LMFDB - Embedded newform 1300.2.bb.a.549.1

Newspace parameters

Level:	$ N $	$=$	$ 1300 = 2^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1300.bb (of order $6$, degree $2$, not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$10.3805522628$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 260)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			549.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1300.549
Dual form			1300.2.bb.a.1049.1

$q$-expansion

$f(q)$	$=$	$q+(-0.866025 + 0.500000i) q^{3} +(0.866025 + 0.500000i) q^{7} +(-1.00000 + 1.73205i) q^{9} +O(q^{10})$	$q+(-0.866025 + 0.500000i) q^{3} +(0.866025 + 0.500000i) q^{7} +(-1.00000 + 1.73205i) q^{9} +(-1.50000 - 2.59808i) q^{11} +(3.46410 - 1.00000i) q^{13} +(2.59808 + 1.50000i) q^{17} +(-3.50000 + 6.06218i) q^{19} -1.00000 q^{21} +(2.59808 - 1.50000i) q^{23} -5.00000i q^{27} +(1.50000 + 2.59808i) q^{29} -4.00000 q^{31} +(2.59808 + 1.50000i) q^{33} +(-6.06218 + 3.50000i) q^{37} +(-2.50000 + 2.59808i) q^{39} +(4.50000 + 7.79423i) q^{41} +(9.52628 + 5.50000i) q^{43} +(-3.00000 - 5.19615i) q^{49} -3.00000 q^{51} +6.00000i q^{53} -7.00000i q^{57} +(-1.50000 + 2.59808i) q^{59} +(-5.50000 + 9.52628i) q^{61} +(-1.73205 + 1.00000i) q^{63} +(-6.06218 + 3.50000i) q^{67} +(-1.50000 + 2.59808i) q^{69} +(1.50000 - 2.59808i) q^{71} -2.00000i q^{73} -3.00000i q^{77} -8.00000 q^{79} +(-0.500000 - 0.866025i) q^{81} +12.0000i q^{83} +(-2.59808 - 1.50000i) q^{87} +(7.50000 + 12.9904i) q^{89} +(3.50000 + 0.866025i) q^{91} +(3.46410 - 2.00000i) q^{93} +(6.06218 + 3.50000i) q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} - 6 q^{11} - 14 q^{19} - 4 q^{21} + 6 q^{29} - 16 q^{31} - 10 q^{39} + 18 q^{41} - 12 q^{49} - 12 q^{51} - 6 q^{59} - 22 q^{61} - 6 q^{69} + 6 q^{71} - 32 q^{79} - 2 q^{81} + 30 q^{89} + 14 q^{91} + 24 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 - 6 * q^11 - 14 * q^19 - 4 * q^21 + 6 * q^29 - 16 * q^31 - 10 * q^39 + 18 * q^41 - 12 * q^49 - 12 * q^51 - 6 * q^59 - 22 * q^61 - 6 * q^69 + 6 * q^71 - 32 * q^79 - 2 * q^81 + 30 * q^89 + 14 * q^91 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$.

$n$	$301$	$651$	$677$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$-1$

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
$3$	−0.866025	+	0.500000i	−0.500000	+	0.288675i	−0.728714	−	0.684819i	$-0.759881\pi$
$3$	−0.866025	+	0.500000i	−0.500000	+	0.288675i	0.228714	+	0.973494i	$0.426548\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	0.866025	+	0.500000i	0.327327	+	0.188982i	0.654654	−	0.755929i	$-0.272814\pi$
$7$	0.866025	+	0.500000i	0.327327	+	0.188982i	−0.327327	+	0.944911i	$0.606148\pi$
$8$		0			0
$9$	−1.00000	+	1.73205i	−0.333333	+	0.577350i
$10$		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
$13$	3.46410	−	1.00000i	0.960769	−	0.277350i
$13$	3.46410	−	1.00000i	0.960769	−	0.277350i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.59808	+	1.50000i	0.630126	+	0.363803i	0.780801	−	0.624780i	$-0.214811\pi$
$17$	2.59808	+	1.50000i	0.630126	+	0.363803i	−0.150675	+	0.988583i	$0.548145\pi$
$18$		0			0
$19$	−3.50000	+	6.06218i	−0.802955	+	1.39076i	0.114708	+	0.993399i	$0.463407\pi$
$19$	−3.50000	+	6.06218i	−0.802955	+	1.39076i	−0.917663	+	0.397360i	$0.869927\pi$
$20$		0			0
$21$	−1.00000			−0.218218
$22$		0			0
$23$	2.59808	−	1.50000i	0.541736	−	0.312772i	−0.204046	−	0.978961i	$-0.565409\pi$
$23$	2.59808	−	1.50000i	0.541736	−	0.312772i	0.745782	+	0.666190i	$0.232076\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	5.00000i		−	0.962250i
$28$		0			0
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	0.971023	−	0.238987i	$-0.0768152\pi$
$29$	1.50000	+	2.59808i	0.278543	+	0.482451i	−0.692480	+	0.721437i	$0.743482\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		0			0
$33$	2.59808	+	1.50000i	0.452267	+	0.261116i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6.06218	+	3.50000i	−0.996616	+	0.575396i	−0.907245	−	0.420602i	$-0.861819\pi$
$37$	−6.06218	+	3.50000i	−0.996616	+	0.575396i	−0.0893706	+	0.995998i	$0.528486\pi$
$38$		0			0
$39$	−2.50000	+	2.59808i	−0.400320	+	0.416025i
$40$		0			0
$41$	4.50000	+	7.79423i	0.702782	+	1.21725i	0.967486	+	0.252924i	$0.0813924\pi$
$41$	4.50000	+	7.79423i	0.702782	+	1.21725i	−0.264704	+	0.964330i	$0.585274\pi$
$42$		0			0
$43$	9.52628	+	5.50000i	1.45274	+	0.838742i	0.998636	−	0.0522047i	$-0.0166248\pi$
$43$	9.52628	+	5.50000i	1.45274	+	0.838742i	0.454108	+	0.890947i	$0.349958\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−3.00000	−	5.19615i	−0.428571	−	0.742307i
$50$		0			0
$51$	−3.00000			−0.420084
$52$		0			0
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	7.00000i		−	0.927173i
