LMFDB - Embedded newform 2052.3.m.a.881.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(881,2052)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2052, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 2]))

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

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

chi := DirichletCharacter("2052.881");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.m (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$55.9129502467$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$40$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			881.15
Character	$\chi$	$=$	2052.881
Dual form			2052.3.m.a.1493.26

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23654i q^{5} +(1.30590 + 2.26189i) q^{7} +O(q^{10})$	$q-2.23654i q^{5} +(1.30590 + 2.26189i) q^{7} +(-9.01402 + 5.20424i) q^{11} +(-0.456388 - 0.790487i) q^{13} +(-9.51759 + 5.49498i) q^{17} +(-18.2448 - 5.30348i) q^{19} +(-4.37454 + 2.52564i) q^{23} +19.9979 q^{25} +26.3219i q^{29} +(20.3381 - 35.2267i) q^{31} +(5.05879 - 2.92070i) q^{35} +65.4978 q^{37} -53.6789i q^{41} +(10.9202 - 18.9143i) q^{43} +16.7173i q^{47} +(21.0893 - 36.5277i) q^{49} +(27.5234 + 15.8907i) q^{53} +(11.6395 + 20.1602i) q^{55} +26.0605i q^{59} -57.3354 q^{61} +(-1.76795 + 1.02073i) q^{65} +(-17.3370 - 30.0286i) q^{67} +(31.4924 - 18.1821i) q^{71} +(-3.45130 - 5.97783i) q^{73} +(-23.5428 - 13.5924i) q^{77} +(29.1943 - 50.5661i) q^{79} +(8.31103 - 4.79838i) q^{83} +(12.2897 + 21.2865i) q^{85} +(135.141 + 78.0237i) q^{89} +(1.19199 - 2.06459i) q^{91} +(-11.8614 + 40.8052i) q^{95} +(-47.6951 + 82.6103i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + q^{7}+O(q^{10}) $ 80 * q + q^7	$ 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) $ 80 * q + q^7 - 18 * q^11 - 5 * q^13 + 9 * q^17 + 20 * q^19 - 72 * q^23 - 400 * q^25 - 8 * q^31 + 22 * q^37 - 44 * q^43 - 267 * q^49 + 36 * q^53 - 14 * q^61 + 144 * q^65 - 77 * q^67 + 135 * q^71 + 43 * q^73 - 216 * q^77 - 17 * q^79 + 171 * q^83 - 216 * q^89 + 122 * q^91 + 288 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times$.

$n$	$325$	$1027$	$1217$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	2.23654i		−	0.447308i	−0.974669	−	0.223654i	$-0.928201\pi$
$5$		−	2.23654i		−	0.447308i	0.974669	−	0.223654i	$-0.0717985\pi$
$6$		0			0
$7$	1.30590	+	2.26189i	0.186557	+	0.323126i	0.944100	−	0.329659i	$-0.106934\pi$
$7$	1.30590	+	2.26189i	0.186557	+	0.323126i	−0.757543	+	0.652785i	$0.773600\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−9.01402	+	5.20424i	−0.819456	+	0.473113i	−0.850229	−	0.526413i	$-0.823537\pi$
$11$	−9.01402	+	5.20424i	−0.819456	+	0.473113i	0.0307728	+	0.999526i	$0.490203\pi$
$12$		0			0
$13$	−0.456388	−	0.790487i	−0.0351067	−	0.0608067i	0.847938	−	0.530095i	$-0.177844\pi$
$13$	−0.456388	−	0.790487i	−0.0351067	−	0.0608067i	−0.883045	+	0.469288i	$0.844510\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−9.51759	+	5.49498i	−0.559858	+	0.323234i	−0.753089	−	0.657919i	$-0.771437\pi$
$17$	−9.51759	+	5.49498i	−0.559858	+	0.323234i	0.193230	+	0.981153i	$0.438103\pi$
$18$		0			0
$19$	−18.2448	−	5.30348i	−0.960253	−	0.279130i
$19$	−18.2448	−	5.30348i	−0.960253	−	0.279130i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.37454	+	2.52564i	−0.190198	+	0.109811i	−0.592075	−	0.805883i	$-0.701691\pi$
$23$	−4.37454	+	2.52564i	−0.190198	+	0.109811i	0.401877	+	0.915693i	$0.368358\pi$
$24$		0			0
$25$	19.9979			0.799916
$26$		0			0
$27$		0			0
$28$		0			0
$29$			26.3219i			0.907653i	0.891090	+	0.453827i	$0.149941\pi$
$29$			26.3219i			0.907653i	−0.891090	+	0.453827i	$0.850059\pi$
$30$		0			0
$31$	20.3381	−	35.2267i	0.656069	−	1.13635i	−0.325555	−	0.945523i	$-0.605551\pi$
$31$	20.3381	−	35.2267i	0.656069	−	1.13635i	0.981625	−	0.190822i	$-0.0611154\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.05879	−	2.92070i	0.144537	−	0.0834485i
$36$		0			0
$37$	65.4978			1.77021			0.885106	−	0.465390i	$-0.154086\pi$
$37$	65.4978			1.77021			0.885106	+	0.465390i	$0.154086\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	53.6789i		−	1.30924i	−0.755957	−	0.654621i	$-0.772828\pi$
$41$		−	53.6789i		−	1.30924i	0.755957	−	0.654621i	$-0.227172\pi$
$42$		0			0
$43$	10.9202	−	18.9143i	0.253957	−	0.439867i	−0.710655	−	0.703541i	$-0.751601\pi$
$43$	10.9202	−	18.9143i	0.253957	−	0.439867i	0.964612	+	0.263674i	$0.0849345\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			16.7173i			0.355687i	0.984059	+	0.177843i	$0.0569121\pi$
$47$			16.7173i			0.355687i	−0.984059	+	0.177843i	$0.943088\pi$
$48$		0			0
$49$	21.0893	−	36.5277i	0.430393	−	0.745462i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	27.5234	+	15.8907i	0.519310	+	0.299824i	0.736652	−	0.676272i	$-0.236405\pi$
$53$	27.5234	+	15.8907i	0.519310	+	0.299824i	−0.217342	+	0.976095i	$0.569739\pi$
$54$		0			0
$55$	11.6395	+	20.1602i	0.211627	+	0.366549i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			26.0605i			0.441703i	0.975307	+	0.220851i	$0.0708836\pi$
$59$			26.0605i			0.441703i	−0.975307	+	0.220851i	$0.929116\pi$
$60$		0			0
$61$	−57.3354			−0.939925			−0.469962	−	0.882686i	$-0.655733\pi$
$61$	−57.3354			−0.939925			−0.469962	+	0.882686i	$0.655733\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.76795	+	1.02073i	−0.0271993	+	0.0157035i
$66$		0			0
$67$	−17.3370	−	30.0286i	−0.258761	−	0.448187i	0.707149	−	0.707064i	$-0.249981\pi$
$67$	−17.3370	−	30.0286i	−0.258761	−	0.448187i	−0.965910	+	0.258877i	$0.916648\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	31.4924	−	18.1821i	0.443555	−	0.256086i	−0.261550	−	0.965190i	$-0.584234\pi$
$71$	31.4924	−	18.1821i	0.443555	−	0.256086i	0.705104	+	0.709104i	$0.250900\pi$
$72$		0			0
$73$	−3.45130	−	5.97783i	−0.0472781	−	0.0818881i	0.841418	−	0.540385i	$-0.181721\pi$
