LMFDB - Embedded newform 224.3.g.a.15.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,3,Mod(15,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("224.15");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	224.g (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:	$6.10355792167$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 6x^{2} + 16 $ x^4 + 6*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			15.1
Root			$0.707107 - 1.87083i$ of defining polynomial
Character	$\chi$	$=$	224.15
Dual form			224.3.g.a.15.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.41421 q^{3} -1.54985i q^{5} +2.64575i q^{7} +2.65685 q^{9} +O(q^{10})$	$q-3.41421 q^{3} -1.54985i q^{5} +2.64575i q^{7} +2.65685 q^{9} +4.48528 q^{11} +1.54985i q^{13} +5.29150i q^{15} +23.6569 q^{17} +24.8701 q^{19} -9.03316i q^{21} +35.2248i q^{23} +22.5980 q^{25} +21.6569 q^{27} +22.4499i q^{29} -46.7156i q^{31} -15.3137 q^{33} +4.10051 q^{35} +58.5826i q^{37} -5.29150i q^{39} -26.9706 q^{41} +17.1716 q^{43} -4.11771i q^{45} -36.1326i q^{47} -7.00000 q^{49} -80.7696 q^{51} -97.8149i q^{53} -6.95149i q^{55} -84.9117 q^{57} -61.5563 q^{59} +37.6825i q^{61} +7.02938i q^{63} +2.40202 q^{65} +33.3726 q^{67} -120.265i q^{69} +102.199i q^{71} +69.3137 q^{73} -77.1543 q^{75} +11.8669i q^{77} +38.7005i q^{79} -97.8528 q^{81} -3.61522 q^{83} -36.6645i q^{85} -76.6489i q^{87} +44.0589 q^{89} -4.10051 q^{91} +159.497i q^{93} -38.5447i q^{95} +96.1076 q^{97} +11.9167 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{3} - 12 q^{9}+O(q^{10}) $ 4 * q - 8 * q^3 - 12 * q^9	$ 4 q - 8 q^{3} - 12 q^{9} - 16 q^{11} + 72 q^{17} - 8 q^{19} - 68 q^{25} + 64 q^{27} - 16 q^{33} + 56 q^{35} - 40 q^{41} + 80 q^{43} - 28 q^{49} - 176 q^{51} - 136 q^{57} - 184 q^{59} + 168 q^{65} + 224 q^{67} + 232 q^{73} - 88 q^{75} - 52 q^{81} - 88 q^{83} + 312 q^{89} - 56 q^{91} - 136 q^{97} + 240 q^{99}+O(q^{100}) $ 4 * q - 8 * q^3 - 12 * q^9 - 16 * q^11 + 72 * q^17 - 8 * q^19 - 68 * q^25 + 64 * q^27 - 16 * q^33 + 56 * q^35 - 40 * q^41 + 80 * q^43 - 28 * q^49 - 176 * q^51 - 136 * q^57 - 184 * q^59 + 168 * q^65 + 224 * q^67 + 232 * q^73 - 88 * q^75 - 52 * q^81 - 88 * q^83 + 312 * q^89 - 56 * q^91 - 136 * q^97 + 240 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/224\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−3.41421	−1.13807	−0.569036	−	0.822313i	$-0.692683\pi$
$3$	−3.41421	−1.13807	−0.569036	+	0.822313i	$0.692683\pi$
$4$	0	0
$5$	− 1.54985i	− 0.309969i	−0.987917	−	0.154985i	$-0.950467\pi$
$5$	− 1.54985i	− 0.309969i	0.987917	−	0.154985i	$-0.0495328\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	2.65685	0.295206
$10$	0	0
$11$	4.48528	0.407753	0.203876	−	0.978997i	$-0.434646\pi$
$11$	4.48528	0.407753	0.203876	+	0.978997i	$0.434646\pi$
$12$	0	0
$13$	1.54985i	0.119219i	0.998222	+	0.0596094i	$0.0189855\pi$
$13$	1.54985i	0.119219i	−0.998222	+	0.0596094i	$0.981014\pi$
$14$	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
$19$	24.8701	1.30895	0.654475	−	0.756083i	$-0.272890\pi$
$19$	24.8701	1.30895	0.654475	+	0.756083i	$0.272890\pi$
$20$	0	0
$21$	− 9.03316i	− 0.430150i
$22$	0	0
$23$	35.2248i	1.53151i	0.643132	+	0.765756i	$0.277635\pi$
$23$	35.2248i	1.53151i	−0.643132	+	0.765756i	$0.722365\pi$
$24$	0	0
$25$	22.5980	0.903919
$26$	0	0
$27$	21.6569	0.802106
$28$	0	0
$29$	22.4499i	0.774136i	0.922051	+	0.387068i	$0.126512\pi$
$29$	22.4499i	0.774136i	−0.922051	+	0.387068i	$0.873488\pi$
$30$	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
$35$	4.10051	0.117157
$36$	0	0
$37$	58.5826i	1.58331i	0.610966	+	0.791657i	$0.290781\pi$
$37$	58.5826i	1.58331i	−0.610966	+	0.791657i	$0.709219\pi$
$38$	0	0
$39$	− 5.29150i	− 0.135680i
$40$	0	0
$41$	−26.9706	−0.657819	−0.328909	−	0.944362i	$-0.606681\pi$
$41$	−26.9706	−0.657819	−0.328909	+	0.944362i	$0.606681\pi$
$42$	0	0
$43$	17.1716	0.399339	0.199669	−	0.979863i	$-0.436013\pi$
$43$	17.1716	0.399339	0.199669	+	0.979863i	$0.436013\pi$
$44$	0	0
$45$	− 4.11771i	− 0.0915047i
$46$	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
$51$	−80.7696	−1.58372
$52$	0	0
$53$	− 97.8149i	− 1.84556i	−0.385322	−	0.922782i	$-0.625910\pi$
$53$	− 97.8149i	− 1.84556i	0.385322	−	0.922782i	$-0.374090\pi$
$54$	0	0
$55$	− 6.95149i	− 0.126391i
$56$	0	0
$57$	−84.9117	−1.48968
$58$	0	0
$59$	−61.5563	−1.04333	−0.521664	−	0.853151i	$-0.674689\pi$
$59$	−61.5563	−1.04333	−0.521664	+	0.853151i	$0.674689\pi$
$60$	0	0
