LMFDB - Embedded newform 1620.3.b.b.809.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(809,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("1620.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.b (of order $2$, degree $1$, 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:	$24$
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.2
Character	$\chi$	$=$	1620.809
Dual form			1620.3.b.b.809.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.90327 + 0.978759i) q^{5} +7.34726i q^{7} +O(q^{10})$	$q+(-4.90327 + 0.978759i) q^{5} +7.34726i q^{7} +9.64692i q^{11} -16.6859i q^{13} -15.4743 q^{17} -17.4865 q^{19} +10.5819 q^{23} +(23.0841 - 9.59823i) q^{25} +41.8058i q^{29} -48.7281 q^{31} +(-7.19120 - 36.0256i) q^{35} -20.4792i q^{37} +27.9021i q^{41} -30.2091i q^{43} -42.1251 q^{47} -4.98225 q^{49} +90.4314 q^{53} +(-9.44201 - 47.3014i) q^{55} -55.4367i q^{59} +116.758 q^{61} +(16.3315 + 81.8154i) q^{65} -17.2837i q^{67} +38.4861i q^{71} -7.38815i q^{73} -70.8784 q^{77} +23.6993 q^{79} +4.26989 q^{83} +(75.8744 - 15.1456i) q^{85} -145.034i q^{89} +122.596 q^{91} +(85.7411 - 17.1151i) q^{95} -102.700i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 6 q^{25} - 60 q^{31} - 216 q^{49} + 42 q^{55} - 96 q^{61} - 228 q^{79} - 96 q^{85} - 84 q^{91}+O(q^{100}) $ 24 * q - 6 * q^25 - 60 * q^31 - 216 * q^49 + 42 * q^55 - 96 * q^61 - 228 * q^79 - 96 * q^85 - 84 * q^91

Display 100 coefficients 10 coefficients

Character values

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

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$-1$

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			0
$3$		0			0
$4$		0			0
$5$	−4.90327	+	0.978759i	−0.980653	+	0.195752i
$5$	−4.90327	+	0.978759i	−0.980653	+	0.195752i
$6$		0			0
$7$			7.34726i			1.04961i	0.851223	+	0.524804i	$0.175862\pi$
$7$			7.34726i			1.04961i	−0.851223	+	0.524804i	$0.824138\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			9.64692i			0.876993i	0.898733	+	0.438496i	$0.144489\pi$
$11$			9.64692i			0.876993i	−0.898733	+	0.438496i	$0.855511\pi$
$12$		0			0
$13$		−	16.6859i		−	1.28353i	−0.766901	−	0.641765i	$-0.778202\pi$
$13$		−	16.6859i		−	1.28353i	0.766901	−	0.641765i	$-0.221798\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−15.4743			−0.910251			−0.455125	−	0.890427i	$-0.650406\pi$
$17$	−15.4743			−0.910251			−0.455125	+	0.890427i	$0.650406\pi$
$18$		0			0
$19$	−17.4865			−0.920344			−0.460172	−	0.887830i	$-0.652212\pi$
$19$	−17.4865			−0.920344			−0.460172	+	0.887830i	$0.652212\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	10.5819			0.460080			0.230040	−	0.973181i	$-0.426114\pi$
$23$	10.5819			0.460080			0.230040	+	0.973181i	$0.426114\pi$
$24$		0			0
$25$	23.0841	−	9.59823i	0.923363	−	0.383929i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			41.8058i			1.44158i	0.693154	+	0.720789i	$0.256220\pi$
$29$			41.8058i			1.44158i	−0.693154	+	0.720789i	$0.743780\pi$
$30$		0			0
$31$	−48.7281			−1.57187			−0.785937	−	0.618306i	$-0.787819\pi$
$31$	−48.7281			−1.57187			−0.785937	+	0.618306i	$0.787819\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−7.19120	−	36.0256i	−0.205463	−	1.02930i
$36$		0			0
$37$		−	20.4792i		−	0.553492i	−0.960943	−	0.276746i	$-0.910744\pi$
$37$		−	20.4792i		−	0.553492i	0.960943	−	0.276746i	$-0.0892561\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			27.9021i			0.680540i	0.940328	+	0.340270i	$0.110518\pi$
$41$			27.9021i			0.680540i	−0.940328	+	0.340270i	$0.889482\pi$
$42$		0			0
$43$		−	30.2091i		−	0.702538i	−0.936275	−	0.351269i	$-0.885750\pi$
$43$		−	30.2091i		−	0.702538i	0.936275	−	0.351269i	$-0.114250\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−42.1251			−0.896279			−0.448139	−	0.893964i	$-0.647913\pi$
$47$	−42.1251			−0.896279			−0.448139	+	0.893964i	$0.647913\pi$
$48$		0			0
$49$	−4.98225			−0.101679
$50$		0			0
$51$		0			0
$52$		0			0
$53$	90.4314			1.70625			0.853127	−	0.521704i	$-0.174703\pi$
$53$	90.4314			1.70625			0.853127	+	0.521704i	$0.174703\pi$
$54$		0			0
$55$	−9.44201	−	47.3014i	−0.171673	−	0.860026i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	55.4367i		−	0.939605i	−0.882771	−	0.469803i	$-0.844325\pi$
$59$		−	55.4367i		−	0.939605i	0.882771	−	0.469803i	$-0.155675\pi$
$60$		0			0
$61$	116.758			1.91407			0.957034	−	0.289977i	$-0.0936478\pi$
$61$	116.758			1.91407			0.957034	+	0.289977i	$0.0936478\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	16.3315	+	81.8154i	0.251253	+	1.25870i
$66$		0			0
$67$		−	17.2837i		−	0.257966i	−0.991647	−	0.128983i	$-0.958829\pi$
$67$		−	17.2837i		−	0.257966i	0.991647	−	0.128983i	$-0.0411713\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			38.4861i			0.542058i	0.962571	+	0.271029i	$0.0873640\pi$
$71$			38.4861i			0.542058i	−0.962571	+	0.271029i	$0.912636\pi$
$72$		0			0
$73$		−	7.38815i		−	0.101208i	−0.998719	−	0.0506038i	$-0.983885\pi$
$73$		−	7.38815i		−	0.101208i	0.998719	−	0.0506038i	$-0.0161146\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−70.8784			−0.920499
$78$		0			0
$79$	23.6993			0.299992			0.149996	−	0.988687i	$-0.452074\pi$
