LMFDB - Embedded newform 1792.3.d.g.1023.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,3,Mod(1023,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1792.1023");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1792.d (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:	$48.8284633734$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1023.1
Root			$-0.581861 + 1.28897i$ of defining polynomial
Character	$\chi$	$=$	1792.1023
Dual form			1792.3.d.g.1023.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.41421i q^{3} -1.54985 q^{5} -2.64575i q^{7} -2.65685 q^{9} +O(q^{10})$	$q-3.41421i q^{3} -1.54985 q^{5} -2.64575i q^{7} -2.65685 q^{9} -4.48528i q^{11} -1.54985 q^{13} +5.29150i q^{15} +23.6569 q^{17} +24.8701i q^{19} -9.03316 q^{21} -35.2248i q^{23} -22.5980 q^{25} -21.6569i q^{27} -22.4499 q^{29} -46.7156i q^{31} -15.3137 q^{33} +4.10051i q^{35} +58.5826 q^{37} +5.29150i q^{39} +26.9706 q^{41} -17.1716i q^{43} +4.11771 q^{45} -36.1326i q^{47} -7.00000 q^{49} -80.7696i q^{51} -97.8149 q^{53} +6.95149i q^{55} +84.9117 q^{57} +61.5563i q^{59} -37.6825 q^{61} +7.02938i q^{63} +2.40202 q^{65} +33.3726i q^{67} -120.265 q^{69} -102.199i q^{71} -69.3137 q^{73} +77.1543i q^{75} -11.8669 q^{77} +38.7005i q^{79} -97.8528 q^{81} -3.61522i q^{83} -36.6645 q^{85} +76.6489i q^{87} -44.0589 q^{89} +4.10051i q^{91} -159.497 q^{93} -38.5447i q^{95} +96.1076 q^{97} +11.9167i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^9	$ 8 q + 24 q^{9} + 144 q^{17} + 136 q^{25} - 32 q^{33} + 80 q^{41} - 56 q^{49} + 272 q^{57} + 336 q^{65} - 464 q^{73} - 104 q^{81} - 624 q^{89} - 272 q^{97}+O(q^{100}) $ 8 * q + 24 * q^9 + 144 * q^17 + 136 * q^25 - 32 * q^33 + 80 * q^41 - 56 * q^49 + 272 * q^57 + 336 * q^65 - 464 * q^73 - 104 * q^81 - 624 * q^89 - 272 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 3.41421i	− 1.13807i	−0.822313	−	0.569036i	$-0.807317\pi$
$3$	− 3.41421i	− 1.13807i	0.822313	−	0.569036i	$-0.192683\pi$
$4$	0	0
$5$	−1.54985	−0.309969	−0.154985	−	0.987917i	$-0.549533\pi$
$5$	−1.54985	−0.309969	−0.154985	+	0.987917i	$0.549533\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	−2.65685	−0.295206
$10$	0	0
$11$	− 4.48528i	− 0.407753i	−0.978997	−	0.203876i	$-0.934646\pi$
$11$	− 4.48528i	− 0.407753i	0.978997	−	0.203876i	$-0.0653541\pi$
$12$	0	0
$13$	−1.54985	−0.119219	−0.0596094	−	0.998222i	$-0.518986\pi$
$13$	−1.54985	−0.119219	−0.0596094	+	0.998222i	$0.518986\pi$
$14$	0	0
$15$	5.29150i	0.352767i
$16$	0	0
$17$	23.6569	1.39158	0.695790	−	0.718245i	$-0.255055\pi$
$17$	23.6569	1.39158	0.695790	+	0.718245i	$0.255055\pi$
$18$	0	0
$19$	24.8701i	1.30895i	0.756083	+	0.654475i	$0.227110\pi$
$19$	24.8701i	1.30895i	−0.756083	+	0.654475i	$0.772890\pi$
$20$	0	0
$21$	−9.03316	−0.430150
$22$	0	0
$23$	− 35.2248i	− 1.53151i	−0.643132	−	0.765756i	$-0.722365\pi$
$23$	− 35.2248i	− 1.53151i	0.643132	−	0.765756i	$-0.277635\pi$
$24$	0	0
$25$	−22.5980	−0.903919
$26$	0	0
$27$	− 21.6569i	− 0.802106i
$28$	0	0
$29$	−22.4499	−0.774136	−0.387068	−	0.922051i	$-0.626512\pi$
$29$	−22.4499	−0.774136	−0.387068	+	0.922051i	$0.626512\pi$
$30$	0	0
$31$	− 46.7156i	− 1.50696i	−0.657473	−	0.753478i	$-0.728375\pi$
$31$	− 46.7156i	− 1.50696i	0.657473	−	0.753478i	$-0.271625\pi$
$32$	0	0
$33$	−15.3137	−0.464052
$34$	0	0
$35$	4.10051i	0.117157i
$36$	0	0
$37$	58.5826	1.58331	0.791657	−	0.610966i	$-0.209219\pi$
$37$	58.5826	1.58331	0.791657	+	0.610966i	$0.209219\pi$
$38$	0	0
$39$	5.29150i	0.135680i
$40$	0	0
$41$	26.9706	0.657819	0.328909	−	0.944362i	$-0.393319\pi$
$41$	26.9706	0.657819	0.328909	+	0.944362i	$0.393319\pi$
$42$	0	0
$43$	− 17.1716i	− 0.399339i	−0.979863	−	0.199669i	$-0.936013\pi$
$43$	− 17.1716i	− 0.399339i	0.979863	−	0.199669i	$-0.0639868\pi$
$44$	0	0
$45$	4.11771	0.0915047
$46$	0	0
$47$	− 36.1326i	− 0.768780i	−0.923171	−	0.384390i	$-0.874412\pi$
$47$	− 36.1326i	− 0.768780i	0.923171	−	0.384390i	$-0.125588\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	− 80.7696i	− 1.58372i
$52$	0	0
$53$	−97.8149	−1.84556	−0.922782	−	0.385322i	$-0.874090\pi$
$53$	−97.8149	−1.84556	−0.922782	+	0.385322i	$0.874090\pi$
$54$	0	0
$55$	6.95149i	0.126391i
$56$	0	0
$57$	84.9117	1.48968
$58$	0	0
$59$	61.5563i	1.04333i	0.853151	+	0.521664i	$0.174689\pi$
$59$	61.5563i	1.04333i	−0.853151	+	0.521664i	$0.825311\pi$
$60$	0	0
$61$	−37.6825	−0.617746	−0.308873	−	0.951103i	$-0.599952\pi$
$61$	−37.6825	−0.617746	−0.308873	+	0.951103i	$0.599952\pi$
$62$	0	0
$63$	7.02938i	0.111577i
$64$	0	0
$65$	2.40202	0.0369542
$66$	0	0
$67$	33.3726i	0.498098i	0.968491	+	0.249049i	$0.0801181\pi$
$67$	33.3726i	0.498098i	−0.968491	+	0.249049i	$0.919882\pi$
$68$	0	0
$69$	−120.265	−1.74297
$70$	0	0
$71$	− 102.199i	− 1.43942i	−0.694277	−	0.719708i	$-0.744276\pi$
$71$	− 102.199i	− 1.43942i	0.694277	−	0.719708i	$-0.255724\pi$
$72$	0	0
$73$	−69.3137	−0.949503	−0.474751	−	0.880120i	$-0.657462\pi$
$73$	−69.3137	−0.949503	−0.474751	+	0.880120i	$0.657462\pi$
