LMFDB - Embedded newform 688.6.a.e.1.6

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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 $ x^8 - 4x^7 - 173x^6 + 462x^5 + 9118x^4 - 14192x^3 - 167688x^2 + 106368*x + 681984
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$7.21373$ 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)

$f(q)$	$=$	$q+11.2683 q^{3} -9.80186 q^{5} +11.1041 q^{7} -116.025 q^{9} +O(q^{10})$	$q+11.2683 q^{3} -9.80186 q^{5} +11.1041 q^{7} -116.025 q^{9} +557.274 q^{11} +107.663 q^{13} -110.450 q^{15} +329.005 q^{17} -2938.10 q^{19} +125.124 q^{21} +385.537 q^{23} -3028.92 q^{25} -4045.61 q^{27} -3309.15 q^{29} +5471.71 q^{31} +6279.54 q^{33} -108.841 q^{35} +4832.22 q^{37} +1213.18 q^{39} +1065.16 q^{41} +1849.00 q^{43} +1137.26 q^{45} +8991.95 q^{47} -16683.7 q^{49} +3707.33 q^{51} -10216.7 q^{53} -5462.32 q^{55} -33107.4 q^{57} +27457.0 q^{59} -36692.9 q^{61} -1288.35 q^{63} -1055.30 q^{65} +26272.2 q^{67} +4344.35 q^{69} -55130.3 q^{71} -9315.14 q^{73} -34130.9 q^{75} +6188.02 q^{77} -56826.5 q^{79} -17393.1 q^{81} +5858.19 q^{83} -3224.86 q^{85} -37288.5 q^{87} -42815.7 q^{89} +1195.50 q^{91} +61657.0 q^{93} +28798.8 q^{95} -231.671 q^{97} -64657.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 26 q^{3} - 212 q^{5} + 136 q^{7} + 546 q^{9}+O(q^{10}) $ 8 * q + 26 * q^3 - 212 * q^5 + 136 * q^7 + 546 * q^9	$ 8 q + 26 q^{3} - 212 q^{5} + 136 q^{7} + 546 q^{9} + 532 q^{11} - 2492 q^{13} + 1780 q^{15} - 2534 q^{17} + 1678 q^{19} - 2256 q^{21} + 2488 q^{23} + 4378 q^{25} + 8960 q^{27} - 4360 q^{29} - 5704 q^{31} - 12852 q^{33} - 5640 q^{35} - 3772 q^{37} - 11120 q^{39} - 10698 q^{41} + 14792 q^{43} - 44912 q^{45} + 77864 q^{47} + 7188 q^{49} + 80246 q^{51} - 62352 q^{53} + 49552 q^{55} - 808 q^{57} + 26224 q^{59} - 82540 q^{61} + 61768 q^{63} - 5000 q^{65} - 27784 q^{67} - 93776 q^{69} + 9504 q^{71} + 14260 q^{73} - 167420 q^{75} - 218140 q^{77} - 160248 q^{79} + 161076 q^{81} + 77176 q^{83} + 141096 q^{85} - 268136 q^{87} - 265692 q^{89} - 401148 q^{91} - 123860 q^{93} - 135884 q^{95} + 144742 q^{97} - 239516 q^{99}+O(q^{100}) $ 8 * q + 26 * q^3 - 212 * q^5 + 136 * q^7 + 546 * q^9 + 532 * q^11 - 2492 * q^13 + 1780 * q^15 - 2534 * q^17 + 1678 * q^19 - 2256 * q^21 + 2488 * q^23 + 4378 * q^25 + 8960 * q^27 - 4360 * q^29 - 5704 * q^31 - 12852 * q^33 - 5640 * q^35 - 3772 * q^37 - 11120 * q^39 - 10698 * q^41 + 14792 * q^43 - 44912 * q^45 + 77864 * q^47 + 7188 * q^49 + 80246 * q^51 - 62352 * q^53 + 49552 * q^55 - 808 * q^57 + 26224 * q^59 - 82540 * q^61 + 61768 * q^63 - 5000 * q^65 - 27784 * q^67 - 93776 * q^69 + 9504 * q^71 + 14260 * q^73 - 167420 * q^75 - 218140 * q^77 - 160248 * q^79 + 161076 * q^81 + 77176 * q^83 + 141096 * q^85 - 268136 * q^87 - 265692 * q^89 - 401148 * q^91 - 123860 * q^93 - 135884 * q^95 + 144742 * q^97 - 239516 * 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
$3$	11.2683	0.722863	0.361431	−	0.932399i	$-0.382288\pi$
$3$	11.2683	0.722863	0.361431	+	0.932399i	$0.382288\pi$
$4$	0	0
$5$	−9.80186	−0.175341	−0.0876705	−	0.996150i	$-0.527942\pi$
$5$	−9.80186	−0.175341	−0.0876705	+	0.996150i	$0.527942\pi$
$6$	0	0
$7$	11.1041	0.0856521	0.0428260	−	0.999083i	$-0.486364\pi$
$7$	11.1041	0.0856521	0.0428260	+	0.999083i	$0.486364\pi$
$8$	0	0
$9$	−116.025	−0.477470
$10$	0	0
$11$	557.274	1.38863	0.694316	−	0.719670i	$-0.255707\pi$
$11$	557.274	1.38863	0.694316	+	0.719670i	$0.255707\pi$
$12$	0	0
$13$	107.663	0.176688	0.0883441	−	0.996090i	$-0.471842\pi$
$13$	107.663	0.176688	0.0883441	+	0.996090i	$0.471842\pi$
$14$	0	0
$15$	−110.450	−0.126747
$16$	0	0
$17$	329.005	0.276109	0.138054	−	0.990425i	$-0.455915\pi$
$17$	329.005	0.276109	0.138054	+	0.990425i	$0.455915\pi$
$18$	0	0
$19$	−2938.10	−1.86716	−0.933582	−	0.358365i	$-0.883334\pi$
$19$	−2938.10	−1.86716	−0.933582	+	0.358365i	$0.883334\pi$
$20$	0	0
$21$	125.124	0.0619147
$22$	0	0
$23$	385.537	0.151966	0.0759830	−	0.997109i	$-0.475791\pi$
$23$	385.537	0.151966	0.0759830	+	0.997109i	$0.475791\pi$
$24$	0	0
$25$	−3028.92	−0.969256
$26$	0	0
$27$	−4045.61	−1.06801
$28$	0	0
$29$	−3309.15	−0.730670	−0.365335	−	0.930876i	$-0.619046\pi$
$29$	−3309.15	−0.730670	−0.365335	+	0.930876i	$0.619046\pi$
$30$	0	0
$31$	5471.71	1.02263	0.511316	−	0.859393i	$-0.329158\pi$
$31$	5471.71	1.02263	0.511316	+	0.859393i	$0.329158\pi$
$32$	0	0
$33$	6279.54	1.00379
$34$	0	0
$35$	−108.841	−0.0150183
$36$	0	0
$37$	4832.22	0.580286	0.290143	−	0.956983i	$-0.406297\pi$
$37$	4832.22	0.580286	0.290143	+	0.956983i	$0.406297\pi$
$38$	0	0
$39$	1213.18	0.127721
$40$	0	0
$41$	1065.16	0.0989587	0.0494793	−	0.998775i	$-0.484244\pi$
$41$	1065.16	0.0989587	0.0494793	+	0.998775i	$0.484244\pi$
$42$	0	0
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	0	0
$45$	1137.26	0.0837200
$46$	0	0
$47$	8991.95	0.593758	0.296879	−	0.954915i	$-0.404054\pi$
$47$	8991.95	0.593758	0.296879	+	0.954915i	$0.404054\pi$
$48$	0	0
$49$	−16683.7	−0.992664
$50$	0	0
$51$	3707.33	0.199589
$52$	0	0
