LMFDB - Embedded newform 784.6.a.bn.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,6,Mod(1,784)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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:	yes
Analytic conductor:	$125.740914733$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 328x^{4} - 1328x^{3} + 25933x^{2} + 205840x + 390334 $ x^6 - 328x^4 - 1328x^3 + 25933x^2 + 205840x + 390334
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 392)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.40535$ 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)

$f(q)$	$=$	$q-9.64432 q^{3} -86.0611 q^{5} -149.987 q^{9} +O(q^{10})$	$q-9.64432 q^{3} -86.0611 q^{5} -149.987 q^{9} -492.475 q^{11} +122.653 q^{13} +830.001 q^{15} +1732.14 q^{17} -1997.05 q^{19} +2579.64 q^{23} +4281.52 q^{25} +3790.09 q^{27} +3059.57 q^{29} +3336.59 q^{31} +4749.58 q^{33} -12078.0 q^{37} -1182.90 q^{39} -252.801 q^{41} +11047.4 q^{43} +12908.1 q^{45} +22500.1 q^{47} -16705.3 q^{51} -39837.5 q^{53} +42382.9 q^{55} +19260.2 q^{57} +23066.2 q^{59} -11273.2 q^{61} -10555.6 q^{65} +33852.6 q^{67} -24878.8 q^{69} -14781.0 q^{71} +32642.0 q^{73} -41292.4 q^{75} -36358.1 q^{79} -106.005 q^{81} +4768.94 q^{83} -149070. q^{85} -29507.5 q^{87} +116259. q^{89} -32179.2 q^{93} +171868. q^{95} -62456.2 q^{97} +73864.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 122 q^{9}+O(q^{10}) $ 6 * q - 122 * q^9	$ 6 q - 122 q^{9} - 248 q^{11} + 376 q^{15} + 2904 q^{23} + 1510 q^{25} - 408 q^{29} - 18648 q^{37} + 15240 q^{39} + 10584 q^{43} - 3016 q^{51} - 66772 q^{53} - 56264 q^{57} - 55316 q^{65} + 169632 q^{67} + 54880 q^{71} - 50576 q^{79} - 270194 q^{81} - 406076 q^{85} - 164944 q^{93} + 551912 q^{95} + 173512 q^{99}+O(q^{100}) $ 6 * q - 122 * q^9 - 248 * q^11 + 376 * q^15 + 2904 * q^23 + 1510 * q^25 - 408 * q^29 - 18648 * q^37 + 15240 * q^39 + 10584 * q^43 - 3016 * q^51 - 66772 * q^53 - 56264 * q^57 - 55316 * q^65 + 169632 * q^67 + 54880 * q^71 - 50576 * q^79 - 270194 * q^81 - 406076 * q^85 - 164944 * q^93 + 551912 * q^95 + 173512 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−9.64432	−0.618683	−0.309342	−	0.950951i	$-0.600109\pi$
$3$	−9.64432	−0.618683	−0.309342	+	0.950951i	$0.600109\pi$
$4$	0	0
$5$	−86.0611	−1.53951	−0.769754	−	0.638340i	$-0.779621\pi$
$5$	−86.0611	−1.53951	−0.769754	+	0.638340i	$0.779621\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−149.987	−0.617231
$10$	0	0
$11$	−492.475	−1.22716	−0.613581	−	0.789632i	$-0.710272\pi$
$11$	−492.475	−1.22716	−0.613581	+	0.789632i	$0.710272\pi$
$12$	0	0
$13$	122.653	0.201288	0.100644	−	0.994922i	$-0.467910\pi$
$13$	122.653	0.201288	0.100644	+	0.994922i	$0.467910\pi$
$14$	0	0
$15$	830.001	0.952468
$16$	0	0
$17$	1732.14	1.45366	0.726828	−	0.686820i	$-0.240994\pi$
$17$	1732.14	1.45366	0.726828	+	0.686820i	$0.240994\pi$
$18$	0	0
$19$	−1997.05	−1.26913	−0.634563	−	0.772871i	$-0.718820\pi$
$19$	−1997.05	−1.26913	−0.634563	+	0.772871i	$0.718820\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2579.64	1.01681	0.508404	−	0.861119i	$-0.330236\pi$
$23$	2579.64	1.01681	0.508404	+	0.861119i	$0.330236\pi$
$24$	0	0
$25$	4281.52	1.37009
$26$	0	0
$27$	3790.09	1.00055
$28$	0	0
$29$	3059.57	0.675563	0.337781	−	0.941225i	$-0.390324\pi$
$29$	3059.57	0.675563	0.337781	+	0.941225i	$0.390324\pi$
$30$	0	0
$31$	3336.59	0.623590	0.311795	−	0.950149i	$-0.399070\pi$
$31$	3336.59	0.623590	0.311795	+	0.950149i	$0.399070\pi$
$32$	0	0
$33$	4749.58	0.759225
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−12078.0	−1.45041	−0.725204	−	0.688534i	$-0.758254\pi$
$37$	−12078.0	−1.45041	−0.725204	+	0.688534i	$0.758254\pi$
$38$	0	0
$39$	−1182.90	−0.124534
$40$	0	0
$41$	−252.801	−0.0234865	−0.0117433	−	0.999931i	$-0.503738\pi$
$41$	−252.801	−0.0234865	−0.0117433	+	0.999931i	$0.503738\pi$
$42$	0	0
$43$	11047.4	0.911146	0.455573	−	0.890198i	$-0.349434\pi$
$43$	11047.4	0.911146	0.455573	+	0.890198i	$0.349434\pi$
$44$	0	0
$45$	12908.1	0.950232
$46$	0	0
$47$	22500.1	1.48573	0.742865	−	0.669442i	$-0.233467\pi$
$47$	22500.1	1.48573	0.742865	+	0.669442i	$0.233467\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−16705.3	−0.899352
$52$	0	0
$53$	−39837.5	−1.94806	−0.974030	−	0.226420i	$-0.927298\pi$
$53$	−39837.5	−1.94806	−0.974030	+	0.226420i	$0.927298\pi$
$54$	0	0
$55$	42382.9	1.88923
$56$	0	0
$57$	19260.2	0.785187
$58$	0	0
$59$	23066.2	0.862672	0.431336	−	0.902191i	$-0.358042\pi$
$59$	23066.2	0.862672	0.431336	+	0.902191i	$0.358042\pi$
$60$	0	0
$61$	−11273.2	−0.387904	−0.193952	−	0.981011i	$-0.562131\pi$
$61$	−11273.2	−0.387904	−0.193952	+	0.981011i	$0.562131\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−10555.6	−0.309885
$66$	0	0
$67$	33852.6	0.921309	0.460654	−	0.887580i	$-0.347615\pi$
