LMFDB - Embedded newform 729.3.b.a.728.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [729,3,Mod(728,729)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(729, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("729.728");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 729 = 3^{6} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	729.b (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:	$19.8638112719$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			728.2
Character	$\chi$	$=$	729.728
Dual form			729.3.b.a.728.29

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.51633i q^{2} -8.36459 q^{4} +4.92652i q^{5} +3.39378 q^{7} +15.3473i q^{8} +O(q^{10})$	\(q-3.51633i q^{2} -8.36459 q^{4} +4.92652i q^{5} +3.39378 q^{7} +15.3473i q^{8} +17.3233 q^{10} -15.8809i q^{11} -3.57802 q^{13} -11.9337i q^{14} +20.5080 q^{16} -11.5458i q^{17} -19.1991 q^{19} -41.2083i q^{20} -55.8424 q^{22} +15.4486i q^{23} +0.729362 q^{25} +12.5815i q^{26} -28.3876 q^{28} -11.1940i q^{29} -24.5121 q^{31} -10.7235i q^{32} -40.5990 q^{34} +16.7196i q^{35} -42.5655 q^{37} +67.5102i q^{38} -75.6090 q^{40} -6.67338i q^{41} -47.8656 q^{43} +132.837i q^{44} +54.3225 q^{46} +51.3341i q^{47} -37.4822 q^{49} -2.56468i q^{50} +29.9286 q^{52} +61.1404i q^{53} +78.2375 q^{55} +52.0856i q^{56} -39.3618 q^{58} -44.9576i q^{59} -4.81824 q^{61} +86.1927i q^{62} +44.3245 q^{64} -17.6272i q^{65} +11.8909 q^{67} +96.5762i q^{68} +58.7915 q^{70} -88.3565i q^{71} -43.1453 q^{73} +149.674i q^{74} +160.592 q^{76} -53.8962i q^{77} -149.058 q^{79} +101.033i q^{80} -23.4658 q^{82} +79.3179i q^{83} +56.8808 q^{85} +168.311i q^{86} +243.729 q^{88} -80.0210i q^{89} -12.1430 q^{91} -129.221i q^{92} +180.508 q^{94} -94.5846i q^{95} -184.123 q^{97} +131.800i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 48 q^{4}+O(q^{10}) $ 30 * q - 48 * q^4	$ 30 q - 48 q^{4} + 6 q^{10} + 48 q^{16} + 6 q^{19} - 24 q^{22} - 30 q^{25} - 12 q^{28} + 6 q^{37} - 24 q^{40} + 6 q^{46} - 42 q^{49} + 96 q^{52} - 12 q^{55} + 48 q^{58} + 18 q^{61} + 102 q^{64} - 90 q^{67} - 150 q^{70} + 132 q^{73} - 24 q^{76} - 12 q^{82} + 96 q^{88} - 192 q^{91} - 24 q^{94}+O(q^{100}) $ 30 * q - 48 * q^4 + 6 * q^10 + 48 * q^16 + 6 * q^19 - 24 * q^22 - 30 * q^25 - 12 * q^28 + 6 * q^37 - 24 * q^40 + 6 * q^46 - 42 * q^49 + 96 * q^52 - 12 * q^55 + 48 * q^58 + 18 * q^61 + 102 * q^64 - 90 * q^67 - 150 * q^70 + 132 * q^73 - 24 * q^76 - 12 * q^82 + 96 * q^88 - 192 * q^91 - 24 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/729\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 3.51633i	− 1.75817i	−0.476669	−	0.879083i	$-0.658156\pi$
$2$	− 3.51633i	− 1.75817i	0.476669	−	0.879083i	$-0.341844\pi$
$3$	0	0
$3$	0	0
$4$	−8.36459	−2.09115
$5$	4.92652i	0.985305i	0.870226	+	0.492652i	$0.163973\pi$
$5$	4.92652i	0.985305i	−0.870226	+	0.492652i	$0.836027\pi$
$6$	0	0
$7$	3.39378	0.484826	0.242413	−	0.970173i	$-0.422061\pi$
$7$	3.39378	0.484826	0.242413	+	0.970173i	$0.422061\pi$
$8$	15.3473i	1.91842i
$9$	0	0
$10$	17.3233	1.73233
$11$	− 15.8809i	− 1.44372i	−0.692042	−	0.721858i	$-0.743289\pi$
$11$	− 15.8809i	− 1.44372i	0.692042	−	0.721858i	$-0.256711\pi$
$12$	0	0
$13$	−3.57802	−0.275232	−0.137616	−	0.990486i	$-0.543944\pi$
$13$	−3.57802	−0.275232	−0.137616	+	0.990486i	$0.543944\pi$
$14$	− 11.9337i	− 0.852405i
$15$	0	0
$16$	20.5080	1.28175
$17$	− 11.5458i	− 0.679167i	−0.940576	−	0.339583i	$-0.889714\pi$
$17$	− 11.5458i	− 0.679167i	0.940576	−	0.339583i	$-0.110286\pi$
$18$	0	0
$19$	−19.1991	−1.01048	−0.505238	−	0.862980i	$-0.668595\pi$
$19$	−19.1991	−1.01048	−0.505238	+	0.862980i	$0.668595\pi$
$20$	− 41.2083i	− 2.06042i
$21$	0	0
$22$	−55.8424	−2.53829
$23$	15.4486i	0.671679i	0.941919	+	0.335840i	$0.109020\pi$
$23$	15.4486i	0.671679i	−0.941919	+	0.335840i	$0.890980\pi$
$24$	0	0
$25$	0.729362	0.0291745
$26$	12.5815i	0.483904i
$27$	0	0
$28$	−28.3876	−1.01384
$29$	− 11.1940i	− 0.386000i	−0.981199	−	0.193000i	$-0.938178\pi$
$29$	− 11.1940i	− 0.386000i	0.981199	−	0.193000i	$-0.0618218\pi$
$30$	0	0
$31$	−24.5121	−0.790713	−0.395357	−	0.918528i	$-0.629379\pi$
$31$	−24.5121	−0.790713	−0.395357	+	0.918528i	$0.629379\pi$
$32$	− 10.7235i	− 0.335110i
$33$	0	0
$34$	−40.5990	−1.19409
$35$	16.7196i	0.477702i
$36$	0	0
$37$	−42.5655	−1.15042	−0.575209	−	0.818006i	$-0.695079\pi$
$37$	−42.5655	−1.15042	−0.575209	+	0.818006i	$0.695079\pi$
$38$	67.5102i	1.77659i
$39$	0	0
$40$	−75.6090	−1.89023
$41$	− 6.67338i	− 0.162765i	−0.996683	−	0.0813827i	$-0.974066\pi$
$41$	− 6.67338i	− 0.162765i	0.996683	−	0.0813827i	$-0.0259336\pi$
$42$	0	0
$43$	−47.8656	−1.11315	−0.556577	−	0.830796i	$-0.687885\pi$
$43$	−47.8656	−1.11315	−0.556577	+	0.830796i	$0.687885\pi$
$44$	132.837i	3.01902i
$45$	0	0
$46$	54.3225	1.18092
$47$	51.3341i	1.09221i	0.837715	+	0.546107i	$0.183891\pi$
$47$	51.3341i	1.09221i	−0.837715	+	0.546107i	$0.816109\pi$
$48$	0	0
$49$	−37.4822	−0.764943
$50$	− 2.56468i	− 0.0512936i
$51$	0	0
$52$	29.9286	0.575551
$53$	61.1404i	1.15359i	0.816888	+	0.576797i	$0.195697\pi$
$53$	61.1404i	1.15359i	−0.816888	+	0.576797i	$0.804303\pi$
$54$	0	0
$55$	78.2375	1.42250
$56$	52.0856i	0.930100i
$57$	0	0
$58$	−39.3618	−0.678652
$59$	− 44.9576i	− 0.761992i	−0.924577	−	0.380996i	$-0.875581\pi$
$59$	− 44.9576i	− 0.761992i	0.924577	−	0.380996i	$-0.124419\pi$
$60$	0	0
