LMFDB - Embedded newform 306.2.b.d.271.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [306,2,Mod(271,306)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(306, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("306.271");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 306 = 2 \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	306.b (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:	$2.44342230185$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 34)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	306.271
Dual form			306.2.b.d.271.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.82843i q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +2.82843i q^{5} +1.00000 q^{8} +2.82843i q^{10} +2.82843i q^{11} +2.00000 q^{13} +1.00000 q^{16} +(3.00000 - 2.82843i) q^{17} -4.00000 q^{19} +2.82843i q^{20} +2.82843i q^{22} -5.65685i q^{23} -3.00000 q^{25} +2.00000 q^{26} +2.82843i q^{29} +1.00000 q^{32} +(3.00000 - 2.82843i) q^{34} -8.48528i q^{37} -4.00000 q^{38} +2.82843i q^{40} -5.65685i q^{41} -4.00000 q^{43} +2.82843i q^{44} -5.65685i q^{46} +7.00000 q^{49} -3.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} -8.00000 q^{55} +2.82843i q^{58} -12.0000 q^{59} +8.48528i q^{61} +1.00000 q^{64} +5.65685i q^{65} -4.00000 q^{67} +(3.00000 - 2.82843i) q^{68} -5.65685i q^{71} -8.48528i q^{74} -4.00000 q^{76} -16.9706i q^{79} +2.82843i q^{80} -5.65685i q^{82} +12.0000 q^{83} +(8.00000 + 8.48528i) q^{85} -4.00000 q^{86} +2.82843i q^{88} -6.00000 q^{89} -5.65685i q^{92} -11.3137i q^{95} +16.9706i q^{97} +7.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 4 q^{13} + 2 q^{16} + 6 q^{17} - 8 q^{19} - 6 q^{25} + 4 q^{26} + 2 q^{32} + 6 q^{34} - 8 q^{38} - 8 q^{43} + 14 q^{49} - 6 q^{50} + 4 q^{52} - 12 q^{53} - 16 q^{55} - 24 q^{59} + 2 q^{64} - 8 q^{67} + 6 q^{68} - 8 q^{76} + 24 q^{83} + 16 q^{85} - 8 q^{86} - 12 q^{89} + 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 4 * q^13 + 2 * q^16 + 6 * q^17 - 8 * q^19 - 6 * q^25 + 4 * q^26 + 2 * q^32 + 6 * q^34 - 8 * q^38 - 8 * q^43 + 14 * q^49 - 6 * q^50 + 4 * q^52 - 12 * q^53 - 16 * q^55 - 24 * q^59 + 2 * q^64 - 8 * q^67 + 6 * q^68 - 8 * q^76 + 24 * q^83 + 16 * q^85 - 8 * q^86 - 12 * q^89 + 14 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/306\mathbb{Z}\right)^\times$.

$n$	$37$	$137$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.00000			0.707107
$2$	1.00000			0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$			2.82843i			1.26491i	0.774597	+	0.632456i	$0.217953\pi$
$5$			2.82843i			1.26491i	−0.774597	+	0.632456i	$0.782047\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	1.00000			0.353553
$9$		0			0
$10$			2.82843i			0.894427i
$11$			2.82843i			0.852803i	0.904534	+	0.426401i	$0.140219\pi$
$11$			2.82843i			0.852803i	−0.904534	+	0.426401i	$0.859781\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$	3.00000	−	2.82843i	0.727607	−	0.685994i
$17$	3.00000	−	2.82843i	0.727607	−	0.685994i
$18$		0			0
$19$	−4.00000			−0.917663			−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000			−0.917663			−0.458831	+	0.888523i	$0.651732\pi$
$20$			2.82843i			0.632456i
$21$		0			0
$22$			2.82843i			0.603023i
$23$		−	5.65685i		−	1.17954i	−0.807573	−	0.589768i	$-0.799219\pi$
$23$		−	5.65685i		−	1.17954i	0.807573	−	0.589768i	$-0.200781\pi$
$24$		0			0
$25$	−3.00000			−0.600000
$26$	2.00000			0.392232
$27$		0			0
$28$		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
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	1.00000			0.176777
$33$		0			0
$34$	3.00000	−	2.82843i	0.514496	−	0.485071i
$35$		0			0
$36$		0			0
$37$		−	8.48528i		−	1.39497i	−0.716599	−	0.697486i	$-0.754302\pi$
$37$		−	8.48528i		−	1.39497i	0.716599	−	0.697486i	$-0.245698\pi$
$38$	−4.00000			−0.648886
$39$		0			0
$40$			2.82843i			0.447214i
$41$		−	5.65685i		−	0.883452i	−0.897150	−	0.441726i	$-0.854366\pi$
$41$		−	5.65685i		−	0.883452i	0.897150	−	0.441726i	$-0.145634\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$			2.82843i			0.426401i
$45$		0			0
$46$		−	5.65685i		−	0.834058i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$	−3.00000			−0.424264
$51$		0			0
$52$	2.00000			0.277350
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	−8.00000			−1.07872
$56$		0			0
$57$		0			0
$58$			2.82843i			0.371391i
$59$	−12.0000			−1.56227			−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000			−1.56227			−0.781133	+	0.624364i	$0.785358\pi$
$60$		0			0
$61$			8.48528i			1.08643i	0.839594	+	0.543214i	$0.182793\pi$
$61$			8.48528i			1.08643i	−0.839594	+	0.543214i	$0.817207\pi$
$62$		0			0
$63$		0			0
$64$	1.00000			0.125000
$65$			5.65685i			0.701646i
$66$		0			0
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$	3.00000	−	2.82843i	0.363803	−	0.342997i
$69$		0			0
$70$		0			0
$71$		−	5.65685i		−	0.671345i	−0.941979	−	0.335673i	$-0.891036\pi$
$71$		−	5.65685i		−	0.671345i	0.941979	−	0.335673i	$-0.108964\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		−	8.48528i		−	0.986394i
$75$		0			0
$76$	−4.00000			−0.458831
$77$		0			0
$78$		0			0
$79$		−	16.9706i		−	1.90934i	−0.297670	−	0.954669i	$-0.596210\pi$
$79$		−	16.9706i		−	1.90934i	0.297670	−	0.954669i	$-0.403790\pi$
$80$			2.82843i			0.316228i
$81$		0			0
$82$		−	5.65685i		−	0.624695i