$67$	33852.6	0.921309	0.460654	+	0.887580i	$0.347615\pi$
$68$	0	0
$69$	−24878.8	−0.629082
$70$	0	0
$71$	−14781.0	−0.347984	−0.173992	−	0.984747i	$-0.555667\pi$
$71$	−14781.0	−0.347984	−0.173992	+	0.984747i	$0.555667\pi$
$72$	0	0
$73$	32642.0	0.716919	0.358459	−	0.933545i	$-0.383302\pi$
$73$	32642.0	0.716919	0.358459	+	0.933545i	$0.383302\pi$
$74$	0	0
$75$	−41292.4	−0.847650
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−36358.1	−0.655441	−0.327721	−	0.944775i	$-0.606280\pi$
$79$	−36358.1	−0.655441	−0.327721	+	0.944775i	$0.606280\pi$
$80$	0	0
$81$	−106.005	−0.00179520
$82$	0	0
$83$	4768.94	0.0759848	0.0379924	−	0.999278i	$-0.487904\pi$
$83$	4768.94	0.0759848	0.0379924	+	0.999278i	$0.487904\pi$
$84$	0	0
$85$	−149070.	−2.23792
$86$	0	0
$87$	−29507.5	−0.417960
$88$	0	0
$89$	116259.	1.55579	0.777896	−	0.628393i	$-0.216287\pi$
$89$	116259.	1.55579	0.777896	+	0.628393i	$0.216287\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−32179.2	−0.385805
$94$	0	0
$95$	171868.	1.95383
$96$	0	0
$97$	−62456.2	−0.673979	−0.336989	−	0.941508i	$-0.609409\pi$
$97$	−62456.2	−0.673979	−0.336989	+	0.941508i	$0.609409\pi$
$98$	0	0
$99$	73864.8	0.757443
$100$	0	0
$101$	144917.	1.41356	0.706782	−	0.707432i	$-0.250146\pi$
$101$	144917.	1.41356	0.706782	+	0.707432i	$0.250146\pi$
$102$	0	0
$103$	−95440.6	−0.886421	−0.443211	−	0.896418i	$-0.646161\pi$
$103$	−95440.6	−0.886421	−0.443211	+	0.896418i	$0.646161\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	149699.	1.26404	0.632020	−	0.774952i	$-0.282226\pi$
$107$	149699.	1.26404	0.632020	+	0.774952i	$0.282226\pi$
$108$	0	0
$109$	75787.8	0.610988	0.305494	−	0.952194i	$-0.401178\pi$
$109$	75787.8	0.610988	0.305494	+	0.952194i	$0.401178\pi$
$110$	0	0
$111$	116484.	0.897343
$112$	0	0
$113$	−192546.	−1.41853	−0.709266	−	0.704941i	$-0.750973\pi$
$113$	−192546.	−1.41853	−0.709266	+	0.704941i	$0.750973\pi$
$114$	0	0
$115$	−222006.	−1.56538
$116$	0	0
$117$	−18396.3	−0.124241
$118$	0	0
$119$	0	0
$120$	0	0
$121$	81480.1	0.505928
$122$	0	0
$123$	2438.09	0.0145307
$124$	0	0
$125$	−99531.6	−0.569752
$126$	0	0
$127$	−155367.	−0.854768	−0.427384	−	0.904070i	$-0.640565\pi$
$127$	−155367.	−0.854768	−0.427384	+	0.904070i	$0.640565\pi$
$128$	0	0
$129$	−106544.	−0.563711
$130$	0	0
$131$	−382087.	−1.94529	−0.972643	−	0.232304i	$-0.925374\pi$
$131$	−382087.	−1.94529	−0.972643	+	0.232304i	$0.925374\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−326180.	−1.54036
$136$	0	0
$137$	−410458.	−1.86839	−0.934193	−	0.356767i	$-0.883879\pi$
$137$	−410458.	−1.86839	−0.934193	+	0.356767i	$0.883879\pi$
$138$	0	0
$139$	344242.	1.51122	0.755608	−	0.655024i	$-0.227342\pi$
$139$	344242.	1.51122	0.755608	+	0.655024i	$0.227342\pi$
$140$	0	0
$141$	−216998.	−0.919196
$142$	0	0
$143$	−60403.3	−0.247013
$144$	0	0
$145$	−263310.	−1.04003
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−337433.	−1.24515	−0.622576	−	0.782560i	$-0.713914\pi$
$149$	−337433.	−1.24515	−0.622576	+	0.782560i	$0.713914\pi$
$150$	0	0
$151$	496010.	1.77030	0.885152	−	0.465303i	$-0.154055\pi$
$151$	496010.	1.77030	0.885152	+	0.465303i	$0.154055\pi$
$152$	0	0
$153$	−259799.	−0.897241
$154$	0	0
$155$	−287151.	−0.960022
$156$	0	0
$157$	−106606.	−0.345170	−0.172585	−	0.984995i	$-0.555212\pi$
$157$	−106606.	−0.345170	−0.172585	+	0.984995i	$0.555212\pi$
$158$	0	0
$159$	384205.	1.20523
$160$	0	0
$161$	0	0
$162$	0	0
$163$	241963.	0.713313	0.356656	−	0.934236i	$-0.383917\pi$
$163$	241963.	0.713313	0.356656	+	0.934236i	$0.383917\pi$
$164$	0	0
$165$	−408754.	−1.16883
$166$	0	0
$167$	291486.	0.808773	0.404386	−	0.914588i	$-0.367485\pi$
$167$	291486.	0.808773	0.404386	+	0.914588i	$0.367485\pi$
$168$	0	0
$169$	−356249.	−0.959483
$170$	0	0
$171$	299531.	0.783343
$172$	0	0
$173$	−394689.	−1.00263	−0.501314	−	0.865266i	$-0.667150\pi$
$173$	−394689.	−1.00263	−0.501314	+	0.865266i	$0.667150\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−222458.	−0.533721
$178$	0	0
$179$	−641833.	−1.49723	−0.748617	−	0.663003i	$-0.769282\pi$
$179$	−641833.	−1.49723	−0.748617	+	0.663003i	$0.769282\pi$
$180$	0	0
$181$	−30068.6	−0.0682208	−0.0341104	−	0.999418i	$-0.510860\pi$
$181$	−30068.6	−0.0682208	−0.0341104	+	0.999418i	$0.510860\pi$
$182$	0	0
$183$	108723.	0.239990
$184$	0	0
$185$	1.03945e6	2.23292
$186$	0	0
$187$	−853036.	−1.78387
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−169468.	−0.336127	−0.168063	−	0.985776i	$-0.553751\pi$
$191$	−169468.	−0.336127	−0.168063	+	0.985776i	$0.553751\pi$
$192$	0	0
$193$	23155.0	0.0447457	0.0223728	−	0.999750i	$-0.492878\pi$
$193$	23155.0	0.0447457	0.0223728	+	0.999750i	$0.492878\pi$
$194$	0	0
$195$	101802.	0.191721
$196$	0	0
$197$	896994.	1.64674	0.823368	−	0.567508i	$-0.192092\pi$
$197$	896994.	1.64674	0.823368	+	0.567508i	$0.192092\pi$
$198$	0	0
