LMFDB - Embedded newform 1216.3.g.e.417.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(417,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1216.417");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.g (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:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.10
Character	$\chi$	$=$	1216.417
Dual form			1216.3.g.e.417.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.58612 q^{3} -0.323607i q^{5} +8.09640 q^{7} +12.0325 q^{9} +O(q^{10})$	$q-4.58612 q^{3} -0.323607i q^{5} +8.09640 q^{7} +12.0325 q^{9} +15.7627i q^{11} -2.70240 q^{13} +1.48410i q^{15} -23.8912 q^{17} +(12.2592 + 14.5160i) q^{19} -37.1310 q^{21} +28.7327 q^{23} +24.8953 q^{25} -13.9072 q^{27} -51.7916 q^{29} -53.3446i q^{31} -72.2897i q^{33} -2.62006i q^{35} +30.8820 q^{37} +12.3935 q^{39} +41.0208i q^{41} +1.24753i q^{43} -3.89379i q^{45} +1.82806 q^{47} +16.5517 q^{49} +109.568 q^{51} +10.5509 q^{53} +5.10094 q^{55} +(-56.2219 - 66.5719i) q^{57} -13.8763 q^{59} -96.8096i q^{61} +97.4197 q^{63} +0.874518i q^{65} -22.4624 q^{67} -131.772 q^{69} +88.9010i q^{71} -34.3165 q^{73} -114.173 q^{75} +127.621i q^{77} +113.440i q^{79} -44.5120 q^{81} +65.8732i q^{83} +7.73138i q^{85} +237.522 q^{87} -120.371i q^{89} -21.8797 q^{91} +244.645i q^{93} +(4.69747 - 3.96715i) q^{95} +88.2932i q^{97} +189.665i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 264 q^{9}+O(q^{10}) $ 48 * q + 264 * q^9	$ 48 q + 264 q^{9} + 24 q^{17} - 160 q^{25} + 328 q^{49} + 8 q^{57} - 472 q^{73} - 736 q^{81}+O(q^{100}) $ 48 * q + 264 * q^9 + 24 * q^17 - 160 * q^25 + 328 * q^49 + 8 * q^57 - 472 * q^73 - 736 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−4.58612			−1.52871			−0.764353	−	0.644798i	$-0.776941\pi$
$3$	−4.58612			−1.52871			−0.764353	+	0.644798i	$0.776941\pi$
$4$		0			0
$5$		−	0.323607i		−	0.0647215i	−0.999476	−	0.0323607i	$-0.989697\pi$
$5$		−	0.323607i		−	0.0647215i	0.999476	−	0.0323607i	$-0.0103025\pi$
$6$		0			0
$7$	8.09640			1.15663			0.578314	−	0.815814i	$-0.303711\pi$
$7$	8.09640			1.15663			0.578314	+	0.815814i	$0.303711\pi$
$8$		0			0
$9$	12.0325			1.33694
$10$		0			0
$11$			15.7627i			1.43298i	0.697600	+	0.716488i	$0.254251\pi$
$11$			15.7627i			1.43298i	−0.697600	+	0.716488i	$0.745749\pi$
$12$		0			0
$13$	−2.70240			−0.207877			−0.103939	−	0.994584i	$-0.533145\pi$
$13$	−2.70240			−0.207877			−0.103939	+	0.994584i	$0.533145\pi$
$14$		0			0
$15$			1.48410i			0.0989401i
$16$		0			0
$17$	−23.8912			−1.40537			−0.702684	−	0.711502i	$-0.748015\pi$
$17$	−23.8912			−1.40537			−0.702684	+	0.711502i	$0.748015\pi$
$18$		0			0
$19$	12.2592	+	14.5160i	0.645219	+	0.763998i
$19$	12.2592	+	14.5160i	0.645219	+	0.763998i
$20$		0			0
$21$	−37.1310			−1.76814
$22$		0			0
$23$	28.7327			1.24925			0.624624	−	0.780925i	$-0.285252\pi$
$23$	28.7327			1.24925			0.624624	+	0.780925i	$0.285252\pi$
$24$		0			0
$25$	24.8953			0.995811
$26$		0			0
$27$	−13.9072			−0.515083
$28$		0			0
$29$	−51.7916			−1.78592			−0.892959	−	0.450139i	$-0.851374\pi$
$29$	−51.7916			−1.78592			−0.892959	+	0.450139i	$0.851374\pi$
$30$		0			0
$31$		−	53.3446i		−	1.72079i	−0.509625	−	0.860397i	$-0.670216\pi$
$31$		−	53.3446i		−	1.72079i	0.509625	−	0.860397i	$-0.329784\pi$
$32$		0			0
$33$		−	72.2897i		−	2.19060i
$34$		0			0
$35$		−	2.62006i		−	0.0748587i
$36$		0			0
$37$	30.8820			0.834649			0.417325	−	0.908757i	$-0.362968\pi$
$37$	30.8820			0.834649			0.417325	+	0.908757i	$0.362968\pi$
$38$		0			0
$39$	12.3935			0.317783
$40$		0			0
$41$			41.0208i			1.00051i	0.865879	+	0.500254i	$0.166760\pi$
$41$			41.0208i			1.00051i	−0.865879	+	0.500254i	$0.833240\pi$
$42$		0			0
$43$			1.24753i			0.0290123i	0.999895	+	0.0145062i	$0.00461761\pi$
$43$			1.24753i			0.0290123i	−0.999895	+	0.0145062i	$0.995382\pi$
$44$		0			0
$45$		−	3.89379i		−	0.0865288i
$46$		0			0
$47$	1.82806			0.0388949			0.0194474	−	0.999811i	$-0.493809\pi$
$47$	1.82806			0.0388949			0.0194474	+	0.999811i	$0.493809\pi$
$48$		0			0
$49$	16.5517			0.337790
$50$		0			0
$51$	109.568			2.14839
$52$		0			0
$53$	10.5509			0.199073			0.0995365	−	0.995034i	$-0.468264\pi$
$53$	10.5509			0.199073			0.0995365	+	0.995034i	$0.468264\pi$
$54$		0			0
$55$	5.10094			0.0927443
$56$		0			0
$57$	−56.2219	−	66.5719i	−0.986349	−	1.16793i
$58$		0			0
$59$	−13.8763			−0.235192			−0.117596	−	0.993062i	$-0.537519\pi$
$59$	−13.8763			−0.235192			−0.117596	+	0.993062i	$0.537519\pi$
$60$		0			0
$61$		−	96.8096i		−	1.58704i	−0.608542	−	0.793522i	$-0.708245\pi$
$61$		−	96.8096i		−	1.58704i	0.608542	−	0.793522i	$-0.291755\pi$
$62$		0			0
$63$	97.4197			1.54634
$64$		0			0
$65$			0.874518i			0.0134541i
$66$		0			0
$67$	−22.4624			−0.335260			−0.167630	−	0.985850i	$-0.553611\pi$
$67$	−22.4624			−0.335260			−0.167630	+	0.985850i	$0.553611\pi$
$68$		0			0
$69$	−131.772			−1.90973
$70$		0			0
$71$			88.9010i			1.25213i	0.779772	+	0.626064i	$0.215335\pi$
$71$			88.9010i			1.25213i	−0.779772	+	0.626064i	$0.784665\pi$
$72$		0			0
$73$	−34.3165			−0.470090			−0.235045	−	0.971985i	$-0.575524\pi$
$73$	−34.3165			−0.470090			−0.235045	+	0.971985i	$0.575524\pi$
$74$		0			0
$75$	−114.173			−1.52230
$76$		0			0
$77$			127.621i			1.65742i
$78$		0			0
$79$			113.440i			1.43595i	0.696067	+	0.717977i	$0.254932\pi$
