LMFDB - Embedded newform 1224.4.c.e.577.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,4,Mod(577,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1224.577");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1224 = 2^{3} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1224.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$72.2183378470$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152 $ x^8 + 95x^6 + 756x^4 + 1780*x^2 + 1152
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.2
Root			$-2.20783i$ of defining polynomial
Character	$\chi$	$=$	1224.577
Dual form			1224.4.c.e.577.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.4090i q^{5} +34.5010i q^{7} +O(q^{10})$	$q-16.4090i q^{5} +34.5010i q^{7} -7.42843i q^{11} -42.0242 q^{13} +(26.0870 - 65.0574i) q^{17} +59.9095 q^{19} -49.4728i q^{23} -144.255 q^{25} +259.369i q^{29} -92.2215i q^{31} +566.126 q^{35} +207.564i q^{37} -176.800i q^{41} -19.0809 q^{43} -80.1389 q^{47} -847.318 q^{49} -319.869 q^{53} -121.893 q^{55} -11.0809 q^{59} +712.648i q^{61} +689.575i q^{65} +484.980 q^{67} +443.968i q^{71} +337.468i q^{73} +256.288 q^{77} +840.905i q^{79} -456.004 q^{83} +(-1067.53 - 428.062i) q^{85} +1205.43 q^{89} -1449.88i q^{91} -983.055i q^{95} -638.484i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 44 q^{13} - 28 q^{17} + 48 q^{19} - 520 q^{25} + 1064 q^{35} + 8 q^{43} - 312 q^{47} - 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 1660 q^{77} - 2472 q^{83} - 2160 q^{85} - 68 q^{89}+O(q^{100}) $ 8 * q + 44 * q^13 - 28 * q^17 + 48 * q^19 - 520 * q^25 + 1064 * q^35 + 8 * q^43 - 312 * q^47 - 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 1660 * q^77 - 2472 * q^83 - 2160 * q^85 - 68 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$.

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	16.4090i		−	1.46766i	−0.679331	−	0.733832i	$-0.737730\pi$
$5$		−	16.4090i		−	1.46766i	0.679331	−	0.733832i	$-0.262270\pi$
$6$		0			0
$7$			34.5010i			1.86288i	0.363898	+	0.931439i	$0.381446\pi$
$7$			34.5010i			1.86288i	−0.363898	+	0.931439i	$0.618554\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	7.42843i		−	0.203614i	−0.994804	−	0.101807i	$-0.967538\pi$
$11$		−	7.42843i		−	0.203614i	0.994804	−	0.101807i	$-0.0324625\pi$
$12$		0			0
$13$	−42.0242			−0.896571			−0.448285	−	0.893890i	$-0.647965\pi$
$13$	−42.0242			−0.896571			−0.448285	+	0.893890i	$0.647965\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	26.0870	−	65.0574i	0.372178	−	0.928161i
$17$	26.0870	−	65.0574i	0.372178	−	0.928161i
$18$		0			0
$19$	59.9095			0.723378			0.361689	−	0.932299i	$-0.382200\pi$
$19$	59.9095			0.723378			0.361689	+	0.932299i	$0.382200\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	49.4728i		−	0.448513i	−0.974530	−	0.224256i	$-0.928005\pi$
$23$		−	49.4728i		−	0.448513i	0.974530	−	0.224256i	$-0.0719953\pi$
$24$		0			0
$25$	−144.255			−1.15404
$26$		0			0
$27$		0			0
$28$		0			0
$29$			259.369i			1.66081i	0.557157	+	0.830407i	$0.311892\pi$
$29$			259.369i			1.66081i	−0.557157	+	0.830407i	$0.688108\pi$
$30$		0			0
$31$		−	92.2215i		−	0.534306i	−0.963654	−	0.267153i	$-0.913917\pi$
$31$		−	92.2215i		−	0.534306i	0.963654	−	0.267153i	$-0.0860829\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	566.126			2.73408
$36$		0			0
$37$			207.564i			0.922252i	0.887335	+	0.461126i	$0.152554\pi$
$37$			207.564i			0.922252i	−0.887335	+	0.461126i	$0.847446\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	176.800i		−	0.673454i	−0.941602	−	0.336727i	$-0.890680\pi$
$41$		−	176.800i		−	0.673454i	0.941602	−	0.336727i	$-0.109320\pi$
$42$		0			0
$43$	−19.0809			−0.0676700			−0.0338350	−	0.999427i	$-0.510772\pi$
$43$	−19.0809			−0.0676700			−0.0338350	+	0.999427i	$0.510772\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−80.1389			−0.248712			−0.124356	−	0.992238i	$-0.539686\pi$
$47$	−80.1389			−0.248712			−0.124356	+	0.992238i	$0.539686\pi$
$48$		0			0
$49$	−847.318			−2.47031
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−319.869			−0.829006			−0.414503	−	0.910048i	$-0.636045\pi$
$53$	−319.869			−0.829006			−0.414503	+	0.910048i	$0.636045\pi$
$54$		0			0
$55$	−121.893			−0.298837
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−11.0809			−0.0244510			−0.0122255	−	0.999925i	$-0.503892\pi$
$59$	−11.0809			−0.0244510			−0.0122255	+	0.999925i	$0.503892\pi$
$60$		0			0
$61$			712.648i			1.49582i	0.663798	+	0.747912i	$0.268943\pi$
$61$			712.648i			1.49582i	−0.663798	+	0.747912i	$0.731057\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			689.575i			1.31587i
$66$		0			0
$67$	484.980			0.884325			0.442163	−	0.896935i	$-0.354211\pi$
$67$	484.980			0.884325			0.442163	+	0.896935i	$0.354211\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			443.968i			0.742103i	0.928612	+	0.371051i	$0.121003\pi$
$71$			443.968i			0.742103i	−0.928612	+	0.371051i	$0.878997\pi$
$72$		0			0
$73$			337.468i			0.541063i	0.962711	+	0.270531i	$0.0871994\pi$