$73$	−3.45130	−	5.97783i	−0.0472781	−	0.0818881i	−0.888696	+	0.458497i	$0.848388\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−23.5428	−	13.5924i	−0.305751	−	0.176525i
$78$		0			0
$79$	29.1943	−	50.5661i	0.369548	−	0.640077i	−0.619947	−	0.784644i	$-0.712846\pi$
$79$	29.1943	−	50.5661i	0.369548	−	0.640077i	0.989495	+	0.144567i	$0.0461791\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.31103	−	4.79838i	0.100133	−	0.0578118i	−0.449097	−	0.893483i	$-0.648254\pi$
$83$	8.31103	−	4.79838i	0.100133	−	0.0578118i	0.549230	+	0.835671i	$0.314921\pi$
$84$		0			0
$85$	12.2897	+	21.2865i	0.144585	+	0.250429i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	135.141	+	78.0237i	1.51844	+	0.876671i	0.999765	+	0.0216992i	$0.00690760\pi$
$89$	135.141	+	78.0237i	1.51844	+	0.876671i	0.518674	+	0.854972i	$0.326426\pi$
$90$		0			0
$91$	1.19199	−	2.06459i	0.0130988	−	0.0226878i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.8614	+	40.8052i	−0.124857	+	0.429529i
$96$		0			0
$97$	−47.6951	+	82.6103i	−0.491702	+	0.851653i	−0.999954	−	0.00955552i	$-0.996958\pi$
$97$	−47.6951	+	82.6103i	−0.491702	+	0.851653i	0.508252	+	0.861208i	$0.330292\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			48.6409i			0.481593i	0.970576	+	0.240796i	$0.0774086\pi$
$101$			48.6409i			0.481593i	−0.970576	+	0.240796i	$0.922591\pi$
$102$		0			0
$103$	102.218	−	177.048i	0.992412	−	1.71891i	0.389725	−	0.920931i	$-0.372570\pi$
$103$	102.218	−	177.048i	0.992412	−	1.71891i	0.602688	−	0.797977i	$-0.294097\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	193.373i		−	1.80722i	−0.428353	−	0.903612i	$-0.640906\pi$
$107$		−	193.373i		−	1.80722i	0.428353	−	0.903612i	$-0.359094\pi$
$108$		0			0
$109$	50.6137	+	87.6656i	0.464346	+	0.804271i	0.999172	−	0.0406912i	$-0.0129560\pi$
$109$	50.6137	+	87.6656i	0.464346	+	0.804271i	−0.534826	+	0.844963i	$0.679623\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−120.372	−	69.4967i	−1.06524	−	0.615015i	−0.138361	−	0.990382i	$-0.544183\pi$
$113$	−120.372	−	69.4967i	−1.06524	−	0.615015i	−0.926876	+	0.375367i	$0.877517\pi$
$114$		0			0
$115$	5.64870	+	9.78384i	0.0491191	+	0.0850768i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−24.8580	−	14.3518i	−0.208891	−	0.120603i
$120$		0			0
$121$	−6.33167	+	10.9668i	−0.0523278	+	0.0906345i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	100.640i		−	0.805116i
$126$		0			0
$127$	32.3301	−	55.9973i	0.254568	−	0.440924i	−0.710210	−	0.703989i	$-0.751400\pi$
$127$	32.3301	−	55.9973i	0.254568	−	0.440924i	0.964778	+	0.263066i	$0.0847335\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	97.5940i		−	0.744992i	−0.928034	−	0.372496i	$-0.878502\pi$
$131$		−	97.5940i		−	0.744992i	0.928034	−	0.372496i	$-0.121498\pi$
$132$		0			0
$133$	−11.8300	−	48.1935i	−0.0889477	−	0.362357i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	215.495i		−	1.57296i	−0.617616	−	0.786480i	$-0.711901\pi$
$137$		−	215.495i		−	1.57296i	0.617616	−	0.786480i	$-0.288099\pi$
$138$		0			0
$139$	47.9409	+	83.0361i	0.344899	+	0.597382i	0.985335	−	0.170629i	$-0.0545799\pi$
$139$	47.9409	+	83.0361i	0.344899	+	0.597382i	−0.640437	+	0.768011i	$0.721247\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.22777	+	4.75031i	0.0575369	+	0.0332189i
$144$		0			0
$145$	58.8701			0.406000
$146$		0			0
$147$		0			0
$148$		0			0
$149$			71.0050i			0.476544i	0.971199	+	0.238272i	$0.0765809\pi$
$149$			71.0050i			0.476544i	−0.971199	+	0.238272i	$0.923419\pi$
$150$		0			0
$151$	19.3277	+	33.4766i	0.127998	+	0.221700i	0.922901	−	0.385037i	$-0.125811\pi$
$151$	19.3277	+	33.4766i	0.127998	+	0.221700i	−0.794903	+	0.606737i	$0.792478\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−78.7859	−	45.4871i	−0.508296	−	0.293465i
$156$		0			0
$157$	249.361			1.58829			0.794143	−	0.607731i	$-0.207920\pi$
$157$	249.361			1.58829			0.794143	+	0.607731i	$0.207920\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−11.4254	−	6.59648i	−0.0709654	−	0.0409719i
$162$		0			0
$163$	−68.7055			−0.421506			−0.210753	−	0.977539i	$-0.567592\pi$
$163$	−68.7055			−0.421506			−0.210753	+	0.977539i	$0.567592\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−30.9360	+	17.8609i	−0.185246	+	0.106952i	−0.589755	−	0.807582i	$-0.700776\pi$
$167$	−30.9360	+	17.8609i	−0.185246	+	0.106952i	0.404509	+	0.914534i	$0.367442\pi$
$168$		0			0
$169$	84.0834	−	145.637i	0.497535	−	0.861756i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	106.420	+	61.4417i	0.615146	+	0.355154i	0.774977	−	0.631990i	$-0.217762\pi$
$173$	106.420	+	61.4417i	0.615146	+	0.355154i	−0.159831	+	0.987144i	$0.551095\pi$
$174$		0			0
$175$	26.1152	+	45.2329i	0.149230	+	0.258474i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	142.388i		−	0.795463i	−0.917502	−	0.397732i	$-0.869798\pi$
$179$		−	142.388i		−	0.795463i	0.917502	−	0.397732i	$-0.130202\pi$
$180$		0			0
$181$	−58.9258	+	102.062i	−0.325557	+	0.563881i	−0.981625	−	0.190821i	$-0.938885\pi$
$181$	−58.9258	+	102.062i	−0.325557	+	0.563881i	0.656068	+	0.754702i	$0.272218\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	146.488i		−	0.791829i
$186$		0			0
$187$	57.1945	−	99.0637i	0.305853	−	0.529752i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−26.7649	+	15.4527i	−0.140130	+	0.0809043i	−0.568426	−	0.822734i	$-0.692447\pi$
$191$	−26.7649	+	15.4527i	−0.140130	+	0.0809043i	0.428296	+	0.903639i	$0.359114\pi$
$192$		0			0
$193$	−43.3676			−0.224702			−0.112351	−	0.993669i	$-0.535838\pi$
$193$	−43.3676			−0.224702			−0.112351	+	0.993669i	$0.535838\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	194.898i		−	0.989332i	−0.869083	−	0.494666i	$-0.835290\pi$
$197$		−	194.898i		−	0.989332i	0.869083	−	0.494666i	$-0.164710\pi$
$198$		0			0
$199$	1.35743	−	2.35113i	0.00682124	−	0.0118147i	−0.862595	−	0.505896i	$-0.831162\pi$
$199$	1.35743	−	2.35113i	0.00682124	−	0.0118147i	0.869416	+	0.494081i	$0.164495\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−59.5372	+	34.3738i	−0.293287	+	0.169329i
