LMFDB - Embedded newform 1764.3.c.g.197.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1764,3,Mod(197,1764)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1764, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1764.197"); 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.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-23})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 17x^{2} - 16x + 18 $ x^4 - 2x^3 + 17x^2 - 16*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.1
Root			$0.500000 + 3.81213i$ of defining polynomial
Character	$\chi$	$=$	1764.197
Dual form			1764.3.c.g.197.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-7.62426i q^{5} +5.10366i q^{11} +20.7823 q^{13} +26.2544i q^{17} -22.3470 q^{19} +39.7809i q^{23} -33.1293 q^{25} +7.62426i q^{29} +11.7823 q^{31} -60.3470 q^{37} -11.8044i q^{41} -30.3470 q^{43} +38.1838i q^{47} +5.90221i q^{53} +38.9116 q^{55} +44.2109i q^{59} +49.4763 q^{61} -158.450i q^{65} +63.1293 q^{67} +41.6279i q^{71} +4.17670 q^{73} -41.7823 q^{79} +145.784i q^{83} +200.170 q^{85} +43.8986i q^{89} +170.379i q^{95} -39.5173 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 56 q^{13} - 8 q^{19} - 24 q^{25} + 20 q^{31} - 160 q^{37} - 40 q^{43} + 20 q^{55} + 8 q^{61} + 144 q^{67} + 288 q^{73} - 140 q^{79} + 448 q^{85} + 276 q^{97}+O(q^{100}) $ 4 * q + 56 * q^13 - 8 * q^19 - 24 * q^25 + 20 * q^31 - 160 * q^37 - 40 * q^43 + 20 * q^55 + 8 * q^61 + 144 * q^67 + 288 * q^73 - 140 * q^79 + 448 * q^85 + 276 * q^97

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
$5$	− 7.62426i	− 1.52485i	−0.647076	−	0.762426i	$-0.724008\pi$
$5$	− 7.62426i	− 1.52485i	0.647076	−	0.762426i	$-0.275992\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.10366i	0.463969i	0.972719	+	0.231985i	$0.0745219\pi$
$11$	5.10366i	0.463969i	−0.972719	+	0.231985i	$0.925478\pi$
$12$	0	0
$13$	20.7823	1.59864	0.799320	−	0.600905i	$-0.205193\pi$
$13$	20.7823	1.59864	0.799320	+	0.600905i	$0.205193\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	26.2544i	1.54438i	0.635394	+	0.772188i	$0.280838\pi$
$17$	26.2544i	1.54438i	−0.635394	+	0.772188i	$0.719162\pi$
$18$	0	0
$19$	−22.3470	−1.17616	−0.588079	−	0.808804i	$-0.700115\pi$
$19$	−22.3470	−1.17616	−0.588079	+	0.808804i	$0.700115\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	39.7809i	1.72960i	0.502114	+	0.864801i	$0.332556\pi$
$23$	39.7809i	1.72960i	−0.502114	+	0.864801i	$0.667444\pi$
$24$	0	0
$25$	−33.1293	−1.32517
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.62426i	0.262905i	0.991322	+	0.131453i	$0.0419642\pi$
$29$	7.62426i	0.262905i	−0.991322	+	0.131453i	$0.958036\pi$
$30$	0	0
$31$	11.7823	0.380075	0.190038	−	0.981777i	$-0.439139\pi$
$31$	11.7823	0.380075	0.190038	+	0.981777i	$0.439139\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−60.3470	−1.63100	−0.815500	−	0.578757i	$-0.803538\pi$
$37$	−60.3470	−1.63100	−0.815500	+	0.578757i	$0.803538\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 11.8044i	− 0.287913i	−0.989584	−	0.143956i	$-0.954017\pi$
$41$	− 11.8044i	− 0.287913i	0.989584	−	0.143956i	$-0.0459825\pi$
$42$	0	0
$43$	−30.3470	−0.705744	−0.352872	−	0.935672i	$-0.614795\pi$
$43$	−30.3470	−0.705744	−0.352872	+	0.935672i	$0.614795\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	38.1838i	0.812421i	0.913780	+	0.406210i	$0.133150\pi$
$47$	38.1838i	0.812421i	−0.913780	+	0.406210i	$0.866850\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.90221i	0.111363i	0.998449	+	0.0556813i	$0.0177331\pi$
$53$	5.90221i	0.111363i	−0.998449	+	0.0556813i	$0.982267\pi$
$54$	0	0
$55$	38.9116	0.707485
$56$	0	0
$57$	0	0
$58$	0	0
$59$	44.2109i	0.749338i	0.927159	+	0.374669i	$0.122244\pi$
$59$	44.2109i	0.749338i	−0.927159	+	0.374669i	$0.877756\pi$
$60$	0	0
$61$	49.4763	0.811087	0.405544	−	0.914076i	$-0.367082\pi$
$61$	49.4763	0.811087	0.405544	+	0.914076i	$0.367082\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 158.450i	− 2.43769i
$66$	0	0
$67$	63.1293	0.942229	0.471114	−	0.882072i	$-0.343852\pi$
$67$	63.1293	0.942229	0.471114	+	0.882072i	$0.343852\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	41.6279i	0.586308i	0.956065	+	0.293154i	$0.0947048\pi$
$71$	41.6279i	0.586308i	−0.956065	+	0.293154i	$0.905295\pi$
$72$	0	0
$73$	4.17670	0.0572151	0.0286075	−	0.999591i	$-0.490893\pi$
$73$	4.17670	0.0572151	0.0286075	+	0.999591i	$0.490893\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−41.7823	−0.528890	−0.264445	−	0.964401i	$-0.585189\pi$
