LMFDB - Embedded newform 1152.4.l.a.863.23

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1152,4,Mod(287,1152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 2])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1152.287"); S:= CuspForms(chi, 4); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [48,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$67.9702003266$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			863.23
Character	$\chi$	$=$	1152.863
Dual form			1152.4.l.a.287.23

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(11.8474 + 11.8474i) q^{5} +12.7523 q^{7} +(15.5373 - 15.5373i) q^{11} +(-42.1911 - 42.1911i) q^{13} +76.6855i q^{17} +(-108.540 + 108.540i) q^{19} -52.0013i q^{23} +155.724i q^{25} +(-89.7171 + 89.7171i) q^{29} +12.9476i q^{31} +(151.083 + 151.083i) q^{35} +(-115.035 + 115.035i) q^{37} -307.899 q^{41} +(342.056 + 342.056i) q^{43} +357.499 q^{47} -180.378 q^{49} +(450.933 + 450.933i) q^{53} +368.154 q^{55} +(-395.827 + 395.827i) q^{59} +(-220.311 - 220.311i) q^{61} -999.712i q^{65} +(-243.024 + 243.024i) q^{67} -414.259i q^{71} -91.5239i q^{73} +(198.137 - 198.137i) q^{77} -236.635i q^{79} +(204.243 + 204.243i) q^{83} +(-908.526 + 908.526i) q^{85} -688.795 q^{89} +(-538.035 - 538.035i) q^{91} -2571.85 q^{95} +968.890 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{19} + 864 q^{43} + 2352 q^{49} + 576 q^{55} - 1824 q^{61} + 816 q^{67} + 480 q^{85} - 3600 q^{91}+O(q^{100}) $ 48 * q - 48 * q^19 + 864 * q^43 + 2352 * q^49 + 576 * q^55 - 1824 * q^61 + 816 * q^67 + 480 * q^85 - 3600 * q^91

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
$5$	11.8474	+	11.8474i	1.05967	+	1.05967i	0.998103	+	0.0615638i	$0.0196088\pi$
$5$	11.8474	+	11.8474i	1.05967	+	1.05967i	0.0615638	+	0.998103i	$0.480391\pi$
$6$		0			0
$7$	12.7523			0.688562			0.344281	−	0.938867i	$-0.388123\pi$
$7$	12.7523			0.688562			0.344281	+	0.938867i	$0.388123\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	15.5373	−	15.5373i	0.425879	−	0.425879i	−0.461343	−	0.887222i	$-0.652632\pi$
$11$	15.5373	−	15.5373i	0.425879	−	0.425879i	0.887222	+	0.461343i	$0.152632\pi$
$12$		0			0
$13$	−42.1911	−	42.1911i	−0.900130	−	0.900130i	0.0953168	−	0.995447i	$-0.469614\pi$
$13$	−42.1911	−	42.1911i	−0.900130	−	0.900130i	−0.995447	+	0.0953168i	$0.969614\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			76.6855i			1.09406i	0.837114	+	0.547028i	$0.184241\pi$
$17$			76.6855i			1.09406i	−0.837114	+	0.547028i	$0.815759\pi$
$18$		0			0
$19$	−108.540	+	108.540i	−1.31057	+	1.31057i	−0.389580	+	0.920993i	$0.627380\pi$
$19$	−108.540	+	108.540i	−1.31057	+	1.31057i	−0.920993	+	0.389580i	$0.872620\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	52.0013i		−	0.471436i	−0.971822	−	0.235718i	$-0.924256\pi$
$23$		−	52.0013i		−	0.471436i	0.971822	−	0.235718i	$-0.0757442\pi$
$24$		0			0
$25$			155.724i			1.24579i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−89.7171	+	89.7171i	−0.574484	+	0.574484i	−0.933378	−	0.358894i	$-0.883154\pi$
$29$	−89.7171	+	89.7171i	−0.574484	+	0.574484i	0.358894	+	0.933378i	$0.383154\pi$
$30$		0			0
$31$			12.9476i			0.0750150i	0.999296	+	0.0375075i	$0.0119418\pi$
$31$			12.9476i			0.0750150i	−0.999296	+	0.0375075i	$0.988058\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	151.083	+	151.083i	0.729646	+	0.729646i
$36$		0			0
$37$	−115.035	+	115.035i	−0.511126	+	0.511126i	−0.914871	−	0.403746i	$-0.867708\pi$
$37$	−115.035	+	115.035i	−0.511126	+	0.511126i	0.403746	+	0.914871i	$0.367708\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−307.899			−1.17282			−0.586411	−	0.810013i	$-0.699460\pi$
$41$	−307.899			−1.17282			−0.586411	+	0.810013i	$0.699460\pi$
$42$		0			0
$43$	342.056	+	342.056i	1.21309	+	1.21309i	0.970003	+	0.243091i	$0.0781615\pi$
$43$	342.056	+	342.056i	1.21309	+	1.21309i	0.243091	+	0.970003i	$0.421839\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	357.499			1.10950			0.554750	−	0.832017i	$-0.312814\pi$
$47$	357.499			1.10950			0.554750	+	0.832017i	$0.312814\pi$
$48$		0			0
$49$	−180.378			−0.525883
$50$		0			0
$51$		0			0
$52$		0			0
$53$	450.933	+	450.933i	1.16869	+	1.16869i	0.982518	+	0.186169i	$0.0596072\pi$
$53$	450.933	+	450.933i	1.16869	+	1.16869i	0.186169	+	0.982518i	$0.440393\pi$
$54$		0			0
$55$	368.154			0.902579
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−395.827	+	395.827i	−0.873428	+	0.873428i	−0.992844	−	0.119417i	$-0.961898\pi$
$59$	−395.827	+	395.827i	−0.873428	+	0.873428i	0.119417	+	0.992844i	$0.461898\pi$
$60$		0			0
$61$	−220.311	−	220.311i	−0.462425	−	0.462425i	0.437025	−	0.899449i	$-0.356032\pi$
$61$	−220.311	−	220.311i	−0.462425	−	0.462425i	−0.899449	+	0.437025i	$0.856032\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	999.712i		−	1.90768i
$66$		0			0
$67$	−243.024	+	243.024i	−0.443137	+	0.443137i	−0.893065	−	0.449928i	$-0.851450\pi$
$67$	−243.024	+	243.024i	−0.443137	+	0.443137i	0.449928	+	0.893065i	$0.351450\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	414.259i		−	0.692444i	−0.938153	−	0.346222i	$-0.887464\pi$
$71$		−	414.259i		−	0.692444i	0.938153	−	0.346222i	$-0.112536\pi$
$72$		0			0
$73$		−	91.5239i		−	0.146741i	−0.997305	−	0.0733703i	$-0.976624\pi$
$73$		−	91.5239i		−	0.146741i	0.997305	−	0.0733703i	$-0.0233755\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	198.137	−	198.137i	0.293244	−	0.293244i
$78$		0			0
$79$		−	236.635i		−	0.337007i	−0.985701	−	0.168503i	$-0.946107\pi$
$79$		−	236.635i		−	0.337007i	0.985701	−	0.168503i	$-0.0538934\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	204.243	+	204.243i	0.270103	+	0.270103i	0.829142	−	0.559039i	$-0.188830\pi$
