LMFDB - Embedded newform 864.3.g.b.703.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,3,Mod(703,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("864.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	864.g (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:	$23.5422948407$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.56070144.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 63x^4 - 74x^3 + 70x^2 - 38*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.5
Root			$0.500000 - 1.56488i$ of defining polynomial
Character	$\chi$	$=$	864.703
Dual form			864.3.g.b.703.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.956810 q^{5} -6.34610i q^{7} +O(q^{10})$	$q+0.956810 q^{5} -6.34610i q^{7} -14.4991i q^{11} +1.76367 q^{13} +1.34309 q^{17} +13.0234i q^{19} -4.19916i q^{23} -24.0845 q^{25} -38.1690 q^{29} +0.172759i q^{31} -6.07202i q^{35} +15.1937 q^{37} -69.3965 q^{41} -5.52734i q^{43} -66.7137i q^{47} +8.72695 q^{49} -31.6556 q^{53} -13.8729i q^{55} -12.2410i q^{59} +92.1017 q^{61} +1.68750 q^{65} -26.4583i q^{67} -85.4493i q^{71} -103.920 q^{73} -92.0126 q^{77} +68.3424i q^{79} -137.277i q^{83} +1.28509 q^{85} -67.3405 q^{89} -11.1924i q^{91} +12.4609i q^{95} -95.6956 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^5	$ 8 q - 8 q^{5} + 8 q^{13} - 24 q^{17} + 24 q^{25} + 128 q^{29} + 24 q^{37} - 160 q^{41} - 144 q^{49} + 48 q^{53} - 136 q^{61} + 280 q^{65} + 72 q^{73} - 520 q^{77} + 96 q^{85} + 168 q^{89} + 104 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 + 8 * q^13 - 24 * q^17 + 24 * q^25 + 128 * q^29 + 24 * q^37 - 160 * q^41 - 144 * q^49 + 48 * q^53 - 136 * q^61 + 280 * q^65 + 72 * q^73 - 520 * q^77 + 96 * q^85 + 168 * q^89 + 104 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	0.956810	0.191362	0.0956810	−	0.995412i	$-0.469497\pi$
$5$	0.956810	0.191362	0.0956810	+	0.995412i	$0.469497\pi$
$6$	0	0
$7$	− 6.34610i	− 0.906586i	−0.891361	−	0.453293i	$-0.850249\pi$
$7$	− 6.34610i	− 0.906586i	0.891361	−	0.453293i	$-0.149751\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 14.4991i	− 1.31810i	−0.752100	−	0.659049i	$-0.770959\pi$
$11$	− 14.4991i	− 1.31810i	0.752100	−	0.659049i	$-0.229041\pi$
$12$	0	0
$13$	1.76367	0.135667	0.0678334	−	0.997697i	$-0.478391\pi$
$13$	1.76367	0.135667	0.0678334	+	0.997697i	$0.478391\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.34309	0.0790056	0.0395028	−	0.999219i	$-0.487423\pi$
$17$	1.34309	0.0790056	0.0395028	+	0.999219i	$0.487423\pi$
$18$	0	0
$19$	13.0234i	0.685442i	0.939437	+	0.342721i	$0.111349\pi$
$19$	13.0234i	0.685442i	−0.939437	+	0.342721i	$0.888651\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.19916i	− 0.182572i	−0.995825	−	0.0912861i	$-0.970902\pi$
$23$	− 4.19916i	− 0.182572i	0.995825	−	0.0912861i	$-0.0290978\pi$
$24$	0	0
$25$	−24.0845	−0.963381
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−38.1690	−1.31617	−0.658087	−	0.752942i	$-0.728634\pi$
$29$	−38.1690	−1.31617	−0.658087	+	0.752942i	$0.728634\pi$
$30$	0	0
$31$	0.172759i	0.00557288i	0.999996	+	0.00278644i	$0.000886953\pi$
$31$	0.172759i	0.00557288i	−0.999996	+	0.00278644i	$0.999113\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 6.07202i	− 0.173486i
$36$	0	0
$37$	15.1937	0.410641	0.205320	−	0.978695i	$-0.434176\pi$
$37$	15.1937	0.410641	0.205320	+	0.978695i	$0.434176\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−69.3965	−1.69260	−0.846298	−	0.532709i	$-0.821174\pi$
$41$	−69.3965	−1.69260	−0.846298	+	0.532709i	$0.821174\pi$
$42$	0	0
$43$	− 5.52734i	− 0.128543i	−0.997932	−	0.0642713i	$-0.979528\pi$
$43$	− 5.52734i	− 0.128543i	0.997932	−	0.0642713i	$-0.0204723\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 66.7137i	− 1.41944i	−0.704484	−	0.709720i	$-0.748821\pi$
$47$	− 66.7137i	− 1.41944i	0.704484	−	0.709720i	$-0.251179\pi$
$48$	0	0
$49$	8.72695	0.178101
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−31.6556	−0.597275	−0.298638	−	0.954367i	$-0.596532\pi$
$53$	−31.6556	−0.597275	−0.298638	+	0.954367i	$0.596532\pi$
$54$	0	0
$55$	− 13.8729i	− 0.252234i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 12.2410i	− 0.207475i	−0.994605	−	0.103738i	$-0.966920\pi$
$59$	− 12.2410i	− 0.207475i	0.994605	−	0.103738i	$-0.0330802\pi$
$60$	0	0
$61$	92.1017	1.50986	0.754932	−	0.655803i	$-0.227670\pi$
$61$	92.1017	1.50986	0.754932	+	0.655803i	$0.227670\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.68750	0.0259615
$66$	0	0
$67$	− 26.4583i	− 0.394900i	−0.980313	−	0.197450i	$-0.936734\pi$
$67$	− 26.4583i	− 0.394900i	0.980313	−	0.197450i	$-0.0632661\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 85.4493i	− 1.20351i	−0.798680	−	0.601755i	$-0.794468\pi$