$199$	859026.	1.53771	0.768854	−	0.639425i	$-0.220827\pi$
$199$	859026.	1.53771	0.768854	+	0.639425i	$0.220827\pi$
$200$	0	0
$201$	−326485.	−0.569998
$202$	0	0
$203$	0	0
$204$	0	0
$205$	21756.3	0.0361577
$206$	0	0
$207$	−386912.	−0.627605
$208$	0	0
$209$	983495.	1.55742
$210$	0	0
$211$	688411.	1.06449	0.532245	−	0.846590i	$-0.321348\pi$
$211$	688411.	1.06449	0.532245	+	0.846590i	$0.321348\pi$
$212$	0	0
$213$	142553.	0.215292
$214$	0	0
$215$	−950750.	−1.40272
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−314810.	−0.443546
$220$	0	0
$221$	212452.	0.292604
$222$	0	0
$223$	830652.	1.11855	0.559277	−	0.828981i	$-0.311079\pi$
$223$	830652.	1.11855	0.559277	+	0.828981i	$0.311079\pi$
$224$	0	0
$225$	−642173.	−0.845660
$226$	0	0
$227$	−947214.	−1.22007	−0.610033	−	0.792376i	$-0.708844\pi$
$227$	−947214.	−1.22007	−0.610033	+	0.792376i	$0.708844\pi$
$228$	0	0
$229$	−404570.	−0.509806	−0.254903	−	0.966967i	$-0.582044\pi$
$229$	−404570.	−0.509806	−0.254903	+	0.966967i	$0.582044\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−553480.	−0.667901	−0.333950	−	0.942591i	$-0.608382\pi$
$233$	−553480.	−0.667901	−0.333950	+	0.942591i	$0.608382\pi$
$234$	0	0
$235$	−1.93638e6	−2.28729
$236$	0	0
$237$	350649.	0.405511
$238$	0	0
$239$	−1.49999e6	−1.69861	−0.849304	−	0.527904i	$-0.822978\pi$
$239$	−1.49999e6	−1.69861	−0.849304	+	0.527904i	$0.822978\pi$
$240$	0	0
$241$	−240446.	−0.266670	−0.133335	−	0.991071i	$-0.542569\pi$
$241$	−240446.	−0.266670	−0.133335	+	0.991071i	$0.542569\pi$
$242$	0	0
$243$	−919970.	−0.999443
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−244943.	−0.255460
$248$	0	0
$249$	−45993.2	−0.0470106
$250$	0	0
$251$	−1.55488e6	−1.55781	−0.778904	−	0.627144i	$-0.784224\pi$
$251$	−1.55488e6	−1.55781	−0.778904	+	0.627144i	$0.784224\pi$
$252$	0	0
$253$	−1.27041e6	−1.24779
$254$	0	0
$255$	1.43768e6	1.38456
$256$	0	0
$257$	−1.18741e6	−1.12142	−0.560708	−	0.828014i	$-0.689471\pi$
$257$	−1.18741e6	−1.12142	−0.560708	+	0.828014i	$0.689471\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−458896.	−0.416978
$262$	0	0
$263$	1.46143e6	1.30284	0.651418	−	0.758719i	$-0.274174\pi$
$263$	1.46143e6	1.30284	0.651418	+	0.758719i	$0.274174\pi$
$264$	0	0
$265$	3.42846e6	2.99905
$266$	0	0
$267$	−1.12124e6	−0.962543
$268$	0	0
$269$	−450060.	−0.379219	−0.189610	−	0.981860i	$-0.560722\pi$
$269$	−450060.	−0.379219	−0.189610	+	0.981860i	$0.560722\pi$
$270$	0	0
$271$	1.11293e6	0.920546	0.460273	−	0.887777i	$-0.347752\pi$
$271$	1.11293e6	0.920546	0.460273	+	0.887777i	$0.347752\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.10854e6	−1.68132
$276$	0	0
$277$	740826.	0.580118	0.290059	−	0.957009i	$-0.406325\pi$
$277$	740826.	0.580118	0.290059	+	0.957009i	$0.406325\pi$
$278$	0	0
$279$	−500446.	−0.384899
$280$	0	0
$281$	−109745.	−0.0829120	−0.0414560	−	0.999140i	$-0.513200\pi$
$281$	−109745.	−0.0829120	−0.0414560	+	0.999140i	$0.513200\pi$
$282$	0	0
$283$	909415.	0.674988	0.337494	−	0.941328i	$-0.390421\pi$
$283$	909415.	0.674988	0.337494	+	0.941328i	$0.390421\pi$
$284$	0	0
$285$	−1.65755e6	−1.20880
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.58046e6	1.11311
$290$	0	0
$291$	602347.	0.416979
$292$	0	0
$293$	−673403.	−0.458254	−0.229127	−	0.973397i	$-0.573587\pi$
$293$	−673403.	−0.458254	−0.229127	+	0.973397i	$0.573587\pi$
$294$	0	0
$295$	−1.98510e6	−1.32809
$296$	0	0
$297$	−1.86652e6	−1.22784
$298$	0	0
$299$	316399.	0.204671
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.39762e6	−0.874548
$304$	0	0
$305$	970187.	0.597181
$306$	0	0
$307$	283055.	0.171405	0.0857027	−	0.996321i	$-0.472686\pi$
$307$	283055.	0.171405	0.0857027	+	0.996321i	$0.472686\pi$
$308$	0	0
$309$	920460.	0.548414
$310$	0	0
$311$	−387696.	−0.227295	−0.113648	−	0.993521i	$-0.536253\pi$
$311$	−387696.	−0.227295	−0.113648	+	0.993521i	$0.536253\pi$
$312$	0	0
$313$	−61332.0	−0.0353856	−0.0176928	−	0.999843i	$-0.505632\pi$
$313$	−61332.0	−0.0353856	−0.0176928	+	0.999843i	$0.505632\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.43575e6	−0.802474	−0.401237	−	0.915974i	$-0.631420\pi$
$317$	−1.43575e6	−0.802474	−0.401237	+	0.915974i	$0.631420\pi$
$318$	0	0
$319$	−1.50676e6	−0.829025
$320$	0	0
$321$	−1.44375e6	−0.782040
$322$	0	0
$323$	−3.45917e6	−1.84487
$324$	0	0
$325$	525140.	0.275782
$326$	0	0
$327$	−730921.	−0.378008
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.53209e6	1.27031	0.635155	−	0.772385i	$-0.280936\pi$
$331$	2.53209e6	1.27031	0.635155	+	0.772385i	$0.280936\pi$
$332$	0	0
$333$	1.81154e6	0.895236
$334$	0	0
$335$	−2.91339e6	−1.41836
$336$	0	0
$337$	−2.25767e6	−1.08289	−0.541446	−	0.840735i	$-0.682123\pi$
$337$	−2.25767e6	−1.08289	−0.541446	+	0.840735i	$0.682123\pi$
$338$	0	0
$339$	1.85698e6	0.877622
$340$	0	0
$341$	−1.64319e6	−0.765246
