LMFDB - Embedded newform 810.4.e.e.541.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,4,Mod(271,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("810.271");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	810.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$47.7915471046$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	810.541
Dual form			810.4.e.e.271.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(2.50000 + 4.33013i) q^{5} +(-16.0000 + 27.7128i) q^{7} +8.00000 q^{8} -10.0000 q^{10} +(-30.0000 + 51.9615i) q^{11} +(17.0000 + 29.4449i) q^{13} +(-32.0000 - 55.4256i) q^{14} +(-8.00000 + 13.8564i) q^{16} -42.0000 q^{17} -76.0000 q^{19} +(10.0000 - 17.3205i) q^{20} +(-60.0000 - 103.923i) q^{22} +(-12.5000 + 21.6506i) q^{25} -68.0000 q^{26} +128.000 q^{28} +(3.00000 - 5.19615i) q^{29} +(116.000 + 200.918i) q^{31} +(-16.0000 - 27.7128i) q^{32} +(42.0000 - 72.7461i) q^{34} -160.000 q^{35} +134.000 q^{37} +(76.0000 - 131.636i) q^{38} +(20.0000 + 34.6410i) q^{40} +(117.000 + 202.650i) q^{41} +(206.000 - 356.802i) q^{43} +240.000 q^{44} +(-180.000 + 311.769i) q^{47} +(-340.500 - 589.763i) q^{49} +(-25.0000 - 43.3013i) q^{50} +(68.0000 - 117.779i) q^{52} -222.000 q^{53} -300.000 q^{55} +(-128.000 + 221.703i) q^{56} +(6.00000 + 10.3923i) q^{58} +(330.000 + 571.577i) q^{59} +(245.000 - 424.352i) q^{61} -464.000 q^{62} +64.0000 q^{64} +(-85.0000 + 147.224i) q^{65} +(-406.000 - 703.213i) q^{67} +(84.0000 + 145.492i) q^{68} +(160.000 - 277.128i) q^{70} -120.000 q^{71} +746.000 q^{73} +(-134.000 + 232.095i) q^{74} +(152.000 + 263.272i) q^{76} +(-960.000 - 1662.77i) q^{77} +(-76.0000 + 131.636i) q^{79} -80.0000 q^{80} -468.000 q^{82} +(-402.000 + 696.284i) q^{83} +(-105.000 - 181.865i) q^{85} +(412.000 + 713.605i) q^{86} +(-240.000 + 415.692i) q^{88} +678.000 q^{89} -1088.00 q^{91} +(-360.000 - 623.538i) q^{94} +(-190.000 - 329.090i) q^{95} +(-97.0000 + 168.009i) q^{97} +1362.00 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 4 q^{4} + 5 q^{5} - 32 q^{7} + 16 q^{8} - 20 q^{10} - 60 q^{11} + 34 q^{13} - 64 q^{14} - 16 q^{16} - 84 q^{17} - 152 q^{19} + 20 q^{20} - 120 q^{22} - 25 q^{25} - 136 q^{26} + 256 q^{28}+ \cdots + 2724 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 4 * q^4 + 5 * q^5 - 32 * q^7 + 16 * q^8 - 20 * q^10 - 60 * q^11 + 34 * q^13 - 64 * q^14 - 16 * q^16 - 84 * q^17 - 152 * q^19 + 20 * q^20 - 120 * q^22 - 25 * q^25 - 136 * q^26 + 256 * q^28 + 6 * q^29 + 232 * q^31 - 32 * q^32 + 84 * q^34 - 320 * q^35 + 268 * q^37 + 152 * q^38 + 40 * q^40 + 234 * q^41 + 412 * q^43 + 480 * q^44 - 360 * q^47 - 681 * q^49 - 50 * q^50 + 136 * q^52 - 444 * q^53 - 600 * q^55 - 256 * q^56 + 12 * q^58 + 660 * q^59 + 490 * q^61 - 928 * q^62 + 128 * q^64 - 170 * q^65 - 812 * q^67 + 168 * q^68 + 320 * q^70 - 240 * q^71 + 1492 * q^73 - 268 * q^74 + 304 * q^76 - 1920 * q^77 - 152 * q^79 - 160 * q^80 - 936 * q^82 - 804 * q^83 - 210 * q^85 + 824 * q^86 - 480 * q^88 + 1356 * q^89 - 2176 * q^91 - 720 * q^94 - 380 * q^95 - 194 * q^97 + 2724 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.00000	+	1.73205i	−0.353553	+	0.612372i
$2$	−1.00000	+	1.73205i	−0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	−	3.46410i	−0.250000	−	0.433013i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$6$		0			0
$7$	−16.0000	+	27.7128i	−0.863919	+	1.49635i	0.00419795	+	0.999991i	$0.498664\pi$
$7$	−16.0000	+	27.7128i	−0.863919	+	1.49635i	−0.868117	+	0.496360i	$0.834670\pi$
$8$	8.00000			0.353553
$9$		0			0
$10$	−10.0000			−0.316228
$11$	−30.0000	+	51.9615i	−0.822304	+	1.42427i	0.0816590	+	0.996660i	$0.473978\pi$
$11$	−30.0000	+	51.9615i	−0.822304	+	1.42427i	−0.903963	+	0.427611i	$0.859355\pi$
$12$		0			0
$13$	17.0000	+	29.4449i	0.362689	+	0.628195i	0.988402	−	0.151858i	$-0.0485255\pi$
$13$	17.0000	+	29.4449i	0.362689	+	0.628195i	−0.625714	+	0.780053i	$0.715192\pi$
$14$	−32.0000	−	55.4256i	−0.610883	−	1.05808i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	−42.0000			−0.599206			−0.299603	−	0.954064i	$-0.596854\pi$
$17$	−42.0000			−0.599206			−0.299603	+	0.954064i	$0.596854\pi$
$18$		0			0
$19$	−76.0000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−76.0000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$	10.0000	−	17.3205i	0.111803	−	0.193649i
$21$		0			0
$22$	−60.0000	−	103.923i	−0.581456	−	1.00711i
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$	−68.0000			−0.512919
$27$		0			0
$28$	128.000			0.863919
$29$	3.00000	−	5.19615i	0.0192099	−	0.0332725i	−0.856261	−	0.516544i	$-0.827218\pi$
$29$	3.00000	−	5.19615i	0.0192099	−	0.0332725i	0.875471	+	0.483272i	$0.160552\pi$
$30$		0			0
$31$	116.000	+	200.918i	0.672071	+	1.16406i	0.977316	+	0.211788i	$0.0679286\pi$
$31$	116.000	+	200.918i	0.672071	+	1.16406i	−0.305244	+	0.952274i	$0.598738\pi$
$32$	−16.0000	−	27.7128i	−0.0883883	−	0.153093i
$33$		0			0
$34$	42.0000	−	72.7461i	0.211851	−	0.366937i
$35$	−160.000			−0.772712
$36$		0			0
$37$	134.000			0.595391			0.297695	−	0.954661i	$-0.403782\pi$
$37$	134.000			0.595391			0.297695	+	0.954661i	$0.403782\pi$
$38$	76.0000	−	131.636i	0.324443	−	0.561951i
$39$		0			0
$40$	20.0000	+	34.6410i	0.0790569	+	0.136931i
$41$	117.000	+	202.650i	0.445667	+	0.771917i	0.998098	−	0.0616409i	$-0.0196334\pi$
$41$	117.000	+	202.650i	0.445667	+	0.771917i	−0.552432	+	0.833558i	$0.686300\pi$
$42$		0			0
$43$	206.000	−	356.802i	0.730575	−	1.26539i	−0.226063	−	0.974113i	$-0.572586\pi$
$43$	206.000	−	356.802i	0.730575	−	1.26539i	0.956638	−	0.291280i	$-0.0940810\pi$
$44$	240.000			0.822304
$45$		0			0
$46$		0			0
$47$	−180.000	+	311.769i	−0.558632	+	0.967579i	0.438979	+	0.898497i	$0.355340\pi$