$74$	0	0
$75$	77.1543i	1.02872i
$76$	0	0
$77$	−11.8669	−0.154116
$78$	0	0
$79$	38.7005i	0.489880i	0.969538	+	0.244940i	$0.0787682\pi$
$79$	38.7005i	0.489880i	−0.969538	+	0.244940i	$0.921232\pi$
$80$	0	0
$81$	−97.8528	−1.20806
$82$	0	0
$83$	− 3.61522i	− 0.0435569i	−0.999763	−	0.0217785i	$-0.993067\pi$
$83$	− 3.61522i	− 0.0435569i	0.999763	−	0.0217785i	$-0.00693285\pi$
$84$	0	0
$85$	−36.6645	−0.431347
$86$	0	0
$87$	76.6489i	0.881022i
$88$	0	0
$89$	−44.0589	−0.495044	−0.247522	−	0.968882i	$-0.579616\pi$
$89$	−44.0589	−0.495044	−0.247522	+	0.968882i	$0.579616\pi$
$90$	0	0
$91$	4.10051i	0.0450605i
$92$	0	0
$93$	−159.497	−1.71502
$94$	0	0
$95$	− 38.5447i	− 0.405734i
$96$	0	0
$97$	96.1076	0.990800	0.495400	−	0.868665i	$-0.335021\pi$
$97$	96.1076	0.990800	0.495400	+	0.868665i	$0.335021\pi$
$98$	0	0
$99$	11.9167i	0.120371i
$100$	0	0
$101$	−19.6162	−0.194219	−0.0971097	−	0.995274i	$-0.530960\pi$
$101$	−19.6162	−0.194219	−0.0971097	+	0.995274i	$0.530960\pi$
$102$	0	0
$103$	− 43.0841i	− 0.418293i	−0.977884	−	0.209146i	$-0.932932\pi$
$103$	− 43.0841i	− 0.418293i	0.977884	−	0.209146i	$-0.0670685\pi$
$104$	0	0
$105$	14.0000	0.133333
$106$	0	0
$107$	15.5980i	0.145776i	0.997340	+	0.0728878i	$0.0232215\pi$
$107$	15.5980i	0.145776i	−0.997340	+	0.0728878i	$0.976779\pi$
$108$	0	0
$109$	3.85180	0.0353376	0.0176688	−	0.999844i	$-0.494376\pi$
$109$	3.85180	0.0353376	0.0176688	+	0.999844i	$0.494376\pi$
$110$	0	0
$111$	− 200.013i	− 1.80192i
$112$	0	0
$113$	−13.7746	−0.121899	−0.0609496	−	0.998141i	$-0.519413\pi$
$113$	−13.7746	−0.121899	−0.0609496	+	0.998141i	$0.519413\pi$
$114$	0	0
$115$	54.5929i	0.474721i
$116$	0	0
$117$	4.11771	0.0351941
$118$	0	0
$119$	− 62.5902i	− 0.525968i
$120$	0	0
$121$	100.882	0.833738
$122$	0	0
$123$	− 92.0833i	− 0.748644i
$124$	0	0
$125$	73.7695	0.590156
$126$	0	0
$127$	125.025i	0.984445i	0.870469	+	0.492223i	$0.163815\pi$
$127$	125.025i	0.984445i	−0.870469	+	0.492223i	$0.836185\pi$
$128$	0	0
$129$	−58.6274	−0.454476
$130$	0	0
$131$	− 100.350i	− 0.766033i	−0.923742	−	0.383016i	$-0.874885\pi$
$131$	− 100.350i	− 0.766033i	0.923742	−	0.383016i	$-0.125115\pi$
$132$	0	0
$133$	65.8000	0.494737
$134$	0	0
$135$	33.5648i	0.248628i
$136$	0	0
$137$	−57.3137	−0.418348	−0.209174	−	0.977878i	$-0.567078\pi$
$137$	−57.3137	−0.418348	−0.209174	+	0.977878i	$0.567078\pi$
$138$	0	0
$139$	− 183.664i	− 1.32132i	−0.750684	−	0.660662i	$-0.770276\pi$
$139$	− 183.664i	− 1.32132i	0.750684	−	0.660662i	$-0.229724\pi$
$140$	0	0
$141$	−123.365	−0.874926
$142$	0	0
$143$	6.95149i	0.0486118i
$144$	0	0
$145$	34.7939	0.239958
$146$	0	0
$147$	23.8995i	0.162582i
$148$	0	0
$149$	−192.310	−1.29067	−0.645335	−	0.763900i	$-0.723282\pi$
$149$	−192.310	−1.29067	−0.645335	+	0.763900i	$0.723282\pi$
$150$	0	0
$151$	114.753i	0.759954i	0.924996	+	0.379977i	$0.124068\pi$
$151$	114.753i	0.759954i	−0.924996	+	0.379977i	$0.875932\pi$
$152$	0	0
$153$	−62.8528	−0.410803
$154$	0	0
$155$	72.4020i	0.467110i
$156$	0	0
$157$	−212.146	−1.35125	−0.675625	−	0.737245i	$-0.736126\pi$
$157$	−212.146	−1.35125	−0.675625	+	0.737245i	$0.736126\pi$
$158$	0	0
$159$	333.961i	2.10038i
$160$	0	0
$161$	−93.1960	−0.578857
$162$	0	0
$163$	240.534i	1.47567i	0.674982	+	0.737835i	$0.264152\pi$
$163$	240.534i	1.47567i	−0.674982	+	0.737835i	$0.735848\pi$
$164$	0	0
$165$	23.7339	0.143842
$166$	0	0
$167$	212.101i	1.27006i	0.772486	+	0.635032i	$0.219013\pi$
$167$	212.101i	1.27006i	−0.772486	+	0.635032i	$0.780987\pi$
$168$	0	0
$169$	−166.598	−0.985787
$170$	0	0
$171$	− 66.0761i	− 0.386410i
$172$	0	0
$173$	−182.213	−1.05325	−0.526627	−	0.850096i	$-0.676544\pi$
$173$	−182.213	−1.05325	−0.526627	+	0.850096i	$0.676544\pi$
$174$	0	0
$175$	59.7886i	0.341649i
$176$	0	0
$177$	210.167	1.18738
$178$	0	0
$179$	− 57.2061i	− 0.319587i	−0.987150	−	0.159793i	$-0.948917\pi$
$179$	− 57.2061i	− 0.319587i	0.987150	−	0.159793i	$-0.0510828\pi$
$180$	0	0
$181$	−326.212	−1.80228	−0.901138	−	0.433533i	$-0.857267\pi$
$181$	−326.212	−1.80228	−0.901138	+	0.433533i	$0.857267\pi$
$182$	0	0
$183$	128.656i	0.703039i
$184$	0	0
$185$	−90.7939	−0.490778
$186$	0	0
$187$	− 106.108i	− 0.567421i
$188$	0	0
$189$	−57.2987	−0.303167
$190$	0	0
$191$	97.0628i	0.508182i	0.967180	+	0.254091i	$0.0817763\pi$
$191$	97.0628i	0.508182i	−0.967180	+	0.254091i	$0.918224\pi$
$192$	0	0
$193$	157.304	0.815045	0.407522	−	0.913195i	$-0.366393\pi$
$193$	157.304	0.815045	0.407522	+	0.913195i	$0.366393\pi$
$194$	0	0
$195$	− 8.20101i	− 0.0420565i
$196$	0	0
$197$	124.117	0.630034	0.315017	−	0.949086i	$-0.397990\pi$
$197$	124.117	0.630034	0.315017	+	0.949086i	$0.397990\pi$
$198$	0	0
$199$	− 180.975i	− 0.909421i	−0.890639	−	0.454710i	$-0.849743\pi$
$199$	− 180.975i	− 0.909421i	0.890639	−	0.454710i	$-0.150257\pi$
$200$	0	0
$201$	113.941	0.566871
$202$	0	0
$203$	59.3970i	0.292596i
$204$	0	0
$205$	−41.8002	−0.203903
$206$	0	0
$207$	93.5871i	0.452111i
$208$	0	0
$209$	111.549	0.533728