$79$	−41.7823	−0.528890	−0.264445	+	0.964401i	$0.585189\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	145.784i	1.75644i	0.478258	+	0.878219i	$0.341268\pi$
$83$	145.784i	1.75644i	−0.478258	+	0.878219i	$0.658732\pi$
$84$	0	0
$85$	200.170	2.35494
$86$	0	0
$87$	0	0
$88$	0	0
$89$	43.8986i	0.493242i	0.969112	+	0.246621i	$0.0793203\pi$
$89$	43.8986i	0.493242i	−0.969112	+	0.246621i	$0.920680\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	170.379i	1.79347i
$96$	0	0
$97$	−39.5173	−0.407395	−0.203697	−	0.979034i	$-0.565296\pi$
$97$	−39.5173	−0.407395	−0.203697	+	0.979034i	$0.565296\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	93.4630i	0.925377i	0.886521	+	0.462688i	$0.153115\pi$
$101$	93.4630i	0.925377i	−0.886521	+	0.462688i	$0.846885\pi$
$102$	0	0
$103$	58.0000	0.563107	0.281553	−	0.959546i	$-0.409150\pi$
$103$	58.0000	0.563107	0.281553	+	0.959546i	$0.409150\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 171.365i	− 1.60154i	−0.598969	−	0.800772i	$-0.704423\pi$
$107$	− 171.365i	− 1.60154i	0.598969	−	0.800772i	$-0.295577\pi$
$108$	0	0
$109$	197.211	1.80928	0.904639	−	0.426179i	$-0.140141\pi$
$109$	197.211	1.80928	0.904639	+	0.426179i	$0.140141\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	135.640i	1.20035i	0.799869	+	0.600175i	$0.204902\pi$
$113$	135.640i	1.20035i	−0.799869	+	0.600175i	$0.795098\pi$
$114$	0	0
$115$	303.300	2.63739
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	94.9526	0.784732
$122$	0	0
$123$	0	0
$124$	0	0
$125$	61.9800i	0.495840i
$126$	0	0
$127$	−186.558	−1.46896	−0.734481	−	0.678629i	$-0.762574\pi$
$127$	−186.558	−1.46896	−0.734481	+	0.678629i	$0.762574\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 50.5993i	− 0.386254i	−0.981174	−	0.193127i	$-0.938137\pi$
$131$	− 50.5993i	− 0.386254i	0.981174	−	0.193127i	$-0.0618630\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 159.123i	− 1.16149i	−0.814087	−	0.580743i	$-0.802762\pi$
$137$	− 159.123i	− 1.16149i	0.814087	−	0.580743i	$-0.197238\pi$
$138$	0	0
$139$	−1.82330	−0.0131173	−0.00655863	−	0.999978i	$-0.502088\pi$
$139$	−1.82330	−0.0131173	−0.00655863	+	0.999978i	$0.502088\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	106.066i	0.741720i
$144$	0	0
$145$	58.1293	0.400892
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 169.706i	− 1.13896i	−0.822004	−	0.569482i	$-0.807144\pi$
$149$	− 169.706i	− 1.13896i	0.822004	−	0.569482i	$-0.192856\pi$
$150$	0	0
$151$	−128.994	−0.854262	−0.427131	−	0.904190i	$-0.640476\pi$
$151$	−128.994	−0.854262	−0.427131	+	0.904190i	$0.640476\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 89.8315i	− 0.579558i
$156$	0	0
$157$	56.6940	0.361108	0.180554	−	0.983565i	$-0.442211\pi$
$157$	56.6940	0.361108	0.180554	+	0.983565i	$0.442211\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−134.959	−0.827970	−0.413985	−	0.910284i	$-0.635863\pi$
$163$	−134.959	−0.827970	−0.413985	+	0.910284i	$0.635863\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	322.191i	1.92929i	0.263560	+	0.964643i	$0.415103\pi$
$167$	322.191i	1.92929i	−0.263560	+	0.964643i	$0.584897\pi$
$168$	0	0
$169$	262.905	1.55565
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 57.4250i	− 0.331937i	−0.986131	−	0.165968i	$-0.946925\pi$
$173$	− 57.4250i	− 0.331937i	0.986131	−	0.165968i	$-0.0530749\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 189.620i	− 1.05933i	−0.848206	−	0.529666i	$-0.822317\pi$
$179$	− 189.620i	− 1.05933i	0.848206	−	0.529666i	$-0.177683\pi$
$180$	0	0
$181$	220.599	1.21878	0.609390	−	0.792871i	$-0.291414\pi$
$181$	220.599	1.21878	0.609390	+	0.792871i	$0.291414\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	460.101i	2.48703i
$186$	0	0
$187$	−133.994	−0.716543
$188$	0	0
$189$	0	0
$190$	0	0
$191$	147.194i	0.770650i	0.922781	+	0.385325i	$0.125911\pi$
$191$	147.194i	0.770650i	−0.922781	+	0.385325i	$0.874089\pi$
$192$	0	0
$193$	−120.470	−0.624196	−0.312098	−	0.950050i	$-0.601032\pi$
$193$	−120.470	−0.624196	−0.312098	+	0.950050i	$0.601032\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 21.6369i	− 0.109832i	−0.998491	−	0.0549160i	$-0.982511\pi$
$197$	− 21.6369i	− 0.109832i	0.998491	−	0.0549160i	$-0.0174891\pi$
$198$	0	0
$199$	19.9181	0.100091	0.0500454	−	0.998747i	$-0.484063\pi$
$199$	19.9181	0.100091	0.0500454	+	0.998747i	$0.484063\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−90.0000	−0.439024
