LMFDB - Embedded newform 688.6.a.h.1.7

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			1.7
Root			$-8.57770$ 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)

$f(q)$	$=$	$q+7.84343 q^{3} +28.1028 q^{5} -195.604 q^{7} -181.481 q^{9} +O(q^{10})$	$q+7.84343 q^{3} +28.1028 q^{5} -195.604 q^{7} -181.481 q^{9} -72.8476 q^{11} +301.666 q^{13} +220.422 q^{15} -1207.20 q^{17} +2350.21 q^{19} -1534.21 q^{21} -516.070 q^{23} -2335.23 q^{25} -3329.38 q^{27} +1531.55 q^{29} -1126.13 q^{31} -571.375 q^{33} -5497.02 q^{35} -9339.18 q^{37} +2366.10 q^{39} +19704.2 q^{41} -1849.00 q^{43} -5100.11 q^{45} +13797.6 q^{47} +21453.9 q^{49} -9468.56 q^{51} -3351.03 q^{53} -2047.22 q^{55} +18433.7 q^{57} -2511.18 q^{59} +49249.3 q^{61} +35498.3 q^{63} +8477.66 q^{65} -9116.54 q^{67} -4047.76 q^{69} -43397.3 q^{71} +80067.4 q^{73} -18316.2 q^{75} +14249.3 q^{77} +65991.7 q^{79} +17986.0 q^{81} +76880.9 q^{83} -33925.6 q^{85} +12012.6 q^{87} -75722.6 q^{89} -59007.1 q^{91} -8832.73 q^{93} +66047.5 q^{95} -67921.7 q^{97} +13220.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	7.84343	0.503156	0.251578	−	0.967837i	$-0.419050\pi$
$3$	7.84343	0.503156	0.251578	+	0.967837i	$0.419050\pi$
$4$	0	0
$5$	28.1028	0.502718	0.251359	−	0.967894i	$-0.419122\pi$
$5$	28.1028	0.502718	0.251359	+	0.967894i	$0.419122\pi$
$6$	0	0
$7$	−195.604	−1.50880	−0.754401	−	0.656413i	$-0.772073\pi$
$7$	−195.604	−1.50880	−0.754401	+	0.656413i	$0.772073\pi$
$8$	0	0
$9$	−181.481	−0.746834
$10$	0	0
$11$	−72.8476	−0.181524	−0.0907619	−	0.995873i	$-0.528930\pi$
$11$	−72.8476	−0.181524	−0.0907619	+	0.995873i	$0.528930\pi$
$12$	0	0
$13$	301.666	0.495072	0.247536	−	0.968879i	$-0.420379\pi$
$13$	301.666	0.495072	0.247536	+	0.968879i	$0.420379\pi$
$14$	0	0
$15$	220.422	0.252946
$16$	0	0
$17$	−1207.20	−1.01311	−0.506554	−	0.862208i	$-0.669081\pi$
$17$	−1207.20	−1.01311	−0.506554	+	0.862208i	$0.669081\pi$
$18$	0	0
$19$	2350.21	1.49356	0.746780	−	0.665071i	$-0.231599\pi$
$19$	2350.21	1.49356	0.746780	+	0.665071i	$0.231599\pi$
$20$	0	0
$21$	−1534.21	−0.759164
$22$	0	0
$23$	−516.070	−0.203418	−0.101709	−	0.994814i	$-0.532431\pi$
$23$	−516.070	−0.203418	−0.101709	+	0.994814i	$0.532431\pi$
$24$	0	0
$25$	−2335.23	−0.747274
$26$	0	0
$27$	−3329.38	−0.878930
$28$	0	0
$29$	1531.55	0.338171	0.169085	−	0.985601i	$-0.445919\pi$
$29$	1531.55	0.338171	0.169085	+	0.985601i	$0.445919\pi$
$30$	0	0
$31$	−1126.13	−0.210467	−0.105234	−	0.994448i	$-0.533559\pi$
$31$	−1126.13	−0.210467	−0.105234	+	0.994448i	$0.533559\pi$
$32$	0	0
$33$	−571.375	−0.0913349
$34$	0	0
$35$	−5497.02	−0.758503
$36$	0	0
$37$	−9339.18	−1.12151	−0.560756	−	0.827981i	$-0.689490\pi$
$37$	−9339.18	−1.12151	−0.560756	+	0.827981i	$0.689490\pi$
$38$	0	0
$39$	2366.10	0.249098
$40$	0	0
$41$	19704.2	1.83062	0.915310	−	0.402750i	$-0.131946\pi$
$41$	19704.2	1.83062	0.915310	+	0.402750i	$0.131946\pi$
$42$	0	0
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	0	0
$45$	−5100.11	−0.375447
$46$	0	0
$47$	13797.6	0.911088	0.455544	−	0.890213i	$-0.349445\pi$
$47$	13797.6	0.911088	0.455544	+	0.890213i	$0.349445\pi$
$48$	0	0
$49$	21453.9	1.27649
$50$	0	0
$51$	−9468.56	−0.509752
$52$	0	0
$53$	−3351.03	−0.163866	−0.0819330	−	0.996638i	$-0.526109\pi$
$53$	−3351.03	−0.163866	−0.0819330	+	0.996638i	$0.526109\pi$
$54$	0	0
$55$	−2047.22	−0.0912554
$56$	0	0
$57$	18433.7	0.751494
$58$	0	0
$59$	−2511.18	−0.0939177	−0.0469588	−	0.998897i	$-0.514953\pi$
$59$	−2511.18	−0.0939177	−0.0469588	+	0.998897i	$0.514953\pi$
$60$	0	0
$61$	49249.3	1.69463	0.847316	−	0.531088i	$-0.178217\pi$
$61$	49249.3	1.69463	0.847316	+	0.531088i	$0.178217\pi$
$62$	0	0
$63$	35498.3	1.12682
$64$	0	0
$65$	8477.66	0.248882
$66$	0	0
$67$	−9116.54	−0.248109	−0.124055	−	0.992275i	$-0.539590\pi$
$67$	−9116.54	−0.248109	−0.124055	+	0.992275i	$0.539590\pi$
$68$	0	0
$69$	−4047.76	−0.102351
$70$	0	0
$71$	−43397.3	−1.02168	−0.510842	−	0.859675i	$-0.670666\pi$
$71$	−43397.3	−1.02168	−0.510842	+	0.859675i	$0.670666\pi$
$72$	0	0
$73$	80067.4	1.75852	0.879262	−	0.476338i	$-0.158036\pi$
$73$	80067.4	1.75852	0.879262	+	0.476338i	$0.158036\pi$
$74$	0	0
$75$	−18316.2	−0.375996
$76$	0	0
$77$	14249.3	0.273884
$78$	0	0
$79$	65991.7	1.18966	0.594828	−	0.803853i	$-0.297220\pi$
$79$	65991.7	1.18966	0.594828	+	0.803853i	$0.297220\pi$
$80$	0	0
$81$	17986.0	0.304594
$82$	0	0
$83$	76880.9	1.22496	0.612482	−	0.790484i	$-0.290171\pi$
$83$	76880.9	1.22496	0.612482	+	0.790484i	$0.290171\pi$
$84$	0	0
$85$	−33925.6	−0.509308
$86$	0	0
$87$	12012.6	0.170153
$88$	0	0
$89$	−75722.6	−1.01333	−0.506665	−	0.862143i	$-0.669122\pi$
