LMFDB - Embedded newform 414.5.b.b.91.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,5,Mod(91,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("414.91");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	414.b (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:	$42.7951647167$
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} + 1428 x^{14} - 600 x^{13} + 788282 x^{12} - 529464 x^{11} + 213396724 x^{10} + \cdots + 274129967370817 $ x^16 + 1428x^14 - 600x^13 + 788282x^12 - 529464x^11 + 213396724x^10 - 175079484x^9 + 29130946113x^8 - 26553654912x^7 + 1790717926120x^6 - 1672125302460x^5 + 32715955056348x^4 - 11255739266544x^3 + 140972261417692x^2 - 70961983286748x + 274129967370817
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 138)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.14
Root			$-0.707107 - 11.7850i$ of defining polynomial
Character	$\chi$	$=$	414.91
Dual form			414.5.b.b.91.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82843 q^{2} +8.00000 q^{4} +26.7045i q^{5} +66.7315i q^{7} +22.6274 q^{8} +O(q^{10})$	\(q+2.82843 q^{2} +8.00000 q^{4} +26.7045i q^{5} +66.7315i q^{7} +22.6274 q^{8} +75.5318i q^{10} -8.81477i q^{11} -136.925 q^{13} +188.745i q^{14} +64.0000 q^{16} +184.077i q^{17} +33.2701i q^{19} +213.636i q^{20} -24.9319i q^{22} +(-493.509 - 190.498i) q^{23} -88.1308 q^{25} -387.283 q^{26} +533.852i q^{28} -1221.54 q^{29} +1220.45 q^{31} +181.019 q^{32} +520.649i q^{34} -1782.03 q^{35} -2059.16i q^{37} +94.1019i q^{38} +604.254i q^{40} -884.746 q^{41} -1572.25i q^{43} -70.5181i q^{44} +(-1395.86 - 538.810i) q^{46} -564.292 q^{47} -2052.10 q^{49} -249.272 q^{50} -1095.40 q^{52} +3033.20i q^{53} +235.394 q^{55} +1509.96i q^{56} -3455.04 q^{58} -5664.52 q^{59} +3378.10i q^{61} +3451.96 q^{62} +512.000 q^{64} -3656.52i q^{65} +6983.41i q^{67} +1472.62i q^{68} -5040.35 q^{70} +3482.47 q^{71} +406.423 q^{73} -5824.17i q^{74} +266.160i q^{76} +588.223 q^{77} -103.984i q^{79} +1709.09i q^{80} -2502.44 q^{82} +10105.6i q^{83} -4915.69 q^{85} -4446.99i q^{86} -199.455i q^{88} +7418.94i q^{89} -9137.24i q^{91} +(-3948.08 - 1523.98i) q^{92} -1596.06 q^{94} -888.461 q^{95} +16181.4i q^{97} -5804.21 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 128 q^{4}+O(q^{10}) $ 16 * q + 128 * q^4	$ 16 q + 128 q^{4} - 208 q^{13} + 1024 q^{16} - 840 q^{23} + 1056 q^{25} - 1920 q^{26} - 3600 q^{29} + 224 q^{31} + 3264 q^{35} + 6144 q^{41} + 1280 q^{46} - 8880 q^{47} - 13888 q^{49} - 7296 q^{50} - 1664 q^{52} + 832 q^{55} + 2944 q^{58} + 18240 q^{59} + 8192 q^{64} + 19584 q^{70} + 30048 q^{71} + 9536 q^{73} - 14160 q^{77} - 19584 q^{82} - 32496 q^{85} - 6720 q^{92} - 21248 q^{94} + 20064 q^{95} - 21504 q^{98}+O(q^{100}) $ 16 * q + 128 * q^4 - 208 * q^13 + 1024 * q^16 - 840 * q^23 + 1056 * q^25 - 1920 * q^26 - 3600 * q^29 + 224 * q^31 + 3264 * q^35 + 6144 * q^41 + 1280 * q^46 - 8880 * q^47 - 13888 * q^49 - 7296 * q^50 - 1664 * q^52 + 832 * q^55 + 2944 * q^58 + 18240 * q^59 + 8192 * q^64 + 19584 * q^70 + 30048 * q^71 + 9536 * q^73 - 14160 * q^77 - 19584 * q^82 - 32496 * q^85 - 6720 * q^92 - 21248 * q^94 + 20064 * q^95 - 21504 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/414\mathbb{Z}\right)^\times$.

$n$	$47$	$235$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.82843			0.707107
$2$	2.82843			0.707107
$3$		0			0
$3$		0			0
$4$	8.00000			0.500000
$5$			26.7045i			1.06818i	0.845428	+	0.534090i	$0.179346\pi$
$5$			26.7045i			1.06818i	−0.845428	+	0.534090i	$0.820654\pi$
$6$		0			0
$7$			66.7315i			1.36187i	0.732345	+	0.680934i	$0.238426\pi$
$7$			66.7315i			1.36187i	−0.732345	+	0.680934i	$0.761574\pi$
$8$	22.6274			0.353553
$9$		0			0
$10$			75.5318i			0.755318i
$11$		−	8.81477i		−	0.0728493i	−0.999336	−	0.0364247i	$-0.988403\pi$
$11$		−	8.81477i		−	0.0728493i	0.999336	−	0.0364247i	$-0.0115969\pi$
$12$		0			0
$13$	−136.925			−0.810209			−0.405105	−	0.914270i	$-0.632765\pi$
$13$	−136.925			−0.810209			−0.405105	+	0.914270i	$0.632765\pi$
$14$			188.745i			0.962986i
$15$		0			0
$16$	64.0000			0.250000
$17$			184.077i			0.636946i	0.947932	+	0.318473i	$0.103170\pi$
$17$			184.077i			0.636946i	−0.947932	+	0.318473i	$0.896830\pi$
$18$		0			0
$19$			33.2701i			0.0921608i	0.998938	+	0.0460804i	$0.0146730\pi$
$19$			33.2701i			0.0921608i	−0.998938	+	0.0460804i	$0.985327\pi$
$20$			213.636i			0.534090i
$21$		0			0
$22$		−	24.9319i		−	0.0515122i
$23$	−493.509	−	190.498i	−0.932910	−	0.360110i
$23$	−493.509	−	190.498i	−0.932910	−	0.360110i
$24$		0			0
$25$	−88.1308			−0.141009
$26$	−387.283			−0.572904
$27$		0			0
$28$			533.852i			0.680934i
$29$	−1221.54			−1.45249			−0.726244	−	0.687437i	$-0.758736\pi$
$29$	−1221.54			−1.45249			−0.726244	+	0.687437i	$0.758736\pi$
$30$		0			0
$31$	1220.45			1.26998			0.634991	−	0.772519i	$-0.281004\pi$
$31$	1220.45			1.26998			0.634991	+	0.772519i	$0.281004\pi$
$32$	181.019			0.176777
$33$		0			0
$34$			520.649i			0.450389i
$35$	−1782.03			−1.45472
$36$		0			0
$37$		−	2059.16i		−	1.50413i	−0.659088	−	0.752066i	$-0.729058\pi$
$37$		−	2059.16i		−	1.50413i	0.659088	−	0.752066i	$-0.270942\pi$
$38$			94.1019i			0.0651676i
$39$		0			0
$40$			604.254i			0.377659i
$41$	−884.746			−0.526321			−0.263160	−	0.964752i	$-0.584765\pi$
$41$	−884.746			−0.526321			−0.263160	+	0.964752i	$0.584765\pi$
$42$		0			0
$43$		−	1572.25i		−	0.850323i	−0.905117	−	0.425162i	$-0.860217\pi$
$43$		−	1572.25i		−	0.850323i	0.905117	−	0.425162i	$-0.139783\pi$
$44$		−	70.5181i		−	0.0364247i
$45$		0			0
$46$	−1395.86	−	538.810i	−0.659667	−	0.254636i
$47$	−564.292			−0.255451			−0.127726	−	0.991810i	$-0.540768\pi$
$47$	−564.292			−0.255451			−0.127726	+	0.991810i	$0.540768\pi$
$48$		0			0
$49$	−2052.10			−0.854685
$50$	−249.272			−0.0997086
$51$		0			0
$52$	−1095.40			−0.405105
$53$			3033.20i			1.07982i	0.841724	+	0.539908i	$0.181541\pi$
