LMFDB - Embedded newform 1152.3.e.b.1025.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,3,Mod(1025,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1152.1025");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1152.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$31.3897264543$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.3
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1152.1025
Dual form			1152.3.e.b.1025.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.04989i q^{5} +7.79796 q^{7} +O(q^{10})$	$q+2.04989i q^{5} +7.79796 q^{7} -4.09978i q^{11} +6.69694 q^{13} +19.3704i q^{17} -1.79796 q^{19} +14.1421i q^{23} +20.7980 q^{25} -28.4914i q^{29} +20.2020 q^{31} +15.9849i q^{35} -41.5959 q^{37} -4.94253i q^{41} +75.1918 q^{43} -13.5707i q^{47} +11.8082 q^{49} -20.8633i q^{53} +8.40408 q^{55} +54.8542i q^{59} -89.1918 q^{61} +13.7280i q^{65} +37.7980 q^{67} +117.937i q^{71} +48.4041 q^{73} -31.9699i q^{77} +92.6061 q^{79} +158.806i q^{83} -39.7071 q^{85} +121.036i q^{89} +52.2225 q^{91} -3.68561i q^{95} +167.373 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7}+O(q^{10}) $ 4 * q - 8 * q^7	$ 4 q - 8 q^{7} - 32 q^{13} + 32 q^{19} + 44 q^{25} + 120 q^{31} - 88 q^{37} + 144 q^{43} + 204 q^{49} + 112 q^{55} - 200 q^{61} + 112 q^{67} + 272 q^{73} + 488 q^{79} - 296 q^{85} + 640 q^{91} + 160 q^{97}+O(q^{100}) $ 4 * q - 8 * q^7 - 32 * q^13 + 32 * q^19 + 44 * q^25 + 120 * q^31 - 88 * q^37 + 144 * q^43 + 204 * q^49 + 112 * q^55 - 200 * q^61 + 112 * q^67 + 272 * q^73 + 488 * q^79 - 296 * q^85 + 640 * q^91 + 160 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	2.04989i	0.409978i	0.978764	+	0.204989i	$0.0657158\pi$
$5$	2.04989i	0.409978i	−0.978764	+	0.204989i	$0.934284\pi$
$6$	0	0
$7$	7.79796	1.11399	0.556997	−	0.830514i	$-0.311953\pi$
$7$	7.79796	1.11399	0.556997	+	0.830514i	$0.311953\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.09978i	− 0.372707i	−0.982483	−	0.186353i	$-0.940333\pi$
$11$	− 4.09978i	− 0.372707i	0.982483	−	0.186353i	$-0.0596670\pi$
$12$	0	0
$13$	6.69694	0.515149	0.257575	−	0.966258i	$-0.417077\pi$
$13$	6.69694	0.515149	0.257575	+	0.966258i	$0.417077\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	19.3704i	1.13944i	0.821841	+	0.569718i	$0.192947\pi$
$17$	19.3704i	1.13944i	−0.821841	+	0.569718i	$0.807053\pi$
$18$	0	0
$19$	−1.79796	−0.0946294	−0.0473147	−	0.998880i	$-0.515066\pi$
$19$	−1.79796	−0.0946294	−0.0473147	+	0.998880i	$0.515066\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	14.1421i	0.614875i	0.951568	+	0.307438i	$0.0994716\pi$
$23$	14.1421i	0.614875i	−0.951568	+	0.307438i	$0.900528\pi$
$24$	0	0
$25$	20.7980	0.831918
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 28.4914i	− 0.982460i	−0.871030	−	0.491230i	$-0.836547\pi$
$29$	− 28.4914i	− 0.982460i	0.871030	−	0.491230i	$-0.163453\pi$
$30$	0	0
$31$	20.2020	0.651679	0.325839	−	0.945425i	$-0.394353\pi$
$31$	20.2020	0.651679	0.325839	+	0.945425i	$0.394353\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	15.9849i	0.456713i
$36$	0	0
$37$	−41.5959	−1.12421	−0.562107	−	0.827065i	$-0.690009\pi$
$37$	−41.5959	−1.12421	−0.562107	+	0.827065i	$0.690009\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 4.94253i	− 0.120550i	−0.998182	−	0.0602748i	$-0.980802\pi$
$41$	− 4.94253i	− 0.120550i	0.998182	−	0.0602748i	$-0.0191977\pi$
$42$	0	0
$43$	75.1918	1.74865	0.874324	−	0.485343i	$-0.161305\pi$
$43$	75.1918	1.74865	0.874324	+	0.485343i	$0.161305\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 13.5707i	− 0.288738i	−0.989524	−	0.144369i	$-0.953885\pi$
$47$	− 13.5707i	− 0.288738i	0.989524	−	0.144369i	$-0.0461152\pi$
$48$	0	0
$49$	11.8082	0.240983
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 20.8633i	− 0.393646i	−0.980439	−	0.196823i	$-0.936938\pi$
$53$	− 20.8633i	− 0.393646i	0.980439	−	0.196823i	$-0.0630625\pi$
$54$	0	0
$55$	8.40408	0.152801
$56$	0	0
$57$	0	0
$58$	0	0
$59$	54.8542i	0.929732i	0.885381	+	0.464866i	$0.153897\pi$
$59$	54.8542i	0.929732i	−0.885381	+	0.464866i	$0.846103\pi$
$60$	0	0
$61$	−89.1918	−1.46216	−0.731081	−	0.682291i	$-0.760984\pi$
$61$	−89.1918	−1.46216	−0.731081	+	0.682291i	$0.760984\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	13.7280i	0.211200i
$66$	0	0
$67$	37.7980	0.564149	0.282074	−	0.959393i	$-0.408978\pi$
$67$	37.7980	0.564149	0.282074	+	0.959393i	$0.408978\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	117.937i	1.66108i	0.556958	+	0.830541i	$0.311968\pi$
$71$	117.937i	1.66108i	−0.556958	+	0.830541i	$0.688032\pi$
$72$	0	0
$73$	48.4041	0.663070	0.331535	−	0.943443i	$-0.392434\pi$
$73$	48.4041	0.663070	0.331535	+	0.943443i	$0.392434\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 31.9699i	− 0.415193i
