LMFDB - Embedded newform 2352.2.h.h.2255.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,2,Mod(2255,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2352.2255");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2352.h (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:	$18.7808145554$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2255.2
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2352.2255
Dual form			2352.2.h.h.2255.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 + 1.00000i) q^{3} +1.41421i q^{5} +(1.00000 - 2.82843i) q^{9} +O(q^{10})$	$q+(-1.41421 + 1.00000i) q^{3} +1.41421i q^{5} +(1.00000 - 2.82843i) q^{9} +1.41421 q^{11} -2.00000 q^{13} +(-1.41421 - 2.00000i) q^{15} +7.07107i q^{17} -4.00000i q^{19} -7.07107 q^{23} +3.00000 q^{25} +(1.41421 + 5.00000i) q^{27} +2.82843i q^{29} -6.00000i q^{31} +(-2.00000 + 1.41421i) q^{33} -4.00000 q^{37} +(2.82843 - 2.00000i) q^{39} -1.41421i q^{41} +4.00000i q^{43} +(4.00000 + 1.41421i) q^{45} -8.48528 q^{47} +(-7.07107 - 10.0000i) q^{51} +11.3137i q^{53} +2.00000i q^{55} +(4.00000 + 5.65685i) q^{57} -11.3137 q^{59} +6.00000 q^{61} -2.82843i q^{65} +12.0000i q^{67} +(10.0000 - 7.07107i) q^{69} -7.07107 q^{71} +2.00000 q^{73} +(-4.24264 + 3.00000i) q^{75} -12.0000i q^{79} +(-7.00000 - 5.65685i) q^{81} +14.1421 q^{83} -10.0000 q^{85} +(-2.82843 - 4.00000i) q^{87} -9.89949i q^{89} +(6.00000 + 8.48528i) q^{93} +5.65685 q^{95} -14.0000 q^{97} +(1.41421 - 4.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^9	$ 4 q + 4 q^{9} - 8 q^{13} + 12 q^{25} - 8 q^{33} - 16 q^{37} + 16 q^{45} + 16 q^{57} + 24 q^{61} + 40 q^{69} + 8 q^{73} - 28 q^{81} - 40 q^{85} + 24 q^{93} - 56 q^{97}+O(q^{100}) $ 4 * q + 4 * q^9 - 8 * q^13 + 12 * q^25 - 8 * q^33 - 16 * q^37 + 16 * q^45 + 16 * q^57 + 24 * q^61 + 40 * q^69 + 8 * q^73 - 28 * q^81 - 40 * q^85 + 24 * q^93 - 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times$.

$n$	$785$	$1471$	$1765$	$2257$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	−1.41421	+	1.00000i	−0.816497	+	0.577350i
$3$	−1.41421	+	1.00000i	−0.816497	+	0.577350i
$4$		0			0
$5$			1.41421i			0.632456i	0.948683	+	0.316228i	$0.102416\pi$
$5$			1.41421i			0.632456i	−0.948683	+	0.316228i	$0.897584\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.00000	−	2.82843i	0.333333	−	0.942809i
$10$		0			0
$11$	1.41421			0.426401			0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421			0.426401			0.213201	+	0.977008i	$0.431611\pi$
$12$		0			0
$13$	−2.00000			−0.554700			−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000			−0.554700			−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$	−1.41421	−	2.00000i	−0.365148	−	0.516398i
$16$		0			0
$17$			7.07107i			1.71499i	0.514496	+	0.857493i	$0.327979\pi$
$17$			7.07107i			1.71499i	−0.514496	+	0.857493i	$0.672021\pi$
$18$		0			0
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−7.07107			−1.47442			−0.737210	−	0.675664i	$-0.763857\pi$
$23$	−7.07107			−1.47442			−0.737210	+	0.675664i	$0.763857\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$	1.41421	+	5.00000i	0.272166	+	0.962250i
$28$		0			0
$29$			2.82843i			0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$			2.82843i			0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$		0			0
$31$		−	6.00000i		−	1.07763i	−0.842424	−	0.538816i	$-0.818872\pi$
$31$		−	6.00000i		−	1.07763i	0.842424	−	0.538816i	$-0.181128\pi$
$32$		0			0
$33$	−2.00000	+	1.41421i	−0.348155	+	0.246183i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$		0			0
$39$	2.82843	−	2.00000i	0.452911	−	0.320256i
$40$		0			0
$41$		−	1.41421i		−	0.220863i	−0.993884	−	0.110432i	$-0.964777\pi$
$41$		−	1.41421i		−	0.220863i	0.993884	−	0.110432i	$-0.0352233\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$		0			0
$45$	4.00000	+	1.41421i	0.596285	+	0.210819i
$46$		0			0
$47$	−8.48528			−1.23771			−0.618853	−	0.785507i	$-0.712402\pi$
$47$	−8.48528			−1.23771			−0.618853	+	0.785507i	$0.712402\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−7.07107	−	10.0000i	−0.990148	−	1.40028i
$52$		0			0
$53$			11.3137i			1.55406i	0.629465	+	0.777029i	$0.283274\pi$
$53$			11.3137i			1.55406i	−0.629465	+	0.777029i	$0.716726\pi$
$54$		0			0
$55$			2.00000i			0.269680i
$56$		0			0
$57$	4.00000	+	5.65685i	0.529813	+	0.749269i
$58$		0			0
$59$	−11.3137			−1.47292			−0.736460	−	0.676481i	$-0.763504\pi$
$59$	−11.3137			−1.47292			−0.736460	+	0.676481i	$0.763504\pi$
$60$		0			0
$61$	6.00000			0.768221			0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000			0.768221			0.384111	+	0.923287i	$0.374508\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	2.82843i		−	0.350823i
$66$		0			0
$67$			12.0000i			1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$			12.0000i			1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$		0			0
$69$	10.0000	−	7.07107i	1.20386	−	0.851257i
$70$		0			0
$71$	−7.07107			−0.839181			−0.419591	−	0.907713i	$-0.637826\pi$
$71$	−7.07107			−0.839181			−0.419591	+	0.907713i	$0.637826\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$		0			0
$75$	−4.24264	+	3.00000i	−0.489898	+	0.346410i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	12.0000i		−	1.35011i	−0.737769	−	0.675053i	$-0.764121\pi$