$83$	204.243	+	204.243i	0.270103	+	0.270103i	−0.559039	+	0.829142i	$0.688830\pi$
$84$		0			0
$85$	−908.526	+	908.526i	−1.15934	+	1.15934i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−688.795			−0.820360			−0.410180	−	0.912005i	$-0.634534\pi$
$89$	−688.795			−0.820360			−0.410180	+	0.912005i	$0.634534\pi$
$90$		0			0
$91$	−538.035	−	538.035i	−0.619795	−	0.619795i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2571.85			−2.77754
$96$		0			0
$97$	968.890			1.01418			0.507092	−	0.861892i	$-0.330720\pi$
$97$	968.890			1.01418			0.507092	+	0.861892i	$0.330720\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1120.79	−	1120.79i	−1.10419	−	1.10419i	−0.993900	−	0.110289i	$-0.964822\pi$
$101$	−1120.79	−	1120.79i	−1.10419	−	1.10419i	−0.110289	−	0.993900i	$-0.535178\pi$
$102$		0			0
$103$	1811.42			1.73286			0.866430	−	0.499298i	$-0.166409\pi$
$103$	1811.42			1.73286			0.866430	+	0.499298i	$0.166409\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−943.403	+	943.403i	−0.852357	+	0.852357i	−0.990423	−	0.138066i	$-0.955911\pi$
$107$	−943.403	+	943.403i	−0.852357	+	0.852357i	0.138066	+	0.990423i	$0.455911\pi$
$108$		0			0
$109$	227.732	+	227.732i	0.200117	+	0.200117i	0.800050	−	0.599933i	$-0.204806\pi$
$109$	227.732	+	227.732i	0.200117	+	0.200117i	−0.599933	+	0.800050i	$0.704806\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1679.37i		−	1.39807i	−0.715090	−	0.699033i	$-0.753614\pi$
$113$		−	1679.37i		−	1.39807i	0.715090	−	0.699033i	$-0.246386\pi$
$114$		0			0
$115$	616.083	−	616.083i	0.499565	−	0.499565i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			977.919i			0.753325i
$120$		0			0
$121$			848.186i			0.637255i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−363.995	+	363.995i	−0.260454	+	0.260454i
$126$		0			0
$127$			1983.72i			1.38604i	0.720921	+	0.693018i	$0.243719\pi$
$127$			1983.72i			1.38604i	−0.720921	+	0.693018i	$0.756281\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	604.660	+	604.660i	0.403277	+	0.403277i	0.879386	−	0.476109i	$-0.157953\pi$
$131$	604.660	+	604.660i	0.403277	+	0.403277i	−0.476109	+	0.879386i	$0.657953\pi$
$132$		0			0
$133$	−1384.14	+	1384.14i	−0.902410	+	0.902410i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	461.883			0.288039			0.144020	−	0.989575i	$-0.453997\pi$
$137$	461.883			0.288039			0.144020	+	0.989575i	$0.453997\pi$
$138$		0			0
$139$	2075.03	+	2075.03i	1.26620	+	1.26620i	0.948037	+	0.318161i	$0.103065\pi$
$139$	2075.03	+	2075.03i	1.26620	+	1.26620i	0.318161	+	0.948037i	$0.396935\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1311.07			−0.766693
$144$		0			0
$145$	−2125.83			−1.21752
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−960.479	−	960.479i	−0.528091	−	0.528091i	0.391912	−	0.920003i	$-0.371814\pi$
$149$	−960.479	−	960.479i	−0.528091	−	0.528091i	−0.920003	+	0.391912i	$0.871814\pi$
$150$		0			0
$151$	1041.85			0.561490			0.280745	−	0.959782i	$-0.409419\pi$
$151$	1041.85			0.561490			0.280745	+	0.959782i	$0.409419\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−153.396	+	153.396i	−0.0794910	+	0.0794910i
$156$		0			0
$157$	1899.04	+	1899.04i	0.965348	+	0.965348i	0.999419	−	0.0340711i	$-0.0108473\pi$
$157$	1899.04	+	1899.04i	0.965348	+	0.965348i	−0.0340711	+	0.999419i	$0.510847\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	663.139i		−	0.324613i
$162$		0			0
$163$	−1647.43	+	1647.43i	−0.791638	+	0.791638i	−0.981760	−	0.190123i	$-0.939111\pi$
$163$	−1647.43	+	1647.43i	−0.791638	+	0.791638i	0.190123	+	0.981760i	$0.439111\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1582.34i		−	0.733205i	−0.930378	−	0.366603i	$-0.880521\pi$
$167$		−	1582.34i		−	0.733205i	0.930378	−	0.366603i	$-0.119479\pi$
$168$		0			0
$169$			1363.17i			0.620469i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1694.26	+	1694.26i	−0.744579	+	0.744579i	−0.973455	−	0.228877i	$-0.926495\pi$
$173$	−1694.26	+	1694.26i	−0.744579	+	0.744579i	0.228877	+	0.973455i	$0.426495\pi$
$174$		0			0
$175$			1985.84i			0.857802i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	538.851	+	538.851i	0.225003	+	0.225003i	0.810602	−	0.585598i	$-0.199140\pi$
$179$	538.851	+	538.851i	0.225003	+	0.225003i	−0.585598	+	0.810602i	$0.699140\pi$
$180$		0			0
$181$	−1601.85	+	1601.85i	−0.657815	+	0.657815i	−0.954863	−	0.297048i	$-0.903998\pi$
$181$	−1601.85	+	1601.85i	−0.657815	+	0.657815i	0.297048	+	0.954863i	$0.403998\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2725.74			−1.08325
$186$		0			0
$187$	1191.48	+	1191.48i	0.465935	+	0.465935i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2604.11			0.986526			0.493263	−	0.869880i	$-0.335804\pi$
$191$	2604.11			0.986526			0.493263	+	0.869880i	$0.335804\pi$
$192$		0			0
$193$	−3986.98			−1.48699			−0.743495	−	0.668742i	$-0.766833\pi$
$193$	−3986.98			−1.48699			−0.743495	+	0.668742i	$0.766833\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	299.734	+	299.734i	0.108402	+	0.108402i	0.759227	−	0.650826i	$-0.225577\pi$
$197$	299.734	+	299.734i	0.108402	+	0.108402i	−0.650826	+	0.759227i	$0.725577\pi$
$198$		0			0
$199$	−2771.54			−0.987283			−0.493641	−	0.869666i	$-0.664334\pi$
$199$	−2771.54			−0.987283			−0.493641	+	0.869666i	$0.664334\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1144.10	+	1144.10i	−0.395568	+	0.395568i
$204$		0			0
$205$	−3647.81	−	3647.81i	−1.24280	−	1.24280i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			3372.85i			1.11629i
$210$		0			0
$211$	3155.78	−	3155.78i	1.02964	−	1.02964i	0.0300882	−	0.999547i	$-0.490421\pi$
