LMFDB - Embedded newform 448.3.c.g.321.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,3,Mod(321,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.321"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.2071158433$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4694952902656.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 20x^{6} + 56x^{4} + 20x^{2} + 1 $ x^8 + 20x^6 + 56x^4 + 20*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.4
Root			$-0.590371i$ of defining polynomial
Character	$\chi$	$=$	448.321
Dual form			448.3.c.g.321.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56056i q^{3} +3.23038i q^{5} +(-3.38162 - 6.12900i) q^{7} +6.56466 q^{9} +3.93481 q^{11} +14.1051i q^{13} +5.04119 q^{15} -26.1633i q^{17} -10.6974i q^{19} +(-9.56466 + 5.27721i) q^{21} +20.9054 q^{23} +14.5647 q^{25} -24.2895i q^{27} -13.1293 q^{29} -39.8951i q^{31} -6.14050i q^{33} +(19.7990 - 10.9239i) q^{35} +1.12932 q^{37} +22.0118 q^{39} -2.36709i q^{41} +59.2720 q^{43} +21.2063i q^{45} -33.6529i q^{47} +(-26.1293 + 41.4519i) q^{49} -40.8293 q^{51} +30.0000 q^{53} +12.7109i q^{55} -16.6940 q^{57} -63.3035i q^{59} -103.149i q^{61} +(-22.1992 - 40.2348i) q^{63} -45.5647 q^{65} -28.7750 q^{67} -32.6240i q^{69} -90.3847 q^{71} +62.5607i q^{73} -22.7290i q^{75} +(-13.3060 - 24.1164i) q^{77} +139.084 q^{79} +21.1767 q^{81} +109.441i q^{83} +84.5173 q^{85} +20.4891i q^{87} -2.04683i q^{89} +(86.4499 - 47.6979i) q^{91} -62.2586 q^{93} +34.5568 q^{95} +90.7708i q^{97} +25.8307 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 56 q^{9} + 32 q^{21} + 8 q^{25} + 112 q^{29} - 208 q^{37} + 8 q^{49} + 240 q^{53} + 192 q^{57} - 256 q^{65} - 432 q^{77} + 712 q^{81} - 192 q^{85} - 64 q^{93}+O(q^{100}) $ 8 * q - 56 * q^9 + 32 * q^21 + 8 * q^25 + 112 * q^29 - 208 * q^37 + 8 * q^49 + 240 * q^53 + 192 * q^57 - 256 * q^65 - 432 * q^77 + 712 * q^81 - 192 * q^85 - 64 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

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

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
$3$		−	1.56056i		−	0.520186i	−0.965584	−	0.260093i	$-0.916247\pi$
$3$		−	1.56056i		−	0.520186i	0.965584	−	0.260093i	$-0.0837532\pi$
$4$		0			0
$5$			3.23038i			0.646076i	0.946386	+	0.323038i	$0.104704\pi$
$5$			3.23038i			0.646076i	−0.946386	+	0.323038i	$0.895296\pi$
$6$		0			0
$7$	−3.38162	−	6.12900i	−0.483088	−	0.875572i
$7$	−3.38162	−	6.12900i	−0.483088	−	0.875572i
$8$		0			0
$9$	6.56466			0.729407
$10$		0			0
$11$	3.93481			0.357710			0.178855	−	0.983875i	$-0.442761\pi$
$11$	3.93481			0.357710			0.178855	+	0.983875i	$0.442761\pi$
$12$		0			0
$13$			14.1051i			1.08500i	0.840054	+	0.542502i	$0.182523\pi$
$13$			14.1051i			1.08500i	−0.840054	+	0.542502i	$0.817477\pi$
$14$		0			0
$15$	5.04119			0.336079
$16$		0			0
$17$		−	26.1633i		−	1.53902i	−0.638637	−	0.769508i	$-0.720501\pi$
$17$		−	26.1633i		−	1.53902i	0.638637	−	0.769508i	$-0.279499\pi$
$18$		0			0
$19$		−	10.6974i		−	0.563023i	−0.959558	−	0.281512i	$-0.909164\pi$
$19$		−	10.6974i		−	0.563023i	0.959558	−	0.281512i	$-0.0908358\pi$
$20$		0			0
$21$	−9.56466	+	5.27721i	−0.455460	+	0.251296i
$22$		0			0
$23$	20.9054			0.908929			0.454465	−	0.890765i	$-0.349831\pi$
$23$	20.9054			0.908929			0.454465	+	0.890765i	$0.349831\pi$
$24$		0			0
$25$	14.5647			0.582586
$26$		0			0
$27$		−	24.2895i		−	0.899613i
$28$		0			0
$29$	−13.1293			−0.452735			−0.226368	−	0.974042i	$-0.572685\pi$
$29$	−13.1293			−0.452735			−0.226368	+	0.974042i	$0.572685\pi$
$30$		0			0
$31$		−	39.8951i		−	1.28694i	−0.765472	−	0.643470i	$-0.777494\pi$
$31$		−	39.8951i		−	1.28694i	0.765472	−	0.643470i	$-0.222506\pi$
$32$		0			0
$33$		−	6.14050i		−	0.186076i
$34$		0			0
$35$	19.7990	−	10.9239i	0.565685	−	0.312112i
$36$		0			0
$37$	1.12932			0.0305222			0.0152611	−	0.999884i	$-0.495142\pi$
$37$	1.12932			0.0305222			0.0152611	+	0.999884i	$0.495142\pi$
$38$		0			0
$39$	22.0118			0.564404
$40$		0			0
$41$		−	2.36709i		−	0.0577339i	−0.999583	−	0.0288670i	$-0.990810\pi$
$41$		−	2.36709i		−	0.0577339i	0.999583	−	0.0288670i	$-0.00918992\pi$
$42$		0			0
$43$	59.2720			1.37842			0.689210	−	0.724562i	$-0.257958\pi$
$43$	59.2720			1.37842			0.689210	+	0.724562i	$0.257958\pi$
$44$		0			0
$45$			21.2063i			0.471252i
$46$		0			0
$47$		−	33.6529i		−	0.716019i	−0.933718	−	0.358010i	$-0.883456\pi$
$47$		−	33.6529i		−	0.716019i	0.933718	−	0.358010i	$-0.116544\pi$
$48$		0			0
$49$	−26.1293	+	41.4519i	−0.533251	+	0.845957i
$50$		0			0
$51$	−40.8293			−0.800575
$52$		0			0
$53$	30.0000			0.566038			0.283019	−	0.959114i	$-0.408664\pi$
$53$	30.0000			0.566038			0.283019	+	0.959114i	$0.408664\pi$
$54$		0			0
$55$			12.7109i			0.231108i
$56$		0			0
$57$	−16.6940			−0.292877
$58$		0			0
$59$		−	63.3035i		−	1.07294i	−0.843919	−	0.536470i	$-0.819757\pi$
$59$		−	63.3035i		−	1.07294i	0.843919	−	0.536470i	$-0.180243\pi$
$60$		0			0
$61$		−	103.149i		−	1.69097i	−0.533997	−	0.845486i	$-0.679311\pi$
$61$		−	103.149i		−	1.69097i	0.533997	−	0.845486i	$-0.320689\pi$
$62$		0			0
$63$	−22.1992	−	40.2348i	−0.352368	−	0.638648i
$64$		0			0
$65$	−45.5647			−0.700995
$66$		0			0
$67$	−28.7750			−0.429477			−0.214739	−	0.976672i	$-0.568890\pi$
$67$	−28.7750			−0.429477			−0.214739	+	0.976672i	$0.568890\pi$
$68$		0			0
$69$		−	32.6240i		−	0.472812i
$70$		0			0
$71$	−90.3847			−1.27302			−0.636512	−	0.771267i	$-0.719623\pi$
$71$	−90.3847			−1.27302			−0.636512	+	0.771267i	$0.719623\pi$
$72$		0			0
