LMFDB - Embedded newform 1152.3.m.f.415.4

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			415.4
Root			$-1.25564 - 1.55672i$ of defining polynomial
Character	$\chi$	$=$	1152.415
Dual form			1152.3.m.f.991.4

$q$-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 $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

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

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	0.792142	−	0.610337i	$-0.208966\pi$
$5$	0.909023	+	0.909023i	0.181805	+	0.181805i	−0.610337	+	0.792142i	$0.708966\pi$
$6$		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	−0.246980	+	0.969021i	$0.579438\pi$
$11$	−13.3760	+	13.3760i	−1.21600	+	1.21600i	−0.969021	+	0.246980i	$0.920562\pi$
$12$		0			0
$13$	−8.32795	+	8.32795i	−0.640612	+	0.640612i	−0.950706	−	0.310094i	$-0.899639\pi$
$13$	−8.32795	+	8.32795i	−0.640612	+	0.640612i	0.310094	+	0.950706i	$0.399639\pi$
$14$		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	0.105123	−	0.994459i	$-0.466476\pi$
$19$	−16.8974	−	16.8974i	−0.889336	−	0.889336i	−0.994459	+	0.105123i	$0.966476\pi$
$20$		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	0.263209	−	0.964739i	$-0.415219\pi$
$29$	35.6105	−	35.6105i	1.22795	−	1.22795i	0.964739	−	0.263209i	$-0.0847809\pi$
$30$		0			0
$31$		−	45.5687i		−	1.46996i	−0.678089	−	0.734980i	$-0.737192\pi$
$31$		−	45.5687i		−	1.46996i	0.678089	−	0.734980i	$-0.262808\pi$
$32$		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	0.556470	−	0.830868i	$-0.312156\pi$
$37$	−10.1527	−	10.1527i	−0.274398	−	0.274398i	−0.830868	+	0.556470i	$0.812156\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	28.4661i		−	0.694295i	−0.937811	−	0.347148i	$-0.887150\pi$
$41$		−	28.4661i		−	0.694295i	0.937811	−	0.347148i	$-0.112850\pi$
$42$		0			0
$43$	−22.7354	+	22.7354i	−0.528730	+	0.528730i	−0.920194	−	0.391464i	$-0.871969\pi$
$43$	−22.7354	+	22.7354i	−0.528730	+	0.528730i	0.391464	+	0.920194i	$0.371969\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			10.7746i			0.229247i	0.993409	+	0.114623i	$0.0365661\pi$
$47$			10.7746i			0.229247i	−0.993409	+	0.114623i	$0.963434\pi$
$48$		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	0.980384	−	0.197097i	$-0.0631514\pi$
$53$	41.5142	+	41.5142i	0.783287	+	0.783287i	−0.197097	+	0.980384i	$0.563151\pi$
$54$		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	−0.862716	−	0.505689i	$-0.831238\pi$
$59$	−21.0646	+	21.0646i	−0.357027	+	0.357027i	0.505689	+	0.862716i	$0.331238\pi$
$60$		0			0
$61$	68.7531	−	68.7531i	1.12710	−	1.12710i	0.136453	−	0.990647i	$-0.456430\pi$
$61$	68.7531	−	68.7531i	1.12710	−	1.12710i	0.990647	−	0.136453i	$-0.0435703\pi$
$62$		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	−0.999923	−	0.0123779i	$-0.996060\pi$
$67$	−67.8242	−	67.8242i	−1.01230	−	1.01230i	−0.0123779	−	0.999923i	$-0.503940\pi$
$68$		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	$0.0407351\pi$
$73$			18.6331i			0.255248i	−0.991823	+	0.127624i	$0.959265\pi$
$74$		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	$-0.987320\pi$
$79$		−	6.29222i		−	0.0796483i	0.999207	−	0.0398242i	$-0.0126798\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−72.0774	−	72.0774i	−0.868402	−	0.868402i	0.123894	−	0.992296i	$-0.460462\pi$
$83$	−72.0774	−	72.0774i	−0.868402	−	0.868402i	−0.992296	+	0.123894i	$0.960462\pi$