$79$	23.6993			0.299992			0.149996	+	0.988687i	$0.452074\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.26989			0.0514444			0.0257222	−	0.999669i	$-0.491811\pi$
$83$	4.26989			0.0514444			0.0257222	+	0.999669i	$0.491811\pi$
$84$		0			0
$85$	75.8744	−	15.1456i	0.892641	−	0.178183i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	145.034i		−	1.62959i	−0.579746	−	0.814797i	$-0.696848\pi$
$89$		−	145.034i		−	1.62959i	0.579746	−	0.814797i	$-0.303152\pi$
$90$		0			0
$91$	122.596			1.34720
$92$		0			0
$93$		0			0
$94$		0			0
$95$	85.7411	−	17.1151i	0.902538	−	0.180159i
$96$		0			0
$97$		−	102.700i		−	1.05876i	−0.848385	−	0.529379i	$-0.822425\pi$
$97$		−	102.700i		−	1.05876i	0.848385	−	0.529379i	$-0.177575\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	110.861i		−	1.09763i	−0.835943	−	0.548816i	$-0.815079\pi$
$101$		−	110.861i		−	1.09763i	0.835943	−	0.548816i	$-0.184921\pi$
$102$		0			0
$103$		−	187.849i		−	1.82378i	−0.410438	−	0.911888i	$-0.634624\pi$
$103$		−	187.849i		−	1.82378i	0.410438	−	0.911888i	$-0.365376\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−121.767			−1.13801			−0.569005	−	0.822334i	$-0.692672\pi$
$107$	−121.767			−1.13801			−0.569005	+	0.822334i	$0.692672\pi$
$108$		0			0
$109$	−45.3350			−0.415918			−0.207959	−	0.978138i	$-0.566682\pi$
$109$	−45.3350			−0.415918			−0.207959	+	0.978138i	$0.566682\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	19.2092			0.169993			0.0849963	−	0.996381i	$-0.472912\pi$
$113$	19.2092			0.169993			0.0849963	+	0.996381i	$0.472912\pi$
$114$		0			0
$115$	−51.8856	+	10.3571i	−0.451179	+	0.0900616i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	113.693i		−	0.955407i
$120$		0			0
$121$	27.9369			0.230884
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−103.793	+	69.6564i	−0.830344	+	0.557251i
$126$		0			0
$127$		−	179.244i		−	1.41137i	−0.708527	−	0.705684i	$-0.750640\pi$
$127$		−	179.244i		−	1.41137i	0.708527	−	0.705684i	$-0.249360\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	84.9928i		−	0.648800i	−0.945920	−	0.324400i	$-0.894838\pi$
$131$		−	84.9928i		−	0.648800i	0.945920	−	0.324400i	$-0.105162\pi$
$132$		0			0
$133$		−	128.478i		−	0.966001i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	66.0739			0.482291			0.241146	−	0.970489i	$-0.422477\pi$
$137$	66.0739			0.482291			0.241146	+	0.970489i	$0.422477\pi$
$138$		0			0
$139$	−9.42041			−0.0677728			−0.0338864	−	0.999426i	$-0.510788\pi$
$139$	−9.42041			−0.0677728			−0.0338864	+	0.999426i	$0.510788\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	160.967			1.12565
$144$		0			0
$145$	−40.9178	−	204.985i	−0.282191	−	1.41369i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			124.758i			0.837305i	0.908146	+	0.418653i	$0.137498\pi$
$149$			124.758i			0.837305i	−0.908146	+	0.418653i	$0.862502\pi$
$150$		0			0
$151$	−291.962			−1.93353			−0.966763	−	0.255674i	$-0.917703\pi$
$151$	−291.962			−1.93353			−0.966763	+	0.255674i	$0.917703\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	238.927	−	47.6931i	1.54146	−	0.307697i
$156$		0			0
$157$		−	143.269i		−	0.912543i	−0.889840	−	0.456272i	$-0.849185\pi$
$157$		−	143.269i		−	0.912543i	0.889840	−	0.456272i	$-0.150815\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			77.7476i			0.482904i
$162$		0			0
$163$			228.204i			1.40002i	0.714132	+	0.700011i	$0.246822\pi$
$163$			228.204i			1.40002i	−0.714132	+	0.700011i	$0.753178\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	60.7552			0.363804			0.181902	−	0.983317i	$-0.441775\pi$
$167$	60.7552			0.363804			0.181902	+	0.983317i	$0.441775\pi$
$168$		0			0
$169$	−109.419			−0.647449
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−277.631			−1.60480			−0.802400	−	0.596786i	$-0.796444\pi$
$173$	−277.631			−1.60480			−0.802400	+	0.596786i	$0.796444\pi$
$174$		0			0
$175$	70.5207	+	169.605i	0.402976	+	0.969169i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			272.749i			1.52374i	0.647731	+	0.761869i	$0.275718\pi$
$179$			272.749i			1.52374i	−0.647731	+	0.761869i	$0.724282\pi$
$180$		0			0
$181$	−110.378			−0.609824			−0.304912	−	0.952380i	$-0.598627\pi$
$181$	−110.378			−0.609824			−0.304912	+	0.952380i	$0.598627\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	20.0442	+	100.415i	0.108347	+	0.542784i
$186$		0			0
$187$		−	149.279i		−	0.798283i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	33.6071i		−	0.175954i	−0.996123	−	0.0879768i	$-0.971960\pi$
$191$		−	33.6071i		−	0.175954i	0.996123	−	0.0879768i	$-0.0280401\pi$
$192$		0			0
$193$		−	270.684i		−	1.40251i	−0.712911	−	0.701255i	$-0.752624\pi$
$193$		−	270.684i		−	1.40251i	0.712911	−	0.701255i	$-0.247376\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−63.0978			−0.320293			−0.160147	−	0.987093i	$-0.551197\pi$
$197$	−63.0978			−0.320293			−0.160147	+	0.987093i	$0.551197\pi$
$198$		0			0
$199$	283.032			1.42227			0.711135	−	0.703056i	$-0.248182\pi$
$199$	283.032			1.42227			0.711135	+	0.703056i	$0.248182\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−307.158			−1.51309
$204$		0			0
$205$	−27.3095	−	136.812i	−0.133217	−	0.667374i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	168.691i		−	0.807135i
$210$		0			0
$211$	113.862			0.539633			0.269816	−	0.962912i	$-0.413037\pi$
