LMFDB - Embedded newform 688.6.a.e.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:	$1$
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			1.7
Root			$10.9591$ 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)

$f(q)$	$=$	$q+25.1057 q^{3} -61.2927 q^{5} +166.001 q^{7} +387.294 q^{9} +O(q^{10})$	$q+25.1057 q^{3} -61.2927 q^{5} +166.001 q^{7} +387.294 q^{9} -607.827 q^{11} -1039.54 q^{13} -1538.79 q^{15} +1439.76 q^{17} +1332.62 q^{19} +4167.57 q^{21} +437.244 q^{23} +631.800 q^{25} +3622.60 q^{27} -87.2656 q^{29} -2654.45 q^{31} -15259.9 q^{33} -10174.7 q^{35} -4671.70 q^{37} -26098.4 q^{39} -9012.91 q^{41} +1849.00 q^{43} -23738.3 q^{45} +8623.49 q^{47} +10749.4 q^{49} +36146.0 q^{51} -28358.3 q^{53} +37255.4 q^{55} +33456.3 q^{57} -48066.0 q^{59} -39750.4 q^{61} +64291.2 q^{63} +63716.5 q^{65} -19233.9 q^{67} +10977.3 q^{69} -17008.5 q^{71} +22236.4 q^{73} +15861.8 q^{75} -100900. q^{77} -36951.1 q^{79} -3164.77 q^{81} +117990. q^{83} -88246.5 q^{85} -2190.86 q^{87} +41537.6 q^{89} -172565. q^{91} -66641.7 q^{93} -81679.9 q^{95} +32593.9 q^{97} -235408. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	25.1057	1.61053	0.805264	−	0.592916i	$-0.202023\pi$
$3$	25.1057	1.61053	0.805264	+	0.592916i	$0.202023\pi$
$4$	0	0
$5$	−61.2927	−1.09644	−0.548219	−	0.836335i	$-0.684694\pi$
$5$	−61.2927	−1.09644	−0.548219	+	0.836335i	$0.684694\pi$
$6$	0	0
$7$	166.001	1.28046	0.640230	−	0.768183i	$-0.278839\pi$
$7$	166.001	1.28046	0.640230	+	0.768183i	$0.278839\pi$
$8$	0	0
$9$	387.294	1.59380
$10$	0	0
$11$	−607.827	−1.51460	−0.757301	−	0.653066i	$-0.773482\pi$
$11$	−607.827	−1.51460	−0.757301	+	0.653066i	$0.773482\pi$
$12$	0	0
$13$	−1039.54	−1.70602	−0.853011	−	0.521894i	$-0.825226\pi$
$13$	−1039.54	−1.70602	−0.853011	+	0.521894i	$0.825226\pi$
$14$	0	0
$15$	−1538.79	−1.76584
$16$	0	0
$17$	1439.76	1.20828	0.604138	−	0.796880i	$-0.293517\pi$
$17$	1439.76	1.20828	0.604138	+	0.796880i	$0.293517\pi$
$18$	0	0
$19$	1332.62	0.846881	0.423440	−	0.905924i	$-0.360822\pi$
$19$	1332.62	0.846881	0.423440	+	0.905924i	$0.360822\pi$
$20$	0	0
$21$	4167.57	2.06222
$22$	0	0
$23$	437.244	0.172347	0.0861737	−	0.996280i	$-0.472536\pi$
$23$	437.244	0.172347	0.0861737	+	0.996280i	$0.472536\pi$
$24$	0	0
$25$	631.800	0.202176
$26$	0	0
$27$	3622.60	0.956336
$28$	0	0
$29$	−87.2656	−0.0192685	−0.00963425	−	0.999954i	$-0.503067\pi$
$29$	−87.2656	−0.0192685	−0.00963425	+	0.999954i	$0.503067\pi$
$30$	0	0
$31$	−2654.45	−0.496101	−0.248050	−	0.968747i	$-0.579790\pi$
$31$	−2654.45	−0.496101	−0.248050	+	0.968747i	$0.579790\pi$
$32$	0	0
$33$	−15259.9	−2.43931
$34$	0	0
$35$	−10174.7	−1.40394
$36$	0	0
$37$	−4671.70	−0.561010	−0.280505	−	0.959853i	$-0.590502\pi$
$37$	−4671.70	−0.561010	−0.280505	+	0.959853i	$0.590502\pi$
$38$	0	0
$39$	−26098.4	−2.74760
$40$	0	0
$41$	−9012.91	−0.837347	−0.418674	−	0.908137i	$-0.637505\pi$
$41$	−9012.91	−0.837347	−0.418674	+	0.908137i	$0.637505\pi$
$42$	0	0
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	0	0
$45$	−23738.3	−1.74751
$46$	0	0
$47$	8623.49	0.569427	0.284714	−	0.958613i	$-0.408101\pi$
$47$	8623.49	0.569427	0.284714	+	0.958613i	$0.408101\pi$
$48$	0	0
$49$	10749.4	0.639577
$50$	0	0
$51$	36146.0	1.94596
$52$	0	0
$53$	−28358.3	−1.38672	−0.693362	−	0.720590i	$-0.743871\pi$
$53$	−28358.3	−1.38672	−0.693362	+	0.720590i	$0.743871\pi$
$54$	0	0
$55$	37255.4	1.66067
$56$	0	0
$57$	33456.3	1.36393
$58$	0	0
$59$	−48066.0	−1.79766	−0.898832	−	0.438294i	$-0.855583\pi$
$59$	−48066.0	−1.79766	−0.898832	+	0.438294i	$0.855583\pi$
$60$	0	0
$61$	−39750.4	−1.36778	−0.683891	−	0.729585i	$-0.739713\pi$
$61$	−39750.4	−1.36778	−0.683891	+	0.729585i	$0.739713\pi$
$62$	0	0
$63$	64291.2	2.04080
$64$	0	0
$65$	63716.5	1.87055
$66$	0	0
$67$	−19233.9	−0.523456	−0.261728	−	0.965142i	$-0.584292\pi$
$67$	−19233.9	−0.523456	−0.261728	+	0.965142i	$0.584292\pi$
$68$	0	0
$69$	10977.3	0.277570
$70$	0	0
$71$	−17008.5	−0.400424	−0.200212	−	0.979753i	$-0.564163\pi$
$71$	−17008.5	−0.400424	−0.200212	+	0.979753i	$0.564163\pi$
$72$	0	0
$73$	22236.4	0.488380	0.244190	−	0.969727i	$-0.421478\pi$
$73$	22236.4	0.488380	0.244190	+	0.969727i	$0.421478\pi$
$74$	0	0
$75$	15861.8	0.325610
$76$	0	0
$77$	−100900.	−1.93939
$78$	0	0
$79$	−36951.1	−0.666131	−0.333066	−	0.942904i	$-0.608083\pi$
$79$	−36951.1	−0.666131	−0.333066	+	0.942904i	$0.608083\pi$
$80$	0	0
$81$	−3164.77	−0.0535957
$82$	0	0
$83$	117990.	1.87997	0.939985	−	0.341216i	$-0.110839\pi$
$83$	117990.	1.87997	0.939985	+	0.341216i	$0.110839\pi$
$84$	0	0
$85$	−88246.5	−1.32480
$86$	0	0
$87$	−2190.86	−0.0310325
$88$	0	0
