LMFDB - Embedded newform 525.6.d.b.274.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,6,Mod(274,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	525.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$84.2015054018$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.6.d.b.274.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.00000i q^{2} -9.00000i q^{3} -4.00000 q^{4} +54.0000 q^{6} -49.0000i q^{7} +168.000i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q+6.00000i q^{2} -9.00000i q^{3} -4.00000 q^{4} +54.0000 q^{6} -49.0000i q^{7} +168.000i q^{8} -81.0000 q^{9} +444.000 q^{11} +36.0000i q^{12} -442.000i q^{13} +294.000 q^{14} -1136.00 q^{16} +126.000i q^{17} -486.000i q^{18} -2684.00 q^{19} -441.000 q^{21} +2664.00i q^{22} +4200.00i q^{23} +1512.00 q^{24} +2652.00 q^{26} +729.000i q^{27} +196.000i q^{28} +5442.00 q^{29} +80.0000 q^{31} -1440.00i q^{32} -3996.00i q^{33} -756.000 q^{34} +324.000 q^{36} +5434.00i q^{37} -16104.0i q^{38} -3978.00 q^{39} +7962.00 q^{41} -2646.00i q^{42} -11524.0i q^{43} -1776.00 q^{44} -25200.0 q^{46} +13920.0i q^{47} +10224.0i q^{48} -2401.00 q^{49} +1134.00 q^{51} +1768.00i q^{52} -9594.00i q^{53} -4374.00 q^{54} +8232.00 q^{56} +24156.0i q^{57} +32652.0i q^{58} -27492.0 q^{59} +49478.0 q^{61} +480.000i q^{62} +3969.00i q^{63} -27712.0 q^{64} +23976.0 q^{66} +59356.0i q^{67} -504.000i q^{68} +37800.0 q^{69} +32040.0 q^{71} -13608.0i q^{72} -61846.0i q^{73} -32604.0 q^{74} +10736.0 q^{76} -21756.0i q^{77} -23868.0i q^{78} +65776.0 q^{79} +6561.00 q^{81} +47772.0i q^{82} +40188.0i q^{83} +1764.00 q^{84} +69144.0 q^{86} -48978.0i q^{87} +74592.0i q^{88} +7974.00 q^{89} -21658.0 q^{91} -16800.0i q^{92} -720.000i q^{93} -83520.0 q^{94} -12960.0 q^{96} +143662. i q^{97} -14406.0i q^{98} -35964.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4} + 108 q^{6} - 162 q^{9}+O(q^{10}) $ 2 * q - 8 * q^4 + 108 * q^6 - 162 * q^9	$ 2 q - 8 q^{4} + 108 q^{6} - 162 q^{9} + 888 q^{11} + 588 q^{14} - 2272 q^{16} - 5368 q^{19} - 882 q^{21} + 3024 q^{24} + 5304 q^{26} + 10884 q^{29} + 160 q^{31} - 1512 q^{34} + 648 q^{36} - 7956 q^{39} + 15924 q^{41} - 3552 q^{44} - 50400 q^{46} - 4802 q^{49} + 2268 q^{51} - 8748 q^{54} + 16464 q^{56} - 54984 q^{59} + 98956 q^{61} - 55424 q^{64} + 47952 q^{66} + 75600 q^{69} + 64080 q^{71} - 65208 q^{74} + 21472 q^{76} + 131552 q^{79} + 13122 q^{81} + 3528 q^{84} + 138288 q^{86} + 15948 q^{89} - 43316 q^{91} - 167040 q^{94} - 25920 q^{96} - 71928 q^{99}+O(q^{100}) $ 2 * q - 8 * q^4 + 108 * q^6 - 162 * q^9 + 888 * q^11 + 588 * q^14 - 2272 * q^16 - 5368 * q^19 - 882 * q^21 + 3024 * q^24 + 5304 * q^26 + 10884 * q^29 + 160 * q^31 - 1512 * q^34 + 648 * q^36 - 7956 * q^39 + 15924 * q^41 - 3552 * q^44 - 50400 * q^46 - 4802 * q^49 + 2268 * q^51 - 8748 * q^54 + 16464 * q^56 - 54984 * q^59 + 98956 * q^61 - 55424 * q^64 + 47952 * q^66 + 75600 * q^69 + 64080 * q^71 - 65208 * q^74 + 21472 * q^76 + 131552 * q^79 + 13122 * q^81 + 3528 * q^84 + 138288 * q^86 + 15948 * q^89 - 43316 * q^91 - 167040 * q^94 - 25920 * q^96 - 71928 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	6.00000i	1.06066i	0.847791	+	0.530330i	$0.177932\pi$
$2$	6.00000i	1.06066i	−0.847791	+	0.530330i	$0.822068\pi$
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	−4.00000	−0.125000
$5$	0	0
$5$	0	0
$6$	54.0000	0.612372
$7$	− 49.0000i	− 0.377964i
$7$	− 49.0000i	− 0.377964i
$8$	168.000i	0.928078i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	444.000	1.10637	0.553186	−	0.833058i	$-0.313412\pi$
$11$	444.000	1.10637	0.553186	+	0.833058i	$0.313412\pi$
$12$	36.0000i	0.0721688i
$13$	− 442.000i	− 0.725377i	−0.931910	−	0.362689i	$-0.881859\pi$
$13$	− 442.000i	− 0.725377i	0.931910	−	0.362689i	$-0.118141\pi$
$14$	294.000	0.400892
$15$	0	0
$16$	−1136.00	−1.10938
$17$	126.000i	0.105742i	0.998601	+	0.0528711i	$0.0168372\pi$
$17$	126.000i	0.105742i	−0.998601	+	0.0528711i	$0.983163\pi$
$18$	− 486.000i	− 0.353553i
$19$	−2684.00	−1.70568	−0.852842	−	0.522169i	$-0.825123\pi$
$19$	−2684.00	−1.70568	−0.852842	+	0.522169i	$0.825123\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	2664.00i	1.17348i
$23$	4200.00i	1.65550i	0.561096	+	0.827751i	$0.310380\pi$
$23$	4200.00i	1.65550i	−0.561096	+	0.827751i	$0.689620\pi$
$24$	1512.00	0.535826
$25$	0	0
$26$	2652.00	0.769379
$27$	729.000i	0.192450i
$28$	196.000i	0.0472456i
$29$	5442.00	1.20161	0.600805	−	0.799396i	$-0.294847\pi$
$29$	5442.00	1.20161	0.600805	+	0.799396i	$0.294847\pi$
$30$	0	0
$31$	80.0000	0.0149515	0.00747577	−	0.999972i	$-0.497620\pi$
$31$	80.0000	0.0149515	0.00747577	+	0.999972i	$0.497620\pi$
$32$	− 1440.00i	− 0.248592i
$33$	− 3996.00i	− 0.638764i
$34$	−756.000	−0.112157
$35$	0	0
$36$	324.000	0.0416667
$37$	5434.00i	0.652552i	0.945274	+	0.326276i	$0.105794\pi$
$37$	5434.00i	0.652552i	−0.945274	+	0.326276i	$0.894206\pi$
$38$	− 16104.0i	− 1.80915i
$39$	−3978.00	−0.418797
$40$	0	0
$41$	7962.00	0.739712	0.369856	−	0.929089i	$-0.379407\pi$
$41$	7962.00	0.739712	0.369856	+	0.929089i	$0.379407\pi$
$42$	− 2646.00i	− 0.231455i
$43$	− 11524.0i	− 0.950456i	−0.879863	−	0.475228i	$-0.842366\pi$
$43$	− 11524.0i	− 0.950456i	0.879863	−	0.475228i	$-0.157634\pi$
$44$	−1776.00	−0.138297
$45$	0	0
$46$	−25200.0	−1.75592
$47$	13920.0i	0.919167i	0.888134	+	0.459584i	$0.152001\pi$
$47$	13920.0i	0.919167i	−0.888134	+	0.459584i	$0.847999\pi$
$48$	10224.0i	0.640498i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	1134.00	0.0610503
$52$	1768.00i	0.0906721i
$53$	− 9594.00i	− 0.469148i	−0.972098	−	0.234574i	$-0.924630\pi$
$53$	− 9594.00i	− 0.469148i	0.972098	−	0.234574i	$-0.0753695\pi$
$54$	−4374.00	−0.204124
$55$	0	0
$56$	8232.00	0.350780
$57$	24156.0i	0.984777i
$58$	32652.0i	1.27450i
$59$	−27492.0	−1.02820	−0.514098	−	0.857731i	$-0.671873\pi$
$59$	−27492.0	−1.02820	−0.514098	+	0.857731i	$0.671873\pi$
$60$	0	0
$61$	49478.0	1.70250	0.851251	−	0.524759i	$-0.175845\pi$