$211$	113.862			0.539633			0.269816	+	0.962912i	$0.413037\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	29.5674	+	148.123i	0.137523	+	0.688946i
$216$		0			0
$217$		−	358.018i		−	1.64985i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			258.202i			1.16833i
$222$		0			0
$223$		−	217.858i		−	0.976941i	−0.872580	−	0.488471i	$-0.837555\pi$
$223$		−	217.858i		−	0.976941i	0.872580	−	0.488471i	$-0.162445\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	52.9198			0.233127			0.116563	−	0.993183i	$-0.462812\pi$
$227$	52.9198			0.233127			0.116563	+	0.993183i	$0.462812\pi$
$228$		0			0
$229$	391.321			1.70883			0.854413	−	0.519595i	$-0.173917\pi$
$229$	391.321			1.70883			0.854413	+	0.519595i	$0.173917\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	178.384			0.765596			0.382798	−	0.923832i	$-0.374961\pi$
$233$	178.384			0.765596			0.382798	+	0.923832i	$0.374961\pi$
$234$		0			0
$235$	206.551	−	41.2303i	0.878939	−	0.175448i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	363.155i		−	1.51948i	−0.650229	−	0.759738i	$-0.725327\pi$
$239$		−	363.155i		−	1.51948i	0.650229	−	0.759738i	$-0.274673\pi$
$240$		0			0
$241$	180.644			0.749559			0.374780	−	0.927114i	$-0.377718\pi$
$241$	180.644			0.749559			0.374780	+	0.927114i	$0.377718\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	24.4293	−	4.87642i	0.0997115	−	0.0199038i
$246$		0			0
$247$			291.778i			1.18129i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			73.5509i			0.293032i	0.989208	+	0.146516i	$0.0468059\pi$
$251$			73.5509i			0.293032i	−0.989208	+	0.146516i	$0.953194\pi$
$252$		0			0
$253$			102.082i			0.403487i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	144.952			0.564014			0.282007	−	0.959412i	$-0.409000\pi$
$257$	144.952			0.564014			0.282007	+	0.959412i	$0.409000\pi$
$258$		0			0
$259$	150.466			0.580950
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−328.862			−1.25043			−0.625213	−	0.780454i	$-0.714988\pi$
$263$	−328.862			−1.25043			−0.625213	+	0.780454i	$0.714988\pi$
$264$		0			0
$265$	−443.409	+	88.5106i	−1.67324	+	0.334002i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	355.832i		−	1.32280i	−0.750035	−	0.661398i	$-0.769963\pi$
$269$		−	355.832i		−	1.32280i	0.750035	−	0.661398i	$-0.230037\pi$
$270$		0			0
$271$	−128.345			−0.473599			−0.236799	−	0.971559i	$-0.576098\pi$
$271$	−128.345			−0.473599			−0.236799	+	0.971559i	$0.576098\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	92.5934	+	222.690i	0.336703	+	0.809782i
$276$		0			0
$277$			270.554i			0.976729i	0.872640	+	0.488365i	$0.162406\pi$
$277$			270.554i			0.976729i	−0.872640	+	0.488365i	$0.837594\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			525.118i			1.86875i	0.356295	+	0.934374i	$0.384040\pi$
$281$			525.118i			1.86875i	−0.356295	+	0.934374i	$0.615960\pi$
$282$		0			0
$283$		−	209.596i		−	0.740623i	−0.928908	−	0.370311i	$-0.879251\pi$
$283$		−	209.596i		−	0.740623i	0.928908	−	0.370311i	$-0.120749\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−205.004			−0.714300
$288$		0			0
$289$	−49.5472			−0.171444
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−119.736			−0.408654			−0.204327	−	0.978903i	$-0.565501\pi$
$293$	−119.736			−0.408654			−0.204327	+	0.978903i	$0.565501\pi$
$294$		0			0
$295$	54.2592	+	271.821i	0.183929	+	0.921427i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	176.568i		−	0.590527i
$300$		0			0
$301$	221.954			0.737390
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−572.496	+	114.278i	−1.87704	+	0.374682i
$306$		0			0
$307$			435.735i			1.41933i	0.704538	+	0.709666i	$0.251154\pi$
$307$			435.735i			1.41933i	−0.704538	+	0.709666i	$0.748846\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			68.4104i			0.219969i	0.993933	+	0.109985i	$0.0350801\pi$
$311$			68.4104i			0.219969i	−0.993933	+	0.109985i	$0.964920\pi$
$312$		0			0
$313$		−	43.9371i		−	0.140374i	−0.997534	−	0.0701870i	$-0.977640\pi$
$313$		−	43.9371i		−	0.140374i	0.997534	−	0.0701870i	$-0.0223596\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	267.533			0.843953			0.421977	−	0.906607i	$-0.361336\pi$
$317$	267.533			0.843953			0.421977	+	0.906607i	$0.361336\pi$
$318$		0			0
$319$	−403.297			−1.26425
$320$		0			0
$321$		0			0
$322$		0			0
$323$	270.591			0.837743
$324$		0			0
$325$	−160.155	−	385.178i	−0.492785	−	1.18516i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	309.504i		−	0.940742i
$330$		0			0
$331$	72.1231			0.217894			0.108947	−	0.994048i	$-0.465252\pi$
$331$	72.1231			0.217894			0.108947	+	0.994048i	$0.465252\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	16.9166	+	84.7468i	0.0504974	+	0.252976i
$336$		0			0
$337$		−	195.792i		−	0.580987i	−0.956877	−	0.290493i	$-0.906181\pi$
$337$		−	195.792i		−	0.580987i	0.956877	−	0.290493i	$-0.0938194\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	470.076i		−	1.37852i
$342$		0			0
$343$			323.410i			0.942886i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	126.817			0.365466			0.182733	−	0.983163i	$-0.441506\pi$
$347$	126.817			0.365466			0.182733	+	0.983163i	$0.441506\pi$
$348$		0			0
$349$	130.553			0.374077			0.187038	−	0.982353i	$-0.440111\pi$