$89$	−75722.6	−1.01333	−0.506665	+	0.862143i	$0.669122\pi$
$90$	0	0
$91$	−59007.1	−0.746965
$92$	0	0
$93$	−8832.73	−0.105898
$94$	0	0
$95$	66047.5	0.750840
$96$	0	0
$97$	−67921.7	−0.732958	−0.366479	−	0.930426i	$-0.619437\pi$
$97$	−67921.7	−0.732958	−0.366479	+	0.930426i	$0.619437\pi$
$98$	0	0
$99$	13220.4	0.135568
$100$	0	0
$101$	130824.	1.27610	0.638051	−	0.769994i	$-0.279741\pi$
$101$	130824.	1.27610	0.638051	+	0.769994i	$0.279741\pi$
$102$	0	0
$103$	−41945.9	−0.389580	−0.194790	−	0.980845i	$-0.562403\pi$
$103$	−41945.9	−0.389580	−0.194790	+	0.980845i	$0.562403\pi$
$104$	0	0
$105$	−43115.5	−0.381645
$106$	0	0
$107$	5277.45	0.0445620	0.0222810	−	0.999752i	$-0.492907\pi$
$107$	5277.45	0.0445620	0.0222810	+	0.999752i	$0.492907\pi$
$108$	0	0
$109$	−138680.	−1.11802	−0.559008	−	0.829163i	$-0.688818\pi$
$109$	−138680.	−1.11802	−0.559008	+	0.829163i	$0.688818\pi$
$110$	0	0
$111$	−73251.2	−0.564296
$112$	0	0
$113$	251325.	1.85157	0.925783	−	0.378055i	$-0.123407\pi$
$113$	251325.	1.85157	0.925783	+	0.378055i	$0.123407\pi$
$114$	0	0
$115$	−14503.0	−0.102262
$116$	0	0
$117$	−54746.5	−0.369736
$118$	0	0
$119$	236132.	1.52858
$120$	0	0
$121$	−155744.	−0.967049
$122$	0	0
$123$	154548.	0.921088
$124$	0	0
$125$	−153448.	−0.878387
$126$	0	0
$127$	175695.	0.966606	0.483303	−	0.875453i	$-0.339437\pi$
$127$	175695.	0.966606	0.483303	+	0.875453i	$0.339437\pi$
$128$	0	0
$129$	−14502.5	−0.0767306
$130$	0	0
$131$	−213758.	−1.08829	−0.544144	−	0.838992i	$-0.683146\pi$
$131$	−213758.	−1.08829	−0.544144	+	0.838992i	$0.683146\pi$
$132$	0	0
$133$	−459711.	−2.25349
$134$	0	0
$135$	−93565.0	−0.441854
$136$	0	0
$137$	126026.	0.573664	0.286832	−	0.957981i	$-0.407398\pi$
$137$	126026.	0.573664	0.286832	+	0.957981i	$0.407398\pi$
$138$	0	0
$139$	−181105.	−0.795049	−0.397524	−	0.917592i	$-0.630131\pi$
$139$	−181105.	−0.795049	−0.397524	+	0.917592i	$0.630131\pi$
$140$	0	0
$141$	108221.	0.458419
$142$	0	0
$143$	−21975.7	−0.0898673
$144$	0	0
$145$	43040.8	0.170004
$146$	0	0
$147$	168272.	0.642272
$148$	0	0
$149$	187084.	0.690354	0.345177	−	0.938538i	$-0.387819\pi$
$149$	187084.	0.690354	0.345177	+	0.938538i	$0.387819\pi$
$150$	0	0
$151$	396158.	1.41392	0.706962	−	0.707252i	$-0.250065\pi$
$151$	396158.	1.41392	0.706962	+	0.707252i	$0.250065\pi$
$152$	0	0
$153$	219083.	0.756623
$154$	0	0
$155$	−31647.4	−0.105806
$156$	0	0
$157$	504286.	1.63278	0.816390	−	0.577501i	$-0.195972\pi$
$157$	504286.	1.63278	0.816390	+	0.577501i	$0.195972\pi$
$158$	0	0
$159$	−26283.6	−0.0824502
$160$	0	0
$161$	100945.	0.306917
$162$	0	0
$163$	466039.	1.37389	0.686946	−	0.726708i	$-0.258951\pi$
$163$	466039.	1.37389	0.686946	+	0.726708i	$0.258951\pi$
$164$	0	0
$165$	−16057.2	−0.0459157
$166$	0	0
$167$	142267.	0.394742	0.197371	−	0.980329i	$-0.436760\pi$
$167$	142267.	0.394742	0.197371	+	0.980329i	$0.436760\pi$
$168$	0	0
$169$	−280291.	−0.754904
$170$	0	0
$171$	−426518.	−1.11544
$172$	0	0
$173$	526184.	1.33666	0.668332	−	0.743863i	$-0.267009\pi$
$173$	526184.	1.33666	0.668332	+	0.743863i	$0.267009\pi$
$174$	0	0
$175$	456781.	1.12749
$176$	0	0
$177$	−19696.2	−0.0472553
$178$	0	0
$179$	−692470.	−1.61536	−0.807678	−	0.589624i	$-0.799276\pi$
$179$	−692470.	−1.61536	−0.807678	+	0.589624i	$0.799276\pi$
$180$	0	0
$181$	286664.	0.650393	0.325197	−	0.945646i	$-0.394569\pi$
$181$	286664.	0.650393	0.325197	+	0.945646i	$0.394569\pi$
$182$	0	0
$183$	386284.	0.852665
$184$	0	0
$185$	−262457.	−0.563805
$186$	0	0
$187$	87941.4	0.183903
$188$	0	0
$189$	651241.	1.32613
$190$	0	0
$191$	692472.	1.37347	0.686734	−	0.726909i	$-0.259044\pi$
$191$	692472.	1.37347	0.686734	+	0.726909i	$0.259044\pi$
$192$	0	0
$193$	855425.	1.65306	0.826530	−	0.562893i	$-0.190312\pi$
$193$	855425.	1.65306	0.826530	+	0.562893i	$0.190312\pi$
$194$	0	0
$195$	66493.9	0.125226
$196$	0	0
$197$	−749649.	−1.37623	−0.688117	−	0.725600i	$-0.741562\pi$
$197$	−749649.	−1.37623	−0.688117	+	0.725600i	$0.741562\pi$
$198$	0	0
$199$	−990082.	−1.77230	−0.886152	−	0.463394i	$-0.846631\pi$
$199$	−990082.	−1.77230	−0.886152	+	0.463394i	$0.846631\pi$
$200$	0	0
$201$	−71504.9	−0.124838
$202$	0	0
$203$	−299577.	−0.510233
$204$	0	0
$205$	553742.	0.920286
$206$	0	0
$207$	93656.7	0.151919
$208$	0	0
$209$	−171207.	−0.271117
$210$	0	0
$211$	274555.	0.424545	0.212272	−	0.977211i	$-0.431914\pi$
$211$	274555.	0.424545	0.212272	+	0.977211i	$0.431914\pi$
$212$	0	0
$213$	−340384.	−0.514067
$214$	0	0
$215$	−51962.1	−0.0766638
$216$	0	0
$217$	220276.	0.317554
$218$	0	0
$219$	628003.	0.884813
$220$	0	0
$221$	−364170.	−0.501561
$222$	0	0