$73$			62.5607i			0.856996i	0.903543	+	0.428498i	$0.140957\pi$
$73$			62.5607i			0.856996i	−0.903543	+	0.428498i	$0.859043\pi$
$74$		0			0
$75$		−	22.7290i		−	0.303053i
$76$		0			0
$77$	−13.3060	−	24.1164i	−0.172805	−	0.313201i
$78$		0			0
$79$	139.084			1.76055			0.880276	−	0.474462i	$-0.157357\pi$
$79$	139.084			1.76055			0.880276	+	0.474462i	$0.157357\pi$
$80$		0			0
$81$	21.1767			0.261441
$82$		0			0
$83$			109.441i			1.31856i	0.751896	+	0.659282i	$0.229140\pi$
$83$			109.441i			1.31856i	−0.751896	+	0.659282i	$0.770860\pi$
$84$		0			0
$85$	84.5173			0.994321
$86$		0			0
$87$			20.4891i			0.235506i
$88$		0			0
$89$		−	2.04683i		−	0.0229981i	−0.999934	−	0.0114991i	$-0.996340\pi$
$89$		−	2.04683i		−	0.0229981i	0.999934	−	0.0114991i	$-0.00366034\pi$
$90$		0			0
$91$	86.4499	−	47.6979i	0.949999	−	0.524153i
$92$		0			0
$93$	−62.2586			−0.669448
$94$		0			0
$95$	34.5568			0.363756
$96$		0			0
$97$			90.7708i			0.935782i	0.883786	+	0.467891i	$0.154986\pi$
$97$			90.7708i			0.935782i	−0.883786	+	0.467891i	$0.845014\pi$
$98$		0			0
$99$	25.8307			0.260916
$100$		0			0
$101$		−	5.91773i		−	0.0585914i	−0.999571	−	0.0292957i	$-0.990674\pi$
$101$		−	5.91773i		−	0.0585914i	0.999571	−	0.0292957i	$-0.00932644\pi$
$102$		0			0
$103$			101.638i			0.986777i	0.869809	+	0.493389i	$0.164242\pi$
$103$			101.638i			0.986777i	−0.869809	+	0.493389i	$0.835758\pi$
$104$		0			0
$105$	−17.0474	−	30.8975i	−0.162356	−	0.294262i
$106$		0			0
$107$	−180.279			−1.68485			−0.842424	−	0.538815i	$-0.818872\pi$
$107$	−180.279			−1.68485			−0.842424	+	0.538815i	$0.818872\pi$
$108$		0			0
$109$	132.259			1.21338			0.606691	−	0.794938i	$-0.292497\pi$
$109$	132.259			1.21338			0.606691	+	0.794938i	$0.292497\pi$
$110$		0			0
$111$		−	1.76237i		−	0.0158772i
$112$		0			0
$113$	111.211			0.984170			0.492085	−	0.870547i	$-0.336235\pi$
$113$	111.211			0.984170			0.492085	+	0.870547i	$0.336235\pi$
$114$		0			0
$115$			67.5322i			0.587237i
$116$		0			0
$117$			92.5949i			0.791409i
$118$		0			0
$119$	−160.355	+	88.4742i	−1.34752	+	0.743481i
$120$		0			0
$121$	−105.517			−0.872044
$122$		0			0
$123$	−3.69398			−0.0300324
$124$		0			0
$125$			127.809i			1.02247i
$126$		0			0
$127$	−110.434			−0.869556			−0.434778	−	0.900538i	$-0.643173\pi$
$127$	−110.434			−0.869556			−0.434778	+	0.900538i	$0.643173\pi$
$128$		0			0
$129$		−	92.4974i		−	0.717034i
$130$		0			0
$131$			200.357i			1.52944i	0.644362	+	0.764721i	$0.277123\pi$
$131$			200.357i			1.52944i	−0.644362	+	0.764721i	$0.722877\pi$
$132$		0			0
$133$	−65.5647	+	36.1747i	−0.492967	+	0.271990i
$134$		0			0
$135$	78.4644			0.581218
$136$		0			0
$137$	214.517			1.56582			0.782910	−	0.622135i	$-0.213735\pi$
$137$	214.517			1.56582			0.782910	+	0.622135i	$0.213735\pi$
$138$		0			0
$139$			47.6979i			0.343150i	0.985171	+	0.171575i	$0.0548857\pi$
$139$			47.6979i			0.343150i	−0.985171	+	0.171575i	$0.945114\pi$
$140$		0			0
$141$	−52.5173			−0.372463
$142$		0			0
$143$			55.5007i			0.388117i
$144$		0			0
$145$		−	42.4127i		−	0.292501i
$146$		0			0
$147$	64.6881	+	40.7763i	0.440055	+	0.277390i
$148$		0			0
$149$	−214.517			−1.43971			−0.719857	−	0.694123i	$-0.755792\pi$
$149$	−214.517			−1.43971			−0.719857	+	0.694123i	$0.755792\pi$
$150$		0			0
$151$	−55.8279			−0.369721			−0.184861	−	0.982765i	$-0.559183\pi$
$151$	−55.8279			−0.369721			−0.184861	+	0.982765i	$0.559183\pi$
$152$		0			0
$153$		−	171.753i		−	1.12257i
$154$		0			0
$155$	128.876			0.831460
$156$		0			0
$157$		−	130.594i		−	0.831807i	−0.909409	−	0.415903i	$-0.863465\pi$
$157$		−	130.594i		−	0.831807i	0.909409	−	0.415903i	$-0.136535\pi$
$158$		0			0
$159$		−	46.8167i		−	0.294445i
$160$		0			0
$161$	−70.6940	−	128.129i	−0.439093	−	0.795833i
$162$		0			0
$163$	−296.610			−1.81969			−0.909847	−	0.414945i	$-0.863801\pi$
$163$	−296.610			−1.81969			−0.909847	+	0.414945i	$0.863801\pi$
$164$		0			0
$165$	19.8361			0.120219
$166$		0			0
$167$			266.102i			1.59343i	0.604358	+	0.796713i	$0.293430\pi$
$167$			266.102i			1.59343i	−0.604358	+	0.796713i	$0.706570\pi$
$168$		0			0
$169$	−29.9526			−0.177234
$170$		0			0
$171$		−	70.2251i		−	0.410673i
$172$		0			0
$173$			292.113i			1.68851i	0.535940	+	0.844256i	$0.319957\pi$
$173$			292.113i			1.68851i	−0.535940	+	0.844256i	$0.680043\pi$
$174$		0			0
$175$	−49.2521	−	89.2668i	−0.281441	−	0.510096i
$176$		0			0
$177$	−98.7887			−0.558128
$178$		0			0
$179$	−219.377			−1.22557			−0.612785	−	0.790250i	$-0.709951\pi$
$179$	−219.377			−1.22557			−0.612785	+	0.790250i	$0.709951\pi$
$180$		0			0
$181$		−	173.577i		−	0.958990i	−0.877545	−	0.479495i	$-0.840820\pi$
$181$		−	173.577i		−	0.958990i	0.877545	−	0.479495i	$-0.159180\pi$
$182$		0			0
$183$	−160.970			−0.879620
$184$		0			0
$185$			3.64813i			0.0197196i
$186$		0			0
$187$		−	102.948i		−	0.550521i
$188$		0			0
$189$	−148.871	+	82.1380i	−0.787676	+	0.434592i
$190$		0			0
$191$	154.823			0.810591			0.405296	−	0.914186i	$-0.367169\pi$
$191$	154.823			0.810591			0.405296	+	0.914186i	$0.367169\pi$
$192$		0			0
$193$	164.435			0.851997			0.425998	−	0.904724i	$-0.359923\pi$
$193$	164.435			0.851997			0.425998	+	0.904724i	$0.359923\pi$
$194$		0			0
$195$			71.1063i			0.364648i
$196$		0			0
$197$	58.8707			0.298836			0.149418	−	0.988774i	$-0.452260\pi$
$197$	58.8707			0.298836			0.149418	+	0.988774i	$0.452260\pi$
$198$		0			0
$199$		−	9.81626i		−	0.0493279i	−0.999696	−	0.0246640i	$-0.992148\pi$
$199$		−	9.81626i		−	0.0493279i	0.999696	−	0.0246640i	$-0.00785158\pi$