$78$	0	0
$79$	92.6061	1.17223	0.586115	−	0.810228i	$-0.300657\pi$
$79$	92.6061	1.17223	0.586115	+	0.810228i	$0.300657\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	158.806i	1.91333i	0.291197	+	0.956663i	$0.405947\pi$
$83$	158.806i	1.91333i	−0.291197	+	0.956663i	$0.594053\pi$
$84$	0	0
$85$	−39.7071	−0.467143
$86$	0	0
$87$	0	0
$88$	0	0
$89$	121.036i	1.35996i	0.733230	+	0.679980i	$0.238012\pi$
$89$	121.036i	1.35996i	−0.733230	+	0.679980i	$0.761988\pi$
$90$	0	0
$91$	52.2225	0.573873
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 3.68561i	− 0.0387959i
$96$	0	0
$97$	167.373	1.72550	0.862750	−	0.505631i	$-0.168740\pi$
$97$	167.373	1.72550	0.862750	+	0.505631i	$0.168740\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 91.7024i	− 0.907944i	−0.891016	−	0.453972i	$-0.850007\pi$
$101$	− 91.7024i	− 0.907944i	0.891016	−	0.453972i	$-0.149993\pi$
$102$	0	0
$103$	58.9898	0.572716	0.286358	−	0.958123i	$-0.407555\pi$
$103$	58.9898	0.572716	0.286358	+	0.958123i	$0.407555\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 182.676i	− 1.70725i	−0.520886	−	0.853626i	$-0.674398\pi$
$107$	− 182.676i	− 1.70725i	0.520886	−	0.853626i	$-0.325602\pi$
$108$	0	0
$109$	−62.6969	−0.575201	−0.287601	−	0.957750i	$-0.592858\pi$
$109$	−62.6969	−0.575201	−0.287601	+	0.957750i	$0.592858\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.7567i	0.0951919i	0.998867	+	0.0475959i	$0.0151560\pi$
$113$	10.7567i	0.0951919i	−0.998867	+	0.0475959i	$0.984844\pi$
$114$	0	0
$115$	−28.9898	−0.252085
$116$	0	0
$117$	0	0
$118$	0	0
$119$	151.050i	1.26932i
$120$	0	0
$121$	104.192	0.861090
$122$	0	0
$123$	0	0
$124$	0	0
$125$	93.8807i	0.751046i
$126$	0	0
$127$	222.182	1.74946	0.874731	−	0.484609i	$-0.161038\pi$
$127$	222.182	1.74946	0.874731	+	0.484609i	$0.161038\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	111.166i	0.848594i	0.905523	+	0.424297i	$0.139479\pi$
$131$	111.166i	0.848594i	−0.905523	+	0.424297i	$0.860521\pi$
$132$	0	0
$133$	−14.0204	−0.105417
$134$	0	0
$135$	0	0
$136$	0	0
$137$	76.5104i	0.558470i	0.960223	+	0.279235i	$0.0900809\pi$
$137$	76.5104i	0.558470i	−0.960223	+	0.279235i	$0.909919\pi$
$138$	0	0
$139$	223.778	1.60991	0.804955	−	0.593336i	$-0.202189\pi$
$139$	223.778	1.60991	0.804955	+	0.593336i	$0.202189\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 27.4559i	− 0.192000i
$144$	0	0
$145$	58.4041	0.402787
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 121.672i	− 0.816592i	−0.912850	−	0.408296i	$-0.866123\pi$
$149$	− 121.672i	− 0.816592i	0.912850	−	0.408296i	$-0.133877\pi$
$150$	0	0
$151$	−82.9898	−0.549601	−0.274801	−	0.961501i	$-0.588612\pi$
$151$	−82.9898	−0.549601	−0.274801	+	0.961501i	$0.588612\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	41.4119i	0.267174i
$156$	0	0
$157$	−32.4245	−0.206525	−0.103263	−	0.994654i	$-0.532928\pi$
$157$	−32.4245	−0.206525	−0.103263	+	0.994654i	$0.532928\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	110.280i	0.684968i
$162$	0	0
$163$	13.1714	0.0808063	0.0404031	−	0.999183i	$-0.487136\pi$
$163$	13.1714	0.0808063	0.0404031	+	0.999183i	$0.487136\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	50.6548i	0.303322i	0.988433	+	0.151661i	$0.0484622\pi$
$167$	50.6548i	0.303322i	−0.988433	+	0.151661i	$0.951538\pi$
$168$	0	0
$169$	−124.151	−0.734621
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 294.335i	− 1.70136i	−0.525687	−	0.850678i	$-0.676192\pi$
$173$	− 294.335i	− 1.70136i	0.525687	−	0.850678i	$-0.323808\pi$
$174$	0	0
$175$	162.182	0.926752
$176$	0	0
$177$	0	0
$178$	0	0
$179$	205.047i	1.14551i	0.819726	+	0.572756i	$0.194126\pi$
$179$	205.047i	1.14551i	−0.819726	+	0.572756i	$0.805874\pi$
$180$	0	0
$181$	−107.464	−0.593725	−0.296863	−	0.954920i	$-0.595940\pi$
$181$	−107.464	−0.593725	−0.296863	+	0.954920i	$0.595940\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 85.2670i	− 0.460903i
$186$	0	0
$187$	79.4143	0.424675
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 219.731i	− 1.15043i	−0.818004	−	0.575213i	$-0.804919\pi$
$191$	− 219.731i	− 1.15043i	0.818004	−	0.575213i	$-0.195081\pi$
$192$	0	0
$193$	−177.151	−0.917881	−0.458940	−	0.888467i	$-0.651771\pi$
$193$	−177.151	−0.917881	−0.458940	+	0.888467i	$0.651771\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	182.055i	0.924136i	0.886845	+	0.462068i	$0.152892\pi$
$197$	182.055i	0.924136i	−0.886845	+	0.462068i	$0.847108\pi$
$198$	0	0
$199$	−267.757	−1.34551	−0.672757	−	0.739864i	$-0.734890\pi$
$199$	−267.757	−1.34551	−0.672757	+	0.739864i	$0.734890\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 222.174i	− 1.09446i
$204$	0	0
$205$	10.1316	0.0494226
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7.37123i	0.0352690i
