LMFDB - Embedded newform 224.3.h.d.209.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,3,Mod(209,224)]

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([0, 1, 1]))

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

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

chi := DirichletCharacter("224.209");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	224.h (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:	$8$
Coefficient field:	8.0.976966189056.51
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{6} + 123x^{4} - 300x^{3} + 86x^{2} + 300x + 526 $ x^8 - 10x^6 + 123x^4 - 300x^3 + 86x^2 + 300*x + 526
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.7
Root			$2.37200 + 0.719687i$ of defining polynomial
Character	$\chi$	$=$	224.209
Dual form			224.3.h.d.209.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.11439 q^{3} -1.10245 q^{5} +(3.73205 - 5.92214i) q^{7} +7.92820 q^{9} +O(q^{10})$	$q+4.11439 q^{3} -1.10245 q^{5} +(3.73205 - 5.92214i) q^{7} +7.92820 q^{9} -10.7436i q^{11} +13.1502 q^{13} -4.53590 q^{15} +20.5149i q^{17} +24.3909 q^{19} +(15.3551 - 24.3660i) q^{21} -16.7846 q^{23} -23.7846 q^{25} -4.40979 q^{27} -4.21478i q^{29} +52.8741i q^{31} -44.2035i q^{33} +(-4.11439 + 6.52885i) q^{35} +9.97227i q^{37} +54.1051 q^{39} +23.6886i q^{41} -65.2331i q^{43} -8.74043 q^{45} -52.8741i q^{47} +(-21.1436 - 44.2035i) q^{49} +84.4063i q^{51} +47.1893i q^{53} +11.8443i q^{55} +100.354 q^{57} -40.2577 q^{59} -1.10245 q^{61} +(29.5885 - 46.9520i) q^{63} -14.4974 q^{65} +65.2331i q^{67} -69.0584 q^{69} -113.033 q^{71} +91.5806i q^{73} -97.8592 q^{75} +(-63.6253 - 40.0958i) q^{77} -27.1769 q^{79} -89.4974 q^{81} -2.34199 q^{83} -22.6166i q^{85} -17.3412i q^{87} +50.5508i q^{89} +(49.0773 - 77.8775i) q^{91} +217.545i q^{93} -26.8897 q^{95} -74.2394i q^{97} -85.1777i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 16 * q^7 + 8 * q^9	$ 8 q + 16 q^{7} + 8 q^{9} - 64 q^{15} + 32 q^{23} - 24 q^{25} + 128 q^{39} - 280 q^{49} + 304 q^{57} + 112 q^{63} + 272 q^{65} - 544 q^{71} + 32 q^{79} - 328 q^{81} + 256 q^{95}+O(q^{100}) $ 8 * q + 16 * q^7 + 8 * q^9 - 64 * q^15 + 32 * q^23 - 24 * q^25 + 128 * q^39 - 280 * q^49 + 304 * q^57 + 112 * q^63 + 272 * q^65 - 544 * q^71 + 32 * q^79 - 328 * q^81 + 256 * q^95

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
$3$	4.11439			1.37146			0.685732	−	0.727854i	$-0.259482\pi$
$3$	4.11439			1.37146			0.685732	+	0.727854i	$0.259482\pi$
$4$		0			0
$5$	−1.10245			−0.220489			−0.110245	−	0.993904i	$-0.535163\pi$
$5$	−1.10245			−0.220489			−0.110245	+	0.993904i	$0.535163\pi$
$6$		0			0
$7$	3.73205	−	5.92214i	0.533150	−	0.846021i
$7$	3.73205	−	5.92214i	0.533150	−	0.846021i
$8$		0			0
$9$	7.92820			0.880911
$10$		0			0
$11$		−	10.7436i		−	0.976694i	−0.872650	−	0.488347i	$-0.837600\pi$
$11$		−	10.7436i		−	0.976694i	0.872650	−	0.488347i	$-0.162400\pi$
$12$		0			0
$13$	13.1502			1.01156			0.505778	−	0.862664i	$-0.331206\pi$
$13$	13.1502			1.01156			0.505778	+	0.862664i	$0.331206\pi$
$14$		0			0
$15$	−4.53590			−0.302393
$16$		0			0
$17$			20.5149i			1.20676i	0.797454	+	0.603380i	$0.206180\pi$
$17$			20.5149i			1.20676i	−0.797454	+	0.603380i	$0.793820\pi$
$18$		0			0
$19$	24.3909			1.28373			0.641867	−	0.766816i	$-0.278160\pi$
$19$	24.3909			1.28373			0.641867	+	0.766816i	$0.278160\pi$
$20$		0			0
$21$	15.3551	−	24.3660i	0.731196	−	1.16029i
$22$		0			0
$23$	−16.7846			−0.729766			−0.364883	−	0.931053i	$-0.618891\pi$
$23$	−16.7846			−0.729766			−0.364883	+	0.931053i	$0.618891\pi$
$24$		0			0
$25$	−23.7846			−0.951384
$26$		0			0
$27$	−4.40979			−0.163326
$28$		0			0
$29$		−	4.21478i		−	0.145337i	−0.997356	−	0.0726686i	$-0.976848\pi$
$29$		−	4.21478i		−	0.145337i	0.997356	−	0.0726686i	$-0.0231515\pi$
$30$		0			0
$31$			52.8741i			1.70562i	0.522224	+	0.852808i	$0.325102\pi$
$31$			52.8741i			1.70562i	−0.522224	+	0.852808i	$0.674898\pi$
$32$		0			0
$33$		−	44.2035i		−	1.33950i
$34$		0			0
$35$	−4.11439	+	6.52885i	−0.117554	+	0.186539i
$36$		0			0
$37$			9.97227i			0.269521i	0.990878	+	0.134760i	$0.0430265\pi$
$37$			9.97227i			0.269521i	−0.990878	+	0.134760i	$0.956974\pi$
$38$		0			0
$39$	54.1051			1.38731
$40$		0			0
$41$			23.6886i			0.577770i	0.957364	+	0.288885i	$0.0932846\pi$
$41$			23.6886i			0.577770i	−0.957364	+	0.288885i	$0.906715\pi$
$42$		0			0
$43$		−	65.2331i		−	1.51705i	−0.651644	−	0.758525i	$-0.725920\pi$
$43$		−	65.2331i		−	1.51705i	0.651644	−	0.758525i	$-0.274080\pi$
$44$		0			0
$45$	−8.74043			−0.194232
$46$		0			0
$47$		−	52.8741i		−	1.12498i	−0.826804	−	0.562491i	$-0.809843\pi$
$47$		−	52.8741i		−	1.12498i	0.826804	−	0.562491i	$-0.190157\pi$
$48$		0			0
$49$	−21.1436	−	44.2035i	−0.431502	−	0.902112i
$50$		0			0
$51$			84.4063i			1.65503i
$52$		0			0
$53$			47.1893i			0.890364i	0.895440	+	0.445182i	$0.146861\pi$
$53$			47.1893i			0.890364i	−0.895440	+	0.445182i	$0.853139\pi$
$54$		0			0
$55$			11.8443i			0.215351i
$56$		0			0
$57$	100.354			1.76059
$58$		0			0
$59$	−40.2577			−0.682334			−0.341167	−	0.940003i	$-0.610822\pi$
$59$	−40.2577			−0.682334			−0.341167	+	0.940003i	$0.610822\pi$
$60$		0			0
$61$	−1.10245			−0.0180729			−0.00903645	−	0.999959i	$-0.502876\pi$
$61$	−1.10245			−0.0180729			−0.00903645	+	0.999959i	$0.502876\pi$
$62$		0			0
$63$	29.5885	−	46.9520i	0.469658	−	0.745269i
$64$		0			0
$65$	−14.4974			−0.223037
$66$		0			0
$67$			65.2331i			0.973629i	0.873505	+	0.486815i	$0.161841\pi$
$67$			65.2331i			0.973629i	−0.873505	+	0.486815i	$0.838159\pi$
$68$		0			0
$69$	−69.0584			−1.00085
$70$		0			0
$71$	−113.033			−1.59202			−0.796009	−	0.605284i	$-0.793059\pi$
