LMFDB - Embedded newform 1224.4.c.e.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,4,Mod(577,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1224.577");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1224 = 2^{3} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1224.c (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:	$72.2183378470$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152 $ x^8 + 95x^6 + 756x^4 + 1780*x^2 + 1152
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.1
Root			$-1.03229i$ of defining polynomial
Character	$\chi$	$=$	1224.577
Dual form			1224.4.c.e.577.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-18.2701i q^{5} +13.8757i q^{7} +O(q^{10})$	$q-18.2701i q^{5} +13.8757i q^{7} +60.8512i q^{11} +61.8450 q^{13} +(-69.2650 - 10.7408i) q^{17} -40.0349 q^{19} +4.88830i q^{23} -208.795 q^{25} -113.141i q^{29} +95.1610i q^{31} +253.510 q^{35} -273.445i q^{37} -446.056i q^{41} -274.325 q^{43} +27.6551 q^{47} +150.464 q^{49} -488.605 q^{53} +1111.76 q^{55} -266.325 q^{59} -502.818i q^{61} -1129.91i q^{65} -1008.84 q^{67} +724.061i q^{71} +188.092i q^{73} -844.355 q^{77} +48.3382i q^{79} -1384.89 q^{83} +(-196.236 + 1265.48i) q^{85} +50.3364 q^{89} +858.144i q^{91} +731.440i q^{95} +1285.79i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 44 q^{13} - 28 q^{17} + 48 q^{19} - 520 q^{25} + 1064 q^{35} + 8 q^{43} - 312 q^{47} - 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 1660 q^{77} - 2472 q^{83} - 2160 q^{85} - 68 q^{89}+O(q^{100}) $ 8 * q + 44 * q^13 - 28 * q^17 + 48 * q^19 - 520 * q^25 + 1064 * q^35 + 8 * q^43 - 312 * q^47 - 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 1660 * q^77 - 2472 * q^83 - 2160 * q^85 - 68 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$.

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	18.2701i		−	1.63412i	−0.576550	−	0.817062i	$-0.695601\pi$
$5$		−	18.2701i		−	1.63412i	0.576550	−	0.817062i	$-0.304399\pi$
$6$		0			0
$7$			13.8757i			0.749219i	0.927183	+	0.374609i	$0.122223\pi$
$7$			13.8757i			0.749219i	−0.927183	+	0.374609i	$0.877777\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			60.8512i			1.66794i	0.551811	+	0.833969i	$0.313937\pi$
$11$			60.8512i			1.66794i	−0.551811	+	0.833969i	$0.686063\pi$
$12$		0			0
$13$	61.8450			1.31944			0.659720	−	0.751512i	$-0.270675\pi$
$13$	61.8450			1.31944			0.659720	+	0.751512i	$0.270675\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−69.2650	−	10.7408i	−0.988189	−	0.153237i
$17$	−69.2650	−	10.7408i	−0.988189	−	0.153237i
$18$		0			0
$19$	−40.0349			−0.483402			−0.241701	−	0.970351i	$-0.577705\pi$
$19$	−40.0349			−0.483402			−0.241701	+	0.970351i	$0.577705\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			4.88830i			0.0443165i	0.999754	+	0.0221583i	$0.00705377\pi$
$23$			4.88830i			0.0443165i	−0.999754	+	0.0221583i	$0.992946\pi$
$24$		0			0
$25$	−208.795			−1.67036
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	113.141i		−	0.724474i	−0.932086	−	0.362237i	$-0.882013\pi$
$29$		−	113.141i		−	0.724474i	0.932086	−	0.362237i	$-0.117987\pi$
$30$		0			0
$31$			95.1610i			0.551336i	0.961253	+	0.275668i	$0.0888991\pi$
$31$			95.1610i			0.551336i	−0.961253	+	0.275668i	$0.911101\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	253.510			1.22432
$36$		0			0
$37$		−	273.445i		−	1.21497i	−0.794330	−	0.607487i	$-0.792178\pi$
$37$		−	273.445i		−	1.21497i	0.794330	−	0.607487i	$-0.207822\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	446.056i		−	1.69908i	−0.527525	−	0.849540i	$-0.676880\pi$
$41$		−	446.056i		−	1.69908i	0.527525	−	0.849540i	$-0.323120\pi$
$42$		0			0
$43$	−274.325			−0.972888			−0.486444	−	0.873712i	$-0.661706\pi$
$43$	−274.325			−0.972888			−0.486444	+	0.873712i	$0.661706\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	27.6551			0.0858280			0.0429140	−	0.999079i	$-0.486336\pi$
$47$	27.6551			0.0858280			0.0429140	+	0.999079i	$0.486336\pi$
$48$		0			0
$49$	150.464			0.438671
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−488.605			−1.26632			−0.633161	−	0.774020i	$-0.718243\pi$
$53$	−488.605			−1.26632			−0.633161	+	0.774020i	$0.718243\pi$
$54$		0			0
$55$	1111.76			2.72562
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−266.325			−0.587671			−0.293835	−	0.955856i	$-0.594932\pi$
$59$	−266.325			−0.587671			−0.293835	+	0.955856i	$0.594932\pi$
$60$		0			0
$61$		−	502.818i		−	1.05540i	−0.849432	−	0.527699i	$-0.823055\pi$
$61$		−	502.818i		−	1.05540i	0.849432	−	0.527699i	$-0.176945\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	1129.91i		−	2.15613i
$66$		0			0
$67$	−1008.84			−1.83955			−0.919776	−	0.392443i	$-0.871630\pi$
$67$	−1008.84			−1.83955			−0.919776	+	0.392443i	$0.871630\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			724.061i			1.21029i	0.796117	+	0.605143i	$0.206884\pi$
$71$			724.061i			1.21029i	−0.796117	+	0.605143i	$0.793116\pi$
$72$		0			0
$73$			188.092i			0.301569i	0.988567	+	0.150785i	$0.0481800\pi$