$84$		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	$0.0189902\pi$
$89$			10.6131i			0.119248i	−0.998221	+	0.0596240i	$0.981010\pi$
$90$		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	0.100753	−	0.994912i	$-0.467875\pi$
$101$	−90.3100	−	90.3100i	−0.894159	−	0.894159i	−0.994912	+	0.100753i	$0.967875\pi$
$102$		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	−0.568197	−	0.822892i	$-0.692359\pi$
$107$	27.2524	−	27.2524i	0.254695	−	0.254695i	0.822892	+	0.568197i	$0.192359\pi$
$108$		0			0
$109$	132.413	−	132.413i	1.21480	−	1.21480i	0.245366	−	0.969430i	$-0.421092\pi$
$109$	132.413	−	132.413i	1.21480	−	1.21480i	0.969430	−	0.245366i	$-0.0789082\pi$
$110$		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	$0.124103\pi$
$127$			96.5399i			0.760157i	−0.924954	+	0.380078i	$0.875897\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−54.5082	−	54.5082i	−0.416093	−	0.416093i	0.467762	−	0.883855i	$-0.345061\pi$
$131$	−54.5082	−	54.5082i	−0.416093	−	0.416093i	−0.883855	+	0.467762i	$0.845061\pi$
$132$		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	$-0.969828\pi$
$137$		−	25.9333i		−	0.189294i	0.995511	−	0.0946471i	$-0.0301723\pi$
$138$		0			0
$139$	−3.64066	+	3.64066i	−0.0261918	+	0.0261918i	−0.720081	−	0.693890i	$-0.755896\pi$
$139$	−3.64066	+	3.64066i	−0.0261918	+	0.0261918i	0.693890	+	0.720081i	$0.255896\pi$
$140$		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	0.640571	−	0.767899i	$-0.278698\pi$
$149$	−18.9718	−	18.9718i	−0.127328	−	0.127328i	−0.767899	+	0.640571i	$0.778698\pi$
$150$		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	−0.929826	−	0.368000i	$-0.880043\pi$
$157$	−88.2067	+	88.2067i	−0.561826	+	0.561826i	0.368000	+	0.929826i	$0.380043\pi$
$158$		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	0.647070	−	0.762431i	$-0.275994\pi$
$163$	−18.8038	−	18.8038i	−0.115361	−	0.115361i	−0.762431	+	0.647070i	$0.775994\pi$
$164$		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	−0.994272	−	0.106876i	$-0.965915\pi$
$173$	−153.520	+	153.520i	−0.887396	+	0.887396i	0.106876	+	0.994272i	$0.465915\pi$
$174$		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	0.962320	−	0.271921i	$-0.0876589\pi$
$179$	123.581	+	123.581i	0.690399	+	0.690399i	−0.271921	+	0.962320i	$0.587659\pi$
$180$		0			0
$181$	−122.965	−	122.965i	−0.679364	−	0.679364i	0.280493	−	0.959856i	$-0.409502\pi$
$181$	−122.965	−	122.965i	−0.679364	−	0.679364i	−0.959856	+	0.280493i	$0.909502\pi$
$182$		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	$0.169553\pi$
$191$			193.992i			1.01566i	−0.861456	+	0.507832i	$0.830447\pi$
$192$		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	0.776758	−	0.629800i	$-0.216863\pi$
$197$	28.9507	+	28.9507i	0.146958	+	0.146958i	−0.629800	+	0.776758i	$0.716863\pi$
$198$		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	0.689474	−	0.724310i	$-0.257842\pi$
$211$	−7.35041	−	7.35041i	−0.0348361	−	0.0348361i	−0.724310	+	0.689474i	$0.757842\pi$
$212$		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	$-0.666869\pi$
$223$		−	386.106i		−	1.73142i	0.500549	−	0.865708i	$-0.333131\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−49.7286	−	49.7286i	−0.219069	−	0.219069i	0.589037	−	0.808106i	$-0.299507\pi$
