LMFDB - Embedded newform 352.6.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [352,6,Mod(1,352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(352, 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("352.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 352 = 2^{5} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	352.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:	$56.4551045742$
Analytic rank:	$1$
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:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-16.7550$ of defining polynomial
Character	$\chi$	$=$	352.1

$q$-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	$-0.856520\pi$
$5$	−100.636	−1.80024	−0.900118	+	0.435645i	$0.856520\pi$
$6$	0	0
$7$	−99.5357	−0.767775	−0.383887	−	0.923380i	$-0.625415\pi$
$7$	−99.5357	−0.767775	−0.383887	+	0.923380i	$0.625415\pi$
$8$	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	$-0.122282\pi$
$13$	1129.85	1.85423	0.927114	+	0.374780i	$0.122282\pi$
$14$	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	$-0.439289\pi$
$23$	961.905	0.379151	0.189576	+	0.981866i	$0.439289\pi$
$24$	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	$-0.571253\pi$
$29$	−2010.70	−0.443968	−0.221984	+	0.975050i	$0.571253\pi$
$30$	0	0
$31$	−5682.00	−1.06193	−0.530967	−	0.847393i	$-0.678171\pi$
$31$	−5682.00	−1.06193	−0.530967	+	0.847393i	$0.678171\pi$
$32$	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	$-0.645896\pi$
$37$	−7369.08	−0.884930	−0.442465	+	0.896786i	$0.645896\pi$
$38$	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	$-0.270102\pi$
$47$	20022.7	1.32214	0.661071	+	0.750323i	$0.270102\pi$
$48$	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	$-0.577264\pi$
$53$	−9830.41	−0.480709	−0.240354	+	0.970685i	$0.577264\pi$
$54$	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	$-0.336062\pi$
$61$	28629.3	0.985114	0.492557	+	0.870280i	$0.336062\pi$
$62$	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	$-0.631550\pi$
$71$	−34117.8	−0.803221	−0.401611	+	0.915810i	$0.631550\pi$
$72$	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	$-0.437671\pi$
$79$	21585.3	0.389127	0.194563	+	0.980890i	$0.437671\pi$
$80$	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	$-0.646246\pi$
$101$	−90924.3	−0.886903	−0.443452	+	0.896298i	$0.646246\pi$
$102$	0	0
$103$	−167606.	−1.55667	−0.778336	−	0.627848i	$-0.783936\pi$
$103$	−167606.	−1.55667	−0.778336	+	0.627848i	$0.783936\pi$
$104$	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	$-0.561004\pi$
$109$	−47254.7	−0.380960	−0.190480	+	0.981691i	$0.561004\pi$
$110$	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	$-0.391563\pi$
$127$	121460.	0.668225	0.334113	+	0.942533i	$0.391563\pi$
$128$	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	$-0.494130\pi$
$149$	9994.87	0.0368817	0.0184409	+	0.999830i	$0.494130\pi$
$150$	0	0
$151$	486380.	1.73593	0.867967	−	0.496621i	$-0.165426\pi$
$151$	486380.	1.73593	0.867967	+	0.496621i	$0.165426\pi$
$152$	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	$-0.578183\pi$
$157$	−150199.	−0.486315	−0.243158	+	0.969987i	$0.578183\pi$
$158$	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	$-0.722998\pi$
$167$	−464673.	−1.28931	−0.644653	+	0.764475i	$0.722998\pi$
$168$	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	$-0.550720\pi$
$173$	−124921.	−0.317338	−0.158669	+	0.987332i	$0.550720\pi$
$174$	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	$-0.716865\pi$
$181$	−555178.	−1.25961	−0.629805	+	0.776753i	$0.716865\pi$
$182$	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	$-0.834106\pi$
$191$	−874483.	−1.73447	−0.867237	+	0.497896i	$0.834106\pi$
$192$	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	$-0.469706\pi$
$197$	103525.	0.190056	0.0950278	+	0.995475i	$0.469706\pi$
$198$	0	0
$199$	163131.	0.292014	0.146007	−	0.989284i	$-0.453358\pi$
$199$	163131.	0.292014	0.146007	+	0.989284i	$0.453358\pi$
$200$	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	$-0.582820\pi$
$223$	−382088.	−0.514519	−0.257260	+	0.966342i	$0.582820\pi$
$224$	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	$-0.406088\pi$
$229$	461501.	0.581545	0.290773	+	0.956792i	$0.406088\pi$
$230$	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	$-0.256817\pi$
