LMFDB - Embedded newform 704.6.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,6,Mod(1,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, 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("704.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 704 = 2^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	704.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:	$112.910209148$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 492x^{4} - 880x^{3} + 32838x^{2} + 13500x - 442908 $ x^6 - 3x^5 - 492x^4 - 880x^3 + 32838x^2 + 13500*x - 442908
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 352)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-16.7550$ of defining polynomial
Character	$\chi$	$=$	704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.0835 q^{3} +100.636 q^{5} +99.5357 q^{7} +15.6776 q^{9} +O(q^{10})$	$q-16.0835 q^{3} +100.636 q^{5} +99.5357 q^{7} +15.6776 q^{9} +121.000 q^{11} -1129.85 q^{13} -1618.58 q^{15} +5.49747 q^{17} +576.534 q^{19} -1600.88 q^{21} -961.905 q^{23} +7002.66 q^{25} +3656.13 q^{27} +2010.70 q^{29} +5682.00 q^{31} -1946.10 q^{33} +10016.9 q^{35} +7369.08 q^{37} +18171.9 q^{39} -18926.0 q^{41} +21955.7 q^{43} +1577.74 q^{45} -20022.7 q^{47} -6899.64 q^{49} -88.4183 q^{51} +9830.41 q^{53} +12177.0 q^{55} -9272.67 q^{57} -3365.53 q^{59} -28629.3 q^{61} +1560.49 q^{63} -113704. q^{65} +42601.9 q^{67} +15470.8 q^{69} +34117.8 q^{71} -5606.36 q^{73} -112627. q^{75} +12043.8 q^{77} -21585.3 q^{79} -62612.9 q^{81} +82926.3 q^{83} +553.245 q^{85} -32338.9 q^{87} +104138. q^{89} -112461. q^{91} -91386.3 q^{93} +58020.3 q^{95} +12807.1 q^{97} +1896.99 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 9 q^{3} + 89 q^{5} + 196 q^{7} + 595 q^{9}+O(q^{10}) $ 6 * q + 9 * q^3 + 89 * q^5 + 196 * q^7 + 595 * q^9	$ 6 q + 9 q^{3} + 89 q^{5} + 196 q^{7} + 595 q^{9} + 726 q^{11} + 1032 q^{13} + 655 q^{15} - 1614 q^{17} + 826 q^{19} + 420 q^{21} + 1507 q^{23} + 10547 q^{25} + 10647 q^{27} + 2146 q^{29} + 9471 q^{31} + 1089 q^{33} - 8644 q^{35} + 3523 q^{37} + 34482 q^{39} - 31020 q^{41} + 8798 q^{43} + 56852 q^{45} - 28240 q^{47} - 46362 q^{49} - 31184 q^{51} + 35568 q^{53} + 10769 q^{55} - 75442 q^{57} + 23763 q^{59} + 61070 q^{61} - 37736 q^{63} - 101942 q^{65} + 60947 q^{67} + 167111 q^{69} + 4421 q^{71} - 40568 q^{73} + 175384 q^{75} + 23716 q^{77} + 204154 q^{79} - 130274 q^{81} + 4450 q^{83} + 144892 q^{85} + 18484 q^{87} - 22415 q^{89} - 70072 q^{91} + 356651 q^{93} + 38962 q^{95} - 454109 q^{97} + 71995 q^{99}+O(q^{100}) $ 6 * q + 9 * q^3 + 89 * q^5 + 196 * q^7 + 595 * q^9 + 726 * q^11 + 1032 * q^13 + 655 * q^15 - 1614 * q^17 + 826 * q^19 + 420 * q^21 + 1507 * q^23 + 10547 * q^25 + 10647 * q^27 + 2146 * q^29 + 9471 * q^31 + 1089 * q^33 - 8644 * q^35 + 3523 * q^37 + 34482 * q^39 - 31020 * q^41 + 8798 * q^43 + 56852 * q^45 - 28240 * q^47 - 46362 * q^49 - 31184 * q^51 + 35568 * q^53 + 10769 * q^55 - 75442 * q^57 + 23763 * q^59 + 61070 * q^61 - 37736 * q^63 - 101942 * q^65 + 60947 * q^67 + 167111 * q^69 + 4421 * q^71 - 40568 * q^73 + 175384 * q^75 + 23716 * q^77 + 204154 * q^79 - 130274 * q^81 + 4450 * q^83 + 144892 * q^85 + 18484 * q^87 - 22415 * q^89 - 70072 * q^91 + 356651 * q^93 + 38962 * q^95 - 454109 * q^97 + 71995 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−16.0835	−1.03175	−0.515877	−	0.856663i	$-0.672534\pi$
$3$	−16.0835	−1.03175	−0.515877	+	0.856663i	$0.672534\pi$
$4$	0	0
$5$	100.636	1.80024	0.900118	−	0.435645i	$-0.143480\pi$
$5$	100.636	1.80024	0.900118	+	0.435645i	$0.143480\pi$
$6$	0	0
$7$	99.5357	0.767775	0.383887	−	0.923380i	$-0.374585\pi$
$7$	99.5357	0.767775	0.383887	+	0.923380i	$0.374585\pi$
$8$	0	0
$9$	15.6776	0.0645170
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−1129.85	−1.85423	−0.927114	−	0.374780i	$-0.877718\pi$
$13$	−1129.85	−1.85423	−0.927114	+	0.374780i	$0.877718\pi$
$14$	0	0
$15$	−1618.58	−1.85740
$16$	0	0
$17$	5.49747	0.00461360	0.00230680	−	0.999997i	$-0.499266\pi$
$17$	5.49747	0.00461360	0.00230680	+	0.999997i	$0.499266\pi$
$18$	0	0
$19$	576.534	0.366388	0.183194	−	0.983077i	$-0.441356\pi$
$19$	576.534	0.366388	0.183194	+	0.983077i	$0.441356\pi$
$20$	0	0
$21$	−1600.88	−0.792155
$22$	0	0
$23$	−961.905	−0.379151	−0.189576	−	0.981866i	$-0.560711\pi$
$23$	−961.905	−0.379151	−0.189576	+	0.981866i	$0.560711\pi$
$24$	0	0
$25$	7002.66	2.24085
$26$	0	0
$27$	3656.13	0.965189
$28$	0	0
$29$	2010.70	0.443968	0.221984	−	0.975050i	$-0.428747\pi$
$29$	2010.70	0.443968	0.221984	+	0.975050i	$0.428747\pi$
$30$	0	0
$31$	5682.00	1.06193	0.530967	−	0.847393i	$-0.321829\pi$
$31$	5682.00	1.06193	0.530967	+	0.847393i	$0.321829\pi$
$32$	0	0
$33$	−1946.10	−0.311086
$34$	0	0
$35$	10016.9	1.38218
$36$	0	0
$37$	7369.08	0.884930	0.442465	−	0.896786i	$-0.354104\pi$
$37$	7369.08	0.884930	0.442465	+	0.896786i	$0.354104\pi$
$38$	0	0
$39$	18171.9	1.91311
$40$	0	0
$41$	−18926.0	−1.75833	−0.879163	−	0.476521i	$-0.841897\pi$
$41$	−18926.0	−1.75833	−0.879163	+	0.476521i	$0.841897\pi$
$42$	0	0