$73$			188.092i			0.301569i	−0.988567	+	0.150785i	$0.951820\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−844.355			−1.24965
$78$		0			0
$79$			48.3382i			0.0688415i	0.999407	+	0.0344207i	$0.0109586\pi$
$79$			48.3382i			0.0688415i	−0.999407	+	0.0344207i	$0.989041\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1384.89			−1.83146			−0.915729	−	0.401797i	$-0.868386\pi$
$83$	−1384.89			−1.83146			−0.915729	+	0.401797i	$0.868386\pi$
$84$		0			0
$85$	−196.236	+	1265.48i	−0.250409	+	1.61482i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	50.3364			0.0599511			0.0299756	−	0.999551i	$-0.490457\pi$
$89$	50.3364			0.0599511			0.0299756	+	0.999551i	$0.490457\pi$
$90$		0			0
$91$			858.144i			0.988549i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			731.440i			0.789938i
$96$		0			0
$97$			1285.79i			1.34589i	0.739691	+	0.672947i	$0.234972\pi$
$97$			1285.79i			1.34589i	−0.739691	+	0.672947i	$0.765028\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	932.025			0.918217			0.459109	−	0.888380i	$-0.348169\pi$
$101$	932.025			0.918217			0.459109	+	0.888380i	$0.348169\pi$
$102$		0			0
$103$	−283.119			−0.270840			−0.135420	−	0.990788i	$-0.543238\pi$
$103$	−283.119			−0.270840			−0.135420	+	0.990788i	$0.543238\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	273.739i		−	0.247321i	−0.992325	−	0.123661i	$-0.960537\pi$
$107$		−	273.739i		−	0.247321i	0.992325	−	0.123661i	$-0.0394634\pi$
$108$		0			0
$109$			834.584i			0.733382i	0.930343	+	0.366691i	$0.119509\pi$
$109$			834.584i			0.733382i	−0.930343	+	0.366691i	$0.880491\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			193.136i			0.160785i	0.996763	+	0.0803927i	$0.0256174\pi$
$113$			193.136i			0.160785i	−0.996763	+	0.0803927i	$0.974383\pi$
$114$		0			0
$115$	89.3095			0.0724187
$116$		0			0
$117$		0			0
$118$		0			0
$119$	149.037	−	961.102i	0.114808	−	0.740370i
$120$		0			0
$121$	−2371.87			−1.78202
$122$		0			0
$123$		0			0
$124$		0			0
$125$			1530.94i			1.09545i
$126$		0			0
$127$	1114.62			0.778793			0.389396	−	0.921070i	$-0.372684\pi$
$127$	1114.62			0.778793			0.389396	+	0.921070i	$0.372684\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			673.762i			0.449365i	0.974432	+	0.224683i	$0.0721345\pi$
$131$			673.762i			0.449365i	−0.974432	+	0.224683i	$0.927865\pi$
$132$		0			0
$133$		−	555.513i		−	0.362174i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1213.64			−0.756849			−0.378425	−	0.925632i	$-0.623534\pi$
$137$	−1213.64			−0.756849			−0.378425	+	0.925632i	$0.623534\pi$
$138$		0			0
$139$		−	818.573i		−	0.499500i	−0.968310	−	0.249750i	$-0.919652\pi$
$139$		−	818.573i		−	0.499500i	0.968310	−	0.249750i	$-0.0803484\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			3763.34i			2.20074i
$144$		0			0
$145$	−2067.09			−1.18388
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2397.27			−1.31807			−0.659033	−	0.752114i	$-0.729034\pi$
$149$	−2397.27			−1.31807			−0.659033	+	0.752114i	$0.729034\pi$
$150$		0			0
$151$	−1197.87			−0.645573			−0.322787	−	0.946472i	$-0.604620\pi$
$151$	−1197.87			−0.645573			−0.322787	+	0.946472i	$0.604620\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1738.60			0.900952
$156$		0			0
$157$	1926.67			0.979393			0.489696	−	0.871893i	$-0.337108\pi$
$157$	1926.67			0.979393			0.489696	+	0.871893i	$0.337108\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−67.8287			−0.0332028
$162$		0			0
$163$		−	1307.62i		−	0.628347i	−0.949366	−	0.314174i	$-0.898273\pi$
$163$		−	1307.62i		−	0.628347i	0.949366	−	0.314174i	$-0.101727\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	2548.82i		−	1.18104i	−0.807023	−	0.590520i	$-0.798923\pi$
$167$		−	2548.82i		−	1.18104i	0.807023	−	0.590520i	$-0.201077\pi$
$168$		0			0
$169$	1627.80			0.740921
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	661.688i		−	0.290793i	−0.989373	−	0.145397i	$-0.953554\pi$
$173$		−	661.688i		−	0.290793i	0.989373	−	0.145397i	$-0.0464458\pi$
$174$		0			0
$175$		−	2897.18i		−	1.25147i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−2927.34			−1.22235			−0.611173	−	0.791497i	$-0.709302\pi$
$179$	−2927.34			−1.22235			−0.611173	+	0.791497i	$0.709302\pi$
$180$		0			0
$181$		−	209.349i		−	0.0859711i	−0.999076	−	0.0429856i	$-0.986313\pi$
$181$		−	209.349i		−	0.0859711i	0.999076	−	0.0429856i	$-0.0136869\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−4995.85			−1.98542
$186$		0			0
$187$	653.592	−	4214.86i	0.255590	−	1.64824i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4259.39			−1.61361			−0.806803	−	0.590820i	$-0.798804\pi$
$191$	−4259.39			−1.61361			−0.806803	+	0.590820i	$0.798804\pi$
$192$		0			0
$193$			72.8650i			0.0271758i	0.999908	+	0.0135879i	$0.00432530\pi$
$193$			72.8650i			0.0271758i	−0.999908	+	0.0135879i	$0.995675\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4200.16i			1.51903i	0.650489	+	0.759516i	$0.274564\pi$
$197$			4200.16i			1.51903i	−0.650489	+	0.759516i	$0.725436\pi$
$198$		0			0
$199$			2013.74i			0.717337i	0.933465	+	0.358669i	$0.116769\pi$