$71$	− 85.4493i	− 1.20351i	0.798680	−	0.601755i	$-0.205532\pi$
$72$	0	0
$73$	−103.920	−1.42356	−0.711781	−	0.702401i	$-0.752111\pi$
$73$	−103.920	−1.42356	−0.711781	+	0.702401i	$0.752111\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−92.0126	−1.19497
$78$	0	0
$79$	68.3424i	0.865093i	0.901612	+	0.432547i	$0.142385\pi$
$79$	68.3424i	0.865093i	−0.901612	+	0.432547i	$0.857615\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 137.277i	− 1.65393i	−0.562251	−	0.826967i	$-0.690064\pi$
$83$	− 137.277i	− 1.65393i	0.562251	−	0.826967i	$-0.309936\pi$
$84$	0	0
$85$	1.28509	0.0151187
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−67.3405	−0.756635	−0.378317	−	0.925676i	$-0.623497\pi$
$89$	−67.3405	−0.756635	−0.378317	+	0.925676i	$0.623497\pi$
$90$	0	0
$91$	− 11.1924i	− 0.122994i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	12.4609i	0.131168i
$96$	0	0
$97$	−95.6956	−0.986553	−0.493276	−	0.869873i	$-0.664201\pi$
$97$	−95.6956	−0.986553	−0.493276	+	0.869873i	$0.664201\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	73.9722	0.732398	0.366199	−	0.930537i	$-0.380659\pi$
$101$	73.9722	0.732398	0.366199	+	0.930537i	$0.380659\pi$
$102$	0	0
$103$	59.6083i	0.578721i	0.957220	+	0.289361i	$0.0934427\pi$
$103$	59.6083i	0.578721i	−0.957220	+	0.289361i	$0.906557\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 137.299i	− 1.28317i	−0.767053	−	0.641583i	$-0.778278\pi$
$107$	− 137.299i	− 1.28317i	0.767053	−	0.641583i	$-0.221722\pi$
$108$	0	0
$109$	122.932	1.12782	0.563909	−	0.825837i	$-0.309297\pi$
$109$	122.932	1.12782	0.563909	+	0.825837i	$0.309297\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−122.117	−1.08068	−0.540340	−	0.841447i	$-0.681704\pi$
$113$	−122.117	−1.08068	−0.540340	+	0.841447i	$0.681704\pi$
$114$	0	0
$115$	− 4.01780i	− 0.0349374i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 8.52342i	− 0.0716254i
$120$	0	0
$121$	−89.2230	−0.737380
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−46.9646	−0.375716
$126$	0	0
$127$	− 91.4995i	− 0.720469i	−0.932862	−	0.360234i	$-0.882697\pi$
$127$	− 91.4995i	− 0.720469i	0.932862	−	0.360234i	$-0.117303\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	117.372i	0.895972i	0.894040	+	0.447986i	$0.147859\pi$
$131$	117.372i	0.895972i	−0.894040	+	0.447986i	$0.852141\pi$
$132$	0	0
$133$	82.6478	0.621412
$134$	0	0
$135$	0	0
$136$	0	0
$137$	56.4879	0.412320	0.206160	−	0.978518i	$-0.433903\pi$
$137$	56.4879	0.412320	0.206160	+	0.978518i	$0.433903\pi$
$138$	0	0
$139$	− 105.929i	− 0.762080i	−0.924559	−	0.381040i	$-0.875566\pi$
$139$	− 105.929i	− 0.762080i	0.924559	−	0.381040i	$-0.124434\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 25.5715i	− 0.178822i
$144$	0	0
$145$	−36.5205	−0.251866
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−290.160	−1.94739	−0.973693	−	0.227865i	$-0.926826\pi$
$149$	−290.160	−1.94739	−0.973693	+	0.227865i	$0.926826\pi$
$150$	0	0
$151$	− 188.866i	− 1.25077i	−0.780316	−	0.625385i	$-0.784942\pi$
$151$	− 188.866i	− 1.25077i	0.780316	−	0.625385i	$-0.215058\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.165298i	0.00106644i
$156$	0	0
$157$	213.852	1.36211	0.681057	−	0.732231i	$-0.261521\pi$
$157$	213.852	1.36211	0.681057	+	0.732231i	$0.261521\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−26.6483	−0.165518
$162$	0	0
$163$	304.944i	1.87082i	0.353566	+	0.935410i	$0.384969\pi$
$163$	304.944i	1.87082i	−0.353566	+	0.935410i	$0.615031\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	166.488i	0.996935i	0.866908	+	0.498467i	$0.166104\pi$
$167$	166.488i	0.996935i	−0.866908	+	0.498467i	$0.833896\pi$
$168$	0	0
$169$	−165.889	−0.981595
$170$	0	0
$171$	0	0
$172$	0	0
$173$	313.389	1.81150	0.905749	−	0.423815i	$-0.139310\pi$
$173$	313.389	1.81150	0.905749	+	0.423815i	$0.139310\pi$
$174$	0	0
$175$	152.843i	0.873388i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	144.278i	0.806021i	0.915195	+	0.403011i	$0.132036\pi$
$179$	144.278i	0.806021i	−0.915195	+	0.403011i	$0.867964\pi$
$180$	0	0
$181$	120.016	0.663074	0.331537	−	0.943442i	$-0.392433\pi$
$181$	120.016	0.663074	0.331537	+	0.943442i	$0.392433\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	14.5375	0.0785810
$186$	0	0
$187$	− 19.4736i	− 0.104137i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 257.420i	− 1.34775i	−0.738846	−	0.673874i	$-0.764629\pi$
$191$	− 257.420i	− 1.34775i	0.738846	−	0.673874i	$-0.235371\pi$
$192$	0	0
$193$	−289.234	−1.49862	−0.749310	−	0.662220i	$-0.769615\pi$
$193$	−289.234	−1.49862	−0.749310	+	0.662220i	$0.769615\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	196.282	0.996357	0.498179	−	0.867074i	$-0.334002\pi$
