LMFDB - Embedded newform 324.4.e.g.217.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,4,Mod(109,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("324.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	324.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$19.1166188419$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	324.217
Dual form			324.4.e.g.109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 + 7.79423i) q^{5} +(0.500000 - 0.866025i) q^{7} +(31.5000 - 54.5596i) q^{11} +(14.0000 + 24.2487i) q^{13} -72.0000 q^{17} +98.0000 q^{19} +(63.0000 + 109.119i) q^{23} +(22.0000 - 38.1051i) q^{25} +(-63.0000 + 109.119i) q^{29} +(129.500 + 224.301i) q^{31} +9.00000 q^{35} +386.000 q^{37} +(-225.000 - 389.711i) q^{41} +(17.0000 - 29.4449i) q^{43} +(-27.0000 + 46.7654i) q^{47} +(171.000 + 296.181i) q^{49} +693.000 q^{53} +567.000 q^{55} +(90.0000 + 155.885i) q^{59} +(140.000 - 242.487i) q^{61} +(-126.000 + 218.238i) q^{65} +(293.000 + 507.491i) q^{67} -504.000 q^{71} +161.000 q^{73} +(-31.5000 - 54.5596i) q^{77} +(-220.000 + 381.051i) q^{79} +(499.500 - 865.159i) q^{83} +(-324.000 - 561.184i) q^{85} -882.000 q^{89} +28.0000 q^{91} +(441.000 + 763.834i) q^{95} +(360.500 - 624.404i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 9 q^{5} + q^{7} + 63 q^{11} + 28 q^{13} - 144 q^{17} + 196 q^{19} + 126 q^{23} + 44 q^{25} - 126 q^{29} + 259 q^{31} + 18 q^{35} + 772 q^{37} - 450 q^{41} + 34 q^{43} - 54 q^{47} + 342 q^{49} + 1386 q^{53}+ \cdots + 721 q^{97}+O(q^{100}) $ 2 * q + 9 * q^5 + q^7 + 63 * q^11 + 28 * q^13 - 144 * q^17 + 196 * q^19 + 126 * q^23 + 44 * q^25 - 126 * q^29 + 259 * q^31 + 18 * q^35 + 772 * q^37 - 450 * q^41 + 34 * q^43 - 54 * q^47 + 342 * q^49 + 1386 * q^53 + 1134 * q^55 + 180 * q^59 + 280 * q^61 - 252 * q^65 + 586 * q^67 - 1008 * q^71 + 322 * q^73 - 63 * q^77 - 440 * q^79 + 999 * q^83 - 648 * q^85 - 1764 * q^89 + 56 * q^91 + 882 * q^95 + 721 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	4.50000	+	7.79423i	0.402492	+	0.697137i	0.994026	−	0.109143i	$-0.0348107\pi$
$5$	4.50000	+	7.79423i	0.402492	+	0.697137i	−0.591534	+	0.806280i	$0.701477\pi$
$6$		0			0
$7$	0.500000	−	0.866025i	0.0269975	−	0.0467610i	−0.852211	−	0.523198i	$-0.824739\pi$
$7$	0.500000	−	0.866025i	0.0269975	−	0.0467610i	0.879208	+	0.476437i	$0.158072\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	31.5000	−	54.5596i	0.863419	−	1.49549i	−0.00518978	−	0.999987i	$-0.501652\pi$
$11$	31.5000	−	54.5596i	0.863419	−	1.49549i	0.868609	−	0.495499i	$-0.165015\pi$
$12$		0			0
$13$	14.0000	+	24.2487i	0.298685	+	0.517337i	0.975835	−	0.218507i	$-0.0701188\pi$
$13$	14.0000	+	24.2487i	0.298685	+	0.517337i	−0.677151	+	0.735844i	$0.736785\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−72.0000			−1.02721			−0.513605	−	0.858027i	$-0.671690\pi$
$17$	−72.0000			−1.02721			−0.513605	+	0.858027i	$0.671690\pi$
$18$		0			0
$19$	98.0000			1.18330			0.591651	−	0.806194i	$-0.298476\pi$
$19$	98.0000			1.18330			0.591651	+	0.806194i	$0.298476\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	63.0000	+	109.119i	0.571148	+	0.989258i	0.996448	+	0.0842053i	$0.0268352\pi$
$23$	63.0000	+	109.119i	0.571148	+	0.989258i	−0.425300	+	0.905052i	$0.639831\pi$
$24$		0			0
$25$	22.0000	−	38.1051i	0.176000	−	0.304841i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−63.0000	+	109.119i	−0.403407	+	0.698722i	−0.994135	−	0.108149i	$-0.965507\pi$
$29$	−63.0000	+	109.119i	−0.403407	+	0.698722i	0.590728	+	0.806871i	$0.298841\pi$
$30$		0			0
$31$	129.500	+	224.301i	0.750287	+	1.29953i	0.947684	+	0.319211i	$0.103418\pi$
$31$	129.500	+	224.301i	0.750287	+	1.29953i	−0.197397	+	0.980324i	$0.563249\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	9.00000			0.0434651
$36$		0			0
$37$	386.000			1.71508			0.857541	−	0.514416i	$-0.171991\pi$
$37$	386.000			1.71508			0.857541	+	0.514416i	$0.171991\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−225.000	−	389.711i	−0.857051	−	1.48446i	−0.874729	−	0.484612i	$-0.838961\pi$
$41$	−225.000	−	389.711i	−0.857051	−	1.48446i	0.0176779	−	0.999844i	$-0.494373\pi$
$42$		0			0
$43$	17.0000	−	29.4449i	0.0602901	−	0.104426i	−0.834305	−	0.551303i	$-0.814131\pi$
$43$	17.0000	−	29.4449i	0.0602901	−	0.104426i	0.894595	+	0.446878i	$0.147464\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−27.0000	+	46.7654i	−0.0837948	+	0.145137i	−0.904877	−	0.425673i	$-0.860037\pi$
$47$	−27.0000	+	46.7654i	−0.0837948	+	0.145137i	0.821082	+	0.570810i	$0.193371\pi$
$48$		0			0
$49$	171.000	+	296.181i	0.498542	+	0.863501i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	693.000			1.79605			0.898027	−	0.439940i	$-0.145000\pi$
$53$	693.000			1.79605			0.898027	+	0.439940i	$0.145000\pi$
$54$		0			0
$55$	567.000			1.39008
$56$		0			0
$57$		0			0
$58$		0			0
$59$	90.0000	+	155.885i	0.198593	+	0.343974i	0.948073	−	0.318054i	$-0.103029\pi$
$59$	90.0000	+	155.885i	0.198593	+	0.343974i	−0.749479	+	0.662028i	$0.769696\pi$
$60$		0			0
$61$	140.000	−	242.487i	0.293855	−	0.508972i	−0.680863	−	0.732411i	$-0.738395\pi$
$61$	140.000	−	242.487i	0.293855	−	0.508972i	0.974718	+	0.223439i	$0.0717283\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−126.000	+	218.238i	−0.240437	+	0.416448i
$66$		0			0
$67$	293.000	+	507.491i	0.534263	+	0.925371i	0.999199	+	0.0400266i	$0.0127443\pi$
$67$	293.000	+	507.491i	0.534263	+	0.925371i	−0.464935	+	0.885345i	$0.653922\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−504.000			−0.842448			−0.421224	−	0.906957i	$-0.638399\pi$
$71$	−504.000			−0.842448			−0.421224	+	0.906957i	$0.638399\pi$
$72$		0			0
$73$	161.000			0.258132			0.129066	−	0.991636i	$-0.458802\pi$
$73$	161.000			0.258132			0.129066	+	0.991636i	$0.458802\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−31.5000	−	54.5596i	−0.0466202	−	0.0807486i
$78$		0			0