$58$		0			0
$59$	−1.50000	+	2.59808i	−0.195283	+	0.338241i	−0.946993	−	0.321253i	$-0.895896\pi$
$59$	−1.50000	+	2.59808i	−0.195283	+	0.338241i	0.751710	+	0.659494i	$0.229229\pi$
$60$		0			0
$61$	−5.50000	+	9.52628i	−0.704203	+	1.21972i	0.262776	+	0.964857i	$0.415362\pi$
$61$	−5.50000	+	9.52628i	−0.704203	+	1.21972i	−0.966978	+	0.254858i	$0.917971\pi$
$62$		0			0
$63$	−1.73205	+	1.00000i	−0.218218	+	0.125988i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−6.06218	+	3.50000i	−0.740613	+	0.427593i	−0.822292	−	0.569066i	$-0.807305\pi$
$67$	−6.06218	+	3.50000i	−0.740613	+	0.427593i	0.0816792	+	0.996659i	$0.473972\pi$
$68$		0			0
$69$	−1.50000	+	2.59808i	−0.180579	+	0.312772i
$70$		0			0
$71$	1.50000	−	2.59808i	0.178017	−	0.308335i	−0.763184	−	0.646181i	$-0.776365\pi$
$71$	1.50000	−	2.59808i	0.178017	−	0.308335i	0.941201	+	0.337846i	$0.109698\pi$
$72$		0			0
$73$		−	2.00000i		−	0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$		−	2.00000i		−	0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	3.00000i		−	0.341882i
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$			12.0000i			1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$			12.0000i			1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−2.59808	−	1.50000i	−0.278543	−	0.160817i
$88$		0			0
$89$	7.50000	+	12.9904i	0.794998	+	1.37698i	0.922840	+	0.385183i	$0.125862\pi$
$89$	7.50000	+	12.9904i	0.794998	+	1.37698i	−0.127842	+	0.991795i	$0.540805\pi$
$90$		0			0
$91$	3.50000	+	0.866025i	0.366900	+	0.0907841i
$92$		0			0
$93$	3.46410	−	2.00000i	0.359211	−	0.207390i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	6.06218	+	3.50000i	0.615521	+	0.355371i	0.775123	−	0.631810i	$-0.217688\pi$
$97$	6.06218	+	3.50000i	0.615521	+	0.355371i	−0.159602	+	0.987181i	$0.551021\pi$
$98$		0			0
$99$	6.00000			0.603023
$100$		0			0
$101$	4.50000	+	7.79423i	0.447767	+	0.775555i	0.998240	−	0.0592978i	$-0.0188862\pi$
$101$	4.50000	+	7.79423i	0.447767	+	0.775555i	−0.550474	+	0.834853i	$0.685553\pi$
$102$		0			0
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.79423	−	4.50000i	0.753497	−	0.435031i	−0.0734594	−	0.997298i	$-0.523404\pi$
$107$	7.79423	−	4.50000i	0.753497	−	0.435031i	0.826956	+	0.562267i	$0.190071\pi$
$108$		0			0
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$	3.50000	−	6.06218i	0.332205	−	0.575396i
$112$		0			0
$113$	7.79423	+	4.50000i	0.733219	+	0.423324i	0.819599	−	0.572938i	$-0.194196\pi$
$113$	7.79423	+	4.50000i	0.733219	+	0.423324i	−0.0863794	+	0.996262i	$0.527530\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.73205	+	7.00000i	−0.160128	+	0.647150i
$118$		0			0
$119$	1.50000	+	2.59808i	0.137505	+	0.238165i
$120$		0			0
$121$	1.00000	−	1.73205i	0.0909091	−	0.157459i
$122$		0			0
$123$	−7.79423	−	4.50000i	−0.702782	−	0.405751i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−16.4545	+	9.50000i	−1.46010	+	0.842989i	−0.999015	−	0.0443678i	$-0.985873\pi$
$127$	−16.4545	+	9.50000i	−1.46010	+	0.842989i	−0.461084	+	0.887357i	$0.652539\pi$
$128$		0			0
$129$	−11.0000			−0.968496
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$	−6.06218	+	3.50000i	−0.525657	+	0.303488i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12.9904	+	7.50000i	1.10984	+	0.640768i	0.938789	−	0.344493i	$-0.111949\pi$
$137$	12.9904	+	7.50000i	1.10984	+	0.640768i	0.171054	+	0.985262i	$0.445283\pi$
$138$		0			0
$139$	2.50000	−	4.33013i	0.212047	−	0.367277i	−0.740308	−	0.672268i	$-0.765320\pi$
$139$	2.50000	−	4.33013i	0.212047	−	0.367277i	0.952355	+	0.304991i	$0.0986536\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.79423	−	7.50000i	−0.651786	−	0.627182i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	5.19615	+	3.00000i	0.428571	+	0.247436i
$148$		0			0
$149$	−10.5000	+	18.1865i	−0.860194	+	1.48990i	0.0115483	+	0.999933i	$0.496324\pi$
$149$	−10.5000	+	18.1865i	−0.860194	+	1.48990i	−0.871742	+	0.489966i	$0.837009\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$	−5.19615	+	3.00000i	−0.420084	+	0.242536i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	10.0000i		−	0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$		−	10.0000i		−	0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$		0			0
$159$	−3.00000	−	5.19615i	−0.237915	−	0.412082i
$160$		0			0
$161$	3.00000			0.236433
$162$		0			0
$163$	−0.866025	−	0.500000i	−0.0678323	−	0.0391630i	0.465700	−	0.884943i	$-0.345802\pi$
$163$	−0.866025	−	0.500000i	−0.0678323	−	0.0391630i	−0.533533	+	0.845780i	$0.679136\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.59808	+	1.50000i	−0.201045	+	0.116073i	−0.597143	−	0.802135i	$-0.703697\pi$
$167$	−2.59808	+	1.50000i	−0.201045	+	0.116073i	0.396098	+	0.918208i	$0.370364\pi$
