LMFDB - Embedded newform 275.6.b.f.199.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [275,6,Mod(199,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("275.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	275.b (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:	$44.1055504486$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 307x^{10} + 32123x^{8} + 1354853x^{6} + 24286348x^{4} + 189182512x^{2} + 529736256 $ x^12 + 307x^10 + 32123x^8 + 1354853x^6 + 24286348x^4 + 189182512*x^2 + 529736256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.12
Root			$11.2169i$ of defining polynomial
Character	$\chi$	$=$	275.199
Dual form			275.6.b.f.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.2169i q^{2} -11.4706i q^{3} -93.8183 q^{4} +128.664 q^{6} -132.518i q^{7} -693.408i q^{8} +111.426 q^{9} +O(q^{10})$	\(q+11.2169i q^{2} -11.4706i q^{3} -93.8183 q^{4} +128.664 q^{6} -132.518i q^{7} -693.408i q^{8} +111.426 q^{9} +121.000 q^{11} +1076.15i q^{12} +916.014i q^{13} +1486.43 q^{14} +4775.68 q^{16} -607.561i q^{17} +1249.85i q^{18} -2342.75 q^{19} -1520.05 q^{21} +1357.24i q^{22} -15.5282i q^{23} -7953.78 q^{24} -10274.8 q^{26} -4065.47i q^{27} +12432.6i q^{28} -837.103 q^{29} +2763.55 q^{31} +31379.2i q^{32} -1387.94i q^{33} +6814.93 q^{34} -10453.8 q^{36} +5914.18i q^{37} -26278.4i q^{38} +10507.2 q^{39} -9477.40 q^{41} -17050.2i q^{42} +3470.90i q^{43} -11352.0 q^{44} +174.178 q^{46} -5638.78i q^{47} -54779.8i q^{48} -753.904 q^{49} -6969.07 q^{51} -85938.8i q^{52} +10588.9i q^{53} +45601.9 q^{54} -91888.7 q^{56} +26872.7i q^{57} -9389.68i q^{58} -29479.8 q^{59} -32344.0 q^{61} +30998.4i q^{62} -14765.9i q^{63} -199154. q^{64} +15568.3 q^{66} +58356.9i q^{67} +57000.3i q^{68} -178.118 q^{69} -1075.01 q^{71} -77263.6i q^{72} -28035.0i q^{73} -66338.6 q^{74} +219793. q^{76} -16034.6i q^{77} +117858. i q^{78} -23231.7 q^{79} -19556.8 q^{81} -106307. i q^{82} -23943.0i q^{83} +142609. q^{84} -38932.6 q^{86} +9602.05i q^{87} -83902.3i q^{88} -95010.4 q^{89} +121388. q^{91} +1456.83i q^{92} -31699.5i q^{93} +63249.5 q^{94} +359937. q^{96} -33677.1i q^{97} -8456.45i q^{98} +13482.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 230 q^{4} + 650 q^{6} - 1664 q^{9}+O(q^{10}) $ 12 * q - 230 * q^4 + 650 * q^6 - 1664 * q^9	$ 12 q - 230 q^{4} + 650 q^{6} - 1664 q^{9} + 1452 q^{11} - 4034 q^{14} + 13350 q^{16} - 1220 q^{19} + 5016 q^{21} - 26894 q^{24} - 15484 q^{26} - 21928 q^{29} + 17736 q^{31} + 85346 q^{34} - 60232 q^{36} - 14416 q^{39} + 320 q^{41} - 27830 q^{44} + 3300 q^{46} - 97024 q^{49} - 86756 q^{51} + 113762 q^{54} - 42594 q^{56} - 241064 q^{59} - 18456 q^{61} - 605678 q^{64} + 78650 q^{66} - 237920 q^{69} + 327896 q^{71} - 267002 q^{74} - 58710 q^{76} + 73424 q^{79} + 404332 q^{81} - 238066 q^{84} + 167992 q^{86} - 509892 q^{89} + 335688 q^{91} - 62828 q^{94} + 468830 q^{96} - 201344 q^{99}+O(q^{100}) $ 12 * q - 230 * q^4 + 650 * q^6 - 1664 * q^9 + 1452 * q^11 - 4034 * q^14 + 13350 * q^16 - 1220 * q^19 + 5016 * q^21 - 26894 * q^24 - 15484 * q^26 - 21928 * q^29 + 17736 * q^31 + 85346 * q^34 - 60232 * q^36 - 14416 * q^39 + 320 * q^41 - 27830 * q^44 + 3300 * q^46 - 97024 * q^49 - 86756 * q^51 + 113762 * q^54 - 42594 * q^56 - 241064 * q^59 - 18456 * q^61 - 605678 * q^64 + 78650 * q^66 - 237920 * q^69 + 327896 * q^71 - 267002 * q^74 - 58710 * q^76 + 73424 * q^79 + 404332 * q^81 - 238066 * q^84 + 167992 * q^86 - 509892 * q^89 + 335688 * q^91 - 62828 * q^94 + 468830 * q^96 - 201344 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$1$	$-1$

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 $

$2$	11.2169i	1.98288i	0.130556	+	0.991441i	$0.458324\pi$
$2$	11.2169i	1.98288i	−0.130556	+	0.991441i	$0.541676\pi$
$3$	− 11.4706i	− 0.735838i	−0.929858	−	0.367919i	$-0.880070\pi$
$3$	− 11.4706i	− 0.735838i	0.929858	−	0.367919i	$-0.119930\pi$
$4$	−93.8183	−2.93182
$5$	0	0
$5$	0	0
$6$	128.664	1.45908
$7$	− 132.518i	− 1.02218i	−0.859527	−	0.511091i	$-0.829241\pi$
$7$	− 132.518i	− 1.02218i	0.859527	−	0.511091i	$-0.170759\pi$
$8$	− 693.408i	− 3.83057i
$9$	111.426	0.458543
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	1076.15i	2.15734i
$13$	916.014i	1.50329i	0.659566	+	0.751646i	$0.270740\pi$
$13$	916.014i	1.50329i	−0.659566	+	0.751646i	$0.729260\pi$
$14$	1486.43	2.02687
$15$	0	0
$16$	4775.68	4.66375
$17$	− 607.561i	− 0.509879i	−0.966957	−	0.254940i	$-0.917944\pi$
$17$	− 607.561i	− 0.509879i	0.966957	−	0.254940i	$-0.0820556\pi$
$18$	1249.85i	0.909236i
$19$	−2342.75	−1.48882	−0.744411	−	0.667722i	$-0.767269\pi$
$19$	−2342.75	−1.48882	−0.744411	+	0.667722i	$0.767269\pi$
$20$	0	0
$21$	−1520.05	−0.752160
$22$	1357.24i	0.597861i
$23$	− 15.5282i	− 0.00612072i	−0.999995	−	0.00306036i	$-0.999026\pi$
$23$	− 15.5282i	− 0.00612072i	0.999995	−	0.00306036i	$-0.000974144\pi$
$24$	−7953.78	−2.81868
$25$	0	0
$26$	−10274.8	−2.98085
$27$	− 4065.47i	− 1.07325i
$28$	12432.6i	2.99686i
$29$	−837.103	−0.184835	−0.0924174	−	0.995720i	$-0.529459\pi$
$29$	−837.103	−0.184835	−0.0924174	+	0.995720i	$0.529459\pi$
$30$	0	0
$31$	2763.55	0.516492	0.258246	−	0.966079i	$-0.416856\pi$
$31$	2763.55	0.516492	0.258246	+	0.966079i	$0.416856\pi$
$32$	31379.2i	5.41710i
$33$	− 1387.94i	− 0.221863i
$34$	6814.93	1.01103
$35$	0	0
$36$	−10453.8	−1.34437
$37$	5914.18i	0.710216i	0.934825	+	0.355108i	$0.115556\pi$
$37$	5914.18i	0.710216i	−0.934825	+	0.355108i	$0.884444\pi$
$38$	− 26278.4i	− 2.95216i
$39$	10507.2	1.10618
$40$	0	0
$41$	−9477.40	−0.880500	−0.440250	−	0.897875i	$-0.645110\pi$
$41$	−9477.40	−0.880500	−0.440250	+	0.897875i	$0.645110\pi$
$42$	− 17050.2i	− 1.49144i
$43$	3470.90i	0.286267i	0.989703	+	0.143133i	$0.0457178\pi$
$43$	3470.90i	0.286267i	−0.989703	+	0.143133i	$0.954282\pi$
$44$	−11352.0	−0.883977
$45$	0	0
$46$	174.178	0.0121367
$47$	− 5638.78i	− 0.372341i	−0.982517	−	0.186170i	$-0.940392\pi$
$47$	− 5638.78i	− 0.372341i	0.982517	−	0.186170i	$-0.0596076\pi$
