LMFDB - Embedded newform 2304.3.b.s.127.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,3,Mod(127,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("2304.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2304.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$62.7794529086$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 1152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.7
Root			$1.28897 + 0.581861i$ of defining polynomial
Character	$\chi$	$=$	2304.127
Dual form			2304.3.b.s.127.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.11993i q^{5} -9.48331i q^{7} +O(q^{10})$	$q+8.11993i q^{5} -9.48331i q^{7} +10.5830 q^{11} +20.9666i q^{13} +4.92615 q^{17} +26.9666 q^{19} -43.7934i q^{23} -40.9333 q^{25} -14.5075i q^{29} +25.4833i q^{31} +77.0038 q^{35} +12.9666i q^{37} +50.1810 q^{41} +5.03337 q^{43} -40.8706i q^{47} -40.9333 q^{49} +27.2826i q^{53} +85.9333i q^{55} -24.0888 q^{59} -28.9666i q^{61} -170.248 q^{65} -65.7998 q^{67} -90.5097i q^{71} +127.800 q^{73} -100.362i q^{77} +35.5501i q^{79} +101.093 q^{83} +40.0000i q^{85} +15.6979 q^{89} +198.833 q^{91} +218.967i q^{95} +73.8665 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 96 q^{19} - 88 q^{25} + 160 q^{43} - 88 q^{49} + 192 q^{67} + 304 q^{73} + 992 q^{91} + 112 q^{97}+O(q^{100}) $ 8 * q + 96 * q^19 - 88 * q^25 + 160 * q^43 - 88 * q^49 + 192 * q^67 + 304 * q^73 + 992 * q^91 + 112 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$.

$n$	$1279$	$1793$	$2053$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	8.11993i	1.62399i	0.583667	+	0.811993i	$0.301617\pi$
$5$	8.11993i	1.62399i	−0.583667	+	0.811993i	$0.698383\pi$
$6$	0	0
$7$	− 9.48331i	− 1.35476i	−0.735634	−	0.677380i	$-0.763115\pi$
$7$	− 9.48331i	− 1.35476i	0.735634	−	0.677380i	$-0.236885\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	10.5830	0.962091	0.481046	−	0.876696i	$-0.340257\pi$
$11$	10.5830	0.962091	0.481046	+	0.876696i	$0.340257\pi$
$12$	0	0
$13$	20.9666i	1.61282i	0.591358	+	0.806409i	$0.298592\pi$
$13$	20.9666i	1.61282i	−0.591358	+	0.806409i	$0.701408\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.92615	0.289774	0.144887	−	0.989448i	$-0.453718\pi$
$17$	4.92615	0.289774	0.144887	+	0.989448i	$0.453718\pi$
$18$	0	0
$19$	26.9666	1.41930	0.709648	−	0.704556i	$-0.248854\pi$
$19$	26.9666	1.41930	0.709648	+	0.704556i	$0.248854\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 43.7934i	− 1.90406i	−0.306001	−	0.952031i	$-0.598991\pi$
$23$	− 43.7934i	− 1.90406i	0.306001	−	0.952031i	$-0.401009\pi$
$24$	0	0
$25$	−40.9333	−1.63733
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 14.5075i	− 0.500258i	−0.968212	−	0.250129i	$-0.919527\pi$
$29$	− 14.5075i	− 0.500258i	0.968212	−	0.250129i	$-0.0804731\pi$
$30$	0	0
$31$	25.4833i	0.822042i	0.911626	+	0.411021i	$0.134828\pi$
$31$	25.4833i	0.822042i	−0.911626	+	0.411021i	$0.865172\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	77.0038	2.20011
$36$	0	0
$37$	12.9666i	0.350449i	0.984528	+	0.175225i	$0.0560652\pi$
$37$	12.9666i	0.350449i	−0.984528	+	0.175225i	$0.943935\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	50.1810	1.22393	0.611963	−	0.790886i	$-0.290380\pi$
$41$	50.1810	1.22393	0.611963	+	0.790886i	$0.290380\pi$
$42$	0	0
$43$	5.03337	0.117055	0.0585276	−	0.998286i	$-0.481359\pi$
$43$	5.03337	0.117055	0.0585276	+	0.998286i	$0.481359\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 40.8706i	− 0.869588i	−0.900530	−	0.434794i	$-0.856821\pi$
$47$	− 40.8706i	− 0.869588i	0.900530	−	0.434794i	$-0.143179\pi$
$48$	0	0
$49$	−40.9333	−0.835373
$50$	0	0
$51$	0	0
$52$	0	0
$53$	27.2826i	0.514766i	0.966309	+	0.257383i	$0.0828602\pi$
$53$	27.2826i	0.514766i	−0.966309	+	0.257383i	$0.917140\pi$
$54$	0	0
$55$	85.9333i	1.56242i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−24.0888	−0.408285	−0.204143	−	0.978941i	$-0.565441\pi$
$59$	−24.0888	−0.408285	−0.204143	+	0.978941i	$0.565441\pi$
$60$	0	0
$61$	− 28.9666i	− 0.474863i	−0.971404	−	0.237431i	$-0.923695\pi$
$61$	− 28.9666i	− 0.474863i	0.971404	−	0.237431i	$-0.0763055\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−170.248	−2.61919
$66$	0	0
$67$	−65.7998	−0.982086	−0.491043	−	0.871135i	$-0.663384\pi$
$67$	−65.7998	−0.982086	−0.491043	+	0.871135i	$0.663384\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 90.5097i	− 1.27478i	−0.770540	−	0.637392i	$-0.780013\pi$
$71$	− 90.5097i	− 1.27478i	0.770540	−	0.637392i	$-0.219987\pi$
$72$	0	0
$73$	127.800	1.75068	0.875341	−	0.483506i	$-0.160637\pi$
$73$	127.800	1.75068	0.875341	+	0.483506i	$0.160637\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 100.362i	− 1.30340i
$78$	0	0
