LMFDB - Embedded newform 1764.4.k.bc.361.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1764,4,Mod(361,1764)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1764.361"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1764, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.k (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$104.079369250$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.7027860559216896.42
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 88x^{6} + 278x^{5} + 3869x^{4} - 8206x^{3} - 71054x^{2} + 75204x + 945876 $ x^8 - 4x^7 - 88x^6 + 278x^5 + 3869x^4 - 8206x^3 - 71054x^2 + 75204*x + 945876
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.3
Root			$7.04303 + 2.29129i$ of defining polynomial
Character	$\chi$	$=$	1764.361
Dual form			1764.4.k.bc.1549.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.29150 + 9.16515i) q^{5} +(-27.6225 + 47.8435i) q^{11} +83.5225 q^{13} +(47.6235 - 82.4864i) q^{17} +(-41.7612 - 72.3326i) q^{19} +(-82.8674 - 143.530i) q^{23} +(6.50000 - 11.2583i) q^{25} -110.490 q^{29} +(41.7612 - 72.3326i) q^{31} +(-39.0000 - 67.5500i) q^{37} -412.737 q^{41} +148.000 q^{43} +(-232.826 - 403.267i) q^{47} +(55.2449 - 95.6870i) q^{53} -584.657 q^{55} +(275.158 - 476.588i) q^{59} +(292.329 + 506.328i) q^{61} +(441.959 + 765.496i) q^{65} +(130.000 - 225.167i) q^{67} +718.184 q^{71} +(-334.090 + 578.661i) q^{73} +(-332.000 - 575.041i) q^{79} -126.996 q^{83} +1008.00 q^{85} +(439.195 + 760.708i) q^{89} +(441.959 - 765.496i) q^{95} +1169.31 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 52 q^{25} - 312 q^{37} + 1184 q^{43} + 1040 q^{67} - 2656 q^{79} + 8064 q^{85}+O(q^{100}) $ 8 * q + 52 * q^25 - 312 * q^37 + 1184 * q^43 + 1040 * q^67 - 2656 * q^79 + 8064 * q^85

Character values

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

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	5.29150	+	9.16515i	0.473286	+	0.819756i	0.999532	−	0.0305763i	$-0.00973425\pi$
$5$	5.29150	+	9.16515i	0.473286	+	0.819756i	−0.526246	+	0.850332i	$0.676401\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−27.6225	+	47.8435i	−0.757135	+	1.31140i	0.187171	+	0.982327i	$0.440068\pi$
$11$	−27.6225	+	47.8435i	−0.757135	+	1.31140i	−0.944306	+	0.329069i	$0.893265\pi$
$12$		0			0
$13$	83.5225			1.78192			0.890960	−	0.454082i	$-0.150033\pi$
$13$	83.5225			1.78192			0.890960	+	0.454082i	$0.150033\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	47.6235	−	82.4864i	0.679435	−	1.17682i	−0.295716	−	0.955276i	$-0.595558\pi$
$17$	47.6235	−	82.4864i	0.679435	−	1.17682i	0.975151	−	0.221541i	$-0.0711085\pi$
$18$		0			0
$19$	−41.7612	−	72.3326i	−0.504246	−	0.873380i	−0.999988	−	0.00491032i	$-0.998437\pi$
$19$	−41.7612	−	72.3326i	−0.504246	−	0.873380i	0.495742	−	0.868470i	$-0.334896\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−82.8674	−	143.530i	−0.751263	−	1.30122i	−0.947211	−	0.320611i	$-0.896112\pi$
$23$	−82.8674	−	143.530i	−0.751263	−	1.30122i	0.195948	−	0.980614i	$-0.437221\pi$
$24$		0			0
$25$	6.50000	−	11.2583i	0.0520000	−	0.0900666i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−110.490			−0.707498			−0.353749	−	0.935340i	$-0.615093\pi$
$29$	−110.490			−0.707498			−0.353749	+	0.935340i	$0.615093\pi$
$30$		0			0
$31$	41.7612	−	72.3326i	0.241953	−	0.419075i	−0.719318	−	0.694681i	$-0.755545\pi$
$31$	41.7612	−	72.3326i	0.241953	−	0.419075i	0.961270	+	0.275607i	$0.0888787\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−39.0000	−	67.5500i	−0.173285	−	0.300139i	0.766281	−	0.642505i	$-0.222105\pi$
$37$	−39.0000	−	67.5500i	−0.173285	−	0.300139i	−0.939567	+	0.342366i	$0.888772\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−412.737			−1.57216			−0.786082	−	0.618122i	$-0.787894\pi$
$41$	−412.737			−1.57216			−0.786082	+	0.618122i	$0.787894\pi$
$42$		0			0
$43$	148.000			0.524879			0.262439	−	0.964948i	$-0.415473\pi$
$43$	148.000			0.524879			0.262439	+	0.964948i	$0.415473\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−232.826	−	403.267i	−0.722578	−	1.25154i	−0.959963	−	0.280127i	$-0.909624\pi$
$47$	−232.826	−	403.267i	−0.722578	−	1.25154i	0.237385	−	0.971416i	$-0.423710\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	55.2449	−	95.6870i	0.143179	−	0.247993i	−0.785513	−	0.618845i	$-0.787601\pi$
$53$	55.2449	−	95.6870i	0.143179	−	0.247993i	0.928692	+	0.370852i	$0.120934\pi$
$54$		0			0
$55$	−584.657			−1.43337
$56$		0			0
$57$		0			0
$58$		0			0
$59$	275.158	−	476.588i	0.607162	−	1.05163i	−0.384544	−	0.923107i	$-0.625641\pi$
$59$	275.158	−	476.588i	0.607162	−	1.05163i	0.991706	−	0.128528i	$-0.0410253\pi$
$60$		0			0
$61$	292.329	+	506.328i	0.613588	+	1.06276i	0.990631	+	0.136569i	$0.0436076\pi$
$61$	292.329	+	506.328i	0.613588	+	1.06276i	−0.377043	+	0.926196i	$0.623059\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	441.959	+	765.496i	0.843358	+	1.46074i
$66$		0			0
$67$	130.000	−	225.167i	0.237045	−	0.410574i	−0.722820	−	0.691036i	$-0.757154\pi$
$67$	130.000	−	225.167i	0.237045	−	0.410574i	0.959865	+	0.280462i	$0.0904877\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	718.184			1.20046			0.600231	−	0.799827i	$-0.295075\pi$
$71$	718.184			1.20046			0.600231	+	0.799827i	$0.295075\pi$
$72$		0			0
$73$	−334.090	+	578.661i	−0.535647	+	0.927768i	0.463484	+	0.886105i	$0.346599\pi$