$349$	130.553			0.374077			0.187038	+	0.982353i	$0.440111\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	261.256			0.740102			0.370051	−	0.929011i	$-0.379340\pi$
$353$	261.256			0.740102			0.370051	+	0.929011i	$0.379340\pi$
$354$		0			0
$355$	−37.6687	−	188.708i	−0.106109	−	0.531571i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	390.386i		−	1.08743i	−0.839271	−	0.543713i	$-0.817018\pi$
$359$		−	390.386i		−	1.08743i	0.839271	−	0.543713i	$-0.182982\pi$
$360$		0			0
$361$	−55.2213			−0.152968
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.23121	+	36.2261i	0.0198115	+	0.0992495i
$366$		0			0
$367$			84.3330i			0.229790i	0.993378	+	0.114895i	$0.0366532\pi$
$367$			84.3330i			0.229790i	−0.993378	+	0.114895i	$0.963347\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			664.423i			1.79090i
$372$		0			0
$373$		−	177.161i		−	0.474963i	−0.971392	−	0.237482i	$-0.923678\pi$
$373$		−	177.161i		−	0.474963i	0.971392	−	0.237482i	$-0.0763219\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	697.566			1.85031
$378$		0			0
$379$	−291.976			−0.770386			−0.385193	−	0.922836i	$-0.625865\pi$
$379$	−291.976			−0.770386			−0.385193	+	0.922836i	$0.625865\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	273.531			0.714179			0.357090	−	0.934070i	$-0.383769\pi$
$383$	273.531			0.714179			0.357090	+	0.934070i	$0.383769\pi$
$384$		0			0
$385$	347.536	−	69.3729i	0.902691	−	0.180189i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	237.978i		−	0.611769i	−0.952069	−	0.305885i	$-0.901048\pi$
$389$		−	237.978i		−	0.611769i	0.952069	−	0.305885i	$-0.0989522\pi$
$390$		0			0
$391$	−163.746			−0.418789
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−116.204	+	23.1959i	−0.294188	+	0.0587239i
$396$		0			0
$397$		−	76.9037i		−	0.193712i	−0.995298	−	0.0968560i	$-0.969121\pi$
$397$		−	76.9037i		−	0.193712i	0.995298	−	0.0968560i	$-0.0308786\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	546.297i		−	1.36234i	−0.732127	−	0.681169i	$-0.761472\pi$
$401$		−	546.297i		−	1.36234i	0.732127	−	0.681169i	$-0.238528\pi$
$402$		0			0
$403$			813.072i			2.01755i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	197.561			0.485409
$408$		0			0
$409$	−562.102			−1.37433			−0.687166	−	0.726500i	$-0.741146\pi$
$409$	−562.102			−1.37433			−0.687166	+	0.726500i	$0.741146\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	407.308			0.986218
$414$		0			0
$415$	−20.9364	+	4.17919i	−0.0504492	+	0.0100703i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	242.077i		−	0.577749i	−0.957367	−	0.288874i	$-0.906719\pi$
$419$		−	242.077i		−	0.577749i	0.957367	−	0.288874i	$-0.0932810\pi$
$420$		0			0
$421$	16.2005			0.0384809			0.0192405	−	0.999815i	$-0.493875\pi$
$421$	16.2005			0.0384809			0.0192405	+	0.999815i	$0.493875\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−357.209	+	148.526i	−0.840491	+	0.349472i
$426$		0			0
$427$			857.852i			2.00902i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	50.0624i		−	0.116154i	−0.998312	−	0.0580770i	$-0.981503\pi$
$431$		−	50.0624i		−	0.116154i	0.998312	−	0.0580770i	$-0.0184969\pi$
$432$		0			0
$433$			452.725i			1.04556i	0.852469	+	0.522778i	$0.175104\pi$
$433$			452.725i			1.04556i	−0.852469	+	0.522778i	$0.824896\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−185.040			−0.423432
$438$		0			0
$439$	230.278			0.524551			0.262275	−	0.964993i	$-0.415527\pi$
$439$	230.278			0.524551			0.262275	+	0.964993i	$0.415527\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−597.848			−1.34954			−0.674772	−	0.738027i	$-0.735758\pi$
$443$	−597.848			−1.34954			−0.674772	+	0.738027i	$0.735758\pi$
$444$		0			0
$445$	141.953	+	711.140i	0.318996	+	1.59807i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	642.091i		−	1.43005i	−0.699101	−	0.715023i	$-0.746416\pi$
$449$		−	642.091i		−	1.43005i	0.699101	−	0.715023i	$-0.253584\pi$
$450$		0			0
$451$	−269.170			−0.596828
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−601.119	+	119.992i	−1.32114	+	0.263718i
$456$		0			0
$457$		−	88.4182i		−	0.193475i	−0.995310	−	0.0967376i	$-0.969159\pi$
$457$		−	88.4182i		−	0.193475i	0.995310	−	0.0967376i	$-0.0308408\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	487.138i		−	1.05670i	−0.849027	−	0.528349i	$-0.822811\pi$
$461$		−	487.138i		−	1.05670i	0.849027	−	0.528349i	$-0.177189\pi$
$462$		0			0
$463$			627.882i			1.35612i	0.735008	+	0.678058i	$0.237178\pi$
$463$			627.882i			1.35612i	−0.735008	+	0.678058i	$0.762822\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−323.410			−0.692527			−0.346263	−	0.938137i	$-0.612550\pi$
$467$	−323.410			−0.692527			−0.346263	+	0.938137i	$0.612550\pi$
$468$		0			0
$469$	126.988			0.270764
$470$		0			0
$471$		0			0
$472$		0			0
$473$	291.425			0.616121
$474$		0			0
$475$	−403.660	+	167.840i	−0.849811	+	0.353347i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	158.394i		−	0.330677i	−0.986237	−	0.165339i	$-0.947128\pi$
$479$		−	158.394i		−	0.330677i	0.986237	−	0.165339i	$-0.0528717\pi$
$480$		0			0
$481$	−341.714			−0.710424
$482$		0			0
$483$		0			0
$484$		0			0
$485$	100.518	+	503.564i	0.207254	+	1.03828i
$486$		0			0
$487$		−	108.301i		−	0.222385i	−0.993799	−	0.111192i	$-0.964533\pi$
