LMFDB - Embedded newform 3850.2.c.q.1849.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3850,2,Mod(1849,3850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3850, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3850.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3850.c (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:	$30.7424047782$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 154)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.3
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	3850.1849
Dual form			3850.2.c.q.1849.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.23607i q^{3} -1.00000 q^{4} +1.23607 q^{6} +1.00000i q^{7} -1.00000i q^{8} +1.47214 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.23607i q^{3} -1.00000 q^{4} +1.23607 q^{6} +1.00000i q^{7} -1.00000i q^{8} +1.47214 q^{9} +1.00000 q^{11} +1.23607i q^{12} +3.23607i q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.47214i q^{17} +1.47214i q^{18} +7.23607 q^{19} +1.23607 q^{21} +1.00000i q^{22} -4.00000i q^{23} -1.23607 q^{24} -3.23607 q^{26} -5.52786i q^{27} -1.00000i q^{28} -4.47214 q^{29} +2.00000 q^{31} +1.00000i q^{32} -1.23607i q^{33} -2.47214 q^{34} -1.47214 q^{36} +6.94427i q^{37} +7.23607i q^{38} +4.00000 q^{39} -2.47214 q^{41} +1.23607i q^{42} +10.4721i q^{43} -1.00000 q^{44} +4.00000 q^{46} -2.00000i q^{47} -1.23607i q^{48} -1.00000 q^{49} +3.05573 q^{51} -3.23607i q^{52} -8.47214i q^{53} +5.52786 q^{54} +1.00000 q^{56} -8.94427i q^{57} -4.47214i q^{58} -2.76393 q^{59} -0.763932 q^{61} +2.00000i q^{62} +1.47214i q^{63} -1.00000 q^{64} +1.23607 q^{66} +11.4164i q^{67} -2.47214i q^{68} -4.94427 q^{69} +6.47214 q^{71} -1.47214i q^{72} -12.9443i q^{73} -6.94427 q^{74} -7.23607 q^{76} +1.00000i q^{77} +4.00000i q^{78} -2.41641 q^{81} -2.47214i q^{82} +12.1803i q^{83} -1.23607 q^{84} -10.4721 q^{86} +5.52786i q^{87} -1.00000i q^{88} -10.0000 q^{89} -3.23607 q^{91} +4.00000i q^{92} -2.47214i q^{93} +2.00000 q^{94} +1.23607 q^{96} +12.4721i q^{97} -1.00000i q^{98} +1.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{6} - 12 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 - 4 * q^6 - 12 * q^9	$ 4 q - 4 q^{4} - 4 q^{6} - 12 q^{9} + 4 q^{11} - 4 q^{14} + 4 q^{16} + 20 q^{19} - 4 q^{21} + 4 q^{24} - 4 q^{26} + 8 q^{31} + 8 q^{34} + 12 q^{36} + 16 q^{39} + 8 q^{41} - 4 q^{44} + 16 q^{46} - 4 q^{49} + 48 q^{51} + 40 q^{54} + 4 q^{56} - 20 q^{59} - 12 q^{61} - 4 q^{64} - 4 q^{66} + 16 q^{69} + 8 q^{71} + 8 q^{74} - 20 q^{76} + 44 q^{81} + 4 q^{84} - 24 q^{86} - 40 q^{89} - 4 q^{91} + 8 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^6 - 12 * q^9 + 4 * q^11 - 4 * q^14 + 4 * q^16 + 20 * q^19 - 4 * q^21 + 4 * q^24 - 4 * q^26 + 8 * q^31 + 8 * q^34 + 12 * q^36 + 16 * q^39 + 8 * q^41 - 4 * q^44 + 16 * q^46 - 4 * q^49 + 48 * q^51 + 40 * q^54 + 4 * q^56 - 20 * q^59 - 12 * q^61 - 4 * q^64 - 4 * q^66 + 16 * q^69 + 8 * q^71 + 8 * q^74 - 20 * q^76 + 44 * q^81 + 4 * q^84 - 24 * q^86 - 40 * q^89 - 4 * q^91 + 8 * q^94 - 4 * q^96 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$.

$n$	$1751$	$2201$	$2927$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	− 1.23607i	− 0.713644i	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	− 1.23607i	− 0.713644i	0.934172	−	0.356822i	$-0.116140\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.23607	0.504623
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	− 1.00000i	− 0.353553i
$9$	1.47214	0.490712
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.23607i	0.356822i
$13$	3.23607i	0.897524i	0.893651	+	0.448762i	$0.148135\pi$
$13$	3.23607i	0.897524i	−0.893651	+	0.448762i	$0.851865\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	2.47214i	0.599581i	0.954005	+	0.299791i	$0.0969168\pi$
$17$	2.47214i	0.599581i	−0.954005	+	0.299791i	$0.903083\pi$
$18$	1.47214i	0.346986i
$19$	7.23607	1.66007	0.830034	−	0.557713i	$-0.188321\pi$
$19$	7.23607	1.66007	0.830034	+	0.557713i	$0.188321\pi$
$20$	0	0
$21$	1.23607	0.269732
$22$	1.00000i	0.213201i
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	−1.23607	−0.252311
$25$	0	0
$26$	−3.23607	−0.634645
$27$	− 5.52786i	− 1.06384i
$28$	− 1.00000i	− 0.188982i
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000i	0.176777i
$33$	− 1.23607i	− 0.215172i
$34$	−2.47214	−0.423968
$35$	0	0
$36$	−1.47214	−0.245356
$37$	6.94427i	1.14163i	0.821078	+	0.570816i	$0.193373\pi$
$37$	6.94427i	1.14163i	−0.821078	+	0.570816i	$0.806627\pi$
$38$	7.23607i	1.17385i
$39$	4.00000	0.640513
$40$	0	0
$41$	−2.47214	−0.386083	−0.193041	−	0.981191i	$-0.561835\pi$
$41$	−2.47214	−0.386083	−0.193041	+	0.981191i	$0.561835\pi$
$42$	1.23607i	0.190729i
$43$	10.4721i	1.59699i	0.602004	+	0.798493i	$0.294369\pi$
$43$	10.4721i	1.59699i	−0.602004	+	0.798493i	$0.705631\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	4.00000	0.589768
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	− 1.23607i	− 0.178411i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	3.05573	0.427888
$52$	− 3.23607i	− 0.448762i
$53$	− 8.47214i	− 1.16374i	−0.813283	−	0.581869i	$-0.802322\pi$
$53$	− 8.47214i	− 1.16374i	0.813283	−	0.581869i	$-0.197678\pi$
$54$	5.52786	0.752247
$55$	0	0
$56$	1.00000	0.133631
$57$	− 8.94427i	− 1.18470i
$58$	− 4.47214i	− 0.587220i
$59$	−2.76393	−0.359833	−0.179917	−	0.983682i	$-0.557583\pi$
$59$	−2.76393	−0.359833	−0.179917	+	0.983682i	$0.557583\pi$
$60$	0	0
$61$	−0.763932	−0.0978115	−0.0489057	−	0.998803i	$-0.515573\pi$
$61$	−0.763932	−0.0978115	−0.0489057	+	0.998803i	$0.515573\pi$
$62$	2.00000i	0.254000i