$61$	−4.81824	−0.0789875	−0.0394938	−	0.999220i	$-0.512575\pi$
$61$	−4.81824	−0.0789875	−0.0394938	+	0.999220i	$0.512575\pi$
$62$	86.1927i	1.39021i
$63$	0	0
$64$	44.3245	0.692570
$65$	− 17.6272i	− 0.271187i
$66$	0	0
$67$	11.8909	0.177477	0.0887384	−	0.996055i	$-0.471716\pi$
$67$	11.8909	0.177477	0.0887384	+	0.996055i	$0.471716\pi$
$68$	96.5762i	1.42024i
$69$	0	0
$70$	58.7915	0.839879
$71$	− 88.3565i	− 1.24446i	−0.782835	−	0.622229i	$-0.786227\pi$
$71$	− 88.3565i	− 1.24446i	0.782835	−	0.622229i	$-0.213773\pi$
$72$	0	0
$73$	−43.1453	−0.591031	−0.295516	−	0.955338i	$-0.595491\pi$
$73$	−43.1453	−0.591031	−0.295516	+	0.955338i	$0.595491\pi$
$74$	149.674i	2.02263i
$75$	0	0
$76$	160.592	2.11306
$77$	− 53.8962i	− 0.699951i
$78$	0	0
$79$	−149.058	−1.88681	−0.943403	−	0.331648i	$-0.892395\pi$
$79$	−149.058	−1.88681	−0.943403	+	0.331648i	$0.892395\pi$
$80$	101.033i	1.26291i
$81$	0	0
$82$	−23.4658	−0.286168
$83$	79.3179i	0.955637i	0.878459	+	0.477819i	$0.158572\pi$
$83$	79.3179i	0.955637i	−0.878459	+	0.477819i	$0.841428\pi$
$84$	0	0
$85$	56.8808	0.669186
$86$	168.311i	1.95711i
$87$	0	0
$88$	243.729	2.76965
$89$	− 80.0210i	− 0.899112i	−0.893252	−	0.449556i	$-0.851582\pi$
$89$	− 80.0210i	− 0.899112i	0.893252	−	0.449556i	$-0.148418\pi$
$90$	0	0
$91$	−12.1430	−0.133440
$92$	− 129.221i	− 1.40458i
$93$	0	0
$94$	180.508	1.92029
$95$	− 94.5846i	− 0.995627i
$96$	0	0
$97$	−184.123	−1.89818	−0.949088	−	0.315010i	$-0.897992\pi$
$97$	−184.123	−1.89818	−0.949088	+	0.315010i	$0.897992\pi$
$98$	131.800i	1.34490i
$99$	0	0
$100$	−6.10082	−0.0610082
$101$	− 102.278i	− 1.01265i	−0.862341	−	0.506327i	$-0.831003\pi$
$101$	− 102.278i	− 1.01265i	0.862341	−	0.506327i	$-0.168997\pi$
$102$	0	0
$103$	131.937	1.28095	0.640473	−	0.767981i	$-0.278738\pi$
$103$	131.937	1.28095	0.640473	+	0.767981i	$0.278738\pi$
$104$	− 54.9130i	− 0.528010i
$105$	0	0
$106$	214.990	2.02821
$107$	− 21.6029i	− 0.201896i	−0.994892	−	0.100948i	$-0.967812\pi$
$107$	− 21.6029i	− 0.201896i	0.994892	−	0.100948i	$-0.0321876\pi$
$108$	0	0
$109$	−149.823	−1.37452	−0.687262	−	0.726410i	$-0.741187\pi$
$109$	−149.823	−1.37452	−0.687262	+	0.726410i	$0.741187\pi$
$110$	− 275.109i	− 2.50099i
$111$	0	0
$112$	69.5997	0.621426
$113$	− 92.9065i	− 0.822182i	−0.911594	−	0.411091i	$-0.865148\pi$
$113$	− 92.9065i	− 0.822182i	0.911594	−	0.411091i	$-0.134852\pi$
$114$	0	0
$115$	−76.1080	−0.661809
$116$	93.6332i	0.807183i
$117$	0	0
$118$	−158.086	−1.33971
$119$	− 39.1841i	− 0.329278i
$120$	0	0
$121$	−131.202	−1.08431
$122$	16.9425i	0.138873i
$123$	0	0
$124$	205.034	1.65350
$125$	126.756i	1.01405i
$126$	0	0
$127$	101.644	0.800348	0.400174	−	0.916439i	$-0.368950\pi$
$127$	101.644	0.800348	0.400174	+	0.916439i	$0.368950\pi$
$128$	− 198.754i	− 1.55276i
$129$	0	0
$130$	−61.9830	−0.476792
$131$	− 31.6232i	− 0.241398i	−0.992689	−	0.120699i	$-0.961486\pi$
$131$	− 31.6232i	− 0.241398i	0.992689	−	0.120699i	$-0.0385137\pi$
$132$	0	0
$133$	−65.1575	−0.489906
$134$	− 41.8125i	− 0.312034i
$135$	0	0
$136$	177.198	1.30293
$137$	27.8536i	0.203311i	0.994820	+	0.101656i	$0.0324140\pi$
$137$	27.8536i	0.203311i	−0.994820	+	0.101656i	$0.967586\pi$
$138$	0	0
$139$	196.579	1.41423	0.707117	−	0.707096i	$-0.249995\pi$
$139$	196.579	1.41423	0.707117	+	0.707096i	$0.249995\pi$
$140$	− 139.852i	− 0.998945i
$141$	0	0
$142$	−310.691	−2.18796
$143$	56.8220i	0.397357i
$144$	0	0
$145$	55.1475	0.380328
$146$	151.713i	1.03913i
$147$	0	0
$148$	356.043	2.40569
$149$	136.118i	0.913543i	0.889584	+	0.456771i	$0.150994\pi$
$149$	136.118i	0.913543i	−0.889584	+	0.456771i	$0.849006\pi$
$150$	0	0
$151$	148.074	0.980621	0.490310	−	0.871548i	$-0.336883\pi$
$151$	148.074	0.980621	0.490310	+	0.871548i	$0.336883\pi$
$152$	− 294.654i	− 1.93852i
$153$	0	0
$154$	−189.517	−1.23063
$155$	− 120.760i	− 0.779094i
$156$	0	0
$157$	35.1711	0.224020	0.112010	−	0.993707i	$-0.464271\pi$
$157$	35.1711	0.224020	0.112010	+	0.993707i	$0.464271\pi$
$158$	524.136i	3.31732i
$159$	0	0
$160$	52.8297	0.330186
$161$	52.4293i	0.325648i
$162$	0	0
$163$	147.146	0.902737	0.451368	−	0.892338i	$-0.350936\pi$
$163$	147.146	0.902737	0.451368	+	0.892338i	$0.350936\pi$
$164$	55.8201i	0.340366i
$165$	0	0
$166$	278.908	1.68017
$167$	− 213.045i	− 1.27572i	−0.770152	−	0.637861i	$-0.779820\pi$
$167$	− 213.045i	− 1.27572i	0.770152	−	0.637861i	$-0.220180\pi$
$168$	0	0
$169$	−156.198	−0.924247
$170$	− 200.012i	− 1.17654i
$171$	0	0
$172$	400.376	2.32777
$173$	52.9212i	0.305903i	0.988234	+	0.152951i	$0.0488778\pi$
$173$	52.9212i	0.305903i	−0.988234	+	0.152951i	$0.951122\pi$
$174$	0	0
$175$	2.47530	0.0141446
$176$	− 325.685i	− 1.85048i
$177$	0	0
$178$	−281.380	−1.58079
$179$	56.3824i	0.314986i	0.987520	+	0.157493i	$0.0503411\pi$
$179$	56.3824i	0.314986i	−0.987520	+	0.157493i	$0.949659\pi$
$180$	0	0
$181$	184.278	1.01811	0.509056	−	0.860733i	$-0.329994\pi$
$181$	184.278	1.01811	0.509056	+	0.860733i	$0.329994\pi$
$182$	42.6989i	0.234609i
$183$	0	0
$184$	−237.095	−1.28856
$185$	− 209.700i	− 1.13351i
$186$	0	0
$187$	−183.358	−0.980523
$188$	− 429.388i	− 2.28398i
$189$	0	0
$190$	−332.591	−1.75048
$191$	29.4886i	0.154391i	0.997016	+	0.0771953i	$0.0245965\pi$
$191$	29.4886i	0.154391i	−0.997016	+	0.0771953i	$0.975404\pi$
$192$	0	0
$193$	−378.169	−1.95942	−0.979712	−	0.200411i	$-0.935772\pi$
$193$	−378.169	−1.95942	−0.979712	+	0.200411i	$0.935772\pi$
$194$	647.438i	3.33731i
$195$	0	0