$73$			337.468i			0.541063i	−0.962711	+	0.270531i	$0.912801\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	256.288			0.379309
$78$		0			0
$79$			840.905i			1.19759i	0.800904	+	0.598793i	$0.204353\pi$
$79$			840.905i			1.19759i	−0.800904	+	0.598793i	$0.795647\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−456.004			−0.603047			−0.301524	−	0.953459i	$-0.597495\pi$
$83$	−456.004			−0.603047			−0.301524	+	0.953459i	$0.597495\pi$
$84$		0			0
$85$	−1067.53	−	428.062i	−1.36223	−	0.546233i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1205.43			1.43568			0.717841	−	0.696207i	$-0.245130\pi$
$89$	1205.43			1.43568			0.717841	+	0.696207i	$0.245130\pi$
$90$		0			0
$91$		−	1449.88i		−	1.67020i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	983.055i		−	1.06168i
$96$		0			0
$97$		−	638.484i		−	0.668332i	−0.942514	−	0.334166i	$-0.891545\pi$
$97$		−	638.484i		−	0.668332i	0.942514	−	0.334166i	$-0.108455\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−739.395			−0.728441			−0.364220	−	0.931313i	$-0.618665\pi$
$101$	−739.395			−0.728441			−0.364220	+	0.931313i	$0.618665\pi$
$102$		0			0
$103$	1588.54			1.51965			0.759824	−	0.650129i	$-0.225285\pi$
$103$	1588.54			1.51965			0.759824	+	0.650129i	$0.225285\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	202.298i		−	0.182775i	−0.995815	−	0.0913875i	$-0.970870\pi$
$107$		−	202.298i		−	0.182775i	0.995815	−	0.0913875i	$-0.0291302\pi$
$108$		0			0
$109$			1244.71i			1.09378i	0.837204	+	0.546890i	$0.184188\pi$
$109$			1244.71i			1.09378i	−0.837204	+	0.546890i	$0.815812\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1492.28i			1.24232i	0.783685	+	0.621159i	$0.213338\pi$
$113$			1492.28i			1.24232i	−0.783685	+	0.621159i	$0.786662\pi$
$114$		0			0
$115$	−811.798			−0.658266
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2244.54	+	900.028i	1.72905	+	0.693323i
$120$		0			0
$121$	1275.82			0.958541
$122$		0			0
$123$		0			0
$124$		0			0
$125$			315.954i			0.226078i
$126$		0			0
$127$	2054.49			1.43548			0.717741	−	0.696311i	$-0.245176\pi$
$127$	2054.49			1.43548			0.717741	+	0.696311i	$0.245176\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			1892.72i			1.26235i	0.775640	+	0.631175i	$0.217427\pi$
$131$			1892.72i			1.26235i	−0.775640	+	0.631175i	$0.782573\pi$
$132$		0			0
$133$			2066.94i			1.34757i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	848.618			0.529214			0.264607	−	0.964356i	$-0.414758\pi$
$137$	848.618			0.529214			0.264607	+	0.964356i	$0.414758\pi$
$138$		0			0
$139$		−	20.6414i		−	0.0125955i	−0.999980	−	0.00629777i	$-0.997995\pi$
$139$		−	20.6414i		−	0.0125955i	0.999980	−	0.00629777i	$-0.00200466\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			312.174i			0.182555i
$144$		0			0
$145$	4255.98			2.43752
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1202.59			0.661206			0.330603	−	0.943770i	$-0.392748\pi$
$149$	1202.59			0.661206			0.330603	+	0.943770i	$0.392748\pi$
$150$		0			0
$151$	−2080.77			−1.12140			−0.560698	−	0.828020i	$-0.689467\pi$
$151$	−2080.77			−1.12140			−0.560698	+	0.828020i	$0.689467\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1513.26			−0.784182
$156$		0			0
$157$	1153.74			0.586486			0.293243	−	0.956038i	$-0.405266\pi$
$157$	1153.74			0.586486			0.293243	+	0.956038i	$0.405266\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1706.86			0.835524
$162$		0			0
$163$			1461.35i			0.702221i	0.936334	+	0.351110i	$0.114196\pi$
$163$			1461.35i			0.702221i	−0.936334	+	0.351110i	$0.885804\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			4157.98i			1.92667i	0.268296	+	0.963337i	$0.413539\pi$
$167$			4157.98i			1.92667i	−0.268296	+	0.963337i	$0.586461\pi$
$168$		0			0
$169$	−430.965			−0.196160
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	2504.08i		−	1.10047i	−0.835010	−	0.550235i	$-0.814538\pi$
$173$		−	2504.08i		−	1.10047i	0.835010	−	0.550235i	$-0.185462\pi$
$174$		0			0
$175$		−	4976.94i		−	2.14983i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−839.685			−0.350620			−0.175310	−	0.984513i	$-0.556093\pi$
$179$	−839.685			−0.350620			−0.175310	+	0.984513i	$0.556093\pi$
$180$		0			0
$181$		−	500.491i		−	0.205532i	−0.994706	−	0.102766i	$-0.967231\pi$
$181$		−	500.491i		−	0.205532i	0.994706	−	0.102766i	$-0.0327692\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3405.92			1.35356
$186$		0			0
$187$	−483.275	−	193.786i	−0.188987	−	0.0757808i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−199.937			−0.0757430			−0.0378715	−	0.999283i	$-0.512058\pi$
$191$	−199.937			−0.0757430			−0.0378715	+	0.999283i	$0.512058\pi$
$192$		0			0
$193$			2758.52i			1.02882i	0.857544	+	0.514411i	$0.171989\pi$
$193$			2758.52i			1.02882i	−0.857544	+	0.514411i	$0.828011\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			3379.87i			1.22236i	0.791490	+	0.611182i	$0.209306\pi$
$197$			3379.87i			1.22236i	−0.791490	+	0.611182i	$0.790694\pi$
$198$		0			0
$199$			2729.46i			0.972293i	0.873877	+	0.486147i	$0.161598\pi$