$63$	1.47214i	0.185472i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	1.23607	0.152149
$67$	11.4164i	1.39474i	0.716713	+	0.697368i	$0.245646\pi$
$67$	11.4164i	1.39474i	−0.716713	+	0.697368i	$0.754354\pi$
$68$	− 2.47214i	− 0.299791i
$69$	−4.94427	−0.595220
$70$	0	0
$71$	6.47214	0.768101	0.384051	−	0.923312i	$-0.374529\pi$
$71$	6.47214	0.768101	0.384051	+	0.923312i	$0.374529\pi$
$72$	− 1.47214i	− 0.173493i
$73$	− 12.9443i	− 1.51501i	−0.652828	−	0.757506i	$-0.726418\pi$
$73$	− 12.9443i	− 1.51501i	0.652828	−	0.757506i	$-0.273582\pi$
$74$	−6.94427	−0.807255
$75$	0	0
$76$	−7.23607	−0.830034
$77$	1.00000i	0.113961i
$78$	4.00000i	0.452911i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	− 2.47214i	− 0.273002i
$83$	12.1803i	1.33697i	0.743727	+	0.668483i	$0.233056\pi$
$83$	12.1803i	1.33697i	−0.743727	+	0.668483i	$0.766944\pi$
$84$	−1.23607	−0.134866
$85$	0	0
$86$	−10.4721	−1.12924
$87$	5.52786i	0.592649i
$88$	− 1.00000i	− 0.106600i
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−3.23607	−0.339232
$92$	4.00000i	0.417029i
$93$	− 2.47214i	− 0.256349i
$94$	2.00000	0.206284
$95$	0	0
$96$	1.23607	0.126156
$97$	12.4721i	1.26635i	0.774007	+	0.633177i	$0.218249\pi$
$97$	12.4721i	1.26635i	−0.774007	+	0.633177i	$0.781751\pi$
$98$	− 1.00000i	− 0.101015i
$99$	1.47214	0.147955
$100$	0	0
$101$	8.18034	0.813974	0.406987	−	0.913434i	$-0.366579\pi$
$101$	8.18034	0.813974	0.406987	+	0.913434i	$0.366579\pi$
$102$	3.05573i	0.302562i
$103$	14.9443i	1.47250i	0.676708	+	0.736251i	$0.263406\pi$
$103$	14.9443i	1.47250i	−0.676708	+	0.736251i	$0.736594\pi$
$104$	3.23607	0.317323
$105$	0	0
$106$	8.47214	0.822887
$107$	2.47214i	0.238990i	0.992835	+	0.119495i	$0.0381276\pi$
$107$	2.47214i	0.238990i	−0.992835	+	0.119495i	$0.961872\pi$
$108$	5.52786i	0.531919i
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	8.58359	0.814719
$112$	1.00000i	0.0944911i
$113$	0.472136i	0.0444148i	0.999753	+	0.0222074i	$0.00706942\pi$
$113$	0.472136i	0.0444148i	−0.999753	+	0.0222074i	$0.992931\pi$
$114$	8.94427	0.837708
$115$	0	0
$116$	4.47214	0.415227
$117$	4.76393i	0.440426i
$118$	− 2.76393i	− 0.254441i
$119$	−2.47214	−0.226620
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 0.763932i	− 0.0691632i
$123$	3.05573i	0.275526i
$124$	−2.00000	−0.179605
$125$	0	0
$126$	−1.47214	−0.131148
$127$	− 12.0000i	− 1.06483i	−0.846484	−	0.532414i	$-0.821285\pi$
$127$	− 12.0000i	− 1.06483i	0.846484	−	0.532414i	$-0.178715\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	12.9443	1.13968
$130$	0	0
$131$	4.76393	0.416227	0.208113	−	0.978105i	$-0.433268\pi$
$131$	4.76393	0.416227	0.208113	+	0.978105i	$0.433268\pi$
$132$	1.23607i	0.107586i
$133$	7.23607i	0.627447i
$134$	−11.4164	−0.986227
$135$	0	0
$136$	2.47214	0.211984
$137$	− 19.8885i	− 1.69919i	−0.527433	−	0.849596i	$-0.676846\pi$
$137$	− 19.8885i	− 1.69919i	0.527433	−	0.849596i	$-0.323154\pi$
$138$	− 4.94427i	− 0.420884i
$139$	21.7082	1.84127	0.920633	−	0.390429i	$-0.127673\pi$
$139$	21.7082	1.84127	0.920633	+	0.390429i	$0.127673\pi$
$140$	0	0
$141$	−2.47214	−0.208191
$142$	6.47214i	0.543130i
$143$	3.23607i	0.270614i
$144$	1.47214	0.122678
$145$	0	0
$146$	12.9443	1.07128
$147$	1.23607i	0.101949i
$148$	− 6.94427i	− 0.570816i
$149$	22.3607	1.83186	0.915929	−	0.401340i	$-0.131455\pi$
$149$	22.3607	1.83186	0.915929	+	0.401340i	$0.131455\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	− 7.23607i	− 0.586923i
$153$	3.63932i	0.294222i
$154$	−1.00000	−0.0805823
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 12.6525i	− 1.00978i	−0.863184	−	0.504889i	$-0.831534\pi$
$157$	− 12.6525i	− 1.00978i	0.863184	−	0.504889i	$-0.168466\pi$
$158$	0	0
$159$	−10.4721	−0.830494
$160$	0	0
$161$	4.00000	0.315244
$162$	− 2.41641i	− 0.189851i
$163$	19.4164i	1.52081i	0.649449	+	0.760405i	$0.275000\pi$
$163$	19.4164i	1.52081i	−0.649449	+	0.760405i	$0.725000\pi$
$164$	2.47214	0.193041
$165$	0	0
$166$	−12.1803	−0.945378
$167$	11.4164i	0.883428i	0.897156	+	0.441714i	$0.145629\pi$
$167$	11.4164i	0.883428i	−0.897156	+	0.441714i	$0.854371\pi$
$168$	− 1.23607i	− 0.0953647i
$169$	2.52786	0.194451
$170$	0	0
$171$	10.6525	0.814615
$172$	− 10.4721i	− 0.798493i
$173$	3.23607i	0.246034i	0.992405	+	0.123017i	$0.0392569\pi$
$173$	3.23607i	0.246034i	−0.992405	+	0.123017i	$0.960743\pi$
$174$	−5.52786	−0.419066
$175$	0	0
$176$	1.00000	0.0753778
$177$	3.41641i	0.256793i
$178$	− 10.0000i	− 0.749532i
$179$	8.94427	0.668526	0.334263	−	0.942480i	$-0.391513\pi$
$179$	8.94427	0.668526	0.334263	+	0.942480i	$0.391513\pi$
$180$	0	0
$181$	9.23607	0.686512	0.343256	−	0.939242i	$-0.388470\pi$
$181$	9.23607	0.686512	0.343256	+	0.939242i	$0.388470\pi$
$182$	− 3.23607i	− 0.239873i
$183$	0.944272i	0.0698026i
$184$	−4.00000	−0.294884
$185$	0	0
$186$	2.47214	0.181266
$187$	2.47214i	0.180780i
$188$	2.00000i	0.145865i
$189$	5.52786	0.402093
$190$	0	0
$191$	−2.47214	−0.178877	−0.0894387	−	0.995992i	$-0.528507\pi$
$191$	−2.47214	−0.178877	−0.0894387	+	0.995992i	$0.528507\pi$
$192$	1.23607i	0.0892055i
$193$	14.9443i	1.07571i	0.843037	+	0.537856i	$0.180766\pi$
$193$	14.9443i	1.07571i	−0.843037	+	0.537856i	$0.819234\pi$
$194$	−12.4721	−0.895447
$195$	0	0
$196$	1.00000	0.0714286
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	1.47214i	0.104620i
$199$	−18.9443	−1.34292	−0.671462	−	0.741039i	$-0.734333\pi$
$199$	−18.9443	−1.34292	−0.671462	+	0.741039i	$0.734333\pi$
$200$	0	0
$201$	14.1115	0.995345
$202$	8.18034i	0.575567i
$203$	− 4.47214i	− 0.313882i
$204$	−3.05573	−0.213944
$205$	0	0
$206$	−14.9443	−1.04122
$207$	− 5.88854i	− 0.409282i
$208$	3.23607i	0.224381i
$209$	7.23607	0.500529
$210$	0	0
$211$	−13.5279	−0.931297	−0.465648	−	0.884970i	$-0.654179\pi$