$227$	−49.7286	−	49.7286i	−0.219069	−	0.219069i	−0.808106	+	0.589037i	$0.799507\pi$
$228$		0			0
$229$	191.870	+	191.870i	0.837861	+	0.837861i	0.988577	−	0.150716i	$-0.0481579\pi$
$229$	191.870	+	191.870i	0.837861	+	0.837861i	−0.150716	+	0.988577i	$0.548158\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	298.610i		−	1.28159i	−0.767712	−	0.640795i	$-0.778605\pi$
$233$		−	298.610i		−	1.28159i	0.767712	−	0.640795i	$-0.221395\pi$
$234$		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	$0.173127\pi$
$239$			247.352i			1.03495i	−0.855700	+	0.517473i	$0.826873\pi$
$240$		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	−0.304303	−	0.952575i	$-0.598424\pi$
$251$	162.716	−	162.716i	0.648272	−	0.648272i	0.952575	+	0.304303i	$0.0984235\pi$
$252$		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	−0.704289	−	0.709913i	$-0.748734\pi$
$269$	1.51275	−	1.51275i	0.00562361	−	0.00562361i	0.709913	+	0.704289i	$0.248734\pi$
$270$		0			0
$271$		−	166.098i		−	0.612909i	−0.951885	−	0.306454i	$-0.900857\pi$
$271$		−	166.098i		−	0.612909i	0.951885	−	0.306454i	$-0.0991427\pi$
$272$		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	−0.987062	−	0.160338i	$-0.948741\pi$
$277$	−317.830	−	317.830i	−1.14740	−	1.14740i	−0.160338	−	0.987062i	$-0.551259\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			402.790i			1.43342i	0.697374	+	0.716708i	$0.254352\pi$
$281$			402.790i			1.43342i	−0.697374	+	0.716708i	$0.745648\pi$
$282$		0			0
$283$	192.406	−	192.406i	0.679881	−	0.679881i	−0.280092	−	0.959973i	$-0.590365\pi$
$283$	192.406	−	192.406i	0.679881	−	0.679881i	0.959973	+	0.280092i	$0.0903649\pi$
$284$		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	0.566704	−	0.823921i	$-0.308218\pi$
$293$	−75.3645	−	75.3645i	−0.257217	−	0.257217i	−0.823921	+	0.566704i	$0.808218\pi$
$294$		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	0.865040	−	0.501703i	$-0.167293\pi$
$307$	111.544	+	111.544i	0.363337	+	0.363337i	−0.501703	+	0.865040i	$0.667293\pi$
$308$		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	$0.284829\pi$
$313$			488.339i			1.56019i	−0.625661	+	0.780095i	$0.715171\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	257.361	−	257.361i	0.811863	−	0.811863i	−0.173050	−	0.984913i	$-0.555362\pi$
$317$	257.361	−	257.361i	0.811863	−	0.811863i	0.984913	+	0.173050i	$0.0553621\pi$
$318$		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	−0.495396	−	0.868667i	$-0.664977\pi$
$331$	123.553	−	123.553i	0.373271	−	0.373271i	0.868667	+	0.495396i	$0.164977\pi$
$332$		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	−0.861994	−	0.506918i	$-0.830785\pi$
$347$	−123.212	+	123.212i	−0.355076	+	0.355076i	0.506918	+	0.861994i	$0.330785\pi$
$348$		0			0
$349$	−115.371	+	115.371i	−0.330575	+	0.330575i	−0.852805	−	0.522230i	$-0.825100\pi$
$349$	−115.371	+	115.371i	−0.330575	+	0.330575i	0.522230	+	0.852805i	$0.325100\pi$
$350$		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	$-0.942717\pi$
$367$		−	131.379i		−	0.357982i	0.983851	−	0.178991i	$-0.0572832\pi$
$368$		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	0.972462	−	0.233062i	$-0.0748745\pi$
$373$	275.796	+	275.796i	0.739400	+	0.739400i	−0.233062	+	0.972462i	$0.574874\pi$
$374$		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	−0.689691	−	0.724104i	$-0.742253\pi$