$79$	−220.000	+	381.051i	−0.313316	+	0.542679i	−0.979078	−	0.203485i	$-0.934773\pi$
$79$	−220.000	+	381.051i	−0.313316	+	0.542679i	0.665762	+	0.746164i	$0.268106\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	499.500	−	865.159i	0.660569	−	1.14414i	−0.319897	−	0.947452i	$-0.603648\pi$
$83$	499.500	−	865.159i	0.660569	−	1.14414i	0.980466	−	0.196687i	$-0.0630183\pi$
$84$		0			0
$85$	−324.000	−	561.184i	−0.413444	−	0.716106i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−882.000			−1.05047			−0.525235	−	0.850957i	$-0.676023\pi$
$89$	−882.000			−1.05047			−0.525235	+	0.850957i	$0.676023\pi$
$90$		0			0
$91$	28.0000			0.0322549
$92$		0			0
$93$		0			0
$94$		0			0
$95$	441.000	+	763.834i	0.476270	+	0.824924i
$96$		0			0
$97$	360.500	−	624.404i	0.377353	−	0.653594i	−0.613323	−	0.789832i	$-0.710168\pi$
$97$	360.500	−	624.404i	0.377353	−	0.653594i	0.990676	+	0.136238i	$0.0435011\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−220.500	+	381.917i	−0.217233	+	0.376259i	−0.953961	−	0.299930i	$-0.903037\pi$
$101$	−220.500	+	381.917i	−0.217233	+	0.376259i	0.736728	+	0.676189i	$0.236370\pi$
$102$		0			0
$103$	266.000	+	460.726i	0.254464	+	0.440744i	0.964750	−	0.263169i	$-0.0847677\pi$
$103$	266.000	+	460.726i	0.254464	+	0.440744i	−0.710286	+	0.703913i	$0.751434\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−819.000			−0.739960			−0.369980	−	0.929040i	$-0.620635\pi$
$107$	−819.000			−0.739960			−0.369980	+	0.929040i	$0.620635\pi$
$108$		0			0
$109$	−1294.00			−1.13709			−0.568545	−	0.822652i	$-0.692493\pi$
$109$	−1294.00			−1.13709			−0.568545	+	0.822652i	$0.692493\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	567.000	+	982.073i	0.472025	+	0.817572i	0.999488	−	0.0320065i	$-0.0101897\pi$
$113$	567.000	+	982.073i	0.472025	+	0.817572i	−0.527462	+	0.849578i	$0.676856\pi$
$114$		0			0
$115$	−567.000	+	982.073i	−0.459765	+	0.796337i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−36.0000	+	62.3538i	−0.0277321	+	0.0480333i
$120$		0			0
$121$	−1319.00	−	2284.58i	−0.990984	−	1.71644i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1521.00			1.08834
$126$		0			0
$127$	−1807.00			−1.26256			−0.631281	−	0.775554i	$-0.717470\pi$
$127$	−1807.00			−1.26256			−0.631281	+	0.775554i	$0.717470\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1102.50	−	1909.59i	−0.735312	−	1.27360i	−0.954586	−	0.297934i	$-0.903702\pi$
$131$	−1102.50	−	1909.59i	−0.735312	−	1.27360i	0.219274	−	0.975663i	$-0.429631\pi$
$132$		0			0
$133$	49.0000	−	84.8705i	0.0319462	−	0.0553324i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	693.000	−	1200.31i	0.432168	−	0.748537i	−0.564892	−	0.825165i	$-0.691082\pi$
$137$	693.000	−	1200.31i	0.432168	−	0.748537i	0.997060	+	0.0766283i	$0.0244155\pi$
$138$		0			0
$139$	−238.000	−	412.228i	−0.145229	−	0.251545i	0.784229	−	0.620471i	$-0.213059\pi$
$139$	−238.000	−	412.228i	−0.145229	−	0.251545i	−0.929459	+	0.368927i	$0.879725\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1764.00			1.03156
$144$		0			0
$145$	−1134.00			−0.649473
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−787.500	−	1363.99i	−0.432983	−	0.749949i	0.564145	−	0.825676i	$-0.309206\pi$
$149$	−787.500	−	1363.99i	−0.432983	−	0.749949i	−0.997129	+	0.0757264i	$0.975872\pi$
$150$		0			0
$151$	−224.500	+	388.845i	−0.120990	+	0.209562i	−0.920159	−	0.391546i	$-0.871940\pi$
$151$	−224.500	+	388.845i	−0.120990	+	0.209562i	0.799168	+	0.601108i	$0.205274\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1165.50	+	2018.71i	−0.603969	+	1.04611i
$156$		0			0
$157$	−910.000	−	1576.17i	−0.462585	−	0.801221i	0.536504	−	0.843898i	$-0.319745\pi$
$157$	−910.000	−	1576.17i	−0.462585	−	0.801221i	−0.999089	+	0.0426767i	$0.986411\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	126.000			0.0616782
$162$		0			0
$163$	−1828.00			−0.878405			−0.439202	−	0.898388i	$-0.644739\pi$
$163$	−1828.00			−0.878405			−0.439202	+	0.898388i	$0.644739\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−405.000	−	701.481i	−0.187664	−	0.325043i	0.756807	−	0.653638i	$-0.226758\pi$
$167$	−405.000	−	701.481i	−0.187664	−	0.325043i	−0.944471	+	0.328595i	$0.893425\pi$
$168$		0			0
$169$	706.500	−	1223.69i	0.321575	−	0.556984i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−661.500	+	1145.75i	−0.290710	+	0.503525i	−0.973978	−	0.226643i	$-0.927225\pi$
$173$	−661.500	+	1145.75i	−0.290710	+	0.503525i	0.683267	+	0.730168i	$0.260558\pi$
$174$		0			0
$175$	−22.0000	−	38.1051i	−0.00950311	−	0.0164599i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	315.000			0.131532			0.0657659	−	0.997835i	$-0.479051\pi$
$179$	315.000			0.131532			0.0657659	+	0.997835i	$0.479051\pi$
$180$		0			0
$181$	−2800.00			−1.14985			−0.574924	−	0.818207i	$-0.694968\pi$
$181$	−2800.00			−1.14985			−0.574924	+	0.818207i	$0.694968\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1737.00	+	3008.57i	0.690307	+	1.19565i
$186$		0			0
$187$	−2268.00	+	3928.29i	−0.886912	+	1.53618i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1638.00	+	2837.10i	−0.620532	+	1.07479i	0.368855	+	0.929487i	$0.379750\pi$
$191$	−1638.00	+	2837.10i	−0.620532	+	1.07479i	−0.989387	+	0.145305i	$0.953584\pi$
$192$		0			0
$193$	−1610.50	−	2789.47i	−0.600655	−	1.04036i	−0.992722	−	0.120428i	$-0.961573\pi$
$193$	−1610.50	−	2789.47i	−0.600655	−	1.04036i	0.392068	−	0.919936i	$-0.371760\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3339.00			−1.20758			−0.603792	−	0.797142i	$-0.706344\pi$
$197$	−3339.00			−1.20758			−0.603792	+	0.797142i	$0.706344\pi$
$198$		0			0
$199$	3689.00			1.31410			0.657051	−	0.753846i	$-0.271804\pi$
$199$	3689.00			1.31410			0.657051	+	0.753846i	$0.271804\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	63.0000	+	109.119i	0.0217819	+	0.0377274i
$204$		0			0