$71$	−113.033			−1.59202			−0.796009	+	0.605284i	$0.793059\pi$
$72$		0			0
$73$			91.5806i			1.25453i	0.778806	+	0.627265i	$0.215826\pi$
$73$			91.5806i			1.25453i	−0.778806	+	0.627265i	$0.784174\pi$
$74$		0			0
$75$	−97.8592			−1.30479
$76$		0			0
$77$	−63.6253	−	40.0958i	−0.826303	−	0.520724i
$78$		0			0
$79$	−27.1769			−0.344012			−0.172006	−	0.985096i	$-0.555025\pi$
$79$	−27.1769			−0.344012			−0.172006	+	0.985096i	$0.555025\pi$
$80$		0			0
$81$	−89.4974			−1.10491
$82$		0			0
$83$	−2.34199			−0.0282168			−0.0141084	−	0.999900i	$-0.504491\pi$
$83$	−2.34199			−0.0282168			−0.0141084	+	0.999900i	$0.504491\pi$
$84$		0			0
$85$		−	22.6166i		−	0.266078i
$86$		0			0
$87$		−	17.3412i		−	0.199325i
$88$		0			0
$89$			50.5508i			0.567987i	0.958826	+	0.283993i	$0.0916594\pi$
$89$			50.5508i			0.567987i	−0.958826	+	0.283993i	$0.908341\pi$
$90$		0			0
$91$	49.0773	−	77.8775i	0.539311	−	0.855797i
$92$		0			0
$93$			217.545i			2.33919i
$94$		0			0
$95$	−26.8897			−0.283050
$96$		0			0
$97$		−	74.2394i		−	0.765355i	−0.923882	−	0.382677i	$-0.875002\pi$
$97$		−	74.2394i		−	0.765355i	0.923882	−	0.382677i	$-0.124998\pi$
$98$		0			0
$99$		−	85.1777i		−	0.860381i
$100$		0			0
$101$	−57.0898			−0.565246			−0.282623	−	0.959231i	$-0.591204\pi$
$101$	−57.0898			−0.565246			−0.282623	+	0.959231i	$0.591204\pi$
$102$		0			0
$103$			17.3412i			0.168362i	0.996450	+	0.0841808i	$0.0268273\pi$
$103$			17.3412i			0.168362i	−0.996450	+	0.0841808i	$0.973173\pi$
$104$		0			0
$105$	−16.9282	+	26.8622i	−0.161221	+	0.255831i
$106$		0			0
$107$			102.037i			0.953615i	0.879008	+	0.476808i	$0.158206\pi$
$107$			102.037i			0.953615i	−0.879008	+	0.476808i	$0.841794\pi$
$108$		0			0
$109$		−	116.279i		−	1.06678i	−0.845869	−	0.533391i	$-0.820917\pi$
$109$		−	116.279i		−	1.06678i	0.845869	−	0.533391i	$-0.179083\pi$
$110$		0			0
$111$			41.0298i			0.369638i
$112$		0			0
$113$	10.3538			0.0916268			0.0458134	−	0.998950i	$-0.485412\pi$
$113$	10.3538			0.0916268			0.0458134	+	0.998950i	$0.485412\pi$
$114$		0			0
$115$	18.5041			0.160906
$116$		0			0
$117$	104.258			0.891091
$118$		0			0
$119$	121.492	+	76.5627i	1.02094	+	0.643384i
$120$		0			0
$121$	5.57437			0.0460692
$122$		0			0
$123$			97.4640i			0.792391i
$124$		0			0
$125$	53.7825			0.430260
$126$		0			0
$127$	111.636			0.879023			0.439511	−	0.898237i	$-0.355152\pi$
$127$	111.636			0.879023			0.439511	+	0.898237i	$0.355152\pi$
$128$		0			0
$129$		−	268.395i		−	2.08058i
$130$		0			0
$131$	77.5826			0.592234			0.296117	−	0.955152i	$-0.404308\pi$
$131$	77.5826			0.592234			0.296117	+	0.955152i	$0.404308\pi$
$132$		0			0
$133$	91.0282	−	144.447i	0.684423	−	1.08607i
$134$		0			0
$135$	4.86156			0.0360116
$136$		0			0
$137$	148.851			1.08651			0.543253	−	0.839569i	$-0.317193\pi$
$137$	148.851			1.08651			0.543253	+	0.839569i	$0.317193\pi$
$138$		0			0
$139$	62.5808			0.450222			0.225111	−	0.974333i	$-0.427726\pi$
$139$	62.5808			0.450222			0.225111	+	0.974333i	$0.427726\pi$
$140$		0			0
$141$		−	217.545i		−	1.54287i
$142$		0			0
$143$		−	141.281i		−	0.987980i
$144$		0			0
$145$			4.64657i			0.0320453i
$146$		0			0
$147$	−86.9930	−	181.870i	−0.591789	−	1.23721i
$148$		0			0
$149$			190.300i			1.27718i	0.769547	+	0.638590i	$0.220482\pi$
$149$			190.300i			1.27718i	−0.769547	+	0.638590i	$0.779518\pi$
$150$		0			0
$151$	−10.3538			−0.0685684			−0.0342842	−	0.999412i	$-0.510915\pi$
$151$	−10.3538			−0.0685684			−0.0342842	+	0.999412i	$0.510915\pi$
$152$		0			0
$153$			162.646i			1.06305i
$154$		0			0
$155$		−	58.2909i		−	0.376070i
$156$		0			0
$157$	311.433			1.98365			0.991824	−	0.127610i	$-0.0407306\pi$
$157$	311.433			1.98365			0.991824	+	0.127610i	$0.0407306\pi$
$158$		0			0
$159$			194.155i			1.22110i
$160$		0			0
$161$	−62.6410	+	99.4009i	−0.389075	+	0.617397i
$162$		0			0
$163$			7.65820i			0.0469828i	0.999724	+	0.0234914i	$0.00747824\pi$
$163$			7.65820i			0.0469828i	−0.999724	+	0.0234914i	$0.992522\pi$
$164$		0			0
$165$			48.7320i			0.295346i
$166$		0			0
$167$		−	135.784i		−	0.813079i	−0.913633	−	0.406539i	$-0.866735\pi$
$167$		−	135.784i		−	0.813079i	0.913633	−	0.406539i	$-0.133265\pi$
$168$		0			0
$169$	3.92820			0.0232438
$170$		0			0
$171$	193.376			1.13086
$172$		0			0
$173$	−283.086			−1.63633			−0.818167	−	0.574980i	$-0.805010\pi$
$173$	−283.086			−1.63633			−0.818167	+	0.574980i	$0.805010\pi$
$174$		0			0
$175$	−88.7654	+	140.856i	−0.507231	+	0.804891i
$176$		0			0
$177$	−165.636			−0.935796
$178$		0			0
$179$		−	56.8036i		−	0.317339i	−0.987332	−	0.158669i	$-0.949280\pi$
$179$		−	56.8036i		−	0.317339i	0.987332	−	0.158669i	$-0.0507204\pi$
$180$		0			0
$181$	171.976			0.950144			0.475072	−	0.879947i	$-0.342422\pi$
$181$	171.976			0.950144			0.475072	+	0.879947i	$0.342422\pi$
$182$		0			0
$183$	−4.53590			−0.0247863
$184$		0			0
$185$		−	10.9939i		−	0.0594265i
$186$		0			0
$187$	220.405			1.17863
$188$		0			0
$189$	−16.4576	+	26.1154i	−0.0870770	+	0.138177i
$190$		0			0
$191$	−119.962			−0.628071			−0.314035	−	0.949411i	$-0.601681\pi$
$191$	−119.962			−0.628071			−0.314035	+	0.949411i	$0.601681\pi$
$192$		0			0
$193$	11.2154			0.0581108			0.0290554	−	0.999578i	$-0.490750\pi$
$193$	11.2154			0.0581108			0.0290554	+	0.999578i	$0.490750\pi$
$194$		0			0
$195$	−59.6480			−0.305887
$196$		0			0
$197$		−	207.986i		−	1.05577i	−0.849317	−	0.527883i	$-0.822986\pi$
$197$		−	207.986i		−	1.05577i	0.849317	−	0.527883i	$-0.177014\pi$
$198$		0			0
$199$		−	101.102i		−	0.508048i	−0.967198	−	0.254024i	$-0.918246\pi$
$199$		−	101.102i		−	0.508048i	0.967198	−	0.254024i	$-0.0817543\pi$
$200$		0			0