$210$	0	0
$211$	−40.2225	−0.190628	−0.0953139	−	0.995447i	$-0.530385\pi$
$211$	−40.2225	−0.190628	−0.0953139	+	0.995447i	$0.530385\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	154.135i	0.716906i
$216$	0	0
$217$	157.535	0.725966
$218$	0	0
$219$	0	0
$220$	0	0
$221$	129.722i	0.586979i
$222$	0	0
$223$	−44.9694	−0.201656	−0.100828	−	0.994904i	$-0.532149\pi$
$223$	−44.9694	−0.201656	−0.100828	+	0.994904i	$0.532149\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	180.605i	0.795618i	0.917468	+	0.397809i	$0.130229\pi$
$227$	180.605i	0.795618i	−0.917468	+	0.397809i	$0.869771\pi$
$228$	0	0
$229$	−129.666	−0.566228	−0.283114	−	0.959086i	$-0.591368\pi$
$229$	−129.666	−0.566228	−0.283114	+	0.959086i	$0.591368\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 209.832i	− 0.900566i	−0.892886	−	0.450283i	$-0.851323\pi$
$233$	− 209.832i	− 0.900566i	0.892886	−	0.450283i	$-0.148677\pi$
$234$	0	0
$235$	27.8184	0.118376
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 334.583i	− 1.39993i	−0.714178	−	0.699964i	$-0.753199\pi$
$239$	− 334.583i	− 1.39993i	0.714178	−	0.699964i	$-0.246801\pi$
$240$	0	0
$241$	−51.8184	−0.215014	−0.107507	−	0.994204i	$-0.534287\pi$
$241$	−51.8184	−0.215014	−0.107507	+	0.994204i	$0.534287\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	24.2054i	0.0987976i
$246$	0	0
$247$	−12.0408	−0.0487483
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 311.541i	− 1.24120i	−0.784127	−	0.620600i	$-0.786889\pi$
$251$	− 311.541i	− 1.24120i	0.784127	−	0.620600i	$-0.213111\pi$
$252$	0	0
$253$	57.9796	0.229168
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 50.0978i	− 0.194933i	−0.995239	−	0.0974665i	$-0.968926\pi$
$257$	− 50.0978i	− 0.194933i	0.995239	−	0.0974665i	$-0.0310739\pi$
$258$	0	0
$259$	−324.363	−1.25237
$260$	0	0
$261$	0	0
$262$	0	0
$263$	334.268i	1.27098i	0.772108	+	0.635491i	$0.219202\pi$
$263$	334.268i	1.27098i	−0.772108	+	0.635491i	$0.780798\pi$
$264$	0	0
$265$	42.7673	0.161386
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 80.1318i	− 0.297888i	−0.988846	−	0.148944i	$-0.952413\pi$
$269$	− 80.1318i	− 0.297888i	0.988846	−	0.148944i	$-0.0475874\pi$
$270$	0	0
$271$	−61.7775	−0.227961	−0.113981	−	0.993483i	$-0.536360\pi$
$271$	−61.7775	−0.227961	−0.113981	+	0.993483i	$0.536360\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 85.2670i	− 0.310062i
$276$	0	0
$277$	423.242	1.52795	0.763974	−	0.645247i	$-0.223245\pi$
$277$	423.242	1.52795	0.763974	+	0.645247i	$0.223245\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	63.6396i	0.226475i	0.993568	+	0.113238i	$0.0361222\pi$
$281$	63.6396i	0.226475i	−0.993568	+	0.113238i	$0.963878\pi$
$282$	0	0
$283$	−98.4245	−0.347790	−0.173895	−	0.984764i	$-0.555635\pi$
$283$	−98.4245	−0.347790	−0.173895	+	0.984764i	$0.555635\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 38.5417i	− 0.134291i
$288$	0	0
$289$	−86.2122	−0.298312
$290$	0	0
$291$	0	0
$292$	0	0
$293$	368.888i	1.25900i	0.776999	+	0.629502i	$0.216741\pi$
$293$	368.888i	1.25900i	−0.776999	+	0.629502i	$0.783259\pi$
$294$	0	0
$295$	−112.445	−0.381169
$296$	0	0
$297$	0	0
$298$	0	0
$299$	94.7090i	0.316753i
$300$	0	0
$301$	586.343	1.94798
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 182.833i	− 0.599453i
$306$	0	0
$307$	−507.737	−1.65387	−0.826933	−	0.562301i	$-0.809916\pi$
$307$	−507.737	−1.65387	−0.826933	+	0.562301i	$0.809916\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 36.7984i	− 0.118323i	−0.998248	−	0.0591614i	$-0.981157\pi$
$311$	− 36.7984i	− 0.118323i	0.998248	−	0.0591614i	$-0.0188427\pi$
$312$	0	0
$313$	−287.576	−0.918772	−0.459386	−	0.888237i	$-0.651930\pi$
$313$	−287.576	−0.918772	−0.459386	+	0.888237i	$0.651930\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	450.726i	1.42185i	0.703268	+	0.710925i	$0.251723\pi$
$317$	450.726i	1.42185i	−0.703268	+	0.710925i	$0.748277\pi$
$318$	0	0
$319$	−116.808	−0.366170
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 34.8272i	− 0.107824i
$324$	0	0
$325$	139.283	0.428562
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 105.824i	− 0.321652i
$330$	0	0
$331$	−133.798	−0.404223	−0.202112	−	0.979362i	$-0.564780\pi$
$331$	−133.798	−0.404223	−0.202112	+	0.979362i	$0.564780\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	77.4816i	0.231288i
$336$	0	0
$337$	−124.808	−0.370351	−0.185175	−	0.982706i	$-0.559285\pi$
$337$	−124.808	−0.370351	−0.185175	+	0.982706i	$0.559285\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 82.8238i	− 0.242885i
$342$	0	0
$343$	−290.020	−0.845541
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 571.699i	− 1.64755i	−0.566919	−	0.823773i	$-0.691865\pi$