$199$			2013.74i			0.717337i	−0.933465	+	0.358669i	$0.883231\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1569.91			0.542790
$204$		0			0
$205$	−8149.47			−2.77651
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	2436.17i		−	0.806284i
$210$		0			0
$211$		−	1431.34i		−	0.467001i	−0.972357	−	0.233500i	$-0.924982\pi$
$211$		−	1431.34i		−	0.467001i	0.972357	−	0.233500i	$-0.0750180\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			5011.94i			1.58982i
$216$		0			0
$217$	−1320.43			−0.413072
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−4283.69	−	664.266i	−1.30386	−	0.202187i
$222$		0			0
$223$	1220.63			0.366544			0.183272	−	0.983062i	$-0.441331\pi$
$223$	1220.63			0.366544			0.183272	+	0.983062i	$0.441331\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	4338.39i		−	1.26850i	−0.773128	−	0.634250i	$-0.781309\pi$
$227$		−	4338.39i		−	1.26850i	0.773128	−	0.634250i	$-0.218691\pi$
$228$		0			0
$229$	−1887.81			−0.544760			−0.272380	−	0.962190i	$-0.587811\pi$
$229$	−1887.81			−0.544760			−0.272380	+	0.962190i	$0.587811\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			4991.46i			1.40344i	0.712453	+	0.701720i	$0.247584\pi$
$233$			4991.46i			1.40344i	−0.712453	+	0.701720i	$0.752416\pi$
$234$		0			0
$235$		−	505.261i		−	0.140254i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2220.06			0.600853			0.300427	−	0.953805i	$-0.402871\pi$
$239$	2220.06			0.600853			0.300427	+	0.953805i	$0.402871\pi$
$240$		0			0
$241$		−	4989.64i		−	1.33366i	−0.745212	−	0.666828i	$-0.767652\pi$
$241$		−	4989.64i		−	1.33366i	0.745212	−	0.666828i	$-0.232348\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	2748.99i		−	0.716843i
$246$		0			0
$247$	−2475.96			−0.637819
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1077.08			0.270855			0.135427	−	0.990787i	$-0.456759\pi$
$251$	1077.08			0.270855			0.135427	+	0.990787i	$0.456759\pi$
$252$		0			0
$253$	−297.459			−0.0739173
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6521.35			1.58284			0.791421	−	0.611271i	$-0.209342\pi$
$257$	6521.35			1.58284			0.791421	+	0.611271i	$0.209342\pi$
$258$		0			0
$259$	3794.24			0.910281
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7879.02			−1.84730			−0.923652	−	0.383231i	$-0.874811\pi$
$263$	−7879.02			−1.84730			−0.923652	+	0.383231i	$0.874811\pi$
$264$		0			0
$265$			8926.85i			2.06933i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			2539.91i			0.575691i	0.957677	+	0.287845i	$0.0929390\pi$
$269$			2539.91i			0.575691i	−0.957677	+	0.287845i	$0.907061\pi$
$270$		0			0
$271$	−919.402			−0.206087			−0.103044	−	0.994677i	$-0.532858\pi$
$271$	−919.402			−0.206087			−0.103044	+	0.994677i	$0.532858\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	12705.4i		−	2.78606i
$276$		0			0
$277$		−	6999.15i		−	1.51819i	−0.650981	−	0.759094i	$-0.725642\pi$
$277$		−	6999.15i		−	1.51819i	0.650981	−	0.759094i	$-0.274358\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1310.63			−0.278242			−0.139121	−	0.990275i	$-0.544428\pi$
$281$	−1310.63			−0.278242			−0.139121	+	0.990275i	$0.544428\pi$
$282$		0			0
$283$			2119.86i			0.445275i	0.974901	+	0.222638i	$0.0714666\pi$
$283$			2119.86i			0.445275i	−0.974901	+	0.222638i	$0.928533\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6189.35			1.27298
$288$		0			0
$289$	4682.27	+	1487.93i	0.953037	+	0.302855i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	4192.22			0.835877			0.417939	−	0.908475i	$-0.362753\pi$
$293$	4192.22			0.835877			0.417939	+	0.908475i	$0.362753\pi$
$294$		0			0
$295$			4865.78i			0.960327i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			302.317i			0.0584730i
$300$		0			0
$301$		−	3806.46i		−	0.728906i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9186.51			−1.72465
$306$		0			0
$307$	899.329			0.167190			0.0835952	−	0.996500i	$-0.473360\pi$
$307$	899.329			0.167190			0.0835952	+	0.996500i	$0.473360\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	8053.67i		−	1.46843i	−0.678917	−	0.734215i	$-0.737550\pi$
$311$		−	8053.67i		−	1.46843i	0.678917	−	0.734215i	$-0.262450\pi$
$312$		0			0
$313$			3940.63i			0.711622i	0.934558	+	0.355811i	$0.115795\pi$
$313$			3940.63i			0.711622i	−0.934558	+	0.355811i	$0.884205\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			5790.24i			1.02591i	0.858416	+	0.512953i	$0.171449\pi$
$317$			5790.24i			1.02591i	−0.858416	+	0.512953i	$0.828551\pi$
$318$		0			0
$319$	6884.76			1.20838
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2773.01	+	430.008i	0.477692	+	0.0740751i
$324$		0			0
$325$	−12912.9			−2.20394
$326$		0			0
$327$		0			0
$328$		0			0
$329$			383.735i			0.0643039i
$330$		0			0
$331$	−6184.41			−1.02697			−0.513484	−	0.858099i	$-0.671645\pi$
$331$	−6184.41			−1.02697			−0.513484	+	0.858099i	$0.671645\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			18431.7i			3.00606i
$336$		0			0
$337$			2895.51i			0.468038i	0.972232	+	0.234019i	$0.0751877\pi$
$337$			2895.51i			0.468038i	−0.972232	+	0.234019i	$0.924812\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−5790.66			−0.919595