$201$			268.395i			1.33530i
$202$		0			0
$203$	−24.9605	−	15.7298i	−0.122958	−	0.0774865i
$204$		0			0
$205$		−	26.1154i		−	0.127392i
$206$		0			0
$207$	−133.072			−0.642859
$208$		0			0
$209$		−	262.047i		−	1.25381i
$210$		0			0
$211$		−	232.503i		−	1.10191i	−0.834535	−	0.550955i	$-0.814263\pi$
$211$		−	232.503i		−	1.10191i	0.834535	−	0.550955i	$-0.185737\pi$
$212$		0			0
$213$	−465.063			−2.18340
$214$		0			0
$215$			71.9161i			0.334494i
$216$		0			0
$217$	313.128	+	197.329i	1.44299	+	0.909350i
$218$		0			0
$219$			376.798i			1.72054i
$220$		0			0
$221$			269.776i			1.22070i
$222$		0			0
$223$		−	140.431i		−	0.629734i	−0.949136	−	0.314867i	$-0.898040\pi$
$223$		−	140.431i		−	0.629734i	0.949136	−	0.314867i	$-0.101960\pi$
$224$		0			0
$225$	−188.569			−0.838085
$226$		0			0
$227$	−180.738			−0.796202			−0.398101	−	0.917342i	$-0.630331\pi$
$227$	−180.738			−0.796202			−0.398101	+	0.917342i	$0.630331\pi$
$228$		0			0
$229$	153.630			0.670875			0.335437	−	0.942063i	$-0.391116\pi$
$229$	153.630			0.670875			0.335437	+	0.942063i	$0.391116\pi$
$230$		0			0
$231$	−261.779	−	164.970i	−1.13324	−	0.714154i
$232$		0			0
$233$	106.287			0.456168			0.228084	−	0.973641i	$-0.426754\pi$
$233$	106.287			0.456168			0.228084	+	0.973641i	$0.426754\pi$
$234$		0			0
$235$			58.2909i			0.248046i
$236$		0			0
$237$	−111.816			−0.471799
$238$		0			0
$239$	371.923			1.55616			0.778082	−	0.628163i	$-0.216193\pi$
$239$	371.923			1.55616			0.778082	+	0.628163i	$0.216193\pi$
$240$		0			0
$241$			7.82024i			0.0324491i	0.999868	+	0.0162246i	$0.00516467\pi$
$241$			7.82024i			0.0324491i	−0.999868	+	0.0162246i	$0.994835\pi$
$242$		0			0
$243$	−328.539			−1.35201
$244$		0			0
$245$	23.3097	+	48.7320i	0.0951416	+	0.198906i
$246$		0			0
$247$	320.746			1.29857
$248$		0			0
$249$	−9.63586			−0.0386982
$250$		0			0
$251$	−211.015			−0.840699			−0.420350	−	0.907362i	$-0.638093\pi$
$251$	−211.015			−0.840699			−0.420350	+	0.907362i	$0.638093\pi$
$252$		0			0
$253$			180.328i			0.712758i
$254$		0			0
$255$		−	93.0536i		−	0.364916i
$256$		0			0
$257$		−	293.556i		−	1.14224i	−0.820866	−	0.571121i	$-0.806509\pi$
$257$		−	293.556i		−	1.14224i	0.820866	−	0.571121i	$-0.193491\pi$
$258$		0			0
$259$	59.0572	+	37.2170i	0.228020	+	0.143695i
$260$		0			0
$261$		−	33.4156i		−	0.128029i
$262$		0			0
$263$	−64.0385			−0.243492			−0.121746	−	0.992561i	$-0.538849\pi$
$263$	−64.0385			−0.243492			−0.121746	+	0.992561i	$0.538849\pi$
$264$		0			0
$265$		−	52.0237i		−	0.196316i
$266$		0			0
$267$			207.986i			0.778973i
$268$		0			0
$269$	333.798			1.24089			0.620443	−	0.784251i	$-0.286953\pi$
$269$	333.798			1.24089			0.620443	+	0.784251i	$0.286953\pi$
$270$		0			0
$271$		−	142.131i		−	0.524470i	−0.965004	−	0.262235i	$-0.915540\pi$
$271$		−	142.131i		−	0.524470i	0.965004	−	0.262235i	$-0.0844596\pi$
$272$		0			0
$273$	201.923	−	320.418i	0.739645	−	1.17369i
$274$		0			0
$275$			255.533i			0.929211i
$276$		0			0
$277$			124.709i			0.450212i	0.974334	+	0.225106i	$0.0722729\pi$
$277$			124.709i			0.450212i	−0.974334	+	0.225106i	$0.927727\pi$
$278$		0			0
$279$			419.197i			1.50250i
$280$		0			0
$281$	−297.138			−1.05743			−0.528716	−	0.848799i	$-0.677326\pi$
$281$	−297.138			−1.05743			−0.528716	+	0.848799i	$0.677326\pi$
$282$		0			0
$283$	−139.277			−0.492146			−0.246073	−	0.969251i	$-0.579140\pi$
$283$	−139.277			−0.492146			−0.246073	+	0.969251i	$0.579140\pi$
$284$		0			0
$285$	−110.635			−0.388192
$286$		0			0
$287$	140.287	+	88.4070i	0.488806	+	0.308038i
$288$		0			0
$289$	−131.862			−0.456268
$290$		0			0
$291$		−	305.450i		−	1.04966i
$292$		0			0
$293$	−372.695			−1.27200			−0.635998	−	0.771690i	$-0.719411\pi$
$293$	−372.695			−1.27200			−0.635998	+	0.771690i	$0.719411\pi$
$294$		0			0
$295$	44.3820			0.150447
$296$		0			0
$297$			47.3772i			0.159519i
$298$		0			0
$299$	−220.721			−0.738198
$300$		0			0
$301$	−386.320	−	243.453i	−1.28346	−	0.808815i
$302$		0			0
$303$	−234.890			−0.775214
$304$		0			0
$305$	1.21539			0.00398489
$306$		0			0
$307$	358.543			1.16789			0.583946	−	0.811793i	$-0.301508\pi$
$307$	358.543			1.16789			0.583946	+	0.811793i	$0.301508\pi$
$308$		0			0
$309$			71.3486i			0.230902i
$310$		0			0
$311$			542.286i			1.74369i	0.489786	+	0.871843i	$0.337075\pi$
$311$			542.286i			1.74369i	−0.489786	+	0.871843i	$0.662925\pi$
$312$		0			0
$313$		−	146.778i		−	0.468939i	−0.972123	−	0.234470i	$-0.924665\pi$
$313$		−	146.778i		−	0.468939i	0.972123	−	0.234470i	$-0.0753354\pi$
$314$		0			0
$315$	−32.6197	+	51.7621i	−0.103555	+	0.164324i
$316$		0			0
$317$		−	101.265i		−	0.319449i	−0.987162	−	0.159725i	$-0.948939\pi$
$317$		−	101.265i		−	0.319449i	0.987162	−	0.159725i	$-0.0510606\pi$
$318$		0			0
$319$	−45.2820			−0.141950
$320$		0			0
$321$			419.819i			1.30785i
$322$		0			0
$323$			500.378i			1.54916i
$324$		0			0
$325$	−312.773			−0.962378
$326$		0			0
$327$		−	478.418i		−	1.46305i
$328$		0			0
$329$	−313.128	−	197.329i	−0.951757	−	0.599784i
$330$		0			0
$331$		−	492.609i		−	1.48824i	−0.668043	−	0.744122i	$-0.732868\pi$
$331$		−	492.609i		−	1.48824i	0.668043	−	0.744122i	$-0.267132\pi$
$332$		0			0
$333$			79.0622i			0.237424i
$334$		0			0
$335$		−	71.9161i		−	0.214675i
$336$		0			0
$337$	−137.626			−0.408384			−0.204192	−	0.978931i	$-0.565457\pi$
$337$	−137.626			−0.408384			−0.204192	+	0.978931i	$0.565457\pi$
$338$		0			0
$339$	42.5997			0.125663
$340$		0			0
$341$	568.060			1.66587
$342$		0			0
$343$	−340.688	−	39.7542i	−0.993261	−	0.115902i
$344$		0			0
$345$	76.1333			0.220676
$346$		0			0
$347$		−	385.999i		−	1.11239i	−0.831052	−	0.556195i	$-0.812261\pi$