$53$	−10216.7	−0.499598	−0.249799	−	0.968298i	$-0.580365\pi$
$53$	−10216.7	−0.499598	−0.249799	+	0.968298i	$0.580365\pi$
$54$	0	0
$55$	−5462.32	−0.243484
$56$	0	0
$57$	−33107.4	−1.34970
$58$	0	0
$59$	27457.0	1.02689	0.513444	−	0.858123i	$-0.328369\pi$
$59$	27457.0	1.02689	0.513444	+	0.858123i	$0.328369\pi$
$60$	0	0
$61$	−36692.9	−1.26258	−0.631288	−	0.775548i	$-0.717473\pi$
$61$	−36692.9	−1.26258	−0.631288	+	0.775548i	$0.717473\pi$
$62$	0	0
$63$	−1288.35	−0.0408963
$64$	0	0
$65$	−1055.30	−0.0309807
$66$	0	0
$67$	26272.2	0.715004	0.357502	−	0.933912i	$-0.383628\pi$
$67$	26272.2	0.715004	0.357502	+	0.933912i	$0.383628\pi$
$68$	0	0
$69$	4344.35	0.109851
$70$	0	0
$71$	−55130.3	−1.29791	−0.648955	−	0.760827i	$-0.724794\pi$
$71$	−55130.3	−1.29791	−0.648955	+	0.760827i	$0.724794\pi$
$72$	0	0
$73$	−9315.14	−0.204589	−0.102294	−	0.994754i	$-0.532618\pi$
$73$	−9315.14	−0.204589	−0.102294	+	0.994754i	$0.532618\pi$
$74$	0	0
$75$	−34130.9	−0.700639
$76$	0	0
$77$	6188.02	0.118939
$78$	0	0
$79$	−56826.5	−1.02443	−0.512216	−	0.858856i	$-0.671175\pi$
$79$	−56826.5	−1.02443	−0.512216	+	0.858856i	$0.671175\pi$
$80$	0	0
$81$	−17393.1	−0.294553
$82$	0	0
$83$	5858.19	0.0933401	0.0466700	−	0.998910i	$-0.485139\pi$
$83$	5858.19	0.0933401	0.0466700	+	0.998910i	$0.485139\pi$
$84$	0	0
$85$	−3224.86	−0.0484132
$86$	0	0
$87$	−37288.5	−0.528174
$88$	0	0
$89$	−42815.7	−0.572965	−0.286483	−	0.958085i	$-0.592486\pi$
$89$	−42815.7	−0.572965	−0.286483	+	0.958085i	$0.592486\pi$
$90$	0	0
$91$	1195.50	0.0151337
$92$	0	0
$93$	61657.0	0.739222
$94$	0	0
$95$	28798.8	0.327390
$96$	0	0
$97$	−231.671	−0.00250002	−0.00125001	−	0.999999i	$-0.500398\pi$
$97$	−231.671	−0.00250002	−0.00125001	+	0.999999i	$0.500398\pi$
$98$	0	0
$99$	−64657.8	−0.663029
$100$	0	0
$101$	−65329.5	−0.637245	−0.318622	−	0.947882i	$-0.603220\pi$
$101$	−65329.5	−0.637245	−0.318622	+	0.947882i	$0.603220\pi$
$102$	0	0
$103$	−210856.	−1.95837	−0.979183	−	0.202980i	$-0.934937\pi$
$103$	−210856.	−1.95837	−0.979183	+	0.202980i	$0.934937\pi$
$104$	0	0
$105$	−1226.45	−0.0108562
$106$	0	0
$107$	158342.	1.33702	0.668508	−	0.743705i	$-0.266933\pi$
$107$	158342.	1.33702	0.668508	+	0.743705i	$0.266933\pi$
$108$	0	0
$109$	−85822.7	−0.691888	−0.345944	−	0.938255i	$-0.612441\pi$
$109$	−85822.7	−0.691888	−0.345944	+	0.938255i	$0.612441\pi$
$110$	0	0
$111$	54450.9	0.419467
$112$	0	0
$113$	101903.	0.750745	0.375372	−	0.926874i	$-0.377515\pi$
$113$	101903.	0.750745	0.375372	+	0.926874i	$0.377515\pi$
$114$	0	0
$115$	−3778.98	−0.0266459
$116$	0	0
$117$	−12491.6	−0.0843633
$118$	0	0
$119$	3653.30	0.0236493
$120$	0	0
$121$	149503.	0.928299
$122$	0	0
$123$	12002.5	0.0715335
$124$	0	0
$125$	60319.9	0.345291
$126$	0	0
$127$	−109901.	−0.604635	−0.302318	−	0.953207i	$-0.597760\pi$
$127$	−109901.	−0.604635	−0.302318	+	0.953207i	$0.597760\pi$
$128$	0	0
$129$	20835.1	0.110236
$130$	0	0
$131$	79310.4	0.403787	0.201893	−	0.979408i	$-0.435291\pi$
$131$	79310.4	0.403787	0.201893	+	0.979408i	$0.435291\pi$
$132$	0	0
$133$	−32624.9	−0.159926
$134$	0	0
$135$	39654.5	0.187266
$136$	0	0
$137$	−360169.	−1.63948	−0.819738	−	0.572738i	$-0.805881\pi$
$137$	−360169.	−1.63948	−0.819738	+	0.572738i	$0.805881\pi$
$138$	0	0
$139$	−330826.	−1.45232	−0.726160	−	0.687525i	$-0.758697\pi$
$139$	−330826.	−1.45232	−0.726160	+	0.687525i	$0.758697\pi$
$140$	0	0
$141$	101324.	0.429205
$142$	0	0
$143$	59997.8	0.245355
$144$	0	0
$145$	32435.8	0.128116
$146$	0	0
$147$	−187997.	−0.717560
$148$	0	0
$149$	24179.7	0.0892249	0.0446125	−	0.999004i	$-0.485795\pi$
$149$	24179.7	0.0892249	0.0446125	+	0.999004i	$0.485795\pi$
$150$	0	0
$151$	−278667.	−0.994589	−0.497295	−	0.867582i	$-0.665673\pi$
$151$	−278667.	−0.994589	−0.497295	+	0.867582i	$0.665673\pi$
$152$	0	0
$153$	−38172.8	−0.131833
$154$	0	0
$155$	−53633.0	−0.179309
$156$	0	0
$157$	−102228.	−0.330995	−0.165497	−	0.986210i	$-0.552923\pi$
$157$	−102228.	−0.330995	−0.165497	+	0.986210i	$0.552923\pi$
$158$	0	0
$159$	−115125.	−0.361141
$160$	0	0
$161$	4281.04	0.0130162
$162$	0	0
$163$	−444216.	−1.30956	−0.654780	−	0.755819i	$-0.727239\pi$
$163$	−444216.	−1.30956	−0.654780	+	0.755819i	$0.727239\pi$
$164$	0	0
$165$	−61551.2	−0.176006
$166$	0	0
$167$	467231.	1.29640	0.648202	−	0.761469i	$-0.275521\pi$
$167$	467231.	1.29640	0.648202	+	0.761469i	$0.275521\pi$
$168$	0	0
$169$	−359702.	−0.968781
$170$	0	0
$171$	340893.	0.891514
$172$	0	0
$173$	296435.	0.753033	0.376516	−	0.926410i	$-0.377122\pi$
$173$	296435.	0.753033	0.376516	+	0.926410i	$0.377122\pi$
$174$	0	0
$175$	−33633.5	−0.0830188
$176$	0	0
$177$	309394.	0.742299
$178$	0	0
$179$	−211576.	−0.493553	−0.246777	−	0.969072i	$-0.579371\pi$
$179$	−211576.	−0.493553	−0.246777	+	0.969072i	$0.579371\pi$
$180$	0	0
$181$	−187769.	−0.426018	−0.213009	−	0.977050i	$-0.568326\pi$
$181$	−187769.	−0.426018	−0.213009	+	0.977050i	$0.568326\pi$
$182$	0	0
$183$	−413467.	−0.912669
$184$	0	0
$185$	−47364.7	−0.101748
$186$	0	0
$187$	183346.	0.383413
$188$	0	0
$189$	−44922.8	−0.0914771
$190$	0	0
