LMFDB - Embedded newform 688.6.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,6,Mod(1,688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, 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("688.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	688.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:	$110.344068031$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + \cdots - 22734604 $ x^10 - 2x^9 - 256x^8 + 266x^7 + 21986x^6 - 10450x^5 - 719484x^4 + 384582x^3 + 8437093x^2 - 5752252*x - 22734604
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.50018$ of defining polynomial
Character	$\chi$	$=$	688.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.8892 q^{3} +47.4635 q^{5} -67.4603 q^{7} +42.2439 q^{9} +O(q^{10})$	$q-16.8892 q^{3} +47.4635 q^{5} -67.4603 q^{7} +42.2439 q^{9} -81.3987 q^{11} +1058.72 q^{13} -801.619 q^{15} +251.378 q^{17} -1612.95 q^{19} +1139.35 q^{21} +32.7403 q^{23} -872.217 q^{25} +3390.60 q^{27} -2583.75 q^{29} +7206.51 q^{31} +1374.76 q^{33} -3201.90 q^{35} -6174.39 q^{37} -17881.0 q^{39} +15514.4 q^{41} -1849.00 q^{43} +2005.04 q^{45} +1692.39 q^{47} -12256.1 q^{49} -4245.56 q^{51} -25612.3 q^{53} -3863.47 q^{55} +27241.3 q^{57} -24532.0 q^{59} +8209.11 q^{61} -2849.78 q^{63} +50250.7 q^{65} +12302.3 q^{67} -552.957 q^{69} -18712.0 q^{71} -12126.5 q^{73} +14731.0 q^{75} +5491.18 q^{77} +52372.8 q^{79} -67529.7 q^{81} -28935.9 q^{83} +11931.3 q^{85} +43637.3 q^{87} +117315. q^{89} -71421.8 q^{91} -121712. q^{93} -76556.1 q^{95} -147143. q^{97} -3438.60 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9}+O(q^{10}) $ 10 * q - 28 * q^3 + 138 * q^5 - 60 * q^7 + 1356 * q^9	$ 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9} - 745 q^{11} + 1917 q^{13} - 1688 q^{15} + 4017 q^{17} + 2404 q^{19} - 228 q^{21} - 1733 q^{23} + 7120 q^{25} + 2324 q^{27} + 6996 q^{29} + 4899 q^{31} - 15734 q^{33} - 7084 q^{35} + 1466 q^{37} + 26542 q^{39} + 10297 q^{41} - 18490 q^{43} + 73822 q^{45} - 48592 q^{47} + 29458 q^{49} - 92972 q^{51} + 127165 q^{53} - 106672 q^{55} + 34060 q^{57} - 99372 q^{59} + 17408 q^{61} - 2244 q^{63} + 54484 q^{65} + 2021 q^{67} + 1654 q^{69} - 11286 q^{71} + 49892 q^{73} + 44662 q^{75} + 98144 q^{77} + 91524 q^{79} - 26450 q^{81} + 105203 q^{83} - 87212 q^{85} - 181200 q^{87} - 62682 q^{89} + 295304 q^{91} - 238430 q^{93} + 305340 q^{95} + 108383 q^{97} + 270499 q^{99}+O(q^{100}) $ 10 * q - 28 * q^3 + 138 * q^5 - 60 * q^7 + 1356 * q^9 - 745 * q^11 + 1917 * q^13 - 1688 * q^15 + 4017 * q^17 + 2404 * q^19 - 228 * q^21 - 1733 * q^23 + 7120 * q^25 + 2324 * q^27 + 6996 * q^29 + 4899 * q^31 - 15734 * q^33 - 7084 * q^35 + 1466 * q^37 + 26542 * q^39 + 10297 * q^41 - 18490 * q^43 + 73822 * q^45 - 48592 * q^47 + 29458 * q^49 - 92972 * q^51 + 127165 * q^53 - 106672 * q^55 + 34060 * q^57 - 99372 * q^59 + 17408 * q^61 - 2244 * q^63 + 54484 * q^65 + 2021 * q^67 + 1654 * q^69 - 11286 * q^71 + 49892 * q^73 + 44662 * q^75 + 98144 * q^77 + 91524 * q^79 - 26450 * q^81 + 105203 * q^83 - 87212 * q^85 - 181200 * q^87 - 62682 * q^89 + 295304 * q^91 - 238430 * q^93 + 305340 * q^95 + 108383 * q^97 + 270499 * 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
$3$	−16.8892	−1.08344	−0.541720	−	0.840559i	$-0.682227\pi$
$3$	−16.8892	−1.08344	−0.541720	+	0.840559i	$0.682227\pi$
$4$	0	0
$5$	47.4635	0.849053	0.424526	−	0.905416i	$-0.360441\pi$
$5$	47.4635	0.849053	0.424526	+	0.905416i	$0.360441\pi$
$6$	0	0
$7$	−67.4603	−0.520359	−0.260180	−	0.965560i	$-0.583782\pi$
$7$	−67.4603	−0.520359	−0.260180	+	0.965560i	$0.583782\pi$
$8$	0	0
$9$	42.2439	0.173843
$10$	0	0
$11$	−81.3987	−0.202832	−0.101416	−	0.994844i	$-0.532337\pi$
$11$	−81.3987	−0.202832	−0.101416	+	0.994844i	$0.532337\pi$
$12$	0	0
$13$	1058.72	1.73750	0.868749	−	0.495253i	$-0.164925\pi$
$13$	1058.72	1.73750	0.868749	+	0.495253i	$0.164925\pi$
$14$	0	0
$15$	−801.619	−0.919898
$16$	0	0
$17$	251.378	0.210962	0.105481	−	0.994421i	$-0.466362\pi$
$17$	251.378	0.210962	0.105481	+	0.994421i	$0.466362\pi$
$18$	0	0
$19$	−1612.95	−1.02503	−0.512515	−	0.858679i	$-0.671286\pi$
$19$	−1612.95	−1.02503	−0.512515	+	0.858679i	$0.671286\pi$
$20$	0	0
$21$	1139.35	0.563778
$22$	0	0
$23$	32.7403	0.0129052	0.00645258	−	0.999979i	$-0.497946\pi$
$23$	32.7403	0.0129052	0.00645258	+	0.999979i	$0.497946\pi$
$24$	0	0
$25$	−872.217	−0.279109
$26$	0	0
$27$	3390.60	0.895092
$28$	0	0
$29$	−2583.75	−0.570499	−0.285249	−	0.958453i	$-0.592076\pi$
$29$	−2583.75	−0.570499	−0.285249	+	0.958453i	$0.592076\pi$
$30$	0	0
$31$	7206.51	1.34685	0.673427	−	0.739254i	$-0.264821\pi$
$31$	7206.51	1.34685	0.673427	+	0.739254i	$0.264821\pi$
$32$	0	0
$33$	1374.76	0.219756
$34$	0	0
$35$	−3201.90	−0.441812
$36$	0	0
$37$	−6174.39	−0.741463	−0.370731	−	0.928740i	$-0.620893\pi$
$37$	−6174.39	−0.741463	−0.370731	+	0.928740i	$0.620893\pi$
$38$	0	0
$39$	−17881.0	−1.88248
$40$	0	0
$41$	15514.4	1.44137	0.720685	−	0.693262i	$-0.243827\pi$
$41$	15514.4	1.44137	0.720685	+	0.693262i	$0.243827\pi$
$42$	0	0
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	0	0
