LMFDB - Embedded newform 688.6.a.e.1.3

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			1.3
Root			$5.65705$ 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)

$f(q)$	$=$	$q-7.84314 q^{3} -107.102 q^{5} +25.5214 q^{7} -181.485 q^{9} +O(q^{10})$	$q-7.84314 q^{3} -107.102 q^{5} +25.5214 q^{7} -181.485 q^{9} -512.073 q^{11} +862.516 q^{13} +840.016 q^{15} -1521.49 q^{17} +1543.11 q^{19} -200.168 q^{21} +3126.31 q^{23} +8345.85 q^{25} +3329.30 q^{27} -947.120 q^{29} -339.499 q^{31} +4016.26 q^{33} -2733.40 q^{35} -7448.67 q^{37} -6764.84 q^{39} +5116.45 q^{41} +1849.00 q^{43} +19437.4 q^{45} +17159.9 q^{47} -16155.7 q^{49} +11933.3 q^{51} +18090.4 q^{53} +54844.0 q^{55} -12102.9 q^{57} -17031.6 q^{59} +10664.9 q^{61} -4631.76 q^{63} -92377.3 q^{65} +8799.60 q^{67} -24520.1 q^{69} +77057.3 q^{71} -7964.75 q^{73} -65457.6 q^{75} -13068.8 q^{77} -68997.0 q^{79} +17988.8 q^{81} -40813.4 q^{83} +162955. q^{85} +7428.39 q^{87} -83692.1 q^{89} +22012.6 q^{91} +2662.74 q^{93} -165271. q^{95} +159824. q^{97} +92933.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−7.84314	−0.503138	−0.251569	−	0.967839i	$-0.580946\pi$
$3$	−7.84314	−0.503138	−0.251569	+	0.967839i	$0.580946\pi$
$4$	0	0
$5$	−107.102	−1.91590	−0.957950	−	0.286936i	$-0.907363\pi$
$5$	−107.102	−1.91590	−0.957950	+	0.286936i	$0.907363\pi$
$6$	0	0
$7$	25.5214	0.196861	0.0984305	−	0.995144i	$-0.468618\pi$
$7$	25.5214	0.196861	0.0984305	+	0.995144i	$0.468618\pi$
$8$	0	0
$9$	−181.485	−0.746853
$10$	0	0
$11$	−512.073	−1.27600	−0.637999	−	0.770037i	$-0.720238\pi$
$11$	−512.073	−1.27600	−0.637999	+	0.770037i	$0.720238\pi$
$12$	0	0
$13$	862.516	1.41550	0.707749	−	0.706464i	$-0.249711\pi$
$13$	862.516	1.41550	0.707749	+	0.706464i	$0.249711\pi$
$14$	0	0
$15$	840.016	0.963961
$16$	0	0
$17$	−1521.49	−1.27687	−0.638435	−	0.769675i	$-0.720418\pi$
$17$	−1521.49	−1.27687	−0.638435	+	0.769675i	$0.720418\pi$
$18$	0	0
$19$	1543.11	0.980650	0.490325	−	0.871540i	$-0.336878\pi$
$19$	1543.11	0.980650	0.490325	+	0.871540i	$0.336878\pi$
$20$	0	0
$21$	−200.168	−0.0990481
$22$	0	0
$23$	3126.31	1.23229	0.616145	−	0.787633i	$-0.288694\pi$
$23$	3126.31	1.23229	0.616145	+	0.787633i	$0.288694\pi$
$24$	0	0
$25$	8345.85	2.67067
$26$	0	0
$27$	3329.30	0.878907
$28$	0	0
$29$	−947.120	−0.209127	−0.104563	−	0.994518i	$-0.533345\pi$
$29$	−947.120	−0.209127	−0.104563	+	0.994518i	$0.533345\pi$
$30$	0	0
$31$	−339.499	−0.0634504	−0.0317252	−	0.999497i	$-0.510100\pi$
$31$	−339.499	−0.0634504	−0.0317252	+	0.999497i	$0.510100\pi$
$32$	0	0
$33$	4016.26	0.642002
$34$	0	0
$35$	−2733.40	−0.377166
$36$	0	0
$37$	−7448.67	−0.894488	−0.447244	−	0.894412i	$-0.647594\pi$
$37$	−7448.67	−0.894488	−0.447244	+	0.894412i	$0.647594\pi$
$38$	0	0
$39$	−6764.84	−0.712190
$40$	0	0
$41$	5116.45	0.475345	0.237672	−	0.971345i	$-0.423616\pi$
$41$	5116.45	0.475345	0.237672	+	0.971345i	$0.423616\pi$
$42$	0	0
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	0	0
$45$	19437.4	1.43089
$46$	0	0
$47$	17159.9	1.13311	0.566553	−	0.824025i	$-0.308276\pi$
$47$	17159.9	1.13311	0.566553	+	0.824025i	$0.308276\pi$
$48$	0	0
$49$	−16155.7	−0.961246
$50$	0	0
$51$	11933.3	0.642442
$52$	0	0
$53$	18090.4	0.884623	0.442312	−	0.896861i	$-0.354159\pi$
$53$	18090.4	0.884623	0.442312	+	0.896861i	$0.354159\pi$
$54$	0	0
$55$	54844.0	2.44468
$56$	0	0
$57$	−12102.9	−0.493402
$58$	0	0
$59$	−17031.6	−0.636980	−0.318490	−	0.947926i	$-0.603176\pi$
$59$	−17031.6	−0.636980	−0.318490	+	0.947926i	$0.603176\pi$
$60$	0	0
$61$	10664.9	0.366972	0.183486	−	0.983022i	$-0.441262\pi$
$61$	10664.9	0.366972	0.183486	+	0.983022i	$0.441262\pi$
$62$	0	0
$63$	−4631.76	−0.147026
$64$	0	0
$65$	−92377.3	−2.71195
$66$	0	0
$67$	8799.60	0.239484	0.119742	−	0.992805i	$-0.461793\pi$
$67$	8799.60	0.239484	0.119742	+	0.992805i	$0.461793\pi$
$68$	0	0
$69$	−24520.1	−0.620011
$70$	0	0
$71$	77057.3	1.81413	0.907064	−	0.420992i	$-0.138318\pi$
$71$	77057.3	1.81413	0.907064	+	0.420992i	$0.138318\pi$
$72$	0	0
$73$	−7964.75	−0.174930	−0.0874652	−	0.996168i	$-0.527877\pi$
$73$	−7964.75	−0.174930	−0.0874652	+	0.996168i	$0.527877\pi$
$74$	0	0
$75$	−65457.6	−1.34371
$76$	0	0
$77$	−13068.8	−0.251194
$78$	0	0
$79$	−68997.0	−1.24383	−0.621917	−	0.783083i	$-0.713646\pi$
$79$	−68997.0	−1.24383	−0.621917	+	0.783083i	$0.713646\pi$
$80$	0	0
$81$	17988.8	0.304641
$82$	0	0
$83$	−40813.4	−0.650290	−0.325145	−	0.945664i	$-0.605413\pi$
$83$	−40813.4	−0.650290	−0.325145	+	0.945664i	$0.605413\pi$
$84$	0	0
$85$	162955.	2.44636
$86$	0	0
$87$	7428.39	0.105220
$88$	0	0
$89$	−83692.1	−1.11998	−0.559989	−	0.828500i	$-0.689195\pi$
