LMFDB - Embedded newform 1764.4.k.bc.361.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(361,1764)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			361.2
Root			$4.39728 - 2.29129i$ of defining polynomial
Character	$\chi$	$=$	1764.361
Dual form			1764.4.k.bc.1549.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.29150 - 9.16515i) q^{5} +O(q^{10})$	$q+(-5.29150 - 9.16515i) q^{5} +(27.6225 - 47.8435i) q^{11} +83.5225 q^{13} +(-47.6235 + 82.4864i) q^{17} +(-41.7612 - 72.3326i) q^{19} +(82.8674 + 143.530i) q^{23} +(6.50000 - 11.2583i) q^{25} +110.490 q^{29} +(41.7612 - 72.3326i) q^{31} +(-39.0000 - 67.5500i) q^{37} +412.737 q^{41} +148.000 q^{43} +(232.826 + 403.267i) q^{47} +(-55.2449 + 95.6870i) q^{53} -584.657 q^{55} +(-275.158 + 476.588i) q^{59} +(292.329 + 506.328i) q^{61} +(-441.959 - 765.496i) q^{65} +(130.000 - 225.167i) q^{67} -718.184 q^{71} +(-334.090 + 578.661i) q^{73} +(-332.000 - 575.041i) q^{79} +126.996 q^{83} +1008.00 q^{85} +(-439.195 - 760.708i) q^{89} +(-441.959 + 765.496i) q^{95} +1169.31 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * 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

Display 100 coefficients 10 coefficients

Character values

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

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

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	0.526246	−	0.850332i	$-0.323599\pi$
$5$	−5.29150	−	9.16515i	−0.473286	−	0.819756i	−0.999532	+	0.0305763i	$0.990266\pi$
$6$		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	$-0.559932\pi$
$11$	27.6225	−	47.8435i	0.757135	−	1.31140i	0.944306	−	0.329069i	$-0.106735\pi$
$12$		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	$0.404442\pi$
$17$	−47.6235	+	82.4864i	−0.679435	+	1.17682i	−0.975151	+	0.221541i	$0.928891\pi$
$18$		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	$0.103888\pi$
$23$	82.8674	+	143.530i	0.751263	+	1.30122i	−0.195948	+	0.980614i	$0.562779\pi$
$24$		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	$-0.384907\pi$
$29$	110.490			0.707498			0.353749	+	0.935340i	$0.384907\pi$
$30$		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	$-0.212106\pi$
$41$	412.737			1.57216			0.786082	+	0.618122i	$0.212106\pi$
$42$		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	$0.0903764\pi$
$47$	232.826	+	403.267i	0.722578	+	1.25154i	−0.237385	+	0.971416i	$0.576290\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−55.2449	+	95.6870i	−0.143179	+	0.247993i	−0.928692	−	0.370852i	$-0.879066\pi$
$53$	−55.2449	+	95.6870i	−0.143179	+	0.247993i	0.785513	+	0.618845i	$0.212399\pi$
$54$		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	$0.374359\pi$
$59$	−275.158	+	476.588i	−0.607162	+	1.05163i	−0.991706	+	0.128528i	$0.958975\pi$
$60$		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	$-0.704925\pi$
$71$	−718.184			−1.20046			−0.600231	+	0.799827i	$0.704925\pi$
$72$		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	$-0.473239\pi$
$83$	126.996			0.167947			0.0839737	+	0.996468i	$0.473239\pi$
$84$		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	$-0.991448\pi$
$89$	−439.195	−	760.708i	−0.523085	−	0.906009i	0.476554	−	0.879145i	$-0.341886\pi$
$90$		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	−0.956068	−	0.293144i	$-0.905298\pi$
$101$	−227.535	+	394.102i	−0.224164	+	0.388263i	0.731904	+	0.681407i	$0.238632\pi$
