LMFDB - Embedded newform 1050.6.g.m.799.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,6,Mod(799,1050)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, 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("1050.799");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1050.g (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:	$168.403010804$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1050.799
Dual form			1050.6.g.m.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +49.0000i q^{7} -64.0000i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +49.0000i q^{7} -64.0000i q^{8} -81.0000 q^{9} +66.0000 q^{11} +144.000i q^{12} -98.0000i q^{13} -196.000 q^{14} +256.000 q^{16} -216.000i q^{17} -324.000i q^{18} +340.000 q^{19} +441.000 q^{21} +264.000i q^{22} +1038.00i q^{23} -576.000 q^{24} +392.000 q^{26} +729.000i q^{27} -784.000i q^{28} +2490.00 q^{29} -7048.00 q^{31} +1024.00i q^{32} -594.000i q^{33} +864.000 q^{34} +1296.00 q^{36} -12238.0i q^{37} +1360.00i q^{38} -882.000 q^{39} +6468.00 q^{41} +1764.00i q^{42} +15412.0i q^{43} -1056.00 q^{44} -4152.00 q^{46} +20604.0i q^{47} -2304.00i q^{48} -2401.00 q^{49} -1944.00 q^{51} +1568.00i q^{52} -32490.0i q^{53} -2916.00 q^{54} +3136.00 q^{56} -3060.00i q^{57} +9960.00i q^{58} -34224.0 q^{59} +35654.0 q^{61} -28192.0i q^{62} -3969.00i q^{63} -4096.00 q^{64} +2376.00 q^{66} +12680.0i q^{67} +3456.00i q^{68} +9342.00 q^{69} -42642.0 q^{71} +5184.00i q^{72} -33734.0i q^{73} +48952.0 q^{74} -5440.00 q^{76} +3234.00i q^{77} -3528.00i q^{78} +85108.0 q^{79} +6561.00 q^{81} +25872.0i q^{82} +106764. i q^{83} -7056.00 q^{84} -61648.0 q^{86} -22410.0i q^{87} -4224.00i q^{88} -34884.0 q^{89} +4802.00 q^{91} -16608.0i q^{92} +63432.0i q^{93} -82416.0 q^{94} +9216.00 q^{96} +18662.0i q^{97} -9604.00i q^{98} -5346.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} + 72 q^{6} - 162 q^{9}+O(q^{10}) $ 2 * q - 32 * q^4 + 72 * q^6 - 162 * q^9	$ 2 q - 32 q^{4} + 72 q^{6} - 162 q^{9} + 132 q^{11} - 392 q^{14} + 512 q^{16} + 680 q^{19} + 882 q^{21} - 1152 q^{24} + 784 q^{26} + 4980 q^{29} - 14096 q^{31} + 1728 q^{34} + 2592 q^{36} - 1764 q^{39} + 12936 q^{41} - 2112 q^{44} - 8304 q^{46} - 4802 q^{49} - 3888 q^{51} - 5832 q^{54} + 6272 q^{56} - 68448 q^{59} + 71308 q^{61} - 8192 q^{64} + 4752 q^{66} + 18684 q^{69} - 85284 q^{71} + 97904 q^{74} - 10880 q^{76} + 170216 q^{79} + 13122 q^{81} - 14112 q^{84} - 123296 q^{86} - 69768 q^{89} + 9604 q^{91} - 164832 q^{94} + 18432 q^{96} - 10692 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 + 72 * q^6 - 162 * q^9 + 132 * q^11 - 392 * q^14 + 512 * q^16 + 680 * q^19 + 882 * q^21 - 1152 * q^24 + 784 * q^26 + 4980 * q^29 - 14096 * q^31 + 1728 * q^34 + 2592 * q^36 - 1764 * q^39 + 12936 * q^41 - 2112 * q^44 - 8304 * q^46 - 4802 * q^49 - 3888 * q^51 - 5832 * q^54 + 6272 * q^56 - 68448 * q^59 + 71308 * q^61 - 8192 * q^64 + 4752 * q^66 + 18684 * q^69 - 85284 * q^71 + 97904 * q^74 - 10880 * q^76 + 170216 * q^79 + 13122 * q^81 - 14112 * q^84 - 123296 * q^86 - 69768 * q^89 + 9604 * q^91 - 164832 * q^94 + 18432 * q^96 - 10692 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	36.0000	0.408248
$7$	49.0000i	0.377964i
$7$	49.0000i	0.377964i
$8$	− 64.0000i	− 0.353553i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	66.0000	0.164461	0.0822304	−	0.996613i	$-0.473796\pi$
$11$	66.0000	0.164461	0.0822304	+	0.996613i	$0.473796\pi$
$12$	144.000i	0.288675i
$13$	− 98.0000i	− 0.160830i	−0.996761	−	0.0804151i	$-0.974375\pi$
$13$	− 98.0000i	− 0.160830i	0.996761	−	0.0804151i	$-0.0256246\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	− 216.000i	− 0.181272i	−0.995884	−	0.0906362i	$-0.971110\pi$
$17$	− 216.000i	− 0.181272i	0.995884	−	0.0906362i	$-0.0288900\pi$
$18$	− 324.000i	− 0.235702i
$19$	340.000	0.216070	0.108035	−	0.994147i	$-0.465544\pi$
$19$	340.000	0.216070	0.108035	+	0.994147i	$0.465544\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	264.000i	0.116291i
$23$	1038.00i	0.409145i	0.978851	+	0.204573i	$0.0655805\pi$
$23$	1038.00i	0.409145i	−0.978851	+	0.204573i	$0.934420\pi$
$24$	−576.000	−0.204124
$25$	0	0
$26$	392.000	0.113724
$27$	729.000i	0.192450i
$28$	− 784.000i	− 0.188982i
$29$	2490.00	0.549800	0.274900	−	0.961473i	$-0.411355\pi$
$29$	2490.00	0.549800	0.274900	+	0.961473i	$0.411355\pi$
$30$	0	0
$31$	−7048.00	−1.31723	−0.658615	−	0.752480i	$-0.728857\pi$
$31$	−7048.00	−1.31723	−0.658615	+	0.752480i	$0.728857\pi$
$32$	1024.00i	0.176777i
$33$	− 594.000i	− 0.0949514i
$34$	864.000	0.128179
$35$	0	0
$36$	1296.00	0.166667
$37$	− 12238.0i	− 1.46962i	−0.678271	−	0.734812i	$-0.737270\pi$
$37$	− 12238.0i	− 1.46962i	0.678271	−	0.734812i	$-0.262730\pi$
$38$	1360.00i	0.152785i
$39$	−882.000	−0.0928554
$40$	0	0
$41$	6468.00	0.600911	0.300456	−	0.953796i	$-0.402861\pi$
$41$	6468.00	0.600911	0.300456	+	0.953796i	$0.402861\pi$
$42$	1764.00i	0.154303i
$43$	15412.0i	1.27112i	0.772050	+	0.635562i	$0.219232\pi$
$43$	15412.0i	1.27112i	−0.772050	+	0.635562i	$0.780768\pi$
$44$	−1056.00	−0.0822304
$45$	0	0
$46$	−4152.00	−0.289310
$47$	20604.0i	1.36053i	0.732968	+	0.680263i	$0.238134\pi$
$47$	20604.0i	1.36053i	−0.732968	+	0.680263i	$0.761866\pi$
$48$	− 2304.00i	− 0.144338i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−1944.00	−0.104658
$52$	1568.00i	0.0804151i
$53$	− 32490.0i	− 1.58877i	−0.607417	−	0.794383i	$-0.707794\pi$
$53$	− 32490.0i	− 1.58877i	0.607417	−	0.794383i	$-0.292206\pi$
$54$	−2916.00	−0.136083
$55$	0	0
$56$	3136.00	0.133631
$57$	− 3060.00i	− 0.124748i
$58$	9960.00i	0.388767i
$59$	−34224.0	−1.27997	−0.639986	−	0.768386i	$-0.721060\pi$
$59$	−34224.0	−1.27997	−0.639986	+	0.768386i	$0.721060\pi$
$60$	0	0
$61$	35654.0	1.22683	0.613414	−	0.789762i	$-0.289796\pi$
$61$	35654.0	1.22683	0.613414	+	0.789762i	$0.289796\pi$
$62$	− 28192.0i	− 0.931422i