$79$			113.440i			1.43595i	−0.696067	+	0.717977i	$0.745068\pi$
$80$		0			0
$81$	−44.5120			−0.549531
$82$		0			0
$83$			65.8732i			0.793653i	0.917894	+	0.396826i	$0.129888\pi$
$83$			65.8732i			0.793653i	−0.917894	+	0.396826i	$0.870112\pi$
$84$		0			0
$85$			7.73138i			0.0909574i
$86$		0			0
$87$	237.522			2.73014
$88$		0			0
$89$		−	120.371i		−	1.35249i	−0.736679	−	0.676243i	$-0.763607\pi$
$89$		−	120.371i		−	1.35249i	0.736679	−	0.676243i	$-0.236393\pi$
$90$		0			0
$91$	−21.8797			−0.240437
$92$		0			0
$93$			244.645i			2.63059i
$94$		0			0
$95$	4.69747	−	3.96715i	0.0494471	−	0.0417595i
$96$		0			0
$97$			88.2932i			0.910239i	0.890430	+	0.455120i	$0.150403\pi$
$97$			88.2932i			0.910239i	−0.890430	+	0.455120i	$0.849597\pi$
$98$		0			0
$99$			189.665i			1.91580i
$100$		0			0
$101$			98.1162i			0.971447i	0.874112	+	0.485724i	$0.161444\pi$
$101$			98.1162i			0.971447i	−0.874112	+	0.485724i	$0.838556\pi$
$102$		0			0
$103$			136.279i			1.32309i	0.749904	+	0.661547i	$0.230100\pi$
$103$			136.279i			1.32309i	−0.749904	+	0.661547i	$0.769900\pi$
$104$		0			0
$105$			12.0159i			0.114437i
$106$		0			0
$107$	−195.523			−1.82732			−0.913659	−	0.406482i	$-0.866756\pi$
$107$	−195.523			−1.82732			−0.913659	+	0.406482i	$0.866756\pi$
$108$		0			0
$109$	−11.6177			−0.106584			−0.0532922	−	0.998579i	$-0.516971\pi$
$109$	−11.6177			−0.106584			−0.0532922	+	0.998579i	$0.516971\pi$
$110$		0			0
$111$	−141.629			−1.27593
$112$		0			0
$113$			97.8220i			0.865681i	0.901470	+	0.432841i	$0.142489\pi$
$113$			97.8220i			0.865681i	−0.901470	+	0.432841i	$0.857511\pi$
$114$		0			0
$115$		−	9.29812i		−	0.0808532i
$116$		0			0
$117$	−32.5166			−0.277919
$118$		0			0
$119$	−193.433			−1.62549
$120$		0			0
$121$	−127.464			−1.05342
$122$		0			0
$123$		−	188.126i		−	1.52948i
$124$		0			0
$125$		−	16.1465i		−	0.129172i
$126$		0			0
$127$			190.390i			1.49913i	0.661930	+	0.749566i	$0.269738\pi$
$127$			190.390i			1.49913i	−0.661930	+	0.749566i	$0.730262\pi$
$128$		0			0
$129$		−	5.72131i		−	0.0443513i
$130$		0			0
$131$			5.14461i			0.0392718i	0.999807	+	0.0196359i	$0.00625070\pi$
$131$			5.14461i			0.0392718i	−0.999807	+	0.0196359i	$0.993749\pi$
$132$		0			0
$133$	99.2550	+	117.527i	0.746278	+	0.883662i
$134$		0			0
$135$			4.50048i			0.0333369i
$136$		0			0
$137$	50.3123			0.367243			0.183621	−	0.982997i	$-0.441218\pi$
$137$	50.3123			0.367243			0.183621	+	0.982997i	$0.441218\pi$
$138$		0			0
$139$		−	114.207i		−	0.821636i	−0.911717	−	0.410818i	$-0.865243\pi$
$139$		−	114.207i		−	0.821636i	0.911717	−	0.410818i	$-0.134757\pi$
$140$		0			0
$141$	−8.38370			−0.0594588
$142$		0			0
$143$		−	42.5973i		−	0.297883i
$144$		0			0
$145$			16.7601i			0.115587i
$146$		0			0
$147$	−75.9081			−0.516382
$148$		0			0
$149$			204.095i			1.36976i	0.728654	+	0.684882i	$0.240146\pi$
$149$			204.095i			1.36976i	−0.728654	+	0.684882i	$0.759854\pi$
$150$		0			0
$151$		−	130.732i		−	0.865773i	−0.901448	−	0.432887i	$-0.857495\pi$
$151$		−	130.732i		−	0.865773i	0.901448	−	0.432887i	$-0.142505\pi$
$152$		0			0
$153$	−287.471			−1.87889
$154$		0			0
$155$	−17.2627			−0.111372
$156$		0			0
$157$		−	29.8681i		−	0.190243i	−0.995466	−	0.0951213i	$-0.969676\pi$
$157$		−	29.8681i		−	0.190243i	0.995466	−	0.0951213i	$-0.0303239\pi$
$158$		0			0
$159$	−48.3875			−0.304324
$160$		0			0
$161$	232.632			1.44492
$162$		0			0
$163$			266.286i			1.63366i	0.576881	+	0.816828i	$0.304270\pi$
$163$			266.286i			1.63366i	−0.576881	+	0.816828i	$0.695730\pi$
$164$		0			0
$165$	−23.3935			−0.141779
$166$		0			0
$167$			17.3715i			0.104021i	0.998647	+	0.0520104i	$0.0165629\pi$
$167$			17.3715i			0.104021i	−0.998647	+	0.0520104i	$0.983437\pi$
$168$		0			0
$169$	−161.697			−0.956787
$170$		0			0
$171$	147.508	+	174.663i	0.862619	+	1.02142i
$172$		0			0
$173$	−49.2803			−0.284857			−0.142429	−	0.989805i	$-0.545491\pi$
$173$	−49.2803			−0.284857			−0.142429	+	0.989805i	$0.545491\pi$
$174$		0			0
$175$	201.562			1.15178
$176$		0			0
$177$	63.6384			0.359539
$178$		0			0
$179$	56.7394			0.316980			0.158490	−	0.987361i	$-0.449337\pi$
$179$	56.7394			0.316980			0.158490	+	0.987361i	$0.449337\pi$
$180$		0			0
$181$	230.448			1.27319			0.636597	−	0.771196i	$-0.280341\pi$
$181$	230.448			1.27319			0.636597	+	0.771196i	$0.280341\pi$
$182$		0			0
$183$			443.980i			2.42612i
$184$		0			0
$185$		−	9.99365i		−	0.0540197i
$186$		0			0
$187$		−	376.591i		−	2.01386i
$188$		0			0
$189$	−112.599			−0.595760
$190$		0			0
$191$	−80.7648			−0.422852			−0.211426	−	0.977394i	$-0.567811\pi$
$191$	−80.7648			−0.422852			−0.211426	+	0.977394i	$0.567811\pi$
$192$		0			0
$193$		−	113.655i		−	0.588885i	−0.955669	−	0.294443i	$-0.904866\pi$
$193$		−	113.655i		−	0.588885i	0.955669	−	0.294443i	$-0.0951340\pi$
$194$		0			0
$195$		−	4.01064i		−	0.0205674i
$196$		0			0
$197$			19.9354i			0.101195i	0.998719	+	0.0505974i	$0.0161125\pi$
$197$			19.9354i			0.101195i	−0.998719	+	0.0505974i	$0.983887\pi$
$198$		0			0
$199$	12.3811			0.0622164			0.0311082	−	0.999516i	$-0.490096\pi$
$199$	12.3811			0.0622164			0.0311082	+	0.999516i	$0.490096\pi$
$200$		0			0
$201$	103.015			0.512514
$202$		0			0
$203$	−419.326			−2.06564
$204$		0			0
$205$	13.2746			0.0647544
$206$		0			0
$207$	345.725			1.67017
$208$		0			0