$197$	196.282	0.996357	0.498179	+	0.867074i	$0.334002\pi$
$198$	0	0
$199$	− 192.265i	− 0.966155i	−0.875578	−	0.483078i	$-0.839519\pi$
$199$	− 192.265i	− 0.966155i	0.875578	−	0.483078i	$-0.160481\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	242.225i	1.19322i
$204$	0	0
$205$	−66.3992	−0.323899
$206$	0	0
$207$	0	0
$208$	0	0
$209$	188.827	0.903479
$210$	0	0
$211$	20.0620i	0.0950804i	0.998869	+	0.0475402i	$0.0151382\pi$
$211$	20.0620i	0.0950804i	−0.998869	+	0.0475402i	$0.984862\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 5.28861i	− 0.0245982i
$216$	0	0
$217$	1.09635	0.00505230
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.36877	0.0107184
$222$	0	0
$223$	319.896i	1.43451i	0.696810	+	0.717256i	$0.254602\pi$
$223$	319.896i	1.43451i	−0.696810	+	0.717256i	$0.745398\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 202.001i	− 0.889872i	−0.895562	−	0.444936i	$-0.853226\pi$
$227$	− 202.001i	− 0.889872i	0.895562	−	0.444936i	$-0.146774\pi$
$228$	0	0
$229$	196.154	0.856568	0.428284	−	0.903644i	$-0.359118\pi$
$229$	196.154	0.856568	0.428284	+	0.903644i	$0.359118\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	395.370	1.69687	0.848434	−	0.529301i	$-0.177546\pi$
$233$	395.370	1.69687	0.848434	+	0.529301i	$0.177546\pi$
$234$	0	0
$235$	− 63.8324i	− 0.271627i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	449.180i	1.87941i	0.341980	+	0.939707i	$0.388902\pi$
$239$	449.180i	1.87941i	−0.341980	+	0.939707i	$0.611098\pi$
$240$	0	0
$241$	187.767	0.779117	0.389558	−	0.921002i	$-0.372628\pi$
$241$	187.767	0.779117	0.389558	+	0.921002i	$0.372628\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	8.35004	0.0340818
$246$	0	0
$247$	22.9689i	0.0929917i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	57.1800i	0.227809i	0.993492	+	0.113904i	$0.0363358\pi$
$251$	57.1800i	0.227809i	−0.993492	+	0.113904i	$0.963664\pi$
$252$	0	0
$253$	−60.8839	−0.240648
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−112.952	−0.439501	−0.219750	−	0.975556i	$-0.570524\pi$
$257$	−112.952	−0.439501	−0.219750	+	0.975556i	$0.570524\pi$
$258$	0	0
$259$	− 96.4208i	− 0.372281i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	48.0573i	0.182727i	0.995818	+	0.0913636i	$0.0291226\pi$
$263$	48.0573i	0.182727i	−0.995818	+	0.0913636i	$0.970877\pi$
$264$	0	0
$265$	−30.2884	−0.114296
$266$	0	0
$267$	0	0
$268$	0	0
$269$	196.360	0.729964	0.364982	−	0.931015i	$-0.381075\pi$
$269$	196.360	0.729964	0.364982	+	0.931015i	$0.381075\pi$
$270$	0	0
$271$	− 242.829i	− 0.896050i	−0.894021	−	0.448025i	$-0.852128\pi$
$271$	− 242.829i	− 0.896050i	0.894021	−	0.448025i	$-0.147872\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	349.203i	1.26983i
$276$	0	0
$277$	228.132	0.823580	0.411790	−	0.911279i	$-0.364904\pi$
$277$	228.132	0.823580	0.411790	+	0.911279i	$0.364904\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	313.999	1.11743	0.558716	−	0.829359i	$-0.311294\pi$
$281$	313.999	1.11743	0.558716	+	0.829359i	$0.311294\pi$
$282$	0	0
$283$	− 522.669i	− 1.84689i	−0.383735	−	0.923443i	$-0.625362\pi$
$283$	− 522.669i	− 1.84689i	0.383735	−	0.923443i	$-0.374638\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	440.397i	1.53449i
$288$	0	0
$289$	−287.196	−0.993758
$290$	0	0
$291$	0	0
$292$	0	0
$293$	173.891	0.593483	0.296742	−	0.954958i	$-0.404100\pi$
$293$	173.891	0.593483	0.296742	+	0.954958i	$0.404100\pi$
$294$	0	0
$295$	− 11.7124i	− 0.0397029i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 7.40593i	− 0.0247690i
$300$	0	0
$301$	−35.0771	−0.116535
$302$	0	0
$303$	0	0
$304$	0	0
$305$	88.1239	0.288931
$306$	0	0
$307$	465.568i	1.51651i	0.651959	+	0.758254i	$0.273947\pi$
$307$	465.568i	1.51651i	−0.651959	+	0.758254i	$0.726053\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	238.054i	0.765448i	0.923863	+	0.382724i	$0.125014\pi$
$311$	238.054i	0.765448i	−0.923863	+	0.382724i	$0.874986\pi$
$312$	0	0
$313$	99.4041	0.317585	0.158793	−	0.987312i	$-0.449240\pi$
$313$	99.4041	0.317585	0.158793	+	0.987312i	$0.449240\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	87.2285	0.275169	0.137584	−	0.990490i	$-0.456066\pi$
$317$	87.2285	0.275169	0.137584	+	0.990490i	$0.456066\pi$
$318$	0	0
$319$	553.415i	1.73484i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	17.4916i	0.0541537i
$324$	0	0
$325$	−42.4771	−0.130699
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−423.372	−1.28685
$330$	0	0
$331$	− 25.0394i	− 0.0756476i	−0.999284	−	0.0378238i	$-0.987957\pi$
$331$	− 25.0394i	− 0.0756476i	0.999284	−	0.0378238i	$-0.0120426\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 25.3156i	− 0.0755689i
$336$	0	0
$337$	−35.5515	−0.105494	−0.0527471	−	0.998608i	$-0.516798\pi$
