LMFDB - Embedded newform 3024.2.k.e.1889.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(1889,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3024.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.k (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:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 378)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.1
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3024.1889
Dual form			3024.2.k.e.1889.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{5} +(-2.00000 - 1.73205i) q^{7} +O(q^{10})$	$q-1.73205 q^{5} +(-2.00000 - 1.73205i) q^{7} -3.00000i q^{11} +3.46410i q^{13} -6.92820 q^{17} -1.73205i q^{19} -3.00000i q^{23} -2.00000 q^{25} +6.00000i q^{29} +5.19615i q^{31} +(3.46410 + 3.00000i) q^{35} +7.00000 q^{37} +12.1244 q^{41} +2.00000 q^{43} -3.46410 q^{47} +(1.00000 + 6.92820i) q^{49} -12.0000i q^{53} +5.19615i q^{55} +3.46410 q^{59} +6.92820i q^{61} -6.00000i q^{65} -2.00000 q^{67} +3.00000i q^{71} -3.46410i q^{73} +(-5.19615 + 6.00000i) q^{77} +10.0000 q^{79} -17.3205 q^{83} +12.0000 q^{85} +5.19615 q^{89} +(6.00000 - 6.92820i) q^{91} +3.00000i q^{95} +13.8564i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7}+O(q^{10}) $ 4 * q - 8 * q^7	$ 4 q - 8 q^{7} - 8 q^{25} + 28 q^{37} + 8 q^{43} + 4 q^{49} - 8 q^{67} + 40 q^{79} + 48 q^{85} + 24 q^{91}+O(q^{100}) $ 4 * q - 8 * q^7 - 8 * q^25 + 28 * q^37 + 8 * q^43 + 4 * q^49 - 8 * q^67 + 40 * q^79 + 48 * q^85 + 24 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$	−1.73205			−0.774597			−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205			−0.774597			−0.387298	+	0.921954i	$0.626592\pi$
$6$		0			0
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$7$	−2.00000	−	1.73205i	−0.755929	−	0.654654i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.00000i		−	0.904534i	−0.891883	−	0.452267i	$-0.850615\pi$
$11$		−	3.00000i		−	0.904534i	0.891883	−	0.452267i	$-0.149385\pi$
$12$		0			0
$13$			3.46410i			0.960769i	0.877058	+	0.480384i	$0.159503\pi$
$13$			3.46410i			0.960769i	−0.877058	+	0.480384i	$0.840497\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6.92820			−1.68034			−0.840168	−	0.542326i	$-0.817544\pi$
$17$	−6.92820			−1.68034			−0.840168	+	0.542326i	$0.817544\pi$
$18$		0			0
$19$		−	1.73205i		−	0.397360i	−0.980064	−	0.198680i	$-0.936335\pi$
$19$		−	1.73205i		−	0.397360i	0.980064	−	0.198680i	$-0.0636654\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	3.00000i		−	0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$		−	3.00000i		−	0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$		0			0
$25$	−2.00000			−0.400000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$			5.19615i			0.933257i	0.884454	+	0.466628i	$0.154531\pi$
$31$			5.19615i			0.933257i	−0.884454	+	0.466628i	$0.845469\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.46410	+	3.00000i	0.585540	+	0.507093i
$36$		0			0
$37$	7.00000			1.15079			0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000			1.15079			0.575396	+	0.817875i	$0.304848\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	12.1244			1.89351			0.946753	−	0.321960i	$-0.104342\pi$
$41$	12.1244			1.89351			0.946753	+	0.321960i	$0.104342\pi$
$42$		0			0
$43$	2.00000			0.304997			0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000			0.304997			0.152499	+	0.988304i	$0.451268\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−3.46410			−0.505291			−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410			−0.505291			−0.252646	+	0.967559i	$0.581301\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	12.0000i		−	1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$		−	12.0000i		−	1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$		0			0
$55$			5.19615i			0.700649i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.46410			0.450988			0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410			0.450988			0.225494	+	0.974245i	$0.427600\pi$
$60$		0			0
$61$			6.92820i			0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$			6.92820i			0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	6.00000i		−	0.744208i
$66$		0			0
$67$	−2.00000			−0.244339			−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000			−0.244339			−0.122169	+	0.992509i	$0.538985\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.00000i			0.356034i	0.984027	+	0.178017i	$0.0569683\pi$
$71$			3.00000i			0.356034i	−0.984027	+	0.178017i	$0.943032\pi$
$72$		0			0
$73$		−	3.46410i		−	0.405442i	−0.979236	−	0.202721i	$-0.935021\pi$
$73$		−	3.46410i		−	0.405442i	0.979236	−	0.202721i	$-0.0649785\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.19615	+	6.00000i	−0.592157	+	0.683763i
$78$		0			0
$79$	10.0000			1.12509			0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000			1.12509			0.562544	+	0.826767i	$0.309823\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−17.3205			−1.90117			−0.950586	−	0.310460i	$-0.899517\pi$