$204$		0			0
$205$	−120.055			−0.585634
$206$		0			0
$207$		0			0
$208$		0			0
$209$	192.060	−	47.1448i	0.918946	−	0.225573i
$210$		0			0
$211$	346.660			1.64294			0.821470	−	0.570251i	$-0.193154\pi$
$211$	346.660			1.64294			0.821470	+	0.570251i	$0.193154\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−42.3025	−	24.4234i	−0.196756	−	0.113597i
$216$		0			0
$217$	106.238			0.489578
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.68742	+	5.01568i	0.0393096	+	0.0226954i
$222$		0			0
$223$	−42.5555	+	73.7082i	−0.190832	+	0.330530i	−0.945526	−	0.325546i	$-0.894452\pi$
$223$	−42.5555	+	73.7082i	−0.190832	+	0.330530i	0.754694	+	0.656076i	$0.227785\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	230.450	−	133.050i	1.01520	−	0.586124i	0.102488	−	0.994734i	$-0.467320\pi$
$227$	230.450	−	133.050i	1.01520	−	0.586124i	0.912709	+	0.408610i	$0.133986\pi$
$228$		0			0
$229$	−47.2542	+	81.8466i	−0.206350	+	0.357409i	−0.950562	−	0.310535i	$-0.899492\pi$
$229$	−47.2542	+	81.8466i	−0.206350	+	0.357409i	0.744212	+	0.667943i	$0.232825\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	238.339	−	137.605i	1.02292	−	0.590580i	0.107968	−	0.994154i	$-0.465566\pi$
$233$	238.339	−	137.605i	1.02292	−	0.590580i	0.914947	+	0.403574i	$0.132232\pi$
$234$		0			0
$235$	37.3889			0.159102
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−154.005	−	88.9146i	−0.644371	−	0.372028i	0.141925	−	0.989877i	$-0.454671\pi$
$239$	−154.005	−	88.9146i	−0.644371	−	0.372028i	−0.786296	+	0.617850i	$0.788004\pi$
$240$		0			0
$241$	−180.423			−0.748645			−0.374322	−	0.927299i	$-0.622125\pi$
$241$	−180.423			−0.748645			−0.374322	+	0.927299i	$0.622125\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−81.6955	−	47.1669i	−0.333451	−	0.192518i
$246$		0			0
$247$	4.13438	+	16.8427i	0.0167384	+	0.0681891i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	204.777	+	118.228i	0.815844	+	0.471028i	0.848981	−	0.528423i	$-0.177216\pi$
$251$	204.777	+	118.228i	0.815844	+	0.471028i	−0.0331370	+	0.999451i	$0.510550\pi$
$252$		0			0
$253$	26.2881	−	45.5324i	0.103906	−	0.179970i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	34.8181	−	20.1022i	0.135479	−	0.0782188i	−0.430729	−	0.902481i	$-0.641743\pi$
$257$	34.8181	−	20.1022i	0.135479	−	0.0782188i	0.566208	+	0.824263i	$0.308410\pi$
$258$		0			0
$259$	85.5336	+	148.149i	0.330246	+	0.572002i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−352.648	−	203.602i	−1.34087	−	0.774150i	−0.353933	−	0.935271i	$-0.615156\pi$
$263$	−352.648	−	203.602i	−1.34087	−	0.774150i	−0.986935	+	0.161120i	$0.948489\pi$
$264$		0			0
$265$	35.5401	−	61.5572i	0.134113	−	0.232291i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	411.714	−	237.703i	1.53054	−	0.883655i	0.531199	−	0.847247i	$-0.321742\pi$
$269$	411.714	−	237.703i	1.53054	−	0.883655i	0.999337	−	0.0364078i	$-0.0115915\pi$
$270$		0			0
$271$	1.43602	+	2.48726i	0.00529897	+	0.00917808i	0.868663	−	0.495404i	$-0.164980\pi$
$271$	1.43602	+	2.48726i	0.00529897	+	0.00917808i	−0.863364	+	0.504582i	$0.831647\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−180.261	+	104.074i	−0.655496	+	0.378451i
$276$		0			0
$277$	52.5685	+	91.0512i	0.189778	+	0.328705i	0.945176	−	0.326561i	$-0.105890\pi$
$277$	52.5685	+	91.0512i	0.189778	+	0.328705i	−0.755398	+	0.655266i	$0.772557\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	278.367i		−	0.990629i	−0.868714	−	0.495314i	$-0.835053\pi$
$281$		−	278.367i		−	0.990629i	0.868714	−	0.495314i	$-0.164947\pi$
$282$		0			0
$283$	209.187			0.739177			0.369588	−	0.929196i	$-0.379499\pi$
$283$	209.187			0.739177			0.369588	+	0.929196i	$0.379499\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	121.416	−	70.0993i	0.423051	−	0.244248i
$288$		0			0
$289$	−84.1104	+	145.683i	−0.291039	+	0.504095i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	403.308	+	232.850i	1.37648	+	0.794710i	0.991734	−	0.128312i	$-0.0409560\pi$
$293$	403.308	+	232.850i	1.37648	+	0.794710i	0.384745	+	0.923023i	$0.374289\pi$
$294$		0			0
$295$	58.2852			0.197577
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.99297	+	2.30535i	0.0133544	+	0.00771018i
$300$		0			0
$301$	57.0425			0.189510
$302$		0			0
$303$		0			0
$304$		0			0
$305$			128.233i			0.420436i
$306$		0			0
$307$	63.2296	+	109.517i	0.205960	+	0.356733i	0.950438	−	0.310914i	$-0.100635\pi$
$307$	63.2296	+	109.517i	0.205960	+	0.356733i	−0.744478	+	0.667647i	$0.767302\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−522.948	−	301.924i	−1.68151	−	0.970817i	−0.960662	−	0.277719i	$-0.910422\pi$
$311$	−522.948	−	301.924i	−1.68151	−	0.970817i	−0.720843	−	0.693099i	$-0.756245\pi$
$312$		0			0
$313$	327.454			1.04618			0.523089	−	0.852278i	$-0.324779\pi$
$313$	327.454			1.04618			0.523089	+	0.852278i	$0.324779\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	42.3035i		−	0.133450i	−0.997771	−	0.0667248i	$-0.978745\pi$
$317$		−	42.3035i		−	0.133450i	0.997771	−	0.0667248i	$-0.0212550\pi$
$318$		0			0
$319$	−136.986	−	237.266i	−0.429423	−	0.743782i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	202.789	−	49.7786i	0.627830	−	0.154113i
$324$		0			0
$325$	−9.12679	−	15.8081i	−0.0280824	−	0.0486402i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−37.8126	+	21.8311i	−0.114932	+	0.0663559i
$330$		0			0
$331$	−228.357	−	395.526i	−0.689901	−	1.19494i	−0.971869	−	0.235520i	$-0.924321\pi$
$331$	−228.357	−	395.526i	−0.689901	−	1.19494i	0.281968	−	0.959424i	$-0.409013\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−67.1600	+	38.7749i	−0.200478	+	0.115746i
$336$		0			0
$337$	−464.351			−1.37790			−0.688948	−	0.724811i	$-0.741927\pi$
$337$	−464.351			−1.37790			−0.688948	+	0.724811i	$0.741927\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			423.379i			1.24158i
$342$		0			0
