LMFDB - Embedded newform 841.4.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [841,4,Mod(1,841)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(841, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("841.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 841 = 29^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	841.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:	$49.6206063148$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.13458092.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 14x^{3} + 18x^{2} + 20x - 8 $ x^5 - x^4 - 14x^3 + 18x^2 + 20*x - 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.27399$ of defining polynomial
Character	$\chi$	$=$	841.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.63099 q^{2} -9.87991 q^{3} -5.33986 q^{4} -16.8209 q^{5} +16.1141 q^{6} +5.21997 q^{7} +21.7572 q^{8} +70.6126 q^{9} +O(q^{10})$	\(q-1.63099 q^{2} -9.87991 q^{3} -5.33986 q^{4} -16.8209 q^{5} +16.1141 q^{6} +5.21997 q^{7} +21.7572 q^{8} +70.6126 q^{9} +27.4348 q^{10} +8.55158 q^{11} +52.7573 q^{12} -11.3429 q^{13} -8.51374 q^{14} +166.189 q^{15} +7.23299 q^{16} -68.4740 q^{17} -115.169 q^{18} -6.93014 q^{19} +89.8214 q^{20} -51.5728 q^{21} -13.9476 q^{22} -132.042 q^{23} -214.959 q^{24} +157.944 q^{25} +18.5001 q^{26} -430.889 q^{27} -27.8739 q^{28} -271.054 q^{30} +0.419319 q^{31} -185.855 q^{32} -84.4888 q^{33} +111.681 q^{34} -87.8048 q^{35} -377.061 q^{36} -395.483 q^{37} +11.3030 q^{38} +112.067 q^{39} -365.977 q^{40} +447.209 q^{41} +84.1150 q^{42} -184.132 q^{43} -45.6642 q^{44} -1187.77 q^{45} +215.360 q^{46} +97.2612 q^{47} -71.4613 q^{48} -315.752 q^{49} -257.605 q^{50} +676.516 q^{51} +60.5693 q^{52} -209.547 q^{53} +702.776 q^{54} -143.845 q^{55} +113.572 q^{56} +68.4691 q^{57} +45.9651 q^{59} -887.427 q^{60} -427.655 q^{61} -0.683906 q^{62} +368.596 q^{63} +245.264 q^{64} +190.798 q^{65} +137.801 q^{66} -405.055 q^{67} +365.641 q^{68} +1304.57 q^{69} +143.209 q^{70} -557.971 q^{71} +1536.33 q^{72} +381.988 q^{73} +645.031 q^{74} -1560.47 q^{75} +37.0060 q^{76} +44.6390 q^{77} -182.780 q^{78} -577.208 q^{79} -121.666 q^{80} +2350.60 q^{81} -729.396 q^{82} -353.745 q^{83} +275.392 q^{84} +1151.80 q^{85} +300.318 q^{86} +186.059 q^{88} +277.871 q^{89} +1937.24 q^{90} -59.2095 q^{91} +705.088 q^{92} -4.14283 q^{93} -158.632 q^{94} +116.571 q^{95} +1836.23 q^{96} -677.917 q^{97} +514.989 q^{98} +603.849 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 8 q^{3} + 26 q^{4} + 10 q^{5} + 34 q^{6} + 40 q^{7} + 84 q^{8} + 33 q^{9}+O(q^{10}) $ 5 * q - 8 * q^3 + 26 * q^4 + 10 * q^5 + 34 * q^6 + 40 * q^7 + 84 * q^8 + 33 * q^9	\( 5 q - 8 q^{3} + 26 q^{4} + 10 q^{5} + 34 q^{6} + 40 q^{7} + 84 q^{8} + 33 q^{9} + 64 q^{10} - 12 q^{11} + 224 q^{12} + 14 q^{13} + 192 q^{14} + 74 q^{15} + 146 q^{16} - 66 q^{17} + 108 q^{18} - 214 q^{19} + 6 q^{20} - 98 q^{22} + 164 q^{23} + 314 q^{24} + 207 q^{25} - 56 q^{26} - 362 q^{27} + 540 q^{28} - 234 q^{30} - 420 q^{31} + 652 q^{32} - 576 q^{33} + 204 q^{34} - 52 q^{35} - 260 q^{36} - 378 q^{37} - 496 q^{38} + 374 q^{39} + 80 q^{40} + 1158 q^{41} + 348 q^{42} + 204 q^{43} - 784 q^{44} - 1506 q^{45} - 580 q^{46} - 248 q^{47} + 1880 q^{48} - 283 q^{49} - 908 q^{50} + 228 q^{51} + 1482 q^{52} - 554 q^{53} + 918 q^{54} - 546 q^{55} + 608 q^{56} + 44 q^{57} + 440 q^{59} - 636 q^{60} - 618 q^{61} + 1250 q^{62} + 804 q^{63} + 2594 q^{64} - 1656 q^{65} - 2940 q^{66} + 1164 q^{67} - 356 q^{68} + 1968 q^{69} + 2184 q^{70} - 692 q^{71} + 2648 q^{72} + 1950 q^{73} - 1832 q^{74} - 3074 q^{75} - 1376 q^{76} + 1616 q^{77} - 1302 q^{78} - 272 q^{79} - 890 q^{80} + 1801 q^{81} + 92 q^{82} + 512 q^{83} + 3208 q^{84} + 1628 q^{85} + 2446 q^{86} - 6954 q^{88} - 866 q^{89} + 2200 q^{90} + 2580 q^{91} + 3468 q^{92} - 40 q^{93} - 5942 q^{94} - 2244 q^{95} + 7386 q^{96} - 1562 q^{97} + 3408 q^{98} + 238 q^{99}+O(q^{100}) \) 5 * q - 8 * q^3 + 26 * q^4 + 10 * q^5 + 34 * q^6 + 40 * q^7 + 84 * q^8 + 33 * q^9 + 64 * q^10 - 12 * q^11 + 224 * q^12 + 14 * q^13 + 192 * q^14 + 74 * q^15 + 146 * q^16 - 66 * q^17 + 108 * q^18 - 214 * q^19 + 6 * q^20 - 98 * q^22 + 164 * q^23 + 314 * q^24 + 207 * q^25 - 56 * q^26 - 362 * q^27 + 540 * q^28 - 234 * q^30 - 420 * q^31 + 652 * q^32 - 576 * q^33 + 204 * q^34 - 52 * q^35 - 260 * q^36 - 378 * q^37 - 496 * q^38 + 374 * q^39 + 80 * q^40 + 1158 * q^41 + 348 * q^42 + 204 * q^43 - 784 * q^44 - 1506 * q^45 - 580 * q^46 - 248 * q^47 + 1880 * q^48 - 283 * q^49 - 908 * q^50 + 228 * q^51 + 1482 * q^52 - 554 * q^53 + 918 * q^54 - 546 * q^55 + 608 * q^56 + 44 * q^57 + 440 * q^59 - 636 * q^60 - 618 * q^61 + 1250 * q^62 + 804 * q^63 + 2594 * q^64 - 1656 * q^65 - 2940 * q^66 + 1164 * q^67 - 356 * q^68 + 1968 * q^69 + 2184 * q^70 - 692 * q^71 + 2648 * q^72 + 1950 * q^73 - 1832 * q^74 - 3074 * q^75 - 1376 * q^76 + 1616 * q^77 - 1302 * q^78 - 272 * q^79 - 890 * q^80 + 1801 * q^81 + 92 * q^82 + 512 * q^83 + 3208 * q^84 + 1628 * q^85 + 2446 * q^86 - 6954 * q^88 - 866 * q^89 + 2200 * q^90 + 2580 * q^91 + 3468 * q^92 - 40 * q^93 - 5942 * q^94 - 2244 * q^95 + 7386 * q^96 - 1562 * q^97 + 3408 * q^98 + 238 * 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 $

$2$	−1.63099	−0.576643	−0.288322	−	0.957534i	$-0.593097\pi$
$2$	−1.63099	−0.576643	−0.288322	+	0.957534i	$0.593097\pi$
$3$	−9.87991	−1.90139	−0.950695	−	0.310128i	$-0.899628\pi$
$3$	−9.87991	−1.90139	−0.950695	+	0.310128i	$0.899628\pi$
$4$	−5.33986	−0.667483
$5$	−16.8209	−1.50451	−0.752255	−	0.658872i	$-0.771034\pi$
$5$	−16.8209	−1.50451	−0.752255	+	0.658872i	$0.771034\pi$
$6$	16.1141	1.09642
$7$	5.21997	0.281852	0.140926	−	0.990020i	$-0.454992\pi$
$7$	5.21997	0.281852	0.140926	+	0.990020i	$0.454992\pi$
$8$	21.7572	0.961543
$9$	70.6126	2.61528
$10$	27.4348	0.867565
$11$	8.55158	0.234400	0.117200	−	0.993108i	$-0.462608\pi$
$11$	8.55158	0.234400	0.117200	+	0.993108i	$0.462608\pi$
$12$	52.7573	1.26914
$13$	−11.3429	−0.241996	−0.120998	−	0.992653i	$-0.538609\pi$
$13$	−11.3429	−0.241996	−0.120998	+	0.992653i	$0.538609\pi$
$14$	−8.51374	−0.162528
$15$	166.189	2.86066
$16$	7.23299	0.113016
$17$	−68.4740	−0.976904	−0.488452	−	0.872591i	$-0.662438\pi$
$17$	−68.4740	−0.976904	−0.488452	+	0.872591i	$0.662438\pi$
$18$	−115.169	−1.50808
$19$	−6.93014	−0.0836780	−0.0418390	−	0.999124i	$-0.513322\pi$