$347$		−	385.999i		−	1.11239i	0.831052	−	0.556195i	$-0.187739\pi$
$348$		0			0
$349$	154.654			0.443133			0.221567	−	0.975145i	$-0.428883\pi$
$349$	154.654			0.443133			0.221567	+	0.975145i	$0.428883\pi$
$350$		0			0
$351$	−57.9897			−0.165213
$352$		0			0
$353$			536.789i			1.52065i	0.649543	+	0.760325i	$0.274960\pi$
$353$			536.789i			1.52065i	−0.649543	+	0.760325i	$0.725040\pi$
$354$		0			0
$355$	124.613			0.351023
$356$		0			0
$357$	499.867	+	315.009i	1.40019	+	0.882377i
$358$		0			0
$359$	−148.918			−0.414813			−0.207407	−	0.978255i	$-0.566502\pi$
$359$	−148.918			−0.414813			−0.207407	+	0.978255i	$0.566502\pi$
$360$		0			0
$361$	233.918			0.647972
$362$		0			0
$363$	22.9351			0.0631822
$364$		0			0
$365$		−	100.963i		−	0.276611i
$366$		0			0
$367$			351.927i			0.958930i	0.877561	+	0.479465i	$0.159169\pi$
$367$			351.927i			0.958930i	−0.877561	+	0.479465i	$0.840831\pi$
$368$		0			0
$369$			187.808i			0.508964i
$370$		0			0
$371$	279.462	+	176.113i	0.753267	+	0.474698i
$372$		0			0
$373$		−	229.776i		−	0.616021i	−0.951383	−	0.308010i	$-0.900337\pi$
$373$		−	229.776i		−	0.616021i	0.951383	−	0.308010i	$-0.0996631\pi$
$374$		0			0
$375$	221.282			0.590085
$376$		0			0
$377$		−	55.4253i		−	0.147017i
$378$		0			0
$379$		−	219.556i		−	0.579304i	−0.957132	−	0.289652i	$-0.906460\pi$
$379$		−	219.556i		−	0.579304i	0.957132	−	0.289652i	$-0.0935396\pi$
$380$		0			0
$381$	459.313			1.20555
$382$		0			0
$383$		−	122.239i		−	0.319162i	−0.987185	−	0.159581i	$-0.948986\pi$
$383$		−	122.239i		−	0.319162i	0.987185	−	0.159581i	$-0.0510143\pi$
$384$		0			0
$385$	70.1436	+	44.2035i	0.182191	+	0.114814i
$386$		0			0
$387$		−	517.182i		−	1.33639i
$388$		0			0
$389$		−	597.015i		−	1.53474i	−0.641202	−	0.767372i	$-0.721564\pi$
$389$		−	597.015i		−	1.53474i	0.641202	−	0.767372i	$-0.278436\pi$
$390$		0			0
$391$		−	344.335i		−	0.880652i
$392$		0			0
$393$	319.205			0.812227
$394$		0			0
$395$	29.9611			0.0758509
$396$		0			0
$397$	−277.969			−0.700175			−0.350087	−	0.936717i	$-0.613848\pi$
$397$	−277.969			−0.700175			−0.350087	+	0.936717i	$0.613848\pi$
$398$		0			0
$399$	374.526	−	594.310i	0.938661	−	1.48950i
$400$		0			0
$401$	511.769			1.27623			0.638116	−	0.769940i	$-0.279714\pi$
$401$	511.769			1.27623			0.638116	+	0.769940i	$0.279714\pi$
$402$		0			0
$403$			695.306i			1.72533i
$404$		0			0
$405$	98.6662			0.243620
$406$		0			0
$407$	107.138			0.263239
$408$		0			0
$409$			310.897i			0.760140i	0.924958	+	0.380070i	$0.124100\pi$
$409$			310.897i			0.760140i	−0.924958	+	0.380070i	$0.875900\pi$
$410$		0			0
$411$	612.432			1.49010
$412$		0			0
$413$	−150.244	+	238.412i	−0.363786	+	0.577269i
$414$		0			0
$415$	2.58192			0.00622150
$416$		0			0
$417$	257.482			0.617463
$418$		0			0
$419$	77.5826			0.185161			0.0925807	−	0.995705i	$-0.470488\pi$
$419$	77.5826			0.185161			0.0925807	+	0.995705i	$0.470488\pi$
$420$		0			0
$421$			305.450i			0.725534i	0.931880	+	0.362767i	$0.118168\pi$
$421$			305.450i			0.725534i	−0.931880	+	0.362767i	$0.881832\pi$
$422$		0			0
$423$		−	419.197i		−	0.991009i
$424$		0			0
$425$		−	487.939i		−	1.14809i
$426$		0			0
$427$	−4.11439	+	6.52885i	−0.00963557	+	0.0152901i
$428$		0			0
$429$		−	581.285i		−	1.35498i
$430$		0			0
$431$	−66.5077			−0.154310			−0.0771551	−	0.997019i	$-0.524584\pi$
$431$	−66.5077			−0.154310			−0.0771551	+	0.997019i	$0.524584\pi$
$432$		0			0
$433$		−	83.5325i		−	0.192916i	−0.995337	−	0.0964579i	$-0.969249\pi$
$433$		−	83.5325i		−	0.192916i	0.995337	−	0.0964579i	$-0.0307513\pi$
$434$		0			0
$435$			19.1178i			0.0439490i
$436$		0			0
$437$	−409.392			−0.936825
$438$		0			0
$439$			507.604i			1.15627i	0.815940	+	0.578136i	$0.196220\pi$
$439$			507.604i			1.15627i	−0.815940	+	0.578136i	$0.803780\pi$
$440$		0			0
$441$	−167.631	−	350.454i	−0.380115	−	0.794681i
$442$		0			0
$443$		−	709.135i		−	1.60076i	−0.599495	−	0.800378i	$-0.704632\pi$
$443$		−	709.135i		−	1.60076i	0.599495	−	0.800378i	$-0.295368\pi$
$444$		0			0
$445$		−	55.7296i		−	0.125235i
$446$		0			0
$447$			782.968i			1.75161i
$448$		0			0
$449$	224.564			0.500143			0.250071	−	0.968227i	$-0.419546\pi$
$449$	224.564			0.500143			0.250071	+	0.968227i	$0.419546\pi$
$450$		0			0
$451$	254.501			0.564305
$452$		0			0
$453$	−42.5997			−0.0940390
$454$		0			0
$455$	−54.1051	+	85.8558i	−0.118912	+	0.188694i
$456$		0			0
$457$	−562.200			−1.23020			−0.615098	−	0.788450i	$-0.710884\pi$
$457$	−562.200			−1.23020			−0.615098	+	0.788450i	$0.710884\pi$
$458$		0			0
$459$		−	90.4664i		−	0.197095i
$460$		0			0
$461$	808.230			1.75321			0.876605	−	0.481211i	$-0.159803\pi$
$461$	808.230			1.75321			0.876605	+	0.481211i	$0.159803\pi$
$462$		0			0
$463$	−545.069			−1.17725			−0.588627	−	0.808404i	$-0.700331\pi$
$463$	−545.069			−1.17725			−0.588627	+	0.808404i	$0.700331\pi$
$464$		0			0
$465$		−	239.832i		−	0.515767i
$466$		0			0
$467$	282.985			0.605965			0.302982	−	0.952996i	$-0.402018\pi$
$467$	282.985			0.605965			0.302982	+	0.952996i	$0.402018\pi$
$468$		0			0
$469$	386.320	+	243.453i	0.823710	+	0.519090i
$470$		0			0
$471$	1281.36			2.72050
$472$		0			0
$473$	−700.841			−1.48169
$474$		0			0
$475$	−580.129			−1.22132
$476$		0			0
$477$			374.126i			0.784332i
$478$		0			0
$479$			335.436i			0.700285i	0.936697	+	0.350142i	$0.113867\pi$
$479$			335.436i			0.700285i	−0.936697	+	0.350142i	$0.886133\pi$
$480$		0			0
$481$			131.138i			0.272635i
$482$		0			0
$483$	−257.730	+	408.974i	−0.533602	+	0.846737i
$484$		0			0
$485$			81.8450i			0.168753i
$486$		0			0
$487$	−331.041			−0.679755			−0.339878	−	0.940470i	$-0.610386\pi$