$89$	41537.6	0.555861	0.277931	−	0.960601i	$-0.410351\pi$
$89$	41537.6	0.555861	0.277931	+	0.960601i	$0.410351\pi$
$90$	0	0
$91$	−172565.	−2.18449
$92$	0	0
$93$	−66641.7	−0.798985
$94$	0	0
$95$	−81679.9	−0.928552
$96$	0	0
$97$	32593.9	0.351728	0.175864	−	0.984415i	$-0.443728\pi$
$97$	32593.9	0.351728	0.175864	+	0.984415i	$0.443728\pi$
$98$	0	0
$99$	−235408.	−2.41398
$100$	0	0
$101$	−7934.83	−0.0773988	−0.0386994	−	0.999251i	$-0.512321\pi$
$101$	−7934.83	−0.0773988	−0.0386994	+	0.999251i	$0.512321\pi$
$102$	0	0
$103$	−132558.	−1.23116	−0.615579	−	0.788075i	$-0.711078\pi$
$103$	−132558.	−1.23116	−0.615579	+	0.788075i	$0.711078\pi$
$104$	0	0
$105$	−255442.	−2.26109
$106$	0	0
$107$	−120642.	−1.01868	−0.509342	−	0.860564i	$-0.670111\pi$
$107$	−120642.	−1.01868	−0.509342	+	0.860564i	$0.670111\pi$
$108$	0	0
$109$	16769.1	0.135190	0.0675950	−	0.997713i	$-0.478467\pi$
$109$	16769.1	0.135190	0.0675950	+	0.997713i	$0.478467\pi$
$110$	0	0
$111$	−117286.	−0.903522
$112$	0	0
$113$	−18201.6	−0.134096	−0.0670478	−	0.997750i	$-0.521358\pi$
$113$	−18201.6	−0.134096	−0.0670478	+	0.997750i	$0.521358\pi$
$114$	0	0
$115$	−26799.9	−0.188968
$116$	0	0
$117$	−402609.	−2.71906
$118$	0	0
$119$	239001.	1.54715
$120$	0	0
$121$	208403.	1.29402
$122$	0	0
$123$	−226275.	−1.34857
$124$	0	0
$125$	152815.	0.874764
$126$	0	0
$127$	155225.	0.853987	0.426994	−	0.904255i	$-0.359573\pi$
$127$	155225.	0.853987	0.426994	+	0.904255i	$0.359573\pi$
$128$	0	0
$129$	46420.4	0.245603
$130$	0	0
$131$	−206749.	−1.05261	−0.526303	−	0.850297i	$-0.676422\pi$
$131$	−206749.	−1.05261	−0.526303	+	0.850297i	$0.676422\pi$
$132$	0	0
$133$	221216.	1.08440
$134$	0	0
$135$	−222039.	−1.04856
$136$	0	0
$137$	−219188.	−0.997736	−0.498868	−	0.866678i	$-0.666251\pi$
$137$	−219188.	−0.997736	−0.498868	+	0.866678i	$0.666251\pi$
$138$	0	0
$139$	−83047.4	−0.364577	−0.182288	−	0.983245i	$-0.558350\pi$
$139$	−83047.4	−0.364577	−0.182288	+	0.983245i	$0.558350\pi$
$140$	0	0
$141$	216498.	0.917079
$142$	0	0
$143$	631863.	2.58394
$144$	0	0
$145$	5348.75	0.0211267
$146$	0	0
$147$	269870.	1.03006
$148$	0	0
$149$	−505431.	−1.86507	−0.932537	−	0.361074i	$-0.882410\pi$
$149$	−505431.	−1.86507	−0.932537	+	0.361074i	$0.882410\pi$
$150$	0	0
$151$	222212.	0.793095	0.396548	−	0.918014i	$-0.370208\pi$
$151$	222212.	0.793095	0.396548	+	0.918014i	$0.370208\pi$
$152$	0	0
$153$	557609.	1.92575
$154$	0	0
$155$	162698.	0.543944
$156$	0	0
$157$	18138.1	0.0587275	0.0293638	−	0.999569i	$-0.490652\pi$
$157$	18138.1	0.0587275	0.0293638	+	0.999569i	$0.490652\pi$
$158$	0	0
$159$	−711953.	−2.23336
$160$	0	0
$161$	72583.1	0.220684
$162$	0	0
$163$	428372.	1.26285	0.631425	−	0.775437i	$-0.282470\pi$
$163$	428372.	1.26285	0.631425	+	0.775437i	$0.282470\pi$
$164$	0	0
$165$	935321.	2.67455
$166$	0	0
$167$	−590110.	−1.63735	−0.818676	−	0.574256i	$-0.805291\pi$
$167$	−590110.	−1.63735	−0.818676	+	0.574256i	$0.805291\pi$
$168$	0	0
$169$	709358.	1.91051
$170$	0	0
$171$	516116.	1.34976
$172$	0	0
$173$	−194001.	−0.492821	−0.246410	−	0.969166i	$-0.579251\pi$
$173$	−194001.	−0.492821	−0.246410	+	0.969166i	$0.579251\pi$
$174$	0	0
$175$	104880.	0.258878
$176$	0	0
$177$	−1.20673e6	−2.89519
$178$	0	0
$179$	434140.	1.01274	0.506369	−	0.862317i	$-0.330988\pi$
$179$	434140.	1.01274	0.506369	+	0.862317i	$0.330988\pi$
$180$	0	0
$181$	−626829.	−1.42217	−0.711087	−	0.703104i	$-0.751797\pi$
$181$	−626829.	−1.42217	−0.711087	+	0.703104i	$0.751797\pi$
$182$	0	0
$183$	−997959.	−2.20285
$184$	0	0
$185$	286341.	0.615112
$186$	0	0
$187$	−875122.	−1.83006
$188$	0	0
$189$	601355.	1.22455
$190$	0	0
$191$	109059.	0.216310	0.108155	−	0.994134i	$-0.465506\pi$
$191$	109059.	0.216310	0.108155	+	0.994134i	$0.465506\pi$
$192$	0	0
$193$	−31563.2	−0.0609941	−0.0304971	−	0.999535i	$-0.509709\pi$
$193$	−31563.2	−0.0609941	−0.0304971	+	0.999535i	$0.509709\pi$
$194$	0	0
$195$	1.59964e6	3.01257
$196$	0	0
$197$	−276843.	−0.508238	−0.254119	−	0.967173i	$-0.581786\pi$
$197$	−276843.	−0.508238	−0.254119	+	0.967173i	$0.581786\pi$
$198$	0	0
$199$	777649.	1.39204	0.696019	−	0.718024i	$-0.254953\pi$
$199$	777649.	1.39204	0.696019	+	0.718024i	$0.254953\pi$
$200$	0	0
$201$	−482879.	−0.843041
$202$	0	0
$203$	−14486.2	−0.0246725
$204$	0	0
$205$	552426.	0.918099
$206$	0	0
$207$	169342.	0.274688
$208$	0	0
$209$	−810002.	−1.28269
$210$	0	0
$211$	89486.6	0.138373	0.0691866	−	0.997604i	$-0.477960\pi$
$211$	89486.6	0.138373	0.0691866	+	0.997604i	$0.477960\pi$
$212$	0	0
$213$	−427009.	−0.644894
$214$	0	0
$215$	−113330.	−0.167205
$216$	0	0
$217$	−440641.	−0.635237
$218$	0	0
$219$	558260.	0.786550