$53$			3033.20i			1.07982i	−0.841724	+	0.539908i	$0.818459\pi$
$54$		0			0
$55$	235.394			0.0778162
$56$			1509.96i			0.481493i
$57$		0			0
$58$	−3455.04			−1.02706
$59$	−5664.52			−1.62727			−0.813633	−	0.581378i	$-0.802514\pi$
$59$	−5664.52			−1.62727			−0.813633	+	0.581378i	$0.802514\pi$
$60$		0			0
$61$			3378.10i			0.907847i	0.891041	+	0.453924i	$0.149976\pi$
$61$			3378.10i			0.907847i	−0.891041	+	0.453924i	$0.850024\pi$
$62$	3451.96			0.898013
$63$		0			0
$64$	512.000			0.125000
$65$		−	3656.52i		−	0.865449i
$66$		0			0
$67$			6983.41i			1.55567i	0.628468	+	0.777836i	$0.283682\pi$
$67$			6983.41i			1.55567i	−0.628468	+	0.777836i	$0.716318\pi$
$68$			1472.62i			0.318473i
$69$		0			0
$70$	−5040.35			−1.02864
$71$	3482.47			0.690829			0.345415	−	0.938450i	$-0.387738\pi$
$71$	3482.47			0.690829			0.345415	+	0.938450i	$0.387738\pi$
$72$		0			0
$73$	406.423			0.0762663			0.0381331	−	0.999273i	$-0.487859\pi$
$73$	406.423			0.0762663			0.0381331	+	0.999273i	$0.487859\pi$
$74$		−	5824.17i		−	1.06358i
$75$		0			0
$76$			266.160i			0.0460804i
$77$	588.223			0.0992112
$78$		0			0
$79$		−	103.984i		−	0.0166614i	−0.999965	−	0.00833069i	$-0.997348\pi$
$79$		−	103.984i		−	0.0166614i	0.999965	−	0.00833069i	$-0.00265177\pi$
$80$			1709.09i			0.267045i
$81$		0			0
$82$	−2502.44			−0.372165
$83$			10105.6i			1.46692i	0.679733	+	0.733460i	$0.262096\pi$
$83$			10105.6i			1.46692i	−0.679733	+	0.733460i	$0.737904\pi$
$84$		0			0
$85$	−4915.69			−0.680373
$86$		−	4446.99i		−	0.601269i
$87$		0			0
$88$		−	199.455i		−	0.0257561i
$89$			7418.94i			0.936617i	0.883565	+	0.468308i	$0.155136\pi$
$89$			7418.94i			0.936617i	−0.883565	+	0.468308i	$0.844864\pi$
$90$		0			0
$91$		−	9137.24i		−	1.10340i
$92$	−3948.08	−	1523.98i	−0.466455	−	0.180055i
$93$		0			0
$94$	−1596.06			−0.180631
$95$	−888.461			−0.0984444
$96$		0			0
$97$			16181.4i			1.71978i	0.510481	+	0.859889i	$0.329467\pi$
$97$			16181.4i			1.71978i	−0.510481	+	0.859889i	$0.670533\pi$
$98$	−5804.21			−0.604354
$99$		0			0
$100$	−705.047			−0.0705047
$101$	−2091.11			−0.204991			−0.102496	−	0.994733i	$-0.532683\pi$
$101$	−2091.11			−0.204991			−0.102496	+	0.994733i	$0.532683\pi$
$102$		0			0
$103$		−	9021.54i		−	0.850367i	−0.905107	−	0.425184i	$-0.860210\pi$
$103$		−	9021.54i		−	0.850367i	0.905107	−	0.425184i	$-0.139790\pi$
$104$	−3098.27			−0.286452
$105$		0			0
$106$			8579.19i			0.763545i
$107$			635.992i			0.0555500i	0.999614	+	0.0277750i	$0.00884219\pi$
$107$			635.992i			0.0555500i	−0.999614	+	0.0277750i	$0.991158\pi$
$108$		0			0
$109$			5742.50i			0.483334i	0.970359	+	0.241667i	$0.0776942\pi$
$109$			5742.50i			0.483334i	−0.970359	+	0.241667i	$0.922306\pi$
$110$	665.795			0.0550244
$111$		0			0
$112$			4270.82i			0.340467i
$113$		−	21000.0i		−	1.64461i	−0.569049	−	0.822303i	$-0.692689\pi$
$113$		−	21000.0i		−	1.64461i	0.569049	−	0.822303i	$-0.307311\pi$
$114$		0			0
$115$	5087.15	−	13178.9i	0.384662	−	0.996516i
$116$	−9772.34			−0.726244
$117$		0			0
$118$	−16021.7			−1.15065
$119$	−12283.8			−0.867436
$120$		0			0
$121$	14563.3			0.994693
$122$			9554.71i			0.641945i
$123$		0			0
$124$	9763.62			0.634991
$125$			14336.8i			0.917557i
$126$		0			0
$127$	−9865.08			−0.611636			−0.305818	−	0.952090i	$-0.598930\pi$
$127$	−9865.08			−0.611636			−0.305818	+	0.952090i	$0.598930\pi$
$128$	1448.15			0.0883883
$129$		0			0
$130$		−	10342.2i		−	0.611965i
$131$	−6324.59			−0.368545			−0.184272	−	0.982875i	$-0.558993\pi$
$131$	−6324.59			−0.368545			−0.184272	+	0.982875i	$0.558993\pi$
$132$		0			0
$133$	−2220.16			−0.125511
$134$			19752.1i			1.10003i
$135$		0			0
$136$			4165.19i			0.225194i
$137$		−	11350.4i		−	0.604739i	−0.953191	−	0.302370i	$-0.902222\pi$
$137$		−	11350.4i		−	0.604739i	0.953191	−	0.302370i	$-0.0977777\pi$
$138$		0			0
$139$	19799.1			1.02474			0.512372	−	0.858764i	$-0.328767\pi$
$139$	19799.1			1.02474			0.512372	+	0.858764i	$0.328767\pi$
$140$	−14256.3			−0.727361
$141$		0			0
$142$	9849.92			0.488490
$143$			1206.96i			0.0590232i
$144$		0			0
$145$		−	32620.7i		−	1.55152i
$146$	1149.54			0.0539284
$147$		0			0
$148$		−	16473.2i		−	0.752066i
$149$		−	1082.56i		−	0.0487616i	−0.999703	−	0.0243808i	$-0.992239\pi$
$149$		−	1082.56i		−	0.0487616i	0.999703	−	0.0243808i	$-0.00776142\pi$
$150$		0			0
$151$	34090.1			1.49512			0.747558	−	0.664197i	$-0.231226\pi$
$151$	34090.1			1.49512			0.747558	+	0.664197i	$0.231226\pi$
$152$			752.816i			0.0325838i
$153$		0			0
$154$	1663.75			0.0701529
$155$			32591.6i			1.35657i
$156$		0			0
$157$			5006.67i			0.203119i	0.994829	+	0.101559i	$0.0323832\pi$
$157$			5006.67i			0.203119i	−0.994829	+	0.101559i	$0.967617\pi$
$158$		−	294.110i		−	0.0117814i
$159$		0			0
$160$			4834.03i			0.188829i
$161$	12712.2	−	32932.6i	0.490422	−	1.27050i
$162$		0			0
$163$	−13052.4			−0.491265			−0.245632	−	0.969363i	$-0.578996\pi$
$163$	−13052.4			−0.491265			−0.245632	+	0.969363i	$0.578996\pi$
$164$	−7077.96			−0.263160
$165$		0			0
$166$			28583.0i			1.03727i
$167$	−30628.9			−1.09824			−0.549122	−	0.835742i	$-0.685038\pi$
$167$	−30628.9			−1.09824			−0.549122	+	0.835742i	$0.685038\pi$
$168$		0			0
$169$	−9812.45			−0.343561
$170$	−13903.7			−0.481096
$171$		0			0
$172$		−	12578.0i		−	0.425162i
$173$	11185.9			0.373747			0.186873	−	0.982384i	$-0.440165\pi$
$173$	11185.9			0.373747			0.186873	+	0.982384i	$0.440165\pi$
$174$		0			0
$175$		−	5881.11i		−	0.192036i
$176$		−	564.145i		−	0.0182123i
$177$		0			0
$178$			20983.9i			0.662288i
$179$	47448.9			1.48088			0.740440	−	0.672123i	$-0.234617\pi$
$179$	47448.9			1.48088			0.740440	+	0.672123i	$0.234617\pi$
$180$		0			0
$181$		−	16174.1i		−	0.493700i	−0.969054	−	0.246850i	$-0.920605\pi$
$181$		−	16174.1i		−	0.493700i	0.969054	−	0.246850i	$-0.0793955\pi$
$182$		−	25844.0i		−	0.780220i
$183$		0			0
$184$	−11166.8	−	4310.48i	−0.329833	−	0.127318i
$185$	54988.7			1.60668
$186$		0			0
$187$	1622.60			0.0464010
