LMFDB - Embedded newform 1620.3.o.g.1241.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(701,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 0]))

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

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

chi := DirichletCharacter("1620.701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.o (of order $6$, 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:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1241.1
Character	$\chi$	$=$	1620.1241
Dual form			1620.3.o.g.701.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.93649 + 1.11803i) q^{5} +(-2.26486 + 3.92285i) q^{7} +O(q^{10})$	$q+(-1.93649 + 1.11803i) q^{5} +(-2.26486 + 3.92285i) q^{7} +(12.9226 + 7.46089i) q^{11} +(-7.82703 - 13.5568i) q^{13} -7.53594i q^{17} +9.77167 q^{19} +(-9.38708 + 5.41963i) q^{23} +(2.50000 - 4.33013i) q^{25} +(3.94676 + 2.27866i) q^{29} +(24.4879 + 42.4143i) q^{31} -10.1288i q^{35} +21.4786 q^{37} +(45.6430 - 26.3520i) q^{41} +(-20.5817 + 35.6486i) q^{43} +(-32.2207 - 18.6026i) q^{47} +(14.2408 + 24.6659i) q^{49} -81.8456i q^{53} -33.3661 q^{55} +(-70.3732 + 40.6300i) q^{59} +(-24.5444 + 42.5121i) q^{61} +(30.3139 + 17.5018i) q^{65} +(23.2834 + 40.3280i) q^{67} +17.4203i q^{71} -101.619 q^{73} +(-58.5359 + 33.7957i) q^{77} +(-28.7070 + 49.7219i) q^{79} +(75.2273 + 43.4325i) q^{83} +(8.42543 + 14.5933i) q^{85} +41.3946i q^{89} +70.9084 q^{91} +(-18.9228 + 10.9251i) q^{95} +(-90.4805 + 156.717i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7}+O(q^{10}) $ 32 * q - 8 * q^7	$ 32 q - 8 q^{7} + 40 q^{13} + 112 q^{19} + 80 q^{25} + 64 q^{31} - 176 q^{37} - 128 q^{43} - 216 q^{49} - 8 q^{61} + 40 q^{67} + 112 q^{73} + 136 q^{79} - 784 q^{91} - 128 q^{97}+O(q^{100}) $ 32 * q - 8 * q^7 + 40 * q^13 + 112 * q^19 + 80 * q^25 + 64 * q^31 - 176 * q^37 - 128 * q^43 - 216 * q^49 - 8 * q^61 + 40 * q^67 + 112 * q^73 + 136 * q^79 - 784 * q^91 - 128 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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
$5$	−1.93649	+	1.11803i	−0.387298	+	0.223607i
$5$	−1.93649	+	1.11803i	−0.387298	+	0.223607i
$6$		0			0
$7$	−2.26486	+	3.92285i	−0.323551	+	0.560407i	−0.981218	−	0.192902i	$-0.938210\pi$
$7$	−2.26486	+	3.92285i	−0.323551	+	0.560407i	0.657667	+	0.753309i	$0.271543\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	12.9226	+	7.46089i	1.17479	+	0.678263i	0.954803	−	0.297241i	$-0.0960664\pi$
$11$	12.9226	+	7.46089i	1.17479	+	0.678263i	0.219983	+	0.975504i	$0.429400\pi$
$12$		0			0
$13$	−7.82703	−	13.5568i	−0.602079	−	1.04283i	−0.992506	−	0.122197i	$-0.961006\pi$
$13$	−7.82703	−	13.5568i	−0.602079	−	1.04283i	0.390427	−	0.920634i	$-0.372327\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	7.53594i		−	0.443290i	−0.975127	−	0.221645i	$-0.928857\pi$
$17$		−	7.53594i		−	0.443290i	0.975127	−	0.221645i	$-0.0711427\pi$
$18$		0			0
$19$	9.77167			0.514299			0.257149	−	0.966372i	$-0.417217\pi$
$19$	9.77167			0.514299			0.257149	+	0.966372i	$0.417217\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−9.38708	+	5.41963i	−0.408134	+	0.235636i	−0.689988	−	0.723821i	$-0.742384\pi$
$23$	−9.38708	+	5.41963i	−0.408134	+	0.235636i	0.281854	+	0.959457i	$0.409051\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.94676	+	2.27866i	0.136095	+	0.0785746i	0.566502	−	0.824061i	$-0.308296\pi$
$29$	3.94676	+	2.27866i	0.136095	+	0.0785746i	−0.430407	+	0.902635i	$0.641630\pi$
$30$		0			0
$31$	24.4879	+	42.4143i	0.789932	+	1.36820i	0.926008	+	0.377503i	$0.123217\pi$
$31$	24.4879	+	42.4143i	0.789932	+	1.36820i	−0.136077	+	0.990698i	$0.543449\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	10.1288i		−	0.289393i
$36$		0			0
$37$	21.4786			0.580502			0.290251	−	0.956951i	$-0.406261\pi$
$37$	21.4786			0.580502			0.290251	+	0.956951i	$0.406261\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	45.6430	−	26.3520i	1.11324	−	0.642731i	0.173576	−	0.984821i	$-0.444468\pi$
$41$	45.6430	−	26.3520i	1.11324	−	0.642731i	0.939667	+	0.342089i	$0.111135\pi$
$42$		0			0
$43$	−20.5817	+	35.6486i	−0.478644	+	0.829036i	−0.999700	−	0.0244863i	$-0.992205\pi$
$43$	−20.5817	+	35.6486i	−0.478644	+	0.829036i	0.521056	+	0.853523i	$0.325538\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−32.2207	−	18.6026i	−0.685547	−	0.395801i	0.116394	−	0.993203i	$-0.462866\pi$
$47$	−32.2207	−	18.6026i	−0.685547	−	0.395801i	−0.801942	+	0.597402i	$0.796200\pi$
$48$		0			0
$49$	14.2408	+	24.6659i	0.290629	+	0.503385i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	81.8456i		−	1.54426i	−0.635466	−	0.772129i	$-0.719192\pi$
$53$		−	81.8456i		−	1.54426i	0.635466	−	0.772129i	$-0.280808\pi$
$54$		0			0
$55$	−33.3661			−0.606657
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−70.3732	+	40.6300i	−1.19277	+	0.688644i	−0.958933	−	0.283632i	$-0.908460\pi$
$59$	−70.3732	+	40.6300i	−1.19277	+	0.688644i	−0.233834	+	0.972277i	$0.575127\pi$
$60$		0			0
$61$	−24.5444	+	42.5121i	−0.402367	+	0.696920i	−0.994011	−	0.109279i	$-0.965146\pi$
$61$	−24.5444	+	42.5121i	−0.402367	+	0.696920i	0.591644	+	0.806199i	$0.298479\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	30.3139	+	17.5018i	0.466368	+	0.269258i
$66$		0			0
$67$	23.2834	+	40.3280i	0.347513	+	0.601910i	0.985807	−	0.167882i	$-0.0536929\pi$
$67$	23.2834	+	40.3280i	0.347513	+	0.601910i	−0.638294	+	0.769793i	$0.720360\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			17.4203i			0.245356i	0.992446	+	0.122678i	$0.0391483\pi$
$71$			17.4203i			0.245356i	−0.992446	+	0.122678i	$0.960852\pi$
$72$		0			0
$73$	−101.619			−1.39205			−0.696023	−	0.718019i	$-0.745049\pi$
$73$	−101.619			−1.39205			−0.696023	+	0.718019i	$0.745049\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−58.5359	+	33.7957i	−0.760207	+	0.438905i