$196$	313.523	1.59961
$197$	− 13.3876i	− 0.0679571i	−0.999423	−	0.0339786i	$-0.989182\pi$
$197$	− 13.3876i	− 0.0679571i	0.999423	−	0.0339786i	$-0.0108178\pi$
$198$	0	0
$199$	−76.1062	−0.382443	−0.191222	−	0.981547i	$-0.561245\pi$
$199$	−76.1062	−0.382443	−0.191222	+	0.981547i	$0.561245\pi$
$200$	11.1938i	0.0559689i
$201$	0	0
$202$	−359.644	−1.78041
$203$	− 37.9900i	− 0.187143i
$204$	0	0
$205$	32.8766	0.160373
$206$	− 463.936i	− 2.25211i
$207$	0	0
$208$	−73.3779	−0.352778
$209$	304.898i	1.45884i
$210$	0	0
$211$	−0.407044	−0.00192912	−0.000964559	−	1.00000i	$-0.500307\pi$
$211$	−0.407044	−0.00192912	−0.000964559		1.00000i	$0.500307\pi$
$212$	− 511.415i	− 2.41233i
$213$	0	0
$214$	−75.9629	−0.354967
$215$	− 235.811i	− 1.09680i
$216$	0	0
$217$	−83.1888	−0.383359
$218$	526.827i	2.41664i
$219$	0	0
$220$	−654.424	−2.97466
$221$	41.3112i	0.186928i
$222$	0	0
$223$	−14.4530	−0.0648118	−0.0324059	−	0.999475i	$-0.510317\pi$
$223$	−14.4530	−0.0648118	−0.0324059	+	0.999475i	$0.510317\pi$
$224$	− 36.3933i	− 0.162470i
$225$	0	0
$226$	−326.690	−1.44553
$227$	347.650i	1.53150i	0.643140	+	0.765748i	$0.277631\pi$
$227$	347.650i	1.53150i	−0.643140	+	0.765748i	$0.722369\pi$
$228$	0	0
$229$	97.8462	0.427276	0.213638	−	0.976913i	$-0.431469\pi$
$229$	97.8462	0.427276	0.213638	+	0.976913i	$0.431469\pi$
$230$	267.621i	1.16357i
$231$	0	0
$232$	171.798	0.740509
$233$	− 69.2176i	− 0.297071i	−0.988907	−	0.148536i	$-0.952544\pi$
$233$	− 69.2176i	− 0.297071i	0.988907	−	0.148536i	$-0.0474560\pi$
$234$	0	0
$235$	−252.899	−1.07616
$236$	376.051i	1.59344i
$237$	0	0
$238$	−137.784	−0.578925
$239$	309.215i	1.29379i	0.762580	+	0.646894i	$0.223932\pi$
$239$	309.215i	1.29379i	−0.762580	+	0.646894i	$0.776068\pi$
$240$	0	0
$241$	199.180	0.826473	0.413237	−	0.910624i	$-0.364398\pi$
$241$	199.180	0.826473	0.413237	+	0.910624i	$0.364398\pi$
$242$	461.350i	1.90640i
$243$	0	0
$244$	40.3026	0.165175
$245$	− 184.657i	− 0.753702i
$246$	0	0
$247$	68.6945	0.278116
$248$	− 376.196i	− 1.51692i
$249$	0	0
$250$	445.717	1.78287
$251$	47.9031i	0.190849i	0.995437	+	0.0954245i	$0.0304209\pi$
$251$	47.9031i	0.190849i	−0.995437	+	0.0954245i	$0.969579\pi$
$252$	0	0
$253$	245.338	0.969714
$254$	− 357.415i	− 1.40714i
$255$	0	0
$256$	−521.586	−2.03745
$257$	− 315.828i	− 1.22890i	−0.788954	−	0.614452i	$-0.789377\pi$
$257$	− 315.828i	− 1.22890i	0.788954	−	0.614452i	$-0.210623\pi$
$258$	0	0
$259$	−144.458	−0.557753
$260$	147.444i	0.567093i
$261$	0	0
$262$	−111.198	−0.424419
$263$	− 149.633i	− 0.568946i	−0.958684	−	0.284473i	$-0.908181\pi$
$263$	− 149.633i	− 0.568946i	0.958684	−	0.284473i	$-0.0918186\pi$
$264$	0	0
$265$	−301.210	−1.13664
$266$	229.115i	0.861336i
$267$	0	0
$268$	−99.4628	−0.371130
$269$	− 60.3653i	− 0.224406i	−0.993685	−	0.112203i	$-0.964209\pi$
$269$	− 60.3653i	− 0.224406i	0.993685	−	0.112203i	$-0.0357907\pi$
$270$	0	0
$271$	127.511	0.470522	0.235261	−	0.971932i	$-0.424406\pi$
$271$	127.511	0.470522	0.235261	+	0.971932i	$0.424406\pi$
$272$	− 236.782i	− 0.870522i
$273$	0	0
$274$	97.9426	0.357455
$275$	− 11.5829i	− 0.0421197i
$276$	0	0
$277$	−177.375	−0.640343	−0.320171	−	0.947360i	$-0.603741\pi$
$277$	−177.375	−0.640343	−0.320171	+	0.947360i	$0.603741\pi$
$278$	− 691.235i	− 2.48646i
$279$	0	0
$280$	−256.601	−0.916431
$281$	− 507.005i	− 1.80429i	−0.431434	−	0.902145i	$-0.641992\pi$
$281$	− 507.005i	− 1.80429i	0.431434	−	0.902145i	$-0.358008\pi$
$282$	0	0
$283$	−91.5725	−0.323578	−0.161789	−	0.986825i	$-0.551726\pi$
$283$	−91.5725	−0.323578	−0.161789	+	0.986825i	$0.551726\pi$
$284$	739.066i	2.60234i
$285$	0	0
$286$	199.805	0.698619
$287$	− 22.6480i	− 0.0789129i
$288$	0	0
$289$	155.694	0.538733
$290$	− 193.917i	− 0.668679i
$291$	0	0
$292$	360.893	1.23593
$293$	− 478.865i	− 1.63435i	−0.576389	−	0.817175i	$-0.695539\pi$
$293$	− 478.865i	− 1.63435i	0.576389	−	0.817175i	$-0.304461\pi$
$294$	0	0
$295$	221.484	0.750795
$296$	− 653.267i	− 2.20698i
$297$	0	0
$298$	478.636	1.60616
$299$	− 55.2754i	− 0.184868i
$300$	0	0
$301$	−162.445	−0.539686
$302$	− 520.676i	− 1.72409i
$303$	0	0
$304$	−393.734	−1.29518
$305$	− 23.7372i	− 0.0778268i
$306$	0	0
$307$	48.3077	0.157354	0.0786770	−	0.996900i	$-0.474930\pi$
$307$	48.3077	0.157354	0.0786770	+	0.996900i	$0.474930\pi$
$308$	450.820i	1.46370i
$309$	0	0
$310$	−424.630	−1.36978
$311$	− 430.212i	− 1.38332i	−0.722223	−	0.691660i	$-0.756880\pi$
$311$	− 430.212i	− 1.38332i	0.722223	−	0.691660i	$-0.243120\pi$
$312$	0	0
$313$	−201.203	−0.642823	−0.321411	−	0.946940i	$-0.604157\pi$
$313$	−201.203	−0.642823	−0.321411	+	0.946940i	$0.604157\pi$
$314$	− 123.673i	− 0.393864i
$315$	0	0
$316$	1246.81	3.94559
$317$	− 18.3708i	− 0.0579521i	−0.999580	−	0.0289760i	$-0.990775\pi$
$317$	− 18.3708i	− 0.0579521i	0.999580	−	0.0289760i	$-0.00922465\pi$
$318$	0	0
$319$	−177.771	−0.557274
$320$	218.366i	0.682393i
$321$	0	0
$322$	184.359	0.572543
$323$	221.669i	0.686282i
$324$	0	0
$325$	−2.60967	−0.00802976
$326$	− 517.415i	− 1.58716i
$327$	0	0
$328$	102.419	0.312252
$329$	174.217i	0.529534i
$330$	0	0
$331$	152.035	0.459320	0.229660	−	0.973271i	$-0.426239\pi$
$331$	152.035	0.459320	0.229660	+	0.973271i	$0.426239\pi$
$332$	− 663.461i	− 1.99838i
$333$	0	0
$334$	−749.138	−2.24293
$335$	58.5810i	0.174869i
$336$	0	0
$337$	479.675	1.42337	0.711684	−	0.702499i	$-0.247933\pi$
$337$	479.675	1.42337	0.711684	+	0.702499i	$0.247933\pi$