$73$	−334.090	+	578.661i	−0.535647	+	0.927768i	−0.999132	+	0.0416633i	$0.986734\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−332.000	−	575.041i	−0.472822	−	0.818951i	0.526694	−	0.850055i	$-0.323431\pi$
$79$	−332.000	−	575.041i	−0.472822	−	0.818951i	−0.999516	+	0.0311034i	$0.990098\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−126.996			−0.167947			−0.0839737	−	0.996468i	$-0.526761\pi$
$83$	−126.996			−0.167947			−0.0839737	+	0.996468i	$0.526761\pi$
$84$		0			0
$85$	1008.00			1.28627
$86$		0			0
$87$		0			0
$88$		0			0
$89$	439.195	+	760.708i	0.523085	+	0.906009i	0.999639	+	0.0268644i	$0.00855223\pi$
$89$	439.195	+	760.708i	0.523085	+	0.906009i	−0.476554	+	0.879145i	$0.658114\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	441.959	−	765.496i	0.477306	−	0.826718i
$96$		0			0
$97$	1169.31			1.22398			0.611989	−	0.790866i	$-0.290370\pi$
$97$	1169.31			1.22398			0.611989	+	0.790866i	$0.290370\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	227.535	−	394.102i	0.224164	−	0.388263i	−0.731904	−	0.681407i	$-0.761368\pi$
$101$	227.535	−	394.102i	0.224164	−	0.388263i	0.956068	+	0.293144i	$0.0947016\pi$
$102$		0			0
$103$	−542.896	−	940.323i	−0.519351	−	0.899542i	−0.999747	−	0.0224903i	$-0.992840\pi$
$103$	−542.896	−	940.323i	−0.519351	−	0.899542i	0.480396	−	0.877052i	$-0.340493\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−469.582	−	813.339i	−0.424263	−	0.734846i	0.572088	−	0.820192i	$-0.306134\pi$
$107$	−469.582	−	813.339i	−0.424263	−	0.734846i	−0.996351	+	0.0853466i	$0.972800\pi$
$108$		0			0
$109$	325.000	−	562.917i	0.285590	−	0.494657i	−0.687162	−	0.726504i	$-0.741144\pi$
$109$	325.000	−	562.917i	0.285590	−	0.494657i	0.972752	+	0.231847i	$0.0744770\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1546.86			−1.28775			−0.643877	−	0.765129i	$-0.722675\pi$
$113$	−1546.86			−1.28775			−0.643877	+	0.765129i	$0.722675\pi$
$114$		0			0
$115$	876.986	−	1518.98i	0.711125	−	1.23170i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−860.500	−	1490.43i	−0.646506	−	1.11978i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1460.45			1.04502
$126$		0			0
$127$	2288.00			1.59864			0.799320	−	0.600906i	$-0.205193\pi$
$127$	2288.00			1.59864			0.799320	+	0.600906i	$0.205193\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−444.486	−	769.873i	−0.296450	−	0.513466i	0.678871	−	0.734257i	$-0.262469\pi$
$131$	−444.486	−	769.873i	−0.296450	−	0.513466i	−0.975321	+	0.220791i	$0.929136\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1436.37	+	2487.86i	−0.895746	+	1.55148i	−0.0628668	+	0.998022i	$0.520024\pi$
$137$	−1436.37	+	2487.86i	−0.895746	+	1.55148i	−0.832879	+	0.553455i	$0.813309\pi$
$138$		0			0
$139$	501.135			0.305796			0.152898	−	0.988242i	$-0.451139\pi$
$139$	501.135			0.305796			0.152898	+	0.988242i	$0.451139\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2307.10	+	3996.01i	−1.34915	+	2.33680i
$144$		0			0
$145$	−584.657	−	1012.66i	−0.334849	−	0.579976i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−939.163	−	1626.68i	−0.516371	−	0.894381i	−0.999819	−	0.0190078i	$-0.993949\pi$
$149$	−939.163	−	1626.68i	−0.516371	−	0.894381i	0.483448	−	0.875373i	$-0.339384\pi$
$150$		0			0
$151$	988.000	−	1711.27i	0.532466	−	0.922257i	−0.466816	−	0.884355i	$-0.654599\pi$
$151$	988.000	−	1711.27i	0.532466	−	0.922257i	0.999281	−	0.0379029i	$-0.0120677\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	883.919			0.458052
$156$		0			0
$157$	−292.329	+	506.328i	−0.148601	+	0.257385i	−0.930711	−	0.365756i	$-0.880810\pi$
$157$	−292.329	+	506.328i	−0.148601	+	0.257385i	0.782110	+	0.623141i	$0.214144\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1674.00	−	2899.45i	−0.804404	−	1.39327i	−0.916693	−	0.399592i	$-0.869152\pi$
$163$	−1674.00	−	2899.45i	−0.804404	−	1.39327i	0.112289	−	0.993676i	$-0.464182\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−84.6640			−0.0392305			−0.0196153	−	0.999808i	$-0.506244\pi$
$167$	−84.6640			−0.0392305			−0.0196153	+	0.999808i	$0.506244\pi$
$168$		0			0
$169$	4779.00			2.17524
$170$		0			0
$171$		0			0
$172$		0			0
$173$	756.685	+	1310.62i	0.332542	+	0.575979i	0.983009	−	0.183555i	$-0.0587605\pi$
$173$	756.685	+	1310.62i	0.332542	+	0.575979i	−0.650468	+	0.759534i	$0.725427\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−82.8674	+	143.530i	−0.0346022	+	0.0599328i	−0.882808	−	0.469734i	$-0.844350\pi$
$179$	−82.8674	+	143.530i	−0.0346022	+	0.0599328i	0.848206	+	0.529667i	$0.177683\pi$
$180$		0			0
$181$	−3591.47			−1.47487			−0.737435	−	0.675418i	$-0.763963\pi$
$181$	−3591.47			−1.47487			−0.737435	+	0.675418i	$0.763963\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	412.737	−	714.882i	0.164027	−	0.284104i
$186$		0			0
$187$	2630.96	+	4556.95i	1.02885	+	1.78202i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1077.28	+	1865.90i	0.408110	+	0.706867i	0.994678	−	0.103033i	$-0.0328548\pi$
$191$	1077.28	+	1865.90i	0.408110	+	0.706867i	−0.586568	+	0.809900i	$0.699521\pi$
$192$		0			0
$193$	689.000	−	1193.38i	0.256970	−	0.445086i	−0.708458	−	0.705753i	$-0.750609\pi$
$193$	689.000	−	1193.38i	0.256970	−	0.445086i	0.965429	+	0.260667i	$0.0839423\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	994.408			0.359638			0.179819	−	0.983700i	$-0.442449\pi$
$197$	994.408			0.359638			0.179819	+	0.983700i	$0.442449\pi$
$198$		0			0
$199$	−2338.63	+	4050.62i	−0.833070	+	1.44292i	0.0625216	+	0.998044i	$0.480086\pi$