$61$	49478.0	1.70250	0.851251	+	0.524759i	$0.175845\pi$
$62$	480.000i	0.0158585i
$63$	3969.00i	0.125988i
$64$	−27712.0	−0.845703
$65$	0	0
$66$	23976.0	0.677512
$67$	59356.0i	1.61539i	0.589600	+	0.807695i	$0.299285\pi$
$67$	59356.0i	1.61539i	−0.589600	+	0.807695i	$0.700715\pi$
$68$	− 504.000i	− 0.0132178i
$69$	37800.0	0.955805
$70$	0	0
$71$	32040.0	0.754304	0.377152	−	0.926151i	$-0.376903\pi$
$71$	32040.0	0.754304	0.377152	+	0.926151i	$0.376903\pi$
$72$	− 13608.0i	− 0.309359i
$73$	− 61846.0i	− 1.35833i	−0.733987	−	0.679164i	$-0.762343\pi$
$73$	− 61846.0i	− 1.35833i	0.733987	−	0.679164i	$-0.237657\pi$
$74$	−32604.0	−0.692136
$75$	0	0
$76$	10736.0	0.213210
$77$	− 21756.0i	− 0.418169i
$78$	− 23868.0i	− 0.444201i
$79$	65776.0	1.18577	0.592884	−	0.805288i	$-0.297989\pi$
$79$	65776.0	1.18577	0.592884	+	0.805288i	$0.297989\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	47772.0i	0.784583i
$83$	40188.0i	0.640326i	0.947362	+	0.320163i	$0.103738\pi$
$83$	40188.0i	0.640326i	−0.947362	+	0.320163i	$0.896262\pi$
$84$	1764.00	0.0272772
$85$	0	0
$86$	69144.0	1.00811
$87$	− 48978.0i	− 0.693750i
$88$	74592.0i	1.02680i
$89$	7974.00	0.106709	0.0533545	−	0.998576i	$-0.483009\pi$
$89$	7974.00	0.106709	0.0533545	+	0.998576i	$0.483009\pi$
$90$	0	0
$91$	−21658.0	−0.274167
$92$	− 16800.0i	− 0.206938i
$93$	− 720.000i	− 0.00863227i
$94$	−83520.0	−0.974924
$95$	0	0
$96$	−12960.0	−0.143525
$97$	143662.i	1.55029i	0.631784	+	0.775144i	$0.282323\pi$
$97$	143662.i	1.55029i	−0.631784	+	0.775144i	$0.717677\pi$
$98$	− 14406.0i	− 0.151523i
$99$	−35964.0	−0.368791
$100$	0	0
$101$	−2706.00	−0.0263952	−0.0131976	−	0.999913i	$-0.504201\pi$
$101$	−2706.00	−0.0263952	−0.0131976	+	0.999913i	$0.504201\pi$
$102$	6804.00i	0.0647536i
$103$	131768.i	1.22382i	0.790928	+	0.611909i	$0.209598\pi$
$103$	131768.i	1.22382i	−0.790928	+	0.611909i	$0.790402\pi$
$104$	74256.0	0.673206
$105$	0	0
$106$	57564.0	0.497607
$107$	128916.i	1.08855i	0.838908	+	0.544274i	$0.183195\pi$
$107$	128916.i	1.08855i	−0.838908	+	0.544274i	$0.816805\pi$
$108$	− 2916.00i	− 0.0240563i
$109$	100978.	0.814068	0.407034	−	0.913413i	$-0.366563\pi$
$109$	100978.	0.814068	0.407034	+	0.913413i	$0.366563\pi$
$110$	0	0
$111$	48906.0	0.376751
$112$	55664.0i	0.419304i
$113$	220146.i	1.62186i	0.585140	+	0.810932i	$0.301040\pi$
$113$	220146.i	1.62186i	−0.585140	+	0.810932i	$0.698960\pi$
$114$	−144936.	−1.04451
$115$	0	0
$116$	−21768.0	−0.150201
$117$	35802.0i	0.241792i
$118$	− 164952.i	− 1.09057i
$119$	6174.00	0.0399668
$120$	0	0
$121$	36085.0	0.224059
$122$	296868.i	1.80578i
$123$	− 71658.0i	− 0.427073i
$124$	−320.000	−0.00186894
$125$	0	0
$126$	−23814.0	−0.133631
$127$	74320.0i	0.408880i	0.978879	+	0.204440i	$0.0655374\pi$
$127$	74320.0i	0.408880i	−0.978879	+	0.204440i	$0.934463\pi$
$128$	− 212352.i	− 1.14560i
$129$	−103716.	−0.548746
$130$	0	0
$131$	−155316.	−0.790748	−0.395374	−	0.918520i	$-0.629385\pi$
$131$	−155316.	−0.790748	−0.395374	+	0.918520i	$0.629385\pi$
$132$	15984.0i	0.0798455i
$133$	131516.i	0.644688i
$134$	−356136.	−1.71338
$135$	0	0
$136$	−21168.0	−0.0981369
$137$	264246.i	1.20284i	0.798934	+	0.601419i	$0.205398\pi$
$137$	264246.i	1.20284i	−0.798934	+	0.601419i	$0.794602\pi$
$138$	226800.i	1.01378i
$139$	−224612.	−0.986043	−0.493022	−	0.870017i	$-0.664108\pi$
$139$	−224612.	−0.986043	−0.493022	+	0.870017i	$0.664108\pi$
$140$	0	0
$141$	125280.	0.530682
$142$	192240.i	0.800061i
$143$	− 196248.i	− 0.802537i
$144$	92016.0	0.369792
$145$	0	0
$146$	371076.	1.44072
$147$	21609.0i	0.0824786i
$148$	− 21736.0i	− 0.0815690i
$149$	82074.0	0.302859	0.151429	−	0.988468i	$-0.451612\pi$
$149$	82074.0	0.302859	0.151429	+	0.988468i	$0.451612\pi$
$150$	0	0
$151$	−287032.	−1.02444	−0.512222	−	0.858853i	$-0.671177\pi$
$151$	−287032.	−1.02444	−0.512222	+	0.858853i	$0.671177\pi$
$152$	− 450912.i	− 1.58301i
$153$	− 10206.0i	− 0.0352474i
$154$	130536.	0.443536
$155$	0	0
$156$	15912.0	0.0523496
$157$	− 129878.i	− 0.420520i	−0.977646	−	0.210260i	$-0.932569\pi$
$157$	− 129878.i	− 0.420520i	0.977646	−	0.210260i	$-0.0674310\pi$
$158$	394656.i	1.25770i
$159$	−86346.0	−0.270863
$160$	0	0
$161$	205800.	0.625721
$162$	39366.0i	0.117851i
$163$	555284.i	1.63699i	0.574513	+	0.818495i	$0.305191\pi$
$163$	555284.i	1.63699i	−0.574513	+	0.818495i	$0.694809\pi$
$164$	−31848.0	−0.0924640
$165$	0	0
$166$	−241128.	−0.679168
$167$	− 43512.0i	− 0.120731i	−0.998176	−	0.0603654i	$-0.980773\pi$
$167$	− 43512.0i	− 0.120731i	0.998176	−	0.0603654i	$-0.0192266\pi$
$168$	− 74088.0i	− 0.202523i
$169$	175929.	0.473828
$170$	0	0
$171$	217404.	0.568561
$172$	46096.0i	0.118807i
$173$	− 18330.0i	− 0.0465637i	−0.999729	−	0.0232818i	$-0.992588\pi$
$173$	− 18330.0i	− 0.0465637i	0.999729	−	0.0232818i	$-0.00741151\pi$
$174$	293868.	0.735833
$175$	0	0
$176$	−504384.	−1.22738
$177$	247428.i	0.593630i
$178$	47844.0i	0.113182i
$179$	153324.	0.357666	0.178833	−	0.983879i	$-0.442768\pi$
$179$	153324.	0.357666	0.178833	+	0.983879i	$0.442768\pi$
$180$	0	0
$181$	−382066.	−0.866846	−0.433423	−	0.901191i	$-0.642694\pi$
$181$	−382066.	−0.866846	−0.433423	+	0.901191i	$0.642694\pi$
$182$	− 129948.i	− 0.290798i
$183$	− 445302.i	− 0.982940i
$184$	−705600.	−1.53643
$185$	0	0
$186$	4320.00	0.00915591
$187$	55944.0i	0.116990i
$188$	− 55680.0i	− 0.114896i
$189$	35721.0	0.0727393
$190$	0	0
$191$	−273408.	−0.542285	−0.271143	−	0.962539i	$-0.587402\pi$
$191$	−273408.	−0.542285	−0.271143	+	0.962539i	$0.587402\pi$
$192$	249408.i	0.488267i
$193$	153602.i	0.296827i	0.988925	+	0.148414i	$0.0474167\pi$
$193$	153602.i	0.296827i	−0.988925	+	0.148414i	$0.952583\pi$
$194$	−861972.	−1.64433
$195$	0	0
$196$	9604.00	0.0178571
$197$	− 154422.i	− 0.283494i	−0.989903	−	0.141747i	$-0.954728\pi$
$197$	− 154422.i	− 0.283494i	0.989903	−	0.141747i	$-0.0452719\pi$
$198$	− 215784.i	− 0.391162i
$199$	366856.	0.656694	0.328347	−	0.944557i	$-0.393508\pi$
