LMFDB - Embedded newform 3724.2.g.f.1861.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3724,2,Mod(1861,3724)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3724, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3724.1861");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3724 = 2^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3724.g (of order $2$, degree $1$, 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:	$29.7362897127$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 25x^{14} + 413x^{12} + 3916x^{10} + 26956x^{8} + 112304x^{6} + 333008x^{4} + 476096x^{2} + 473344 $ x^16 + 25x^14 + 413x^12 + 3916x^10 + 26956x^8 + 112304x^6 + 333008x^4 + 476096*x^2 + 473344
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 532)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1861.1
Root			$-1.61383 + 2.79524i$ of defining polynomial
Character	$\chi$	$=$	3724.1861
Dual form			3724.2.g.f.1861.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.22766 q^{3} -4.02258i q^{5} +7.41779 q^{9} +O(q^{10})$	$q-3.22766 q^{3} -4.02258i q^{5} +7.41779 q^{9} +2.81086 q^{11} -5.18661 q^{13} +12.9835i q^{15} -1.59544i q^{17} +(3.90190 - 1.94299i) q^{19} +0.810861 q^{23} -11.1812 q^{25} -14.2591 q^{27} -5.10348i q^{29} -3.22766 q^{31} -9.07250 q^{33} -3.83394i q^{37} +16.7406 q^{39} +6.45532 q^{41} +1.26133 q^{43} -29.8387i q^{45} -1.40445i q^{47} +5.14954i q^{51} -11.2270i q^{53} -11.3069i q^{55} +(-12.5940 + 6.27130i) q^{57} +11.4567 q^{59} -8.68879i q^{61} +20.8636i q^{65} -2.24352i q^{67} -2.61718 q^{69} -7.83399i q^{71} +9.77722i q^{73} +36.0890 q^{75} -9.10353i q^{79} +23.7702 q^{81} +4.14493i q^{83} -6.41779 q^{85} +16.4723i q^{87} +6.27006 q^{89} +10.4178 q^{93} +(-7.81582 - 15.6957i) q^{95} -12.8052 q^{97} +20.8504 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 52 q^{9}+O(q^{10}) $ 16 * q + 52 * q^9	$ 16 q + 52 q^{9} + 8 q^{11} - 24 q^{23} - 20 q^{25} + 24 q^{39} + 16 q^{43} - 2 q^{57} + 48 q^{81} - 36 q^{85} + 100 q^{93} - 62 q^{95} + 156 q^{99}+O(q^{100}) $ 16 * q + 52 * q^9 + 8 * q^11 - 24 * q^23 - 20 * q^25 + 24 * q^39 + 16 * q^43 - 2 * q^57 + 48 * q^81 - 36 * q^85 + 100 * q^93 - 62 * q^95 + 156 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3724\mathbb{Z}\right)^\times$.

$n$	$1863$	$3041$	$3137$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−3.22766			−1.86349			−0.931745	−	0.363113i	$-0.881714\pi$
$3$	−3.22766			−1.86349			−0.931745	+	0.363113i	$0.881714\pi$
$4$		0			0
$5$		−	4.02258i		−	1.79895i	−0.436969	−	0.899477i	$-0.643948\pi$
$5$		−	4.02258i		−	1.79895i	0.436969	−	0.899477i	$-0.356052\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	7.41779			2.47260
$10$		0			0
$11$	2.81086			0.847507			0.423753	−	0.905778i	$-0.360712\pi$
$11$	2.81086			0.847507			0.423753	+	0.905778i	$0.360712\pi$
$12$		0			0
$13$	−5.18661			−1.43851			−0.719254	−	0.694748i	$-0.755516\pi$
$13$	−5.18661			−1.43851			−0.719254	+	0.694748i	$0.755516\pi$
$14$		0			0
$15$			12.9835i			3.35233i
$16$		0			0
$17$		−	1.59544i		−	0.386951i	−0.981105	−	0.193475i	$-0.938024\pi$
$17$		−	1.59544i		−	0.386951i	0.981105	−	0.193475i	$-0.0619760\pi$
$18$		0			0
$19$	3.90190	−	1.94299i	0.895157	−	0.445751i
$19$	3.90190	−	1.94299i	0.895157	−	0.445751i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.810861			0.169076			0.0845381	−	0.996420i	$-0.473059\pi$
$23$	0.810861			0.169076			0.0845381	+	0.996420i	$0.473059\pi$
$24$		0			0
$25$	−11.1812			−2.23623
$26$		0			0
$27$	−14.2591			−2.74417
$28$		0			0
$29$		−	5.10348i		−	0.947693i	−0.880608	−	0.473846i	$-0.842865\pi$
$29$		−	5.10348i		−	0.947693i	0.880608	−	0.473846i	$-0.157135\pi$
$30$		0			0
$31$	−3.22766			−0.579705			−0.289852	−	0.957071i	$-0.593606\pi$
$31$	−3.22766			−0.579705			−0.289852	+	0.957071i	$0.593606\pi$
$32$		0			0
$33$	−9.07250			−1.57932
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	3.83394i		−	0.630296i	−0.949043	−	0.315148i	$-0.897946\pi$
$37$		−	3.83394i		−	0.630296i	0.949043	−	0.315148i	$-0.102054\pi$
$38$		0			0
$39$	16.7406			2.68064
$40$		0			0
$41$	6.45532			1.00815			0.504076	−	0.863659i	$-0.331833\pi$
$41$	6.45532			1.00815			0.504076	+	0.863659i	$0.331833\pi$
$42$		0			0
$43$	1.26133			0.192351			0.0961756	−	0.995364i	$-0.469339\pi$
$43$	1.26133			0.192351			0.0961756	+	0.995364i	$0.469339\pi$
$44$		0			0
$45$		−	29.8387i		−	4.44809i
$46$		0			0
$47$		−	1.40445i		−	0.204860i	−0.994740	−	0.102430i	$-0.967338\pi$
$47$		−	1.40445i		−	0.204860i	0.994740	−	0.102430i	$-0.0326618\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$			5.14954i			0.721079i
$52$		0			0
$53$		−	11.2270i		−	1.54215i	−0.636746	−	0.771073i	$-0.719720\pi$
$53$		−	11.2270i		−	1.54215i	0.636746	−	0.771073i	$-0.280280\pi$
$54$		0			0
$55$		−	11.3069i		−	1.52462i
$56$		0			0
$57$	−12.5940	+	6.27130i	−1.66812	+	0.830654i
$58$		0			0
$59$	11.4567			1.49153			0.745766	−	0.666208i	$-0.232084\pi$
$59$	11.4567			1.49153			0.745766	+	0.666208i	$0.232084\pi$
$60$		0			0
$61$		−	8.68879i		−	1.11249i	−0.831020	−	0.556243i	$-0.812242\pi$
$61$		−	8.68879i		−	1.11249i	0.831020	−	0.556243i	$-0.187758\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			20.8636i			2.58781i
$66$		0			0