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 114.052i	− 0.545701i
$210$	0	0
$211$	−31.6530	−0.150014	−0.0750071	−	0.997183i	$-0.523898\pi$
$211$	−31.6530	−0.150014	−0.0750071	+	0.997183i	$0.523898\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	231.373i	1.07615i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	545.627i	2.46890i
$222$	0	0
$223$	250.653	1.12400	0.562002	−	0.827136i	$-0.310031\pi$
$223$	250.653	1.12400	0.562002	+	0.827136i	$0.310031\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	227.443i	1.00195i	0.865461	+	0.500976i	$0.167025\pi$
$227$	227.443i	1.00195i	−0.865461	+	0.500976i	$0.832975\pi$
$228$	0	0
$229$	−225.129	−0.983097	−0.491549	−	0.870850i	$-0.663569\pi$
$229$	−225.129	−0.983097	−0.491549	+	0.870850i	$0.663569\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	53.0574i	0.227714i	0.993497	+	0.113857i	$0.0363206\pi$
$233$	53.0574i	0.227714i	−0.993497	+	0.113857i	$0.963679\pi$
$234$	0	0
$235$	291.123	1.23882
$236$	0	0
$237$	0	0
$238$	0	0
$239$	451.366i	1.88856i	0.329143	+	0.944280i	$0.393240\pi$
$239$	451.366i	1.88856i	−0.329143	+	0.944280i	$0.606760\pi$
$240$	0	0
$241$	88.3880	0.366755	0.183378	−	0.983043i	$-0.441297\pi$
$241$	88.3880	0.366755	0.183378	+	0.983043i	$0.441297\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−464.423	−1.88025
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 233.706i	− 0.931102i	−0.885021	−	0.465551i	$-0.845856\pi$
$251$	− 233.706i	− 0.931102i	0.885021	−	0.465551i	$-0.154144\pi$
$252$	0	0
$253$	−203.028	−0.802483
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 374.825i	− 1.45846i	−0.684268	−	0.729231i	$-0.739878\pi$
$257$	− 374.825i	− 1.45846i	0.684268	−	0.729231i	$-0.260122\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	400.829i	1.52406i	0.647539	+	0.762032i	$0.275798\pi$
$263$	400.829i	1.52406i	−0.647539	+	0.762032i	$0.724202\pi$
$264$	0	0
$265$	45.0000	0.169811
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 337.801i	− 1.25576i	−0.778308	−	0.627882i	$-0.783922\pi$
$269$	− 337.801i	− 1.25576i	0.778308	−	0.627882i	$-0.216078\pi$
$270$	0	0
$271$	354.558	1.30833	0.654167	−	0.756351i	$-0.273020\pi$
$271$	354.558	1.30833	0.654167	+	0.756351i	$0.273020\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 169.081i	− 0.614840i
$276$	0	0
$277$	−275.306	−0.993885	−0.496942	−	0.867784i	$-0.665544\pi$
$277$	−275.306	−0.993885	−0.496942	+	0.867784i	$0.665544\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	190.120i	0.676585i	0.941041	+	0.338292i	$0.109849\pi$
$281$	190.120i	0.676585i	−0.941041	+	0.338292i	$0.890151\pi$
$282$	0	0
$283$	145.830	0.515299	0.257650	−	0.966238i	$-0.417052\pi$
$283$	145.830	0.515299	0.257650	+	0.966238i	$0.417052\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−400.293	−1.38510
$290$	0	0
$291$	0	0
$292$	0	0
$293$	340.870i	1.16338i	0.813411	+	0.581689i	$0.197608\pi$
$293$	340.870i	1.16338i	−0.813411	+	0.581689i	$0.802392\pi$
$294$	0	0
$295$	337.076	1.14263
$296$	0	0
$297$	0	0
$298$	0	0
$299$	826.739i	2.76501i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 377.220i	− 1.23679i
$306$	0	0
$307$	115.300	0.375569	0.187784	−	0.982210i	$-0.439869\pi$
$307$	115.300	0.375569	0.187784	+	0.982210i	$0.439869\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	135.889i	0.436944i	0.975843	+	0.218472i	$0.0701072\pi$
$311$	135.889i	0.436944i	−0.975843	+	0.218472i	$0.929893\pi$
$312$	0	0
$313$	19.1767	0.0612674	0.0306337	−	0.999531i	$-0.490247\pi$
$313$	19.1767	0.0612674	0.0306337	+	0.999531i	$0.490247\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 267.099i	− 0.842583i	−0.906925	−	0.421292i	$-0.861577\pi$
$317$	− 267.099i	− 0.842583i	0.906925	−	0.421292i	$-0.138423\pi$
$318$	0	0
$319$	−38.9116	−0.121980
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 586.707i	− 1.81643i
$324$	0	0
$325$	−688.504	−2.11848
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	480.694	1.45225	0.726124	−	0.687564i	$-0.241320\pi$
$331$	480.694	1.45225	0.726124	+	0.687564i	$0.241320\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 481.314i	− 1.43676i
$336$	0	0
$337$	−234.823	−0.696805	−0.348403	−	0.937345i	$-0.613276\pi$
$337$	−234.823	−0.696805	−0.348403	+	0.937345i	$0.613276\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	60.1330i	0.176343i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 61.2440i	− 0.176496i	−0.996099	−	0.0882478i	$-0.971873\pi$
$347$	− 61.2440i	− 0.176496i	0.996099	−	0.0882478i	$-0.0281267\pi$