$200$		0			0
$201$			44.9050i			0.223408i
$202$		0			0
$203$	44.3983	+	80.4696i	0.218711	+	0.396402i
$204$		0			0
$205$	7.64660			0.0373005
$206$		0			0
$207$	137.237			0.662979
$208$		0			0
$209$		−	42.0924i		−	0.201399i
$210$		0			0
$211$	167.484			0.793762			0.396881	−	0.917870i	$-0.370093\pi$
$211$	167.484			0.793762			0.396881	+	0.917870i	$0.370093\pi$
$212$		0			0
$213$			141.051i			0.662209i
$214$		0			0
$215$			191.471i			0.890563i
$216$		0			0
$217$	−244.517	+	134.910i	−1.12681	+	0.621705i
$218$		0			0
$219$	97.6296			0.445797
$220$		0			0
$221$	369.035			1.66984
$222$		0			0
$223$		−	172.971i		−	0.775654i	−0.921732	−	0.387827i	$-0.873226\pi$
$223$		−	172.971i		−	0.775654i	0.921732	−	0.387827i	$-0.126774\pi$
$224$		0			0
$225$	95.6120			0.424942
$226$		0			0
$227$			358.854i			1.58086i	0.612555	+	0.790428i	$0.290142\pi$
$227$			358.854i			1.58086i	−0.612555	+	0.790428i	$0.709858\pi$
$228$		0			0
$229$		−	239.786i		−	1.04710i	−0.851995	−	0.523550i	$-0.824607\pi$
$229$		−	239.786i		−	1.04710i	0.851995	−	0.523550i	$-0.175393\pi$
$230$		0			0
$231$	−37.6351	+	20.7648i	−0.162923	+	0.0898910i
$232$		0			0
$233$	−25.4827			−0.109368			−0.0546839	−	0.998504i	$-0.517415\pi$
$233$	−25.4827			−0.109368			−0.0546839	+	0.998504i	$0.517415\pi$
$234$		0			0
$235$	108.712			0.462602
$236$		0			0
$237$		−	217.048i		−	0.915815i
$238$		0			0
$239$	−40.0887			−0.167735			−0.0838676	−	0.996477i	$-0.526727\pi$
$239$	−40.0887			−0.167735			−0.0838676	+	0.996477i	$0.526727\pi$
$240$		0			0
$241$			399.356i			1.65708i	0.559932	+	0.828538i	$0.310827\pi$
$241$			399.356i			1.65708i	−0.559932	+	0.828538i	$0.689173\pi$
$242$		0			0
$243$		−	251.653i		−	1.03561i
$244$		0			0
$245$	−133.905	−	84.4076i	−0.546552	−	0.344521i
$246$		0			0
$247$	150.888			0.610883
$248$		0			0
$249$	170.789			0.685899
$250$		0			0
$251$		−	373.781i		−	1.48917i	−0.667530	−	0.744583i	$-0.732649\pi$
$251$		−	373.781i		−	1.48917i	0.667530	−	0.744583i	$-0.267351\pi$
$252$		0			0
$253$	82.2586			0.325133
$254$		0			0
$255$		−	131.894i		−	0.517232i
$256$		0			0
$257$			189.729i			0.738245i	0.929381	+	0.369123i	$0.120342\pi$
$257$			189.729i			0.738245i	−0.929381	+	0.369123i	$0.879658\pi$
$258$		0			0
$259$	−3.81893	−	6.92160i	−0.0147449	−	0.0267243i
$260$		0			0
$261$	−86.1895			−0.330228
$262$		0			0
$263$	−28.8909			−0.109851			−0.0549256	−	0.998490i	$-0.517492\pi$
$263$	−28.8909			−0.109851			−0.0549256	+	0.998490i	$0.517492\pi$
$264$		0			0
$265$			96.9113i			0.365703i
$266$		0			0
$267$	−3.19420			−0.0119633
$268$		0			0
$269$		−	132.766i		−	0.493553i	−0.969072	−	0.246776i	$-0.920629\pi$
$269$		−	132.766i		−	0.493553i	0.969072	−	0.246776i	$-0.0793713\pi$
$270$		0			0
$271$		−	333.004i		−	1.22880i	−0.788996	−	0.614399i	$-0.789399\pi$
$271$		−	333.004i		−	1.22880i	0.788996	−	0.614399i	$-0.210601\pi$
$272$		0			0
$273$	−74.4353	−	134.910i	−0.272657	−	0.494176i
$274$		0			0
$275$	57.3092			0.208397
$276$		0			0
$277$	436.940			1.57740			0.788700	−	0.614778i	$-0.210755\pi$
$277$	436.940			1.57740			0.788700	+	0.614778i	$0.210755\pi$
$278$		0			0
$279$		−	261.898i		−	0.938702i
$280$		0			0
$281$	−154.517			−0.549884			−0.274942	−	0.961461i	$-0.588659\pi$
$281$	−154.517			−0.549884			−0.274942	+	0.961461i	$0.588659\pi$
$282$		0			0
$283$		−	421.907i		−	1.49084i	−0.666597	−	0.745418i	$-0.732250\pi$
$283$		−	421.907i		−	1.49084i	0.666597	−	0.745418i	$-0.267750\pi$
$284$		0			0
$285$		−	53.9279i		−	0.189221i
$286$		0			0
$287$	−14.5079	+	8.00460i	−0.0505502	+	0.0278906i
$288$		0			0
$289$	−395.517			−1.36857
$290$		0			0
$291$	141.653			0.486780
$292$		0			0
$293$			170.639i			0.582386i	0.956664	+	0.291193i	$0.0940522\pi$
$293$			170.639i			0.582386i	−0.956664	+	0.291193i	$0.905948\pi$
$294$		0			0
$295$	204.494			0.693201
$296$		0			0
$297$		−	95.5747i		−	0.321800i
$298$		0			0
$299$			294.871i			0.986192i
$300$		0			0
$301$	−200.435	−	363.278i	−0.665898	−	1.20690i
$302$		0			0
$303$	−9.23496			−0.0304784
$304$		0			0
$305$	333.211			1.09250
$306$		0			0
$307$		−	261.470i		−	0.851693i	−0.904795	−	0.425846i	$-0.859976\pi$
$307$		−	261.470i		−	0.851693i	0.904795	−	0.425846i	$-0.140024\pi$
$308$		0			0
$309$	158.612			0.513308
$310$		0			0
$311$		−	6.92160i		−	0.0222560i	−0.999938	−	0.0111280i	$-0.996458\pi$
$311$		−	6.92160i		−	0.0222560i	0.999938	−	0.0111280i	$-0.00354222\pi$
$312$		0			0
$313$		−	65.0531i		−	0.207837i	−0.994586	−	0.103919i	$-0.966862\pi$
$313$		−	65.0531i		−	0.207837i	0.994586	−	0.103919i	$-0.0331382\pi$
$314$		0			0
$315$	129.974	−	71.7117i	0.412615	−	0.227656i
$316$		0			0
$317$	−65.6466			−0.207087			−0.103544	−	0.994625i	$-0.533018\pi$
$317$	−65.6466			−0.207087			−0.103544	+	0.994625i	$0.533018\pi$
$318$		0			0
$319$	−51.6614			−0.161948
$320$		0			0
$321$			281.335i			0.876434i
$322$		0			0
$323$	−279.880			−0.866502
$324$		0			0
$325$			205.435i			0.632109i
$326$		0			0
$327$		−	206.397i		−	0.631184i
$328$		0			0
$329$	−206.259	+	113.801i	−0.626926	+	0.345900i
$330$		0			0
$331$	76.2426			0.230340			0.115170	−	0.993346i	$-0.463259\pi$
$331$	76.2426			0.230340			0.115170	+	0.993346i	$0.463259\pi$
$332$		0			0
$333$	7.41360			0.0222631
$334$		0			0
$335$		−	92.9541i		−	0.277475i
$336$		0			0
$337$	182.177			0.540584			0.270292	−	0.962778i	$-0.412880\pi$
$337$	182.177			0.540584			0.270292	+	0.962778i	$0.412880\pi$
$338$		0			0
$339$		−	173.552i		−	0.511952i
$340$		0			0
$341$		−	156.980i		−	0.460351i