$61$	37.6825i	0.617746i	0.951103	+	0.308873i	$0.0999518\pi$
$61$	37.6825i	0.617746i	−0.951103	+	0.308873i	$0.900048\pi$
$62$	0	0
$63$	7.02938i	0.111577i
$64$	0	0
$65$	2.40202	0.0369542
$66$	0	0
$67$	33.3726	0.498098	0.249049	−	0.968491i	$-0.419882\pi$
$67$	33.3726	0.498098	0.249049	+	0.968491i	$0.419882\pi$
$68$	0	0
$69$	− 120.265i	− 1.74297i
$70$	0	0
$71$	102.199i	1.43942i	0.694277	+	0.719708i	$0.255724\pi$
$71$	102.199i	1.43942i	−0.694277	+	0.719708i	$0.744276\pi$
$72$	0	0
$73$	69.3137	0.949503	0.474751	−	0.880120i	$-0.342538\pi$
$73$	69.3137	0.949503	0.474751	+	0.880120i	$0.342538\pi$
$74$	0	0
$75$	−77.1543	−1.02872
$76$	0	0
$77$	11.8669i	0.154116i
$78$	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
$83$	−3.61522	−0.0435569	−0.0217785	−	0.999763i	$-0.506933\pi$
$83$	−3.61522	−0.0435569	−0.0217785	+	0.999763i	$0.506933\pi$
$84$	0	0
$85$	− 36.6645i	− 0.431347i
$86$	0	0
$87$	− 76.6489i	− 0.881022i
$88$	0	0
$89$	44.0589	0.495044	0.247522	−	0.968882i	$-0.420384\pi$
$89$	44.0589	0.495044	0.247522	+	0.968882i	$0.420384\pi$
$90$	0	0
$91$	−4.10051	−0.0450605
$92$	0	0
$93$	159.497i	1.71502i
$94$	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
$99$	11.9167	0.120371
$100$	0	0
$101$	− 19.6162i	− 0.194219i	−0.995274	−	0.0971097i	$-0.969040\pi$
$101$	− 19.6162i	− 0.194219i	0.995274	−	0.0971097i	$-0.0309598\pi$
$102$	0	0
$103$	43.0841i	0.418293i	0.977884	+	0.209146i	$0.0670685\pi$
$103$	43.0841i	0.418293i	−0.977884	+	0.209146i	$0.932932\pi$
$104$	0	0
$105$	−14.0000	−0.133333
$106$	0	0
$107$	−15.5980	−0.145776	−0.0728878	−	0.997340i	$-0.523221\pi$
$107$	−15.5980	−0.145776	−0.0728878	+	0.997340i	$0.523221\pi$
$108$	0	0
$109$	− 3.85180i	− 0.0353376i	−0.999844	−	0.0176688i	$-0.994376\pi$
$109$	− 3.85180i	− 0.0353376i	0.999844	−	0.0176688i	$-0.00562445\pi$
$110$	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
$115$	54.5929	0.474721
$116$	0	0
$117$	4.11771i	0.0351941i
$118$	0	0
$119$	62.5902i	0.525968i
$120$	0	0
$121$	−100.882	−0.833738
$122$	0	0
$123$	92.0833	0.748644
$124$	0	0
$125$	− 73.7695i	− 0.590156i
$126$	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
$131$	−100.350	−0.766033	−0.383016	−	0.923742i	$-0.625115\pi$
$131$	−100.350	−0.766033	−0.383016	+	0.923742i	$0.625115\pi$
$132$	0	0
$133$	65.8000i	0.494737i
$134$	0	0
$135$	− 33.5648i	− 0.248628i
$136$	0	0
$137$	57.3137	0.418348	0.209174	−	0.977878i	$-0.432922\pi$
$137$	57.3137	0.418348	0.209174	+	0.977878i	$0.432922\pi$
$138$	0	0
$139$	183.664	1.32132	0.660662	−	0.750684i	$-0.270276\pi$
$139$	183.664	1.32132	0.660662	+	0.750684i	$0.270276\pi$
$140$	0	0
$141$	123.365i	0.874926i
$142$	0	0
$143$	6.95149i	0.0486118i
$144$	0	0
$145$	34.7939	0.239958
$146$	0	0
$147$	23.8995	0.162582
$148$	0	0
$149$	− 192.310i	− 1.29067i	−0.763900	−	0.645335i	$-0.776718\pi$
$149$	− 192.310i	− 1.29067i	0.763900	−	0.645335i	$-0.223282\pi$
$150$	0	0
$151$	− 114.753i	− 0.759954i	−0.924996	−	0.379977i	$-0.875932\pi$
$151$	− 114.753i	− 0.759954i	0.924996	−	0.379977i	$-0.124068\pi$
$152$	0	0
$153$	62.8528	0.410803
$154$	0	0
$155$	−72.4020	−0.467110
$156$	0	0
$157$	212.146i	1.35125i	0.737245	+	0.675625i	$0.236126\pi$
$157$	212.146i	1.35125i	−0.737245	+	0.675625i	$0.763874\pi$
$158$	0	0
$159$	333.961i	2.10038i
$160$	0	0
$161$	−93.1960	−0.578857
$162$	0	0
$163$	240.534	1.47567	0.737835	−	0.674982i	$-0.235848\pi$
$163$	240.534	1.47567	0.737835	+	0.674982i	$0.235848\pi$
$164$	0	0
$165$	23.7339i	0.143842i
$166$	0	0
$167$	− 212.101i	− 1.27006i	−0.772486	−	0.635032i	$-0.780987\pi$
$167$	− 212.101i	− 1.27006i	0.772486	−	0.635032i	$-0.219013\pi$
$168$	0	0
$169$	166.598	0.985787
$170$	0	0
$171$	66.0761	0.386410
$172$	0	0
$173$	182.213i	1.05325i	0.850096	+	0.526627i	$0.176544\pi$
$173$	182.213i	1.05325i	−0.850096	+	0.526627i	$0.823456\pi$
$174$	0	0
$175$	59.7886i	0.341649i
$176$	0	0
$177$	210.167	1.18738
$178$	0	0
$179$	−57.2061	−0.319587	−0.159793	−	0.987150i	$-0.551083\pi$
$179$	−57.2061	−0.319587	−0.159793	+	0.987150i	$0.551083\pi$
$180$	0	0
$181$	− 326.212i	− 1.80228i	−0.433533	−	0.901138i	$-0.642733\pi$
$181$	− 326.212i	− 1.80228i	0.433533	−	0.901138i	$-0.357267\pi$
$182$	0	0
$183$	− 128.656i	− 0.703039i
$184$	0	0
$185$	90.7939	0.490778
$186$	0	0
$187$	106.108	0.567421
$188$	0	0
$189$	57.2987i	0.303167i
$190$	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
$195$	−8.20101	−0.0420565
$196$	0	0
$197$	124.117i	0.630034i	0.949086	+	0.315017i	$0.102010\pi$
$197$	124.117i	0.630034i	−0.949086	+	0.315017i	$0.897990\pi$