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$		0			0
$85$	8.00000	+	8.48528i	0.867722	+	0.920358i
$86$	−4.00000			−0.431331
$87$		0			0
$88$			2.82843i			0.301511i
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$		0			0
$92$		−	5.65685i		−	0.589768i
$93$		0			0
$94$		0			0
$95$		−	11.3137i		−	1.16076i
$96$		0			0
$97$			16.9706i			1.72310i	0.507673	+	0.861550i	$0.330506\pi$
$97$			16.9706i			1.72310i	−0.507673	+	0.861550i	$0.669494\pi$
$98$	7.00000			0.707107
$99$		0			0
$100$	−3.00000			−0.300000
$101$	6.00000			0.597022			0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000			0.597022			0.298511	+	0.954406i	$0.403510\pi$
$102$		0			0
$103$	8.00000			0.788263			0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000			0.788263			0.394132	+	0.919054i	$0.371045\pi$
$104$	2.00000			0.196116
$105$		0			0
$106$	−6.00000			−0.582772
$107$			2.82843i			0.273434i	0.990610	+	0.136717i	$0.0436552\pi$
$107$			2.82843i			0.273434i	−0.990610	+	0.136717i	$0.956345\pi$
$108$		0			0
$109$		−	8.48528i		−	0.812743i	−0.913708	−	0.406371i	$-0.866794\pi$
$109$		−	8.48528i		−	0.812743i	0.913708	−	0.406371i	$-0.133206\pi$
$110$	−8.00000			−0.762770
$111$		0			0
$112$		0			0
$113$			11.3137i			1.06430i	0.846649	+	0.532152i	$0.178617\pi$
$113$			11.3137i			1.06430i	−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$	16.0000			1.49201
$116$			2.82843i			0.262613i
$117$		0			0
$118$	−12.0000			−1.10469
$119$		0			0
$120$		0			0
$121$	3.00000			0.272727
$122$			8.48528i			0.768221i
$123$		0			0
$124$		0			0
$125$			5.65685i			0.505964i
$126$		0			0
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000			0.0883883
$129$		0			0
$130$			5.65685i			0.496139i
$131$			2.82843i			0.247121i	0.992337	+	0.123560i	$0.0394313\pi$
$131$			2.82843i			0.247121i	−0.992337	+	0.123560i	$0.960569\pi$
$132$		0			0
$133$		0			0
$134$	−4.00000			−0.345547
$135$		0			0
$136$	3.00000	−	2.82843i	0.257248	−	0.242536i
$137$	18.0000			1.53784			0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000			1.53784			0.768922	+	0.639343i	$0.220793\pi$
$138$		0			0
$139$			8.48528i			0.719712i	0.933008	+	0.359856i	$0.117174\pi$
$139$			8.48528i			0.719712i	−0.933008	+	0.359856i	$0.882826\pi$
$140$		0			0
$141$		0			0
$142$		−	5.65685i		−	0.474713i
$143$			5.65685i			0.473050i
$144$		0			0
$145$	−8.00000			−0.664364
$146$		0			0
$147$		0			0
$148$		−	8.48528i		−	0.697486i
$149$	−6.00000			−0.491539			−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000			−0.491539			−0.245770	+	0.969328i	$0.579041\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$	−4.00000			−0.324443
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$		−	16.9706i		−	1.35011i
$159$		0			0
$160$			2.82843i			0.223607i
$161$		0			0
$162$		0			0
$163$		−	8.48528i		−	0.664619i	−0.943170	−	0.332309i	$-0.892172\pi$
$163$		−	8.48528i		−	0.664619i	0.943170	−	0.332309i	$-0.107828\pi$
$164$		−	5.65685i		−	0.441726i
$165$		0			0
$166$	12.0000			0.931381
$167$			11.3137i			0.875481i	0.899101	+	0.437741i	$0.144221\pi$
$167$			11.3137i			0.875481i	−0.899101	+	0.437741i	$0.855779\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$	8.00000	+	8.48528i	0.613572	+	0.650791i
$171$		0			0
$172$	−4.00000			−0.304997
$173$			2.82843i			0.215041i	0.994203	+	0.107521i	$0.0342912\pi$
$173$			2.82843i			0.215041i	−0.994203	+	0.107521i	$0.965709\pi$
$174$		0			0
$175$		0			0
$176$			2.82843i			0.213201i
$177$		0			0
$178$	−6.00000			−0.449719
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$			8.48528i			0.630706i	0.948974	+	0.315353i	$0.102123\pi$
$181$			8.48528i			0.630706i	−0.948974	+	0.315353i	$0.897877\pi$
$182$		0			0
$183$		0			0
$184$		−	5.65685i		−	0.417029i
$185$	24.0000			1.76452
$186$		0			0
$187$	8.00000	+	8.48528i	0.585018	+	0.620505i
$188$		0			0
$189$		0			0
$190$		−	11.3137i		−	0.820783i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$			16.9706i			1.22157i	0.791797	+	0.610784i	$0.209146\pi$
$193$			16.9706i			1.22157i	−0.791797	+	0.610784i	$0.790854\pi$
$194$			16.9706i			1.21842i
$195$		0			0
$196$	7.00000			0.500000
$197$			2.82843i			0.201517i	0.994911	+	0.100759i	$0.0321270\pi$
$197$			2.82843i			0.201517i	−0.994911	+	0.100759i	$0.967873\pi$
$198$		0			0
$199$		−	16.9706i		−	1.20301i	−0.798869	−	0.601506i	$-0.794568\pi$
$199$		−	16.9706i		−	1.20301i	0.798869	−	0.601506i	$-0.205432\pi$
$200$	−3.00000			−0.212132
$201$		0			0
$202$	6.00000			0.422159
$203$		0			0
$204$		0			0
$205$	16.0000			1.11749
$206$	8.00000			0.557386
$207$		0			0
$208$	2.00000			0.138675
$209$		−	11.3137i		−	0.782586i
$210$		0			0
$211$			8.48528i			0.584151i	0.956395	+	0.292075i	$0.0943458\pi$
$211$			8.48528i			0.584151i	−0.956395	+	0.292075i	$0.905654\pi$
$212$	−6.00000			−0.412082
$213$		0			0
$214$			2.82843i			0.193347i
$215$		−	11.3137i		−	0.771589i
$216$		0			0
$217$		0			0
$218$		−	8.48528i		−	0.574696i
$219$		0			0
$220$	−8.00000			−0.539360
$221$	6.00000	−	5.65685i	0.403604	−	0.380521i
$222$		0			0
$223$	−16.0000			−1.07144			−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000			−1.07144			−0.535720	+	0.844396i	$0.679960\pi$
$224$		0			0
$225$		0			0
$226$			11.3137i			0.752577i
$227$			19.7990i			1.31411i	0.753845	+	0.657053i	$0.228197\pi$