$487$		−	108.301i		−	0.222385i	0.993799	−	0.111192i	$-0.0354670\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			679.140i			1.38318i	0.722291	+	0.691589i	$0.243089\pi$
$491$			679.140i			1.38318i	−0.722291	+	0.691589i	$0.756911\pi$
$492$		0			0
$493$		−	646.913i		−	1.31220i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−282.768			−0.568949
$498$		0			0
$499$	−74.3129			−0.148924			−0.0744618	−	0.997224i	$-0.523724\pi$
$499$	−74.3129			−0.148924			−0.0744618	+	0.997224i	$0.523724\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−278.694			−0.554063			−0.277031	−	0.960861i	$-0.589351\pi$
$503$	−278.694			−0.554063			−0.277031	+	0.960861i	$0.589351\pi$
$504$		0			0
$505$	108.506	+	543.580i	0.214863	+	1.07640i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			276.468i			0.543160i	0.962416	+	0.271580i	$0.0875461\pi$
$509$			276.468i			0.543160i	−0.962416	+	0.271580i	$0.912454\pi$
$510$		0			0
$511$	54.2827			0.106228
$512$		0			0
$513$		0			0
$514$		0			0
$515$	183.859	+	921.074i	0.357008	+	1.78849i
$516$		0			0
$517$		−	406.378i		−	0.786030i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			929.478i			1.78403i	0.452008	+	0.892014i	$0.350708\pi$
$521$			929.478i			1.78403i	−0.452008	+	0.892014i	$0.649292\pi$
$522$		0			0
$523$			217.275i			0.415439i	0.978188	+	0.207719i	$0.0666041\pi$
$523$			217.275i			0.415439i	−0.978188	+	0.207719i	$0.933396\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	754.032			1.43080
$528$		0			0
$529$	−417.024			−0.788326
$530$		0			0
$531$		0			0
$532$		0			0
$533$	465.572			0.873493
$534$		0			0
$535$	597.057	−	119.181i	1.11599	−	0.222768i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	48.0634i		−	0.0891714i
$540$		0			0
$541$	393.139			0.726690			0.363345	−	0.931655i	$-0.381635\pi$
$541$	393.139			0.726690			0.363345	+	0.931655i	$0.381635\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	222.290	−	44.3721i	0.407871	−	0.0814166i
$546$		0			0
$547$			876.683i			1.60271i	0.598188	+	0.801356i	$0.295888\pi$
$547$			876.683i			1.60271i	−0.598188	+	0.801356i	$0.704112\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	731.038i		−	1.32675i
$552$		0			0
$553$			174.125i			0.314874i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−546.180			−0.980575			−0.490287	−	0.871561i	$-0.663108\pi$
$557$	−546.180			−0.980575			−0.490287	+	0.871561i	$0.663108\pi$
$558$		0			0
$559$	−504.066			−0.901728
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−948.666			−1.68502			−0.842510	−	0.538681i	$-0.818923\pi$
$563$	−948.666			−1.68502			−0.842510	+	0.538681i	$0.818923\pi$
$564$		0			0
$565$	−94.1877	+	18.8011i	−0.166704	+	0.0332764i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	116.957i		−	0.205548i	−0.994705	−	0.102774i	$-0.967228\pi$
$569$		−	116.957i		−	0.205548i	0.994705	−	0.102774i	$-0.0327719\pi$
$570$		0			0
$571$	−143.844			−0.251916			−0.125958	−	0.992036i	$-0.540200\pi$
$571$	−143.844			−0.251916			−0.125958	+	0.992036i	$0.540200\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	244.272	−	101.567i	0.424821	−	0.176638i
$576$		0			0
$577$			665.087i			1.15266i	0.817216	+	0.576332i	$0.195516\pi$
$577$			665.087i			1.15266i	−0.817216	+	0.576332i	$0.804484\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			31.3720i			0.0539965i
$582$		0			0
$583$			872.385i			1.49637i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−223.307			−0.380421			−0.190210	−	0.981743i	$-0.560917\pi$
$587$	−223.307			−0.380421			−0.190210	+	0.981743i	$0.560917\pi$
$588$		0			0
$589$	852.086			1.44666
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−320.932			−0.541200			−0.270600	−	0.962692i	$-0.587222\pi$
$593$	−320.932			−0.541200			−0.270600	+	0.962692i	$0.587222\pi$
$594$		0			0
$595$	111.278	+	557.469i	0.187023	+	0.936923i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	619.547i		−	1.03430i	−0.855894	−	0.517151i	$-0.826992\pi$
$599$		−	619.547i		−	1.03430i	0.855894	−	0.517151i	$-0.173008\pi$
$600$		0			0
$601$	−583.311			−0.970567			−0.485283	−	0.874357i	$-0.661284\pi$
$601$	−583.311			−0.970567			−0.485283	+	0.874357i	$0.661284\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−136.982	+	27.3435i	−0.226417	+	0.0451959i
$606$		0			0
$607$			149.646i			0.246533i	0.992374	+	0.123267i	$0.0393370\pi$
$607$			149.646i			0.246533i	−0.992374	+	0.123267i	$0.960663\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			702.895i			1.15040i
$612$		0			0
$613$		−	775.281i		−	1.26473i	−0.774669	−	0.632367i	$-0.782084\pi$
$613$		−	775.281i		−	1.26473i	0.774669	−	0.632367i	$-0.217916\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1195.51			−1.93761			−0.968807	−	0.247817i	$-0.920287\pi$
$617$	−1195.51			−1.93761			−0.968807	+	0.247817i	$0.920287\pi$
$618$		0			0
$619$	−835.072			−1.34907			−0.674533	−	0.738244i	$-0.735655\pi$
$619$	−835.072			−1.34907			−0.674533	+	0.738244i	$0.735655\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1065.60			1.71044
$624$		0			0
$625$	440.748	−	443.132i	0.705197	−	0.709012i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			316.901i			0.503817i
$630$		0			0
$631$	378.090			0.599191			0.299596	−	0.954066i	$-0.403148\pi$
