LMFDB - Embedded newform 49.6.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,6,Mod(1,49)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("49.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 49 = 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	49.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:	$7.85880717084$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.27492$ of defining polynomial
Character	$\chi$	$=$	49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.725083 q^{2} -19.6495 q^{3} -31.4743 q^{4} +46.7492 q^{5} -14.2475 q^{6} -46.0241 q^{8} +143.103 q^{9} +O(q^{10})$	\(q+0.725083 q^{2} -19.6495 q^{3} -31.4743 q^{4} +46.7492 q^{5} -14.2475 q^{6} -46.0241 q^{8} +143.103 q^{9} +33.8970 q^{10} +666.090 q^{11} +618.453 q^{12} +650.640 q^{13} -918.598 q^{15} +973.805 q^{16} -1186.89 q^{17} +103.762 q^{18} +1565.05 q^{19} -1471.40 q^{20} +482.970 q^{22} -1100.15 q^{23} +904.350 q^{24} -939.515 q^{25} +471.768 q^{26} +1962.93 q^{27} +2396.72 q^{29} -666.060 q^{30} +2048.46 q^{31} +2178.86 q^{32} -13088.3 q^{33} -860.596 q^{34} -4504.06 q^{36} +1077.54 q^{37} +1134.79 q^{38} -12784.7 q^{39} -2151.59 q^{40} -1098.21 q^{41} +16564.3 q^{43} -20964.7 q^{44} +6689.95 q^{45} -797.702 q^{46} +8298.39 q^{47} -19134.8 q^{48} -681.226 q^{50} +23321.9 q^{51} -20478.4 q^{52} +5519.18 q^{53} +1423.28 q^{54} +31139.1 q^{55} -30752.5 q^{57} +1737.82 q^{58} +14230.4 q^{59} +28912.2 q^{60} +14234.7 q^{61} +1485.30 q^{62} -29581.9 q^{64} +30416.9 q^{65} -9490.12 q^{66} +19730.4 q^{67} +37356.6 q^{68} +21617.5 q^{69} +64562.7 q^{71} -6586.18 q^{72} -28567.0 q^{73} +781.309 q^{74} +18461.0 q^{75} -49258.8 q^{76} -9270.00 q^{78} -30633.4 q^{79} +45524.6 q^{80} -73344.6 q^{81} -796.293 q^{82} +675.946 q^{83} -55486.3 q^{85} +12010.5 q^{86} -47094.4 q^{87} -30656.2 q^{88} -125971. q^{89} +4850.76 q^{90} +34626.5 q^{92} -40251.1 q^{93} +6017.02 q^{94} +73164.9 q^{95} -42813.5 q^{96} +22906.8 q^{97} +95319.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 9 q^{2} + 6 q^{3} + 5 q^{4} + 18 q^{5} + 198 q^{6} - 9 q^{8} + 558 q^{9}+O(q^{10}) $ 2 * q + 9 * q^2 + 6 * q^3 + 5 * q^4 + 18 * q^5 + 198 * q^6 - 9 * q^8 + 558 * q^9	\( 2 q + 9 q^{2} + 6 q^{3} + 5 q^{4} + 18 q^{5} + 198 q^{6} - 9 q^{8} + 558 q^{9} - 204 q^{10} + 396 q^{11} + 1554 q^{12} + 350 q^{13} - 1656 q^{15} + 113 q^{16} - 1800 q^{17} + 3537 q^{18} + 3266 q^{19} - 2520 q^{20} - 1752 q^{22} + 2088 q^{23} + 1854 q^{24} - 3238 q^{25} - 2016 q^{26} + 6372 q^{27} + 6696 q^{29} - 6768 q^{30} + 20 q^{31} - 6129 q^{32} - 20016 q^{33} - 5934 q^{34} + 10629 q^{36} + 6232 q^{37} + 15210 q^{38} - 20496 q^{39} - 3216 q^{40} + 6048 q^{41} - 3020 q^{43} - 30816 q^{44} - 5238 q^{45} + 25584 q^{46} - 11700 q^{47} - 41214 q^{48} - 19701 q^{50} + 7596 q^{51} - 31444 q^{52} + 9468 q^{53} + 37908 q^{54} + 38904 q^{55} + 12876 q^{57} + 37314 q^{58} + 43938 q^{59} + 2016 q^{60} + 64754 q^{61} - 15300 q^{62} - 70783 q^{64} + 39060 q^{65} - 66816 q^{66} + 24784 q^{67} + 14994 q^{68} + 103392 q^{69} + 97416 q^{71} + 8775 q^{72} - 17452 q^{73} + 43434 q^{74} - 40494 q^{75} + 12782 q^{76} - 73080 q^{78} + 51256 q^{79} + 70272 q^{80} - 61074 q^{81} + 58338 q^{82} - 117558 q^{83} - 37860 q^{85} - 150048 q^{86} + 63180 q^{87} - 40656 q^{88} - 84276 q^{89} - 93852 q^{90} + 150912 q^{92} - 92280 q^{93} - 159468 q^{94} + 24264 q^{95} - 255906 q^{96} - 20776 q^{97} - 16740 q^{99}+O(q^{100}) \) 2 * q + 9 * q^2 + 6 * q^3 + 5 * q^4 + 18 * q^5 + 198 * q^6 - 9 * q^8 + 558 * q^9 - 204 * q^10 + 396 * q^11 + 1554 * q^12 + 350 * q^13 - 1656 * q^15 + 113 * q^16 - 1800 * q^17 + 3537 * q^18 + 3266 * q^19 - 2520 * q^20 - 1752 * q^22 + 2088 * q^23 + 1854 * q^24 - 3238 * q^25 - 2016 * q^26 + 6372 * q^27 + 6696 * q^29 - 6768 * q^30 + 20 * q^31 - 6129 * q^32 - 20016 * q^33 - 5934 * q^34 + 10629 * q^36 + 6232 * q^37 + 15210 * q^38 - 20496 * q^39 - 3216 * q^40 + 6048 * q^41 - 3020 * q^43 - 30816 * q^44 - 5238 * q^45 + 25584 * q^46 - 11700 * q^47 - 41214 * q^48 - 19701 * q^50 + 7596 * q^51 - 31444 * q^52 + 9468 * q^53 + 37908 * q^54 + 38904 * q^55 + 12876 * q^57 + 37314 * q^58 + 43938 * q^59 + 2016 * q^60 + 64754 * q^61 - 15300 * q^62 - 70783 * q^64 + 39060 * q^65 - 66816 * q^66 + 24784 * q^67 + 14994 * q^68 + 103392 * q^69 + 97416 * q^71 + 8775 * q^72 - 17452 * q^73 + 43434 * q^74 - 40494 * q^75 + 12782 * q^76 - 73080 * q^78 + 51256 * q^79 + 70272 * q^80 - 61074 * q^81 + 58338 * q^82 - 117558 * q^83 - 37860 * q^85 - 150048 * q^86 + 63180 * q^87 - 40656 * q^88 - 84276 * q^89 - 93852 * q^90 + 150912 * q^92 - 92280 * q^93 - 159468 * q^94 + 24264 * q^95 - 255906 * q^96 - 20776 * q^97 - 16740 * 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 $

$2$	0.725083	0.128178	0.0640889	−	0.997944i	$-0.479586\pi$
$2$	0.725083	0.128178	0.0640889	+	0.997944i	$0.479586\pi$
$3$	−19.6495	−1.26052	−0.630258	−	0.776386i	$-0.717051\pi$
$3$	−19.6495	−1.26052	−0.630258	+	0.776386i	$0.717051\pi$
$4$	−31.4743	−0.983570
$5$	46.7492	0.836275	0.418137	−	0.908384i	$-0.362683\pi$
$5$	46.7492	0.836275	0.418137	+	0.908384i	$0.362683\pi$
$6$	−14.2475	−0.161570
$7$	0	0
$7$	0	0
$8$	−46.0241	−0.254250
$9$	143.103	0.588901
$10$	33.8970	0.107192
$11$	666.090	1.65978	0.829891	−	0.557926i	$-0.188403\pi$
$11$	666.090	1.65978	0.829891	+	0.557926i	$0.188403\pi$
$12$	618.453	1.23981
$13$	650.640	1.06778	0.533890	−	0.845554i	$-0.320729\pi$
$13$	650.640	1.06778	0.533890	+	0.845554i	$0.320729\pi$
$14$	0	0
$15$	−918.598	−1.05414
$16$	973.805	0.950981
$17$	−1186.89	−0.996069	−0.498035	−	0.867157i	$-0.665945\pi$
$17$	−1186.89	−0.996069	−0.498035	+	0.867157i	$0.665945\pi$
$18$	103.762	0.0754840
$19$	1565.05	0.994591	0.497296	−	0.867581i	$-0.334326\pi$
$19$	1565.05	0.994591	0.497296	+	0.867581i	$0.334326\pi$
$20$	−1471.40	−0.822535
$21$	0	0
$22$	482.970	0.212747
$23$	−1100.15	−0.433644	−0.216822	−	0.976211i	$-0.569569\pi$
$23$	−1100.15	−0.433644	−0.216822	+	0.976211i	$0.569569\pi$
$24$	904.350	0.320486
$25$	−939.515	−0.300645
$26$	471.768	0.136866
$27$	1962.93	0.518197
$28$	0	0
$29$	2396.72	0.529203	0.264602	−	0.964358i	$-0.414760\pi$
$29$	2396.72	0.529203	0.264602	+	0.964358i	$0.414760\pi$
$30$	−666.060	−0.135117
$31$	2048.46	0.382844	0.191422	−	0.981508i	$-0.438690\pi$
