LMFDB - Embedded newform 912.2.k.a.607.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,2,Mod(607,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("912.607");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	912.k (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:	$7.28235666434$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			607.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	912.607
Dual form			912.2.k.a.607.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.00000 q^{5} +2.82843i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.00000 q^{5} +2.82843i q^{7} +1.00000 q^{9} -1.41421i q^{11} -2.82843i q^{13} +2.00000 q^{15} -2.00000 q^{17} +(-1.00000 + 4.24264i) q^{19} -2.82843i q^{21} -1.41421i q^{23} -1.00000 q^{25} -1.00000 q^{27} -7.07107i q^{29} +6.00000 q^{31} +1.41421i q^{33} -5.65685i q^{35} -11.3137i q^{37} +2.82843i q^{39} -4.24264i q^{41} -2.00000 q^{45} -7.07107i q^{47} -1.00000 q^{49} +2.00000 q^{51} -9.89949i q^{53} +2.82843i q^{55} +(1.00000 - 4.24264i) q^{57} -8.00000 q^{59} -8.00000 q^{61} +2.82843i q^{63} +5.65685i q^{65} -2.00000 q^{67} +1.41421i q^{69} -8.00000 q^{71} +1.00000 q^{75} +4.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -12.7279i q^{83} +4.00000 q^{85} +7.07107i q^{87} +9.89949i q^{89} +8.00000 q^{91} -6.00000 q^{93} +(2.00000 - 8.48528i) q^{95} +2.82843i q^{97} -1.41421i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} + 4 q^{15} - 4 q^{17} - 2 q^{19} - 2 q^{25} - 2 q^{27} + 12 q^{31} - 4 q^{45} - 2 q^{49} + 4 q^{51} + 2 q^{57} - 16 q^{59} - 16 q^{61} - 4 q^{67} - 16 q^{71} + 2 q^{75} + 8 q^{77} - 16 q^{79} + 2 q^{81} + 8 q^{85} + 16 q^{91} - 12 q^{93} + 4 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 + 4 * q^15 - 4 * q^17 - 2 * q^19 - 2 * q^25 - 2 * q^27 + 12 * q^31 - 4 * q^45 - 2 * q^49 + 4 * q^51 + 2 * q^57 - 16 * q^59 - 16 * q^61 - 4 * q^67 - 16 * q^71 + 2 * q^75 + 8 * q^77 - 16 * q^79 + 2 * q^81 + 8 * q^85 + 16 * q^91 - 12 * q^93 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	−2.00000			−0.894427			−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000			−0.894427			−0.447214	+	0.894427i	$0.647584\pi$
$6$		0			0
$7$			2.82843i			1.06904i	0.845154	+	0.534522i	$0.179509\pi$
$7$			2.82843i			1.06904i	−0.845154	+	0.534522i	$0.820491\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	1.41421i		−	0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$		−	1.41421i		−	0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$		0			0
$13$		−	2.82843i		−	0.784465i	−0.919866	−	0.392232i	$-0.871703\pi$
$13$		−	2.82843i		−	0.784465i	0.919866	−	0.392232i	$-0.128297\pi$
$14$		0			0
$15$	2.00000			0.516398
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$	−1.00000	+	4.24264i	−0.229416	+	0.973329i
$19$	−1.00000	+	4.24264i	−0.229416	+	0.973329i
$20$		0			0
$21$		−	2.82843i		−	0.617213i
$22$		0			0
$23$		−	1.41421i		−	0.294884i	−0.989071	−	0.147442i	$-0.952896\pi$
$23$		−	1.41421i		−	0.294884i	0.989071	−	0.147442i	$-0.0471040\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$		−	7.07107i		−	1.31306i	−0.754298	−	0.656532i	$-0.772023\pi$
$29$		−	7.07107i		−	1.31306i	0.754298	−	0.656532i	$-0.227977\pi$
$30$		0			0
$31$	6.00000			1.07763			0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000			1.07763			0.538816	+	0.842424i	$0.318872\pi$
$32$		0			0
$33$			1.41421i			0.246183i
$34$		0			0
$35$		−	5.65685i		−	0.956183i
$36$		0			0
$37$		−	11.3137i		−	1.85996i	−0.367607	−	0.929981i	$-0.619823\pi$
$37$		−	11.3137i		−	1.85996i	0.367607	−	0.929981i	$-0.380177\pi$
$38$		0			0
$39$			2.82843i			0.452911i
$40$		0			0
$41$		−	4.24264i		−	0.662589i	−0.943527	−	0.331295i	$-0.892515\pi$
$41$		−	4.24264i		−	0.662589i	0.943527	−	0.331295i	$-0.107485\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−2.00000			−0.298142
$46$		0			0
$47$		−	7.07107i		−	1.03142i	−0.856763	−	0.515711i	$-0.827528\pi$
$47$		−	7.07107i		−	1.03142i	0.856763	−	0.515711i	$-0.172472\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	2.00000			0.280056
$52$		0			0
$53$		−	9.89949i		−	1.35980i	−0.733305	−	0.679900i	$-0.762023\pi$