$79$		−	12.0000i		−	1.35011i	0.737769	−	0.675053i	$-0.235879\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$	14.1421			1.55230			0.776151	−	0.630548i	$-0.217170\pi$
$83$	14.1421			1.55230			0.776151	+	0.630548i	$0.217170\pi$
$84$		0			0
$85$	−10.0000			−1.08465
$86$		0			0
$87$	−2.82843	−	4.00000i	−0.303239	−	0.428845i
$88$		0			0
$89$		−	9.89949i		−	1.04934i	−0.851304	−	0.524672i	$-0.824188\pi$
$89$		−	9.89949i		−	1.04934i	0.851304	−	0.524672i	$-0.175812\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	6.00000	+	8.48528i	0.622171	+	0.879883i
$94$		0			0
$95$	5.65685			0.580381
$96$		0			0
$97$	−14.0000			−1.42148			−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000			−1.42148			−0.710742	+	0.703452i	$0.751641\pi$
$98$		0			0
$99$	1.41421	−	4.00000i	0.142134	−	0.402015i
$100$		0			0
$101$		−	9.89949i		−	0.985037i	−0.870302	−	0.492518i	$-0.836076\pi$
$101$		−	9.89949i		−	0.985037i	0.870302	−	0.492518i	$-0.163924\pi$
$102$		0			0
$103$		−	10.0000i		−	0.985329i	−0.870219	−	0.492665i	$-0.836023\pi$
$103$		−	10.0000i		−	0.985329i	0.870219	−	0.492665i	$-0.163977\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−18.3848			−1.77732			−0.888662	−	0.458563i	$-0.848364\pi$
$107$	−18.3848			−1.77732			−0.888662	+	0.458563i	$0.848364\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$	5.65685	−	4.00000i	0.536925	−	0.379663i
$112$		0			0
$113$		−	5.65685i		−	0.532152i	−0.963952	−	0.266076i	$-0.914273\pi$
$113$		−	5.65685i		−	0.532152i	0.963952	−	0.266076i	$-0.0857272\pi$
$114$		0			0
$115$		−	10.0000i		−	0.932505i
$116$		0			0
$117$	−2.00000	+	5.65685i	−0.184900	+	0.522976i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−9.00000			−0.818182
$122$		0			0
$123$	1.41421	+	2.00000i	0.127515	+	0.180334i
$124$		0			0
$125$			11.3137i			1.01193i
$126$		0			0
$127$		−	20.0000i		−	1.77471i	−0.461084	−	0.887357i	$-0.652539\pi$
$127$		−	20.0000i		−	1.77471i	0.461084	−	0.887357i	$-0.347461\pi$
$128$		0			0
$129$	−4.00000	−	5.65685i	−0.352180	−	0.498058i
$130$		0			0
$131$	−2.82843			−0.247121			−0.123560	−	0.992337i	$-0.539431\pi$
$131$	−2.82843			−0.247121			−0.123560	+	0.992337i	$0.539431\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−7.07107	+	2.00000i	−0.608581	+	0.172133i
$136$		0			0
$137$		−	14.1421i		−	1.20824i	−0.796892	−	0.604122i	$-0.793524\pi$
$137$		−	14.1421i		−	1.20824i	0.796892	−	0.604122i	$-0.206476\pi$
$138$		0			0
$139$			2.00000i			0.169638i	0.996396	+	0.0848189i	$0.0270312\pi$
$139$			2.00000i			0.169638i	−0.996396	+	0.0848189i	$0.972969\pi$
$140$		0			0
$141$	12.0000	−	8.48528i	1.01058	−	0.714590i
$142$		0			0
$143$	−2.82843			−0.236525
$144$		0			0
$145$	−4.00000			−0.332182
$146$		0			0
$147$		0			0
$148$		0			0
$149$			5.65685i			0.463428i	0.972784	+	0.231714i	$0.0744333\pi$
$149$			5.65685i			0.463428i	−0.972784	+	0.231714i	$0.925567\pi$
$150$		0			0
$151$		−	8.00000i		−	0.651031i	−0.945537	−	0.325515i	$-0.894462\pi$
$151$		−	8.00000i		−	0.651031i	0.945537	−	0.325515i	$-0.105538\pi$
$152$		0			0
$153$	20.0000	+	7.07107i	1.61690	+	0.571662i
$154$		0			0
$155$	8.48528			0.681554
$156$		0			0
$157$	−18.0000			−1.43656			−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000			−1.43656			−0.718278	+	0.695756i	$0.755069\pi$
$158$		0			0
$159$	−11.3137	−	16.0000i	−0.897235	−	1.26888i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	16.0000i		−	1.25322i	−0.779334	−	0.626608i	$-0.784443\pi$
$163$		−	16.0000i		−	1.25322i	0.779334	−	0.626608i	$-0.215557\pi$
$164$		0			0
$165$	−2.00000	−	2.82843i	−0.155700	−	0.220193i
$166$		0			0
$167$	−2.82843			−0.218870			−0.109435	−	0.993994i	$-0.534904\pi$
$167$	−2.82843			−0.218870			−0.109435	+	0.993994i	$0.534904\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	−11.3137	−	4.00000i	−0.865181	−	0.305888i
$172$		0			0
$173$			12.7279i			0.967686i	0.875155	+	0.483843i	$0.160759\pi$
$173$			12.7279i			0.967686i	−0.875155	+	0.483843i	$0.839241\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	16.0000	−	11.3137i	1.20263	−	0.850390i
$178$		0			0
$179$	9.89949			0.739923			0.369961	−	0.929047i	$-0.379371\pi$
$179$	9.89949			0.739923			0.369961	+	0.929047i	$0.379371\pi$
$180$		0			0
$181$	−18.0000			−1.33793			−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000			−1.33793			−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$	−8.48528	+	6.00000i	−0.627250	+	0.443533i
$184$		0			0
$185$		−	5.65685i		−	0.415900i
$186$		0			0
$187$			10.0000i			0.731272i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.24264			−0.306987			−0.153493	−	0.988150i	$-0.549052\pi$
$191$	−4.24264			−0.306987			−0.153493	+	0.988150i	$0.549052\pi$
$192$		0			0
$193$	12.0000			0.863779			0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000			0.863779			0.431889	+	0.901927i	$0.357847\pi$
$194$		0			0
$195$	2.82843	+	4.00000i	0.202548	+	0.286446i
$196$		0			0
$197$		−	16.9706i		−	1.20910i	−0.796566	−	0.604551i	$-0.793352\pi$
$197$		−	16.9706i		−	1.20910i	0.796566	−	0.604551i	$-0.206648\pi$
$198$		0			0
$199$			16.0000i			1.13421i	0.823646	+	0.567105i	$0.191937\pi$
$199$			16.0000i			1.13421i	−0.823646	+	0.567105i	$0.808063\pi$
$200$		0			0
$201$	−12.0000	−	16.9706i	−0.846415	−	1.19701i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	2.00000			0.139686
$206$		0			0
$207$	−7.07107	+	20.0000i	−0.491473	+	1.39010i
$208$		0			0