$83$	−17.3205			−1.90117			−0.950586	+	0.310460i	$0.899517\pi$
$84$		0			0
$85$	12.0000			1.30158
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.19615			0.550791			0.275396	−	0.961331i	$-0.411191\pi$
$89$	5.19615			0.550791			0.275396	+	0.961331i	$0.411191\pi$
$90$		0			0
$91$	6.00000	−	6.92820i	0.628971	−	0.726273i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			3.00000i			0.307794i
$96$		0			0
$97$			13.8564i			1.40690i	0.710742	+	0.703452i	$0.248359\pi$
$97$			13.8564i			1.40690i	−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$			19.0526i			1.87730i	0.344865	+	0.938652i	$0.387925\pi$
$103$			19.0526i			1.87730i	−0.344865	+	0.938652i	$0.612075\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	13.0000			1.24517			0.622587	−	0.782551i	$-0.286082\pi$
$109$	13.0000			1.24517			0.622587	+	0.782551i	$0.286082\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	12.0000i		−	1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$		−	12.0000i		−	1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$		0			0
$115$			5.19615i			0.484544i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	13.8564	+	12.0000i	1.27021	+	1.10004i
$120$		0			0
$121$	2.00000			0.181818
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.1244			1.08444
$126$		0			0
$127$	22.0000			1.95218			0.976092	−	0.217357i	$-0.0697436\pi$
$127$	22.0000			1.95218			0.976092	+	0.217357i	$0.0697436\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.3923			−0.907980			−0.453990	−	0.891007i	$-0.650000\pi$
$131$	−10.3923			−0.907980			−0.453990	+	0.891007i	$0.650000\pi$
$132$		0			0
$133$	−3.00000	+	3.46410i	−0.260133	+	0.300376i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			18.0000i			1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$			18.0000i			1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$		0			0
$139$			3.46410i			0.293821i	0.989150	+	0.146911i	$0.0469330\pi$
$139$			3.46410i			0.293821i	−0.989150	+	0.146911i	$0.953067\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	10.3923			0.869048
$144$		0			0
$145$		−	10.3923i		−	0.863034i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			12.0000i			0.983078i	0.870855	+	0.491539i	$0.163566\pi$
$149$			12.0000i			0.983078i	−0.870855	+	0.491539i	$0.836434\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	9.00000i		−	0.722897i
$156$		0			0
$157$		−	10.3923i		−	0.829396i	−0.909959	−	0.414698i	$-0.863887\pi$
$157$		−	10.3923i		−	0.829396i	0.909959	−	0.414698i	$-0.136113\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−5.19615	+	6.00000i	−0.409514	+	0.472866i
$162$		0			0
$163$	−2.00000			−0.156652			−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000			−0.156652			−0.0783260	+	0.996928i	$0.524958\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.92820			0.536120			0.268060	−	0.963402i	$-0.413617\pi$
$167$	6.92820			0.536120			0.268060	+	0.963402i	$0.413617\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.19615			−0.395056			−0.197528	−	0.980297i	$-0.563291\pi$
$173$	−5.19615			−0.395056			−0.197528	+	0.980297i	$0.563291\pi$
$174$		0			0
$175$	4.00000	+	3.46410i	0.302372	+	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	12.0000i		−	0.896922i	−0.893802	−	0.448461i	$-0.851972\pi$
$179$		−	12.0000i		−	0.896922i	0.893802	−	0.448461i	$-0.148028\pi$
$180$		0			0
$181$			13.8564i			1.02994i	0.857209	+	0.514969i	$0.172197\pi$
$181$			13.8564i			1.02994i	−0.857209	+	0.514969i	$0.827803\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−12.1244			−0.891400
$186$		0			0
$187$			20.7846i			1.51992i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			21.0000i			1.51951i	0.650211	+	0.759753i	$0.274680\pi$
$191$			21.0000i			1.51951i	−0.650211	+	0.759753i	$0.725320\pi$
$192$		0			0
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			12.0000i			0.854965i	0.904024	+	0.427482i	$0.140599\pi$
$197$			12.0000i			0.854965i	−0.904024	+	0.427482i	$0.859401\pi$
$198$		0			0
$199$			1.73205i			0.122782i	0.998114	+	0.0613909i	$0.0195536\pi$
$199$			1.73205i			0.122782i	−0.998114	+	0.0613909i	$0.980446\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.3923	−	12.0000i	0.729397	−	0.842235i
$204$		0			0
$205$	−21.0000			−1.46670
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.19615			−0.359425
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.46410			−0.236250
$216$		0			0
$217$	9.00000	−	10.3923i	0.610960	−	0.705476i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	24.0000i		−	1.61441i
$222$		0			0
$223$			15.5885i			1.04388i	0.852982	+	0.521940i	$0.174792\pi$
$223$			15.5885i			1.04388i	−0.852982	+	0.521940i	$0.825208\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	20.7846			1.37952			0.689761	−	0.724037i	$-0.257715\pi$