$205$	2025.00	−	3507.40i	0.689913	−	1.19496i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3087.00	−	5346.84i	1.02169	−	1.76961i
$210$		0			0
$211$	3011.00	+	5215.20i	0.982397	+	1.70156i	0.652976	+	0.757379i	$0.273520\pi$
$211$	3011.00	+	5215.20i	0.982397	+	1.70156i	0.329421	+	0.944183i	$0.393146\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	306.000			0.0970652
$216$		0			0
$217$	259.000			0.0810233
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1008.00	−	1745.91i	−0.306812	−	0.531414i
$222$		0			0
$223$	476.000	−	824.456i	0.142939	−	0.247577i	−0.785663	−	0.618654i	$-0.787678\pi$
$223$	476.000	−	824.456i	0.142939	−	0.247577i	0.928602	+	0.371077i	$0.121012\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2646.00	−	4583.01i	0.773662	−	1.34002i	−0.161882	−	0.986810i	$-0.551756\pi$
$227$	2646.00	−	4583.01i	0.773662	−	1.34002i	0.935544	−	0.353211i	$-0.114910\pi$
$228$		0			0
$229$	−1099.00	−	1903.52i	−0.317135	−	0.549294i	0.662754	−	0.748837i	$-0.269388\pi$
$229$	−1099.00	−	1903.52i	−0.317135	−	0.549294i	−0.979889	+	0.199543i	$0.936054\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5166.00			−1.45251			−0.726257	−	0.687423i	$-0.758742\pi$
$233$	−5166.00			−1.45251			−0.726257	+	0.687423i	$0.758742\pi$
$234$		0			0
$235$	−486.000			−0.134907
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1701.00	+	2946.22i	0.460370	+	0.797385i	0.998979	−	0.0451709i	$-0.0143832\pi$
$239$	1701.00	+	2946.22i	0.460370	+	0.797385i	−0.538609	+	0.842556i	$0.681050\pi$
$240$		0			0
$241$	−931.000	+	1612.54i	−0.248842	+	0.431007i	−0.963205	−	0.268768i	$-0.913383\pi$
$241$	−931.000	+	1612.54i	−0.248842	+	0.431007i	0.714363	+	0.699776i	$0.246717\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1539.00	+	2665.63i	−0.401319	+	0.695105i
$246$		0			0
$247$	1372.00	+	2376.37i	0.353434	+	0.612166i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5472.00			1.37605			0.688027	−	0.725685i	$-0.258477\pi$
$251$	5472.00			1.37605			0.688027	+	0.725685i	$0.258477\pi$
$252$		0			0
$253$	7938.00			1.97256
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2646.00	+	4583.01i	0.642229	+	1.11237i	0.984934	+	0.172931i	$0.0553237\pi$
$257$	2646.00	+	4583.01i	0.642229	+	1.11237i	−0.342705	+	0.939443i	$0.611343\pi$
$258$		0			0
$259$	193.000	−	334.286i	0.0463028	−	0.0801989i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	819.000	−	1418.55i	0.192022	−	0.332591i	−0.753898	−	0.656991i	$-0.771829\pi$
$263$	819.000	−	1418.55i	0.192022	−	0.332591i	0.945920	+	0.324400i	$0.105162\pi$
$264$		0			0
$265$	3118.50	+	5401.40i	0.722898	+	1.25210i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1206.00			0.273350			0.136675	−	0.990616i	$-0.456358\pi$
$269$	1206.00			0.273350			0.136675	+	0.990616i	$0.456358\pi$
$270$		0			0
$271$	4319.00			0.968120			0.484060	−	0.875035i	$-0.339162\pi$
$271$	4319.00			0.968120			0.484060	+	0.875035i	$0.339162\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1386.00	−	2400.62i	−0.303923	−	0.526411i
$276$		0			0
$277$	1124.00	−	1946.83i	0.243807	−	0.422287i	−0.717988	−	0.696055i	$-0.754937\pi$
$277$	1124.00	−	1946.83i	0.243807	−	0.422287i	0.961796	+	0.273769i	$0.0882702\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1008.00	+	1745.91i	−0.213994	+	0.370648i	−0.952961	−	0.303094i	$-0.901981\pi$
$281$	−1008.00	+	1745.91i	−0.213994	+	0.370648i	0.738967	+	0.673742i	$0.235314\pi$
$282$		0			0
$283$	1169.00	+	2024.77i	0.245547	+	0.425300i	0.962285	−	0.272042i	$-0.0876991\pi$
$283$	1169.00	+	2024.77i	0.245547	+	0.425300i	−0.716738	+	0.697342i	$0.754366\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−450.000			−0.0925528
$288$		0			0
$289$	271.000			0.0551598
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4257.00	−	7373.34i	−0.848794	−	1.47015i	−0.882285	−	0.470715i	$-0.843996\pi$
$293$	−4257.00	−	7373.34i	−0.848794	−	1.47015i	0.0334915	−	0.999439i	$-0.489337\pi$
$294$		0			0
$295$	−810.000	+	1402.96i	−0.159864	+	0.276893i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1764.00	+	3055.34i	−0.341186	+	0.590952i
$300$		0			0
$301$	−17.0000	−	29.4449i	−0.00325536	−	0.00563845i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2520.00			0.473098
$306$		0			0
$307$	6104.00			1.13477			0.567384	−	0.823453i	$-0.307956\pi$
$307$	6104.00			1.13477			0.567384	+	0.823453i	$0.307956\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2169.00	−	3756.82i	−0.395475	−	0.684983i	0.597687	−	0.801730i	$-0.296087\pi$
$311$	−2169.00	−	3756.82i	−0.395475	−	0.684983i	−0.993162	+	0.116747i	$0.962753\pi$
$312$		0			0
$313$	4077.50	−	7062.44i	0.736338	−	1.27538i	−0.217795	−	0.975994i	$-0.569886\pi$
$313$	4077.50	−	7062.44i	0.736338	−	1.27538i	0.954134	−	0.299381i	$-0.0967802\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−2488.50	+	4310.21i	−0.440909	+	0.763676i	−0.997757	−	0.0669378i	$-0.978677\pi$
$317$	−2488.50	+	4310.21i	−0.440909	+	0.763676i	0.556848	+	0.830614i	$0.312010\pi$
$318$		0			0
$319$	3969.00	+	6874.51i	0.696619	+	1.20658i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7056.00			−1.21550
$324$		0			0
$325$	1232.00			0.210274
$326$		0			0
$327$		0			0
$328$		0			0
$329$	27.0000	+	46.7654i	0.00452449	+	0.00783665i
$330$		0			0
$331$	−2839.00	+	4917.29i	−0.471437	+	0.816552i	−0.999466	−	0.0326738i	$-0.989598\pi$
$331$	−2839.00	+	4917.29i	−0.471437	+	0.816552i	0.528029	+	0.849226i	$0.322931\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2637.00	+	4567.42i	−0.430074	+	0.744910i
$336$		0			0
$337$	−1453.00	−	2516.67i	−0.234866	−	0.406800i	0.724367	−	0.689414i	$-0.242132\pi$
$337$	−1453.00	−	2516.67i	−0.234866	−	0.406800i	−0.959234	+	0.282614i	$0.908799\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16317.0			2.59125
$342$		0			0
$343$	685.000			0.107832
$344$		0			0
$345$		0			0
$346$		0			0