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2.14110e6	0.968477
$346$	0	0
$347$	−1.91281e6	−0.852804	−0.426402	−	0.904534i	$-0.640219\pi$
$347$	−1.91281e6	−0.852804	−0.426402	+	0.904534i	$0.640219\pi$
$348$	0	0
$349$	2.58959e6	1.13807	0.569033	−	0.822315i	$-0.307318\pi$
$349$	2.58959e6	1.13807	0.569033	+	0.822315i	$0.307318\pi$
$350$	0	0
$351$	464865.	0.201400
$352$	0	0
$353$	532261.	0.227346	0.113673	−	0.993518i	$-0.463738\pi$
$353$	532261.	0.227346	0.113673	+	0.993518i	$0.463738\pi$
$354$	0	0
$355$	1.27207e6	0.535724
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.82610e6	0.747805	0.373903	−	0.927468i	$-0.378019\pi$
$359$	1.82610e6	0.747805	0.373903	+	0.927468i	$0.378019\pi$
$360$	0	0
$361$	1.51210e6	0.610679
$362$	0	0
$363$	−785821.	−0.313009
$364$	0	0
$365$	−2.80921e6	−1.10370
$366$	0	0
$367$	−1.13686e6	−0.440598	−0.220299	−	0.975432i	$-0.570703\pi$
$367$	−1.13686e6	−0.440598	−0.220299	+	0.975432i	$0.570703\pi$
$368$	0	0
$369$	37916.9	0.0144966
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−4.23016e6	−1.57429	−0.787145	−	0.616768i	$-0.788442\pi$
$373$	−4.23016e6	−1.57429	−0.787145	+	0.616768i	$0.788442\pi$
$374$	0	0
$375$	959914.	0.352496
$376$	0	0
$377$	375265.	0.135983
$378$	0	0
$379$	−2.00144e6	−0.715722	−0.357861	−	0.933775i	$-0.616494\pi$
$379$	−2.00144e6	−0.715722	−0.357861	+	0.933775i	$0.616494\pi$
$380$	0	0
$381$	1.49841e6	0.528831
$382$	0	0
$383$	−4.23325e6	−1.47461	−0.737305	−	0.675560i	$-0.763902\pi$
$383$	−4.23325e6	−1.47461	−0.737305	+	0.675560i	$0.763902\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.65696e6	−0.562388
$388$	0	0
$389$	4.19599e6	1.40592	0.702960	−	0.711230i	$-0.251861\pi$
$389$	4.19599e6	1.40592	0.702960	+	0.711230i	$0.251861\pi$
$390$	0	0
$391$	4.46830e6	1.47809
$392$	0	0
$393$	3.68497e6	1.20352
$394$	0	0
$395$	3.12902e6	1.00906
$396$	0	0
$397$	1.37572e6	0.438080	0.219040	−	0.975716i	$-0.429708\pi$
$397$	1.37572e6	0.438080	0.219040	+	0.975716i	$0.429708\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.04265e6	−1.25547	−0.627734	−	0.778428i	$-0.716017\pi$
$401$	−4.04265e6	−1.25547	−0.627734	+	0.778428i	$0.716017\pi$
$402$	0	0
$403$	409242.	0.125521
$404$	0	0
$405$	9122.87	0.00276372
$406$	0	0
$407$	5.94810e6	1.77989
$408$	0	0
$409$	3.06429e6	0.905777	0.452889	−	0.891567i	$-0.350393\pi$
$409$	3.06429e6	0.905777	0.452889	+	0.891567i	$0.350393\pi$
$410$	0	0
$411$	3.95858e6	1.15594
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−410421.	−0.116979
$416$	0	0
$417$	−3.31998e6	−0.934964
$418$	0	0
$419$	3.59596e6	1.00064	0.500322	−	0.865839i	$-0.333215\pi$
$419$	3.59596e6	1.00064	0.500322	+	0.865839i	$0.333215\pi$
$420$	0	0
$421$	2.36536e6	0.650418	0.325209	−	0.945642i	$-0.394565\pi$
$421$	2.36536e6	0.650418	0.325209	+	0.945642i	$0.394565\pi$
$422$	0	0
$423$	−3.37472e6	−0.917038
$424$	0	0
$425$	7.41621e6	1.99163
$426$	0	0
$427$	0	0
$428$	0	0
$429$	582549.	0.152823
$430$	0	0
$431$	−2.35167e6	−0.609795	−0.304897	−	0.952385i	$-0.598622\pi$
$431$	−2.35167e6	−0.609795	−0.304897	+	0.952385i	$0.598622\pi$
$432$	0	0
$433$	197540.	0.0506333	0.0253166	−	0.999679i	$-0.491941\pi$
$433$	197540.	0.0506333	0.0253166	+	0.999679i	$0.491941\pi$
$434$	0	0
$435$	2.53945e6	0.643452
$436$	0	0
$437$	−5.15166e6	−1.29046
$438$	0	0
$439$	1.55953e6	0.386219	0.193109	−	0.981177i	$-0.438143\pi$
$439$	1.55953e6	0.386219	0.193109	+	0.981177i	$0.438143\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.47818e6	−1.32625	−0.663127	−	0.748507i	$-0.730771\pi$
$443$	−5.47818e6	−1.32625	−0.663127	+	0.748507i	$0.730771\pi$
$444$	0	0
$445$	−1.00054e7	−2.39515
$446$	0	0
$447$	3.25431e6	0.770354
$448$	0	0
$449$	2.69245e6	0.630278	0.315139	−	0.949046i	$-0.397949\pi$
$449$	2.69245e6	0.630278	0.315139	+	0.949046i	$0.397949\pi$
$450$	0	0
$451$	124498.	0.0288218
$452$	0	0
$453$	−4.78367e6	−1.09526
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.33045e6	1.86586	0.932928	−	0.360063i	$-0.117245\pi$
$457$	8.33045e6	1.86586	0.932928	+	0.360063i	$0.117245\pi$
$458$	0	0
$459$	6.56498e6	1.45446
$460$	0	0
$461$	−3.89041e6	−0.852595	−0.426297	−	0.904583i	$-0.640182\pi$
$461$	−3.89041e6	−0.852595	−0.426297	+	0.904583i	$0.640182\pi$
$462$	0	0
$463$	3.67786e6	0.797338	0.398669	−	0.917095i	$-0.369472\pi$
$463$	3.67786e6	0.797338	0.398669	+	0.917095i	$0.369472\pi$
$464$	0	0
$465$	2.76938e6	0.593950
$466$	0	0
$467$	3.10332e6	0.658468	0.329234	−	0.944248i	$-0.393209\pi$
$467$	3.10332e6	0.658468	0.329234	+	0.944248i	$0.393209\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1.02814e6	0.213551
$472$	0	0
$473$	−5.44055e6	−1.11812
$474$	0	0
$475$	−8.55040e6	−1.73881
$476$	0	0
$477$	5.97511e6	1.20240
$478$	0	0
$479$	−2.99359e6	−0.596147	−0.298074	−	0.954543i	$-0.596344\pi$
$479$	−2.99359e6	−0.596147	−0.298074	+	0.954543i	$0.596344\pi$
$480$	0	0