$338$	549.243i	1.62498i
$339$	0	0
$340$	−475.785	−1.39937
$341$	389.274i	1.14156i
$342$	0	0
$343$	−293.502	−0.855691
$344$	− 734.610i	− 2.13549i
$345$	0	0
$346$	186.088	0.537828
$347$	− 409.420i	− 1.17989i	−0.807445	−	0.589943i	$-0.799150\pi$
$347$	− 409.420i	− 1.17989i	0.807445	−	0.589943i	$-0.200850\pi$
$348$	0	0
$349$	107.002	0.306597	0.153299	−	0.988180i	$-0.451010\pi$
$349$	107.002	0.306597	0.153299	+	0.988180i	$0.451010\pi$
$350$	− 8.70397i	− 0.0248685i
$351$	0	0
$352$	−170.299	−0.483804
$353$	− 558.336i	− 1.58169i	−0.612018	−	0.790844i	$-0.709642\pi$
$353$	− 558.336i	− 1.58169i	0.612018	−	0.790844i	$-0.290358\pi$
$354$	0	0
$355$	435.290	1.22617
$356$	669.343i	1.88018i
$357$	0	0
$358$	198.259	0.553797
$359$	221.670i	0.617464i	0.951149	+	0.308732i	$0.0999047\pi$
$359$	221.670i	0.617464i	−0.951149	+	0.308732i	$0.900095\pi$
$360$	0	0
$361$	7.60375	0.0210630
$362$	− 647.984i	− 1.79001i
$363$	0	0
$364$	101.571	0.279042
$365$	− 212.556i	− 0.582346i
$366$	0	0
$367$	172.035	0.468760	0.234380	−	0.972145i	$-0.424694\pi$
$367$	172.035	0.468760	0.234380	+	0.972145i	$0.424694\pi$
$368$	316.820i	0.860925i
$369$	0	0
$370$	−737.374	−1.99290
$371$	207.497i	0.559292i
$372$	0	0
$373$	125.029	0.335198	0.167599	−	0.985855i	$-0.446399\pi$
$373$	125.029	0.335198	0.167599	+	0.985855i	$0.446399\pi$
$374$	644.747i	1.72392i
$375$	0	0
$376$	−787.842	−2.09532
$377$	40.0523i	0.106240i
$378$	0	0
$379$	−613.387	−1.61843	−0.809217	−	0.587510i	$-0.800108\pi$
$379$	−613.387	−1.61843	−0.809217	+	0.587510i	$0.800108\pi$
$380$	791.161i	2.08200i
$381$	0	0
$382$	103.692	0.271444
$383$	73.1952i	0.191110i	0.995424	+	0.0955550i	$0.0304626\pi$
$383$	73.1952i	0.191110i	−0.995424	+	0.0955550i	$0.969537\pi$
$384$	0	0
$385$	265.521	0.689665
$386$	1329.77i	3.44499i
$387$	0	0
$388$	1540.11	3.96937
$389$	17.1195i	0.0440090i	0.999758	+	0.0220045i	$0.00700482\pi$
$389$	17.1195i	0.0440090i	−0.999758	+	0.0220045i	$0.992995\pi$
$390$	0	0
$391$	178.367	0.456182
$392$	− 575.253i	− 1.46748i
$393$	0	0
$394$	−47.0751	−0.119480
$395$	− 734.336i	− 1.85908i
$396$	0	0
$397$	−165.177	−0.416063	−0.208032	−	0.978122i	$-0.566706\pi$
$397$	−165.177	−0.416063	−0.208032	+	0.978122i	$0.566706\pi$
$398$	267.615i	0.672398i
$399$	0	0
$400$	14.9578	0.0373944
$401$	− 113.477i	− 0.282986i	−0.989939	−	0.141493i	$-0.954810\pi$
$401$	− 113.477i	− 0.282986i	0.989939	−	0.141493i	$-0.0451903\pi$
$402$	0	0
$403$	87.7047	0.217630
$404$	855.514i	2.11761i
$405$	0	0
$406$	−133.586	−0.329028
$407$	675.977i	1.66088i
$408$	0	0
$409$	−549.490	−1.34350	−0.671748	−	0.740780i	$-0.734456\pi$
$409$	−549.490	−1.34350	−0.671748	+	0.740780i	$0.734456\pi$
$410$	− 115.605i	− 0.281963i
$411$	0	0
$412$	−1103.60	−2.67865
$413$	− 152.576i	− 0.369434i
$414$	0	0
$415$	−390.761	−0.941594
$416$	38.3690i	0.0922331i
$417$	0	0
$418$	1072.12	2.56488
$419$	− 362.062i	− 0.864109i	−0.901847	−	0.432055i	$-0.857789\pi$
$419$	− 362.062i	− 0.864109i	0.901847	−	0.432055i	$-0.142211\pi$
$420$	0	0
$421$	639.984	1.52015	0.760076	−	0.649834i	$-0.225162\pi$
$421$	639.984	1.52015	0.760076	+	0.649834i	$0.225162\pi$
$422$	1.43130i	0.00339171i
$423$	0	0
$424$	−938.343	−2.21307
$425$	− 8.42110i	− 0.0198143i
$426$	0	0
$427$	−16.3521	−0.0382952
$428$	180.699i	0.422195i
$429$	0	0
$430$	−829.190	−1.92835
$431$	140.062i	0.324969i	0.986711	+	0.162484i	$0.0519507\pi$
$431$	140.062i	0.324969i	−0.986711	+	0.162484i	$0.948049\pi$
$432$	0	0
$433$	28.4373	0.0656750	0.0328375	−	0.999461i	$-0.489546\pi$
$433$	28.4373	0.0656750	0.0328375	+	0.999461i	$0.489546\pi$
$434$	292.520i	0.674008i
$435$	0	0
$436$	1253.21	2.87433
$437$	− 296.599i	− 0.678716i
$438$	0	0
$439$	652.653	1.48668	0.743341	−	0.668913i	$-0.233240\pi$
$439$	652.653	1.48668	0.743341	+	0.668913i	$0.233240\pi$
$440$	1200.74i	2.72895i
$441$	0	0
$442$	145.264	0.328651
$443$	671.334i	1.51543i	0.652588	+	0.757713i	$0.273683\pi$
$443$	671.334i	1.51543i	−0.652588	+	0.757713i	$0.726317\pi$
$444$	0	0
$445$	394.225	0.885900
$446$	50.8216i	0.113950i
$447$	0	0
$448$	150.428	0.335776
$449$	− 9.85152i	− 0.0219410i	−0.999940	−	0.0109705i	$-0.996508\pi$
$449$	− 9.85152i	− 0.0219410i	0.999940	−	0.0109705i	$-0.00349209\pi$
$450$	0	0
$451$	−105.979	−0.234987
$452$	777.125i	1.71930i
$453$	0	0
$454$	1222.45	2.69263
$455$	− 59.8229i	− 0.131479i
$456$	0	0
$457$	770.621	1.68626	0.843130	−	0.537710i	$-0.180710\pi$
$457$	770.621	1.68626	0.843130	+	0.537710i	$0.180710\pi$
$458$	− 344.060i	− 0.751222i
$459$	0	0
$460$	636.612	1.38394
$461$	− 437.345i	− 0.948687i	−0.880340	−	0.474343i	$-0.842685\pi$
$461$	− 437.345i	− 0.948687i	0.880340	−	0.474343i	$-0.157315\pi$
$462$	0	0
$463$	209.886	0.453317	0.226658	−	0.973974i	$-0.427220\pi$
$463$	209.886	0.453317	0.226658	+	0.973974i	$0.427220\pi$
$464$	− 229.567i	− 0.494755i
$465$	0	0
$466$	−243.392	−0.522301
$467$	179.408i	0.384170i	0.981378	+	0.192085i	$0.0615250\pi$
$467$	179.408i	0.384170i	−0.981378	+	0.192085i	$0.938475\pi$
$468$	0	0
$469$	40.3553	0.0860454
$470$	889.275i	1.89207i
$471$	0	0
$472$	689.979	1.46182
$473$	760.147i	1.60708i
$474$	0	0
$475$	−14.0031	−0.0294801
$476$	327.759i	0.688569i
$477$	0	0
$478$	1087.30	2.27469
$479$	96.4860i	0.201432i	0.994915	+	0.100716i	$0.0321134\pi$
$479$	96.4860i	0.201432i	−0.994915	+	0.100716i	$0.967887\pi$
$480$	0	0
$481$	152.300	0.316632
$482$	− 700.383i	− 1.45308i
$483$	0	0