$198$	0	0
$199$	180.975i	0.909421i	0.890639	+	0.454710i	$0.150257\pi$
$199$	180.975i	0.909421i	−0.890639	+	0.454710i	$0.849743\pi$
$200$	0	0
$201$	−113.941	−0.566871
$202$	0	0
$203$	−59.3970	−0.292596
$204$	0	0
$205$	41.8002i	0.203903i
$206$	0	0
$207$	93.5871i	0.452111i
$208$	0	0
$209$	111.549	0.533728
$210$	0	0
$211$	−164.049	−0.777482	−0.388741	−	0.921347i	$-0.627090\pi$
$211$	−164.049	−0.777482	−0.388741	+	0.921347i	$0.627090\pi$
$212$	0	0
$213$	− 348.928i	− 1.63816i
$214$	0	0
$215$	− 26.6133i	− 0.123783i
$216$	0	0
$217$	123.598	0.569576
$218$	0	0
$219$	−236.652	−1.08060
$220$	0	0
$221$	36.6645i	0.165903i
$222$	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
$227$	−105.806	−0.466106	−0.233053	−	0.972464i	$-0.574872\pi$
$227$	−105.806	−0.466106	−0.233053	+	0.972464i	$0.574872\pi$
$228$	0	0
$229$	− 74.8788i	− 0.326982i	−0.986545	−	0.163491i	$-0.947725\pi$
$229$	− 74.8788i	− 0.326982i	0.986545	−	0.163491i	$-0.0522754\pi$
$230$	0	0
$231$	− 40.5163i	− 0.175395i
$232$	0	0
$233$	−419.137	−1.79887	−0.899436	−	0.437053i	$-0.856022\pi$
$233$	−419.137	−1.79887	−0.899436	+	0.437053i	$0.856022\pi$
$234$	0	0
$235$	−56.0000	−0.238298
$236$	0	0
$237$	− 132.132i	− 0.557518i
$238$	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
$243$	139.179	0.572752
$244$	0	0
$245$	10.8489i	0.0442813i
$246$	0	0
$247$	38.5447i	0.156052i
$248$	0	0
$249$	12.3431	0.0495709
$250$	0	0
$251$	−124.919	−0.497685	−0.248842	−	0.968544i	$-0.580050\pi$
$251$	−124.919	−0.497685	−0.248842	+	0.968544i	$0.580050\pi$
$252$	0	0
$253$	157.993i	0.624478i
$254$	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
$259$	−154.995	−0.598436
$260$	0	0
$261$	59.6462i	0.228530i
$262$	0	0
$263$	− 257.624i	− 0.979558i	−0.871847	−	0.489779i	$-0.837078\pi$
$263$	− 257.624i	− 0.979558i	0.871847	−	0.489779i	$-0.162922\pi$
$264$	0	0
$265$	−151.598	−0.572068
$266$	0	0
$267$	−150.426	−0.563395
$268$	0	0
$269$	− 215.246i	− 0.800171i	−0.916478	−	0.400085i	$-0.868980\pi$
$269$	− 215.246i	− 0.800171i	0.916478	−	0.400085i	$-0.131020\pi$
$270$	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
$275$	101.358	0.368576
$276$	0	0
$277$	− 166.449i	− 0.600898i	−0.953798	−	0.300449i	$-0.902864\pi$
$277$	− 166.449i	− 0.600898i	0.953798	−	0.300449i	$-0.0971365\pi$
$278$	0	0
$279$	− 124.117i	− 0.444863i
$280$	0	0
$281$	−421.765	−1.50094	−0.750471	−	0.660904i	$-0.770173\pi$
$281$	−421.765	−1.50094	−0.750471	+	0.660904i	$0.770173\pi$
$282$	0	0
$283$	345.439	1.22063	0.610316	−	0.792158i	$-0.291043\pi$
$283$	345.439	1.22063	0.610316	+	0.792158i	$0.291043\pi$
$284$	0	0
$285$	131.600i	0.461754i
$286$	0	0
$287$	− 71.3574i	− 0.248632i
$288$	0	0
$289$	270.647	0.936494
$290$	0	0
$291$	−328.132	−1.12760
$292$	0	0
$293$	511.038i	1.74416i	0.489365	+	0.872079i	$0.337229\pi$
$293$	511.038i	1.74416i	−0.489365	+	0.872079i	$0.662771\pi$
$294$	0	0
$295$	95.4028i	0.323399i
$296$	0	0
$297$	97.1371	0.327061
$298$	0	0
$299$	−54.5929	−0.182585
$300$	0	0
$301$	45.4317i	0.150936i
$302$	0	0
$303$	66.9738i	0.221036i
$304$	0	0
$305$	58.4020	0.191482
$306$	0	0
$307$	−223.331	−0.727462	−0.363731	−	0.931504i	$-0.618497\pi$
$307$	−223.331	−0.727462	−0.363731	+	0.931504i	$0.618497\pi$
$308$	0	0
$309$	− 147.098i	− 0.476047i
$310$	0	0
$311$	12.3988i	0.0398674i	0.999801	+	0.0199337i	$0.00634551\pi$
$311$	12.3988i	0.0398674i	−0.999801	+	0.0199337i	$0.993654\pi$
$312$	0	0
$313$	410.049	1.31006	0.655030	−	0.755603i	$-0.272656\pi$
$313$	410.049	1.31006	0.655030	+	0.755603i	$0.272656\pi$
$314$	0	0
$315$	10.8944	0.0345855
$316$	0	0
$317$	− 130.316i	− 0.411092i	−0.978647	−	0.205546i	$-0.934103\pi$
$317$	− 130.316i	− 0.411092i	0.978647	−	0.205546i	$-0.0658970\pi$
$318$	0	0
$319$	100.694i	0.315656i
$320$	0	0
$321$	53.2548	0.165903
$322$	0	0
$323$	588.347	1.82151
$324$	0	0
$325$	35.0234i	0.107764i
$326$	0	0
$327$	13.1509i	0.0402167i
$328$	0	0
$329$	95.5980	0.290571
$330$	0	0
$331$	−214.260	−0.647311	−0.323655	−	0.946175i	$-0.604912\pi$
$331$	−214.260	−0.647311	−0.323655	+	0.946175i	$0.604912\pi$
$332$	0	0
$333$	155.645i	0.467404i
$334$	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
$339$	47.0294	0.138730
$340$	0	0
$341$	− 209.533i	− 0.614466i
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	−186.392	−0.540266
$346$	0	0
$347$	109.691	0.316113	0.158057	−	0.987430i	$-0.449477\pi$
$347$	109.691	0.316113	0.158057	+	0.987430i	$0.449477\pi$
$348$	0	0
$349$	463.479i	1.32802i	0.747723	+	0.664010i	$0.231147\pi$