$223$	568751.	0.765878	0.382939	−	0.923774i	$-0.374912\pi$
$223$	568751.	0.765878	0.382939	+	0.923774i	$0.374912\pi$
$224$	0	0
$225$	423799.	0.558090
$226$	0	0
$227$	−607473.	−0.782461	−0.391230	−	0.920293i	$-0.627950\pi$
$227$	−607473.	−0.782461	−0.391230	+	0.920293i	$0.627950\pi$
$228$	0	0
$229$	104213.	0.131320	0.0656600	−	0.997842i	$-0.479085\pi$
$229$	104213.	0.131320	0.0656600	+	0.997842i	$0.479085\pi$
$230$	0	0
$231$	111763.	0.137806
$232$	0	0
$233$	1.28050e6	1.54521	0.772606	−	0.634885i	$-0.218953\pi$
$233$	1.28050e6	1.54521	0.772606	+	0.634885i	$0.218953\pi$
$234$	0	0
$235$	387752.	0.458020
$236$	0	0
$237$	517601.	0.598583
$238$	0	0
$239$	−221000.	−0.250264	−0.125132	−	0.992140i	$-0.539935\pi$
$239$	−221000.	−0.250264	−0.125132	+	0.992140i	$0.539935\pi$
$240$	0	0
$241$	−480522.	−0.532931	−0.266465	−	0.963845i	$-0.585856\pi$
$241$	−480522.	−0.532931	−0.266465	+	0.963845i	$0.585856\pi$
$242$	0	0
$243$	950112.	1.03219
$244$	0	0
$245$	602915.	0.641713
$246$	0	0
$247$	708979.	0.739420
$248$	0	0
$249$	603010.	0.616348
$250$	0	0
$251$	−722732.	−0.724091	−0.362045	−	0.932161i	$-0.617921\pi$
$251$	−722732.	−0.724091	−0.362045	+	0.932161i	$0.617921\pi$
$252$	0	0
$253$	37594.5	0.0369252
$254$	0	0
$255$	−266093.	−0.256261
$256$	0	0
$257$	−1.34492e6	−1.27018	−0.635089	−	0.772439i	$-0.719037\pi$
$257$	−1.34492e6	−1.27018	−0.635089	+	0.772439i	$0.719037\pi$
$258$	0	0
$259$	1.82678e6	1.69214
$260$	0	0
$261$	−277946.	−0.252557
$262$	0	0
$263$	18066.1	0.0161055	0.00805275	−	0.999968i	$-0.497437\pi$
$263$	18066.1	0.0161055	0.00805275	+	0.999968i	$0.497437\pi$
$264$	0	0
$265$	−94173.4	−0.0823784
$266$	0	0
$267$	−593925.	−0.509863
$268$	0	0
$269$	−549502.	−0.463008	−0.231504	−	0.972834i	$-0.574365\pi$
$269$	−549502.	−0.463008	−0.231504	+	0.972834i	$0.574365\pi$
$270$	0	0
$271$	−13732.4	−0.0113586	−0.00567929	−	0.999984i	$-0.501808\pi$
$271$	−13732.4	−0.0113586	−0.00567929	+	0.999984i	$0.501808\pi$
$272$	0	0
$273$	−462818.	−0.375840
$274$	0	0
$275$	170116.	0.135648
$276$	0	0
$277$	−501657.	−0.392833	−0.196416	−	0.980521i	$-0.562930\pi$
$277$	−501657.	−0.392833	−0.196416	+	0.980521i	$0.562930\pi$
$278$	0	0
$279$	204371.	0.157184
$280$	0	0
$281$	605657.	0.457574	0.228787	−	0.973477i	$-0.426524\pi$
$281$	605657.	0.457574	0.228787	+	0.973477i	$0.426524\pi$
$282$	0	0
$283$	1.85484e6	1.37671	0.688353	−	0.725376i	$-0.258334\pi$
$283$	1.85484e6	1.37671	0.688353	+	0.725376i	$0.258334\pi$
$284$	0	0
$285$	518039.	0.377790
$286$	0	0
$287$	−3.85421e6	−2.76204
$288$	0	0
$289$	37467.4	0.0263882
$290$	0	0
$291$	−532739.	−0.368792
$292$	0	0
$293$	950968.	0.647138	0.323569	−	0.946205i	$-0.395117\pi$
$293$	950968.	0.647138	0.323569	+	0.946205i	$0.395117\pi$
$294$	0	0
$295$	−70571.1	−0.0472141
$296$	0	0
$297$	242538.	0.159547
$298$	0	0
$299$	−155681.	−0.100706
$300$	0	0
$301$	361672.	0.230090
$302$	0	0
$303$	1.02611e6	0.642079
$304$	0	0
$305$	1.38404e6	0.851923
$306$	0	0
$307$	736219.	0.445822	0.222911	−	0.974839i	$-0.428444\pi$
$307$	736219.	0.445822	0.222911	+	0.974839i	$0.428444\pi$
$308$	0	0
$309$	−329000.	−0.196020
$310$	0	0
$311$	1.02220e6	0.599286	0.299643	−	0.954051i	$-0.403133\pi$
$311$	1.02220e6	0.599286	0.299643	+	0.954051i	$0.403133\pi$
$312$	0	0
$313$	−60474.7	−0.0348909	−0.0174455	−	0.999848i	$-0.505553\pi$
$313$	−60474.7	−0.0348909	−0.0174455	+	0.999848i	$0.505553\pi$
$314$	0	0
$315$	997602.	0.566475
$316$	0	0
$317$	842265.	0.470761	0.235381	−	0.971903i	$-0.424366\pi$
$317$	842265.	0.470761	0.235381	+	0.971903i	$0.424366\pi$
$318$	0	0
$319$	−111570.	−0.0613860
$320$	0	0
$321$	41393.3	0.0224216
$322$	0	0
$323$	−2.83717e6	−1.51314
$324$	0	0
$325$	−704460.	−0.369954
$326$	0	0
$327$	−1.08773e6	−0.562536
$328$	0	0
$329$	−2.69887e6	−1.37465
$330$	0	0
$331$	3.50942e6	1.76062	0.880308	−	0.474402i	$-0.157336\pi$
$331$	3.50942e6	1.76062	0.880308	+	0.474402i	$0.157336\pi$
$332$	0	0
$333$	1.69488e6	0.837584
$334$	0	0
$335$	−256200.	−0.124729
$336$	0	0
$337$	−1.49987e6	−0.719414	−0.359707	−	0.933065i	$-0.617123\pi$
$337$	−1.49987e6	−0.719414	−0.359707	+	0.933065i	$0.617123\pi$
$338$	0	0
$339$	1.97125e6	0.931627
$340$	0	0
$341$	82036.0	0.0382048
$342$	0	0
$343$	−908952.	−0.417163
$344$	0	0
$345$	−113753.	−0.0514537
$346$	0	0
$347$	−2.16958e6	−0.967280	−0.483640	−	0.875267i	$-0.660686\pi$
$347$	−2.16958e6	−0.967280	−0.483640	+	0.875267i	$0.660686\pi$
$348$	0	0
$349$	393576.	0.172968	0.0864838	−	0.996253i	$-0.472437\pi$
$349$	393576.	0.172968	0.0864838	+	0.996253i	$0.472437\pi$
$350$	0	0
$351$	−1.00436e6	−0.435133
$352$	0	0
$353$	272744.	0.116498	0.0582491	−	0.998302i	$-0.481448\pi$