$220$	0	0
$221$	−1.49669e6	−2.06135
$222$	0	0
$223$	952284.	1.28234	0.641172	−	0.767398i	$-0.278449\pi$
$223$	952284.	1.28234	0.641172	+	0.767398i	$0.278449\pi$
$224$	0	0
$225$	244693.	0.322229
$226$	0	0
$227$	465809.	0.599988	0.299994	−	0.953941i	$-0.403015\pi$
$227$	465809.	0.599988	0.299994	+	0.953941i	$0.403015\pi$
$228$	0	0
$229$	477440.	0.601631	0.300815	−	0.953682i	$-0.402741\pi$
$229$	477440.	0.601631	0.300815	+	0.953682i	$0.402741\pi$
$230$	0	0
$231$	−2.53316e6	−3.12344
$232$	0	0
$233$	743589.	0.897312	0.448656	−	0.893705i	$-0.351903\pi$
$233$	743589.	0.897312	0.448656	+	0.893705i	$0.351903\pi$
$234$	0	0
$235$	−528557.	−0.624342
$236$	0	0
$237$	−927682.	−1.07282
$238$	0	0
$239$	−1.27115e6	−1.43947	−0.719734	−	0.694250i	$-0.755736\pi$
$239$	−1.27115e6	−1.43947	−0.719734	+	0.694250i	$0.755736\pi$
$240$	0	0
$241$	−875960.	−0.971497	−0.485749	−	0.874099i	$-0.661453\pi$
$241$	−875960.	−0.971497	−0.485749	+	0.874099i	$0.661453\pi$
$242$	0	0
$243$	−959745.	−1.04265
$244$	0	0
$245$	−658858.	−0.701256
$246$	0	0
$247$	−1.38532e6	−1.44480
$248$	0	0
$249$	2.96222e6	3.02774
$250$	0	0
$251$	1.19674e6	1.19899	0.599494	−	0.800380i	$-0.295369\pi$
$251$	1.19674e6	1.19899	0.599494	+	0.800380i	$0.295369\pi$
$252$	0	0
$253$	−265769.	−0.261038
$254$	0	0
$255$	−2.21549e6	−2.13363
$256$	0	0
$257$	−1.31630e6	−1.24315	−0.621573	−	0.783357i	$-0.713506\pi$
$257$	−1.31630e6	−1.24315	−0.621573	+	0.783357i	$0.713506\pi$
$258$	0	0
$259$	−775507.	−0.718350
$260$	0	0
$261$	−33797.4	−0.0307102
$262$	0	0
$263$	−1.74612e6	−1.55663	−0.778314	−	0.627876i	$-0.783925\pi$
$263$	−1.74612e6	−1.55663	−0.778314	+	0.627876i	$0.783925\pi$
$264$	0	0
$265$	1.73816e6	1.52046
$266$	0	0
$267$	1.04283e6	0.895231
$268$	0	0
$269$	−470733.	−0.396637	−0.198319	−	0.980138i	$-0.563548\pi$
$269$	−470733.	−0.396637	−0.198319	+	0.980138i	$0.563548\pi$
$270$	0	0
$271$	−29459.5	−0.0243670	−0.0121835	−	0.999926i	$-0.503878\pi$
$271$	−29459.5	−0.0243670	−0.0121835	+	0.999926i	$0.503878\pi$
$272$	0	0
$273$	−4.33237e6	−3.51819
$274$	0	0
$275$	−384025.	−0.306216
$276$	0	0
$277$	1.01172e6	0.792244	0.396122	−	0.918198i	$-0.370356\pi$
$277$	1.01172e6	0.792244	0.396122	+	0.918198i	$0.370356\pi$
$278$	0	0
$279$	−1.02805e6	−0.790687
$280$	0	0
$281$	−74592.9	−0.0563549	−0.0281775	−	0.999603i	$-0.508970\pi$
$281$	−74592.9	−0.0563549	−0.0281775	+	0.999603i	$0.508970\pi$
$282$	0	0
$283$	1.14789e6	0.851993	0.425996	−	0.904725i	$-0.359924\pi$
$283$	1.14789e6	0.851993	0.425996	+	0.904725i	$0.359924\pi$
$284$	0	0
$285$	−2.05063e6	−1.49546
$286$	0	0
$287$	−1.49615e6	−1.07219
$288$	0	0
$289$	653038.	0.459932
$290$	0	0
$291$	818290.	0.566467
$292$	0	0
$293$	−1.30694e6	−0.889376	−0.444688	−	0.895686i	$-0.646685\pi$
$293$	−1.30694e6	−0.889376	−0.444688	+	0.895686i	$0.646685\pi$
$294$	0	0
$295$	2.94610e6	1.97103
$296$	0	0
$297$	−2.20191e6	−1.44847
$298$	0	0
$299$	−454535.	−0.294028
$300$	0	0
$301$	306936.	0.195268
$302$	0	0
$303$	−199209.	−0.124653
$304$	0	0
$305$	2.43641e6	1.49969
$306$	0	0
$307$	1.50999e6	0.914380	0.457190	−	0.889369i	$-0.348856\pi$
$307$	1.50999e6	0.914380	0.457190	+	0.889369i	$0.348856\pi$
$308$	0	0
$309$	−3.32796e6	−1.98282
$310$	0	0
$311$	−1.69979e6	−0.996539	−0.498269	−	0.867022i	$-0.666031\pi$
$311$	−1.69979e6	−0.996539	−0.498269	+	0.867022i	$0.666031\pi$
$312$	0	0
$313$	2.85190e6	1.64541	0.822703	−	0.568471i	$-0.192465\pi$
$313$	2.85190e6	1.64541	0.822703	+	0.568471i	$0.192465\pi$
$314$	0	0
$315$	−3.94059e6	−2.23761
$316$	0	0
$317$	−158403.	−0.0885351	−0.0442676	−	0.999020i	$-0.514095\pi$
$317$	−158403.	−0.0885351	−0.0442676	+	0.999020i	$0.514095\pi$
$318$	0	0
$319$	53042.4	0.0291841
$320$	0	0
$321$	−3.02880e6	−1.64062
$322$	0	0
$323$	1.91865e6	1.02327
$324$	0	0
$325$	−656784.	−0.344917
$326$	0	0
$327$	421000.	0.217727
$328$	0	0
$329$	1.43151e6	0.729129
$330$	0	0
$331$	−1.38726e6	−0.695968	−0.347984	−	0.937501i	$-0.613134\pi$
$331$	−1.38726e6	−0.695968	−0.347984	+	0.937501i	$0.613134\pi$
$332$	0	0
$333$	−1.80932e6	−0.894139
$334$	0	0
$335$	1.17890e6	0.573937
$336$	0	0
$337$	602990.	0.289225	0.144612	−	0.989488i	$-0.453807\pi$
$337$	602990.	0.289225	0.144612	+	0.989488i	$0.453807\pi$
$338$	0	0
$339$	−456964.	−0.215965
$340$	0	0
$341$	1.61345e6	0.751395
$342$	0	0
$343$	−1.00557e6	−0.461507
$344$	0	0
$345$	−672829.	−0.304339
$346$	0	0
$347$	−1.38847e6	−0.619030	−0.309515	−	0.950895i	$-0.600167\pi$
$347$	−1.38847e6	−0.619030	−0.309515	+	0.950895i	$0.600167\pi$
$348$	0	0
$349$	1.90607e6	0.837676	0.418838	−	0.908061i	$-0.362437\pi$
$349$	1.90607e6	0.837676	0.418838	+	0.908061i	$0.362437\pi$
