LMFDB - Embedded newform 1056.3.e.a.241.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1056,3,Mod(241,1056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1056, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1056.241");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1056 = 2^{5} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1056.e (of order $2$, degree $1$, 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:	$28.7739159164$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 264)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.7
Character	$\chi$	$=$	1056.241
Dual form			1056.3.e.a.241.42

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -4.86020i q^{5} -11.9744i q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} -4.86020i q^{5} -11.9744i q^{7} -3.00000 q^{9} +(-1.44010 - 10.9053i) q^{11} +20.4277 q^{13} -8.41811 q^{15} -26.3096i q^{17} -12.5628 q^{19} -20.7402 q^{21} +35.3899 q^{23} +1.37849 q^{25} +5.19615i q^{27} -13.2504 q^{29} +19.6674 q^{31} +(-18.8886 + 2.49433i) q^{33} -58.1978 q^{35} +9.27077i q^{37} -35.3818i q^{39} +13.7607i q^{41} +23.0956 q^{43} +14.5806i q^{45} +63.7164 q^{47} -94.3858 q^{49} -45.5696 q^{51} -21.7323i q^{53} +(-53.0020 + 6.99917i) q^{55} +21.7595i q^{57} +66.8708i q^{59} -50.9978 q^{61} +35.9231i q^{63} -99.2826i q^{65} +87.6588i q^{67} -61.2970i q^{69} -13.1449 q^{71} +47.7609i q^{73} -2.38762i q^{75} +(-130.584 + 17.2443i) q^{77} +39.2459i q^{79} +9.00000 q^{81} -60.8874 q^{83} -127.870 q^{85} +22.9504i q^{87} +25.2453 q^{89} -244.609i q^{91} -34.0650i q^{93} +61.0579i q^{95} +107.717 q^{97} +(4.32030 + 32.7160i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 144 q^{9}+O(q^{10}) $ 48 * q - 144 * q^9	$ 48 q - 144 q^{9} + 128 q^{23} - 240 q^{25} - 336 q^{49} + 128 q^{55} - 512 q^{71} + 432 q^{81} - 32 q^{97}+O(q^{100}) $ 48 * q - 144 * q^9 + 128 * q^23 - 240 * q^25 - 336 * q^49 + 128 * q^55 - 512 * q^71 + 432 * q^81 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1056\mathbb{Z}\right)^\times$.

$n$	$133$	$353$	$673$	$991$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$		−	1.73205i		−	0.577350i
$3$		−	1.73205i		−	0.577350i
$4$		0			0
$5$		−	4.86020i		−	0.972039i	−0.873948	−	0.486020i	$-0.838448\pi$
$5$		−	4.86020i		−	0.972039i	0.873948	−	0.486020i	$-0.161552\pi$
$6$		0			0
$7$		−	11.9744i		−	1.71063i	−0.518112	−	0.855313i	$-0.673365\pi$
$7$		−	11.9744i		−	1.71063i	0.518112	−	0.855313i	$-0.326635\pi$
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	−1.44010	−	10.9053i	−0.130918	−	0.991393i
$11$	−1.44010	−	10.9053i	−0.130918	−	0.991393i
$12$		0			0
$13$	20.4277			1.57136			0.785680	−	0.618633i	$-0.212313\pi$
$13$	20.4277			1.57136			0.785680	+	0.618633i	$0.212313\pi$
$14$		0			0
$15$	−8.41811			−0.561207
$16$		0			0
$17$		−	26.3096i		−	1.54763i	−0.633414	−	0.773813i	$-0.718347\pi$
$17$		−	26.3096i		−	1.54763i	0.633414	−	0.773813i	$-0.281653\pi$
$18$		0			0
$19$	−12.5628			−0.661202			−0.330601	−	0.943771i	$-0.607251\pi$
$19$	−12.5628			−0.661202			−0.330601	+	0.943771i	$0.607251\pi$
$20$		0			0
$21$	−20.7402			−0.987630
$22$		0			0
$23$	35.3899			1.53869			0.769345	−	0.638834i	$-0.220583\pi$
$23$	35.3899			1.53869			0.769345	+	0.638834i	$0.220583\pi$
$24$		0			0
$25$	1.37849			0.0551397
$26$		0			0
$27$			5.19615i			0.192450i
$28$		0			0
$29$	−13.2504			−0.456912			−0.228456	−	0.973554i	$-0.573368\pi$
$29$	−13.2504			−0.456912			−0.228456	+	0.973554i	$0.573368\pi$
$30$		0			0
$31$	19.6674			0.634434			0.317217	−	0.948353i	$-0.397252\pi$
$31$	19.6674			0.634434			0.317217	+	0.948353i	$0.397252\pi$
$32$		0			0
$33$	−18.8886	+	2.49433i	−0.572381	+	0.0755857i
$34$		0			0
$35$	−58.1978			−1.66280
$36$		0			0
$37$			9.27077i			0.250561i	0.992121	+	0.125281i	$0.0399832\pi$
$37$			9.27077i			0.250561i	−0.992121	+	0.125281i	$0.960017\pi$
$38$		0			0
$39$		−	35.3818i		−	0.907226i
$40$		0			0
$41$			13.7607i			0.335627i	0.985819	+	0.167814i	$0.0536707\pi$
$41$			13.7607i			0.335627i	−0.985819	+	0.167814i	$0.946329\pi$
$42$		0			0
$43$	23.0956			0.537107			0.268554	−	0.963265i	$-0.413454\pi$
$43$	23.0956			0.537107			0.268554	+	0.963265i	$0.413454\pi$
$44$		0			0
$45$			14.5806i			0.324013i
$46$		0			0
$47$	63.7164			1.35567			0.677834	−	0.735215i	$-0.262919\pi$
$47$	63.7164			1.35567			0.677834	+	0.735215i	$0.262919\pi$
$48$		0			0
$49$	−94.3858			−1.92624
$50$		0			0
$51$	−45.5696			−0.893522
$52$		0			0
$53$		−	21.7323i		−	0.410043i	−0.978757	−	0.205022i	$-0.934274\pi$
$53$		−	21.7323i		−	0.410043i	0.978757	−	0.205022i	$-0.0657265\pi$
$54$		0			0
$55$	−53.0020	+	6.99917i	−0.963673	+	0.127258i
$56$		0			0
$57$			21.7595i			0.381745i
$58$		0			0
$59$			66.8708i			1.13340i	0.823923	+	0.566702i	$0.191781\pi$
$59$			66.8708i			1.13340i	−0.823923	+	0.566702i	$0.808219\pi$
$60$		0			0
$61$	−50.9978			−0.836029			−0.418015	−	0.908440i	$-0.637274\pi$
$61$	−50.9978			−0.836029			−0.418015	+	0.908440i	$0.637274\pi$
$62$		0			0
$63$			35.9231i			0.570209i
$64$		0			0
$65$		−	99.2826i		−	1.52742i
$66$		0			0
$67$			87.6588i			1.30834i	0.756347	+	0.654171i	$0.226982\pi$
$67$			87.6588i			1.30834i	−0.756347	+	0.654171i	$0.773018\pi$
$68$		0			0
$69$		−	61.2970i		−	0.888363i
$70$		0			0
$71$	−13.1449			−0.185140			−0.0925700	−	0.995706i	$-0.529508\pi$
$71$	−13.1449			−0.185140			−0.0925700	+	0.995706i	$0.529508\pi$
