LMFDB - Embedded newform 1028.6.a.b.1.17

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:	$0$
Dimension:	$57$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q-15.1335 q^{3} +45.8699 q^{5} +223.098 q^{7} -13.9785 q^{9} +O(q^{10})$	$q-15.1335 q^{3} +45.8699 q^{5} +223.098 q^{7} -13.9785 q^{9} -100.145 q^{11} +780.695 q^{13} -694.170 q^{15} -2204.67 q^{17} +950.856 q^{19} -3376.24 q^{21} +1715.14 q^{23} -1020.95 q^{25} +3888.97 q^{27} +4622.57 q^{29} +5701.47 q^{31} +1515.53 q^{33} +10233.5 q^{35} -9148.54 q^{37} -11814.6 q^{39} +2675.33 q^{41} +15649.5 q^{43} -641.192 q^{45} +20497.2 q^{47} +32965.6 q^{49} +33364.3 q^{51} -5420.18 q^{53} -4593.62 q^{55} -14389.7 q^{57} +29419.4 q^{59} -55916.4 q^{61} -3118.57 q^{63} +35810.4 q^{65} -22142.6 q^{67} -25956.0 q^{69} -82016.1 q^{71} +55187.4 q^{73} +15450.6 q^{75} -22342.0 q^{77} -45888.7 q^{79} -55456.8 q^{81} -58892.9 q^{83} -101128. q^{85} -69955.5 q^{87} +87313.4 q^{89} +174171. q^{91} -86283.0 q^{93} +43615.7 q^{95} +113496. q^{97} +1399.87 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * 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
$3$	−15.1335	−0.970812	−0.485406	−	0.874289i	$-0.661328\pi$
$3$	−15.1335	−0.970812	−0.485406	+	0.874289i	$0.661328\pi$
$4$	0	0
$5$	45.8699	0.820545	0.410273	−	0.911963i	$-0.365434\pi$
$5$	45.8699	0.820545	0.410273	+	0.911963i	$0.365434\pi$
$6$	0	0
$7$	223.098	1.72088	0.860439	−	0.509554i	$-0.170190\pi$
$7$	223.098	1.72088	0.860439	+	0.509554i	$0.170190\pi$
$8$	0	0
$9$	−13.9785	−0.0575246
$10$	0	0
$11$	−100.145	−0.249543	−0.124772	−	0.992185i	$-0.539820\pi$
$11$	−100.145	−0.249543	−0.124772	+	0.992185i	$0.539820\pi$
$12$	0	0
$13$	780.695	1.28122	0.640609	−	0.767867i	$-0.278682\pi$
$13$	780.695	1.28122	0.640609	+	0.767867i	$0.278682\pi$
$14$	0	0
$15$	−694.170	−0.796595
$16$	0	0
$17$	−2204.67	−1.85021	−0.925105	−	0.379712i	$-0.876023\pi$
$17$	−2204.67	−1.85021	−0.925105	+	0.379712i	$0.876023\pi$
$18$	0	0
$19$	950.856	0.604270	0.302135	−	0.953265i	$-0.402301\pi$
$19$	950.856	0.604270	0.302135	+	0.953265i	$0.402301\pi$
$20$	0	0
$21$	−3376.24	−1.67065
$22$	0	0
$23$	1715.14	0.676052	0.338026	−	0.941137i	$-0.390241\pi$
$23$	1715.14	0.676052	0.338026	+	0.941137i	$0.390241\pi$
$24$	0	0
$25$	−1020.95	−0.326705
$26$	0	0
$27$	3888.97	1.02666
$28$	0	0
$29$	4622.57	1.02068	0.510339	−	0.859973i	$-0.329520\pi$
$29$	4622.57	1.02068	0.510339	+	0.859973i	$0.329520\pi$
$30$	0	0
$31$	5701.47	1.06557	0.532786	−	0.846250i	$-0.321145\pi$
$31$	5701.47	1.06557	0.532786	+	0.846250i	$0.321145\pi$
$32$	0	0
$33$	1515.53	0.242260
$34$	0	0
$35$	10233.5	1.41206
$36$	0	0
$37$	−9148.54	−1.09862	−0.549310	−	0.835618i	$-0.685109\pi$
$37$	−9148.54	−1.09862	−0.549310	+	0.835618i	$0.685109\pi$
$38$	0	0
$39$	−11814.6	−1.24382
$40$	0	0
$41$	2675.33	0.248552	0.124276	−	0.992248i	$-0.460339\pi$
$41$	2675.33	0.248552	0.124276	+	0.992248i	$0.460339\pi$
$42$	0	0
$43$	15649.5	1.29072	0.645358	−	0.763880i	$-0.276708\pi$
$43$	15649.5	1.29072	0.645358	+	0.763880i	$0.276708\pi$
$44$	0	0
$45$	−641.192	−0.0472016
$46$	0	0
$47$	20497.2	1.35348	0.676738	−	0.736224i	$-0.263393\pi$
$47$	20497.2	1.35348	0.676738	+	0.736224i	$0.263393\pi$
$48$	0	0
$49$	32965.6	1.96142
$50$	0	0
$51$	33364.3	1.79621
$52$	0	0
$53$	−5420.18	−0.265048	−0.132524	−	0.991180i	$-0.542308\pi$
$53$	−5420.18	−0.265048	−0.132524	+	0.991180i	$0.542308\pi$
$54$	0	0
$55$	−4593.62	−0.204762
$56$	0	0
$57$	−14389.7	−0.586632
$58$	0	0
$59$	29419.4	1.10028	0.550141	−	0.835072i	$-0.314574\pi$
$59$	29419.4	1.10028	0.550141	+	0.835072i	$0.314574\pi$
$60$	0	0
$61$	−55916.4	−1.92404	−0.962020	−	0.272978i	$-0.911992\pi$
$61$	−55916.4	−1.92404	−0.962020	+	0.272978i	$0.911992\pi$
$62$	0	0
$63$	−3118.57	−0.0989928
$64$	0	0
$65$	35810.4	1.05130
$66$	0	0
$67$	−22142.6	−0.602618	−0.301309	−	0.953527i	$-0.597424\pi$
$67$	−22142.6	−0.602618	−0.301309	+	0.953527i	$0.597424\pi$
$68$	0	0
$69$	−25956.0	−0.656319
$70$	0	0
$71$	−82016.1	−1.93087	−0.965436	−	0.260640i	$-0.916066\pi$
$71$	−82016.1	−1.93087	−0.965436	+	0.260640i	$0.916066\pi$
$72$	0	0
$73$	55187.4	1.21208	0.606042	−	0.795433i	$-0.292756\pi$
$73$	55187.4	1.21208	0.606042	+	0.795433i	$0.292756\pi$
$74$	0	0
$75$	15450.6	0.317169
$76$	0	0
$77$	−22342.0	−0.429434
$78$	0	0
$79$	−45888.7	−0.827252	−0.413626	−	0.910447i	$-0.635738\pi$
$79$	−45888.7	−0.827252	−0.413626	+	0.910447i	$0.635738\pi$
$80$	0	0
$81$	−55456.8	−0.939166
$82$	0	0
$83$	−58892.9	−0.938355	−0.469178	−	0.883104i	$-0.655450\pi$
$83$	−58892.9	−0.938355	−0.469178	+	0.883104i	$0.655450\pi$