$89$	−83692.1	−1.11998	−0.559989	+	0.828500i	$0.689195\pi$
$90$	0	0
$91$	22012.6	0.278656
$92$	0	0
$93$	2662.74	0.0319243
$94$	0	0
$95$	−165271.	−1.87883
$96$	0	0
$97$	159824.	1.72469	0.862346	−	0.506320i	$-0.168994\pi$
$97$	159824.	1.72469	0.862346	+	0.506320i	$0.168994\pi$
$98$	0	0
$99$	92933.6	0.952982
$100$	0	0
$101$	−168968.	−1.64817	−0.824085	−	0.566466i	$-0.808310\pi$
$101$	−168968.	−1.64817	−0.824085	+	0.566466i	$0.808310\pi$
$102$	0	0
$103$	145084.	1.34749	0.673747	−	0.738962i	$-0.264684\pi$
$103$	145084.	1.34749	0.673747	+	0.738962i	$0.264684\pi$
$104$	0	0
$105$	21438.4	0.189766
$106$	0	0
$107$	−72693.1	−0.613810	−0.306905	−	0.951740i	$-0.599293\pi$
$107$	−72693.1	−0.613810	−0.306905	+	0.951740i	$0.599293\pi$
$108$	0	0
$109$	−3254.39	−0.0262364	−0.0131182	−	0.999914i	$-0.504176\pi$
$109$	−3254.39	−0.0262364	−0.0131182	+	0.999914i	$0.504176\pi$
$110$	0	0
$111$	58420.9	0.450050
$112$	0	0
$113$	116394.	0.857503	0.428752	−	0.903422i	$-0.358954\pi$
$113$	116394.	0.857503	0.428752	+	0.903422i	$0.358954\pi$
$114$	0	0
$115$	−334834.	−2.36094
$116$	0	0
$117$	−156534.	−1.05717
$118$	0	0
$119$	−38830.6	−0.251366
$120$	0	0
$121$	101167.	0.628170
$122$	0	0
$123$	−40129.0	−0.239164
$124$	0	0
$125$	−559163.	−3.20084
$126$	0	0
$127$	6451.01	0.0354910	0.0177455	−	0.999843i	$-0.494351\pi$
$127$	6451.01	0.0354910	0.0177455	+	0.999843i	$0.494351\pi$
$128$	0	0
$129$	−14502.0	−0.0767278
$130$	0	0
$131$	48048.9	0.244628	0.122314	−	0.992491i	$-0.460969\pi$
$131$	48048.9	0.244628	0.122314	+	0.992491i	$0.460969\pi$
$132$	0	0
$133$	39382.5	0.193052
$134$	0	0
$135$	−356574.	−1.68390
$136$	0	0
$137$	−103084.	−0.469233	−0.234616	−	0.972088i	$-0.575383\pi$
$137$	−103084.	−0.469233	−0.234616	+	0.972088i	$0.575383\pi$
$138$	0	0
$139$	−219626.	−0.964156	−0.482078	−	0.876128i	$-0.660118\pi$
$139$	−219626.	−0.964156	−0.482078	+	0.876128i	$0.660118\pi$
$140$	0	0
$141$	−134588.	−0.570109
$142$	0	0
$143$	−441671.	−1.80617
$144$	0	0
$145$	101438.	0.400666
$146$	0	0
$147$	126711.	0.483639
$148$	0	0
$149$	−335627.	−1.23849	−0.619243	−	0.785200i	$-0.712560\pi$
$149$	−335627.	−1.23849	−0.619243	+	0.785200i	$0.712560\pi$
$150$	0	0
$151$	84920.3	0.303088	0.151544	−	0.988450i	$-0.451575\pi$
$151$	84920.3	0.303088	0.151544	+	0.988450i	$0.451575\pi$
$152$	0	0
$153$	276128.	0.953634
$154$	0	0
$155$	36361.0	0.121565
$156$	0	0
$157$	313767.	1.01592	0.507958	−	0.861382i	$-0.330401\pi$
$157$	313767.	1.01592	0.507958	+	0.861382i	$0.330401\pi$
$158$	0	0
$159$	−141885.	−0.445087
$160$	0	0
$161$	79787.9	0.242590
$162$	0	0
$163$	−239642.	−0.706469	−0.353235	−	0.935535i	$-0.614918\pi$
$163$	−239642.	−0.706469	−0.353235	+	0.935535i	$0.614918\pi$
$164$	0	0
$165$	−430149.	−1.23001
$166$	0	0
$167$	506675.	1.40585	0.702923	−	0.711266i	$-0.251878\pi$
$167$	506675.	1.40585	0.702923	+	0.711266i	$0.251878\pi$
$168$	0	0
$169$	372641.	1.00363
$170$	0	0
$171$	−280052.	−0.732401
$172$	0	0
$173$	−427174.	−1.08515	−0.542575	−	0.840007i	$-0.682551\pi$
$173$	−427174.	−1.08515	−0.542575	+	0.840007i	$0.682551\pi$
$174$	0	0
$175$	212998.	0.525751
$176$	0	0
$177$	133581.	0.320489
$178$	0	0
$179$	−11674.9	−0.0272345	−0.0136172	−	0.999907i	$-0.504335\pi$
$179$	−11674.9	−0.0272345	−0.0136172	+	0.999907i	$0.504335\pi$
$180$	0	0
$181$	71691.3	0.162656	0.0813280	−	0.996687i	$-0.474084\pi$
$181$	71691.3	0.162656	0.0813280	+	0.996687i	$0.474084\pi$
$182$	0	0
$183$	−83646.4	−0.184637
$184$	0	0
$185$	797768.	1.71375
$186$	0	0
$187$	779114.	1.62928
$188$	0	0
$189$	84968.3	0.173023
$190$	0	0
$191$	−212639.	−0.421755	−0.210877	−	0.977513i	$-0.567632\pi$
$191$	−212639.	−0.421755	−0.210877	+	0.977513i	$0.567632\pi$
$192$	0	0
$193$	−137469.	−0.265651	−0.132825	−	0.991139i	$-0.542405\pi$
$193$	−137469.	−0.265651	−0.132825	+	0.991139i	$0.542405\pi$
$194$	0	0
$195$	724528.	1.36448
$196$	0	0
$197$	−28517.3	−0.0523531	−0.0261766	−	0.999657i	$-0.508333\pi$
$197$	−28517.3	−0.0523531	−0.0261766	+	0.999657i	$0.508333\pi$
$198$	0	0
$199$	353575.	0.632921	0.316460	−	0.948606i	$-0.397506\pi$
$199$	353575.	0.632921	0.316460	+	0.948606i	$0.397506\pi$
$200$	0	0
$201$	−69016.5	−0.120493
$202$	0	0
$203$	−24171.8	−0.0411689
$204$	0	0
$205$	−547982.	−0.910713
$206$	0	0
$207$	−567379.	−0.920339
$208$	0	0
$209$	−790187.	−1.25131
$210$	0	0
$211$	−327840.	−0.506939	−0.253469	−	0.967343i	$-0.581572\pi$
$211$	−327840.	−0.506939	−0.253469	+	0.967343i	$0.581572\pi$
$212$	0	0
$213$	−604371.	−0.912756
$214$	0	0
$215$	−198032.	−0.292172
$216$	0	0
$217$	−8664.49	−0.0124909
$218$	0	0
$219$	62468.7	0.0880140
$220$	0	0
$221$	−1.31231e6	−1.80741