$342$		0			0
$343$	342.418	+	19.9722i	0.998303	+	0.0582280i
$344$		0			0
$345$	105.388			0.305472
$346$		0			0
$347$	−13.5355			−0.0390073			−0.0195037	−	0.999810i	$-0.506209\pi$
$347$	−13.5355			−0.0390073			−0.0195037	+	0.999810i	$0.506209\pi$
$348$		0			0
$349$			350.134i			1.00325i	0.865085	+	0.501625i	$0.167264\pi$
$349$			350.134i			1.00325i	−0.865085	+	0.501625i	$0.832736\pi$
$350$		0			0
$351$	342.605			0.976084
$352$		0			0
$353$		−	572.390i		−	1.62150i	−0.585392	−	0.810750i	$-0.699060\pi$
$353$		−	572.390i		−	1.62150i	0.585392	−	0.810750i	$-0.300940\pi$
$354$		0			0
$355$		−	291.977i		−	0.822470i
$356$		0			0
$357$	138.069	+	250.243i	0.386748	+	0.700960i
$358$		0			0
$359$	−458.196			−1.27631			−0.638156	−	0.769907i	$-0.720303\pi$
$359$	−458.196			−1.27631			−0.638156	+	0.769907i	$0.720303\pi$
$360$		0			0
$361$	246.565			0.683005
$362$		0			0
$363$			164.666i			0.453625i
$364$		0			0
$365$	−202.095			−0.553684
$366$		0			0
$367$			272.394i			0.742217i	0.928590	+	0.371108i	$0.121022\pi$
$367$			272.394i			0.742217i	−0.928590	+	0.371108i	$0.878978\pi$
$368$		0			0
$369$		−	15.5392i		−	0.0421115i
$370$		0			0
$371$	−101.449	−	183.870i	−0.273446	−	0.495607i
$372$		0			0
$373$	−705.974			−1.89269			−0.946346	−	0.323154i	$-0.895257\pi$
$373$	−705.974			−1.89269			−0.946346	+	0.323154i	$0.895257\pi$
$374$		0			0
$375$	199.453			0.531875
$376$		0			0
$377$		−	185.190i		−	0.491220i
$378$		0			0
$379$	−5.41603			−0.0142903			−0.00714516	−	0.999974i	$-0.502274\pi$
$379$	−5.41603			−0.0142903			−0.00714516	+	0.999974i	$0.502274\pi$
$380$		0			0
$381$			172.338i			0.452331i
$382$		0			0
$383$			127.286i			0.332340i	0.986097	+	0.166170i	$0.0531401\pi$
$383$			127.286i			0.332340i	−0.986097	+	0.166170i	$0.946860\pi$
$384$		0			0
$385$	77.9052	−	42.9835i	0.202351	−	0.111645i
$386$		0			0
$387$	389.101			1.00543
$388$		0			0
$389$	49.8361			0.128113			0.0640567	−	0.997946i	$-0.479596\pi$
$389$	49.8361			0.128113			0.0640567	+	0.997946i	$0.479596\pi$
$390$		0			0
$391$		−	546.953i		−	1.39886i
$392$		0			0
$393$	312.668			0.795594
$394$		0			0
$395$			449.293i			1.13745i
$396$		0			0
$397$			112.172i			0.282549i	0.989970	+	0.141275i	$0.0451201\pi$
$397$			112.172i			0.282549i	−0.989970	+	0.141275i	$0.954880\pi$
$398$		0			0
$399$	56.4527	+	102.317i	0.141485	+	0.256435i
$400$		0			0
$401$	−583.375			−1.45480			−0.727400	−	0.686213i	$-0.759272\pi$
$401$	−583.375			−1.45480			−0.727400	+	0.686213i	$0.759272\pi$
$402$		0			0
$403$	562.723			1.39633
$404$		0			0
$405$			68.4087i			0.168910i
$406$		0			0
$407$	4.44366			0.0109181
$408$		0			0
$409$		−	179.815i		−	0.439646i	−0.975540	−	0.219823i	$-0.929452\pi$
$409$		−	179.815i		−	0.439646i	0.975540	−	0.219823i	$-0.0705479\pi$
$410$		0			0
$411$		−	334.767i		−	0.814517i
$412$		0			0
$413$	−387.987	+	214.068i	−0.939436	+	0.518325i
$414$		0			0
$415$	−353.535			−0.851892
$416$		0			0
$417$	74.4353			0.178502
$418$		0			0
$419$			297.515i			0.710060i	0.934855	+	0.355030i	$0.115529\pi$
$419$			297.515i			0.710060i	−0.934855	+	0.355030i	$0.884471\pi$
$420$		0			0
$421$	198.707			0.471988			0.235994	−	0.971755i	$-0.424165\pi$
$421$	198.707			0.471988			0.235994	+	0.971755i	$0.424165\pi$
$422$		0			0
$423$		−	220.920i		−	0.522269i
$424$		0			0
$425$		−	381.059i		−	0.896610i
$426$		0			0
$427$	−632.202	+	348.812i	−1.48057	+	0.816889i
$428$		0			0
$429$	86.6120			0.201893
$430$		0			0
$431$	−128.136			−0.297299			−0.148649	−	0.988890i	$-0.547493\pi$
$431$	−128.136			−0.297299			−0.148649	+	0.988890i	$0.547493\pi$
$432$		0			0
$433$		−	593.568i		−	1.37083i	−0.728154	−	0.685414i	$-0.759621\pi$
$433$		−	593.568i		−	1.37083i	0.728154	−	0.685414i	$-0.240379\pi$
$434$		0			0
$435$	−66.1874			−0.152155
$436$		0			0
$437$		−	223.634i		−	0.511748i
$438$		0			0
$439$			88.9764i			0.202680i	0.994852	+	0.101340i	$0.0323130\pi$
$439$			88.9764i			0.202680i	−0.994852	+	0.101340i	$0.967687\pi$
$440$		0			0
$441$	−171.530	+	272.118i	−0.388957	+	0.617047i
$442$		0			0
$443$	508.626			1.14814			0.574070	−	0.818806i	$-0.305364\pi$
$443$	508.626			1.14814			0.574070	+	0.818806i	$0.305364\pi$
$444$		0			0
$445$	6.61204			0.0148585
$446$		0			0
$447$			334.767i			0.748919i
$448$		0			0
$449$	23.2241			0.0517240			0.0258620	−	0.999666i	$-0.491767\pi$
$449$	23.2241			0.0517240			0.0258620	+	0.999666i	$0.491767\pi$
$450$		0			0
$451$		−	9.31405i		−	0.0206520i
$452$		0			0
$453$			87.1227i			0.192324i
$454$		0			0
$455$	154.082	+	279.266i	0.338642	+	0.613771i
$456$		0			0
$457$	273.306			0.598044			0.299022	−	0.954246i	$-0.403340\pi$
$457$	273.306			0.598044			0.299022	+	0.954246i	$0.403340\pi$
$458$		0			0
$459$	−635.494			−1.38452
$460$		0			0
$461$		−	141.273i		−	0.306450i	−0.988191	−	0.153225i	$-0.951034\pi$
$461$		−	141.273i		−	0.306450i	0.988191	−	0.153225i	$-0.0489659\pi$
$462$		0			0
$463$	17.5953			0.0380028			0.0190014	−	0.999819i	$-0.493951\pi$
$463$	17.5953			0.0380028			0.0190014	+	0.999819i	$0.493951\pi$
$464$		0			0
$465$		−	201.119i		−	0.432514i
$466$		0			0
$467$			295.073i			0.631849i	0.948784	+	0.315924i	$0.102315\pi$
$467$			295.073i			0.631849i	−0.948784	+	0.315924i	$0.897685\pi$
$468$		0			0
$469$	97.3060	+	176.362i	0.207476	+	0.376038i
$470$		0			0
$471$	−203.799			−0.432694
$472$		0			0
$473$	233.224			0.493074
$474$		0			0
$475$		−	155.805i		−	0.328010i
$476$		0			0
$477$	196.940			0.412872
$478$		0			0
$479$			785.694i			1.64028i	0.572163	+	0.820140i	$0.306104\pi$