$487$	−331.041			−0.679755			−0.339878	+	0.940470i	$0.610386\pi$
$488$		0			0
$489$			31.5088i			0.0644352i
$490$		0			0
$491$			189.529i			0.386005i	0.981198	+	0.193003i	$0.0618226\pi$
$491$			189.529i			0.386005i	−0.981198	+	0.193003i	$0.938177\pi$
$492$		0			0
$493$	86.4658			0.175387
$494$		0			0
$495$			93.9039i			0.189705i
$496$		0			0
$497$	−421.846	+	669.400i	−0.848785	+	1.34688i
$498$		0			0
$499$		−	139.556i		−	0.279672i	−0.990175	−	0.139836i	$-0.955342\pi$
$499$		−	139.556i		−	0.279672i	0.990175	−	0.139836i	$-0.0446576\pi$
$500$		0			0
$501$		−	558.669i		−	1.11511i
$502$		0			0
$503$		−	243.628i		−	0.484349i	−0.970233	−	0.242175i	$-0.922139\pi$
$503$		−	243.628i		−	0.484349i	0.970233	−	0.242175i	$-0.0778607\pi$
$504$		0			0
$505$	62.9385			0.124631
$506$		0			0
$507$	16.1622			0.0318780
$508$		0			0
$509$	−542.704			−1.06622			−0.533108	−	0.846047i	$-0.678976\pi$
$509$	−542.704			−1.06622			−0.533108	+	0.846047i	$0.678976\pi$
$510$		0			0
$511$	542.354	+	341.784i	1.06136	+	0.668852i
$512$		0			0
$513$	−107.559			−0.209666
$514$		0			0
$515$		−	19.1178i		−	0.0371220i
$516$		0			0
$517$	−568.060			−1.09876
$518$		0			0
$519$	−1164.73			−2.24417
$520$		0			0
$521$		−	639.592i		−	1.22762i	−0.789453	−	0.613812i	$-0.789635\pi$
$521$		−	639.592i		−	1.22762i	0.789453	−	0.613812i	$-0.210365\pi$
$522$		0			0
$523$	−830.769			−1.58847			−0.794234	−	0.607612i	$-0.792128\pi$
$523$	−830.769			−1.58847			−0.794234	+	0.607612i	$0.792128\pi$
$524$		0			0
$525$	−365.215	+	579.536i	−0.695648	+	1.10388i
$526$		0			0
$527$	−1084.71			−2.05827
$528$		0			0
$529$	−247.277			−0.467442
$530$		0			0
$531$	−319.171			−0.601076
$532$		0			0
$533$			311.510i			0.584446i
$534$		0			0
$535$		−	112.490i		−	0.210262i
$536$		0			0
$537$		−	233.712i		−	0.435218i
$538$		0			0
$539$	−474.906	+	227.159i	−0.881087	+	0.421445i
$540$		0			0
$541$		−	630.733i		−	1.16587i	−0.812520	−	0.582933i	$-0.801905\pi$
$541$		−	630.733i		−	1.16587i	0.812520	−	0.582933i	$-0.198095\pi$
$542$		0			0
$543$	707.577			1.30309
$544$		0			0
$545$			128.192i			0.235214i
$546$		0			0
$547$			561.588i			1.02667i	0.858188	+	0.513335i	$0.171590\pi$
$547$			561.588i			1.02667i	−0.858188	+	0.513335i	$0.828410\pi$
$548$		0			0
$549$	−8.74043			−0.0159206
$550$		0			0
$551$		−	102.802i		−	0.186574i
$552$		0			0
$553$	−101.426	+	160.946i	−0.183410	+	0.291041i
$554$		0			0
$555$		−	45.2332i		−	0.0815013i
$556$		0			0
$557$		−	350.270i		−	0.628850i	−0.949282	−	0.314425i	$-0.898188\pi$
$557$		−	350.270i		−	0.628850i	0.949282	−	0.314425i	$-0.101812\pi$
$558$		0			0
$559$		−	857.830i		−	1.53458i
$560$		0			0
$561$	906.831			1.61645
$562$		0			0
$563$	−196.562			−0.349133			−0.174567	−	0.984645i	$-0.555852\pi$
$563$	−196.562			−0.349133			−0.174567	+	0.984645i	$0.555852\pi$
$564$		0			0
$565$	−11.4146			−0.0202027
$566$		0			0
$567$	−334.009	+	530.017i	−0.589081	+	0.934774i
$568$		0			0
$569$	679.174			1.19363			0.596814	−	0.802380i	$-0.296433\pi$
$569$	679.174			1.19363			0.596814	+	0.802380i	$0.296433\pi$
$570$		0			0
$571$			676.022i			1.18393i	0.805965	+	0.591963i	$0.201647\pi$
$571$			676.022i			1.18393i	−0.805965	+	0.591963i	$0.798353\pi$
$572$		0			0
$573$	−493.568			−0.861376
$574$		0			0
$575$	399.215			0.694288
$576$		0			0
$577$		−	104.047i		−	0.180325i	−0.995927	−	0.0901624i	$-0.971261\pi$
$577$		−	104.047i		−	0.180325i	0.995927	−	0.0901624i	$-0.0287386\pi$
$578$		0			0
$579$	46.1445			0.0796969
$580$		0			0
$581$	−8.74043	+	13.8696i	−0.0150438	+	0.0238720i
$582$		0			0
$583$	506.985			0.869613
$584$		0			0
$585$	−114.939			−0.196476
$586$		0			0
$587$	1027.12			1.74978			0.874890	−	0.484322i	$-0.160934\pi$
$587$	1027.12			1.74978			0.874890	+	0.484322i	$0.160934\pi$
$588$		0			0
$589$			1289.65i			2.18956i
$590$		0			0
$591$		−	855.735i		−	1.44794i
$592$		0			0
$593$			1057.94i			1.78404i	0.451992	+	0.892022i	$0.350713\pi$
$593$			1057.94i			1.78404i	−0.451992	+	0.892022i	$0.649287\pi$
$594$		0			0
$595$	−133.939	−	84.4063i	−0.225107	−	0.141859i
$596$		0			0
$597$		−	415.972i		−	0.696770i
$598$		0			0
$599$	650.295			1.08563			0.542817	−	0.839851i	$-0.317358\pi$
$599$	650.295			1.08563			0.542817	+	0.839851i	$0.317358\pi$
$600$		0			0
$601$		−	779.794i		−	1.29749i	−0.761004	−	0.648747i	$-0.775293\pi$
$601$		−	779.794i		−	1.29749i	0.761004	−	0.648747i	$-0.224707\pi$
$602$		0			0
$603$			517.182i			0.857681i
$604$		0			0
$605$	−6.14545			−0.0101578
$606$		0			0
$607$		−	209.796i		−	0.345627i	−0.984955	−	0.172814i	$-0.944714\pi$
$607$		−	209.796i		−	0.345627i	0.984955	−	0.172814i	$-0.0552858\pi$
$608$		0			0
$609$	−102.697	−	64.7184i	−0.168633	−	0.106270i
$610$		0			0
$611$		−	695.306i		−	1.13798i
$612$		0			0
$613$			1082.68i			1.76620i	0.469183	+	0.883101i	$0.344549\pi$
$613$			1082.68i			1.76620i	−0.469183	+	0.883101i	$0.655451\pi$
$614$		0			0
$615$		−	107.449i		−	0.174714i
$616$		0			0
$617$	−435.902			−0.706487			−0.353243	−	0.935531i	$-0.614921\pi$
$617$	−435.902			−0.706487			−0.353243	+	0.935531i	$0.614921\pi$
$618$		0			0
$619$	−560.717			−0.905844			−0.452922	−	0.891550i	$-0.649618\pi$
$619$	−560.717			−0.905844			−0.452922	+	0.891550i	$0.649618\pi$
$620$		0			0
$621$	74.0166			0.119189
$622$		0			0
$623$	299.369	+	188.658i	0.480529	+	0.302822i
$624$		0			0
$625$	535.323			0.856517
$626$		0			0
$627$		−	1078.16i		−	1.71956i
$628$		0			0
$629$	−204.580			−0.325247
$630$		0			0
$631$	−333.972			−0.529274			−0.264637	−	0.964348i	$-0.585252\pi$
$631$	−333.972			−0.529274			−0.264637	+	0.964348i	$0.585252\pi$
$632$		0			0
$633$		−	956.608i		−	1.51123i
$634$		0			0
$635$	−123.073			−0.193815