$342$		0			0
$343$			6847.17i			1.07788i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	1555.79i		−	0.240690i	−0.992732	−	0.120345i	$-0.961600\pi$
$347$		−	1555.79i		−	0.240690i	0.992732	−	0.120345i	$-0.0384000\pi$
$348$		0			0
$349$	1103.98			0.169325			0.0846627	−	0.996410i	$-0.473019\pi$
$349$	1103.98			0.169325			0.0846627	+	0.996410i	$0.473019\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9622.07			−1.45080			−0.725398	−	0.688330i	$-0.758344\pi$
$353$	−9622.07			−1.45080			−0.725398	+	0.688330i	$0.758344\pi$
$354$		0			0
$355$	13228.6			1.97776
$356$		0			0
$357$		0			0
$358$		0			0
$359$	3095.00			0.455008			0.227504	−	0.973777i	$-0.426944\pi$
$359$	3095.00			0.455008			0.227504	+	0.973777i	$0.426944\pi$
$360$		0			0
$361$	−5256.21			−0.766323
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3436.46			0.492802
$366$		0			0
$367$			1805.83i			0.256849i	0.991719	+	0.128425i	$0.0409921\pi$
$367$			1805.83i			0.256849i	−0.991719	+	0.128425i	$0.959008\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	6779.75i		−	0.948752i
$372$		0			0
$373$	6265.65			0.869767			0.434884	−	0.900487i	$-0.356789\pi$
$373$	6265.65			0.869767			0.434884	+	0.900487i	$0.356789\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	6997.20i		−	0.955900i
$378$		0			0
$379$			1997.53i			0.270729i	0.990796	+	0.135364i	$0.0432205\pi$
$379$			1997.53i			0.270729i	−0.990796	+	0.135364i	$0.956779\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−13879.9			−1.85178			−0.925889	−	0.377797i	$-0.876682\pi$
$383$	−13879.9			−1.85178			−0.925889	+	0.377797i	$0.876682\pi$
$384$		0			0
$385$			15426.4i			2.04208i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7261.69			0.946483			0.473242	−	0.880933i	$-0.343084\pi$
$389$	7261.69			0.946483			0.473242	+	0.880933i	$0.343084\pi$
$390$		0			0
$391$	52.5044	−	338.588i	0.00679095	−	0.0437931i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	883.143			0.112496
$396$		0			0
$397$		−	7324.04i		−	0.925902i	−0.886384	−	0.462951i	$-0.846791\pi$
$397$		−	7324.04i		−	0.925902i	0.886384	−	0.462951i	$-0.153209\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	2017.73i		−	0.251274i	−0.992076	−	0.125637i	$-0.959903\pi$
$401$		−	2017.73i		−	0.251274i	0.992076	−	0.125637i	$-0.0400975\pi$
$402$		0			0
$403$			5885.23i			0.727455i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	16639.4			2.02650
$408$		0			0
$409$	−837.375			−0.101236			−0.0506180	−	0.998718i	$-0.516119\pi$
$409$	−837.375			−0.101236			−0.0506180	+	0.998718i	$0.516119\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	3695.45i		−	0.440294i
$414$		0			0
$415$			25301.9i			2.99283i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			6292.34i			0.733653i	0.930289	+	0.366827i	$0.119556\pi$
$419$			6292.34i			0.733653i	−0.930289	+	0.366827i	$0.880444\pi$
$420$		0			0
$421$	−3436.25			−0.397797			−0.198899	−	0.980020i	$-0.563736\pi$
$421$	−3436.25			−0.397797			−0.198899	+	0.980020i	$0.563736\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	14462.2	+	2242.63i	1.65063	+	0.255962i
$426$		0			0
$427$	6976.96			0.790724
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	1198.92i		−	0.133991i	−0.997753	−	0.0669955i	$-0.978659\pi$
$431$		−	1198.92i		−	0.133991i	0.997753	−	0.0669955i	$-0.0213413\pi$
$432$		0			0
$433$	−1921.84			−0.213297			−0.106649	−	0.994297i	$-0.534012\pi$
$433$	−1921.84			−0.213297			−0.106649	+	0.994297i	$0.534012\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	195.702i		−	0.0214227i
$438$		0			0
$439$		−	17813.5i		−	1.93666i	−0.249681	−	0.968328i	$-0.580326\pi$
$439$		−	17813.5i		−	1.93666i	0.249681	−	0.968328i	$-0.419674\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	10707.1			1.14833			0.574165	−	0.818739i	$-0.305327\pi$
$443$	10707.1			1.14833			0.574165	+	0.818739i	$0.305327\pi$
$444$		0			0
$445$		−	919.650i		−	0.0979676i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			9542.98i			1.00303i	0.865149	+	0.501516i	$0.167224\pi$
$449$			9542.98i			1.00303i	−0.865149	+	0.501516i	$0.832776\pi$
$450$		0			0
$451$	27143.0			2.83396
$452$		0			0
$453$		0			0
$454$		0			0
$455$	15678.4			1.61541
$456$		0			0
$457$	−17234.9			−1.76415			−0.882075	−	0.471109i	$-0.843854\pi$
$457$	−17234.9			−1.76415			−0.882075	+	0.471109i	$0.843854\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−11358.1			−1.14751			−0.573755	−	0.819027i	$-0.694514\pi$
$461$	−11358.1			−1.14751			−0.573755	+	0.819027i	$0.694514\pi$
$462$		0			0
$463$	5407.62			0.542793			0.271397	−	0.962468i	$-0.412514\pi$
$463$	5407.62			0.542793			0.271397	+	0.962468i	$0.412514\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5059.61			0.501350			0.250675	−	0.968071i	$-0.419347\pi$
$467$	5059.61			0.501350			0.250675	+	0.968071i	$0.419347\pi$
$468$		0			0
$469$		−	13998.5i		−	1.37823i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	16693.0i		−	1.62272i
$474$		0			0
$475$	8359.09			0.807455