$379$	13.0427	−	13.0427i	0.0344135	−	0.0344135i	0.724104	+	0.689691i	$0.242253\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			121.974i			0.318470i	0.987241	+	0.159235i	$0.0509027\pi$
$383$			121.974i			0.318470i	−0.987241	+	0.159235i	$0.949097\pi$
$384$		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	0.340563	−	0.940222i	$-0.389382\pi$
$389$	−233.267	−	233.267i	−0.599659	−	0.599659i	−0.940222	+	0.340563i	$0.889382\pi$
$390$		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	−0.593689	−	0.804695i	$-0.702329\pi$
$397$	83.7693	−	83.7693i	0.211006	−	0.211006i	0.804695	+	0.593689i	$0.202329\pi$
$398$		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	$-0.814914\pi$
$409$		−	449.285i		−	1.09850i	0.835659	−	0.549248i	$-0.185086\pi$
$410$		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	0.396307	−	0.918118i	$-0.370292\pi$
$419$	−218.639	−	218.639i	−0.521811	−	0.521811i	−0.918118	+	0.396307i	$0.870292\pi$
$420$		0			0
$421$	61.2101	+	61.2101i	0.145392	+	0.145392i	0.776056	−	0.630664i	$-0.217217\pi$
$421$	61.2101	+	61.2101i	0.145392	+	0.145392i	−0.630664	+	0.776056i	$0.717217\pi$
$422$		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	$-0.802527\pi$
$431$		−	501.119i		−	1.16269i	0.813657	−	0.581345i	$-0.197473\pi$
$432$		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	−0.283746	−	0.958899i	$-0.591577\pi$
$443$	299.093	−	299.093i	0.675153	−	0.675153i	0.958899	+	0.283746i	$0.0915773\pi$
$444$		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	$0.247514\pi$
$457$			641.227i			1.40312i	−0.712609	+	0.701562i	$0.752486\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−393.690	+	393.690i	−0.853991	+	0.853991i	−0.990622	−	0.136631i	$-0.956373\pi$
$461$	−393.690	+	393.690i	−0.853991	+	0.853991i	0.136631	+	0.990622i	$0.456373\pi$
$462$		0			0
$463$			395.861i			0.854991i	0.904018	+	0.427495i	$0.140604\pi$
$463$			395.861i			0.854991i	−0.904018	+	0.427495i	$0.859396\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−83.1457	−	83.1457i	−0.178042	−	0.178042i	0.612460	−	0.790502i	$-0.290180\pi$
$467$	−83.1457	−	83.1457i	−0.178042	−	0.178042i	−0.790502	+	0.612460i	$0.790180\pi$
$468$		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	$0.148183\pi$
$479$			430.043i			0.897793i	−0.893584	+	0.448896i	$0.851817\pi$
$480$		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	−0.999993	−	0.00379588i	$-0.998792\pi$
$491$	−489.133	+	489.133i	−0.996197	+	0.996197i	0.00379588	+	0.999993i	$0.498792\pi$
$492$		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	0.396188	−	0.918170i	$-0.370333\pi$
$499$	−260.469	−	260.469i	−0.521982	−	0.521982i	−0.918170	+	0.396188i	$0.870333\pi$
$500$		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	−0.986894	−	0.161371i	$-0.948408\pi$
$509$	−420.191	+	420.191i	−0.825523	+	0.825523i	0.161371	+	0.986894i	$0.448408\pi$
$510$		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	$0.124199\pi$
$521$			396.333i			0.760716i	−0.924839	+	0.380358i	$0.875801\pi$
$522$		0			0
$523$	−564.600	+	564.600i	−1.07954	+	1.07954i	−0.0829913	+	0.996550i	$0.526447\pi$
$523$	−564.600	+	564.600i	−1.07954	+	1.07954i	−0.996550	+	0.0829913i	$0.973553\pi$
$524$		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	−0.679259	−	0.733899i	$-0.737699\pi$
$541$	29.5601	−	29.5601i	0.0546398	−	0.0546398i	0.733899	+	0.679259i	$0.237699\pi$