$43$	21955.7	1.81083	0.905413	−	0.424532i	$-0.139561\pi$
$43$	21955.7	1.81083	0.905413	+	0.424532i	$0.139561\pi$
$44$	0	0
$45$	1577.74	0.116146
$46$	0	0
$47$	−20022.7	−1.32214	−0.661071	−	0.750323i	$-0.729898\pi$
$47$	−20022.7	−1.32214	−0.661071	+	0.750323i	$0.729898\pi$
$48$	0	0
$49$	−6899.64	−0.410522
$50$	0	0
$51$	−88.4183	−0.00476011
$52$	0	0
$53$	9830.41	0.480709	0.240354	−	0.970685i	$-0.422736\pi$
$53$	9830.41	0.480709	0.240354	+	0.970685i	$0.422736\pi$
$54$	0	0
$55$	12177.0	0.542792
$56$	0	0
$57$	−9272.67	−0.378022
$58$	0	0
$59$	−3365.53	−0.125870	−0.0629352	−	0.998018i	$-0.520046\pi$
$59$	−3365.53	−0.125870	−0.0629352	+	0.998018i	$0.520046\pi$
$60$	0	0
$61$	−28629.3	−0.985114	−0.492557	−	0.870280i	$-0.663938\pi$
$61$	−28629.3	−0.985114	−0.492557	+	0.870280i	$0.663938\pi$
$62$	0	0
$63$	1560.49	0.0495346
$64$	0	0
$65$	−113704.	−3.33805
$66$	0	0
$67$	42601.9	1.15942	0.579712	−	0.814821i	$-0.303165\pi$
$67$	42601.9	1.15942	0.579712	+	0.814821i	$0.303165\pi$
$68$	0	0
$69$	15470.8	0.391191
$70$	0	0
$71$	34117.8	0.803221	0.401611	−	0.915810i	$-0.368450\pi$
$71$	34117.8	0.803221	0.401611	+	0.915810i	$0.368450\pi$
$72$	0	0
$73$	−5606.36	−0.123133	−0.0615664	−	0.998103i	$-0.519610\pi$
$73$	−5606.36	−0.123133	−0.0615664	+	0.998103i	$0.519610\pi$
$74$	0	0
$75$	−112627.	−2.31201
$76$	0	0
$77$	12043.8	0.231493
$78$	0	0
$79$	−21585.3	−0.389127	−0.194563	−	0.980890i	$-0.562329\pi$
$79$	−21585.3	−0.389127	−0.194563	+	0.980890i	$0.562329\pi$
$80$	0	0
$81$	−62612.9	−1.06035
$82$	0	0
$83$	82926.3	1.32129	0.660643	−	0.750700i	$-0.270284\pi$
$83$	82926.3	1.32129	0.660643	+	0.750700i	$0.270284\pi$
$84$	0	0
$85$	553.245	0.00830558
$86$	0	0
$87$	−32338.9	−0.458065
$88$	0	0
$89$	104138.	1.39358	0.696792	−	0.717273i	$-0.254610\pi$
$89$	104138.	1.39358	0.696792	+	0.717273i	$0.254610\pi$
$90$	0	0
$91$	−112461.	−1.42363
$92$	0	0
$93$	−91386.3	−1.09565
$94$	0	0
$95$	58020.3	0.659585
$96$	0	0
$97$	12807.1	0.138205	0.0691024	−	0.997610i	$-0.477986\pi$
$97$	12807.1	0.138205	0.0691024	+	0.997610i	$0.477986\pi$
$98$	0	0
$99$	1896.99	0.0194526
$100$	0	0
$101$	90924.3	0.886903	0.443452	−	0.896298i	$-0.353754\pi$
$101$	90924.3	0.886903	0.443452	+	0.896298i	$0.353754\pi$
$102$	0	0
$103$	167606.	1.55667	0.778336	−	0.627848i	$-0.216064\pi$
$103$	167606.	1.55667	0.778336	+	0.627848i	$0.216064\pi$
$104$	0	0
$105$	−161107.	−1.42607
$106$	0	0
$107$	−131979.	−1.11441	−0.557206	−	0.830374i	$-0.688127\pi$
$107$	−131979.	−1.11441	−0.557206	+	0.830374i	$0.688127\pi$
$108$	0	0
$109$	47254.7	0.380960	0.190480	−	0.981691i	$-0.438996\pi$
$109$	47254.7	0.380960	0.190480	+	0.981691i	$0.438996\pi$
$110$	0	0
$111$	−118520.	−0.913031
$112$	0	0
$113$	10173.1	0.0749476	0.0374738	−	0.999298i	$-0.488069\pi$
$113$	10173.1	0.0749476	0.0374738	+	0.999298i	$0.488069\pi$
$114$	0	0
$115$	−96802.6	−0.682562
$116$	0	0
$117$	−17713.4	−0.119629
$118$	0	0
$119$	547.194	0.00354221
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	304396.	1.81416
$124$	0	0
$125$	390234.	2.23383
$126$	0	0
$127$	−121460.	−0.668225	−0.334113	−	0.942533i	$-0.608437\pi$
$127$	−121460.	−0.668225	−0.334113	+	0.942533i	$0.608437\pi$
$128$	0	0
$129$	−353124.	−1.86833
$130$	0	0
$131$	322828.	1.64359	0.821794	−	0.569785i	$-0.192973\pi$
$131$	322828.	1.64359	0.821794	+	0.569785i	$0.192973\pi$
$132$	0	0
$133$	57385.8	0.281303
$134$	0	0
$135$	367939.	1.73757
$136$	0	0
$137$	−415124.	−1.88963	−0.944814	−	0.327608i	$-0.893758\pi$
$137$	−415124.	−1.88963	−0.944814	+	0.327608i	$0.893758\pi$
$138$	0	0
$139$	−96216.7	−0.422390	−0.211195	−	0.977444i	$-0.567735\pi$
$139$	−96216.7	−0.422390	−0.211195	+	0.977444i	$0.567735\pi$
$140$	0	0
$141$	322035.	1.36413
$142$	0	0
$143$	−136712.	−0.559071
$144$	0	0
$145$	202349.	0.799247
$146$	0	0
$147$	110970.	0.423557
$148$	0	0
$149$	−9994.87	−0.0368817	−0.0184409	−	0.999830i	$-0.505870\pi$
$149$	−9994.87	−0.0368817	−0.0184409	+	0.999830i	$0.505870\pi$
$150$	0	0
$151$	−486380.	−1.73593	−0.867967	−	0.496621i	$-0.834574\pi$
$151$	−486380.	−1.73593	−0.867967	+	0.496621i	$0.834574\pi$
$152$	0	0
$153$	86.1873	0.000297656 0
$154$	0	0
$155$	571816.	1.91173
$156$	0	0
$157$	150199.	0.486315	0.243158	−	0.969987i	$-0.421817\pi$
$157$	150199.	0.486315	0.243158	+	0.969987i	$0.421817\pi$
$158$	0	0
$159$	−158107.	−0.495973
$160$	0	0
$161$	−95743.9	−0.291103
$162$	0	0
$163$	−339066.	−0.999573	−0.499787	−	0.866148i	$-0.666588\pi$
$163$	−339066.	−0.999573	−0.499787	+	0.866148i	$0.666588\pi$
$164$	0	0
$165$	−195848.	−0.560028
$166$	0	0
$167$	464673.	1.28931	0.644653	−	0.764475i	$-0.277002\pi$
$167$	464673.	1.28931	0.644653	+	0.764475i	$0.277002\pi$
$168$	0	0
$169$	905271.	2.43816
$170$	0	0
$171$	9038.69	0.0236383
$172$	0	0
$173$	124921.	0.317338	0.158669	−	0.987332i	$-0.449280\pi$
$173$	124921.	0.317338	0.158669	+	0.987332i	$0.449280\pi$
$174$	0	0
$175$	697015.	1.72047
$176$	0	0
$177$	54129.4	0.129867
$178$	0	0
$179$	391469.	0.913198	0.456599	−	0.889673i	$-0.349067\pi$
$179$	391469.	0.913198	0.456599	+	0.889673i	$0.349067\pi$
$180$	0	0
