LMFDB - Embedded newform 495.3.g.a.89.25

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [495,3,Mod(89,495)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("495.89"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	495.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4877730858$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			89.25
Character	$\chi$	$=$	495.89
Dual form			495.3.g.a.89.26

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.881621 q^{2} -3.22275 q^{4} +(-4.45405 + 2.27188i) q^{5} -4.31988i q^{7} -6.36772 q^{8} +(-3.92678 + 2.00293i) q^{10} +3.31662i q^{11} -0.805140i q^{13} -3.80850i q^{14} +7.27707 q^{16} +23.4538 q^{17} +24.5089 q^{19} +(14.3543 - 7.32168i) q^{20} +2.92400i q^{22} +9.68854 q^{23} +(14.6771 - 20.2381i) q^{25} -0.709828i q^{26} +13.9219i q^{28} -11.7672i q^{29} -26.1512 q^{31} +31.8865 q^{32} +20.6773 q^{34} +(9.81424 + 19.2410i) q^{35} -16.0917i q^{37} +21.6076 q^{38} +(28.3622 - 14.4667i) q^{40} +72.3595i q^{41} -44.7144i q^{43} -10.6886i q^{44} +8.54162 q^{46} +51.7564 q^{47} +30.3386 q^{49} +(12.9397 - 17.8423i) q^{50} +2.59476i q^{52} +39.9939 q^{53} +(-7.53497 - 14.7724i) q^{55} +27.5078i q^{56} -10.3742i q^{58} +72.6424i q^{59} -51.9859 q^{61} -23.0555 q^{62} -0.996477 q^{64} +(1.82918 + 3.58613i) q^{65} -80.0624i q^{67} -75.5856 q^{68} +(8.65244 + 16.9632i) q^{70} -6.50151i q^{71} -48.8081i q^{73} -14.1868i q^{74} -78.9861 q^{76} +14.3274 q^{77} +65.8002 q^{79} +(-32.4124 + 16.5326i) q^{80} +63.7936i q^{82} +54.1991 q^{83} +(-104.464 + 53.2841i) q^{85} -39.4211i q^{86} -21.1193i q^{88} -147.434i q^{89} -3.47811 q^{91} -31.2237 q^{92} +45.6295 q^{94} +(-109.164 + 55.6813i) q^{95} -155.533i q^{97} +26.7472 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 72 q^{4} + 8 q^{10} + 184 q^{16} - 80 q^{19} + 32 q^{25} - 16 q^{31} - 160 q^{34} - 136 q^{40} + 560 q^{46} - 104 q^{49} - 96 q^{61} + 264 q^{64} - 872 q^{70} - 176 q^{76} - 672 q^{79} + 16 q^{85}+ \cdots + 400 q^{94}+O(q^{100}) $ 40 * q + 72 * q^4 + 8 * q^10 + 184 * q^16 - 80 * q^19 + 32 * q^25 - 16 * q^31 - 160 * q^34 - 136 * q^40 + 560 * q^46 - 104 * q^49 - 96 * q^61 + 264 * q^64 - 872 * q^70 - 176 * q^76 - 672 * q^79 + 16 * q^85 - 864 * q^91 + 400 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$.

$n$	$46$	$56$	$397$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	0.881621			0.440810			0.220405	−	0.975408i	$-0.429262\pi$
$2$	0.881621			0.440810			0.220405	+	0.975408i	$0.429262\pi$
$3$		0			0
$3$		0			0
$4$	−3.22275			−0.805686
$5$	−4.45405	+	2.27188i	−0.890810	+	0.454375i
$5$	−4.45405	+	2.27188i	−0.890810	+	0.454375i
$6$		0			0
$7$		−	4.31988i		−	0.617126i	−0.951204	−	0.308563i	$-0.900152\pi$
$7$		−	4.31988i		−	0.617126i	0.951204	−	0.308563i	$-0.0998480\pi$
$8$	−6.36772			−0.795965
$9$		0			0
$10$	−3.92678	+	2.00293i	−0.392678	+	0.200293i
$11$			3.31662i			0.301511i
$11$			3.31662i			0.301511i
$12$		0			0
$13$		−	0.805140i		−	0.0619338i	−0.999520	−	0.0309669i	$-0.990141\pi$
$13$		−	0.805140i		−	0.0619338i	0.999520	−	0.0309669i	$-0.00985865\pi$
$14$		−	3.80850i		−	0.272035i
$15$		0			0
$16$	7.27707			0.454817
$17$	23.4538			1.37963			0.689817	−	0.723983i	$-0.257691\pi$
$17$	23.4538			1.37963			0.689817	+	0.723983i	$0.257691\pi$
$18$		0			0
$19$	24.5089			1.28994			0.644972	−	0.764206i	$-0.276869\pi$
$19$	24.5089			1.28994			0.644972	+	0.764206i	$0.276869\pi$
$20$	14.3543	−	7.32168i	0.717714	−	0.366084i
$21$		0			0
$22$			2.92400i			0.132909i
$23$	9.68854			0.421241			0.210620	−	0.977568i	$-0.432452\pi$
$23$	9.68854			0.421241			0.210620	+	0.977568i	$0.432452\pi$
$24$		0			0
$25$	14.6771	−	20.2381i	0.587086	−	0.809525i
$26$		−	0.709828i		−	0.0273011i
$27$		0			0
$28$			13.9219i			0.497210i
$29$		−	11.7672i		−	0.405766i	−0.979203	−	0.202883i	$-0.934969\pi$
$29$		−	11.7672i		−	0.405766i	0.979203	−	0.202883i	$-0.0650311\pi$
$30$		0			0
$31$	−26.1512			−0.843588			−0.421794	−	0.906692i	$-0.638599\pi$
$31$	−26.1512			−0.843588			−0.421794	+	0.906692i	$0.638599\pi$
$32$	31.8865			0.996453
$33$		0			0
$34$	20.6773			0.608157
$35$	9.81424	+	19.2410i	0.280407	+	0.549742i
$36$		0			0
$37$		−	16.0917i		−	0.434911i	−0.976070	−	0.217455i	$-0.930224\pi$
$37$		−	16.0917i		−	0.434911i	0.976070	−	0.217455i	$-0.0697757\pi$
$38$	21.6076			0.568621
$39$		0			0
$40$	28.3622	−	14.4667i	0.709054	−	0.361667i
$41$			72.3595i			1.76486i	0.470439	+	0.882432i	$0.344095\pi$
$41$			72.3595i			1.76486i	−0.470439	+	0.882432i	$0.655905\pi$
$42$		0			0
$43$		−	44.7144i		−	1.03987i	−0.854206	−	0.519935i	$-0.825956\pi$
$43$		−	44.7144i		−	1.03987i	0.854206	−	0.519935i	$-0.174044\pi$
$44$		−	10.6886i		−	0.242924i
$45$		0			0
$46$	8.54162			0.185687
$47$	51.7564			1.10120			0.550600	−	0.834769i	$-0.314399\pi$
$47$	51.7564			1.10120			0.550600	+	0.834769i	$0.314399\pi$
$48$		0			0
$49$	30.3386			0.619156
$50$	12.9397	−	17.8423i	0.258793	−	0.356847i
$51$		0			0
$52$			2.59476i			0.0498992i
$53$	39.9939			0.754602			0.377301	−	0.926091i	$-0.376852\pi$
$53$	39.9939			0.754602			0.377301	+	0.926091i	$0.376852\pi$
$54$		0			0
$55$	−7.53497	−	14.7724i	−0.136999	−	0.268589i
$56$			27.5078i			0.491211i
$57$		0			0
$58$		−	10.3742i		−	0.178866i
$59$			72.6424i			1.23123i	0.788048	+	0.615613i	$0.211092\pi$
$59$			72.6424i			1.23123i	−0.788048	+	0.615613i	$0.788908\pi$
$60$		0			0
$61$	−51.9859			−0.852228			−0.426114	−	0.904669i	$-0.640118\pi$
$61$	−51.9859			−0.852228			−0.426114	+	0.904669i	$0.640118\pi$
$62$	−23.0555			−0.371862
$63$		0			0
$64$	−0.996477			−0.0155699
$65$	1.82918	+	3.58613i	0.0281412	+	0.0551713i
$66$		0			0
$67$		−	80.0624i		−	1.19496i	−0.801884	−	0.597480i	$-0.796169\pi$
$67$		−	80.0624i		−	1.19496i	0.801884	−	0.597480i	$-0.203831\pi$
$68$	−75.5856			−1.11155
$69$		0			0
$70$	8.65244	+	16.9632i	0.123606	+	0.242332i
$71$		−	6.50151i		−	0.0915706i	−0.998951	−	0.0457853i	$-0.985421\pi$
$71$		−	6.50151i		−	0.0915706i	0.998951	−	0.0457853i	$-0.0145790\pi$