$78$		0			0
$79$	−28.7070	+	49.7219i	−0.363379	+	0.629391i	−0.988515	−	0.151125i	$-0.951710\pi$
$79$	−28.7070	+	49.7219i	−0.363379	+	0.629391i	0.625135	+	0.780516i	$0.285044\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	75.2273	+	43.4325i	0.906353	+	0.523283i	0.879256	−	0.476350i	$-0.158040\pi$
$83$	75.2273	+	43.4325i	0.906353	+	0.523283i	0.0270968	+	0.999633i	$0.491374\pi$
$84$		0			0
$85$	8.42543	+	14.5933i	0.0991227	+	0.171686i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			41.3946i			0.465108i	0.972584	+	0.232554i	$0.0747082\pi$
$89$			41.3946i			0.465108i	−0.972584	+	0.232554i	$0.925292\pi$
$90$		0			0
$91$	70.9084			0.779213
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−18.9228	+	10.9251i	−0.199187	+	0.115001i
$96$		0			0
$97$	−90.4805	+	156.717i	−0.932789	+	1.61564i	−0.154259	+	0.988031i	$0.549299\pi$
$97$	−90.4805	+	156.717i	−0.932789	+	1.61564i	−0.778530	+	0.627607i	$0.784034\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	55.2853	+	31.9190i	0.547379	+	0.316029i	0.748064	−	0.663626i	$-0.230983\pi$
$101$	55.2853	+	31.9190i	0.547379	+	0.316029i	−0.200685	+	0.979656i	$0.564317\pi$
$102$		0			0
$103$	13.6921	+	23.7155i	0.132933	+	0.230247i	0.924806	−	0.380439i	$-0.124227\pi$
$103$	13.6921	+	23.7155i	0.132933	+	0.230247i	−0.791873	+	0.610686i	$0.790894\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	3.87078i		−	0.0361755i	−0.999836	−	0.0180878i	$-0.994242\pi$
$107$		−	3.87078i		−	0.0361755i	0.999836	−	0.0180878i	$-0.00575783\pi$
$108$		0			0
$109$	−65.8864			−0.604462			−0.302231	−	0.953235i	$-0.597731\pi$
$109$	−65.8864			−0.604462			−0.302231	+	0.953235i	$0.597731\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−18.3675	+	10.6045i	−0.162544	+	0.0938450i	−0.579066	−	0.815281i	$-0.696582\pi$
$113$	−18.3675	+	10.6045i	−0.162544	+	0.0938450i	0.416521	+	0.909126i	$0.363249\pi$
$114$		0			0
$115$	12.1187	−	20.9902i	0.105380	−	0.182523i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	29.5623	+	17.0678i	0.248423	+	0.143427i
$120$		0			0
$121$	50.8298	+	88.0398i	0.420081	+	0.727602i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	46.5874			0.366830			0.183415	−	0.983036i	$-0.441285\pi$
$127$	46.5874			0.366830			0.183415	+	0.983036i	$0.441285\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−120.460	+	69.5474i	−0.919539	+	0.530896i	−0.883488	−	0.468454i	$-0.844811\pi$
$131$	−120.460	+	69.5474i	−0.919539	+	0.530896i	−0.0360513	+	0.999350i	$0.511478\pi$
$132$		0			0
$133$	−22.1315	+	38.3328i	−0.166402	+	0.288217i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	184.463	+	106.500i	1.34645	+	0.777371i	0.987744	−	0.156081i	$-0.0498862\pi$
$137$	184.463	+	106.500i	1.34645	+	0.777371i	0.358702	+	0.933452i	$0.383220\pi$
$138$		0			0
$139$	126.328	+	218.807i	0.908835	+	1.57415i	0.815686	+	0.578495i	$0.196360\pi$
$139$	126.328	+	218.807i	0.908835	+	1.57415i	0.0931489	+	0.995652i	$0.470307\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	233.586i		−	1.63347i
$144$		0			0
$145$	−10.1905			−0.0702793
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−77.7390	+	44.8827i	−0.521738	+	0.301226i	−0.737646	−	0.675188i	$-0.764062\pi$
$149$	−77.7390	+	44.8827i	−0.521738	+	0.301226i	0.215907	+	0.976414i	$0.430729\pi$
$150$		0			0
$151$	−94.4037	+	163.512i	−0.625190	+	1.08286i	0.363314	+	0.931667i	$0.381645\pi$
$151$	−94.4037	+	163.512i	−0.625190	+	1.08286i	−0.988504	+	0.151194i	$0.951688\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−94.8411	−	54.7566i	−0.611878	−	0.353268i
$156$		0			0
$157$	65.9966	+	114.309i	0.420361	+	0.728086i	0.995975	−	0.0896356i	$-0.0285703\pi$
$157$	65.9966	+	114.309i	0.420361	+	0.728086i	−0.575614	+	0.817722i	$0.695237\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	49.0988i		−	0.304962i
$162$		0			0
$163$	173.331			1.06338			0.531691	−	0.846939i	$-0.321557\pi$
$163$	173.331			1.06338			0.531691	+	0.846939i	$0.321557\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	22.6383	−	13.0702i	0.135559	−	0.0782648i	−0.430687	−	0.902501i	$-0.641729\pi$
$167$	22.6383	−	13.0702i	0.135559	−	0.0782648i	0.566246	+	0.824237i	$0.308395\pi$
$168$		0			0
$169$	−38.0247	+	65.8607i	−0.224998	+	0.389709i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−203.324	−	117.389i	−1.17529	−	0.678551i	−0.220366	−	0.975417i	$-0.570725\pi$
$173$	−203.324	−	117.389i	−1.17529	−	0.678551i	−0.954919	+	0.296866i	$0.904059\pi$
$174$		0			0
$175$	11.3243	+	19.6142i	0.0647102	+	0.112081i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	170.441i		−	0.952184i	−0.879395	−	0.476092i	$-0.842053\pi$
$179$		−	170.441i		−	0.952184i	0.879395	−	0.476092i	$-0.157947\pi$
$180$		0			0
$181$	−21.4034			−0.118251			−0.0591254	−	0.998251i	$-0.518831\pi$
$181$	−21.4034			−0.118251			−0.0591254	+	0.998251i	$0.518831\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−41.5931	+	24.0138i	−0.224827	+	0.129804i
$186$		0			0
$187$	56.2248	−	97.3842i	0.300667	−	0.520771i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	160.046	+	92.4025i	0.837936	+	0.483783i	0.856562	−	0.516044i	$-0.172596\pi$
$191$	160.046	+	92.4025i	0.837936	+	0.483783i	−0.0186261	+	0.999827i	$0.505929\pi$
$192$		0			0
$193$	190.160	+	329.367i	0.985286	+	1.70657i	0.640658	+	0.767826i	$0.278661\pi$
$193$	190.160	+	329.367i	0.985286	+	1.70657i	0.344628	+	0.938740i	$0.388005\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	79.3079i		−	0.402578i	−0.979532	−	0.201289i	$-0.935487\pi$
$197$		−	79.3079i		−	0.402578i	0.979532	−	0.201289i	$-0.0645131\pi$
$198$		0			0
$199$	−87.2272			−0.438328			−0.219164	−	0.975688i	$-0.570333\pi$
$199$	−87.2272			−0.438328			−0.219164	+	0.975688i	$0.570333\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−17.8777	+	10.3217i	−0.0880675	+	0.0508458i
