LMFDB - Embedded newform 784.6.a.bl.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,6,Mod(1,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,5,0,81,0,0,0,390] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$125.740914733$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 200x^{3} - 99x^{2} + 5803x - 3615 $ x^5 - 200x^3 - 99x^2 + 5803*x - 3615
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 7 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.72951$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-12.4590 q^{3} +97.6805 q^{5} -87.7730 q^{9} -89.9322 q^{11} -849.795 q^{13} -1217.00 q^{15} -316.838 q^{17} -2130.80 q^{19} -3878.96 q^{23} +6416.47 q^{25} +4121.11 q^{27} +4643.20 q^{29} -3495.64 q^{31} +1120.47 q^{33} +5600.99 q^{37} +10587.6 q^{39} +20507.1 q^{41} +4823.62 q^{43} -8573.70 q^{45} +305.649 q^{47} +3947.49 q^{51} +6975.95 q^{53} -8784.62 q^{55} +26547.7 q^{57} +5949.91 q^{59} -29452.5 q^{61} -83008.4 q^{65} -2162.72 q^{67} +48328.1 q^{69} -20514.0 q^{71} +68147.4 q^{73} -79942.9 q^{75} +72515.9 q^{79} -30016.1 q^{81} +41952.3 q^{83} -30948.9 q^{85} -57849.7 q^{87} +39734.6 q^{89} +43552.2 q^{93} -208138. q^{95} -26298.1 q^{97} +7893.62 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 81 q^{5} + 390 q^{9} - 361 q^{11} + 342 q^{13} + 1049 q^{15} + 1809 q^{17} - 1277 q^{19} - 911 q^{23} + 3940 q^{25} + 4751 q^{27} + 5442 q^{29} - 2187 q^{31} - 5553 q^{33} - 8181 q^{37}+ \cdots - 249798 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 81 * q^5 + 390 * q^9 - 361 * q^11 + 342 * q^13 + 1049 * q^15 + 1809 * q^17 - 1277 * q^19 - 911 * q^23 + 3940 * q^25 + 4751 * q^27 + 5442 * q^29 - 2187 * q^31 - 5553 * q^33 - 8181 * q^37 + 3422 * q^39 + 16578 * q^41 - 6332 * q^43 + 41310 * q^45 - 16101 * q^47 + 67865 * q^51 + 16047 * q^53 - 45629 * q^55 + 22347 * q^57 - 71027 * q^59 + 31093 * q^61 - 64370 * q^65 - 47981 * q^67 + 137249 * q^69 - 22512 * q^71 + 123333 * q^73 + 45460 * q^75 + 212481 * q^79 + 52917 * q^81 - 87460 * q^83 + 222141 * q^85 - 318070 * q^87 + 129045 * q^89 - 252835 * q^93 - 300417 * q^95 + 328274 * q^97 - 249798 * q^99

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
$3$	−12.4590	−0.799246	−0.399623	−	0.916680i	$-0.630859\pi$
$3$	−12.4590	−0.799246	−0.399623	+	0.916680i	$0.630859\pi$
$4$	0	0
$5$	97.6805	1.74736	0.873681	−	0.486500i	$-0.161727\pi$
$5$	97.6805	1.74736	0.873681	+	0.486500i	$0.161727\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−87.7730	−0.361206
$10$	0	0
$11$	−89.9322	−0.224096	−0.112048	−	0.993703i	$-0.535741\pi$
$11$	−89.9322	−0.224096	−0.112048	+	0.993703i	$0.535741\pi$
$12$	0	0
$13$	−849.795	−1.39462	−0.697310	−	0.716770i	$-0.745620\pi$
$13$	−849.795	−1.39462	−0.697310	+	0.716770i	$0.745620\pi$
$14$	0	0
$15$	−1217.00	−1.39657
$16$	0	0
$17$	−316.838	−0.265898	−0.132949	−	0.991123i	$-0.542445\pi$
$17$	−316.838	−0.265898	−0.132949	+	0.991123i	$0.542445\pi$
$18$	0	0
$19$	−2130.80	−1.35413	−0.677063	−	0.735925i	$-0.736748\pi$
$19$	−2130.80	−1.35413	−0.677063	+	0.735925i	$0.736748\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3878.96	−1.52896	−0.764480	−	0.644647i	$-0.777004\pi$
$23$	−3878.96	−1.52896	−0.764480	+	0.644647i	$0.777004\pi$
$24$	0	0
$25$	6416.47	2.05327
$26$	0	0
$27$	4121.11	1.08794
$28$	0	0
$29$	4643.20	1.02523	0.512617	−	0.858618i	$-0.328676\pi$
$29$	4643.20	1.02523	0.512617	+	0.858618i	$0.328676\pi$
$30$	0	0
$31$	−3495.64	−0.653314	−0.326657	−	0.945143i	$-0.605922\pi$
$31$	−3495.64	−0.653314	−0.326657	+	0.945143i	$0.605922\pi$
$32$	0	0
$33$	1120.47	0.179108
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5600.99	0.672606	0.336303	−	0.941754i	$-0.390823\pi$
$37$	5600.99	0.672606	0.336303	+	0.941754i	$0.390823\pi$
$38$	0	0
$39$	10587.6	1.11464
$40$	0	0
$41$	20507.1	1.90522	0.952610	−	0.304195i	$-0.0983875\pi$
$41$	20507.1	1.90522	0.952610	+	0.304195i	$0.0983875\pi$
$42$	0	0
$43$	4823.62	0.397834	0.198917	−	0.980016i	$-0.436258\pi$
$43$	4823.62	0.397834	0.198917	+	0.980016i	$0.436258\pi$
$44$	0	0
$45$	−8573.70	−0.631157
$46$	0	0
$47$	305.649	0.0201827	0.0100913	−	0.999949i	$-0.496788\pi$
$47$	305.649	0.0201827	0.0100913	+	0.999949i	$0.496788\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3947.49	0.212518
$52$	0	0
$53$	6975.95	0.341125	0.170562	−	0.985347i	$-0.445442\pi$
$53$	6975.95	0.341125	0.170562	+	0.985347i	$0.445442\pi$
$54$	0	0
$55$	−8784.62	−0.391576
$56$	0	0
$57$	26547.7	1.08228
$58$	0	0
$59$	5949.91	0.222526	0.111263	−	0.993791i	$-0.464510\pi$
$59$	5949.91	0.222526	0.111263	+	0.993791i	$0.464510\pi$
$60$	0	0
$61$	−29452.5	−1.01344	−0.506720	−	0.862111i	$-0.669142\pi$
$61$	−29452.5	−1.01344	−0.506720	+	0.862111i	$0.669142\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−83008.4	−2.43690
$66$	0	0
$67$	−2162.72	−0.0588591	−0.0294296	−	0.999567i	$-0.509369\pi$