$481$	−1.48140e6	−0.291950
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.37505e6	1.03760
$486$	0	0
$487$	394673.	0.0754075	0.0377037	−	0.999289i	$-0.487996\pi$
$487$	394673.	0.0754075	0.0377037	+	0.999289i	$0.487996\pi$
$488$	0	0
$489$	−2.33357e6	−0.441315
$490$	0	0
$491$	6.87661e6	1.28727	0.643636	−	0.765332i	$-0.277425\pi$
$491$	6.87661e6	1.28727	0.643636	+	0.765332i	$0.277425\pi$
$492$	0	0
$493$	5.29962e6	0.982036
$494$	0	0
$495$	−6.35689e6	−1.16609
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9.99724e6	1.79733	0.898667	−	0.438631i	$-0.144536\pi$
$499$	9.99724e6	1.79733	0.898667	+	0.438631i	$0.144536\pi$
$500$	0	0
$501$	−2.81118e6	−0.500374
$502$	0	0
$503$	6.29767e6	1.10984	0.554920	−	0.831904i	$-0.312749\pi$
$503$	6.29767e6	1.10984	0.554920	+	0.831904i	$0.312749\pi$
$504$	0	0
$505$	−1.24717e7	−2.17619
$506$	0	0
$507$	3.43578e6	0.593616
$508$	0	0
$509$	5.41178e6	0.925861	0.462931	−	0.886395i	$-0.346798\pi$
$509$	5.41178e6	0.925861	0.462931	+	0.886395i	$0.346798\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−7.56900e6	−1.26983
$514$	0	0
$515$	8.21373e6	1.36465
$516$	0	0
$517$	−1.10807e7	−1.82323
$518$	0	0
$519$	3.80650e6	0.620309
$520$	0	0
$521$	−5.40153e6	−0.871811	−0.435905	−	0.899992i	$-0.643572\pi$
$521$	−5.40153e6	−0.871811	−0.435905	+	0.899992i	$0.643572\pi$
$522$	0	0
$523$	2.96309e6	0.473686	0.236843	−	0.971548i	$-0.423887\pi$
$523$	2.96309e6	0.473686	0.236843	+	0.971548i	$0.423887\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.77946e6	0.906485
$528$	0	0
$529$	218181.	0.0338982
$530$	0	0
$531$	−3.45963e6	−0.532468
$532$	0	0
$533$	−31006.7	−0.00472757
$534$	0	0
$535$	−1.28833e7	−1.94600
$536$	0	0
$537$	6.19004e6	0.926313
$538$	0	0
$539$	0	0
$540$	0	0
$541$	6.73637e6	0.989539	0.494770	−	0.869024i	$-0.335252\pi$
$541$	6.73637e6	0.989539	0.494770	+	0.869024i	$0.335252\pi$
$542$	0	0
$543$	289991.	0.0422071
$544$	0	0
$545$	−6.52238e6	−0.940622
$546$	0	0
$547$	−4.77831e6	−0.682820	−0.341410	−	0.939915i	$-0.610904\pi$
$547$	−4.77831e6	−0.682820	−0.341410	+	0.939915i	$0.610904\pi$
$548$	0	0
$549$	1.69084e6	0.239426
$550$	0	0
$551$	−6.11011e6	−0.857374
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.00247e7	−1.38147
$556$	0	0
$557$	5.46318e6	0.746118	0.373059	−	0.927808i	$-0.378309\pi$
$557$	5.46318e6	0.746118	0.373059	+	0.927808i	$0.378309\pi$
$558$	0	0
$559$	1.35499e6	0.183403
$560$	0	0
$561$	8.22696e6	1.10365
$562$	0	0
$563$	−2.95176e6	−0.392473	−0.196236	−	0.980557i	$-0.562872\pi$
$563$	−2.95176e6	−0.392473	−0.196236	+	0.980557i	$0.562872\pi$
$564$	0	0
$565$	1.65708e7	2.18384
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.11631e6	0.921455	0.460728	−	0.887542i	$-0.347589\pi$
$569$	7.11631e6	0.921455	0.460728	+	0.887542i	$0.347589\pi$
$570$	0	0
$571$	−5.67239e6	−0.728075	−0.364037	−	0.931384i	$-0.618602\pi$
$571$	−5.67239e6	−0.728075	−0.364037	+	0.931384i	$0.618602\pi$
$572$	0	0
$573$	1.63440e6	0.207956
$574$	0	0
$575$	1.10448e7	1.39312
$576$	0	0
$577$	−1.41107e6	−0.176445	−0.0882225	−	0.996101i	$-0.528119\pi$
$577$	−1.41107e6	−0.176445	−0.0882225	+	0.996101i	$0.528119\pi$
$578$	0	0
$579$	−223314.	−0.0276834
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.96189e7	2.39059
$584$	0	0
$585$	1.58321e6	0.191271
$586$	0	0
$587$	1.01832e6	0.121980	0.0609901	−	0.998138i	$-0.480574\pi$
$587$	1.01832e6	0.121980	0.0609901	+	0.998138i	$0.480574\pi$
$588$	0	0
$589$	−6.66334e6	−0.791414
$590$	0	0
$591$	−8.65090e6	−1.01881
$592$	0	0
$593$	−8.55229e6	−0.998724	−0.499362	−	0.866393i	$-0.666432\pi$
$593$	−8.55229e6	−0.998724	−0.499362	+	0.866393i	$0.666432\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.28472e6	−0.951354
$598$	0	0
$599$	2.41242e6	0.274717	0.137359	−	0.990521i	$-0.456139\pi$
$599$	2.41242e6	0.274717	0.137359	+	0.990521i	$0.456139\pi$
$600$	0	0
$601$	−2.19294e6	−0.247652	−0.123826	−	0.992304i	$-0.539516\pi$
$601$	−2.19294e6	−0.247652	−0.123826	+	0.992304i	$0.539516\pi$
$602$	0	0
$603$	−5.07745e6	−0.568660
$604$	0	0
$605$	−7.01227e6	−0.778880
$606$	0	0
$607$	1.21147e7	1.33457	0.667283	−	0.744804i	$-0.267457\pi$
$607$	1.21147e7	1.33457	0.667283	+	0.744804i	$0.267457\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.75970e6	0.299060
$612$	0	0
$613$	1.64943e6	0.177289	0.0886444	−	0.996063i	$-0.471747\pi$
$613$	1.64943e6	0.177289	0.0886444	+	0.996063i	$0.471747\pi$
$614$	0	0
$615$	−209825.	−0.0223702
$616$	0	0
$617$	349418.	0.0369515	0.0184757	−	0.999829i	$-0.494119\pi$
$617$	349418.	0.0369515	0.0184757	+	0.999829i	$0.494119\pi$
$618$	0	0
$619$	−1.13692e7	−1.19263	−0.596314	−	0.802751i	$-0.703369\pi$
$619$	−1.13692e7	−1.19263	−0.596314	+	0.802751i	$0.703369\pi$
$620$	0	0
$621$	9.77706e6	1.01737
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4.81395e6	−0.492949
$626$	0	0