$350$	0	0
$351$	−3.76585e6	−1.63153
$352$	0	0
$353$	−2.53782e6	−1.08399	−0.541993	−	0.840383i	$-0.682330\pi$
$353$	−2.53782e6	−1.08399	−0.541993	+	0.840383i	$0.682330\pi$
$354$	0	0
$355$	1.04250e6	0.439040
$356$	0	0
$357$	6.00028e6	2.49173
$358$	0	0
$359$	589213.	0.241288	0.120644	−	0.992696i	$-0.461504\pi$
$359$	589213.	0.241288	0.120644	+	0.992696i	$0.461504\pi$
$360$	0	0
$361$	−700223.	−0.282793
$362$	0	0
$363$	5.23209e6	2.08405
$364$	0	0
$365$	−1.36293e6	−0.535478
$366$	0	0
$367$	−1.67673e6	−0.649826	−0.324913	−	0.945744i	$-0.605335\pi$
$367$	−1.67673e6	−0.649826	−0.324913	+	0.945744i	$0.605335\pi$
$368$	0	0
$369$	−3.49065e6	−1.33457
$370$	0	0
$371$	−4.70750e6	−1.77564
$372$	0	0
$373$	3.12665e6	1.16361	0.581804	−	0.813329i	$-0.302347\pi$
$373$	3.12665e6	1.16361	0.581804	+	0.813329i	$0.302347\pi$
$374$	0	0
$375$	3.83652e6	1.40883
$376$	0	0
$377$	90716.4	0.0328725
$378$	0	0
$379$	1.98431e6	0.709596	0.354798	−	0.934943i	$-0.384550\pi$
$379$	1.98431e6	0.709596	0.354798	+	0.934943i	$0.384550\pi$
$380$	0	0
$381$	3.89702e6	1.37537
$382$	0	0
$383$	−321813.	−0.112100	−0.0560502	−	0.998428i	$-0.517851\pi$
$383$	−321813.	−0.112100	−0.0560502	+	0.998428i	$0.517851\pi$
$384$	0	0
$385$	6.18444e6	2.12642
$386$	0	0
$387$	716107.	0.243053
$388$	0	0
$389$	−1.34564e6	−0.450874	−0.225437	−	0.974258i	$-0.572381\pi$
$389$	−1.34564e6	−0.450874	−0.225437	+	0.974258i	$0.572381\pi$
$390$	0	0
$391$	629525.	0.208243
$392$	0	0
$393$	−5.19057e6	−1.69525
$394$	0	0
$395$	2.26484e6	0.730372
$396$	0	0
$397$	1.40299e6	0.446763	0.223382	−	0.974731i	$-0.428290\pi$
$397$	1.40299e6	0.446763	0.223382	+	0.974731i	$0.428290\pi$
$398$	0	0
$399$	5.55378e6	1.74645
$400$	0	0
$401$	3.44258e6	1.06911	0.534556	−	0.845133i	$-0.320479\pi$
$401$	3.44258e6	1.06911	0.534556	+	0.845133i	$0.320479\pi$
$402$	0	0
$403$	2.75942e6	0.846359
$404$	0	0
$405$	193977.	0.0587643
$406$	0	0
$407$	2.83958e6	0.849706
$408$	0	0
$409$	6.44614e6	1.90542	0.952711	−	0.303877i	$-0.0982811\pi$
$409$	6.44614e6	1.90542	0.952711	+	0.303877i	$0.0982811\pi$
$410$	0	0
$411$	−5.50286e6	−1.60688
$412$	0	0
$413$	−7.97902e6	−2.30184
$414$	0	0
$415$	−7.23194e6	−2.06127
$416$	0	0
$417$	−2.08496e6	−0.587161
$418$	0	0
$419$	1.77865e6	0.494943	0.247472	−	0.968895i	$-0.420400\pi$
$419$	1.77865e6	0.494943	0.247472	+	0.968895i	$0.420400\pi$
$420$	0	0
$421$	−3.37157e6	−0.927100	−0.463550	−	0.886071i	$-0.653425\pi$
$421$	−3.37157e6	−0.927100	−0.463550	+	0.886071i	$0.653425\pi$
$422$	0	0
$423$	3.33983e6	0.907555
$424$	0	0
$425$	909638.	0.244285
$426$	0	0
$427$	−6.59861e6	−1.75139
$428$	0	0
$429$	1.58633e7	4.16151
$430$	0	0
$431$	−1.38644e6	−0.359508	−0.179754	−	0.983712i	$-0.557530\pi$
$431$	−1.38644e6	−0.359508	−0.179754	+	0.983712i	$0.557530\pi$
$432$	0	0
$433$	5.74536e6	1.47264	0.736322	−	0.676632i	$-0.236561\pi$
$433$	5.74536e6	1.47264	0.736322	+	0.676632i	$0.236561\pi$
$434$	0	0
$435$	134284.	0.0340252
$436$	0	0
$437$	582681.	0.145958
$438$	0	0
$439$	5.14484e6	1.27412	0.637060	−	0.770814i	$-0.280150\pi$
$439$	5.14484e6	1.27412	0.637060	+	0.770814i	$0.280150\pi$
$440$	0	0
$441$	4.16317e6	1.01936
$442$	0	0
$443$	5.65845e6	1.36990	0.684949	−	0.728591i	$-0.259824\pi$
$443$	5.65845e6	1.36990	0.684949	+	0.728591i	$0.259824\pi$
$444$	0	0
$445$	−2.54595e6	−0.609468
$446$	0	0
$447$	−1.26892e7	−3.00376
$448$	0	0
$449$	−956361.	−0.223875	−0.111938	−	0.993715i	$-0.535706\pi$
$449$	−956361.	−0.223875	−0.111938	+	0.993715i	$0.535706\pi$
$450$	0	0
$451$	5.47829e6	1.26825
$452$	0	0
$453$	5.57878e6	1.27730
$454$	0	0
$455$	1.05770e7	2.39516
$456$	0	0
$457$	−5.50698e6	−1.23345	−0.616727	−	0.787177i	$-0.711542\pi$
$457$	−5.50698e6	−1.23345	−0.616727	+	0.787177i	$0.711542\pi$
$458$	0	0
$459$	5.21565e6	1.15552
$460$	0	0
$461$	2.10202e6	0.460665	0.230333	−	0.973112i	$-0.426019\pi$
$461$	2.10202e6	0.460665	0.230333	+	0.973112i	$0.426019\pi$
$462$	0	0
$463$	−2.36108e6	−0.511868	−0.255934	−	0.966694i	$-0.582383\pi$
$463$	−2.36108e6	−0.511868	−0.255934	+	0.966694i	$0.582383\pi$
$464$	0	0
$465$	4.08465e6	0.876037
$466$	0	0
$467$	8.56485e6	1.81730	0.908652	−	0.417554i	$-0.137113\pi$
$467$	8.56485e6	1.81730	0.908652	+	0.417554i	$0.137113\pi$
$468$	0	0
$469$	−3.19285e6	−0.670264
$470$	0	0
$471$	455368.	0.0945823
$472$	0	0
$473$	−1.12387e6	−0.230975
$474$	0	0
$475$	841950.	0.171219
$476$	0	0
$477$	−1.09830e7	−2.21016
$478$	0	0
$479$	−3.09570e6	−0.616481	−0.308241	−	0.951308i	$-0.599740\pi$
$479$	−3.09570e6	−0.616481	−0.308241	+	0.951308i	$0.599740\pi$
$480$	0	0
$481$	4.85643e6	0.957095
$482$	0	0
$483$	1.82225e6	0.355418
$484$	0	0