$353$	272744.	0.116498	0.0582491	+	0.998302i	$0.481448\pi$
$354$	0	0
$355$	−1.21959e6	−0.513619
$356$	0	0
$357$	1.85209e6	0.769115
$358$	0	0
$359$	3.96827e6	1.62504	0.812521	−	0.582932i	$-0.198094\pi$
$359$	3.96827e6	1.62504	0.812521	+	0.582932i	$0.198094\pi$
$360$	0	0
$361$	3.04739e6	1.23072
$362$	0	0
$363$	−1.22157e6	−0.486577
$364$	0	0
$365$	2.25012e6	0.884042
$366$	0	0
$367$	4.90395e6	1.90056	0.950278	−	0.311404i	$-0.100799\pi$
$367$	4.90395e6	1.90056	0.950278	+	0.311404i	$0.100799\pi$
$368$	0	0
$369$	−3.57592e6	−1.36717
$370$	0	0
$371$	655475.	0.247242
$372$	0	0
$373$	2.52633e6	0.940196	0.470098	−	0.882614i	$-0.344219\pi$
$373$	2.52633e6	0.940196	0.470098	+	0.882614i	$0.344219\pi$
$374$	0	0
$375$	−1.20356e6	−0.441966
$376$	0	0
$377$	462016.	0.167419
$378$	0	0
$379$	607631.	0.217291	0.108646	−	0.994081i	$-0.465349\pi$
$379$	607631.	0.217291	0.108646	+	0.994081i	$0.465349\pi$
$380$	0	0
$381$	1.37805e6	0.486354
$382$	0	0
$383$	3.94370e6	1.37375	0.686874	−	0.726777i	$-0.258982\pi$
$383$	3.94370e6	1.37375	0.686874	+	0.726777i	$0.258982\pi$
$384$	0	0
$385$	400445.	0.137686
$386$	0	0
$387$	335558.	0.113891
$388$	0	0
$389$	−2.42464e6	−0.812405	−0.406203	−	0.913783i	$-0.633147\pi$
$389$	−2.42464e6	−0.812405	−0.406203	+	0.913783i	$0.633147\pi$
$390$	0	0
$391$	622998.	0.206084
$392$	0	0
$393$	−1.67660e6	−0.547579
$394$	0	0
$395$	1.85455e6	0.598062
$396$	0	0
$397$	−2.94322e6	−0.937232	−0.468616	−	0.883402i	$-0.655247\pi$
$397$	−2.94322e6	−0.937232	−0.468616	+	0.883402i	$0.655247\pi$
$398$	0	0
$399$	−3.60571e6	−1.13386
$400$	0	0
$401$	−5.82714e6	−1.80965	−0.904824	−	0.425785i	$-0.859998\pi$
$401$	−5.82714e6	−1.80965	−0.904824	+	0.425785i	$0.859998\pi$
$402$	0	0
$403$	−339715.	−0.104196
$404$	0	0
$405$	505457.	0.153125
$406$	0	0
$407$	680337.	0.203581
$408$	0	0
$409$	1.22650e6	0.362542	0.181271	−	0.983433i	$-0.441979\pi$
$409$	1.22650e6	0.362542	0.181271	+	0.983433i	$0.441979\pi$
$410$	0	0
$411$	988473.	0.288642
$412$	0	0
$413$	491196.	0.141703
$414$	0	0
$415$	2.16057e6	0.615812
$416$	0	0
$417$	−1.42049e6	−0.400034
$418$	0	0
$419$	−1.13490e6	−0.315809	−0.157904	−	0.987454i	$-0.550474\pi$
$419$	−1.13490e6	−0.315809	−0.157904	+	0.987454i	$0.550474\pi$
$420$	0	0
$421$	−5.21555e6	−1.43415	−0.717075	−	0.696996i	$-0.754520\pi$
$421$	−5.21555e6	−1.43415	−0.717075	+	0.696996i	$0.754520\pi$
$422$	0	0
$423$	−2.50400e6	−0.680431
$424$	0	0
$425$	2.81909e6	0.757070
$426$	0	0
$427$	−9.63336e6	−2.55687
$428$	0	0
$429$	−172365.	−0.0452173
$430$	0	0
$431$	−4.94938e6	−1.28339	−0.641693	−	0.766961i	$-0.721768\pi$
$431$	−4.94938e6	−1.28339	−0.641693	+	0.766961i	$0.721768\pi$
$432$	0	0
$433$	−2.02994e6	−0.520311	−0.260156	−	0.965567i	$-0.583774\pi$
$433$	−2.02994e6	−0.520311	−0.260156	+	0.965567i	$0.583774\pi$
$434$	0	0
$435$	337588.	0.0855388
$436$	0	0
$437$	−1.21287e6	−0.303817
$438$	0	0
$439$	4.05767e6	1.00488	0.502442	−	0.864611i	$-0.332435\pi$
$439$	4.05767e6	1.00488	0.502442	+	0.864611i	$0.332435\pi$
$440$	0	0
$441$	−3.89347e6	−0.953323
$442$	0	0
$443$	−3.73717e6	−0.904761	−0.452380	−	0.891825i	$-0.649425\pi$
$443$	−3.73717e6	−0.904761	−0.452380	+	0.891825i	$0.649425\pi$
$444$	0	0
$445$	−2.12802e6	−0.509419
$446$	0	0
$447$	1.46738e6	0.347356
$448$	0	0
$449$	−2.08643e6	−0.488414	−0.244207	−	0.969723i	$-0.578528\pi$
$449$	−2.08643e6	−0.488414	−0.244207	+	0.969723i	$0.578528\pi$
$450$	0	0
$451$	−1.43540e6	−0.332301
$452$	0	0
$453$	3.10724e6	0.711424
$454$	0	0
$455$	−1.65826e6	−0.375513
$456$	0	0
$457$	383012.	0.0857871	0.0428936	−	0.999080i	$-0.486342\pi$
$457$	383012.	0.0857871	0.0428936	+	0.999080i	$0.486342\pi$
$458$	0	0
$459$	4.01922e6	0.890452
$460$	0	0
$461$	−767599.	−0.168222	−0.0841109	−	0.996456i	$-0.526805\pi$
$461$	−767599.	−0.168222	−0.0841109	+	0.996456i	$0.526805\pi$
$462$	0	0
$463$	−2.87956e6	−0.624272	−0.312136	−	0.950037i	$-0.601045\pi$
$463$	−2.87956e6	−0.624272	−0.312136	+	0.950037i	$0.601045\pi$
$464$	0	0
$465$	−248224.	−0.0532368
$466$	0	0
$467$	−4.73971e6	−1.00568	−0.502840	−	0.864379i	$-0.667712\pi$
$467$	−4.73971e6	−1.00568	−0.502840	+	0.864379i	$0.667712\pi$
$468$	0	0
$469$	1.78323e6	0.374348
$470$	0	0
$471$	3.95533e6	0.821543
$472$	0	0
$473$	134695.	0.0276821
$474$	0	0
$475$	−5.48829e6	−1.11610
$476$	0	0
$477$	608147.	0.122381
$478$	0	0
$479$	−5.41230e6	−1.07781	−0.538906	−	0.842366i	$-0.681162\pi$
$479$	−5.41230e6	−1.07781	−0.538906	+	0.842366i	$0.681162\pi$
$480$	0	0
$481$	−2.81731e6	−0.555229
$482$	0	0
$483$	791758.	0.154427
$484$	0	0
$485$	−1.90879e6	−0.368471
$486$	0	0