$204$		0			0
$205$	−58.9248	+	102.061i	−0.287438	+	0.497857i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	126.276	+	72.9054i	0.604191	+	0.348830i
$210$		0			0
$211$	−168.236	−	291.393i	−0.797327	−	1.38101i	−0.921351	−	0.388732i	$-0.872913\pi$
$211$	−168.236	−	291.393i	−0.797327	−	1.38101i	0.124024	−	0.992279i	$-0.460420\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	92.0442i		−	0.428113i
$216$		0			0
$217$	−221.846			−1.02233
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−102.163	+	58.9840i	−0.462277	+	0.266896i
$222$		0			0
$223$	−41.8581	+	72.5004i	−0.187705	+	0.325114i	−0.944485	−	0.328556i	$-0.893438\pi$
$223$	−41.8581	+	72.5004i	−0.187705	+	0.325114i	0.756780	+	0.653670i	$0.226771\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	271.315	+	156.644i	1.19522	+	0.690060i	0.959486	−	0.281757i	$-0.0909173\pi$
$227$	271.315	+	156.644i	1.19522	+	0.690060i	0.235734	+	0.971818i	$0.424251\pi$
$228$		0			0
$229$	−112.802	−	195.379i	−0.492586	−	0.853184i	0.507377	−	0.861724i	$-0.330615\pi$
$229$	−112.802	−	195.379i	−0.492586	−	0.853184i	−0.999964	+	0.00853978i	$0.997282\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	62.6406i		−	0.268844i	−0.990924	−	0.134422i	$-0.957082\pi$
$233$		−	62.6406i		−	0.268844i	0.990924	−	0.134422i	$-0.0429177\pi$
$234$		0			0
$235$	83.1936			0.354015
$236$		0			0
$237$		0			0
$238$		0			0
$239$	91.1510	−	52.6260i	0.381385	−	0.220193i	−0.297036	−	0.954866i	$-0.595998\pi$
$239$	91.1510	−	52.6260i	0.381385	−	0.220193i	0.678421	+	0.734674i	$0.262665\pi$
$240$		0			0
$241$	−135.295	+	234.337i	−0.561389	+	0.972354i	0.435987	+	0.899953i	$0.356399\pi$
$241$	−135.295	+	234.337i	−0.561389	+	0.972354i	−0.997376	+	0.0724007i	$0.976934\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−55.1545	−	31.8435i	−0.225121	−	0.129973i
$246$		0			0
$247$	−76.4832	−	132.473i	−0.309648	−	0.536327i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	4.05439i		−	0.0161529i	−0.999967	−	0.00807647i	$-0.997429\pi$
$251$		−	4.05439i		−	0.0161529i	0.999967	−	0.00807647i	$-0.00257085\pi$
$252$		0			0
$253$	−161.741			−0.639293
$254$		0			0
$255$		0			0
$256$		0			0
$257$	33.6710	−	19.4400i	0.131016	−	0.0756418i	−0.433060	−	0.901365i	$-0.642566\pi$
$257$	33.6710	−	19.4400i	0.131016	−	0.0756418i	0.564075	+	0.825723i	$0.309233\pi$
$258$		0			0
$259$	−48.6459	+	84.2572i	−0.187822	+	0.325317i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−113.246	−	65.3825i	−0.430593	−	0.248603i	0.269006	−	0.963138i	$-0.413305\pi$
$263$	−113.246	−	65.3825i	−0.430593	−	0.248603i	−0.699599	+	0.714536i	$0.746638\pi$
$264$		0			0
$265$	91.5062	+	158.493i	0.345306	+	0.598088i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			327.701i			1.21822i	0.793086	+	0.609109i	$0.208473\pi$
$269$			327.701i			1.21822i	−0.793086	+	0.609109i	$0.791527\pi$
$270$		0			0
$271$	151.222			0.558016			0.279008	−	0.960289i	$-0.409994\pi$
$271$	151.222			0.558016			0.279008	+	0.960289i	$0.409994\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	64.6132	−	37.3045i	0.234957	−	0.135653i
$276$		0			0
$277$	−182.096	+	315.400i	−0.657388	+	1.13863i	0.323902	+	0.946091i	$0.395005\pi$
$277$	−182.096	+	315.400i	−0.657388	+	1.13863i	−0.981289	+	0.192538i	$0.938328\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	103.822	+	59.9418i	0.369474	+	0.213316i	0.673229	−	0.739434i	$-0.264907\pi$
$281$	103.822	+	59.9418i	0.369474	+	0.213316i	−0.303755	+	0.952750i	$0.598240\pi$
$282$		0			0
$283$	210.787	+	365.094i	0.744831	+	1.29008i	0.950274	+	0.311416i	$0.100803\pi$
$283$	210.787	+	365.094i	0.744831	+	1.29008i	−0.205443	+	0.978669i	$0.565864\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			238.734i			0.831826i
$288$		0			0
$289$	232.210			0.803494
$290$		0			0
$291$		0			0
$292$		0			0
$293$	283.065	−	163.428i	0.966093	−	0.557774i	0.0680502	−	0.997682i	$-0.478322\pi$
$293$	283.065	−	163.428i	0.966093	−	0.557774i	0.898043	+	0.439908i	$0.144989\pi$
$294$		0			0
$295$	90.8515	−	157.359i	0.307971	−	0.533421i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	146.946	+	84.8393i	0.491458	+	0.283743i
$300$		0			0
$301$	−93.2293	−	161.478i	−0.309732	−	0.536471i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	109.766i		−	0.359888i
$306$		0			0
$307$	157.856			0.514190			0.257095	−	0.966386i	$-0.417235\pi$
$307$	157.856			0.514190			0.257095	+	0.966386i	$0.417235\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−380.038	+	219.415i	−1.22199	+	0.705514i	−0.965341	−	0.260992i	$-0.915950\pi$
$311$	−380.038	+	219.415i	−1.22199	+	0.705514i	−0.256645	+	0.966506i	$0.582617\pi$
$312$		0			0
$313$	203.852	−	353.082i	0.651284	−	1.12806i	−0.331528	−	0.943446i	$-0.607564\pi$
$313$	203.852	−	353.082i	0.651284	−	1.12806i	0.982812	−	0.184611i	$-0.0591026\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−380.871	−	219.896i	−1.20149	−	0.693678i	−0.240600	−	0.970624i	$-0.577344\pi$
$317$	−380.871	−	219.896i	−1.20149	−	0.693678i	−0.960885	+	0.276946i	$0.910678\pi$
$318$		0			0
$319$	34.0017	+	58.8927i	0.106588	+	0.184617i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	73.6387i		−	0.227984i
$324$		0			0
$325$	−78.2703			−0.240832
$326$		0			0
$327$		0			0
$328$		0			0
$329$	145.951	−	84.2647i	0.443619	−	0.256124i
$330$		0			0
$331$	0.744429	−	1.28939i	0.00224903	−	0.00389543i	−0.864899	−	0.501946i	$-0.832617\pi$
$331$	0.744429	−	1.28939i	0.00224903	−	0.00389543i	0.867148	+	0.498051i	$0.165951\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−90.1761	−	52.0632i	−0.269183	−	0.155413i
$336$		0			0
$337$	−62.7444	−	108.677i	−0.186185	−	0.322482i	0.757790	−	0.652499i	$-0.226279\pi$
$337$	−62.7444	−	108.677i	−0.186185	−	0.322482i	−0.943975	+	0.330016i	$0.892946\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			730.806i			2.14313i