$67$	−2162.72	−0.0588591	−0.0294296	+	0.999567i	$0.509369\pi$
$68$	0	0
$69$	48328.1	1.22202
$70$	0	0
$71$	−20514.0	−0.482953	−0.241477	−	0.970407i	$-0.577632\pi$
$71$	−20514.0	−0.482953	−0.241477	+	0.970407i	$0.577632\pi$
$72$	0	0
$73$	68147.4	1.49672	0.748362	−	0.663290i	$-0.230840\pi$
$73$	68147.4	1.49672	0.748362	+	0.663290i	$0.230840\pi$
$74$	0	0
$75$	−79942.9	−1.64107
$76$	0	0
$77$	0	0
$78$	0	0
$79$	72515.9	1.30727	0.653635	−	0.756810i	$-0.273243\pi$
$79$	72515.9	1.30727	0.653635	+	0.756810i	$0.273243\pi$
$80$	0	0
$81$	−30016.1	−0.508325
$82$	0	0
$83$	41952.3	0.668437	0.334218	−	0.942496i	$-0.391528\pi$
$83$	41952.3	0.668437	0.334218	+	0.942496i	$0.391528\pi$
$84$	0	0
$85$	−30948.9	−0.464620
$86$	0	0
$87$	−57849.7	−0.819414
$88$	0	0
$89$	39734.6	0.531733	0.265867	−	0.964010i	$-0.414342\pi$
$89$	39734.6	0.531733	0.265867	+	0.964010i	$0.414342\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	43552.2	0.522159
$94$	0	0
$95$	−208138.	−2.36615
$96$	0	0
$97$	−26298.1	−0.283789	−0.141894	−	0.989882i	$-0.545319\pi$
$97$	−26298.1	−0.283789	−0.141894	+	0.989882i	$0.545319\pi$
$98$	0	0
$99$	7893.62	0.0809446
$100$	0	0
$101$	154511.	1.50715	0.753574	−	0.657363i	$-0.228328\pi$
$101$	154511.	1.50715	0.753574	+	0.657363i	$0.228328\pi$
$102$	0	0
$103$	152351.	1.41498	0.707492	−	0.706721i	$-0.249826\pi$
$103$	152351.	1.41498	0.707492	+	0.706721i	$0.249826\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−94753.4	−0.800084	−0.400042	−	0.916497i	$-0.631004\pi$
$107$	−94753.4	−0.800084	−0.400042	+	0.916497i	$0.631004\pi$
$108$	0	0
$109$	−161726.	−1.30381	−0.651905	−	0.758301i	$-0.726030\pi$
$109$	−161726.	−1.30381	−0.651905	+	0.758301i	$0.726030\pi$
$110$	0	0
$111$	−69782.8	−0.537578
$112$	0	0
$113$	36424.4	0.268346	0.134173	−	0.990958i	$-0.457162\pi$
$113$	36424.4	0.268346	0.134173	+	0.990958i	$0.457162\pi$
$114$	0	0
$115$	−378899.	−2.67165
$116$	0	0
$117$	74589.0	0.503744
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−152963.	−0.949781
$122$	0	0
$123$	−255498.	−1.52274
$124$	0	0
$125$	321513.	1.84045
$126$	0	0
$127$	−80695.1	−0.443954	−0.221977	−	0.975052i	$-0.571251\pi$
$127$	−80695.1	−0.443954	−0.221977	+	0.975052i	$0.571251\pi$
$128$	0	0
$129$	−60097.5	−0.317967
$130$	0	0
$131$	156726.	0.797925	0.398963	−	0.916967i	$-0.369370\pi$
$131$	156726.	0.797925	0.398963	+	0.916967i	$0.369370\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	402551.	1.90102
$136$	0	0
$137$	88122.7	0.401131	0.200566	−	0.979680i	$-0.435722\pi$
$137$	88122.7	0.401131	0.200566	+	0.979680i	$0.435722\pi$
$138$	0	0
$139$	90339.0	0.396587	0.198293	−	0.980143i	$-0.436460\pi$
$139$	90339.0	0.396587	0.198293	+	0.980143i	$0.436460\pi$
$140$	0	0
$141$	−3808.09	−0.0161309
$142$	0	0
$143$	76423.9	0.312528
$144$	0	0
$145$	453550.	1.79145
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−267141.	−0.985767	−0.492883	−	0.870095i	$-0.664057\pi$
$149$	−267141.	−0.985767	−0.492883	+	0.870095i	$0.664057\pi$
$150$	0	0
$151$	548228.	1.95668	0.978339	−	0.207011i	$-0.0663735\pi$
$151$	548228.	1.95668	0.978339	+	0.207011i	$0.0663735\pi$
$152$	0	0
$153$	27809.8	0.0960439
$154$	0	0
$155$	−341456.	−1.14158
$156$	0	0
$157$	3287.67	0.0106448	0.00532242	−	0.999986i	$-0.498306\pi$
$157$	3287.67	0.0106448	0.00532242	+	0.999986i	$0.498306\pi$
$158$	0	0
$159$	−86913.4	−0.272643
$160$	0	0
$161$	0	0
$162$	0	0
$163$	75208.1	0.221715	0.110858	−	0.993836i	$-0.464640\pi$
$163$	75208.1	0.221715	0.110858	+	0.993836i	$0.464640\pi$
$164$	0	0
$165$	109448.	0.312966
$166$	0	0
$167$	−489398.	−1.35791	−0.678955	−	0.734180i	$-0.737567\pi$
$167$	−489398.	−1.35791	−0.678955	+	0.734180i	$0.737567\pi$
$168$	0	0
$169$	350859.	0.944965
$170$	0	0
$171$	187027.	0.489118
$172$	0	0
$173$	−134063.	−0.340561	−0.170280	−	0.985396i	$-0.554467\pi$
$173$	−134063.	−0.340561	−0.170280	+	0.985396i	$0.554467\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−74130.0	−0.177853
$178$	0	0
$179$	−831307.	−1.93923	−0.969614	−	0.244639i	$-0.921331\pi$
$179$	−831307.	−1.93923	−0.969614	+	0.244639i	$0.921331\pi$
$180$	0	0
$181$	711957.	1.61532	0.807658	−	0.589652i	$-0.200735\pi$
$181$	711957.	1.61532	0.807658	+	0.589652i	$0.200735\pi$
$182$	0	0
$183$	366950.	0.809988
$184$	0	0
$185$	547107.	1.17529
$186$	0	0
$187$	28494.0	0.0595867
$188$	0	0
$189$	0	0
$190$	0	0
$191$	580389.	1.15116	0.575580	−	0.817745i	$-0.304776\pi$
$191$	580389.	1.15116	0.575580	+	0.817745i	$0.304776\pi$
$192$	0	0
$193$	−287860.	−0.556273	−0.278137	−	0.960542i	$-0.589717\pi$
$193$	−287860.	−0.556273	−0.278137	+	0.960542i	$0.589717\pi$
$194$	0	0
$195$	1.03420e6	1.94769
$196$	0	0
$197$	859783.	1.57842	0.789211	−	0.614121i	$-0.210490\pi$
$197$	859783.	1.57842	0.789211	+	0.614121i	$0.210490\pi$
$198$	0	0
$199$	418333.	0.748840	0.374420	−	0.927259i	$-0.377842\pi$
