LMFDB - Embedded newform 1152.3.m.f.991.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(415,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

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

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

chi := DirichletCharacter("1152.415");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.m (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + \cdots + 65536 $ x^16 - 6x^14 - 4x^13 + 10x^12 + 56x^11 + 88x^10 - 128x^9 - 496x^8 - 512x^7 + 1408x^6 + 3584x^5 + 2560x^4 - 4096x^3 - 24576*x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			991.4
Root			$-1.25564 + 1.55672i$ of defining polynomial
Character	$\chi$	$=$	1152.991
Dual form			1152.3.m.f.415.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.909023 - 0.909023i) q^{5} -0.654713 q^{7} +O(q^{10})$	$q+(0.909023 - 0.909023i) q^{5} -0.654713 q^{7} +(-13.3760 - 13.3760i) q^{11} +(-8.32795 - 8.32795i) q^{13} +3.93529 q^{17} +(-16.8974 + 16.8974i) q^{19} +23.1787 q^{23} +23.3474i q^{25} +(35.6105 + 35.6105i) q^{29} +45.5687i q^{31} +(-0.595149 + 0.595149i) q^{35} +(-10.1527 + 10.1527i) q^{37} +28.4661i q^{41} +(-22.7354 - 22.7354i) q^{43} -10.7746i q^{47} -48.5714 q^{49} +(41.5142 - 41.5142i) q^{53} -24.3182 q^{55} +(-21.0646 - 21.0646i) q^{59} +(68.7531 + 68.7531i) q^{61} -15.1406 q^{65} +(-67.8242 + 67.8242i) q^{67} -33.3094 q^{71} -18.6331i q^{73} +(8.75745 + 8.75745i) q^{77} +6.29222i q^{79} +(-72.0774 + 72.0774i) q^{83} +(3.57727 - 3.57727i) q^{85} -10.6131i q^{89} +(5.45242 + 5.45242i) q^{91} +30.7202i q^{95} +143.631 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 32 q^{11} + 32 q^{19} + 128 q^{23} + 32 q^{29} + 96 q^{35} + 96 q^{37} - 160 q^{43} + 112 q^{49} - 160 q^{53} - 256 q^{55} - 128 q^{59} + 32 q^{61} + 32 q^{65} - 320 q^{67} - 512 q^{71} + 224 q^{77} - 160 q^{83} - 160 q^{85} + 480 q^{91}+O(q^{100}) $ 16 * q + 32 * q^11 + 32 * q^19 + 128 * q^23 + 32 * q^29 + 96 * q^35 + 96 * q^37 - 160 * q^43 + 112 * q^49 - 160 * q^53 - 256 * q^55 - 128 * q^59 + 32 * q^61 + 32 * q^65 - 320 * q^67 - 512 * q^71 + 224 * q^77 - 160 * q^83 - 160 * q^85 + 480 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.909023	−	0.909023i	0.181805	−	0.181805i	−0.610337	−	0.792142i	$-0.708966\pi$
$5$	0.909023	−	0.909023i	0.181805	−	0.181805i	0.792142	+	0.610337i	$0.208966\pi$
$6$		0			0
$7$	−0.654713			−0.0935305			−0.0467652	−	0.998906i	$-0.514891\pi$
$7$	−0.654713			−0.0935305			−0.0467652	+	0.998906i	$0.514891\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−13.3760	−	13.3760i	−1.21600	−	1.21600i	−0.969021	−	0.246980i	$-0.920562\pi$
$11$	−13.3760	−	13.3760i	−1.21600	−	1.21600i	−0.246980	−	0.969021i	$-0.579438\pi$
$12$		0			0
$13$	−8.32795	−	8.32795i	−0.640612	−	0.640612i	0.310094	−	0.950706i	$-0.399639\pi$
$13$	−8.32795	−	8.32795i	−0.640612	−	0.640612i	−0.950706	+	0.310094i	$0.899639\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.93529			0.231488			0.115744	−	0.993279i	$-0.463075\pi$
$17$	3.93529			0.231488			0.115744	+	0.993279i	$0.463075\pi$
$18$		0			0
$19$	−16.8974	+	16.8974i	−0.889336	+	0.889336i	−0.994459	−	0.105123i	$-0.966476\pi$
$19$	−16.8974	+	16.8974i	−0.889336	+	0.889336i	0.105123	+	0.994459i	$0.466476\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	23.1787			1.00777			0.503884	−	0.863771i	$-0.331904\pi$
$23$	23.1787			1.00777			0.503884	+	0.863771i	$0.331904\pi$
$24$		0			0
$25$			23.3474i			0.933894i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	35.6105	+	35.6105i	1.22795	+	1.22795i	0.964739	+	0.263209i	$0.0847809\pi$
$29$	35.6105	+	35.6105i	1.22795	+	1.22795i	0.263209	+	0.964739i	$0.415219\pi$
$30$		0			0
$31$			45.5687i			1.46996i	0.678089	+	0.734980i	$0.262808\pi$
$31$			45.5687i			1.46996i	−0.678089	+	0.734980i	$0.737192\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.595149	+	0.595149i	−0.0170043	+	0.0170043i
$36$		0			0
$37$	−10.1527	+	10.1527i	−0.274398	+	0.274398i	−0.830868	−	0.556470i	$-0.812156\pi$
$37$	−10.1527	+	10.1527i	−0.274398	+	0.274398i	0.556470	+	0.830868i	$0.312156\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			28.4661i			0.694295i	0.937811	+	0.347148i	$0.112850\pi$
$41$			28.4661i			0.694295i	−0.937811	+	0.347148i	$0.887150\pi$
$42$		0			0
$43$	−22.7354	−	22.7354i	−0.528730	−	0.528730i	0.391464	−	0.920194i	$-0.371969\pi$
$43$	−22.7354	−	22.7354i	−0.528730	−	0.528730i	−0.920194	+	0.391464i	$0.871969\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	10.7746i		−	0.229247i	−0.993409	−	0.114623i	$-0.963434\pi$
$47$		−	10.7746i		−	0.229247i	0.993409	−	0.114623i	$-0.0365661\pi$
$48$		0			0
$49$	−48.5714			−0.991252
$50$		0			0
$51$		0			0
$52$		0			0
$53$	41.5142	−	41.5142i	0.783287	−	0.783287i	−0.197097	−	0.980384i	$-0.563151\pi$
$53$	41.5142	−	41.5142i	0.783287	−	0.783287i	0.980384	+	0.197097i	$0.0631514\pi$
$54$		0			0
$55$	−24.3182			−0.442149
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−21.0646	−	21.0646i	−0.357027	−	0.357027i	0.505689	−	0.862716i	$-0.331238\pi$
$59$	−21.0646	−	21.0646i	−0.357027	−	0.357027i	−0.862716	+	0.505689i	$0.831238\pi$
$60$		0			0
$61$	68.7531	+	68.7531i	1.12710	+	1.12710i	0.990647	+	0.136453i	$0.0435703\pi$
$61$	68.7531	+	68.7531i	1.12710	+	1.12710i	0.136453	+	0.990647i	$0.456430\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−15.1406			−0.232932
$66$		0			0
$67$	−67.8242	+	67.8242i	−1.01230	+	1.01230i	−0.0123779	+	0.999923i	$0.503940\pi$
$67$	−67.8242	+	67.8242i	−1.01230	+	1.01230i	−0.999923	+	0.0123779i	$0.996060\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−33.3094			−0.469147			−0.234573	−	0.972098i	$-0.575369\pi$
$71$	−33.3094			−0.469147			−0.234573	+	0.972098i	$0.575369\pi$
$72$		0			0