$485$	−1.99777e6	−0.385648
$486$	0	0
$487$	4.69724e6	0.897471	0.448736	−	0.893665i	$-0.351875\pi$
$487$	4.69724e6	0.897471	0.448736	+	0.893665i	$0.351875\pi$
$488$	0	0
$489$	1.07546e7	2.03386
$490$	0	0
$491$	−4.40851e6	−0.825255	−0.412628	−	0.910900i	$-0.635389\pi$
$491$	−4.40851e6	−0.825255	−0.412628	+	0.910900i	$0.635389\pi$
$492$	0	0
$493$	−125641.	−0.0232817
$494$	0	0
$495$	1.44288e7	2.64677
$496$	0	0
$497$	−2.82343e6	−0.512726
$498$	0	0
$499$	−3.52789e6	−0.634255	−0.317127	−	0.948383i	$-0.602718\pi$
$499$	−3.52789e6	−0.634255	−0.317127	+	0.948383i	$0.602718\pi$
$500$	0	0
$501$	−1.48151e7	−2.63700
$502$	0	0
$503$	−79054.9	−0.0139319	−0.00696593	−	0.999976i	$-0.502217\pi$
$503$	−79054.9	−0.0139319	−0.00696593	+	0.999976i	$0.502217\pi$
$504$	0	0
$505$	486347.	0.0848629
$506$	0	0
$507$	1.78089e7	3.07693
$508$	0	0
$509$	155909.	0.0266733	0.0133366	−	0.999911i	$-0.495755\pi$
$509$	155909.	0.0266733	0.0133366	+	0.999911i	$0.495755\pi$
$510$	0	0
$511$	3.69127e6	0.625351
$512$	0	0
$513$	4.82754e6	0.809903
$514$	0	0
$515$	8.12486e6	1.34989
$516$	0	0
$517$	−5.24159e6	−0.862456
$518$	0	0
$519$	−4.87052e6	−0.793702
$520$	0	0
$521$	−4.24897e6	−0.685787	−0.342893	−	0.939374i	$-0.611407\pi$
$521$	−4.24897e6	−0.685787	−0.342893	+	0.939374i	$0.611407\pi$
$522$	0	0
$523$	−1.30285e6	−0.208276	−0.104138	−	0.994563i	$-0.533208\pi$
$523$	−1.30285e6	−0.208276	−0.104138	+	0.994563i	$0.533208\pi$
$524$	0	0
$525$	2.63307e6	0.416931
$526$	0	0
$527$	−3.82176e6	−0.599427
$528$	0	0
$529$	−6.24516e6	−0.970296
$530$	0	0
$531$	−1.86157e7	−2.86512
$532$	0	0
$533$	9.36932e6	1.42853
$534$	0	0
$535$	7.39449e6	1.11692
$536$	0	0
$537$	1.08994e7	1.63104
$538$	0	0
$539$	−6.53376e6	−0.968704
$540$	0	0
$541$	1.58581e6	0.232948	0.116474	−	0.993194i	$-0.462841\pi$
$541$	1.58581e6	0.232948	0.116474	+	0.993194i	$0.462841\pi$
$542$	0	0
$543$	−1.57370e7	−2.29045
$544$	0	0
$545$	−1.02783e6	−0.148227
$546$	0	0
$547$	5.60569e6	0.801052	0.400526	−	0.916285i	$-0.368827\pi$
$547$	5.60569e6	0.801052	0.400526	+	0.916285i	$0.368827\pi$
$548$	0	0
$549$	−1.53951e7	−2.17997
$550$	0	0
$551$	−116292.	−0.0163181
$552$	0	0
$553$	−6.13393e6	−0.852954
$554$	0	0
$555$	7.18878e6	0.990656
$556$	0	0
$557$	−44346.9	−0.00605655	−0.00302828	−	0.999995i	$-0.500964\pi$
$557$	−44346.9	−0.00605655	−0.00302828	+	0.999995i	$0.500964\pi$
$558$	0	0
$559$	−1.92212e6	−0.260166
$560$	0	0
$561$	−2.19705e7	−2.94736
$562$	0	0
$563$	−4.30007e6	−0.571748	−0.285874	−	0.958267i	$-0.592284\pi$
$563$	−4.30007e6	−0.571748	−0.285874	+	0.958267i	$0.592284\pi$
$564$	0	0
$565$	1.11563e6	0.147027
$566$	0	0
$567$	−525355.	−0.0686271
$568$	0	0
$569$	6.54339e6	0.847271	0.423635	−	0.905833i	$-0.360754\pi$
$569$	6.54339e6	0.847271	0.423635	+	0.905833i	$0.360754\pi$
$570$	0	0
$571$	−8.65292e6	−1.11064	−0.555319	−	0.831637i	$-0.687404\pi$
$571$	−8.65292e6	−1.11064	−0.555319	+	0.831637i	$0.687404\pi$
$572$	0	0
$573$	2.73799e6	0.348374
$574$	0	0
$575$	276251.	0.0348445
$576$	0	0
$577$	3.48407e6	0.435660	0.217830	−	0.975987i	$-0.430102\pi$
$577$	3.48407e6	0.435660	0.217830	+	0.975987i	$0.430102\pi$
$578$	0	0
$579$	−792415.	−0.0982328
$580$	0	0
$581$	1.95865e7	2.40723
$582$	0	0
$583$	1.72369e7	2.10033
$584$	0	0
$585$	2.46770e7	2.98128
$586$	0	0
$587$	−6.56208e6	−0.786043	−0.393021	−	0.919529i	$-0.628570\pi$
$587$	−6.56208e6	−0.786043	−0.393021	+	0.919529i	$0.628570\pi$
$588$	0	0
$589$	−3.53737e6	−0.420138
$590$	0	0
$591$	−6.95032e6	−0.818532
$592$	0	0
$593$	8.08785e6	0.944488	0.472244	−	0.881468i	$-0.343444\pi$
$593$	8.08785e6	0.944488	0.472244	+	0.881468i	$0.343444\pi$
$594$	0	0
$595$	−1.46490e7	−1.69635
$596$	0	0
$597$	1.95234e7	2.24192
$598$	0	0
$599$	9.16029e6	1.04314	0.521569	−	0.853209i	$-0.325347\pi$
$599$	9.16029e6	1.04314	0.521569	+	0.853209i	$0.325347\pi$
$600$	0	0
$601$	−1.25891e7	−1.42171	−0.710853	−	0.703341i	$-0.751691\pi$
$601$	−1.25891e7	−1.42171	−0.710853	+	0.703341i	$0.751691\pi$
$602$	0	0
$603$	−7.44917e6	−0.834285
$604$	0	0
$605$	−1.27736e7	−1.41881
$606$	0	0
$607$	1.05315e7	1.16017	0.580083	−	0.814558i	$-0.303020\pi$
$607$	1.05315e7	1.16017	0.580083	+	0.814558i	$0.303020\pi$
$608$	0	0
$609$	−363685.	−0.0397358
$610$	0	0
$611$	−8.96450e6	−0.971455
$612$	0	0
$613$	2.50815e6	0.269589	0.134794	−	0.990874i	$-0.456963\pi$
$613$	2.50815e6	0.269589	0.134794	+	0.990874i	$0.456963\pi$
$614$	0	0
$615$	1.38690e7	1.47863
$616$	0	0
$617$	1.04940e7	1.10975	0.554876	−	0.831933i	$-0.312766\pi$
$617$	1.04940e7	1.10975	0.554876	+	0.831933i	$0.312766\pi$
$618$	0	0
$619$	7.31973e6	0.767836	0.383918	−	0.923367i	$-0.374574\pi$