$47$	−180.000	+	311.769i	−0.558632	+	0.967579i	−0.997611	+	0.0690815i	$0.977993\pi$
$48$		0			0
$49$	−340.500	−	589.763i	−0.992711	−	1.71943i
$50$	−25.0000	−	43.3013i	−0.0707107	−	0.122474i
$51$		0			0
$52$	68.0000	−	117.779i	0.181344	−	0.314098i
$53$	−222.000			−0.575359			−0.287680	−	0.957727i	$-0.592884\pi$
$53$	−222.000			−0.575359			−0.287680	+	0.957727i	$0.592884\pi$
$54$		0			0
$55$	−300.000			−0.735491
$56$	−128.000	+	221.703i	−0.305441	+	0.529040i
$57$		0			0
$58$	6.00000	+	10.3923i	0.0135834	+	0.0235272i
$59$	330.000	+	571.577i	0.728175	+	1.26124i	0.957654	+	0.287923i	$0.0929647\pi$
$59$	330.000	+	571.577i	0.728175	+	1.26124i	−0.229478	+	0.973314i	$0.573702\pi$
$60$		0			0
$61$	245.000	−	424.352i	0.514246	−	0.890701i	−0.485617	−	0.874172i	$-0.661405\pi$
$61$	245.000	−	424.352i	0.514246	−	0.890701i	0.999863	−	0.0165293i	$-0.00526168\pi$
$62$	−464.000			−0.950453
$63$		0			0
$64$	64.0000			0.125000
$65$	−85.0000	+	147.224i	−0.162199	+	0.280937i
$66$		0			0
$67$	−406.000	−	703.213i	−0.740310	−	1.28226i	−0.952354	−	0.304995i	$-0.901345\pi$
$67$	−406.000	−	703.213i	−0.740310	−	1.28226i	0.212044	−	0.977260i	$-0.431988\pi$
$68$	84.0000	+	145.492i	0.149801	+	0.259464i
$69$		0			0
$70$	160.000	−	277.128i	0.273195	−	0.473188i
$71$	−120.000			−0.200583			−0.100291	−	0.994958i	$-0.531978\pi$
$71$	−120.000			−0.200583			−0.100291	+	0.994958i	$0.531978\pi$
$72$		0			0
$73$	746.000			1.19606			0.598032	−	0.801472i	$-0.295949\pi$
$73$	746.000			1.19606			0.598032	+	0.801472i	$0.295949\pi$
$74$	−134.000	+	232.095i	−0.210502	+	0.364601i
$75$		0			0
$76$	152.000	+	263.272i	0.229416	+	0.397360i
$77$	−960.000	−	1662.77i	−1.42081	−	2.46091i
$78$		0			0
$79$	−76.0000	+	131.636i	−0.108236	+	0.187471i	−0.915056	−	0.403327i	$-0.867854\pi$
$79$	−76.0000	+	131.636i	−0.108236	+	0.187471i	0.806820	+	0.590798i	$0.201187\pi$
$80$	−80.0000			−0.111803
$81$		0			0
$82$	−468.000			−0.630268
$83$	−402.000	+	696.284i	−0.531629	+	0.920809i	0.467689	+	0.883893i	$0.345087\pi$
$83$	−402.000	+	696.284i	−0.531629	+	0.920809i	−0.999318	+	0.0369159i	$0.988247\pi$
$84$		0			0
$85$	−105.000	−	181.865i	−0.133986	−	0.232071i
$86$	412.000	+	713.605i	0.516594	+	0.894767i
$87$		0			0
$88$	−240.000	+	415.692i	−0.290728	+	0.503556i
$89$	678.000			0.807504			0.403752	−	0.914868i	$-0.367706\pi$
$89$	678.000			0.807504			0.403752	+	0.914868i	$0.367706\pi$
$90$		0			0
$91$	−1088.00			−1.25333
$92$		0			0
$93$		0			0
$94$	−360.000	−	623.538i	−0.395012	−	0.684182i
$95$	−190.000	−	329.090i	−0.205196	−	0.355409i
$96$		0			0
$97$	−97.0000	+	168.009i	−0.101535	+	0.175863i	−0.912317	−	0.409484i	$-0.865709\pi$
$97$	−97.0000	+	168.009i	−0.101535	+	0.175863i	0.810782	+	0.585348i	$0.199042\pi$
$98$	1362.00			1.40391
$99$		0			0
$100$	100.000			0.100000
$101$	399.000	−	691.088i	0.393089	−	0.680850i	−0.599766	−	0.800175i	$-0.704740\pi$
$101$	399.000	−	691.088i	0.393089	−	0.680850i	0.992855	+	0.119325i	$0.0380731\pi$
$102$		0			0
$103$	−544.000	−	942.236i	−0.520407	−	0.901371i	−0.999718	−	0.0237264i	$-0.992447\pi$
$103$	−544.000	−	942.236i	−0.520407	−	0.901371i	0.479312	−	0.877645i	$-0.340886\pi$
$104$	136.000	+	235.559i	0.128230	+	0.222100i
$105$		0			0
$106$	222.000	−	384.515i	0.203420	−	0.352334i
$107$	−1716.00			−1.55039			−0.775196	−	0.631721i	$-0.782349\pi$
$107$	−1716.00			−1.55039			−0.775196	+	0.631721i	$0.782349\pi$
$108$		0			0
$109$	−970.000			−0.852378			−0.426189	−	0.904634i	$-0.640144\pi$
$109$	−970.000			−0.852378			−0.426189	+	0.904634i	$0.640144\pi$
$110$	300.000	−	519.615i	0.260035	−	0.450394i
$111$		0			0
$112$	−256.000	−	443.405i	−0.215980	−	0.374088i
$113$	213.000	+	368.927i	0.177322	+	0.307130i	0.940962	−	0.338511i	$-0.109923\pi$
$113$	213.000	+	368.927i	0.177322	+	0.307130i	−0.763641	+	0.645642i	$0.776590\pi$
$114$		0			0
$115$		0			0
$116$	−24.0000			−0.0192099
$117$		0			0
$118$	−1320.00			−1.02980
$119$	672.000	−	1163.94i	0.517665	−	0.896622i
$120$		0			0
$121$	−1134.50	−	1965.01i	−0.852367	−	1.47634i
$122$	490.000	+	848.705i	0.363627	+	0.629821i
$123$		0			0
$124$	464.000	−	803.672i	0.336036	−	0.582031i
$125$	−125.000			−0.0894427
$126$		0			0
$127$	200.000			0.139741			0.0698706	−	0.997556i	$-0.477741\pi$
$127$	200.000			0.139741			0.0698706	+	0.997556i	$0.477741\pi$
$128$	−64.0000	+	110.851i	−0.0441942	+	0.0765466i
$129$		0			0
$130$	−170.000	−	294.449i	−0.114692	−	0.198653i
$131$	30.0000	+	51.9615i	0.0200085	+	0.0346557i	0.875856	−	0.482572i	$-0.160297\pi$
$131$	30.0000	+	51.9615i	0.0200085	+	0.0346557i	−0.855848	+	0.517228i	$0.826964\pi$
$132$		0			0
$133$	1216.00	−	2106.17i	0.792786	−	1.37315i
$134$	1624.00			1.04696
$135$		0			0
$136$	−336.000			−0.211851
$137$	321.000	−	555.988i	0.200182	−	0.346725i	−0.748405	−	0.663242i	$-0.769180\pi$
$137$	321.000	−	555.988i	0.200182	−	0.346725i	0.948587	+	0.316517i	$0.102513\pi$
$138$		0			0
$139$	1418.00	+	2456.05i	0.865275	+	1.49870i	0.866774	+	0.498701i	$0.166189\pi$
$139$	1418.00	+	2456.05i	0.865275	+	1.49870i	−0.00149936	+	0.999999i	$0.500477\pi$
$140$	320.000	+	554.256i	0.193178	+	0.334594i
$141$		0			0
$142$	120.000	−	207.846i	0.0709167	−	0.122831i
$143$	−2040.00			−1.19296
$144$		0			0
$145$	30.0000			0.0171818
$146$	−746.000	+	1292.11i	−0.422873	+	0.732437i
$147$		0			0
$148$	−268.000	−	464.190i	−0.148848	−	0.257812i
$149$	−777.000	−	1345.80i	−0.427210	−	0.739950i	0.569414	−	0.822051i	$-0.307170\pi$
$149$	−777.000	−	1345.80i	−0.427210	−	0.739950i	−0.996624	+	0.0821013i	$0.973837\pi$
$150$		0			0
$151$	1136.00	−	1967.61i	0.612228	−	1.06041i	−0.378637	−	0.925545i	$-0.623607\pi$
$151$	1136.00	−	1967.61i	0.612228	−	1.06041i	0.990864	−	0.134864i	$-0.0430597\pi$