$487$	−5.16701e6	−0.987227	−0.493614	−	0.869681i	$-0.664324\pi$
$487$	−5.16701e6	−0.987227	−0.493614	+	0.869681i	$0.664324\pi$
$488$	0	0
$489$	3.65534e6	0.691283
$490$	0	0
$491$	−1.68112e6	−0.314698	−0.157349	−	0.987543i	$-0.550295\pi$
$491$	−1.68112e6	−0.314698	−0.157349	+	0.987543i	$0.550295\pi$
$492$	0	0
$493$	−1.84888e6	−0.342603
$494$	0	0
$495$	371531.	0.0681526
$496$	0	0
$497$	8.48868e6	1.54152
$498$	0	0
$499$	−3.57956e6	−0.643545	−0.321772	−	0.946817i	$-0.604279\pi$
$499$	−3.57956e6	−0.643545	−0.321772	+	0.946817i	$0.604279\pi$
$500$	0	0
$501$	1.11586e6	0.198617
$502$	0	0
$503$	1.07972e6	0.190279	0.0951397	−	0.995464i	$-0.469670\pi$
$503$	1.07972e6	0.190279	0.0951397	+	0.995464i	$0.469670\pi$
$504$	0	0
$505$	3.67653e6	0.641520
$506$	0	0
$507$	−2.19844e6	−0.379835
$508$	0	0
$509$	1.11760e6	0.191202	0.0956012	−	0.995420i	$-0.469523\pi$
$509$	1.11760e6	0.191202	0.0956012	+	0.995420i	$0.469523\pi$
$510$	0	0
$511$	−1.56615e7	−2.65327
$512$	0	0
$513$	−7.82475e6	−1.31274
$514$	0	0
$515$	−1.17880e6	−0.195849
$516$	0	0
$517$	−1.00513e6	−0.165384
$518$	0	0
$519$	4.12709e6	0.672551
$520$	0	0
$521$	−6.50399e6	−1.04975	−0.524874	−	0.851180i	$-0.675888\pi$
$521$	−6.50399e6	−1.04975	−0.524874	+	0.851180i	$0.675888\pi$
$522$	0	0
$523$	−437675.	−0.0699677	−0.0349838	−	0.999388i	$-0.511138\pi$
$523$	−437675.	−0.0699677	−0.0349838	+	0.999388i	$0.511138\pi$
$524$	0	0
$525$	3.58273e6	0.567304
$526$	0	0
$527$	1.35946e6	0.213226
$528$	0	0
$529$	−6.17001e6	−0.958621
$530$	0	0
$531$	455730.	0.0701409
$532$	0	0
$533$	5.94407e6	0.906288
$534$	0	0
$535$	148311.	0.0224021
$536$	0	0
$537$	−5.43134e6	−0.812777
$538$	0	0
$539$	−1.56287e6	−0.231713
$540$	0	0
$541$	6.56956e6	0.965035	0.482518	−	0.875886i	$-0.339722\pi$
$541$	6.56956e6	0.965035	0.482518	+	0.875886i	$0.339722\pi$
$542$	0	0
$543$	2.24843e6	0.327249
$544$	0	0
$545$	−3.89730e6	−0.562047
$546$	0	0
$547$	2.84947e6	0.407189	0.203595	−	0.979055i	$-0.434738\pi$
$547$	2.84947e6	0.407189	0.203595	+	0.979055i	$0.434738\pi$
$548$	0	0
$549$	−8.93780e6	−1.26561
$550$	0	0
$551$	3.59946e6	0.505078
$552$	0	0
$553$	−1.29082e7	−1.79496
$554$	0	0
$555$	−2.05856e6	−0.283682
$556$	0	0
$557$	1.14478e7	1.56344	0.781722	−	0.623627i	$-0.214342\pi$
$557$	1.14478e7	1.56344	0.781722	+	0.623627i	$0.214342\pi$
$558$	0	0
$559$	−557781.	−0.0754977
$560$	0	0
$561$	689763.	0.0925321
$562$	0	0
$563$	−3.98082e6	−0.529300	−0.264650	−	0.964345i	$-0.585256\pi$
$563$	−3.98082e6	−0.529300	−0.264650	+	0.964345i	$0.585256\pi$
$564$	0	0
$565$	7.06293e6	0.930816
$566$	0	0
$567$	−3.51813e6	−0.459573
$568$	0	0
$569$	8.49999e6	1.10062	0.550311	−	0.834960i	$-0.314509\pi$
$569$	8.49999e6	1.10062	0.550311	+	0.834960i	$0.314509\pi$
$570$	0	0
$571$	−5.24847e6	−0.673663	−0.336831	−	0.941565i	$-0.609355\pi$
$571$	−5.24847e6	−0.673663	−0.336831	+	0.941565i	$0.609355\pi$
$572$	0	0
$573$	5.43135e6	0.691069
$574$	0	0
$575$	1.20514e6	0.152009
$576$	0	0
$577$	1.94309e6	0.242970	0.121485	−	0.992593i	$-0.461234\pi$
$577$	1.94309e6	0.242970	0.121485	+	0.992593i	$0.461234\pi$
$578$	0	0
$579$	6.70946e6	0.831747
$580$	0	0
$581$	−1.50382e7	−1.84823
$582$	0	0
$583$	244115.	0.0297456
$584$	0	0
$585$	−1.53853e6	−0.185873
$586$	0	0
$587$	−924441.	−0.110735	−0.0553674	−	0.998466i	$-0.517633\pi$
$587$	−924441.	−0.110735	−0.0553674	+	0.998466i	$0.517633\pi$
$588$	0	0
$589$	−2.64665e6	−0.314346
$590$	0	0
$591$	−5.87982e6	−0.692461
$592$	0	0
$593$	−1.09140e7	−1.27453	−0.637263	−	0.770646i	$-0.719934\pi$
$593$	−1.09140e7	−1.27453	−0.637263	+	0.770646i	$0.719934\pi$
$594$	0	0
$595$	6.63598e6	0.768445
$596$	0	0
$597$	−7.76564e6	−0.891746
$598$	0	0
$599$	6.47200e6	0.737006	0.368503	−	0.929626i	$-0.379870\pi$
$599$	6.47200e6	0.737006	0.368503	+	0.929626i	$0.379870\pi$
$600$	0	0
$601$	−9.66608e6	−1.09160	−0.545801	−	0.837915i	$-0.683775\pi$
$601$	−9.66608e6	−1.09160	−0.545801	+	0.837915i	$0.683775\pi$
$602$	0	0
$603$	1.65447e6	0.185296
$604$	0	0
$605$	−4.37685e6	−0.486153
$606$	0	0
$607$	−1.50825e7	−1.66151	−0.830753	−	0.556641i	$-0.812090\pi$
$607$	−1.50825e7	−1.66151	−0.830753	+	0.556641i	$0.812090\pi$
$608$	0	0
$609$	−2.34971e6	−0.256727
$610$	0	0
$611$	4.16228e6	0.451054
$612$	0	0
$613$	−2.07592e6	−0.223131	−0.111565	−	0.993757i	$-0.535586\pi$
$613$	−2.07592e6	−0.223131	−0.111565	+	0.993757i	$0.535586\pi$
$614$	0	0
$615$	4.34324e6	0.463048
$616$	0	0
$617$	1.05100e7	1.11145	0.555727	−	0.831365i	$-0.312440\pi$
$617$	1.05100e7	1.11145	0.555727	+	0.831365i	$0.312440\pi$
$618$	0	0
$619$	6.33902e6	0.664960	0.332480	−	0.943110i	$-0.392115\pi$
$619$	6.33902e6	0.664960	0.332480	+	0.943110i	$0.392115\pi$