$79$	35.5501i	0.450001i	0.974359	+	0.225000i	$0.0722383\pi$
$79$	35.5501i	0.450001i	−0.974359	+	0.225000i	$0.927762\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	101.093	1.21798	0.608992	−	0.793176i	$-0.291574\pi$
$83$	101.093	1.21798	0.608992	+	0.793176i	$0.291574\pi$
$84$	0	0
$85$	40.0000i	0.470588i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	15.6979	0.176381	0.0881906	−	0.996104i	$-0.471892\pi$
$89$	15.6979	0.176381	0.0881906	+	0.996104i	$0.471892\pi$
$90$	0	0
$91$	198.833	2.18498
$92$	0	0
$93$	0	0
$94$	0	0
$95$	218.967i	2.30492i
$96$	0	0
$97$	73.8665	0.761510	0.380755	−	0.924676i	$-0.375664\pi$
$97$	73.8665	0.761510	0.380755	+	0.924676i	$0.375664\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	114.869i	1.13732i	0.822572	+	0.568661i	$0.192538\pi$
$101$	114.869i	1.13732i	−0.822572	+	0.568661i	$0.807462\pi$
$102$	0	0
$103$	121.483i	1.17945i	0.807604	+	0.589725i	$0.200764\pi$
$103$	121.483i	1.17945i	−0.807604	+	0.589725i	$0.799236\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−156.931	−1.46664	−0.733320	−	0.679883i	$-0.762030\pi$
$107$	−156.931	−1.46664	−0.733320	+	0.679883i	$0.762030\pi$
$108$	0	0
$109$	− 114.766i	− 1.05290i	−0.850205	−	0.526451i	$-0.823522\pi$
$109$	− 114.766i	− 1.05290i	0.850205	−	0.526451i	$-0.176478\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	179.180	1.58567	0.792834	−	0.609438i	$-0.208605\pi$
$113$	179.180	1.58567	0.792834	+	0.609438i	$0.208605\pi$
$114$	0	0
$115$	355.600	3.09217
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 46.7162i	− 0.392573i
$120$	0	0
$121$	−9.00000	−0.0743802
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 129.377i	− 1.03502i
$126$	0	0
$127$	96.5834i	0.760499i	0.924884	+	0.380250i	$0.124162\pi$
$127$	96.5834i	0.760499i	−0.924884	+	0.380250i	$0.875838\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−169.328	−1.29258	−0.646290	−	0.763092i	$-0.723681\pi$
$131$	−169.328	−1.29258	−0.646290	+	0.763092i	$0.723681\pi$
$132$	0	0
$133$	− 255.733i	− 1.92280i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	235.207	1.71684	0.858420	−	0.512948i	$-0.171447\pi$
$137$	235.207	1.71684	0.858420	+	0.512948i	$0.171447\pi$
$138$	0	0
$139$	30.2002	0.217268	0.108634	−	0.994082i	$-0.465352\pi$
$139$	30.2002	0.217268	0.108634	+	0.994082i	$0.465352\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	221.890i	1.55168i
$144$	0	0
$145$	117.800	0.812412
$146$	0	0
$147$	0	0
$148$	0	0
$149$	170.519i	1.14442i	0.820107	+	0.572210i	$0.193914\pi$
$149$	170.519i	1.14442i	−0.820107	+	0.572210i	$0.806086\pi$
$150$	0	0
$151$	− 87.1496i	− 0.577150i	−0.957457	−	0.288575i	$-0.906819\pi$
$151$	− 87.1496i	− 0.577150i	0.957457	−	0.288575i	$-0.0931814\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−206.923	−1.33499
$156$	0	0
$157$	98.7664i	0.629085i	0.949243	+	0.314543i	$0.101851\pi$
$157$	98.7664i	0.629085i	−0.949243	+	0.314543i	$0.898149\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−415.307	−2.57955
$162$	0	0
$163$	67.2336	0.412476	0.206238	−	0.978502i	$-0.433878\pi$
$163$	67.2336	0.412476	0.206238	+	0.978502i	$0.433878\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	216.044i	1.29368i	0.762626	+	0.646839i	$0.223910\pi$
$167$	216.044i	1.29368i	−0.762626	+	0.646839i	$0.776090\pi$
$168$	0	0
$169$	−270.600	−1.60118
$170$	0	0
$171$	0	0
$172$	0	0
$173$	118.876i	0.687145i	0.939126	+	0.343573i	$0.111637\pi$
$173$	118.876i	0.687145i	−0.939126	+	0.343573i	$0.888363\pi$
$174$	0	0
$175$	388.183i	2.21819i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	24.0888	0.134574	0.0672872	−	0.997734i	$-0.478566\pi$
$179$	24.0888	0.134574	0.0672872	+	0.997734i	$0.478566\pi$
$180$	0	0
$181$	131.033i	0.723941i	0.932189	+	0.361971i	$0.117896\pi$
$181$	131.033i	0.723941i	−0.932189	+	0.361971i	$0.882104\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−105.288	−0.569125
$186$	0	0
$187$	52.1335	0.278789
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 128.457i	− 0.672552i	−0.941763	−	0.336276i	$-0.890832\pi$
$191$	− 128.457i	− 0.672552i	0.941763	−	0.336276i	$-0.109168\pi$
$192$	0	0
$193$	−167.666	−0.868737	−0.434369	−	0.900735i	$-0.643028\pi$
$193$	−167.666	−0.868737	−0.434369	+	0.900735i	$0.643028\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 122.341i	− 0.621020i	−0.950570	−	0.310510i	$-0.899500\pi$
$197$	− 122.341i	− 0.621020i	0.950570	−	0.310510i	$-0.100500\pi$
$198$	0	0
$199$	363.283i	1.82554i	0.408470	+	0.912772i	$0.366062\pi$
$199$	363.283i	1.82554i	−0.408470	+	0.912772i	$0.633938\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−137.579	−0.677729
$204$	0	0
$205$	407.466i	1.98764i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	285.388	1.36549
$210$	0	0
