LMFDB - Embedded newform 980.2.q.f.569.1

Newspace parameters

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.q (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$7.82533939809$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			569.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	980.569
Dual form			980.2.q.f.949.1

$q$-expansion

$f(q)$	$=$	$q+(-2.59808 + 1.50000i) q^{3} +(1.86603 - 1.23205i) q^{5} +(3.00000 - 5.19615i) q^{9} +O(q^{10})$	\(q+(-2.59808 + 1.50000i) q^{3} +(1.86603 - 1.23205i) q^{5} +(3.00000 - 5.19615i) q^{9} +(-1.50000 - 2.59808i) q^{11} -1.00000i q^{13} +(-3.00000 + 6.00000i) q^{15} +(-4.33013 + 2.50000i) q^{17} +(-4.00000 + 6.92820i) q^{19} +(1.73205 + 1.00000i) q^{23} +(1.96410 - 4.59808i) q^{25} +9.00000i q^{27} +1.00000 q^{29} +(1.00000 + 1.73205i) q^{31} +(7.79423 + 4.50000i) q^{33} +(-8.66025 - 5.00000i) q^{37} +(1.50000 + 2.59808i) q^{39} -6.00000 q^{41} +4.00000i q^{43} +(-0.803848 - 13.3923i) q^{45} +(-9.52628 - 5.50000i) q^{47} +(7.50000 - 12.9904i) q^{51} +(-5.19615 + 3.00000i) q^{53} +(-6.00000 - 3.00000i) q^{55} -24.0000i q^{57} +(-5.00000 - 8.66025i) q^{59} +(-1.23205 - 1.86603i) q^{65} +(-8.66025 + 5.00000i) q^{67} -6.00000 q^{69} +(8.66025 - 5.00000i) q^{73} +(1.79423 + 14.8923i) q^{75} +(-3.50000 + 6.06218i) q^{79} +(-4.50000 - 7.79423i) q^{81} -12.0000i q^{83} +(-5.00000 + 10.0000i) q^{85} +(-2.59808 + 1.50000i) q^{87} +(4.00000 - 6.92820i) q^{89} +(-5.19615 - 3.00000i) q^{93} +(1.07180 + 17.8564i) q^{95} +3.00000i q^{97} -18.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 12 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 + 12 * q^9	$ 4 q + 4 q^{5} + 12 q^{9} - 6 q^{11} - 12 q^{15} - 16 q^{19} - 6 q^{25} + 4 q^{29} + 4 q^{31} + 6 q^{39} - 24 q^{41} - 24 q^{45} + 30 q^{51} - 24 q^{55} - 20 q^{59} + 2 q^{65} - 24 q^{69} - 24 q^{75} - 14 q^{79} - 18 q^{81} - 20 q^{85} + 16 q^{89} + 32 q^{95} - 72 q^{99}+O(q^{100}) $ 4 * q + 4 * q^5 + 12 * q^9 - 6 * q^11 - 12 * q^15 - 16 * q^19 - 6 * q^25 + 4 * q^29 + 4 * q^31 + 6 * q^39 - 24 * q^41 - 24 * q^45 + 30 * q^51 - 24 * q^55 - 20 * q^59 + 2 * q^65 - 24 * q^69 - 24 * q^75 - 14 * q^79 - 18 * q^81 - 20 * q^85 + 16 * q^89 + 32 * q^95 - 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$	$1$

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
$3$	−2.59808	+	1.50000i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$3$	−2.59808	+	1.50000i	−1.50000	+	0.866025i	−1.00000			$\pi$
$4$		0			0
$5$	1.86603	−	1.23205i	0.834512	−	0.550990i
$5$	1.86603	−	1.23205i	0.834512	−	0.550990i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	3.00000	−	5.19615i	1.00000	−	1.73205i
$10$		0			0
$11$	−1.50000	−	2.59808i	−0.452267	−	0.783349i	0.546259	−	0.837616i	$-0.316051\pi$
$11$	−1.50000	−	2.59808i	−0.452267	−	0.783349i	−0.998526	+	0.0542666i	$0.982718\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i	−0.990338	−	0.138675i	$-0.955716\pi$
$13$		−	1.00000i		−	0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$		0			0
$15$	−3.00000	+	6.00000i	−0.774597	+	1.54919i
$16$		0			0
$17$	−4.33013	+	2.50000i	−1.05021	+	0.606339i	−0.922708	−	0.385499i	$-0.874029\pi$
$17$	−4.33013	+	2.50000i	−1.05021	+	0.606339i	−0.127502	+	0.991838i	$0.540696\pi$
$18$		0			0
$19$	−4.00000	+	6.92820i	−0.917663	+	1.58944i	−0.114708	+	0.993399i	$0.536593\pi$
$19$	−4.00000	+	6.92820i	−0.917663	+	1.58944i	−0.802955	+	0.596040i	$0.796740\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.73205	+	1.00000i	0.361158	+	0.208514i	0.669588	−	0.742732i	$-0.266471\pi$
$23$	1.73205	+	1.00000i	0.361158	+	0.208514i	−0.308431	+	0.951247i	$0.599804\pi$
$24$		0			0
$25$	1.96410	−	4.59808i	0.392820	−	0.919615i
$26$		0			0
$27$			9.00000i			1.73205i
$28$		0			0
$29$	1.00000			0.185695			0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000			0.185695			0.0928477	+	0.995680i	$0.470403\pi$
$30$		0			0
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	0.941745	−	0.336327i	$-0.109185\pi$
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	−0.762140	+	0.647412i	$0.775851\pi$
$32$		0			0
$33$	7.79423	+	4.50000i	1.35680	+	0.783349i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−8.66025	−	5.00000i	−1.42374	−	0.821995i	−0.427121	−	0.904194i	$-0.640472\pi$
$37$	−8.66025	−	5.00000i	−1.42374	−	0.821995i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$		0			0
$39$	1.50000	+	2.59808i	0.240192	+	0.416025i
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$		0			0
$45$	−0.803848	−	13.3923i	−0.119831	−	1.99641i
$46$		0			0
$47$	−9.52628	−	5.50000i	−1.38955	−	0.802257i	−0.396286	−	0.918127i	$-0.629701\pi$
$47$	−9.52628	−	5.50000i	−1.38955	−	0.802257i	−0.993264	+	0.115870i	$0.963035\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	7.50000	−	12.9904i	1.05021	−	1.81902i
$52$		0			0
$53$	−5.19615	+	3.00000i	−0.713746	+	0.412082i	−0.812447	−	0.583036i	$-0.801865\pi$