$209$	−228.811	+	193.238i	−1.09479	+	0.924582i
$210$		0			0
$211$	−265.822			−1.25982			−0.629909	−	0.776669i	$-0.716908\pi$
$211$	−265.822			−1.25982			−0.629909	+	0.776669i	$0.716908\pi$
$212$		0			0
$213$		−	407.711i		−	1.91413i
$214$		0			0
$215$	0.403710			0.00187772
$216$		0			0
$217$		−	431.899i		−	1.99032i
$218$		0			0
$219$	157.380			0.718629
$220$		0			0
$221$	64.5638			0.292144
$222$		0			0
$223$			181.609i			0.814390i	0.913341	+	0.407195i	$0.133493\pi$
$223$			181.609i			0.814390i	−0.913341	+	0.407195i	$0.866507\pi$
$224$		0			0
$225$	299.552			1.33134
$226$		0			0
$227$	413.152			1.82005			0.910025	−	0.414553i	$-0.136062\pi$
$227$	413.152			1.82005			0.910025	+	0.414553i	$0.136062\pi$
$228$		0			0
$229$			142.290i			0.621356i	0.950515	+	0.310678i	$0.100556\pi$
$229$			142.290i			0.621356i	−0.950515	+	0.310678i	$0.899444\pi$
$230$		0			0
$231$		−	585.287i		−	2.53371i
$232$		0			0
$233$	−222.260			−0.953905			−0.476952	−	0.878929i	$-0.658259\pi$
$233$	−222.260			−0.953905			−0.476952	+	0.878929i	$0.658259\pi$
$234$		0			0
$235$		−	0.591574i		−	0.00251733i
$236$		0			0
$237$		−	520.250i		−	2.19515i
$238$		0			0
$239$	391.140			1.63657			0.818286	−	0.574812i	$-0.194925\pi$
$239$	391.140			1.63657			0.818286	+	0.574812i	$0.194925\pi$
$240$		0			0
$241$			300.859i			1.24838i	0.781274	+	0.624188i	$0.214570\pi$
$241$			300.859i			1.24838i	−0.781274	+	0.624188i	$0.785430\pi$
$242$		0			0
$243$	329.302			1.35515
$244$		0			0
$245$		−	5.35626i		−	0.0218623i
$246$		0			0
$247$	−33.1292	−	39.2280i	−0.134126	−	0.158818i
$248$		0			0
$249$		−	302.102i		−	1.21326i
$250$		0			0
$251$		−	120.770i		−	0.481154i	−0.970630	−	0.240577i	$-0.922663\pi$
$251$		−	120.770i		−	0.481154i	0.970630	−	0.240577i	$-0.0773367\pi$
$252$		0			0
$253$			452.906i			1.79014i
$254$		0			0
$255$		−	35.4570i		−	0.139047i
$256$		0			0
$257$		−	1.62541i		−	0.00632457i	−0.999995	−	0.00316228i	$-0.998993\pi$
$257$		−	1.62541i		−	0.00632457i	0.999995	−	0.00316228i	$-0.00100659\pi$
$258$		0			0
$259$	250.033			0.965379
$260$		0			0
$261$	−623.181			−2.38766
$262$		0			0
$263$	−136.060			−0.517337			−0.258669	−	0.965966i	$-0.583284\pi$
$263$	−136.060			−0.517337			−0.258669	+	0.965966i	$0.583284\pi$
$264$		0			0
$265$		−	3.41434i		−	0.0128843i
$266$		0			0
$267$			552.037i			2.06755i
$268$		0			0
$269$	−0.387264			−0.00143964			−0.000719822	−	1.00000i	$-0.500229\pi$
$269$	−0.387264			−0.00143964			−0.000719822		1.00000i	$0.500229\pi$
$270$		0			0
$271$	−431.503			−1.59226			−0.796130	−	0.605125i	$-0.793123\pi$
$271$	−431.503			−1.59226			−0.796130	+	0.605125i	$0.793123\pi$
$272$		0			0
$273$	100.343			0.367557
$274$		0			0
$275$			392.418i			1.42697i
$276$		0			0
$277$		−	143.616i		−	0.518470i	−0.965814	−	0.259235i	$-0.916530\pi$
$277$		−	143.616i		−	0.518470i	0.965814	−	0.259235i	$-0.0834704\pi$
$278$		0			0
$279$		−	641.867i		−	2.30060i
$280$		0			0
$281$			197.212i			0.701822i	0.936409	+	0.350911i	$0.114128\pi$
$281$			197.212i			0.701822i	−0.936409	+	0.350911i	$0.885872\pi$
$282$		0			0
$283$		−	47.0745i		−	0.166341i	−0.996535	−	0.0831705i	$-0.973495\pi$
$283$		−	47.0745i		−	0.166341i	0.996535	−	0.0831705i	$-0.0265046\pi$
$284$		0			0
$285$	−21.5432	+	18.1938i	−0.0755900	+	0.0638380i
$286$		0			0
$287$			332.121i			1.15722i
$288$		0			0
$289$	281.792			0.975057
$290$		0			0
$291$		−	404.923i		−	1.39149i
$292$		0			0
$293$	−544.484			−1.85831			−0.929154	−	0.369692i	$-0.879463\pi$
$293$	−544.484			−1.85831			−0.929154	+	0.369692i	$0.879463\pi$
$294$		0			0
$295$			4.49048i			0.0152220i
$296$		0			0
$297$		−	219.216i		−	0.738101i
$298$		0			0
$299$	−77.6474			−0.259690
$300$		0			0
$301$			10.1005i			0.0335565i
$302$		0			0
$303$		−	449.972i		−	1.48506i
$304$		0			0
$305$	−31.3283			−0.102716
$306$		0			0
$307$	80.0883			0.260874			0.130437	−	0.991457i	$-0.458362\pi$
$307$	80.0883			0.260874			0.130437	+	0.991457i	$0.458362\pi$
$308$		0			0
$309$		−	624.990i		−	2.02262i
$310$		0			0
$311$	−149.245			−0.479889			−0.239944	−	0.970787i	$-0.577129\pi$
$311$	−149.245			−0.479889			−0.239944	+	0.970787i	$0.577129\pi$
$312$		0			0
$313$	104.148			0.332743			0.166371	−	0.986063i	$-0.446795\pi$
$313$	104.148			0.332743			0.166371	+	0.986063i	$0.446795\pi$
$314$		0			0
$315$		−	31.5257i		−	0.100082i
$316$		0			0
$317$	32.7397			0.103280			0.0516399	−	0.998666i	$-0.483555\pi$
$317$	32.7397			0.103280			0.0516399	+	0.998666i	$0.483555\pi$
$318$		0			0
$319$		−	816.377i		−	2.55918i
$320$		0			0
$321$	896.691			2.79343
$322$		0			0
$323$	−292.886	−	346.804i	−0.906769	−	1.07370i
$324$		0			0
$325$	−67.2771			−0.207006
$326$		0			0
$327$	53.2801			0.162936
$328$		0			0
$329$	14.8007			0.0449870
$330$		0			0
$331$	−171.550			−0.518277			−0.259139	−	0.965840i	$-0.583439\pi$
$331$	−171.550			−0.518277			−0.259139	+	0.965840i	$0.583439\pi$
$332$		0			0
$333$	371.587			1.11588
$334$		0			0
$335$			7.26901i			0.0216985i
$336$		0			0
$337$			182.851i			0.542585i	0.962497	+	0.271292i	$0.0874510\pi$
$337$			182.851i			0.542585i	−0.962497	+	0.271292i	$0.912549\pi$
$338$		0			0
$339$		−	448.623i		−	1.32337i
$340$		0			0
$341$	840.857			2.46585
$342$		0			0
$343$	−262.714			−0.765931
$344$		0			0
$345$			42.6423i			0.123601i
$346$		0			0
$347$			459.359i			1.32380i	0.749591	+	0.661901i	$0.230250\pi$