$211$	3155.78	−	3155.78i	1.02964	−	1.02964i	0.999547	−	0.0300882i	$-0.00957881\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			8104.98i			2.57095i
$216$		0			0
$217$			165.113i			0.0516525i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	3235.44	−	3235.44i	0.984793	−	0.984793i
$222$		0			0
$223$			4954.34i			1.48775i	0.668321	+	0.743873i	$0.267013\pi$
$223$			4954.34i			1.48775i	−0.668321	+	0.743873i	$0.732987\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4344.90	−	4344.90i	−1.27040	−	1.27040i	−0.945878	−	0.324522i	$-0.894796\pi$
$227$	−4344.90	−	4344.90i	−1.27040	−	1.27040i	−0.324522	−	0.945878i	$-0.605204\pi$
$228$		0			0
$229$	−1409.26	+	1409.26i	−0.406667	+	0.406667i	−0.880575	−	0.473907i	$-0.842843\pi$
$229$	−1409.26	+	1409.26i	−0.406667	+	0.406667i	0.473907	+	0.880575i	$0.342843\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	315.304			0.0886534			0.0443267	−	0.999017i	$-0.485886\pi$
$233$	315.304			0.0886534			0.0443267	+	0.999017i	$0.485886\pi$
$234$		0			0
$235$	4235.44	+	4235.44i	1.17570	+	1.17570i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3855.01			1.04335			0.521674	−	0.853145i	$-0.325308\pi$
$239$	3855.01			1.04335			0.521674	+	0.853145i	$0.325308\pi$
$240$		0			0
$241$	−2826.65			−0.755520			−0.377760	−	0.925903i	$-0.623306\pi$
$241$	−2826.65			−0.755520			−0.377760	+	0.925903i	$0.623306\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2137.01	−	2137.01i	−0.557260	−	0.557260i
$246$		0			0
$247$	9158.87			2.35937
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1055.38	−	1055.38i	0.265399	−	0.265399i	−0.561844	−	0.827243i	$-0.689908\pi$
$251$	1055.38	−	1055.38i	0.265399	−	0.265399i	0.827243	+	0.561844i	$0.189908\pi$
$252$		0			0
$253$	−807.959	−	807.959i	−0.200775	−	0.200775i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			3642.49i			0.884094i	0.896992	+	0.442047i	$0.145748\pi$
$257$			3642.49i			0.884094i	−0.896992	+	0.442047i	$0.854252\pi$
$258$		0			0
$259$	−1466.97	+	1466.97i	−0.351942	+	0.351942i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			2762.58i			0.647712i	0.946106	+	0.323856i	$0.104979\pi$
$263$			2762.58i			0.647712i	−0.946106	+	0.323856i	$0.895021\pi$
$264$		0			0
$265$			10684.8i			2.47684i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	687.157	−	687.157i	0.155750	−	0.155750i	−0.624930	−	0.780680i	$-0.714873\pi$
$269$	687.157	−	687.157i	0.155750	−	0.155750i	0.780680	+	0.624930i	$0.214873\pi$
$270$		0			0
$271$		−	8340.64i		−	1.86959i	−0.355194	−	0.934793i	$-0.615585\pi$
$271$		−	8340.64i		−	1.86959i	0.355194	−	0.934793i	$-0.384415\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2419.52	+	2419.52i	0.530555	+	0.530555i
$276$		0			0
$277$	3316.47	−	3316.47i	0.719377	−	0.719377i	−0.249100	−	0.968478i	$-0.580135\pi$
$277$	3316.47	−	3316.47i	0.719377	−	0.719377i	0.968478	+	0.249100i	$0.0801349\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1959.56			−0.416006			−0.208003	−	0.978128i	$-0.566696\pi$
$281$	−1959.56			−0.416006			−0.208003	+	0.978128i	$0.566696\pi$
$282$		0			0
$283$	98.9689	+	98.9689i	0.0207883	+	0.0207883i	0.717425	−	0.696636i	$-0.245321\pi$
$283$	98.9689	+	98.9689i	0.0207883	+	0.0207883i	−0.696636	+	0.717425i	$0.745321\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3926.43			−0.807561
$288$		0			0
$289$	−967.662			−0.196960
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−5930.48	−	5930.48i	−1.18247	−	1.18247i	−0.979104	−	0.203362i	$-0.934813\pi$
$293$	−5930.48	−	5930.48i	−1.18247	−	1.18247i	−0.203362	−	0.979104i	$-0.565187\pi$
$294$		0			0
$295$	−9379.06			−1.85108
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2193.99	+	2193.99i	−0.424354	+	0.424354i
$300$		0			0
$301$	4362.02	+	4362.02i	0.835291	+	0.835291i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	5220.23i		−	0.980032i
$306$		0			0
$307$	−370.094	+	370.094i	−0.0688025	+	0.0688025i	−0.740671	−	0.671868i	$-0.765492\pi$
$307$	−370.094	+	370.094i	−0.0688025	+	0.0688025i	0.671868	+	0.740671i	$0.265492\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	3969.20i		−	0.723706i	−0.932235	−	0.361853i	$-0.882144\pi$
$311$		−	3969.20i		−	0.723706i	0.932235	−	0.361853i	$-0.117856\pi$
$312$		0			0
$313$		−	1293.74i		−	0.233630i	−0.993154	−	0.116815i	$-0.962732\pi$
$313$		−	1293.74i		−	0.233630i	0.993154	−	0.116815i	$-0.0372685\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	471.086	−	471.086i	0.0834663	−	0.0834663i	−0.664141	−	0.747607i	$-0.731203\pi$
$317$	471.086	−	471.086i	0.0834663	−	0.0834663i	0.747607	+	0.664141i	$0.231203\pi$
$318$		0			0
$319$			2787.92i			0.489321i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8323.47	−	8323.47i	−1.43384	−	1.43384i
$324$		0			0
$325$	6570.14	−	6570.14i	1.12137	−	1.12137i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4558.94			0.763960
$330$		0			0
$331$	−1232.13	−	1232.13i	−0.204605	−	0.204605i	0.597365	−	0.801970i	$-0.296214\pi$
$331$	−1232.13	−	1232.13i	−0.204605	−	0.204605i	−0.801970	+	0.597365i	$0.796214\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5758.43			−0.939154
$336$		0			0
$337$	5124.30			0.828303			0.414152	−	0.910208i	$-0.364078\pi$
$337$	5124.30			0.828303			0.414152	+	0.910208i	$0.364078\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	201.171	+	201.171i	0.0319473	+	0.0319473i
$342$		0			0
$343$	−6674.29			−1.05066
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1034.16	+	1034.16i	−0.159990	+	0.159990i	−0.782562	−	0.622573i	$-0.786088\pi$
$347$	−1034.16	+	1034.16i	−0.159990	+	0.159990i	0.622573	+	0.782562i	$0.286088\pi$
$348$		0			0