$31$	2048.46	0.382844	0.191422	+	0.981508i	$0.438690\pi$
$32$	2178.86	0.376144
$33$	−13088.3	−2.09218
$34$	−860.596	−0.127674
$35$	0	0
$36$	−4504.06	−0.579226
$37$	1077.54	0.129399	0.0646995	−	0.997905i	$-0.479391\pi$
$37$	1077.54	0.129399	0.0646995	+	0.997905i	$0.479391\pi$
$38$	1134.79	0.127484
$39$	−12784.7	−1.34595
$40$	−2151.59	−0.212622
$41$	−1098.21	−0.102029	−0.0510147	−	0.998698i	$-0.516246\pi$
$41$	−1098.21	−0.102029	−0.0510147	+	0.998698i	$0.516246\pi$
$42$	0	0
$43$	16564.3	1.36616	0.683081	−	0.730343i	$-0.260640\pi$
$43$	16564.3	1.36616	0.683081	+	0.730343i	$0.260640\pi$
$44$	−20964.7	−1.63251
$45$	6689.95	0.492483
$46$	−797.702	−0.0555835
$47$	8298.39	0.547960	0.273980	−	0.961735i	$-0.411660\pi$
$47$	8298.39	0.547960	0.273980	+	0.961735i	$0.411660\pi$
$48$	−19134.8	−1.19873
$49$	0	0
$50$	−681.226	−0.0385360
$51$	23321.9	1.25556
$52$	−20478.4	−1.05024
$53$	5519.18	0.269889	0.134944	−	0.990853i	$-0.456914\pi$
$53$	5519.18	0.269889	0.134944	+	0.990853i	$0.456914\pi$
$54$	1423.28	0.0664213
$55$	31139.1	1.38803
$56$	0	0
$57$	−30752.5	−1.25370
$58$	1737.82	0.0678321
$59$	14230.4	0.532216	0.266108	−	0.963943i	$-0.414262\pi$
$59$	14230.4	0.532216	0.266108	+	0.963943i	$0.414262\pi$
$60$	28912.2	1.03682
$61$	14234.7	0.489807	0.244904	−	0.969547i	$-0.421244\pi$
$61$	14234.7	0.489807	0.244904	+	0.969547i	$0.421244\pi$
$62$	1485.30	0.0490721
$63$	0	0
$64$	−29581.9	−0.902768
$65$	30416.9	0.892958
$66$	−9490.12	−0.268171
$67$	19730.4	0.536970	0.268485	−	0.963284i	$-0.413477\pi$
$67$	19730.4	0.536970	0.268485	+	0.963284i	$0.413477\pi$
$68$	37356.6	0.979704
$69$	21617.5	0.546615
$70$	0	0
$71$	64562.7	1.51997	0.759986	−	0.649940i	$-0.225206\pi$
$71$	64562.7	1.51997	0.759986	+	0.649940i	$0.225206\pi$
$72$	−6586.18	−0.149728
$73$	−28567.0	−0.627418	−0.313709	−	0.949519i	$-0.601572\pi$
$73$	−28567.0	−0.627418	−0.313709	+	0.949519i	$0.601572\pi$
$74$	781.309	0.0165861
$75$	18461.0	0.378968
$76$	−49258.8	−0.978251
$77$	0	0
$78$	−9270.00	−0.172521
$79$	−30633.4	−0.552239	−0.276119	−	0.961123i	$-0.589049\pi$
$79$	−30633.4	−0.552239	−0.276119	+	0.961123i	$0.589049\pi$
$80$	45524.6	0.795282
$81$	−73344.6	−1.24210
$82$	−796.293	−0.0130779
$83$	675.946	0.0107700	0.00538501	−	0.999986i	$-0.498286\pi$
$83$	675.946	0.0107700	0.00538501	+	0.999986i	$0.498286\pi$
$84$	0	0
$85$	−55486.3	−0.832987
$86$	12010.5	0.175111
$87$	−47094.4	−0.667069
$88$	−30656.2	−0.421999
$89$	−125971.	−1.68576	−0.842882	−	0.538098i	$-0.819143\pi$
$89$	−125971.	−1.68576	−0.842882	+	0.538098i	$0.819143\pi$
$90$	4850.76	0.0631254
$91$	0	0
$92$	34626.5	0.426520
$93$	−40251.1	−0.482582
$94$	6017.02	0.0702363
$95$	73164.9	0.831751
$96$	−42813.5	−0.474136
$97$	22906.8	0.247192	0.123596	−	0.992333i	$-0.460557\pi$
$97$	22906.8	0.247192	0.123596	+	0.992333i	$0.460557\pi$
$98$	0	0
$99$	95319.4	0.977447
$100$	29570.5	0.295705
$101$	181474.	1.77015	0.885077	−	0.465444i	$-0.154105\pi$
$101$	181474.	1.77015	0.885077	+	0.465444i	$0.154105\pi$
$102$	16910.3	0.160935
$103$	−64772.0	−0.601581	−0.300791	−	0.953690i	$-0.597251\pi$
$103$	−64772.0	−0.601581	−0.300791	+	0.953690i	$0.597251\pi$
$104$	−29945.1	−0.271483
$105$	0	0
$106$	4001.86	0.0345938
$107$	−148170.	−1.25112	−0.625562	−	0.780175i	$-0.715130\pi$
$107$	−148170.	−1.25112	−0.625562	+	0.780175i	$0.715130\pi$
$108$	−61781.7	−0.509683
$109$	−111294.	−0.897237	−0.448618	−	0.893723i	$-0.648084\pi$
$109$	−111294.	−0.897237	−0.448618	+	0.893723i	$0.648084\pi$
$110$	22578.5	0.177915
$111$	−21173.2	−0.163110
$112$	0	0
$113$	−43175.5	−0.318084	−0.159042	−	0.987272i	$-0.550840\pi$
$113$	−43175.5	−0.318084	−0.159042	+	0.987272i	$0.550840\pi$
$114$	−22298.1	−0.160696
$115$	−51431.2	−0.362646
$116$	−75435.0	−0.520509
$117$	93108.5	0.628817
$118$	10318.2	0.0682182
$119$	0	0
$120$	42277.6	0.268014
$121$	282625.	1.75488
$122$	10321.4	0.0627824
$123$	21579.3	0.128610
$124$	−64473.6	−0.376554
$125$	−190013.	−1.08770
$126$	0	0
$127$	−131449.	−0.723182	−0.361591	−	0.932337i	$-0.617766\pi$
$127$	−131449.	−0.723182	−0.361591	+	0.932337i	$0.617766\pi$
$128$	−91172.8	−0.491859
$129$	−325480.	−1.72207
$130$	22054.7	0.114457
$131$	349458.	1.77916	0.889582	−	0.456775i	$-0.150995\pi$
$131$	349458.	1.77916	0.889582	+	0.456775i	$0.150995\pi$
$132$	411946.	2.05781
$133$	0	0
$134$	14306.2	0.0688276
$135$	91765.2	0.433355
$136$	54625.7	0.253250
$137$	386434.	1.75903	0.879516	−	0.475869i	$-0.157866\pi$
$137$	386434.	1.75903	0.879516	+	0.475869i	$0.157866\pi$
$138$	15674.4	0.0700639
$139$	−17289.3	−0.0758997	−0.0379498	−	0.999280i	$-0.512083\pi$
$139$	−17289.3	−0.0758997	−0.0379498	+	0.999280i	$0.512083\pi$
$140$	0	0
$141$	−163059.	−0.690713
$142$	46813.3	0.194827
$143$	433384.	1.77228
$144$	139354.	0.560034
$145$	112045.	0.442559
$146$	−20713.4	−0.0804210
$147$	0	0
$148$	−33914.9	−0.127273
$149$	−112171.	−0.413917	−0.206959	−	0.978350i	$-0.566357\pi$
$149$	−112171.	−0.413917	−0.206959	+	0.978350i	$0.566357\pi$
$150$	13385.8	0.0485752
$151$	30495.4	0.108841	0.0544205	−	0.998518i	$-0.482669\pi$
$151$	30495.4	0.108841	0.0544205	+	0.998518i	$0.482669\pi$
$152$	−72030.1	−0.252874
$153$	−169848.	−0.586586
$154$	0	0
$155$	95763.6	0.320163
$156$	402390.	1.32384
$157$	−523509.	−1.69502	−0.847510	−	0.530780i	$-0.821899\pi$
$157$	−523509.	−1.69502	−0.847510	+	0.530780i	$0.821899\pi$
$158$	−22211.7	−0.0707847
$159$	−108449.	−0.340199
$160$	101860.	0.314560
$161$	0	0
$162$	−53180.9	−0.159209
$163$	−439646.	−1.29609	−0.648043	−	0.761604i	$-0.724412\pi$
$163$	−439646.	−1.29609	−0.648043	+	0.761604i	$0.724412\pi$
$164$	34565.3	0.100353
$165$	−611869.	−1.74964
$166$	490.117	0.00138048
$167$	−279353.	−0.775107	−0.387554	−	0.921847i	$-0.626680\pi$
$167$	−279353.	−0.775107	−0.387554	+	0.921847i	$0.626680\pi$
$168$	0	0
$169$	52038.8	0.140156
$170$	−40232.2	−0.106770
$171$	223964.	0.585716
$172$	−521349.	−1.34372
$173$	−99699.4	−0.253266	−0.126633	−	0.991950i	$-0.540417\pi$
$173$	−99699.4	−0.253266	−0.126633	+	0.991950i	$0.540417\pi$
$174$	−34147.3	−0.0855034
$175$	0	0
$176$	648641.	1.57842
$177$	−279621.	−0.670867
$178$	−91339.7	−0.216077
$179$	329980.	0.769760	0.384880	−	0.922967i	$-0.374243\pi$
$179$	329980.	0.769760	0.384880	+	0.922967i	$0.374243\pi$