$210$	0	0
$211$	− 164.049i	− 0.777482i	−0.921347	−	0.388741i	$-0.872910\pi$
$211$	− 164.049i	− 0.777482i	0.921347	−	0.388741i	$-0.127090\pi$
$212$	0	0
$213$	−348.928	−1.63816
$214$	0	0
$215$	26.6133i	0.123783i
$216$	0	0
$217$	−123.598	−0.569576
$218$	0	0
$219$	236.652i	1.08060i
$220$	0	0
$221$	−36.6645	−0.165903
$222$	0	0
$223$	10.5830i	0.0474574i	0.999718	+	0.0237287i	$0.00755379\pi$
$223$	10.5830i	0.0474574i	−0.999718	+	0.0237287i	$0.992446\pi$
$224$	0	0
$225$	60.0395	0.266842
$226$	0	0
$227$	− 105.806i	− 0.466106i	−0.972464	−	0.233053i	$-0.925128\pi$
$227$	− 105.806i	− 0.466106i	0.972464	−	0.233053i	$-0.0748716\pi$
$228$	0	0
$229$	−74.8788	−0.326982	−0.163491	−	0.986545i	$-0.552275\pi$
$229$	−74.8788	−0.326982	−0.163491	+	0.986545i	$0.552275\pi$
$230$	0	0
$231$	40.5163i	0.175395i
$232$	0	0
$233$	419.137	1.79887	0.899436	−	0.437053i	$-0.143978\pi$
$233$	419.137	1.79887	0.899436	+	0.437053i	$0.143978\pi$
$234$	0	0
$235$	56.0000i	0.238298i
$236$	0	0
$237$	132.132	0.557518
$238$	0	0
$239$	− 148.318i	− 0.620577i	−0.950642	−	0.310288i	$-0.899574\pi$
$239$	− 148.318i	− 0.620577i	0.950642	−	0.310288i	$-0.100426\pi$
$240$	0	0
$241$	−459.872	−1.90818	−0.954092	−	0.299515i	$-0.903175\pi$
$241$	−459.872	−1.90818	−0.954092	+	0.299515i	$0.903175\pi$
$242$	0	0
$243$	139.179i	0.572752i
$244$	0	0
$245$	10.8489	0.0442813
$246$	0	0
$247$	− 38.5447i	− 0.156052i
$248$	0	0
$249$	−12.3431	−0.0495709
$250$	0	0
$251$	124.919i	0.497685i	0.968544	+	0.248842i	$0.0800501\pi$
$251$	124.919i	0.497685i	−0.968544	+	0.248842i	$0.919950\pi$
$252$	0	0
$253$	−157.993	−0.624478
$254$	0	0
$255$	125.180i	0.490903i
$256$	0	0
$257$	−427.352	−1.66285	−0.831425	−	0.555637i	$-0.812474\pi$
$257$	−427.352	−1.66285	−0.831425	+	0.555637i	$0.812474\pi$
$258$	0	0
$259$	− 154.995i	− 0.598436i
$260$	0	0
$261$	59.6462	0.228530
$262$	0	0
$263$	257.624i	0.979558i	0.871847	+	0.489779i	$0.162922\pi$
$263$	257.624i	0.979558i	−0.871847	+	0.489779i	$0.837078\pi$
$264$	0	0
$265$	151.598	0.572068
$266$	0	0
$267$	150.426i	0.563395i
$268$	0	0
$269$	215.246	0.800171	0.400085	−	0.916478i	$-0.368980\pi$
$269$	215.246	0.800171	0.400085	+	0.916478i	$0.368980\pi$
$270$	0	0
$271$	− 378.549i	− 1.39686i	−0.715678	−	0.698431i	$-0.753882\pi$
$271$	− 378.549i	− 1.39686i	0.715678	−	0.698431i	$-0.246118\pi$
$272$	0	0
$273$	14.0000	0.0512821
$274$	0	0
$275$	101.358i	0.368576i
$276$	0	0
$277$	−166.449	−0.600898	−0.300449	−	0.953798i	$-0.597136\pi$
$277$	−166.449	−0.600898	−0.300449	+	0.953798i	$0.597136\pi$
$278$	0	0
$279$	124.117i	0.444863i
$280$	0	0
$281$	421.765	1.50094	0.750471	−	0.660904i	$-0.229827\pi$
$281$	421.765	1.50094	0.750471	+	0.660904i	$0.229827\pi$
$282$	0	0
$283$	− 345.439i	− 1.22063i	−0.792158	−	0.610316i	$-0.791043\pi$
$283$	− 345.439i	− 1.22063i	0.792158	−	0.610316i	$-0.208957\pi$
$284$	0	0
$285$	−131.600	−0.461754
$286$	0	0
$287$	− 71.3574i	− 0.248632i
$288$	0	0
$289$	270.647	0.936494
$290$	0	0
$291$	− 328.132i	− 1.12760i
$292$	0	0
$293$	511.038	1.74416	0.872079	−	0.489365i	$-0.162771\pi$
$293$	511.038	1.74416	0.872079	+	0.489365i	$0.162771\pi$
$294$	0	0
$295$	− 95.4028i	− 0.323399i
$296$	0	0
$297$	−97.1371	−0.327061
$298$	0	0
$299$	54.5929i	0.182585i
$300$	0	0
$301$	−45.4317	−0.150936
$302$	0	0
$303$	66.9738i	0.221036i
$304$	0	0
$305$	58.4020	0.191482
$306$	0	0
$307$	− 223.331i	− 0.727462i	−0.931504	−	0.363731i	$-0.881503\pi$
$307$	− 223.331i	− 0.727462i	0.931504	−	0.363731i	$-0.118497\pi$
$308$	0	0
$309$	−147.098	−0.476047
$310$	0	0
$311$	− 12.3988i	− 0.0398674i	−0.999801	−	0.0199337i	$-0.993654\pi$
$311$	− 12.3988i	− 0.0398674i	0.999801	−	0.0199337i	$-0.00634551\pi$
$312$	0	0
$313$	−410.049	−1.31006	−0.655030	−	0.755603i	$-0.727344\pi$
$313$	−410.049	−1.31006	−0.655030	+	0.755603i	$0.727344\pi$
$314$	0	0
$315$	− 10.8944i	− 0.0345855i
$316$	0	0
$317$	130.316	0.411092	0.205546	−	0.978647i	$-0.434103\pi$
$317$	130.316	0.411092	0.205546	+	0.978647i	$0.434103\pi$
$318$	0	0
$319$	100.694i	0.315656i
$320$	0	0
$321$	53.2548	0.165903
$322$	0	0
$323$	588.347i	1.82151i
$324$	0	0
$325$	35.0234	0.107764
$326$	0	0
$327$	− 13.1509i	− 0.0402167i
$328$	0	0
$329$	−95.5980	−0.290571
$330$	0	0
$331$	214.260i	0.647311i	0.946175	+	0.323655i	$0.104912\pi$
$331$	214.260i	0.647311i	−0.946175	+	0.323655i	$0.895088\pi$
$332$	0	0
$333$	−155.645	−0.467404
$334$	0	0
$335$	− 51.7223i	− 0.154395i
$336$	0	0
$337$	164.049	0.486792	0.243396	−	0.969927i	$-0.421739\pi$
$337$	164.049	0.486792	0.243396	+	0.969927i	$0.421739\pi$
$338$	0	0
$339$	47.0294i	0.138730i
$340$	0	0
$341$	−209.533	−0.614466
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	186.392	0.540266
$346$	0	0
$347$	− 109.691i	− 0.316113i	−0.987430	−	0.158057i	$-0.949477\pi$
$347$	− 109.691i	− 0.316113i	0.987430	−	0.158057i	$-0.0505229\pi$
$348$	0	0
$349$	−463.479	−1.32802	−0.664010	−	0.747723i	$-0.731147\pi$
$349$	−463.479	−1.32802	−0.664010	+	0.747723i	$0.731147\pi$
$350$	0	0