$349$	463.479i	1.32802i	−0.747723	+	0.664010i	$0.768853\pi$
$350$	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
$355$	158.392	0.446174
$356$	0	0
$357$	− 213.696i	− 0.598589i
$358$	0	0
$359$	365.114i	1.01703i	0.861053	+	0.508515i	$0.169805\pi$
$359$	365.114i	1.01703i	−0.861053	+	0.508515i	$0.830195\pi$
$360$	0	0
$361$	257.520	0.713351
$362$	0	0
$363$	344.434	0.948853
$364$	0	0
$365$	− 107.426i	− 0.294316i
$366$	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
$371$	258.794	0.697558
$372$	0	0
$373$	251.553i	0.674406i	0.941432	+	0.337203i	$0.109481\pi$
$373$	251.553i	0.674406i	−0.941432	+	0.337203i	$0.890519\pi$
$374$	0	0
$375$	251.865i	0.671640i
$376$	0	0
$377$	−34.7939	−0.0922916
$378$	0	0
$379$	−286.024	−0.754682	−0.377341	−	0.926074i	$-0.623161\pi$
$379$	−286.024	−0.754682	−0.377341	+	0.926074i	$0.623161\pi$
$380$	0	0
$381$	− 426.860i	− 1.12037i
$382$	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
$387$	45.6224	0.117887
$388$	0	0
$389$	77.1807i	0.198408i	0.995067	+	0.0992040i	$0.0316296\pi$
$389$	77.1807i	0.198408i	−0.995067	+	0.0992040i	$0.968370\pi$
$390$	0	0
$391$	833.307i	2.13122i
$392$	0	0
$393$	342.617	0.871800
$394$	0	0
$395$	59.9798	0.151848
$396$	0	0
$397$	− 657.514i	− 1.65621i	−0.560576	−	0.828103i	$-0.689420\pi$
$397$	− 657.514i	− 1.65621i	0.560576	−	0.828103i	$-0.310580\pi$
$398$	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
$403$	72.4020	0.179658
$404$	0	0
$405$	151.657i	0.374461i
$406$	0	0
$407$	262.759i	0.645600i
$408$	0	0
$409$	145.265	0.355171	0.177585	−	0.984105i	$-0.443171\pi$
$409$	145.265	0.355171	0.177585	+	0.984105i	$0.443171\pi$
$410$	0	0
$411$	−195.681	−0.476110
$412$	0	0
$413$	− 162.863i	− 0.394341i
$414$	0	0
$415$	5.60304i	0.0135013i
$416$	0	0
$417$	−627.068	−1.50376
$418$	0	0
$419$	−707.012	−1.68738	−0.843690	−	0.536831i	$-0.819621\pi$
$419$	−707.012	−1.68738	−0.843690	+	0.536831i	$0.819621\pi$
$420$	0	0
$421$	− 121.989i	− 0.289761i	−0.989449	−	0.144880i	$-0.953720\pi$
$421$	− 121.989i	− 0.289761i	0.989449	−	0.144880i	$-0.0462798\pi$
$422$	0	0
$423$	− 95.9992i	− 0.226948i
$424$	0	0
$425$	534.597	1.25788
$426$	0	0
$427$	−99.6985	−0.233486
$428$	0	0
$429$	− 23.7339i	− 0.0553237i
$430$	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
$435$	−118.794	−0.273090
$436$	0	0
$437$	876.042i	2.00467i
$438$	0	0
$439$	− 440.543i	− 1.00352i	−0.865008	−	0.501758i	$-0.832687\pi$
$439$	− 440.543i	− 1.00352i	0.865008	−	0.501758i	$-0.167313\pi$
$440$	0	0
$441$	−18.5980	−0.0421723
$442$	0	0
$443$	487.058	1.09945	0.549727	−	0.835344i	$-0.314732\pi$
$443$	487.058	1.09945	0.549727	+	0.835344i	$0.314732\pi$
$444$	0	0
$445$	− 68.2844i	− 0.153448i
$446$	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
$451$	−120.971	−0.268227
$452$	0	0
$453$	391.791i	0.864882i
$454$	0	0
$455$	6.35515i	0.0139674i
$456$	0	0
$457$	−514.323	−1.12543	−0.562717	−	0.826650i	$-0.690244\pi$
$457$	−514.323	−1.12543	−0.562717	+	0.826650i	$0.690244\pi$
$458$	0	0
$459$	512.333	1.11619
$460$	0	0
$461$	202.224i	0.438664i	0.975650	+	0.219332i	$0.0703878\pi$
$461$	202.224i	0.438664i	−0.975650	+	0.219332i	$0.929612\pi$
$462$	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
$467$	347.282	0.743645	0.371822	−	0.928304i	$-0.378733\pi$
$467$	347.282	0.743645	0.371822	+	0.928304i	$0.378733\pi$
$468$	0	0
$469$	88.2956i	0.188263i
$470$	0	0
$471$	− 724.313i	− 1.53782i
$472$	0	0
$473$	77.0193	0.162832
$474$	0	0
$475$	562.013	1.18319
$476$	0	0
$477$	− 259.880i	− 0.544822i
$478$	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
$483$	318.191	0.658780
$484$	0	0
$485$	− 148.952i	− 0.307117i
$486$	0	0
$487$	701.643i	1.44074i	0.693588	+	0.720372i	$0.256029\pi$
$487$	701.643i	1.44074i	−0.693588	+	0.720372i	$0.743971\pi$
$488$	0	0
$489$	−821.235	−1.67942
$490$	0	0
$491$	59.9512	0.122100	0.0610501	−	0.998135i	$-0.480555\pi$
$491$	59.9512	0.122100	0.0610501	+	0.998135i	$0.480555\pi$
$492$	0	0
$493$	531.095i	1.07727i
$494$	0	0
$495$	− 18.4691i	− 0.0373113i
$496$	0	0
$497$	−270.392	−0.544048
$498$	0	0
$499$	84.2843	0.168906	0.0844532	−	0.996427i	$-0.473086\pi$
$499$	84.2843	0.168906	0.0844532	+	0.996427i	$0.473086\pi$
$500$	0	0
$501$	724.157i	1.44542i
$502$	0	0
$503$	409.987i	0.815083i	0.913187	+	0.407542i	$0.133614\pi$
$503$	409.987i	0.815083i	−0.913187	+	0.407542i	$0.866386\pi$
$504$	0	0
$505$	−30.4020	−0.0602020
$506$	0	0
$507$	−568.801	−1.12190
$508$	0	0
$509$	− 477.033i	− 0.937196i	−0.883411	−	0.468598i	$-0.844759\pi$