$209$		−	5.65685i		−	0.391293i
$210$		0			0
$211$			20.0000i			1.37686i	0.725304	+	0.688428i	$0.241699\pi$
$211$			20.0000i			1.37686i	−0.725304	+	0.688428i	$0.758301\pi$
$212$		0			0
$213$	10.0000	−	7.07107i	0.685189	−	0.484502i
$214$		0			0
$215$	−5.65685			−0.385794
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−2.82843	+	2.00000i	−0.191127	+	0.135147i
$220$		0			0
$221$		−	14.1421i		−	0.951303i
$222$		0			0
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$		0			0
$225$	3.00000	−	8.48528i	0.200000	−	0.565685i
$226$		0			0
$227$	−5.65685			−0.375459			−0.187729	−	0.982221i	$-0.560113\pi$
$227$	−5.65685			−0.375459			−0.187729	+	0.982221i	$0.560113\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			14.1421i			0.926482i	0.886232	+	0.463241i	$0.153314\pi$
$233$			14.1421i			0.926482i	−0.886232	+	0.463241i	$0.846686\pi$
$234$		0			0
$235$		−	12.0000i		−	0.782794i
$236$		0			0
$237$	12.0000	+	16.9706i	0.779484	+	1.10236i
$238$		0			0
$239$	−24.0416			−1.55512			−0.777562	−	0.628806i	$-0.783544\pi$
$239$	−24.0416			−1.55512			−0.777562	+	0.628806i	$0.783544\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$		0			0
$243$	15.5563	+	1.00000i	0.997940	+	0.0641500i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			8.00000i			0.509028i
$248$		0			0
$249$	−20.0000	+	14.1421i	−1.26745	+	0.896221i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−10.0000			−0.628695
$254$		0			0
$255$	14.1421	−	10.0000i	0.885615	−	0.626224i
$256$		0			0
$257$		−	1.41421i		−	0.0882162i	−0.999027	−	0.0441081i	$-0.985955\pi$
$257$		−	1.41421i		−	0.0882162i	0.999027	−	0.0441081i	$-0.0140446\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	8.00000	+	2.82843i	0.495188	+	0.175075i
$262$		0			0
$263$	−9.89949			−0.610429			−0.305215	−	0.952284i	$-0.598728\pi$
$263$	−9.89949			−0.610429			−0.305215	+	0.952284i	$0.598728\pi$
$264$		0			0
$265$	−16.0000			−0.982872
$266$		0			0
$267$	9.89949	+	14.0000i	0.605839	+	0.856786i
$268$		0			0
$269$		−	1.41421i		−	0.0862261i	−0.999070	−	0.0431131i	$-0.986272\pi$
$269$		−	1.41421i		−	0.0862261i	0.999070	−	0.0431131i	$-0.0137276\pi$
$270$		0			0
$271$			10.0000i			0.607457i	0.952759	+	0.303728i	$0.0982315\pi$
$271$			10.0000i			0.607457i	−0.952759	+	0.303728i	$0.901768\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.24264			0.255841
$276$		0			0
$277$	8.00000			0.480673			0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000			0.480673			0.240337	+	0.970690i	$0.422742\pi$
$278$		0			0
$279$	−16.9706	−	6.00000i	−1.01600	−	0.359211i
$280$		0			0
$281$			31.1127i			1.85603i	0.372545	+	0.928014i	$0.378485\pi$
$281$			31.1127i			1.85603i	−0.372545	+	0.928014i	$0.621515\pi$
$282$		0			0
$283$		−	12.0000i		−	0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$		−	12.0000i		−	0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$		0			0
$285$	−8.00000	+	5.65685i	−0.473879	+	0.335083i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−33.0000			−1.94118
$290$		0			0
$291$	19.7990	−	14.0000i	1.16064	−	0.820695i
$292$		0			0
$293$		−	15.5563i		−	0.908812i	−0.890795	−	0.454406i	$-0.849852\pi$
$293$		−	15.5563i		−	0.908812i	0.890795	−	0.454406i	$-0.150148\pi$
$294$		0			0
$295$		−	16.0000i		−	0.931556i
$296$		0			0
$297$	2.00000	+	7.07107i	0.116052	+	0.410305i
$298$		0			0
$299$	14.1421			0.817861
$300$		0			0
$301$		0			0
$302$		0			0
$303$	9.89949	+	14.0000i	0.568711	+	0.804279i
$304$		0			0
$305$			8.48528i			0.485866i
$306$		0			0
$307$			12.0000i			0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$			12.0000i			0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$		0			0
$309$	10.0000	+	14.1421i	0.568880	+	0.804518i
$310$		0			0
$311$	8.48528			0.481156			0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528			0.481156			0.240578	+	0.970630i	$0.422663\pi$
$312$		0			0
$313$	30.0000			1.69570			0.847850	−	0.530236i	$-0.177897\pi$
$313$	30.0000			1.69570			0.847850	+	0.530236i	$0.177897\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	16.9706i		−	0.953162i	−0.879131	−	0.476581i	$-0.841876\pi$
$317$		−	16.9706i		−	0.953162i	0.879131	−	0.476581i	$-0.158124\pi$
$318$		0			0
$319$			4.00000i			0.223957i
$320$		0			0
$321$	26.0000	−	18.3848i	1.45118	−	1.02614i
$322$		0			0
$323$	28.2843			1.57378
$324$		0			0
$325$	−6.00000			−0.332820
$326$		0			0
$327$	−2.82843	+	2.00000i	−0.156412	+	0.110600i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			8.00000i			0.439720i	0.975531	+	0.219860i	$0.0705600\pi$
$331$			8.00000i			0.439720i	−0.975531	+	0.219860i	$0.929440\pi$
$332$		0			0
$333$	−4.00000	+	11.3137i	−0.219199	+	0.619987i
$334$		0			0
$335$	−16.9706			−0.927201
$336$		0			0
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$		0			0
$339$	5.65685	+	8.00000i	0.307238	+	0.434500i
$340$		0			0
$341$		−	8.48528i		−	0.459504i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	10.0000	+	14.1421i	0.538382	+	0.761387i
$346$		0			0
$347$	−12.7279			−0.683271			−0.341635	−	0.939833i	$-0.610981\pi$
$347$	−12.7279			−0.683271			−0.341635	+	0.939833i	$0.610981\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$	−2.82843	−	10.0000i	−0.150970	−	0.533761i
$352$		0			0
$353$		−	1.41421i		−	0.0752710i	−0.999292	−	0.0376355i	$-0.988017\pi$
$353$		−	1.41421i		−	0.0752710i	0.999292	−	0.0376355i	$-0.0119826\pi$
