LMFDB - Embedded newform 378.2.d.b.377.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(377,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("378.377");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.d (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:	$3.01834519640$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			377.3
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	378.377
Dual form			378.2.d.b.377.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{4} -1.73205 q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} -1.73205i q^{10} +3.00000i q^{11} +3.46410i q^{13} +(-1.73205 + 2.00000i) q^{14} +1.00000 q^{16} -6.92820 q^{17} +1.73205i q^{19} +1.73205 q^{20} -3.00000 q^{22} +3.00000i q^{23} -2.00000 q^{25} -3.46410 q^{26} +(-2.00000 - 1.73205i) q^{28} +6.00000i q^{29} -5.19615i q^{31} +1.00000i q^{32} -6.92820i q^{34} +(-3.46410 - 3.00000i) q^{35} +7.00000 q^{37} -1.73205 q^{38} +1.73205i q^{40} +12.1244 q^{41} -2.00000 q^{43} -3.00000i q^{44} -3.00000 q^{46} +3.46410 q^{47} +(1.00000 + 6.92820i) q^{49} -2.00000i q^{50} -3.46410i q^{52} -12.0000i q^{53} -5.19615i q^{55} +(1.73205 - 2.00000i) q^{56} -6.00000 q^{58} -3.46410 q^{59} +6.92820i q^{61} +5.19615 q^{62} -1.00000 q^{64} -6.00000i q^{65} +2.00000 q^{67} +6.92820 q^{68} +(3.00000 - 3.46410i) q^{70} -3.00000i q^{71} -3.46410i q^{73} +7.00000i q^{74} -1.73205i q^{76} +(-5.19615 + 6.00000i) q^{77} -10.0000 q^{79} -1.73205 q^{80} +12.1244i q^{82} +17.3205 q^{83} +12.0000 q^{85} -2.00000i q^{86} +3.00000 q^{88} +5.19615 q^{89} +(-6.00000 + 6.92820i) q^{91} -3.00000i q^{92} +3.46410i q^{94} -3.00000i q^{95} +13.8564i q^{97} +(-6.92820 + 1.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 8 q^{7} + 4 q^{16} - 12 q^{22} - 8 q^{25} - 8 q^{28} + 28 q^{37} - 8 q^{43} - 12 q^{46} + 4 q^{49} - 24 q^{58} - 4 q^{64} + 8 q^{67} + 12 q^{70} - 40 q^{79} + 48 q^{85} + 12 q^{88} - 24 q^{91}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^7 + 4 * q^16 - 12 * q^22 - 8 * q^25 - 8 * q^28 + 28 * q^37 - 8 * q^43 - 12 * q^46 + 4 * q^49 - 24 * q^58 - 4 * q^64 + 8 * q^67 + 12 * q^70 - 40 * q^79 + 48 * q^85 + 12 * q^88 - 24 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$-1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	−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
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$		−	1.73205i		−	0.547723i
$11$			3.00000i			0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$			3.00000i			0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$		0			0
$13$			3.46410i			0.960769i	0.877058	+	0.480384i	$0.159503\pi$
$13$			3.46410i			0.960769i	−0.877058	+	0.480384i	$0.840497\pi$
$14$	−1.73205	+	2.00000i	−0.462910	+	0.534522i
$15$		0			0
$16$	1.00000			0.250000
$17$	−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	$0.0636654\pi$
$19$			1.73205i			0.397360i	−0.980064	+	0.198680i	$0.936335\pi$
$20$	1.73205			0.387298
$21$		0			0
$22$	−3.00000			−0.639602
$23$			3.00000i			0.625543i	0.949828	+	0.312772i	$0.101257\pi$
$23$			3.00000i			0.625543i	−0.949828	+	0.312772i	$0.898743\pi$
$24$		0			0
$25$	−2.00000			−0.400000
$26$	−3.46410			−0.679366
$27$		0			0
$28$	−2.00000	−	1.73205i	−0.377964	−	0.327327i
$29$			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	$-0.845469\pi$
$31$		−	5.19615i		−	0.933257i	0.884454	−	0.466628i	$-0.154531\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	6.92820i		−	1.18818i
$35$	−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	$0.304848\pi$
$38$	−1.73205			−0.280976
$39$		0			0
$40$			1.73205i			0.273861i
$41$	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	$-0.548732\pi$
$43$	−2.00000			−0.304997			−0.152499	+	0.988304i	$0.548732\pi$
$44$		−	3.00000i		−	0.452267i
$45$		0			0
$46$	−3.00000			−0.442326
$47$	3.46410			0.505291			0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410			0.505291			0.252646	+	0.967559i	$0.418699\pi$
$48$		0			0
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		−	2.00000i		−	0.282843i
$51$		0			0
$52$		−	3.46410i		−	0.480384i
$53$		−	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
$56$	1.73205	−	2.00000i	0.231455	−	0.267261i
$57$		0			0
$58$	−6.00000			−0.787839
$59$	−3.46410			−0.450988			−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410			−0.450988			−0.225494	+	0.974245i	$0.572400\pi$
$60$		0			0
$61$			6.92820i			0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$			6.92820i			0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$	5.19615			0.659912
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	6.00000i		−	0.744208i
$66$		0			0
$67$	2.00000			0.244339			0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000			0.244339			0.122169	+	0.992509i	$0.461015\pi$
$68$	6.92820			0.840168
$69$		0			0
