LMFDB - Embedded newform 176.6.e.d.175.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,6,Mod(175,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("176.175");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$N$	$=$	$176 = 2^{4} \cdot 11$
Weight:	$k$	$=$	$6$
Character orbit:	$[\chi]$	$=$	176.e (of order $2$ , degree $1$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$28.2275522871$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} + 5899 x^{14} + 31909903 x^{12} + 16672338574 x^{10} + 7262078908132 x^{8} + \cdots + 21\!\cdots\!96$ x^16 + 5899x^14 + 31909903x^12 + 16672338574x^10 + 7262078908132x^8 + 510923309868544x^6 + 29217515417325568x^4 + 267327060014792704*x^2 + 2137083591258013696
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{50}\cdot 3^{2}$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			175.15
Root			$10.7118 + 18.5534i$ of defining polynomial
Character	$\chi$	$=$	176.175
Dual form			176.6.e.d.175.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+21.5024i q^{3} +24.8335 q^{5} -17.3632 q^{7} -219.355 q^{9} +(-379.780 - 129.684i) q^{11} -1002.39i q^{13} +533.980i q^{15} -846.763i q^{17} -1735.80 q^{19} -373.351i q^{21} +1110.11i q^{23} -2508.30 q^{25} +508.426i q^{27} +758.494i q^{29} -4938.46i q^{31} +(2788.51 - 8166.21i) q^{33} -431.189 q^{35} -1944.38 q^{37} +21553.9 q^{39} -11226.6i q^{41} +18052.7 q^{43} -5447.35 q^{45} -12212.7i q^{47} -16505.5 q^{49} +18207.5 q^{51} +11536.3 q^{53} +(-9431.27 - 3220.49i) q^{55} -37324.0i q^{57} -19586.2i q^{59} -684.354i q^{61} +3808.70 q^{63} -24892.9i q^{65} +25381.7i q^{67} -23870.2 q^{69} +57269.5i q^{71} -66380.2i q^{73} -53934.5i q^{75} +(6594.20 + 2251.72i) q^{77} -28098.7 q^{79} -64235.7 q^{81} -47152.5 q^{83} -21028.1i q^{85} -16309.5 q^{87} -54807.7 q^{89} +17404.8i q^{91} +106189. q^{93} -43106.0 q^{95} +89682.6 q^{97} +(83306.7 + 28446.7i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 100 q^{5} + 892 q^{9} + 12396 q^{25} - 8948 q^{33} - 6644 q^{37} + 16840 q^{45} + 123856 q^{49} + 672 q^{53} - 45108 q^{69} + 155968 q^{77} - 191512 q^{81} + 252348 q^{89} + 125852 q^{93} + 45180 q^{97}+O(q^{100})$ 16 * q - 100 * q^5 + 892 * q^9 + 12396 * q^25 - 8948 * q^33 - 6644 * q^37 + 16840 * q^45 + 123856 * q^49 + 672 * q^53 - 45108 * q^69 + 155968 * q^77 - 191512 * q^81 + 252348 * q^89 + 125852 * q^93 + 45180 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/176\mathbb{Z}\right)^\times$ .

$n$	$111$	$133$	$145$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$			21.5024i			1.37938i	0.724104	+	0.689691i	$0.242254\pi$
$3$			21.5024i			1.37938i	−0.724104	+	0.689691i	$0.757746\pi$
$4$		0			0
$5$	24.8335			0.444235			0.222117	−	0.975020i	$-0.428703\pi$
$5$	24.8335			0.444235			0.222117	+	0.975020i	$0.428703\pi$
$6$		0			0
$7$	−17.3632			−0.133932			−0.0669660	−	0.997755i	$-0.521332\pi$
$7$	−17.3632			−0.133932			−0.0669660	+	0.997755i	$0.521332\pi$
$8$		0			0
$9$	−219.355			−0.902695
$10$		0			0
$11$	−379.780	−	129.684i	−0.946348	−	0.323149i
$11$	−379.780	−	129.684i	−0.946348	−	0.323149i
$12$		0			0
$13$		−	1002.39i		−	1.64506i	−0.568725	−	0.822528i	$-0.692563\pi$
$13$		−	1002.39i		−	1.64506i	0.568725	−	0.822528i	$-0.307437\pi$
$14$		0			0
$15$			533.980i			0.612769i
$16$		0			0
$17$		−	846.763i		−	0.710624i	−0.934748	−	0.355312i	$-0.884375\pi$
$17$		−	846.763i		−	0.710624i	0.934748	−	0.355312i	$-0.115625\pi$
$18$		0			0
$19$	−1735.80			−1.10310			−0.551552	−	0.834140i	$-0.685964\pi$
$19$	−1735.80			−1.10310			−0.551552	+	0.834140i	$0.685964\pi$
$20$		0			0
$21$		−	373.351i		−	0.184744i
$22$		0			0
$23$			1110.11i			0.437571i	0.975773	+	0.218785i	$0.0702094\pi$
$23$			1110.11i			0.437571i	−0.975773	+	0.218785i	$0.929791\pi$
$24$		0			0
$25$	−2508.30			−0.802656
$26$		0			0
$27$			508.426i			0.134220i
$28$		0			0
$29$			758.494i			0.167478i	0.996488	+	0.0837389i	$0.0266862\pi$
$29$			758.494i			0.167478i	−0.996488	+	0.0837389i	$0.973314\pi$
$30$		0			0
$31$		−	4938.46i		−	0.922969i	−0.887148	−	0.461484i	$-0.847317\pi$
$31$		−	4938.46i		−	0.922969i	0.887148	−	0.461484i	$-0.152683\pi$
$32$		0			0
$33$	2788.51	−	8166.21i	0.445746	−	1.30538i
$34$		0			0
$35$	−431.189			−0.0594973
$36$		0			0
$37$	−1944.38			−0.233494			−0.116747	−	0.993162i	$-0.537247\pi$
$37$	−1944.38			−0.233494			−0.116747	+	0.993162i	$0.537247\pi$
$38$		0			0
$39$	21553.9			2.26916
$40$		0			0
$41$		−	11226.6i		−	1.04301i	−0.853249	−	0.521503i	$-0.825371\pi$
$41$		−	11226.6i		−	1.04301i	0.853249	−	0.521503i	$-0.174629\pi$
$42$		0			0
$43$	18052.7			1.48892			0.744461	−	0.667666i	$-0.232707\pi$
$43$	18052.7			1.48892			0.744461	+	0.667666i	$0.232707\pi$
$44$		0			0
$45$	−5447.35			−0.401009
$46$		0			0
$47$		−	12212.7i		−	0.806431i	−0.915105	−	0.403215i	$-0.867893\pi$
$47$		−	12212.7i		−	0.806431i	0.915105	−	0.403215i	$-0.132107\pi$
$48$		0			0
$49$	−16505.5			−0.982062
$50$		0			0
$51$	18207.5			0.980222
$52$		0			0
$53$	11536.3			0.564128			0.282064	−	0.959396i	$-0.408981\pi$
$53$	11536.3			0.564128			0.282064	+	0.959396i	$0.408981\pi$
$54$		0			0
$55$	−9431.27	−	3220.49i	−0.420401	−	0.143554i
$56$		0			0
$57$		−	37324.0i		−	1.52160i
$58$		0			0
$59$		−	19586.2i		−	0.732520i	−0.930513	−	0.366260i	$-0.880638\pi$
$59$		−	19586.2i		−	0.732520i	0.930513	−	0.366260i	$-0.119362\pi$
$60$		0			0
$61$		−	684.354i		−	0.0235481i	−0.999931	−	0.0117741i	$-0.996252\pi$