$67$		−	2.24352i		−	0.274090i	−0.990565	−	0.137045i	$-0.956240\pi$
$67$		−	2.24352i		−	0.274090i	0.990565	−	0.137045i	$-0.0437604\pi$
$68$		0			0
$69$	−2.61718			−0.315072
$70$		0			0
$71$		−	7.83399i		−	0.929724i	−0.885383	−	0.464862i	$-0.846104\pi$
$71$		−	7.83399i		−	0.929724i	0.885383	−	0.464862i	$-0.153896\pi$
$72$		0			0
$73$			9.77722i			1.14434i	0.820136	+	0.572168i	$0.193898\pi$
$73$			9.77722i			1.14434i	−0.820136	+	0.572168i	$0.806102\pi$
$74$		0			0
$75$	36.0890			4.16720
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	9.10353i		−	1.02423i	−0.858918	−	0.512114i	$-0.828863\pi$
$79$		−	9.10353i		−	1.02423i	0.858918	−	0.512114i	$-0.171137\pi$
$80$		0			0
$81$	23.7702			2.64113
$82$		0			0
$83$			4.14493i			0.454965i	0.973782	+	0.227482i	$0.0730494\pi$
$83$			4.14493i			0.454965i	−0.973782	+	0.227482i	$0.926951\pi$
$84$		0			0
$85$	−6.41779			−0.696107
$86$		0			0
$87$			16.4723i			1.76602i
$88$		0			0
$89$	6.27006			0.664625			0.332313	−	0.943169i	$-0.392171\pi$
$89$	6.27006			0.664625			0.332313	+	0.943169i	$0.392171\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	10.4178			1.08027
$94$		0			0
$95$	−7.81582	−	15.6957i	−0.801886	−	1.61035i
$96$		0			0
$97$	−12.8052			−1.30017			−0.650083	−	0.759863i	$-0.725266\pi$
$97$	−12.8052			−1.30017			−0.650083	+	0.759863i	$0.725266\pi$
$98$		0			0
$99$	20.8504			2.09554
$100$		0			0
$101$		−	0.746558i		−	0.0742853i	−0.999310	−	0.0371426i	$-0.988174\pi$
$101$		−	0.746558i		−	0.0742853i	0.999310	−	0.0371426i	$-0.0118256\pi$
$102$		0			0
$103$	5.00135			0.492798			0.246399	−	0.969168i	$-0.420753\pi$
$103$	5.00135			0.492798			0.246399	+	0.969168i	$0.420753\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			2.77656i			0.268421i	0.990953	+	0.134210i	$0.0428498\pi$
$107$			2.77656i			0.268421i	−0.990953	+	0.134210i	$0.957150\pi$
$108$		0			0
$109$			2.24352i			0.214890i	0.994211	+	0.107445i	$0.0342670\pi$
$109$			2.24352i			0.214890i	−0.994211	+	0.107445i	$0.965733\pi$
$110$		0			0
$111$			12.3746i			1.17455i
$112$		0			0
$113$			9.95745i			0.936718i	0.883538	+	0.468359i	$0.155155\pi$
$113$			9.95745i			0.936718i	−0.883538	+	0.468359i	$0.844845\pi$
$114$		0			0
$115$		−	3.26176i		−	0.304160i
$116$		0			0
$117$	−38.4732			−3.55685
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−3.09906			−0.281733
$122$		0			0
$123$	−20.8356			−1.87868
$124$		0			0
$125$			24.8643i			2.22393i
$126$		0			0
$127$		−	10.7400i		−	0.953022i	−0.879169	−	0.476511i	$-0.841901\pi$
$127$		−	10.7400i		−	0.953022i	0.879169	−	0.476511i	$-0.158099\pi$
$128$		0			0
$129$	−4.07115			−0.358445
$130$		0			0
$131$		−	20.3347i		−	1.77665i	−0.459217	−	0.888324i	$-0.651870\pi$
$131$		−	20.3347i		−	1.77665i	0.459217	−	0.888324i	$-0.348130\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$			57.3585i			4.93663i
$136$		0			0
$137$	−8.93463			−0.763337			−0.381669	−	0.924299i	$-0.624650\pi$
$137$	−8.93463			−0.763337			−0.381669	+	0.924299i	$0.624650\pi$
$138$		0			0
$139$		−	5.56365i		−	0.471902i	−0.971765	−	0.235951i	$-0.924179\pi$
$139$		−	5.56365i		−	0.471902i	0.971765	−	0.235951i	$-0.0758205\pi$
$140$		0			0
$141$			4.53310i			0.381756i
$142$		0			0
$143$	−14.5788			−1.21914
$144$		0			0
$145$	−20.5292			−1.70486
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−12.7038			−1.04074			−0.520369	−	0.853941i	$-0.674206\pi$
$149$	−12.7038			−1.04074			−0.520369	+	0.853941i	$0.674206\pi$
$150$		0			0
$151$			5.05743i			0.411568i	0.978597	+	0.205784i	$0.0659744\pi$
$151$			5.05743i			0.411568i	−0.978597	+	0.205784i	$0.934026\pi$
$152$		0			0
$153$		−	11.8346i		−	0.956773i
$154$		0			0
$155$			12.9835i			1.04286i
$156$		0			0
$157$		−	3.15077i		−	0.251459i	−0.992065	−	0.125729i	$-0.959873\pi$
$157$		−	3.15077i		−	0.251459i	0.992065	−	0.125729i	$-0.0401271\pi$
$158$		0			0
$159$			36.2369i			2.87377i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.69607			0.211172			0.105586	−	0.994410i	$-0.466328\pi$
$163$	2.69607			0.211172			0.105586	+	0.994410i	$0.466328\pi$
$164$		0			0
$165$			36.4949i			2.84112i
$166$		0			0
$167$	5.79709			0.448592			0.224296	−	0.974521i	$-0.427992\pi$
$167$	5.79709			0.448592			0.224296	+	0.974521i	$0.427992\pi$
$168$		0			0
$169$	13.9009			1.06930
$170$		0			0
$171$	28.9434	−	14.4127i	2.21336	−	1.10216i
$172$		0			0
$173$	7.53877			0.573162			0.286581	−	0.958056i	$-0.407481\pi$
$173$	7.53877			0.573162			0.286581	+	0.958056i	$0.407481\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−36.9782			−2.77946
$178$		0			0
$179$			16.3305i			1.22060i	0.792171	+	0.610299i	$0.208951\pi$
$179$			16.3305i			1.22060i	−0.792171	+	0.610299i	$0.791049\pi$
$180$		0			0
$181$	−7.98905			−0.593822			−0.296911	−	0.954905i	$-0.595956\pi$
$181$	−7.98905			−0.593822			−0.296911	+	0.954905i	$0.595956\pi$
$182$		0			0
$183$			28.0445i			2.07311i
$184$		0			0
$185$	−15.4223			−1.13387
$186$		0			0
$187$		−	4.48456i		−	0.327943i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.32088			0.312648			0.156324	−	0.987706i	$-0.450036\pi$
$191$	4.32088			0.312648			0.156324	+	0.987706i	$0.450036\pi$
$192$		0			0
$193$		−	14.7401i		−	1.06101i	−0.847681	−	0.530506i	$-0.822002\pi$
$193$		−	14.7401i		−	1.06101i	0.847681	−	0.530506i	$-0.177998\pi$