$84$	0	0
$85$	−101128.	−1.51818
$86$	0	0
$87$	−69955.5	−0.990886
$88$	0	0
$89$	87313.4	1.16844	0.584219	−	0.811596i	$-0.301401\pi$
$89$	87313.4	1.16844	0.584219	+	0.811596i	$0.301401\pi$
$90$	0	0
$91$	174171.	2.20482
$92$	0	0
$93$	−86283.0	−1.03447
$94$	0	0
$95$	43615.7	0.495831
$96$	0	0
$97$	113496.	1.22476	0.612378	−	0.790565i	$-0.290213\pi$
$97$	113496.	1.22476	0.612378	+	0.790565i	$0.290213\pi$
$98$	0	0
$99$	1399.87	0.0143549
$100$	0	0
$101$	117458.	1.14572	0.572861	−	0.819653i	$-0.305834\pi$
$101$	117458.	1.14572	0.572861	+	0.819653i	$0.305834\pi$
$102$	0	0
$103$	−6719.64	−0.0624099	−0.0312049	−	0.999513i	$-0.509934\pi$
$103$	−6719.64	−0.0624099	−0.0312049	+	0.999513i	$0.509934\pi$
$104$	0	0
$105$	−154868.	−1.37084
$106$	0	0
$107$	−13815.5	−0.116656	−0.0583281	−	0.998297i	$-0.518577\pi$
$107$	−13815.5	−0.116656	−0.0583281	+	0.998297i	$0.518577\pi$
$108$	0	0
$109$	20840.8	0.168015	0.0840074	−	0.996465i	$-0.473228\pi$
$109$	20840.8	0.168015	0.0840074	+	0.996465i	$0.473228\pi$
$110$	0	0
$111$	138449.	1.06655
$112$	0	0
$113$	111165.	0.818976	0.409488	−	0.912315i	$-0.365707\pi$
$113$	111165.	0.818976	0.409488	+	0.912315i	$0.365707\pi$
$114$	0	0
$115$	78673.3	0.554732
$116$	0	0
$117$	−10912.9	−0.0737016
$118$	0	0
$119$	−491856.	−3.18398
$120$	0	0
$121$	−151022.	−0.937728
$122$	0	0
$123$	−40486.9	−0.241297
$124$	0	0
$125$	−190174.	−1.08862
$126$	0	0
$127$	−115345.	−0.634583	−0.317292	−	0.948328i	$-0.602773\pi$
$127$	−115345.	−0.634583	−0.317292	+	0.948328i	$0.602773\pi$
$128$	0	0
$129$	−236832.	−1.25304
$130$	0	0
$131$	−336836.	−1.71491	−0.857453	−	0.514562i	$-0.827955\pi$
$131$	−336836.	−1.71491	−0.857453	+	0.514562i	$0.827955\pi$
$132$	0	0
$133$	212134.	1.03987
$134$	0	0
$135$	178387.	0.842419
$136$	0	0
$137$	226546.	1.03123	0.515615	−	0.856820i	$-0.327563\pi$
$137$	226546.	1.03123	0.515615	+	0.856820i	$0.327563\pi$
$138$	0	0
$139$	20235.5	0.0888335	0.0444167	−	0.999013i	$-0.485857\pi$
$139$	20235.5	0.0888335	0.0444167	+	0.999013i	$0.485857\pi$
$140$	0	0
$141$	−310194.	−1.31397
$142$	0	0
$143$	−78182.4	−0.319719
$144$	0	0
$145$	212037.	0.837513
$146$	0	0
$147$	−498883.	−1.90417
$148$	0	0
$149$	151790.	0.560115	0.280058	−	0.959983i	$-0.409646\pi$
$149$	151790.	0.560115	0.280058	+	0.959983i	$0.409646\pi$
$150$	0	0
$151$	326159.	1.16409	0.582046	−	0.813156i	$-0.302252\pi$
$151$	326159.	1.16409	0.582046	+	0.813156i	$0.302252\pi$
$152$	0	0
$153$	30817.9	0.106433
$154$	0	0
$155$	261526.	0.874350
$156$	0	0
$157$	137958.	0.446683	0.223341	−	0.974740i	$-0.428304\pi$
$157$	137958.	0.446683	0.223341	+	0.974740i	$0.428304\pi$
$158$	0	0
$159$	82026.1	0.257312
$160$	0	0
$161$	382644.	1.16340
$162$	0	0
$163$	−201619.	−0.594378	−0.297189	−	0.954819i	$-0.596049\pi$
$163$	−201619.	−0.594378	−0.297189	+	0.954819i	$0.596049\pi$
$164$	0	0
$165$	69517.4	0.198785
$166$	0	0
$167$	−251096.	−0.696704	−0.348352	−	0.937364i	$-0.613259\pi$
$167$	−251096.	−0.696704	−0.348352	+	0.937364i	$0.613259\pi$
$168$	0	0
$169$	238191.	0.641518
$170$	0	0
$171$	−13291.5	−0.0347604
$172$	0	0
$173$	114681.	0.291325	0.145662	−	0.989334i	$-0.453469\pi$
$173$	114681.	0.291325	0.145662	+	0.989334i	$0.453469\pi$
$174$	0	0
$175$	−227772.	−0.562220
$176$	0	0
$177$	−445217.	−1.06817
$178$	0	0
$179$	−628451.	−1.46602	−0.733008	−	0.680220i	$-0.761884\pi$
$179$	−628451.	−1.46602	−0.733008	+	0.680220i	$0.761884\pi$
$180$	0	0
$181$	10144.3	0.0230158	0.0115079	−	0.999934i	$-0.496337\pi$
$181$	10144.3	0.0230158	0.0115079	+	0.999934i	$0.496337\pi$
$182$	0	0
$183$	846208.	1.86788
$184$	0	0
$185$	−419643.	−0.901468
$186$	0	0
$187$	220786.	0.461708
$188$	0	0
$189$	867621.	1.76675
$190$	0	0
$191$	353383.	0.700910	0.350455	−	0.936580i	$-0.386027\pi$
$191$	353383.	0.700910	0.350455	+	0.936580i	$0.386027\pi$
$192$	0	0
$193$	155411.	0.300323	0.150162	−	0.988661i	$-0.452021\pi$
$193$	155411.	0.300323	0.150162	+	0.988661i	$0.452021\pi$
$194$	0	0
$195$	−541935.	−1.02061
$196$	0	0
$197$	−1.08825e6	−1.99785	−0.998926	−	0.0463326i	$-0.985247\pi$
$197$	−1.08825e6	−1.99785	−0.998926	+	0.0463326i	$0.985247\pi$
$198$	0	0
$199$	897879.	1.60726	0.803628	−	0.595132i	$-0.202900\pi$
$199$	897879.	1.60726	0.803628	+	0.595132i	$0.202900\pi$
$200$	0	0
$201$	335094.	0.585028
$202$	0	0
$203$	1.03129e6	1.75646
$204$	0	0
$205$	122717.	0.203948
$206$	0	0
$207$	−23975.1	−0.0388897
$208$	0	0
$209$	−95223.2	−0.150791
$210$	0	0
$211$	1.18641e6	1.83455	0.917275	−	0.398256i	$-0.130384\pi$
$211$	1.18641e6	1.83455	0.917275	+	0.398256i	$0.130384\pi$
$212$	0	0
$213$	1.24119e6	1.87451