$53$		−	9.89949i		−	1.35980i	0.733305	−	0.679900i	$-0.237977\pi$
$54$		0			0
$55$			2.82843i			0.381385i
$56$		0			0
$57$	1.00000	−	4.24264i	0.132453	−	0.561951i
$58$		0			0
$59$	−8.00000			−1.04151			−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000			−1.04151			−0.520756	+	0.853706i	$0.674350\pi$
$60$		0			0
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$			2.82843i			0.356348i
$64$		0			0
$65$			5.65685i			0.701646i
$66$		0			0
$67$	−2.00000			−0.244339			−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000			−0.244339			−0.122169	+	0.992509i	$0.538985\pi$
$68$		0			0
$69$			1.41421i			0.170251i
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	1.00000			0.115470
$76$		0			0
$77$	4.00000			0.455842
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	12.7279i		−	1.39707i	−0.715575	−	0.698535i	$-0.753835\pi$
$83$		−	12.7279i		−	1.39707i	0.715575	−	0.698535i	$-0.246165\pi$
$84$		0			0
$85$	4.00000			0.433861
$86$		0			0
$87$			7.07107i			0.758098i
$88$		0			0
$89$			9.89949i			1.04934i	0.851304	+	0.524672i	$0.175812\pi$
$89$			9.89949i			1.04934i	−0.851304	+	0.524672i	$0.824188\pi$
$90$		0			0
$91$	8.00000			0.838628
$92$		0			0
$93$	−6.00000			−0.622171
$94$		0			0
$95$	2.00000	−	8.48528i	0.205196	−	0.870572i
$96$		0			0
$97$			2.82843i			0.287183i	0.989637	+	0.143592i	$0.0458652\pi$
$97$			2.82843i			0.287183i	−0.989637	+	0.143592i	$0.954135\pi$
$98$		0			0
$99$		−	1.41421i		−	0.142134i
$100$		0			0
$101$	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	$-0.628955\pi$
$103$	−8.00000			−0.788263			−0.394132	+	0.919054i	$0.628955\pi$
$104$		0			0
$105$			5.65685i			0.552052i
$106$		0			0
$107$	20.0000			1.93347			0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000			1.93347			0.966736	+	0.255774i	$0.0823304\pi$
$108$		0			0
$109$			8.48528i			0.812743i	0.913708	+	0.406371i	$0.133206\pi$
$109$			8.48528i			0.812743i	−0.913708	+	0.406371i	$0.866794\pi$
$110$		0			0
$111$			11.3137i			1.07385i
$112$		0			0
$113$		−	1.41421i		−	0.133038i	−0.997785	−	0.0665190i	$-0.978811\pi$
$113$		−	1.41421i		−	0.133038i	0.997785	−	0.0665190i	$-0.0211893\pi$
$114$		0			0
$115$			2.82843i			0.263752i
$116$		0			0
$117$		−	2.82843i		−	0.261488i
$118$		0			0
$119$		−	5.65685i		−	0.518563i
$120$		0			0
$121$	9.00000			0.818182
$122$		0			0
$123$			4.24264i			0.382546i
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$	14.0000			1.24230			0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000			1.24230			0.621150	+	0.783692i	$0.286666\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			4.24264i			0.370681i	0.982674	+	0.185341i	$0.0593388\pi$
$131$			4.24264i			0.370681i	−0.982674	+	0.185341i	$0.940661\pi$
$132$		0			0
$133$	−12.0000	−	2.82843i	−1.04053	−	0.245256i
$134$		0			0
$135$	2.00000			0.172133
$136$		0			0
$137$	−6.00000			−0.512615			−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000			−0.512615			−0.256307	+	0.966595i	$0.582506\pi$
$138$		0			0
$139$		−	2.82843i		−	0.239904i	−0.992780	−	0.119952i	$-0.961726\pi$
$139$		−	2.82843i		−	0.239904i	0.992780	−	0.119952i	$-0.0382741\pi$
$140$		0			0
$141$			7.07107i			0.595491i
$142$		0			0
$143$	−4.00000			−0.334497
$144$		0			0
$145$			14.1421i			1.17444i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$	−14.0000			−1.14692			−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000			−1.14692			−0.573462	+	0.819232i	$0.694400\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$	−2.00000			−0.161690
$154$		0			0
$155$	−12.0000			−0.963863
$156$		0			0
$157$	8.00000			0.638470			0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000			0.638470			0.319235	+	0.947676i	$0.396574\pi$
$158$		0			0
$159$			9.89949i			0.785081i
$160$		0			0
$161$	4.00000			0.315244
$162$		0			0
$163$			8.48528i			0.664619i	0.943170	+	0.332309i	$0.107828\pi$
$163$			8.48528i			0.664619i	−0.943170	+	0.332309i	$0.892172\pi$
$164$		0			0
$165$		−	2.82843i		−	0.220193i
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	5.00000			0.384615
$170$		0			0