$484$	1097.45	2.26746
$485$	− 907.087i	− 1.87028i
$486$	0	0
$487$	392.647	0.806257	0.403128	−	0.915143i	$-0.367923\pi$
$487$	392.647	0.806257	0.403128	+	0.915143i	$0.367923\pi$
$488$	− 73.9472i	− 0.151531i
$489$	0	0
$490$	−649.316	−1.32513
$491$	417.226i	0.849747i	0.905253	+	0.424873i	$0.139681\pi$
$491$	417.226i	0.849747i	−0.905253	+	0.424873i	$0.860319\pi$
$492$	0	0
$493$	−129.244	−0.262158
$494$	− 241.553i	− 0.488973i
$495$	0	0
$496$	−502.694	−1.01350
$497$	− 299.863i	− 0.603346i
$498$	0	0
$499$	−814.814	−1.63289	−0.816447	−	0.577421i	$-0.804059\pi$
$499$	−814.814	−1.63289	−0.816447	+	0.577421i	$0.804059\pi$
$500$	− 1060.26i	− 2.12053i
$501$	0	0
$502$	168.443	0.335544
$503$	585.691i	1.16439i	0.813048	+	0.582197i	$0.197807\pi$
$503$	585.691i	1.16439i	−0.813048	+	0.582197i	$0.802193\pi$
$504$	0	0
$505$	503.876	0.997773
$506$	− 862.688i	− 1.70492i
$507$	0	0
$508$	−850.212	−1.67365
$509$	827.866i	1.62646i	0.581946	+	0.813228i	$0.302292\pi$
$509$	827.866i	1.62646i	−0.581946	+	0.813228i	$0.697708\pi$
$510$	0	0
$511$	−146.426	−0.286548
$512$	1039.05i	2.02940i
$513$	0	0
$514$	−1110.56	−2.16062
$515$	649.993i	1.26212i
$516$	0	0
$517$	815.230	1.57685
$518$	507.963i	0.980623i
$519$	0	0
$520$	270.530	0.520251
$521$	− 656.699i	− 1.26046i	−0.776409	−	0.630230i	$-0.782961\pi$
$521$	− 656.699i	− 1.26046i	0.776409	−	0.630230i	$-0.217039\pi$
$522$	0	0
$523$	521.774	0.997655	0.498827	−	0.866701i	$-0.333764\pi$
$523$	521.774	0.997655	0.498827	+	0.866701i	$0.333764\pi$
$524$	264.515i	0.504800i
$525$	0	0
$526$	−526.158	−1.00030
$527$	283.013i	0.537026i
$528$	0	0
$529$	290.340	0.548847
$530$	1059.15i	1.99840i
$531$	0	0
$532$	545.015	1.02446
$533$	23.8775i	0.0447982i
$534$	0	0
$535$	106.427	0.198929
$536$	182.494i	0.340475i
$537$	0	0
$538$	−212.264	−0.394543
$539$	595.250i	1.10436i
$540$	0	0
$541$	−10.3822	−0.0191908	−0.00959538	−	0.999954i	$-0.503054\pi$
$541$	−10.3822	−0.0191908	−0.00959538	+	0.999954i	$0.503054\pi$
$542$	− 448.372i	− 0.827255i
$543$	0	0
$544$	−123.812	−0.227596
$545$	− 738.107i	− 1.35432i
$546$	0	0
$547$	243.089	0.444403	0.222202	−	0.975001i	$-0.428676\pi$
$547$	243.089	0.444403	0.222202	+	0.975001i	$0.428676\pi$
$548$	− 232.984i	− 0.425154i
$549$	0	0
$550$	−40.7293	−0.0740534
$551$	214.914i	0.390044i
$552$	0	0
$553$	−505.870	−0.914774
$554$	623.709i	1.12583i
$555$	0	0
$556$	−1644.30	−2.95737
$557$	485.496i	0.871627i	0.900037	+	0.435813i	$0.143539\pi$
$557$	485.496i	0.871627i	−0.900037	+	0.435813i	$0.856461\pi$
$558$	0	0
$559$	171.264	0.306375
$560$	342.885i	0.612294i
$561$	0	0
$562$	−1782.80	−3.17224
$563$	443.448i	0.787653i	0.919185	+	0.393826i	$0.128849\pi$
$563$	443.448i	0.787653i	−0.919185	+	0.393826i	$0.871151\pi$
$564$	0	0
$565$	457.706	0.810099
$566$	321.999i	0.568904i
$567$	0	0
$568$	1356.04	2.38739
$569$	229.818i	0.403898i	0.979396	+	0.201949i	$0.0647276\pi$
$569$	229.818i	0.403898i	−0.979396	+	0.201949i	$0.935272\pi$
$570$	0	0
$571$	−298.606	−0.522952	−0.261476	−	0.965210i	$-0.584209\pi$
$571$	−298.606	−0.522952	−0.261476	+	0.965210i	$0.584209\pi$
$572$	− 475.293i	− 0.830931i
$573$	0	0
$574$	−79.6379	−0.138742
$575$	11.2676i	0.0195959i
$576$	0	0
$577$	−639.484	−1.10829	−0.554145	−	0.832420i	$-0.686955\pi$
$577$	−639.484	−1.10829	−0.554145	+	0.832420i	$0.686955\pi$
$578$	− 547.471i	− 0.947181i
$579$	0	0
$580$	−461.286	−0.795321
$581$	269.188i	0.463318i
$582$	0	0
$583$	970.963	1.66546
$584$	− 662.166i	− 1.13385i
$585$	0	0
$586$	−1683.85	−2.87346
$587$	− 639.431i	− 1.08932i	−0.838657	−	0.544660i	$-0.816659\pi$
$587$	− 639.431i	− 1.08932i	0.838657	−	0.544660i	$-0.183341\pi$
$588$	0	0
$589$	470.609	0.798997
$590$	− 778.813i	− 1.32002i
$591$	0	0
$592$	−872.933	−1.47455
$593$	576.408i	0.972021i	0.873953	+	0.486010i	$0.161548\pi$
$593$	576.408i	0.972021i	−0.873953	+	0.486010i	$0.838452\pi$
$594$	0	0
$595$	193.041	0.324439
$596$	− 1138.57i	− 1.91035i
$597$	0	0
$598$	−194.367	−0.325028
$599$	− 627.769i	− 1.04803i	−0.851710	−	0.524014i	$-0.824434\pi$
$599$	− 627.769i	− 1.04803i	0.851710	−	0.524014i	$-0.175566\pi$
$600$	0	0
$601$	−148.184	−0.246563	−0.123281	−	0.992372i	$-0.539342\pi$
$601$	−148.184	−0.246563	−0.123281	+	0.992372i	$0.539342\pi$
$602$	571.212i	0.948858i
$603$	0	0
$604$	−1238.58	−2.05062
$605$	− 646.370i	− 1.06838i
$606$	0	0
$607$	31.8127	0.0524097	0.0262048	−	0.999657i	$-0.491658\pi$
$607$	31.8127	0.0524097	0.0262048	+	0.999657i	$0.491658\pi$
$608$	205.882i	0.338621i
$609$	0	0
$610$	−83.4678	−0.136832
$611$	− 183.674i	− 0.300612i
$612$	0	0
$613$	−90.0158	−0.146845	−0.0734223	−	0.997301i	$-0.523392\pi$
$613$	−90.0158	−0.146845	−0.0734223	+	0.997301i	$0.523392\pi$
$614$	− 169.866i	− 0.276655i
$615$	0	0
$616$	827.164	1.34280
$617$	− 514.590i	− 0.834020i	−0.908902	−	0.417010i	$-0.863078\pi$
$617$	− 514.590i	− 0.834020i	0.908902	−	0.417010i	$-0.136922\pi$
$618$	0	0
$619$	−901.624	−1.45658	−0.728291	−	0.685268i	$-0.759685\pi$
$619$	−901.624	−1.45658	−0.728291	+	0.685268i	$0.759685\pi$
$620$	1010.10i	1.62920i
$621$	0	0
$622$	−1512.77	−2.43211
$623$	− 271.574i	− 0.435913i
$624$	0	0
$625$	−606.234	−0.969974
$626$	707.498i	1.13019i
$627$	0	0
$628$	−294.192	−0.468459
$629$	491.454i	0.781326i
$630$	0	0
$631$	158.107	0.250566	0.125283	−	0.992121i	$-0.460016\pi$
$631$	158.107	0.250566	0.125283	+	0.992121i	$0.460016\pi$
$632$	− 2287.64i	− 3.61968i
$633$	0	0
$634$	−64.5979	−0.101889
