LMFDB - Embedded newform 1028.6.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1028,6,Mod(1,1028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1028.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1028 = 2^{2} \cdot 257 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1028.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:	$164.874566768$
Analytic rank:	$1$
Dimension:	$49$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	1028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-29.3150 q^{3} +39.9945 q^{5} -134.016 q^{7} +616.369 q^{9} +O(q^{10})$	$q-29.3150 q^{3} +39.9945 q^{5} -134.016 q^{7} +616.369 q^{9} +497.286 q^{11} -1121.31 q^{13} -1172.44 q^{15} +1150.34 q^{17} -589.786 q^{19} +3928.69 q^{21} +656.510 q^{23} -1525.44 q^{25} -10945.3 q^{27} -1750.47 q^{29} +3502.76 q^{31} -14577.9 q^{33} -5359.92 q^{35} +925.163 q^{37} +32871.3 q^{39} -8622.25 q^{41} -11157.4 q^{43} +24651.4 q^{45} +23415.7 q^{47} +1153.38 q^{49} -33722.4 q^{51} -30649.4 q^{53} +19888.7 q^{55} +17289.6 q^{57} +51745.8 q^{59} -15055.1 q^{61} -82603.5 q^{63} -44846.4 q^{65} +19007.0 q^{67} -19245.6 q^{69} +49627.2 q^{71} -20845.2 q^{73} +44718.3 q^{75} -66644.4 q^{77} +61081.9 q^{79} +171084. q^{81} +50146.4 q^{83} +46007.5 q^{85} +51315.0 q^{87} +10621.4 q^{89} +150274. q^{91} -102683. q^{93} -23588.2 q^{95} +8802.78 q^{97} +306512. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) $ 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9	$ 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9} - 484 q^{11} - 2922 q^{13} - 244 q^{15} - 3638 q^{17} - 3567 q^{19} - 3811 q^{21} + 3712 q^{23} + 13568 q^{25} - 8814 q^{27} - 5911 q^{29} - 3539 q^{31} - 17215 q^{33} - 10524 q^{35} - 31531 q^{37} - 14986 q^{39} - 35365 q^{41} - 44830 q^{43} - 34403 q^{45} + 12410 q^{47} + 68891 q^{49} - 48608 q^{51} - 79861 q^{53} - 34265 q^{55} - 109824 q^{57} - 51564 q^{59} - 66228 q^{61} + 24981 q^{63} - 31903 q^{65} - 80135 q^{67} + 50903 q^{69} + 162703 q^{71} - 58717 q^{73} + 207827 q^{75} + 89720 q^{77} + 125647 q^{79} + 420081 q^{81} - 28722 q^{83} - 125617 q^{85} + 65848 q^{87} + 93837 q^{89} - 66437 q^{91} - 278711 q^{93} - 83170 q^{95} - 347060 q^{97} + 7934 q^{99}+O(q^{100}) $ 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9 - 484 * q^11 - 2922 * q^13 - 244 * q^15 - 3638 * q^17 - 3567 * q^19 - 3811 * q^21 + 3712 * q^23 + 13568 * q^25 - 8814 * q^27 - 5911 * q^29 - 3539 * q^31 - 17215 * q^33 - 10524 * q^35 - 31531 * q^37 - 14986 * q^39 - 35365 * q^41 - 44830 * q^43 - 34403 * q^45 + 12410 * q^47 + 68891 * q^49 - 48608 * q^51 - 79861 * q^53 - 34265 * q^55 - 109824 * q^57 - 51564 * q^59 - 66228 * q^61 + 24981 * q^63 - 31903 * q^65 - 80135 * q^67 + 50903 * q^69 + 162703 * q^71 - 58717 * q^73 + 207827 * q^75 + 89720 * q^77 + 125647 * q^79 + 420081 * q^81 - 28722 * q^83 - 125617 * q^85 + 65848 * q^87 + 93837 * q^89 - 66437 * q^91 - 278711 * q^93 - 83170 * q^95 - 347060 * q^97 + 7934 * 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
$3$	−29.3150	−1.88056	−0.940279	−	0.340405i	$-0.889436\pi$
$3$	−29.3150	−1.88056	−0.940279	+	0.340405i	$0.889436\pi$
$4$	0	0
$5$	39.9945	0.715443	0.357722	−	0.933828i	$-0.383554\pi$
$5$	39.9945	0.715443	0.357722	+	0.933828i	$0.383554\pi$
$6$	0	0
$7$	−134.016	−1.03374	−0.516872	−	0.856063i	$-0.672904\pi$
$7$	−134.016	−1.03374	−0.516872	+	0.856063i	$0.672904\pi$
$8$	0	0
$9$	616.369	2.53650
$10$	0	0
$11$	497.286	1.23915	0.619576	−	0.784937i	$-0.287305\pi$
$11$	497.286	1.23915	0.619576	+	0.784937i	$0.287305\pi$
$12$	0	0
$13$	−1121.31	−1.84022	−0.920108	−	0.391665i	$-0.871899\pi$
$13$	−1121.31	−1.84022	−0.920108	+	0.391665i	$0.871899\pi$
$14$	0	0
$15$	−1172.44	−1.34543
$16$	0	0
$17$	1150.34	0.965397	0.482698	−	0.875787i	$-0.339657\pi$
$17$	1150.34	0.965397	0.482698	+	0.875787i	$0.339657\pi$
$18$	0	0
$19$	−589.786	−0.374810	−0.187405	−	0.982283i	$-0.560008\pi$
$19$	−589.786	−0.374810	−0.187405	+	0.982283i	$0.560008\pi$
$20$	0	0
$21$	3928.69	1.94401
$22$	0	0
$23$	656.510	0.258775	0.129387	−	0.991594i	$-0.458699\pi$
$23$	656.510	0.258775	0.129387	+	0.991594i	$0.458699\pi$
$24$	0	0
$25$	−1525.44	−0.488141
$26$	0	0
$27$	−10945.3	−2.88947
$28$	0	0
$29$	−1750.47	−0.386509	−0.193255	−	0.981149i	$-0.561904\pi$
$29$	−1750.47	−0.386509	−0.193255	+	0.981149i	$0.561904\pi$
$30$	0	0
$31$	3502.76	0.654646	0.327323	−	0.944913i	$-0.393854\pi$
$31$	3502.76	0.654646	0.327323	+	0.944913i	$0.393854\pi$
$32$	0	0
$33$	−14577.9	−2.33030
$34$	0	0
$35$	−5359.92	−0.739585
$36$	0	0
$37$	925.163	0.111100	0.0555500	−	0.998456i	$-0.482309\pi$
$37$	925.163	0.111100	0.0555500	+	0.998456i	$0.482309\pi$
$38$	0	0
$39$	32871.3	3.46063
$40$	0	0
$41$	−8622.25	−0.801053	−0.400526	−	0.916285i	$-0.631173\pi$
$41$	−8622.25	−0.801053	−0.400526	+	0.916285i	$0.631173\pi$
$42$	0	0
$43$	−11157.4	−0.920219	−0.460110	−	0.887862i	$-0.652190\pi$
$43$	−11157.4	−0.920219	−0.460110	+	0.887862i	$0.652190\pi$
$44$	0	0
$45$	24651.4	1.81472