$211$	138.067	0.654345	0.327172	−	0.944965i	$-0.393904\pi$
$211$	138.067	0.654345	0.327172	+	0.944965i	$0.393904\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	40.8706i	0.190096i
$216$	0	0
$217$	241.666	1.11367
$218$	0	0
$219$	0	0
$220$	0	0
$221$	103.285i	0.467352i
$222$	0	0
$223$	264.316i	1.18528i	0.805469	+	0.592638i	$0.201913\pi$
$223$	264.316i	1.18528i	−0.805469	+	0.592638i	$0.798087\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	243.409	1.07229	0.536143	−	0.844127i	$-0.319881\pi$
$227$	243.409	1.07229	0.536143	+	0.844127i	$0.319881\pi$
$228$	0	0
$229$	234.232i	1.02285i	0.859328	+	0.511425i	$0.170882\pi$
$229$	234.232i	1.02285i	−0.859328	+	0.511425i	$0.829118\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	35.4025	0.151942	0.0759711	−	0.997110i	$-0.475794\pi$
$233$	35.4025	0.151942	0.0759711	+	0.997110i	$0.475794\pi$
$234$	0	0
$235$	331.867	1.41220
$236$	0	0
$237$	0	0
$238$	0	0
$239$	32.1022i	0.134319i	0.997742	+	0.0671594i	$0.0213936\pi$
$239$	32.1022i	0.134319i	−0.997742	+	0.0671594i	$0.978606\pi$
$240$	0	0
$241$	−103.800	−0.430704	−0.215352	−	0.976536i	$-0.569090\pi$
$241$	−103.800	−0.430704	−0.215352	+	0.976536i	$0.569090\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 332.375i	− 1.35663i
$246$	0	0
$247$	565.399i	2.28907i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	312.753	1.24603	0.623014	−	0.782211i	$-0.285908\pi$
$251$	312.753	1.24603	0.623014	+	0.782211i	$0.285908\pi$
$252$	0	0
$253$	− 463.466i	− 1.83188i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−33.5636	−0.130598	−0.0652988	−	0.997866i	$-0.520800\pi$
$257$	−33.5636	−0.130598	−0.0652988	+	0.997866i	$0.520800\pi$
$258$	0	0
$259$	122.967	0.474775
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 338.656i	− 1.28767i	−0.765166	−	0.643833i	$-0.777343\pi$
$263$	− 338.656i	− 1.28767i	0.765166	−	0.643833i	$-0.222657\pi$
$264$	0	0
$265$	−221.533	−0.835973
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 310.832i	− 1.15551i	−0.816211	−	0.577754i	$-0.803929\pi$
$269$	− 310.832i	− 1.15551i	0.816211	−	0.577754i	$-0.196071\pi$
$270$	0	0
$271$	− 266.749i	− 0.984314i	−0.870506	−	0.492157i	$-0.836209\pi$
$271$	− 266.749i	− 0.984314i	0.870506	−	0.492157i	$-0.163791\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−433.197	−1.57526
$276$	0	0
$277$	458.766i	1.65620i	0.560583	+	0.828098i	$0.310577\pi$
$277$	458.766i	1.65620i	−0.560583	+	0.828098i	$0.689423\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−395.602	−1.40784	−0.703919	−	0.710281i	$-0.748568\pi$
$281$	−395.602	−1.40784	−0.703919	+	0.710281i	$0.748568\pi$
$282$	0	0
$283$	375.733	1.32768	0.663839	−	0.747875i	$-0.268926\pi$
$283$	375.733	1.32768	0.663839	+	0.747875i	$0.268926\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 475.882i	− 1.65813i
$288$	0	0
$289$	−264.733	−0.916031
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.74572i	0.0332619i	0.999862	+	0.0166309i	$0.00529403\pi$
$293$	9.74572i	0.0332619i	−0.999862	+	0.0166309i	$0.994706\pi$
$294$	0	0
$295$	− 195.600i	− 0.663049i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	918.201	3.07090
$300$	0	0
$301$	− 47.7330i	− 0.158582i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	235.207	0.771170
$306$	0	0
$307$	−89.5328	−0.291638	−0.145819	−	0.989311i	$-0.546582\pi$
$307$	−89.5328	−0.291638	−0.145819	+	0.989311i	$0.546582\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	274.452i	0.882482i	0.897389	+	0.441241i	$0.145462\pi$
$311$	274.452i	0.882482i	−0.897389	+	0.441241i	$0.854538\pi$
$312$	0	0
$313$	151.666	0.484557	0.242278	−	0.970207i	$-0.422105\pi$
$313$	151.666	0.484557	0.242278	+	0.970207i	$0.422105\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	201.914i	0.636954i	0.947931	+	0.318477i	$0.103171\pi$
$317$	201.914i	0.636954i	−0.947931	+	0.318477i	$0.896829\pi$
$318$	0	0
$319$	− 153.533i	− 0.481294i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	132.842	0.411275
$324$	0	0
$325$	− 858.232i	− 2.64072i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−387.589	−1.17808
$330$	0	0
$331$	195.600	0.590935	0.295468	−	0.955353i	$-0.404525\pi$
$331$	195.600	0.590935	0.295468	+	0.955353i	$0.404525\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 534.290i	− 1.59489i
$336$	0	0
$337$	23.9333	0.0710186	0.0355093	−	0.999369i	$-0.488695\pi$
$337$	23.9333	0.0710186	0.0355093	+	0.999369i	$0.488695\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	269.690i	0.790880i
$342$	0	0
$343$	− 76.4994i	− 0.223030i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	139.796	0.402869	0.201435	−	0.979502i	$-0.435440\pi$
$347$	139.796	0.402869	0.201435	+	0.979502i	$0.435440\pi$
$348$	0	0