$635$	500.753i	0.788587i
$636$	0	0
$637$	134.112	0.210537
$638$	625.100i	0.979781i
$639$	0	0
$640$	979.165	1.52994
$641$	− 770.333i	− 1.20177i	−0.799336	−	0.600884i	$-0.794815\pi$
$641$	− 770.333i	− 1.20177i	0.799336	−	0.600884i	$-0.205185\pi$
$642$	0	0
$643$	234.152	0.364155	0.182078	−	0.983284i	$-0.441718\pi$
$643$	234.152	0.364155	0.182078	+	0.983284i	$0.441718\pi$
$644$	− 438.550i	− 0.680978i
$645$	0	0
$646$	779.462	1.20660
$647$	− 1162.87i	− 1.79733i	−0.438638	−	0.898664i	$-0.644539\pi$
$647$	− 1162.87i	− 1.79733i	0.438638	−	0.898664i	$-0.355461\pi$
$648$	0	0
$649$	−713.965	−1.10010
$650$	9.17647i	0.0141176i
$651$	0	0
$652$	−1230.82	−1.88776
$653$	897.644i	1.37465i	0.726352	+	0.687323i	$0.241214\pi$
$653$	897.644i	1.37465i	−0.726352	+	0.687323i	$0.758786\pi$
$654$	0	0
$655$	155.792	0.237851
$656$	− 136.858i	− 0.208624i
$657$	0	0
$658$	612.604	0.931009
$659$	− 204.046i	− 0.309630i	−0.987944	−	0.154815i	$-0.950522\pi$
$659$	− 204.046i	− 0.309630i	0.987944	−	0.154815i	$-0.0494781\pi$
$660$	0	0
$661$	−647.461	−0.979517	−0.489758	−	0.871858i	$-0.662915\pi$
$661$	−647.461	−0.979517	−0.489758	+	0.871858i	$0.662915\pi$
$662$	− 534.605i	− 0.807560i
$663$	0	0
$664$	−1217.32	−1.83331
$665$	− 321.000i	− 0.482706i
$666$	0	0
$667$	172.932	0.259268
$668$	1782.04i	2.66772i
$669$	0	0
$670$	205.990	0.307448
$671$	76.5178i	0.114035i
$672$	0	0
$673$	−444.140	−0.659940	−0.329970	−	0.943991i	$-0.607039\pi$
$673$	−444.140	−0.659940	−0.329970	+	0.943991i	$0.607039\pi$
$674$	− 1686.70i	− 2.50252i
$675$	0	0
$676$	1306.53	1.93274
$677$	− 273.250i	− 0.403618i	−0.979425	−	0.201809i	$-0.935318\pi$
$677$	− 273.250i	− 0.403618i	0.979425	−	0.201809i	$-0.0646821\pi$
$678$	0	0
$679$	−624.874	−0.920286
$680$	872.970i	1.28378i
$681$	0	0
$682$	1368.82	2.00706
$683$	239.261i	0.350309i	0.984541	+	0.175155i	$0.0560426\pi$
$683$	239.261i	0.350309i	−0.984541	+	0.175155i	$0.943957\pi$
$684$	0	0
$685$	−137.222	−0.200323
$686$	1032.05i	1.50445i
$687$	0	0
$688$	−981.627	−1.42678
$689$	− 218.761i	− 0.317506i
$690$	0	0
$691$	−928.844	−1.34420	−0.672101	−	0.740459i	$-0.734608\pi$
$691$	−928.844	−1.34420	−0.672101	+	0.740459i	$0.734608\pi$
$692$	− 442.664i	− 0.639688i
$693$	0	0
$694$	−1439.66	−2.07444
$695$	968.449i	1.39345i
$696$	0	0
$697$	−77.0497	−0.110545
$698$	− 376.256i	− 0.539049i
$699$	0	0
$700$	−20.7049	−0.0295784
$701$	211.750i	0.302068i	0.988529	+	0.151034i	$0.0482604\pi$
$701$	211.750i	0.302068i	−0.988529	+	0.151034i	$0.951740\pi$
$702$	0	0
$703$	817.217	1.16247
$704$	− 703.911i	− 0.999874i
$705$	0	0
$706$	−1963.29	−2.78087
$707$	− 347.110i	− 0.490962i
$708$	0	0
$709$	112.135	0.158159	0.0790794	−	0.996868i	$-0.474802\pi$
$709$	112.135	0.158159	0.0790794	+	0.996868i	$0.474802\pi$
$710$	− 1530.63i	− 2.15581i
$711$	0	0
$712$	1228.11	1.72487
$713$	− 378.678i	− 0.531106i
$714$	0	0
$715$	−279.935	−0.391517
$716$	− 471.616i	− 0.658682i
$717$	0	0
$718$	779.464	1.08560
$719$	− 681.350i	− 0.947636i	−0.880623	−	0.473818i	$-0.842875\pi$
$719$	− 681.350i	− 0.947636i	0.880623	−	0.473818i	$-0.157125\pi$
$720$	0	0
$721$	447.767	0.621036
$722$	− 26.7373i	− 0.0370323i
$723$	0	0
$724$	−1541.41	−2.12902
$725$	− 8.16449i	− 0.0112614i
$726$	0	0
$727$	409.756	0.563626	0.281813	−	0.959469i	$-0.409064\pi$
$727$	409.756	0.563626	0.281813	+	0.959469i	$0.409064\pi$
$728$	− 186.363i	− 0.255993i
$729$	0	0
$730$	−747.419	−1.02386
$731$	552.648i	0.756017i
$732$	0	0
$733$	804.754	1.09789	0.548945	−	0.835858i	$-0.315030\pi$
$733$	804.754	1.09789	0.548945	+	0.835858i	$0.315030\pi$
$734$	− 604.932i	− 0.824158i
$735$	0	0
$736$	165.664	0.225087
$737$	− 188.838i	− 0.256226i
$738$	0	0
$739$	27.5580	0.0372910	0.0186455	−	0.999826i	$-0.494065\pi$
$739$	27.5580	0.0372910	0.0186455	+	0.999826i	$0.494065\pi$
$740$	1754.05i	2.37034i
$741$	0	0
$742$	729.630	0.983329
$743$	261.542i	0.352008i	0.984389	+	0.176004i	$0.0563171\pi$
$743$	261.542i	0.352008i	−0.984389	+	0.176004i	$0.943683\pi$
$744$	0	0
$745$	−670.588	−0.900118
$746$	− 439.643i	− 0.589334i
$747$	0	0
$748$	1533.71	2.05042
$749$	− 73.3155i	− 0.0978846i
$750$	0	0
$751$	−651.035	−0.866891	−0.433446	−	0.901180i	$-0.642702\pi$
$751$	−651.035	−0.866891	−0.433446	+	0.901180i	$0.642702\pi$
$752$	1052.76i	1.39995i
$753$	0	0
$754$	140.837	0.186787
$755$	729.489i	0.966210i
$756$	0	0
$757$	636.818	0.841240	0.420620	−	0.907237i	$-0.361813\pi$
$757$	636.818	0.841240	0.420620	+	0.907237i	$0.361813\pi$
$758$	2156.87i	2.84548i
$759$	0	0
$760$	1451.62	1.91003
$761$	− 883.496i	− 1.16097i	−0.814272	−	0.580484i	$-0.802863\pi$
$761$	− 883.496i	− 1.16097i	0.814272	−	0.580484i	$-0.197137\pi$
$762$	0	0
$763$	−508.467	−0.666405
$764$	− 246.660i	− 0.322853i
$765$	0	0
$766$	257.378	0.336003
$767$	160.859i	0.209725i
$768$	0	0
$769$	543.786	0.707134	0.353567	−	0.935409i	$-0.384969\pi$
$769$	543.786	0.707134	0.353567	+	0.935409i	$0.384969\pi$
$770$	− 933.660i	− 1.21255i
$771$	0	0
$772$	3163.23	4.09744
$773$	− 1159.87i	− 1.50048i	−0.661165	−	0.750241i	$-0.729938\pi$
$773$	− 1159.87i	− 1.50048i	0.661165	−	0.750241i	$-0.270062\pi$
$774$	0	0
$775$	−17.8782	−0.0230687
$776$	− 2825.80i	− 3.64150i
$777$	0	0
$778$	60.1979	0.0773752
$779$	128.123i	0.164471i
$780$	0	0
$781$	−1403.18	−1.79664
$782$	− 627.199i	− 0.802044i
$783$	0	0
$784$	−768.685	−0.980466
$785$	173.271i	0.220728i
$786$	0	0
$787$	169.032	0.214780	0.107390	−	0.994217i	$-0.465751\pi$