$152$	−608.000			−0.324443
$153$		0			0
$154$	3840.00			2.00932
$155$	−580.000	+	1004.59i	−0.300559	+	0.520584i
$156$		0			0
$157$	−847.000	−	1467.05i	−0.430560	−	0.745752i	0.566361	−	0.824157i	$-0.308351\pi$
$157$	−847.000	−	1467.05i	−0.430560	−	0.745752i	−0.996922	+	0.0784048i	$0.975017\pi$
$158$	−152.000	−	263.272i	−0.0765346	−	0.132562i
$159$		0			0
$160$	80.0000	−	138.564i	0.0395285	−	0.0684653i
$161$		0			0
$162$		0			0
$163$	−52.0000			−0.0249874			−0.0124937	−	0.999922i	$-0.503977\pi$
$163$	−52.0000			−0.0249874			−0.0124937	+	0.999922i	$0.503977\pi$
$164$	468.000	−	810.600i	0.222833	−	0.385959i
$165$		0			0
$166$	−804.000	−	1392.57i	−0.375919	−	0.651110i
$167$	−600.000	−	1039.23i	−0.278020	−	0.481545i	0.692872	−	0.721060i	$-0.256345\pi$
$167$	−600.000	−	1039.23i	−0.278020	−	0.481545i	−0.970893	+	0.239515i	$0.923012\pi$
$168$		0			0
$169$	520.500	−	901.532i	0.236914	−	0.410347i
$170$	420.000			0.189485
$171$		0			0
$172$	−1648.00			−0.730575
$173$	27.0000	−	46.7654i	0.0118657	−	0.0205521i	−0.860032	−	0.510241i	$-0.829556\pi$
$173$	27.0000	−	46.7654i	0.0118657	−	0.0205521i	0.871897	+	0.489689i	$0.162890\pi$
$174$		0			0
$175$	−400.000	−	692.820i	−0.172784	−	0.299270i
$176$	−480.000	−	831.384i	−0.205576	−	0.356068i
$177$		0			0
$178$	−678.000	+	1174.33i	−0.285496	+	0.494493i
$179$	−876.000			−0.365784			−0.182892	−	0.983133i	$-0.558546\pi$
$179$	−876.000			−0.365784			−0.182892	+	0.983133i	$0.558546\pi$
$180$		0			0
$181$	3854.00			1.58268			0.791341	−	0.611375i	$-0.209383\pi$
$181$	3854.00			1.58268			0.791341	+	0.611375i	$0.209383\pi$
$182$	1088.00	−	1884.47i	0.443120	−	0.767507i
$183$		0			0
$184$		0			0
$185$	335.000	+	580.237i	0.133133	+	0.230594i
$186$		0			0
$187$	1260.00	−	2182.38i	0.492729	−	0.853432i
$188$	1440.00			0.558632
$189$		0			0
$190$	760.000			0.290191
$191$	−1392.00	+	2411.01i	−0.527338	+	0.913376i	0.472154	+	0.881516i	$0.343477\pi$
$191$	−1392.00	+	2411.01i	−0.527338	+	0.913376i	−0.999492	+	0.0318605i	$0.989857\pi$
$192$		0			0
$193$	−457.000	−	791.547i	−0.170443	−	0.295217i	0.768132	−	0.640292i	$-0.221187\pi$
$193$	−457.000	−	791.547i	−0.170443	−	0.295217i	−0.938575	+	0.345075i	$0.887853\pi$
$194$	−194.000	−	336.018i	−0.0717958	−	0.124354i
$195$		0			0
$196$	−1362.00	+	2359.05i	−0.496356	+	0.859713i
$197$	5202.00			1.88136			0.940678	−	0.339300i	$-0.110190\pi$
$197$	5202.00			1.88136			0.940678	+	0.339300i	$0.110190\pi$
$198$		0			0
$199$	3152.00			1.12281			0.561405	−	0.827541i	$-0.310261\pi$
$199$	3152.00			1.12281			0.561405	+	0.827541i	$0.310261\pi$
$200$	−100.000	+	173.205i	−0.0353553	+	0.0612372i
$201$		0			0
$202$	798.000	+	1382.18i	0.277956	+	0.481434i
$203$	96.0000	+	166.277i	0.0331915	+	0.0574894i
$204$		0			0
$205$	−585.000	+	1013.25i	−0.199308	+	0.345212i
$206$	2176.00			0.735967
$207$		0			0
$208$	−544.000			−0.181344
$209$	2280.00	−	3949.08i	0.754598	−	1.30700i
$210$		0			0
$211$	−370.000	−	640.859i	−0.120720	−	0.209093i	0.799332	−	0.600890i	$-0.205187\pi$
$211$	−370.000	−	640.859i	−0.120720	−	0.209093i	−0.920052	+	0.391797i	$0.871854\pi$
$212$	444.000	+	769.031i	0.143840	+	0.249138i
$213$		0			0
$214$	1716.00	−	2972.20i	0.548146	−	0.949418i
$215$	2060.00			0.653446
$216$		0			0
$217$	−7424.00			−2.32246
$218$	970.000	−	1680.09i	0.301361	−	0.521972i
$219$		0			0
$220$	600.000	+	1039.23i	0.183873	+	0.318477i
$221$	−714.000	−	1236.68i	−0.217325	−	0.376418i
$222$		0			0
$223$	260.000	−	450.333i	0.0780757	−	0.135231i	−0.824344	−	0.566089i	$-0.808456\pi$
$223$	260.000	−	450.333i	0.0780757	−	0.135231i	0.902420	+	0.430858i	$0.141789\pi$
$224$	1024.00			0.305441
$225$		0			0
$226$	−852.000			−0.250771
$227$	198.000	−	342.946i	0.0578930	−	0.100274i	−0.835626	−	0.549298i	$-0.814895\pi$
$227$	198.000	−	342.946i	0.0578930	−	0.100274i	0.893519	+	0.449025i	$0.148228\pi$
$228$		0			0
$229$	665.000	+	1151.81i	0.191897	+	0.332376i	0.945879	−	0.324520i	$-0.105203\pi$
$229$	665.000	+	1151.81i	0.191897	+	0.332376i	−0.753982	+	0.656895i	$0.771869\pi$
$230$		0			0
$231$		0			0
$232$	24.0000	−	41.5692i	0.00679171	−	0.0117636i
$233$	−4866.00			−1.36816			−0.684082	−	0.729405i	$-0.739797\pi$
$233$	−4866.00			−1.36816			−0.684082	+	0.729405i	$0.739797\pi$
$234$		0			0
$235$	−1800.00			−0.499656
$236$	1320.00	−	2286.31i	0.364088	−	0.630618i
$237$		0			0
$238$	1344.00	+	2327.88i	0.366044	+	0.634008i
$239$	−912.000	−	1579.63i	−0.246830	−	0.427522i	0.715815	−	0.698290i	$-0.246056\pi$
$239$	−912.000	−	1579.63i	−0.246830	−	0.427522i	−0.962645	+	0.270768i	$0.912722\pi$
$240$		0			0
$241$	−3241.00	+	5613.58i	−0.866270	+	1.50042i	−0.000490169		1.00000i	$0.500156\pi$
$241$	−3241.00	+	5613.58i	−0.866270	+	1.50042i	−0.865780	+	0.500424i	$0.833177\pi$
$242$	4538.00			1.20543
$243$		0			0
$244$	−1960.00			−0.514246
$245$	1702.50	−	2948.82i	0.443954	−	0.768951i
$246$		0			0
$247$	−1292.00	−	2237.81i	−0.332826	−	0.576471i
$248$	928.000	+	1607.34i	0.237613	+	0.411558i
$249$		0			0
$250$	125.000	−	216.506i	0.0316228	−	0.0547723i
$251$	−1476.00			−0.371172			−0.185586	−	0.982628i	$-0.559418\pi$
$251$	−1476.00			−0.371172			−0.185586	+	0.982628i	$0.559418\pi$
$252$		0			0
$253$		0			0
$254$	−200.000	+	346.410i	−0.0494060	+	0.0855736i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	2157.00	+	3736.03i	0.523541	+	0.906799i	0.999625	+	0.0273992i	$0.00872254\pi$
$257$	2157.00	+	3736.03i	0.523541	+	0.906799i	−0.476084	+	0.879400i	$0.657944\pi$
$258$		0			0
$259$	−2144.00	+	3713.52i	−0.514369	+	0.890914i
$260$	680.000			0.162199
$261$		0			0
$262$	−120.000			−0.0282963
$263$	−2640.00	+	4572.61i	−0.618971	+	1.07209i	0.370703	+	0.928752i	$0.379117\pi$