$53$	−5.19615	+	3.00000i	−0.713746	+	0.412082i	0.0987002	+	0.995117i	$0.468532\pi$
$54$		0			0
$55$	−6.00000	−	3.00000i	−0.809040	−	0.404520i
$56$		0			0
$57$		−	24.0000i		−	3.17888i
$58$		0			0
$59$	−5.00000	−	8.66025i	−0.650945	−	1.12747i	−0.982894	−	0.184172i	$-0.941040\pi$
$59$	−5.00000	−	8.66025i	−0.650945	−	1.12747i	0.331949	−	0.943297i	$-0.392294\pi$
$60$		0			0
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.23205	−	1.86603i	−0.152817	−	0.231452i
$66$		0			0
$67$	−8.66025	+	5.00000i	−1.05802	+	0.610847i	−0.924883	−	0.380251i	$-0.875838\pi$
$67$	−8.66025	+	5.00000i	−1.05802	+	0.610847i	−0.133135	+	0.991098i	$0.542504\pi$
$68$		0			0
$69$	−6.00000			−0.722315
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	8.66025	−	5.00000i	1.01361	−	0.585206i	0.101361	−	0.994850i	$-0.467680\pi$
$73$	8.66025	−	5.00000i	1.01361	−	0.585206i	0.912245	+	0.409644i	$0.134347\pi$
$74$		0			0
$75$	1.79423	+	14.8923i	0.207180	+	1.71962i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−3.50000	+	6.06218i	−0.393781	+	0.682048i	−0.992945	−	0.118578i	$-0.962166\pi$
$79$	−3.50000	+	6.06218i	−0.393781	+	0.682048i	0.599164	+	0.800626i	$0.295500\pi$
$80$		0			0
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$		−	12.0000i		−	1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$		0			0
$85$	−5.00000	+	10.0000i	−0.542326	+	1.08465i
$86$		0			0
$87$	−2.59808	+	1.50000i	−0.278543	+	0.160817i
$88$		0			0
$89$	4.00000	−	6.92820i	0.423999	−	0.734388i	−0.572327	−	0.820025i	$-0.693959\pi$
$89$	4.00000	−	6.92820i	0.423999	−	0.734388i	0.996326	+	0.0856373i	$0.0272926\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−5.19615	−	3.00000i	−0.538816	−	0.311086i
$94$		0			0
$95$	1.07180	+	17.8564i	0.109964	+	1.83203i
$96$		0			0
$97$			3.00000i			0.304604i	0.988334	+	0.152302i	$0.0486686\pi$
$97$			3.00000i			0.304604i	−0.988334	+	0.152302i	$0.951331\pi$
$98$		0			0
$99$	−18.0000			−1.80907
$100$		0			0
$101$	6.00000	+	10.3923i	0.597022	+	1.03407i	0.993258	+	0.115924i	$0.0369830\pi$
$101$	6.00000	+	10.3923i	0.597022	+	1.03407i	−0.396236	+	0.918149i	$0.629684\pi$
$102$		0			0
$103$	−4.33013	−	2.50000i	−0.426660	−	0.246332i	0.271263	−	0.962505i	$-0.412559\pi$
$103$	−4.33013	−	2.50000i	−0.426660	−	0.246332i	−0.697923	+	0.716173i	$0.745892\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.92820	−	4.00000i	−0.669775	−	0.386695i	0.126217	−	0.992003i	$-0.459717\pi$
$107$	−6.92820	−	4.00000i	−0.669775	−	0.386695i	−0.795991	+	0.605308i	$0.793050\pi$
$108$		0			0
$109$	−3.50000	−	6.06218i	−0.335239	−	0.580651i	0.648292	−	0.761392i	$-0.275484\pi$
$109$	−3.50000	−	6.06218i	−0.335239	−	0.580651i	−0.983531	+	0.180741i	$0.942150\pi$
$110$		0			0
$111$	30.0000			2.84747
$112$		0			0
$113$			10.0000i			0.940721i	0.882474	+	0.470360i	$0.155876\pi$
$113$			10.0000i			0.940721i	−0.882474	+	0.470360i	$0.844124\pi$
$114$		0			0
$115$	4.46410	−	0.267949i	0.416280	−	0.0249864i
$116$		0			0
$117$	−5.19615	−	3.00000i	−0.480384	−	0.277350i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.00000	−	1.73205i	0.0909091	−	0.157459i
$122$		0			0
$123$	15.5885	−	9.00000i	1.40556	−	0.811503i
$124$		0			0
$125$	−2.00000	−	11.0000i	−0.178885	−	0.983870i
$126$		0			0
$127$		−	2.00000i		−	0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$		−	2.00000i		−	0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$		0			0
$129$	−6.00000	−	10.3923i	−0.528271	−	0.914991i
$130$		0			0
$131$	−1.00000	+	1.73205i	−0.0873704	+	0.151330i	−0.906399	−	0.422423i	$-0.861180\pi$
$131$	−1.00000	+	1.73205i	−0.0873704	+	0.151330i	0.819028	+	0.573753i	$0.194513\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	11.0885	+	16.7942i	0.954342	+	1.44542i
$136$		0			0
$137$	3.46410	−	2.00000i	0.295958	−	0.170872i	−0.344668	−	0.938725i	$-0.612008\pi$
$137$	3.46410	−	2.00000i	0.295958	−	0.170872i	0.640626	+	0.767853i	$0.278675\pi$
$138$		0			0
$139$	−10.0000			−0.848189			−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000			−0.848189			−0.424094	+	0.905618i	$0.639408\pi$
$140$		0			0
$141$	33.0000			2.77910
$142$		0			0
$143$	−2.59808	+	1.50000i	−0.217262	+	0.125436i
$144$		0			0
$145$	1.86603	−	1.23205i	0.154965	−	0.102316i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	−0.962348	−	0.271821i	$-0.912374\pi$
$149$	−3.00000	+	5.19615i	−0.245770	+	0.425685i	0.716578	+	0.697507i	$0.245707\pi$
$150$		0			0
$151$	−4.50000	−	7.79423i	−0.366205	−	0.634285i	0.622764	−	0.782410i	$-0.286010\pi$
$151$	−4.50000	−	7.79423i	−0.366205	−	0.634285i	−0.988969	+	0.148124i	$0.952676\pi$
$152$		0			0
$153$			30.0000i			2.42536i
$154$		0			0
$155$	4.00000	+	2.00000i	0.321288	+	0.160644i
$156$		0			0
$157$	−15.5885	+	9.00000i	−1.24409	+	0.718278i	−0.969925	−	0.243403i	$-0.921736\pi$
$157$	−15.5885	+	9.00000i	−1.24409	+	0.718278i	−0.274169	+	0.961681i	$0.588403\pi$
$158$		0			0