$349$	857.250	+	857.250i	0.131483	+	0.131483i	0.769786	−	0.638303i	$-0.220363\pi$
$349$	857.250	+	857.250i	0.131483	+	0.131483i	−0.638303	+	0.769786i	$0.720363\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	6438.74i		−	0.970820i	−0.874287	−	0.485410i	$-0.838670\pi$
$353$		−	6438.74i		−	0.970820i	0.874287	−	0.485410i	$-0.161330\pi$
$354$		0			0
$355$	4907.91	−	4907.91i	0.733760	−	0.733760i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	3177.72i		−	0.467169i	−0.972336	−	0.233585i	$-0.924954\pi$
$359$		−	3177.72i		−	0.467169i	0.972336	−	0.233585i	$-0.0750456\pi$
$360$		0			0
$361$		−	16703.0i		−	2.43520i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1084.32	−	1084.32i	0.155496	−	0.155496i
$366$		0			0
$367$		−	3014.10i		−	0.428705i	−0.976756	−	0.214353i	$-0.931236\pi$
$367$		−	3014.10i		−	0.428705i	0.976756	−	0.214353i	$-0.0687642\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	5750.45	+	5750.45i	0.804713	+	0.804713i
$372$		0			0
$373$	−3831.87	+	3831.87i	−0.531922	+	0.531922i	−0.921144	−	0.389222i	$-0.872744\pi$
$373$	−3831.87	+	3831.87i	−0.531922	+	0.531922i	0.389222	+	0.921144i	$0.372744\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7570.52			1.03422
$378$		0			0
$379$	−7178.18	−	7178.18i	−0.972871	−	0.972871i	0.0267701	−	0.999642i	$-0.491478\pi$
$379$	−7178.18	−	7178.18i	−0.972871	−	0.972871i	−0.999642	+	0.0267701i	$0.991478\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	896.953			0.119666			0.0598331	−	0.998208i	$-0.480943\pi$
$383$	896.953			0.119666			0.0598331	+	0.998208i	$0.480943\pi$
$384$		0			0
$385$	4694.82			0.621482
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5572.80	+	5572.80i	0.726355	+	0.726355i	0.969892	−	0.243536i	$-0.0783076\pi$
$389$	5572.80	+	5572.80i	0.726355	+	0.726355i	−0.243536	+	0.969892i	$0.578308\pi$
$390$		0			0
$391$	3987.75			0.515778
$392$		0			0
$393$		0			0
$394$		0			0
$395$	2803.52	−	2803.52i	0.357115	−	0.357115i
$396$		0			0
$397$	5204.36	+	5204.36i	0.657933	+	0.657933i	0.954891	−	0.296958i	$-0.0959721\pi$
$397$	5204.36	+	5204.36i	0.657933	+	0.657933i	−0.296958	+	0.954891i	$0.595972\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			14787.8i			1.84157i	0.390075	+	0.920783i	$0.372449\pi$
$401$			14787.8i			1.84157i	−0.390075	+	0.920783i	$0.627551\pi$
$402$		0			0
$403$	546.275	−	546.275i	0.0675233	−	0.0675233i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3574.66i			0.435355i
$408$		0			0
$409$			4498.56i			0.543862i	0.962317	+	0.271931i	$0.0876622\pi$
$409$			4498.56i			0.543862i	−0.962317	+	0.271931i	$0.912338\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5047.72	+	5047.72i	−0.601409	+	0.601409i
$414$		0			0
$415$			4839.50i			0.572438i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−212.986	−	212.986i	−0.0248331	−	0.0248331i	0.694581	−	0.719414i	$-0.255590\pi$
$419$	−212.986	−	212.986i	−0.0248331	−	0.0248331i	−0.719414	+	0.694581i	$0.755590\pi$
$420$		0			0
$421$	−1934.26	+	1934.26i	−0.223919	+	0.223919i	−0.810147	−	0.586227i	$-0.800613\pi$
$421$	−1934.26	+	1934.26i	−0.223919	+	0.223919i	0.586227	+	0.810147i	$0.300613\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−11941.7			−1.36296
$426$		0			0
$427$	−2809.48	−	2809.48i	−0.318408	−	0.318408i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4971.77			0.555643			0.277821	−	0.960633i	$-0.410388\pi$
$431$	4971.77			0.555643			0.277821	+	0.960633i	$0.410388\pi$
$432$		0			0
$433$	1279.52			0.142009			0.0710044	−	0.997476i	$-0.477380\pi$
$433$	1279.52			0.142009			0.0710044	+	0.997476i	$0.477380\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5644.25	+	5644.25i	0.617851	+	0.617851i
$438$		0			0
$439$	5331.83			0.579668			0.289834	−	0.957077i	$-0.406400\pi$
$439$	5331.83			0.579668			0.289834	+	0.957077i	$0.406400\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10618.8	−	10618.8i	1.13886	−	1.13886i	0.150199	−	0.988656i	$-0.452008\pi$
$443$	10618.8	−	10618.8i	1.13886	−	1.13886i	0.988656	−	0.150199i	$-0.0479915\pi$
$444$		0			0
$445$	−8160.45	−	8160.45i	−0.869309	−	0.869309i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	1471.81i		−	0.154697i	−0.997004	−	0.0773485i	$-0.975355\pi$
$449$		−	1471.81i		−	0.154697i	0.997004	−	0.0773485i	$-0.0246454\pi$
$450$		0			0
$451$	−4783.91	+	4783.91i	−0.499480	+	0.499480i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	12748.7i		−	1.31355i
$456$		0			0
$457$		−	4489.34i		−	0.459524i	−0.973247	−	0.229762i	$-0.926205\pi$
$457$		−	4489.34i		−	0.459524i	0.973247	−	0.229762i	$-0.0737947\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2175.66	+	2175.66i	−0.219806	+	0.219806i	−0.808417	−	0.588610i	$-0.799675\pi$
$461$	−2175.66	+	2175.66i	−0.219806	+	0.219806i	0.588610	+	0.808417i	$0.299675\pi$
$462$		0			0
$463$		−	3759.95i		−	0.377408i	−0.982034	−	0.188704i	$-0.939571\pi$
$463$		−	3759.95i		−	0.377408i	0.982034	−	0.188704i	$-0.0604287\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5824.44	+	5824.44i	0.577137	+	0.577137i	0.934113	−	0.356976i	$-0.116192\pi$
$467$	5824.44	+	5824.44i	0.577137	+	0.577137i	−0.356976	+	0.934113i	$0.616192\pi$
$468$		0			0
$469$	−3099.13	+	3099.13i	−0.305127	+	0.305127i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10629.2			1.03326
$474$		0			0
$475$	−16902.3	−	16902.3i	−1.63270	−	1.63270i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12175.2			1.16138			0.580690	−	0.814125i	$-0.302783\pi$
$479$	12175.2			1.16138			0.580690	+	0.814125i	$0.302783\pi$
$480$		0			0
$481$	9706.90			0.920159
$482$		0			0
$483$		0			0
$484$		0			0
$485$	11478.9	+	11478.9i	1.07470	+	1.07470i
$486$		0			0