$19$	−6.93014	−0.0836780	−0.0418390	+	0.999124i	$0.513322\pi$
$20$	89.8214	1.00423
$21$	−51.5728	−0.535910
$22$	−13.9476	−0.135165
$23$	−132.042	−1.19708	−0.598538	−	0.801095i	$-0.704251\pi$
$23$	−132.042	−1.19708	−0.598538	+	0.801095i	$0.704251\pi$
$24$	−214.959	−1.82827
$25$	157.944	1.26355
$26$	18.5001	0.139545
$27$	−430.889	−3.07128
$28$	−27.8739	−0.188131
$29$	0	0
$29$	0	0
$30$	−271.054	−1.64958
$31$	0.419319	0.00242941	0.00121471	−	0.999999i	$-0.499613\pi$
$31$	0.419319	0.00242941	0.00121471	+	0.999999i	$0.499613\pi$
$32$	−185.855	−1.02671
$33$	−84.4888	−0.445685
$34$	111.681	0.563325
$35$	−87.8048	−0.424049
$36$	−377.061	−1.74565
$37$	−395.483	−1.75722	−0.878609	−	0.477542i	$-0.841528\pi$
$37$	−395.483	−1.75722	−0.878609	+	0.477542i	$0.841528\pi$
$38$	11.3030	0.0482524
$39$	112.067	0.460128
$40$	−365.977	−1.44665
$41$	447.209	1.70347	0.851736	−	0.523971i	$-0.175550\pi$
$41$	447.209	1.70347	0.851736	+	0.523971i	$0.175550\pi$
$42$	84.1150	0.309029
$43$	−184.132	−0.653020	−0.326510	−	0.945194i	$-0.605873\pi$
$43$	−184.132	−0.653020	−0.326510	+	0.945194i	$0.605873\pi$
$44$	−45.6642	−0.156458
$45$	−1187.77	−3.93472
$46$	215.360	0.690285
$47$	97.2612	0.301851	0.150926	−	0.988545i	$-0.451775\pi$
$47$	97.2612	0.301851	0.150926	+	0.988545i	$0.451775\pi$
$48$	−71.4613	−0.214886
$49$	−315.752	−0.920559
$50$	−257.605	−0.728617
$51$	676.516	1.85748
$52$	60.5693	0.161528
$53$	−209.547	−0.543086	−0.271543	−	0.962426i	$-0.587534\pi$
$53$	−209.547	−0.543086	−0.271543	+	0.962426i	$0.587534\pi$
$54$	702.776	1.77103
$55$	−143.845	−0.352657
$56$	113.572	0.271013
$57$	68.4691	0.159105
$58$	0	0
$59$	45.9651	0.101426	0.0507131	−	0.998713i	$-0.483851\pi$
$59$	45.9651	0.101426	0.0507131	+	0.998713i	$0.483851\pi$
$60$	−887.427	−1.90944
$61$	−427.655	−0.897634	−0.448817	−	0.893624i	$-0.648154\pi$
$61$	−427.655	−0.897634	−0.448817	+	0.893624i	$0.648154\pi$
$62$	−0.683906	−0.00140091
$63$	368.596	0.737122
$64$	245.264	0.479031
$65$	190.798	0.364085
$66$	137.801	0.257001
$67$	−405.055	−0.738588	−0.369294	−	0.929313i	$-0.620400\pi$
$67$	−405.055	−0.738588	−0.369294	+	0.929313i	$0.620400\pi$
$68$	365.641	0.652067
$69$	1304.57	2.27611
$70$	143.209	0.244525
$71$	−557.971	−0.932662	−0.466331	−	0.884610i	$-0.654424\pi$
$71$	−557.971	−0.932662	−0.466331	+	0.884610i	$0.654424\pi$
$72$	1536.33	2.51470
$73$	381.988	0.612443	0.306222	−	0.951960i	$-0.400935\pi$
$73$	381.988	0.612443	0.306222	+	0.951960i	$0.400935\pi$
$74$	645.031	1.01329
$75$	−1560.47	−2.40250
$76$	37.0060	0.0558536
$77$	44.6390	0.0660660
$78$	−182.780	−0.265330
$79$	−577.208	−0.822038	−0.411019	−	0.911627i	$-0.634827\pi$
$79$	−577.208	−0.822038	−0.411019	+	0.911627i	$0.634827\pi$
$80$	−121.666	−0.170033
$81$	2350.60	3.22442
$82$	−729.396	−0.982296
$83$	−353.745	−0.467813	−0.233907	−	0.972259i	$-0.575151\pi$
$83$	−353.745	−0.467813	−0.233907	+	0.972259i	$0.575151\pi$
$84$	275.392	0.357711
$85$	1151.80	1.46976
$86$	300.318	0.376560
$87$	0	0
$88$	186.059	0.225385
$89$	277.871	0.330947	0.165473	−	0.986214i	$-0.447085\pi$
$89$	277.871	0.330947	0.165473	+	0.986214i	$0.447085\pi$
$90$	1937.24	2.26893
$91$	−59.2095	−0.0682070
$92$	705.088	0.799027
$93$	−4.14283	−0.00461926
$94$	−158.632	−0.174060
$95$	116.571	0.125894
$96$	1836.23	1.95218
$97$	−677.917	−0.709609	−0.354804	−	0.934941i	$-0.615453\pi$
$97$	−677.917	−0.709609	−0.354804	+	0.934941i	$0.615453\pi$
$98$	514.989	0.530834
$99$	603.849	0.613022
$100$	−843.397	−0.843397
$101$	−567.816	−0.559404	−0.279702	−	0.960087i	$-0.590236\pi$
$101$	−567.816	−0.559404	−0.279702	+	0.960087i	$0.590236\pi$
$102$	−1103.39	−1.07110
$103$	319.205	0.305362	0.152681	−	0.988276i	$-0.451209\pi$
$103$	319.205	0.305362	0.152681	+	0.988276i	$0.451209\pi$
$104$	−246.789	−0.232689
$105$	867.503	0.806282
$106$	341.771	0.313167
$107$	−79.6547	−0.0719674	−0.0359837	−	0.999352i	$-0.511456\pi$
$107$	−79.6547	−0.0719674	−0.0359837	+	0.999352i	$0.511456\pi$
$108$	2300.88	2.05003
$109$	−1708.43	−1.50126	−0.750632	−	0.660721i	$-0.770251\pi$
$109$	−1708.43	−1.50126	−0.750632	+	0.660721i	$0.770251\pi$
$110$	234.611	0.203357
$111$	3907.34	3.34116
$112$	37.7560	0.0318536
$113$	−1050.31	−0.874383	−0.437191	−	0.899369i	$-0.644027\pi$
$113$	−1050.31	−0.874383	−0.437191	+	0.899369i	$0.644027\pi$
$114$	−111.673	−0.0917465
$115$	2221.08	1.80101
$116$	0	0
$117$	−800.950	−0.632887
$118$	−74.9687	−0.0584867
$119$	−357.432	−0.275342
$120$	3615.82	2.75065
$121$	−1257.87	−0.945057
$122$	697.503	0.517615
$123$	−4418.39	−3.23896
$124$	−2.23910	−0.00162159
$125$	−554.143	−0.396513
$126$	−601.177	−0.425057
$127$	−366.926	−0.256373	−0.128187	−	0.991750i	$-0.540916\pi$
$127$	−366.926	−0.256373	−0.128187	+	0.991750i	$0.540916\pi$
$128$	1086.81	0.750482
$129$	1819.21	1.24165
$130$	−311.190	−0.209947
$131$	−2310.86	−1.54123	−0.770614	−	0.637302i	$-0.780050\pi$
$131$	−2310.86	−1.54123	−0.770614	+	0.637302i	$0.780050\pi$
$132$	451.158	0.297487
$133$	−36.1751	−0.0235848
$134$	660.643	0.425902
$135$	7247.95	4.62077
$136$	−1489.80	−0.939335
$137$	−1899.65	−1.18466	−0.592329	−	0.805696i	$-0.701791\pi$
$137$	−1899.65	−1.18466	−0.592329	+	0.805696i	$0.701791\pi$
$138$	−2127.74	−1.31250
$139$	1309.49	0.799061	0.399531	−	0.916720i	$-0.369173\pi$
$139$	1309.49	0.799061	0.399531	+	0.916720i	$0.369173\pi$
$140$	468.865	0.283045
$141$	−960.932	−0.573937
$142$	910.047	0.537813
$143$	−96.9994	−0.0567238
$144$	510.740	0.295567
$145$	0	0
$146$	−623.020	−0.353161
$147$	3119.60	1.75034
$148$	2111.83	1.17291
$149$	1782.98	0.980320	0.490160	−	0.871632i	$-0.336938\pi$
$149$	1782.98	0.980320	0.490160	+	0.871632i	$0.336938\pi$
$150$	2545.12	1.38539
$151$	−1631.96	−0.879515	−0.439758	−	0.898116i	$-0.644936\pi$
$151$	−1631.96	−0.879515	−0.439758	+	0.898116i	$0.644936\pi$
$152$	−150.781	−0.0804600
$153$	−4835.12	−2.55488
$154$	−72.8059	−0.0380965
$155$	−7.05333	−0.00365508
$156$	−598.420	−0.307128
$157$	−852.817	−0.433517	−0.216759	−	0.976225i	$-0.569548\pi$
$157$	−852.817	−0.433517	−0.216759	+	0.976225i	$0.569548\pi$
$158$	941.422	0.474022
$159$	2070.31	1.03262
$160$	3126.25	1.54470
$161$	−689.257	−0.337398
$162$	−3833.81	−1.85934
$163$	−3280.24	−1.57625	−0.788123	−	0.615518i	$-0.788947\pi$
$163$	−3280.24	−1.57625	−0.788123	+	0.615518i	$0.788947\pi$