$72$		0			0
$73$		−	48.8081i		−	0.668604i	−0.942466	−	0.334302i	$-0.891500\pi$
$73$		−	48.8081i		−	0.668604i	0.942466	−	0.334302i	$-0.108500\pi$
$74$		−	14.1868i		−	0.191713i
$75$		0			0
$76$	−78.9861			−1.03929
$77$	14.3274			0.186070
$78$		0			0
$79$	65.8002			0.832915			0.416457	−	0.909155i	$-0.363272\pi$
$79$	65.8002			0.832915			0.416457	+	0.909155i	$0.363272\pi$
$80$	−32.4124	+	16.5326i	−0.405155	+	0.206658i
$81$		0			0
$82$			63.7936i			0.777971i
$83$	54.1991			0.653001			0.326501	−	0.945197i	$-0.394130\pi$
$83$	54.1991			0.653001			0.326501	+	0.945197i	$0.394130\pi$
$84$		0			0
$85$	−104.464	+	53.2841i	−1.22899	+	0.626872i
$86$		−	39.4211i		−	0.458385i
$87$		0			0
$88$		−	21.1193i		−	0.239993i
$89$		−	147.434i		−	1.65656i	−0.560317	−	0.828278i	$-0.689321\pi$
$89$		−	147.434i		−	1.65656i	0.560317	−	0.828278i	$-0.310679\pi$
$90$		0			0
$91$	−3.47811			−0.0382210
$92$	−31.2237			−0.339388
$93$		0			0
$94$	45.6295			0.485421
$95$	−109.164	+	55.6813i	−1.14910	+	0.586119i
$96$		0			0
$97$		−	155.533i		−	1.60343i	−0.597706	−	0.801715i	$-0.703921\pi$
$97$		−	155.533i		−	1.60343i	0.597706	−	0.801715i	$-0.296079\pi$
$98$	26.7472			0.272930
$99$		0			0
$100$	−47.3007	+	65.2223i	−0.473007	+	0.652223i
$101$			60.8521i			0.602496i	0.953546	+	0.301248i	$0.0974032\pi$
$101$			60.8521i			0.602496i	−0.953546	+	0.301248i	$0.902597\pi$
$102$		0			0
$103$			165.708i			1.60881i	0.594080	+	0.804406i	$0.297516\pi$
$103$			165.708i			1.60881i	−0.594080	+	0.804406i	$0.702484\pi$
$104$			5.12691i			0.0492972i
$105$		0			0
$106$	35.2595			0.332636
$107$	38.8630			0.363206			0.181603	−	0.983372i	$-0.441871\pi$
$107$	38.8630			0.363206			0.181603	+	0.983372i	$0.441871\pi$
$108$		0			0
$109$	−3.33758			−0.0306200			−0.0153100	−	0.999883i	$-0.504874\pi$
$109$	−3.33758			−0.0306200			−0.0153100	+	0.999883i	$0.504874\pi$
$110$	−6.64298	−	13.0237i	−0.0603907	−	0.118397i
$111$		0			0
$112$		−	31.4361i		−	0.280679i
$113$	−8.23449			−0.0728716			−0.0364358	−	0.999336i	$-0.511600\pi$
$113$	−8.23449			−0.0728716			−0.0364358	+	0.999336i	$0.511600\pi$
$114$		0			0
$115$	−43.1533	+	22.0112i	−0.375246	+	0.191402i
$116$			37.9227i			0.326920i
$117$		0			0
$118$			64.0430i			0.542737i
$119$		−	101.318i		−	0.851408i
$120$		0			0
$121$	−11.0000			−0.0909091
$122$	−45.8318			−0.375671
$123$		0			0
$124$	84.2787			0.679667
$125$	−19.3942	+	123.486i	−0.155154	+	0.987890i
$126$		0			0
$127$			199.289i			1.56921i	0.619998	+	0.784603i	$0.287133\pi$
$127$			199.289i			1.56921i	−0.619998	+	0.784603i	$0.712867\pi$
$128$	−128.424			−1.00332
$129$		0			0
$130$	1.61264	+	3.16161i	0.0124049	+	0.0243201i
$131$		−	17.4859i		−	0.133480i	−0.997770	−	0.0667402i	$-0.978740\pi$
$131$		−	17.4859i		−	0.133480i	0.997770	−	0.0667402i	$-0.0212599\pi$
$132$		0			0
$133$		−	105.876i		−	0.796058i
$134$		−	70.5846i		−	0.526751i
$135$		0			0
$136$	−149.347			−1.09814
$137$	148.470			1.08373			0.541863	−	0.840467i	$-0.317719\pi$
$137$	148.470			1.08373			0.541863	+	0.840467i	$0.317719\pi$
$138$		0			0
$139$	119.272			0.858075			0.429038	−	0.903287i	$-0.358853\pi$
$139$	119.272			0.858075			0.429038	+	0.903287i	$0.358853\pi$
$140$	−31.6288	−	62.0087i	−0.225920	−	0.442920i
$141$		0			0
$142$		−	5.73186i		−	0.0403652i
$143$	2.67035			0.0186738
$144$		0			0
$145$	26.7337	+	52.4118i	0.184370	+	0.361460i
$146$		−	43.0302i		−	0.294727i
$147$		0			0
$148$			51.8595i			0.350402i
$149$			71.6881i			0.481128i	0.970633	+	0.240564i	$0.0773324\pi$
$149$			71.6881i			0.481128i	−0.970633	+	0.240564i	$0.922668\pi$
$150$		0			0
$151$	168.913			1.11863			0.559313	−	0.828957i	$-0.311065\pi$
$151$	168.913			1.11863			0.559313	+	0.828957i	$0.311065\pi$
$152$	−156.066			−1.02675
$153$		0			0
$154$	12.6314			0.0820218
$155$	116.479	−	59.4124i	0.751477	−	0.383306i
$156$		0			0
$157$		−	183.252i		−	1.16721i	−0.812039	−	0.583604i	$-0.801642\pi$
$157$		−	183.252i		−	1.16721i	0.812039	−	0.583604i	$-0.198358\pi$
$158$	58.0109			0.367157
$159$		0			0
$160$	−142.024	+	72.4422i	−0.887651	+	0.452764i
$161$		−	41.8533i		−	0.259959i
$162$		0			0
$163$			100.620i			0.617303i	0.951175	+	0.308651i	$0.0998776\pi$
$163$			100.620i			0.617303i	−0.951175	+	0.308651i	$0.900122\pi$
$164$		−	233.196i		−	1.42193i
$165$		0			0
$166$	47.7831			0.287850
$167$	−214.819			−1.28634			−0.643169	−	0.765724i	$-0.722381\pi$
$167$	−214.819			−1.28634			−0.643169	+	0.765724i	$0.722381\pi$
$168$		0			0
$169$	168.352			0.996164
$170$	−92.0980	+	46.9764i	−0.541753	+	0.276332i
$171$		0			0
$172$			144.103i			0.837808i
$173$	−143.522			−0.829606			−0.414803	−	0.909911i	$-0.636150\pi$
$173$	−143.522			−0.829606			−0.414803	+	0.909911i	$0.636150\pi$
$174$		0			0
$175$	−87.4263	−	63.4035i	−0.499579	−	0.362306i
$176$			24.1353i			0.137132i
$177$		0			0
$178$		−	129.980i		−	0.730227i
$179$		−	132.148i		−	0.738256i	−0.929379	−	0.369128i	$-0.879656\pi$
$179$		−	132.148i		−	0.738256i	0.929379	−	0.369128i	$-0.120344\pi$
$180$		0			0
$181$	−248.395			−1.37235			−0.686174	−	0.727438i	$-0.740711\pi$
$181$	−248.395			−1.37235			−0.686174	+	0.727438i	$0.740711\pi$
$182$	−3.06637			−0.0168482
$183$		0			0
$184$	−61.6939			−0.335293
$185$	36.5584	+	71.6733i	0.197613	+	0.387423i
$186$		0			0
$187$			77.7874i			0.415976i
$188$	−166.798			−0.887222
$189$		0			0
$190$	−96.2413	+	49.0898i	−0.506533	+	0.258367i
$191$		−	184.466i		−	0.965793i	−0.875678	−	0.482896i	$-0.839585\pi$
$191$		−	184.466i		−	0.965793i	0.875678	−	0.482896i	$-0.160415\pi$
$192$		0			0
$193$			242.874i			1.25841i	0.777238	+	0.629207i	$0.216620\pi$
$193$			242.874i			1.25841i	−0.777238	+	0.629207i	$0.783380\pi$
$194$		−	137.121i		−	0.706809i
$195$		0			0
$196$	−97.7737			−0.498845
$197$	115.056			0.584041			0.292021	−	0.956412i	$-0.405672\pi$
$197$	115.056			0.584041			0.292021	+	0.956412i	$0.405672\pi$
$198$		0			0
$199$	−66.2275			−0.332802			−0.166401	−	0.986058i	$-0.553215\pi$
$199$	−66.2275			−0.332802			−0.166401	+	0.986058i	$0.553215\pi$