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	2151.09i		−	0.205190i	−0.994723	−	0.102595i	$-0.967285\pi$
$479$		−	2151.09i		−	0.205190i	0.994723	−	0.102595i	$-0.0327146\pi$
$480$		0			0
$481$		−	16911.2i		−	1.60308i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	23491.4			2.19936
$486$		0			0
$487$		−	12035.6i		−	1.11988i	−0.828532	−	0.559942i	$-0.810823\pi$
$487$		−	12035.6i		−	1.11988i	0.828532	−	0.559942i	$-0.189177\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−16672.1			−1.53238			−0.766191	−	0.642613i	$-0.777850\pi$
$491$	−16672.1			−1.53238			−0.766191	+	0.642613i	$0.777850\pi$
$492$		0			0
$493$	−1215.23	+	7836.70i	−0.111016	+	0.715918i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−10046.9			−0.906769
$498$		0			0
$499$		−	6097.31i		−	0.547000i	−0.961872	−	0.273500i	$-0.911819\pi$
$499$		−	6097.31i		−	0.547000i	0.961872	−	0.273500i	$-0.0881814\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	4307.43i		−	0.381827i	−0.981607	−	0.190913i	$-0.938855\pi$
$503$		−	4307.43i		−	0.381827i	0.981607	−	0.190913i	$-0.0611450\pi$
$504$		0			0
$505$		−	17028.2i		−	1.50048i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12433.5			−1.08272			−0.541362	−	0.840790i	$-0.682091\pi$
$509$	−12433.5			−1.08272			−0.541362	+	0.840790i	$0.682091\pi$
$510$		0			0
$511$	−2609.92			−0.225941
$512$		0			0
$513$		0			0
$514$		0			0
$515$			5172.60i			0.442587i
$516$		0			0
$517$			1682.85i			0.143156i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			628.628i			0.0528612i	0.999651	+	0.0264306i	$0.00841410\pi$
$521$			628.628i			0.0528612i	−0.999651	+	0.0264306i	$0.991586\pi$
$522$		0			0
$523$	−10333.5			−0.863963			−0.431981	−	0.901883i	$-0.642185\pi$
$523$	−10333.5			−0.863963			−0.431981	+	0.901883i	$0.642185\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1022.11	−	6591.32i	0.0844852	−	0.544825i
$528$		0			0
$529$	12143.1			0.998036
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	27586.3i		−	2.24183i
$534$		0			0
$535$	−5001.24			−0.404154
$536$		0			0
$537$		0			0
$538$		0			0
$539$			9155.92i			0.731676i
$540$		0			0
$541$			4122.30i			0.327599i	0.986494	+	0.163800i	$0.0523751\pi$
$541$			4122.30i			0.327599i	−0.986494	+	0.163800i	$0.947625\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	15247.9			1.19844
$546$		0			0
$547$		−	21268.5i		−	1.66248i	−0.555913	−	0.831241i	$-0.687631\pi$
$547$		−	21268.5i		−	1.66248i	0.555913	−	0.831241i	$-0.312369\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			4529.58i			0.350212i
$552$		0			0
$553$	−670.728			−0.0515773
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8773.96			−0.667441			−0.333721	−	0.942672i	$-0.608304\pi$
$557$	−8773.96			−0.667441			−0.333721	+	0.942672i	$0.608304\pi$
$558$		0			0
$559$	−16965.6			−1.28367
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5507.46			0.412276			0.206138	−	0.978523i	$-0.433910\pi$
$563$	5507.46			0.412276			0.206138	+	0.978523i	$0.433910\pi$
$564$		0			0
$565$	3528.62			0.262743
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13023.4			0.959528			0.479764	−	0.877398i	$-0.340722\pi$
$569$	13023.4			0.959528			0.479764	+	0.877398i	$0.340722\pi$
$570$		0			0
$571$		−	13591.0i		−	0.996089i	−0.867152	−	0.498044i	$-0.834052\pi$
$571$		−	13591.0i		−	0.996089i	0.867152	−	0.498044i	$-0.165948\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	1020.65i		−	0.0740246i
$576$		0			0
$577$	2171.91			0.156703			0.0783517	−	0.996926i	$-0.475034\pi$
$577$	2171.91			0.156703			0.0783517	+	0.996926i	$0.475034\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	19216.3i		−	1.37216i
$582$		0			0
$583$		−	29732.2i		−	2.11215i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	25902.3			1.82130			0.910651	−	0.413176i	$-0.135581\pi$
$587$	25902.3			1.82130			0.910651	+	0.413176i	$0.135581\pi$
$588$		0			0
$589$		−	3809.76i		−	0.266517i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	24167.0			1.67356			0.836780	−	0.547539i	$-0.184435\pi$
$593$	24167.0			1.67356			0.836780	+	0.547539i	$0.184435\pi$
$594$		0			0
$595$	−17559.4	−	2722.91i	−1.20986	−	0.187611i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	14936.3			1.01883			0.509415	−	0.860521i	$-0.329862\pi$
$599$	14936.3			1.01883			0.509415	+	0.860521i	$0.329862\pi$
$600$		0			0
$601$			15669.9i			1.06354i	0.846889	+	0.531770i	$0.178473\pi$
$601$			15669.9i			1.06354i	−0.846889	+	0.531770i	$0.821527\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			43334.2i			2.91204i
$606$		0			0
$607$			16641.8i			1.11280i	0.830914	+	0.556401i	$0.187818\pi$
$607$			16641.8i			1.11280i	−0.830914	+	0.556401i	$0.812182\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1710.33			0.113245
$612$		0			0
$613$	17114.6			1.12766			0.563828	−	0.825892i	$-0.309328\pi$
$613$	17114.6			1.12766			0.563828	+	0.825892i	$0.309328\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			21652.6i			1.41280i	0.707811	+	0.706402i	$0.249683\pi$
$617$			21652.6i			1.41280i	−0.707811	+	0.706402i	$0.750317\pi$
$618$		0			0