$479$			785.694i			1.64028i	−0.572163	+	0.820140i	$0.693896\pi$
$480$		0			0
$481$			15.9291i			0.0331167i
$482$		0			0
$483$	−199.953	+	110.322i	−0.413981	+	0.228410i
$484$		0			0
$485$	−293.224			−0.604586
$486$		0			0
$487$	−503.451			−1.03378			−0.516890	−	0.856052i	$-0.672910\pi$
$487$	−503.451			−1.03378			−0.516890	+	0.856052i	$0.672910\pi$
$488$		0			0
$489$			462.877i			0.946579i
$490$		0			0
$491$	−851.213			−1.73363			−0.866816	−	0.498628i	$-0.833837\pi$
$491$	−851.213			−1.73363			−0.866816	+	0.498628i	$0.833837\pi$
$492$		0			0
$493$			343.506i			0.696767i
$494$		0			0
$495$			83.4429i			0.168571i
$496$		0			0
$497$	305.647	+	553.968i	0.614983	+	1.11462i
$498$		0			0
$499$	393.990			0.789559			0.394779	−	0.918776i	$-0.370821\pi$
$499$	393.990			0.789559			0.394779	+	0.918776i	$0.370821\pi$
$500$		0			0
$501$	415.268			0.828877
$502$		0			0
$503$			389.312i			0.773980i	0.922084	+	0.386990i	$0.126485\pi$
$503$			389.312i			0.773980i	−0.922084	+	0.386990i	$0.873515\pi$
$504$		0			0
$505$	19.1165			0.0378545
$506$		0			0
$507$			46.7428i			0.0921948i
$508$		0			0
$509$		−	519.840i		−	1.02130i	−0.859790	−	0.510649i	$-0.829405\pi$
$509$		−	519.840i		−	1.02130i	0.859790	−	0.510649i	$-0.170595\pi$
$510$		0			0
$511$	383.435	−	211.556i	0.750362	−	0.414005i
$512$		0			0
$513$	−259.836			−0.506503
$514$		0			0
$515$	−328.329			−0.637533
$516$		0			0
$517$		−	132.418i		−	0.256127i
$518$		0			0
$519$	455.858			0.878340
$520$		0			0
$521$			930.043i			1.78511i	0.450938	+	0.892555i	$0.351090\pi$
$521$			930.043i			1.78511i	−0.450938	+	0.892555i	$0.648910\pi$
$522$		0			0
$523$			1007.90i			1.92715i	0.267442	+	0.963574i	$0.413822\pi$
$523$			1007.90i			1.92715i	−0.267442	+	0.963574i	$0.586178\pi$
$524$		0			0
$525$	−139.306	+	76.8608i	−0.265345	+	0.146401i
$526$		0			0
$527$	−1043.79			−1.98062
$528$		0			0
$529$	−91.9654			−0.173848
$530$		0			0
$531$		−	415.566i		−	0.782610i
$532$		0			0
$533$	33.3880			0.0626416
$534$		0			0
$535$		−	582.368i		−	1.08854i
$536$		0			0
$537$			342.350i			0.637524i
$538$		0			0
$539$	−102.814	+	163.105i	−0.190749	+	0.302607i
$540$		0			0
$541$	629.293			1.16320			0.581602	−	0.813474i	$-0.302426\pi$
$541$	629.293			1.16320			0.581602	+	0.813474i	$0.302426\pi$
$542$		0			0
$543$	−270.877			−0.498853
$544$		0			0
$545$			427.245i			0.783936i
$546$		0			0
$547$	−227.728			−0.416322			−0.208161	−	0.978095i	$-0.566748\pi$
$547$	−227.728			−0.416322			−0.208161	+	0.978095i	$0.566748\pi$
$548$		0			0
$549$		−	677.140i		−	1.23341i
$550$		0			0
$551$			140.450i			0.254901i
$552$		0			0
$553$	−470.328	−	852.444i	−0.850502	−	1.54149i
$554$		0			0
$555$	5.69312			0.0102579
$556$		0			0
$557$	452.259			0.811954			0.405977	−	0.913883i	$-0.366931\pi$
$557$	452.259			0.811954			0.405977	+	0.913883i	$0.366931\pi$
$558$		0			0
$559$			836.035i			1.49559i
$560$		0			0
$561$	−160.656			−0.286373
$562$		0			0
$563$			123.008i			0.218487i	0.994015	+	0.109244i	$0.0348429\pi$
$563$			123.008i			0.218487i	−0.994015	+	0.109244i	$0.965157\pi$
$564$		0			0
$565$			359.254i			0.635848i
$566$		0			0
$567$	−71.6115	−	129.792i	−0.126299	−	0.228910i
$568$		0			0
$569$	317.823			0.558565			0.279282	−	0.960209i	$-0.409903\pi$
$569$	317.823			0.558565			0.279282	+	0.960209i	$0.409903\pi$
$570$		0			0
$571$	934.353			1.63635			0.818173	−	0.574972i	$-0.194987\pi$
$571$	934.353			1.63635			0.818173	+	0.574972i	$0.194987\pi$
$572$		0			0
$573$		−	241.610i		−	0.421658i
$574$		0			0
$575$	304.480			0.529530
$576$		0			0
$577$			382.661i			0.663190i	0.943422	+	0.331595i	$0.107587\pi$
$577$			382.661i			0.663190i	−0.943422	+	0.331595i	$0.892413\pi$
$578$		0			0
$579$		−	256.611i		−	0.443197i
$580$		0			0
$581$	670.763	−	370.087i	1.15450	−	0.636983i
$582$		0			0
$583$	118.044			0.202477
$584$		0			0
$585$	−299.116			−0.511310
$586$		0			0
$587$			962.844i			1.64028i	0.572163	+	0.820140i	$0.306104\pi$
$587$			962.844i			1.64028i	−0.572163	+	0.820140i	$0.693896\pi$
$588$		0			0
$589$	−426.776			−0.724577
$590$		0			0
$591$		−	91.8711i		−	0.155450i
$592$		0			0
$593$		−	136.511i		−	0.230205i	−0.993354	−	0.115102i	$-0.963280\pi$
$593$		−	136.511i		−	0.230205i	0.993354	−	0.115102i	$-0.0367196\pi$
$594$		0			0
$595$	−285.805	−	518.007i	−0.480345	−	0.870599i
$596$		0			0
$597$	−15.3188			−0.0256597
$598$		0			0
$599$	120.150			0.200585			0.100292	−	0.994958i	$-0.468022\pi$
$599$	120.150			0.200585			0.100292	+	0.994958i	$0.468022\pi$
$600$		0			0
$601$			145.464i			0.242037i	0.992650	+	0.121019i	$0.0386161\pi$
$601$			145.464i			0.242037i	−0.992650	+	0.121019i	$0.961384\pi$
$602$		0			0
$603$	−188.898			−0.313264
$604$		0			0
$605$		−	340.861i		−	0.563406i
$606$		0			0
$607$			839.885i			1.38367i	0.722058	+	0.691833i	$0.243196\pi$
$607$			839.885i			1.38367i	−0.722058	+	0.691833i	$0.756804\pi$
$608$		0			0
$609$	125.577	−	69.2862i	0.206203	−	0.113770i
$610$		0			0
$611$	474.676			0.776884
$612$		0			0
$613$	−948.397			−1.54714			−0.773570	−	0.633711i	$-0.781531\pi$
$613$	−948.397			−1.54714			−0.773570	+	0.633711i	$0.781531\pi$
$614$		0			0
$615$		−	11.9330i		−	0.0194032i
$616$		0			0
$617$	−1029.28			−1.66820			−0.834101	−	0.551612i	$-0.814013\pi$
$617$	−1029.28			−1.66820			−0.834101	+	0.551612i	$0.814013\pi$
$618$		0			0
$619$			729.440i			1.17842i	0.807981	+	0.589208i	$0.200560\pi$
$619$			729.440i			1.17842i	−0.807981	+	0.589208i	$0.799440\pi$
$620$		0			0
$621$		−	507.782i		−	0.817684i