$159$	9.00000	−	15.5885i	0.713746	−	1.23625i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	5.19615	+	3.00000i	0.406994	+	0.234978i	0.689497	−	0.724288i	$-0.257831\pi$
$163$	5.19615	+	3.00000i	0.406994	+	0.234978i	−0.282503	+	0.959266i	$0.591165\pi$
$164$		0			0
$165$	20.0885	−	1.20577i	1.56388	−	0.0938692i
$166$		0			0
$167$			3.00000i			0.232147i	0.993241	+	0.116073i	$0.0370308\pi$
$167$			3.00000i			0.232147i	−0.993241	+	0.116073i	$0.962969\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$	24.0000	+	41.5692i	1.83533	+	3.17888i
$172$		0			0
$173$	−7.79423	−	4.50000i	−0.592584	−	0.342129i	0.173534	−	0.984828i	$-0.444481\pi$
$173$	−7.79423	−	4.50000i	−0.592584	−	0.342129i	−0.766119	+	0.642699i	$0.777815\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	25.9808	+	15.0000i	1.95283	+	1.12747i
$178$		0			0
$179$	2.00000	+	3.46410i	0.149487	+	0.258919i	0.931038	−	0.364922i	$-0.118904\pi$
$179$	2.00000	+	3.46410i	0.149487	+	0.258919i	−0.781551	+	0.623841i	$0.785571\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−22.3205	+	1.33975i	−1.64104	+	0.0985001i
$186$		0			0
$187$	12.9904	+	7.50000i	0.949951	+	0.548454i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.50000	+	6.06218i	−0.253251	+	0.438644i	−0.964419	−	0.264378i	$-0.914833\pi$
$191$	−3.50000	+	6.06218i	−0.253251	+	0.438644i	0.711168	+	0.703022i	$0.248167\pi$
$192$		0			0
$193$	−6.92820	+	4.00000i	−0.498703	+	0.287926i	−0.728178	−	0.685388i	$-0.759632\pi$
$193$	−6.92820	+	4.00000i	−0.498703	+	0.287926i	0.229475	+	0.973315i	$0.426299\pi$
$194$		0			0
$195$	6.00000	+	3.00000i	0.429669	+	0.214834i
$196$		0			0
$197$			10.0000i			0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$			10.0000i			0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$		0			0
$199$	9.00000	+	15.5885i	0.637993	+	1.10504i	0.985873	+	0.167497i	$0.0535685\pi$
$199$	9.00000	+	15.5885i	0.637993	+	1.10504i	−0.347879	+	0.937539i	$0.613098\pi$
$200$		0			0
$201$	15.0000	−	25.9808i	1.05802	−	1.83254i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−11.1962	+	7.39230i	−0.781973	+	0.516301i
$206$		0			0
$207$	10.3923	−	6.00000i	0.722315	−	0.417029i
$208$		0			0
$209$	24.0000			1.66011
$210$		0			0
$211$	−3.00000			−0.206529			−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000			−0.206529			−0.103264	+	0.994654i	$0.532929\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.92820	+	7.46410i	0.336101	+	0.509048i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−15.0000	+	25.9808i	−1.01361	+	1.75562i
$220$		0			0
$221$	2.50000	+	4.33013i	0.168168	+	0.291276i
$222$		0			0
$223$		−	19.0000i		−	1.27233i	−0.771551	−	0.636167i	$-0.780519\pi$
$223$		−	19.0000i		−	1.27233i	0.771551	−	0.636167i	$-0.219481\pi$
$224$		0			0
$225$	−18.0000	−	24.0000i	−1.20000	−	1.60000i
$226$		0			0
$227$	23.3827	−	13.5000i	1.55196	−	0.896026i	0.553981	−	0.832529i	$-0.313108\pi$
$227$	23.3827	−	13.5000i	1.55196	−	0.896026i	0.997982	−	0.0634974i	$-0.0202255\pi$
$228$		0			0
$229$	−13.0000	+	22.5167i	−0.859064	+	1.48794i	0.0137585	+	0.999905i	$0.495620\pi$
$229$	−13.0000	+	22.5167i	−0.859064	+	1.48794i	−0.872823	+	0.488037i	$0.837713\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.8564	+	8.00000i	0.907763	+	0.524097i	0.879711	−	0.475509i	$-0.157736\pi$
$233$	13.8564	+	8.00000i	0.907763	+	0.524097i	0.0280525	+	0.999606i	$0.491069\pi$
$234$		0			0
$235$	−24.5526	+	1.47372i	−1.60163	+	0.0961349i
$236$		0			0
$237$		−	21.0000i		−	1.36410i
$238$		0			0
$239$	−5.00000			−0.323423			−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000			−0.323423			−0.161712	+	0.986838i	$0.551701\pi$
$240$		0			0
$241$	9.00000	+	15.5885i	0.579741	+	1.00414i	0.995509	+	0.0946700i	$0.0301796\pi$
$241$	9.00000	+	15.5885i	0.579741	+	1.00414i	−0.415768	+	0.909471i	$0.636487\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	6.92820	+	4.00000i	0.440831	+	0.254514i
$248$		0			0
$249$	18.0000	+	31.1769i	1.14070	+	1.97576i
$250$		0			0
$251$	2.00000			0.126239			0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000			0.126239			0.0631194	+	0.998006i	$0.479895\pi$
$252$		0			0
$253$		−	6.00000i		−	0.377217i
$254$		0			0
$255$	−2.00962	−	33.4808i	−0.125847	−	2.09665i
$256$		0			0
$257$	−5.19615	−	3.00000i	−0.324127	−	0.187135i	0.329104	−	0.944294i	$-0.393253\pi$
$257$	−5.19615	−	3.00000i	−0.324127	−	0.187135i	−0.653231	+	0.757159i	$0.726587\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	3.00000	−	5.19615i	0.185695	−	0.321634i
$262$		0			0
$263$	20.7846	−	12.0000i	1.28163	−	0.739952i	0.304487	−	0.952517i	$-0.401515\pi$
$263$	20.7846	−	12.0000i	1.28163	−	0.739952i	0.977147	+	0.212565i	$0.0681817\pi$
$264$		0			0
$265$	−6.00000	+	12.0000i	−0.368577	+	0.737154i
$266$		0			0
$267$			24.0000i			1.46878i
$268$		0			0
$269$	1.00000	+	1.73205i	0.0609711	+	0.105605i	0.894900	−	0.446267i	$-0.147247\pi$