$48$	− 54779.8i	− 3.43176i
$49$	−753.904	−0.0448566
$50$	0	0
$51$	−6969.07	−0.375188
$52$	− 85938.8i	− 4.40739i
$53$	10588.9i	0.517797i	0.965905	+	0.258898i	$0.0833595\pi$
$53$	10588.9i	0.517797i	−0.965905	+	0.258898i	$0.916641\pi$
$54$	45601.9	2.12813
$55$	0	0
$56$	−91888.7	−3.91554
$57$	26872.7i	1.09553i
$58$	− 9389.68i	− 0.366506i
$59$	−29479.8	−1.10254	−0.551270	−	0.834327i	$-0.685856\pi$
$59$	−29479.8	−1.10254	−0.551270	+	0.834327i	$0.685856\pi$
$60$	0	0
$61$	−32344.0	−1.11294	−0.556468	−	0.830869i	$-0.687844\pi$
$61$	−32344.0	−1.11294	−0.556468	+	0.830869i	$0.687844\pi$
$62$	30998.4i	1.02414i
$63$	− 14765.9i	− 0.468714i
$64$	−199154.	−6.07771
$65$	0	0
$66$	15568.3	0.439929
$67$	58356.9i	1.58820i	0.607787	+	0.794100i	$0.292058\pi$
$67$	58356.9i	1.58820i	−0.607787	+	0.794100i	$0.707942\pi$
$68$	57000.3i	1.49487i
$69$	−178.118	−0.00450386
$70$	0	0
$71$	−1075.01	−0.0253086	−0.0126543	−	0.999920i	$-0.504028\pi$
$71$	−1075.01	−0.0253086	−0.0126543	+	0.999920i	$0.504028\pi$
$72$	− 77263.6i	− 1.75648i
$73$	− 28035.0i	− 0.615734i	−0.951429	−	0.307867i	$-0.900385\pi$
$73$	− 28035.0i	− 0.615734i	0.951429	−	0.307867i	$-0.0996152\pi$
$74$	−66338.6	−1.40827
$75$	0	0
$76$	219793.	4.36496
$77$	− 16034.6i	− 0.308200i
$78$	117858.i	2.19342i
$79$	−23231.7	−0.418806	−0.209403	−	0.977829i	$-0.567152\pi$
$79$	−23231.7	−0.418806	−0.209403	+	0.977829i	$0.567152\pi$
$80$	0	0
$81$	−19556.8	−0.331196
$82$	− 106307.i	− 1.74593i
$83$	− 23943.0i	− 0.381491i	−0.981640	−	0.190745i	$-0.938910\pi$
$83$	− 23943.0i	− 0.381491i	0.981640	−	0.190745i	$-0.0610905\pi$
$84$	142609.	2.20520
$85$	0	0
$86$	−38932.6	−0.567633
$87$	9602.05i	0.136008i
$88$	− 83902.3i	− 1.15496i
$89$	−95010.4	−1.27144	−0.635721	−	0.771919i	$-0.719297\pi$
$89$	−95010.4	−1.27144	−0.635721	+	0.771919i	$0.719297\pi$
$90$	0	0
$91$	121388.	1.53664
$92$	1456.83i	0.0179449i
$93$	− 31699.5i	− 0.380054i
$94$	63249.5	0.738307
$95$	0	0
$96$	359937.	3.98610
$97$	− 33677.1i	− 0.363418i	−0.983352	−	0.181709i	$-0.941837\pi$
$97$	− 33677.1i	− 0.363418i	0.983352	−	0.181709i	$-0.0581628\pi$
$98$	− 8456.45i	− 0.0889453i
$99$	13482.5	0.138256
$100$	0	0
$101$	−78905.4	−0.769668	−0.384834	−	0.922986i	$-0.625741\pi$
$101$	−78905.4	−0.769668	−0.384834	+	0.922986i	$0.625741\pi$
$102$	− 78171.2i	− 0.743954i
$103$	56918.2i	0.528638i	0.964435	+	0.264319i	$0.0851471\pi$
$103$	56918.2i	0.528638i	−0.964435	+	0.264319i	$0.914853\pi$
$104$	635171.	5.75847
$105$	0	0
$106$	−118774.	−1.02673
$107$	190035.i	1.60463i	0.596904	+	0.802313i	$0.296397\pi$
$107$	190035.i	1.60463i	−0.596904	+	0.802313i	$0.703603\pi$
$108$	381415.i	3.14658i
$109$	−108794.	−0.877078	−0.438539	−	0.898712i	$-0.644504\pi$
$109$	−108794.	−0.877078	−0.438539	+	0.898712i	$0.644504\pi$
$110$	0	0
$111$	67839.0	0.522603
$112$	− 632862.i	− 4.76720i
$113$	208092.i	1.53306i	0.642207	+	0.766532i	$0.278019\pi$
$113$	208092.i	1.53306i	−0.642207	+	0.766532i	$0.721981\pi$
$114$	−301428.	−2.17231
$115$	0	0
$116$	78535.5	0.541903
$117$	102068.i	0.689324i
$118$	− 330671.i	− 2.18621i
$119$	−80512.5	−0.521189
$120$	0	0
$121$	14641.0	0.0909091
$122$	− 362799.i	− 2.20682i
$123$	108711.i	0.647905i
$124$	−259272.	−1.51426
$125$	0	0
$126$	165627.	0.929405
$127$	− 308832.i	− 1.69908i	−0.527527	−	0.849538i	$-0.676881\pi$
$127$	− 308832.i	− 1.69908i	0.527527	−	0.849538i	$-0.323119\pi$
$128$	− 1.22976e6i	− 6.63429i
$129$	39813.2	0.210646
$130$	0	0
$131$	−63060.0	−0.321052	−0.160526	−	0.987032i	$-0.551319\pi$
$131$	−63060.0	−0.321052	−0.160526	+	0.987032i	$0.551319\pi$
$132$	130214.i	0.650464i
$133$	310456.i	1.52185i
$134$	−654582.	−3.14921
$135$	0	0
$136$	−421287.	−1.95313
$137$	− 41313.4i	− 0.188057i	−0.995570	−	0.0940284i	$-0.970026\pi$
$137$	− 41313.4i	− 0.188057i	0.995570	−	0.0940284i	$-0.0299745\pi$
$138$	− 1997.92i	− 0.00893062i
$139$	257986.	1.13256	0.566278	−	0.824214i	$-0.308383\pi$
$139$	257986.	1.13256	0.566278	+	0.824214i	$0.308383\pi$
$140$	0	0
$141$	−64680.0	−0.273982
$142$	− 12058.3i	− 0.0501839i
$143$	110838.i	0.453260i
$144$	532135.	2.13853
$145$	0	0
$146$	314465.	1.22093
$147$	8647.72i	0.0330072i
$148$	− 554858.i	− 2.08222i
$149$	−233787.	−0.862689	−0.431344	−	0.902187i	$-0.641961\pi$
$149$	−233787.	−0.862689	−0.431344	+	0.902187i	$0.641961\pi$
$150$	0	0
$151$	−398038.	−1.42063	−0.710317	−	0.703882i	$-0.751448\pi$
$151$	−398038.	−1.42063	−0.710317	+	0.703882i	$0.751448\pi$
$152$	1.62448e6i	5.70304i
$153$	− 67698.0i	− 0.233801i
$154$	179858.	0.611123
$155$	0	0
$156$	−985768.	−3.24312
$157$	283315.i	0.917318i	0.888612	+	0.458659i	$0.151670\pi$
$157$	283315.i	0.917318i	−0.888612	+	0.458659i	$0.848330\pi$
$158$	− 260587.i	− 0.830444i
$159$	121460.	0.381014
$160$	0	0
$161$	−2057.76	−0.00625649
$162$	− 219366.i	− 0.656722i
$163$	491644.i	1.44938i	0.689077	+	0.724688i	$0.258016\pi$
$163$	491644.i	1.44938i	−0.689077	+	0.724688i	$0.741984\pi$
$164$	889153.	2.58147
$165$	0	0
$166$	268566.	0.756451
$167$	− 454917.i	− 1.26224i	−0.775687	−	0.631118i	$-0.782596\pi$
$167$	− 454917.i	− 1.26224i	0.775687	−	0.631118i	$-0.217404\pi$
$168$	1.05402e6i	2.88120i
$169$	−467788.	−1.25989
$170$	0	0
$171$	−261043.	−0.682688
$172$	− 325634.i	− 0.839282i
$173$	275876.i	0.700807i	0.936599	+	0.350404i	$0.113956\pi$
$173$	275876.i	0.700807i	−0.936599	+	0.350404i	$0.886044\pi$
$174$	−107705.	−0.269689
$175$	0	0
$176$	577858.	1.40617
$177$	338150.i	0.811290i
$178$	− 1.06572e6i	− 2.52112i
$179$	19479.7	0.0454411	0.0227206	−	0.999742i	$-0.492767\pi$
$179$	19479.7	0.0454411	0.0227206	+	0.999742i	$0.492767\pi$
$180$	0	0
$181$	−542896.	−1.23174	−0.615871	−	0.787847i	$-0.711196\pi$
$181$	−542896.	−1.23174	−0.615871	+	0.787847i	$0.711196\pi$
$182$	1.36159e6i	3.04697i
$183$	371005.i	0.818940i
$184$	−10767.4	−0.0234459
$185$	0	0
$186$	355570.	0.753603
$187$	− 73514.8i	− 0.153734i
$188$	529020.i	1.09164i
$189$	−538746.	−1.09706
$190$	0	0
$191$	242935.	0.481845	0.240922	−	0.970544i	$-0.422550\pi$