$619$	7.31973e6	0.767836	0.383918	+	0.923367i	$0.374574\pi$
$620$	0	0
$621$	1.58396e6	0.164822
$622$	0	0
$623$	6.89529e6	0.711758
$624$	0	0
$625$	−1.13408e7	−1.16130
$626$	0	0
$627$	−2.03356e7	−2.06580
$628$	0	0
$629$	−6.72610e6	−0.677855
$630$	0	0
$631$	6.30990e6	0.630883	0.315442	−	0.948945i	$-0.397847\pi$
$631$	6.30990e6	0.630883	0.315442	+	0.948945i	$0.397847\pi$
$632$	0	0
$633$	2.24662e6	0.222854
$634$	0	0
$635$	−9.51414e6	−0.936344
$636$	0	0
$637$	−1.11744e7	−1.09113
$638$	0	0
$639$	−6.58729e6	−0.638196
$640$	0	0
$641$	176304.	0.0169479	0.00847395	−	0.999964i	$-0.497303\pi$
$641$	176304.	0.0169479	0.00847395	+	0.999964i	$0.497303\pi$
$642$	0	0
$643$	4.73829e6	0.451954	0.225977	−	0.974133i	$-0.427443\pi$
$643$	4.73829e6	0.451954	0.225977	+	0.974133i	$0.427443\pi$
$644$	0	0
$645$	−2.84523e6	−0.269289
$646$	0	0
$647$	1.27894e7	1.20113	0.600563	−	0.799577i	$-0.294943\pi$
$647$	1.27894e7	1.20113	0.600563	+	0.799577i	$0.294943\pi$
$648$	0	0
$649$	2.92158e7	2.72274
$650$	0	0
$651$	−1.10626e7	−1.02307
$652$	0	0
$653$	2.31597e6	0.212545	0.106272	−	0.994337i	$-0.466108\pi$
$653$	2.31597e6	0.212545	0.106272	+	0.994337i	$0.466108\pi$
$654$	0	0
$655$	1.26722e7	1.15412
$656$	0	0
$657$	8.61203e6	0.778381
$658$	0	0
$659$	4.32880e6	0.388288	0.194144	−	0.980973i	$-0.437807\pi$
$659$	4.32880e6	0.388288	0.194144	+	0.980973i	$0.437807\pi$
$660$	0	0
$661$	1.34429e7	1.19671	0.598355	−	0.801231i	$-0.295821\pi$
$661$	1.34429e7	1.19671	0.598355	+	0.801231i	$0.295821\pi$
$662$	0	0
$663$	−3.75754e7	−3.31986
$664$	0	0
$665$	−1.35590e7	−1.18897
$666$	0	0
$667$	−38156.4	−0.00332088
$668$	0	0
$669$	2.39077e7	2.06525
$670$	0	0
$671$	2.41614e7	2.07164
$672$	0	0
$673$	−1.30893e7	−1.11398	−0.556991	−	0.830518i	$-0.688044\pi$
$673$	−1.30893e7	−1.11398	−0.556991	+	0.830518i	$0.688044\pi$
$674$	0	0
$675$	2.28876e6	0.193348
$676$	0	0
$677$	1.26746e6	0.106283	0.0531415	−	0.998587i	$-0.483077\pi$
$677$	1.26746e6	0.106283	0.0531415	+	0.998587i	$0.483077\pi$
$678$	0	0
$679$	5.41062e6	0.450373
$680$	0	0
$681$	1.16944e7	0.966298
$682$	0	0
$683$	1.77529e7	1.45619	0.728093	−	0.685479i	$-0.240407\pi$
$683$	1.77529e7	1.45619	0.728093	+	0.685479i	$0.240407\pi$
$684$	0	0
$685$	1.34346e7	1.09396
$686$	0	0
$687$	1.19864e7	0.968944
$688$	0	0
$689$	2.94797e7	2.36578
$690$	0	0
$691$	−5.47133e6	−0.435911	−0.217955	−	0.975959i	$-0.569939\pi$
$691$	−5.47133e6	−0.435911	−0.217955	+	0.975959i	$0.569939\pi$
$692$	0	0
$693$	−3.90780e7	−3.09100
$694$	0	0
$695$	5.09020e6	0.399736
$696$	0	0
$697$	−1.29764e7	−1.01175
$698$	0	0
$699$	1.86683e7	1.44515
$700$	0	0
$701$	−1.16242e7	−0.893443	−0.446721	−	0.894673i	$-0.647408\pi$
$701$	−1.16242e7	−0.893443	−0.446721	+	0.894673i	$0.647408\pi$
$702$	0	0
$703$	−6.22560e6	−0.475108
$704$	0	0
$705$	−1.32698e7	−1.00552
$706$	0	0
$707$	−1.31719e6	−0.0991060
$708$	0	0
$709$	−4.30447e6	−0.321591	−0.160795	−	0.986988i	$-0.551406\pi$
$709$	−4.30447e6	−0.321591	−0.160795	+	0.986988i	$0.551406\pi$
$710$	0	0
$711$	−1.43110e7	−1.06168
$712$	0	0
$713$	−1.16064e6	−0.0855017
$714$	0	0
$715$	−3.87286e7	−2.83313
$716$	0	0
$717$	−3.19130e7	−2.31830
$718$	0	0
$719$	−1.35107e7	−0.974664	−0.487332	−	0.873217i	$-0.662030\pi$
$719$	−1.35107e7	−0.974664	−0.487332	+	0.873217i	$0.662030\pi$
$720$	0	0
$721$	−2.20048e7	−1.57645
$722$	0	0
$723$	−2.19915e7	−1.56462
$724$	0	0
$725$	−55134.4	−0.00389563
$726$	0	0
$727$	−2.62118e7	−1.83934	−0.919668	−	0.392697i	$-0.871542\pi$
$727$	−2.62118e7	−1.83934	−0.919668	+	0.392697i	$0.871542\pi$
$728$	0	0
$729$	−2.33260e7	−1.62563
$730$	0	0
$731$	2.66211e6	0.184260
$732$	0	0
$733$	−2.19136e7	−1.50644	−0.753222	−	0.657766i	$-0.771501\pi$
$733$	−2.19136e7	−1.50644	−0.753222	+	0.657766i	$0.771501\pi$
$734$	0	0
$735$	−1.65411e7	−1.12939
$736$	0	0
$737$	1.16909e7	0.792827
$738$	0	0
$739$	2.72080e6	0.183267	0.0916337	−	0.995793i	$-0.470791\pi$
$739$	2.72080e6	0.183267	0.0916337	+	0.995793i	$0.470791\pi$
$740$	0	0
$741$	−3.47793e7	−2.32689
$742$	0	0
$743$	5.79913e6	0.385381	0.192691	−	0.981260i	$-0.438279\pi$
$743$	5.79913e6	0.385381	0.192691	+	0.981260i	$0.438279\pi$
$744$	0	0
$745$	3.09792e7	2.04494
$746$	0	0
$747$	4.56969e7	2.99630
$748$	0	0
$749$	−2.00267e7	−1.30438
$750$	0	0
$751$	2.22708e6	0.144091	0.0720455	−	0.997401i	$-0.477047\pi$
$751$	2.22708e6	0.144091	0.0720455	+	0.997401i	$0.477047\pi$
$752$	0	0
$753$	3.00449e7	1.93100
$754$	0	0
$755$	−1.36200e7	−0.869579
$756$	0	0
$757$	1.68706e7	1.07002	0.535009	−	0.844846i	$-0.320308\pi$
$757$	1.68706e7	1.07002	0.535009	+	0.844846i	$0.320308\pi$
$758$	0	0
$759$	−6.67231e6	−0.420409
$760$	0	0