$263$	−2640.00	+	4572.61i	−0.618971	+	1.07209i	−0.989674	+	0.143338i	$0.954216\pi$
$264$		0			0
$265$	−555.000	−	961.288i	−0.128654	−	0.222836i
$266$	2432.00	+	4212.35i	0.560585	+	0.970961i
$267$		0			0
$268$	−1624.00	+	2812.85i	−0.370155	+	0.641128i
$269$	−5526.00			−1.25251			−0.626257	−	0.779617i	$-0.715414\pi$
$269$	−5526.00			−1.25251			−0.626257	+	0.779617i	$0.715414\pi$
$270$		0			0
$271$	2024.00			0.453687			0.226844	−	0.973931i	$-0.427159\pi$
$271$	2024.00			0.453687			0.226844	+	0.973931i	$0.427159\pi$
$272$	336.000	−	581.969i	0.0749007	−	0.129732i
$273$		0			0
$274$	642.000	+	1111.98i	0.141550	+	0.245171i
$275$	−750.000	−	1299.04i	−0.164461	−	0.284854i
$276$		0			0
$277$	−1027.00	+	1778.82i	−0.222767	+	0.385844i	−0.955647	−	0.294514i	$-0.904842\pi$
$277$	−1027.00	+	1778.82i	−0.222767	+	0.385844i	0.732880	+	0.680358i	$0.238176\pi$
$278$	−5672.00			−1.22368
$279$		0			0
$280$	−1280.00			−0.273195
$281$	−3651.00	+	6323.72i	−0.775090	+	1.34250i	0.159653	+	0.987173i	$0.448962\pi$
$281$	−3651.00	+	6323.72i	−0.775090	+	1.34250i	−0.934744	+	0.355323i	$0.884371\pi$
$282$		0			0
$283$	1862.00	+	3225.08i	0.391111	+	0.677424i	0.992596	−	0.121460i	$-0.0387575\pi$
$283$	1862.00	+	3225.08i	0.391111	+	0.677424i	−0.601485	+	0.798884i	$0.705424\pi$
$284$	240.000	+	415.692i	0.0501457	+	0.0868549i
$285$		0			0
$286$	2040.00	−	3533.38i	0.421775	−	0.730536i
$287$	−7488.00			−1.54008
$288$		0			0
$289$	−3149.00			−0.640953
$290$	−30.0000	+	51.9615i	−0.00607469	+	0.0105217i
$291$		0			0
$292$	−1492.00	−	2584.22i	−0.299016	−	0.517911i
$293$	−3609.00	−	6250.97i	−0.719591	−	1.24637i	−0.961162	−	0.275984i	$-0.910996\pi$
$293$	−3609.00	−	6250.97i	−0.719591	−	1.24637i	0.241572	−	0.970383i	$-0.422337\pi$
$294$		0			0
$295$	−1650.00	+	2857.88i	−0.325650	+	0.564042i
$296$	1072.00			0.210502
$297$		0			0
$298$	3108.00			0.604166
$299$		0			0
$300$		0			0
$301$	6592.00	+	11417.7i	1.26231	+	2.18639i
$302$	2272.00	+	3935.22i	0.432910	+	0.749823i
$303$		0			0
$304$	608.000	−	1053.09i	0.114708	−	0.198680i
$305$	2450.00			0.459956
$306$		0			0
$307$	2540.00			0.472200			0.236100	−	0.971729i	$-0.424131\pi$
$307$	2540.00			0.472200			0.236100	+	0.971729i	$0.424131\pi$
$308$	−3840.00	+	6651.08i	−0.710404	+	1.23046i
$309$		0			0
$310$	−1160.00	−	2009.18i	−0.212528	−	0.368109i
$311$	780.000	+	1351.00i	0.142218	+	0.246328i	0.928332	−	0.371753i	$-0.121243\pi$
$311$	780.000	+	1351.00i	0.142218	+	0.246328i	−0.786114	+	0.618082i	$0.787910\pi$
$312$		0			0
$313$	467.000	−	808.868i	0.0843335	−	0.146070i	−0.820773	−	0.571254i	$-0.806457\pi$
$313$	467.000	−	808.868i	0.0843335	−	0.146070i	0.905107	+	0.425184i	$0.139791\pi$
$314$	3388.00			0.608904
$315$		0			0
$316$	608.000			0.108236
$317$	−837.000	+	1449.73i	−0.148298	+	0.256860i	−0.930599	−	0.366041i	$-0.880713\pi$
$317$	−837.000	+	1449.73i	−0.148298	+	0.256860i	0.782300	+	0.622902i	$0.214046\pi$
$318$		0			0
$319$	180.000	+	311.769i	0.0315927	+	0.0547201i
$320$	160.000	+	277.128i	0.0279508	+	0.0484123i
$321$		0			0
$322$		0			0
$323$	3192.00			0.549869
$324$		0			0
$325$	−850.000			−0.145075
$326$	52.0000	−	90.0666i	0.00883440	−	0.0153016i
$327$		0			0
$328$	936.000	+	1621.20i	0.157567	+	0.272914i
$329$	−5760.00	−	9976.61i	−0.965225	−	1.67182i
$330$		0			0
$331$	1994.00	−	3453.71i	0.331118	−	0.573514i	−0.651613	−	0.758551i	$-0.725907\pi$
$331$	1994.00	−	3453.71i	0.331118	−	0.573514i	0.982731	+	0.185038i	$0.0592408\pi$
$332$	3216.00			0.531629
$333$		0			0
$334$	2400.00			0.393180
$335$	2030.00	−	3516.06i	0.331077	−	0.573442i
$336$		0			0
$337$	−1.00000	−	1.73205i	−0.000161642	−	0.000279973i	0.865945	−	0.500140i	$-0.166718\pi$
$337$	−1.00000	−	1.73205i	−0.000161642	−	0.000279973i	−0.866106	+	0.499860i	$0.833385\pi$
$338$	1041.00	+	1803.06i	0.167523	+	0.290159i
$339$		0			0
$340$	−420.000	+	727.461i	−0.0669932	+	0.116036i
$341$	−13920.0			−2.21059
$342$		0			0
$343$	10816.0			1.70265
$344$	1648.00	−	2854.42i	0.258297	−	0.447384i
$345$		0			0
$346$	54.0000	+	93.5307i	0.00839034	+	0.0145325i
$347$	882.000	+	1527.67i	0.136450	+	0.236339i	0.926151	−	0.377154i	$-0.123097\pi$
$347$	882.000	+	1527.67i	0.136450	+	0.236339i	−0.789700	+	0.613493i	$0.789764\pi$
$348$		0			0
$349$	−2155.00	+	3732.57i	−0.330529	+	0.572492i	−0.982616	−	0.185652i	$-0.940560\pi$
$349$	−2155.00	+	3732.57i	−0.330529	+	0.572492i	0.652087	+	0.758144i	$0.273894\pi$
$350$	1600.00			0.244353
$351$		0			0
$352$	1920.00			0.290728
$353$	69.0000	−	119.512i	0.0104037	−	0.0180197i	−0.860777	−	0.508983i	$-0.830022\pi$
$353$	69.0000	−	119.512i	0.0104037	−	0.0180197i	0.871180	+	0.490963i	$0.163355\pi$
$354$		0			0
$355$	−300.000	−	519.615i	−0.0448517	−	0.0776854i
$356$	−1356.00	−	2348.66i	−0.201876	−	0.349659i
$357$		0			0
$358$	876.000	−	1517.28i	0.129324	−	0.223996i
$359$	11976.0			1.76064			0.880319	−	0.474382i	$-0.157328\pi$
$359$	11976.0			1.76064			0.880319	+	0.474382i	$0.157328\pi$
$360$		0			0
$361$	−1083.00			−0.157895
$362$	−3854.00	+	6675.32i	−0.559563	+	0.969191i
$363$		0			0
$364$	2176.00	+	3768.94i	0.313333	+	0.542710i
$365$	1865.00	+	3230.27i	0.267448	+	0.463234i
$366$		0			0
$367$	−4852.00	+	8403.91i	−0.690115	+	1.19531i	0.281684	+	0.959507i	$0.409107\pi$
$367$	−4852.00	+	8403.91i	−0.690115	+	1.19531i	−0.971800	+	0.235808i	$0.924226\pi$
$368$		0			0
$369$		0			0
$370$	−1340.00			−0.188279
$371$	3552.00	−	6152.24i	0.497064	−	0.860940i
$372$		0			0
$373$	4061.00	+	7033.86i	0.563728	+	0.976406i	0.997167	+	0.0752227i	$0.0239668\pi$
$373$	4061.00	+	7033.86i	0.563728	+	0.976406i	−0.433439	+	0.901183i	$0.642700\pi$
$374$	2520.00	+	4364.77i	0.348412	+	0.603467i
$375$		0			0
$376$	−1440.00	+	2494.15i	−0.197506	+	0.342091i