$188$	−4514.34			−0.127726
$189$		0			0
$190$	−2512.95			−0.0696107
$191$		−	3552.05i		−	0.0973672i	−0.998814	−	0.0486836i	$-0.984497\pi$
$191$		−	3552.05i		−	0.0973672i	0.998814	−	0.0486836i	$-0.0155026\pi$
$192$		0			0
$193$	28418.4			0.762929			0.381465	−	0.924383i	$-0.375420\pi$
$193$	28418.4			0.762929			0.381465	+	0.924383i	$0.375420\pi$
$194$			45767.9i			1.21607i
$195$		0			0
$196$	−16416.8			−0.427343
$197$	14923.4			0.384534			0.192267	−	0.981343i	$-0.438416\pi$
$197$	14923.4			0.384534			0.192267	+	0.981343i	$0.438416\pi$
$198$		0			0
$199$			71331.2i			1.80125i	0.434599	+	0.900624i	$0.356890\pi$
$199$			71331.2i			1.80125i	−0.434599	+	0.900624i	$0.643110\pi$
$200$	−1994.17			−0.0498543
$201$		0			0
$202$	−5914.57			−0.144951
$203$		−	81515.4i		−	1.97810i
$204$		0			0
$205$		−	23626.7i		−	0.562206i
$206$		−	25516.8i		−	0.601300i
$207$		0			0
$208$	−8763.22			−0.202552
$209$	293.268			0.00671385
$210$		0			0
$211$	−20415.9			−0.458568			−0.229284	−	0.973360i	$-0.573638\pi$
$211$	−20415.9			−0.458568			−0.229284	+	0.973360i	$0.573638\pi$
$212$			24265.6i			0.539908i
$213$		0			0
$214$			1798.86i			0.0392798i
$215$	41986.1			0.908299
$216$		0			0
$217$			81442.7i			1.72955i
$218$			16242.2i			0.341769i
$219$		0			0
$220$	1883.15			0.0389081
$221$		−	25204.8i		−	0.516059i
$222$		0			0
$223$	10531.8			0.211783			0.105892	−	0.994378i	$-0.466230\pi$
$223$	10531.8			0.211783			0.105892	+	0.994378i	$0.466230\pi$
$224$			12079.7i			0.240747i
$225$		0			0
$226$		−	59396.9i		−	1.16291i
$227$			43696.8i			0.848004i	0.905661	+	0.424002i	$0.139375\pi$
$227$			43696.8i			0.848004i	−0.905661	+	0.424002i	$0.860625\pi$
$228$		0			0
$229$		−	71464.4i		−	1.36276i	−0.731931	−	0.681379i	$-0.761381\pi$
$229$		−	71464.4i		−	1.36276i	0.731931	−	0.681379i	$-0.238619\pi$
$230$	14388.6	−	37275.6i	0.271997	−	0.704643i
$231$		0			0
$232$	−27640.4			−0.513532
$233$	26628.4			0.490494			0.245247	−	0.969461i	$-0.421131\pi$
$233$	26628.4			0.490494			0.245247	+	0.969461i	$0.421131\pi$
$234$		0			0
$235$		−	15069.1i		−	0.272868i
$236$	−45316.1			−0.813633
$237$		0			0
$238$	−34743.7			−0.613370
$239$	−23532.5			−0.411977			−0.205988	−	0.978554i	$-0.566041\pi$
$239$	−23532.5			−0.411977			−0.205988	+	0.978554i	$0.566041\pi$
$240$		0			0
$241$		−	6120.68i		−	0.105382i	−0.998611	−	0.0526909i	$-0.983220\pi$
$241$		−	6120.68i		−	0.105382i	0.998611	−	0.0526909i	$-0.0167798\pi$
$242$	41191.2			0.703354
$243$		0			0
$244$			27024.8i			0.453924i
$245$		−	54800.3i		−	0.912958i
$246$		0			0
$247$		−	4555.51i		−	0.0746696i
$248$	27615.7			0.449007
$249$		0			0
$250$			40550.7i			0.648811i
$251$			124768.i			1.98041i	0.139627	+	0.990204i	$0.455410\pi$
$251$			124768.i			1.98041i	−0.139627	+	0.990204i	$0.544590\pi$
$252$		0			0
$253$	−1679.19	+	4350.17i	−0.0262337	+	0.0679618i
$254$	−27902.7			−0.432492
$255$		0			0
$256$	4096.00			0.0625000
$257$	67049.4			1.01515			0.507573	−	0.861608i	$-0.330543\pi$
$257$	67049.4			1.01515			0.507573	+	0.861608i	$0.330543\pi$
$258$		0			0
$259$	137411.			2.04843
$260$		−	29252.2i		−	0.432725i
$261$		0			0
$262$	−17888.6			−0.260600
$263$		−	50313.6i		−	0.727402i	−0.931516	−	0.363701i	$-0.881513\pi$
$263$		−	50313.6i		−	0.727402i	0.931516	−	0.363701i	$-0.118487\pi$
$264$		0			0
$265$	−81000.2			−1.15344
$266$	−6279.57			−0.0887496
$267$		0			0
$268$			55867.3i			0.777836i
$269$	31067.8			0.429344			0.214672	−	0.976686i	$-0.431132\pi$
$269$	31067.8			0.429344			0.214672	+	0.976686i	$0.431132\pi$
$270$		0			0
$271$	10.6034			0.000144379			7.21896e−5	−	1.00000i	$-0.499977\pi$
$271$	10.6034			0.000144379			7.21896e−5		1.00000i	$0.499977\pi$
$272$			11780.9i			0.159236i
$273$		0			0
$274$		−	32103.6i		−	0.427615i
$275$			776.852i			0.0102724i
$276$		0			0
$277$	−110743.			−1.44330			−0.721650	−	0.692258i	$-0.756616\pi$
$277$	−110743.			−1.44330			−0.721650	+	0.692258i	$0.756616\pi$
$278$	56000.3			0.724604
$279$		0			0
$280$	−40322.8			−0.514322
$281$			95523.4i			1.20975i	0.796319	+	0.604877i	$0.206778\pi$
$281$			95523.4i			1.20975i	−0.796319	+	0.604877i	$0.793222\pi$
$282$		0			0
$283$			39396.5i			0.491910i	0.969281	+	0.245955i	$0.0791015\pi$
$283$			39396.5i			0.491910i	−0.969281	+	0.245955i	$0.920899\pi$
$284$	27859.8			0.345415
$285$		0			0
$286$			3413.81i			0.0417357i
$287$		−	59040.4i		−	0.716780i
$288$		0			0
$289$	49636.6			0.594300
$290$		−	92265.3i		−	1.09709i
$291$		0			0
$292$	3251.38			0.0381331
$293$		−	81900.7i		−	0.954009i	−0.878901	−	0.477005i	$-0.841722\pi$
$293$		−	81900.7i		−	0.954009i	0.878901	−	0.477005i	$-0.158278\pi$
$294$		0			0
$295$		−	151268.i		−	1.73821i
$296$		−	46593.4i		−	0.531791i
$297$		0			0
$298$		−	3061.93i		−	0.0344796i
$299$	67573.9	+	26084.0i	0.755852	+	0.291764i
$300$		0			0
$301$	104919.			1.15803
$302$	96421.5			1.05721
$303$		0			0
$304$			2129.28i			0.0230402i
$305$	−90210.5			−0.969744
$306$		0			0
$307$	160322.			1.70105			0.850524	−	0.525936i	$-0.176285\pi$
$307$	160322.			1.70105			0.850524	+	0.525936i	$0.176285\pi$
$308$	4705.78			0.0496056
$309$		0			0
$310$			92183.0i			0.959240i
$311$	50635.7			0.523523			0.261762	−	0.965133i	$-0.415697\pi$
$311$	50635.7			0.523523			0.261762	+	0.965133i	$0.415697\pi$
$312$		0			0
$313$		−	175913.i		−	1.79560i	−0.440408	−	0.897798i	$-0.645166\pi$
$313$		−	175913.i		−	1.79560i	0.440408	−	0.897798i	$-0.354834\pi$
$314$			14161.0i			0.143627i
$315$		0			0
$316$		−	831.869i		−	0.00833069i
$317$	15238.4			0.151642			0.0758212	−	0.997121i	$-0.475842\pi$
$317$	15238.4			0.151642			0.0758212	+	0.997121i	$0.475842\pi$
$318$		0			0
$319$			10767.6i			0.105813i
$320$			13672.7i			0.133523i
$321$		0			0
$322$	35955.6	−	93147.6i	0.346781	−	0.898380i
$323$	−6124.26			−0.0587014
$324$		0			0
$325$	12067.3			0.114247
$326$	−36917.8			−0.347376
$327$		0			0
$328$	−20019.5			−0.186083
$329$		−	37656.1i		−	0.347891i
$330$		0			0