$102$		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	0.996351	−	0.0853466i	$-0.0271997\pi$
$107$	469.582	+	813.339i	0.424263	+	0.734846i	−0.572088	+	0.820192i	$0.693866\pi$
$108$		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	$-0.277325\pi$
$113$	1546.86			1.28775			0.643877	+	0.765129i	$0.277325\pi$
$114$		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	0.975321	−	0.220791i	$-0.0708639\pi$
$131$	444.486	+	769.873i	0.296450	+	0.513466i	−0.678871	+	0.734257i	$0.737531\pi$
$132$		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	$-0.479976\pi$
$137$	1436.37	−	2487.86i	0.895746	−	1.55148i	0.832879	−	0.553455i	$-0.186691\pi$
$138$		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	$0.00605074\pi$
$149$	939.163	+	1626.68i	0.516371	+	0.894381i	−0.483448	+	0.875373i	$0.660616\pi$
$150$		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	$-0.493756\pi$
$167$	84.6640			0.0392305			0.0196153	+	0.999808i	$0.493756\pi$
$168$		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	0.650468	−	0.759534i	$-0.274573\pi$
$173$	−756.685	−	1310.62i	−0.332542	−	0.575979i	−0.983009	+	0.183555i	$0.941239\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	82.8674	−	143.530i	0.0346022	−	0.0599328i	−0.848206	−	0.529667i	$-0.822317\pi$
$179$	82.8674	−	143.530i	0.0346022	−	0.0599328i	0.882808	+	0.469734i	$0.155650\pi$
$180$		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	0.586568	−	0.809900i	$-0.300479\pi$
$191$	−1077.28	−	1865.90i	−0.408110	−	0.706867i	−0.994678	+	0.103033i	$0.967145\pi$
$192$		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	$-0.557551\pi$
$197$	−994.408			−0.359638			−0.179819	+	0.983700i	$0.557551\pi$
$198$		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	−0.947892	−	0.318590i	$-0.896791\pi$
$227$	−677.312	+	1173.14i	−0.198039	+	0.343013i	0.749854	+	0.661604i	$0.230124\pi$
$228$		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	$-0.989562\pi$
$233$	−1878.33	−	3253.36i	−0.528126	−	0.914741i	0.471337	−	0.881953i	$-0.343772\pi$
$234$		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	$-0.359877\pi$
$239$	3148.96			0.852256			0.426128	+	0.904663i	$0.359877\pi$
$240$		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	$-0.227240\pi$
$251$	6011.15			1.51163			0.755817	+	0.654783i	$0.227240\pi$
$252$		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	0.798549	−	0.601930i	$-0.205601\pi$
$257$	−502.693	−	870.689i	−0.122012	−	0.211331i	−0.920561	+	0.390599i	$0.872268\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1850.70	−	3205.51i	0.433914	−	0.751561i	−0.563293	−	0.826258i	$-0.690466\pi$
$263$	1850.70	−	3205.51i	0.433914	−	0.751561i	0.997206	+	0.0746968i	$0.0237989\pi$
$264$		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	−0.999627	−	0.0273274i	$-0.991300\pi$
$269$	−2100.73	+	3638.57i	−0.476147	+	0.824711i	0.523479	+	0.852038i	$0.324634\pi$
$270$		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	$-0.867654\pi$
$281$	−8618.21			−1.82961			−0.914803	+	0.403901i	$0.867654\pi$
$282$		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	$-0.295875\pi$
$293$	6000.56			1.19644			0.598220	+	0.801332i	$0.295875\pi$
$294$		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	−0.839850	−	0.542819i	$-0.817357\pi$
$311$	275.158	−	476.588i	0.0501697	−	0.0868965i	0.890020	+	0.455922i	$0.150690\pi$
$312$		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	$0.157685\pi$