$61$		−	684.354i		−	0.0235481i	0.999931	−	0.0117741i	$-0.00374789\pi$
$62$		0			0
$63$	3808.70			0.120900
$64$		0			0
$65$		−	24892.9i		−	0.730791i
$66$		0			0
$67$			25381.7i			0.690770i	0.938461	+	0.345385i	$0.112252\pi$
$67$			25381.7i			0.690770i	−0.938461	+	0.345385i	$0.887748\pi$
$68$		0			0
$69$	−23870.2			−0.603577
$70$		0			0
$71$			57269.5i			1.34827i	0.738607	+	0.674136i	$0.235484\pi$
$71$			57269.5i			1.34827i	−0.738607	+	0.674136i	$0.764516\pi$
$72$		0			0
$73$		−	66380.2i		−	1.45791i	−0.684561	−	0.728956i	$-0.740006\pi$
$73$		−	66380.2i		−	1.45791i	0.684561	−	0.728956i	$-0.259994\pi$
$74$		0			0
$75$		−	53934.5i		−	1.10717i
$76$		0			0
$77$	6594.20	+	2251.72i	0.126746	+	0.0432800i
$78$		0			0
$79$	−28098.7			−0.506545			−0.253273	−	0.967395i	$-0.581507\pi$
$79$	−28098.7			−0.506545			−0.253273	+	0.967395i	$0.581507\pi$
$80$		0			0
$81$	−64235.7			−1.08784
$82$		0			0
$83$	−47152.5			−0.751293			−0.375647	−	0.926763i	$-0.622579\pi$
$83$	−47152.5			−0.751293			−0.375647	+	0.926763i	$0.622579\pi$
$84$		0			0
$85$		−	21028.1i		−	0.315684i
$86$		0			0
$87$	−16309.5			−0.231016
$88$		0			0
$89$	−54807.7			−0.733444			−0.366722	−	0.930331i	$-0.619520\pi$
$89$	−54807.7			−0.733444			−0.366722	+	0.930331i	$0.619520\pi$
$90$		0			0
$91$			17404.8i			0.220326i
$92$		0			0
$93$	106189.			1.27313
$94$		0			0
$95$	−43106.0			−0.490037
$96$		0			0
$97$	89682.6			0.967785			0.483893	−	0.875127i	$-0.339223\pi$
$97$	89682.6			0.967785			0.483893	+	0.875127i	$0.339223\pi$
$98$		0			0
$99$	83306.7	+	28446.7i	0.854264	+	0.291705i
$100$		0			0
$101$			135403.i			1.32076i	0.750931	+	0.660381i	$0.229605\pi$
$101$			135403.i			1.32076i	−0.750931	+	0.660381i	$0.770395\pi$
$102$		0			0
$103$		−	55268.7i		−	0.513318i	−0.966502	−	0.256659i	$-0.917378\pi$
$103$		−	55268.7i		−	0.513318i	0.966502	−	0.256659i	$-0.0826217\pi$
$104$		0			0
$105$		−	9271.61i		−	0.0820695i
$106$		0			0
$107$	79805.4			0.673865			0.336932	−	0.941529i	$-0.390611\pi$
$107$	79805.4			0.673865			0.336932	+	0.941529i	$0.390611\pi$
$108$		0			0
$109$			20374.2i			0.164254i	0.996622	+	0.0821268i	$0.0261712\pi$
$109$			20374.2i			0.164254i	−0.996622	+	0.0821268i	$0.973829\pi$
$110$		0			0
$111$		−	41808.8i		−	0.322078i
$112$		0			0
$113$	−69352.1			−0.510932			−0.255466	−	0.966818i	$-0.582229\pi$
$113$	−69352.1			−0.510932			−0.255466	+	0.966818i	$0.582229\pi$
$114$		0			0
$115$			27568.0i			0.194384i
$116$		0			0
$117$			219880.i			1.48498i
$118$		0			0
$119$			14702.5i			0.0951753i
$120$		0			0
$121$	127415.	+	98502.5i	0.791149	+	0.611623i
$122$		0			0
$123$	241398.			1.43870
$124$		0			0
$125$	−139894.			−0.800802
$126$		0			0
$127$	−189808.			−1.04425			−0.522125	−	0.852869i	$-0.674860\pi$
$127$	−189808.			−1.04425			−0.522125	+	0.852869i	$0.674860\pi$
$128$		0			0
$129$			388178.i			2.05379i
$130$		0			0
$131$	−177122.			−0.901766			−0.450883	−	0.892583i	$-0.648891\pi$
$131$	−177122.			−0.901766			−0.450883	+	0.892583i	$0.648891\pi$
$132$		0			0
$133$	30139.1			0.147741
$134$		0			0
$135$			12626.0i			0.0596253i
$136$		0			0
$137$	340956.			1.55202			0.776009	−	0.630722i	$-0.217241\pi$
$137$	340956.			1.55202			0.776009	+	0.630722i	$0.217241\pi$
$138$		0			0
$139$	−64495.5			−0.283134			−0.141567	−	0.989929i	$-0.545214\pi$
$139$	−64495.5			−0.283134			−0.141567	+	0.989929i	$0.545214\pi$
$140$		0			0
$141$	262603.			1.11238
$142$		0			0
$143$	−129994.	+	380690.i	−0.531598	+	1.55679i
$144$		0			0
$145$			18836.1i			0.0743995i
$146$		0			0
$147$		−	354909.i		−	1.35464i
$148$		0			0
$149$			20917.1i			0.0771855i	0.999255	+	0.0385927i	$0.0122875\pi$
$149$			20917.1i			0.0771855i	−0.999255	+	0.0385927i	$0.987712\pi$
$150$		0			0
$151$	−433488.			−1.54716			−0.773580	−	0.633699i	$-0.781536\pi$
$151$	−433488.			−1.54716			−0.773580	+	0.633699i	$0.781536\pi$
$152$		0			0
$153$			185742.i			0.641477i
$154$		0			0
$155$		−	122639.i		−	0.410015i
$156$		0			0
$157$	−581511.			−1.88282			−0.941410	−	0.337266i	$-0.890498\pi$
$157$	−581511.			−1.88282			−0.941410	+	0.337266i	$0.890498\pi$
$158$		0			0
$159$			248059.i			0.778148i
$160$		0			0
$161$		−	19275.1i		−	0.0586047i
$162$		0			0
$163$			322467.i			0.950640i	0.879813	+	0.475320i	$0.157668\pi$
$163$			322467.i			0.950640i	−0.879813	+	0.475320i	$0.842332\pi$
$164$		0			0
$165$	69248.4	−	202795.i	0.198016	−	0.579893i
$166$		0			0
$167$	−552878.			−1.53405			−0.767023	−	0.641620i	$-0.778263\pi$
$167$	−552878.			−1.53405			−0.767023	+	0.641620i	$0.778263\pi$
$168$		0			0
$169$	−633502.			−1.70621
$170$		0			0
$171$	380757.			0.995767
$172$		0			0
$173$		−	713189.i		−	1.81171i	−0.423586	−	0.905856i	$-0.639229\pi$
$173$		−	713189.i		−	1.81171i	0.423586	−	0.905856i	$-0.360771\pi$
$174$		0			0
$175$	43552.1			0.107501
$176$		0			0
$177$	421150.			1.01043
$178$		0			0
$179$		−	332381.i		−	0.775360i	−0.921794	−	0.387680i	$-0.873277\pi$
$179$		−	332381.i		−	0.775360i	0.921794	−	0.387680i	$-0.126723\pi$
$180$		0			0
$181$	658032.			1.49297			0.746484	−	0.665403i	$-0.231740\pi$
$181$	658032.			1.49297			0.746484	+	0.665403i	$0.231740\pi$
$182$		0			0
$183$	14715.3			0.0324819
$184$		0			0
$185$	−48285.6			−0.103726
$186$		0			0
$187$	−109811.	+	321584.i	−0.229637	+	0.672497i
$188$		0			0
$189$		−	8827.90i		−	0.0179764i
$190$		0			0
$191$		−	90836.0i		−	0.180167i	−0.995934	−	0.0900834i	$-0.971287\pi$