$331$	−10413.5			−0.0950473			−0.0475237	−	0.998870i	$-0.515133\pi$
$331$	−10413.5			−0.0950473			−0.0475237	+	0.998870i	$0.515133\pi$
$332$			80844.9i			0.733460i
$333$		0			0
$334$	−86631.6			−0.776575
$335$	−186488.			−1.66174
$336$		0			0
$337$			45745.9i			0.402802i	0.979509	+	0.201401i	$0.0645495\pi$
$337$			45745.9i			0.402802i	−0.979509	+	0.201401i	$0.935451\pi$
$338$	−27753.8			−0.242934
$339$		0			0
$340$	−39325.5			−0.340186
$341$		−	10758.0i		−	0.0925173i
$342$		0			0
$343$			23282.7i			0.197899i
$344$		−	35575.9i		−	0.300635i
$345$		0			0
$346$	31638.4			0.264279
$347$	−202424.			−1.68114			−0.840570	−	0.541704i	$-0.817780\pi$
$347$	−202424.			−1.68114			−0.840570	+	0.541704i	$0.817780\pi$
$348$		0			0
$349$	160230.			1.31550			0.657752	−	0.753234i	$-0.271507\pi$
$349$	160230.			1.31550			0.657752	+	0.753234i	$0.271507\pi$
$350$		−	16634.3i		−	0.135790i
$351$		0			0
$352$		−	1595.64i		−	0.0128781i
$353$	96178.5			0.771842			0.385921	−	0.922532i	$-0.373884\pi$
$353$	96178.5			0.771842			0.385921	+	0.922532i	$0.373884\pi$
$354$		0			0
$355$			92997.7i			0.737930i
$356$			59351.5i			0.468308i
$357$		0			0
$358$	134206.			1.04714
$359$			14946.4i			0.115970i	0.998317	+	0.0579851i	$0.0184676\pi$
$359$			14946.4i			0.115970i	−0.998317	+	0.0579851i	$0.981532\pi$
$360$		0			0
$361$	129214.			0.991506
$362$		−	45747.3i		−	0.349099i
$363$		0			0
$364$		−	73097.9i		−	0.551699i
$365$			10853.3i			0.0814661i
$366$		0			0
$367$		−	133014.i		−	0.987566i	−0.869585	−	0.493783i	$-0.835614\pi$
$367$		−	133014.i		−	0.987566i	0.869585	−	0.493783i	$-0.164386\pi$
$368$	−31584.6	−	12191.9i	−0.233228	−	0.0900274i
$369$		0			0
$370$	155532.			1.13610
$371$	−202410.			−1.47057
$372$		0			0
$373$			178722.i			1.28458i	0.766462	+	0.642290i	$0.222016\pi$
$373$			178722.i			1.28458i	−0.766462	+	0.642290i	$0.777984\pi$
$374$	4589.40			0.0328105
$375$		0			0
$376$	−12768.5			−0.0903157
$377$	167260.			1.17682
$378$		0			0
$379$			199044.i			1.38571i	0.721079	+	0.692853i	$0.243647\pi$
$379$			199044.i			1.38571i	−0.721079	+	0.692853i	$0.756353\pi$
$380$	−7107.69			−0.0492222
$381$		0			0
$382$		−	10046.7i		−	0.0688490i
$383$			116619.i			0.795007i	0.917601	+	0.397504i	$0.130123\pi$
$383$			116619.i			0.795007i	−0.917601	+	0.397504i	$0.869877\pi$
$384$		0			0
$385$			15708.2i			0.105975i
$386$	80379.3			0.539473
$387$		0			0
$388$			129451.i			0.859889i
$389$			283865.i			1.87591i	0.346758	+	0.937955i	$0.387283\pi$
$389$			283865.i			1.87591i	−0.346758	+	0.937955i	$0.612717\pi$
$390$		0			0
$391$	35066.3	−	90843.9i	0.229370	−	0.594213i
$392$	−46433.7			−0.302177
$393$		0			0
$394$	42209.6			0.271906
$395$	2776.83			0.0177973
$396$		0			0
$397$	161458.			1.02442			0.512212	−	0.858859i	$-0.328826\pi$
$397$	161458.			1.02442			0.512212	+	0.858859i	$0.328826\pi$
$398$			201755.i			1.27368i
$399$		0			0
$400$	−5640.37			−0.0352523
$401$			172741.i			1.07426i	0.843501	+	0.537128i	$0.180491\pi$
$401$			172741.i			1.07426i	−0.843501	+	0.537128i	$0.819509\pi$
$402$		0			0
$403$	−167111.			−1.02895
$404$	−16728.9			−0.102496
$405$		0			0
$406$		−	230560.i		−	1.39873i
$407$	−18151.0			−0.109575
$408$		0			0
$409$	−305480.			−1.82615			−0.913073	−	0.407796i	$-0.866297\pi$
$409$	−305480.			−1.82615			−0.913073	+	0.407796i	$0.866297\pi$
$410$		−	66826.4i		−	0.397539i
$411$		0			0
$412$		−	72172.3i		−	0.425184i
$413$		−	378002.i		−	2.21612i
$414$		0			0
$415$	−269865.			−1.56694
$416$	−24786.1			−0.143226
$417$		0			0
$418$	829.487			0.00474741
$419$			181747.i			1.03524i	0.855612	+	0.517619i	$0.173181\pi$
$419$			181747.i			1.03524i	−0.855612	+	0.517619i	$0.826819\pi$
$420$		0			0
$421$			92369.5i			0.521152i	0.965453	+	0.260576i	$0.0839125\pi$
$421$			92369.5i			0.521152i	−0.965453	+	0.260576i	$0.916088\pi$
$422$	−57744.9			−0.324257
$423$		0			0
$424$			68633.5i			0.381772i
$425$		−	16222.9i		−	0.0898152i
$426$		0			0
$427$	−225426.			−1.23637
$428$			5087.93i			0.0277750i
$429$		0			0
$430$	118755.			0.642264
$431$		−	313163.i		−	1.68584i	−0.538041	−	0.842918i	$-0.680836\pi$
$431$		−	313163.i		−	1.68584i	0.538041	−	0.842918i	$-0.319164\pi$
$432$		0			0
$433$		−	102902.i		−	0.548844i	−0.961609	−	0.274422i	$-0.911514\pi$
$433$		−	102902.i		−	0.548844i	0.961609	−	0.274422i	$-0.0884865\pi$
$434$			230355.i			1.22298i
$435$		0			0
$436$			45940.0i			0.241667i
$437$	6337.88	−	16419.1i	0.0331880	−	0.0859778i
$438$		0			0
$439$	−195379.			−1.01379			−0.506895	−	0.862008i	$-0.669207\pi$
$439$	−195379.			−1.01379			−0.506895	+	0.862008i	$0.669207\pi$
$440$	5326.36			0.0275122
$441$		0			0
$442$		−	71290.1i		−	0.364909i
$443$	−228229.			−1.16296			−0.581478	−	0.813562i	$-0.697525\pi$
$443$	−228229.			−1.16296			−0.581478	+	0.813562i	$0.697525\pi$
$444$		0			0
$445$	−198119.			−1.00048
$446$	29788.3			0.149753
$447$		0			0
$448$			34166.6i			0.170234i
$449$	201941.			1.00169			0.500843	−	0.865538i	$-0.333024\pi$
$449$	201941.			1.00169			0.500843	+	0.865538i	$0.333024\pi$
$450$		0			0
$451$			7798.82i			0.0383421i
$452$		−	168000.i		−	0.822303i
$453$		0			0
$454$			123593.i			0.599629i
$455$	244006.			1.17863
$456$		0			0
$457$			126293.i			0.604711i	0.953195	+	0.302356i	$0.0977730\pi$
$457$			126293.i			0.604711i	−0.953195	+	0.302356i	$0.902227\pi$
$458$		−	202132.i		−	0.963615i
$459$		0			0
$460$	40697.2	−	105431.i	0.192331	−	0.498258i
$461$	−329484.			−1.55036			−0.775180	−	0.631740i	$-0.782341\pi$
$461$	−329484.			−1.55036			−0.775180	+	0.631740i	$0.782341\pi$
$462$		0			0
$463$	−188711.			−0.880311			−0.440155	−	0.897922i	$-0.645077\pi$
$463$	−188711.			−0.880311			−0.440155	+	0.897922i	$0.645077\pi$
$464$	−78178.7			−0.363122
$465$		0			0
$466$	75316.5			0.346832
$467$		−	164612.i		−	0.754792i	−0.926052	−	0.377396i	$-0.876820\pi$
$467$		−	164612.i		−	0.754792i	0.926052	−	0.377396i	$-0.123180\pi$
$468$		0			0
$469$	−466014.			−2.11862