$199$	−2338.63	+	4050.62i	−0.833070	+	1.44292i	−0.895592	+	0.444876i	$0.853248\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−2184.00	−	3782.80i	−0.744084	−	1.28879i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4614.19			1.52713
$210$		0			0
$211$	5108.00			1.66658			0.833292	−	0.552833i	$-0.186453\pi$
$211$	5108.00			1.66658			0.833292	+	0.552833i	$0.186453\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	783.142	+	1356.44i	0.248418	+	0.430273i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3977.63	−	6889.46i	1.21070	−	2.09699i
$222$		0			0
$223$	1169.31			0.351135			0.175567	−	0.984467i	$-0.443824\pi$
$223$	1169.31			0.351135			0.175567	+	0.984467i	$0.443824\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	677.312	−	1173.14i	0.198039	−	0.343013i	−0.749854	−	0.661604i	$-0.769876\pi$
$227$	677.312	−	1173.14i	0.198039	−	0.343013i	0.947892	+	0.318590i	$0.103209\pi$
$228$		0			0
$229$	−542.896	−	940.323i	−0.156662	−	0.271346i	0.777001	−	0.629499i	$-0.216740\pi$
$229$	−542.896	−	940.323i	−0.156662	−	0.271346i	−0.933663	+	0.358153i	$0.883407\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1878.33	+	3253.36i	0.528126	+	0.914741i	0.999462	+	0.0327872i	$0.0104384\pi$
$233$	1878.33	+	3253.36i	0.528126	+	0.914741i	−0.471337	+	0.881953i	$0.656228\pi$
$234$		0			0
$235$	2464.00	−	4267.77i	0.683973	−	1.18468i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3148.96			−0.852256			−0.426128	−	0.904663i	$-0.640123\pi$
$239$	−3148.96			−0.852256			−0.426128	+	0.904663i	$0.640123\pi$
$240$		0			0
$241$	2923.29	−	5063.28i	0.781350	−	1.35334i	−0.149805	−	0.988716i	$-0.547865\pi$
$241$	2923.29	−	5063.28i	0.781350	−	1.35334i	0.931155	−	0.364623i	$-0.118802\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−3488.00	−	6041.39i	−0.898527	−	1.55629i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6011.15			−1.51163			−0.755817	−	0.654783i	$-0.772760\pi$
$251$	−6011.15			−1.51163			−0.755817	+	0.654783i	$0.772760\pi$
$252$		0			0
$253$	9156.00			2.27523
$254$		0			0
$255$		0			0
$256$		0			0
$257$	502.693	+	870.689i	0.122012	+	0.211331i	0.920561	−	0.390599i	$-0.127732\pi$
$257$	502.693	+	870.689i	0.122012	+	0.211331i	−0.798549	+	0.601930i	$0.794399\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1850.70	+	3205.51i	−0.433914	+	0.751561i	−0.997206	−	0.0746968i	$-0.976201\pi$
$263$	−1850.70	+	3205.51i	−0.433914	+	0.751561i	0.563293	+	0.826258i	$0.309534\pi$
$264$		0			0
$265$	1169.31			0.271058
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2100.73	−	3638.57i	0.476147	−	0.824711i	−0.523479	−	0.852038i	$-0.675366\pi$
$269$	2100.73	−	3638.57i	0.476147	−	0.824711i	0.999627	+	0.0273274i	$0.00869966\pi$
$270$		0			0
$271$	−3800.27	−	6582.26i	−0.851845	−	1.47544i	−0.879541	−	0.475823i	$-0.842150\pi$
$271$	−3800.27	−	6582.26i	−0.851845	−	1.47544i	0.0276959	−	0.999616i	$-0.491183\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	359.092	+	621.965i	0.0787420	+	0.136385i
$276$		0			0
$277$	529.000	−	916.255i	0.114746	−	0.198745i	−0.802932	−	0.596070i	$-0.796728\pi$
$277$	529.000	−	916.255i	0.114746	−	0.198745i	0.917678	+	0.397325i	$0.130061\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8618.21			1.82961			0.914803	−	0.403901i	$-0.132346\pi$
$281$	8618.21			1.82961			0.914803	+	0.403901i	$0.132346\pi$
$282$		0			0
$283$	542.896	−	940.323i	0.114035	−	0.197514i	−0.803359	−	0.595495i	$-0.796956\pi$
$283$	542.896	−	940.323i	0.114035	−	0.197514i	0.917393	+	0.397981i	$0.130289\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2079.50	−	3601.80i	−0.423265	−	0.733116i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−6000.56			−1.19644			−0.598220	−	0.801332i	$-0.704125\pi$
$293$	−6000.56			−1.19644			−0.598220	+	0.801332i	$0.704125\pi$
$294$		0			0
$295$	5824.00			1.14945
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6921.29	−	11988.0i	−1.33869	−	2.31868i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3093.71	+	5358.47i	−0.580805	+	1.00598i
$306$		0			0
$307$	6932.36			1.28877			0.644383	−	0.764703i	$-0.277114\pi$
$307$	6932.36			1.28877			0.644383	+	0.764703i	$0.277114\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−275.158	+	476.588i	−0.0501697	+	0.0868965i	−0.890020	−	0.455922i	$-0.849310\pi$
$311$	−275.158	+	476.588i	−0.0501697	+	0.0868965i	0.839850	+	0.542819i	$0.182643\pi$
$312$		0			0
$313$	−250.567	−	433.995i	−0.0452489	−	0.0783734i	0.842514	−	0.538675i	$-0.181075\pi$
$313$	−250.567	−	433.995i	−0.0452489	−	0.0783734i	−0.887763	+	0.460301i	$0.847741\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4806.31	−	8324.77i	−0.851574	−	1.47497i	−0.879787	−	0.475368i	$-0.842315\pi$
$317$	−4806.31	−	8324.77i	−0.851574	−	1.47497i	0.0282125	−	0.999602i	$-0.491019\pi$
$318$		0			0
$319$	3052.00	−	5286.22i	0.535671	−	0.927810i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7955.27			−1.37041
$324$		0			0
$325$	542.896	−	940.323i	0.0926598	−	0.160492i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4162.00	−	7208.80i	−0.691131	−	1.19707i	−0.971468	−	0.237172i	$-0.923780\pi$
$331$	−4162.00	−	7208.80i	−0.691131	−	1.19707i	0.280337	−	0.959902i	$-0.409554\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	2751.58			0.448761
$336$		0			0
$337$	9794.00			1.58313			0.791563	−	0.611088i	$-0.209268\pi$
$337$	9794.00			1.58313			0.791563	+	0.611088i	$0.209268\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2307.10	+	3996.01i	0.366382	+	0.634592i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1077.28	+	1865.90i	−0.166660	+	0.288664i	−0.937244	−	0.348675i	$-0.886632\pi$