$317$	4806.31	+	8324.77i	0.851574	+	1.47497i	−0.0282125	+	0.999602i	$0.508981\pi$
$318$		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	−0.770583	−	0.637339i	$-0.780035\pi$
$347$	1077.28	−	1865.90i	0.166660	−	0.288664i	0.937244	+	0.348675i	$0.113368\pi$
$348$		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	−0.999278	−	0.0379913i	$-0.987904\pi$
$353$	−3095.53	+	5361.61i	−0.466738	+	0.808413i	0.532540	+	0.846405i	$0.321237\pi$
$354$		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	0.953343	−	0.301890i	$-0.0976176\pi$
$359$	1463.99	+	2535.71i	0.215227	+	0.372784i	−0.738116	+	0.674674i	$0.764284\pi$
$360$		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	$-0.981047\pi$
$383$	−4127.37	−	7148.82i	−0.550650	−	0.953753i	0.447578	−	0.894245i	$-0.352287\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2154.55	+	3731.79i	−0.280823	+	0.486399i	−0.971588	−	0.236680i	$-0.923941\pi$
$389$	−2154.55	+	3731.79i	−0.280823	+	0.486399i	0.690765	+	0.723080i	$0.257274\pi$
$390$		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	0.844648	−	0.535322i	$-0.179810\pi$
$401$	−331.469	−	574.122i	−0.0412788	−	0.0714970i	−0.885927	+	0.463825i	$0.846477\pi$
$402$		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	$-0.785957\pi$
$419$	−13419.3			−1.56461			−0.782307	+	0.622893i	$0.785957\pi$
$420$		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	−0.999253	−	0.0386580i	$-0.987692\pi$
$431$	−4170.99	+	7224.37i	−0.466147	+	0.807391i	0.533105	+	0.846049i	$0.321025\pi$
$432$		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	$-0.902598\pi$
$443$	−6877.99	−	11913.0i	−0.737660	−	1.27766i	0.215887	−	0.976418i	$-0.430736\pi$
$444$		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	$-0.662184\pi$
$449$	−9281.14			−0.975511			−0.487755	+	0.872980i	$0.662184\pi$
$450$		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	$-0.754438\pi$
$461$	−14191.8			−1.43379			−0.716896	+	0.697180i	$0.754438\pi$
$462$		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	0.736113	−	0.676859i	$-0.236659\pi$
$467$	−2201.27	−	3812.70i	−0.218121	−	0.377796i	−0.954233	+	0.299063i	$0.903326\pi$
$468$		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	−0.555689	−	0.831390i	$-0.687546\pi$
$479$	4635.36	−	8028.67i	0.442160	−	0.765844i	0.997850	+	0.0655459i	$0.0208789\pi$
$480$		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	$-0.850095\pi$
$491$	−19391.0			−1.78228			−0.891142	+	0.453724i	$0.850095\pi$
$492$		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	$-0.234299\pi$
$503$	16721.1			1.48222			0.741112	+	0.671381i	$0.234299\pi$
$504$		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	0.999703	−	0.0243545i	$-0.00775306\pi$
$509$	5497.87	+	9522.59i	0.478760	+	0.829237i	−0.520943	+	0.853591i	$0.674420\pi$
$510$		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	$-0.495205\pi$
$521$	10387.2	−	17991.2i	0.873459	−	1.51288i	0.858395	−	0.512990i	$-0.171462\pi$
$522$		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	−0.936263	−	0.351299i	$-0.885740\pi$
$557$	−2154.55	+	3731.79i	−0.163898	+	0.283880i	0.772365	+	0.635179i	$0.219074\pi$
$558$		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	−0.865232	−	0.501372i	$-0.832829\pi$
$563$	21.1660	−	36.6606i	0.00158444	−	0.00274433i	0.866817	+	0.498627i	$0.166162\pi$
$564$		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	0.961911	−	0.273363i	$-0.0881359\pi$