$487$	1409.76			0.131175			0.0655876	−	0.997847i	$-0.479108\pi$
$487$	1409.76			0.131175			0.0655876	+	0.997847i	$0.479108\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3130.87	−	3130.87i	0.287769	−	0.287769i	−0.548429	−	0.836197i	$-0.684774\pi$
$491$	3130.87	−	3130.87i	0.287769	−	0.287769i	0.836197	+	0.548429i	$0.184774\pi$
$492$		0			0
$493$	−6880.00	−	6880.00i	−0.628518	−	0.628518i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	5282.78i		−	0.476791i
$498$		0			0
$499$	−5590.00	+	5590.00i	−0.501489	+	0.501489i	−0.911900	−	0.410412i	$-0.865385\pi$
$499$	−5590.00	+	5590.00i	−0.501489	+	0.501489i	0.410412	+	0.911900i	$0.365385\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	5278.26i		−	0.467885i	−0.972250	−	0.233943i	$-0.924837\pi$
$503$		−	5278.26i		−	0.467885i	0.972250	−	0.233943i	$-0.0751628\pi$
$504$		0			0
$505$		−	26557.0i		−	2.34014i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1789.92	+	1789.92i	−0.155868	+	0.155868i	−0.780733	−	0.624865i	$-0.785154\pi$
$509$	−1789.92	+	1789.92i	−0.155868	+	0.155868i	0.624865	+	0.780733i	$0.285154\pi$
$510$		0			0
$511$		−	1167.14i		−	0.101040i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	21460.7	+	21460.7i	1.83626	+	1.83626i
$516$		0			0
$517$	5554.56	−	5554.56i	0.472513	−	0.472513i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3939.70			0.331289			0.165645	−	0.986186i	$-0.447030\pi$
$521$	3939.70			0.331289			0.165645	+	0.986186i	$0.447030\pi$
$522$		0			0
$523$	10743.7	+	10743.7i	0.898260	+	0.898260i	0.995282	−	0.0970218i	$-0.0309316\pi$
$523$	10743.7	+	10743.7i	0.898260	+	0.898260i	−0.0970218	+	0.995282i	$0.530932\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−992.897			−0.0820707
$528$		0			0
$529$	9462.86			0.777748
$530$		0			0
$531$		0			0
$532$		0			0
$533$	12990.6	+	12990.6i	1.05569	+	1.05569i
$534$		0			0
$535$	−22353.8			−1.80643
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−2802.58	+	2802.58i	−0.223962	+	0.223962i
$540$		0			0
$541$	845.856	+	845.856i	0.0672203	+	0.0672203i	0.739918	−	0.672697i	$-0.234864\pi$
$541$	845.856	+	845.856i	0.0672203	+	0.0672203i	−0.672697	+	0.739918i	$0.734864\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			5396.08i			0.424115i
$546$		0			0
$547$	−2207.08	+	2207.08i	−0.172519	+	0.172519i	−0.788085	−	0.615566i	$-0.788927\pi$
$547$	−2207.08	+	2207.08i	−0.172519	+	0.172519i	0.615566	+	0.788085i	$0.288927\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	19475.9i		−	1.50581i
$552$		0			0
$553$		−	3017.65i		−	0.232050i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	981.595	−	981.595i	0.0746706	−	0.0746706i	−0.668785	−	0.743456i	$-0.733185\pi$
$557$	981.595	−	981.595i	0.0746706	−	0.0746706i	0.743456	+	0.668785i	$0.233185\pi$
$558$		0			0
$559$		−	28863.4i		−	2.18389i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4253.32	−	4253.32i	−0.318394	−	0.318394i	0.529756	−	0.848150i	$-0.322284\pi$
$563$	−4253.32	−	4253.32i	−0.318394	−	0.318394i	−0.848150	+	0.529756i	$0.822284\pi$
$564$		0			0
$565$	19896.2	−	19896.2i	1.48148	−	1.48148i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5343.19			0.393670			0.196835	−	0.980437i	$-0.436934\pi$
$569$	5343.19			0.393670			0.196835	+	0.980437i	$0.436934\pi$
$570$		0			0
$571$	1599.99	+	1599.99i	0.117263	+	0.117263i	0.763303	−	0.646040i	$-0.223576\pi$
$571$	1599.99	+	1599.99i	0.117263	+	0.117263i	−0.646040	+	0.763303i	$0.723576\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8097.83			0.587309
$576$		0			0
$577$	−6250.00			−0.450938			−0.225469	−	0.974250i	$-0.572391\pi$
$577$	−6250.00			−0.450938			−0.225469	+	0.974250i	$0.572391\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2604.57	+	2604.57i	0.185983	+	0.185983i
$582$		0			0
$583$	14012.5			0.995438
$584$		0			0
$585$		0			0
$586$		0			0
$587$	10453.2	−	10453.2i	0.735009	−	0.735009i	−0.236598	−	0.971608i	$-0.576033\pi$
$587$	10453.2	−	10453.2i	0.735009	−	0.735009i	0.971608	+	0.236598i	$0.0760325\pi$
$588$		0			0
$589$	−1405.34	−	1405.34i	−0.0983126	−	0.0983126i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	6756.56i		−	0.467890i	−0.972250	−	0.233945i	$-0.924836\pi$
$593$		−	6756.56i		−	0.467890i	0.972250	−	0.233945i	$-0.0751635\pi$
$594$		0			0
$595$	−11585.8	+	11585.8i	−0.798274	+	0.798274i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	10583.3i		−	0.721907i	−0.932584	−	0.360953i	$-0.882451\pi$
$599$		−	10583.3i		−	0.721907i	0.932584	−	0.360953i	$-0.117549\pi$
$600$		0			0
$601$			27897.7i			1.89346i	0.322029	+	0.946730i	$0.395635\pi$
$601$			27897.7i			1.89346i	−0.322029	+	0.946730i	$0.604365\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10048.8	+	10048.8i	−0.675278	+	0.675278i
$606$		0			0
$607$		−	15554.4i		−	1.04009i	−0.854139	−	0.520045i	$-0.825915\pi$
$607$		−	15554.4i		−	1.04009i	0.854139	−	0.520045i	$-0.174085\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−15083.2	−	15083.2i	−0.998695	−	0.998695i
$612$		0			0
$613$	−1060.22	+	1060.22i	−0.0698561	+	0.0698561i	−0.741172	−	0.671316i	$-0.765730\pi$
$613$	−1060.22	+	1060.22i	−0.0698561	+	0.0698561i	0.671316	+	0.741172i	$0.265730\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1588.09			0.103621			0.0518103	−	0.998657i	$-0.483501\pi$
$617$	1588.09			0.103621			0.0518103	+	0.998657i	$0.483501\pi$
$618$		0			0
$619$	−16739.9	−	16739.9i	−1.08697	−	1.08697i	−0.995839	−	0.0911258i	$-0.970953\pi$
$619$	−16739.9	−	16739.9i	−1.08697	−	1.08697i	−0.0911258	−	0.995839i	$-0.529047\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−8783.74			−0.564869
$624$		0			0
$625$	10840.6			0.693800