$347$	−1077.28	+	1865.90i	−0.166660	+	0.288664i	0.770583	+	0.637339i	$0.219965\pi$
$348$		0			0
$349$	−1753.97			−0.269020			−0.134510	−	0.990912i	$-0.542946\pi$
$349$	−1753.97			−0.269020			−0.134510	+	0.990912i	$0.542946\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3095.53	−	5361.61i	0.466738	−	0.808413i	−0.532540	−	0.846405i	$-0.678763\pi$
$353$	3095.53	−	5361.61i	0.466738	−	0.808413i	0.999278	+	0.0379913i	$0.0120959\pi$
$354$		0			0
$355$	3800.27	+	6582.26i	0.568162	+	0.984085i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1463.99	−	2535.71i	−0.215227	−	0.372784i	0.738116	−	0.674674i	$-0.235716\pi$
$359$	−1463.99	−	2535.71i	−0.215227	−	0.372784i	−0.953343	+	0.301890i	$0.902382\pi$
$360$		0			0
$361$	−58.5000	+	101.325i	−0.00852894	+	0.0147726i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−7071.35			−1.01406
$366$		0			0
$367$	−4343.17	+	7522.59i	−0.617743	+	1.06996i	0.372154	+	0.928171i	$0.378619\pi$
$367$	−4343.17	+	7522.59i	−0.617743	+	1.06996i	−0.989897	+	0.141790i	$0.954714\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−2327.00	−	4030.48i	−0.323023	−	0.559492i	0.658087	−	0.752942i	$-0.271366\pi$
$373$	−2327.00	−	4030.48i	−0.323023	−	0.559492i	−0.981110	+	0.193450i	$0.938032\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−9228.38			−1.26070
$378$		0			0
$379$	2388.00			0.323650			0.161825	−	0.986819i	$-0.448262\pi$
$379$	2388.00			0.323650			0.161825	+	0.986819i	$0.448262\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4127.37	+	7148.82i	0.550650	+	0.953753i	0.998228	+	0.0595085i	$0.0189533\pi$
$383$	4127.37	+	7148.82i	0.550650	+	0.953753i	−0.447578	+	0.894245i	$0.647713\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2154.55	−	3731.79i	0.280823	−	0.486399i	−0.690765	−	0.723080i	$-0.742726\pi$
$389$	2154.55	−	3731.79i	0.280823	−	0.486399i	0.971588	+	0.236680i	$0.0760593\pi$
$390$		0			0
$391$	−15785.7			−2.04174
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3513.56	−	6085.66i	0.447560	−	0.775197i
$396$		0			0
$397$	5220.15	+	9041.57i	0.659929	+	1.14303i	0.980633	+	0.195852i	$0.0627472\pi$
$397$	5220.15	+	9041.57i	0.659929	+	1.14303i	−0.320704	+	0.947179i	$0.603919\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	331.469	+	574.122i	0.0412788	+	0.0714970i	0.885927	−	0.463825i	$-0.153523\pi$
$401$	331.469	+	574.122i	0.0412788	+	0.0714970i	−0.844648	+	0.535322i	$0.820190\pi$
$402$		0			0
$403$	3488.00	−	6041.39i	0.431141	−	0.746757i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4309.10			0.524802
$408$		0			0
$409$	−3758.51	+	6509.93i	−0.454392	+	0.787030i	−0.998653	−	0.0518858i	$-0.983477\pi$
$409$	−3758.51	+	6509.93i	−0.454392	+	0.787030i	0.544261	+	0.838916i	$0.316810\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−672.000	−	1163.94i	−0.0794872	−	0.137676i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	13419.3			1.56461			0.782307	−	0.622893i	$-0.214043\pi$
$419$	13419.3			1.56461			0.782307	+	0.622893i	$0.214043\pi$
$420$		0			0
$421$	8238.00			0.953671			0.476836	−	0.878993i	$-0.341784\pi$
$421$	8238.00			0.953671			0.476836	+	0.878993i	$0.341784\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−619.106	−	1072.32i	−0.0706613	−	0.122389i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4170.99	−	7224.37i	0.466147	−	0.807391i	−0.533105	−	0.846049i	$-0.678975\pi$
$431$	4170.99	−	7224.37i	0.466147	−	0.807391i	0.999253	+	0.0386580i	$0.0123083\pi$
$432$		0			0
$433$	1837.49			0.203936			0.101968	−	0.994788i	$-0.467486\pi$
$433$	1837.49			0.203936			0.101968	+	0.994788i	$0.467486\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6921.29	+	11988.0i	−0.757643	+	1.31228i
$438$		0			0
$439$	−5261.91	−	9113.90i	−0.572067	−	0.990849i	−0.996353	−	0.0853213i	$-0.972808\pi$
$439$	−5261.91	−	9113.90i	−0.572067	−	0.990849i	0.424286	−	0.905528i	$-0.360525\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6877.99	+	11913.0i	0.737660	+	1.27766i	0.953547	+	0.301246i	$0.0974024\pi$
$443$	6877.99	+	11913.0i	0.737660	+	1.27766i	−0.215887	+	0.976418i	$0.569264\pi$
$444$		0			0
$445$	−4648.00	+	8050.57i	−0.495138	+	0.857604i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	9281.14			0.975511			0.487755	−	0.872980i	$-0.337816\pi$
$449$	9281.14			0.975511			0.487755	+	0.872980i	$0.337816\pi$
$450$		0			0
$451$	11400.8	−	19746.8i	1.19034	−	2.06173i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1445.00	+	2502.81i	0.147909	+	0.256185i	0.930454	−	0.366408i	$-0.119413\pi$
$457$	1445.00	+	2502.81i	0.147909	+	0.256185i	−0.782546	+	0.622593i	$0.786079\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	14191.8			1.43379			0.716896	−	0.697180i	$-0.245562\pi$
$461$	14191.8			1.43379			0.716896	+	0.697180i	$0.245562\pi$
$462$		0			0
$463$	−8632.00			−0.866443			−0.433221	−	0.901288i	$-0.642623\pi$
$463$	−8632.00			−0.866443			−0.433221	+	0.901288i	$0.642623\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	2201.27	+	3812.70i	0.218121	+	0.377796i	0.954233	−	0.299063i	$-0.0966740\pi$
$467$	2201.27	+	3812.70i	0.218121	+	0.377796i	−0.736113	+	0.676859i	$0.763341\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4088.12	+	7080.84i	−0.397404	+	0.688324i
$474$		0			0
$475$	−1085.79			−0.104883