$73$		−	18.6331i		−	0.255248i	−0.991823	−	0.127624i	$-0.959265\pi$
$73$		−	18.6331i		−	0.255248i	0.991823	−	0.127624i	$-0.0407351\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.75745	+	8.75745i	0.113733	+	0.113733i
$78$		0			0
$79$			6.29222i			0.0796483i	0.999207	+	0.0398242i	$0.0126798\pi$
$79$			6.29222i			0.0796483i	−0.999207	+	0.0398242i	$0.987320\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−72.0774	+	72.0774i	−0.868402	+	0.868402i	−0.992296	−	0.123894i	$-0.960462\pi$
$83$	−72.0774	+	72.0774i	−0.868402	+	0.868402i	0.123894	+	0.992296i	$0.460462\pi$
$84$		0			0
$85$	3.57727	−	3.57727i	0.0420855	−	0.0420855i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	10.6131i		−	0.119248i	−0.998221	−	0.0596240i	$-0.981010\pi$
$89$		−	10.6131i		−	0.119248i	0.998221	−	0.0596240i	$-0.0189902\pi$
$90$		0			0
$91$	5.45242	+	5.45242i	0.0599167	+	0.0599167i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			30.7202i			0.323371i
$96$		0			0
$97$	143.631			1.48073			0.740366	−	0.672204i	$-0.234652\pi$
$97$	143.631			1.48073			0.740366	+	0.672204i	$0.234652\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−90.3100	+	90.3100i	−0.894159	+	0.894159i	−0.994912	−	0.100753i	$-0.967875\pi$
$101$	−90.3100	+	90.3100i	−0.894159	+	0.894159i	0.100753	+	0.994912i	$0.467875\pi$
$102$		0			0
$103$	−95.1656			−0.923938			−0.461969	−	0.886896i	$-0.652857\pi$
$103$	−95.1656			−0.923938			−0.461969	+	0.886896i	$0.652857\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	27.2524	+	27.2524i	0.254695	+	0.254695i	0.822892	−	0.568197i	$-0.192359\pi$
$107$	27.2524	+	27.2524i	0.254695	+	0.254695i	−0.568197	+	0.822892i	$0.692359\pi$
$108$		0			0
$109$	132.413	+	132.413i	1.21480	+	1.21480i	0.969430	+	0.245366i	$0.0789082\pi$
$109$	132.413	+	132.413i	1.21480	+	1.21480i	0.245366	+	0.969430i	$0.421092\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−37.9551			−0.335886			−0.167943	−	0.985797i	$-0.553712\pi$
$113$	−37.9551			−0.335886			−0.167943	+	0.985797i	$0.553712\pi$
$114$		0			0
$115$	21.0699	−	21.0699i	0.183217	−	0.183217i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2.57649			−0.0216512
$120$		0			0
$121$			236.835i			1.95731i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	43.9488	+	43.9488i	0.351591	+	0.351591i
$126$		0			0
$127$		−	96.5399i		−	0.760157i	−0.924954	−	0.380078i	$-0.875897\pi$
$127$		−	96.5399i		−	0.760157i	0.924954	−	0.380078i	$-0.124103\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−54.5082	+	54.5082i	−0.416093	+	0.416093i	−0.883855	−	0.467762i	$-0.845061\pi$
$131$	−54.5082	+	54.5082i	−0.416093	+	0.416093i	0.467762	+	0.883855i	$0.345061\pi$
$132$		0			0
$133$	11.0629	−	11.0629i	0.0831801	−	0.0831801i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			25.9333i			0.189294i	0.995511	+	0.0946471i	$0.0301723\pi$
$137$			25.9333i			0.189294i	−0.995511	+	0.0946471i	$0.969828\pi$
$138$		0			0
$139$	−3.64066	−	3.64066i	−0.0261918	−	0.0261918i	0.693890	−	0.720081i	$-0.255896\pi$
$139$	−3.64066	−	3.64066i	−0.0261918	−	0.0261918i	−0.720081	+	0.693890i	$0.755896\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			222.789i			1.55797i
$144$		0			0
$145$	64.7415			0.446493
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−18.9718	+	18.9718i	−0.127328	+	0.127328i	−0.767899	−	0.640571i	$-0.778698\pi$
$149$	−18.9718	+	18.9718i	−0.127328	+	0.127328i	0.640571	+	0.767899i	$0.278698\pi$
$150$		0			0
$151$	−103.209			−0.683503			−0.341751	−	0.939790i	$-0.611020\pi$
$151$	−103.209			−0.683503			−0.341751	+	0.939790i	$0.611020\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	41.4230	+	41.4230i	0.267245	+	0.267245i
$156$		0			0
$157$	−88.2067	−	88.2067i	−0.561826	−	0.561826i	0.368000	−	0.929826i	$-0.380043\pi$
$157$	−88.2067	−	88.2067i	−0.561826	−	0.561826i	−0.929826	+	0.368000i	$0.880043\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−15.1754			−0.0942571
$162$		0			0
$163$	−18.8038	+	18.8038i	−0.115361	+	0.115361i	−0.762431	−	0.647070i	$-0.775994\pi$
$163$	−18.8038	+	18.8038i	−0.115361	+	0.115361i	0.647070	+	0.762431i	$0.275994\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−267.105			−1.59943			−0.799715	−	0.600380i	$-0.795016\pi$
$167$	−267.105			−1.59943			−0.799715	+	0.600380i	$0.795016\pi$
$168$		0			0
$169$		−	30.2905i		−	0.179234i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−153.520	−	153.520i	−0.887396	−	0.887396i	0.106876	−	0.994272i	$-0.465915\pi$
$173$	−153.520	−	153.520i	−0.887396	−	0.887396i	−0.994272	+	0.106876i	$0.965915\pi$
$174$		0			0
$175$		−	15.2858i		−	0.0873476i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	123.581	−	123.581i	0.690399	−	0.690399i	−0.271921	−	0.962320i	$-0.587659\pi$
$179$	123.581	−	123.581i	0.690399	−	0.690399i	0.962320	+	0.271921i	$0.0876589\pi$
$180$		0			0
$181$	−122.965	+	122.965i	−0.679364	+	0.679364i	−0.959856	−	0.280493i	$-0.909502\pi$
$181$	−122.965	+	122.965i	−0.679364	+	0.679364i	0.280493	+	0.959856i	$0.409502\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			18.4581i			0.0997737i
$186$		0			0
$187$	−52.6385	−	52.6385i	−0.281489	−	0.281489i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	193.992i		−	1.01566i	−0.861456	−	0.507832i	$-0.830447\pi$
$191$		−	193.992i		−	1.01566i	0.861456	−	0.507832i	$-0.169553\pi$
$192$		0			0
$193$	141.555			0.733444			0.366722	−	0.930331i	$-0.380480\pi$
$193$	141.555			0.733444			0.366722	+	0.930331i	$0.380480\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	28.9507	−	28.9507i	0.146958	−	0.146958i	−0.629800	−	0.776758i	$-0.716863\pi$
$197$	28.9507	−	28.9507i	0.146958	−	0.146958i	0.776758	+	0.629800i	$0.216863\pi$
$198$		0			0
$199$	−27.6253			−0.138821			−0.0694104	−	0.997588i	$-0.522112\pi$