$222$	0	0
$223$	−247951.	−0.333890	−0.166945	−	0.985966i	$-0.553390\pi$
$223$	−247951.	−0.333890	−0.166945	+	0.985966i	$0.553390\pi$
$224$	0	0
$225$	−1.51465e6	−1.99460
$226$	0	0
$227$	868910.	1.11921	0.559603	−	0.828761i	$-0.310954\pi$
$227$	868910.	1.11921	0.559603	+	0.828761i	$0.310954\pi$
$228$	0	0
$229$	238721.	0.300817	0.150408	−	0.988624i	$-0.451941\pi$
$229$	238721.	0.300817	0.150408	+	0.988624i	$0.451941\pi$
$230$	0	0
$231$	102501.	0.126385
$232$	0	0
$233$	507726.	0.612688	0.306344	−	0.951921i	$-0.400894\pi$
$233$	507726.	0.612688	0.306344	+	0.951921i	$0.400894\pi$
$234$	0	0
$235$	−1.83786e6	−2.17092
$236$	0	0
$237$	541153.	0.625819
$238$	0	0
$239$	−143880.	−0.162932	−0.0814660	−	0.996676i	$-0.525960\pi$
$239$	−143880.	−0.162932	−0.0814660	+	0.996676i	$0.525960\pi$
$240$	0	0
$241$	398237.	0.441671	0.220835	−	0.975311i	$-0.429122\pi$
$241$	398237.	0.441671	0.220835	+	0.975311i	$0.429122\pi$
$242$	0	0
$243$	−950107.	−1.03218
$244$	0	0
$245$	1.73030e6	1.84165
$246$	0	0
$247$	1.33096e6	1.38811
$248$	0	0
$249$	320105.	0.327185
$250$	0	0
$251$	−1.65625e6	−1.65937	−0.829684	−	0.558233i	$-0.811480\pi$
$251$	−1.65625e6	−1.65937	−0.829684	+	0.558233i	$0.811480\pi$
$252$	0	0
$253$	−1.60090e6	−1.57240
$254$	0	0
$255$	−1.27808e6	−1.23085
$256$	0	0
$257$	32691.4	0.0308746	0.0154373	−	0.999881i	$-0.495086\pi$
$257$	32691.4	0.0308746	0.0154373	+	0.999881i	$0.495086\pi$
$258$	0	0
$259$	−190101.	−0.176090
$260$	0	0
$261$	171888.	0.156187
$262$	0	0
$263$	−1.02727e6	−0.915791	−0.457895	−	0.889006i	$-0.651397\pi$
$263$	−1.02727e6	−0.915791	−0.457895	+	0.889006i	$0.651397\pi$
$264$	0	0
$265$	−1.93752e6	−1.69485
$266$	0	0
$267$	656409.	0.563503
$268$	0	0
$269$	−1.88917e6	−1.59180	−0.795902	−	0.605426i	$-0.793003\pi$
$269$	−1.88917e6	−1.59180	−0.795902	+	0.605426i	$0.793003\pi$
$270$	0	0
$271$	−605259.	−0.500631	−0.250316	−	0.968164i	$-0.580534\pi$
$271$	−605259.	−0.500631	−0.250316	+	0.968164i	$0.580534\pi$
$272$	0	0
$273$	−172648.	−0.140202
$274$	0	0
$275$	−4.27368e6	−3.40777
$276$	0	0
$277$	−1.16555e6	−0.912710	−0.456355	−	0.889798i	$-0.650845\pi$
$277$	−1.16555e6	−0.912710	−0.456355	+	0.889798i	$0.650845\pi$
$278$	0	0
$279$	61614.0	0.0473881
$280$	0	0
$281$	938873.	0.709318	0.354659	−	0.934996i	$-0.384597\pi$
$281$	938873.	0.709318	0.354659	+	0.934996i	$0.384597\pi$
$282$	0	0
$283$	2.54537e6	1.88923	0.944616	−	0.328179i	$-0.106435\pi$
$283$	2.54537e6	1.88923	0.944616	+	0.328179i	$0.106435\pi$
$284$	0	0
$285$	1.29624e6	0.945309
$286$	0	0
$287$	130579.	0.0935768
$288$	0	0
$289$	895076.	0.630399
$290$	0	0
$291$	−1.25352e6	−0.867757
$292$	0	0
$293$	−1.61560e6	−1.09942	−0.549712	−	0.835354i	$-0.685263\pi$
$293$	−1.61560e6	−1.09942	−0.549712	+	0.835354i	$0.685263\pi$
$294$	0	0
$295$	1.82412e6	1.22039
$296$	0	0
$297$	−1.70484e6	−1.12148
$298$	0	0
$299$	2.69650e6	1.74430
$300$	0	0
$301$	47189.1	0.0300210
$302$	0	0
$303$	1.32524e6	0.829257
$304$	0	0
$305$	−1.14223e6	−0.703081
$306$	0	0
$307$	1.49055e6	0.902613	0.451307	−	0.892369i	$-0.350958\pi$
$307$	1.49055e6	0.902613	0.451307	+	0.892369i	$0.350958\pi$
$308$	0	0
$309$	−1.13792e6	−0.677975
$310$	0	0
$311$	754959.	0.442611	0.221305	−	0.975205i	$-0.428968\pi$
$311$	754959.	0.442611	0.221305	+	0.975205i	$0.428968\pi$
$312$	0	0
$313$	1.31681e6	0.759735	0.379867	−	0.925041i	$-0.375970\pi$
$313$	1.31681e6	0.759735	0.379867	+	0.925041i	$0.375970\pi$
$314$	0	0
$315$	496071.	0.281687
$316$	0	0
$317$	−2.19577e6	−1.22727	−0.613633	−	0.789591i	$-0.710293\pi$
$317$	−2.19577e6	−1.22727	−0.613633	+	0.789591i	$0.710293\pi$
$318$	0	0
$319$	484994.	0.266845
$320$	0	0
$321$	570142.	0.308831
$322$	0	0
$323$	−2.34783e6	−1.25216
$324$	0	0
$325$	7.19843e6	3.78033
$326$	0	0
$327$	25524.7	0.0132005
$328$	0	0
$329$	437946.	0.223065
$330$	0	0
$331$	−727999.	−0.365225	−0.182613	−	0.983185i	$-0.558455\pi$
$331$	−727999.	−0.365225	−0.182613	+	0.983185i	$0.558455\pi$
$332$	0	0
$333$	1.35182e6	0.668050
$334$	0	0
$335$	−942455.	−0.458827
$336$	0	0
$337$	626937.	0.300711	0.150355	−	0.988632i	$-0.451958\pi$
$337$	626937.	0.300711	0.150355	+	0.988632i	$0.451958\pi$
$338$	0	0
$339$	−912897.	−0.431442
$340$	0	0
$341$	173848.	0.0809625
$342$	0	0
$343$	−841254.	−0.386093
$344$	0	0
$345$	2.62615e6	1.18788
$346$	0	0
$347$	3.30393e6	1.47302	0.736508	−	0.676428i	$-0.236473\pi$
$347$	3.30393e6	1.47302	0.736508	+	0.676428i	$0.236473\pi$
$348$	0	0
$349$	−2.93486e6	−1.28980	−0.644902	−	0.764265i	$-0.723102\pi$
$349$	−2.93486e6	−1.28980	−0.644902	+	0.764265i	$0.723102\pi$
$350$	0	0
$351$	2.87157e6	1.24409
$352$	0	0
$353$	633691.	0.270670	0.135335	−	0.990800i	$-0.456789\pi$