$45$	2005.04	0.147602
$46$	0	0
$47$	1692.39	0.111752	0.0558760	−	0.998438i	$-0.482205\pi$
$47$	1692.39	0.111752	0.0558760	+	0.998438i	$0.482205\pi$
$48$	0	0
$49$	−12256.1	−0.729226
$50$	0	0
$51$	−4245.56	−0.228565
$52$	0	0
$53$	−25612.3	−1.25245	−0.626223	−	0.779644i	$-0.715400\pi$
$53$	−25612.3	−1.25245	−0.626223	+	0.779644i	$0.715400\pi$
$54$	0	0
$55$	−3863.47	−0.172215
$56$	0	0
$57$	27241.3	1.11056
$58$	0	0
$59$	−24532.0	−0.917495	−0.458748	−	0.888567i	$-0.651702\pi$
$59$	−24532.0	−0.917495	−0.458748	+	0.888567i	$0.651702\pi$
$60$	0	0
$61$	8209.11	0.282470	0.141235	−	0.989976i	$-0.454893\pi$
$61$	8209.11	0.282470	0.141235	+	0.989976i	$0.454893\pi$
$62$	0	0
$63$	−2849.78	−0.0904608
$64$	0	0
$65$	50250.7	1.47523
$66$	0	0
$67$	12302.3	0.334810	0.167405	−	0.985888i	$-0.446461\pi$
$67$	12302.3	0.334810	0.167405	+	0.985888i	$0.446461\pi$
$68$	0	0
$69$	−552.957	−0.0139820
$70$	0	0
$71$	−18712.0	−0.440530	−0.220265	−	0.975440i	$-0.570692\pi$
$71$	−18712.0	−0.440530	−0.220265	+	0.975440i	$0.570692\pi$
$72$	0	0
$73$	−12126.5	−0.266334	−0.133167	−	0.991094i	$-0.542515\pi$
$73$	−12126.5	−0.266334	−0.133167	+	0.991094i	$0.542515\pi$
$74$	0	0
$75$	14731.0	0.302398
$76$	0	0
$77$	5491.18	0.105545
$78$	0	0
$79$	52372.8	0.944144	0.472072	−	0.881560i	$-0.343506\pi$
$79$	52372.8	0.944144	0.472072	+	0.881560i	$0.343506\pi$
$80$	0	0
$81$	−67529.7	−1.14362
$82$	0	0
$83$	−28935.9	−0.461043	−0.230521	−	0.973067i	$-0.574043\pi$
$83$	−28935.9	−0.461043	−0.230521	+	0.973067i	$0.574043\pi$
$84$	0	0
$85$	11931.3	0.179118
$86$	0	0
$87$	43637.3	0.618101
$88$	0	0
$89$	117315.	1.56992	0.784962	−	0.619544i	$-0.212683\pi$
$89$	117315.	1.56992	0.784962	+	0.619544i	$0.212683\pi$
$90$	0	0
$91$	−71421.8	−0.904123
$92$	0	0
$93$	−121712.	−1.45924
$94$	0	0
$95$	−76556.1	−0.870304
$96$	0	0
$97$	−147143.	−1.58785	−0.793925	−	0.608016i	$-0.791966\pi$
$97$	−147143.	−1.58785	−0.793925	+	0.608016i	$0.791966\pi$
$98$	0	0
$99$	−3438.60	−0.0352609
$100$	0	0
$101$	−118561.	−1.15648	−0.578239	−	0.815867i	$-0.696260\pi$
$101$	−118561.	−1.15648	−0.578239	+	0.815867i	$0.696260\pi$
$102$	0	0
$103$	−49153.5	−0.456522	−0.228261	−	0.973600i	$-0.573304\pi$
$103$	−49153.5	−0.456522	−0.228261	+	0.973600i	$0.573304\pi$
$104$	0	0
$105$	54077.4	0.478677
$106$	0	0
$107$	156632.	1.32258	0.661290	−	0.750130i	$-0.270009\pi$
$107$	156632.	1.32258	0.661290	+	0.750130i	$0.270009\pi$
$108$	0	0
$109$	208919.	1.68427	0.842135	−	0.539267i	$-0.181299\pi$
$109$	208919.	1.68427	0.842135	+	0.539267i	$0.181299\pi$
$110$	0	0
$111$	104280.	0.803331
$112$	0	0
$113$	33491.2	0.246737	0.123369	−	0.992361i	$-0.460630\pi$
$113$	33491.2	0.246737	0.123369	+	0.992361i	$0.460630\pi$
$114$	0	0
$115$	1553.97	0.0109572
$116$	0	0
$117$	44724.6	0.302052
$118$	0	0
$119$	−16958.0	−0.109776
$120$	0	0
$121$	−154425.	−0.958859
$122$	0	0
$123$	−262025.	−1.56164
$124$	0	0
$125$	−189722.	−1.08603
$126$	0	0
$127$	−72430.2	−0.398483	−0.199242	−	0.979950i	$-0.563848\pi$
$127$	−72430.2	−0.398483	−0.199242	+	0.979950i	$0.563848\pi$
$128$	0	0
$129$	31228.1	0.165223
$130$	0	0
$131$	73042.1	0.371873	0.185937	−	0.982562i	$-0.440468\pi$
$131$	73042.1	0.371873	0.185937	+	0.982562i	$0.440468\pi$
$132$	0	0
$133$	108810.	0.533383
$134$	0	0
$135$	160930.	0.759980
$136$	0	0
$137$	290889.	1.32412	0.662058	−	0.749453i	$-0.269683\pi$
$137$	290889.	1.32412	0.662058	+	0.749453i	$0.269683\pi$
$138$	0	0
$139$	375146.	1.64689	0.823443	−	0.567399i	$-0.192050\pi$
$139$	375146.	1.64689	0.823443	+	0.567399i	$0.192050\pi$
$140$	0	0
$141$	−28583.0	−0.121077
$142$	0	0
$143$	−86178.7	−0.352420
$144$	0	0
$145$	−122634.	−0.484384
$146$	0	0
$147$	206995.	0.790073
$148$	0	0
$149$	431136.	1.59092	0.795460	−	0.606006i	$-0.207229\pi$
$149$	431136.	1.59092	0.795460	+	0.606006i	$0.207229\pi$
$150$	0	0
$151$	−17252.4	−0.0615755	−0.0307878	−	0.999526i	$-0.509802\pi$
$151$	−17252.4	−0.0615755	−0.0307878	+	0.999526i	$0.509802\pi$
$152$	0	0
$153$	10619.2	0.0366743
$154$	0	0
$155$	342046.	1.14355
$156$	0	0
$157$	−139497.	−0.451663	−0.225832	−	0.974166i	$-0.572510\pi$
$157$	−139497.	−0.451663	−0.225832	+	0.974166i	$0.572510\pi$
$158$	0	0
$159$	432570.	1.35695
$160$	0	0
$161$	−2208.67	−0.00671532
$162$	0	0
$163$	556197.	1.63968	0.819841	−	0.572591i	$-0.194062\pi$
$163$	556197.	1.63968	0.819841	+	0.572591i	$0.194062\pi$
$164$	0	0
$165$	65250.7	0.186584
$166$	0	0
$167$	239206.	0.663713	0.331857	−	0.943330i	$-0.392325\pi$
$167$	239206.	0.663713	0.331857	+	0.943330i	$0.392325\pi$
$168$	0	0
$169$	749603.	2.01890
$170$	0	0
$171$	−68137.1	−0.178194
$172$	0	0
$173$	79563.7	0.202116	0.101058	−	0.994881i	$-0.467777\pi$
$173$	79563.7	0.202116	0.101058	+	0.994881i	$0.467777\pi$
$174$	0	0
$175$	58840.0	0.145237
$176$	0	0
$177$	414326.	0.994051
$178$	0	0
$179$	95838.9	0.223568	0.111784	−	0.993733i	$-0.464344\pi$
$179$	95838.9	0.223568	0.111784	+	0.993733i	$0.464344\pi$
$180$	0	0
$181$	−219413.	−0.497813	−0.248906	−	0.968528i	$-0.580071\pi$
$181$	−219413.	−0.497813	−0.248906	+	0.968528i	$0.580071\pi$