$199$			2729.46i			0.972293i	−0.873877	+	0.486147i	$0.838402\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−8948.48			−3.09389
$204$		0			0
$205$	−2901.12			−0.988404
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	445.034i		−	0.147290i
$210$		0			0
$211$			2149.01i			0.701157i	0.936533	+	0.350579i	$0.114015\pi$
$211$			2149.01i			0.701157i	−0.936533	+	0.350579i	$0.885985\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			313.098i			0.0993168i
$216$		0			0
$217$	3181.73			0.995346
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1096.29	+	2733.99i	−0.333684	+	0.832162i
$222$		0			0
$223$	−115.665			−0.0347332			−0.0173666	−	0.999849i	$-0.505528\pi$
$223$	−115.665			−0.0347332			−0.0173666	+	0.999849i	$0.505528\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			150.739i			0.0440743i	0.999757	+	0.0220372i	$0.00701521\pi$
$227$			150.739i			0.0440743i	−0.999757	+	0.0220372i	$0.992985\pi$
$228$		0			0
$229$	−252.262			−0.0727946			−0.0363973	−	0.999337i	$-0.511588\pi$
$229$	−252.262			−0.0727946			−0.0363973	+	0.999337i	$0.511588\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			1025.45i			0.288323i	0.989554	+	0.144161i	$0.0460484\pi$
$233$			1025.45i			0.288323i	−0.989554	+	0.144161i	$0.953952\pi$
$234$		0			0
$235$			1315.00i			0.365026i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3803.45			−1.02939			−0.514696	−	0.857373i	$-0.672095\pi$
$239$	−3803.45			−1.02939			−0.514696	+	0.857373i	$0.672095\pi$
$240$		0			0
$241$		−	2073.99i		−	0.554346i	−0.960820	−	0.277173i	$-0.910603\pi$
$241$		−	2073.99i		−	0.554346i	0.960820	−	0.277173i	$-0.0893975\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			13903.6i			3.62559i
$246$		0			0
$247$	−2517.65			−0.648560
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1242.05			0.312341			0.156170	−	0.987730i	$-0.450085\pi$
$251$	1242.05			0.312341			0.156170	+	0.987730i	$0.450085\pi$
$252$		0			0
$253$	−367.505			−0.0913236
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1469.82			−0.356751			−0.178375	−	0.983963i	$-0.557084\pi$
$257$	−1469.82			−0.356751			−0.178375	+	0.983963i	$0.557084\pi$
$258$		0			0
$259$	−7161.16			−1.71804
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4308.70			1.01021			0.505106	−	0.863057i	$-0.331453\pi$
$263$	4308.70			1.01021			0.505106	+	0.863057i	$0.331453\pi$
$264$		0			0
$265$			5248.72i			1.21670i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	3812.72i		−	0.864184i	−0.901830	−	0.432092i	$-0.857776\pi$
$269$		−	3812.72i		−	0.864184i	0.901830	−	0.432092i	$-0.142224\pi$
$270$		0			0
$271$	2859.05			0.640868			0.320434	−	0.947271i	$-0.396171\pi$
$271$	2859.05			0.640868			0.320434	+	0.947271i	$0.396171\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			1071.59i			0.234979i
$276$		0			0
$277$			5377.92i			1.16653i	0.812283	+	0.583263i	$0.198224\pi$
$277$			5377.92i			1.16653i	−0.812283	+	0.583263i	$0.801776\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1867.29			−0.396418			−0.198209	−	0.980160i	$-0.563512\pi$
$281$	−1867.29			−0.396418			−0.198209	+	0.980160i	$0.563512\pi$
$282$		0			0
$283$		−	4296.76i		−	0.902529i	−0.892390	−	0.451265i	$-0.850973\pi$
$283$		−	4296.76i		−	0.902529i	0.892390	−	0.451265i	$-0.149027\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6099.79			1.25456
$288$		0			0
$289$	−3551.94	−	3394.31i	−0.722967	−	0.690883i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−8816.10			−1.75782			−0.878911	−	0.476985i	$-0.841730\pi$
$293$	−8816.10			−1.75782			−0.878911	+	0.476985i	$0.841730\pi$
$294$		0			0
$295$			181.826i			0.0358859i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			2079.06i			0.402123i
$300$		0			0
$301$		−	658.309i		−	0.126061i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	11693.8			2.19537
$306$		0			0
$307$	6143.84			1.14217			0.571087	−	0.820890i	$-0.306522\pi$
$307$	6143.84			1.14217			0.571087	+	0.820890i	$0.306522\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	7409.80i		−	1.35103i	−0.737345	−	0.675516i	$-0.763921\pi$
$311$		−	7409.80i		−	1.35103i	0.737345	−	0.675516i	$-0.236079\pi$
$312$		0			0
$313$			9455.98i			1.70761i	0.520589	+	0.853807i	$0.325712\pi$
$313$			9455.98i			1.70761i	−0.520589	+	0.853807i	$0.674288\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	1045.23i		−	0.185191i	−0.995704	−	0.0925957i	$-0.970484\pi$
$317$		−	1045.23i		−	0.185191i	0.995704	−	0.0925957i	$-0.0295164\pi$
$318$		0			0
$319$	1926.71			0.338165
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1562.86	−	3897.56i	0.269226	−	0.671412i
$324$		0			0
$325$	6062.20			1.03468
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	2764.87i		−	0.463320i
$330$		0			0
$331$	−6370.04			−1.05779			−0.528896	−	0.848687i	$-0.677394\pi$
$331$	−6370.04			−1.05779			−0.528896	+	0.848687i	$0.677394\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	7958.04i		−	1.29789i
$336$		0			0
$337$			9216.37i			1.48976i	0.667201	+	0.744878i	$0.267492\pi$
$337$			9216.37i			1.48976i	−0.667201	+	0.744878i	$0.732508\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−685.062			−0.108792