$787$	169.032	0.214780	0.107390	+	0.994217i	$0.465751\pi$
$788$	111.981i	0.142108i
$789$	0	0
$790$	−2582.17	−3.26857
$791$	− 315.305i	− 0.398615i
$792$	0	0
$793$	17.2397	0.0217399
$794$	580.817i	0.731508i
$795$	0	0
$796$	636.597	0.799745
$797$	7.38904i	0.00927107i	0.999989	+	0.00463554i	$0.00147554\pi$
$797$	7.38904i	0.00927107i	−0.999989	+	0.00463554i	$0.998524\pi$
$798$	0	0
$799$	592.695	0.741796
$800$	− 7.82134i	− 0.00977667i
$801$	0	0
$802$	−399.025	−0.497537
$803$	685.185i	0.853281i
$804$	0	0
$805$	−258.294	−0.320862
$806$	− 308.399i	− 0.382629i
$807$	0	0
$808$	1569.70	1.94269
$809$	1006.43i	1.24404i	0.783001	+	0.622021i	$0.213688\pi$
$809$	1006.43i	1.24404i	−0.783001	+	0.622021i	$0.786312\pi$
$810$	0	0
$811$	662.217	0.816543	0.408272	−	0.912861i	$-0.366132\pi$
$811$	662.217	0.816543	0.408272	+	0.912861i	$0.366132\pi$
$812$	317.771i	0.391344i
$813$	0	0
$814$	2376.96	2.92010
$815$	724.919i	0.889471i
$816$	0	0
$817$	918.974	1.12482
$818$	1932.19i	2.36209i
$819$	0	0
$820$	−274.999	−0.335364
$821$	698.654i	0.850979i	0.904963	+	0.425489i	$0.139898\pi$
$821$	698.654i	0.850979i	−0.904963	+	0.425489i	$0.860102\pi$
$822$	0	0
$823$	−477.199	−0.579828	−0.289914	−	0.957053i	$-0.593627\pi$
$823$	−477.199	−0.579828	−0.289914	+	0.957053i	$0.593627\pi$
$824$	2024.89i	2.45739i
$825$	0	0
$826$	−536.509	−0.649526
$827$	1609.23i	1.94586i	0.231096	+	0.972931i	$0.425769\pi$
$827$	1609.23i	1.94586i	−0.231096	+	0.972931i	$0.574231\pi$
$828$	0	0
$829$	−1425.02	−1.71896	−0.859481	−	0.511167i	$-0.829213\pi$
$829$	−1425.02	−1.71896	−0.859481	+	0.511167i	$0.829213\pi$
$830$	1374.05i	1.65548i
$831$	0	0
$832$	−158.594	−0.190617
$833$	432.764i	0.519524i
$834$	0	0
$835$	1049.57	1.25697
$836$	− 2550.34i	− 3.05065i
$837$	0	0
$838$	−1273.13	−1.51925
$839$	602.142i	0.717690i	0.933397	+	0.358845i	$0.116829\pi$
$839$	602.142i	0.717690i	−0.933397	+	0.358845i	$0.883171\pi$
$840$	0	0
$841$	715.694	0.851004
$842$	− 2250.40i	− 2.67268i
$843$	0	0
$844$	3.40475	0.00403407
$845$	− 769.512i	− 0.910665i
$846$	0	0
$847$	−445.271	−0.525704
$848$	1253.87i	1.47862i
$849$	0	0
$850$	−29.6114	−0.0348369
$851$	− 657.578i	− 0.772712i
$852$	0	0
$853$	−1396.23	−1.63684	−0.818422	−	0.574618i	$-0.805151\pi$
$853$	−1396.23	−1.63684	−0.818422	+	0.574618i	$0.805151\pi$
$854$	57.4993i	0.0673294i
$855$	0	0
$856$	331.547	0.387321
$857$	− 843.341i	− 0.984062i	−0.870578	−	0.492031i	$-0.836255\pi$
$857$	− 843.341i	− 0.984062i	0.870578	−	0.492031i	$-0.163745\pi$
$858$	0	0
$859$	−663.761	−0.772713	−0.386357	−	0.922349i	$-0.626267\pi$
$859$	−663.761	−0.772713	−0.386357	+	0.922349i	$0.626267\pi$
$860$	1972.46i	2.29356i
$861$	0	0
$862$	492.503	0.571349
$863$	437.898i	0.507414i	0.967281	+	0.253707i	$0.0816499\pi$
$863$	437.898i	0.507414i	−0.967281	+	0.253707i	$0.918350\pi$
$864$	0	0
$865$	−260.717	−0.301408
$866$	− 99.9949i	− 0.115468i
$867$	0	0
$868$	695.840	0.801659
$869$	2367.17i	2.72401i
$870$	0	0
$871$	−42.5460	−0.0488473
$872$	− 2299.39i	− 2.63691i
$873$	0	0
$874$	−1042.94	−1.19330
$875$	430.184i	0.491638i
$876$	0	0
$877$	1482.31	1.69021	0.845104	−	0.534602i	$-0.179539\pi$
$877$	1482.31	1.69021	0.845104	+	0.534602i	$0.179539\pi$
$878$	− 2294.95i	− 2.61383i
$879$	0	0
$880$	1604.49	1.82329
$881$	359.938i	0.408556i	0.978913	+	0.204278i	$0.0654846\pi$
$881$	359.938i	0.408556i	−0.978913	+	0.204278i	$0.934515\pi$
$882$	0	0
$883$	524.614	0.594127	0.297064	−	0.954858i	$-0.403993\pi$
$883$	524.614	0.594127	0.297064	+	0.954858i	$0.403993\pi$
$884$	− 345.551i	− 0.390895i
$885$	0	0
$886$	2360.63	2.66437
$887$	1208.08i	1.36198i	0.732293	+	0.680990i	$0.238450\pi$
$887$	1208.08i	1.36198i	−0.732293	+	0.680990i	$0.761550\pi$
$888$	0	0
$889$	344.958	0.388030
$890$	− 1386.23i	− 1.55756i
$891$	0	0
$892$	120.894	0.135531
$893$	− 985.566i	− 1.10366i
$894$	0	0
$895$	−277.769	−0.310357
$896$	− 674.527i	− 0.752821i
$897$	0	0
$898$	−34.6412	−0.0385760
$899$	274.389i	0.305215i
$900$	0	0
$901$	705.917	0.783482
$902$	372.657i	0.413146i
$903$	0	0
$904$	1425.87	1.57729
$905$	907.852i	1.00315i
$906$	0	0
$907$	−1405.88	−1.55003	−0.775016	−	0.631941i	$-0.782258\pi$
$907$	−1405.88	−1.55003	−0.775016	+	0.631941i	$0.782258\pi$
$908$	− 2907.95i	− 3.20259i
$909$	0	0
$910$	−210.357	−0.231162
$911$	1555.14i	1.70707i	0.521033	+	0.853537i	$0.325547\pi$
$911$	1555.14i	1.70707i	−0.521033	+	0.853537i	$0.674453\pi$
$912$	0	0
$913$	1259.64	1.37967
$914$	− 2709.76i	− 2.96473i
$915$	0	0
$916$	−818.443	−0.893497
$917$	− 107.322i	− 0.117036i
$918$	0	0
$919$	−1361.80	−1.48183	−0.740916	−	0.671597i	$-0.765609\pi$
$919$	−1361.80	−1.48183	−0.740916	+	0.671597i	$0.765609\pi$
$920$	− 1168.06i	− 1.26963i
$921$	0	0
$922$	−1537.85	−1.66795
$923$	316.141i	0.342515i
$924$	0	0
$925$	−31.0457	−0.0335629
$926$	− 738.027i	− 0.797006i
$927$	0	0
$928$	−120.039	−0.129353
$929$	− 1551.80i	− 1.67039i	−0.549952	−	0.835197i	$-0.685354\pi$
$929$	− 1551.80i	− 1.67039i	0.549952	−	0.835197i	$-0.314646\pi$
$930$	0	0
$931$	719.623	0.772957
$932$	578.977i	0.621220i
$933$	0	0
$934$	630.856	0.675435
$935$	− 903.317i	− 0.966114i
$936$	0	0
$937$	−219.145	−0.233879	−0.116939	−	0.993139i	$-0.537308\pi$
$937$	−219.145	−0.233879	−0.116939	+	0.993139i	$0.537308\pi$
$938$	− 141.903i	− 0.151282i
$939$	0	0
$940$	2115.39	2.25042
$941$	93.6544i	0.0995264i	0.998761	+	0.0497632i	$0.0158467\pi$