$46$	0	0
$47$	23415.7	1.54619	0.773093	−	0.634293i	$-0.218709\pi$
$47$	23415.7	1.54619	0.773093	+	0.634293i	$0.218709\pi$
$48$	0	0
$49$	1153.38	0.0686249
$50$	0	0
$51$	−33722.4	−1.81548
$52$	0	0
$53$	−30649.4	−1.49876	−0.749381	−	0.662139i	$-0.769649\pi$
$53$	−30649.4	−1.49876	−0.749381	+	0.662139i	$0.769649\pi$
$54$	0	0
$55$	19888.7	0.886543
$56$	0	0
$57$	17289.6	0.704851
$58$	0	0
$59$	51745.8	1.93528	0.967642	−	0.252325i	$-0.0811953\pi$
$59$	51745.8	1.93528	0.967642	+	0.252325i	$0.0811953\pi$
$60$	0	0
$61$	−15055.1	−0.518036	−0.259018	−	0.965873i	$-0.583399\pi$
$61$	−15055.1	−0.518036	−0.259018	+	0.965873i	$0.583399\pi$
$62$	0	0
$63$	−82603.5	−2.62209
$64$	0	0
$65$	−44846.4	−1.31657
$66$	0	0
$67$	19007.0	0.517282	0.258641	−	0.965974i	$-0.416725\pi$
$67$	19007.0	0.517282	0.258641	+	0.965974i	$0.416725\pi$
$68$	0	0
$69$	−19245.6	−0.486640
$70$	0	0
$71$	49627.2	1.16835	0.584176	−	0.811627i	$-0.301418\pi$
$71$	49627.2	1.16835	0.584176	+	0.811627i	$0.301418\pi$
$72$	0	0
$73$	−20845.2	−0.457825	−0.228912	−	0.973447i	$-0.573517\pi$
$73$	−20845.2	−0.457825	−0.228912	+	0.973447i	$0.573517\pi$
$74$	0	0
$75$	44718.3	0.917977
$76$	0	0
$77$	−66644.4	−1.28096
$78$	0	0
$79$	61081.9	1.10115	0.550573	−	0.834787i	$-0.314409\pi$
$79$	61081.9	1.10115	0.550573	+	0.834787i	$0.314409\pi$
$80$	0	0
$81$	171084.	2.89732
$82$	0	0
$83$	50146.4	0.798997	0.399498	−	0.916734i	$-0.369184\pi$
$83$	50146.4	0.798997	0.399498	+	0.916734i	$0.369184\pi$
$84$	0	0
$85$	46007.5	0.690687
$86$	0	0
$87$	51315.0	0.726853
$88$	0	0
$89$	10621.4	0.142137	0.0710683	−	0.997471i	$-0.477359\pi$
$89$	10621.4	0.142137	0.0710683	+	0.997471i	$0.477359\pi$
$90$	0	0
$91$	150274.	1.90231
$92$	0	0
$93$	−102683.	−1.23110
$94$	0	0
$95$	−23588.2	−0.268155
$96$	0	0
$97$	8802.78	0.0949928	0.0474964	−	0.998871i	$-0.484876\pi$
$97$	8802.78	0.0949928	0.0474964	+	0.998871i	$0.484876\pi$
$98$	0	0
$99$	306512.	3.14310
$100$	0	0
$101$	156203.	1.52365	0.761827	−	0.647780i	$-0.224302\pi$
$101$	156203.	1.52365	0.761827	+	0.647780i	$0.224302\pi$
$102$	0	0
$103$	113297.	1.05226	0.526131	−	0.850403i	$-0.323642\pi$
$103$	113297.	1.05226	0.526131	+	0.850403i	$0.323642\pi$
$104$	0	0
$105$	157126.	1.39083
$106$	0	0
$107$	−75287.6	−0.635718	−0.317859	−	0.948138i	$-0.602964\pi$
$107$	−75287.6	−0.635718	−0.317859	+	0.948138i	$0.602964\pi$
$108$	0	0
$109$	12357.7	0.0996257	0.0498128	−	0.998759i	$-0.484138\pi$
$109$	12357.7	0.0996257	0.0498128	+	0.998759i	$0.484138\pi$
$110$	0	0
$111$	−27121.2	−0.208930
$112$	0	0
$113$	253891.	1.87047	0.935235	−	0.354026i	$-0.115188\pi$
$113$	253891.	1.87047	0.935235	+	0.354026i	$0.115188\pi$
$114$	0	0
$115$	26256.8	0.185138
$116$	0	0
$117$	−691143.	−4.66770
$118$	0	0
$119$	−154165.	−0.997972
$120$	0	0
$121$	86242.2	0.535496
$122$	0	0
$123$	252761.	1.50643
$124$	0	0
$125$	−185992.	−1.06468
$126$	0	0
$127$	−197923.	−1.08890	−0.544448	−	0.838795i	$-0.683261\pi$
$127$	−197923.	−1.08890	−0.544448	+	0.838795i	$0.683261\pi$
$128$	0	0
$129$	327079.	1.73053
$130$	0	0
$131$	−50468.7	−0.256947	−0.128474	−	0.991713i	$-0.541008\pi$
$131$	−50468.7	−0.256947	−0.128474	+	0.991713i	$0.541008\pi$
$132$	0	0
$133$	79041.0	0.387457
$134$	0	0
$135$	−437752.	−2.06725
$136$	0	0
$137$	−235499.	−1.07198	−0.535990	−	0.844224i	$-0.680062\pi$
$137$	−235499.	−1.07198	−0.535990	+	0.844224i	$0.680062\pi$
$138$	0	0
$139$	−58501.9	−0.256822	−0.128411	−	0.991721i	$-0.540988\pi$
$139$	−58501.9	−0.256822	−0.128411	+	0.991721i	$0.540988\pi$
$140$	0	0
$141$	−686430.	−2.90769
$142$	0	0
$143$	−557613.	−2.28031
$144$	0	0
$145$	−70009.2	−0.276525
$146$	0	0
$147$	−33811.3	−0.129053
$148$	0	0
$149$	113859.	0.420146	0.210073	−	0.977686i	$-0.432630\pi$
$149$	113859.	0.420146	0.210073	+	0.977686i	$0.432630\pi$
$150$	0	0
$151$	136345.	0.486628	0.243314	−	0.969948i	$-0.421765\pi$
$151$	136345.	0.486628	0.243314	+	0.969948i	$0.421765\pi$
$152$	0	0
$153$	709037.	2.44873
$154$	0	0
$155$	140091.	0.468362
$156$	0	0
$157$	31096.0	0.100683	0.0503415	−	0.998732i	$-0.483969\pi$
$157$	31096.0	0.100683	0.0503415	+	0.998732i	$0.483969\pi$
$158$	0	0
$159$	898488.	2.81851
$160$	0	0
$161$	−87983.0	−0.267506
$162$	0	0
$163$	−282871.	−0.833911	−0.416955	−	0.908927i	$-0.636903\pi$
$163$	−282871.	−0.833911	−0.416955	+	0.908927i	$0.636903\pi$
$164$	0	0
$165$	−583037.	−1.66719
$166$	0	0
$167$	−44550.6	−0.123613	−0.0618063	−	0.998088i	$-0.519686\pi$
$167$	−44550.6	−0.123613	−0.0618063	+	0.998088i	$0.519686\pi$
$168$	0	0
$169$	886051.	2.38639
$170$	0	0
$171$	−363526.	−0.950703
$172$	0	0
$173$	−333528.	−0.847259	−0.423630	−	0.905835i	$-0.639244\pi$
$173$	−333528.	−0.847259	−0.423630	+	0.905835i	$0.639244\pi$
$174$	0	0
$175$	204434.	0.504612
$176$	0	0
$177$	−1.51693e6	−3.63942
$178$	0	0
$179$	−527307.	−1.23007	−0.615036	−	0.788499i	$-0.710859\pi$
$179$	−527307.	−1.23007	−0.615036	+	0.788499i	$0.710859\pi$
$180$	0	0
$181$	−508109.	−1.15282	−0.576408	−	0.817162i	$-0.695546\pi$
$181$	−508109.	−1.15282	−0.576408	+	0.817162i	$0.695546\pi$