$622$		0			0
$623$	−12.5450	+	6.92160i	−0.0201365	+	0.0111101i
$624$		0			0
$625$	−48.7542			−0.0780067
$626$		0			0
$627$	−65.6876			−0.104765
$628$		0			0
$629$		−	29.5467i		−	0.0469741i
$630$		0			0
$631$	528.907			0.838204			0.419102	−	0.907939i	$-0.362345\pi$
$631$	528.907			0.838204			0.419102	+	0.907939i	$0.362345\pi$
$632$		0			0
$633$		−	261.368i		−	0.412904i
$634$		0			0
$635$		−	356.742i		−	0.561799i
$636$		0			0
$637$	−584.681	−	368.556i	−0.917867	−	0.578580i
$638$		0			0
$639$	−593.345			−0.928552
$640$		0			0
$641$	−191.047			−0.298046			−0.149023	−	0.988834i	$-0.547613\pi$
$641$	−191.047			−0.298046			−0.149023	+	0.988834i	$0.547613\pi$
$642$		0			0
$643$		−	113.192i		−	0.176037i	−0.996119	−	0.0880187i	$-0.971946\pi$
$643$		−	113.192i		−	0.176037i	0.996119	−	0.0880187i	$-0.0280535\pi$
$644$		0			0
$645$	298.802			0.463258
$646$		0			0
$647$			11.5293i			0.0178197i	0.999960	+	0.00890984i	$0.00283613\pi$
$647$			11.5293i			0.0178197i	−0.999960	+	0.00890984i	$0.997164\pi$
$648$		0			0
$649$		−	249.087i		−	0.383801i
$650$		0			0
$651$	210.535	+	381.583i	0.323402	+	0.586149i
$652$		0			0
$653$	714.845			1.09471			0.547355	−	0.836901i	$-0.315635\pi$
$653$	714.845			1.09471			0.547355	+	0.836901i	$0.315635\pi$
$654$		0			0
$655$	−647.228			−0.988135
$656$		0			0
$657$			410.690i			0.625099i
$658$		0			0
$659$	−30.7379			−0.0466432			−0.0233216	−	0.999728i	$-0.507424\pi$
$659$	−30.7379			−0.0466432			−0.0233216	+	0.999728i	$0.507424\pi$
$660$		0			0
$661$			177.796i			0.268980i	0.990915	+	0.134490i	$0.0429397\pi$
$661$			177.796i			0.268980i	−0.990915	+	0.134490i	$0.957060\pi$
$662$		0			0
$663$		−	575.900i		−	0.868627i
$664$		0			0
$665$	−116.858	−	211.799i	−0.175726	−	0.318494i
$666$		0			0
$667$	−274.473			−0.411504
$668$		0			0
$669$	−269.931			−0.403484
$670$		0			0
$671$		−	405.873i		−	0.604878i
$672$		0			0
$673$	899.362			1.33635			0.668174	−	0.744005i	$-0.267076\pi$
$673$	899.362			1.33635			0.668174	+	0.744005i	$0.267076\pi$
$674$		0			0
$675$		−	353.769i		−	0.524102i
$676$		0			0
$677$			639.977i			0.945313i	0.881247	+	0.472657i	$0.156705\pi$
$677$			639.977i			0.945313i	−0.881247	+	0.472657i	$0.843295\pi$
$678$		0			0
$679$	556.335	−	306.952i	0.819344	−	0.452065i
$680$		0			0
$681$	560.013			0.822339
$682$		0			0
$683$	−459.196			−0.672322			−0.336161	−	0.941805i	$-0.609129\pi$
$683$	−459.196			−0.672322			−0.336161	+	0.941805i	$0.609129\pi$
$684$		0			0
$685$			692.972i			1.01164i
$686$		0			0
$687$	−374.200			−0.544687
$688$		0			0
$689$			423.152i			0.614153i
$690$		0			0
$691$		−	795.387i		−	1.15107i	−0.817778	−	0.575533i	$-0.804795\pi$
$691$		−	795.387i		−	1.15107i	0.817778	−	0.575533i	$-0.195205\pi$
$692$		0			0
$693$	−87.3495	−	158.316i	−0.126045	−	0.228451i
$694$		0			0
$695$	−154.082			−0.221701
$696$		0			0
$697$	−61.9309			−0.0888535
$698$		0			0
$699$			39.7673i			0.0568916i
$700$		0			0
$701$	−645.810			−0.921270			−0.460635	−	0.887590i	$-0.652378\pi$
$701$	−645.810			−0.921270			−0.460635	+	0.887590i	$0.652378\pi$
$702$		0			0
$703$		−	12.0808i		−	0.0171847i
$704$		0			0
$705$		−	169.651i		−	0.240639i
$706$		0			0
$707$	−36.2698	+	20.0115i	−0.0513009	+	0.0283048i
$708$		0			0
$709$	−418.000			−0.589563			−0.294781	−	0.955565i	$-0.595247\pi$
$709$	−418.000			−0.589563			−0.294781	+	0.955565i	$0.595247\pi$
$710$		0			0
$711$	913.037			1.28416
$712$		0			0
$713$		−	834.022i		−	1.16974i
$714$		0			0
$715$	−179.288			−0.250753
$716$		0			0
$717$			62.5607i			0.0872534i
$718$		0			0
$719$		−	258.373i		−	0.359351i	−0.983726	−	0.179675i	$-0.942495\pi$
$719$		−	258.373i		−	0.359351i	0.983726	−	0.179675i	$-0.0575047\pi$
$720$		0			0
$721$	622.940	−	343.701i	0.863994	−	0.476701i
$722$		0			0
$723$	623.217			0.861988
$724$		0			0
$725$	−191.224			−0.263757
$726$		0			0
$727$		−	345.892i		−	0.475780i	−0.971292	−	0.237890i	$-0.923544\pi$
$727$		−	345.892i		−	0.475780i	0.971292	−	0.237890i	$-0.0764558\pi$
$728$		0			0
$729$	−202.129			−0.277269
$730$		0			0
$731$		−	1550.75i		−	2.12141i
$732$		0			0
$733$			1100.86i			1.50186i	0.660382	+	0.750930i	$0.270394\pi$
$733$			1100.86i			1.50186i	−0.660382	+	0.750930i	$0.729606\pi$
$734$		0			0
$735$	−131.723	+	208.967i	−0.179215	+	0.284309i
$736$		0			0
$737$	−113.224			−0.153628
$738$		0			0
$739$	538.142			0.728203			0.364101	−	0.931359i	$-0.381376\pi$
$739$	538.142			0.728203			0.364101	+	0.931359i	$0.381376\pi$
$740$		0			0
$741$		−	235.470i		−	0.317773i
$742$		0			0
$743$	−73.5300			−0.0989637			−0.0494819	−	0.998775i	$-0.515757\pi$
$743$	−73.5300			−0.0989637			−0.0494819	+	0.998775i	$0.515757\pi$
$744$		0			0
$745$		−	692.972i		−	0.930164i
$746$		0			0
$747$			718.442i			0.961770i
$748$		0			0
$749$	609.634	+	1104.93i	0.813930	+	1.47521i
$750$		0			0
$751$	1198.49			1.59586			0.797932	−	0.602747i	$-0.205927\pi$
$751$	1198.49			1.59586			0.797932	+	0.602747i	$0.205927\pi$
$752$		0			0
$753$	−583.306			−0.774643
$754$		0			0
$755$		−	180.345i		−	0.238868i
$756$		0			0
$757$	172.259			0.227554			0.113777	−	0.993506i	$-0.463705\pi$
$757$	172.259			0.227554			0.113777	+	0.993506i	$0.463705\pi$
$758$		0			0
$759$		−	128.369i		−	0.169130i
$760$		0			0
$761$			899.967i			1.18261i	0.806448	+	0.591305i	$0.201387\pi$
$761$			899.967i			1.18261i	−0.806448	+	0.591305i	$0.798613\pi$
$762$		0			0
$763$	−447.248	−	810.613i	−0.586171	−	1.06240i
$764$		0			0
$765$	554.827			0.725264
$766$		0			0
$767$	892.899			1.16415