$347$			459.359i			1.32380i	−0.749591	+	0.661901i	$0.769750\pi$
$348$		0			0
$349$		−	308.948i		−	0.885239i	−0.896710	−	0.442619i	$-0.854049\pi$
$349$		−	308.948i		−	0.885239i	0.896710	−	0.442619i	$-0.145951\pi$
$350$		0			0
$351$	37.5830			0.107074
$352$		0			0
$353$	−138.347			−0.391917			−0.195958	−	0.980612i	$-0.562782\pi$
$353$	−138.347			−0.391917			−0.195958	+	0.980612i	$0.562782\pi$
$354$		0			0
$355$	28.7690			0.0810395
$356$		0			0
$357$	887.107			2.48489
$358$		0			0
$359$	572.384			1.59439			0.797193	−	0.603725i	$-0.206317\pi$
$359$	572.384			1.59439			0.797193	+	0.603725i	$0.206317\pi$
$360$		0			0
$361$	−60.4264	+	355.907i	−0.167386	+	0.985891i
$362$		0			0
$363$	584.563			1.61037
$364$		0			0
$365$			11.1051i			0.0304249i
$366$		0			0
$367$	−569.204			−1.55096			−0.775482	−	0.631370i	$-0.782493\pi$
$367$	−569.204			−1.55096			−0.775482	+	0.631370i	$0.782493\pi$
$368$		0			0
$369$			493.582i			1.33762i
$370$		0			0
$371$	85.4241			0.230254
$372$		0			0
$373$	−426.308			−1.14292			−0.571458	−	0.820631i	$-0.693622\pi$
$373$	−426.308			−1.14292			−0.571458	+	0.820631i	$0.693622\pi$
$374$		0			0
$375$			74.0496i			0.197466i
$376$		0			0
$377$	139.962			0.371251
$378$		0			0
$379$	411.263			1.08513			0.542563	−	0.840015i	$-0.317454\pi$
$379$	411.263			1.08513			0.542563	+	0.840015i	$0.317454\pi$
$380$		0			0
$381$		−	873.150i		−	2.29173i
$382$		0			0
$383$			485.648i			1.26801i	0.773329	+	0.634005i	$0.218590\pi$
$383$			485.648i			1.26801i	−0.773329	+	0.634005i	$0.781410\pi$
$384$		0			0
$385$	41.2992			0.107271
$386$		0			0
$387$			15.0109i			0.0387877i
$388$		0			0
$389$		−	151.447i		−	0.389324i	−0.980870	−	0.194662i	$-0.937639\pi$
$389$		−	151.447i		−	0.389324i	0.980870	−	0.194662i	$-0.0623610\pi$
$390$		0			0
$391$	−686.460			−1.75565
$392$		0			0
$393$		−	23.5938i		−	0.0600350i
$394$		0			0
$395$	36.7101			0.0929370
$396$		0			0
$397$		−	391.307i		−	0.985661i	−0.870125	−	0.492830i	$-0.835962\pi$
$397$		−	391.307i		−	0.985661i	0.870125	−	0.492830i	$-0.164038\pi$
$398$		0			0
$399$	−455.195	−	538.993i	−1.14084	−	1.35086i
$400$		0			0
$401$		−	531.100i		−	1.32444i	−0.749310	−	0.662220i	$-0.769615\pi$
$401$		−	531.100i		−	1.32444i	0.749310	−	0.662220i	$-0.230385\pi$
$402$		0			0
$403$			144.159i			0.357714i
$404$		0			0
$405$			14.4044i			0.0355664i
$406$		0			0
$407$			486.785i			1.19603i
$408$		0			0
$409$			6.97313i			0.0170492i	0.999964	+	0.00852461i	$0.00271350\pi$
$409$			6.97313i			0.0170492i	−0.999964	+	0.00852461i	$0.997286\pi$
$410$		0			0
$411$	−230.738			−0.561406
$412$		0			0
$413$	−112.348			−0.272030
$414$		0			0
$415$	21.3170			0.0513664
$416$		0			0
$417$			523.769i			1.25604i
$418$		0			0
$419$		−	244.894i		−	0.584473i	−0.956346	−	0.292236i	$-0.905601\pi$
$419$		−	244.894i		−	0.584473i	0.956346	−	0.292236i	$-0.0943994\pi$
$420$		0			0
$421$	669.982			1.59141			0.795703	−	0.605687i	$-0.207102\pi$
$421$	669.982			1.59141			0.795703	+	0.605687i	$0.207102\pi$
$422$		0			0
$423$	21.9961			0.0520002
$424$		0			0
$425$	−594.779			−1.39948
$426$		0			0
$427$		−	783.810i		−	1.83562i
$428$		0			0
$429$			195.356i			0.455375i
$430$		0			0
$431$			603.062i			1.39922i	0.714526	+	0.699608i	$0.246642\pi$
$431$			603.062i			1.39922i	−0.714526	+	0.699608i	$0.753358\pi$
$432$		0			0
$433$			614.205i			1.41849i	0.704964	+	0.709243i	$0.250963\pi$
$433$			614.205i			1.41849i	−0.704964	+	0.709243i	$0.749037\pi$
$434$		0			0
$435$		−	76.8640i		−	0.176699i
$436$		0			0
$437$	352.239	+	417.083i	0.806038	+	0.954424i
$438$		0			0
$439$			111.058i			0.252980i	0.991968	+	0.126490i	$0.0403711\pi$
$439$			111.058i			0.252980i	−0.991968	+	0.126490i	$0.959629\pi$
$440$		0			0
$441$	199.158			0.451605
$442$		0			0
$443$		−	254.595i		−	0.574708i	−0.957825	−	0.287354i	$-0.907224\pi$
$443$		−	254.595i		−	0.574708i	0.957825	−	0.287354i	$-0.0927756\pi$
$444$		0			0
$445$	−38.9530			−0.0875349
$446$		0			0
$447$		−	936.002i		−	2.09397i
$448$		0			0
$449$		−	637.170i		−	1.41909i	−0.704662	−	0.709543i	$-0.748901\pi$
$449$		−	637.170i		−	1.41909i	0.704662	−	0.709543i	$-0.251099\pi$
$450$		0			0
$451$	−646.600			−1.43370
$452$		0			0
$453$			599.551i			1.32351i
$454$		0			0
$455$			7.08045i			0.0155614i
$456$		0			0
$457$	−619.257			−1.35505			−0.677524	−	0.735500i	$-0.736947\pi$
$457$	−619.257			−1.35505			−0.677524	+	0.735500i	$0.736947\pi$
$458$		0			0
$459$	332.261			0.723881
$460$		0			0
$461$			693.390i			1.50410i	0.659106	+	0.752050i	$0.270935\pi$
$461$			693.390i			1.50410i	−0.659106	+	0.752050i	$0.729065\pi$
$462$		0			0
$463$	−221.733			−0.478905			−0.239453	−	0.970908i	$-0.576968\pi$
$463$	−221.733			−0.478905			−0.239453	+	0.970908i	$0.576968\pi$
$464$		0			0
$465$	79.1688			0.170255
$466$		0			0
$467$		−	386.716i		−	0.828085i	−0.910258	−	0.414042i	$-0.864117\pi$
$467$		−	386.716i		−	0.828085i	0.910258	−	0.414042i	$-0.135883\pi$
$468$		0			0
$469$	−181.865			−0.387772
$470$		0			0
$471$			136.979i			0.290825i
$472$		0			0
$473$	−19.6645			−0.0415739
$474$		0			0
$475$	305.195	+	361.379i	0.642516	+	0.760798i
$476$		0			0
$477$	126.953			0.266149
$478$		0			0
$479$	−263.608			−0.550330			−0.275165	−	0.961397i	$-0.588733\pi$
$479$	−263.608			−0.550330			−0.275165	+	0.961397i	$0.588733\pi$