$470$		−	42622.0i		−	0.192947i
$471$		0			0
$472$	−128173.			−0.575326
$473$	−13859.0			−0.0619455
$474$		0			0
$475$		−	2932.12i		−	0.0129955i
$476$	−98270.1			−0.433718
$477$		0			0
$478$	−66560.0			−0.291311
$479$		−	306234.i		−	1.33470i	−0.744745	−	0.667349i	$-0.767429\pi$
$479$		−	306234.i		−	1.33470i	0.744745	−	0.667349i	$-0.232571\pi$
$480$		0			0
$481$			281951.i			1.21866i
$482$		−	17311.9i		−	0.0745162i
$483$		0			0
$484$	116506.			0.497346
$485$	−432116.			−1.83703
$486$		0			0
$487$	−365176.			−1.53973			−0.769864	−	0.638208i	$-0.779676\pi$
$487$	−365176.			−1.53973			−0.769864	+	0.638208i	$0.779676\pi$
$488$			76437.6i			0.320972i
$489$		0			0
$490$		−	154999.i		−	0.645559i
$491$	−380595.			−1.57870			−0.789351	−	0.613942i	$-0.789583\pi$
$491$	−380595.			−1.57870			−0.789351	+	0.613942i	$0.789583\pi$
$492$		0			0
$493$		−	224858.i		−	0.925156i
$494$		−	12884.9i		−	0.0527993i
$495$		0			0
$496$	78109.0			0.317496
$497$			232391.i			0.940819i
$498$		0			0
$499$	−15929.0			−0.0639716			−0.0319858	−	0.999488i	$-0.510183\pi$
$499$	−15929.0			−0.0639716			−0.0319858	+	0.999488i	$0.510183\pi$
$500$			114695.i			0.458779i
$501$		0			0
$502$			352896.i			1.40036i
$503$			239167.i			0.945289i	0.881253	+	0.472645i	$0.156701\pi$
$503$			239167.i			0.945289i	−0.881253	+	0.472645i	$0.843299\pi$
$504$		0			0
$505$		−	55842.2i		−	0.218968i
$506$	−4749.48	+	12304.1i	−0.0185500	+	0.0480563i
$507$		0			0
$508$	−78920.6			−0.305818
$509$	297194.			1.14711			0.573555	−	0.819167i	$-0.305564\pi$
$509$	297194.			1.14711			0.573555	+	0.819167i	$0.305564\pi$
$510$		0			0
$511$			27121.2i			0.103865i
$512$	11585.2			0.0441942
$513$		0			0
$514$	189644.			0.717817
$515$	240916.			0.908345
$516$		0			0
$517$			4974.10i			0.0186094i
$518$	388656.			1.44846
$519$		0			0
$520$		−	82737.7i		−	0.305983i
$521$			490440.i			1.80680i	0.428797	+	0.903401i	$0.358938\pi$
$521$			490440.i			1.80680i	−0.428797	+	0.903401i	$0.641062\pi$
$522$		0			0
$523$			54355.5i			0.198719i	0.995052	+	0.0993596i	$0.0316794\pi$
$523$			54355.5i			0.198719i	−0.995052	+	0.0993596i	$0.968321\pi$
$524$	−50596.7			−0.184272
$525$		0			0
$526$		−	142308.i		−	0.514351i
$527$			224658.i			0.808910i
$528$		0			0
$529$	207262.	+	188025.i	0.740642	+	0.671900i
$530$	−229103.			−0.815604
$531$		0			0
$532$	−17761.3			−0.0627555
$533$	121144.			0.426430
$534$		0			0
$535$	−16983.8			−0.0593374
$536$			158016.i			0.550013i
$537$		0			0
$538$	87873.0			0.303592
$539$			18088.8i			0.0622632i
$540$		0			0
$541$	290778.			0.993498			0.496749	−	0.867894i	$-0.334527\pi$
$541$	290778.			0.993498			0.496749	+	0.867894i	$0.334527\pi$
$542$	29.9908			0.000102092
$543$		0			0
$544$			33321.5i			0.112597i
$545$	−153351.			−0.516288
$546$		0			0
$547$	−361239.			−1.20731			−0.603656	−	0.797245i	$-0.706290\pi$
$547$	−361239.			−1.20731			−0.603656	+	0.797245i	$0.706290\pi$
$548$		−	90802.8i		−	0.302370i
$549$		0			0
$550$			2197.27i			0.00726370i
$551$		−	40640.8i		−	0.133863i
$552$		0			0
$553$	6938.99			0.0226906
$554$	−313229.			−1.02057
$555$		0			0
$556$	158393.			0.512372
$557$		−	423652.i		−	1.36552i	−0.730641	−	0.682762i	$-0.760779\pi$
$557$		−	423652.i		−	1.36552i	0.730641	−	0.682762i	$-0.239221\pi$
$558$		0			0
$559$			215281.i			0.688940i
$560$	−114050.			−0.363680
$561$		0			0
$562$			270181.i			0.855425i
$563$			278440.i			0.878446i	0.898378	+	0.439223i	$0.144746\pi$
$563$			278440.i			0.878446i	−0.898378	+	0.439223i	$0.855254\pi$
$564$		0			0
$565$	560794.			1.75674
$566$			111430.i			0.347833i
$567$		0			0
$568$	78799.3			0.244245
$569$		−	22300.7i		−	0.0688803i	−0.999407	−	0.0344401i	$-0.989035\pi$
$569$		−	22300.7i		−	0.0688803i	0.999407	−	0.0344401i	$-0.0109648\pi$
$570$		0			0
$571$			351483.i			1.07803i	0.842295	+	0.539017i	$0.181204\pi$
$571$			351483.i			1.07803i	−0.842295	+	0.539017i	$0.818796\pi$
$572$			9655.72i			0.0295116i
$573$		0			0
$574$		−	166992.i		−	0.506840i
$575$	43493.4	+	16788.7i	0.131549	+	0.0507788i
$576$		0			0
$577$	331198.			0.994801			0.497401	−	0.867521i	$-0.334288\pi$
$577$	331198.			0.994801			0.497401	+	0.867521i	$0.334288\pi$
$578$	140393.			0.420234
$579$		0			0
$580$		−	260966.i		−	0.775760i
$581$	−674363.			−1.99775
$582$		0			0
$583$	26737.0			0.0786638
$584$	9196.30			0.0269642
$585$		0			0
$586$		−	231650.i		−	0.674586i
$587$	−243067.			−0.705423			−0.352711	−	0.935732i	$-0.614740\pi$
$587$	−243067.			−0.705423			−0.352711	+	0.935732i	$0.614740\pi$
$588$		0			0
$589$			40604.5i			0.117043i
$590$		−	427851.i		−	1.22910i
$591$		0			0
$592$		−	131786.i		−	0.376033i
$593$	103094.			0.293172			0.146586	−	0.989198i	$-0.453171\pi$
$593$	103094.			0.293172			0.146586	+	0.989198i	$0.453171\pi$
$594$		0			0
$595$		−	328032.i		−	0.926578i
$596$		−	8660.45i		−	0.0243808i
$597$		0			0
$598$	191128.	+	73776.7i	0.534468	+	0.206308i
$599$	354671.			0.988490			0.494245	−	0.869323i	$-0.335445\pi$
$599$	354671.			0.988490			0.494245	+	0.869323i	$0.335445\pi$
$600$		0			0
$601$	681636.			1.88714			0.943569	−	0.331175i	$-0.107445\pi$
$601$	681636.			1.88714			0.943569	+	0.331175i	$0.107445\pi$
$602$	296754.			0.818850
$603$		0			0
$604$	272721.			0.747558
$605$			388906.i			1.06251i
$606$		0			0
$607$	−674049.			−1.82942			−0.914711	−	0.404109i	$-0.867582\pi$
$607$	−674049.			−1.82942			−0.914711	+	0.404109i	$0.867582\pi$
$608$			6022.52i			0.0162919i
$609$		0			0
$610$	−255154.			−0.685713
$611$	77265.9			0.206969
$612$		0			0
$613$		−	432868.i		−	1.15195i	−0.817467	−	0.575976i	$-0.804622\pi$
$613$		−	432868.i		−	1.15195i	0.817467	−	0.575976i	$-0.195378\pi$
$614$	453459.			1.20282
$615$		0			0
$616$	13310.0			0.0350764
$617$			366413.i			0.962500i	0.876583	+	0.481250i	$0.159817\pi$
$617$			366413.i			0.962500i	−0.876583	+	0.481250i	$0.840183\pi$
$618$		0			0
$619$			575223.i			1.50126i	0.660724	+	0.750629i	$0.270249\pi$