$191$	655959.	1.30105	0.650524	−	0.759486i	$-0.274550\pi$
$191$	655959.	1.30105	0.650524	+	0.759486i	$0.274550\pi$
$192$	0	0
$193$	450394.	0.870361	0.435180	−	0.900343i	$-0.356685\pi$
$193$	450394.	0.870361	0.435180	+	0.900343i	$0.356685\pi$
$194$	0	0
$195$	−11891.4	−0.0223948
$196$	0	0
$197$	−973133.	−1.78652	−0.893258	−	0.449545i	$-0.851586\pi$
$197$	−973133.	−1.78652	−0.893258	+	0.449545i	$0.851586\pi$
$198$	0	0
$199$	−259840.	−0.465129	−0.232564	−	0.972581i	$-0.574712\pi$
$199$	−259840.	−0.465129	−0.232564	+	0.972581i	$0.574712\pi$
$200$	0	0
$201$	296043.	0.516850
$202$	0	0
$203$	−36745.1	−0.0625834
$204$	0	0
$205$	−10440.5	−0.0173515
$206$	0	0
$207$	−44732.0	−0.0725591
$208$	0	0
$209$	−1.63733e6	−2.59280
$210$	0	0
$211$	621161.	0.960502	0.480251	−	0.877131i	$-0.340546\pi$
$211$	621161.	0.960502	0.480251	+	0.877131i	$0.340546\pi$
$212$	0	0
$213$	−621226.	−0.938211
$214$	0	0
$215$	−18123.6	−0.0267392
$216$	0	0
$217$	60758.4	0.0875905
$218$	0	0
$219$	−104966.	−0.147890
$220$	0	0
$221$	35421.6	0.0487852
$222$	0	0
$223$	779614.	1.04983	0.524913	−	0.851156i	$-0.324098\pi$
$223$	779614.	1.04983	0.524913	+	0.851156i	$0.324098\pi$
$224$	0	0
$225$	351431.	0.462790
$226$	0	0
$227$	−795424.	−1.02455	−0.512276	−	0.858821i	$-0.671197\pi$
$227$	−795424.	−1.02455	−0.512276	+	0.858821i	$0.671197\pi$
$228$	0	0
$229$	−702980.	−0.885838	−0.442919	−	0.896562i	$-0.646057\pi$
$229$	−702980.	−0.885838	−0.442919	+	0.896562i	$0.646057\pi$
$230$	0	0
$231$	69728.6	0.0859767
$232$	0	0
$233$	−219327.	−0.264669	−0.132334	−	0.991205i	$-0.542247\pi$
$233$	−219327.	−0.264669	−0.132334	+	0.991205i	$0.542247\pi$
$234$	0	0
$235$	−88137.9	−0.104110
$236$	0	0
$237$	−640339.	−0.740524
$238$	0	0
$239$	−1.75148e6	−1.98340	−0.991701	−	0.128565i	$-0.958963\pi$
$239$	−1.75148e6	−1.98340	−0.991701	+	0.128565i	$0.958963\pi$
$240$	0	0
$241$	985600.	1.09310	0.546548	−	0.837428i	$-0.315942\pi$
$241$	985600.	1.09310	0.546548	+	0.837428i	$0.315942\pi$
$242$	0	0
$243$	787092.	0.855086
$244$	0	0
$245$	163531.	0.174055
$246$	0	0
$247$	−316324.	−0.329906
$248$	0	0
$249$	66011.9	0.0674721
$250$	0	0
$251$	−126142.	−0.126379	−0.0631894	−	0.998002i	$-0.520127\pi$
$251$	−126142.	−0.126379	−0.0631894	+	0.998002i	$0.520127\pi$
$252$	0	0
$253$	214850.	0.211025
$254$	0	0
$255$	−36338.7	−0.0349961
$256$	0	0
$257$	1.55933e6	1.47267	0.736335	−	0.676617i	$-0.236555\pi$
$257$	1.55933e6	1.47267	0.736335	+	0.676617i	$0.236555\pi$
$258$	0	0
$259$	53657.4	0.0497027
$260$	0	0
$261$	383944.	0.348873
$262$	0	0
$263$	107289.	0.0956460	0.0478230	−	0.998856i	$-0.484772\pi$
$263$	107289.	0.0956460	0.0478230	+	0.998856i	$0.484772\pi$
$264$	0	0
$265$	100143.	0.0876001
$266$	0	0
$267$	−482461.	−0.414175
$268$	0	0
$269$	829468.	0.698906	0.349453	−	0.936954i	$-0.386367\pi$
$269$	829468.	0.698906	0.349453	+	0.936954i	$0.386367\pi$
$270$	0	0
$271$	−2.22835e6	−1.84315	−0.921575	−	0.388200i	$-0.873097\pi$
$271$	−2.22835e6	−1.84315	−0.921575	+	0.388200i	$0.873097\pi$
$272$	0	0
$273$	13471.3	0.0109396
$274$	0	0
$275$	−1.68794e6	−1.34594
$276$	0	0
$277$	−1.57467e6	−1.23308	−0.616538	−	0.787325i	$-0.711465\pi$
$277$	−1.57467e6	−1.23308	−0.616538	+	0.787325i	$0.711465\pi$
$278$	0	0
$279$	−634856.	−0.488275
$280$	0	0
$281$	−2.26025e6	−1.70762	−0.853809	−	0.520586i	$-0.825713\pi$
$281$	−2.26025e6	−1.70762	−0.853809	+	0.520586i	$0.825713\pi$
$282$	0	0
$283$	1.81290e6	1.34558	0.672789	−	0.739835i	$-0.265096\pi$
$283$	1.81290e6	1.34558	0.672789	+	0.739835i	$0.265096\pi$
$284$	0	0
$285$	324514.	0.236658
$286$	0	0
$287$	11827.6	0.00847602
$288$	0	0
$289$	−1.31161e6	−0.923764
$290$	0	0
$291$	−2610.55	−0.00180717
$292$	0	0
$293$	−1.92999e6	−1.31336	−0.656682	−	0.754168i	$-0.728041\pi$
$293$	−1.92999e6	−1.31336	−0.656682	+	0.754168i	$0.728041\pi$
$294$	0	0
$295$	−269130.	−0.180056
$296$	0	0
$297$	−2.25451e6	−1.48307
$298$	0	0
$299$	41508.0	0.0268506
$300$	0	0
$301$	20531.5	0.0130618
$302$	0	0
$303$	−736154.	−0.460640
$304$	0	0
$305$	359659.	0.221381
$306$	0	0
$307$	2.35372e6	1.42531	0.712654	−	0.701515i	$-0.247493\pi$
$307$	2.35372e6	1.42531	0.712654	+	0.701515i	$0.247493\pi$
$308$	0	0
$309$	−2.37600e6	−1.41563
$310$	0	0
$311$	−1.25933e6	−0.738312	−0.369156	−	0.929367i	$-0.620353\pi$
$311$	−1.25933e6	−0.738312	−0.369156	+	0.929367i	$0.620353\pi$
$312$	0	0
$313$	−1.21791e6	−0.702676	−0.351338	−	0.936249i	$-0.614273\pi$
$313$	−1.21791e6	−0.702676	−0.351338	+	0.936249i	$0.614273\pi$
$314$	0	0
$315$	12628.3	0.00717079
$316$	0	0
$317$	689890.	0.385595	0.192798	−	0.981239i	$-0.438244\pi$
$317$	689890.	0.385595	0.192798	+	0.981239i	$0.438244\pi$
$318$	0	0
$319$	−1.84410e6	−1.01463
$320$	0	0
$321$	1.78425e6	0.966480
$322$	0	0
$323$	−966649.	−0.515540
$324$	0	0
$325$	−326103.	−0.171256
$326$	0	0
$327$	−967077.	−0.500140
$328$	0	0
$329$	99847.5	0.0508566
$330$	0	0
$331$	−1.93331e6	−0.969912	−0.484956	−	0.874538i	$-0.661164\pi$
$331$	−1.93331e6	−0.969912	−0.484956	+	0.874538i	$0.661164\pi$
$332$	0	0
$333$	−560658.	−0.277069
$334$	0	0
$335$	−257516.	−0.125370
$336$	0	0