$63$	− 3969.00i	− 0.125988i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	2376.00	0.0671408
$67$	12680.0i	0.345090i	0.985002	+	0.172545i	$0.0551990\pi$
$67$	12680.0i	0.345090i	−0.985002	+	0.172545i	$0.944801\pi$
$68$	3456.00i	0.0906362i
$69$	9342.00	0.236220
$70$	0	0
$71$	−42642.0	−1.00390	−0.501951	−	0.864896i	$-0.667384\pi$
$71$	−42642.0	−1.00390	−0.501951	+	0.864896i	$0.667384\pi$
$72$	5184.00i	0.117851i
$73$	− 33734.0i	− 0.740902i	−0.928852	−	0.370451i	$-0.879203\pi$
$73$	− 33734.0i	− 0.740902i	0.928852	−	0.370451i	$-0.120797\pi$
$74$	48952.0	1.03918
$75$	0	0
$76$	−5440.00	−0.108035
$77$	3234.00i	0.0621603i
$78$	− 3528.00i	− 0.0656587i
$79$	85108.0	1.53427	0.767137	−	0.641484i	$-0.221681\pi$
$79$	85108.0	1.53427	0.767137	+	0.641484i	$0.221681\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	25872.0i	0.424908i
$83$	106764.i	1.70110i	0.525895	+	0.850550i	$0.323730\pi$
$83$	106764.i	1.70110i	−0.525895	+	0.850550i	$0.676270\pi$
$84$	−7056.00	−0.109109
$85$	0	0
$86$	−61648.0	−0.898820
$87$	− 22410.0i	− 0.317427i
$88$	− 4224.00i	− 0.0581456i
$89$	−34884.0	−0.466822	−0.233411	−	0.972378i	$-0.574989\pi$
$89$	−34884.0	−0.466822	−0.233411	+	0.972378i	$0.574989\pi$
$90$	0	0
$91$	4802.00	0.0607881
$92$	− 16608.0i	− 0.204573i
$93$	63432.0i	0.760503i
$94$	−82416.0	−0.962037
$95$	0	0
$96$	9216.00	0.102062
$97$	18662.0i	0.201386i	0.994918	+	0.100693i	$0.0321060\pi$
$97$	18662.0i	0.201386i	−0.994918	+	0.100693i	$0.967894\pi$
$98$	− 9604.00i	− 0.101015i
$99$	−5346.00	−0.0548202
$100$	0	0
$101$	153084.	1.49323	0.746614	−	0.665257i	$-0.231678\pi$
$101$	153084.	1.49323	0.746614	+	0.665257i	$0.231678\pi$
$102$	− 7776.00i	− 0.0740041i
$103$	− 35864.0i	− 0.333093i	−0.986034	−	0.166547i	$-0.946738\pi$
$103$	− 35864.0i	− 0.333093i	0.986034	−	0.166547i	$-0.0532616\pi$
$104$	−6272.00	−0.0568621
$105$	0	0
$106$	129960.	1.12343
$107$	− 95454.0i	− 0.805999i	−0.915200	−	0.403000i	$-0.867968\pi$
$107$	− 95454.0i	− 0.805999i	0.915200	−	0.403000i	$-0.132032\pi$
$108$	− 11664.0i	− 0.0962250i
$109$	−212222.	−1.71090	−0.855449	−	0.517887i	$-0.826719\pi$
$109$	−212222.	−1.71090	−0.855449	+	0.517887i	$0.826719\pi$
$110$	0	0
$111$	−110142.	−0.848488
$112$	12544.0i	0.0944911i
$113$	− 62106.0i	− 0.457549i	−0.973479	−	0.228774i	$-0.926528\pi$
$113$	− 62106.0i	− 0.457549i	0.973479	−	0.228774i	$-0.0734718\pi$
$114$	12240.0	0.0882103
$115$	0	0
$116$	−39840.0	−0.274900
$117$	7938.00i	0.0536101i
$118$	− 136896.i	− 0.905077i
$119$	10584.0	0.0685145
$120$	0	0
$121$	−156695.	−0.972953
$122$	142616.i	0.867498i
$123$	− 58212.0i	− 0.346936i
$124$	112768.	0.658615
$125$	0	0
$126$	15876.0	0.0890871
$127$	− 53044.0i	− 0.291828i	−0.989297	−	0.145914i	$-0.953388\pi$
$127$	− 53044.0i	− 0.291828i	0.989297	−	0.145914i	$-0.0466123\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	138708.	0.733884
$130$	0	0
$131$	69324.0	0.352944	0.176472	−	0.984306i	$-0.443532\pi$
$131$	69324.0	0.352944	0.176472	+	0.984306i	$0.443532\pi$
$132$	9504.00i	0.0474757i
$133$	16660.0i	0.0816669i
$134$	−50720.0	−0.244015
$135$	0	0
$136$	−13824.0	−0.0640894
$137$	129846.i	0.591054i	0.955334	+	0.295527i	$0.0954952\pi$
$137$	129846.i	0.591054i	−0.955334	+	0.295527i	$0.904505\pi$
$138$	37368.0i	0.167033i
$139$	104356.	0.458121	0.229061	−	0.973412i	$-0.426435\pi$
$139$	104356.	0.458121	0.229061	+	0.973412i	$0.426435\pi$
$140$	0	0
$141$	185436.	0.785500
$142$	− 170568.i	− 0.709867i
$143$	− 6468.00i	− 0.0264503i
$144$	−20736.0	−0.0833333
$145$	0	0
$146$	134936.	0.523897
$147$	21609.0i	0.0824786i
$148$	195808.i	0.734812i
$149$	−217194.	−0.801461	−0.400730	−	0.916196i	$-0.631244\pi$
$149$	−217194.	−0.801461	−0.400730	+	0.916196i	$0.631244\pi$
$150$	0	0
$151$	221000.	0.788769	0.394385	−	0.918945i	$-0.370958\pi$
$151$	221000.	0.788769	0.394385	+	0.918945i	$0.370958\pi$
$152$	− 21760.0i	− 0.0763924i
$153$	17496.0i	0.0604241i
$154$	−12936.0	−0.0439540
$155$	0	0
$156$	14112.0	0.0464277
$157$	− 378370.i	− 1.22509i	−0.790436	−	0.612544i	$-0.790146\pi$
$157$	− 378370.i	− 1.22509i	0.790436	−	0.612544i	$-0.209854\pi$
$158$	340432.i	1.08489i
$159$	−292410.	−0.917275
$160$	0	0
$161$	−50862.0	−0.154642
$162$	26244.0i	0.0785674i
$163$	− 104816.i	− 0.309000i	−0.987993	−	0.154500i	$-0.950623\pi$
$163$	− 104816.i	− 0.309000i	0.987993	−	0.154500i	$-0.0493767\pi$
$164$	−103488.	−0.300456
$165$	0	0
$166$	−427056.	−1.20286
$167$	− 426972.i	− 1.18470i	−0.805681	−	0.592350i	$-0.798200\pi$
$167$	− 426972.i	− 1.18470i	0.805681	−	0.592350i	$-0.201800\pi$
$168$	− 28224.0i	− 0.0771517i
$169$	361689.	0.974134
$170$	0	0
$171$	−27540.0	−0.0720234
$172$	− 246592.i	− 0.635562i
$173$	− 331068.i	− 0.841012i	−0.907290	−	0.420506i	$-0.861853\pi$
$173$	− 331068.i	− 0.841012i	0.907290	−	0.420506i	$-0.138147\pi$
$174$	89640.0	0.224455
$175$	0	0
$176$	16896.0	0.0411152
$177$	308016.i	0.738993i
$178$	− 139536.i	− 0.330093i
$179$	400194.	0.933551	0.466775	−	0.884376i	$-0.345416\pi$
$179$	400194.	0.933551	0.466775	+	0.884376i	$0.345416\pi$
$180$	0	0
$181$	588098.	1.33430	0.667150	−	0.744924i	$-0.267514\pi$
$181$	588098.	1.33430	0.667150	+	0.744924i	$0.267514\pi$
$182$	19208.0i	0.0429837i
$183$	− 320886.i	− 0.708309i
$184$	66432.0	0.144655
$185$	0	0
$186$	−253728.	−0.537757
$187$	− 14256.0i	− 0.0298122i
$188$	− 329664.i	− 0.680263i
$189$	−35721.0	−0.0727393
$190$	0	0
$191$	939342.	1.86312	0.931559	−	0.363590i	$-0.118449\pi$
$191$	939342.	1.86312	0.931559	+	0.363590i	$0.118449\pi$
$192$	36864.0i	0.0721688i
$193$	− 338390.i	− 0.653919i	−0.945038	−	0.326960i	$-0.893976\pi$
$193$	− 338390.i	− 0.653919i	0.945038	−	0.326960i	$-0.106024\pi$
$194$	−74648.0	−0.142401
$195$	0	0
$196$	38416.0	0.0714286
$197$	− 237942.i	− 0.436823i	−0.975857	−	0.218412i	$-0.929912\pi$
$197$	− 237942.i	− 0.436823i	0.975857	−	0.218412i	$-0.0700875\pi$
$198$	− 21384.0i	− 0.0387638i
$199$	−204464.	−0.366003	−0.183001	−	0.983113i	$-0.558581\pi$