$182$	0	0
$183$	441341.	0.974196
$184$	0	0
$185$	37001.4	0.0794858
$186$	0	0
$187$	572050.	1.19627
$188$	0	0
$189$	1.46685e6	2.98697
$190$	0	0
$191$	−608908.	−1.20773	−0.603863	−	0.797088i	$-0.706373\pi$
$191$	−608908.	−1.20773	−0.603863	+	0.797088i	$0.706373\pi$
$192$	0	0
$193$	750609.	1.45051	0.725255	−	0.688481i	$-0.241722\pi$
$193$	750609.	1.45051	0.725255	+	0.688481i	$0.241722\pi$
$194$	0	0
$195$	1.31467e6	2.47589
$196$	0	0
$197$	733413.	1.34643	0.673214	−	0.739448i	$-0.264913\pi$
$197$	733413.	1.34643	0.673214	+	0.739448i	$0.264913\pi$
$198$	0	0
$199$	468603.	0.838827	0.419414	−	0.907795i	$-0.362236\pi$
$199$	468603.	0.838827	0.419414	+	0.907795i	$0.362236\pi$
$200$	0	0
$201$	−557191.	−0.972778
$202$	0	0
$203$	234592.	0.399551
$204$	0	0
$205$	−344843.	−0.573108
$206$	0	0
$207$	404652.	0.656381
$208$	0	0
$209$	−293292.	−0.464446
$210$	0	0
$211$	637093.	0.985137	0.492569	−	0.870274i	$-0.336058\pi$
$211$	637093.	0.985137	0.492569	+	0.870274i	$0.336058\pi$
$212$	0	0
$213$	−1.45482e6	−2.19715
$214$	0	0
$215$	−446234.	−0.658365
$216$	0	0
$217$	−469427.	−0.676735
$218$	0	0
$219$	611078.	0.860966
$220$	0	0
$221$	−1.28990e6	−1.77654
$222$	0	0
$223$	942094.	1.26862	0.634311	−	0.773078i	$-0.281284\pi$
$223$	942094.	1.26862	0.634311	+	0.773078i	$0.281284\pi$
$224$	0	0
$225$	−940234.	−1.23817
$226$	0	0
$227$	168800.	0.217424	0.108712	−	0.994073i	$-0.465327\pi$
$227$	168800.	0.217424	0.108712	+	0.994073i	$0.465327\pi$
$228$	0	0
$229$	−227578.	−0.286775	−0.143387	−	0.989667i	$-0.545799\pi$
$229$	−227578.	−0.286775	−0.143387	+	0.989667i	$0.545799\pi$
$230$	0	0
$231$	1.95368e6	2.40893
$232$	0	0
$233$	−1.63859e6	−1.97734	−0.988669	−	0.150111i	$-0.952037\pi$
$233$	−1.63859e6	−1.97734	−0.988669	+	0.150111i	$0.952037\pi$
$234$	0	0
$235$	936498.	1.10621
$236$	0	0
$237$	−1.79062e6	−2.07077
$238$	0	0
$239$	−1.21758e6	−1.37880	−0.689400	−	0.724381i	$-0.742126\pi$
$239$	−1.21758e6	−1.37880	−0.689400	+	0.724381i	$0.742126\pi$
$240$	0	0
$241$	364488.	0.404242	0.202121	−	0.979361i	$-0.435217\pi$
$241$	364488.	0.404242	0.202121	+	0.979361i	$0.435217\pi$
$242$	0	0
$243$	−2.35562e6	−2.55911
$244$	0	0
$245$	46128.8	0.0490972
$246$	0	0
$247$	661335.	0.689730
$248$	0	0
$249$	−1.47004e6	−1.50256
$250$	0	0
$251$	−888075.	−0.889745	−0.444872	−	0.895594i	$-0.646751\pi$
$251$	−888075.	−0.889745	−0.444872	+	0.895594i	$0.646751\pi$
$252$	0	0
$253$	326473.	0.320661
$254$	0	0
$255$	−1.34871e6	−1.29888
$256$	0	0
$257$	66049.0	0.0623783
$257$	66049.0	0.0623783
$258$	0	0
$259$	−123987.	−0.114849
$260$	0	0
$261$	−1.07894e6	−0.980380
$262$	0	0
$263$	−1.45296e6	−1.29528	−0.647640	−	0.761947i	$-0.724244\pi$
$263$	−1.45296e6	−1.29528	−0.647640	+	0.761947i	$0.724244\pi$
$264$	0	0
$265$	−1.22581e6	−1.07228
$266$	0	0
$267$	−311365.	−0.267296
$268$	0	0
$269$	−240509.	−0.202652	−0.101326	−	0.994853i	$-0.532308\pi$
$269$	−240509.	−0.202652	−0.101326	+	0.994853i	$0.532308\pi$
$270$	0	0
$271$	−617043.	−0.510379	−0.255189	−	0.966891i	$-0.582138\pi$
$271$	−617043.	−0.510379	−0.255189	+	0.966891i	$0.582138\pi$
$272$	0	0
$273$	−4.40529e6	−3.57740
$274$	0	0
$275$	−758580.	−0.604881
$276$	0	0
$277$	−1.99266e6	−1.56040	−0.780198	−	0.625533i	$-0.784882\pi$
$277$	−1.99266e6	−1.56040	−0.780198	+	0.625533i	$0.784882\pi$
$278$	0	0
$279$	2.15899e6	1.66051
$280$	0	0
$281$	−266178.	−0.201097	−0.100548	−	0.994932i	$-0.532060\pi$
$281$	−266178.	−0.201097	−0.100548	+	0.994932i	$0.532060\pi$
$282$	0	0
$283$	831036.	0.616814	0.308407	−	0.951255i	$-0.400204\pi$
$283$	831036.	0.616814	0.308407	+	0.951255i	$0.400204\pi$
$284$	0	0
$285$	691488.	0.504281
$286$	0	0
$287$	1.15552e6	0.828083
$288$	0	0
$289$	−96563.4	−0.0680093
$290$	0	0
$291$	−258054.	−0.178639
$292$	0	0
$293$	231235.	0.157357	0.0786783	−	0.996900i	$-0.474930\pi$
$293$	231235.	0.157357	0.0786783	+	0.996900i	$0.474930\pi$
$294$	0	0
$295$	2.06955e6	1.38459
$296$	0	0
$297$	−5.44295e6	−3.58049
$298$	0	0
$299$	−736153.	−0.476201
$300$	0	0
$301$	1.49527e6	0.951270
$302$	0	0
$303$	−4.57910e6	−2.86532
$304$	0	0
$305$	−602122.	−0.370625
$306$	0	0
$307$	−2.71566e6	−1.64448	−0.822241	−	0.569139i	$-0.807277\pi$
$307$	−2.71566e6	−1.64448	−0.822241	+	0.569139i	$0.807277\pi$
$308$	0	0
$309$	−3.32129e6	−1.97884
$310$	0	0
$311$	−508503.	−0.298121	−0.149060	−	0.988828i	$-0.547625\pi$
$311$	−508503.	−0.298121	−0.149060	+	0.988828i	$0.547625\pi$
$312$	0	0
$313$	−1.10899e6	−0.639834	−0.319917	−	0.947446i	$-0.603655\pi$
$313$	−1.10899e6	−0.639834	−0.319917	+	0.947446i	$0.603655\pi$
$314$	0	0
$315$	−3.30369e6	−1.87595
$316$	0	0
$317$	1.81212e6	1.01283	0.506417	−	0.862289i	$-0.330970\pi$
$317$	1.81212e6	1.01283	0.506417	+	0.862289i	$0.330970\pi$
$318$	0	0
$319$	−870485.	−0.478944
$320$	0	0
$321$	2.20706e6	1.19550
$322$	0	0
$323$	−678458.	−0.361840
$324$	0	0
$325$	1.71050e6	0.898284
$326$	0	0
$327$	−362266.	−0.187352
$328$	0	0
$329$	−3.13808e6	−1.59836
$330$	0	0
$331$	−341054.	−0.171101	−0.0855507	−	0.996334i	$-0.527265\pi$