$269$	1.00000	+	1.73205i	0.0609711	+	0.105605i	−0.833929	+	0.551872i	$0.813914\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−14.8923	+	1.79423i	−0.898040	+	0.108196i
$276$		0			0
$277$	−12.1244	+	7.00000i	−0.728482	+	0.420589i	−0.817867	−	0.575408i	$-0.804843\pi$
$277$	−12.1244	+	7.00000i	−0.728482	+	0.420589i	0.0893846	+	0.995997i	$0.471510\pi$
$278$		0			0
$279$	12.0000			0.718421
$280$		0			0
$281$	15.0000			0.894825			0.447412	−	0.894328i	$-0.352346\pi$
$281$	15.0000			0.894825			0.447412	+	0.894328i	$0.352346\pi$
$282$		0			0
$283$	6.06218	−	3.50000i	0.360359	−	0.208053i	−0.308879	−	0.951101i	$-0.599954\pi$
$283$	6.06218	−	3.50000i	0.360359	−	0.208053i	0.669238	+	0.743048i	$0.266621\pi$
$284$		0			0
$285$	−29.5692	−	44.7846i	−1.75153	−	2.65281i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	4.00000	−	6.92820i	0.235294	−	0.407541i
$290$		0			0
$291$	−4.50000	−	7.79423i	−0.263795	−	0.456906i
$292$		0			0
$293$		−	15.0000i		−	0.876309i	−0.898900	−	0.438155i	$-0.855632\pi$
$293$		−	15.0000i		−	0.876309i	0.898900	−	0.438155i	$-0.144368\pi$
$294$		0			0
$295$	−20.0000	−	10.0000i	−1.16445	−	0.582223i
$296$		0			0
$297$	23.3827	−	13.5000i	1.35680	−	0.783349i
$298$		0			0
$299$	1.00000	−	1.73205i	0.0578315	−	0.100167i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−31.1769	−	18.0000i	−1.79107	−	1.03407i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	19.0000i		−	1.08439i	−0.840254	−	0.542194i	$-0.817594\pi$
$307$		−	19.0000i		−	1.08439i	0.840254	−	0.542194i	$-0.182406\pi$
$308$		0			0
$309$	15.0000			0.853320
$310$		0			0
$311$	−6.00000	−	10.3923i	−0.340229	−	0.589294i	0.644246	−	0.764818i	$-0.277171\pi$
$311$	−6.00000	−	10.3923i	−0.340229	−	0.589294i	−0.984475	+	0.175525i	$0.943838\pi$
$312$		0			0
$313$	−6.06218	−	3.50000i	−0.342655	−	0.197832i	0.318791	−	0.947825i	$-0.396723\pi$
$313$	−6.06218	−	3.50000i	−0.342655	−	0.197832i	−0.661445	+	0.749993i	$0.730057\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−24.2487	−	14.0000i	−1.36194	−	0.786318i	−0.372061	−	0.928208i	$-0.621349\pi$
$317$	−24.2487	−	14.0000i	−1.36194	−	0.786318i	−0.989882	+	0.141890i	$0.954682\pi$
$318$		0			0
$319$	−1.50000	−	2.59808i	−0.0839839	−	0.145464i
$320$		0			0
$321$	24.0000			1.33955
$322$		0			0
$323$		−	40.0000i		−	2.22566i
$324$		0			0
$325$	−4.59808	−	1.96410i	−0.255055	−	0.108949i
$326$		0			0
$327$	18.1865	+	10.5000i	1.00572	+	0.580651i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−10.0000	+	17.3205i	−0.549650	+	0.952021i	0.448649	+	0.893708i	$0.351905\pi$
$331$	−10.0000	+	17.3205i	−0.549650	+	0.952021i	−0.998298	+	0.0583130i	$0.981428\pi$
$332$		0			0
$333$	−51.9615	+	30.0000i	−2.84747	+	1.64399i
$334$		0			0
$335$	−10.0000	+	20.0000i	−0.546358	+	1.09272i
$336$		0			0
$337$			22.0000i			1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$			22.0000i			1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$		0			0
$339$	−15.0000	−	25.9808i	−0.814688	−	1.41108i
$340$		0			0
$341$	3.00000	−	5.19615i	0.162459	−	0.281387i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−11.1962	+	7.39230i	−0.602781	+	0.397988i
$346$		0			0
$347$	15.5885	−	9.00000i	0.836832	−	0.483145i	−0.0193540	−	0.999813i	$-0.506161\pi$
$347$	15.5885	−	9.00000i	0.836832	−	0.483145i	0.856186	+	0.516667i	$0.172828\pi$
$348$		0			0
$349$	−36.0000			−1.92704			−0.963518	−	0.267644i	$-0.913755\pi$
$349$	−36.0000			−1.92704			−0.963518	+	0.267644i	$0.913755\pi$
$350$		0			0
$351$	9.00000			0.480384
$352$		0			0
$353$	7.79423	−	4.50000i	0.414845	−	0.239511i	−0.278024	−	0.960574i	$-0.589680\pi$
$353$	7.79423	−	4.50000i	0.414845	−	0.239511i	0.692869	+	0.721063i	$0.256346\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	14.0000	−	24.2487i	0.738892	−	1.27980i	−0.214103	−	0.976811i	$-0.568683\pi$
$359$	14.0000	−	24.2487i	0.738892	−	1.27980i	0.952995	−	0.302987i	$-0.0979839\pi$
$360$		0			0
$361$	−22.5000	−	38.9711i	−1.18421	−	2.05111i
$362$		0			0
$363$			6.00000i			0.314918i
$364$		0			0
$365$	10.0000	−	20.0000i	0.523424	−	1.04685i
$366$		0			0
$367$	16.4545	−	9.50000i	0.858917	−	0.495896i	−0.00473247	−	0.999989i	$-0.501506\pi$
$367$	16.4545	−	9.50000i	0.858917	−	0.495896i	0.863649	+	0.504093i	$0.168173\pi$
$368$		0			0
$369$	−18.0000	+	31.1769i	−0.937043	+	1.62301i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	27.7128	+	16.0000i	1.43492	+	0.828449i	0.997490	−	0.0708063i	$-0.0225572\pi$
$373$	27.7128	+	16.0000i	1.43492	+	0.828449i	0.437425	+	0.899255i	$0.355891\pi$
$374$		0			0
$375$	21.6962	+	25.5788i	1.12038	+	1.32089i
$376$		0			0
$377$		−	1.00000i		−	0.0515026i
$378$		0			0
$379$	−28.0000			−1.43826			−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000			−1.43826			−0.719132	+	0.694874i	$0.755460\pi$
$380$		0			0
$381$	3.00000	+	5.19615i	0.153695	+	0.266207i
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	20.7846	+	12.0000i	1.05654	+	0.609994i
$388$		0			0
$389$	4.50000	+	7.79423i	0.228159	+	0.395183i	0.957263	−	0.289220i	$-0.0933960\pi$