$227$	20.7846			1.37952			0.689761	+	0.724037i	$0.257715\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			24.0000i			1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$			24.0000i			1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$		0			0
$235$	6.00000			0.391397
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$			17.3205i			1.11571i	0.829938	+	0.557856i	$0.188376\pi$
$241$			17.3205i			1.11571i	−0.829938	+	0.557856i	$0.811624\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.73205	−	12.0000i	−0.110657	−	0.766652i
$246$		0			0
$247$	6.00000			0.381771
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−13.8564			−0.874609			−0.437304	−	0.899314i	$-0.644067\pi$
$251$	−13.8564			−0.874609			−0.437304	+	0.899314i	$0.644067\pi$
$252$		0			0
$253$	−9.00000			−0.565825
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−5.19615			−0.324127			−0.162064	−	0.986780i	$-0.551815\pi$
$257$	−5.19615			−0.324127			−0.162064	+	0.986780i	$0.551815\pi$
$258$		0			0
$259$	−14.0000	−	12.1244i	−0.869918	−	0.753371i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	27.0000i		−	1.66489i	−0.554107	−	0.832446i	$-0.686940\pi$
$263$		−	27.0000i		−	1.66489i	0.554107	−	0.832446i	$-0.313060\pi$
$264$		0			0
$265$			20.7846i			1.27679i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.73205			0.105605			0.0528025	−	0.998605i	$-0.483185\pi$
$269$	1.73205			0.105605			0.0528025	+	0.998605i	$0.483185\pi$
$270$		0			0
$271$			17.3205i			1.05215i	0.850439	+	0.526073i	$0.176336\pi$
$271$			17.3205i			1.05215i	−0.850439	+	0.526073i	$0.823664\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			6.00000i			0.361814i
$276$		0			0
$277$	11.0000			0.660926			0.330463	−	0.943819i	$-0.392795\pi$
$277$	11.0000			0.660926			0.330463	+	0.943819i	$0.392795\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	30.0000i		−	1.78965i	−0.446417	−	0.894825i	$-0.647300\pi$
$281$		−	30.0000i		−	1.78965i	0.446417	−	0.894825i	$-0.352700\pi$
$282$		0			0
$283$		−	31.1769i		−	1.85328i	−0.375956	−	0.926638i	$-0.622686\pi$
$283$		−	31.1769i		−	1.85328i	0.375956	−	0.926638i	$-0.377314\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−24.2487	−	21.0000i	−1.43136	−	1.23959i
$288$		0			0
$289$	31.0000			1.82353
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.92820			0.404750			0.202375	−	0.979308i	$-0.435134\pi$
$293$	6.92820			0.404750			0.202375	+	0.979308i	$0.435134\pi$
$294$		0			0
$295$	−6.00000			−0.349334
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.3923			0.601003
$300$		0			0
$301$	−4.00000	−	3.46410i	−0.230556	−	0.199667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	12.0000i		−	0.687118i
$306$		0			0
$307$		−	15.5885i		−	0.889680i	−0.895610	−	0.444840i	$-0.853260\pi$
$307$		−	15.5885i		−	0.889680i	0.895610	−	0.444840i	$-0.146740\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−24.2487			−1.37502			−0.687509	−	0.726176i	$-0.741296\pi$
$311$	−24.2487			−1.37502			−0.687509	+	0.726176i	$0.741296\pi$
$312$		0			0
$313$		−	17.3205i		−	0.979013i	−0.872000	−	0.489506i	$-0.837177\pi$
$313$		−	17.3205i		−	0.979013i	0.872000	−	0.489506i	$-0.162823\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	18.0000i		−	1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$		−	18.0000i		−	1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$		0			0
$319$	18.0000			1.00781
$320$		0			0
$321$		0			0
$322$		0			0
$323$			12.0000i			0.667698i
$324$		0			0
$325$		−	6.92820i		−	0.384308i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.92820	+	6.00000i	0.381964	+	0.330791i
$330$		0			0
$331$	−28.0000			−1.53902			−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000			−1.53902			−0.769510	+	0.638635i	$0.779499\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	3.46410			0.189264
$336$		0			0
$337$	−5.00000			−0.272367			−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000			−0.272367			−0.136184	+	0.990684i	$0.543484\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	15.5885			0.844162
$342$		0			0
$343$	10.0000	−	15.5885i	0.539949	−	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			27.0000i			1.44944i	0.689046	+	0.724718i	$0.258030\pi$
$347$			27.0000i			1.44944i	−0.689046	+	0.724718i	$0.741970\pi$
$348$		0			0
$349$			27.7128i			1.48343i	0.670714	+	0.741716i	$0.265988\pi$
$349$			27.7128i			1.48343i	−0.670714	+	0.741716i	$0.734012\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	19.0526			1.01407			0.507033	−	0.861927i	$-0.330742\pi$
$353$	19.0526			1.01407			0.507033	+	0.861927i	$0.330742\pi$
$354$		0			0
$355$		−	5.19615i		−	0.275783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	16.0000			0.842105
$362$		0			0
$363$		0			0
$364$		0			0
$365$			6.00000i			0.314054i
$366$		0			0