$337$	−35.5515	−0.105494	−0.0527471	+	0.998608i	$0.516798\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.50485	0.00734560
$342$	0	0
$343$	− 366.341i	− 1.06805i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	277.688i	0.800254i	0.916460	+	0.400127i	$0.131034\pi$
$347$	277.688i	0.800254i	−0.916460	+	0.400127i	$0.868966\pi$
$348$	0	0
$349$	−382.657	−1.09644	−0.548219	−	0.836335i	$-0.684694\pi$
$349$	−382.657	−1.09644	−0.548219	+	0.836335i	$0.684694\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	444.310	1.25867	0.629335	−	0.777134i	$-0.283327\pi$
$353$	444.310	1.25867	0.629335	+	0.777134i	$0.283327\pi$
$354$	0	0
$355$	− 81.7587i	− 0.230306i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 10.9605i	− 0.0305306i	−0.999883	−	0.0152653i	$-0.995141\pi$
$359$	− 10.9605i	− 0.0305306i	0.999883	−	0.0152653i	$-0.00485929\pi$
$360$	0	0
$361$	191.391	0.530170
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−99.4318	−0.272416
$366$	0	0
$367$	166.081i	0.452538i	0.974065	+	0.226269i	$0.0726528\pi$
$367$	166.081i	0.452538i	−0.974065	+	0.226269i	$0.927347\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	200.890i	0.541482i
$372$	0	0
$373$	−240.406	−0.644519	−0.322259	−	0.946651i	$-0.604442\pi$
$373$	−240.406	−0.644519	−0.322259	+	0.946651i	$0.604442\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−67.3175	−0.178561
$378$	0	0
$379$	224.504i	0.592359i	0.955132	+	0.296179i	$0.0957127\pi$
$379$	224.504i	0.592359i	−0.955132	+	0.296179i	$0.904287\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 126.068i	− 0.329160i	−0.986364	−	0.164580i	$-0.947373\pi$
$383$	− 126.068i	− 0.329160i	0.986364	−	0.164580i	$-0.0526268\pi$
$384$	0	0
$385$	−88.0386	−0.228672
$386$	0	0
$387$	0	0
$388$	0	0
$389$	435.033	1.11834	0.559169	−	0.829054i	$-0.311120\pi$
$389$	435.033	1.11834	0.559169	+	0.829054i	$0.311120\pi$
$390$	0	0
$391$	− 5.63987i	− 0.0144242i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	65.3907i	0.165546i
$396$	0	0
$397$	−34.0620	−0.0857985	−0.0428993	−	0.999079i	$-0.513659\pi$
$397$	−34.0620	−0.0857985	−0.0428993	+	0.999079i	$0.513659\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	152.431	0.380126	0.190063	−	0.981772i	$-0.439131\pi$
$401$	152.431	0.380126	0.190063	+	0.981772i	$0.439131\pi$
$402$	0	0
$403$	0.304690i	0 0.000756055i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 220.294i	− 0.541264i
$408$	0	0
$409$	714.512	1.74697	0.873486	−	0.486849i	$-0.161854\pi$
$409$	714.512	1.74697	0.873486	+	0.486849i	$0.161854\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−77.6830	−0.188094
$414$	0	0
$415$	− 131.348i	− 0.316500i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 131.654i	− 0.314211i	−0.987582	−	0.157106i	$-0.949784\pi$
$419$	− 131.654i	− 0.314211i	0.987582	−	0.157106i	$-0.0502163\pi$
$420$	0	0
$421$	−55.5081	−0.131848	−0.0659241	−	0.997825i	$-0.521000\pi$
$421$	−55.5081	−0.131848	−0.0659241	+	0.997825i	$0.521000\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−32.3478	−0.0761124
$426$	0	0
$427$	− 584.487i	− 1.36882i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 743.495i	− 1.72505i	−0.506018	−	0.862523i	$-0.668883\pi$
$431$	− 743.495i	− 1.72505i	0.506018	−	0.862523i	$-0.331117\pi$
$432$	0	0
$433$	526.775	1.21657	0.608285	−	0.793719i	$-0.291858\pi$
$433$	526.775	1.21657	0.608285	+	0.793719i	$0.291858\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	54.6873	0.125143
$438$	0	0
$439$	− 684.577i	− 1.55940i	−0.626152	−	0.779701i	$-0.715371\pi$
$439$	− 684.577i	− 1.55940i	0.626152	−	0.779701i	$-0.284629\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 59.9868i	− 0.135410i	−0.997705	−	0.0677052i	$-0.978432\pi$
$443$	− 59.9868i	− 0.135410i	0.997705	−	0.0677052i	$-0.0215677\pi$
$444$	0	0
$445$	−64.4321	−0.144791
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−78.3940	−0.174597	−0.0872985	−	0.996182i	$-0.527823\pi$
$449$	−78.3940	−0.174597	−0.0872985	+	0.996182i	$0.527823\pi$
$450$	0	0
$451$	1006.18i	2.23101i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 10.7090i	− 0.0235363i
$456$	0	0
$457$	−64.6748	−0.141520	−0.0707602	−	0.997493i	$-0.522543\pi$
$457$	−64.6748	−0.141520	−0.0707602	+	0.997493i	$0.522543\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−321.909	−0.698285	−0.349143	−	0.937070i	$-0.613527\pi$
$461$	−321.909	−0.698285	−0.349143	+	0.937070i	$0.613527\pi$
$462$	0	0
$463$	520.291i	1.12374i	0.827226	+	0.561869i	$0.189917\pi$
$463$	520.291i	1.12374i	−0.827226	+	0.561869i	$0.810083\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 454.944i	− 0.974185i	−0.873350	−	0.487093i	$-0.838057\pi$
$467$	− 454.944i	− 0.974185i	0.873350	−	0.487093i	$-0.161943\pi$
$468$	0	0