$191$		−	90836.0i		−	0.180167i	0.995934	−	0.0900834i	$-0.0287134\pi$
$192$		0			0
$193$			420632.i			0.812848i	0.913685	+	0.406424i	$0.133224\pi$
$193$			420632.i			0.812848i	−0.913685	+	0.406424i	$0.866776\pi$
$194$		0			0
$195$	535259.			1.00804
$196$		0			0
$197$		−	205037.i		−	0.376415i	−0.982129	−	0.188208i	$-0.939732\pi$
$197$		−	205037.i		−	0.376415i	0.982129	−	0.188208i	$-0.0602678\pi$
$198$		0			0
$199$		−	328466.i		−	0.587973i	−0.955810	−	0.293987i	$-0.905018\pi$
$199$		−	328466.i		−	0.587973i	0.955810	−	0.293987i	$-0.0949821\pi$
$200$		0			0
$201$	−545768.			−0.952836
$202$		0			0
$203$		−	13169.9i		−	0.0224307i
$204$		0			0
$205$		−	278794.i		−	0.463340i
$206$		0			0
$207$		−	243509.i		−	0.394993i
$208$		0			0
$209$	659224.	+	225105.i	1.04392	+	0.356467i
$210$		0			0
$211$	52624.3			0.0813730			0.0406865	−	0.999172i	$-0.487046\pi$
$211$	52624.3			0.0813730			0.0406865	+	0.999172i	$0.487046\pi$
$212$		0			0
$213$	−1.23143e6			−1.85978
$214$		0			0
$215$	448312.			0.661431
$216$		0			0
$217$			85747.4i			0.123615i
$218$		0			0
$219$	1.42734e6			2.01102
$220$		0			0
$221$	−848791.			−1.16902
$222$		0			0
$223$		−	434352.i		−	0.584897i	−0.956281	−	0.292448i	$-0.905530\pi$
$223$		−	434352.i		−	0.584897i	0.956281	−	0.292448i	$-0.0944700\pi$
$224$		0			0
$225$	550208.			0.724553
$226$		0			0
$227$	1.01255e6			1.30422			0.652111	−	0.758124i	$-0.273884\pi$
$227$	1.01255e6			1.30422			0.652111	+	0.758124i	$0.273884\pi$
$228$		0			0
$229$	1.20382e6			1.51695			0.758477	−	0.651699i	$-0.225944\pi$
$229$	1.20382e6			1.51695			0.758477	+	0.651699i	$0.225944\pi$
$230$		0			0
$231$	−48417.5	+	141791.i	−0.0596997	+	0.174832i
$232$		0			0
$233$		−	1.52503e6i		−	1.84030i	−0.391563	−	0.920151i	$-0.628066\pi$
$233$		−	1.52503e6i		−	1.84030i	0.391563	−	0.920151i	$-0.371934\pi$
$234$		0			0
$235$		−	303284.i		−	0.358245i
$236$		0			0
$237$		−	604190.i		−	0.698720i
$238$		0			0
$239$	340228.			0.385279			0.192640	−	0.981270i	$-0.438295\pi$
$239$	340228.			0.385279			0.192640	+	0.981270i	$0.438295\pi$
$240$		0			0
$241$			30423.5i			0.0337417i	0.999858	+	0.0168708i	$0.00537041\pi$
$241$			30423.5i			0.0337417i	−0.999858	+	0.0168708i	$0.994630\pi$
$242$		0			0
$243$		−	1.25768e6i		−	1.36632i
$244$		0			0
$245$	−409889.			−0.436266
$246$		0			0
$247$			1.73996e6i			1.81467i
$248$		0			0
$249$		−	1.01389e6i		−	1.03632i
$250$		0			0
$251$		−	138093.i		−	0.138353i	−0.997604	−	0.0691763i	$-0.977963\pi$
$251$		−	138093.i		−	0.138353i	0.997604	−	0.0691763i	$-0.0220371\pi$
$252$		0			0
$253$	143964.	−	421600.i	0.141401	−	0.414094i
$254$		0			0
$255$	452155.			0.435449
$256$		0			0
$257$	−1.93844e6			−1.83071			−0.915357	−	0.402644i	$-0.868091\pi$
$257$	−1.93844e6			−1.83071			−0.915357	+	0.402644i	$0.868091\pi$
$258$		0			0
$259$	33760.6			0.0312723
$260$		0			0
$261$		−	166380.i		−	0.151181i
$262$		0			0
$263$	−2.00327e6			−1.78587			−0.892934	−	0.450187i	$-0.851357\pi$
$263$	−2.00327e6			−1.78587			−0.892934	+	0.450187i	$0.851357\pi$
$264$		0			0
$265$	286487.			0.250605
$266$		0			0
$267$		−	1.17850e6i		−	1.01170i
$268$		0			0
$269$	237551.			0.200160			0.100080	−	0.994979i	$-0.468090\pi$
$269$	237551.			0.200160			0.100080	+	0.994979i	$0.468090\pi$
$270$		0			0
$271$	−1.89294e6			−1.56572			−0.782860	−	0.622198i	$-0.786240\pi$
$271$	−1.89294e6			−1.56572			−0.782860	+	0.622198i	$0.786240\pi$
$272$		0			0
$273$	−374245.			−0.303913
$274$		0			0
$275$	952603.	+	325285.i	0.759591	+	0.259377i
$276$		0			0
$277$			1.63161e6i			1.27766i	0.769347	+	0.638831i	$0.220582\pi$
$277$			1.63161e6i			1.27766i	−0.769347	+	0.638831i	$0.779418\pi$
$278$		0			0
$279$			1.08328e6i			0.833160i
$280$		0			0
$281$		−	2.01749e6i		−	1.52422i	−0.647450	−	0.762108i	$-0.724164\pi$
$281$		−	2.01749e6i		−	1.52422i	0.647450	−	0.762108i	$-0.275836\pi$
$282$		0			0
$283$	−393320.			−0.291931			−0.145965	−	0.989290i	$-0.546629\pi$
$283$	−393320.			−0.291931			−0.145965	+	0.989290i	$0.546629\pi$
$284$		0			0
$285$		−	926885.i		−	0.675949i
$286$		0			0
$287$			194929.i			0.139692i
$288$		0			0
$289$	702849.			0.495014
$290$		0			0
$291$			1.92840e6i			1.33495i
$292$		0			0
$293$			1.71699e6i			1.16842i	0.811603	+	0.584209i	$0.198595\pi$
$293$			1.71699e6i			1.16842i	−0.811603	+	0.584209i	$0.801405\pi$
$294$		0			0
$295$		−	486393.i		−	0.325411i
$296$		0			0
$297$	65934.4	−	193090.i	0.0433732	−	0.127019i
$298$		0			0
$299$	1.11277e6			0.719828
$300$		0			0
$301$	−313453.			−0.199414
$302$		0			0
$303$	−2.91149e6			−1.82183
$304$		0			0
$305$		−	16994.9i		−	0.0104609i
$306$		0			0
$307$	−365235.			−0.221170			−0.110585	−	0.993867i	$-0.535272\pi$
$307$	−365235.			−0.221170			−0.110585	+	0.993867i	$0.535272\pi$
$308$		0			0
$309$	1.18841e6			0.708062
$310$		0			0
$311$		−	1.43559e6i		−	0.841647i	−0.907143	−	0.420823i	$-0.861741\pi$
$311$		−	1.43559e6i		−	0.841647i	0.907143	−	0.420823i	$-0.138259\pi$
$312$		0			0
$313$	−939798.			−0.542218			−0.271109	−	0.962549i	$-0.587390\pi$
$313$	−939798.			−0.542218			−0.271109	+	0.962549i	$0.587390\pi$
$314$		0			0
$315$	94583.3			0.0537079
$316$		0			0
$317$	1.73291e6			0.968562			0.484281	−	0.874913i	$-0.339081\pi$
$317$	1.73291e6			0.968562			0.484281	+	0.874913i	$0.339081\pi$
$318$		0			0
$319$	98364.2	−	288061.i	0.0541203	−	0.158492i
$320$		0			0
$321$			1.71601e6i			0.929517i
$322$		0			0
$323$			1.46981e6i			0.783892i