$367$			1.73205i			0.0904123i	0.998978	+	0.0452062i	$0.0143945\pi$
$367$			1.73205i			0.0904123i	−0.998978	+	0.0452062i	$0.985606\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−20.7846	+	24.0000i	−1.07908	+	1.24602i
$372$		0			0
$373$	5.00000			0.258890			0.129445	−	0.991587i	$-0.458680\pi$
$373$	5.00000			0.258890			0.129445	+	0.991587i	$0.458680\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−20.7846			−1.07046
$378$		0			0
$379$	2.00000			0.102733			0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000			0.102733			0.0513665	+	0.998680i	$0.483642\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	13.8564			0.708029			0.354015	−	0.935240i	$-0.384816\pi$
$383$	13.8564			0.708029			0.354015	+	0.935240i	$0.384816\pi$
$384$		0			0
$385$	9.00000	−	10.3923i	0.458682	−	0.529641i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			24.0000i			1.21685i	0.793612	+	0.608424i	$0.208198\pi$
$389$			24.0000i			1.21685i	−0.793612	+	0.608424i	$0.791802\pi$
$390$		0			0
$391$			20.7846i			1.05112i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−17.3205			−0.871489
$396$		0			0
$397$		−	20.7846i		−	1.04315i	−0.853206	−	0.521575i	$-0.825345\pi$
$397$		−	20.7846i		−	1.04315i	0.853206	−	0.521575i	$-0.174655\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			18.0000i			0.898877i	0.893311	+	0.449439i	$0.148376\pi$
$401$			18.0000i			0.898877i	−0.893311	+	0.449439i	$0.851624\pi$
$402$		0			0
$403$	−18.0000			−0.896644
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	21.0000i		−	1.04093i
$408$		0			0
$409$		−	3.46410i		−	0.171289i	−0.996326	−	0.0856444i	$-0.972705\pi$
$409$		−	3.46410i		−	0.171289i	0.996326	−	0.0856444i	$-0.0272949\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.92820	−	6.00000i	−0.340915	−	0.295241i
$414$		0			0
$415$	30.0000			1.47264
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6.92820			0.338465			0.169232	−	0.985576i	$-0.445871\pi$
$419$	6.92820			0.338465			0.169232	+	0.985576i	$0.445871\pi$
$420$		0			0
$421$	−13.0000			−0.633581			−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000			−0.633581			−0.316791	+	0.948495i	$0.602605\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	13.8564			0.672134
$426$		0			0
$427$	12.0000	−	13.8564i	0.580721	−	0.670559i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			21.0000i			1.01153i	0.862670	+	0.505767i	$0.168791\pi$
$431$			21.0000i			1.01153i	−0.862670	+	0.505767i	$0.831209\pi$
$432$		0			0
$433$			10.3923i			0.499422i	0.968320	+	0.249711i	$0.0803357\pi$
$433$			10.3923i			0.499422i	−0.968320	+	0.249711i	$0.919664\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.19615			−0.248566
$438$		0			0
$439$			10.3923i			0.495998i	0.968760	+	0.247999i	$0.0797729\pi$
$439$			10.3923i			0.495998i	−0.968760	+	0.247999i	$0.920227\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			9.00000i			0.427603i	0.976877	+	0.213801i	$0.0685846\pi$
$443$			9.00000i			0.427603i	−0.976877	+	0.213801i	$0.931415\pi$
$444$		0			0
$445$	−9.00000			−0.426641
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	12.0000i		−	0.566315i	−0.959073	−	0.283158i	$-0.908618\pi$
$449$		−	12.0000i		−	0.566315i	0.959073	−	0.283158i	$-0.0913819\pi$
$450$		0			0
$451$		−	36.3731i		−	1.71274i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−10.3923	+	12.0000i	−0.487199	+	0.562569i
$456$		0			0
$457$	−17.0000			−0.795226			−0.397613	−	0.917553i	$-0.630161\pi$
$457$	−17.0000			−0.795226			−0.397613	+	0.917553i	$0.630161\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−19.0526			−0.887366			−0.443683	−	0.896184i	$-0.646328\pi$
$461$	−19.0526			−0.887366			−0.443683	+	0.896184i	$0.646328\pi$
$462$		0			0
$463$	−32.0000			−1.48717			−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000			−1.48717			−0.743583	+	0.668644i	$0.766875\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	31.1769			1.44270			0.721348	−	0.692573i	$-0.243523\pi$
$467$	31.1769			1.44270			0.721348	+	0.692573i	$0.243523\pi$
$468$		0			0
$469$	4.00000	+	3.46410i	0.184703	+	0.159957i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	6.00000i		−	0.275880i
$474$		0			0
$475$			3.46410i			0.158944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−27.7128			−1.26623			−0.633115	−	0.774057i	$-0.718224\pi$
$479$	−27.7128			−1.26623			−0.633115	+	0.774057i	$0.718224\pi$
$480$		0			0
$481$			24.2487i			1.10565i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	24.0000i		−	1.08978i
$486$		0			0
$487$	−10.0000			−0.453143			−0.226572	−	0.973995i	$-0.572752\pi$
$487$	−10.0000			−0.453143			−0.226572	+	0.973995i	$0.572752\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	9.00000i		−	0.406164i	−0.979162	−	0.203082i	$-0.934904\pi$
$491$		−	9.00000i		−	0.406164i	0.979162	−	0.203082i	$-0.0650959\pi$
$492$		0			0