$353$	633691.	0.270670	0.135335	+	0.990800i	$0.456789\pi$
$354$	0	0
$355$	−8.25300e6	−3.47569
$356$	0	0
$357$	304554.	0.126472
$358$	0	0
$359$	−3.08062e6	−1.26154	−0.630771	−	0.775969i	$-0.717261\pi$
$359$	−3.08062e6	−1.26154	−0.630771	+	0.775969i	$0.717261\pi$
$360$	0	0
$361$	−94896.2	−0.0383249
$362$	0	0
$363$	−793470.	−0.316056
$364$	0	0
$365$	853042.	0.335149
$366$	0	0
$367$	1.32881e6	0.514988	0.257494	−	0.966280i	$-0.417103\pi$
$367$	1.32881e6	0.514988	0.257494	+	0.966280i	$0.417103\pi$
$368$	0	0
$369$	−928559.	−0.355012
$370$	0	0
$371$	461692.	0.174148
$372$	0	0
$373$	−1.91319e6	−0.712011	−0.356006	−	0.934484i	$-0.615862\pi$
$373$	−1.91319e6	−0.712011	−0.356006	+	0.934484i	$0.615862\pi$
$374$	0	0
$375$	4.38560e6	1.61046
$376$	0	0
$377$	−816906.	−0.296019
$378$	0	0
$379$	−703691.	−0.251642	−0.125821	−	0.992053i	$-0.540157\pi$
$379$	−703691.	−0.251642	−0.125821	+	0.992053i	$0.540157\pi$
$380$	0	0
$381$	−50596.2	−0.0178569
$382$	0	0
$383$	−1.73748e6	−0.605234	−0.302617	−	0.953112i	$-0.597860\pi$
$383$	−1.73748e6	−0.605234	−0.302617	+	0.953112i	$0.597860\pi$
$384$	0	0
$385$	1.39970e6	0.481263
$386$	0	0
$387$	−335566.	−0.113894
$388$	0	0
$389$	3.93096e6	1.31712	0.658559	−	0.752529i	$-0.271166\pi$
$389$	3.93096e6	1.31712	0.658559	+	0.752529i	$0.271166\pi$
$390$	0	0
$391$	−4.75666e6	−1.57347
$392$	0	0
$393$	−376854.	−0.123081
$394$	0	0
$395$	7.38971e6	2.38306
$396$	0	0
$397$	3.99784e6	1.27306	0.636530	−	0.771252i	$-0.280369\pi$
$397$	3.99784e6	1.27306	0.636530	+	0.771252i	$0.280369\pi$
$398$	0	0
$399$	−308882.	−0.0971316
$400$	0	0
$401$	4.39985e6	1.36640	0.683198	−	0.730233i	$-0.260588\pi$
$401$	4.39985e6	1.36640	0.683198	+	0.730233i	$0.260588\pi$
$402$	0	0
$403$	−292823.	−0.0898138
$404$	0	0
$405$	−1.92663e6	−0.583662
$406$	0	0
$407$	3.81426e6	1.14136
$408$	0	0
$409$	−1.64907e6	−0.487451	−0.243726	−	0.969844i	$-0.578370\pi$
$409$	−1.64907e6	−0.487451	−0.243726	+	0.969844i	$0.578370\pi$
$410$	0	0
$411$	808499.	0.236089
$412$	0	0
$413$	−434671.	−0.125396
$414$	0	0
$415$	4.37120e6	1.24589
$416$	0	0
$417$	1.72256e6	0.485103
$418$	0	0
$419$	5.93011e6	1.65017	0.825083	−	0.565012i	$-0.191129\pi$
$419$	5.93011e6	1.65017	0.825083	+	0.565012i	$0.191129\pi$
$420$	0	0
$421$	4.85544e6	1.33513	0.667565	−	0.744552i	$-0.267337\pi$
$421$	4.85544e6	1.33513	0.667565	+	0.744552i	$0.267337\pi$
$422$	0	0
$423$	−3.11427e6	−0.846264
$424$	0	0
$425$	−1.26981e7	−3.41010
$426$	0	0
$427$	272184.	0.0722424
$428$	0	0
$429$	3.46409e6	0.908752
$430$	0	0
$431$	−2.59168e6	−0.672029	−0.336015	−	0.941857i	$-0.609079\pi$
$431$	−2.59168e6	−0.672029	−0.336015	+	0.941857i	$0.609079\pi$
$432$	0	0
$433$	−5.08691e6	−1.30387	−0.651934	−	0.758275i	$-0.726042\pi$
$433$	−5.08691e6	−1.30387	−0.651934	+	0.758275i	$0.726042\pi$
$434$	0	0
$435$	−795596.	−0.201590
$436$	0	0
$437$	4.82426e6	1.20845
$438$	0	0
$439$	5.71552e6	1.41545	0.707725	−	0.706488i	$-0.249722\pi$
$439$	5.71552e6	1.41545	0.707725	+	0.706488i	$0.249722\pi$
$440$	0	0
$441$	2.93201e6	0.717909
$442$	0	0
$443$	4.81790e6	1.16640	0.583201	−	0.812328i	$-0.301800\pi$
$443$	4.81790e6	1.16640	0.583201	+	0.812328i	$0.301800\pi$
$444$	0	0
$445$	8.96359e6	2.14576
$446$	0	0
$447$	2.63237e6	0.623128
$448$	0	0
$449$	−2.36680e6	−0.554046	−0.277023	−	0.960863i	$-0.589348\pi$
$449$	−2.36680e6	−0.554046	−0.277023	+	0.960863i	$0.589348\pi$
$450$	0	0
$451$	−2.61999e6	−0.606539
$452$	0	0
$453$	−666042.	−0.152495
$454$	0	0
$455$	−2.35760e6	−0.533877
$456$	0	0
$457$	1.78378e6	0.399531	0.199766	−	0.979844i	$-0.435982\pi$
$457$	1.78378e6	0.399531	0.199766	+	0.979844i	$0.435982\pi$
$458$	0	0
$459$	−5.06549e6	−1.12225
$460$	0	0
$461$	−4.26575e6	−0.934852	−0.467426	−	0.884032i	$-0.654819\pi$
$461$	−4.26575e6	−0.934852	−0.467426	+	0.884032i	$0.654819\pi$
$462$	0	0
$463$	2.33897e6	0.507074	0.253537	−	0.967326i	$-0.418406\pi$
$463$	2.33897e6	0.507074	0.253537	+	0.967326i	$0.418406\pi$
$464$	0	0
$465$	−285185.	−0.0611637
$466$	0	0
$467$	6.84641e6	1.45268	0.726342	−	0.687334i	$-0.241219\pi$
$467$	6.84641e6	1.45268	0.726342	+	0.687334i	$0.241219\pi$
$468$	0	0
$469$	224578.	0.0471450
$470$	0	0
$471$	−2.46091e6	−0.511145
$472$	0	0
$473$	−946822.	−0.194588
$474$	0	0
$475$	1.28786e7	2.61899
$476$	0	0
$477$	−3.28314e6	−0.660683
$478$	0	0
$479$	−9.12451e6	−1.81707	−0.908533	−	0.417814i	$-0.862797\pi$
$479$	−9.12451e6	−1.81707	−0.908533	+	0.417814i	$0.862797\pi$
$480$	0	0
$481$	−6.42460e6	−1.26614
$482$	0	0
$483$	−625788.	−0.122056
$484$	0	0
$485$	−1.71174e7	−3.30434
$486$	0	0
$487$	−2.68151e6	−0.512338	−0.256169	−	0.966632i	$-0.582460\pi$