$347$	3496.50	+	6056.12i	0.540928	+	0.936914i	0.998851	+	0.0479227i	$0.0152601\pi$
$347$	3496.50	+	6056.12i	0.540928	+	0.936914i	−0.457923	+	0.888992i	$0.651407\pi$
$348$		0			0
$349$	−3955.00	+	6850.26i	−0.606608	+	1.05068i	0.385187	+	0.922839i	$0.374137\pi$
$349$	−3955.00	+	6850.26i	−0.606608	+	1.05068i	−0.991795	+	0.127838i	$0.959196\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1233.00	−	2135.62i	0.185909	−	0.322004i	−0.757973	−	0.652286i	$-0.773810\pi$
$353$	1233.00	−	2135.62i	0.185909	−	0.322004i	0.943883	+	0.330281i	$0.107144\pi$
$354$		0			0
$355$	−2268.00	−	3928.29i	−0.339079	−	0.587302i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7182.00			−1.05585			−0.527927	−	0.849290i	$-0.677030\pi$
$359$	−7182.00			−1.05585			−0.527927	+	0.849290i	$0.677030\pi$
$360$		0			0
$361$	2745.00			0.400204
$362$		0			0
$363$		0			0
$364$		0			0
$365$	724.500	+	1254.87i	0.103896	+	0.179953i
$366$		0			0
$367$	5715.50	−	9899.54i	0.812934	−	1.40804i	−0.0978684	−	0.995199i	$-0.531202\pi$
$367$	5715.50	−	9899.54i	0.812934	−	1.40804i	0.910802	−	0.412843i	$-0.135464\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	346.500	−	600.156i	0.0484889	−	0.0839852i
$372$		0			0
$373$	3308.00	+	5729.62i	0.459200	+	0.795358i	0.998919	−	0.0464871i	$-0.0148027\pi$
$373$	3308.00	+	5729.62i	0.459200	+	0.795358i	−0.539718	+	0.841846i	$0.681469\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3528.00			−0.481966
$378$		0			0
$379$	−9820.00			−1.33092			−0.665461	−	0.746433i	$-0.731765\pi$
$379$	−9820.00			−1.33092			−0.665461	+	0.746433i	$0.731765\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	720.000	+	1247.08i	0.0960582	+	0.166378i	0.910050	−	0.414499i	$-0.136043\pi$
$383$	720.000	+	1247.08i	0.0960582	+	0.166378i	−0.813992	+	0.580877i	$0.802710\pi$
$384$		0			0
$385$	283.500	−	491.036i	0.0375286	−	0.0650014i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2992.50	−	5183.16i	0.390041	−	0.675570i	−0.602414	−	0.798184i	$-0.705794\pi$
$389$	2992.50	−	5183.16i	0.390041	−	0.675570i	0.992454	+	0.122614i	$0.0391277\pi$
$390$		0			0
$391$	−4536.00	−	7856.58i	−0.586689	−	1.01618i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3960.00			−0.504428
$396$		0			0
$397$	−11284.0			−1.42652			−0.713259	−	0.700900i	$-0.752782\pi$
$397$	−11284.0			−1.42652			−0.713259	+	0.700900i	$0.752782\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3654.00	−	6328.91i	−0.455043	−	0.788157i	0.543648	−	0.839313i	$-0.317043\pi$
$401$	−3654.00	−	6328.91i	−0.455043	−	0.788157i	−0.998691	+	0.0511565i	$0.983709\pi$
$402$		0			0
$403$	−3626.00	+	6280.42i	−0.448198	+	0.776302i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12159.0	−	21060.0i	1.48083	−	2.56488i
$408$		0			0
$409$	−3167.50	−	5486.27i	−0.382941	−	0.663273i	0.608540	−	0.793523i	$-0.291755\pi$
$409$	−3167.50	−	5486.27i	−0.382941	−	0.663273i	−0.991481	+	0.130250i	$0.958422\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	180.000			0.0214461
$414$		0			0
$415$	8991.00			1.06350
$416$		0			0
$417$		0			0
$418$		0			0
$419$	3186.00	+	5518.31i	0.371471	+	0.643406i	0.989792	−	0.142519i	$-0.0455202\pi$
$419$	3186.00	+	5518.31i	0.371471	+	0.643406i	−0.618321	+	0.785926i	$0.712187\pi$
$420$		0			0
$421$	−1660.00	+	2875.20i	−0.192170	+	0.332848i	−0.945969	−	0.324257i	$-0.894886\pi$
$421$	−1660.00	+	2875.20i	−0.192170	+	0.332848i	0.753799	+	0.657105i	$0.228219\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1584.00	+	2743.57i	−0.180789	+	0.313136i
$426$		0			0
$427$	−140.000	−	242.487i	−0.0158667	−	0.0274819i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2898.00			0.323879			0.161939	−	0.986801i	$-0.448225\pi$
$431$	2898.00			0.323879			0.161939	+	0.986801i	$0.448225\pi$
$432$		0			0
$433$	−4291.00			−0.476241			−0.238120	−	0.971236i	$-0.576531\pi$
$433$	−4291.00			−0.476241			−0.238120	+	0.971236i	$0.576531\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	6174.00	+	10693.7i	0.675841	+	1.17059i
$438$		0			0
$439$	4161.50	−	7207.93i	0.452432	−	0.783635i	−0.546105	−	0.837717i	$-0.683890\pi$
$439$	4161.50	−	7207.93i	0.452432	−	0.783635i	0.998536	+	0.0540821i	$0.0172233\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1890.00	+	3273.58i	−0.202701	+	0.351089i	−0.949398	−	0.314076i	$-0.898305\pi$
$443$	−1890.00	+	3273.58i	−0.202701	+	0.351089i	0.746697	+	0.665165i	$0.231639\pi$
$444$		0			0
$445$	−3969.00	−	6874.51i	−0.422806	−	0.732321i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12474.0			−1.31110			−0.655551	−	0.755151i	$-0.727563\pi$
$449$	−12474.0			−1.31110			−0.655551	+	0.755151i	$0.727563\pi$
$450$		0			0
$451$	−28350.0			−2.95998
$452$		0			0
$453$		0			0
$454$		0			0
$455$	126.000	+	218.238i	0.0129824	+	0.0224861i
$456$		0			0
$457$	−8339.50	+	14444.4i	−0.853622	+	1.47852i	0.0242951	+	0.999705i	$0.492266\pi$
$457$	−8339.50	+	14444.4i	−0.853622	+	1.47852i	−0.877917	+	0.478812i	$0.841067\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8635.50	+	14957.1i	−0.872441	+	1.51111i	−0.0129772	+	0.999916i	$0.504131\pi$
$461$	−8635.50	+	14957.1i	−0.872441	+	1.51111i	−0.859464	+	0.511196i	$0.829202\pi$
$462$		0			0
$463$	−8693.50	−	15057.6i	−0.872616	−	1.51142i	−0.859281	−	0.511504i	$-0.829088\pi$
$463$	−8693.50	−	15057.6i	−0.872616	−	1.51142i	−0.0133352	−	0.999911i	$-0.504245\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3087.00			−0.305887			−0.152944	−	0.988235i	$-0.548875\pi$
$467$	−3087.00			−0.305887			−0.152944	+	0.988235i	$0.548875\pi$
$468$		0			0
$469$	586.000			0.0576950
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1071.00	−	1855.03i	−0.104111	−	0.180326i
$474$		0			0
$475$	2156.00	−	3734.30i	0.208261	−	0.360719i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2619.00	+	4536.24i	−0.249823	+	0.432706i	−0.963477	−	0.267793i	$-0.913706\pi$
$479$	−2619.00	+	4536.24i	−0.249823	+	0.432706i	0.713654	+	0.700499i	$0.247039\pi$