$199$	−204464.	−0.366003	−0.183001	+	0.983113i	$0.558581\pi$
$200$	0	0
$201$	114120.	0.199238
$202$	612336.i	1.05587i
$203$	122010.i	0.207805i
$204$	31104.0	0.0523288
$205$	0	0
$206$	143456.	0.235532
$207$	− 84078.0i	− 0.136382i
$208$	− 25088.0i	− 0.0402076i
$209$	22440.0	0.0355351
$210$	0	0
$211$	−348724.	−0.539232	−0.269616	−	0.962968i	$-0.586897\pi$
$211$	−348724.	−0.539232	−0.269616	+	0.962968i	$0.586897\pi$
$212$	519840.i	0.794383i
$213$	383778.i	0.579604i
$214$	381816.	0.569928
$215$	0	0
$216$	46656.0	0.0680414
$217$	− 345352.i	− 0.497866i
$218$	− 848888.i	− 1.20979i
$219$	−303606.	−0.427760
$220$	0	0
$221$	−21168.0	−0.0291541
$222$	− 440568.i	− 0.599971i
$223$	− 1.47006e6i	− 1.97957i	−0.142554	−	0.989787i	$-0.545532\pi$
$223$	− 1.47006e6i	− 1.97957i	0.142554	−	0.989787i	$-0.454468\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	248424.	0.323536
$227$	− 589560.i	− 0.759387i	−0.925112	−	0.379694i	$-0.876029\pi$
$227$	− 589560.i	− 0.759387i	0.925112	−	0.379694i	$-0.123971\pi$
$228$	48960.0i	0.0623741i
$229$	1.04534e6	1.31725	0.658627	−	0.752469i	$-0.271137\pi$
$229$	1.04534e6	1.31725	0.658627	+	0.752469i	$0.271137\pi$
$230$	0	0
$231$	29106.0	0.0358883
$232$	− 159360.i	− 0.194383i
$233$	− 651222.i	− 0.785849i	−0.919571	−	0.392925i	$-0.871463\pi$
$233$	− 651222.i	− 0.785849i	0.919571	−	0.392925i	$-0.128537\pi$
$234$	−31752.0	−0.0379080
$235$	0	0
$236$	547584.	0.639986
$237$	− 765972.i	− 0.885813i
$238$	42336.0i	0.0484471i
$239$	513462.	0.581452	0.290726	−	0.956806i	$-0.406103\pi$
$239$	513462.	0.581452	0.290726	+	0.956806i	$0.406103\pi$
$240$	0	0
$241$	−694714.	−0.770484	−0.385242	−	0.922816i	$-0.625882\pi$
$241$	−694714.	−0.770484	−0.385242	+	0.922816i	$0.625882\pi$
$242$	− 626780.i	− 0.687981i
$243$	− 59049.0i	− 0.0641500i
$244$	−570464.	−0.613414
$245$	0	0
$246$	232848.	0.245321
$247$	− 33320.0i	− 0.0347506i
$248$	451072.i	0.465711i
$249$	960876.	0.982130
$250$	0	0
$251$	−1.39608e6	−1.39870	−0.699352	−	0.714777i	$-0.746528\pi$
$251$	−1.39608e6	−1.39870	−0.699352	+	0.714777i	$0.746528\pi$
$252$	63504.0i	0.0629941i
$253$	68508.0i	0.0672884i
$254$	212176.	0.206354
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 1.00520e6i	− 0.949339i	−0.880164	−	0.474670i	$-0.842568\pi$
$257$	− 1.00520e6i	− 0.949339i	0.880164	−	0.474670i	$-0.157432\pi$
$258$	554832.i	0.518934i
$259$	599662.	0.555466
$260$	0	0
$261$	−201690.	−0.183267
$262$	277296.i	0.249569i
$263$	− 1.25301e6i	− 1.11703i	−0.829494	−	0.558515i	$-0.811371\pi$
$263$	− 1.25301e6i	− 1.11703i	0.829494	−	0.558515i	$-0.188629\pi$
$264$	−38016.0	−0.0335704
$265$	0	0
$266$	−66640.0	−0.0577472
$267$	313956.i	0.269520i
$268$	− 202880.i	− 0.172545i
$269$	1.76069e6	1.48355	0.741774	−	0.670650i	$-0.233985\pi$
$269$	1.76069e6	1.48355	0.741774	+	0.670650i	$0.233985\pi$
$270$	0	0
$271$	770528.	0.637331	0.318666	−	0.947867i	$-0.396765\pi$
$271$	770528.	0.637331	0.318666	+	0.947867i	$0.396765\pi$
$272$	− 55296.0i	− 0.0453181i
$273$	− 43218.0i	− 0.0350960i
$274$	−519384.	−0.417938
$275$	0	0
$276$	−149472.	−0.118110
$277$	707738.i	0.554208i	0.960840	+	0.277104i	$0.0893747\pi$
$277$	707738.i	0.554208i	−0.960840	+	0.277104i	$0.910625\pi$
$278$	417424.i	0.323941i
$279$	570888.	0.439077
$280$	0	0
$281$	2.30432e6	1.74091	0.870456	−	0.492247i	$-0.163824\pi$
$281$	2.30432e6	1.74091	0.870456	+	0.492247i	$0.163824\pi$
$282$	741744.i	0.555432i
$283$	− 1.60903e6i	− 1.19426i	−0.802146	−	0.597128i	$-0.796308\pi$
$283$	− 1.60903e6i	− 1.19426i	0.802146	−	0.597128i	$-0.203692\pi$
$284$	682272.	0.501951
$285$	0	0
$286$	25872.0	0.0187032
$287$	316932.i	0.227123i
$288$	− 82944.0i	− 0.0589256i
$289$	1.37320e6	0.967140
$290$	0	0
$291$	167958.	0.116270
$292$	539744.i	0.370451i
$293$	− 517020.i	− 0.351834i	−0.984405	−	0.175917i	$-0.943711\pi$
$293$	− 517020.i	− 0.351834i	0.984405	−	0.175917i	$-0.0562891\pi$
$294$	−86436.0	−0.0583212
$295$	0	0
$296$	−783232.	−0.519590
$297$	48114.0i	0.0316505i
$298$	− 868776.i	− 0.566718i
$299$	101724.	0.0658030
$300$	0	0
$301$	−755188.	−0.480440
$302$	884000.i	0.557744i
$303$	− 1.37776e6i	− 0.862116i
$304$	87040.0	0.0540176
$305$	0	0
$306$	−69984.0	−0.0427263
$307$	1.35002e6i	0.817512i	0.912644	+	0.408756i	$0.134037\pi$
$307$	1.35002e6i	0.817512i	−0.912644	+	0.408756i	$0.865963\pi$
$308$	− 51744.0i	− 0.0310802i
$309$	−322776.	−0.192311
$310$	0	0
$311$	1.34538e6	0.788758	0.394379	−	0.918948i	$-0.370960\pi$
$311$	1.34538e6	0.788758	0.394379	+	0.918948i	$0.370960\pi$
$312$	56448.0i	0.0328293i
$313$	− 256154.i	− 0.147788i	−0.997266	−	0.0738942i	$-0.976457\pi$
$313$	− 256154.i	− 0.147788i	0.997266	−	0.0738942i	$-0.0235427\pi$
$314$	1.51348e6	0.866269
$315$	0	0
$316$	−1.36173e6	−0.767137
$317$	1.84629e6i	1.03193i	0.856609	+	0.515967i	$0.172567\pi$
$317$	1.84629e6i	1.03193i	−0.856609	+	0.515967i	$0.827433\pi$
$318$	− 1.16964e6i	− 0.648611i
$319$	164340.	0.0904204
$320$	0	0
$321$	−859086.	−0.465344
$322$	− 203448.i	− 0.109349i
$323$	− 73440.0i	− 0.0391675i
$324$	−104976.	−0.0555556
$325$	0	0
$326$	419264.	0.218496
$327$	1.91000e6i	0.987788i
$328$	− 413952.i	− 0.212454i
$329$	−1.00960e6	−0.514231
$330$	0	0
$331$	−3.33238e6	−1.67180	−0.835900	−	0.548881i	$-0.815054\pi$
$331$	−3.33238e6	−1.67180	−0.835900	+	0.548881i	$0.815054\pi$
$332$	− 1.70822e6i	− 0.850550i
$333$	991278.i	0.489875i
$334$	1.70789e6	0.837709
$335$	0	0
$336$	112896.	0.0545545
$337$	− 1.63481e6i	− 0.784136i	−0.919936	−	0.392068i	$-0.871760\pi$
$337$	− 1.63481e6i	− 0.784136i	0.919936	−	0.392068i	$-0.128240\pi$
$338$	1.44676e6i	0.688816i
$339$	−558954.	−0.264166
$340$	0	0
$341$	−465168.	−0.216633
$342$	− 110160.i	− 0.0509282i
$343$	− 117649.i	− 0.0539949i
$344$	986368.	0.449410
$345$	0	0
$346$	1.32427e6	0.594685
$347$	− 841530.i	− 0.375185i	−0.982247	−	0.187593i	$-0.939932\pi$
$347$	− 841530.i	− 0.375185i	0.982247	−	0.187593i	$-0.0600685\pi$
$348$	358560.i	0.158713i