$211$	−13.5279	−0.931297	−0.465648	+	0.884970i	$0.654179\pi$
$212$	8.47214i	0.581869i
$213$	− 8.00000i	− 0.548151i
$214$	−2.47214	−0.168992
$215$	0	0
$216$	−5.52786	−0.376124
$217$	2.00000i	0.135769i
$218$	10.0000i	0.677285i
$219$	−16.0000	−1.08118
$220$	0	0
$221$	−8.00000	−0.538138
$222$	8.58359i	0.576093i
$223$	0.472136i	0.0316166i	0.999875	+	0.0158083i	$0.00503214\pi$
$223$	0.472136i	0.0316166i	−0.999875	+	0.0158083i	$0.994968\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−0.472136	−0.0314060
$227$	− 19.2361i	− 1.27674i	−0.769729	−	0.638371i	$-0.779608\pi$
$227$	− 19.2361i	− 1.27674i	0.769729	−	0.638371i	$-0.220392\pi$
$228$	8.94427i	0.592349i
$229$	−17.2361	−1.13899	−0.569496	−	0.821994i	$-0.692861\pi$
$229$	−17.2361	−1.13899	−0.569496	+	0.821994i	$0.692861\pi$
$230$	0	0
$231$	1.23607	0.0813273
$232$	4.47214i	0.293610i
$233$	14.9443i	0.979032i	0.871994	+	0.489516i	$0.162826\pi$
$233$	14.9443i	0.979032i	−0.871994	+	0.489516i	$0.837174\pi$
$234$	−4.76393	−0.311428
$235$	0	0
$236$	2.76393	0.179917
$237$	0	0
$238$	− 2.47214i	− 0.160245i
$239$	−20.0000	−1.29369	−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000	−1.29369	−0.646846	+	0.762620i	$0.723912\pi$
$240$	0	0
$241$	15.4164	0.993058	0.496529	−	0.868020i	$-0.334608\pi$
$241$	15.4164	0.993058	0.496529	+	0.868020i	$0.334608\pi$
$242$	1.00000i	0.0642824i
$243$	− 13.5967i	− 0.872232i
$244$	0.763932	0.0489057
$245$	0	0
$246$	−3.05573	−0.194826
$247$	23.4164i	1.48995i
$248$	− 2.00000i	− 0.127000i
$249$	15.0557	0.954118
$250$	0	0
$251$	29.2361	1.84536	0.922682	−	0.385562i	$-0.125992\pi$
$251$	29.2361	1.84536	0.922682	+	0.385562i	$0.125992\pi$
$252$	− 1.47214i	− 0.0927358i
$253$	− 4.00000i	− 0.251478i
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.94427i	0.433172i	0.976264	+	0.216586i	$0.0694921\pi$
$257$	6.94427i	0.433172i	−0.976264	+	0.216586i	$0.930508\pi$
$258$	12.9443i	0.805875i
$259$	−6.94427	−0.431496
$260$	0	0
$261$	−6.58359	−0.407514
$262$	4.76393i	0.294317i
$263$	4.94427i	0.304877i	0.988313	+	0.152438i	$0.0487126\pi$
$263$	4.94427i	0.304877i	−0.988313	+	0.152438i	$0.951287\pi$
$264$	−1.23607	−0.0760747
$265$	0	0
$266$	−7.23607	−0.443672
$267$	12.3607i	0.756461i
$268$	− 11.4164i	− 0.697368i
$269$	22.7639	1.38794	0.693971	−	0.720003i	$-0.255860\pi$
$269$	22.7639	1.38794	0.693971	+	0.720003i	$0.255860\pi$
$270$	0	0
$271$	0.944272	0.0573604	0.0286802	−	0.999589i	$-0.490870\pi$
$271$	0.944272	0.0573604	0.0286802	+	0.999589i	$0.490870\pi$
$272$	2.47214i	0.149895i
$273$	4.00000i	0.242091i
$274$	19.8885	1.20151
$275$	0	0
$276$	4.94427	0.297610
$277$	3.52786i	0.211969i	0.994368	+	0.105984i	$0.0337994\pi$
$277$	3.52786i	0.211969i	−0.994368	+	0.105984i	$0.966201\pi$
$278$	21.7082i	1.30197i
$279$	2.94427	0.176269
$280$	0	0
$281$	28.8328	1.72002	0.860011	−	0.510276i	$-0.170457\pi$
$281$	28.8328	1.72002	0.860011	+	0.510276i	$0.170457\pi$
$282$	− 2.47214i	− 0.147214i
$283$	− 14.6525i	− 0.870999i	−0.900189	−	0.435500i	$-0.856572\pi$
$283$	− 14.6525i	− 0.870999i	0.900189	−	0.435500i	$-0.143428\pi$
$284$	−6.47214	−0.384051
$285$	0	0
$286$	−3.23607	−0.191353
$287$	− 2.47214i	− 0.145926i
$288$	1.47214i	0.0867464i
$289$	10.8885	0.640503
$290$	0	0
$291$	15.4164	0.903726
$292$	12.9443i	0.757506i
$293$	26.6525i	1.55705i	0.627611	+	0.778527i	$0.284033\pi$
$293$	26.6525i	1.55705i	−0.627611	+	0.778527i	$0.715967\pi$
$294$	−1.23607	−0.0720889
$295$	0	0
$296$	6.94427	0.403628
$297$	− 5.52786i	− 0.320759i
$298$	22.3607i	1.29532i
$299$	12.9443	0.748587
$300$	0	0
$301$	−10.4721	−0.603604
$302$	12.0000i	0.690522i
$303$	− 10.1115i	− 0.580888i
$304$	7.23607	0.415017
$305$	0	0
$306$	−3.63932	−0.208046
$307$	− 26.0689i	− 1.48783i	−0.668274	−	0.743915i	$-0.732967\pi$
$307$	− 26.0689i	− 1.48783i	0.668274	−	0.743915i	$-0.267033\pi$
$308$	− 1.00000i	− 0.0569803i
$309$	18.4721	1.05084
$310$	0	0
$311$	−21.4164	−1.21441	−0.607207	−	0.794544i	$-0.707710\pi$
$311$	−21.4164	−1.21441	−0.607207	+	0.794544i	$0.707710\pi$
$312$	− 4.00000i	− 0.226455i
$313$	− 19.5279i	− 1.10378i	−0.833917	−	0.551890i	$-0.813907\pi$
$313$	− 19.5279i	− 1.10378i	0.833917	−	0.551890i	$-0.186093\pi$
$314$	12.6525	0.714021
$315$	0	0
$316$	0	0
$317$	− 30.9443i	− 1.73800i	−0.494809	−	0.869002i	$-0.664762\pi$
$317$	− 30.9443i	− 1.73800i	0.494809	−	0.869002i	$-0.335238\pi$
$318$	− 10.4721i	− 0.587248i
$319$	−4.47214	−0.250392
$320$	0	0
$321$	3.05573	0.170554
$322$	4.00000i	0.222911i
$323$	17.8885i	0.995345i
$324$	2.41641	0.134245
$325$	0	0
$326$	−19.4164	−1.07538
$327$	− 12.3607i	− 0.683547i
$328$	2.47214i	0.136501i
$329$	2.00000	0.110264
$330$	0	0
$331$	−16.9443	−0.931341	−0.465671	−	0.884958i	$-0.654187\pi$
$331$	−16.9443	−0.931341	−0.465671	+	0.884958i	$0.654187\pi$
$332$	− 12.1803i	− 0.668483i
$333$	10.2229i	0.560212i
$334$	−11.4164	−0.624678
$335$	0	0
$336$	1.23607	0.0674330
$337$	18.0000i	0.980522i	0.871576	+	0.490261i	$0.163099\pi$
$337$	18.0000i	0.980522i	−0.871576	+	0.490261i	$0.836901\pi$
$338$	2.52786i	0.137498i
$339$	0.583592	0.0316964
$340$	0	0
$341$	2.00000	0.108306
$342$	10.6525i	0.576020i
$343$	− 1.00000i	− 0.0539949i
$344$	10.4721	0.564620
$345$	0	0
$346$	−3.23607	−0.173972
$347$	2.47214i	0.132711i	0.997796	+	0.0663556i	$0.0211372\pi$
$347$	2.47214i	0.132711i	−0.997796	+	0.0663556i	$0.978863\pi$
$348$	− 5.52786i	− 0.296325i
$349$	−21.7082	−1.16201	−0.581007	−	0.813899i	$-0.697341\pi$
$349$	−21.7082	−1.16201	−0.581007	+	0.813899i	$0.697341\pi$
$350$	0	0
$351$	17.8885	0.954820
$352$	1.00000i	0.0533002i
$353$	17.0557i	0.907785i	0.891056	+	0.453892i	$0.149965\pi$