$631$	378.090			0.599191			0.299596	+	0.954066i	$0.403148\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	175.436	+	878.880i	0.276278	+	1.38406i
$636$		0			0
$637$			83.1333i			0.130508i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	81.4809i		−	0.127115i	−0.997978	−	0.0635577i	$-0.979755\pi$
$641$		−	81.4809i		−	0.127115i	0.997978	−	0.0635577i	$-0.0202447\pi$
$642$		0			0
$643$			849.766i			1.32157i	0.750577	+	0.660783i	$0.229775\pi$
$643$			849.766i			1.32157i	−0.750577	+	0.660783i	$0.770225\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	569.387			0.880042			0.440021	−	0.897988i	$-0.354971\pi$
$647$	569.387			0.880042			0.440021	+	0.897988i	$0.354971\pi$
$648$		0			0
$649$	534.793			0.824027
$650$		0			0
$651$		0			0
$652$		0			0
$653$	940.392			1.44011			0.720055	−	0.693917i	$-0.244117\pi$
$653$	940.392			1.44011			0.720055	+	0.693917i	$0.244117\pi$
$654$		0			0
$655$	83.1875	+	416.742i	0.127004	+	0.636248i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	1263.48i		−	1.91726i	−0.284651	−	0.958631i	$-0.591878\pi$
$659$		−	1263.48i		−	1.91726i	0.284651	−	0.958631i	$-0.408122\pi$
$660$		0			0
$661$	131.621			0.199125			0.0995623	−	0.995031i	$-0.468256\pi$
$661$	131.621			0.199125			0.0995623	+	0.995031i	$0.468256\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	125.749	+	629.962i	0.189096	+	0.947312i
$666$		0			0
$667$			442.382i			0.663242i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			1126.36i			1.67862i
$672$		0			0
$673$			238.795i			0.354822i	0.984137	+	0.177411i	$0.0567722\pi$
$673$			238.795i			0.354822i	−0.984137	+	0.177411i	$0.943228\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	351.521			0.519233			0.259617	−	0.965712i	$-0.416404\pi$
$677$	351.521			0.519233			0.259617	+	0.965712i	$0.416404\pi$
$678$		0			0
$679$	754.561			1.11128
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1067.52			−1.56299			−0.781495	−	0.623911i	$-0.785543\pi$
$683$	−1067.52			−1.56299			−0.781495	+	0.623911i	$0.785543\pi$
$684$		0			0
$685$	−323.978	+	64.6704i	−0.472961	+	0.0944094i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1508.93i		−	2.19003i
$690$		0			0
$691$	462.586			0.669444			0.334722	−	0.942317i	$-0.391358\pi$
$691$	462.586			0.669444			0.334722	+	0.942317i	$0.391358\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	46.1908	−	9.22031i	0.0664616	−	0.0132666i
$696$		0			0
$697$		−	431.765i		−	0.619462i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			519.309i			0.740812i	0.928870	+	0.370406i	$0.120781\pi$
$701$			519.309i			0.740812i	−0.928870	+	0.370406i	$0.879219\pi$
$702$		0			0
$703$			358.110i			0.509403i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	814.523			1.15208
$708$		0			0
$709$	−625.144			−0.881727			−0.440863	−	0.897574i	$-0.645328\pi$
$709$	−625.144			−0.881727			−0.440863	+	0.897574i	$0.645328\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−515.634			−0.723189
$714$		0			0
$715$	−789.266	+	157.548i	−1.10387	+	0.220347i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1385.24i		−	1.92662i	−0.268400	−	0.963308i	$-0.586495\pi$
$719$		−	1385.24i		−	1.92662i	0.268400	−	0.963308i	$-0.413505\pi$
$720$		0			0
$721$	1380.18			1.91425
$722$		0			0
$723$		0			0
$724$		0			0
$725$	401.261	+	965.047i	0.553464	+	1.33110i
$726$		0			0
$727$		−	1358.49i		−	1.86863i	−0.356451	−	0.934314i	$-0.616013\pi$
$727$		−	1358.49i		−	1.86863i	0.356451	−	0.934314i	$-0.383987\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			467.464i			0.639486i
$732$		0			0
$733$			926.431i			1.26389i	0.775013	+	0.631945i	$0.217743\pi$
$733$			926.431i			1.26389i	−0.775013	+	0.631945i	$0.782257\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	166.735			0.226235
$738$		0			0
$739$	−693.908			−0.938983			−0.469491	−	0.882937i	$-0.655563\pi$
$739$	−693.908			−0.938983			−0.469491	+	0.882937i	$0.655563\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−100.379			−0.135099			−0.0675497	−	0.997716i	$-0.521518\pi$
$743$	−100.379			−0.135099			−0.0675497	+	0.997716i	$0.521518\pi$
$744$		0			0
$745$	−122.108	−	611.724i	−0.163904	−	0.821106i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	894.655i		−	1.19447i
$750$		0			0
$751$	4.71806			0.00628237			0.00314118	−	0.999995i	$-0.499000\pi$
$751$	4.71806			0.00628237			0.00314118	+	0.999995i	$0.499000\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1431.57	−	285.761i	1.89612	−	0.378491i
$756$		0			0
$757$			133.356i			0.176163i	0.996113	+	0.0880816i	$0.0280736\pi$
$757$			133.356i			0.176163i	−0.996113	+	0.0880816i	$0.971926\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	219.732i		−	0.288741i	−0.989524	−	0.144371i	$-0.953884\pi$
$761$		−	219.732i		−	0.288741i	0.989524	−	0.144371i	$-0.0461157\pi$
$762$		0			0
$763$		−	333.088i		−	0.436551i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−925.011			−1.20601
$768$		0			0
$769$	−638.833			−0.830732			−0.415366	−	0.909654i	$-0.636346\pi$
$769$	−638.833			−0.830732			−0.415366	+	0.909654i	$0.636346\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1244.36			−1.60978			−0.804888	−	0.593426i	$-0.797775\pi$
$773$	−1244.36			−1.60978			−0.804888	+	0.593426i	$0.797775\pi$
$774$		0			0