$199$	366856.	0.656694	0.328347	+	0.944557i	$0.393508\pi$
$200$	0	0
$201$	534204.	0.932646
$202$	− 16236.0i	− 0.0279963i
$203$	− 266658.i	− 0.454166i
$204$	−4536.00	−0.00763128
$205$	0	0
$206$	−790608.	−1.29806
$207$	− 340200.i	− 0.551834i
$208$	502112.i	0.804715i
$209$	−1.19170e6	−1.88712
$210$	0	0
$211$	520244.	0.804453	0.402227	−	0.915540i	$-0.368236\pi$
$211$	520244.	0.804453	0.402227	+	0.915540i	$0.368236\pi$
$212$	38376.0i	0.0586435i
$213$	− 288360.i	− 0.435498i
$214$	−773496.	−1.15458
$215$	0	0
$216$	−122472.	−0.178609
$217$	− 3920.00i	− 0.00565115i
$218$	605868.i	0.863449i
$219$	−556614.	−0.784231
$220$	0	0
$221$	55692.0	0.0767030
$222$	293436.i	0.399605i
$223$	304736.i	0.410357i	0.978725	+	0.205178i	$0.0657775\pi$
$223$	304736.i	0.410357i	−0.978725	+	0.205178i	$0.934223\pi$
$224$	−70560.0	−0.0939590
$225$	0	0
$226$	−1.32088e6	−1.72025
$227$	− 288588.i	− 0.371718i	−0.982576	−	0.185859i	$-0.940493\pi$
$227$	− 288588.i	− 0.371718i	0.982576	−	0.185859i	$-0.0595068\pi$
$228$	− 96624.0i	− 0.123097i
$229$	−772190.	−0.973051	−0.486525	−	0.873666i	$-0.661736\pi$
$229$	−772190.	−0.973051	−0.486525	+	0.873666i	$0.661736\pi$
$230$	0	0
$231$	−195804.	−0.241430
$232$	914256.i	1.11519i
$233$	252234.i	0.304378i	0.988351	+	0.152189i	$0.0486323\pi$
$233$	252234.i	0.304378i	−0.988351	+	0.152189i	$0.951368\pi$
$234$	−214812.	−0.256460
$235$	0	0
$236$	109968.	0.128525
$237$	− 591984.i	− 0.684603i
$238$	37044.0i	0.0423912i
$239$	1.45114e6	1.64329	0.821643	−	0.570002i	$-0.193058\pi$
$239$	1.45114e6	1.64329	0.821643	+	0.570002i	$0.193058\pi$
$240$	0	0
$241$	−146398.	−0.162365	−0.0811825	−	0.996699i	$-0.525870\pi$
$241$	−146398.	−0.162365	−0.0811825	+	0.996699i	$0.525870\pi$
$242$	216510.i	0.237651i
$243$	− 59049.0i	− 0.0641500i
$244$	−197912.	−0.212813
$245$	0	0
$246$	429948.	0.452979
$247$	1.18633e6i	1.23726i
$248$	13440.0i	0.0138762i
$249$	361692.	0.369692
$250$	0	0
$251$	607860.	0.609003	0.304501	−	0.952512i	$-0.401510\pi$
$251$	607860.	0.609003	0.304501	+	0.952512i	$0.401510\pi$
$252$	− 15876.0i	− 0.0157485i
$253$	1.86480e6i	1.83160i
$254$	−445920.	−0.433683
$255$	0	0
$256$	387328.	0.369385
$257$	− 95586.0i	− 0.0902737i	−0.998981	−	0.0451369i	$-0.985628\pi$
$257$	− 95586.0i	− 0.0902737i	0.998981	−	0.0451369i	$-0.0143724\pi$
$258$	− 622296.i	− 0.582033i
$259$	266266.	0.246642
$260$	0	0
$261$	−440802.	−0.400537
$262$	− 931896.i	− 0.838715i
$263$	− 2.20034e6i	− 1.96156i	−0.195121	−	0.980779i	$-0.562510\pi$
$263$	− 2.20034e6i	− 1.96156i	0.195121	−	0.980779i	$-0.437490\pi$
$264$	671328.	0.592823
$265$	0	0
$266$	−789096.	−0.683795
$267$	− 71766.0i	− 0.0616085i
$268$	− 237424.i	− 0.201924i
$269$	−1.77025e6	−1.49160	−0.745801	−	0.666169i	$-0.767933\pi$
$269$	−1.77025e6	−1.49160	−0.745801	+	0.666169i	$0.767933\pi$
$270$	0	0
$271$	−223504.	−0.184868	−0.0924341	−	0.995719i	$-0.529465\pi$
$271$	−223504.	−0.184868	−0.0924341	+	0.995719i	$0.529465\pi$
$272$	− 143136.i	− 0.117308i
$273$	194922.i	0.158290i
$274$	−1.58548e6	−1.27580
$275$	0	0
$276$	−151200.	−0.119476
$277$	342778.i	0.268419i	0.990953	+	0.134210i	$0.0428495\pi$
$277$	342778.i	0.268419i	−0.990953	+	0.134210i	$0.957150\pi$
$278$	− 1.34767e6i	− 1.04586i
$279$	−6480.00	−0.00498384
$280$	0	0
$281$	480378.	0.362925	0.181463	−	0.983398i	$-0.441917\pi$
$281$	480378.	0.362925	0.181463	+	0.983398i	$0.441917\pi$
$282$	751680.i	0.562873i
$283$	− 29980.0i	− 0.0222518i	−0.999938	−	0.0111259i	$-0.996458\pi$
$283$	− 29980.0i	− 0.0222518i	0.999938	−	0.0111259i	$-0.00354156\pi$
$284$	−128160.	−0.0942880
$285$	0	0
$286$	1.17749e6	0.851219
$287$	− 390138.i	− 0.279585i
$288$	116640.i	0.0828641i
$289$	1.40398e6	0.988819
$290$	0	0
$291$	1.29296e6	0.895060
$292$	247384.i	0.169791i
$293$	− 198066.i	− 0.134785i	−0.997727	−	0.0673924i	$-0.978532\pi$
$293$	− 198066.i	− 0.134785i	0.997727	−	0.0673924i	$-0.0214679\pi$
$294$	−129654.	−0.0874818
$295$	0	0
$296$	−912912.	−0.605619
$297$	323676.i	0.212921i
$298$	492444.i	0.321230i
$299$	1.85640e6	1.20086
$300$	0	0
$301$	−564676.	−0.359239
$302$	− 1.72219e6i	− 1.08659i
$303$	24354.0i	0.0152393i
$304$	3.04902e6	1.89224
$305$	0	0
$306$	61236.0	0.0373855
$307$	1.04564e6i	0.633191i	0.948561	+	0.316595i	$0.102540\pi$
$307$	1.04564e6i	0.633191i	−0.948561	+	0.316595i	$0.897460\pi$
$308$	87024.0i	0.0522712i
$309$	1.18591e6	0.706572
$310$	0	0
$311$	1.83718e6	1.07708	0.538542	−	0.842598i	$-0.318975\pi$
$311$	1.83718e6	1.07708	0.538542	+	0.842598i	$0.318975\pi$
$312$	− 668304.i	− 0.388676i
$313$	− 365494.i	− 0.210872i	−0.994426	−	0.105436i	$-0.966376\pi$
$313$	− 365494.i	− 0.210872i	0.994426	−	0.105436i	$-0.0336239\pi$
$314$	779268.	0.446029
$315$	0	0
$316$	−263104.	−0.148221
$317$	28338.0i	0.0158388i	0.999969	+	0.00791938i	$0.00252084\pi$
$317$	28338.0i	0.0158388i	−0.999969	+	0.00791938i	$0.997479\pi$
$318$	− 518076.i	− 0.287293i
$319$	2.41625e6	1.32943
$320$	0	0
$321$	1.16024e6	0.628473
$322$	1.23480e6i	0.663677i
$323$	− 338184.i	− 0.180363i
$324$	−26244.0	−0.0138889
$325$	0	0
$326$	−3.33170e6	−1.73629
$327$	− 908802.i	− 0.470002i
$328$	1.33762e6i	0.686510i
$329$	682080.	0.347413
$330$	0	0
$331$	1.93392e6	0.970214	0.485107	−	0.874455i	$-0.338781\pi$
$331$	1.93392e6	0.970214	0.485107	+	0.874455i	$0.338781\pi$
$332$	− 160752.i	− 0.0800408i
$333$	− 440154.i	− 0.217517i
$334$	261072.	0.128054
$335$	0	0
$336$	500976.	0.242085
$337$	1.88817e6i	0.905664i	0.891596	+	0.452832i	$0.149586\pi$
$337$	1.88817e6i	0.905664i	−0.891596	+	0.452832i	$0.850414\pi$
$338$	1.05557e6i	0.502570i
$339$	1.98131e6	0.936384
$340$	0	0
$341$	35520.0	0.0165420
$342$	1.30442e6i	0.603050i
$343$	117649.i	0.0539949i
$344$	1.93603e6	0.882097
$345$	0	0
$346$	109980.	0.0493882
$347$	− 2.91937e6i	− 1.30156i	−0.759264	−	0.650782i	$-0.774441\pi$
$347$	− 2.91937e6i	− 1.30156i	0.759264	−	0.650782i	$-0.225559\pi$
$348$	195912.i	0.0867187i
$349$	780682.	0.343092	0.171546	−	0.985176i	$-0.445124\pi$