$480$		0			0
$481$	−83.4557			−0.173505
$482$		0			0
$483$	−1066.88			−2.20885
$484$		0			0
$485$	28.5723			0.0589120
$486$		0			0
$487$			312.880i			0.642464i	0.947000	+	0.321232i	$0.104097\pi$
$487$			312.880i			0.642464i	−0.947000	+	0.321232i	$0.895903\pi$
$488$		0			0
$489$		−	1221.22i		−	2.49738i
$490$		0			0
$491$		−	67.1921i		−	0.136847i	−0.997656	−	0.0684237i	$-0.978203\pi$
$491$		−	67.1921i		−	0.136847i	0.997656	−	0.0684237i	$-0.0217970\pi$
$492$		0			0
$493$	1237.37			2.50987
$494$		0			0
$495$	61.3768			0.123994
$496$		0			0
$497$			719.779i			1.44825i
$498$		0			0
$499$		−	638.944i		−	1.28045i	−0.768188	−	0.640224i	$-0.778841\pi$
$499$		−	638.944i		−	1.28045i	0.768188	−	0.640224i	$-0.221159\pi$
$500$		0			0
$501$		−	79.6676i		−	0.159017i
$502$		0			0
$503$	47.2084			0.0938537			0.0469269	−	0.998898i	$-0.485057\pi$
$503$	47.2084			0.0938537			0.0469269	+	0.998898i	$0.485057\pi$
$504$		0			0
$505$	31.7511			0.0628735
$506$		0			0
$507$	741.561			1.46265
$508$		0			0
$509$	152.130			0.298879			0.149440	−	0.988771i	$-0.452253\pi$
$509$	152.130			0.298879			0.149440	+	0.988771i	$0.452253\pi$
$510$		0			0
$511$	−277.840			−0.543719
$512$		0			0
$513$	−170.491	−	201.877i	−0.332341	−	0.393522i
$514$		0			0
$515$	44.1008			0.0856326
$516$		0			0
$517$			28.8152i			0.0557354i
$518$		0			0
$519$	226.005			0.435463
$520$		0			0
$521$			273.833i			0.525591i	0.964852	+	0.262795i	$0.0846445\pi$
$521$			273.833i			0.525591i	−0.964852	+	0.262795i	$0.915356\pi$
$522$		0			0
$523$	598.807			1.14495			0.572473	−	0.819923i	$-0.305984\pi$
$523$	598.807			1.14495			0.572473	+	0.819923i	$0.305984\pi$
$524$		0			0
$525$	−924.388			−1.76074
$526$		0			0
$527$			1274.47i			2.41835i
$528$		0			0
$529$	296.569			0.560622
$530$		0			0
$531$	−166.966			−0.314437
$532$		0			0
$533$		−	110.855i		−	0.207983i
$534$		0			0
$535$			63.2727i			0.118267i
$536$		0			0
$537$	−260.213			−0.484569
$538$		0			0
$539$			260.900i			0.484045i
$540$		0			0
$541$			1004.48i			1.85671i	0.371694	+	0.928355i	$0.378777\pi$
$541$			1004.48i			1.85671i	−0.371694	+	0.928355i	$0.621223\pi$
$542$		0			0
$543$	−1056.86			−1.94634
$544$		0			0
$545$			3.75957i			0.00689830i
$546$		0			0
$547$	−221.020			−0.404059			−0.202029	−	0.979379i	$-0.564754\pi$
$547$	−221.020			−0.404059			−0.202029	+	0.979379i	$0.564754\pi$
$548$		0			0
$549$		−	1164.86i		−	2.12178i
$550$		0			0
$551$	−634.921	−	751.805i	−1.15231	−	1.36444i
$552$		0			0
$553$			918.458i			1.66086i
$554$		0			0
$555$			45.8320i			0.0825803i
$556$		0			0
$557$		−	403.065i		−	0.723636i	−0.932249	−	0.361818i	$-0.882156\pi$
$557$		−	403.065i		−	0.723636i	0.932249	−	0.361818i	$-0.117844\pi$
$558$		0			0
$559$		−	3.37133i		−	0.00603100i
$560$		0			0
$561$			1727.09i			3.07859i
$562$		0			0
$563$	−892.295			−1.58489			−0.792447	−	0.609941i	$-0.791193\pi$
$563$	−892.295			−1.58489			−0.792447	+	0.609941i	$0.791193\pi$
$564$		0			0
$565$	31.6559			0.0560282
$566$		0			0
$567$	−360.387			−0.635603
$568$		0			0
$569$		−	276.824i		−	0.486510i	−0.969962	−	0.243255i	$-0.921785\pi$
$569$		−	276.824i		−	0.486510i	0.969962	−	0.243255i	$-0.0782152\pi$
$570$		0			0
$571$			606.699i			1.06252i	0.847209	+	0.531260i	$0.178281\pi$
$571$			606.699i			1.06252i	−0.847209	+	0.531260i	$0.821719\pi$
$572$		0			0
$573$	370.397			0.646417
$574$		0			0
$575$	715.309			1.24402
$576$		0			0
$577$	804.482			1.39425			0.697125	−	0.716950i	$-0.254462\pi$
$577$	804.482			1.39425			0.697125	+	0.716950i	$0.254462\pi$
$578$		0			0
$579$			521.234i			0.900232i
$580$		0			0
$581$			533.336i			0.917961i
$582$		0			0
$583$			166.310i			0.285267i
$584$		0			0
$585$			10.5226i			0.0179874i
$586$		0			0
$587$		−	138.544i		−	0.236020i	−0.993012	−	0.118010i	$-0.962348\pi$
$587$		−	138.544i		−	0.236020i	0.993012	−	0.118010i	$-0.0376515\pi$
$588$		0			0
$589$	774.348	−	653.960i	1.31468	−	1.11029i
$590$		0			0
$591$		−	91.4259i		−	0.154697i
$592$		0			0
$593$	−680.758			−1.14799			−0.573995	−	0.818859i	$-0.694607\pi$
$593$	−680.758			−1.14799			−0.573995	+	0.818859i	$0.694607\pi$
$594$		0			0
$595$			62.5964i			0.105204i
$596$		0			0
$597$	−56.7810			−0.0951105
$598$		0			0
$599$		−	529.164i		−	0.883413i	−0.897160	−	0.441706i	$-0.854373\pi$
$599$		−	529.164i		−	0.883413i	0.897160	−	0.441706i	$-0.145627\pi$
$600$		0			0
$601$		−	448.349i		−	0.746004i	−0.927830	−	0.373002i	$-0.878328\pi$
$601$		−	448.349i		−	0.746004i	0.927830	−	0.373002i	$-0.121672\pi$
$602$		0			0
$603$	−270.279			−0.448223
$604$		0			0
$605$			41.2482i			0.0681788i
$606$		0			0
$607$		−	1053.81i		−	1.73609i	−0.496482	−	0.868047i	$-0.665375\pi$
$607$		−	1053.81i		−	1.73609i	0.496482	−	0.868047i	$-0.334625\pi$
$608$		0			0
$609$	1923.08			3.15776
$610$		0			0
$611$	−4.94016			−0.00808536
$612$		0			0
$613$		−	673.805i		−	1.09919i	−0.835430	−	0.549596i	$-0.814782\pi$
$613$		−	673.805i		−	1.09919i	0.835430	−	0.549596i	$-0.185218\pi$
$614$		0			0
$615$	−60.8791			−0.0989904
$616$		0			0
$617$	−938.504			−1.52108			−0.760538	−	0.649293i	$-0.775065\pi$
$617$	−938.504			−1.52108			−0.760538	+	0.649293i	$0.775065\pi$
$618$		0			0
$619$		−	873.689i		−	1.41145i	−0.708484	−	0.705726i	$-0.750621\pi$
$619$		−	873.689i		−	1.41145i	0.708484	−	0.705726i	$-0.249379\pi$