$342$		0			0
$343$		−	17399.4i		−	2.73901i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			7810.63i			1.20835i	0.796852	+	0.604174i	$0.206497\pi$
$347$			7810.63i			1.20835i	−0.796852	+	0.604174i	$0.793503\pi$
$348$		0			0
$349$	8377.96			1.28499			0.642495	−	0.766290i	$-0.277899\pi$
$349$	8377.96			1.28499			0.642495	+	0.766290i	$0.277899\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−500.551			−0.0754721			−0.0377360	−	0.999288i	$-0.512015\pi$
$353$	−500.551			−0.0754721			−0.0377360	+	0.999288i	$0.512015\pi$
$354$		0			0
$355$	7285.06			1.08916
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12643.0			1.85870			0.929351	−	0.369198i	$-0.120368\pi$
$359$	12643.0			1.85870			0.929351	+	0.369198i	$0.120368\pi$
$360$		0			0
$361$	−3269.85			−0.476724
$362$		0			0
$363$		0			0
$364$		0			0
$365$	5537.50			0.794099
$366$		0			0
$367$		−	640.446i		−	0.0910927i	−0.998962	−	0.0455464i	$-0.985497\pi$
$367$		−	640.446i		−	0.0910927i	0.998962	−	0.0455464i	$-0.0145029\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	11035.8i		−	1.54434i
$372$		0			0
$373$	−1138.01			−0.157972			−0.0789861	−	0.996876i	$-0.525168\pi$
$373$	−1138.01			−0.157972			−0.0789861	+	0.996876i	$0.525168\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	10899.8i		−	1.48904i
$378$		0			0
$379$		−	12320.8i		−	1.66987i	−0.550352	−	0.834933i	$-0.685507\pi$
$379$		−	12320.8i		−	1.66987i	0.550352	−	0.834933i	$-0.314493\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9934.77			1.32544			0.662719	−	0.748868i	$-0.269402\pi$
$383$	9934.77			1.32544			0.662719	+	0.748868i	$0.269402\pi$
$384$		0			0
$385$		−	4205.43i		−	0.556698i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3323.46			−0.433178			−0.216589	−	0.976263i	$-0.569493\pi$
$389$	−3323.46			−0.433178			−0.216589	+	0.976263i	$0.569493\pi$
$390$		0			0
$391$	−3218.57	−	1290.60i	−0.416292	−	0.166927i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	13798.4			1.75765
$396$		0			0
$397$		−	3094.45i		−	0.391199i	−0.980684	−	0.195599i	$-0.937335\pi$
$397$		−	3094.45i		−	0.391199i	0.980684	−	0.195599i	$-0.0626652\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			6261.50i			0.779761i	0.920865	+	0.389881i	$0.127484\pi$
$401$			6261.50i			0.779761i	−0.920865	+	0.389881i	$0.872516\pi$
$402$		0			0
$403$			3875.54i			0.479043i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1541.88			0.187784
$408$		0			0
$409$	13966.7			1.68853			0.844264	−	0.535927i	$-0.180038\pi$
$409$	13966.7			1.68853			0.844264	+	0.535927i	$0.180038\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	382.302i		−	0.0455492i
$414$		0			0
$415$			7482.56i			0.885071i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			6308.88i			0.735582i	0.929908	+	0.367791i	$0.119886\pi$
$419$			6308.88i			0.735582i	−0.929908	+	0.367791i	$0.880114\pi$
$420$		0			0
$421$	−3142.63			−0.363806			−0.181903	−	0.983317i	$-0.558226\pi$
$421$	−3142.63			−0.363806			−0.181903	+	0.983317i	$0.558226\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3763.18	+	9384.85i	−0.429508	+	1.07113i
$426$		0			0
$427$	−24587.1			−2.78654
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	6900.34i		−	0.771178i	−0.922671	−	0.385589i	$-0.873998\pi$
$431$		−	6900.34i		−	0.771178i	0.922671	−	0.385589i	$-0.126002\pi$
$432$		0			0
$433$	−15236.0			−1.69098			−0.845490	−	0.533991i	$-0.820692\pi$
$433$	−15236.0			−1.69098			−0.845490	+	0.533991i	$0.820692\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	2963.89i		−	0.324444i
$438$		0			0
$439$			9594.64i			1.04311i	0.853216	+	0.521557i	$0.174649\pi$
$439$			9594.64i			1.04311i	−0.853216	+	0.521557i	$0.825351\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3375.02			0.361968			0.180984	−	0.983486i	$-0.442072\pi$
$443$	3375.02			0.361968			0.180984	+	0.983486i	$0.442072\pi$
$444$		0			0
$445$		−	19780.0i		−	2.10710i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			2918.05i			0.306707i	0.988171	+	0.153354i	$0.0490073\pi$
$449$			2918.05i			0.306707i	−0.988171	+	0.153354i	$0.950993\pi$
$450$		0			0
$451$	−1313.35			−0.137125
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−23791.0			−2.45130
$456$		0			0
$457$	−3829.50			−0.391983			−0.195992	−	0.980606i	$-0.562793\pi$
$457$	−3829.50			−0.391983			−0.195992	+	0.980606i	$0.562793\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6988.84			−0.706080			−0.353040	−	0.935608i	$-0.614852\pi$
$461$	−6988.84			−0.706080			−0.353040	+	0.935608i	$0.614852\pi$
$462$		0			0
$463$	8669.67			0.870224			0.435112	−	0.900376i	$-0.356709\pi$
$463$	8669.67			0.870224			0.435112	+	0.900376i	$0.356709\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12728.0			−1.26120			−0.630600	−	0.776108i	$-0.717191\pi$
$467$	−12728.0			−1.26120			−0.630600	+	0.776108i	$0.717191\pi$
$468$		0			0
$469$			16732.3i			1.64739i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			141.741i			0.0137786i
$474$		0			0
$475$	−8642.24			−0.834807
