.361289\pi$
$613$	517.508	0.844221	0.422111	+	0.906544i	$0.361289\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−681.421	−1.10441	−0.552205	−	0.833708i	$-0.686214\pi$
$617$	−681.421	−1.10441	−0.552205	+	0.833708i	$0.686214\pi$
$618$	0	0
$619$	893.835i	1.44400i	0.691894	+	0.721999i	$0.256777\pi$
$619$	893.835i	1.44400i	−0.691894	+	0.721999i	$0.743223\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−404.910	−0.647855
$626$	0	0
$627$	0	0
$628$	0	0
$629$	583.268i	0.927295i
$630$	0	0
$631$	−949.588	−1.50489	−0.752447	−	0.658653i	$-0.771127\pi$
$631$	−949.588	−1.50489	−0.752447	+	0.658653i	$0.771127\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	97.8313i	0.154065i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−430.788	−0.672056	−0.336028	−	0.941852i	$-0.609084\pi$
$641$	−430.788	−0.672056	−0.336028	+	0.941852i	$0.609084\pi$
$642$	0	0
$643$	540.999i	0.841368i	0.907207	+	0.420684i	$0.138210\pi$
$643$	540.999i	0.841368i	−0.907207	+	0.420684i	$0.861790\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	99.1141i	0.153190i	0.997062	+	0.0765951i	$0.0244049\pi$
$647$	99.1141i	0.153190i	−0.997062	+	0.0765951i	$0.975595\pi$
$648$	0	0
$649$	− 1283.79i	− 1.97811i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	829.244	1.26990	0.634950	−	0.772553i	$-0.281021\pi$
$653$	829.244	1.26990	0.634950	+	0.772553i	$0.281021\pi$
$654$	0	0
$655$	−611.779	−0.934013
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−584.716	−0.887278	−0.443639	−	0.896206i	$-0.646313\pi$
$659$	−584.716	−0.887278	−0.443639	+	0.896206i	$0.646313\pi$
$660$	0	0
$661$	− 385.274i	− 0.582865i	−0.956591	−	0.291433i	$-0.905868\pi$
$661$	− 385.274i	− 0.582865i	0.956591	−	0.291433i	$-0.0941319\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−356.553	−0.534562
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1202.71i	1.79241i
$672$	0	0
$673$	797.266	1.18465	0.592323	−	0.805701i	$-0.298211\pi$
$673$	797.266	1.18465	0.592323	+	0.805701i	$0.298211\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	479.900i	0.708862i	0.935082	+	0.354431i	$0.115325\pi$
$677$	479.900i	0.708862i	−0.935082	+	0.354431i	$0.884675\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1070.34	−1.56712	−0.783559	−	0.621318i	$-0.786598\pi$
$683$	−1070.34	−1.56712	−0.783559	+	0.621318i	$0.786598\pi$
$684$	0	0
$685$	282.809i	0.412860i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	169.324i	0.245753i
$690$	0	0
$691$	29.9465i	0.0433379i	0.999765	+	0.0216690i	$0.00689798\pi$
$691$	29.9465i	0.0433379i	−0.999765	+	0.0216690i	$0.993102\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1047.74	−1.50754
$696$	0	0
$697$	−809.065	−1.16078
$698$	0	0
$699$	0	0
$700$	0	0
$701$	396.453	0.565554	0.282777	−	0.959186i	$-0.408744\pi$
$701$	396.453	0.565554	0.282777	+	0.959186i	$0.408744\pi$
$702$	0	0
$703$	428.200i	0.609104i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−347.497	−0.490123	−0.245062	−	0.969507i	$-0.578808\pi$
$709$	−347.497	−0.490123	−0.245062	+	0.969507i	$0.578808\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 266.343i	− 0.373552i
$714$	0	0
$715$	−360.191	−0.503764
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1050.83i	1.46152i	0.682635	+	0.730760i	$0.260834\pi$
$719$	1050.83i	1.46152i	−0.682635	+	0.730760i	$0.739166\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−144.104	−0.198764