$171$	−1.00000	+	4.24264i	−0.0764719	+	0.324443i
$172$		0			0
$173$			7.07107i			0.537603i	0.963196	+	0.268802i	$0.0866276\pi$
$173$			7.07107i			0.537603i	−0.963196	+	0.268802i	$0.913372\pi$
$174$		0			0
$175$		−	2.82843i		−	0.213809i
$176$		0			0
$177$	8.00000			0.601317
$178$		0			0
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$		0			0
$181$		−	8.48528i		−	0.630706i	−0.948974	−	0.315353i	$-0.897877\pi$
$181$		−	8.48528i		−	0.630706i	0.948974	−	0.315353i	$-0.102123\pi$
$182$		0			0
$183$	8.00000			0.591377
$184$		0			0
$185$			22.6274i			1.66360i
$186$		0			0
$187$			2.82843i			0.206835i
$188$		0			0
$189$		−	2.82843i		−	0.205738i
$190$		0			0
$191$		−	12.7279i		−	0.920960i	−0.887670	−	0.460480i	$-0.847677\pi$
$191$		−	12.7279i		−	0.920960i	0.887670	−	0.460480i	$-0.152323\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		−	5.65685i		−	0.405096i
$196$		0			0
$197$	−14.0000			−0.997459			−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000			−0.997459			−0.498729	+	0.866758i	$0.666200\pi$
$198$		0			0
$199$			11.3137i			0.802008i	0.916076	+	0.401004i	$0.131339\pi$
$199$			11.3137i			0.802008i	−0.916076	+	0.401004i	$0.868661\pi$
$200$		0			0
$201$	2.00000			0.141069
$202$		0			0
$203$	20.0000			1.40372
$204$		0			0
$205$			8.48528i			0.592638i
$206$		0			0
$207$		−	1.41421i		−	0.0982946i
$208$		0			0
$209$	6.00000	+	1.41421i	0.415029	+	0.0978232i
$210$		0			0
$211$	−10.0000			−0.688428			−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000			−0.688428			−0.344214	+	0.938891i	$0.611855\pi$
$212$		0			0
$213$	8.00000			0.548151
$214$		0			0
$215$		0			0
$216$		0			0
$217$			16.9706i			1.15204i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			5.65685i			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
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	−12.0000			−0.796468			−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000			−0.796468			−0.398234	+	0.917284i	$0.630377\pi$
$228$		0			0
$229$	22.0000			1.45380			0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000			1.45380			0.726900	+	0.686743i	$0.240960\pi$
$230$		0			0
$231$	−4.00000			−0.263181
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$			14.1421i			0.922531i
$236$		0			0
$237$	8.00000			0.519656
$238$		0			0
$239$		−	26.8701i		−	1.73808i	−0.494742	−	0.869040i	$-0.664738\pi$
$239$		−	26.8701i		−	1.73808i	0.494742	−	0.869040i	$-0.335262\pi$
$240$		0			0
$241$			2.82843i			0.182195i	0.995842	+	0.0910975i	$0.0290375\pi$
$241$			2.82843i			0.182195i	−0.995842	+	0.0910975i	$0.970963\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	2.00000			0.127775
$246$		0			0
$247$	12.0000	+	2.82843i	0.763542	+	0.179969i
$248$		0			0
$249$			12.7279i			0.806599i
$250$		0			0
$251$			24.0416i			1.51749i	0.651385	+	0.758747i	$0.274188\pi$
$251$			24.0416i			1.51749i	−0.651385	+	0.758747i	$0.725812\pi$
$252$		0			0
$253$	−2.00000			−0.125739
$254$		0			0
$255$	−4.00000			−0.250490
$256$		0			0
$257$			9.89949i			0.617514i	0.951141	+	0.308757i	$0.0999129\pi$
$257$			9.89949i			0.617514i	−0.951141	+	0.308757i	$0.900087\pi$
$258$		0			0
$259$	32.0000			1.98838
$260$		0			0
$261$		−	7.07107i		−	0.437688i
$262$		0			0
$263$		−	24.0416i		−	1.48247i	−0.671245	−	0.741235i	$-0.734240\pi$
$263$		−	24.0416i		−	1.48247i	0.671245	−	0.741235i	$-0.265760\pi$
$264$		0			0
$265$			19.7990i			1.21624i
$266$		0			0
$267$		−	9.89949i		−	0.605839i
$268$		0			0
$269$			24.0416i			1.46584i	0.680313	+	0.732922i	$0.261844\pi$
$269$			24.0416i			1.46584i	−0.680313	+	0.732922i	$0.738156\pi$
$270$		0			0
$271$			11.3137i			0.687259i	0.939105	+	0.343629i	$0.111656\pi$
$271$			11.3137i			0.687259i	−0.939105	+	0.343629i	$0.888344\pi$
$272$		0			0
$273$	−8.00000			−0.484182
$274$		0			0
$275$			1.41421i			0.0852803i
$276$		0			0
$277$	−22.0000			−1.32185			−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000			−1.32185			−0.660926	+	0.750451i	$0.729836\pi$
$278$		0			0
$279$	6.00000			0.359211
$280$		0			0
$281$			1.41421i			0.0843649i	0.999110	+	0.0421825i	$0.0134311\pi$