$509$	− 477.033i	− 0.937196i	0.883411	−	0.468598i	$-0.155241\pi$
$510$	0	0
$511$	183.387i	0.358878i
$512$	0	0
$513$	538.607	1.04992
$514$	0	0
$515$	66.7737	0.129658
$516$	0	0
$517$	− 162.065i	− 0.313472i
$518$	0	0
$519$	− 622.114i	− 1.19868i
$520$	0	0
$521$	210.873	0.404747	0.202373	−	0.979308i	$-0.435135\pi$
$521$	210.873	0.404747	0.202373	+	0.979308i	$0.435135\pi$
$522$	0	0
$523$	−511.566	−0.978139	−0.489069	−	0.872245i	$-0.662663\pi$
$523$	−511.566	−0.978139	−0.489069	+	0.872245i	$0.662663\pi$
$524$	0	0
$525$	− 204.131i	− 0.388821i
$526$	0	0
$527$	− 1105.15i	− 2.09705i
$528$	0	0
$529$	−711.784	−1.34553
$530$	0	0
$531$	−163.546	−0.307997
$532$	0	0
$533$	− 41.8002i	− 0.0784244i
$534$	0	0
$535$	24.1745i	0.0451859i
$536$	0	0
$537$	195.314	0.363713
$538$	0	0
$539$	−31.3970	−0.0582504
$540$	0	0
$541$	− 342.417i	− 0.632933i	−0.948604	−	0.316466i	$-0.897504\pi$
$541$	− 342.417i	− 0.632933i	0.948604	−	0.316466i	$-0.102496\pi$
$542$	0	0
$543$	1113.76i	2.05112i
$544$	0	0
$545$	−5.96970	−0.0109536
$546$	0	0
$547$	441.976	0.807999	0.404000	−	0.914759i	$-0.367620\pi$
$547$	441.976	0.807999	0.404000	+	0.914759i	$0.367620\pi$
$548$	0	0
$549$	100.117i	0.182362i
$550$	0	0
$551$	558.331i	1.01331i
$552$	0	0
$553$	−102.392	−0.185157
$554$	0	0
$555$	−309.990	−0.558540
$556$	0	0
$557$	− 365.710i	− 0.656571i	−0.944579	−	0.328285i	$-0.893529\pi$
$557$	− 365.710i	− 0.656571i	0.944579	−	0.328285i	$-0.106471\pi$
$558$	0	0
$559$	26.6133i	0.0476087i
$560$	0	0
$561$	−362.274	−0.645765
$562$	0	0
$563$	−806.389	−1.43231	−0.716154	−	0.697943i	$-0.754099\pi$
$563$	−806.389	−1.43231	−0.716154	+	0.697943i	$0.754099\pi$
$564$	0	0
$565$	21.3485i	0.0377850i
$566$	0	0
$567$	− 258.894i	− 0.456604i
$568$	0	0
$569$	222.891	0.391725	0.195862	−	0.980631i	$-0.437249\pi$
$569$	222.891	0.391725	0.195862	+	0.980631i	$0.437249\pi$
$570$	0	0
$571$	−573.082	−1.00365	−0.501823	−	0.864970i	$-0.667337\pi$
$571$	−573.082	−1.00365	−0.501823	+	0.864970i	$0.667337\pi$
$572$	0	0
$573$	− 331.393i	− 0.578348i
$574$	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
$579$	−537.068	−0.927579
$580$	0	0
$581$	− 9.56498i	− 0.0164630i
$582$	0	0
$583$	− 438.727i	− 0.752534i
$584$	0	0
$585$	6.38182	0.0109091
$586$	0	0
$587$	−21.1198	−0.0359793	−0.0179896	−	0.999838i	$-0.505727\pi$
$587$	−21.1198	−0.0359793	−0.0179896	+	0.999838i	$0.505727\pi$
$588$	0	0
$589$	− 1161.82i	− 1.97253i
$590$	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
$595$	97.0051	0.163034
$596$	0	0
$597$	− 617.886i	− 1.03499i
$598$	0	0
$599$	− 324.130i	− 0.541119i	−0.962703	−	0.270559i	$-0.912791\pi$
$599$	− 324.130i	− 0.541119i	0.962703	−	0.270559i	$-0.0872086\pi$
$600$	0	0
$601$	−721.862	−1.20110	−0.600551	−	0.799587i	$-0.705052\pi$
$601$	−721.862	−1.20110	−0.600551	+	0.799587i	$0.705052\pi$
$602$	0	0
$603$	88.6661	0.147042
$604$	0	0
$605$	156.352i	0.258433i
$606$	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
$611$	56.0000	0.0916530
$612$	0	0
$613$	21.8269i	0.0356066i	0.999842	+	0.0178033i	$0.00566727\pi$
$613$	21.8269i	0.0356066i	−0.999842	+	0.0178033i	$0.994333\pi$
$614$	0	0
$615$	− 142.715i	− 0.232057i
$616$	0	0
$617$	−699.578	−1.13384	−0.566919	−	0.823774i	$-0.691865\pi$
$617$	−699.578	−1.13384	−0.566919	+	0.823774i	$0.691865\pi$
$618$	0	0
$619$	−96.1981	−0.155409	−0.0777044	−	0.996976i	$-0.524759\pi$
$619$	−96.1981	−0.155409	−0.0777044	+	0.996976i	$0.524759\pi$
$620$	0	0
$621$	762.858i	1.22843i
$622$	0	0
$623$	116.569i	0.187109i
$624$	0	0
$625$	450.618	0.720989
$626$	0	0
$627$	−380.853	−0.607421
$628$	0	0
$629$	1385.88i	2.20331i
$630$	0	0
$631$	− 269.399i	− 0.426940i	−0.976950	−	0.213470i	$-0.931523\pi$
$631$	− 269.399i	− 0.426940i	0.976950	−	0.213470i	$-0.0684766\pi$
$632$	0	0
$633$	560.098	0.884830
$634$	0	0
$635$	193.769	0.305148
$636$	0	0
$637$	− 10.8489i	− 0.0170313i
$638$	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
$643$	−1281.70	−1.99332	−0.996658	−	0.0816828i	$-0.973971\pi$
$643$	−1281.70	−1.99332	−0.996658	+	0.0816828i	$0.973971\pi$
$644$	0	0
$645$	90.8634i	0.140874i
$646$	0	0
$647$	260.761i	0.403031i	0.979485	+	0.201516i	$0.0645867\pi$
$647$	260.761i	0.403031i	−0.979485	+	0.201516i	$0.935413\pi$
$648$	0	0
$649$	−276.098	−0.425420
$650$	0	0
$651$	−421.990	−0.648218
$652$	0	0
$653$	− 1090.58i	− 1.67011i	−0.550167	−	0.835055i	$-0.685436\pi$
$653$	− 1090.58i	− 1.67011i	0.550167	−	0.835055i	$-0.314564\pi$
$654$	0	0
$655$	155.527i	0.237446i
$656$	0	0