$480$		0			0
$481$	5404.00	+	9360.00i	0.512269	+	0.887275i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6489.00			0.607526
$486$		0			0
$487$	−4384.00			−0.407922			−0.203961	−	0.978979i	$-0.565382\pi$
$487$	−4384.00			−0.407922			−0.203961	+	0.978979i	$0.565382\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1921.50	−	3328.14i	−0.176611	−	0.305900i	0.764107	−	0.645090i	$-0.223180\pi$
$491$	−1921.50	−	3328.14i	−0.176611	−	0.305900i	−0.940718	+	0.339191i	$0.889847\pi$
$492$		0			0
$493$	4536.00	−	7856.58i	0.414384	−	0.717734i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−252.000	+	436.477i	−0.0227440	+	0.0393937i
$498$		0			0
$499$	−6283.00	−	10882.5i	−0.563659	−	0.976286i	−0.997173	−	0.0751389i	$-0.976060\pi$
$499$	−6283.00	−	10882.5i	−0.563659	−	0.976286i	0.433514	−	0.901147i	$-0.357273\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	5238.00			0.464316			0.232158	−	0.972678i	$-0.425421\pi$
$503$	5238.00			0.464316			0.232158	+	0.972678i	$0.425421\pi$
$504$		0			0
$505$	−3969.00			−0.349739
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3154.50	−	5463.75i	−0.274697	−	0.475789i	0.695362	−	0.718660i	$-0.255244\pi$
$509$	−3154.50	−	5463.75i	−0.274697	−	0.475789i	−0.970059	+	0.242871i	$0.921911\pi$
$510$		0			0
$511$	80.5000	−	139.430i	0.00696890	−	0.0120705i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2394.00	+	4146.53i	−0.204839	+	0.354792i
$516$		0			0
$517$	1701.00	+	2946.22i	0.144700	+	0.250628i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	5868.00			0.493439			0.246720	−	0.969087i	$-0.420647\pi$
$521$	5868.00			0.493439			0.246720	+	0.969087i	$0.420647\pi$
$522$		0			0
$523$	6776.00			0.566527			0.283264	−	0.959042i	$-0.408583\pi$
$523$	6776.00			0.566527			0.283264	+	0.959042i	$0.408583\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9324.00	−	16149.6i	−0.770702	−	1.33489i
$528$		0			0
$529$	−1854.50	+	3212.09i	−0.152420	+	0.264000i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6300.00	−	10911.9i	0.511976	−	0.886769i
$534$		0			0
$535$	−3685.50	−	6383.47i	−0.297828	−	0.515853i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	21546.0			1.72180
$540$		0			0
$541$	−9388.00			−0.746066			−0.373033	−	0.927818i	$-0.621682\pi$
$541$	−9388.00			−0.746066			−0.373033	+	0.927818i	$0.621682\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5823.00	−	10085.7i	−0.457670	−	0.792707i
$546$		0			0
$547$	−7528.00	+	13038.9i	−0.588435	+	1.01920i	0.406002	+	0.913872i	$0.366922\pi$
$547$	−7528.00	+	13038.9i	−0.588435	+	1.01920i	−0.994438	+	0.105328i	$0.966411\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6174.00	+	10693.7i	−0.477353	+	0.826799i
$552$		0			0
$553$	220.000	+	381.051i	0.0169175	+	0.0293019i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−16569.0			−1.26041			−0.630207	−	0.776427i	$-0.717030\pi$
$557$	−16569.0			−1.26041			−0.630207	+	0.776427i	$0.717030\pi$
$558$		0			0
$559$	952.000			0.0720310
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7276.50	+	12603.3i	0.544703	+	0.943454i	0.998626	+	0.0524124i	$0.0166910\pi$
$563$	7276.50	+	12603.3i	0.544703	+	0.943454i	−0.453922	+	0.891041i	$0.649976\pi$
$564$		0			0
$565$	−5103.00	+	8838.66i	−0.379973	+	0.658133i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2142.00	−	3710.05i	0.157816	−	0.273345i	−0.776265	−	0.630407i	$-0.782888\pi$
$569$	2142.00	−	3710.05i	0.157816	−	0.273345i	0.934081	+	0.357062i	$0.116221\pi$
$570$		0			0
$571$	−346.000	−	599.290i	−0.0253584	−	0.0439220i	0.853068	−	0.521800i	$-0.174739\pi$
$571$	−346.000	−	599.290i	−0.0253584	−	0.0439220i	−0.878426	+	0.477878i	$0.841406\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	5544.00			0.402088
$576$		0			0
$577$	4886.00			0.352525			0.176262	−	0.984343i	$-0.443599\pi$
$577$	4886.00			0.352525			0.176262	+	0.984343i	$0.443599\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−499.500	−	865.159i	−0.0356674	−	0.0617777i
$582$		0			0
$583$	21829.5	−	37809.8i	1.55075	−	2.68597i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5512.50	+	9547.93i	−0.387607	+	0.671355i	−0.992127	−	0.125235i	$-0.960032\pi$
$587$	−5512.50	+	9547.93i	−0.387607	+	0.671355i	0.604520	+	0.796590i	$0.293365\pi$
$588$		0			0
$589$	12691.0	+	21981.5i	0.887816	+	1.53774i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14562.0			1.00841			0.504207	−	0.863583i	$-0.331785\pi$
$593$	14562.0			1.00841			0.504207	+	0.863583i	$0.331785\pi$
$594$		0			0
$595$	−648.000			−0.0446477
$596$		0			0
$597$		0			0
$598$		0			0
$599$	5733.00	+	9929.85i	0.391058	+	0.677333i	0.992589	−	0.121516i	$-0.0387757\pi$
$599$	5733.00	+	9929.85i	0.391058	+	0.677333i	−0.601531	+	0.798849i	$0.705442\pi$
$600$		0			0
$601$	−5477.50	+	9487.31i	−0.371767	+	0.643919i	−0.989837	−	0.142204i	$-0.954581\pi$
$601$	−5477.50	+	9487.31i	−0.371767	+	0.643919i	0.618071	+	0.786123i	$0.287915\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11871.0	−	20561.2i	0.797727	−	1.38170i
$606$		0			0
$607$	−616.000	−	1066.94i	−0.0411906	−	0.0713441i	0.844695	−	0.535248i	$-0.179782\pi$
$607$	−616.000	−	1066.94i	−0.0411906	−	0.0713441i	−0.885886	+	0.463904i	$0.846448\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1512.00			−0.100113
$612$		0			0
$613$	25622.0			1.68819			0.844097	−	0.536191i	$-0.180137\pi$
$613$	25622.0			1.68819			0.844097	+	0.536191i	$0.180137\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	10521.0	+	18222.9i	0.686482	+	1.18902i	0.972969	+	0.230938i	$0.0741794\pi$
$617$	10521.0	+	18222.9i	0.686482	+	1.18902i	−0.286486	+	0.958084i	$0.592487\pi$
$618$		0			0
$619$	−3262.00	+	5649.95i	−0.211811	+	0.366867i	−0.952281	−	0.305222i	$-0.901269\pi$
$619$	−3262.00	+	5649.95i	−0.211811	+	0.366867i	0.740471	+	0.672089i	$0.234603\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−441.000	+	763.834i	−0.0283600	+	0.0491210i