$569$	3314.69	+	5741.22i	0.244217	+	0.422995i	−0.717694	+	0.696358i	$0.754803\pi$
$570$		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	$-0.860031\pi$
$587$	−25737.9			−1.80974			−0.904868	+	0.425691i	$0.860031\pi$
$588$		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	0.862152	−	0.506649i	$-0.169116\pi$
$593$	−111.122	−	192.468i	−0.00769514	−	0.0133284i	−0.869847	+	0.493321i	$0.835783\pi$
$594$		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	$-0.557014\pi$
$599$	11187.1	−	19376.6i	0.763092	−	1.32171i	0.941249	−	0.337713i	$-0.109653\pi$
$600$		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	$-0.617382\pi$
$617$	−11049.0			−0.720932			−0.360466	+	0.932772i	$0.617382\pi$
$618$		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	−0.721935	−	0.691961i	$-0.756747\pi$
$641$	3867.14	−	6698.09i	0.238289	−	0.412728i	0.960223	+	0.279233i	$0.0900803\pi$
$642$		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	−0.993880	−	0.110469i	$-0.964765\pi$
$647$	−6603.80	+	11438.1i	−0.401270	+	0.695021i	0.592609	+	0.805490i	$0.298098\pi$
$648$		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	$-0.977655\pi$
$653$	−9336.39	−	16171.1i	−0.559512	−	0.969103i	0.438025	−	0.898963i	$-0.355678\pi$
$654$		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	$-0.439873\pi$
$659$	6353.16			0.375545			0.187772	+	0.982213i	$0.439873\pi$
$660$		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	$0.120639\pi$
$677$	13826.7	+	23948.5i	0.784938	+	1.35955i	−0.144098	+	0.989563i	$0.546028\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	3950.01	−	6841.62i	0.221293	−	0.383290i	−0.733908	−	0.679249i	$-0.762306\pi$
$683$	3950.01	−	6841.62i	0.221293	−	0.383290i	0.955201	+	0.295959i	$0.0956390\pi$
$684$		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	$-0.605178\pi$
$701$	−12043.4			−0.648891			−0.324445	+	0.945904i	$0.605178\pi$
$702$		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	0.794627	−	0.607098i	$-0.207667\pi$
$719$	−2476.42	−	4289.29i	−0.128449	−	0.222481i	−0.923076	+	0.384618i	$0.874333\pi$
$720$		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	$-0.509118\pi$
$743$	−1160.14			−0.0572833			−0.0286417	+	0.999590i	$0.509118\pi$
$744$		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	$0.122053\pi$
$761$	16535.9	+	28641.1i	0.787684	+	1.36431i	−0.139698	+	0.990194i	$0.544613\pi$
$762$		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	$-0.641617\pi$
$773$	12175.7	−	21089.0i	0.566535	−	0.981267i	0.996905	−	0.0786143i	$-0.0250496\pi$
$774$		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	$-0.233441\pi$
$797$	33431.7			1.48584			0.742918	+	0.669382i	$0.233441\pi$
$798$		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	$0.450373\pi$
$809$	−17899.4	+	31002.6i	−0.777883	+	1.34733i	−0.933160	+	0.359462i	$0.882960\pi$
$810$		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	$0.0811233\pi$
$821$	16518.2	+	28610.4i	0.702180	+	1.21621i	−0.265519	+	0.964106i	$0.585543\pi$
$822$		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	$-0.431824\pi$
$827$	10109.8			0.425094			0.212547	+	0.977151i	$0.431824\pi$
$828$		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	$-0.351633\pi$
$839$	21843.3			0.898826			0.449413	+	0.893324i	$0.351633\pi$
$840$		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	−0.858657	−	0.512550i	$-0.828701\pi$