$620$		0			0
$621$	−399.593			−0.643467
$622$		0			0
$623$		−	974.574i		−	1.56432i
$624$		0			0
$625$	617.157			0.987451
$626$		0			0
$627$	1049.35	−	886.211i	1.67361	−	1.41341i
$628$		0			0
$629$	−737.810			−1.17299
$630$		0			0
$631$	888.811			1.40858			0.704288	−	0.709915i	$-0.251267\pi$
$631$	888.811			1.40858			0.704288	+	0.709915i	$0.251267\pi$
$632$		0			0
$633$	1219.09			1.92589
$634$		0			0
$635$	61.6115			0.0970260
$636$		0			0
$637$	−44.7294			−0.0702189
$638$		0			0
$639$			1069.70i			1.67402i
$640$		0			0
$641$		−	491.699i		−	0.767081i	−0.923524	−	0.383540i	$-0.874705\pi$
$641$		−	491.699i		−	0.767081i	0.923524	−	0.383540i	$-0.125295\pi$
$642$		0			0
$643$			852.032i			1.32509i	0.749023	+	0.662544i	$0.230523\pi$
$643$			852.032i			1.32509i	−0.749023	+	0.662544i	$0.769477\pi$
$644$		0			0
$645$	−1.85146			−0.00287048
$646$		0			0
$647$	510.498			0.789023			0.394511	−	0.918891i	$-0.370914\pi$
$647$	510.498			0.789023			0.394511	+	0.918891i	$0.370914\pi$
$648$		0			0
$649$		−	218.729i		−	0.337024i
$650$		0			0
$651$			1980.74i			3.04261i
$652$		0			0
$653$			1068.09i			1.63567i	0.575452	+	0.817836i	$0.304826\pi$
$653$			1068.09i			1.63567i	−0.575452	+	0.817836i	$0.695174\pi$
$654$		0			0
$655$	1.66483			0.00254173
$656$		0			0
$657$	−412.913			−0.628482
$658$		0			0
$659$	42.0110			0.0637496			0.0318748	−	0.999492i	$-0.489852\pi$
$659$	42.0110			0.0637496			0.0318748	+	0.999492i	$0.489852\pi$
$660$		0			0
$661$	−141.464			−0.214014			−0.107007	−	0.994258i	$-0.534127\pi$
$661$	−141.464			−0.214014			−0.107007	+	0.994258i	$0.534127\pi$
$662$		0			0
$663$	−296.097			−0.446602
$664$		0			0
$665$	38.0326	−	32.1197i	0.0571919	−	0.0483002i
$666$		0			0
$667$	−1488.11			−2.23105
$668$		0			0
$669$		−	832.880i		−	1.24496i
$670$		0			0
$671$	1525.98			2.27419
$672$		0			0
$673$		−	1032.50i		−	1.53417i	−0.641546	−	0.767085i	$-0.721706\pi$
$673$		−	1032.50i		−	1.53417i	0.641546	−	0.767085i	$-0.278294\pi$
$674$		0			0
$675$	−346.225			−0.512925
$676$		0			0
$677$	−662.404			−0.978440			−0.489220	−	0.872160i	$-0.662718\pi$
$677$	−662.404			−0.978440			−0.489220	+	0.872160i	$0.662718\pi$
$678$		0			0
$679$			714.857i			1.05281i
$680$		0			0
$681$	−1894.76			−2.78232
$682$		0			0
$683$	828.996			1.21376			0.606879	−	0.794795i	$-0.292421\pi$
$683$	828.996			1.21376			0.606879	+	0.794795i	$0.292421\pi$
$684$		0			0
$685$		−	16.2814i		−	0.0237685i
$686$		0			0
$687$		−	652.561i		−	0.949870i
$688$		0			0
$689$	−28.5127			−0.0413827
$690$		0			0
$691$		−	685.276i		−	0.991716i	−0.868404	−	0.495858i	$-0.834854\pi$
$691$		−	685.276i		−	0.991716i	0.868404	−	0.495858i	$-0.165146\pi$
$692$		0			0
$693$			1535.60i			2.21587i
$694$		0			0
$695$	−36.9584			−0.0531775
$696$		0			0
$697$		−	980.039i		−	1.40608i
$698$		0			0
$699$	1019.31			1.45824
$700$		0			0
$701$			76.5531i			0.109206i	0.998508	+	0.0546028i	$0.0173893\pi$
$701$			76.5531i			0.109206i	−0.998508	+	0.0546028i	$0.982611\pi$
$702$		0			0
$703$	378.587	+	448.282i	0.538531	+	0.637671i
$704$		0			0
$705$			2.71303i			0.00384826i
$706$		0			0
$707$			794.388i			1.12360i
$708$		0			0
$709$		−	893.900i		−	1.26079i	−0.776275	−	0.630395i	$-0.782893\pi$
$709$		−	893.900i		−	1.26079i	0.776275	−	0.630395i	$-0.217107\pi$
$710$		0			0
$711$			1364.97i			1.91978i
$712$		0			0
$713$		−	1532.74i		−	2.14970i
$714$		0			0
$715$	−13.7848			−0.0192794
$716$		0			0
$717$	−1793.82			−2.50184
$718$		0			0
$719$	735.713			1.02325			0.511623	−	0.859210i	$-0.329045\pi$
$719$	735.713			1.02325			0.511623	+	0.859210i	$0.329045\pi$
$720$		0			0
$721$			1103.37i			1.53033i
$722$		0			0
$723$		−	1379.77i		−	1.90840i
$724$		0			0
$725$	−1289.37			−1.77844
$726$		0			0
$727$	1366.64			1.87984			0.939919	−	0.341396i	$-0.110900\pi$
$727$	1366.64			1.87984			0.939919	+	0.341396i	$0.110900\pi$
$728$		0			0
$729$	−1109.61			−1.52210
$730$		0			0
$731$		−	29.8050i		−	0.0407729i
$732$		0			0
$733$			231.571i			0.315923i	0.987445	+	0.157961i	$0.0504921\pi$
$733$			231.571i			0.315923i	−0.987445	+	0.157961i	$0.949508\pi$
$734$		0			0
$735$			24.5644i			0.0334210i
$736$		0			0
$737$		−	354.069i		−	0.480420i
$738$		0			0
$739$			1118.87i			1.51404i	0.653394	+	0.757018i	$0.273345\pi$
$739$			1118.87i			1.51404i	−0.653394	+	0.757018i	$0.726655\pi$
$740$		0			0
$741$	151.934	+	179.904i	0.205040	+	0.242786i
$742$		0			0
$743$			541.335i			0.728580i	0.931286	+	0.364290i	$0.118688\pi$
$743$			541.335i			0.728580i	−0.931286	+	0.364290i	$0.881312\pi$
$744$		0			0
$745$	66.0466			0.0886531
$746$		0			0
$747$			792.617i			1.06107i
$748$		0			0
$749$	−1583.03			−2.11353
$750$		0			0
$751$			567.078i			0.755098i	0.925990	+	0.377549i	$0.123233\pi$
$751$			567.078i			0.755098i	−0.925990	+	0.377549i	$0.876767\pi$
$752$		0			0
$753$			553.864i			0.735543i
$754$		0			0
$755$	−42.3058			−0.0560341
$756$		0			0
$757$			360.370i			0.476050i	0.971259	+	0.238025i	$0.0765000\pi$
$757$			360.370i			0.476050i	−0.971259	+	0.238025i	$0.923500\pi$
$758$		0			0
$759$		−	2077.08i		−	2.73660i
$760$		0			0
$761$	40.9265			0.0537798			0.0268899	−	0.999638i	$-0.491440\pi$
$761$	40.9265			0.0537798			0.0268899	+	0.999638i	$0.491440\pi$
$762$		0			0
$763$	−94.0615			−0.123279
$764$		0			0
$765$			93.0276i			0.121605i
$766$		0			0