$469$	−167.907	−0.358011
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−80.1412	−0.169432
$474$	0	0
$475$	− 313.662i	− 0.660341i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	341.743i	0.713451i	0.934209	+	0.356726i	$0.116107\pi$
$479$	341.743i	0.713451i	−0.934209	+	0.356726i	$0.883893\pi$
$480$	0	0
$481$	26.7966	0.0557103
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−91.5625	−0.188789
$486$	0	0
$487$	− 683.859i	− 1.40423i	−0.712065	−	0.702113i	$-0.752240\pi$
$487$	− 683.859i	− 1.40423i	0.712065	−	0.702113i	$-0.247760\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 419.194i	− 0.853756i	−0.904309	−	0.426878i	$-0.859613\pi$
$491$	− 419.194i	− 0.853756i	0.904309	−	0.426878i	$-0.140387\pi$
$492$	0	0
$493$	−51.2646	−0.103985
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−542.270	−1.09109
$498$	0	0
$499$	402.803i	0.807221i	0.914931	+	0.403610i	$0.132245\pi$
$499$	402.803i	0.807221i	−0.914931	+	0.403610i	$0.867755\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	321.360i	0.638887i	0.947605	+	0.319444i	$0.103496\pi$
$503$	321.360i	0.638887i	−0.947605	+	0.319444i	$0.896504\pi$
$504$	0	0
$505$	70.7773	0.140153
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−366.245	−0.719539	−0.359769	−	0.933041i	$-0.617145\pi$
$509$	−366.245	−0.719539	−0.359769	+	0.933041i	$0.617145\pi$
$510$	0	0
$511$	659.488i	1.29058i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	57.0338i	0.110745i
$516$	0	0
$517$	−967.287	−1.87096
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−660.982	−1.26868	−0.634340	−	0.773054i	$-0.718728\pi$
$521$	−660.982	−1.26868	−0.634340	+	0.773054i	$0.718728\pi$
$522$	0	0
$523$	− 197.073i	− 0.376812i	−0.982091	−	0.188406i	$-0.939668\pi$
$523$	− 197.073i	− 0.376812i	0.982091	−	0.188406i	$-0.0603321\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.232032i	0 0.000440289i
$528$	0	0
$529$	511.367	0.966667
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−122.392	−0.229629
$534$	0	0
$535$	− 131.369i	− 0.245549i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 126.533i	− 0.234755i
$540$	0	0
$541$	−777.149	−1.43650	−0.718252	−	0.695783i	$-0.755058\pi$
$541$	−777.149	−1.43650	−0.718252	+	0.695783i	$0.755058\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	117.623	0.215821
$546$	0	0
$547$	400.931i	0.732963i	0.930425	+	0.366482i	$0.119438\pi$
$547$	400.931i	0.732963i	−0.930425	+	0.366482i	$0.880562\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 497.090i	− 0.902160i
$552$	0	0
$553$	433.708	0.784282
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−553.074	−0.992951	−0.496476	−	0.868051i	$-0.665373\pi$
$557$	−553.074	−0.992951	−0.496476	+	0.868051i	$0.665373\pi$
$558$	0	0
$559$	− 9.74839i	− 0.0174390i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 548.074i	− 0.973489i	−0.873545	−	0.486744i	$-0.838184\pi$
$563$	− 548.074i	− 0.973489i	0.873545	−	0.486744i	$-0.161816\pi$
$564$	0	0
$565$	−116.843	−0.206801
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−627.765	−1.10328	−0.551639	−	0.834083i	$-0.685997\pi$
$569$	−627.765	−1.10328	−0.551639	+	0.834083i	$0.685997\pi$
$570$	0	0
$571$	− 652.428i	− 1.14261i	−0.820739	−	0.571303i	$-0.806438\pi$
$571$	− 652.428i	− 1.14261i	0.820739	−	0.571303i	$-0.193562\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	101.135i	0.175887i
$576$	0	0
$577$	300.791	0.521301	0.260651	−	0.965433i	$-0.416063\pi$
$577$	300.791	0.521301	0.260651	+	0.965433i	$0.416063\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−871.171	−1.49943
$582$	0	0
$583$	458.977i	0.787267i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 513.527i	− 0.874833i	−0.899259	−	0.437417i	$-0.855893\pi$
$587$	− 513.527i	− 0.874833i	0.899259	−	0.437417i	$-0.144107\pi$
$588$	0	0
$589$	−2.24991	−0.00381989
$590$	0	0
$591$	0	0
$592$	0	0
$593$	114.700	0.193423	0.0967116	−	0.995312i	$-0.469168\pi$
$593$	114.700	0.193423	0.0967116	+	0.995312i	$0.469168\pi$
$594$	0	0
$595$	− 8.15529i	− 0.0137064i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 926.005i	− 1.54592i	−0.634456	−	0.772959i	$-0.718776\pi$
$599$	− 926.005i	− 1.54592i	0.634456	−	0.772959i	$-0.281224\pi$
$600$	0	0
$601$	−67.6176	−0.112509	−0.0562543	−	0.998416i	$-0.517916\pi$
$601$	−67.6176	−0.112509	−0.0562543	+	0.998416i	$0.517916\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−85.3694	−0.141106
$606$	0	0
$607$	634.634i	1.04552i	0.852478	+	0.522762i	$0.175098\pi$
$607$	634.634i	1.04552i	−0.852478	+	0.522762i	$0.824902\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 117.661i	− 0.192571i
$612$	0	0
$613$	115.485	0.188394	0.0941969	−	0.995554i	$-0.469972\pi$