$347$	− 571.699i	− 1.64755i	0.566919	−	0.823773i	$-0.308135\pi$
$348$	0	0
$349$	−499.898	−1.43237	−0.716186	−	0.697909i	$-0.754114\pi$
$349$	−499.898	−1.43237	−0.716186	+	0.697909i	$0.754114\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 5.64242i	− 0.0159842i	−0.999968	−	0.00799210i	$-0.997456\pi$
$353$	− 5.64242i	− 0.0159842i	0.999968	−	0.00799210i	$-0.00254399\pi$
$354$	0	0
$355$	−241.757	−0.681006
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 586.769i	− 1.63445i	−0.576316	−	0.817227i	$-0.695510\pi$
$359$	− 586.769i	− 1.63445i	0.576316	−	0.817227i	$-0.304490\pi$
$360$	0	0
$361$	−357.767	−0.991045
$362$	0	0
$363$	0	0
$364$	0	0
$365$	99.2229i	0.271844i
$366$	0	0
$367$	−26.6265	−0.0725519	−0.0362759	−	0.999342i	$-0.511550\pi$
$367$	−26.6265	−0.0725519	−0.0362759	+	0.999342i	$0.511550\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 162.691i	− 0.438520i
$372$	0	0
$373$	226.041	0.606008	0.303004	−	0.952989i	$-0.402011\pi$
$373$	226.041	0.606008	0.303004	+	0.952989i	$0.402011\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 190.805i	− 0.506114i
$378$	0	0
$379$	343.292	0.905783	0.452892	−	0.891566i	$-0.350392\pi$
$379$	343.292	0.905783	0.452892	+	0.891566i	$0.350392\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 654.680i	− 1.70935i	−0.519166	−	0.854673i	$-0.673757\pi$
$383$	− 654.680i	− 1.70935i	0.519166	−	0.854673i	$-0.326243\pi$
$384$	0	0
$385$	65.5347	0.170220
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 298.948i	− 0.768504i	−0.923228	−	0.384252i	$-0.874459\pi$
$389$	− 298.948i	− 0.768504i	0.923228	−	0.384252i	$-0.125541\pi$
$390$	0	0
$391$	−273.939	−0.700611
$392$	0	0
$393$	0	0
$394$	0	0
$395$	189.832i	0.480588i
$396$	0	0
$397$	529.596	1.33399	0.666997	−	0.745060i	$-0.267579\pi$
$397$	529.596	1.33399	0.666997	+	0.745060i	$0.267579\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 447.034i	− 1.11480i	−0.830244	−	0.557399i	$-0.811799\pi$
$401$	− 447.034i	− 1.11480i	0.830244	−	0.557399i	$-0.188201\pi$
$402$	0	0
$403$	135.292	0.335712
$404$	0	0
$405$	0	0
$406$	0	0
$407$	170.534i	0.419002i
$408$	0	0
$409$	−503.292	−1.23054	−0.615271	−	0.788316i	$-0.710953\pi$
$409$	−503.292	−1.23054	−0.615271	+	0.788316i	$0.710953\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	427.751i	1.03572i
$414$	0	0
$415$	−325.535	−0.784421
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 142.722i	− 0.340624i	−0.985390	−	0.170312i	$-0.945522\pi$
$419$	− 142.722i	− 0.340624i	0.985390	−	0.170312i	$-0.0544776\pi$
$420$	0	0
$421$	41.3031	0.0981070	0.0490535	−	0.998796i	$-0.484380\pi$
$421$	41.3031	0.0981070	0.0490535	+	0.998796i	$0.484380\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	402.865i	0.947917i
$426$	0	0
$427$	−695.514	−1.62884
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 441.719i	− 1.02487i	−0.858726	−	0.512436i	$-0.828743\pi$
$431$	− 441.719i	− 1.02487i	0.858726	−	0.512436i	$-0.171257\pi$
$432$	0	0
$433$	−460.343	−1.06315	−0.531574	−	0.847012i	$-0.678399\pi$
$433$	−460.343	−1.06315	−0.531574	+	0.847012i	$0.678399\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 25.4270i	− 0.0581853i
$438$	0	0
$439$	−219.353	−0.499665	−0.249833	−	0.968289i	$-0.580376\pi$
$439$	−219.353	−0.499665	−0.249833	+	0.968289i	$0.580376\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 19.2982i	− 0.0435626i	−0.999763	−	0.0217813i	$-0.993066\pi$
$443$	− 19.2982i	− 0.0435626i	0.999763	−	0.0217813i	$-0.00693375\pi$
$444$	0	0
$445$	−248.111	−0.557553
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 573.627i	− 1.27756i	−0.769387	−	0.638782i	$-0.779438\pi$
$449$	− 573.627i	− 1.27756i	0.769387	−	0.638782i	$-0.220562\pi$
$450$	0	0
$451$	−20.2633	−0.0449296
$452$	0	0
$453$	0	0
$454$	0	0
$455$	107.050i	0.235275i
$456$	0	0
$457$	−71.0102	−0.155383	−0.0776917	−	0.996977i	$-0.524755\pi$
$457$	−71.0102	−0.155383	−0.0776917	+	0.996977i	$0.524755\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 751.869i	− 1.63095i	−0.578791	−	0.815476i	$-0.696475\pi$
$461$	− 751.869i	− 1.63095i	0.578791	−	0.815476i	$-0.303525\pi$
$462$	0	0
$463$	−638.182	−1.37836	−0.689181	−	0.724589i	$-0.742030\pi$
$463$	−638.182	−1.37836	−0.689181	+	0.724589i	$0.742030\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 225.032i	− 0.481867i	−0.970542	−	0.240933i	$-0.922547\pi$
$467$	− 225.032i	− 0.481867i	0.970542	−	0.240933i	$-0.0774535\pi$
$468$	0	0
$469$	294.747	0.628458
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 308.270i	− 0.651733i
$474$	0	0
$475$	−37.3939	−0.0787240
$476$	0	0
$477$	0	0
$478$	0	0
$479$	757.675i	1.58178i	0.611955	+	0.790892i	$0.290383\pi$
$479$	757.675i	1.58178i	−0.611955	+	0.790892i	$0.709617\pi$
$480$	0	0
$481$	−278.565	−0.579138
$482$	0	0