$627$	−9.48514e6	−0.963552
$628$	0	0
$629$	−2.09208e7	−2.10839
$630$	0	0
$631$	7.40595e6	0.740470	0.370235	−	0.928938i	$-0.379277\pi$
$631$	7.40595e6	0.740470	0.370235	+	0.928938i	$0.379277\pi$
$632$	0	0
$633$	−6.63926e6	−0.658583
$634$	0	0
$635$	1.33710e7	1.31592
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.21697e6	0.214786
$640$	0	0
$641$	7.14412e6	0.686758	0.343379	−	0.939197i	$-0.388428\pi$
$641$	7.14412e6	0.686758	0.343379	+	0.939197i	$0.388428\pi$
$642$	0	0
$643$	−1.43543e7	−1.36916	−0.684580	−	0.728938i	$-0.740014\pi$
$643$	−1.43543e7	−1.36916	−0.684580	+	0.728938i	$0.740014\pi$
$644$	0	0
$645$	9.16934e6	0.867838
$646$	0	0
$647$	−5.64893e6	−0.530524	−0.265262	−	0.964176i	$-0.585459\pi$
$647$	−5.64893e6	−0.530524	−0.265262	+	0.964176i	$0.585459\pi$
$648$	0	0
$649$	−1.13595e7	−1.05864
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.27302e7	1.16830	0.584149	−	0.811646i	$-0.301428\pi$
$653$	1.27302e7	1.16830	0.584149	+	0.811646i	$0.301428\pi$
$654$	0	0
$655$	3.28828e7	2.99479
$656$	0	0
$657$	−4.89588e6	−0.442504
$658$	0	0
$659$	−6.85048e6	−0.614479	−0.307240	−	0.951632i	$-0.599405\pi$
$659$	−6.85048e6	−0.614479	−0.307240	+	0.951632i	$0.599405\pi$
$660$	0	0
$661$	−1.38491e7	−1.23287	−0.616434	−	0.787406i	$-0.711423\pi$
$661$	−1.38491e7	−1.23287	−0.616434	+	0.787406i	$0.711423\pi$
$662$	0	0
$663$	−2.04895e6	−0.181029
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.89258e6	0.686918
$668$	0	0
$669$	−8.01107e6	−0.692030
$670$	0	0
$671$	5.55178e6	0.476021
$672$	0	0
$673$	1.61044e7	1.37059	0.685294	−	0.728266i	$-0.259674\pi$
$673$	1.61044e7	1.37059	0.685294	+	0.728266i	$0.259674\pi$
$674$	0	0
$675$	1.62274e7	1.37085
$676$	0	0
$677$	−1.81191e6	−0.151937	−0.0759686	−	0.997110i	$-0.524205\pi$
$677$	−1.81191e6	−0.151937	−0.0759686	+	0.997110i	$0.524205\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	9.13523e6	0.754835
$682$	0	0
$683$	−1.49131e7	−1.22326	−0.611628	−	0.791146i	$-0.709485\pi$
$683$	−1.49131e7	−1.22326	−0.611628	+	0.791146i	$0.709485\pi$
$684$	0	0
$685$	3.53244e7	2.87640
$686$	0	0
$687$	3.90180e6	0.315408
$688$	0	0
$689$	−4.88617e6	−0.392122
$690$	0	0
$691$	2.17649e7	1.73405	0.867024	−	0.498266i	$-0.166030\pi$
$691$	2.17649e7	1.73405	0.867024	+	0.498266i	$0.166030\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.96258e7	−2.32653
$696$	0	0
$697$	−437888.	−0.0341414
$698$	0	0
$699$	5.33794e6	0.413219
$700$	0	0
$701$	5.79995e6	0.445789	0.222894	−	0.974843i	$-0.428450\pi$
$701$	5.79995e6	0.445789	0.222894	+	0.974843i	$0.428450\pi$
$702$	0	0
$703$	2.41203e7	1.84075
$704$	0	0
$705$	1.86751e7	1.41511
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−5.28791e6	−0.395065	−0.197532	−	0.980296i	$-0.563293\pi$
$709$	−5.28791e6	−0.395065	−0.197532	+	0.980296i	$0.563293\pi$
$710$	0	0
$711$	5.45325e6	0.404559
$712$	0	0
$713$	8.60720e6	0.634071
$714$	0	0
$715$	5.19838e6	0.380279
$716$	0	0
$717$	1.44664e7	1.05090
$718$	0	0
$719$	779118.	0.0562058	0.0281029	−	0.999605i	$-0.491053\pi$
$719$	779118.	0.0562058	0.0281029	+	0.999605i	$0.491053\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	2.31894e6	0.164985
$724$	0	0
$725$	1.30996e7	0.925580
$726$	0	0
$727$	−1.97579e6	−0.138645	−0.0693224	−	0.997594i	$-0.522084\pi$
$727$	−1.97579e6	−0.138645	−0.0693224	+	0.997594i	$0.522084\pi$
$728$	0	0
$729$	8.89825e6	0.620134
$730$	0	0
$731$	1.91356e7	1.32449
$732$	0	0
$733$	−1.18659e7	−0.815719	−0.407859	−	0.913045i	$-0.633725\pi$
$733$	−1.18659e7	−0.815719	−0.407859	+	0.913045i	$0.633725\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.66715e7	−1.13060
$738$	0	0
$739$	−2.53439e7	−1.70711	−0.853556	−	0.521001i	$-0.825559\pi$
$739$	−2.53439e7	−1.70711	−0.853556	+	0.521001i	$0.825559\pi$
$740$	0	0
$741$	2.36231e6	0.158049
$742$	0	0
$743$	−2.10785e7	−1.40077	−0.700386	−	0.713764i	$-0.746989\pi$
$743$	−2.10785e7	−1.40077	−0.700386	+	0.713764i	$0.746989\pi$
$744$	0	0
$745$	2.90399e7	1.91692
$746$	0	0
$747$	−715280.	−0.0469002
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−3.70657e6	−0.239812	−0.119906	−	0.992785i	$-0.538259\pi$
$751$	−3.70657e6	−0.239812	−0.119906	+	0.992785i	$0.538259\pi$
$752$	0	0
$753$	1.49958e7	0.963789
$754$	0	0
$755$	−4.26871e7	−2.72540
$756$	0	0
$757$	−1.71834e6	−0.108986	−0.0544928	−	0.998514i	$-0.517354\pi$
$757$	−1.71834e6	−0.108986	−0.0544928	+	0.998514i	$0.517354\pi$
$758$	0	0
$759$	1.22522e7	0.771986
$760$	0	0
$761$	2.47983e7	1.55225	0.776123	−	0.630582i	$-0.217184\pi$
$761$	2.47983e7	1.55225	0.776123	+	0.630582i	$0.217184\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.23586e7	1.38131
$766$	0	0
$767$	2.82913e6	0.173646
$768$	0	0
$769$	−9.97437e6	−0.608232	−0.304116	−	0.952635i	$-0.598361\pi$
$769$	−9.97437e6	−0.608232	−0.304116	+	0.952635i	$0.598361\pi$
$770$	0	0