$70$	3.00000	−	3.46410i	0.358569	−	0.414039i
$71$		−	3.00000i		−	0.356034i	−0.984027	−	0.178017i	$-0.943032\pi$
$71$		−	3.00000i		−	0.356034i	0.984027	−	0.178017i	$-0.0569683\pi$
$72$		0			0
$73$		−	3.46410i		−	0.405442i	−0.979236	−	0.202721i	$-0.935021\pi$
$73$		−	3.46410i		−	0.405442i	0.979236	−	0.202721i	$-0.0649785\pi$
$74$			7.00000i			0.813733i
$75$		0			0
$76$		−	1.73205i		−	0.198680i
$77$	−5.19615	+	6.00000i	−0.592157	+	0.683763i
$78$		0			0
$79$	−10.0000			−1.12509			−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000			−1.12509			−0.562544	+	0.826767i	$0.690177\pi$
$80$	−1.73205			−0.193649
$81$		0			0
$82$			12.1244i			1.33891i
$83$	17.3205			1.90117			0.950586	−	0.310460i	$-0.100483\pi$
$83$	17.3205			1.90117			0.950586	+	0.310460i	$0.100483\pi$
$84$		0			0
$85$	12.0000			1.30158
$86$		−	2.00000i		−	0.215666i
$87$		0			0
$88$	3.00000			0.319801
$89$	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
$92$		−	3.00000i		−	0.312772i
$93$		0			0
$94$			3.46410i			0.357295i
$95$		−	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	$0.751641\pi$
$98$	−6.92820	+	1.00000i	−0.699854	+	0.101015i
$99$		0			0
$100$	2.00000			0.200000
$101$		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	$-0.612075\pi$
$103$		−	19.0526i		−	1.87730i	0.344865	−	0.938652i	$-0.387925\pi$
$104$	3.46410			0.339683
$105$		0			0
$106$	12.0000			1.16554
$107$		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	$0.286082\pi$
$110$	5.19615			0.495434
$111$		0			0
$112$	2.00000	+	1.73205i	0.188982	+	0.163663i
$113$		−	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
$116$		−	6.00000i		−	0.557086i
$117$		0			0
$118$		−	3.46410i		−	0.318896i
$119$	−13.8564	−	12.0000i	−1.27021	−	1.10004i
$120$		0			0
$121$	2.00000			0.181818
$122$	−6.92820			−0.627250
$123$		0			0
$124$			5.19615i			0.466628i
$125$	12.1244			1.08444
$126$		0			0
$127$	−22.0000			−1.95218			−0.976092	−	0.217357i	$-0.930256\pi$
$127$	−22.0000			−1.95218			−0.976092	+	0.217357i	$0.930256\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	6.00000			0.526235
$131$	10.3923			0.907980			0.453990	−	0.891007i	$-0.350000\pi$
$131$	10.3923			0.907980			0.453990	+	0.891007i	$0.350000\pi$
$132$		0			0
$133$	−3.00000	+	3.46410i	−0.260133	+	0.300376i
$134$			2.00000i			0.172774i
$135$		0			0
$136$			6.92820i			0.594089i
$137$			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	$-0.953067\pi$
$139$		−	3.46410i		−	0.293821i	0.989150	−	0.146911i	$-0.0469330\pi$
$140$	3.46410	+	3.00000i	0.292770	+	0.253546i
$141$		0			0
$142$	3.00000			0.251754
$143$	−10.3923			−0.869048
$144$		0			0
$145$		−	10.3923i		−	0.863034i
$146$	3.46410			0.286691
$147$		0			0
$148$	−7.00000			−0.575396
$149$			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	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$	1.73205			0.140488
$153$		0			0
$154$	−6.00000	−	5.19615i	−0.483494	−	0.418718i
$155$			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	$-0.136113\pi$
$158$		−	10.0000i		−	0.795557i
$159$		0			0
$160$		−	1.73205i		−	0.136931i
$161$	−5.19615	+	6.00000i	−0.409514	+	0.472866i
$162$		0			0
$163$	2.00000			0.156652			0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000			0.156652			0.0783260	+	0.996928i	$0.475042\pi$
$164$	−12.1244			−0.946753
$165$		0			0
$166$			17.3205i			1.34433i
$167$	−6.92820			−0.536120			−0.268060	−	0.963402i	$-0.586383\pi$
$167$	−6.92820			−0.536120			−0.268060	+	0.963402i	$0.586383\pi$
$168$		0			0
$169$	1.00000			0.0769231
$170$			12.0000i			0.920358i
$171$		0			0
$172$	2.00000			0.152499
$173$	−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
$176$			3.00000i			0.226134i
$177$		0			0
$178$			5.19615i			0.389468i
$179$			12.0000i			0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$			12.0000i			0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$		0			0
$181$			13.8564i			1.02994i	0.857209	+	0.514969i	$0.172197\pi$
$181$			13.8564i			1.02994i	−0.857209	+	0.514969i	$0.827803\pi$
$182$	−6.92820	−	6.00000i	−0.513553	−	0.444750i
$183$		0			0
$184$	3.00000			0.221163
$185$	−12.1244			−0.891400
$186$		0			0
$187$		−	20.7846i		−	1.51992i
$188$	−3.46410			−0.252646
$189$		0			0
$190$	3.00000			0.217643
$191$		−	21.0000i		−	1.51951i	−0.650211	−	0.759753i	$-0.725320\pi$
$191$		−	21.0000i		−	1.51951i	0.650211	−	0.759753i	$-0.274680\pi$
$192$		0			0
$193$	−14.0000			−1.00774			−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000			−1.00774			−0.503871	+	0.863779i	$0.668091\pi$
$194$	−13.8564			−0.994832
$195$		0			0
$196$	−1.00000	−	6.92820i	−0.0714286	−	0.494872i
$197$			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	$-0.980446\pi$
$199$		−	1.73205i		−	0.122782i	0.998114	−	0.0613909i	$-0.0195536\pi$
$200$			2.00000i			0.141421i
$201$		0			0
$202$		0			0
$203$	−10.3923	+	12.0000i	−0.729397	+	0.842235i