$377$	204.000			0.0278688
$378$		0			0
$379$	3404.00			0.461350			0.230675	−	0.973031i	$-0.425907\pi$
$379$	3404.00			0.461350			0.230675	+	0.973031i	$0.425907\pi$
$380$	−760.000	+	1316.36i	−0.102598	+	0.177705i
$381$		0			0
$382$	−2784.00	−	4822.03i	−0.372884	−	0.645855i
$383$	−1260.00	−	2182.38i	−0.168102	−	0.291161i	0.769651	−	0.638465i	$-0.220430\pi$
$383$	−1260.00	−	2182.38i	−0.168102	−	0.291161i	−0.937752	+	0.347304i	$0.887097\pi$
$384$		0			0
$385$	4800.00	−	8313.84i	0.635404	−	1.10055i
$386$	1828.00			0.241043
$387$		0			0
$388$	776.000			0.101535
$389$	783.000	−	1356.20i	0.102056	−	0.176766i	−0.810476	−	0.585772i	$-0.800791\pi$
$389$	783.000	−	1356.20i	0.102056	−	0.176766i	0.912531	+	0.409006i	$0.134125\pi$
$390$		0			0
$391$		0			0
$392$	−2724.00	−	4718.11i	−0.350976	−	0.607909i
$393$		0			0
$394$	−5202.00	+	9010.13i	−0.665160	+	1.15209i
$395$	−760.000			−0.0968095
$396$		0			0
$397$	−4354.00			−0.550431			−0.275215	−	0.961383i	$-0.588749\pi$
$397$	−4354.00			−0.550431			−0.275215	+	0.961383i	$0.588749\pi$
$398$	−3152.00	+	5459.42i	−0.396974	+	0.687578i
$399$		0			0
$400$	−200.000	−	346.410i	−0.0250000	−	0.0433013i
$401$	−4023.00	−	6968.04i	−0.500995	−	0.867749i	−0.999999	−	0.00114942i	$-0.999634\pi$
$401$	−4023.00	−	6968.04i	−0.500995	−	0.867749i	0.499004	−	0.866600i	$-0.333699\pi$
$402$		0			0
$403$	−3944.00	+	6831.21i	−0.487505	+	0.844384i
$404$	−3192.00			−0.393089
$405$		0			0
$406$	−384.000			−0.0469399
$407$	−4020.00	+	6962.84i	−0.489592	+	0.847998i
$408$		0			0
$409$	1403.00	+	2430.07i	0.169618	+	0.293788i	0.938286	−	0.345861i	$-0.112413\pi$
$409$	1403.00	+	2430.07i	0.169618	+	0.293788i	−0.768667	+	0.639649i	$0.779080\pi$
$410$	−1170.00	−	2026.50i	−0.140932	−	0.244102i
$411$		0			0
$412$	−2176.00	+	3768.94i	−0.260203	+	0.450686i
$413$	−21120.0			−2.51634
$414$		0			0
$415$	−4020.00			−0.475504
$416$	544.000	−	942.236i	0.0641149	−	0.111050i
$417$		0			0
$418$	4560.00	+	7898.15i	0.533581	+	0.924190i
$419$	5790.00	+	10028.6i	0.675084	+	1.16928i	0.976444	+	0.215769i	$0.0692258\pi$
$419$	5790.00	+	10028.6i	0.675084	+	1.16928i	−0.301361	+	0.953510i	$0.597441\pi$
$420$		0			0
$421$	185.000	−	320.429i	0.0214165	−	0.0370945i	−0.855119	−	0.518433i	$-0.826516\pi$
$421$	185.000	−	320.429i	0.0214165	−	0.0370945i	0.876535	+	0.481338i	$0.159849\pi$
$422$	1480.00			0.170723
$423$		0			0
$424$	−1776.00			−0.203420
$425$	525.000	−	909.327i	0.0599206	−	0.103785i
$426$		0			0
$427$	7840.00	+	13579.3i	0.888534	+	1.53899i
$428$	3432.00	+	5944.40i	0.387598	+	0.671340i
$429$		0			0
$430$	−2060.00	+	3568.02i	−0.231028	+	0.400152i
$431$	−5040.00			−0.563267			−0.281634	−	0.959522i	$-0.590876\pi$
$431$	−5040.00			−0.563267			−0.281634	+	0.959522i	$0.590876\pi$
$432$		0			0
$433$	−3742.00			−0.415310			−0.207655	−	0.978202i	$-0.566583\pi$
$433$	−3742.00			−0.415310			−0.207655	+	0.978202i	$0.566583\pi$
$434$	7424.00	−	12858.7i	0.821114	−	1.42221i
$435$		0			0
$436$	1940.00	+	3360.18i	0.213094	+	0.369090i
$437$		0			0
$438$		0			0
$439$	3104.00	−	5376.29i	0.337462	−	0.584501i	−0.646493	−	0.762920i	$-0.723765\pi$
$439$	3104.00	−	5376.29i	0.337462	−	0.584501i	0.983955	+	0.178419i	$0.0570982\pi$
$440$	−2400.00			−0.260035
$441$		0			0
$442$	2856.00			0.307344
$443$	−7782.00	+	13478.8i	−0.834614	+	1.44559i	0.0597304	+	0.998215i	$0.480976\pi$
$443$	−7782.00	+	13478.8i	−0.834614	+	1.44559i	−0.894344	+	0.447379i	$0.852357\pi$
$444$		0			0
$445$	1695.00	+	2935.83i	0.180563	+	0.312745i
$446$	520.000	+	900.666i	0.0552079	+	0.0956228i
$447$		0			0
$448$	−1024.00	+	1773.62i	−0.107990	+	0.187044i
$449$	15774.0			1.65795			0.828977	−	0.559283i	$-0.188924\pi$
$449$	15774.0			1.65795			0.828977	+	0.559283i	$0.188924\pi$
$450$		0			0
$451$	−14040.0			−1.46589
$452$	852.000	−	1475.71i	0.0886609	−	0.153565i
$453$		0			0
$454$	396.000	+	685.892i	0.0409366	+	0.0709042i
$455$	−2720.00	−	4711.18i	−0.280254	−	0.485414i
$456$		0			0
$457$	−4861.00	+	8419.50i	−0.497567	+	0.861811i	−0.999996	−	0.00280744i	$-0.999106\pi$
$457$	−4861.00	+	8419.50i	−0.497567	+	0.861811i	0.502429	+	0.864618i	$0.332440\pi$
$458$	−2660.00			−0.271383
$459$		0			0
$460$		0			0
$461$	−5445.00	+	9431.02i	−0.550106	+	0.952812i	0.448160	+	0.893953i	$0.352079\pi$
$461$	−5445.00	+	9431.02i	−0.550106	+	0.952812i	−0.998266	+	0.0588585i	$0.981254\pi$
$462$		0			0
$463$	−7564.00	−	13101.2i	−0.759242	−	1.31505i	−0.943238	−	0.332118i	$-0.892237\pi$
$463$	−7564.00	−	13101.2i	−0.759242	−	1.31505i	0.183996	−	0.982927i	$-0.441097\pi$
$464$	48.0000	+	83.1384i	0.00480247	+	0.00831811i
$465$		0			0
$466$	4866.00	−	8428.16i	0.483719	−	0.837826i
$467$	−10668.0			−1.05708			−0.528540	−	0.848909i	$-0.677260\pi$
$467$	−10668.0			−1.05708			−0.528540	+	0.848909i	$0.677260\pi$
$468$		0			0
$469$	25984.0			2.55827
$470$	1800.00	−	3117.69i	0.176655	−	0.305975i
$471$		0			0
$472$	2640.00	+	4572.61i	0.257449	+	0.445914i
$473$	12360.0	+	21408.1i	1.20151	+	2.08107i
$474$		0			0
$475$	950.000	−	1645.45i	0.0917663	−	0.158944i
$476$	−5376.00			−0.517665
$477$		0			0
$478$	3648.00			0.349070
$479$	7632.00	−	13219.0i	0.728006	−	1.26094i	−0.229718	−	0.973257i	$-0.573781\pi$
$479$	7632.00	−	13219.0i	0.728006	−	1.26094i	0.957725	−	0.287687i	$-0.0928862\pi$
$480$		0			0
$481$	2278.00	+	3945.61i	0.215941	+	0.374022i
$482$	−6482.00	−	11227.2i	−0.612546	−	1.06096i
$483$		0			0
$484$	−4538.00	+	7860.05i	−0.426183	+	0.738171i
$485$	−970.000			−0.0908153
$486$		0			0
$487$	−5776.00			−0.537445			−0.268722	−	0.963218i	$-0.586601\pi$
$487$	−5776.00			−0.537445			−0.268722	+	0.963218i	$0.586601\pi$
$488$	1960.00	−	3394.82i	0.181814	−	0.314910i
$489$		0			0