$636$		0			0
$637$	−278.043	−	581.285i	−0.436488	−	0.912536i
$638$		0			0
$639$	−896.151			−1.40243
$640$		0			0
$641$	670.354			1.04579			0.522897	−	0.852396i	$-0.324851\pi$
$641$	670.354			1.04579			0.522897	+	0.852396i	$0.324851\pi$
$642$		0			0
$643$	−484.612			−0.753673			−0.376837	−	0.926280i	$-0.622988\pi$
$643$	−484.612			−0.753673			−0.376837	+	0.926280i	$0.622988\pi$
$644$		0			0
$645$			295.891i			0.458746i
$646$		0			0
$647$		−	975.878i		−	1.50831i	−0.656695	−	0.754156i	$-0.728046\pi$
$647$		−	975.878i		−	1.50831i	0.656695	−	0.754156i	$-0.271954\pi$
$648$		0			0
$649$			432.514i			0.666431i
$650$		0			0
$651$	1288.33	+	811.888i	1.97900	+	1.24714i
$652$		0			0
$653$			813.541i			1.24585i	0.782281	+	0.622926i	$0.214056\pi$
$653$			813.541i			1.24585i	−0.782281	+	0.622926i	$0.785944\pi$
$654$		0			0
$655$	−85.5307			−0.130581
$656$		0			0
$657$			726.070i			1.10513i
$658$		0			0
$659$		−	382.914i		−	0.581053i	−0.956867	−	0.290527i	$-0.906170\pi$
$659$		−	382.914i		−	0.581053i	0.956867	−	0.290527i	$-0.0938304\pi$
$660$		0			0
$661$	−537.587			−0.813294			−0.406647	−	0.913585i	$-0.633302\pi$
$661$	−537.587			−0.813294			−0.406647	+	0.913585i	$0.633302\pi$
$662$		0			0
$663$			1109.96i			1.67415i
$664$		0			0
$665$	−100.354	+	159.245i	−0.150908	+	0.239466i
$666$		0			0
$667$			70.7434i			0.106062i
$668$		0			0
$669$		−	577.787i		−	0.863657i
$670$		0			0
$671$			11.8443i			0.0176517i
$672$		0			0
$673$	−34.8616			−0.0518002			−0.0259001	−	0.999665i	$-0.508245\pi$
$673$	−34.8616			−0.0518002			−0.0259001	+	0.999665i	$0.508245\pi$
$674$		0			0
$675$	104.885			0.155385
$676$		0			0
$677$	799.093			1.18034			0.590172	−	0.807277i	$-0.299060\pi$
$677$	799.093			1.18034			0.590172	+	0.807277i	$0.299060\pi$
$678$		0			0
$679$	−439.656	−	277.065i	−0.647506	−	0.408049i
$680$		0			0
$681$	−743.626			−1.09196
$682$		0			0
$683$		−	464.235i		−	0.679700i	−0.940480	−	0.339850i	$-0.889624\pi$
$683$		−	464.235i		−	0.679700i	0.940480	−	0.339850i	$-0.110376\pi$
$684$		0			0
$685$	−164.101			−0.239563
$686$		0			0
$687$	632.095			0.920080
$688$		0			0
$689$			620.550i			0.900653i
$690$		0			0
$691$	−670.097			−0.969750			−0.484875	−	0.874584i	$-0.661135\pi$
$691$	−670.097			−0.969750			−0.484875	+	0.874584i	$0.661135\pi$
$692$		0			0
$693$	−504.435	−	317.887i	−0.727900	−	0.458712i
$694$		0			0
$695$	−68.9921			−0.0992692
$696$		0			0
$697$	−485.969			−0.697230
$698$		0			0
$699$	437.307			0.625618
$700$		0			0
$701$		−	1239.37i		−	1.76801i	−0.467478	−	0.884005i	$-0.654837\pi$
$701$		−	1239.37i		−	1.76801i	0.467478	−	0.884005i	$-0.345163\pi$
$702$		0			0
$703$			243.233i			0.345993i
$704$		0			0
$705$			239.832i			0.340187i
$706$		0			0
$707$	−213.062	+	338.094i	−0.301361	+	0.478209i
$708$		0			0
$709$		−	1213.67i		−	1.71181i	−0.517134	−	0.855904i	$-0.673001\pi$
$709$		−	1213.67i		−	1.71181i	0.517134	−	0.855904i	$-0.326999\pi$
$710$		0			0
$711$	−215.464			−0.303044
$712$		0			0
$713$		−	887.471i		−	1.24470i
$714$		0			0
$715$			155.755i			0.217839i
$716$		0			0
$717$	1530.24			2.13422
$718$		0			0
$719$			295.652i			0.411198i	0.978636	+	0.205599i	$0.0659143\pi$
$719$			295.652i			0.411198i	−0.978636	+	0.205599i	$0.934086\pi$
$720$		0			0
$721$	102.697	+	64.7184i	0.142437	+	0.0897620i
$722$		0			0
$723$			32.1755i			0.0445028i
$724$		0			0
$725$			100.247i			0.138272i
$726$		0			0
$727$			1349.79i			1.85666i	0.371755	+	0.928331i	$0.378756\pi$
$727$			1349.79i			1.85666i	−0.371755	+	0.928331i	$0.621244\pi$
$728$		0			0
$729$	−546.261			−0.749330
$730$		0			0
$731$	1338.25			1.83071
$732$		0			0
$733$	243.166			0.331741			0.165870	−	0.986148i	$-0.446957\pi$
$733$	243.166			0.331741			0.165870	+	0.986148i	$0.446957\pi$
$734$		0			0
$735$	95.9052	+	200.503i	0.130483	+	0.272793i
$736$		0			0
$737$	700.841			0.950938
$738$		0			0
$739$			177.903i			0.240735i	0.992729	+	0.120367i	$0.0384072\pi$
$739$			177.903i			0.240735i	−0.992729	+	0.120367i	$0.961593\pi$
$740$		0			0
$741$	1319.67			1.78094
$742$		0			0
$743$	−338.928			−0.456162			−0.228081	−	0.973642i	$-0.573245\pi$
$743$	−338.928			−0.456162			−0.228081	+	0.973642i	$0.573245\pi$
$744$		0			0
$745$		−	209.796i		−	0.281605i
$746$		0			0
$747$	−18.5678			−0.0248565
$748$		0			0
$749$	604.277	+	380.807i	0.806778	+	0.508420i
$750$		0			0
$751$	96.3229			0.128260			0.0641298	−	0.997942i	$-0.479573\pi$
$751$	96.3229			0.128260			0.0641298	+	0.997942i	$0.479573\pi$
$752$		0			0
$753$	−868.200			−1.15299
$754$		0			0
$755$	11.4146			0.0151186
$756$		0			0
$757$		−	326.221i		−	0.430939i	−0.976511	−	0.215470i	$-0.930872\pi$
$757$		−	326.221i		−	0.430939i	0.976511	−	0.215470i	$-0.0691283\pi$
$758$		0			0
$759$			741.938i			0.977521i
$760$		0			0
$761$		−	93.5093i		−	0.122877i	−0.998111	−	0.0614384i	$-0.980431\pi$
$761$		−	93.5093i		−	0.122877i	0.998111	−	0.0614384i	$-0.0195688\pi$
$762$		0			0
$763$	−688.623	−	433.960i	−0.902520	−	0.568755i
$764$		0			0
$765$		−	179.309i		−	0.234391i
$766$		0			0
$767$	−529.397			−0.690218
$768$		0			0
$769$		−	311.125i		−	0.404584i	−0.979325	−	0.202292i	$-0.935161\pi$
$769$		−	311.125i		−	0.404584i	0.979325	−	0.202292i	$-0.0648390\pi$
$770$		0			0
$771$		−	1207.80i		−	1.56654i
$772$		0			0
$773$	192.295			0.248765			0.124382	−	0.992234i	$-0.460305\pi$
$773$	192.295			0.248765			0.124382	+	0.992234i	$0.460305\pi$
$774$		0			0
$775$		−	1257.59i		−	1.62270i
$776$		0			0
$777$	242.985	+	153.125i	0.312721	+	0.197073i
$778$		0			0
$779$			577.787i			0.741703i
$780$		0			0
$781$			1214.39i			1.55491i
$782$		0			0
$783$			18.5863i			0.0237373i
$784$		0			0