$199$	418333.	0.748840	0.374420	+	0.927259i	$0.377842\pi$
$200$	0	0
$201$	26945.4	0.0470429
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.00314e6	3.32911
$206$	0	0
$207$	340468.	0.552269
$208$	0	0
$209$	191628.	0.303454
$210$	0	0
$211$	787076.	1.21706	0.608528	−	0.793532i	$-0.291760\pi$
$211$	787076.	1.21706	0.608528	+	0.793532i	$0.291760\pi$
$212$	0	0
$213$	255585.	0.385999
$214$	0	0
$215$	471173.	0.695160
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−849049.	−1.19625
$220$	0	0
$221$	269248.	0.370827
$222$	0	0
$223$	766758.	1.03252	0.516258	−	0.856433i	$-0.327325\pi$
$223$	766758.	1.03252	0.516258	+	0.856433i	$0.327325\pi$
$224$	0	0
$225$	−563193.	−0.741653
$226$	0	0
$227$	245220.	0.315857	0.157929	−	0.987451i	$-0.449518\pi$
$227$	245220.	0.315857	0.157929	+	0.987451i	$0.449518\pi$
$228$	0	0
$229$	182839.	0.230399	0.115199	−	0.993342i	$-0.463249\pi$
$229$	182839.	0.230399	0.115199	+	0.993342i	$0.463249\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	655948.	0.791552	0.395776	−	0.918347i	$-0.370476\pi$
$233$	655948.	0.791552	0.395776	+	0.918347i	$0.370476\pi$
$234$	0	0
$235$	29856.0	0.0352664
$236$	0	0
$237$	−903476.	−1.04483
$238$	0	0
$239$	875734.	0.991694	0.495847	−	0.868410i	$-0.334858\pi$
$239$	875734.	0.991694	0.495847	+	0.868410i	$0.334858\pi$
$240$	0	0
$241$	113437.	0.125809	0.0629046	−	0.998020i	$-0.479964\pi$
$241$	113437.	0.125809	0.0629046	+	0.998020i	$0.479964\pi$
$242$	0	0
$243$	−627458.	−0.681662
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.81075e6	1.88849
$248$	0	0
$249$	−522684.	−0.534246
$250$	0	0
$251$	598368.	0.599493	0.299746	−	0.954019i	$-0.403098\pi$
$251$	598368.	0.599493	0.299746	+	0.954019i	$0.403098\pi$
$252$	0	0
$253$	348844.	0.342633
$254$	0	0
$255$	385593.	0.371346
$256$	0	0
$257$	−845203.	−0.798231	−0.399115	−	0.916901i	$-0.630683\pi$
$257$	−845203.	−0.798231	−0.399115	+	0.916901i	$0.630683\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−407548.	−0.370320
$262$	0	0
$263$	−1.29754e6	−1.15673	−0.578366	−	0.815778i	$-0.696309\pi$
$263$	−1.29754e6	−1.15673	−0.578366	+	0.815778i	$0.696309\pi$
$264$	0	0
$265$	681414.	0.596068
$266$	0	0
$267$	−495054.	−0.424986
$268$	0	0
$269$	−1.12453e6	−0.947524	−0.473762	−	0.880653i	$-0.657104\pi$
$269$	−1.12453e6	−0.947524	−0.473762	+	0.880653i	$0.657104\pi$
$270$	0	0
$271$	858010.	0.709691	0.354845	−	0.934925i	$-0.384533\pi$
$271$	858010.	0.709691	0.354845	+	0.934925i	$0.384533\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−577047.	−0.460129
$276$	0	0
$277$	−2.00294e6	−1.56844	−0.784219	−	0.620484i	$-0.786936\pi$
$277$	−2.00294e6	−1.56844	−0.784219	+	0.620484i	$0.786936\pi$
$278$	0	0
$279$	306822.	0.235981
$280$	0	0
$281$	−1.19565e6	−0.903312	−0.451656	−	0.892192i	$-0.649166\pi$
$281$	−1.19565e6	−0.903312	−0.451656	+	0.892192i	$0.649166\pi$
$282$	0	0
$283$	848458.	0.629744	0.314872	−	0.949134i	$-0.398038\pi$
$283$	848458.	0.629744	0.314872	+	0.949134i	$0.398038\pi$
$284$	0	0
$285$	2.59319e6	1.89114
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.31947e6	−0.929298
$290$	0	0
$291$	327648.	0.226817
$292$	0	0
$293$	273221.	0.185928	0.0929640	−	0.995669i	$-0.470366\pi$
$293$	273221.	0.185928	0.0929640	+	0.995669i	$0.470366\pi$
$294$	0	0
$295$	581190.	0.388833
$296$	0	0
$297$	−370620.	−0.243802
$298$	0	0
$299$	3.29633e6	2.13232
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.92506e6	−1.20458
$304$	0	0
$305$	−2.87694e6	−1.77085
$306$	0	0
$307$	2.57136e6	1.55710	0.778552	−	0.627581i	$-0.215955\pi$
$307$	2.57136e6	1.55710	0.778552	+	0.627581i	$0.215955\pi$
$308$	0	0
$309$	−1.89814e6	−1.13092
$310$	0	0
$311$	1.14728e6	0.672617	0.336309	−	0.941752i	$-0.390821\pi$
$311$	1.14728e6	0.672617	0.336309	+	0.941752i	$0.390821\pi$
$312$	0	0
$313$	275870.	0.159163	0.0795817	−	0.996828i	$-0.474642\pi$
$313$	275870.	0.159163	0.0795817	+	0.996828i	$0.474642\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−424172.	−0.237079	−0.118540	−	0.992949i	$-0.537821\pi$
$317$	−424172.	−0.237079	−0.118540	+	0.992949i	$0.537821\pi$
$318$	0	0
$319$	−417573.	−0.229750
$320$	0	0
$321$	1.18053e6	0.639464
$322$	0	0
$323$	675120.	0.360060
$324$	0	0
$325$	−5.45269e6	−2.86353
$326$	0	0
$327$	2.01495e6	1.04206
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.09545e6	−0.549569	−0.274785	−	0.961506i	$-0.588607\pi$
$331$	−1.09545e6	−0.549569	−0.274785	+	0.961506i	$0.588607\pi$
$332$	0	0
$333$	−491615.	−0.242949
$334$	0	0
$335$	−211256.	−0.102848
$336$	0	0
$337$	−1.43125e6	−0.686502	−0.343251	−	0.939244i	$-0.611528\pi$
$337$	−1.43125e6	−0.686502	−0.343251	+	0.939244i	$0.611528\pi$
$338$	0	0
$339$	−453812.	−0.214475
$340$	0	0
$341$	314370.	0.146405
$342$	0	0
$343$	0	0
$344$	0	0
$345$	4.72071e6	2.13530
$346$	0	0
$347$	−924631.	−0.412235	−0.206118	−	0.978527i	$-0.566083\pi$