$493$		−	41.5692i		−	1.87218i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.19615	−	6.00000i	0.233079	−	0.269137i
$498$		0			0
$499$	−10.0000			−0.447661			−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000			−0.447661			−0.223831	+	0.974628i	$0.571856\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	24.2487			1.08120			0.540598	−	0.841281i	$-0.318198\pi$
$503$	24.2487			1.08120			0.540598	+	0.841281i	$0.318198\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−13.8564			−0.614174			−0.307087	−	0.951681i	$-0.599354\pi$
$509$	−13.8564			−0.614174			−0.307087	+	0.951681i	$0.599354\pi$
$510$		0			0
$511$	−6.00000	+	6.92820i	−0.265424	+	0.306486i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	33.0000i		−	1.45415i
$516$		0			0
$517$			10.3923i			0.457053i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	22.5167			0.986473			0.493236	−	0.869895i	$-0.335814\pi$
$521$	22.5167			0.986473			0.493236	+	0.869895i	$0.335814\pi$
$522$		0			0
$523$		−	39.8372i		−	1.74196i	−0.491320	−	0.870979i	$-0.663486\pi$
$523$		−	39.8372i		−	1.74196i	0.491320	−	0.870979i	$-0.336514\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	36.0000i		−	1.56818i
$528$		0			0
$529$	14.0000			0.608696
$530$		0			0
$531$		0			0
$532$		0			0
$533$			42.0000i			1.81922i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	20.7846	−	3.00000i	0.895257	−	0.129219i
$540$		0			0
$541$	−25.0000			−1.07483			−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000			−1.07483			−0.537417	+	0.843317i	$0.680600\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−22.5167			−0.964508
$546$		0			0
$547$	−8.00000			−0.342055			−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000			−0.342055			−0.171028	+	0.985266i	$0.554709\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	10.3923			0.442727
$552$		0			0
$553$	−20.0000	−	17.3205i	−0.850487	−	0.736543i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			12.0000i			0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$			12.0000i			0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$		0			0
$559$			6.92820i			0.293032i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.3923			0.437983			0.218992	−	0.975727i	$-0.429723\pi$
$563$	10.3923			0.437983			0.218992	+	0.975727i	$0.429723\pi$
$564$		0			0
$565$			20.7846i			0.874415i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−32.0000			−1.33916			−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000			−1.33916			−0.669579	+	0.742741i	$0.733526\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			6.00000i			0.250217i
$576$		0			0
$577$		−	34.6410i		−	1.44212i	−0.692870	−	0.721062i	$-0.743654\pi$
$577$		−	34.6410i		−	1.44212i	0.692870	−	0.721062i	$-0.256346\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	34.6410	+	30.0000i	1.43715	+	1.24461i
$582$		0			0
$583$	−36.0000			−1.49097
$584$		0			0
$585$		0			0
$586$		0			0
$587$	41.5692			1.71575			0.857873	−	0.513862i	$-0.171786\pi$
$587$	41.5692			1.71575			0.857873	+	0.513862i	$0.171786\pi$
$588$		0			0
$589$	9.00000			0.370839
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−19.0526			−0.782395			−0.391197	−	0.920307i	$-0.627939\pi$
$593$	−19.0526			−0.782395			−0.391197	+	0.920307i	$0.627939\pi$
$594$		0			0
$595$	−24.0000	−	20.7846i	−0.983904	−	0.852086i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			9.00000i			0.367730i	0.982952	+	0.183865i	$0.0588609\pi$
$599$			9.00000i			0.367730i	−0.982952	+	0.183865i	$0.941139\pi$
$600$		0			0
$601$		−	10.3923i		−	0.423911i	−0.977279	−	0.211955i	$-0.932017\pi$
$601$		−	10.3923i		−	0.423911i	0.977279	−	0.211955i	$-0.0679832\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.46410			−0.140836
$606$		0			0
$607$			17.3205i			0.703018i	0.936185	+	0.351509i	$0.114331\pi$
$607$			17.3205i			0.703018i	−0.936185	+	0.351509i	$0.885669\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	12.0000i		−	0.485468i
$612$		0			0
$613$	−19.0000			−0.767403			−0.383701	−	0.923457i	$-0.625351\pi$
$613$	−19.0000			−0.767403			−0.383701	+	0.923457i	$0.625351\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			18.0000i			0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$			18.0000i			0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$		0			0
$619$		−	19.0526i		−	0.765787i	−0.923792	−	0.382893i	$-0.874928\pi$
$619$		−	19.0526i		−	0.765787i	0.923792	−	0.382893i	$-0.125072\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−10.3923	−	9.00000i	−0.416359	−	0.360577i
$624$		0			0
$625$	−11.0000			−0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−48.4974			−1.93372
$630$		0			0
$631$	34.0000			1.35352			0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000			1.35352			0.676759	+	0.736204i	$0.263384\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−38.1051			−1.51216
$636$		0			0