$72$		0			0
$73$			47.7609i			0.654259i	0.944980	+	0.327130i	$0.106081\pi$
$73$			47.7609i			0.654259i	−0.944980	+	0.327130i	$0.893919\pi$
$74$		0			0
$75$		−	2.38762i		−	0.0318349i
$76$		0			0
$77$	−130.584	+	17.2443i	−1.69590	+	0.223952i
$78$		0			0
$79$			39.2459i			0.496784i	0.968660	+	0.248392i	$0.0799021\pi$
$79$			39.2459i			0.496784i	−0.968660	+	0.248392i	$0.920098\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	−60.8874			−0.733583			−0.366791	−	0.930303i	$-0.619544\pi$
$83$	−60.8874			−0.733583			−0.366791	+	0.930303i	$0.619544\pi$
$84$		0			0
$85$	−127.870			−1.50435
$86$		0			0
$87$			22.9504i			0.263798i
$88$		0			0
$89$	25.2453			0.283655			0.141827	−	0.989891i	$-0.454702\pi$
$89$	25.2453			0.283655			0.141827	+	0.989891i	$0.454702\pi$
$90$		0			0
$91$		−	244.609i		−	2.68801i
$92$		0			0
$93$		−	34.0650i		−	0.366291i
$94$		0			0
$95$			61.0579i			0.642714i
$96$		0			0
$97$	107.717			1.11048			0.555240	−	0.831690i	$-0.312626\pi$
$97$	107.717			1.11048			0.555240	+	0.831690i	$0.312626\pi$
$98$		0			0
$99$	4.32030	+	32.7160i	0.0436394	+	0.330464i
$100$		0			0
$101$	81.8762			0.810655			0.405328	−	0.914171i	$-0.367157\pi$
$101$	81.8762			0.810655			0.405328	+	0.914171i	$0.367157\pi$
$102$		0			0
$103$	176.532			1.71390			0.856951	−	0.515398i	$-0.172356\pi$
$103$	176.532			1.71390			0.856951	+	0.515398i	$0.172356\pi$
$104$		0			0
$105$			100.802i			0.960015i
$106$		0			0
$107$	−29.2228			−0.273111			−0.136555	−	0.990632i	$-0.543603\pi$
$107$	−29.2228			−0.273111			−0.136555	+	0.990632i	$0.543603\pi$
$108$		0			0
$109$	−29.8090			−0.273477			−0.136738	−	0.990607i	$-0.543662\pi$
$109$	−29.8090			−0.273477			−0.136738	+	0.990607i	$0.543662\pi$
$110$		0			0
$111$	16.0575			0.144662
$112$		0			0
$113$	128.899			1.14070			0.570349	−	0.821403i	$-0.306808\pi$
$113$	128.899			1.14070			0.570349	+	0.821403i	$0.306808\pi$
$114$		0			0
$115$		−	172.002i		−	1.49567i
$116$		0			0
$117$	−61.2831			−0.523787
$118$		0			0
$119$	−315.042			−2.64741
$120$		0			0
$121$	−116.852	+	31.4095i	−0.965721	+	0.259583i
$122$		0			0
$123$	23.8343			0.193775
$124$		0			0
$125$		−	128.205i		−	1.02564i
$126$		0			0
$127$		−	88.8698i		−	0.699763i	−0.936794	−	0.349881i	$-0.886222\pi$
$127$		−	88.8698i		−	0.699763i	0.936794	−	0.349881i	$-0.113778\pi$
$128$		0			0
$129$		−	40.0028i		−	0.310099i
$130$		0			0
$131$	−4.84006			−0.0369470			−0.0184735	−	0.999829i	$-0.505881\pi$
$131$	−4.84006			−0.0369470			−0.0184735	+	0.999829i	$0.505881\pi$
$132$		0			0
$133$			150.432i			1.13107i
$134$		0			0
$135$	25.2543			0.187069
$136$		0			0
$137$	−185.886			−1.35683			−0.678416	−	0.734678i	$-0.737333\pi$
$137$	−185.886			−1.35683			−0.678416	+	0.734678i	$0.737333\pi$
$138$		0			0
$139$	−225.347			−1.62120			−0.810601	−	0.585599i	$-0.800859\pi$
$139$	−225.347			−1.62120			−0.810601	+	0.585599i	$0.800859\pi$
$140$		0			0
$141$		−	110.360i		−	0.782695i
$142$		0			0
$143$	−29.4179	−	222.771i	−0.205720	−	1.55784i
$144$		0			0
$145$			64.3998i			0.444136i
$146$		0			0
$147$			163.481i			1.11212i
$148$		0			0
$149$	−167.306			−1.12286			−0.561428	−	0.827525i	$-0.689748\pi$
$149$	−167.306			−1.12286			−0.561428	+	0.827525i	$0.689748\pi$
$150$		0			0
$151$			201.617i			1.33521i	0.744515	+	0.667606i	$0.232681\pi$
$151$			201.617i			1.33521i	−0.744515	+	0.667606i	$0.767319\pi$
$152$		0			0
$153$			78.9289i			0.515875i
$154$		0			0
$155$		−	95.5876i		−	0.616695i
$156$		0			0
$157$			303.154i			1.93091i	0.260562	+	0.965457i	$0.416092\pi$
$157$			303.154i			1.93091i	−0.260562	+	0.965457i	$0.583908\pi$
$158$		0			0
$159$	−37.6414			−0.236739
$160$		0			0
$161$		−	423.772i		−	2.63212i
$162$		0			0
$163$			131.153i			0.804617i	0.915504	+	0.402308i	$0.131792\pi$
$163$			131.153i			0.804617i	−0.915504	+	0.402308i	$0.868208\pi$
$164$		0			0
$165$	12.1229	+	91.8022i	0.0734723	+	0.556377i
$166$		0			0
$167$		−	208.910i		−	1.25096i	−0.780242	−	0.625478i	$-0.784904\pi$
$167$		−	208.910i		−	1.25096i	0.780242	−	0.625478i	$-0.215096\pi$
$168$		0			0
$169$	248.290			1.46917
$170$		0			0
$171$	37.6885			0.220401
$172$		0			0
$173$	68.6927			0.397068			0.198534	−	0.980094i	$-0.436382\pi$
$173$	68.6927			0.397068			0.198534	+	0.980094i	$0.436382\pi$
$174$		0			0
$175$		−	16.5066i		−	0.0943234i
$176$		0			0
$177$	115.824			0.654371
$178$		0			0
$179$			98.8446i			0.552204i	0.961128	+	0.276102i	$0.0890428\pi$
$179$			98.8446i			0.552204i	−0.961128	+	0.276102i	$0.910957\pi$
$180$		0			0
$181$		−	79.2186i		−	0.437672i	−0.975762	−	0.218836i	$-0.929774\pi$
$181$		−	79.2186i		−	0.437672i	0.975762	−	0.218836i	$-0.0702260\pi$
$182$		0			0
$183$			88.3308i			0.482682i
$184$		0			0
$185$	45.0578			0.243556
$186$		0			0
$187$	−286.915	+	37.8885i	−1.53431	+	0.202613i
$188$		0			0
$189$	62.2207			0.329210
$190$		0			0
$191$	188.156			0.985111			0.492555	−	0.870281i	$-0.336063\pi$
$191$	188.156			0.985111			0.492555	+	0.870281i	$0.336063\pi$
$192$		0			0
$193$			152.773i			0.791572i	0.918343	+	0.395786i	$0.129528\pi$
$193$			152.773i			0.791572i	−0.918343	+	0.395786i	$0.870472\pi$
$194$		0			0
$195$	−171.962			−0.881859
$196$		0			0
$197$	268.063			1.36073			0.680364	−	0.732874i	$-0.261822\pi$
$197$	268.063			1.36073			0.680364	+	0.732874i	$0.261822\pi$
$198$		0			0
$199$	−314.609			−1.58095			−0.790475	−	0.612494i	$-0.790166\pi$