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4635.36	+	8028.67i	−0.442160	+	0.765844i	−0.997850	−	0.0655459i	$-0.979121\pi$
$479$	−4635.36	+	8028.67i	−0.442160	+	0.765844i	0.555689	+	0.831390i	$0.312454\pi$
$480$		0			0
$481$	−3257.38	−	5641.94i	−0.308781	−	0.534824i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6187.43	+	10716.9i	0.579292	+	1.00336i
$486$		0			0
$487$	6268.00	−	10856.5i	0.583224	−	1.01017i	−0.411870	−	0.911243i	$-0.635124\pi$
$487$	6268.00	−	10856.5i	0.583224	−	1.01017i	0.995094	−	0.0989314i	$-0.0315424\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	19391.0			1.78228			0.891142	−	0.453724i	$-0.149905\pi$
$491$	19391.0			1.78228			0.891142	+	0.453724i	$0.149905\pi$
$492$		0			0
$493$	−5261.91	+	9113.90i	−0.480699	+	0.832595i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−3378.00	−	5850.87i	−0.303046	−	0.524891i	0.673778	−	0.738934i	$-0.264670\pi$
$499$	−3378.00	−	5850.87i	−0.303046	−	0.524891i	−0.976824	+	0.214042i	$0.931337\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−16721.1			−1.48222			−0.741112	−	0.671381i	$-0.765701\pi$
$503$	−16721.1			−1.48222			−0.741112	+	0.671381i	$0.765701\pi$
$504$		0			0
$505$	4816.00			0.424375
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5497.87	−	9522.59i	−0.478760	−	0.829237i	0.520943	−	0.853591i	$-0.325580\pi$
$509$	−5497.87	−	9522.59i	−0.478760	−	0.829237i	−0.999703	+	0.0243545i	$0.992247\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5745.47	−	9951.45i	0.491603	−	0.851482i
$516$		0			0
$517$	25724.9			2.18836
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10387.2	+	17991.2i	−0.873459	+	1.51288i	−0.0150646	+	0.999887i	$0.504795\pi$
$521$	−10387.2	+	17991.2i	−0.873459	+	1.51288i	−0.858395	+	0.512990i	$0.828538\pi$
$522$		0			0
$523$	835.225	+	1446.65i	0.0698314	+	0.120952i	0.898827	−	0.438304i	$-0.144421\pi$
$523$	835.225	+	1446.65i	0.0698314	+	0.120952i	−0.828996	+	0.559255i	$0.811087\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3977.63	−	6889.46i	−0.328783	−	0.569468i
$528$		0			0
$529$	−7650.50	+	13251.1i	−0.628791	+	1.08910i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−34472.8			−2.80147
$534$		0			0
$535$	4969.59	−	8607.58i	0.401596	−	0.695585i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	9685.00	+	16774.9i	0.769669	+	1.33310i	0.937743	+	0.347331i	$0.112912\pi$
$541$	9685.00	+	16774.9i	0.769669	+	1.33310i	−0.168074	+	0.985774i	$0.553755\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6878.95			0.540664
$546$		0			0
$547$	−20396.0			−1.59428			−0.797139	−	0.603796i	$-0.793654\pi$
$547$	−20396.0			−1.59428			−0.797139	+	0.603796i	$0.793654\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4614.19	+	7992.01i	0.356753	+	0.617915i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2154.55	−	3731.79i	0.163898	−	0.283880i	−0.772365	−	0.635179i	$-0.780926\pi$
$557$	2154.55	−	3731.79i	0.163898	−	0.283880i	0.936263	+	0.351299i	$0.114260\pi$
$558$		0			0
$559$	12361.3			0.935292
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−21.1660	+	36.6606i	−0.00158444	+	0.00274433i	−0.866817	−	0.498627i	$-0.833838\pi$
$563$	−21.1660	+	36.6606i	−0.00158444	+	0.00274433i	0.865232	+	0.501372i	$0.167171\pi$
$564$		0			0
$565$	−8185.20	−	14177.2i	−0.609476	−	1.05564i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3314.69	−	5741.22i	−0.244217	−	0.422995i	0.717694	−	0.696358i	$-0.245197\pi$
$569$	−3314.69	−	5741.22i	−0.244217	−	0.422995i	−0.961911	+	0.273363i	$0.911864\pi$
$570$		0			0
$571$	−654.000	+	1132.76i	−0.0479318	+	0.0830203i	−0.888996	−	0.457915i	$-0.848596\pi$
$571$	−654.000	+	1132.76i	−0.0479318	+	0.0830203i	0.841064	+	0.540935i	$0.181930\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2154.55			−0.156263
$576$		0			0
$577$	−4343.17	+	7522.59i	−0.313359	+	0.542755i	−0.979087	−	0.203440i	$-0.934788\pi$
$577$	−4343.17	+	7522.59i	−0.313359	+	0.542755i	0.665728	+	0.746195i	$0.268121\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	3052.00	+	5286.22i	0.216811	+	0.375528i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	25737.9			1.80974			0.904868	−	0.425691i	$-0.139969\pi$
$587$	25737.9			1.80974			0.904868	+	0.425691i	$0.139969\pi$
$588$		0			0
$589$	−6976.00			−0.488015
$590$		0			0
$591$		0			0
$592$		0			0
$593$	111.122	+	192.468i	0.00769514	+	0.0133284i	0.869847	−	0.493321i	$-0.164217\pi$
$593$	111.122	+	192.468i	0.00769514	+	0.0133284i	−0.862152	+	0.506649i	$0.830884\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11187.1	+	19376.6i	−0.763092	+	1.32171i	0.178157	+	0.984002i	$0.442986\pi$
$599$	−11187.1	+	19376.6i	−0.763092	+	1.32171i	−0.941249	+	0.337713i	$0.890347\pi$
$600$		0			0
$601$	−21715.8			−1.47389			−0.736945	−	0.675953i	$-0.763732\pi$
$601$	−21715.8			−1.47389			−0.736945	+	0.675953i	$0.763732\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9106.68	−	15773.2i	0.611965	−	1.05996i
$606$		0			0
$607$	6765.32	+	11717.9i	0.452382	+	0.783548i	0.998533	−	0.0541379i	$-0.0172411\pi$
$607$	6765.32	+	11717.9i	0.452382	+	0.783548i	−0.546152	+	0.837686i	$0.683908\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−19446.2	−	33681.8i	−1.28758	−	2.23015i
$612$		0			0
$613$	−6385.00	+	11059.1i	−0.420698	+	0.728670i	−0.996008	−	0.0892655i	$-0.971548\pi$
$613$	−6385.00	+	11059.1i	−0.420698	+	0.728670i	0.575310	+	0.817935i	$0.304881\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	11049.0			0.720932			0.360466	−	0.932772i	$-0.382618\pi$