$487$	−2.68151e6	−0.512338	−0.256169	+	0.966632i	$0.582460\pi$
$488$	0	0
$489$	1.87954e6	0.355451
$490$	0	0
$491$	−4.83155e6	−0.904446	−0.452223	−	0.891905i	$-0.649369\pi$
$491$	−4.83155e6	−0.904446	−0.452223	+	0.891905i	$0.649369\pi$
$492$	0	0
$493$	1.44103e6	0.267028
$494$	0	0
$495$	−9.95338e6	−1.82582
$496$	0	0
$497$	1.96661e6	0.357131
$498$	0	0
$499$	−8.26537e6	−1.48597	−0.742987	−	0.669306i	$-0.766591\pi$
$499$	−8.26537e6	−1.48597	−0.742987	+	0.669306i	$0.766591\pi$
$500$	0	0
$501$	−3.97392e6	−0.707334
$502$	0	0
$503$	1.06078e7	1.86941	0.934703	−	0.355429i	$-0.115665\pi$
$503$	1.06078e7	1.86941	0.934703	+	0.355429i	$0.115665\pi$
$504$	0	0
$505$	1.80969e7	3.15773
$506$	0	0
$507$	−2.92268e6	−0.504965
$508$	0	0
$509$	−1.89756e6	−0.324639	−0.162320	−	0.986738i	$-0.551898\pi$
$509$	−1.89756e6	−0.324639	−0.162320	+	0.986738i	$0.551898\pi$
$510$	0	0
$511$	−203272.	−0.0344370
$512$	0	0
$513$	5.13749e6	0.861901
$514$	0	0
$515$	−1.55388e7	−2.58166
$516$	0	0
$517$	−8.78713e6	−1.44584
$518$	0	0
$519$	3.35039e6	0.545980
$520$	0	0
$521$	−1.76155e6	−0.284316	−0.142158	−	0.989844i	$-0.545404\pi$
$521$	−1.76155e6	−0.284316	−0.142158	+	0.989844i	$0.545404\pi$
$522$	0	0
$523$	7.12167e6	1.13849	0.569243	−	0.822170i	$-0.307236\pi$
$523$	7.12167e6	1.13849	0.569243	+	0.822170i	$0.307236\pi$
$524$	0	0
$525$	−1.67057e6	−0.264525
$526$	0	0
$527$	516544.	0.0810179
$528$	0	0
$529$	3.33749e6	0.518538
$530$	0	0
$531$	3.09099e6	0.475730
$532$	0	0
$533$	4.41302e6	0.672849
$534$	0	0
$535$	7.78558e6	1.17600
$536$	0	0
$537$	91567.6	0.0137027
$538$	0	0
$539$	8.27287e6	1.22655
$540$	0	0
$541$	6.26443e6	0.920214	0.460107	−	0.887864i	$-0.347811\pi$
$541$	6.26443e6	0.920214	0.460107	+	0.887864i	$0.347811\pi$
$542$	0	0
$543$	−562285.	−0.0818383
$544$	0	0
$545$	348552.	0.0502663
$546$	0	0
$547$	3.19734e6	0.456899	0.228449	−	0.973556i	$-0.426634\pi$
$547$	3.19734e6	0.456899	0.228449	+	0.973556i	$0.426634\pi$
$548$	0	0
$549$	−1.93552e6	−0.274074
$550$	0	0
$551$	−1.46151e6	−0.205080
$552$	0	0
$553$	−1.76090e6	−0.244862
$554$	0	0
$555$	−6.25700e6	−0.862251
$556$	0	0
$557$	1.13299e7	1.54735	0.773677	−	0.633581i	$-0.218416\pi$
$557$	1.13299e7	1.54735	0.773677	+	0.633581i	$0.218416\pi$
$558$	0	0
$559$	1.59479e6	0.215861
$560$	0	0
$561$	−6.11070e6	−0.819754
$562$	0	0
$563$	4.20790e6	0.559492	0.279746	−	0.960074i	$-0.409750\pi$
$563$	4.20790e6	0.559492	0.279746	+	0.960074i	$0.409750\pi$
$564$	0	0
$565$	−1.24661e7	−1.64289
$566$	0	0
$567$	459099.	0.0599720
$568$	0	0
$569$	−3.28836e6	−0.425794	−0.212897	−	0.977075i	$-0.568290\pi$
$569$	−3.28836e6	−0.425794	−0.212897	+	0.977075i	$0.568290\pi$
$570$	0	0
$571$	2.54135e6	0.326192	0.163096	−	0.986610i	$-0.447852\pi$
$571$	2.54135e6	0.326192	0.163096	+	0.986610i	$0.447852\pi$
$572$	0	0
$573$	1.66776e6	0.212201
$574$	0	0
$575$	2.60917e7	3.29104
$576$	0	0
$577$	−1.69533e6	−0.211990	−0.105995	−	0.994367i	$-0.533803\pi$
$577$	−1.69533e6	−0.211990	−0.105995	+	0.994367i	$0.533803\pi$
$578$	0	0
$579$	1.07819e6	0.133659
$580$	0	0
$581$	−1.04161e6	−0.128017
$582$	0	0
$583$	−9.26360e6	−1.12878
$584$	0	0
$585$	1.67651e7	2.02543
$586$	0	0
$587$	−6.20781e6	−0.743607	−0.371803	−	0.928311i	$-0.621260\pi$
$587$	−6.20781e6	−0.743607	−0.371803	+	0.928311i	$0.621260\pi$
$588$	0	0
$589$	−523886.	−0.0622226
$590$	0	0
$591$	223665.	0.0263408
$592$	0	0
$593$	3.01403e6	0.351974	0.175987	−	0.984393i	$-0.443688\pi$
$593$	3.01403e6	0.351974	0.175987	+	0.984393i	$0.443688\pi$
$594$	0	0
$595$	4.15884e6	0.481592
$596$	0	0
$597$	−2.77314e6	−0.318446
$598$	0	0
$599$	−1.21463e7	−1.38317	−0.691586	−	0.722294i	$-0.743088\pi$
$599$	−1.21463e7	−1.38317	−0.691586	+	0.722294i	$0.743088\pi$
$600$	0	0
$601$	1.68363e6	0.190134	0.0950671	−	0.995471i	$-0.469693\pi$
$601$	1.68363e6	0.190134	0.0950671	+	0.995471i	$0.469693\pi$
$602$	0	0
$603$	−1.59700e6	−0.178859
$604$	0	0
$605$	−1.08352e7	−1.20351
$606$	0	0
$607$	5.11718e6	0.563714	0.281857	−	0.959456i	$-0.409050\pi$
$607$	5.11718e6	0.563714	0.281857	+	0.959456i	$0.409050\pi$
$608$	0	0
$609$	189583.	0.0207136
$610$	0	0
$611$	1.48007e7	1.60391
$612$	0	0
$613$	4.10152e6	0.440853	0.220426	−	0.975404i	$-0.429255\pi$
$613$	4.10152e6	0.440853	0.220426	+	0.975404i	$0.429255\pi$
$614$	0	0
$615$	4.29790e6	0.458214
$616$	0	0
$617$	1.52311e7	1.61071	0.805355	−	0.592792i	$-0.201975\pi$
$617$	1.52311e7	1.61071	0.805355	+	0.592792i	$0.201975\pi$
$618$	0	0
$619$	−5.27961e6	−0.553829	−0.276914	−	0.960895i	$-0.589312\pi$
$619$	−5.27961e6	−0.553829	−0.276914	+	0.960895i	$0.589312\pi$
$620$	0	0
$621$	1.04084e7	1.08307
$622$	0	0