$619$			17883.9i			1.16125i	0.814171	+	0.580625i	$0.197192\pi$
$619$			17883.9i			1.16125i	−0.814171	+	0.580625i	$0.802808\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			698.455i			0.0449165i
$624$		0			0
$625$	1871.02			0.119745
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−2937.02	+	18940.1i	−0.186179	+	1.20062i
$630$		0			0
$631$	18222.4			1.14964			0.574820	−	0.818280i	$-0.305072\pi$
$631$	18222.4			1.14964			0.574820	+	0.818280i	$0.305072\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	20364.2i		−	1.27264i
$636$		0			0
$637$	9305.46			0.578800
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	17105.1i		−	1.05399i	−0.849868	−	0.526996i	$-0.823318\pi$
$641$		−	17105.1i		−	1.05399i	0.849868	−	0.526996i	$-0.176682\pi$
$642$		0			0
$643$			155.386i			0.00953008i	0.999989	+	0.00476504i	$0.00151677\pi$
$643$			155.386i			0.00953008i	−0.999989	+	0.00476504i	$0.998483\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−17013.9			−1.03383			−0.516914	−	0.856037i	$-0.672919\pi$
$647$	−17013.9			−1.03383			−0.516914	+	0.856037i	$0.672919\pi$
$648$		0			0
$649$		−	16206.2i		−	0.980199i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			31766.8i			1.90373i	0.306524	+	0.951863i	$0.400834\pi$
$653$			31766.8i			1.90373i	−0.306524	+	0.951863i	$0.599166\pi$
$654$		0			0
$655$	12309.7			0.734318
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5545.66			−0.327812			−0.163906	−	0.986476i	$-0.552409\pi$
$659$	−5545.66			−0.327812			−0.163906	+	0.986476i	$0.552409\pi$
$660$		0			0
$661$	12808.2			0.753676			0.376838	−	0.926279i	$-0.377011\pi$
$661$	12808.2			0.753676			0.376838	+	0.926279i	$0.377011\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−10149.3			−0.591836
$666$		0			0
$667$	553.067			0.0321062
$668$		0			0
$669$		0			0
$670$		0			0
$671$	30597.1			1.76034
$672$		0			0
$673$			13921.9i			0.797399i	0.917082	+	0.398699i	$0.130538\pi$
$673$			13921.9i			0.797399i	−0.917082	+	0.398699i	$0.869462\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			15570.9i			0.883954i	0.897026	+	0.441977i	$0.145723\pi$
$677$			15570.9i			0.883954i	−0.897026	+	0.441977i	$0.854277\pi$
$678$		0			0
$679$	−17841.2			−1.00837
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	6547.76i		−	0.366827i	−0.983036	−	0.183414i	$-0.941285\pi$
$683$		−	6547.76i		−	0.366827i	0.983036	−	0.183414i	$-0.0587148\pi$
$684$		0			0
$685$			22173.3i			1.23679i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−30217.8			−1.67084
$690$		0			0
$691$			12241.5i			0.673936i	0.941516	+	0.336968i	$0.109401\pi$
$691$			12241.5i			0.673936i	−0.941516	+	0.336968i	$0.890599\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−14955.4			−0.816244
$696$		0			0
$697$	−4791.01	+	30896.1i	−0.260362	+	1.67901i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	20002.0			1.07770			0.538849	−	0.842403i	$-0.318859\pi$
$701$	20002.0			1.07770			0.538849	+	0.842403i	$0.318859\pi$
$702$		0			0
$703$			10947.3i			0.587320i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			12932.5i			0.687946i
$708$		0			0
$709$			781.933i			0.0414191i	0.999786	+	0.0207095i	$0.00659252\pi$
$709$			781.933i			0.0414191i	−0.999786	+	0.0207095i	$0.993407\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−465.175			−0.0244333
$714$		0			0
$715$	68756.5			3.59629
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	3489.63i		−	0.181003i	−0.995896	−	0.0905017i	$-0.971153\pi$
$719$		−	3489.63i		−	0.181003i	0.995896	−	0.0905017i	$-0.0288470\pi$
$720$		0			0
$721$		−	3928.48i		−	0.202919i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			23623.3i			1.21013i
$726$		0			0
$727$	−4894.00			−0.249667			−0.124834	−	0.992178i	$-0.539840\pi$
$727$	−4894.00			−0.249667			−0.124834	+	0.992178i	$0.539840\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	19001.1	+	2946.48i	0.961398	+	0.149083i
$732$		0			0
$733$	−5187.08			−0.261377			−0.130688	−	0.991423i	$-0.541719\pi$
$733$	−5187.08			−0.261377			−0.130688	+	0.991423i	$0.541719\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	61389.4i		−	3.06826i
$738$		0			0
$739$	10603.0			0.527793			0.263896	−	0.964551i	$-0.414992\pi$
$739$	10603.0			0.527793			0.263896	+	0.964551i	$0.414992\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			4784.54i			0.236242i	0.992999	+	0.118121i	$0.0376871\pi$
$743$			4784.54i			0.236242i	−0.992999	+	0.118121i	$0.962313\pi$
$744$		0			0
$745$			43798.2i			2.15388i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3798.33			0.185298
$750$		0			0
$751$		−	6361.82i		−	0.309116i	−0.987984	−	0.154558i	$-0.950605\pi$
$751$		−	6361.82i		−	0.309116i	0.987984	−	0.154558i	$-0.0493953\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			21885.2i			1.05495i
$756$		0			0
$757$	−28528.8			−1.36975			−0.684873	−	0.728662i	$-0.740142\pi$
$757$	−28528.8			−1.36975			−0.684873	+	0.728662i	$0.740142\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	18097.5			0.862069			0.431035	−	0.902335i	$-0.358149\pi$