$180$	−210561.	−0.484392
$181$	505810.	1.14760	0.573800	−	0.818995i	$-0.305469\pi$
$181$	505810.	1.14760	0.573800	+	0.818995i	$0.305469\pi$
$182$	0	0
$183$	−279706.	−0.617410
$184$	50633.5	0.110254
$185$	50374.3	0.108213
$186$	−29185.4	−0.0618562
$187$	−790578.	−1.65326
$188$	−261186.	−0.538958
$189$	0	0
$190$	53050.6	0.106612
$191$	−63835.6	−0.126613	−0.0633067	−	0.997994i	$-0.520165\pi$
$191$	−63835.6	−0.126613	−0.0633067	+	0.997994i	$0.520165\pi$
$192$	581270.	1.13795
$193$	469355.	0.907001	0.453501	−	0.891256i	$-0.350175\pi$
$193$	469355.	0.907001	0.453501	+	0.891256i	$0.350175\pi$
$194$	16609.3	0.0316845
$195$	−597676.	−1.12559
$196$	0	0
$197$	268021.	0.492043	0.246021	−	0.969264i	$-0.420877\pi$
$197$	268021.	0.492043	0.246021	+	0.969264i	$0.420877\pi$
$198$	69114.5	0.125287
$199$	605167.	1.08328	0.541642	−	0.840609i	$-0.317803\pi$
$199$	605167.	1.08328	0.541642	+	0.840609i	$0.317803\pi$
$200$	43240.3	0.0764388
$201$	−387693.	−0.676859
$202$	131584.	0.226894
$203$	0	0
$204$	−734039.	−1.23493
$205$	−51340.4	−0.0853246
$206$	−46965.1	−0.0771094
$207$	−157435.	−0.255374
$208$	633596.	1.01544
$209$	1.04246e6	1.65080
$210$	0	0
$211$	335389.	0.518612	0.259306	−	0.965795i	$-0.416506\pi$
$211$	335389.	0.518612	0.259306	+	0.965795i	$0.416506\pi$
$212$	−173712.	−0.265455
$213$	−1.26862e6	−1.91595
$214$	−107435.	−0.160366
$215$	774367.	1.14249
$216$	−90341.9	−0.131751
$217$	0	0
$218$	−80697.7	−0.115006
$219$	561327.	0.790871
$220$	−980081.	−1.36523
$221$	−772240.	−1.06358
$222$	−15352.3	−0.0209070
$223$	−1.02526e6	−1.38061	−0.690305	−	0.723518i	$-0.742524\pi$
$223$	−1.02526e6	−1.38061	−0.690305	+	0.723518i	$0.742524\pi$
$224$	0	0
$225$	−134447.	−0.177050
$226$	−31305.8	−0.0407713
$227$	504226.	0.649473	0.324736	−	0.945805i	$-0.394724\pi$
$227$	504226.	0.649473	0.324736	+	0.945805i	$0.394724\pi$
$228$	967912.	1.23310
$229$	1.11939e6	1.41057	0.705283	−	0.708925i	$-0.250820\pi$
$229$	1.11939e6	1.41057	0.705283	+	0.708925i	$0.250820\pi$
$230$	−37291.9	−0.0464831
$231$	0	0
$232$	−110307.	−0.134550
$233$	770703.	0.930031	0.465015	−	0.885303i	$-0.346049\pi$
$233$	770703.	0.930031	0.465015	+	0.885303i	$0.346049\pi$
$234$	67511.3	0.0806004
$235$	387943.	0.458245
$236$	−447892.	−0.523472
$237$	601930.	0.696106
$238$	0	0
$239$	−171646.	−0.194374	−0.0971871	−	0.995266i	$-0.530985\pi$
$239$	−171646.	−0.194374	−0.0971871	+	0.995266i	$0.530985\pi$
$240$	−894535.	−1.00247
$241$	383779.	0.425637	0.212818	−	0.977092i	$-0.431736\pi$
$241$	383779.	0.425637	0.212818	+	0.977092i	$0.431736\pi$
$242$	204926.	0.224936
$243$	964193.	1.04749
$244$	−448028.	−0.481760
$245$	0	0
$246$	15646.8	0.0164849
$247$	1.01828e6	1.06201
$248$	−94278.3	−0.0973380
$249$	−13282.0	−0.0135758
$250$	−137775.	−0.139418
$251$	1.57046e6	1.57342	0.786708	−	0.617325i	$-0.211784\pi$
$251$	1.57046e6	1.57342	0.786708	+	0.617325i	$0.211784\pi$
$252$	0	0
$253$	−732801.	−0.719755
$254$	−95311.4	−0.0926959
$255$	1.09028e6	1.04999
$256$	880513.	0.839723
$257$	−790656.	−0.746715	−0.373357	−	0.927688i	$-0.621793\pi$
$257$	−790656.	−0.746715	−0.373357	+	0.927688i	$0.621793\pi$
$258$	−236000.	−0.220731
$259$	0	0
$260$	−957348.	−0.878287
$261$	342978.	0.311648
$262$	253386.	0.228049
$263$	464416.	0.414017	0.207008	−	0.978339i	$-0.433627\pi$
$263$	464416.	0.414017	0.207008	+	0.978339i	$0.433627\pi$
$264$	602379.	0.531936
$265$	258017.	0.225701
$266$	0	0
$267$	2.47527e6	2.12493
$268$	−621001.	−0.528147
$269$	−1.99959e6	−1.68484	−0.842422	−	0.538818i	$-0.818871\pi$
$269$	−1.99959e6	−1.68484	−0.842422	+	0.538818i	$0.818871\pi$
$270$	66537.4	0.0555464
$271$	−1.61296e6	−1.33414	−0.667070	−	0.744995i	$-0.732452\pi$
$271$	−1.61296e6	−1.33414	−0.667070	+	0.744995i	$0.732452\pi$
$272$	−1.15580e6	−0.947243
$273$	0	0
$274$	280197.	0.225469
$275$	−625801.	−0.499005
$276$	−680393.	−0.537635
$277$	−2.08119e6	−1.62972	−0.814860	−	0.579658i	$-0.803186\pi$
$277$	−2.08119e6	−1.62972	−0.814860	+	0.579658i	$0.803186\pi$
$278$	−12536.2	−0.00972865
$279$	293140.	0.225457
$280$	0	0
$281$	−982035.	−0.741927	−0.370964	−	0.928647i	$-0.620973\pi$
$281$	−982035.	−0.741927	−0.370964	+	0.928647i	$0.620973\pi$
$282$	−118231.	−0.0885340
$283$	1.39622e6	1.03630	0.518152	−	0.855289i	$-0.326620\pi$
$283$	1.39622e6	1.03630	0.518152	+	0.855289i	$0.326620\pi$
$284$	−2.03206e6	−1.49500
$285$	−1.43765e6	−1.04844
$286$	314240.	0.227167
$287$	0	0
$288$	311801.	0.221512
$289$	−11140.3	−0.00784609
$290$	81241.7	0.0567262
$291$	−450107.	−0.311590
$292$	899124.	0.617110
$293$	−2.56205e6	−1.74348	−0.871742	−	0.489965i	$-0.837010\pi$
$293$	−2.56205e6	−1.74348	−0.871742	+	0.489965i	$0.837010\pi$
$294$	0	0
$295$	665260.	0.445078
$296$	−49593.0	−0.0328996
$297$	1.30749e6	0.860094
$298$	−81333.0	−0.0530550
$299$	−715803.	−0.463037
$300$	−581046.	−0.372741
$301$	0	0
$302$	22111.7	0.0139510
$303$	−3.56588e6	−2.23131
$304$	1.52405e6	0.945838
$305$	665463.	0.409613
$306$	−123154.	−0.0751873
$307$	884855.	0.535829	0.267915	−	0.963443i	$-0.413666\pi$
$307$	884855.	0.535829	0.267915	+	0.963443i	$0.413666\pi$
$308$	0	0
$309$	1.27274e6	0.758303
$310$	69436.5	0.0410378
$311$	−1.80120e6	−1.05599	−0.527997	−	0.849246i	$-0.677057\pi$
$311$	−1.80120e6	−1.05599	−0.527997	+	0.849246i	$0.677057\pi$
$312$	588406.	0.342208
$313$	−950366.	−0.548315	−0.274158	−	0.961685i	$-0.588399\pi$
$313$	−950366.	−0.548315	−0.274158	+	0.961685i	$0.588399\pi$
$314$	−379587.	−0.217264
$315$	0	0
$316$	964162.	0.543166
$317$	3.04277e6	1.70068	0.850338	−	0.526237i	$-0.176398\pi$
$317$	3.04277e6	1.70068	0.850338	+	0.526237i	$0.176398\pi$
$318$	−78634.6	−0.0436060
$319$	1.59643e6	0.878362
$320$	−1.38293e6	−0.754962
$321$	2.91146e6	1.57706
$322$	0	0
$323$	−1.85755e6	−0.990682
$324$	2.30847e6	1.22169
$325$	−611286.	−0.321023
$326$	−318780.	−0.166129
$327$	2.18688e6	1.13098
$328$	50544.1	0.0259409
$329$	0	0
$330$	−443655.	−0.224265
$331$	−2.19616e6	−1.10178	−0.550889	−	0.834579i	$-0.685711\pi$
$331$	−2.19616e6	−1.10178	−0.550889	+	0.834579i	$0.685711\pi$
$332$	−21274.9	−0.0105931
$333$	154200.	0.0762032
$334$	−202554.	−0.0993515
$335$	922382.	0.449054
$336$	0	0
$337$	−2.41491e6	−1.15832	−0.579158	−	0.815216i	$-0.696618\pi$
$337$	−2.41491e6	−1.15832	−0.579158	+	0.815216i	$0.696618\pi$