$337$	347418.	0.166639	0.0833196	−	0.996523i	$-0.473448\pi$
$337$	347418.	0.166639	0.0833196	+	0.996523i	$0.473448\pi$
$338$	0	0
$339$	1.14828e6	0.542686
$340$	0	0
$341$	3.04924e6	1.42006
$342$	0	0
$343$	−371884.	−0.170676
$344$	0	0
$345$	−42582.7	−0.0192613
$346$	0	0
$347$	4.28700e6	1.91130	0.955652	−	0.294498i	$-0.0951526\pi$
$347$	4.28700e6	1.91130	0.955652	+	0.294498i	$0.0951526\pi$
$348$	0	0
$349$	2.58234e6	1.13488	0.567439	−	0.823415i	$-0.307934\pi$
$349$	2.58234e6	1.13488	0.567439	+	0.823415i	$0.307934\pi$
$350$	0	0
$351$	−435562.	−0.188704
$352$	0	0
$353$	−2.31876e6	−0.990419	−0.495210	−	0.868774i	$-0.664909\pi$
$353$	−2.31876e6	−0.990419	−0.495210	+	0.868774i	$0.664909\pi$
$354$	0	0
$355$	540380.	0.227577
$356$	0	0
$357$	41166.5	0.0170952
$358$	0	0
$359$	1.41572e6	0.579752	0.289876	−	0.957064i	$-0.406386\pi$
$359$	1.41572e6	0.579752	0.289876	+	0.957064i	$0.406386\pi$
$360$	0	0
$361$	6.15632e6	2.48630
$362$	0	0
$363$	1.68465e6	0.671033
$364$	0	0
$365$	91305.7	0.0358728
$366$	0	0
$367$	−2.64620e6	−1.02555	−0.512776	−	0.858523i	$-0.671383\pi$
$367$	−2.64620e6	−1.02555	−0.512776	+	0.858523i	$0.671383\pi$
$368$	0	0
$369$	−123585.	−0.0472498
$370$	0	0
$371$	−113447.	−0.0427916
$372$	0	0
$373$	3.49265e6	1.29982	0.649910	−	0.760011i	$-0.274806\pi$
$373$	3.49265e6	1.29982	0.649910	+	0.760011i	$0.274806\pi$
$374$	0	0
$375$	679703.	0.249598
$376$	0	0
$377$	−356273.	−0.129101
$378$	0	0
$379$	−1.42204e6	−0.508526	−0.254263	−	0.967135i	$-0.581833\pi$
$379$	−1.42204e6	−0.508526	−0.254263	+	0.967135i	$0.581833\pi$
$380$	0	0
$381$	−1.23840e6	−0.437068
$382$	0	0
$383$	2.81442e6	0.980373	0.490187	−	0.871618i	$-0.336929\pi$
$383$	2.81442e6	0.980373	0.490187	+	0.871618i	$0.336929\pi$
$384$	0	0
$385$	−60654.1	−0.0208549
$386$	0	0
$387$	−214530.	−0.0728134
$388$	0	0
$389$	−165325.	−0.0553943	−0.0276971	−	0.999616i	$-0.508817\pi$
$389$	−165325.	−0.0553943	−0.0276971	+	0.999616i	$0.508817\pi$
$390$	0	0
$391$	126844.	0.0419591
$392$	0	0
$393$	893695.	0.291882
$394$	0	0
$395$	557006.	0.179625
$396$	0	0
$397$	−3.79445e6	−1.20830	−0.604148	−	0.796872i	$-0.706486\pi$
$397$	−3.79445e6	−1.20830	−0.604148	+	0.796872i	$0.706486\pi$
$398$	0	0
$399$	−367628.	−0.115605
$400$	0	0
$401$	4.59396e6	1.42668	0.713339	−	0.700819i	$-0.247182\pi$
$401$	4.59396e6	1.42668	0.713339	+	0.700819i	$0.247182\pi$
$402$	0	0
$403$	589100.	0.180687
$404$	0	0
$405$	170485.	0.0516473
$406$	0	0
$407$	2.69287e6	0.805804
$408$	0	0
$409$	3.83956e6	1.13494	0.567471	−	0.823393i	$-0.307922\pi$
$409$	3.83956e6	1.13494	0.567471	+	0.823393i	$0.307922\pi$
$410$	0	0
$411$	−4.05850e6	−1.18512
$412$	0	0
$413$	304885.	0.0879551
$414$	0	0
$415$	−57421.2	−0.0163663
$416$	0	0
$417$	−3.72785e6	−1.04983
$418$	0	0
$419$	−2.87922e6	−0.801199	−0.400600	−	0.916253i	$-0.631198\pi$
$419$	−2.87922e6	−0.801199	−0.400600	+	0.916253i	$0.631198\pi$
$420$	0	0
$421$	1.41982e6	0.390417	0.195208	−	0.980762i	$-0.437462\pi$
$421$	1.41982e6	0.390417	0.195208	+	0.980762i	$0.437462\pi$
$422$	0	0
$423$	−1.04329e6	−0.283501
$424$	0	0
$425$	−996531.	−0.267620
$426$	0	0
$427$	−407441.	−0.108142
$428$	0	0
$429$	676074.	0.177358
$430$	0	0
$431$	4.77968e6	1.23938	0.619691	−	0.784846i	$-0.287258\pi$
$431$	4.77968e6	1.23938	0.619691	+	0.784846i	$0.287258\pi$
$432$	0	0
$433$	213516.	0.0547282	0.0273641	−	0.999626i	$-0.491289\pi$
$433$	213516.	0.0547282	0.0273641	+	0.999626i	$0.491289\pi$
$434$	0	0
$435$	365497.	0.0926106
$436$	0	0
$437$	−1.13275e6	−0.283745
$438$	0	0
$439$	−555761.	−0.137634	−0.0688172	−	0.997629i	$-0.521923\pi$
$439$	−555761.	−0.137634	−0.0688172	+	0.997629i	$0.521923\pi$
$440$	0	0
$441$	1.93573e6	0.473967
$442$	0	0
$443$	1.23542e6	0.299092	0.149546	−	0.988755i	$-0.452219\pi$
$443$	1.23542e6	0.299092	0.149546	+	0.988755i	$0.452219\pi$
$444$	0	0
$445$	419674.	0.100464
$446$	0	0
$447$	272465.	0.0644974
$448$	0	0
$449$	1.61043e6	0.376986	0.188493	−	0.982074i	$-0.439640\pi$
$449$	1.61043e6	0.376986	0.188493	+	0.982074i	$0.439640\pi$
$450$	0	0
$451$	593584.	0.137417
$452$	0	0
$453$	−3.14011e6	−0.718951
$454$	0	0
$455$	−11718.1	−0.00265356
$456$	0	0
$457$	4.44189e6	0.994894	0.497447	−	0.867494i	$-0.334271\pi$
$457$	4.44189e6	0.994894	0.497447	+	0.867494i	$0.334271\pi$
$458$	0	0
$459$	−1.33102e6	−0.294886
$460$	0	0
$461$	−8.65627e6	−1.89705	−0.948524	−	0.316705i	$-0.897423\pi$
$461$	−8.65627e6	−1.89705	−0.948524	+	0.316705i	$0.897423\pi$
$462$	0	0
$463$	8.50880e6	1.84466	0.922329	−	0.386404i	$-0.126283\pi$
$463$	8.50880e6	1.84466	0.922329	+	0.386404i	$0.126283\pi$
$464$	0	0
$465$	−604353.	−0.129616
$466$	0	0
$467$	692449.	0.146925	0.0734625	−	0.997298i	$-0.476595\pi$
$467$	692449.	0.146925	0.0734625	+	0.997298i	$0.476595\pi$
$468$	0	0
$469$	291728.	0.0612416
$470$	0	0
$471$	−1.15194e6	−0.239264
$472$	0	0
$473$	1.03040e6	0.211764
$474$	0	0
$475$	8.89927e6	1.80976
$476$	0	0
$477$	1.18539e6	0.238543
$478$	0	0
$479$	7.63881e6	1.52120	0.760601	−	0.649220i	$-0.224905\pi$
$479$	7.63881e6	1.52120	0.760601	+	0.649220i	$0.224905\pi$
$480$	0	0