$200$	−93.4600	+	128.871i	−0.467300	+	0.644353i
$201$		0			0
$202$			53.6485i			0.265586i
$203$	−50.8329			−0.250409
$204$		0			0
$205$	−164.392	−	322.293i	−0.801911	−	1.57216i
$206$			146.091i			0.709181i
$207$		0			0
$208$		−	5.85906i		−	0.0281685i
$209$			81.2869i			0.388933i
$210$		0			0
$211$	222.115			1.05268			0.526339	−	0.850275i	$-0.323564\pi$
$211$	222.115			1.05268			0.526339	+	0.850275i	$0.323564\pi$
$212$	−128.890			−0.607973
$213$		0			0
$214$	34.2624			0.160105
$215$	101.586	+	199.160i	0.472491	+	0.926326i
$216$		0			0
$217$			112.970i			0.520600i
$218$	−2.94248			−0.0134976
$219$		0			0
$220$	24.2833	+	47.6077i	0.110379	+	0.216399i
$221$		−	18.8836i		−	0.0854461i
$222$		0			0
$223$			82.8808i			0.371663i	0.982582	+	0.185831i	$0.0594978\pi$
$223$			82.8808i			0.371663i	−0.982582	+	0.185831i	$0.940502\pi$
$224$		−	137.746i		−	0.614937i
$225$		0			0
$226$	−7.25970			−0.0321226
$227$	−38.0897			−0.167796			−0.0838979	−	0.996474i	$-0.526737\pi$
$227$	−38.0897			−0.167796			−0.0838979	+	0.996474i	$0.526737\pi$
$228$		0			0
$229$	331.747			1.44868			0.724339	−	0.689443i	$-0.242145\pi$
$229$	331.747			1.44868			0.724339	+	0.689443i	$0.242145\pi$
$230$	−38.0448	+	19.4055i	−0.165412	+	0.0843718i
$231$		0			0
$232$			74.9303i			0.322975i
$233$	−119.256			−0.511828			−0.255914	−	0.966700i	$-0.582376\pi$
$233$	−119.256			−0.511828			−0.255914	+	0.966700i	$0.582376\pi$
$234$		0			0
$235$	−230.526	+	117.584i	−0.980961	+	0.500359i
$236$		−	234.108i		−	0.991983i
$237$		0			0
$238$		−	89.3237i		−	0.375310i
$239$		−	53.3917i		−	0.223396i	−0.993742	−	0.111698i	$-0.964371\pi$
$239$		−	53.3917i		−	0.223396i	0.993742	−	0.111698i	$-0.0356289\pi$
$240$		0			0
$241$	192.817			0.800072			0.400036	−	0.916499i	$-0.368998\pi$
$241$	192.817			0.800072			0.400036	+	0.916499i	$0.368998\pi$
$242$	−9.69783			−0.0400737
$243$		0			0
$244$	167.537			0.686628
$245$	−135.130	+	68.9256i	−0.551550	+	0.281329i
$246$		0			0
$247$		−	19.7331i		−	0.0798912i
$248$	166.524			0.671467
$249$		0			0
$250$	−17.0984	+	108.868i	−0.0683934	+	0.435472i
$251$		−	148.869i		−	0.593105i	−0.955017	−	0.296553i	$-0.904163\pi$
$251$		−	148.869i		−	0.593105i	0.955017	−	0.296553i	$-0.0958370\pi$
$252$		0			0
$253$			32.1333i			0.127009i
$254$			175.698i			0.691723i
$255$		0			0
$256$	−109.236			−0.426702
$257$	437.976			1.70419			0.852093	−	0.523390i	$-0.175333\pi$
$257$	437.976			1.70419			0.852093	+	0.523390i	$0.175333\pi$
$258$		0			0
$259$	−69.5142			−0.268395
$260$	−5.89498	−	11.5572i	−0.0226730	−	0.0444508i
$261$		0			0
$262$		−	15.4160i		−	0.0588395i
$263$	−271.978			−1.03414			−0.517069	−	0.855944i	$-0.672977\pi$
$263$	−271.978			−1.03414			−0.517069	+	0.855944i	$0.672977\pi$
$264$		0			0
$265$	−178.135	+	90.8613i	−0.672207	+	0.342873i
$266$		−	93.3422i		−	0.350910i
$267$		0			0
$268$			258.021i			0.962763i
$269$		−	59.8748i		−	0.222583i	−0.993788	−	0.111291i	$-0.964501\pi$
$269$		−	59.8748i		−	0.222583i	0.993788	−	0.111291i	$-0.0354987\pi$
$270$		0			0
$271$	107.993			0.398499			0.199250	−	0.979949i	$-0.436150\pi$
$271$	107.993			0.398499			0.199250	+	0.979949i	$0.436150\pi$
$272$	170.675			0.627481
$273$		0			0
$274$	130.895			0.477717
$275$	67.1222	+	48.6786i	0.244081	+	0.177013i
$276$		0			0
$277$		−	421.095i		−	1.52020i	−0.649808	−	0.760098i	$-0.725151\pi$
$277$		−	421.095i		−	1.52020i	0.649808	−	0.760098i	$-0.274849\pi$
$278$	105.153			0.378248
$279$		0			0
$280$	−62.4943	−	122.521i	−0.223194	−	0.437575i
$281$			477.167i			1.69810i	0.528311	+	0.849051i	$0.322825\pi$
$281$			477.167i			1.69810i	−0.528311	+	0.849051i	$0.677175\pi$
$282$		0			0
$283$		−	487.972i		−	1.72428i	−0.506667	−	0.862142i	$-0.669123\pi$
$283$		−	487.972i		−	1.72428i	0.506667	−	0.862142i	$-0.330877\pi$
$284$			20.9527i			0.0737771i
$285$		0			0
$286$	2.35423			0.00823158
$287$	312.584			1.08914
$288$		0			0
$289$	261.080			0.903392
$290$	23.5689	+	46.2073i	0.0812722	+	0.159335i
$291$		0			0
$292$			157.296i			0.538685i
$293$	183.025			0.624658			0.312329	−	0.949974i	$-0.398891\pi$
$293$	183.025			0.624658			0.312329	+	0.949974i	$0.398891\pi$
$294$		0			0
$295$	−165.035	−	323.553i	−0.559439	−	1.09679i
$296$			102.467i			0.346174i
$297$		0			0
$298$			63.2017i			0.212086i
$299$		−	7.80063i		−	0.0260891i
$300$		0			0
$301$	−193.161			−0.641730
$302$	148.917			0.493102
$303$		0			0
$304$	178.353			0.586688
$305$	231.548	−	118.106i	0.759173	−	0.387232i
$306$		0			0
$307$			63.3011i			0.206193i	0.994671	+	0.103096i	$0.0328750\pi$
$307$			63.3011i			0.206193i	−0.994671	+	0.103096i	$0.967125\pi$
$308$	−46.1736			−0.149914
$309$		0			0
$310$	102.690	−	52.3792i	0.331259	−	0.168965i
$311$			576.118i			1.85247i	0.376948	+	0.926234i	$0.376974\pi$
$311$			576.118i			1.85247i	−0.376948	+	0.926234i	$0.623026\pi$
$312$		0			0
$313$		−	59.5453i		−	0.190240i	−0.995466	−	0.0951202i	$-0.969676\pi$
$313$		−	59.5453i		−	0.190240i	0.995466	−	0.0951202i	$-0.0303236\pi$
$314$		−	161.558i		−	0.514517i
$315$		0			0
$316$	−212.057			−0.671068
$317$	−342.936			−1.08182			−0.540908	−	0.841082i	$-0.681919\pi$
$317$	−342.936			−1.08182			−0.540908	+	0.841082i	$0.681919\pi$
$318$		0			0
$319$	39.0274			0.122343
$320$	4.43836	−	2.26387i	0.0138699	−	0.00707460i
$321$		0			0
$322$		−	36.8988i		−	0.114592i
$323$	574.827			1.77965
$324$		0			0
$325$	−16.2945	−	11.8172i	−0.0501370	−	0.0363605i
$326$			88.7089i			0.272113i
$327$		0			0
$328$		−	460.765i		−	1.40477i
$329$		−	223.582i		−	0.679579i
$330$		0			0
$331$	−413.679			−1.24978			−0.624892	−	0.780711i	$-0.714857\pi$
$331$	−413.679			−1.24978			−0.624892	+	0.780711i	$0.714857\pi$
$332$	−174.670			−0.526114
$333$		0			0
$334$	−189.388			−0.567031
$335$	181.892	+	356.602i	0.542961	+	1.06448i
$336$		0			0
$337$		−	605.660i		−	1.79721i	−0.438759	−	0.898605i	$-0.644582\pi$
$337$		−	605.660i		−	1.79721i	0.438759	−	0.898605i	$-0.355418\pi$
$338$	148.422			0.439119
$339$		0			0
$340$	336.662	−	171.721i	0.990183	−	0.505062i