$637$	−24.0000	+	3.46410i	−0.950915	+	0.137253i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$			5.19615i			0.204916i	0.994737	+	0.102458i	$0.0326708\pi$
$643$			5.19615i			0.204916i	−0.994737	+	0.102458i	$0.967329\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	10.3923			0.408564			0.204282	−	0.978912i	$-0.434514\pi$
$647$	10.3923			0.408564			0.204282	+	0.978912i	$0.434514\pi$
$648$		0			0
$649$		−	10.3923i		−	0.407934i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	18.0000i		−	0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$		−	18.0000i		−	0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$		0			0
$655$	18.0000			0.703318
$656$		0			0
$657$		0			0
$658$		0			0
$659$			21.0000i			0.818044i	0.912525	+	0.409022i	$0.134130\pi$
$659$			21.0000i			0.818044i	−0.912525	+	0.409022i	$0.865870\pi$
$660$		0			0
$661$		−	38.1051i		−	1.48212i	−0.671440	−	0.741059i	$-0.734324\pi$
$661$		−	38.1051i		−	1.48212i	0.671440	−	0.741059i	$-0.265676\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	5.19615	−	6.00000i	0.201498	−	0.232670i
$666$		0			0
$667$	18.0000			0.696963
$668$		0			0
$669$		0			0
$670$		0			0
$671$	20.7846			0.802381
$672$		0			0
$673$	−14.0000			−0.539660			−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000			−0.539660			−0.269830	+	0.962908i	$0.586968\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	25.9808			0.998522			0.499261	−	0.866452i	$-0.333605\pi$
$677$	25.9808			0.998522			0.499261	+	0.866452i	$0.333605\pi$
$678$		0			0
$679$	24.0000	−	27.7128i	0.921035	−	1.06352i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			33.0000i			1.26271i	0.775494	+	0.631355i	$0.217501\pi$
$683$			33.0000i			1.26271i	−0.775494	+	0.631355i	$0.782499\pi$
$684$		0			0
$685$		−	31.1769i		−	1.19121i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	41.5692			1.58366
$690$		0			0
$691$		−	10.3923i		−	0.395342i	−0.980268	−	0.197671i	$-0.936662\pi$
$691$		−	10.3923i		−	0.395342i	0.980268	−	0.197671i	$-0.0633378\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	6.00000i		−	0.227593i
$696$		0			0
$697$	−84.0000			−3.18173
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	6.00000i		−	0.226617i	−0.993560	−	0.113308i	$-0.963855\pi$
$701$		−	6.00000i		−	0.226617i	0.993560	−	0.113308i	$-0.0361448\pi$
$702$		0			0
$703$		−	12.1244i		−	0.457279i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.00000			0.0375558			0.0187779	−	0.999824i	$-0.494022\pi$
$709$	1.00000			0.0375558			0.0187779	+	0.999824i	$0.494022\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	15.5885			0.583792
$714$		0			0
$715$	−18.0000			−0.673162
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3.46410			0.129189			0.0645946	−	0.997912i	$-0.479425\pi$
$719$	3.46410			0.129189			0.0645946	+	0.997912i	$0.479425\pi$
$720$		0			0
$721$	33.0000	−	38.1051i	1.22898	−	1.41911i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	12.0000i		−	0.445669i
$726$		0			0
$727$			51.9615i			1.92715i	0.267445	+	0.963573i	$0.413821\pi$
$727$			51.9615i			1.92715i	−0.267445	+	0.963573i	$0.586179\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−13.8564			−0.512498
$732$		0			0
$733$		−	10.3923i		−	0.383849i	−0.981410	−	0.191924i	$-0.938527\pi$
$733$		−	10.3923i		−	0.383849i	0.981410	−	0.191924i	$-0.0614728\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			6.00000i			0.221013i
$738$		0			0
$739$	−20.0000			−0.735712			−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000			−0.735712			−0.367856	+	0.929883i	$0.619908\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			27.0000i			0.990534i	0.868741	+	0.495267i	$0.164930\pi$
$743$			27.0000i			0.990534i	−0.868741	+	0.495267i	$0.835070\pi$
$744$		0			0
$745$		−	20.7846i		−	0.761489i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	2.00000			0.0729810			0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000			0.0729810			0.0364905	+	0.999334i	$0.488382\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−13.8564			−0.504286
$756$		0			0
$757$	−2.00000			−0.0726912			−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000			−0.0726912			−0.0363456	+	0.999339i	$0.511572\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13.8564			−0.502294			−0.251147	−	0.967949i	$-0.580808\pi$
$761$	−13.8564			−0.502294			−0.251147	+	0.967949i	$0.580808\pi$
$762$		0			0
$763$	−26.0000	−	22.5167i	−0.941263	−	0.815158i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			12.0000i			0.433295i
$768$		0			0
$769$			38.1051i			1.37411i	0.726607	+	0.687053i	$0.241096\pi$
$769$			38.1051i			1.37411i	−0.726607	+	0.687053i	$0.758904\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	19.0526			0.685273			0.342636	−	0.939468i	$-0.388680\pi$
$773$	19.0526			0.685273			0.342636	+	0.939468i	$0.388680\pi$
$774$		0			0
$775$		−	10.3923i		−	0.373303i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	21.0000i		−	0.752403i