$181$	555178.	1.25961	0.629805	−	0.776753i	$-0.283135\pi$
$181$	555178.	1.25961	0.629805	+	0.776753i	$0.283135\pi$
$182$	0	0
$183$	460459.	1.01640
$184$	0	0
$185$	741597.	1.59308
$186$	0	0
$187$	665.194	0.00139105
$188$	0	0
$189$	363916.	0.741048
$190$	0	0
$191$	874483.	1.73447	0.867237	−	0.497896i	$-0.165894\pi$
$191$	874483.	1.73447	0.867237	+	0.497896i	$0.165894\pi$
$192$	0	0
$193$	29380.7	0.0567765	0.0283883	−	0.999597i	$-0.490963\pi$
$193$	29380.7	0.0567765	0.0283883	+	0.999597i	$0.490963\pi$
$194$	0	0
$195$	1.82875e6	3.44405
$196$	0	0
$197$	−103525.	−0.190056	−0.0950278	−	0.995475i	$-0.530294\pi$
$197$	−103525.	−0.190056	−0.0950278	+	0.995475i	$0.530294\pi$
$198$	0	0
$199$	−163131.	−0.292014	−0.146007	−	0.989284i	$-0.546642\pi$
$199$	−163131.	−0.292014	−0.146007	+	0.989284i	$0.546642\pi$
$200$	0	0
$201$	−685187.	−1.19624
$202$	0	0
$203$	200136.	0.340867
$204$	0	0
$205$	−1.90464e6	−3.16540
$206$	0	0
$207$	−15080.4	−0.0244617
$208$	0	0
$209$	69760.7	0.110470
$210$	0	0
$211$	848872.	1.31261	0.656306	−	0.754495i	$-0.272118\pi$
$211$	848872.	1.31261	0.656306	+	0.754495i	$0.272118\pi$
$212$	0	0
$213$	−548732.	−0.828727
$214$	0	0
$215$	2.20954e6	3.25991
$216$	0	0
$217$	565562.	0.815326
$218$	0	0
$219$	90169.6	0.127043
$220$	0	0
$221$	−6211.32	−0.00855467
$222$	0	0
$223$	382088.	0.514519	0.257260	−	0.966342i	$-0.417180\pi$
$223$	382088.	0.514519	0.257260	+	0.966342i	$0.417180\pi$
$224$	0	0
$225$	109785.	0.144573
$226$	0	0
$227$	−850742.	−1.09581	−0.547903	−	0.836542i	$-0.684574\pi$
$227$	−850742.	−1.09581	−0.547903	+	0.836542i	$0.684574\pi$
$228$	0	0
$229$	−461501.	−0.581545	−0.290773	−	0.956792i	$-0.593912\pi$
$229$	−461501.	−0.581545	−0.290773	+	0.956792i	$0.593912\pi$
$230$	0	0
$231$	−193706.	−0.238844
$232$	0	0
$233$	874178.	1.05490	0.527449	−	0.849587i	$-0.323149\pi$
$233$	874178.	1.05490	0.527449	+	0.849587i	$0.323149\pi$
$234$	0	0
$235$	−2.01501e6	−2.38017
$236$	0	0
$237$	347167.	0.401483
$238$	0	0
$239$	−1.22182e6	−1.38360	−0.691802	−	0.722087i	$-0.743183\pi$
$239$	−1.22182e6	−1.38360	−0.691802	+	0.722087i	$0.743183\pi$
$240$	0	0
$241$	385792.	0.427869	0.213935	−	0.976848i	$-0.431372\pi$
$241$	385792.	0.427869	0.213935	+	0.976848i	$0.431372\pi$
$242$	0	0
$243$	118592.	0.128837
$244$	0	0
$245$	−694354.	−0.739036
$246$	0	0
$247$	−651398.	−0.679366
$248$	0	0
$249$	−1.33374e6	−1.36324
$250$	0	0
$251$	283850.	0.284384	0.142192	−	0.989839i	$-0.454585\pi$
$251$	283850.	0.284384	0.142192	+	0.989839i	$0.454585\pi$
$252$	0	0
$253$	−116391.	−0.114318
$254$	0	0
$255$	−8898.09	−0.00856932
$256$	0	0
$257$	−1.57262e6	−1.48522	−0.742610	−	0.669724i	$-0.766412\pi$
$257$	−1.57262e6	−1.48522	−0.742610	+	0.669724i	$0.766412\pi$
$258$	0	0
$259$	733487.	0.679427
$260$	0	0
$261$	31522.9	0.0286435
$262$	0	0
$263$	282196.	0.251571	0.125786	−	0.992057i	$-0.459855\pi$
$263$	282196.	0.251571	0.125786	+	0.992057i	$0.459855\pi$
$264$	0	0
$265$	989296.	0.865389
$266$	0	0
$267$	−1.67489e6	−1.43784
$268$	0	0
$269$	1.58865e6	1.33859	0.669297	−	0.742995i	$-0.266595\pi$
$269$	1.58865e6	1.33859	0.669297	+	0.742995i	$0.266595\pi$
$270$	0	0
$271$	1.26811e6	1.04890	0.524448	−	0.851443i	$-0.324272\pi$
$271$	1.26811e6	1.04890	0.524448	+	0.851443i	$0.324272\pi$
$272$	0	0
$273$	1.80876e6	1.46884
$274$	0	0
$275$	847322.	0.675642
$276$	0	0
$277$	716430.	0.561015	0.280507	−	0.959852i	$-0.409497\pi$
$277$	716430.	0.561015	0.280507	+	0.959852i	$0.409497\pi$
$278$	0	0
$279$	89080.4	0.0685128
$280$	0	0
$281$	1.80960e6	1.36715	0.683576	−	0.729880i	$-0.260424\pi$
$281$	1.80960e6	1.36715	0.683576	+	0.729880i	$0.260424\pi$
$282$	0	0
$283$	972811.	0.722042	0.361021	−	0.932558i	$-0.382428\pi$
$283$	972811.	0.722042	0.361021	+	0.932558i	$0.382428\pi$
$284$	0	0
$285$	−933167.	−0.680530
$286$	0	0
$287$	−1.88381e6	−1.35000
$288$	0	0
$289$	−1.41983e6	−0.999979
$290$	0	0
$291$	−205983.	−0.142593
$292$	0	0
$293$	−1.32696e6	−0.903000	−0.451500	−	0.892271i	$-0.649111\pi$
$293$	−1.32696e6	−0.903000	−0.451500	+	0.892271i	$0.649111\pi$
$294$	0	0
$295$	−338694.	−0.226596
$296$	0	0
$297$	442392.	0.291015
$298$	0	0
$299$	1.08681e6	0.703033
$300$	0	0
$301$	2.18538e6	1.39031
$302$	0	0
$303$	−1.46238e6	−0.915066
$304$	0	0
$305$	−2.88115e6	−1.77344
$306$	0	0
$307$	−564125.	−0.341609	−0.170805	−	0.985305i	$-0.554637\pi$
$307$	−564125.	−0.341609	−0.170805	+	0.985305i	$0.554637\pi$
$308$	0	0
$309$	−2.69569e6	−1.60610
$310$	0	0
$311$	2.99389e6	1.75523	0.877617	−	0.479363i	$-0.159132\pi$
$311$	2.99389e6	1.75523	0.877617	+	0.479363i	$0.159132\pi$
$312$	0	0
$313$	2.69271e6	1.55356	0.776781	−	0.629771i	$-0.216851\pi$
$313$	2.69271e6	1.55356	0.776781	+	0.629771i	$0.216851\pi$
$314$	0	0
$315$	157041.	0.0891739
$316$	0	0
$317$	−671680.	−0.375417	−0.187709	−	0.982225i	$-0.560106\pi$
$317$	−671680.	−0.375417	−0.187709	+	0.982225i	$0.560106\pi$
$318$	0	0
$319$	243294.	0.133861
$320$	0	0
$321$	2.12268e6	1.14980
$322$	0	0
$323$	3169.48	0.00169037
$324$	0	0
$325$	−7.91197e6	−4.15505
$326$	0	0
$327$	−760019.	−0.393057
$328$	0	0
$329$	−1.99298e6	−1.01511
$330$	0	0