$347$	−924631.	−0.412235	−0.206118	+	0.978527i	$0.566083\pi$
$348$	0	0
$349$	2.32854e6	1.02334	0.511669	−	0.859182i	$-0.329027\pi$
$349$	2.32854e6	1.02334	0.511669	+	0.859182i	$0.329027\pi$
$350$	0	0
$351$	−3.50209e6	−1.51726
$352$	0	0
$353$	2.16381e6	0.924236	0.462118	−	0.886818i	$-0.347090\pi$
$353$	2.16381e6	0.924236	0.462118	+	0.886818i	$0.347090\pi$
$354$	0	0
$355$	−2.00382e6	−0.843894
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.72484e6	−1.52536	−0.762680	−	0.646777i	$-0.776117\pi$
$359$	−3.72484e6	−1.52536	−0.762680	+	0.646777i	$0.776117\pi$
$360$	0	0
$361$	2.06422e6	0.833660
$362$	0	0
$363$	1.90577e6	0.759109
$364$	0	0
$365$	6.65667e6	2.61532
$366$	0	0
$367$	−3.80685e6	−1.47537	−0.737684	−	0.675146i	$-0.764081\pi$
$367$	−3.80685e6	−1.47537	−0.737684	+	0.675146i	$0.764081\pi$
$368$	0	0
$369$	−1.79997e6	−0.688176
$370$	0	0
$371$	0	0
$372$	0	0
$373$	237891.	0.0885330	0.0442665	−	0.999020i	$-0.485905\pi$
$373$	237891.	0.0885330	0.0442665	+	0.999020i	$0.485905\pi$
$374$	0	0
$375$	−4.00573e6	−1.47097
$376$	0	0
$377$	−3.94577e6	−1.42981
$378$	0	0
$379$	4.19409e6	1.49982	0.749911	−	0.661539i	$-0.230096\pi$
$379$	4.19409e6	1.49982	0.749911	+	0.661539i	$0.230096\pi$
$380$	0	0
$381$	1.00538e6	0.354828
$382$	0	0
$383$	−4.33905e6	−1.51146	−0.755732	−	0.654881i	$-0.772719\pi$
$383$	−4.33905e6	−1.51146	−0.755732	+	0.654881i	$0.772719\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−423383.	−0.143700
$388$	0	0
$389$	522439.	0.175050	0.0875250	−	0.996162i	$-0.472104\pi$
$389$	522439.	0.175050	0.0875250	+	0.996162i	$0.472104\pi$
$390$	0	0
$391$	1.22900e6	0.406548
$392$	0	0
$393$	−1.95265e6	−0.637739
$394$	0	0
$395$	7.08338e6	2.28427
$396$	0	0
$397$	2.96716e6	0.944854	0.472427	−	0.881370i	$-0.343378\pi$
$397$	2.96716e6	0.944854	0.472427	+	0.881370i	$0.343378\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.43448e6	−0.445486	−0.222743	−	0.974877i	$-0.571501\pi$
$401$	−1.43448e6	−0.445486	−0.222743	+	0.974877i	$0.571501\pi$
$402$	0	0
$403$	2.97058e6	0.911125
$404$	0	0
$405$	−2.93198e6	−0.888227
$406$	0	0
$407$	−503709.	−0.150728
$408$	0	0
$409$	−2.07226e6	−0.612542	−0.306271	−	0.951944i	$-0.599081\pi$
$409$	−2.07226e6	−0.612542	−0.306271	+	0.951944i	$0.599081\pi$
$410$	0	0
$411$	−1.09792e6	−0.320603
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.09792e6	1.16800
$416$	0	0
$417$	−1.12554e6	−0.316971
$418$	0	0
$419$	−3.33983e6	−0.929373	−0.464686	−	0.885475i	$-0.653833\pi$
$419$	−3.33983e6	−0.929373	−0.464686	+	0.885475i	$0.653833\pi$
$420$	0	0
$421$	3.37062e6	0.926839	0.463419	−	0.886139i	$-0.346622\pi$
$421$	3.37062e6	0.926839	0.463419	+	0.886139i	$0.346622\pi$
$422$	0	0
$423$	−26827.7	−0.00729010
$424$	0	0
$425$	−2.03298e6	−0.545961
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−952167.	−0.249787
$430$	0	0
$431$	3.15789e6	0.818849	0.409425	−	0.912344i	$-0.365729\pi$
$431$	3.15789e6	0.818849	0.409425	+	0.912344i	$0.365729\pi$
$432$	0	0
$433$	5.90699e6	1.51407	0.757035	−	0.653374i	$-0.226647\pi$
$433$	5.90699e6	1.51407	0.757035	+	0.653374i	$0.226647\pi$
$434$	0	0
$435$	−5.65079e6	−1.43181
$436$	0	0
$437$	8.26531e6	2.07041
$438$	0	0
$439$	−1.85888e6	−0.460351	−0.230176	−	0.973149i	$-0.573930\pi$
$439$	−1.85888e6	−0.460351	−0.230176	+	0.973149i	$0.573930\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.64300e6	1.85035	0.925176	−	0.379539i	$-0.123917\pi$
$443$	7.64300e6	1.85035	0.925176	+	0.379539i	$0.123917\pi$
$444$	0	0
$445$	3.88129e6	0.929130
$446$	0	0
$447$	3.32831e6	0.787870
$448$	0	0
$449$	−6.33168e6	−1.48219	−0.741093	−	0.671402i	$-0.765692\pi$
$449$	−6.33168e6	−1.48219	−0.741093	+	0.671402i	$0.765692\pi$
$450$	0	0
$451$	−1.84425e6	−0.426951
$452$	0	0
$453$	−6.83039e6	−1.56387
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2.94801e6	−0.660295	−0.330148	−	0.943929i	$-0.607099\pi$
$457$	−2.94801e6	−0.660295	−0.330148	+	0.943929i	$0.607099\pi$
$458$	0	0
$459$	−1.30572e6	−0.289281
$460$	0	0
$461$	2.89750e6	0.634996	0.317498	−	0.948259i	$-0.397157\pi$
$461$	2.89750e6	0.634996	0.317498	+	0.948259i	$0.397157\pi$
$462$	0	0
$463$	2.53742e6	0.550098	0.275049	−	0.961430i	$-0.411306\pi$
$463$	2.53742e6	0.550098	0.275049	+	0.961430i	$0.411306\pi$
$464$	0	0
$465$	4.25420e6	0.912400
$466$	0	0
$467$	1.08250e6	0.229686	0.114843	−	0.993384i	$-0.463364\pi$
$467$	1.08250e6	0.229686	0.114843	+	0.993384i	$0.463364\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−40961.1	−0.00850784
$472$	0	0
$473$	−433799.	−0.0891529
$474$	0	0
$475$	−1.36722e7	−2.78039
$476$	0	0
$477$	−612299.	−0.123216
$478$	0	0
$479$	−5.65011e6	−1.12517	−0.562585	−	0.826739i	$-0.690193\pi$
$479$	−5.65011e6	−1.12517	−0.562585	+	0.826739i	$0.690193\pi$
$480$	0	0
$481$	−4.75969e6	−0.938029
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.56881e6	−0.495881