$619$			575223.i			1.50126i	−0.660724	+	0.750629i	$0.729751\pi$
$620$			260733.i			0.678285i
$621$		0			0
$622$	143219.			0.370187
$623$	−495078.			−1.27555
$624$		0			0
$625$	−437940.			−1.12113
$626$		−	497556.i		−	1.26968i
$627$		0			0
$628$			40053.4i			0.101559i
$629$	379044.			0.958050
$630$		0			0
$631$		−	544431.i		−	1.36736i	−0.729780	−	0.683682i	$-0.760378\pi$
$631$		−	544431.i		−	1.36736i	0.729780	−	0.683682i	$-0.239622\pi$
$632$		−	2352.88i		−	0.00589068i
$633$		0			0
$634$	43100.7			0.107227
$635$		−	263442.i		−	0.653338i
$636$		0			0
$637$	280984.			0.692474
$638$			30455.4i			0.0748209i
$639$		0			0
$640$			38672.3i			0.0944147i
$641$		−	437564.i		−	1.06494i	−0.846449	−	0.532470i	$-0.821264\pi$
$641$		−	437564.i		−	1.06494i	0.846449	−	0.532470i	$-0.178736\pi$
$642$		0			0
$643$		−	9226.04i		−	0.0223148i	−0.999938	−	0.0111574i	$-0.996448\pi$
$643$		−	9226.04i		−	0.0223148i	0.999938	−	0.0111574i	$-0.00355159\pi$
$644$	101698.	−	263461.i	0.245211	−	0.635250i
$645$		0			0
$646$	−17322.0			−0.0415082
$647$	78818.7			0.188287			0.0941436	−	0.995559i	$-0.469989\pi$
$647$	78818.7			0.188287			0.0941436	+	0.995559i	$0.469989\pi$
$648$		0			0
$649$			49931.4i			0.118545i
$650$	34131.6			0.0807848
$651$		0			0
$652$	−104419.			−0.245632
$653$	483227.			1.13325			0.566624	−	0.823977i	$-0.308249\pi$
$653$	483227.			1.13325			0.566624	+	0.823977i	$0.308249\pi$
$654$		0			0
$655$		−	168895.i		−	0.393672i
$656$	−56623.7			−0.131580
$657$		0			0
$658$		−	106507.i		−	0.245996i
$659$		−	657389.i		−	1.51374i	−0.653564	−	0.756871i	$-0.726727\pi$
$659$		−	657389.i		−	1.51374i	0.653564	−	0.756871i	$-0.273273\pi$
$660$		0			0
$661$		−	783821.i		−	1.79397i	−0.442066	−	0.896983i	$-0.645754\pi$
$661$		−	783821.i		−	1.79397i	0.442066	−	0.896983i	$-0.354246\pi$
$662$	−29453.8			−0.0672086
$663$		0			0
$664$			228664.i			0.518635i
$665$		−	59288.4i		−	0.134068i
$666$		0			0
$667$	602843.	+	232701.i	1.35504	+	0.523055i
$668$	−245031.			−0.549122
$669$		0			0
$670$	−527469.			−1.17503
$671$	29777.1			0.0661360
$672$		0			0
$673$	626018.			1.38215			0.691077	−	0.722781i	$-0.257136\pi$
$673$	626018.			1.38215			0.691077	+	0.722781i	$0.257136\pi$
$674$			129389.i			0.284824i
$675$		0			0
$676$	−78499.6			−0.171781
$677$		−	2298.21i		−	0.00501432i	−0.999997	−	0.00250716i	$-0.999202\pi$
$677$		−	2298.21i		−	0.00501432i	0.999997	−	0.00250716i	$-0.000798054\pi$
$678$		0			0
$679$	−1.07981e6			−2.34211
$680$	−111229.			−0.240548
$681$		0			0
$682$		−	30428.2i		−	0.0654196i
$683$	815691.			1.74857			0.874287	−	0.485409i	$-0.161329\pi$
$683$	815691.			1.74857			0.874287	+	0.485409i	$0.161329\pi$
$684$		0			0
$685$	303106.			0.645971
$686$			65853.3i			0.139936i
$687$		0			0
$688$		−	100624.i		−	0.212581i
$689$		−	415322.i		−	0.874876i
$690$		0			0
$691$	849576.			1.77929			0.889644	−	0.456655i	$-0.150953\pi$
$691$	849576.			1.77929			0.889644	+	0.456655i	$0.150953\pi$
$692$	89486.9			0.186873
$693$		0			0
$694$	−572542.			−1.18874
$695$			528725.i			1.09461i
$696$		0			0
$697$		−	162862.i		−	0.335238i
$698$	453198.			0.930202
$699$		0			0
$700$		−	47048.8i		−	0.0960181i
$701$		−	488828.i		−	0.994763i	−0.867532	−	0.497382i	$-0.834295\pi$
$701$		−	488828.i		−	0.994763i	0.867532	−	0.497382i	$-0.165705\pi$
$702$		0			0
$703$	68508.2			0.138622
$704$		−	4513.16i		−	0.00910616i
$705$		0			0
$706$	272034.			0.545775
$707$		−	139543.i		−	0.279171i
$708$		0			0
$709$		−	932925.i		−	1.85590i	−0.372706	−	0.927950i	$-0.621570\pi$
$709$		−	932925.i		−	1.85590i	0.372706	−	0.927950i	$-0.378430\pi$
$710$			263037.i			0.521796i
$711$		0			0
$712$			167872.i			0.331144i
$713$	−602305.	−	232494.i	−1.18478	−	0.457333i
$714$		0			0
$715$	−32231.4			−0.0630474
$716$	379591.			0.740440
$717$		0			0
$718$			42274.7i			0.0820033i
$719$	74115.8			0.143368			0.0716841	−	0.997427i	$-0.477163\pi$
$719$	74115.8			0.143368			0.0716841	+	0.997427i	$0.477163\pi$
$720$		0			0
$721$	602022.			1.15809
$722$	365473.			0.701101
$723$		0			0
$724$		−	129393.i		−	0.246850i
$725$	107656.			0.204814
$726$		0			0
$727$			864443.i			1.63556i	0.575528	+	0.817782i	$0.304796\pi$
$727$			864443.i			1.63556i	−0.575528	+	0.817782i	$0.695204\pi$
$728$		−	206752.i		−	0.390110i
$729$		0			0
$730$			30697.8i			0.0576052i
$731$	289415.			0.541610
$732$		0			0
$733$			526886.i			0.980637i	0.871543	+	0.490319i	$0.163120\pi$
$733$			526886.i			0.980637i	−0.871543	+	0.490319i	$0.836880\pi$
$734$		−	376221.i		−	0.698314i
$735$		0			0
$736$	−89334.7	−	34483.8i	−0.164917	−	0.0636590i
$737$	61557.1			0.113330
$738$		0			0
$739$	90319.3			0.165383			0.0826917	−	0.996575i	$-0.473648\pi$
$739$	90319.3			0.165383			0.0826917	+	0.996575i	$0.473648\pi$
$740$	439910.			0.803342
$741$		0			0
$742$	−572503.			−1.03985
$743$		−	79340.7i		−	0.143720i	−0.997415	−	0.0718602i	$-0.977106\pi$
$743$		−	79340.7i		−	0.143720i	0.997415	−	0.0718602i	$-0.0228935\pi$
$744$		0			0
$745$	28909.1			0.0520862
$746$			505503.i			0.908335i
$747$		0			0
$748$	12980.8			0.0232005
$749$	−42440.7			−0.0756518
$750$		0			0
$751$			825836.i			1.46424i	0.681173	+	0.732122i	$0.261470\pi$
$751$			825836.i			1.46424i	−0.681173	+	0.732122i	$0.738530\pi$
$752$	−36114.7			−0.0638628
$753$		0			0
$754$	473083.			0.832137
$755$			910360.i			1.59705i
$756$		0			0
$757$		−	391369.i		−	0.682960i	−0.939889	−	0.341480i	$-0.889072\pi$
$757$		−	391369.i		−	0.682960i	0.939889	−	0.341480i	$-0.110928\pi$
$758$			562982.i			0.979842i
$759$		0			0
$760$	−20103.6			−0.0348053
$761$	−48843.0			−0.0843398			−0.0421699	−	0.999110i	$-0.513427\pi$
$761$	−48843.0			−0.0843398			−0.0421699	+	0.999110i	$0.513427\pi$
$762$		0			0
$763$	−383206.			−0.658238
$764$		−	28416.4i		−	0.0486836i
$765$		0			0
$766$			329848.i			0.562155i
$767$	775616.			1.31843
$768$		0			0
$769$		−	32257.6i		−	0.0545480i	−0.999628	−	0.0272740i	$-0.991317\pi$