$199$	−27.6253			−0.138821			−0.0694104	+	0.997588i	$0.522112\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−23.3147	−	23.3147i	−0.114851	−	0.114851i
$204$		0			0
$205$	25.8763	+	25.8763i	0.126226	+	0.126226i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	452.039			2.16287
$210$		0			0
$211$	−7.35041	+	7.35041i	−0.0348361	+	0.0348361i	−0.724310	−	0.689474i	$-0.757842\pi$
$211$	−7.35041	+	7.35041i	−0.0348361	+	0.0348361i	0.689474	+	0.724310i	$0.257842\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−41.3340			−0.192251
$216$		0			0
$217$		−	29.8345i		−	0.137486i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−32.7729	−	32.7729i	−0.148294	−	0.148294i
$222$		0			0
$223$			386.106i			1.73142i	0.500549	+	0.865708i	$0.333131\pi$
$223$			386.106i			1.73142i	−0.500549	+	0.865708i	$0.666869\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−49.7286	+	49.7286i	−0.219069	+	0.219069i	−0.808106	−	0.589037i	$-0.799507\pi$
$227$	−49.7286	+	49.7286i	−0.219069	+	0.219069i	0.589037	+	0.808106i	$0.299507\pi$
$228$		0			0
$229$	191.870	−	191.870i	0.837861	−	0.837861i	−0.150716	−	0.988577i	$-0.548158\pi$
$229$	191.870	−	191.870i	0.837861	−	0.837861i	0.988577	+	0.150716i	$0.0481579\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			298.610i			1.28159i	0.767712	+	0.640795i	$0.221395\pi$
$233$			298.610i			1.28159i	−0.767712	+	0.640795i	$0.778605\pi$
$234$		0			0
$235$	−9.79435	−	9.79435i	−0.0416781	−	0.0416781i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	247.352i		−	1.03495i	−0.855700	−	0.517473i	$-0.826873\pi$
$239$		−	247.352i		−	1.03495i	0.855700	−	0.517473i	$-0.173127\pi$
$240$		0			0
$241$	−220.337			−0.914260			−0.457130	−	0.889400i	$-0.651123\pi$
$241$	−220.337			−0.914260			−0.457130	+	0.889400i	$0.651123\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−44.1525	+	44.1525i	−0.180214	+	0.180214i
$246$		0			0
$247$	281.441			1.13944
$248$		0			0
$249$		0			0
$250$		0			0
$251$	162.716	+	162.716i	0.648272	+	0.648272i	0.952575	−	0.304303i	$-0.0984235\pi$
$251$	162.716	+	162.716i	0.648272	+	0.648272i	−0.304303	+	0.952575i	$0.598424\pi$
$252$		0			0
$253$	−310.038	−	310.038i	−1.22545	−	1.22545i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−101.165			−0.393637			−0.196819	−	0.980440i	$-0.563061\pi$
$257$	−101.165			−0.393637			−0.196819	+	0.980440i	$0.563061\pi$
$258$		0			0
$259$	6.64713	−	6.64713i	0.0256646	−	0.0256646i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	323.635			1.23055			0.615276	−	0.788312i	$-0.289045\pi$
$263$	323.635			1.23055			0.615276	+	0.788312i	$0.289045\pi$
$264$		0			0
$265$		−	75.4747i		−	0.284810i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.51275	+	1.51275i	0.00562361	+	0.00562361i	0.709913	−	0.704289i	$-0.248734\pi$
$269$	1.51275	+	1.51275i	0.00562361	+	0.00562361i	−0.704289	+	0.709913i	$0.748734\pi$
$270$		0			0
$271$			166.098i			0.612909i	0.951885	+	0.306454i	$0.0991427\pi$
$271$			166.098i			0.612909i	−0.951885	+	0.306454i	$0.900857\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	312.294	−	312.294i	1.13562	−	1.13562i
$276$		0			0
$277$	−317.830	+	317.830i	−1.14740	+	1.14740i	−0.160338	+	0.987062i	$0.551259\pi$
$277$	−317.830	+	317.830i	−1.14740	+	1.14740i	−0.987062	+	0.160338i	$0.948741\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	402.790i		−	1.43342i	−0.697374	−	0.716708i	$-0.745648\pi$
$281$		−	402.790i		−	1.43342i	0.697374	−	0.716708i	$-0.254352\pi$
$282$		0			0
$283$	192.406	+	192.406i	0.679881	+	0.679881i	0.959973	−	0.280092i	$-0.0903649\pi$
$283$	192.406	+	192.406i	0.679881	+	0.679881i	−0.280092	+	0.959973i	$0.590365\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	18.6371i		−	0.0649378i
$288$		0			0
$289$	−273.513			−0.946413
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−75.3645	+	75.3645i	−0.257217	+	0.257217i	−0.823921	−	0.566704i	$-0.808218\pi$
$293$	−75.3645	+	75.3645i	−0.257217	+	0.257217i	0.566704	+	0.823921i	$0.308218\pi$
$294$		0			0
$295$	−38.2964			−0.129818
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−193.031	−	193.031i	−0.645588	−	0.645588i
$300$		0			0
$301$	14.8852	+	14.8852i	0.0494524	+	0.0494524i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	124.996			0.409824
$306$		0			0
$307$	111.544	−	111.544i	0.363337	−	0.363337i	−0.501703	−	0.865040i	$-0.667293\pi$
$307$	111.544	−	111.544i	0.363337	−	0.363337i	0.865040	+	0.501703i	$0.167293\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	224.484			0.721813			0.360906	−	0.932602i	$-0.382467\pi$
$311$	224.484			0.721813			0.360906	+	0.932602i	$0.382467\pi$
$312$		0			0
$313$		−	488.339i		−	1.56019i	−0.625661	−	0.780095i	$-0.715171\pi$
$313$		−	488.339i		−	1.56019i	0.625661	−	0.780095i	$-0.284829\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	257.361	+	257.361i	0.811863	+	0.811863i	0.984913	−	0.173050i	$-0.0553621\pi$
$317$	257.361	+	257.361i	0.811863	+	0.811863i	−0.173050	+	0.984913i	$0.555362\pi$
$318$		0			0
$319$		−	952.652i		−	2.98637i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−66.4962	+	66.4962i	−0.205871	+	0.205871i
$324$		0			0
$325$	194.436	−	194.436i	0.598263	−	0.598263i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			7.05427i			0.0214416i
$330$		0			0
$331$	123.553	+	123.553i	0.373271	+	0.373271i	0.868667	−	0.495396i	$-0.164977\pi$
$331$	123.553	+	123.553i	0.373271	+	0.373271i	−0.495396	+	0.868667i	$0.664977\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			123.307i			0.368082i
$336$		0			0
$337$	−246.234			−0.730665			−0.365333	−	0.930877i	$-0.619045\pi$
$337$	−246.234			−0.730665			−0.365333	+	0.930877i	$0.619045\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	609.528	−	609.528i	1.78747	−	1.78747i
$342$		0			0