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−8821.52	−	8821.52i	−0.559200	−	0.559200i
$630$		0			0
$631$	−24451.3			−1.54262			−0.771308	−	0.636462i	$-0.780397\pi$
$631$	−24451.3			−1.54262			−0.771308	+	0.636462i	$0.780397\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−23502.0	+	23502.0i	−1.46874	+	1.46874i
$636$		0			0
$637$	7610.33	+	7610.33i	0.473363	+	0.473363i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			21231.7i			1.30827i	0.756376	+	0.654137i	$0.226968\pi$
$641$			21231.7i			1.30827i	−0.756376	+	0.654137i	$0.773032\pi$
$642$		0			0
$643$	6155.73	−	6155.73i	0.377540	−	0.377540i	−0.492674	−	0.870214i	$-0.663980\pi$
$643$	6155.73	−	6155.73i	0.377540	−	0.377540i	0.870214	+	0.492674i	$0.163980\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	2628.40i		−	0.159711i	−0.996806	−	0.0798556i	$-0.974554\pi$
$647$		−	2628.40i		−	0.159711i	0.996806	−	0.0798556i	$-0.0254459\pi$
$648$		0			0
$649$			12300.1i			0.743949i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	16580.7	−	16580.7i	0.993651	−	0.993651i	−0.00632934	−	0.999980i	$-0.502015\pi$
$653$	16580.7	−	16580.7i	0.993651	−	0.993651i	0.999980	+	0.00632934i	$0.00201470\pi$
$654$		0			0
$655$			14327.3i			0.854680i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10829.7	−	10829.7i	−0.640160	−	0.640160i	0.310435	−	0.950595i	$-0.399525\pi$
$659$	−10829.7	−	10829.7i	−0.640160	−	0.640160i	−0.950595	+	0.310435i	$0.899525\pi$
$660$		0			0
$661$	7601.63	−	7601.63i	0.447306	−	0.447306i	−0.447152	−	0.894458i	$-0.647562\pi$
$661$	7601.63	−	7601.63i	0.447306	−	0.447306i	0.894458	+	0.447152i	$0.147562\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−32797.1			−1.91251
$666$		0			0
$667$	4665.41	+	4665.41i	0.270833	+	0.270833i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6846.06			−0.393874
$672$		0			0
$673$	17454.0			0.999704			0.499852	−	0.866111i	$-0.333388\pi$
$673$	17454.0			0.999704			0.499852	+	0.866111i	$0.333388\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8654.19	+	8654.19i	0.491296	+	0.491296i	0.908714	−	0.417418i	$-0.137065\pi$
$677$	8654.19	+	8654.19i	0.491296	+	0.491296i	−0.417418	+	0.908714i	$0.637065\pi$
$678$		0			0
$679$	12355.6			0.698329
$680$		0			0
$681$		0			0
$682$		0			0
$683$	17026.8	−	17026.8i	0.953898	−	0.953898i	−0.0450852	−	0.998983i	$-0.514356\pi$
$683$	17026.8	−	17026.8i	0.953898	−	0.953898i	0.998983	+	0.0450852i	$0.0143559\pi$
$684$		0			0
$685$	5472.13	+	5472.13i	0.305225	+	0.305225i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	38050.7i		−	2.10394i
$690$		0			0
$691$	4489.84	−	4489.84i	0.247180	−	0.247180i	−0.572632	−	0.819813i	$-0.694078\pi$
$691$	4489.84	−	4489.84i	0.247180	−	0.247180i	0.819813	+	0.572632i	$0.194078\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			49167.5i			2.68350i
$696$		0			0
$697$		−	23611.4i		−	1.28313i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18090.5	−	18090.5i	0.974706	−	0.974706i	−0.0249815	−	0.999688i	$-0.507953\pi$
$701$	18090.5	−	18090.5i	0.974706	−	0.974706i	0.999688	+	0.0249815i	$0.00795270\pi$
$702$		0			0
$703$		−	24971.9i		−	1.33973i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−14292.7	−	14292.7i	−0.760302	−	0.760302i
$708$		0			0
$709$	−5021.47	+	5021.47i	−0.265988	+	0.265988i	−0.827481	−	0.561493i	$-0.810227\pi$
$709$	−5021.47	+	5021.47i	−0.265988	+	0.265988i	0.561493	+	0.827481i	$0.310227\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	673.295			0.0353648
$714$		0			0
$715$	−15532.8	−	15532.8i	−0.812439	−	0.812439i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4631.53			−0.240232			−0.120116	−	0.992760i	$-0.538327\pi$
$719$	−4631.53			−0.240232			−0.120116	+	0.992760i	$0.538327\pi$
$720$		0			0
$721$	23099.9			1.19318
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−13971.1	−	13971.1i	−0.715686	−	0.715686i
$726$		0			0
$727$	7016.43			0.357944			0.178972	−	0.983854i	$-0.442723\pi$
$727$	7016.43			0.357944			0.178972	+	0.983854i	$0.442723\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−26230.7	+	26230.7i	−1.32719	+	1.32719i
$732$		0			0
$733$	−9830.27	−	9830.27i	−0.495347	−	0.495347i	0.414639	−	0.909986i	$-0.363908\pi$
$733$	−9830.27	−	9830.27i	−0.495347	−	0.495347i	−0.909986	+	0.414639i	$0.863908\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			7551.88i			0.377445i
$738$		0			0
$739$	25624.1	−	25624.1i	1.27551	−	1.27551i	0.332350	−	0.943156i	$-0.392158\pi$
$739$	25624.1	−	25624.1i	1.27551	−	1.27551i	0.943156	−	0.332350i	$-0.107842\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1936.75i		−	0.0956294i	−0.998856	−	0.0478147i	$-0.984774\pi$
$743$		−	1936.75i		−	0.0956294i	0.998856	−	0.0478147i	$-0.0152257\pi$
$744$		0			0
$745$		−	22758.4i		−	1.11920i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−12030.6	+	12030.6i	−0.586901	+	0.586901i
$750$		0			0
$751$			32386.7i			1.57365i	0.617178	+	0.786823i	$0.288276\pi$
$751$			32386.7i			1.57365i	−0.617178	+	0.786823i	$0.711724\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12343.3	+	12343.3i	0.594992	+	0.594992i
$756$		0			0
$757$	−1721.80	+	1721.80i	−0.0826681	+	0.0826681i	−0.747232	−	0.664564i	$-0.768617\pi$
$757$	−1721.80	+	1721.80i	−0.0826681	+	0.0826681i	0.664564	+	0.747232i	$0.268617\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	29323.2			1.39680			0.698400	−	0.715708i	$-0.253896\pi$
$761$	29323.2			1.39680			0.698400	+	0.715708i	$0.253896\pi$
$762$		0			0
$763$	2904.12	+	2904.12i	0.137793	+	0.137793i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	33400.7			1.57240
$768$		0			0
$769$	−22066.6			−1.03478			−0.517388	−	0.855751i	$-0.673096\pi$