$613$	115.485	0.188394	0.0941969	+	0.995554i	$0.469972\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	692.513	1.12239	0.561193	−	0.827685i	$-0.310342\pi$
$617$	692.513	1.12239	0.561193	+	0.827685i	$0.310342\pi$
$618$	0	0
$619$	− 711.629i	− 1.14964i	−0.818279	−	0.574821i	$-0.805072\pi$
$619$	− 711.629i	− 1.14964i	0.818279	−	0.574821i	$-0.194928\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	427.350i	0.685955i
$624$	0	0
$625$	557.177	0.891483
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.4066	0.0324429
$630$	0	0
$631$	− 1036.16i	− 1.64210i	−0.570859	−	0.821048i	$-0.693390\pi$
$631$	− 1036.16i	− 1.64210i	0.570859	−	0.821048i	$-0.306610\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 87.5477i	− 0.137870i
$636$	0	0
$637$	15.3915	0.0241624
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1024.87	−1.59886	−0.799431	−	0.600757i	$-0.794866\pi$
$641$	−1024.87	−1.59886	−0.799431	+	0.600757i	$0.794866\pi$
$642$	0	0
$643$	− 824.393i	− 1.28210i	−0.767498	−	0.641052i	$-0.778498\pi$
$643$	− 824.393i	− 1.28210i	0.767498	−	0.641052i	$-0.221502\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 414.268i	− 0.640290i	−0.947369	−	0.320145i	$-0.896268\pi$
$647$	− 414.268i	− 0.640290i	0.947369	−	0.320145i	$-0.103732\pi$
$648$	0	0
$649$	−177.484	−0.273473
$650$	0	0
$651$	0	0
$652$	0	0
$653$	796.975	1.22048	0.610241	−	0.792216i	$-0.291072\pi$
$653$	796.975	1.22048	0.610241	+	0.792216i	$0.291072\pi$
$654$	0	0
$655$	112.303i	0.171455i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 53.1521i	− 0.0806557i	−0.999187	−	0.0403279i	$-0.987160\pi$
$659$	− 53.1521i	− 0.0806557i	0.999187	−	0.0403279i	$-0.0128402\pi$
$660$	0	0
$661$	−649.763	−0.983000	−0.491500	−	0.870878i	$-0.663551\pi$
$661$	−649.763	−0.983000	−0.491500	+	0.870878i	$0.663551\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	79.0783	0.118915
$666$	0	0
$667$	160.278i	0.240297i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1335.39i	− 1.99015i
$672$	0	0
$673$	829.302	1.23225	0.616123	−	0.787650i	$-0.288702\pi$
$673$	829.302	1.23225	0.616123	+	0.787650i	$0.288702\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1166.76	−1.72343	−0.861716	−	0.507392i	$-0.830610\pi$
$677$	−1166.76	−1.72343	−0.861716	+	0.507392i	$0.830610\pi$
$678$	0	0
$679$	607.294i	0.894395i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	690.100i	1.01039i	0.863004	+	0.505197i	$0.168580\pi$
$683$	690.100i	1.01039i	−0.863004	+	0.505197i	$0.831420\pi$
$684$	0	0
$685$	54.0482	0.0789024
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−55.8300	−0.0810304
$690$	0	0
$691$	819.624i	1.18614i	0.805150	+	0.593071i	$0.202085\pi$
$691$	819.624i	1.18614i	−0.805150	+	0.593071i	$0.797915\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 101.354i	− 0.145833i
$696$	0	0
$697$	−93.2060	−0.133725
$698$	0	0
$699$	0	0
$700$	0	0
$701$	378.889	0.540498	0.270249	−	0.962791i	$-0.412894\pi$
$701$	378.889	0.540498	0.270249	+	0.962791i	$0.412894\pi$
$702$	0	0
$703$	197.874i	0.281470i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 469.435i	− 0.663982i
$708$	0	0
$709$	−296.838	−0.418672	−0.209336	−	0.977844i	$-0.567130\pi$
$709$	−296.838	−0.418672	−0.209336	+	0.977844i	$0.567130\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0.725445	0.00101745
$714$	0	0
$715$	− 24.4671i	− 0.0342197i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 523.144i	− 0.727600i	−0.931477	−	0.363800i	$-0.881479\pi$
$719$	− 523.144i	− 0.727600i	0.931477	−	0.363800i	$-0.118521\pi$
$720$	0	0
$721$	378.280	0.524661
$722$	0	0
$723$	0	0
$724$	0	0
$725$	919.283	1.26798
$726$	0	0
$727$	− 636.546i	− 0.875579i	−0.899077	−	0.437790i	$-0.855761\pi$
$727$	− 636.546i	− 0.875579i	0.899077	−	0.437790i	$-0.144239\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 7.42373i	− 0.0101556i
$732$	0	0
$733$	420.326	0.573432	0.286716	−	0.958016i	$-0.407436\pi$
$733$	420.326	0.573432	0.286716	+	0.958016i	$0.407436\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−383.621	−0.520517
$738$	0	0
$739$	987.498i	1.33626i	0.744044	+	0.668131i	$0.232905\pi$
$739$	987.498i	1.33626i	−0.744044	+	0.668131i	$0.767095\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	463.675i	0.624058i	0.950073	+	0.312029i	$0.101009\pi$
$743$	463.675i	0.624058i	−0.950073	+	0.312029i	$0.898991\pi$
$744$	0	0
$745$	−277.628	−0.372656
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−871.313	−1.16330
$750$	0	0
$751$	326.135i	0.434268i	0.976142	+	0.217134i	$0.0696709\pi$
$751$	326.135i	0.434268i	−0.976142	+	0.217134i	$0.930329\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 180.709i	− 0.239350i
$756$	0	0
$757$	−404.389	−0.534199	−0.267100	−	0.963669i	$-0.586065\pi$