$343$	63.8813			0.186243
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−123.212	−	123.212i	−0.355076	−	0.355076i	0.506918	−	0.861994i	$-0.330785\pi$
$347$	−123.212	−	123.212i	−0.355076	−	0.355076i	−0.861994	+	0.506918i	$0.830785\pi$
$348$		0			0
$349$	−115.371	−	115.371i	−0.330575	−	0.330575i	0.522230	−	0.852805i	$-0.325100\pi$
$349$	−115.371	−	115.371i	−0.330575	−	0.330575i	−0.852805	+	0.522230i	$0.825100\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−650.544			−1.84290			−0.921451	−	0.388495i	$-0.872995\pi$
$353$	−650.544			−1.84290			−0.921451	+	0.388495i	$0.872995\pi$
$354$		0			0
$355$	−30.2790	+	30.2790i	−0.0852930	+	0.0852930i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−94.4878			−0.263197			−0.131599	−	0.991303i	$-0.542011\pi$
$359$	−94.4878			−0.263197			−0.131599	+	0.991303i	$0.542011\pi$
$360$		0			0
$361$		−	210.044i		−	0.581838i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−16.9379	−	16.9379i	−0.0464053	−	0.0464053i
$366$		0			0
$367$			131.379i			0.357982i	0.983851	+	0.178991i	$0.0572832\pi$
$367$			131.379i			0.357982i	−0.983851	+	0.178991i	$0.942717\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−27.1799	+	27.1799i	−0.0732612	+	0.0732612i
$372$		0			0
$373$	275.796	−	275.796i	0.739400	−	0.739400i	−0.233062	−	0.972462i	$-0.574874\pi$
$373$	275.796	−	275.796i	0.739400	−	0.739400i	0.972462	+	0.233062i	$0.0748745\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	593.125i		−	1.57328i
$378$		0			0
$379$	13.0427	+	13.0427i	0.0344135	+	0.0344135i	0.724104	−	0.689691i	$-0.242253\pi$
$379$	13.0427	+	13.0427i	0.0344135	+	0.0344135i	−0.689691	+	0.724104i	$0.742253\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	121.974i		−	0.318470i	−0.987241	−	0.159235i	$-0.949097\pi$
$383$		−	121.974i		−	0.318470i	0.987241	−	0.159235i	$-0.0509027\pi$
$384$		0			0
$385$	15.9214			0.0413544
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−233.267	+	233.267i	−0.599659	+	0.599659i	−0.940222	−	0.340563i	$-0.889382\pi$
$389$	−233.267	+	233.267i	−0.599659	+	0.599659i	0.340563	+	0.940222i	$0.389382\pi$
$390$		0			0
$391$	91.2149			0.233286
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.71977	+	5.71977i	0.0144804	+	0.0144804i
$396$		0			0
$397$	83.7693	+	83.7693i	0.211006	+	0.211006i	0.804695	−	0.593689i	$-0.202329\pi$
$397$	83.7693	+	83.7693i	0.211006	+	0.211006i	−0.593689	+	0.804695i	$0.702329\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−589.134			−1.46916			−0.734581	−	0.678521i	$-0.762621\pi$
$401$	−589.134			−1.46916			−0.734581	+	0.678521i	$0.762621\pi$
$402$		0			0
$403$	379.494	−	379.494i	0.941673	−	0.941673i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	271.606			0.667337
$408$		0			0
$409$			449.285i			1.09850i	0.835659	+	0.549248i	$0.185086\pi$
$409$			449.285i			1.09850i	−0.835659	+	0.549248i	$0.814914\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	13.7913	+	13.7913i	0.0333929	+	0.0333929i
$414$		0			0
$415$			131.040i			0.315759i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−218.639	+	218.639i	−0.521811	+	0.521811i	−0.918118	−	0.396307i	$-0.870292\pi$
$419$	−218.639	+	218.639i	−0.521811	+	0.521811i	0.396307	+	0.918118i	$0.370292\pi$
$420$		0			0
$421$	61.2101	−	61.2101i	0.145392	−	0.145392i	−0.630664	−	0.776056i	$-0.717217\pi$
$421$	61.2101	−	61.2101i	0.145392	−	0.145392i	0.776056	+	0.630664i	$0.217217\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			91.8787i			0.216185i
$426$		0			0
$427$	−45.0136	−	45.0136i	−0.105418	−	0.105418i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			501.119i			1.16269i	0.813657	+	0.581345i	$0.197473\pi$
$431$			501.119i			1.16269i	−0.813657	+	0.581345i	$0.802527\pi$
$432$		0			0
$433$	75.5505			0.174482			0.0872408	−	0.996187i	$-0.472195\pi$
$433$	75.5505			0.174482			0.0872408	+	0.996187i	$0.472195\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−391.659	+	391.659i	−0.896245	+	0.896245i
$438$		0			0
$439$	717.251			1.63383			0.816915	−	0.576758i	$-0.195682\pi$
$439$	717.251			1.63383			0.816915	+	0.576758i	$0.195682\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	299.093	+	299.093i	0.675153	+	0.675153i	0.958899	−	0.283746i	$-0.0915773\pi$
$443$	299.093	+	299.093i	0.675153	+	0.675153i	−0.283746	+	0.958899i	$0.591577\pi$
$444$		0			0
$445$	−9.64753	−	9.64753i	−0.0216798	−	0.0216798i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	44.5560			0.0992339			0.0496170	−	0.998768i	$-0.484200\pi$
$449$	44.5560			0.0992339			0.0496170	+	0.998768i	$0.484200\pi$
$450$		0			0
$451$	380.763	−	380.763i	0.844263	−	0.844263i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	9.91275			0.0217863
$456$		0			0
$457$		−	641.227i		−	1.40312i	−0.712609	−	0.701562i	$-0.752486\pi$
$457$		−	641.227i		−	1.40312i	0.712609	−	0.701562i	$-0.247514\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−393.690	−	393.690i	−0.853991	−	0.853991i	0.136631	−	0.990622i	$-0.456373\pi$
$461$	−393.690	−	393.690i	−0.853991	−	0.853991i	−0.990622	+	0.136631i	$0.956373\pi$
$462$		0			0
$463$		−	395.861i		−	0.854991i	−0.904018	−	0.427495i	$-0.859396\pi$
$463$		−	395.861i		−	0.854991i	0.904018	−	0.427495i	$-0.140604\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−83.1457	+	83.1457i	−0.178042	+	0.178042i	−0.790502	−	0.612460i	$-0.790180\pi$
$467$	−83.1457	+	83.1457i	−0.178042	+	0.178042i	0.612460	+	0.790502i	$0.290180\pi$
$468$		0			0
$469$	44.4054	−	44.4054i	0.0946810	−	0.0946810i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			608.217i			1.28587i
$474$		0			0
$475$	−394.509	−	394.509i	−0.830546	−	0.830546i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	430.043i		−	0.897793i	−0.893584	−	0.448896i	$-0.851817\pi$