$542$		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	0.822362	−	0.568965i	$-0.192656\pi$
$547$	138.608	+	138.608i	0.253397	+	0.253397i	−0.568965	+	0.822362i	$0.692656\pi$
$548$		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	−0.650767	−	0.759277i	$-0.725553\pi$
$557$	60.4400	−	60.4400i	0.108510	−	0.108510i	0.759277	+	0.650767i	$0.225553\pi$
$558$		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	0.903471	−	0.428649i	$-0.141010\pi$
$563$	267.325	+	267.325i	0.474822	+	0.474822i	−0.428649	+	0.903471i	$0.641010\pi$
$564$		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	$-0.910518\pi$
$569$		−	315.715i		−	0.554859i	0.960746	−	0.277429i	$-0.0894825\pi$
$570$		0			0
$571$	670.572	−	670.572i	1.17438	−	1.17438i	0.193228	−	0.981154i	$-0.438104\pi$
$571$	670.572	−	670.572i	1.17438	−	1.17438i	0.981154	−	0.193228i	$-0.0618956\pi$
$572$		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	−0.967711	−	0.252064i	$-0.918891\pi$
$587$	−420.085	+	420.085i	−0.715647	+	0.715647i	0.252064	+	0.967711i	$0.418891\pi$
$588$		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	$-0.897613\pi$
$601$		−	380.001i		−	0.632280i	0.948712	−	0.316140i	$-0.102387\pi$
$602$		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	$-0.952149\pi$
$607$		−	181.813i		−	0.299527i	0.988722	−	0.149763i	$-0.0478512\pi$
$608$		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	0.660693	−	0.750657i	$-0.270263\pi$
$613$	−55.1479	−	55.1479i	−0.0899640	−	0.0899640i	−0.750657	+	0.660693i	$0.770263\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			579.674i			0.939504i	0.882798	+	0.469752i	$0.155657\pi$
$617$			579.674i			0.939504i	−0.882798	+	0.469752i	$0.844343\pi$
$618$		0			0
$619$	−91.1070	+	91.1070i	−0.147184	+	0.147184i	−0.776859	−	0.629675i	$-0.783188\pi$
$619$	−91.1070	+	91.1070i	−0.147184	+	0.147184i	0.629675	+	0.776859i	$0.283188\pi$
$620$		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	−0.844353	−	0.535787i	$-0.820015\pi$
$643$	−887.430	−	887.430i	−1.38014	−	1.38014i	−0.535787	−	0.844353i	$-0.679985\pi$
$644$		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	−0.160549	−	0.987028i	$-0.551326\pi$
$653$	539.691	−	539.691i	0.826479	−	0.826479i	0.987028	+	0.160549i	$0.0513264\pi$
$654$		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	0.998745	−	0.0500862i	$-0.0159496\pi$
$659$	625.166	+	625.166i	0.948659	+	0.948659i	−0.0500862	+	0.998745i	$0.515950\pi$
$660$		0			0
$661$	326.893	+	326.893i	0.494544	+	0.494544i	0.909734	−	0.415191i	$-0.136285\pi$
$661$	326.893	+	326.893i	0.494544	+	0.494544i	−0.415191	+	0.909734i	$0.636285\pi$
$662$		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	0.607885	−	0.794025i	$-0.292018\pi$
$677$	−126.017	−	126.017i	−0.186140	−	0.186140i	−0.794025	+	0.607885i	$0.792018\pi$
$678$		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	−0.0857647	−	0.996315i	$-0.527333\pi$
$683$	621.906	−	621.906i	0.910551	−	0.910551i	0.996315	+	0.0857647i	$0.0273333\pi$
$684$		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	0.935934	−	0.352176i	$-0.114558\pi$
$691$	403.376	+	403.376i	0.583758	+	0.583758i	−0.352176	+	0.935934i	$0.614558\pi$
$692$		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	−0.291086	−	0.956697i	$-0.594017\pi$