$191$	242935.	0.481845	0.240922	+	0.970544i	$0.422550\pi$
$192$	2.28442e6i	4.47221i
$193$	207159.i	0.400322i	0.979763	+	0.200161i	$0.0641466\pi$
$193$	207159.i	0.400322i	−0.979763	+	0.200161i	$0.935853\pi$
$194$	377752.	0.720614
$195$	0	0
$196$	70730.0	0.131511
$197$	253518.i	0.465417i	0.972547	+	0.232709i	$0.0747588\pi$
$197$	253518.i	0.465417i	−0.972547	+	0.232709i	$0.925241\pi$
$198$	151232.i	0.274145i
$199$	−756118.	−1.35350	−0.676748	−	0.736215i	$-0.736611\pi$
$199$	−756118.	−1.35350	−0.676748	+	0.736215i	$0.736611\pi$
$200$	0	0
$201$	669387.	1.16866
$202$	− 885072.i	− 1.52616i
$203$	110931.i	0.188935i
$204$	653826.	1.09999
$205$	0	0
$206$	−638444.	−1.04823
$207$	− 1730.25i	− 0.00280661i
$208$	4.37459e6i	7.01098i
$209$	−283473.	−0.448896
$210$	0	0
$211$	1.01855e6	1.57499	0.787494	−	0.616322i	$-0.211378\pi$
$211$	1.01855e6	1.57499	0.787494	+	0.616322i	$0.211378\pi$
$212$	− 993428.i	− 1.51809i
$213$	12331.0i	0.0186230i
$214$	−2.13160e6	−3.18178
$215$	0	0
$216$	−2.81903e6	−4.11116
$217$	− 366219.i	− 0.527949i
$218$	− 1.22033e6i	− 1.73914i
$219$	−321578.	−0.453081
$220$	0	0
$221$	556534.	0.766498
$222$	760942.i	1.03626i
$223$	612261.i	0.824469i	0.911078	+	0.412235i	$0.135252\pi$
$223$	612261.i	0.824469i	−0.911078	+	0.412235i	$0.864748\pi$
$224$	4.15829e6	5.53726
$225$	0	0
$226$	−2.33415e6	−3.03988
$227$	− 625281.i	− 0.805398i	−0.915332	−	0.402699i	$-0.868072\pi$
$227$	− 625281.i	− 0.805398i	0.915332	−	0.402699i	$-0.131928\pi$
$228$	− 2.52115e6i	− 3.21190i
$229$	−1.11167e6	−1.40083	−0.700415	−	0.713736i	$-0.747002\pi$
$229$	−1.11167e6	−1.40083	−0.700415	+	0.713736i	$0.747002\pi$
$230$	0	0
$231$	−183926.	−0.226785
$232$	580454.i	0.708023i
$233$	− 927589.i	− 1.11935i	−0.828712	−	0.559675i	$-0.810926\pi$
$233$	− 927589.i	− 1.11935i	0.828712	−	0.559675i	$-0.189074\pi$
$234$	−1.14488e6	−1.36685
$235$	0	0
$236$	2.76574e6	3.23245
$237$	266481.i	0.308174i
$238$	− 903098.i	− 1.03346i
$239$	225927.	0.255843	0.127921	−	0.991784i	$-0.459169\pi$
$239$	225927.	0.255843	0.127921	+	0.991784i	$0.459169\pi$
$240$	0	0
$241$	−1.32668e6	−1.47138	−0.735688	−	0.677320i	$-0.763141\pi$
$241$	−1.32668e6	−1.47138	−0.735688	+	0.677320i	$0.763141\pi$
$242$	164226.i	0.180262i
$243$	− 763582.i	− 0.829545i
$244$	3.03446e6	3.26293
$245$	0	0
$246$	−1.21940e6	−1.28472
$247$	− 2.14599e6i	− 2.23813i
$248$	− 1.91627e6i	− 1.97846i
$249$	−274640.	−0.280715
$250$	0	0
$251$	717486.	0.718834	0.359417	−	0.933177i	$-0.382975\pi$
$251$	717486.	0.718834	0.359417	+	0.933177i	$0.382975\pi$
$252$	1.38531e6i	1.37419i
$253$	− 1878.92i	− 0.00184547i
$254$	3.46413e6	3.36907
$255$	0	0
$256$	7.42108e6	7.07730
$257$	716981.i	0.677135i	0.940942	+	0.338567i	$0.109942\pi$
$257$	716981.i	0.677135i	−0.940942	+	0.338567i	$0.890058\pi$
$258$	446580.i	0.417686i
$259$	783733.	0.725970
$260$	0	0
$261$	−93275.0	−0.0847547
$262$	− 707336.i	− 0.636608i
$263$	1.58342e6i	1.41158i	0.708420	+	0.705791i	$0.249408\pi$
$263$	1.58342e6i	1.41158i	−0.708420	+	0.705791i	$0.750592\pi$
$264$	−962408.	−0.849864
$265$	0	0
$266$	−3.48235e6	−3.01764
$267$	1.08982e6i	0.935574i
$268$	− 5.47494e6i	− 4.65632i
$269$	−148436.	−0.125072	−0.0625359	−	0.998043i	$-0.519919\pi$
$269$	−148436.	−0.125072	−0.0625359	+	0.998043i	$0.519919\pi$
$270$	0	0
$271$	−703775.	−0.582117	−0.291059	−	0.956705i	$-0.594007\pi$
$271$	−703775.	−0.582117	−0.291059	+	0.956705i	$0.594007\pi$
$272$	− 2.90152e6i	− 2.37795i
$273$	− 1.39239e6i	− 1.13072i
$274$	463407.	0.372894
$275$	0	0
$276$	16710.7	0.0132045
$277$	− 917597.i	− 0.718542i	−0.933233	−	0.359271i	$-0.883025\pi$
$277$	− 917597.i	− 0.718542i	0.933233	−	0.359271i	$-0.116975\pi$
$278$	2.89380e6i	2.24572i
$279$	307931.	0.236834
$280$	0	0
$281$	2.27113e6	1.71584	0.857921	−	0.513782i	$-0.171756\pi$
$281$	2.27113e6	1.71584	0.857921	+	0.513782i	$0.171756\pi$
$282$	− 725508.i	− 0.543274i
$283$	576587.i	0.427956i	0.976839	+	0.213978i	$0.0686420\pi$
$283$	576587.i	0.427956i	−0.976839	+	0.213978i	$0.931358\pi$
$284$	100856.	0.0742001
$285$	0	0
$286$	−1.24325e6	−0.898761
$287$	1.25592e6i	0.900032i
$288$	3.49645e6i	2.48397i
$289$	1.05073e6	0.740023
$290$	0	0
$291$	−386296.	−0.267416
$292$	2.63019e6i	1.80522i
$293$	954488.i	0.649533i	0.945794	+	0.324767i	$0.105286\pi$
$293$	954488.i	0.649533i	−0.945794	+	0.324767i	$0.894714\pi$
$294$	−97000.3	−0.0654493
$295$	0	0
$296$	4.10094e6	2.72053
$297$	− 491922.i	− 0.323597i
$298$	− 2.62236e6i	− 1.71061i
$299$	14224.1	0.00920123
$300$	0	0
$301$	459955.	0.292617
$302$	− 4.46474e6i	− 2.81695i
$303$	905090.i	0.566351i
$304$	−1.11882e7	−6.94349
$305$	0	0
$306$	759360.	0.463601
$307$	252381.i	0.152831i	0.997076	+	0.0764153i	$0.0243475\pi$
$307$	252381.i	0.152831i	−0.997076	+	0.0764153i	$0.975653\pi$
$308$	1.50434e6i	0.903586i
$309$	652884.	0.388992
$310$	0	0
$311$	784287.	0.459805	0.229903	−	0.973214i	$-0.426159\pi$
$311$	784287.	0.459805	0.229903	+	0.973214i	$0.426159\pi$
$312$	− 7.28578e6i	− 4.23730i
$313$	2.16086e6i	1.24671i	0.781938	+	0.623356i	$0.214231\pi$
$313$	2.16086e6i	1.24671i	−0.781938	+	0.623356i	$0.785769\pi$
$314$	−3.17790e6	−1.81893
$315$	0	0
$316$	2.17956e6	1.22787
$317$	3.39464e6i	1.89734i	0.316269	+	0.948670i	$0.397570\pi$
$317$	3.39464e6i	1.89734i	−0.316269	+	0.948670i	$0.602430\pi$
$318$	1.36240e6i	0.755507i
$319$	−101289.	−0.0557298
$320$	0	0
$321$	2.17981e6	1.18074
$322$	− 23081.7i	− 0.0124059i
$323$	1.42336e6i	0.759119i
$324$	1.83478e6	0.971006
$325$	0	0
$326$	−5.51470e6	−2.87394
$327$	1.24793e6i	0.645387i
$328$	6.57170e6i	3.37282i
$329$	−747237.	−0.380600
$330$	0	0
$331$	−1.50736e6	−0.756216	−0.378108	−	0.925762i	$-0.623425\pi$
$331$	−1.50736e6	−0.756216	−0.378108	+	0.925762i	$0.623425\pi$
$332$	2.24629e6i	1.11846i
$333$	658993.i	0.325664i
$334$	5.10274e6	2.50287
$335$	0	0
$336$	−7.25929e6	−3.50789
$337$	− 301876.i	− 0.144795i	−0.997376	−	0.0723976i	$-0.976935\pi$
$337$	− 301876.i	− 0.144795i	0.997376	−	0.0723976i	$-0.0230651\pi$