$349$	− 21.2336i	− 0.0608412i	−0.999537	−	0.0304206i	$-0.990315\pi$
$349$	− 21.2336i	− 0.0608412i	0.999537	−	0.0304206i	$-0.00968468\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	515.669	1.46082	0.730409	−	0.683010i	$-0.239329\pi$
$353$	515.669	1.46082	0.730409	+	0.683010i	$0.239329\pi$
$354$	0	0
$355$	734.932	2.07023
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 253.992i	− 0.707499i	−0.935340	−	0.353749i	$-0.884907\pi$
$359$	− 253.992i	− 0.707499i	0.935340	−	0.353749i	$-0.115093\pi$
$360$	0	0
$361$	366.199	1.01440
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1037.73i	2.84308i
$366$	0	0
$367$	384.682i	1.04818i	0.851663	+	0.524090i	$0.175595\pi$
$367$	384.682i	1.04818i	−0.851663	+	0.524090i	$0.824405\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	258.730	0.697384
$372$	0	0
$373$	197.501i	0.529492i	0.964318	+	0.264746i	$0.0852881\pi$
$373$	197.501i	0.529492i	−0.964318	+	0.264746i	$0.914712\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	304.173	0.806825
$378$	0	0
$379$	−234.433	−0.618556	−0.309278	−	0.950972i	$-0.600087\pi$
$379$	−234.433	−0.618556	−0.309278	+	0.950972i	$0.600087\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 315.322i	− 0.823296i	−0.911343	−	0.411648i	$-0.864953\pi$
$383$	− 315.322i	− 0.823296i	0.911343	−	0.411648i	$-0.135047\pi$
$384$	0	0
$385$	814.932	2.11671
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 486.760i	− 1.25131i	−0.780099	−	0.625656i	$-0.784831\pi$
$389$	− 486.760i	− 1.25131i	0.780099	−	0.625656i	$-0.215169\pi$
$390$	0	0
$391$	− 215.733i	− 0.551747i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−288.664	−0.730795
$396$	0	0
$397$	115.300i	0.290429i	0.989400	+	0.145215i	$0.0463872\pi$
$397$	115.300i	0.290429i	−0.989400	+	0.145215i	$0.953613\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−52.3487	−0.130545	−0.0652727	−	0.997867i	$-0.520792\pi$
$401$	−52.3487	−0.130545	−0.0652727	+	0.997867i	$0.520792\pi$
$402$	0	0
$403$	−534.299	−1.32580
$404$	0	0
$405$	0	0
$406$	0	0
$407$	137.226i	0.337164i
$408$	0	0
$409$	−345.333	−0.844334	−0.422167	−	0.906518i	$-0.638730\pi$
$409$	−345.333	−0.844334	−0.422167	+	0.906518i	$0.638730\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	228.442i	0.553128i
$414$	0	0
$415$	820.865i	1.97799i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−644.151	−1.53735	−0.768676	−	0.639638i	$-0.779084\pi$
$419$	−644.151	−1.53735	−0.768676	+	0.639638i	$0.779084\pi$
$420$	0	0
$421$	− 756.700i	− 1.79739i	−0.438578	−	0.898693i	$-0.644518\pi$
$421$	− 756.700i	− 1.79739i	0.438578	−	0.898693i	$-0.355482\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−201.643	−0.474455
$426$	0	0
$427$	−274.700	−0.643325
$428$	0	0
$429$	0	0
$430$	0	0
$431$	49.6391i	0.115172i	0.998341	+	0.0575859i	$0.0183403\pi$
$431$	49.6391i	0.115172i	−0.998341	+	0.0575859i	$0.981660\pi$
$432$	0	0
$433$	581.066	1.34195	0.670976	−	0.741479i	$-0.265875\pi$
$433$	581.066	1.34195	0.670976	+	0.741479i	$0.265875\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1180.96i	− 2.70243i
$438$	0	0
$439$	274.482i	0.625244i	0.949878	+	0.312622i	$0.101207\pi$
$439$	274.482i	0.625244i	−0.949878	+	0.312622i	$0.898793\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	466.760	1.05364	0.526818	−	0.849978i	$-0.323385\pi$
$443$	466.760	1.05364	0.526818	+	0.849978i	$0.323385\pi$
$444$	0	0
$445$	127.466i	0.286441i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	479.347	1.06759	0.533794	−	0.845615i	$-0.320766\pi$
$449$	479.347	1.06759	0.533794	+	0.845615i	$0.320766\pi$
$450$	0	0
$451$	531.066	1.17753
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1614.51i	3.54838i
$456$	0	0
$457$	405.600	0.887526	0.443763	−	0.896144i	$-0.353643\pi$
$457$	405.600	0.887526	0.443763	+	0.896144i	$0.353643\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 636.171i	− 1.37998i	−0.723819	−	0.689990i	$-0.757615\pi$
$461$	− 636.171i	− 1.37998i	0.723819	−	0.689990i	$-0.242385\pi$
$462$	0	0
$463$	− 323.016i	− 0.697659i	−0.937186	−	0.348830i	$-0.886579\pi$
$463$	− 323.016i	− 0.697659i	0.937186	−	0.348830i	$-0.113421\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	128.104	0.274313	0.137157	−	0.990549i	$-0.456204\pi$
$467$	128.104	0.274313	0.137157	+	0.990549i	$0.456204\pi$
$468$	0	0
$469$	624.000i	1.33049i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	53.2682	0.112618
$474$	0	0
$475$	−1103.83	−2.32386
$476$	0	0
$477$	0	0
$478$	0	0
$479$	67.1272i	0.140140i	0.997542	+	0.0700701i	$0.0223223\pi$
$479$	67.1272i	0.140140i	−0.997542	+	0.0700701i	$0.977678\pi$
$480$	0	0
$481$	−271.867	−0.565211
$482$	0	0
$483$	0	0
$484$	0	0
$485$	599.791i	1.23668i
$486$	0	0