$342$		0			0
$343$	−350.970			−1.02324
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−171.620	+	99.0848i	−0.494582	+	0.285547i	−0.726473	−	0.687195i	$-0.758842\pi$
$347$	−171.620	+	99.0848i	−0.494582	+	0.285547i	0.231892	+	0.972742i	$0.425509\pi$
$348$		0			0
$349$	−33.8274	+	58.5908i	−0.0969267	+	0.167882i	−0.910411	−	0.413705i	$-0.864235\pi$
$349$	−33.8274	+	58.5908i	−0.0969267	+	0.167882i	0.813484	+	0.581587i	$0.197568\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−495.143	−	285.871i	−1.40267	−	0.809833i	−0.408005	−	0.912980i	$-0.633776\pi$
$353$	−495.143	−	285.871i	−1.40267	−	0.809833i	−0.994666	+	0.103147i	$0.967109\pi$
$354$		0			0
$355$	−19.4765	−	33.7343i	−0.0548634	−	0.0950262i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	648.758i		−	1.80713i	−0.428455	−	0.903563i	$-0.640942\pi$
$359$		−	648.758i		−	1.80713i	0.428455	−	0.903563i	$-0.359058\pi$
$360$		0			0
$361$	−265.514			−0.735497
$362$		0			0
$363$		0			0
$364$		0			0
$365$	196.785	−	113.614i	0.539137	−	0.311271i
$366$		0			0
$367$	256.297	−	443.919i	0.698356	−	1.20959i	−0.270680	−	0.962669i	$-0.587249\pi$
$367$	256.297	−	443.919i	0.698356	−	1.20959i	0.969036	−	0.246919i	$-0.0794181\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	321.068	+	185.369i	0.865413	+	0.499646i
$372$		0			0
$373$	−93.2229	−	161.467i	−0.249927	−	0.432887i	0.713578	−	0.700576i	$-0.247073\pi$
$373$	−93.2229	−	161.467i	−0.249927	−	0.432887i	−0.963505	+	0.267689i	$0.913740\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	71.3406i		−	0.189232i
$378$		0			0
$379$	257.240			0.678735			0.339367	−	0.940654i	$-0.389787\pi$
$379$	257.240			0.678735			0.339367	+	0.940654i	$0.389787\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−226.437	+	130.733i	−0.591219	+	0.341341i	−0.765579	−	0.643341i	$-0.777548\pi$
$383$	−226.437	+	130.733i	−0.591219	+	0.341341i	0.174360	+	0.984682i	$0.444214\pi$
$384$		0			0
$385$	75.5695	−	130.890i	0.196284	−	0.339975i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−350.519	−	202.372i	−0.901077	−	0.520237i	−0.0235274	−	0.999723i	$-0.507490\pi$
$389$	−350.519	−	202.372i	−0.901077	−	0.520237i	−0.877549	+	0.479486i	$0.840823\pi$
$390$		0			0
$391$	40.8420	+	70.7405i	0.104455	+	0.180922i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	128.381i		−	0.325016i
$396$		0			0
$397$	289.798			0.729970			0.364985	−	0.931013i	$-0.381074\pi$
$397$	289.798			0.729970			0.364985	+	0.931013i	$0.381074\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	666.871	−	385.018i	1.66302	−	0.960146i	0.691760	−	0.722127i	$-0.256835\pi$
$401$	666.871	−	385.018i	1.66302	−	0.960146i	0.971261	−	0.238019i	$-0.0764979\pi$
$402$		0			0
$403$	383.335	−	663.955i	0.951202	−	1.64753i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	277.560	+	160.249i	0.681966	+	0.393733i
$408$		0			0
$409$	−253.671	−	439.371i	−0.620222	−	1.07426i	−0.989444	−	0.144915i	$-0.953709\pi$
$409$	−253.671	−	439.371i	−0.620222	−	1.07426i	0.369222	−	0.929341i	$-0.379624\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	368.085i		−	0.891246i
$414$		0			0
$415$	−194.236			−0.468038
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−189.672	+	109.507i	−0.452677	+	0.261353i	−0.708960	−	0.705249i	$-0.750835\pi$
$419$	−189.672	+	109.507i	−0.452677	+	0.261353i	0.256283	+	0.966602i	$0.417502\pi$
$420$		0			0
$421$	181.027	−	313.547i	0.429992	−	0.744768i	−0.566880	−	0.823800i	$-0.691850\pi$
$421$	181.027	−	313.547i	0.429992	−	0.744768i	0.996872	+	0.0790326i	$0.0251831\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−32.6316	−	18.8398i	−0.0767801	−	0.0443290i
$426$		0			0
$427$	−111.179	−	192.568i	−0.260372	−	0.450978i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			619.958i			1.43842i	0.694794	+	0.719209i	$0.255496\pi$
$431$			619.958i			1.43842i	−0.694794	+	0.719209i	$0.744504\pi$
$432$		0			0
$433$	−506.921			−1.17072			−0.585359	−	0.810774i	$-0.699047\pi$
$433$	−506.921			−1.17072			−0.585359	+	0.810774i	$0.699047\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−91.7275	+	52.9589i	−0.209903	+	0.121187i
$438$		0			0
$439$	287.794	−	498.475i	0.655568	−	1.13548i	−0.326183	−	0.945307i	$-0.605762\pi$
$439$	287.794	−	498.475i	0.655568	−	1.13548i	0.981751	−	0.190171i	$-0.0609042\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−419.097	−	241.966i	−0.946042	−	0.546198i	−0.0541930	−	0.998530i	$-0.517259\pi$
$443$	−419.097	−	241.966i	−0.946042	−	0.546198i	−0.891849	+	0.452333i	$0.850592\pi$
$444$		0			0
$445$	−46.2805	−	80.1602i	−0.104001	−	0.180135i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			560.721i			1.24882i	0.781096	+	0.624411i	$0.214661\pi$
$449$			560.721i			1.24882i	−0.781096	+	0.624411i	$0.785339\pi$
$450$		0			0
$451$	786.437			1.74376
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−137.314	+	79.2780i	−0.301788	+	0.174237i
$456$		0			0
$457$	−254.138	+	440.180i	−0.556101	+	0.963195i	0.441716	+	0.897155i	$0.354370\pi$
$457$	−254.138	+	440.180i	−0.556101	+	0.963195i	−0.997817	+	0.0660398i	$0.978964\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	508.162	+	293.388i	1.10230	+	0.636416i	0.936825	−	0.349797i	$-0.113750\pi$
$461$	508.162	+	293.388i	1.10230	+	0.636416i	0.165479	+	0.986213i	$0.447083\pi$
$462$		0			0
$463$	−83.9079	−	145.333i	−0.181226	−	0.313893i	0.761072	−	0.648667i	$-0.224673\pi$
$463$	−83.9079	−	145.333i	−0.181226	−	0.313893i	−0.942298	+	0.334774i	$0.891340\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			896.380i			1.91944i	0.280953	+	0.959721i	$0.409349\pi$
$467$			896.380i			1.91944i	−0.280953	+	0.959721i	$0.590651\pi$
$468$		0			0
$469$	−210.934			−0.449753
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−531.940	+	307.116i	−1.12461	+	0.649293i