$617$	11049.0			0.720932			0.360466	+	0.932772i	$0.382618\pi$
$618$		0			0
$619$	−9103.95	+	15768.5i	−0.591145	+	1.02389i	0.402934	+	0.915229i	$0.367990\pi$
$619$	−9103.95	+	15768.5i	−0.591145	+	1.02389i	−0.994079	+	0.108663i	$0.965343\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	6915.50	+	11978.0i	0.442592	+	0.766592i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−7429.27			−0.470945
$630$		0			0
$631$	−23952.0			−1.51112			−0.755558	−	0.655082i	$-0.772634\pi$
$631$	−23952.0			−1.51112			−0.755558	+	0.655082i	$0.772634\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	12107.0	+	20969.9i	0.756614	+	1.31049i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3867.14	+	6698.09i	−0.238289	+	0.412728i	−0.960223	−	0.279233i	$-0.909920\pi$
$641$	−3867.14	+	6698.09i	−0.238289	+	0.412728i	0.721935	+	0.691961i	$0.243253\pi$
$642$		0			0
$643$	−16286.9			−0.998899			−0.499449	−	0.866343i	$-0.666464\pi$
$643$	−16286.9			−0.998899			−0.499449	+	0.866343i	$0.666464\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6603.80	−	11438.1i	0.401270	−	0.695021i	−0.592609	−	0.805490i	$-0.701902\pi$
$647$	6603.80	−	11438.1i	0.401270	−	0.695021i	0.993880	+	0.110469i	$0.0352354\pi$
$648$		0			0
$649$	15201.1	+	26329.1i	0.919407	+	1.59246i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9336.39	+	16171.1i	0.559512	+	0.969103i	0.997537	+	0.0701402i	$0.0223447\pi$
$653$	9336.39	+	16171.1i	0.559512	+	0.969103i	−0.438025	+	0.898963i	$0.644322\pi$
$654$		0			0
$655$	4704.00	−	8147.57i	0.280611	−	0.486033i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6353.16			−0.375545			−0.187772	−	0.982213i	$-0.560127\pi$
$659$	−6353.16			−0.375545			−0.187772	+	0.982213i	$0.560127\pi$
$660$		0			0
$661$	9980.93	−	17287.5i	0.587312	−	1.01725i	−0.407271	−	0.913307i	$-0.633519\pi$
$661$	9980.93	−	17287.5i	0.587312	−	1.01725i	0.994583	−	0.103947i	$-0.0331472\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	9156.00	+	15858.7i	0.531517	+	0.920614i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−32299.3			−1.85827
$672$		0			0
$673$	−12046.0			−0.689954			−0.344977	−	0.938611i	$-0.612113\pi$
$673$	−12046.0			−0.689954			−0.344977	+	0.938611i	$0.612113\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13826.7	−	23948.5i	−0.784938	−	1.35955i	−0.929036	−	0.369989i	$-0.879361\pi$
$677$	−13826.7	−	23948.5i	−0.784938	−	1.35955i	0.144098	−	0.989563i	$-0.453972\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−3950.01	+	6841.62i	−0.221293	+	0.383290i	−0.955201	−	0.295959i	$-0.904361\pi$
$683$	−3950.01	+	6841.62i	−0.221293	+	0.383290i	0.733908	+	0.679249i	$0.237694\pi$
$684$		0			0
$685$	−30402.2			−1.69578
$686$		0			0
$687$		0			0
$688$		0			0
$689$	4614.19	−	7992.01i	0.255133	−	0.441903i
$690$		0			0
$691$	−3842.03	−	6654.60i	−0.211516	−	0.366357i	0.740673	−	0.671866i	$-0.234507\pi$
$691$	−3842.03	−	6654.60i	−0.211516	−	0.366357i	−0.952189	+	0.305509i	$0.901174\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2651.76	+	4592.98i	0.144729	+	0.250678i
$696$		0			0
$697$	−19656.0	+	34045.2i	−1.06818	+	1.85015i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	12043.4			0.648891			0.324445	−	0.945904i	$-0.394822\pi$
$701$	12043.4			0.648891			0.324445	+	0.945904i	$0.394822\pi$
$702$		0			0
$703$	−3257.38	+	5641.94i	−0.174757	+	0.302688i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	12769.0	+	22116.6i	0.676375	+	1.17152i	0.976065	+	0.217479i	$0.0697834\pi$
$709$	12769.0	+	22116.6i	0.676375	+	1.17152i	−0.299690	+	0.954037i	$0.596883\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−13842.6			−0.727080
$714$		0			0
$715$	−48832.0			−2.55414
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2476.42	+	4289.29i	0.128449	+	0.222481i	0.923076	−	0.384618i	$-0.125667\pi$
$719$	2476.42	+	4289.29i	0.128449	+	0.222481i	−0.794627	+	0.607098i	$0.792333\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−718.184	+	1243.93i	−0.0367899	+	0.0637220i
$726$		0			0
$727$	14115.3			0.720093			0.360046	−	0.932934i	$-0.382761\pi$
$727$	14115.3			0.720093			0.360046	+	0.932934i	$0.382761\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	7048.28	−	12208.0i	0.356621	−	0.617686i
$732$		0			0
$733$	−6472.99	−	11211.5i	−0.326174	−	0.564949i	0.655575	−	0.755130i	$-0.272426\pi$
$733$	−6472.99	−	11211.5i	−0.326174	−	0.564949i	−0.981749	+	0.190180i	$0.939093\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	7181.84	+	12439.3i	0.358950	+	0.621720i
$738$		0			0
$739$	7270.00	−	12592.0i	0.361883	−	0.626799i	−0.626388	−	0.779511i	$-0.715467\pi$
$739$	7270.00	−	12592.0i	0.361883	−	0.626799i	0.988271	+	0.152712i	$0.0488008\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1160.14			0.0572833			0.0286417	−	0.999590i	$-0.490882\pi$
$743$	1160.14			0.0572833			0.0286417	+	0.999590i	$0.490882\pi$
$744$		0			0
$745$	9939.17	−	17215.2i	0.488783	−	0.846596i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	3384.00	+	5861.26i	0.164426	+	0.284794i	0.936451	−	0.350798i	$-0.114089\pi$
$751$	3384.00	+	5861.26i	0.164426	+	0.284794i	−0.772025	+	0.635592i	$0.780756\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	20912.0			1.00803
$756$		0			0
$757$	−2354.00			−0.113022			−0.0565110	−	0.998402i	$-0.517998\pi$
$757$	−2354.00			−0.113022			−0.0565110	+	0.998402i	$0.517998\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−16535.9	−	28641.1i	−0.787684	−	1.36431i	−0.927382	−	0.374115i	$-0.877947\pi$