$338$	37732.5	0.0179648
$339$	848377.	0.400950
$340$	1.74639e6	0.819302
$341$	1.36446e6	0.635438
$342$	162392.	0.0750757
$343$	0	0
$344$	−762357.	−0.347346
$345$	1.01060e6	0.457121
$346$	−72290.3	−0.0324631
$347$	−1.08833e6	−0.485219	−0.242609	−	0.970124i	$-0.578003\pi$
$347$	−1.08833e6	−0.485219	−0.242609	+	0.970124i	$0.578003\pi$
$348$	1.48226e6	0.656110
$349$	−2.79267e6	−1.22731	−0.613657	−	0.789573i	$-0.710302\pi$
$349$	−2.79267e6	−1.22731	−0.613657	+	0.789573i	$0.710302\pi$
$350$	0	0
$351$	1.27716e6	0.553321
$352$	1.45132e6	0.624317
$353$	−2.53134e6	−1.08122	−0.540610	−	0.841273i	$-0.681807\pi$
$353$	−2.53134e6	−1.08122	−0.540610	+	0.841273i	$0.681807\pi$
$354$	−202748.	−0.0859902
$355$	3.01825e6	1.27111
$356$	3.96485e6	1.65807
$357$	0	0
$358$	239263.	0.0986661
$359$	1.09028e6	0.446480	0.223240	−	0.974763i	$-0.428337\pi$
$359$	1.09028e6	0.446480	0.223240	+	0.974763i	$0.428337\pi$
$360$	−307899.	−0.125214
$361$	−26712.8	−0.0107883
$362$	366754.	0.147097
$363$	−5.55343e6	−2.21205
$364$	0	0
$365$	−1.33548e6	−0.524694
$366$	−202810.	−0.0791382
$367$	−188070.	−0.0728879	−0.0364439	−	0.999336i	$-0.511603\pi$
$367$	−188070.	−0.0728879	−0.0364439	+	0.999336i	$0.511603\pi$
$368$	−1.07133e6	−0.412387
$369$	−157157.	−0.0600853
$370$	36525.6	0.0138705
$371$	0	0
$372$	1.26687e6	0.474653
$373$	−1.79371e6	−0.667545	−0.333772	−	0.942654i	$-0.608322\pi$
$373$	−1.79371e6	−0.667545	−0.333772	+	0.942654i	$0.608322\pi$
$374$	−573234.	−0.211911
$375$	3.73366e6	1.37106
$376$	−381926.	−0.139319
$377$	1.55940e6	0.565073
$378$	0	0
$379$	3.58806e6	1.28310	0.641551	−	0.767080i	$-0.278291\pi$
$379$	3.58806e6	1.28310	0.641551	+	0.767080i	$0.278291\pi$
$380$	−2.30281e6	−0.818086
$381$	2.58291e6	0.911583
$382$	−46286.1	−0.0162290
$383$	3.42457e6	1.19291	0.596457	−	0.802645i	$-0.296575\pi$
$383$	3.42457e6	1.19291	0.596457	+	0.802645i	$0.296575\pi$
$384$	1.79150e6	0.619996
$385$	0	0
$386$	340321.	0.116257
$387$	2.37040e6	0.804534
$388$	−720974.	−0.243131
$389$	8625.27	0.00289001	0.00144500	−	0.999999i	$-0.499540\pi$
$389$	8625.27	0.00289001	0.00144500	+	0.999999i	$0.499540\pi$
$390$	−433365.	−0.144275
$391$	1.30576e6	0.431940
$392$	0	0
$393$	−6.86667e6	−2.24267
$394$	194337.	0.0630690
$395$	−1.43208e6	−0.461823
$396$	−3.00011e6	−0.961388
$397$	−1.25709e6	−0.400306	−0.200153	−	0.979765i	$-0.564144\pi$
$397$	−1.25709e6	−0.400306	−0.200153	+	0.979765i	$0.564144\pi$
$398$	438796.	0.138853
$399$	0	0
$400$	−914904.	−0.285908
$401$	1.42670e6	0.443070	0.221535	−	0.975152i	$-0.428893\pi$
$401$	1.42670e6	0.443070	0.221535	+	0.975152i	$0.428893\pi$
$402$	−281110.	−0.0867583
$403$	1.33281e6	0.408794
$404$	−5.71176e6	−1.74107
$405$	−3.42880e6	−1.03873
$406$	0	0
$407$	717741.	0.214774
$408$	−1.07337e6	−0.319226
$409$	3.06529e6	0.906073	0.453036	−	0.891492i	$-0.350341\pi$
$409$	3.06529e6	0.906073	0.453036	+	0.891492i	$0.350341\pi$
$410$	−37226.0	−0.0109367
$411$	−7.59324e6	−2.21729
$412$	2.03865e6	0.591698
$413$	0	0
$414$	−114154.	−0.0327332
$415$	31599.9	0.00900670
$416$	1.41765e6	0.401640
$417$	339726.	0.0956728
$418$	755873.	0.211596
$419$	248240.	0.0690776	0.0345388	−	0.999403i	$-0.489004\pi$
$419$	248240.	0.0690776	0.0345388	+	0.999403i	$0.489004\pi$
$420$	0	0
$421$	5.96280e6	1.63963	0.819814	−	0.572630i	$-0.194077\pi$
$421$	5.96280e6	1.63963	0.819814	+	0.572630i	$0.194077\pi$
$422$	243185.	0.0664746
$423$	1.18752e6	0.322695
$424$	−254015.	−0.0686192
$425$	1.11510e6	0.299463
$426$	−919857.	−0.245582
$427$	0	0
$428$	4.66353e6	1.23057
$429$	−8.51579e6	−2.23399
$430$	561481.	0.146441
$431$	−4.93538e6	−1.27976	−0.639879	−	0.768476i	$-0.721015\pi$
$431$	−4.93538e6	−1.27976	−0.639879	+	0.768476i	$0.721015\pi$
$432$	1.91151e6	0.492795
$433$	−4.15513e6	−1.06504	−0.532519	−	0.846418i	$-0.678755\pi$
$433$	−4.15513e6	−1.06504	−0.532519	+	0.846418i	$0.678755\pi$
$434$	0	0
$435$	−2.20162e6	−0.557853
$436$	3.50291e6	0.882496
$437$	−1.72180e6	−0.431299
$438$	407008.	0.101372
$439$	227955.	0.0564531	0.0282265	−	0.999602i	$-0.491014\pi$
$439$	227955.	0.0564531	0.0282265	+	0.999602i	$0.491014\pi$
$440$	−1.43315e6	−0.352907
$441$	0	0
$442$	−559938.	−0.136328
$443$	−1.98462e6	−0.480472	−0.240236	−	0.970715i	$-0.577225\pi$
$443$	−1.98462e6	−0.480472	−0.240236	+	0.970715i	$0.577225\pi$
$444$	666411.	0.160430
$445$	−5.88906e6	−1.40976
$446$	−743397.	−0.176963
$447$	2.20410e6	0.521749
$448$	0	0
$449$	2.61077e6	0.611157	0.305579	−	0.952167i	$-0.401150\pi$
$449$	2.61077e6	0.611157	0.305579	+	0.952167i	$0.401150\pi$
$450$	−97485.5	−0.0226939
$451$	−731506.	−0.169347
$452$	1.35892e6	0.312858
$453$	−599220.	−0.137196
$454$	365606.	0.0832480
$455$	0	0
$456$	1.41536e6	0.318752
$457$	−4.09917e6	−0.918132	−0.459066	−	0.888402i	$-0.651816\pi$
$457$	−4.09917e6	−0.918132	−0.459066	+	0.888402i	$0.651816\pi$
$458$	811652.	0.180803
$459$	−2.32979e6	−0.516160
$460$	1.61876e6	0.356687
$461$	2.62378e6	0.575009	0.287505	−	0.957779i	$-0.407174\pi$
$461$	2.62378e6	0.575009	0.287505	+	0.957779i	$0.407174\pi$
$462$	0	0
$463$	−4.28563e6	−0.929100	−0.464550	−	0.885547i	$-0.653784\pi$
$463$	−4.28563e6	−0.929100	−0.464550	+	0.885547i	$0.653784\pi$
$464$	2.33394e6	0.503262
$465$	−1.88171e6	−0.403571
$466$	558824.	0.119209
$467$	−990118.	−0.210085	−0.105042	−	0.994468i	$-0.533498\pi$
$467$	−990118.	−0.210085	−0.105042	+	0.994468i	$0.533498\pi$
$468$	−2.93052e6	−0.618486
$469$	0	0
$470$	281291.	0.0587369
$471$	1.02867e7	2.13660
$472$	−654942.	−0.135316
$473$	1.10333e7	2.26753
$474$	436449.	0.0892253
$475$	−1.47039e6	−0.299019
$476$	0	0
$477$	789812.	0.158938
$478$	−124457.	−0.0249144
$479$	−2.57976e6	−0.513737	−0.256868	−	0.966446i	$-0.582691\pi$
$479$	−2.57976e6	−0.513737	−0.256868	+	0.966446i	$0.582691\pi$
$480$	−2.00150e6	−0.396508
$481$	701093.	0.138170
$482$	278272.	0.0545571
$483$	0	0
$484$	−8.89540e6	−1.72604
$485$	1.07087e6	0.206720
$486$	699120.	0.134264
$487$	3.21474e6	0.614219	0.307109	−	0.951674i	$-0.400638\pi$
$487$	3.21474e6	0.614219	0.307109	+	0.951674i	$0.400638\pi$
$488$	−655141.	−0.124533
$489$	8.63882e6	1.63374
$490$	0	0
$491$	−7.86108e6	−1.47156	−0.735781	−	0.677220i	$-0.763185\pi$
$491$	−7.86108e6	−1.47156	−0.735781	+	0.677220i	$0.763185\pi$
$492$	−679192.	−0.126497