$623$	−2.13594e6	−0.220480
$624$	0	0
$625$	3.38068e7	3.46181
$626$	0	0
$627$	6.19754e6	0.629580
$628$	0	0
$629$	1.13331e7	1.14215
$630$	0	0
$631$	1.84888e7	1.84857	0.924283	−	0.381709i	$-0.124664\pi$
$631$	1.84888e7	1.84857	0.924283	+	0.381709i	$0.124664\pi$
$632$	0	0
$633$	2.57129e6	0.255060
$634$	0	0
$635$	−690917.	−0.0679973
$636$	0	0
$637$	−1.39345e7	−1.36064
$638$	0	0
$639$	−1.39848e7	−1.35489
$640$	0	0
$641$	−3.87212e6	−0.372223	−0.186112	−	0.982529i	$-0.559589\pi$
$641$	−3.87212e6	−0.372223	−0.186112	+	0.982529i	$0.559589\pi$
$642$	0	0
$643$	−3.24879e6	−0.309880	−0.154940	−	0.987924i	$-0.549518\pi$
$643$	−3.24879e6	−0.309880	−0.154940	+	0.987924i	$0.549518\pi$
$644$	0	0
$645$	1.55319e6	0.147003
$646$	0	0
$647$	−1.85942e7	−1.74629	−0.873147	−	0.487456i	$-0.837925\pi$
$647$	−1.85942e7	−1.74629	−0.873147	+	0.487456i	$0.837925\pi$
$648$	0	0
$649$	8.72142e6	0.812785
$650$	0	0
$651$	67956.8	0.00628464
$652$	0	0
$653$	9.77909e6	0.897461	0.448730	−	0.893667i	$-0.351876\pi$
$653$	9.77909e6	0.897461	0.448730	+	0.893667i	$0.351876\pi$
$654$	0	0
$655$	−5.14614e6	−0.468682
$656$	0	0
$657$	1.44548e6	0.130647
$658$	0	0
$659$	−1.92292e7	−1.72484	−0.862418	−	0.506196i	$-0.831051\pi$
$659$	−1.92292e7	−1.72484	−0.862418	+	0.506196i	$0.831051\pi$
$660$	0	0
$661$	−1.39614e7	−1.24287	−0.621434	−	0.783467i	$-0.713449\pi$
$661$	−1.39614e7	−1.24287	−0.621434	+	0.783467i	$0.713449\pi$
$662$	0	0
$663$	1.02926e7	0.909374
$664$	0	0
$665$	−4.21794e6	−0.369868
$666$	0	0
$667$	−2.96099e6	−0.257705
$668$	0	0
$669$	1.94471e6	0.167993
$670$	0	0
$671$	−5.46121e6	−0.468255
$672$	0	0
$673$	1.13333e7	0.964537	0.482269	−	0.876023i	$-0.339813\pi$
$673$	1.13333e7	0.964537	0.482269	+	0.876023i	$0.339813\pi$
$674$	0	0
$675$	2.77858e7	2.34727
$676$	0	0
$677$	−9.84009e6	−0.825140	−0.412570	−	0.910926i	$-0.635369\pi$
$677$	−9.84009e6	−0.825140	−0.412570	+	0.910926i	$0.635369\pi$
$678$	0	0
$679$	4.07892e6	0.339524
$680$	0	0
$681$	−6.81498e6	−0.563115
$682$	0	0
$683$	−4282.68	−0.000351288 0	−0.000175644	−	1.00000i	$-0.500056\pi$
$683$	−4282.68	−0.000351288 0	−0.000175644		1.00000i	$0.500056\pi$
$684$	0	0
$685$	1.10405e7	0.899003
$686$	0	0
$687$	−1.87232e6	−0.151352
$688$	0	0
$689$	1.56033e7	1.25218
$690$	0	0
$691$	−5.24701e6	−0.418039	−0.209019	−	0.977911i	$-0.567027\pi$
$691$	−5.24701e6	−0.418039	−0.209019	+	0.977911i	$0.567027\pi$
$692$	0	0
$693$	2.37180e6	0.187605
$694$	0	0
$695$	2.35224e7	1.84723
$696$	0	0
$697$	−7.78462e6	−0.606954
$698$	0	0
$699$	−3.98217e6	−0.308267
$700$	0	0
$701$	−3.73379e6	−0.286982	−0.143491	−	0.989652i	$-0.545833\pi$
$701$	−3.73379e6	−0.286982	−0.143491	+	0.989652i	$0.545833\pi$
$702$	0	0
$703$	−1.14942e7	−0.877180
$704$	0	0
$705$	1.44146e7	1.09227
$706$	0	0
$707$	−4.31231e6	−0.324461
$708$	0	0
$709$	−1.21242e7	−0.905810	−0.452905	−	0.891559i	$-0.649612\pi$
$709$	−1.21242e7	−0.905810	−0.452905	+	0.891559i	$0.649612\pi$
$710$	0	0
$711$	1.25219e7	0.928960
$712$	0	0
$713$	−1.06138e6	−0.0781892
$714$	0	0
$715$	4.73039e7	3.46044
$716$	0	0
$717$	1.12847e6	0.0819772
$718$	0	0
$719$	−1.59176e7	−1.14830	−0.574149	−	0.818751i	$-0.694667\pi$
$719$	−1.59176e7	−1.14830	−0.574149	+	0.818751i	$0.694667\pi$
$720$	0	0
$721$	3.70275e6	0.265269
$722$	0	0
$723$	−3.12342e6	−0.222221
$724$	0	0
$725$	−7.90452e6	−0.558509
$726$	0	0
$727$	633198.	0.0444328	0.0222164	−	0.999753i	$-0.492928\pi$
$727$	633198.	0.0444328	0.0222164	+	0.999753i	$0.492928\pi$
$728$	0	0
$729$	3.08055e6	0.214689
$730$	0	0
$731$	−2.81324e6	−0.194721
$732$	0	0
$733$	−2.64288e7	−1.81684	−0.908421	−	0.418057i	$-0.862711\pi$
$733$	−2.64288e7	−1.81684	−0.908421	+	0.418057i	$0.862711\pi$
$734$	0	0
$735$	−1.35710e7	−0.926603
$736$	0	0
$737$	−4.50603e6	−0.305581
$738$	0	0
$739$	−2.15407e6	−0.145094	−0.0725470	−	0.997365i	$-0.523113\pi$
$739$	−2.15407e6	−0.145094	−0.0725470	+	0.997365i	$0.523113\pi$
$740$	0	0
$741$	−1.04389e7	−0.698409
$742$	0	0
$743$	−1.76951e7	−1.17593	−0.587964	−	0.808887i	$-0.700070\pi$
$743$	−1.76951e7	−1.17593	−0.587964	+	0.808887i	$0.700070\pi$
$744$	0	0
$745$	3.59463e7	2.37281
$746$	0	0
$747$	7.40702e6	0.485671
$748$	0	0
$749$	−1.85523e6	−0.120835
$750$	0	0
$751$	−2.58345e7	−1.67148	−0.835739	−	0.549127i	$-0.814960\pi$
$751$	−2.58345e7	−1.67148	−0.835739	+	0.549127i	$0.814960\pi$
$752$	0	0
$753$	1.29902e7	0.834891
$754$	0	0
$755$	−9.09514e6	−0.580687
$756$	0	0
$757$	−2.47558e7	−1.57013	−0.785067	−	0.619411i	$-0.787371\pi$
$757$	−2.47558e7	−1.57013	−0.785067	+	0.619411i	$0.787371\pi$
$758$	0	0
$759$	1.25561e7	0.791133
$760$	0	0
$761$	−1.12818e7	−0.706183	−0.353092	−	0.935589i	$-0.614870\pi$