$389$	4.50000	+	7.79423i	0.228159	+	0.395183i	−0.729103	+	0.684403i	$0.760063\pi$
$390$		0			0
$391$	−10.0000			−0.505722
$392$		0			0
$393$		−	6.00000i		−	0.302660i
$394$		0			0
$395$	0.937822	+	15.6244i	0.0471870	+	0.786147i
$396$		0			0
$397$	14.7224	+	8.50000i	0.738898	+	0.426603i	0.821668	−	0.569966i	$-0.193044\pi$
$397$	14.7224	+	8.50000i	0.738898	+	0.426603i	−0.0827707	+	0.996569i	$0.526377\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	13.5000	−	23.3827i	0.674158	−	1.16768i	−0.302556	−	0.953131i	$-0.597840\pi$
$401$	13.5000	−	23.3827i	0.674158	−	1.16768i	0.976714	−	0.214544i	$-0.0688266\pi$
$402$		0			0
$403$	1.73205	−	1.00000i	0.0862796	−	0.0498135i
$404$		0			0
$405$	−18.0000	−	9.00000i	−0.894427	−	0.447214i
$406$		0			0
$407$			30.0000i			1.48704i
$408$		0			0
$409$	2.00000	+	3.46410i	0.0988936	+	0.171289i	0.911227	−	0.411905i	$-0.135136\pi$
$409$	2.00000	+	3.46410i	0.0988936	+	0.171289i	−0.812333	+	0.583193i	$0.801803\pi$
$410$		0			0
$411$	−6.00000	+	10.3923i	−0.295958	+	0.512615i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−14.7846	−	22.3923i	−0.725748	−	1.09920i
$416$		0			0
$417$	25.9808	−	15.0000i	1.27228	−	0.734553i
$418$		0			0
$419$	−2.00000			−0.0977064			−0.0488532	−	0.998806i	$-0.515557\pi$
$419$	−2.00000			−0.0977064			−0.0488532	+	0.998806i	$0.515557\pi$
$420$		0			0
$421$	−23.0000			−1.12095			−0.560476	−	0.828171i	$-0.689382\pi$
$421$	−23.0000			−1.12095			−0.560476	+	0.828171i	$0.689382\pi$
$422$		0			0
$423$	−57.1577	+	33.0000i	−2.77910	+	1.60451i
$424$		0			0
$425$	2.99038	+	24.8205i	0.145055	+	1.20397i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	4.50000	−	7.79423i	0.217262	−	0.376309i
$430$		0			0
$431$	18.5000	+	32.0429i	0.891114	+	1.54345i	0.838542	+	0.544837i	$0.183408\pi$
$431$	18.5000	+	32.0429i	0.891114	+	1.54345i	0.0525716	+	0.998617i	$0.483258\pi$
$432$		0			0
$433$			38.0000i			1.82616i	0.407777	+	0.913082i	$0.366304\pi$
$433$			38.0000i			1.82616i	−0.407777	+	0.913082i	$0.633696\pi$
$434$		0			0
$435$	−3.00000	+	6.00000i	−0.143839	+	0.287678i
$436$		0			0
$437$	−13.8564	+	8.00000i	−0.662842	+	0.382692i
$438$		0			0
$439$	13.0000	−	22.5167i	0.620456	−	1.07466i	−0.368945	−	0.929451i	$-0.620281\pi$
$439$	13.0000	−	22.5167i	0.620456	−	1.07466i	0.989401	−	0.145210i	$-0.0463858\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10.3923	+	6.00000i	0.493753	+	0.285069i	0.726130	−	0.687557i	$-0.241317\pi$
$443$	10.3923	+	6.00000i	0.493753	+	0.285069i	−0.232377	+	0.972626i	$0.574650\pi$
$444$		0			0
$445$	−1.07180	−	17.8564i	−0.0508080	−	0.846475i
$446$		0			0
$447$		−	18.0000i		−	0.851371i
$448$		0			0
$449$	−11.0000			−0.519122			−0.259561	−	0.965727i	$-0.583578\pi$
$449$	−11.0000			−0.519122			−0.259561	+	0.965727i	$0.583578\pi$
$450$		0			0
$451$	9.00000	+	15.5885i	0.423793	+	0.734032i
$452$		0			0
$453$	23.3827	+	13.5000i	1.09861	+	0.634285i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	19.0526	+	11.0000i	0.891241	+	0.514558i	0.874348	−	0.485299i	$-0.161289\pi$
$457$	19.0526	+	11.0000i	0.891241	+	0.514558i	0.0168929	+	0.999857i	$0.494623\pi$
$458$		0			0
$459$	−22.5000	−	38.9711i	−1.05021	−	1.81902i
$460$		0			0
$461$	28.0000			1.30409			0.652045	−	0.758180i	$-0.273911\pi$
$461$	28.0000			1.30409			0.652045	+	0.758180i	$0.273911\pi$
$462$		0			0
$463$		−	4.00000i		−	0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$		−	4.00000i		−	0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$		0			0
$465$	−13.3923	+	0.803848i	−0.621053	+	0.0372775i
$466$		0			0
$467$	19.9186	+	11.5000i	0.921722	+	0.532157i	0.884184	−	0.467139i	$-0.154715\pi$
$467$	19.9186	+	11.5000i	0.921722	+	0.532157i	0.0375381	+	0.999295i	$0.488048\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	27.0000	−	46.7654i	1.24409	−	2.15483i
$472$		0			0
$473$	10.3923	−	6.00000i	0.477839	−	0.275880i
$474$		0			0
$475$	24.0000	+	32.0000i	1.10120	+	1.46826i
$476$		0			0
$477$			36.0000i			1.64833i
$478$		0			0
$479$	−9.00000	−	15.5885i	−0.411220	−	0.712255i	0.583803	−	0.811895i	$-0.301564\pi$
$479$	−9.00000	−	15.5885i	−0.411220	−	0.712255i	−0.995023	+	0.0996406i	$0.968231\pi$
$480$		0			0
$481$	−5.00000	+	8.66025i	−0.227980	+	0.394874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.69615	+	5.59808i	0.167834	+	0.254196i
$486$		0			0
$487$	−22.5167	+	13.0000i	−1.02033	+	0.589086i	−0.914199	−	0.405266i	$-0.867179\pi$
$487$	−22.5167	+	13.0000i	−1.02033	+	0.589086i	−0.106129	+	0.994352i	$0.533846\pi$
$488$		0			0
$489$	−18.0000			−0.813988
$490$		0			0
$491$	33.0000			1.48927			0.744635	−	0.667472i	$-0.232624\pi$
$491$	33.0000			1.48927			0.744635	+	0.667472i	$0.232624\pi$
$492$		0			0
$493$	−4.33013	+	2.50000i	−0.195019	+	0.112594i
$494$		0			0
$495$	−33.5885	+	22.1769i	−1.50969	+	0.996778i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−14.5000	+	25.1147i	−0.649109	+	1.12429i	0.334227	+	0.942493i	$0.391525\pi$