$351$	33.5648i	0.0956261i
$352$	0	0
$353$	78.0975	0.221240	0.110620	−	0.993863i	$-0.464716\pi$
$353$	78.0975	0.221240	0.110620	+	0.993863i	$0.464716\pi$
$354$	0	0
$355$	158.392i	0.446174i
$356$	0	0
$357$	−213.696	−0.598589
$358$	0	0
$359$	− 365.114i	− 1.01703i	−0.861053	−	0.508515i	$-0.830195\pi$
$359$	− 365.114i	− 1.01703i	0.861053	−	0.508515i	$-0.169805\pi$
$360$	0	0
$361$	−257.520	−0.713351
$362$	0	0
$363$	− 344.434i	− 0.948853i
$364$	0	0
$365$	107.426	0.294316
$366$	0	0
$367$	− 220.739i	− 0.601468i	−0.953708	−	0.300734i	$-0.902768\pi$
$367$	− 220.739i	− 0.601468i	0.953708	−	0.300734i	$-0.0972317\pi$
$368$	0	0
$369$	−71.6569	−0.194192
$370$	0	0
$371$	258.794i	0.697558i
$372$	0	0
$373$	251.553	0.674406	0.337203	−	0.941432i	$-0.390519\pi$
$373$	251.553	0.674406	0.337203	+	0.941432i	$0.390519\pi$
$374$	0	0
$375$	− 251.865i	− 0.671640i
$376$	0	0
$377$	34.7939	0.0922916
$378$	0	0
$379$	286.024i	0.754682i	0.926074	+	0.377341i	$0.123161\pi$
$379$	286.024i	0.754682i	−0.926074	+	0.377341i	$0.876839\pi$
$380$	0	0
$381$	426.860	1.12037
$382$	0	0
$383$	106.894i	0.279096i	0.990215	+	0.139548i	$0.0445649\pi$
$383$	106.894i	0.279096i	−0.990215	+	0.139548i	$0.955435\pi$
$384$	0	0
$385$	18.3919	0.0477712
$386$	0	0
$387$	45.6224i	0.117887i
$388$	0	0
$389$	77.1807	0.198408	0.0992040	−	0.995067i	$-0.468370\pi$
$389$	77.1807	0.198408	0.0992040	+	0.995067i	$0.468370\pi$
$390$	0	0
$391$	− 833.307i	− 2.13122i
$392$	0	0
$393$	−342.617	−0.871800
$394$	0	0
$395$	− 59.9798i	− 0.151848i
$396$	0	0
$397$	657.514	1.65621	0.828103	−	0.560576i	$-0.189420\pi$
$397$	657.514	1.65621	0.828103	+	0.560576i	$0.189420\pi$
$398$	0	0
$399$	− 224.655i	− 0.563046i
$400$	0	0
$401$	318.794	0.794997	0.397499	−	0.917603i	$-0.369878\pi$
$401$	318.794	0.794997	0.397499	+	0.917603i	$0.369878\pi$
$402$	0	0
$403$	72.4020i	0.179658i
$404$	0	0
$405$	151.657	0.374461
$406$	0	0
$407$	− 262.759i	− 0.645600i
$408$	0	0
$409$	−145.265	−0.355171	−0.177585	−	0.984105i	$-0.556829\pi$
$409$	−145.265	−0.355171	−0.177585	+	0.984105i	$0.556829\pi$
$410$	0	0
$411$	195.681i	0.476110i
$412$	0	0
$413$	162.863	0.394341
$414$	0	0
$415$	5.60304i	0.0135013i
$416$	0	0
$417$	−627.068	−1.50376
$418$	0	0
$419$	− 707.012i	− 1.68738i	−0.536831	−	0.843690i	$-0.680379\pi$
$419$	− 707.012i	− 1.68738i	0.536831	−	0.843690i	$-0.319621\pi$
$420$	0	0
$421$	−121.989	−0.289761	−0.144880	−	0.989449i	$-0.546280\pi$
$421$	−121.989	−0.289761	−0.144880	+	0.989449i	$0.546280\pi$
$422$	0	0
$423$	95.9992i	0.226948i
$424$	0	0
$425$	−534.597	−1.25788
$426$	0	0
$427$	99.6985i	0.233486i
$428$	0	0
$429$	23.7339	0.0553237
$430$	0	0
$431$	− 588.861i	− 1.36627i	−0.730294	−	0.683133i	$-0.760617\pi$
$431$	− 588.861i	− 1.36627i	0.730294	−	0.683133i	$-0.239383\pi$
$432$	0	0
$433$	137.696	0.318004	0.159002	−	0.987278i	$-0.449172\pi$
$433$	137.696	0.318004	0.159002	+	0.987278i	$0.449172\pi$
$434$	0	0
$435$	− 118.794i	− 0.273090i
$436$	0	0
$437$	876.042	2.00467
$438$	0	0
$439$	440.543i	1.00352i	0.865008	+	0.501758i	$0.167313\pi$
$439$	440.543i	1.00352i	−0.865008	+	0.501758i	$0.832687\pi$
$440$	0	0
$441$	18.5980	0.0421723
$442$	0	0
$443$	− 487.058i	− 1.09945i	−0.835344	−	0.549727i	$-0.814732\pi$
$443$	− 487.058i	− 1.09945i	0.835344	−	0.549727i	$-0.185268\pi$
$444$	0	0
$445$	68.2844	0.153448
$446$	0	0
$447$	656.587i	1.46887i
$448$	0	0
$449$	264.039	0.588059	0.294030	−	0.955796i	$-0.405004\pi$
$449$	264.039	0.588059	0.294030	+	0.955796i	$0.405004\pi$
$450$	0	0
$451$	− 120.971i	− 0.268227i
$452$	0	0
$453$	391.791	0.864882
$454$	0	0
$455$	− 6.35515i	− 0.0139674i
$456$	0	0
$457$	514.323	1.12543	0.562717	−	0.826650i	$-0.309756\pi$
$457$	514.323	1.12543	0.562717	+	0.826650i	$0.309756\pi$
$458$	0	0
$459$	− 512.333i	− 1.11619i
$460$	0	0
$461$	−202.224	−0.438664	−0.219332	−	0.975650i	$-0.570388\pi$
$461$	−202.224	−0.438664	−0.219332	+	0.975650i	$0.570388\pi$
$462$	0	0
$463$	− 722.653i	− 1.56081i	−0.625277	−	0.780403i	$-0.715014\pi$
$463$	− 722.653i	− 1.56081i	0.625277	−	0.780403i	$-0.284986\pi$
$464$	0	0
$465$	247.196	0.531604
$466$	0	0
$467$	347.282i	0.743645i	0.928304	+	0.371822i	$0.121267\pi$
$467$	347.282i	0.743645i	−0.928304	+	0.371822i	$0.878733\pi$
$468$	0	0
$469$	88.2956	0.188263
$470$	0	0
$471$	724.313i	1.53782i
$472$	0	0
$473$	−77.0193	−0.162832
$474$	0	0
$475$	− 562.013i	− 1.18319i
$476$	0	0
$477$	259.880	0.544822
$478$	0	0
$479$	29.1811i	0.0609210i	0.999536	+	0.0304605i	$0.00969737\pi$
$479$	29.1811i	0.0609210i	−0.999536	+	0.0304605i	$0.990303\pi$
$480$	0	0
$481$	−90.7939	−0.188761
$482$	0	0
$483$	318.191i	0.658780i
$484$	0	0
$485$	−148.952	−0.307117
$486$	0	0
$487$	− 701.643i	− 1.44074i	−0.693588	−	0.720372i	$-0.743971\pi$
$487$	− 701.643i	− 1.44074i	0.693588	−	0.720372i	$-0.256029\pi$
$488$	0	0
$489$	821.235	1.67942
$490$	0	0
$491$	− 59.9512i	− 0.122100i	−0.998135	−	0.0610501i	$-0.980555\pi$
$491$	− 59.9512i	− 0.122100i	0.998135	−	0.0610501i	$-0.0194450\pi$
$492$	0	0