$657$	184.156	0.280299
$658$	0	0
$659$	362.780	0.550500	0.275250	−	0.961373i	$-0.411239\pi$
$659$	362.780	0.550500	0.275250	+	0.961373i	$0.411239\pi$
$660$	0	0
$661$	117.834i	0.178266i	0.996020	+	0.0891330i	$0.0284096\pi$
$661$	117.834i	0.178266i	−0.996020	+	0.0891330i	$0.971590\pi$
$662$	0	0
$663$	− 125.180i	− 0.188809i
$664$	0	0
$665$	101.980	0.153353
$666$	0	0
$667$	−790.794	−1.18560
$668$	0	0
$669$	− 36.1326i	− 0.0540099i
$670$	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
$675$	489.401	0.725039
$676$	0	0
$677$	125.796i	0.185813i	0.995675	+	0.0929066i	$0.0296158\pi$
$677$	125.796i	0.185813i	−0.995675	+	0.0929066i	$0.970384\pi$
$678$	0	0
$679$	254.277i	0.374487i
$680$	0	0
$681$	361.245	0.530462
$682$	0	0
$683$	553.775	0.810797	0.405399	−	0.914140i	$-0.367133\pi$
$683$	553.775	0.810797	0.405399	+	0.914140i	$0.367133\pi$
$684$	0	0
$685$	− 88.8274i	− 0.129675i
$686$	0	0
$687$	255.652i	0.372128i
$688$	0	0
$689$	151.598	0.220026
$690$	0	0
$691$	1046.83	1.51494	0.757471	−	0.652868i	$-0.226435\pi$
$691$	1046.83	1.51494	0.757471	+	0.652868i	$0.226435\pi$
$692$	0	0
$693$	31.5287i	0.0454960i
$694$	0	0
$695$	− 284.651i	− 0.409569i
$696$	0	0
$697$	−638.039	−0.915407
$698$	0	0
$699$	1431.02	2.04724
$700$	0	0
$701$	− 625.993i	− 0.893000i	−0.894784	−	0.446500i	$-0.852670\pi$
$701$	− 625.993i	− 0.893000i	0.894784	−	0.446500i	$-0.147330\pi$
$702$	0	0
$703$	1456.95i	2.07248i
$704$	0	0
$705$	191.196	0.271200
$706$	0	0
$707$	51.8995	0.0734081
$708$	0	0
$709$	593.492i	0.837083i	0.908198	+	0.418541i	$0.137459\pi$
$709$	593.492i	0.837083i	−0.908198	+	0.418541i	$0.862541\pi$
$710$	0	0
$711$	102.822i	0.144615i
$712$	0	0
$713$	1645.55	2.30792
$714$	0	0
$715$	10.7737	0.0150682
$716$	0	0
$717$	506.389i	0.706261i
$718$	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
$723$	1570.10	2.17165
$724$	0	0
$725$	507.323i	0.699756i
$726$	0	0
$727$	− 944.144i	− 1.29868i	−0.760496	−	0.649342i	$-0.775044\pi$
$727$	− 944.144i	− 1.29868i	0.760496	−	0.649342i	$-0.224956\pi$
$728$	0	0
$729$	405.489	0.556227
$730$	0	0
$731$	406.225	0.555712
$732$	0	0
$733$	− 218.254i	− 0.297755i	−0.988856	−	0.148878i	$-0.952434\pi$
$733$	− 218.254i	− 0.297755i	0.988856	−	0.148878i	$-0.0475660\pi$
$734$	0	0
$735$	− 37.0405i	− 0.0503953i
$736$	0	0
$737$	149.685	0.203101
$738$	0	0
$739$	7.29942	0.00987743	0.00493872	−	0.999988i	$-0.498428\pi$
$739$	7.29942	0.00987743	0.00493872	+	0.999988i	$0.498428\pi$
$740$	0	0
$741$	− 131.600i	− 0.177598i
$742$	0	0
$743$	106.867i	0.143832i	0.997411	+	0.0719159i	$0.0229113\pi$
$743$	106.867i	0.143832i	−0.997411	+	0.0719159i	$0.977089\pi$
$744$	0	0
$745$	−298.051	−0.400068
$746$	0	0
$747$	−9.60512	−0.0128583
$748$	0	0
$749$	− 41.2684i	− 0.0550980i
$750$	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
$755$	−177.849	−0.235562
$756$	0	0
$757$	704.275i	0.930350i	0.885219	+	0.465175i	$0.154009\pi$
$757$	704.275i	0.930350i	−0.885219	+	0.465175i	$0.845991\pi$
$758$	0	0
$759$	− 539.422i	− 0.710701i
$760$	0	0
$761$	−1002.93	−1.31791	−0.658955	−	0.752182i	$-0.729001\pi$
$761$	−1002.93	−1.31791	−0.658955	+	0.752182i	$0.729001\pi$
$762$	0	0
$763$	10.1909	0.0133564
$764$	0	0
$765$	− 97.4121i	− 0.127336i
$766$	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
$771$	1459.07	1.89244
$772$	0	0
$773$	− 564.265i	− 0.729968i	−0.931014	−	0.364984i	$-0.881075\pi$
$773$	− 564.265i	− 0.729968i	0.931014	−	0.364984i	$-0.118925\pi$
$774$	0	0
$775$	− 1055.68i	− 1.36217i
$776$	0	0
$777$	529.186	0.681063
$778$	0	0
$779$	−670.759	−0.861052
$780$	0	0
$781$	458.389i	0.586926i
$782$	0	0
$783$	486.195i	0.620939i
$784$	0	0
$785$	328.794	0.418846
$786$	0	0
$787$	−923.345	−1.17325	−0.586623	−	0.809860i	$-0.699543\pi$
$787$	−923.345	−1.17325	−0.586623	+	0.809860i	$0.699543\pi$
$788$	0	0
$789$	879.582i	1.11481i
$790$	0	0
$791$	− 36.4442i	− 0.0460735i
$792$	0	0
$793$	−58.4020	−0.0736469
$794$	0	0
$795$	517.588	0.651054
$796$	0	0
$797$	− 207.983i	− 0.260957i	−0.991451	−	0.130479i	$-0.958349\pi$
$797$	− 207.983i	− 0.260957i	0.991451	−	0.130479i	$-0.0416514\pi$
$798$	0	0
$799$	− 854.785i	− 1.06982i
$800$	0	0
$801$	117.058	0.146140
$802$	0	0
$803$	310.891	0.387162
$804$	0	0
$805$	144.439i	0.179428i
$806$	0	0
$807$	734.896i	0.910652i
$808$	0	0
$809$	340.540	0.420939	0.210470	−	0.977600i	$-0.432501\pi$
$809$	340.540	0.420939	0.210470	+	0.977600i	$0.432501\pi$
$810$	0	0
$811$	907.380	1.11884	0.559420	−	0.828884i	$-0.311024\pi$
$811$	907.380	1.11884	0.559420	+	0.828884i	$0.311024\pi$