$338$	− 5.24712e6i	− 2.49821i
$339$	2.38694e6	1.12809
$340$	0	0
$341$	334390.	0.155728
$342$	− 2.92809e6i	− 1.35369i
$343$	− 2.12732e6i	− 0.976331i
$344$	2.40675e6	1.09656
$345$	0	0
$346$	−3.09447e6	−1.38962
$347$	325201.i	0.144987i	0.997369	+	0.0724934i	$0.0230956\pi$
$347$	325201.i	0.144987i	−0.997369	+	0.0724934i	$0.976904\pi$
$348$	− 900848.i	− 0.398752i
$349$	2.73891e6	1.20369	0.601845	−	0.798613i	$-0.294432\pi$
$349$	2.73891e6	1.20369	0.601845	+	0.798613i	$0.294432\pi$
$350$	0	0
$351$	3.72403e6	1.61341
$352$	3.79688e6i	1.63332i
$353$	− 2.86184e6i	− 1.22238i	−0.791482	−	0.611192i	$-0.790690\pi$
$353$	− 2.86184e6i	− 1.22238i	0.791482	−	0.611192i	$-0.209310\pi$
$354$	−3.79298e6	−1.60869
$355$	0	0
$356$	8.91371e6	3.72764
$357$	923524.i	0.383511i
$358$	218501.i	0.0901043i
$359$	3.49224e6	1.43011	0.715053	−	0.699071i	$-0.246403\pi$
$359$	3.49224e6	1.43011	0.715053	+	0.699071i	$0.246403\pi$
$360$	0	0
$361$	3.01239e6	1.21659
$362$	− 6.08959e6i	− 2.44240i
$363$	− 167941.i	− 0.0668943i
$364$	−1.13884e7	−4.50515
$365$	0	0
$366$	−4.16151e6	−1.62386
$367$	− 420097.i	− 0.162811i	−0.996681	−	0.0814056i	$-0.974059\pi$
$367$	− 420097.i	− 0.162811i	0.996681	−	0.0814056i	$-0.0259409\pi$
$368$	− 74157.9i	− 0.0285455i
$369$	−1.05603e6	−0.403747
$370$	0	0
$371$	1.40321e6	0.529283
$372$	2.97400e6i	1.11425i
$373$	281554.i	0.104783i	0.998627	+	0.0523914i	$0.0166843\pi$
$373$	281554.i	0.104783i	−0.998627	+	0.0523914i	$0.983316\pi$
$374$	824607.	0.304837
$375$	0	0
$376$	−3.90997e6	−1.42628
$377$	− 766798.i	− 0.277861i
$378$	− 6.04305e6i	− 2.17534i
$379$	1.41256e6	0.505138	0.252569	−	0.967579i	$-0.418725\pi$
$379$	1.41256e6	0.505138	0.252569	+	0.967579i	$0.418725\pi$
$380$	0	0
$381$	−3.54248e6	−1.25024
$382$	2.72497e6i	0.955441i
$383$	− 2.23548e6i	− 0.778708i	−0.921088	−	0.389354i	$-0.872698\pi$
$383$	− 2.23548e6i	− 0.778708i	0.921088	−	0.389354i	$-0.127302\pi$
$384$	−1.41060e7	−4.88176
$385$	0	0
$386$	−2.32367e6	−0.793792
$387$	386748.i	0.131265i
$388$	3.15953e6i	1.06548i
$389$	−5.10434e6	−1.71027	−0.855136	−	0.518403i	$-0.826527\pi$
$389$	−5.10434e6	−1.71027	−0.855136	+	0.518403i	$0.826527\pi$
$390$	0	0
$391$	−9434.34	−0.00312083
$392$	522763.i	0.171826i
$393$	723334.i	0.236242i
$394$	−2.84367e6	−0.922867
$395$	0	0
$396$	−1.26491e6	−0.405341
$397$	− 252110.i	− 0.0802813i	−0.999194	−	0.0401406i	$-0.987219\pi$
$397$	− 252110.i	− 0.0802813i	0.999194	−	0.0401406i	$-0.0127806\pi$
$398$	− 8.48128e6i	− 2.68382i
$399$	3.56111e6	1.11983
$400$	0	0
$401$	1.61321e6	0.500991	0.250496	−	0.968118i	$-0.419406\pi$
$401$	1.61321e6	0.500991	0.250496	+	0.968118i	$0.419406\pi$
$402$	7.50843e6i	2.31731i
$403$	2.53145e6i	0.776439i
$404$	7.40277e6	2.25653
$405$	0	0
$406$	−1.24430e6	−0.374636
$407$	715616.i	0.214138i
$408$	4.83241e6i	1.43719i
$409$	−1.08687e6	−0.321270	−0.160635	−	0.987014i	$-0.551354\pi$
$409$	−1.08687e6	−0.321270	−0.160635	+	0.987014i	$0.551354\pi$
$410$	0	0
$411$	−473888.	−0.138379
$412$	− 5.33997e6i	− 1.54987i
$413$	3.90659e6i	1.12700i
$414$	19408.0	0.00556518
$415$	0	0
$416$	−2.87438e7	−8.14348
$417$	− 2.95925e6i	− 0.833377i
$418$	− 3.17968e6i	− 0.890109i
$419$	740488.	0.206055	0.103027	−	0.994679i	$-0.467147\pi$
$419$	740488.	0.206055	0.103027	+	0.994679i	$0.467147\pi$
$420$	0	0
$421$	6.54748e6	1.80040	0.900199	−	0.435478i	$-0.143421\pi$
$421$	6.54748e6	1.80040	0.900199	+	0.435478i	$0.143421\pi$
$422$	1.14250e7i	3.12302i
$423$	− 628306.i	− 0.170734i
$424$	7.34239e6	1.98346
$425$	0	0
$426$	−138315.	−0.0369272
$427$	4.28615e6i	1.13762i
$428$	− 1.78287e7i	− 4.70447i
$429$	1.27137e6	0.333526
$430$	0	0
$431$	−2.75325e6	−0.713925	−0.356963	−	0.934119i	$-0.616188\pi$
$431$	−2.75325e6	−0.713925	−0.356963	+	0.934119i	$0.616188\pi$
$432$	− 1.94154e7i	− 5.00538i
$433$	1.26541e6i	0.324349i	0.986762	+	0.162175i	$0.0518508\pi$
$433$	1.26541e6i	0.324349i	−0.986762	+	0.162175i	$0.948149\pi$
$434$	4.10784e6	1.04686
$435$	0	0
$436$	1.02069e7	2.57144
$437$	36378.8i	0.00911266i
$438$	− 3.60709e6i	− 0.898405i
$439$	1.77205e6	0.438850	0.219425	−	0.975629i	$-0.429582\pi$
$439$	1.77205e6	0.438850	0.219425	+	0.975629i	$0.429582\pi$
$440$	0	0
$441$	−84004.5	−0.0205687
$442$	6.24257e6i	1.51987i
$443$	− 3.87503e6i	− 0.938135i	−0.883162	−	0.469068i	$-0.844590\pi$
$443$	− 3.87503e6i	− 0.938135i	0.883162	−	0.469068i	$-0.155410\pi$
$444$	−6.36454e6	−1.53218
$445$	0	0
$446$	−6.86766e6	−1.63483
$447$	2.68167e6i	0.634799i
$448$	2.63915e7i	6.21253i
$449$	−5.14887e6	−1.20530	−0.602651	−	0.798005i	$-0.705889\pi$
$449$	−5.14887e6	−1.20530	−0.602651	+	0.798005i	$0.705889\pi$
$450$	0	0
$451$	−1.14677e6	−0.265481
$452$	− 1.95229e7i	− 4.49467i
$453$	4.56572e6i	1.04536i
$454$	7.01370e6	1.59701
$455$	0	0
$456$	1.86338e7	4.19651
$457$	− 2.37155e6i	− 0.531179i	−0.964086	−	0.265590i	$-0.914433\pi$
$457$	− 2.37155e6i	− 0.531179i	0.964086	−	0.265590i	$-0.0855666\pi$
$458$	− 1.24694e7i	− 2.77768i
$459$	−2.47002e6	−0.547228
$460$	0	0
$461$	−2.82106e6	−0.618244	−0.309122	−	0.951022i	$-0.600035\pi$
$461$	−2.82106e6	−0.618244	−0.309122	+	0.951022i	$0.600035\pi$
$462$	− 2.06308e6i	− 0.449688i
$463$	− 5.26742e6i	− 1.14194i	−0.820969	−	0.570972i	$-0.806566\pi$
$463$	− 5.26742e6i	− 1.14194i	0.820969	−	0.570972i	$-0.193434\pi$
$464$	−3.99774e6	−0.862024
$465$	0	0
$466$	1.04047e7	2.21954
$467$	1.18996e6i	0.252487i	0.991999	+	0.126243i	$0.0402920\pi$
$467$	1.18996e6i	0.252487i	−0.991999	+	0.126243i	$0.959708\pi$
$468$	− 9.57581e6i	− 2.02097i
$469$	7.73332e6	1.62343
$470$	0	0
$471$	3.24978e6	0.674997
$472$	2.04415e7i	4.22336i
$473$	419979.i	0.0863126i
$474$	−2.98908e6	−0.611072
$475$	0	0
$476$	7.55354e6	1.52803
$477$	1.17987e6i	0.237432i
$478$	2.53419e6i	0.507306i
$479$	5.62901e6	1.12097	0.560484	−	0.828165i	$-0.310615\pi$
$479$	5.62901e6	1.12097	0.560484	+	0.828165i	$0.310615\pi$
$480$	0	0
$481$	−5.41747e6	−1.06766
$482$	− 1.48812e7i	− 2.91757i
$483$	23603.7i	0.00460376i