$354$		0			0
$355$		−	10.0000i		−	0.530745i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	32.5269			1.71670			0.858352	−	0.513061i	$-0.171488\pi$
$359$	32.5269			1.71670			0.858352	+	0.513061i	$0.171488\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$	12.7279	−	9.00000i	0.668043	−	0.472377i
$364$		0			0
$365$			2.82843i			0.148047i
$366$		0			0
$367$			32.0000i			1.67039i	0.549957	+	0.835193i	$0.314644\pi$
$367$			32.0000i			1.67039i	−0.549957	+	0.835193i	$0.685356\pi$
$368$		0			0
$369$	−4.00000	−	1.41421i	−0.208232	−	0.0736210i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$		0			0
$375$	−11.3137	−	16.0000i	−0.584237	−	0.826236i
$376$		0			0
$377$		−	5.65685i		−	0.291343i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	20.0000	+	28.2843i	1.02463	+	1.44905i
$382$		0			0
$383$	−5.65685			−0.289052			−0.144526	−	0.989501i	$-0.546166\pi$
$383$	−5.65685			−0.289052			−0.144526	+	0.989501i	$0.546166\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	11.3137	+	4.00000i	0.575108	+	0.203331i
$388$		0			0
$389$			14.1421i			0.717035i	0.933523	+	0.358517i	$0.116718\pi$
$389$			14.1421i			0.717035i	−0.933523	+	0.358517i	$0.883282\pi$
$390$		0			0
$391$		−	50.0000i		−	2.52861i
$392$		0			0
$393$	4.00000	−	2.82843i	0.201773	−	0.142675i
$394$		0			0
$395$	16.9706			0.853882
$396$		0			0
$397$	−22.0000			−1.10415			−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000			−1.10415			−0.552074	+	0.833795i	$0.686163\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	19.7990i		−	0.988714i	−0.869259	−	0.494357i	$-0.835403\pi$
$401$		−	19.7990i		−	0.988714i	0.869259	−	0.494357i	$-0.164597\pi$
$402$		0			0
$403$			12.0000i			0.597763i
$404$		0			0
$405$	8.00000	−	9.89949i	0.397523	−	0.491910i
$406$		0			0
$407$	−5.65685			−0.280400
$408$		0			0
$409$	18.0000			0.890043			0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000			0.890043			0.445021	+	0.895520i	$0.353196\pi$
$410$		0			0
$411$	14.1421	+	20.0000i	0.697580	+	0.986527i
$412$		0			0
$413$		0			0
$414$		0			0
$415$			20.0000i			0.981761i
$416$		0			0
$417$	−2.00000	−	2.82843i	−0.0979404	−	0.138509i
$418$		0			0
$419$	5.65685			0.276355			0.138178	−	0.990407i	$-0.455875\pi$
$419$	5.65685			0.276355			0.138178	+	0.990407i	$0.455875\pi$
$420$		0			0
$421$	−16.0000			−0.779792			−0.389896	−	0.920859i	$-0.627489\pi$
$421$	−16.0000			−0.779792			−0.389896	+	0.920859i	$0.627489\pi$
$422$		0			0
$423$	−8.48528	+	24.0000i	−0.412568	+	1.16692i
$424$		0			0
$425$			21.2132i			1.02899i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	4.00000	−	2.82843i	0.193122	−	0.136558i
$430$		0			0
$431$	−9.89949			−0.476842			−0.238421	−	0.971162i	$-0.576630\pi$
$431$	−9.89949			−0.476842			−0.238421	+	0.971162i	$0.576630\pi$
$432$		0			0
$433$	−34.0000			−1.63394			−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000			−1.63394			−0.816968	+	0.576683i	$0.804347\pi$
$434$		0			0
$435$	5.65685	−	4.00000i	0.271225	−	0.191785i
$436$		0			0
$437$			28.2843i			1.35302i
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	12.7279			0.604722			0.302361	−	0.953194i	$-0.402225\pi$
$443$	12.7279			0.604722			0.302361	+	0.953194i	$0.402225\pi$
$444$		0			0
$445$	14.0000			0.663664
$446$		0			0
$447$	−5.65685	−	8.00000i	−0.267560	−	0.378387i
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		−	2.00000i		−	0.0941763i
$452$		0			0
$453$	8.00000	+	11.3137i	0.375873	+	0.531564i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	6.00000			0.280668			0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000			0.280668			0.140334	+	0.990104i	$0.455182\pi$
$458$		0			0
$459$	−35.3553	+	10.0000i	−1.65025	+	0.466760i
$460$		0			0
$461$			15.5563i			0.724531i	0.932075	+	0.362266i	$0.117997\pi$
$461$			15.5563i			0.724531i	−0.932075	+	0.362266i	$0.882003\pi$
$462$		0			0
$463$			16.0000i			0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$			16.0000i			0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$		0			0
$465$	−12.0000	+	8.48528i	−0.556487	+	0.393496i
$466$		0			0
$467$	28.2843			1.30884			0.654420	−	0.756131i	$-0.272913\pi$
$467$	28.2843			1.30884			0.654420	+	0.756131i	$0.272913\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	25.4558	−	18.0000i	1.17294	−	0.829396i
$472$		0			0
$473$			5.65685i			0.260102i
$474$		0			0
$475$		−	12.0000i		−	0.550598i
$476$		0			0
$477$	32.0000	+	11.3137i	1.46518	+	0.518019i
$478$		0			0
$479$	36.7696			1.68004			0.840022	−	0.542553i	$-0.182542\pi$
$479$	36.7696			1.68004			0.840022	+	0.542553i	$0.182542\pi$
$480$		0			0
$481$	8.00000			0.364769
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	19.7990i		−	0.899026i
$486$		0			0
$487$		−	16.0000i		−	0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$		−	16.0000i		−	0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$		0			0
$489$	16.0000	+	22.6274i	0.723545	+	1.02325i
$490$		0			0
$491$	35.3553			1.59556			0.797782	−	0.602946i	$-0.206007\pi$
$491$	35.3553			1.59556			0.797782	+	0.602946i	$0.206007\pi$
$492$		0			0
$493$	−20.0000			−0.900755
$494$		0			0
$495$	5.65685	+	2.00000i	0.254257	+	0.0898933i
$496$		0			0
$497$		0			0
$498$		0			0
$499$			40.0000i			1.79065i	0.445418	+	0.895323i	$0.353055\pi$
$499$			40.0000i			1.79065i	−0.445418	+	0.895323i	$0.646945\pi$
$500$		0			0