$812$	0	0
$813$	1292.45i	1.58973i
$814$	0	0
$815$	− 372.791i	− 0.457412i
$816$	0	0
$817$	427.058	0.522715
$818$	0	0
$819$	−10.8944	−0.0133021
$820$	0	0
$821$	633.423i	0.771526i	0.922598	+	0.385763i	$0.126062\pi$
$821$	633.423i	0.771526i	−0.922598	+	0.385763i	$0.873938\pi$
$822$	0	0
$823$	143.649i	0.174544i	0.996185	+	0.0872718i	$0.0278149\pi$
$823$	143.649i	0.174544i	−0.996185	+	0.0872718i	$0.972185\pi$
$824$	0	0
$825$	−346.059	−0.419465
$826$	0	0
$827$	−1545.57	−1.86888	−0.934442	−	0.356114i	$-0.884101\pi$
$827$	−1545.57	−1.86888	−0.934442	+	0.356114i	$0.884101\pi$
$828$	0	0
$829$	− 743.956i	− 0.897413i	−0.893679	−	0.448707i	$-0.851885\pi$
$829$	− 743.956i	− 0.897413i	0.893679	−	0.448707i	$-0.148115\pi$
$830$	0	0
$831$	568.291i	0.683864i
$832$	0	0
$833$	−165.598	−0.198797
$834$	0	0
$835$	−328.723	−0.393681
$836$	0	0
$837$	− 1011.71i	− 1.20874i
$838$	0	0
$839$	− 96.3107i	− 0.114792i	−0.998351	−	0.0573961i	$-0.981720\pi$
$839$	− 96.3107i	− 0.114792i	0.998351	−	0.0573961i	$-0.0182798\pi$
$840$	0	0
$841$	337.000	0.400713
$842$	0	0
$843$	1439.99	1.70818
$844$	0	0
$845$	− 258.201i	− 0.305563i
$846$	0	0
$847$	− 266.909i	− 0.315123i
$848$	0	0
$849$	−1179.40	−1.38917
$850$	0	0
$851$	−2063.56	−2.42486
$852$	0	0
$853$	− 904.866i	− 1.06080i	−0.847746	−	0.530402i	$-0.822041\pi$
$853$	− 904.866i	− 1.06080i	0.847746	−	0.530402i	$-0.177959\pi$
$854$	0	0
$855$	− 102.408i	− 0.119775i
$856$	0	0
$857$	160.932	0.187785	0.0938926	−	0.995582i	$-0.470069\pi$
$857$	160.932	0.187785	0.0938926	+	0.995582i	$0.470069\pi$
$858$	0	0
$859$	−231.693	−0.269724	−0.134862	−	0.990864i	$-0.543059\pi$
$859$	−231.693	−0.269724	−0.134862	+	0.990864i	$0.543059\pi$
$860$	0	0
$861$	243.629i	0.282961i
$862$	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
$867$	−924.046	−1.06580
$868$	0	0
$869$	173.583i	0.199750i
$870$	0	0
$871$	51.7223i	0.0593827i
$872$	0	0
$873$	255.344	0.292490
$874$	0	0
$875$	195.176	0.223058
$876$	0	0
$877$	1436.14i	1.63755i	0.574111	+	0.818777i	$0.305348\pi$
$877$	1436.14i	1.63755i	−0.574111	+	0.818777i	$0.694652\pi$
$878$	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
$883$	−1277.99	−1.44733	−0.723664	−	0.690153i	$-0.757543\pi$
$883$	−1277.99	−1.44733	−0.723664	+	0.690153i	$0.757543\pi$
$884$	0	0
$885$	− 325.726i	− 0.368052i
$886$	0	0
$887$	980.717i	1.10566i	0.833295	+	0.552828i	$0.186451\pi$
$887$	980.717i	1.10566i	−0.833295	+	0.552828i	$0.813549\pi$
$888$	0	0
$889$	−330.784	−0.372085
$890$	0	0
$891$	−438.897	−0.492590
$892$	0	0
$893$	− 898.621i	− 1.00629i
$894$	0	0
$895$	88.6605i	0.0990621i
$896$	0	0
$897$	186.392	0.207795
$898$	0	0
$899$	1048.76	1.16659
$900$	0	0
$901$	− 2313.99i	− 2.56825i
$902$	0	0
$903$	− 155.114i	− 0.171776i
$904$	0	0
$905$	−505.578	−0.558649
$906$	0	0
$907$	658.372	0.725878	0.362939	−	0.931813i	$-0.381773\pi$
$907$	658.372	0.725878	0.362939	+	0.931813i	$0.381773\pi$
$908$	0	0
$909$	− 52.1173i	− 0.0573348i
$910$	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
$915$	−199.397	−0.217920
$916$	0	0
$917$	− 265.502i	− 0.289533i
$918$	0	0
$919$	1339.73i	1.45782i	0.684611	+	0.728908i	$0.259972\pi$
$919$	1339.73i	1.45782i	−0.684611	+	0.728908i	$0.740028\pi$
$920$	0	0
$921$	762.500	0.827904
$922$	0	0
$923$	−158.392	−0.171606
$924$	0	0
$925$	1323.85i	1.43119i
$926$	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
$931$	−174.090	−0.186993
$932$	0	0
$933$	− 42.3320i	− 0.0453719i
$934$	0	0
$935$	− 164.450i	− 0.175883i
$936$	0	0
$937$	610.235	0.651265	0.325633	−	0.945496i	$-0.394423\pi$
$937$	610.235	0.651265	0.325633	+	0.945496i	$0.394423\pi$
$938$	0	0
$939$	−1399.99	−1.49094
$940$	0	0
$941$	− 1852.90i	− 1.96907i	−0.175175	−	0.984537i	$-0.556049\pi$
$941$	− 1852.90i	− 1.96907i	0.175175	−	0.984537i	$-0.443951\pi$
$942$	0	0
$943$	− 950.032i	− 1.00746i
$944$	0	0
$945$	88.8040	0.0939725
$946$	0	0
$947$	1832.90	1.93548	0.967738	−	0.251959i	$-0.0810748\pi$
$947$	1832.90	1.93548	0.967738	+	0.251959i	$0.0810748\pi$
$948$	0	0
$949$	107.426i	0.113199i
$950$	0	0
$951$	444.927i	0.467852i
$952$	0	0
$953$	349.687	0.366933	0.183467	−	0.983026i	$-0.441268\pi$
$953$	349.687	0.366933	0.183467	+	0.983026i	$0.441268\pi$
$954$	0	0
$955$	150.432	0.157521
$956$	0	0
$957$	− 343.792i	− 0.359239i
$958$	0	0
$959$	151.638i	0.158121i
$960$	0	0
$961$	−1221.35	−1.27092
$962$	0	0
$963$	−41.4416	−0.0430338
$964$	0	0
$965$	− 243.796i	− 0.252639i
$966$	0	0
$967$	632.128i	0.653700i	0.945076	+	0.326850i	$0.105987\pi$