$726$	0	0
$727$	47.3417i	0.0651193i	0.999470	+	0.0325596i	$0.0103659\pi$
$727$	47.3417i	0.0651193i	−0.999470	+	0.0325596i	$0.989634\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	28.0924i	0.0384301i
$732$	0	0
$733$	62.3420i	0.0850505i	0.999095	+	0.0425252i	$0.0135403\pi$
$733$	62.3420i	0.0850505i	−0.999095	+	0.0425252i	$0.986460\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	519.959	0.705507
$738$	0	0
$739$	1338.35	1.81103	0.905515	−	0.424314i	$-0.139485\pi$
$739$	1338.35	1.81103	0.905515	+	0.424314i	$0.139485\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	242.908	0.326929	0.163464	−	0.986549i	$-0.447733\pi$
$743$	242.908	0.326929	0.163464	+	0.986549i	$0.447733\pi$
$744$	0	0
$745$	− 339.208i	− 0.455312i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−992.583	−1.32168	−0.660841	−	0.750526i	$-0.729800\pi$
$751$	−992.583	−1.32168	−0.660841	+	0.750526i	$0.729800\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 138.666i	− 0.183663i
$756$	0	0
$757$	780.060	1.03046	0.515231	−	0.857051i	$-0.327706\pi$
$757$	780.060	1.03046	0.515231	+	0.857051i	$0.327706\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	464.592i	0.610502i	0.952272	+	0.305251i	$0.0987404\pi$
$761$	464.592i	0.610502i	−0.952272	+	0.305251i	$0.901260\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−301.752	−0.393418
$768$	0	0
$769$	611.409i	0.795071i	0.917587	+	0.397535i	$0.130134\pi$
$769$	611.409i	0.795071i	−0.917587	+	0.397535i	$0.869866\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 235.706i	− 0.304924i	−0.988309	−	0.152462i	$-0.951280\pi$
$773$	− 235.706i	− 0.304924i	0.988309	−	0.152462i	$-0.0487201\pi$
$774$	0	0
$775$	− 107.645i	− 0.138896i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−593.966	−0.762473
$780$	0	0
$781$	−287.789	−0.368488
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1083.31	−1.38001
$786$	0	0
$787$	594.978i	0.756008i	0.925804	+	0.378004i	$0.123389\pi$
$787$	594.978i	0.756008i	−0.925804	+	0.378004i	$0.876611\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	282.693	0.356486
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 547.544i	− 0.687006i	−0.939152	−	0.343503i	$-0.888386\pi$
$797$	− 547.544i	− 0.687006i	0.939152	−	0.343503i	$-0.111614\pi$
$798$	0	0
$799$	−646.161	−0.808712
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1590.70i	− 1.98095i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	512.137	0.633050	0.316525	−	0.948584i	$-0.397484\pi$
$809$	512.137	0.633050	0.316525	+	0.948584i	$0.397484\pi$
$810$	0	0
$811$	− 1615.33i	− 1.99178i	−0.0905851	−	0.995889i	$-0.528874\pi$
$811$	− 1615.33i	− 1.99178i	0.0905851	−	0.995889i	$-0.471126\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1033.81i	− 1.26848i
$816$	0	0
$817$	20.6237i	0.0252432i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	981.553	1.19556	0.597779	−	0.801661i	$-0.296050\pi$
$821$	981.553	1.19556	0.597779	+	0.801661i	$0.296050\pi$
$822$	0	0
$823$	−1356.55	−1.64830	−0.824151	−	0.566370i	$-0.808347\pi$
$823$	−1356.55	−1.64830	−0.824151	+	0.566370i	$0.808347\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	601.883	0.727791	0.363896	−	0.931440i	$-0.381446\pi$
$827$	601.883	0.727791	0.363896	+	0.931440i	$0.381446\pi$
$828$	0	0
$829$	178.728i	0.215595i	0.994173	+	0.107798i	$0.0343798\pi$