$486$	0	0
$487$	3.82581e6	0.730973	0.365486	−	0.930817i	$-0.380903\pi$
$487$	3.82581e6	0.730973	0.365486	+	0.930817i	$0.380903\pi$
$488$	0	0
$489$	−937019.	−0.177205
$490$	0	0
$491$	8.53724e6	1.59814	0.799068	−	0.601240i	$-0.205327\pi$
$491$	8.53724e6	1.59814	0.799068	+	0.601240i	$0.205327\pi$
$492$	0	0
$493$	−1.47114e6	−0.272608
$494$	0	0
$495$	771052.	0.141439
$496$	0	0
$497$	0	0
$498$	0	0
$499$	2.52909e6	0.454687	0.227344	−	0.973815i	$-0.426996\pi$
$499$	2.52909e6	0.454687	0.227344	+	0.973815i	$0.426996\pi$
$500$	0	0
$501$	6.09742e6	1.08530
$502$	0	0
$503$	−4.58634e6	−0.808251	−0.404126	−	0.914703i	$-0.632424\pi$
$503$	−4.58634e6	−0.808251	−0.404126	+	0.914703i	$0.632424\pi$
$504$	0	0
$505$	1.50927e7	2.63353
$506$	0	0
$507$	−4.37135e6	−0.755259
$508$	0	0
$509$	2.37089e6	0.405617	0.202809	−	0.979218i	$-0.434993\pi$
$509$	2.37089e6	0.405617	0.202809	+	0.979218i	$0.434993\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−8.78126e6	−1.47321
$514$	0	0
$515$	1.48817e7	2.47249
$516$	0	0
$517$	−27487.7	−0.00452285
$518$	0	0
$519$	1.67030e6	0.272192
$520$	0	0
$521$	−626890.	−0.101181	−0.0505903	−	0.998719i	$-0.516110\pi$
$521$	−626890.	−0.101181	−0.0505903	+	0.998719i	$0.516110\pi$
$522$	0	0
$523$	4.12840e6	0.659976	0.329988	−	0.943985i	$-0.392955\pi$
$523$	4.12840e6	0.659976	0.329988	+	0.943985i	$0.392955\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.10755e6	0.173715
$528$	0	0
$529$	8.61003e6	1.33772
$530$	0	0
$531$	−522241.	−0.0803776
$532$	0	0
$533$	−1.74268e7	−2.65706
$534$	0	0
$535$	−9.25556e6	−1.39804
$536$	0	0
$537$	1.03573e7	1.54992
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.22629e6	1.20840	0.604200	−	0.796833i	$-0.293493\pi$
$541$	8.22629e6	1.20840	0.604200	+	0.796833i	$0.293493\pi$
$542$	0	0
$543$	−8.87028e6	−1.29103
$544$	0	0
$545$	−1.57975e7	−2.27823
$546$	0	0
$547$	−1.05898e7	−1.51328	−0.756642	−	0.653830i	$-0.773161\pi$
$547$	−1.05898e7	−1.51328	−0.756642	+	0.653830i	$0.773161\pi$
$548$	0	0
$549$	2.58514e6	0.366060
$550$	0	0
$551$	−9.89375e6	−1.38830
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−6.81642e6	−0.939342
$556$	0	0
$557$	3.16577e6	0.432356	0.216178	−	0.976354i	$-0.430641\pi$
$557$	3.16577e6	0.432356	0.216178	+	0.976354i	$0.430641\pi$
$558$	0	0
$559$	−4.09909e6	−0.554827
$560$	0	0
$561$	−355007.	−0.0476244
$562$	0	0
$563$	1.64690e6	0.218975	0.109488	−	0.993988i	$-0.465079\pi$
$563$	1.64690e6	0.218975	0.109488	+	0.993988i	$0.465079\pi$
$564$	0	0
$565$	3.55795e6	0.468898
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−8.81019e6	−1.14079	−0.570394	−	0.821371i	$-0.693209\pi$
$569$	−8.81019e6	−1.14079	−0.570394	+	0.821371i	$0.693209\pi$
$570$	0	0
$571$	−4.43285e6	−0.568974	−0.284487	−	0.958680i	$-0.591823\pi$
$571$	−4.43285e6	−0.568974	−0.284487	+	0.958680i	$0.591823\pi$
$572$	0	0
$573$	−7.23108e6	−0.920061
$574$	0	0
$575$	−2.48893e7	−3.13937
$576$	0	0
$577$	8.84105e6	1.10551	0.552757	−	0.833343i	$-0.313576\pi$
$577$	8.84105e6	1.10551	0.552757	+	0.833343i	$0.313576\pi$
$578$	0	0
$579$	3.58645e6	0.444599
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−627362.	−0.0764446
$584$	0	0
$585$	7.28589e6	0.880224
$586$	0	0
$587$	3.17377e6	0.380172	0.190086	−	0.981767i	$-0.439123\pi$
$587$	3.17377e6	0.380172	0.190086	+	0.981767i	$0.439123\pi$
$588$	0	0
$589$	7.44852e6	0.884671
$590$	0	0
$591$	−1.07121e7	−1.26155
$592$	0	0
$593$	7.20524e6	0.841418	0.420709	−	0.907196i	$-0.361781\pi$
$593$	7.20524e6	0.841418	0.420709	+	0.907196i	$0.361781\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.21201e6	−0.598508
$598$	0	0
$599$	−8.39924e6	−0.956474	−0.478237	−	0.878231i	$-0.658724\pi$
$599$	−8.39924e6	−0.956474	−0.478237	+	0.878231i	$0.658724\pi$
$600$	0	0
$601$	3.87350e6	0.437438	0.218719	−	0.975788i	$-0.429812\pi$
$601$	3.87350e6	0.437438	0.218719	+	0.975788i	$0.429812\pi$
$602$	0	0
$603$	189829.	0.0212602
$604$	0	0
$605$	−1.49415e7	−1.65961
$606$	0	0
$607$	599025.	0.0659892	0.0329946	−	0.999456i	$-0.489496\pi$
$607$	599025.	0.0659892	0.0329946	+	0.999456i	$0.489496\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−259739.	−0.0281472
$612$	0	0
$613$	−1.42766e7	−1.53452	−0.767260	−	0.641336i	$-0.778380\pi$
$613$	−1.42766e7	−1.53452	−0.767260	+	0.641336i	$0.778380\pi$
$614$	0	0
$615$	−2.49572e7	−2.66078
$616$	0	0
$617$	808348.	0.0854841	0.0427421	−	0.999086i	$-0.486391\pi$
$617$	808348.	0.0854841	0.0427421	+	0.999086i	$0.486391\pi$
$618$	0	0
$619$	7.85443e6	0.823926	0.411963	−	0.911201i	$-0.364843\pi$
$619$	7.85443e6	0.823926	0.411963	+	0.911201i	$0.364843\pi$
$620$	0	0
$621$	−1.59856e7	−1.66341
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.13540e7	1.16265
$626$	0	0
$627$	−2.38749e6	−0.242534
$628$	0	0
$629$	−1.77461e6	−0.178845
$630$	0	0
$631$	5.49768e6	0.549675	0.274837	−	0.961491i	$-0.411376\pi$