$701$	466.593	−	466.593i	0.665611	−	0.665611i	0.956697	+	0.291086i	$0.0940166\pi$
$702$		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	−0.984376	−	0.176081i	$-0.943658\pi$
$709$	−822.764	−	822.764i	−1.16046	−	1.16046i	−0.176081	−	0.984376i	$-0.556342\pi$
$710$		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	$-0.835593\pi$
$719$		−	710.142i		−	0.987681i	0.869553	−	0.493840i	$-0.164407\pi$
$720$		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	−0.638452	−	0.769662i	$-0.720425\pi$
$733$	96.1768	−	96.1768i	0.131210	−	0.131210i	0.769662	+	0.638452i	$0.220425\pi$
$734$		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	−0.974757	−	0.223268i	$-0.928327\pi$
$739$	−885.341	−	885.341i	−1.19803	−	1.19803i	−0.223268	−	0.974757i	$-0.571673\pi$
$740$		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	$-0.723400\pi$
$751$		−	1147.02i		−	1.52732i	0.645618	−	0.763661i	$-0.276600\pi$
$752$		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	0.268869	−	0.963177i	$-0.413350\pi$
$757$	−525.591	−	525.591i	−0.694308	−	0.694308i	−0.963177	+	0.268869i	$0.913350\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	788.107i		−	1.03562i	−0.855495	−	0.517810i	$-0.826747\pi$
$761$		−	788.107i		−	1.03562i	0.855495	−	0.517810i	$-0.173253\pi$
$762$		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	0.862684	−	0.505743i	$-0.168782\pi$
$773$	275.915	+	275.915i	0.356941	+	0.356941i	−0.505743	+	0.862684i	$0.668782\pi$
$774$		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	0.537354	−	0.843356i	$-0.319424\pi$
$787$	−240.824	−	240.824i	−0.306002	−	0.306002i	−0.843356	+	0.537354i	$0.819424\pi$
$788$		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	−0.990683	−	0.136187i	$-0.956515\pi$
$797$	−681.033	+	681.033i	−0.854496	+	0.854496i	0.136187	+	0.990683i	$0.456515\pi$
$798$		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	$0.0887539\pi$
$809$			445.322i			0.550459i	−0.961379	+	0.275230i	$0.911246\pi$
$810$		0			0
$811$	−223.787	+	223.787i	−0.275939	+	0.275939i	−0.831485	−	0.555546i	$-0.812509\pi$
$811$	−223.787	+	223.787i	−0.275939	+	0.275939i	0.555546	+	0.831485i	$0.312509\pi$
$812$		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	0.900593	−	0.434664i	$-0.143133\pi$
$821$	382.528	+	382.528i	0.465929	+	0.465929i	−0.434664	+	0.900593i	$0.643133\pi$
$822$		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	−0.538071	−	0.842900i	$-0.680847\pi$
$827$	252.094	−	252.094i	0.304829	−	0.304829i	0.842900	+	0.538071i	$0.180847\pi$
$828$		0			0
$829$	870.285	−	870.285i	1.04980	−	1.04980i	0.0511072	−	0.998693i	$-0.483725\pi$
$829$	870.285	−	870.285i	1.04980	−	1.04980i	0.998693	−	0.0511072i	$-0.0162750\pi$
$830$		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	0.981950	−	0.189141i	$-0.0605703\pi$
$853$	676.266	+	676.266i	0.792809	+	0.792809i	−0.189141	+	0.981950i	$0.560570\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			291.272i			0.339874i	0.985455	+	0.169937i	$0.0543563\pi$
$857$			291.272i			0.339874i	−0.985455	+	0.169937i	$0.945644\pi$
$858$		0			0
$859$	−988.357	+	988.357i	−1.15059	+	1.15059i	−0.164156	+	0.986434i	$0.552490\pi$
$859$	−988.357	+	988.357i	−1.15059	+	1.15059i	−0.986434	+	0.164156i	$0.947510\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			922.512i			1.06896i	0.845181	+	0.534480i	$0.179492\pi$