$857$	365.114	−	632.395i	0.0145531	−	0.0252068i	0.873210	+	0.487344i	$0.162034\pi$
$858$		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	$-0.946683\pi$
$863$	−16159.1	−	27988.4i	−0.637385	−	1.10398i	0.348619	−	0.937264i	$-0.386651\pi$
$864$		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	$-0.622762\pi$
$881$	−19673.8			−0.752358			−0.376179	+	0.926547i	$0.622762\pi$
$882$		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	0.677636	−	0.735398i	$-0.263005\pi$
$887$	−7873.76	−	13637.7i	−0.298055	−	0.516247i	−0.975691	+	0.219151i	$0.929671\pi$
$888$		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	$-0.528497\pi$
$911$	−4916.80			−0.178815			−0.0894077	+	0.995995i	$0.528497\pi$
$912$		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	$-0.890383\pi$
$929$	−21605.2	−	37421.3i	−0.763018	−	1.32159i	0.178270	−	0.983982i	$-0.442950\pi$
$930$		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	−0.903612	−	0.428353i	$-0.859094\pi$
$941$	−2333.55	+	4041.83i	−0.0808413	+	0.140021i	0.822770	+	0.568374i	$0.192427\pi$
$942$		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	0.872121	−	0.489291i	$-0.162744\pi$
$947$	359.092	+	621.965i	0.0123220	+	0.0213423i	−0.859799	+	0.510633i	$0.829411\pi$
$948$		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	$-0.287263\pi$
$953$	36461.6			1.23936			0.619679	+	0.784855i	$0.287263\pi$
$954$		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	$0.0506810\pi$
$971$	19091.7	+	33067.9i	0.630982	+	1.09289i	−0.356370	+	0.934345i	$0.615986\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5524.49	+	9568.70i	−0.180905	+	0.313337i	−0.942189	−	0.335082i	$-0.891236\pi$
$977$	−5524.49	+	9568.70i	−0.180905	+	0.313337i	0.761284	+	0.648419i	$0.224569\pi$
$978$		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	$-0.562750\pi$
$983$	23155.6	−	40106.7i	0.751322	−	1.30133i	0.947182	−	0.320696i	$-0.103917\pi$
$984$		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.361.2		8
3.2	odd	2	inner	1764.4.k.bc.361.3		8
7.2	even	3	inner	1764.4.k.bc.1549.2		8
7.3	odd	6		1764.4.a.bb.1.1	✓	4
7.4	even	3		1764.4.a.bb.1.3	yes	4
7.5	odd	6	inner	1764.4.k.bc.1549.4		8
7.6	odd	2	inner	1764.4.k.bc.361.4		8
21.2	odd	6	inner	1764.4.k.bc.1549.3		8
21.5	even	6	inner	1764.4.k.bc.1549.1		8
21.11	odd	6		1764.4.a.bb.1.2	yes	4
21.17	even	6		1764.4.a.bb.1.4	yes	4
21.20	even	2	inner	1764.4.k.bc.361.1		8

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

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)

\(f(q)\)	\(=\)	\(q+(-5.29150 - 9.16515i) q^{5} +O(q^{10})\)	\(q+(-5.29150 - 9.16515i) q^{5} +(27.6225 - 47.8435i) q^{11} +83.5225 q^{13} +(-47.6235 + 82.4864i) q^{17} +(-41.7612 - 72.3326i) q^{19} +(82.8674 + 143.530i) q^{23} +(6.50000 - 11.2583i) q^{25} +110.490 q^{29} +(41.7612 - 72.3326i) q^{31} +(-39.0000 - 67.5500i) q^{37} +412.737 q^{41} +148.000 q^{43} +(232.826 + 403.267i) q^{47} +(-55.2449 + 95.6870i) q^{53} -584.657 q^{55} +(-275.158 + 476.588i) q^{59} +(292.329 + 506.328i) q^{61} +(-441.959 - 765.496i) q^{65} +(130.000 - 225.167i) q^{67} -718.184 q^{71} +(-334.090 + 578.661i) q^{73} +(-332.000 - 575.041i) q^{79} +126.996 q^{83} +1008.00 q^{85} +(-439.195 - 760.708i) q^{89} +(-441.959 + 765.496i) q^{95} +1169.31 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * 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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