$483$	0	0
$484$	0	0
$485$	343.097i	0.707416i
$486$	0	0
$487$	266.100	0.546407	0.273203	−	0.961956i	$-0.411917\pi$
$487$	266.100	0.546407	0.273203	+	0.961956i	$0.411917\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	14.3701i	0.0292671i	0.999893	+	0.0146335i	$0.00465817\pi$
$491$	14.3701i	0.0292671i	−0.999893	+	0.0146335i	$0.995342\pi$
$492$	0	0
$493$	551.889	1.11945
$494$	0	0
$495$	0	0
$496$	0	0
$497$	919.666i	1.85043i
$498$	0	0
$499$	603.151	1.20872	0.604360	−	0.796712i	$-0.293429\pi$
$499$	603.151	1.20872	0.604360	+	0.796712i	$0.293429\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 781.388i	− 1.55345i	−0.629837	−	0.776727i	$-0.716878\pi$
$503$	− 781.388i	− 1.55345i	0.629837	−	0.776727i	$-0.283122\pi$
$504$	0	0
$505$	187.980	0.372237
$506$	0	0
$507$	0	0
$508$	0	0
$509$	930.002i	1.82712i	0.406709	+	0.913558i	$0.366676\pi$
$509$	930.002i	1.82712i	−0.406709	+	0.913558i	$0.633324\pi$
$510$	0	0
$511$	377.453	0.738656
$512$	0	0
$513$	0	0
$514$	0	0
$515$	120.922i	0.234801i
$516$	0	0
$517$	−55.6367	−0.107615
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 463.947i	− 0.890494i	−0.895408	−	0.445247i	$-0.853116\pi$
$521$	− 463.947i	− 0.890494i	0.895408	−	0.445247i	$-0.146884\pi$
$522$	0	0
$523$	−426.524	−0.815534	−0.407767	−	0.913086i	$-0.633693\pi$
$523$	−426.524	−0.815534	−0.407767	+	0.913086i	$0.633693\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	391.322i	0.742546i
$528$	0	0
$529$	329.000	0.621928
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 33.0998i	− 0.0621010i
$534$	0	0
$535$	374.465	0.699935
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 48.4108i	− 0.0898160i
$540$	0	0
$541$	−281.930	−0.521127	−0.260563	−	0.965457i	$-0.583908\pi$
$541$	−281.930	−0.521127	−0.260563	+	0.965457i	$0.583908\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 128.522i	− 0.235820i
$546$	0	0
$547$	−204.727	−0.374272	−0.187136	−	0.982334i	$-0.559920\pi$
$547$	−204.727	−0.374272	−0.187136	+	0.982334i	$0.559920\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	51.2263i	0.0929697i
$552$	0	0
$553$	722.139	1.30586
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 813.465i	− 1.46044i	−0.683212	−	0.730220i	$-0.739418\pi$
$557$	− 813.465i	− 1.46044i	0.683212	−	0.730220i	$-0.260582\pi$
$558$	0	0
$559$	503.555	0.900814
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 666.523i	− 1.18388i	−0.805983	−	0.591939i	$-0.798363\pi$
$563$	− 666.523i	− 1.18388i	0.805983	−	0.591939i	$-0.201637\pi$
$564$	0	0
$565$	−22.0500	−0.0390265
$566$	0	0
$567$	0	0
$568$	0	0
$569$	308.756i	0.542629i	0.962491	+	0.271315i	$0.0874584\pi$
$569$	308.756i	0.542629i	−0.962491	+	0.271315i	$0.912542\pi$
$570$	0	0
$571$	−534.747	−0.936510	−0.468255	−	0.883594i	$-0.655117\pi$
$571$	−534.747	−0.936510	−0.468255	+	0.883594i	$0.655117\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	294.128i	0.511526i
$576$	0	0
$577$	158.000	0.273830	0.136915	−	0.990583i	$-0.456281\pi$
$577$	158.000	0.273830	0.136915	+	0.990583i	$0.456281\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1238.36i	2.13143i
$582$	0	0
$583$	−85.5347	−0.146715
$584$	0	0
$585$	0	0
$586$	0	0
$587$	783.503i	1.33476i	0.744718	+	0.667379i	$0.232584\pi$
$587$	783.503i	1.33476i	−0.744718	+	0.667379i	$0.767416\pi$
$588$	0	0
$589$	−36.3224	−0.0616680
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 26.1283i	− 0.0440613i	−0.999757	−	0.0220306i	$-0.992987\pi$
$593$	− 26.1283i	− 0.0440613i	0.999757	−	0.0220306i	$-0.00701313\pi$
$594$	0	0
$595$	−309.635	−0.520394
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 440.063i	− 0.734663i	−0.930090	−	0.367331i	$-0.880272\pi$
$599$	− 440.063i	− 0.734663i	0.930090	−	0.367331i	$-0.119728\pi$
$600$	0	0
$601$	66.0000	0.109817	0.0549085	−	0.998491i	$-0.482513\pi$
$601$	66.0000	0.109817	0.0549085	+	0.998491i	$0.482513\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	213.582i	0.353027i
$606$	0	0
$607$	−465.373	−0.766678	−0.383339	−	0.923608i	$-0.625226\pi$
$607$	−465.373	−0.766678	−0.383339	+	0.923608i	$0.625226\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 90.8820i	− 0.148743i
$612$	0	0
$613$	−580.706	−0.947318	−0.473659	−	0.880708i	$-0.657067\pi$
$613$	−580.706	−0.947318	−0.473659	+	0.880708i	$0.657067\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	43.8118i	0.0710077i	0.999370	+	0.0355039i	$0.0113036\pi$
$617$	43.8118i	0.0710077i	−0.999370	+	0.0355039i	$0.988696\pi$
$618$	0	0
$619$	401.980	0.649402	0.324701	−	0.945817i	$-0.394736\pi$
$619$	401.980	0.649402	0.324701	+	0.945817i	$0.394736\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	943.837i	1.51499i
$624$	0	0
$625$	327.504	0.524007
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 805.729i	− 1.28097i