$199$	−314.609			−1.58095			−0.790475	+	0.612494i	$0.790166\pi$
$200$		0			0
$201$	151.830			0.755371
$202$		0			0
$203$			158.666i			0.781605i
$204$		0			0
$205$	66.8798			0.326243
$206$		0			0
$207$	−106.170			−0.512896
$208$		0			0
$209$	18.0918	+	137.002i	0.0865634	+	0.655511i
$210$		0			0
$211$	365.685			1.73310			0.866552	−	0.499086i	$-0.166331\pi$
$211$	365.685			1.73310			0.866552	+	0.499086i	$0.166331\pi$
$212$		0			0
$213$			22.7677i			0.106891i
$214$		0			0
$215$		−	112.249i		−	0.522089i
$216$		0			0
$217$		−	235.505i		−	1.08528i
$218$		0			0
$219$	82.7243			0.377737
$220$		0			0
$221$		−	537.445i		−	2.43188i
$222$		0			0
$223$	74.2015			0.332742			0.166371	−	0.986063i	$-0.446795\pi$
$223$	74.2015			0.332742			0.166371	+	0.986063i	$0.446795\pi$
$224$		0			0
$225$	−4.13548			−0.0183799
$226$		0			0
$227$	133.968			0.590166			0.295083	−	0.955472i	$-0.404653\pi$
$227$	133.968			0.590166			0.295083	+	0.955472i	$0.404653\pi$
$228$		0			0
$229$			165.628i			0.723266i	0.932321	+	0.361633i	$0.117781\pi$
$229$			165.628i			0.723266i	−0.932321	+	0.361633i	$0.882219\pi$
$230$		0			0
$231$	29.8680	+	226.179i	0.129299	+	0.979130i
$232$		0			0
$233$			287.599i			1.23433i	0.786834	+	0.617165i	$0.211719\pi$
$233$			287.599i			1.23433i	−0.786834	+	0.617165i	$0.788281\pi$
$234$		0			0
$235$		−	309.674i		−	1.31776i
$236$		0			0
$237$	67.9759			0.286818
$238$		0			0
$239$		−	122.493i		−	0.512524i	−0.966607	−	0.256262i	$-0.917509\pi$
$239$		−	122.493i		−	0.512524i	0.966607	−	0.256262i	$-0.0824910\pi$
$240$		0			0
$241$			159.428i			0.661527i	0.943714	+	0.330764i	$0.107306\pi$
$241$			159.428i			0.661527i	−0.943714	+	0.330764i	$0.892694\pi$
$242$		0			0
$243$		−	15.5885i		−	0.0641500i
$244$		0			0
$245$			458.733i			1.87238i
$246$		0			0
$247$	−256.630			−1.03899
$248$		0			0
$249$			105.460i			0.423534i
$250$		0			0
$251$		−	180.790i		−	0.720278i	−0.932899	−	0.360139i	$-0.882729\pi$
$251$		−	180.790i		−	0.720278i	0.932899	−	0.360139i	$-0.117271\pi$
$252$		0			0
$253$	−50.9650	−	385.938i	−0.201443	−	1.52545i
$254$		0			0
$255$			221.477i			0.868539i
$256$		0			0
$257$	227.534			0.885348			0.442674	−	0.896683i	$-0.354030\pi$
$257$	227.534			0.885348			0.442674	+	0.896683i	$0.354030\pi$
$258$		0			0
$259$	111.012			0.428617
$260$		0			0
$261$	39.7513			0.152304
$262$		0			0
$263$		−	379.305i		−	1.44222i	−0.692818	−	0.721112i	$-0.743631\pi$
$263$		−	379.305i		−	1.44222i	0.692818	−	0.721112i	$-0.256369\pi$
$264$		0			0
$265$	−105.623			−0.398578
$266$		0			0
$267$		−	43.7261i		−	0.163768i
$268$		0			0
$269$			412.994i			1.53529i	0.640874	+	0.767646i	$0.278572\pi$
$269$			412.994i			1.53529i	−0.640874	+	0.767646i	$0.721428\pi$
$270$		0			0
$271$			297.160i			1.09653i	0.836305	+	0.548265i	$0.184711\pi$
$271$			297.160i			1.09653i	−0.836305	+	0.548265i	$0.815289\pi$
$272$		0			0
$273$	−423.675			−1.55192
$274$		0			0
$275$	−1.98517	−	15.0329i	−0.00721879	−	0.0546651i
$276$		0			0
$277$	−84.8580			−0.306347			−0.153173	−	0.988199i	$-0.548949\pi$
$277$	−84.8580			−0.306347			−0.153173	+	0.988199i	$0.548949\pi$
$278$		0			0
$279$	−59.0023			−0.211478
$280$		0			0
$281$		−	376.397i		−	1.33949i	−0.742591	−	0.669745i	$-0.766404\pi$
$281$		−	376.397i		−	1.33949i	0.742591	−	0.669745i	$-0.233596\pi$
$282$		0			0
$283$	22.1706			0.0783415			0.0391708	−	0.999233i	$-0.487528\pi$
$283$	22.1706			0.0783415			0.0391708	+	0.999233i	$0.487528\pi$
$284$		0			0
$285$	105.755			0.371071
$286$		0			0
$287$	164.776			0.574133
$288$		0			0
$289$	−403.197			−1.39515
$290$		0			0
$291$		−	186.571i		−	0.641136i
$292$		0			0
$293$	63.7547			0.217593			0.108796	−	0.994064i	$-0.465300\pi$
$293$	63.7547			0.217593			0.108796	+	0.994064i	$0.465300\pi$
$294$		0			0
$295$	325.005			1.10171
$296$		0			0
$297$	56.6657	−	7.48299i	0.190794	−	0.0251952i
$298$		0			0
$299$	722.933			2.41784
$300$		0			0
$301$		−	276.556i		−	0.918789i
$302$		0			0
$303$		−	141.814i		−	0.468032i
$304$		0			0
$305$			247.859i			0.812653i
$306$		0			0
$307$	162.646			0.529793			0.264896	−	0.964277i	$-0.414662\pi$
$307$	162.646			0.529793			0.264896	+	0.964277i	$0.414662\pi$
$308$		0			0
$309$		−	305.762i		−	0.989522i
$310$		0			0
$311$	116.979			0.376139			0.188069	−	0.982156i	$-0.439777\pi$
$311$	116.979			0.376139			0.188069	+	0.982156i	$0.439777\pi$
$312$		0			0
$313$	270.622			0.864606			0.432303	−	0.901729i	$-0.357701\pi$
$313$	270.622			0.864606			0.432303	+	0.901729i	$0.357701\pi$
$314$		0			0
$315$	174.593			0.554265
$316$		0			0
$317$			32.6056i			0.102857i	0.998677	+	0.0514284i	$0.0163774\pi$
$317$			32.6056i			0.102857i	−0.998677	+	0.0514284i	$0.983623\pi$
$318$		0			0
$319$	19.0820	+	144.500i	0.0598181	+	0.452979i
$320$		0			0
$321$			50.6154i			0.157680i
$322$		0			0
$323$			330.524i			1.02329i
$324$		0			0
$325$	28.1594			0.0866443
$326$		0			0
$327$			51.6306i			0.157892i
$328$		0			0
$329$		−	762.964i		−	2.31904i
$330$		0			0
$331$			450.105i			1.35983i	0.733290	+	0.679917i	$0.237984\pi$
$331$			450.105i			1.35983i	−0.733290	+	0.679917i	$0.762016\pi$
$332$		0			0
$333$		−	27.8123i		−	0.0835205i
$334$		0			0
$335$	426.039			1.27176
$336$		0			0
$337$		−	300.954i		−	0.893039i	−0.894774	−	0.446519i	$-0.852663\pi$
$337$		−	300.954i		−	0.893039i	0.894774	−	0.446519i	$-0.147337\pi$
$338$		0			0
$339$		−	223.259i		−	0.658582i
$340$		0			0
$341$	−28.3231	−	214.480i	−0.0830590	−	0.628973i