$624$		0			0
$625$	4094.50	+	7091.88i	0.262048	+	0.453880i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−27792.0			−1.76175
$630$		0			0
$631$	20663.0			1.30361			0.651807	−	0.758384i	$-0.274011\pi$
$631$	20663.0			1.30361			0.651807	+	0.758384i	$0.274011\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−8131.50	−	14084.2i	−0.508171	−	0.880178i
$636$		0			0
$637$	−4788.00	+	8293.06i	−0.297814	+	0.515829i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−441.000	+	763.834i	−0.0271739	+	0.0470665i	−0.879293	−	0.476282i	$-0.841984\pi$
$641$	−441.000	+	763.834i	−0.0271739	+	0.0470665i	0.852119	+	0.523349i	$0.175317\pi$
$642$		0			0
$643$	3626.00	+	6280.42i	0.222388	+	0.385187i	0.955533	−	0.294885i	$-0.0952815\pi$
$643$	3626.00	+	6280.42i	0.222388	+	0.385187i	−0.733145	+	0.680073i	$0.761948\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−21168.0			−1.28624			−0.643122	−	0.765764i	$-0.722361\pi$
$647$	−21168.0			−1.28624			−0.643122	+	0.765764i	$0.722361\pi$
$648$		0			0
$649$	11340.0			0.685877
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7150.50	+	12385.0i	0.428516	+	0.742211i	0.996742	−	0.0806618i	$-0.0257034\pi$
$653$	7150.50	+	12385.0i	0.428516	+	0.742211i	−0.568226	+	0.822873i	$0.692370\pi$
$654$		0			0
$655$	9922.50	−	17186.3i	0.591915	−	1.02523i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7528.50	−	13039.7i	0.445021	−	0.770799i	−0.553033	−	0.833159i	$-0.686530\pi$
$659$	7528.50	−	13039.7i	0.445021	−	0.770799i	0.998054	+	0.0623608i	$0.0198629\pi$
$660$		0			0
$661$	2345.00	+	4061.66i	0.137988	+	0.239002i	0.926735	−	0.375716i	$-0.122603\pi$
$661$	2345.00	+	4061.66i	0.137988	+	0.239002i	−0.788747	+	0.614718i	$0.789270\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	882.000			0.0514323
$666$		0			0
$667$	−15876.0			−0.921621
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−8820.00	−	15276.7i	−0.507440	−	0.878912i
$672$		0			0
$673$	2601.50	−	4505.93i	0.149005	−	0.258084i	−0.781855	−	0.623460i	$-0.785726\pi$
$673$	2601.50	−	4505.93i	0.149005	−	0.258084i	0.930860	+	0.365376i	$0.119060\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3087.00	−	5346.84i	0.175248	−	0.303539i	−0.764999	−	0.644032i	$-0.777261\pi$
$677$	3087.00	−	5346.84i	0.175248	−	0.303539i	0.940247	+	0.340493i	$0.110594\pi$
$678$		0			0
$679$	−360.500	−	624.404i	−0.0203751	−	0.0352908i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−756.000			−0.0423536			−0.0211768	−	0.999776i	$-0.506741\pi$
$683$	−756.000			−0.0423536			−0.0211768	+	0.999776i	$0.506741\pi$
$684$		0			0
$685$	12474.0			0.695777
$686$		0			0
$687$		0			0
$688$		0			0
$689$	9702.00	+	16804.4i	0.536454	+	0.929165i
$690$		0			0
$691$	−8974.00	+	15543.4i	−0.494048	+	0.855716i	−0.999976	−	0.00685939i	$-0.997817\pi$
$691$	−8974.00	+	15543.4i	−0.494048	+	0.855716i	0.505929	+	0.862575i	$0.331150\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2142.00	−	3710.05i	0.116907	−	0.202490i
$696$		0			0
$697$	16200.0	+	28059.2i	0.880371	+	1.52485i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−34083.0			−1.83637			−0.918186	−	0.396149i	$-0.870346\pi$
$701$	−34083.0			−1.83637			−0.918186	+	0.396149i	$0.870346\pi$
$702$		0			0
$703$	37828.0			2.02946
$704$		0			0
$705$		0			0
$706$		0			0
$707$	220.500	+	381.917i	0.0117295	+	0.0203161i
$708$		0			0
$709$	2642.00	−	4576.08i	0.139947	−	0.242395i	−0.787529	−	0.616277i	$-0.788640\pi$
$709$	2642.00	−	4576.08i	0.139947	−	0.242395i	0.927476	+	0.373882i	$0.121973\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−16317.0	+	28261.9i	−0.857050	+	1.48445i
$714$		0			0
$715$	7938.00	+	13749.0i	0.415195	+	0.719139i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	24516.0			1.27162			0.635808	−	0.771847i	$-0.280667\pi$
$719$	24516.0			1.27162			0.635808	+	0.771847i	$0.280667\pi$
$720$		0			0
$721$	532.000			0.0274795
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2772.00	+	4801.24i	0.141999	+	0.245950i
$726$		0			0
$727$	6240.50	−	10808.9i	0.318359	−	0.551415i	−0.661786	−	0.749692i	$-0.730201\pi$
$727$	6240.50	−	10808.9i	0.318359	−	0.551415i	0.980146	+	0.198278i	$0.0635348\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1224.00	+	2120.03i	−0.0619306	+	0.107267i
$732$		0			0
$733$	1547.00	+	2679.48i	0.0779533	+	0.135019i	0.902367	−	0.430969i	$-0.141828\pi$
$733$	1547.00	+	2679.48i	0.0779533	+	0.135019i	−0.824413	+	0.565988i	$0.808495\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	36918.0			1.84517
$738$		0			0
$739$	−376.000			−0.0187164			−0.00935818	−	0.999956i	$-0.502979\pi$
$739$	−376.000			−0.0187164			−0.00935818	+	0.999956i	$0.502979\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	14742.0	+	25533.9i	0.727902	+	1.26076i	0.957768	+	0.287542i	$0.0928380\pi$
$743$	14742.0	+	25533.9i	0.727902	+	1.26076i	−0.229866	+	0.973222i	$0.573829\pi$
$744$		0			0
$745$	7087.50	−	12275.9i	0.348545	−	0.603697i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−409.500	+	709.275i	−0.0199770	+	0.0346013i
$750$		0			0
$751$	−5840.50	−	10116.0i	−0.283785	−	0.491531i	0.688528	−	0.725209i	$-0.258257\pi$
$751$	−5840.50	−	10116.0i	−0.283785	−	0.491531i	−0.972314	+	0.233678i	$0.924924\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−4041.00			−0.194791
$756$		0			0
$757$	890.000			0.0427313			0.0213657	−	0.999772i	$-0.493199\pi$
$757$	890.000			0.0427313			0.0213657	+	0.999772i	$0.493199\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−16416.0	−	28433.3i	−0.781970	−	1.35441i	−0.930792	−	0.365549i	$-0.880881\pi$
$761$	−16416.0	−	28433.3i	−0.781970	−	1.35441i	0.148822	−	0.988864i	$-0.452452\pi$
$762$		0			0
$763$	−647.000	+	1120.64i	−0.0306985	+	0.0531714i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−2520.00	+	4364.77i	−0.118634	+	0.205479i