$757$	−404.389	−0.534199	−0.267100	+	0.963669i	$0.586065\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	322.469	0.423744	0.211872	−	0.977297i	$-0.432044\pi$
$761$	322.469	0.423744	0.211872	+	0.977297i	$0.432044\pi$
$762$	0	0
$763$	− 780.140i	− 1.02246i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 21.5891i	− 0.0281475i
$768$	0	0
$769$	−520.611	−0.676997	−0.338499	−	0.940967i	$-0.609919\pi$
$769$	−520.611	−0.676997	−0.338499	+	0.940967i	$0.609919\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−664.521	−0.859666	−0.429833	−	0.902909i	$-0.641427\pi$
$773$	−664.521	−0.859666	−0.429833	+	0.902909i	$0.641427\pi$
$774$	0	0
$775$	− 4.16083i	− 0.00536881i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 903.777i	− 1.16018i
$780$	0	0
$781$	−1238.93	−1.58634
$782$	0	0
$783$	0	0
$784$	0	0
$785$	204.616	0.260657
$786$	0	0
$787$	711.537i	0.904113i	0.891989	+	0.452057i	$0.149310\pi$
$787$	711.537i	0.904113i	−0.891989	+	0.452057i	$0.850690\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	774.966i	0.979729i
$792$	0	0
$793$	162.437	0.204838
$794$	0	0
$795$	0	0
$796$	0	0
$797$	691.498	0.867626	0.433813	−	0.901003i	$-0.357168\pi$
$797$	691.498	0.867626	0.433813	+	0.901003i	$0.357168\pi$
$798$	0	0
$799$	− 89.6028i	− 0.112144i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1506.74i	1.87639i
$804$	0	0
$805$	−25.4974	−0.0316738
$806$	0	0
$807$	0	0
$808$	0	0
$809$	646.526	0.799167	0.399583	−	0.916697i	$-0.369155\pi$
$809$	646.526	0.799167	0.399583	+	0.916697i	$0.369155\pi$
$810$	0	0
$811$	391.295i	0.482485i	0.970465	+	0.241243i	$0.0775549\pi$
$811$	391.295i	0.482485i	−0.970465	+	0.241243i	$0.922445\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	291.773i	0.358004i
$816$	0	0
$817$	71.9847	0.0881085
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−271.408	−0.330582	−0.165291	−	0.986245i	$-0.552856\pi$
$821$	−271.408	−0.330582	−0.165291	+	0.986245i	$0.552856\pi$
$822$	0	0
$823$	− 1135.15i	− 1.37928i	−0.724152	−	0.689641i	$-0.757769\pi$
$823$	− 1135.15i	− 1.37928i	0.724152	−	0.689641i	$-0.242231\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	266.279i	0.321982i	0.986956	+	0.160991i	$0.0514690\pi$
$827$	266.279i	0.321982i	−0.986956	+	0.160991i	$0.948531\pi$
$828$	0	0
$829$	723.241	0.872426	0.436213	−	0.899844i	$-0.356319\pi$
$829$	723.241	0.872426	0.436213	+	0.899844i	$0.356319\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	11.7211	0.0140710
$834$	0	0
$835$	159.298i	0.190776i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1162.52i	1.38560i	0.721131	+	0.692799i	$0.243623\pi$
$839$	1162.52i	1.38560i	−0.721131	+	0.692799i	$0.756377\pi$
$840$	0	0
$841$	615.875	0.732312
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−158.725	−0.187840
$846$	0	0
$847$	566.218i	0.668498i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 63.8008i	− 0.0749716i
$852$	0	0
$853$	770.807	0.903642	0.451821	−	0.892109i	$-0.350775\pi$
$853$	770.807	0.903642	0.451821	+	0.892109i	$0.350775\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−383.535	−0.447532	−0.223766	−	0.974643i	$-0.571835\pi$
$857$	−383.535	−0.447532	−0.223766	+	0.974643i	$0.571835\pi$
$858$	0	0
$859$	− 338.150i	− 0.393655i	−0.980438	−	0.196828i	$-0.936936\pi$
$859$	− 338.150i	− 0.393655i	0.980438	−	0.196828i	$-0.0630640\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	783.026i	0.907331i	0.891172	+	0.453665i	$0.149884\pi$
$863$	783.026i	0.907331i	−0.891172	+	0.453665i	$0.850116\pi$
$864$	0	0
$865$	299.854	0.346652
$866$	0	0
$867$	0	0
$868$	0	0
$869$	990.901	1.14028
$870$	0	0
$871$	− 46.6637i	− 0.0535749i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	298.042i	0.340619i
$876$	0	0
$877$	1668.78	1.90283	0.951414	−	0.307916i	$-0.0996314\pi$
$877$	1668.78	1.90283	0.951414	+	0.307916i	$0.0996314\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1269.86	−1.44139	−0.720695	−	0.693252i	$-0.756177\pi$
$881$	−1269.86	−1.44139	−0.720695	+	0.693252i	$0.756177\pi$
$882$	0	0
$883$	− 471.136i	− 0.533562i	−0.963757	−	0.266781i	$-0.914040\pi$
$883$	− 471.136i	− 0.533562i	0.963757	−	0.266781i	$-0.0859601\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 686.179i	− 0.773596i	−0.922165	−	0.386798i	$-0.873581\pi$
$887$	− 686.179i	− 0.773596i	0.922165	−	0.386798i	$-0.126419\pi$
$888$	0	0
$889$	−580.666	−0.653167
$890$	0	0
$891$	0	0
$892$	0	0
$893$	868.839	0.972944
$894$	0	0
$895$	138.046i	0.154242i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 6.59406i	− 0.00733488i
$900$	0	0
$901$	−42.5165	−0.0471881
$902$	0	0
$903$	0	0
$904$	0	0
$905$	114.833	0.126887
$906$	0	0
$907$	− 406.300i	− 0.447960i	−0.974594	−	0.223980i	$-0.928095\pi$