$349$	977242.	0.429476	0.214738	−	0.976672i	$-0.431110\pi$
$349$	977242.	0.429476	0.214738	+	0.976672i	$0.431110\pi$
$350$	0	0
$351$	71442.0	0.0309518
$352$	67584.0i	0.0290728i
$353$	− 3.45857e6i	− 1.47727i	−0.674106	−	0.738634i	$-0.735471\pi$
$353$	− 3.45857e6i	− 1.47727i	0.674106	−	0.738634i	$-0.264529\pi$
$354$	−1.23206e6	−0.522547
$355$	0	0
$356$	558144.	0.233411
$357$	− 95256.0i	− 0.0395569i
$358$	1.60078e6i	0.660120i
$359$	3.47301e6	1.42223	0.711115	−	0.703076i	$-0.248190\pi$
$359$	3.47301e6	1.42223	0.711115	+	0.703076i	$0.248190\pi$
$360$	0	0
$361$	−2.36050e6	−0.953314
$362$	2.35239e6i	0.943492i
$363$	1.41026e6i	0.561734i
$364$	−76832.0	−0.0303941
$365$	0	0
$366$	1.28354e6	0.500850
$367$	3.11994e6i	1.20915i	0.796548	+	0.604575i	$0.206657\pi$
$367$	3.11994e6i	1.20915i	−0.796548	+	0.604575i	$0.793343\pi$
$368$	265728.i	0.102286i
$369$	−523908.	−0.200304
$370$	0	0
$371$	1.59201e6	0.600497
$372$	− 1.01491e6i	− 0.380252i
$373$	2.01673e6i	0.750543i	0.926915	+	0.375272i	$0.122451\pi$
$373$	2.01673e6i	0.750543i	−0.926915	+	0.375272i	$0.877549\pi$
$374$	57024.0	0.0210804
$375$	0	0
$376$	1.31866e6	0.481019
$377$	− 244020.i	− 0.0884244i
$378$	− 142884.i	− 0.0514344i
$379$	5.38083e6	1.92420	0.962102	−	0.272690i	$-0.0879134\pi$
$379$	5.38083e6	1.92420	0.962102	+	0.272690i	$0.0879134\pi$
$380$	0	0
$381$	−477396.	−0.168487
$382$	3.75737e6i	1.31742i
$383$	− 807432.i	− 0.281261i	−0.990062	−	0.140630i	$-0.955087\pi$
$383$	− 807432.i	− 0.281261i	0.990062	−	0.140630i	$-0.0449129\pi$
$384$	−147456.	−0.0510310
$385$	0	0
$386$	1.35356e6	0.462391
$387$	− 1.24837e6i	− 0.423708i
$388$	− 298592.i	− 0.100693i
$389$	−891390.	−0.298671	−0.149336	−	0.988787i	$-0.547714\pi$
$389$	−891390.	−0.298671	−0.149336	+	0.988787i	$0.547714\pi$
$390$	0	0
$391$	224208.	0.0741667
$392$	153664.i	0.0505076i
$393$	− 623916.i	− 0.203772i
$394$	951768.	0.308881
$395$	0	0
$396$	85536.0	0.0274101
$397$	1.12345e6i	0.357749i	0.983872	+	0.178875i	$0.0572457\pi$
$397$	1.12345e6i	0.357749i	−0.983872	+	0.178875i	$0.942754\pi$
$398$	− 817856.i	− 0.258803i
$399$	149940.	0.0471504
$400$	0	0
$401$	1.72037e6	0.534271	0.267136	−	0.963659i	$-0.413923\pi$
$401$	1.72037e6	0.534271	0.267136	+	0.963659i	$0.413923\pi$
$402$	456480.i	0.140882i
$403$	690704.i	0.211850i
$404$	−2.44934e6	−0.746614
$405$	0	0
$406$	−488040.	−0.146940
$407$	− 807708.i	− 0.241695i
$408$	124416.i	0.0370021i
$409$	−77246.0	−0.0228332	−0.0114166	−	0.999935i	$-0.503634\pi$
$409$	−77246.0	−0.0228332	−0.0114166	+	0.999935i	$0.503634\pi$
$410$	0	0
$411$	1.16861e6	0.341245
$412$	573824.i	0.166547i
$413$	− 1.67698e6i	− 0.483784i
$414$	336312.	0.0964365
$415$	0	0
$416$	100352.	0.0284310
$417$	− 939204.i	− 0.264496i
$418$	89760.0i	0.0251271i
$419$	5.20615e6	1.44871	0.724356	−	0.689427i	$-0.242137\pi$
$419$	5.20615e6	1.44871	0.724356	+	0.689427i	$0.242137\pi$
$420$	0	0
$421$	1.71847e6	0.472539	0.236270	−	0.971688i	$-0.424075\pi$
$421$	1.71847e6	0.472539	0.236270	+	0.971688i	$0.424075\pi$
$422$	− 1.39490e6i	− 0.381295i
$423$	− 1.66892e6i	− 0.453509i
$424$	−2.07936e6	−0.561714
$425$	0	0
$426$	−1.53511e6	−0.409842
$427$	1.74705e6i	0.463697i
$428$	1.52726e6i	0.403000i
$429$	−58212.0	−0.0152711
$430$	0	0
$431$	−580626.	−0.150558	−0.0752789	−	0.997163i	$-0.523985\pi$
$431$	−580626.	−0.150558	−0.0752789	+	0.997163i	$0.523985\pi$
$432$	186624.i	0.0481125i
$433$	− 4.15087e6i	− 1.06395i	−0.846761	−	0.531973i	$-0.821451\pi$
$433$	− 4.15087e6i	− 1.06395i	0.846761	−	0.531973i	$-0.178549\pi$
$434$	1.38141e6	0.352045
$435$	0	0
$436$	3.39555e6	0.855449
$437$	352920.i	0.0884042i
$438$	− 1.21442e6i	− 0.302472i
$439$	−3.88407e6	−0.961891	−0.480946	−	0.876750i	$-0.659707\pi$
$439$	−3.88407e6	−0.961891	−0.480946	+	0.876750i	$0.659707\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	− 84672.0i	− 0.0206150i
$443$	2.31499e6i	0.560453i	0.959934	+	0.280226i	$0.0904095\pi$
$443$	2.31499e6i	0.560453i	−0.959934	+	0.280226i	$0.909590\pi$
$444$	1.76227e6	0.424244
$445$	0	0
$446$	5.88022e6	1.39977
$447$	1.95475e6i	0.462723i
$448$	− 200704.i	− 0.0472456i
$449$	1.92281e6	0.450113	0.225056	−	0.974346i	$-0.427743\pi$
$449$	1.92281e6	0.450113	0.225056	+	0.974346i	$0.427743\pi$
$450$	0	0
$451$	426888.	0.0988263
$452$	993696.i	0.228774i
$453$	− 1.98900e6i	− 0.455396i
$454$	2.35824e6	0.536968
$455$	0	0
$456$	−195840.	−0.0441051
$457$	6.86215e6i	1.53699i	0.639858	+	0.768493i	$0.278993\pi$
$457$	6.86215e6i	1.53699i	−0.639858	+	0.768493i	$0.721007\pi$
$458$	4.18137e6i	0.931440i
$459$	157464.	0.0348859
$460$	0	0
$461$	2.97167e6	0.651250	0.325625	−	0.945499i	$-0.394425\pi$
$461$	2.97167e6	0.651250	0.325625	+	0.945499i	$0.394425\pi$
$462$	116424.i	0.0253768i
$463$	− 4.87423e6i	− 1.05670i	−0.849025	−	0.528352i	$-0.822810\pi$
$463$	− 4.87423e6i	− 1.05670i	0.849025	−	0.528352i	$-0.177190\pi$
$464$	637440.	0.137450
$465$	0	0
$466$	2.60489e6	0.555679
$467$	− 8.17301e6i	− 1.73416i	−0.498167	−	0.867081i	$-0.665993\pi$
$467$	− 8.17301e6i	− 1.73416i	0.498167	−	0.867081i	$-0.334007\pi$
$468$	− 127008.i	− 0.0268050i
$469$	−621320.	−0.130432
$470$	0	0
$471$	−3.40533e6	−0.707305
$472$	2.19034e6i	0.452539i
$473$	1.01719e6i	0.209050i
$474$	3.06389e6	0.626364
$475$	0	0
$476$	−169344.	−0.0342572
$477$	2.63169e6i	0.529589i
$478$	2.05385e6i	0.411148i
$479$	−2.34397e6	−0.466782	−0.233391	−	0.972383i	$-0.574982\pi$
$479$	−2.34397e6	−0.466782	−0.233391	+	0.972383i	$0.574982\pi$
$480$	0	0
$481$	−1.19932e6	−0.236360
$482$	− 2.77886e6i	− 0.544814i
$483$	457758.i	0.0892829i
$484$	2.50712e6	0.486476
$485$	0	0
$486$	236196.	0.0453609
$487$	316928.i	0.0605534i	0.999542	+	0.0302767i	$0.00963884\pi$
$487$	316928.i	0.0605534i	−0.999542	+	0.0302767i	$0.990361\pi$
$488$	− 2.28186e6i	− 0.433749i
$489$	−943344.	−0.178401
$490$	0	0
$491$	−5.20041e6	−0.973495	−0.486748	−	0.873543i	$-0.661817\pi$
$491$	−5.20041e6	−0.973495	−0.486748	+	0.873543i	$0.661817\pi$