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	13282.8i		−	1.26703i	−0.773730	−	0.633516i	$-0.781611\pi$
$479$		−	13282.8i		−	1.26703i	0.773730	−	0.633516i	$-0.218389\pi$
$480$		0			0
$481$		−	8722.72i		−	0.826864i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10476.9			−0.980888
$486$		0			0
$487$			5014.37i			0.466577i	0.972408	+	0.233288i	$0.0749486\pi$
$487$			5014.37i			0.466577i	−0.972408	+	0.233288i	$0.925051\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−14518.4			−1.33443			−0.667217	−	0.744863i	$-0.732515\pi$
$491$	−14518.4			−1.33443			−0.667217	+	0.744863i	$0.732515\pi$
$492$		0			0
$493$	16873.9	+	6766.16i	1.54150	+	0.618119i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−15317.3			−1.38245
$498$		0			0
$499$		−	18158.2i		−	1.62901i	−0.580158	−	0.814504i	$-0.697009\pi$
$499$		−	18158.2i		−	1.62901i	0.580158	−	0.814504i	$-0.302991\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	12331.2i		−	1.09309i	−0.837431	−	0.546543i	$-0.815943\pi$
$503$		−	12331.2i		−	1.09309i	0.837431	−	0.546543i	$-0.184057\pi$
$504$		0			0
$505$			12132.7i			1.06911i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	4051.78			0.352833			0.176417	−	0.984316i	$-0.443549\pi$
$509$	4051.78			0.352833			0.176417	+	0.984316i	$0.443549\pi$
$510$		0			0
$511$	−11643.0			−1.00793
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	26066.4i		−	2.23033i
$516$		0			0
$517$			595.307i			0.0506413i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	6101.23i		−	0.513052i	−0.966537	−	0.256526i	$-0.917422\pi$
$521$		−	6101.23i		−	0.513052i	0.966537	−	0.256526i	$-0.0825779\pi$
$522$		0			0
$523$	−1233.70			−0.103147			−0.0515734	−	0.998669i	$-0.516424\pi$
$523$	−1233.70			−0.103147			−0.0515734	+	0.998669i	$0.516424\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5999.69	−	2405.78i	−0.495922	−	0.198857i
$528$		0			0
$529$	9719.44			0.798836
$530$		0			0
$531$		0			0
$532$		0			0
$533$			7429.90i			0.603799i
$534$		0			0
$535$	−3319.51			−0.268252
$536$		0			0
$537$		0			0
$538$		0			0
$539$			6294.24i			0.502991i
$540$		0			0
$541$			8968.57i			0.712734i	0.934346	+	0.356367i	$0.115985\pi$
$541$			8968.57i			0.712734i	−0.934346	+	0.356367i	$0.884015\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	20424.5			1.60530
$546$		0			0
$547$		−	7187.71i		−	0.561836i	−0.959732	−	0.280918i	$-0.909361\pi$
$547$		−	7187.71i		−	0.561836i	0.959732	−	0.280918i	$-0.0906389\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			15538.7i			1.20140i
$552$		0			0
$553$	−29012.0			−2.23095
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12502.2			−0.951047			−0.475524	−	0.879703i	$-0.657741\pi$
$557$	−12502.2			−0.951047			−0.475524	+	0.879703i	$0.657741\pi$
$558$		0			0
$559$	801.860			0.0606709
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20391.1			−1.52643			−0.763216	−	0.646143i	$-0.776381\pi$
$563$	−20391.1			−1.52643			−0.763216	+	0.646143i	$0.776381\pi$
$564$		0			0
$565$	24486.8			1.82331
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−3407.53			−0.251057			−0.125528	−	0.992090i	$-0.540063\pi$
$569$	−3407.53			−0.251057			−0.125528	+	0.992090i	$0.540063\pi$
$570$		0			0
$571$			9682.41i			0.709626i	0.934937	+	0.354813i	$0.115455\pi$
$571$			9682.41i			0.709626i	−0.934937	+	0.354813i	$0.884545\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			7136.69i			0.517601i
$576$		0			0
$577$	−10357.7			−0.747309			−0.373654	−	0.927568i	$-0.621895\pi$
$577$	−10357.7			−0.747309			−0.373654	+	0.927568i	$0.621895\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	15732.6i		−	1.12340i
$582$		0			0
$583$			2376.12i			0.168798i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	760.786			0.0534940			0.0267470	−	0.999642i	$-0.491485\pi$
$587$	760.786			0.0534940			0.0267470	+	0.999642i	$0.491485\pi$
$588$		0			0
$589$		−	5524.95i		−	0.386505i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−23861.5			−1.65241			−0.826203	−	0.563373i	$-0.809503\pi$
$593$	−23861.5			−1.65241			−0.826203	+	0.563373i	$0.809503\pi$
$594$		0			0
$595$	14768.5	−	36830.7i	1.01757	−	2.53767i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−40.2985			−0.00274884			−0.00137442	−	0.999999i	$-0.500437\pi$
$599$	−40.2985			−0.00274884			−0.00137442	+	0.999999i	$0.500437\pi$
$600$		0			0
$601$		−	28351.4i		−	1.92426i	−0.272594	−	0.962129i	$-0.587882\pi$
$601$		−	28351.4i		−	1.92426i	0.272594	−	0.962129i	$-0.412118\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	20934.9i		−	1.40682i
$606$		0			0
$607$			380.879i			0.0254686i	0.999919	+	0.0127343i	$0.00405356\pi$
$607$			380.879i			0.0254686i	−0.999919	+	0.0127343i	$0.995946\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3367.78			0.222988
$612$		0			0
$613$	−18449.5			−1.21561			−0.607804	−	0.794087i	$-0.707949\pi$
$613$	−18449.5			−1.21561			−0.607804	+	0.794087i	$0.707949\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	13692.2i		−	0.893397i	−0.894685	−	0.446698i	$-0.852600\pi$