$214$	0	0
$215$	717843.	1.05909
$216$	0	0
$217$	1.27199e6	1.83372
$218$	0	0
$219$	−835176.	−1.17671
$220$	0	0
$221$	−1.72117e6	−2.37052
$222$	0	0
$223$	838970.	1.12976	0.564878	−	0.825175i	$-0.308923\pi$
$223$	838970.	1.12976	0.564878	+	0.825175i	$0.308923\pi$
$224$	0	0
$225$	14271.4	0.0187936
$226$	0	0
$227$	576631.	0.742734	0.371367	−	0.928486i	$-0.378889\pi$
$227$	576631.	0.742734	0.371367	+	0.928486i	$0.378889\pi$
$228$	0	0
$229$	343402.	0.432727	0.216363	−	0.976313i	$-0.430580\pi$
$229$	343402.	0.432727	0.216363	+	0.976313i	$0.430580\pi$
$230$	0	0
$231$	338112.	0.416899
$232$	0	0
$233$	−707648.	−0.853940	−0.426970	−	0.904266i	$-0.640419\pi$
$233$	−707648.	−0.853940	−0.426970	+	0.904266i	$0.640419\pi$
$234$	0	0
$235$	940205.	1.11059
$236$	0	0
$237$	694454.	0.803106
$238$	0	0
$239$	628463.	0.711680	0.355840	−	0.934547i	$-0.384195\pi$
$239$	628463.	0.711680	0.355840	+	0.934547i	$0.384195\pi$
$240$	0	0
$241$	1.12915e6	1.25230	0.626151	−	0.779702i	$-0.284629\pi$
$241$	1.12915e6	1.25230	0.626151	+	0.779702i	$0.284629\pi$
$242$	0	0
$243$	−105767.	−0.114904
$244$	0	0
$245$	1.51213e6	1.60943
$246$	0	0
$247$	742328.	0.774201
$248$	0	0
$249$	891252.	0.910966
$250$	0	0
$251$	447305.	0.448146	0.224073	−	0.974572i	$-0.428065\pi$
$251$	447305.	0.448146	0.224073	+	0.974572i	$0.428065\pi$
$252$	0	0
$253$	−171762.	−0.168704
$254$	0	0
$255$	1.53041e6	1.47387
$256$	0	0
$257$	−66049.0	−0.0623783
$257$	−66049.0	−0.0623783
$258$	0	0
$259$	−2.04102e6	−1.89059
$260$	0	0
$261$	−64616.6	−0.0587141
$262$	0	0
$263$	−156383.	−0.139412	−0.0697060	−	0.997568i	$-0.522206\pi$
$263$	−156383.	−0.139412	−0.0697060	+	0.997568i	$0.522206\pi$
$264$	0	0
$265$	−248623.	−0.217484
$266$	0	0
$267$	−1.32135e6	−1.13433
$268$	0	0
$269$	762796.	0.642729	0.321364	−	0.946956i	$-0.395859\pi$
$269$	762796.	0.642729	0.321364	+	0.946956i	$0.395859\pi$
$270$	0	0
$271$	−1.06176e6	−0.878216	−0.439108	−	0.898434i	$-0.644705\pi$
$271$	−1.06176e6	−0.878216	−0.439108	+	0.898434i	$0.644705\pi$
$272$	0	0
$273$	−2.63581e6	−2.14046
$274$	0	0
$275$	102243.	0.0815271
$276$	0	0
$277$	−202690.	−0.158720	−0.0793601	−	0.996846i	$-0.525288\pi$
$277$	−202690.	−0.158720	−0.0793601	+	0.996846i	$0.525288\pi$
$278$	0	0
$279$	−79698.0	−0.0612966
$280$	0	0
$281$	57582.6	0.0435037	0.0217518	−	0.999763i	$-0.493076\pi$
$281$	57582.6	0.0435037	0.0217518	+	0.999763i	$0.493076\pi$
$282$	0	0
$283$	2.41875e6	1.79525	0.897625	−	0.440759i	$-0.145291\pi$
$283$	2.41875e6	1.79525	0.897625	+	0.440759i	$0.145291\pi$
$284$	0	0
$285$	−660056.	−0.481358
$286$	0	0
$287$	596859.	0.427727
$288$	0	0
$289$	3.44071e6	2.42328
$290$	0	0
$291$	−1.71758e6	−1.18901
$292$	0	0
$293$	994224.	0.676574	0.338287	−	0.941043i	$-0.390153\pi$
$293$	994224.	0.676574	0.338287	+	0.941043i	$0.390153\pi$
$294$	0	0
$295$	1.34946e6	0.902831
$296$	0	0
$297$	−389460.	−0.256196
$298$	0	0
$299$	1.33900e6	0.866170
$300$	0	0
$301$	3.49138e6	2.22116
$302$	0	0
$303$	−1.77754e6	−1.11228
$304$	0	0
$305$	−2.56488e6	−1.57876
$306$	0	0
$307$	1.22531e6	0.741996	0.370998	−	0.928634i	$-0.379016\pi$
$307$	1.22531e6	0.741996	0.370998	+	0.928634i	$0.379016\pi$
$308$	0	0
$309$	101691.	0.0605882
$310$	0	0
$311$	−308104.	−0.180632	−0.0903162	−	0.995913i	$-0.528788\pi$
$311$	−308104.	−0.180632	−0.0903162	+	0.995913i	$0.528788\pi$
$312$	0	0
$313$	−1.05302e6	−0.607541	−0.303771	−	0.952745i	$-0.598246\pi$
$313$	−1.05302e6	−0.607541	−0.303771	+	0.952745i	$0.598246\pi$
$314$	0	0
$315$	−143048.	−0.0812281
$316$	0	0
$317$	−2.41452e6	−1.34953	−0.674765	−	0.738033i	$-0.735755\pi$
$317$	−2.41452e6	−1.34953	−0.674765	+	0.738033i	$0.735755\pi$
$318$	0	0
$319$	−462926.	−0.254703
$320$	0	0
$321$	209077.	0.113251
$322$	0	0
$323$	−2.09632e6	−1.11803
$324$	0	0
$325$	−797053.	−0.418580
$326$	0	0
$327$	−315393.	−0.163111
$328$	0	0
$329$	4.57288e6	2.32916
$330$	0	0
$331$	1.49811e6	0.751578	0.375789	−	0.926705i	$-0.377372\pi$
$331$	1.49811e6	0.751578	0.375789	+	0.926705i	$0.377372\pi$
$332$	0	0
$333$	127883.	0.0631977
$334$	0	0
$335$	−1.01568e6	−0.494475
$336$	0	0
$337$	−1.13235e6	−0.543133	−0.271566	−	0.962420i	$-0.587542\pi$
$337$	−1.13235e6	−0.543133	−0.271566	+	0.962420i	$0.587542\pi$
$338$	0	0
$339$	−1.68231e6	−0.795072
$340$	0	0
$341$	−570972.	−0.265906
$342$	0	0
$343$	3.60494e6	1.65448
$344$	0	0
$345$	−1.19060e6	−0.538540
$346$	0	0
$347$	−595027.	−0.265285	−0.132643	−	0.991164i	$-0.542346\pi$
$347$	−595027.	−0.265285	−0.132643	+	0.991164i	$0.542346\pi$
$348$	0	0
$349$	1.17975e6	0.518472	0.259236	−	0.965814i	$-0.416529\pi$