$168$		0			0
$169$	11.0000	−	6.92820i	0.846154	−	0.532939i
$170$		0			0
$171$	−7.00000	−	12.1244i	−0.535303	−	0.927173i
$172$		0			0
$173$	−2.59808	−	1.50000i	−0.197528	−	0.114043i	0.397974	−	0.917397i	$-0.369713\pi$
$173$	−2.59808	−	1.50000i	−0.197528	−	0.114043i	−0.595502	+	0.803354i	$0.703047\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	3.00000i		−	0.225494i
$178$		0			0
$179$	−10.5000	−	18.1865i	−0.784807	−	1.35933i	−0.929114	−	0.369792i	$-0.879429\pi$
$179$	−10.5000	−	18.1865i	−0.784807	−	1.35933i	0.144308	−	0.989533i	$-0.453905\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		−	11.0000i		−	0.813143i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	9.00000i		−	0.658145i
$188$		0			0
$189$	2.50000	−	4.33013i	0.181848	−	0.314970i
$190$		0			0
$191$	1.50000	−	2.59808i	0.108536	−	0.187990i	−0.806641	−	0.591041i	$-0.798717\pi$
$191$	1.50000	−	2.59808i	0.108536	−	0.187990i	0.915177	+	0.403051i	$0.132050\pi$
$192$		0			0
$193$	−4.33013	+	2.50000i	−0.311689	+	0.179954i	−0.647682	−	0.761911i	$-0.724262\pi$
$193$	−4.33013	+	2.50000i	−0.311689	+	0.179954i	0.335993	+	0.941865i	$0.390928\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	18.1865	−	10.5000i	1.29574	−	0.748094i	0.316072	−	0.948735i	$-0.397636\pi$
$197$	18.1865	−	10.5000i	1.29574	−	0.748094i	0.979665	+	0.200641i	$0.0643025\pi$
$198$		0			0
$199$	8.50000	−	14.7224i	0.602549	−	1.04365i	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	8.50000	−	14.7224i	0.602549	−	1.04365i	0.992434	−	0.122782i	$-0.0391815\pi$
$200$		0			0
$201$	3.50000	−	6.06218i	0.246871	−	0.427593i
$202$		0			0
$203$			3.00000i			0.210559i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			6.00000i			0.417029i
$208$		0			0
$209$	21.0000			1.45260
$210$		0			0
$211$	−5.50000	−	9.52628i	−0.378636	−	0.655816i	0.612228	−	0.790681i	$-0.290273\pi$
$211$	−5.50000	−	9.52628i	−0.378636	−	0.655816i	−0.990864	+	0.134865i	$0.956940\pi$
$212$		0			0
$213$			3.00000i			0.205557i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.46410	−	2.00000i	−0.235159	−	0.135769i
$218$		0			0
$219$	1.00000	+	1.73205i	0.0675737	+	0.117041i
$220$		0			0
$221$	10.5000	+	2.59808i	0.706306	+	0.174766i
$222$		0			0
$223$	16.4545	−	9.50000i	1.10187	−	0.636167i	0.165161	−	0.986267i	$-0.447186\pi$
$223$	16.4545	−	9.50000i	1.10187	−	0.636167i	0.936713	+	0.350100i	$0.113852\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−23.3827	−	13.5000i	−1.55196	−	0.896026i	−0.997982	−	0.0634974i	$-0.979775\pi$
$227$	−23.3827	−	13.5000i	−1.55196	−	0.896026i	−0.553981	−	0.832529i	$-0.686892\pi$
$228$		0			0
$229$	22.0000			1.45380			0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000			1.45380			0.726900	+	0.686743i	$0.240960\pi$
$230$		0			0
$231$	1.50000	+	2.59808i	0.0986928	+	0.170941i
$232$		0			0
$233$		−	18.0000i		−	1.17922i	−0.807688	−	0.589610i	$-0.799282\pi$
$233$		−	18.0000i		−	1.17922i	0.807688	−	0.589610i	$-0.200718\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	6.92820	−	4.00000i	0.450035	−	0.259828i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	0.500000	−	0.866025i	0.0322078	−	0.0557856i	−0.849472	−	0.527633i	$-0.823079\pi$
$241$	0.500000	−	0.866025i	0.0322078	−	0.0557856i	0.881680	+	0.471848i	$0.156413\pi$
$242$		0			0
$243$	13.8564	+	8.00000i	0.888889	+	0.513200i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−6.06218	+	24.5000i	−0.385727	+	1.55890i
$248$		0			0
$249$	−6.00000	−	10.3923i	−0.380235	−	0.658586i
$250$		0			0
$251$	−10.5000	+	18.1865i	−0.662754	+	1.14792i	0.317135	+	0.948380i	$0.397279\pi$
$251$	−10.5000	+	18.1865i	−0.662754	+	1.14792i	−0.979889	+	0.199543i	$0.936054\pi$
$252$		0			0
$253$	−7.79423	−	4.50000i	−0.490019	−	0.282913i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.79423	−	4.50000i	0.486191	−	0.280702i	−0.236802	−	0.971558i	$-0.576099\pi$
$257$	7.79423	−	4.50000i	0.486191	−	0.280702i	0.722993	+	0.690856i	$0.242766\pi$
$258$		0			0
$259$	−7.00000			−0.434959
$260$		0			0
$261$	−6.00000			−0.371391
$262$		0			0
$263$	2.59808	−	1.50000i	0.160204	−	0.0924940i	−0.417755	−	0.908560i	$-0.637183\pi$
$263$	2.59808	−	1.50000i	0.160204	−	0.0924940i	0.577959	+	0.816066i	$0.303849\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−12.9904	−	7.50000i	−0.794998	−	0.458993i
$268$		0			0
$269$	13.5000	−	23.3827i	0.823110	−	1.42567i	−0.0802460	−	0.996775i	$-0.525571\pi$
$269$	13.5000	−	23.3827i	0.823110	−	1.42567i	0.903356	−	0.428892i	$-0.141096\pi$
$270$		0			0
$271$	−11.5000	−	19.9186i	−0.698575	−	1.20997i	−0.968960	−	0.247216i	$-0.920484\pi$
$271$	−11.5000	−	19.9186i	−0.698575	−	1.20997i	0.270385	−	0.962752i	$-0.412849\pi$
$272$		0			0