$617$		−	13692.2i		−	0.893397i	0.894685	−	0.446698i	$-0.147400\pi$
$618$		0			0
$619$		−	15000.2i		−	0.974004i	−0.873401	−	0.487002i	$-0.838090\pi$
$619$		−	15000.2i		−	0.974004i	0.873401	−	0.487002i	$-0.161910\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			41588.7i			2.67450i
$624$		0			0
$625$	−12847.4			−0.822232
$626$		0			0
$627$		0			0
$628$		0			0
$629$	13503.6	+	5414.73i	0.855998	+	0.343242i
$630$		0			0
$631$	−25659.3			−1.61883			−0.809413	−	0.587240i	$-0.800215\pi$
$631$	−25659.3			−1.61883			−0.809413	+	0.587240i	$0.800215\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	33712.0i		−	2.10680i
$636$		0			0
$637$	35607.9			2.21481
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	6709.60i		−	0.413437i	−0.978400	−	0.206719i	$-0.933722\pi$
$641$		−	6709.60i		−	0.413437i	0.978400	−	0.206719i	$-0.0662785\pi$
$642$		0			0
$643$		−	11976.4i		−	0.734528i	−0.930117	−	0.367264i	$-0.880295\pi$
$643$		−	11976.4i		−	0.734528i	0.930117	−	0.367264i	$-0.119705\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−21265.3			−1.29215			−0.646077	−	0.763272i	$-0.723592\pi$
$647$	−21265.3			−1.29215			−0.646077	+	0.763272i	$0.723592\pi$
$648$		0			0
$649$			82.3137i			0.00497857i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			19671.9i			1.17890i	0.807806	+	0.589449i	$0.200655\pi$
$653$			19671.9i			1.17890i	−0.807806	+	0.589449i	$0.799345\pi$
$654$		0			0
$655$	31057.6			1.85271
$656$		0			0
$657$		0			0
$658$		0			0
$659$	14782.6			0.873824			0.436912	−	0.899504i	$-0.356072\pi$
$659$	14782.6			0.873824			0.436912	+	0.899504i	$0.356072\pi$
$660$		0			0
$661$	27770.8			1.63413			0.817063	−	0.576548i	$-0.195601\pi$
$661$	27770.8			1.63413			0.817063	+	0.576548i	$0.195601\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	33916.4			1.97777
$666$		0			0
$667$	12831.7			0.744896
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5293.86			0.304571
$672$		0			0
$673$			21891.3i			1.25386i	0.779077	+	0.626929i	$0.215688\pi$
$673$			21891.3i			1.25386i	−0.779077	+	0.626929i	$0.784312\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			1946.42i			0.110498i	0.998473	+	0.0552488i	$0.0175952\pi$
$677$			1946.42i			0.110498i	−0.998473	+	0.0552488i	$0.982405\pi$
$678$		0			0
$679$	22028.3			1.24502
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	10516.5i		−	0.589169i	−0.955625	−	0.294584i	$-0.904819\pi$
$683$		−	10516.5i		−	0.589169i	0.955625	−	0.294584i	$-0.0951812\pi$
$684$		0			0
$685$		−	13925.0i		−	0.776709i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	13442.2			0.743263
$690$		0			0
$691$			26695.9i			1.46970i	0.678232	+	0.734848i	$0.262747\pi$
$691$			26695.9i			1.46970i	−0.678232	+	0.734848i	$0.737253\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−338.704			−0.0184860
$696$		0			0
$697$	−11502.2	−	4612.20i	−0.625074	−	0.250645i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	22317.0			1.20243			0.601214	−	0.799088i	$-0.294684\pi$
$701$	22317.0			1.20243			0.601214	+	0.799088i	$0.294684\pi$
$702$		0			0
$703$			12435.1i			0.667137i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	25509.8i		−	1.35700i
$708$		0			0
$709$		−	28249.7i		−	1.49639i	−0.663478	−	0.748196i	$-0.730920\pi$
$709$		−	28249.7i		−	1.49639i	0.663478	−	0.748196i	$-0.269080\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4562.46			−0.239643
$714$		0			0
$715$	5122.46			0.267929
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	30991.2i		−	1.60748i	−0.594982	−	0.803739i	$-0.702841\pi$
$719$		−	30991.2i		−	1.60748i	0.594982	−	0.803739i	$-0.297159\pi$
$720$		0			0
$721$			54806.3i			2.83092i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	37415.2i		−	1.91664i
$726$		0			0
$727$	13328.0			0.679929			0.339965	−	0.940438i	$-0.389585\pi$
$727$	13328.0			0.679929			0.339965	+	0.940438i	$0.389585\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−497.764	+	1241.35i	−0.0251853	+	0.0628086i
$732$		0			0
$733$	−27976.7			−1.40974			−0.704872	−	0.709334i	$-0.748996\pi$
$733$	−27976.7			−1.40974			−0.704872	+	0.709334i	$0.748996\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	3602.64i		−	0.180061i
$738$		0			0
$739$	33971.2			1.69100			0.845501	−	0.533974i	$-0.179302\pi$
$739$	33971.2			1.69100			0.845501	+	0.533974i	$0.179302\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			25922.4i			1.27995i	0.768397	+	0.639974i	$0.221055\pi$
$743$			25922.4i			1.27995i	−0.768397	+	0.639974i	$0.778945\pi$
$744$		0			0
$745$		−	19733.2i		−	0.970428i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6979.49			0.340488
$750$		0			0
$751$		−	18950.5i		−	0.920792i	−0.887714	−	0.460396i	$-0.847707\pi$
$751$		−	18950.5i		−	0.920792i	0.887714	−	0.460396i	$-0.152293\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			34143.4i			1.64583i
$756$		0			0
$757$	−8184.07			−0.392940			−0.196470	−	0.980510i	$-0.562948\pi$
$757$	−8184.07			−0.392940			−0.196470	+	0.980510i	$0.562948\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	25803.9			1.22916			0.614579	−	0.788855i	$-0.289326\pi$