$501$	4.00000	−	2.82843i	0.178707	−	0.126365i
$502$		0			0
$503$	−16.9706			−0.756680			−0.378340	−	0.925667i	$-0.623505\pi$
$503$	−16.9706			−0.756680			−0.378340	+	0.925667i	$0.623505\pi$
$504$		0			0
$505$	14.0000			0.622992
$506$		0			0
$507$	12.7279	−	9.00000i	0.565267	−	0.399704i
$508$		0			0
$509$		−	18.3848i		−	0.814891i	−0.913230	−	0.407445i	$-0.866420\pi$
$509$		−	18.3848i		−	0.814891i	0.913230	−	0.407445i	$-0.133580\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	20.0000	−	5.65685i	0.883022	−	0.249756i
$514$		0			0
$515$	14.1421			0.623177
$516$		0			0
$517$	−12.0000			−0.527759
$518$		0			0
$519$	−12.7279	−	18.0000i	−0.558694	−	0.790112i
$520$		0			0
$521$			15.5563i			0.681536i	0.940147	+	0.340768i	$0.110687\pi$
$521$			15.5563i			0.681536i	−0.940147	+	0.340768i	$0.889313\pi$
$522$		0			0
$523$			38.0000i			1.66162i	0.556553	+	0.830812i	$0.312124\pi$
$523$			38.0000i			1.66162i	−0.556553	+	0.830812i	$0.687876\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	42.4264			1.84812
$528$		0			0
$529$	27.0000			1.17391
$530$		0			0
$531$	−11.3137	+	32.0000i	−0.490973	+	1.38868i
$532$		0			0
$533$			2.82843i			0.122513i
$534$		0			0
$535$		−	26.0000i		−	1.12408i
$536$		0			0
$537$	−14.0000	+	9.89949i	−0.604145	+	0.427195i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−8.00000			−0.343947			−0.171973	−	0.985102i	$-0.555014\pi$
$541$	−8.00000			−0.343947			−0.171973	+	0.985102i	$0.555014\pi$
$542$		0			0
$543$	25.4558	−	18.0000i	1.09241	−	0.772454i
$544$		0			0
$545$			2.82843i			0.121157i
$546$		0			0
$547$			20.0000i			0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$			20.0000i			0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$		0			0
$549$	6.00000	−	16.9706i	0.256074	−	0.724286i
$550$		0			0
$551$	11.3137			0.481980
$552$		0			0
$553$		0			0
$554$		0			0
$555$	5.65685	+	8.00000i	0.240120	+	0.339581i
$556$		0			0
$557$		−	28.2843i		−	1.19844i	−0.800583	−	0.599222i	$-0.795477\pi$
$557$		−	28.2843i		−	1.19844i	0.800583	−	0.599222i	$-0.204523\pi$
$558$		0			0
$559$		−	8.00000i		−	0.338364i
$560$		0			0
$561$	−10.0000	−	14.1421i	−0.422200	−	0.597081i
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$	8.00000			0.336563
$566$		0			0
$567$		0			0
$568$		0			0
$569$			25.4558i			1.06716i	0.845748	+	0.533582i	$0.179155\pi$
$569$			25.4558i			1.06716i	−0.845748	+	0.533582i	$0.820845\pi$
$570$		0			0
$571$			4.00000i			0.167395i	0.996491	+	0.0836974i	$0.0266729\pi$
$571$			4.00000i			0.167395i	−0.996491	+	0.0836974i	$0.973327\pi$
$572$		0			0
$573$	6.00000	−	4.24264i	0.250654	−	0.177239i
$574$		0			0
$575$	−21.2132			−0.884652
$576$		0			0
$577$	10.0000			0.416305			0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000			0.416305			0.208153	+	0.978096i	$0.433255\pi$
$578$		0			0
$579$	−16.9706	+	12.0000i	−0.705273	+	0.498703i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			16.0000i			0.662652i
$584$		0			0
$585$	−8.00000	−	2.82843i	−0.330759	−	0.116941i
$586$		0			0
$587$	−22.6274			−0.933933			−0.466967	−	0.884275i	$-0.654653\pi$
$587$	−22.6274			−0.933933			−0.466967	+	0.884275i	$0.654653\pi$
$588$		0			0
$589$	−24.0000			−0.988903
$590$		0			0
$591$	16.9706	+	24.0000i	0.698076	+	0.987228i
$592$		0			0
$593$			35.3553i			1.45187i	0.687763	+	0.725935i	$0.258593\pi$
$593$			35.3553i			1.45187i	−0.687763	+	0.725935i	$0.741407\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−16.0000	−	22.6274i	−0.654836	−	0.926079i
$598$		0			0
$599$	−29.6985			−1.21345			−0.606724	−	0.794913i	$-0.707517\pi$
$599$	−29.6985			−1.21345			−0.606724	+	0.794913i	$0.707517\pi$
$600$		0			0
$601$	−2.00000			−0.0815817			−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000			−0.0815817			−0.0407909	+	0.999168i	$0.512988\pi$
$602$		0			0
$603$	33.9411	+	12.0000i	1.38219	+	0.488678i
$604$		0			0
$605$		−	12.7279i		−	0.517464i
$606$		0			0
$607$		−	16.0000i		−	0.649420i	−0.945814	−	0.324710i	$-0.894733\pi$
$607$		−	16.0000i		−	0.649420i	0.945814	−	0.324710i	$-0.105267\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	16.9706			0.686555
$612$		0			0
$613$	−38.0000			−1.53481			−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000			−1.53481			−0.767403	+	0.641165i	$0.778451\pi$
$614$		0			0
$615$	−2.82843	+	2.00000i	−0.114053	+	0.0806478i
$616$		0			0
$617$		−	36.7696i		−	1.48029i	−0.672449	−	0.740143i	$-0.734758\pi$
$617$		−	36.7696i		−	1.48029i	0.672449	−	0.740143i	$-0.265242\pi$
$618$		0			0
$619$			14.0000i			0.562708i	0.959604	+	0.281354i	$0.0907834\pi$
$619$			14.0000i			0.562708i	−0.959604	+	0.281354i	$0.909217\pi$
$620$		0			0
$621$	−10.0000	−	35.3553i	−0.401286	−	1.41876i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$	5.65685	+	8.00000i	0.225913	+	0.319489i
$628$		0			0
$629$		−	28.2843i		−	1.12777i
$630$		0			0
$631$			24.0000i			0.955425i	0.878516	+	0.477712i	$0.158534\pi$
$631$			24.0000i			0.955425i	−0.878516	+	0.477712i	$0.841466\pi$
$632$		0			0
$633$	−20.0000	−	28.2843i	−0.794929	−	1.12420i
$634$		0			0
$635$	28.2843			1.12243
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−7.07107	+	20.0000i	−0.279727	+	0.791188i
$640$		0			0
$641$			19.7990i			0.782013i	0.920388	+	0.391007i	$0.127873\pi$
$641$			19.7990i			0.782013i	−0.920388	+	0.391007i	$0.872127\pi$
$642$		0			0