$273$	−3.46410	+	1.00000i	−0.209657	+	0.0605228i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	16.4545	+	9.50000i	0.988654	+	0.570800i	0.904872	−	0.425684i	$-0.139967\pi$
$277$	16.4545	+	9.50000i	0.988654	+	0.570800i	0.0837823	+	0.996484i	$0.473300\pi$
$278$		0			0
$279$	4.00000	−	6.92820i	0.239474	−	0.414781i
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	−4.33013	+	2.50000i	−0.257399	+	0.148610i	−0.623148	−	0.782104i	$-0.714146\pi$
$283$	−4.33013	+	2.50000i	−0.257399	+	0.148610i	0.365748	+	0.930714i	$0.380813\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			9.00000i			0.531253i
$288$		0			0
$289$	−4.00000	−	6.92820i	−0.235294	−	0.407541i
$290$		0			0
$291$	−7.00000			−0.410347
$292$		0			0
$293$	−23.3827	−	13.5000i	−1.36603	−	0.788678i	−0.375613	−	0.926777i	$-0.622568\pi$
$293$	−23.3827	−	13.5000i	−1.36603	−	0.788678i	−0.990419	+	0.138098i	$0.955901\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−12.9904	+	7.50000i	−0.753778	+	0.435194i
$298$		0			0
$299$	7.50000	−	7.79423i	0.433736	−	0.450752i
$300$		0			0
$301$	5.50000	+	9.52628i	0.317015	+	0.549086i
$302$		0			0
$303$	−7.79423	−	4.50000i	−0.447767	−	0.258518i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			20.0000i			1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$			20.0000i			1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$		0			0
$309$	4.00000	+	6.92820i	0.227552	+	0.394132i
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$			22.0000i			1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$			22.0000i			1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	18.0000i		−	1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$		0			0
$319$	4.50000	−	7.79423i	0.251952	−	0.436393i
$320$		0			0
$321$	−4.50000	+	7.79423i	−0.251166	+	0.435031i
$322$		0			0
$323$	−18.1865	+	10.5000i	−1.01193	+	0.584236i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	1.73205	−	1.00000i	0.0957826	−	0.0553001i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	9.50000	−	16.4545i	0.522167	−	0.904420i	−0.477500	−	0.878632i	$-0.658457\pi$
$331$	9.50000	−	16.4545i	0.522167	−	0.904420i	0.999667	−	0.0257885i	$-0.00820965\pi$
$332$		0			0
$333$		−	14.0000i		−	0.767195i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	34.0000i		−	1.85210i	−0.377403	−	0.926049i	$-0.623183\pi$
$337$		−	34.0000i		−	1.85210i	0.377403	−	0.926049i	$-0.376817\pi$
$338$		0			0
$339$	−9.00000			−0.488813
$340$		0			0
$341$	6.00000	+	10.3923i	0.324918	+	0.562775i
$342$		0			0
$343$		−	13.0000i		−	0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	28.5788	+	16.5000i	1.53419	+	0.885766i	0.999162	+	0.0409337i	$0.0130332\pi$
$347$	28.5788	+	16.5000i	1.53419	+	0.885766i	0.535031	+	0.844833i	$0.320300\pi$
$348$		0			0
$349$	−0.500000	−	0.866025i	−0.0267644	−	0.0463573i	0.852333	−	0.523000i	$-0.175187\pi$
$349$	−0.500000	−	0.866025i	−0.0267644	−	0.0463573i	−0.879097	+	0.476642i	$0.841854\pi$
$350$		0			0
$351$	−5.00000	−	17.3205i	−0.266880	−	0.924500i
$352$		0			0
$353$	−7.79423	+	4.50000i	−0.414845	+	0.239511i	−0.692869	−	0.721063i	$-0.743654\pi$
$353$	−7.79423	+	4.50000i	−0.414845	+	0.239511i	0.278024	+	0.960574i	$0.410320\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−2.59808	−	1.50000i	−0.137505	−	0.0793884i
$358$		0			0
$359$	24.0000			1.26667			0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000			1.26667			0.633336	+	0.773877i	$0.281685\pi$
$360$		0			0
$361$	−15.0000	−	25.9808i	−0.789474	−	1.36741i
$362$		0			0
$363$			2.00000i			0.104973i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	4.33013	−	2.50000i	0.226031	−	0.130499i	−0.382709	−	0.923869i	$-0.625009\pi$
$367$	4.33013	−	2.50000i	0.226031	−	0.130499i	0.608740	+	0.793370i	$0.291675\pi$
$368$		0			0
$369$	−18.0000			−0.937043
$370$		0			0
$371$	−3.00000	+	5.19615i	−0.155752	+	0.269771i
$372$		0			0
$373$	−26.8468	−	15.5000i	−1.39007	−	0.802560i	−0.396751	−	0.917926i	$-0.629862\pi$
$373$	−26.8468	−	15.5000i	−1.39007	−	0.802560i	−0.993323	+	0.115367i	$0.963196\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7.79423	+	7.50000i	0.401423	+	0.386270i
$378$		0			0
$379$	−0.500000	−	0.866025i	−0.0256833	−	0.0444847i	0.852898	−	0.522077i	$-0.174843\pi$
$379$	−0.500000	−	0.866025i	−0.0256833	−	0.0444847i	−0.878581	+	0.477593i	$0.841509\pi$
$380$		0			0
$381$	9.50000	−	16.4545i	0.486700	−	0.842989i
$382$		0			0
$383$	−7.79423	−	4.50000i	−0.398266	−	0.229939i	0.287469	−	0.957790i	$-0.407186\pi$
$383$	−7.79423	−	4.50000i	−0.398266	−	0.229939i	−0.685736	+	0.727851i	$0.740519\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−19.0526	+	11.0000i	−0.968496	+	0.559161i
$388$		0			0
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		0			0