$484$	−1.37359e6	−0.266529
$485$	0	0
$486$	8.56500e6	1.64489
$487$	− 1.09746e6i	− 0.209684i	−0.994489	−	0.104842i	$-0.966566\pi$
$487$	− 1.09746e6i	− 0.209684i	0.994489	−	0.104842i	$-0.0334336\pi$
$488$	2.24276e7i	4.26318i
$489$	5.63943e6	1.06651
$490$	0	0
$491$	1.33860e6	0.250580	0.125290	−	0.992120i	$-0.460014\pi$
$491$	1.33860e6	0.250580	0.125290	+	0.992120i	$0.460014\pi$
$492$	− 1.01991e7i	− 1.89954i
$493$	508591.i	0.0942435i
$494$	2.40713e7	4.43796
$495$	0	0
$496$	1.31978e7	2.40879
$497$	142458.i	0.0258700i
$498$	− 3.08061e6i	− 0.556625i
$499$	−1.29736e6	−0.233243	−0.116622	−	0.993176i	$-0.537206\pi$
$499$	−1.29736e6	−0.233243	−0.116622	+	0.993176i	$0.537206\pi$
$500$	0	0
$501$	−5.21816e6	−0.928801
$502$	8.04795e6i	1.42536i
$503$	− 5.44945e6i	− 0.960357i	−0.877171	−	0.480179i	$-0.840572\pi$
$503$	− 5.44945e6i	− 0.960357i	0.877171	−	0.480179i	$-0.159428\pi$
$504$	−1.02388e7	−1.79544
$505$	0	0
$506$	21075.6	0.00365934
$507$	5.36580e6i	0.927074i
$508$	2.89741e7i	4.98139i
$509$	−9.81233e6	−1.67872	−0.839359	−	0.543578i	$-0.817069\pi$
$509$	−9.81233e6	−1.67872	−0.839359	+	0.543578i	$0.817069\pi$
$510$	0	0
$511$	−3.71513e6	−0.629393
$512$	4.38892e7i	7.39916i
$513$	9.52439e6i	1.59788i
$514$	−8.04229e6	−1.34268
$515$	0	0
$516$	−3.73520e6	−0.617576
$517$	− 682292.i	− 0.112265i
$518$	8.79103e6i	1.43951i
$519$	3.16446e6	0.515680
$520$	0	0
$521$	8.84276e6	1.42723	0.713614	−	0.700539i	$-0.247057\pi$
$521$	8.84276e6	1.42723	0.713614	+	0.700539i	$0.247057\pi$
$522$	− 1.04625e6i	− 0.168059i
$523$	− 5.57320e6i	− 0.890944i	−0.895296	−	0.445472i	$-0.853036\pi$
$523$	− 5.57320e6i	− 0.890944i	0.895296	−	0.445472i	$-0.146964\pi$
$524$	5.91618e6	0.941267
$525$	0	0
$526$	−1.77610e7	−2.79900
$527$	− 1.67903e6i	− 0.263348i
$528$	− 6.62836e6i	− 1.03472i
$529$	6.43610e6	0.999963
$530$	0	0
$531$	−3.28481e6	−0.505561
$532$	− 2.91264e7i	− 4.46178i
$533$	− 8.68143e6i	− 1.32365i
$534$	−1.22244e7	−1.85513
$535$	0	0
$536$	4.04651e7	6.08372
$537$	− 223443.i	− 0.0334373i
$538$	− 1.66499e6i	− 0.248003i
$539$	−91222.4	−0.0135248
$540$	0	0
$541$	−7.05506e6	−1.03635	−0.518176	−	0.855274i	$-0.673389\pi$
$541$	−7.05506e6	−1.03635	−0.518176	+	0.855274i	$0.673389\pi$
$542$	− 7.89415e6i	− 1.15427i
$543$	6.22732e6i	0.906362i
$544$	1.90648e7	2.76207
$545$	0	0
$546$	1.56183e7	2.24208
$547$	− 1.22859e7i	− 1.75565i	−0.478985	−	0.877823i	$-0.658995\pi$
$547$	− 1.22859e7i	− 1.75565i	0.478985	−	0.877823i	$-0.341005\pi$
$548$	3.87595e6i	0.551349i
$549$	−3.60397e6	−0.510328
$550$	0	0
$551$	1.96113e6	0.275186
$552$	123508.i	0.0172523i
$553$	3.07861e6i	0.428097i
$554$	1.02926e7	1.42478
$555$	0	0
$556$	−2.42038e7	−3.32045
$557$	− 8.42444e6i	− 1.15054i	−0.817962	−	0.575272i	$-0.804896\pi$
$557$	− 8.42444e6i	− 1.15054i	0.817962	−	0.575272i	$-0.195104\pi$
$558$	3.45403e6i	0.469613i
$559$	−3.17939e6	−0.430342
$560$	0	0
$561$	−843257.	−0.113124
$562$	2.54750e7i	3.40231i
$563$	− 7.88182e6i	− 1.04799i	−0.851723	−	0.523993i	$-0.824442\pi$
$563$	− 7.88182e6i	− 1.04799i	0.851723	−	0.523993i	$-0.175558\pi$
$564$	6.06817e6	0.803267
$565$	0	0
$566$	−6.46751e6	−0.848586
$567$	2.59162e6i	0.338542i
$568$	745421.i	0.0969462i
$569$	−7.11408e6	−0.921167	−0.460583	−	0.887616i	$-0.652360\pi$
$569$	−7.11408e6	−0.921167	−0.460583	+	0.887616i	$0.652360\pi$
$570$	0	0
$571$	−71565.4	−0.00918572	−0.00459286	−	0.999989i	$-0.501462\pi$
$571$	−71565.4	−0.00918572	−0.00459286	+	0.999989i	$0.501462\pi$
$572$	− 1.03986e7i	− 1.32888i
$573$	− 2.78661e6i	− 0.354559i
$574$	−1.40875e7	−1.78466
$575$	0	0
$576$	−2.21910e7	−2.78689
$577$	− 2.17667e6i	− 0.272179i	−0.990697	−	0.136089i	$-0.956547\pi$
$577$	− 2.17667e6i	− 0.272179i	0.990697	−	0.136089i	$-0.0434534\pi$
$578$	1.17859e7i	1.46738i
$579$	2.37623e6	0.294572
$580$	0	0
$581$	−3.17287e6	−0.389953
$582$	− 4.33304e6i	− 0.530255i
$583$	1.28125e6i	0.156122i
$584$	−1.94397e7	−2.35861
$585$	0	0
$586$	−1.07064e7	−1.28795
$587$	− 852228.i	− 0.102085i	−0.998696	−	0.0510423i	$-0.983746\pi$
$587$	− 852228.i	− 0.102085i	0.998696	−	0.0510423i	$-0.0162543\pi$
$588$	− 811314.i	− 0.0967711i
$589$	−6.47432e6	−0.768964
$590$	0	0
$591$	2.90799e6	0.342472
$592$	2.82442e7i	3.31227i
$593$	− 9.10178e6i	− 1.06289i	−0.847092	−	0.531446i	$-0.821649\pi$
$593$	− 9.10178e6i	− 1.06289i	0.847092	−	0.531446i	$-0.178351\pi$
$594$	5.51782e6	0.641655
$595$	0	0
$596$	2.19335e7	2.52925
$597$	8.67311e6i	0.995954i
$598$	159550.i	0.0182450i
$599$	3.68657e6	0.419813	0.209906	−	0.977721i	$-0.432684\pi$
$599$	3.68657e6	0.419813	0.209906	+	0.977721i	$0.432684\pi$
$600$	0	0
$601$	1.18735e7	1.34088	0.670442	−	0.741962i	$-0.266104\pi$
$601$	1.18735e7	1.34088	0.670442	+	0.741962i	$0.266104\pi$
$602$	5.15926e6i	0.580224i
$603$	6.50247e6i	0.728258i
$604$	3.73432e7	4.16504
$605$	0	0
$606$	−1.01523e7	−1.12301
$607$	8.77821e6i	0.967017i	0.875340	+	0.483509i	$0.160638\pi$
$607$	8.77821e6i	0.967017i	−0.875340	+	0.483509i	$0.839362\pi$
$608$	− 7.35137e7i	− 8.06509i
$609$	1.27244e6	0.139025
$610$	0	0
$611$	5.16520e6	0.559737
$612$	6.35131e6i	0.685464i
$613$	− 1.52076e7i	− 1.63459i	−0.576220	−	0.817294i	$-0.695473\pi$
$613$	− 1.52076e7i	− 1.63459i	0.576220	−	0.817294i	$-0.304527\pi$
$614$	−2.83092e6	−0.303045
$615$	0	0
$616$	−1.11185e7	−1.18058
$617$	− 1.13759e7i	− 1.20302i	−0.798865	−	0.601511i	$-0.794566\pi$
$617$	− 1.13759e7i	− 1.20302i	0.798865	−	0.601511i	$-0.205434\pi$
$618$	7.32332e6i	0.771324i
$619$	−7.14541e6	−0.749550	−0.374775	−	0.927116i	$-0.622280\pi$
$619$	−7.14541e6	−0.749550	−0.374775	+	0.927116i	$0.622280\pi$
$620$	0	0
$621$	−63129.6	−0.00656907
$622$	8.79725e6i	0.911739i
$623$	1.25905e7i	1.29964i
$624$	5.01791e7	5.15895
$625$	0	0
$626$	−2.42381e7	−2.47208
$627$	3.25160e6i	0.330315i
$628$	− 2.65801e7i	− 2.68941i
$629$	3.59322e6	0.362124
$630$	0	0
$631$	4.11229e6	0.411159	0.205580	−	0.978640i	$-0.434092\pi$
$631$	4.11229e6	0.411159	0.205580	+	0.978640i	$0.434092\pi$
$632$	1.61090e7i	1.60427i