$780$		0			0
$781$	9.00000			0.322045
$782$		0			0
$783$		0			0
$784$		0			0
$785$			18.0000i			0.642448i
$786$		0			0
$787$			17.3205i			0.617409i	0.951158	+	0.308705i	$0.0998955\pi$
$787$			17.3205i			0.617409i	−0.951158	+	0.308705i	$0.900105\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−20.7846	+	24.0000i	−0.739016	+	0.853342i
$792$		0			0
$793$	−24.0000			−0.852265
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15.5885			−0.552171			−0.276086	−	0.961133i	$-0.589037\pi$
$797$	−15.5885			−0.552171			−0.276086	+	0.961133i	$0.589037\pi$
$798$		0			0
$799$	24.0000			0.849059
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−10.3923			−0.366736
$804$		0			0
$805$	9.00000	−	10.3923i	0.317208	−	0.366281i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	6.00000i		−	0.210949i	−0.994422	−	0.105474i	$-0.966364\pi$
$809$		−	6.00000i		−	0.210949i	0.994422	−	0.105474i	$-0.0336361\pi$
$810$		0			0
$811$			36.3731i			1.27723i	0.769526	+	0.638616i	$0.220493\pi$
$811$			36.3731i			1.27723i	−0.769526	+	0.638616i	$0.779507\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.46410			0.121342
$816$		0			0
$817$		−	3.46410i		−	0.121194i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	30.0000i		−	1.04701i	−0.852023	−	0.523504i	$-0.824625\pi$
$821$		−	30.0000i		−	1.04701i	0.852023	−	0.523504i	$-0.175375\pi$
$822$		0			0
$823$	4.00000			0.139431			0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000			0.139431			0.0697156	+	0.997567i	$0.477791\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			3.00000i			0.104320i	0.998639	+	0.0521601i	$0.0166106\pi$
$827$			3.00000i			0.104320i	−0.998639	+	0.0521601i	$0.983389\pi$
$828$		0			0
$829$		−	38.1051i		−	1.32345i	−0.749749	−	0.661723i	$-0.769826\pi$
$829$		−	38.1051i		−	1.32345i	0.749749	−	0.661723i	$-0.230174\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−6.92820	−	48.0000i	−0.240048	−	1.66310i
$834$		0			0
$835$	−12.0000			−0.415277
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−24.2487			−0.837158			−0.418579	−	0.908180i	$-0.637472\pi$
$839$	−24.2487			−0.837158			−0.418579	+	0.908180i	$0.637472\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.73205			−0.0595844
$846$		0			0
$847$	−4.00000	−	3.46410i	−0.137442	−	0.119028i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	21.0000i		−	0.719871i
$852$		0			0
$853$			13.8564i			0.474434i	0.971457	+	0.237217i	$0.0762353\pi$
$853$			13.8564i			0.474434i	−0.971457	+	0.237217i	$0.923765\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−43.3013			−1.47914			−0.739572	−	0.673078i	$-0.764972\pi$
$857$	−43.3013			−1.47914			−0.739572	+	0.673078i	$0.764972\pi$
$858$		0			0
$859$		−	29.4449i		−	1.00465i	−0.864680	−	0.502323i	$-0.832479\pi$
$859$		−	29.4449i		−	1.00465i	0.864680	−	0.502323i	$-0.167521\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	9.00000			0.306009
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	30.0000i		−	1.01768i
$870$		0			0
$871$		−	6.92820i		−	0.234753i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−24.2487	−	21.0000i	−0.819756	−	0.709930i
$876$		0			0
$877$	50.0000			1.68838			0.844190	−	0.536044i	$-0.180082\pi$
$877$	50.0000			1.68838			0.844190	+	0.536044i	$0.180082\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	25.9808			0.875314			0.437657	−	0.899142i	$-0.355808\pi$
$881$	25.9808			0.875314			0.437657	+	0.899142i	$0.355808\pi$
$882$		0			0
$883$	−16.0000			−0.538443			−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000			−0.538443			−0.269221	+	0.963078i	$0.586766\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−24.2487			−0.814192			−0.407096	−	0.913385i	$-0.633459\pi$
$887$	−24.2487			−0.814192			−0.407096	+	0.913385i	$0.633459\pi$
$888$		0			0
$889$	−44.0000	−	38.1051i	−1.47571	−	1.27800i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			6.00000i			0.200782i
$894$		0			0
$895$			20.7846i			0.694753i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−31.1769			−1.03981
$900$		0			0
$901$			83.1384i			2.76974i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	24.0000i		−	0.797787i
$906$		0			0
$907$	8.00000			0.265636			0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000			0.265636			0.132818	+	0.991140i	$0.457597\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			12.0000i			0.397578i	0.980042	+	0.198789i	$0.0637008\pi$
$911$			12.0000i			0.397578i	−0.980042	+	0.198789i	$0.936299\pi$
$912$		0			0
$913$			51.9615i			1.71968i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	20.7846	+	18.0000i	0.686368	+	0.594412i
$918$		0			0
$919$	−4.00000			−0.131948			−0.0659739	−	0.997821i	$-0.521015\pi$
$919$	−4.00000			−0.131948			−0.0659739	+	0.997821i	$0.521015\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−10.3923			−0.342067