$490$	3405.00	+	5897.63i	0.313923	+	0.543730i
$491$	7122.00	+	12335.7i	0.654606	+	1.13381i	0.981993	+	0.188920i	$0.0604986\pi$
$491$	7122.00	+	12335.7i	0.654606	+	1.13381i	−0.327387	+	0.944890i	$0.606168\pi$
$492$		0			0
$493$	−126.000	+	218.238i	−0.0115107	+	0.0199370i
$494$	5168.00			0.470687
$495$		0			0
$496$	−3712.00			−0.336036
$497$	1920.00	−	3325.54i	0.173287	−	0.300142i
$498$		0			0
$499$	8558.00	+	14822.9i	0.767753	+	1.32979i	0.938779	+	0.344520i	$0.111958\pi$
$499$	8558.00	+	14822.9i	0.767753	+	1.32979i	−0.171026	+	0.985267i	$0.554708\pi$
$500$	250.000	+	433.013i	0.0223607	+	0.0387298i
$501$		0			0
$502$	1476.00	−	2556.51i	0.131229	−	0.227296i
$503$	16848.0			1.49347			0.746735	−	0.665122i	$-0.231620\pi$
$503$	16848.0			1.49347			0.746735	+	0.665122i	$0.231620\pi$
$504$		0			0
$505$	3990.00			0.351589
$506$		0			0
$507$		0			0
$508$	−400.000	−	692.820i	−0.0349353	−	0.0605097i
$509$	−1917.00	−	3320.34i	−0.166934	−	0.289139i	0.770406	−	0.637553i	$-0.220053\pi$
$509$	−1917.00	−	3320.34i	−0.166934	−	0.289139i	−0.937340	+	0.348415i	$0.886720\pi$
$510$		0			0
$511$	−11936.0	+	20673.8i	−1.03330	+	1.78973i
$512$	512.000			0.0441942
$513$		0			0
$514$	−8628.00			−0.740398
$515$	2720.00	−	4711.18i	0.232733	−	0.403105i
$516$		0			0
$517$	−10800.0	−	18706.1i	−0.918730	−	1.59129i
$518$	−4288.00	−	7427.03i	−0.363714	−	0.629971i
$519$		0			0
$520$	−680.000	+	1177.79i	−0.0573461	+	0.0993264i
$521$	18822.0			1.58274			0.791369	−	0.611338i	$-0.209369\pi$
$521$	18822.0			1.58274			0.791369	+	0.611338i	$0.209369\pi$
$522$		0			0
$523$	−15340.0			−1.28255			−0.641273	−	0.767313i	$-0.721593\pi$
$523$	−15340.0			−1.28255			−0.641273	+	0.767313i	$0.721593\pi$
$524$	120.000	−	207.846i	0.0100042	−	0.0173279i
$525$		0			0
$526$	−5280.00	−	9145.23i	−0.437679	−	0.758082i
$527$	−4872.00	−	8438.55i	−0.402709	−	0.697512i
$528$		0			0
$529$	6083.50	−	10536.9i	0.500000	−	0.866025i
$530$	2220.00			0.181945
$531$		0			0
$532$	−9728.00			−0.792786
$533$	−3978.00	+	6890.10i	−0.323276	+	0.559931i
$534$		0			0
$535$	−4290.00	−	7430.50i	−0.346678	−	0.600464i
$536$	−3248.00	−	5625.70i	−0.261739	−	0.453346i
$537$		0			0
$538$	5526.00	−	9571.31i	0.442830	−	0.767005i
$539$	40860.0			3.26524
$540$		0			0
$541$	18950.0			1.50596			0.752980	−	0.658044i	$-0.228616\pi$
$541$	18950.0			1.50596			0.752980	+	0.658044i	$0.228616\pi$
$542$	−2024.00	+	3505.67i	−0.160403	+	0.277826i
$543$		0			0
$544$	672.000	+	1163.94i	0.0529628	+	0.0917343i
$545$	−2425.00	−	4200.22i	−0.190597	−	0.330124i
$546$		0			0
$547$	5018.00	−	8691.43i	0.392238	−	0.679376i	−0.600506	−	0.799620i	$-0.705034\pi$
$547$	5018.00	−	8691.43i	0.392238	−	0.679376i	0.992744	+	0.120244i	$0.0383676\pi$
$548$	−2568.00			−0.200182
$549$		0			0
$550$	3000.00			0.232583
$551$	−228.000	+	394.908i	−0.0176282	+	0.0305329i
$552$		0			0
$553$	−2432.00	−	4212.35i	−0.187015	−	0.323919i
$554$	−2054.00	−	3557.63i	−0.157520	−	0.272833i
$555$		0			0
$556$	5672.00	−	9824.19i	0.432637	−	0.749350i
$557$	−10326.0			−0.785506			−0.392753	−	0.919644i	$-0.628477\pi$
$557$	−10326.0			−0.785506			−0.392753	+	0.919644i	$0.628477\pi$
$558$		0			0
$559$	14008.0			1.05988
$560$	1280.00	−	2217.03i	0.0965891	−	0.167297i
$561$		0			0
$562$	−7302.00	−	12647.4i	−0.548072	−	0.949288i
$563$	2262.00	+	3917.90i	0.169328	+	0.293286i	0.938184	−	0.346137i	$-0.112507\pi$
$563$	2262.00	+	3917.90i	0.169328	+	0.293286i	−0.768855	+	0.639423i	$0.779173\pi$
$564$		0			0
$565$	−1065.00	+	1844.63i	−0.0793007	+	0.137353i
$566$	−7448.00			−0.553114
$567$		0			0
$568$	−960.000			−0.0709167
$569$	8181.00	−	14169.9i	0.602751	−	1.04400i	−0.389651	−	0.920962i	$-0.627405\pi$
$569$	8181.00	−	14169.9i	0.602751	−	1.04400i	0.992403	−	0.123033i	$-0.0392621\pi$
$570$		0			0
$571$	−3310.00	−	5733.09i	−0.242591	−	0.420179i	0.718861	−	0.695154i	$-0.244664\pi$
$571$	−3310.00	−	5733.09i	−0.242591	−	0.420179i	−0.961451	+	0.274975i	$0.911330\pi$
$572$	4080.00	+	7066.77i	0.298240	+	0.516567i
$573$		0			0
$574$	7488.00	−	12969.6i	0.544500	−	0.943102i
$575$		0			0
$576$		0			0
$577$	8834.00			0.637373			0.318687	−	0.947860i	$-0.396758\pi$
$577$	8834.00			0.637373			0.318687	+	0.947860i	$0.396758\pi$
$578$	3149.00	−	5454.23i	0.226611	−	0.392502i
$579$		0			0
$580$	−60.0000	−	103.923i	−0.00429546	−	0.00743995i
$581$	−12864.0	−	22281.1i	−0.918569	−	1.59101i
$582$		0			0
$583$	6660.00	−	11535.5i	0.473120	−	0.819468i
$584$	5968.00			0.422873
$585$		0			0
$586$	14436.0			1.01765
$587$	1818.00	−	3148.87i	0.127831	−	0.221410i	−0.795005	−	0.606603i	$-0.792532\pi$
$587$	1818.00	−	3148.87i	0.127831	−	0.221410i	0.922836	+	0.385193i	$0.125865\pi$
$588$		0			0
$589$	−8816.00	−	15269.8i	−0.616735	−	1.06822i
$590$	−3300.00	−	5715.77i	−0.230269	−	0.398838i
$591$		0			0
$592$	−1072.00	+	1856.76i	−0.0744239	+	0.128906i
$593$	−6570.00			−0.454971			−0.227485	−	0.973782i	$-0.573050\pi$
$593$	−6570.00			−0.454971			−0.227485	+	0.973782i	$0.573050\pi$
$594$		0			0
$595$	6720.00			0.463014
$596$	−3108.00	+	5383.21i	−0.213605	+	0.369975i
$597$		0			0
$598$		0			0
$599$	8292.00	+	14362.2i	0.565613	+	0.979670i	0.996992	+	0.0774993i	$0.0246936\pi$
$599$	8292.00	+	14362.2i	0.565613	+	0.979670i	−0.431380	+	0.902170i	$0.641973\pi$
$600$		0			0
$601$	251.000	−	434.745i	0.0170358	−	0.0295068i	−0.857382	−	0.514681i	$-0.827910\pi$
$601$	251.000	−	434.745i	0.0170358	−	0.0295068i	0.874418	+	0.485174i	$0.161244\pi$
$602$	−26368.0			−1.78518
$603$		0			0
$604$	−9088.00			−0.612228
$605$	5672.50	−	9825.06i	0.381190	−	0.660240i
$606$		0			0
$607$	9284.00	+	16080.4i	0.620801	+	1.07526i	0.989337	+	0.145645i	$0.0465257\pi$
$607$	9284.00	+	16080.4i	0.620801	+	1.07526i	−0.368536	+	0.929613i	$0.620141\pi$