$499$	−14.5000	+	25.1147i	−0.649109	+	1.12429i	−0.983336	+	0.181797i	$0.941809\pi$
$500$		0			0
$501$	−4.50000	−	7.79423i	−0.201045	−	0.348220i
$502$		0			0
$503$		−	1.00000i		−	0.0445878i	−0.999751	−	0.0222939i	$-0.992903\pi$
$503$		−	1.00000i		−	0.0445878i	0.999751	−	0.0222939i	$-0.00709696\pi$
$504$		0			0
$505$	24.0000	+	12.0000i	1.06799	+	0.533993i
$506$		0			0
$507$	−31.1769	+	18.0000i	−1.38462	+	0.799408i
$508$		0			0
$509$	13.0000	−	22.5167i	0.576215	−	0.998033i	−0.419694	−	0.907666i	$-0.637862\pi$
$509$	13.0000	−	22.5167i	0.576215	−	0.998033i	0.995908	−	0.0903676i	$-0.0288042\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−62.3538	−	36.0000i	−2.75299	−	1.58944i
$514$		0			0
$515$	−11.1603	+	0.669873i	−0.491780	+	0.0295181i
$516$		0			0
$517$			33.0000i			1.45134i
$518$		0			0
$519$	27.0000			1.18517
$520$		0			0
$521$	−6.00000	−	10.3923i	−0.262865	−	0.455295i	0.704137	−	0.710064i	$-0.251334\pi$
$521$	−6.00000	−	10.3923i	−0.262865	−	0.455295i	−0.967002	+	0.254769i	$0.918001\pi$
$522$		0			0
$523$	−17.3205	−	10.0000i	−0.757373	−	0.437269i	0.0709788	−	0.997478i	$-0.477388\pi$
$523$	−17.3205	−	10.0000i	−0.757373	−	0.437269i	−0.828352	+	0.560208i	$0.810721\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.66025	−	5.00000i	−0.377247	−	0.217803i
$528$		0			0
$529$	−9.50000	−	16.4545i	−0.413043	−	0.715412i
$530$		0			0
$531$	−60.0000			−2.60378
$532$		0			0
$533$			6.00000i			0.259889i
$534$		0			0
$535$	−17.8564	+	1.07180i	−0.772000	+	0.0463378i
$536$		0			0
$537$	−10.3923	−	6.00000i	−0.448461	−	0.258919i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	12.5000	−	21.6506i	0.537417	−	0.930834i	−0.461625	−	0.887075i	$-0.652733\pi$
$541$	12.5000	−	21.6506i	0.537417	−	0.930834i	0.999042	−	0.0437584i	$-0.0139332\pi$
$542$		0			0
$543$	25.9808	−	15.0000i	1.11494	−	0.643712i
$544$		0			0
$545$	−14.0000	−	7.00000i	−0.599694	−	0.299847i
$546$		0			0
$547$		−	8.00000i		−	0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$		−	8.00000i		−	0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4.00000	+	6.92820i	−0.170406	+	0.295151i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	55.9808	−	36.9615i	2.37625	−	1.56893i
$556$		0			0
$557$	17.3205	−	10.0000i	0.733893	−	0.423714i	−0.0859514	−	0.996299i	$-0.527393\pi$
$557$	17.3205	−	10.0000i	0.733893	−	0.423714i	0.819845	+	0.572586i	$0.194060\pi$
$558$		0			0
$559$	4.00000			0.169182
$560$		0			0
$561$	−45.0000			−1.89990
$562$		0			0
$563$	13.8564	−	8.00000i	0.583978	−	0.337160i	−0.178735	−	0.983897i	$-0.557200\pi$
$563$	13.8564	−	8.00000i	0.583978	−	0.337160i	0.762713	+	0.646737i	$0.223867\pi$
$564$		0			0
$565$	12.3205	+	18.6603i	0.518328	+	0.785043i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	9.00000	−	15.5885i	0.377300	−	0.653502i	−0.613369	−	0.789797i	$-0.710186\pi$
$569$	9.00000	−	15.5885i	0.377300	−	0.653502i	0.990668	+	0.136295i	$0.0435194\pi$
$570$		0			0
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	0.577301	−	0.816532i	$-0.304106\pi$
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	−0.995788	+	0.0916910i	$0.970773\pi$
$572$		0			0
$573$		−	21.0000i		−	0.877288i
$574$		0			0
$575$	8.00000	−	6.00000i	0.333623	−	0.250217i
$576$		0			0
$577$	14.7224	−	8.50000i	0.612903	−	0.353860i	−0.161198	−	0.986922i	$-0.551536\pi$
$577$	14.7224	−	8.50000i	0.612903	−	0.353860i	0.774101	+	0.633062i	$0.218202\pi$
$578$		0			0
$579$	12.0000	−	20.7846i	0.498703	−	0.863779i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	15.5885	+	9.00000i	0.645608	+	0.372742i
$584$		0			0
$585$	−13.3923	+	0.803848i	−0.553704	+	0.0332350i
$586$		0			0
$587$			28.0000i			1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$587$			28.0000i			1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$588$		0			0
$589$	−16.0000			−0.659269
$590$		0			0
$591$	−15.0000	−	25.9808i	−0.617018	−	1.06871i
$592$		0			0
$593$	−2.59808	−	1.50000i	−0.106690	−	0.0615976i	0.445705	−	0.895180i	$-0.352953\pi$
$593$	−2.59808	−	1.50000i	−0.106690	−	0.0615976i	−0.552396	+	0.833582i	$0.686286\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−46.7654	−	27.0000i	−1.91398	−	1.10504i
$598$		0			0
$599$	−10.5000	−	18.1865i	−0.429018	−	0.743082i	0.567768	−	0.823189i	$-0.307807\pi$
$599$	−10.5000	−	18.1865i	−0.429018	−	0.743082i	−0.996786	+	0.0801071i	$0.974474\pi$
$600$		0			0
$601$	8.00000			0.326327			0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000			0.326327			0.163163	+	0.986599i	$0.447830\pi$
$602$		0			0
$603$			60.0000i			2.44339i
$604$		0			0
$605$	−0.267949	−	4.46410i	−0.0108937	−	0.181492i
$606$		0			0
$607$	−4.33013	−	2.50000i	−0.175754	−	0.101472i	0.409542	−	0.912291i	$-0.365689\pi$
$607$	−4.33013	−	2.50000i	−0.175754	−	0.101472i	−0.585296	+	0.810819i	$0.699022\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.50000	+	9.52628i	−0.222506	+	0.385392i