$761$	18097.5			0.862069			0.431035	+	0.902335i	$0.358149\pi$
$762$		0			0
$763$	−11580.5			−0.549463
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−16470.9			−0.775396
$768$		0			0
$769$	7189.93			0.337159			0.168580	−	0.985688i	$-0.446082\pi$
$769$	7189.93			0.337159			0.168580	+	0.985688i	$0.446082\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5316.50			0.247375			0.123688	−	0.992321i	$-0.460528\pi$
$773$	5316.50			0.247375			0.123688	+	0.992321i	$0.460528\pi$
$774$		0			0
$775$		−	19869.2i		−	0.920931i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			17857.8i			0.821338i
$780$		0			0
$781$	−44060.0			−2.01868
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	35200.3i		−	1.60045i
$786$		0			0
$787$		−	8498.14i		−	0.384912i	−0.981306	−	0.192456i	$-0.938355\pi$
$787$		−	8498.14i		−	0.384912i	0.981306	−	0.192456i	$-0.0616453\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2679.91			−0.120463
$792$		0			0
$793$		−	31096.8i		−	1.39253i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	19009.1			0.844841			0.422420	−	0.906400i	$-0.361181\pi$
$797$	19009.1			0.844841			0.422420	+	0.906400i	$0.361181\pi$
$798$		0			0
$799$	−1915.53	−	297.039i	−0.0848143	−	0.0131520i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11445.7			−0.502999
$804$		0			0
$805$			1239.23i			0.0542575i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			16963.6i			0.737215i	0.929585	+	0.368608i	$0.120165\pi$
$809$			16963.6i			0.737215i	−0.929585	+	0.368608i	$0.879835\pi$
$810$		0			0
$811$			24943.1i			1.07999i	0.841669	+	0.539994i	$0.181574\pi$
$811$			24943.1i			1.07999i	−0.841669	+	0.539994i	$0.818426\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−23890.3			−1.02680
$816$		0			0
$817$	10982.6			0.470296
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	24317.6i		−	1.03373i	−0.856068	−	0.516863i	$-0.827100\pi$
$821$		−	24317.6i		−	1.03373i	0.856068	−	0.516863i	$-0.172900\pi$
$822$		0			0
$823$			4969.93i			0.210499i	0.994446	+	0.105250i	$0.0335642\pi$
$823$			4969.93i			0.210499i	−0.994446	+	0.105250i	$0.966436\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	13711.0i		−	0.576516i	−0.957553	−	0.288258i	$-0.906924\pi$
$827$		−	13711.0i		−	0.576516i	0.957553	−	0.288258i	$-0.0930761\pi$
$828$		0			0
$829$	35645.2			1.49338			0.746689	−	0.665173i	$-0.231642\pi$
$829$	35645.2			1.49338			0.746689	+	0.665173i	$0.231642\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−10421.9	−	1616.11i	−0.433490	−	0.0672207i
$834$		0			0
$835$	−46567.1			−1.92996
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	39704.8i		−	1.63380i	−0.576777	−	0.816902i	$-0.695690\pi$
$839$		−	39704.8i		−	1.63380i	0.576777	−	0.816902i	$-0.304310\pi$
$840$		0			0
$841$	11588.1			0.475137
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	29740.1i		−	1.21076i
$846$		0			0
$847$		−	32911.4i		−	1.33512i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1336.68			0.0538434
$852$		0			0
$853$		−	4378.77i		−	0.175763i	−0.996131	−	0.0878817i	$-0.971990\pi$
$853$		−	4378.77i		−	0.175763i	0.996131	−	0.0878817i	$-0.0280097\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			14706.1i			0.586173i	0.956086	+	0.293086i	$0.0946824\pi$
$857$			14706.1i			0.586173i	−0.956086	+	0.293086i	$0.905318\pi$
$858$		0			0
$859$	−22056.6			−0.876092			−0.438046	−	0.898953i	$-0.644329\pi$
$859$	−22056.6			−0.876092			−0.438046	+	0.898953i	$0.644329\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−13451.2			−0.530571			−0.265286	−	0.964170i	$-0.585466\pi$
$863$	−13451.2			−0.530571			−0.265286	+	0.964170i	$0.585466\pi$
$864$		0			0
$865$	−12089.1			−0.475192
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2941.44			−0.114823
$870$		0			0
$871$	−62392.0			−2.42718
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−21242.9			−0.820734
$876$		0			0
$877$		−	36821.0i		−	1.41774i	−0.705339	−	0.708870i	$-0.749205\pi$
$877$		−	36821.0i		−	1.41774i	0.705339	−	0.708870i	$-0.250795\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	20929.4i		−	0.800372i	−0.916434	−	0.400186i	$-0.868945\pi$
$881$		−	20929.4i		−	0.800372i	0.916434	−	0.400186i	$-0.131055\pi$
$882$		0			0
$883$	14362.9			0.547394			0.273697	−	0.961816i	$-0.411753\pi$
$883$	14362.9			0.547394			0.273697	+	0.961816i	$0.411753\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	15832.4i		−	0.599323i	−0.954046	−	0.299662i	$-0.903126\pi$
$887$		−	15832.4i		−	0.599323i	0.954046	−	0.299662i	$-0.0968738\pi$
$888$		0			0
$889$			15466.2i			0.583486i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1107.17			−0.0414894
$894$		0			0
$895$			53482.7i			1.99746i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	10766.6			0.399429
$900$		0			0
$901$	33843.2	+	5248.02i	1.25137	+	0.194048i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3824.82			−0.140487
$906$		0			0
$907$		−	34058.7i		−	1.24686i	−0.781880	−	0.623429i	$-0.785739\pi$
$907$		−	34058.7i		−	1.24686i	0.781880	−	0.623429i	$-0.214261\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	19317.1i		−	0.702531i	−0.936276	−	0.351265i	$-0.885752\pi$