$194$		0			0
$195$		−	67.3405i		−	4.82235i
$196$		0			0
$197$	9.99418			0.712056			0.356028	−	0.934475i	$-0.384131\pi$
$197$	9.99418			0.712056			0.356028	+	0.934475i	$0.384131\pi$
$198$		0			0
$199$			10.9535i			0.776472i	0.921560	+	0.388236i	$0.126915\pi$
$199$			10.9535i			0.776472i	−0.921560	+	0.388236i	$0.873085\pi$
$200$		0			0
$201$			7.24132i			0.510763i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		−	25.9671i		−	1.81362i
$206$		0			0
$207$	6.01480			0.418057
$208$		0			0
$209$	10.9677	−	5.46146i	0.758651	−	0.377777i
$210$		0			0
$211$			22.1792i			1.52688i	0.645880	+	0.763439i	$0.276490\pi$
$211$			22.1792i			1.52688i	−0.645880	+	0.763439i	$0.723510\pi$
$212$		0			0
$213$			25.2855i			1.73253i
$214$		0			0
$215$		−	5.07381i		−	0.346031i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		−	31.5575i		−	2.13246i
$220$		0			0
$221$			8.27493i			0.556632i
$222$		0			0
$223$	14.6843			0.983336			0.491668	−	0.870783i	$-0.336387\pi$
$223$	14.6843			0.983336			0.491668	+	0.870783i	$0.336387\pi$
$224$		0			0
$225$	−82.9395			−5.52930
$226$		0			0
$227$	−0.425219			−0.0282228			−0.0141114	−	0.999900i	$-0.504492\pi$
$227$	−0.425219			−0.0282228			−0.0141114	+	0.999900i	$0.504492\pi$
$228$		0			0
$229$		−	12.2610i		−	0.810227i	−0.914266	−	0.405114i	$-0.867232\pi$
$229$		−	12.2610i		−	0.810227i	0.914266	−	0.405114i	$-0.132768\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.763383			−0.0500109			−0.0250054	−	0.999687i	$-0.507960\pi$
$233$	−0.763383			−0.0500109			−0.0250054	+	0.999687i	$0.507960\pi$
$234$		0			0
$235$	−5.64953			−0.368535
$236$		0			0
$237$			29.3831i			1.90864i
$238$		0			0
$239$	−24.3950			−1.57798			−0.788991	−	0.614404i	$-0.789396\pi$
$239$	−24.3950			−1.57798			−0.788991	+	0.614404i	$0.789396\pi$
$240$		0			0
$241$	−23.0986			−1.48791			−0.743956	−	0.668229i	$-0.767053\pi$
$241$	−23.0986			−1.48791			−0.743956	+	0.668229i	$0.767053\pi$
$242$		0			0
$243$	−33.9448			−2.17756
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−20.2376	+	10.0775i	−1.28769	+	0.641217i
$248$		0			0
$249$		−	13.3784i		−	0.847822i
$250$		0			0
$251$		−	13.2893i		−	0.838811i	−0.907799	−	0.419406i	$-0.862239\pi$
$251$		−	13.2893i		−	0.838811i	0.907799	−	0.419406i	$-0.137761\pi$
$252$		0			0
$253$	2.27922			0.143293
$254$		0			0
$255$	20.7144			1.29719
$256$		0			0
$257$	−20.6089			−1.28555			−0.642775	−	0.766055i	$-0.722217\pi$
$257$	−20.6089			−1.28555			−0.642775	+	0.766055i	$0.722217\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		−	37.8565i		−	2.34326i
$262$		0			0
$263$	0.828748			0.0511028			0.0255514	−	0.999674i	$-0.491866\pi$
$263$	0.828748			0.0511028			0.0255514	+	0.999674i	$0.491866\pi$
$264$		0			0
$265$	−45.1615			−2.77425
$266$		0			0
$267$	−20.2376			−1.23852
$268$		0			0
$269$	−24.6323			−1.50186			−0.750930	−	0.660382i	$-0.770394\pi$
$269$	−24.6323			−1.50186			−0.750930	+	0.660382i	$0.770394\pi$
$270$		0			0
$271$		−	9.31815i		−	0.566037i	−0.959114	−	0.283019i	$-0.908664\pi$
$271$		−	9.31815i		−	0.566037i	0.959114	−	0.283019i	$-0.0913358\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−31.4287			−1.89522
$276$		0			0
$277$	−14.1287			−0.848908			−0.424454	−	0.905449i	$-0.639534\pi$
$277$	−14.1287			−0.848908			−0.424454	+	0.905449i	$0.639534\pi$
$278$		0			0
$279$	−23.9421			−1.43338
$280$		0			0
$281$			20.1644i			1.20291i	0.798907	+	0.601454i	$0.205412\pi$
$281$			20.1644i			1.20291i	−0.798907	+	0.601454i	$0.794588\pi$
$282$		0			0
$283$			21.7563i			1.29328i	0.762796	+	0.646639i	$0.223826\pi$
$283$			21.7563i			1.29328i	−0.762796	+	0.646639i	$0.776174\pi$
$284$		0			0
$285$	25.2268	+	50.6604i	1.49431	+	3.00086i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	14.4546			0.850269
$290$		0			0
$291$	41.3307			2.42285
$292$		0			0
$293$	6.45532			0.377124			0.188562	−	0.982061i	$-0.439617\pi$
$293$	6.45532			0.377124			0.188562	+	0.982061i	$0.439617\pi$
$294$		0			0
$295$		−	46.0854i		−	2.68320i
$296$		0			0
$297$	−40.0804			−2.32570
$298$		0			0
$299$	−4.20562			−0.243217
$300$		0			0
$301$		0			0
$302$		0			0
$303$			2.40963i			0.138430i
$304$		0			0
$305$	−34.9514			−2.00131
$306$		0			0
$307$	−2.80244			−0.159944			−0.0799719	−	0.996797i	$-0.525483\pi$
$307$	−2.80244			−0.159944			−0.0799719	+	0.996797i	$0.525483\pi$
$308$		0			0
$309$	−16.1427			−0.918324
$310$		0			0
$311$			19.0016i			1.07748i	0.842472	+	0.538740i	$0.181099\pi$
$311$			19.0016i			1.07748i	−0.842472	+	0.538740i	$0.818901\pi$
$312$		0			0
$313$			33.8098i			1.91104i	0.294924	+	0.955521i	$0.404706\pi$
$313$			33.8098i			1.91104i	−0.294924	+	0.955521i	$0.595294\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			22.5000i			1.26373i	0.775079	+	0.631864i	$0.217710\pi$
$317$			22.5000i			1.26373i	−0.775079	+	0.631864i	$0.782290\pi$
$318$		0			0
$319$		−	14.3452i		−	0.803176i
$320$		0			0
$321$		−	8.96180i		−	0.500199i
$322$		0			0
$323$	−3.09992	−	6.22524i	−0.172484	−	0.346382i
$324$		0			0
$325$	57.9924			3.21684
$326$		0			0
$327$		−	7.24132i		−	0.400446i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			16.3305i			0.897604i	0.893631	+	0.448802i	$0.148149\pi$
$331$			16.3305i			0.897604i	−0.893631	+	0.448802i	$0.851851\pi$