$349$	780682.	0.343092	0.171546	+	0.985176i	$0.445124\pi$
$350$	0	0
$351$	322218.	0.139599
$352$	− 639360.i	− 0.275036i
$353$	1.33437e6i	0.569954i	0.958534	+	0.284977i	$0.0919859\pi$
$353$	1.33437e6i	0.569954i	−0.958534	+	0.284977i	$0.908014\pi$
$354$	−1.48457e6	−0.629639
$355$	0	0
$356$	−31896.0	−0.0133386
$357$	− 55566.0i	− 0.0230748i
$358$	919944.i	0.379362i
$359$	−1.01743e6	−0.416648	−0.208324	−	0.978060i	$-0.566801\pi$
$359$	−1.01743e6	−0.416648	−0.208324	+	0.978060i	$0.566801\pi$
$360$	0	0
$361$	4.72776e6	1.90936
$362$	− 2.29240e6i	− 0.919429i
$363$	− 324765.i	− 0.129361i
$364$	86632.0	0.0342709
$365$	0	0
$366$	2.67181e6	1.04257
$367$	− 837680.i	− 0.324648i	−0.986737	−	0.162324i	$-0.948101\pi$
$367$	− 837680.i	− 0.324648i	0.986737	−	0.162324i	$-0.0518990\pi$
$368$	− 4.77120e6i	− 1.83657i
$369$	−644922.	−0.246571
$370$	0	0
$371$	−470106.	−0.177321
$372$	2880.00i	0.00107903i
$373$	− 1.51993e6i	− 0.565655i	−0.959171	−	0.282827i	$-0.908728\pi$
$373$	− 1.51993e6i	− 0.565655i	0.959171	−	0.282827i	$-0.0912724\pi$
$374$	−335664.	−0.124087
$375$	0	0
$376$	−2.33856e6	−0.853059
$377$	− 2.40536e6i	− 0.871620i
$378$	214326.i	0.0771517i
$379$	−2.64465e6	−0.945737	−0.472869	−	0.881133i	$-0.656781\pi$
$379$	−2.64465e6	−0.945737	−0.472869	+	0.881133i	$0.656781\pi$
$380$	0	0
$381$	668880.	0.236067
$382$	− 1.64045e6i	− 0.575180i
$383$	2.01336e6i	0.701333i	0.936500	+	0.350667i	$0.114045\pi$
$383$	2.01336e6i	0.701333i	−0.936500	+	0.350667i	$0.885955\pi$
$384$	−1.91117e6	−0.661410
$385$	0	0
$386$	−921612.	−0.314833
$387$	933444.i	0.316819i
$388$	− 574648.i	− 0.193786i
$389$	726234.	0.243334	0.121667	−	0.992571i	$-0.461176\pi$
$389$	726234.	0.243334	0.121667	+	0.992571i	$0.461176\pi$
$390$	0	0
$391$	−529200.	−0.175056
$392$	− 403368.i	− 0.132583i
$393$	1.39784e6i	0.456538i
$394$	926532.	0.300691
$395$	0	0
$396$	143856.	0.0460988
$397$	− 4.57578e6i	− 1.45710i	−0.684993	−	0.728549i	$-0.740195\pi$
$397$	− 4.57578e6i	− 1.45710i	0.684993	−	0.728549i	$-0.259805\pi$
$398$	2.20114e6i	0.696529i
$399$	1.18364e6	0.372211
$400$	0	0
$401$	−33870.0	−0.0105185	−0.00525926	−	0.999986i	$-0.501674\pi$
$401$	−33870.0	−0.0105185	−0.00525926	+	0.999986i	$0.501674\pi$
$402$	3.20522e6i	0.989221i
$403$	− 35360.0i	− 0.0108455i
$404$	10824.0	0.00329940
$405$	0	0
$406$	1.59995e6	0.481716
$407$	2.41270e6i	0.721966i
$408$	190512.i	0.0566594i
$409$	5.86178e6	1.73269	0.866346	−	0.499444i	$-0.166462\pi$
$409$	5.86178e6	1.73269	0.866346	+	0.499444i	$0.166462\pi$
$410$	0	0
$411$	2.37821e6	0.694459
$412$	− 527072.i	− 0.152977i
$413$	1.34711e6i	0.388622i
$414$	2.04120e6	0.585308
$415$	0	0
$416$	−636480.	−0.180323
$417$	2.02151e6i	0.569292i
$418$	− 7.15018e6i	− 2.00159i
$419$	−302748.	−0.0842454	−0.0421227	−	0.999112i	$-0.513412\pi$
$419$	−302748.	−0.0842454	−0.0421227	+	0.999112i	$0.513412\pi$
$420$	0	0
$421$	−5.36708e6	−1.47582	−0.737909	−	0.674900i	$-0.764187\pi$
$421$	−5.36708e6	−1.47582	−0.737909	+	0.674900i	$0.764187\pi$
$422$	3.12146e6i	0.853252i
$423$	− 1.12752e6i	− 0.306389i
$424$	1.61179e6	0.435406
$425$	0	0
$426$	1.73016e6	0.461915
$427$	− 2.42442e6i	− 0.643485i
$428$	− 515664.i	− 0.136068i
$429$	−1.76623e6	−0.463345
$430$	0	0
$431$	1.17706e6	0.305214	0.152607	−	0.988287i	$-0.451233\pi$
$431$	1.17706e6	0.305214	0.152607	+	0.988287i	$0.451233\pi$
$432$	− 828144.i	− 0.213499i
$433$	− 3.66249e6i	− 0.938766i	−0.882995	−	0.469383i	$-0.844476\pi$
$433$	− 3.66249e6i	− 0.938766i	0.882995	−	0.469383i	$-0.155524\pi$
$434$	23520.0	0.00599395
$435$	0	0
$436$	−403912.	−0.101758
$437$	− 1.12728e7i	− 2.82376i
$438$	− 3.33968e6i	− 0.831802i
$439$	2.53674e6	0.628225	0.314113	−	0.949386i	$-0.398293\pi$
$439$	2.53674e6	0.628225	0.314113	+	0.949386i	$0.398293\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	334152.i	0.0813558i
$443$	6.01504e6i	1.45623i	0.685457	+	0.728113i	$0.259603\pi$
$443$	6.01504e6i	1.45623i	−0.685457	+	0.728113i	$0.740397\pi$
$444$	−195624.	−0.0470939
$445$	0	0
$446$	−1.82842e6	−0.435249
$447$	− 738666.i	− 0.174856i
$448$	1.35789e6i	0.319646i
$449$	−5.65965e6	−1.32487	−0.662436	−	0.749119i	$-0.730477\pi$
$449$	−5.65965e6	−1.32487	−0.662436	+	0.749119i	$0.730477\pi$
$450$	0	0
$451$	3.53513e6	0.818397
$452$	− 880584.i	− 0.202733i
$453$	2.58329e6i	0.591463i
$454$	1.73153e6	0.394267
$455$	0	0
$456$	−4.05821e6	−0.913949
$457$	6.46159e6i	1.44727i	0.690184	+	0.723634i	$0.257530\pi$
$457$	6.46159e6i	1.44727i	−0.690184	+	0.723634i	$0.742470\pi$
$458$	− 4.63314e6i	− 1.03208i
$459$	−91854.0	−0.0203501
$460$	0	0
$461$	−3.37353e6	−0.739320	−0.369660	−	0.929167i	$-0.620526\pi$
$461$	−3.37353e6	−0.739320	−0.369660	+	0.929167i	$0.620526\pi$
$462$	− 1.17482e6i	− 0.256075i
$463$	− 4.54974e6i	− 0.986358i	−0.869928	−	0.493179i	$-0.835835\pi$
$463$	− 4.54974e6i	− 0.986358i	0.869928	−	0.493179i	$-0.164165\pi$
$464$	−6.18211e6	−1.33304
$465$	0	0
$466$	−1.51340e6	−0.322842
$467$	− 2.01136e6i	− 0.426773i	−0.976968	−	0.213386i	$-0.931551\pi$
$467$	− 2.01136e6i	− 0.426773i	0.976968	−	0.213386i	$-0.0684493\pi$
$468$	− 143208.i	− 0.0302240i
$469$	2.90844e6	0.610560
$470$	0	0
$471$	−1.16890e6	−0.242787
$472$	− 4.61866e6i	− 0.954247i
$473$	− 5.11666e6i	− 1.05156i
$474$	3.55190e6	0.726132
$475$	0	0
$476$	−24696.0	−0.00499585
$477$	777114.i	0.156383i
$478$	8.70682e6i	1.74297i
$479$	7.60402e6	1.51427	0.757137	−	0.653257i	$-0.226598\pi$
$479$	7.60402e6	1.51427	0.757137	+	0.653257i	$0.226598\pi$
$480$	0	0
$481$	2.40183e6	0.473347
$482$	− 878388.i	− 0.172214i
$483$	− 1.85220e6i	− 0.361260i
$484$	−144340.	−0.0280074
$485$	0	0
$486$	354294.	0.0680414
$487$	− 673112.i	− 0.128607i	−0.997930	−	0.0643035i	$-0.979517\pi$
$487$	− 673112.i	− 0.128607i	0.997930	−	0.0643035i	$-0.0204826\pi$
$488$	8.31230e6i	1.58005i
$489$	4.99756e6	0.945117
$490$	0	0
$491$	−2.47170e6	−0.462692	−0.231346	−	0.972872i	$-0.574313\pi$
$491$	−2.47170e6	−0.462692	−0.231346	+	0.972872i	$0.574313\pi$