$612$		0			0
$613$	10.3923	−	6.00000i	0.419741	−	0.242338i	−0.275225	−	0.961380i	$-0.588752\pi$
$613$	10.3923	−	6.00000i	0.419741	−	0.242338i	0.694967	+	0.719042i	$0.255419\pi$
$614$		0			0
$615$	18.0000	−	36.0000i	0.725830	−	1.45166i
$616$		0			0
$617$		−	34.0000i		−	1.36879i	−0.729112	−	0.684394i	$-0.760067\pi$
$617$		−	34.0000i		−	1.36879i	0.729112	−	0.684394i	$-0.239933\pi$
$618$		0			0
$619$	−1.00000	−	1.73205i	−0.0401934	−	0.0696170i	0.845229	−	0.534404i	$-0.179464\pi$
$619$	−1.00000	−	1.73205i	−0.0401934	−	0.0696170i	−0.885422	+	0.464787i	$0.846131\pi$
$620$		0			0
$621$	−9.00000	+	15.5885i	−0.361158	+	0.625543i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−17.2846	−	18.0622i	−0.691384	−	0.722487i
$626$		0			0
$627$	−62.3538	+	36.0000i	−2.49017	+	1.43770i
$628$		0			0
$629$	50.0000			1.99363
$630$		0			0
$631$	15.0000			0.597141			0.298570	−	0.954388i	$-0.403490\pi$
$631$	15.0000			0.597141			0.298570	+	0.954388i	$0.403490\pi$
$632$		0			0
$633$	7.79423	−	4.50000i	0.309793	−	0.178859i
$634$		0			0
$635$	−2.46410	−	3.73205i	−0.0977849	−	0.148102i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	13.0000	+	22.5167i	0.513469	+	0.889355i	0.999878	+	0.0156233i	$0.00497325\pi$
$641$	13.0000	+	22.5167i	0.513469	+	0.889355i	−0.486409	+	0.873731i	$0.661693\pi$
$642$		0			0
$643$			5.00000i			0.197181i	0.995128	+	0.0985904i	$0.0314334\pi$
$643$			5.00000i			0.197181i	−0.995128	+	0.0985904i	$0.968567\pi$
$644$		0			0
$645$	−24.0000	−	12.0000i	−0.944999	−	0.472500i
$646$		0			0
$647$	−20.7846	+	12.0000i	−0.817127	+	0.471769i	−0.849425	−	0.527710i	$-0.823051\pi$
$647$	−20.7846	+	12.0000i	−0.817127	+	0.471769i	0.0322975	+	0.999478i	$0.489718\pi$
$648$		0			0
$649$	−15.0000	+	25.9808i	−0.588802	+	1.01983i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	31.1769	+	18.0000i	1.22005	+	0.704394i	0.964928	−	0.262515i	$-0.0845520\pi$
$653$	31.1769	+	18.0000i	1.22005	+	0.704394i	0.255119	+	0.966910i	$0.417885\pi$
$654$		0			0
$655$	0.267949	+	4.46410i	0.0104696	+	0.174427i
$656$		0			0
$657$		−	60.0000i		−	2.34082i
$658$		0			0
$659$	−39.0000			−1.51922			−0.759612	−	0.650376i	$-0.774611\pi$
$659$	−39.0000			−1.51922			−0.759612	+	0.650376i	$0.774611\pi$
$660$		0			0
$661$	−14.0000	−	24.2487i	−0.544537	−	0.943166i	−0.998636	−	0.0522143i	$-0.983372\pi$
$661$	−14.0000	−	24.2487i	−0.544537	−	0.943166i	0.454099	−	0.890951i	$-0.349961\pi$
$662$		0			0
$663$	−12.9904	−	7.50000i	−0.504505	−	0.291276i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	1.73205	+	1.00000i	0.0670653	+	0.0387202i
$668$		0			0
$669$	28.5000	+	49.3634i	1.10187	+	1.90850i
$670$		0			0
$671$		0			0
$672$		0			0
$673$			16.0000i			0.616755i	0.951264	+	0.308377i	$0.0997859\pi$
$673$			16.0000i			0.616755i	−0.951264	+	0.308377i	$0.900214\pi$
$674$		0			0
$675$	41.3827	+	17.6769i	1.59282	+	0.680385i
$676$		0			0
$677$	9.52628	+	5.50000i	0.366125	+	0.211382i	0.671764	−	0.740765i	$-0.265537\pi$
$677$	9.52628	+	5.50000i	0.366125	+	0.211382i	−0.305639	+	0.952147i	$0.598870\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−40.5000	+	70.1481i	−1.55196	+	2.68808i
$682$		0			0
$683$	−34.6410	+	20.0000i	−1.32550	+	0.765279i	−0.984600	−	0.174820i	$-0.944066\pi$
$683$	−34.6410	+	20.0000i	−1.32550	+	0.765279i	−0.340901	+	0.940099i	$0.610732\pi$
$684$		0			0
$685$	4.00000	−	8.00000i	0.152832	−	0.305664i
$686$		0			0
$687$		−	78.0000i		−	2.97589i
$688$		0			0
$689$	3.00000	+	5.19615i	0.114291	+	0.197958i
$690$		0			0
$691$	−20.0000	+	34.6410i	−0.760836	+	1.31781i	0.181584	+	0.983375i	$0.441877\pi$
$691$	−20.0000	+	34.6410i	−0.760836	+	1.31781i	−0.942420	+	0.334431i	$0.891456\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−18.6603	+	12.3205i	−0.707824	+	0.467344i
$696$		0			0
$697$	25.9808	−	15.0000i	0.984092	−	0.568166i
$698$		0			0
$699$	−48.0000			−1.81553
$700$		0			0
$701$	−25.0000			−0.944237			−0.472118	−	0.881535i	$-0.656511\pi$
$701$	−25.0000			−0.944237			−0.472118	+	0.881535i	$0.656511\pi$
$702$		0			0
$703$	69.2820	−	40.0000i	2.61302	−	1.50863i
$704$		0			0
$705$	61.5788	−	40.6577i	2.31919	−	1.53126i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	7.50000	−	12.9904i	0.281668	−	0.487864i	−0.690127	−	0.723688i	$-0.742446\pi$
$709$	7.50000	−	12.9904i	0.281668	−	0.487864i	0.971796	+	0.235824i	$0.0757789\pi$
$710$		0			0
$711$	21.0000	+	36.3731i	0.787562	+	1.36410i
$712$		0			0
$713$			4.00000i			0.149801i
$714$		0			0
$715$	−3.00000	+	6.00000i	−0.112194	+	0.224387i
$716$		0			0
$717$	12.9904	−	7.50000i	0.485135	−	0.280093i
$718$		0			0
$719$	1.00000	−	1.73205i	0.0372937	−	0.0645946i	−0.846776	−	0.531949i	$-0.821460\pi$
$719$	1.00000	−	1.73205i	0.0372937	−	0.0645946i	0.884070	+	0.467355i	$0.154793\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−46.7654	−	27.0000i	−1.73922	−	1.00414i
$724$		0			0
$725$	1.96410	−	4.59808i	0.0729449	−	0.170768i
$726$		0			0