$492$	931392.i	0.173468i
$493$	− 537840.i	− 0.0996634i
$494$	133280.	0.0245724
$495$	0	0
$496$	−1.80429e6	−0.329308
$497$	− 2.08946e6i	− 0.379440i
$498$	3.84350e6i	0.694471i
$499$	4.86773e6	0.875135	0.437568	−	0.899185i	$-0.355840\pi$
$499$	4.86773e6	0.875135	0.437568	+	0.899185i	$0.355840\pi$
$500$	0	0
$501$	−3.84275e6	−0.683987
$502$	− 5.58432e6i	− 0.989034i
$503$	− 426888.i	− 0.0752305i	−0.999292	−	0.0376153i	$-0.988024\pi$
$503$	− 426888.i	− 0.0752305i	0.999292	−	0.0376153i	$-0.0119761\pi$
$504$	−254016.	−0.0445435
$505$	0	0
$506$	−274032.	−0.0475801
$507$	− 3.25520e6i	− 0.562416i
$508$	848704.i	0.145914i
$509$	9.41621e6	1.61095	0.805474	−	0.592631i	$-0.201911\pi$
$509$	9.41621e6	1.61095	0.805474	+	0.592631i	$0.201911\pi$
$510$	0	0
$511$	1.65297e6	0.280035
$512$	262144.i	0.0441942i
$513$	247860.i	0.0415827i
$514$	4.02082e6	0.671284
$515$	0	0
$516$	−2.21933e6	−0.366942
$517$	1.35986e6i	0.223753i
$518$	2.39865e6i	0.392773i
$519$	−2.97961e6	−0.485558
$520$	0	0
$521$	1.84039e6	0.297041	0.148520	−	0.988909i	$-0.452549\pi$
$521$	1.84039e6	0.297041	0.148520	+	0.988909i	$0.452549\pi$
$522$	− 806760.i	− 0.129589i
$523$	979108.i	0.156522i	0.996933	+	0.0782612i	$0.0249368\pi$
$523$	979108.i	0.156522i	−0.996933	+	0.0782612i	$0.975063\pi$
$524$	−1.10918e6	−0.176472
$525$	0	0
$526$	5.01204e6	0.789860
$527$	1.52237e6i	0.238777i
$528$	− 152064.i	− 0.0237379i
$529$	5.35890e6	0.832600
$530$	0	0
$531$	2.77214e6	0.426658
$532$	− 266560.i	− 0.0408334i
$533$	− 633864.i	− 0.0966447i
$534$	−1.25582e6	−0.190579
$535$	0	0
$536$	811520.	0.122008
$537$	− 3.60175e6i	− 0.538986i
$538$	7.04275e6i	1.04903i
$539$	−158466.	−0.0234944
$540$	0	0
$541$	5.96117e6	0.875666	0.437833	−	0.899056i	$-0.355746\pi$
$541$	5.96117e6	0.875666	0.437833	+	0.899056i	$0.355746\pi$
$542$	3.08211e6i	0.450661i
$543$	− 5.29288e6i	− 0.770358i
$544$	221184.	0.0320447
$545$	0	0
$546$	172872.	0.0248166
$547$	8.73025e6i	1.24755i	0.781604	+	0.623775i	$0.214402\pi$
$547$	8.73025e6i	1.24755i	−0.781604	+	0.623775i	$0.785598\pi$
$548$	− 2.07754e6i	− 0.295527i
$549$	−2.88797e6	−0.408943
$550$	0	0
$551$	846600.	0.118795
$552$	− 597888.i	− 0.0835165i
$553$	4.17029e6i	0.579901i
$554$	−2.83095e6	−0.391885
$555$	0	0
$556$	−1.66970e6	−0.229061
$557$	− 3.01066e6i	− 0.411172i	−0.978639	−	0.205586i	$-0.934090\pi$
$557$	− 3.01066e6i	− 0.411172i	0.978639	−	0.205586i	$-0.0659101\pi$
$558$	2.28355e6i	0.310474i
$559$	1.51038e6	0.204435
$560$	0	0
$561$	−128304.	−0.0172121
$562$	9.21727e6i	1.23101i
$563$	− 1.17573e7i	− 1.56327i	−0.623733	−	0.781637i	$-0.714385\pi$
$563$	− 1.17573e7i	− 1.56327i	0.623733	−	0.781637i	$-0.285615\pi$
$564$	−2.96698e6	−0.392750
$565$	0	0
$566$	6.43611e6	0.844467
$567$	321489.i	0.0419961i
$568$	2.72909e6i	0.354933i
$569$	−1.31578e7	−1.70374	−0.851870	−	0.523754i	$-0.824531\pi$
$569$	−1.31578e7	−1.70374	−0.851870	+	0.523754i	$0.824531\pi$
$570$	0	0
$571$	−1.03344e7	−1.32647	−0.663234	−	0.748412i	$-0.730817\pi$
$571$	−1.03344e7	−1.32647	−0.663234	+	0.748412i	$0.730817\pi$
$572$	103488.i	0.0132251i
$573$	− 8.45408e6i	− 1.07567i
$574$	−1.26773e6	−0.160600
$575$	0	0
$576$	331776.	0.0416667
$577$	− 7.88133e6i	− 0.985508i	−0.870169	−	0.492754i	$-0.835990\pi$
$577$	− 7.88133e6i	− 0.985508i	0.870169	−	0.492754i	$-0.164010\pi$
$578$	5.49280e6i	0.683872i
$579$	−3.04551e6	−0.377541
$580$	0	0
$581$	−5.23144e6	−0.642955
$582$	671832.i	0.0822154i
$583$	− 2.14434e6i	− 0.261290i
$584$	−2.15898e6	−0.261948
$585$	0	0
$586$	2.06808e6	0.248784
$587$	− 554568.i	− 0.0664293i	−0.999448	−	0.0332146i	$-0.989426\pi$
$587$	− 554568.i	− 0.0664293i	0.999448	−	0.0332146i	$-0.0105745\pi$
$588$	− 345744.i	− 0.0412393i
$589$	−2.39632e6	−0.284614
$590$	0	0
$591$	−2.14148e6	−0.252200
$592$	− 3.13293e6i	− 0.367406i
$593$	9.20369e6i	1.07479i	0.843329	+	0.537397i	$0.180592\pi$
$593$	9.20369e6i	1.07479i	−0.843329	+	0.537397i	$0.819408\pi$
$594$	−192456.	−0.0223803
$595$	0	0
$596$	3.47510e6	0.400730
$597$	1.84018e6i	0.211312i
$598$	406896.i	0.0465297i
$599$	−8.54295e6	−0.972839	−0.486419	−	0.873725i	$-0.661697\pi$
$599$	−8.54295e6	−0.972839	−0.486419	+	0.873725i	$0.661697\pi$
$600$	0	0
$601$	−9.61555e6	−1.08590	−0.542948	−	0.839767i	$-0.682692\pi$
$601$	−9.61555e6	−1.08590	−0.542948	+	0.839767i	$0.682692\pi$
$602$	− 3.02075e6i	− 0.339722i
$603$	− 1.02708e6i	− 0.115030i
$604$	−3.53600e6	−0.394385
$605$	0	0
$606$	5.51102e6	0.609608
$607$	2.21264e6i	0.243747i	0.992546	+	0.121873i	$0.0388902\pi$
$607$	2.21264e6i	0.243747i	−0.992546	+	0.121873i	$0.961110\pi$
$608$	348160.i	0.0381962i
$609$	1.09809e6	0.119976
$610$	0	0
$611$	2.01919e6	0.218814
$612$	− 279936.i	− 0.0302121i
$613$	7.96215e6i	0.855814i	0.903823	+	0.427907i	$0.140749\pi$
$613$	7.96215e6i	0.855814i	−0.903823	+	0.427907i	$0.859251\pi$
$614$	−5.40008e6	−0.578068
$615$	0	0
$616$	206976.	0.0219770
$617$	− 1.37397e7i	− 1.45299i	−0.687170	−	0.726497i	$-0.741147\pi$
$617$	− 1.37397e7i	− 1.45299i	0.687170	−	0.726497i	$-0.258853\pi$
$618$	− 1.29110e6i	− 0.135985i
$619$	8.70113e6	0.912744	0.456372	−	0.889789i	$-0.349149\pi$
$619$	8.70113e6	0.912744	0.456372	+	0.889789i	$0.349149\pi$
$620$	0	0
$621$	−756702.	−0.0787401
$622$	5.38152e6i	0.557736i
$623$	− 1.70932e6i	− 0.176442i
$624$	−225792.	−0.0232138
$625$	0	0
$626$	1.02462e6	0.104502
$627$	− 201960.i	− 0.0205162i
$628$	6.05392e6i	0.612544i
$629$	−2.64341e6	−0.266402
$630$	0	0
$631$	445412.	0.0445337	0.0222668	−	0.999752i	$-0.492912\pi$
$631$	445412.	0.0445337	0.0222668	+	0.999752i	$0.492912\pi$
$632$	− 5.44691e6i	− 0.542447i
$633$	3.13852e6i	0.311326i
$634$	−7.38516e6	−0.729687
$635$	0	0
$636$	4.67856e6	0.458637
$637$	235298.i	0.0229757i
$638$	657360.i	0.0639369i
$639$	3.45400e6	0.334634
$640$	0	0
$641$	−8.00119e6	−0.769147	−0.384573	−	0.923094i	$-0.625651\pi$
$641$	−8.00119e6	−0.769147	−0.384573	+	0.923094i	$0.625651\pi$
$642$	− 3.43634e6i	− 0.329048i