$493$	−531.095	−1.07727
$494$	0	0
$495$	− 18.4691i	− 0.0373113i
$496$	0	0
$497$	−270.392	−0.544048
$498$	0	0
$499$	84.2843i	0.168906i	0.996427	+	0.0844532i	$0.0269143\pi$
$499$	84.2843i	0.168906i	−0.996427	+	0.0844532i	$0.973086\pi$
$500$	0	0
$501$	724.157	1.44542
$502$	0	0
$503$	− 409.987i	− 0.815083i	−0.913187	−	0.407542i	$-0.866386\pi$
$503$	− 409.987i	− 0.815083i	0.913187	−	0.407542i	$-0.133614\pi$
$504$	0	0
$505$	30.4020	0.0602020
$506$	0	0
$507$	568.801i	1.12190i
$508$	0	0
$509$	477.033	0.937196	0.468598	−	0.883411i	$-0.344759\pi$
$509$	477.033	0.937196	0.468598	+	0.883411i	$0.344759\pi$
$510$	0	0
$511$	183.387i	0.358878i
$512$	0	0
$513$	538.607	1.04992
$514$	0	0
$515$	66.7737i	0.129658i
$516$	0	0
$517$	−162.065	−0.313472
$518$	0	0
$519$	622.114i	1.19868i
$520$	0	0
$521$	−210.873	−0.404747	−0.202373	−	0.979308i	$-0.564865\pi$
$521$	−210.873	−0.404747	−0.202373	+	0.979308i	$0.564865\pi$
$522$	0	0
$523$	511.566i	0.978139i	0.872245	+	0.489069i	$0.162663\pi$
$523$	511.566i	0.978139i	−0.872245	+	0.489069i	$0.837337\pi$
$524$	0	0
$525$	204.131	0.388821
$526$	0	0
$527$	− 1105.15i	− 2.09705i
$528$	0	0
$529$	−711.784	−1.34553
$530$	0	0
$531$	− 163.546i	− 0.307997i
$532$	0	0
$533$	−41.8002	−0.0784244
$534$	0	0
$535$	− 24.1745i	− 0.0451859i
$536$	0	0
$537$	−195.314	−0.363713
$538$	0	0
$539$	31.3970i	0.0582504i
$540$	0	0
$541$	342.417	0.632933	0.316466	−	0.948604i	$-0.397504\pi$
$541$	342.417	0.632933	0.316466	+	0.948604i	$0.397504\pi$
$542$	0	0
$543$	1113.76i	2.05112i
$544$	0	0
$545$	−5.96970	−0.0109536
$546$	0	0
$547$	441.976i	0.807999i	0.914759	+	0.404000i	$0.132380\pi$
$547$	441.976i	0.807999i	−0.914759	+	0.404000i	$0.867620\pi$
$548$	0	0
$549$	100.117	0.182362
$550$	0	0
$551$	− 558.331i	− 1.01331i
$552$	0	0
$553$	102.392	0.185157
$554$	0	0
$555$	309.990i	0.558540i
$556$	0	0
$557$	365.710	0.656571	0.328285	−	0.944579i	$-0.393529\pi$
$557$	365.710	0.656571	0.328285	+	0.944579i	$0.393529\pi$
$558$	0	0
$559$	26.6133i	0.0476087i
$560$	0	0
$561$	−362.274	−0.645765
$562$	0	0
$563$	− 806.389i	− 1.43231i	−0.697943	−	0.716154i	$-0.745901\pi$
$563$	− 806.389i	− 1.43231i	0.697943	−	0.716154i	$-0.254099\pi$
$564$	0	0
$565$	21.3485	0.0377850
$566$	0	0
$567$	258.894i	0.456604i
$568$	0	0
$569$	−222.891	−0.391725	−0.195862	−	0.980631i	$-0.562751\pi$
$569$	−222.891	−0.391725	−0.195862	+	0.980631i	$0.562751\pi$
$570$	0	0
$571$	573.082i	1.00365i	0.864970	+	0.501823i	$0.167337\pi$
$571$	573.082i	1.00365i	−0.864970	+	0.501823i	$0.832663\pi$
$572$	0	0
$573$	331.393	0.578348
$574$	0	0
$575$	796.008i	1.38436i
$576$	0	0
$577$	−723.901	−1.25459	−0.627297	−	0.778780i	$-0.715839\pi$
$577$	−723.901	−1.25459	−0.627297	+	0.778780i	$0.715839\pi$
$578$	0	0
$579$	− 537.068i	− 0.927579i
$580$	0	0
$581$	−9.56498	−0.0164630
$582$	0	0
$583$	438.727i	0.752534i
$584$	0	0
$585$	−6.38182	−0.0109091
$586$	0	0
$587$	21.1198i	0.0359793i	0.999838	+	0.0179896i	$0.00572659\pi$
$587$	21.1198i	0.0359793i	−0.999838	+	0.0179896i	$0.994273\pi$
$588$	0	0
$589$	1161.82	1.97253
$590$	0	0
$591$	− 423.761i	− 0.717023i
$592$	0	0
$593$	−128.745	−0.217108	−0.108554	−	0.994091i	$-0.534622\pi$
$593$	−128.745	−0.217108	−0.108554	+	0.994091i	$0.534622\pi$
$594$	0	0
$595$	97.0051i	0.163034i
$596$	0	0
$597$	−617.886	−1.03499
$598$	0	0
$599$	324.130i	0.541119i	0.962703	+	0.270559i	$0.0872086\pi$
$599$	324.130i	0.541119i	−0.962703	+	0.270559i	$0.912791\pi$
$600$	0	0
$601$	721.862	1.20110	0.600551	−	0.799587i	$-0.294948\pi$
$601$	721.862	1.20110	0.600551	+	0.799587i	$0.294948\pi$
$602$	0	0
$603$	− 88.6661i	− 0.147042i
$604$	0	0
$605$	−156.352	−0.258433
$606$	0	0
$607$	− 705.999i	− 1.16310i	−0.813512	−	0.581548i	$-0.802447\pi$
$607$	− 705.999i	− 1.16310i	0.813512	−	0.581548i	$-0.197553\pi$
$608$	0	0
$609$	202.794	0.332995
$610$	0	0
$611$	56.0000i	0.0916530i
$612$	0	0
$613$	21.8269	0.0356066	0.0178033	−	0.999842i	$-0.494333\pi$
$613$	21.8269	0.0356066	0.0178033	+	0.999842i	$0.494333\pi$
$614$	0	0
$615$	142.715i	0.232057i
$616$	0	0
$617$	699.578	1.13384	0.566919	−	0.823774i	$-0.308135\pi$
$617$	699.578	1.13384	0.566919	+	0.823774i	$0.308135\pi$
$618$	0	0
$619$	96.1981i	0.155409i	0.996976	+	0.0777044i	$0.0247590\pi$
$619$	96.1981i	0.155409i	−0.996976	+	0.0777044i	$0.975241\pi$
$620$	0	0
$621$	−762.858	−1.22843
$622$	0	0
$623$	116.569i	0.187109i
$624$	0	0
$625$	450.618	0.720989
$626$	0	0
$627$	− 380.853i	− 0.607421i
$628$	0	0
$629$	1385.88	2.20331
$630$	0	0
$631$	269.399i	0.426940i	0.976950	+	0.213470i	$0.0684766\pi$
$631$	269.399i	0.426940i	−0.976950	+	0.213470i	$0.931523\pi$
$632$	0	0
$633$	−560.098	−0.884830
$634$	0	0
$635$	− 193.769i	− 0.305148i
$636$	0	0
$637$	10.8489	0.0170313
$638$	0	0
$639$	271.527i	0.424924i
$640$	0	0
$641$	635.813	0.991908	0.495954	−	0.868349i	$-0.334818\pi$
$641$	635.813	0.991908	0.495954	+	0.868349i	$0.334818\pi$
$642$	0	0
$643$	− 1281.70i	− 1.99332i	−0.0816828	−	0.996658i	$-0.526029\pi$