$348$	0	0
$349$	−84.5173	−0.242170	−0.121085	−	0.992642i	$-0.538637\pi$
$349$	−84.5173	−0.242170	−0.121085	+	0.992642i	$0.538637\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 551.543i	− 1.56245i	−0.624252	−	0.781223i	$-0.714596\pi$
$353$	− 551.543i	− 1.56245i	0.624252	−	0.781223i	$-0.285404\pi$
$354$	0	0
$355$	317.382	0.894033
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 285.653i	− 0.795691i	−0.917452	−	0.397845i	$-0.869758\pi$
$359$	− 285.653i	− 0.795691i	0.917452	−	0.397845i	$-0.130242\pi$
$360$	0	0
$361$	138.388	0.383346
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 31.8442i	− 0.0872445i
$366$	0	0
$367$	−11.2651	−0.0306950	−0.0153475	−	0.999882i	$-0.504885\pi$
$367$	−11.2651	−0.0306950	−0.0153475	+	0.999882i	$0.504885\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	159.375	0.427279	0.213640	−	0.976913i	$-0.431468\pi$
$373$	159.375	0.427279	0.213640	+	0.976913i	$0.431468\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	158.450i	0.420291i
$378$	0	0
$379$	315.375	0.832124	0.416062	−	0.909336i	$-0.363410\pi$
$379$	315.375	0.832124	0.416062	+	0.909336i	$0.363410\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 499.257i	− 1.30354i	−0.758415	−	0.651772i	$-0.774026\pi$
$383$	− 499.257i	− 1.30354i	0.758415	−	0.651772i	$-0.225974\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 546.127i	− 1.40393i	−0.712213	−	0.701963i	$-0.752307\pi$
$389$	− 546.127i	− 1.40393i	0.712213	−	0.701963i	$-0.247693\pi$
$390$	0	0
$391$	−1044.42	−2.67116
$392$	0	0
$393$	0	0
$394$	0	0
$395$	318.559i	0.806479i
$396$	0	0
$397$	−523.647	−1.31901	−0.659505	−	0.751701i	$-0.729234\pi$
$397$	−523.647	−1.31901	−0.659505	+	0.751701i	$0.729234\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	547.100i	1.36434i	0.731194	+	0.682169i	$0.238963\pi$
$401$	547.100i	1.36434i	−0.731194	+	0.682169i	$0.761037\pi$
$402$	0	0
$403$	244.864	0.607604
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 307.991i	− 0.756734i
$408$	0	0
$409$	644.798	1.57652	0.788261	−	0.615341i	$-0.210982\pi$
$409$	644.798	1.57652	0.788261	+	0.615341i	$0.210982\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1111.50	2.67831
$416$	0	0
$417$	0	0
$418$	0	0
$419$	64.9379i	0.154983i	0.996993	+	0.0774916i	$0.0246911\pi$
$419$	64.9379i	0.154983i	−0.996993	+	0.0774916i	$0.975309\pi$
$420$	0	0
$421$	−283.722	−0.673924	−0.336962	−	0.941518i	$-0.609399\pi$
$421$	−283.722	−0.673924	−0.336962	+	0.941518i	$0.609399\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 869.790i	− 2.04657i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 36.5106i	− 0.0847114i	−0.999103	−	0.0423557i	$-0.986514\pi$
$431$	− 36.5106i	− 0.0847114i	0.999103	−	0.0423557i	$-0.0134863\pi$
$432$	0	0
$433$	−382.082	−0.882406	−0.441203	−	0.897407i	$-0.645448\pi$
$433$	−382.082	−0.882406	−0.441203	+	0.897407i	$0.645448\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 888.983i	− 2.03429i
$438$	0	0
$439$	579.946	1.32106	0.660531	−	0.750799i	$-0.270331\pi$
$439$	579.946	1.32106	0.660531	+	0.750799i	$0.270331\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	705.313i	1.59213i	0.605212	+	0.796065i	$0.293088\pi$
$443$	705.313i	1.59213i	−0.605212	+	0.796065i	$0.706912\pi$
$444$	0	0
$445$	334.694	0.752121
$446$	0	0
$447$	0	0
$448$	0	0
$449$	532.400i	1.18575i	0.805296	+	0.592873i	$0.202006\pi$
$449$	532.400i	1.18575i	−0.805296	+	0.592873i	$0.797994\pi$
$450$	0	0
$451$	60.2458	0.133583
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	444.905	0.973534	0.486767	−	0.873532i	$-0.338176\pi$
$457$	444.905	0.973534	0.486767	+	0.873532i	$0.338176\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	380.463i	0.825300i	0.910890	+	0.412650i	$0.135397\pi$
$461$	380.463i	0.825300i	−0.910890	+	0.412650i	$0.864603\pi$
$462$	0	0
$463$	−847.035	−1.82945	−0.914724	−	0.404079i	$-0.867592\pi$
$463$	−847.035	−1.82945	−0.914724	+	0.404079i	$0.867592\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	175.372i	0.375528i	0.982214	+	0.187764i	$0.0601240\pi$
$467$	175.372i	0.375528i	−0.982214	+	0.187764i	$0.939876\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 154.881i	− 0.327444i
$474$	0	0
$475$	740.341	1.55861
$476$	0	0
$477$	0	0
$478$	0	0
$479$	215.500i	0.449896i	0.974371	+	0.224948i	$0.0722212\pi$
$479$	215.500i	0.449896i	−0.974371	+	0.224948i	$0.927779\pi$
$480$	0	0
$481$	−1254.15	−2.60738
$482$	0	0
$483$	0	0
$484$	0	0
$485$	301.290i	0.621216i
$486$	0	0