$487$	180.183i	0.369986i	0.982740	+	0.184993i	$0.0592262\pi$
$487$	180.183i	0.369986i	−0.982740	+	0.184993i	$0.940774\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−423.320	−0.862159	−0.431080	−	0.902314i	$-0.641867\pi$
$491$	−423.320	−0.862159	−0.431080	+	0.902314i	$0.641867\pi$
$492$	0	0
$493$	− 71.4661i	− 0.144962i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−858.332	−1.72703
$498$	0	0
$499$	−830.932	−1.66519	−0.832597	−	0.553879i	$-0.813147\pi$
$499$	−830.932	−1.66519	−0.832597	+	0.553879i	$0.813147\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 359.116i	− 0.713948i	−0.934114	−	0.356974i	$-0.883808\pi$
$503$	− 359.116i	− 0.713948i	0.934114	−	0.356974i	$-0.116192\pi$
$504$	0	0
$505$	−932.732	−1.84699
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 43.3093i	− 0.0850870i	−0.999095	−	0.0425435i	$-0.986454\pi$
$509$	− 43.3093i	− 0.0850870i	0.999095	−	0.0425435i	$-0.0135461\pi$
$510$	0	0
$511$	− 1211.97i	− 2.37175i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−986.436	−1.91541
$516$	0	0
$517$	− 432.534i	− 0.836623i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−426.407	−0.818440	−0.409220	−	0.912436i	$-0.634199\pi$
$521$	−426.407	−0.818440	−0.409220	+	0.912436i	$0.634199\pi$
$522$	0	0
$523$	384.366	0.734925	0.367463	−	0.930038i	$-0.380227\pi$
$523$	384.366	0.734925	0.367463	+	0.930038i	$0.380227\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	125.535i	0.238206i
$528$	0	0
$529$	−1388.86	−2.62545
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1052.13i	1.97397i
$534$	0	0
$535$	− 1274.26i	− 2.38180i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−433.197	−0.803705
$540$	0	0
$541$	− 403.033i	− 0.744979i	−0.928037	−	0.372489i	$-0.878504\pi$
$541$	− 403.033i	− 0.744979i	0.928037	−	0.372489i	$-0.121496\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	931.895	1.70990
$546$	0	0
$547$	−1066.43	−1.94960	−0.974801	−	0.223075i	$-0.928391\pi$
$547$	−1066.43	−1.94960	−0.974801	+	0.223075i	$0.928391\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 391.218i	− 0.710015i
$552$	0	0
$553$	337.132	0.609643
$554$	0	0
$555$	0	0
$556$	0	0
$557$	221.406i	0.397497i	0.980051	+	0.198749i	$0.0636877\pi$
$557$	221.406i	0.397497i	−0.980051	+	0.198749i	$0.936312\pi$
$558$	0	0
$559$	105.533i	0.188789i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−747.764	−1.32818	−0.664089	−	0.747654i	$-0.731180\pi$
$563$	−747.764	−1.32818	−0.664089	+	0.747654i	$0.731180\pi$
$564$	0	0
$565$	1454.93i	2.57510i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−489.199	−0.859752	−0.429876	−	0.902888i	$-0.641443\pi$
$569$	−489.199	−0.859752	−0.429876	+	0.902888i	$0.641443\pi$
$570$	0	0
$571$	784.198	1.37338	0.686688	−	0.726952i	$-0.259064\pi$
$571$	784.198	1.37338	0.686688	+	0.726952i	$0.259064\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1792.61i	3.11758i
$576$	0	0
$577$	−239.132	−0.414441	−0.207220	−	0.978294i	$-0.566442\pi$
$577$	−239.132	−0.414441	−0.207220	+	0.978294i	$0.566442\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 958.694i	− 1.65008i
$582$	0	0
$583$	288.732i	0.495252i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	84.6640	0.144232	0.0721159	−	0.997396i	$-0.477025\pi$
$587$	84.6640	0.144232	0.0721159	+	0.997396i	$0.477025\pi$
$588$	0	0
$589$	687.199i	1.16672i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1135.38	−1.91463	−0.957317	−	0.289041i	$-0.906664\pi$
$593$	−1135.38	−1.91463	−0.957317	+	0.289041i	$0.906664\pi$
$594$	0	0
$595$	379.333	0.637534
$596$	0	0
$597$	0	0
$598$	0	0
$599$	411.678i	0.687275i	0.939102	+	0.343638i	$0.111659\pi$
$599$	411.678i	0.687275i	−0.939102	+	0.343638i	$0.888341\pi$
$600$	0	0
$601$	463.666	0.771491	0.385746	−	0.922605i	$-0.373944\pi$
$601$	463.666	0.771491	0.385746	+	0.922605i	$0.373944\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 73.0794i	− 0.120792i
$606$	0	0
$607$	− 803.016i	− 1.32293i	−0.749978	−	0.661463i	$-0.769936\pi$
$607$	− 803.016i	− 1.32293i	0.749978	−	0.661463i	$-0.230064\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	856.919	1.40249
$612$	0	0
$613$	− 459.567i	− 0.749702i	−0.927085	−	0.374851i	$-0.877694\pi$
$613$	− 459.567i	− 0.749702i	0.927085	−	0.374851i	$-0.122306\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−370.052	−0.599760	−0.299880	−	0.953977i	$-0.596947\pi$
$617$	−370.052	−0.599760	−0.299880	+	0.953977i	$0.596947\pi$
$618$	0	0
$619$	10.7987	0.0174453	0.00872267	−	0.999962i	$-0.497223\pi$
$619$	10.7987	0.0174453	0.00872267	+	0.999962i	$0.497223\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 148.868i	− 0.238954i
$624$	0	0
$625$	27.2002	0.0435204
$626$	0	0
$627$	0	0
$628$	0	0
$629$	63.8756i	0.101551i
$630$	0	0