$631$	5.49768e6	0.549675	0.274837	+	0.961491i	$0.411376\pi$
$632$	0	0
$633$	−9.80619e6	−0.972728
$634$	0	0
$635$	−7.88233e6	−0.775747
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.80058e6	0.174445
$640$	0	0
$641$	2.03482e6	0.195606	0.0978029	−	0.995206i	$-0.468819\pi$
$641$	2.03482e6	0.195606	0.0978029	+	0.995206i	$0.468819\pi$
$642$	0	0
$643$	5.87725e6	0.560592	0.280296	−	0.959914i	$-0.409567\pi$
$643$	5.87725e6	0.560592	0.280296	+	0.959914i	$0.409567\pi$
$644$	0	0
$645$	−5.87036e6	−0.555604
$646$	0	0
$647$	5.82452e6	0.547015	0.273508	−	0.961870i	$-0.411816\pi$
$647$	5.82452e6	0.547015	0.273508	+	0.961870i	$0.411816\pi$
$648$	0	0
$649$	−535089.	−0.0498671
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.30966e6	0.120192	0.0600958	−	0.998193i	$-0.480859\pi$
$653$	1.30966e6	0.120192	0.0600958	+	0.998193i	$0.480859\pi$
$654$	0	0
$655$	1.53091e7	1.39426
$656$	0	0
$657$	−5.98150e6	−0.540625
$658$	0	0
$659$	1.02215e7	0.916857	0.458429	−	0.888731i	$-0.348412\pi$
$659$	1.02215e7	0.916857	0.458429	+	0.888731i	$0.348412\pi$
$660$	0	0
$661$	1.89747e7	1.68916	0.844580	−	0.535430i	$-0.179850\pi$
$661$	1.89747e7	1.68916	0.844580	+	0.535430i	$0.179850\pi$
$662$	0	0
$663$	−3.35456e6	−0.296382
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.80108e7	−1.56754
$668$	0	0
$669$	−9.55305e6	−0.825234
$670$	0	0
$671$	2.64873e6	0.227108
$672$	0	0
$673$	−7.91921e6	−0.673976	−0.336988	−	0.941509i	$-0.609408\pi$
$673$	−7.91921e6	−0.673976	−0.336988	+	0.941509i	$0.609408\pi$
$674$	0	0
$675$	2.64430e7	2.23383
$676$	0	0
$677$	−2.12395e7	−1.78104	−0.890519	−	0.454946i	$-0.849658\pi$
$677$	−2.12395e7	−1.78104	−0.890519	+	0.454946i	$0.849658\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−3.05520e6	−0.252448
$682$	0	0
$683$	1.00250e7	0.822307	0.411153	−	0.911566i	$-0.365126\pi$
$683$	1.00250e7	0.822307	0.411153	+	0.911566i	$0.365126\pi$
$684$	0	0
$685$	8.60787e6	0.700921
$686$	0	0
$687$	−2.27800e6	−0.184145
$688$	0	0
$689$	−5.92813e6	−0.475740
$690$	0	0
$691$	−9.08309e6	−0.723667	−0.361833	−	0.932243i	$-0.617849\pi$
$691$	−9.08309e6	−0.723667	−0.361833	+	0.932243i	$0.617849\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.82436e6	0.692981
$696$	0	0
$697$	−6.49744e6	−0.506594
$698$	0	0
$699$	−8.17246e6	−0.632645
$700$	0	0
$701$	−3.36984e6	−0.259009	−0.129504	−	0.991579i	$-0.541339\pi$
$701$	−3.36984e6	−0.259009	−0.129504	+	0.991579i	$0.541339\pi$
$702$	0	0
$703$	−1.19346e7	−0.910793
$704$	0	0
$705$	−371976.	−0.0281866
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.06576e7	−0.796242	−0.398121	−	0.917333i	$-0.630338\pi$
$709$	−1.06576e7	−0.796242	−0.398121	+	0.917333i	$0.630338\pi$
$710$	0	0
$711$	−6.36493e6	−0.472193
$712$	0	0
$713$	1.35595e7	0.998892
$714$	0	0
$715$	7.46513e6	0.546100
$716$	0	0
$717$	−1.09108e7	−0.792607
$718$	0	0
$719$	1.18689e7	0.856226	0.428113	−	0.903725i	$-0.359179\pi$
$719$	1.18689e7	0.856226	0.428113	+	0.903725i	$0.359179\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.41331e6	−0.100553
$724$	0	0
$725$	2.97930e7	2.10508
$726$	0	0
$727$	−1.17898e7	−0.827311	−0.413656	−	0.910433i	$-0.635748\pi$
$727$	−1.17898e7	−0.827311	−0.413656	+	0.910433i	$0.635748\pi$
$728$	0	0
$729$	1.51114e7	1.05314
$730$	0	0
$731$	−1.52831e6	−0.105783
$732$	0	0
$733$	−396045.	−0.0272260	−0.0136130	−	0.999907i	$-0.504333\pi$
$733$	−396045.	−0.0272260	−0.0136130	+	0.999907i	$0.504333\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	194498.	0.0131901
$738$	0	0
$739$	2.39493e7	1.61318	0.806589	−	0.591112i	$-0.201311\pi$
$739$	2.39493e7	1.61318	0.806589	+	0.591112i	$0.201311\pi$
$740$	0	0
$741$	−2.25601e7	−1.50937
$742$	0	0
$743$	7.45015e6	0.495100	0.247550	−	0.968875i	$-0.420375\pi$
$743$	7.45015e6	0.495100	0.247550	+	0.968875i	$0.420375\pi$
$744$	0	0
$745$	−2.60944e7	−1.72249
$746$	0	0
$747$	−3.68228e6	−0.241443
$748$	0	0
$749$	0	0
$750$	0	0
$751$	6.08615e6	0.393770	0.196885	−	0.980427i	$-0.436917\pi$
$751$	6.08615e6	0.393770	0.196885	+	0.980427i	$0.436917\pi$
$752$	0	0
$753$	−7.45508e6	−0.479142
$754$	0	0
$755$	5.35512e7	3.41902
$756$	0	0
$757$	−9.81022e6	−0.622213	−0.311107	−	0.950375i	$-0.600700\pi$
$757$	−9.81022e6	−0.622213	−0.311107	+	0.950375i	$0.600700\pi$
$758$	0	0
$759$	−4.34625e6	−0.273848
$760$	0	0
$761$	−1.91742e7	−1.20021	−0.600103	−	0.799922i	$-0.704874\pi$
$761$	−1.91742e7	−1.20021	−0.600103	+	0.799922i	$0.704874\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.71648e6	0.167823
$766$	0	0
$767$	−5.05621e6	−0.310339
$768$	0	0
$769$	−2.30199e7	−1.40374	−0.701872	−	0.712303i	$-0.747652\pi$
$769$	−2.30199e7	−1.40374	−0.701872	+	0.712303i	$0.747652\pi$
$770$	0	0
$771$	1.05304e7	0.637983
$772$	0	0
$773$	8.25290e6	0.496773	0.248386	−	0.968661i	$-0.420100\pi$
$773$	8.25290e6	0.496773	0.248386	+	0.968661i	$0.420100\pi$
$774$	0	0