$768$		0			0
$769$	−1592.50	−	2758.29i	−0.0746775	−	0.129345i	0.826268	−	0.563277i	$-0.190459\pi$
$769$	−1592.50	−	2758.29i	−0.0746775	−	0.129345i	−0.900946	+	0.433931i	$0.857126\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	21222.0			0.987454			0.493727	−	0.869617i	$-0.335634\pi$
$773$	21222.0			0.987454			0.493727	+	0.869617i	$0.335634\pi$
$774$		0			0
$775$	11396.0			0.528202
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−22050.0	−	38191.7i	−1.01415	−	1.75656i
$780$		0			0
$781$	−15876.0	+	27498.0i	−0.727385	+	1.25987i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8190.00	−	14185.5i	0.372374	−	0.644971i
$786$		0			0
$787$	−6160.00	−	10669.4i	−0.279009	−	0.483258i	0.692130	−	0.721773i	$-0.256673\pi$
$787$	−6160.00	−	10669.4i	−0.279009	−	0.483258i	−0.971139	+	0.238515i	$0.923339\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1134.00			0.0509740
$792$		0			0
$793$	7840.00			0.351080
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−5953.50	−	10311.8i	−0.264597	−	0.458295i	0.702861	−	0.711327i	$-0.251906\pi$
$797$	−5953.50	−	10311.8i	−0.264597	−	0.458295i	−0.967458	+	0.253032i	$0.918572\pi$
$798$		0			0
$799$	1944.00	−	3367.11i	0.0860748	−	0.149086i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	5071.50	−	8784.10i	0.222876	−	0.386032i
$804$		0			0
$805$	567.000	+	982.073i	0.0248250	+	0.0429982i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	14994.0			0.651620			0.325810	−	0.945435i	$-0.394363\pi$
$809$	14994.0			0.651620			0.325810	+	0.945435i	$0.394363\pi$
$810$		0			0
$811$	−38878.0			−1.68334			−0.841672	−	0.539990i	$-0.818428\pi$
$811$	−38878.0			−1.68334			−0.841672	+	0.539990i	$0.818428\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−8226.00	−	14247.8i	−0.353551	−	0.612369i
$816$		0			0
$817$	1666.00	−	2885.60i	0.0713414	−	0.123567i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1953.00	−	3382.70i	0.0830209	−	0.143796i	−0.821525	−	0.570172i	$-0.806877\pi$
$821$	1953.00	−	3382.70i	0.0830209	−	0.143796i	0.904546	+	0.426376i	$0.140210\pi$
$822$		0			0
$823$	5103.50	+	8839.52i	0.216157	+	0.374394i	0.953630	−	0.300982i	$-0.0973145\pi$
$823$	5103.50	+	8839.52i	0.216157	+	0.374394i	−0.737473	+	0.675376i	$0.763981\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−8064.00			−0.339072			−0.169536	−	0.985524i	$-0.554227\pi$
$827$	−8064.00			−0.339072			−0.169536	+	0.985524i	$0.554227\pi$
$828$		0			0
$829$	−10486.0			−0.439317			−0.219659	−	0.975577i	$-0.570494\pi$
$829$	−10486.0			−0.439317			−0.219659	+	0.975577i	$0.570494\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−12312.0	−	21325.0i	−0.512107	−	0.886996i
$834$		0			0
$835$	3645.00	−	6313.33i	0.151066	−	0.261655i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−16848.0	+	29181.6i	−0.693275	+	1.20079i	0.277484	+	0.960730i	$0.410499\pi$
$839$	−16848.0	+	29181.6i	−0.693275	+	1.20079i	−0.970759	+	0.240057i	$0.922834\pi$
$840$		0			0
$841$	4256.50	+	7372.47i	0.174525	+	0.302287i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12717.0			0.517726
$846$		0			0
$847$	−2638.00			−0.107016
$848$		0			0
$849$		0			0
$850$		0			0
$851$	24318.0	+	42120.0i	0.979566	+	1.69666i
$852$		0			0
$853$	7889.00	−	13664.1i	0.316664	−	0.548478i	−0.663126	−	0.748508i	$-0.730771\pi$
$853$	7889.00	−	13664.1i	0.316664	−	0.548478i	0.979790	+	0.200030i	$0.0641040\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−13068.0	+	22634.4i	−0.520880	+	0.902191i	0.478825	+	0.877910i	$0.341063\pi$
$857$	−13068.0	+	22634.4i	−0.520880	+	0.902191i	−0.999705	+	0.0242807i	$0.992270\pi$
$858$		0			0
$859$	−11956.0	−	20708.4i	−0.474893	−	0.822540i	0.524693	−	0.851291i	$-0.324180\pi$
$859$	−11956.0	−	20708.4i	−0.474893	−	0.822540i	−0.999587	+	0.0287519i	$0.990847\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−17892.0			−0.705737			−0.352868	−	0.935673i	$-0.614794\pi$
$863$	−17892.0			−0.705737			−0.352868	+	0.935673i	$0.614794\pi$
$864$		0			0
$865$	−11907.0			−0.468035
$866$		0			0
$867$		0			0
$868$		0			0
$869$	13860.0	+	24006.2i	0.541045	+	0.937118i
$870$		0			0
$871$	−8204.00	+	14209.7i	−0.319153	+	0.552789i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	760.500	−	1317.22i	0.0293824	−	0.0508918i
$876$		0			0
$877$	−22081.0	−	38245.4i	−0.850197	−	1.47258i	−0.881031	−	0.473059i	$-0.843150\pi$
$877$	−22081.0	−	38245.4i	−0.850197	−	1.47258i	0.0308341	−	0.999525i	$-0.490184\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	8820.00			0.337291			0.168645	−	0.985677i	$-0.446061\pi$
$881$	8820.00			0.337291			0.168645	+	0.985677i	$0.446061\pi$
$882$		0			0
$883$	−4654.00			−0.177372			−0.0886861	−	0.996060i	$-0.528267\pi$
$883$	−4654.00			−0.177372			−0.0886861	+	0.996060i	$0.528267\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	16542.0	+	28651.6i	0.626185	+	1.08458i	0.988310	+	0.152455i	$0.0487178\pi$
$887$	16542.0	+	28651.6i	0.626185	+	1.08458i	−0.362126	+	0.932129i	$0.617949\pi$
$888$		0			0
$889$	−903.500	+	1564.91i	−0.0340860	+	0.0590386i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2646.00	+	4583.01i	−0.0991546	+	0.171741i
$894$		0			0
$895$	1417.50	+	2455.18i	0.0529406	+	0.0916957i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−32634.0			−1.21068
$900$		0			0
$901$	−49896.0			−1.84492
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−12600.0	−	21823.8i	−0.462805	−	0.801601i
$906$		0			0
$907$	−23629.0	+	40926.6i	−0.865036	+	1.49829i	0.00197452	+	0.999998i	$0.499371\pi$
$907$	−23629.0	+	40926.6i	−0.865036	+	1.49829i	−0.867011	+	0.498289i	$0.833962\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	18333.0	−	31753.7i	0.666739	−	1.15483i	−0.312071	−	0.950059i	$-0.601023\pi$
$911$	18333.0	−	31753.7i	0.666739	−	1.15483i	0.978811	−	0.204767i	$-0.0656438\pi$
$912$		0			0