$631$	− 63.9505i	− 0.101348i	−0.998715	−	0.0506739i	$-0.983863\pi$
$631$	− 63.9505i	− 0.101348i	0.998715	−	0.0506739i	$-0.0161369\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−784.251	−1.23504
$636$	0	0
$637$	− 858.232i	− 1.34730i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	772.748	1.20554	0.602768	−	0.797917i	$-0.294065\pi$
$641$	772.748	1.20554	0.602768	+	0.797917i	$0.294065\pi$
$642$	0	0
$643$	−23.3671	−0.0363407	−0.0181704	−	0.999835i	$-0.505784\pi$
$643$	−23.3671	−0.0363407	−0.0181704	+	0.999835i	$0.505784\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 686.130i	− 1.06048i	−0.847848	−	0.530239i	$-0.822102\pi$
$647$	− 686.130i	− 1.06048i	0.847848	−	0.530239i	$-0.177898\pi$
$648$	0	0
$649$	−254.932	−0.392808
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 899.686i	− 1.37777i	−0.724869	−	0.688887i	$-0.758100\pi$
$653$	− 899.686i	− 1.37777i	0.724869	−	0.688887i	$-0.241900\pi$
$654$	0	0
$655$	− 1374.93i	− 2.09913i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	217.506	0.330054	0.165027	−	0.986289i	$-0.447229\pi$
$659$	217.506	0.330054	0.165027	+	0.986289i	$0.447229\pi$
$660$	0	0
$661$	829.234i	1.25451i	0.778812	+	0.627257i	$0.215822\pi$
$661$	829.234i	1.25451i	−0.778812	+	0.627257i	$0.784178\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2076.53	3.12261
$666$	0	0
$667$	−635.333	−0.952523
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 306.554i	− 0.456861i
$672$	0	0
$673$	−234.400	−0.348292	−0.174146	−	0.984720i	$-0.555716\pi$
$673$	−234.400	−0.348292	−0.174146	+	0.984720i	$0.555716\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	772.690i	1.14134i	0.821178	+	0.570672i	$0.193317\pi$
$677$	772.690i	1.14134i	−0.821178	+	0.570672i	$0.806683\pi$
$678$	0	0
$679$	− 700.499i	− 1.03166i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−640.522	−0.937806	−0.468903	−	0.883250i	$-0.655351\pi$
$683$	−640.522	−0.937806	−0.468903	+	0.883250i	$0.655351\pi$
$684$	0	0
$685$	1909.86i	2.78812i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−572.024	−0.830224
$690$	0	0
$691$	−973.300	−1.40854	−0.704269	−	0.709933i	$-0.748725\pi$
$691$	−973.300	−1.40854	−0.704269	+	0.709933i	$0.748725\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	245.224i	0.352840i
$696$	0	0
$697$	247.199	0.354662
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 759.045i	− 1.08280i	−0.840764	−	0.541401i	$-0.817894\pi$
$701$	− 759.045i	− 1.08280i	0.840764	−	0.541401i	$-0.182106\pi$
$702$	0	0
$703$	349.666i	0.497392i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1089.34	1.54080
$708$	0	0
$709$	− 334.034i	− 0.471135i	−0.971858	−	0.235567i	$-0.924305\pi$
$709$	− 334.034i	− 0.471135i	0.971858	−	0.235567i	$-0.0756948\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1116.00	1.56522
$714$	0	0
$715$	−1801.73	−2.51990
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 318.196i	− 0.442554i	−0.975211	−	0.221277i	$-0.928977\pi$
$719$	− 318.196i	− 0.442554i	0.975211	−	0.221277i	$-0.0710225\pi$
$720$	0	0
$721$	1152.06	1.59787
$722$	0	0
$723$	0	0
$724$	0	0
$725$	593.839i	0.819088i
$726$	0	0
$727$	− 877.617i	− 1.20718i	−0.797296	−	0.603588i	$-0.793737\pi$
$727$	− 877.617i	− 1.20718i	0.797296	−	0.603588i	$-0.206263\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	24.7951	0.0339195
$732$	0	0
$733$	− 1001.70i	− 1.36657i	−0.730150	−	0.683287i	$-0.760550\pi$
$733$	− 1001.70i	− 1.36657i	0.730150	−	0.683287i	$-0.239450\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−696.359	−0.944857
$738$	0	0
$739$	−648.999	−0.878212	−0.439106	−	0.898435i	$-0.644705\pi$
$739$	−648.999	−0.878212	−0.439106	+	0.898435i	$0.644705\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 411.678i	− 0.554075i	−0.960859	−	0.277038i	$-0.910647\pi$
$743$	− 411.678i	− 0.554075i	0.960859	−	0.277038i	$-0.0893526\pi$
$744$	0	0
$745$	−1384.60	−1.85852
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1488.22i	1.98694i
$750$	0	0
$751$	471.882i	0.628338i	0.949367	+	0.314169i	$0.101726\pi$
$751$	471.882i	0.628338i	−0.949367	+	0.314169i	$0.898274\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	707.649	0.937283
$756$	0	0
$757$	164.101i	0.216778i	0.994109	+	0.108389i	$0.0345693\pi$
$757$	164.101i	0.216778i	−0.994109	+	0.108389i	$0.965431\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	331.562	0.435693	0.217846	−	0.975983i	$-0.430097\pi$
$761$	331.562	0.435693	0.217846	+	0.975983i	$0.430097\pi$
$762$	0	0
$763$	−1088.37	−1.42643
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 505.061i	− 0.658489i
$768$	0	0
$769$	847.533	1.10212	0.551062	−	0.834465i	$-0.314223\pi$