$633$	− 1.16834e7i	− 1.15894i
$634$	−3.80772e7	−3.76220
$635$	0	0
$636$	−1.13952e7	−1.11707
$637$	− 690587.i	− 0.0674326i
$638$	− 1.13615e6i	− 0.110506i
$639$	−119784.	−0.0116051
$640$	0	0
$641$	2.34166e6	0.225102	0.112551	−	0.993646i	$-0.464098\pi$
$641$	2.34166e6	0.225102	0.112551	+	0.993646i	$0.464098\pi$
$642$	2.44506e7i	2.34128i
$643$	3.71655e6i	0.354497i	0.984166	+	0.177248i	$0.0567196\pi$
$643$	3.71655e6i	0.354497i	−0.984166	+	0.177248i	$0.943280\pi$
$644$	193056.	0.0183429
$645$	0	0
$646$	−1.59657e7	−1.50524
$647$	8.96177e6i	0.841653i	0.907141	+	0.420826i	$0.138260\pi$
$647$	8.96177e6i	0.841653i	−0.907141	+	0.420826i	$0.861740\pi$
$648$	1.35608e7i	1.26867i
$649$	−3.56705e6	−0.332428
$650$	0	0
$651$	−4.20075e6	−0.388485
$652$	− 4.61251e7i	− 4.24931i
$653$	1.79019e7i	1.64292i	0.570267	+	0.821460i	$0.306840\pi$
$653$	1.79019e7i	1.64292i	−0.570267	+	0.821460i	$0.693160\pi$
$654$	−1.39979e7	−1.27973
$655$	0	0
$656$	−4.52610e7	−4.10643
$657$	− 3.12382e6i	− 0.282341i
$658$	− 8.38166e6i	− 0.754685i
$659$	−1.90122e7	−1.70537	−0.852686	−	0.522423i	$-0.825028\pi$
$659$	−1.90122e7	−1.70537	−0.852686	+	0.522423i	$0.825028\pi$
$660$	0	0
$661$	−4.58472e6	−0.408140	−0.204070	−	0.978956i	$-0.565417\pi$
$661$	−4.58472e6	−0.408140	−0.204070	+	0.978956i	$0.565417\pi$
$662$	− 1.69078e7i	− 1.49949i
$663$	− 6.38376e6i	− 0.564018i
$664$	−1.66023e7	−1.46133
$665$	0	0
$666$	−7.39184e6	−0.645754
$667$	12998.7i	0.00113132i
$668$	4.26795e7i	3.70065i
$669$	7.02299e6	0.606676
$670$	0	0
$671$	−3.91363e6	−0.335563
$672$	− 4.76980e7i	− 4.07453i
$673$	1.51723e7i	1.29126i	0.763651	+	0.645630i	$0.223405\pi$
$673$	1.51723e7i	1.29126i	−0.763651	+	0.645630i	$0.776595\pi$
$674$	3.38611e6	0.287112
$675$	0	0
$676$	4.38871e7	3.69377
$677$	− 1.08298e7i	− 0.908129i	−0.890969	−	0.454064i	$-0.849974\pi$
$677$	− 1.08298e7i	− 0.908129i	0.890969	−	0.454064i	$-0.150026\pi$
$678$	2.67740e7i	2.23686i
$679$	−4.46281e6	−0.371479
$680$	0	0
$681$	−7.17233e6	−0.592642
$682$	3.75081e6i	0.308791i
$683$	1.41968e7i	1.16450i	0.813012	+	0.582248i	$0.197827\pi$
$683$	1.41968e7i	1.16450i	−0.813012	+	0.582248i	$0.802173\pi$
$684$	2.44906e7	2.00152
$685$	0	0
$686$	2.38618e7	1.93595
$687$	1.27514e7i	1.03078i
$688$	1.65759e7i	1.33508i
$689$	−9.69954e6	−0.778400
$690$	0	0
$691$	9.99651e6	0.796440	0.398220	−	0.917290i	$-0.369628\pi$
$691$	9.99651e6	0.796440	0.398220	+	0.917290i	$0.369628\pi$
$692$	− 2.58822e7i	− 2.05464i
$693$	− 1.78667e6i	− 0.141323i
$694$	−3.64774e6	−0.287492
$695$	0	0
$696$	6.65814e6	0.520990
$697$	5.75809e6i	0.448949i
$698$	3.07221e7i	2.38678i
$699$	−1.06400e7	−0.823660
$700$	0	0
$701$	−3.82320e6	−0.293854	−0.146927	−	0.989147i	$-0.546938\pi$
$701$	−3.82320e6	−0.293854	−0.146927	+	0.989147i	$0.546938\pi$
$702$	4.17719e7i	3.19920i
$703$	− 1.38555e7i	− 1.05738i
$704$	−2.40977e7	−1.83250
$705$	0	0
$706$	3.21009e7	2.42385
$707$	1.04564e7i	0.786741i
$708$	− 3.17246e7i	− 2.37856i
$709$	−5.38358e6	−0.402213	−0.201106	−	0.979569i	$-0.564454\pi$
$709$	−5.38358e6	−0.402213	−0.201106	+	0.979569i	$0.564454\pi$
$710$	0	0
$711$	−2.58861e6	−0.192041
$712$	6.58810e7i	4.87035i
$713$	− 42913.1i	− 0.00316130i
$714$	−1.03591e7	−0.760457
$715$	0	0
$716$	−1.82755e6	−0.133225
$717$	− 2.59151e6i	− 0.188259i
$718$	3.91720e7i	2.83573i
$719$	1.66113e7	1.19834	0.599171	−	0.800621i	$-0.295497\pi$
$719$	1.66113e7	1.19834	0.599171	+	0.800621i	$0.295497\pi$
$720$	0	0
$721$	7.54266e6	0.540364
$722$	3.37896e7i	2.41235i
$723$	1.52178e7i	1.08269i
$724$	5.09335e7	3.61125
$725$	0	0
$726$	1.88377e6	0.132644
$727$	1.47622e7i	1.03589i	0.855413	+	0.517947i	$0.173303\pi$
$727$	1.47622e7i	1.03589i	−0.855413	+	0.517947i	$0.826697\pi$
$728$	− 8.41713e7i	− 5.88621i
$729$	−1.35110e7	−0.941606
$730$	0	0
$731$	2.10878e6	0.145961
$732$	− 3.48070e7i	− 2.40098i
$733$	8.48819e6i	0.583519i	0.956492	+	0.291760i	$0.0942408\pi$
$733$	8.48819e6i	0.583519i	−0.956492	+	0.291760i	$0.905759\pi$
$734$	4.71217e6	0.322835
$735$	0	0
$736$	487263.	0.0331565
$737$	7.06119e6i	0.478861i
$738$	− 1.18453e7i	− 0.800583i
$739$	−2.60406e7	−1.75404	−0.877022	−	0.480450i	$-0.840473\pi$
$739$	−2.60406e7	−1.75404	−0.877022	+	0.480450i	$0.840473\pi$
$740$	0	0
$741$	−2.46158e7	−1.64690
$742$	1.57396e7i	1.04951i
$743$	791476.i	0.0525975i	0.999654	+	0.0262988i	$0.00837212\pi$
$743$	791476.i	0.0525975i	−0.999654	+	0.0262988i	$0.991628\pi$
$744$	−2.19807e7	−1.45583
$745$	0	0
$746$	−3.15816e6	−0.207772
$747$	− 2.66787e6i	− 0.174930i
$748$	6.89703e6i	0.450722i
$749$	2.51829e7	1.64022
$750$	0	0
$751$	−3.80752e6	−0.246344	−0.123172	−	0.992385i	$-0.539307\pi$
$751$	−3.80752e6	−0.246344	−0.123172	+	0.992385i	$0.539307\pi$
$752$	− 2.69290e7i	− 1.73650i
$753$	− 8.22997e6i	− 0.528946i
$754$	8.60107e6	0.550965
$755$	0	0
$756$	5.05442e7	3.21638
$757$	− 2.46270e7i	− 1.56197i	−0.624552	−	0.780983i	$-0.714719\pi$
$757$	− 2.46270e7i	− 1.56197i	0.624552	−	0.780983i	$-0.285281\pi$
$758$	1.58445e7i	1.00163i
$759$	−21552.3	−0.00135796
$760$	0	0
$761$	−1.49393e7	−0.935123	−0.467561	−	0.883961i	$-0.654867\pi$
$761$	−1.49393e7	−0.935123	−0.467561	+	0.883961i	$0.654867\pi$
$762$	− 3.97355e7i	− 2.47909i
$763$	1.44171e7i	0.896534i
$764$	−2.27918e7	−1.41268
$765$	0	0
$766$	2.50752e7	1.54409
$767$	− 2.70039e7i	− 1.65744i
$768$	− 8.51241e7i	− 5.20774i
$769$	−9.19320e6	−0.560597	−0.280299	−	0.959913i	$-0.590434\pi$
$769$	−9.19320e6	−0.560597	−0.280299	+	0.959913i	$0.590434\pi$
$770$	0	0
$771$	8.22419e6	0.498261
$772$	− 1.94353e7i	− 1.17367i
$773$	2.65966e7i	1.60095i	0.599367	+	0.800474i	$0.295419\pi$
$773$	2.65966e7i	1.60095i	−0.599367	+	0.800474i	$0.704581\pi$
$774$	−4.33810e6	−0.260284
$775$	0	0
$776$	−2.33520e7	−1.39210
$777$	− 8.98986e6i	− 0.534196i
$778$	− 5.72547e7i	− 3.39127i
$779$	2.22032e7	1.31091
$780$	0	0
$781$	−130076.	−0.00763082
$782$	− 105824.i	− 0.00618823i
$783$	3.40322e6i	0.198374i
$784$	−3.60041e6	−0.209200
$785$	0	0
$786$	−8.11355e6	−0.468441