$479$		−	430.043i		−	0.897793i	0.893584	−	0.448896i	$-0.148183\pi$
$480$		0			0
$481$	169.103			0.351565
$482$		0			0
$483$		0			0
$484$		0			0
$485$	130.564	−	130.564i	0.269204	−	0.269204i
$486$		0			0
$487$	573.790			1.17821			0.589107	−	0.808055i	$-0.299480\pi$
$487$	573.790			1.17821			0.589107	+	0.808055i	$0.299480\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−489.133	−	489.133i	−0.996197	−	0.996197i	0.00379588	−	0.999993i	$-0.498792\pi$
$491$	−489.133	−	489.133i	−0.996197	−	0.996197i	−0.999993	+	0.00379588i	$0.998792\pi$
$492$		0			0
$493$	140.138	+	140.138i	0.284255	+	0.284255i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	21.8081			0.0438795
$498$		0			0
$499$	−260.469	+	260.469i	−0.521982	+	0.521982i	−0.918170	−	0.396188i	$-0.870333\pi$
$499$	−260.469	+	260.469i	−0.521982	+	0.521982i	0.396188	+	0.918170i	$0.370333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	975.416			1.93920			0.969598	−	0.244701i	$-0.0786900\pi$
$503$	975.416			1.93920			0.969598	+	0.244701i	$0.0786900\pi$
$504$		0			0
$505$			164.188i			0.325124i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−420.191	−	420.191i	−0.825523	−	0.825523i	0.161371	−	0.986894i	$-0.448408\pi$
$509$	−420.191	−	420.191i	−0.825523	−	0.825523i	−0.986894	+	0.161371i	$0.948408\pi$
$510$		0			0
$511$			12.1994i			0.0238735i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−86.5077	+	86.5077i	−0.167976	+	0.167976i
$516$		0			0
$517$	−144.121	+	144.121i	−0.278764	+	0.278764i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	396.333i		−	0.760716i	−0.924839	−	0.380358i	$-0.875801\pi$
$521$		−	396.333i		−	0.760716i	0.924839	−	0.380358i	$-0.124199\pi$
$522$		0			0
$523$	−564.600	−	564.600i	−1.07954	−	1.07954i	−0.996550	−	0.0829913i	$-0.973553\pi$
$523$	−564.600	−	564.600i	−1.07954	−	1.07954i	−0.0829913	−	0.996550i	$-0.526447\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			179.326i			0.340278i
$528$		0			0
$529$	8.25115			0.0155976
$530$		0			0
$531$		0			0
$532$		0			0
$533$	237.064	−	237.064i	0.444773	−	0.444773i
$534$		0			0
$535$	49.5461			0.0926095
$536$		0			0
$537$		0			0
$538$		0			0
$539$	649.691	+	649.691i	1.20536	+	1.20536i
$540$		0			0
$541$	29.5601	+	29.5601i	0.0546398	+	0.0546398i	0.733899	−	0.679259i	$-0.237699\pi$
$541$	29.5601	+	29.5601i	0.0546398	+	0.0546398i	−0.679259	+	0.733899i	$0.737699\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	240.733			0.441711
$546$		0			0
$547$	138.608	−	138.608i	0.253397	−	0.253397i	−0.568965	−	0.822362i	$-0.692656\pi$
$547$	138.608	−	138.608i	0.253397	−	0.253397i	0.822362	+	0.568965i	$0.192656\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1203.45			−2.18412
$552$		0			0
$553$		−	4.11960i		−	0.00744955i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	60.4400	+	60.4400i	0.108510	+	0.108510i	0.759277	−	0.650767i	$-0.225553\pi$
$557$	60.4400	+	60.4400i	0.108510	+	0.108510i	−0.650767	+	0.759277i	$0.725553\pi$
$558$		0			0
$559$			378.678i			0.677421i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	267.325	−	267.325i	0.474822	−	0.474822i	−0.428649	−	0.903471i	$-0.641010\pi$
$563$	267.325	−	267.325i	0.474822	−	0.474822i	0.903471	+	0.428649i	$0.141010\pi$
$564$		0			0
$565$	−34.5021	+	34.5021i	−0.0610656	+	0.0610656i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			315.715i			0.554859i	0.960746	+	0.277429i	$0.0894825\pi$
$569$			315.715i			0.554859i	−0.960746	+	0.277429i	$0.910518\pi$
$570$		0			0
$571$	670.572	+	670.572i	1.17438	+	1.17438i	0.981154	+	0.193228i	$0.0618956\pi$
$571$	670.572	+	670.572i	1.17438	+	1.17438i	0.193228	+	0.981154i	$0.438104\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			541.161i			0.941149i
$576$		0			0
$577$	413.628			0.716859			0.358430	−	0.933557i	$-0.383312\pi$
$577$	413.628			0.716859			0.358430	+	0.933557i	$0.383312\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	47.1900	−	47.1900i	0.0812220	−	0.0812220i
$582$		0			0
$583$	−1110.59			−1.90495
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−420.085	−	420.085i	−0.715647	−	0.715647i	0.252064	−	0.967711i	$-0.418891\pi$
$587$	−420.085	−	420.085i	−0.715647	−	0.715647i	−0.967711	+	0.252064i	$0.918891\pi$
$588$		0			0
$589$	−769.993	−	769.993i	−1.30729	−	1.30729i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	740.798			1.24924			0.624619	−	0.780930i	$-0.285254\pi$
$593$	740.798			1.24924			0.624619	+	0.780930i	$0.285254\pi$
$594$		0			0
$595$	−2.34209	+	2.34209i	−0.00393628	+	0.00393628i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−435.161			−0.726479			−0.363240	−	0.931696i	$-0.618329\pi$
$599$	−435.161			−0.726479			−0.363240	+	0.931696i	$0.618329\pi$
$600$		0			0
$601$			380.001i			0.632280i	0.948712	+	0.316140i	$0.102387\pi$
$601$			380.001i			0.632280i	−0.948712	+	0.316140i	$0.897613\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	215.288	+	215.288i	0.355849	+	0.355849i
$606$		0			0
$607$			181.813i			0.299527i	0.988722	+	0.149763i	$0.0478512\pi$
$607$			181.813i			0.299527i	−0.988722	+	0.149763i	$0.952149\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−89.7303	+	89.7303i	−0.146858	+	0.146858i
$612$		0			0
$613$	−55.1479	+	55.1479i	−0.0899640	+	0.0899640i	−0.750657	−	0.660693i	$-0.770263\pi$
$613$	−55.1479	+	55.1479i	−0.0899640	+	0.0899640i	0.660693	+	0.750657i	$0.270263\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	579.674i		−	0.939504i	−0.882798	−	0.469752i	$-0.844343\pi$
$617$		−	579.674i		−	0.939504i	0.882798	−	0.469752i	$-0.155657\pi$
$618$		0			0
$619$	−91.1070	−	91.1070i	−0.147184	−	0.147184i	0.629675	−	0.776859i	$-0.283188\pi$
$619$	−91.1070	−	91.1070i	−0.147184	−	0.147184i	−0.776859	+	0.629675i	$0.783188\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			6.94852i			0.0111533i