$342$		0			0
$343$			543.466i			1.58445i
$344$		0			0
$345$	−297.916			−0.863523
$346$		0			0
$347$	110.465			0.318342			0.159171	−	0.987251i	$-0.449118\pi$
$347$	110.465			0.318342			0.159171	+	0.987251i	$0.449118\pi$
$348$		0			0
$349$	396.482			1.13605			0.568026	−	0.823010i	$-0.307707\pi$
$349$	396.482			1.13605			0.568026	+	0.823010i	$0.307707\pi$
$350$		0			0
$351$			106.145i			0.302409i
$352$		0			0
$353$	68.7998			0.194900			0.0974502	−	0.995240i	$-0.468931\pi$
$353$	68.7998			0.194900			0.0974502	+	0.995240i	$0.468931\pi$
$354$		0			0
$355$			63.8870i			0.179963i
$356$		0			0
$357$			545.668i			1.52848i
$358$		0			0
$359$		−	289.584i		−	0.806642i	−0.915059	−	0.403321i	$-0.867856\pi$
$359$		−	289.584i		−	0.806642i	0.915059	−	0.403321i	$-0.132144\pi$
$360$		0			0
$361$	−203.175			−0.562812
$362$		0			0
$363$	54.4029	+	202.394i	0.149870	+	0.557559i
$364$		0			0
$365$	232.127			0.635965
$366$		0			0
$367$	177.690			0.484169			0.242084	−	0.970255i	$-0.422169\pi$
$367$	177.690			0.484169			0.242084	+	0.970255i	$0.422169\pi$
$368$		0			0
$369$		−	41.2822i		−	0.111876i
$370$		0			0
$371$	−260.231			−0.701431
$372$		0			0
$373$	250.697			0.672109			0.336055	−	0.941843i	$-0.390907\pi$
$373$	250.697			0.672109			0.336055	+	0.941843i	$0.390907\pi$
$374$		0			0
$375$	−222.057			−0.592152
$376$		0			0
$377$	−270.676			−0.717973
$378$		0			0
$379$			76.5238i			0.201910i	0.994891	+	0.100955i	$0.0321898\pi$
$379$			76.5238i			0.201910i	−0.994891	+	0.100955i	$0.967810\pi$
$380$		0			0
$381$	−153.927			−0.404008
$382$		0			0
$383$	276.986			0.723201			0.361600	−	0.932333i	$-0.382230\pi$
$383$	276.986			0.723201			0.361600	+	0.932333i	$0.382230\pi$
$384$		0			0
$385$	83.8108	+	634.666i	0.217690	+	1.64848i
$386$		0			0
$387$	−69.2868			−0.179036
$388$		0			0
$389$			199.141i			0.511929i	0.966686	+	0.255965i	$0.0823931\pi$
$389$			199.141i			0.511929i	−0.966686	+	0.255965i	$0.917607\pi$
$390$		0			0
$391$		−	931.094i		−	2.38131i
$392$		0			0
$393$			8.38322i			0.0213314i
$394$		0			0
$395$	190.743			0.482893
$396$		0			0
$397$			325.529i			0.819972i	0.912092	+	0.409986i	$0.134466\pi$
$397$			325.529i			0.819972i	−0.912092	+	0.409986i	$0.865534\pi$
$398$		0			0
$399$	260.556			0.653023
$400$		0			0
$401$	−187.031			−0.466411			−0.233205	−	0.972428i	$-0.574921\pi$
$401$	−187.031			−0.466411			−0.233205	+	0.972428i	$0.574921\pi$
$402$		0			0
$403$	401.760			0.996924
$404$		0			0
$405$		−	43.7418i		−	0.108004i
$406$		0			0
$407$	101.101	−	13.3509i	0.248405	−	0.0328031i
$408$		0			0
$409$			495.269i			1.21093i	0.795874	+	0.605463i	$0.207012\pi$
$409$			495.269i			1.21093i	−0.795874	+	0.605463i	$0.792988\pi$
$410$		0			0
$411$			321.964i			0.783368i
$412$		0			0
$413$	800.737			1.93883
$414$		0			0
$415$			295.925i			0.713071i
$416$		0			0
$417$			390.312i			0.936001i
$418$		0			0
$419$			390.401i			0.931745i	0.884852	+	0.465872i	$0.154259\pi$
$419$			390.401i			0.931745i	−0.884852	+	0.465872i	$0.845741\pi$
$420$		0			0
$421$		−	87.5019i		−	0.207843i	−0.994585	−	0.103922i	$-0.966861\pi$
$421$		−	87.5019i		−	0.207843i	0.994585	−	0.103922i	$-0.0331391\pi$
$422$		0			0
$423$	−191.149			−0.451889
$424$		0			0
$425$		−	36.2676i		−	0.0853356i
$426$		0			0
$427$			610.667i			1.43013i
$428$		0			0
$429$	−385.850	+	50.9534i	−0.899417	+	0.118772i
$430$		0			0
$431$			165.880i			0.384873i	0.981309	+	0.192436i	$0.0616389\pi$
$431$			165.880i			0.384873i	−0.981309	+	0.192436i	$0.938361\pi$
$432$		0			0
$433$	−639.331			−1.47651			−0.738257	−	0.674519i	$-0.764351\pi$
$433$	−639.331			−1.47651			−0.738257	+	0.674519i	$0.764351\pi$
$434$		0			0
$435$	111.544			0.256422
$436$		0			0
$437$	−444.597			−1.01738
$438$		0			0
$439$			242.448i			0.552274i	0.961118	+	0.276137i	$0.0890544\pi$
$439$			242.448i			0.552274i	−0.961118	+	0.276137i	$0.910946\pi$
$440$		0			0
$441$	283.157			0.642080
$442$		0			0
$443$		−	704.372i		−	1.59000i	−0.606606	−	0.795002i	$-0.707470\pi$
$443$		−	704.372i		−	1.59000i	0.606606	−	0.795002i	$-0.292530\pi$
$444$		0			0
$445$		−	122.697i		−	0.275724i
$446$		0			0
$447$			289.782i			0.648282i
$448$		0			0
$449$	608.936			1.35621			0.678103	−	0.734967i	$-0.262802\pi$
$449$	608.936			1.35621			0.678103	+	0.734967i	$0.262802\pi$
$450$		0			0
$451$	150.065	−	19.8168i	0.332739	−	0.0439398i
$452$		0			0
$453$	349.211			0.770885
$454$		0			0
$455$	−1188.85			−2.61285
$456$		0			0
$457$		−	234.175i		−	0.512417i	−0.966621	−	0.256209i	$-0.917527\pi$
$457$		−	234.175i		−	0.512417i	0.966621	−	0.256209i	$-0.0824734\pi$
$458$		0			0
$459$	136.709			0.297841
$460$		0			0
$461$	−238.716			−0.517822			−0.258911	−	0.965901i	$-0.583364\pi$
$461$	−238.716			−0.517822			−0.258911	+	0.965901i	$0.583364\pi$
$462$		0			0
$463$	42.6873			0.0921972			0.0460986	−	0.998937i	$-0.485321\pi$
$463$	42.6873			0.0921972			0.0460986	+	0.998937i	$0.485321\pi$
$464$		0			0
$465$	−165.563			−0.356049
$466$		0			0
$467$		−	604.273i		−	1.29395i	−0.762513	−	0.646973i	$-0.776035\pi$
$467$		−	604.273i		−	1.29395i	0.762513	−	0.646973i	$-0.223965\pi$
$468$		0			0
$469$	1049.66			2.23808
$470$		0			0
$471$	525.077			1.11481
$472$		0			0
$473$	−33.2600	−	251.865i	−0.0703172	−	0.532484i
$474$		0			0
$475$	−17.3178			−0.0364585
$476$		0			0
$477$			65.1969i			0.136681i
$478$		0			0
$479$		−	295.948i		−	0.617845i	−0.951087	−	0.308923i	$-0.900032\pi$