$787$	− 1.01592e7i	− 0.584684i	−0.956314	−	0.292342i	$-0.905565\pi$
$787$	− 1.01592e7i	− 0.584684i	0.956314	−	0.292342i	$-0.0944345\pi$
$788$	− 2.37846e7i	− 1.36452i
$789$	1.81627e7	1.03870
$790$	0	0
$791$	2.75759e7	1.56707
$792$	− 9.34889e6i	− 0.529599i
$793$	− 2.96276e7i	− 1.67307i
$794$	2.82789e6	0.159188
$795$	0	0
$796$	7.09377e7	3.96821
$797$	− 1.13291e7i	− 0.631756i	−0.948800	−	0.315878i	$-0.897701\pi$
$797$	− 1.13291e7i	− 0.631756i	0.948800	−	0.315878i	$-0.102299\pi$
$798$	3.99445e7i	2.22049i
$799$	−3.42590e6	−0.189849
$800$	0	0
$801$	−1.05866e7	−0.583010
$802$	1.80952e7i	0.993406i
$803$	− 3.39223e6i	− 0.185651i
$804$	−6.28008e7	−3.42630
$805$	0	0
$806$	−2.83950e7	−1.53959
$807$	1.70265e6i	0.0920326i
$808$	5.47136e7i	2.94827i
$809$	−2.78519e7	−1.49618	−0.748091	−	0.663597i	$-0.769029\pi$
$809$	−2.78519e7	−1.49618	−0.748091	+	0.663597i	$0.769029\pi$
$810$	0	0
$811$	−3.02449e7	−1.61473	−0.807365	−	0.590052i	$-0.799107\pi$
$811$	−3.02449e7	−1.61473	−0.807365	+	0.590052i	$0.799107\pi$
$812$	− 1.04073e7i	− 0.553923i
$813$	8.07270e6i	0.428344i
$814$	−8.02697e6	−0.424611
$815$	0	0
$816$	−3.32821e7	−1.74979
$817$	− 8.13146e6i	− 0.426200i
$818$	− 1.21913e7i	− 0.637040i
$819$	1.35258e7	0.704615
$820$	0	0
$821$	−1.98523e7	−1.02790	−0.513952	−	0.857819i	$-0.671819\pi$
$821$	−1.98523e7	−1.02790	−0.513952	+	0.857819i	$0.671819\pi$
$822$	− 5.31554e6i	− 0.274390i
$823$	− 1.25066e7i	− 0.643636i	−0.946802	−	0.321818i	$-0.895706\pi$
$823$	− 1.25066e7i	− 0.643636i	0.946802	−	0.321818i	$-0.104294\pi$
$824$	3.94675e7	2.02498
$825$	0	0
$826$	−4.38197e7	−2.23470
$827$	− 8.51692e6i	− 0.433031i	−0.976279	−	0.216515i	$-0.930531\pi$
$827$	− 8.51692e6i	− 0.433031i	0.976279	−	0.216515i	$-0.0694691\pi$
$828$	162329.i	0.00822848i
$829$	−2.94676e6	−0.148922	−0.0744609	−	0.997224i	$-0.523724\pi$
$829$	−2.94676e6	−0.148922	−0.0744609	+	0.997224i	$0.523724\pi$
$830$	0	0
$831$	−1.05254e7	−0.528731
$832$	− 1.82428e8i	− 9.13658i
$833$	458043.i	0.0228714i
$834$	3.31935e7	1.65249
$835$	0	0
$836$	2.65950e7	1.31608
$837$	− 1.12351e7i	− 0.554325i
$838$	8.30596e6i	0.408583i
$839$	5.40609e6	0.265142	0.132571	−	0.991174i	$-0.457677\pi$
$839$	5.40609e6	0.265142	0.132571	+	0.991174i	$0.457677\pi$
$840$	0	0
$841$	−1.98104e7	−0.965836
$842$	7.34422e7i	3.56998i
$843$	− 2.60512e7i	− 1.26258i
$844$	−9.55589e7	−4.61758
$845$	0	0
$846$	7.04763e6	0.338546
$847$	− 1.94019e6i	− 0.0929257i
$848$	5.05690e7i	2.41488i
$849$	6.61379e6	0.314906
$850$	0	0
$851$	91836.8	0.00434703
$852$	− 1.15687e6i	− 0.0545993i
$853$	473034.i	0.0222597i	0.999938	+	0.0111299i	$0.00354282\pi$
$853$	473034.i	0.0222597i	−0.999938	+	0.0111299i	$0.996457\pi$
$854$	−4.80773e7	−2.25577
$855$	0	0
$856$	1.31772e8	6.14663
$857$	6.34036e6i	0.294891i	0.989070	+	0.147446i	$0.0471051\pi$
$857$	6.34036e6i	0.294891i	−0.989070	+	0.147446i	$0.952895\pi$
$858$	1.42608e7i	0.661342i
$859$	2.40671e7	1.11286	0.556430	−	0.830895i	$-0.312171\pi$
$859$	2.40671e7	1.11286	0.556430	+	0.830895i	$0.312171\pi$
$860$	0	0
$861$	1.44061e7	0.662277
$862$	− 3.08829e7i	− 1.41563i
$863$	1.06639e6i	0.0487402i	0.999703	+	0.0243701i	$0.00775802\pi$
$863$	1.06639e6i	0.0487402i	−0.999703	+	0.0243701i	$0.992242\pi$
$864$	1.27571e8	5.81390
$865$	0	0
$866$	−1.41940e7	−0.643146
$867$	− 1.20524e7i	− 0.544537i
$868$	3.43581e7i	1.54785i
$869$	−2.81104e6	−0.126275
$870$	0	0
$871$	−5.34557e7	−2.38753
$872$	7.54385e7i	3.35971i
$873$	− 3.75251e6i	− 0.166643i
$874$	−408057.	−0.0180693
$875$	0	0
$876$	3.01698e7	1.32835
$877$	− 1.28432e7i	− 0.563864i	−0.959434	−	0.281932i	$-0.909025\pi$
$877$	− 1.28432e7i	− 0.563864i	0.959434	−	0.281932i	$-0.0909752\pi$
$878$	1.98769e7i	0.870187i
$879$	1.09485e7	0.477951
$880$	0	0
$881$	2.79947e6	0.121517	0.0607583	−	0.998153i	$-0.480648\pi$
$881$	2.79947e6	0.121517	0.0607583	+	0.998153i	$0.480648\pi$
$882$	− 942268.i	− 0.0407852i
$883$	− 2.81700e7i	− 1.21587i	−0.793988	−	0.607933i	$-0.791999\pi$
$883$	− 2.81700e7i	− 1.21587i	0.793988	−	0.607933i	$-0.208001\pi$
$884$	−5.22130e7	−2.24723
$885$	0	0
$886$	4.34657e7	1.86021
$887$	4.42238e7i	1.88733i	0.330905	+	0.943664i	$0.392646\pi$
$887$	4.42238e7i	1.88733i	−0.330905	+	0.943664i	$0.607354\pi$
$888$	− 4.70401e7i	− 2.00187i
$889$	−4.09257e7	−1.73677
$890$	0	0
$891$	−2.36637e6	−0.0998592
$892$	− 5.74413e7i	− 2.41720i
$893$	1.32103e7i	0.554348i
$894$	−3.00799e7	−1.25873
$895$	0	0
$896$	−1.62964e8	−6.78145
$897$	− 163158.i	− 0.00677062i
$898$	− 5.77542e7i	− 2.38997i
$899$	−2.31338e6	−0.0954657
$900$	0	0
$901$	6.43337e6	0.264014
$902$	− 1.28631e7i	− 0.526417i
$903$	− 5.27595e6i	− 0.215318i
$904$	1.44293e8	5.87251
$905$	0	0
$906$	−5.12131e7	−2.07282
$907$	6.22787e6i	0.251375i	0.992070	+	0.125687i	$0.0401136\pi$
$907$	6.22787e6i	0.251375i	−0.992070	+	0.125687i	$0.959886\pi$
$908$	5.86628e7i	2.36128i
$909$	−8.79211e6	−0.352926
$910$	0	0
$911$	1.08864e7	0.434599	0.217300	−	0.976105i	$-0.430275\pi$
$911$	1.08864e7	0.434599	0.217300	+	0.976105i	$0.430275\pi$
$912$	1.28336e8i	5.10928i
$913$	− 2.89711e6i	− 0.115024i
$914$	2.66013e7	1.05327
$915$	0	0
$916$	1.04295e8	4.10698
$917$	8.35655e6i	0.328174i
$918$	− 2.77059e7i	− 1.08509i
$919$	1.61046e7	0.629014	0.314507	−	0.949255i	$-0.398161\pi$
$919$	1.61046e7	0.629014	0.314507	+	0.949255i	$0.398161\pi$
$920$	0	0
$921$	2.89495e6	0.112458
$922$	− 3.16435e7i	− 1.22590i
$923$	− 984726.i	− 0.0380462i
$924$	1.72557e7	0.664893
$925$	0	0
$926$	5.90839e7	2.26434
$927$	6.34216e6i	0.242403i
$928$	− 2.62676e7i	− 1.00127i
$929$	−1.97673e6	−0.0751465	−0.0375732	−	0.999294i	$-0.511963\pi$
$929$	−1.97673e6	−0.0751465	−0.0375732	+	0.999294i	$0.511963\pi$
$930$	0	0
$931$	1.76621e6	0.0667834
$932$	8.70248e7i	3.28173i
$933$	− 8.99622e6i	− 0.338342i
$934$	−1.33476e7	−0.500652
$935$	0	0
$936$	7.07745e7	2.64051
$937$	1.38945e7i	0.517003i	0.966011	+	0.258502i	$0.0832287\pi$
$937$	1.38945e7i	0.517003i	−0.966011	+	0.258502i	$0.916771\pi$
$938$	8.67436e7i	3.21907i
$939$	2.47863e7	0.917377