$941$	93.6544i	0.0995264i	−0.998761	+	0.0497632i	$0.984153\pi$
$942$	0	0
$943$	103.095	0.109326
$944$	− 921.989i	− 0.976683i
$945$	0	0
$946$	2672.93	2.82551
$947$	− 500.370i	− 0.528373i	−0.964472	−	0.264187i	$-0.914896\pi$
$947$	− 500.370i	− 0.528373i	0.964472	−	0.264187i	$-0.0851035\pi$
$948$	0	0
$949$	154.375	0.162671
$950$	49.2394i	0.0518310i
$951$	0	0
$952$	601.371	0.631693
$953$	1342.70i	1.40892i	0.709744	+	0.704459i	$0.248810\pi$
$953$	1342.70i	1.40892i	−0.709744	+	0.704459i	$0.751190\pi$
$954$	0	0
$955$	−145.276	−0.152122
$956$	− 2586.46i	− 2.70550i
$957$	0	0
$958$	339.277	0.354151
$959$	94.5292i	0.0985706i
$960$	0	0
$961$	−360.156	−0.374772
$962$	− 535.537i	− 0.556692i
$963$	0	0
$964$	−1666.06	−1.72828
$965$	− 1863.06i	− 1.93063i
$966$	0	0
$967$	718.141	0.742648	0.371324	−	0.928503i	$-0.378904\pi$
$967$	718.141	0.742648	0.371324	+	0.928503i	$0.378904\pi$
$968$	− 2013.60i	− 2.08017i
$969$	0	0
$970$	−3189.62	−3.28827
$971$	− 1118.48i	− 1.15189i	−0.817489	−	0.575944i	$-0.804635\pi$
$971$	− 1118.48i	− 1.15189i	0.817489	−	0.575944i	$-0.195365\pi$
$972$	0	0
$973$	667.145	0.685658
$974$	− 1380.68i	− 1.41753i
$975$	0	0
$976$	−98.8124	−0.101242
$977$	643.697i	0.658850i	0.944182	+	0.329425i	$0.106855\pi$
$977$	643.697i	0.658850i	−0.944182	+	0.329425i	$0.893145\pi$
$978$	0	0
$979$	−1270.80	−1.29806
$980$	1544.58i	1.57610i
$981$	0	0
$982$	1467.10	1.49400
$983$	− 346.589i	− 0.352583i	−0.984338	−	0.176292i	$-0.943590\pi$
$983$	− 346.589i	− 0.352583i	0.984338	−	0.176292i	$-0.0564102\pi$
$984$	0	0
$985$	65.9541	0.0669585
$986$	454.465i	0.460918i
$987$	0	0
$988$	−574.602	−0.581581
$989$	− 739.458i	− 0.747682i
$990$	0	0
$991$	408.198	0.411905	0.205953	−	0.978562i	$-0.433971\pi$
$991$	408.198	0.411905	0.205953	+	0.978562i	$0.433971\pi$
$992$	262.856i	0.264976i
$993$	0	0
$994$	−1054.42	−1.06078
$995$	− 374.939i	− 0.376823i
$996$	0	0
$997$	−875.283	−0.877917	−0.438958	−	0.898507i	$-0.644652\pi$
$997$	−875.283	−0.877917	−0.438958	+	0.898507i	$0.644652\pi$
$998$	2865.16i	2.87090i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	729.3.b.a.728.2		30
3.2	odd	2	inner	729.3.b.a.728.29		30
27.2	odd	18		243.3.f.c.53.5		30
27.4	even	9		243.3.f.d.107.5		30
27.5	odd	18		81.3.f.a.8.5		30
27.7	even	9		243.3.f.a.134.1		30
27.11	odd	18		27.3.f.a.14.1	yes	30
27.13	even	9		243.3.f.c.188.5		30
27.14	odd	18		243.3.f.b.188.1		30
27.16	even	9		81.3.f.a.71.5		30
27.20	odd	18		243.3.f.d.134.5		30
27.22	even	9		27.3.f.a.2.1	✓	30
27.23	odd	18		243.3.f.a.107.1		30
27.25	even	9		243.3.f.b.53.1		30
108.11	even	18		432.3.bc.a.257.2		30
108.103	odd	18		432.3.bc.a.353.2		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.3.f.a.2.1	✓	30	27.22	even	9
27.3.f.a.14.1	yes	30	27.11	odd	18
81.3.f.a.8.5		30	27.5	odd	18
81.3.f.a.71.5		30	27.16	even	9
243.3.f.a.107.1		30	27.23	odd	18
243.3.f.a.134.1		30	27.7	even	9
243.3.f.b.53.1		30	27.25	even	9
243.3.f.b.188.1		30	27.14	odd	18
243.3.f.c.53.5		30	27.2	odd	18
243.3.f.c.188.5		30	27.13	even	9
243.3.f.d.107.5		30	27.4	even	9
243.3.f.d.134.5		30	27.20	odd	18
432.3.bc.a.257.2		30	108.11	even	18
432.3.bc.a.353.2		30	108.103	odd	18
729.3.b.a.728.2		30	1.1	even	1	trivial
729.3.b.a.728.29		30	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 729 = 3^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	729.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.51633i q^{2} -8.36459 q^{4} +4.92652i q^{5} +3.39378 q^{7} +15.3473i q^{8} +O(q^{10})\)	\(q-3.51633i q^{2} -8.36459 q^{4} +4.92652i q^{5} +3.39378 q^{7} +15.3473i q^{8} +17.3233 q^{10} -15.8809i q^{11} -3.57802 q^{13} -11.9337i q^{14} +20.5080 q^{16} -11.5458i q^{17} -19.1991 q^{19} -41.2083i q^{20} -55.8424 q^{22} +15.4486i q^{23} +0.729362 q^{25} +12.5815i q^{26} -28.3876 q^{28} -11.1940i q^{29} -24.5121 q^{31} -10.7235i q^{32} -40.5990 q^{34} +16.7196i q^{35} -42.5655 q^{37} +67.5102i q^{38} -75.6090 q^{40} -6.67338i q^{41} -47.8656 q^{43} +132.837i q^{44} +54.3225 q^{46} +51.3341i q^{47} -37.4822 q^{49} -2.56468i q^{50} +29.9286 q^{52} +61.1404i q^{53} +78.2375 q^{55} +52.0856i q^{56} -39.3618 q^{58} -44.9576i q^{59} -4.81824 q^{61} +86.1927i q^{62} +44.3245 q^{64} -17.6272i q^{65} +11.8909 q^{67} +96.5762i q^{68} +58.7915 q^{70} -88.3565i q^{71} -43.1453 q^{73} +149.674i q^{74} +160.592 q^{76} -53.8962i q^{77} -149.058 q^{79} +101.033i q^{80} -23.4658 q^{82} +79.3179i q^{83} +56.8808 q^{85} +168.311i q^{86} +243.729 q^{88} -80.0210i q^{89} -12.1430 q^{91} -129.221i q^{92} +180.508 q^{94} -94.5846i q^{95} -184.123 q^{97} +131.800i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 48 q^{4}+O(q^{10}) \) 30 * q - 48 * q^4	\( 30 q - 48 q^{4} + 6 q^{10} + 48 q^{16} + 6 q^{19} - 24 q^{22} - 30 q^{25} - 12 q^{28} + 6 q^{37} - 24 q^{40} + 6 q^{46} - 42 q^{49} + 96 q^{52} - 12 q^{55} + 48 q^{58} + 18 q^{61} + 102 q^{64} - 90 q^{67} - 150 q^{70} + 132 q^{73} - 24 q^{76} - 12 q^{82} + 96 q^{88} - 192 q^{91} - 24 q^{94}+O(q^{100}) \) 30 * q - 48 * q^4 + 6 * q^10 + 48 * q^16 + 6 * q^19 - 24 * q^22 - 30 * q^25 - 12 * q^28 + 6 * q^37 - 24 * q^40 + 6 * q^46 - 42 * q^49 + 96 * q^52 - 12 * q^55 + 48 * q^58 + 18 * q^61 + 102 * q^64 - 90 * q^67 - 150 * q^70 + 132 * q^73 - 24 * q^76 - 12 * q^82 + 96 * q^88 - 192 * q^91 - 24 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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