$911$		−	19317.1i		−	0.702531i	0.936276	−	0.351265i	$-0.114248\pi$
$912$		0			0
$913$		−	84271.9i		−	3.05476i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9348.93			−0.336673
$918$		0			0
$919$	−19221.2			−0.689934			−0.344967	−	0.938615i	$-0.612110\pi$
$919$	−19221.2			−0.689934			−0.344967	+	0.938615i	$0.612110\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			44779.6i			1.59690i
$924$		0			0
$925$			57093.9i			2.02945i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	34753.9i		−	1.22738i	−0.789546	−	0.613692i	$-0.789684\pi$
$929$		−	34753.9i		−	1.22738i	0.789546	−	0.613692i	$-0.210316\pi$
$930$		0			0
$931$	−6023.81			−0.212054
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−77005.7	−	11941.2i	−2.69343	−	0.417666i
$936$		0			0
$937$	649.069			0.0226298			0.0113149	−	0.999936i	$-0.496398\pi$
$937$	649.069			0.0226298			0.0113149	+	0.999936i	$0.496398\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	13716.6i		−	0.475183i	−0.971365	−	0.237591i	$-0.923642\pi$
$941$		−	13716.6i		−	0.475183i	0.971365	−	0.237591i	$-0.0763579\pi$
$942$		0			0
$943$	2180.46			0.0752973
$944$		0			0
$945$		0			0
$946$		0			0
$947$			43131.8i			1.48004i	0.672586	+	0.740019i	$0.265184\pi$
$947$			43131.8i			1.48004i	−0.672586	+	0.740019i	$0.734816\pi$
$948$		0			0
$949$			11632.6i			0.397903i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	4974.39			0.169083			0.0845415	−	0.996420i	$-0.473057\pi$
$953$	4974.39			0.169083			0.0845415	+	0.996420i	$0.473057\pi$
$954$		0			0
$955$			77819.4i			2.63683i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	16840.2i		−	0.567046i
$960$		0			0
$961$	20735.4			0.696028
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1331.25			0.0444087
$966$		0			0
$967$	−83.3000			−0.00277016			−0.00138508	−	0.999999i	$-0.500441\pi$
$967$	−83.3000			−0.00277016			−0.00138508	+	0.999999i	$0.500441\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1577.72			0.0521438			0.0260719	−	0.999660i	$-0.491700\pi$
$971$	1577.72			0.0521438			0.0260719	+	0.999660i	$0.491700\pi$
$972$		0			0
$973$	11358.3			0.374235
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−30161.6			−0.987672			−0.493836	−	0.869555i	$-0.664406\pi$
$977$	−30161.6			−0.987672			−0.493836	+	0.869555i	$0.664406\pi$
$978$		0			0
$979$			3063.03i			0.0999948i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			3072.86i			0.0997040i	0.998757	+	0.0498520i	$0.0158750\pi$
$983$			3072.86i			0.0997040i	−0.998757	+	0.0498520i	$0.984125\pi$
$984$		0			0
$985$	76737.3			2.48229
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1340.98i		−	0.0431150i
$990$		0			0
$991$		−	22582.4i		−	0.723869i	−0.932204	−	0.361934i	$-0.882116\pi$
$991$		−	22582.4i		−	0.723869i	0.932204	−	0.361934i	$-0.117884\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	36791.1			1.17222
$996$		0			0
$997$		−	17226.0i		−	0.547195i	−0.961844	−	0.273597i	$-0.911786\pi$
$997$		−	17226.0i		−	0.547195i	0.961844	−	0.273597i	$-0.0882136\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1224.4.c.e.577.1		8
3.2	odd	2		136.4.b.b.33.7	yes	8
12.11	even	2		272.4.b.f.33.2		8
17.16	even	2	inner	1224.4.c.e.577.8		8
51.38	odd	4		2312.4.a.k.1.7		8
51.47	odd	4		2312.4.a.k.1.2		8
51.50	odd	2		136.4.b.b.33.2	✓	8
204.203	even	2		272.4.b.f.33.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.b.b.33.2	✓	8	51.50	odd	2
136.4.b.b.33.7	yes	8	3.2	odd	2
272.4.b.f.33.2		8	12.11	even	2
272.4.b.f.33.7		8	204.203	even	2
1224.4.c.e.577.1		8	1.1	even	1	trivial
1224.4.c.e.577.8		8	17.16	even	2	inner
2312.4.a.k.1.2		8	51.47	odd	4
2312.4.a.k.1.7		8	51.38	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:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1224.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-18.2701i q^{5} +13.8757i q^{7} +O(q^{10})\)	\(q-18.2701i q^{5} +13.8757i q^{7} +60.8512i q^{11} +61.8450 q^{13} +(-69.2650 - 10.7408i) q^{17} -40.0349 q^{19} +4.88830i q^{23} -208.795 q^{25} -113.141i q^{29} +95.1610i q^{31} +253.510 q^{35} -273.445i q^{37} -446.056i q^{41} -274.325 q^{43} +27.6551 q^{47} +150.464 q^{49} -488.605 q^{53} +1111.76 q^{55} -266.325 q^{59} -502.818i q^{61} -1129.91i q^{65} -1008.84 q^{67} +724.061i q^{71} +188.092i q^{73} -844.355 q^{77} +48.3382i q^{79} -1384.89 q^{83} +(-196.236 + 1265.48i) q^{85} +50.3364 q^{89} +858.144i q^{91} +731.440i q^{95} +1285.79i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 44 q^{13} - 28 q^{17} + 48 q^{19} - 520 q^{25} + 1064 q^{35} + 8 q^{43} - 312 q^{47} - 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 1660 q^{77} - 2472 q^{83} - 2160 q^{85} - 68 q^{89}+O(q^{100}) \) 8 * q + 44 * q^13 - 28 * q^17 + 48 * q^19 - 520 * q^25 + 1064 * q^35 + 8 * q^43 - 312 * q^47 - 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 1660 * q^77 - 2472 * q^83 - 2160 * q^85 - 68 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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