$608$	1216.00	+	2106.17i	0.0811107	+	0.140488i
$609$		0			0
$610$	−2450.00	+	4243.52i	−0.162619	+	0.281664i
$611$	−12240.0			−0.810438
$612$		0			0
$613$	−13114.0			−0.864061			−0.432031	−	0.901859i	$-0.642203\pi$
$613$	−13114.0			−0.864061			−0.432031	+	0.901859i	$0.642203\pi$
$614$	−2540.00	+	4399.41i	−0.166948	+	0.289162i
$615$		0			0
$616$	−7680.00	−	13302.2i	−0.502331	−	0.870063i
$617$	2625.00	+	4546.63i	0.171278	+	0.296662i	0.938867	−	0.344280i	$-0.111877\pi$
$617$	2625.00	+	4546.63i	0.171278	+	0.296662i	−0.767589	+	0.640942i	$0.778544\pi$
$618$		0			0
$619$	5402.00	−	9356.54i	0.350767	−	0.607546i	−0.635617	−	0.772004i	$-0.719254\pi$
$619$	5402.00	−	9356.54i	0.350767	−	0.607546i	0.986384	+	0.164458i	$0.0525876\pi$
$620$	4640.00			0.300559
$621$		0			0
$622$	−3120.00			−0.201126
$623$	−10848.0	+	18789.3i	−0.697618	+	1.20831i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$	934.000	+	1617.74i	0.0596328	+	0.103287i
$627$		0			0
$628$	−3388.00	+	5868.19i	−0.215280	+	0.372876i
$629$	−5628.00			−0.356762
$630$		0			0
$631$	−27088.0			−1.70896			−0.854482	−	0.519481i	$-0.826125\pi$
$631$	−27088.0			−1.70896			−0.854482	+	0.519481i	$0.826125\pi$
$632$	−608.000	+	1053.09i	−0.0382673	+	0.0662809i
$633$		0			0
$634$	−1674.00	−	2899.45i	−0.104863	−	0.181628i
$635$	500.000	+	866.025i	0.0312471	+	0.0541215i
$636$		0			0
$637$	11577.0	−	20052.0i	0.720090	−	1.24723i
$638$	−720.000			−0.0446788
$639$		0			0
$640$	−640.000			−0.0395285
$641$	9465.00	−	16393.9i	0.583222	−	1.01017i	−0.411873	−	0.911241i	$-0.635125\pi$
$641$	9465.00	−	16393.9i	0.583222	−	1.01017i	0.995095	−	0.0989281i	$-0.0315414\pi$
$642$		0			0
$643$	−10054.0	−	17414.0i	−0.616627	−	1.06803i	−0.990097	−	0.140387i	$-0.955165\pi$
$643$	−10054.0	−	17414.0i	−0.616627	−	1.06803i	0.373470	−	0.927642i	$-0.378168\pi$
$644$		0			0
$645$		0			0
$646$	−3192.00	+	5528.71i	−0.194408	+	0.336725i
$647$	7152.00			0.434581			0.217291	−	0.976107i	$-0.430278\pi$
$647$	7152.00			0.434581			0.217291	+	0.976107i	$0.430278\pi$
$648$		0			0
$649$	−39600.0			−2.39512
$650$	850.000	−	1472.24i	0.0512919	−	0.0888402i
$651$		0			0
$652$	104.000	+	180.133i	0.00624686	+	0.0108199i
$653$	−15813.0	−	27388.9i	−0.947642	−	1.64136i	−0.750372	−	0.661016i	$-0.770126\pi$
$653$	−15813.0	−	27388.9i	−0.947642	−	1.64136i	−0.197270	−	0.980349i	$-0.563208\pi$
$654$		0			0
$655$	−150.000	+	259.808i	−0.00894807	+	0.0154985i
$656$	−3744.00			−0.222833
$657$		0			0
$658$	23040.0			1.36503
$659$	14046.0	−	24328.4i	0.830280	−	1.43809i	−0.0675363	−	0.997717i	$-0.521514\pi$
$659$	14046.0	−	24328.4i	0.830280	−	1.43809i	0.897816	−	0.440370i	$-0.145153\pi$
$660$		0			0
$661$	6593.00	+	11419.4i	0.387955	+	0.671957i	0.992174	−	0.124861i	$-0.0398484\pi$
$661$	6593.00	+	11419.4i	0.387955	+	0.671957i	−0.604220	+	0.796818i	$0.706515\pi$
$662$	3988.00	+	6907.42i	0.234136	+	0.405535i
$663$		0			0
$664$	−3216.00	+	5570.28i	−0.187959	+	0.325555i
$665$	12160.0			0.709090
$666$		0			0
$667$		0			0
$668$	−2400.00	+	4156.92i	−0.139010	+	0.240773i
$669$		0			0
$670$	4060.00	+	7032.13i	0.234107	+	0.405485i
$671$	14700.0	+	25461.1i	0.845734	+	1.46485i
$672$		0			0
$673$	−2569.00	+	4449.64i	−0.147144	+	0.254860i	−0.930171	−	0.367127i	$-0.880341\pi$
$673$	−2569.00	+	4449.64i	−0.147144	+	0.254860i	0.783027	+	0.621988i	$0.213675\pi$
$674$	4.00000			0.000228597
$675$		0			0
$676$	−4164.00			−0.236914
$677$	3039.00	−	5263.70i	0.172523	−	0.298819i	−0.766778	−	0.641912i	$-0.778141\pi$
$677$	3039.00	−	5263.70i	0.172523	−	0.298819i	0.939301	+	0.343093i	$0.111475\pi$
$678$		0			0
$679$	−3104.00	−	5376.29i	−0.175435	−	0.303863i
$680$	−840.000	−	1454.92i	−0.0473714	−	0.0820496i
$681$		0			0
$682$	13920.0	−	24110.1i	0.781561	−	1.35370i
$683$	−32244.0			−1.80642			−0.903208	−	0.429203i	$-0.858795\pi$
$683$	−32244.0			−1.80642			−0.903208	+	0.429203i	$0.858795\pi$
$684$		0			0
$685$	3210.00			0.179048
$686$	−10816.0	+	18733.9i	−0.601978	+	1.04266i
$687$		0			0
$688$	3296.00	+	5708.84i	0.182644	+	0.316348i
$689$	−3774.00	−	6536.76i	−0.208676	−	0.361438i
$690$		0			0
$691$	−2242.00	+	3883.26i	−0.123429	+	0.213786i	−0.921118	−	0.389284i	$-0.872723\pi$
$691$	−2242.00	+	3883.26i	−0.123429	+	0.213786i	0.797689	+	0.603070i	$0.206056\pi$
$692$	−216.000			−0.0118657
$693$		0			0
$694$	−3528.00			−0.192970
$695$	−7090.00	+	12280.2i	−0.386963	+	0.670239i
$696$		0			0
$697$	−4914.00	−	8511.30i	−0.267046	−	0.462537i
$698$	−4310.00	−	7465.14i	−0.233719	−	0.404813i
$699$		0			0
$700$	−1600.00	+	2771.28i	−0.0863919	+	0.149635i
$701$	30426.0			1.63934			0.819668	−	0.572839i	$-0.194158\pi$
$701$	30426.0			1.63934			0.819668	+	0.572839i	$0.194158\pi$
$702$		0			0
$703$	−10184.0			−0.546368
$704$	−1920.00	+	3325.54i	−0.102788	+	0.178034i
$705$		0			0
$706$	138.000	+	239.023i	0.00735651	+	0.0127419i
$707$	12768.0	+	22114.8i	0.679194	+	1.17640i
$708$		0			0
$709$	−6631.00	+	11485.2i	−0.351245	+	0.608374i	−0.986468	−	0.163955i	$-0.947575\pi$
$709$	−6631.00	+	11485.2i	−0.351245	+	0.608374i	0.635223	+	0.772329i	$0.280908\pi$
$710$	1200.00			0.0634299
$711$		0			0
$712$	5424.00			0.285496
$713$		0			0
$714$		0			0
$715$	−5100.00	−	8833.46i	−0.266754	−	0.462032i
$716$	1752.00	+	3034.55i	0.0914460	+	0.158389i
$717$		0			0
$718$	−11976.0	+	20743.0i	−0.622480	+	1.07817i
$719$	−13920.0			−0.722014			−0.361007	−	0.932563i	$-0.617567\pi$
$719$	−13920.0			−0.722014			−0.361007	+	0.932563i	$0.617567\pi$
$720$		0			0
$721$	34816.0			1.79836
$722$	1083.00	−	1875.81i	0.0558242	−	0.0966904i
$723$		0			0
$724$	−7708.00	−	13350.6i	−0.395671	−	0.685322i
$725$	75.0000	+	129.904i	0.00384197	+	0.00665449i
$726$		0			0