$331$	618419.	0.310251	0.155125	−	0.987895i	$-0.450422\pi$
$331$	618419.	0.310251	0.155125	+	0.987895i	$0.450422\pi$
$332$	0	0
$333$	115530.	0.0570931
$334$	0	0
$335$	4.28730e6	2.08724
$336$	0	0
$337$	1.62745e6	0.780606	0.390303	−	0.920686i	$-0.372370\pi$
$337$	1.62745e6	0.780606	0.390303	+	0.920686i	$0.372370\pi$
$338$	0	0
$339$	−163619.	−0.0773275
$340$	0	0
$341$	687522.	0.320185
$342$	0	0
$343$	−2.35966e6	−1.08296
$344$	0	0
$345$	1.55692e6	0.704236
$346$	0	0
$347$	1.80110e6	0.802996	0.401498	−	0.915860i	$-0.368490\pi$
$347$	1.80110e6	0.802996	0.401498	+	0.915860i	$0.368490\pi$
$348$	0	0
$349$	−2.29014e6	−1.00647	−0.503233	−	0.864151i	$-0.667856\pi$
$349$	−2.29014e6	−1.00647	−0.503233	+	0.864151i	$0.667856\pi$
$350$	0	0
$351$	−4.13088e6	−1.78968
$352$	0	0
$353$	−2.59156e6	−1.10694	−0.553470	−	0.832869i	$-0.686697\pi$
$353$	−2.59156e6	−1.10694	−0.553470	+	0.832869i	$0.686697\pi$
$354$	0	0
$355$	3.43349e6	1.44599
$356$	0	0
$357$	−8800.78	−0.00365469
$358$	0	0
$359$	−1.04765e6	−0.429024	−0.214512	−	0.976721i	$-0.568816\pi$
$359$	−1.04765e6	−0.429024	−0.214512	+	0.976721i	$0.568816\pi$
$360$	0	0
$361$	−2.14371e6	−0.865760
$362$	0	0
$363$	−235478.	−0.0937958
$364$	0	0
$365$	−564203.	−0.221668
$366$	0	0
$367$	3.72232e6	1.44261	0.721304	−	0.692618i	$-0.243543\pi$
$367$	3.72232e6	1.44261	0.721304	+	0.692618i	$0.243543\pi$
$368$	0	0
$369$	−296715.	−0.113442
$370$	0	0
$371$	978477.	0.369076
$372$	0	0
$373$	4.97449e6	1.85130	0.925650	−	0.378381i	$-0.123519\pi$
$373$	4.97449e6	1.85130	0.925650	+	0.378381i	$0.123519\pi$
$374$	0	0
$375$	−6.27631e6	−2.30476
$376$	0	0
$377$	−2.27179e6	−0.823217
$378$	0	0
$379$	1.49702e6	0.535341	0.267670	−	0.963511i	$-0.413746\pi$
$379$	1.49702e6	0.535341	0.267670	+	0.963511i	$0.413746\pi$
$380$	0	0
$381$	1.95349e6	0.689444
$382$	0	0
$383$	5.26503e6	1.83402	0.917009	−	0.398866i	$-0.130596\pi$
$383$	5.26503e6	1.83402	0.917009	+	0.398866i	$0.130596\pi$
$384$	0	0
$385$	1.21205e6	0.416742
$386$	0	0
$387$	344214.	0.116829
$388$	0	0
$389$	3.47145e6	1.16315	0.581576	−	0.813492i	$-0.302436\pi$
$389$	3.47145e6	1.16315	0.581576	+	0.813492i	$0.302436\pi$
$390$	0	0
$391$	−5288.04	−0.00174925
$392$	0	0
$393$	−5.19219e6	−1.69578
$394$	0	0
$395$	−2.17227e6	−0.700521
$396$	0	0
$397$	5.69270e6	1.81277	0.906383	−	0.422456i	$-0.138832\pi$
$397$	5.69270e6	1.81277	0.906383	+	0.422456i	$0.138832\pi$
$398$	0	0
$399$	−922962.	−0.290236
$400$	0	0
$401$	1.32139e6	0.410364	0.205182	−	0.978724i	$-0.434221\pi$
$401$	1.32139e6	0.410364	0.205182	+	0.978724i	$0.434221\pi$
$402$	0	0
$403$	−6.41982e6	−1.96907
$404$	0	0
$405$	−6.30113e6	−1.90889
$406$	0	0
$407$	891659.	0.266817
$408$	0	0
$409$	147690.	0.0436558	0.0218279	−	0.999762i	$-0.493051\pi$
$409$	147690.	0.0436558	0.0218279	+	0.999762i	$0.493051\pi$
$410$	0	0
$411$	6.67663e6	1.94963
$412$	0	0
$413$	−334991.	−0.0966401
$414$	0	0
$415$	8.34539e6	2.37863
$416$	0	0
$417$	1.54750e6	0.435803
$418$	0	0
$419$	−3.96934e6	−1.10454	−0.552272	−	0.833664i	$-0.686239\pi$
$419$	−3.96934e6	−1.10454	−0.552272	+	0.833664i	$0.686239\pi$
$420$	0	0
$421$	4.22916e6	1.16292	0.581459	−	0.813576i	$-0.302482\pi$
$421$	4.22916e6	1.16292	0.581459	+	0.813576i	$0.302482\pi$
$422$	0	0
$423$	−313909.	−0.0853007
$424$	0	0
$425$	38496.9	0.0103384
$426$	0	0
$427$	−2.84964e6	−0.756346
$428$	0	0
$429$	2.19880e6	0.576823
$430$	0	0
$431$	2.80166e6	0.726479	0.363239	−	0.931696i	$-0.381671\pi$
$431$	2.80166e6	0.726479	0.363239	+	0.931696i	$0.381671\pi$
$432$	0	0
$433$	−4.99859e6	−1.28123	−0.640615	−	0.767862i	$-0.721321\pi$
$433$	−4.99859e6	−1.28123	−0.640615	+	0.767862i	$0.721321\pi$
$434$	0	0
$435$	−3.25447e6	−0.824626
$436$	0	0
$437$	−554571.	−0.138916
$438$	0	0
$439$	617298.	0.152874	0.0764370	−	0.997074i	$-0.475646\pi$
$439$	617298.	0.152874	0.0764370	+	0.997074i	$0.475646\pi$
$440$	0	0
$441$	−108170.	−0.0264856
$442$	0	0
$443$	−3.10619e6	−0.752002	−0.376001	−	0.926619i	$-0.622701\pi$
$443$	−3.10619e6	−0.752002	−0.376001	+	0.926619i	$0.622701\pi$
$444$	0	0
$445$	1.04800e7	2.50878
$446$	0	0
$447$	160752.	0.0380529
$448$	0	0
$449$	5.73212e6	1.34184	0.670919	−	0.741531i	$-0.265900\pi$
$449$	5.73212e6	1.34184	0.670919	+	0.741531i	$0.265900\pi$
$450$	0	0
$451$	−2.29005e6	−0.530155
$452$	0	0
$453$	7.82267e6	1.79106
$454$	0	0
$455$	−1.13176e7	−2.56287
$456$	0	0
$457$	−481316.	−0.107805	−0.0539026	−	0.998546i	$-0.517166\pi$
$457$	−481316.	−0.107805	−0.0539026	+	0.998546i	$0.517166\pi$
$458$	0	0
$459$	20099.5	0.00445300
$460$	0	0
$461$	−4.21739e6	−0.924254	−0.462127	−	0.886814i	$-0.652914\pi$
$461$	−4.21739e6	−0.924254	−0.462127	+	0.886814i	$0.652914\pi$
$462$	0	0
$463$	−989813.	−0.214586	−0.107293	−	0.994227i	$-0.534218\pi$
$463$	−989813.	−0.214586	−0.107293	+	0.994227i	$0.534218\pi$
$464$	0	0
$465$	−9.19678e6	−1.97244
$466$	0	0
$467$	−2.02797e6	−0.430298	−0.215149	−	0.976581i	$-0.569024\pi$
$467$	−2.02797e6	−0.430298	−0.215149	+	0.976581i	$0.569024\pi$
$468$	0	0
$469$	4.24042e6	0.890177
$470$	0	0
$471$	−2.41572e6	−0.501758
$472$	0	0
$473$	2.65664e6	0.545984
$474$	0	0