$487$	−392.287	−0.805517	−0.402759	−	0.915306i	$-0.631949\pi$
$487$	−392.287	−0.805517	−0.402759	+	0.915306i	$0.631949\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 491.584i	− 1.00119i	−0.865682	−	0.500595i	$-0.833115\pi$
$491$	− 491.584i	− 1.00119i	0.865682	−	0.500595i	$-0.166885\pi$
$492$	0	0
$493$	−200.170	−0.406025
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	124.965	0.250432	0.125216	−	0.992130i	$-0.460038\pi$
$499$	124.965	0.250432	0.125216	+	0.992130i	$0.460038\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	21.9357i	0.0436097i	0.999762	+	0.0218049i	$0.00694125\pi$
$503$	21.9357i	0.0436097i	−0.999762	+	0.0218049i	$0.993059\pi$
$504$	0	0
$505$	712.586	1.41106
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 291.832i	− 0.573344i	−0.958029	−	0.286672i	$-0.907451\pi$
$509$	− 291.832i	− 0.573344i	0.958029	−	0.286672i	$-0.0925491\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 442.207i	− 0.858654i
$516$	0	0
$517$	−194.877	−0.376938
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 525.539i	− 1.00871i	−0.863496	−	0.504356i	$-0.831730\pi$
$521$	− 525.539i	− 1.00871i	0.863496	−	0.504356i	$-0.168270\pi$
$522$	0	0
$523$	−240.675	−0.460181	−0.230091	−	0.973169i	$-0.573902\pi$
$523$	−240.675	−0.460181	−0.230091	+	0.973169i	$0.573902\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	309.338i	0.586979i
$528$	0	0
$529$	−1053.52	−1.99153
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 245.323i	− 0.460269i
$534$	0	0
$535$	−1306.53	−2.44212
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−342.416	−0.632932	−0.316466	−	0.948604i	$-0.602496\pi$
$541$	−342.416	−0.632932	−0.316466	+	0.948604i	$0.602496\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1503.59i	− 2.75888i
$546$	0	0
$547$	94.5237	0.172804	0.0864019	−	0.996260i	$-0.472463\pi$
$547$	94.5237	0.172804	0.0864019	+	0.996260i	$0.472463\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 170.379i	− 0.309218i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	67.7437i	0.121622i	0.998149	+	0.0608112i	$0.0193688\pi$
$557$	67.7437i	0.121622i	−0.998149	+	0.0608112i	$0.980631\pi$
$558$	0	0
$559$	−630.681	−1.12823
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 517.165i	− 0.918588i	−0.888284	−	0.459294i	$-0.848103\pi$
$563$	− 517.165i	− 0.918588i	0.888284	−	0.459294i	$-0.151897\pi$
$564$	0	0
$565$	1034.15	1.83036
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 214.875i	− 0.377637i	−0.982012	−	0.188818i	$-0.939534\pi$
$569$	− 214.875i	− 0.377637i	0.982012	−	0.188818i	$-0.0604657\pi$
$570$	0	0
$571$	434.933	0.761705	0.380852	−	0.924636i	$-0.375631\pi$
$571$	434.933	0.761705	0.380852	+	0.924636i	$0.375631\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 1317.91i	− 2.29202i
$576$	0	0
$577$	−307.965	−0.533736	−0.266868	−	0.963733i	$-0.585989\pi$
$577$	−307.965	−0.533736	−0.266868	+	0.963733i	$0.585989\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−30.1229	−0.0516688
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 342.668i	− 0.583761i	−0.956455	−	0.291881i	$-0.905719\pi$
$587$	− 342.668i	− 0.583761i	0.956455	−	0.291881i	$-0.0942810\pi$
$588$	0	0
$589$	−263.300	−0.447028
$590$	0	0
$591$	0	0
$592$	0	0
$593$	580.117i	0.978275i	0.872207	+	0.489138i	$0.162688\pi$
$593$	580.117i	0.978275i	−0.872207	+	0.489138i	$0.837312\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	533.073i	0.889939i	0.895546	+	0.444969i	$0.146785\pi$
$599$	533.073i	0.889939i	−0.895546	+	0.444969i	$0.853215\pi$
$600$	0	0
$601$	244.388	0.406636	0.203318	−	0.979113i	$-0.434828\pi$
$601$	244.388	0.406636	0.203318	+	0.979113i	$0.434828\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 723.943i	− 1.19660i
$606$	0	0
$607$	489.416	0.806287	0.403143	−	0.915137i	$-0.367918\pi$
$607$	489.416	0.806287	0.403143	+	0.915137i	$0.367918\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	793.548i	1.29877i
$612$	0	0
$613$	−399.558	−0.651808	−0.325904	−	0.945403i	$-0.605669\pi$
$613$	−399.558	−0.651808	−0.325904	+	0.945403i	$0.605669\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	800.485i	1.29738i	0.761052	+	0.648691i	$0.224683\pi$
$617$	800.485i	1.29738i	−0.761052	+	0.648691i	$0.775317\pi$
$618$	0	0
$619$	−737.192	−1.19094	−0.595470	−	0.803377i	$-0.703034\pi$
$619$	−737.192	−1.19094	−0.595470	+	0.803377i	$0.703034\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−355.681	−0.569090