$620$	0	0
$621$	1.71820e6	0.178790
$622$	0	0
$623$	1.48116e7	1.52891
$624$	0	0
$625$	2.98529e6	0.305694
$626$	0	0
$627$	−1.34285e6	−0.136414
$628$	0	0
$629$	1.12742e7	1.13621
$630$	0	0
$631$	1.15541e7	1.15521	0.577606	−	0.816316i	$-0.303987\pi$
$631$	1.15541e7	1.15521	0.577606	+	0.816316i	$0.303987\pi$
$632$	0	0
$633$	2.15345e6	0.213612
$634$	0	0
$635$	4.93751e6	0.485930
$636$	0	0
$637$	6.47191e6	0.631952
$638$	0	0
$639$	7.87577e6	0.763028
$640$	0	0
$641$	−3.92627e6	−0.377429	−0.188714	−	0.982032i	$-0.560432\pi$
$641$	−3.92627e6	−0.377429	−0.188714	+	0.982032i	$0.560432\pi$
$642$	0	0
$643$	1.87570e7	1.78911	0.894553	−	0.446961i	$-0.147494\pi$
$643$	1.87570e7	1.78911	0.894553	+	0.446961i	$0.147494\pi$
$644$	0	0
$645$	−407561.	−0.0385739
$646$	0	0
$647$	9.03463e6	0.848496	0.424248	−	0.905546i	$-0.360538\pi$
$647$	9.03463e6	0.848496	0.424248	+	0.905546i	$0.360538\pi$
$648$	0	0
$649$	182933.	0.0170483
$650$	0	0
$651$	1.72772e6	0.159779
$652$	0	0
$653$	1.74423e7	1.60074	0.800371	−	0.599505i	$-0.204636\pi$
$653$	1.74423e7	1.60074	0.800371	+	0.599505i	$0.204636\pi$
$654$	0	0
$655$	−6.00720e6	−0.547103
$656$	0	0
$657$	−1.45307e7	−1.31333
$658$	0	0
$659$	−1.74636e7	−1.56646	−0.783230	−	0.621732i	$-0.786429\pi$
$659$	−1.74636e7	−1.56646	−0.783230	+	0.621732i	$0.786429\pi$
$660$	0	0
$661$	5.12626e6	0.456348	0.228174	−	0.973620i	$-0.426724\pi$
$661$	5.12626e6	0.456348	0.228174	+	0.973620i	$0.426724\pi$
$662$	0	0
$663$	−2.85634e6	−0.252364
$664$	0	0
$665$	−1.29192e7	−1.13287
$666$	0	0
$667$	−790386.	−0.0687899
$668$	0	0
$669$	4.46096e6	0.385356
$670$	0	0
$671$	−3.58770e6	−0.307616
$672$	0	0
$673$	4.98529e6	0.424280	0.212140	−	0.977239i	$-0.431957\pi$
$673$	4.98529e6	0.424280	0.212140	+	0.977239i	$0.431957\pi$
$674$	0	0
$675$	7.77489e6	0.656802
$676$	0	0
$677$	8.15170e6	0.683560	0.341780	−	0.939780i	$-0.388970\pi$
$677$	8.15170e6	0.683560	0.341780	+	0.939780i	$0.388970\pi$
$678$	0	0
$679$	1.32857e7	1.10589
$680$	0	0
$681$	−4.76467e6	−0.393700
$682$	0	0
$683$	3.28051e6	0.269085	0.134543	−	0.990908i	$-0.457043\pi$
$683$	3.28051e6	0.269085	0.134543	+	0.990908i	$0.457043\pi$
$684$	0	0
$685$	3.54167e6	0.288391
$686$	0	0
$687$	817384.	0.0660745
$688$	0	0
$689$	−1.01089e6	−0.0811254
$690$	0	0
$691$	−1.54825e7	−1.23352	−0.616758	−	0.787153i	$-0.711554\pi$
$691$	−1.54825e7	−1.23352	−0.616758	+	0.787153i	$0.711554\pi$
$692$	0	0
$693$	−2.58597e6	−0.204546
$694$	0	0
$695$	−5.08956e6	−0.399685
$696$	0	0
$697$	−2.37868e7	−1.85462
$698$	0	0
$699$	1.00435e7	0.777483
$700$	0	0
$701$	−4.91809e6	−0.378008	−0.189004	−	0.981976i	$-0.560526\pi$
$701$	−4.91809e6	−0.378008	−0.189004	+	0.981976i	$0.560526\pi$
$702$	0	0
$703$	−2.19490e7	−1.67505
$704$	0	0
$705$	3.04131e6	0.230456
$706$	0	0
$707$	−2.55898e7	−1.92539
$708$	0	0
$709$	−8.69925e6	−0.649930	−0.324965	−	0.945726i	$-0.605352\pi$
$709$	−8.69925e6	−0.649930	−0.324965	+	0.945726i	$0.605352\pi$
$710$	0	0
$711$	−1.19762e7	−0.888476
$712$	0	0
$713$	581162.	0.0428128
$714$	0	0
$715$	−617578.	−0.0451779
$716$	0	0
$717$	−1.73340e6	−0.125922
$718$	0	0
$719$	8.92512e6	0.643861	0.321930	−	0.946763i	$-0.395668\pi$
$719$	8.92512e6	0.643861	0.321930	+	0.946763i	$0.395668\pi$
$720$	0	0
$721$	8.20479e6	0.587799
$722$	0	0
$723$	−3.76894e6	−0.268147
$724$	0	0
$725$	−3.57652e6	−0.252706
$726$	0	0
$727$	−1.33205e7	−0.934729	−0.467365	−	0.884065i	$-0.654796\pi$
$727$	−1.33205e7	−0.934729	−0.467365	+	0.884065i	$0.654796\pi$
$728$	0	0
$729$	3.08154e6	0.214758
$730$	0	0
$731$	2.23211e6	0.154498
$732$	0	0
$733$	1.41244e7	0.970981	0.485490	−	0.874242i	$-0.338641\pi$
$733$	1.41244e7	0.970981	0.485490	+	0.874242i	$0.338641\pi$
$734$	0	0
$735$	4.72892e6	0.322882
$736$	0	0
$737$	664118.	0.0450377
$738$	0	0
$739$	1.65506e7	1.11482	0.557408	−	0.830239i	$-0.311796\pi$
$739$	1.65506e7	1.11482	0.557408	+	0.830239i	$0.311796\pi$
$740$	0	0
$741$	5.56083e6	0.372044
$742$	0	0
$743$	−1.32085e7	−0.877771	−0.438885	−	0.898543i	$-0.644627\pi$
$743$	−1.32085e7	−0.877771	−0.438885	+	0.898543i	$0.644627\pi$
$744$	0	0
$745$	5.25759e6	0.347054
$746$	0	0
$747$	−1.39524e7	−0.914845
$748$	0	0
$749$	−1.03229e6	−0.0672353
$750$	0	0
$751$	−1.62740e6	−0.105291	−0.0526457	−	0.998613i	$-0.516765\pi$
$751$	−1.62740e6	−0.105291	−0.0526457	+	0.998613i	$0.516765\pi$
$752$	0	0
$753$	−5.66870e6	−0.364331
$754$	0	0
$755$	1.11331e7	0.710805
$756$	0	0
$757$	−1.41817e7	−0.899472	−0.449736	−	0.893162i	$-0.648482\pi$
$757$	−1.41817e7	−0.899472	−0.449736	+	0.893162i	$0.648482\pi$
$758$	0	0
$759$	294870.	0.0185791
$760$	0	0