$391$	9.00000			0.455150
$392$		0			0
$393$	10.3923	−	6.00000i	0.524222	−	0.302660i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−4.33013	−	2.50000i	−0.217323	−	0.125471i	0.387387	−	0.921917i	$-0.373378\pi$
$397$	−4.33013	−	2.50000i	−0.217323	−	0.125471i	−0.604710	+	0.796446i	$0.706711\pi$
$398$		0			0
$399$	3.50000	−	6.06218i	0.175219	−	0.303488i
$400$		0			0
$401$	16.5000	+	28.5788i	0.823971	+	1.42716i	0.902703	+	0.430263i	$0.141579\pi$
$401$	16.5000	+	28.5788i	0.823971	+	1.42716i	−0.0787327	+	0.996896i	$0.525087\pi$
$402$		0			0
$403$	−13.8564	+	4.00000i	−0.690237	+	0.199254i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.1865	+	10.5000i	0.901473	+	0.520466i
$408$		0			0
$409$	−12.5000	+	21.6506i	−0.618085	+	1.07056i	0.371750	+	0.928333i	$0.378758\pi$
$409$	−12.5000	+	21.6506i	−0.618085	+	1.07056i	−0.989835	+	0.142222i	$0.954575\pi$
$410$		0			0
$411$	−15.0000			−0.739895
$412$		0			0
$413$	−2.59808	+	1.50000i	−0.127843	+	0.0738102i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			5.00000i			0.244851i
$418$		0			0
$419$	−4.50000	−	7.79423i	−0.219839	−	0.380773i	0.734919	−	0.678155i	$-0.237220\pi$
$419$	−4.50000	−	7.79423i	−0.219839	−	0.380773i	−0.954759	+	0.297382i	$0.903887\pi$
$420$		0			0
$421$	2.00000			0.0974740			0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000			0.0974740			0.0487370	+	0.998812i	$0.484480\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−9.52628	+	5.50000i	−0.461009	+	0.266164i
$428$		0			0
$429$	10.5000	+	2.59808i	0.506945	+	0.125436i
$430$		0			0
$431$	4.50000	+	7.79423i	0.216757	+	0.375435i	0.953815	−	0.300395i	$-0.0971186\pi$
$431$	4.50000	+	7.79423i	0.216757	+	0.375435i	−0.737057	+	0.675830i	$0.763785\pi$
$432$		0			0
$433$	25.1147	+	14.5000i	1.20694	+	0.696826i	0.962089	−	0.272736i	$-0.0879285\pi$
$433$	25.1147	+	14.5000i	1.20694	+	0.696826i	0.244848	+	0.969561i	$0.421262\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			21.0000i			1.00457i
$438$		0			0
$439$	5.50000	+	9.52628i	0.262501	+	0.454665i	0.966906	−	0.255134i	$-0.0821195\pi$
$439$	5.50000	+	9.52628i	0.262501	+	0.454665i	−0.704405	+	0.709798i	$0.748786\pi$
$440$		0			0
$441$	12.0000			0.571429
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	21.0000i		−	0.993266i
$448$		0			0
$449$	1.50000	−	2.59808i	0.0707894	−	0.122611i	−0.828458	−	0.560051i	$-0.810782\pi$
$449$	1.50000	−	2.59808i	0.0707894	−	0.122611i	0.899247	+	0.437440i	$0.144115\pi$
$450$		0			0
$451$	13.5000	−	23.3827i	0.635690	−	1.10105i
$452$		0			0
$453$	−6.92820	+	4.00000i	−0.325515	+	0.187936i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	4.33013	−	2.50000i	0.202555	−	0.116945i	−0.395292	−	0.918556i	$-0.629357\pi$
$457$	4.33013	−	2.50000i	0.202555	−	0.116945i	0.597847	+	0.801611i	$0.296023\pi$
$458$		0			0
$459$	7.50000	−	12.9904i	0.350070	−	0.606339i
$460$		0			0
$461$	−13.5000	+	23.3827i	−0.628758	+	1.08904i	0.359044	+	0.933321i	$0.383103\pi$
$461$	−13.5000	+	23.3827i	−0.628758	+	1.08904i	−0.987801	+	0.155719i	$0.950230\pi$
$462$		0			0
$463$			4.00000i			0.185896i	0.995671	+	0.0929479i	$0.0296290\pi$
$463$			4.00000i			0.185896i	−0.995671	+	0.0929479i	$0.970371\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	36.0000i		−	1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$		−	36.0000i		−	1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$		0			0
$469$	−7.00000			−0.323230
$470$		0			0
$471$	5.00000	+	8.66025i	0.230388	+	0.399043i
$472$		0			0
$473$		−	33.0000i		−	1.51734i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−10.3923	−	6.00000i	−0.475831	−	0.274721i
$478$		0			0
$479$	19.5000	+	33.7750i	0.890978	+	1.54322i	0.838705	+	0.544586i	$0.183313\pi$
$479$	19.5000	+	33.7750i	0.890978	+	1.54322i	0.0522726	+	0.998633i	$0.483354\pi$
$480$		0			0
$481$	−17.5000	+	18.1865i	−0.797931	+	0.829235i
$482$		0			0
$483$	−2.59808	+	1.50000i	−0.118217	+	0.0682524i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−9.52628	−	5.50000i	−0.431677	−	0.249229i	0.268384	−	0.963312i	$-0.413510\pi$
$487$	−9.52628	−	5.50000i	−0.431677	−	0.249229i	−0.700061	+	0.714083i	$0.746844\pi$
$488$		0			0
$489$	1.00000			0.0452216
$490$		0			0
$491$	4.50000	+	7.79423i	0.203082	+	0.351749i	0.949520	−	0.313707i	$-0.101571\pi$
$491$	4.50000	+	7.79423i	0.203082	+	0.351749i	−0.746438	+	0.665455i	$0.768237\pi$
$492$		0			0
$493$			9.00000i			0.405340i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.59808	−	1.50000i	0.116540	−	0.0672842i
$498$		0			0
$499$	−32.0000			−1.43252			−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000			−1.43252			−0.716258	+	0.697835i	$0.754147\pi$
$500$		0			0
$501$	1.50000	−	2.59808i	0.0670151	−	0.116073i