$624$		0			0
$625$	−503.783			−0.806053
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−39.9540	+	39.9540i	−0.0635199	+	0.0635199i
$630$		0			0
$631$	−693.474			−1.09901			−0.549504	−	0.835491i	$-0.685183\pi$
$631$	−693.474			−1.09901			−0.549504	+	0.835491i	$0.685183\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−87.7570	−	87.7570i	−0.138200	−	0.138200i
$636$		0			0
$637$	404.500	+	404.500i	0.635007	+	0.635007i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	218.329			0.340607			0.170304	−	0.985392i	$-0.445525\pi$
$641$	218.329			0.340607			0.170304	+	0.985392i	$0.445525\pi$
$642$		0			0
$643$	−887.430	+	887.430i	−1.38014	+	1.38014i	−0.535787	+	0.844353i	$0.679985\pi$
$643$	−887.430	+	887.430i	−1.38014	+	1.38014i	−0.844353	+	0.535787i	$0.820015\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	223.177			0.344941			0.172470	−	0.985015i	$-0.444825\pi$
$647$	223.177			0.344941			0.172470	+	0.985015i	$0.444825\pi$
$648$		0			0
$649$			563.520i			0.868290i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	539.691	+	539.691i	0.826479	+	0.826479i	0.987028	−	0.160549i	$-0.0513264\pi$
$653$	539.691	+	539.691i	0.826479	+	0.826479i	−0.160549	+	0.987028i	$0.551326\pi$
$654$		0			0
$655$			99.0983i			0.151295i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	625.166	−	625.166i	0.948659	−	0.948659i	−0.0500862	−	0.998745i	$-0.515950\pi$
$659$	625.166	−	625.166i	0.948659	−	0.948659i	0.998745	+	0.0500862i	$0.0159496\pi$
$660$		0			0
$661$	326.893	−	326.893i	0.494544	−	0.494544i	−0.415191	−	0.909734i	$-0.636285\pi$
$661$	326.893	−	326.893i	0.494544	−	0.494544i	0.909734	+	0.415191i	$0.136285\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	20.1129i		−	0.0302450i
$666$		0			0
$667$	825.404	+	825.404i	1.23749	+	1.23749i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	1839.28i		−	2.74111i
$672$		0			0
$673$	422.147			0.627262			0.313631	−	0.949545i	$-0.398455\pi$
$673$	422.147			0.627262			0.313631	+	0.949545i	$0.398455\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−126.017	+	126.017i	−0.186140	+	0.186140i	−0.794025	−	0.607885i	$-0.792018\pi$
$677$	−126.017	+	126.017i	−0.186140	+	0.186140i	0.607885	+	0.794025i	$0.292018\pi$
$678$		0			0
$679$	−94.0372			−0.138494
$680$		0			0
$681$		0			0
$682$		0			0
$683$	621.906	+	621.906i	0.910551	+	0.910551i	0.996315	−	0.0857647i	$-0.0273333\pi$
$683$	621.906	+	621.906i	0.910551	+	0.910551i	−0.0857647	+	0.996315i	$0.527333\pi$
$684$		0			0
$685$	23.5740	+	23.5740i	0.0344145	+	0.0344145i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−691.456			−1.00357
$690$		0			0
$691$	403.376	−	403.376i	0.583758	−	0.583758i	−0.352176	−	0.935934i	$-0.614558\pi$
$691$	403.376	−	403.376i	0.583758	−	0.583758i	0.935934	+	0.352176i	$0.114558\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.61889			−0.00952359
$696$		0			0
$697$			112.022i			0.160721i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	466.593	+	466.593i	0.665611	+	0.665611i	0.956697	−	0.291086i	$-0.0940166\pi$
$701$	466.593	+	466.593i	0.665611	+	0.665611i	−0.291086	+	0.956697i	$0.594017\pi$
$702$		0			0
$703$		−	343.109i		−	0.488065i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	59.1272	−	59.1272i	0.0836311	−	0.0836311i
$708$		0			0
$709$	−822.764	+	822.764i	−1.16046	+	1.16046i	−0.176081	+	0.984376i	$0.556342\pi$
$709$	−822.764	+	822.764i	−1.16046	+	1.16046i	−0.984376	+	0.176081i	$0.943658\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			1056.22i			1.48138i
$714$		0			0
$715$	202.521	+	202.521i	0.283246	+	0.283246i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			710.142i			0.987681i	0.869553	+	0.493840i	$0.164407\pi$
$719$			710.142i			0.987681i	−0.869553	+	0.493840i	$0.835593\pi$
$720$		0			0
$721$	62.3062			0.0864164
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−831.411	+	831.411i	−1.14677	+	1.14677i
$726$		0			0
$727$	−214.095			−0.294490			−0.147245	−	0.989100i	$-0.547041\pi$
$727$	−214.095			−0.294490			−0.147245	+	0.989100i	$0.547041\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−89.4704	−	89.4704i	−0.122395	−	0.122395i
$732$		0			0
$733$	96.1768	+	96.1768i	0.131210	+	0.131210i	0.769662	−	0.638452i	$-0.220425\pi$
$733$	96.1768	+	96.1768i	0.131210	+	0.131210i	−0.638452	+	0.769662i	$0.720425\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1814.43			2.46192
$738$		0			0
$739$	−885.341	+	885.341i	−1.19803	+	1.19803i	−0.223268	+	0.974757i	$0.571673\pi$
$739$	−885.341	+	885.341i	−1.19803	+	1.19803i	−0.974757	+	0.223268i	$0.928327\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−906.258			−1.21973			−0.609864	−	0.792506i	$-0.708776\pi$
$743$	−906.258			−1.21973			−0.609864	+	0.792506i	$0.708776\pi$
$744$		0			0
$745$			34.4917i			0.0462976i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−17.8425	−	17.8425i	−0.0238218	−	0.0238218i
$750$		0			0
$751$			1147.02i			1.52732i	0.645618	+	0.763661i	$0.276600\pi$
$751$			1147.02i			1.52732i	−0.645618	+	0.763661i	$0.723400\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−93.8192	+	93.8192i	−0.124264	+	0.124264i
$756$		0			0
$757$	−525.591	+	525.591i	−0.694308	+	0.694308i	−0.963177	−	0.268869i	$-0.913350\pi$
$757$	−525.591	+	525.591i	−0.694308	+	0.694308i	0.268869	+	0.963177i	$0.413350\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			788.107i			1.03562i	0.855495	+	0.517810i	$0.173253\pi$
$761$			788.107i			1.03562i	−0.855495	+	0.517810i	$0.826747\pi$
$762$		0			0
$763$	−86.6925	−	86.6925i	−0.113621	−	0.113621i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			350.850i			0.457431i
$768$		0			0
$769$	−768.187			−0.998943			−0.499471	−	0.866330i	$-0.666472\pi$