$331$	−341054.	−0.171101	−0.0855507	+	0.996334i	$0.527265\pi$
$332$	0	0
$333$	570242.	0.281805
$334$	0	0
$335$	760176.	0.370086
$336$	0	0
$337$	−170300.	−0.0816846	−0.0408423	−	0.999166i	$-0.513004\pi$
$337$	−170300.	−0.0816846	−0.0408423	+	0.999166i	$0.513004\pi$
$338$	0	0
$339$	−7.44281e6	−3.51753
$340$	0	0
$341$	1.74187e6	0.811205
$342$	0	0
$343$	2.09784e6	0.962803
$344$	0	0
$345$	−769717.	−0.348164
$346$	0	0
$347$	−4.00376e6	−1.78502	−0.892512	−	0.451025i	$-0.851059\pi$
$347$	−4.00376e6	−1.78502	−0.892512	+	0.451025i	$0.851059\pi$
$348$	0	0
$349$	767002.	0.337080	0.168540	−	0.985695i	$-0.446095\pi$
$349$	767002.	0.337080	0.168540	+	0.985695i	$0.446095\pi$
$350$	0	0
$351$	1.22731e7	5.31725
$352$	0	0
$353$	−1.85262e6	−0.791317	−0.395659	−	0.918398i	$-0.629484\pi$
$353$	−1.85262e6	−0.791317	−0.395659	+	0.918398i	$0.629484\pi$
$354$	0	0
$355$	1.98481e6	0.835889
$356$	0	0
$357$	4.51935e6	1.87674
$358$	0	0
$359$	782213.	0.320324	0.160162	−	0.987091i	$-0.448798\pi$
$359$	782213.	0.320324	0.160162	+	0.987091i	$0.448798\pi$
$360$	0	0
$361$	−2.12825e6	−0.859518
$362$	0	0
$363$	−2.52819e6	−1.00703
$364$	0	0
$365$	−833694.	−0.327548
$366$	0	0
$367$	−3.84550e6	−1.49035	−0.745175	−	0.666869i	$-0.767634\pi$
$367$	−3.84550e6	−1.49035	−0.745175	+	0.666869i	$0.767634\pi$
$368$	0	0
$369$	−5.31449e6	−2.03187
$370$	0	0
$371$	4.10752e6	1.54933
$372$	0	0
$373$	791056.	0.294398	0.147199	−	0.989107i	$-0.452974\pi$
$373$	791056.	0.294398	0.147199	+	0.989107i	$0.452974\pi$
$374$	0	0
$375$	5.45235e6	2.00219
$376$	0	0
$377$	1.96283e6	0.711260
$378$	0	0
$379$	3.92084e6	1.40211	0.701054	−	0.713108i	$-0.252713\pi$
$379$	3.92084e6	1.40211	0.701054	+	0.713108i	$0.252713\pi$
$380$	0	0
$381$	5.80210e6	2.04773
$382$	0	0
$383$	309877.	0.107943	0.0539713	−	0.998542i	$-0.482812\pi$
$383$	309877.	0.107943	0.0539713	+	0.998542i	$0.482812\pi$
$384$	0	0
$385$	−2.66541e6	−0.916457
$386$	0	0
$387$	−6.87706e6	−2.33413
$388$	0	0
$389$	5.43802e6	1.82208	0.911039	−	0.412320i	$-0.135282\pi$
$389$	5.43802e6	1.82208	0.911039	+	0.412320i	$0.135282\pi$
$390$	0	0
$391$	755213.	0.249820
$392$	0	0
$393$	1.47949e6	0.483204
$394$	0	0
$395$	2.44294e6	0.787807
$396$	0	0
$397$	−1.30755e6	−0.416373	−0.208187	−	0.978089i	$-0.566756\pi$
$397$	−1.30755e6	−0.416373	−0.208187	+	0.978089i	$0.566756\pi$
$398$	0	0
$399$	−2.31709e6	−0.728635
$400$	0	0
$401$	−4.52126e6	−1.40410	−0.702051	−	0.712127i	$-0.747732\pi$
$401$	−4.52126e6	−1.40410	−0.702051	+	0.712127i	$0.747732\pi$
$402$	0	0
$403$	−3.92769e6	−1.20469
$404$	0	0
$405$	6.84242e6	2.07287
$406$	0	0
$407$	460071.	0.137670
$408$	0	0
$409$	2.29253e6	0.677653	0.338826	−	0.940849i	$-0.389970\pi$
$409$	2.29253e6	0.677653	0.338826	+	0.940849i	$0.389970\pi$
$410$	0	0
$411$	6.90364e6	2.01592
$412$	0	0
$413$	−6.93478e6	−2.00059
$414$	0	0
$415$	2.00558e6	0.571637
$416$	0	0
$417$	1.71498e6	0.482969
$418$	0	0
$419$	−1.27236e6	−0.354059	−0.177029	−	0.984206i	$-0.556649\pi$
$419$	−1.27236e6	−0.354059	−0.177029	+	0.984206i	$0.556649\pi$
$420$	0	0
$421$	2.77183e6	0.762188	0.381094	−	0.924536i	$-0.375547\pi$
$421$	2.77183e6	0.762188	0.381094	+	0.924536i	$0.375547\pi$
$422$	0	0
$423$	1.44327e7	3.92190
$424$	0	0
$425$	−1.75478e6	−0.471250
$426$	0	0
$427$	2.01763e6	0.535516
$428$	0	0
$429$	1.63464e7	4.28825
$430$	0	0
$431$	2.07773e6	0.538761	0.269380	−	0.963034i	$-0.413181\pi$
$431$	2.07773e6	0.538761	0.269380	+	0.963034i	$0.413181\pi$
$432$	0	0
$433$	2.16080e6	0.553854	0.276927	−	0.960891i	$-0.410684\pi$
$433$	2.16080e6	0.553854	0.276927	+	0.960891i	$0.410684\pi$
$434$	0	0
$435$	2.05232e6	0.520022
$436$	0	0
$437$	−387200.	−0.0969911
$438$	0	0
$439$	1.01238e6	0.250716	0.125358	−	0.992112i	$-0.459992\pi$
$439$	1.01238e6	0.250716	0.125358	+	0.992112i	$0.459992\pi$
$440$	0	0
$441$	710906.	0.174067
$442$	0	0
$443$	3.18472e6	0.771013	0.385506	−	0.922705i	$-0.374027\pi$
$443$	3.18472e6	0.771013	0.385506	+	0.922705i	$0.374027\pi$
$444$	0	0
$445$	424796.	0.101691
$446$	0	0
$447$	−3.33777e6	−0.790109
$448$	0	0
$449$	3.50095e6	0.819541	0.409770	−	0.912189i	$-0.365609\pi$
$449$	3.50095e6	0.819541	0.409770	+	0.912189i	$0.365609\pi$
$450$	0	0
$451$	−4.28772e6	−0.992626
$452$	0	0
$453$	−3.99696e6	−0.915133
$454$	0	0
$455$	6.01015e6	1.36099
$456$	0	0
$457$	4.98577e6	1.11671	0.558356	−	0.829601i	$-0.311432\pi$
$457$	4.98577e6	1.11671	0.558356	+	0.829601i	$0.311432\pi$
$458$	0	0
$459$	−1.25909e7	−2.78949
$460$	0	0
$461$	−8.19363e6	−1.79566	−0.897830	−	0.440343i	$-0.854857\pi$
$461$	−8.19363e6	−1.79566	−0.897830	+	0.440343i	$0.854857\pi$
$462$	0	0
$463$	−8.87667e6	−1.92441	−0.962204	−	0.272328i	$-0.912206\pi$
$463$	−8.87667e6	−1.92441	−0.962204	+	0.272328i	$0.912206\pi$
$464$	0	0
$465$	−4.10677e6	−0.880781
$466$	0	0
$467$	−1.61920e6	−0.343564	−0.171782	−	0.985135i	$-0.554952\pi$
$467$	−1.61920e6	−0.343564	−0.171782	+	0.985135i	$0.554952\pi$
$468$	0	0
$469$	−2.54725e6	−0.534737
$470$	0	0
$471$	−911580.	−0.189340
$472$	0	0
$473$	−5.54841e6	−1.14029
$474$	0	0