$643$	− 1281.70i	− 1.99332i	0.0816828	−	0.996658i	$-0.473971\pi$
$644$	0	0
$645$	90.8634	0.140874
$646$	0	0
$647$	− 260.761i	− 0.403031i	−0.979485	−	0.201516i	$-0.935413\pi$
$647$	− 260.761i	− 0.403031i	0.979485	−	0.201516i	$-0.0645867\pi$
$648$	0	0
$649$	276.098	0.425420
$650$	0	0
$651$	421.990i	0.648218i
$652$	0	0
$653$	1090.58	1.67011	0.835055	−	0.550167i	$-0.185436\pi$
$653$	1090.58	1.67011	0.835055	+	0.550167i	$0.185436\pi$
$654$	0	0
$655$	155.527i	0.237446i
$656$	0	0
$657$	184.156	0.280299
$658$	0	0
$659$	362.780i	0.550500i	0.961373	+	0.275250i	$0.0887607\pi$
$659$	362.780i	0.550500i	−0.961373	+	0.275250i	$0.911239\pi$
$660$	0	0
$661$	117.834	0.178266	0.0891330	−	0.996020i	$-0.471590\pi$
$661$	117.834	0.178266	0.0891330	+	0.996020i	$0.471590\pi$
$662$	0	0
$663$	125.180i	0.188809i
$664$	0	0
$665$	−101.980	−0.153353
$666$	0	0
$667$	790.794i	1.18560i
$668$	0	0
$669$	36.1326	0.0540099
$670$	0	0
$671$	169.017i	0.251888i
$672$	0	0
$673$	−6.56854	−0.00976009	−0.00488005	−	0.999988i	$-0.501553\pi$
$673$	−6.56854	−0.00976009	−0.00488005	+	0.999988i	$0.501553\pi$
$674$	0	0
$675$	489.401i	0.725039i
$676$	0	0
$677$	125.796	0.185813	0.0929066	−	0.995675i	$-0.470384\pi$
$677$	125.796	0.185813	0.0929066	+	0.995675i	$0.470384\pi$
$678$	0	0
$679$	− 254.277i	− 0.374487i
$680$	0	0
$681$	−361.245	−0.530462
$682$	0	0
$683$	− 553.775i	− 0.810797i	−0.914140	−	0.405399i	$-0.867133\pi$
$683$	− 553.775i	− 0.810797i	0.914140	−	0.405399i	$-0.132867\pi$
$684$	0	0
$685$	88.8274	0.129675
$686$	0	0
$687$	255.652i	0.372128i
$688$	0	0
$689$	151.598	0.220026
$690$	0	0
$691$	1046.83i	1.51494i	0.652868	+	0.757471i	$0.273565\pi$
$691$	1046.83i	1.51494i	−0.652868	+	0.757471i	$0.726435\pi$
$692$	0	0
$693$	31.5287	0.0454960
$694$	0	0
$695$	284.651i	0.409569i
$696$	0	0
$697$	638.039	0.915407
$698$	0	0
$699$	− 1431.02i	− 2.04724i
$700$	0	0
$701$	625.993	0.893000	0.446500	−	0.894784i	$-0.352670\pi$
$701$	625.993	0.893000	0.446500	+	0.894784i	$0.352670\pi$
$702$	0	0
$703$	1456.95i	2.07248i
$704$	0	0
$705$	191.196	0.271200
$706$	0	0
$707$	51.8995i	0.0734081i
$708$	0	0
$709$	593.492	0.837083	0.418541	−	0.908198i	$-0.362541\pi$
$709$	593.492	0.837083	0.418541	+	0.908198i	$0.362541\pi$
$710$	0	0
$711$	− 102.822i	− 0.144615i
$712$	0	0
$713$	−1645.55	−2.30792
$714$	0	0
$715$	− 10.7737i	− 0.0150682i
$716$	0	0
$717$	−506.389	−0.706261
$718$	0	0
$719$	611.505i	0.850493i	0.905078	+	0.425247i	$0.139813\pi$
$719$	611.505i	0.850493i	−0.905078	+	0.425247i	$0.860187\pi$
$720$	0	0
$721$	−113.990	−0.158100
$722$	0	0
$723$	1570.10i	2.17165i
$724$	0	0
$725$	507.323	0.699756
$726$	0	0
$727$	944.144i	1.29868i	0.760496	+	0.649342i	$0.224956\pi$
$727$	944.144i	1.29868i	−0.760496	+	0.649342i	$0.775044\pi$
$728$	0	0
$729$	−405.489	−0.556227
$730$	0	0
$731$	− 406.225i	− 0.555712i
$732$	0	0
$733$	218.254	0.297755	0.148878	−	0.988856i	$-0.452434\pi$
$733$	218.254	0.297755	0.148878	+	0.988856i	$0.452434\pi$
$734$	0	0
$735$	− 37.0405i	− 0.0503953i
$736$	0	0
$737$	149.685	0.203101
$738$	0	0
$739$	7.29942i	0.00987743i	0.999988	+	0.00493872i	$0.00157205\pi$
$739$	7.29942i	0.00987743i	−0.999988	+	0.00493872i	$0.998428\pi$
$740$	0	0
$741$	−131.600	−0.177598
$742$	0	0
$743$	− 106.867i	− 0.143832i	−0.997411	−	0.0719159i	$-0.977089\pi$
$743$	− 106.867i	− 0.143832i	0.997411	−	0.0719159i	$-0.0229113\pi$
$744$	0	0
$745$	298.051	0.400068
$746$	0	0
$747$	9.60512i	0.0128583i
$748$	0	0
$749$	41.2684	0.0550980
$750$	0	0
$751$	− 127.463i	− 0.169725i	−0.996393	−	0.0848624i	$-0.972955\pi$
$751$	− 127.463i	− 0.169725i	0.996393	−	0.0848624i	$-0.0270451\pi$
$752$	0	0
$753$	426.500	0.566400
$754$	0	0
$755$	− 177.849i	− 0.235562i
$756$	0	0
$757$	704.275	0.930350	0.465175	−	0.885219i	$-0.345991\pi$
$757$	704.275	0.930350	0.465175	+	0.885219i	$0.345991\pi$
$758$	0	0
$759$	539.422i	0.710701i
$760$	0	0
$761$	1002.93	1.31791	0.658955	−	0.752182i	$-0.270999\pi$
$761$	1002.93	1.31791	0.658955	+	0.752182i	$0.270999\pi$
$762$	0	0
$763$	− 10.1909i	− 0.0133564i
$764$	0	0
$765$	97.4121	0.127336
$766$	0	0
$767$	− 95.4028i	− 0.124384i
$768$	0	0
$769$	−646.950	−0.841288	−0.420644	−	0.907226i	$-0.638196\pi$
$769$	−646.950	−0.841288	−0.420644	+	0.907226i	$0.638196\pi$
$770$	0	0
$771$	1459.07i	1.89244i
$772$	0	0
$773$	−564.265	−0.729968	−0.364984	−	0.931014i	$-0.618925\pi$
$773$	−564.265	−0.729968	−0.364984	+	0.931014i	$0.618925\pi$
$774$	0	0
$775$	1055.68i	1.36217i
$776$	0	0
$777$	−529.186	−0.681063
$778$	0	0
$779$	670.759i	0.861052i
$780$	0	0
$781$	−458.389	−0.586926
$782$	0	0
$783$	486.195i	0.620939i
$784$	0	0
$785$	328.794	0.418846
$786$	0	0
$787$	− 923.345i	− 1.17325i	−0.809860	−	0.586623i	$-0.800457\pi$
$787$	− 923.345i	− 1.17325i	0.809860	−	0.586623i	$-0.199543\pi$
$788$	0	0
$789$	879.582	1.11481
$790$	0	0
$791$	36.4442i	0.0460735i
$792$	0	0
$793$	58.4020	0.0736469
$794$	0	0
$795$	− 517.588i	− 0.651054i
$796$	0	0
$797$	207.983	0.260957	0.130479	−	0.991451i	$-0.458349\pi$