$492$	286632.i	0.0533841i
$493$	685692.i	0.127061i
$494$	−7.11797e6	−1.31232
$495$	0	0
$496$	−90880.0	−0.0165869
$497$	− 1.56996e6i	− 0.285100i
$498$	2.17015e6i	0.392118i
$499$	−6.08152e6	−1.09335	−0.546677	−	0.837343i	$-0.684108\pi$
$499$	−6.08152e6	−1.09335	−0.546677	+	0.837343i	$0.684108\pi$
$500$	0	0
$501$	−391608.	−0.0697039
$502$	3.64716e6i	0.645945i
$503$	− 846216.i	− 0.149129i	−0.997216	−	0.0745644i	$-0.976243\pi$
$503$	− 846216.i	− 0.149129i	0.997216	−	0.0745644i	$-0.0237566\pi$
$504$	−666792.	−0.116927
$505$	0	0
$506$	−1.11888e7	−1.94271
$507$	− 1.58336e6i	− 0.273565i
$508$	− 297280.i	− 0.0511101i
$509$	7.66785e6	1.31183	0.655917	−	0.754833i	$-0.272282\pi$
$509$	7.66785e6	1.31183	0.655917	+	0.754833i	$0.272282\pi$
$510$	0	0
$511$	−3.03045e6	−0.513400
$512$	− 4.47130e6i	− 0.753804i
$513$	− 1.95664e6i	− 0.328259i
$514$	573516.	0.0957498
$515$	0	0
$516$	414864.	0.0685933
$517$	6.18048e6i	1.01694i
$518$	1.59760e6i	0.261603i
$519$	−164970.	−0.0268835
$520$	0	0
$521$	−9.68938e6	−1.56387	−0.781937	−	0.623357i	$-0.785768\pi$
$521$	−9.68938e6	−1.56387	−0.781937	+	0.623357i	$0.785768\pi$
$522$	− 2.64481e6i	− 0.424833i
$523$	− 7.51678e6i	− 1.20165i	−0.799381	−	0.600824i	$-0.794839\pi$
$523$	− 7.51678e6i	− 1.20165i	0.799381	−	0.600824i	$-0.205161\pi$
$524$	621264.	0.0988435
$525$	0	0
$526$	1.32021e7	2.08055
$527$	10080.0i	0.00158101i
$528$	4.53946e6i	0.708629i
$529$	−1.12037e7	−1.74069
$530$	0	0
$531$	2.22685e6	0.342732
$532$	− 526064.i	− 0.0805860i
$533$	− 3.51920e6i	− 0.536570i
$534$	430596.	0.0653457
$535$	0	0
$536$	−9.97181e6	−1.49921
$537$	− 1.37992e6i	− 0.206499i
$538$	− 1.06215e7i	− 1.58208i
$539$	−1.06604e6	−0.158053
$540$	0	0
$541$	7.34325e6	1.07869	0.539343	−	0.842086i	$-0.318673\pi$
$541$	7.34325e6	1.07869	0.539343	+	0.842086i	$0.318673\pi$
$542$	− 1.34102e6i	− 0.196082i
$543$	3.43859e6i	0.500474i
$544$	181440.	0.0262867
$545$	0	0
$546$	−1.16953e6	−0.167892
$547$	− 2.18296e6i	− 0.311945i	−0.987761	−	0.155973i	$-0.950149\pi$
$547$	− 2.18296e6i	− 0.311945i	0.987761	−	0.155973i	$-0.0498512\pi$
$548$	− 1.05698e6i	− 0.150355i
$549$	−4.00772e6	−0.567501
$550$	0	0
$551$	−1.46063e7	−2.04957
$552$	6.35040e6i	0.887061i
$553$	− 3.22302e6i	− 0.448178i
$554$	−2.05667e6	−0.284702
$555$	0	0
$556$	898448.	0.123255
$557$	− 1.25466e7i	− 1.71351i	−0.515724	−	0.856755i	$-0.672477\pi$
$557$	− 1.25466e7i	− 1.71351i	0.515724	−	0.856755i	$-0.327523\pi$
$558$	− 38880.0i	− 0.00528617i
$559$	−5.09361e6	−0.689439
$560$	0	0
$561$	503496.	0.0675443
$562$	2.88227e6i	0.384940i
$563$	5.15972e6i	0.686050i	0.939326	+	0.343025i	$0.111451\pi$
$563$	5.15972e6i	0.686050i	−0.939326	+	0.343025i	$0.888549\pi$
$564$	−501120.	−0.0663352
$565$	0	0
$566$	179880.	0.0236016
$567$	− 321489.i	− 0.0419961i
$568$	5.38272e6i	0.700053i
$569$	−1.17452e7	−1.52083	−0.760414	−	0.649439i	$-0.775004\pi$
$569$	−1.17452e7	−1.52083	−0.760414	+	0.649439i	$0.775004\pi$
$570$	0	0
$571$	−7.54728e6	−0.968725	−0.484362	−	0.874867i	$-0.660948\pi$
$571$	−7.54728e6	−0.968725	−0.484362	+	0.874867i	$0.660948\pi$
$572$	784992.i	0.100317i
$573$	2.46067e6i	0.313089i
$574$	2.34083e6	0.296544
$575$	0	0
$576$	2.24467e6	0.281901
$577$	− 9.28483e6i	− 1.16101i	−0.814258	−	0.580503i	$-0.802856\pi$
$577$	− 9.28483e6i	− 1.16101i	0.814258	−	0.580503i	$-0.197144\pi$
$578$	8.42389e6i	1.04880i
$579$	1.38242e6	0.171373
$580$	0	0
$581$	1.96921e6	0.242020
$582$	7.75775e6i	0.949354i
$583$	− 4.25974e6i	− 0.519053i
$584$	1.03901e7	1.26063
$585$	0	0
$586$	1.18840e6	0.142961
$587$	− 1.47623e6i	− 0.176831i	−0.996084	−	0.0884155i	$-0.971820\pi$
$587$	− 1.47623e6i	− 0.176831i	0.996084	−	0.0884155i	$-0.0281803\pi$
$588$	− 86436.0i	− 0.0103098i
$589$	−214720.	−0.0255026
$590$	0	0
$591$	−1.38980e6	−0.163675
$592$	− 6.17302e6i	− 0.723925i
$593$	− 1.24007e7i	− 1.44813i	−0.689729	−	0.724067i	$-0.742270\pi$
$593$	− 1.24007e7i	− 1.44813i	0.689729	−	0.724067i	$-0.257730\pi$
$594$	−1.94206e6	−0.225837
$595$	0	0
$596$	−328296.	−0.0378573
$597$	− 3.30170e6i	− 0.379142i
$598$	1.11384e7i	1.27371i
$599$	3.69127e6	0.420348	0.210174	−	0.977664i	$-0.432597\pi$
$599$	3.69127e6	0.420348	0.210174	+	0.977664i	$0.432597\pi$
$600$	0	0
$601$	9.12223e6	1.03018	0.515092	−	0.857135i	$-0.327758\pi$
$601$	9.12223e6	1.03018	0.515092	+	0.857135i	$0.327758\pi$
$602$	− 3.38806e6i	− 0.381030i
$603$	− 4.80784e6i	− 0.538464i
$604$	1.14813e6	0.128055
$605$	0	0
$606$	−146124.	−0.0161637
$607$	5.67914e6i	0.625620i	0.949816	+	0.312810i	$0.101270\pi$
$607$	5.67914e6i	0.625620i	−0.949816	+	0.312810i	$0.898730\pi$
$608$	3.86496e6i	0.424020i
$609$	−2.39992e6	−0.262213
$610$	0	0
$611$	6.15264e6	0.666743
$612$	40824.0i	0.00440592i
$613$	− 1.40106e7i	− 1.50593i	−0.658060	−	0.752966i	$-0.728623\pi$
$613$	− 1.40106e7i	− 1.50593i	0.658060	−	0.752966i	$-0.271377\pi$
$614$	−6.27382e6	−0.671600
$615$	0	0
$616$	3.65501e6	0.388094
$617$	253686.i	0.0268277i	0.999910	+	0.0134139i	$0.00426989\pi$
$617$	253686.i	0.0268277i	−0.999910	+	0.0134139i	$0.995730\pi$
$618$	7.11547e6i	0.749433i
$619$	−4.30034e6	−0.451103	−0.225552	−	0.974231i	$-0.572418\pi$
$619$	−4.30034e6	−0.451103	−0.225552	+	0.974231i	$0.572418\pi$
$620$	0	0
$621$	−3.06180e6	−0.318602
$622$	1.10231e7i	1.14242i
$623$	− 390726.i	− 0.0403322i
$624$	4.51901e6	0.464603
$625$	0	0
$626$	2.19296e6	0.223664
$627$	1.07253e7i	1.08953i
$628$	519512.i	0.0525650i
$629$	−684684.	−0.0690023
$630$	0	0
$631$	1.04150e7	1.04132	0.520662	−	0.853763i	$-0.325685\pi$
$631$	1.04150e7	1.04132	0.520662	+	0.853763i	$0.325685\pi$
$632$	1.10504e7i	1.10048i
$633$	− 4.68220e6i	− 0.464451i
$634$	−170028.	−0.0167995
$635$	0	0
$636$	345384.	0.0338579
$637$	1.06124e6i	0.103625i
$638$	1.44975e7i	1.41007i
$639$	−2.59524e6	−0.251435
$640$	0	0
$641$	4.52714e6	0.435190	0.217595	−	0.976039i	$-0.430179\pi$
$641$	4.52714e6	0.435190	0.217595	+	0.976039i	$0.430179\pi$
$642$	6.96146e6i	0.666596i
$643$	1.49687e7i	1.42776i	0.700266	+	0.713882i	$0.253065\pi$