$761$	−631434.	−0.0395245	−0.0197622	−	0.999805i	$-0.506291\pi$
$761$	−631434.	−0.0395245	−0.0197622	+	0.999805i	$0.506291\pi$
$762$	0	0
$763$	2.71264e7	1.68686
$764$	0	0
$765$	6.15684e6	0.380368
$766$	0	0
$767$	−757537.	−0.0464960
$768$	0	0
$769$	2920.58	0.000178096 0	8.90480e−5	−	1.00000i	$-0.499972\pi$
$769$	2920.58	0.000178096 0	8.90480e−5		1.00000i	$0.499972\pi$
$770$	0	0
$771$	−1.05488e7	−0.639098
$772$	0	0
$773$	−8.64224e6	−0.520209	−0.260104	−	0.965580i	$-0.583757\pi$
$773$	−8.64224e6	−0.520209	−0.260104	+	0.965580i	$0.583757\pi$
$774$	0	0
$775$	2.62978e6	0.157277
$776$	0	0
$777$	1.43282e7	0.851412
$778$	0	0
$779$	4.63089e7	2.73414
$780$	0	0
$781$	3.16139e6	0.185460
$782$	0	0
$783$	−5.09911e6	−0.297228
$784$	0	0
$785$	1.41718e7	0.820828
$786$	0	0
$787$	−6.88838e6	−0.396443	−0.198221	−	0.980157i	$-0.563516\pi$
$787$	−6.88838e6	−0.396443	−0.198221	+	0.980157i	$0.563516\pi$
$788$	0	0
$789$	141700.	0.00810358
$790$	0	0
$791$	−4.91601e7	−2.79365
$792$	0	0
$793$	1.48568e7	0.838965
$794$	0	0
$795$	−738642.	−0.0414492
$796$	0	0
$797$	2.93617e7	1.63733	0.818664	−	0.574273i	$-0.194715\pi$
$797$	2.93617e7	1.63733	0.818664	+	0.574273i	$0.194715\pi$
$798$	0	0
$799$	−1.66565e7	−0.923030
$800$	0	0
$801$	1.37422e7	0.756788
$802$	0	0
$803$	−5.83272e6	−0.319214
$804$	0	0
$805$	2.83685e6	0.154293
$806$	0	0
$807$	−4.30998e6	−0.232965
$808$	0	0
$809$	−2.84195e6	−0.152667	−0.0763334	−	0.997082i	$-0.524321\pi$
$809$	−2.84195e6	−0.152667	−0.0763334	+	0.997082i	$0.524321\pi$
$810$	0	0
$811$	−1.67390e6	−0.0893670	−0.0446835	−	0.999001i	$-0.514228\pi$
$811$	−1.67390e6	−0.0893670	−0.0446835	+	0.999001i	$0.514228\pi$
$812$	0	0
$813$	−107709.	−0.00571515
$814$	0	0
$815$	1.30970e7	0.690681
$816$	0	0
$817$	−4.34554e6	−0.227766
$818$	0	0
$819$	1.07086e7	0.557859
$820$	0	0
$821$	3.75970e7	1.94668	0.973340	−	0.229365i	$-0.0736650\pi$
$821$	3.75970e7	1.94668	0.973340	+	0.229365i	$0.0736650\pi$
$822$	0	0
$823$	−1.67263e7	−0.860797	−0.430398	−	0.902639i	$-0.641627\pi$
$823$	−1.67263e7	−0.860797	−0.430398	+	0.902639i	$0.641627\pi$
$824$	0	0
$825$	1.33429e6	0.0682522
$826$	0	0
$827$	2.69145e7	1.36843	0.684216	−	0.729279i	$-0.260144\pi$
$827$	2.69145e7	1.36843	0.684216	+	0.729279i	$0.260144\pi$
$828$	0	0
$829$	−7.34223e6	−0.371058	−0.185529	−	0.982639i	$-0.559400\pi$
$829$	−7.34223e6	−0.371058	−0.185529	+	0.982639i	$0.559400\pi$
$830$	0	0
$831$	−3.93471e6	−0.197656
$832$	0	0
$833$	−2.58991e7	−1.29322
$834$	0	0
$835$	3.99811e6	0.198444
$836$	0	0
$837$	3.74932e6	0.184986
$838$	0	0
$839$	1.44619e7	0.709284	0.354642	−	0.935002i	$-0.384603\pi$
$839$	1.44619e7	0.709284	0.354642	+	0.935002i	$0.384603\pi$
$840$	0	0
$841$	−1.81655e7	−0.885641
$842$	0	0
$843$	4.75043e6	0.230231
$844$	0	0
$845$	−7.87695e6	−0.379504
$846$	0	0
$847$	3.04642e7	1.45909
$848$	0	0
$849$	1.45483e7	0.692699
$850$	0	0
$851$	4.81967e6	0.228136
$852$	0	0
$853$	−3.00835e6	−0.141565	−0.0707825	−	0.997492i	$-0.522550\pi$
$853$	−3.00835e6	−0.141565	−0.0707825	+	0.997492i	$0.522550\pi$
$854$	0	0
$855$	−1.19863e7	−0.560753
$856$	0	0
$857$	5.35883e6	0.249240	0.124620	−	0.992205i	$-0.460229\pi$
$857$	5.35883e6	0.249240	0.124620	+	0.992205i	$0.460229\pi$
$858$	0	0
$859$	−2.59898e7	−1.20176	−0.600882	−	0.799338i	$-0.705184\pi$
$859$	−2.59898e7	−1.20176	−0.600882	+	0.799338i	$0.705184\pi$
$860$	0	0
$861$	−3.02302e7	−1.38974
$862$	0	0
$863$	4.32743e7	1.97789	0.988947	−	0.148266i	$-0.0473693\pi$
$863$	4.32743e7	1.97789	0.988947	+	0.148266i	$0.0473693\pi$
$864$	0	0
$865$	1.47872e7	0.671965
$866$	0	0
$867$	293873.	0.0132774
$868$	0	0
$869$	−4.80734e6	−0.215951
$870$	0	0
$871$	−2.75015e6	−0.122832
$872$	0	0
$873$	1.23265e7	0.547398
$874$	0	0
$875$	3.00150e7	1.32531
$876$	0	0
$877$	1.03640e7	0.455017	0.227508	−	0.973776i	$-0.426942\pi$
$877$	1.03640e7	0.455017	0.227508	+	0.973776i	$0.426942\pi$
$878$	0	0
$879$	7.45885e6	0.325611
$880$	0	0
$881$	−2.43175e7	−1.05555	−0.527776	−	0.849384i	$-0.676974\pi$
$881$	−2.43175e7	−1.05555	−0.527776	+	0.849384i	$0.676974\pi$
$882$	0	0
$883$	−1.78896e7	−0.772147	−0.386073	−	0.922468i	$-0.626169\pi$
$883$	−1.78896e7	−0.772147	−0.386073	+	0.922468i	$0.626169\pi$
$884$	0	0
$885$	−553520.	−0.0237561
$886$	0	0
$887$	2.17492e7	0.928185	0.464092	−	0.885787i	$-0.346381\pi$
$887$	2.17492e7	0.928185	0.464092	+	0.885787i	$0.346381\pi$
$888$	0	0
$889$	−3.43666e7	−1.45842
$890$	0	0
$891$	−1.31024e6	−0.0552912
$892$	0	0
$893$	3.24274e7	1.36076
$894$	0	0
$895$	−1.94603e7	−0.812069
$896$	0	0
$897$	−1.22107e6	−0.0506711
$898$	0	0
$899$	−1.72472e6	−0.0711738
$900$	0	0
$901$	4.04536e6	0.166014
$902$	0	0