$797$	207.983	0.260957	0.130479	+	0.991451i	$0.458349\pi$
$798$	0	0
$799$	− 854.785i	− 1.06982i
$800$	0	0
$801$	117.058	0.146140
$802$	0	0
$803$	310.891i	0.387162i
$804$	0	0
$805$	144.439	0.179428
$806$	0	0
$807$	− 734.896i	− 0.910652i
$808$	0	0
$809$	−340.540	−0.420939	−0.210470	−	0.977600i	$-0.567499\pi$
$809$	−340.540	−0.420939	−0.210470	+	0.977600i	$0.567499\pi$
$810$	0	0
$811$	− 907.380i	− 1.11884i	−0.828884	−	0.559420i	$-0.811024\pi$
$811$	− 907.380i	− 1.11884i	0.828884	−	0.559420i	$-0.188976\pi$
$812$	0	0
$813$	−1292.45	−1.58973
$814$	0	0
$815$	− 372.791i	− 0.457412i
$816$	0	0
$817$	427.058	0.522715
$818$	0	0
$819$	− 10.8944i	− 0.0133021i
$820$	0	0
$821$	633.423	0.771526	0.385763	−	0.922598i	$-0.373938\pi$
$821$	633.423	0.771526	0.385763	+	0.922598i	$0.373938\pi$
$822$	0	0
$823$	− 143.649i	− 0.174544i	−0.996185	−	0.0872718i	$-0.972185\pi$
$823$	− 143.649i	− 0.174544i	0.996185	−	0.0872718i	$-0.0278149\pi$
$824$	0	0
$825$	346.059	0.419465
$826$	0	0
$827$	1545.57i	1.86888i	0.356114	+	0.934442i	$0.384101\pi$
$827$	1545.57i	1.86888i	−0.356114	+	0.934442i	$0.615899\pi$
$828$	0	0
$829$	743.956	0.897413	0.448707	−	0.893679i	$-0.351885\pi$
$829$	743.956	0.897413	0.448707	+	0.893679i	$0.351885\pi$
$830$	0	0
$831$	568.291i	0.683864i
$832$	0	0
$833$	−165.598	−0.198797
$834$	0	0
$835$	− 328.723i	− 0.393681i
$836$	0	0
$837$	−1011.71	−1.20874
$838$	0	0
$839$	96.3107i	0.114792i	0.998351	+	0.0573961i	$0.0182798\pi$
$839$	96.3107i	0.114792i	−0.998351	+	0.0573961i	$0.981720\pi$
$840$	0	0
$841$	−337.000	−0.400713
$842$	0	0
$843$	− 1439.99i	− 1.70818i
$844$	0	0
$845$	258.201	0.305563
$846$	0	0
$847$	− 266.909i	− 0.315123i
$848$	0	0
$849$	−1179.40	−1.38917
$850$	0	0
$851$	− 2063.56i	− 2.42486i
$852$	0	0
$853$	−904.866	−1.06080	−0.530402	−	0.847746i	$-0.677959\pi$
$853$	−904.866	−1.06080	−0.530402	+	0.847746i	$0.677959\pi$
$854$	0	0
$855$	102.408i	0.119775i
$856$	0	0
$857$	−160.932	−0.187785	−0.0938926	−	0.995582i	$-0.529931\pi$
$857$	−160.932	−0.187785	−0.0938926	+	0.995582i	$0.529931\pi$
$858$	0	0
$859$	231.693i	0.269724i	0.990864	+	0.134862i	$0.0430590\pi$
$859$	231.693i	0.269724i	−0.990864	+	0.134862i	$0.956941\pi$
$860$	0	0
$861$	−243.629	−0.282961
$862$	0	0
$863$	1337.35i	1.54965i	0.632176	+	0.774825i	$0.282162\pi$
$863$	1337.35i	1.54965i	−0.632176	+	0.774825i	$0.717838\pi$
$864$	0	0
$865$	282.402	0.326476
$866$	0	0
$867$	− 924.046i	− 1.06580i
$868$	0	0
$869$	173.583	0.199750
$870$	0	0
$871$	− 51.7223i	− 0.0593827i
$872$	0	0
$873$	−255.344	−0.292490
$874$	0	0
$875$	− 195.176i	− 0.223058i
$876$	0	0
$877$	−1436.14	−1.63755	−0.818777	−	0.574111i	$-0.805348\pi$
$877$	−1436.14	−1.63755	−0.818777	+	0.574111i	$0.805348\pi$
$878$	0	0
$879$	− 1744.79i	− 1.98498i
$880$	0	0
$881$	−186.706	−0.211926	−0.105963	−	0.994370i	$-0.533792\pi$
$881$	−186.706	−0.211926	−0.105963	+	0.994370i	$0.533792\pi$
$882$	0	0
$883$	− 1277.99i	− 1.44733i	−0.690153	−	0.723664i	$-0.742457\pi$
$883$	− 1277.99i	− 1.44733i	0.690153	−	0.723664i	$-0.257543\pi$
$884$	0	0
$885$	−325.726	−0.368052
$886$	0	0
$887$	− 980.717i	− 1.10566i	−0.833295	−	0.552828i	$-0.813549\pi$
$887$	− 980.717i	− 1.10566i	0.833295	−	0.552828i	$-0.186451\pi$
$888$	0	0
$889$	330.784	0.372085
$890$	0	0
$891$	438.897i	0.492590i
$892$	0	0
$893$	898.621	1.00629
$894$	0	0
$895$	88.6605i	0.0990621i
$896$	0	0
$897$	186.392	0.207795
$898$	0	0
$899$	1048.76i	1.16659i
$900$	0	0
$901$	−2313.99	−2.56825
$902$	0	0
$903$	155.114i	0.171776i
$904$	0	0
$905$	505.578	0.558649
$906$	0	0
$907$	− 658.372i	− 0.725878i	−0.931813	−	0.362939i	$-0.881773\pi$
$907$	− 658.372i	− 0.725878i	0.931813	−	0.362939i	$-0.118227\pi$
$908$	0	0
$909$	52.1173	0.0573348
$910$	0	0
$911$	276.507i	0.303520i	0.988417	+	0.151760i	$0.0484941\pi$
$911$	276.507i	0.303520i	−0.988417	+	0.151760i	$0.951506\pi$
$912$	0	0
$913$	−16.2153	−0.0177605
$914$	0	0
$915$	− 199.397i	− 0.217920i
$916$	0	0
$917$	−265.502	−0.289533
$918$	0	0
$919$	− 1339.73i	− 1.45782i	−0.684611	−	0.728908i	$-0.740028\pi$
$919$	− 1339.73i	− 1.45782i	0.684611	−	0.728908i	$-0.259972\pi$
$920$	0	0
$921$	−762.500	−0.827904
$922$	0	0
$923$	158.392i	0.171606i
$924$	0	0
$925$	−1323.85	−1.43119
$926$	0	0
$927$	114.468i	0.123482i
$928$	0	0
$929$	35.4012	0.0381067	0.0190534	−	0.999818i	$-0.493935\pi$
$929$	35.4012	0.0381067	0.0190534	+	0.999818i	$0.493935\pi$
$930$	0	0
$931$	− 174.090i	− 0.186993i
$932$	0	0
$933$	−42.3320	−0.0453719
$934$	0	0
$935$	164.450i	0.175883i
$936$	0	0
$937$	−610.235	−0.651265	−0.325633	−	0.945496i	$-0.605577\pi$
$937$	−610.235	−0.651265	−0.325633	+	0.945496i	$0.605577\pi$
$938$	0	0
$939$	1399.99i	1.49094i
$940$	0	0
$941$	1852.90	1.96907	0.984537	−	0.175175i	$-0.0560491\pi$
$941$	1852.90	1.96907	0.984537	+	0.175175i	$0.0560491\pi$
$942$	0	0
$943$	− 950.032i	− 1.00746i
$944$	0	0
$945$	88.8040	0.0939725
$946$	0	0
$947$	1832.90i	1.93548i	0.251959	+	0.967738i	$0.418925\pi$
$947$	1832.90i	1.93548i	−0.251959	+	0.967738i	$0.581075\pi$