$474$		0			0
$475$	24.4292	−	42.3126i	0.0514299	−	0.0890791i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	42.4303	+	24.4971i	0.0885810	+	0.0511423i	0.543636	−	0.839321i	$-0.317047\pi$
$479$	42.4303	+	24.4971i	0.0885810	+	0.0511423i	−0.455055	+	0.890463i	$0.650380\pi$
$480$		0			0
$481$	−168.113	−	291.181i	−0.349508	−	0.605366i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	404.641i		−	0.834312i
$486$		0			0
$487$	338.170			0.694393			0.347197	−	0.937792i	$-0.387134\pi$
$487$	338.170			0.694393			0.347197	+	0.937792i	$0.387134\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	645.469	−	372.662i	1.31460	−	0.758986i	0.331747	−	0.943368i	$-0.392362\pi$
$491$	645.469	−	372.662i	1.31460	−	0.758986i	0.982855	+	0.184383i	$0.0590286\pi$
$492$		0			0
$493$	17.1719	−	29.7425i	0.0348314	−	0.0603297i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−68.3372	−	39.4545i	−0.137499	−	0.0793854i
$498$		0			0
$499$	−224.102	−	388.157i	−0.449103	−	0.777869i	0.549225	−	0.835675i	$-0.314923\pi$
$499$	−224.102	−	388.157i	−0.449103	−	0.777869i	−0.998328	+	0.0578054i	$0.981590\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	869.698i		−	1.72902i	−0.502615	−	0.864511i	$-0.667629\pi$
$503$		−	869.698i		−	1.72902i	0.502615	−	0.864511i	$-0.332371\pi$
$504$		0			0
$505$	−142.746			−0.282665
$506$		0			0
$507$		0			0
$508$		0			0
$509$	627.704	−	362.405i	1.23321	−	0.711994i	0.265513	−	0.964107i	$-0.414459\pi$
$509$	627.704	−	362.405i	1.23321	−	0.711994i	0.967698	+	0.252113i	$0.0811255\pi$
$510$		0			0
$511$	230.153	−	398.638i	0.450398	−	0.780113i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−53.0295	−	30.6166i	−0.102970	−	0.0594496i
$516$		0			0
$517$	−277.585	−	480.791i	−0.536914	−	0.929963i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			847.907i			1.62746i	0.581242	+	0.813731i	$0.302567\pi$
$521$			847.907i			1.62746i	−0.581242	+	0.813731i	$0.697433\pi$
$522$		0			0
$523$	−208.461			−0.398588			−0.199294	−	0.979940i	$-0.563865\pi$
$523$	−208.461			−0.398588			−0.199294	+	0.979940i	$0.563865\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	319.631	−	184.539i	0.606511	−	0.350169i
$528$		0			0
$529$	−205.755	+	356.378i	−0.388951	+	0.673683i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−714.498	−	412.515i	−1.34052	−	0.773950i
$534$		0			0
$535$	4.32767	+	7.49574i	0.00808910	+	0.0140107i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			424.997i			0.788492i
$540$		0			0
$541$	427.550			0.790295			0.395147	−	0.918618i	$-0.370694\pi$
$541$	427.550			0.790295			0.395147	+	0.918618i	$0.370694\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	127.588	−	73.6632i	0.234107	−	0.135162i
$546$		0			0
$547$	165.569	−	286.774i	0.302685	−	0.524266i	−0.674058	−	0.738678i	$-0.735450\pi$
$547$	165.569	−	286.774i	0.302685	−	0.524266i	0.976743	+	0.214412i	$0.0687836\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	38.5665	+	22.2664i	0.0699936	+	0.0404108i
$552$		0			0
$553$	−130.034	−	225.226i	−0.235144	−	0.407281i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			889.916i			1.59769i	0.601534	+	0.798847i	$0.294556\pi$
$557$			889.916i			1.59769i	−0.601534	+	0.798847i	$0.705444\pi$
$558$		0			0
$559$	644.374			1.15273
$560$		0			0
$561$		0			0
$562$		0			0
$563$	501.920	−	289.784i	0.891511	−	0.514714i	0.0170743	−	0.999854i	$-0.494565\pi$
$563$	501.920	−	289.784i	0.891511	−	0.514714i	0.874436	+	0.485140i	$0.161231\pi$
$564$		0			0
$565$	23.7124	−	41.0710i	0.0419688	−	0.0726920i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−428.063	−	247.142i	−0.752307	−	0.434345i	0.0742198	−	0.997242i	$-0.476353\pi$
$569$	−428.063	−	247.142i	−0.752307	−	0.434345i	−0.826527	+	0.562897i	$0.809687\pi$
$570$		0			0
$571$	−164.165	−	284.342i	−0.287504	−	0.497972i	0.685709	−	0.727875i	$-0.259492\pi$
$571$	−164.165	−	284.342i	−0.287504	−	0.497972i	−0.973213	+	0.229904i	$0.926159\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			54.1963i			0.0942545i
$576$		0			0
$577$	271.036			0.469734			0.234867	−	0.972028i	$-0.424535\pi$
$577$	271.036			0.469734			0.234867	+	0.972028i	$0.424535\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−340.758	+	196.737i	−0.586503	+	0.338618i
$582$		0			0
$583$	610.642	−	1057.66i	1.04741	−	1.81417i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−376.662	−	217.466i	−0.641673	−	0.370470i	0.143585	−	0.989638i	$-0.454137\pi$
$587$	−376.662	−	217.466i	−0.641673	−	0.370470i	−0.785259	+	0.619168i	$0.787470\pi$
$588$		0			0
$589$	239.288	+	414.458i	0.406261	+	0.703664i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	735.060i		−	1.23956i	−0.784775	−	0.619780i	$-0.787222\pi$
$593$		−	735.060i		−	1.23956i	0.784775	−	0.619780i	$-0.212778\pi$
$594$		0			0
$595$	−76.3296			−0.128285
$596$		0			0
$597$		0			0
$598$		0			0
$599$	438.530	−	253.185i	0.732103	−	0.422680i	−0.0870882	−	0.996201i	$-0.527756\pi$
$599$	438.530	−	253.185i	0.732103	−	0.422680i	0.819191	+	0.573521i	$0.194423\pi$
$600$		0			0
$601$	−432.447	+	749.020i	−0.719546	+	1.24629i	0.241634	+	0.970367i	$0.422317\pi$
$601$	−432.447	+	749.020i	−0.719546	+	1.24629i	−0.961180	+	0.275922i	$0.911017\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−196.863	−	113.659i	−0.325393	−	0.187866i
$606$		0			0
$607$	−141.662	−	245.365i	−0.233380	−	0.404226i	0.725421	−	0.688306i	$-0.241645\pi$
$607$	−141.662	−	245.365i	−0.233380	−	0.404226i	−0.958801	+	0.284080i	$0.908312\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			582.414i			0.953214i
$612$		0			0
$613$	−1170.79			−1.90994			−0.954971	−	0.296698i	$-0.904114\pi$
$613$	−1170.79			−1.90994			−0.954971	+	0.296698i	$0.904114\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	543.687	−	313.898i	0.881178	−	0.508748i	0.0101314	−	0.999949i	$-0.496775\pi$