$775$	−2.24297e7	−1.34143
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.36966e7	−2.57991
$780$	0	0
$781$	1.84487e6	0.108228
$782$	0	0
$783$	1.91351e7	1.11539
$784$	0	0
$785$	321141.	0.0186004
$786$	0	0
$787$	5.57084e6	0.320615	0.160307	−	0.987067i	$-0.448751\pi$
$787$	5.57084e6	0.320615	0.160307	+	0.987067i	$0.448751\pi$
$788$	0	0
$789$	1.61661e7	0.924513
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.50286e7	1.41336
$794$	0	0
$795$	−8.48974e6	−0.476405
$796$	0	0
$797$	1.34823e7	0.751827	0.375914	−	0.926655i	$-0.377329\pi$
$797$	1.34823e7	0.751827	0.375914	+	0.926655i	$0.377329\pi$
$798$	0	0
$799$	−96841.4	−0.00536654
$800$	0	0
$801$	−3.48762e6	−0.192065
$802$	0	0
$803$	−6.12864e6	−0.335410
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.40105e7	0.757305
$808$	0	0
$809$	1.45843e7	0.783455	0.391727	−	0.920081i	$-0.371878\pi$
$809$	1.45843e7	0.783455	0.391727	+	0.920081i	$0.371878\pi$
$810$	0	0
$811$	1.27768e7	0.682133	0.341067	−	0.940039i	$-0.389212\pi$
$811$	1.27768e7	0.682133	0.341067	+	0.940039i	$0.389212\pi$
$812$	0	0
$813$	−1.06900e7	−0.567218
$814$	0	0
$815$	7.34636e6	0.387417
$816$	0	0
$817$	−1.02782e7	−0.538718
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−5.05627e6	−0.261802	−0.130901	−	0.991395i	$-0.541787\pi$
$821$	−5.05627e6	−0.261802	−0.130901	+	0.991395i	$0.541787\pi$
$822$	0	0
$823$	3.03045e7	1.55958	0.779791	−	0.626040i	$-0.215325\pi$
$823$	3.03045e7	1.55958	0.779791	+	0.626040i	$0.215325\pi$
$824$	0	0
$825$	7.18944e6	0.367757
$826$	0	0
$827$	−4.98199e6	−0.253302	−0.126651	−	0.991947i	$-0.540423\pi$
$827$	−4.98199e6	−0.253302	−0.126651	+	0.991947i	$0.540423\pi$
$828$	0	0
$829$	1.35609e7	0.685331	0.342666	−	0.939457i	$-0.388670\pi$
$829$	1.35609e7	0.685331	0.342666	+	0.939457i	$0.388670\pi$
$830$	0	0
$831$	2.49546e7	1.25357
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−4.78046e7	−2.37276
$836$	0	0
$837$	−1.44059e7	−0.710766
$838$	0	0
$839$	2.69853e6	0.132350	0.0661748	−	0.997808i	$-0.478920\pi$
$839$	2.69853e6	0.132350	0.0661748	+	0.997808i	$0.478920\pi$
$840$	0	0
$841$	1.04818e6	0.0511030
$842$	0	0
$843$	1.48966e7	0.721968
$844$	0	0
$845$	3.42720e7	1.65119
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.05709e7	−0.503321
$850$	0	0
$851$	−2.17260e7	−1.02839
$852$	0	0
$853$	−1.11719e7	−0.525722	−0.262861	−	0.964834i	$-0.584666\pi$
$853$	−1.11719e7	−0.525722	−0.262861	+	0.964834i	$0.584666\pi$
$854$	0	0
$855$	1.82689e7	0.854666
$856$	0	0
$857$	−8.53589e6	−0.397006	−0.198503	−	0.980100i	$-0.563608\pi$
$857$	−8.53589e6	−0.397006	−0.198503	+	0.980100i	$0.563608\pi$
$858$	0	0
$859$	1.98762e7	0.919076	0.459538	−	0.888158i	$-0.348015\pi$
$859$	1.98762e7	0.919076	0.459538	+	0.888158i	$0.348015\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.55367e7	1.16718	0.583590	−	0.812049i	$-0.301648\pi$
$863$	2.55367e7	1.16718	0.583590	+	0.812049i	$0.301648\pi$
$864$	0	0
$865$	−1.30954e7	−0.595083
$866$	0	0
$867$	1.64393e7	0.742738
$868$	0	0
$869$	−6.52151e6	−0.292954
$870$	0	0
$871$	1.83787e6	0.0820861
$872$	0	0
$873$	2.30826e6	0.102506
$874$	0	0
$875$	0	0
$876$	0	0
$877$	79097.2	0.00347266	0.00173633	−	0.999998i	$-0.499447\pi$
$877$	79097.2	0.00347266	0.00173633	+	0.999998i	$0.499447\pi$
$878$	0	0
$879$	−3.40406e6	−0.148602
$880$	0	0
$881$	2.13111e7	0.925053	0.462526	−	0.886605i	$-0.346943\pi$
$881$	2.13111e7	0.925053	0.462526	+	0.886605i	$0.346943\pi$
$882$	0	0
$883$	−1.77648e6	−0.0766760	−0.0383380	−	0.999265i	$-0.512206\pi$
$883$	−1.77648e6	−0.0766760	−0.0383380	+	0.999265i	$0.512206\pi$
$884$	0	0
$885$	−7.24106e6	−0.310773
$886$	0	0
$887$	3.16331e7	1.35000	0.674998	−	0.737819i	$-0.264144\pi$
$887$	3.16331e7	1.35000	0.674998	+	0.737819i	$0.264144\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.69941e6	0.113913
$892$	0	0
$893$	−651279.	−0.0273299
$894$	0	0
$895$	−8.12025e7	−3.38853
$896$	0	0
$897$	−4.10690e7	−1.70425
$898$	0	0
$899$	−1.62310e7	−0.669800
$900$	0	0
$901$	−2.21025e6	−0.0907045
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.95443e7	2.82254
$906$	0	0
$907$	−3.87208e7	−1.56288	−0.781442	−	0.623978i	$-0.785515\pi$
$907$	−3.87208e7	−1.56288	−0.781442	+	0.623978i	$0.785515\pi$
$908$	0	0
$909$	−1.35619e7	−0.544391
$910$	0	0
$911$	−3.02596e7	−1.20800	−0.604000	−	0.796985i	$-0.706427\pi$
$911$	−3.02596e7	−1.20800	−0.604000	+	0.796985i	$0.706427\pi$
$912$	0	0
$913$	−3.77286e6	−0.149794
$914$	0	0
$915$	3.58438e7	1.41534
$916$	0	0
$917$	0	0
$918$	0	0
$919$	2.94529e7	1.15037	0.575187	−	0.818022i	$-0.304929\pi$
$919$	2.94529e7	1.15037	0.575187	+	0.818022i	$0.304929\pi$
$920$	0	0
$921$	−3.20366e7	−1.24451
$922$	0	0
$923$	1.74327e7	0.673536
$924$	0	0
$925$	3.59386e7	1.38104
$926$	0	0
$927$	−1.33723e7	−0.511100
$928$	0	0
$929$	−1.14539e7	−0.435426	−0.217713	−	0.976013i	$-0.569860\pi$