$475$	899684.	0.182960
$476$	0	0
$477$	−1.88914e7	−3.80161
$478$	0	0
$479$	−7.62140e6	−1.51774	−0.758868	−	0.651244i	$-0.774247\pi$
$479$	−7.62140e6	−1.51774	−0.758868	+	0.651244i	$0.774247\pi$
$480$	0	0
$481$	−1.03740e6	−0.204448
$482$	0	0
$483$	2.57922e6	0.503061
$484$	0	0
$485$	352063.	0.0679620
$486$	0	0
$487$	−6.89131e6	−1.31668	−0.658339	−	0.752722i	$-0.728741\pi$
$487$	−6.89131e6	−1.31668	−0.658339	+	0.752722i	$0.728741\pi$
$488$	0	0
$489$	8.29236e6	1.56822
$490$	0	0
$491$	4.51838e6	0.845822	0.422911	−	0.906171i	$-0.361008\pi$
$491$	4.51838e6	0.845822	0.422911	+	0.906171i	$0.361008\pi$
$492$	0	0
$493$	−2.01365e6	−0.373135
$494$	0	0
$495$	1.22588e7	2.24871
$496$	0	0
$497$	−6.65085e6	−1.20778
$498$	0	0
$499$	−7.89342e6	−1.41910	−0.709552	−	0.704653i	$-0.751102\pi$
$499$	−7.89342e6	−1.41910	−0.709552	+	0.704653i	$0.751102\pi$
$500$	0	0
$501$	1.30600e6	0.232461
$502$	0	0
$503$	6.69455e6	1.17978	0.589890	−	0.807483i	$-0.299171\pi$
$503$	6.69455e6	1.17978	0.589890	+	0.807483i	$0.299171\pi$
$504$	0	0
$505$	6.24727e6	1.09009
$506$	0	0
$507$	−2.59746e7	−4.48775
$508$	0	0
$509$	556096.	0.0951383	0.0475692	−	0.998868i	$-0.484853\pi$
$509$	556096.	0.0951383	0.0475692	+	0.998868i	$0.484853\pi$
$510$	0	0
$511$	2.79360e6	0.473273
$512$	0	0
$513$	6.45539e6	1.08300
$514$	0	0
$515$	4.53124e6	0.752834
$516$	0	0
$517$	1.16443e7	1.91596
$518$	0	0
$519$	9.77736e6	1.59332
$520$	0	0
$521$	2.46344e6	0.397601	0.198801	−	0.980040i	$-0.436295\pi$
$521$	2.46344e6	0.397601	0.198801	+	0.980040i	$0.436295\pi$
$522$	0	0
$523$	−1.70990e6	−0.273348	−0.136674	−	0.990616i	$-0.543641\pi$
$523$	−1.70990e6	−0.273348	−0.136674	+	0.990616i	$0.543641\pi$
$524$	0	0
$525$	−5.99298e6	−0.948953
$526$	0	0
$527$	4.02938e6	0.631993
$528$	0	0
$529$	−6.00534e6	−0.933036
$530$	0	0
$531$	3.18945e7	4.90885
$532$	0	0
$533$	9.66825e6	1.47411
$534$	0	0
$535$	−3.01109e6	−0.454820
$536$	0	0
$537$	1.54580e7	2.31322
$538$	0	0
$539$	573559.	0.0850366
$540$	0	0
$541$	−5.70132e6	−0.837495	−0.418747	−	0.908103i	$-0.637531\pi$
$541$	−5.70132e6	−0.837495	−0.418747	+	0.908103i	$0.637531\pi$
$542$	0	0
$543$	1.48952e7	2.16794
$544$	0	0
$545$	494240.	0.0712765
$546$	0	0
$547$	4.60090e6	0.657468	0.328734	−	0.944423i	$-0.393378\pi$
$547$	4.60090e6	0.657468	0.328734	+	0.944423i	$0.393378\pi$
$548$	0	0
$549$	−9.27951e6	−1.31400
$550$	0	0
$551$	1.03240e6	0.144867
$552$	0	0
$553$	−8.18597e6	−1.13830
$554$	0	0
$555$	−1.08470e6	−0.149478
$556$	0	0
$557$	−4.64983e6	−0.635038	−0.317519	−	0.948252i	$-0.602850\pi$
$557$	−4.64983e6	−0.635038	−0.317519	+	0.948252i	$0.602850\pi$
$558$	0	0
$559$	1.25109e7	1.69340
$560$	0	0
$561$	−1.67697e7	−2.24966
$562$	0	0
$563$	−1.43341e7	−1.90590	−0.952951	−	0.303125i	$-0.901970\pi$
$563$	−1.43341e7	−1.90590	−0.952951	+	0.303125i	$0.901970\pi$
$564$	0	0
$565$	1.01542e7	1.33822
$566$	0	0
$567$	−2.29280e7	−2.99509
$568$	0	0
$569$	−4.45099e6	−0.576336	−0.288168	−	0.957580i	$-0.593046\pi$
$569$	−4.45099e6	−0.576336	−0.288168	+	0.957580i	$0.593046\pi$
$570$	0	0
$571$	1.83795e6	0.235908	0.117954	−	0.993019i	$-0.462366\pi$
$571$	1.83795e6	0.235908	0.117954	+	0.993019i	$0.462366\pi$
$572$	0	0
$573$	1.78501e7	2.27120
$574$	0	0
$575$	−1.00147e6	−0.126318
$576$	0	0
$577$	1.12760e7	1.40998	0.704991	−	0.709216i	$-0.250951\pi$
$577$	1.12760e7	1.40998	0.704991	+	0.709216i	$0.250951\pi$
$578$	0	0
$579$	−2.20041e7	−2.72777
$580$	0	0
$581$	−6.72044e6	−0.825957
$582$	0	0
$583$	−1.52415e7	−1.85719
$584$	0	0
$585$	−2.76419e7	−3.33948
$586$	0	0
$587$	1.21846e7	1.45954	0.729768	−	0.683695i	$-0.239628\pi$
$587$	1.21846e7	1.45954	0.729768	+	0.683695i	$0.239628\pi$
$588$	0	0
$589$	−2.06588e6	−0.245367
$590$	0	0
$591$	−2.15000e7	−2.53204
$592$	0	0
$593$	−2.91916e6	−0.340895	−0.170448	−	0.985367i	$-0.554521\pi$
$593$	−2.91916e6	−0.340895	−0.170448	+	0.985367i	$0.554521\pi$
$594$	0	0
$595$	−6.16575e6	−0.713993
$596$	0	0
$597$	−1.37371e7	−1.57746
$598$	0	0
$599$	1.28390e7	1.46206	0.731030	−	0.682346i	$-0.239040\pi$
$599$	1.28390e7	1.46206	0.731030	+	0.682346i	$0.239040\pi$
$600$	0	0
$601$	−506880.	−0.0572425	−0.0286213	−	0.999590i	$-0.509112\pi$
$601$	−506880.	−0.0572425	−0.0286213	+	0.999590i	$0.509112\pi$
$602$	0	0
$603$	1.17153e7	1.31208
$604$	0	0
$605$	3.44921e6	0.383117
$606$	0	0
$607$	1.38806e7	1.52910	0.764552	−	0.644562i	$-0.222960\pi$
$607$	1.38806e7	1.52910	0.764552	+	0.644562i	$0.222960\pi$
$608$	0	0
$609$	−6.87705e6	−0.751379
$610$	0	0
$611$	−2.62563e7	−2.84532
$612$	0	0
$613$	−9.48319e6	−1.01930	−0.509651	−	0.860381i	$-0.670226\pi$
$613$	−9.48319e6	−1.01930	−0.509651	+	0.860381i	$0.670226\pi$
$614$	0	0
$615$	1.01091e7	1.07776
$616$	0	0
$617$	−4.01718e6	−0.424823	−0.212411	−	0.977180i	$-0.568132\pi$
$617$	−4.01718e6	−0.424823	−0.212411	+	0.977180i	$0.568132\pi$
$618$	0	0
$619$	8.25652e6	0.866105	0.433052	−	0.901369i	$-0.357437\pi$
$619$	8.25652e6	0.866105	0.433052	+	0.901369i	$0.357437\pi$
$620$	0	0
$621$	−7.18570e6	−0.747722
$622$	0	0
$623$	−1.42344e6	−0.146933
$624$	0	0