$502$		0			0
$503$	12.9904	+	7.50000i	0.579212	+	0.334408i	0.760820	−	0.648963i	$-0.224797\pi$
$503$	12.9904	+	7.50000i	0.579212	+	0.334408i	−0.181608	+	0.983371i	$0.558130\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−6.06218	+	11.5000i	−0.269231	+	0.510733i
$508$		0			0
$509$	−4.50000	−	7.79423i	−0.199459	−	0.345473i	0.748894	−	0.662690i	$-0.230585\pi$
$509$	−4.50000	−	7.79423i	−0.199459	−	0.345473i	−0.948353	+	0.317217i	$0.897252\pi$
$510$		0			0
$511$	1.00000	−	1.73205i	0.0442374	−	0.0766214i
$512$		0			0
$513$	30.3109	+	17.5000i	1.33826	+	0.772644i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	3.00000			0.131685
$520$		0			0
$521$	30.0000			1.31432			0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000			1.31432			0.657162	+	0.753749i	$0.271757\pi$
$522$		0			0
$523$	−25.1147	+	14.5000i	−1.09819	+	0.634041i	−0.935745	−	0.352677i	$-0.885272\pi$
$523$	−25.1147	+	14.5000i	−1.09819	+	0.634041i	−0.162446	+	0.986718i	$0.551938\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−10.3923	−	6.00000i	−0.452696	−	0.261364i
$528$		0			0
$529$	−7.00000	+	12.1244i	−0.304348	+	0.527146i
$530$		0			0
$531$	−3.00000	−	5.19615i	−0.130189	−	0.225494i
$532$		0			0
$533$	23.3827	+	22.5000i	1.01282	+	0.974583i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	18.1865	+	10.5000i	0.784807	+	0.453108i
$538$		0			0
$539$	−9.00000	+	15.5885i	−0.387657	+	0.671442i
$540$		0			0
$541$	−22.0000			−0.945854			−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000			−0.945854			−0.472927	+	0.881102i	$0.656803\pi$
$542$		0			0
$543$	−1.73205	+	1.00000i	−0.0743294	+	0.0429141i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			8.00000i			0.342055i	0.985266	+	0.171028i	$0.0547087\pi$
$547$			8.00000i			0.342055i	−0.985266	+	0.171028i	$0.945291\pi$
$548$		0			0
$549$	−11.0000	−	19.0526i	−0.469469	−	0.813143i
$550$		0			0
$551$	−21.0000			−0.894630
$552$		0			0
$553$	−6.92820	−	4.00000i	−0.294617	−	0.170097i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−33.7750	+	19.5000i	−1.43109	+	0.826242i	−0.997204	−	0.0747252i	$-0.976192\pi$
$557$	−33.7750	+	19.5000i	−1.43109	+	0.826242i	−0.433888	+	0.900967i	$0.642859\pi$
$558$		0			0
$559$	38.5000	+	9.52628i	1.62838	+	0.402919i
$560$		0			0
$561$	4.50000	+	7.79423i	0.189990	+	0.329073i
$562$		0			0
$563$	33.7750	+	19.5000i	1.42345	+	0.821827i	0.996592	−	0.0824933i	$-0.0262883\pi$
$563$	33.7750	+	19.5000i	1.42345	+	0.821827i	0.426855	+	0.904320i	$0.359622\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	13.5000	+	23.3827i	0.565949	+	0.980253i	0.996961	+	0.0779066i	$0.0248236\pi$
$569$	13.5000	+	23.3827i	0.565949	+	0.980253i	−0.431011	+	0.902347i	$0.641843\pi$
$570$		0			0
$571$	−40.0000			−1.67395			−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000			−1.67395			−0.836974	+	0.547243i	$0.815677\pi$
$572$		0			0
$573$			3.00000i			0.125327i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			2.00000i			0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$			2.00000i			0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$		0			0
$579$	2.50000	−	4.33013i	0.103896	−	0.179954i
$580$		0			0
$581$	−6.00000	+	10.3923i	−0.248922	+	0.431145i
$582$		0			0
$583$	15.5885	−	9.00000i	0.645608	−	0.372742i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	28.5788	−	16.5000i	1.17957	−	0.681028i	0.223659	−	0.974668i	$-0.428200\pi$
$587$	28.5788	−	16.5000i	1.17957	−	0.681028i	0.955916	+	0.293640i	$0.0948666\pi$
$588$		0			0
$589$	14.0000	−	24.2487i	0.576860	−	0.999151i
$590$		0			0
$591$	−10.5000	+	18.1865i	−0.431912	+	0.748094i
$592$		0			0
$593$			6.00000i			0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$			6.00000i			0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			17.0000i			0.695764i
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	−17.5000	−	30.3109i	−0.713840	−	1.23641i	−0.963405	−	0.268049i	$-0.913621\pi$
$601$	−17.5000	−	30.3109i	−0.713840	−	1.23641i	0.249565	−	0.968358i	$-0.419712\pi$
$602$		0			0
$603$		−	14.0000i		−	0.570124i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	11.2583	+	6.50000i	0.456962	+	0.263827i	0.710766	−	0.703429i	$-0.248349\pi$
$607$	11.2583	+	6.50000i	0.456962	+	0.263827i	−0.253804	+	0.967256i	$0.581682\pi$
$608$		0			0
$609$	−1.50000	−	2.59808i	−0.0607831	−	0.105279i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	37.2391	−	21.5000i	1.50407	−	0.868377i	0.504084	−	0.863655i	$-0.331830\pi$
$613$	37.2391	−	21.5000i	1.50407	−	0.868377i	0.999989	−	0.00472215i	$-0.00150311\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−28.5788	−	16.5000i	−1.15054	−	0.664265i	−0.201522	−	0.979484i	$-0.564589\pi$