$493$	−2.84465e6	−0.527123
$494$	738341.	0.136125
$495$	4.45610e6	0.817415
$496$	1.99480e6	0.364078
$497$	0	0
$498$	−9630.55	−0.00174011
$499$	−1.35382e6	−0.243395	−0.121697	−	0.992567i	$-0.538834\pi$
$499$	−1.35382e6	−0.243395	−0.121697	+	0.992567i	$0.538834\pi$
$500$	5.98051e6	1.06983
$501$	5.48914e6	0.977035
$502$	1.13872e6	0.201677
$503$	−3.85775e6	−0.679851	−0.339926	−	0.940452i	$-0.610402\pi$
$503$	−3.85775e6	−0.679851	−0.339926	+	0.940452i	$0.610402\pi$
$504$	0	0
$505$	8.48376e6	1.48034
$506$	−531341.	−0.0922565
$507$	−1.02254e6	−0.176669
$508$	4.13726e6	0.711301
$509$	1.06060e7	1.81451	0.907253	−	0.420585i	$-0.138175\pi$
$509$	1.06060e7	1.81451	0.907253	+	0.420585i	$0.138175\pi$
$510$	790542.	0.134586
$511$	0	0
$512$	3.55598e6	0.599493
$513$	3.07208e6	0.515394
$514$	−573291.	−0.0957122
$515$	−3.02804e6	−0.503087
$516$	1.02443e7	1.69378
$517$	5.52747e6	0.909495
$518$	0	0
$519$	1.95904e6	0.319246
$520$	−1.39991e6	−0.227034
$521$	9.17989e6	1.48164	0.740821	−	0.671703i	$-0.234437\pi$
$521$	9.17989e6	1.48164	0.740821	+	0.671703i	$0.234437\pi$
$522$	248687.	0.0399464
$523$	−9.05585e6	−1.44769	−0.723844	−	0.689964i	$-0.757627\pi$
$523$	−9.05585e6	−1.44769	−0.723844	+	0.689964i	$0.757627\pi$
$524$	−1.09989e7	−1.74993
$525$	0	0
$526$	336740.	0.0530677
$527$	−2.43130e6	−0.381339
$528$	−1.27455e7	−1.98963
$529$	−5.22601e6	−0.811953
$530$	187084.	0.0289299
$531$	2.03642e6	0.313422
$532$	0	0
$533$	−714539.	−0.108945
$534$	1.79478e6	0.272369
$535$	−6.92681e6	−1.04628
$536$	−908075.	−0.136524
$537$	−6.48394e6	−0.970295
$538$	−1.44987e6	−0.215960
$539$	0	0
$540$	−2.88824e6	−0.426235
$541$	1.21783e7	1.78894	0.894468	−	0.447132i	$-0.147555\pi$
$541$	1.21783e7	1.78894	0.894468	+	0.447132i	$0.147555\pi$
$542$	−1.16953e6	−0.171007
$543$	−9.93891e6	−1.44657
$544$	−2.58608e6	−0.374666
$545$	−5.20292e6	−0.750336
$546$	0	0
$547$	−9.00451e6	−1.28674	−0.643372	−	0.765554i	$-0.722465\pi$
$547$	−9.00451e6	−1.28674	−0.643372	+	0.765554i	$0.722465\pi$
$548$	−1.21627e7	−1.73013
$549$	2.03703e6	0.288448
$550$	−453758.	−0.0639613
$551$	3.75099e6	0.526341
$552$	−994924.	−0.138977
$553$	0	0
$554$	−1.50904e6	−0.208894
$555$	−989830.	−0.136404
$556$	544167.	0.0746527
$557$	1.54461e6	0.210950	0.105475	−	0.994422i	$-0.466364\pi$
$557$	1.54461e6	0.210950	0.105475	+	0.994422i	$0.466364\pi$
$558$	212551.	0.0288986
$559$	1.07774e7	1.45876
$560$	0	0
$561$	1.55345e7	2.08396
$562$	−712057.	−0.0950986
$563$	1.18748e7	1.57890	0.789449	−	0.613816i	$-0.210366\pi$
$563$	1.18748e7	1.57890	0.789449	+	0.613816i	$0.210366\pi$
$564$	5.13217e6	0.679365
$565$	−2.01842e6	−0.266005
$566$	1.01237e6	0.132831
$567$	0	0
$568$	−2.97144e6	−0.386452
$569$	−1.54308e6	−0.199806	−0.0999031	−	0.994997i	$-0.531853\pi$
$569$	−1.54308e6	−0.199806	−0.0999031	+	0.994997i	$0.531853\pi$
$570$	−1.04242e6	−0.134386
$571$	−7.01812e6	−0.900804	−0.450402	−	0.892826i	$-0.648719\pi$
$571$	−7.01812e6	−0.900804	−0.450402	+	0.892826i	$0.648719\pi$
$572$	−1.36404e7	−1.74317
$573$	1.25434e6	0.159598
$574$	0	0
$575$	1.03361e6	0.130373
$576$	−4.23326e6	−0.531641
$577$	5.37543e6	0.672162	0.336081	−	0.941833i	$-0.390898\pi$
$577$	5.37543e6	0.672162	0.336081	+	0.941833i	$0.390898\pi$
$578$	−8077.66	−0.00100569
$579$	−9.22259e6	−1.14329
$580$	−3.52652e6	−0.435288
$581$	0	0
$582$	−326365.	−0.0399389
$583$	3.67627e6	0.447957
$584$	1.31477e6	0.159521
$585$	4.35274e6	0.525864
$586$	−1.85770e6	−0.223476
$587$	2.06682e6	0.247575	0.123788	−	0.992309i	$-0.460496\pi$
$587$	2.06682e6	0.247575	0.123788	+	0.992309i	$0.460496\pi$
$588$	0	0
$589$	3.20594e6	0.380774
$590$	482369.	0.0570492
$591$	−5.26648e6	−0.620228
$592$	1.04932e6	0.123056
$593$	5.46947e6	0.638717	0.319358	−	0.947634i	$-0.396533\pi$
$593$	5.46947e6	0.638717	0.319358	+	0.947634i	$0.396533\pi$
$594$	948035.	0.110245
$595$	0	0
$596$	3.53049e6	0.407117
$597$	−1.18912e7	−1.36550
$598$	−519016.	−0.0593510
$599$	1.09943e7	1.25199	0.625996	−	0.779826i	$-0.284693\pi$
$599$	1.09943e7	1.25199	0.625996	+	0.779826i	$0.284693\pi$
$600$	−849651.	−0.0963524
$601$	−1.58788e7	−1.79322	−0.896608	−	0.442826i	$-0.853976\pi$
$601$	−1.58788e7	−1.79322	−0.896608	+	0.442826i	$0.853976\pi$
$602$	0	0
$603$	2.82348e6	0.316222
$604$	−959820.	−0.107053
$605$	1.32125e7	1.46756
$606$	−2.58555e6	−0.286004
$607$	−5.33262e6	−0.587447	−0.293724	−	0.955890i	$-0.594895\pi$
$607$	−5.33262e6	−0.587447	−0.293724	+	0.955890i	$0.594895\pi$
$608$	3.41003e6	0.374110
$609$	0	0
$610$	482516.	0.0525033
$611$	5.39926e6	0.585102
$612$	5.34584e6	0.576949
$613$	−8.91838e6	−0.958594	−0.479297	−	0.877653i	$-0.659108\pi$
$613$	−8.91838e6	−0.958594	−0.479297	+	0.877653i	$0.659108\pi$
$614$	641593.	0.0686814
$615$	1.00881e6	0.107553
$616$	0	0
$617$	9.63586e6	1.01901	0.509504	−	0.860468i	$-0.329829\pi$
$617$	9.63586e6	1.01901	0.509504	+	0.860468i	$0.329829\pi$
$618$	922841.	0.0971976
$619$	−1.21747e7	−1.27712	−0.638560	−	0.769572i	$-0.720470\pi$
$619$	−1.21747e7	−1.27712	−0.638560	+	0.769572i	$0.720470\pi$
$620$	−3.01409e6	−0.314903
$621$	−2.15952e6	−0.224713
$622$	−1.30602e6	−0.135355
$623$	0	0
$624$	−1.24498e7	−1.27998
$625$	−5.94695e6	−0.608968
$626$	−689094.	−0.0702818
$627$	−2.04839e7	−2.08087
$628$	1.64770e7	1.66717
$629$	−1.27893e6	−0.128890
$630$	0	0
$631$	−1.30854e7	−1.30832	−0.654161	−	0.756356i	$-0.726978\pi$
$631$	−1.30854e7	−1.30832	−0.654161	+	0.756356i	$0.726978\pi$
$632$	1.40987e6	0.140407
$633$	−6.59023e6	−0.653719
$634$	2.20626e6	0.217989
$635$	−6.14513e6	−0.604779
$636$	3.41336e6	0.334610
$637$	0	0
$638$	1.15754e6	0.112586
$639$	9.23911e6	0.895113
$640$	−4.26226e6	−0.411329
$641$	−441107.	−0.0424032	−0.0212016	−	0.999775i	$-0.506749\pi$
$641$	−441107.	−0.0424032	−0.0212016	+	0.999775i	$0.506749\pi$
$642$	2.11105e6	0.202144
$643$	4.18888e6	0.399550	0.199775	−	0.979842i	$-0.435979\pi$
$643$	4.18888e6	0.399550	0.199775	+	0.979842i	$0.435979\pi$
$644$	0	0
$645$	−1.52159e7	−1.44012
$646$	−1.34688e6	−0.126983
$647$	−1.87822e7	−1.76395	−0.881973	−	0.471300i	$-0.843785\pi$
$647$	−1.87822e7	−1.76395	−0.881973	+	0.471300i	$0.843785\pi$
$648$	3.37562e6	0.315803
$649$	9.47874e6	0.883362
$650$	−443233.	−0.0411480
$651$	0	0
$652$	1.38375e7	1.27479
$653$	−1.51733e6	−0.139251	−0.0696254	−	0.997573i	$-0.522180\pi$