$768$		0			0
$769$		−	1170.05i		−	1.52152i	−0.649031	−	0.760762i	$-0.724826\pi$
$769$		−	1170.05i		−	1.52152i	0.649031	−	0.760762i	$-0.275174\pi$
$770$		0			0
$771$	296.083			0.384025
$772$		0			0
$773$			684.116i			0.885014i	0.896765	+	0.442507i	$0.145911\pi$
$773$			684.116i			0.885014i	−0.896765	+	0.442507i	$0.854089\pi$
$774$		0			0
$775$		−	581.059i		−	0.749753i
$776$		0			0
$777$	−10.8016	+	5.95966i	−0.0139016	+	0.00767009i
$778$		0			0
$779$	−25.3218			−0.0325056
$780$		0			0
$781$	−355.647			−0.455373
$782$		0			0
$783$			318.905i			0.407286i
$784$		0			0
$785$	421.867			0.537410
$786$		0			0
$787$		−	709.630i		−	0.901690i	−0.892602	−	0.450845i	$-0.851123\pi$
$787$		−	709.630i		−	0.901690i	0.892602	−	0.450845i	$-0.148877\pi$
$788$		0			0
$789$			45.0859i			0.0571430i
$790$		0			0
$791$	−376.074	−	681.614i	−0.475441	−	0.861712i
$792$		0			0
$793$	1454.93			1.83471
$794$		0			0
$795$	151.236			0.190234
$796$		0			0
$797$		−	1173.92i		−	1.47293i	−0.676478	−	0.736463i	$-0.736494\pi$
$797$		−	1173.92i		−	1.47293i	0.676478	−	0.736463i	$-0.263506\pi$
$798$		0			0
$799$	−880.470			−1.10197
$800$		0			0
$801$		−	13.4368i		−	0.0167750i
$802$		0			0
$803$			246.164i			0.306556i
$804$		0			0
$805$	413.905	−	228.368i	0.514168	−	0.283687i
$806$		0			0
$807$	−207.189			−0.256739
$808$		0			0
$809$	−123.211			−0.152301			−0.0761503	−	0.997096i	$-0.524263\pi$
$809$	−123.211			−0.152301			−0.0761503	+	0.997096i	$0.524263\pi$
$810$		0			0
$811$			295.753i			0.364676i	0.983236	+	0.182338i	$0.0583666\pi$
$811$			295.753i			0.364676i	−0.983236	+	0.182338i	$0.941633\pi$
$812$		0			0
$813$	−519.672			−0.639203
$814$		0			0
$815$		−	958.162i		−	1.17566i
$816$		0			0
$817$		−	634.059i		−	0.776082i
$818$		0			0
$819$	567.514	−	313.121i	0.692936	−	0.382321i
$820$		0			0
$821$	−692.750			−0.843788			−0.421894	−	0.906645i	$-0.638635\pi$
$821$	−692.750			−0.843788			−0.421894	+	0.906645i	$0.638635\pi$
$822$		0			0
$823$	−631.980			−0.767897			−0.383949	−	0.923354i	$-0.625436\pi$
$823$	−631.980			−0.767897			−0.383949	+	0.923354i	$0.625436\pi$
$824$		0			0
$825$		−	89.4342i		−	0.108405i
$826$		0			0
$827$	1074.28			1.29900			0.649501	−	0.760360i	$-0.274978\pi$
$827$	1074.28			1.29900			0.649501	+	0.760360i	$0.274978\pi$
$828$		0			0
$829$			474.866i			0.572817i	0.958108	+	0.286409i	$0.0924615\pi$
$829$			474.866i			0.572817i	−0.958108	+	0.286409i	$0.907539\pi$
$830$		0			0
$831$		−	681.870i		−	0.820541i
$832$		0			0
$833$	1084.52	+	683.629i	1.30194	+	0.820683i
$834$		0			0
$835$	−859.610			−1.02947
$836$		0			0
$837$	−969.035			−1.15775
$838$		0			0
$839$		−	137.181i		−	0.163506i	−0.996653	−	0.0817528i	$-0.973948\pi$
$839$		−	137.181i		−	0.163506i	0.996653	−	0.0817528i	$-0.0260518\pi$
$840$		0			0
$841$	−668.621			−0.795031
$842$		0			0
$843$			241.133i			0.286042i
$844$		0			0
$845$		−	96.7583i		−	0.114507i
$846$		0			0
$847$	356.819	+	646.716i	0.421274	+	0.763537i
$848$		0			0
$849$	−658.410			−0.775512
$850$		0			0
$851$	23.6089			0.0277425
$852$		0			0
$853$			586.940i			0.688089i	0.938953	+	0.344045i	$0.111797\pi$
$853$			586.940i			0.688089i	−0.938953	+	0.344045i	$0.888203\pi$
$854$		0			0
$855$	226.854			0.265326
$856$		0			0
$857$		−	861.341i		−	1.00507i	−0.864558	−	0.502533i	$-0.832402\pi$
$857$		−	861.341i		−	1.00507i	0.864558	−	0.502533i	$-0.167598\pi$
$858$		0			0
$859$		−	253.465i		−	0.295070i	−0.989057	−	0.147535i	$-0.952866\pi$
$859$		−	253.465i		−	0.295070i	0.989057	−	0.147535i	$-0.0471339\pi$
$860$		0			0
$861$	12.4916	+	22.6404i	0.0145083	+	0.0262955i
$862$		0			0
$863$	600.447			0.695767			0.347883	−	0.937538i	$-0.386901\pi$
$863$	600.447			0.695767			0.347883	+	0.937538i	$0.386901\pi$
$864$		0			0
$865$	−943.634			−1.09091
$866$		0			0
$867$			617.227i			0.711912i
$868$		0			0
$869$	547.268			0.629767
$870$		0			0
$871$		−	405.873i		−	0.465985i
$872$		0			0
$873$			595.880i			0.682565i
$874$		0			0
$875$	783.340	−	432.201i	0.895246	−	0.493943i
$876$		0			0
$877$	1156.61			1.31883			0.659414	−	0.751780i	$-0.270804\pi$
$877$	1156.61			1.31883			0.659414	+	0.751780i	$0.270804\pi$
$878$		0			0
$879$	266.292			0.302949
$880$		0			0
$881$		−	57.7013i		−	0.0654952i	−0.999464	−	0.0327476i	$-0.989574\pi$
$881$		−	57.7013i		−	0.0654952i	0.999464	−	0.0327476i	$-0.0104257\pi$
$882$		0			0
$883$	590.053			0.668237			0.334118	−	0.942531i	$-0.391561\pi$
$883$	590.053			0.668237			0.334118	+	0.942531i	$0.391561\pi$
$884$		0			0
$885$		−	319.125i		−	0.360593i
$886$		0			0
$887$			1077.85i			1.21516i	0.794258	+	0.607580i	$0.207860\pi$
$887$			1077.85i			1.21516i	−0.794258	+	0.607580i	$0.792140\pi$
$888$		0			0
$889$	373.444	+	676.848i	0.420072	+	0.761359i
$890$		0			0
$891$	83.3263			0.0935199
$892$		0			0
$893$	−360.000			−0.403135
$894$		0			0
$895$		−	708.670i		−	0.791810i
$896$		0			0
$897$	460.164			0.513003
$898$		0			0
$899$			523.796i			0.582643i
$900$		0			0
$901$		−	784.898i		−	0.871141i
$902$		0			0
$903$	−566.917	+	312.791i	−0.627815	+	0.346391i
$904$		0			0
$905$	560.720			0.619580
$906$		0			0
$907$	−132.793			−0.146409			−0.0732045	−	0.997317i	$-0.523323\pi$
$907$	−132.793			−0.146409			−0.0732045	+	0.997317i	$0.523323\pi$
$908$		0			0
$909$		−	38.8479i		−	0.0427369i
$910$		0			0
$911$	1173.87			1.28855			0.644274	−	0.764794i	$-0.277160\pi$
$911$	1173.87			1.28855			0.644274	+	0.764794i	$0.277160\pi$
$912$		0			0
$913$			430.629i			0.471664i
$914$		0			0
$915$		−	519.995i		−	0.568301i