$727$			28.0000i			1.03846i	0.854634	+	0.519231i	$0.173782\pi$
$727$			28.0000i			1.03846i	−0.854634	+	0.519231i	$0.826218\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	−10.0000	−	17.3205i	−0.369863	−	0.640622i
$732$		0			0
$733$	35.5070	+	20.5000i	1.31148	+	0.757185i	0.982342	−	0.187096i	$-0.0599076\pi$
$733$	35.5070	+	20.5000i	1.31148	+	0.757185i	0.329141	+	0.944281i	$0.393241\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	25.9808	+	15.0000i	0.957014	+	0.552532i
$738$		0			0
$739$	−2.50000	−	4.33013i	−0.0919640	−	0.159286i	0.816373	−	0.577524i	$-0.195981\pi$
$739$	−2.50000	−	4.33013i	−0.0919640	−	0.159286i	−0.908337	+	0.418238i	$0.862648\pi$
$740$		0			0
$741$	−24.0000			−0.881662
$742$		0			0
$743$		−	30.0000i		−	1.10059i	−0.834969	−	0.550297i	$-0.814515\pi$
$743$		−	30.0000i		−	1.10059i	0.834969	−	0.550297i	$-0.185485\pi$
$744$		0			0
$745$	0.803848	+	13.3923i	0.0294507	+	0.490656i
$746$		0			0
$747$	−62.3538	−	36.0000i	−2.28141	−	1.31717i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−6.50000	+	11.2583i	−0.237188	+	0.410822i	−0.959906	−	0.280321i	$-0.909559\pi$
$751$	−6.50000	+	11.2583i	−0.237188	+	0.410822i	0.722718	+	0.691143i	$0.242893\pi$
$752$		0			0
$753$	−5.19615	+	3.00000i	−0.189358	+	0.109326i
$754$		0			0
$755$	−18.0000	−	9.00000i	−0.655087	−	0.327544i
$756$		0			0
$757$		−	48.0000i		−	1.74459i	−0.488980	−	0.872295i	$-0.662631\pi$
$757$		−	48.0000i		−	1.74459i	0.488980	−	0.872295i	$-0.337369\pi$
$758$		0			0
$759$	9.00000	+	15.5885i	0.326679	+	0.565825i
$760$		0			0
$761$	19.0000	−	32.9090i	0.688749	−	1.19295i	−0.283493	−	0.958974i	$-0.591493\pi$
$761$	19.0000	−	32.9090i	0.688749	−	1.19295i	0.972243	−	0.233975i	$-0.0751733\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	36.9615	+	55.9808i	1.33635	+	2.02399i
$766$		0			0
$767$	−8.66025	+	5.00000i	−0.312704	+	0.180540i
$768$		0			0
$769$	−16.0000			−0.576975			−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000			−0.576975			−0.288487	+	0.957484i	$0.593152\pi$
$770$		0			0
$771$	18.0000			0.648254
$772$		0			0
$773$	23.3827	−	13.5000i	0.841017	−	0.485561i	−0.0165929	−	0.999862i	$-0.505282\pi$
$773$	23.3827	−	13.5000i	0.841017	−	0.485561i	0.857610	+	0.514301i	$0.171949\pi$
$774$		0			0
$775$	9.92820	−	1.19615i	0.356632	−	0.0429671i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	24.0000	−	41.5692i	0.859889	−	1.48937i
$780$		0			0
$781$		0			0
$782$		0			0
$783$			9.00000i			0.321634i
$784$		0			0
$785$	−18.0000	+	36.0000i	−0.642448	+	1.28490i
$786$		0			0
$787$	2.59808	−	1.50000i	0.0926114	−	0.0534692i	−0.452979	−	0.891521i	$-0.649639\pi$
$787$	2.59808	−	1.50000i	0.0926114	−	0.0534692i	0.545590	+	0.838052i	$0.316305\pi$
$788$		0			0
$789$	−36.0000	+	62.3538i	−1.28163	+	2.21986i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−2.41154	−	40.1769i	−0.0855286	−	1.42493i
$796$		0			0
$797$			43.0000i			1.52314i

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:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.59808 + 1.50000i) q^{3} +(1.86603 - 1.23205i) q^{5} +(3.00000 - 5.19615i) q^{9} +O(q^{10})\)	\(q+(-2.59808 + 1.50000i) q^{3} +(1.86603 - 1.23205i) q^{5} +(3.00000 - 5.19615i) q^{9} +(-1.50000 - 2.59808i) q^{11} -1.00000i q^{13} +(-3.00000 + 6.00000i) q^{15} +(-4.33013 + 2.50000i) q^{17} +(-4.00000 + 6.92820i) q^{19} +(1.73205 + 1.00000i) q^{23} +(1.96410 - 4.59808i) q^{25} +9.00000i q^{27} +1.00000 q^{29} +(1.00000 + 1.73205i) q^{31} +(7.79423 + 4.50000i) q^{33} +(-8.66025 - 5.00000i) q^{37} +(1.50000 + 2.59808i) q^{39} -6.00000 q^{41} +4.00000i q^{43} +(-0.803848 - 13.3923i) q^{45} +(-9.52628 - 5.50000i) q^{47} +(7.50000 - 12.9904i) q^{51} +(-5.19615 + 3.00000i) q^{53} +(-6.00000 - 3.00000i) q^{55} -24.0000i q^{57} +(-5.00000 - 8.66025i) q^{59} +(-1.23205 - 1.86603i) q^{65} +(-8.66025 + 5.00000i) q^{67} -6.00000 q^{69} +(8.66025 - 5.00000i) q^{73} +(1.79423 + 14.8923i) q^{75} +(-3.50000 + 6.06218i) q^{79} +(-4.50000 - 7.79423i) q^{81} -12.0000i q^{83} +(-5.00000 + 10.0000i) q^{85} +(-2.59808 + 1.50000i) q^{87} +(4.00000 - 6.92820i) q^{89} +(-5.19615 - 3.00000i) q^{93} +(1.07180 + 17.8564i) q^{95} +3.00000i q^{97} -18.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 12 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 + 12 * q^9	\( 4 q + 4 q^{5} + 12 q^{9} - 6 q^{11} - 12 q^{15} - 16 q^{19} - 6 q^{25} + 4 q^{29} + 4 q^{31} + 6 q^{39} - 24 q^{41} - 24 q^{45} + 30 q^{51} - 24 q^{55} - 20 q^{59} + 2 q^{65} - 24 q^{69} - 24 q^{75} - 14 q^{79} - 18 q^{81} - 20 q^{85} + 16 q^{89} + 32 q^{95} - 72 q^{99}+O(q^{100}) \) 4 * q + 4 * q^5 + 12 * q^9 - 6 * q^11 - 12 * q^15 - 16 * q^19 - 6 * q^25 + 4 * q^29 + 4 * q^31 + 6 * q^39 - 24 * q^41 - 24 * q^45 + 30 * q^51 - 24 * q^55 - 20 * q^59 + 2 * q^65 - 24 * q^69 - 24 * q^75 - 14 * q^79 - 18 * q^81 - 20 * q^85 + 16 * q^89 + 32 * q^95 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more