$475$	4.03728e6	0.821021
$476$	0	0
$477$	154118.	0.0310139
$478$	0	0
$479$	3.70289e6	0.737398	0.368699	−	0.929549i	$-0.379803\pi$
$479$	3.70289e6	0.737398	0.368699	+	0.929549i	$0.379803\pi$
$480$	0	0
$481$	−8.32597e6	−1.64086
$482$	0	0
$483$	1.53989e6	0.300347
$484$	0	0
$485$	1.28886e6	0.248801
$486$	0	0
$487$	−8.22043e6	−1.57062	−0.785311	−	0.619101i	$-0.787497\pi$
$487$	−8.22043e6	−1.57062	−0.785311	+	0.619101i	$0.787497\pi$
$488$	0	0
$489$	5.45335e6	1.03131
$490$	0	0
$491$	−977146.	−0.182918	−0.0914588	−	0.995809i	$-0.529153\pi$
$491$	−977146.	−0.182918	−0.0914588	+	0.995809i	$0.529153\pi$
$492$	0	0
$493$	11053.7	0.00204829
$494$	0	0
$495$	190906.	0.0350193
$496$	0	0
$497$	3.39594e6	0.616693
$498$	0	0
$499$	−8.97522e6	−1.61359	−0.806796	−	0.590830i	$-0.798800\pi$
$499$	−8.97522e6	−1.61359	−0.806796	+	0.590830i	$0.798800\pi$
$500$	0	0
$501$	−7.47355e6	−1.33025
$502$	0	0
$503$	1.40004e6	0.246729	0.123365	−	0.992361i	$-0.460632\pi$
$503$	1.40004e6	0.246729	0.123365	+	0.992361i	$0.460632\pi$
$504$	0	0
$505$	9.15028e6	1.59664
$506$	0	0
$507$	−1.45599e7	−2.51558
$508$	0	0
$509$	2.68974e6	0.460167	0.230083	−	0.973171i	$-0.426100\pi$
$509$	2.68974e6	0.460167	0.230083	+	0.973171i	$0.426100\pi$
$510$	0	0
$511$	−558033.	−0.0945382
$512$	0	0
$513$	2.10788e6	0.353633
$514$	0	0
$515$	1.68673e7	2.80238
$516$	0	0
$517$	−2.42275e6	−0.398641
$518$	0	0
$519$	−2.00917e6	−0.327415
$520$	0	0
$521$	6.78645e6	1.09534	0.547670	−	0.836695i	$-0.315515\pi$
$521$	6.78645e6	1.09534	0.547670	+	0.836695i	$0.315515\pi$
$522$	0	0
$523$	−202631.	−0.0323931	−0.0161965	−	0.999869i	$-0.505156\pi$
$523$	−202631.	−0.0323931	−0.0161965	+	0.999869i	$0.505156\pi$
$524$	0	0
$525$	−1.12104e7	−1.77510
$526$	0	0
$527$	31236.6	0.00489934
$528$	0	0
$529$	−5.51108e6	−0.856244
$530$	0	0
$531$	−52763.6	−0.00812078
$532$	0	0
$533$	2.13836e7	3.26034
$534$	0	0
$535$	−1.32819e7	−2.00621
$536$	0	0
$537$	−6.29618e6	−0.942196
$538$	0	0
$539$	−834856.	−0.123777
$540$	0	0
$541$	−3.08517e6	−0.453196	−0.226598	−	0.973988i	$-0.572760\pi$
$541$	−3.08517e6	−0.453196	−0.226598	+	0.973988i	$0.572760\pi$
$542$	0	0
$543$	−8.92919e6	−1.29961
$544$	0	0
$545$	4.75554e6	0.685817
$546$	0	0
$547$	−5.25255e6	−0.750589	−0.375294	−	0.926906i	$-0.622458\pi$
$547$	−5.25255e6	−0.750589	−0.375294	+	0.926906i	$0.622458\pi$
$548$	0	0
$549$	−448840.	−0.0635566
$550$	0	0
$551$	1.15923e6	0.162664
$552$	0	0
$553$	−2.14851e6	−0.298762
$554$	0	0
$555$	−1.19274e7	−1.64367
$556$	0	0
$557$	−3.43928e6	−0.469709	−0.234855	−	0.972030i	$-0.575461\pi$
$557$	−3.43928e6	−0.469709	−0.234855	+	0.972030i	$0.575461\pi$
$558$	0	0
$559$	−2.48067e7	−3.35768
$560$	0	0
$561$	−10698.6	−0.00143523
$562$	0	0
$563$	−1.10226e7	−1.46559	−0.732796	−	0.680448i	$-0.761785\pi$
$563$	−1.10226e7	−1.46559	−0.732796	+	0.680448i	$0.761785\pi$
$564$	0	0
$565$	1.02378e6	0.134923
$566$	0	0
$567$	−6.23222e6	−0.814114
$568$	0	0
$569$	−2.65040e6	−0.343187	−0.171593	−	0.985168i	$-0.554892\pi$
$569$	−2.65040e6	−0.343187	−0.171593	+	0.985168i	$0.554892\pi$
$570$	0	0
$571$	−8.31534e6	−1.06731	−0.533654	−	0.845703i	$-0.679182\pi$
$571$	−8.31534e6	−1.06731	−0.533654	+	0.845703i	$0.679182\pi$
$572$	0	0
$573$	−1.40647e7	−1.78955
$574$	0	0
$575$	−6.73590e6	−0.849622
$576$	0	0
$577$	6.23879e6	0.780119	0.390060	−	0.920790i	$-0.372454\pi$
$577$	6.23879e6	0.780119	0.390060	+	0.920790i	$0.372454\pi$
$578$	0	0
$579$	−472543.	−0.0585794
$580$	0	0
$581$	8.25413e6	1.01445
$582$	0	0
$583$	1.18948e6	0.144939
$584$	0	0
$585$	−1.78261e6	−0.215361
$586$	0	0
$587$	9.45705e6	1.13282	0.566409	−	0.824124i	$-0.308332\pi$
$587$	9.45705e6	1.13282	0.566409	+	0.824124i	$0.308332\pi$
$588$	0	0
$589$	3.27587e6	0.389080
$590$	0	0
$591$	1.66504e6	0.196091
$592$	0	0
$593$	−1.24162e7	−1.44995	−0.724973	−	0.688778i	$-0.758148\pi$
$593$	−1.24162e7	−1.44995	−0.724973	+	0.688778i	$0.758148\pi$
$594$	0	0
$595$	55067.6	0.00637682
$596$	0	0
$597$	2.62371e6	0.301287
$598$	0	0
$599$	2.42313e6	0.275937	0.137969	−	0.990437i	$-0.455943\pi$
$599$	2.42313e6	0.275937	0.137969	+	0.990437i	$0.455943\pi$
$600$	0	0
$601$	−8.03836e6	−0.907781	−0.453890	−	0.891057i	$-0.649964\pi$
$601$	−8.03836e6	−0.907781	−0.453890	+	0.891057i	$0.649964\pi$
$602$	0	0
$603$	667898.	0.0748026
$604$	0	0
$605$	1.47342e6	0.163658
$606$	0	0
$607$	2.56220e6	0.282255	0.141127	−	0.989991i	$-0.454927\pi$
$607$	2.56220e6	0.282255	0.141127	+	0.989991i	$0.454927\pi$
$608$	0	0
$609$	−3.21888e6	−0.351691
$610$	0	0
$611$	2.26227e7	2.45155
$612$	0	0
$613$	−2.69197e6	−0.289347	−0.144673	−	0.989479i	$-0.546213\pi$
$613$	−2.69197e6	−0.289347	−0.144673	+	0.989479i	$0.546213\pi$
$614$	0	0
$615$	3.06332e7	3.26592
$616$	0	0
$617$	−1.39002e7	−1.46997	−0.734984	−	0.678084i	$-0.762811\pi$
$617$	−1.39002e7	−1.46997	−0.734984	+	0.678084i	$0.762811\pi$
$618$	0	0
$619$	−1.63082e6	−0.171072	−0.0855360	−	0.996335i	$-0.527260\pi$
$619$	−1.63082e6	−0.171072	−0.0855360	+	0.996335i	$0.527260\pi$
$620$	0	0
$621$	−3.51685e6	−0.365953
$622$	0	0
$623$	1.03654e7	1.06996
$624$	0	0
$625$	1.73883e7	1.78057
$626$	0	0