$481$	520250.	0.102530
$482$	0	0
$483$	48240.1	0.00940893
$484$	0	0
$485$	2270.81	0.000438356 0
$486$	0	0
$487$	−4.82019e6	−0.920962	−0.460481	−	0.887670i	$-0.652323\pi$
$487$	−4.82019e6	−0.920962	−0.460481	+	0.887670i	$0.652323\pi$
$488$	0	0
$489$	−5.00557e6	−0.946633
$490$	0	0
$491$	−66664.0	−0.0124792	−0.00623961	−	0.999981i	$-0.501986\pi$
$491$	−66664.0	−0.0124792	−0.00623961	+	0.999981i	$0.501986\pi$
$492$	0	0
$493$	−1.08873e6	−0.201744
$494$	0	0
$495$	633767.	0.116256
$496$	0	0
$497$	−612172.	−0.111169
$498$	0	0
$499$	2.93211e6	0.527144	0.263572	−	0.964640i	$-0.415099\pi$
$499$	2.93211e6	0.527144	0.263572	+	0.964640i	$0.415099\pi$
$500$	0	0
$501$	5.26490e6	0.937122
$502$	0	0
$503$	3.90093e6	0.687462	0.343731	−	0.939068i	$-0.388309\pi$
$503$	3.90093e6	0.687462	0.343731	+	0.939068i	$0.388309\pi$
$504$	0	0
$505$	640351.	0.111735
$506$	0	0
$507$	−4.05323e6	−0.700296
$508$	0	0
$509$	2.09591e6	0.358573	0.179287	−	0.983797i	$-0.442621\pi$
$509$	2.09591e6	0.358573	0.179287	+	0.983797i	$0.442621\pi$
$510$	0	0
$511$	−103436.	−0.0175235
$512$	0	0
$513$	1.18864e7	1.99414
$514$	0	0
$515$	2.06678e6	0.343382
$516$	0	0
$517$	5.01098e6	0.824511
$518$	0	0
$519$	3.34032e6	0.544339
$520$	0	0
$521$	3.34502e6	0.539889	0.269944	−	0.962876i	$-0.412995\pi$
$521$	3.34502e6	0.539889	0.269944	+	0.962876i	$0.412995\pi$
$522$	0	0
$523$	1.04450e7	1.66976	0.834878	−	0.550434i	$-0.185538\pi$
$523$	1.04450e7	1.66976	0.834878	+	0.550434i	$0.185538\pi$
$524$	0	0
$525$	−378992.	−0.0600112
$526$	0	0
$527$	1.80022e6	0.282357
$528$	0	0
$529$	−6.28770e6	−0.976906
$530$	0	0
$531$	−3.18570e6	−0.490308
$532$	0	0
$533$	114678.	0.0174848
$534$	0	0
$535$	−1.55205e6	−0.234434
$536$	0	0
$537$	−2.38411e6	−0.356771
$538$	0	0
$539$	−9.29739e6	−1.37844
$540$	0	0
$541$	5.88573e6	0.864584	0.432292	−	0.901734i	$-0.357705\pi$
$541$	5.88573e6	0.864584	0.432292	+	0.901734i	$0.357705\pi$
$542$	0	0
$543$	−2.11584e6	−0.307952
$544$	0	0
$545$	841222.	0.121316
$546$	0	0
$547$	2.94922e6	0.421444	0.210722	−	0.977546i	$-0.432419\pi$
$547$	2.94922e6	0.421444	0.210722	+	0.977546i	$0.432419\pi$
$548$	0	0
$549$	4.25730e6	0.602842
$550$	0	0
$551$	9.72260e6	1.36428
$552$	0	0
$553$	−631007.	−0.0877448
$554$	0	0
$555$	−533720.	−0.0735498
$556$	0	0
$557$	−1.44836e7	−1.97806	−0.989029	−	0.147723i	$-0.952805\pi$
$557$	−1.44836e7	−1.97806	−0.989029	+	0.147723i	$0.952805\pi$
$558$	0	0
$559$	199069.	0.0269447
$560$	0	0
$561$	2.06600e6	0.277155
$562$	0	0
$563$	−6.77740e6	−0.901139	−0.450570	−	0.892741i	$-0.648779\pi$
$563$	−6.77740e6	−0.901139	−0.450570	+	0.892741i	$0.648779\pi$
$564$	0	0
$565$	−998842.	−0.131636
$566$	0	0
$567$	−193134.	−0.0252291
$568$	0	0
$569$	1.03288e7	1.33742	0.668710	−	0.743523i	$-0.266847\pi$
$569$	1.03288e7	1.33742	0.668710	+	0.743523i	$0.266847\pi$
$570$	0	0
$571$	9.52868e6	1.22304	0.611522	−	0.791227i	$-0.290557\pi$
$571$	9.52868e6	1.22304	0.611522	+	0.791227i	$0.290557\pi$
$572$	0	0
$573$	7.39155e6	0.940478
$574$	0	0
$575$	−1.16776e6	−0.147294
$576$	0	0
$577$	5.58573e6	0.698458	0.349229	−	0.937037i	$-0.386444\pi$
$577$	5.58573e6	0.698458	0.349229	+	0.937037i	$0.386444\pi$
$578$	0	0
$579$	5.07518e6	0.629151
$580$	0	0
$581$	65049.9	0.00799478
$582$	0	0
$583$	−5.69350e6	−0.693758
$584$	0	0
$585$	122441.	0.0147923
$586$	0	0
$587$	1.34089e7	1.60619	0.803096	−	0.595849i	$-0.203184\pi$
$587$	1.34089e7	1.60619	0.803096	+	0.595849i	$0.203184\pi$
$588$	0	0
$589$	−1.60764e7	−1.90942
$590$	0	0
$591$	−1.09656e7	−1.29141
$592$	0	0
$593$	−1.46278e7	−1.70821	−0.854106	−	0.520099i	$-0.825895\pi$
$593$	−1.46278e7	−1.70821	−0.854106	+	0.520099i	$0.825895\pi$
$594$	0	0
$595$	−35809.1	−0.00414669
$596$	0	0
$597$	−2.92796e6	−0.336224
$598$	0	0
$599$	−1.02632e7	−1.16873	−0.584367	−	0.811489i	$-0.698657\pi$
$599$	−1.02632e7	−1.16873	−0.584367	+	0.811489i	$0.698657\pi$
$600$	0	0
$601$	9.07707e6	1.02508	0.512542	−	0.858662i	$-0.328704\pi$
$601$	9.07707e6	1.02508	0.512542	+	0.858662i	$0.328704\pi$
$602$	0	0
$603$	−3.04823e6	−0.341393
$604$	0	0
$605$	−1.46541e6	−0.162769
$606$	0	0
$607$	1.20707e7	1.32972	0.664862	−	0.746966i	$-0.268490\pi$
$607$	1.20707e7	1.32972	0.664862	+	0.746966i	$0.268490\pi$
$608$	0	0
$609$	−414055.	−0.0452392
$610$	0	0
$611$	968100.	0.104910
$612$	0	0
$613$	1.40958e6	0.151509	0.0757544	−	0.997127i	$-0.475864\pi$
$613$	1.40958e6	0.151509	0.0757544	+	0.997127i	$0.475864\pi$
$614$	0	0
$615$	−117647.	−0.0125428
$616$	0	0
$617$	1.51488e7	1.60201	0.801005	−	0.598658i	$-0.204299\pi$
$617$	1.51488e7	1.60201	0.801005	+	0.598658i	$0.204299\pi$
$618$	0	0
$619$	−1.58021e7	−1.65763	−0.828816	−	0.559521i	$-0.810985\pi$
$619$	−1.58021e7	−1.65763	−0.828816	+	0.559521i	$0.810985\pi$
$620$	0	0
$621$	−1.55973e6	−0.162301
$622$	0	0
$623$	−475430.	−0.0490757
$624$	0	0
$625$	8.87414e6	0.908712
$626$	0	0
$627$	−1.84499e7	−1.87424
$628$	0	0
$629$	1.58982e6	0.160222
$630$	0	0
$631$	1.08922e7	1.08904	0.544519	−	0.838749i	$-0.316712\pi$
$631$	1.08922e7	1.08904	0.544519	+	0.838749i	$0.316712\pi$
$632$	0	0