$341$		−	86.7338i		−	0.254351i
$342$		0			0
$343$		−	342.733i		−	0.999223i
$344$			284.729i			0.827700i
$345$		0			0
$346$	−126.532			−0.365699
$347$	−357.744			−1.03096			−0.515481	−	0.856901i	$-0.672387\pi$
$347$	−357.744			−1.03096			−0.515481	+	0.856901i	$0.672387\pi$
$348$		0			0
$349$	293.440			0.840802			0.420401	−	0.907338i	$-0.361889\pi$
$349$	293.440			0.840802			0.420401	+	0.907338i	$0.361889\pi$
$350$	−77.0768	−	55.8978i	−0.220219	−	0.159708i
$351$		0			0
$352$			105.756i			0.300442i
$353$	560.030			1.58649			0.793243	−	0.608905i	$-0.208391\pi$
$353$	560.030			1.58649			0.793243	+	0.608905i	$0.208391\pi$
$354$		0			0
$355$	14.7706	+	28.9581i	0.0416074	+	0.0815720i
$356$			475.141i			1.33466i
$357$		0			0
$358$		−	116.504i		−	0.325431i
$359$		−	217.132i		−	0.604824i	−0.953177	−	0.302412i	$-0.902208\pi$
$359$		−	217.132i		−	0.604824i	0.953177	−	0.302412i	$-0.0977918\pi$
$360$		0			0
$361$	239.688			0.663956
$362$	−218.990			−0.604945
$363$		0			0
$364$	11.2091			0.0307941
$365$	110.886	+	217.394i	0.303797	+	0.595599i
$366$		0			0
$367$			547.500i			1.49183i	0.666044	+	0.745913i	$0.267986\pi$
$367$			547.500i			1.49183i	−0.666044	+	0.745913i	$0.732014\pi$
$368$	70.5042			0.191587
$369$		0			0
$370$	32.2306	+	63.1886i	0.0871098	+	0.170780i
$371$		−	172.769i		−	0.465685i
$372$		0			0
$373$		−	462.335i		−	1.23950i	−0.784798	−	0.619752i	$-0.787233\pi$
$373$		−	462.335i		−	1.23950i	0.784798	−	0.619752i	$-0.212767\pi$
$374$			68.5790i			0.183366i
$375$		0			0
$376$	−329.570			−0.876517
$377$	−9.47425			−0.0251306
$378$		0			0
$379$	105.655			0.278772			0.139386	−	0.990238i	$-0.455487\pi$
$379$	105.655			0.278772			0.139386	+	0.990238i	$0.455487\pi$
$380$	351.808	−	179.447i	0.925810	−	0.472228i
$381$		0			0
$382$		−	162.629i		−	0.425731i
$383$	45.8064			0.119599			0.0597995	−	0.998210i	$-0.480954\pi$
$383$	45.8064			0.119599			0.0597995	+	0.998210i	$0.480954\pi$
$384$		0			0
$385$	−63.8151	+	32.5502i	−0.165753	+	0.0845459i
$386$			214.123i			0.554722i
$387$		0			0
$388$			501.242i			1.29186i
$389$			387.639i			0.996501i	0.867033	+	0.498250i	$0.166024\pi$
$389$			387.639i			0.996501i	−0.867033	+	0.498250i	$0.833976\pi$
$390$		0			0
$391$	227.233			0.581159
$392$	−193.188			−0.492826
$393$		0			0
$394$	101.436			0.257451
$395$	−293.078	+	149.490i	−0.741969	+	0.378456i
$396$		0			0
$397$			559.989i			1.41055i	0.708933	+	0.705276i	$0.249177\pi$
$397$			559.989i			1.41055i	−0.708933	+	0.705276i	$0.750823\pi$
$398$	−58.3875			−0.146702
$399$		0			0
$400$	106.807	−	147.274i	0.267016	−	0.368185i
$401$		−	202.648i		−	0.505358i	−0.967550	−	0.252679i	$-0.918688\pi$
$401$		−	202.648i		−	0.505358i	0.967550	−	0.252679i	$-0.0813116\pi$
$402$		0			0
$403$			21.0554i			0.0522466i
$404$		−	196.111i		−	0.485423i
$405$		0			0
$406$	−44.8154			−0.110383
$407$	53.3701			0.131131
$408$		0			0
$409$	379.874			0.928788			0.464394	−	0.885629i	$-0.346272\pi$
$409$	379.874			0.928788			0.464394	+	0.885629i	$0.346272\pi$
$410$	−144.931	−	284.140i	−0.353491	−	0.693024i
$411$		0			0
$412$		−	534.033i		−	1.29620i
$413$	313.806			0.759822
$414$		0			0
$415$	−241.406	+	123.134i	−0.581700	+	0.296708i
$416$		−	25.6731i		−	0.0617141i
$417$		0			0
$418$			71.6642i			0.171446i
$419$		−	493.630i		−	1.17811i	−0.808091	−	0.589057i	$-0.799499\pi$
$419$		−	493.630i		−	1.17811i	0.808091	−	0.589057i	$-0.200501\pi$
$420$		0			0
$421$	675.728			1.60505			0.802527	−	0.596616i	$-0.203488\pi$
$421$	675.728			1.60505			0.802527	+	0.596616i	$0.203488\pi$
$422$	195.821			0.464031
$423$		0			0
$424$	−254.670			−0.600637
$425$	344.235	−	474.661i	0.809964	−	1.11685i
$426$		0			0
$427$			224.573i			0.525932i
$428$	−125.246			−0.292630
$429$		0			0
$430$	89.5599	+	175.584i	0.208279	+	0.408334i
$431$		−	656.908i		−	1.52415i	−0.647489	−	0.762075i	$-0.724181\pi$
$431$		−	656.908i		−	1.52415i	0.647489	−	0.762075i	$-0.275819\pi$
$432$		0			0
$433$			767.919i			1.77348i	0.462265	+	0.886742i	$0.347037\pi$
$433$			767.919i			1.77348i	−0.462265	+	0.886742i	$0.652963\pi$
$434$			99.5969i			0.229486i
$435$		0			0
$436$	10.7562			0.0246701
$437$	237.456			0.543377
$438$		0			0
$439$	221.614			0.504815			0.252408	−	0.967621i	$-0.418778\pi$
$439$	221.614			0.504815			0.252408	+	0.967621i	$0.418778\pi$
$440$	47.9806	+	94.0666i	0.109047	+	0.213788i
$441$		0			0
$442$		−	16.6482i		−	0.0376655i
$443$	−705.229			−1.59194			−0.795970	−	0.605336i	$-0.793039\pi$
$443$	−705.229			−1.59194			−0.795970	+	0.605336i	$0.793039\pi$
$444$		0			0
$445$	334.951	+	656.676i	0.752699	+	1.47568i
$446$			73.0694i			0.163833i
$447$		0			0
$448$			4.30466i			0.00960862i
$449$			210.018i			0.467746i	0.972267	+	0.233873i	$0.0751400\pi$
$449$			210.018i			0.467746i	−0.972267	+	0.233873i	$0.924860\pi$
$450$		0			0
$451$	−239.989			−0.532127
$452$	26.5377			0.0587116
$453$		0			0
$454$	−33.5806			−0.0739661
$455$	15.4917	−	7.90183i	0.0340476	−	0.0173667i
$456$		0			0
$457$			142.090i			0.310918i	0.987842	+	0.155459i	$0.0496857\pi$
$457$			142.090i			0.310918i	−0.987842	+	0.155459i	$0.950314\pi$
$458$	292.475			0.638593
$459$		0			0
$460$	139.072	−	70.9364i	0.302330	−	0.154210i
$461$		−	249.655i		−	0.541552i	−0.962642	−	0.270776i	$-0.912720\pi$
$461$		−	249.655i		−	0.541552i	0.962642	−	0.270776i	$-0.0872803\pi$
$462$		0			0
$463$			176.885i			0.382041i	0.981586	+	0.191021i	$0.0611797\pi$
$463$			176.885i			0.382041i	−0.981586	+	0.191021i	$0.938820\pi$
$464$		−	85.6308i		−	0.184549i
$465$		0			0
$466$	−105.139			−0.225619
$467$	−555.406			−1.18931			−0.594653	−	0.803983i	$-0.702710\pi$
$467$	−555.406			−1.18931			−0.594653	+	0.803983i	$0.702710\pi$
$468$		0			0
$469$	−345.860			−0.737441
$470$	−203.236	+	103.665i	−0.432418	+	0.220563i
$471$		0			0
$472$		−	462.566i		−	0.980014i
$473$	148.301			0.313532
$474$		0			0
$475$	359.721	−	496.015i	0.757308	−	1.04424i
$476$			326.521i			0.685968i
$477$		0			0
$478$		−	47.0712i		−	0.0984753i