$332$		0			0
$333$		−	28.4393i		−	1.55847i
$334$		0			0
$335$	−9.02475			−0.493075
$336$		0			0
$337$			3.26355i			0.177777i	0.996042	+	0.0888885i	$0.0283315\pi$
$337$			3.26355i			0.177777i	−0.996042	+	0.0888885i	$0.971669\pi$
$338$		0			0
$339$		−	32.1393i		−	1.74557i
$340$		0			0
$341$	−9.07250			−0.491304
$342$		0			0
$343$		0			0
$344$		0			0
$345$			10.5278i			0.566800i
$346$		0			0
$347$	2.58006			0.138505			0.0692524	−	0.997599i	$-0.477939\pi$
$347$	2.58006			0.138505			0.0692524	+	0.997599i	$0.477939\pi$
$348$		0			0
$349$			24.7341i			1.32399i	0.749510	+	0.661993i	$0.230289\pi$
$349$			24.7341i			1.32399i	−0.749510	+	0.661993i	$0.769711\pi$
$350$		0			0
$351$	73.9565			3.94751
$352$		0			0
$353$		−	22.2490i		−	1.18420i	−0.805866	−	0.592099i	$-0.798300\pi$
$353$		−	22.2490i		−	1.18420i	0.805866	−	0.592099i	$-0.201700\pi$
$354$		0			0
$355$	−31.5129			−1.67253
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−11.0584			−0.583640			−0.291820	−	0.956473i	$-0.594261\pi$
$359$	−11.0584			−0.583640			−0.291820	+	0.956473i	$0.594261\pi$
$360$		0			0
$361$	11.4496	−	15.1627i	0.602611	−	0.798035i
$362$		0			0
$363$	10.0027			0.525006
$364$		0			0
$365$	39.3297			2.05861
$366$		0			0
$367$			18.7506i			0.978773i	0.872067	+	0.489386i	$0.162779\pi$
$367$			18.7506i			0.978773i	−0.872067	+	0.489386i	$0.837221\pi$
$368$		0			0
$369$	47.8842			2.49275
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	28.5775i		−	1.47969i	−0.672779	−	0.739844i	$-0.734899\pi$
$373$		−	28.5775i		−	1.47969i	0.672779	−	0.739844i	$-0.265101\pi$
$374$		0			0
$375$		−	80.2534i		−	4.14427i
$376$		0			0
$377$			26.4698i			1.36326i
$378$		0			0
$379$			2.98002i			0.153073i	0.997067	+	0.0765367i	$0.0243862\pi$
$379$			2.98002i			0.153073i	−0.997067	+	0.0765367i	$0.975614\pi$
$380$		0			0
$381$			34.6651i			1.77595i
$382$		0			0
$383$	−2.37722			−0.121470			−0.0607352	−	0.998154i	$-0.519345\pi$
$383$	−2.37722			−0.121470			−0.0607352	+	0.998154i	$0.519345\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	9.35629			0.475607
$388$		0			0
$389$	19.8624			1.00707			0.503533	−	0.863976i	$-0.332033\pi$
$389$	19.8624			1.00707			0.503533	+	0.863976i	$0.332033\pi$
$390$		0			0
$391$		−	1.29368i		−	0.0654242i
$392$		0			0
$393$			65.6334i			3.31077i
$394$		0			0
$395$	−36.6197			−1.84254
$396$		0			0
$397$		−	4.62631i		−	0.232188i	−0.993238	−	0.116094i	$-0.962963\pi$
$397$		−	4.62631i		−	0.232188i	0.993238	−	0.116094i	$-0.0370374\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	27.5114i		−	1.37385i	−0.726726	−	0.686927i	$-0.758959\pi$
$401$		−	27.5114i		−	1.37385i	0.726726	−	0.686927i	$-0.241041\pi$
$402$		0			0
$403$	16.7406			0.833910
$404$		0			0
$405$		−	95.6176i		−	4.75128i
$406$		0			0
$407$		−	10.7767i		−	0.534180i
$408$		0			0
$409$	1.18894			0.0587893			0.0293946	−	0.999568i	$-0.490642\pi$
$409$	1.18894			0.0587893			0.0293946	+	0.999568i	$0.490642\pi$
$410$		0			0
$411$	28.8380			1.42247
$412$		0			0
$413$		0			0
$414$		0			0
$415$	16.6733			0.818460
$416$		0			0
$417$			17.9576i			0.879385i
$418$		0			0
$419$			25.4257i			1.24212i	0.783761	+	0.621062i	$0.213299\pi$
$419$			25.4257i			1.24212i	−0.783761	+	0.621062i	$0.786701\pi$
$420$		0			0
$421$		−	38.7384i		−	1.88799i	−0.329953	−	0.943997i	$-0.607033\pi$
$421$		−	38.7384i		−	1.88799i	0.329953	−	0.943997i	$-0.392967\pi$
$422$		0			0
$423$		−	10.4179i		−	0.506537i
$424$		0			0
$425$			17.8389i			0.865313i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	47.0556			2.27186
$430$		0			0
$431$			20.5593i			0.990306i	0.868806	+	0.495153i	$0.164888\pi$
$431$			20.5593i			0.990306i	−0.868806	+	0.495153i	$0.835112\pi$
$432$		0			0
$433$	−5.13886			−0.246958			−0.123479	−	0.992347i	$-0.539405\pi$
$433$	−5.13886			−0.246958			−0.123479	+	0.992347i	$0.539405\pi$
$434$		0			0
$435$	66.2612			3.17698
$436$		0			0
$437$	3.16390	−	1.57549i	0.151350	−	0.0753660i
$438$		0			0
$439$	19.4935			0.930373			0.465187	−	0.885213i	$-0.345987\pi$
$439$	19.4935			0.930373			0.465187	+	0.885213i	$0.345987\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−7.84261			−0.372613			−0.186307	−	0.982492i	$-0.559652\pi$
$443$	−7.84261			−0.372613			−0.186307	+	0.982492i	$0.559652\pi$
$444$		0			0
$445$		−	25.2218i		−	1.19563i
$446$		0			0
$447$	41.0037			1.93941
$448$		0			0
$449$		−	3.97210i		−	0.187455i	−0.995598	−	0.0937275i	$-0.970122\pi$
$449$		−	3.97210i		−	0.187455i	0.995598	−	0.0937275i	$-0.0298783\pi$
$450$		0			0
$451$	18.1450			0.854415
$452$		0			0
$453$		−	16.3237i		−	0.766952i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−10.2633			−0.480096			−0.240048	−	0.970761i	$-0.577163\pi$
$457$	−10.2633			−0.480096			−0.240048	+	0.970761i	$0.577163\pi$
$458$		0			0
$459$			22.7496i			1.06186i
$460$		0			0
$461$		−	5.99418i		−	0.279177i	−0.990210	−	0.139588i	$-0.955422\pi$
$461$		−	5.99418i		−	0.279177i	0.990210	−	0.139588i	$-0.0445779\pi$
$462$		0			0
$463$	−10.1296			−0.470762			−0.235381	−	0.971903i	$-0.575634\pi$
$463$	−10.1296			−0.470762			−0.235381	+	0.971903i	$0.575634\pi$
$464$		0			0
$465$		−	41.9064i		−	1.94336i
$466$		0			0
$467$			32.6384i			1.51033i	0.655537	+	0.755163i	$0.272443\pi$