$643$			4.00000i			0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$			4.00000i			0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$		0			0
$645$	8.00000	−	5.65685i	0.315000	−	0.222738i
$646$		0			0
$647$	−2.82843			−0.111197			−0.0555985	−	0.998453i	$-0.517707\pi$
$647$	−2.82843			−0.111197			−0.0555985	+	0.998453i	$0.517707\pi$
$648$		0			0
$649$	−16.0000			−0.628055
$650$		0			0
$651$		0			0
$652$		0			0
$653$			48.0833i			1.88164i	0.338902	+	0.940822i	$0.389945\pi$
$653$			48.0833i			1.88164i	−0.338902	+	0.940822i	$0.610055\pi$
$654$		0			0
$655$		−	4.00000i		−	0.156293i
$656$		0			0
$657$	2.00000	−	5.65685i	0.0780274	−	0.220695i
$658$		0			0
$659$	−24.0416			−0.936529			−0.468264	−	0.883588i	$-0.655121\pi$
$659$	−24.0416			−0.936529			−0.468264	+	0.883588i	$0.655121\pi$
$660$		0			0
$661$	−18.0000			−0.700119			−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000			−0.700119			−0.350059	+	0.936727i	$0.613839\pi$
$662$		0			0
$663$	14.1421	+	20.0000i	0.549235	+	0.776736i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	20.0000i		−	0.774403i
$668$		0			0
$669$	−16.0000	−	22.6274i	−0.618596	−	0.874826i
$670$		0			0
$671$	8.48528			0.327571
$672$		0			0
$673$	−8.00000			−0.308377			−0.154189	−	0.988041i	$-0.549276\pi$
$673$	−8.00000			−0.308377			−0.154189	+	0.988041i	$0.549276\pi$
$674$		0			0
$675$	4.24264	+	15.0000i	0.163299	+	0.577350i
$676$		0			0
$677$		−	29.6985i		−	1.14141i	−0.821157	−	0.570703i	$-0.806671\pi$
$677$		−	29.6985i		−	1.14141i	0.821157	−	0.570703i	$-0.193329\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	8.00000	−	5.65685i	0.306561	−	0.216771i
$682$		0			0
$683$	−9.89949			−0.378794			−0.189397	−	0.981901i	$-0.560653\pi$
$683$	−9.89949			−0.378794			−0.189397	+	0.981901i	$0.560653\pi$
$684$		0			0
$685$	20.0000			0.764161
$686$		0			0
$687$	−14.1421	+	10.0000i	−0.539556	+	0.381524i
$688$		0			0
$689$		−	22.6274i		−	0.862036i
$690$		0			0
$691$		−	6.00000i		−	0.228251i	−0.993466	−	0.114125i	$-0.963593\pi$
$691$		−	6.00000i		−	0.228251i	0.993466	−	0.114125i	$-0.0364066\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.82843			−0.107288
$696$		0			0
$697$	10.0000			0.378777
$698$		0			0
$699$	−14.1421	−	20.0000i	−0.534905	−	0.756469i
$700$		0			0
$701$			31.1127i			1.17511i	0.809184	+	0.587555i	$0.199909\pi$
$701$			31.1127i			1.17511i	−0.809184	+	0.587555i	$0.800091\pi$
$702$		0			0
$703$			16.0000i			0.603451i
$704$		0			0
$705$	12.0000	+	16.9706i	0.451946	+	0.639148i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	6.00000			0.225335			0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000			0.225335			0.112667	+	0.993633i	$0.464061\pi$
$710$		0			0
$711$	−33.9411	−	12.0000i	−1.27289	−	0.450035i
$712$		0			0
$713$			42.4264i			1.58888i
$714$		0			0
$715$		−	4.00000i		−	0.149592i
$716$		0			0
$717$	34.0000	−	24.0416i	1.26975	−	0.897851i
$718$		0			0
$719$	−28.2843			−1.05483			−0.527413	−	0.849609i	$-0.676838\pi$
$719$	−28.2843			−1.05483			−0.527413	+	0.849609i	$0.676838\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−2.82843	+	2.00000i	−0.105190	+	0.0743808i
$724$		0			0
$725$			8.48528i			0.315135i
$726$		0			0
$727$			2.00000i			0.0741759i	0.999312	+	0.0370879i	$0.0118082\pi$
$727$			2.00000i			0.0741759i	−0.999312	+	0.0370879i	$0.988192\pi$
$728$		0			0
$729$	−23.0000	+	14.1421i	−0.851852	+	0.523783i
$730$		0			0
$731$	−28.2843			−1.04613
$732$		0			0
$733$	26.0000			0.960332			0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000			0.960332			0.480166	+	0.877178i	$0.340576\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			16.9706i			0.625119i
$738$		0			0
$739$		−	8.00000i		−	0.294285i	−0.989115	−	0.147142i	$-0.952992\pi$
$739$		−	8.00000i		−	0.294285i	0.989115	−	0.147142i	$-0.0470076\pi$
$740$		0			0
$741$	−8.00000	−	11.3137i	−0.293887	−	0.415619i
$742$		0			0
$743$	18.3848			0.674472			0.337236	−	0.941420i	$-0.390508\pi$
$743$	18.3848			0.674472			0.337236	+	0.941420i	$0.390508\pi$
$744$		0			0
$745$	−8.00000			−0.293097
$746$		0			0
$747$	14.1421	−	40.0000i	0.517434	−	1.46352i
$748$		0			0
$749$		0			0
$750$		0			0
$751$			28.0000i			1.02173i	0.859660	+	0.510867i	$0.170676\pi$
$751$			28.0000i			1.02173i	−0.859660	+	0.510867i	$0.829324\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	11.3137			0.411748
$756$		0			0
$757$	6.00000			0.218074			0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000			0.218074			0.109037	+	0.994038i	$0.465223\pi$
$758$		0			0
$759$	14.1421	−	10.0000i	0.513327	−	0.362977i
$760$		0			0
$761$		−	35.3553i		−	1.28163i	−0.767695	−	0.640815i	$-0.778597\pi$
$761$		−	35.3553i		−	1.28163i	0.767695	−	0.640815i	$-0.221403\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−10.0000	+	28.2843i	−0.361551	+	1.02262i
$766$		0			0
$767$	22.6274			0.817029
$768$		0			0
$769$	10.0000			0.360609			0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000			0.360609			0.180305	+	0.983611i	$0.442292\pi$
$770$		0			0
$771$	1.41421	+	2.00000i	0.0509317	+	0.0720282i
$772$		0			0
$773$		−	41.0122i		−	1.47511i	−0.675289	−	0.737553i	$-0.735981\pi$
$773$		−	41.0122i		−	1.47511i	0.675289	−	0.737553i	$-0.264019\pi$
$774$		0			0
$775$		−	18.0000i		−	0.646579i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.65685			−0.202678
$780$		0			0