$349$	1.17975e6	0.518472	0.259236	+	0.965814i	$0.416529\pi$
$350$	0	0
$351$	3.03610e6	1.31537
$352$	0	0
$353$	1.98720e6	0.848797	0.424399	−	0.905475i	$-0.360486\pi$
$353$	1.98720e6	0.848797	0.424399	+	0.905475i	$0.360486\pi$
$354$	0	0
$355$	−3.76207e6	−1.58437
$356$	0	0
$357$	7.44349e6	3.09105
$358$	0	0
$359$	1.43461e6	0.587488	0.293744	−	0.955884i	$-0.405099\pi$
$359$	1.43461e6	0.587488	0.293744	+	0.955884i	$0.405099\pi$
$360$	0	0
$361$	−1.57197e6	−0.634858
$362$	0	0
$363$	2.28549e6	0.910357
$364$	0	0
$365$	2.53144e6	0.994570
$366$	0	0
$367$	−3.30824e6	−1.28213	−0.641065	−	0.767487i	$-0.721507\pi$
$367$	−3.30824e6	−1.28213	−0.641065	+	0.767487i	$0.721507\pi$
$368$	0	0
$369$	−37397.0	−0.0142979
$370$	0	0
$371$	−1.20923e6	−0.456115
$372$	0	0
$373$	1.80954e6	0.673434	0.336717	−	0.941606i	$-0.390683\pi$
$373$	1.80954e6	0.673434	0.336717	+	0.941606i	$0.390683\pi$
$374$	0	0
$375$	2.87800e6	1.05685
$376$	0	0
$377$	3.60882e6	1.30771
$378$	0	0
$379$	−309566.	−0.110702	−0.0553510	−	0.998467i	$-0.517628\pi$
$379$	−309566.	−0.110702	−0.0553510	+	0.998467i	$0.517628\pi$
$380$	0	0
$381$	1.74556e6	0.616061
$382$	0	0
$383$	−1.16602e6	−0.406171	−0.203086	−	0.979161i	$-0.565097\pi$
$383$	−1.16602e6	−0.406171	−0.203086	+	0.979161i	$0.565097\pi$
$384$	0	0
$385$	−1.02483e6	−0.352370
$386$	0	0
$387$	−218757.	−0.0742480
$388$	0	0
$389$	−2.40881e6	−0.807101	−0.403551	−	0.914957i	$-0.632224\pi$
$389$	−2.40881e6	−0.807101	−0.403551	+	0.914957i	$0.632224\pi$
$390$	0	0
$391$	−3.78132e6	−1.25084
$392$	0	0
$393$	5.09750e6	1.66485
$394$	0	0
$395$	−2.10491e6	−0.678798
$396$	0	0
$397$	1.65385e6	0.526648	0.263324	−	0.964707i	$-0.415181\pi$
$397$	1.65385e6	0.526648	0.263324	+	0.964707i	$0.415181\pi$
$398$	0	0
$399$	−3.21032e6	−1.00952
$400$	0	0
$401$	4.37776e6	1.35954	0.679768	−	0.733427i	$-0.262080\pi$
$401$	4.37776e6	1.35954	0.679768	+	0.733427i	$0.262080\pi$
$402$	0	0
$403$	4.45111e6	1.36523
$404$	0	0
$405$	−2.54380e6	−0.770629
$406$	0	0
$407$	916178.	0.274153
$408$	0	0
$409$	1.98075e6	0.585494	0.292747	−	0.956190i	$-0.405431\pi$
$409$	1.98075e6	0.585494	0.292747	+	0.956190i	$0.405431\pi$
$410$	0	0
$411$	−3.42843e6	−1.00113
$412$	0	0
$413$	6.56340e6	1.89345
$414$	0	0
$415$	−2.70141e6	−0.769963
$416$	0	0
$417$	−306233.	−0.0862406
$418$	0	0
$419$	803046.	0.223463	0.111731	−	0.993738i	$-0.464360\pi$
$419$	803046.	0.223463	0.111731	+	0.993738i	$0.464360\pi$
$420$	0	0
$421$	1.56823e6	0.431227	0.215613	−	0.976479i	$-0.430825\pi$
$421$	1.56823e6	0.431227	0.215613	+	0.976479i	$0.430825\pi$
$422$	0	0
$423$	−286520.	−0.0778582
$424$	0	0
$425$	2.25086e6	0.604473
$426$	0	0
$427$	−1.24748e7	−3.31104
$428$	0	0
$429$	1.18317e6	0.310387
$430$	0	0
$431$	3.91630e6	1.01551	0.507753	−	0.861503i	$-0.330476\pi$
$431$	3.91630e6	1.01551	0.507753	+	0.861503i	$0.330476\pi$
$432$	0	0
$433$	6.42159e6	1.64597	0.822987	−	0.568060i	$-0.192306\pi$
$433$	6.42159e6	1.64597	0.822987	+	0.568060i	$0.192306\pi$
$434$	0	0
$435$	−3.20885e6	−0.813067
$436$	0	0
$437$	1.63085e6	0.408518
$438$	0	0
$439$	−1.10210e6	−0.272936	−0.136468	−	0.990644i	$-0.543575\pi$
$439$	−1.10210e6	−0.272936	−0.136468	+	0.990644i	$0.543575\pi$
$440$	0	0
$441$	−460809.	−0.112830
$442$	0	0
$443$	−6.85293e6	−1.65908	−0.829540	−	0.558448i	$-0.811397\pi$
$443$	−6.85293e6	−1.65908	−0.829540	+	0.558448i	$0.811397\pi$
$444$	0	0
$445$	4.00506e6	0.958757
$446$	0	0
$447$	−2.29711e6	−0.543767
$448$	0	0
$449$	−2.30067e6	−0.538565	−0.269283	−	0.963061i	$-0.586787\pi$
$449$	−2.30067e6	−0.538565	−0.269283	+	0.963061i	$0.586787\pi$
$450$	0	0
$451$	−267920.	−0.0620245
$452$	0	0
$453$	−4.93592e6	−1.13011
$454$	0	0
$455$	7.98921e6	1.80915
$456$	0	0
$457$	1.05761e6	0.236884	0.118442	−	0.992961i	$-0.462210\pi$
$457$	1.05761e6	0.236884	0.118442	+	0.992961i	$0.462210\pi$
$458$	0	0
$459$	−8.57390e6	−1.89953
$460$	0	0
$461$	7.33120e6	1.60666	0.803328	−	0.595537i	$-0.203061\pi$
$461$	7.33120e6	1.60666	0.803328	+	0.595537i	$0.203061\pi$
$462$	0	0
$463$	3.42888e6	0.743360	0.371680	−	0.928361i	$-0.378782\pi$
$463$	3.42888e6	0.743360	0.371680	+	0.928361i	$0.378782\pi$
$464$	0	0
$465$	−3.95779e6	−0.848829
$466$	0	0
$467$	1.17342e6	0.248978	0.124489	−	0.992221i	$-0.460271\pi$
$467$	1.17342e6	0.248978	0.124489	+	0.992221i	$0.460271\pi$
$468$	0	0
$469$	−4.93997e6	−1.03703
$470$	0	0
$471$	−2.08779e6	−0.433645
$472$	0	0
$473$	−1.56722e6	−0.322090
$474$	0	0
$475$	−970780.	−0.197418
$476$	0	0
$477$	75766.0	0.0152468
$478$	0	0
$479$	−8.21492e6	−1.63593	−0.817965	−	0.575268i	$-0.804898\pi$
$479$	−8.21492e6	−1.63593	−0.817965	+	0.575268i	$0.804898\pi$