$204$		0			0
$205$	−21.0000			−1.46670
$206$	19.0526			1.32745
$207$		0			0
$208$			3.46410i			0.240192i
$209$	−5.19615			−0.359425
$210$		0			0
$211$	16.0000			1.10149			0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000			1.10149			0.550743	+	0.834675i	$0.314345\pi$
$212$			12.0000i			0.824163i
$213$		0			0
$214$		0			0
$215$	3.46410			0.236250
$216$		0			0
$217$	9.00000	−	10.3923i	0.610960	−	0.705476i
$218$			13.0000i			0.880471i
$219$		0			0
$220$			5.19615i			0.350325i
$221$		−	24.0000i		−	1.61441i
$222$		0			0
$223$		−	15.5885i		−	1.04388i	−0.852982	−	0.521940i	$-0.825208\pi$
$223$		−	15.5885i		−	1.04388i	0.852982	−	0.521940i	$-0.174792\pi$
$224$	−1.73205	+	2.00000i	−0.115728	+	0.133631i
$225$		0			0
$226$	12.0000			0.798228
$227$	−20.7846			−1.37952			−0.689761	−	0.724037i	$-0.742285\pi$
$227$	−20.7846			−1.37952			−0.689761	+	0.724037i	$0.742285\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$	5.19615			0.342624
$231$		0			0
$232$	6.00000			0.393919
$233$			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
$236$	3.46410			0.225494
$237$		0			0
$238$	12.0000	−	13.8564i	0.777844	−	0.898177i
$239$		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	$0.811624\pi$
$242$			2.00000i			0.128565i
$243$		0			0
$244$		−	6.92820i		−	0.443533i
$245$	−1.73205	−	12.0000i	−0.110657	−	0.766652i
$246$		0			0
$247$	−6.00000			−0.381771
$248$	−5.19615			−0.329956
$249$		0			0
$250$			12.1244i			0.766812i
$251$	13.8564			0.874609			0.437304	−	0.899314i	$-0.355933\pi$
$251$	13.8564			0.874609			0.437304	+	0.899314i	$0.355933\pi$
$252$		0			0
$253$	−9.00000			−0.565825
$254$		−	22.0000i		−	1.38040i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−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
$260$			6.00000i			0.372104i
$261$		0			0
$262$			10.3923i			0.642039i
$263$			27.0000i			1.66489i	0.554107	+	0.832446i	$0.313060\pi$
$263$			27.0000i			1.66489i	−0.554107	+	0.832446i	$0.686940\pi$
$264$		0			0
$265$			20.7846i			1.27679i
$266$	−3.46410	−	3.00000i	−0.212398	−	0.183942i
$267$		0			0
$268$	−2.00000			−0.122169
$269$	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	$-0.823664\pi$
$271$		−	17.3205i		−	1.05215i	0.850439	−	0.526073i	$-0.176336\pi$
$272$	−6.92820			−0.420084
$273$		0			0
$274$	−18.0000			−1.08742
$275$		−	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	$0.392795\pi$
$278$	3.46410			0.207763
$279$		0			0
$280$	−3.00000	+	3.46410i	−0.179284	+	0.207020i
$281$		−	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	$0.377314\pi$
$283$			31.1769i			1.85328i	−0.375956	+	0.926638i	$0.622686\pi$
$284$			3.00000i			0.178017i
$285$		0			0
$286$		−	10.3923i		−	0.614510i
$287$	24.2487	+	21.0000i	1.43136	+	1.23959i
$288$		0			0
$289$	31.0000			1.82353
$290$	10.3923			0.610257
$291$		0			0
$292$			3.46410i			0.202721i
$293$	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
$296$		−	7.00000i		−	0.406867i
$297$		0			0
$298$	−12.0000			−0.695141
$299$	−10.3923			−0.601003
$300$		0			0
$301$	−4.00000	−	3.46410i	−0.230556	−	0.199667i
$302$		−	8.00000i		−	0.460348i
$303$		0			0
$304$			1.73205i			0.0993399i
$305$		−	12.0000i		−	0.687118i
$306$		0			0
$307$			15.5885i			0.889680i	0.895610	+	0.444840i	$0.146740\pi$
$307$			15.5885i			0.889680i	−0.895610	+	0.444840i	$0.853260\pi$
$308$	5.19615	−	6.00000i	0.296078	−	0.341882i
$309$		0			0
$310$	−9.00000			−0.511166
$311$	24.2487			1.37502			0.687509	−	0.726176i	$-0.258704\pi$
$311$	24.2487			1.37502			0.687509	+	0.726176i	$0.258704\pi$
$312$		0			0
$313$		−	17.3205i		−	0.979013i	−0.872000	−	0.489506i	$-0.837177\pi$
$313$		−	17.3205i		−	0.979013i	0.872000	−	0.489506i	$-0.162823\pi$
$314$	10.3923			0.586472
$315$		0			0
$316$	10.0000			0.562544
$317$		−	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
$320$	1.73205			0.0968246
$321$		0			0
$322$	−6.00000	−	5.19615i	−0.334367	−	0.289570i
$323$		−	12.0000i		−	0.667698i
$324$		0			0
$325$		−	6.92820i		−	0.384308i
$326$			2.00000i			0.110770i
$327$		0			0
$328$		−	12.1244i		−	0.669456i
$329$	6.92820	+	6.00000i	0.381964	+	0.330791i
$330$		0			0
$331$	28.0000			1.53902			0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000			1.53902			0.769510	+	0.638635i	$0.220501\pi$
$332$	−17.3205			−0.950586
$333$		0			0
$334$		−	6.92820i		−	0.379094i
$335$	−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	$0.543484\pi$
$338$			1.00000i			0.0543928i
$339$		0			0
$340$	−12.0000			−0.650791
$341$	15.5885			0.844162
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$			2.00000i			0.107833i
$345$		0			0
$346$		−	5.19615i		−	0.279347i
$347$		−	27.0000i		−	1.44944i	−0.689046	−	0.724718i	$-0.741970\pi$
$347$		−	27.0000i		−	1.44944i	0.689046	−	0.724718i	$-0.258030\pi$
$348$		0			0
$349$			27.7128i			1.48343i	0.670714	+	0.741716i	$0.265988\pi$