$479$		−	295.948i		−	0.617845i	0.951087	−	0.308923i	$-0.0999684\pi$
$480$		0			0
$481$			189.381i			0.393722i
$482$		0			0
$483$	−733.994			−1.51966
$484$		0			0
$485$		−	523.524i		−	1.07943i
$486$		0			0
$487$	−797.217			−1.63700			−0.818498	−	0.574510i	$-0.805193\pi$
$487$	−797.217			−1.63700			−0.818498	+	0.574510i	$0.805193\pi$
$488$		0			0
$489$	227.163			0.464546
$490$		0			0
$491$	−446.042			−0.908436			−0.454218	−	0.890891i	$-0.650081\pi$
$491$	−446.042			−0.908436			−0.454218	+	0.890891i	$0.650081\pi$
$492$		0			0
$493$			348.614i			0.707129i
$494$		0			0
$495$	159.006	−	20.9975i	0.321224	−	0.0424192i
$496$		0			0
$497$			157.403i			0.316705i
$498$		0			0
$499$		−	536.083i		−	1.07432i	−0.843482	−	0.537158i	$-0.819498\pi$
$499$		−	536.083i		−	1.07432i	0.843482	−	0.537158i	$-0.180502\pi$
$500$		0			0
$501$	−361.842			−0.722240
$502$		0			0
$503$			955.799i			1.90020i	0.311950	+	0.950099i	$0.399018\pi$
$503$			955.799i			1.90020i	−0.311950	+	0.950099i	$0.600982\pi$
$504$		0			0
$505$		−	397.934i		−	0.787989i
$506$		0			0
$507$		−	430.052i		−	0.848228i
$508$		0			0
$509$		−	310.852i		−	0.610710i	−0.952239	−	0.305355i	$-0.901225\pi$
$509$		−	310.852i		−	0.610710i	0.952239	−	0.305355i	$-0.0987752\pi$
$510$		0			0
$511$	571.907			1.11919
$512$		0			0
$513$		−	65.2784i		−	0.127248i
$514$		0			0
$515$		−	857.980i		−	1.66598i
$516$		0			0
$517$	−91.7580	−	694.848i	−0.177482	−	1.34400i
$518$		0			0
$519$		−	118.979i		−	0.229247i
$520$		0			0
$521$	−677.923			−1.30120			−0.650598	−	0.759422i	$-0.725482\pi$
$521$	−677.923			−1.30120			−0.650598	+	0.759422i	$0.725482\pi$
$522$		0			0
$523$	−673.578			−1.28791			−0.643956	−	0.765062i	$-0.722708\pi$
$523$	−673.578			−1.28791			−0.643956	+	0.765062i	$0.722708\pi$
$524$		0			0
$525$	−28.5903			−0.0544576
$526$		0			0
$527$		−	517.443i		−	0.981866i
$528$		0			0
$529$	723.442			1.36756
$530$		0			0
$531$		−	200.613i		−	0.377801i
$532$		0			0
$533$			281.100i			0.527392i
$534$		0			0
$535$			142.029i			0.265474i
$536$		0			0
$537$	171.204			0.318815
$538$		0			0
$539$	135.925	+	1029.31i	0.252180	+	1.90966i
$540$		0			0
$541$	140.980			0.260592			0.130296	−	0.991475i	$-0.458407\pi$
$541$	140.980			0.260592			0.130296	+	0.991475i	$0.458407\pi$
$542$		0			0
$543$	−137.211			−0.252690
$544$		0			0
$545$			144.877i			0.265830i
$546$		0			0
$547$	116.001			0.212067			0.106033	−	0.994363i	$-0.466185\pi$
$547$	116.001			0.212067			0.106033	+	0.994363i	$0.466185\pi$
$548$		0			0
$549$	152.993			0.278676
$550$		0			0
$551$	166.463			0.302111
$552$		0			0
$553$	469.945			0.849811
$554$		0			0
$555$		−	78.0424i		−	0.140617i
$556$		0			0
$557$	−300.194			−0.538947			−0.269474	−	0.963008i	$-0.586850\pi$
$557$	−300.194			−0.538947			−0.269474	+	0.963008i	$0.586850\pi$
$558$		0			0
$559$	471.790			0.843989
$560$		0			0
$561$	65.6249	+	496.952i	0.116978	+	0.885832i
$562$		0			0
$563$	−1050.47			−1.86584			−0.932920	−	0.360085i	$-0.882748\pi$
$563$	−1050.47			−1.86584			−0.932920	+	0.360085i	$0.882748\pi$
$564$		0			0
$565$		−	626.473i		−	1.10880i
$566$		0			0
$567$		−	107.769i		−	0.190070i
$568$		0			0
$569$		−	354.672i		−	0.623326i	−0.950193	−	0.311663i	$-0.899114\pi$
$569$		−	354.672i		−	0.623326i	0.950193	−	0.311663i	$-0.100886\pi$
$570$		0			0
$571$	752.067			1.31711			0.658553	−	0.752535i	$-0.271169\pi$
$571$	752.067			1.31711			0.658553	+	0.752535i	$0.271169\pi$
$572$		0			0
$573$		−	325.896i		−	0.568754i
$574$		0			0
$575$	48.7846			0.0848429
$576$		0			0
$577$	652.998			1.13171			0.565856	−	0.824504i	$-0.308546\pi$
$577$	652.998			1.13171			0.565856	+	0.824504i	$0.308546\pi$
$578$		0			0
$579$	264.611			0.457014
$580$		0			0
$581$			729.089i			1.25489i
$582$		0			0
$583$	−236.998	+	31.2967i	−0.406514	+	0.0536822i
$584$		0			0
$585$			297.848i			0.509141i
$586$		0			0
$587$			235.350i			0.400936i	0.979700	+	0.200468i	$0.0642463\pi$
$587$			235.350i			0.400936i	−0.979700	+	0.200468i	$0.935754\pi$
$588$		0			0
$589$	−247.079			−0.419489
$590$		0			0
$591$		−	464.299i		−	0.785617i
$592$		0			0
$593$		−	544.105i		−	0.917546i	−0.888554	−	0.458773i	$-0.848289\pi$
$593$		−	544.105i		−	0.917546i	0.888554	−	0.458773i	$-0.151711\pi$
$594$		0			0
$595$			1531.16i			2.57338i
$596$		0			0
$597$			544.919i			0.912762i
$598$		0			0
$599$	−15.9242			−0.0265846			−0.0132923	−	0.999912i	$-0.504231\pi$
$599$	−15.9242			−0.0265846			−0.0132923	+	0.999912i	$0.504231\pi$
$600$		0			0
$601$			1122.93i			1.86843i	0.356705	+	0.934217i	$0.383900\pi$
$601$			1122.93i			1.86843i	−0.356705	+	0.934217i	$0.616100\pi$
$602$		0			0
$603$		−	262.977i		−	0.436114i
$604$		0			0
$605$	152.657	+	567.925i	0.252325	+	0.938719i
$606$		0			0
$607$		−	191.642i		−	0.315720i	−0.987462	−	0.157860i	$-0.949541\pi$
$607$		−	191.642i		−	0.315720i	0.987462	−	0.157860i	$-0.0504595\pi$
$608$		0			0
$609$	274.817			0.451260
$610$		0			0
$611$	1301.58			2.13024
$612$		0			0
$613$	−830.770			−1.35525			−0.677626	−	0.735406i	$-0.736991\pi$
$613$	−830.770			−1.35525			−0.677626	+	0.735406i	$0.736991\pi$
$614$		0			0
$615$		−	115.839i		−	0.188356i
$616$		0			0
$617$	−708.354			−1.14806			−0.574031	−	0.818834i	$-0.694621\pi$
$617$	−708.354			−1.14806			−0.574031	+	0.818834i	$0.694621\pi$
$618$		0			0
$619$		−	235.751i		−	0.380858i	−0.981701	−	0.190429i	$-0.939012\pi$
$619$		−	235.751i		−	0.380858i	0.981701	−	0.190429i	$-0.0609878\pi$