$633$	6.99944e6	0.694311
$634$	0	0
$635$	1.07724e6	0.106017
$636$	0	0
$637$	−1.79622e6	−0.175392
$638$	0	0
$639$	6.39650e6	0.619712
$640$	0	0
$641$	2.41942e6	0.232577	0.116289	−	0.993215i	$-0.462900\pi$
$641$	2.41942e6	0.232577	0.116289	+	0.993215i	$0.462900\pi$
$642$	0	0
$643$	−1.38769e7	−1.32363	−0.661814	−	0.749668i	$-0.730213\pi$
$643$	−1.38769e7	−1.32363	−0.661814	+	0.749668i	$0.730213\pi$
$644$	0	0
$645$	−204223.	−0.0193288
$646$	0	0
$647$	7.58249e6	0.712116	0.356058	−	0.934464i	$-0.384120\pi$
$647$	7.58249e6	0.712116	0.356058	+	0.934464i	$0.384120\pi$
$648$	0	0
$649$	1.53011e7	1.42597
$650$	0	0
$651$	684645.	0.0633159
$652$	0	0
$653$	3.26919e6	0.300025	0.150012	−	0.988684i	$-0.452069\pi$
$653$	3.26919e6	0.300025	0.150012	+	0.988684i	$0.452069\pi$
$654$	0	0
$655$	−777389.	−0.0708003
$656$	0	0
$657$	1.08079e6	0.0976850
$658$	0	0
$659$	−1.94787e6	−0.174721	−0.0873606	−	0.996177i	$-0.527843\pi$
$659$	−1.94787e6	−0.174721	−0.0873606	+	0.996177i	$0.527843\pi$
$660$	0	0
$661$	−1.04897e7	−0.933810	−0.466905	−	0.884307i	$-0.654631\pi$
$661$	−1.04897e7	−0.933810	−0.466905	+	0.884307i	$0.654631\pi$
$662$	0	0
$663$	399142.	0.0352650
$664$	0	0
$665$	319785.	0.0280417
$666$	0	0
$667$	−1.27580e6	−0.111037
$668$	0	0
$669$	8.78494e6	0.758881
$670$	0	0
$671$	−2.04480e7	−1.75325
$672$	0	0
$673$	−1.81198e6	−0.154211	−0.0771054	−	0.997023i	$-0.524568\pi$
$673$	−1.81198e6	−0.154211	−0.0771054	+	0.997023i	$0.524568\pi$
$674$	0	0
$675$	1.22538e7	1.03517
$676$	0	0
$677$	−1.71538e7	−1.43843	−0.719216	−	0.694786i	$-0.755499\pi$
$677$	−1.71538e7	−1.43843	−0.719216	+	0.694786i	$0.755499\pi$
$678$	0	0
$679$	−2572.50	−0.000214132 0
$680$	0	0
$681$	−8.96308e6	−0.740610
$682$	0	0
$683$	−1.80610e7	−1.48146	−0.740732	−	0.671801i	$-0.765521\pi$
$683$	−1.80610e7	−1.48146	−0.740732	+	0.671801i	$0.765521\pi$
$684$	0	0
$685$	3.53033e6	0.287467
$686$	0	0
$687$	−7.92140e6	−0.640339
$688$	0	0
$689$	−1.09996e6	−0.0882732
$690$	0	0
$691$	1.44368e7	1.15021	0.575105	−	0.818080i	$-0.304961\pi$
$691$	1.44368e7	1.15021	0.575105	+	0.818080i	$0.304961\pi$
$692$	0	0
$693$	−717966.	−0.0567899
$694$	0	0
$695$	3.24271e6	0.254651
$696$	0	0
$697$	350442.	0.0273233
$698$	0	0
$699$	−2.47145e6	−0.191319
$700$	0	0
$701$	−1.39926e7	−1.07549	−0.537743	−	0.843109i	$-0.680723\pi$
$701$	−1.39926e7	−1.07549	−0.537743	+	0.843109i	$0.680723\pi$
$702$	0	0
$703$	−1.41975e7	−1.08349
$704$	0	0
$705$	−993165.	−0.0752573
$706$	0	0
$707$	−725425.	−0.0545813
$708$	0	0
$709$	−1.37678e7	−1.02860	−0.514302	−	0.857609i	$-0.671949\pi$
$709$	−1.37678e7	−1.02860	−0.514302	+	0.857609i	$0.671949\pi$
$710$	0	0
$711$	6.59331e6	0.489136
$712$	0	0
$713$	2.10955e6	0.155405
$714$	0	0
$715$	−588090.	−0.0430208
$716$	0	0
$717$	−1.97362e7	−1.43373
$718$	0	0
$719$	8.26505e6	0.596243	0.298121	−	0.954528i	$-0.403640\pi$
$719$	8.26505e6	0.596243	0.298121	+	0.954528i	$0.403640\pi$
$720$	0	0
$721$	−2.34137e6	−0.167738
$722$	0	0
$723$	1.11060e7	0.790158
$724$	0	0
$725$	1.00232e7	0.708206
$726$	0	0
$727$	−7.18688e6	−0.504318	−0.252159	−	0.967686i	$-0.581141\pi$
$727$	−7.18688e6	−0.504318	−0.252159	+	0.967686i	$0.581141\pi$
$728$	0	0
$729$	1.30957e7	0.912663
$730$	0	0
$731$	608330.	0.0421062
$732$	0	0
$733$	−1.10728e6	−0.0761196	−0.0380598	−	0.999275i	$-0.512118\pi$
$733$	−1.10728e6	−0.0761196	−0.0380598	+	0.999275i	$0.512118\pi$
$734$	0	0
$735$	1.84272e6	0.125818
$736$	0	0
$737$	1.46408e7	0.992878
$738$	0	0
$739$	−1.98449e7	−1.33671	−0.668356	−	0.743841i	$-0.733002\pi$
$739$	−1.98449e7	−1.33671	−0.668356	+	0.743841i	$0.733002\pi$
$740$	0	0
$741$	−3.56444e6	−0.238477
$742$	0	0
$743$	2.19204e7	1.45672	0.728360	−	0.685194i	$-0.240283\pi$
$743$	2.19204e7	1.45672	0.728360	+	0.685194i	$0.240283\pi$
$744$	0	0
$745$	−237006.	−0.0156448
$746$	0	0
$747$	−679697.	−0.0445671
$748$	0	0
$749$	1.75825e6	0.114518
$750$	0	0
$751$	1.62135e7	1.04900	0.524502	−	0.851409i	$-0.324252\pi$
$751$	1.62135e7	1.04900	0.524502	+	0.851409i	$0.324252\pi$
$752$	0	0
$753$	−1.42140e6	−0.0913545
$754$	0	0
$755$	2.73146e6	0.174392
$756$	0	0
$757$	1.31863e7	0.836342	0.418171	−	0.908368i	$-0.362671\pi$
$757$	1.31863e7	0.836342	0.418171	+	0.908368i	$0.362671\pi$
$758$	0	0
$759$	2.42099e6	0.152542
$760$	0	0
$761$	−1.92620e7	−1.20570	−0.602851	−	0.797854i	$-0.705969\pi$
$761$	−1.92620e7	−1.20570	−0.602851	+	0.797854i	$0.705969\pi$
$762$	0	0
$763$	−952983.	−0.0592617
$764$	0	0
$765$	374165.	0.0231158
$766$	0	0
$767$	2.95610e6	0.181439
$768$	0	0
$769$	−456438.	−0.0278334	−0.0139167	−	0.999903i	$-0.504430\pi$
$769$	−456438.	−0.0278334	−0.0139167	+	0.999903i	$0.504430\pi$
$770$	0	0
$771$	1.75710e7	1.06454
$772$	0	0
$773$	−6.14461e6	−0.369867	−0.184933	−	0.982751i	$-0.559207\pi$
$773$	−6.14461e6	−0.369867	−0.184933	+	0.982751i	$0.559207\pi$
$774$	0	0
$775$	−1.65734e7	−0.991191
$776$	0	0
$777$	604628.	0.0359282
$778$	0	0
$779$	−3.12953e6	−0.184772
$780$	0	0
$781$	−3.07227e7	−1.80232
$782$	0	0
$783$	1.33875e7	0.780361
$784$	0	0
$785$	1.00203e6	0.0580369
$786$	0	0