$349$			27.7128i			1.48343i	−0.670714	+	0.741716i	$0.734012\pi$
$350$	3.46410	−	4.00000i	0.185164	−	0.213809i
$351$		0			0
$352$	−3.00000			−0.159901
$353$	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
$356$	−5.19615			−0.275396
$357$		0			0
$358$	−12.0000			−0.634220
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	16.0000			0.842105
$362$	−13.8564			−0.728277
$363$		0			0
$364$	6.00000	−	6.92820i	0.314485	−	0.363137i
$365$			6.00000i			0.314054i
$366$		0			0
$367$		−	1.73205i		−	0.0904123i	−0.998978	−	0.0452062i	$-0.985606\pi$
$367$		−	1.73205i		−	0.0904123i	0.998978	−	0.0452062i	$-0.0143945\pi$
$368$			3.00000i			0.156386i
$369$		0			0
$370$		−	12.1244i		−	0.630315i
$371$	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	$0.458680\pi$
$374$	20.7846			1.07475
$375$		0			0
$376$		−	3.46410i		−	0.178647i
$377$	−20.7846			−1.07046
$378$		0			0
$379$	−2.00000			−0.102733			−0.0513665	−	0.998680i	$-0.516358\pi$
$379$	−2.00000			−0.102733			−0.0513665	+	0.998680i	$0.516358\pi$
$380$			3.00000i			0.153897i
$381$		0			0
$382$	21.0000			1.07445
$383$	−13.8564			−0.708029			−0.354015	−	0.935240i	$-0.615184\pi$
$383$	−13.8564			−0.708029			−0.354015	+	0.935240i	$0.615184\pi$
$384$		0			0
$385$	9.00000	−	10.3923i	0.458682	−	0.529641i
$386$		−	14.0000i		−	0.712581i
$387$		0			0
$388$		−	13.8564i		−	0.703452i
$389$			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
$392$	6.92820	−	1.00000i	0.349927	−	0.0505076i
$393$		0			0
$394$	−12.0000			−0.604551
$395$	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	$-0.174655\pi$
$398$	1.73205			0.0868199
$399$		0			0
$400$	−2.00000			−0.100000
$401$			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
$406$	−12.0000	−	10.3923i	−0.595550	−	0.515761i
$407$			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	$-0.0272949\pi$
$410$		−	21.0000i		−	1.03712i
$411$		0			0
$412$			19.0526i			0.938652i
$413$	−6.92820	−	6.00000i	−0.340915	−	0.295241i
$414$		0			0
$415$	−30.0000			−1.47264
$416$	−3.46410			−0.169842
$417$		0			0
$418$		−	5.19615i		−	0.254152i
$419$	−6.92820			−0.338465			−0.169232	−	0.985576i	$-0.554129\pi$
$419$	−6.92820			−0.338465			−0.169232	+	0.985576i	$0.554129\pi$
$420$		0			0
$421$	−13.0000			−0.633581			−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000			−0.633581			−0.316791	+	0.948495i	$0.602605\pi$
$422$			16.0000i			0.778868i
$423$		0			0
$424$	−12.0000			−0.582772
$425$	13.8564			0.672134
$426$		0			0
$427$	−12.0000	+	13.8564i	−0.580721	+	0.670559i
$428$		0			0
$429$		0			0
$430$			3.46410i			0.167054i
$431$		−	21.0000i		−	1.01153i	−0.862670	−	0.505767i	$-0.831209\pi$
$431$		−	21.0000i		−	1.01153i	0.862670	−	0.505767i	$-0.168791\pi$
$432$		0			0
$433$			10.3923i			0.499422i	0.968320	+	0.249711i	$0.0803357\pi$
$433$			10.3923i			0.499422i	−0.968320	+	0.249711i	$0.919664\pi$
$434$	10.3923	+	9.00000i	0.498847	+	0.432014i
$435$		0			0
$436$	−13.0000			−0.622587
$437$	−5.19615			−0.248566
$438$		0			0
$439$		−	10.3923i		−	0.495998i	−0.968760	−	0.247999i	$-0.920227\pi$
$439$		−	10.3923i		−	0.495998i	0.968760	−	0.247999i	$-0.0797729\pi$
$440$	−5.19615			−0.247717
$441$		0			0
$442$	24.0000			1.14156
$443$		−	9.00000i		−	0.427603i	−0.976877	−	0.213801i	$-0.931415\pi$
$443$		−	9.00000i		−	0.427603i	0.976877	−	0.213801i	$-0.0685846\pi$
$444$		0			0
$445$	−9.00000			−0.426641
$446$	15.5885			0.738135
$447$		0			0
$448$	−2.00000	−	1.73205i	−0.0944911	−	0.0818317i
$449$		−	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
$452$			12.0000i			0.564433i
$453$		0			0
$454$		−	20.7846i		−	0.975470i
$455$	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
$460$			5.19615i			0.242272i
$461$	−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	$-0.233125\pi$
$463$	32.0000			1.48717			0.743583	+	0.668644i	$0.233125\pi$
$464$			6.00000i			0.278543i
$465$		0			0
$466$	−24.0000			−1.11178
$467$	−31.1769			−1.44270			−0.721348	−	0.692573i	$-0.756477\pi$
$467$	−31.1769			−1.44270			−0.721348	+	0.692573i	$0.756477\pi$
$468$		0			0
$469$	4.00000	+	3.46410i	0.184703	+	0.159957i
$470$		−	6.00000i		−	0.276759i
$471$		0			0
$472$			3.46410i			0.159448i
$473$		−	6.00000i		−	0.275880i
$474$		0			0
$475$		−	3.46410i		−	0.158944i
$476$	13.8564	+	12.0000i	0.635107	+	0.550019i
$477$		0			0
$478$		0			0
$479$	27.7128			1.26623			0.633115	−	0.774057i	$-0.281776\pi$
$479$	27.7128			1.26623			0.633115	+	0.774057i	$0.281776\pi$
$480$		0			0
$481$			24.2487i			1.10565i
$482$	−17.3205			−0.788928
$483$		0			0
$484$	−2.00000			−0.0909091
$485$		−	24.0000i		−	1.08978i
$486$		0			0
$487$	10.0000			0.453143			0.226572	−	0.973995i	$-0.427248\pi$
$487$	10.0000			0.453143			0.226572	+	0.973995i	$0.427248\pi$
$488$	6.92820			0.313625
$489$		0			0
$490$	12.0000	−	1.73205i	0.542105	−	0.0782461i