$467$			32.6384i			1.51033i	−0.655537	+	0.755163i	$0.727557\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$			10.1696i			0.468591i
$472$		0			0
$473$	3.54543			0.163019
$474$		0			0
$475$	−43.6278	+	21.7249i	−2.00178	+	0.996805i
$476$		0			0
$477$		−	83.2795i		−	3.81310i
$478$		0			0
$479$			26.1951i			1.19689i	0.801165	+	0.598443i	$0.204214\pi$
$479$			26.1951i			1.19689i	−0.801165	+	0.598443i	$0.795786\pi$
$480$		0			0
$481$			19.8852i			0.906685i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			51.5098i			2.33894i
$486$		0			0
$487$			37.6263i			1.70501i	0.522720	+	0.852505i	$0.324917\pi$
$487$			37.6263i			1.70501i	−0.522720	+	0.852505i	$0.675083\pi$
$488$		0			0
$489$	−8.70199			−0.393518
$490$		0			0
$491$	−39.1633			−1.76741			−0.883707	−	0.468040i	$-0.844960\pi$
$491$	−39.1633			−1.76741			−0.883707	+	0.468040i	$0.844960\pi$
$492$		0			0
$493$	−8.14230			−0.366711
$494$		0			0
$495$		−	83.8723i		−	3.76978i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−24.7782			−1.10922			−0.554612	−	0.832109i	$-0.687133\pi$
$499$	−24.7782			−1.10922			−0.554612	+	0.832109i	$0.687133\pi$
$500$		0			0
$501$	−18.7110			−0.835947
$502$		0			0
$503$			27.3235i			1.21829i	0.793058	+	0.609147i	$0.208488\pi$
$503$			27.3235i			1.21829i	−0.793058	+	0.609147i	$0.791512\pi$
$504$		0			0
$505$	−3.00309			−0.133636
$506$		0			0
$507$	−44.8675			−1.99264
$508$		0			0
$509$	11.3512			0.503132			0.251566	−	0.967840i	$-0.419054\pi$
$509$	11.3512			0.503132			0.251566	+	0.967840i	$0.419054\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−55.6376	+	27.7053i	−2.45646	+	1.22322i
$514$		0			0
$515$		−	20.1184i		−	0.886521i
$516$		0			0
$517$		−	3.94772i		−	0.173621i
$518$		0			0
$519$	−24.3326			−1.06808
$520$		0			0
$521$	16.3783			0.717545			0.358772	−	0.933425i	$-0.383195\pi$
$521$	16.3783			0.717545			0.358772	+	0.933425i	$0.383195\pi$
$522$		0			0
$523$	10.3412			0.452190			0.226095	−	0.974105i	$-0.427404\pi$
$523$	10.3412			0.452190			0.226095	+	0.974105i	$0.427404\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			5.14954i			0.224317i
$528$		0			0
$529$	−22.3425			−0.971413
$530$		0			0
$531$	84.9832			3.68796
$532$		0			0
$533$	−33.4812			−1.45023
$534$		0			0
$535$	11.1690			0.482876
$536$		0			0
$537$		−	52.7092i		−	2.27457i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	30.1652			1.29690			0.648452	−	0.761255i	$-0.275417\pi$
$541$	30.1652			1.29690			0.648452	+	0.761255i	$0.275417\pi$
$542$		0			0
$543$	25.7859			1.10658
$544$		0			0
$545$	9.02475			0.386578
$546$		0			0
$547$		−	36.7992i		−	1.57342i	−0.617323	−	0.786709i	$-0.711783\pi$
$547$		−	36.7992i		−	1.57342i	0.617323	−	0.786709i	$-0.288217\pi$
$548$		0			0
$549$		−	64.4516i		−	2.75073i
$550$		0			0
$551$	−9.91599	−	19.9133i	−0.422435	−	0.848334i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	49.7780			2.11296
$556$		0			0
$557$	−4.75757			−0.201585			−0.100792	−	0.994907i	$-0.532138\pi$
$557$	−4.75757			−0.201585			−0.100792	+	0.994907i	$0.532138\pi$
$558$		0			0
$559$	−6.54204			−0.276699
$560$		0			0
$561$			14.4746i			0.611119i
$562$		0			0
$563$	3.38720			0.142753			0.0713767	−	0.997449i	$-0.477261\pi$
$563$	3.38720			0.142753			0.0713767	+	0.997449i	$0.477261\pi$
$564$		0			0
$565$	40.0547			1.68511
$566$		0			0
$567$		0			0
$568$		0			0
$569$			17.6714i			0.740823i	0.928868	+	0.370411i	$0.120783\pi$
$569$			17.6714i			0.740823i	−0.928868	+	0.370411i	$0.879217\pi$
$570$		0			0
$571$	25.2386			1.05620			0.528101	−	0.849182i	$-0.322904\pi$
$571$	25.2386			1.05620			0.528101	+	0.849182i	$0.322904\pi$
$572$		0			0
$573$	−13.9463			−0.582616
$574$		0			0
$575$	−9.06638			−0.378094
$576$		0			0
$577$		−	34.1859i		−	1.42318i	−0.702595	−	0.711590i	$-0.747976\pi$
$577$		−	34.1859i		−	1.42318i	0.702595	−	0.711590i	$-0.252024\pi$
$578$		0			0
$579$			47.5759i			1.97719i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	31.5575i		−	1.30698i
$584$		0			0
$585$			154.762i			6.39860i
$586$		0			0
$587$		−	17.4627i		−	0.720764i	−0.932805	−	0.360382i	$-0.882646\pi$
$587$		−	17.4627i		−	0.720764i	0.932805	−	0.360382i	$-0.117354\pi$
$588$		0			0
$589$	−12.5940	+	6.27130i	−0.518927	+	0.258404i
$590$		0			0
$591$	−32.2578			−1.32691
$592$		0			0
$593$		−	25.2182i		−	1.03559i	−0.855506	−	0.517793i	$-0.826754\pi$
$593$		−	25.2182i		−	1.03559i	0.855506	−	0.517793i	$-0.173246\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	35.3541i		−	1.44695i
$598$		0			0
$599$		−	29.7923i		−	1.21728i	−0.793447	−	0.608640i	$-0.791715\pi$
$599$		−	29.7923i		−	1.21728i	0.793447	−	0.608640i	$-0.208285\pi$
$600$		0			0
$601$	21.9097			0.893714			0.446857	−	0.894605i	$-0.352543\pi$
$601$	21.9097			0.893714			0.446857	+	0.894605i	$0.352543\pi$
$602$		0			0
$603$		−	16.6420i		−	0.677713i
$604$		0			0
$605$			12.4662i			0.506824i
$606$		0			0
$607$	−45.8775			−1.86211			−0.931055	−	0.364880i	$-0.881110\pi$
$607$	−45.8775			−1.86211			−0.931055	+	0.364880i	$0.881110\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			7.28435i			0.294693i
$612$		0			0
$613$	33.3694			1.34778			0.673888	−	0.738833i	$-0.264623\pi$
$613$	33.3694			1.34778			0.673888	+	0.738833i	$0.264623\pi$
$614$		0			0
$615$			83.8128i			3.37966i