$164$	−2388.04	−1.13704
$165$	1421.18	0.670538
$166$	576.955	0.269761
$167$	−1682.26	−0.779504	−0.389752	−	0.920920i	$-0.627439\pi$
$167$	−1682.26	−0.779504	−0.389752	+	0.920920i	$0.627439\pi$
$168$	−1122.08	−0.515301
$169$	−2068.34	−0.941438
$170$	−1878.57	−0.847528
$171$	−489.355	−0.218842
$172$	983.239	0.435879
$173$	−1590.22	−0.698854	−0.349427	−	0.936964i	$-0.613624\pi$
$173$	−1590.22	−0.698854	−0.349427	+	0.936964i	$0.613624\pi$
$174$	0	0
$175$	824.462	0.356134
$176$	61.8535	0.0264908
$177$	−454.131	−0.192851
$178$	−453.206	−0.190838
$179$	−1794.60	−0.749356	−0.374678	−	0.927155i	$-0.622247\pi$
$179$	−1794.60	−0.749356	−0.374678	+	0.927155i	$0.622247\pi$
$180$	6342.52	2.62635
$181$	2353.41	0.966450	0.483225	−	0.875496i	$-0.339465\pi$
$181$	2353.41	0.966450	0.483225	+	0.875496i	$0.339465\pi$
$182$	96.5702	0.0393311
$183$	4225.20	1.70675
$184$	−2872.88	−1.15104
$185$	6652.40	2.64375
$186$	6.75693	0.00266367
$187$	−585.560	−0.228986
$188$	−519.361	−0.201480
$189$	−2249.23	−0.865646
$190$	−190.127	−0.0725962
$191$	2184.35	0.827508	0.413754	−	0.910389i	$-0.364217\pi$
$191$	2184.35	0.827508	0.413754	+	0.910389i	$0.364217\pi$
$192$	−2423.19	−0.910825
$193$	3109.71	1.15980	0.579901	−	0.814687i	$-0.303091\pi$
$193$	3109.71	1.15980	0.579901	+	0.814687i	$0.303091\pi$
$194$	1105.68	0.409191
$195$	−1885.06	−0.692267
$196$	1686.07	0.614457
$197$	923.756	0.334086	0.167043	−	0.985950i	$-0.446578\pi$
$197$	923.756	0.334086	0.167043	+	0.985950i	$0.446578\pi$
$198$	−984.874	−0.353495
$199$	4548.71	1.62035	0.810174	−	0.586189i	$-0.199372\pi$
$199$	4548.71	1.62035	0.810174	+	0.586189i	$0.199372\pi$
$200$	3436.42	1.21496
$201$	4001.91	1.40434
$202$	926.103	0.322576
$203$	0	0
$204$	−3612.50	−1.23983
$205$	−7522.48	−2.56289
$206$	−520.622	−0.176085
$207$	−9323.85	−3.13069
$208$	−82.0429	−0.0273493
$209$	−59.2636	−0.0196141
$210$	−1414.89	−0.464937
$211$	318.664	0.103970	0.0519851	−	0.998648i	$-0.483445\pi$
$211$	318.664	0.103970	0.0519851	+	0.998648i	$0.483445\pi$
$212$	1118.95	0.362500
$213$	5512.70	1.77335
$214$	129.916	0.0414995
$215$	3097.27	0.982475
$216$	−9374.94	−2.95317
$217$	2.18883	0.000684735 0
$218$	2786.44	0.865694
$219$	−3774.01	−1.16449
$220$	768.115	0.235392
$221$	776.691	0.236407
$222$	−6372.85	−1.92666
$223$	1706.70	0.512509	0.256254	−	0.966609i	$-0.417512\pi$
$223$	1706.70	0.512509	0.256254	+	0.966609i	$0.417512\pi$
$224$	−970.157	−0.289381
$225$	11152.8	3.30454
$226$	1713.06	0.504207
$227$	3043.63	0.889925	0.444962	−	0.895549i	$-0.353217\pi$
$227$	3043.63	0.889925	0.444962	+	0.895549i	$0.353217\pi$
$228$	−365.616	−0.106199
$229$	2621.57	0.756500	0.378250	−	0.925704i	$-0.376526\pi$
$229$	2621.57	0.756500	0.378250	+	0.925704i	$0.376526\pi$
$230$	−3622.56	−1.03854
$231$	−441.029	−0.125617
$232$	0	0
$233$	−2778.73	−0.781291	−0.390646	−	0.920541i	$-0.627748\pi$
$233$	−2778.73	−0.781291	−0.390646	+	0.920541i	$0.627748\pi$
$234$	1306.34	0.364950
$235$	−1636.02	−0.454138
$236$	−245.447	−0.0677002
$237$	5702.76	1.56301
$238$	582.969	0.158774
$239$	4722.67	1.27818	0.639088	−	0.769133i	$-0.279312\pi$
$239$	4722.67	1.27818	0.639088	+	0.769133i	$0.279312\pi$
$240$	1202.05	0.323299
$241$	−4704.13	−1.25734	−0.628672	−	0.777671i	$-0.716401\pi$
$241$	−4704.13	−1.25734	−0.628672	+	0.777671i	$0.716401\pi$
$242$	2051.58	0.544961
$243$	−11589.7	−3.05959
$244$	2283.62	0.599155
$245$	5311.24	1.38499
$246$	7206.36	1.86773
$247$	78.6076	0.0202497
$248$	9.12321	0.00233599
$249$	3494.96	0.889496
$250$	903.804	0.228646
$251$	−4449.07	−1.11882	−0.559408	−	0.828893i	$-0.688971\pi$
$251$	−4449.07	−1.11882	−0.559408	+	0.828893i	$0.688971\pi$
$252$	−1968.25	−0.492016
$253$	−1129.17	−0.280594
$254$	598.454	0.147836
$255$	−11379.6	−2.79459
$256$	−3734.70	−0.911792
$257$	7226.68	1.75404	0.877019	−	0.480456i	$-0.159529\pi$
$257$	7226.68	1.75404	0.877019	+	0.480456i	$0.159529\pi$
$258$	−2967.11	−0.715986
$259$	−2064.41	−0.495275
$260$	−1018.83	−0.243020
$261$	0	0
$262$	3769.00	0.888739
$263$	4789.59	1.12296	0.561481	−	0.827490i	$-0.310232\pi$
$263$	4789.59	1.12296	0.561481	+	0.827490i	$0.310232\pi$
$264$	−1838.24	−0.428545
$265$	3524.78	0.817078
$266$	59.0014	0.0136000
$267$	−2745.34	−0.629259
$268$	2162.94	0.492995
$269$	47.0448	0.0106631	0.00533154	−	0.999986i	$-0.498303\pi$
$269$	47.0448	0.0106631	0.00533154	+	0.999986i	$0.498303\pi$
$270$	−11821.4	−2.66454
$271$	−7721.78	−1.73087	−0.865433	−	0.501025i	$-0.832956\pi$
$271$	−7721.78	−1.73087	−0.865433	+	0.501025i	$0.832956\pi$
$272$	−495.272	−0.110405
$273$	584.984	0.129688
$274$	3098.32	0.683125
$275$	1350.67	0.296176
$276$	−6966.20	−1.51926
$277$	−7554.66	−1.63868	−0.819342	−	0.573305i	$-0.805661\pi$
$277$	−7554.66	−1.63868	−0.819342	+	0.573305i	$0.805661\pi$
$278$	−2135.77	−0.460773
$279$	29.6092	0.00635360
$280$	−1910.39	−0.407741
$281$	−2094.83	−0.444723	−0.222361	−	0.974964i	$-0.571377\pi$
$281$	−2094.83	−0.444723	−0.222361	+	0.974964i	$0.571377\pi$
$282$	1567.27	0.330957
$283$	−8200.83	−1.72257	−0.861287	−	0.508119i	$-0.830341\pi$
$283$	−8200.83	−1.72257	−0.861287	+	0.508119i	$0.830341\pi$
$284$	2979.49	0.622536
$285$	−1151.71	−0.239374
$286$	158.205	0.0327094
$287$	2334.42	0.480127
$288$	−13123.7	−2.68514
$289$	−224.318	−0.0456580
$290$	0	0
$291$	6697.76	1.34924
$292$	−2039.76	−0.408795
$293$	−3518.87	−0.701620	−0.350810	−	0.936447i	$-0.614094\pi$
$293$	−3518.87	−0.701620	−0.350810	+	0.936447i	$0.614094\pi$
$294$	−5088.05	−1.00932
$295$	−773.175	−0.152597
$296$	−8604.62	−1.68964
$297$	−3684.78	−0.719907
$298$	−2908.03	−0.565295
$299$	1497.74	0.289687
$300$	8332.69	1.60363
$301$	−961.163	−0.184055
$302$	2661.71	0.507167
$303$	5609.97	1.06364
$304$	−50.1256	−0.00945691
$305$	7193.56	1.35050
$306$	7886.06	1.47325
$307$	8725.36	1.62209	0.811046	−	0.584982i	$-0.198898\pi$
$307$	8725.36	1.62209	0.811046	+	0.584982i	$0.198898\pi$
$308$	−238.366	−0.0440979
$309$	−3153.72	−0.580611
$310$	11.5039	0.00210768
$311$	−2640.09	−0.481370	−0.240685	−	0.970603i	$-0.577372\pi$
$311$	−2640.09	−0.481370	−0.240685	+	0.970603i	$0.577372\pi$
$312$	2438.26	0.442433
$313$	−1938.00	−0.349976	−0.174988	−	0.984571i	$-0.555989\pi$
$313$	−1938.00	−0.349976	−0.174988	+	0.984571i	$0.555989\pi$
$314$	1390.94	0.249985
$315$	−6200.12	−1.10901
$316$	3082.21	0.548696
$317$	5546.11	0.982651	0.491326	−	0.870976i	$-0.336512\pi$