$324$		0			0
$325$			2.51431e6i			1.32041i
$326$		0			0
$327$	−438096.			−0.226568
$328$		0			0
$329$			212052.i			0.108007i
$330$		0			0
$331$		−	1.92231e6i		−	0.964393i	−0.876063	−	0.482197i	$-0.839839\pi$
$331$		−	1.92231e6i		−	0.964393i	0.876063	−	0.482197i	$-0.160161\pi$
$332$		0			0
$333$	426508.			0.210774
$334$		0			0
$335$			630315.i			0.306864i
$336$		0			0
$337$			1.85537e6i			0.889931i	0.895548	+	0.444965i	$0.146784\pi$
$337$			1.85537e6i			0.889931i	−0.895548	+	0.444965i	$0.853216\pi$
$338$		0			0
$339$		−	1.49124e6i		−	0.704771i
$340$		0			0
$341$	−640436.	+	1.87553e6i	−0.298257	+	0.873450i
$342$		0			0
$343$	578412.			0.265462
$344$		0			0
$345$	−592779.			−0.268130
$346$		0			0
$347$	1.52856e6			0.681487			0.340743	−	0.940156i	$-0.389321\pi$
$347$	1.52856e6			0.681487			0.340743	+	0.940156i	$0.389321\pi$
$348$		0			0
$349$			4.10112e6i			1.80235i	0.433457	+	0.901174i	$0.357294\pi$
$349$			4.10112e6i			1.80235i	−0.433457	+	0.901174i	$0.642706\pi$
$350$		0			0
$351$	509643.			0.220800
$352$		0			0
$353$	3.11061e6			1.32865			0.664323	−	0.747446i	$-0.268720\pi$
$353$	3.11061e6			1.32865			0.664323	+	0.747446i	$0.268720\pi$
$354$		0			0
$355$			1.42220e6i			0.598949i
$356$		0			0
$357$	−316140.			−0.131283
$358$		0			0
$359$	3.51575e6			1.43973			0.719867	−	0.694112i	$-0.244203\pi$
$359$	3.51575e6			1.43973			0.719867	+	0.694112i	$0.244203\pi$
$360$		0			0
$361$	536915.			0.216839
$362$		0			0
$363$	−2.11804e6	+	2.73974e6i	−0.843662	+	1.09130i
$364$		0			0
$365$		−	1.64845e6i		−	0.647655i
$366$		0			0
$367$			4.03985e6i			1.56567i	0.622229	+	0.782835i	$0.286227\pi$
$367$			4.03985e6i			1.56567i	−0.622229	+	0.782835i	$0.713773\pi$
$368$		0			0
$369$			2.46260e6i			0.941517i
$370$		0			0
$371$	−200307.			−0.0755548
$372$		0			0
$373$			3.77771e6i			1.40591i	0.711236	+	0.702953i	$0.248136\pi$
$373$			3.77771e6i			1.40591i	−0.711236	+	0.702953i	$0.751864\pi$
$374$		0			0
$375$		−	3.00807e6i		−	1.10461i
$376$		0			0
$377$	760311.			0.275510
$378$		0			0
$379$		−	4.10485e6i		−	1.46791i	−0.679198	−	0.733955i	$-0.737672\pi$
$379$		−	4.10485e6i		−	1.46791i	0.679198	−	0.733955i	$-0.262328\pi$
$380$		0			0
$381$		−	4.08133e6i		−	1.44042i
$382$		0			0
$383$		−	1.96710e6i		−	0.685218i	−0.939478	−	0.342609i	$-0.888689\pi$
$383$		−	1.96710e6i		−	0.685218i	0.939478	−	0.342609i	$-0.111311\pi$
$384$		0			0
$385$	163757.	+	55918.0i	0.0563051	+	0.0192265i
$386$		0			0
$387$	−3.95996e6			−1.34404
$388$		0			0
$389$	1.60422e6			0.537515			0.268758	−	0.963208i	$-0.413387\pi$
$389$	1.60422e6			0.537515			0.268758	+	0.963208i	$0.413387\pi$
$390$		0			0
$391$	940004.			0.310948
$392$		0			0
$393$		−	3.80855e6i		−	1.24388i
$394$		0			0
$395$	−697788.			−0.225025
$396$		0			0
$397$	1.57123e6			0.500338			0.250169	−	0.968202i	$-0.419514\pi$
$397$	1.57123e6			0.500338			0.250169	+	0.968202i	$0.419514\pi$
$398$		0			0
$399$			648064.i			0.203791i
$400$		0			0
$401$	−4.93351e6			−1.53213			−0.766064	−	0.642764i	$-0.777787\pi$
$401$	−4.93351e6			−1.53213			−0.766064	+	0.642764i	$0.777787\pi$
$402$		0			0
$403$	−4.95028e6			−1.51833
$404$		0			0
$405$	−1.59519e6			−0.483255
$406$		0			0
$407$	738436.	+	252153.i	0.220967	+	0.0754534i
$408$		0			0
$409$			3.30820e6i			0.977876i	0.872318	+	0.488938i	$0.162616\pi$
$409$			3.30820e6i			0.977876i	−0.872318	+	0.488938i	$0.837384\pi$
$410$		0			0
$411$			7.33138e6i			2.14083i
$412$		0			0
$413$			340079.i			0.0981079i
$414$		0			0
$415$	−1.17096e6			−0.333751
$416$		0			0
$417$		−	1.38681e6i		−	0.390550i
$418$		0			0
$419$		−	6.24278e6i		−	1.73717i	−0.495537	−	0.868587i	$-0.665029\pi$
$419$		−	6.24278e6i		−	1.73717i	0.495537	−	0.868587i	$-0.334971\pi$
$420$		0			0
$421$	2.84586e6			0.782543			0.391271	−	0.920275i	$-0.372035\pi$
$421$	2.84586e6			0.782543			0.391271	+	0.920275i	$0.372035\pi$
$422$		0			0
$423$			2.67892e6i			0.727961i
$424$		0			0
$425$			2.12394e6i			0.570386i
$426$		0			0
$427$			11882.6i			0.00315385i
$428$		0			0
$429$	−8.18576e6	−	2.79519e6i	−2.14741	−	0.733277i
$430$		0			0
$431$	−5.13347e6			−1.33112			−0.665561	−	0.746343i	$-0.731808\pi$
$431$	−5.13347e6			−1.33112			−0.665561	+	0.746343i	$0.731808\pi$
$432$		0			0
$433$	4.79632e6			1.22939			0.614694	−	0.788766i	$-0.289280\pi$
$433$	4.79632e6			1.22939			0.614694	+	0.788766i	$0.289280\pi$
$434$		0			0
$435$	−405021.			−0.102625
$436$		0			0
$437$		−	1.92694e6i		−	0.482686i
$438$		0			0
$439$	5.42633e6			1.34383			0.671916	−	0.740628i	$-0.265472\pi$
$439$	5.42633e6			1.34383			0.671916	+	0.740628i	$0.265472\pi$
$440$		0			0
$441$	3.62057e6			0.886503
$442$		0			0
$443$			3.88802e6i			0.941282i	0.882325	+	0.470641i	$0.155977\pi$
$443$			3.88802e6i			0.941282i	−0.882325	+	0.470641i	$0.844023\pi$
$444$		0			0
$445$	−1.36107e6			−0.325821
$446$		0			0
$447$	−449769.			−0.106468
$448$		0			0
$449$	−1.62518e6			−0.380439			−0.190220	−	0.981742i	$-0.560920\pi$
$449$	−1.62518e6			−0.380439			−0.190220	+	0.981742i	$0.560920\pi$
$450$		0			0
$451$	−1.45590e6	+	4.26363e6i	−0.337047	+	0.987047i
$452$		0			0
$453$		−	9.32106e6i		−	2.13412i
$454$		0			0
$455$			432221.i			0.0978763i
$456$		0			0
$457$		−	158589.i		−	0.0355209i	−0.999842	−	0.0177604i	$-0.994346\pi$
$457$		−	158589.i		−	0.0355209i	0.999842	−	0.0177604i	$-0.00565362\pi$
$458$		0			0
$459$	430516.			0.0953801
$460$		0			0