$781$	−10.0000			−0.357828
$782$		0			0
$783$	−14.1421	+	4.00000i	−0.505399	+	0.142948i
$784$		0			0
$785$		−	25.4558i		−	0.908558i
$786$		0			0
$787$			30.0000i			1.06938i	0.845047	+	0.534692i	$0.179572\pi$
$787$			30.0000i			1.06938i	−0.845047	+	0.534692i	$0.820428\pi$
$788$		0			0
$789$	14.0000	−	9.89949i	0.498413	−	0.352431i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−12.0000			−0.426132
$794$		0			0
$795$	22.6274	−	16.0000i	0.802512	−	0.567462i
$796$		0			0
$797$			15.5563i			0.551034i	0.961296	+	0.275517i	$0.0888491\pi$
$797$			15.5563i			0.551034i	−0.961296	+	0.275517i	$0.911151\pi$
$798$		0			0
$799$		−	60.0000i		−	2.12265i
$800$		0			0
$801$	−28.0000	−	9.89949i	−0.989331	−	0.349781i
$802$		0			0
$803$	2.82843			0.0998130
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1.41421	+	2.00000i	0.0497827	+	0.0704033i
$808$		0			0
$809$		−	22.6274i		−	0.795538i	−0.917486	−	0.397769i	$-0.869785\pi$
$809$		−	22.6274i		−	0.795538i	0.917486	−	0.397769i	$-0.130215\pi$
$810$		0			0
$811$			30.0000i			1.05344i	0.850038	+	0.526721i	$0.176579\pi$
$811$			30.0000i			1.05344i	−0.850038	+	0.526721i	$0.823421\pi$
$812$		0			0
$813$	−10.0000	−	14.1421i	−0.350715	−	0.495986i
$814$		0			0
$815$	22.6274			0.792604
$816$		0			0
$817$	16.0000			0.559769
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	22.6274i		−	0.789702i	−0.918745	−	0.394851i	$-0.870796\pi$
$821$		−	22.6274i		−	0.789702i	0.918745	−	0.394851i	$-0.129204\pi$
$822$		0			0
$823$			12.0000i			0.418294i	0.977884	+	0.209147i	$0.0670687\pi$
$823$			12.0000i			0.418294i	−0.977884	+	0.209147i	$0.932931\pi$
$824$		0			0
$825$	−6.00000	+	4.24264i	−0.208893	+	0.147710i
$826$		0			0
$827$	12.7279			0.442593			0.221297	−	0.975207i	$-0.428971\pi$
$827$	12.7279			0.442593			0.221297	+	0.975207i	$0.428971\pi$
$828$		0			0
$829$	10.0000			0.347314			0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000			0.347314			0.173657	+	0.984806i	$0.444442\pi$
$830$		0			0
$831$	−11.3137	+	8.00000i	−0.392468	+	0.277517i
$832$		0			0
$833$		0			0
$834$		0			0
$835$		−	4.00000i		−	0.138426i
$836$		0			0
$837$	30.0000	−	8.48528i	1.03695	−	0.293294i
$838$		0			0
$839$	−31.1127			−1.07413			−0.537065	−	0.843541i	$-0.680467\pi$
$839$	−31.1127			−1.07413			−0.537065	+	0.843541i	$0.680467\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$	−31.1127	−	44.0000i	−1.07158	−	1.51544i
$844$		0			0
$845$		−	12.7279i		−	0.437854i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	12.0000	+	16.9706i	0.411839	+	0.582428i
$850$		0			0
$851$	28.2843			0.969572
$852$		0			0
$853$	26.0000			0.890223			0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000			0.890223			0.445112	+	0.895475i	$0.353164\pi$
$854$		0			0
$855$	5.65685	−	16.0000i	0.193460	−	0.547188i
$856$		0			0
$857$		−	1.41421i		−	0.0483086i	−0.999708	−	0.0241543i	$-0.992311\pi$
$857$		−	1.41421i		−	0.0483086i	0.999708	−	0.0241543i	$-0.00768930\pi$
$858$		0			0
$859$			20.0000i			0.682391i	0.939992	+	0.341196i	$0.110832\pi$
$859$			20.0000i			0.682391i	−0.939992	+	0.341196i	$0.889168\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−24.0416			−0.818387			−0.409193	−	0.912448i	$-0.634190\pi$
$863$	−24.0416			−0.818387			−0.409193	+	0.912448i	$0.634190\pi$
$864$		0			0
$865$	−18.0000			−0.612018
$866$		0			0
$867$	46.6690	−	33.0000i	1.58496	−	1.12074i
$868$		0			0
$869$		−	16.9706i		−	0.575687i
$870$		0			0
$871$		−	24.0000i		−	0.813209i
$872$		0			0
$873$	−14.0000	+	39.5980i	−0.473828	+	1.34019i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	34.0000			1.14810			0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000			1.14810			0.574049	+	0.818821i	$0.305372\pi$
$878$		0			0
$879$	15.5563	+	22.0000i	0.524703	+	0.742042i
$880$		0			0
$881$		−	12.7279i		−	0.428815i	−0.976744	−	0.214407i	$-0.931218\pi$
$881$		−	12.7279i		−	0.428815i	0.976744	−	0.214407i	$-0.0687820\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	16.0000	+	22.6274i	0.537834	+	0.760612i
$886$		0			0
$887$	2.82843			0.0949693			0.0474846	−	0.998872i	$-0.484879\pi$
$887$	2.82843			0.0949693			0.0474846	+	0.998872i	$0.484879\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−9.89949	−	8.00000i	−0.331646	−	0.268010i
$892$		0			0
$893$			33.9411i			1.13580i
$894$		0			0
$895$			14.0000i			0.467968i
$896$		0			0
$897$	−20.0000	+	14.1421i	−0.667781	+	0.472192i
$898$		0			0
$899$	16.9706			0.566000
$900$		0			0
$901$	−80.0000			−2.66519
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	25.4558i		−	0.846181i
$906$		0			0
$907$		−	40.0000i		−	1.32818i	−0.747653	−	0.664089i	$-0.768820\pi$
$907$		−	40.0000i		−	1.32818i	0.747653	−	0.664089i	$-0.231180\pi$
$908$		0			0
$909$	−28.0000	−	9.89949i	−0.928701	−	0.328346i
$910$		0			0
$911$	9.89949			0.327985			0.163992	−	0.986462i	$-0.447563\pi$
$911$	9.89949			0.327985			0.163992	+	0.986462i	$0.447563\pi$
$912$		0			0
$913$	20.0000			0.661903
$914$		0			0
$915$	−8.48528	−	12.0000i	−0.280515	−	0.396708i
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	28.0000i		−	0.923635i	−0.886975	−	0.461817i	$-0.847198\pi$
$919$		−	28.0000i		−	0.923635i	0.886975	−	0.461817i	$-0.152802\pi$
$920$		0			0
$921$	−12.0000	−	16.9706i	−0.395413	−	0.559199i
$922$		0			0
$923$	14.1421			0.465494
$924$		0			0
$925$	−12.0000			−0.394558
$926$		0			0