$967$	632.128i	0.653700i	−0.945076	+	0.326850i	$0.894013\pi$
$968$	0	0
$969$	−2008.74	−2.07301
$970$	0	0
$971$	−656.497	−0.676104	−0.338052	−	0.941128i	$-0.609768\pi$
$971$	−656.497	−0.676104	−0.338052	+	0.941128i	$0.609768\pi$
$972$	0	0
$973$	485.929i	0.499413i
$974$	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
$979$	197.616	0.201855
$980$	0	0
$981$	− 10.2337i	− 0.0104319i
$982$	0	0
$983$	− 698.607i	− 0.710689i	−0.934735	−	0.355345i	$-0.884364\pi$
$983$	− 698.607i	− 0.710689i	0.934735	−	0.355345i	$-0.115636\pi$
$984$	0	0
$985$	192.362	0.195291
$986$	0	0
$987$	−326.392	−0.330691
$988$	0	0
$989$	604.865i	0.611592i
$990$	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
$995$	280.483	0.281892
$996$	0	0
$997$	− 52.3910i	− 0.0525487i	−0.999655	−	0.0262743i	$-0.991636\pi$
$997$	− 52.3910i	− 0.0525487i	0.999655	−	0.0262743i	$-0.00836434\pi$
$998$	0	0
$999$	1268.71i	1.26998i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.g.a.15.1		4
3.2	odd	2		2016.3.g.a.1135.3		4
4.3	odd	2		56.3.g.a.43.4	yes	4
7.6	odd	2		1568.3.g.h.687.4		4
8.3	odd	2	inner	224.3.g.a.15.2		4
8.5	even	2		56.3.g.a.43.3	✓	4
12.11	even	2		504.3.g.a.379.1		4
16.3	odd	4		1792.3.d.g.1023.7		8
16.5	even	4		1792.3.d.g.1023.8		8
16.11	odd	4		1792.3.d.g.1023.2		8
16.13	even	4		1792.3.d.g.1023.1		8
24.5	odd	2		504.3.g.a.379.2		4
24.11	even	2		2016.3.g.a.1135.2		4
28.3	even	6		392.3.k.j.275.1		8
28.11	odd	6		392.3.k.i.275.1		8
28.19	even	6		392.3.k.j.67.3		8
28.23	odd	6		392.3.k.i.67.3		8
28.27	even	2		392.3.g.h.99.4		4
56.5	odd	6		392.3.k.j.67.1		8
56.13	odd	2		392.3.g.h.99.3		4
56.27	even	2		1568.3.g.h.687.3		4
56.37	even	6		392.3.k.i.67.1		8
56.45	odd	6		392.3.k.j.275.3		8
56.53	even	6		392.3.k.i.275.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.g.a.43.3	✓	4	8.5	even	2
56.3.g.a.43.4	yes	4	4.3	odd	2
224.3.g.a.15.1		4	1.1	even	1	trivial
224.3.g.a.15.2		4	8.3	odd	2	inner
392.3.g.h.99.3		4	56.13	odd	2
392.3.g.h.99.4		4	28.27	even	2
392.3.k.i.67.1		8	56.37	even	6
392.3.k.i.67.3		8	28.23	odd	6
392.3.k.i.275.1		8	28.11	odd	6
392.3.k.i.275.3		8	56.53	even	6
392.3.k.j.67.1		8	56.5	odd	6
392.3.k.j.67.3		8	28.19	even	6
392.3.k.j.275.1		8	28.3	even	6
392.3.k.j.275.3		8	56.45	odd	6
504.3.g.a.379.1		4	12.11	even	2
504.3.g.a.379.2		4	24.5	odd	2
1568.3.g.h.687.3		4	56.27	even	2
1568.3.g.h.687.4		4	7.6	odd	2
1792.3.d.g.1023.1		8	16.13	even	4
1792.3.d.g.1023.2		8	16.11	odd	4
1792.3.d.g.1023.7		8	16.3	odd	4
1792.3.d.g.1023.8		8	16.5	even	4
2016.3.g.a.1135.2		4	24.11	even	2
2016.3.g.a.1135.3		4	3.2	odd	2

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:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	224.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.41421 q^{3} -1.54985i q^{5} +2.64575i q^{7} +2.65685 q^{9} +O(q^{10})\)	\(q-3.41421 q^{3} -1.54985i q^{5} +2.64575i q^{7} +2.65685 q^{9} +4.48528 q^{11} +1.54985i q^{13} +5.29150i q^{15} +23.6569 q^{17} +24.8701 q^{19} -9.03316i q^{21} +35.2248i q^{23} +22.5980 q^{25} +21.6569 q^{27} +22.4499i q^{29} -46.7156i q^{31} -15.3137 q^{33} +4.10051 q^{35} +58.5826i q^{37} -5.29150i q^{39} -26.9706 q^{41} +17.1716 q^{43} -4.11771i q^{45} -36.1326i q^{47} -7.00000 q^{49} -80.7696 q^{51} -97.8149i q^{53} -6.95149i q^{55} -84.9117 q^{57} -61.5563 q^{59} +37.6825i q^{61} +7.02938i q^{63} +2.40202 q^{65} +33.3726 q^{67} -120.265i q^{69} +102.199i q^{71} +69.3137 q^{73} -77.1543 q^{75} +11.8669i q^{77} +38.7005i q^{79} -97.8528 q^{81} -3.61522 q^{83} -36.6645i q^{85} -76.6489i q^{87} +44.0589 q^{89} -4.10051 q^{91} +159.497i q^{93} -38.5447i q^{95} +96.1076 q^{97} +11.9167 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{3} - 12 q^{9}+O(q^{10}) \) 4 * q - 8 * q^3 - 12 * q^9	\( 4 q - 8 q^{3} - 12 q^{9} - 16 q^{11} + 72 q^{17} - 8 q^{19} - 68 q^{25} + 64 q^{27} - 16 q^{33} + 56 q^{35} - 40 q^{41} + 80 q^{43} - 28 q^{49} - 176 q^{51} - 136 q^{57} - 184 q^{59} + 168 q^{65} + 224 q^{67} + 232 q^{73} - 88 q^{75} - 52 q^{81} - 88 q^{83} + 312 q^{89} - 56 q^{91} - 136 q^{97} + 240 q^{99}+O(q^{100}) \) 4 * q - 8 * q^3 - 12 * q^9 - 16 * q^11 + 72 * q^17 - 8 * q^19 - 68 * q^25 + 64 * q^27 - 16 * q^33 + 56 * q^35 - 40 * q^41 + 80 * q^43 - 28 * q^49 - 176 * q^51 - 136 * q^57 - 184 * q^59 + 168 * q^65 + 224 * q^67 + 232 * q^73 - 88 * q^75 - 52 * q^81 - 88 * q^83 + 312 * q^89 - 56 * q^91 - 136 * q^97 + 240 * q^99

Introduction

L-functions

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