$643$	1.58402e7i	1.51090i	0.655209	+	0.755448i	$0.272581\pi$
$643$	1.58402e7i	1.51090i	−0.655209	+	0.755448i	$0.727419\pi$
$644$	813792.	0.0773212
$645$	0	0
$646$	293760.	0.0276956
$647$	1.30187e6i	0.122266i	0.998130	+	0.0611331i	$0.0194714\pi$
$647$	1.30187e6i	0.122266i	−0.998130	+	0.0611331i	$0.980529\pi$
$648$	− 419904.i	− 0.0392837i
$649$	−2.25878e6	−0.210505
$650$	0	0
$651$	−3.10817e6	−0.287443
$652$	1.67706e6i	0.154500i
$653$	− 7.34149e6i	− 0.673753i	−0.941549	−	0.336877i	$-0.890629\pi$
$653$	− 7.34149e6i	− 0.673753i	0.941549	−	0.336877i	$-0.109371\pi$
$654$	−7.63999e6	−0.698471
$655$	0	0
$656$	1.65581e6	0.150228
$657$	2.73245e6i	0.246967i
$658$	− 4.03838e6i	− 0.363616i
$659$	6.18934e6	0.555176	0.277588	−	0.960700i	$-0.410465\pi$
$659$	6.18934e6	0.555176	0.277588	+	0.960700i	$0.410465\pi$
$660$	0	0
$661$	−1.96690e7	−1.75097	−0.875484	−	0.483248i	$-0.839457\pi$
$661$	−1.96690e7	−1.75097	−0.875484	+	0.483248i	$0.839457\pi$
$662$	− 1.33295e7i	− 1.18214i
$663$	190512.i	0.0168321i
$664$	6.83290e6	0.601429
$665$	0	0
$666$	−3.96511e6	−0.346394
$667$	2.58462e6i	0.224948i
$668$	6.83155e6i	0.592350i
$669$	−1.32305e7	−1.14291
$670$	0	0
$671$	2.35316e6	0.201765
$672$	451584.i	0.0385758i
$673$	− 7.18259e6i	− 0.611285i	−0.952146	−	0.305642i	$-0.901129\pi$
$673$	− 7.18259e6i	− 0.611285i	0.952146	−	0.305642i	$-0.0988712\pi$
$674$	6.53922e6	0.554468
$675$	0	0
$676$	−5.78702e6	−0.487067
$677$	− 1.89192e7i	− 1.58647i	−0.608917	−	0.793234i	$-0.708396\pi$
$677$	− 1.89192e7i	− 1.58647i	0.608917	−	0.793234i	$-0.291604\pi$
$678$	− 2.23582e6i	− 0.186794i
$679$	−914438.	−0.0761167
$680$	0	0
$681$	−5.30604e6	−0.438432
$682$	− 1.86067e6i	− 0.153182i
$683$	− 2.12204e7i	− 1.74061i	−0.492512	−	0.870306i	$-0.663921\pi$
$683$	− 2.12204e7i	− 1.74061i	0.492512	−	0.870306i	$-0.336079\pi$
$684$	440640.	0.0360117
$685$	0	0
$686$	470596.	0.0381802
$687$	− 9.40808e6i	− 0.760517i
$688$	3.94547e6i	0.317781i
$689$	−3.18402e6	−0.255522
$690$	0	0
$691$	1.63276e7	1.30085	0.650424	−	0.759571i	$-0.274591\pi$
$691$	1.63276e7	1.30085	0.650424	+	0.759571i	$0.274591\pi$
$692$	5.29709e6i	0.420506i
$693$	− 261954.i	− 0.0207201i
$694$	3.36612e6	0.265296
$695$	0	0
$696$	−1.43424e6	−0.112227
$697$	− 1.39709e6i	− 0.108929i
$698$	3.90897e6i	0.303685i
$699$	−5.86100e6	−0.453710
$700$	0	0
$701$	−5.40470e6	−0.415409	−0.207705	−	0.978192i	$-0.566599\pi$
$701$	−5.40470e6	−0.415409	−0.207705	+	0.978192i	$0.566599\pi$
$702$	285768.i	0.0218862i
$703$	− 4.16092e6i	− 0.317542i
$704$	−270336.	−0.0205576
$705$	0	0
$706$	1.38343e7	1.04459
$707$	7.50112e6i	0.564387i
$708$	− 4.92826e6i	− 0.369496i
$709$	−2.21195e7	−1.65257	−0.826284	−	0.563253i	$-0.809550\pi$
$709$	−2.21195e7	−1.65257	−0.826284	+	0.563253i	$0.809550\pi$
$710$	0	0
$711$	−6.89375e6	−0.511424
$712$	2.23258e6i	0.165046i
$713$	− 7.31582e6i	− 0.538939i
$714$	381024.	0.0279709
$715$	0	0
$716$	−6.40310e6	−0.466775
$717$	− 4.62116e6i	− 0.335701i
$718$	1.38920e7i	1.00567i
$719$	−2.55819e7	−1.84548	−0.922742	−	0.385418i	$-0.874057\pi$
$719$	−2.55819e7	−1.84548	−0.922742	+	0.385418i	$0.874057\pi$
$720$	0	0
$721$	1.75734e6	0.125897
$722$	− 9.44200e6i	− 0.674095i
$723$	6.25243e6i	0.444839i
$724$	−9.40957e6	−0.667150
$725$	0	0
$726$	−5.64102e6	−0.397206
$727$	− 9.29438e6i	− 0.652205i	−0.945334	−	0.326103i	$-0.894265\pi$
$727$	− 9.29438e6i	− 0.652205i	0.945334	−	0.326103i	$-0.105735\pi$
$728$	− 307328.i	− 0.0214918i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	3.32899e6	0.230420
$732$	5.13418e6i	0.354155i
$733$	− 3.40699e6i	− 0.234213i	−0.993119	−	0.117107i	$-0.962638\pi$
$733$	− 3.40699e6i	− 0.234213i	0.993119	−	0.117107i	$-0.0373619\pi$
$734$	−1.24797e7	−0.854999
$735$	0	0
$736$	−1.06291e6	−0.0723274
$737$	836880.i	0.0567537i
$738$	− 2.09563e6i	− 0.141636i
$739$	−2.18135e7	−1.46932	−0.734658	−	0.678438i	$-0.762657\pi$
$739$	−2.18135e7	−1.46932	−0.734658	+	0.678438i	$0.762657\pi$
$740$	0	0
$741$	−299880.	−0.0200633
$742$	6.36804e6i	0.424616i
$743$	− 3.79246e6i	− 0.252028i	−0.992028	−	0.126014i	$-0.959782\pi$
$743$	− 3.79246e6i	− 0.252028i	0.992028	−	0.126014i	$-0.0402185\pi$
$744$	4.05965e6	0.268878
$745$	0	0
$746$	−8.06692e6	−0.530714
$747$	− 8.64788e6i	− 0.567033i
$748$	228096.i	0.0149061i
$749$	4.67725e6	0.304639
$750$	0	0
$751$	−2.01483e7	−1.30358	−0.651790	−	0.758400i	$-0.725982\pi$
$751$	−2.01483e7	−1.30358	−0.651790	+	0.758400i	$0.725982\pi$
$752$	5.27462e6i	0.340132i
$753$	1.25647e7i	0.807542i
$754$	976080.	0.0625255
$755$	0	0
$756$	571536.	0.0363696
$757$	1.18427e7i	0.751126i	0.926797	+	0.375563i	$0.122551\pi$
$757$	1.18427e7i	0.751126i	−0.926797	+	0.375563i	$0.877449\pi$
$758$	2.15233e7i	1.36062i
$759$	616572.	0.0388490
$760$	0	0
$761$	2.97791e6	0.186402	0.0932008	−	0.995647i	$-0.470290\pi$
$761$	2.97791e6	0.186402	0.0932008	+	0.995647i	$0.470290\pi$
$762$	− 1.90958e6i	− 0.119138i
$763$	− 1.03989e7i	− 0.646659i
$764$	−1.50295e7	−0.931559
$765$	0	0
$766$	3.22973e6	0.198881
$767$	3.35395e6i	0.205858i
$768$	− 589824.i	− 0.0360844i
$769$	2.02441e7	1.23447	0.617237	−	0.786777i	$-0.288252\pi$
$769$	2.02441e7	1.23447	0.617237	+	0.786777i	$0.288252\pi$
$770$	0	0
$771$	−9.04684e6	−0.548101
$772$	5.41424e6i	0.326960i
$773$	7.37953e6i	0.444202i	0.975024	+	0.222101i	$0.0712914\pi$
$773$	7.37953e6i	0.444202i	−0.975024	+	0.222101i	$0.928709\pi$
$774$	4.99349e6	0.299607
$775$	0	0
$776$	1.19437e6	0.0712006
$777$	− 5.39696e6i	− 0.320698i
$778$	− 3.56556e6i	− 0.211193i
$779$	2.19912e6	0.129839
$780$	0	0
$781$	−2.81437e6	−0.165103
$782$	896832.i	0.0524438i
$783$	1.81521e6i	0.105809i
$784$	−614656.	−0.0357143
$785$	0	0
$786$	2.49566e6	0.144089
$787$	1.36289e7i	0.784377i	0.919885	+	0.392188i	$0.128282\pi$
$787$	1.36289e7i	0.784377i	−0.919885	+	0.392188i	$0.871718\pi$
$788$	3.80707e6i	0.218412i
$789$	−1.12771e7	−0.644918
$790$	0	0
$791$	3.04319e6	0.172937
$792$	342144.i	0.0193819i
$793$	− 3.49409e6i	− 0.197311i