$627$	−1.12199e6	−0.113978
$628$	0	0
$629$	40511.3	0.00408272
$630$	0	0
$631$	−7.26684e6	−0.726561	−0.363280	−	0.931680i	$-0.618343\pi$
$631$	−7.26684e6	−0.726561	−0.363280	+	0.931680i	$0.618343\pi$
$632$	0	0
$633$	−1.36528e7	−1.35429
$634$	0	0
$635$	−1.22233e7	−1.20296
$636$	0	0
$637$	7.79556e6	0.761200
$638$	0	0
$639$	534887.	0.0518215
$640$	0	0
$641$	−2.03214e6	−0.195347	−0.0976737	−	0.995218i	$-0.531140\pi$
$641$	−2.03214e6	−0.195347	−0.0976737	+	0.995218i	$0.531140\pi$
$642$	0	0
$643$	8.46142e6	0.807079	0.403539	−	0.914962i	$-0.367780\pi$
$643$	8.46142e6	0.807079	0.403539	+	0.914962i	$0.367780\pi$
$644$	0	0
$645$	−3.55371e7	−3.36343
$646$	0	0
$647$	−1.72450e7	−1.61958	−0.809788	−	0.586722i	$-0.800418\pi$
$647$	−1.72450e7	−1.61958	−0.809788	+	0.586722i	$0.800418\pi$
$648$	0	0
$649$	−407229.	−0.0379513
$650$	0	0
$651$	−9.09620e6	−0.841216
$652$	0	0
$653$	−218182.	−0.0200233	−0.0100116	−	0.999950i	$-0.503187\pi$
$653$	−218182.	−0.0200233	−0.0100116	+	0.999950i	$0.503187\pi$
$654$	0	0
$655$	3.24882e7	2.95885
$656$	0	0
$657$	−87894.4	−0.00794416
$658$	0	0
$659$	1.09287e7	0.980287	0.490143	−	0.871642i	$-0.336944\pi$
$659$	1.09287e7	0.980287	0.490143	+	0.871642i	$0.336944\pi$
$660$	0	0
$661$	1.01145e6	0.0900413	0.0450206	−	0.998986i	$-0.485665\pi$
$661$	1.01145e6	0.0900413	0.0450206	+	0.998986i	$0.485665\pi$
$662$	0	0
$663$	99899.5	0.00882632
$664$	0	0
$665$	5.77509e6	0.506413
$666$	0	0
$667$	−1.93410e6	−0.168331
$668$	0	0
$669$	−6.14530e6	−0.530857
$670$	0	0
$671$	−3.46415e6	−0.297023
$672$	0	0
$673$	1.74638e7	1.48628	0.743142	−	0.669134i	$-0.233335\pi$
$673$	1.74638e7	1.48628	0.743142	+	0.669134i	$0.233335\pi$
$674$	0	0
$675$	2.56026e7	2.16285
$676$	0	0
$677$	−7.41726e6	−0.621974	−0.310987	−	0.950414i	$-0.600660\pi$
$677$	−7.41726e6	−0.621974	−0.310987	+	0.950414i	$0.600660\pi$
$678$	0	0
$679$	1.27477e6	0.106110
$680$	0	0
$681$	1.36829e7	1.13060
$682$	0	0
$683$	−1.42076e7	−1.16538	−0.582691	−	0.812694i	$-0.698000\pi$
$683$	−1.42076e7	−1.16538	−0.582691	+	0.812694i	$0.698000\pi$
$684$	0	0
$685$	−4.17765e7	−3.40178
$686$	0	0
$687$	7.42253e6	0.600012
$688$	0	0
$689$	−1.11069e7	−0.891343
$690$	0	0
$691$	6.44515e6	0.513497	0.256749	−	0.966478i	$-0.417349\pi$
$691$	6.44515e6	0.513497	0.256749	+	0.966478i	$0.417349\pi$
$692$	0	0
$693$	188819.	0.0149352
$694$	0	0
$695$	−9.68289e6	−0.760402
$696$	0	0
$697$	−104045.	−0.00811222
$698$	0	0
$699$	−1.40598e7	−1.08839
$700$	0	0
$701$	1.64220e7	1.26221	0.631105	−	0.775697i	$-0.282602\pi$
$701$	1.64220e7	1.26221	0.631105	+	0.775697i	$0.282602\pi$
$702$	0	0
$703$	4.24853e6	0.324228
$704$	0	0
$705$	3.24084e7	2.45575
$706$	0	0
$707$	9.05021e6	0.680942
$708$	0	0
$709$	−2.34022e7	−1.74840	−0.874200	−	0.485566i	$-0.838614\pi$
$709$	−2.34022e7	−1.74840	−0.874200	+	0.485566i	$0.838614\pi$
$710$	0	0
$711$	−338407.	−0.0251053
$712$	0	0
$713$	−5.46555e6	−0.402633
$714$	0	0
$715$	−1.37582e7	−1.00646
$716$	0	0
$717$	1.96511e7	1.42754
$718$	0	0
$719$	−512196.	−0.0369500	−0.0184750	−	0.999829i	$-0.505881\pi$
$719$	−512196.	−0.0369500	−0.0184750	+	0.999829i	$0.505881\pi$
$720$	0	0
$721$	1.66828e7	1.19517
$722$	0	0
$723$	−6.20488e6	−0.441456
$724$	0	0
$725$	1.40802e7	0.994866
$726$	0	0
$727$	2.41905e6	0.169750	0.0848749	−	0.996392i	$-0.472951\pi$
$727$	2.41905e6	0.169750	0.0848749	+	0.996392i	$0.472951\pi$
$728$	0	0
$729$	1.33076e7	0.927427
$730$	0	0
$731$	120701.	0.00835443
$732$	0	0
$733$	−2.52577e6	−0.173633	−0.0868167	−	0.996224i	$-0.527669\pi$
$733$	−2.52577e6	−0.173633	−0.0868167	+	0.996224i	$0.527669\pi$
$734$	0	0
$735$	1.11676e7	0.762504
$736$	0	0
$737$	5.15484e6	0.349580
$738$	0	0
$739$	2.14494e7	1.44479	0.722393	−	0.691483i	$-0.243042\pi$
$739$	2.14494e7	1.44479	0.722393	+	0.691483i	$0.243042\pi$
$740$	0	0
$741$	1.04767e7	0.700939
$742$	0	0
$743$	−5.97440e6	−0.397029	−0.198514	−	0.980098i	$-0.563612\pi$
$743$	−5.97440e6	−0.397029	−0.198514	+	0.980098i	$0.563612\pi$
$744$	0	0
$745$	−1.00585e6	−0.0663959
$746$	0	0
$747$	1.30009e6	0.0852455
$748$	0	0
$749$	−1.31366e7	−0.855618
$750$	0	0
$751$	−1.90539e7	−1.23277	−0.616387	−	0.787443i	$-0.711404\pi$
$751$	−1.90539e7	−1.23277	−0.616387	+	0.787443i	$0.711404\pi$
$752$	0	0
$753$	−4.56530e6	−0.293414
$754$	0	0
$755$	−4.89475e7	−3.12509
$756$	0	0
$757$	−1.24379e7	−0.788874	−0.394437	−	0.918923i	$-0.629060\pi$
$757$	−1.24379e7	−0.788874	−0.394437	+	0.918923i	$0.629060\pi$
$758$	0	0
$759$	1.87196e6	0.117949
$760$	0	0
$761$	7.50491e6	0.469768	0.234884	−	0.972023i	$-0.424529\pi$
$761$	7.50491e6	0.469768	0.234884	+	0.972023i	$0.424529\pi$
$762$	0	0
$763$	4.70353e6	0.292491
$764$	0	0
$765$	8673.57	0.000535851 0
$766$	0	0
$767$	3.80255e6	0.233392
$768$	0	0
$769$	−1.13597e7	−0.692710	−0.346355	−	0.938104i	$-0.612581\pi$
$769$	−1.13597e7	−0.692710	−0.346355	+	0.938104i	$0.612581\pi$
$770$	0	0
$771$	2.52932e7	1.53238
$772$	0	0
$773$	1.57906e7	0.950493	0.475247	−	0.879853i	$-0.342359\pi$
$773$	1.57906e7	0.950493	0.475247	+	0.879853i	$0.342359\pi$
$774$	0	0
$775$	3.97892e7	2.37964
$776$	0	0
$777$	−1.17970e7	−0.701002