$761$	−1.12818e7	−0.706183	−0.353092	+	0.935589i	$0.614870\pi$
$762$	0	0
$763$	−83056.7	−0.00516492
$764$	0	0
$765$	−2.95739e7	−1.82707
$766$	0	0
$767$	−1.46900e7	−0.901643
$768$	0	0
$769$	−1.45315e7	−0.886124	−0.443062	−	0.896491i	$-0.646108\pi$
$769$	−1.45315e7	−0.886124	−0.443062	+	0.896491i	$0.646108\pi$
$770$	0	0
$771$	−256403.	−0.0155341
$772$	0	0
$773$	−5.49667e6	−0.330865	−0.165433	−	0.986221i	$-0.552902\pi$
$773$	−5.49667e6	−0.330865	−0.165433	+	0.986221i	$0.552902\pi$
$774$	0	0
$775$	−2.83341e6	−0.169455
$776$	0	0
$777$	1.49099e6	0.0885974
$778$	0	0
$779$	7.89526e6	0.466147
$780$	0	0
$781$	−3.94590e7	−2.31482
$782$	0	0
$783$	−3.15324e6	−0.183803
$784$	0	0
$785$	−3.36050e7	−1.94639
$786$	0	0
$787$	−1.62860e7	−0.937297	−0.468649	−	0.883385i	$-0.655259\pi$
$787$	−1.62860e7	−0.937297	−0.468649	+	0.883385i	$0.655259\pi$
$788$	0	0
$789$	8.05704e6	0.460769
$790$	0	0
$791$	2.97055e6	0.168809
$792$	0	0
$793$	9.19866e6	0.519447
$794$	0	0
$795$	1.51962e7	0.852742
$796$	0	0
$797$	1.65610e7	0.923506	0.461753	−	0.887009i	$-0.347221\pi$
$797$	1.65610e7	0.923506	0.461753	+	0.887009i	$0.347221\pi$
$798$	0	0
$799$	−2.61087e7	−1.44683
$800$	0	0
$801$	1.51889e7	0.836458
$802$	0	0
$803$	4.07853e6	0.223211
$804$	0	0
$805$	−8.54545e6	−0.464778
$806$	0	0
$807$	1.48170e7	0.800896
$808$	0	0
$809$	−1.93881e7	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$809$	−1.93881e7	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$810$	0	0
$811$	−1.99327e7	−1.06418	−0.532089	−	0.846688i	$-0.678593\pi$
$811$	−1.99327e7	−1.06418	−0.532089	+	0.846688i	$0.678593\pi$
$812$	0	0
$813$	4.74713e6	0.251886
$814$	0	0
$815$	2.56661e7	1.35352
$816$	0	0
$817$	2.85322e6	0.149548
$818$	0	0
$819$	−3.99497e6	−0.208115
$820$	0	0
$821$	−2.97106e7	−1.53834	−0.769172	−	0.639041i	$-0.779331\pi$
$821$	−2.97106e7	−1.53834	−0.769172	+	0.639041i	$0.779331\pi$
$822$	0	0
$823$	3.70742e6	0.190797	0.0953987	−	0.995439i	$-0.469587\pi$
$823$	3.70742e6	0.190797	0.0953987	+	0.995439i	$0.469587\pi$
$824$	0	0
$825$	3.35191e7	1.71458
$826$	0	0
$827$	5.42093e6	0.275620	0.137810	−	0.990459i	$-0.455994\pi$
$827$	5.42093e6	0.275620	0.137810	+	0.990459i	$0.455994\pi$
$828$	0	0
$829$	−3.24596e7	−1.64043	−0.820214	−	0.572056i	$-0.806146\pi$
$829$	−3.24596e7	−1.64043	−0.820214	+	0.572056i	$0.806146\pi$
$830$	0	0
$831$	9.14159e6	0.459218
$832$	0	0
$833$	2.45807e7	1.22739
$834$	0	0
$835$	−5.42659e7	−2.69346
$836$	0	0
$837$	−1.13029e6	−0.0557670
$838$	0	0
$839$	6.99699e6	0.343168	0.171584	−	0.985170i	$-0.445112\pi$
$839$	6.99699e6	0.343168	0.171584	+	0.985170i	$0.445112\pi$
$840$	0	0
$841$	−1.96141e7	−0.956266
$842$	0	0
$843$	−7.36371e6	−0.356885
$844$	0	0
$845$	−3.99107e7	−1.92286
$846$	0	0
$847$	2.58194e6	0.123662
$848$	0	0
$849$	−1.99637e7	−0.950543
$850$	0	0
$851$	−2.32869e7	−1.10227
$852$	0	0
$853$	−5.68399e6	−0.267474	−0.133737	−	0.991017i	$-0.542698\pi$
$853$	−5.68399e6	−0.267474	−0.133737	+	0.991017i	$0.542698\pi$
$854$	0	0
$855$	2.99942e7	1.40321
$856$	0	0
$857$	−8.71203e6	−0.405198	−0.202599	−	0.979262i	$-0.564939\pi$
$857$	−8.71203e6	−0.405198	−0.202599	+	0.979262i	$0.564939\pi$
$858$	0	0
$859$	−1.41568e7	−0.654609	−0.327304	−	0.944919i	$-0.606140\pi$
$859$	−1.41568e7	−0.654609	−0.327304	+	0.944919i	$0.606140\pi$
$860$	0	0
$861$	−1.02415e6	−0.0470820
$862$	0	0
$863$	−9.82195e6	−0.448922	−0.224461	−	0.974483i	$-0.572062\pi$
$863$	−9.82195e6	−0.448922	−0.224461	+	0.974483i	$0.572062\pi$
$864$	0	0
$865$	4.57513e7	2.07904
$866$	0	0
$867$	−7.02021e6	−0.317177
$868$	0	0
$869$	3.53315e7	1.58713
$870$	0	0
$871$	7.58980e6	0.338988
$872$	0	0
$873$	−2.90056e7	−1.28809
$874$	0	0
$875$	−1.42706e7	−0.630120
$876$	0	0
$877$	2.13034e7	0.935298	0.467649	−	0.883914i	$-0.345101\pi$
$877$	2.13034e7	0.935298	0.467649	+	0.883914i	$0.345101\pi$
$878$	0	0
$879$	1.26714e7	0.553162
$880$	0	0
$881$	3.10088e7	1.34600	0.673000	−	0.739642i	$-0.265005\pi$
$881$	3.10088e7	1.34600	0.673000	+	0.739642i	$0.265005\pi$
$882$	0	0
$883$	2.21207e7	0.954766	0.477383	−	0.878695i	$-0.341585\pi$
$883$	2.21207e7	0.954766	0.477383	+	0.878695i	$0.341585\pi$
$884$	0	0
$885$	−1.43068e7	−0.614024
$886$	0	0
$887$	−4.29991e7	−1.83506	−0.917530	−	0.397666i	$-0.869820\pi$
$887$	−4.29991e7	−1.83506	−0.917530	+	0.397666i	$0.869820\pi$
$888$	0	0
$889$	164639.	0.00698680
$890$	0	0
$891$	−9.21156e6	−0.388722
$892$	0	0
$893$	2.64797e7	1.11118
$894$	0	0
$895$	1.25040e6	0.0521786
$896$	0	0
$897$	−2.11490e7	−0.877624
$898$	0	0
$899$	321546.	0.0132692
$900$	0	0
$901$	−2.75244e7	−1.12955
$902$	0	0
$903$	−370111.	−0.0151047
$904$	0	0
$905$	−7.67828e6	−0.311632