$769$	847.533	1.10212	0.551062	+	0.834465i	$0.314223\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	980.557i	1.26851i	0.773125	+	0.634254i	$0.218693\pi$
$773$	980.557i	1.26851i	−0.773125	+	0.634254i	$0.781307\pi$
$774$	0	0
$775$	− 1043.12i	− 1.34596i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1353.21	1.73711
$780$	0	0
$781$	− 957.864i	− 1.22646i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−801.976	−1.02163
$786$	0	0
$787$	335.832	0.426724	0.213362	−	0.976973i	$-0.431559\pi$
$787$	335.832	0.426724	0.213362	+	0.976973i	$0.431559\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1699.22i	− 2.14820i
$792$	0	0
$793$	607.333	0.765867
$794$	0	0
$795$	0	0
$796$	0	0
$797$	283.114i	0.355224i	0.984101	+	0.177612i	$0.0568372\pi$
$797$	283.114i	0.355224i	−0.984101	+	0.177612i	$0.943163\pi$
$798$	0	0
$799$	− 201.335i	− 0.251984i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1352.51	1.68432
$804$	0	0
$805$	− 3372.26i	− 4.18915i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−117.308	−0.145004	−0.0725019	−	0.997368i	$-0.523098\pi$
$809$	−117.308	−0.145004	−0.0725019	+	0.997368i	$0.523098\pi$
$810$	0	0
$811$	−1235.43	−1.52334	−0.761672	−	0.647963i	$-0.775621\pi$
$811$	−1235.43	−1.52334	−0.761672	+	0.647963i	$0.775621\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	545.932i	0.669855i
$816$	0	0
$817$	135.733	0.166136
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 823.364i	− 1.00288i	−0.865192	−	0.501440i	$-0.832804\pi$
$821$	− 823.364i	− 1.00288i	0.865192	−	0.501440i	$-0.167196\pi$
$822$	0	0
$823$	743.249i	0.903097i	0.892247	+	0.451548i	$0.149128\pi$
$823$	743.249i	0.903097i	−0.892247	+	0.451548i	$0.850872\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−542.352	−0.655806	−0.327903	−	0.944711i	$-0.606342\pi$
$827$	−542.352	−0.655806	−0.327903	+	0.944711i	$0.606342\pi$
$828$	0	0
$829$	906.564i	1.09356i	0.837275	+	0.546782i	$0.184147\pi$
$829$	906.564i	1.09356i	−0.837275	+	0.546782i	$0.815853\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−201.643	−0.242069
$834$	0	0
$835$	−1754.26	−2.10092
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1015.97i	− 1.21093i	−0.795873	−	0.605464i	$-0.792988\pi$
$839$	− 1015.97i	− 1.21093i	0.795873	−	0.605464i	$-0.207012\pi$
$840$	0	0
$841$	630.533	0.749742
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 2197.25i	− 2.60030i
$846$	0	0
$847$	85.3498i	0.100767i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	567.853	0.667277
$852$	0	0
$853$	102.568i	0.120244i	0.998191	+	0.0601222i	$0.0191490\pi$
$853$	102.568i	0.120244i	−0.998191	+	0.0601222i	$0.980851\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1081.85	−1.26237	−0.631183	−	0.775634i	$-0.717430\pi$
$857$	−1081.85	−1.26237	−0.631183	+	0.775634i	$0.717430\pi$
$858$	0	0
$859$	597.231	0.695264	0.347632	−	0.937631i	$-0.386986\pi$
$859$	597.231	0.695264	0.347632	+	0.937631i	$0.386986\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	537.261i	0.622551i	0.950320	+	0.311275i	$0.100756\pi$
$863$	537.261i	0.622551i	−0.950320	+	0.311275i	$0.899244\pi$
$864$	0	0
$865$	−965.266	−1.11591
$866$	0	0
$867$	0	0
$868$	0	0
$869$	376.226i	0.432942i
$870$	0	0
$871$	− 1379.60i	− 1.58393i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1226.92	−1.40220
$876$	0	0
$877$	− 395.899i	− 0.451424i	−0.974194	−	0.225712i	$-0.927529\pi$
$877$	− 395.899i	− 0.451424i	0.974194	−	0.225712i	$-0.0724708\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−289.066	−0.328111	−0.164056	−	0.986451i	$-0.552458\pi$
$881$	−289.066	−0.328111	−0.164056	+	0.986451i	$0.552458\pi$
$882$	0	0
$883$	1548.96	1.75421	0.877103	−	0.480302i	$-0.159473\pi$
$883$	1548.96	1.75421	0.877103	+	0.480302i	$0.159473\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1544.41i	− 1.74116i	−0.492023	−	0.870582i	$-0.663743\pi$
$887$	− 1544.41i	− 1.74116i	0.492023	−	0.870582i	$-0.336257\pi$
$888$	0	0
$889$	915.931	1.03029
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1102.14i	− 1.23420i
$894$	0	0
$895$	195.600i	0.218547i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	369.699	0.411233
$900$	0	0
$901$	134.398i	0.149166i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1063.98	−1.17567
$906$	0	0
$907$	472.831	0.521313	0.260657	−	0.965432i	$-0.416061\pi$
$907$	472.831	0.521313	0.260657	+	0.965432i	$0.416061\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1094.84i	1.20180i	0.799326	+	0.600898i	$0.205190\pi$
$911$	1094.84i	1.20180i	−0.799326	+	0.600898i	$0.794810\pi$
$912$	0	0
$913$	1069.86	1.17181
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1605.79i	1.75114i
$918$	0	0
$919$	152.850i	0.166323i	0.996536	+	0.0831613i	$0.0265017\pi$