$794$	−4.49382e6	−0.252967
$795$	0	0
$796$	3.27142e6	0.183001
$797$	− 1.49548e7i	− 0.833938i	−0.908921	−	0.416969i	$-0.863092\pi$
$797$	− 1.49548e7i	− 0.833938i	0.908921	−	0.416969i	$-0.136908\pi$
$798$	599760.i	0.0333404i
$799$	4.45046e6	0.246626
$800$	0	0
$801$	2.82560e6	0.155607
$802$	6.88150e6i	0.377787i
$803$	− 2.22644e6i	− 0.121849i
$804$	−1.82592e6	−0.0996189
$805$	0	0
$806$	−2.76282e6	−0.149801
$807$	− 1.58462e7i	− 0.856527i
$808$	− 9.79738e6i	− 0.527936i
$809$	−2.87242e7	−1.54304	−0.771519	−	0.636206i	$-0.780503\pi$
$809$	−2.87242e7	−1.54304	−0.771519	+	0.636206i	$0.780503\pi$
$810$	0	0
$811$	−1.52265e7	−0.812922	−0.406461	−	0.913668i	$-0.633237\pi$
$811$	−1.52265e7	−0.812922	−0.406461	+	0.913668i	$0.633237\pi$
$812$	− 1.95216e6i	− 0.103902i
$813$	− 6.93475e6i	− 0.367963i
$814$	3.23083e6	0.170904
$815$	0	0
$816$	−497664.	−0.0261644
$817$	5.24008e6i	0.274652i
$818$	− 308984.i	− 0.0161455i
$819$	−388962.	−0.0202627
$820$	0	0
$821$	−3.31001e7	−1.71384	−0.856921	−	0.515447i	$-0.827626\pi$
$821$	−3.31001e7	−1.71384	−0.856921	+	0.515447i	$0.827626\pi$
$822$	4.67446e6i	0.241297i
$823$	1.35915e7i	0.699470i	0.936849	+	0.349735i	$0.113728\pi$
$823$	1.35915e7i	0.699470i	−0.936849	+	0.349735i	$0.886272\pi$
$824$	−2.29530e6	−0.117766
$825$	0	0
$826$	6.70790e6	0.342087
$827$	3.13936e6i	0.159616i	0.996810	+	0.0798082i	$0.0254308\pi$
$827$	3.13936e6i	0.159616i	−0.996810	+	0.0798082i	$0.974569\pi$
$828$	1.34525e6i	0.0681909i
$829$	−1.27081e7	−0.642234	−0.321117	−	0.947040i	$-0.604058\pi$
$829$	−1.27081e7	−0.642234	−0.321117	+	0.947040i	$0.604058\pi$
$830$	0	0
$831$	6.36964e6	0.319972
$832$	401408.i	0.0201038i
$833$	518616.i	0.0258960i
$834$	3.75682e6	0.187027
$835$	0	0
$836$	−359040.	−0.0177675
$837$	− 5.13799e6i	− 0.253501i
$838$	2.08246e7i	1.02439i
$839$	2.98312e7	1.46307	0.731536	−	0.681803i	$-0.238804\pi$
$839$	2.98312e7	1.46307	0.731536	+	0.681803i	$0.238804\pi$
$840$	0	0
$841$	−1.43110e7	−0.697720
$842$	6.87390e6i	0.334136i
$843$	− 2.07389e7i	− 1.00512i
$844$	5.57958e6	0.269616
$845$	0	0
$846$	6.67570e6	0.320679
$847$	− 7.67806e6i	− 0.367742i
$848$	− 8.31744e6i	− 0.397192i
$849$	−1.44813e7	−0.689504
$850$	0	0
$851$	1.27030e7	0.601290
$852$	− 6.14045e6i	− 0.289802i
$853$	1.92215e7i	0.904515i	0.891888	+	0.452257i	$0.149381\pi$
$853$	1.92215e7i	0.904515i	−0.891888	+	0.452257i	$0.850619\pi$
$854$	−6.98818e6	−0.327884
$855$	0	0
$856$	−6.10906e6	−0.284964
$857$	− 2.65655e7i	− 1.23556i	−0.786349	−	0.617782i	$-0.788031\pi$
$857$	− 2.65655e7i	− 1.23556i	0.786349	−	0.617782i	$-0.211969\pi$
$858$	− 232848.i	− 0.0107983i
$859$	9.16844e6	0.423948	0.211974	−	0.977275i	$-0.432011\pi$
$859$	9.16844e6	0.423948	0.211974	+	0.977275i	$0.432011\pi$
$860$	0	0
$861$	2.85239e6	0.131130
$862$	− 2.32250e6i	− 0.106460i
$863$	2.92196e7i	1.33551i	0.744381	+	0.667755i	$0.232745\pi$
$863$	2.92196e7i	1.33551i	−0.744381	+	0.667755i	$0.767255\pi$
$864$	−746496.	−0.0340207
$865$	0	0
$866$	1.66035e7	0.752324
$867$	− 1.23588e7i	− 0.558379i
$868$	5.52563e6i	0.248933i
$869$	5.61713e6	0.252328
$870$	0	0
$871$	1.24264e6	0.0555009
$872$	1.35822e7i	0.604894i
$873$	− 1.51162e6i	− 0.0671286i
$874$	−1.41168e6	−0.0625112
$875$	0	0
$876$	4.85770e6	0.213880
$877$	9.71286e6i	0.426430i	0.977005	+	0.213215i	$0.0683936\pi$
$877$	9.71286e6i	0.426430i	−0.977005	+	0.213215i	$0.931606\pi$
$878$	− 1.55363e7i	− 0.680160i
$879$	−4.65318e6	−0.203132
$880$	0	0
$881$	1.65372e7	0.717833	0.358917	−	0.933370i	$-0.383146\pi$
$881$	1.65372e7	0.717833	0.358917	+	0.933370i	$0.383146\pi$
$882$	777924.i	0.0336718i
$883$	2.39487e7i	1.03367i	0.856086	+	0.516833i	$0.172889\pi$
$883$	2.39487e7i	1.03367i	−0.856086	+	0.516833i	$0.827111\pi$
$884$	338688.	0.0145770
$885$	0	0
$886$	−9.25994e6	−0.396300
$887$	− 4.62846e6i	− 0.197527i	−0.995111	−	0.0987637i	$-0.968511\pi$
$887$	− 4.62846e6i	− 0.197527i	0.995111	−	0.0987637i	$-0.0314888\pi$
$888$	7.04909e6i	0.299986i
$889$	2.59916e6	0.110301
$890$	0	0
$891$	433026.	0.0182734
$892$	2.35209e7i	0.989787i
$893$	7.00536e6i	0.293969i
$894$	−7.81898e6	−0.327195
$895$	0	0
$896$	802816.	0.0334077
$897$	− 915516.i	− 0.0379914i
$898$	7.69126e6i	0.318278i
$899$	−1.75495e7	−0.724212
$900$	0	0
$901$	−7.01784e6	−0.287999
$902$	1.70755e6i	0.0698808i
$903$	6.79669e6i	0.277382i
$904$	−3.97478e6	−0.161768
$905$	0	0
$906$	7.95600e6	0.322014
$907$	2.06126e7i	0.831983i	0.909369	+	0.415991i	$0.136565\pi$
$907$	2.06126e7i	0.831983i	−0.909369	+	0.415991i	$0.863435\pi$
$908$	9.43296e6i	0.379694i
$909$	−1.23998e7	−0.497743
$910$	0	0
$911$	−3.46749e6	−0.138427	−0.0692133	−	0.997602i	$-0.522049\pi$
$911$	−3.46749e6	−0.138427	−0.0692133	+	0.997602i	$0.522049\pi$
$912$	− 783360.i	− 0.0311870i
$913$	7.04642e6i	0.279764i
$914$	−2.74486e7	−1.08681
$915$	0	0
$916$	−1.67255e7	−0.658627
$917$	3.39688e6i	0.133400i
$918$	629856.i	0.0246680i
$919$	3.61227e7	1.41088	0.705442	−	0.708767i	$-0.250748\pi$
$919$	3.61227e7	1.41088	0.705442	+	0.708767i	$0.250748\pi$
$920$	0	0
$921$	1.21502e7	0.471991
$922$	1.18867e7i	0.460504i
$923$	4.17892e6i	0.161458i
$924$	−465696.	−0.0179441
$925$	0	0
$926$	1.94969e7	0.747203
$927$	2.90498e6i	0.111031i
$928$	2.54976e6i	0.0971917i
$929$	−1.29366e7	−0.491792	−0.245896	−	0.969296i	$-0.579082\pi$
$929$	−1.29366e7	−0.491792	−0.245896	+	0.969296i	$0.579082\pi$
$930$	0	0
$931$	−816340.	−0.0308672
$932$	1.04196e7i	0.392925i
$933$	− 1.21084e7i	− 0.455390i
$934$	3.26920e7	1.22624
$935$	0	0
$936$	508032.	0.0189540
$937$	5.01394e7i	1.86565i	0.360332	+	0.932824i	$0.382664\pi$
$937$	5.01394e7i	1.86565i	−0.360332	+	0.932824i	$0.617336\pi$
$938$	− 2.48528e6i	− 0.0922292i
$939$	−2.30539e6	−0.0853257
$940$	0	0
$941$	−1.05568e7	−0.388651	−0.194325	−	0.980937i	$-0.562252\pi$
$941$	−1.05568e7	−0.388651	−0.194325	+	0.980937i	$0.562252\pi$
$942$	− 1.36213e7i	− 0.500140i
$943$	6.71378e6i	0.245860i
$944$	−8.76134e6	−0.319993
$945$	0	0