$785$	−343.338			−0.437374
$786$		0			0
$787$	58.4877			0.0743172			0.0371586	−	0.999309i	$-0.488169\pi$
$787$	58.4877			0.0743172			0.0371586	+	0.999309i	$0.488169\pi$
$788$		0			0
$789$	−263.479			−0.333941
$790$		0			0
$791$	38.6410	−	61.3169i	0.0488508	−	0.0775182i
$792$		0			0
$793$	−14.4974			−0.0182817
$794$		0			0
$795$		−	214.046i		−	0.269240i
$796$		0			0
$797$	−195.523			−0.245324			−0.122662	−	0.992449i	$-0.539143\pi$
$797$	−195.523			−0.245324			−0.122662	+	0.992449i	$0.539143\pi$
$798$		0			0
$799$	1084.71			1.35758
$800$		0			0
$801$			400.777i			0.500346i
$802$		0			0
$803$	983.909			1.22529
$804$		0			0
$805$	69.0584	−	109.584i	0.0857869	−	0.136130i
$806$		0			0
$807$	1373.38			1.70183
$808$		0			0
$809$	557.184			0.688732			0.344366	−	0.938835i	$-0.388094\pi$
$809$	557.184			0.688732			0.344366	+	0.938835i	$0.388094\pi$
$810$		0			0
$811$	460.284			0.567552			0.283776	−	0.958891i	$-0.408413\pi$
$811$	460.284			0.567552			0.283776	+	0.958891i	$0.408413\pi$
$812$		0			0
$813$		−	584.784i		−	0.719292i
$814$		0			0
$815$		−	8.44276i		−	0.0103592i
$816$		0			0
$817$		−	1591.10i		−	1.94749i
$818$		0			0
$819$	389.095	−	617.429i	0.475085	−	0.753881i
$820$		0			0
$821$			130.053i			0.158408i	0.996858	+	0.0792040i	$0.0252378\pi$
$821$			130.053i			0.158408i	−0.996858	+	0.0792040i	$0.974762\pi$
$822$		0			0
$823$	967.233			1.17525			0.587627	−	0.809132i	$-0.300062\pi$
$823$	967.233			1.17525			0.587627	+	0.809132i	$0.300062\pi$
$824$		0			0
$825$			1051.36i			1.27438i
$826$		0			0
$827$		−	138.014i		−	0.166885i	−0.996513	−	0.0834424i	$-0.973409\pi$
$827$		−	138.014i		−	0.166885i	0.996513	−	0.0834424i	$-0.0265915\pi$
$828$		0			0
$829$	−1134.08			−1.36801			−0.684004	−	0.729478i	$-0.739763\pi$
$829$	−1134.08			−1.36801			−0.684004	+	0.729478i	$0.739763\pi$
$830$		0			0
$831$			513.101i			0.617450i
$832$		0			0
$833$	906.831	−	433.759i	1.08863	−	0.520719i
$834$		0			0
$835$			149.695i			0.179275i
$836$		0			0
$837$		−	233.164i		−	0.278571i
$838$		0			0
$839$		−	1023.26i		−	1.21961i	−0.792550	−	0.609807i	$-0.791247\pi$
$839$		−	1023.26i		−	1.21961i	0.792550	−	0.609807i	$-0.208753\pi$
$840$		0			0
$841$	823.236			0.978877
$842$		0			0
$843$	−1222.54			−1.45023
$844$		0			0
$845$	−4.33064			−0.00512501
$846$		0			0
$847$	20.8038	−	33.0122i	0.0245618	−	0.0389755i
$848$		0			0
$849$	−573.041			−0.674960
$850$		0			0
$851$		−	167.381i		−	0.196687i
$852$		0			0
$853$	−50.0002			−0.0586169			−0.0293084	−	0.999570i	$-0.509331\pi$
$853$	−50.0002			−0.0586169			−0.0293084	+	0.999570i	$0.509331\pi$
$854$		0			0
$855$	−213.187			−0.249342
$856$		0			0
$857$			307.496i			0.358805i	0.983776	+	0.179402i	$0.0574164\pi$
$857$			307.496i			0.358805i	−0.983776	+	0.179402i	$0.942584\pi$
$858$		0			0
$859$	1360.72			1.58408			0.792039	−	0.610470i	$-0.209019\pi$
$859$	1360.72			1.58408			0.792039	+	0.610470i	$0.209019\pi$
$860$		0			0
$861$	577.196	+	363.741i	0.670379	+	0.422463i
$862$		0			0
$863$	−940.403			−1.08969			−0.544845	−	0.838537i	$-0.683412\pi$
$863$	−940.403			−1.08969			−0.544845	+	0.838537i	$0.683412\pi$
$864$		0			0
$865$	312.087			0.360795
$866$		0			0
$867$	−542.530			−0.625755
$868$		0			0
$869$			291.979i			0.335994i
$870$		0			0
$871$			857.830i			0.984880i
$872$		0			0
$873$		−	588.585i		−	0.674210i
$874$		0			0
$875$	200.719	−	318.508i	0.229393	−	0.364009i
$876$		0			0
$877$			418.946i			0.477704i	0.971056	+	0.238852i	$0.0767711\pi$
$877$			418.946i			0.477704i	−0.971056	+	0.238852i	$0.923229\pi$
$878$		0			0
$879$	−1533.41			−1.74450
$880$		0			0
$881$			855.735i			0.971322i	0.874147	+	0.485661i	$0.161421\pi$
$881$			855.735i			0.971322i	−0.874147	+	0.485661i	$0.838579\pi$
$882$		0			0
$883$			1056.84i			1.19688i	0.801168	+	0.598439i	$0.204212\pi$
$883$			1056.84i			1.19688i	−0.801168	+	0.598439i	$0.795788\pi$
$884$		0			0
$885$	182.605			0.206333
$886$		0			0
$887$		−	885.315i		−	0.998100i	−0.866573	−	0.499050i	$-0.833682\pi$
$887$		−	885.315i		−	0.998100i	0.866573	−	0.499050i	$-0.166318\pi$
$888$		0			0
$889$	416.631	−	661.124i	0.468651	−	0.743671i
$890$		0			0
$891$			961.527i			1.07916i
$892$		0			0
$893$		−	1289.65i		−	1.44418i
$894$		0			0
$895$			62.6230i			0.0699698i
$896$		0			0
$897$	−908.133			−1.01241
$898$		0			0
$899$	222.853			0.247890
$900$		0			0
$901$	−968.084			−1.07446
$902$		0			0
$903$	−1589.47	−	1001.66i	−1.76021	−	1.10926i
$904$		0			0
$905$	−189.595			−0.209497
$906$		0			0
$907$			790.234i			0.871262i	0.900125	+	0.435631i	$0.143475\pi$
$907$			790.234i			0.871262i	−0.900125	+	0.435631i	$0.856525\pi$
$908$		0			0
$909$	−452.620			−0.497931
$910$		0			0
$911$	1360.17			1.49305			0.746525	−	0.665357i	$-0.231721\pi$
$911$	1360.17			1.49305			0.746525	+	0.665357i	$0.231721\pi$
$912$		0			0
$913$			25.1615i			0.0275591i
$914$		0			0
$915$	5.00059			0.00546513
$916$		0			0
$917$	289.542	−	459.455i	0.315749	−	0.501042i
$918$		0			0
$919$	927.121			1.00884			0.504418	−	0.863460i	$-0.331707\pi$
$919$	927.121			1.00884			0.504418	+	0.863460i	$0.331707\pi$
$920$		0			0
$921$	1475.18			1.60172
$922$		0			0
$923$	−1486.41			−1.61041
$924$		0			0
$925$		−	237.187i		−	0.256418i
$926$		0			0
$927$			137.485i			0.148312i
$928$		0			0
$929$			907.531i			0.976890i	0.872595	+	0.488445i	$0.162436\pi$
$929$			907.531i			0.976890i	−0.872595	+	0.488445i	$0.837564\pi$
$930$		0			0
$931$	−515.712	−	1078.16i	−0.553934	−	1.15807i
$932$		0			0
$933$			2231.18i			2.39140i
$934$		0			0
$935$	−242.985			−0.259877
$936$		0			0
$937$			1376.66i			1.46922i	0.678492	+	0.734608i	$0.262634\pi$
$937$			1376.66i			1.46922i	−0.678492	+	0.734608i	$0.737366\pi$