$829$	178.728i	0.215595i	−0.994173	+	0.107798i	$0.965620\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	613.598	0.734848
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 852.989i	− 1.01667i	−0.861158	−	0.508337i	$-0.830261\pi$
$839$	− 852.989i	− 1.01667i	0.861158	−	0.508337i	$-0.169739\pi$
$840$	0	0
$841$	−404.769	−0.481295
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 634.343i	− 0.750702i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	780.130	0.916721
$852$	0	0
$853$	855.303i	1.00270i	0.865245	+	0.501350i	$0.167163\pi$
$853$	855.303i	1.00270i	−0.865245	+	0.501350i	$0.832837\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1444.79i	1.68587i	0.538018	+	0.842933i	$0.319173\pi$
$857$	1444.79i	1.68587i	−0.538018	+	0.842933i	$0.680827\pi$
$858$	0	0
$859$	424.075i	0.493684i	0.969056	+	0.246842i	$0.0793929\pi$
$859$	424.075i	0.493684i	−0.969056	+	0.246842i	$0.920607\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1349.20	−1.56339	−0.781694	−	0.623663i	$-0.785644\pi$
$863$	−1349.20	−1.56339	−0.781694	+	0.623663i	$0.785644\pi$
$864$	0	0
$865$	1422.66	1.64470
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2527.89	2.90896
$870$	0	0
$871$	− 122.215i	− 0.140315i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	643.658	0.733932	0.366966	−	0.930234i	$-0.380397\pi$
$877$	643.658	0.733932	0.366966	+	0.930234i	$0.380397\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1206.54i	− 1.36951i	−0.728774	−	0.684754i	$-0.759909\pi$
$881$	− 1206.54i	− 1.36951i	0.728774	−	0.684754i	$-0.240091\pi$
$882$	0	0
$883$	652.754	0.739245	0.369623	−	0.929182i	$-0.379487\pi$
$883$	652.754	0.739245	0.369623	+	0.929182i	$0.379487\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1051.26i	− 1.18519i	−0.805502	−	0.592593i	$-0.798104\pi$
$887$	− 1051.26i	− 1.18519i	0.805502	−	0.592593i	$-0.201896\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−474.372	−0.531211
$894$	0	0
$895$	500.288i	0.558981i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	325.862i	0.362472i
$900$	0	0
$901$	− 484.466i	− 0.537698i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−738.070	−0.815547
$906$	0	0
$907$	286.161	0.315502	0.157751	−	0.987479i	$-0.449576\pi$
$907$	286.161	0.315502	0.157751	+	0.987479i	$0.449576\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−713.465	−0.783167	−0.391583	−	0.920143i	$-0.628073\pi$
$911$	−713.465	−0.783167	−0.391583	+	0.920143i	$0.628073\pi$
$912$	0	0
$913$	− 2083.12i	− 2.28162i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	341.397	0.371487	0.185744	−	0.982598i	$-0.440531\pi$
$919$	341.397	0.371487	0.185744	+	0.982598i	$0.440531\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	67.6439i	0.0732870i
$924$	0	0
$925$	315.296	0.340861
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 91.8664i	− 0.0988875i	−0.998777	−	0.0494437i	$-0.984255\pi$
$929$	− 91.8664i	− 0.0988875i	0.998777	−	0.0494437i	$-0.0157448\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1030.57	1.10222
$936$	0	0
$937$	− 906.902i	− 0.967879i	−0.875102	−	0.483939i	$-0.839206\pi$
$937$	− 906.902i	− 0.967879i	0.875102	−	0.483939i	$-0.160794\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 390.150i	− 0.414612i	−0.978276	−	0.207306i	$-0.933530\pi$
$941$	− 390.150i	− 0.414612i	0.978276	−	0.207306i	$-0.0664696\pi$