$643$	1.49687e7i	1.42776i	−0.700266	+	0.713882i	$0.746935\pi$
$644$	−823200.	−0.0782151
$645$	0	0
$646$	2.02910e6	0.191304
$647$	1.73020e7i	1.62493i	0.583010	+	0.812465i	$0.301875\pi$
$647$	1.73020e7i	1.62493i	−0.583010	+	0.812465i	$0.698125\pi$
$648$	1.10225e6i	0.103120i
$649$	−1.22064e7	−1.13757
$650$	0	0
$651$	−35280.0	−0.00326269
$652$	− 2.22114e6i	− 0.204624i
$653$	4.07470e6i	0.373949i	0.982365	+	0.186975i	$0.0598683\pi$
$653$	4.07470e6i	0.373949i	−0.982365	+	0.186975i	$0.940132\pi$
$654$	5.45281e6	0.498513
$655$	0	0
$656$	−9.04483e6	−0.820618
$657$	5.00953e6i	0.452776i
$658$	4.09248e6i	0.368487i
$659$	3.79475e6	0.340384	0.170192	−	0.985411i	$-0.445561\pi$
$659$	3.79475e6	0.340384	0.170192	+	0.985411i	$0.445561\pi$
$660$	0	0
$661$	1.64261e7	1.46228	0.731142	−	0.682225i	$-0.238988\pi$
$661$	1.64261e7	1.46228	0.731142	+	0.682225i	$0.238988\pi$
$662$	1.16035e7i	1.02907i
$663$	− 501228.i	− 0.0442845i
$664$	−6.75158e6	−0.594272
$665$	0	0
$666$	2.64092e6	0.230712
$667$	2.28564e7i	1.98927i
$668$	174048.i	0.0150913i
$669$	2.74262e6	0.236920
$670$	0	0
$671$	2.19682e7	1.88360
$672$	635040.i	0.0542473i
$673$	5.50675e6i	0.468660i	0.972157	+	0.234330i	$0.0752896\pi$
$673$	5.50675e6i	0.468660i	−0.972157	+	0.234330i	$0.924710\pi$
$674$	−1.13290e7	−0.960602
$675$	0	0
$676$	−703716.	−0.0592285
$677$	− 1.83957e7i	− 1.54257i	−0.636488	−	0.771286i	$-0.719614\pi$
$677$	− 1.83957e7i	− 1.54257i	0.636488	−	0.771286i	$-0.280386\pi$
$678$	1.18879e7i	0.993185i
$679$	7.03944e6	0.585954
$680$	0	0
$681$	−2.59729e6	−0.214612
$682$	213120.i	0.0175454i
$683$	1.75835e6i	0.144229i	0.997396	+	0.0721146i	$0.0229747\pi$
$683$	1.75835e6i	0.144229i	−0.997396	+	0.0721146i	$0.977025\pi$
$684$	−869616.	−0.0710702
$685$	0	0
$686$	−705894.	−0.0572703
$687$	6.94971e6i	0.561791i
$688$	1.30913e7i	1.05441i
$689$	−4.24055e6	−0.340309
$690$	0	0
$691$	−5.36314e6	−0.427291	−0.213646	−	0.976911i	$-0.568534\pi$
$691$	−5.36314e6	−0.427291	−0.213646	+	0.976911i	$0.568534\pi$
$692$	73320.0i	0.00582046i
$693$	1.76224e6i	0.139390i
$694$	1.75162e7	1.38052
$695$	0	0
$696$	8.22830e6	0.643854
$697$	1.00321e6i	0.0782187i
$698$	4.68409e6i	0.363904i
$699$	2.27011e6	0.175733
$700$	0	0
$701$	−2.12606e7	−1.63411	−0.817054	−	0.576561i	$-0.804394\pi$
$701$	−2.12606e7	−1.63411	−0.817054	+	0.576561i	$0.804394\pi$
$702$	1.93331e6i	0.148067i
$703$	− 1.45849e7i	− 1.11305i
$704$	−1.23041e7	−0.935662
$705$	0	0
$706$	−8.00622e6	−0.604527
$707$	132594.i	0.00997643i
$708$	− 989712.i	− 0.0742037i
$709$	−2.07729e6	−0.155196	−0.0775980	−	0.996985i	$-0.524725\pi$
$709$	−2.07729e6	−0.155196	−0.0775980	+	0.996985i	$0.524725\pi$
$710$	0	0
$711$	−5.32786e6	−0.395256
$712$	1.33963e6i	0.0990343i
$713$	336000.i	0.0247523i
$714$	333396.	0.0244746
$715$	0	0
$716$	−613296.	−0.0447082
$717$	− 1.30602e7i	− 0.948752i
$718$	− 6.10459e6i	− 0.441922i
$719$	−4.23619e6	−0.305600	−0.152800	−	0.988257i	$-0.548829\pi$
$719$	−4.23619e6	−0.305600	−0.152800	+	0.988257i	$0.548829\pi$
$720$	0	0
$721$	6.45663e6	0.462560
$722$	2.83665e7i	2.02518i
$723$	1.31758e6i	0.0937415i
$724$	1.52826e6	0.108356
$725$	0	0
$726$	1.94859e6	0.137208
$727$	− 2.14524e7i	− 1.50536i	−0.658389	−	0.752678i	$-0.728762\pi$
$727$	− 2.14524e7i	− 1.50536i	0.658389	−	0.752678i	$-0.271238\pi$
$728$	− 3.63854e6i	− 0.254448i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	1.45202e6	0.100503
$732$	1.78121e6i	0.122867i
$733$	− 1.48892e7i	− 1.02355i	−0.859118	−	0.511777i	$-0.828987\pi$
$733$	− 1.48892e7i	− 1.02355i	0.859118	−	0.511777i	$-0.171013\pi$
$734$	5.02608e6	0.344341
$735$	0	0
$736$	6.04800e6	0.411545
$737$	2.63541e7i	1.78722i
$738$	− 3.86953e6i	− 0.261528i
$739$	−6.99324e6	−0.471050	−0.235525	−	0.971868i	$-0.575681\pi$
$739$	−6.99324e6	−0.471050	−0.235525	+	0.971868i	$0.575681\pi$
$740$	0	0
$741$	1.06770e7	0.714335
$742$	− 2.82064e6i	− 0.188078i
$743$	1.90428e6i	0.126549i	0.997996	+	0.0632745i	$0.0201544\pi$
$743$	1.90428e6i	0.126549i	−0.997996	+	0.0632745i	$0.979846\pi$
$744$	120960.	0.00801142
$745$	0	0
$746$	9.11958e6	0.599968
$747$	− 3.25523e6i	− 0.213442i
$748$	− 223776.i	− 0.0146238i
$749$	6.31688e6	0.411432
$750$	0	0
$751$	1.95361e7	1.26398	0.631988	−	0.774978i	$-0.282239\pi$
$751$	1.95361e7	1.26398	0.631988	+	0.774978i	$0.282239\pi$
$752$	− 1.58131e7i	− 1.01970i
$753$	− 5.47074e6i	− 0.351608i
$754$	1.44322e7	0.924493
$755$	0	0
$756$	−142884.	−0.00909241
$757$	− 1.25183e6i	− 0.0793973i	−0.999212	−	0.0396986i	$-0.987360\pi$
$757$	− 1.25183e6i	− 0.0793973i	0.999212	−	0.0396986i	$-0.0126398\pi$
$758$	− 1.58679e7i	− 1.00311i
$759$	1.67832e7	1.05748
$760$	0	0
$761$	2.04472e7	1.27989	0.639944	−	0.768422i	$-0.278958\pi$
$761$	2.04472e7	1.27989	0.639944	+	0.768422i	$0.278958\pi$
$762$	4.01328e6i	0.250387i
$763$	− 4.94792e6i	− 0.307689i
$764$	1.09363e6	0.0677857
$765$	0	0
$766$	−1.20802e7	−0.743876
$767$	1.21515e7i	0.745831i
$768$	− 3.48595e6i	− 0.213264i
$769$	−2.21064e6	−0.134804	−0.0674020	−	0.997726i	$-0.521471\pi$
$769$	−2.21064e6	−0.134804	−0.0674020	+	0.997726i	$0.521471\pi$
$770$	0	0
$771$	−860274.	−0.0521196
$772$	− 614408.i	− 0.0371034i
$773$	1.29151e7i	0.777405i	0.921363	+	0.388703i	$0.127077\pi$
$773$	1.29151e7i	0.777405i	−0.921363	+	0.388703i	$0.872923\pi$
$774$	−5.60066e6	−0.336037
$775$	0	0
$776$	−2.41352e7	−1.43879
$777$	− 2.39639e6i	− 0.142399i
$778$	4.35740e6i	0.258095i
$779$	−2.13700e7	−1.26171
$780$	0	0
$781$	1.42258e7	0.834541
$782$	− 3.17520e6i	− 0.185675i
$783$	3.96722e6i	0.231250i
$784$	2.72754e6	0.158482
$785$	0	0
$786$	−8.38706e6	−0.484232
$787$	1.35499e7i	0.779830i	0.920851	+	0.389915i	$0.127496\pi$
$787$	1.35499e7i	0.779830i	−0.920851	+	0.389915i	$0.872504\pi$
$788$	617688.i	0.0354367i
$789$	−1.98031e7	−1.13251
$790$	0	0
$791$	1.07872e7	0.613007
$792$	− 6.04195e6i	− 0.342266i
$793$	− 2.18693e7i	− 1.23496i
$794$	2.74547e7	1.54549
$795$	0	0
$796$	−1.46742e6	−0.0820867