$903$	2.83675e6	0.115771
$904$	0	0
$905$	8.05605e6	0.326965
$906$	0	0
$907$	−2.98499e7	−1.20483	−0.602414	−	0.798184i	$-0.705794\pi$
$907$	−2.98499e7	−1.20483	−0.602414	+	0.798184i	$0.705794\pi$
$908$	0	0
$909$	−2.37421e7	−0.953036
$910$	0	0
$911$	1.09193e7	0.435912	0.217956	−	0.975959i	$-0.430061\pi$
$911$	1.09193e7	0.435912	0.217956	+	0.975959i	$0.430061\pi$
$912$	0	0
$913$	−5.60060e6	−0.222360
$914$	0	0
$915$	1.08557e7	0.428650
$916$	0	0
$917$	4.18119e7	1.64201
$918$	0	0
$919$	−9.16828e6	−0.358096	−0.179048	−	0.983840i	$-0.557302\pi$
$919$	−9.16828e6	−0.358096	−0.179048	+	0.983840i	$0.557302\pi$
$920$	0	0
$921$	5.77448e6	0.224318
$922$	0	0
$923$	−1.30915e7	−0.505807
$924$	0	0
$925$	2.18091e7	0.838078
$926$	0	0
$927$	7.61237e6	0.290952
$928$	0	0
$929$	−1.38913e6	−0.0528083	−0.0264041	−	0.999651i	$-0.508406\pi$
$929$	−1.38913e6	−0.0528083	−0.0264041	+	0.999651i	$0.508406\pi$
$930$	0	0
$931$	5.04212e7	1.90651
$932$	0	0
$933$	8.01753e6	0.301534
$934$	0	0
$935$	2.47140e6	0.0924515
$936$	0	0
$937$	−2.52918e7	−0.941090	−0.470545	−	0.882376i	$-0.655943\pi$
$937$	−2.52918e7	−0.941090	−0.470545	+	0.882376i	$0.655943\pi$
$938$	0	0
$939$	−474329.	−0.0175556
$940$	0	0
$941$	2.69608e7	0.992563	0.496282	−	0.868162i	$-0.334698\pi$
$941$	2.69608e7	0.992563	0.496282	+	0.868162i	$0.334698\pi$
$942$	0	0
$943$	−1.01687e7	−0.372381
$944$	0	0
$945$	1.83017e7	0.666671
$946$	0	0
$947$	3.19275e6	0.115689	0.0578443	−	0.998326i	$-0.481577\pi$
$947$	3.19275e6	0.115689	0.0578443	+	0.998326i	$0.481577\pi$
$948$	0	0
$949$	2.41536e7	0.870596
$950$	0	0
$951$	6.60625e6	0.236866
$952$	0	0
$953$	−3.59618e7	−1.28265	−0.641327	−	0.767268i	$-0.721616\pi$
$953$	−3.59618e7	−1.28265	−0.641327	+	0.767268i	$0.721616\pi$
$954$	0	0
$955$	1.94604e7	0.690468
$956$	0	0
$957$	−875089.	−0.0308868
$958$	0	0
$959$	−2.46511e7	−0.865545
$960$	0	0
$961$	−2.73610e7	−0.955704
$962$	0	0
$963$	−957755.	−0.0332804
$964$	0	0
$965$	2.40398e7	0.831023
$966$	0	0
$967$	1.05454e7	0.362659	0.181330	−	0.983422i	$-0.441960\pi$
$967$	1.05454e7	0.362659	0.181330	+	0.983422i	$0.441960\pi$
$968$	0	0
$969$	−2.22531e7	−0.761345
$970$	0	0
$971$	1.47340e7	0.501500	0.250750	−	0.968052i	$-0.419323\pi$
$971$	1.47340e7	0.501500	0.250750	+	0.968052i	$0.419323\pi$
$972$	0	0
$973$	3.54249e7	1.19957
$974$	0	0
$975$	−5.52539e6	−0.186145
$976$	0	0
$977$	4.83492e7	1.62051	0.810257	−	0.586075i	$-0.199327\pi$
$977$	4.83492e7	1.62051	0.810257	+	0.586075i	$0.199327\pi$
$978$	0	0
$979$	5.51621e6	0.183943
$980$	0	0
$981$	2.51677e7	0.834971
$982$	0	0
$983$	1.25654e7	0.414754	0.207377	−	0.978261i	$-0.433507\pi$
$983$	1.25654e7	0.414754	0.207377	+	0.978261i	$0.433507\pi$
$984$	0	0
$985$	−2.10672e7	−0.691858
$986$	0	0
$987$	−2.11684e7	−0.691665
$988$	0	0
$989$	954214.	0.0310209
$990$	0	0
$991$	4.25363e7	1.37586	0.687932	−	0.725776i	$-0.258519\pi$
$991$	4.25363e7	1.37586	0.687932	+	0.725776i	$0.258519\pi$
$992$	0	0
$993$	2.75259e7	0.885865
$994$	0	0
$995$	−2.78241e7	−0.890970
$996$	0	0
$997$	−3.70772e7	−1.18132	−0.590662	−	0.806919i	$-0.701133\pi$
$997$	−3.70772e7	−1.18132	−0.590662	+	0.806919i	$0.701133\pi$
$998$	0	0
$999$	3.10937e7	0.985732

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.6.a.h.1.7		10
4.3	odd	2		43.6.a.b.1.9	✓	10
12.11	even	2		387.6.a.e.1.2		10
20.19	odd	2		1075.6.a.b.1.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.9	✓	10	4.3	odd	2
387.6.a.e.1.2		10	12.11	even	2
688.6.a.h.1.7		10	1.1	even	1	trivial
1075.6.a.b.1.2		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)

\(f(q)\)	\(=\)	\(q+7.84343 q^{3} +28.1028 q^{5} -195.604 q^{7} -181.481 q^{9} +O(q^{10})\)	\(q+7.84343 q^{3} +28.1028 q^{5} -195.604 q^{7} -181.481 q^{9} -72.8476 q^{11} +301.666 q^{13} +220.422 q^{15} -1207.20 q^{17} +2350.21 q^{19} -1534.21 q^{21} -516.070 q^{23} -2335.23 q^{25} -3329.38 q^{27} +1531.55 q^{29} -1126.13 q^{31} -571.375 q^{33} -5497.02 q^{35} -9339.18 q^{37} +2366.10 q^{39} +19704.2 q^{41} -1849.00 q^{43} -5100.11 q^{45} +13797.6 q^{47} +21453.9 q^{49} -9468.56 q^{51} -3351.03 q^{53} -2047.22 q^{55} +18433.7 q^{57} -2511.18 q^{59} +49249.3 q^{61} +35498.3 q^{63} +8477.66 q^{65} -9116.54 q^{67} -4047.76 q^{69} -43397.3 q^{71} +80067.4 q^{73} -18316.2 q^{75} +14249.3 q^{77} +65991.7 q^{79} +17986.0 q^{81} +76880.9 q^{83} -33925.6 q^{85} +12012.6 q^{87} -75722.6 q^{89} -59007.1 q^{91} -8832.73 q^{93} +66047.5 q^{95} -67921.7 q^{97} +13220.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9}+O(q^{10}) \) 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

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