$630$	0	0
$631$	94.5857	0.149898	0.0749491	−	0.997187i	$-0.476121\pi$
$631$	94.5857	0.149898	0.0749491	+	0.997187i	$0.476121\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	455.447i	0.717240i
$636$	0	0
$637$	79.0785	0.124142
$638$	0	0
$639$	0	0
$640$	0	0
$641$	866.698i	1.35210i	0.736854	+	0.676051i	$0.236310\pi$
$641$	866.698i	1.35210i	−0.736854	+	0.676051i	$0.763690\pi$
$642$	0	0
$643$	802.020	1.24731	0.623655	−	0.781700i	$-0.285647\pi$
$643$	802.020	1.24731	0.623655	+	0.781700i	$0.285647\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 109.223i	− 0.168815i	−0.996431	−	0.0844076i	$-0.973100\pi$
$647$	− 109.223i	− 0.168815i	0.996431	−	0.0844076i	$-0.0268998\pi$
$648$	0	0
$649$	224.890	0.346517
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 976.727i	− 1.49575i	−0.663837	−	0.747877i	$-0.731073\pi$
$653$	− 976.727i	− 1.49575i	0.663837	−	0.747877i	$-0.268927\pi$
$654$	0	0
$655$	−227.878	−0.347905
$656$	0	0
$657$	0	0
$658$	0	0
$659$	528.887i	0.802560i	0.915955	+	0.401280i	$0.131435\pi$
$659$	528.887i	0.802560i	−0.915955	+	0.401280i	$0.868565\pi$
$660$	0	0
$661$	−287.535	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$661$	−287.535	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 28.7403i	− 0.0432185i
$666$	0	0
$667$	402.929	0.604091
$668$	0	0
$669$	0	0
$670$	0	0
$671$	365.667i	0.544958i
$672$	0	0
$673$	−597.029	−0.887115	−0.443558	−	0.896246i	$-0.646284\pi$
$673$	−597.029	−0.887115	−0.443558	+	0.896246i	$0.646284\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 652.630i	− 0.964003i	−0.876170	−	0.482001i	$-0.839910\pi$
$677$	− 652.630i	− 0.964003i	0.876170	−	0.482001i	$-0.160090\pi$
$678$	0	0
$679$	1305.17	1.92220
$680$	0	0
$681$	0	0
$682$	0	0
$683$	288.971i	0.423091i	0.977368	+	0.211546i	$0.0678497\pi$
$683$	288.971i	0.423091i	−0.977368	+	0.211546i	$0.932150\pi$
$684$	0	0
$685$	−156.838	−0.228960
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 139.720i	− 0.202787i
$690$	0	0
$691$	752.241	1.08863	0.544313	−	0.838882i	$-0.316790\pi$
$691$	752.241	1.08863	0.544313	+	0.838882i	$0.316790\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	458.719i	0.660027i
$696$	0	0
$697$	95.7388	0.137358
$698$	0	0
$699$	0	0
$700$	0	0
$701$	808.152i	1.15286i	0.817148	+	0.576428i	$0.195554\pi$
$701$	808.152i	1.15286i	−0.817148	+	0.576428i	$0.804446\pi$
$702$	0	0
$703$	74.7878	0.106384
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 715.091i	− 1.01144i
$708$	0	0
$709$	−737.748	−1.04055	−0.520274	−	0.854000i	$-0.674170\pi$
$709$	−737.748	−1.04055	−0.520274	+	0.854000i	$0.674170\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	285.700i	0.400701i
$714$	0	0
$715$	56.2816	0.0787156
$716$	0	0
$717$	0	0
$718$	0	0
$719$	236.275i	0.328616i	0.986409	+	0.164308i	$0.0525390\pi$
$719$	236.275i	0.328616i	−0.986409	+	0.164308i	$0.947461\pi$
$720$	0	0
$721$	460.000	0.638003
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 592.562i	− 0.817327i
$726$	0	0
$727$	444.606	0.611563	0.305781	−	0.952102i	$-0.401082\pi$
$727$	444.606	0.611563	0.305781	+	0.952102i	$0.401082\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1456.50i	1.99247i
$732$	0	0
$733$	−1196.74	−1.63265	−0.816327	−	0.577590i	$-0.803993\pi$
$733$	−1196.74	−1.63265	−0.816327	+	0.577590i	$0.803993\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 154.963i	− 0.210262i
$738$	0	0
$739$	1437.78	1.94557	0.972784	−	0.231712i	$-0.0744326\pi$
$739$	1437.78	1.94557	0.972784	+	0.231712i	$0.0744326\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1265.05i	− 1.70262i	−0.524661	−	0.851311i	$-0.675808\pi$
$743$	− 1265.05i	− 1.70262i	0.524661	−	0.851311i	$-0.324192\pi$
$744$	0	0
$745$	249.414	0.334784
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1424.50i	− 1.90187i
$750$	0	0
$751$	−578.990	−0.770958	−0.385479	−	0.922717i	$-0.625964\pi$
$751$	−578.990	−0.770958	−0.385479	+	0.922717i	$0.625964\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 170.120i	− 0.225324i
$756$	0	0
$757$	82.6561	0.109189	0.0545945	−	0.998509i	$-0.482613\pi$
$757$	82.6561	0.109189	0.0545945	+	0.998509i	$0.482613\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 512.088i	− 0.672915i	−0.941699	−	0.336457i	$-0.890771\pi$
$761$	− 512.088i	− 0.672915i	0.941699	−	0.336457i	$-0.109229\pi$
$762$	0	0
$763$	−488.908	−0.640771
$764$	0	0
$765$	0	0
$766$	0	0
$767$	367.355i	0.478950i
$768$	0	0
$769$	1267.49	1.64824	0.824118	−	0.566418i	$-0.191671\pi$
$769$	1267.49	1.64824	0.824118	+	0.566418i	$0.191671\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	437.342i	0.565772i	0.959154	+	0.282886i	$0.0912918\pi$
$773$	437.342i	0.565772i	−0.959154	+	0.282886i	$0.908708\pi$
$774$	0	0