$653$	−1.51733e6	−0.139251	−0.0696254	+	0.997573i	$0.522180\pi$
$654$	1.58567e6	0.144967
$655$	1.63368e7	1.48787
$656$	−1.06944e6	−0.0970281
$657$	−4.08802e6	−0.369487
$658$	0	0
$659$	1.84809e7	1.65772	0.828859	−	0.559458i	$-0.188991\pi$
$659$	1.84809e7	1.65772	0.828859	+	0.559458i	$0.188991\pi$
$660$	1.92581e7	1.72089
$661$	1.03952e7	0.925403	0.462702	−	0.886514i	$-0.346880\pi$
$661$	1.03952e7	0.925403	0.462702	+	0.886514i	$0.346880\pi$
$662$	−1.59240e6	−0.141223
$663$	1.51741e7	1.34066
$664$	−31109.8	−0.00273828
$665$	0	0
$666$	111808.	0.00976756
$667$	−2.63676e6	−0.229486
$668$	8.79242e6	0.762372
$669$	2.01458e7	1.74028
$670$	668803.	0.0575587
$671$	9.48162e6	0.812973
$672$	0	0
$673$	−1.10398e7	−0.939556	−0.469778	−	0.882785i	$-0.655666\pi$
$673$	−1.10398e7	−0.939556	−0.469778	+	0.882785i	$0.655666\pi$
$674$	−1.75101e6	−0.148470
$675$	−1.84420e6	−0.155793
$676$	−1.63788e6	−0.137853
$677$	8.23485e6	0.690532	0.345266	−	0.938505i	$-0.387789\pi$
$677$	8.23485e6	0.690532	0.345266	+	0.938505i	$0.387789\pi$
$678$	615144.	0.0513928
$679$	0	0
$680$	2.55371e6	0.211787
$681$	−9.90780e6	−0.818671
$682$	989343.	0.0814490
$683$	1.98437e7	1.62768	0.813842	−	0.581086i	$-0.197372\pi$
$683$	1.98437e7	1.62768	0.813842	+	0.581086i	$0.197372\pi$
$684$	−7.04909e6	−0.576093
$685$	1.80655e7	1.47103
$686$	0	0
$687$	−2.19955e7	−1.77804
$688$	1.61304e7	1.29919
$689$	3.59100e6	0.288182
$690$	732767.	0.0585927
$691$	2.37019e6	0.188837	0.0944185	−	0.995533i	$-0.469901\pi$
$691$	2.37019e6	0.188837	0.0944185	+	0.995533i	$0.469901\pi$
$692$	3.13796e6	0.249105
$693$	0	0
$694$	−789131.	−0.0621943
$695$	−808259.	−0.0634730
$696$	2.16748e6	0.169602
$697$	1.30346e6	0.101628
$698$	−2.02491e6	−0.157314
$699$	−1.51439e7	−1.17232
$700$	0	0
$701$	−1.56833e7	−1.20543	−0.602714	−	0.797957i	$-0.705914\pi$
$701$	−1.56833e7	−1.20543	−0.602714	+	0.797957i	$0.705914\pi$
$702$	926045.	0.0709234
$703$	1.68641e6	0.128699
$704$	−1.97042e7	−1.49840
$705$	−7.62288e6	−0.577626
$706$	−1.83543e6	−0.138588
$707$	0	0
$708$	8.80085e6	0.659844
$709$	3.68544e6	0.275343	0.137671	−	0.990478i	$-0.456038\pi$
$709$	3.68544e6	0.275343	0.137671	+	0.990478i	$0.456038\pi$
$710$	2.18848e6	0.162928
$711$	−4.38373e6	−0.325214
$712$	5.79772e6	0.428605
$713$	−2.25361e6	−0.166018
$714$	0	0
$715$	2.02604e7	1.48212
$716$	−1.03859e7	−0.757113
$717$	3.37276e6	0.245012
$718$	790544.	0.0572289
$719$	1.56427e7	1.12847	0.564233	−	0.825615i	$-0.309172\pi$
$719$	1.56427e7	1.12847	0.564233	+	0.825615i	$0.309172\pi$
$720$	6.51470e6	0.468342
$721$	0	0
$722$	−19369.0	−0.00138282
$723$	−7.54107e6	−0.536522
$724$	−1.59200e7	−1.12875
$725$	−2.25175e6	−0.159102
$726$	−4.02670e6	−0.283536
$727$	−1.85908e7	−1.30456	−0.652279	−	0.757979i	$-0.726187\pi$
$727$	−1.85908e7	−1.30456	−0.652279	+	0.757979i	$0.726187\pi$
$728$	0	0
$729$	−1.12318e6	−0.0782767
$730$	−968335.	−0.0672541
$731$	−1.96601e7	−1.36079
$732$	8.80353e6	0.607266
$733$	−2.49466e7	−1.71495	−0.857476	−	0.514524i	$-0.827969\pi$
$733$	−2.49466e7	−1.71495	−0.857476	+	0.514524i	$0.827969\pi$
$734$	−136367.	−0.00934260
$735$	0	0
$736$	−2.39708e6	−0.163113
$737$	1.31422e7	0.891253
$738$	−113952.	−0.00770159
$739$	−2.42944e7	−1.63642	−0.818211	−	0.574918i	$-0.805034\pi$
$739$	−2.42944e7	−1.63642	−0.818211	+	0.574918i	$0.805034\pi$
$740$	−1.58549e6	−0.106435
$741$	−2.00088e7	−1.33868
$742$	0	0
$743$	4.16541e6	0.276812	0.138406	−	0.990376i	$-0.455802\pi$
$743$	4.16541e6	0.276812	0.138406	+	0.990376i	$0.455802\pi$
$744$	1.85252e6	0.122696
$745$	−5.24388e6	−0.346148
$746$	−1.30059e6	−0.0855644
$747$	96729.9	0.00634248
$748$	2.48828e7	1.62610
$749$	0	0
$750$	2.70721e6	0.175739
$751$	−2.36434e7	−1.52972	−0.764858	−	0.644199i	$-0.777191\pi$
$751$	−2.36434e7	−1.52972	−0.764858	+	0.644199i	$0.777191\pi$
$752$	8.08101e6	0.521100
$753$	−3.08588e7	−1.98332
$754$	1.13070e6	0.0724298
$755$	1.42564e6	0.0910209
$756$	0	0
$757$	1.50108e7	0.952062	0.476031	−	0.879429i	$-0.342075\pi$
$757$	1.50108e7	0.952062	0.476031	+	0.879429i	$0.342075\pi$
$758$	2.60164e6	0.164465
$759$	1.43992e7	0.907262
$760$	−3.36735e6	−0.211472
$761$	−2.92191e6	−0.182897	−0.0914483	−	0.995810i	$-0.529150\pi$
$761$	−2.92191e6	−0.182897	−0.0914483	+	0.995810i	$0.529150\pi$
$762$	1.87282e6	0.116845
$763$	0	0
$764$	2.00918e6	0.124533
$765$	−7.94025e6	−0.490547
$766$	2.48310e6	0.152905
$767$	9.25887e6	0.568290
$768$	−1.73016e7	−1.05848
$769$	−1.42847e7	−0.871073	−0.435536	−	0.900171i	$-0.643441\pi$
$769$	−1.42847e7	−0.871073	−0.435536	+	0.900171i	$0.643441\pi$
$770$	0	0
$771$	1.55360e7	0.941246
$772$	−1.47726e7	−0.892100
$773$	1.09012e7	0.656186	0.328093	−	0.944645i	$-0.393594\pi$
$773$	1.09012e7	0.656186	0.328093	+	0.944645i	$0.393594\pi$
$774$	1.71874e6	0.103123
$775$	−1.92455e6	−0.115100
$776$	−1.05426e6	−0.0628485
$777$	0	0
$778$	6254.04	0.000370434 0
$779$	−1.71875e6	−0.101478
$780$	1.88114e7	1.10710
$781$	4.30045e7	2.52282
$782$	946787.	0.0553650
$783$	4.70459e6	0.274231
$784$	0	0
$785$	−2.44736e7	−1.41750
$786$	−4.97890e6	−0.287460
$787$	2.56449e7	1.47593	0.737963	−	0.674841i	$-0.235788\pi$
$787$	2.56449e7	1.47593	0.737963	+	0.674841i	$0.235788\pi$
$788$	−8.43576e6	−0.483959
$789$	−9.12555e6	−0.521875
$790$	−1.03838e6	−0.0591955
$791$	0	0
$792$	−4.38699e6	−0.248516
$793$	9.26169e6	0.523007
$794$	−911498.	−0.0513103
$795$	−5.06991e6	−0.284500
$796$	−1.90472e7	−1.06549
$797$	−7.73086e6	−0.431104	−0.215552	−	0.976492i	$-0.569155\pi$
$797$	−7.73086e6	−0.431104	−0.215552	+	0.976492i	$0.569155\pi$
$798$	0	0
$799$	−9.84931e6	−0.545807
$800$	−2.04707e6	−0.113086
$801$	−1.80269e7	−0.992748
$802$	1.03448e6	0.0567917
$803$	−1.90282e7	−1.04138
$804$	1.22024e7	0.665738
$805$	0	0
$806$	966395.	0.0523983
$807$	3.92909e7	2.12377
$808$	−8.35218e6	−0.450061
$809$	−1.87811e7	−1.00890	−0.504452	−	0.863440i	$-0.668305\pi$
$809$	−1.87811e7	−1.00890	−0.504452	+	0.863440i	$0.668305\pi$
$810$	−2.48616e6	−0.133143
$811$	9.00729e6	0.480886	0.240443	−	0.970663i	$-0.422707\pi$
$811$	9.00729e6	0.480886	0.240443	+	0.970663i	$0.422707\pi$
$812$	0	0
$813$	3.16939e7	1.68170
$814$	520422.	0.0275293
$815$	−2.05531e7	−1.08388
$816$	2.27110e7	1.19402
$817$	2.59240e7	1.35877
$818$	2.22259e6	0.116138
$819$	0	0