$461$		−	4.89741e6i		−	1.07328i	−0.843810	−	0.536641i	$-0.819693\pi$
$461$		−	4.89741e6i		−	1.07328i	0.843810	−	0.536641i	$-0.180307\pi$
$462$		0			0
$463$		−	1.08899e6i		−	0.236087i	−0.993008	−	0.118043i	$-0.962338\pi$
$463$		−	1.08899e6i		−	0.236087i	0.993008	−	0.118043i	$-0.0376622\pi$
$464$		0			0
$465$	2.63704e6			0.565567
$466$		0			0
$467$		−	3.70383e6i		−	0.785884i	−0.919563	−	0.392942i	$-0.871457\pi$
$467$		−	3.70383e6i		−	0.785884i	0.919563	−	0.392942i	$-0.128543\pi$
$468$		0			0
$469$		−	440707.i		−	0.0925163i
$470$		0			0
$471$		−	1.25039e7i		−	2.59713i
$472$		0			0
$473$	−6.85608e6	−	2.34114e6i	−1.40904	−	0.481144i
$474$		0			0
$475$	4.35391e6			0.885413
$476$		0			0
$477$	−2.53055e6			−0.509236
$478$		0			0
$479$	8.72832e6			1.73817			0.869085	−	0.494663i	$-0.164709\pi$
$479$	8.72832e6			1.73817			0.869085	+	0.494663i	$0.164709\pi$
$480$		0			0
$481$			1.94903e6i			0.384111i
$482$		0			0
$483$	414463.			0.0808383
$484$		0			0
$485$	2.22713e6			0.429924
$486$		0			0
$487$			406543.i			0.0776754i	0.999246	+	0.0388377i	$0.0123655\pi$
$487$			406543.i			0.0776754i	−0.999246	+	0.0388377i	$0.987634\pi$
$488$		0			0
$489$	−6.93383e6			−1.31130
$490$		0			0
$491$	6.45829e6			1.20896			0.604482	−	0.796619i	$-0.293380\pi$
$491$	6.45829e6			1.20896			0.604482	+	0.796619i	$0.293380\pi$
$492$		0			0
$493$	642265.			0.119014
$494$		0			0
$495$	2.06880e6	+	706431.i	0.379494	+	0.129586i
$496$		0			0
$497$		−	994382.i		−	0.180577i
$498$		0			0
$499$			7.32952e6i			1.31772i	0.752264	+	0.658861i	$0.228962\pi$
$499$			7.32952e6i			1.31772i	−0.752264	+	0.658861i	$0.771038\pi$
$500$		0			0
$501$		−	1.18882e7i		−	2.11604i
$502$		0			0
$503$	5.25123e6			0.925425			0.462712	−	0.886508i	$-0.346876\pi$
$503$	5.25123e6			0.925425			0.462712	+	0.886508i	$0.346876\pi$
$504$		0			0
$505$			3.36252e6i			0.586728i
$506$		0			0
$507$		−	1.36218e7i		−	2.35351i
$508$		0			0
$509$	5.95648e6			1.01905			0.509525	−	0.860456i	$-0.329821\pi$
$509$	5.95648e6			1.01905			0.509525	+	0.860456i	$0.329821\pi$
$510$		0			0
$511$			1.15257e6i			0.195261i
$512$		0			0
$513$		−	882527.i		−	0.148059i
$514$		0			0
$515$		−	1.37251e6i		−	0.228034i
$516$		0			0
$517$	−1.58379e6	+	4.63815e6i	−0.260597	+	0.763164i
$518$		0			0
$519$	1.53353e7			2.49904
$520$		0			0
$521$	4.01452e6			0.647947			0.323973	−	0.946066i	$-0.394981\pi$
$521$	4.01452e6			0.647947			0.323973	+	0.946066i	$0.394981\pi$
$522$		0			0
$523$	−9.80948e6			−1.56817			−0.784083	−	0.620656i	$-0.786866\pi$
$523$	−9.80948e6			−1.56817			−0.784083	+	0.620656i	$0.786866\pi$
$524$		0			0
$525$			936476.i			0.148285i
$526$		0			0
$527$	−4.18170e6			−0.655884
$528$		0			0
$529$	5.20399e6			0.808532
$530$		0			0
$531$			4.29632e6i			0.661242i
$532$		0			0
$533$	−1.12534e7			−1.71580
$534$		0			0
$535$	1.98184e6			0.299354
$536$		0			0
$537$	7.14699e6			1.06952
$538$		0			0
$539$	6.26847e6	+	2.14049e6i	0.929373	+	0.317353i
$540$		0			0
$541$			7.61090e6i			1.11800i	0.829167	+	0.559001i	$0.188815\pi$
$541$			7.61090e6i			1.11800i	−0.829167	+	0.559001i	$0.811185\pi$
$542$		0			0
$543$			1.41493e7i			2.05937i
$544$		0			0
$545$			505963.i			0.0729671i
$546$		0			0
$547$	−3.68175e6			−0.526121			−0.263060	−	0.964779i	$-0.584732\pi$
$547$	−3.68175e6			−0.526121			−0.263060	+	0.964779i	$0.584732\pi$
$548$		0			0
$549$			150116.i			0.0212568i
$550$		0			0
$551$		−	1.31660e6i		−	0.184746i
$552$		0			0
$553$	487883.			0.0678427
$554$		0			0
$555$		−	1.03826e6i		−	0.143078i
$556$		0			0
$557$		−	3.63337e6i		−	0.496217i	−0.968732	−	0.248109i	$-0.920191\pi$
$557$		−	3.63337e6i		−	0.496217i	0.968732	−	0.248109i	$-0.0798090\pi$
$558$		0			0
$559$		−	1.80960e7i		−	2.44936i
$560$		0			0
$561$	−6.91484e6	−	2.36121e6i	−0.927631	−	0.316758i
$562$		0			0
$563$	−3.05729e6			−0.406504			−0.203252	−	0.979126i	$-0.565151\pi$
$563$	−3.05729e6			−0.406504			−0.203252	+	0.979126i	$0.565151\pi$
$564$		0			0
$565$	−1.72225e6			−0.226974
$566$		0			0
$567$	1.11534e6			0.145696
$568$		0			0
$569$		−	6.39647e6i		−	0.828246i	−0.910221	−	0.414123i	$-0.864088\pi$
$569$		−	6.39647e6i		−	0.828246i	0.910221	−	0.414123i	$-0.135912\pi$
$570$		0			0
$571$	−1.07558e7			−1.38055			−0.690276	−	0.723546i	$-0.742511\pi$
$571$	−1.07558e7			−1.38055			−0.690276	+	0.723546i	$0.742511\pi$
$572$		0			0
$573$	1.95320e6			0.248519
$574$		0			0
$575$		−	2.78450e6i		−	0.351219i
$576$		0			0
$577$	−6.82517e6			−0.853442			−0.426721	−	0.904383i	$-0.640331\pi$
$577$	−6.82517e6			−0.853442			−0.426721	+	0.904383i	$0.640331\pi$
$578$		0			0
$579$	−9.04462e6			−1.12123
$580$		0			0
$581$	818718.			0.100622
$582$		0			0
$583$	−4.38127e6	−	1.49607e6i	−0.533861	−	0.182297i
$584$		0			0
$585$			5.46039e6i			0.659681i
$586$		0			0
$587$			9.12801e6i			1.09340i	0.837327	+	0.546702i	$0.184117\pi$
$587$			9.12801e6i			1.09340i	−0.837327	+	0.546702i	$0.815883\pi$
$588$		0			0
$589$			8.57219e6i			1.01813i
$590$		0			0
$591$	4.40880e6			0.519220
$592$		0			0
$593$		−	1.59316e7i		−	1.86047i	−0.366967	−	0.930234i	$-0.619604\pi$
$593$		−	1.59316e7i		−	1.86047i	0.366967	−	0.930234i	$-0.380396\pi$
$594$		0			0
$595$			365115.i			0.0422802i
$596$		0			0
$597$	7.06282e6			0.811040
$598$		0			0
$599$			1.48485e7i			1.69089i	0.534066	+	0.845443i	$0.320663\pi$
$599$			1.48485e7i			1.69089i	−0.534066	+	0.845443i	$0.679337\pi$
$600$		0			0
$601$			2.29423e6i			0.259090i	0.991574	+	0.129545i	$0.0413517\pi$