$479$		−	321.546i		−	0.671287i	−0.941989	−	0.335643i	$-0.891046\pi$
$479$		−	321.546i		−	0.671287i	0.941989	−	0.335643i	$-0.108954\pi$
$480$		0			0
$481$	−12.9561			−0.0269357
$482$	169.992			0.352680
$483$		0			0
$484$	35.4502			0.0732442
$485$	353.351	+	692.751i	0.728559	+	1.42835i
$486$		0			0
$487$			59.1900i			0.121540i	0.998152	+	0.0607700i	$0.0193556\pi$
$487$			59.1900i			0.121540i	−0.998152	+	0.0607700i	$0.980644\pi$
$488$	331.032			0.678344
$489$		0			0
$490$	−119.133	+	60.7663i	−0.243129	+	0.124013i
$491$			139.798i			0.284721i	0.989815	+	0.142360i	$0.0454692\pi$
$491$			139.798i			0.284721i	−0.989815	+	0.142360i	$0.954531\pi$
$492$		0			0
$493$		−	275.986i		−	0.559809i
$494$		−	17.3971i		−	0.0352168i
$495$		0			0
$496$	−190.304			−0.383678
$497$	−28.0857			−0.0565106
$498$		0			0
$499$	−800.429			−1.60407			−0.802033	−	0.597280i	$-0.796248\pi$
$499$	−800.429			−1.60407			−0.802033	+	0.597280i	$0.796248\pi$
$500$	62.5027	−	397.965i	0.125005	−	0.795930i
$501$		0			0
$502$		−	131.246i		−	0.261447i
$503$	−951.201			−1.89106			−0.945528	−	0.325541i	$-0.894454\pi$
$503$	−951.201			−1.89106			−0.945528	+	0.325541i	$0.894454\pi$
$504$		0			0
$505$	−138.248	−	271.038i	−0.273759	−	0.536710i
$506$			28.3293i			0.0559868i
$507$		0			0
$508$		−	642.259i		−	1.26429i
$509$		−	70.3645i		−	0.138241i	−0.997608	−	0.0691203i	$-0.977981\pi$
$509$		−	70.3645i		−	0.138241i	0.997608	−	0.0691203i	$-0.0220192\pi$
$510$		0			0
$511$	−210.845			−0.412613
$512$	417.393			0.815222
$513$		0			0
$514$	386.129			0.751223
$515$	−376.467	−	738.070i	−0.731005	−	1.43315i
$516$		0			0
$517$			171.657i			0.332024i
$518$	−61.2852			−0.118311
$519$		0			0
$520$	−11.6477	−	22.8355i	−0.0223994	−	0.0439144i
$521$			211.223i			0.405419i	0.979239	+	0.202710i	$0.0649747\pi$
$521$			211.223i			0.405419i	−0.979239	+	0.202710i	$0.935025\pi$
$522$		0			0
$523$			929.551i			1.77734i	0.458543	+	0.888672i	$0.348371\pi$
$523$			929.551i			1.77734i	−0.458543	+	0.888672i	$0.651629\pi$
$524$			56.3527i			0.107543i
$525$		0			0
$526$	−239.782			−0.455859
$527$	−613.346			−1.16384
$528$		0			0
$529$	−435.132			−0.822556
$530$	−157.047	+	80.1052i	−0.296316	+	0.151142i
$531$		0			0
$532$			341.210i			0.641373i
$533$	58.2595			0.109305
$534$		0			0
$535$	−173.098	+	88.2920i	−0.323547	+	0.165032i
$536$			509.815i			0.951147i
$537$		0			0
$538$		−	52.7869i		−	0.0981169i
$539$			100.622i			0.186682i
$540$		0			0
$541$	−161.592			−0.298691			−0.149346	−	0.988785i	$-0.547717\pi$
$541$	−161.592			−0.298691			−0.149346	+	0.988785i	$0.547717\pi$
$542$	95.2091			0.175663
$543$		0			0
$544$	747.859			1.37474
$545$	14.8658	−	7.58258i	0.0272766	−	0.0139130i
$546$		0			0
$547$			32.3116i			0.0590706i	0.999564	+	0.0295353i	$0.00940274\pi$
$547$			32.3116i			0.0590706i	−0.999564	+	0.0295353i	$0.990597\pi$
$548$	−478.482			−0.873143
$549$		0			0
$550$	59.1764	+	42.9160i	0.107593	+	0.0780292i
$551$		−	288.402i		−	0.523415i
$552$		0			0
$553$		−	284.249i		−	0.514013i
$554$		−	371.246i		−	0.670118i
$555$		0			0
$556$	−384.385			−0.691340
$557$	−742.165			−1.33243			−0.666216	−	0.745759i	$-0.732087\pi$
$557$	−742.165			−1.33243			−0.666216	+	0.745759i	$0.732087\pi$
$558$		0			0
$559$	−36.0013			−0.0644031
$560$	71.4189	+	140.018i	0.127534	+	0.250032i
$561$		0			0
$562$			420.680i			0.748541i
$563$	−944.546			−1.67770			−0.838851	−	0.544361i	$-0.816772\pi$
$563$	−944.546			−1.67770			−0.838851	+	0.544361i	$0.816772\pi$
$564$		0			0
$565$	36.6768	−	18.7078i	0.0649148	−	0.0331111i
$566$		−	430.206i		−	0.760082i
$567$		0			0
$568$			41.3998i			0.0728870i
$569$			764.699i			1.34393i	0.740581	+	0.671967i	$0.234551\pi$
$569$			764.699i			1.34393i	−0.740581	+	0.671967i	$0.765449\pi$
$570$		0			0
$571$	199.849			0.349999			0.174999	−	0.984569i	$-0.444008\pi$
$571$	199.849			0.349999			0.174999	+	0.984569i	$0.444008\pi$
$572$	−8.60585			−0.0150452
$573$		0			0
$574$	275.581			0.480106
$575$	142.200	−	196.078i	0.247305	−	0.341005i
$576$		0			0
$577$		−	266.197i		−	0.461347i	−0.973031	−	0.230674i	$-0.925907\pi$
$577$		−	266.197i		−	0.461347i	0.973031	−	0.230674i	$-0.0740930\pi$
$578$	230.174			0.398225
$579$		0			0
$580$	−86.1558	−	168.910i	−0.148544	−	0.291224i
$581$		−	234.134i		−	0.402984i
$582$		0			0
$583$			132.645i			0.227521i
$584$			310.796i			0.532185i
$585$		0			0
$586$	161.358			0.275356
$587$	734.572			1.25140			0.625700	−	0.780064i	$-0.284813\pi$
$587$	734.572			1.25140			0.625700	+	0.780064i	$0.284813\pi$
$588$		0			0
$589$	−640.939			−1.08818
$590$	−145.498	−	285.251i	−0.246607	−	0.483476i
$591$		0			0
$592$		−	117.100i		−	0.197805i
$593$	−204.161			−0.344285			−0.172143	−	0.985072i	$-0.555069\pi$
$593$	−204.161			−0.344285			−0.172143	+	0.985072i	$0.555069\pi$
$594$		0			0
$595$	230.181	+	451.274i	0.386859	+	0.758443i
$596$		−	231.033i		−	0.387638i
$597$		0			0
$598$		−	6.87720i		−	0.0115003i
$599$			811.743i			1.35516i	0.735447	+	0.677582i	$0.236972\pi$
$599$			811.743i			1.35516i	−0.735447	+	0.677582i	$0.763028\pi$
$600$		0			0
$601$	614.601			1.02263			0.511316	−	0.859393i	$-0.329158\pi$
$601$	614.601			1.02263			0.511316	+	0.859393i	$0.329158\pi$
$602$	−170.295			−0.282881
$603$		0			0
$604$	−544.362			−0.901262
$605$	48.9946	−	24.9907i	0.0809828	−	0.0413069i
$606$		0			0
$607$			382.001i			0.629327i	0.949203	+	0.314663i	$0.101892\pi$
$607$			382.001i			0.629327i	−0.949203	+	0.314663i	$0.898108\pi$
$608$	781.504			1.28537
$609$		0			0
$610$	204.137	−	104.124i	0.334651	−	0.170696i
$611$		−	41.6712i		−	0.0682016i
$612$		0			0
$613$		−	35.8793i		−	0.0585306i	−0.999572	−	0.0292653i	$-0.990683\pi$
$613$		−	35.8793i		−	0.0585306i	0.999572	−	0.0292653i	$-0.00931676\pi$
$614$			55.8076i			0.0908918i
$615$		0			0
$616$	−91.2330			−0.148106
$617$	41.8557			0.0678375			0.0339187	−	0.999425i	$-0.489201\pi$
$617$	41.8557			0.0678375			0.0339187	+	0.999425i	$0.489201\pi$
$618$		0			0
$619$	−969.366			−1.56602			−0.783010	−	0.622010i	$-0.786316\pi$