$343$	238.140			0.694286
$344$		0			0
$345$		0			0
$346$		0			0
$347$			479.950i			1.38314i	0.722310	+	0.691570i	$0.243081\pi$
$347$			479.950i			1.38314i	−0.722310	+	0.691570i	$0.756919\pi$
$348$		0			0
$349$	115.688	+	200.378i	0.331484	+	0.574148i	0.982803	−	0.184657i	$-0.0591173\pi$
$349$	115.688	+	200.378i	0.331484	+	0.574148i	−0.651319	+	0.758804i	$0.725784\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.54484	+	0.891915i	−0.00437632	+	0.00252667i	−0.502187	−	0.864759i	$-0.667471\pi$
$353$	−1.54484	+	0.891915i	−0.00437632	+	0.00252667i	0.497810	+	0.867286i	$0.334138\pi$
$354$		0			0
$355$	−40.6650	−	70.4339i	−0.114549	−	0.198405i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−85.5489	+	49.3917i	−0.238298	+	0.137581i	−0.614394	−	0.788999i	$-0.710599\pi$
$359$	−85.5489	+	49.3917i	−0.238298	+	0.137581i	0.376096	+	0.926581i	$0.377266\pi$
$360$		0			0
$361$	304.746	+	193.522i	0.844172	+	0.536072i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−13.3697	+	7.71897i	−0.0366292	+	0.0211479i
$366$		0			0
$367$	22.2671			0.0606734			0.0303367	−	0.999540i	$-0.490342\pi$
$367$	22.2671			0.0606734			0.0303367	+	0.999540i	$0.490342\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			83.0064i			0.223737i
$372$		0			0
$373$	−293.673	+	508.657i	−0.787328	+	1.36369i	0.140271	+	0.990113i	$0.455203\pi$
$373$	−293.673	+	508.657i	−0.787328	+	1.36369i	−0.927599	+	0.373578i	$0.878131\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	20.8071	−	12.0130i	0.0551914	−	0.0318647i
$378$		0			0
$379$	−366.161			−0.966123			−0.483062	−	0.875586i	$-0.660475\pi$
$379$	−366.161			−0.966123			−0.483062	+	0.875586i	$0.660475\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			606.201i			1.58277i	0.611319	+	0.791385i	$0.290639\pi$
$383$			606.201i			1.58277i	−0.611319	+	0.791385i	$0.709361\pi$
$384$		0			0
$385$	−30.4000	+	52.6544i	−0.0789611	+	0.136765i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	609.633i		−	1.56718i	−0.621278	−	0.783590i	$-0.713386\pi$
$389$		−	609.633i		−	1.56718i	0.621278	−	0.783590i	$-0.286614\pi$
$390$		0			0
$391$	27.7567	−	48.0761i	0.0709891	−	0.122957i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−113.093	−	65.2942i	−0.286311	−	0.165302i
$396$		0			0
$397$	4.32872	+	7.49756i	0.0109036	+	0.0188855i	0.871426	−	0.490528i	$-0.163196\pi$
$397$	4.32872	+	7.49756i	0.0109036	+	0.0188855i	−0.860522	+	0.509413i	$0.829863\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	114.436i		−	0.285377i	−0.989768	−	0.142688i	$-0.954425\pi$
$401$		−	114.436i		−	0.285377i	0.989768	−	0.142688i	$-0.0455746\pi$
$402$		0			0
$403$	−37.1283			−0.0921298
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−590.399	+	340.867i	−1.45061	+	0.837510i
$408$		0			0
$409$	−11.9356	−	20.6730i	−0.0291823	−	0.0505453i	0.851065	−	0.525060i	$-0.175957\pi$
$409$	−11.9356	−	20.6730i	−0.0291823	−	0.0505453i	−0.880248	+	0.474514i	$0.842624\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−58.9458	+	34.0324i	−0.142726	+	0.0824028i
$414$		0			0
$415$	−10.7318	−	18.5880i	−0.0258597	−	0.0447902i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−243.314	−	140.478i	−0.580703	−	0.335269i	0.180710	−	0.983536i	$-0.442161\pi$
$419$	−243.314	−	140.478i	−0.580703	−	0.335269i	−0.761413	+	0.648268i	$0.775494\pi$
$420$		0			0
$421$	257.145	−	445.389i	0.610797	−	1.05793i	−0.380310	−	0.924859i	$-0.624183\pi$
$421$	257.145	−	445.389i	0.610797	−	1.05793i	0.991106	−	0.133072i	$-0.0424841\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−190.332	+	109.888i	−0.447839	+	0.258560i
$426$		0			0
$427$	−74.8743	−	129.686i	−0.175350	−	0.303715i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	263.180	+	151.947i	0.610625	+	0.352545i	0.773210	−	0.634150i	$-0.218650\pi$
$431$	263.180	+	151.947i	0.610625	+	0.352545i	−0.162585	+	0.986695i	$0.551983\pi$
$432$		0			0
$433$	236.247	−	409.191i	0.545604	−	0.945014i	−0.452965	−	0.891528i	$-0.649634\pi$
$433$	236.247	−	409.191i	0.545604	−	0.945014i	0.998569	−	0.0534853i	$-0.0170330\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	93.2074	−	22.8796i	0.213289	−	0.0523561i
$438$		0			0
$439$	169.379	−	293.372i	0.385828	−	0.668274i	−0.606055	−	0.795422i	$-0.707249\pi$
$439$	169.379	−	293.372i	0.385828	−	0.668274i	0.991884	+	0.127148i	$0.0405823\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			70.5923i			0.159351i	0.996821	+	0.0796753i	$0.0253883\pi$
$443$			70.5923i			0.159351i	−0.996821	+	0.0796753i	$0.974612\pi$
$444$		0			0
$445$	174.503	−	302.248i	0.392142	−	0.679209i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	195.529i		−	0.435476i	−0.976007	−	0.217738i	$-0.930132\pi$
$449$		−	195.529i		−	0.435476i	0.976007	−	0.217738i	$-0.0698678\pi$
$450$		0			0
$451$	279.358	+	483.863i	0.619420	+	1.07287i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.61754	−	2.66594i	−0.0101484	−	0.00585921i
$456$		0			0
$457$	49.0934	+	85.0323i	0.107425	+	0.186066i	0.914727	−	0.404073i	$-0.132406\pi$
$457$	49.0934	+	85.0323i	0.107425	+	0.186066i	−0.807301	+	0.590140i	$0.799073\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	162.878	+	94.0379i	0.353315	+	0.203987i	0.666145	−	0.745823i	$-0.267943\pi$
$461$	162.878	+	94.0379i	0.353315	+	0.203987i	−0.312829	+	0.949809i	$0.601277\pi$
$462$		0			0
$463$	205.254	−	355.510i	0.443313	−	0.767840i	−0.554620	−	0.832104i	$-0.687136\pi$
$463$	205.254	−	355.510i	0.443313	−	0.767840i	0.997933	+	0.0642632i	$0.0204697\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	352.331i		−	0.754457i	−0.926120	−	0.377229i	$-0.876877\pi$
$467$		−	352.331i		−	0.754457i	0.926120	−	0.377229i	$-0.123123\pi$
$468$		0			0
$469$	45.2808	−	78.4286i	0.0965475	−	0.167225i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			227.325i			0.480602i
$474$		0			0
$475$	−364.858	−	106.058i	−0.768122	−	0.223281i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	34.0644i		−	0.0711156i	−0.999368	−	0.0355578i	$-0.988679\pi$