$601$			2.29423e6i			0.259090i	−0.991574	+	0.129545i	$0.958648\pi$
$602$		0			0
$603$		−	5.56760e6i		−	0.623555i
$604$		0			0
$605$	3.16417e6	+	2.44616e6i	0.351456	+	0.271704i
$606$		0			0
$607$	−1.12310e7			−1.23722			−0.618608	−	0.785700i	$-0.712303\pi$
$607$	−1.12310e7			−1.23722			−0.618608	+	0.785700i	$0.712303\pi$
$608$		0			0
$609$	283185.			0.0309404
$610$		0			0
$611$	−1.22419e7			−1.32662
$612$		0			0
$613$		−	2.53211e6i		−	0.272165i	−0.990698	−	0.136082i	$-0.956549\pi$
$613$		−	2.53211e6i		−	0.272165i	0.990698	−	0.136082i	$-0.0434512\pi$
$614$		0			0
$615$	5.99476e6			0.639122
$616$		0			0
$617$	6.45127e6			0.682233			0.341116	−	0.940021i	$-0.389195\pi$
$617$	6.45127e6			0.682233			0.341116	+	0.940021i	$0.389195\pi$
$618$		0			0
$619$			2.02338e6i			0.212252i	0.994353	+	0.106126i	$0.0338447\pi$
$619$			2.02338e6i			0.212252i	−0.994353	+	0.106126i	$0.966155\pi$
$620$		0			0
$621$	−564411.			−0.0587309
$622$		0			0
$623$	951637.			0.0982316
$624$		0			0
$625$	4.36437e6			0.446911
$626$		0			0
$627$	−4.84031e6	+	1.41749e7i	−0.491705	+	1.43997i
$628$		0			0
$629$			1.64643e6i			0.165926i
$630$		0			0
$631$			1.49941e7i			1.49915i	0.661917	+	0.749577i	$0.269743\pi$
$631$			1.49941e7i			1.49915i	−0.661917	+	0.749577i	$0.730257\pi$
$632$		0			0
$633$			1.13155e6i			0.112245i
$634$		0			0
$635$	−4.71358e6			−0.463892
$636$		0			0
$637$			1.65450e7i			1.61555i
$638$		0			0
$639$		−	1.25623e7i		−	1.21708i
$640$		0			0
$641$	−1.19189e7			−1.14576			−0.572879	−	0.819640i	$-0.694173\pi$
$641$	−1.19189e7			−1.14576			−0.572879	+	0.819640i	$0.694173\pi$
$642$		0			0
$643$			1.25106e6i			0.119330i	0.998218	+	0.0596650i	$0.0190033\pi$
$643$			1.25106e6i			0.119330i	−0.998218	+	0.0596650i	$0.980997\pi$
$644$		0			0
$645$			9.63981e6i			0.912366i
$646$		0			0
$647$		−	1.34042e7i		−	1.25887i	−0.777052	−	0.629436i	$-0.783286\pi$
$647$		−	1.34042e7i		−	1.25887i	0.777052	−	0.629436i	$-0.216714\pi$
$648$		0			0
$649$	−2.54000e6	+	7.43844e6i	−0.236713	+	0.693219i
$650$		0			0
$651$	−1.84378e6			−0.170513
$652$		0			0
$653$	−5.76704e6			−0.529261			−0.264630	−	0.964350i	$-0.585250\pi$
$653$	−5.76704e6			−0.529261			−0.264630	+	0.964350i	$0.585250\pi$
$654$		0			0
$655$	−4.39855e6			−0.400596
$656$		0			0
$657$			1.45608e7i			1.31605i
$658$		0			0
$659$	2.97388e6			0.266754			0.133377	−	0.991065i	$-0.457418\pi$
$659$	2.97388e6			0.266754			0.133377	+	0.991065i	$0.457418\pi$
$660$		0			0
$661$	1.48821e7			1.32483			0.662415	−	0.749137i	$-0.269532\pi$
$661$	1.48821e7			1.32483			0.662415	+	0.749137i	$0.269532\pi$
$662$		0			0
$663$		−	1.82511e7i		−	1.61252i
$664$		0			0
$665$	748459.			0.0656317
$666$		0			0
$667$	−842016.			−0.0732834
$668$		0			0
$669$	9.33962e6			0.806796
$670$		0			0
$671$	−88749.4	+	259904.i	−0.00760956	+	0.0222847i
$672$		0			0
$673$		−	1.01247e7i		−	0.861681i	−0.902428	−	0.430841i	$-0.858217\pi$
$673$		−	1.01247e7i		−	0.861681i	0.902428	−	0.430841i	$-0.141783\pi$
$674$		0			0
$675$		−	1.27528e6i		−	0.107733i
$676$		0			0
$677$			9.01121e6i			0.755634i	0.925880	+	0.377817i	$0.123325\pi$
$677$			9.01121e6i			0.755634i	−0.925880	+	0.377817i	$0.876675\pi$
$678$		0			0
$679$	−1.55718e6			−0.129617
$680$		0			0
$681$			2.17723e7i			1.79902i
$682$		0			0
$683$		−	1.77893e7i		−	1.45917i	−0.683888	−	0.729587i	$-0.739712\pi$
$683$		−	1.77893e7i		−	1.45917i	0.683888	−	0.729587i	$-0.260288\pi$
$684$		0			0
$685$	8.46712e6			0.689460
$686$		0			0
$687$			2.58851e7i			2.09246i
$688$		0			0
$689$		−	1.15639e7i		−	0.928022i
$690$		0			0
$691$			1.25810e7i			1.00235i	0.865346	+	0.501175i	$0.167099\pi$
$691$			1.25810e7i			1.00235i	−0.865346	+	0.501175i	$0.832901\pi$
$692$		0			0
$693$	−1.44647e6	−	493926.i	−0.114413	−	0.0390687i
$694$		0			0
$695$	−1.60165e6			−0.125778
$696$		0			0
$697$	−9.50624e6			−0.741185
$698$		0			0
$699$	3.27919e7			2.53848
$700$		0			0
$701$			1.14760e7i			0.882055i	0.897494	+	0.441027i	$0.145386\pi$
$701$			1.14760e7i			0.882055i	−0.897494	+	0.441027i	$0.854614\pi$
$702$		0			0
$703$	3.37505e6			0.257568
$704$		0			0
$705$	6.52134e6			0.494156
$706$		0			0
$707$		−	2.35103e6i		−	0.176892i
$708$		0			0
$709$	7.02721e6			0.525010			0.262505	−	0.964931i	$-0.415451\pi$
$709$	7.02721e6			0.525010			0.262505	+	0.964931i	$0.415451\pi$
$710$		0			0
$711$	6.16359e6			0.457256
$712$		0			0
$713$	5.48225e6			0.403864
$714$		0			0
$715$	−3.22820e6	+	9.45385e6i	−0.236154	+	0.691582i
$716$		0			0
$717$			7.31574e6i			0.531447i
$718$		0			0
$719$			9.13945e6i			0.659323i	0.944099	+	0.329661i	$0.106935\pi$
$719$			9.13945e6i			0.659323i	−0.944099	+	0.329661i	$0.893065\pi$
$720$		0			0
$721$			959642.i			0.0687497i
$722$		0			0
$723$	−654179.			−0.0465427
$724$		0			0
$725$		−	1.90253e6i		−	0.134427i
$726$		0			0
$727$		−	6.11257e6i		−	0.428931i	−0.976732	−	0.214466i	$-0.931199\pi$
$727$		−	6.11257e6i		−	0.428931i	0.976732	−	0.214466i	$-0.0688010\pi$
$728$		0			0
$729$	1.14338e7			0.796844
$730$		0			0
$731$		−	1.52864e7i		−	1.05806i
$732$		0			0
$733$			1.88446e7i			1.29547i	0.761865	+	0.647735i	$0.224284\pi$
$733$			1.88446e7i			1.29547i	−0.761865	+	0.647735i	$0.775716\pi$
$734$		0			0
$735$		−	8.81362e6i		−	0.601778i
$736$		0			0
$737$	3.29159e6	−	9.63947e6i	0.223222	−	0.653709i
$738$		0			0
$739$	−1.46890e7			−0.989423			−0.494712	−	0.869057i	$-0.664726\pi$