$353$	17.0557i	0.907785i	−0.891056	+	0.453892i	$0.850035\pi$
$354$	−3.41641	−0.181580
$355$	0	0
$356$	10.0000	0.529999
$357$	3.05573i	0.161726i
$358$	8.94427i	0.472719i
$359$	−26.8328	−1.41618	−0.708091	−	0.706121i	$-0.750443\pi$
$359$	−26.8328	−1.41618	−0.708091	+	0.706121i	$0.750443\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	9.23607i	0.485437i
$363$	− 1.23607i	− 0.0648767i
$364$	3.23607	0.169616
$365$	0	0
$366$	−0.944272	−0.0493579
$367$	− 5.41641i	− 0.282734i	−0.989957	−	0.141367i	$-0.954850\pi$
$367$	− 5.41641i	− 0.282734i	0.989957	−	0.141367i	$-0.0451498\pi$
$368$	− 4.00000i	− 0.208514i
$369$	−3.63932	−0.189455
$370$	0	0
$371$	8.47214	0.439851
$372$	2.47214i	0.128174i
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	−2.47214	−0.127831
$375$	0	0
$376$	−2.00000	−0.103142
$377$	− 14.4721i	− 0.745353i
$378$	5.52786i	0.284323i
$379$	−14.4721	−0.743384	−0.371692	−	0.928356i	$-0.621222\pi$
$379$	−14.4721	−0.743384	−0.371692	+	0.928356i	$0.621222\pi$
$380$	0	0
$381$	−14.8328	−0.759908
$382$	− 2.47214i	− 0.126485i
$383$	23.8885i	1.22065i	0.792152	+	0.610324i	$0.208961\pi$
$383$	23.8885i	1.22065i	−0.792152	+	0.610324i	$0.791039\pi$
$384$	−1.23607	−0.0630778
$385$	0	0
$386$	−14.9443	−0.760643
$387$	15.4164i	0.783660i
$388$	− 12.4721i	− 0.633177i
$389$	−33.4164	−1.69428	−0.847140	−	0.531370i	$-0.821677\pi$
$389$	−33.4164	−1.69428	−0.847140	+	0.531370i	$0.821677\pi$
$390$	0	0
$391$	9.88854	0.500085
$392$	1.00000i	0.0505076i
$393$	− 5.88854i	− 0.297038i
$394$	−18.0000	−0.906827
$395$	0	0
$396$	−1.47214	−0.0739776
$397$	− 23.7082i	− 1.18988i	−0.803770	−	0.594940i	$-0.797176\pi$
$397$	− 23.7082i	− 1.18988i	0.803770	−	0.594940i	$-0.202824\pi$
$398$	− 18.9443i	− 0.949591i
$399$	8.94427	0.447774
$400$	0	0
$401$	14.3607	0.717138	0.358569	−	0.933503i	$-0.383265\pi$
$401$	14.3607	0.717138	0.358569	+	0.933503i	$0.383265\pi$
$402$	14.1115i	0.703815i
$403$	6.47214i	0.322400i
$404$	−8.18034	−0.406987
$405$	0	0
$406$	4.47214	0.221948
$407$	6.94427i	0.344215i
$408$	− 3.05573i	− 0.151281i
$409$	−3.41641	−0.168930	−0.0844652	−	0.996426i	$-0.526918\pi$
$409$	−3.41641	−0.168930	−0.0844652	+	0.996426i	$0.526918\pi$
$410$	0	0
$411$	−24.5836	−1.21262
$412$	− 14.9443i	− 0.736251i
$413$	− 2.76393i	− 0.136004i
$414$	5.88854	0.289406
$415$	0	0
$416$	−3.23607	−0.158661
$417$	− 26.8328i	− 1.31401i
$418$	7.23607i	0.353928i
$419$	−17.2361	−0.842037	−0.421019	−	0.907052i	$-0.638327\pi$
$419$	−17.2361	−0.842037	−0.421019	+	0.907052i	$0.638327\pi$
$420$	0	0
$421$	16.4721	0.802803	0.401401	−	0.915902i	$-0.368523\pi$
$421$	16.4721	0.802803	0.401401	+	0.915902i	$0.368523\pi$
$422$	− 13.5279i	− 0.658526i
$423$	− 2.94427i	− 0.143155i
$424$	−8.47214	−0.411443
$425$	0	0
$426$	8.00000	0.387601
$427$	− 0.763932i	− 0.0369693i
$428$	− 2.47214i	− 0.119495i
$429$	4.00000	0.193122
$430$	0	0
$431$	23.0557	1.11056	0.555278	−	0.831665i	$-0.312612\pi$
$431$	23.0557	1.11056	0.555278	+	0.831665i	$0.312612\pi$
$432$	− 5.52786i	− 0.265959i
$433$	− 28.4721i	− 1.36828i	−0.729349	−	0.684142i	$-0.760177\pi$
$433$	− 28.4721i	− 1.36828i	0.729349	−	0.684142i	$-0.239823\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 28.9443i	− 1.38459i
$438$	− 16.0000i	− 0.764510i
$439$	−8.94427	−0.426887	−0.213443	−	0.976955i	$-0.568468\pi$
$439$	−8.94427	−0.426887	−0.213443	+	0.976955i	$0.568468\pi$
$440$	0	0
$441$	−1.47214	−0.0701017
$442$	− 8.00000i	− 0.380521i
$443$	24.9443i	1.18514i	0.805520	+	0.592569i	$0.201886\pi$
$443$	24.9443i	1.18514i	−0.805520	+	0.592569i	$0.798114\pi$
$444$	−8.58359	−0.407359
$445$	0	0
$446$	−0.472136	−0.0223563
$447$	− 27.6393i	− 1.30729i
$448$	− 1.00000i	− 0.0472456i
$449$	−18.9443	−0.894035	−0.447018	−	0.894525i	$-0.647514\pi$
$449$	−18.9443	−0.894035	−0.447018	+	0.894525i	$0.647514\pi$
$450$	0	0
$451$	−2.47214	−0.116408
$452$	− 0.472136i	− 0.0222074i
$453$	− 14.8328i	− 0.696906i
$454$	19.2361	0.902793
$455$	0	0
$456$	−8.94427	−0.418854
$457$	26.9443i	1.26040i	0.776433	+	0.630200i	$0.217027\pi$
$457$	26.9443i	1.26040i	−0.776433	+	0.630200i	$0.782973\pi$
$458$	− 17.2361i	− 0.805389i
$459$	13.6656	0.637857
$460$	0	0
$461$	24.7639	1.15337	0.576686	−	0.816966i	$-0.304346\pi$
$461$	24.7639	1.15337	0.576686	+	0.816966i	$0.304346\pi$
$462$	1.23607i	0.0575071i
$463$	30.4721i	1.41616i	0.706132	+	0.708080i	$0.250438\pi$
$463$	30.4721i	1.41616i	−0.706132	+	0.708080i	$0.749562\pi$
$464$	−4.47214	−0.207614
$465$	0	0
$466$	−14.9443	−0.692280
$467$	− 27.1246i	− 1.25518i	−0.778545	−	0.627589i	$-0.784042\pi$
$467$	− 27.1246i	− 1.25518i	0.778545	−	0.627589i	$-0.215958\pi$
$468$	− 4.76393i	− 0.220213i
$469$	−11.4164	−0.527161
$470$	0	0
$471$	−15.6393	−0.720622
$472$	2.76393i	0.127220i
$473$	10.4721i	0.481509i
$474$	0	0
$475$	0	0
$476$	2.47214	0.113310
$477$	− 12.4721i	− 0.571060i
$478$	− 20.0000i	− 0.914779i
$479$	−12.3607	−0.564774	−0.282387	−	0.959301i	$-0.591126\pi$
$479$	−12.3607	−0.564774	−0.282387	+	0.959301i	$0.591126\pi$
$480$	0	0
$481$	−22.4721	−1.02464
$482$	15.4164i	0.702198i
$483$	− 4.94427i	− 0.224972i
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	13.5967	0.616761
$487$	16.9443i	0.767818i	0.923371	+	0.383909i	$0.125422\pi$
$487$	16.9443i	0.767818i	−0.923371	+	0.383909i	$0.874578\pi$
$488$	0.763932i	0.0345816i
$489$	24.0000	1.08532
$490$	0	0
$491$	−16.9443	−0.764684	−0.382342	−	0.924021i	$-0.624882\pi$
$491$	−16.9443	−0.764684	−0.382342	+	0.924021i	$0.624882\pi$
$492$	− 3.05573i	− 0.137763i
$493$	− 11.0557i	− 0.497925i
$494$	−23.4164	−1.05355
$495$	0	0
$496$	2.00000	0.0898027
$497$	6.47214i	0.290315i
$498$	15.0557i	0.674663i
$499$	32.3607	1.44866	0.724331	−	0.689452i	$-0.242149\pi$