$317$	5546.11	0.982651	0.491326	+	0.870976i	$0.336512\pi$
$318$	−3376.66	−0.595452
$319$	0	0
$320$	−4125.57	−0.720707
$321$	786.981	0.136838
$322$	1124.17	0.194558
$323$	474.534	0.0817454
$324$	−12551.9	−2.15224
$325$	−1791.53	−0.305774
$326$	5350.04	0.908931
$327$	16879.1	2.85449
$328$	9730.04	1.63796
$329$	507.701	0.0850773
$330$	−2317.94	−0.386661
$331$	−185.492	−0.0308023	−0.0154012	−	0.999881i	$-0.504903\pi$
$331$	−185.492	−0.0308023	−0.0154012	+	0.999881i	$0.504903\pi$
$332$	1888.95	0.312257
$333$	−27926.1	−4.59562
$334$	2743.76	0.449496
$335$	6813.41	1.11121
$336$	−373.026	−0.0605662
$337$	8508.32	1.37530	0.687652	−	0.726040i	$-0.258641\pi$
$337$	8508.32	1.37530	0.687652	+	0.726040i	$0.258641\pi$
$338$	3373.45	0.542874
$339$	10377.0	1.66254
$340$	−6150.43	−0.981041
$341$	3.58584	0.000569454 0
$342$	798.135	0.126194
$343$	−3438.67	−0.541313
$344$	−4006.20	−0.627906
$345$	−21944.0	−3.42442
$346$	2593.63	0.402990
$347$	7853.53	1.21498	0.607492	−	0.794326i	$-0.292176\pi$
$347$	7853.53	1.21498	0.607492	+	0.794326i	$0.292176\pi$
$348$	0	0
$349$	6328.33	0.970624	0.485312	−	0.874341i	$-0.338706\pi$
$349$	6328.33	0.970624	0.485312	+	0.874341i	$0.338706\pi$
$350$	−1344.69	−0.205362
$351$	4887.51	0.743237
$352$	−1589.35	−0.240661
$353$	−7973.45	−1.20222	−0.601111	−	0.799166i	$-0.705275\pi$
$353$	−7973.45	−1.20222	−0.601111	+	0.799166i	$0.705275\pi$
$354$	740.684	0.111206
$355$	9385.59	1.40320
$356$	−1483.79	−0.220901
$357$	3531.40	0.523533
$358$	2926.98	0.432111
$359$	10059.4	1.47887	0.739435	−	0.673228i	$-0.235093\pi$
$359$	10059.4	1.47887	0.739435	+	0.673228i	$0.235093\pi$
$360$	−25842.6	−3.78340
$361$	−6810.97	−0.992998
$362$	−3838.39	−0.557297
$363$	12427.6	1.79692
$364$	316.170	0.0455270
$365$	−6425.40	−0.921426
$366$	−6891.27	−0.984187
$367$	3032.67	0.431346	0.215673	−	0.976466i	$-0.430805\pi$
$367$	3032.67	0.431346	0.215673	+	0.976466i	$0.430805\pi$
$368$	−955.061	−0.135288
$369$	31578.6	4.45506
$370$	−10850.0	−1.52450
$371$	−1093.83	−0.153070
$372$	22.1221	0.00308328
$373$	−11109.4	−1.54215	−0.771077	−	0.636742i	$-0.780282\pi$
$373$	−11109.4	−1.54215	−0.771077	+	0.636742i	$0.780282\pi$
$374$	955.045	0.132043
$375$	5474.89	0.753925
$376$	2116.13	0.290243
$377$	0	0
$378$	3668.47	0.499169
$379$	−4510.20	−0.611275	−0.305638	−	0.952148i	$-0.598870\pi$
$379$	−4510.20	−0.611275	−0.305638	+	0.952148i	$0.598870\pi$
$380$	−622.475	−0.0840323
$381$	3625.20	0.487466
$382$	−3562.66	−0.477177
$383$	7810.96	1.04209	0.521047	−	0.853528i	$-0.325542\pi$
$383$	7810.96	1.04209	0.521047	+	0.853528i	$0.325542\pi$
$384$	−10737.6	−1.42696
$385$	−750.869	−0.0993970
$386$	−5071.92	−0.668792
$387$	−13002.0	−1.70783
$388$	3619.98	0.473652
$389$	−9221.90	−1.20198	−0.600989	−	0.799258i	$-0.705226\pi$
$389$	−9221.90	−1.20198	−0.600989	+	0.799258i	$0.705226\pi$
$390$	3074.53	0.399191
$391$	9041.46	1.16943
$392$	−6869.88	−0.885157
$393$	22831.1	2.93047
$394$	−1506.64	−0.192648
$395$	9709.17	1.23676
$396$	−3224.47	−0.409181
$397$	−10034.3	−1.26854	−0.634268	−	0.773113i	$-0.718699\pi$
$397$	−10034.3	−1.26854	−0.634268	+	0.773113i	$0.718699\pi$
$398$	−7418.91	−0.934363
$399$	357.407	0.0448439
$400$	1142.41	0.142801
$401$	−8193.64	−1.02038	−0.510188	−	0.860063i	$-0.670424\pi$
$401$	−8193.64	−1.02038	−0.510188	+	0.860063i	$0.670424\pi$
$402$	−6527.09	−0.809805
$403$	−4.75628	−0.000587908 0
$404$	3032.06	0.373392
$405$	−39539.3	−4.85117
$406$	0	0
$407$	−3382.01	−0.411892
$408$	14719.1	1.78604
$409$	−3921.69	−0.474120	−0.237060	−	0.971495i	$-0.576184\pi$
$409$	−3921.69	−0.474120	−0.237060	+	0.971495i	$0.576184\pi$
$410$	12269.1	1.47787
$411$	18768.4	2.25250
$412$	−1704.51	−0.203824
$413$	239.936	0.0285872
$414$	15207.1	1.80529
$415$	5950.31	0.703830
$416$	2108.13	0.248460
$417$	−12937.6	−1.51933
$418$	96.6586	0.0113103
$419$	3026.66	0.352892	0.176446	−	0.984310i	$-0.443540\pi$
$419$	3026.66	0.352892	0.176446	+	0.984310i	$0.443540\pi$
$420$	−4632.35	−0.538179
$421$	−4823.43	−0.558384	−0.279192	−	0.960235i	$-0.590067\pi$
$421$	−4823.43	−0.558384	−0.279192	+	0.960235i	$0.590067\pi$
$422$	−519.739	−0.0599538
$423$	6867.87	0.789426
$424$	−4559.17	−0.522200
$425$	−10815.0	−1.23437
$426$	−8991.19	−1.02259
$427$	−2232.35	−0.253000
$428$	425.345	0.0480370
$429$	958.346	0.107854
$430$	−5051.63	−0.566537
$431$	−2029.51	−0.226816	−0.113408	−	0.993548i	$-0.536177\pi$
$431$	−2029.51	−0.226816	−0.113408	+	0.993548i	$0.536177\pi$
$432$	−3116.61	−0.347102
$433$	535.808	0.0594672	0.0297336	−	0.999558i	$-0.490534\pi$
$433$	535.808	0.0594672	0.0297336	+	0.999558i	$0.490534\pi$
$434$	−3.56997	−0.000394848 0
$435$	0	0
$436$	9122.77	1.00207
$437$	915.072	0.100169
$438$	6155.38	0.671497
$439$	8791.74	0.955824	0.477912	−	0.878408i	$-0.341394\pi$
$439$	8791.74	0.955824	0.477912	+	0.878408i	$0.341394\pi$
$440$	−3129.68	−0.339095
$441$	−22296.1	−2.40752
$442$	−1266.78	−0.136322
$443$	−13916.9	−1.49258	−0.746291	−	0.665619i	$-0.768167\pi$
$443$	−13916.9	−1.49258	−0.746291	+	0.665619i	$0.768167\pi$
$444$	−20864.7	−2.23016
$445$	−4674.05	−0.497912
$446$	−2783.62	−0.295535
$447$	−17615.7	−1.86397
$448$	1280.27	0.135016
$449$	−2129.54	−0.223829	−0.111915	−	0.993718i	$-0.535698\pi$
$449$	−2129.54	−0.223829	−0.111915	+	0.993718i	$0.535698\pi$
$450$	−18190.2	−1.90554
$451$	3824.35	0.399294
$452$	5608.53	0.583635
$453$	16123.6	1.67230
$454$	−4964.14	−0.513169
$455$	995.958	0.102618
$456$	1489.70	0.152986
$457$	1932.25	0.197783	0.0988915	−	0.995098i	$-0.468470\pi$
$457$	1932.25	0.197783	0.0988915	+	0.995098i	$0.468470\pi$
$458$	−4275.77	−0.436231
$459$	29504.6	3.00035
$460$	−11860.2	−1.20214
$461$	16518.3	1.66884	0.834418	−	0.551132i	$-0.185804\pi$
$461$	16518.3	1.66884	0.834418	+	0.551132i	$0.185804\pi$
$462$	719.316	0.0724364
$463$	11535.2	1.15785	0.578926	−	0.815380i	$-0.303472\pi$
$463$	11535.2	1.15785	0.578926	+	0.815380i	$0.303472\pi$
$464$	0	0
$465$	69.6863	0.00694973
$466$	4532.10	0.450526
$467$	−15667.8	−1.55250	−0.776250	−	0.630425i	$-0.782880\pi$
$467$	−15667.8	−1.55250	−0.776250	+	0.630425i	$0.782880\pi$
$468$	4276.96	0.422441
$469$	−2114.38	−0.208172
$470$	2668.34	0.261876
$471$	8425.75	0.824285
$472$	1000.07	0.0975255
$473$	−1574.62	−0.153068
$474$	−9301.17	−0.901301
$475$	−1094.57	−0.105731
$476$	1908.64	0.183786