$769$	−768.187			−0.998943			−0.499471	+	0.866330i	$0.666472\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	275.915	−	275.915i	0.356941	−	0.356941i	−0.505743	−	0.862684i	$-0.668782\pi$
$773$	275.915	−	275.915i	0.356941	−	0.356941i	0.862684	+	0.505743i	$0.168782\pi$
$774$		0			0
$775$	−1063.91			−1.37279
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−481.003	−	481.003i	−0.617462	−	0.617462i
$780$		0			0
$781$	445.547	+	445.547i	0.570483	+	0.570483i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−160.364			−0.204285
$786$		0			0
$787$	−240.824	+	240.824i	−0.306002	+	0.306002i	−0.843356	−	0.537354i	$-0.819424\pi$
$787$	−240.824	+	240.824i	−0.306002	+	0.306002i	0.537354	+	0.843356i	$0.319424\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	24.8497			0.0314156
$792$		0			0
$793$		−	1145.14i		−	1.44407i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−681.033	−	681.033i	−0.854496	−	0.854496i	0.136187	−	0.990683i	$-0.456515\pi$
$797$	−681.033	−	681.033i	−0.854496	−	0.854496i	−0.990683	+	0.136187i	$0.956515\pi$
$798$		0			0
$799$		−	42.4012i		−	0.0530678i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−249.237	+	249.237i	−0.310382	+	0.310382i
$804$		0			0
$805$	−13.7948	+	13.7948i	−0.0171364	+	0.0171364i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	445.322i		−	0.550459i	−0.961379	−	0.275230i	$-0.911246\pi$
$809$		−	445.322i		−	0.550459i	0.961379	−	0.275230i	$-0.0887539\pi$
$810$		0			0
$811$	−223.787	−	223.787i	−0.275939	−	0.275939i	0.555546	−	0.831485i	$-0.312509\pi$
$811$	−223.787	−	223.787i	−0.275939	−	0.275939i	−0.831485	+	0.555546i	$0.812509\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			34.1861i			0.0419462i
$816$		0			0
$817$	768.337			0.940437
$818$		0			0
$819$		0			0
$820$		0			0
$821$	382.528	−	382.528i	0.465929	−	0.465929i	−0.434664	−	0.900593i	$-0.643133\pi$
$821$	382.528	−	382.528i	0.465929	−	0.465929i	0.900593	+	0.434664i	$0.143133\pi$
$822$		0			0
$823$	−730.046			−0.887055			−0.443527	−	0.896261i	$-0.646273\pi$
$823$	−730.046			−0.887055			−0.443527	+	0.896261i	$0.646273\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	252.094	+	252.094i	0.304829	+	0.304829i	0.842900	−	0.538071i	$-0.180847\pi$
$827$	252.094	+	252.094i	0.304829	+	0.304829i	−0.538071	+	0.842900i	$0.680847\pi$
$828$		0			0
$829$	870.285	+	870.285i	1.04980	+	1.04980i	0.998693	+	0.0511072i	$0.0162750\pi$
$829$	870.285	+	870.285i	1.04980	+	1.04980i	0.0511072	+	0.998693i	$0.483725\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−191.143			−0.229463
$834$		0			0
$835$	−242.804	+	242.804i	−0.290784	+	0.290784i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−550.145			−0.655715			−0.327858	−	0.944727i	$-0.606327\pi$
$839$	−550.145			−0.655715			−0.327858	+	0.944727i	$0.606327\pi$
$840$		0			0
$841$			1695.21i			2.01571i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−27.5348	−	27.5348i	−0.0325855	−	0.0325855i
$846$		0			0
$847$		−	155.059i		−	0.183069i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−235.327	+	235.327i	−0.276530	+	0.276530i
$852$		0			0
$853$	676.266	−	676.266i	0.792809	−	0.792809i	−0.189141	−	0.981950i	$-0.560570\pi$
$853$	676.266	−	676.266i	0.792809	−	0.792809i	0.981950	+	0.189141i	$0.0605703\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	291.272i		−	0.339874i	−0.985455	−	0.169937i	$-0.945644\pi$
$857$		−	291.272i		−	0.339874i	0.985455	−	0.169937i	$-0.0543563\pi$
$858$		0			0
$859$	−988.357	−	988.357i	−1.15059	−	1.15059i	−0.986434	−	0.164156i	$-0.947510\pi$
$859$	−988.357	−	988.357i	−1.15059	−	1.15059i	−0.164156	−	0.986434i	$-0.552490\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	922.512i		−	1.06896i	−0.845181	−	0.534480i	$-0.820508\pi$
$863$		−	922.512i		−	1.06896i	0.845181	−	0.534480i	$-0.179492\pi$
$864$		0			0
$865$	−279.105			−0.322665
$866$		0			0
$867$		0			0
$868$		0			0
$869$	84.1648	−	84.1648i	0.0968524	−	0.0968524i
$870$		0			0
$871$	1129.67			1.29698
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−28.7739	−	28.7739i	−0.0328845	−	0.0328845i
$876$		0			0
$877$	406.278	+	406.278i	0.463259	+	0.463259i	0.899722	−	0.436463i	$-0.143769\pi$
$877$	406.278	+	406.278i	0.463259	+	0.463259i	−0.436463	+	0.899722i	$0.643769\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1009.42			1.14577			0.572884	−	0.819637i	$-0.305825\pi$
$881$	1009.42			1.14577			0.572884	+	0.819637i	$0.305825\pi$
$882$		0			0
$883$	−56.3792	+	56.3792i	−0.0638496	+	0.0638496i	−0.738311	−	0.674461i	$-0.764376\pi$
$883$	−56.3792	+	56.3792i	−0.0638496	+	0.0638496i	0.674461	+	0.738311i	$0.264376\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−21.0431			−0.0237239			−0.0118620	−	0.999930i	$-0.503776\pi$
$887$	−21.0431			−0.0237239			−0.0118620	+	0.999930i	$0.503776\pi$
$888$		0			0
$889$			63.2060i			0.0710978i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	182.063	+	182.063i	0.203877	+	0.203877i
$894$		0			0
$895$		−	224.677i		−	0.251035i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1622.72	+	1622.72i	−1.80503	+	1.80503i
$900$		0			0
$901$	163.371	−	163.371i	0.181321	−	0.181321i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			223.556i			0.247023i
$906$		0			0
$907$	−496.202	−	496.202i	−0.547080	−	0.547080i	0.378515	−	0.925595i	$-0.376435\pi$
$907$	−496.202	−	496.202i	−0.547080	−	0.547080i	−0.925595	+	0.378515i	$0.876435\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1532.15i			1.68183i	0.541163	+	0.840917i	$0.317984\pi$
$911$			1532.15i			1.68183i	−0.541163	+	0.840917i	$0.682016\pi$
$912$		0			0
$913$	1928.21			2.11195
$914$		0			0
$915$		0			0
$916$		0			0
$917$	35.6872	−	35.6872i	0.0389174	−	0.0389174i
$918$		0			0
$919$	727.639			0.791773			0.395886	−	0.918299i	$-0.370437\pi$