$616$		0			0
$617$	20.5990			0.829283			0.414641	−	0.909985i	$-0.363907\pi$
$617$	20.5990			0.829283			0.414641	+	0.909985i	$0.363907\pi$
$618$		0			0
$619$			30.8279i			1.23908i	0.784966	+	0.619539i	$0.212680\pi$
$619$			30.8279i			1.23908i	−0.784966	+	0.619539i	$0.787320\pi$
$620$		0			0
$621$	−11.5622			−0.463974
$622$		0			0
$623$		0			0
$624$		0			0
$625$	44.1127			1.76451
$626$		0			0
$627$	−35.4000	+	17.6277i	−1.41374	+	0.703984i
$628$		0			0
$629$	−6.11682			−0.243893
$630$		0			0
$631$	39.1800			1.55973			0.779866	−	0.625946i	$-0.215287\pi$
$631$	39.1800			1.55973			0.779866	+	0.625946i	$0.215287\pi$
$632$		0			0
$633$		−	71.5868i		−	2.84532i
$634$		0			0
$635$	−43.2026			−1.71444
$636$		0			0
$637$		0			0
$638$		0			0
$639$		−	58.1109i		−	2.29883i
$640$		0			0
$641$			27.6956i			1.09391i	0.837161	+	0.546956i	$0.184213\pi$
$641$			27.6956i			1.09391i	−0.837161	+	0.546956i	$0.815787\pi$
$642$		0			0
$643$		−	43.4733i		−	1.71442i	−0.514967	−	0.857210i	$-0.672196\pi$
$643$		−	43.4733i		−	1.71442i	0.514967	−	0.857210i	$-0.327804\pi$
$644$		0			0
$645$			16.3765i			0.644825i
$646$		0			0
$647$		−	4.49193i		−	0.176596i	−0.996094	−	0.0882981i	$-0.971857\pi$
$647$		−	4.49193i		−	0.176596i	0.996094	−	0.0882981i	$-0.0281428\pi$
$648$		0			0
$649$	32.2031			1.26408
$650$		0			0
$651$		0			0
$652$		0			0
$653$	42.4544			1.66137			0.830684	−	0.556744i	$-0.187950\pi$
$653$	42.4544			1.66137			0.830684	+	0.556744i	$0.187950\pi$
$654$		0			0
$655$	−81.7979			−3.19611
$656$		0			0
$657$			72.5253i			2.82948i
$658$		0			0
$659$		−	8.01821i		−	0.312345i	−0.987730	−	0.156173i	$-0.950084\pi$
$659$		−	8.01821i		−	0.312345i	0.987730	−	0.156173i	$-0.0499156\pi$
$660$		0			0
$661$	−49.6648			−1.93174			−0.965869	−	0.259031i	$-0.916597\pi$
$661$	−49.6648			−1.93174			−0.965869	+	0.259031i	$0.916597\pi$
$662$		0			0
$663$		−	26.7086i		−	1.03728i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	4.13822i		−	0.160232i
$668$		0			0
$669$	−47.3960			−1.83244
$670$		0			0
$671$		−	24.4230i		−	0.942839i
$672$		0			0
$673$			18.2992i			0.705381i	0.935740	+	0.352691i	$0.114733\pi$
$673$			18.2992i			0.705381i	−0.935740	+	0.352691i	$0.885267\pi$
$674$		0			0
$675$	159.434			6.13660
$676$		0			0
$677$	−3.88589			−0.149347			−0.0746735	−	0.997208i	$-0.523791\pi$
$677$	−3.88589			−0.149347			−0.0746735	+	0.997208i	$0.523791\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	1.37246			0.0525928
$682$		0			0
$683$			2.61045i			0.0998861i	0.998752	+	0.0499430i	$0.0159040\pi$
$683$			2.61045i			0.0998861i	−0.998752	+	0.0499430i	$0.984096\pi$
$684$		0			0
$685$			35.9403i			1.37321i
$686$		0			0
$687$			39.5742i			1.50985i
$688$		0			0
$689$			58.2301i			2.21839i
$690$		0			0
$691$		−	40.5670i		−	1.54324i	−0.636083	−	0.771621i	$-0.719446\pi$
$691$		−	40.5670i		−	1.54324i	0.636083	−	0.771621i	$-0.280554\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−22.3802			−0.848930
$696$		0			0
$697$		−	10.2991i		−	0.390105i
$698$		0			0
$699$	2.46394			0.0931947
$700$		0			0
$701$	27.8296			1.05111			0.525554	−	0.850760i	$-0.323858\pi$
$701$	27.8296			1.05111			0.525554	+	0.850760i	$0.323858\pi$
$702$		0			0
$703$	−7.44929	−	14.9596i	−0.280955	−	0.564213i
$704$		0			0
$705$	18.2348			0.686760
$706$		0			0
$707$		0			0
$708$		0			0
$709$	34.6552			1.30150			0.650752	−	0.759290i	$-0.274454\pi$
$709$	34.6552			1.30150			0.650752	+	0.759290i	$0.274454\pi$
$710$		0			0
$711$		−	67.5281i		−	2.53250i
$712$		0			0
$713$	−2.61718			−0.0980143
$714$		0			0
$715$			58.6446i			2.19318i
$716$		0			0
$717$	78.7388			2.94056
$718$		0			0
$719$			19.3777i			0.722667i	0.932437	+	0.361333i	$0.117678\pi$
$719$			19.3777i			0.722667i	−0.932437	+	0.361333i	$0.882322\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	74.5544			2.77271
$724$		0			0
$725$			57.0629i			2.11926i
$726$		0			0
$727$			34.1207i			1.26547i	0.774370	+	0.632733i	$0.218067\pi$
$727$			34.1207i			1.26547i	−0.774370	+	0.632733i	$0.781933\pi$
$728$		0			0
$729$	38.2516			1.41673
$730$		0			0
$731$		−	2.01238i		−	0.0744305i
$732$		0			0
$733$		−	22.3288i		−	0.824734i	−0.911018	−	0.412367i	$-0.864702\pi$
$733$		−	22.3288i		−	0.824734i	0.911018	−	0.412367i	$-0.135298\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	6.30623i		−	0.232293i
$738$		0			0
$739$	12.6031			0.463611			0.231806	−	0.972762i	$-0.425537\pi$
$739$	12.6031			0.463611			0.231806	+	0.972762i	$0.425537\pi$
$740$		0			0
$741$	65.3202	−	32.5268i	2.39960	−	1.19490i
$742$		0			0
$743$		−	14.5279i		−	0.532977i	−0.963838	−	0.266488i	$-0.914137\pi$
$743$		−	14.5279i		−	0.532977i	0.963838	−	0.266488i	$-0.0858634\pi$
$744$		0			0
$745$			51.1022i			1.87224i
$746$		0			0
$747$			30.7462i			1.12494i
$748$		0			0
$749$		0			0
$750$		0			0
$751$			35.5115i			1.29583i	0.761711	+	0.647917i	$0.224360\pi$
$751$			35.5115i			1.29583i	−0.761711	+	0.647917i	$0.775640\pi$
$752$		0			0
$753$			42.8932i			1.56312i
$754$		0			0
$755$	20.3439			0.740391
$756$		0			0
$757$	25.3427			0.921095			0.460547	−	0.887635i	$-0.347653\pi$
$757$	25.3427			0.921095			0.460547	+	0.887635i	$0.347653\pi$
$758$		0			0
$759$	−7.35654			−0.267026
$760$		0			0
$761$			12.4497i			0.451302i	0.974208	+	0.225651i	$0.0724509\pi$