$480$	0	0
$481$	−7.14222e6	−1.40757
$482$	0	0
$483$	−5.79073e6	−1.12945
$484$	0	0
$485$	5.20603e6	1.00497
$486$	0	0
$487$	−1.01242e7	−1.93436	−0.967178	−	0.254100i	$-0.918221\pi$
$487$	−1.01242e7	−1.93436	−0.967178	+	0.254100i	$0.918221\pi$
$488$	0	0
$489$	3.05119e6	0.577029
$490$	0	0
$491$	9.33525e6	1.74752	0.873760	−	0.486357i	$-0.161675\pi$
$491$	9.33525e6	1.74752	0.873760	+	0.486357i	$0.161675\pi$
$492$	0	0
$493$	−1.01912e7	−1.88847
$494$	0	0
$495$	64211.9	0.0117788
$496$	0	0
$497$	−1.82976e7	−3.32279
$498$	0	0
$499$	2.30424e6	0.414263	0.207132	−	0.978313i	$-0.433587\pi$
$499$	2.30424e6	0.414263	0.207132	+	0.978313i	$0.433587\pi$
$500$	0	0
$501$	3.79995e6	0.676369
$502$	0	0
$503$	−1.26858e6	−0.223562	−0.111781	−	0.993733i	$-0.535655\pi$
$503$	−1.26858e6	−0.223562	−0.111781	+	0.993733i	$0.535655\pi$
$504$	0	0
$505$	5.38778e6	0.940116
$506$	0	0
$507$	−3.60466e6	−0.622793
$508$	0	0
$509$	6.00596e6	1.02752	0.513758	−	0.857935i	$-0.328253\pi$
$509$	6.00596e6	1.02752	0.513758	+	0.857935i	$0.328253\pi$
$510$	0	0
$511$	1.23122e7	2.08585
$512$	0	0
$513$	3.69785e6	0.620378
$514$	0	0
$515$	−308229.	−0.0512101
$516$	0	0
$517$	−2.05269e6	−0.337751
$518$	0	0
$519$	−1.73553e6	−0.282822
$520$	0	0
$521$	−16152.2	−0.00260698	−0.00130349	−	0.999999i	$-0.500415\pi$
$521$	−16152.2	−0.00260698	−0.00130349	+	0.999999i	$0.500415\pi$
$522$	0	0
$523$	1.05014e7	1.67877	0.839387	−	0.543534i	$-0.182914\pi$
$523$	1.05014e7	1.67877	0.839387	+	0.543534i	$0.182914\pi$
$524$	0	0
$525$	3.44698e6	0.545809
$526$	0	0
$527$	−1.25699e7	−1.97153
$528$	0	0
$529$	−3.49463e6	−0.542953
$530$	0	0
$531$	−411239.	−0.0632933
$532$	0	0
$533$	2.08861e6	0.318449
$534$	0	0
$535$	−633717.	−0.0957218
$536$	0	0
$537$	9.51063e6	1.42323
$538$	0	0
$539$	−3.30132e6	−0.489459
$540$	0	0
$541$	132045.	0.0193968	0.00969839	−	0.999953i	$-0.496913\pi$
$541$	132045.	0.0193968	0.00969839	+	0.999953i	$0.496913\pi$
$542$	0	0
$543$	−153519.	−0.0223440
$544$	0	0
$545$	955964.	0.137864
$546$	0	0
$547$	6.61654e6	0.945502	0.472751	−	0.881196i	$-0.343261\pi$
$547$	6.61654e6	0.945502	0.472751	+	0.881196i	$0.343261\pi$
$548$	0	0
$549$	781626.	0.110680
$550$	0	0
$551$	4.39540e6	0.616765
$552$	0	0
$553$	−1.02377e7	−1.42360
$554$	0	0
$555$	6.35064e6	0.875156
$556$	0	0
$557$	−1.55249e6	−0.212027	−0.106014	−	0.994365i	$-0.533809\pi$
$557$	−1.55249e6	−0.212027	−0.106014	+	0.994365i	$0.533809\pi$
$558$	0	0
$559$	1.22175e7	1.65369
$560$	0	0
$561$	−3.34125e6	−0.448231
$562$	0	0
$563$	−6.16477e6	−0.819682	−0.409841	−	0.912157i	$-0.634416\pi$
$563$	−6.16477e6	−0.819682	−0.409841	+	0.912157i	$0.634416\pi$
$564$	0	0
$565$	5.09912e6	0.672007
$566$	0	0
$567$	−1.23723e7	−1.61619
$568$	0	0
$569$	1.52317e6	0.197227	0.0986137	−	0.995126i	$-0.468559\pi$
$569$	1.52317e6	0.197227	0.0986137	+	0.995126i	$0.468559\pi$
$570$	0	0
$571$	8.58185e6	1.10152	0.550758	−	0.834665i	$-0.314339\pi$
$571$	8.58185e6	1.10152	0.550758	+	0.834665i	$0.314339\pi$
$572$	0	0
$573$	−5.34790e6	−0.680451
$574$	0	0
$575$	−1.75108e6	−0.220870
$576$	0	0
$577$	1.28548e7	1.60741	0.803706	−	0.595027i	$-0.202859\pi$
$577$	1.28548e7	1.60741	0.803706	+	0.595027i	$0.202859\pi$
$578$	0	0
$579$	−2.35191e6	−0.291557
$580$	0	0
$581$	−1.31389e7	−1.61479
$582$	0	0
$583$	542802.	0.0661409
$584$	0	0
$585$	−500575.	−0.0604755
$586$	0	0
$587$	−481152.	−0.0576351	−0.0288175	−	0.999585i	$-0.509174\pi$
$587$	−481152.	−0.0576351	−0.0288175	+	0.999585i	$0.509174\pi$
$588$	0	0
$589$	5.42128e6	0.643893
$590$	0	0
$591$	1.64690e7	1.93954
$592$	0	0
$593$	1.19191e7	1.39190	0.695948	−	0.718092i	$-0.254984\pi$
$593$	1.19191e7	1.39190	0.695948	+	0.718092i	$0.254984\pi$
$594$	0	0
$595$	−2.25614e7	−2.61260
$596$	0	0
$597$	−1.35880e7	−1.56034
$598$	0	0
$599$	7.84801e6	0.893701	0.446851	−	0.894609i	$-0.352546\pi$
$599$	7.84801e6	0.893701	0.446851	+	0.894609i	$0.352546\pi$
$600$	0	0
$601$	4.90036e6	0.553403	0.276702	−	0.960956i	$-0.410759\pi$
$601$	4.90036e6	0.553403	0.276702	+	0.960956i	$0.410759\pi$
$602$	0	0
$603$	309520.	0.0346654
$604$	0	0
$605$	−6.92736e6	−0.769449
$606$	0	0
$607$	−1.48980e7	−1.64118	−0.820591	−	0.571516i	$-0.806356\pi$
$607$	−1.48980e7	−1.64118	−0.820591	+	0.571516i	$0.806356\pi$
$608$	0	0
$609$	−1.56069e7	−1.70519
$610$	0	0
$611$	1.60021e7	1.73410
$612$	0	0
$613$	1.24789e7	1.34129	0.670647	−	0.741776i	$-0.266016\pi$
$613$	1.24789e7	1.34129	0.670647	+	0.741776i	$0.266016\pi$
$614$	0	0
$615$	−1.85713e6	−0.197995
$616$	0	0
$617$	−7.69981e6	−0.814268	−0.407134	−	0.913368i	$-0.633472\pi$