$182$	0	0
$183$	−138645.	−0.306039
$184$	0	0
$185$	−293058.	−0.629541
$186$	0	0
$187$	−20461.8	−0.0427898
$188$	0	0
$189$	−228731.	−0.465769
$190$	0	0
$191$	−328729.	−0.652011	−0.326005	−	0.945368i	$-0.605703\pi$
$191$	−328729.	−0.652011	−0.326005	+	0.945368i	$0.605703\pi$
$192$	0	0
$193$	575743.	1.11259	0.556295	−	0.830985i	$-0.312222\pi$
$193$	575743.	1.11259	0.556295	+	0.830985i	$0.312222\pi$
$194$	0	0
$195$	−848693.	−1.59832
$196$	0	0
$197$	962886.	1.76770	0.883852	−	0.467768i	$-0.154942\pi$
$197$	962886.	1.76770	0.883852	+	0.467768i	$0.154942\pi$
$198$	0	0
$199$	760147.	1.36071	0.680354	−	0.732884i	$-0.261826\pi$
$199$	760147.	1.36071	0.680354	+	0.732884i	$0.261826\pi$
$200$	0	0
$201$	−207775.	−0.362747
$202$	0	0
$203$	174300.	0.296864
$204$	0	0
$205$	736368.	1.22380
$206$	0	0
$207$	1383.08	0.00224347
$208$	0	0
$209$	131292.	0.207908
$210$	0	0
$211$	979789.	1.51505	0.757524	−	0.652807i	$-0.226409\pi$
$211$	979789.	1.51505	0.757524	+	0.652807i	$0.226409\pi$
$212$	0	0
$213$	316031.	0.477288
$214$	0	0
$215$	−87760.0	−0.129479
$216$	0	0
$217$	−486153.	−0.700848
$218$	0	0
$219$	204806.	0.288557
$220$	0	0
$221$	266140.	0.366547
$222$	0	0
$223$	−668754.	−0.900542	−0.450271	−	0.892892i	$-0.648673\pi$
$223$	−668754.	−0.900542	−0.450271	+	0.892892i	$0.648673\pi$
$224$	0	0
$225$	−36845.8	−0.0485212
$226$	0	0
$227$	1.40720e6	1.81256	0.906280	−	0.422677i	$-0.138910\pi$
$227$	1.40720e6	1.81256	0.906280	+	0.422677i	$0.138910\pi$
$228$	0	0
$229$	706746.	0.890583	0.445292	−	0.895386i	$-0.353100\pi$
$229$	706746.	0.890583	0.445292	+	0.895386i	$0.353100\pi$
$230$	0	0
$231$	−92741.4	−0.114352
$232$	0	0
$233$	810873.	0.978505	0.489252	−	0.872142i	$-0.337270\pi$
$233$	810873.	0.978505	0.489252	+	0.872142i	$0.337270\pi$
$234$	0	0
$235$	80326.7	0.0948834
$236$	0	0
$237$	−884533.	−1.02292
$238$	0	0
$239$	−81145.7	−0.0918905	−0.0459453	−	0.998944i	$-0.514630\pi$
$239$	−81145.7	−0.0918905	−0.0459453	+	0.998944i	$0.514630\pi$
$240$	0	0
$241$	538808.	0.597574	0.298787	−	0.954320i	$-0.403418\pi$
$241$	538808.	0.597574	0.298787	+	0.954320i	$0.403418\pi$
$242$	0	0
$243$	316604.	0.343954
$244$	0	0
$245$	−581718.	−0.619152
$246$	0	0
$247$	−1.70767e6	−1.78099
$248$	0	0
$249$	488703.	0.499512
$250$	0	0
$251$	228910.	0.229340	0.114670	−	0.993404i	$-0.463419\pi$
$251$	228910.	0.229340	0.114670	+	0.993404i	$0.463419\pi$
$252$	0	0
$253$	−2665.02	−0.00261758
$254$	0	0
$255$	−201509.	−0.194064
$256$	0	0
$257$	1.42445e6	1.34528	0.672642	−	0.739968i	$-0.265160\pi$
$257$	1.42445e6	1.34528	0.672642	+	0.739968i	$0.265160\pi$
$258$	0	0
$259$	416526.	0.385827
$260$	0	0
$261$	−109147.	−0.0991773
$262$	0	0
$263$	−60691.4	−0.0541050	−0.0270525	−	0.999634i	$-0.508612\pi$
$263$	−60691.4	−0.0541050	−0.0270525	+	0.999634i	$0.508612\pi$
$264$	0	0
$265$	−1.21565e6	−1.06339
$266$	0	0
$267$	−1.98135e6	−1.70092
$268$	0	0
$269$	−1.85520e6	−1.56319	−0.781594	−	0.623788i	$-0.785593\pi$
$269$	−1.85520e6	−1.56319	−0.781594	+	0.623788i	$0.785593\pi$
$270$	0	0
$271$	1.57226e6	1.30047	0.650236	−	0.759732i	$-0.274670\pi$
$271$	1.57226e6	1.30047	0.650236	+	0.759732i	$0.274670\pi$
$272$	0	0
$273$	1.20625e6	0.979563
$274$	0	0
$275$	70997.3	0.0566122
$276$	0	0
$277$	321513.	0.251767	0.125883	−	0.992045i	$-0.459823\pi$
$277$	321513.	0.251767	0.125883	+	0.992045i	$0.459823\pi$
$278$	0	0
$279$	304431.	0.234141
$280$	0	0
$281$	−501565.	−0.378932	−0.189466	−	0.981887i	$-0.560676\pi$
$281$	−501565.	−0.378932	−0.189466	+	0.981887i	$0.560676\pi$
$282$	0	0
$283$	−347650.	−0.258034	−0.129017	−	0.991642i	$-0.541182\pi$
$283$	−347650.	−0.258034	−0.129017	+	0.991642i	$0.541182\pi$
$284$	0	0
$285$	1.29297e6	0.942922
$286$	0	0
$287$	−1.04661e6	−0.750030
$288$	0	0
$289$	−1.35667e6	−0.955495
$290$	0	0
$291$	2.48512e6	1.72034
$292$	0	0
$293$	−1.60874e6	−1.09475	−0.547376	−	0.836886i	$-0.684373\pi$
$293$	−1.60874e6	−1.09475	−0.547376	+	0.836886i	$0.684373\pi$
$294$	0	0
$295$	−1.16438e6	−0.779002
$296$	0	0
$297$	−275991.	−0.181553
$298$	0	0
$299$	34663.0	0.0224227
$300$	0	0
$301$	124734.	0.0793540
$302$	0	0
$303$	2.00239e6	1.25298
$304$	0	0
$305$	389633.	0.239832
$306$	0	0
$307$	−2.50047e6	−1.51417	−0.757085	−	0.653316i	$-0.773377\pi$
$307$	−2.50047e6	−1.51417	−0.757085	+	0.653316i	$0.773377\pi$
$308$	0	0
$309$	830162.	0.494614
$310$	0	0
$311$	−2.80353e6	−1.64363	−0.821816	−	0.569753i	$-0.807039\pi$
$311$	−2.80353e6	−1.64363	−0.821816	+	0.569753i	$0.807039\pi$
$312$	0	0
$313$	−102581.	−0.0591841	−0.0295920	−	0.999562i	$-0.509421\pi$
$313$	−102581.	−0.0591841	−0.0295920	+	0.999562i	$0.509421\pi$
$314$	0	0
$315$	−135261.	−0.0768060
$316$	0	0
$317$	−263420.	−0.147232	−0.0736158	−	0.997287i	$-0.523454\pi$
$317$	−263420.	−0.147232	−0.0736158	+	0.997287i	$0.523454\pi$
$318$	0	0
$319$	210313.	0.115715
$320$	0	0
$321$	−2.64539e6	−1.43294
$322$	0	0
$323$	−405459.	−0.216243
$324$	0	0
$325$	−923437.	−0.484952
$326$	0	0
$327$	−3.52847e6	−1.82481
$328$	0	0
$329$	−114169.	−0.0581512
$330$	0	0
$331$	2.31706e6	1.16243	0.581216	−	0.813749i	$-0.302577\pi$