$491$			9.00000i			0.406164i	0.979162	+	0.203082i	$0.0650959\pi$
$491$			9.00000i			0.406164i	−0.979162	+	0.203082i	$0.934904\pi$
$492$		0			0
$493$		−	41.5692i		−	1.87218i
$494$		−	6.00000i		−	0.269953i
$495$		0			0
$496$		−	5.19615i		−	0.233314i
$497$	5.19615	−	6.00000i	0.233079	−	0.269137i
$498$		0			0
$499$	10.0000			0.447661			0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000			0.447661			0.223831	+	0.974628i	$0.428144\pi$
$500$	−12.1244			−0.542218
$501$		0			0
$502$			13.8564i			0.618442i
$503$	−24.2487			−1.08120			−0.540598	−	0.841281i	$-0.681802\pi$
$503$	−24.2487			−1.08120			−0.540598	+	0.841281i	$0.681802\pi$
$504$		0			0
$505$		0			0
$506$		−	9.00000i		−	0.400099i
$507$		0			0
$508$	22.0000			0.976092
$509$	−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
$512$			1.00000i			0.0441942i
$513$		0			0
$514$		−	5.19615i		−	0.229192i
$515$			33.0000i			1.45415i
$516$		0			0
$517$			10.3923i			0.457053i
$518$	−12.1244	+	14.0000i	−0.532714	+	0.615125i
$519$		0			0
$520$	−6.00000			−0.263117
$521$	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	$0.336514\pi$
$523$			39.8372i			1.74196i	−0.491320	+	0.870979i	$0.663486\pi$
$524$	−10.3923			−0.453990
$525$		0			0
$526$	−27.0000			−1.17726
$527$			36.0000i			1.56818i
$528$		0			0
$529$	14.0000			0.608696
$530$	−20.7846			−0.902826
$531$		0			0
$532$	3.00000	−	3.46410i	0.130066	−	0.150188i
$533$			42.0000i			1.81922i
$534$		0			0
$535$		0			0
$536$		−	2.00000i		−	0.0863868i
$537$		0			0
$538$			1.73205i			0.0746740i
$539$	−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	$0.680600\pi$
$542$	17.3205			0.743980
$543$		0			0
$544$		−	6.92820i		−	0.297044i
$545$	−22.5167			−0.964508
$546$		0			0
$547$	8.00000			0.342055			0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000			0.342055			0.171028	+	0.985266i	$0.445291\pi$
$548$		−	18.0000i		−	0.768922i
$549$		0			0
$550$	6.00000			0.255841
$551$	−10.3923			−0.442727
$552$		0			0
$553$	−20.0000	−	17.3205i	−0.850487	−	0.736543i
$554$			11.0000i			0.467345i
$555$		0			0
$556$			3.46410i			0.146911i
$557$			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
$560$	−3.46410	−	3.00000i	−0.146385	−	0.126773i
$561$		0			0
$562$	30.0000			1.26547
$563$	−10.3923			−0.437983			−0.218992	−	0.975727i	$-0.570277\pi$
$563$	−10.3923			−0.437983			−0.218992	+	0.975727i	$0.570277\pi$
$564$		0			0
$565$			20.7846i			0.874415i
$566$	−31.1769			−1.31046
$567$		0			0
$568$	−3.00000			−0.125877
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$	10.3923			0.434524
$573$		0			0
$574$	−21.0000	+	24.2487i	−0.876523	+	1.01212i
$575$		−	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	$-0.256346\pi$
$578$			31.0000i			1.28943i
$579$		0			0
$580$			10.3923i			0.431517i
$581$	34.6410	+	30.0000i	1.43715	+	1.24461i
$582$		0			0
$583$	36.0000			1.49097
$584$	−3.46410			−0.143346
$585$		0			0
$586$			6.92820i			0.286201i
$587$	−41.5692			−1.71575			−0.857873	−	0.513862i	$-0.828214\pi$
$587$	−41.5692			−1.71575			−0.857873	+	0.513862i	$0.828214\pi$
$588$		0			0
$589$	9.00000			0.370839
$590$			6.00000i			0.247016i
$591$		0			0
$592$	7.00000			0.287698
$593$	−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
$596$		−	12.0000i		−	0.491539i
$597$		0			0
$598$		−	10.3923i		−	0.424973i
$599$		−	9.00000i		−	0.367730i	−0.982952	−	0.183865i	$-0.941139\pi$
$599$		−	9.00000i		−	0.367730i	0.982952	−	0.183865i	$-0.0588609\pi$
$600$		0			0
$601$		−	10.3923i		−	0.423911i	−0.977279	−	0.211955i	$-0.932017\pi$
$601$		−	10.3923i		−	0.423911i	0.977279	−	0.211955i	$-0.0679832\pi$
$602$	3.46410	−	4.00000i	0.141186	−	0.163028i
$603$		0			0
$604$	8.00000			0.325515
$605$	−3.46410			−0.140836
$606$		0			0
$607$		−	17.3205i		−	0.703018i	−0.936185	−	0.351509i	$-0.885669\pi$
$607$		−	17.3205i		−	0.703018i	0.936185	−	0.351509i	$-0.114331\pi$
$608$	−1.73205			−0.0702439
$609$		0			0
$610$	12.0000			0.485866
$611$			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	$0.625351\pi$
$614$	−15.5885			−0.629099
$615$		0			0
$616$	6.00000	+	5.19615i	0.241747	+	0.209359i
$617$			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	$0.125072\pi$
$619$			19.0526i			0.765787i	−0.923792	+	0.382893i	$0.874928\pi$
$620$		−	9.00000i		−	0.361449i
$621$		0			0
$622$			24.2487i			0.972285i
$623$	10.3923	+	9.00000i	0.416359	+	0.360577i
$624$		0			0
$625$	−11.0000			−0.440000
$626$	17.3205			0.692267
$627$		0			0
$628$			10.3923i			0.414698i
$629$	−48.4974			−1.93372
$630$		0			0
$631$	−34.0000			−1.35352			−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000			−1.35352			−0.676759	+	0.736204i	$0.736616\pi$
$632$			10.0000i			0.397779i
$633$		0			0
$634$	18.0000			0.714871
$635$	38.1051			1.51216
$636$		0			0
$637$	−24.0000	+	3.46410i	−0.950915	+	0.137253i
$638$		−	18.0000i		−	0.712627i
$639$		0			0