$787$	−8.50558e6	−0.489516	−0.244758	−	0.969584i	$-0.578709\pi$
$787$	−8.50558e6	−0.489516	−0.244758	+	0.969584i	$0.578709\pi$
$788$	0	0
$789$	1.20897e6	0.0691389
$790$	0	0
$791$	1.13154e6	0.0643029
$792$	0	0
$793$	−3.95047e6	−0.223082
$794$	0	0
$795$	1.12844e6	0.0633228
$796$	0	0
$797$	1.28439e7	0.716227	0.358113	−	0.933678i	$-0.383420\pi$
$797$	1.28439e7	0.716227	0.358113	+	0.933678i	$0.383420\pi$
$798$	0	0
$799$	2.95840e6	0.163942
$800$	0	0
$801$	4.96770e6	0.273573
$802$	0	0
$803$	−5.19108e6	−0.284099
$804$	0	0
$805$	−41962.1	−0.00228227
$806$	0	0
$807$	9.34670e6	0.505213
$808$	0	0
$809$	2.98801e7	1.60513	0.802567	−	0.596562i	$-0.203467\pi$
$809$	2.98801e7	1.60513	0.802567	+	0.596562i	$0.203467\pi$
$810$	0	0
$811$	2.05001e7	1.09447	0.547234	−	0.836980i	$-0.315681\pi$
$811$	2.05001e7	1.09447	0.547234	+	0.836980i	$0.315681\pi$
$812$	0	0
$813$	−2.51098e7	−1.33234
$814$	0	0
$815$	4.35415e6	0.229620
$816$	0	0
$817$	−5.43254e6	−0.284740
$818$	0	0
$819$	−138708.	−0.00722589
$820$	0	0
$821$	2.20698e7	1.14272	0.571360	−	0.820699i	$-0.306416\pi$
$821$	2.20698e7	1.14272	0.571360	+	0.820699i	$0.306416\pi$
$822$	0	0
$823$	−1.59817e6	−0.0822476	−0.0411238	−	0.999154i	$-0.513094\pi$
$823$	−1.59817e6	−0.0822476	−0.0411238	+	0.999154i	$0.513094\pi$
$824$	0	0
$825$	−1.90202e7	−0.972929
$826$	0	0
$827$	1.49218e7	0.758680	0.379340	−	0.925257i	$-0.376151\pi$
$827$	1.49218e7	0.758680	0.379340	+	0.925257i	$0.376151\pi$
$828$	0	0
$829$	7.71986e6	0.390142	0.195071	−	0.980789i	$-0.437506\pi$
$829$	7.71986e6	0.390142	0.195071	+	0.980789i	$0.437506\pi$
$830$	0	0
$831$	−1.77439e7	−0.891344
$832$	0	0
$833$	−5.48902e6	−0.274083
$834$	0	0
$835$	−4.57973e6	−0.227313
$836$	0	0
$837$	−2.21364e7	−1.09218
$838$	0	0
$839$	−3.50556e7	−1.71930	−0.859652	−	0.510880i	$-0.829320\pi$
$839$	−3.50556e7	−1.71930	−0.859652	+	0.510880i	$0.829320\pi$
$840$	0	0
$841$	−9.56068e6	−0.466121
$842$	0	0
$843$	−2.54692e7	−1.23437
$844$	0	0
$845$	3.52575e6	0.169867
$846$	0	0
$847$	1.66010e6	0.0795108
$848$	0	0
$849$	2.04284e7	0.972668
$850$	0	0
$851$	1.86300e6	0.0881837
$852$	0	0
$853$	−2.21072e7	−1.04031	−0.520153	−	0.854073i	$-0.674125\pi$
$853$	−2.21072e7	−1.04031	−0.520153	+	0.854073i	$0.674125\pi$
$854$	0	0
$855$	−3.34139e6	−0.156319
$856$	0	0
$857$	1.60053e7	0.744410	0.372205	−	0.928151i	$-0.378602\pi$
$857$	1.60053e7	0.744410	0.372205	+	0.928151i	$0.378602\pi$
$858$	0	0
$859$	2.09057e7	0.966677	0.483338	−	0.875434i	$-0.339424\pi$
$859$	2.09057e7	0.966677	0.483338	+	0.875434i	$0.339424\pi$
$860$	0	0
$861$	133277.	0.00612700
$862$	0	0
$863$	−1.41874e7	−0.648449	−0.324224	−	0.945980i	$-0.605103\pi$
$863$	−1.41874e7	−0.648449	−0.324224	+	0.945980i	$0.605103\pi$
$864$	0	0
$865$	−2.90561e6	−0.132037
$866$	0	0
$867$	−1.47797e7	−0.667755
$868$	0	0
$869$	−3.16680e7	−1.42256
$870$	0	0
$871$	2.82854e6	0.126333
$872$	0	0
$873$	26879.7	0.00119368
$874$	0	0
$875$	669798.	0.0295749
$876$	0	0
$877$	2.06566e7	0.906901	0.453450	−	0.891282i	$-0.350193\pi$
$877$	2.06566e7	0.906901	0.453450	+	0.891282i	$0.350193\pi$
$878$	0	0
$879$	−2.17477e7	−0.949382
$880$	0	0
$881$	−1.35902e7	−0.589909	−0.294955	−	0.955511i	$-0.595305\pi$
$881$	−1.35902e7	−0.589909	−0.294955	+	0.955511i	$0.595305\pi$
$882$	0	0
$883$	−1.20900e7	−0.521824	−0.260912	−	0.965363i	$-0.584023\pi$
$883$	−1.20900e7	−0.521824	−0.260912	+	0.965363i	$0.584023\pi$
$884$	0	0
$885$	−3.03264e6	−0.130155
$886$	0	0
$887$	−1.83559e7	−0.783371	−0.391685	−	0.920099i	$-0.628108\pi$
$887$	−1.83559e7	−0.783371	−0.391685	+	0.920099i	$0.628108\pi$
$888$	0	0
$889$	−1.22035e6	−0.0517883
$890$	0	0
$891$	−9.69271e6	−0.409026
$892$	0	0
$893$	−2.64192e7	−1.10864
$894$	0	0
$895$	2.07384e6	0.0865401
$896$	0	0
$897$	467725.	0.0194093
$898$	0	0
$899$	−1.81067e7	−0.747206
$900$	0	0
$901$	−3.36134e6	−0.137943
$902$	0	0
$903$	231355.	0.00944190
$904$	0	0
$905$	1.84049e6	0.0746984
$906$	0	0
$907$	−1.78761e7	−0.721531	−0.360765	−	0.932657i	$-0.617485\pi$
$907$	−1.78761e7	−0.721531	−0.360765	+	0.932657i	$0.617485\pi$
$908$	0	0
$909$	7.57987e6	0.304265
$910$	0	0
$911$	1.47994e7	0.590809	0.295405	−	0.955372i	$-0.404546\pi$
$911$	1.47994e7	0.590809	0.295405	+	0.955372i	$0.404546\pi$
$912$	0	0
$913$	3.26462e6	0.129615
$914$	0	0
$915$	4.05275e6	0.160028
$916$	0	0
$917$	880670.	0.0345852
$918$	0	0
$919$	2.50852e6	0.0979780	0.0489890	−	0.998799i	$-0.484400\pi$
$919$	2.50852e6	0.0979780	0.0489890	+	0.998799i	$0.484400\pi$
$920$	0	0
$921$	2.65225e7	1.03030
$922$	0	0
$923$	−5.93549e6	−0.229325
$924$	0	0
$925$	−1.46364e7	−0.562445
$926$	0	0
$927$	2.44646e7	0.935060
$928$	0	0
$929$	−9.51733e6	−0.361806	−0.180903	−	0.983501i	$-0.557902\pi$
$929$	−9.51733e6	−0.361806	−0.180903	+	0.983501i	$0.557902\pi$
$930$	0	0
$931$	4.90183e7	1.85347
$932$	0	0
$933$	−1.41906e7	−0.533698
$934$	0	0
$935$	−1.79713e6	−0.0672281
$936$	0	0
$937$	4.27838e6	0.159195	0.0795977	−	0.996827i	$-0.474636\pi$
$937$	4.27838e6	0.159195	0.0795977	+	0.996827i	$0.474636\pi$
$938$	0	0
$939$	−1.37238e7	−0.507939
$940$	0	0