$620$		0			0
$621$			183.891i			0.296121i
$622$		0			0
$623$		−	302.297i		−	0.485227i
$624$		0			0
$625$	−588.637			−0.941820
$626$		0			0
$627$	237.294	−	31.3358i	0.378459	−	0.0499774i
$628$		0			0
$629$	243.911			0.387775
$630$		0			0
$631$	−922.285			−1.46162			−0.730812	−	0.682579i	$-0.760858\pi$
$631$	−922.285			−1.46162			−0.730812	+	0.682579i	$0.760858\pi$
$632$		0			0
$633$		−	633.385i		−	1.00061i
$634$		0			0
$635$	−431.925			−0.680197
$636$		0			0
$637$	−1928.08			−3.02682
$638$		0			0
$639$	39.4348			0.0617133
$640$		0			0
$641$	−640.847			−0.999761			−0.499881	−	0.866094i	$-0.666623\pi$
$641$	−640.847			−0.999761			−0.499881	+	0.866094i	$0.666623\pi$
$642$		0			0
$643$		−	784.730i		−	1.22042i	−0.792240	−	0.610210i	$-0.791085\pi$
$643$		−	784.730i		−	1.22042i	0.792240	−	0.610210i	$-0.208915\pi$
$644$		0			0
$645$	−194.421			−0.301428
$646$		0			0
$647$	−926.595			−1.43214			−0.716070	−	0.698028i	$-0.754061\pi$
$647$	−926.595			−1.43214			−0.716070	+	0.698028i	$0.754061\pi$
$648$		0			0
$649$	729.248	−	96.3008i	1.12365	−	0.148383i
$650$		0			0
$651$	−407.907			−0.626586
$652$		0			0
$653$			325.852i			0.499007i	0.968374	+	0.249504i	$0.0802675\pi$
$653$			325.852i			0.499007i	−0.968374	+	0.249504i	$0.919733\pi$
$654$		0			0
$655$			23.5236i			0.0359139i
$656$		0			0
$657$		−	143.283i		−	0.218086i
$658$		0			0
$659$	−1131.91			−1.71761			−0.858805	−	0.512302i	$-0.828793\pi$
$659$	−1131.91			−1.71761			−0.858805	+	0.512302i	$0.828793\pi$
$660$		0			0
$661$		−	681.797i		−	1.03146i	−0.856750	−	0.515732i	$-0.827520\pi$
$661$		−	681.797i		−	1.03146i	0.856750	−	0.515732i	$-0.172480\pi$
$662$		0			0
$663$	−930.882			−1.40405
$664$		0			0
$665$	731.130			1.09944
$666$		0			0
$667$	−468.931			−0.703045
$668$		0			0
$669$		−	128.521i		−	0.192109i
$670$		0			0
$671$	73.4420	+	556.148i	0.109452	+	0.828834i
$672$		0			0
$673$		−	357.156i		−	0.530692i	−0.964153	−	0.265346i	$-0.914514\pi$
$673$		−	357.156i		−	0.530692i	0.964153	−	0.265346i	$-0.0854862\pi$
$674$		0			0
$675$			7.16286i			0.0106116i
$676$		0			0
$677$	−824.839			−1.21837			−0.609187	−	0.793027i	$-0.708504\pi$
$677$	−824.839			−1.21837			−0.609187	+	0.793027i	$0.708504\pi$
$678$		0			0
$679$		−	1289.84i		−	1.89962i
$680$		0			0
$681$		−	232.039i		−	0.340733i
$682$		0			0
$683$			710.849i			1.04078i	0.853930	+	0.520388i	$0.174213\pi$
$683$			710.849i			1.04078i	−0.853930	+	0.520388i	$0.825787\pi$
$684$		0			0
$685$			903.443i			1.31889i
$686$		0			0
$687$	286.876			0.417578
$688$		0			0
$689$		−	443.941i		−	0.644326i
$690$		0			0
$691$			237.467i			0.343657i	0.985127	+	0.171828i	$0.0549674\pi$
$691$			237.467i			0.343657i	−0.985127	+	0.171828i	$0.945033\pi$
$692$		0			0
$693$	391.753	−	51.7330i	0.565301	−	0.0746507i
$694$		0			0
$695$			1095.23i			1.57587i
$696$		0			0
$697$	362.040			0.519425
$698$		0			0
$699$	498.136			0.712641
$700$		0			0
$701$	−516.111			−0.736250			−0.368125	−	0.929776i	$-0.620000\pi$
$701$	−516.111			−0.736250			−0.368125	+	0.929776i	$0.620000\pi$
$702$		0			0
$703$		−	116.467i		−	0.165672i
$704$		0			0
$705$	−536.371			−0.760810
$706$		0			0
$707$		−	980.416i		−	1.38673i
$708$		0			0
$709$		−	160.225i		−	0.225988i	−0.993596	−	0.112994i	$-0.963956\pi$
$709$		−	160.225i		−	0.225988i	0.993596	−	0.112994i	$-0.0360440\pi$
$710$		0			0
$711$		−	117.738i		−	0.165595i
$712$		0			0
$713$	696.028			0.976196
$714$		0			0
$715$	−1082.71	+	142.977i	−1.51428	+	0.199968i
$716$		0			0
$717$	−212.164			−0.295906
$718$		0			0
$719$	140.837			0.195878			0.0979392	−	0.995192i	$-0.468775\pi$
$719$	140.837			0.195878			0.0979392	+	0.995192i	$0.468775\pi$
$720$		0			0
$721$		−	2113.86i		−	2.93184i
$722$		0			0
$723$	276.137			0.381933
$724$		0			0
$725$	−18.2656			−0.0251940
$726$		0			0
$727$	−525.474			−0.722798			−0.361399	−	0.932411i	$-0.617701\pi$
$727$	−525.474			−0.722798			−0.361399	+	0.932411i	$0.617701\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$		−	607.637i		−	0.831241i
$732$		0			0
$733$	537.444			0.733211			0.366606	−	0.930376i	$-0.380520\pi$
$733$	537.444			0.733211			0.366606	+	0.930376i	$0.380520\pi$
$734$		0			0
$735$	794.549			1.08102
$736$		0			0
$737$	955.948	−	126.238i	1.29708	−	0.171286i
$738$		0			0
$739$	−1006.41			−1.36186			−0.680928	−	0.732350i	$-0.738423\pi$
$739$	−1006.41			−1.36186			−0.680928	+	0.732350i	$0.738423\pi$
$740$		0			0
$741$			444.496i			0.599859i
$742$		0			0
$743$		−	556.837i		−	0.749445i	−0.927137	−	0.374722i	$-0.877738\pi$
$743$		−	556.837i		−	0.749445i	0.927137	−	0.374722i	$-0.122262\pi$
$744$		0			0
$745$			813.138i			1.09146i
$746$		0			0
$747$	182.662			0.244528
$748$		0			0
$749$			349.925i			0.467190i
$750$		0			0
$751$	768.649			1.02350			0.511750	−	0.859134i	$-0.328997\pi$
$751$	768.649			1.02350			0.511750	+	0.859134i	$0.328997\pi$
$752$		0			0
$753$	−313.137			−0.415853
$754$		0			0
$755$	979.899			1.29788
$756$		0			0
$757$		−	532.098i		−	0.702903i	−0.936206	−	0.351452i	$-0.885688\pi$
$757$		−	532.098i		−	0.702903i	0.936206	−	0.351452i	$-0.114312\pi$
$758$		0			0
$759$	−668.464	+	88.2739i	−0.880717	+	0.116303i
$760$		0			0
$761$			1136.37i			1.49326i	0.665242	+	0.746628i	$0.268328\pi$
$761$			1136.37i			1.49326i	−0.665242	+	0.746628i	$0.731672\pi$
$762$		0			0
$763$			356.944i			0.467816i
$764$		0			0
$765$	383.610			0.501451