$924$		0			0
$925$	−14.0000			−0.460317
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−27.7128			−0.909228			−0.454614	−	0.890689i	$-0.650223\pi$
$929$	−27.7128			−0.909228			−0.454614	+	0.890689i	$0.650223\pi$
$930$		0			0
$931$	12.0000	−	1.73205i	0.393284	−	0.0567657i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	36.0000i		−	1.17733i
$936$		0			0
$937$			6.92820i			0.226335i	0.993576	+	0.113167i	$0.0360996\pi$
$937$			6.92820i			0.226335i	−0.993576	+	0.113167i	$0.963900\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.1244			0.395243			0.197621	−	0.980278i	$-0.436678\pi$
$941$	12.1244			0.395243			0.197621	+	0.980278i	$0.436678\pi$
$942$		0			0
$943$		−	36.3731i		−	1.18447i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			15.0000i			0.487435i	0.969846	+	0.243717i	$0.0783669\pi$
$947$			15.0000i			0.487435i	−0.969846	+	0.243717i	$0.921633\pi$
$948$		0			0
$949$	12.0000			0.389536
$950$		0			0
$951$		0			0
$952$		0			0
$953$			18.0000i			0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$			18.0000i			0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$		0			0
$955$		−	36.3731i		−	1.17700i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	31.1769	−	36.0000i	1.00676	−	1.16250i
$960$		0			0
$961$	4.00000			0.129032
$962$		0			0
$963$		0			0
$964$		0			0
$965$	24.2487			0.780594
$966$		0			0
$967$	16.0000			0.514525			0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000			0.514525			0.257263	+	0.966342i	$0.417179\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	3.46410			0.111168			0.0555842	−	0.998454i	$-0.482298\pi$
$971$	3.46410			0.111168			0.0555842	+	0.998454i	$0.482298\pi$
$972$		0			0
$973$	6.00000	−	6.92820i	0.192351	−	0.222108i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			48.0000i			1.53566i	0.640656	+	0.767828i	$0.278662\pi$
$977$			48.0000i			1.53566i	−0.640656	+	0.767828i	$0.721338\pi$
$978$		0			0
$979$		−	15.5885i		−	0.498209i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.46410			0.110488			0.0552438	−	0.998473i	$-0.482406\pi$
$983$	3.46410			0.110488			0.0552438	+	0.998473i	$0.482406\pi$
$984$		0			0
$985$		−	20.7846i		−	0.662253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	6.00000i		−	0.190789i
$990$		0			0
$991$	2.00000			0.0635321			0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000			0.0635321			0.0317660	+	0.999495i	$0.489887\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	3.00000i		−	0.0951064i
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.k.e.1889.1		4
3.2	odd	2	inner	3024.2.k.e.1889.3		4
4.3	odd	2		378.2.d.b.377.3	yes	4
7.6	odd	2	inner	3024.2.k.e.1889.4		4
12.11	even	2		378.2.d.b.377.2	yes	4
21.20	even	2	inner	3024.2.k.e.1889.2		4
28.27	even	2		378.2.d.b.377.4	yes	4
36.7	odd	6		1134.2.m.c.377.2		4
36.11	even	6		1134.2.m.c.377.1		4
36.23	even	6		1134.2.m.b.755.2		4
36.31	odd	6		1134.2.m.b.755.1		4
84.83	odd	2		378.2.d.b.377.1	✓	4
252.83	odd	6		1134.2.m.b.377.1		4
252.139	even	6		1134.2.m.c.755.1		4
252.167	odd	6		1134.2.m.c.755.2		4
252.223	even	6		1134.2.m.b.377.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.d.b.377.1	✓	4	84.83	odd	2
378.2.d.b.377.2	yes	4	12.11	even	2
378.2.d.b.377.3	yes	4	4.3	odd	2
378.2.d.b.377.4	yes	4	28.27	even	2
1134.2.m.b.377.1		4	252.83	odd	6
1134.2.m.b.377.2		4	252.223	even	6
1134.2.m.b.755.1		4	36.31	odd	6
1134.2.m.b.755.2		4	36.23	even	6
1134.2.m.c.377.1		4	36.11	even	6
1134.2.m.c.377.2		4	36.7	odd	6
1134.2.m.c.755.1		4	252.139	even	6
1134.2.m.c.755.2		4	252.167	odd	6
3024.2.k.e.1889.1		4	1.1	even	1	trivial
3024.2.k.e.1889.2		4	21.20	even	2	inner
3024.2.k.e.1889.3		4	3.2	odd	2	inner
3024.2.k.e.1889.4		4	7.6	odd	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:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{5} +(-2.00000 - 1.73205i) q^{7} +O(q^{10})\)	\(q-1.73205 q^{5} +(-2.00000 - 1.73205i) q^{7} -3.00000i q^{11} +3.46410i q^{13} -6.92820 q^{17} -1.73205i q^{19} -3.00000i q^{23} -2.00000 q^{25} +6.00000i q^{29} +5.19615i q^{31} +(3.46410 + 3.00000i) q^{35} +7.00000 q^{37} +12.1244 q^{41} +2.00000 q^{43} -3.46410 q^{47} +(1.00000 + 6.92820i) q^{49} -12.0000i q^{53} +5.19615i q^{55} +3.46410 q^{59} +6.92820i q^{61} -6.00000i q^{65} -2.00000 q^{67} +3.00000i q^{71} -3.46410i q^{73} +(-5.19615 + 6.00000i) q^{77} +10.0000 q^{79} -17.3205 q^{83} +12.0000 q^{85} +5.19615 q^{89} +(6.00000 - 6.92820i) q^{91} +3.00000i q^{95} +13.8564i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7}+O(q^{10}) \) 4 * q - 8 * q^7	\( 4 q - 8 q^{7} - 8 q^{25} + 28 q^{37} + 8 q^{43} + 4 q^{49} - 8 q^{67} + 40 q^{79} + 48 q^{85} + 24 q^{91}+O(q^{100}) \) 4 * q - 8 * q^7 - 8 * q^25 + 28 * q^37 + 8 * q^43 + 4 * q^49 - 8 * q^67 + 40 * q^79 + 48 * q^85 + 24 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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