$640$			1.73205i			0.0684653i
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		−	5.19615i		−	0.204916i	−0.994737	−	0.102458i	$-0.967329\pi$
$643$		−	5.19615i		−	0.204916i	0.994737	−	0.102458i	$-0.0326708\pi$
$644$	5.19615	−	6.00000i	0.204757	−	0.236433i
$645$		0			0
$646$	12.0000			0.472134
$647$	−10.3923			−0.408564			−0.204282	−	0.978912i	$-0.565486\pi$
$647$	−10.3923			−0.408564			−0.204282	+	0.978912i	$0.565486\pi$
$648$		0			0
$649$		−	10.3923i		−	0.407934i
$650$	6.92820			0.271746
$651$		0			0
$652$	−2.00000			−0.0783260
$653$		−	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
$656$	12.1244			0.473377
$657$		0			0
$658$	−6.00000	+	6.92820i	−0.233904	+	0.270089i
$659$		−	21.0000i		−	0.818044i	−0.912525	−	0.409022i	$-0.865870\pi$
$659$		−	21.0000i		−	0.818044i	0.912525	−	0.409022i	$-0.134130\pi$
$660$		0			0
$661$		−	38.1051i		−	1.48212i	−0.671440	−	0.741059i	$-0.734324\pi$
$661$		−	38.1051i		−	1.48212i	0.671440	−	0.741059i	$-0.265676\pi$
$662$			28.0000i			1.08825i
$663$		0			0
$664$		−	17.3205i		−	0.672166i
$665$	5.19615	−	6.00000i	0.201498	−	0.232670i
$666$		0			0
$667$	−18.0000			−0.696963
$668$	6.92820			0.268060
$669$		0			0
$670$		−	3.46410i		−	0.133830i
$671$	−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	$0.586968\pi$
$674$		−	5.00000i		−	0.192593i
$675$		0			0
$676$	−1.00000			−0.0384615
$677$	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
$680$		−	12.0000i		−	0.460179i
$681$		0			0
$682$			15.5885i			0.596913i
$683$		−	33.0000i		−	1.26271i	−0.775494	−	0.631355i	$-0.782499\pi$
$683$		−	33.0000i		−	1.26271i	0.775494	−	0.631355i	$-0.217501\pi$
$684$		0			0
$685$		−	31.1769i		−	1.19121i
$686$	−15.5885	−	10.0000i	−0.595170	−	0.381802i
$687$		0			0
$688$	−2.00000			−0.0762493
$689$	41.5692			1.58366
$690$		0			0
$691$			10.3923i			0.395342i	0.980268	+	0.197671i	$0.0633378\pi$
$691$			10.3923i			0.395342i	−0.980268	+	0.197671i	$0.936662\pi$
$692$	5.19615			0.197528
$693$		0			0
$694$	27.0000			1.02491
$695$			6.00000i			0.227593i
$696$		0			0
$697$	−84.0000			−3.18173
$698$	−27.7128			−1.04895
$699$		0			0
$700$	4.00000	+	3.46410i	0.151186	+	0.130931i
$701$		−	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
$704$		−	3.00000i		−	0.113067i
$705$		0			0
$706$			19.0526i			0.717053i
$707$		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	$0.494022\pi$
$710$	−5.19615			−0.195008
$711$		0			0
$712$		−	5.19615i		−	0.194734i
$713$	15.5885			0.583792
$714$		0			0
$715$	18.0000			0.673162
$716$		−	12.0000i		−	0.448461i
$717$		0			0
$718$		0			0
$719$	−3.46410			−0.129189			−0.0645946	−	0.997912i	$-0.520575\pi$
$719$	−3.46410			−0.129189			−0.0645946	+	0.997912i	$0.520575\pi$
$720$		0			0
$721$	33.0000	−	38.1051i	1.22898	−	1.41911i
$722$			16.0000i			0.595458i
$723$		0			0
$724$		−	13.8564i		−	0.514969i
$725$		−	12.0000i		−	0.445669i
$726$		0			0
$727$		−	51.9615i		−	1.92715i	−0.267445	−	0.963573i	$-0.586179\pi$
$727$		−	51.9615i		−	1.92715i	0.267445	−	0.963573i	$-0.413821\pi$
$728$	6.92820	+	6.00000i	0.256776	+	0.222375i
$729$		0			0
$730$	−6.00000			−0.222070
$731$	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	$-0.0614728\pi$
$734$	1.73205			0.0639312
$735$		0			0
$736$	−3.00000			−0.110581
$737$			6.00000i			0.221013i
$738$		0			0
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$	12.1244			0.445700
$741$		0			0
$742$	24.0000	+	20.7846i	0.881068	+	0.763027i
$743$		−	27.0000i		−	0.990534i	−0.868741	−	0.495267i	$-0.835070\pi$
$743$		−	27.0000i		−	0.990534i	0.868741	−	0.495267i	$-0.164930\pi$
$744$		0			0
$745$		−	20.7846i		−	0.761489i
$746$			5.00000i			0.183063i
$747$		0			0
$748$			20.7846i			0.759961i
$749$		0			0
$750$		0			0
$751$	−2.00000			−0.0729810			−0.0364905	−	0.999334i	$-0.511618\pi$
$751$	−2.00000			−0.0729810			−0.0364905	+	0.999334i	$0.511618\pi$
$752$	3.46410			0.126323
$753$		0			0
$754$		−	20.7846i		−	0.756931i
$755$	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	$0.511572\pi$
$758$		−	2.00000i		−	0.0726433i
$759$		0			0
$760$	−3.00000			−0.108821
$761$	−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
$764$			21.0000i			0.759753i
$765$		0			0
$766$		−	13.8564i		−	0.500652i
$767$		−	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	$0.758904\pi$
$770$	10.3923	+	9.00000i	0.374513	+	0.324337i
$771$		0			0
$772$	14.0000			0.503871
$773$	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
$776$	13.8564			0.497416
$777$		0			0
$778$	−24.0000			−0.860442
$779$			21.0000i			0.752403i
$780$		0			0
$781$	9.00000			0.322045
$782$	20.7846			0.743256
$783$		0			0
$784$	1.00000	+	6.92820i	0.0357143	+	0.247436i
$785$			18.0000i			0.642448i
$786$		0			0
$787$		−	17.3205i		−	0.617409i	−0.951158	−	0.308705i	$-0.900105\pi$
$787$		−	17.3205i		−	0.617409i	0.951158	−	0.308705i	$-0.0998955\pi$