$919$	727.639			0.791773			0.395886	+	0.918299i	$0.370437\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	277.399	+	277.399i	0.300541	+	0.300541i
$924$		0			0
$925$	−237.040	−	237.040i	−0.256259	−	0.256259i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1544.59			−1.66264			−0.831318	−	0.555797i	$-0.812413\pi$
$929$	−1544.59			−1.66264			−0.831318	+	0.555797i	$0.812413\pi$
$930$		0			0
$931$	820.729	−	820.729i	0.881556	−	0.881556i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−95.6992			−0.102352
$936$		0			0
$937$			716.111i			0.764259i	0.924109	+	0.382130i	$0.124809\pi$
$937$			716.111i			0.764259i	−0.924109	+	0.382130i	$0.875191\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	537.759	+	537.759i	0.571476	+	0.571476i	0.932541	−	0.361065i	$-0.117587\pi$
$941$	537.759	+	537.759i	0.571476	+	0.571476i	−0.361065	+	0.932541i	$0.617587\pi$
$942$		0			0
$943$			659.807i			0.699689i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	734.420	−	734.420i	0.775523	−	0.775523i	−0.203543	−	0.979066i	$-0.565246\pi$
$947$	734.420	−	734.420i	0.775523	−	0.775523i	0.979066	+	0.203543i	$0.0652457\pi$
$948$		0			0
$949$	−155.176	+	155.176i	−0.163515	+	0.163515i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	138.512i		−	0.145343i	−0.997356	−	0.0726714i	$-0.976848\pi$
$953$		−	138.512i		−	0.145343i	0.997356	−	0.0726714i	$-0.0231524\pi$
$954$		0			0
$955$	−176.343	−	176.343i	−0.184652	−	0.184652i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	16.9789i		−	0.0177048i
$960$		0			0
$961$	−1115.51			−1.16078
$962$		0			0
$963$		0			0
$964$		0			0
$965$	128.676	−	128.676i	0.133343	−	0.133343i
$966$		0			0
$967$	−701.820			−0.725770			−0.362885	−	0.931834i	$-0.618208\pi$
$967$	−701.820			−0.725770			−0.362885	+	0.931834i	$0.618208\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−62.5684	−	62.5684i	−0.0644371	−	0.0644371i	0.674154	−	0.738591i	$-0.264508\pi$
$971$	−62.5684	−	62.5684i	−0.0644371	−	0.0644371i	−0.738591	+	0.674154i	$0.764508\pi$
$972$		0			0
$973$	2.38359	+	2.38359i	0.00244973	+	0.00244973i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1740.21			−1.78117			−0.890587	−	0.454814i	$-0.849706\pi$
$977$	−1740.21			−1.78117			−0.890587	+	0.454814i	$0.849706\pi$
$978$		0			0
$979$	−141.961	+	141.961i	−0.145006	+	0.145006i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1451.81			1.47692			0.738460	−	0.674298i	$-0.235554\pi$
$983$	1451.81			1.47692			0.738460	+	0.674298i	$0.235554\pi$
$984$		0			0
$985$		−	52.6337i		−	0.0534353i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−526.976	−	526.976i	−0.532837	−	0.532837i
$990$		0			0
$991$		−	22.1684i		−	0.0223698i	−0.999937	−	0.0111849i	$-0.996440\pi$
$991$		−	22.1684i		−	0.0223698i	0.999937	−	0.0111849i	$-0.00356033\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−25.1121	+	25.1121i	−0.0252382	+	0.0252382i
$996$		0			0
$997$	−13.5020	+	13.5020i	−0.0135426	+	0.0135426i	−0.713846	−	0.700303i	$-0.753048\pi$
$997$	−13.5020	+	13.5020i	−0.0135426	+	0.0135426i	0.700303	+	0.713846i	$0.253048\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.m.f.991.4		16
3.2	odd	2		384.3.l.a.223.3		16
4.3	odd	2		1152.3.m.c.991.4		16
8.3	odd	2		576.3.m.c.559.5		16
8.5	even	2		144.3.m.c.19.3		16
12.11	even	2		384.3.l.b.223.7		16
16.3	odd	4		144.3.m.c.91.3		16
16.5	even	4		1152.3.m.c.415.4		16
16.11	odd	4	inner	1152.3.m.f.415.4		16
16.13	even	4		576.3.m.c.271.5		16
24.5	odd	2		48.3.l.a.19.6	✓	16
24.11	even	2		192.3.l.a.175.2		16
48.5	odd	4		384.3.l.b.31.7		16
48.11	even	4		384.3.l.a.31.3		16
48.29	odd	4		192.3.l.a.79.2		16
48.35	even	4		48.3.l.a.43.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.l.a.19.6	✓	16	24.5	odd	2
48.3.l.a.43.6	yes	16	48.35	even	4
144.3.m.c.19.3		16	8.5	even	2
144.3.m.c.91.3		16	16.3	odd	4
192.3.l.a.79.2		16	48.29	odd	4
192.3.l.a.175.2		16	24.11	even	2
384.3.l.a.31.3		16	48.11	even	4
384.3.l.a.223.3		16	3.2	odd	2
384.3.l.b.31.7		16	48.5	odd	4
384.3.l.b.223.7		16	12.11	even	2
576.3.m.c.271.5		16	16.13	even	4
576.3.m.c.559.5		16	8.3	odd	2
1152.3.m.c.415.4		16	16.5	even	4
1152.3.m.c.991.4		16	4.3	odd	2
1152.3.m.f.415.4		16	16.11	odd	4	inner
1152.3.m.f.991.4		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1152.m (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.909023 - 0.909023i) q^{5} -0.654713 q^{7} +O(q^{10})\)	\(q+(0.909023 - 0.909023i) q^{5} -0.654713 q^{7} +(-13.3760 - 13.3760i) q^{11} +(-8.32795 - 8.32795i) q^{13} +3.93529 q^{17} +(-16.8974 + 16.8974i) q^{19} +23.1787 q^{23} +23.3474i q^{25} +(35.6105 + 35.6105i) q^{29} +45.5687i q^{31} +(-0.595149 + 0.595149i) q^{35} +(-10.1527 + 10.1527i) q^{37} +28.4661i q^{41} +(-22.7354 - 22.7354i) q^{43} -10.7746i q^{47} -48.5714 q^{49} +(41.5142 - 41.5142i) q^{53} -24.3182 q^{55} +(-21.0646 - 21.0646i) q^{59} +(68.7531 + 68.7531i) q^{61} -15.1406 q^{65} +(-67.8242 + 67.8242i) q^{67} -33.3094 q^{71} -18.6331i q^{73} +(8.75745 + 8.75745i) q^{77} +6.29222i q^{79} +(-72.0774 + 72.0774i) q^{83} +(3.57727 - 3.57727i) q^{85} -10.6131i q^{89} +(5.45242 + 5.45242i) q^{91} +30.7202i q^{95} +143.631 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 32 q^{11} + 32 q^{19} + 128 q^{23} + 32 q^{29} + 96 q^{35} + 96 q^{37} - 160 q^{43} + 112 q^{49} - 160 q^{53} - 256 q^{55} - 128 q^{59} + 32 q^{61} + 32 q^{65} - 320 q^{67} - 512 q^{71} + 224 q^{77} - 160 q^{83} - 160 q^{85} + 480 q^{91}+O(q^{100}) \) 16 * q + 32 * q^11 + 32 * q^19 + 128 * q^23 + 32 * q^29 + 96 * q^35 + 96 * q^37 - 160 * q^43 + 112 * q^49 - 160 * q^53 - 256 * q^55 - 128 * q^59 + 32 * q^61 + 32 * q^65 - 320 * q^67 - 512 * q^71 + 224 * q^77 - 160 * q^83 - 160 * q^85 + 480 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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