$617$	−7.69981e6	−0.814268	−0.407134	+	0.913368i	$0.633472\pi$
$618$	0	0
$619$	3.17458e6	0.333012	0.166506	−	0.986040i	$-0.446751\pi$
$619$	3.17458e6	0.333012	0.166506	+	0.986040i	$0.446751\pi$
$620$	0	0
$621$	6.67014e6	0.694074
$622$	0	0
$623$	1.94794e7	2.01074
$624$	0	0
$625$	−5.53280e6	−0.566559
$626$	0	0
$627$	1.44106e6	0.146390
$628$	0	0
$629$	2.01695e7	2.03268
$630$	0	0
$631$	−5.29625e6	−0.529535	−0.264768	−	0.964312i	$-0.585295\pi$
$631$	−5.29625e6	−0.529535	−0.264768	+	0.964312i	$0.585295\pi$
$632$	0	0
$633$	−1.79545e7	−1.78100
$634$	0	0
$635$	−5.29085e6	−0.520704
$636$	0	0
$637$	2.57360e7	2.51300
$638$	0	0
$639$	1.14646e6	0.111073
$640$	0	0
$641$	−1.48289e7	−1.42549	−0.712746	−	0.701422i	$-0.752549\pi$
$641$	−1.48289e7	−1.42549	−0.712746	+	0.701422i	$0.752549\pi$
$642$	0	0
$643$	2.13878e6	0.204004	0.102002	−	0.994784i	$-0.467475\pi$
$643$	2.13878e6	0.204004	0.102002	+	0.994784i	$0.467475\pi$
$644$	0	0
$645$	−1.08634e7	−1.02818
$646$	0	0
$647$	−3.05302e6	−0.286727	−0.143364	−	0.989670i	$-0.545792\pi$
$647$	−3.05302e6	−0.286727	−0.143364	+	0.989670i	$0.545792\pi$
$648$	0	0
$649$	−2.94620e6	−0.274568
$650$	0	0
$651$	−1.92495e7	−1.78020
$652$	0	0
$653$	−607481.	−0.0557506	−0.0278753	−	0.999611i	$-0.508874\pi$
$653$	−607481.	−0.0557506	−0.0278753	+	0.999611i	$0.508874\pi$
$654$	0	0
$655$	−1.54506e7	−1.40716
$656$	0	0
$657$	−771436.	−0.0697247
$658$	0	0
$659$	−1.02191e7	−0.916640	−0.458320	−	0.888787i	$-0.651549\pi$
$659$	−1.02191e7	−0.916640	−0.458320	+	0.888787i	$0.651549\pi$
$660$	0	0
$661$	−1.03360e7	−0.920126	−0.460063	−	0.887886i	$-0.652173\pi$
$661$	−1.03360e7	−0.920126	−0.460063	+	0.887886i	$0.652173\pi$
$662$	0	0
$663$	2.60473e7	2.30133
$664$	0	0
$665$	9.73055e6	0.853264
$666$	0	0
$667$	7.92836e6	0.690032
$668$	0	0
$669$	−1.26965e7	−1.09678
$670$	0	0
$671$	5.59972e6	0.480132
$672$	0	0
$673$	−1.05228e7	−0.895556	−0.447778	−	0.894145i	$-0.647784\pi$
$673$	−1.05228e7	−0.895556	−0.447778	+	0.894145i	$0.647784\pi$
$674$	0	0
$675$	−3.97046e6	−0.335414
$676$	0	0
$677$	7.70720e6	0.646286	0.323143	−	0.946350i	$-0.395261\pi$
$677$	7.70720e6	0.646286	0.323143	+	0.946350i	$0.395261\pi$
$678$	0	0
$679$	2.53206e7	2.10765
$680$	0	0
$681$	−8.72642e6	−0.721055
$682$	0	0
$683$	−6.38616e6	−0.523827	−0.261914	−	0.965091i	$-0.584354\pi$
$683$	−6.38616e6	−0.523827	−0.261914	+	0.965091i	$0.584354\pi$
$684$	0	0
$685$	1.03917e7	0.846172
$686$	0	0
$687$	−5.19686e6	−0.420096
$688$	0	0
$689$	−4.23151e6	−0.339584
$690$	0	0
$691$	1.20095e7	0.956823	0.478411	−	0.878136i	$-0.341213\pi$
$691$	1.20095e7	0.956823	0.478411	+	0.878136i	$0.341213\pi$
$692$	0	0
$693$	312308.	0.0247030
$694$	0	0
$695$	928199.	0.0728919
$696$	0	0
$697$	−5.89821e6	−0.459873
$698$	0	0
$699$	1.07092e7	0.829015
$700$	0	0
$701$	2.07214e7	1.59266	0.796332	−	0.604860i	$-0.206771\pi$
$701$	2.07214e7	1.59266	0.796332	+	0.604860i	$0.206771\pi$
$702$	0	0
$703$	−8.69895e6	−0.663863
$704$	0	0
$705$	−1.42286e7	−1.07817
$706$	0	0
$707$	2.62046e7	1.97165
$708$	0	0
$709$	−1.43371e7	−1.07114	−0.535569	−	0.844491i	$-0.679903\pi$
$709$	−1.43371e7	−1.07114	−0.535569	+	0.844491i	$0.679903\pi$
$710$	0	0
$711$	641454.	0.0475874
$712$	0	0
$713$	9.77883e6	0.720382
$714$	0	0
$715$	−3.58622e6	−0.262344
$716$	0	0
$717$	−9.51082e6	−0.690908
$718$	0	0
$719$	−5.85027e6	−0.422040	−0.211020	−	0.977482i	$-0.567679\pi$
$719$	−5.85027e6	−0.422040	−0.211020	+	0.977482i	$0.567679\pi$
$720$	0	0
$721$	−1.49914e6	−0.107400
$722$	0	0
$723$	−1.70879e7	−1.21575
$724$	0	0
$725$	−4.71943e6	−0.333461
$726$	0	0
$727$	1.25030e7	0.877362	0.438681	−	0.898643i	$-0.355446\pi$
$727$	1.25030e7	0.877362	0.438681	+	0.898643i	$0.355446\pi$
$728$	0	0
$729$	1.50766e7	1.05072
$730$	0	0
$731$	−3.45021e7	−2.38809
$732$	0	0
$733$	−1.37932e7	−0.948209	−0.474105	−	0.880469i	$-0.657228\pi$
$733$	−1.37932e7	−0.948209	−0.474105	+	0.880469i	$0.657228\pi$
$734$	0	0
$735$	−2.28837e7	−1.56246
$736$	0	0
$737$	2.21746e6	0.150379
$738$	0	0
$739$	1.25371e6	0.0844473	0.0422236	−	0.999108i	$-0.486556\pi$
$739$	1.25371e6	0.0844473	0.0422236	+	0.999108i	$0.486556\pi$
$740$	0	0
$741$	−1.12340e7	−0.751603
$742$	0	0
$743$	−9.29850e6	−0.617932	−0.308966	−	0.951073i	$-0.599983\pi$
$743$	−9.29850e6	−0.617932	−0.308966	+	0.951073i	$0.599983\pi$
$744$	0	0
$745$	6.96259e6	0.459600
$746$	0	0
$747$	823233.	0.0539786
$748$	0	0
$749$	−3.08221e6	−0.200751
$750$	0	0
$751$	−2.30007e7	−1.48813	−0.744064	−	0.668108i	$-0.767104\pi$
$751$	−2.30007e7	−1.48813	−0.744064	+	0.668108i	$0.767104\pi$
$752$	0	0
$753$	−6.76928e6	−0.435066
$754$	0	0