$281$			1.41421i			0.0843649i	−0.999110	+	0.0421825i	$0.986569\pi$
$282$		0			0
$283$		−	28.2843i		−	1.68133i	−0.541559	−	0.840663i	$-0.682166\pi$
$283$		−	28.2843i		−	1.68133i	0.541559	−	0.840663i	$-0.317834\pi$
$284$		0			0
$285$	−2.00000	+	8.48528i	−0.118470	+	0.502625i
$286$		0			0
$287$	12.0000			0.708338
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		−	2.82843i		−	0.165805i
$292$		0			0
$293$		−	29.6985i		−	1.73500i	−0.497434	−	0.867502i	$-0.665724\pi$
$293$		−	29.6985i		−	1.73500i	0.497434	−	0.867502i	$-0.334276\pi$
$294$		0			0
$295$	16.0000			0.931556
$296$		0			0
$297$			1.41421i			0.0820610i
$298$		0			0
$299$	−4.00000			−0.231326
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−6.00000			−0.344691
$304$		0			0
$305$	16.0000			0.916157
$306$		0			0
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$		0			0
$309$	8.00000			0.455104
$310$		0			0
$311$		−	9.89949i		−	0.561349i	−0.959803	−	0.280674i	$-0.909442\pi$
$311$		−	9.89949i		−	0.561349i	0.959803	−	0.280674i	$-0.0905581\pi$
$312$		0			0
$313$	20.0000			1.13047			0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000			1.13047			0.565233	+	0.824931i	$0.308786\pi$
$314$		0			0
$315$		−	5.65685i		−	0.318728i
$316$		0			0
$317$		−	7.07107i		−	0.397151i	−0.980086	−	0.198575i	$-0.936369\pi$
$317$		−	7.07107i		−	0.397151i	0.980086	−	0.198575i	$-0.0636315\pi$
$318$		0			0
$319$	−10.0000			−0.559893
$320$		0			0
$321$	−20.0000			−1.11629
$322$		0			0
$323$	2.00000	−	8.48528i	0.111283	−	0.472134i
$324$		0			0
$325$			2.82843i			0.156893i
$326$		0			0
$327$		−	8.48528i		−	0.469237i
$328$		0			0
$329$	20.0000			1.10264
$330$		0			0
$331$	−28.0000			−1.53902			−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000			−1.53902			−0.769510	+	0.638635i	$0.779499\pi$
$332$		0			0
$333$		−	11.3137i		−	0.619987i
$334$		0			0
$335$	4.00000			0.218543
$336$		0			0
$337$		−	25.4558i		−	1.38667i	−0.720616	−	0.693334i	$-0.756141\pi$
$337$		−	25.4558i		−	1.38667i	0.720616	−	0.693334i	$-0.243859\pi$
$338$		0			0
$339$			1.41421i			0.0768095i
$340$		0			0
$341$		−	8.48528i		−	0.459504i
$342$		0			0
$343$			16.9706i			0.916324i
$344$		0			0
$345$		−	2.82843i		−	0.152277i
$346$		0			0
$347$			32.5269i			1.74614i	0.487598	+	0.873068i	$0.337873\pi$
$347$			32.5269i			1.74614i	−0.487598	+	0.873068i	$0.662127\pi$
$348$		0			0
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$		0			0
$351$			2.82843i			0.150970i
$352$		0			0
$353$	18.0000			0.958043			0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000			0.958043			0.479022	+	0.877803i	$0.340992\pi$
$354$		0			0
$355$	16.0000			0.849192
$356$		0			0
$357$			5.65685i			0.299392i
$358$		0			0
$359$			18.3848i			0.970311i	0.874428	+	0.485156i	$0.161237\pi$
$359$			18.3848i			0.970311i	−0.874428	+	0.485156i	$0.838763\pi$
$360$		0			0
$361$	−17.0000	−	8.48528i	−0.894737	−	0.446594i
$362$		0			0
$363$	−9.00000			−0.472377
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	16.9706i		−	0.885856i	−0.896557	−	0.442928i	$-0.853940\pi$
$367$		−	16.9706i		−	0.885856i	0.896557	−	0.442928i	$-0.146060\pi$
$368$		0			0
$369$		−	4.24264i		−	0.220863i
$370$		0			0
$371$	28.0000			1.45369
$372$		0			0
$373$		−	22.6274i		−	1.17160i	−0.810454	−	0.585802i	$-0.800780\pi$
$373$		−	22.6274i		−	1.17160i	0.810454	−	0.585802i	$-0.199220\pi$
$374$		0			0
$375$	−12.0000			−0.619677
$376$		0			0
$377$	−20.0000			−1.03005
$378$		0			0
$379$	−28.0000			−1.43826			−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000			−1.43826			−0.719132	+	0.694874i	$0.755460\pi$
$380$		0			0
$381$	−14.0000			−0.717242
$382$		0			0
$383$	−32.0000			−1.63512			−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000			−1.63512			−0.817562	+	0.575841i	$0.804675\pi$
$384$		0			0
$385$	−8.00000			−0.407718
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−22.0000			−1.11544			−0.557722	−	0.830028i	$-0.688325\pi$
$389$	−22.0000			−1.11544			−0.557722	+	0.830028i	$0.688325\pi$
$390$		0			0
$391$			2.82843i			0.143040i
$392$		0			0
$393$		−	4.24264i		−	0.214013i
$394$		0			0
$395$	16.0000			0.805047
$396$		0			0