$739$	−1.46890e7			−0.989423			−0.494712	+	0.869057i	$0.664726\pi$
$740$		0			0
$741$	−3.74134e7			−2.50312
$742$		0			0
$743$	3.25028e6			0.215998			0.107999	−	0.994151i	$-0.465556\pi$
$743$	3.25028e6			0.215998			0.107999	+	0.994151i	$0.465556\pi$
$744$		0			0
$745$			519444.i			0.0342885i
$746$		0			0
$747$	1.03431e7			0.678189
$748$		0			0
$749$	−1.38568e6			−0.0902521
$750$		0			0
$751$			1.08732e6i			0.0703491i	0.999381	+	0.0351745i	$0.0111987\pi$
$751$			1.08732e6i			0.0703491i	−0.999381	+	0.0351745i	$0.988801\pi$
$752$		0			0
$753$	2.96934e6			0.190841
$754$		0			0
$755$	−1.07650e7			−0.687302
$756$		0			0
$757$	5.59676e6			0.354974			0.177487	−	0.984123i	$-0.443203\pi$
$757$	5.59676e6			0.354974			0.177487	+	0.984123i	$0.443203\pi$
$758$		0			0
$759$	9.06543e6	+	3.09557e6i	0.571194	+	0.195045i
$760$		0			0
$761$		−	1.20859e7i		−	0.756514i	−0.925701	−	0.378257i	$-0.876524\pi$
$761$		−	1.20859e7i		−	0.756514i	0.925701	−	0.378257i	$-0.123476\pi$
$762$		0			0
$763$		−	353762.i		−	0.0219988i
$764$		0			0
$765$			4.61261e6i			0.284966i
$766$		0			0
$767$	−1.96331e7			−1.20504
$768$		0			0
$769$		−	1.50738e7i		−	0.919192i	−0.888128	−	0.459596i	$-0.847994\pi$
$769$		−	1.50738e7i		−	0.919192i	0.888128	−	0.459596i	$-0.152006\pi$
$770$		0			0
$771$		−	4.16813e7i		−	2.52525i
$772$		0			0
$773$	−2.17962e7			−1.31199			−0.655996	−	0.754764i	$-0.727751\pi$
$773$	−2.17962e7			−1.31199			−0.655996	+	0.754764i	$0.727751\pi$
$774$		0			0
$775$			1.23871e7i			0.740826i
$776$		0			0
$777$			725935.i			0.0431365i
$778$		0			0
$779$			1.94871e7i			1.15054i
$780$		0			0
$781$	7.42691e6	−	2.17498e7i	0.435693	−	1.27593i
$782$		0			0
$783$	−385638.			−0.0224789
$784$		0			0
$785$	−1.44409e7			−0.836414
$786$		0			0
$787$	−625459.			−0.0359966			−0.0179983	−	0.999838i	$-0.505729\pi$
$787$	−625459.			−0.0359966			−0.0179983	+	0.999838i	$0.505729\pi$
$788$		0			0
$789$		−	4.30751e7i		−	2.46340i
$790$		0			0
$791$	1.20417e6			0.0684302
$792$		0			0
$793$	−685993.			−0.0387380
$794$		0			0
$795$			6.16017e6i			0.345680i
$796$		0			0
$797$	3.01329e7			1.68033			0.840167	−	0.542327i	$-0.182457\pi$
$797$	3.01329e7			1.68033			0.840167	+	0.542327i	$0.182457\pi$
$798$		0			0
$799$	−1.03413e7			−0.573069
$800$		0			0
$801$	1.20223e7			0.662076
$802$		0			0
$803$	−8.60842e6	+	2.52099e7i	−0.471123	+	1.37969i
$804$		0			0
$805$		−	478669.i		−	0.0260343i
$806$		0			0
$807$			5.10793e6i			0.276097i
$808$		0			0
$809$		−	2.15254e7i		−	1.15632i	−0.815923	−	0.578161i	$-0.803770\pi$
$809$		−	2.15254e7i		−	1.15632i	0.815923	−	0.578161i	$-0.196230\pi$
$810$		0			0
$811$	2.04421e7			1.09137			0.545687	−	0.837989i	$-0.316269\pi$
$811$	2.04421e7			1.09137			0.545687	+	0.837989i	$0.316269\pi$
$812$		0			0
$813$		−	4.07029e7i		−	2.15973i
$814$		0			0
$815$			8.00797e6i			0.422307i
$816$		0			0
$817$	−3.13360e7			−1.64244
$818$		0			0
$819$		−	3.81782e6i		−	0.198887i
$820$		0			0
$821$		−	1.75635e7i		−	0.909398i	−0.890645	−	0.454699i	$-0.849747\pi$
$821$		−	1.75635e7i		−	0.909398i	0.890645	−	0.454699i	$-0.150253\pi$
$822$		0			0
$823$		−	8.42240e6i		−	0.433447i	−0.976233	−	0.216724i	$-0.930463\pi$
$823$		−	8.42240e6i		−	0.433447i	0.976233	−	0.216724i	$-0.0695370\pi$
$824$		0			0
$825$	−6.99442e6	+	2.04833e7i	−0.357781	+	1.04777i
$826$		0			0
$827$	2.90903e7			1.47905			0.739527	−	0.673127i	$-0.235049\pi$
$827$	2.90903e7			1.47905			0.739527	+	0.673127i	$0.235049\pi$
$828$		0			0
$829$	−3.46016e7			−1.74868			−0.874340	−	0.485314i	$-0.838705\pi$
$829$	−3.46016e7			−1.74868			−0.874340	+	0.485314i	$0.838705\pi$
$830$		0			0
$831$	−3.50835e7			−1.76238
$832$		0			0
$833$			1.39763e7i			0.697877i
$834$		0			0
$835$	−1.37299e7			−0.681476
$836$		0			0
$837$	2.51084e6			0.123881
$838$		0			0
$839$		−	3.45570e7i		−	1.69485i	−0.530916	−	0.847424i	$-0.678152\pi$
$839$		−	3.45570e7i		−	1.69485i	0.530916	−	0.847424i	$-0.321848\pi$
$840$		0			0
$841$	1.99358e7			0.971951
$842$		0			0
$843$	4.33811e7			2.10248
$844$		0			0
$845$	−1.57321e7			−0.757956
$846$		0			0
$847$	−2.21234e6	−	1.71032e6i	−0.105960	−	0.0819160i
$848$		0			0
$849$		−	8.45734e6i		−	0.402684i
$850$		0			0
$851$		−	2.15848e6i		−	0.102170i
$852$		0			0
$853$		−	2.70989e7i		−	1.27520i	−0.770366	−	0.637602i	$-0.779926\pi$
$853$		−	2.70989e7i		−	1.27520i	0.770366	−	0.637602i	$-0.220074\pi$
$854$		0			0
$855$	9.45552e6			0.442354
$856$		0			0
$857$			2.63162e7i			1.22397i	0.790868	+	0.611987i	$0.209629\pi$
$857$			2.63162e7i			1.22397i	−0.790868	+	0.611987i	$0.790371\pi$
$858$		0			0
$859$		−	2.68729e7i		−	1.24260i	−0.783573	−	0.621299i	$-0.786605\pi$
$859$		−	2.68729e7i		−	1.24260i	0.783573	−	0.621299i	$-0.213395\pi$
$860$		0			0
$861$	−4.19145e6			−0.192689
$862$		0			0
$863$			3.85777e7i			1.76323i	0.471969	+	0.881615i	$0.343543\pi$
$863$			3.85777e7i			1.76323i	−0.471969	+	0.881615i	$0.656457\pi$
$864$		0			0
$865$		−	1.77109e7i		−	0.804825i
$866$		0			0
$867$			1.51130e7i			0.682813i
$868$		0			0
$869$	1.06713e7	+	3.64394e6i	0.479368	+	0.163690i
$870$		0			0
$871$	2.54425e7			1.13635
$872$		0			0
$873$	−1.96723e7			−0.873615
$874$		0			0
$875$	2.42901e6			0.107253
$876$		0			0
$877$			8.42238e6i			0.369773i	0.982760	+	0.184887i	$0.0591918\pi$
$877$			8.42238e6i			0.369773i	−0.982760	+	0.184887i	$0.940808\pi$
$878$		0			0
$879$	−3.69194e7			−1.61169
$880$		0			0
$881$	1.13583e7			0.493030			0.246515	−	0.969139i	$-0.420714\pi$