$907$	− 406.300i	− 0.447960i	0.974594	−	0.223980i	$-0.0719051\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1340.86i	1.47186i	0.677060	+	0.735928i	$0.263254\pi$
$911$	1340.86i	1.47186i	−0.677060	+	0.735928i	$0.736746\pi$
$912$	0	0
$913$	−1990.38	−2.18005
$914$	0	0
$915$	0	0
$916$	0	0
$917$	744.857	0.812276
$918$	0	0
$919$	− 1568.33i	− 1.70656i	−0.521452	−	0.853281i	$-0.674609\pi$
$919$	− 1568.33i	− 1.70656i	0.521452	−	0.853281i	$-0.325391\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 150.704i	− 0.163276i
$924$	0	0
$925$	−365.933	−0.395603
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1252.87	1.34862	0.674312	−	0.738446i	$-0.264440\pi$
$929$	1252.87	1.34862	0.674312	+	0.738446i	$0.264440\pi$
$930$	0	0
$931$	113.655i	0.122078i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 18.6326i	− 0.0199279i
$936$	0	0
$937$	−423.086	−0.451532	−0.225766	−	0.974182i	$-0.572489\pi$
$937$	−423.086	−0.451532	−0.225766	+	0.974182i	$0.572489\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	449.433	0.477612	0.238806	−	0.971067i	$-0.423244\pi$
$941$	449.433	0.477612	0.238806	+	0.971067i	$0.423244\pi$
$942$	0	0
$943$	291.407i	0.309021i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	723.129i	0.763599i	0.924245	+	0.381800i	$0.124696\pi$
$947$	723.129i	0.763599i	−0.924245	+	0.381800i	$0.875304\pi$
$948$	0	0
$949$	−183.280	−0.193130
$950$	0	0
$951$	0	0
$952$	0	0
$953$	921.421	0.966863	0.483432	−	0.875382i	$-0.339390\pi$
$953$	921.421	0.966863	0.483432	+	0.875382i	$0.339390\pi$
$954$	0	0
$955$	− 246.302i	− 0.257908i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 358.478i	− 0.373804i
$960$	0	0
$961$	960.970	0.999969
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−276.742	−0.286779
$966$	0	0
$967$	317.463i	0.328297i	0.986436	+	0.164148i	$0.0524876\pi$
$967$	317.463i	0.328297i	−0.986436	+	0.164148i	$0.947512\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 559.532i	− 0.576243i	−0.957594	−	0.288122i	$-0.906969\pi$
$971$	− 559.532i	− 0.576243i	0.957594	−	0.288122i	$-0.0930307\pi$
$972$	0	0
$973$	−672.237	−0.690891
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1742.63	1.78366	0.891829	−	0.452372i	$-0.149422\pi$
$977$	1742.63	1.78366	0.891829	+	0.452372i	$0.149422\pi$
$978$	0	0
$979$	976.374i	0.997318i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1071.37i	1.08989i	0.838470	+	0.544947i	$0.183450\pi$
$983$	1071.37i	1.08989i	−0.838470	+	0.544947i	$0.816550\pi$
$984$	0	0
$985$	187.805	0.190665
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−23.2102	−0.0234683
$990$	0	0
$991$	1167.78i	1.17839i	0.807992	+	0.589193i	$0.200554\pi$
$991$	1167.78i	1.17839i	−0.807992	+	0.589193i	$0.799446\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 183.961i	− 0.184885i
$996$	0	0
$997$	1320.67	1.32464	0.662321	−	0.749220i	$-0.269571\pi$
$997$	1320.67	1.32464	0.662321	+	0.749220i	$0.269571\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.3.g.b.703.5	✓	8
3.2	odd	2		864.3.g.d.703.3	yes	8
4.3	odd	2	inner	864.3.g.b.703.6	yes	8
8.3	odd	2		1728.3.g.m.703.4		8
8.5	even	2		1728.3.g.m.703.3		8
12.11	even	2		864.3.g.d.703.4	yes	8
24.5	odd	2		1728.3.g.j.703.5		8
24.11	even	2		1728.3.g.j.703.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.g.b.703.5	✓	8	1.1	even	1	trivial
864.3.g.b.703.6	yes	8	4.3	odd	2	inner
864.3.g.d.703.3	yes	8	3.2	odd	2
864.3.g.d.703.4	yes	8	12.11	even	2
1728.3.g.j.703.5		8	24.5	odd	2
1728.3.g.j.703.6		8	24.11	even	2
1728.3.g.m.703.3		8	8.5	even	2
1728.3.g.m.703.4		8	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	864.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.956810 q^{5} -6.34610i q^{7} +O(q^{10})\)	\(q+0.956810 q^{5} -6.34610i q^{7} -14.4991i q^{11} +1.76367 q^{13} +1.34309 q^{17} +13.0234i q^{19} -4.19916i q^{23} -24.0845 q^{25} -38.1690 q^{29} +0.172759i q^{31} -6.07202i q^{35} +15.1937 q^{37} -69.3965 q^{41} -5.52734i q^{43} -66.7137i q^{47} +8.72695 q^{49} -31.6556 q^{53} -13.8729i q^{55} -12.2410i q^{59} +92.1017 q^{61} +1.68750 q^{65} -26.4583i q^{67} -85.4493i q^{71} -103.920 q^{73} -92.0126 q^{77} +68.3424i q^{79} -137.277i q^{83} +1.28509 q^{85} -67.3405 q^{89} -11.1924i q^{91} +12.4609i q^{95} -95.6956 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} + 8 q^{13} - 24 q^{17} + 24 q^{25} + 128 q^{29} + 24 q^{37} - 160 q^{41} - 144 q^{49} + 48 q^{53} - 136 q^{61} + 280 q^{65} + 72 q^{73} - 520 q^{77} + 96 q^{85} + 168 q^{89} + 104 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 + 8 * q^13 - 24 * q^17 + 24 * q^25 + 128 * q^29 + 24 * q^37 - 160 * q^41 - 144 * q^49 + 48 * q^53 - 136 * q^61 + 280 * q^65 + 72 * q^73 - 520 * q^77 + 96 * q^85 + 168 * q^89 + 104 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more