$499$	32.3607	1.44866	0.724331	+	0.689452i	$0.242149\pi$
$500$	0	0
$501$	14.1115	0.630453
$502$	29.2361i	1.30487i
$503$	− 4.00000i	− 0.178351i	−0.996016	−	0.0891756i	$-0.971577\pi$
$503$	− 4.00000i	− 0.178351i	0.996016	−	0.0891756i	$-0.0284232\pi$
$504$	1.47214	0.0655741
$505$	0	0
$506$	4.00000	0.177822
$507$	− 3.12461i	− 0.138769i
$508$	12.0000i	0.532414i
$509$	−24.0689	−1.06683	−0.533417	−	0.845852i	$-0.679092\pi$
$509$	−24.0689	−1.06683	−0.533417	+	0.845852i	$0.679092\pi$
$510$	0	0
$511$	12.9443	0.572621
$512$	1.00000i	0.0441942i
$513$	− 40.0000i	− 1.76604i
$514$	−6.94427	−0.306299
$515$	0	0
$516$	−12.9443	−0.569840
$517$	− 2.00000i	− 0.0879599i
$518$	− 6.94427i	− 0.305114i
$519$	4.00000	0.175581
$520$	0	0
$521$	−10.3607	−0.453910	−0.226955	−	0.973905i	$-0.572877\pi$
$521$	−10.3607	−0.453910	−0.226955	+	0.973905i	$0.572877\pi$
$522$	− 6.58359i	− 0.288156i
$523$	14.2918i	0.624937i	0.949928	+	0.312468i	$0.101156\pi$
$523$	14.2918i	0.624937i	−0.949928	+	0.312468i	$0.898844\pi$
$524$	−4.76393	−0.208113
$525$	0	0
$526$	−4.94427	−0.215580
$527$	4.94427i	0.215376i
$528$	− 1.23607i	− 0.0537930i
$529$	7.00000	0.304348
$530$	0	0
$531$	−4.06888	−0.176575
$532$	− 7.23607i	− 0.313723i
$533$	− 8.00000i	− 0.346518i
$534$	−12.3607	−0.534899
$535$	0	0
$536$	11.4164	0.493114
$537$	− 11.0557i	− 0.477090i
$538$	22.7639i	0.981423i
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−26.9443	−1.15842	−0.579212	−	0.815177i	$-0.696640\pi$
$541$	−26.9443	−1.15842	−0.579212	+	0.815177i	$0.696640\pi$
$542$	0.944272i	0.0405600i
$543$	− 11.4164i	− 0.489925i
$544$	−2.47214	−0.105992
$545$	0	0
$546$	−4.00000	−0.171184
$547$	− 0.944272i	− 0.0403742i	−0.999796	−	0.0201871i	$-0.993574\pi$
$547$	− 0.944272i	− 0.0403742i	0.999796	−	0.0201871i	$-0.00642618\pi$
$548$	19.8885i	0.849596i
$549$	−1.12461	−0.0479973
$550$	0	0
$551$	−32.3607	−1.37861
$552$	4.94427i	0.210442i
$553$	0	0
$554$	−3.52786	−0.149885
$555$	0	0
$556$	−21.7082	−0.920633
$557$	24.8328i	1.05220i	0.850423	+	0.526100i	$0.176346\pi$
$557$	24.8328i	1.05220i	−0.850423	+	0.526100i	$0.823654\pi$
$558$	2.94427i	0.124641i
$559$	−33.8885	−1.43333
$560$	0	0
$561$	3.05573	0.129013
$562$	28.8328i	1.21624i
$563$	− 31.2361i	− 1.31644i	−0.752824	−	0.658222i	$-0.771309\pi$
$563$	− 31.2361i	− 1.31644i	0.752824	−	0.658222i	$-0.228691\pi$
$564$	2.47214	0.104096
$565$	0	0
$566$	14.6525	0.615889
$567$	− 2.41641i	− 0.101480i
$568$	− 6.47214i	− 0.271565i
$569$	36.8328	1.54411	0.772056	−	0.635555i	$-0.219228\pi$
$569$	36.8328	1.54411	0.772056	+	0.635555i	$0.219228\pi$
$570$	0	0
$571$	−10.1115	−0.423151	−0.211576	−	0.977362i	$-0.567859\pi$
$571$	−10.1115	−0.423151	−0.211576	+	0.977362i	$0.567859\pi$
$572$	− 3.23607i	− 0.135307i
$573$	3.05573i	0.127655i
$574$	2.47214	0.103185
$575$	0	0
$576$	−1.47214	−0.0613390
$577$	26.9443i	1.12170i	0.827916	+	0.560852i	$0.189526\pi$
$577$	26.9443i	1.12170i	−0.827916	+	0.560852i	$0.810474\pi$
$578$	10.8885i	0.452904i
$579$	18.4721	0.767676
$580$	0	0
$581$	−12.1803	−0.505326
$582$	15.4164i	0.639031i
$583$	− 8.47214i	− 0.350880i
$584$	−12.9443	−0.535638
$585$	0	0
$586$	−26.6525	−1.10100
$587$	− 5.81966i	− 0.240203i	−0.992762	−	0.120102i	$-0.961678\pi$
$587$	− 5.81966i	− 0.240203i	0.992762	−	0.120102i	$-0.0383220\pi$
$588$	− 1.23607i	− 0.0509746i
$589$	14.4721	0.596314
$590$	0	0
$591$	22.2492	0.915211
$592$	6.94427i	0.285408i
$593$	− 24.0000i	− 0.985562i	−0.870153	−	0.492781i	$-0.835980\pi$
$593$	− 24.0000i	− 0.985562i	0.870153	−	0.492781i	$-0.164020\pi$
$594$	5.52786	0.226811
$595$	0	0
$596$	−22.3607	−0.915929
$597$	23.4164i	0.958370i
$598$	12.9443i	0.529331i
$599$	32.3607	1.32222	0.661111	−	0.750288i	$-0.270085\pi$
$599$	32.3607	1.32222	0.661111	+	0.750288i	$0.270085\pi$
$600$	0	0
$601$	−34.8328	−1.42086	−0.710430	−	0.703768i	$-0.751500\pi$
$601$	−34.8328	−1.42086	−0.710430	+	0.703768i	$0.751500\pi$
$602$	− 10.4721i	− 0.426812i
$603$	16.8065i	0.684414i
$604$	−12.0000	−0.488273
$605$	0	0
$606$	10.1115	0.410750
$607$	− 32.0000i	− 1.29884i	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	− 32.0000i	− 1.29884i	0.760430	−	0.649420i	$-0.224988\pi$
$608$	7.23607i	0.293461i
$609$	−5.52786	−0.224000
$610$	0	0
$611$	6.47214	0.261835
$612$	− 3.63932i	− 0.147111i
$613$	− 28.4721i	− 1.14998i	−0.818161	−	0.574989i	$-0.805006\pi$
$613$	− 28.4721i	− 1.14998i	0.818161	−	0.574989i	$-0.194994\pi$
$614$	26.0689	1.05205
$615$	0	0
$616$	1.00000	0.0402911
$617$	21.4164i	0.862192i	0.902306	+	0.431096i	$0.141873\pi$
$617$	21.4164i	0.862192i	−0.902306	+	0.431096i	$0.858127\pi$
$618$	18.4721i	0.743058i
$619$	−18.5410	−0.745227	−0.372613	−	0.927987i	$-0.621538\pi$
$619$	−18.5410	−0.745227	−0.372613	+	0.927987i	$0.621538\pi$
$620$	0	0
$621$	−22.1115	−0.887302
$622$	− 21.4164i	− 0.858720i
$623$	− 10.0000i	− 0.400642i
$624$	4.00000	0.160128
$625$	0	0
$626$	19.5279	0.780490
$627$	− 8.94427i	− 0.357200i
$628$	12.6525i	0.504889i
$629$	−17.1672	−0.684500
$630$	0	0
$631$	−31.4164	−1.25067	−0.625334	−	0.780357i	$-0.715037\pi$
$631$	−31.4164	−1.25067	−0.625334	+	0.780357i	$0.715037\pi$
$632$	0	0
$633$	16.7214i	0.664614i
$634$	30.9443	1.22895
$635$	0	0
$636$	10.4721	0.415247
$637$	− 3.23607i	− 0.128218i
$638$	− 4.47214i	− 0.177054i
$639$	9.52786	0.376916
$640$	0	0
$641$	27.5279	1.08729	0.543643	−	0.839317i	$-0.317045\pi$
$641$	27.5279	1.08729	0.543643	+	0.839317i	$0.317045\pi$
$642$	3.05573i	0.120600i
$643$	18.7639i	0.739977i	0.929036	+	0.369989i	$0.120638\pi$
$643$	18.7639i	0.739977i	−0.929036	+	0.369989i	$0.879362\pi$
$644$	−4.00000	−0.157622
$645$	0	0
$646$	−17.8885	−0.703815
$647$	− 28.8328i	− 1.13353i	−0.823878	−	0.566767i	$-0.808194\pi$