$619$	−969.366			−1.56602			−0.783010	+	0.622010i	$0.786316\pi$
$620$	−375.382	+	191.471i	−0.605455	+	0.308824i
$621$		0			0
$622$			507.917i			0.816587i
$623$	−636.895			−1.02230
$624$		0			0
$625$	−194.163	−	594.076i	−0.310660	−	0.950521i
$626$		−	52.4963i		−	0.0838600i
$627$		0			0
$628$			590.573i			0.940403i
$629$		−	377.411i		−	0.600018i
$630$		0			0
$631$	−405.690			−0.642933			−0.321466	−	0.946921i	$-0.604176\pi$
$631$	−405.690			−0.642933			−0.321466	+	0.946921i	$0.604176\pi$
$632$	−418.998			−0.662971
$633$		0			0
$634$	−302.339			−0.476876
$635$	−452.761	−	887.645i	−0.713009	−	1.39787i
$636$		0			0
$637$		−	24.4268i		−	0.0383467i
$638$	34.4074			0.0539301
$639$		0			0
$640$	572.009	−	291.765i	0.893765	−	0.455882i
$641$		−	775.204i		−	1.20937i	−0.796466	−	0.604683i	$-0.793300\pi$
$641$		−	775.204i		−	1.20937i	0.796466	−	0.604683i	$-0.206700\pi$
$642$		0			0
$643$			1105.14i			1.71872i	0.511369	+	0.859361i	$0.329138\pi$
$643$			1105.14i			1.71872i	−0.511369	+	0.859361i	$0.670862\pi$
$644$			134.883i			0.209445i
$645$		0			0
$646$	506.780			0.784489
$647$	−471.466			−0.728695			−0.364348	−	0.931263i	$-0.618708\pi$
$647$	−471.466			−0.728695			−0.364348	+	0.931263i	$0.618708\pi$
$648$		0			0
$649$	−240.928			−0.371229
$650$	−14.3656	−	10.4182i	−0.0221009	−	0.0160281i
$651$		0			0
$652$		−	324.274i		−	0.497352i
$653$	−720.386			−1.10320			−0.551598	−	0.834110i	$-0.685982\pi$
$653$	−720.386			−1.10320			−0.551598	+	0.834110i	$0.685982\pi$
$654$		0			0
$655$	39.7259	+	77.8832i	0.0606502	+	0.118906i
$656$			526.565i			0.802690i
$657$		0			0
$658$		−	197.114i		−	0.299566i
$659$			111.043i			0.168503i	0.996445	+	0.0842514i	$0.0268499\pi$
$659$			111.043i			0.168503i	−0.996445	+	0.0842514i	$0.973150\pi$
$660$		0			0
$661$	627.984			0.950051			0.475025	−	0.879972i	$-0.342439\pi$
$661$	627.984			0.950051			0.475025	+	0.879972i	$0.342439\pi$
$662$	−364.708			−0.550918
$663$		0			0
$664$	−345.125			−0.519766
$665$	240.537	+	471.576i	0.361709	+	0.709136i
$666$		0			0
$667$		−	114.007i		−	0.170925i
$668$	692.306			1.03639
$669$		0			0
$670$	160.360	+	314.388i	0.239343	+	0.469235i
$671$		−	172.418i		−	0.256956i
$672$		0			0
$673$		−	83.2679i		−	0.123726i	−0.998085	−	0.0618632i	$-0.980296\pi$
$673$		−	83.2679i		−	0.123726i	0.998085	−	0.0618632i	$-0.0197043\pi$
$674$		−	533.962i		−	0.792228i
$675$		0			0
$676$	−542.555			−0.802596
$677$	−305.320			−0.450990			−0.225495	−	0.974244i	$-0.572400\pi$
$677$	−305.320			−0.450990			−0.225495	+	0.974244i	$0.572400\pi$
$678$		0			0
$679$	−671.883			−0.989518
$680$	665.200	−	339.299i	0.978235	−	0.498968i
$681$		0			0
$682$		−	76.4663i		−	0.112121i
$683$	−377.711			−0.553017			−0.276508	−	0.961011i	$-0.589177\pi$
$683$	−377.711			−0.553017			−0.276508	+	0.961011i	$0.589177\pi$
$684$		0			0
$685$	−661.295	+	337.306i	−0.965393	+	0.492418i
$686$		−	302.161i		−	0.440468i
$687$		0			0
$688$		−	325.390i		−	0.472950i
$689$		−	32.2007i		−	0.0467354i
$690$		0			0
$691$	662.708			0.959057			0.479528	−	0.877526i	$-0.340808\pi$
$691$	662.708			0.959057			0.479528	+	0.877526i	$0.340808\pi$
$692$	462.534			0.668402
$693$		0			0
$694$	−315.394			−0.454459
$695$	−531.246	+	270.972i	−0.764382	+	0.389888i
$696$		0			0
$697$			1697.10i			2.43487i
$698$	258.703			0.370634
$699$		0			0
$700$	281.753	+	204.333i	0.402504	+	0.291905i
$701$		−	491.514i		−	0.701162i	−0.936532	−	0.350581i	$-0.885984\pi$
$701$		−	491.514i		−	0.701162i	0.936532	−	0.350581i	$-0.114016\pi$
$702$		0			0
$703$		−	394.391i		−	0.561011i
$704$		−	3.30494i		−	0.00469452i
$705$		0			0
$706$	493.734			0.699340
$707$	262.874			0.371816
$708$		0			0
$709$	−990.345			−1.39682			−0.698410	−	0.715698i	$-0.746109\pi$
$709$	−990.345			−1.39682			−0.698410	+	0.715698i	$0.746109\pi$
$710$	13.0221	+	25.5300i	0.0183410	+	0.0359578i
$711$		0			0
$712$			938.815i			1.31856i
$713$	−253.367			−0.355354
$714$		0			0
$715$	−11.8939	+	6.06670i	−0.0166348	+	0.00848490i
$716$			425.879i			0.594803i
$717$		0			0
$718$		−	191.428i		−	0.266613i
$719$			740.697i			1.03018i	0.857137	+	0.515088i	$0.172241\pi$
$719$			740.697i			1.03018i	−0.857137	+	0.515088i	$0.827759\pi$
$720$		0			0
$721$	715.837			0.992839
$722$	211.314			0.292678
$723$		0			0
$724$	800.514			1.10568
$725$	−238.146	−	172.709i	−0.328477	−	0.238219i
$726$		0			0
$727$		−	1038.96i		−	1.42910i	−0.699582	−	0.714552i	$-0.746630\pi$
$727$		−	1038.96i		−	1.42910i	0.699582	−	0.714552i	$-0.253370\pi$
$728$	22.1476			0.0304226
$729$		0			0
$730$	97.7593	+	191.659i	0.133917	+	0.262546i
$731$		−	1048.72i		−	1.43464i
$732$		0			0
$733$		−	653.980i		−	0.892196i	−0.894984	−	0.446098i	$-0.852813\pi$
$733$		−	653.980i		−	0.892196i	0.894984	−	0.446098i	$-0.147187\pi$
$734$			482.687i			0.657612i
$735$		0			0
$736$	308.934			0.419747
$737$	265.537			0.360294
$738$		0			0
$739$	−1190.97			−1.61160			−0.805799	−	0.592189i	$-0.798264\pi$
$739$	−1190.97			−1.61160			−0.805799	+	0.592189i	$0.798264\pi$
$740$	−117.818	−	230.985i	−0.159214	−	0.312141i
$741$		0			0
$742$		−	152.317i		−	0.205279i
$743$	210.179			0.282879			0.141439	−	0.989947i	$-0.454827\pi$
$743$	210.179			0.282879			0.141439	+	0.989947i	$0.454827\pi$
$744$		0			0
$745$	−162.867	−	319.303i	−0.218613	−	0.428594i
$746$		−	407.604i		−	0.546386i
$747$		0			0
$748$		−	250.689i		−	0.335146i
$749$		−	167.884i		−	0.224144i
$750$		0			0
$751$	145.894			0.194266			0.0971331	−	0.995271i	$-0.469033\pi$
$751$	145.894			0.194266			0.0971331	+	0.995271i	$0.469033\pi$
$752$	376.635			0.500844
$753$		0			0
$754$	−8.35269			−0.0110778
$755$	−752.345	+	383.749i	−0.996483	+	0.508276i
$756$		0			0
$757$		−	630.188i		−	0.832480i	−0.909255	−	0.416240i	$-0.863348\pi$
$757$		−	630.188i		−	0.832480i	0.909255	−	0.416240i	$-0.136652\pi$
$758$	93.1473			0.122886
$759$		0			0
$760$	695.126	−	354.563i	0.914640	−	0.466530i
$761$		−	711.376i		−	0.934790i	−0.884048	−	0.467395i	$-0.845193\pi$