$788$		−	12.0000i		−	0.427482i
$789$		0			0
$790$			17.3205i			0.616236i
$791$	20.7846	−	24.0000i	0.739016	−	0.853342i
$792$		0			0
$793$	−24.0000			−0.852265
$794$	20.7846			0.737618
$795$		0			0
$796$			1.73205i			0.0613909i
$797$	−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
$800$		−	2.00000i		−	0.0707107i
$801$		0			0
$802$	−18.0000			−0.635602
$803$	10.3923			0.366736
$804$		0			0
$805$	9.00000	−	10.3923i	0.317208	−	0.366281i
$806$			18.0000i			0.634023i
$807$		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	$-0.779507\pi$
$811$		−	36.3731i		−	1.27723i	0.769526	−	0.638616i	$-0.220493\pi$
$812$	10.3923	−	12.0000i	0.364698	−	0.421117i
$813$		0			0
$814$	−21.0000			−0.736050
$815$	−3.46410			−0.121342
$816$		0			0
$817$		−	3.46410i		−	0.121194i
$818$	3.46410			0.121119
$819$		0			0
$820$	21.0000			0.733352
$821$		−	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	$-0.522209\pi$
$823$	−4.00000			−0.139431			−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−19.0526			−0.663727
$825$		0			0
$826$	6.00000	−	6.92820i	0.208767	−	0.241063i
$827$		−	3.00000i		−	0.104320i	−0.998639	−	0.0521601i	$-0.983389\pi$
$827$		−	3.00000i		−	0.104320i	0.998639	−	0.0521601i	$-0.0166106\pi$
$828$		0			0
$829$		−	38.1051i		−	1.32345i	−0.749749	−	0.661723i	$-0.769826\pi$
$829$		−	38.1051i		−	1.32345i	0.749749	−	0.661723i	$-0.230174\pi$
$830$		−	30.0000i		−	1.04132i
$831$		0			0
$832$		−	3.46410i		−	0.120096i
$833$	−6.92820	−	48.0000i	−0.240048	−	1.66310i
$834$		0			0
$835$	12.0000			0.415277
$836$	5.19615			0.179713
$837$		0			0
$838$		−	6.92820i		−	0.239331i
$839$	24.2487			0.837158			0.418579	−	0.908180i	$-0.362528\pi$
$839$	24.2487			0.837158			0.418579	+	0.908180i	$0.362528\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		−	13.0000i		−	0.448010i
$843$		0			0
$844$	−16.0000			−0.550743
$845$	−1.73205			−0.0595844
$846$		0			0
$847$	4.00000	+	3.46410i	0.137442	+	0.119028i
$848$		−	12.0000i		−	0.412082i
$849$		0			0
$850$			13.8564i			0.475271i
$851$			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	$0.923765\pi$
$854$	−13.8564	−	12.0000i	−0.474156	−	0.410632i
$855$		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	$0.167521\pi$
$859$			29.4449i			1.00465i	−0.864680	+	0.502323i	$0.832479\pi$
$860$	−3.46410			−0.118125
$861$		0			0
$862$	21.0000			0.715263
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	9.00000			0.306009
$866$	−10.3923			−0.353145
$867$		0			0
$868$	−9.00000	+	10.3923i	−0.305480	+	0.352738i
$869$		−	30.0000i		−	1.01768i
$870$		0			0
$871$			6.92820i			0.234753i
$872$		−	13.0000i		−	0.440236i
$873$		0			0
$874$		−	5.19615i		−	0.175762i
$875$	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	$0.180082\pi$
$878$	10.3923			0.350723
$879$		0			0
$880$		−	5.19615i		−	0.175162i
$881$	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	$-0.413234\pi$
$883$	16.0000			0.538443			0.269221	+	0.963078i	$0.413234\pi$
$884$			24.0000i			0.807207i
$885$		0			0
$886$	9.00000			0.302361
$887$	24.2487			0.814192			0.407096	−	0.913385i	$-0.366541\pi$
$887$	24.2487			0.814192			0.407096	+	0.913385i	$0.366541\pi$
$888$		0			0
$889$	−44.0000	−	38.1051i	−1.47571	−	1.27800i
$890$		−	9.00000i		−	0.301681i
$891$		0			0
$892$			15.5885i			0.521940i
$893$			6.00000i			0.200782i
$894$		0			0
$895$		−	20.7846i		−	0.694753i
$896$	1.73205	−	2.00000i	0.0578638	−	0.0668153i
$897$		0			0
$898$	12.0000			0.400445
$899$	31.1769			1.03981
$900$		0			0
$901$			83.1384i			2.76974i
$902$	−36.3731			−1.21109
$903$		0			0
$904$	−12.0000			−0.399114
$905$		−	24.0000i		−	0.797787i
$906$		0			0
$907$	−8.00000			−0.265636			−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000			−0.265636			−0.132818	+	0.991140i	$0.542403\pi$
$908$	20.7846			0.689761
$909$		0			0
$910$	12.0000	+	10.3923i	0.397796	+	0.344502i
$911$		−	12.0000i		−	0.397578i	−0.980042	−	0.198789i	$-0.936299\pi$
$911$		−	12.0000i		−	0.397578i	0.980042	−	0.198789i	$-0.0637008\pi$
$912$		0			0
$913$			51.9615i			1.71968i
$914$		−	17.0000i		−	0.562310i
$915$		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	$-0.478985\pi$
$919$	4.00000			0.131948			0.0659739	+	0.997821i	$0.478985\pi$
$920$	−5.19615			−0.171312
$921$		0			0
$922$		−	19.0526i		−	0.627463i
$923$	10.3923			0.342067
$924$		0			0
$925$	−14.0000			−0.460317
$926$			32.0000i			1.05159i
$927$		0			0
$928$	−6.00000			−0.196960
$929$	−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
$932$		−	24.0000i		−	0.786146i
$933$		0			0
$934$		−	31.1769i		−	1.02014i
$935$			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	$0.963900\pi$
$938$	−3.46410	+	4.00000i	−0.113107	+	0.130605i
$939$		0			0
$940$	6.00000			0.195698
$941$	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
$944$	−3.46410			−0.112747
$945$		0			0
$946$	6.00000			0.195077