$397$	10.0000			0.501886			0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000			0.501886			0.250943	+	0.968002i	$0.419259\pi$
$398$		0			0
$399$	12.0000	+	2.82843i	0.600751	+	0.141598i
$400$		0			0
$401$		−	26.8701i		−	1.34183i	−0.741536	−	0.670913i	$-0.765902\pi$
$401$		−	26.8701i		−	1.34183i	0.741536	−	0.670913i	$-0.234098\pi$
$402$		0			0
$403$		−	16.9706i		−	0.845364i
$404$		0			0
$405$	−2.00000			−0.0993808
$406$		0			0
$407$	−16.0000			−0.793091
$408$		0			0
$409$			25.4558i			1.25871i	0.777118	+	0.629355i	$0.216681\pi$
$409$			25.4558i			1.25871i	−0.777118	+	0.629355i	$0.783319\pi$
$410$		0			0
$411$	6.00000			0.295958
$412$		0			0
$413$		−	22.6274i		−	1.11342i
$414$		0			0
$415$			25.4558i			1.24958i
$416$		0			0
$417$			2.82843i			0.138509i
$418$		0			0
$419$			18.3848i			0.898155i	0.893493	+	0.449078i	$0.148247\pi$
$419$			18.3848i			0.898155i	−0.893493	+	0.449078i	$0.851753\pi$
$420$		0			0
$421$		−	14.1421i		−	0.689246i	−0.938741	−	0.344623i	$-0.888007\pi$
$421$		−	14.1421i		−	0.689246i	0.938741	−	0.344623i	$-0.111993\pi$
$422$		0			0
$423$		−	7.07107i		−	0.343807i
$424$		0			0
$425$	2.00000			0.0970143
$426$		0			0
$427$		−	22.6274i		−	1.09502i
$428$		0			0
$429$	4.00000			0.193122
$430$		0			0
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$		−	33.9411i		−	1.63111i	−0.578682	−	0.815553i	$-0.696433\pi$
$433$		−	33.9411i		−	1.63111i	0.578682	−	0.815553i	$-0.303567\pi$
$434$		0			0
$435$		−	14.1421i		−	0.678064i
$436$		0			0
$437$	6.00000	+	1.41421i	0.287019	+	0.0676510i
$438$		0			0
$439$	32.0000			1.52728			0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000			1.52728			0.763638	+	0.645644i	$0.223411\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$			26.8701i			1.27663i	0.769773	+	0.638317i	$0.220369\pi$
$443$			26.8701i			1.27663i	−0.769773	+	0.638317i	$0.779631\pi$
$444$		0			0
$445$		−	19.7990i		−	0.938562i
$446$		0			0
$447$	14.0000			0.662177
$448$		0			0
$449$		−	15.5563i		−	0.734150i	−0.930191	−	0.367075i	$-0.880359\pi$
$449$		−	15.5563i		−	0.734150i	0.930191	−	0.367075i	$-0.119641\pi$
$450$		0			0
$451$	−6.00000			−0.282529
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−16.0000			−0.750092
$456$		0			0
$457$	8.00000			0.374224			0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000			0.374224			0.187112	+	0.982339i	$0.440087\pi$
$458$		0			0
$459$	2.00000			0.0933520
$460$		0			0
$461$	−14.0000			−0.652045			−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000			−0.652045			−0.326023	+	0.945362i	$0.605709\pi$
$462$		0			0
$463$		−	16.9706i		−	0.788689i	−0.918963	−	0.394344i	$-0.870972\pi$
$463$		−	16.9706i		−	0.788689i	0.918963	−	0.394344i	$-0.129028\pi$
$464$		0			0
$465$	12.0000			0.556487
$466$		0			0
$467$			24.0416i			1.11251i	0.831010	+	0.556257i	$0.187763\pi$
$467$			24.0416i			1.11251i	−0.831010	+	0.556257i	$0.812237\pi$
$468$		0			0
$469$		−	5.65685i		−	0.261209i
$470$		0			0
$471$	−8.00000			−0.368621
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.00000	−	4.24264i	0.0458831	−	0.194666i
$476$		0			0
$477$		−	9.89949i		−	0.453267i
$478$		0			0
$479$			7.07107i			0.323085i	0.986866	+	0.161543i	$0.0516469\pi$
$479$			7.07107i			0.323085i	−0.986866	+	0.161543i	$0.948353\pi$
$480$		0			0
$481$	−32.0000			−1.45907
$482$		0			0
$483$	−4.00000			−0.182006
$484$		0			0
$485$		−	5.65685i		−	0.256865i
$486$		0			0
$487$	10.0000			0.453143			0.226572	−	0.973995i	$-0.427248\pi$
$487$	10.0000			0.453143			0.226572	+	0.973995i	$0.427248\pi$
$488$		0			0
$489$		−	8.48528i		−	0.383718i
$490$		0			0
$491$			24.0416i			1.08498i	0.840061	+	0.542492i	$0.182519\pi$
$491$			24.0416i			1.08498i	−0.840061	+	0.542492i	$0.817481\pi$
$492$		0			0
$493$			14.1421i			0.636930i
$494$		0			0
$495$			2.82843i			0.127128i
$496$		0			0
$497$		−	22.6274i		−	1.01498i
$498$		0			0
$499$		−	25.4558i		−	1.13956i	−0.821797	−	0.569780i	$-0.807028\pi$
$499$		−	25.4558i		−	1.13956i	0.821797	−	0.569780i	$-0.192972\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			12.7279i			0.567510i	0.958897	+	0.283755i	$0.0915802\pi$