$617$	−28.5788	−	16.5000i	−1.15054	−	0.664265i	−0.949019	+	0.315219i	$0.897922\pi$
$618$		0			0
$619$	−44.0000			−1.76851			−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000			−1.76851			−0.884255	+	0.467005i	$0.845333\pi$
$620$		0			0
$621$	−7.50000	−	12.9904i	−0.300965	−	0.521286i
$622$		0			0
$623$			15.0000i			0.600962i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−18.1865	+	10.5000i	−0.726300	+	0.419330i
$628$		0			0
$629$	−21.0000			−0.837325
$630$		0			0
$631$	−8.50000	+	14.7224i	−0.338380	+	0.586091i	−0.984128	−	0.177459i	$-0.943212\pi$
$631$	−8.50000	+	14.7224i	−0.338380	+	0.586091i	0.645748	+	0.763550i	$0.276545\pi$
$632$		0			0
$633$	9.52628	+	5.50000i	0.378636	+	0.218605i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−15.5885	−	15.0000i	−0.617637	−	0.594322i
$638$		0			0
$639$	3.00000	+	5.19615i	0.118678	+	0.205557i
$640$		0			0
$641$	−13.5000	+	23.3827i	−0.533218	+	0.923561i	0.466029	+	0.884769i	$0.345684\pi$
$641$	−13.5000	+	23.3827i	−0.533218	+	0.923561i	−0.999247	+	0.0387913i	$0.987649\pi$
$642$		0			0
$643$	9.52628	+	5.50000i	0.375680	+	0.216899i	0.675937	−	0.736959i	$-0.263739\pi$
$643$	9.52628	+	5.50000i	0.375680	+	0.216899i	−0.300257	+	0.953858i	$0.597072\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	38.9711	−	22.5000i	1.53211	−	0.884566i	0.532850	−	0.846210i	$-0.321121\pi$
$647$	38.9711	−	22.5000i	1.53211	−	0.884566i	0.999264	−	0.0383563i	$-0.0122122\pi$
$648$		0			0
$649$	9.00000			0.353281
$650$		0			0
$651$	4.00000			0.156772
$652$		0			0
$653$	33.7750	−	19.5000i	1.32172	−	0.763094i	0.337715	−	0.941248i	$-0.390346\pi$
$653$	33.7750	−	19.5000i	1.32172	−	0.763094i	0.984003	+	0.178154i	$0.0570127\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	3.46410	+	2.00000i	0.135147	+	0.0780274i
$658$		0			0
$659$	−19.5000	+	33.7750i	−0.759612	+	1.31569i	0.183436	+	0.983032i	$0.441278\pi$
$659$	−19.5000	+	33.7750i	−0.759612	+	1.31569i	−0.943049	+	0.332655i	$0.892055\pi$
$660$		0			0
$661$	0.500000	+	0.866025i	0.0194477	+	0.0336845i	0.875585	−	0.483063i	$-0.160476\pi$
$661$	0.500000	+	0.866025i	0.0194477	+	0.0336845i	−0.856138	+	0.516748i	$0.827143\pi$
$662$		0			0
$663$	−10.3923	+	3.00000i	−0.403604	+	0.116510i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	7.79423	+	4.50000i	0.301794	+	0.174241i
$668$		0			0
$669$	−9.50000	+	16.4545i	−0.367291	+	0.636167i
$670$		0			0
$671$	33.0000			1.27395
$672$		0			0
$673$	−14.7224	+	8.50000i	−0.567508	+	0.327651i	−0.756153	−	0.654394i	$-0.772924\pi$
$673$	−14.7224	+	8.50000i	−0.567508	+	0.327651i	0.188645	+	0.982045i	$0.439590\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	42.0000i		−	1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$		−	42.0000i		−	1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$		0			0
$679$	3.50000	+	6.06218i	0.134318	+	0.232645i
$680$		0			0
$681$	27.0000			1.03464
$682$		0			0
$683$	44.1673	+	25.5000i	1.69001	+	0.975730i	0.954495	+	0.298227i	$0.0963952\pi$
$683$	44.1673	+	25.5000i	1.69001	+	0.975730i	0.735520	+	0.677503i	$0.236938\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−19.0526	+	11.0000i	−0.726900	+	0.419676i
$688$		0			0
$689$	6.00000	+	20.7846i	0.228582	+	0.791831i
$690$		0			0
$691$	−11.5000	−	19.9186i	−0.437481	−	0.757739i	0.560014	−	0.828483i	$-0.310796\pi$
$691$	−11.5000	−	19.9186i	−0.437481	−	0.757739i	−0.997494	+	0.0707446i	$0.977462\pi$
$692$		0			0
$693$	5.19615	+	3.00000i	0.197386	+	0.113961i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			27.0000i			1.02270i
$698$		0			0
$699$	9.00000	+	15.5885i	0.340411	+	0.589610i
$700$		0			0
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$		−	49.0000i		−	1.84807i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			9.00000i			0.338480i
$708$		0			0
$709$	−18.5000	+	32.0429i	−0.694782	+	1.20340i	0.275472	+	0.961309i	$0.411166\pi$
$709$	−18.5000	+	32.0429i	−0.694782	+	1.20340i	−0.970254	+	0.242089i	$0.922167\pi$
$710$		0			0
$711$	8.00000	−	13.8564i	0.300023	−	0.519656i
$712$		0			0
$713$	−10.3923	+	6.00000i	−0.389195	+	0.224702i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	4.50000	−	7.79423i	0.167822	−	0.290676i	−0.769832	−	0.638247i	$-0.779660\pi$
$719$	4.50000	−	7.79423i	0.167822	−	0.290676i	0.937654	+	0.347571i	$0.112993\pi$
$720$		0			0
$721$	4.00000	−	6.92820i	0.148968	−	0.258020i
$722$		0			0
$723$			1.00000i			0.0371904i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	52.0000i		−	1.92857i	−0.264861	−	0.964287i	$-0.585326\pi$
$727$		−	52.0000i		−	1.92857i	0.264861	−	0.964287i	$-0.414674\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	16.5000	+	28.5788i	0.610275	+	1.05703i
$732$		0			0
$733$			34.0000i			1.25582i	0.778287	+	0.627909i	$0.216089\pi$