$761$	25803.9			1.22916			0.614579	+	0.788855i	$0.289326\pi$
$762$		0			0
$763$	−42943.9			−2.03758
$764$		0			0
$765$		0			0
$766$		0			0
$767$	465.666			0.0219221
$768$		0			0
$769$	5222.53			0.244901			0.122451	−	0.992475i	$-0.460925\pi$
$769$	5222.53			0.244901			0.122451	+	0.992475i	$0.460925\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−32636.6			−1.51857			−0.759286	−	0.650757i	$-0.774452\pi$
$773$	−32636.6			−1.51857			−0.759286	+	0.650757i	$0.774452\pi$
$774$		0			0
$775$			13303.4i			0.616610i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	10592.0i		−	0.487162i
$780$		0			0
$781$	3297.99			0.151103
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	18931.7i		−	0.860764i
$786$		0			0
$787$		−	17014.8i		−	0.770664i	−0.922778	−	0.385332i	$-0.874087\pi$
$787$		−	17014.8i		−	0.770664i	0.922778	−	0.385332i	$-0.125913\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−51485.1			−2.31429
$792$		0			0
$793$		−	29948.5i		−	1.34111i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	20826.6			0.925616			0.462808	−	0.886458i	$-0.346842\pi$
$797$	20826.6			0.925616			0.462808	+	0.886458i	$0.346842\pi$
$798$		0			0
$799$	−2090.59	+	5213.63i	−0.0925652	+	0.230845i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2506.86			0.110168
$804$		0			0
$805$		−	28007.8i		−	1.22627i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			8639.25i			0.375451i	0.982222	+	0.187726i	$0.0601115\pi$
$809$			8639.25i			0.375451i	−0.982222	+	0.187726i	$0.939888\pi$
$810$		0			0
$811$			22319.6i			0.966397i	0.875511	+	0.483199i	$0.160525\pi$
$811$			22319.6i			0.966397i	−0.875511	+	0.483199i	$0.839475\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	23979.3			1.03062
$816$		0			0
$817$	−1143.13			−0.0489510
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	9361.94i		−	0.397971i	−0.980002	−	0.198985i	$-0.936235\pi$
$821$		−	9361.94i		−	0.397971i	0.980002	−	0.198985i	$-0.0637646\pi$
$822$		0			0
$823$		−	19128.8i		−	0.810192i	−0.914274	−	0.405096i	$-0.867238\pi$
$823$		−	19128.8i		−	0.810192i	0.914274	−	0.405096i	$-0.132762\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			40640.9i			1.70886i	0.519570	+	0.854428i	$0.326092\pi$
$827$			40640.9i			1.70886i	−0.519570	+	0.854428i	$0.673908\pi$
$828$		0			0
$829$	−25220.3			−1.05662			−0.528310	−	0.849052i	$-0.677174\pi$
$829$	−25220.3			−1.05662			−0.528310	+	0.849052i	$0.677174\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−22104.0	+	55124.3i	−0.919397	+	2.29285i
$834$		0			0
$835$	68228.3			2.82771
$836$		0			0
$837$		0			0
$838$		0			0
$839$			4544.50i			0.187001i	0.995619	+	0.0935004i	$0.0298056\pi$
$839$			4544.50i			0.187001i	−0.995619	+	0.0935004i	$0.970194\pi$
$840$		0			0
$841$	−42883.3			−1.75830
$842$		0			0
$843$		0			0
$844$		0			0
$845$			7071.69i			0.287898i
$846$		0			0
$847$			44017.0i			1.78565i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10268.8			0.413641
$852$		0			0
$853$			22395.8i			0.898964i	0.893289	+	0.449482i	$0.148391\pi$
$853$			22395.8i			0.898964i	−0.893289	+	0.449482i	$0.851609\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	5735.42i		−	0.228610i	−0.993446	−	0.114305i	$-0.963536\pi$
$857$		−	5735.42i		−	0.228610i	0.993446	−	0.114305i	$-0.0364640\pi$
$858$		0			0
$859$	−34718.6			−1.37903			−0.689513	−	0.724273i	$-0.742175\pi$
$859$	−34718.6			−1.37903			−0.689513	+	0.724273i	$0.742175\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	22239.9			0.877237			0.438619	−	0.898673i	$-0.355468\pi$
$863$	22239.9			0.877237			0.438619	+	0.898673i	$0.355468\pi$
$864$		0			0
$865$	−41089.4			−1.61512
$866$		0			0
$867$		0			0
$868$		0			0
$869$	6246.61			0.243845
$870$		0			0
$871$	−20380.9			−0.792860
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−10900.7			−0.421156
$876$		0			0
$877$			10379.6i			0.399650i	0.979832	+	0.199825i	$0.0640373\pi$
$877$			10379.6i			0.399650i	−0.979832	+	0.199825i	$0.935963\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	44934.3i		−	1.71836i	−0.511672	−	0.859181i	$-0.670974\pi$
$881$		−	44934.3i		−	1.71836i	0.511672	−	0.859181i	$-0.329026\pi$
$882$		0			0
$883$	−15846.0			−0.603918			−0.301959	−	0.953321i	$-0.597640\pi$
$883$	−15846.0			−0.603918			−0.301959	+	0.953321i	$0.597640\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			30415.0i			1.15134i	0.817683	+	0.575669i	$0.195258\pi$
$887$			30415.0i			1.15134i	−0.817683	+	0.575669i	$0.804742\pi$
$888$		0			0
$889$			70881.8i			2.67413i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−4801.08			−0.179913
$894$		0			0
$895$			13778.4i			0.514593i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	23919.4			0.887382
$900$		0			0
$901$	−8344.42	+	20809.8i	−0.308538	+	0.769451i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−8212.55			−0.301651
$906$		0			0
$907$			7288.75i			0.266835i	0.991060	+	0.133417i	$0.0425951\pi$