$239$	1.22182e6	1.38360	0.691802	+	0.722087i	$0.256817\pi$
$240$	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	$-0.540145\pi$
$263$	−282196.	−0.251571	−0.125786	+	0.992057i	$0.540145\pi$
$264$	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	$-0.733405\pi$
$269$	−1.58865e6	−1.33859	−0.669297	+	0.742995i	$0.733405\pi$
$270$	0	0
$271$	−1.26811e6	−1.04890	−0.524448	−	0.851443i	$-0.675728\pi$
$271$	−1.26811e6	−1.04890	−0.524448	+	0.851443i	$0.675728\pi$
$272$	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	$-0.590503\pi$
$277$	−716430.	−0.561015	−0.280507	+	0.959852i	$0.590503\pi$
$278$	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	$-0.350889\pi$
$293$	1.32696e6	0.903000	0.451500	+	0.892271i	$0.350889\pi$
$294$	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	$-0.840868\pi$
$311$	−2.99389e6	−1.75523	−0.877617	+	0.479363i	$0.840868\pi$
$312$	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	$-0.439894\pi$
$317$	671680.	0.375417	0.187709	+	0.982225i	$0.439894\pi$
$318$	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	$-0.332144\pi$
$349$	2.29014e6	1.00647	0.503233	+	0.864151i	$0.332144\pi$
$350$	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	$-0.431184\pi$
$359$	1.04765e6	0.429024	0.214512	+	0.976721i	$0.431184\pi$
$360$	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	$-0.756457\pi$
$367$	−3.72232e6	−1.44261	−0.721304	+	0.692618i	$0.756457\pi$
$368$	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	$-0.876481\pi$
$373$	−4.97449e6	−1.85130	−0.925650	+	0.378381i	$0.876481\pi$
$374$	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	$-0.869404\pi$
$383$	−5.26503e6	−1.83402	−0.917009	+	0.398866i	$0.869404\pi$
$384$	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	$-0.697564\pi$
$389$	−3.47145e6	−1.16315	−0.581576	+	0.813492i	$0.697564\pi$
$390$	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	$-0.861168\pi$
$397$	−5.69270e6	−1.81277	−0.906383	+	0.422456i	$0.861168\pi$
$398$	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	$-0.697518\pi$
$421$	−4.22916e6	−1.16292	−0.581459	+	0.813576i	$0.697518\pi$
$422$	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	$-0.618329\pi$
$431$	−2.80166e6	−0.726479	−0.363239	+	0.931696i	$0.618329\pi$
$432$	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	$-0.524354\pi$
$439$	−617298.	−0.152874	−0.0764370	+	0.997074i	$0.524354\pi$
$440$	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	$-0.347086\pi$
$461$	4.21739e6	0.924254	0.462127	+	0.886814i	$0.347086\pi$
$462$	0	0
$463$	989813.	0.214586	0.107293	−	0.994227i	$-0.465782\pi$
$463$	989813.	0.214586	0.107293	+	0.994227i	$0.465782\pi$
$464$	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	$-0.620197\pi$
$479$	−3.70289e6	−0.737398	−0.368699	+	0.929549i	$0.620197\pi$
$480$	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	$-0.212503\pi$
$487$	8.22043e6	1.57062	0.785311	+	0.619101i	$0.212503\pi$
$488$	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	$-0.539368\pi$
$503$	−1.40004e6	−0.246729	−0.123365	+	0.992361i	$0.539368\pi$
$504$	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	$-0.573900\pi$
$509$	−2.68974e6	−0.460167	−0.230083	+	0.973171i	$0.573900\pi$
$510$	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	$-0.427240\pi$
$541$	3.08517e6	0.453196	0.226598	+	0.973988i	$0.427240\pi$
$542$	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	$-0.424539\pi$
$557$	3.43928e6	0.469709	0.234855	+	0.972030i	$0.424539\pi$
$558$	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	$-0.544057\pi$
$599$	−2.42313e6	−0.275937	−0.137969	+	0.990437i	$0.544057\pi$
$600$	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	$-0.545073\pi$
$607$	−2.56220e6	−0.282255	−0.141127	+	0.989991i	$0.545073\pi$
$608$	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	$-0.453787\pi$
$613$	2.69197e6	0.289347	0.144673	+	0.989479i	$0.453787\pi$
$614$	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	$-0.381657\pi$
$631$	7.26684e6	0.726561	0.363280	+	0.931680i	$0.381657\pi$
$632$	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	$-0.199582\pi$
$647$	1.72450e7	1.61958	0.809788	+	0.586722i	$0.199582\pi$
$648$	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	$-0.496813\pi$