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1584.37i	− 2.51888i
$630$	0	0
$631$	−1046.82	−1.65898	−0.829490	−	0.558521i	$-0.811369\pi$
$631$	−1046.82	−1.65898	−0.829490	+	0.558521i	$0.811369\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1422.37i	2.23995i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 176.420i	− 0.275226i	−0.990486	−	0.137613i	$-0.956057\pi$
$641$	− 176.420i	− 0.275226i	0.990486	−	0.137613i	$-0.0439431\pi$
$642$	0	0
$643$	865.349	1.34580	0.672900	−	0.739733i	$-0.265048\pi$
$643$	865.349	1.34580	0.672900	+	0.739733i	$0.265048\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 544.704i	− 0.841892i	−0.907086	−	0.420946i	$-0.861698\pi$
$647$	− 544.704i	− 0.841892i	0.907086	−	0.420946i	$-0.138302\pi$
$648$	0	0
$649$	−225.638	−0.347670
$650$	0	0
$651$	0	0
$652$	0	0
$653$	647.090i	0.990949i	0.868623	+	0.495474i	$0.165006\pi$
$653$	647.090i	0.990949i	−0.868623	+	0.495474i	$0.834994\pi$
$654$	0	0
$655$	−385.782	−0.588981
$656$	0	0
$657$	0	0
$658$	0	0
$659$	265.738i	0.403245i	0.979463	+	0.201622i	$0.0646213\pi$
$659$	265.738i	0.403245i	−0.979463	+	0.201622i	$0.935379\pi$
$660$	0	0
$661$	−865.558	−1.30947	−0.654734	−	0.755859i	$-0.727219\pi$
$661$	−865.558	−1.30947	−0.654734	+	0.755859i	$0.727219\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−303.300	−0.454722
$668$	0	0
$669$	0	0
$670$	0	0
$671$	252.510i	0.376320i
$672$	0	0
$673$	−344.022	−0.511176	−0.255588	−	0.966786i	$-0.582269\pi$
$673$	−344.022	−0.511176	−0.255588	+	0.966786i	$0.582269\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 100.365i	− 0.148249i	−0.997249	−	0.0741247i	$-0.976384\pi$
$677$	− 100.365i	− 0.148249i	0.997249	−	0.0741247i	$-0.0236163\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	267.273i	0.391322i	0.980672	+	0.195661i	$0.0626852\pi$
$683$	267.273i	0.391322i	−0.980672	+	0.195661i	$0.937315\pi$
$684$	0	0
$685$	−1213.20	−1.77109
$686$	0	0
$687$	0	0
$688$	0	0
$689$	122.662i	0.178029i
$690$	0	0
$691$	−539.880	−0.781302	−0.390651	−	0.920539i	$-0.627750\pi$
$691$	−539.880	−0.781302	−0.390651	+	0.920539i	$0.627750\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.9013i	0.0200019i
$696$	0	0
$697$	309.918	0.444646
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1083.03i	1.54498i	0.635025	+	0.772492i	$0.280990\pi$
$701$	1083.03i	1.54498i	−0.635025	+	0.772492i	$0.719010\pi$
$702$	0	0
$703$	1348.57	1.91831
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−573.962	−0.809537	−0.404768	−	0.914419i	$-0.632648\pi$
$709$	−573.962	−0.809537	−0.404768	+	0.914419i	$0.632648\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	468.711i	0.657379i
$714$	0	0
$715$	808.675	1.13101
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 672.907i	− 0.935892i	−0.883757	−	0.467946i	$-0.844994\pi$
$719$	− 672.907i	− 0.935892i	0.883757	−	0.467946i	$-0.155006\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 252.587i	− 0.348395i
$726$	0	0
$727$	924.627	1.27184	0.635920	−	0.771755i	$-0.280621\pi$
$727$	924.627	1.27184	0.635920	+	0.771755i	$0.280621\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 796.742i	− 1.08993i
$732$	0	0
$733$	−982.239	−1.34003	−0.670013	−	0.742349i	$-0.733712\pi$
$733$	−982.239	−1.34003	−0.670013	+	0.742349i	$0.733712\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	322.191i	0.437165i
$738$	0	0
$739$	−104.233	−0.141046	−0.0705230	−	0.997510i	$-0.522467\pi$
$739$	−104.233	−0.141046	−0.0705230	+	0.997510i	$0.522467\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1381.60i	− 1.85949i	−0.368203	−	0.929745i	$-0.620027\pi$
$743$	− 1381.60i	− 1.85949i	0.368203	−	0.929745i	$-0.379973\pi$
$744$	0	0
$745$	−1293.88	−1.73675
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−318.312	−0.423851	−0.211926	−	0.977286i	$-0.567973\pi$
$751$	−318.312	−0.423851	−0.211926	+	0.977286i	$0.567973\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	983.480i	1.30262i
$756$	0	0
$757$	541.097	0.714792	0.357396	−	0.933953i	$-0.383665\pi$
$757$	541.097	0.714792	0.357396	+	0.933953i	$0.383665\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 686.134i	− 0.901622i	−0.892619	−	0.450811i	$-0.851135\pi$
$761$	− 686.134i	− 0.901622i	0.892619	−	0.450811i	$-0.148865\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	918.806i	1.19792i
$768$	0	0
$769$	−1233.76	−1.60437	−0.802187	−	0.597073i	$-0.796330\pi$
$769$	−1233.76	−1.60437	−0.802187	+	0.597073i	$0.796330\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	355.382i	0.459744i	0.973221	+	0.229872i	$0.0738308\pi$