$625$	−2.67166e6	−0.273578
$626$	0	0
$627$	8.59786e6	0.873417
$628$	0	0
$629$	1.06426e6	0.107256
$630$	0	0
$631$	−1.81107e7	−1.81077	−0.905383	−	0.424595i	$-0.860416\pi$
$631$	−1.81107e7	−1.81077	−0.905383	+	0.424595i	$0.860416\pi$
$632$	0	0
$633$	−1.86764e7	−1.85261
$634$	0	0
$635$	−7.91581e6	−0.779043
$636$	0	0
$637$	−1.29330e6	−0.126285
$638$	0	0
$639$	3.05886e7	2.96352
$640$	0	0
$641$	1.52312e6	0.146416	0.0732081	−	0.997317i	$-0.476676\pi$
$641$	1.52312e6	0.146416	0.0732081	+	0.997317i	$0.476676\pi$
$642$	0	0
$643$	1.95832e7	1.86791	0.933955	−	0.357390i	$-0.116333\pi$
$643$	1.95832e7	1.86791	0.933955	+	0.357390i	$0.116333\pi$
$644$	0	0
$645$	1.30813e7	1.23809
$646$	0	0
$647$	1.13079e7	1.06199	0.530997	−	0.847374i	$-0.321818\pi$
$647$	1.13079e7	1.06199	0.530997	+	0.847374i	$0.321818\pi$
$648$	0	0
$649$	2.57324e7	2.39811
$650$	0	0
$651$	1.37613e7	1.27264
$652$	0	0
$653$	−1.18803e7	−1.09029	−0.545147	−	0.838341i	$-0.683526\pi$
$653$	−1.18803e7	−1.09029	−0.545147	+	0.838341i	$0.683526\pi$
$654$	0	0
$655$	−2.01847e6	−0.183831
$656$	0	0
$657$	−1.28483e7	−1.16127
$658$	0	0
$659$	1.27025e7	1.13940	0.569699	−	0.821854i	$-0.307060\pi$
$659$	1.27025e7	1.13940	0.569699	+	0.821854i	$0.307060\pi$
$660$	0	0
$661$	1.70254e7	1.51563	0.757814	−	0.652471i	$-0.226267\pi$
$661$	1.70254e7	1.51563	0.757814	+	0.652471i	$0.226267\pi$
$662$	0	0
$663$	3.78133e7	3.34088
$664$	0	0
$665$	3.16120e6	0.277203
$666$	0	0
$667$	−1.14920e6	−0.100019
$668$	0	0
$669$	−2.76175e7	−2.38572
$670$	0	0
$671$	−7.48670e6	−0.641925
$672$	0	0
$673$	−1.96920e7	−1.67591	−0.837956	−	0.545738i	$-0.816249\pi$
$673$	−1.96920e7	−1.67591	−0.837956	+	0.545738i	$0.816249\pi$
$674$	0	0
$675$	1.66964e7	1.41047
$676$	0	0
$677$	−5.50571e6	−0.461681	−0.230840	−	0.972992i	$-0.574148\pi$
$677$	−5.50571e6	−0.461681	−0.230840	+	0.972992i	$0.574148\pi$
$678$	0	0
$679$	−1.17972e6	−0.0981981
$680$	0	0
$681$	−4.94837e6	−0.408879
$682$	0	0
$683$	−3.89767e6	−0.319708	−0.159854	−	0.987141i	$-0.551102\pi$
$683$	−3.89767e6	−0.319708	−0.159854	+	0.987141i	$0.551102\pi$
$684$	0	0
$685$	−9.41865e6	−0.766941
$686$	0	0
$687$	6.67144e6	0.539297
$688$	0	0
$689$	3.43676e7	2.75804
$690$	0	0
$691$	1.76278e6	0.140444	0.0702220	−	0.997531i	$-0.477629\pi$
$691$	1.76278e6	0.140444	0.0702220	+	0.997531i	$0.477629\pi$
$692$	0	0
$693$	−4.10775e7	−3.24916
$694$	0	0
$695$	−2.33975e6	−0.183742
$696$	0	0
$697$	−9.91857e6	−0.773334
$698$	0	0
$699$	4.80353e7	3.71850
$700$	0	0
$701$	−9.36382e6	−0.719710	−0.359855	−	0.933008i	$-0.617174\pi$
$701$	−9.36382e6	−0.719710	−0.359855	+	0.933008i	$0.617174\pi$
$702$	0	0
$703$	−545649.	−0.0416413
$704$	0	0
$705$	−2.74534e7	−2.08029
$706$	0	0
$707$	−2.09338e7	−1.57507
$708$	0	0
$709$	−1.18038e7	−0.881870	−0.440935	−	0.897539i	$-0.645353\pi$
$709$	−1.18038e7	−0.881870	−0.440935	+	0.897539i	$0.645353\pi$
$710$	0	0
$711$	3.76490e7	2.79305
$712$	0	0
$713$	2.29960e6	0.169406
$714$	0	0
$715$	−2.23015e7	−1.63143
$716$	0	0
$717$	3.56932e7	2.59291
$718$	0	0
$719$	1.93976e7	1.39935	0.699673	−	0.714463i	$-0.253329\pi$
$719$	1.93976e7	1.39935	0.699673	+	0.714463i	$0.253329\pi$
$720$	0	0
$721$	−1.51836e7	−1.08777
$722$	0	0
$723$	−1.06850e7	−0.760200
$724$	0	0
$725$	2.67024e6	0.188671
$726$	0	0
$727$	−6.54240e6	−0.459094	−0.229547	−	0.973298i	$-0.573724\pi$
$727$	−6.54240e6	−0.459094	−0.229547	+	0.973298i	$0.573724\pi$
$728$	0	0
$729$	2.74814e7	1.91523
$730$	0	0
$731$	−1.28348e7	−0.888376
$732$	0	0
$733$	2.03647e7	1.39997	0.699983	−	0.714160i	$-0.253191\pi$
$733$	2.03647e7	1.39997	0.699983	+	0.714160i	$0.253191\pi$
$734$	0	0
$735$	−1.35226e6	−0.0923301
$736$	0	0
$737$	9.45193e6	0.640990
$738$	0	0
$739$	−1.47292e7	−0.992126	−0.496063	−	0.868287i	$-0.665221\pi$
$739$	−1.47292e7	−0.992126	−0.496063	+	0.868287i	$0.665221\pi$
$740$	0	0
$741$	−1.93870e7	−1.29708
$742$	0	0
$743$	1.95134e7	1.29676	0.648382	−	0.761315i	$-0.275446\pi$
$743$	1.95134e7	1.29676	0.648382	+	0.761315i	$0.275446\pi$
$744$	0	0
$745$	4.55372e6	0.300591
$746$	0	0
$747$	3.09087e7	2.02665
$748$	0	0
$749$	1.00898e7	0.657169
$750$	0	0
$751$	−8.90593e6	−0.576208	−0.288104	−	0.957599i	$-0.593025\pi$
$751$	−8.90593e6	−0.576208	−0.288104	+	0.957599i	$0.593025\pi$
$752$	0	0
$753$	2.60339e7	1.67322
$754$	0	0
$755$	5.45306e6	0.348155
$756$	0	0
$757$	−2.09018e7	−1.32570	−0.662848	−	0.748754i	$-0.730652\pi$
$757$	−2.09018e7	−1.32570	−0.662848	+	0.748754i	$0.730652\pi$
$758$	0	0
$759$	−9.57055e6	−0.603021
$760$	0	0
$761$	−4.63799e6	−0.290314	−0.145157	−	0.989409i	$-0.546369\pi$
$761$	−4.63799e6	−0.290314	−0.145157	+	0.989409i	$0.546369\pi$
$762$	0	0
$763$	−1.65613e6	−0.102987
$764$	0	0
$765$	2.83576e7	1.75192
$766$	0	0
$767$	−5.80232e7	−3.56134
$768$	0	0
$769$	−1.20413e7	−0.734271	−0.367135	−	0.930168i	$-0.619661\pi$
$769$	−1.20413e7	−0.734271	−0.367135	+	0.930168i	$0.619661\pi$
$770$	0	0
$771$	−1.93623e6	−0.117306
$772$	0	0
$773$	−408366.	−0.0245811	−0.0122905	−	0.999924i	$-0.503912\pi$
$773$	−408366.	−0.0245811	−0.0122905	+	0.999924i	$0.503912\pi$