$477$	−14796.7	−1.42032
$478$	−7702.65	−0.737052
$479$	−4672.28	−0.445682	−0.222841	−	0.974855i	$-0.571533\pi$
$479$	−4672.28	−0.445682	−0.222841	+	0.974855i	$0.571533\pi$
$480$	−30887.1	−2.93707
$481$	4485.92	0.425239
$482$	7672.41	0.725038
$483$	6809.80	0.641525
$484$	6716.85	0.630809
$485$	11403.2	1.06761
$486$	18902.8	1.76429
$487$	−8465.76	−0.787722	−0.393861	−	0.919170i	$-0.628861\pi$
$487$	−8465.76	−0.787722	−0.393861	+	0.919170i	$0.628861\pi$
$488$	−9304.60	−0.863113
$489$	32408.4	2.99706
$490$	−8662.60	−0.798646
$491$	12302.6	1.13077	0.565385	−	0.824827i	$-0.308728\pi$
$491$	12302.6	1.13077	0.565385	+	0.824827i	$0.308728\pi$
$492$	23593.6	2.16195
$493$	0	0
$494$	−128.209	−0.0116769
$495$	−10157.3	−0.922297
$496$	3.03293	0.000274562 0
$497$	−2912.59	−0.262873
$498$	−5700.26	−0.512922
$499$	−2781.31	−0.249516	−0.124758	−	0.992187i	$-0.539815\pi$
$499$	−2781.31	−0.249516	−0.124758	+	0.992187i	$0.539815\pi$
$500$	2959.05	0.264665
$501$	16620.6	1.48214
$502$	7256.40	0.645158
$503$	13673.9	1.21211	0.606054	−	0.795424i	$-0.292752\pi$
$503$	13673.9	1.21211	0.606054	+	0.795424i	$0.292752\pi$
$504$	8019.62	0.708774
$505$	9551.18	0.841628
$506$	1841.67	0.161803
$507$	20435.0	1.79004
$508$	1959.33	0.171125
$509$	7431.68	0.647158	0.323579	−	0.946201i	$-0.395114\pi$
$509$	7431.68	0.647158	0.323579	+	0.946201i	$0.395114\pi$
$510$	18560.1	1.61148
$511$	1993.97	0.172618
$512$	−2603.24	−0.224704
$513$	2986.12	0.256999
$514$	−11786.7	−1.01145
$515$	−5369.33	−0.459419
$516$	−9714.31	−0.828776
$517$	831.737	0.0707538
$518$	3367.04	0.285597
$519$	15711.2	1.32879
$520$	4151.23	0.350083
$521$	7797.86	0.655721	0.327860	−	0.944726i	$-0.393672\pi$
$521$	7797.86	0.655721	0.327860	+	0.944726i	$0.393672\pi$
$522$	0	0
$523$	−16670.9	−1.39382	−0.696908	−	0.717160i	$-0.745442\pi$
$523$	−16670.9	−1.39382	−0.696908	+	0.717160i	$0.745442\pi$
$524$	12339.7	1.02874
$525$	−8145.61	−0.677149
$526$	−7811.79	−0.647548
$527$	−28.7124	−0.00237331
$528$	−611.107	−0.0503693
$529$	5268.18	0.432990
$530$	−5748.90	−0.471163
$531$	3245.71	0.265258
$532$	193.170	0.0157425
$533$	−5072.64	−0.412233
$534$	4477.63	0.362858
$535$	1339.87	0.108276
$536$	−8812.88	−0.710184
$537$	17730.5	1.42482
$538$	−76.7297	−0.00614880
$539$	−2700.18	−0.215779
$540$	−38703.0	−3.08428
$541$	−13400.1	−1.06491	−0.532455	−	0.846458i	$-0.678730\pi$
$541$	−13400.1	−1.06491	−0.532455	+	0.846458i	$0.678730\pi$
$542$	12594.2	0.998092
$543$	−23251.5	−1.83760
$544$	12726.2	1.00300
$545$	28737.4	2.25867
$546$	−954.105	−0.0747838
$547$	−17339.4	−1.35535	−0.677677	−	0.735360i	$-0.737013\pi$
$547$	−17339.4	−1.35535	−0.677677	+	0.735360i	$0.737013\pi$
$548$	10143.9	0.790739
$549$	−30197.9	−2.34757
$550$	−2202.93	−0.170788
$551$	0	0
$552$	28383.7	2.18857
$553$	−3013.01	−0.231693
$554$	12321.6	0.944936
$555$	−65725.1	−5.02680
$556$	−6992.50	−0.533360
$557$	−7790.90	−0.592659	−0.296330	−	0.955086i	$-0.595763\pi$
$557$	−7790.90	−0.592659	−0.296330	+	0.955086i	$0.595763\pi$
$558$	−48.2924	−0.00366376
$559$	2088.58	0.158028
$560$	−635.091	−0.0479241
$561$	5785.28	0.435392
$562$	3416.66	0.256447
$563$	21514.2	1.61051	0.805253	−	0.592932i	$-0.202030\pi$
$563$	21514.2	1.61051	0.805253	+	0.592932i	$0.202030\pi$
$564$	5131.24	0.383093
$565$	17667.3	1.31552
$566$	13375.5	0.993311
$567$	12270.1	0.908808
$568$	−12139.9	−0.896794
$569$	−16993.5	−1.25203	−0.626015	−	0.779811i	$-0.715315\pi$
$569$	−16993.5	−1.25203	−0.626015	+	0.779811i	$0.715315\pi$
$570$	1878.44	0.138034
$571$	16791.4	1.23064	0.615321	−	0.788277i	$-0.289027\pi$
$571$	16791.4	1.23064	0.615321	+	0.788277i	$0.289027\pi$
$572$	517.963	0.0378621
$573$	−21581.2	−1.57342
$574$	−3807.42	−0.276862
$575$	−20855.3	−1.51256
$576$	17318.7	1.25280
$577$	−8108.87	−0.585055	−0.292527	−	0.956257i	$-0.594496\pi$
$577$	−8108.87	−0.585055	−0.292527	+	0.956257i	$0.594496\pi$
$578$	365.860	0.0263284
$579$	−30723.7	−2.20524
$580$	0	0
$581$	−1846.54	−0.131854
$582$	−10924.0	−0.778032
$583$	−1791.96	−0.127299
$584$	8311.00	0.588890
$585$	13472.7	0.952185
$586$	5739.25	0.404584
$587$	16076.3	1.13039	0.565197	−	0.824956i	$-0.308800\pi$
$587$	16076.3	1.13039	0.565197	+	0.824956i	$0.308800\pi$
$588$	−16658.2	−1.16832
$589$	−2.90594	−0.000203289 0
$590$	1261.04	0.0879938
$591$	−9126.63	−0.635227
$592$	−2860.53	−0.198593
$593$	−4341.83	−0.300671	−0.150335	−	0.988635i	$-0.548035\pi$
$593$	−4341.83	−0.300671	−0.150335	+	0.988635i	$0.548035\pi$
$594$	6009.85	0.415130
$595$	6012.34	0.414255
$596$	−9520.88	−0.654346
$597$	−44940.8	−3.08091
$598$	−2442.80	−0.167046
$599$	10540.1	0.718960	0.359480	−	0.933153i	$-0.382954\pi$
$599$	10540.1	0.718960	0.359480	+	0.933153i	$0.382954\pi$
$600$	−33951.5	−2.31011
$601$	−16485.6	−1.11890	−0.559451	−	0.828863i	$-0.688988\pi$
$601$	−16485.6	−1.11890	−0.559451	+	0.828863i	$0.688988\pi$
$602$	1567.65	0.106134
$603$	−28602.0	−1.93162
$604$	8714.43	0.587061
$605$	21158.6	1.42185
$606$	−9149.82	−0.613343
$607$	−13326.5	−0.891112	−0.445556	−	0.895254i	$-0.646994\pi$
$607$	−13326.5	−0.891112	−0.445556	+	0.895254i	$0.646994\pi$
$608$	1288.00	0.0859132
$609$	0	0
$610$	−11732.7	−0.778756
$611$	−1103.22	−0.0730467
$612$	25818.9	1.70534
$613$	20459.4	1.34804	0.674019	−	0.738714i	$-0.264567\pi$
$613$	20459.4	1.34804	0.674019	+	0.738714i	$0.264567\pi$
$614$	−14231.0	−0.935369
$615$	74321.4	4.87305
$616$	971.221	0.0635253
$617$	3108.50	0.202826	0.101413	−	0.994844i	$-0.467664\pi$
$617$	3108.50	0.202826	0.101413	+	0.994844i	$0.467664\pi$
$618$	5143.70	0.334806
$619$	−10426.3	−0.677012	−0.338506	−	0.940964i	$-0.609922\pi$
$619$	−10426.3	−0.677012	−0.338506	+	0.940964i	$0.609922\pi$
$620$	37.6638	0.00243970
$621$	56895.5	3.67655
$622$	4305.97	0.277579
$623$	1450.48	0.0932780
$624$	810.576	0.0520016
$625$	−10421.8	−0.666992
$626$	3160.87	0.201811
$627$	585.519	0.0372941
$628$	4553.92	0.289365
$629$	27080.3	1.71663
$630$	10112.4	0.639502
$631$	18018.2	1.13675	0.568377	−	0.822768i	$-0.307572\pi$
$631$	18018.2	1.13675	0.568377	+	0.822768i	$0.307572\pi$
$632$	−12558.4	−0.790424
$633$	−3148.37	−0.197688
$634$	−9045.67	−0.566639
$635$	6172.04	0.385716
$636$	−11055.2	−0.689255
$637$	3581.53	0.222772
$638$	0	0
$639$	−39399.8	−2.43917
$640$	−18281.2	−1.12911
$641$	−15178.7	−0.935295	−0.467648	−	0.883915i	$-0.654898\pi$