$503$			12.7279i			0.567510i	−0.958897	+	0.283755i	$0.908420\pi$
$504$		0			0
$505$	−12.0000			−0.533993
$506$		0			0
$507$	−5.00000			−0.222058
$508$		0			0
$509$			24.0416i			1.06563i	0.846233	+	0.532813i	$0.178865\pi$
$509$			24.0416i			1.06563i	−0.846233	+	0.532813i	$0.821135\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	1.00000	−	4.24264i	0.0441511	−	0.187317i
$514$		0			0
$515$	16.0000			0.705044
$516$		0			0
$517$	−10.0000			−0.439799
$518$		0			0
$519$		−	7.07107i		−	0.310385i
$520$		0			0
$521$		−	1.41421i		−	0.0619578i	−0.999520	−	0.0309789i	$-0.990138\pi$
$521$		−	1.41421i		−	0.0619578i	0.999520	−	0.0309789i	$-0.00986247\pi$
$522$		0			0
$523$	34.0000			1.48672			0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000			1.48672			0.743358	+	0.668894i	$0.233232\pi$
$524$		0			0
$525$			2.82843i			0.123443i
$526$		0			0
$527$	−12.0000			−0.522728
$528$		0			0
$529$	21.0000			0.913043
$530$		0			0
$531$	−8.00000			−0.347170
$532$		0			0
$533$	−12.0000			−0.519778
$534$		0			0
$535$	−40.0000			−1.72935
$536$		0			0
$537$	20.0000			0.863064
$538$		0			0
$539$			1.41421i			0.0609145i
$540$		0			0
$541$	12.0000			0.515920			0.257960	−	0.966156i	$-0.416950\pi$
$541$	12.0000			0.515920			0.257960	+	0.966156i	$0.416950\pi$
$542$		0			0
$543$			8.48528i			0.364138i
$544$		0			0
$545$		−	16.9706i		−	0.726939i
$546$		0			0
$547$	−12.0000			−0.513083			−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000			−0.513083			−0.256541	+	0.966533i	$0.582583\pi$
$548$		0			0
$549$	−8.00000			−0.341432
$550$		0			0
$551$	30.0000	+	7.07107i	1.27804	+	0.301238i
$552$		0			0
$553$		−	22.6274i		−	0.962216i
$554$		0			0
$555$		−	22.6274i		−	0.960480i
$556$		0			0
$557$	18.0000			0.762684			0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000			0.762684			0.381342	+	0.924434i	$0.375462\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		−	2.82843i		−	0.119416i
$562$		0			0
$563$	36.0000			1.51722			0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000			1.51722			0.758610	+	0.651546i	$0.225879\pi$
$564$		0			0
$565$			2.82843i			0.118993i
$566$		0			0
$567$			2.82843i			0.118783i
$568$		0			0
$569$			32.5269i			1.36360i	0.731539	+	0.681800i	$0.238802\pi$
$569$			32.5269i			1.36360i	−0.731539	+	0.681800i	$0.761198\pi$
$570$		0			0
$571$		−	45.2548i		−	1.89386i	−0.321446	−	0.946928i	$-0.604169\pi$
$571$		−	45.2548i		−	1.89386i	0.321446	−	0.946928i	$-0.395831\pi$
$572$		0			0
$573$			12.7279i			0.531717i
$574$		0			0
$575$			1.41421i			0.0589768i
$576$		0			0
$577$	20.0000			0.832611			0.416305	−	0.909225i	$-0.363325\pi$
$577$	20.0000			0.832611			0.416305	+	0.909225i	$0.363325\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	36.0000			1.49353
$582$		0			0
$583$	−14.0000			−0.579821
$584$		0			0
$585$			5.65685i			0.233882i
$586$		0			0
$587$			35.3553i			1.45927i	0.683836	+	0.729636i	$0.260310\pi$
$587$			35.3553i			1.45927i	−0.683836	+	0.729636i	$0.739690\pi$
$588$		0			0
$589$	−6.00000	+	25.4558i	−0.247226	+	1.04889i
$590$		0			0
$591$	14.0000			0.575883
$592$		0			0
$593$	−42.0000			−1.72473			−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000			−1.72473			−0.862367	+	0.506284i	$0.831019\pi$
$594$		0			0
$595$			11.3137i			0.463817i
$596$		0			0
$597$		−	11.3137i		−	0.463039i
$598$		0			0
$599$	44.0000			1.79779			0.898896	−	0.438163i	$-0.144371\pi$
$599$	44.0000			1.79779			0.898896	+	0.438163i	$0.144371\pi$
$600$		0			0
$601$		−	28.2843i		−	1.15374i	−0.816836	−	0.576870i	$-0.804274\pi$
$601$		−	28.2843i		−	1.15374i	0.816836	−	0.576870i	$-0.195726\pi$
$602$		0			0
$603$	−2.00000			−0.0814463
$604$		0			0
$605$	−18.0000			−0.731804
$606$		0			0
$607$	−34.0000			−1.38002			−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000			−1.38002			−0.690009	+	0.723801i	$0.742393\pi$
$608$		0			0
$609$	−20.0000			−0.810441
$610$		0			0
$611$	−20.0000			−0.809113
$612$		0			0
$613$	16.0000			0.646234			0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000			0.646234			0.323117	+	0.946359i	$0.395269\pi$
$614$		0			0
$615$		−	8.48528i		−	0.342160i