$774$	0	0
$775$	−5.34325e6	−0.319559
$776$	0	0
$777$	3.63468e6	0.215980
$778$	0	0
$779$	5.08529e6	0.300242
$780$	0	0
$781$	2.46789e7	1.44776
$782$	0	0
$783$	1.91594e7	1.11681
$784$	0	0
$785$	1.24367e6	0.0720330
$786$	0	0
$787$	−2.66265e7	−1.53242	−0.766209	−	0.642592i	$-0.777859\pi$
$787$	−2.66265e7	−1.53242	−0.766209	+	0.642592i	$0.777859\pi$
$788$	0	0
$789$	4.25934e7	2.43585
$790$	0	0
$791$	−3.40255e7	−1.93359
$792$	0	0
$793$	1.68815e7	0.953298
$794$	0	0
$795$	3.59346e7	2.01648
$796$	0	0
$797$	−2.12406e6	−0.118446	−0.0592232	−	0.998245i	$-0.518862\pi$
$797$	−2.12406e6	−0.118446	−0.0592232	+	0.998245i	$0.518862\pi$
$798$	0	0
$799$	2.69361e7	1.49268
$800$	0	0
$801$	6.54668e6	0.360529
$802$	0	0
$803$	−1.03660e7	−0.567314
$804$	0	0
$805$	−3.51884e6	−0.191386
$806$	0	0
$807$	7.05052e6	0.381098
$808$	0	0
$809$	−2.14429e7	−1.15189	−0.575946	−	0.817488i	$-0.695366\pi$
$809$	−2.14429e7	−1.15189	−0.575946	+	0.817488i	$0.695366\pi$
$810$	0	0
$811$	−2.61318e7	−1.39514	−0.697568	−	0.716519i	$-0.745734\pi$
$811$	−2.61318e7	−1.39514	−0.697568	+	0.716519i	$0.745734\pi$
$812$	0	0
$813$	1.80886e7	0.959796
$814$	0	0
$815$	−1.13133e7	−0.596616
$816$	0	0
$817$	6.58047e6	0.344907
$818$	0	0
$819$	9.26244e7	4.82520
$820$	0	0
$821$	2.00658e7	1.03896	0.519481	−	0.854482i	$-0.326125\pi$
$821$	2.00658e7	1.03896	0.519481	+	0.854482i	$0.326125\pi$
$822$	0	0
$823$	−1.74128e7	−0.896128	−0.448064	−	0.894001i	$-0.647886\pi$
$823$	−1.74128e7	−0.896128	−0.448064	+	0.894001i	$0.647886\pi$
$824$	0	0
$825$	2.22378e7	1.13751
$826$	0	0
$827$	−3.52436e7	−1.79191	−0.895955	−	0.444145i	$-0.853507\pi$
$827$	−3.52436e7	−1.79191	−0.895955	+	0.444145i	$0.853507\pi$
$828$	0	0
$829$	−1.91347e7	−0.967019	−0.483509	−	0.875339i	$-0.660638\pi$
$829$	−1.91347e7	−0.967019	−0.483509	+	0.875339i	$0.660638\pi$
$830$	0	0
$831$	5.84149e7	2.93441
$832$	0	0
$833$	1.32678e6	0.0662502
$834$	0	0
$835$	−1.78178e6	−0.0884378
$836$	0	0
$837$	−3.83388e7	−1.89158
$838$	0	0
$839$	−1.79744e7	−0.881557	−0.440779	−	0.897616i	$-0.645298\pi$
$839$	−1.79744e7	−0.881557	−0.440779	+	0.897616i	$0.645298\pi$
$840$	0	0
$841$	−1.74470e7	−0.850611
$842$	0	0
$843$	7.80299e6	0.378175
$844$	0	0
$845$	3.54372e7	1.70733
$846$	0	0
$847$	−1.15579e7	−0.553566
$848$	0	0
$849$	−2.43618e7	−1.15995
$850$	0	0
$851$	607379.	0.0287499
$852$	0	0
$853$	−3.28705e7	−1.54680	−0.773398	−	0.633920i	$-0.781445\pi$
$853$	−3.28705e7	−1.54680	−0.773398	+	0.633920i	$0.781445\pi$
$854$	0	0
$855$	−1.45390e7	−0.680174
$856$	0	0
$857$	4.29585e6	0.199801	0.0999003	−	0.994997i	$-0.468148\pi$
$857$	4.29585e6	0.199801	0.0999003	+	0.994997i	$0.468148\pi$
$858$	0	0
$859$	−3.98478e6	−0.184256	−0.0921280	−	0.995747i	$-0.529367\pi$
$859$	−3.98478e6	−0.184256	−0.0921280	+	0.995747i	$0.529367\pi$
$860$	0	0
$861$	−3.38741e7	−1.55726
$862$	0	0
$863$	−1.40511e7	−0.642218	−0.321109	−	0.947042i	$-0.604056\pi$
$863$	−1.40511e7	−0.642218	−0.321109	+	0.947042i	$0.604056\pi$
$864$	0	0
$865$	−1.33393e7	−0.606166
$866$	0	0
$867$	2.83076e6	0.127895
$868$	0	0
$869$	3.03752e7	1.36449
$870$	0	0
$871$	−2.13128e7	−0.951910
$872$	0	0
$873$	5.42576e6	0.240949
$874$	0	0
$875$	2.49260e7	1.10061
$876$	0	0
$877$	2.87797e7	1.26354	0.631768	−	0.775158i	$-0.282330\pi$
$877$	2.87797e7	1.26354	0.631768	+	0.775158i	$0.282330\pi$
$878$	0	0
$879$	−6.77866e6	−0.295918
$880$	0	0
$881$	−1.05521e7	−0.458033	−0.229017	−	0.973423i	$-0.573551\pi$
$881$	−1.05521e7	−0.458033	−0.229017	+	0.973423i	$0.573551\pi$
$882$	0	0
$883$	−3.25543e7	−1.40510	−0.702549	−	0.711635i	$-0.747955\pi$
$883$	−3.25543e7	−1.40510	−0.702549	+	0.711635i	$0.747955\pi$
$884$	0	0
$885$	−6.06687e7	−2.60380
$886$	0	0
$887$	−4.02203e7	−1.71647	−0.858236	−	0.513256i	$-0.828439\pi$
$887$	−4.02203e7	−1.71647	−0.858236	+	0.513256i	$0.828439\pi$
$888$	0	0
$889$	2.65249e7	1.12564
$890$	0	0
$891$	8.50776e7	3.59022
$892$	0	0
$893$	−1.38102e7	−0.579525
$894$	0	0
$895$	−2.10894e7	−0.880047
$896$	0	0
$897$	2.15803e7	0.895523
$898$	0	0
$899$	−6.13148e6	−0.253027
$900$	0	0
$901$	−3.52574e7	−1.44690
$902$	0	0
$903$	−4.38339e7	−1.78892
$904$	0	0
$905$	−2.03215e7	−0.824775
$906$	0	0
$907$	−1.90163e7	−0.767551	−0.383776	−	0.923426i	$-0.625376\pi$
$907$	−1.90163e7	−0.767551	−0.383776	+	0.923426i	$0.625376\pi$
$908$	0	0
$909$	9.62788e7	3.86475
$910$	0	0
$911$	1.56033e7	0.622904	0.311452	−	0.950262i	$-0.399185\pi$
$911$	1.56033e7	0.622904	0.311452	+	0.950262i	$0.399185\pi$
$912$	0	0
$913$	2.49371e7	0.990078
$914$	0	0
$915$	1.76512e7	0.696982
$916$	0	0
$917$	6.76363e6	0.265617
$918$	0	0
$919$	3.22713e7	1.26045	0.630227	−	0.776411i	$-0.282962\pi$
$919$	3.22713e7	1.26045	0.630227	+	0.776411i	$0.282962\pi$
$920$	0	0
$921$	7.96095e7	3.09255
$922$	0	0
$923$	−5.56476e7	−2.15002
$924$	0	0
$925$	−1.41128e6	−0.0542325
$926$	0	0
$927$	6.98325e7	2.66906
$928$	0	0
$929$	−2.68290e7	−1.01992	−0.509959	−	0.860199i	$-0.670340\pi$
$929$	−2.68290e7	−1.01992	−0.509959	+	0.860199i	$0.670340\pi$
$930$	0	0