$761$	−16535.9	−	28641.1i	−0.787684	−	1.36431i	0.139698	−	0.990194i	$-0.455387\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	22981.9	−	39805.8i	1.08191	−	1.87393i
$768$		0			0
$769$	22885.2			1.07316			0.536580	−	0.843850i	$-0.319716\pi$
$769$	22885.2			1.07316			0.536580	+	0.843850i	$0.319716\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12175.7	+	21089.0i	−0.566535	+	0.981267i	0.430371	+	0.902652i	$0.358383\pi$
$773$	−12175.7	+	21089.0i	−0.566535	+	0.981267i	−0.996905	+	0.0786143i	$0.974950\pi$
$774$		0			0
$775$	−542.896	−	940.323i	−0.0251631	−	0.0435838i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	17236.4	+	29854.3i	0.792758	+	1.37310i
$780$		0			0
$781$	−19838.0	+	34360.4i	−0.908911	+	1.57428i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6187.43			−0.281323
$786$		0			0
$787$	−18124.4	+	31392.3i	−0.820920	+	1.42188i	0.0840778	+	0.996459i	$0.473206\pi$
$787$	−18124.4	+	31392.3i	−0.820920	+	1.42188i	−0.904998	+	0.425416i	$0.860128\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	24416.0	+	42289.8i	1.09336	+	1.89376i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−33431.7			−1.48584			−0.742918	−	0.669382i	$-0.766559\pi$
$797$	−33431.7			−1.48584			−0.742918	+	0.669382i	$0.766559\pi$
$798$		0			0
$799$	−44352.0			−1.96378
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−18456.8	−	31968.0i	−0.811115	−	1.40489i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	17899.4	−	31002.6i	0.777883	−	1.34733i	−0.155276	−	0.987871i	$-0.549627\pi$
$809$	17899.4	−	31002.6i	0.777883	−	1.34733i	0.933160	−	0.359462i	$-0.117040\pi$
$810$		0			0
$811$	30402.2			1.31636			0.658178	−	0.752862i	$-0.271327\pi$
$811$	30402.2			1.31636			0.658178	+	0.752862i	$0.271327\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	17716.0	−	30684.9i	0.761427	−	1.31883i
$816$		0			0
$817$	−6180.66	−	10705.2i	−0.264668	−	0.458419i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−16518.2	−	28610.4i	−0.702180	−	1.21621i	−0.967699	−	0.252107i	$-0.918877\pi$
$821$	−16518.2	−	28610.4i	−0.702180	−	1.21621i	0.265519	−	0.964106i	$-0.414457\pi$
$822$		0			0
$823$	5200.00	−	9006.66i	0.220244	−	0.381473i	−0.734638	−	0.678459i	$-0.762648\pi$
$823$	5200.00	−	9006.66i	0.220244	−	0.381473i	0.954882	+	0.296986i	$0.0959814\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−10109.8			−0.425094			−0.212547	−	0.977151i	$-0.568176\pi$
$827$	−10109.8			−0.425094			−0.212547	+	0.977151i	$0.568176\pi$
$828$		0			0
$829$	7057.65	−	12224.2i	0.295684	−	0.512140i	−0.679460	−	0.733713i	$-0.737786\pi$
$829$	7057.65	−	12224.2i	0.295684	−	0.512140i	0.975144	+	0.221573i	$0.0711191\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−448.000	−	775.959i	−0.0185673	−	0.0321595i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−21843.3			−0.898826			−0.449413	−	0.893324i	$-0.648367\pi$
$839$	−21843.3			−0.898826			−0.449413	+	0.893324i	$0.648367\pi$
$840$		0			0
$841$	−12181.0			−0.499446
$842$		0			0
$843$		0			0
$844$		0			0
$845$	25288.1	+	43800.3i	1.02951	+	1.78317i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6463.65	+	11195.4i	−0.260366	+	0.450967i
$852$		0			0
$853$	25641.4			1.02924			0.514622	−	0.857417i	$-0.327932\pi$
$853$	25641.4			1.02924			0.514622	+	0.857417i	$0.327932\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−365.114	+	632.395i	−0.0145531	+	0.0252068i	−0.873210	−	0.487344i	$-0.837966\pi$
$857$	−365.114	+	632.395i	−0.0145531	+	0.0252068i	0.858657	+	0.512550i	$0.171299\pi$
$858$		0			0
$859$	−11400.8	−	19746.8i	−0.452841	−	0.784344i	0.545720	−	0.837968i	$-0.316256\pi$
$859$	−11400.8	−	19746.8i	−0.452841	−	0.784344i	−0.998561	+	0.0536234i	$0.982923\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	16159.1	+	27988.4i	0.637385	+	1.10398i	0.986004	+	0.166719i	$0.0533173\pi$
$863$	16159.1	+	27988.4i	0.637385	+	1.10398i	−0.348619	+	0.937264i	$0.613349\pi$
$864$		0			0
$865$	−8008.00	+	13870.3i	−0.314775	+	0.545206i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	36682.6			1.43196
$870$		0			0
$871$	10857.9	−	18806.5i	0.422396	−	0.731611i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	7083.00	+	12268.1i	0.272721	+	0.472366i	0.969558	−	0.244864i	$-0.0787432\pi$
$877$	7083.00	+	12268.1i	0.272721	+	0.472366i	−0.696837	+	0.717230i	$0.745410\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	19673.8			0.752358			0.376179	−	0.926547i	$-0.377238\pi$
$881$	19673.8			0.752358			0.376179	+	0.926547i	$0.377238\pi$
$882$		0			0
$883$	−44876.0			−1.71030			−0.855152	−	0.518378i	$-0.826536\pi$
$883$	−44876.0			−1.71030			−0.855152	+	0.518378i	$0.826536\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	7873.76	+	13637.7i	0.298055	+	0.516247i	0.975691	−	0.219151i	$-0.0703288\pi$
$887$	7873.76	+	13637.7i	0.298055	+	0.516247i	−0.677636	+	0.735398i	$0.736995\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−19446.2	+	33681.8i	−0.728715	+	1.26217i
$894$		0			0
$895$	−1753.97			−0.0655070
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4614.19	+	7992.01i	−0.171181	+	0.296494i
$900$		0			0
$901$	−5261.91	−	9113.90i	−0.194561	−	0.336990i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−19004.2	−	32916.3i	−0.698036	−	1.20903i
$906$		0			0
$907$	178.000	−	308.305i	0.00651642	−	0.0112868i	−0.862749	−	0.505633i	$-0.831259\pi$