$755$	1.49609e7	0.955190
$756$	0	0
$757$	2.43938e7	1.54718	0.773589	−	0.633687i	$-0.218459\pi$
$757$	2.43938e7	1.54718	0.773589	+	0.633687i	$0.218459\pi$
$758$	0	0
$759$	2.59936e6	0.163780
$760$	0	0
$761$	−2.52000e7	−1.57739	−0.788695	−	0.614785i	$-0.789243\pi$
$761$	−2.52000e7	−1.57739	−0.788695	+	0.614785i	$0.789243\pi$
$762$	0	0
$763$	4.64953e6	0.289133
$764$	0	0
$765$	1.41361e6	0.0873328
$766$	0	0
$767$	2.29676e7	1.40970
$768$	0	0
$769$	−3.77603e6	−0.230261	−0.115130	−	0.993350i	$-0.536729\pi$
$769$	−3.77603e6	−0.230261	−0.115130	+	0.993350i	$0.536729\pi$
$770$	0	0
$771$	999550.	0.0605576
$772$	0	0
$773$	2.42931e7	1.46229	0.731147	−	0.682220i	$-0.238985\pi$
$773$	2.42931e7	1.46229	0.731147	+	0.682220i	$0.238985\pi$
$774$	0	0
$775$	−5.82094e6	−0.348128
$776$	0	0
$777$	3.08877e7	1.83541
$778$	0	0
$779$	2.54385e6	0.150192
$780$	0	0
$781$	8.21348e6	0.481836
$782$	0	0
$783$	1.79771e7	1.04789
$784$	0	0
$785$	6.32814e6	0.366524
$786$	0	0
$787$	7.82065e6	0.450097	0.225048	−	0.974348i	$-0.427746\pi$
$787$	7.82065e6	0.450097	0.225048	+	0.974348i	$0.427746\pi$
$788$	0	0
$789$	2.36661e6	0.135343
$790$	0	0
$791$	2.48006e7	1.40936
$792$	0	0
$793$	−4.36536e7	−2.46511
$794$	0	0
$795$	3.76253e6	0.211136
$796$	0	0
$797$	1.95592e7	1.09070	0.545351	−	0.838208i	$-0.316397\pi$
$797$	1.95592e7	1.09070	0.545351	+	0.838208i	$0.316397\pi$
$798$	0	0
$799$	−4.51896e7	−2.50421
$800$	0	0
$801$	−1.22051e6	−0.0672140
$802$	0	0
$803$	−5.52672e6	−0.302468
$804$	0	0
$805$	1.75518e7	0.954625
$806$	0	0
$807$	−1.15437e7	−0.623969
$808$	0	0
$809$	−2.50008e6	−0.134302	−0.0671509	−	0.997743i	$-0.521391\pi$
$809$	−2.50008e6	−0.134302	−0.0671509	+	0.997743i	$0.521391\pi$
$810$	0	0
$811$	5.72282e6	0.305533	0.152767	−	0.988262i	$-0.451182\pi$
$811$	5.72282e6	0.305533	0.152767	+	0.988262i	$0.451182\pi$
$812$	0	0
$813$	1.60680e7	0.852582
$814$	0	0
$815$	−9.24825e6	−0.487714
$816$	0	0
$817$	1.48805e7	0.779940
$818$	0	0
$819$	−2.43465e6	−0.126831
$820$	0	0
$821$	−3.32993e7	−1.72416	−0.862079	−	0.506773i	$-0.830838\pi$
$821$	−3.32993e7	−1.72416	−0.862079	+	0.506773i	$0.830838\pi$
$822$	0	0
$823$	−1.88982e7	−0.972570	−0.486285	−	0.873800i	$-0.661648\pi$
$823$	−1.88982e7	−0.972570	−0.486285	+	0.873800i	$0.661648\pi$
$824$	0	0
$825$	−1.54729e6	−0.0791475
$826$	0	0
$827$	−1.91865e7	−0.975511	−0.487755	−	0.872980i	$-0.662184\pi$
$827$	−1.91865e7	−0.975511	−0.487755	+	0.872980i	$0.662184\pi$
$828$	0	0
$829$	3.23125e7	1.63299	0.816497	−	0.577350i	$-0.195913\pi$
$829$	3.23125e7	1.63299	0.816497	+	0.577350i	$0.195913\pi$
$830$	0	0
$831$	3.06739e6	0.154087
$832$	0	0
$833$	−7.26781e7	−3.62904
$834$	0	0
$835$	−1.15177e7	−0.571677
$836$	0	0
$837$	2.21729e7	1.09398
$838$	0	0
$839$	−2.31814e6	−0.113693	−0.0568467	−	0.998383i	$-0.518105\pi$
$839$	−2.31814e6	−0.113693	−0.0568467	+	0.998383i	$0.518105\pi$
$840$	0	0
$841$	857031.	0.0417836
$842$	0	0
$843$	−871424.	−0.0422339
$844$	0	0
$845$	1.09258e7	0.526395
$846$	0	0
$847$	−3.36927e7	−1.61372
$848$	0	0
$849$	−3.66041e7	−1.74285
$850$	0	0
$851$	−1.56910e7	−0.742725
$852$	0	0
$853$	−3.68565e7	−1.73437	−0.867184	−	0.497987i	$-0.834073\pi$
$853$	−3.68565e7	−1.73437	−0.867184	+	0.497987i	$0.834073\pi$
$854$	0	0
$855$	−609681.	−0.0285225
$856$	0	0
$857$	−3.75163e7	−1.74489	−0.872444	−	0.488715i	$-0.837466\pi$
$857$	−3.75163e7	−1.74489	−0.872444	+	0.488715i	$0.837466\pi$
$858$	0	0
$859$	2.37280e7	1.09718	0.548591	−	0.836091i	$-0.315165\pi$
$859$	2.37280e7	1.09718	0.548591	+	0.836091i	$0.315165\pi$
$860$	0	0
$861$	−9.03254e6	−0.415243
$862$	0	0
$863$	4.26203e7	1.94800	0.974000	−	0.226546i	$-0.0727435\pi$
$863$	4.26203e7	1.94800	0.974000	+	0.226546i	$0.0727435\pi$
$864$	0	0
$865$	5.26042e6	0.239045
$866$	0	0
$867$	−5.20698e7	−2.35254
$868$	0	0
$869$	4.59551e6	0.206435
$870$	0	0
$871$	−1.72866e7	−0.772084
$872$	0	0
$873$	−1.58650e6	−0.0704536
$874$	0	0
$875$	−4.24275e7	−1.87338
$876$	0	0
$877$	1.27763e7	0.560926	0.280463	−	0.959865i	$-0.409512\pi$
$877$	1.27763e7	0.560926	0.280463	+	0.959865i	$0.409512\pi$
$878$	0	0
$879$	−1.50460e7	−0.656826
$880$	0	0
$881$	−1.84273e7	−0.799873	−0.399936	−	0.916543i	$-0.630968\pi$
$881$	−1.84273e7	−0.799873	−0.399936	+	0.916543i	$0.630968\pi$
$882$	0	0
$883$	−6.89074e6	−0.297416	−0.148708	−	0.988881i	$-0.547511\pi$
$883$	−6.89074e6	−0.297416	−0.148708	+	0.988881i	$0.547511\pi$
$884$	0	0
$885$	−2.04221e7	−0.876479
$886$	0	0
$887$	−2.70353e7	−1.15378	−0.576888	−	0.816823i	$-0.695733\pi$
$887$	−2.70353e7	−1.15378	−0.576888	+	0.816823i	$0.695733\pi$
$888$	0	0
$889$	−2.57331e7	−1.09204