$331$	2.31706e6	1.16243	0.581216	+	0.813749i	$0.302577\pi$
$332$	0	0
$333$	−260830.	−0.128898
$334$	0	0
$335$	583909.	0.284271
$336$	0	0
$337$	1.40113e6	0.672055	0.336028	−	0.941852i	$-0.390916\pi$
$337$	1.40113e6	0.672055	0.336028	+	0.941852i	$0.390916\pi$
$338$	0	0
$339$	−565639.	−0.267325
$340$	0	0
$341$	−586600.	−0.273185
$342$	0	0
$343$	1.96061e6	0.899819
$344$	0	0
$345$	−26245.3	−0.0118714
$346$	0	0
$347$	3.64231e6	1.62388	0.811940	−	0.583741i	$-0.198412\pi$
$347$	3.64231e6	1.62388	0.811940	+	0.583741i	$0.198412\pi$
$348$	0	0
$349$	2.59872e6	1.14208	0.571039	−	0.820923i	$-0.306540\pi$
$349$	2.59872e6	1.14208	0.571039	+	0.820923i	$0.306540\pi$
$350$	0	0
$351$	3.58971e6	1.55522
$352$	0	0
$353$	−4.35884e6	−1.86180	−0.930902	−	0.365269i	$-0.880977\pi$
$353$	−4.35884e6	−1.86180	−0.930902	+	0.365269i	$0.880977\pi$
$354$	0	0
$355$	−888139.	−0.374033
$356$	0	0
$357$	286407.	0.118936
$358$	0	0
$359$	−3.30615e6	−1.35390	−0.676949	−	0.736029i	$-0.736698\pi$
$359$	−3.30615e6	−1.35390	−0.676949	+	0.736029i	$0.736698\pi$
$360$	0	0
$361$	125500.	0.0506845
$362$	0	0
$363$	2.60811e6	1.03887
$364$	0	0
$365$	−575564.	−0.226132
$366$	0	0
$367$	548156.	0.212441	0.106221	−	0.994343i	$-0.466125\pi$
$367$	548156.	0.212441	0.106221	+	0.994343i	$0.466125\pi$
$368$	0	0
$369$	655389.	0.250572
$370$	0	0
$371$	1.72781e6	0.651722
$372$	0	0
$373$	−549721.	−0.204583	−0.102292	−	0.994754i	$-0.532617\pi$
$373$	−549721.	−0.204583	−0.102292	+	0.994754i	$0.532617\pi$
$374$	0	0
$375$	3.20424e6	1.17665
$376$	0	0
$377$	−2.73547e6	−0.991240
$378$	0	0
$379$	2.24483e6	0.802760	0.401380	−	0.915912i	$-0.368531\pi$
$379$	2.24483e6	0.802760	0.401380	+	0.915912i	$0.368531\pi$
$380$	0	0
$381$	1.22329e6	0.431733
$382$	0	0
$383$	−5.10313e6	−1.77762	−0.888812	−	0.458272i	$-0.848469\pi$
$383$	−5.10313e6	−1.77762	−0.888812	+	0.458272i	$0.848469\pi$
$384$	0	0
$385$	260631.	0.0896135
$386$	0	0
$387$	−78108.9	−0.0265108
$388$	0	0
$389$	4.70768e6	1.57737	0.788683	−	0.614800i	$-0.210763\pi$
$389$	4.70768e6	1.57737	0.788683	+	0.614800i	$0.210763\pi$
$390$	0	0
$391$	8230.20	0.00272250
$392$	0	0
$393$	−1.23362e6	−0.402902
$394$	0	0
$395$	2.48580e6	0.801628
$396$	0	0
$397$	3.81755e6	1.21565	0.607825	−	0.794071i	$-0.292042\pi$
$397$	3.81755e6	1.21565	0.607825	+	0.794071i	$0.292042\pi$
$398$	0	0
$399$	−1.83771e6	−0.577889
$400$	0	0
$401$	1.31578e6	0.408622	0.204311	−	0.978906i	$-0.434505\pi$
$401$	1.31578e6	0.408622	0.204311	+	0.978906i	$0.434505\pi$
$402$	0	0
$403$	7.62970e6	2.34016
$404$	0	0
$405$	−3.20520e6	−0.970995
$406$	0	0
$407$	502587.	0.150392
$408$	0	0
$409$	3.92071e6	1.15893	0.579464	−	0.814998i	$-0.303262\pi$
$409$	3.92071e6	1.15893	0.579464	+	0.814998i	$0.303262\pi$
$410$	0	0
$411$	−4.91287e6	−1.43460
$412$	0	0
$413$	1.65494e6	0.477427
$414$	0	0
$415$	−1.37340e6	−0.391450
$416$	0	0
$417$	−6.33591e6	−1.78430
$418$	0	0
$419$	−2.92842e6	−0.814890	−0.407445	−	0.913230i	$-0.633580\pi$
$419$	−2.92842e6	−0.814890	−0.407445	+	0.913230i	$0.633580\pi$
$420$	0	0
$421$	4.43337e6	1.21907	0.609535	−	0.792759i	$-0.291356\pi$
$421$	4.43337e6	1.21907	0.609535	+	0.792759i	$0.291356\pi$
$422$	0	0
$423$	71493.0	0.0194273
$424$	0	0
$425$	−219256.	−0.0588816
$426$	0	0
$427$	−553789.	−0.146986
$428$	0	0
$429$	1.45549e6	0.381826
$430$	0	0
$431$	−4.27188e6	−1.10771	−0.553854	−	0.832614i	$-0.686844\pi$
$431$	−4.27188e6	−1.10771	−0.553854	+	0.832614i	$0.686844\pi$
$432$	0	0
$433$	−1.65833e6	−0.425061	−0.212531	−	0.977154i	$-0.568171\pi$
$433$	−1.65833e6	−0.425061	−0.212531	+	0.977154i	$0.568171\pi$
$434$	0	0
$435$	2.07118e6	0.524801
$436$	0	0
$437$	−52808.5	−0.0132282
$438$	0	0
$439$	1.44252e6	0.357240	0.178620	−	0.983918i	$-0.442837\pi$
$439$	1.44252e6	0.357240	0.178620	+	0.983918i	$0.442837\pi$
$440$	0	0
$441$	−517745.	−0.126771
$442$	0	0
$443$	−5.80251e6	−1.40477	−0.702387	−	0.711795i	$-0.747882\pi$
$443$	−5.80251e6	−1.40477	−0.702387	+	0.711795i	$0.747882\pi$
$444$	0	0
$445$	5.56818e6	1.33295
$446$	0	0
$447$	−7.28152e6	−1.72367
$448$	0	0
$449$	−1.86453e6	−0.436470	−0.218235	−	0.975896i	$-0.570030\pi$
$449$	−1.86453e6	−0.436470	−0.218235	+	0.975896i	$0.570030\pi$
$450$	0	0
$451$	−1.26285e6	−0.292355
$452$	0	0
$453$	291379.	0.0667134
$454$	0	0
$455$	−3.38993e6	−0.767648
$456$	0	0
$457$	3.61109e6	0.808812	0.404406	−	0.914580i	$-0.367478\pi$
$457$	3.61109e6	0.808812	0.404406	+	0.914580i	$0.367478\pi$
$458$	0	0
$459$	852323.	0.188831
$460$	0	0
$461$	5.82273e6	1.27607	0.638034	−	0.770008i	$-0.279748\pi$
$461$	5.82273e6	1.27607	0.638034	+	0.770008i	$0.279748\pi$
$462$	0	0
$463$	−759569.	−0.164670	−0.0823351	−	0.996605i	$-0.526238\pi$
$463$	−759569.	−0.164670	−0.0823351	+	0.996605i	$0.526238\pi$
$464$	0	0
$465$	−5.77687e6	−1.23897
$466$	0	0
$467$	−3.30533e6	−0.701330	−0.350665	−	0.936501i	$-0.614044\pi$
$467$	−3.30533e6	−0.701330	−0.350665	+	0.936501i	$0.614044\pi$
$468$	0	0
$469$	−829915.	−0.174221
$470$	0	0
$471$	2.35598e6	0.489350
$472$	0	0
$473$	150506.	0.0309315
$474$	0	0
$475$	1.40684e6	0.286095
$476$	0	0
$477$	−1.08196e6	−0.217729
$478$	0	0