$641$	−15178.7	−0.935295	−0.467648	+	0.883915i	$0.654898\pi$
$642$	−1283.56	−0.0789067
$643$	23957.1	1.46932	0.734662	−	0.678433i	$-0.237341\pi$
$643$	23957.1	1.46932	0.734662	+	0.678433i	$0.237341\pi$
$644$	3680.54	0.225207
$645$	−30600.7	−1.86807
$646$	−773.962	−0.0471379
$647$	−12835.2	−0.779915	−0.389958	−	0.920833i	$-0.627510\pi$
$647$	−12835.2	−0.779915	−0.389958	+	0.920833i	$0.627510\pi$
$648$	51142.5	3.10041
$649$	393.074	0.0237743
$650$	2921.98	0.176322
$651$	−21.6255	−0.00130195
$652$	17516.0	1.05212
$653$	4355.97	0.261045	0.130523	−	0.991445i	$-0.458335\pi$
$653$	4355.97	0.261045	0.130523	+	0.991445i	$0.458335\pi$
$654$	−27529.7	−1.64602
$655$	38870.8	2.31879
$656$	3234.66	0.192519
$657$	26973.2	1.60171
$658$	−828.056	−0.0490593
$659$	32704.1	1.93319	0.966594	−	0.256313i	$-0.0825079\pi$
$659$	32704.1	1.93319	0.966594	+	0.256313i	$0.0825079\pi$
$660$	−7588.91	−0.447572
$661$	−2839.66	−0.167095	−0.0835475	−	0.996504i	$-0.526625\pi$
$661$	−2839.66	−0.167095	−0.0835475	+	0.996504i	$0.526625\pi$
$662$	302.536	0.0177620
$663$	−7673.64	−0.449501
$664$	−7696.50	−0.449823
$665$	608.499	0.0354836
$666$	45547.3	2.65003
$667$	0	0
$668$	8983.04	0.520306
$669$	−16862.1	−0.974478
$670$	−11112.6	−0.640773
$671$	−3657.13	−0.210405
$672$	9585.06	0.550226
$673$	−18569.9	−1.06362	−0.531810	−	0.846864i	$-0.678488\pi$
$673$	−18569.9	−1.06362	−0.531810	+	0.846864i	$0.678488\pi$
$674$	−13877.0	−0.793060
$675$	−68056.1	−3.88071
$676$	11044.6	0.628393
$677$	−18106.5	−1.02790	−0.513952	−	0.857819i	$-0.671819\pi$
$677$	−18106.5	−1.02790	−0.513952	+	0.857819i	$0.671819\pi$
$678$	−16924.8	−0.958693
$679$	−3538.71	−0.200005
$680$	25059.9	1.41324
$681$	−30070.8	−1.69209
$682$	−5.84848	−0.000328372 0
$683$	−5510.97	−0.308743	−0.154371	−	0.988013i	$-0.549335\pi$
$683$	−5510.97	−0.308743	−0.154371	+	0.988013i	$0.549335\pi$
$684$	2613.09	0.146073
$685$	31953.9	1.78233
$686$	5608.44	0.312145
$687$	−25900.9	−1.43840
$688$	−1331.82	−0.0738014
$689$	2376.87	0.131425
$690$	35790.6	1.97467
$691$	24652.0	1.35717	0.678587	−	0.734520i	$-0.262592\pi$
$691$	24652.0	1.35717	0.678587	+	0.734520i	$0.262592\pi$
$692$	8491.53	0.466473
$693$	3152.08	0.172781
$694$	−12809.1	−0.700613
$695$	−22026.9	−1.20220
$696$	0	0
$697$	−30622.2	−1.66413
$698$	−10321.5	−0.559704
$699$	27453.6	1.48554
$700$	−4402.51	−0.237713
$701$	−3399.10	−0.183142	−0.0915708	−	0.995799i	$-0.529189\pi$
$701$	−3399.10	−0.183142	−0.0915708	+	0.995799i	$0.529189\pi$
$702$	−7971.50	−0.428582
$703$	2740.75	0.147041
$704$	2097.39	0.112285
$705$	16163.8	0.863493
$706$	13004.7	0.693253
$707$	−2963.98	−0.157669
$708$	2424.99	0.128724
$709$	26329.1	1.39465	0.697327	−	0.716753i	$-0.254373\pi$
$709$	26329.1	1.39465	0.697327	+	0.716753i	$0.254373\pi$
$710$	−15307.8	−0.809145
$711$	−40758.2	−2.14986
$712$	6045.70	0.318219
$713$	−55.3678	−0.00290819
$714$	−5759.68	−0.301892
$715$	1631.62	0.0853415
$716$	9582.91	0.500182
$717$	−46659.6	−2.43031
$718$	−16406.8	−0.852781
$719$	22345.6	1.15904	0.579521	−	0.814957i	$-0.303240\pi$
$719$	22345.6	1.15904	0.579521	+	0.814957i	$0.303240\pi$
$720$	−8591.13	−0.444684
$721$	1666.24	0.0860668
$722$	11108.7	0.572606
$723$	46476.4	2.39070
$724$	−12566.9	−0.645089
$725$	0	0
$726$	−20269.4	−1.03618
$727$	29862.8	1.52345	0.761727	−	0.647898i	$-0.224352\pi$
$727$	29862.8	1.52345	0.761727	+	0.647898i	$0.224352\pi$
$728$	−1288.23	−0.0655839
$729$	51039.2	2.59306
$730$	10479.8	0.531334
$731$	12608.2	0.637938
$732$	−22562.0	−1.13923
$733$	987.919	0.0497812	0.0248906	−	0.999690i	$-0.492076\pi$
$733$	987.919	0.0497812	0.0248906	+	0.999690i	$0.492076\pi$
$734$	−4946.27	−0.248733
$735$	−52474.6	−2.63341
$736$	24540.7	1.22905
$737$	−3463.86	−0.173125
$738$	−51504.5	−2.56898
$739$	35870.9	1.78556	0.892782	−	0.450489i	$-0.148750\pi$
$739$	35870.9	1.78556	0.892782	+	0.450489i	$0.148750\pi$
$740$	−35522.9	−1.76466
$741$	−776.636	−0.0385026
$742$	1784.03	0.0882667
$743$	−8322.20	−0.410918	−0.205459	−	0.978666i	$-0.565869\pi$
$743$	−8322.20	−0.410918	−0.205459	+	0.978666i	$0.565869\pi$
$744$	−90.1365	−0.00444162
$745$	−29991.4	−1.47490
$746$	18119.4	0.889273
$747$	−24978.8	−1.22346
$748$	3126.81	0.152844
$749$	−415.795	−0.0202841
$750$	−8929.50	−0.434746
$751$	−20781.2	−1.00974	−0.504872	−	0.863194i	$-0.668460\pi$
$751$	−20781.2	−1.00974	−0.504872	+	0.863194i	$0.668460\pi$
$752$	703.489	0.0341139
$753$	43956.4	2.12730
$754$	0	0
$755$	27451.0	1.32324
$756$	12010.6	0.577804
$757$	16424.7	0.788596	0.394298	−	0.918983i	$-0.370988\pi$
$757$	16424.7	0.788596	0.394298	+	0.918983i	$0.370988\pi$
$758$	7356.10	0.352488
$759$	11156.1	0.533519
$760$	2536.27	0.121053
$761$	416.312	0.0198309	0.00991543	−	0.999951i	$-0.496844\pi$
$761$	416.312	0.0198309	0.00991543	+	0.999951i	$0.496844\pi$
$762$	−5912.67	−0.281094
$763$	−8917.95	−0.423134
$764$	−11664.1	−0.552347
$765$	81331.3	3.84384
$766$	−12739.6	−0.600916
$767$	−521.376	−0.0245447
$768$	36898.5	1.73367
$769$	37129.2	1.74111	0.870554	−	0.492073i	$-0.163761\pi$
$769$	37129.2	1.74111	0.870554	+	0.492073i	$0.163761\pi$
$770$	1224.66	0.0573166
$771$	−71398.9	−3.33511
$772$	−16605.4	−0.774148
$773$	2182.83	0.101567	0.0507833	−	0.998710i	$-0.483828\pi$
$773$	2182.83	0.101567	0.0507833	+	0.998710i	$0.483828\pi$
$774$	21206.2	0.984809
$775$	66.2287	0.00306969
$776$	−14749.6	−0.682319
$777$	20396.2	0.941711
$778$	15040.9	0.693112
$779$	−3099.22	−0.142543
$780$	10066.0	0.462076
$781$	−4771.53	−0.218616
$782$	−14746.6	−0.674343
$783$	0	0
$784$	−2283.83	−0.104037
$785$	14345.2	0.652231
$786$	−37237.4	−1.68984
$787$	7068.15	0.320143	0.160071	−	0.987105i	$-0.448828\pi$
$787$	7068.15	0.320143	0.160071	+	0.987105i	$0.448828\pi$
$788$	−4932.73	−0.222996
$789$	−47320.7	−2.13519
$790$	−15835.6	−0.713171
$791$	−5482.61	−0.246446
$792$	13138.1	0.589446
$793$	4850.84	0.217224
$794$	16365.9	0.731493
$795$	−34824.5	−1.55358
$796$	−24289.5	−1.08155
$797$	−15086.0	−0.670482	−0.335241	−	0.942132i	$-0.608818\pi$
$797$	−15086.0	−0.670482	−0.335241	+	0.942132i	$0.608818\pi$
$798$	−582.928	−0.0258589
$799$	−6659.86	−0.294880
$800$	−29354.6	−1.29730
$801$	19621.2	0.865519
$802$	13363.8	0.588393
$803$	3266.60	0.143557
$804$	−21369.6	−0.937375
$805$	11593.9	0.507619
$806$	7.75746	0.000339013 0
$807$	−464.798	−0.0202747
$808$	−12354.1	−0.537890