$907$	178.000	−	308.305i	0.00651642	−	0.0112868i	0.869265	+	0.494346i	$0.164592\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	4916.80			0.178815			0.0894077	−	0.995995i	$-0.471503\pi$
$911$	4916.80			0.178815			0.0894077	+	0.995995i	$0.471503\pi$
$912$		0			0
$913$	3507.94	−	6075.94i	0.127159	−	0.220245i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	21528.0	+	37287.6i	0.772735	+	1.33842i	0.936059	+	0.351843i	$0.114445\pi$
$919$	21528.0	+	37287.6i	0.772735	+	1.33842i	−0.163324	+	0.986572i	$0.552222\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	59984.5			2.13913
$924$		0			0
$925$	−1014.00			−0.0360434
$926$		0			0
$927$		0			0
$928$		0			0
$929$	21605.2	+	37421.3i	0.763018	+	1.32159i	0.941288	+	0.337604i	$0.109617\pi$
$929$	21605.2	+	37421.3i	0.763018	+	1.32159i	−0.178270	+	0.983982i	$0.557050\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−27843.4	+	48226.2i	−0.973880	+	1.68681i
$936$		0			0
$937$	−52118.0			−1.81710			−0.908549	−	0.417778i	$-0.862809\pi$
$937$	−52118.0			−1.81710			−0.908549	+	0.417778i	$0.862809\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	2333.55	−	4041.83i	0.0808413	−	0.140021i	−0.822770	−	0.568374i	$-0.807573\pi$
$941$	2333.55	−	4041.83i	0.0808413	−	0.140021i	0.903612	+	0.428353i	$0.140906\pi$
$942$		0			0
$943$	34202.4	+	59240.4i	1.18111	+	2.04574i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−359.092	−	621.965i	−0.0123220	−	0.0213423i	0.859799	−	0.510633i	$-0.170589\pi$
$947$	−359.092	−	621.965i	−0.0123220	−	0.0213423i	−0.872121	+	0.489291i	$0.837256\pi$
$948$		0			0
$949$	−27904.0	+	48331.1i	−0.954481	+	1.65321i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−36461.6			−1.23936			−0.619679	−	0.784855i	$-0.712737\pi$
$953$	−36461.6			−1.23936			−0.619679	+	0.784855i	$0.712737\pi$
$954$		0			0
$955$	−11400.8	+	19746.8i	−0.386305	+	0.669101i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	11407.5	+	19758.4i	0.382918	+	0.663233i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	14583.4			0.486483
$966$		0			0
$967$	−39992.0			−1.32994			−0.664972	−	0.746868i	$-0.731557\pi$
$967$	−39992.0			−1.32994			−0.664972	+	0.746868i	$0.731557\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−19091.7	−	33067.9i	−0.630982	−	1.09289i	−0.987351	−	0.158547i	$-0.949319\pi$
$971$	−19091.7	−	33067.9i	−0.630982	−	1.09289i	0.356370	−	0.934345i	$-0.384014\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5524.49	−	9568.70i	0.180905	−	0.313337i	−0.761284	−	0.648419i	$-0.775431\pi$
$977$	5524.49	−	9568.70i	0.180905	−	0.313337i	0.942189	+	0.335082i	$0.108764\pi$
$978$		0			0
$979$	−48526.5			−1.58418
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−23155.6	+	40106.7i	−0.751322	+	1.30133i	0.195860	+	0.980632i	$0.437250\pi$
$983$	−23155.6	+	40106.7i	−0.751322	+	1.30133i	−0.947182	+	0.320696i	$0.896083\pi$
$984$		0			0
$985$	5261.91	+	9113.90i	0.170212	+	0.294815i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−12264.4	−	21242.5i	−0.394322	−	0.682985i
$990$		0			0
$991$	25532.0	−	44222.7i	0.818416	−	1.41754i	−0.0884321	−	0.996082i	$-0.528186\pi$
$991$	25532.0	−	44222.7i	0.818416	−	1.41754i	0.906849	−	0.421457i	$-0.138481\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−49499.4			−1.57712
$996$		0			0
$997$	−7057.65	+	12224.2i	−0.224191	+	0.388309i	−0.956076	−	0.293118i	$-0.905307\pi$
$997$	−7057.65	+	12224.2i	−0.224191	+	0.388309i	0.731886	+	0.681427i	$0.238640\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.k.bc.361.3		8
3.2	odd	2	inner	1764.4.k.bc.361.2		8
7.2	even	3	inner	1764.4.k.bc.1549.3		8
7.3	odd	6		1764.4.a.bb.1.4	yes	4
7.4	even	3		1764.4.a.bb.1.2	yes	4
7.5	odd	6	inner	1764.4.k.bc.1549.1		8
7.6	odd	2	inner	1764.4.k.bc.361.1		8
21.2	odd	6	inner	1764.4.k.bc.1549.2		8
21.5	even	6	inner	1764.4.k.bc.1549.4		8
21.11	odd	6		1764.4.a.bb.1.3	yes	4
21.17	even	6		1764.4.a.bb.1.1	✓	4
21.20	even	2	inner	1764.4.k.bc.361.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.4.a.bb.1.1	✓	4	21.17	even	6
1764.4.a.bb.1.2	yes	4	7.4	even	3
1764.4.a.bb.1.3	yes	4	21.11	odd	6
1764.4.a.bb.1.4	yes	4	7.3	odd	6
1764.4.k.bc.361.1		8	7.6	odd	2	inner
1764.4.k.bc.361.2		8	3.2	odd	2	inner
1764.4.k.bc.361.3		8	1.1	even	1	trivial
1764.4.k.bc.361.4		8	21.20	even	2	inner
1764.4.k.bc.1549.1		8	7.5	odd	6	inner
1764.4.k.bc.1549.2		8	21.2	odd	6	inner
1764.4.k.bc.1549.3		8	7.2	even	3	inner
1764.4.k.bc.1549.4		8	21.5	even	6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(5.29150 + 9.16515i) q^{5} +(-27.6225 + 47.8435i) q^{11} +83.5225 q^{13} +(47.6235 - 82.4864i) q^{17} +(-41.7612 - 72.3326i) q^{19} +(-82.8674 - 143.530i) q^{23} +(6.50000 - 11.2583i) q^{25} -110.490 q^{29} +(41.7612 - 72.3326i) q^{31} +(-39.0000 - 67.5500i) q^{37} -412.737 q^{41} +148.000 q^{43} +(-232.826 - 403.267i) q^{47} +(55.2449 - 95.6870i) q^{53} -584.657 q^{55} +(275.158 - 476.588i) q^{59} +(292.329 + 506.328i) q^{61} +(441.959 + 765.496i) q^{65} +(130.000 - 225.167i) q^{67} +718.184 q^{71} +(-334.090 + 578.661i) q^{73} +(-332.000 - 575.041i) q^{79} -126.996 q^{83} +1008.00 q^{85} +(439.195 + 760.708i) q^{89} +(441.959 - 765.496i) q^{95} +1169.31 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 52 q^{25} - 312 q^{37} + 1184 q^{43} + 1040 q^{67} - 2656 q^{79} + 8064 q^{85}+O(q^{100}) \) 8 * q + 52 * q^25 - 312 * q^37 + 1184 * q^43 + 1040 * q^67 - 2656 * q^79 + 8064 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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