$931$	−680247.	−0.0257213
$932$	0	0
$933$	1.49068e7	0.560634
$934$	0	0
$935$	2.28789e7	0.855865
$936$	0	0
$937$	4.46199e7	1.66027	0.830137	−	0.557559i	$-0.188262\pi$
$937$	4.46199e7	1.66027	0.830137	+	0.557559i	$0.188262\pi$
$938$	0	0
$939$	3.25101e7	1.20325
$940$	0	0
$941$	−3.59329e7	−1.32287	−0.661437	−	0.750001i	$-0.730053\pi$
$941$	−3.59329e7	−1.32287	−0.661437	+	0.750001i	$0.730053\pi$
$942$	0	0
$943$	−5.66059e6	−0.207292
$944$	0	0
$945$	5.86659e7	2.13701
$946$	0	0
$947$	−5.22375e7	−1.89281	−0.946406	−	0.322978i	$-0.895316\pi$
$947$	−5.22375e7	−1.89281	−0.946406	+	0.322978i	$0.895316\pi$
$948$	0	0
$949$	2.33740e7	0.842496
$950$	0	0
$951$	−5.31222e7	−1.90469
$952$	0	0
$953$	1.63455e7	0.582997	0.291498	−	0.956571i	$-0.405846\pi$
$953$	1.63455e7	0.582997	0.291498	+	0.956571i	$0.405846\pi$
$954$	0	0
$955$	−2.43530e7	−0.864059
$956$	0	0
$957$	2.55182e7	0.900681
$958$	0	0
$959$	3.15607e7	1.10815
$960$	0	0
$961$	−1.63598e7	−0.571439
$962$	0	0
$963$	−4.64050e7	−1.61250
$964$	0	0
$965$	3.00202e7	1.03776
$966$	0	0
$967$	−4.88527e7	−1.68005	−0.840025	−	0.542548i	$-0.817460\pi$
$967$	−4.88527e7	−1.68005	−0.840025	+	0.542548i	$0.817460\pi$
$968$	0	0
$969$	1.98890e7	0.680461
$970$	0	0
$971$	1.44057e7	0.490328	0.245164	−	0.969482i	$-0.421158\pi$
$971$	1.44057e7	0.490328	0.245164	+	0.969482i	$0.421158\pi$
$972$	0	0
$973$	7.84021e6	0.265488
$974$	0	0
$975$	−5.01432e7	−1.68928
$976$	0	0
$977$	−3.42859e7	−1.14916	−0.574578	−	0.818450i	$-0.694834\pi$
$977$	−3.42859e7	−1.14916	−0.574578	+	0.818450i	$0.694834\pi$
$978$	0	0
$979$	5.28186e6	0.176129
$980$	0	0
$981$	7.61690e6	0.252700
$982$	0	0
$983$	2.77814e7	0.917001	0.458500	−	0.888694i	$-0.348387\pi$
$983$	2.77814e7	0.917001	0.458500	+	0.888694i	$0.348387\pi$
$984$	0	0
$985$	2.93325e7	0.963293
$986$	0	0
$987$	9.19928e7	3.00581
$988$	0	0
$989$	−7.32493e6	−0.238129
$990$	0	0
$991$	4.01124e7	1.29746	0.648731	−	0.761018i	$-0.275300\pi$
$991$	4.01124e7	1.29746	0.648731	+	0.761018i	$0.275300\pi$
$992$	0	0
$993$	9.99800e6	0.321766
$994$	0	0
$995$	1.87415e7	0.600133
$996$	0	0
$997$	−4.08262e7	−1.30077	−0.650386	−	0.759604i	$-0.725393\pi$
$997$	−4.08262e7	−1.30077	−0.650386	+	0.759604i	$0.725393\pi$
$998$	0	0
$999$	−1.01262e7	−0.321020

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.a.1.2	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.a.1.2	✓	49	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1028 = 2^{2} \cdot 257 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1028.a (trivial)

\(f(q)\)	\(=\)	\(q-29.3150 q^{3} +39.9945 q^{5} -134.016 q^{7} +616.369 q^{9} +O(q^{10})\)	\(q-29.3150 q^{3} +39.9945 q^{5} -134.016 q^{7} +616.369 q^{9} +497.286 q^{11} -1121.31 q^{13} -1172.44 q^{15} +1150.34 q^{17} -589.786 q^{19} +3928.69 q^{21} +656.510 q^{23} -1525.44 q^{25} -10945.3 q^{27} -1750.47 q^{29} +3502.76 q^{31} -14577.9 q^{33} -5359.92 q^{35} +925.163 q^{37} +32871.3 q^{39} -8622.25 q^{41} -11157.4 q^{43} +24651.4 q^{45} +23415.7 q^{47} +1153.38 q^{49} -33722.4 q^{51} -30649.4 q^{53} +19888.7 q^{55} +17289.6 q^{57} +51745.8 q^{59} -15055.1 q^{61} -82603.5 q^{63} -44846.4 q^{65} +19007.0 q^{67} -19245.6 q^{69} +49627.2 q^{71} -20845.2 q^{73} +44718.3 q^{75} -66644.4 q^{77} +61081.9 q^{79} +171084. q^{81} +50146.4 q^{83} +46007.5 q^{85} +51315.0 q^{87} +10621.4 q^{89} +150274. q^{91} -102683. q^{93} -23588.2 q^{95} +8802.78 q^{97} +306512. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9} - 484 q^{11} - 2922 q^{13} - 244 q^{15} - 3638 q^{17} - 3567 q^{19} - 3811 q^{21} + 3712 q^{23} + 13568 q^{25} - 8814 q^{27} - 5911 q^{29} - 3539 q^{31} - 17215 q^{33} - 10524 q^{35} - 31531 q^{37} - 14986 q^{39} - 35365 q^{41} - 44830 q^{43} - 34403 q^{45} + 12410 q^{47} + 68891 q^{49} - 48608 q^{51} - 79861 q^{53} - 34265 q^{55} - 109824 q^{57} - 51564 q^{59} - 66228 q^{61} + 24981 q^{63} - 31903 q^{65} - 80135 q^{67} + 50903 q^{69} + 162703 q^{71} - 58717 q^{73} + 207827 q^{75} + 89720 q^{77} + 125647 q^{79} + 420081 q^{81} - 28722 q^{83} - 125617 q^{85} + 65848 q^{87} + 93837 q^{89} - 66437 q^{91} - 278711 q^{93} - 83170 q^{95} - 347060 q^{97} + 7934 q^{99}+O(q^{100}) \) 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9 - 484 * q^11 - 2922 * q^13 - 244 * q^15 - 3638 * q^17 - 3567 * q^19 - 3811 * q^21 + 3712 * q^23 + 13568 * q^25 - 8814 * q^27 - 5911 * q^29 - 3539 * q^31 - 17215 * q^33 - 10524 * q^35 - 31531 * q^37 - 14986 * q^39 - 35365 * q^41 - 44830 * q^43 - 34403 * q^45 + 12410 * q^47 + 68891 * q^49 - 48608 * q^51 - 79861 * q^53 - 34265 * q^55 - 109824 * q^57 - 51564 * q^59 - 66228 * q^61 + 24981 * q^63 - 31903 * q^65 - 80135 * q^67 + 50903 * q^69 + 162703 * q^71 - 58717 * q^73 + 207827 * q^75 + 89720 * q^77 + 125647 * q^79 + 420081 * q^81 - 28722 * q^83 - 125617 * q^85 + 65848 * q^87 + 93837 * q^89 - 66437 * q^91 - 278711 * q^93 - 83170 * q^95 - 347060 * q^97 + 7934 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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