$797$	2.45956e7i	1.37155i	0.727813	+	0.685776i	$0.240537\pi$
$797$	2.45956e7i	1.37155i	−0.727813	+	0.685776i	$0.759463\pi$
$798$	7.10186e6i	0.394789i
$799$	−1.75392e6	−0.0971948
$800$	0	0
$801$	−645894.	−0.0355697
$802$	− 203220.i	− 0.0111566i
$803$	− 2.74596e7i	− 1.50282i
$804$	−2.13682e6	−0.116581
$805$	0	0
$806$	212160.	0.0115034
$807$	1.59322e7i	0.861177i
$808$	− 454608.i	− 0.0244968i
$809$	−1.55237e7	−0.833920	−0.416960	−	0.908925i	$-0.636905\pi$
$809$	−1.55237e7	−0.833920	−0.416960	+	0.908925i	$0.636905\pi$
$810$	0	0
$811$	−2.66262e7	−1.42153	−0.710766	−	0.703429i	$-0.751651\pi$
$811$	−2.66262e7	−1.42153	−0.710766	+	0.703429i	$0.751651\pi$
$812$	1.06663e6i	0.0567707i
$813$	2.01154e6i	0.106734i
$814$	−1.44762e7	−0.765760
$815$	0	0
$816$	−1.28822e6	−0.0677276
$817$	3.09304e7i	1.62118i
$818$	3.51707e7i	1.83780i
$819$	1.75430e6	0.0913889
$820$	0	0
$821$	−1.23891e7	−0.641477	−0.320739	−	0.947168i	$-0.603931\pi$
$821$	−1.23891e7	−0.641477	−0.320739	+	0.947168i	$0.603931\pi$
$822$	1.42693e7i	0.736585i
$823$	− 3.65630e6i	− 0.188166i	−0.995564	−	0.0940831i	$-0.970008\pi$
$823$	− 3.65630e6i	− 0.188166i	0.995564	−	0.0940831i	$-0.0299919\pi$
$824$	−2.21370e7	−1.13580
$825$	0	0
$826$	−8.08265e6	−0.412196
$827$	− 2.80463e7i	− 1.42597i	−0.701178	−	0.712987i	$-0.747342\pi$
$827$	− 2.80463e7i	− 1.42597i	0.701178	−	0.712987i	$-0.252658\pi$
$828$	1.36080e6i	0.0689792i
$829$	−2.11153e7	−1.06712	−0.533558	−	0.845763i	$-0.679145\pi$
$829$	−2.11153e7	−1.06712	−0.533558	+	0.845763i	$0.679145\pi$
$830$	0	0
$831$	3.08500e6	0.154972
$832$	1.22487e7i	0.613454i
$833$	− 302526.i	− 0.0151060i
$834$	−1.21290e7	−0.603826
$835$	0	0
$836$	4.76678e6	0.235890
$837$	58320.0i	0.00287742i
$838$	− 1.81649e6i	− 0.0893557i
$839$	−1.33947e7	−0.656944	−0.328472	−	0.944514i	$-0.606534\pi$
$839$	−1.33947e7	−0.656944	−0.328472	+	0.944514i	$0.606534\pi$
$840$	0	0
$841$	9.10422e6	0.443867
$842$	− 3.22025e7i	− 1.56534i
$843$	− 4.32340e6i	− 0.209535i
$844$	−2.08098e6	−0.100557
$845$	0	0
$846$	6.76512e6	0.324975
$847$	− 1.76816e6i	− 0.0846865i
$848$	1.08988e7i	0.520461i
$849$	−269820.	−0.0128471
$850$	0	0
$851$	−2.28228e7	−1.08030
$852$	1.15344e6i	0.0544372i
$853$	3.01513e7i	1.41884i	0.704786	+	0.709420i	$0.251043\pi$
$853$	3.01513e7i	1.41884i	−0.704786	+	0.709420i	$0.748957\pi$
$854$	1.45465e7	0.682519
$855$	0	0
$856$	−2.16579e7	−1.01026
$857$	− 2.39894e7i	− 1.11575i	−0.829925	−	0.557875i	$-0.811617\pi$
$857$	− 2.39894e7i	− 1.11575i	0.829925	−	0.557875i	$-0.188383\pi$
$858$	− 1.05974e7i	− 0.491452i
$859$	8.87576e6	0.410414	0.205207	−	0.978719i	$-0.434213\pi$
$859$	8.87576e6	0.410414	0.205207	+	0.978719i	$0.434213\pi$
$860$	0	0
$861$	−3.51124e6	−0.161418
$862$	7.06234e6i	0.323728i
$863$	− 8.71286e6i	− 0.398230i	−0.979976	−	0.199115i	$-0.936193\pi$
$863$	− 8.71286e6i	− 0.398230i	0.979976	−	0.199115i	$-0.0638067\pi$
$864$	1.04976e6	0.0478416
$865$	0	0
$866$	2.19750e7	0.995711
$867$	− 1.26358e7i	− 0.570895i
$868$	15680.0i	0 0.000706394i
$869$	2.92045e7	1.31190
$870$	0	0
$871$	2.62354e7	1.17177
$872$	1.69643e7i	0.755518i
$873$	− 1.16366e7i	− 0.516763i
$874$	6.76368e7	2.99505
$875$	0	0
$876$	2.22646e6	0.0980288
$877$	2.95788e7i	1.29862i	0.760524	+	0.649310i	$0.224942\pi$
$877$	2.95788e7i	1.29862i	−0.760524	+	0.649310i	$0.775058\pi$
$878$	1.52205e7i	0.666333i
$879$	−1.78259e6	−0.0778180
$880$	0	0
$881$	2.45670e7	1.06638	0.533190	−	0.845995i	$-0.320993\pi$
$881$	2.45670e7	1.06638	0.533190	+	0.845995i	$0.320993\pi$
$882$	1.16689e6i	0.0505076i
$883$	1.45682e7i	0.628788i	0.949293	+	0.314394i	$0.101801\pi$
$883$	1.45682e7i	0.628788i	−0.949293	+	0.314394i	$0.898199\pi$
$884$	−222768.	−0.00958787
$885$	0	0
$886$	−3.60902e7	−1.54456
$887$	− 1.61714e7i	− 0.690141i	−0.938577	−	0.345070i	$-0.887855\pi$
$887$	− 1.61714e7i	− 0.690141i	0.938577	−	0.345070i	$-0.112145\pi$
$888$	8.21621e6i	0.349654i
$889$	3.64168e6	0.154542
$890$	0	0
$891$	2.91308e6	0.122930
$892$	− 1.21894e6i	− 0.0512946i
$893$	− 3.73613e7i	− 1.56781i
$894$	4.43200e6	0.185462
$895$	0	0
$896$	−1.04052e7	−0.432995
$897$	− 1.67076e7i	− 0.693319i
$898$	− 3.39579e7i	− 1.40524i
$899$	435360.	0.0179659
$900$	0	0
$901$	1.20884e6	0.0496087
$902$	2.12108e7i	0.868041i
$903$	5.08208e6i	0.207407i
$904$	−3.69845e7	−1.50522
$905$	0	0
$906$	−1.54997e7	−0.627341
$907$	− 3.14446e7i	− 1.26919i	−0.772844	−	0.634596i	$-0.781167\pi$
$907$	− 3.14446e7i	− 1.26919i	0.772844	−	0.634596i	$-0.218833\pi$
$908$	1.15435e6i	0.0464648i
$909$	219186.	0.00879839
$910$	0	0
$911$	1.51427e7	0.604514	0.302257	−	0.953227i	$-0.402260\pi$
$911$	1.51427e7	0.604514	0.302257	+	0.953227i	$0.402260\pi$
$912$	− 2.74412e7i	− 1.09249i
$913$	1.78435e7i	0.708439i
$914$	−3.87695e7	−1.53506
$915$	0	0
$916$	3.08876e6	0.121631
$917$	7.61048e6i	0.298875i
$918$	− 551124.i	− 0.0215845i
$919$	−4.14876e7	−1.62043	−0.810214	−	0.586134i	$-0.800649\pi$
$919$	−4.14876e7	−1.62043	−0.810214	+	0.586134i	$0.800649\pi$
$920$	0	0
$921$	9.41072e6	0.365573
$922$	− 2.02412e7i	− 0.784167i
$923$	− 1.41617e7i	− 0.547155i
$924$	783216.	0.0301788
$925$	0	0
$926$	2.72985e7	1.04619
$927$	− 1.06732e7i	− 0.407939i
$928$	− 7.83648e6i	− 0.298711i
$929$	1.78495e7	0.678556	0.339278	−	0.940686i	$-0.389817\pi$
$929$	1.78495e7	0.678556	0.339278	+	0.940686i	$0.389817\pi$
$930$	0	0
$931$	6.44428e6	0.243669
$932$	− 1.00894e6i	− 0.0380473i
$933$	− 1.65346e7i	− 0.621855i
$934$	1.20681e7	0.452661
$935$	0	0
$936$	−6.01474e6	−0.224402
$937$	− 2.96399e7i	− 1.10288i	−0.834215	−	0.551439i	$-0.814079\pi$
$937$	− 2.96399e7i	− 1.10288i	0.834215	−	0.551439i	$-0.185921\pi$
$938$	1.74507e7i	0.647597i
$939$	−3.28945e6	−0.121747
$940$	0	0
$941$	−3.22282e7	−1.18648	−0.593242	−	0.805024i	$-0.702152\pi$
$941$	−3.22282e7	−1.18648	−0.593242	+	0.805024i	$0.702152\pi$
$942$	− 7.01341e6i	− 0.257515i
$943$	3.34404e7i	1.22459i
$944$	3.12309e7	1.14066
$945$	0	0
$946$	3.06999e7	1.11535
$947$	− 4.84885e7i	− 1.75697i	−0.477772	−	0.878484i	$-0.658556\pi$