$647$	− 28.8328i	− 1.13353i	0.823878	−	0.566767i	$-0.191806\pi$
$648$	2.41641i	0.0949255i
$649$	−2.76393	−0.108494
$650$	0	0
$651$	2.47214	0.0968906
$652$	− 19.4164i	− 0.760405i
$653$	− 46.3607i	− 1.81423i	−0.420879	−	0.907117i	$-0.638278\pi$
$653$	− 46.3607i	− 1.81423i	0.420879	−	0.907117i	$-0.361722\pi$
$654$	12.3607	0.483341
$655$	0	0
$656$	−2.47214	−0.0965207
$657$	− 19.0557i	− 0.743435i
$658$	2.00000i	0.0779681i
$659$	−16.5836	−0.646005	−0.323003	−	0.946398i	$-0.604692\pi$
$659$	−16.5836	−0.646005	−0.323003	+	0.946398i	$0.604692\pi$
$660$	0	0
$661$	−3.12461	−0.121533	−0.0607667	−	0.998152i	$-0.519355\pi$
$661$	−3.12461	−0.121533	−0.0607667	+	0.998152i	$0.519355\pi$
$662$	− 16.9443i	− 0.658558i
$663$	9.88854i	0.384039i
$664$	12.1803	0.472689
$665$	0	0
$666$	−10.2229	−0.396130
$667$	17.8885i	0.692647i
$668$	− 11.4164i	− 0.441714i
$669$	0.583592	0.0225630
$670$	0	0
$671$	−0.763932	−0.0294913
$672$	1.23607i	0.0476824i
$673$	3.88854i	0.149892i	0.997188	+	0.0749462i	$0.0238785\pi$
$673$	3.88854i	0.149892i	−0.997188	+	0.0749462i	$0.976122\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	−2.52786	−0.0972255
$677$	− 26.0689i	− 1.00191i	−0.865474	−	0.500954i	$-0.832982\pi$
$677$	− 26.0689i	− 1.00191i	0.865474	−	0.500954i	$-0.167018\pi$
$678$	0.583592i	0.0224127i
$679$	−12.4721	−0.478637
$680$	0	0
$681$	−23.7771	−0.911140
$682$	2.00000i	0.0765840i
$683$	− 32.9443i	− 1.26058i	−0.776361	−	0.630289i	$-0.782936\pi$
$683$	− 32.9443i	− 1.26058i	0.776361	−	0.630289i	$-0.217064\pi$
$684$	−10.6525	−0.407308
$685$	0	0
$686$	1.00000	0.0381802
$687$	21.3050i	0.812835i
$688$	10.4721i	0.399246i
$689$	27.4164	1.04448
$690$	0	0
$691$	12.6525	0.481323	0.240661	−	0.970609i	$-0.422636\pi$
$691$	12.6525	0.481323	0.240661	+	0.970609i	$0.422636\pi$
$692$	− 3.23607i	− 0.123017i
$693$	1.47214i	0.0559218i
$694$	−2.47214	−0.0938410
$695$	0	0
$696$	5.52786	0.209533
$697$	− 6.11146i	− 0.231488i
$698$	− 21.7082i	− 0.821668i
$699$	18.4721	0.698680
$700$	0	0
$701$	−42.7214	−1.61356	−0.806782	−	0.590850i	$-0.798793\pi$
$701$	−42.7214	−1.61356	−0.806782	+	0.590850i	$0.798793\pi$
$702$	17.8885i	0.675160i
$703$	50.2492i	1.89519i
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−17.0557	−0.641901
$707$	8.18034i	0.307653i
$708$	− 3.41641i	− 0.128396i
$709$	−4.47214	−0.167955	−0.0839773	−	0.996468i	$-0.526762\pi$
$709$	−4.47214	−0.167955	−0.0839773	+	0.996468i	$0.526762\pi$
$710$	0	0
$711$	0	0
$712$	10.0000i	0.374766i
$713$	− 8.00000i	− 0.299602i
$714$	−3.05573	−0.114358
$715$	0	0
$716$	−8.94427	−0.334263
$717$	24.7214i	0.923236i
$718$	− 26.8328i	− 1.00139i
$719$	−16.8328	−0.627758	−0.313879	−	0.949463i	$-0.601629\pi$
$719$	−16.8328	−0.627758	−0.313879	+	0.949463i	$0.601629\pi$
$720$	0	0
$721$	−14.9443	−0.556554
$722$	33.3607i	1.24156i
$723$	− 19.0557i	− 0.708690i
$724$	−9.23607	−0.343256
$725$	0	0
$726$	1.23607	0.0458748
$727$	18.0000i	0.667583i	0.942647	+	0.333792i	$0.108328\pi$
$727$	18.0000i	0.667583i	−0.942647	+	0.333792i	$0.891672\pi$
$728$	3.23607i	0.119937i
$729$	−24.0557	−0.890953
$730$	0	0
$731$	−25.8885	−0.957522
$732$	− 0.944272i	− 0.0349013i
$733$	− 49.1246i	− 1.81446i	−0.420636	−	0.907229i	$-0.638193\pi$
$733$	− 49.1246i	− 1.81446i	0.420636	−	0.907229i	$-0.361807\pi$
$734$	5.41641	0.199923
$735$	0	0
$736$	4.00000	0.147442
$737$	11.4164i	0.420529i
$738$	− 3.63932i	− 0.133965i
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	28.9443	1.06329
$742$	8.47214i	0.311022i
$743$	− 21.8885i	− 0.803013i	−0.915856	−	0.401506i	$-0.868487\pi$
$743$	− 21.8885i	− 0.803013i	0.915856	−	0.401506i	$-0.131513\pi$
$744$	−2.47214	−0.0906329
$745$	0	0
$746$	−6.00000	−0.219676
$747$	17.9311i	0.656065i
$748$	− 2.47214i	− 0.0903902i
$749$	−2.47214	−0.0903299
$750$	0	0
$751$	−16.9443	−0.618305	−0.309153	−	0.951012i	$-0.600045\pi$
$751$	−16.9443	−0.618305	−0.309153	+	0.951012i	$0.600045\pi$
$752$	− 2.00000i	− 0.0729325i
$753$	− 36.1378i	− 1.31693i
$754$	14.4721	0.527044
$755$	0	0
$756$	−5.52786	−0.201046
$757$	− 23.3050i	− 0.847033i	−0.905888	−	0.423516i	$-0.860796\pi$
$757$	− 23.3050i	− 0.847033i	0.905888	−	0.423516i	$-0.139204\pi$
$758$	− 14.4721i	− 0.525652i
$759$	−4.94427	−0.179466
$760$	0	0
$761$	−11.4164	−0.413844	−0.206922	−	0.978357i	$-0.566345\pi$
$761$	−11.4164	−0.413844	−0.206922	+	0.978357i	$0.566345\pi$
$762$	− 14.8328i	− 0.537336i
$763$	10.0000i	0.362024i
$764$	2.47214	0.0894387
$765$	0	0
$766$	−23.8885	−0.863128
$767$	− 8.94427i	− 0.322959i
$768$	− 1.23607i	− 0.0446028i
$769$	43.4164	1.56564	0.782818	−	0.622251i	$-0.213782\pi$
$769$	43.4164	1.56564	0.782818	+	0.622251i	$0.213782\pi$
$770$	0	0
$771$	8.58359	0.309131
$772$	− 14.9443i	− 0.537856i
$773$	− 15.7082i	− 0.564985i	−0.959270	−	0.282492i	$-0.908839\pi$
$773$	− 15.7082i	− 0.564985i	0.959270	−	0.282492i	$-0.0911612\pi$
$774$	−15.4164	−0.554131
$775$	0	0
$776$	12.4721	0.447724
$777$	8.58359i	0.307935i
$778$	− 33.4164i	− 1.19804i
$779$	−17.8885	−0.640924
$780$	0	0
$781$	6.47214	0.231591
$782$	9.88854i	0.353614i
$783$	24.7214i	0.883469i
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	5.88854	0.210037
$787$	− 28.1803i	− 1.00452i	−0.864716	−	0.502260i	$-0.832502\pi$
$787$	− 28.1803i	− 1.00452i	0.864716	−	0.502260i	$-0.167498\pi$
$788$	− 18.0000i	− 0.641223i
$789$	6.11146	0.217574
$790$	0	0
$791$	−0.472136	−0.0167872
$792$	− 1.47214i	− 0.0523101i
$793$	− 2.47214i	− 0.0877881i
$794$	23.7082	0.841373
$795$	0	0
$796$	18.9443	0.671462
$797$	− 41.5967i	− 1.47343i	−0.676202	−	0.736716i	$-0.736375\pi$
$797$	− 41.5967i	− 1.47343i	0.676202	−	0.736716i	$-0.263625\pi$
$798$	8.94427i	0.316624i