$771$	1.14517e7	0.693801
$772$	0	0
$773$	1.10753e7	0.666663	0.333331	−	0.942810i	$-0.391827\pi$
$773$	1.10753e7	0.666663	0.333331	+	0.942810i	$0.391827\pi$
$774$	0	0
$775$	1.42857e7	0.854372
$776$	0	0
$777$	0	0
$778$	0	0
$779$	504856.	0.0298074
$780$	0	0
$781$	7.27929e6	0.427033
$782$	0	0
$783$	1.15961e7	0.675937
$784$	0	0
$785$	9.17465e6	0.531392
$786$	0	0
$787$	−1.28344e7	−0.738648	−0.369324	−	0.929301i	$-0.620411\pi$
$787$	−1.28344e7	−0.738648	−0.369324	+	0.929301i	$0.620411\pi$
$788$	0	0
$789$	−1.40945e7	−0.806044
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−1.38269e6	−0.0780804
$794$	0	0
$795$	−3.30652e7	−1.85547
$796$	0	0
$797$	3.09073e7	1.72352	0.861759	−	0.507319i	$-0.169363\pi$
$797$	3.09073e7	1.72352	0.861759	+	0.507319i	$0.169363\pi$
$798$	0	0
$799$	3.89734e7	2.15974
$800$	0	0
$801$	−1.74373e7	−0.960283
$802$	0	0
$803$	−1.60754e7	−0.879776
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.34053e6	0.234617
$808$	0	0
$809$	−5.92648e6	−0.318365	−0.159183	−	0.987249i	$-0.550886\pi$
$809$	−5.92648e6	−0.318365	−0.159183	+	0.987249i	$0.550886\pi$
$810$	0	0
$811$	−2.19883e7	−1.17392	−0.586962	−	0.809615i	$-0.699676\pi$
$811$	−2.19883e7	−1.17392	−0.586962	+	0.809615i	$0.699676\pi$
$812$	0	0
$813$	−1.07335e7	−0.569526
$814$	0	0
$815$	−2.08236e7	−1.09815
$816$	0	0
$817$	−2.20621e7	−1.15636
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.14095e7	−0.590755	−0.295378	−	0.955381i	$-0.595445\pi$
$821$	−1.14095e7	−0.590755	−0.295378	+	0.955381i	$0.595445\pi$
$822$	0	0
$823$	1.20186e7	0.618519	0.309259	−	0.950978i	$-0.399919\pi$
$823$	1.20186e7	0.618519	0.309259	+	0.950978i	$0.399919\pi$
$824$	0	0
$825$	2.03354e7	1.04020
$826$	0	0
$827$	−2.15657e7	−1.09648	−0.548239	−	0.836322i	$-0.684701\pi$
$827$	−2.15657e7	−1.09648	−0.548239	+	0.836322i	$0.684701\pi$
$828$	0	0
$829$	−2.10157e7	−1.06208	−0.531041	−	0.847346i	$-0.678199\pi$
$829$	−2.10157e7	−1.06208	−0.531041	+	0.847346i	$0.678199\pi$
$830$	0	0
$831$	−7.14476e6	−0.358910
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.50856e7	−1.24511
$836$	0	0
$837$	1.26460e7	0.623935
$838$	0	0
$839$	4.16220e6	0.204135	0.102068	−	0.994777i	$-0.467454\pi$
$839$	4.16220e6	0.204135	0.102068	+	0.994777i	$0.467454\pi$
$840$	0	0
$841$	−1.11502e7	−0.543615
$842$	0	0
$843$	1.05841e6	0.0512963
$844$	0	0
$845$	3.06592e7	1.47713
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−8.77068e6	−0.417604
$850$	0	0
$851$	−3.11568e7	−1.47479
$852$	0	0
$853$	2.38464e7	1.12215	0.561075	−	0.827765i	$-0.310388\pi$
$853$	2.38464e7	1.12215	0.561075	+	0.827765i	$0.310388\pi$
$854$	0	0
$855$	−2.57780e7	−1.20596
$856$	0	0
$857$	−3.42055e7	−1.59090	−0.795451	−	0.606018i	$-0.792766\pi$
$857$	−3.42055e7	−1.59090	−0.795451	+	0.606018i	$0.792766\pi$
$858$	0	0
$859$	−1.54170e7	−0.712881	−0.356440	−	0.934318i	$-0.616010\pi$
$859$	−1.54170e7	−0.712881	−0.356440	+	0.934318i	$0.616010\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.46408e7	0.669174	0.334587	−	0.942365i	$-0.391403\pi$
$863$	1.46408e7	0.669174	0.334587	+	0.942365i	$0.391403\pi$
$864$	0	0
$865$	3.39674e7	1.54355
$866$	0	0
$867$	−1.52425e7	−0.688665
$868$	0	0
$869$	1.79055e7	0.804333
$870$	0	0
$871$	4.15211e6	0.185449
$872$	0	0
$873$	9.36762e6	0.416000
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−127278.	−0.00558799	−0.00279399	−	0.999996i	$-0.500889\pi$
$877$	−127278.	−0.00558799	−0.00279399	+	0.999996i	$0.500889\pi$
$878$	0	0
$879$	6.49451e6	0.283514
$880$	0	0
$881$	−2.27483e7	−0.987436	−0.493718	−	0.869622i	$-0.664362\pi$
$881$	−2.27483e7	−0.987436	−0.493718	+	0.869622i	$0.664362\pi$
$882$	0	0
$883$	−2.89959e7	−1.25151	−0.625756	−	0.780019i	$-0.715209\pi$
$883$	−2.89959e7	−1.25151	−0.625756	+	0.780019i	$0.715209\pi$
$884$	0	0
$885$	1.91450e7	0.821668
$886$	0	0
$887$	1.10781e7	0.472778	0.236389	−	0.971658i	$-0.424036\pi$
$887$	1.10781e7	0.472778	0.236389	+	0.971658i	$0.424036\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	52204.5	0.00220300
$892$	0	0
$893$	−4.49338e7	−1.88558
$894$	0	0
$895$	5.52369e7	2.30500
$896$	0	0
$897$	−3.05145e6	−0.126627
$898$	0	0
$899$	1.02086e7	0.421274
$900$	0	0
$901$	−6.90042e7	−2.83181
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.58774e6	0.105027
$906$	0	0
$907$	6.13014e6	0.247430	0.123715	−	0.992318i	$-0.460519\pi$
$907$	6.13014e6	0.247430	0.123715	+	0.992318i	$0.460519\pi$
$908$	0	0
$909$	−2.17356e7	−0.872495
$910$	0	0
$911$	−4.18984e7	−1.67264	−0.836318	−	0.548245i	$-0.815296\pi$
$911$	−4.18984e7	−1.67264	−0.836318	+	0.548245i	$0.815296\pi$
$912$	0	0
$913$	−2.34858e6	−0.0932457
$914$	0	0
$915$	−9.35680e6	−0.369466
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.20908e7	−0.472243	−0.236122	−	0.971723i	$-0.575876\pi$
$919$	−1.20908e7	−0.472243	−0.236122	+	0.971723i	$0.575876\pi$