$775$	420.161	0.542144
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.88647i	0.0114075i
$780$	0	0
$781$	483.514	0.619096
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 66.4666i	− 0.0846708i
$786$	0	0
$787$	965.031	1.22621	0.613107	−	0.790000i	$-0.289919\pi$
$787$	965.031	1.22621	0.613107	+	0.790000i	$0.289919\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	83.8802i	0.106043i
$792$	0	0
$793$	−597.312	−0.753231
$794$	0	0
$795$	0	0
$796$	0	0
$797$	996.513i	1.25033i	0.780492	+	0.625165i	$0.214968\pi$
$797$	996.513i	1.25033i	−0.780492	+	0.625165i	$0.785032\pi$
$798$	0	0
$799$	262.869	0.328998
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 198.446i	− 0.247131i
$804$	0	0
$805$	−226.061	−0.280821
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1530.32i	1.89162i	0.324723	+	0.945809i	$0.394729\pi$
$809$	1530.32i	1.89162i	−0.324723	+	0.945809i	$0.605271\pi$
$810$	0	0
$811$	833.716	1.02801	0.514005	−	0.857787i	$-0.328161\pi$
$811$	833.716	1.02801	0.514005	+	0.857787i	$0.328161\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	26.9999i	0.0331288i
$816$	0	0
$817$	−135.192	−0.165473
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1414.40i	− 1.72278i	−0.507942	−	0.861391i	$-0.669593\pi$
$821$	− 1414.40i	− 1.72278i	0.507942	−	0.861391i	$-0.330407\pi$
$822$	0	0
$823$	1168.16	1.41939	0.709697	−	0.704507i	$-0.248832\pi$
$823$	1168.16	1.41939	0.709697	+	0.704507i	$0.248832\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	624.110i	0.754667i	0.926077	+	0.377334i	$0.123159\pi$
$827$	624.110i	0.754667i	−0.926077	+	0.377334i	$0.876841\pi$
$828$	0	0
$829$	−1531.46	−1.84736	−0.923682	−	0.383161i	$-0.874836\pi$
$829$	−1531.46	−1.84736	−0.923682	+	0.383161i	$0.874836\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	228.729i	0.274584i
$834$	0	0
$835$	−103.837	−0.124355
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1013.95i	1.20852i	0.796787	+	0.604260i	$0.206531\pi$
$839$	1013.95i	1.20852i	−0.796787	+	0.604260i	$0.793469\pi$
$840$	0	0
$841$	29.2429	0.0347715
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 254.496i	− 0.301178i
$846$	0	0
$847$	812.484	0.959249
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 588.255i	− 0.691252i
$852$	0	0
$853$	258.808	0.303409	0.151705	−	0.988426i	$-0.451524\pi$
$853$	258.808	0.303409	0.151705	+	0.988426i	$0.451524\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1453.90i	1.69650i	0.529600	+	0.848248i	$0.322342\pi$
$857$	1453.90i	1.69650i	−0.529600	+	0.848248i	$0.677658\pi$
$858$	0	0
$859$	−740.645	−0.862218	−0.431109	−	0.902300i	$-0.641877\pi$
$859$	−740.645	−0.862218	−0.431109	+	0.902300i	$0.641877\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	675.593i	0.782842i	0.920212	+	0.391421i	$0.128016\pi$
$863$	675.593i	0.782842i	−0.920212	+	0.391421i	$0.871984\pi$
$864$	0	0
$865$	603.353	0.697518
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 379.664i	− 0.436898i
$870$	0	0
$871$	253.131	0.290621
$872$	0	0
$873$	0	0
$874$	0	0
$875$	732.078i	0.836660i
$876$	0	0
$877$	321.678	0.366793	0.183397	−	0.983039i	$-0.441291\pi$
$877$	321.678	0.366793	0.183397	+	0.983039i	$0.441291\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	586.899i	0.666173i	0.942896	+	0.333087i	$0.108090\pi$
$881$	586.899i	0.666173i	−0.942896	+	0.333087i	$0.891910\pi$
$882$	0	0
$883$	−1186.95	−1.34422	−0.672110	−	0.740451i	$-0.734612\pi$
$883$	−1186.95	−1.34422	−0.672110	+	0.740451i	$0.734612\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 682.707i	− 0.769681i	−0.922983	−	0.384841i	$-0.874256\pi$
$887$	− 682.707i	− 0.769681i	0.922983	−	0.384841i	$-0.125744\pi$
$888$	0	0
$889$	1732.56	1.94889
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24.3995i	0.0273231i
$894$	0	0
$895$	−420.322	−0.469634
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 575.583i	− 0.640249i
$900$	0	0
$901$	404.130	0.448534
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 220.290i	− 0.243414i
$906$	0	0
$907$	803.837	0.886259	0.443129	−	0.896458i	$-0.353868\pi$
$907$	803.837	0.886259	0.443129	+	0.896458i	$0.353868\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1090.77i	1.19733i	0.800998	+	0.598667i	$0.204303\pi$
$911$	1090.77i	1.19733i	−0.800998	+	0.598667i	$0.795697\pi$
$912$	0	0
$913$	651.069	0.713110
$914$	0	0
$915$	0	0
$916$	0	0
$917$	866.867i	0.945329i
$918$	0	0
$919$	362.141	0.394060	0.197030	−	0.980397i	$-0.436870\pi$
$919$	362.141	0.394060	0.197030	+	0.980397i	$0.436870\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	789.815i	0.855704i
$924$	0	0
$925$	−865.110	−0.935254
$926$	0	0
$927$	0	0
$928$	0	0
$929$	421.437i	0.453646i	0.973936	+	0.226823i	$0.0728339\pi$