$775$	−1124.84	+	467.704i	−1.45141	+	0.603489i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	487.911i		−	0.626330i
$780$		0			0
$781$	−371.273			−0.475381
$782$		0			0
$783$		0			0
$784$		0			0
$785$	140.226	+	702.488i	0.178632	+	0.894889i
$786$		0			0
$787$			1138.73i			1.44693i	0.690360	+	0.723466i	$0.257452\pi$
$787$			1138.73i			1.44693i	−0.690360	+	0.723466i	$0.742548\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			141.135i			0.178426i
$792$		0			0
$793$		−	1948.21i		−	2.45676i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1365.72			−1.71357			−0.856785	−	0.515673i	$-0.827542\pi$
$797$	−1365.72			−1.71357			−0.856785	+	0.515673i	$0.827542\pi$
$798$		0			0
$799$	651.855			0.815839
$800$		0			0
$801$		0			0
$802$		0			0
$803$	71.2729			0.0887582
$804$		0			0
$805$	−76.0962	−	381.217i	−0.0945294	−	0.473562i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			504.881i			0.624080i	0.950069	+	0.312040i	$0.101012\pi$
$809$			504.881i			0.624080i	−0.950069	+	0.312040i	$0.898988\pi$
$810$		0			0
$811$	−1484.09			−1.82995			−0.914974	−	0.403514i	$-0.867789\pi$
$811$	−1484.09			−1.82995			−0.914974	+	0.403514i	$0.867789\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−223.356	−	1118.94i	−0.274057	−	1.37294i
$816$		0			0
$817$			528.253i			0.646576i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	396.424i		−	0.482855i	−0.970419	−	0.241428i	$-0.922384\pi$
$821$		−	396.424i		−	0.482855i	0.970419	−	0.241428i	$-0.0776156\pi$
$822$		0			0
$823$		−	711.138i		−	0.864080i	−0.901854	−	0.432040i	$-0.857794\pi$
$823$		−	711.138i		−	0.864080i	0.901854	−	0.432040i	$-0.142206\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	296.720			0.358791			0.179396	−	0.983777i	$-0.442586\pi$
$827$	296.720			0.358791			0.179396	+	0.983777i	$0.442586\pi$
$828$		0			0
$829$	252.785			0.304928			0.152464	−	0.988309i	$-0.451279\pi$
$829$	252.785			0.304928			0.152464	+	0.988309i	$0.451279\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	77.0967			0.0925530
$834$		0			0
$835$	−297.899	+	59.4647i	−0.356765	+	0.0712152i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			291.010i			0.346854i	0.984847	+	0.173427i	$0.0554840\pi$
$839$			291.010i			0.346854i	−0.984847	+	0.173427i	$0.944516\pi$
$840$		0			0
$841$	−906.722			−1.07815
$842$		0			0
$843$		0			0
$844$		0			0
$845$	536.510	−	107.095i	0.634923	−	0.126739i
$846$		0			0
$847$			205.260i			0.242338i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	216.708i		−	0.254651i
$852$		0			0
$853$		−	43.6198i		−	0.0511370i	−0.999673	−	0.0255685i	$-0.991860\pi$
$853$		−	43.6198i		−	0.0511370i	0.999673	−	0.0255685i	$-0.00813959\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−343.361			−0.400654			−0.200327	−	0.979729i	$-0.564201\pi$
$857$	−343.361			−0.400654			−0.200327	+	0.979729i	$0.564201\pi$
$858$		0			0
$859$	15.4882			0.0180305			0.00901526	−	0.999959i	$-0.497130\pi$
$859$	15.4882			0.0180305			0.00901526	+	0.999959i	$0.497130\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	880.604			1.02040			0.510199	−	0.860056i	$-0.329572\pi$
$863$	880.604			1.02040			0.510199	+	0.860056i	$0.329572\pi$
$864$		0			0
$865$	1361.30	−	271.733i	1.57375	−	0.314143i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			228.626i			0.263091i
$870$		0			0
$871$	−288.395			−0.331108
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−511.784	−	762.594i	−0.584896	−	0.871536i
$876$		0			0
$877$		−	1243.43i		−	1.41783i	−0.705296	−	0.708913i	$-0.749186\pi$
$877$		−	1243.43i		−	1.41783i	0.705296	−	0.708913i	$-0.250814\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1006.63i		−	1.14260i	−0.820742	−	0.571299i	$-0.806440\pi$
$881$		−	1006.63i		−	1.14260i	0.820742	−	0.571299i	$-0.193560\pi$
$882$		0			0
$883$		−	765.263i		−	0.866662i	−0.901235	−	0.433331i	$-0.857338\pi$
$883$		−	765.263i		−	0.866662i	0.901235	−	0.433331i	$-0.142662\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−676.959			−0.763200			−0.381600	−	0.924328i	$-0.624627\pi$
$887$	−676.959			−0.763200			−0.381600	+	0.924328i	$0.624627\pi$
$888$		0			0
$889$	1316.95			1.48138
$890$		0			0
$891$		0			0
$892$		0			0
$893$	736.622			0.824885
$894$		0			0
$895$	−266.956	−	1337.36i	−0.298274	−	1.49426i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	2037.12i		−	2.26598i
$900$		0			0
$901$	−1399.36			−1.55312
$902$		0			0
$903$		0			0
$904$		0			0
$905$	541.214	−	108.034i	0.598026	−	0.119374i
$906$		0			0
$907$			23.4353i			0.0258383i	0.999917	+	0.0129191i	$0.00411240\pi$
$907$			23.4353i			0.0258383i	−0.999917	+	0.0129191i	$0.995888\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1253.82i			1.37631i	0.725563	+	0.688156i	$0.241580\pi$
$911$			1253.82i			1.37631i	−0.725563	+	0.688156i	$0.758420\pi$
$912$		0			0
$913$			41.1913i			0.0451164i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	624.464			0.680986
$918$		0			0
$919$	1113.99			1.21217			0.606087	−	0.795398i	$-0.292738\pi$
$919$	1113.99			1.21217			0.606087	+	0.795398i	$0.292738\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	642.176			0.695748
$924$		0			0
$925$	−196.564	−	472.744i	−0.212502	−	0.511074i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			635.968i			0.684573i	0.939596	+	0.342286i	$0.111201\pi$