$948$	0	0
$949$	107.426	0.113199
$950$	0	0
$951$	− 444.927i	− 0.467852i
$952$	0	0
$953$	−349.687	−0.366933	−0.183467	−	0.983026i	$-0.558732\pi$
$953$	−349.687	−0.366933	−0.183467	+	0.983026i	$0.558732\pi$
$954$	0	0
$955$	− 150.432i	− 0.157521i
$956$	0	0
$957$	343.792	0.359239
$958$	0	0
$959$	151.638i	0.158121i
$960$	0	0
$961$	−1221.35	−1.27092
$962$	0	0
$963$	− 41.4416i	− 0.0430338i
$964$	0	0
$965$	−243.796	−0.252639
$966$	0	0
$967$	− 632.128i	− 0.653700i	−0.945076	−	0.326850i	$-0.894013\pi$
$967$	− 632.128i	− 0.653700i	0.945076	−	0.326850i	$-0.105987\pi$
$968$	0	0
$969$	2008.74	2.07301
$970$	0	0
$971$	656.497i	0.676104i	0.941128	+	0.338052i	$0.109768\pi$
$971$	656.497i	0.676104i	−0.941128	+	0.338052i	$0.890232\pi$
$972$	0	0
$973$	−485.929	−0.499413
$974$	0	0
$975$	− 119.577i	− 0.122643i
$976$	0	0
$977$	169.314	0.173300	0.0866498	−	0.996239i	$-0.472384\pi$
$977$	169.314	0.173300	0.0866498	+	0.996239i	$0.472384\pi$
$978$	0	0
$979$	197.616i	0.201855i
$980$	0	0
$981$	−10.2337	−0.0104319
$982$	0	0
$983$	698.607i	0.710689i	0.934735	+	0.355345i	$0.115636\pi$
$983$	698.607i	0.710689i	−0.934735	+	0.355345i	$0.884364\pi$
$984$	0	0
$985$	−192.362	−0.195291
$986$	0	0
$987$	326.392i	0.330691i
$988$	0	0
$989$	−604.865	−0.611592
$990$	0	0
$991$	− 429.702i	− 0.433605i	−0.976216	−	0.216802i	$-0.930437\pi$
$991$	− 429.702i	− 0.433605i	0.976216	−	0.216802i	$-0.0695627\pi$
$992$	0	0
$993$	731.529	0.736686
$994$	0	0
$995$	280.483i	0.281892i
$996$	0	0
$997$	−52.3910	−0.0525487	−0.0262743	−	0.999655i	$-0.508364\pi$
$997$	−52.3910	−0.0525487	−0.0262743	+	0.999655i	$0.508364\pi$
$998$	0	0
$999$	− 1268.71i	− 1.26998i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.3.d.g.1023.1		8
4.3	odd	2	inner	1792.3.d.g.1023.7		8
8.3	odd	2	inner	1792.3.d.g.1023.2		8
8.5	even	2	inner	1792.3.d.g.1023.8		8
16.3	odd	4		224.3.g.a.15.2		4
16.5	even	4		224.3.g.a.15.1		4
16.11	odd	4		56.3.g.a.43.4	yes	4
16.13	even	4		56.3.g.a.43.3	✓	4
48.5	odd	4		2016.3.g.a.1135.3		4
48.11	even	4		504.3.g.a.379.1		4
48.29	odd	4		504.3.g.a.379.2		4
48.35	even	4		2016.3.g.a.1135.2		4
112.11	odd	12		392.3.k.i.275.1		8
112.13	odd	4		392.3.g.h.99.3		4
112.27	even	4		392.3.g.h.99.4		4
112.45	odd	12		392.3.k.j.275.3		8
112.59	even	12		392.3.k.j.275.1		8
112.61	odd	12		392.3.k.j.67.1		8
112.69	odd	4		1568.3.g.h.687.4		4
112.75	even	12		392.3.k.j.67.3		8
112.83	even	4		1568.3.g.h.687.3		4
112.93	even	12		392.3.k.i.67.1		8
112.107	odd	12		392.3.k.i.67.3		8
112.109	even	12		392.3.k.i.275.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.g.a.43.3	✓	4	16.13	even	4
56.3.g.a.43.4	yes	4	16.11	odd	4
224.3.g.a.15.1		4	16.5	even	4
224.3.g.a.15.2		4	16.3	odd	4
392.3.g.h.99.3		4	112.13	odd	4
392.3.g.h.99.4		4	112.27	even	4
392.3.k.i.67.1		8	112.93	even	12
392.3.k.i.67.3		8	112.107	odd	12
392.3.k.i.275.1		8	112.11	odd	12
392.3.k.i.275.3		8	112.109	even	12
392.3.k.j.67.1		8	112.61	odd	12
392.3.k.j.67.3		8	112.75	even	12
392.3.k.j.275.1		8	112.59	even	12
392.3.k.j.275.3		8	112.45	odd	12
504.3.g.a.379.1		4	48.11	even	4
504.3.g.a.379.2		4	48.29	odd	4
1568.3.g.h.687.3		4	112.83	even	4
1568.3.g.h.687.4		4	112.69	odd	4
1792.3.d.g.1023.1		8	1.1	even	1	trivial
1792.3.d.g.1023.2		8	8.3	odd	2	inner
1792.3.d.g.1023.7		8	4.3	odd	2	inner
1792.3.d.g.1023.8		8	8.5	even	2	inner
2016.3.g.a.1135.2		4	48.35	even	4
2016.3.g.a.1135.3		4	48.5	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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1792.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.41421i q^{3} -1.54985 q^{5} -2.64575i q^{7} -2.65685 q^{9} +O(q^{10})\)	\(q-3.41421i q^{3} -1.54985 q^{5} -2.64575i q^{7} -2.65685 q^{9} -4.48528i q^{11} -1.54985 q^{13} +5.29150i q^{15} +23.6569 q^{17} +24.8701i q^{19} -9.03316 q^{21} -35.2248i q^{23} -22.5980 q^{25} -21.6569i q^{27} -22.4499 q^{29} -46.7156i q^{31} -15.3137 q^{33} +4.10051i q^{35} +58.5826 q^{37} +5.29150i q^{39} +26.9706 q^{41} -17.1716i q^{43} +4.11771 q^{45} -36.1326i q^{47} -7.00000 q^{49} -80.7696i q^{51} -97.8149 q^{53} +6.95149i q^{55} +84.9117 q^{57} +61.5563i q^{59} -37.6825 q^{61} +7.02938i q^{63} +2.40202 q^{65} +33.3726i q^{67} -120.265 q^{69} -102.199i q^{71} -69.3137 q^{73} +77.1543i q^{75} -11.8669 q^{77} +38.7005i q^{79} -97.8528 q^{81} -3.61522i q^{83} -36.6645 q^{85} +76.6489i q^{87} -44.0589 q^{89} +4.10051i q^{91} -159.497 q^{93} -38.5447i q^{95} +96.1076 q^{97} +11.9167i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^9	\( 8 q + 24 q^{9} + 144 q^{17} + 136 q^{25} - 32 q^{33} + 80 q^{41} - 56 q^{49} + 272 q^{57} + 336 q^{65} - 464 q^{73} - 104 q^{81} - 624 q^{89} - 272 q^{97}+O(q^{100}) \) 8 * q + 24 * q^9 + 144 * q^17 + 136 * q^25 - 32 * q^33 + 80 * q^41 - 56 * q^49 + 272 * q^57 + 336 * q^65 - 464 * q^73 - 104 * q^81 - 624 * q^89 - 272 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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