$479$		−	34.0644i		−	0.0711156i	0.999368	−	0.0355578i	$-0.0113208\pi$
$480$		0			0
$481$	−29.8924	−	51.7752i	−0.0621464	−	0.107641i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	184.761	+	106.672i	0.380951	+	0.219942i
$486$		0			0
$487$	−815.065			−1.67364			−0.836822	−	0.547475i	$-0.815589\pi$
$487$	−815.065			−1.67364			−0.836822	+	0.547475i	$0.815589\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			655.809i			1.33566i	0.744314	+	0.667830i	$0.232776\pi$
$491$			655.809i			1.33566i	−0.744314	+	0.667830i	$0.767224\pi$
$492$		0			0
$493$	−144.639	−	250.521i	−0.293385	−	0.508157i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	82.2518	+	47.4881i	0.165497	+	0.0955495i
$498$		0			0
$499$	−303.820			−0.608857			−0.304429	−	0.952535i	$-0.598466\pi$
$499$	−303.820			−0.608857			−0.304429	+	0.952535i	$0.598466\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	220.195	+	127.130i	0.437764	+	0.252743i	0.702649	−	0.711537i	$-0.252001\pi$
$503$	220.195	+	127.130i	0.437764	+	0.252743i	−0.264885	+	0.964280i	$0.585334\pi$
$504$		0			0
$505$	108.787			0.215420
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−234.866	+	135.600i	−0.461426	+	0.266404i	−0.712644	−	0.701526i	$-0.752502\pi$
$509$	−234.866	+	135.600i	−0.461426	+	0.266404i	0.251218	+	0.967931i	$0.419169\pi$
$510$		0			0
$511$	9.01411	−	15.6129i	0.0176401	−	0.0305536i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−395.974	−	228.616i	−0.768881	−	0.443914i
$516$		0			0
$517$	−87.0008	−	150.690i	−0.168280	−	0.291470i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			764.752i			1.46785i	0.679229	+	0.733927i	$0.262315\pi$
$521$			764.752i			1.46785i	−0.679229	+	0.733927i	$0.737685\pi$
$522$		0			0
$523$	234.866	−	406.799i	0.449074	−	0.777819i	−0.549252	−	0.835657i	$-0.685087\pi$
$523$	234.866	−	406.799i	0.449074	−	0.777819i	0.998326	+	0.0578377i	$0.0184206\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			447.031i			0.848256i
$528$		0			0
$529$	−251.742	+	436.030i	−0.475883	+	0.824254i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−42.4325	+	24.4984i	−0.0796106	+	0.0459632i
$534$		0			0
$535$	−432.486			−0.808385
$536$		0			0
$537$		0			0
$538$		0			0
$539$			439.015i			0.814498i
$540$		0			0
$541$	241.908	−	418.997i	0.447150	−	0.774486i	−0.551049	−	0.834473i	$-0.685772\pi$
$541$	241.908	−	418.997i	0.447150	−	0.774486i	0.998199	+	0.0599864i	$0.0191057\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	196.067	−	113.200i	0.359757	−	0.207706i
$546$		0			0
$547$	−415.197			−0.759043			−0.379522	−	0.925183i	$-0.623911\pi$
$547$	−415.197			−0.759043			−0.379522	+	0.925183i	$0.623911\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	139.598	−	480.239i	0.253354	−	0.871577i
$552$		0			0
$553$	152.499			0.275768
$554$		0			0
$555$		0			0
$556$		0			0
$557$	831.480	+	480.055i	1.49278	+	0.861859i	0.999966	−	0.00827418i	$-0.00263378\pi$
$557$	831.480	+	480.055i	1.49278	+	0.861859i	0.492817	+	0.870133i	$0.335967\pi$
$558$		0			0
$559$	−19.9353			−0.0356624
$560$		0			0
$561$		0			0
$562$		0			0
$563$	31.2391	+	18.0359i	0.0554868	+	0.0320353i	0.527487	−	0.849563i	$-0.323134\pi$
$563$	31.2391	+	18.0359i	0.0554868	+	0.0320353i	−0.472000	+	0.881599i	$0.656468\pi$
$564$		0			0
$565$	−155.432	+	269.216i	−0.275101	+	0.476489i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	141.307	−	81.5839i	0.248344	−	0.143381i	−0.370662	−	0.928768i	$-0.620869\pi$
$569$	141.307	−	81.5839i	0.248344	−	0.143381i	0.619006	+	0.785387i	$0.287536\pi$
$570$		0			0
$571$	−229.873	+	398.152i	−0.402580	+	0.697289i	−0.994036	−	0.109048i	$-0.965220\pi$
$571$	−229.873	+	398.152i	−0.402580	+	0.697289i	0.591457	+	0.806337i	$0.298553\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−87.4817	+	50.5076i	−0.152142	+	0.0878392i
$576$		0			0
$577$	−421.023			−0.729675			−0.364838	−	0.931071i	$-0.618875\pi$
$577$	−421.023			−0.729675			−0.364838	+	0.931071i	$0.618875\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	21.7068	+	12.5324i	0.0373610	+	0.0215704i
$582$		0			0
$583$	−330.796			−0.567402
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−747.269	−	431.436i	−1.27303	−	0.734984i	−0.297473	−	0.954730i	$-0.596144\pi$
$587$	−747.269	−	431.436i	−1.27303	−	0.734984i	−0.975557	+	0.219746i	$0.929477\pi$
$588$		0			0
$589$	−557.890	+	534.842i	−0.947181	+	0.908050i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	101.112	+	58.3770i	0.170509	+	0.0984435i	0.582826	−	0.812597i	$-0.301947\pi$
$593$	101.112	+	58.3770i	0.170509	+	0.0984435i	−0.412317	+	0.911041i	$0.635280\pi$
$594$		0			0
$595$	−32.0983	+	55.5960i	−0.0539468	+	0.0934386i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−395.700	+	228.458i	−0.660601	+	0.381398i	−0.792506	−	0.609864i	$-0.791224\pi$
$599$	−395.700	+	228.458i	−0.660601	+	0.381398i	0.131905	+	0.991262i	$0.457891\pi$
$600$		0			0
$601$	452.634	+	783.985i	0.753134	+	1.30447i	0.946297	+	0.323300i	$0.104792\pi$
$601$	452.634	+	783.985i	0.753134	+	1.30447i	−0.193162	+	0.981167i	$0.561874\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	24.5276	+	14.1610i	0.0405415	+	0.0234066i
$606$		0			0
$607$	−128.308	+	222.237i	−0.211381	+	0.366123i	−0.952147	−	0.305640i	$-0.901130\pi$
$607$	−128.308	+	222.237i	−0.211381	+	0.366123i	0.740766	+	0.671763i	$0.234463\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	13.2148	−	7.62956i	0.0216281	−	0.0124870i
$612$		0			0
$613$	−224.574	−	388.974i	−0.366353	−	0.634542i	0.622639	−	0.782509i	$-0.286060\pi$
$613$	−224.574	−	388.974i	−0.366353	−	0.634542i	−0.988992	+	0.147967i	$0.952727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−171.165	+	98.8223i	−0.277415	+	0.160166i	−0.632253	−	0.774762i	$-0.717870\pi$
$617$	−171.165	+	98.8223i	−0.277415	+	0.160166i	0.354837	+	0.934928i	$0.384536\pi$