$947$		−	15.0000i		−	0.487435i	−0.969846	−	0.243717i	$-0.921633\pi$
$947$		−	15.0000i		−	0.487435i	0.969846	−	0.243717i	$-0.0783669\pi$
$948$		0			0
$949$	12.0000			0.389536
$950$	3.46410			0.112390
$951$		0			0
$952$	−12.0000	+	13.8564i	−0.388922	+	0.449089i
$953$			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
$958$			27.7128i			0.895360i
$959$	−31.1769	+	36.0000i	−1.00676	+	1.16250i
$960$		0			0
$961$	4.00000			0.129032
$962$	−24.2487			−0.781810
$963$		0			0
$964$		−	17.3205i		−	0.557856i
$965$	24.2487			0.780594
$966$		0			0
$967$	−16.0000			−0.514525			−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000			−0.514525			−0.257263	+	0.966342i	$0.582821\pi$
$968$		−	2.00000i		−	0.0642824i
$969$		0			0
$970$	24.0000			0.770594
$971$	−3.46410			−0.111168			−0.0555842	−	0.998454i	$-0.517702\pi$
$971$	−3.46410			−0.111168			−0.0555842	+	0.998454i	$0.517702\pi$
$972$		0			0
$973$	6.00000	−	6.92820i	0.192351	−	0.222108i
$974$			10.0000i			0.320421i
$975$		0			0
$976$			6.92820i			0.221766i
$977$			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
$980$	1.73205	+	12.0000i	0.0553283	+	0.383326i
$981$		0			0
$982$	−9.00000			−0.287202
$983$	−3.46410			−0.110488			−0.0552438	−	0.998473i	$-0.517594\pi$
$983$	−3.46410			−0.110488			−0.0552438	+	0.998473i	$0.517594\pi$
$984$		0			0
$985$		−	20.7846i		−	0.662253i
$986$	41.5692			1.32383
$987$		0			0
$988$	6.00000			0.190885
$989$		−	6.00000i		−	0.190789i
$990$		0			0
$991$	−2.00000			−0.0635321			−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000			−0.0635321			−0.0317660	+	0.999495i	$0.510113\pi$
$992$	5.19615			0.164978
$993$		0			0
$994$	6.00000	+	5.19615i	0.190308	+	0.164812i
$995$			3.00000i			0.0951064i
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$			10.0000i			0.316544i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.d.b.377.3	yes	4
3.2	odd	2	inner	378.2.d.b.377.2	yes	4
4.3	odd	2		3024.2.k.e.1889.1		4
7.6	odd	2	inner	378.2.d.b.377.4	yes	4
9.2	odd	6		1134.2.m.c.377.1		4
9.4	even	3		1134.2.m.b.755.1		4
9.5	odd	6		1134.2.m.b.755.2		4
9.7	even	3		1134.2.m.c.377.2		4
12.11	even	2		3024.2.k.e.1889.3		4
21.20	even	2	inner	378.2.d.b.377.1	✓	4
28.27	even	2		3024.2.k.e.1889.4		4
63.13	odd	6		1134.2.m.c.755.1		4
63.20	even	6		1134.2.m.b.377.1		4
63.34	odd	6		1134.2.m.b.377.2		4
63.41	even	6		1134.2.m.c.755.2		4
84.83	odd	2		3024.2.k.e.1889.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.d.b.377.1	✓	4	21.20	even	2	inner
378.2.d.b.377.2	yes	4	3.2	odd	2	inner
378.2.d.b.377.3	yes	4	1.1	even	1	trivial
378.2.d.b.377.4	yes	4	7.6	odd	2	inner
1134.2.m.b.377.1		4	63.20	even	6
1134.2.m.b.377.2		4	63.34	odd	6
1134.2.m.b.755.1		4	9.4	even	3
1134.2.m.b.755.2		4	9.5	odd	6
1134.2.m.c.377.1		4	9.2	odd	6
1134.2.m.c.377.2		4	9.7	even	3
1134.2.m.c.755.1		4	63.13	odd	6
1134.2.m.c.755.2		4	63.41	even	6
3024.2.k.e.1889.1		4	4.3	odd	2
3024.2.k.e.1889.2		4	84.83	odd	2
3024.2.k.e.1889.3		4	12.11	even	2
3024.2.k.e.1889.4		4	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} -1.73205 q^{5} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} -1.73205i q^{10} +3.00000i q^{11} +3.46410i q^{13} +(-1.73205 + 2.00000i) q^{14} +1.00000 q^{16} -6.92820 q^{17} +1.73205i q^{19} +1.73205 q^{20} -3.00000 q^{22} +3.00000i q^{23} -2.00000 q^{25} -3.46410 q^{26} +(-2.00000 - 1.73205i) q^{28} +6.00000i q^{29} -5.19615i q^{31} +1.00000i q^{32} -6.92820i q^{34} +(-3.46410 - 3.00000i) q^{35} +7.00000 q^{37} -1.73205 q^{38} +1.73205i q^{40} +12.1244 q^{41} -2.00000 q^{43} -3.00000i q^{44} -3.00000 q^{46} +3.46410 q^{47} +(1.00000 + 6.92820i) q^{49} -2.00000i q^{50} -3.46410i q^{52} -12.0000i q^{53} -5.19615i q^{55} +(1.73205 - 2.00000i) q^{56} -6.00000 q^{58} -3.46410 q^{59} +6.92820i q^{61} +5.19615 q^{62} -1.00000 q^{64} -6.00000i q^{65} +2.00000 q^{67} +6.92820 q^{68} +(3.00000 - 3.46410i) q^{70} -3.00000i q^{71} -3.46410i q^{73} +7.00000i q^{74} -1.73205i q^{76} +(-5.19615 + 6.00000i) q^{77} -10.0000 q^{79} -1.73205 q^{80} +12.1244i q^{82} +17.3205 q^{83} +12.0000 q^{85} -2.00000i q^{86} +3.00000 q^{88} +5.19615 q^{89} +(-6.00000 + 6.92820i) q^{91} -3.00000i q^{92} +3.46410i q^{94} -3.00000i q^{95} +13.8564i q^{97} +(-6.92820 + 1.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 8 q^{7} + 4 q^{16} - 12 q^{22} - 8 q^{25} - 8 q^{28} + 28 q^{37} - 8 q^{43} - 12 q^{46} + 4 q^{49} - 24 q^{58} - 4 q^{64} + 8 q^{67} + 12 q^{70} - 40 q^{79} + 48 q^{85} + 12 q^{88} - 24 q^{91}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^7 + 4 * q^16 - 12 * q^22 - 8 * q^25 - 8 * q^28 + 28 * q^37 - 8 * q^43 - 12 * q^46 + 4 * q^49 - 24 * q^58 - 4 * q^64 + 8 * q^67 + 12 * q^70 - 40 * q^79 + 48 * q^85 + 12 * q^88 - 24 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

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