$890$	0	0
$891$	5.55370e6	0.234363
$892$	0	0
$893$	1.94899e7	0.817864
$894$	0	0
$895$	−2.88270e7	−1.20293
$896$	0	0
$897$	−2.02637e7	−0.840888
$898$	0	0
$899$	2.63555e7	1.08761
$900$	0	0
$901$	1.19497e7	0.490394
$902$	0	0
$903$	−5.28366e7	−2.15633
$904$	0	0
$905$	465318.	0.0188855
$906$	0	0
$907$	4.44074e7	1.79241	0.896204	−	0.443643i	$-0.146314\pi$
$907$	4.44074e7	1.79241	0.896204	+	0.443643i	$0.146314\pi$
$908$	0	0
$909$	−1.64188e6	−0.0659072
$910$	0	0
$911$	1.88450e7	0.752315	0.376157	−	0.926556i	$-0.377245\pi$
$911$	1.88450e7	0.752315	0.376157	+	0.926556i	$0.377245\pi$
$912$	0	0
$913$	5.89780e6	0.234160
$914$	0	0
$915$	3.88155e7	1.53268
$916$	0	0
$917$	−7.51474e7	−2.95114
$918$	0	0
$919$	−2.41199e7	−0.942078	−0.471039	−	0.882112i	$-0.656121\pi$
$919$	−2.41199e7	−0.942078	−0.471039	+	0.882112i	$0.656121\pi$
$920$	0	0
$921$	−1.85432e7	−0.720338
$922$	0	0
$923$	−6.40296e7	−2.47387
$924$	0	0
$925$	9.34024e6	0.358925
$926$	0	0
$927$	93930.4	0.00359010
$928$	0	0
$929$	2.23104e7	0.848141	0.424071	−	0.905629i	$-0.360601\pi$
$929$	2.23104e7	0.848141	0.424071	+	0.905629i	$0.360601\pi$
$930$	0	0
$931$	3.13455e7	1.18523
$932$	0	0
$933$	4.66267e6	0.175360
$934$	0	0
$935$	1.01274e7	0.378852
$936$	0	0
$937$	1.34054e7	0.498806	0.249403	−	0.968400i	$-0.419766\pi$
$937$	1.34054e7	0.498806	0.249403	+	0.968400i	$0.419766\pi$
$938$	0	0
$939$	1.59358e7	0.589808
$940$	0	0
$941$	5.29861e6	0.195069	0.0975344	−	0.995232i	$-0.468904\pi$
$941$	5.29861e6	0.195069	0.0975344	+	0.995232i	$0.468904\pi$
$942$	0	0
$943$	4.58856e6	0.168034
$944$	0	0
$945$	3.97977e7	1.44970
$946$	0	0
$947$	−3.26296e7	−1.18232	−0.591162	−	0.806553i	$-0.701330\pi$
$947$	−3.26296e7	−1.18232	−0.591162	+	0.806553i	$0.701330\pi$
$948$	0	0
$949$	4.30845e7	1.55294
$950$	0	0
$951$	3.65400e7	1.31014
$952$	0	0
$953$	1.66407e7	0.593524	0.296762	−	0.954951i	$-0.404093\pi$
$953$	1.66407e7	0.593524	0.296762	+	0.954951i	$0.404093\pi$
$954$	0	0
$955$	1.62096e7	0.575128
$956$	0	0
$957$	7.00567e6	0.247269
$958$	0	0
$959$	5.05420e7	1.77462
$960$	0	0
$961$	3.87764e6	0.135444
$962$	0	0
$963$	193120.	0.00671061
$964$	0	0
$965$	7.12869e6	0.246429
$966$	0	0
$967$	4.66522e6	0.160437	0.0802187	−	0.996777i	$-0.474438\pi$
$967$	4.66522e6	0.160437	0.0802187	+	0.996777i	$0.474438\pi$
$968$	0	0
$969$	3.17246e7	1.08539
$970$	0	0
$971$	−5.33590e6	−0.181618	−0.0908092	−	0.995868i	$-0.528945\pi$
$971$	−5.33590e6	−0.181618	−0.0908092	+	0.995868i	$0.528945\pi$
$972$	0	0
$973$	4.51449e6	0.152871
$974$	0	0
$975$	1.20622e7	0.406363
$976$	0	0
$977$	4.38184e7	1.46866	0.734329	−	0.678794i	$-0.237497\pi$
$977$	4.38184e7	1.46866	0.734329	+	0.678794i	$0.237497\pi$
$978$	0	0
$979$	−8.74397e6	−0.291576
$980$	0	0
$981$	−291323.	−0.00966500
$982$	0	0
$983$	−3.27581e7	−1.08127	−0.540636	−	0.841257i	$-0.681816\pi$
$983$	−3.27581e7	−1.08127	−0.540636	+	0.841257i	$0.681816\pi$
$984$	0	0
$985$	−4.99179e7	−1.63933
$986$	0	0
$987$	−6.92035e7	−2.26118
$988$	0	0
$989$	2.68412e7	0.872591
$990$	0	0
$991$	−9.88829e6	−0.319843	−0.159922	−	0.987130i	$-0.551124\pi$
$991$	−9.88829e6	−0.319843	−0.159922	+	0.987130i	$0.551124\pi$
$992$	0	0
$993$	−2.26716e7	−0.729640
$994$	0	0
$995$	4.11856e7	1.31883
$996$	0	0
$997$	−2.97146e7	−0.946742	−0.473371	−	0.880863i	$-0.656963\pi$
$997$	−2.97146e7	−0.946742	−0.473371	+	0.880863i	$0.656963\pi$
$998$	0	0
$999$	−3.55784e7	−1.12791

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.b.1.17	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.b.1.17	✓	57	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)

\(f(q)\)	\(=\)	\(q-15.1335 q^{3} +45.8699 q^{5} +223.098 q^{7} -13.9785 q^{9} +O(q^{10})\)	\(q-15.1335 q^{3} +45.8699 q^{5} +223.098 q^{7} -13.9785 q^{9} -100.145 q^{11} +780.695 q^{13} -694.170 q^{15} -2204.67 q^{17} +950.856 q^{19} -3376.24 q^{21} +1715.14 q^{23} -1020.95 q^{25} +3888.97 q^{27} +4622.57 q^{29} +5701.47 q^{31} +1515.53 q^{33} +10233.5 q^{35} -9148.54 q^{37} -11814.6 q^{39} +2675.33 q^{41} +15649.5 q^{43} -641.192 q^{45} +20497.2 q^{47} +32965.6 q^{49} +33364.3 q^{51} -5420.18 q^{53} -4593.62 q^{55} -14389.7 q^{57} +29419.4 q^{59} -55916.4 q^{61} -3118.57 q^{63} +35810.4 q^{65} -22142.6 q^{67} -25956.0 q^{69} -82016.1 q^{71} +55187.4 q^{73} +15450.6 q^{75} -22342.0 q^{77} -45888.7 q^{79} -55456.8 q^{81} -58892.9 q^{83} -101128. q^{85} -69955.5 q^{87} +87313.4 q^{89} +174171. q^{91} -86283.0 q^{93} +43615.7 q^{95} +113496. q^{97} +1399.87 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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