$927$	−28.2843	−	10.0000i	−0.928977	−	0.328443i
$928$		0			0
$929$			9.89949i			0.324792i	0.986726	+	0.162396i	$0.0519222\pi$
$929$			9.89949i			0.324792i	−0.986726	+	0.162396i	$0.948078\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−12.0000	+	8.48528i	−0.392862	+	0.277796i
$934$		0			0
$935$	−14.1421			−0.462497
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		0			0
$939$	−42.4264	+	30.0000i	−1.38453	+	0.979013i
$940$		0			0
$941$			29.6985i			0.968143i	0.875028	+	0.484071i	$0.160843\pi$
$941$			29.6985i			0.968143i	−0.875028	+	0.484071i	$0.839157\pi$
$942$		0			0
$943$			10.0000i			0.325645i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−21.2132			−0.689336			−0.344668	−	0.938725i	$-0.612009\pi$
$947$	−21.2132			−0.689336			−0.344668	+	0.938725i	$0.612009\pi$
$948$		0			0
$949$	−4.00000			−0.129845
$950$		0			0
$951$	16.9706	+	24.0000i	0.550308	+	0.778253i
$952$		0			0
$953$		−	33.9411i		−	1.09946i	−0.835342	−	0.549730i	$-0.814730\pi$
$953$		−	33.9411i		−	1.09946i	0.835342	−	0.549730i	$-0.185270\pi$
$954$		0			0
$955$		−	6.00000i		−	0.194155i
$956$		0			0
$957$	−4.00000	−	5.65685i	−0.129302	−	0.182860i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−5.00000			−0.161290
$962$		0			0
$963$	−18.3848	+	52.0000i	−0.592441	+	1.67568i
$964$		0			0
$965$			16.9706i			0.546302i
$966$		0			0
$967$			4.00000i			0.128631i	0.997930	+	0.0643157i	$0.0204865\pi$
$967$			4.00000i			0.128631i	−0.997930	+	0.0643157i	$0.979514\pi$
$968$		0			0
$969$	−40.0000	+	28.2843i	−1.28499	+	0.908622i
$970$		0			0
$971$	−19.7990			−0.635380			−0.317690	−	0.948195i	$-0.602907\pi$
$971$	−19.7990			−0.635380			−0.317690	+	0.948195i	$0.602907\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	8.48528	−	6.00000i	0.271746	−	0.192154i
$976$		0			0
$977$			14.1421i			0.452447i	0.974075	+	0.226224i	$0.0726380\pi$
$977$			14.1421i			0.452447i	−0.974075	+	0.226224i	$0.927362\pi$
$978$		0			0
$979$		−	14.0000i		−	0.447442i
$980$		0			0
$981$	2.00000	−	5.65685i	0.0638551	−	0.180609i
$982$		0			0
$983$	−28.2843			−0.902128			−0.451064	−	0.892492i	$-0.648955\pi$
$983$	−28.2843			−0.902128			−0.451064	+	0.892492i	$0.648955\pi$
$984$		0			0
$985$	24.0000			0.764704
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	28.2843i		−	0.899388i
$990$		0			0
$991$			44.0000i			1.39771i	0.715265	+	0.698853i	$0.246306\pi$
$991$			44.0000i			1.39771i	−0.715265	+	0.698853i	$0.753694\pi$
$992$		0			0
$993$	−8.00000	−	11.3137i	−0.253872	−	0.359030i
$994$		0			0
$995$	−22.6274			−0.717337
$996$		0			0
$997$	−46.0000			−1.45683			−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000			−1.45683			−0.728417	+	0.685134i	$0.759744\pi$
$998$		0			0
$999$	−5.65685	−	20.0000i	−0.178975	−	0.632772i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.2.h.h.2255.2		4
3.2	odd	2	inner	2352.2.h.h.2255.4		4
4.3	odd	2	inner	2352.2.h.h.2255.3		4
7.6	odd	2		336.2.h.a.239.3	yes	4
12.11	even	2	inner	2352.2.h.h.2255.1		4
21.20	even	2		336.2.h.a.239.1	✓	4
28.27	even	2		336.2.h.a.239.2	yes	4
56.13	odd	2		1344.2.h.c.575.2		4
56.27	even	2		1344.2.h.c.575.3		4
84.83	odd	2		336.2.h.a.239.4	yes	4
168.83	odd	2		1344.2.h.c.575.1		4
168.125	even	2		1344.2.h.c.575.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.2.h.a.239.1	✓	4	21.20	even	2
336.2.h.a.239.2	yes	4	28.27	even	2
336.2.h.a.239.3	yes	4	7.6	odd	2
336.2.h.a.239.4	yes	4	84.83	odd	2
1344.2.h.c.575.1		4	168.83	odd	2
1344.2.h.c.575.2		4	56.13	odd	2
1344.2.h.c.575.3		4	56.27	even	2
1344.2.h.c.575.4		4	168.125	even	2
2352.2.h.h.2255.1		4	12.11	even	2	inner
2352.2.h.h.2255.2		4	1.1	even	1	trivial
2352.2.h.h.2255.3		4	4.3	odd	2	inner
2352.2.h.h.2255.4		4	3.2	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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2352.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 + 1.00000i) q^{3} +1.41421i q^{5} +(1.00000 - 2.82843i) q^{9} +O(q^{10})\)	\(q+(-1.41421 + 1.00000i) q^{3} +1.41421i q^{5} +(1.00000 - 2.82843i) q^{9} +1.41421 q^{11} -2.00000 q^{13} +(-1.41421 - 2.00000i) q^{15} +7.07107i q^{17} -4.00000i q^{19} -7.07107 q^{23} +3.00000 q^{25} +(1.41421 + 5.00000i) q^{27} +2.82843i q^{29} -6.00000i q^{31} +(-2.00000 + 1.41421i) q^{33} -4.00000 q^{37} +(2.82843 - 2.00000i) q^{39} -1.41421i q^{41} +4.00000i q^{43} +(4.00000 + 1.41421i) q^{45} -8.48528 q^{47} +(-7.07107 - 10.0000i) q^{51} +11.3137i q^{53} +2.00000i q^{55} +(4.00000 + 5.65685i) q^{57} -11.3137 q^{59} +6.00000 q^{61} -2.82843i q^{65} +12.0000i q^{67} +(10.0000 - 7.07107i) q^{69} -7.07107 q^{71} +2.00000 q^{73} +(-4.24264 + 3.00000i) q^{75} -12.0000i q^{79} +(-7.00000 - 5.65685i) q^{81} +14.1421 q^{83} -10.0000 q^{85} +(-2.82843 - 4.00000i) q^{87} -9.89949i q^{89} +(6.00000 + 8.48528i) q^{93} +5.65685 q^{95} -14.0000 q^{97} +(1.41421 - 4.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^9	\( 4 q + 4 q^{9} - 8 q^{13} + 12 q^{25} - 8 q^{33} - 16 q^{37} + 16 q^{45} + 16 q^{57} + 24 q^{61} + 40 q^{69} + 8 q^{73} - 28 q^{81} - 40 q^{85} + 24 q^{93} - 56 q^{97}+O(q^{100}) \) 4 * q + 4 * q^9 - 8 * q^13 + 12 * q^25 - 8 * q^33 - 16 * q^37 + 16 * q^45 + 16 * q^57 + 24 * q^61 + 40 * q^69 + 8 * q^73 - 28 * q^81 - 40 * q^85 + 24 * q^93 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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