$929$			635.968i			0.684573i	−0.939596	+	0.342286i	$0.888799\pi$
$930$		0			0
$931$	87.1223			0.0935792
$932$		0			0
$933$		0			0
$934$		0			0
$935$	146.108	+	731.955i	0.156265	+	0.782839i
$936$		0			0
$937$			81.1236i			0.0865780i	0.999063	+	0.0432890i	$0.0137836\pi$
$937$			81.1236i			0.0865780i	−0.999063	+	0.0432890i	$0.986216\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	338.123i		−	0.359323i	−0.983729	−	0.179661i	$-0.942500\pi$
$941$		−	338.123i		−	0.359323i	0.983729	−	0.179661i	$-0.0575002\pi$
$942$		0			0
$943$			295.256i			0.313103i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	389.778			0.411592			0.205796	−	0.978595i	$-0.434022\pi$
$947$	389.778			0.411592			0.205796	+	0.978595i	$0.434022\pi$
$948$		0			0
$949$	−123.278			−0.129903
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1439.63			−1.51063			−0.755314	−	0.655363i	$-0.772516\pi$
$953$	−1439.63			−1.51063			−0.755314	+	0.655363i	$0.772516\pi$
$954$		0			0
$955$	32.8933	+	164.785i	0.0344432	+	0.172550i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			485.462i			0.506217i
$960$		0			0
$961$	1413.43			1.47079
$962$		0			0
$963$		0			0
$964$		0			0
$965$	264.935	+	1327.24i	0.274544	+	1.37538i
$966$		0			0
$967$		−	665.617i		−	0.688332i	−0.938909	−	0.344166i	$-0.888162\pi$
$967$		−	665.617i		−	0.688332i	0.938909	−	0.344166i	$-0.111838\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	437.415i		−	0.450479i	−0.974303	−	0.225239i	$-0.927684\pi$
$971$		−	437.415i		−	0.450479i	0.974303	−	0.225239i	$-0.0723164\pi$
$972$		0			0
$973$		−	69.2142i		−	0.0711349i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1236.61			1.26572			0.632862	−	0.774265i	$-0.281880\pi$
$977$	1236.61			1.26572			0.632862	+	0.774265i	$0.281880\pi$
$978$		0			0
$979$	1399.13			1.42914
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1273.34			1.29536			0.647680	−	0.761913i	$-0.275739\pi$
$983$	1273.34			1.29536			0.647680	+	0.761913i	$0.275739\pi$
$984$		0			0
$985$	309.385	−	61.7575i	0.314097	−	0.0626980i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	319.668i		−	0.323224i
$990$		0			0
$991$	443.540			0.447568			0.223784	−	0.974639i	$-0.428159\pi$
$991$	443.540			0.447568			0.223784	+	0.974639i	$0.428159\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1387.78	+	277.020i	−1.39475	+	0.278412i
$996$		0			0
$997$			557.980i			0.559659i	0.960050	+	0.279830i	$0.0902780\pi$
$997$			557.980i			0.559659i	−0.960050	+	0.279830i	$0.909722\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.b.b.809.2		24
3.2	odd	2	inner	1620.3.b.b.809.23		24
5.4	even	2	inner	1620.3.b.b.809.24		24
9.2	odd	6		180.3.t.a.149.12	yes	24
9.4	even	3		180.3.t.a.29.1	✓	24
9.5	odd	6		540.3.t.a.89.3		24
9.7	even	3		540.3.t.a.449.7		24
15.14	odd	2	inner	1620.3.b.b.809.1		24
45.2	even	12		900.3.p.f.401.6		24
45.4	even	6		180.3.t.a.29.12	yes	24
45.7	odd	12		2700.3.p.f.2501.10		24
45.13	odd	12		900.3.p.f.101.7		24
45.14	odd	6		540.3.t.a.89.7		24
45.22	odd	12		900.3.p.f.101.6		24
45.23	even	12		2700.3.p.f.1601.3		24
45.29	odd	6		180.3.t.a.149.1	yes	24
45.32	even	12		2700.3.p.f.1601.10		24
45.34	even	6		540.3.t.a.449.3		24
45.38	even	12		900.3.p.f.401.7		24
45.43	odd	12		2700.3.p.f.2501.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.t.a.29.1	✓	24	9.4	even	3
180.3.t.a.29.12	yes	24	45.4	even	6
180.3.t.a.149.1	yes	24	45.29	odd	6
180.3.t.a.149.12	yes	24	9.2	odd	6
540.3.t.a.89.3		24	9.5	odd	6
540.3.t.a.89.7		24	45.14	odd	6
540.3.t.a.449.3		24	45.34	even	6
540.3.t.a.449.7		24	9.7	even	3
900.3.p.f.101.6		24	45.22	odd	12
900.3.p.f.101.7		24	45.13	odd	12
900.3.p.f.401.6		24	45.2	even	12
900.3.p.f.401.7		24	45.38	even	12
1620.3.b.b.809.1		24	15.14	odd	2	inner
1620.3.b.b.809.2		24	1.1	even	1	trivial
1620.3.b.b.809.23		24	3.2	odd	2	inner
1620.3.b.b.809.24		24	5.4	even	2	inner
2700.3.p.f.1601.3		24	45.23	even	12
2700.3.p.f.1601.10		24	45.32	even	12
2700.3.p.f.2501.3		24	45.43	odd	12
2700.3.p.f.2501.10		24	45.7	odd	12

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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.90327 + 0.978759i) q^{5} +7.34726i q^{7} +O(q^{10})\)	\(q+(-4.90327 + 0.978759i) q^{5} +7.34726i q^{7} +9.64692i q^{11} -16.6859i q^{13} -15.4743 q^{17} -17.4865 q^{19} +10.5819 q^{23} +(23.0841 - 9.59823i) q^{25} +41.8058i q^{29} -48.7281 q^{31} +(-7.19120 - 36.0256i) q^{35} -20.4792i q^{37} +27.9021i q^{41} -30.2091i q^{43} -42.1251 q^{47} -4.98225 q^{49} +90.4314 q^{53} +(-9.44201 - 47.3014i) q^{55} -55.4367i q^{59} +116.758 q^{61} +(16.3315 + 81.8154i) q^{65} -17.2837i q^{67} +38.4861i q^{71} -7.38815i q^{73} -70.8784 q^{77} +23.6993 q^{79} +4.26989 q^{83} +(75.8744 - 15.1456i) q^{85} -145.034i q^{89} +122.596 q^{91} +(85.7411 - 17.1151i) q^{95} -102.700i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 6 q^{25} - 60 q^{31} - 216 q^{49} + 42 q^{55} - 96 q^{61} - 228 q^{79} - 96 q^{85} - 84 q^{91}+O(q^{100}) \) 24 * q - 6 * q^25 - 60 * q^31 - 216 * q^49 + 42 * q^55 - 96 * q^61 - 228 * q^79 - 96 * q^85 - 84 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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