$616$		0			0
$617$	−34.0000			−1.36879			−0.684394	−	0.729112i	$-0.739933\pi$
$617$	−34.0000			−1.36879			−0.684394	+	0.729112i	$0.739933\pi$
$618$		0			0
$619$		−	25.4558i		−	1.02316i	−0.859237	−	0.511578i	$-0.829061\pi$
$619$		−	25.4558i		−	1.02316i	0.859237	−	0.511578i	$-0.170939\pi$
$620$		0			0
$621$			1.41421i			0.0567504i
$622$		0			0
$623$	−28.0000			−1.12180
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$	−6.00000	−	1.41421i	−0.239617	−	0.0564782i
$628$		0			0
$629$			22.6274i			0.902214i
$630$		0			0
$631$			14.1421i			0.562990i	0.959563	+	0.281495i	$0.0908302\pi$
$631$			14.1421i			0.562990i	−0.959563	+	0.281495i	$0.909170\pi$
$632$		0			0
$633$	10.0000			0.397464
$634$		0			0
$635$	−28.0000			−1.11115
$636$		0			0
$637$			2.82843i			0.112066i
$638$		0			0
$639$	−8.00000			−0.316475
$640$		0			0
$641$		−	41.0122i		−	1.61988i	−0.586510	−	0.809942i	$-0.699498\pi$
$641$		−	41.0122i		−	1.61988i	0.586510	−	0.809942i	$-0.300502\pi$
$642$		0			0
$643$		−	39.5980i		−	1.56159i	−0.624786	−	0.780796i	$-0.714814\pi$
$643$		−	39.5980i		−	1.56159i	0.624786	−	0.780796i	$-0.285186\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			7.07107i			0.277992i	0.990293	+	0.138996i	$0.0443876\pi$
$647$			7.07107i			0.277992i	−0.990293	+	0.138996i	$0.955612\pi$
$648$		0			0
$649$			11.3137i			0.444102i
$650$		0			0
$651$		−	16.9706i		−	0.665129i
$652$		0			0
$653$	−6.00000			−0.234798			−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000			−0.234798			−0.117399	+	0.993085i	$0.537456\pi$
$654$		0			0
$655$		−	8.48528i		−	0.331547i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	8.00000			0.311636			0.155818	−	0.987786i	$-0.450199\pi$
$659$	8.00000			0.311636			0.155818	+	0.987786i	$0.450199\pi$
$660$		0			0
$661$			39.5980i			1.54018i	0.637934	+	0.770091i	$0.279789\pi$
$661$			39.5980i			1.54018i	−0.637934	+	0.770091i	$0.720211\pi$
$662$		0			0
$663$		−	5.65685i		−	0.219694i
$664$		0			0
$665$	24.0000	+	5.65685i	0.930680	+	0.219363i
$666$		0			0
$667$	−10.0000			−0.387202
$668$		0			0
$669$	16.0000			0.618596
$670$		0			0
$671$			11.3137i			0.436761i
$672$		0			0
$673$		−	28.2843i		−	1.09028i	−0.838346	−	0.545139i	$-0.816477\pi$
$673$		−	28.2843i		−	1.09028i	0.838346	−	0.545139i	$-0.183523\pi$
$674$		0			0
$675$	1.00000			0.0384900
$676$		0			0
$677$		−	9.89949i		−	0.380468i	−0.981739	−	0.190234i	$-0.939075\pi$
$677$		−	9.89949i		−	0.380468i	0.981739	−	0.190234i	$-0.0609248\pi$
$678$		0			0
$679$	−8.00000			−0.307012
$680$		0			0
$681$	12.0000			0.459841
$682$		0			0
$683$	−32.0000			−1.22445			−0.612223	−	0.790685i	$-0.709725\pi$
$683$	−32.0000			−1.22445			−0.612223	+	0.790685i	$0.709725\pi$
$684$		0			0
$685$	12.0000			0.458496
$686$		0			0
$687$	−22.0000			−0.839352
$688$		0			0
$689$	−28.0000			−1.06672
$690$		0			0
$691$			16.9706i			0.645591i	0.946469	+	0.322795i	$0.104623\pi$
$691$			16.9706i			0.645591i	−0.946469	+	0.322795i	$0.895377\pi$
$692$		0			0
$693$	4.00000			0.151947
$694$		0			0
$695$			5.65685i			0.214577i
$696$		0			0
$697$			8.48528i			0.321403i
$698$		0			0
$699$	6.00000			0.226941
$700$		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
$703$	48.0000	+	11.3137i	1.81035	+	0.426705i
$704$		0			0
$705$		−	14.1421i		−	0.532624i
$706$		0			0
$707$			16.9706i			0.638244i
$708$		0			0
$709$	4.00000			0.150223			0.0751116	−	0.997175i	$-0.476069\pi$
$709$	4.00000			0.150223			0.0751116	+	0.997175i	$0.476069\pi$
$710$		0			0
$711$	−8.00000			−0.300023
$712$		0			0
$713$		−	8.48528i		−	0.317776i
$714$		0			0
$715$	8.00000			0.299183
$716$		0			0
$717$			26.8701i			1.00348i
$718$		0			0
$719$		−	32.5269i		−	1.21305i	−0.795065	−	0.606525i	$-0.792563\pi$
$719$		−	32.5269i		−	1.21305i	0.795065	−	0.606525i	$-0.207437\pi$
$720$		0			0
$721$		−	22.6274i		−	0.842689i
$722$		0			0
$723$		−	2.82843i		−	0.105190i
$724$		0			0
$725$			7.07107i			0.262613i
$726$		0			0
$727$		−	28.2843i		−	1.04901i	−0.851409	−	0.524503i	$-0.824251\pi$
$727$		−	28.2843i		−	1.04901i	0.851409	−	0.524503i	$-0.175749\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−52.0000			−1.92066			−0.960332	−	0.278859i	$-0.910044\pi$