$947$	− 4.84885e7i	− 1.75697i	0.477772	−	0.878484i	$-0.341444\pi$
$948$	2.36794e6i	0.0855754i
$949$	−2.73359e7	−0.985300
$950$	0	0
$951$	255042.	0.00914451
$952$	1.03723e6i	0.0370923i
$953$	− 2.03264e7i	− 0.724983i	−0.931987	−	0.362491i	$-0.881926\pi$
$953$	− 2.03264e7i	− 0.724983i	0.931987	−	0.362491i	$-0.118074\pi$
$954$	−4.66268e6	−0.165869
$955$	0	0
$956$	−5.80454e6	−0.205411
$957$	− 2.17462e7i	− 0.767546i
$958$	4.56241e7i	1.60613i
$959$	1.29481e7	0.454630
$960$	0	0
$961$	−2.86228e7	−0.999776
$962$	1.44110e7i	0.502060i
$963$	− 1.04422e7i	− 0.362849i
$964$	585592.	0.0202956
$965$	0	0
$966$	1.11132e7	0.383174
$967$	3.66292e6i	0.125968i	0.998015	+	0.0629841i	$0.0200618\pi$
$967$	3.66292e6i	0.125968i	−0.998015	+	0.0629841i	$0.979938\pi$
$968$	6.06228e6i	0.207945i
$969$	−3.04366e6	−0.104132
$970$	0	0
$971$	1.48741e6	0.0506271	0.0253136	−	0.999680i	$-0.491942\pi$
$971$	1.48741e6	0.0506271	0.0253136	+	0.999680i	$0.491942\pi$
$972$	236196.i	0.00801875i
$973$	1.10060e7i	0.372689i
$974$	4.03867e6	0.136408
$975$	0	0
$976$	−5.62070e7	−1.88871
$977$	− 4.07930e7i	− 1.36725i	−0.729831	−	0.683627i	$-0.760401\pi$
$977$	− 4.07930e7i	− 1.36725i	0.729831	−	0.683627i	$-0.239599\pi$
$978$	2.99853e7i	1.00245i
$979$	3.54046e6	0.118060
$980$	0	0
$981$	−8.17922e6	−0.271356
$982$	− 1.48302e7i	− 0.490759i
$983$	− 9.26326e6i	− 0.305759i	−0.988245	−	0.152880i	$-0.951145\pi$
$983$	− 9.26326e6i	− 0.305759i	0.988245	−	0.152880i	$-0.0488547\pi$
$984$	1.20385e7	0.396357
$985$	0	0
$986$	−4.11415e6	−0.134768
$987$	− 6.13872e6i	− 0.200579i
$988$	− 4.74531e6i	− 0.154658i
$989$	4.84008e7	1.57348
$990$	0	0
$991$	−5.22051e7	−1.68861	−0.844303	−	0.535866i	$-0.819985\pi$
$991$	−5.22051e7	−1.68861	−0.844303	+	0.535866i	$0.819985\pi$
$992$	− 115200.i	− 0.00371684i
$993$	− 1.74052e7i	− 0.560153i
$994$	9.41976e6	0.302394
$995$	0	0
$996$	−1.44677e6	−0.0462116
$997$	1.86609e7i	0.594560i	0.954790	+	0.297280i	$0.0960795\pi$
$997$	1.86609e7i	0.594560i	−0.954790	+	0.297280i	$0.903921\pi$
$998$	− 3.64891e7i	− 1.15968i
$999$	−3.96139e6	−0.125584

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.6.d.b.274.2		2
5.2	odd	4		21.6.a.a.1.1	✓	1
5.3	odd	4		525.6.a.d.1.1		1
5.4	even	2	inner	525.6.d.b.274.1		2
15.2	even	4		63.6.a.d.1.1		1
20.7	even	4		336.6.a.r.1.1		1
35.2	odd	12		147.6.e.j.67.1		2
35.12	even	12		147.6.e.i.67.1		2
35.17	even	12		147.6.e.i.79.1		2
35.27	even	4		147.6.a.b.1.1		1
35.32	odd	12		147.6.e.j.79.1		2
60.47	odd	4		1008.6.a.c.1.1		1
105.62	odd	4		441.6.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.a.a.1.1	✓	1	5.2	odd	4
63.6.a.d.1.1		1	15.2	even	4
147.6.a.b.1.1		1	35.27	even	4
147.6.e.i.67.1		2	35.12	even	12
147.6.e.i.79.1		2	35.17	even	12
147.6.e.j.67.1		2	35.2	odd	12
147.6.e.j.79.1		2	35.32	odd	12
336.6.a.r.1.1		1	20.7	even	4
441.6.a.j.1.1		1	105.62	odd	4
525.6.a.d.1.1		1	5.3	odd	4
525.6.d.b.274.1		2	5.4	even	2	inner
525.6.d.b.274.2		2	1.1	even	1	trivial
1008.6.a.c.1.1		1	60.47	odd	4

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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+6.00000i q^{2} -9.00000i q^{3} -4.00000 q^{4} +54.0000 q^{6} -49.0000i q^{7} +168.000i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q+6.00000i q^{2} -9.00000i q^{3} -4.00000 q^{4} +54.0000 q^{6} -49.0000i q^{7} +168.000i q^{8} -81.0000 q^{9} +444.000 q^{11} +36.0000i q^{12} -442.000i q^{13} +294.000 q^{14} -1136.00 q^{16} +126.000i q^{17} -486.000i q^{18} -2684.00 q^{19} -441.000 q^{21} +2664.00i q^{22} +4200.00i q^{23} +1512.00 q^{24} +2652.00 q^{26} +729.000i q^{27} +196.000i q^{28} +5442.00 q^{29} +80.0000 q^{31} -1440.00i q^{32} -3996.00i q^{33} -756.000 q^{34} +324.000 q^{36} +5434.00i q^{37} -16104.0i q^{38} -3978.00 q^{39} +7962.00 q^{41} -2646.00i q^{42} -11524.0i q^{43} -1776.00 q^{44} -25200.0 q^{46} +13920.0i q^{47} +10224.0i q^{48} -2401.00 q^{49} +1134.00 q^{51} +1768.00i q^{52} -9594.00i q^{53} -4374.00 q^{54} +8232.00 q^{56} +24156.0i q^{57} +32652.0i q^{58} -27492.0 q^{59} +49478.0 q^{61} +480.000i q^{62} +3969.00i q^{63} -27712.0 q^{64} +23976.0 q^{66} +59356.0i q^{67} -504.000i q^{68} +37800.0 q^{69} +32040.0 q^{71} -13608.0i q^{72} -61846.0i q^{73} -32604.0 q^{74} +10736.0 q^{76} -21756.0i q^{77} -23868.0i q^{78} +65776.0 q^{79} +6561.00 q^{81} +47772.0i q^{82} +40188.0i q^{83} +1764.00 q^{84} +69144.0 q^{86} -48978.0i q^{87} +74592.0i q^{88} +7974.00 q^{89} -21658.0 q^{91} -16800.0i q^{92} -720.000i q^{93} -83520.0 q^{94} -12960.0 q^{96} +143662. i q^{97} -14406.0i q^{98} -35964.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4} + 108 q^{6} - 162 q^{9}+O(q^{10}) \) 2 * q - 8 * q^4 + 108 * q^6 - 162 * q^9	\( 2 q - 8 q^{4} + 108 q^{6} - 162 q^{9} + 888 q^{11} + 588 q^{14} - 2272 q^{16} - 5368 q^{19} - 882 q^{21} + 3024 q^{24} + 5304 q^{26} + 10884 q^{29} + 160 q^{31} - 1512 q^{34} + 648 q^{36} - 7956 q^{39} + 15924 q^{41} - 3552 q^{44} - 50400 q^{46} - 4802 q^{49} + 2268 q^{51} - 8748 q^{54} + 16464 q^{56} - 54984 q^{59} + 98956 q^{61} - 55424 q^{64} + 47952 q^{66} + 75600 q^{69} + 64080 q^{71} - 65208 q^{74} + 21472 q^{76} + 131552 q^{79} + 13122 q^{81} + 3528 q^{84} + 138288 q^{86} + 15948 q^{89} - 43316 q^{91} - 167040 q^{94} - 25920 q^{96} - 71928 q^{99}+O(q^{100}) \) 2 * q - 8 * q^4 + 108 * q^6 - 162 * q^9 + 888 * q^11 + 588 * q^14 - 2272 * q^16 - 5368 * q^19 - 882 * q^21 + 3024 * q^24 + 5304 * q^26 + 10884 * q^29 + 160 * q^31 - 1512 * q^34 + 648 * q^36 - 7956 * q^39 + 15924 * q^41 - 3552 * q^44 - 50400 * q^46 - 4802 * q^49 + 2268 * q^51 - 8748 * q^54 + 16464 * q^56 - 54984 * q^59 + 98956 * q^61 - 55424 * q^64 + 47952 * q^66 + 75600 * q^69 + 64080 * q^71 - 65208 * q^74 + 21472 * q^76 + 131552 * q^79 + 13122 * q^81 + 3528 * q^84 + 138288 * q^86 + 15948 * q^89 - 43316 * q^91 - 167040 * q^94 - 25920 * q^96 - 71928 * q^99

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