$227$			19.7990i			1.31411i	−0.753845	+	0.657053i	$0.771803\pi$
$228$		0			0
$229$	−22.0000			−1.45380			−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000			−1.45380			−0.726900	+	0.686743i	$0.759040\pi$
$230$	16.0000			1.05501
$231$		0			0
$232$			2.82843i			0.185695i
$233$		−	22.6274i		−	1.48237i	−0.671300	−	0.741186i	$-0.734264\pi$
$233$		−	22.6274i		−	1.48237i	0.671300	−	0.741186i	$-0.265736\pi$
$234$		0			0
$235$		0			0
$236$	−12.0000			−0.781133
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$			16.9706i			1.09317i	0.837404	+	0.546585i	$0.184072\pi$
$241$			16.9706i			1.09317i	−0.837404	+	0.546585i	$0.815928\pi$
$242$	3.00000			0.192847
$243$		0			0
$244$			8.48528i			0.543214i
$245$			19.7990i			1.26491i
$246$		0			0
$247$	−8.00000			−0.509028
$248$		0			0
$249$		0			0
$250$			5.65685i			0.357771i
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$	16.0000			1.00591
$254$	−16.0000			−1.00393
$255$		0			0
$256$	1.00000			0.0625000
$257$	−6.00000			−0.374270			−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000			−0.374270			−0.187135	+	0.982334i	$0.559920\pi$
$258$		0			0
$259$		0			0
$260$			5.65685i			0.350823i
$261$		0			0
$262$			2.82843i			0.174741i
$263$	−24.0000			−1.47990			−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000			−1.47990			−0.739952	+	0.672660i	$0.765152\pi$
$264$		0			0
$265$		−	16.9706i		−	1.04249i
$266$		0			0
$267$		0			0
$268$	−4.00000			−0.244339
$269$		−	31.1127i		−	1.89697i	−0.316815	−	0.948487i	$-0.602613\pi$
$269$		−	31.1127i		−	1.89697i	0.316815	−	0.948487i	$-0.397387\pi$
$270$		0			0
$271$	−16.0000			−0.971931			−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000			−0.971931			−0.485965	+	0.873978i	$0.661532\pi$
$272$	3.00000	−	2.82843i	0.181902	−	0.171499i
$273$		0			0
$274$	18.0000			1.08742
$275$		−	8.48528i		−	0.511682i
$276$		0			0
$277$		−	25.4558i		−	1.52949i	−0.644331	−	0.764747i	$-0.722864\pi$
$277$		−	25.4558i		−	1.52949i	0.644331	−	0.764747i	$-0.277136\pi$
$278$			8.48528i			0.508913i
$279$		0			0
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$			8.48528i			0.504398i	0.967675	+	0.252199i	$0.0811537\pi$
$283$			8.48528i			0.504398i	−0.967675	+	0.252199i	$0.918846\pi$
$284$		−	5.65685i		−	0.335673i
$285$		0			0
$286$			5.65685i			0.334497i
$287$		0			0
$288$		0			0
$289$	1.00000	−	16.9706i	0.0588235	−	0.998268i
$290$	−8.00000			−0.469776
$291$		0			0
$292$		0			0
$293$	−6.00000			−0.350524			−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000			−0.350524			−0.175262	+	0.984522i	$0.556077\pi$
$294$		0			0
$295$		−	33.9411i		−	1.97613i
$296$		−	8.48528i		−	0.493197i
$297$		0			0
$298$	−6.00000			−0.347571
$299$		−	11.3137i		−	0.654289i
$300$		0			0
$301$		0			0
$302$	8.00000			0.460348
$303$		0			0
$304$	−4.00000			−0.229416
$305$	−24.0000			−1.37424
$306$		0			0
$307$	20.0000			1.14146			0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000			1.14146			0.570730	+	0.821138i	$0.306660\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			11.3137i			0.641542i	0.947157	+	0.320771i	$0.103942\pi$
$311$			11.3137i			0.641542i	−0.947157	+	0.320771i	$0.896058\pi$
$312$		0			0
$313$		−	16.9706i		−	0.959233i	−0.877478	−	0.479616i	$-0.840776\pi$
$313$		−	16.9706i		−	0.959233i	0.877478	−	0.479616i	$-0.159224\pi$
$314$	14.0000			0.790066
$315$		0			0
$316$		−	16.9706i		−	0.954669i
$317$			2.82843i			0.158860i	0.996840	+	0.0794301i	$0.0253101\pi$
$317$			2.82843i			0.158860i	−0.996840	+	0.0794301i	$0.974690\pi$
$318$		0			0
$319$	−8.00000			−0.447914
$320$			2.82843i			0.158114i
$321$		0			0
$322$		0			0
$323$	−12.0000	+	11.3137i	−0.667698	+	0.629512i
$324$		0			0
$325$	−6.00000			−0.332820
$326$		−	8.48528i		−	0.469956i
$327$		0			0
$328$		−	5.65685i		−	0.312348i
$329$		0			0
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$	12.0000			0.658586
$333$		0			0
$334$			11.3137i			0.619059i
$335$		−	11.3137i		−	0.618134i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−9.00000			−0.489535
$339$		0			0
$340$	8.00000	+	8.48528i	0.433861	+	0.460179i
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−4.00000			−0.215666
$345$		0			0
$346$			2.82843i			0.152057i
$347$		−	14.1421i		−	0.759190i	−0.925153	−	0.379595i	$-0.876063\pi$
$347$		−	14.1421i		−	0.759190i	0.925153	−	0.379595i	$-0.123937\pi$
$348$		0			0
$349$	−34.0000			−1.81998			−0.909989	−	0.414632i	$-0.863910\pi$
$349$	−34.0000			−1.81998			−0.909989	+	0.414632i	$0.863910\pi$
$350$		0			0
$351$		0			0
$352$			2.82843i			0.150756i
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$		0			0
$355$	16.0000			0.849192
$356$	−6.00000			−0.317999
$357$		0			0
$358$	−12.0000			−0.634220
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$			8.48528i			0.445976i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$			16.9706i			0.885856i	0.896557	+	0.442928i	$0.146060\pi$
$367$			16.9706i			0.885856i	−0.896557	+	0.442928i	$0.853940\pi$
$368$		−	5.65685i		−	0.294884i
$369$		0			0
$370$	24.0000			1.24770
$371$		0			0
$372$		0			0
$373$	−22.0000			−1.13912			−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000			−1.13912			−0.569558	+	0.821951i	$0.692886\pi$
$374$	8.00000	+	8.48528i	0.413670	+	0.438763i
$375$		0			0
$376$		0			0
$377$			5.65685i			0.291343i
$378$		0			0