$773$	355.382i	0.459744i	−0.973221	+	0.229872i	$0.926169\pi$
$774$	0	0
$775$	−390.341	−0.503665
$776$	0	0
$777$	0	0
$778$	0	0
$779$	263.793i	0.338631i
$780$	0	0
$781$	−212.455	−0.272029
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 432.250i	− 0.550636i
$786$	0	0
$787$	62.7823	0.0797742	0.0398871	−	0.999204i	$-0.487300\pi$
$787$	62.7823	0.0797742	0.0398871	+	0.999204i	$0.487300\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1028.23	1.29664
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 294.027i	− 0.368917i	−0.982840	−	0.184459i	$-0.940947\pi$
$797$	− 294.027i	− 0.368917i	0.982840	−	0.184459i	$-0.0590531\pi$
$798$	0	0
$799$	−1002.49	−1.25468
$800$	0	0
$801$	0	0
$802$	0	0
$803$	21.3165i	0.0265460i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 172.400i	− 0.213103i	−0.994307	−	0.106551i	$-0.966019\pi$
$809$	− 172.400i	− 0.213103i	0.994307	−	0.106551i	$-0.0339808\pi$
$810$	0	0
$811$	−28.1511	−0.0347115	−0.0173558	−	0.999849i	$-0.505525\pi$
$811$	−28.1511	−0.0347115	−0.0173558	+	0.999849i	$0.505525\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1028.96i	1.26253i
$816$	0	0
$817$	678.164	0.830066
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 895.260i	− 1.09045i	−0.838290	−	0.545225i	$-0.816444\pi$
$821$	− 895.260i	− 1.09045i	0.838290	−	0.545225i	$-0.183556\pi$
$822$	0	0
$823$	1337.99	1.62574	0.812872	−	0.582442i	$-0.197903\pi$
$823$	1337.99	1.62574	0.812872	+	0.582442i	$0.197903\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	330.739i	0.399926i	0.979803	+	0.199963i	$0.0640821\pi$
$827$	330.739i	0.399926i	−0.979803	+	0.199963i	$0.935918\pi$
$828$	0	0
$829$	1070.57	1.29140	0.645698	−	0.763593i	$-0.276567\pi$
$829$	1070.57	1.29140	0.645698	+	0.763593i	$0.276567\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2456.47	2.94188
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 818.205i	− 0.975214i	−0.873063	−	0.487607i	$-0.837870\pi$
$839$	− 818.205i	− 0.975214i	0.873063	−	0.487607i	$-0.162130\pi$
$840$	0	0
$841$	782.871	0.930881
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 2004.46i	− 2.37214i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 2400.66i	− 2.82098i
$852$	0	0
$853$	1455.47	1.70630	0.853148	−	0.521670i	$-0.174691\pi$
$853$	1455.47	1.70630	0.853148	+	0.521670i	$0.174691\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.1706i	0.0363718i	0.999835	+	0.0181859i	$0.00578907\pi$
$857$	31.1706i	0.0363718i	−0.999835	+	0.0181859i	$0.994211\pi$
$858$	0	0
$859$	1178.67	1.37215	0.686074	−	0.727532i	$-0.259333\pi$
$859$	1178.67	1.37215	0.686074	+	0.727532i	$0.259333\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	540.684i	0.626517i	0.949668	+	0.313258i	$0.101421\pi$
$863$	540.684i	0.626517i	−0.949668	+	0.313258i	$0.898579\pi$
$864$	0	0
$865$	−437.823	−0.506154
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 213.243i	− 0.245389i
$870$	0	0
$871$	1311.97	1.50629
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1475.95	1.68295	0.841476	−	0.540295i	$-0.181687\pi$
$877$	1475.95	1.68295	0.841476	+	0.540295i	$0.181687\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	919.702i	1.04393i	0.852967	+	0.521965i	$0.174801\pi$
$881$	919.702i	1.04393i	−0.852967	+	0.521965i	$0.825199\pi$
$882$	0	0
$883$	−938.164	−1.06247	−0.531237	−	0.847223i	$-0.678272\pi$
$883$	−938.164	−1.06247	−0.531237	+	0.847223i	$0.678272\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	756.836i	0.853254i	0.904428	+	0.426627i	$0.140298\pi$
$887$	756.836i	0.853254i	−0.904428	+	0.426627i	$0.859702\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 853.292i	− 0.955534i
$894$	0	0
$895$	−1445.72	−1.61532
$896$	0	0
$897$	0	0
$898$	0	0
$899$	89.8315i	0.0999238i
$900$	0	0
$901$	−154.959	−0.171986
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1681.91i	− 1.85846i
$906$	0	0
$907$	−383.679	−0.423019	−0.211510	−	0.977376i	$-0.567838\pi$
$907$	−383.679	−0.423019	−0.211510	+	0.977376i	$0.567838\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	25.2548i	0.0277221i	0.999904	+	0.0138610i	$0.00441225\pi$
$911$	25.2548i	0.0277221i	−0.999904	+	0.0138610i	$0.995588\pi$
$912$	0	0
$913$	−744.035	−0.814934
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−813.116	−0.884784	−0.442392	−	0.896822i	$-0.645870\pi$
$919$	−813.116	−0.884784	−0.442392	+	0.896822i	$0.645870\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	865.124i	0.937296i