$940$	0	0
$941$	4.33771e7	1.59693	0.798466	−	0.602040i	$-0.205645\pi$
$941$	4.33771e7	1.59693	0.798466	+	0.602040i	$0.205645\pi$
$942$	3.64524e7i	1.33844i
$943$	147167.i	0.00538930i
$944$	−1.40786e8	−5.14197
$945$	0	0
$946$	−4.71085e6	−0.171148
$947$	− 1.79958e7i	− 0.652075i	−0.945357	−	0.326037i	$-0.894286\pi$
$947$	− 1.79958e7i	− 0.652075i	0.945357	−	0.326037i	$-0.105714\pi$
$948$	− 2.50008e7i	− 0.903510i
$949$	2.56804e7	0.925629
$950$	0	0
$951$	3.89384e7	1.39613
$952$	5.58280e7i	1.99645i
$953$	− 2.95957e7i	− 1.05559i	−0.849371	−	0.527796i	$-0.823018\pi$
$953$	− 2.95957e7i	− 1.05559i	0.849371	−	0.527796i	$-0.176982\pi$
$954$	−1.32345e7	−0.470800
$955$	0	0
$956$	−2.11961e7	−0.750085
$957$	1.16185e6i	0.0410081i
$958$	6.31399e7i	2.22275i
$959$	−5.47475e6	−0.192228
$960$	0	0
$961$	−2.09919e7	−0.733236
$962$	− 6.07671e7i	− 2.11705i
$963$	2.11748e7i	0.735789i
$964$	1.24467e8	4.31381
$965$	0	0
$966$	−264760.	−0.00912872
$967$	5.50973e7i	1.89480i	0.320046	+	0.947402i	$0.396301\pi$
$967$	5.50973e7i	1.89480i	−0.320046	+	0.947402i	$0.603699\pi$
$968$	− 1.01522e7i	− 0.348234i
$969$	1.63268e7	0.558588
$970$	0	0
$971$	1.86888e7	0.636112	0.318056	−	0.948072i	$-0.396970\pi$
$971$	1.86888e7	0.636112	0.318056	+	0.948072i	$0.396970\pi$
$972$	7.16379e7i	2.43208i
$973$	− 3.41877e7i	− 1.15768i
$974$	1.23100e7	0.415778
$975$	0	0
$976$	−1.54465e8	−5.19045
$977$	− 7.09223e6i	− 0.237710i	−0.992912	−	0.118855i	$-0.962078\pi$
$977$	− 7.09223e6i	− 0.237710i	0.992912	−	0.118855i	$-0.0379223\pi$
$978$	6.32568e7i	2.11476i
$979$	−1.14963e7	−0.383354
$980$	0	0
$981$	−1.21225e7	−0.402178
$982$	1.50149e7i	0.496870i
$983$	− 2.97902e7i	− 0.983307i	−0.870791	−	0.491653i	$-0.836393\pi$
$983$	− 2.97902e7i	− 0.983307i	0.870791	−	0.491653i	$-0.163607\pi$
$984$	7.53812e7	2.48185
$985$	0	0
$986$	−5.70480e6	−0.186874
$987$	8.57124e6i	0.280060i
$988$	2.01333e8i	6.56181i
$989$	53896.9	0.00175216
$990$	0	0
$991$	−2.60948e7	−0.844054	−0.422027	−	0.906583i	$-0.638681\pi$
$991$	−2.60948e7	−0.844054	−0.422027	+	0.906583i	$0.638681\pi$
$992$	8.67180e7i	2.79789i
$993$	1.72902e7i	0.556452i
$994$	−1.59793e6	−0.0512971
$995$	0	0
$996$	2.57663e7	0.823007
$997$	− 2.68998e7i	− 0.857061i	−0.903527	−	0.428531i	$-0.859031\pi$
$997$	− 2.68998e7i	− 0.857061i	0.903527	−	0.428531i	$-0.140969\pi$
$998$	− 1.45523e7i	− 0.462494i
$999$	2.40439e7	0.762240

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.6.b.f.199.12		12
5.2	odd	4		275.6.a.f.1.1		6
5.3	odd	4		55.6.a.d.1.6	✓	6
5.4	even	2	inner	275.6.b.f.199.1		12
15.8	even	4		495.6.a.l.1.1		6
20.3	even	4		880.6.a.u.1.3		6
55.43	even	4		605.6.a.e.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.6.a.d.1.6	✓	6	5.3	odd	4
275.6.a.f.1.1		6	5.2	odd	4
275.6.b.f.199.1		12	5.4	even	2	inner
275.6.b.f.199.12		12	1.1	even	1	trivial
495.6.a.l.1.1		6	15.8	even	4
605.6.a.e.1.1		6	55.43	even	4
880.6.a.u.1.3		6	20.3	even	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:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	275.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+11.2169i q^{2} -11.4706i q^{3} -93.8183 q^{4} +128.664 q^{6} -132.518i q^{7} -693.408i q^{8} +111.426 q^{9} +O(q^{10})\)	\(q+11.2169i q^{2} -11.4706i q^{3} -93.8183 q^{4} +128.664 q^{6} -132.518i q^{7} -693.408i q^{8} +111.426 q^{9} +121.000 q^{11} +1076.15i q^{12} +916.014i q^{13} +1486.43 q^{14} +4775.68 q^{16} -607.561i q^{17} +1249.85i q^{18} -2342.75 q^{19} -1520.05 q^{21} +1357.24i q^{22} -15.5282i q^{23} -7953.78 q^{24} -10274.8 q^{26} -4065.47i q^{27} +12432.6i q^{28} -837.103 q^{29} +2763.55 q^{31} +31379.2i q^{32} -1387.94i q^{33} +6814.93 q^{34} -10453.8 q^{36} +5914.18i q^{37} -26278.4i q^{38} +10507.2 q^{39} -9477.40 q^{41} -17050.2i q^{42} +3470.90i q^{43} -11352.0 q^{44} +174.178 q^{46} -5638.78i q^{47} -54779.8i q^{48} -753.904 q^{49} -6969.07 q^{51} -85938.8i q^{52} +10588.9i q^{53} +45601.9 q^{54} -91888.7 q^{56} +26872.7i q^{57} -9389.68i q^{58} -29479.8 q^{59} -32344.0 q^{61} +30998.4i q^{62} -14765.9i q^{63} -199154. q^{64} +15568.3 q^{66} +58356.9i q^{67} +57000.3i q^{68} -178.118 q^{69} -1075.01 q^{71} -77263.6i q^{72} -28035.0i q^{73} -66338.6 q^{74} +219793. q^{76} -16034.6i q^{77} +117858. i q^{78} -23231.7 q^{79} -19556.8 q^{81} -106307. i q^{82} -23943.0i q^{83} +142609. q^{84} -38932.6 q^{86} +9602.05i q^{87} -83902.3i q^{88} -95010.4 q^{89} +121388. q^{91} +1456.83i q^{92} -31699.5i q^{93} +63249.5 q^{94} +359937. q^{96} -33677.1i q^{97} -8456.45i q^{98} +13482.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 230 q^{4} + 650 q^{6} - 1664 q^{9}+O(q^{10}) \) 12 * q - 230 * q^4 + 650 * q^6 - 1664 * q^9	\( 12 q - 230 q^{4} + 650 q^{6} - 1664 q^{9} + 1452 q^{11} - 4034 q^{14} + 13350 q^{16} - 1220 q^{19} + 5016 q^{21} - 26894 q^{24} - 15484 q^{26} - 21928 q^{29} + 17736 q^{31} + 85346 q^{34} - 60232 q^{36} - 14416 q^{39} + 320 q^{41} - 27830 q^{44} + 3300 q^{46} - 97024 q^{49} - 86756 q^{51} + 113762 q^{54} - 42594 q^{56} - 241064 q^{59} - 18456 q^{61} - 605678 q^{64} + 78650 q^{66} - 237920 q^{69} + 327896 q^{71} - 267002 q^{74} - 58710 q^{76} + 73424 q^{79} + 404332 q^{81} - 238066 q^{84} + 167992 q^{86} - 509892 q^{89} + 335688 q^{91} - 62828 q^{94} + 468830 q^{96} - 201344 q^{99}+O(q^{100}) \) 12 * q - 230 * q^4 + 650 * q^6 - 1664 * q^9 + 1452 * q^11 - 4034 * q^14 + 13350 * q^16 - 1220 * q^19 + 5016 * q^21 - 26894 * q^24 - 15484 * q^26 - 21928 * q^29 + 17736 * q^31 + 85346 * q^34 - 60232 * q^36 - 14416 * q^39 + 320 * q^41 - 27830 * q^44 + 3300 * q^46 - 97024 * q^49 - 86756 * q^51 + 113762 * q^54 - 42594 * q^56 - 241064 * q^59 - 18456 * q^61 - 605678 * q^64 + 78650 * q^66 - 237920 * q^69 + 327896 * q^71 - 267002 * q^74 - 58710 * q^76 + 73424 * q^79 + 404332 * q^81 - 238066 * q^84 + 167992 * q^86 - 509892 * q^89 + 335688 * q^91 - 62828 * q^94 + 468830 * q^96 - 201344 * q^99

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