$727$	4688.00	−	8119.85i	0.239159	−	0.414235i	−0.721315	−	0.692608i	$-0.756462\pi$
$727$	4688.00	−	8119.85i	0.239159	−	0.414235i	0.960473	+	0.278373i	$0.0897951\pi$
$728$	−8704.00			−0.443120
$729$		0			0
$730$	−7460.00			−0.378229
$731$	−8652.00	+	14985.7i	−0.437764	+	0.758230i
$732$		0			0
$733$	−3007.00	−	5208.28i	−0.151523	−	0.262445i	0.780265	−	0.625449i	$-0.215084\pi$
$733$	−3007.00	−	5208.28i	−0.151523	−	0.262445i	−0.931787	+	0.363004i	$0.881751\pi$
$734$	−9704.00	−	16807.8i	−0.487985	−	0.845215i
$735$		0			0
$736$		0			0
$737$	48720.0			2.43504
$738$		0			0
$739$	−7468.00			−0.371739			−0.185869	−	0.982574i	$-0.559510\pi$
$739$	−7468.00			−0.371739			−0.185869	+	0.982574i	$0.559510\pi$
$740$	1340.00	−	2320.95i	0.0665667	−	0.115297i
$741$		0			0
$742$	7104.00	+	12304.5i	0.351477	+	0.608776i
$743$	15624.0	+	27061.6i	0.771452	+	1.33619i	0.936767	+	0.349954i	$0.113803\pi$
$743$	15624.0	+	27061.6i	0.771452	+	1.33619i	−0.165315	+	0.986241i	$0.552864\pi$
$744$		0			0
$745$	3885.00	−	6729.02i	0.191054	−	0.330916i
$746$	−16244.0			−0.797232
$747$		0			0
$748$	−10080.0			−0.492729
$749$	27456.0	−	47555.2i	1.33941	−	2.31993i
$750$		0			0
$751$	−16420.0	−	28440.3i	−0.797835	−	1.38189i	−0.921023	−	0.389508i	$-0.872645\pi$
$751$	−16420.0	−	28440.3i	−0.797835	−	1.38189i	0.123188	−	0.992383i	$-0.460688\pi$
$752$	−2880.00	−	4988.31i	−0.139658	−	0.241895i
$753$		0			0
$754$	−204.000	+	353.338i	−0.00985311	+	0.0170661i
$755$	11360.0			0.547593
$756$		0			0
$757$	−19066.0			−0.915410			−0.457705	−	0.889104i	$-0.651328\pi$
$757$	−19066.0			−0.915410			−0.457705	+	0.889104i	$0.651328\pi$
$758$	−3404.00	+	5895.90i	−0.163112	+	0.282518i
$759$		0			0
$760$	−1520.00	−	2632.72i	−0.0725476	−	0.125656i
$761$	3429.00	+	5939.20i	0.163339	+	0.282912i	0.936064	−	0.351829i	$-0.114440\pi$
$761$	3429.00	+	5939.20i	0.163339	+	0.282912i	−0.772725	+	0.634741i	$0.781107\pi$
$762$		0			0
$763$	15520.0	−	26881.4i	0.736385	−	1.27546i
$764$	11136.0			0.527338
$765$		0			0
$766$	5040.00			0.237732
$767$	−11220.0	+	19433.6i	−0.528202	+	0.914872i
$768$		0			0
$769$	−11089.0	−	19206.7i	−0.519999	−	0.900665i	−0.999730	−	0.0232494i	$-0.992599\pi$
$769$	−11089.0	−	19206.7i	−0.519999	−	0.900665i	0.479730	−	0.877416i	$-0.340735\pi$
$770$	9600.00	+	16627.7i	0.449299	+	0.778208i
$771$		0			0
$772$	−1828.00	+	3166.19i	−0.0852217	+	0.147608i
$773$	−14286.0			−0.664724			−0.332362	−	0.943152i	$-0.607846\pi$
$773$	−14286.0			−0.664724			−0.332362	+	0.943152i	$0.607846\pi$
$774$		0			0
$775$	−5800.00			−0.268829
$776$	−776.000	+	1344.07i	−0.0358979	+	0.0621770i
$777$		0			0
$778$	1566.00	+	2712.39i	0.0721643	+	0.124992i
$779$	−8892.00	−	15401.4i	−0.408972	−	0.708360i
$780$		0			0
$781$	3600.00	−	6235.38i	0.164940	−	0.285684i
$782$		0			0
$783$		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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	810.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(2.50000 + 4.33013i) q^{5} +(-16.0000 + 27.7128i) q^{7} +8.00000 q^{8} -10.0000 q^{10} +(-30.0000 + 51.9615i) q^{11} +(17.0000 + 29.4449i) q^{13} +(-32.0000 - 55.4256i) q^{14} +(-8.00000 + 13.8564i) q^{16} -42.0000 q^{17} -76.0000 q^{19} +(10.0000 - 17.3205i) q^{20} +(-60.0000 - 103.923i) q^{22} +(-12.5000 + 21.6506i) q^{25} -68.0000 q^{26} +128.000 q^{28} +(3.00000 - 5.19615i) q^{29} +(116.000 + 200.918i) q^{31} +(-16.0000 - 27.7128i) q^{32} +(42.0000 - 72.7461i) q^{34} -160.000 q^{35} +134.000 q^{37} +(76.0000 - 131.636i) q^{38} +(20.0000 + 34.6410i) q^{40} +(117.000 + 202.650i) q^{41} +(206.000 - 356.802i) q^{43} +240.000 q^{44} +(-180.000 + 311.769i) q^{47} +(-340.500 - 589.763i) q^{49} +(-25.0000 - 43.3013i) q^{50} +(68.0000 - 117.779i) q^{52} -222.000 q^{53} -300.000 q^{55} +(-128.000 + 221.703i) q^{56} +(6.00000 + 10.3923i) q^{58} +(330.000 + 571.577i) q^{59} +(245.000 - 424.352i) q^{61} -464.000 q^{62} +64.0000 q^{64} +(-85.0000 + 147.224i) q^{65} +(-406.000 - 703.213i) q^{67} +(84.0000 + 145.492i) q^{68} +(160.000 - 277.128i) q^{70} -120.000 q^{71} +746.000 q^{73} +(-134.000 + 232.095i) q^{74} +(152.000 + 263.272i) q^{76} +(-960.000 - 1662.77i) q^{77} +(-76.0000 + 131.636i) q^{79} -80.0000 q^{80} -468.000 q^{82} +(-402.000 + 696.284i) q^{83} +(-105.000 - 181.865i) q^{85} +(412.000 + 713.605i) q^{86} +(-240.000 + 415.692i) q^{88} +678.000 q^{89} -1088.00 q^{91} +(-360.000 - 623.538i) q^{94} +(-190.000 - 329.090i) q^{95} +(-97.0000 + 168.009i) q^{97} +1362.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{4} + 5 q^{5} - 32 q^{7} + 16 q^{8} - 20 q^{10} - 60 q^{11} + 34 q^{13} - 64 q^{14} - 16 q^{16} - 84 q^{17} - 152 q^{19} + 20 q^{20} - 120 q^{22} - 25 q^{25} - 136 q^{26} + 256 q^{28}+ \cdots + 2724 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 4 * q^4 + 5 * q^5 - 32 * q^7 + 16 * q^8 - 20 * q^10 - 60 * q^11 + 34 * q^13 - 64 * q^14 - 16 * q^16 - 84 * q^17 - 152 * q^19 + 20 * q^20 - 120 * q^22 - 25 * q^25 - 136 * q^26 + 256 * q^28 + 6 * q^29 + 232 * q^31 - 32 * q^32 + 84 * q^34 - 320 * q^35 + 268 * q^37 + 152 * q^38 + 40 * q^40 + 234 * q^41 + 412 * q^43 + 480 * q^44 - 360 * q^47 - 681 * q^49 - 50 * q^50 + 136 * q^52 - 444 * q^53 - 600 * q^55 - 256 * q^56 + 12 * q^58 + 660 * q^59 + 490 * q^61 - 928 * q^62 + 128 * q^64 - 170 * q^65 - 812 * q^67 + 168 * q^68 + 320 * q^70 - 240 * q^71 + 1492 * q^73 - 268 * q^74 + 304 * q^76 - 1920 * q^77 - 152 * q^79 - 160 * q^80 - 936 * q^82 - 804 * q^83 - 210 * q^85 + 824 * q^86 - 480 * q^88 + 1356 * q^89 - 2176 * q^91 - 720 * q^94 - 380 * q^95 - 194 * q^97 + 2724 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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