$379$			25.4558i			1.30758i	0.756677	+	0.653789i	$0.226822\pi$
$379$			25.4558i			1.30758i	−0.756677	+	0.653789i	$0.773178\pi$
$380$		−	11.3137i		−	0.580381i
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$			16.9706i			0.863779i
$387$		0			0
$388$			16.9706i			0.861550i
$389$	6.00000			0.304212			0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000			0.304212			0.152106	+	0.988364i	$0.451394\pi$
$390$		0			0
$391$	−16.0000	−	16.9706i	−0.809155	−	0.858238i
$392$	7.00000			0.353553
$393$		0			0
$394$			2.82843i			0.142494i
$395$	48.0000			2.41514
$396$		0			0
$397$			8.48528i			0.425864i	0.977067	+	0.212932i	$0.0683013\pi$
$397$			8.48528i			0.425864i	−0.977067	+	0.212932i	$0.931699\pi$
$398$		−	16.9706i		−	0.850657i
$399$		0			0
$400$	−3.00000			−0.150000
$401$			28.2843i			1.41245i	0.707988	+	0.706225i	$0.249603\pi$
$401$			28.2843i			1.41245i	−0.707988	+	0.706225i	$0.750397\pi$
$402$		0			0
$403$		0			0
$404$	6.00000			0.298511
$405$		0			0
$406$		0			0
$407$	24.0000			1.18964
$408$		0			0
$409$	2.00000			0.0988936			0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000			0.0988936			0.0494468	+	0.998777i	$0.484254\pi$
$410$	16.0000			0.790184
$411$		0			0
$412$	8.00000			0.394132
$413$		0			0
$414$		0			0
$415$			33.9411i			1.66610i
$416$	2.00000			0.0980581
$417$		0			0
$418$		−	11.3137i		−	0.553372i
$419$		−	14.1421i		−	0.690889i	−0.938439	−	0.345444i	$-0.887728\pi$
$419$		−	14.1421i		−	0.690889i	0.938439	−	0.345444i	$-0.112272\pi$
$420$		0			0
$421$	26.0000			1.26716			0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000			1.26716			0.633581	+	0.773676i	$0.281584\pi$
$422$			8.48528i			0.413057i
$423$		0			0
$424$	−6.00000			−0.291386
$425$	−9.00000	+	8.48528i	−0.436564	+	0.411597i
$426$		0			0
$427$		0			0
$428$			2.82843i			0.136717i
$429$		0			0
$430$		−	11.3137i		−	0.545595i
$431$		−	5.65685i		−	0.272481i	−0.990676	−	0.136241i	$-0.956498\pi$
$431$		−	5.65685i		−	0.272481i	0.990676	−	0.136241i	$-0.0435020\pi$
$432$		0			0
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$		0			0
$435$		0			0
$436$		−	8.48528i		−	0.406371i
$437$			22.6274i			1.08242i
$438$		0			0
$439$			16.9706i			0.809961i	0.914325	+	0.404980i	$0.132722\pi$
$439$			16.9706i			0.809961i	−0.914325	+	0.404980i	$0.867278\pi$
$440$	−8.00000			−0.381385
$441$		0			0
$442$	6.00000	−	5.65685i	0.285391	−	0.269069i
$443$	12.0000			0.570137			0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000			0.570137			0.285069	+	0.958507i	$0.407984\pi$
$444$		0			0
$445$		−	16.9706i		−	0.804482i
$446$	−16.0000			−0.757622
$447$		0			0
$448$		0			0
$449$		−	5.65685i		−	0.266963i	−0.991051	−	0.133482i	$-0.957384\pi$
$449$		−	5.65685i		−	0.266963i	0.991051	−	0.133482i	$-0.0426157\pi$
$450$		0			0
$451$	16.0000			0.753411
$452$			11.3137i			0.532152i
$453$		0			0
$454$			19.7990i			0.929213i
$455$		0			0
$456$		0			0
$457$	−10.0000			−0.467780			−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000			−0.467780			−0.233890	+	0.972263i	$0.575146\pi$
$458$	−22.0000			−1.02799
$459$		0			0
$460$	16.0000			0.746004
$461$	18.0000			0.838344			0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000			0.838344			0.419172	+	0.907907i	$0.362320\pi$
$462$		0			0
$463$	32.0000			1.48717			0.743583	−	0.668644i	$-0.233125\pi$
$463$	32.0000			1.48717			0.743583	+	0.668644i	$0.233125\pi$
$464$			2.82843i			0.131306i
$465$		0			0
$466$		−	22.6274i		−	1.04819i
$467$	36.0000			1.66588			0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000			1.66588			0.832941	+	0.553362i	$0.186655\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−12.0000			−0.552345
$473$		−	11.3137i		−	0.520205i
$474$		0			0
$475$	12.0000			0.550598
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	5.65685i		−	0.258468i	−0.991614	−	0.129234i	$-0.958748\pi$
$479$		−	5.65685i		−	0.258468i	0.991614	−	0.129234i	$-0.0412519\pi$
$480$		0			0
$481$		−	16.9706i		−	0.773791i
$482$			16.9706i			0.772988i
$483$		0			0
$484$	3.00000			0.136364
$485$	−48.0000			−2.17957
$486$		0			0
$487$			16.9706i			0.769010i	0.923123	+	0.384505i	$0.125628\pi$
$487$			16.9706i			0.769010i	−0.923123	+	0.384505i	$0.874372\pi$
$488$			8.48528i			0.384111i
$489$		0			0
$490$			19.7990i			0.894427i
$491$	12.0000			0.541552			0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000			0.541552			0.270776	+	0.962642i	$0.412720\pi$
$492$		0			0
$493$	8.00000	+	8.48528i	0.360302	+	0.382158i
$494$	−8.00000			−0.359937
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	25.4558i		−	1.13956i	−0.821797	−	0.569780i	$-0.807028\pi$
$499$		−	25.4558i		−	1.13956i	0.821797	−	0.569780i	$-0.192972\pi$
$500$			5.65685i			0.252982i
$501$		0			0
$502$	12.0000			0.535586
$503$			28.2843i			1.26113i	0.776135	+	0.630567i	$0.217177\pi$
$503$			28.2843i			1.26113i	−0.776135	+	0.630567i	$0.782823\pi$
$504$		0			0
$505$			16.9706i			0.755180i
$506$	16.0000			0.711287
$507$		0			0
$508$	−16.0000			−0.709885
$509$	18.0000			0.797836			0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000			0.797836			0.398918	+	0.916987i	$0.369386\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			0.0441942
$513$		0			0
$514$	−6.00000			−0.264649
$515$			22.6274i			0.997083i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$			5.65685i			0.248069i