$881$	1.13583e7			0.493030			0.246515	+	0.969139i	$0.420714\pi$
$882$		0			0
$883$		−	1.60767e7i		−	0.693895i	−0.937884	−	0.346948i	$-0.887218\pi$
$883$		−	1.60767e7i		−	0.693895i	0.937884	−	0.346948i	$-0.112782\pi$
$884$		0			0
$885$	1.04586e7			0.448866
$886$		0			0
$887$	−1.08235e7			−0.461910			−0.230955	−	0.972964i	$-0.574185\pi$
$887$	−1.08235e7			−0.461910			−0.230955	+	0.972964i	$0.574185\pi$
$888$		0			0
$889$	3.29567e6			0.139859
$890$		0			0
$891$	2.43954e7	+	8.33030e6i	1.02947	+	0.351533i
$892$		0			0
$893$			2.11989e7i			0.889577i
$894$		0			0
$895$		−	8.25417e6i		−	0.344442i
$896$		0			0
$897$			2.39273e7i			0.992918i
$898$		0			0
$899$	3.74579e6			0.154577
$900$		0			0
$901$		−	9.76853e6i		−	0.400883i
$902$		0			0
$903$		−	6.74001e6i		−	0.275069i
$904$		0			0
$905$	1.63412e7			0.663228
$906$		0			0
$907$		−	2.07738e7i		−	0.838491i	−0.907873	−	0.419246i	$-0.862295\pi$
$907$		−	2.07738e7i		−	0.838491i	0.907873	−	0.419246i	$-0.137705\pi$
$908$		0			0
$909$		−	2.97013e7i		−	1.19224i
$910$		0			0
$911$		−	2.04650e7i		−	0.816990i	−0.912761	−	0.408495i	$-0.866054\pi$
$911$		−	2.04650e7i		−	0.816990i	0.912761	−	0.408495i	$-0.133946\pi$
$912$		0			0
$913$	1.79076e7	+	6.11490e6i	0.710985	+	0.242780i
$914$		0			0
$915$	365432.			0.0144296
$916$		0			0
$917$	3.07540e6			0.120775
$918$		0			0
$919$	1.64857e7			0.643902			0.321951	−	0.946756i	$-0.395661\pi$
$919$	1.64857e7			0.643902			0.321951	+	0.946756i	$0.395661\pi$
$920$		0			0
$921$		−	7.85344e6i		−	0.305078i
$922$		0			0
$923$	5.74066e7			2.21798
$924$		0			0
$925$	4.87707e6			0.187415
$926$		0			0
$927$			1.21235e7i			0.463370i
$928$		0			0
$929$	−1.98337e7			−0.753987			−0.376994	−	0.926216i	$-0.623042\pi$
$929$	−1.98337e7			−0.753987			−0.376994	+	0.926216i	$0.623042\pi$
$930$		0			0
$931$	2.86503e7			1.08332
$932$		0			0
$933$	3.08687e7			1.16095
$934$		0			0
$935$	−2.72699e6	+	7.98605e6i	−0.102013	+	0.298747i
$936$		0			0
$937$			4.78808e6i			0.178161i	0.996024	+	0.0890804i	$0.0283928\pi$
$937$			4.78808e6i			0.178161i	−0.996024	+	0.0890804i	$0.971607\pi$
$938$		0			0
$939$		−	2.02079e7i		−	0.747925i
$940$		0			0
$941$			3.97857e6i			0.146472i	0.997315	+	0.0732358i	$0.0233326\pi$
$941$			3.97857e6i			0.146472i	−0.997315	+	0.0732358i	$0.976667\pi$
$942$		0			0
$943$	1.24628e7			0.456389
$944$		0			0
$945$		−	219227.i		−	0.00798574i
$946$		0			0
$947$			6.12959e6i			0.222104i	0.993815	+	0.111052i	$0.0354220\pi$
$947$			6.12959e6i			0.222104i	−0.993815	+	0.111052i	$0.964578\pi$
$948$		0			0
$949$	−6.65392e7			−2.39835
$950$		0			0
$951$			3.72617e7i			1.33602i
$952$		0			0
$953$			2.77177e7i			0.988611i	0.869288	+	0.494305i	$0.164578\pi$
$953$			2.77177e7i			0.988611i	−0.869288	+	0.494305i	$0.835422\pi$
$954$		0			0
$955$		−	2.25577e6i		−	0.0800363i
$956$		0			0
$957$	6.19402e6	+	2.11507e6i	0.218622	+	0.0746526i
$958$		0			0
$959$	−5.92009e6			−0.207865
$960$		0			0
$961$	4.24080e6			0.148129
$962$		0			0
$963$	−1.75057e7			−0.608294
$964$		0			0
$965$			1.04458e7i			0.361095i
$966$		0			0
$967$	−2.03671e6			−0.0700428			−0.0350214	−	0.999387i	$-0.511150\pi$
$967$	−2.03671e6			−0.0700428			−0.0350214	+	0.999387i	$0.511150\pi$
$968$		0			0
$969$	−3.16046e7			−1.08129
$970$		0			0
$971$		−	1.99075e7i		−	0.677593i	−0.940860	−	0.338797i	$-0.889980\pi$
$971$		−	1.99075e7i		−	0.677593i	0.940860	−	0.338797i	$-0.110020\pi$
$972$		0			0
$973$	1.11985e6			0.0379208
$974$		0			0
$975$	−5.40637e7			−1.82135
$976$		0			0
$977$	1.08541e7			0.363795			0.181897	−	0.983318i	$-0.441776\pi$
$977$	1.08541e7			0.363795			0.181897	+	0.983318i	$0.441776\pi$
$978$		0			0
$979$	2.08149e7	+	7.10766e6i	0.694093	+	0.237012i
$980$		0			0
$981$		−	4.46919e6i		−	0.148271i
$982$		0			0
$983$		−	2.24671e7i		−	0.741587i	−0.928715	−	0.370794i	$-0.879086\pi$
$983$		−	2.24671e7i		−	0.741587i	0.928715	−	0.370794i	$-0.120914\pi$
$984$		0			0
$985$		−	5.09178e6i		−	0.167217i
$986$		0			0
$987$	−4.55963e6			−0.148983
$988$		0			0
$989$			2.00406e7i			0.651509i
$990$		0			0
$991$			5.54826e7i			1.79462i	0.441402	+	0.897310i	$0.354481\pi$
$991$			5.54826e7i			1.79462i	−0.441402	+	0.897310i	$0.645519\pi$
$992$		0			0
$993$	4.13344e7			1.33027
$994$		0			0
$995$		−	8.15695e6i		−	0.261198i
$996$		0			0
$997$			3.87606e7i			1.23496i	0.786588	+	0.617479i	$0.211846\pi$
$997$			3.87606e7i			1.23496i	−0.786588	+	0.617479i	$0.788154\pi$
$998$		0			0
$999$		−	988571.i		−	0.0313396i

Display

a_p

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p

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

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n

up to: 50 250 1000 (See only

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a_p

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)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	176.6.e.d.175.15	yes	16
4.3	odd	2	inner	176.6.e.d.175.2	yes	16
11.10	odd	2	inner	176.6.e.d.175.16	yes	16
44.43	even	2	inner	176.6.e.d.175.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
176.6.e.d.175.1	✓	16	44.43	even	2	inner
176.6.e.d.175.2	yes	16	4.3	odd	2	inner
176.6.e.d.175.15	yes	16	1.1	even	1	trivial
176.6.e.d.175.16	yes	16	11.10	odd	2	inner

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	176.6.e.d.175.15

Level	$176$
Weight	$6$
Character	176.175
Analytic conductor	$28.228$
Analytic rank	$0$
Dimension	$16$
Inner twists	$4$