$929$	−1.14539e7	−0.435426	−0.217713	+	0.976013i	$0.569860\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.42940e7	−0.537587
$934$	0	0
$935$	2.78330e6	0.104119
$936$	0	0
$937$	3.69132e7	1.37351	0.686757	−	0.726887i	$-0.259034\pi$
$937$	3.69132e7	1.37351	0.686757	+	0.726887i	$0.259034\pi$
$938$	0	0
$939$	−3.43707e6	−0.127211
$940$	0	0
$941$	−3.75064e7	−1.38080	−0.690401	−	0.723427i	$-0.742566\pi$
$941$	−3.75064e7	−1.38080	−0.690401	+	0.723427i	$0.742566\pi$
$942$	0	0
$943$	−7.95464e7	−2.91301
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.36067e7	0.493036	0.246518	−	0.969138i	$-0.420714\pi$
$947$	1.36067e7	0.493036	0.246518	+	0.969138i	$0.420714\pi$
$948$	0	0
$949$	−5.79113e7	−2.08736
$950$	0	0
$951$	5.28476e6	0.189485
$952$	0	0
$953$	3.67074e7	1.30925	0.654624	−	0.755955i	$-0.272827\pi$
$953$	3.67074e7	1.30925	0.654624	+	0.755955i	$0.272827\pi$
$954$	0	0
$955$	5.66927e7	2.01149
$956$	0	0
$957$	5.20255e6	0.183627
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.64097e7	−0.573180
$962$	0	0
$963$	8.31679e6	0.288995
$964$	0	0
$965$	−2.81183e7	−0.972010
$966$	0	0
$967$	−8.58815e6	−0.295348	−0.147674	−	0.989036i	$-0.547179\pi$
$967$	−8.58815e6	−0.295348	−0.147674	+	0.989036i	$0.547179\pi$
$968$	0	0
$969$	−8.41133e6	−0.287777
$970$	0	0
$971$	4.03948e7	1.37492	0.687459	−	0.726223i	$-0.258726\pi$
$971$	4.03948e7	1.37492	0.687459	+	0.726223i	$0.258726\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	6.79351e7	2.28867
$976$	0	0
$977$	1.99067e7	0.667212	0.333606	−	0.942713i	$-0.391734\pi$
$977$	1.99067e7	0.667212	0.333606	+	0.942713i	$0.391734\pi$
$978$	0	0
$979$	−3.57342e6	−0.119159
$980$	0	0
$981$	1.41952e7	0.470943
$982$	0	0
$983$	−1.68676e7	−0.556761	−0.278380	−	0.960471i	$-0.589798\pi$
$983$	−1.68676e7	−0.556761	−0.278380	+	0.960471i	$0.589798\pi$
$984$	0	0
$985$	8.39840e7	2.75808
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.87107e7	−0.608273
$990$	0	0
$991$	−4.90834e7	−1.58763	−0.793817	−	0.608157i	$-0.791909\pi$
$991$	−4.90834e7	−1.58763	−0.793817	+	0.608157i	$0.791909\pi$
$992$	0	0
$993$	1.36482e7	0.439241
$994$	0	0
$995$	4.08629e7	1.30849
$996$	0	0
$997$	2.79855e7	0.891650	0.445825	−	0.895120i	$-0.352910\pi$
$997$	2.79855e7	0.891650	0.445825	+	0.895120i	$0.352910\pi$
$998$	0	0
$999$	2.30823e7	0.731754

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.bl.1.2		5
4.3	odd	2		392.6.a.j.1.4		5
7.3	odd	6		112.6.i.f.65.2		10
7.5	odd	6		112.6.i.f.81.2		10
7.6	odd	2		784.6.a.bk.1.4		5
28.3	even	6		56.6.i.b.9.4	✓	10
28.11	odd	6		392.6.i.o.177.2		10
28.19	even	6		56.6.i.b.25.4	yes	10
28.23	odd	6		392.6.i.o.361.2		10
28.27	even	2		392.6.a.k.1.2		5
84.47	odd	6		504.6.s.b.361.1		10
84.59	odd	6		504.6.s.b.289.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.i.b.9.4	✓	10	28.3	even	6
56.6.i.b.25.4	yes	10	28.19	even	6
112.6.i.f.65.2		10	7.3	odd	6
112.6.i.f.81.2		10	7.5	odd	6
392.6.a.j.1.4		5	4.3	odd	2
392.6.a.k.1.2		5	28.27	even	2
392.6.i.o.177.2		10	28.11	odd	6
392.6.i.o.361.2		10	28.23	odd	6
504.6.s.b.289.1		10	84.59	odd	6
504.6.s.b.361.1		10	84.47	odd	6
784.6.a.bk.1.4		5	7.6	odd	2
784.6.a.bl.1.2		5	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q-12.4590 q^{3} +97.6805 q^{5} -87.7730 q^{9} -89.9322 q^{11} -849.795 q^{13} -1217.00 q^{15} -316.838 q^{17} -2130.80 q^{19} -3878.96 q^{23} +6416.47 q^{25} +4121.11 q^{27} +4643.20 q^{29} -3495.64 q^{31} +1120.47 q^{33} +5600.99 q^{37} +10587.6 q^{39} +20507.1 q^{41} +4823.62 q^{43} -8573.70 q^{45} +305.649 q^{47} +3947.49 q^{51} +6975.95 q^{53} -8784.62 q^{55} +26547.7 q^{57} +5949.91 q^{59} -29452.5 q^{61} -83008.4 q^{65} -2162.72 q^{67} +48328.1 q^{69} -20514.0 q^{71} +68147.4 q^{73} -79942.9 q^{75} +72515.9 q^{79} -30016.1 q^{81} +41952.3 q^{83} -30948.9 q^{85} -57849.7 q^{87} +39734.6 q^{89} +43552.2 q^{93} -208138. q^{95} -26298.1 q^{97} +7893.62 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 81 q^{5} + 390 q^{9} - 361 q^{11} + 342 q^{13} + 1049 q^{15} + 1809 q^{17} - 1277 q^{19} - 911 q^{23} + 3940 q^{25} + 4751 q^{27} + 5442 q^{29} - 2187 q^{31} - 5553 q^{33} - 8181 q^{37}+ \cdots - 249798 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 81 * q^5 + 390 * q^9 - 361 * q^11 + 342 * q^13 + 1049 * q^15 + 1809 * q^17 - 1277 * q^19 - 911 * q^23 + 3940 * q^25 + 4751 * q^27 + 5442 * q^29 - 2187 * q^31 - 5553 * q^33 - 8181 * q^37 + 3422 * q^39 + 16578 * q^41 - 6332 * q^43 + 41310 * q^45 - 16101 * q^47 + 67865 * q^51 + 16047 * q^53 - 45629 * q^55 + 22347 * q^57 - 71027 * q^59 + 31093 * q^61 - 64370 * q^65 - 47981 * q^67 + 137249 * q^69 - 22512 * q^71 + 123333 * q^73 + 45460 * q^75 + 212481 * q^79 + 52917 * q^81 - 87460 * q^83 + 222141 * q^85 - 318070 * q^87 + 129045 * q^89 - 252835 * q^93 - 300417 * q^95 + 328274 * q^97 - 249798 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more