$761$			12.4497i			0.451302i	−0.974208	+	0.225651i	$0.927549\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−47.6058			−1.72119
$766$		0			0
$767$	−59.4213			−2.14558
$768$		0			0
$769$			20.2616i			0.730651i	0.930880	+	0.365325i	$0.119042\pi$
$769$			20.2616i			0.730651i	−0.930880	+	0.365325i	$0.880958\pi$
$770$		0			0
$771$	66.5187			2.39561
$772$		0			0
$773$	1.19523			0.0429895			0.0214947	−	0.999769i	$-0.493157\pi$
$773$	1.19523			0.0429895			0.0214947	+	0.999769i	$0.493157\pi$
$774$		0			0
$775$	36.0890			1.29636
$776$		0			0
$777$		0			0
$778$		0			0
$779$	25.1880	−	12.5426i	0.902454	−	0.449385i
$780$		0			0
$781$		−	22.0203i		−	0.787947i
$782$		0			0
$783$			72.7711i			2.60063i
$784$		0			0
$785$	−12.6742			−0.452363
$786$		0			0
$787$	29.2085			1.04117			0.520585	−	0.853810i	$-0.325714\pi$
$787$	29.2085			1.04117			0.520585	+	0.853810i	$0.325714\pi$
$788$		0			0
$789$	−2.67492			−0.0952295
$790$		0			0
$791$		0			0
$792$		0			0
$793$			45.0654i			1.60032i
$794$		0			0
$795$	145.766			5.16979
$796$		0			0
$797$	−51.2783			−1.81637			−0.908186	−	0.418567i	$-0.862532\pi$
$797$	−51.2783			−1.81637			−0.908186	+	0.418567i	$0.862532\pi$
$798$		0			0
$799$	−2.24072			−0.0792710
$800$		0			0
$801$	46.5100			1.64335
$802$		0			0
$803$			27.4824i			0.969833i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	79.5048			2.79870
$808$		0			0
$809$	−34.2800			−1.20522			−0.602611	−	0.798035i	$-0.705873\pi$
$809$	−34.2800			−1.20522			−0.602611	+	0.798035i	$0.705873\pi$
$810$		0			0
$811$	27.6678			0.971548			0.485774	−	0.874085i	$-0.338538\pi$
$811$	27.6678			0.971548			0.485774	+	0.874085i	$0.338538\pi$
$812$		0			0
$813$			30.0758i			1.05480i
$814$		0			0
$815$		−	10.8452i		−	0.379889i
$816$		0			0
$817$	4.92159	−	2.45075i	0.172185	−	0.0857409i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−41.5309			−1.44944			−0.724719	−	0.689045i	$-0.758030\pi$
$821$	−41.5309			−1.44944			−0.724719	+	0.689045i	$0.758030\pi$
$822$		0			0
$823$	−1.10581			−0.0385462			−0.0192731	−	0.999814i	$-0.506135\pi$
$823$	−1.10581			−0.0385462			−0.0192731	+	0.999814i	$0.506135\pi$
$824$		0			0
$825$	101.441			3.53173
$826$		0			0
$827$		−	34.0019i		−	1.18236i	−0.806539	−	0.591180i	$-0.798662\pi$
$827$		−	34.0019i		−	1.18236i	0.806539	−	0.591180i	$-0.201338\pi$
$828$		0			0
$829$	−44.1038			−1.53179			−0.765894	−	0.642967i	$-0.777703\pi$
$829$	−44.1038			−1.53179			−0.765894	+	0.642967i	$0.777703\pi$
$830$		0			0
$831$	45.6025			1.58193
$832$		0			0
$833$		0			0
$834$		0			0
$835$		−	23.3193i		−	0.806996i
$836$		0			0
$837$	46.0236			1.59081
$838$		0			0
$839$	−26.8313			−0.926318			−0.463159	−	0.886275i	$-0.653284\pi$
$839$	−26.8313			−0.926318			−0.463159	+	0.886275i	$0.653284\pi$
$840$		0			0
$841$	2.95447			0.101878
$842$		0			0
$843$		−	65.0839i		−	2.24161i
$844$		0			0
$845$		−	55.9177i		−	1.92363i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		−	70.2219i		−	2.41001i
$850$		0			0
$851$		−	3.10879i		−	0.106568i
$852$		0			0
$853$		−	11.2933i		−	0.386676i	−0.981132	−	0.193338i	$-0.938069\pi$
$853$		−	11.2933i		−	0.386676i	0.981132	−	0.193338i	$-0.0619314\pi$
$854$		0			0
$855$	−57.9761	−	116.427i	−1.98274	−	3.98173i
$856$		0			0
$857$	15.5856			0.532392			0.266196	−	0.963919i	$-0.414233\pi$
$857$	15.5856			0.532392			0.266196	+	0.963919i	$0.414233\pi$
$858$		0			0
$859$		−	4.82223i		−	0.164532i	−0.996610	−	0.0822662i	$-0.973784\pi$
$859$		−	4.82223i		−	0.164532i	0.996610	−	0.0822662i	$-0.0262158\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	41.5818i		−	1.41546i	−0.706483	−	0.707730i	$-0.749719\pi$
$863$		−	41.5818i		−	1.41546i	0.706483	−	0.707730i	$-0.250281\pi$
$864$		0			0
$865$		−	30.3253i		−	1.03109i
$866$		0			0
$867$	−46.6544			−1.58447
$868$		0			0
$869$		−	25.5888i		−	0.868040i
$870$		0			0
$871$			11.6363i			0.394280i
$872$		0			0
$873$	−94.9859			−3.21479
$874$		0			0
$875$		0			0
$876$		0			0
$877$			32.2480i			1.08894i	0.838781	+	0.544468i	$0.183269\pi$
$877$			32.2480i			1.08894i	−0.838781	+	0.544468i	$0.816731\pi$
$878$		0			0
$879$	−20.8356			−0.702767
$880$		0			0
$881$		−	54.5320i		−	1.83723i	−0.395153	−	0.918615i	$-0.629308\pi$
$881$		−	54.5320i		−	1.83723i	0.395153	−	0.918615i	$-0.370692\pi$
$882$		0			0
$883$	−54.2600			−1.82599			−0.912997	−	0.407966i	$-0.866238\pi$
$883$	−54.2600			−1.82599			−0.912997	+	0.407966i	$0.866238\pi$
$884$		0			0
$885$			148.748i			5.00011i
$886$		0			0
$887$	−36.8019			−1.23569			−0.617844	−	0.786300i	$-0.711994\pi$
$887$	−36.8019			−1.23569			−0.617844	+	0.786300i	$0.711994\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	66.8148			2.23838
$892$		0			0
$893$	−2.72883	−	5.48003i	−0.0913169	−	0.183382i
$894$		0			0
$895$	65.6907			2.19580
$896$		0			0
$897$	13.5743			0.453233
$898$		0			0
$899$			16.4723i			0.549382i
$900$		0			0
$901$	−17.9120			−0.596735
$902$		0			0
$903$		0			0
$904$		0			0
$905$			32.1366i			1.06826i
$906$		0			0
$907$			5.68258i			0.188687i	0.995540	+	0.0943435i	$0.0300752\pi$
$907$			5.68258i			0.188687i	−0.995540	+	0.0943435i	$0.969925\pi$
$908$		0			0
$909$		−	5.53781i		−	0.183677i
$910$		0			0
$911$			23.6688i			0.784181i	0.919927	+	0.392091i	$0.128248\pi$