$820$	1.61590e6	0.0839228
$821$	−9.27965e6	−0.480478	−0.240239	−	0.970714i	$-0.577226\pi$
$821$	−9.27965e6	−0.480478	−0.240239	+	0.970714i	$0.577226\pi$
$822$	−5.50572e6	−0.284207
$823$	−1.08308e7	−0.557393	−0.278697	−	0.960379i	$-0.589902\pi$
$823$	−1.08308e7	−0.557393	−0.278697	+	0.960379i	$0.589902\pi$
$824$	2.98107e6	0.152952
$825$	1.22967e7	0.629004
$826$	0	0
$827$	−2.05230e7	−1.04346	−0.521731	−	0.853110i	$-0.674714\pi$
$827$	−2.05230e7	−1.04346	−0.521731	+	0.853110i	$0.674714\pi$
$828$	4.95515e6	0.251178
$829$	1.42216e7	0.718724	0.359362	−	0.933198i	$-0.382994\pi$
$829$	1.42216e7	0.718724	0.359362	+	0.933198i	$0.382994\pi$
$830$	22912.6	0.00115446
$831$	4.08944e7	2.05429
$832$	−1.92472e7	−0.963958
$833$	0	0
$834$	246329.	0.0122631
$835$	−1.30595e7	−0.648202
$836$	−3.28108e7	−1.62368
$837$	4.02097e6	0.198389
$838$	179995.	0.00885421
$839$	9.29934e6	0.456087	0.228043	−	0.973651i	$-0.426767\pi$
$839$	9.29934e6	0.456087	0.228043	+	0.973651i	$0.426767\pi$
$840$	0	0
$841$	−1.47669e7	−0.719944
$842$	4.32353e6	0.210164
$843$	1.92965e7	0.935211
$844$	−1.05561e7	−0.510092
$845$	2.43277e6	0.117209
$846$	861053.	0.0413623
$847$	0	0
$848$	5.37461e6	0.256659
$849$	−2.74350e7	−1.30628
$850$	808543.	0.0383845
$851$	−1.18546e6	−0.0561131
$852$	3.99290e7	1.88447
$853$	3.07436e6	0.144671	0.0723357	−	0.997380i	$-0.476955\pi$
$853$	3.07436e6	0.144671	0.0723357	+	0.997380i	$0.476955\pi$
$854$	0	0
$855$	1.04701e7	0.489819
$856$	6.81938e6	0.318098
$857$	−3.45835e7	−1.60848	−0.804242	−	0.594302i	$-0.797428\pi$
$857$	−3.45835e7	−1.60848	−0.804242	+	0.594302i	$0.797428\pi$
$858$	−6.17465e6	−0.286348
$859$	−1.63022e7	−0.753814	−0.376907	−	0.926251i	$-0.623012\pi$
$859$	−1.63022e7	−0.753814	−0.376907	+	0.926251i	$0.623012\pi$
$860$	−2.43726e7	−1.12372
$861$	0	0
$862$	−3.57856e6	−0.164036
$863$	2.56962e7	1.17447	0.587235	−	0.809416i	$-0.300216\pi$
$863$	2.56962e7	1.17447	0.587235	+	0.809416i	$0.300216\pi$
$864$	4.27694e6	0.194917
$865$	−4.66087e6	−0.211800
$866$	−3.01281e6	−0.136514
$867$	218902.	0.00989012
$868$	0	0
$869$	−2.04046e7	−0.916596
$870$	−1.59636e6	−0.0715043
$871$	1.28374e7	0.573366
$872$	5.12222e6	0.228122
$873$	3.27803e6	0.145572
$874$	−1.24844e6	−0.0552829
$875$	0	0
$876$	−1.76673e7	−0.777877
$877$	3.30060e7	1.44908	0.724542	−	0.689230i	$-0.242051\pi$
$877$	3.30060e7	1.44908	0.724542	+	0.689230i	$0.242051\pi$
$878$	165286.	0.00723603
$879$	5.03429e7	2.19769
$880$	3.03235e7	1.31999
$881$	−2.26705e6	−0.0984060	−0.0492030	−	0.998789i	$-0.515668\pi$
$881$	−2.26705e6	−0.0984060	−0.0492030	+	0.998789i	$0.515668\pi$
$882$	0	0
$883$	1.97779e7	0.853649	0.426825	−	0.904334i	$-0.359632\pi$
$883$	1.97779e7	0.853649	0.426825	+	0.904334i	$0.359632\pi$
$884$	2.43057e7	1.04611
$885$	−1.30720e7	−0.561029
$886$	−1.43901e6	−0.0615858
$887$	36468.3	0.00155635	0.000778173	−	1.00000i	$-0.499752\pi$
$887$	36468.3	0.00155635	0.000778173		1.00000i	$0.499752\pi$
$888$	974478.	0.0414705
$889$	0	0
$890$	−4.27005e6	−0.180700
$891$	−4.88541e7	−2.06161
$892$	3.22692e7	1.35793
$893$	1.29874e7	0.544997
$894$	1.59815e6	0.0668767
$895$	1.54263e7	0.643730
$896$	0	0
$897$	1.40652e7	0.583665
$898$	1.89303e6	0.0783367
$899$	4.90958e6	0.202602
$900$	4.23163e6	0.174141
$901$	−6.55068e6	−0.268828
$902$	−530403.	−0.0217065
$903$	0	0
$904$	1.98711e6	0.0808727
$905$	2.36462e7	0.959709
$906$	−434484.	−0.0175854
$907$	1.62298e7	0.655079	0.327540	−	0.944837i	$-0.393781\pi$
$907$	1.62298e7	0.655079	0.327540	+	0.944837i	$0.393781\pi$
$908$	−1.58702e7	−0.638802
$909$	2.59695e7	1.04245
$910$	0	0
$911$	−2.61699e7	−1.04474	−0.522368	−	0.852720i	$-0.674951\pi$
$911$	−2.61699e7	−1.04474	−0.522368	+	0.852720i	$0.674951\pi$
$912$	−2.99469e7	−1.19224
$913$	450241.	0.0178759
$914$	−2.97224e6	−0.117684
$915$	−1.30760e7	−0.516324
$916$	−3.52320e7	−1.38739
$917$	0	0
$918$	−1.68929e6	−0.0661602
$919$	4.05973e6	0.158565	0.0792826	−	0.996852i	$-0.474737\pi$
$919$	4.05973e6	0.158565	0.0792826	+	0.996852i	$0.474737\pi$
$920$	2.36708e6	0.0922025
$921$	−1.73870e7	−0.675421
$922$	1.90246e6	0.0737034
$923$	4.20070e7	1.62300
$924$	0	0
$925$	−1.01237e6	−0.0389031
$926$	−3.10744e6	−0.119090
$927$	−9.26907e6	−0.354272
$928$	5.22212e6	0.199057
$929$	−3.69518e6	−0.140474	−0.0702370	−	0.997530i	$-0.522376\pi$
$929$	−3.69518e6	−0.140474	−0.0702370	+	0.997530i	$0.522376\pi$
$930$	−1.36439e6	−0.0517288
$931$	0	0
$932$	−2.42573e7	−0.914751
$933$	3.53927e7	1.33110
$934$	−717918.	−0.0269282
$935$	−3.69589e7	−1.38258
$936$	−4.28523e6	−0.159877
$937$	1.91384e7	0.712126	0.356063	−	0.934462i	$-0.384119\pi$
$937$	1.91384e7	0.712126	0.356063	+	0.934462i	$0.384119\pi$
$938$	0	0
$939$	1.86742e7	0.691160
$940$	−1.22102e7	−0.450717
$941$	−1.20954e7	−0.445294	−0.222647	−	0.974899i	$-0.571470\pi$
$941$	−1.20954e7	−0.445294	−0.222647	+	0.974899i	$0.571470\pi$
$942$	7.45870e6	0.273865
$943$	1.20820e6	0.0442445
$944$	1.38577e7	0.506127
$945$	0	0
$946$	8.00007e6	0.290647
$947$	−1.95969e7	−0.710087	−0.355044	−	0.934850i	$-0.615534\pi$
$947$	−1.95969e7	−0.710087	−0.355044	+	0.934850i	$0.615534\pi$
$948$	−1.89453e7	−0.684669
$949$	−1.85868e7	−0.669945
$950$	−1.06615e6	−0.0383275
$951$	−5.97890e7	−2.14373
$952$	0	0
$953$	4.23265e7	1.50966	0.754832	−	0.655918i	$-0.227718\pi$
$953$	4.23265e7	1.50966	0.754832	+	0.655918i	$0.227718\pi$
$954$	572679.	0.0203723
$955$	−2.98426e6	−0.105884
$956$	5.40243e6	0.191181
$957$	−3.13691e7	−1.10719
$958$	−1.87054e6	−0.0658496
$959$	0	0
$960$	2.71739e7	0.951642
$961$	−2.44330e7	−0.853430
$962$	508351.	0.0177103
$963$	−2.12035e7	−0.736788
$964$	−1.20792e7	−0.418644
$965$	2.19419e7	0.758502
$966$	0	0
$967$	1.39211e6	0.0478749	0.0239375	−	0.999713i	$-0.492380\pi$
$967$	1.39211e6	0.0478749	0.0239375	+	0.999713i	$0.492380\pi$
$968$	−1.30075e7	−0.446176
$969$	3.64999e7	1.24877
$970$	776471.	0.0264970
$971$	−4.41877e7	−1.50402	−0.752009	−	0.659153i	$-0.770915\pi$
$971$	−4.41877e7	−1.50402	−0.752009	+	0.659153i	$0.770915\pi$
$972$	−3.03473e7	−1.03028
$973$	0	0
$974$	2.33095e6	0.0787292
$975$	1.20115e7	0.404654
$976$	1.38619e7	0.465798
$977$	−5.60457e7	−1.87848	−0.939239	−	0.343264i	$-0.888467\pi$
$977$	−5.60457e7	−1.87848	−0.939239	+	0.343264i	$0.888467\pi$
$978$	6.26386e6	0.209409
$979$	−8.39082e7	−2.79800