$778$	0	0
$779$	−1.09115e7	−0.644229
$780$	0	0
$781$	4.12826e6	0.242180
$782$	0	0
$783$	7.35136e6	0.428512
$784$	0	0
$785$	1.51155e7	0.875483
$786$	0	0
$787$	5.96895e6	0.343527	0.171764	−	0.985138i	$-0.445054\pi$
$787$	5.96895e6	0.343527	0.171764	+	0.985138i	$0.445054\pi$
$788$	0	0
$789$	−4.53868e6	−0.259560
$790$	0	0
$791$	1.01259e6	0.0575429
$792$	0	0
$793$	3.23469e7	1.82663
$794$	0	0
$795$	−1.59113e7	−0.892869
$796$	0	0
$797$	−2.04962e7	−1.14295	−0.571476	−	0.820619i	$-0.693629\pi$
$797$	−2.04962e7	−1.14295	−0.571476	+	0.820619i	$0.693629\pi$
$798$	0	0
$799$	−110074.	−0.00609984
$800$	0	0
$801$	1.63263e6	0.0899099
$802$	0	0
$803$	−678369.	−0.0371259
$804$	0	0
$805$	−9.63531e6	−0.524054
$806$	0	0
$807$	−2.55511e7	−1.38110
$808$	0	0
$809$	−3.68444e7	−1.97925	−0.989625	−	0.143674i	$-0.954108\pi$
$809$	−3.68444e7	−1.97925	−0.989625	+	0.143674i	$0.954108\pi$
$810$	0	0
$811$	2.94791e7	1.57384	0.786922	−	0.617053i	$-0.211674\pi$
$811$	2.94791e7	1.57384	0.786922	+	0.617053i	$0.211674\pi$
$812$	0	0
$813$	−2.03955e7	−1.08220
$814$	0	0
$815$	−3.41223e7	−1.79947
$816$	0	0
$817$	1.26582e7	0.663465
$818$	0	0
$819$	−1.76312e6	−0.0918483
$820$	0	0
$821$	1.59725e7	0.827020	0.413510	−	0.910500i	$-0.364303\pi$
$821$	1.59725e7	0.827020	0.413510	+	0.910500i	$0.364303\pi$
$822$	0	0
$823$	4.71603e6	0.242704	0.121352	−	0.992610i	$-0.461277\pi$
$823$	4.71603e6	0.242704	0.121352	+	0.992610i	$0.461277\pi$
$824$	0	0
$825$	−1.36279e7	−0.697097
$826$	0	0
$827$	1.07195e7	0.545016	0.272508	−	0.962154i	$-0.412147\pi$
$827$	1.07195e7	0.545016	0.272508	+	0.962154i	$0.412147\pi$
$828$	0	0
$829$	−3.14806e7	−1.59095	−0.795475	−	0.605987i	$-0.792778\pi$
$829$	−3.14806e7	−1.59095	−0.795475	+	0.605987i	$0.792778\pi$
$830$	0	0
$831$	−1.15227e7	−0.578829
$832$	0	0
$833$	−37930.5	−0.00189398
$834$	0	0
$835$	4.67630e7	2.32106
$836$	0	0
$837$	2.07741e7	1.02497
$838$	0	0
$839$	6.69007e6	0.328115	0.164057	−	0.986451i	$-0.447542\pi$
$839$	6.69007e6	0.328115	0.164057	+	0.986451i	$0.447542\pi$
$840$	0	0
$841$	−1.64683e7	−0.802893
$842$	0	0
$843$	−2.91046e7	−1.41056
$844$	0	0
$845$	9.11031e7	4.38926
$846$	0	0
$847$	1.45730e6	0.0697977
$848$	0	0
$849$	−1.56462e7	−0.744970
$850$	0	0
$851$	−7.08836e6	−0.335522
$852$	0	0
$853$	2.91412e7	1.37131	0.685654	−	0.727928i	$-0.259516\pi$
$853$	2.91412e7	1.37131	0.685654	+	0.727928i	$0.259516\pi$
$854$	0	0
$855$	909621.	0.0425545
$856$	0	0
$857$	−2.11821e7	−0.985184	−0.492592	−	0.870260i	$-0.663950\pi$
$857$	−2.11821e7	−0.985184	−0.492592	+	0.870260i	$0.663950\pi$
$858$	0	0
$859$	9.16226e6	0.423662	0.211831	−	0.977306i	$-0.432057\pi$
$859$	9.16226e6	0.423662	0.211831	+	0.977306i	$0.432057\pi$
$860$	0	0
$861$	3.02982e7	1.39287
$862$	0	0
$863$	3.11789e7	1.42506	0.712532	−	0.701640i	$-0.247548\pi$
$863$	3.11789e7	1.42506	0.712532	+	0.701640i	$0.247548\pi$
$864$	0	0
$865$	1.25716e7	0.571283
$866$	0	0
$867$	2.28357e7	1.03173
$868$	0	0
$869$	−2.61183e6	−0.117326
$870$	0	0
$871$	−4.81339e7	−2.14984
$872$	0	0
$873$	200786.	0.00891656
$874$	0	0
$875$	3.88422e7	1.71508
$876$	0	0
$877$	−4.32059e7	−1.89690	−0.948449	−	0.316931i	$-0.897348\pi$
$877$	−4.32059e7	−1.89690	−0.948449	+	0.316931i	$0.897348\pi$
$878$	0	0
$879$	2.13420e7	0.931674
$880$	0	0
$881$	−2.40150e7	−1.04242	−0.521210	−	0.853428i	$-0.674519\pi$
$881$	−2.40150e7	−1.04242	−0.521210	+	0.853428i	$0.674519\pi$
$882$	0	0
$883$	−2.76263e7	−1.19240	−0.596200	−	0.802836i	$-0.703323\pi$
$883$	−2.76263e7	−1.19240	−0.596200	+	0.802836i	$0.703323\pi$
$884$	0	0
$885$	5.44738e6	0.233792
$886$	0	0
$887$	−1.53944e7	−0.656984	−0.328492	−	0.944507i	$-0.606540\pi$
$887$	−1.53944e7	−0.656984	−0.328492	+	0.944507i	$0.606540\pi$
$888$	0	0
$889$	−1.20896e7	−0.513047
$890$	0	0
$891$	−7.57616e6	−0.319709
$892$	0	0
$893$	−1.15438e7	−0.484417
$894$	0	0
$895$	3.93960e7	1.64397
$896$	0	0
$897$	−1.74797e7	−0.725357
$898$	0	0
$899$	1.14248e7	0.471464
$900$	0	0
$901$	54042.3	0.00221780
$902$	0	0
$903$	−3.51485e7	−1.43445
$904$	0	0
$905$	5.58711e7	2.26760
$906$	0	0
$907$	−1.15688e7	−0.466951	−0.233475	−	0.972363i	$-0.575010\pi$
$907$	−1.15688e7	−0.466951	−0.233475	+	0.972363i	$0.575010\pi$
$908$	0	0
$909$	1.42548e6	0.0572204
$910$	0	0
$911$	2.43936e6	0.0973824	0.0486912	−	0.998814i	$-0.484495\pi$
$911$	2.43936e6	0.0973824	0.0486912	+	0.998814i	$0.484495\pi$
$912$	0	0
$913$	1.00341e7	0.398383
$914$	0	0
$915$	4.63389e7	1.82975
$916$	0	0
$917$	3.21329e7	1.26191
$918$	0	0
$919$	4.19368e7	1.63797	0.818986	−	0.573814i	$-0.194537\pi$
$919$	4.19368e7	1.63797	0.818986	+	0.573814i	$0.194537\pi$
$920$	0	0
$921$	9.07309e6	0.352457
$922$	0	0
$923$	−3.85481e7	−1.48936
$924$	0	0
$925$	5.16032e7	1.98300
$926$	0	0
$927$	2.62767e6	0.100432
$928$	0	0
$929$	−2.44846e7	−0.930796	−0.465398	−	0.885101i	$-0.654089\pi$
$929$	−2.44846e7	−0.930796	−0.465398	+	0.885101i	$0.654089\pi$
$930$	0	0
$931$	−3.97788e6	−0.150410
$932$	0	0
$933$	−4.81521e7	−1.81097
$934$	0	0
$935$	66942.6	0.00250423
$936$	0	0
$937$	1.50232e7	0.559002	0.279501	−	0.960145i	$-0.409831\pi$