$479$	7.63918e6	1.52128	0.760638	−	0.649176i	$-0.224886\pi$
$479$	7.63918e6	1.52128	0.760638	+	0.649176i	$0.224886\pi$
$480$	0	0
$481$	−6.53697e6	−1.28829
$482$	0	0
$483$	37302.7	0.00727565
$484$	0	0
$485$	−6.98391e6	−1.34817
$486$	0	0
$487$	4.03922e6	0.771747	0.385873	−	0.922552i	$-0.373900\pi$
$487$	4.03922e6	0.771747	0.385873	+	0.922552i	$0.373900\pi$
$488$	0	0
$489$	−9.39371e6	−1.77650
$490$	0	0
$491$	−7.62584e6	−1.42753	−0.713763	−	0.700387i	$-0.753011\pi$
$491$	−7.62584e6	−1.42753	−0.713763	+	0.700387i	$0.753011\pi$
$492$	0	0
$493$	−649497.	−0.120354
$494$	0	0
$495$	−163208.	−0.0299383
$496$	0	0
$497$	1.26232e6	0.229234
$498$	0	0
$499$	−9.55168e6	−1.71723	−0.858615	−	0.512621i	$-0.828675\pi$
$499$	−9.55168e6	−1.71723	−0.858615	+	0.512621i	$0.828675\pi$
$500$	0	0
$501$	−4.03999e6	−0.719094
$502$	0	0
$503$	4.69119e6	0.826730	0.413365	−	0.910566i	$-0.364353\pi$
$503$	4.69119e6	0.826730	0.413365	+	0.910566i	$0.364353\pi$
$504$	0	0
$505$	−5.62731e6	−0.981911
$506$	0	0
$507$	−1.26602e7	−2.18736
$508$	0	0
$509$	2.37623e6	0.406531	0.203265	−	0.979124i	$-0.434845\pi$
$509$	2.37623e6	0.406531	0.203265	+	0.979124i	$0.434845\pi$
$510$	0	0
$511$	818054.	0.138589
$512$	0	0
$513$	−5.46886e6	−0.917495
$514$	0	0
$515$	−2.33300e6	−0.387611
$516$	0	0
$517$	−137758.	−0.0226668
$518$	0	0
$519$	−1.34376e6	−0.218980
$520$	0	0
$521$	6.31208e6	1.01878	0.509388	−	0.860537i	$-0.329872\pi$
$521$	6.31208e6	1.01878	0.509388	+	0.860537i	$0.329872\pi$
$522$	0	0
$523$	1.03668e7	1.65725	0.828627	−	0.559801i	$-0.189122\pi$
$523$	1.03668e7	1.65725	0.828627	+	0.559801i	$0.189122\pi$
$524$	0	0
$525$	−993758.	−0.157356
$526$	0	0
$527$	1.81156e6	0.284136
$528$	0	0
$529$	−6.43527e6	−0.999833
$530$	0	0
$531$	−1.03633e6	−0.159500
$532$	0	0
$533$	1.64255e7	2.50438
$534$	0	0
$535$	7.43432e6	1.12294
$536$	0	0
$537$	−1.61864e6	−0.242222
$538$	0	0
$539$	997631.	0.147910
$540$	0	0
$541$	−5.71736e6	−0.839851	−0.419926	−	0.907559i	$-0.637944\pi$
$541$	−5.71736e6	−0.839851	−0.419926	+	0.907559i	$0.637944\pi$
$542$	0	0
$543$	3.70570e6	0.539350
$544$	0	0
$545$	9.91603e6	1.43003
$546$	0	0
$547$	−4.37335e6	−0.624950	−0.312475	−	0.949926i	$-0.601158\pi$
$547$	−4.37335e6	−0.624950	−0.312475	+	0.949926i	$0.601158\pi$
$548$	0	0
$549$	346785.	0.0491054
$550$	0	0
$551$	4.16744e6	0.584778
$552$	0	0
$553$	−3.53309e6	−0.491294
$554$	0	0
$555$	4.94950e6	0.682070
$556$	0	0
$557$	−5.52441e6	−0.754481	−0.377240	−	0.926115i	$-0.623127\pi$
$557$	−5.52441e6	−0.754481	−0.377240	+	0.926115i	$0.623127\pi$
$558$	0	0
$559$	−1.95758e6	−0.264966
$560$	0	0
$561$	345583.	0.0463602
$562$	0	0
$563$	−1.38558e7	−1.84230	−0.921149	−	0.389211i	$-0.872748\pi$
$563$	−1.38558e7	−1.84230	−0.921149	+	0.389211i	$0.872748\pi$
$564$	0	0
$565$	1.58961e6	0.209493
$566$	0	0
$567$	4.55557e6	0.595094
$568$	0	0
$569$	8.96028e6	1.16022	0.580111	−	0.814538i	$-0.303009\pi$
$569$	8.96028e6	1.16022	0.580111	+	0.814538i	$0.303009\pi$
$570$	0	0
$571$	−1.86281e6	−0.239099	−0.119550	−	0.992828i	$-0.538145\pi$
$571$	−1.86281e6	−0.239099	−0.119550	+	0.992828i	$0.538145\pi$
$572$	0	0
$573$	5.55196e6	0.706415
$574$	0	0
$575$	−28556.7	−0.00360195
$576$	0	0
$577$	−8.48683e6	−1.06122	−0.530610	−	0.847616i	$-0.678037\pi$
$577$	−8.48683e6	−1.06122	−0.530610	+	0.847616i	$0.678037\pi$
$578$	0	0
$579$	−9.72382e6	−1.20543
$580$	0	0
$581$	1.95202e6	0.239908
$582$	0	0
$583$	2.08481e6	0.254036
$584$	0	0
$585$	2.12279e6	0.256458
$586$	0	0
$587$	9.06080e6	1.08535	0.542677	−	0.839942i	$-0.317411\pi$
$587$	9.06080e6	1.08535	0.542677	+	0.839942i	$0.317411\pi$
$588$	0	0
$589$	−1.16237e7	−1.38056
$590$	0	0
$591$	−1.62623e7	−1.91520
$592$	0	0
$593$	1.56414e7	1.82658	0.913290	−	0.407310i	$-0.133533\pi$
$593$	1.56414e7	1.82658	0.913290	+	0.407310i	$0.133533\pi$
$594$	0	0
$595$	−804887.	−0.0932058
$596$	0	0
$597$	−1.28382e7	−1.47425
$598$	0	0
$599$	1.40306e6	0.159775	0.0798877	−	0.996804i	$-0.474544\pi$
$599$	1.40306e6	0.159775	0.0798877	+	0.996804i	$0.474544\pi$
$600$	0	0
$601$	−1.02325e6	−0.115557	−0.0577786	−	0.998329i	$-0.518402\pi$
$601$	−1.02325e6	−0.115557	−0.0577786	+	0.998329i	$0.518402\pi$
$602$	0	0
$603$	519696.	0.0582044
$604$	0	0
$605$	−7.32956e6	−0.814122
$606$	0	0
$607$	−6.79951e6	−0.749042	−0.374521	−	0.927219i	$-0.622193\pi$
$607$	−6.79951e6	−0.749042	−0.374521	+	0.927219i	$0.622193\pi$
$608$	0	0
$609$	−2.94378e6	−0.321635
$610$	0	0
$611$	1.79177e6	0.194169
$612$	0	0
$613$	3.70194e6	0.397904	0.198952	−	0.980009i	$-0.436246\pi$
$613$	3.70194e6	0.397904	0.198952	+	0.980009i	$0.436246\pi$
$614$	0	0
$615$	−1.24366e7	−1.32591
$616$	0	0
$617$	1.14193e7	1.20761	0.603806	−	0.797131i	$-0.293650\pi$
$617$	1.14193e7	1.20761	0.603806	+	0.797131i	$0.293650\pi$
$618$	0	0
$619$	8.12725e6	0.852545	0.426272	−	0.904595i	$-0.359827\pi$
$619$	8.12725e6	0.852545	0.426272	+	0.904595i	$0.359827\pi$
$620$	0	0
$621$	111010.	0.0115513
$622$	0	0
$623$	−7.91410e6	−0.816924
$624$	0	0
$625$	−6.27919e6	−0.642989
$626$	0	0
$627$	−2.21741e6	−0.225256
$628$	0	0
$629$	−1.55210e6	−0.156421
$630$	0	0