$521$		−	5.65685i		−	0.247831i	−0.992293	−	0.123916i	$-0.960455\pi$
$521$		−	5.65685i		−	0.247831i	0.992293	−	0.123916i	$-0.0395452\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$			2.82843i			0.123560i
$525$		0			0
$526$	−24.0000			−1.04645
$527$		0			0
$528$		0			0
$529$	−9.00000			−0.391304
$530$		−	16.9706i		−	0.737154i
$531$		0			0
$532$		0			0
$533$		−	11.3137i		−	0.490051i
$534$		0			0
$535$	−8.00000			−0.345870
$536$	−4.00000			−0.172774
$537$		0			0
$538$		−	31.1127i		−	1.34136i
$539$			19.7990i			0.852803i
$540$		0			0
$541$		−	8.48528i		−	0.364811i	−0.983223	−	0.182405i	$-0.941612\pi$
$541$		−	8.48528i		−	0.364811i	0.983223	−	0.182405i	$-0.0583883\pi$
$542$	−16.0000			−0.687259
$543$		0			0
$544$	3.00000	−	2.82843i	0.128624	−	0.121268i
$545$	24.0000			1.02805
$546$		0			0
$547$			42.4264i			1.81402i	0.421107	+	0.907011i	$0.361642\pi$
$547$			42.4264i			1.81402i	−0.421107	+	0.907011i	$0.638358\pi$
$548$	18.0000			0.768922
$549$		0			0
$550$		−	8.48528i		−	0.361814i
$551$		−	11.3137i		−	0.481980i
$552$		0			0
$553$		0			0
$554$		−	25.4558i		−	1.08152i
$555$		0			0
$556$			8.48528i			0.359856i
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$	−18.0000			−0.759284
$563$	−36.0000			−1.51722			−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000			−1.51722			−0.758610	+	0.651546i	$0.774121\pi$
$564$		0			0
$565$	−32.0000			−1.34625
$566$			8.48528i			0.356663i
$567$		0			0
$568$		−	5.65685i		−	0.237356i
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$			25.4558i			1.06529i	0.846338	+	0.532647i	$0.178803\pi$
$571$			25.4558i			1.06529i	−0.846338	+	0.532647i	$0.821197\pi$
$572$			5.65685i			0.236525i
$573$		0			0
$574$		0			0
$575$			16.9706i			0.707721i
$576$		0			0
$577$	38.0000			1.58196			0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000			1.58196			0.790980	+	0.611842i	$0.209571\pi$
$578$	1.00000	−	16.9706i	0.0415945	−	0.705882i
$579$		0			0
$580$	−8.00000			−0.332182
$581$		0			0
$582$		0			0
$583$		−	16.9706i		−	0.702849i
$584$		0			0
$585$		0			0
$586$	−6.00000			−0.247858
$587$	12.0000			0.495293			0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000			0.495293			0.247647	+	0.968850i	$0.420343\pi$
$588$		0			0
$589$		0			0
$590$		−	33.9411i		−	1.39733i
$591$		0			0
$592$		−	8.48528i		−	0.348743i
$593$	−18.0000			−0.739171			−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000			−0.739171			−0.369586	+	0.929197i	$0.620500\pi$
$594$		0			0
$595$		0			0
$596$	−6.00000			−0.245770
$597$		0			0
$598$		−	11.3137i		−	0.462652i
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	8.00000			0.325515
$605$			8.48528i			0.344976i
$606$		0			0
$607$		−	33.9411i		−	1.37763i	−0.724938	−	0.688814i	$-0.758132\pi$
$607$		−	33.9411i		−	1.37763i	0.724938	−	0.688814i	$-0.241868\pi$
$608$	−4.00000			−0.162221
$609$		0			0
$610$	−24.0000			−0.971732
$611$		0			0
$612$		0			0
$613$	−10.0000			−0.403896			−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000			−0.403896			−0.201948	+	0.979396i	$0.564727\pi$
$614$	20.0000			0.807134
$615$		0			0
$616$		0			0
$617$			11.3137i			0.455473i	0.973723	+	0.227736i	$0.0731324\pi$
$617$			11.3137i			0.455473i	−0.973723	+	0.227736i	$0.926868\pi$
$618$		0			0
$619$		−	8.48528i		−	0.341052i	−0.985353	−	0.170526i	$-0.945453\pi$
$619$		−	8.48528i		−	0.341052i	0.985353	−	0.170526i	$-0.0545467\pi$
$620$		0			0
$621$		0			0
$622$			11.3137i			0.453638i
$623$		0			0
$624$		0			0
$625$	−31.0000			−1.24000
$626$		−	16.9706i		−	0.678280i
$627$		0			0
$628$	14.0000			0.558661
$629$	−24.0000	−	25.4558i	−0.956943	−	1.01499i
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$		−	16.9706i		−	0.675053i
$633$		0			0
$634$			2.82843i			0.112331i
$635$		−	45.2548i		−	1.79588i
$636$		0			0
$637$	14.0000			0.554700
$638$	−8.00000			−0.316723
$639$		0			0
$640$			2.82843i			0.111803i
$641$		−	5.65685i		−	0.223432i	−0.993740	−	0.111716i	$-0.964365\pi$
$641$		−	5.65685i		−	0.223432i	0.993740	−	0.111716i	$-0.0356347\pi$
$642$		0			0
$643$		−	8.48528i		−	0.334627i	−0.985904	−	0.167313i	$-0.946491\pi$
$643$		−	8.48528i		−	0.334627i	0.985904	−	0.167313i	$-0.0535092\pi$
$644$		0			0
$645$		0			0
$646$	−12.0000	+	11.3137i	−0.472134	+	0.445132i
$647$	24.0000			0.943537			0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000			0.943537			0.471769	+	0.881722i	$0.343616\pi$
$648$		0			0
$649$		−	33.9411i		−	1.33231i
$650$	−6.00000			−0.235339
$651$		0			0
$652$		−	8.48528i		−	0.332309i
$653$		−	14.1421i		−	0.553425i	−0.960953	−	0.276712i	$-0.910755\pi$
$653$		−	14.1421i		−	0.553425i	0.960953	−	0.276712i	$-0.0892449\pi$
$654$		0			0
$655$	−8.00000			−0.312586
$656$		−	5.65685i		−	0.220863i
$657$		0			0
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$	−10.0000			−0.388955			−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000			−0.388955			−0.194477	+	0.980907i	$0.562301\pi$
$662$	−4.00000			−0.155464
$663$		0			0
$664$	12.0000			0.465690
$665$		0			0
$666$		0			0
$667$	16.0000			0.619522
$668$			11.3137i			0.437741i
$669$		0			0
$670$		−	11.3137i		−	0.437087i
$671$	−24.0000			−0.926510
$672$		0			0
$673$		−	16.9706i		−	0.654167i	−0.944995	−	0.327084i	$-0.893934\pi$
$673$		−	16.9706i		−	0.654167i	0.944995	−	0.327084i	$-0.106066\pi$