$938$		0			0
$939$		−	603.902i		−	0.643133i
$940$		0			0
$941$	−1266.59			−1.34601			−0.673004	−	0.739639i	$-0.734996\pi$
$941$	−1266.59			−1.34601			−0.673004	+	0.739639i	$0.734996\pi$
$942$		0			0
$943$		−	397.604i		−	0.421637i
$944$		0			0
$945$	18.1436	−	28.7909i	0.0191996	−	0.0304665i
$946$		0			0
$947$			1058.05i			1.11727i	0.829414	+	0.558635i	$0.188675\pi$
$947$			1058.05i			1.11727i	−0.829414	+	0.558635i	$0.811325\pi$
$948$		0			0
$949$			1204.31i			1.26903i
$950$		0			0
$951$		−	416.646i		−	0.438113i
$952$		0			0
$953$	−1151.72			−1.20852			−0.604262	−	0.796786i	$-0.706532\pi$
$953$	−1151.72			−1.20852			−0.604262	+	0.796786i	$0.706532\pi$
$954$		0			0
$955$	132.251			0.138483
$956$		0			0
$957$	−186.308			−0.194679
$958$		0			0
$959$	555.520	−	881.519i	0.579271	−	0.919206i
$960$		0			0
$961$	−1834.67			−1.90913
$962$		0			0
$963$			808.969i			0.840050i
$964$		0			0
$965$	−12.3644			−0.0128128
$966$		0			0
$967$	−1443.46			−1.49272			−0.746361	−	0.665542i	$-0.768201\pi$
$967$	−1443.46			−1.49272			−0.746361	+	0.665542i	$0.768201\pi$
$968$		0			0
$969$			2058.75i			2.12461i
$970$		0			0
$971$	−1604.91			−1.65284			−0.826420	−	0.563054i	$-0.809626\pi$
$971$	−1604.91			−1.65284			−0.826420	+	0.563054i	$0.809626\pi$
$972$		0			0
$973$	233.555	−	370.613i	0.240036	−	0.380897i
$974$		0			0
$975$	−1286.87			−1.31987
$976$		0			0
$977$	1239.07			1.26824			0.634118	−	0.773236i	$-0.281363\pi$
$977$	1239.07			1.26824			0.634118	+	0.773236i	$0.281363\pi$
$978$		0			0
$979$	543.099			0.554749
$980$		0			0
$981$		−	921.885i		−	0.939741i
$982$		0			0
$983$			1584.98i			1.61239i	0.591651	+	0.806194i	$0.298476\pi$
$983$			1584.98i			1.61239i	−0.591651	+	0.806194i	$0.701524\pi$
$984$		0			0
$985$			229.293i			0.232785i
$986$		0			0
$987$	−1288.33	−	811.888i	−1.30530	−	0.822581i
$988$		0			0
$989$			1094.91i			1.10709i
$990$		0			0
$991$	−1479.12			−1.49255			−0.746274	−	0.665639i	$-0.768159\pi$
$991$	−1479.12			−1.49255			−0.746274	+	0.665639i	$0.768159\pi$
$992$		0			0
$993$		−	2026.79i		−	2.04107i
$994$		0			0
$995$			111.459i			0.112019i
$996$		0			0
$997$	599.409			0.601213			0.300606	−	0.953748i	$-0.402811\pi$
$997$	599.409			0.601213			0.300606	+	0.953748i	$0.402811\pi$
$998$		0			0
$999$		−	43.9756i		−	0.0440196i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.3.h.d.209.7		8
3.2	odd	2		2016.3.l.f.433.5		8
4.3	odd	2		56.3.h.d.13.7	yes	8
7.6	odd	2	inner	224.3.h.d.209.2		8
8.3	odd	2		56.3.h.d.13.6	yes	8
8.5	even	2	inner	224.3.h.d.209.1		8
12.11	even	2		504.3.l.f.181.2		8
21.20	even	2		2016.3.l.f.433.4		8
24.5	odd	2		2016.3.l.f.433.3		8
24.11	even	2		504.3.l.f.181.3		8
28.3	even	6		392.3.j.d.117.1		16
28.11	odd	6		392.3.j.d.117.2		16
28.19	even	6		392.3.j.d.325.5		16
28.23	odd	6		392.3.j.d.325.6		16
28.27	even	2		56.3.h.d.13.8	yes	8
56.3	even	6		392.3.j.d.117.6		16
56.11	odd	6		392.3.j.d.117.5		16
56.13	odd	2	inner	224.3.h.d.209.8		8
56.19	even	6		392.3.j.d.325.2		16
56.27	even	2		56.3.h.d.13.5	✓	8
56.51	odd	6		392.3.j.d.325.1		16
84.83	odd	2		504.3.l.f.181.1		8
168.83	odd	2		504.3.l.f.181.4		8
168.125	even	2		2016.3.l.f.433.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.h.d.13.5	✓	8	56.27	even	2
56.3.h.d.13.6	yes	8	8.3	odd	2
56.3.h.d.13.7	yes	8	4.3	odd	2
56.3.h.d.13.8	yes	8	28.27	even	2
224.3.h.d.209.1		8	8.5	even	2	inner
224.3.h.d.209.2		8	7.6	odd	2	inner
224.3.h.d.209.7		8	1.1	even	1	trivial
224.3.h.d.209.8		8	56.13	odd	2	inner
392.3.j.d.117.1		16	28.3	even	6
392.3.j.d.117.2		16	28.11	odd	6
392.3.j.d.117.5		16	56.11	odd	6
392.3.j.d.117.6		16	56.3	even	6
392.3.j.d.325.1		16	56.51	odd	6
392.3.j.d.325.2		16	56.19	even	6
392.3.j.d.325.5		16	28.19	even	6
392.3.j.d.325.6		16	28.23	odd	6
504.3.l.f.181.1		8	84.83	odd	2
504.3.l.f.181.2		8	12.11	even	2
504.3.l.f.181.3		8	24.11	even	2
504.3.l.f.181.4		8	168.83	odd	2
2016.3.l.f.433.3		8	24.5	odd	2
2016.3.l.f.433.4		8	21.20	even	2
2016.3.l.f.433.5		8	3.2	odd	2
2016.3.l.f.433.6		8	168.125	even	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.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.11439 q^{3} -1.10245 q^{5} +(3.73205 - 5.92214i) q^{7} +7.92820 q^{9} +O(q^{10})\)	\(q+4.11439 q^{3} -1.10245 q^{5} +(3.73205 - 5.92214i) q^{7} +7.92820 q^{9} -10.7436i q^{11} +13.1502 q^{13} -4.53590 q^{15} +20.5149i q^{17} +24.3909 q^{19} +(15.3551 - 24.3660i) q^{21} -16.7846 q^{23} -23.7846 q^{25} -4.40979 q^{27} -4.21478i q^{29} +52.8741i q^{31} -44.2035i q^{33} +(-4.11439 + 6.52885i) q^{35} +9.97227i q^{37} +54.1051 q^{39} +23.6886i q^{41} -65.2331i q^{43} -8.74043 q^{45} -52.8741i q^{47} +(-21.1436 - 44.2035i) q^{49} +84.4063i q^{51} +47.1893i q^{53} +11.8443i q^{55} +100.354 q^{57} -40.2577 q^{59} -1.10245 q^{61} +(29.5885 - 46.9520i) q^{63} -14.4974 q^{65} +65.2331i q^{67} -69.0584 q^{69} -113.033 q^{71} +91.5806i q^{73} -97.8592 q^{75} +(-63.6253 - 40.0958i) q^{77} -27.1769 q^{79} -89.4974 q^{81} -2.34199 q^{83} -22.6166i q^{85} -17.3412i q^{87} +50.5508i q^{89} +(49.0773 - 77.8775i) q^{91} +217.545i q^{93} -26.8897 q^{95} -74.2394i q^{97} -85.1777i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 16 * q^7 + 8 * q^9	\( 8 q + 16 q^{7} + 8 q^{9} - 64 q^{15} + 32 q^{23} - 24 q^{25} + 128 q^{39} - 280 q^{49} + 304 q^{57} + 112 q^{63} + 272 q^{65} - 544 q^{71} + 32 q^{79} - 328 q^{81} + 256 q^{95}+O(q^{100}) \) 8 * q + 16 * q^7 + 8 * q^9 - 64 * q^15 + 32 * q^23 - 24 * q^25 + 128 * q^39 - 280 * q^49 + 304 * q^57 + 112 * q^63 + 272 * q^65 - 544 * q^71 + 32 * q^79 - 328 * q^81 + 256 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more