$769$		−	32257.6i		−	0.0545480i	0.999628	−	0.0272740i	$-0.00868266\pi$
$770$			44429.5i			0.0749359i
$771$		0			0
$772$	227347.			0.381465
$773$		−	261048.i		−	0.436879i	−0.975850	−	0.218440i	$-0.929903\pi$
$773$		−	261048.i		−	0.436879i	0.975850	−	0.218440i	$-0.0700967\pi$
$774$		0			0
$775$	−107560.			−0.179079
$776$			366143.i			0.608033i
$777$		0			0
$778$			802890.i			1.32647i
$779$		−	29435.5i		−	0.0485062i
$780$		0			0
$781$		−	30697.2i		−	0.0503264i
$782$	99182.6	−	256945.i	0.162189	−	0.420172i
$783$		0			0
$784$	−131334.			−0.213671
$785$	−133701.			−0.216967
$786$		0			0
$787$		−	769153.i		−	1.24183i	−0.783877	−	0.620917i	$-0.786761\pi$
$787$		−	769153.i		−	1.24183i	0.783877	−	0.620917i	$-0.213239\pi$
$788$	119387.			0.192267
$789$		0			0
$790$	7854.07			0.0125846
$791$	1.40136e6			2.23974
$792$		0			0
$793$		−	462547.i		−	0.735546i
$794$	456673.			0.724377
$795$		0			0
$796$			570650.i			0.900624i
$797$		−	358409.i		−	0.564238i	−0.959379	−	0.282119i	$-0.908963\pi$
$797$		−	358409.i		−	0.564238i	0.959379	−	0.282119i	$-0.0910373\pi$
$798$		0			0
$799$		−	103873.i		−	0.162709i
$800$	−15953.4			−0.0249272
$801$		0			0
$802$			488586.i			0.759613i
$803$		−	3582.52i		−	0.00555594i
$804$		0			0
$805$	879450.	+	339474.i	1.35712	+	0.523859i
$806$	−472661.			−0.727578
$807$		0			0
$808$	−47316.5			−0.0724753
$809$	−427855.			−0.653732			−0.326866	−	0.945071i	$-0.605993\pi$
$809$	−427855.			−0.653732			−0.326866	+	0.945071i	$0.605993\pi$
$810$		0			0
$811$	364799.			0.554641			0.277321	−	0.960777i	$-0.410554\pi$
$811$	364799.			0.554641			0.277321	+	0.960777i	$0.410554\pi$
$812$		−	652123.i		−	0.989049i
$813$		0			0
$814$	−51338.7			−0.0774812
$815$		−	348558.i		−	0.524759i
$816$		0			0
$817$	52308.8			0.0783665
$818$	−864027.			−1.29128
$819$		0			0
$820$		−	189014.i		−	0.281103i
$821$	1.11083e6			1.64801			0.824006	−	0.566581i	$-0.191734\pi$
$821$	1.11083e6			1.64801			0.824006	+	0.566581i	$0.191734\pi$
$822$		0			0
$823$	−775829.			−1.14542			−0.572712	−	0.819756i	$-0.694109\pi$
$823$	−775829.			−1.14542			−0.572712	+	0.819756i	$0.694109\pi$
$824$		−	204134.i		−	0.300650i
$825$		0			0
$826$		−	1.06915e6i		−	1.56704i
$827$			671515.i			0.981848i	0.871202	+	0.490924i	$0.163341\pi$
$827$			671515.i			0.981848i	−0.871202	+	0.490924i	$0.836659\pi$
$828$		0			0
$829$	−1.00171e6			−1.45758			−0.728789	−	0.684739i	$-0.759916\pi$
$829$	−1.00171e6			−1.45758			−0.728789	+	0.684739i	$0.759916\pi$
$830$	−763295.			−1.10799
$831$		0			0
$832$	−70105.8			−0.101276
$833$		−	377745.i		−	0.544388i
$834$		0			0
$835$		−	817930.i		−	1.17312i
$836$	2346.14			0.00335693
$837$		0			0
$838$			514059.i			0.732023i
$839$		−	704933.i		−	1.00144i	−0.865610	−	0.500719i	$-0.833069\pi$
$839$		−	704933.i		−	1.00144i	0.865610	−	0.500719i	$-0.166931\pi$
$840$		0			0
$841$	784885.			1.10972
$842$			261260.i			0.368510i
$843$		0			0
$844$	−163327.			−0.229284
$845$		−	262037.i		−	0.366985i
$846$		0			0
$847$			971832.i			1.35464i
$848$			194125.i			0.269954i
$849$		0			0
$850$		−	45885.2i		−	0.0635090i
$851$	−392265.	+	1.01621e6i	−0.541652	+	1.40322i
$852$		0			0
$853$	412976.			0.567579			0.283790	−	0.958887i	$-0.408408\pi$
$853$	412976.			0.567579			0.283790	+	0.958887i	$0.408408\pi$
$854$	−637600.			−0.874244
$855$		0			0
$856$			14390.8i			0.0196399i
$857$	566697.			0.771595			0.385798	−	0.922583i	$-0.373926\pi$
$857$	566697.			0.771595			0.385798	+	0.922583i	$0.373926\pi$
$858$		0			0
$859$	−1.36198e6			−1.84580			−0.922901	−	0.385036i	$-0.874189\pi$
$859$	−1.36198e6			−1.84580			−0.922901	+	0.385036i	$0.874189\pi$
$860$	335889.			0.454149
$861$		0			0
$862$		−	885758.i		−	1.19207i
$863$	−64405.4			−0.0864771			−0.0432385	−	0.999065i	$-0.513768\pi$
$863$	−64405.4			−0.0864771			−0.0432385	+	0.999065i	$0.513768\pi$
$864$		0			0
$865$			298713.i			0.399229i
$866$		−	291051.i		−	0.388091i
$867$		0			0
$868$			651542.i			0.864774i
$869$	−916.591			−0.00121377
$870$		0			0
$871$		−	956206.i		−	1.26042i
$872$			129938.i			0.170885i
$873$		0			0
$874$	17926.2	−	46440.2i	0.0234675	−	0.0607955i
$875$	−956719.			−1.24959
$876$		0			0
$877$	335218.			0.435841			0.217921	−	0.975967i	$-0.430073\pi$
$877$	335218.			0.435841			0.217921	+	0.975967i	$0.430073\pi$
$878$	−552614.			−0.716858
$879$		0			0
$880$	15065.2			0.0194540
$881$			168848.i			0.217543i	0.994067	+	0.108771i	$0.0346916\pi$
$881$			168848.i			0.217543i	−0.994067	+	0.108771i	$0.965308\pi$
$882$		0			0
$883$	155798.			0.199821			0.0999103	−	0.994996i	$-0.468144\pi$
$883$	155798.			0.199821			0.0999103	+	0.994996i	$0.468144\pi$
$884$		−	201639.i		−	0.258030i
$885$		0			0
$886$	−645529.			−0.822334
$887$	−817194.			−1.03867			−0.519335	−	0.854570i	$-0.673820\pi$
$887$	−817194.			−1.03867			−0.519335	+	0.854570i	$0.673820\pi$
$888$		0			0
$889$		−	658312.i		−	0.832968i
$890$	−560366.			−0.707443
$891$		0			0
$892$	84254.1			0.105892
$893$		−	18774.0i		−	0.0235426i
$894$		0			0
$895$			1.26710e6i			1.58185i
$896$			96637.6i			0.120373i
$897$		0			0
$898$	571175.			0.708298
$899$	−1.49084e6			−1.84463
$900$		0			0
$901$	−558344.			−0.687784
$902$			22058.4i			0.0271120i
$903$		0			0
$904$		−	475175.i		−	0.581456i
$905$	431922.			0.527361
$906$		0			0
$907$			426923.i			0.518961i	0.965748	+	0.259481i	$0.0835514\pi$
$907$			426923.i			0.518961i	−0.965748	+	0.259481i	$0.916449\pi$
$908$			349574.i			0.424002i
$909$		0			0
$910$	690152.			0.833416
$911$			120577.i			0.145288i	0.997358	+	0.0726438i	$0.0231436\pi$
$911$			120577.i			0.145288i	−0.997358	+	0.0726438i	$0.976856\pi$
$912$		0			0
$913$	89078.6			0.106864
$914$			357212.i			0.427595i
$915$		0			0
$916$		−	571715.i		−	0.681379i
$917$		−	422050.i		−	0.501909i
$918$		0			0
$919$			1.26268e6i			1.49507i	0.664221	+	0.747537i	$0.268764\pi$
$919$			1.26268e6i			1.49507i	−0.664221	+	0.747537i	$0.731236\pi$
$920$	115109.	−	298205.i	0.135999	−	0.352322i