$674$		0			0
$675$		0			0
$676$	−9.00000			−0.346154
$677$		−	14.1421i		−	0.543526i	−0.962364	−	0.271763i	$-0.912393\pi$
$677$		−	14.1421i		−	0.543526i	0.962364	−	0.271763i	$-0.0876068\pi$
$678$		0			0
$679$		0			0
$680$	8.00000	+	8.48528i	0.306786	+	0.325396i
$681$		0			0
$682$		0			0
$683$			19.7990i			0.757587i	0.925481	+	0.378794i	$0.123661\pi$
$683$			19.7990i			0.757587i	−0.925481	+	0.378794i	$0.876339\pi$
$684$		0			0
$685$			50.9117i			1.94524i
$686$		0			0
$687$		0			0
$688$	−4.00000			−0.152499
$689$	−12.0000			−0.457164
$690$		0			0
$691$			8.48528i			0.322795i	0.986889	+	0.161398i	$0.0516002\pi$
$691$			8.48528i			0.322795i	−0.986889	+	0.161398i	$0.948400\pi$
$692$			2.82843i			0.107521i
$693$		0			0
$694$		−	14.1421i		−	0.536828i
$695$	−24.0000			−0.910372
$696$		0			0
$697$	−16.0000	−	16.9706i	−0.606043	−	0.642806i
$698$	−34.0000			−1.28692
$699$		0			0
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$			33.9411i			1.28011i
$704$			2.82843i			0.106600i
$705$		0			0
$706$	6.00000			0.225813
$707$		0			0
$708$		0			0
$709$			8.48528i			0.318671i	0.987224	+	0.159336i	$0.0509352\pi$
$709$			8.48528i			0.318671i	−0.987224	+	0.159336i	$0.949065\pi$
$710$	16.0000			0.600469
$711$		0			0
$712$	−6.00000			−0.224860
$713$		0			0
$714$		0			0
$715$	−16.0000			−0.598366
$716$	−12.0000			−0.448461
$717$		0			0
$718$	−24.0000			−0.895672
$719$		−	39.5980i		−	1.47676i	−0.674387	−	0.738378i	$-0.735592\pi$
$719$		−	39.5980i		−	1.47676i	0.674387	−	0.738378i	$-0.264408\pi$
$720$		0			0
$721$		0			0
$722$	−3.00000			−0.111648
$723$		0			0
$724$			8.48528i			0.315353i
$725$		−	8.48528i		−	0.315135i
$726$		0			0
$727$	8.00000			0.296704			0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000			0.296704			0.148352	+	0.988935i	$0.452603\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12.0000	+	11.3137i	−0.443836	+	0.418453i
$732$		0			0
$733$	14.0000			0.517102			0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000			0.517102			0.258551	+	0.965998i	$0.416755\pi$
$734$			16.9706i			0.626395i
$735$		0			0
$736$		−	5.65685i		−	0.208514i
$737$		−	11.3137i		−	0.416746i
$738$		0			0
$739$	44.0000			1.61857			0.809283	−	0.587419i	$-0.199856\pi$
$739$	44.0000			1.61857			0.809283	+	0.587419i	$0.199856\pi$
$740$	24.0000			0.882258
$741$		0			0
$742$		0			0
$743$			11.3137i			0.415060i	0.978229	+	0.207530i	$0.0665424\pi$
$743$			11.3137i			0.415060i	−0.978229	+	0.207530i	$0.933458\pi$
$744$		0			0
$745$		−	16.9706i		−	0.621753i
$746$	−22.0000			−0.805477
$747$		0			0
$748$	8.00000	+	8.48528i	0.292509	+	0.310253i
$749$		0			0
$750$		0			0
$751$		−	16.9706i		−	0.619265i	−0.950856	−	0.309632i	$-0.899794\pi$
$751$		−	16.9706i		−	0.619265i	0.950856	−	0.309632i	$-0.100206\pi$
$752$		0			0
$753$		0			0
$754$			5.65685i			0.206010i
$755$			22.6274i			0.823496i
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$			25.4558i			0.924598i
$759$		0			0
$760$		−	11.3137i		−	0.410391i
$761$	−30.0000			−1.08750			−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000			−1.08750			−0.543750	+	0.839248i	$0.682996\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−24.0000			−0.866590
$768$		0			0
$769$	−10.0000			−0.360609			−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000			−0.360609			−0.180305	+	0.983611i	$0.557708\pi$
$770$		0			0
$771$		0			0
$772$			16.9706i			0.610784i
$773$	54.0000			1.94225			0.971123	−	0.238581i	$-0.0766824\pi$
$773$	54.0000			1.94225			0.971123	+	0.238581i	$0.0766824\pi$
$774$		0			0
$775$		0			0
$776$			16.9706i			0.609208i
$777$		0			0
$778$	6.00000			0.215110
$779$			22.6274i			0.810711i
$780$		0			0
$781$	16.0000			0.572525
$782$	−16.0000	−	16.9706i	−0.572159	−	0.606866i
$783$		0			0
$784$	7.00000			0.250000
$785$			39.5980i			1.41331i
$786$		0			0
$787$			42.4264i			1.51234i	0.654376	+	0.756169i	$0.272931\pi$
$787$			42.4264i			1.51234i	−0.654376	+	0.756169i	$0.727069\pi$
$788$			2.82843i			0.100759i
$789$		0			0
$790$	48.0000			1.70776
$791$		0			0
$792$		0			0
$793$			16.9706i			0.602642i
$794$			8.48528i			0.301131i
$795$		0			0
$796$		−	16.9706i		−	0.601506i
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$		0			0
$800$	−3.00000			−0.106066
$801$		0			0
$802$			28.2843i			0.998752i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	6.00000			0.211079
$809$		−	39.5980i		−	1.39219i	−0.717949	−	0.696095i	$-0.754919\pi$
$809$		−	39.5980i		−	1.39219i	0.717949	−	0.696095i	$-0.245081\pi$
$810$		0			0
$811$		−	42.4264i		−	1.48979i	−0.667180	−	0.744896i	$-0.732499\pi$
$811$		−	42.4264i		−	1.48979i	0.667180	−	0.744896i	$-0.267501\pi$
$812$		0			0
$813$		0			0
$814$	24.0000			0.841200
$815$	24.0000			0.840683
$816$		0			0
$817$	16.0000			0.559769
$818$	2.00000			0.0699284
$819$		0			0
$820$	16.0000			0.558744
$821$		−	48.0833i		−	1.67812i	−0.544041	−	0.839059i	$-0.683106\pi$
$821$		−	48.0833i		−	1.67812i	0.544041	−	0.839059i	$-0.316894\pi$
$822$		0			0
$823$		−	16.9706i		−	0.591557i	−0.955257	−	0.295778i	$-0.904421\pi$
$823$		−	16.9706i		−	0.591557i	0.955257	−	0.295778i	$-0.0955790\pi$
$824$	8.00000			0.278693
$825$		0			0
$826$		0			0
$827$			36.7696i			1.27860i	0.768956	+	0.639301i	$0.220776\pi$
$827$			36.7696i			1.27860i	−0.768956	+	0.639301i	$0.779224\pi$