$916$		0			0
$917$	1227.99	−	677.530i	1.33914	−	0.738855i
$918$		0			0
$919$	1166.90			1.26975			0.634875	−	0.772615i	$-0.281052\pi$
$919$	1166.90			1.26975			0.634875	+	0.772615i	$0.281052\pi$
$920$		0			0
$921$	−408.038			−0.443039
$922$		0			0
$923$		−	1274.88i		−	1.38124i
$924$		0			0
$925$	16.4482			0.0177818
$926$		0			0
$927$			667.219i			0.719762i
$928$		0			0
$929$			679.675i			0.731619i	0.930690	+	0.365810i	$0.119208\pi$
$929$			679.675i			0.731619i	−0.930690	+	0.365810i	$0.880792\pi$
$930$		0			0
$931$	443.429	+	279.517i	0.476294	+	0.300233i
$932$		0			0
$933$	−10.8016			−0.0115772
$934$		0			0
$935$	332.559			0.355678
$936$		0			0
$937$			1411.66i			1.50658i	0.657691	+	0.753288i	$0.271533\pi$
$937$			1411.66i			1.50658i	−0.657691	+	0.753288i	$0.728467\pi$
$938$		0			0
$939$	−101.519			−0.108114
$940$		0			0
$941$		−	607.408i		−	0.645493i	−0.946486	−	0.322746i	$-0.895394\pi$
$941$		−	607.408i		−	0.645493i	0.946486	−	0.322746i	$-0.104606\pi$
$942$		0			0
$943$		−	49.4849i		−	0.0524761i
$944$		0			0
$945$	−265.337	−	480.909i	−0.280780	−	0.508898i
$946$		0			0
$947$	−954.000			−1.00739			−0.503696	−	0.863881i	$-0.668027\pi$
$947$	−954.000			−1.00739			−0.503696	+	0.863881i	$0.668027\pi$
$948$		0			0
$949$	−882.423			−0.929845
$950$		0			0
$951$			102.445i			0.107724i
$952$		0			0
$953$	−1105.48			−1.16000			−0.580001	−	0.814615i	$-0.696948\pi$
$953$	−1105.48			−1.16000			−0.580001	+	0.814615i	$0.696948\pi$
$954$		0			0
$955$			500.136i			0.523703i
$956$		0			0
$957$			80.6205i			0.0842430i
$958$		0			0
$959$	−725.415	−	1314.78i	−0.756429	−	1.37099i
$960$		0			0
$961$	−630.621			−0.656213
$962$		0			0
$963$	−1183.47			−1.22894
$964$		0			0
$965$			531.188i			0.550454i
$966$		0			0
$967$	−1508.91			−1.56040			−0.780201	−	0.625529i	$-0.784883\pi$
$967$	−1508.91			−1.56040			−0.780201	+	0.625529i	$0.784883\pi$
$968$		0			0
$969$			436.769i			0.450742i
$970$		0			0
$971$		−	1059.05i		−	1.09068i	−0.838216	−	0.545338i	$-0.816401\pi$
$971$		−	1059.05i		−	1.09068i	0.838216	−	0.545338i	$-0.183599\pi$
$972$		0			0
$973$	292.341	−	161.296i	0.300453	−	0.165772i
$974$		0			0
$975$	320.594			0.328814
$976$		0			0
$977$	−1164.21			−1.19161			−0.595807	−	0.803127i	$-0.703168\pi$
$977$	−1164.21			−1.19161			−0.595807	+	0.803127i	$0.703168\pi$
$978$		0			0
$979$		−	8.05389i		−	0.00822665i
$980$		0			0
$981$	868.233			0.885049
$982$		0			0
$983$			674.013i			0.685669i	0.939396	+	0.342835i	$0.111387\pi$
$983$			674.013i			0.685669i	−0.939396	+	0.342835i	$0.888613\pi$
$984$		0			0
$985$			190.175i			0.193071i
$986$		0			0
$987$	177.593	+	321.878i	0.179932	+	0.326118i
$988$		0			0
$989$	1239.10			1.25289
$990$		0			0
$991$	−1482.70			−1.49617			−0.748084	−	0.663604i	$-0.769026\pi$
$991$	−1482.70			−1.49617			−0.748084	+	0.663604i	$0.769026\pi$
$992$		0			0
$993$		−	118.981i		−	0.119820i
$994$		0			0
$995$	31.7102			0.0318696
$996$		0			0
$997$		−	102.564i		−	0.102873i	−0.998676	−	0.0514365i	$-0.983620\pi$
$997$		−	102.564i		−	0.102873i	0.998676	−	0.0514365i	$-0.0163800\pi$
$998$		0			0
$999$		−	27.4307i		−	0.0274581i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.c.g.321.4		8
4.3	odd	2	inner	448.3.c.g.321.6		8
7.6	odd	2	inner	448.3.c.g.321.5		8
8.3	odd	2		224.3.c.a.97.3	✓	8
8.5	even	2		224.3.c.a.97.5	yes	8
24.5	odd	2		2016.3.f.c.1441.5		8
24.11	even	2		2016.3.f.c.1441.6		8
28.27	even	2	inner	448.3.c.g.321.3		8
56.13	odd	2		224.3.c.a.97.4	yes	8
56.27	even	2		224.3.c.a.97.6	yes	8
168.83	odd	2		2016.3.f.c.1441.4		8
168.125	even	2		2016.3.f.c.1441.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.c.a.97.3	✓	8	8.3	odd	2
224.3.c.a.97.4	yes	8	56.13	odd	2
224.3.c.a.97.5	yes	8	8.5	even	2
224.3.c.a.97.6	yes	8	56.27	even	2
448.3.c.g.321.3		8	28.27	even	2	inner
448.3.c.g.321.4		8	1.1	even	1	trivial
448.3.c.g.321.5		8	7.6	odd	2	inner
448.3.c.g.321.6		8	4.3	odd	2	inner
2016.3.f.c.1441.3		8	168.125	even	2
2016.3.f.c.1441.4		8	168.83	odd	2
2016.3.f.c.1441.5		8	24.5	odd	2
2016.3.f.c.1441.6		8	24.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.56056i q^{3} +3.23038i q^{5} +(-3.38162 - 6.12900i) q^{7} +6.56466 q^{9} +3.93481 q^{11} +14.1051i q^{13} +5.04119 q^{15} -26.1633i q^{17} -10.6974i q^{19} +(-9.56466 + 5.27721i) q^{21} +20.9054 q^{23} +14.5647 q^{25} -24.2895i q^{27} -13.1293 q^{29} -39.8951i q^{31} -6.14050i q^{33} +(19.7990 - 10.9239i) q^{35} +1.12932 q^{37} +22.0118 q^{39} -2.36709i q^{41} +59.2720 q^{43} +21.2063i q^{45} -33.6529i q^{47} +(-26.1293 + 41.4519i) q^{49} -40.8293 q^{51} +30.0000 q^{53} +12.7109i q^{55} -16.6940 q^{57} -63.3035i q^{59} -103.149i q^{61} +(-22.1992 - 40.2348i) q^{63} -45.5647 q^{65} -28.7750 q^{67} -32.6240i q^{69} -90.3847 q^{71} +62.5607i q^{73} -22.7290i q^{75} +(-13.3060 - 24.1164i) q^{77} +139.084 q^{79} +21.1767 q^{81} +109.441i q^{83} +84.5173 q^{85} +20.4891i q^{87} -2.04683i q^{89} +(86.4499 - 47.6979i) q^{91} -62.2586 q^{93} +34.5568 q^{95} +90.7708i q^{97} +25.8307 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 56 q^{9} + 32 q^{21} + 8 q^{25} + 112 q^{29} - 208 q^{37} + 8 q^{49} + 240 q^{53} + 192 q^{57} - 256 q^{65} - 432 q^{77} + 712 q^{81} - 192 q^{85} - 64 q^{93}+O(q^{100}) \) 8 * q - 56 * q^9 + 32 * q^21 + 8 * q^25 + 112 * q^29 - 208 * q^37 + 8 * q^49 + 240 * q^53 + 192 * q^57 - 256 * q^65 - 432 * q^77 + 712 * q^81 - 192 * q^85 - 64 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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