$863$			922.512i			1.06896i	−0.845181	+	0.534480i	$0.820508\pi$
$864$		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	−0.436463	−	0.899722i	$-0.643769\pi$
$877$	406.278	−	406.278i	0.463259	−	0.463259i	0.899722	+	0.436463i	$0.143769\pi$
$878$		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	0.674461	−	0.738311i	$-0.264376\pi$
$883$	−56.3792	−	56.3792i	−0.0638496	−	0.0638496i	−0.738311	+	0.674461i	$0.764376\pi$
$884$		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	−0.925595	−	0.378515i	$-0.876435\pi$
$907$	−496.202	+	496.202i	−0.547080	+	0.547080i	0.378515	+	0.925595i	$0.376435\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1532.15i		−	1.68183i	−0.541163	−	0.840917i	$-0.682016\pi$
$911$		−	1532.15i		−	1.68183i	0.541163	−	0.840917i	$-0.317984\pi$
$912$		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	$-0.875191\pi$
$937$		−	716.111i		−	0.764259i	0.924109	−	0.382130i	$-0.124809\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	537.759	−	537.759i	0.571476	−	0.571476i	−0.361065	−	0.932541i	$-0.617587\pi$
$941$	537.759	−	537.759i	0.571476	−	0.571476i	0.932541	+	0.361065i	$0.117587\pi$
$942$		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	0.979066	−	0.203543i	$-0.0652457\pi$
$947$	734.420	+	734.420i	0.775523	+	0.775523i	−0.203543	+	0.979066i	$0.565246\pi$
$948$		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	$0.0231524\pi$
$953$			138.512i			0.145343i	−0.997356	+	0.0726714i	$0.976848\pi$
$954$		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	−0.738591	−	0.674154i	$-0.764508\pi$
$971$	−62.5684	+	62.5684i	−0.0644371	+	0.0644371i	0.674154	+	0.738591i	$0.264508\pi$
$972$		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	$0.00356033\pi$
$991$			22.1684i			0.0223698i	−0.999937	+	0.0111849i	$0.996440\pi$
$992$		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	0.700303	−	0.713846i	$-0.253048\pi$
$997$	−13.5020	−	13.5020i	−0.0135426	−	0.0135426i	−0.713846	+	0.700303i	$0.753048\pi$
$998$		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.415.4		16
3.2	odd	2		384.3.l.a.31.3		16
4.3	odd	2		1152.3.m.c.415.4		16
8.3	odd	2		576.3.m.c.271.5		16
8.5	even	2		144.3.m.c.91.3		16
12.11	even	2		384.3.l.b.31.7		16
16.3	odd	4	inner	1152.3.m.f.991.4		16
16.5	even	4		576.3.m.c.559.5		16
16.11	odd	4		144.3.m.c.19.3		16
16.13	even	4		1152.3.m.c.991.4		16
24.5	odd	2		48.3.l.a.43.6	yes	16
24.11	even	2		192.3.l.a.79.2		16
48.5	odd	4		192.3.l.a.175.2		16
48.11	even	4		48.3.l.a.19.6	✓	16
48.29	odd	4		384.3.l.b.223.7		16
48.35	even	4		384.3.l.a.223.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.3.l.a.19.6	✓	16	48.11	even	4
48.3.l.a.43.6	yes	16	24.5	odd	2
144.3.m.c.19.3		16	16.11	odd	4
144.3.m.c.91.3		16	8.5	even	2
192.3.l.a.79.2		16	24.11	even	2
192.3.l.a.175.2		16	48.5	odd	4
384.3.l.a.31.3		16	3.2	odd	2
384.3.l.a.223.3		16	48.35	even	4
384.3.l.b.31.7		16	12.11	even	2
384.3.l.b.223.7		16	48.29	odd	4
576.3.m.c.271.5		16	8.3	odd	2
576.3.m.c.559.5		16	16.5	even	4
1152.3.m.c.415.4		16	4.3	odd	2
1152.3.m.c.991.4		16	16.13	even	4
1152.3.m.f.415.4		16	1.1	even	1	trivial
1152.3.m.f.991.4		16	16.3	odd	4	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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
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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