$761$	−6.73683e6	−0.421691	−0.210845	−	0.977519i	$-0.567622\pi$
$761$	−6.73683e6	−0.421691	−0.210845	+	0.977519i	$0.567622\pi$
$762$	0	0
$763$	2.78370e6	0.173105
$764$	0	0
$765$	−3.41774e7	−2.11147
$766$	0	0
$767$	4.99668e7	3.06685
$768$	0	0
$769$	2.99965e7	1.82917	0.914587	−	0.404390i	$-0.132516\pi$
$769$	2.99965e7	1.82917	0.914587	+	0.404390i	$0.132516\pi$
$770$	0	0
$771$	−3.30466e7	−2.00212
$772$	0	0
$773$	2.03964e7	1.22773	0.613867	−	0.789409i	$-0.289613\pi$
$773$	2.03964e7	1.22773	0.613867	+	0.789409i	$0.289613\pi$
$774$	0	0
$775$	−1.67708e6	−0.100300
$776$	0	0
$777$	−1.94696e7	−1.15692
$778$	0	0
$779$	−1.20108e7	−0.709133
$780$	0	0
$781$	1.03382e7	0.606482
$782$	0	0
$783$	−316128.	−0.0184272
$784$	0	0
$785$	−1.11173e6	−0.0643911
$786$	0	0
$787$	9.96285e6	0.573386	0.286693	−	0.958023i	$-0.407444\pi$
$787$	9.96285e6	0.573386	0.286693	+	0.958023i	$0.407444\pi$
$788$	0	0
$789$	−4.38375e7	−2.50699
$790$	0	0
$791$	−3.02149e6	−0.171704
$792$	0	0
$793$	4.13223e7	2.33346
$794$	0	0
$795$	4.36375e7	2.44874
$796$	0	0
$797$	−4.33007e6	−0.241462	−0.120731	−	0.992685i	$-0.538524\pi$
$797$	−4.33007e6	−0.241462	−0.120731	+	0.992685i	$0.538524\pi$
$798$	0	0
$799$	1.24157e7	0.688026
$800$	0	0
$801$	1.60873e7	0.885933
$802$	0	0
$803$	−1.35159e7	−0.739701
$804$	0	0
$805$	−4.44881e6	−0.241966
$806$	0	0
$807$	−1.18181e7	−0.638796
$808$	0	0
$809$	4.65045e6	0.249818	0.124909	−	0.992168i	$-0.460136\pi$
$809$	4.65045e6	0.249818	0.124909	+	0.992168i	$0.460136\pi$
$810$	0	0
$811$	2.23521e7	1.19334	0.596672	−	0.802486i	$-0.296490\pi$
$811$	2.23521e7	1.19334	0.596672	+	0.802486i	$0.296490\pi$
$812$	0	0
$813$	−739599.	−0.0392437
$814$	0	0
$815$	−2.62561e7	−1.38464
$816$	0	0
$817$	2.46401e6	0.129148
$818$	0	0
$819$	−6.68336e7	−3.48165
$820$	0	0
$821$	6.86450e6	0.355427	0.177714	−	0.984082i	$-0.443130\pi$
$821$	6.86450e6	0.355427	0.177714	+	0.984082i	$0.443130\pi$
$822$	0	0
$823$	3.78159e7	1.94614	0.973072	−	0.230502i	$-0.0740369\pi$
$823$	3.78159e7	1.94614	0.973072	+	0.230502i	$0.0740369\pi$
$824$	0	0
$825$	−9.64121e6	−0.493170
$826$	0	0
$827$	−3.11812e7	−1.58536	−0.792681	−	0.609637i	$-0.791315\pi$
$827$	−3.11812e7	−1.58536	−0.792681	+	0.609637i	$0.791315\pi$
$828$	0	0
$829$	−7.56161e6	−0.382145	−0.191072	−	0.981576i	$-0.561196\pi$
$829$	−7.56161e6	−0.382145	−0.191072	+	0.981576i	$0.561196\pi$
$830$	0	0
$831$	2.53998e7	1.27593
$832$	0	0
$833$	1.54765e7	0.772786
$834$	0	0
$835$	3.61695e7	1.79525
$836$	0	0
$837$	−9.61600e6	−0.474439
$838$	0	0
$839$	−1.20663e7	−0.591792	−0.295896	−	0.955220i	$-0.595618\pi$
$839$	−1.20663e7	−0.591792	−0.295896	+	0.955220i	$0.595618\pi$
$840$	0	0
$841$	−2.05035e7	−0.999629
$842$	0	0
$843$	−1.87270e6	−0.0907612
$844$	0	0
$845$	−4.34785e7	−2.09475
$846$	0	0
$847$	3.45951e7	1.65694
$848$	0	0
$849$	2.88186e7	1.37216
$850$	0	0
$851$	−2.04267e6	−0.0966886
$852$	0	0
$853$	2.73307e7	1.28611	0.643055	−	0.765820i	$-0.277667\pi$
$853$	2.73307e7	1.28611	0.643055	+	0.765820i	$0.277667\pi$
$854$	0	0
$855$	−3.16341e7	−1.47993
$856$	0	0
$857$	−3.10213e7	−1.44280	−0.721402	−	0.692516i	$-0.756502\pi$
$857$	−3.10213e7	−1.44280	−0.721402	+	0.692516i	$0.756502\pi$
$858$	0	0
$859$	−2.98878e7	−1.38201	−0.691006	−	0.722849i	$-0.742832\pi$
$859$	−2.98878e7	−1.38201	−0.691006	+	0.722849i	$0.742832\pi$
$860$	0	0
$861$	−3.75619e7	−1.72679
$862$	0	0
$863$	2.37019e7	1.08332	0.541660	−	0.840598i	$-0.317796\pi$
$863$	2.37019e7	1.08332	0.541660	+	0.840598i	$0.317796\pi$
$864$	0	0
$865$	1.18909e7	0.540347
$866$	0	0
$867$	1.63949e7	0.740734
$868$	0	0
$869$	2.24599e7	1.00892
$870$	0	0
$871$	1.99945e7	0.893027
$872$	0	0
$873$	1.26234e7	0.560584
$874$	0	0
$875$	2.53675e7	1.12010
$876$	0	0
$877$	3.85645e7	1.69312	0.846562	−	0.532290i	$-0.178668\pi$
$877$	3.85645e7	1.69312	0.846562	+	0.532290i	$0.178668\pi$
$878$	0	0
$879$	−3.28115e7	−1.43237
$880$	0	0
$881$	−3.45951e7	−1.50167	−0.750836	−	0.660489i	$-0.770349\pi$
$881$	−3.45951e7	−1.50167	−0.750836	+	0.660489i	$0.770349\pi$
$882$	0	0
$883$	8.71550e6	0.376175	0.188088	−	0.982152i	$-0.439771\pi$
$883$	8.71550e6	0.376175	0.188088	+	0.982152i	$0.439771\pi$
$884$	0	0
$885$	7.39638e7	3.17439
$886$	0	0
$887$	−4.58909e6	−0.195847	−0.0979237	−	0.995194i	$-0.531220\pi$
$887$	−4.58909e6	−0.195847	−0.0979237	+	0.995194i	$0.531220\pi$
$888$	0	0
$889$	2.57675e7	1.09350
$890$	0	0
$891$	1.92363e6	0.0811761
$892$	0	0
$893$	1.14918e7	0.482237
$894$	0	0
$895$	−2.66096e7	−1.11040
$896$	0	0
$897$	−1.14114e7	−0.473541
$898$	0	0
$899$	231642.	0.00955912
$900$	0	0
$901$	−4.08290e7	−1.67555
$902$	0	0
$903$	7.70583e6	0.314485
$904$	0	0
$905$	3.84201e7	1.55933