$761$		−	711.376i		−	0.934790i	0.884048	−	0.467395i	$-0.154807\pi$
$762$		0			0
$763$			14.4180i			0.0188964i
$764$			594.488i			0.778126i
$765$		0			0
$766$	40.3839			0.0527205
$767$	58.4873			0.0762546
$768$		0			0
$769$	−732.462			−0.952486			−0.476243	−	0.879314i	$-0.658002\pi$
$769$	−732.462			−0.952486			−0.476243	+	0.879314i	$0.658002\pi$
$770$	−56.2607	+	28.6969i	−0.0730658	+	0.0372687i
$771$		0			0
$772$		−	782.721i		−	1.01389i
$773$	573.547			0.741975			0.370988	−	0.928638i	$-0.379019\pi$
$773$	573.547			0.741975			0.370988	+	0.928638i	$0.379019\pi$
$774$		0			0
$775$	−383.825	+	529.252i	−0.495259	+	0.682905i
$776$			990.389i			1.27627i
$777$		0			0
$778$			341.750i			0.439268i
$779$			1773.45i			2.27658i
$780$		0			0
$781$	21.5631			0.0276096
$782$	200.333			0.256181
$783$		0			0
$784$	220.776			0.281602
$785$	416.325	+	816.212i	0.530350	+	1.03976i
$786$		0			0
$787$		−	569.340i		−	0.723430i	−0.932289	−	0.361715i	$-0.882191\pi$
$787$		−	569.340i		−	0.723430i	0.932289	−	0.361715i	$-0.117809\pi$
$788$	−370.796			−0.470554
$789$		0			0
$790$	−258.383	+	131.794i	−0.327067	+	0.166827i
$791$			35.5720i			0.0449709i
$792$		0			0
$793$			41.8559i			0.0527817i
$794$			493.698i			0.621786i
$795$		0			0
$796$	213.434			0.268134
$797$	525.152			0.658911			0.329456	−	0.944171i	$-0.393135\pi$
$797$	525.152			0.658911			0.329456	+	0.944171i	$0.393135\pi$
$798$		0			0
$799$	1213.88			1.51925
$800$	468.003	−	645.323i	0.585003	−	0.806653i
$801$		0			0
$802$		−	178.659i		−	0.222767i
$803$	161.878			0.201592
$804$		0			0
$805$	95.0857	+	186.417i	0.118119	+	0.231574i
$806$			18.5629i			0.0230309i
$807$		0			0
$808$		−	387.489i		−	0.479566i
$809$			55.7383i			0.0688978i	0.999406	+	0.0344489i	$0.0109676\pi$
$809$			55.7383i			0.0688978i	−0.999406	+	0.0344489i	$0.989032\pi$
$810$		0			0
$811$	466.175			0.574816			0.287408	−	0.957808i	$-0.407207\pi$
$811$	466.175			0.574816			0.287408	+	0.957808i	$0.407207\pi$
$812$	163.822			0.201751
$813$		0			0
$814$	47.0522			0.0578037
$815$	−228.597	−	448.168i	−0.280487	−	0.549899i
$816$		0			0
$817$		−	1095.90i		−	1.34137i
$818$	334.905			0.409419
$819$		0			0
$820$	529.793	+	1038.67i	0.646089	+	1.26667i
$821$			48.0796i			0.0585622i	0.999571	+	0.0292811i	$0.00932180\pi$
$821$			48.0796i			0.0585622i	−0.999571	+	0.0292811i	$0.990678\pi$
$822$		0			0
$823$			442.369i			0.537508i	0.963209	+	0.268754i	$0.0866119\pi$
$823$			442.369i			0.537508i	−0.963209	+	0.268754i	$0.913388\pi$
$824$		−	1055.18i		−	1.28056i
$825$		0			0
$826$	276.658			0.334937
$827$	504.480			0.610012			0.305006	−	0.952350i	$-0.401341\pi$
$827$	504.480			0.610012			0.305006	+	0.952350i	$0.401341\pi$
$828$		0			0
$829$	754.176			0.909742			0.454871	−	0.890557i	$-0.349685\pi$
$829$	754.176			0.909742			0.454871	+	0.890557i	$0.349685\pi$
$830$	−212.828	+	108.557i	−0.256420	+	0.130792i
$831$		0			0
$832$			0.802303i			0.000964306i
$833$	711.556			0.854209
$834$		0			0
$835$	956.813	−	488.042i	1.14588	−	0.584481i
$836$		−	261.967i		−	0.313358i
$837$		0			0
$838$		−	435.194i		−	0.519325i
$839$		−	84.1954i		−	0.100352i	−0.998740	−	0.0501760i	$-0.984022\pi$
$839$		−	84.1954i		−	0.100352i	0.998740	−	0.0501760i	$-0.0159782\pi$
$840$		0			0
$841$	702.533			0.835354
$842$	595.735			0.707524
$843$		0			0
$844$	−715.820			−0.848128
$845$	−749.847	+	382.475i	−0.887393	+	0.452633i
$846$		0			0
$847$			47.5187i			0.0561023i
$848$	291.038			0.343206
$849$		0			0
$850$	303.484	−	418.471i	0.357040	−	0.492318i
$851$		−	155.905i		−	0.183202i
$852$		0			0
$853$			616.420i			0.722650i	0.932440	+	0.361325i	$0.117676\pi$
$853$			616.420i			0.722650i	−0.932440	+	0.361325i	$0.882324\pi$
$854$			197.988i			0.231836i
$855$		0			0
$856$	−247.469			−0.289099
$857$	1479.76			1.72668			0.863338	−	0.504626i	$-0.168370\pi$
$857$	1479.76			1.72668			0.863338	+	0.504626i	$0.168370\pi$
$858$		0			0
$859$	−1415.85			−1.64825			−0.824127	−	0.566405i	$-0.808334\pi$
$859$	−1415.85			−1.64825			−0.824127	+	0.566405i	$0.808334\pi$
$860$	−327.384	−	641.842i	−0.380680	−	0.746328i
$861$		0			0
$862$		−	579.144i		−	0.671861i
$863$	371.205			0.430133			0.215067	−	0.976599i	$-0.431003\pi$
$863$	371.205			0.430133			0.215067	+	0.976599i	$0.431003\pi$
$864$		0			0
$865$	639.254	−	326.064i	0.739022	−	0.376953i
$866$			677.013i			0.781770i
$867$		0			0
$868$		−	364.074i		−	0.419440i
$869$			218.235i			0.251133i
$870$		0			0
$871$	−64.4614			−0.0740085
$872$	21.2528			0.0243725
$873$		0			0
$874$	209.346			0.239526
$875$	533.446	+	83.7808i	0.609653	+	0.0957495i
$876$		0			0
$877$		−	741.354i		−	0.845330i	−0.906286	−	0.422665i	$-0.861095\pi$
$877$		−	741.354i		−	0.845330i	0.906286	−	0.422665i	$-0.138905\pi$
$878$	195.379			0.222528
$879$		0			0
$880$	−54.8324	−	107.500i	−0.0623096	−	0.122159i
$881$			685.220i			0.777776i	0.921285	+	0.388888i	$0.127141\pi$
$881$			685.220i			0.777776i	−0.921285	+	0.388888i	$0.872859\pi$
$882$		0			0
$883$		−	1053.40i		−	1.19298i	−0.802622	−	0.596488i	$-0.796562\pi$
$883$		−	1053.40i		−	1.19298i	0.802622	−	0.596488i	$-0.203438\pi$
$884$			60.8570i			0.0688427i
$885$		0			0
$886$	−621.745			−0.701744
$887$	−390.220			−0.439932			−0.219966	−	0.975508i	$-0.570595\pi$
$887$	−390.220			−0.439932			−0.219966	+	0.975508i	$0.570595\pi$
$888$		0			0
$889$	860.906			0.968398
$890$	295.300	+	578.939i	0.331797	+	0.650494i
$891$		0			0
$892$		−	267.104i		−	0.299443i
$893$	1268.50			1.42049
$894$		0			0
$895$	300.224	+	588.593i	0.335446	+	0.657646i
$896$			554.779i			0.619172i
$897$		0			0
$898$			185.156i			0.206187i
$899$			307.727i			0.342299i
$900$		0			0
$901$	938.009			1.04108
$902$	−211.579			−0.234567
$903$		0			0
$904$	52.4349			0.0580032
$905$	1106.36	−	564.323i	1.22250	−	0.623561i
$906$		0			0
$907$			414.594i			0.457105i	0.973532	+	0.228553i	$0.0733993\pi$
$907$			414.594i			0.457105i	−0.973532	+	0.228553i	$0.926601\pi$
$908$	122.753			0.135191
$909$		0			0
$910$	13.6578	−	6.96642i	0.0150085	−	0.00765541i