$799$	4.94427	0.174916
$800$	0	0
$801$	−14.7214	−0.520154
$802$	14.3607i	0.507093i
$803$	− 12.9443i	− 0.456793i
$804$	−14.1115	−0.497673
$805$	0	0
$806$	−6.47214	−0.227971
$807$	− 28.1378i	− 0.990496i
$808$	− 8.18034i	− 0.287783i
$809$	21.0557	0.740280	0.370140	−	0.928976i	$-0.379310\pi$
$809$	21.0557	0.740280	0.370140	+	0.928976i	$0.379310\pi$
$810$	0	0
$811$	4.76393	0.167284	0.0836421	−	0.996496i	$-0.473345\pi$
$811$	4.76393	0.167284	0.0836421	+	0.996496i	$0.473345\pi$
$812$	4.47214i	0.156941i
$813$	− 1.16718i	− 0.0409349i
$814$	−6.94427	−0.243397
$815$	0	0
$816$	3.05573	0.106972
$817$	75.7771i	2.65110i
$818$	− 3.41641i	− 0.119452i
$819$	−4.76393	−0.166465
$820$	0	0
$821$	−1.41641	−0.0494330	−0.0247165	−	0.999695i	$-0.507868\pi$
$821$	−1.41641	−0.0494330	−0.0247165	+	0.999695i	$0.507868\pi$
$822$	− 24.5836i	− 0.857451i
$823$	46.2492i	1.61215i	0.591816	+	0.806073i	$0.298411\pi$
$823$	46.2492i	1.61215i	−0.591816	+	0.806073i	$0.701589\pi$
$824$	14.9443	0.520608
$825$	0	0
$826$	2.76393	0.0961695
$827$	16.9443i	0.589210i	0.955619	+	0.294605i	$0.0951881\pi$
$827$	16.9443i	0.589210i	−0.955619	+	0.294605i	$0.904812\pi$
$828$	5.88854i	0.204641i
$829$	−11.7082	−0.406643	−0.203321	−	0.979112i	$-0.565174\pi$
$829$	−11.7082	−0.406643	−0.203321	+	0.979112i	$0.565174\pi$
$830$	0	0
$831$	4.36068	0.151270
$832$	− 3.23607i	− 0.112190i
$833$	− 2.47214i	− 0.0856544i
$834$	26.8328	0.929144
$835$	0	0
$836$	−7.23607	−0.250265
$837$	− 11.0557i	− 0.382142i
$838$	− 17.2361i	− 0.595410i
$839$	−16.8328	−0.581133	−0.290567	−	0.956855i	$-0.593844\pi$
$839$	−16.8328	−0.581133	−0.290567	+	0.956855i	$0.593844\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	16.4721i	0.567667i
$843$	− 35.6393i	− 1.22748i
$844$	13.5279	0.465648
$845$	0	0
$846$	2.94427	0.101226
$847$	1.00000i	0.0343604i
$848$	− 8.47214i	− 0.290934i
$849$	−18.1115	−0.621584
$850$	0	0
$851$	27.7771	0.952186
$852$	8.00000i	0.274075i
$853$	− 32.5410i	− 1.11418i	−0.830451	−	0.557092i	$-0.811917\pi$
$853$	− 32.5410i	− 1.11418i	0.830451	−	0.557092i	$-0.188083\pi$
$854$	0.763932	0.0261412
$855$	0	0
$856$	2.47214	0.0844959
$857$	− 46.4721i	− 1.58746i	−0.608272	−	0.793729i	$-0.708137\pi$
$857$	− 46.4721i	− 1.58746i	0.608272	−	0.793729i	$-0.291863\pi$
$858$	4.00000i	0.136558i
$859$	−15.1246	−0.516045	−0.258023	−	0.966139i	$-0.583071\pi$
$859$	−15.1246	−0.516045	−0.258023	+	0.966139i	$0.583071\pi$
$860$	0	0
$861$	−3.05573	−0.104139
$862$	23.0557i	0.785281i
$863$	− 0.583592i	− 0.0198657i	−0.999951	−	0.00993285i	$-0.996838\pi$
$863$	− 0.583592i	− 0.0198657i	0.999951	−	0.00993285i	$-0.00316178\pi$
$864$	5.52786	0.188062
$865$	0	0
$866$	28.4721	0.967523
$867$	− 13.4590i	− 0.457091i
$868$	− 2.00000i	− 0.0678844i
$869$	0	0
$870$	0	0
$871$	−36.9443	−1.25181
$872$	− 10.0000i	− 0.338643i
$873$	18.3607i	0.621415i
$874$	28.9443	0.979055
$875$	0	0
$876$	16.0000	0.540590
$877$	9.05573i	0.305790i	0.988242	+	0.152895i	$0.0488597\pi$
$877$	9.05573i	0.305790i	−0.988242	+	0.152895i	$0.951140\pi$
$878$	− 8.94427i	− 0.301855i
$879$	32.9443	1.11118
$880$	0	0
$881$	28.8328	0.971402	0.485701	−	0.874125i	$-0.338564\pi$
$881$	28.8328	0.971402	0.485701	+	0.874125i	$0.338564\pi$
$882$	− 1.47214i	− 0.0495694i
$883$	2.83282i	0.0953318i	0.998863	+	0.0476659i	$0.0151783\pi$
$883$	2.83282i	0.0953318i	−0.998863	+	0.0476659i	$0.984822\pi$
$884$	8.00000	0.269069
$885$	0	0
$886$	−24.9443	−0.838019
$887$	− 44.3607i	− 1.48949i	−0.667351	−	0.744743i	$-0.732572\pi$
$887$	− 44.3607i	− 1.48949i	0.667351	−	0.744743i	$-0.267428\pi$
$888$	− 8.58359i	− 0.288046i
$889$	12.0000	0.402467
$890$	0	0
$891$	−2.41641	−0.0809527
$892$	− 0.472136i	− 0.0158083i
$893$	− 14.4721i	− 0.484292i
$894$	27.6393	0.924397
$895$	0	0
$896$	1.00000	0.0334077
$897$	− 16.0000i	− 0.534224i
$898$	− 18.9443i	− 0.632179i
$899$	−8.94427	−0.298308
$900$	0	0
$901$	20.9443	0.697755
$902$	− 2.47214i	− 0.0823131i
$903$	12.9443i	0.430758i
$904$	0.472136	0.0157030
$905$	0	0
$906$	14.8328	0.492787
$907$	− 24.3607i	− 0.808883i	−0.914564	−	0.404442i	$-0.867466\pi$
$907$	− 24.3607i	− 0.808883i	0.914564	−	0.404442i	$-0.132534\pi$
$908$	19.2361i	0.638371i
$909$	12.0426	0.399427
$910$	0	0
$911$	−28.0000	−0.927681	−0.463841	−	0.885919i	$-0.653529\pi$
$911$	−28.0000	−0.927681	−0.463841	+	0.885919i	$0.653529\pi$
$912$	− 8.94427i	− 0.296174i
$913$	12.1803i	0.403110i
$914$	−26.9443	−0.891237
$915$	0	0
$916$	17.2361	0.569496
$917$	4.76393i	0.157319i
$918$	13.6656i	0.451033i
$919$	22.1115	0.729390	0.364695	−	0.931127i	$-0.381173\pi$
$919$	22.1115	0.729390	0.364695	+	0.931127i	$0.381173\pi$
$920$	0	0
$921$	−32.2229	−1.06178
$922$	24.7639i	0.815557i
$923$	20.9443i	0.689389i
$924$	−1.23607	−0.0406637
$925$	0	0
$926$	−30.4721	−1.00138
$927$	22.0000i	0.722575i
$928$	− 4.47214i	− 0.146805i
$929$	−40.2492	−1.32053	−0.660267	−	0.751031i	$-0.729557\pi$
$929$	−40.2492	−1.32053	−0.660267	+	0.751031i	$0.729557\pi$
$930$	0	0
$931$	−7.23607	−0.237153
$932$	− 14.9443i	− 0.489516i
$933$	26.4721i	0.866659i
$934$	27.1246	0.887544
$935$	0	0
$936$	4.76393	0.155714
$937$	− 3.05573i	− 0.0998263i	−0.998754	−	0.0499131i	$-0.984106\pi$
$937$	− 3.05573i	− 0.0998263i	0.998754	−	0.0499131i	$-0.0158945\pi$
$938$	− 11.4164i	− 0.372759i
$939$	−24.1378	−0.787706
$940$	0	0
$941$	−11.8197	−0.385310	−0.192655	−	0.981267i	$-0.561710\pi$
$941$	−11.8197	−0.385310	−0.192655	+	0.981267i	$0.561710\pi$
$942$	− 15.6393i	− 0.509557i
$943$	9.88854i	0.322015i
$944$	−2.76393	−0.0899583
$945$	0	0
$946$	−10.4721	−0.340479
$947$	16.9443i	0.550615i	0.961356	+	0.275307i	$0.0887796\pi$
$947$	16.9443i	0.550615i	−0.961356	+	0.275307i	$0.911220\pi$