$907$			7288.75i			0.266835i	−0.991060	+	0.133417i	$0.957405\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	10232.6i		−	0.372140i	−0.982536	−	0.186070i	$-0.940425\pi$
$911$		−	10232.6i		−	0.372140i	0.982536	−	0.186070i	$-0.0595752\pi$
$912$		0			0
$913$			3387.39i			0.122789i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−65300.8			−2.35160
$918$		0			0
$919$	−26533.8			−0.952415			−0.476207	−	0.879333i	$-0.657989\pi$
$919$	−26533.8			−0.952415			−0.476207	+	0.879333i	$0.657989\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	18657.4i		−	0.665348i
$924$		0			0
$925$		−	29942.1i		−	1.06431i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			7037.69i			0.248546i	0.992248	+	0.124273i	$0.0396599\pi$
$929$			7037.69i			0.248546i	−0.992248	+	0.124273i	$0.960340\pi$
$930$		0			0
$931$	−50762.4			−1.78697
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3179.83	+	7930.05i	−0.111221	+	0.277369i
$936$		0			0
$937$	16895.0			0.589047			0.294523	−	0.955644i	$-0.404839\pi$
$937$	16895.0			0.589047			0.294523	+	0.955644i	$0.404839\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			51103.6i			1.77038i	0.465229	+	0.885190i	$0.345972\pi$
$941$			51103.6i			1.77038i	−0.465229	+	0.885190i	$0.654028\pi$
$942$		0			0
$943$	−8746.81			−0.302052
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	22510.4i		−	0.772429i	−0.922409	−	0.386215i	$-0.873782\pi$
$947$		−	22510.4i		−	0.772429i	0.922409	−	0.386215i	$-0.126218\pi$
$948$		0			0
$949$		−	14181.8i		−	0.485101i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	4260.98			0.144834			0.0724169	−	0.997374i	$-0.476929\pi$
$953$	4260.98			0.144834			0.0724169	+	0.997374i	$0.476929\pi$
$954$		0			0
$955$			3280.76i			0.111165i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			29278.1i			0.985861i
$960$		0			0
$961$	21286.2			0.714517
$962$		0			0
$963$		0			0
$964$		0			0
$965$	45264.5			1.50996
$966$		0			0
$967$	5646.95			0.187791			0.0938953	−	0.995582i	$-0.470068\pi$
$967$	5646.95			0.187791			0.0938953	+	0.995582i	$0.470068\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3320.02			−0.109726			−0.0548632	−	0.998494i	$-0.517472\pi$
$971$	−3320.02			−0.109726			−0.0548632	+	0.998494i	$0.517472\pi$
$972$		0			0
$973$	712.148			0.0234639
$974$		0			0
$975$		0			0
$976$		0			0
$977$	5821.02			0.190615			0.0953076	−	0.995448i	$-0.469617\pi$
$977$	5821.02			0.190615			0.0953076	+	0.995448i	$0.469617\pi$
$978$		0			0
$979$		−	8954.49i		−	0.292326i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	234.506i		−	0.00760894i	−0.999993	−	0.00380447i	$-0.998789\pi$
$983$		−	234.506i		−	0.00760894i	0.999993	−	0.00380447i	$-0.00121100\pi$
$984$		0			0
$985$	55460.3			1.79402
$986$		0			0
$987$		0			0
$988$		0			0
$989$			943.985i			0.0303508i
$990$		0			0
$991$			16708.2i			0.535573i	0.963478	+	0.267787i	$0.0862922\pi$
$991$			16708.2i			0.535573i	−0.963478	+	0.267787i	$0.913708\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	44787.7			1.42700
$996$		0			0
$997$			10281.9i			0.326611i	0.986576	+	0.163305i	$0.0522156\pi$
$997$			10281.9i			0.326611i	−0.986576	+	0.163305i	$0.947784\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1224.4.c.e.577.2		8
3.2	odd	2		136.4.b.b.33.1	✓	8
12.11	even	2		272.4.b.f.33.8		8
17.16	even	2	inner	1224.4.c.e.577.7		8
51.38	odd	4		2312.4.a.k.1.1		8
51.47	odd	4		2312.4.a.k.1.8		8
51.50	odd	2		136.4.b.b.33.8	yes	8
204.203	even	2		272.4.b.f.33.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.b.b.33.1	✓	8	3.2	odd	2
136.4.b.b.33.8	yes	8	51.50	odd	2
272.4.b.f.33.1		8	204.203	even	2
272.4.b.f.33.8		8	12.11	even	2
1224.4.c.e.577.2		8	1.1	even	1	trivial
1224.4.c.e.577.7		8	17.16	even	2	inner
2312.4.a.k.1.1		8	51.38	odd	4
2312.4.a.k.1.8		8	51.47	odd	4

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:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1224.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-16.4090i q^{5} +34.5010i q^{7} +O(q^{10})\)	\(q-16.4090i q^{5} +34.5010i q^{7} -7.42843i q^{11} -42.0242 q^{13} +(26.0870 - 65.0574i) q^{17} +59.9095 q^{19} -49.4728i q^{23} -144.255 q^{25} +259.369i q^{29} -92.2215i q^{31} +566.126 q^{35} +207.564i q^{37} -176.800i q^{41} -19.0809 q^{43} -80.1389 q^{47} -847.318 q^{49} -319.869 q^{53} -121.893 q^{55} -11.0809 q^{59} +712.648i q^{61} +689.575i q^{65} +484.980 q^{67} +443.968i q^{71} +337.468i q^{73} +256.288 q^{77} +840.905i q^{79} -456.004 q^{83} +(-1067.53 - 428.062i) q^{85} +1205.43 q^{89} -1449.88i q^{91} -983.055i q^{95} -638.484i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 44 q^{13} - 28 q^{17} + 48 q^{19} - 520 q^{25} + 1064 q^{35} + 8 q^{43} - 312 q^{47} - 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 1660 q^{77} - 2472 q^{83} - 2160 q^{85} - 68 q^{89}+O(q^{100}) \) 8 * q + 44 * q^13 - 28 * q^17 + 48 * q^19 - 520 * q^25 + 1064 * q^35 + 8 * q^43 - 312 * q^47 - 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 1660 * q^77 - 2472 * q^83 - 2160 * q^85 - 68 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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