$942$	0	0
$943$	1082.14i	1.14755i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	609.513	0.643625	0.321813	−	0.946803i	$-0.395708\pi$
$947$	609.513	0.643625	0.321813	+	0.946803i	$0.395708\pi$
$948$	0	0
$949$	−373.889	−0.393983
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−110.441	−0.115887	−0.0579437	−	0.998320i	$-0.518454\pi$
$953$	−110.441	−0.115887	−0.0579437	+	0.998320i	$0.518454\pi$
$954$	0	0
$955$	1114.60i	1.16712i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	717.583	0.746704
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 125.924i	− 0.130491i
$966$	0	0
$967$	−1125.53	−1.16394	−0.581969	−	0.813211i	$-0.697717\pi$
$967$	−1125.53	−1.16394	−0.581969	+	0.813211i	$0.697717\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1129.12i	1.16285i	0.813602	+	0.581423i	$0.197504\pi$
$971$	1129.12i	1.16285i	−0.813602	+	0.581423i	$0.802496\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1512.86	−1.54848	−0.774238	−	0.632895i	$-0.781867\pi$
$977$	−1512.86	−1.54848	−0.774238	+	0.632895i	$0.781867\pi$
$978$	0	0
$979$	− 1517.91i	− 1.55047i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	579.463i	0.589484i	0.955577	+	0.294742i	$0.0952337\pi$
$983$	579.463i	0.589484i	−0.955577	+	0.294742i	$0.904766\pi$
$984$	0	0
$985$	1009.51i	1.02488i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	37.5740	0.0379919
$990$	0	0
$991$	172.754	0.174322	0.0871612	−	0.996194i	$-0.472220\pi$
$991$	172.754	0.174322	0.0871612	+	0.996194i	$0.472220\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	125.700	0.126332
$996$	0	0
$997$	− 384.823i	− 0.385981i	−0.981201	−	0.192991i	$-0.938181\pi$
$997$	− 384.823i	− 0.385981i	0.981201	−	0.192991i	$-0.0618187\pi$
$998$	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.685.4	yes	8
3.2	odd	2	inner	1764.3.d.g.685.5	yes	8
7.2	even	3		1764.3.z.n.325.3		16
7.3	odd	6		1764.3.z.n.901.3		16
7.4	even	3		1764.3.z.n.901.5		16
7.5	odd	6		1764.3.z.n.325.5		16
7.6	odd	2	inner	1764.3.d.g.685.6	yes	8
21.2	odd	6		1764.3.z.n.325.6		16
21.5	even	6		1764.3.z.n.325.4		16
21.11	odd	6		1764.3.z.n.901.4		16
21.17	even	6		1764.3.z.n.901.6		16
21.20	even	2	inner	1764.3.d.g.685.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.3.d.g.685.3	✓	8	21.20	even	2	inner
1764.3.d.g.685.4	yes	8	1.1	even	1	trivial
1764.3.d.g.685.5	yes	8	3.2	odd	2	inner
1764.3.d.g.685.6	yes	8	7.6	odd	2	inner
1764.3.z.n.325.3		16	7.2	even	3
1764.3.z.n.325.4		16	21.5	even	6
1764.3.z.n.325.5		16	7.5	odd	6
1764.3.z.n.325.6		16	21.2	odd	6
1764.3.z.n.901.3		16	7.3	odd	6
1764.3.z.n.901.4		16	21.11	odd	6
1764.3.z.n.901.5		16	7.4	even	3
1764.3.z.n.901.6		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)

\(f(q)\)	\(=\)	\(q-4.25447i q^{5} +O(q^{10})\)	\(q-4.25447i q^{5} +18.9787 q^{11} -4.46088i q^{13} +12.7634i q^{17} +9.37011i q^{19} +17.0712 q^{23} +6.89949 q^{25} -20.8862 q^{29} -15.6018i q^{31} +45.6985 q^{37} +63.3894i q^{41} +2.20101 q^{43} +50.6260i q^{47} -37.9574 q^{53} -80.7443i q^{55} -67.6439i q^{59} +63.3716i q^{61} -18.9787 q^{65} +27.3970 q^{67} -15.1638 q^{71} -83.8151i q^{73} +133.196 q^{79} -109.761i q^{83} +54.3015 q^{85} -79.9797i q^{89} +39.8648 q^{95} -119.547i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 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

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