$631$	−8.36884e6	−0.836743	−0.418371	−	0.908276i	$-0.637399\pi$
$631$	−8.36884e6	−0.836743	−0.418371	+	0.908276i	$0.637399\pi$
$632$	0	0
$633$	−1.65478e7	−1.64146
$634$	0	0
$635$	−3.43779e6	−0.338334
$636$	0	0
$637$	−1.29758e7	−1.26703
$638$	0	0
$639$	−790469.	−0.0765831
$640$	0	0
$641$	−1.04858e7	−1.00799	−0.503996	−	0.863706i	$-0.668137\pi$
$641$	−1.04858e7	−1.00799	−0.503996	+	0.863706i	$0.668137\pi$
$642$	0	0
$643$	−1.22946e7	−1.17270	−0.586349	−	0.810058i	$-0.699435\pi$
$643$	−1.22946e7	−1.17270	−0.586349	+	0.810058i	$0.699435\pi$
$644$	0	0
$645$	1.48219e6	0.140283
$646$	0	0
$647$	1.39774e7	1.31270	0.656349	−	0.754457i	$-0.272100\pi$
$647$	1.39774e7	1.31270	0.656349	+	0.754457i	$0.272100\pi$
$648$	0	0
$649$	1.99688e6	0.186097
$650$	0	0
$651$	8.21072e6	0.759327
$652$	0	0
$653$	−5.00675e6	−0.459487	−0.229743	−	0.973251i	$-0.573789\pi$
$653$	−5.00675e6	−0.459487	−0.229743	+	0.973251i	$0.573789\pi$
$654$	0	0
$655$	3.46683e6	0.315740
$656$	0	0
$657$	−512268.	−0.0463003
$658$	0	0
$659$	−5.50845e6	−0.494101	−0.247050	−	0.969003i	$-0.579461\pi$
$659$	−5.50845e6	−0.494101	−0.247050	+	0.969003i	$0.579461\pi$
$660$	0	0
$661$	−6.69305e6	−0.595827	−0.297914	−	0.954593i	$-0.596291\pi$
$661$	−6.69305e6	−0.595827	−0.297914	+	0.954593i	$0.596291\pi$
$662$	0	0
$663$	−4.49488e6	−0.397131
$664$	0	0
$665$	5.16450e6	0.452870
$666$	0	0
$667$	−84592.7	−0.00736238
$668$	0	0
$669$	1.12947e7	0.975684
$670$	0	0
$671$	−668211.	−0.0572938
$672$	0	0
$673$	−7.54961e6	−0.642521	−0.321260	−	0.946991i	$-0.604106\pi$
$673$	−7.54961e6	−0.642521	−0.321260	+	0.946991i	$0.604106\pi$
$674$	0	0
$675$	−2.95734e6	−0.249828
$676$	0	0
$677$	−1.51658e6	−0.127173	−0.0635863	−	0.997976i	$-0.520254\pi$
$677$	−1.51658e6	−0.127173	−0.0635863	+	0.997976i	$0.520254\pi$
$678$	0	0
$679$	9.92629e6	0.826252
$680$	0	0
$681$	−2.37665e7	−1.96380
$682$	0	0
$683$	2.30637e7	1.89181	0.945904	−	0.324448i	$-0.105178\pi$
$683$	2.30637e7	1.89181	0.945904	+	0.324448i	$0.105178\pi$
$684$	0	0
$685$	1.38066e7	1.12424
$686$	0	0
$687$	−1.19363e7	−0.964894
$688$	0	0
$689$	−2.71164e7	−2.17612
$690$	0	0
$691$	1.55625e7	1.23990	0.619948	−	0.784643i	$-0.287154\pi$
$691$	1.55625e7	1.23990	0.619948	+	0.784643i	$0.287154\pi$
$692$	0	0
$693$	231969.	0.0183483
$694$	0	0
$695$	1.78058e7	1.39829
$696$	0	0
$697$	3.89998e6	0.304075
$698$	0	0
$699$	−1.36950e7	−1.06015
$700$	0	0
$701$	1.48227e6	0.113929	0.0569643	−	0.998376i	$-0.481858\pi$
$701$	1.48227e6	0.113929	0.0569643	+	0.998376i	$0.481858\pi$
$702$	0	0
$703$	9.95896e6	0.760021
$704$	0	0
$705$	−1.35665e6	−0.102800
$706$	0	0
$707$	7.99814e6	0.601784
$708$	0	0
$709$	1.23095e7	0.919652	0.459826	−	0.888009i	$-0.347912\pi$
$709$	1.23095e7	0.919652	0.459826	+	0.888009i	$0.347912\pi$
$710$	0	0
$711$	2.21243e6	0.164133
$712$	0	0
$713$	235944.	0.0173814
$714$	0	0
$715$	−4.09034e6	−0.299223
$716$	0	0
$717$	1.37048e6	0.0995579
$718$	0	0
$719$	2.07641e7	1.49793	0.748963	−	0.662612i	$-0.230552\pi$
$719$	2.07641e7	1.49793	0.748963	+	0.662612i	$0.230552\pi$
$720$	0	0
$721$	3.31591e6	0.237555
$722$	0	0
$723$	−9.10002e6	−0.647435
$724$	0	0
$725$	2.25359e6	0.159232
$726$	0	0
$727$	7.15312e6	0.501949	0.250974	−	0.967994i	$-0.419249\pi$
$727$	7.15312e6	0.501949	0.250974	+	0.967994i	$0.419249\pi$
$728$	0	0
$729$	1.10625e7	0.770968
$730$	0	0
$731$	−464798.	−0.0321715
$732$	0	0
$733$	1.65135e7	1.13521	0.567607	−	0.823299i	$-0.307869\pi$
$733$	1.65135e7	1.13521	0.567607	+	0.823299i	$0.307869\pi$
$734$	0	0
$735$	9.82473e6	0.670814
$736$	0	0
$737$	−1.00139e6	−0.0679100
$738$	0	0
$739$	7.25575e6	0.488733	0.244366	−	0.969683i	$-0.421420\pi$
$739$	7.25575e6	0.488733	0.244366	+	0.969683i	$0.421420\pi$
$740$	0	0
$741$	2.88410e7	1.92959
$742$	0	0
$743$	1.59618e7	1.06074	0.530372	−	0.847765i	$-0.322052\pi$
$743$	1.59618e7	1.06074	0.530372	+	0.847765i	$0.322052\pi$
$744$	0	0
$745$	2.04632e7	1.35077
$746$	0	0
$747$	−1.22236e6	−0.0801491
$748$	0	0
$749$	−1.05665e7	−0.688217
$750$	0	0
$751$	−2.56869e7	−1.66192	−0.830962	−	0.556329i	$-0.812209\pi$
$751$	−2.56869e7	−1.66192	−0.830962	+	0.556329i	$0.812209\pi$
$752$	0	0
$753$	−3.86610e6	−0.248476
$754$	0	0
$755$	−818861.	−0.0522809
$756$	0	0
$757$	7.97201e6	0.505624	0.252812	−	0.967515i	$-0.418645\pi$
$757$	7.97201e6	0.505624	0.252812	+	0.967515i	$0.418645\pi$
$758$	0	0
$759$	45010.0	0.00283599
$760$	0	0
$761$	2.21020e7	1.38347	0.691736	−	0.722150i	$-0.256846\pi$
$761$	2.21020e7	1.38347	0.691736	+	0.722150i	$0.256846\pi$
$762$	0	0
$763$	−1.40937e7	−0.876425
$764$	0	0
$765$	504023.	0.0311385
$766$	0	0
$767$	−2.59727e7	−1.59415
$768$	0	0
$769$	−2.64726e7	−1.61428	−0.807142	−	0.590357i	$-0.798987\pi$
$769$	−2.64726e7	−1.61428	−0.807142	+	0.590357i	$0.798987\pi$
$770$	0	0
$771$	−2.40577e7	−1.45753
$772$	0	0
$773$	2.48033e7	1.49301	0.746503	−	0.665382i	$-0.231732\pi$
$773$	2.48033e7	1.49301	0.746503	+	0.665382i	$0.231732\pi$
$774$	0	0
$775$	−6.28564e6	−0.375920
$776$	0	0
$777$	−7.03477e6	−0.418021
$778$	0	0
$779$	−2.50239e7	−1.47745
$780$	0	0
$781$	1.52314e6	0.0893534
$782$	0	0