$769$	−22066.6			−1.03478			−0.517388	+	0.855751i	$0.673096\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−13267.0	−	13267.0i	−0.617313	−	0.617313i	0.327529	−	0.944841i	$-0.393784\pi$
$773$	−13267.0	−	13267.0i	−0.617313	−	0.617313i	−0.944841	+	0.327529i	$0.893784\pi$
$774$		0			0
$775$	−2016.25			−0.0934528
$776$		0			0
$777$		0			0
$778$		0			0
$779$	33419.5	−	33419.5i	1.53707	−	1.53707i
$780$		0			0
$781$	−6436.46	−	6436.46i	−0.294897	−	0.294897i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			44997.5i			2.04590i
$786$		0			0
$787$	−14004.3	+	14004.3i	−0.634306	+	0.634306i	−0.949145	−	0.314839i	$-0.898049\pi$
$787$	−14004.3	+	14004.3i	−0.634306	+	0.634306i	0.314839	+	0.949145i	$0.398049\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	21415.8i		−	0.962655i
$792$		0			0
$793$			18590.3i			0.832485i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12572.2	−	12572.2i	0.558760	−	0.558760i	−0.370194	−	0.928954i	$-0.620709\pi$
$797$	12572.2	−	12572.2i	0.558760	−	0.558760i	0.928954	+	0.370194i	$0.120709\pi$
$798$		0			0
$799$			27414.9i			1.21386i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1422.03	−	1422.03i	−0.0624937	−	0.0624937i
$804$		0			0
$805$	7856.50	−	7856.50i	0.343981	−	0.343981i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	40623.8			1.76546			0.882729	−	0.469883i	$-0.155704\pi$
$809$	40623.8			1.76546			0.882729	+	0.469883i	$0.155704\pi$
$810$		0			0
$811$	27924.9	+	27924.9i	1.20910	+	1.20910i	0.971320	+	0.237777i	$0.0764187\pi$
$811$	27924.9	+	27924.9i	1.20910	+	1.20910i	0.237777	+	0.971320i	$0.423581\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−39035.7			−1.67774
$816$		0			0
$817$	−74253.8			−3.17970
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−19156.4	−	19156.4i	−0.814327	−	0.814327i	0.170952	−	0.985279i	$-0.445316\pi$
$821$	−19156.4	−	19156.4i	−0.814327	−	0.814327i	−0.985279	+	0.170952i	$0.945316\pi$
$822$		0			0
$823$	−39729.9			−1.68274			−0.841371	−	0.540459i	$-0.818251\pi$
$823$	−39729.9			−1.68274			−0.841371	+	0.540459i	$0.818251\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	9676.86	−	9676.86i	0.406889	−	0.406889i	−0.473763	−	0.880652i	$-0.657105\pi$
$827$	9676.86	−	9676.86i	0.406889	−	0.406889i	0.880652	+	0.473763i	$0.157105\pi$
$828$		0			0
$829$	31914.5	+	31914.5i	1.33708	+	1.33708i	0.898878	+	0.438199i	$0.144383\pi$
$829$	31914.5	+	31914.5i	1.33708	+	1.33708i	0.438199	+	0.898878i	$0.355617\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	13832.4i		−	0.575345i
$834$		0			0
$835$	18746.7	−	18746.7i	0.776953	−	0.776953i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			21568.7i			0.887525i	0.896145	+	0.443762i	$0.146357\pi$
$839$			21568.7i			0.887525i	−0.896145	+	0.443762i	$0.853643\pi$
$840$		0			0
$841$			8290.69i			0.339936i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−16150.1	+	16150.1i	−0.657490	+	0.657490i
$846$		0			0
$847$			10816.4i			0.438789i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5981.98	+	5981.98i	0.240963	+	0.240963i
$852$		0			0
$853$	−9556.39	+	9556.39i	−0.383593	+	0.383593i	−0.872395	−	0.488802i	$-0.837434\pi$
$853$	−9556.39	+	9556.39i	−0.383593	+	0.383593i	0.488802	+	0.872395i	$0.337434\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−14819.6			−0.590697			−0.295349	−	0.955390i	$-0.595436\pi$
$857$	−14819.6			−0.590697			−0.295349	+	0.955390i	$0.595436\pi$
$858$		0			0
$859$	4446.63	+	4446.63i	0.176621	+	0.176621i	0.789881	−	0.613260i	$-0.210142\pi$
$859$	4446.63	+	4446.63i	0.176621	+	0.176621i	−0.613260	+	0.789881i	$0.710142\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	32985.3			1.30108			0.650541	−	0.759471i	$-0.274542\pi$
$863$	32985.3			1.30108			0.650541	+	0.759471i	$0.274542\pi$
$864$		0			0
$865$	−40145.2			−1.57801
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3676.67	−	3676.67i	−0.143524	−	0.143524i
$870$		0			0
$871$	20506.9			0.797761
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4641.79	+	4641.79i	−0.179338	+	0.179338i
$876$		0			0
$877$	−4904.79	−	4904.79i	−0.188852	−	0.188852i	0.606348	−	0.795200i	$-0.292634\pi$
$877$	−4904.79	−	4904.79i	−0.188852	−	0.188852i	−0.795200	+	0.606348i	$0.792634\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			5133.26i			0.196304i	0.995171	+	0.0981520i	$0.0312931\pi$
$881$			5133.26i			0.196304i	−0.995171	+	0.0981520i	$0.968707\pi$
$882$		0			0
$883$	−11367.4	+	11367.4i	−0.433231	+	0.433231i	−0.889726	−	0.456495i	$-0.849105\pi$
$883$	−11367.4	+	11367.4i	−0.433231	+	0.433231i	0.456495	+	0.889726i	$0.349105\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			33679.0i			1.27489i	0.770494	+	0.637447i	$0.220009\pi$
$887$			33679.0i			1.27489i	−0.770494	+	0.637447i	$0.779991\pi$
$888$		0			0
$889$			25297.1i			0.954371i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−38803.0	+	38803.0i	−1.45408	+	1.45408i
$894$		0			0
$895$			12768.0i			0.476857i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1161.63	−	1161.63i	−0.0430950	−	0.0430950i
$900$		0			0
$901$	−34580.0	+	34580.0i	−1.27861	+	1.27861i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−37955.6			−1.39413
$906$		0			0
$907$	24197.4	+	24197.4i	0.885844	+	0.885844i	0.994121	−	0.108276i	$-0.0345332\pi$
$907$	24197.4	+	24197.4i	0.885844	+	0.885844i	−0.108276	+	0.994121i	$0.534533\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−6916.65			−0.251546			−0.125773	−	0.992059i	$-0.540141\pi$
$911$	−6916.65			−0.251546			−0.125773	+	0.992059i	$0.540141\pi$
$912$		0			0
$913$	6346.75			0.230062
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7710.83	+	7710.83i	0.277681	+	0.277681i