$921$		0			0
$922$	−931922.			−1.09627
$923$	−476839.			−0.559716
$924$		0			0
$925$			181475.i			0.212096i
$926$	−533756.			−0.622474
$927$		0			0
$928$	−221123.			−0.256766
$929$	778978.			0.902597			0.451298	−	0.892373i	$-0.350961\pi$
$929$	778978.			0.902597			0.451298	+	0.892373i	$0.350961\pi$
$930$		0			0
$931$		−	68273.5i		−	0.0787685i
$932$	213027.			0.245247
$933$		0			0
$934$		−	465592.i		−	0.533718i
$935$			43330.7i			0.0495647i
$936$		0			0
$937$		−	1.53056e6i		−	1.74329i	−0.490137	−	0.871645i	$-0.663053\pi$
$937$		−	1.53056e6i		−	1.74329i	0.490137	−	0.871645i	$-0.336947\pi$
$938$	−1.31809e6			−1.49809
$939$		0			0
$940$		−	120553.i		−	0.136434i
$941$			230300.i			0.260084i	0.991508	+	0.130042i	$0.0415113\pi$
$941$			230300.i			0.260084i	−0.991508	+	0.130042i	$0.958489\pi$
$942$		0			0
$943$	436630.	+	168542.i	0.491010	+	0.189533i
$944$	−362529.			−0.406817
$945$		0			0
$946$	−39199.2			−0.0438021
$947$	−301394.			−0.336074			−0.168037	−	0.985781i	$-0.553743\pi$
$947$	−301394.			−0.336074			−0.168037	+	0.985781i	$0.553743\pi$
$948$		0			0
$949$	−55649.6			−0.0617916
$950$		−	8293.28i		−	0.00918923i
$951$		0			0
$952$	−277950.			−0.306685
$953$		−	178700.i		−	0.196761i	−0.995149	−	0.0983803i	$-0.968634\pi$
$953$		−	178700.i		−	0.196761i	0.995149	−	0.0983803i	$-0.0313662\pi$
$954$		0			0
$955$	94855.8			0.104006
$956$	−188260.			−0.205988
$957$		0			0
$958$		−	866162.i		−	0.943774i
$959$	757427.			0.823575
$960$		0			0
$961$	565985.			0.612855
$962$			797477.i			0.861723i
$963$		0			0
$964$		−	48965.5i		−	0.0526909i
$965$			758898.i			0.814946i
$966$		0			0
$967$	−331589.			−0.354607			−0.177304	−	0.984156i	$-0.556737\pi$
$967$	−331589.			−0.354607			−0.177304	+	0.984156i	$0.556737\pi$
$968$	329530.			0.351677
$969$		0			0
$970$	−1.22221e6			−1.29898
$971$			1.13672e6i			1.20563i	0.797881	+	0.602815i	$0.205954\pi$
$971$			1.13672e6i			1.20563i	−0.797881	+	0.602815i	$0.794046\pi$
$972$		0			0
$973$			1.32122e6i			1.39557i
$974$	−1.03287e6			−1.08875
$975$		0			0
$976$			216198.i			0.226962i
$977$		−	123255.i		−	0.129126i	−0.997914	−	0.0645631i	$-0.979435\pi$
$977$		−	123255.i		−	0.129126i	0.997914	−	0.0645631i	$-0.0205654\pi$
$978$		0			0
$979$	65396.2			0.0682319
$980$		−	438403.i		−	0.456479i
$981$		0			0
$982$	−1.07649e6			−1.11631
$983$			381404.i			0.394710i	0.980332	+	0.197355i	$0.0632351\pi$
$983$			381404.i			0.394710i	−0.980332	+	0.197355i	$0.936765\pi$
$984$		0			0
$985$			398521.i			0.410751i
$986$		−	635995.i		−	0.654184i
$987$		0			0
$988$		−	36444.1i		−	0.0373348i
$989$	−299510.	+	775919.i	−0.306210	+	0.793275i
$990$		0			0
$991$	−551610.			−0.561675			−0.280837	−	0.959755i	$-0.590612\pi$
$991$	−551610.			−0.561675			−0.280837	+	0.959755i	$0.590612\pi$
$992$	220926.			0.224503
$993$		0			0
$994$			657300.i			0.665259i
$995$	−1.90487e6			−1.92406
$996$		0			0
$997$	−443006.			−0.445676			−0.222838	−	0.974856i	$-0.571532\pi$
$997$	−443006.			−0.445676			−0.222838	+	0.974856i	$0.571532\pi$
$998$	−45054.0			−0.0452348
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.5.b.b.91.14		16
3.2	odd	2		138.5.b.a.91.2	✓	16
12.11	even	2		1104.5.c.a.1057.10		16
23.22	odd	2	inner	414.5.b.b.91.11		16
69.68	even	2		138.5.b.a.91.3	yes	16
276.275	odd	2		1104.5.c.a.1057.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.b.a.91.2	✓	16	3.2	odd	2
138.5.b.a.91.3	yes	16	69.68	even	2
414.5.b.b.91.11		16	23.22	odd	2	inner
414.5.b.b.91.14		16	1.1	even	1	trivial
1104.5.c.a.1057.10		16	12.11	even	2
1104.5.c.a.1057.15		16	276.275	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	414.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.82843 q^{2} +8.00000 q^{4} +26.7045i q^{5} +66.7315i q^{7} +22.6274 q^{8} +O(q^{10})\)	\(q+2.82843 q^{2} +8.00000 q^{4} +26.7045i q^{5} +66.7315i q^{7} +22.6274 q^{8} +75.5318i q^{10} -8.81477i q^{11} -136.925 q^{13} +188.745i q^{14} +64.0000 q^{16} +184.077i q^{17} +33.2701i q^{19} +213.636i q^{20} -24.9319i q^{22} +(-493.509 - 190.498i) q^{23} -88.1308 q^{25} -387.283 q^{26} +533.852i q^{28} -1221.54 q^{29} +1220.45 q^{31} +181.019 q^{32} +520.649i q^{34} -1782.03 q^{35} -2059.16i q^{37} +94.1019i q^{38} +604.254i q^{40} -884.746 q^{41} -1572.25i q^{43} -70.5181i q^{44} +(-1395.86 - 538.810i) q^{46} -564.292 q^{47} -2052.10 q^{49} -249.272 q^{50} -1095.40 q^{52} +3033.20i q^{53} +235.394 q^{55} +1509.96i q^{56} -3455.04 q^{58} -5664.52 q^{59} +3378.10i q^{61} +3451.96 q^{62} +512.000 q^{64} -3656.52i q^{65} +6983.41i q^{67} +1472.62i q^{68} -5040.35 q^{70} +3482.47 q^{71} +406.423 q^{73} -5824.17i q^{74} +266.160i q^{76} +588.223 q^{77} -103.984i q^{79} +1709.09i q^{80} -2502.44 q^{82} +10105.6i q^{83} -4915.69 q^{85} -4446.99i q^{86} -199.455i q^{88} +7418.94i q^{89} -9137.24i q^{91} +(-3948.08 - 1523.98i) q^{92} -1596.06 q^{94} -888.461 q^{95} +16181.4i q^{97} -5804.21 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 128 q^{4}+O(q^{10}) \) 16 * q + 128 * q^4	\( 16 q + 128 q^{4} - 208 q^{13} + 1024 q^{16} - 840 q^{23} + 1056 q^{25} - 1920 q^{26} - 3600 q^{29} + 224 q^{31} + 3264 q^{35} + 6144 q^{41} + 1280 q^{46} - 8880 q^{47} - 13888 q^{49} - 7296 q^{50} - 1664 q^{52} + 832 q^{55} + 2944 q^{58} + 18240 q^{59} + 8192 q^{64} + 19584 q^{70} + 30048 q^{71} + 9536 q^{73} - 14160 q^{77} - 19584 q^{82} - 32496 q^{85} - 6720 q^{92} - 21248 q^{94} + 20064 q^{95} - 21504 q^{98}+O(q^{100}) \) 16 * q + 128 * q^4 - 208 * q^13 + 1024 * q^16 - 840 * q^23 + 1056 * q^25 - 1920 * q^26 - 3600 * q^29 + 224 * q^31 + 3264 * q^35 + 6144 * q^41 + 1280 * q^46 - 8880 * q^47 - 13888 * q^49 - 7296 * q^50 - 1664 * q^52 + 832 * q^55 + 2944 * q^58 + 18240 * q^59 + 8192 * q^64 + 19584 * q^70 + 30048 * q^71 + 9536 * q^73 - 14160 * q^77 - 19584 * q^82 - 32496 * q^85 - 6720 * q^92 - 21248 * q^94 + 20064 * q^95 - 21504 * q^98

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