$937$	1.50232e7	0.559002	0.279501	+	0.960145i	$0.409831\pi$
$938$	0	0
$939$	−4.33081e7	−1.60289
$940$	0	0
$941$	2.07793e7	0.764990	0.382495	−	0.923957i	$-0.375065\pi$
$941$	2.07793e7	0.764990	0.382495	+	0.923957i	$0.375065\pi$
$942$	0	0
$943$	1.82050e7	0.666672
$944$	0	0
$945$	3.66231e7	1.33406
$946$	0	0
$947$	4.09922e7	1.48534	0.742670	−	0.669657i	$-0.233559\pi$
$947$	4.09922e7	1.48534	0.742670	+	0.669657i	$0.233559\pi$
$948$	0	0
$949$	6.33435e6	0.228316
$950$	0	0
$951$	1.08029e7	0.387338
$952$	0	0
$953$	−1.32155e7	−0.471359	−0.235680	−	0.971831i	$-0.575732\pi$
$953$	−1.32155e7	−0.471359	−0.235680	+	0.971831i	$0.575732\pi$
$954$	0	0
$955$	8.80047e7	3.12246
$956$	0	0
$957$	−3.91301e6	−0.138112
$958$	0	0
$959$	−4.13197e7	−1.45081
$960$	0	0
$961$	3.65602e6	0.127703
$962$	0	0
$963$	−2.06912e6	−0.0718986
$964$	0	0
$965$	2.95676e6	0.102211
$966$	0	0
$967$	−2.42092e7	−0.832557	−0.416279	−	0.909237i	$-0.636666\pi$
$967$	−2.42092e7	−0.832557	−0.416279	+	0.909237i	$0.636666\pi$
$968$	0	0
$969$	−50976.2	−0.00174405
$970$	0	0
$971$	−4.87361e7	−1.65883	−0.829417	−	0.558630i	$-0.811327\pi$
$971$	−4.87361e7	−1.65883	−0.829417	+	0.558630i	$0.811327\pi$
$972$	0	0
$973$	−9.57700e6	−0.324300
$974$	0	0
$975$	1.27252e8	4.28699
$976$	0	0
$977$	8.53236e6	0.285978	0.142989	−	0.989724i	$-0.454329\pi$
$977$	8.53236e6	0.285978	0.142989	+	0.989724i	$0.454329\pi$
$978$	0	0
$979$	1.26007e7	0.420181
$980$	0	0
$981$	740842.	0.0245784
$982$	0	0
$983$	−1.37548e7	−0.454015	−0.227008	−	0.973893i	$-0.572894\pi$
$983$	−1.37548e7	−0.454015	−0.227008	+	0.973893i	$0.572894\pi$
$984$	0	0
$985$	−1.04184e7	−0.342145
$986$	0	0
$987$	3.20539e7	1.04734
$988$	0	0
$989$	−2.11193e7	−0.686577
$990$	0	0
$991$	−7.12861e6	−0.230580	−0.115290	−	0.993332i	$-0.536780\pi$
$991$	−7.12861e6	−0.230580	−0.115290	+	0.993332i	$0.536780\pi$
$992$	0	0
$993$	−9.94631e6	−0.320102
$994$	0	0
$995$	−1.64169e7	−0.525694
$996$	0	0
$997$	1.71603e7	0.546746	0.273373	−	0.961908i	$-0.411861\pi$
$997$	1.71603e7	0.546746	0.273373	+	0.961908i	$0.411861\pi$
$998$	0	0
$999$	2.69423e7	0.854125

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	704.6.a.bc.1.2		6
4.3	odd	2		704.6.a.ba.1.5		6
8.3	odd	2		352.6.a.d.1.2	yes	6
8.5	even	2		352.6.a.b.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.6.a.b.1.5	✓	6	8.5	even	2
352.6.a.d.1.2	yes	6	8.3	odd	2
704.6.a.ba.1.5		6	4.3	odd	2
704.6.a.bc.1.2		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	704.a (trivial)

\(f(q)\)	\(=\)	\(q-16.0835 q^{3} +100.636 q^{5} +99.5357 q^{7} +15.6776 q^{9} +O(q^{10})\)	\(q-16.0835 q^{3} +100.636 q^{5} +99.5357 q^{7} +15.6776 q^{9} +121.000 q^{11} -1129.85 q^{13} -1618.58 q^{15} +5.49747 q^{17} +576.534 q^{19} -1600.88 q^{21} -961.905 q^{23} +7002.66 q^{25} +3656.13 q^{27} +2010.70 q^{29} +5682.00 q^{31} -1946.10 q^{33} +10016.9 q^{35} +7369.08 q^{37} +18171.9 q^{39} -18926.0 q^{41} +21955.7 q^{43} +1577.74 q^{45} -20022.7 q^{47} -6899.64 q^{49} -88.4183 q^{51} +9830.41 q^{53} +12177.0 q^{55} -9272.67 q^{57} -3365.53 q^{59} -28629.3 q^{61} +1560.49 q^{63} -113704. q^{65} +42601.9 q^{67} +15470.8 q^{69} +34117.8 q^{71} -5606.36 q^{73} -112627. q^{75} +12043.8 q^{77} -21585.3 q^{79} -62612.9 q^{81} +82926.3 q^{83} +553.245 q^{85} -32338.9 q^{87} +104138. q^{89} -112461. q^{91} -91386.3 q^{93} +58020.3 q^{95} +12807.1 q^{97} +1896.99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 9 q^{3} + 89 q^{5} + 196 q^{7} + 595 q^{9}+O(q^{10}) \) 6 * q + 9 * q^3 + 89 * q^5 + 196 * q^7 + 595 * q^9	\( 6 q + 9 q^{3} + 89 q^{5} + 196 q^{7} + 595 q^{9} + 726 q^{11} + 1032 q^{13} + 655 q^{15} - 1614 q^{17} + 826 q^{19} + 420 q^{21} + 1507 q^{23} + 10547 q^{25} + 10647 q^{27} + 2146 q^{29} + 9471 q^{31} + 1089 q^{33} - 8644 q^{35} + 3523 q^{37} + 34482 q^{39} - 31020 q^{41} + 8798 q^{43} + 56852 q^{45} - 28240 q^{47} - 46362 q^{49} - 31184 q^{51} + 35568 q^{53} + 10769 q^{55} - 75442 q^{57} + 23763 q^{59} + 61070 q^{61} - 37736 q^{63} - 101942 q^{65} + 60947 q^{67} + 167111 q^{69} + 4421 q^{71} - 40568 q^{73} + 175384 q^{75} + 23716 q^{77} + 204154 q^{79} - 130274 q^{81} + 4450 q^{83} + 144892 q^{85} + 18484 q^{87} - 22415 q^{89} - 70072 q^{91} + 356651 q^{93} + 38962 q^{95} - 454109 q^{97} + 71995 q^{99}+O(q^{100}) \) 6 * q + 9 * q^3 + 89 * q^5 + 196 * q^7 + 595 * q^9 + 726 * q^11 + 1032 * q^13 + 655 * q^15 - 1614 * q^17 + 826 * q^19 + 420 * q^21 + 1507 * q^23 + 10547 * q^25 + 10647 * q^27 + 2146 * q^29 + 9471 * q^31 + 1089 * q^33 - 8644 * q^35 + 3523 * q^37 + 34482 * q^39 - 31020 * q^41 + 8798 * q^43 + 56852 * q^45 - 28240 * q^47 - 46362 * q^49 - 31184 * q^51 + 35568 * q^53 + 10769 * q^55 - 75442 * q^57 + 23763 * q^59 + 61070 * q^61 - 37736 * q^63 - 101942 * q^65 + 60947 * q^67 + 167111 * q^69 + 4421 * q^71 - 40568 * q^73 + 175384 * q^75 + 23716 * q^77 + 204154 * q^79 - 130274 * q^81 + 4450 * q^83 + 144892 * q^85 + 18484 * q^87 - 22415 * q^89 - 70072 * q^91 + 356651 * q^93 + 38962 * q^95 - 454109 * q^97 + 71995 * q^99

Introduction

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