$618$		0			0
$619$	−198.212	−	343.312i	−0.320213	−	0.554624i	0.660319	−	0.750985i	$-0.270421\pi$
$619$	−198.212	−	343.312i	−0.320213	−	0.554624i	−0.980532	+	0.196361i	$0.937088\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			407.565i			0.654197i
$624$		0			0
$625$	274.863			0.439781
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−623.381	+	359.909i	−0.991067	+	0.572193i
$630$		0			0
$631$	−154.867	+	268.238i	−0.245432	+	0.425100i	−0.962253	−	0.272157i	$-0.912263\pi$
$631$	−154.867	+	268.238i	−0.245432	+	0.425100i	0.716821	+	0.697257i	$0.245596\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−125.240	−	72.3075i	−0.197229	−	0.113870i
$636$		0			0
$637$	−38.4995			−0.0604388
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−939.193	−	542.243i	−1.46520	−	0.845933i	−0.465956	−	0.884808i	$-0.654289\pi$
$641$	−939.193	−	542.243i	−1.46520	−	0.845933i	−0.999244	+	0.0388748i	$0.987623\pi$
$642$		0			0
$643$	−521.261			−0.810671			−0.405335	−	0.914168i	$-0.632845\pi$
$643$	−521.261			−0.810671			−0.405335	+	0.914168i	$0.632845\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	109.502i		−	0.169246i	−0.996413	−	0.0846232i	$-0.973031\pi$
$647$		−	109.502i		−	0.169246i	0.996413	−	0.0846232i	$-0.0269687\pi$
$648$		0			0
$649$	−135.625	−	234.909i	−0.208975	−	0.361956i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	421.496	+	243.351i	0.645476	+	0.372666i	0.786721	−	0.617309i	$-0.211777\pi$
$653$	421.496	+	243.351i	0.645476	+	0.372666i	−0.141245	+	0.989975i	$0.545110\pi$
$654$		0			0
$655$	−218.273			−0.333241
$656$		0			0
$657$		0			0
$658$		0			0
$659$			738.615i			1.12081i	0.828218	+	0.560406i	$0.189355\pi$
$659$			738.615i			1.12081i	−0.828218	+	0.560406i	$0.810645\pi$
$660$		0			0
$661$	194.464	+	336.822i	0.294197	+	0.509564i	0.974798	−	0.223090i	$-0.0716145\pi$
$661$	194.464	+	336.822i	0.294197	+	0.509564i	−0.680601	+	0.732655i	$0.738281\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−107.787	+	26.4583i	−0.162085	+	0.0397870i
$666$		0			0
$667$	−66.4799	−	115.146i	−0.0996700	−	0.172633i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	516.822	−	298.388i	0.770227	−	0.444691i
$672$		0			0
$673$	435.096	+	753.608i	0.646502	+	1.11977i	0.983952	+	0.178431i	$0.0571022\pi$
$673$	435.096	+	753.608i	0.646502	+	1.11977i	−0.337450	+	0.941343i	$0.609564\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	859.080	−	495.990i	1.26895	−	0.732629i	0.294162	−	0.955756i	$-0.404960\pi$
$677$	859.080	−	495.990i	1.26895	−	0.732629i	0.974790	+	0.223126i	$0.0716262\pi$
$678$		0			0
$679$	−249.140			−0.366922
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	260.321i		−	0.381144i	−0.981673	−	0.190572i	$-0.938966\pi$
$683$		−	260.321i		−	0.381144i	0.981673	−	0.190572i	$-0.0610343\pi$
$684$		0			0
$685$	−481.964			−0.703597
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	29.0092i		−	0.0421033i
$690$		0			0
$691$	97.0956	+	168.175i	0.140515	+	0.243378i	0.927691	−	0.373350i	$-0.121791\pi$
$691$	97.0956	+	168.175i	0.140515	+	0.243378i	−0.787176	+	0.616729i	$0.788458\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	185.713	−	107.222i	0.267214	−	0.154276i
$696$		0			0
$697$	294.965	+	510.894i	0.423192	+	0.732990i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1199.67	+	692.631i	−1.71137	+	0.988061i	−0.778656	+	0.627451i	$0.784098\pi$
$701$	−1199.67	+	692.631i	−1.71137	+	0.988061i	−0.932716	+	0.360611i	$0.882568\pi$
$702$		0			0
$703$	−1195.00	−	347.366i	−1.69985	−	0.494120i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−110.020	+	63.5201i	−0.155615	+	0.0898446i
$708$		0			0
$709$	1132.54			1.59737			0.798685	−	0.601749i	$-0.205529\pi$
$709$	1132.54			1.59737			0.798685	+	0.601749i	$0.205529\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			205.468i			0.288173i
$714$		0			0
$715$	10.6242	−	18.4017i	0.0148591	−	0.0257367i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−212.442	+	122.653i	−0.295468	+	0.170589i	−0.640405	−	0.768037i	$-0.721234\pi$
$719$	−212.442	+	122.653i	−0.295468	+	0.170589i	0.344937	+	0.938626i	$0.387900\pi$
$720$		0			0
$721$	533.948			0.740566
$722$		0			0
$723$		0			0
$724$		0			0
$725$			526.383i			0.726046i
$726$		0			0
$727$	−197.407	+	341.920i	−0.271537	+	0.470316i	−0.969256	−	0.246056i	$-0.920865\pi$
$727$	−197.407	+	341.920i	−0.271537	+	0.470316i	0.697719	+	0.716372i	$0.254199\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			240.024i			0.328351i
$732$		0			0
$733$	−441.511	+	764.719i	−0.602334	+	1.04327i	0.390133	+	0.920759i	$0.372429\pi$
$733$	−441.511	+	764.719i	−0.602334	+	1.04327i	−0.992467	+	0.122515i	$0.960904\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	312.552	+	180.452i	0.424087	+	0.244847i
$738$		0			0
$739$	167.647	+	290.374i	0.226857	+	0.392928i	0.956875	−	0.290500i	$-0.0938217\pi$
$739$	167.647	+	290.374i	0.226857	+	0.392928i	−0.730018	+	0.683428i	$0.760488\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			471.896i			0.635122i	0.948238	+	0.317561i	$0.102864\pi$
$743$			471.896i			0.635122i	−0.948238	+	0.317561i	$0.897136\pi$
$744$		0			0
$745$	158.805			0.213162
$746$		0			0
$747$		0			0
$748$		0			0
$749$	437.387	−	252.526i	0.583962	−	0.337150i
$750$		0			0
$751$	−508.867	−	881.384i	−0.677586	−	1.17361i	−0.975706	−	0.219085i	$-0.929693\pi$
$751$	−508.867	−	881.384i	−0.677586	−	1.17361i	0.298120	−	0.954528i	$-0.403641\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	74.8718	−	43.2273i	0.0991680	−	0.0572546i
$756$		0			0
$757$	−187.856	−	325.377i	−0.248159	−	0.429824i	0.714856	−	0.699272i	$-0.246492\pi$
$757$	−187.856	−	325.377i	−0.248159	−	0.429824i	−0.963015	+	0.269448i	$0.913159\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	554.683	+	320.246i	0.728887	+	0.420823i	0.818015	−	0.575197i	$-0.195075\pi$
$761$	554.683	+	320.246i	0.728887	+	0.420823i	−0.0891278	+	0.996020i	$0.528408\pi$