$809$	21798.3	0.947326	0.473663	−	0.880706i	$-0.342931\pi$
$809$	21798.3	0.947326	0.473663	+	0.880706i	$0.342931\pi$
$810$	64488.3	2.79739
$811$	5569.61	0.241153	0.120577	−	0.992704i	$-0.461526\pi$
$811$	5569.61	0.241153	0.120577	+	0.992704i	$0.461526\pi$
$812$	0	0
$813$	76290.4	3.29105
$814$	5516.03	0.237515
$815$	55176.6	2.37148
$816$	4893.24	0.209924
$817$	1276.06	0.0546434
$818$	6396.25	0.273398
$819$	−4180.93	−0.178381
$820$	40169.0	1.71069
$821$	21281.8	0.904676	0.452338	−	0.891847i	$-0.350590\pi$
$821$	21281.8	0.904676	0.452338	+	0.891847i	$0.350590\pi$
$822$	−30611.1	−1.29889
$823$	−16799.0	−0.711514	−0.355757	−	0.934578i	$-0.615777\pi$
$823$	−16799.0	−0.711514	−0.355757	+	0.934578i	$0.615777\pi$
$824$	6945.02	0.293618
$825$	−13344.5	−0.563145
$826$	−391.335	−0.0164846
$827$	−7360.62	−0.309497	−0.154748	−	0.987954i	$-0.549457\pi$
$827$	−7360.62	−0.309497	−0.154748	+	0.987954i	$0.549457\pi$
$828$	49788.1	2.08968
$829$	11634.6	0.487438	0.243719	−	0.969846i	$-0.421633\pi$
$829$	11634.6	0.487438	0.243719	+	0.969846i	$0.421633\pi$
$830$	−9704.92	−0.405859
$831$	74639.4	3.11578
$832$	−2782.00	−0.115924
$833$	21620.8	0.899299
$834$	21101.2	0.876110
$835$	28297.2	1.17277
$836$	316.459	0.0130921
$837$	−180.680	−0.00746141
$838$	−4936.46	−0.203493
$839$	32151.0	1.32297	0.661486	−	0.749957i	$-0.269926\pi$
$839$	32151.0	1.32297	0.661486	+	0.749957i	$0.269926\pi$
$840$	18874.5	0.775275
$841$	0	0
$842$	7866.99	0.321988
$843$	20696.7	0.845592
$844$	−1701.62	−0.0693984
$845$	34791.4	1.41640
$846$	−11201.4	−0.455217
$847$	−6566.05	−0.266366
$848$	−1515.66	−0.0613771
$849$	81023.4	3.27528
$850$	17639.2	0.711789
$851$	52220.6	2.10352
$852$	−29437.1	−1.18368
$853$	−35044.9	−1.40670	−0.703349	−	0.710844i	$-0.748313\pi$
$853$	−35044.9	−1.40670	−0.703349	+	0.710844i	$0.748313\pi$
$854$	3640.95	0.145891
$855$	8231.41	0.329249
$856$	−1733.07	−0.0691997
$857$	7143.05	0.284716	0.142358	−	0.989815i	$-0.454532\pi$
$857$	7143.05	0.284716	0.142358	+	0.989815i	$0.454532\pi$
$858$	−1563.06	−0.0621933
$859$	−33960.3	−1.34891	−0.674454	−	0.738317i	$-0.735621\pi$
$859$	−33960.3	−1.34891	−0.674454	+	0.738317i	$0.735621\pi$
$860$	−16539.0	−0.655785
$861$	−23063.9	−0.912909
$862$	3310.11	0.130792
$863$	45823.4	1.80747	0.903736	−	0.428090i	$-0.140814\pi$
$863$	45823.4	1.80747	0.903736	+	0.428090i	$0.140814\pi$
$864$	80082.7	3.15332
$865$	26748.9	1.05143
$866$	−873.900	−0.0342914
$867$	2216.24	0.0868136
$868$	−11.6881	−0.000457049 0
$869$	−4936.04	−0.192685
$870$	0	0
$871$	4594.49	0.178735
$872$	−37170.7	−1.44353
$873$	−47869.5	−1.85583
$874$	−1492.48	−0.0577617
$875$	−2892.61	−0.111758
$876$	20152.7	0.777279
$877$	−21281.3	−0.819406	−0.409703	−	0.912219i	$-0.634368\pi$
$877$	−21281.3	−0.819406	−0.409703	+	0.912219i	$0.634368\pi$
$878$	−14339.3	−0.551169
$879$	34766.1	1.33405
$880$	−1040.43	−0.0398557
$881$	−1359.32	−0.0519826	−0.0259913	−	0.999662i	$-0.508274\pi$
$881$	−1359.32	−0.0519826	−0.0259913	+	0.999662i	$0.508274\pi$
$882$	36364.7	1.38828
$883$	−47928.2	−1.82663	−0.913313	−	0.407257i	$-0.866485\pi$
$883$	−47928.2	−1.82663	−0.913313	+	0.407257i	$0.866485\pi$
$884$	−4147.42	−0.157797
$885$	7638.90	0.290146
$886$	22698.5	0.860688
$887$	−3100.58	−0.117370	−0.0586851	−	0.998277i	$-0.518691\pi$
$887$	−3100.58	−0.117370	−0.0586851	+	0.998277i	$0.518691\pi$
$888$	85012.9	3.21266
$889$	−1915.34	−0.0722593
$890$	7623.34	0.287118
$891$	20101.3	0.755803
$892$	−9113.56	−0.342090
$893$	−674.033	−0.0252583
$894$	28731.1	1.07485
$895$	30186.8	1.12741
$896$	5673.14	0.211525
$897$	−14797.5	−0.550808
$898$	3473.27	0.129070
$899$	0	0
$900$	−59554.5	−2.20572
$901$	14348.5	0.530543
$902$	−6237.48	−0.230250
$903$	9496.21	0.349960
$904$	−22851.9	−0.840756
$905$	−39586.5	−1.45403
$906$	−26297.5	−0.964321
$907$	−23428.1	−0.857680	−0.428840	−	0.903380i	$-0.641078\pi$
$907$	−23428.1	−0.857680	−0.428840	+	0.903380i	$0.641078\pi$
$908$	−16252.6	−0.594009
$909$	−40094.9	−1.46300
$910$	−1624.40	−0.0591740
$911$	12868.3	0.467999	0.234000	−	0.972237i	$-0.424819\pi$
$911$	12868.3	0.467999	0.234000	+	0.972237i	$0.424819\pi$
$912$	495.237	0.0179813
$913$	−3025.07	−0.109655
$914$	−3151.49	−0.114050
$915$	−71071.7	−2.56782
$916$	−13998.8	−0.504950
$917$	−12062.6	−0.434398
$918$	−48121.9	−1.73013
$919$	3938.41	0.141367	0.0706834	−	0.997499i	$-0.477482\pi$
$919$	3938.41	0.141367	0.0706834	+	0.997499i	$0.477482\pi$
$920$	48324.4	1.73175
$921$	−86205.7	−3.08423
$922$	−26941.2	−0.962323
$923$	6328.99	0.225700
$924$	2355.03	0.0838473
$925$	−62464.1	−2.22033
$926$	−18813.8	−0.667667
$927$	22539.9	0.798607
$928$	0	0
$929$	−15856.7	−0.560001	−0.280001	−	0.960000i	$-0.590335\pi$
$929$	−15856.7	−0.560001	−0.280001	+	0.960000i	$0.590335\pi$
$930$	−113.658	−0.00400751
$931$	2188.20	0.0770306
$932$	14838.0	0.521498
$933$	26083.9	0.915271
$934$	25554.0	0.895238
$935$	9849.67	0.344512
$936$	−17426.4	−0.608548
$937$	−18559.7	−0.647085	−0.323542	−	0.946214i	$-0.604874\pi$
$937$	−18559.7	−0.647085	−0.323542	+	0.946214i	$0.604874\pi$
$938$	3448.54	0.120041
$939$	19147.3	0.665441
$940$	8736.14	0.303129
$941$	24125.6	0.835785	0.417892	−	0.908497i	$-0.362769\pi$
$941$	24125.6	0.835785	0.417892	+	0.908497i	$0.362769\pi$
$942$	−13742.4	−0.475318
$943$	−59050.6	−2.03919
$944$	332.465	0.0114627
$945$	37834.1	1.30237
$946$	2568.19	0.0882655
$947$	6883.31	0.236196	0.118098	−	0.993002i	$-0.462320\pi$
$947$	6883.31	0.236196	0.118098	+	0.993002i	$0.462320\pi$
$948$	−30452.0	−1.04328
$949$	−4332.84	−0.148209
$950$	1785.24	0.0609693
$951$	−54795.0	−1.86840
$952$	−7776.73	−0.264753
$953$	37573.8	1.27716	0.638580	−	0.769555i	$-0.279522\pi$
$953$	37573.8	1.27716	0.638580	+	0.769555i	$0.279522\pi$
$954$	24133.3	0.819020
$955$	−36742.8	−1.24499
$956$	−25218.4	−0.853161
$957$	0	0
$958$	7620.45	0.257000
$959$	−9916.13	−0.333898
$960$	40760.2	1.37034
$961$	−29790.8	−0.999994
$962$	−7316.50	−0.245211
$963$	−5624.63	−0.188215
$964$	25119.4	0.839255
$965$	−52308.2	−1.74493
$966$	−11106.7	−0.369931
$967$	1584.28	0.0526856	0.0263428	−	0.999653i	$-0.491614\pi$
$967$	1584.28	0.0526856	0.0263428	+	0.999653i	$0.491614\pi$
$968$	−27367.8	−0.908712
$969$	−4688.35	−0.155430
$970$	−18598.5	−0.615632
$971$	−36569.1	−1.20861	−0.604304	−	0.796754i	$-0.706549\pi$
$971$	−36569.1	−1.20861	−0.604304	+	0.796754i	$0.706549\pi$