$766$		0			0
$767$			1366.02i			1.78099i
$768$		0			0
$769$		−	874.120i		−	1.13670i	−0.822788	−	0.568349i	$-0.807582\pi$
$769$		−	874.120i		−	1.13670i	0.822788	−	0.568349i	$-0.192418\pi$
$770$		0			0
$771$		−	394.101i		−	0.511156i
$772$		0			0
$773$		−	417.756i		−	0.540434i	−0.962799	−	0.270217i	$-0.912905\pi$
$773$		−	417.756i		−	0.540434i	0.962799	−	0.270217i	$-0.0870955\pi$
$774$		0			0
$775$	27.1114			0.0349825
$776$		0			0
$777$		−	192.278i		−	0.247462i
$778$		0			0
$779$		−	172.874i		−	0.221917i
$780$		0			0
$781$	18.9300	+	143.350i	0.0242382	+	0.183547i
$782$		0			0
$783$		−	68.8513i		−	0.0879327i
$784$		0			0
$785$	1473.39			1.87692
$786$		0			0
$787$	−847.047			−1.07630			−0.538150	−	0.842849i	$-0.680876\pi$
$787$	−847.047			−1.07630			−0.538150	+	0.842849i	$0.680876\pi$
$788$		0			0
$789$	−656.976			−0.832669
$790$		0			0
$791$		−	1543.48i		−	1.95131i
$792$		0			0
$793$	−1041.77			−1.31370
$794$		0			0
$795$			182.945i			0.230119i
$796$		0			0
$797$		−	847.736i		−	1.06366i	−0.846852	−	0.531829i	$-0.821505\pi$
$797$		−	847.736i		−	1.06366i	0.846852	−	0.531829i	$-0.178495\pi$
$798$		0			0
$799$		−	1676.35i		−	2.09807i
$800$		0			0
$801$	−75.7358			−0.0945516
$802$		0			0
$803$	520.848	−	68.7805i	0.648628	−	0.0856545i
$804$		0			0
$805$	−2059.61			−2.55853
$806$		0			0
$807$	715.326			0.886402
$808$		0			0
$809$		−	0.607771i		−	0.000751262i	−1.00000		0.000375631i	$-0.999880\pi$
$809$		−	0.607771i		−	0.000751262i	1.00000		0.000375631i	$-0.000119567\pi$
$810$		0			0
$811$	146.430			0.180554			0.0902772	−	0.995917i	$-0.471225\pi$
$811$	146.430			0.180554			0.0902772	+	0.995917i	$0.471225\pi$
$812$		0			0
$813$	514.695			0.633082
$814$		0			0
$815$	637.427			0.782119
$816$		0			0
$817$	−290.146			−0.355136
$818$		0			0
$819$			733.827i			0.896003i
$820$		0			0
$821$	−440.937			−0.537073			−0.268537	−	0.963269i	$-0.586540\pi$
$821$	−440.937			−0.537073			−0.268537	+	0.963269i	$0.586540\pi$
$822$		0			0
$823$	−960.312			−1.16684			−0.583421	−	0.812170i	$-0.698286\pi$
$823$	−960.312			−1.16684			−0.583421	+	0.812170i	$0.698286\pi$
$824$		0			0
$825$	−26.0378	+	3.43841i	−0.0315609	+	0.00416777i
$826$		0			0
$827$	−215.542			−0.260631			−0.130316	−	0.991473i	$-0.541599\pi$
$827$	−215.542			−0.260631			−0.130316	+	0.991473i	$0.541599\pi$
$828$		0			0
$829$		−	745.907i		−	0.899767i	−0.893087	−	0.449883i	$-0.851466\pi$
$829$		−	745.907i		−	0.899767i	0.893087	−	0.449883i	$-0.148534\pi$
$830$		0			0
$831$			146.978i			0.176869i
$832$		0			0
$833$			2483.25i			2.98110i
$834$		0			0
$835$	−1015.34			−1.21598
$836$		0			0
$837$			102.195i			0.122097i
$838$		0			0
$839$	874.084			1.04182			0.520908	−	0.853613i	$-0.325593\pi$
$839$	874.084			1.04182			0.520908	+	0.853613i	$0.325593\pi$
$840$		0			0
$841$	−665.426			−0.791232
$842$		0			0
$843$	−651.938			−0.773355
$844$		0			0
$845$		−	1206.74i		−	1.42810i
$846$		0			0
$847$	376.110	+	1399.23i	0.444049	+	1.65199i
$848$		0			0
$849$		−	38.4007i		−	0.0452305i
$850$		0			0
$851$			328.091i			0.385536i
$852$		0			0
$853$	1467.81			1.72076			0.860382	−	0.509650i	$-0.170225\pi$
$853$	1467.81			1.72076			0.860382	+	0.509650i	$0.170225\pi$
$854$		0			0
$855$		−	183.174i		−	0.214238i
$856$		0			0
$857$			967.730i			1.12921i	0.825362	+	0.564603i	$0.190971\pi$
$857$			967.730i			1.12921i	−0.825362	+	0.564603i	$0.809029\pi$
$858$		0			0
$859$		−	481.940i		−	0.561047i	−0.959847	−	0.280524i	$-0.909492\pi$
$859$		−	481.940i		−	0.561047i	0.959847	−	0.280524i	$-0.0905081\pi$
$860$		0			0
$861$		−	285.401i		−	0.331476i
$862$		0			0
$863$	762.815			0.883910			0.441955	−	0.897037i	$-0.354285\pi$
$863$	762.815			0.883910			0.441955	+	0.897037i	$0.354285\pi$
$864$		0			0
$865$		−	333.860i		−	0.385965i
$866$		0			0
$867$			698.358i			0.805487i
$868$		0			0
$869$	427.989	−	56.5181i	0.492508	−	0.0650381i
$870$		0			0
$871$			1790.67i			2.05588i
$872$		0			0
$873$	−323.150			−0.370160
$874$		0			0
$875$	−1535.17			−1.75448
$876$		0			0
$877$	1184.53			1.35066			0.675330	−	0.737515i	$-0.264001\pi$
$877$	1184.53			1.35066			0.675330	+	0.737515i	$0.264001\pi$
$878$		0			0
$879$		−	110.426i		−	0.125627i
$880$		0			0
$881$	−1031.60			−1.17094			−0.585471	−	0.810693i	$-0.699090\pi$
$881$	−1031.60			−1.17094			−0.585471	+	0.810693i	$0.699090\pi$
$882$		0			0
$883$		−	773.139i		−	0.875582i	−0.899077	−	0.437791i	$-0.855761\pi$
$883$		−	773.139i		−	0.875582i	0.899077	−	0.437791i	$-0.144239\pi$
$884$		0			0
$885$		−	562.926i		−	0.636075i
$886$		0			0
$887$		−	379.167i		−	0.427471i	−0.976892	−	0.213736i	$-0.931437\pi$
$887$		−	379.167i		−	0.427471i	0.976892	−	0.213736i	$-0.0685631\pi$
$888$		0			0
$889$	−1064.16			−1.19703
$890$		0			0
$891$	−12.9609	−	98.1479i	−0.0145465	−	0.110155i
$892$		0			0
$893$	−800.458			−0.896370
$894$		0			0
$895$	480.404			0.536764
$896$		0			0
$897$		−	1252.16i		−	1.39594i
$898$		0			0
$899$	−260.602			−0.289880
$900$		0			0
$901$	−571.769			−0.634594
$902$		0			0
$903$	−479.008			−0.530463
$904$		0			0
$905$	−385.018			−0.425434
$906$		0			0
$907$		−	404.375i		−	0.445837i	−0.974837	−	0.222919i	$-0.928442\pi$
$907$		−	404.375i		−	0.445837i	0.974837	−	0.222919i	$-0.0715585\pi$
$908$		0			0
$909$	−245.629			−0.270218
$910$		0			0
$911$	−693.987			−0.761786			−0.380893	−	0.924619i	$-0.624383\pi$
$911$	−693.987			−0.761786			−0.380893	+	0.924619i	$0.624383\pi$