$948$	0	0
$949$	41.8885	1.35976
$950$	0	0
$951$	−38.2492	−1.24032
$952$	2.47214i	0.0801224i
$953$	− 22.9443i	− 0.743238i	−0.928385	−	0.371619i	$-0.878803\pi$
$953$	− 22.9443i	− 0.743238i	0.928385	−	0.371619i	$-0.121197\pi$
$954$	12.4721	0.403800
$955$	0	0
$956$	20.0000	0.646846
$957$	5.52786i	0.178690i
$958$	− 12.3607i	− 0.399355i
$959$	19.8885	0.642235
$960$	0	0
$961$	−27.0000	−0.870968
$962$	− 22.4721i	− 0.724531i
$963$	3.63932i	0.117275i
$964$	−15.4164	−0.496529
$965$	0	0
$966$	4.94427	0.159079
$967$	45.8885i	1.47568i	0.674978	+	0.737838i	$0.264153\pi$
$967$	45.8885i	1.47568i	−0.674978	+	0.737838i	$0.735847\pi$
$968$	− 1.00000i	− 0.0321412i
$969$	22.1115	0.710322
$970$	0	0
$971$	50.5410	1.62194	0.810969	−	0.585089i	$-0.198940\pi$
$971$	50.5410	1.62194	0.810969	+	0.585089i	$0.198940\pi$
$972$	13.5967i	0.436116i
$973$	21.7082i	0.695933i
$974$	−16.9443	−0.542929
$975$	0	0
$976$	−0.763932	−0.0244529
$977$	− 28.8328i	− 0.922443i	−0.887285	−	0.461222i	$-0.847411\pi$
$977$	− 28.8328i	− 0.922443i	0.887285	−	0.461222i	$-0.152589\pi$
$978$	24.0000i	0.767435i
$979$	−10.0000	−0.319601
$980$	0	0
$981$	14.7214	0.470017
$982$	− 16.9443i	− 0.540713i
$983$	− 14.0000i	− 0.446531i	−0.974758	−	0.223265i	$-0.928328\pi$
$983$	− 14.0000i	− 0.446531i	0.974758	−	0.223265i	$-0.0716716\pi$
$984$	3.05573	0.0974131
$985$	0	0
$986$	11.0557	0.352086
$987$	− 2.47214i	− 0.0786890i
$988$	− 23.4164i	− 0.744975i
$989$	41.8885	1.33198
$990$	0	0
$991$	−0.360680	−0.0114574	−0.00572869	−	0.999984i	$-0.501824\pi$
$991$	−0.360680	−0.0114574	−0.00572869	+	0.999984i	$0.501824\pi$
$992$	2.00000i	0.0635001i
$993$	20.9443i	0.664646i
$994$	−6.47214	−0.205284
$995$	0	0
$996$	−15.0557	−0.477059
$997$	24.1803i	0.765799i	0.923790	+	0.382900i	$0.125074\pi$
$997$	24.1803i	0.765799i	−0.923790	+	0.382900i	$0.874926\pi$
$998$	32.3607i	1.02436i
$999$	38.3870	1.21451

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3850.2.c.q.1849.3		4
5.2	odd	4		3850.2.a.bj.1.1		2
5.3	odd	4		154.2.a.d.1.2	✓	2
5.4	even	2	inner	3850.2.c.q.1849.2		4
15.8	even	4		1386.2.a.m.1.2		2
20.3	even	4		1232.2.a.p.1.1		2
35.3	even	12		1078.2.e.n.177.2		4
35.13	even	4		1078.2.a.w.1.1		2
35.18	odd	12		1078.2.e.q.177.1		4
35.23	odd	12		1078.2.e.q.67.1		4
35.33	even	12		1078.2.e.n.67.2		4
40.3	even	4		4928.2.a.bk.1.2		2
40.13	odd	4		4928.2.a.bt.1.1		2
55.43	even	4		1694.2.a.l.1.2		2
105.83	odd	4		9702.2.a.cu.1.1		2
140.83	odd	4		8624.2.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.a.d.1.2	✓	2	5.3	odd	4
1078.2.a.w.1.1		2	35.13	even	4
1078.2.e.n.67.2		4	35.33	even	12
1078.2.e.n.177.2		4	35.3	even	12
1078.2.e.q.67.1		4	35.23	odd	12
1078.2.e.q.177.1		4	35.18	odd	12
1232.2.a.p.1.1		2	20.3	even	4
1386.2.a.m.1.2		2	15.8	even	4
1694.2.a.l.1.2		2	55.43	even	4
3850.2.a.bj.1.1		2	5.2	odd	4
3850.2.c.q.1849.2		4	5.4	even	2	inner
3850.2.c.q.1849.3		4	1.1	even	1	trivial
4928.2.a.bk.1.2		2	40.3	even	4
4928.2.a.bt.1.1		2	40.13	odd	4
8624.2.a.bf.1.2		2	140.83	odd	4
9702.2.a.cu.1.1		2	105.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3850.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.23607i q^{3} -1.00000 q^{4} +1.23607 q^{6} +1.00000i q^{7} -1.00000i q^{8} +1.47214 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.23607i q^{3} -1.00000 q^{4} +1.23607 q^{6} +1.00000i q^{7} -1.00000i q^{8} +1.47214 q^{9} +1.00000 q^{11} +1.23607i q^{12} +3.23607i q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.47214i q^{17} +1.47214i q^{18} +7.23607 q^{19} +1.23607 q^{21} +1.00000i q^{22} -4.00000i q^{23} -1.23607 q^{24} -3.23607 q^{26} -5.52786i q^{27} -1.00000i q^{28} -4.47214 q^{29} +2.00000 q^{31} +1.00000i q^{32} -1.23607i q^{33} -2.47214 q^{34} -1.47214 q^{36} +6.94427i q^{37} +7.23607i q^{38} +4.00000 q^{39} -2.47214 q^{41} +1.23607i q^{42} +10.4721i q^{43} -1.00000 q^{44} +4.00000 q^{46} -2.00000i q^{47} -1.23607i q^{48} -1.00000 q^{49} +3.05573 q^{51} -3.23607i q^{52} -8.47214i q^{53} +5.52786 q^{54} +1.00000 q^{56} -8.94427i q^{57} -4.47214i q^{58} -2.76393 q^{59} -0.763932 q^{61} +2.00000i q^{62} +1.47214i q^{63} -1.00000 q^{64} +1.23607 q^{66} +11.4164i q^{67} -2.47214i q^{68} -4.94427 q^{69} +6.47214 q^{71} -1.47214i q^{72} -12.9443i q^{73} -6.94427 q^{74} -7.23607 q^{76} +1.00000i q^{77} +4.00000i q^{78} -2.41641 q^{81} -2.47214i q^{82} +12.1803i q^{83} -1.23607 q^{84} -10.4721 q^{86} +5.52786i q^{87} -1.00000i q^{88} -10.0000 q^{89} -3.23607 q^{91} +4.00000i q^{92} -2.47214i q^{93} +2.00000 q^{94} +1.23607 q^{96} +12.4721i q^{97} -1.00000i q^{98} +1.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{6} - 12 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 - 4 * q^6 - 12 * q^9	\( 4 q - 4 q^{4} - 4 q^{6} - 12 q^{9} + 4 q^{11} - 4 q^{14} + 4 q^{16} + 20 q^{19} - 4 q^{21} + 4 q^{24} - 4 q^{26} + 8 q^{31} + 8 q^{34} + 12 q^{36} + 16 q^{39} + 8 q^{41} - 4 q^{44} + 16 q^{46} - 4 q^{49} + 48 q^{51} + 40 q^{54} + 4 q^{56} - 20 q^{59} - 12 q^{61} - 4 q^{64} - 4 q^{66} + 16 q^{69} + 8 q^{71} + 8 q^{74} - 20 q^{76} + 44 q^{81} + 4 q^{84} - 24 q^{86} - 40 q^{89} - 4 q^{91} + 8 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^6 - 12 * q^9 + 4 * q^11 - 4 * q^14 + 4 * q^16 + 20 * q^19 - 4 * q^21 + 4 * q^24 - 4 * q^26 + 8 * q^31 + 8 * q^34 + 12 * q^36 + 16 * q^39 + 8 * q^41 - 4 * q^44 + 16 * q^46 - 4 * q^49 + 48 * q^51 + 40 * q^54 + 4 * q^56 - 20 * q^59 - 12 * q^61 - 4 * q^64 - 4 * q^66 + 16 * q^69 + 8 * q^71 + 8 * q^74 - 20 * q^76 + 44 * q^81 + 4 * q^84 - 24 * q^86 - 40 * q^89 - 4 * q^91 + 8 * q^94 - 4 * q^96 - 12 * q^99

Introduction

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