$617$	543.687	−	313.898i	0.881178	−	0.508748i	0.871047	+	0.491200i	$0.163442\pi$
$618$		0			0
$619$	130.204	−	225.520i	0.210345	−	0.364329i	−0.741477	−	0.670978i	$-0.765874\pi$
$619$	130.204	−	225.520i	0.210345	−	0.364329i	0.951823	+	0.306649i	$0.0992078\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−162.385	−	93.7528i	−0.260650	−	0.150486i
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	161.861i		−	0.257331i
$630$		0			0
$631$	604.324			0.957723			0.478862	−	0.877890i	$-0.341050\pi$
$631$	604.324			0.957723			0.478862	+	0.877890i	$0.341050\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−90.2162	+	52.0863i	−0.142073	+	0.0820257i
$636$		0			0
$637$	222.927	−	386.121i	0.349964	−	0.606155i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−237.935	−	137.372i	−0.371193	−	0.214309i	0.302786	−	0.953058i	$-0.402083\pi$
$641$	−237.935	−	137.372i	−0.371193	−	0.214309i	−0.673980	+	0.738750i	$0.735417\pi$
$642$		0			0
$643$	66.9700	+	115.995i	0.104152	+	0.180397i	0.913392	−	0.407082i	$-0.133454\pi$
$643$	66.9700	+	115.995i	0.104152	+	0.180397i	−0.809239	+	0.587479i	$0.800120\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	918.547i		−	1.41970i	−0.704352	−	0.709851i	$-0.748762\pi$
$647$		−	918.547i		−	1.41970i	0.704352	−	0.709851i	$-0.251238\pi$
$648$		0			0
$649$	−1212.54			−1.86833
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1014.31	−	585.613i	1.55331	−	0.896804i	0.555441	−	0.831556i	$-0.312549\pi$
$653$	1014.31	−	585.613i	1.55331	−	0.896804i	0.997869	−	0.0652482i	$-0.0207839\pi$
$654$		0			0
$655$	155.513	−	269.356i	0.237424	−	0.411230i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	332.558	+	192.002i	0.504640	+	0.291354i	0.730628	−	0.682776i	$-0.239228\pi$
$659$	332.558	+	192.002i	0.504640	+	0.291354i	−0.225987	+	0.974130i	$0.572561\pi$
$660$		0			0
$661$	−252.259	−	436.925i	−0.381632	−	0.661006i	0.609664	−	0.792660i	$-0.291304\pi$
$661$	−252.259	−	436.925i	−0.381632	−	0.661006i	−0.991296	+	0.131654i	$0.957971\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	98.9749i		−	0.148834i
$666$		0			0
$667$	−49.3981			−0.0740601
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−634.356	+	366.246i	−0.945390	+	0.545821i
$672$		0			0
$673$	303.578	−	525.813i	0.451082	−	0.781298i	−0.547371	−	0.836890i	$-0.684371\pi$
$673$	303.578	−	525.813i	0.451082	−	0.781298i	0.998454	+	0.0555924i	$0.0177047\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	701.540	+	405.035i	1.03625	+	0.598278i	0.918768	−	0.394797i	$-0.129185\pi$
$677$	701.540	+	405.035i	1.03625	+	0.598278i	0.117480	+	0.993075i	$0.462518\pi$
$678$		0			0
$679$	−409.851	−	709.883i	−0.603610	−	1.04548i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1095.97i		−	1.60465i	−0.596889	−	0.802324i	$-0.703597\pi$
$683$		−	1095.97i		−	1.60465i	0.596889	−	0.802324i	$-0.296403\pi$
$684$		0			0
$685$	−476.282			−0.695302
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−1109.57	+	640.608i	−1.61040	+	0.929765i
$690$		0			0
$691$	−495.310	+	857.902i	−0.716802	+	1.24154i	0.245458	+	0.969407i	$0.421062\pi$
$691$	−495.310	+	857.902i	−0.716802	+	1.24154i	−0.962260	+	0.272130i	$0.912272\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−489.266	−	282.478i	−0.703980	−	0.406443i
$696$		0			0
$697$	−198.587	−	343.962i	−0.284917	−	0.493490i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			302.260i			0.431184i	0.976484	+	0.215592i	$0.0691680\pi$
$701$			302.260i			0.431184i	−0.976484	+	0.215592i	$0.930832\pi$
$702$		0			0
$703$	209.882			0.298551
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−250.427	+	144.584i	−0.354210	+	0.204503i
$708$		0			0
$709$	454.761	−	787.670i	0.641412	−	1.11096i	−0.343705	−	0.939078i	$-0.611682\pi$
$709$	454.761	−	787.670i	0.641412	−	1.11096i	0.985118	−	0.171881i	$-0.0549845\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−459.739	−	265.431i	−0.644796	−	0.372273i
$714$		0			0
$715$	261.158	+	452.338i	0.365255	+	0.632641i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			779.361i			1.08395i	0.840394	+	0.541976i	$0.182323\pi$
$719$			779.361i			1.08395i	−0.840394	+	0.541976i	$0.817677\pi$
$720$		0			0
$721$	−124.043			−0.172043
$722$		0			0
$723$		0			0
$724$		0			0
$725$	19.7338	−	11.3933i	0.0272190	−	0.0157149i
$726$		0			0
$727$	−96.5145	+	167.168i	−0.132757	+	0.229942i	−0.924738	−	0.380603i	$-0.875716\pi$
$727$	−96.5145	+	167.168i	−0.132757	+	0.229942i	0.791981	+	0.610545i	$0.209050\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	268.645	+	155.102i	0.367504	+	0.212178i
$732$		0			0
$733$	429.154	+	743.317i	0.585477	+	1.01408i	0.994816	+	0.101693i	$0.0324259\pi$
$733$	429.154	+	743.317i	0.585477	+	1.01408i	−0.409339	+	0.912382i	$0.634241\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			694.859i			0.942821i
$738$		0			0
$739$	977.178			1.32230			0.661149	−	0.750255i	$-0.270069\pi$
$739$	977.178			1.32230			0.661149	+	0.750255i	$0.270069\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−156.349	+	90.2680i	−0.210429	+	0.121491i	−0.601511	−	0.798865i	$-0.705434\pi$
$743$	−156.349	+	90.2680i	−0.210429	+	0.121491i	0.391082	+	0.920356i	$0.372101\pi$
$744$		0			0
$745$	100.361	−	173.830i	0.134712	−	0.233329i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	15.1845	+	8.76677i	0.0202730	+	0.0117046i
$750$		0			0
$751$	−700.877	−	1213.95i	−0.933258	−	1.61645i	−0.777711	−	0.628622i	$-0.783619\pi$
$751$	−700.877	−	1213.95i	−0.933258	−	1.61645i	−0.155547	−	0.987829i	$-0.549714\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	422.186i		−	0.559187i
$756$		0			0
$757$	−69.2097			−0.0914262			−0.0457131	−	0.998955i	$-0.514556\pi$
$757$	−69.2097			−0.0914262			−0.0457131	+	0.998955i	$0.514556\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−236.592	+	136.597i	−0.310897	+	0.179496i	−0.647328	−	0.762212i	$-0.724113\pi$