$946$	−4.06877e6	−0.147821
$947$	− 3.14684e6i	− 0.114025i	−0.998373	−	0.0570124i	$-0.981843\pi$
$947$	− 3.14684e6i	− 0.114025i	0.998373	−	0.0570124i	$-0.0181575\pi$
$948$	1.22556e7i	0.442906i
$949$	−3.30593e6	−0.119159
$950$	0	0
$951$	1.66166e7	0.595787
$952$	− 677376.i	− 0.0242235i
$953$	− 5.22829e7i	− 1.86478i	−0.361455	−	0.932389i	$-0.617720\pi$
$953$	− 5.22829e7i	− 1.86478i	0.361455	−	0.932389i	$-0.382280\pi$
$954$	−1.05268e7	−0.374476
$955$	0	0
$956$	−8.21539e6	−0.290726
$957$	− 1.47906e6i	− 0.0522043i
$958$	− 9.37589e6i	− 0.330064i
$959$	−6.36245e6	−0.223397
$960$	0	0
$961$	2.10452e7	0.735095
$962$	− 4.79730e6i	− 0.167132i
$963$	7.73177e6i	0.268666i
$964$	1.11154e7	0.385242
$965$	0	0
$966$	−1.83103e6	−0.0631325
$967$	− 2.48235e7i	− 0.853682i	−0.904327	−	0.426841i	$-0.859626\pi$
$967$	− 2.48235e7i	− 0.853682i	0.904327	−	0.426841i	$-0.140374\pi$
$968$	1.00285e7i	0.343991i
$969$	−660960.	−0.0226134
$970$	0	0
$971$	1.33077e7	0.452956	0.226478	−	0.974016i	$-0.427279\pi$
$971$	1.33077e7	0.452956	0.226478	+	0.974016i	$0.427279\pi$
$972$	944784.i	0.0320750i
$973$	5.11344e6i	0.173154i
$974$	−1.26771e6	−0.0428177
$975$	0	0
$976$	9.12742e6	0.306707
$977$	8.17705e6i	0.274069i	0.990566	+	0.137035i	$0.0437571\pi$
$977$	8.17705e6i	0.274069i	−0.990566	+	0.137035i	$0.956243\pi$
$978$	− 3.77338e6i	− 0.126149i
$979$	−2.30234e6	−0.0767739
$980$	0	0
$981$	1.71900e7	0.570299
$982$	− 2.08016e7i	− 0.688365i
$983$	1.32465e7i	0.437238i	0.975810	+	0.218619i	$0.0701552\pi$
$983$	1.32465e7i	0.437238i	−0.975810	+	0.218619i	$0.929845\pi$
$984$	−3.72557e6	−0.122661
$985$	0	0
$986$	2.15136e6	0.0704727
$987$	9.08636e6i	0.296891i
$988$	533120.i	0.0173753i
$989$	−1.59977e7	−0.520075
$990$	0	0
$991$	−1.48550e7	−0.480494	−0.240247	−	0.970712i	$-0.577228\pi$
$991$	−1.48550e7	−0.480494	−0.240247	+	0.970712i	$0.577228\pi$
$992$	− 7.21715e6i	− 0.232856i
$993$	2.99914e7i	0.965215i
$994$	8.35783e6	0.268304
$995$	0	0
$996$	−1.53740e7	−0.491065
$997$	− 3.33769e6i	− 0.106343i	−0.998585	−	0.0531714i	$-0.983067\pi$
$997$	− 3.33769e6i	− 0.106343i	0.998585	−	0.0531714i	$-0.0169330\pi$
$998$	1.94709e7i	0.618814i
$999$	8.92150e6	0.282829

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.6.g.m.799.2		2
5.2	odd	4		1050.6.a.a.1.1		1
5.3	odd	4		42.6.a.f.1.1	✓	1
5.4	even	2	inner	1050.6.g.m.799.1		2
15.8	even	4		126.6.a.b.1.1		1
20.3	even	4		336.6.a.g.1.1		1
35.3	even	12		294.6.e.f.79.1		2
35.13	even	4		294.6.a.i.1.1		1
35.18	odd	12		294.6.e.b.79.1		2
35.23	odd	12		294.6.e.b.67.1		2
35.33	even	12		294.6.e.f.67.1		2
60.23	odd	4		1008.6.a.k.1.1		1
105.83	odd	4		882.6.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.f.1.1	✓	1	5.3	odd	4
126.6.a.b.1.1		1	15.8	even	4
294.6.a.i.1.1		1	35.13	even	4
294.6.e.b.67.1		2	35.23	odd	12
294.6.e.b.79.1		2	35.18	odd	12
294.6.e.f.67.1		2	35.33	even	12
294.6.e.f.79.1		2	35.3	even	12
336.6.a.g.1.1		1	20.3	even	4
882.6.a.i.1.1		1	105.83	odd	4
1008.6.a.k.1.1		1	60.23	odd	4
1050.6.a.a.1.1		1	5.2	odd	4
1050.6.g.m.799.1		2	5.4	even	2	inner
1050.6.g.m.799.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1050.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +49.0000i q^{7} -64.0000i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{2} -9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +49.0000i q^{7} -64.0000i q^{8} -81.0000 q^{9} +66.0000 q^{11} +144.000i q^{12} -98.0000i q^{13} -196.000 q^{14} +256.000 q^{16} -216.000i q^{17} -324.000i q^{18} +340.000 q^{19} +441.000 q^{21} +264.000i q^{22} +1038.00i q^{23} -576.000 q^{24} +392.000 q^{26} +729.000i q^{27} -784.000i q^{28} +2490.00 q^{29} -7048.00 q^{31} +1024.00i q^{32} -594.000i q^{33} +864.000 q^{34} +1296.00 q^{36} -12238.0i q^{37} +1360.00i q^{38} -882.000 q^{39} +6468.00 q^{41} +1764.00i q^{42} +15412.0i q^{43} -1056.00 q^{44} -4152.00 q^{46} +20604.0i q^{47} -2304.00i q^{48} -2401.00 q^{49} -1944.00 q^{51} +1568.00i q^{52} -32490.0i q^{53} -2916.00 q^{54} +3136.00 q^{56} -3060.00i q^{57} +9960.00i q^{58} -34224.0 q^{59} +35654.0 q^{61} -28192.0i q^{62} -3969.00i q^{63} -4096.00 q^{64} +2376.00 q^{66} +12680.0i q^{67} +3456.00i q^{68} +9342.00 q^{69} -42642.0 q^{71} +5184.00i q^{72} -33734.0i q^{73} +48952.0 q^{74} -5440.00 q^{76} +3234.00i q^{77} -3528.00i q^{78} +85108.0 q^{79} +6561.00 q^{81} +25872.0i q^{82} +106764. i q^{83} -7056.00 q^{84} -61648.0 q^{86} -22410.0i q^{87} -4224.00i q^{88} -34884.0 q^{89} +4802.00 q^{91} -16608.0i q^{92} +63432.0i q^{93} -82416.0 q^{94} +9216.00 q^{96} +18662.0i q^{97} -9604.00i q^{98} -5346.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} + 72 q^{6} - 162 q^{9}+O(q^{10}) \) 2 * q - 32 * q^4 + 72 * q^6 - 162 * q^9	\( 2 q - 32 q^{4} + 72 q^{6} - 162 q^{9} + 132 q^{11} - 392 q^{14} + 512 q^{16} + 680 q^{19} + 882 q^{21} - 1152 q^{24} + 784 q^{26} + 4980 q^{29} - 14096 q^{31} + 1728 q^{34} + 2592 q^{36} - 1764 q^{39} + 12936 q^{41} - 2112 q^{44} - 8304 q^{46} - 4802 q^{49} - 3888 q^{51} - 5832 q^{54} + 6272 q^{56} - 68448 q^{59} + 71308 q^{61} - 8192 q^{64} + 4752 q^{66} + 18684 q^{69} - 85284 q^{71} + 97904 q^{74} - 10880 q^{76} + 170216 q^{79} + 13122 q^{81} - 14112 q^{84} - 123296 q^{86} - 69768 q^{89} + 9604 q^{91} - 164832 q^{94} + 18432 q^{96} - 10692 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 + 72 * q^6 - 162 * q^9 + 132 * q^11 - 392 * q^14 + 512 * q^16 + 680 * q^19 + 882 * q^21 - 1152 * q^24 + 784 * q^26 + 4980 * q^29 - 14096 * q^31 + 1728 * q^34 + 2592 * q^36 - 1764 * q^39 + 12936 * q^41 - 2112 * q^44 - 8304 * q^46 - 4802 * q^49 - 3888 * q^51 - 5832 * q^54 + 6272 * q^56 - 68448 * q^59 + 71308 * q^61 - 8192 * q^64 + 4752 * q^66 + 18684 * q^69 - 85284 * q^71 + 97904 * q^74 - 10880 * q^76 + 170216 * q^79 + 13122 * q^81 - 14112 * q^84 - 123296 * q^86 - 69768 * q^89 + 9604 * q^91 - 164832 * q^94 + 18432 * q^96 - 10692 * q^99

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