$906$	0	0
$907$	−3.34186e7	−1.34887	−0.674435	−	0.738334i	$-0.735613\pi$
$907$	−3.34186e7	−1.34887	−0.674435	+	0.738334i	$0.735613\pi$
$908$	0	0
$909$	3.06653e7	1.23094
$910$	0	0
$911$	3.77821e7	1.50831	0.754155	−	0.656697i	$-0.228047\pi$
$911$	3.77821e7	1.50831	0.754155	+	0.656697i	$0.228047\pi$
$912$	0	0
$913$	2.08994e7	0.829769
$914$	0	0
$915$	8.95869e6	0.353746
$916$	0	0
$917$	1.22628e6	0.0481576
$918$	0	0
$919$	−3.71228e7	−1.44995	−0.724973	−	0.688777i	$-0.758148\pi$
$919$	−3.71228e7	−1.44995	−0.724973	+	0.688777i	$0.758148\pi$
$920$	0	0
$921$	−1.16906e7	−0.454139
$922$	0	0
$923$	6.64632e7	2.56789
$924$	0	0
$925$	−6.21655e7	−2.38888
$926$	0	0
$927$	−2.63306e7	−1.00638
$928$	0	0
$929$	4.73143e6	0.179868	0.0899339	−	0.995948i	$-0.471334\pi$
$929$	4.73143e6	0.179868	0.0899339	+	0.995948i	$0.471334\pi$
$930$	0	0
$931$	−2.49300e7	−0.942646
$932$	0	0
$933$	−5.92124e6	−0.222694
$934$	0	0
$935$	−8.34447e7	−3.12154
$936$	0	0
$937$	−1.74328e7	−0.648660	−0.324330	−	0.945944i	$-0.605139\pi$
$937$	−1.74328e7	−0.648660	−0.324330	+	0.945944i	$0.605139\pi$
$938$	0	0
$939$	−1.03279e7	−0.382251
$940$	0	0
$941$	3.82735e7	1.40904	0.704521	−	0.709683i	$-0.251162\pi$
$941$	3.82735e7	1.40904	0.704521	+	0.709683i	$0.251162\pi$
$942$	0	0
$943$	1.59956e7	0.585762
$944$	0	0
$945$	−9.10028e6	−0.331494
$946$	0	0
$947$	1.65041e7	0.598021	0.299011	−	0.954250i	$-0.403343\pi$
$947$	1.65041e7	0.598021	0.299011	+	0.954250i	$0.403343\pi$
$948$	0	0
$949$	−6.86973e6	−0.247613
$950$	0	0
$951$	1.72217e7	0.617484
$952$	0	0
$953$	−2.15575e7	−0.768894	−0.384447	−	0.923147i	$-0.625608\pi$
$953$	−2.15575e7	−0.768894	−0.384447	+	0.923147i	$0.625608\pi$
$954$	0	0
$955$	2.27741e7	0.808040
$956$	0	0
$957$	−3.80388e6	−0.134260
$958$	0	0
$959$	−2.63084e6	−0.0923736
$960$	0	0
$961$	−2.85139e7	−0.995974
$962$	0	0
$963$	1.31927e7	0.458425
$964$	0	0
$965$	1.47232e7	0.508960
$966$	0	0
$967$	3.38528e7	1.16420	0.582101	−	0.813116i	$-0.302231\pi$
$967$	3.38528e7	1.16420	0.582101	+	0.813116i	$0.302231\pi$
$968$	0	0
$969$	1.84144e7	0.630011
$970$	0	0
$971$	−1.12966e7	−0.384502	−0.192251	−	0.981346i	$-0.561579\pi$
$971$	−1.12966e7	−0.384502	−0.192251	+	0.981346i	$0.561579\pi$
$972$	0	0
$973$	−5.60517e6	−0.189805
$974$	0	0
$975$	−5.64583e7	−1.90202
$976$	0	0
$977$	−2.28369e7	−0.765423	−0.382711	−	0.923868i	$-0.625010\pi$
$977$	−2.28369e7	−0.765423	−0.382711	+	0.923868i	$0.625010\pi$
$978$	0	0
$979$	4.28564e7	1.42909
$980$	0	0
$981$	590624.	0.0195947
$982$	0	0
$983$	−1.60143e7	−0.528598	−0.264299	−	0.964441i	$-0.585140\pi$
$983$	−1.60143e7	−0.528598	−0.264299	+	0.964441i	$0.585140\pi$
$984$	0	0
$985$	3.05426e6	0.100303
$986$	0	0
$987$	−3.43487e6	−0.112232
$988$	0	0
$989$	5.78055e6	0.187922
$990$	0	0
$991$	−1.70581e7	−0.551755	−0.275878	−	0.961193i	$-0.588968\pi$
$991$	−1.70581e7	−0.551755	−0.275878	+	0.961193i	$0.588968\pi$
$992$	0	0
$993$	5.70980e6	0.183759
$994$	0	0
$995$	−3.78687e7	−1.21261
$996$	0	0
$997$	−2.65006e7	−0.844341	−0.422171	−	0.906516i	$-0.638732\pi$
$997$	−2.65006e7	−0.844341	−0.422171	+	0.906516i	$0.638732\pi$
$998$	0	0
$999$	−2.47988e7	−0.786172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.6.a.e.1.3		8
4.3	odd	2		43.6.a.a.1.6	✓	8
12.11	even	2		387.6.a.c.1.3		8
20.19	odd	2		1075.6.a.a.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.a.1.6	✓	8	4.3	odd	2
387.6.a.c.1.3		8	12.11	even	2
688.6.a.e.1.3		8	1.1	even	1	trivial
1075.6.a.a.1.3		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)

\(f(q)\)	\(=\)	\(q-7.84314 q^{3} -107.102 q^{5} +25.5214 q^{7} -181.485 q^{9} +O(q^{10})\)	\(q-7.84314 q^{3} -107.102 q^{5} +25.5214 q^{7} -181.485 q^{9} -512.073 q^{11} +862.516 q^{13} +840.016 q^{15} -1521.49 q^{17} +1543.11 q^{19} -200.168 q^{21} +3126.31 q^{23} +8345.85 q^{25} +3329.30 q^{27} -947.120 q^{29} -339.499 q^{31} +4016.26 q^{33} -2733.40 q^{35} -7448.67 q^{37} -6764.84 q^{39} +5116.45 q^{41} +1849.00 q^{43} +19437.4 q^{45} +17159.9 q^{47} -16155.7 q^{49} +11933.3 q^{51} +18090.4 q^{53} +54844.0 q^{55} -12102.9 q^{57} -17031.6 q^{59} +10664.9 q^{61} -4631.76 q^{63} -92377.3 q^{65} +8799.60 q^{67} -24520.1 q^{69} +77057.3 q^{71} -7964.75 q^{73} -65457.6 q^{75} -13068.8 q^{77} -68997.0 q^{79} +17988.8 q^{81} -40813.4 q^{83} +162955. q^{85} +7428.39 q^{87} -83692.1 q^{89} +22012.6 q^{91} +2662.74 q^{93} -165271. q^{95} +159824. q^{97} +92933.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 26 q^{3} - 212 q^{5} + 136 q^{7} + 546 q^{9}+O(q^{10}) \) 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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