$767$	37.4994			0.0488910
$768$		0			0
$769$	−170.819			−0.222132			−0.111066	−	0.993813i	$-0.535426\pi$
$769$	−170.819			−0.222132			−0.111066	+	0.993813i	$0.535426\pi$
$770$		0			0
$771$			7.45433i			0.00966840i
$772$		0			0
$773$	1027.67			1.32945			0.664725	−	0.747088i	$-0.268549\pi$
$773$	1027.67			1.32945			0.664725	+	0.747088i	$0.268549\pi$
$774$		0			0
$775$		−	1328.03i		−	1.71359i
$776$		0			0
$777$	−1146.68			−1.47578
$778$		0			0
$779$	−595.457	+	502.881i	−0.764386	+	0.645546i
$780$		0			0
$781$	−1401.32			−1.79427
$782$		0			0
$783$	720.278			0.919895
$784$		0			0
$785$	−9.66554			−0.0123128
$786$		0			0
$787$	1370.72			1.74171			0.870853	−	0.491544i	$-0.163567\pi$
$787$	1370.72			1.74171			0.870853	+	0.491544i	$0.163567\pi$
$788$		0			0
$789$	623.985			0.790856
$790$		0			0
$791$			792.006i			1.00127i
$792$		0			0
$793$			261.619i			0.329910i
$794$		0			0
$795$			15.6586i			0.0196963i
$796$		0			0
$797$	−507.322			−0.636539			−0.318269	−	0.948000i	$-0.603102\pi$
$797$	−507.322			−0.636539			−0.318269	+	0.948000i	$0.603102\pi$
$798$		0			0
$799$	−43.6746			−0.0546616
$800$		0			0
$801$		−	1448.36i		−	1.80819i
$802$		0			0
$803$		−	540.922i		−	0.673627i
$804$		0			0
$805$		−	75.2813i		−	0.0935172i
$806$		0			0
$807$	1.77604			0.00220079
$808$		0			0
$809$	808.551			0.999445			0.499722	−	0.866186i	$-0.333435\pi$
$809$	808.551			0.999445			0.499722	+	0.866186i	$0.333435\pi$
$810$		0			0
$811$	96.4414			0.118917			0.0594583	−	0.998231i	$-0.481063\pi$
$811$	96.4414			0.118917			0.0594583	+	0.998231i	$0.481063\pi$
$812$		0			0
$813$	1978.92			2.43410
$814$		0			0
$815$	86.1721			0.105733
$816$		0			0
$817$	−18.1091	+	15.2936i	−0.0221653	+	0.0187193i
$818$		0			0
$819$	−263.267			−0.321450
$820$		0			0
$821$			1175.04i			1.43123i	0.698496	+	0.715614i	$0.253853\pi$
$821$			1175.04i			1.43123i	−0.698496	+	0.715614i	$0.746147\pi$
$822$		0			0
$823$	486.014			0.590539			0.295270	−	0.955414i	$-0.404591\pi$
$823$	486.014			0.590539			0.295270	+	0.955414i	$0.404591\pi$
$824$		0			0
$825$		−	1799.67i		−	2.18142i
$826$		0			0
$827$	−706.002			−0.853690			−0.426845	−	0.904325i	$-0.640375\pi$
$827$	−706.002			−0.853690			−0.426845	+	0.904325i	$0.640375\pi$
$828$		0			0
$829$	−264.350			−0.318878			−0.159439	−	0.987208i	$-0.550968\pi$
$829$	−264.350			−0.318878			−0.159439	+	0.987208i	$0.550968\pi$
$830$		0			0
$831$			658.640i			0.792588i
$832$		0			0
$833$	−395.441			−0.474719
$834$		0			0
$835$	5.62154			0.00673238
$836$		0			0
$837$			741.876i			0.886351i
$838$		0			0
$839$		−	350.939i		−	0.418283i	−0.977885	−	0.209141i	$-0.932933\pi$
$839$		−	350.939i		−	0.418283i	0.977885	−	0.209141i	$-0.0670669\pi$
$840$		0			0
$841$	1841.37			2.18950
$842$		0			0
$843$		−	904.437i		−	1.07288i
$844$		0			0
$845$			52.3263i			0.0619247i
$846$		0			0
$847$	−1032.00			−1.21841
$848$		0			0
$849$			215.889i			0.254287i
$850$		0			0
$851$	887.325			1.04268
$852$		0			0
$853$		−	918.300i		−	1.07655i	−0.842768	−	0.538277i	$-0.819075\pi$
$853$		−	918.300i		−	1.07655i	0.842768	−	0.538277i	$-0.180925\pi$
$854$		0			0
$855$	56.5222	−	47.7346i	0.0661078	−	0.0558300i
$856$		0			0
$857$		−	755.091i		−	0.881087i	−0.897731	−	0.440543i	$-0.854786\pi$
$857$		−	755.091i		−	0.881087i	0.897731	−	0.440543i	$-0.145214\pi$
$858$		0			0
$859$			893.583i			1.04026i	0.854087	+	0.520130i	$0.174116\pi$
$859$			893.583i			1.04026i	−0.854087	+	0.520130i	$0.825884\pi$
$860$		0			0
$861$		−	1523.15i		−	1.76904i
$862$		0			0
$863$			610.340i			0.707230i	0.935391	+	0.353615i	$0.115048\pi$
$863$			610.340i			0.707230i	−0.935391	+	0.353615i	$0.884952\pi$
$864$		0			0
$865$			15.9475i			0.0184364i
$866$		0			0
$867$	−1292.33			−1.49058
$868$		0			0
$869$	−1788.13			−2.05769
$870$		0			0
$871$	60.7026			0.0696930
$872$		0			0
$873$			1062.38i			1.21694i
$874$		0			0
$875$		−	130.728i		−	0.149404i
$876$		0			0
$877$	103.356			0.117852			0.0589259	−	0.998262i	$-0.481232\pi$
$877$	103.356			0.117852			0.0589259	+	0.998262i	$0.481232\pi$
$878$		0			0
$879$	2497.07			2.84081
$880$		0			0
$881$	−875.630			−0.993904			−0.496952	−	0.867778i	$-0.665548\pi$
$881$	−875.630			−0.993904			−0.496952	+	0.867778i	$0.665548\pi$
$882$		0			0
$883$		−	446.254i		−	0.505384i	−0.967547	−	0.252692i	$-0.918684\pi$
$883$		−	446.254i		−	0.505384i	0.967547	−	0.252692i	$-0.0813159\pi$
$884$		0			0
$885$		−	20.5938i		−	0.0232699i
$886$		0			0
$887$			680.106i			0.766748i	0.923593	+	0.383374i	$0.125238\pi$
$887$			680.106i			0.766748i	−0.923593	+	0.383374i	$0.874762\pi$
$888$		0			0
$889$			1541.47i			1.73394i
$890$		0			0
$891$		−	701.630i		−	0.787464i
$892$		0			0
$893$	22.4105	+	26.5361i	0.0250957	+	0.0297156i
$894$		0			0
$895$		−	18.3613i		−	0.0205154i
$896$		0			0
$897$	356.100			0.396990
$898$		0			0
$899$			2762.80i			3.07319i
$900$		0			0
$901$	−252.073			−0.279771
$902$		0			0
$903$		−	46.3221i		−	0.0512980i
$904$		0			0
$905$		−	74.5747i		−	0.0824030i
$906$		0			0
$907$	−1241.68			−1.36899			−0.684496	−	0.729016i	$-0.739978\pi$
$907$	−1241.68			−1.36899			−0.684496	+	0.729016i	$0.739978\pi$
$908$		0			0
$909$			1180.58i			1.29877i
$910$		0			0
$911$			673.760i			0.739583i	0.929115	+	0.369791i	$0.120571\pi$
$911$			673.760i			0.739583i	−0.929115	+	0.369791i	$0.879429\pi$