$941$	−3.20489e7	−1.17988	−0.589941	−	0.807446i	$-0.700849\pi$
$941$	−3.20489e7	−1.17988	−0.589941	+	0.807446i	$0.700849\pi$
$942$	0	0
$943$	410657.	0.0150384
$944$	0	0
$945$	440327.	0.0160397
$946$	0	0
$947$	−1.34081e7	−0.485841	−0.242920	−	0.970046i	$-0.578105\pi$
$947$	−1.34081e7	−0.485841	−0.242920	+	0.970046i	$0.578105\pi$
$948$	0	0
$949$	−1.00289e6	−0.0361485
$950$	0	0
$951$	7.77389e6	0.278732
$952$	0	0
$953$	4.23589e7	1.51082	0.755409	−	0.655254i	$-0.227438\pi$
$953$	4.23589e7	1.51082	0.755409	+	0.655254i	$0.227438\pi$
$954$	0	0
$955$	−6.42961e6	−0.228127
$956$	0	0
$957$	−2.07799e7	−0.733440
$958$	0	0
$959$	−3.99935e6	−0.140425
$960$	0	0
$961$	1.31048e6	0.0457745
$962$	0	0
$963$	−1.83717e7	−0.638385
$964$	0	0
$965$	−4.41470e6	−0.152610
$966$	0	0
$967$	4.12202e7	1.41757	0.708785	−	0.705425i	$-0.249244\pi$
$967$	4.12202e7	1.41757	0.708785	+	0.705425i	$0.249244\pi$
$968$	0	0
$969$	−1.08925e7	−0.372665
$970$	0	0
$971$	1.13026e7	0.384708	0.192354	−	0.981326i	$-0.438388\pi$
$971$	1.13026e7	0.384708	0.192354	+	0.981326i	$0.438388\pi$
$972$	0	0
$973$	−3.67352e6	−0.124394
$974$	0	0
$975$	−3.67463e6	−0.123795
$976$	0	0
$977$	−2.79073e7	−0.935367	−0.467684	−	0.883896i	$-0.654911\pi$
$977$	−2.79073e7	−0.935367	−0.467684	+	0.883896i	$0.654911\pi$
$978$	0	0
$979$	−2.38601e7	−0.795638
$980$	0	0
$981$	9.95758e6	0.330355
$982$	0	0
$983$	1.22161e7	0.403226	0.201613	−	0.979465i	$-0.435382\pi$
$983$	1.22161e7	0.403226	0.201613	+	0.979465i	$0.435382\pi$
$984$	0	0
$985$	9.53852e6	0.313249
$986$	0	0
$987$	1.12511e6	0.0367623
$988$	0	0
$989$	712858.	0.0231746
$990$	0	0
$991$	−2.47723e7	−0.801277	−0.400638	−	0.916236i	$-0.631212\pi$
$991$	−2.47723e7	−0.801277	−0.400638	+	0.916236i	$0.631212\pi$
$992$	0	0
$993$	−2.17852e7	−0.701113
$994$	0	0
$995$	2.54691e6	0.0815561
$996$	0	0
$997$	2.12381e7	0.676670	0.338335	−	0.941026i	$-0.390136\pi$
$997$	2.12381e7	0.676670	0.338335	+	0.941026i	$0.390136\pi$
$998$	0	0
$999$	−1.95493e7	−0.619750

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.6.a.e.1.6		8
4.3	odd	2		43.6.a.a.1.7	✓	8
12.11	even	2		387.6.a.c.1.2		8
20.19	odd	2		1075.6.a.a.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.a.1.7	✓	8	4.3	odd	2
387.6.a.c.1.2		8	12.11	even	2
688.6.a.e.1.6		8	1.1	even	1	trivial
1075.6.a.a.1.2		8	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)

\(f(q)\)	\(=\)	\(q+11.2683 q^{3} -9.80186 q^{5} +11.1041 q^{7} -116.025 q^{9} +O(q^{10})\)	\(q+11.2683 q^{3} -9.80186 q^{5} +11.1041 q^{7} -116.025 q^{9} +557.274 q^{11} +107.663 q^{13} -110.450 q^{15} +329.005 q^{17} -2938.10 q^{19} +125.124 q^{21} +385.537 q^{23} -3028.92 q^{25} -4045.61 q^{27} -3309.15 q^{29} +5471.71 q^{31} +6279.54 q^{33} -108.841 q^{35} +4832.22 q^{37} +1213.18 q^{39} +1065.16 q^{41} +1849.00 q^{43} +1137.26 q^{45} +8991.95 q^{47} -16683.7 q^{49} +3707.33 q^{51} -10216.7 q^{53} -5462.32 q^{55} -33107.4 q^{57} +27457.0 q^{59} -36692.9 q^{61} -1288.35 q^{63} -1055.30 q^{65} +26272.2 q^{67} +4344.35 q^{69} -55130.3 q^{71} -9315.14 q^{73} -34130.9 q^{75} +6188.02 q^{77} -56826.5 q^{79} -17393.1 q^{81} +5858.19 q^{83} -3224.86 q^{85} -37288.5 q^{87} -42815.7 q^{89} +1195.50 q^{91} +61657.0 q^{93} +28798.8 q^{95} -231.671 q^{97} -64657.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 26 q^{3} - 212 q^{5} + 136 q^{7} + 546 q^{9}+O(q^{10}) \) 8 * q + 26 * q^3 - 212 * q^5 + 136 * q^7 + 546 * q^9	\( 8 q + 26 q^{3} - 212 q^{5} + 136 q^{7} + 546 q^{9} + 532 q^{11} - 2492 q^{13} + 1780 q^{15} - 2534 q^{17} + 1678 q^{19} - 2256 q^{21} + 2488 q^{23} + 4378 q^{25} + 8960 q^{27} - 4360 q^{29} - 5704 q^{31} - 12852 q^{33} - 5640 q^{35} - 3772 q^{37} - 11120 q^{39} - 10698 q^{41} + 14792 q^{43} - 44912 q^{45} + 77864 q^{47} + 7188 q^{49} + 80246 q^{51} - 62352 q^{53} + 49552 q^{55} - 808 q^{57} + 26224 q^{59} - 82540 q^{61} + 61768 q^{63} - 5000 q^{65} - 27784 q^{67} - 93776 q^{69} + 9504 q^{71} + 14260 q^{73} - 167420 q^{75} - 218140 q^{77} - 160248 q^{79} + 161076 q^{81} + 77176 q^{83} + 141096 q^{85} - 268136 q^{87} - 265692 q^{89} - 401148 q^{91} - 123860 q^{93} - 135884 q^{95} + 144742 q^{97} - 239516 q^{99}+O(q^{100}) \) 8 * q + 26 * q^3 - 212 * q^5 + 136 * q^7 + 546 * q^9 + 532 * q^11 - 2492 * q^13 + 1780 * q^15 - 2534 * q^17 + 1678 * q^19 - 2256 * q^21 + 2488 * q^23 + 4378 * q^25 + 8960 * q^27 - 4360 * q^29 - 5704 * q^31 - 12852 * q^33 - 5640 * q^35 - 3772 * q^37 - 11120 * q^39 - 10698 * q^41 + 14792 * q^43 - 44912 * q^45 + 77864 * q^47 + 7188 * q^49 + 80246 * q^51 - 62352 * q^53 + 49552 * q^55 - 808 * q^57 + 26224 * q^59 - 82540 * q^61 + 61768 * q^63 - 5000 * q^65 - 27784 * q^67 - 93776 * q^69 + 9504 * q^71 + 14260 * q^73 - 167420 * q^75 - 218140 * q^77 - 160248 * q^79 + 161076 * q^81 + 77176 * q^83 + 141096 * q^85 - 268136 * q^87 - 265692 * q^89 - 401148 * q^91 - 123860 * q^93 - 135884 * q^95 + 144742 * q^97 - 239516 * q^99

Introduction

L-functions

Modular forms

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Database

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