$761$	−236.592	+	136.597i	−0.310897	+	0.179496i	0.336431	+	0.941708i	$0.390780\pi$
$762$		0			0
$763$	149.223	−	258.462i	0.195574	−	0.338745i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1101.63	+	636.024i	1.43628	+	0.829236i
$768$		0			0
$769$	154.379	+	267.393i	0.200753	+	0.347715i	0.948771	−	0.315963i	$-0.102328\pi$
$769$	154.379	+	267.393i	0.200753	+	0.347715i	−0.748018	+	0.663678i	$0.768994\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			764.959i			0.989597i	0.869008	+	0.494799i	$0.164758\pi$
$773$			764.959i			0.989597i	−0.869008	+	0.494799i	$0.835242\pi$
$774$		0			0
$775$	244.879			0.315973
$776$		0			0
$777$		0			0
$778$		0			0
$779$	446.008	−	257.503i	0.572539	−	0.330556i
$780$		0			0
$781$	−129.971	+	225.116i	−0.166416	+	0.288241i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−255.604	−	147.573i	−0.325610	−	0.187991i
$786$		0			0
$787$	566.734	+	981.612i	0.720120	+	1.24728i	0.960952	+	0.276717i	$0.0892464\pi$
$787$	566.734	+	981.612i	0.720120	+	1.24728i	−0.240832	+	0.970567i	$0.577420\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	96.0706i		−	0.121455i
$792$		0			0
$793$	768.438			0.969026
$794$		0			0
$795$		0			0
$796$		0			0
$797$	452.278	−	261.123i	0.567476	−	0.327632i	−0.188665	−	0.982042i	$-0.560416\pi$
$797$	452.278	−	261.123i	0.567476	−	0.327632i	0.756141	+	0.654409i	$0.227083\pi$
$798$		0			0
$799$	−140.188	+	242.813i	−0.175455	+	0.303897i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1313.19	−	758.171i	−1.63536	−	0.944173i
$804$		0			0
$805$	54.8941	+	95.0794i	0.0681915	+	0.118111i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			198.946i			0.245916i	0.992412	+	0.122958i	$0.0392380\pi$
$809$			198.946i			0.245916i	−0.992412	+	0.122958i	$0.960762\pi$
$810$		0			0
$811$	1256.10			1.54883			0.774415	−	0.632678i	$-0.218044\pi$
$811$	1256.10			1.54883			0.774415	+	0.632678i	$0.218044\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−335.654	+	193.790i	−0.411846	+	0.237779i
$816$		0			0
$817$	−201.118	+	348.346i	−0.246166	+	0.426372i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1201.66	+	693.781i	1.46366	+	0.845044i	0.999178	−	0.0405384i	$-0.0129073\pi$
$821$	1201.66	+	693.781i	1.46366	+	0.845044i	0.464482	+	0.885583i	$0.346241\pi$
$822$		0			0
$823$	503.241	+	871.639i	0.611471	+	1.05910i	0.990993	+	0.133916i	$0.0427554\pi$
$823$	503.241	+	871.639i	0.611471	+	1.05910i	−0.379521	+	0.925183i	$0.623911\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			768.926i			0.929778i	0.885369	+	0.464889i	$0.153906\pi$
$827$			768.926i			0.929778i	−0.885369	+	0.464889i	$0.846094\pi$
$828$		0			0
$829$	−602.188			−0.726403			−0.363201	−	0.931711i	$-0.618316\pi$
$829$	−602.188			−0.726403			−0.363201	+	0.931711i	$0.618316\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	185.880	−	107.318i	0.223146	−	0.128833i
$834$		0			0
$835$	−29.2259	+	50.6208i	−0.0350011	+	0.0606237i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1111.05	+	641.464i	1.32425	+	0.764558i	0.984404	−	0.175922i	$-0.0562905\pi$
$839$	1111.05	+	641.464i	1.32425	+	0.764558i	0.339850	+	0.940480i	$0.389624\pi$
$840$		0			0
$841$	−410.115	−	710.341i	−0.487652	−	0.844638i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	170.052i		−	0.201245i
$846$		0			0
$847$	−460.489			−0.543671
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−201.621	+	116.406i	−0.236923	+	0.136787i
$852$		0			0
$853$	−599.045	+	1037.58i	−0.702281	+	1.21639i	0.265383	+	0.964143i	$0.414502\pi$
$853$	−599.045	+	1037.58i	−0.702281	+	1.21639i	−0.967664	+	0.252243i	$0.918832\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−481.135	−	277.783i	−0.561418	−	0.324135i	0.192297	−	0.981337i	$-0.438406\pi$
$857$	−481.135	−	277.783i	−0.561418	−	0.324135i	−0.753714	+	0.657202i	$0.771740\pi$
$858$		0			0
$859$	290.615	+	503.360i	0.338318	+	0.585984i	0.984116	−	0.177524i	$-0.0568088\pi$
$859$	290.615	+	503.360i	0.338318	+	0.585984i	−0.645799	+	0.763508i	$0.723475\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1487.60i		−	1.72375i	−0.507120	−	0.861876i	$-0.669290\pi$
$863$		−	1487.60i		−	1.72375i	0.507120	−	0.861876i	$-0.330710\pi$
$864$		0			0
$865$	524.981			0.606915
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−741.940	+	428.359i	−0.853786	+	0.492933i
$870$		0			0
$871$	364.479	−	631.297i	0.418461	−	0.724795i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−43.8588	−	25.3219i	−0.0501243	−	0.0289393i
$876$		0			0
$877$	−520.859	−	902.155i	−0.593910	−	1.02868i	−0.993700	−	0.112076i	$-0.964250\pi$
$877$	−520.859	−	902.155i	−0.593910	−	1.02868i	0.399789	−	0.916607i	$-0.369083\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1205.55i			1.36839i	0.729299	+	0.684196i	$0.239847\pi$
$881$			1205.55i			1.36839i	−0.729299	+	0.684196i	$0.760153\pi$
$882$		0			0
$883$	75.6957			0.0857256			0.0428628	−	0.999081i	$-0.486352\pi$
$883$	75.6957			0.0857256			0.0428628	+	0.999081i	$0.486352\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1307.91	+	755.124i	−1.47454	+	0.851324i	−0.999588	−	0.0286903i	$-0.990866\pi$
$887$	−1307.91	+	755.124i	−1.47454	+	0.851324i	−0.474948	+	0.880014i	$0.657533\pi$
$888$		0			0
$889$	−105.514	+	182.756i	−0.118688	+	0.205574i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−314.850	−	181.779i	−0.352576	−	0.203560i
$894$		0			0
$895$	190.559	+	330.057i	0.212915	+	0.368779i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			223.199i			0.248274i
$900$		0			0
$901$	−616.784			−0.684554
$902$		0			0
$903$		0			0
$904$		0			0
$905$	41.4475	−	23.9297i	0.0457983	−	0.0264417i
$906$		0			0
$907$	176.596	−	305.873i	0.194703	−	0.337235i	−0.752100	−	0.659049i	$-0.770959\pi$