$912$		0			0
$913$	−1038.34			−1.13728
$914$		0			0
$915$	143.675			0.157022
$916$		0			0
$917$			41.6528i			0.0454229i
$918$		0			0
$919$	−1399.96			−1.52335			−0.761677	−	0.647957i	$-0.775624\pi$
$919$	−1399.96			−1.52335			−0.761677	+	0.647957i	$0.775624\pi$
$920$		0			0
$921$	−367.294			−0.398799
$922$		0			0
$923$		−	240.247i		−	0.260289i
$924$		0			0
$925$	768.817			0.831153
$926$		0			0
$927$			1639.77i			1.76890i
$928$		0			0
$929$	479.876			0.516551			0.258275	−	0.966071i	$-0.416846\pi$
$929$	479.876			0.516551			0.258275	+	0.966071i	$0.416846\pi$
$930$		0			0
$931$	202.910	+	240.264i	0.217949	+	0.258071i
$932$		0			0
$933$	684.457			0.733609
$934$		0			0
$935$	−121.868			−0.130340
$936$		0			0
$937$	−482.124			−0.514540			−0.257270	−	0.966340i	$-0.582823\pi$
$937$	−482.124			−0.514540			−0.257270	+	0.966340i	$0.582823\pi$
$938$		0			0
$939$	−477.637			−0.508665
$940$		0			0
$941$	1233.93			1.31129			0.655647	−	0.755068i	$-0.272396\pi$
$941$	1233.93			1.31129			0.655647	+	0.755068i	$0.272396\pi$
$942$		0			0
$943$			1178.64i			1.24988i
$944$		0			0
$945$			36.4377i			0.0385584i
$946$		0			0
$947$			488.859i			0.516219i	0.966116	+	0.258109i	$0.0830995\pi$
$947$			488.859i			0.516219i	−0.966116	+	0.258109i	$0.916901\pi$
$948$		0			0
$949$	92.7371			0.0977209
$950$		0			0
$951$	−150.148			−0.157885
$952$		0			0
$953$			1830.89i			1.92118i	0.277965	+	0.960591i	$0.410340\pi$
$953$			1830.89i			1.92118i	−0.277965	+	0.960591i	$0.589660\pi$
$954$		0			0
$955$			26.1361i			0.0273676i
$956$		0			0
$957$			3744.00i			3.91223i
$958$		0			0
$959$	407.348			0.424764
$960$		0			0
$961$	−1884.65			−1.96113
$962$		0			0
$963$	−2352.62			−2.44302
$964$		0			0
$965$	−36.7796			−0.0381135
$966$		0			0
$967$	−529.813			−0.547893			−0.273947	−	0.961745i	$-0.588329\pi$
$967$	−529.813			−0.547893			−0.273947	+	0.961745i	$0.588329\pi$
$968$		0			0
$969$	1343.21	+	1590.49i	1.38618	+	1.64137i
$970$		0			0
$971$	−1335.82			−1.37571			−0.687856	−	0.725847i	$-0.741448\pi$
$971$	−1335.82			−1.37571			−0.687856	+	0.725847i	$0.741448\pi$
$972$		0			0
$973$		−	924.670i		−	0.950328i
$974$		0			0
$975$	308.541			0.316452
$976$		0			0
$977$		−	803.318i		−	0.822229i	−0.911584	−	0.411115i	$-0.865140\pi$
$977$		−	803.318i		−	0.822229i	0.911584	−	0.411115i	$-0.134860\pi$
$978$		0			0
$979$	1897.38			1.93808
$980$		0			0
$981$	−139.790			−0.142497
$982$		0			0
$983$			784.639i			0.798209i	0.916906	+	0.399104i	$0.130679\pi$
$983$			784.639i			0.798209i	−0.916906	+	0.399104i	$0.869321\pi$
$984$		0			0
$985$	6.45123			0.00654947
$986$		0			0
$987$	−67.8778			−0.0687718
$988$		0			0
$989$			35.8449i			0.0362436i
$990$		0			0
$991$		−	1505.00i		−	1.51867i	−0.650699	−	0.759336i	$-0.725524\pi$
$991$		−	1505.00i		−	1.51867i	0.650699	−	0.759336i	$-0.274476\pi$
$992$		0			0
$993$	786.747			0.792293
$994$		0			0
$995$		−	4.00660i		−	0.00402674i
$996$		0			0
$997$			1376.43i			1.38057i	0.723539	+	0.690284i	$0.242514\pi$
$997$			1376.43i			1.38057i	−0.723539	+	0.690284i	$0.757486\pi$
$998$		0			0
$999$	−429.484			−0.429914

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.g.e.417.10	yes	48
4.3	odd	2	inner	1216.3.g.e.417.38	yes	48
8.3	odd	2	inner	1216.3.g.e.417.12	yes	48
8.5	even	2	inner	1216.3.g.e.417.40	yes	48
19.18	odd	2	inner	1216.3.g.e.417.37	yes	48
76.75	even	2	inner	1216.3.g.e.417.9	✓	48
152.37	odd	2	inner	1216.3.g.e.417.11	yes	48
152.75	even	2	inner	1216.3.g.e.417.39	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.3.g.e.417.9	✓	48	76.75	even	2	inner
1216.3.g.e.417.10	yes	48	1.1	even	1	trivial
1216.3.g.e.417.11	yes	48	152.37	odd	2	inner
1216.3.g.e.417.12	yes	48	8.3	odd	2	inner
1216.3.g.e.417.37	yes	48	19.18	odd	2	inner
1216.3.g.e.417.38	yes	48	4.3	odd	2	inner
1216.3.g.e.417.39	yes	48	152.75	even	2	inner
1216.3.g.e.417.40	yes	48	8.5	even	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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.58612 q^{3} -0.323607i q^{5} +8.09640 q^{7} +12.0325 q^{9} +O(q^{10})\)	\(q-4.58612 q^{3} -0.323607i q^{5} +8.09640 q^{7} +12.0325 q^{9} +15.7627i q^{11} -2.70240 q^{13} +1.48410i q^{15} -23.8912 q^{17} +(12.2592 + 14.5160i) q^{19} -37.1310 q^{21} +28.7327 q^{23} +24.8953 q^{25} -13.9072 q^{27} -51.7916 q^{29} -53.3446i q^{31} -72.2897i q^{33} -2.62006i q^{35} +30.8820 q^{37} +12.3935 q^{39} +41.0208i q^{41} +1.24753i q^{43} -3.89379i q^{45} +1.82806 q^{47} +16.5517 q^{49} +109.568 q^{51} +10.5509 q^{53} +5.10094 q^{55} +(-56.2219 - 66.5719i) q^{57} -13.8763 q^{59} -96.8096i q^{61} +97.4197 q^{63} +0.874518i q^{65} -22.4624 q^{67} -131.772 q^{69} +88.9010i q^{71} -34.3165 q^{73} -114.173 q^{75} +127.621i q^{77} +113.440i q^{79} -44.5120 q^{81} +65.8732i q^{83} +7.73138i q^{85} +237.522 q^{87} -120.371i q^{89} -21.8797 q^{91} +244.645i q^{93} +(4.69747 - 3.96715i) q^{95} +88.2932i q^{97} +189.665i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 264 q^{9}+O(q^{10}) \) 48 * q + 264 * q^9	\( 48 q + 264 q^{9} + 24 q^{17} - 160 q^{25} + 328 q^{49} + 8 q^{57} - 472 q^{73} - 736 q^{81}+O(q^{100}) \) 48 * q + 264 * q^9 + 24 * q^17 - 160 * q^25 + 328 * q^49 + 8 * q^57 - 472 * q^73 - 736 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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