$929$	421.437i	0.453646i	−0.973936	+	0.226823i	$0.927166\pi$
$930$	0	0
$931$	−21.2306	−0.0228041
$932$	0	0
$933$	0	0
$934$	0	0
$935$	162.790i	0.174107i
$936$	0	0
$937$	9.23266	0.00985342	0.00492671	−	0.999988i	$-0.498432\pi$
$937$	9.23266	0.00985342	0.00492671	+	0.999988i	$0.498432\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 175.499i	− 0.186503i	−0.995643	−	0.0932513i	$-0.970274\pi$
$941$	− 175.499i	− 0.186503i	0.995643	−	0.0932513i	$-0.0297260\pi$
$942$	0	0
$943$	69.8979	0.0741230
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1185.99i	1.25237i	0.779675	+	0.626185i	$0.215384\pi$
$947$	1185.99i	1.25237i	−0.779675	+	0.626185i	$0.784616\pi$
$948$	0	0
$949$	324.159	0.341580
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 929.581i	− 0.975426i	−0.873004	−	0.487713i	$-0.837831\pi$
$953$	− 929.581i	− 0.975426i	0.873004	−	0.487713i	$-0.162169\pi$
$954$	0	0
$955$	450.424	0.471649
$956$	0	0
$957$	0	0
$958$	0	0
$959$	596.625i	0.622132i
$960$	0	0
$961$	−552.878	−0.575315
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 363.140i	− 0.376311i
$966$	0	0
$967$	272.284	0.281576	0.140788	−	0.990040i	$-0.455036\pi$
$967$	272.284	0.281576	0.140788	+	0.990040i	$0.455036\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1439.00i	− 1.48197i	−0.671520	−	0.740986i	$-0.734358\pi$
$971$	− 1439.00i	− 1.48197i	0.671520	−	0.740986i	$-0.265642\pi$
$972$	0	0
$973$	1745.01	1.79343
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1448.98i	− 1.48309i	−0.670902	−	0.741546i	$-0.734093\pi$
$977$	− 1448.98i	− 1.48309i	0.670902	−	0.741546i	$-0.265907\pi$
$978$	0	0
$979$	496.222	0.506867
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 289.013i	− 0.294011i	−0.989136	−	0.147006i	$-0.953036\pi$
$983$	− 289.013i	− 0.294011i	0.989136	−	0.147006i	$-0.0469636\pi$
$984$	0	0
$985$	−373.192	−0.378875
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1063.37i	1.07520i
$990$	0	0
$991$	340.202	0.343292	0.171646	−	0.985159i	$-0.445092\pi$
$991$	340.202	0.343292	0.171646	+	0.985159i	$0.445092\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 548.872i	− 0.551630i
$996$	0	0
$997$	−1664.06	−1.66906	−0.834532	−	0.550959i	$-0.814262\pi$
$997$	−1664.06	−1.66906	−0.834532	+	0.550959i	$0.814262\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.3.e.b.1025.3	yes	4
3.2	odd	2	inner	1152.3.e.b.1025.2	✓	4
4.3	odd	2		1152.3.e.f.1025.3	yes	4
8.3	odd	2		1152.3.e.h.1025.2	yes	4
8.5	even	2		1152.3.e.d.1025.2	yes	4
12.11	even	2		1152.3.e.f.1025.2	yes	4
16.3	odd	4		2304.3.h.i.2177.6		8
16.5	even	4		2304.3.h.k.2177.3		8
16.11	odd	4		2304.3.h.i.2177.3		8
16.13	even	4		2304.3.h.k.2177.6		8
24.5	odd	2		1152.3.e.d.1025.3	yes	4
24.11	even	2		1152.3.e.h.1025.3	yes	4
48.5	odd	4		2304.3.h.k.2177.5		8
48.11	even	4		2304.3.h.i.2177.5		8
48.29	odd	4		2304.3.h.k.2177.4		8
48.35	even	4		2304.3.h.i.2177.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.3.e.b.1025.2	✓	4	3.2	odd	2	inner
1152.3.e.b.1025.3	yes	4	1.1	even	1	trivial
1152.3.e.d.1025.2	yes	4	8.5	even	2
1152.3.e.d.1025.3	yes	4	24.5	odd	2
1152.3.e.f.1025.2	yes	4	12.11	even	2
1152.3.e.f.1025.3	yes	4	4.3	odd	2
1152.3.e.h.1025.2	yes	4	8.3	odd	2
1152.3.e.h.1025.3	yes	4	24.11	even	2
2304.3.h.i.2177.3		8	16.11	odd	4
2304.3.h.i.2177.4		8	48.35	even	4
2304.3.h.i.2177.5		8	48.11	even	4
2304.3.h.i.2177.6		8	16.3	odd	4
2304.3.h.k.2177.3		8	16.5	even	4
2304.3.h.k.2177.4		8	48.29	odd	4
2304.3.h.k.2177.5		8	48.5	odd	4
2304.3.h.k.2177.6		8	16.13	even	4

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.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.04989i q^{5} +7.79796 q^{7} +O(q^{10})\)	\(q+2.04989i q^{5} +7.79796 q^{7} -4.09978i q^{11} +6.69694 q^{13} +19.3704i q^{17} -1.79796 q^{19} +14.1421i q^{23} +20.7980 q^{25} -28.4914i q^{29} +20.2020 q^{31} +15.9849i q^{35} -41.5959 q^{37} -4.94253i q^{41} +75.1918 q^{43} -13.5707i q^{47} +11.8082 q^{49} -20.8633i q^{53} +8.40408 q^{55} +54.8542i q^{59} -89.1918 q^{61} +13.7280i q^{65} +37.7980 q^{67} +117.937i q^{71} +48.4041 q^{73} -31.9699i q^{77} +92.6061 q^{79} +158.806i q^{83} -39.7071 q^{85} +121.036i q^{89} +52.2225 q^{91} -3.68561i q^{95} +167.373 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7}+O(q^{10}) \) 4 * q - 8 * q^7	\( 4 q - 8 q^{7} - 32 q^{13} + 32 q^{19} + 44 q^{25} + 120 q^{31} - 88 q^{37} + 144 q^{43} + 204 q^{49} + 112 q^{55} - 200 q^{61} + 112 q^{67} + 272 q^{73} + 488 q^{79} - 296 q^{85} + 640 q^{91} + 160 q^{97}+O(q^{100}) \) 4 * q - 8 * q^7 - 32 * q^13 + 32 * q^19 + 44 * q^25 + 120 * q^31 - 88 * q^37 + 144 * q^43 + 204 * q^49 + 112 * q^55 - 200 * q^61 + 112 * q^67 + 272 * q^73 + 488 * q^79 - 296 * q^85 + 640 * q^91 + 160 * q^97

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