$733$			34.0000i			1.25582i	−0.778287	+	0.627909i	$0.783911\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	18.1865	+	10.5000i	0.669910	+	0.386772i
$738$		0			0
$739$	23.5000	+	40.7032i	0.864461	+	1.49729i	0.867581	+	0.497296i	$0.165674\pi$
$739$	23.5000	+	40.7032i	0.864461	+	1.49729i	−0.00311943	+	0.999995i	$0.500993\pi$
$740$		0			0
$741$	−7.00000	−	24.2487i	−0.257151	−	0.890799i
$742$		0			0
$743$	−18.1865	+	10.5000i	−0.667199	+	0.385208i	−0.795015	−	0.606590i	$-0.792537\pi$
$743$	−18.1865	+	10.5000i	−0.667199	+	0.385208i	0.127815	+	0.991798i	$0.459204\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−20.7846	−	12.0000i	−0.760469	−	0.439057i
$748$		0			0
$749$	9.00000			0.328853
$750$		0			0
$751$	6.50000	+	11.2583i	0.237188	+	0.410822i	0.959906	−	0.280321i	$-0.0904408\pi$
$751$	6.50000	+	11.2583i	0.237188	+	0.410822i	−0.722718	+	0.691143i	$0.757107\pi$
$752$		0			0
$753$		−	21.0000i		−	0.765283i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	25.1147	−	14.5000i	0.912811	−	0.527011i	0.0314762	−	0.999505i	$-0.489979\pi$
$757$	25.1147	−	14.5000i	0.912811	−	0.527011i	0.881334	+	0.472493i	$0.156646\pi$
$758$		0			0
$759$	9.00000			0.326679
$760$		0			0
$761$	−1.50000	+	2.59808i	−0.0543750	+	0.0941802i	−0.891932	−	0.452170i	$-0.850650\pi$
$761$	−1.50000	+	2.59808i	−0.0543750	+	0.0941802i	0.837557	+	0.546350i	$0.183983\pi$
$762$		0			0
$763$	−1.73205	−	1.00000i	−0.0627044	−	0.0362024i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2.59808	+	10.5000i	−0.0938111	+	0.379133i
$768$		0			0
$769$	−6.50000	−	11.2583i	−0.234396	−	0.405986i	0.724701	−	0.689063i	$-0.241978\pi$
$769$	−6.50000	−	11.2583i	−0.234396	−	0.405986i	−0.959097	+	0.283078i	$0.908645\pi$
$770$		0			0
$771$	−4.50000	+	7.79423i	−0.162064	+	0.280702i
$772$		0			0
$773$	−23.3827	−	13.5000i	−0.841017	−	0.485561i	0.0165929	−	0.999862i	$-0.494718\pi$
$773$	−23.3827	−	13.5000i	−0.841017	−	0.485561i	−0.857610	+	0.514301i	$0.828051\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	6.06218	−	3.50000i	0.217479	−	0.125562i
$778$		0			0
$779$	−63.0000			−2.25721
$780$		0			0
$781$	−9.00000			−0.322045
$782$		0			0
$783$	12.9904	−	7.50000i	0.464238	−	0.268028i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	32.0429	+	18.5000i	1.14221	+	0.659454i	0.946976	−	0.321303i	$-0.104121\pi$
$787$	32.0429	+	18.5000i	1.14221	+	0.659454i	0.195231	+	0.980757i	$0.437454\pi$
$788$		0			0
$789$	−1.50000	+	2.59808i	−0.0534014	+	0.0924940i
$790$		0			0
$791$	4.50000	+	7.79423i	0.160002	+	0.277131i
$792$		0			0
$793$	−9.52628	+	38.5000i	−0.338288	+	1.36718i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	44.1673	+	25.5000i	1.56449	+	0.903256i	0.996794	+	0.0800155i	$0.0254970\pi$
$797$	44.1673	+	25.5000i	1.56449	+	0.903256i	0.567692	+	0.823241i	$0.307836\pi$
$798$		0			0
$799$		0

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:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.bb (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 0.500000i) q^{3} +(0.866025 + 0.500000i) q^{7} +(-1.00000 + 1.73205i) q^{9} +O(q^{10})\)	\(q+(-0.866025 + 0.500000i) q^{3} +(0.866025 + 0.500000i) q^{7} +(-1.00000 + 1.73205i) q^{9} +(-1.50000 - 2.59808i) q^{11} +(3.46410 - 1.00000i) q^{13} +(2.59808 + 1.50000i) q^{17} +(-3.50000 + 6.06218i) q^{19} -1.00000 q^{21} +(2.59808 - 1.50000i) q^{23} -5.00000i q^{27} +(1.50000 + 2.59808i) q^{29} -4.00000 q^{31} +(2.59808 + 1.50000i) q^{33} +(-6.06218 + 3.50000i) q^{37} +(-2.50000 + 2.59808i) q^{39} +(4.50000 + 7.79423i) q^{41} +(9.52628 + 5.50000i) q^{43} +(-3.00000 - 5.19615i) q^{49} -3.00000 q^{51} +6.00000i q^{53} -7.00000i q^{57} +(-1.50000 + 2.59808i) q^{59} +(-5.50000 + 9.52628i) q^{61} +(-1.73205 + 1.00000i) q^{63} +(-6.06218 + 3.50000i) q^{67} +(-1.50000 + 2.59808i) q^{69} +(1.50000 - 2.59808i) q^{71} -2.00000i q^{73} -3.00000i q^{77} -8.00000 q^{79} +(-0.500000 - 0.866025i) q^{81} +12.0000i q^{83} +(-2.59808 - 1.50000i) q^{87} +(7.50000 + 12.9904i) q^{89} +(3.50000 + 0.866025i) q^{91} +(3.46410 - 2.00000i) q^{93} +(6.06218 + 3.50000i) q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} - 6 q^{11} - 14 q^{19} - 4 q^{21} + 6 q^{29} - 16 q^{31} - 10 q^{39} + 18 q^{41} - 12 q^{49} - 12 q^{51} - 6 q^{59} - 22 q^{61} - 6 q^{69} + 6 q^{71} - 32 q^{79} - 2 q^{81} + 30 q^{89} + 14 q^{91} + 24 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 - 6 * q^11 - 14 * q^19 - 4 * q^21 + 6 * q^29 - 16 * q^31 - 10 * q^39 + 18 * q^41 - 12 * q^49 - 12 * q^51 - 6 * q^59 - 22 * q^61 - 6 * q^69 + 6 * q^71 - 32 * q^79 - 2 * q^81 + 30 * q^89 + 14 * q^91 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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