$911$			23.6688i			0.784181i	−0.919927	+	0.392091i	$0.871752\pi$
$912$		0			0
$913$			11.6508i			0.385585i
$914$		0			0
$915$	112.811			3.72942
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−50.4365			−1.66375			−0.831873	−	0.554966i	$-0.812731\pi$
$919$	−50.4365			−1.66375			−0.831873	+	0.554966i	$0.812731\pi$
$920$		0			0
$921$	9.04533			0.298054
$922$		0			0
$923$			40.6319i			1.33741i
$924$		0			0
$925$			42.8679i			1.40949i
$926$		0			0
$927$	37.0990			1.21849
$928$		0			0
$929$		−	50.1862i		−	1.64656i	−0.567639	−	0.823278i	$-0.692143\pi$
$929$		−	50.1862i		−	1.64656i	0.567639	−	0.823278i	$-0.307857\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		−	61.3306i		−	2.00787i
$934$		0			0
$935$	−18.0395			−0.589955
$936$		0			0
$937$		−	1.38951i		−	0.0453932i	−0.999742	−	0.0226966i	$-0.992775\pi$
$937$		−	1.38951i		−	0.0453932i	0.999742	−	0.0226966i	$-0.00722517\pi$
$938$		0			0
$939$		−	109.126i		−	3.56121i
$940$		0			0
$941$	23.2041			0.756432			0.378216	−	0.925717i	$-0.376538\pi$
$941$	23.2041			0.756432			0.378216	+	0.925717i	$0.376538\pi$
$942$		0			0
$943$	5.23437			0.170454
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−13.0930			−0.425467			−0.212733	−	0.977110i	$-0.568237\pi$
$947$	−13.0930			−0.425467			−0.212733	+	0.977110i	$0.568237\pi$
$948$		0			0
$949$		−	50.7106i		−	1.64614i
$950$		0			0
$951$		−	72.6225i		−	2.35495i
$952$		0			0
$953$		−	51.2256i		−	1.65936i	−0.558240	−	0.829679i	$-0.688523\pi$
$953$		−	51.2256i		−	1.65936i	0.558240	−	0.829679i	$-0.311477\pi$
$954$		0			0
$955$		−	17.3811i		−	0.562439i
$956$		0			0
$957$			46.3014i			1.49671i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−20.5822			−0.663942
$962$		0			0
$963$			20.5960i			0.663695i
$964$		0			0
$965$	−59.2931			−1.90871
$966$		0			0
$967$	−50.5857			−1.62673			−0.813364	−	0.581755i	$-0.802366\pi$
$967$	−50.5857			−1.62673			−0.813364	+	0.581755i	$0.802366\pi$
$968$		0			0
$969$	10.0055	+	20.0930i	0.321422	+	0.645479i
$970$		0			0
$971$	−40.6331			−1.30398			−0.651990	−	0.758228i	$-0.726065\pi$
$971$	−40.6331			−1.30398			−0.651990	+	0.758228i	$0.726065\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−187.180			−5.99455
$976$		0			0
$977$			42.4818i			1.35911i	0.733624	+	0.679556i	$0.237827\pi$
$977$			42.4818i			1.35911i	−0.733624	+	0.679556i	$0.762173\pi$
$978$		0			0
$979$	17.6243			0.563274
$980$		0			0
$981$			16.6420i			0.531337i
$982$		0			0
$983$	47.4817			1.51443			0.757215	−	0.653166i	$-0.226560\pi$
$983$	47.4817			1.51443			0.757215	+	0.653166i	$0.226560\pi$
$984$		0			0
$985$		−	40.2024i		−	1.28096i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.02276			0.0325220
$990$		0			0
$991$		−	9.39646i		−	0.298488i	−0.988800	−	0.149244i	$-0.952316\pi$
$991$		−	9.39646i		−	0.298488i	0.988800	−	0.149244i	$-0.0476840\pi$
$992$		0			0
$993$		−	52.7092i		−	1.67268i
$994$		0			0
$995$	44.0613			1.39684
$996$		0			0
$997$		−	37.5006i		−	1.18765i	−0.804592	−	0.593827i	$-0.797616\pi$
$997$		−	37.5006i		−	1.18765i	0.804592	−	0.593827i	$-0.202384\pi$
$998$		0			0
$999$			54.6686i			1.72964i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3724.2.g.f.1861.1		16
7.2	even	3		532.2.v.e.493.8	yes	16
7.3	odd	6		532.2.v.e.341.1	✓	16
7.6	odd	2	inner	3724.2.g.f.1861.16		16
19.18	odd	2	inner	3724.2.g.f.1861.15		16
133.37	odd	6		532.2.v.e.493.1	yes	16
133.94	even	6		532.2.v.e.341.8	yes	16
133.132	even	2	inner	3724.2.g.f.1861.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.v.e.341.1	✓	16	7.3	odd	6
532.2.v.e.341.8	yes	16	133.94	even	6
532.2.v.e.493.1	yes	16	133.37	odd	6
532.2.v.e.493.8	yes	16	7.2	even	3
3724.2.g.f.1861.1		16	1.1	even	1	trivial
3724.2.g.f.1861.2		16	133.132	even	2	inner
3724.2.g.f.1861.15		16	19.18	odd	2	inner
3724.2.g.f.1861.16		16	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3724.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.22766 q^{3} -4.02258i q^{5} +7.41779 q^{9} +O(q^{10})\)	\(q-3.22766 q^{3} -4.02258i q^{5} +7.41779 q^{9} +2.81086 q^{11} -5.18661 q^{13} +12.9835i q^{15} -1.59544i q^{17} +(3.90190 - 1.94299i) q^{19} +0.810861 q^{23} -11.1812 q^{25} -14.2591 q^{27} -5.10348i q^{29} -3.22766 q^{31} -9.07250 q^{33} -3.83394i q^{37} +16.7406 q^{39} +6.45532 q^{41} +1.26133 q^{43} -29.8387i q^{45} -1.40445i q^{47} +5.14954i q^{51} -11.2270i q^{53} -11.3069i q^{55} +(-12.5940 + 6.27130i) q^{57} +11.4567 q^{59} -8.68879i q^{61} +20.8636i q^{65} -2.24352i q^{67} -2.61718 q^{69} -7.83399i q^{71} +9.77722i q^{73} +36.0890 q^{75} -9.10353i q^{79} +23.7702 q^{81} +4.14493i q^{83} -6.41779 q^{85} +16.4723i q^{87} +6.27006 q^{89} +10.4178 q^{93} +(-7.81582 - 15.6957i) q^{95} -12.8052 q^{97} +20.8504 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 52 q^{9}+O(q^{10}) \) 16 * q + 52 * q^9	\( 16 q + 52 q^{9} + 8 q^{11} - 24 q^{23} - 20 q^{25} + 24 q^{39} + 16 q^{43} - 2 q^{57} + 48 q^{81} - 36 q^{85} + 100 q^{93} - 62 q^{95} + 156 q^{99}+O(q^{100}) \) 16 * q + 52 * q^9 + 8 * q^11 - 24 * q^23 - 20 * q^25 + 24 * q^39 + 16 * q^43 - 2 * q^57 + 48 * q^81 - 36 * q^85 + 100 * q^93 - 62 * q^95 + 156 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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