$828$		0			0
$829$	−34.0000			−1.18087			−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000			−1.18087			−0.590434	+	0.807086i	$0.701044\pi$
$830$			33.9411i			1.17811i
$831$		0			0
$832$	2.00000			0.0693375
$833$	21.0000	−	19.7990i	0.727607	−	0.685994i
$834$		0			0
$835$	−32.0000			−1.10741
$836$		−	11.3137i		−	0.391293i
$837$		0			0
$838$		−	14.1421i		−	0.488532i
$839$			45.2548i			1.56237i	0.624299	+	0.781185i	$0.285385\pi$
$839$			45.2548i			1.56237i	−0.624299	+	0.781185i	$0.714615\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$	26.0000			0.896019
$843$		0			0
$844$			8.48528i			0.292075i
$845$		−	25.4558i		−	0.875708i
$846$		0			0
$847$		0			0
$848$	−6.00000			−0.206041
$849$		0			0
$850$	−9.00000	+	8.48528i	−0.308697	+	0.291043i
$851$	−48.0000			−1.64542
$852$		0			0
$853$		−	25.4558i		−	0.871592i	−0.900046	−	0.435796i	$-0.856467\pi$
$853$		−	25.4558i		−	0.871592i	0.900046	−	0.435796i	$-0.143533\pi$
$854$		0			0
$855$		0			0
$856$			2.82843i			0.0966736i
$857$			28.2843i			0.966172i	0.875573	+	0.483086i	$0.160484\pi$
$857$			28.2843i			0.966172i	−0.875573	+	0.483086i	$0.839516\pi$
$858$		0			0
$859$	−4.00000			−0.136478			−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000			−0.136478			−0.0682391	+	0.997669i	$0.521738\pi$
$860$		−	11.3137i		−	0.385794i
$861$		0			0
$862$		−	5.65685i		−	0.192673i
$863$	48.0000			1.63394			0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000			1.63394			0.816970	+	0.576681i	$0.195652\pi$
$864$		0			0
$865$	−8.00000			−0.272008
$866$	14.0000			0.475739
$867$		0			0
$868$		0			0
$869$	48.0000			1.62829
$870$		0			0
$871$	−8.00000			−0.271070
$872$		−	8.48528i		−	0.287348i
$873$		0			0
$874$			22.6274i			0.765384i
$875$		0			0
$876$		0			0
$877$		−	8.48528i		−	0.286528i	−0.989685	−	0.143264i	$-0.954240\pi$
$877$		−	8.48528i		−	0.286528i	0.989685	−	0.143264i	$-0.0457597\pi$
$878$			16.9706i			0.572729i
$879$		0			0
$880$	−8.00000			−0.269680
$881$			45.2548i			1.52467i	0.647180	+	0.762337i	$0.275948\pi$
$881$			45.2548i			1.52467i	−0.647180	+	0.762337i	$0.724052\pi$
$882$		0			0
$883$	44.0000			1.48072			0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000			1.48072			0.740359	+	0.672212i	$0.234656\pi$
$884$	6.00000	−	5.65685i	0.201802	−	0.190261i
$885$		0			0
$886$	12.0000			0.403148
$887$		−	39.5980i		−	1.32957i	−0.747035	−	0.664785i	$-0.768523\pi$
$887$		−	39.5980i		−	1.32957i	0.747035	−	0.664785i	$-0.231477\pi$
$888$		0			0
$889$		0			0
$890$		−	16.9706i		−	0.568855i
$891$		0			0
$892$	−16.0000			−0.535720
$893$		0			0
$894$		0			0
$895$		−	33.9411i		−	1.13453i
$896$		0			0
$897$		0			0
$898$		−	5.65685i		−	0.188772i
$899$		0			0
$900$		0			0
$901$	−18.0000	+	16.9706i	−0.599667	+	0.565371i
$902$	16.0000			0.532742
$903$		0			0
$904$			11.3137i			0.376288i
$905$	−24.0000			−0.797787
$906$		0			0
$907$			42.4264i			1.40875i	0.709830	+	0.704373i	$0.248772\pi$
$907$			42.4264i			1.40875i	−0.709830	+	0.704373i	$0.751228\pi$
$908$			19.7990i			0.657053i
$909$		0			0
$910$		0			0
$911$		−	56.5685i		−	1.87420i	−0.349062	−	0.937100i	$-0.613500\pi$
$911$		−	56.5685i		−	1.87420i	0.349062	−	0.937100i	$-0.386500\pi$
$912$		0			0
$913$			33.9411i			1.12329i
$914$	−10.0000			−0.330771
$915$		0			0
$916$	−22.0000			−0.726900
$917$		0			0
$918$		0			0
$919$	−40.0000			−1.31948			−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000			−1.31948			−0.659739	+	0.751495i	$0.729333\pi$
$920$	16.0000			0.527504
$921$		0			0
$922$	18.0000			0.592798
$923$		−	11.3137i		−	0.372395i
$924$		0			0
$925$			25.4558i			0.836983i
$926$	32.0000			1.05159
$927$		0			0
$928$			2.82843i			0.0928477i
$929$		−	5.65685i		−	0.185595i	−0.995685	−	0.0927977i	$-0.970419\pi$
$929$		−	5.65685i		−	0.185595i	0.995685	−	0.0927977i	$-0.0295810\pi$
$930$		0			0
$931$	−28.0000			−0.917663
$932$		−	22.6274i		−	0.741186i
$933$		0			0
$934$	36.0000			1.17796
$935$	−24.0000	+	22.6274i	−0.784884	+	0.739996i
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			36.7696i			1.19865i	0.800505	+	0.599327i	$0.204565\pi$
$941$			36.7696i			1.19865i	−0.800505	+	0.599327i	$0.795435\pi$
$942$		0			0
$943$	−32.0000			−1.04206
$944$	−12.0000			−0.390567
$945$		0			0
$946$		−	11.3137i		−	0.367840i
$947$			2.82843i			0.0919115i	0.998943	+	0.0459558i	$0.0146333\pi$
$947$			2.82843i			0.0919115i	−0.998943	+	0.0459558i	$0.985367\pi$
$948$		0			0
$949$		0			0
$950$	12.0000			0.389331
$951$		0			0
$952$		0			0
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		−	5.65685i		−	0.182765i
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		−	16.9706i		−	0.547153i
$963$		0			0
$964$			16.9706i			0.546585i
$965$	−48.0000			−1.54517
$966$		0			0
$967$	8.00000			0.257263			0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000			0.257263			0.128631	+	0.991692i	$0.458942\pi$
$968$	3.00000			0.0964237
$969$		0			0
$970$	−48.0000			−1.54119
$971$	−12.0000			−0.385098			−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000			−0.385098			−0.192549	+	0.981287i	$0.561675\pi$
$972$		0			0
$973$		0			0
$974$			16.9706i			0.543772i
$975$		0			0
$976$			8.48528i			0.271607i
$977$	−18.0000			−0.575871			−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000			−0.575871			−0.287936	+	0.957650i	$0.592969\pi$