$653$	218182.	0.0200233	0.0100116	+	0.999950i	$0.496813\pi$
$654$	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	$-0.514335\pi$
$661$	−1.01145e6	−0.0900413	−0.0450206	+	0.998986i	$0.514335\pi$
$662$	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	$-0.399340\pi$
$677$	7.41726e6	0.621974	0.310987	+	0.950414i	$0.399340\pi$
$678$	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	$-0.717398\pi$
$701$	−1.64220e7	−1.26221	−0.631105	+	0.775697i	$0.717398\pi$
$702$	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	$-0.161386\pi$
$709$	2.34022e7	1.74840	0.874200	+	0.485566i	$0.161386\pi$
$710$	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	$-0.494119\pi$
$719$	512196.	0.0369500	0.0184750	+	0.999829i	$0.494119\pi$
$720$	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	$-0.527049\pi$
$727$	−2.41905e6	−0.169750	−0.0848749	+	0.996392i	$0.527049\pi$
$728$	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	$-0.472331\pi$
$733$	2.52577e6	0.173633	0.0868167	+	0.996224i	$0.472331\pi$
$734$	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	$-0.436388\pi$
$743$	5.97440e6	0.397029	0.198514	+	0.980098i	$0.436388\pi$
$744$	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	$-0.288596\pi$
$751$	1.90539e7	1.23277	0.616387	+	0.787443i	$0.288596\pi$
$752$	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	$-0.370940\pi$
$757$	1.24379e7	0.788874	0.394437	+	0.918923i	$0.370940\pi$
$758$	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	$-0.657641\pi$
$773$	−1.57906e7	−0.950493	−0.475247	+	0.879853i	$0.657641\pi$
$774$	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	$-0.306371\pi$
$797$	2.04962e7	1.14295	0.571476	+	0.820619i	$0.306371\pi$
$798$	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	$-0.635697\pi$
$821$	−1.59725e7	−0.827020	−0.413510	+	0.910500i	$0.635697\pi$
$822$	0	0
$823$	−4.71603e6	−0.242704	−0.121352	−	0.992610i	$-0.538723\pi$
$823$	−4.71603e6	−0.242704	−0.121352	+	0.992610i	$0.538723\pi$
$824$	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	$-0.207222\pi$
$829$	3.14806e7	1.59095	0.795475	+	0.605987i	$0.207222\pi$
$830$	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	$-0.552458\pi$
$839$	−6.69007e6	−0.328115	−0.164057	+	0.986451i	$0.552458\pi$
$840$	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	$-0.740484\pi$
$853$	−2.91412e7	−1.37131	−0.685654	+	0.727928i	$0.740484\pi$
$854$	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	$-0.752452\pi$
$863$	−3.11789e7	−1.42506	−0.712532	+	0.701640i	$0.752452\pi$
$864$	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	$-0.102652\pi$
$877$	4.32059e7	1.89690	0.948449	+	0.316931i	$0.102652\pi$
$878$	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	$-0.393460\pi$
$887$	1.53944e7	0.656984	0.328492	+	0.944507i	$0.393460\pi$
$888$	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	$-0.515505\pi$
$911$	−2.43936e6	−0.0973824	−0.0486912	+	0.998814i	$0.515505\pi$
$912$	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	$-0.805463\pi$
$919$	−4.19368e7	−1.63797	−0.818986	+	0.573814i	$0.805463\pi$
$920$	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	$-0.624935\pi$
$941$	−2.07793e7	−0.764990	−0.382495	+	0.923957i	$0.624935\pi$
$942$	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	$-0.363334\pi$
$967$	2.42092e7	0.832557	0.416279	+	0.909237i	$0.363334\pi$
$968$	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	$-0.427106\pi$
$983$	1.37548e7	0.454015	0.227008	+	0.973893i	$0.427106\pi$
$984$	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	$-0.463220\pi$
$991$	7.12861e6	0.230580	0.115290	+	0.993332i	$0.463220\pi$
$992$	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	$-0.588139\pi$
$997$	−1.71603e7	−0.546746	−0.273373	+	0.961908i	$0.588139\pi$
$998$	0	0
$999$	−2.69423e7	−0.854125

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.6.a.b.1.5	✓	6	4.3	odd	2
352.6.a.d.1.2	yes	6	1.1	even	1	trivial
704.6.a.ba.1.5		6	8.5	even	2
704.6.a.bc.1.2		6	8.3	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:	\( N \)	\(=\)	\( 352 = 2^{5} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	352.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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