$733$	−52.0000			−1.92066			−0.960332	+	0.278859i	$0.910044\pi$
$734$		0			0
$735$	−2.00000			−0.0737711
$736$		0			0
$737$			2.82843i			0.104186i
$738$		0			0
$739$			36.7696i			1.35259i	0.736631	+	0.676295i	$0.236415\pi$
$739$			36.7696i			1.35259i	−0.736631	+	0.676295i	$0.763585\pi$
$740$		0			0
$741$	−12.0000	−	2.82843i	−0.440831	−	0.103905i
$742$		0			0
$743$	36.0000			1.32071			0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000			1.32071			0.660356	+	0.750953i	$0.270405\pi$
$744$		0			0
$745$	28.0000			1.02584
$746$		0			0
$747$		−	12.7279i		−	0.465690i
$748$		0			0
$749$			56.5685i			2.06697i
$750$		0			0
$751$	40.0000			1.45962			0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000			1.45962			0.729810	+	0.683650i	$0.239608\pi$
$752$		0			0
$753$		−	24.0416i		−	0.876126i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	22.0000			0.799604			0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000			0.799604			0.399802	+	0.916602i	$0.369079\pi$
$758$		0			0
$759$	2.00000			0.0725954
$760$		0			0
$761$	18.0000			0.652499			0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000			0.652499			0.326250	+	0.945284i	$0.394215\pi$
$762$		0			0
$763$	−24.0000			−0.868858
$764$		0			0
$765$	4.00000			0.144620
$766$		0			0
$767$			22.6274i			0.817029i
$768$		0			0
$769$	20.0000			0.721218			0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000			0.721218			0.360609	+	0.932717i	$0.382569\pi$
$770$		0			0
$771$		−	9.89949i		−	0.356522i
$772$		0			0
$773$			21.2132i			0.762986i	0.924372	+	0.381493i	$0.124590\pi$
$773$			21.2132i			0.762986i	−0.924372	+	0.381493i	$0.875410\pi$
$774$		0			0
$775$	−6.00000			−0.215526
$776$		0			0
$777$	−32.0000			−1.14799
$778$		0			0
$779$	18.0000	+	4.24264i	0.644917	+	0.152008i
$780$		0			0
$781$			11.3137i			0.404836i
$782$		0			0
$783$			7.07107i			0.252699i
$784$		0			0
$785$	−16.0000			−0.571064
$786$		0			0
$787$	−42.0000			−1.49714			−0.748569	−	0.663057i	$-0.769259\pi$
$787$	−42.0000			−1.49714			−0.748569	+	0.663057i	$0.769259\pi$
$788$		0			0
$789$			24.0416i			0.855905i
$790$		0			0
$791$	4.00000			0.142224
$792$		0			0
$793$			22.6274i			0.803523i
$794$		0			0
$795$		−	19.7990i		−	0.702198i
$796$		0			0
$797$		−	7.07107i		−	0.250470i	−0.992127	&m

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:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	912.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.00000 q^{5} +2.82843i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.00000 q^{5} +2.82843i q^{7} +1.00000 q^{9} -1.41421i q^{11} -2.82843i q^{13} +2.00000 q^{15} -2.00000 q^{17} +(-1.00000 + 4.24264i) q^{19} -2.82843i q^{21} -1.41421i q^{23} -1.00000 q^{25} -1.00000 q^{27} -7.07107i q^{29} +6.00000 q^{31} +1.41421i q^{33} -5.65685i q^{35} -11.3137i q^{37} +2.82843i q^{39} -4.24264i q^{41} -2.00000 q^{45} -7.07107i q^{47} -1.00000 q^{49} +2.00000 q^{51} -9.89949i q^{53} +2.82843i q^{55} +(1.00000 - 4.24264i) q^{57} -8.00000 q^{59} -8.00000 q^{61} +2.82843i q^{63} +5.65685i q^{65} -2.00000 q^{67} +1.41421i q^{69} -8.00000 q^{71} +1.00000 q^{75} +4.00000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -12.7279i q^{83} +4.00000 q^{85} +7.07107i q^{87} +9.89949i q^{89} +8.00000 q^{91} -6.00000 q^{93} +(2.00000 - 8.48528i) q^{95} +2.82843i q^{97} -1.41421i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} + 4 q^{15} - 4 q^{17} - 2 q^{19} - 2 q^{25} - 2 q^{27} + 12 q^{31} - 4 q^{45} - 2 q^{49} + 4 q^{51} + 2 q^{57} - 16 q^{59} - 16 q^{61} - 4 q^{67} - 16 q^{71} + 2 q^{75} + 8 q^{77} - 16 q^{79} + 2 q^{81} + 8 q^{85} + 16 q^{91} - 12 q^{93} + 4 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 + 4 * q^15 - 4 * q^17 - 2 * q^19 - 2 * q^25 - 2 * q^27 + 12 * q^31 - 4 * q^45 - 2 * q^49 + 4 * q^51 + 2 * q^57 - 16 * q^59 - 16 * q^61 - 4 * q^67 - 16 * q^71 + 2 * q^75 + 8 * q^77 - 16 * q^79 + 2 * q^81 + 8 * q^85 + 16 * q^91 - 12 * q^93 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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