$919$	152.850i	0.166323i	−0.996536	+	0.0831613i	$0.973498\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1897.68	2.05599
$924$	0	0
$925$	− 530.766i	− 0.573802i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	63.7112	0.0685804	0.0342902	−	0.999412i	$-0.489083\pi$
$929$	63.7112	0.0685804	0.0342902	+	0.999412i	$0.489083\pi$
$930$	0	0
$931$	−1103.83	−1.18564
$932$	0	0
$933$	0	0
$934$	0	0
$935$	423.320i	0.452749i
$936$	0	0
$937$	−1320.67	−1.40946	−0.704731	−	0.709475i	$-0.748932\pi$
$937$	−1320.67	−1.40946	−0.704731	+	0.709475i	$0.748932\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.41027i	0.00574949i	0.999996	+	0.00287475i	$0.000915061\pi$
$941$	5.41027i	0.00574949i	−0.999996	+	0.00287475i	$0.999085\pi$
$942$	0	0
$943$	− 2197.60i	− 2.33043i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	83.9577	0.0886565	0.0443283	−	0.999017i	$-0.485885\pi$
$947$	83.9577	0.0886565	0.0443283	+	0.999017i	$0.485885\pi$
$948$	0	0
$949$	2679.53i	2.82353i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1577.15	1.65494	0.827468	−	0.561513i	$-0.189780\pi$
$953$	1577.15	1.65494	0.827468	+	0.561513i	$0.189780\pi$
$954$	0	0
$955$	1043.07	1.09222
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 2230.54i	− 2.32590i
$960$	0	0
$961$	311.601	0.324246
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1361.44i	− 1.41082i
$966$	0	0
$967$	278.616i	0.288124i	0.989569	+	0.144062i	$0.0460164\pi$
$967$	278.616i	0.288124i	−0.989569	+	0.144062i	$0.953984\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−380.684	−0.392053	−0.196027	−	0.980599i	$-0.562804\pi$
$971$	−380.684	−0.392053	−0.196027	+	0.980599i	$0.562804\pi$
$972$	0	0
$973$	− 286.398i	− 0.294346i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1630.75	1.66914	0.834571	−	0.550901i	$-0.185716\pi$
$977$	1630.75	1.66914	0.834571	+	0.550901i	$0.185716\pi$
$978$	0	0
$979$	166.131	0.169695
$980$	0	0
$981$	0	0
$982$	0	0
$983$	928.382i	0.944437i	0.881482	+	0.472219i	$0.156547\pi$
$983$	928.382i	0.944437i	−0.881482	+	0.472219i	$0.843453\pi$
$984$	0	0
$985$	993.399	1.00853
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 220.429i	− 0.222880i
$990$	0	0
$991$	− 1399.98i	− 1.41269i	−0.707865	−	0.706347i	$-0.750342\pi$
$991$	− 1399.98i	− 1.41269i	0.707865	−	0.706347i	$-0.249658\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2949.83	−2.96466
$996$	0	0
$997$	− 595.567i	− 0.597359i	−0.954353	−	0.298680i	$-0.903454\pi$
$997$	− 595.567i	− 0.597359i	0.954353	−	0.298680i	$-0.0965462\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2304.3.b.s.127.7		8
3.2	odd	2	inner	2304.3.b.s.127.1		8
4.3	odd	2		2304.3.b.r.127.8		8
8.3	odd	2	inner	2304.3.b.s.127.2		8
8.5	even	2		2304.3.b.r.127.1		8
12.11	even	2		2304.3.b.r.127.2		8
16.3	odd	4		1152.3.g.d.127.7	yes	8
16.5	even	4		1152.3.g.e.127.2	yes	8
16.11	odd	4		1152.3.g.e.127.1	yes	8
16.13	even	4		1152.3.g.d.127.8	yes	8
24.5	odd	2		2304.3.b.r.127.7		8
24.11	even	2	inner	2304.3.b.s.127.8		8
48.5	odd	4		1152.3.g.e.127.8	yes	8
48.11	even	4		1152.3.g.e.127.7	yes	8
48.29	odd	4		1152.3.g.d.127.2	yes	8
48.35	even	4		1152.3.g.d.127.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.g.d.127.1	✓	8	48.35	even	4
1152.3.g.d.127.2	yes	8	48.29	odd	4
1152.3.g.d.127.7	yes	8	16.3	odd	4
1152.3.g.d.127.8	yes	8	16.13	even	4
1152.3.g.e.127.1	yes	8	16.11	odd	4
1152.3.g.e.127.2	yes	8	16.5	even	4
1152.3.g.e.127.7	yes	8	48.11	even	4
1152.3.g.e.127.8	yes	8	48.5	odd	4
2304.3.b.r.127.1		8	8.5	even	2
2304.3.b.r.127.2		8	12.11	even	2
2304.3.b.r.127.7		8	24.5	odd	2
2304.3.b.r.127.8		8	4.3	odd	2
2304.3.b.s.127.1		8	3.2	odd	2	inner
2304.3.b.s.127.2		8	8.3	odd	2	inner
2304.3.b.s.127.7		8	1.1	even	1	trivial
2304.3.b.s.127.8		8	24.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2304.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.11993i q^{5} -9.48331i q^{7} +O(q^{10})\)	\(q+8.11993i q^{5} -9.48331i q^{7} +10.5830 q^{11} +20.9666i q^{13} +4.92615 q^{17} +26.9666 q^{19} -43.7934i q^{23} -40.9333 q^{25} -14.5075i q^{29} +25.4833i q^{31} +77.0038 q^{35} +12.9666i q^{37} +50.1810 q^{41} +5.03337 q^{43} -40.8706i q^{47} -40.9333 q^{49} +27.2826i q^{53} +85.9333i q^{55} -24.0888 q^{59} -28.9666i q^{61} -170.248 q^{65} -65.7998 q^{67} -90.5097i q^{71} +127.800 q^{73} -100.362i q^{77} +35.5501i q^{79} +101.093 q^{83} +40.0000i q^{85} +15.6979 q^{89} +198.833 q^{91} +218.967i q^{95} +73.8665 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 96 q^{19} - 88 q^{25} + 160 q^{43} - 88 q^{49} + 192 q^{67} + 304 q^{73} + 992 q^{91} + 112 q^{97}+O(q^{100}) \) 8 * q + 96 * q^19 - 88 * q^25 + 160 * q^43 - 88 * q^49 + 192 * q^67 + 304 * q^73 + 992 * q^91 + 112 * q^97

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