$978$		0			0
$979$		−	16.9706i		−	0.542382i
$980$			19.7990i			0.632456i
$981$		0			0
$982$	12.0000			0.382935
$983$		−	5.65685i		−	0.180426i	−0.995923	−	0.0902128i	$-0.971245\pi$
$983$		−	5.65685i		−	0.180426i	0.995923	−	0.0902128i	$-0.0287547\pi$
$984$		0			0
$985$	−8.00000			−0.254901
$986$	8.00000	+	8.48528i	0.254772	+	0.270226i
$987$		0			0
$988$	−8.00000			−0.254514
$989$			22.6274i			0.719510i
$990$		0			0
$991$		−	50.9117i		−	1.61726i	−0.588315	−	0.808632i	$-0.700209\pi$
$991$		−	50.9117i		−	1.61726i	0.588315	−	0.808632i	$-0.299791\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	48.0000			1.52170
$996$		0			0
$997$		−	59.3970i		−	1.88112i	−0.339626	−	0.940560i	$-0.610301\pi$
$997$		−	59.3970i		−	1.88112i	0.339626	−	0.940560i	$-0.389699\pi$
$998$		−	25.4558i		−	0.805791i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	306.2.b.d.271.2		2
3.2	odd	2		34.2.b.a.33.2	yes	2
4.3	odd	2		2448.2.c.n.577.2		2
12.11	even	2		272.2.b.a.33.1		2
15.2	even	4		850.2.d.i.849.2		4
15.8	even	4		850.2.d.i.849.3		4
15.14	odd	2		850.2.b.f.101.1		2
17.4	even	4		5202.2.a.u.1.2		2
17.13	even	4		5202.2.a.u.1.1		2
17.16	even	2	inner	306.2.b.d.271.1		2
21.20	even	2		1666.2.b.c.883.1		2
24.5	odd	2		1088.2.b.b.577.1		2
24.11	even	2		1088.2.b.a.577.2		2
51.2	odd	8		578.2.c.d.251.1		2
51.5	even	16		578.2.d.g.179.1		8
51.8	odd	8		578.2.c.d.327.1		2
51.11	even	16		578.2.d.g.423.1		8
51.14	even	16		578.2.d.g.399.2		8
51.20	even	16		578.2.d.g.399.1		8
51.23	even	16		578.2.d.g.423.2		8
51.26	odd	8		578.2.c.a.327.1		2
51.29	even	16		578.2.d.g.179.2		8
51.32	odd	8		578.2.c.a.251.1		2
51.38	odd	4		578.2.a.d.1.2		2
51.41	even	16		578.2.d.g.155.2		8
51.44	even	16		578.2.d.g.155.1		8
51.47	odd	4		578.2.a.d.1.1		2
51.50	odd	2		34.2.b.a.33.1	✓	2
68.67	odd	2		2448.2.c.n.577.1		2
204.47	even	4		4624.2.a.s.1.2		2
204.191	even	4		4624.2.a.s.1.1		2
204.203	even	2		272.2.b.a.33.2		2
255.152	even	4		850.2.d.i.849.1		4
255.203	even	4		850.2.d.i.849.4		4
255.254	odd	2		850.2.b.f.101.2		2
357.356	even	2		1666.2.b.c.883.2		2
408.101	odd	2		1088.2.b.b.577.2		2
408.203	even	2		1088.2.b.a.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
34.2.b.a.33.1	✓	2	51.50	odd	2
34.2.b.a.33.2	yes	2	3.2	odd	2
272.2.b.a.33.1		2	12.11	even	2
272.2.b.a.33.2		2	204.203	even	2
306.2.b.d.271.1		2	17.16	even	2	inner
306.2.b.d.271.2		2	1.1	even	1	trivial
578.2.a.d.1.1		2	51.47	odd	4
578.2.a.d.1.2		2	51.38	odd	4
578.2.c.a.251.1		2	51.32	odd	8
578.2.c.a.327.1		2	51.26	odd	8
578.2.c.d.251.1		2	51.2	odd	8
578.2.c.d.327.1		2	51.8	odd	8
578.2.d.g.155.1		8	51.44	even	16
578.2.d.g.155.2		8	51.41	even	16
578.2.d.g.179.1		8	51.5	even	16
578.2.d.g.179.2		8	51.29	even	16
578.2.d.g.399.1		8	51.20	even	16
578.2.d.g.399.2		8	51.14	even	16
578.2.d.g.423.1		8	51.11	even	16
578.2.d.g.423.2		8	51.23	even	16
850.2.b.f.101.1		2	15.14	odd	2
850.2.b.f.101.2		2	255.254	odd	2
850.2.d.i.849.1		4	255.152	even	4
850.2.d.i.849.2		4	15.2	even	4
850.2.d.i.849.3		4	15.8	even	4
850.2.d.i.849.4		4	255.203	even	4
1088.2.b.a.577.1		2	408.203	even	2
1088.2.b.a.577.2		2	24.11	even	2
1088.2.b.b.577.1		2	24.5	odd	2
1088.2.b.b.577.2		2	408.101	odd	2
1666.2.b.c.883.1		2	21.20	even	2
1666.2.b.c.883.2		2	357.356	even	2
2448.2.c.n.577.1		2	68.67	odd	2
2448.2.c.n.577.2		2	4.3	odd	2
4624.2.a.s.1.1		2	204.191	even	4
4624.2.a.s.1.2		2	204.47	even	4
5202.2.a.u.1.1		2	17.13	even	4
5202.2.a.u.1.2		2	17.4	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 306 = 2 \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	306.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.82843i q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.82843i q^{5} +1.00000 q^{8} +2.82843i q^{10} +2.82843i q^{11} +2.00000 q^{13} +1.00000 q^{16} +(3.00000 - 2.82843i) q^{17} -4.00000 q^{19} +2.82843i q^{20} +2.82843i q^{22} -5.65685i q^{23} -3.00000 q^{25} +2.00000 q^{26} +2.82843i q^{29} +1.00000 q^{32} +(3.00000 - 2.82843i) q^{34} -8.48528i q^{37} -4.00000 q^{38} +2.82843i q^{40} -5.65685i q^{41} -4.00000 q^{43} +2.82843i q^{44} -5.65685i q^{46} +7.00000 q^{49} -3.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} -8.00000 q^{55} +2.82843i q^{58} -12.0000 q^{59} +8.48528i q^{61} +1.00000 q^{64} +5.65685i q^{65} -4.00000 q^{67} +(3.00000 - 2.82843i) q^{68} -5.65685i q^{71} -8.48528i q^{74} -4.00000 q^{76} -16.9706i q^{79} +2.82843i q^{80} -5.65685i q^{82} +12.0000 q^{83} +(8.00000 + 8.48528i) q^{85} -4.00000 q^{86} +2.82843i q^{88} -6.00000 q^{89} -5.65685i q^{92} -11.3137i q^{95} +16.9706i q^{97} +7.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 4 q^{13} + 2 q^{16} + 6 q^{17} - 8 q^{19} - 6 q^{25} + 4 q^{26} + 2 q^{32} + 6 q^{34} - 8 q^{38} - 8 q^{43} + 14 q^{49} - 6 q^{50} + 4 q^{52} - 12 q^{53} - 16 q^{55} - 24 q^{59} + 2 q^{64} - 8 q^{67} + 6 q^{68} - 8 q^{76} + 24 q^{83} + 16 q^{85} - 8 q^{86} - 12 q^{89} + 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 4 * q^13 + 2 * q^16 + 6 * q^17 - 8 * q^19 - 6 * q^25 + 4 * q^26 + 2 * q^32 + 6 * q^34 - 8 * q^38 - 8 * q^43 + 14 * q^49 - 6 * q^50 + 4 * q^52 - 12 * q^53 - 16 * q^55 - 24 * q^59 + 2 * q^64 - 8 * q^67 + 6 * q^68 - 8 * q^76 + 24 * q^83 + 16 * q^85 - 8 * q^86 - 12 * q^89 + 14 * q^98

Introduction

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