$911$		−	717.772i		−	0.787895i	−0.919133	−	0.393947i	$-0.871109\pi$
$911$		−	717.772i		−	0.787895i	0.919133	−	0.393947i	$-0.128891\pi$
$912$		0			0
$913$			179.758i			0.196887i
$914$			125.269i			0.137056i
$915$		0			0
$916$	−1069.14			−1.16718
$917$	−75.5371			−0.0823742
$918$		0			0
$919$	−1719.54			−1.87110			−0.935548	−	0.353199i	$-0.885094\pi$
$919$	−1719.54			−1.87110			−0.935548	+	0.353199i	$0.885094\pi$
$920$	274.788	−	140.161i	0.298682	−	0.152349i
$921$		0			0
$922$		−	220.101i		−	0.238722i
$923$	−5.23462			−0.00567132
$924$		0			0
$925$	−325.666	−	236.180i	−0.352071	−	0.255330i
$926$			155.945i			0.168408i
$927$		0			0
$928$		−	375.215i		−	0.404327i
$929$			744.595i			0.801502i	0.916187	+	0.400751i	$0.131251\pi$
$929$			744.595i			0.801502i	−0.916187	+	0.400751i	$0.868749\pi$
$930$		0			0
$931$	743.568			0.798676
$932$	384.332			0.412373
$933$		0			0
$934$	−489.657			−0.524258
$935$	−176.724	−	346.469i	−0.189009	−	0.370555i
$936$		0			0
$937$		−	1420.79i		−	1.51631i	−0.652072	−	0.758157i	$-0.726100\pi$
$937$		−	1420.79i		−	1.51631i	0.652072	−	0.758157i	$-0.273900\pi$
$938$	−304.917			−0.325072
$939$		0			0
$940$	742.926	−	378.944i	0.790347	−	0.403132i
$941$			448.376i			0.476489i	0.971205	+	0.238245i	$0.0765720\pi$
$941$			448.376i			0.476489i	−0.971205	+	0.238245i	$0.923428\pi$
$942$		0			0
$943$			701.057i			0.743433i
$944$			528.623i			0.559982i
$945$		0			0
$946$	130.745			0.138208
$947$	−1062.83			−1.12231			−0.561154	−	0.827711i	$-0.689643\pi$
$947$	−1062.83			−1.12231			−0.561154	+	0.827711i	$0.689643\pi$
$948$		0			0
$949$	−39.2973			−0.0414092
$950$	317.138	−	437.297i	0.333829	−	0.460312i
$951$		0			0
$952$			645.162i			0.677691i
$953$	−425.667			−0.446661			−0.223330	−	0.974743i	$-0.571693\pi$
$953$	−425.667			−0.446661			−0.223330	+	0.974743i	$0.571693\pi$
$954$		0			0
$955$	419.085	+	821.623i	0.438832	+	0.860338i
$956$			172.068i			0.179987i
$957$		0			0
$958$		−	283.482i		−	0.295910i
$959$		−	641.374i		−	0.668795i
$960$		0			0
$961$	−277.113			−0.288359
$962$	−11.4223			−0.0118735
$963$		0			0
$964$	−621.401			−0.644607
$965$	−551.780	−	1081.77i	−0.571793	−	1.12101i
$966$		0			0
$967$			1846.11i			1.90911i	0.298030	+	0.954557i	$0.403670\pi$
$967$			1846.11i			1.90911i	−0.298030	+	0.954557i	$0.596330\pi$
$968$	70.0449			0.0723605
$969$		0			0
$970$	311.522	+	610.743i	0.321157	+	0.629632i
$971$		−	745.297i		−	0.767556i	−0.923425	−	0.383778i	$-0.874623\pi$
$971$		−	745.297i		−	0.767556i	0.923425	−	0.383778i	$-0.125377\pi$
$972$		0			0
$973$		−	515.243i		−	0.529541i
$974$			52.1831i			0.0535761i
$975$		0			0
$976$	−378.305			−0.387608
$977$	780.193			0.798559			0.399280	−	0.916829i	$-0.369260\pi$
$977$	780.193			0.798559			0.399280	+	0.916829i	$0.369260\pi$
$978$		0			0
$979$	488.982			0.499471
$980$	435.489	−	222.130i	0.444376	−	0.226663i
$981$		0			0
$982$			123.249i			0.125508i
$983$	58.9362			0.0599555			0.0299777	−	0.999551i	$-0.490456\pi$
$983$	58.9362			0.0599555			0.0299777	+	0.999551i	$0.490456\pi$
$984$		0			0
$985$	−512.466	+	261.393i	−0.520270	+	0.265374i
$986$		−	243.315i		−	0.246769i
$987$		0			0
$988$			63.5948i			0.0643672i
$989$		−	433.217i		−	0.438035i
$990$		0			0
$991$	1627.29			1.64207			0.821034	−	0.570880i	$-0.193398\pi$
$991$	1627.29			1.64207			0.821034	+	0.570880i	$0.193398\pi$
$992$	−833.871			−0.840596
$993$		0			0
$994$	−24.7610			−0.0249104
$995$	294.981	−	150.461i	0.296463	−	0.151217i
$996$		0			0
$997$			439.025i			0.440346i	0.975461	+	0.220173i	$0.0706623\pi$
$997$			439.025i			0.440346i	−0.975461	+	0.220173i	$0.929338\pi$
$998$	−705.675			−0.707089
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.3.g.a.89.25	yes	40
3.2	odd	2	inner	495.3.g.a.89.15	✓	40
5.4	even	2	inner	495.3.g.a.89.16	yes	40
15.14	odd	2	inner	495.3.g.a.89.26	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.3.g.a.89.15	✓	40	3.2	odd	2	inner
495.3.g.a.89.16	yes	40	5.4	even	2	inner
495.3.g.a.89.25	yes	40	1.1	even	1	trivial
495.3.g.a.89.26	yes	40	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	495.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.881621 q^{2} -3.22275 q^{4} +(-4.45405 + 2.27188i) q^{5} -4.31988i q^{7} -6.36772 q^{8} +(-3.92678 + 2.00293i) q^{10} +3.31662i q^{11} -0.805140i q^{13} -3.80850i q^{14} +7.27707 q^{16} +23.4538 q^{17} +24.5089 q^{19} +(14.3543 - 7.32168i) q^{20} +2.92400i q^{22} +9.68854 q^{23} +(14.6771 - 20.2381i) q^{25} -0.709828i q^{26} +13.9219i q^{28} -11.7672i q^{29} -26.1512 q^{31} +31.8865 q^{32} +20.6773 q^{34} +(9.81424 + 19.2410i) q^{35} -16.0917i q^{37} +21.6076 q^{38} +(28.3622 - 14.4667i) q^{40} +72.3595i q^{41} -44.7144i q^{43} -10.6886i q^{44} +8.54162 q^{46} +51.7564 q^{47} +30.3386 q^{49} +(12.9397 - 17.8423i) q^{50} +2.59476i q^{52} +39.9939 q^{53} +(-7.53497 - 14.7724i) q^{55} +27.5078i q^{56} -10.3742i q^{58} +72.6424i q^{59} -51.9859 q^{61} -23.0555 q^{62} -0.996477 q^{64} +(1.82918 + 3.58613i) q^{65} -80.0624i q^{67} -75.5856 q^{68} +(8.65244 + 16.9632i) q^{70} -6.50151i q^{71} -48.8081i q^{73} -14.1868i q^{74} -78.9861 q^{76} +14.3274 q^{77} +65.8002 q^{79} +(-32.4124 + 16.5326i) q^{80} +63.7936i q^{82} +54.1991 q^{83} +(-104.464 + 53.2841i) q^{85} -39.4211i q^{86} -21.1193i q^{88} -147.434i q^{89} -3.47811 q^{91} -31.2237 q^{92} +45.6295 q^{94} +(-109.164 + 55.6813i) q^{95} -155.533i q^{97} +26.7472 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 72 q^{4} + 8 q^{10} + 184 q^{16} - 80 q^{19} + 32 q^{25} - 16 q^{31} - 160 q^{34} - 136 q^{40} + 560 q^{46} - 104 q^{49} - 96 q^{61} + 264 q^{64} - 872 q^{70} - 176 q^{76} - 672 q^{79} + 16 q^{85}+ \cdots + 400 q^{94}+O(q^{100}) \) 40 * q + 72 * q^4 + 8 * q^10 + 184 * q^16 - 80 * q^19 + 32 * q^25 - 16 * q^31 - 160 * q^34 - 136 * q^40 + 560 * q^46 - 104 * q^49 - 96 * q^61 + 264 * q^64 - 872 * q^70 - 176 * q^76 - 672 * q^79 + 16 * q^85 - 864 * q^91 + 400 * q^94

Introduction

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