$972$	61887.5	2.04222
$973$	6835.50	0.225217
$974$	13807.6	0.454234
$975$	17700.2	0.581395
$976$	−3093.23	−0.101447
$977$	−33583.0	−1.09971	−0.549854	−	0.835261i	$-0.685317\pi$
$977$	−33583.0	−1.09971	−0.549854	+	0.835261i	$0.685317\pi$
$978$	−52858.0	−1.72823
$979$	2376.23	0.0775738
$980$	−28361.3	−0.924457
$981$	−120637.	−3.92623
$982$	−20065.4	−0.652051
$983$	25900.6	0.840387	0.420193	−	0.907435i	$-0.361962\pi$
$983$	25900.6	0.840387	0.420193	+	0.907435i	$0.361962\pi$
$984$	−96131.9	−3.11440
$985$	−15538.4	−0.502635
$986$	0	0
$987$	−5016.04	−0.161765
$988$	−419.754	−0.0135163
$989$	24313.2	0.781714
$990$	16566.5	0.531836
$991$	29604.3	0.948951	0.474475	−	0.880269i	$-0.342638\pi$
$991$	29604.3	0.948951	0.474475	+	0.880269i	$0.342638\pi$
$992$	−77.9324	−0.00249431
$993$	1832.65	0.0585672
$994$	4750.42	0.151584
$995$	−76513.5	−2.43783
$996$	−18662.6	−0.593723
$997$	−39267.2	−1.24735	−0.623673	−	0.781685i	$-0.714360\pi$
$997$	−39267.2	−1.24735	−0.623673	+	0.781685i	$0.714360\pi$
$998$	4536.30	0.143882
$999$	170409.	5.39691

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	841.4.a.b.1.3		5
29.28	even	2		29.4.a.b.1.3	✓	5
87.86	odd	2		261.4.a.f.1.3		5
116.115	odd	2		464.4.a.l.1.1		5
145.144	even	2		725.4.a.c.1.3		5
203.202	odd	2		1421.4.a.e.1.3		5
232.115	odd	2		1856.4.a.bb.1.5		5
232.173	even	2		1856.4.a.y.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.4.a.b.1.3	✓	5	29.28	even	2
261.4.a.f.1.3		5	87.86	odd	2
464.4.a.l.1.1		5	116.115	odd	2
725.4.a.c.1.3		5	145.144	even	2
841.4.a.b.1.3		5	1.1	even	1	trivial
1421.4.a.e.1.3		5	203.202	odd	2
1856.4.a.y.1.1		5	232.173	even	2
1856.4.a.bb.1.5		5	232.115	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:	\( N \)	\(=\)	\( 841 = 29^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	841.a (trivial)

\(f(q)\)	\(=\)	\(q-1.63099 q^{2} -9.87991 q^{3} -5.33986 q^{4} -16.8209 q^{5} +16.1141 q^{6} +5.21997 q^{7} +21.7572 q^{8} +70.6126 q^{9} +O(q^{10})\)	\(q-1.63099 q^{2} -9.87991 q^{3} -5.33986 q^{4} -16.8209 q^{5} +16.1141 q^{6} +5.21997 q^{7} +21.7572 q^{8} +70.6126 q^{9} +27.4348 q^{10} +8.55158 q^{11} +52.7573 q^{12} -11.3429 q^{13} -8.51374 q^{14} +166.189 q^{15} +7.23299 q^{16} -68.4740 q^{17} -115.169 q^{18} -6.93014 q^{19} +89.8214 q^{20} -51.5728 q^{21} -13.9476 q^{22} -132.042 q^{23} -214.959 q^{24} +157.944 q^{25} +18.5001 q^{26} -430.889 q^{27} -27.8739 q^{28} -271.054 q^{30} +0.419319 q^{31} -185.855 q^{32} -84.4888 q^{33} +111.681 q^{34} -87.8048 q^{35} -377.061 q^{36} -395.483 q^{37} +11.3030 q^{38} +112.067 q^{39} -365.977 q^{40} +447.209 q^{41} +84.1150 q^{42} -184.132 q^{43} -45.6642 q^{44} -1187.77 q^{45} +215.360 q^{46} +97.2612 q^{47} -71.4613 q^{48} -315.752 q^{49} -257.605 q^{50} +676.516 q^{51} +60.5693 q^{52} -209.547 q^{53} +702.776 q^{54} -143.845 q^{55} +113.572 q^{56} +68.4691 q^{57} +45.9651 q^{59} -887.427 q^{60} -427.655 q^{61} -0.683906 q^{62} +368.596 q^{63} +245.264 q^{64} +190.798 q^{65} +137.801 q^{66} -405.055 q^{67} +365.641 q^{68} +1304.57 q^{69} +143.209 q^{70} -557.971 q^{71} +1536.33 q^{72} +381.988 q^{73} +645.031 q^{74} -1560.47 q^{75} +37.0060 q^{76} +44.6390 q^{77} -182.780 q^{78} -577.208 q^{79} -121.666 q^{80} +2350.60 q^{81} -729.396 q^{82} -353.745 q^{83} +275.392 q^{84} +1151.80 q^{85} +300.318 q^{86} +186.059 q^{88} +277.871 q^{89} +1937.24 q^{90} -59.2095 q^{91} +705.088 q^{92} -4.14283 q^{93} -158.632 q^{94} +116.571 q^{95} +1836.23 q^{96} -677.917 q^{97} +514.989 q^{98} +603.849 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 8 q^{3} + 26 q^{4} + 10 q^{5} + 34 q^{6} + 40 q^{7} + 84 q^{8} + 33 q^{9}+O(q^{10}) \) 5 * q - 8 * q^3 + 26 * q^4 + 10 * q^5 + 34 * q^6 + 40 * q^7 + 84 * q^8 + 33 * q^9	\( 5 q - 8 q^{3} + 26 q^{4} + 10 q^{5} + 34 q^{6} + 40 q^{7} + 84 q^{8} + 33 q^{9} + 64 q^{10} - 12 q^{11} + 224 q^{12} + 14 q^{13} + 192 q^{14} + 74 q^{15} + 146 q^{16} - 66 q^{17} + 108 q^{18} - 214 q^{19} + 6 q^{20} - 98 q^{22} + 164 q^{23} + 314 q^{24} + 207 q^{25} - 56 q^{26} - 362 q^{27} + 540 q^{28} - 234 q^{30} - 420 q^{31} + 652 q^{32} - 576 q^{33} + 204 q^{34} - 52 q^{35} - 260 q^{36} - 378 q^{37} - 496 q^{38} + 374 q^{39} + 80 q^{40} + 1158 q^{41} + 348 q^{42} + 204 q^{43} - 784 q^{44} - 1506 q^{45} - 580 q^{46} - 248 q^{47} + 1880 q^{48} - 283 q^{49} - 908 q^{50} + 228 q^{51} + 1482 q^{52} - 554 q^{53} + 918 q^{54} - 546 q^{55} + 608 q^{56} + 44 q^{57} + 440 q^{59} - 636 q^{60} - 618 q^{61} + 1250 q^{62} + 804 q^{63} + 2594 q^{64} - 1656 q^{65} - 2940 q^{66} + 1164 q^{67} - 356 q^{68} + 1968 q^{69} + 2184 q^{70} - 692 q^{71} + 2648 q^{72} + 1950 q^{73} - 1832 q^{74} - 3074 q^{75} - 1376 q^{76} + 1616 q^{77} - 1302 q^{78} - 272 q^{79} - 890 q^{80} + 1801 q^{81} + 92 q^{82} + 512 q^{83} + 3208 q^{84} + 1628 q^{85} + 2446 q^{86} - 6954 q^{88} - 866 q^{89} + 2200 q^{90} + 2580 q^{91} + 3468 q^{92} - 40 q^{93} - 5942 q^{94} - 2244 q^{95} + 7386 q^{96} - 1562 q^{97} + 3408 q^{98} + 238 q^{99}+O(q^{100}) \) 5 * q - 8 * q^3 + 26 * q^4 + 10 * q^5 + 34 * q^6 + 40 * q^7 + 84 * q^8 + 33 * q^9 + 64 * q^10 - 12 * q^11 + 224 * q^12 + 14 * q^13 + 192 * q^14 + 74 * q^15 + 146 * q^16 - 66 * q^17 + 108 * q^18 - 214 * q^19 + 6 * q^20 - 98 * q^22 + 164 * q^23 + 314 * q^24 + 207 * q^25 - 56 * q^26 - 362 * q^27 + 540 * q^28 - 234 * q^30 - 420 * q^31 + 652 * q^32 - 576 * q^33 + 204 * q^34 - 52 * q^35 - 260 * q^36 - 378 * q^37 - 496 * q^38 + 374 * q^39 + 80 * q^40 + 1158 * q^41 + 348 * q^42 + 204 * q^43 - 784 * q^44 - 1506 * q^45 - 580 * q^46 - 248 * q^47 + 1880 * q^48 - 283 * q^49 - 908 * q^50 + 228 * q^51 + 1482 * q^52 - 554 * q^53 + 918 * q^54 - 546 * q^55 + 608 * q^56 + 44 * q^57 + 440 * q^59 - 636 * q^60 - 618 * q^61 + 1250 * q^62 + 804 * q^63 + 2594 * q^64 - 1656 * q^65 - 2940 * q^66 + 1164 * q^67 - 356 * q^68 + 1968 * q^69 + 2184 * q^70 - 692 * q^71 + 2648 * q^72 + 1950 * q^73 - 1832 * q^74 - 3074 * q^75 - 1376 * q^76 + 1616 * q^77 - 1302 * q^78 - 272 * q^79 - 890 * q^80 + 1801 * q^81 + 92 * q^82 + 512 * q^83 + 3208 * q^84 + 1628 * q^85 + 2446 * q^86 - 6954 * q^88 - 866 * q^89 + 2200 * q^90 + 2580 * q^91 + 3468 * q^92 - 40 * q^93 - 5942 * q^94 - 2244 * q^95 + 7386 * q^96 - 1562 * q^97 + 3408 * q^98 + 238 * q^99

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