$913$	−31468.5	−	54505.0i	−1.14070	−	1.97574i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2205.00			−0.0794062
$918$		0			0
$919$	−2995.00			−0.107504			−0.0537519	−	0.998554i	$-0.517118\pi$
$919$	−2995.00			−0.107504			−0.0537519	+	0.998554i	$0.517118\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−7056.00	−	12221.4i	−0.251626	−	0.435830i
$924$		0			0
$925$	8492.00	−	14708.6i	0.301854	−	0.522827i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6642.00	+	11504.3i	−0.234572	+	0.406290i	−0.959148	−	0.282905i	$-0.908702\pi$
$929$	−6642.00	+	11504.3i	−0.234572	+	0.406290i	0.724577	+	0.689194i	$0.242035\pi$
$930$		0			0
$931$	16758.0	+	29025.7i	0.589926	+	1.02178i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−40824.0			−1.42790
$936$		0			0
$937$	−2989.00			−0.104212			−0.0521059	−	0.998642i	$-0.516593\pi$
$937$	−2989.00			−0.104212			−0.0521059	+	0.998642i	$0.516593\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	23242.5	+	40257.2i	0.805190	+	1.39463i	0.916163	+	0.400807i	$0.131270\pi$
$941$	23242.5	+	40257.2i	0.805190	+	1.39463i	−0.110973	+	0.993823i	$0.535397\pi$
$942$		0			0
$943$	28350.0	−	49103.6i	0.979006	−	1.69569i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	5764.50	−	9984.41i	0.197805	−	0.342608i	−0.750012	−	0.661425i	$-0.769952\pi$
$947$	5764.50	−	9984.41i	0.197805	−	0.342608i	0.947816	+	0.318817i	$0.103285\pi$
$948$		0			0
$949$	2254.00	+	3904.04i	0.0771000	+	0.133541i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−37044.0			−1.25915			−0.629577	−	0.776938i	$-0.716772\pi$
$953$	−37044.0			−1.25915			−0.629577	+	0.776938i	$0.716772\pi$
$954$		0			0
$955$	−29484.0			−0.999036
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−693.000	−	1200.31i	−0.0233349	−	0.0404172i
$960$		0			0
$961$	−18645.0	+	32294.1i	−0.625860	+	1.08402i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	14494.5	−	25105.2i	0.483518	−	0.837477i
$966$		0			0
$967$	−6227.50	−	10786.3i	−0.207097	−	0.358703i	0.743702	−	0.668512i	$-0.233068\pi$
$967$	−6227.50	−	10786.3i	−0.207097	−	0.358703i	−0.950799	+	0.309809i	$0.899735\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	20169.0			0.666585			0.333292	−	0.942823i	$-0.391840\pi$
$971$	20169.0			0.666585			0.333292	+	0.942823i	$0.391840\pi$
$972$		0			0
$973$	−476.000			−0.0156833
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19971.0	−	34590.8i	−0.653970	−	1.13271i	−0.982151	−	0.188095i	$-0.939769\pi$
$977$	−19971.0	−	34590.8i	−0.653970	−	1.13271i	0.328181	−	0.944615i	$-0.393565\pi$
$978$		0			0
$979$	−27783.0	+	48121.6i	−0.906995	+	1.57096i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	27117.0	−	46968.0i	0.879856	−	1.52395i	0.0283568	−	0.999598i	$-0.490973\pi$
$983$	27117.0	−	46968.0i	0.879856	−	1.52395i	0.851499	−	0.524357i	$-0.175694\pi$
$984$		0			0
$985$	−15025.5	−	26024.9i	−0.486043	−	0.841851i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	4284.00			0.137738
$990$		0			0
$991$	−26137.0			−0.837809			−0.418905	−	0.908030i	$-0.637586\pi$
$991$	−26137.0			−0.837809			−0.418905	+	0.908030i	$0.637586\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	16600.5	+	28752.9i	0.528916	+	0.916109i
$996$		0			0
$997$	16730.0	−	28977.2i	0.531439	−	0.920479i	−0.467888	−	0.883788i	$-0.654985\pi$
$997$	16730.0	−	28977.2i	0.531439	−	0.920479i	0.999327	−	0.0366911i	$-0.0116818\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.4.e.g.217.1		2
3.2	odd	2		324.4.e.b.217.1		2
9.2	odd	6		108.4.a.d.1.1	yes	1
9.4	even	3	inner	324.4.e.g.109.1		2
9.5	odd	6		324.4.e.b.109.1		2
9.7	even	3		108.4.a.a.1.1	✓	1
36.7	odd	6		432.4.a.c.1.1		1
36.11	even	6		432.4.a.l.1.1		1
72.11	even	6		1728.4.a.h.1.1		1
72.29	odd	6		1728.4.a.g.1.1		1
72.43	odd	6		1728.4.a.z.1.1		1
72.61	even	6		1728.4.a.y.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.4.a.a.1.1	✓	1	9.7	even	3
108.4.a.d.1.1	yes	1	9.2	odd	6
324.4.e.b.109.1		2	9.5	odd	6
324.4.e.b.217.1		2	3.2	odd	2
324.4.e.g.109.1		2	9.4	even	3	inner
324.4.e.g.217.1		2	1.1	even	1	trivial
432.4.a.c.1.1		1	36.7	odd	6
432.4.a.l.1.1		1	36.11	even	6
1728.4.a.g.1.1		1	72.29	odd	6
1728.4.a.h.1.1		1	72.11	even	6
1728.4.a.y.1.1		1	72.61	even	6
1728.4.a.z.1.1		1	72.43	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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.50000 + 7.79423i) q^{5} +(0.500000 - 0.866025i) q^{7} +(31.5000 - 54.5596i) q^{11} +(14.0000 + 24.2487i) q^{13} -72.0000 q^{17} +98.0000 q^{19} +(63.0000 + 109.119i) q^{23} +(22.0000 - 38.1051i) q^{25} +(-63.0000 + 109.119i) q^{29} +(129.500 + 224.301i) q^{31} +9.00000 q^{35} +386.000 q^{37} +(-225.000 - 389.711i) q^{41} +(17.0000 - 29.4449i) q^{43} +(-27.0000 + 46.7654i) q^{47} +(171.000 + 296.181i) q^{49} +693.000 q^{53} +567.000 q^{55} +(90.0000 + 155.885i) q^{59} +(140.000 - 242.487i) q^{61} +(-126.000 + 218.238i) q^{65} +(293.000 + 507.491i) q^{67} -504.000 q^{71} +161.000 q^{73} +(-31.5000 - 54.5596i) q^{77} +(-220.000 + 381.051i) q^{79} +(499.500 - 865.159i) q^{83} +(-324.000 - 561.184i) q^{85} -882.000 q^{89} +28.0000 q^{91} +(441.000 + 763.834i) q^{95} +(360.500 - 624.404i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 9 q^{5} + q^{7} + 63 q^{11} + 28 q^{13} - 144 q^{17} + 196 q^{19} + 126 q^{23} + 44 q^{25} - 126 q^{29} + 259 q^{31} + 18 q^{35} + 772 q^{37} - 450 q^{41} + 34 q^{43} - 54 q^{47} + 342 q^{49} + 1386 q^{53}+ \cdots + 721 q^{97}+O(q^{100}) \) 2 * q + 9 * q^5 + q^7 + 63 * q^11 + 28 * q^13 - 144 * q^17 + 196 * q^19 + 126 * q^23 + 44 * q^25 - 126 * q^29 + 259 * q^31 + 18 * q^35 + 772 * q^37 - 450 * q^41 + 34 * q^43 - 54 * q^47 + 342 * q^49 + 1386 * q^53 + 1134 * q^55 + 180 * q^59 + 280 * q^61 - 252 * q^65 + 586 * q^67 - 1008 * q^71 + 322 * q^73 - 63 * q^77 - 440 * q^79 + 999 * q^83 - 648 * q^85 - 1764 * q^89 + 56 * q^91 + 882 * q^95 + 721 * q^97

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