$906$	0	0
$907$	−1.10880e7	−0.447544	−0.223772	−	0.974642i	$-0.571837\pi$
$907$	−1.10880e7	−0.447544	−0.223772	+	0.974642i	$0.571837\pi$
$908$	0	0
$909$	−3.07311e6	−0.123358
$910$	0	0
$911$	−3.54367e7	−1.41468	−0.707339	−	0.706875i	$-0.750104\pi$
$911$	−3.54367e7	−1.41468	−0.707339	+	0.706875i	$0.750104\pi$
$912$	0	0
$913$	−7.17177e7	−2.84740
$914$	0	0
$915$	6.11677e7	2.41529
$916$	0	0
$917$	−3.43206e7	−1.34782
$918$	0	0
$919$	−3.32990e7	−1.30060	−0.650298	−	0.759679i	$-0.725356\pi$
$919$	−3.32990e7	−1.30060	−0.650298	+	0.759679i	$0.725356\pi$
$920$	0	0
$921$	3.79092e7	1.47264
$922$	0	0
$923$	1.76811e7	0.683131
$924$	0	0
$925$	−2.95158e6	−0.113423
$926$	0	0
$927$	−5.13390e7	−1.96222
$928$	0	0
$929$	3.43816e7	1.30703	0.653517	−	0.756912i	$-0.273293\pi$
$929$	3.43816e7	1.30703	0.653517	+	0.756912i	$0.273293\pi$
$930$	0	0
$931$	1.43248e7	0.541645
$932$	0	0
$933$	−4.26743e7	−1.60495
$934$	0	0
$935$	5.36386e7	2.00654
$936$	0	0
$937$	3.81989e7	1.42135	0.710677	−	0.703518i	$-0.248389\pi$
$937$	3.81989e7	1.42135	0.710677	+	0.703518i	$0.248389\pi$
$938$	0	0
$939$	7.15988e7	2.64997
$940$	0	0
$941$	2.92661e7	1.07744	0.538718	−	0.842486i	$-0.318909\pi$
$941$	2.92661e7	1.07744	0.538718	+	0.842486i	$0.318909\pi$
$942$	0	0
$943$	−3.94085e6	−0.144315
$944$	0	0
$945$	−3.68587e7	−1.34264
$946$	0	0
$947$	−4.49447e6	−0.162856	−0.0814280	−	0.996679i	$-0.525948\pi$
$947$	−4.49447e6	−0.162856	−0.0814280	+	0.996679i	$0.525948\pi$
$948$	0	0
$949$	−2.31157e7	−0.833186
$950$	0	0
$951$	−3.97681e6	−0.142588
$952$	0	0
$953$	−7.96508e6	−0.284091	−0.142046	−	0.989860i	$-0.545368\pi$
$953$	−7.96508e6	−0.284091	−0.142046	+	0.989860i	$0.545368\pi$
$954$	0	0
$955$	−6.68451e6	−0.237171
$956$	0	0
$957$	1.33166e6	0.0470018
$958$	0	0
$959$	−3.63855e7	−1.27756
$960$	0	0
$961$	−2.15831e7	−0.753884
$962$	0	0
$963$	−4.67240e7	−1.62358
$964$	0	0
$965$	1.93460e6	0.0668763
$966$	0	0
$967$	−3.91114e6	−0.134505	−0.0672523	−	0.997736i	$-0.521423\pi$
$967$	−3.91114e6	−0.134505	−0.0672523	+	0.997736i	$0.521423\pi$
$968$	0	0
$969$	4.81689e7	1.64800
$970$	0	0
$971$	2.98964e7	1.01759	0.508793	−	0.860889i	$-0.330092\pi$
$971$	2.98964e7	1.01759	0.508793	+	0.860889i	$0.330092\pi$
$972$	0	0
$973$	−1.37860e7	−0.466826
$974$	0	0
$975$	−1.64890e7	−0.555498
$976$	0	0
$977$	−4.36187e7	−1.46196	−0.730981	−	0.682398i	$-0.760937\pi$
$977$	−4.36187e7	−1.46196	−0.730981	+	0.682398i	$0.760937\pi$
$978$	0	0
$979$	−2.52477e7	−0.841909
$980$	0	0
$981$	6.49459e6	0.215466
$982$	0	0
$983$	1.95377e7	0.644895	0.322448	−	0.946587i	$-0.395494\pi$
$983$	1.95377e7	0.644895	0.322448	+	0.946587i	$0.395494\pi$
$984$	0	0
$985$	1.69684e7	0.557252
$986$	0	0
$987$	3.59390e7	1.17428
$988$	0	0
$989$	808465.	0.0262827
$990$	0	0
$991$	−3.45462e7	−1.11742	−0.558709	−	0.829364i	$-0.688703\pi$
$991$	−3.45462e7	−1.11742	−0.558709	+	0.829364i	$0.688703\pi$
$992$	0	0
$993$	−3.48282e7	−1.12088
$994$	0	0
$995$	−4.76642e7	−1.52628
$996$	0	0
$997$	3.34457e7	1.06562	0.532810	−	0.846235i	$-0.321136\pi$
$997$	3.34457e7	1.06562	0.532810	+	0.846235i	$0.321136\pi$
$998$	0	0
$999$	−1.69237e7	−0.536514

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.6.a.e.1.7		8
4.3	odd	2		43.6.a.a.1.8	✓	8
12.11	even	2		387.6.a.c.1.1		8
20.19	odd	2		1075.6.a.a.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.a.1.8	✓	8	4.3	odd	2
387.6.a.c.1.1		8	12.11	even	2
688.6.a.e.1.7		8	1.1	even	1	trivial
1075.6.a.a.1.1		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)

\(f(q)\)	\(=\)	\(q+25.1057 q^{3} -61.2927 q^{5} +166.001 q^{7} +387.294 q^{9} +O(q^{10})\)	\(q+25.1057 q^{3} -61.2927 q^{5} +166.001 q^{7} +387.294 q^{9} -607.827 q^{11} -1039.54 q^{13} -1538.79 q^{15} +1439.76 q^{17} +1332.62 q^{19} +4167.57 q^{21} +437.244 q^{23} +631.800 q^{25} +3622.60 q^{27} -87.2656 q^{29} -2654.45 q^{31} -15259.9 q^{33} -10174.7 q^{35} -4671.70 q^{37} -26098.4 q^{39} -9012.91 q^{41} +1849.00 q^{43} -23738.3 q^{45} +8623.49 q^{47} +10749.4 q^{49} +36146.0 q^{51} -28358.3 q^{53} +37255.4 q^{55} +33456.3 q^{57} -48066.0 q^{59} -39750.4 q^{61} +64291.2 q^{63} +63716.5 q^{65} -19233.9 q^{67} +10977.3 q^{69} -17008.5 q^{71} +22236.4 q^{73} +15861.8 q^{75} -100900. q^{77} -36951.1 q^{79} -3164.77 q^{81} +117990. q^{83} -88246.5 q^{85} -2190.86 q^{87} +41537.6 q^{89} -172565. q^{91} -66641.7 q^{93} -81679.9 q^{95} +32593.9 q^{97} -235408. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 26 q^{3} - 212 q^{5} + 136 q^{7} + 546 q^{9}+O(q^{10}) \) 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

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