$912$		0			0
$913$	87.6840	+	663.997i	0.0960394	+	0.727269i
$914$		0			0
$915$	429.305			0.469186
$916$		0			0
$917$			57.9567i			0.0632025i
$918$		0			0
$919$		−	166.808i		−	0.181511i	−0.995873	−	0.0907554i	$-0.971072\pi$
$919$		−	166.808i		−	0.181511i	0.995873	−	0.0907554i	$-0.0289281\pi$
$920$		0			0
$921$		−	281.712i		−	0.305876i
$922$		0			0
$923$	−268.521			−0.290922
$924$		0			0
$925$			12.7797i			0.0138159i
$926$		0			0
$927$	−529.596			−0.571301
$928$		0			0
$929$	−1163.16			−1.25205			−0.626027	−	0.779802i	$-0.715320\pi$
$929$	−1163.16			−1.25205			−0.626027	+	0.779802i	$0.715320\pi$
$930$		0			0
$931$	1185.75			1.27363
$932$		0			0
$933$		−	202.614i		−	0.217164i
$934$		0			0
$935$	184.146	+	1394.46i	0.196947	+	1.49141i
$936$		0			0
$937$			1454.88i			1.55270i	0.630301	+	0.776351i	$0.282931\pi$
$937$			1454.88i			1.55270i	−0.630301	+	0.776351i	$0.717069\pi$
$938$		0			0
$939$		−	468.730i		−	0.499180i
$940$		0			0
$941$	−270.375			−0.287328			−0.143664	−	0.989627i	$-0.545888\pi$
$941$	−270.375			−0.287328			−0.143664	+	0.989627i	$0.545888\pi$
$942$		0			0
$943$			486.990i			0.516426i
$944$		0			0
$945$		−	302.405i		−	0.320005i
$946$		0			0
$947$		−	362.572i		−	0.382864i	−0.981506	−	0.191432i	$-0.938687\pi$
$947$		−	362.572i		−	0.382864i	0.981506	−	0.191432i	$-0.0613131\pi$
$948$		0			0
$949$			975.645i			1.02808i
$950$		0			0
$951$	56.4746			0.0593844
$952$		0			0
$953$			1341.52i			1.40768i	0.710360	+	0.703839i	$0.248532\pi$
$953$			1341.52i			1.40768i	−0.710360	+	0.703839i	$0.751468\pi$
$954$		0			0
$955$		−	914.476i		−	0.957566i
$956$		0			0
$957$	250.282	−	33.0510i	0.261528	−	0.0345360i
$958$		0			0
$959$			2225.87i			2.32103i
$960$		0			0
$961$	−574.192			−0.597494
$962$		0			0
$963$	87.6685			0.0910369
$964$		0			0
$965$	742.508			0.769439
$966$		0			0
$967$		−	1394.99i		−	1.44259i	−0.692627	−	0.721296i	$-0.743547\pi$
$967$		−	1394.99i		−	1.44259i	0.692627	−	0.721296i	$-0.256453\pi$
$968$		0			0
$969$	572.484			0.590799
$970$		0			0
$971$			827.553i			0.852269i	0.904660	+	0.426134i	$0.140125\pi$
$971$			827.553i			0.852269i	−0.904660	+	0.426134i	$0.859875\pi$
$972$		0			0
$973$			2698.39i			2.77327i
$974$		0			0
$975$		−	48.7735i		−	0.0500241i
$976$		0			0
$977$	564.845			0.578142			0.289071	−	0.957308i	$-0.406654\pi$
$977$	564.845			0.578142			0.289071	+	0.957308i	$0.406654\pi$
$978$		0			0
$979$	−36.3558	−	275.308i	−0.0371356	−	0.281213i
$980$		0			0
$981$	89.4269			0.0911589
$982$		0			0
$983$	540.105			0.549445			0.274723	−	0.961524i	$-0.411414\pi$
$983$	540.105			0.549445			0.274723	+	0.961524i	$0.411414\pi$
$984$		0			0
$985$		−	1302.84i		−	1.32268i
$986$		0			0
$987$	−1321.49			−1.33890
$988$		0			0
$989$	817.350			0.826441
$990$		0			0
$991$	1701.26			1.71671			0.858357	−	0.513053i	$-0.171486\pi$
$991$	1701.26			1.71671			0.858357	+	0.513053i	$0.171486\pi$
$992$		0			0
$993$	779.604			0.785100
$994$		0			0
$995$			1529.06i			1.53675i
$996$		0			0
$997$	251.928			0.252686			0.126343	−	0.991987i	$-0.459676\pi$
$997$	251.928			0.252686			0.126343	+	0.991987i	$0.459676\pi$
$998$		0			0
$999$	−48.1724			−0.0482206

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1056.3.e.a.241.7		48
4.3	odd	2		264.3.e.a.109.47	yes	48
8.3	odd	2		264.3.e.a.109.1	✓	48
8.5	even	2	inner	1056.3.e.a.241.41		48
11.10	odd	2	inner	1056.3.e.a.241.8		48
44.43	even	2		264.3.e.a.109.2	yes	48
88.21	odd	2	inner	1056.3.e.a.241.42		48
88.43	even	2		264.3.e.a.109.48	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.3.e.a.109.1	✓	48	8.3	odd	2
264.3.e.a.109.2	yes	48	44.43	even	2
264.3.e.a.109.47	yes	48	4.3	odd	2
264.3.e.a.109.48	yes	48	88.43	even	2
1056.3.e.a.241.7		48	1.1	even	1	trivial
1056.3.e.a.241.8		48	11.10	odd	2	inner
1056.3.e.a.241.41		48	8.5	even	2	inner
1056.3.e.a.241.42		48	88.21	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1056 = 2^{5} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1056.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -4.86020i q^{5} -11.9744i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} -4.86020i q^{5} -11.9744i q^{7} -3.00000 q^{9} +(-1.44010 - 10.9053i) q^{11} +20.4277 q^{13} -8.41811 q^{15} -26.3096i q^{17} -12.5628 q^{19} -20.7402 q^{21} +35.3899 q^{23} +1.37849 q^{25} +5.19615i q^{27} -13.2504 q^{29} +19.6674 q^{31} +(-18.8886 + 2.49433i) q^{33} -58.1978 q^{35} +9.27077i q^{37} -35.3818i q^{39} +13.7607i q^{41} +23.0956 q^{43} +14.5806i q^{45} +63.7164 q^{47} -94.3858 q^{49} -45.5696 q^{51} -21.7323i q^{53} +(-53.0020 + 6.99917i) q^{55} +21.7595i q^{57} +66.8708i q^{59} -50.9978 q^{61} +35.9231i q^{63} -99.2826i q^{65} +87.6588i q^{67} -61.2970i q^{69} -13.1449 q^{71} +47.7609i q^{73} -2.38762i q^{75} +(-130.584 + 17.2443i) q^{77} +39.2459i q^{79} +9.00000 q^{81} -60.8874 q^{83} -127.870 q^{85} +22.9504i q^{87} +25.2453 q^{89} -244.609i q^{91} -34.0650i q^{93} +61.0579i q^{95} +107.717 q^{97} +(4.32030 + 32.7160i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 144 q^{9}+O(q^{10}) \) 48 * q - 144 * q^9	\( 48 q - 144 q^{9} + 128 q^{23} - 240 q^{25} - 336 q^{49} + 128 q^{55} - 512 q^{71} + 432 q^{81} - 32 q^{97}+O(q^{100}) \) 48 * q - 144 * q^9 + 128 * q^23 - 240 * q^25 - 336 * q^49 + 128 * q^55 - 512 * q^71 + 432 * q^81 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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