$920$	0	0
$921$	−2.72987e6	−0.106046
$922$	0	0
$923$	−1.81293e6	−0.0700451
$924$	0	0
$925$	−5.17121e7	−1.98718
$926$	0	0
$927$	1.43149e7	0.547127
$928$	0	0
$929$	−4.73579e7	−1.80033	−0.900167	−	0.435546i	$-0.856555\pi$
$929$	−4.73579e7	−1.80033	−0.900167	+	0.435546i	$0.856555\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.73906e6	0.140624
$934$	0	0
$935$	7.34133e7	2.74629
$936$	0	0
$937$	4.38649e6	0.163218	0.0816089	−	0.996664i	$-0.473994\pi$
$937$	4.38649e6	0.163218	0.0816089	+	0.996664i	$0.473994\pi$
$938$	0	0
$939$	591505.	0.0218925
$940$	0	0
$941$	−1.01622e7	−0.374121	−0.187060	−	0.982348i	$-0.559896\pi$
$941$	−1.01622e7	−0.374121	−0.187060	+	0.982348i	$0.559896\pi$
$942$	0	0
$943$	−652135.	−0.0238813
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.56083e6	0.310199	0.155100	−	0.987899i	$-0.450430\pi$
$947$	8.56083e6	0.310199	0.155100	+	0.987899i	$0.450430\pi$
$948$	0	0
$949$	4.00363e6	0.144307
$950$	0	0
$951$	1.38468e7	0.496477
$952$	0	0
$953$	1.00507e6	0.0358478	0.0179239	−	0.999839i	$-0.494294\pi$
$953$	1.00507e6	0.0358478	0.0179239	+	0.999839i	$0.494294\pi$
$954$	0	0
$955$	1.45846e7	0.517470
$956$	0	0
$957$	1.45317e7	0.512904
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.74963e7	−0.611136
$962$	0	0
$963$	−2.24530e7	−0.780204
$964$	0	0
$965$	−1.99274e6	−0.0688863
$966$	0	0
$967$	3.86477e7	1.32910	0.664550	−	0.747244i	$-0.268623\pi$
$967$	3.86477e7	1.32910	0.664550	+	0.747244i	$0.268623\pi$
$968$	0	0
$969$	3.33614e7	1.14139
$970$	0	0
$971$	−2.23871e7	−0.761990	−0.380995	−	0.924577i	$-0.624418\pi$
$971$	−2.23871e7	−0.761990	−0.380995	+	0.924577i	$0.624418\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−5.06462e6	−0.170622
$976$	0	0
$977$	−3.76382e7	−1.26151	−0.630757	−	0.775981i	$-0.717255\pi$
$977$	−3.76382e7	−1.26151	−0.630757	+	0.775981i	$0.717255\pi$
$978$	0	0
$979$	−5.72546e7	−1.90921
$980$	0	0
$981$	−1.13672e7	−0.377121
$982$	0	0
$983$	−5.99323e7	−1.97823	−0.989115	−	0.147143i	$-0.952992\pi$
$983$	−5.99323e7	−1.97823	−0.989115	+	0.147143i	$0.952992\pi$
$984$	0	0
$985$	−7.71963e7	−2.53516
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.84982e7	0.926461
$990$	0	0
$991$	3.68170e6	0.119087	0.0595434	−	0.998226i	$-0.481036\pi$
$991$	3.68170e6	0.119087	0.0595434	+	0.998226i	$0.481036\pi$
$992$	0	0
$993$	−2.44203e7	−0.785919
$994$	0	0
$995$	−7.39288e7	−2.36731
$996$	0	0
$997$	−5.39837e6	−0.171999	−0.0859993	−	0.996295i	$-0.527408\pi$
$997$	−5.39837e6	−0.171999	−0.0859993	+	0.996295i	$0.527408\pi$
$998$	0	0
$999$	−4.57767e7	−1.45121

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.bn.1.3		6
4.3	odd	2		392.6.a.m.1.4	yes	6
7.6	odd	2	inner	784.6.a.bn.1.4		6
28.3	even	6		392.6.i.q.177.4		12
28.11	odd	6		392.6.i.q.177.3		12
28.19	even	6		392.6.i.q.361.4		12
28.23	odd	6		392.6.i.q.361.3		12
28.27	even	2		392.6.a.m.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.6.a.m.1.3	✓	6	28.27	even	2
392.6.a.m.1.4	yes	6	4.3	odd	2
392.6.i.q.177.3		12	28.11	odd	6
392.6.i.q.177.4		12	28.3	even	6
392.6.i.q.361.3		12	28.23	odd	6
392.6.i.q.361.4		12	28.19	even	6
784.6.a.bn.1.3		6	1.1	even	1	trivial
784.6.a.bn.1.4		6	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-9.64432 q^{3} -86.0611 q^{5} -149.987 q^{9} +O(q^{10})\)	\(q-9.64432 q^{3} -86.0611 q^{5} -149.987 q^{9} -492.475 q^{11} +122.653 q^{13} +830.001 q^{15} +1732.14 q^{17} -1997.05 q^{19} +2579.64 q^{23} +4281.52 q^{25} +3790.09 q^{27} +3059.57 q^{29} +3336.59 q^{31} +4749.58 q^{33} -12078.0 q^{37} -1182.90 q^{39} -252.801 q^{41} +11047.4 q^{43} +12908.1 q^{45} +22500.1 q^{47} -16705.3 q^{51} -39837.5 q^{53} +42382.9 q^{55} +19260.2 q^{57} +23066.2 q^{59} -11273.2 q^{61} -10555.6 q^{65} +33852.6 q^{67} -24878.8 q^{69} -14781.0 q^{71} +32642.0 q^{73} -41292.4 q^{75} -36358.1 q^{79} -106.005 q^{81} +4768.94 q^{83} -149070. q^{85} -29507.5 q^{87} +116259. q^{89} -32179.2 q^{93} +171868. q^{95} -62456.2 q^{97} +73864.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 122 q^{9}+O(q^{10}) \) 6 * q - 122 * q^9	\( 6 q - 122 q^{9} - 248 q^{11} + 376 q^{15} + 2904 q^{23} + 1510 q^{25} - 408 q^{29} - 18648 q^{37} + 15240 q^{39} + 10584 q^{43} - 3016 q^{51} - 66772 q^{53} - 56264 q^{57} - 55316 q^{65} + 169632 q^{67} + 54880 q^{71} - 50576 q^{79} - 270194 q^{81} - 406076 q^{85} - 164944 q^{93} + 551912 q^{95} + 173512 q^{99}+O(q^{100}) \) 6 * q - 122 * q^9 - 248 * q^11 + 376 * q^15 + 2904 * q^23 + 1510 * q^25 - 408 * q^29 - 18648 * q^37 + 15240 * q^39 + 10584 * q^43 - 3016 * q^51 - 66772 * q^53 - 56264 * q^57 - 55316 * q^65 + 169632 * q^67 + 54880 * q^71 - 50576 * q^79 - 270194 * q^81 - 406076 * q^85 - 164944 * q^93 + 551912 * q^95 + 173512 * q^99

Introduction

L-functions

Modular forms

Varieties

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