$924$	0	0
$925$	1999.25	2.16136
$926$	0	0
$927$	0	0
$928$	0	0
$929$	322.989i	0.347674i	0.984774	+	0.173837i	$0.0556166\pi$
$929$	322.989i	0.347674i	−0.984774	+	0.173837i	$0.944383\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1021.60i	1.09262i
$936$	0	0
$937$	−649.858	−0.693552	−0.346776	−	0.937948i	$-0.612724\pi$
$937$	−649.858	−0.693552	−0.346776	+	0.937948i	$0.612724\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 665.782i	− 0.707526i	−0.935335	−	0.353763i	$-0.884902\pi$
$941$	− 665.782i	− 0.707526i	0.935335	−	0.353763i	$-0.115098\pi$
$942$	0	0
$943$	469.590	0.497975
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1450.83i	1.53203i	0.642824	+	0.766014i	$0.277763\pi$
$947$	1450.83i	1.53203i	−0.642824	+	0.766014i	$0.722237\pi$
$948$	0	0
$949$	86.8016	0.0914663
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 562.272i	− 0.590002i	−0.955497	−	0.295001i	$-0.904680\pi$
$953$	− 562.272i	− 0.590002i	0.955497	−	0.295001i	$-0.0953200\pi$
$954$	0	0
$955$	1122.25	1.17513
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−822.177	−0.855543
$962$	0	0
$963$	0	0
$964$	0	0
$965$	918.494i	0.951807i
$966$	0	0
$967$	−1238.53	−1.28080	−0.640399	−	0.768042i	$-0.721231\pi$
$967$	−1238.53	−1.28080	−0.640399	+	0.768042i	$0.721231\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	439.923i	0.453061i	0.974004	+	0.226531i	$0.0727384\pi$
$971$	439.923i	0.453061i	−0.974004	+	0.226531i	$0.927262\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 775.654i	− 0.793914i	−0.917837	−	0.396957i	$-0.870066\pi$
$977$	− 775.654i	− 0.793914i	0.917837	−	0.396957i	$-0.129934\pi$
$978$	0	0
$979$	−224.043	−0.228849
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1353.23i	1.37663i	0.725411	+	0.688316i	$0.241650\pi$
$983$	1353.23i	1.37663i	−0.725411	+	0.688316i	$0.758350\pi$
$984$	0	0
$985$	−164.965	−0.167478
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1207.23i	− 1.22066i
$990$	0	0
$991$	−214.615	−0.216564	−0.108282	−	0.994120i	$-0.534535\pi$
$991$	−214.615	−0.216564	−0.108282	+	0.994120i	$0.534535\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 151.860i	− 0.152624i
$996$	0	0
$997$	1038.61	1.04174	0.520869	−	0.853637i	$-0.325608\pi$
$997$	1038.61	1.04174	0.520869	+	0.853637i	$0.325608\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.c.g.197.1		4
3.2	odd	2	inner	1764.3.c.g.197.4		4
7.2	even	3		1764.3.bk.f.557.1		8
7.3	odd	6		252.3.bk.b.233.1	yes	8
7.4	even	3		1764.3.bk.f.1745.4		8
7.5	odd	6		252.3.bk.b.53.4	yes	8
7.6	odd	2		1764.3.c.e.197.4		4
21.2	odd	6		1764.3.bk.f.557.4		8
21.5	even	6		252.3.bk.b.53.1	✓	8
21.11	odd	6		1764.3.bk.f.1745.1		8
21.17	even	6		252.3.bk.b.233.4	yes	8
21.20	even	2		1764.3.c.e.197.1		4
28.3	even	6		1008.3.dc.d.737.1		8
28.19	even	6		1008.3.dc.d.305.4		8
84.47	odd	6		1008.3.dc.d.305.1		8
84.59	odd	6		1008.3.dc.d.737.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.bk.b.53.1	✓	8	21.5	even	6
252.3.bk.b.53.4	yes	8	7.5	odd	6
252.3.bk.b.233.1	yes	8	7.3	odd	6
252.3.bk.b.233.4	yes	8	21.17	even	6
1008.3.dc.d.305.1		8	84.47	odd	6
1008.3.dc.d.305.4		8	28.19	even	6
1008.3.dc.d.737.1		8	28.3	even	6
1008.3.dc.d.737.4		8	84.59	odd	6
1764.3.c.e.197.1		4	21.20	even	2
1764.3.c.e.197.4		4	7.6	odd	2
1764.3.c.g.197.1		4	1.1	even	1	trivial
1764.3.c.g.197.4		4	3.2	odd	2	inner
1764.3.bk.f.557.1		8	7.2	even	3
1764.3.bk.f.557.4		8	21.2	odd	6
1764.3.bk.f.1745.1		8	21.11	odd	6
1764.3.bk.f.1745.4		8	7.4	even	3

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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-7.62426i q^{5} +5.10366i q^{11} +20.7823 q^{13} +26.2544i q^{17} -22.3470 q^{19} +39.7809i q^{23} -33.1293 q^{25} +7.62426i q^{29} +11.7823 q^{31} -60.3470 q^{37} -11.8044i q^{41} -30.3470 q^{43} +38.1838i q^{47} +5.90221i q^{53} +38.9116 q^{55} +44.2109i q^{59} +49.4763 q^{61} -158.450i q^{65} +63.1293 q^{67} +41.6279i q^{71} +4.17670 q^{73} -41.7823 q^{79} +145.784i q^{83} +200.170 q^{85} +43.8986i q^{89} +170.379i q^{95} -39.5173 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 56 q^{13} - 8 q^{19} - 24 q^{25} + 20 q^{31} - 160 q^{37} - 40 q^{43} + 20 q^{55} + 8 q^{61} + 144 q^{67} + 288 q^{73} - 140 q^{79} + 448 q^{85} + 276 q^{97}+O(q^{100}) \) 4 * q + 56 * q^13 - 8 * q^19 - 24 * q^25 + 20 * q^31 - 160 * q^37 - 40 * q^43 + 20 * q^55 + 8 * q^61 + 144 * q^67 + 288 * q^73 - 140 * q^79 + 448 * q^85 + 276 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more