$783$	−8.76045e6	−0.510649
$784$	0	0
$785$	−6.62100e6	−0.383486
$786$	0	0
$787$	−8.09631e6	−0.465962	−0.232981	−	0.972481i	$-0.574848\pi$
$787$	−8.09631e6	−0.465962	−0.232981	+	0.972481i	$0.574848\pi$
$788$	0	0
$789$	1.02503e6	0.0586196
$790$	0	0
$791$	−2.25933e6	−0.128392
$792$	0	0
$793$	8.69118e6	0.490790
$794$	0	0
$795$	2.05313e7	1.15212
$796$	0	0
$797$	1.30377e7	0.727034	0.363517	−	0.931587i	$-0.381576\pi$
$797$	1.30377e7	0.727034	0.363517	+	0.931587i	$0.381576\pi$
$798$	0	0
$799$	425429.	0.0235755
$800$	0	0
$801$	4.95584e6	0.272920
$802$	0	0
$803$	987077.	0.0540210
$804$	0	0
$805$	−104831.	−0.00570166
$806$	0	0
$807$	3.13329e7	1.69362
$808$	0	0
$809$	−3.29258e6	−0.176874	−0.0884371	−	0.996082i	$-0.528187\pi$
$809$	−3.29258e6	−0.176874	−0.0884371	+	0.996082i	$0.528187\pi$
$810$	0	0
$811$	−8.58147e6	−0.458152	−0.229076	−	0.973409i	$-0.573570\pi$
$811$	−8.58147e6	−0.458152	−0.229076	+	0.973409i	$0.573570\pi$
$812$	0	0
$813$	−2.65541e7	−1.40898
$814$	0	0
$815$	2.63991e7	1.39218
$816$	0	0
$817$	2.98234e6	0.156315
$818$	0	0
$819$	−3.01713e6	−0.157175
$820$	0	0
$821$	−7.34065e6	−0.380081	−0.190041	−	0.981776i	$-0.560862\pi$
$821$	−7.34065e6	−0.380081	−0.190041	+	0.981776i	$0.560862\pi$
$822$	0	0
$823$	−3.32707e7	−1.71223	−0.856116	−	0.516783i	$-0.827129\pi$
$823$	−3.32707e7	−1.71223	−0.856116	+	0.516783i	$0.827129\pi$
$824$	0	0
$825$	−1.19908e6	−0.0613359
$826$	0	0
$827$	3.77399e7	1.91883	0.959417	−	0.281992i	$-0.0909951\pi$
$827$	3.77399e7	1.91883	0.959417	+	0.281992i	$0.0909951\pi$
$828$	0	0
$829$	−2.81993e7	−1.42512	−0.712560	−	0.701611i	$-0.752465\pi$
$829$	−2.81993e7	−1.42512	−0.712560	+	0.701611i	$0.752465\pi$
$830$	0	0
$831$	−5.43008e6	−0.272775
$832$	0	0
$833$	−3.08092e6	−0.153839
$834$	0	0
$835$	1.13535e7	0.563528
$836$	0	0
$837$	2.44344e7	1.20556
$838$	0	0
$839$	9.00386e6	0.441594	0.220797	−	0.975320i	$-0.429134\pi$
$839$	9.00386e6	0.441594	0.220797	+	0.975320i	$0.429134\pi$
$840$	0	0
$841$	−1.38354e7	−0.674531
$842$	0	0
$843$	8.47102e6	0.410551
$844$	0	0
$845$	3.55788e7	1.71415
$846$	0	0
$847$	1.04176e7	0.498951
$848$	0	0
$849$	5.87152e6	0.279564
$850$	0	0
$851$	−202152.	−0.00956870
$852$	0	0
$853$	2.76660e7	1.30189	0.650944	−	0.759126i	$-0.274373\pi$
$853$	2.76660e7	1.30189	0.650944	+	0.759126i	$0.274373\pi$
$854$	0	0
$855$	−3.23403e6	−0.151296
$856$	0	0
$857$	−3.77780e7	−1.75706	−0.878531	−	0.477686i	$-0.841476\pi$
$857$	−3.77780e7	−1.75706	−0.878531	+	0.477686i	$0.841476\pi$
$858$	0	0
$859$	−2.02066e7	−0.934352	−0.467176	−	0.884164i	$-0.654729\pi$
$859$	−2.02066e7	−0.934352	−0.467176	+	0.884164i	$0.654729\pi$
$860$	0	0
$861$	1.76763e7	0.812613
$862$	0	0
$863$	992399.	0.0453586	0.0226793	−	0.999743i	$-0.492780\pi$
$863$	992399.	0.0453586	0.0226793	+	0.999743i	$0.492780\pi$
$864$	0	0
$865$	3.77637e6	0.171607
$866$	0	0
$867$	2.29130e7	1.03522
$868$	0	0
$869$	−4.26308e6	−0.191502
$870$	0	0
$871$	1.30247e7	0.581732
$872$	0	0
$873$	−6.21588e6	−0.276037
$874$	0	0
$875$	1.27987e7	0.565126
$876$	0	0
$877$	3.48750e7	1.53114	0.765572	−	0.643351i	$-0.222456\pi$
$877$	3.48750e7	1.53114	0.765572	+	0.643351i	$0.222456\pi$
$878$	0	0
$879$	2.71702e7	1.18610
$880$	0	0
$881$	2.68921e7	1.16731	0.583654	−	0.812002i	$-0.301622\pi$
$881$	2.68921e7	1.16731	0.583654	+	0.812002i	$0.301622\pi$
$882$	0	0
$883$	2.40336e6	0.103733	0.0518666	−	0.998654i	$-0.483483\pi$
$883$	2.40336e6	0.103733	0.0518666	+	0.998654i	$0.483483\pi$
$884$	0	0
$885$	1.96653e7	0.844002
$886$	0	0
$887$	2.56525e7	1.09477	0.547383	−	0.836882i	$-0.315624\pi$
$887$	2.56525e7	1.09477	0.547383	+	0.836882i	$0.315624\pi$
$888$	0	0
$889$	4.88616e6	0.207355
$890$	0	0
$891$	5.49683e6	0.231963
$892$	0	0
$893$	−2.72973e6	−0.114549
$894$	0	0
$895$	4.54885e6	0.189821
$896$	0	0
$897$	−585429.	−0.0242937
$898$	0	0
$899$	−1.86198e7	−0.768379
$900$	0	0
$901$	−6.43837e6	−0.264219
$902$	0	0
$903$	−2.10665e6	−0.0859753
$904$	0	0
$905$	−1.04141e7	−0.422669
$906$	0	0
$907$	2.68686e7	1.08450	0.542248	−	0.840219i	$-0.317574\pi$
$907$	2.68686e7	1.08450	0.542248	+	0.840219i	$0.317574\pi$
$908$	0	0
$909$	−5.00847e6	−0.201046
$910$	0	0
$911$	−1.69356e7	−0.676091	−0.338045	−	0.941130i	$-0.609766\pi$
$911$	−1.69356e7	−0.676091	−0.338045	+	0.941130i	$0.609766\pi$
$912$	0	0
$913$	2.35534e6	0.0935141
$914$	0	0
$915$	−6.58058e6	−0.259843
$916$	0	0
$917$	−4.92744e6	−0.193508
$918$	0	0
$919$	2.20166e7	0.859927	0.429964	−	0.902846i	$-0.358526\pi$
$919$	2.20166e7	0.859927	0.429964	+	0.902846i	$0.358526\pi$
$920$	0	0
$921$	4.22308e7	1.64051
$922$	0	0
$923$	−1.98109e7	−0.765420
$924$	0	0
$925$	5.38540e6	0.206949
$926$	0	0
$927$	−2.07643e6	−0.0793631
$928$	0	0
$929$	−8.11263e6	−0.308406	−0.154203	−	0.988039i	$-0.549281\pi$
$929$	−8.11263e6	−0.308406	−0.154203	+	0.988039i	$0.549281\pi$
$930$	0	0
$931$	1.97685e7	0.747478
$932$	0	0
$933$	4.73493e7	1.78078
$934$	0	0
$935$	−971190.	−0.0363308
$936$	0	0
$937$	−1.80064e7	−0.670004	−0.335002	−	0.942217i	$-0.608737\pi$
$937$	−1.80064e7	−0.670004	−0.335002	+	0.942217i	$0.608737\pi$
$938$	0	0
$939$	1.73250e6	0.0641224