$907$	176.596	−	305.873i	0.194703	−	0.337235i	0.946803	+	0.321813i	$0.104292\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	849.700	+	490.575i	0.932712	+	0.538501i	0.887668	−	0.460484i	$-0.152324\pi$
$911$	849.700	+	490.575i	0.932712	+	0.538501i	0.0450436	+	0.998985i	$0.485657\pi$
$912$		0			0
$913$	648.090	+	1122.52i	0.709847	+	1.22949i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	630.060i		−	0.687088i
$918$		0			0
$919$	−303.202			−0.329926			−0.164963	−	0.986300i	$-0.552750\pi$
$919$	−303.202			−0.329926			−0.164963	+	0.986300i	$0.552750\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	236.164	−	136.349i	0.255865	−	0.147724i
$924$		0			0
$925$	53.6964	−	93.0050i	0.0580502	−	0.100546i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−69.3559	−	40.0426i	−0.0746565	−	0.0431030i	0.462207	−	0.886772i	$-0.347058\pi$
$929$	−69.3559	−	40.0426i	−0.0746565	−	0.0431030i	−0.536864	+	0.843669i	$0.680391\pi$
$930$		0			0
$931$	139.157	+	241.027i	0.149470	+	0.258890i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			251.445i			0.268925i
$936$		0			0
$937$	−367.445			−0.392150			−0.196075	−	0.980589i	$-0.562820\pi$
$937$	−367.445			−0.392150			−0.196075	+	0.980589i	$0.562820\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	583.005	−	336.598i	0.619559	−	0.357702i	−0.157138	−	0.987577i	$-0.550227\pi$
$941$	583.005	−	336.598i	0.619559	−	0.357702i	0.776697	+	0.629874i	$0.216894\pi$
$942$		0			0
$943$	−285.636	+	494.736i	−0.302902	+	0.524641i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	866.108	+	500.048i	0.914581	+	0.528034i	0.881902	−	0.471432i	$-0.156263\pi$
$947$	866.108	+	500.048i	0.914581	+	0.528034i	0.0326790	+	0.999466i	$0.489596\pi$
$948$		0			0
$949$	795.378	+	1377.63i	0.838122	+	1.45167i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1800.63i		−	1.88943i	−0.327889	−	0.944716i	$-0.606337\pi$
$953$		−	1800.63i		−	1.88943i	0.327889	−	0.944716i	$-0.393663\pi$
$954$		0			0
$955$	−413.236			−0.432708
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−835.565	+	482.414i	−0.871288	+	0.503038i
$960$		0			0
$961$	−718.812	+	1245.02i	−0.747984	+	1.29555i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−736.487	−	425.211i	−0.763199	−	0.440633i
$966$		0			0
$967$	236.137	+	409.001i	0.244196	+	0.422959i	0.961905	−	0.273383i	$-0.0881428\pi$
$967$	236.137	+	409.001i	0.244196	+	0.422959i	−0.717710	+	0.696343i	$0.754809\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	350.248i		−	0.360708i	−0.983602	−	0.180354i	$-0.942276\pi$
$971$		−	350.248i		−	0.360708i	0.983602	−	0.180354i	$-0.0577244\pi$
$972$		0			0
$973$	−1144.46			−1.17622
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1157.45	−	668.251i	1.18469	−	0.683983i	0.227598	−	0.973755i	$-0.426913\pi$
$977$	1157.45	−	668.251i	1.18469	−	0.683983i	0.957096	+	0.289772i	$0.0935795\pi$
$978$		0			0
$979$	−308.840	+	534.927i	−0.315465	+	0.546402i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1174.62	−	678.167i	−1.19493	−	0.689895i	−0.235512	−	0.971871i	$-0.575677\pi$
$983$	−1174.62	−	678.167i	−1.19493	−	0.689895i	−0.959421	+	0.281976i	$0.909010\pi$
$984$		0			0
$985$	88.6690	+	153.579i	0.0900193	+	0.155918i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	446.181i		−	0.451144i
$990$		0			0
$991$	−1699.68			−1.71512			−0.857558	−	0.514388i	$-0.828019\pi$
$991$	−1699.68			−1.71512			−0.857558	+	0.514388i	$0.828019\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	168.915	−	97.5229i	0.169764	−	0.0980130i
$996$		0			0
$997$	−816.024	+	1413.40i	−0.818480	+	1.41765i	0.0883222	+	0.996092i	$0.471849\pi$
$997$	−816.024	+	1413.40i	−0.818480	+	1.41765i	−0.906802	+	0.421557i	$0.861484\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.o.g.1241.1		32
3.2	odd	2	inner	1620.3.o.g.1241.16		32
9.2	odd	6		1620.3.g.c.161.14	yes	16
9.4	even	3	inner	1620.3.o.g.701.16		32
9.5	odd	6	inner	1620.3.o.g.701.1		32
9.7	even	3		1620.3.g.c.161.6	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.g.c.161.6	✓	16	9.7	even	3
1620.3.g.c.161.14	yes	16	9.2	odd	6
1620.3.o.g.701.1		32	9.5	odd	6	inner
1620.3.o.g.701.16		32	9.4	even	3	inner
1620.3.o.g.1241.1		32	1.1	even	1	trivial
1620.3.o.g.1241.16		32	3.2	odd	2	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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.93649 + 1.11803i) q^{5} +(-2.26486 + 3.92285i) q^{7} +O(q^{10})\)	\(q+(-1.93649 + 1.11803i) q^{5} +(-2.26486 + 3.92285i) q^{7} +(12.9226 + 7.46089i) q^{11} +(-7.82703 - 13.5568i) q^{13} -7.53594i q^{17} +9.77167 q^{19} +(-9.38708 + 5.41963i) q^{23} +(2.50000 - 4.33013i) q^{25} +(3.94676 + 2.27866i) q^{29} +(24.4879 + 42.4143i) q^{31} -10.1288i q^{35} +21.4786 q^{37} +(45.6430 - 26.3520i) q^{41} +(-20.5817 + 35.6486i) q^{43} +(-32.2207 - 18.6026i) q^{47} +(14.2408 + 24.6659i) q^{49} -81.8456i q^{53} -33.3661 q^{55} +(-70.3732 + 40.6300i) q^{59} +(-24.5444 + 42.5121i) q^{61} +(30.3139 + 17.5018i) q^{65} +(23.2834 + 40.3280i) q^{67} +17.4203i q^{71} -101.619 q^{73} +(-58.5359 + 33.7957i) q^{77} +(-28.7070 + 49.7219i) q^{79} +(75.2273 + 43.4325i) q^{83} +(8.42543 + 14.5933i) q^{85} +41.3946i q^{89} +70.9084 q^{91} +(-18.9228 + 10.9251i) q^{95} +(-90.4805 + 156.717i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7}+O(q^{10}) \) 32 * q - 8 * q^7	\( 32 q - 8 q^{7} + 40 q^{13} + 112 q^{19} + 80 q^{25} + 64 q^{31} - 176 q^{37} - 128 q^{43} - 216 q^{49} - 8 q^{61} + 40 q^{67} + 112 q^{73} + 136 q^{79} - 784 q^{91} - 128 q^{97}+O(q^{100}) \) 32 * q - 8 * q^7 + 40 * q^13 + 112 * q^19 + 80 * q^25 + 64 * q^31 - 176 * q^37 - 128 * q^43 - 216 * q^49 - 8 * q^61 + 40 * q^67 + 112 * q^73 + 136 * q^79 - 784 * q^91 - 128 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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