$762$		0			0
$763$	−132.193	+	228.965i	−0.173254	+	0.300085i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	20.6004	−	11.8937i	0.0268585	−	0.0155067i
$768$		0			0
$769$	443.085	+	767.446i	0.576183	+	0.997979i	0.995912	+	0.0903292i	$0.0287919\pi$
$769$	443.085	+	767.446i	0.576183	+	0.997979i	−0.419729	+	0.907650i	$0.637875\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	356.974	+	206.099i	0.461803	+	0.266622i	0.712802	−	0.701365i	$-0.247426\pi$
$773$	356.974	+	206.099i	0.461803	+	0.266622i	−0.250999	+	0.967987i	$0.580759\pi$
$774$		0			0
$775$	406.720	−	704.460i	0.524800	−	0.908980i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−284.685	+	979.362i	−0.365449	+	1.25720i
$780$		0			0
$781$	−189.249	+	327.788i	−0.242316	+	0.419703i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	557.705i		−	0.710453i
$786$		0			0
$787$	262.887	−	455.334i	0.334037	−	0.578569i	−0.649262	−	0.760564i	$-0.724922\pi$
$787$	262.887	−	455.334i	0.334037	−	0.578569i	0.983299	+	0.181996i	$0.0582557\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	363.023i		−	0.458942i
$792$		0			0
$793$	26.1672	+	45.3229i	0.0329977	+	0.0571537i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	425.382	+	245.594i	0.533729	+	0.308149i	0.742534	−	0.669809i	$-0.233624\pi$
$797$	425.382	+	245.594i	0.533729	+	0.308149i	−0.208805	+	0.977957i	$0.566957\pi$
$798$		0			0
$799$	−91.8612	−	159.108i	−0.114970	−	0.199134i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	62.2202	+	35.9229i	0.0774847	+	0.0447358i
$804$		0			0
$805$	−14.7533	+	25.5534i	−0.0183270	+	0.0317434i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1024.63i		−	1.26654i	−0.773931	−	0.633270i	$-0.781712\pi$
$809$		−	1024.63i		−	1.26654i	0.773931	−	0.633270i	$-0.218288\pi$
$810$		0			0
$811$	−427.052	+	739.675i	−0.526574	+	0.912053i	0.472947	+	0.881091i	$0.343190\pi$
$811$	−427.052	+	739.675i	−0.526574	+	0.912053i	−0.999521	+	0.0309618i	$0.990143\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			153.663i			0.188543i
$816$		0			0
$817$	−299.548	+	287.172i	−0.366643	+	0.351496i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	1489.41i		−	1.81415i	−0.420974	−	0.907073i	$-0.638312\pi$
$821$		−	1489.41i		−	1.81415i	0.420974	−	0.907073i	$-0.361688\pi$
$822$		0			0
$823$	209.211	+	362.364i	0.254205	+	0.440296i	0.964679	−	0.263427i	$-0.0848529\pi$
$823$	209.211	+	362.364i	0.254205	+	0.440296i	−0.710474	+	0.703723i	$0.751520\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	840.873	+	485.478i	1.01678	+	0.587036i	0.913168	−	0.407583i	$-0.133628\pi$
$827$	840.873	+	485.478i	1.01678	+	0.587036i	0.103607	+	0.994618i	$0.466962\pi$
$828$		0			0
$829$	−1204.76			−1.45327			−0.726637	−	0.687021i	$-0.758918\pi$
$829$	−1204.76			−1.45327			−0.726637	+	0.687021i	$0.758918\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			463.540i			0.556471i
$834$		0			0
$835$	39.9466	+	69.1896i	0.0478403	+	0.0828618i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−957.435	−	552.775i	−1.14116	−	0.658850i	−0.194444	−	0.980914i	$-0.562290\pi$
$839$	−957.435	−	552.775i	−1.14116	−	0.658850i	−0.946718	+	0.322063i	$0.895624\pi$
$840$		0			0
$841$	148.155			0.176165
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−325.722	−	188.056i	−0.385470	−	0.222551i
$846$		0			0
$847$	−33.0741			−0.0390485
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−286.523	+	165.424i	−0.336690	+	0.194388i
$852$		0			0
$853$	412.332	−	714.181i	0.483391	−	0.837257i	−0.516427	−	0.856331i	$-0.672738\pi$
$853$	412.332	−	714.181i	0.483391	−	0.837257i	0.999818	+	0.0190736i	$0.00607169\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−500.718	−	289.090i	−0.584268	−	0.337327i	0.178560	−	0.983929i	$-0.442856\pi$
$857$	−500.718	−	289.090i	−0.584268	−	0.337327i	−0.762828	+	0.646602i	$0.776190\pi$
$858$		0			0
$859$	459.542	+	795.950i	0.534973	+	0.926601i	0.999165	+	0.0408657i	$0.0130116\pi$
$859$	459.542	+	795.950i	0.534973	+	0.926601i	−0.464192	+	0.885735i	$0.653655\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			102.762i			0.119075i	0.998226	+	0.0595377i	$0.0189626\pi$
$863$			102.762i			0.119075i	−0.998226	+	0.0595377i	$0.981037\pi$
$864$		0			0
$865$	137.417	−	238.013i	0.158863	−	0.275159i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			607.738i			0.699353i
$870$		0			0
$871$	−15.8248	+	27.4093i	−0.0181685	+	0.0314688i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	227.635	−	131.425i	0.260154	−	0.150200i
$876$		0			0
$877$	−713.518			−0.813589			−0.406795	−	0.913520i	$-0.633354\pi$
$877$	−713.518			−0.813589			−0.406795	+	0.913520i	$0.633354\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			132.789i			0.150725i	0.997156	+	0.0753626i	$0.0240114\pi$
$881$			132.789i			0.150725i	−0.997156	+	0.0753626i	$0.975989\pi$
$882$		0			0
$883$	−26.8525	+	46.5099i	−0.0304105	+	0.0526726i	−0.880830	−	0.473432i	$-0.843015\pi$
$883$	−26.8525	+	46.5099i	−0.0304105	+	0.0526726i	0.850420	+	0.526105i	$0.176348\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−291.575	+	168.341i	−0.328720	+	0.189787i	−0.655273	−	0.755392i	$-0.727446\pi$
$887$	−291.575	+	168.341i	−0.328720	+	0.189787i	0.326552	+	0.945179i	$0.394113\pi$
$888$		0			0
$889$	168.879			0.189966
$890$		0			0
$891$		0			0
$892$		0			0
$893$	88.6598	−	305.004i	0.0992830	−	0.341549i
$894$		0			0
$895$	−318.456			−0.355817
$896$		0			0
$897$		0			0
$898$		0			0
$899$	927.235	+	535.340i	1.03141	+	0.595483i
$900$		0			0
$901$	−349.276			−0.387653
$902$		0			0
$903$		0			0
$904$		0			0
$905$	228.267	+	131.790i	0.252228	+	0.145624i
$906$		0			0
$907$	75.8361	−	131.352i	0.0836120	−	0.144820i	−0.821187	−	0.570659i	$-0.806688\pi$
$907$	75.8361	−	131.352i	0.0836120	−	0.144820i	0.904799	+	0.425839i	$0.140021\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−601.083	+	347.036i	−0.659806	+	0.380939i	−0.792203	−	0.610257i	$-0.791066\pi$