$980$	0	0
$981$	−1.59266e7	−0.528384
$982$	−5.69993e6	−0.188621
$983$	5.00560e7	1.65224	0.826118	−	0.563497i	$-0.190544\pi$
$983$	5.00560e7	1.65224	0.826118	+	0.563497i	$0.190544\pi$
$984$	−993166.	−0.0326990
$985$	1.25298e7	0.411483
$986$	−2.06261e6	−0.0675654
$987$	0	0
$988$	−3.20497e7	−1.04456
$989$	−1.82233e7	−0.592428
$990$	3.23104e6	0.104774
$991$	1.59116e7	0.514670	0.257335	−	0.966322i	$-0.417156\pi$
$991$	1.59116e7	0.514670	0.257335	+	0.966322i	$0.417156\pi$
$992$	4.46330e6	0.144005
$993$	4.31534e7	1.38881
$994$	0	0
$995$	2.82911e7	0.905924
$996$	418041.	0.0133528
$997$	−4.25995e7	−1.35727	−0.678635	−	0.734476i	$-0.737428\pi$
$997$	−4.25995e7	−1.35727	−0.678635	+	0.734476i	$0.737428\pi$
$998$	−981634.	−0.0311978
$999$	2.11514e6	0.0670542

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.6.a.f.1.1		2
3.2	odd	2		441.6.a.l.1.2		2
4.3	odd	2		784.6.a.v.1.2		2
7.2	even	3		49.6.c.d.18.2		4
7.3	odd	6		49.6.c.e.30.2		4
7.4	even	3		49.6.c.d.30.2		4
7.5	odd	6		49.6.c.e.18.2		4
7.6	odd	2		7.6.a.b.1.1	✓	2
21.20	even	2		63.6.a.f.1.2		2
28.27	even	2		112.6.a.h.1.1		2
35.13	even	4		175.6.b.c.99.2		4
35.27	even	4		175.6.b.c.99.3		4
35.34	odd	2		175.6.a.c.1.2		2
56.13	odd	2		448.6.a.w.1.1		2
56.27	even	2		448.6.a.u.1.2		2
77.76	even	2		847.6.a.c.1.2		2
84.83	odd	2		1008.6.a.bq.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.a.b.1.1	✓	2	7.6	odd	2
49.6.a.f.1.1		2	1.1	even	1	trivial
49.6.c.d.18.2		4	7.2	even	3
49.6.c.d.30.2		4	7.4	even	3
49.6.c.e.18.2		4	7.5	odd	6
49.6.c.e.30.2		4	7.3	odd	6
63.6.a.f.1.2		2	21.20	even	2
112.6.a.h.1.1		2	28.27	even	2
175.6.a.c.1.2		2	35.34	odd	2
175.6.b.c.99.2		4	35.13	even	4
175.6.b.c.99.3		4	35.27	even	4
441.6.a.l.1.2		2	3.2	odd	2
448.6.a.u.1.2		2	56.27	even	2
448.6.a.w.1.1		2	56.13	odd	2
784.6.a.v.1.2		2	4.3	odd	2
847.6.a.c.1.2		2	77.76	even	2
1008.6.a.bq.1.2		2	84.83	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:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	49.a (trivial)

\(f(q)\)	\(=\)	\(q+0.725083 q^{2} -19.6495 q^{3} -31.4743 q^{4} +46.7492 q^{5} -14.2475 q^{6} -46.0241 q^{8} +143.103 q^{9} +O(q^{10})\)	\(q+0.725083 q^{2} -19.6495 q^{3} -31.4743 q^{4} +46.7492 q^{5} -14.2475 q^{6} -46.0241 q^{8} +143.103 q^{9} +33.8970 q^{10} +666.090 q^{11} +618.453 q^{12} +650.640 q^{13} -918.598 q^{15} +973.805 q^{16} -1186.89 q^{17} +103.762 q^{18} +1565.05 q^{19} -1471.40 q^{20} +482.970 q^{22} -1100.15 q^{23} +904.350 q^{24} -939.515 q^{25} +471.768 q^{26} +1962.93 q^{27} +2396.72 q^{29} -666.060 q^{30} +2048.46 q^{31} +2178.86 q^{32} -13088.3 q^{33} -860.596 q^{34} -4504.06 q^{36} +1077.54 q^{37} +1134.79 q^{38} -12784.7 q^{39} -2151.59 q^{40} -1098.21 q^{41} +16564.3 q^{43} -20964.7 q^{44} +6689.95 q^{45} -797.702 q^{46} +8298.39 q^{47} -19134.8 q^{48} -681.226 q^{50} +23321.9 q^{51} -20478.4 q^{52} +5519.18 q^{53} +1423.28 q^{54} +31139.1 q^{55} -30752.5 q^{57} +1737.82 q^{58} +14230.4 q^{59} +28912.2 q^{60} +14234.7 q^{61} +1485.30 q^{62} -29581.9 q^{64} +30416.9 q^{65} -9490.12 q^{66} +19730.4 q^{67} +37356.6 q^{68} +21617.5 q^{69} +64562.7 q^{71} -6586.18 q^{72} -28567.0 q^{73} +781.309 q^{74} +18461.0 q^{75} -49258.8 q^{76} -9270.00 q^{78} -30633.4 q^{79} +45524.6 q^{80} -73344.6 q^{81} -796.293 q^{82} +675.946 q^{83} -55486.3 q^{85} +12010.5 q^{86} -47094.4 q^{87} -30656.2 q^{88} -125971. q^{89} +4850.76 q^{90} +34626.5 q^{92} -40251.1 q^{93} +6017.02 q^{94} +73164.9 q^{95} -42813.5 q^{96} +22906.8 q^{97} +95319.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 9 q^{2} + 6 q^{3} + 5 q^{4} + 18 q^{5} + 198 q^{6} - 9 q^{8} + 558 q^{9}+O(q^{10}) \) 2 * q + 9 * q^2 + 6 * q^3 + 5 * q^4 + 18 * q^5 + 198 * q^6 - 9 * q^8 + 558 * q^9	\( 2 q + 9 q^{2} + 6 q^{3} + 5 q^{4} + 18 q^{5} + 198 q^{6} - 9 q^{8} + 558 q^{9} - 204 q^{10} + 396 q^{11} + 1554 q^{12} + 350 q^{13} - 1656 q^{15} + 113 q^{16} - 1800 q^{17} + 3537 q^{18} + 3266 q^{19} - 2520 q^{20} - 1752 q^{22} + 2088 q^{23} + 1854 q^{24} - 3238 q^{25} - 2016 q^{26} + 6372 q^{27} + 6696 q^{29} - 6768 q^{30} + 20 q^{31} - 6129 q^{32} - 20016 q^{33} - 5934 q^{34} + 10629 q^{36} + 6232 q^{37} + 15210 q^{38} - 20496 q^{39} - 3216 q^{40} + 6048 q^{41} - 3020 q^{43} - 30816 q^{44} - 5238 q^{45} + 25584 q^{46} - 11700 q^{47} - 41214 q^{48} - 19701 q^{50} + 7596 q^{51} - 31444 q^{52} + 9468 q^{53} + 37908 q^{54} + 38904 q^{55} + 12876 q^{57} + 37314 q^{58} + 43938 q^{59} + 2016 q^{60} + 64754 q^{61} - 15300 q^{62} - 70783 q^{64} + 39060 q^{65} - 66816 q^{66} + 24784 q^{67} + 14994 q^{68} + 103392 q^{69} + 97416 q^{71} + 8775 q^{72} - 17452 q^{73} + 43434 q^{74} - 40494 q^{75} + 12782 q^{76} - 73080 q^{78} + 51256 q^{79} + 70272 q^{80} - 61074 q^{81} + 58338 q^{82} - 117558 q^{83} - 37860 q^{85} - 150048 q^{86} + 63180 q^{87} - 40656 q^{88} - 84276 q^{89} - 93852 q^{90} + 150912 q^{92} - 92280 q^{93} - 159468 q^{94} + 24264 q^{95} - 255906 q^{96} - 20776 q^{97} - 16740 q^{99}+O(q^{100}) \) 2 * q + 9 * q^2 + 6 * q^3 + 5 * q^4 + 18 * q^5 + 198 * q^6 - 9 * q^8 + 558 * q^9 - 204 * q^10 + 396 * q^11 + 1554 * q^12 + 350 * q^13 - 1656 * q^15 + 113 * q^16 - 1800 * q^17 + 3537 * q^18 + 3266 * q^19 - 2520 * q^20 - 1752 * q^22 + 2088 * q^23 + 1854 * q^24 - 3238 * q^25 - 2016 * q^26 + 6372 * q^27 + 6696 * q^29 - 6768 * q^30 + 20 * q^31 - 6129 * q^32 - 20016 * q^33 - 5934 * q^34 + 10629 * q^36 + 6232 * q^37 + 15210 * q^38 - 20496 * q^39 - 3216 * q^40 + 6048 * q^41 - 3020 * q^43 - 30816 * q^44 - 5238 * q^45 + 25584 * q^46 - 11700 * q^47 - 41214 * q^48 - 19701 * q^50 + 7596 * q^51 - 31444 * q^52 + 9468 * q^53 + 37908 * q^54 + 38904 * q^55 + 12876 * q^57 + 37314 * q^58 + 43938 * q^59 + 2016 * q^60 + 64754 * q^61 - 15300 * q^62 - 70783 * q^64 + 39060 * q^65 - 66816 * q^66 + 24784 * q^67 + 14994 * q^68 + 103392 * q^69 + 97416 * q^71 + 8775 * q^72 - 17452 * q^73 + 43434 * q^74 - 40494 * q^75 + 12782 * q^76 - 73080 * q^78 + 51256 * q^79 + 70272 * q^80 - 61074 * q^81 + 58338 * q^82 - 117558 * q^83 - 37860 * q^85 - 150048 * q^86 + 63180 * q^87 - 40656 * q^88 - 84276 * q^89 - 93852 * q^90 + 150912 * q^92 - 92280 * q^93 - 159468 * q^94 + 24264 * q^95 - 255906 * q^96 - 20776 * q^97 - 16740 * q^99

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