$918$		0			0
$919$	26324.8			0.944913			0.472457	−	0.881354i	$-0.343367\pi$
$919$	26324.8			0.944913			0.472457	+	0.881354i	$0.343367\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−17478.0	+	17478.0i	−0.623290	+	0.623290i
$924$		0			0
$925$	−17913.7	−	17913.7i	−0.636754	−	0.636754i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			26972.1i			0.952557i	0.879295	+	0.476278i	$0.158015\pi$
$929$			26972.1i			0.952557i	−0.879295	+	0.476278i	$0.841985\pi$
$930$		0			0
$931$	19578.3	−	19578.3i	0.689207	−	0.689207i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			28232.1i			0.987473i
$936$		0			0
$937$		−	2342.01i		−	0.0816545i	−0.999166	−	0.0408272i	$-0.987001\pi$
$937$		−	2342.01i		−	0.0816545i	0.999166	−	0.0408272i	$-0.0129993\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12461.8	−	12461.8i	0.431715	−	0.431715i	−0.457496	−	0.889212i	$-0.651254\pi$
$941$	12461.8	−	12461.8i	0.431715	−	0.431715i	0.889212	+	0.457496i	$0.151254\pi$
$942$		0			0
$943$			16011.2i			0.552911i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	23195.0	+	23195.0i	0.795919	+	0.795919i	0.982449	−	0.186530i	$-0.0597243\pi$
$947$	23195.0	+	23195.0i	0.795919	+	0.795919i	−0.186530	+	0.982449i	$0.559724\pi$
$948$		0			0
$949$	−3861.49	+	3861.49i	−0.132086	+	0.132086i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	19212.6			0.653049			0.326525	−	0.945189i	$-0.394122\pi$
$953$	19212.6			0.653049			0.326525	+	0.945189i	$0.394122\pi$
$954$		0			0
$955$	30852.0	+	30852.0i	1.04539	+	1.04539i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5890.09			0.198333
$960$		0			0
$961$	29623.4			0.994373
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−47235.5	−	47235.5i	−1.57571	−	1.57571i
$966$		0			0
$967$	40032.4			1.33129			0.665644	−	0.746269i	$-0.268157\pi$
$967$	40032.4			1.33129			0.665644	+	0.746269i	$0.268157\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12435.1	+	12435.1i	−0.410981	+	0.410981i	−0.882080	−	0.471099i	$-0.843857\pi$
$971$	−12435.1	+	12435.1i	−0.410981	+	0.410981i	0.471099	+	0.882080i	$0.343857\pi$
$972$		0			0
$973$	26461.4	+	26461.4i	0.871855	+	0.871855i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			8094.29i			0.265055i	0.991179	+	0.132528i	$0.0423094\pi$
$977$			8094.29i			0.265055i	−0.991179	+	0.132528i	$0.957691\pi$
$978$		0			0
$979$	−10702.0	+	10702.0i	−0.349374	+	0.349374i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			3321.21i			0.107762i	0.998547	+	0.0538811i	$0.0171592\pi$
$983$			3321.21i			0.107762i	−0.998547	+	0.0538811i	$0.982841\pi$
$984$		0			0
$985$			7102.15i			0.229739i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	17787.4	−	17787.4i	0.571897	−	0.571897i
$990$		0			0
$991$			33606.0i			1.07722i	0.842554	+	0.538612i	$0.181051\pi$
$991$			33606.0i			1.07722i	−0.842554	+	0.538612i	$0.818949\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−32835.6	−	32835.6i	−1.04619	−	1.04619i
$996$		0			0
$997$	−2809.08	+	2809.08i	−0.0892323	+	0.0892323i	−0.750314	−	0.661082i	$-0.770098\pi$
$997$	−2809.08	+	2809.08i	−0.0892323	+	0.0892323i	0.661082	+	0.750314i	$0.270098\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.4.l.a.863.23		48
3.2	odd	2	inner	1152.4.l.a.863.2		48
4.3	odd	2		1152.4.l.b.863.23		48
8.3	odd	2		144.4.l.a.35.10	✓	48
8.5	even	2		576.4.l.a.431.2		48
12.11	even	2		1152.4.l.b.863.2		48
16.3	odd	4		576.4.l.a.143.23		48
16.5	even	4		1152.4.l.b.287.2		48
16.11	odd	4	inner	1152.4.l.a.287.2		48
16.13	even	4		144.4.l.a.107.15	yes	48
24.5	odd	2		576.4.l.a.431.23		48
24.11	even	2		144.4.l.a.35.15	yes	48
48.5	odd	4		1152.4.l.b.287.23		48
48.11	even	4	inner	1152.4.l.a.287.23		48
48.29	odd	4		144.4.l.a.107.10	yes	48
48.35	even	4		576.4.l.a.143.2		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.4.l.a.35.10	✓	48	8.3	odd	2
144.4.l.a.35.15	yes	48	24.11	even	2
144.4.l.a.107.10	yes	48	48.29	odd	4
144.4.l.a.107.15	yes	48	16.13	even	4
576.4.l.a.143.2		48	48.35	even	4
576.4.l.a.143.23		48	16.3	odd	4
576.4.l.a.431.2		48	8.5	even	2
576.4.l.a.431.23		48	24.5	odd	2
1152.4.l.a.287.2		48	16.11	odd	4	inner
1152.4.l.a.287.23		48	48.11	even	4	inner
1152.4.l.a.863.2		48	3.2	odd	2	inner
1152.4.l.a.863.23		48	1.1	even	1	trivial
1152.4.l.b.287.2		48	16.5	even	4
1152.4.l.b.287.23		48	48.5	odd	4
1152.4.l.b.863.2		48	12.11	even	2
1152.4.l.b.863.23		48	4.3	odd	2

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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(11.8474 + 11.8474i) q^{5} +12.7523 q^{7} +(15.5373 - 15.5373i) q^{11} +(-42.1911 - 42.1911i) q^{13} +76.6855i q^{17} +(-108.540 + 108.540i) q^{19} -52.0013i q^{23} +155.724i q^{25} +(-89.7171 + 89.7171i) q^{29} +12.9476i q^{31} +(151.083 + 151.083i) q^{35} +(-115.035 + 115.035i) q^{37} -307.899 q^{41} +(342.056 + 342.056i) q^{43} +357.499 q^{47} -180.378 q^{49} +(450.933 + 450.933i) q^{53} +368.154 q^{55} +(-395.827 + 395.827i) q^{59} +(-220.311 - 220.311i) q^{61} -999.712i q^{65} +(-243.024 + 243.024i) q^{67} -414.259i q^{71} -91.5239i q^{73} +(198.137 - 198.137i) q^{77} -236.635i q^{79} +(204.243 + 204.243i) q^{83} +(-908.526 + 908.526i) q^{85} -688.795 q^{89} +(-538.035 - 538.035i) q^{91} -2571.85 q^{95} +968.890 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{19} + 864 q^{43} + 2352 q^{49} + 576 q^{55} - 1824 q^{61} + 816 q^{67} + 480 q^{85} - 3600 q^{91}+O(q^{100}) \) 48 * q - 48 * q^19 + 864 * q^43 + 2352 * q^49 + 576 * q^55 - 1824 * q^61 + 816 * q^67 + 480 * q^85 - 3600 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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