$911$	−601.083	+	347.036i	−0.659806	+	0.380939i	0.132397	+	0.991197i	$0.457733\pi$
$912$		0			0
$913$	−49.9439	+	86.5053i	−0.0547030	+	0.0947484i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	220.746	−	127.448i	0.240727	−	0.138984i
$918$		0			0
$919$	654.831			0.712547			0.356274	−	0.934382i	$-0.384047\pi$
$919$	654.831			0.712547			0.356274	+	0.934382i	$0.384047\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−28.7455	−	16.5962i	−0.0311435	−	0.0179807i
$924$		0			0
$925$	1309.82			1.41602
$926$		0			0
$927$		0			0
$928$		0			0
$929$	30.2452	+	17.4621i	0.0325568	+	0.0187967i	0.516190	−	0.856474i	$-0.327350\pi$
$929$	30.2452	+	17.4621i	0.0325568	+	0.0187967i	−0.483633	+	0.875271i	$0.660683\pi$
$930$		0			0
$931$	−578.493	+	554.594i	−0.621367	+	0.595697i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−221.560	−	127.918i	−0.236962	−	0.136810i
$936$		0			0
$937$	−19.5119	+	33.7956i	−0.0208238	+	0.0360678i	−0.876250	−	0.481858i	$-0.839962\pi$
$937$	−19.5119	+	33.7956i	−0.0208238	+	0.0360678i	0.855426	+	0.517926i	$0.173296\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.1587	−	7.01985i	0.0129211	−	0.00745999i	−0.493526	−	0.869731i	$-0.664292\pi$
$941$	12.1587	−	7.01985i	0.0129211	−	0.00745999i	0.506447	+	0.862271i	$0.330959\pi$
$942$		0			0
$943$	135.574	+	234.821i	0.143769	+	0.249015i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	533.838	+	308.211i	0.563714	+	0.325461i	0.754635	−	0.656145i	$-0.227814\pi$
$947$	533.838	+	308.211i	0.563714	+	0.325461i	−0.190921	+	0.981605i	$0.561147\pi$
$948$		0			0
$949$	−3.15026	+	5.45642i	−0.00331956	+	0.00574965i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−697.352	+	402.617i	−0.731744	+	0.422473i	−0.819060	−	0.573708i	$-0.805504\pi$
$953$	−697.352	+	402.617i	−0.731744	+	0.422473i	0.0873157	+	0.996181i	$0.472171\pi$
$954$		0			0
$955$	34.5606	+	59.8607i	0.0361891	+	0.0626814i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	487.426	−	281.415i	0.508265	−	0.293447i
$960$		0			0
$961$	−346.780	−	600.641i	−0.360854	−	0.625017i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			96.9933i			0.100511i
$966$		0			0
$967$	−244.947			−0.253306			−0.126653	−	0.991947i	$-0.540423\pi$
$967$	−244.947			−0.253306			−0.126653	+	0.991947i	$0.540423\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	903.998	−	521.923i	0.930997	−	0.537511i	0.0438700	−	0.999037i	$-0.486031\pi$
$971$	903.998	−	521.923i	0.930997	−	0.537511i	0.887127	+	0.461526i	$0.152698\pi$
$972$		0			0
$973$	−125.212	+	216.874i	−0.128687	+	0.222892i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1293.85	−	747.007i	−1.32431	−	0.764592i	−0.339899	−	0.940462i	$-0.610393\pi$
$977$	−1293.85	−	747.007i	−1.32431	−	0.764592i	−0.984413	+	0.175870i	$0.943726\pi$
$978$		0			0
$979$	−1624.22			−1.65906
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−379.230	−	218.949i	−0.385789	−	0.222735i	0.294545	−	0.955638i	$-0.404832\pi$
$983$	−379.230	−	218.949i	−0.385789	−	0.222735i	−0.680334	+	0.732902i	$0.738165\pi$
$984$		0			0
$985$	−435.898			−0.442536
$986$		0			0
$987$		0			0
$988$		0			0
$989$			110.322i			0.111549i
$990$		0			0
$991$	98.5250	+	170.650i	0.0994198	+	0.172200i	0.911445	−	0.411423i	$-0.134968\pi$
$991$	98.5250	+	170.650i	0.0994198	+	0.172200i	−0.812025	+	0.583623i	$0.801635\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−5.25840	−	3.03594i	−0.00528482	−	0.00305119i
$996$		0			0
$997$	−287.539			−0.288405			−0.144202	−	0.989548i	$-0.546062\pi$
$997$	−287.539			−0.288405			−0.144202	+	0.989548i	$0.546062\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.m.a.881.15		80
3.2	odd	2		684.3.m.a.653.27	yes	80
9.2	odd	6		2052.3.be.a.197.15		80
9.7	even	3		684.3.be.a.425.1	yes	80
19.11	even	3		2052.3.be.a.125.15		80
57.11	odd	6		684.3.be.a.581.1	yes	80
171.11	odd	6	inner	2052.3.m.a.1493.26		80
171.106	even	3		684.3.m.a.353.27	✓	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.27	✓	80	171.106	even	3
684.3.m.a.653.27	yes	80	3.2	odd	2
684.3.be.a.425.1	yes	80	9.7	even	3
684.3.be.a.581.1	yes	80	57.11	odd	6
2052.3.m.a.881.15		80	1.1	even	1	trivial
2052.3.m.a.1493.26		80	171.11	odd	6	inner
2052.3.be.a.125.15		80	19.11	even	3
2052.3.be.a.197.15		80	9.2	odd	6

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:	\( N \)	\(=\)	\( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2052.m (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-2.23654i q^{5} +(1.30590 + 2.26189i) q^{7} +O(q^{10})\)	\(q-2.23654i q^{5} +(1.30590 + 2.26189i) q^{7} +(-9.01402 + 5.20424i) q^{11} +(-0.456388 - 0.790487i) q^{13} +(-9.51759 + 5.49498i) q^{17} +(-18.2448 - 5.30348i) q^{19} +(-4.37454 + 2.52564i) q^{23} +19.9979 q^{25} +26.3219i q^{29} +(20.3381 - 35.2267i) q^{31} +(5.05879 - 2.92070i) q^{35} +65.4978 q^{37} -53.6789i q^{41} +(10.9202 - 18.9143i) q^{43} +16.7173i q^{47} +(21.0893 - 36.5277i) q^{49} +(27.5234 + 15.8907i) q^{53} +(11.6395 + 20.1602i) q^{55} +26.0605i q^{59} -57.3354 q^{61} +(-1.76795 + 1.02073i) q^{65} +(-17.3370 - 30.0286i) q^{67} +(31.4924 - 18.1821i) q^{71} +(-3.45130 - 5.97783i) q^{73} +(-23.5428 - 13.5924i) q^{77} +(29.1943 - 50.5661i) q^{79} +(8.31103 - 4.79838i) q^{83} +(12.2897 + 21.2865i) q^{85} +(135.141 + 78.0237i) q^{89} +(1.19199 - 2.06459i) q^{91} +(-11.8614 + 40.8052i) q^{95} +(-47.6951 + 82.6103i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 80 * q + q^7	\( 80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97}+O(q^{100}) \) 80 * q + q^7 - 18 * q^11 - 5 * q^13 + 9 * q^17 + 20 * q^19 - 72 * q^23 - 400 * q^25 - 8 * q^31 + 22 * q^37 - 44 * q^43 - 267 * q^49 + 36 * q^53 - 14 * q^61 + 144 * q^65 - 77 * q^67 + 135 * q^71 + 43 * q^73 - 216 * q^77 - 17 * q^79 + 171 * q^83 - 216 * q^89 + 122 * q^91 + 288 * q^95 - 8 * q^97

Introduction

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