$940$	0	0
$941$	2.90643e7	1.07000	0.535002	−	0.844851i	$-0.320311\pi$
$941$	2.90643e7	1.07000	0.535002	+	0.844851i	$0.320311\pi$
$942$	0	0
$943$	507947.	0.0186011
$944$	0	0
$945$	−1.08564e7	−0.395463
$946$	0	0
$947$	1.97188e7	0.714504	0.357252	−	0.934008i	$-0.383714\pi$
$947$	1.97188e7	0.714504	0.357252	+	0.934008i	$0.383714\pi$
$948$	0	0
$949$	−1.28386e7	−0.462755
$950$	0	0
$951$	4.44895e6	0.159517
$952$	0	0
$953$	−5.34568e6	−0.190665	−0.0953325	−	0.995445i	$-0.530391\pi$
$953$	−5.34568e6	−0.190665	−0.0953325	+	0.995445i	$0.530391\pi$
$954$	0	0
$955$	−1.56026e7	−0.553592
$956$	0	0
$957$	−3.55202e6	−0.125370
$958$	0	0
$959$	−1.96235e7	−0.689016
$960$	0	0
$961$	2.33046e7	0.814017
$962$	0	0
$963$	6.61676e6	0.229922
$964$	0	0
$965$	2.73268e7	0.944648
$966$	0	0
$967$	605934.	0.0208381	0.0104191	−	0.999946i	$-0.496683\pi$
$967$	605934.	0.0208381	0.0104191	+	0.999946i	$0.496683\pi$
$968$	0	0
$969$	6.84787e6	0.234286
$970$	0	0
$971$	−2.31358e7	−0.787475	−0.393737	−	0.919223i	$-0.628818\pi$
$971$	−2.31358e7	−0.787475	−0.393737	+	0.919223i	$0.628818\pi$
$972$	0	0
$973$	−2.53075e7	−0.856972
$974$	0	0
$975$	1.55961e7	0.525416
$976$	0	0
$977$	2.22967e6	0.0747315	0.0373658	−	0.999302i	$-0.488103\pi$
$977$	2.22967e6	0.0747315	0.0373658	+	0.999302i	$0.488103\pi$
$978$	0	0
$979$	−9.54928e6	−0.318430
$980$	0	0
$981$	8.82555e6	0.292799
$982$	0	0
$983$	2.21182e6	0.0730071	0.0365036	−	0.999334i	$-0.488378\pi$
$983$	2.21182e6	0.0730071	0.0365036	+	0.999334i	$0.488378\pi$
$984$	0	0
$985$	4.57019e7	1.50087
$986$	0	0
$987$	1.92822e6	0.0630033
$988$	0	0
$989$	−60536.9	−0.00196802
$990$	0	0
$991$	1.66837e7	0.539645	0.269823	−	0.962910i	$-0.413035\pi$
$991$	1.66837e7	0.539645	0.269823	+	0.962910i	$0.413035\pi$
$992$	0	0
$993$	−3.91332e7	−1.25943
$994$	0	0
$995$	3.60792e7	1.15531
$996$	0	0
$997$	1.01549e7	0.323547	0.161774	−	0.986828i	$-0.448279\pi$
$997$	1.01549e7	0.323547	0.161774	+	0.986828i	$0.448279\pi$
$998$	0	0
$999$	−2.09349e7	−0.663677

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.6.a.h.1.4		10
4.3	odd	2		43.6.a.b.1.4	✓	10
12.11	even	2		387.6.a.e.1.7		10
20.19	odd	2		1075.6.a.b.1.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.4	✓	10	4.3	odd	2
387.6.a.e.1.7		10	12.11	even	2
688.6.a.h.1.4		10	1.1	even	1	trivial
1075.6.a.b.1.7		10	20.19	odd	2

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:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	688.a (trivial)

\(f(q)\)	\(=\)	\(q-16.8892 q^{3} +47.4635 q^{5} -67.4603 q^{7} +42.2439 q^{9} +O(q^{10})\)	\(q-16.8892 q^{3} +47.4635 q^{5} -67.4603 q^{7} +42.2439 q^{9} -81.3987 q^{11} +1058.72 q^{13} -801.619 q^{15} +251.378 q^{17} -1612.95 q^{19} +1139.35 q^{21} +32.7403 q^{23} -872.217 q^{25} +3390.60 q^{27} -2583.75 q^{29} +7206.51 q^{31} +1374.76 q^{33} -3201.90 q^{35} -6174.39 q^{37} -17881.0 q^{39} +15514.4 q^{41} -1849.00 q^{43} +2005.04 q^{45} +1692.39 q^{47} -12256.1 q^{49} -4245.56 q^{51} -25612.3 q^{53} -3863.47 q^{55} +27241.3 q^{57} -24532.0 q^{59} +8209.11 q^{61} -2849.78 q^{63} +50250.7 q^{65} +12302.3 q^{67} -552.957 q^{69} -18712.0 q^{71} -12126.5 q^{73} +14731.0 q^{75} +5491.18 q^{77} +52372.8 q^{79} -67529.7 q^{81} -28935.9 q^{83} +11931.3 q^{85} +43637.3 q^{87} +117315. q^{89} -71421.8 q^{91} -121712. q^{93} -76556.1 q^{95} -147143. q^{97} -3438.60 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9}+O(q^{10}) \) 10 * q - 28 * q^3 + 138 * q^5 - 60 * q^7 + 1356 * q^9	\( 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9} - 745 q^{11} + 1917 q^{13} - 1688 q^{15} + 4017 q^{17} + 2404 q^{19} - 228 q^{21} - 1733 q^{23} + 7120 q^{25} + 2324 q^{27} + 6996 q^{29} + 4899 q^{31} - 15734 q^{33} - 7084 q^{35} + 1466 q^{37} + 26542 q^{39} + 10297 q^{41} - 18490 q^{43} + 73822 q^{45} - 48592 q^{47} + 29458 q^{49} - 92972 q^{51} + 127165 q^{53} - 106672 q^{55} + 34060 q^{57} - 99372 q^{59} + 17408 q^{61} - 2244 q^{63} + 54484 q^{65} + 2021 q^{67} + 1654 q^{69} - 11286 q^{71} + 49892 q^{73} + 44662 q^{75} + 98144 q^{77} + 91524 q^{79} - 26450 q^{81} + 105203 q^{83} - 87212 q^{85} - 181200 q^{87} - 62682 q^{89} + 295304 q^{91} - 238430 q^{93} + 305340 q^{95} + 108383 q^{97} + 270499 q^{99}+O(q^{100}) \) 10 * q - 28 * q^3 + 138 * q^5 - 60 * q^7 + 1356 * q^9 - 745 * q^11 + 1917 * q^13 - 1688 * q^15 + 4017 * q^17 + 2404 * q^19 - 228 * q^21 - 1733 * q^23 + 7120 * q^25 + 2324 * q^27 + 6996 * q^29 + 4899 * q^31 - 15734 * q^33 - 7084 * q^35 + 1466 * q^37 + 26542 * q^39 + 10297 * q^41 - 18490 * q^43 + 73822 * q^45 - 48592 * q^47 + 29458 * q^49 - 92972 * q^51 + 127165 * q^53 - 106672 * q^55 + 34060 * q^57 - 99372 * q^59 + 17408 * q^61 - 2244 * q^63 + 54484 * q^65 + 2021 * q^67 + 1654 * q^69 - 11286 * q^71 + 49892 * q^73 + 44662 * q^75 + 98144 * q^77 + 91524 * q^79 - 26450 * q^81 + 105203 * q^83 - 87212 * q^85 - 181200 * q^87 - 62682 * q^89 + 295304 * q^91 - 238430 * q^93 + 305340 * q^95 + 108383 * q^97 + 270499 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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