LMFDB - Embedded newform 414.5.b.b.91.1

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			91.1
Root			$0.707107 - 14.6559i$ of defining polynomial
Character	$\chi$	$=$	414.91
Dual form			414.5.b.b.91.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{2} +8.00000 q^{4} -33.4782i q^{5} +19.6774i q^{7} -22.6274 q^{8} +O(q^{10})$	\(q-2.82843 q^{2} +8.00000 q^{4} -33.4782i q^{5} +19.6774i q^{7} -22.6274 q^{8} +94.6906i q^{10} -54.0138i q^{11} -159.682 q^{13} -55.6561i q^{14} +64.0000 q^{16} -77.2301i q^{17} +379.657i q^{19} -267.826i q^{20} +152.774i q^{22} +(-142.258 + 509.513i) q^{23} -495.789 q^{25} +451.648 q^{26} +157.419i q^{28} -1624.58 q^{29} +651.991 q^{31} -181.019 q^{32} +218.440i q^{34} +658.763 q^{35} -974.907i q^{37} -1073.83i q^{38} +757.525i q^{40} +171.454 q^{41} +531.633i q^{43} -432.110i q^{44} +(402.367 - 1441.12i) q^{46} +2572.53 q^{47} +2013.80 q^{49} +1402.30 q^{50} -1277.45 q^{52} +2079.78i q^{53} -1808.28 q^{55} -445.248i q^{56} +4595.01 q^{58} -8.57641 q^{59} -1060.41i q^{61} -1844.11 q^{62} +512.000 q^{64} +5345.85i q^{65} -4746.67i q^{67} -617.840i q^{68} -1863.26 q^{70} -1386.31 q^{71} -2617.74 q^{73} +2757.45i q^{74} +3037.26i q^{76} +1062.85 q^{77} +3921.65i q^{79} -2142.60i q^{80} -484.944 q^{82} +10557.1i q^{83} -2585.52 q^{85} -1503.68i q^{86} +1222.19i q^{88} +11318.8i q^{89} -3142.12i q^{91} +(-1138.07 + 4076.10i) q^{92} -7276.22 q^{94} +12710.2 q^{95} +2774.76i q^{97} -5695.89 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	33.4782i		−	1.33913i	−0.742755	−	0.669564i	$-0.766481\pi$
$5$		−	33.4782i		−	1.33913i	0.742755	−	0.669564i	$-0.233519\pi$
$6$		0			0
$7$			19.6774i			0.401579i	0.979634	+	0.200790i	$0.0643508\pi$
$7$			19.6774i			0.401579i	−0.979634	+	0.200790i	$0.935649\pi$
$8$	−22.6274			−0.353553
$9$		0			0
$10$			94.6906i			0.946906i
$11$		−	54.0138i		−	0.446395i	−0.974773	−	0.223198i	$-0.928350\pi$
$11$		−	54.0138i		−	0.446395i	0.974773	−	0.223198i	$-0.0716495\pi$
$12$		0			0
$13$	−159.682			−0.944862			−0.472431	−	0.881368i	$-0.656623\pi$
$13$	−159.682			−0.944862			−0.472431	+	0.881368i	$0.656623\pi$
$14$		−	55.6561i		−	0.283959i
$15$		0			0
$16$	64.0000			0.250000
$17$		−	77.2301i		−	0.267232i	−0.991033	−	0.133616i	$-0.957341\pi$
$17$		−	77.2301i		−	0.267232i	0.991033	−	0.133616i	$-0.0426589\pi$
$18$		0			0
$19$			379.657i			1.05168i	0.850583	+	0.525841i	$0.176249\pi$
$19$			379.657i			1.05168i	−0.850583	+	0.525841i	$0.823751\pi$
$20$		−	267.826i		−	0.669564i
$21$		0			0
$22$			152.774i			0.315649i
$23$	−142.258	+	509.513i	−0.268919	+	0.963163i
$23$	−142.258	+	509.513i	−0.268919	+	0.963163i
$24$		0			0
$25$	−495.789			−0.793263
$26$	451.648			0.668118
$27$		0			0
$28$			157.419i			0.200790i
$29$	−1624.58			−1.93173			−0.965863	−	0.259052i	$-0.916590\pi$
$29$	−1624.58			−1.93173			−0.965863	+	0.259052i	$0.916590\pi$
$30$		0			0
$31$	651.991			0.678451			0.339225	−	0.940705i	$-0.389835\pi$
$31$	651.991			0.678451			0.339225	+	0.940705i	$0.389835\pi$
$32$	−181.019			−0.176777
$33$		0			0
$34$			218.440i			0.188962i
$35$	658.763			0.537766
$36$		0			0
$37$		−	974.907i		−	0.712131i	−0.934461	−	0.356065i	$-0.884118\pi$
$37$		−	974.907i		−	0.712131i	0.934461	−	0.356065i	$-0.115882\pi$
$38$		−	1073.83i		−	0.743652i
$39$		0			0
$40$			757.525i			0.473453i
$41$	171.454			0.101995			0.0509975	−	0.998699i	$-0.483760\pi$
$41$	171.454			0.101995			0.0509975	+	0.998699i	$0.483760\pi$
$42$		0			0
$43$			531.633i			0.287524i	0.989612	+	0.143762i	$0.0459201\pi$
$43$			531.633i			0.287524i	−0.989612	+	0.143762i	$0.954080\pi$
$44$		−	432.110i		−	0.223198i
$45$		0			0
$46$	402.367	−	1441.12i	0.190155	−	0.681059i
$47$	2572.53			1.16457			0.582284	−	0.812985i	$-0.302159\pi$
$47$	2572.53			1.16457			0.582284	+	0.812985i	$0.302159\pi$
$48$		0			0
$49$	2013.80			0.838734
$50$	1402.30			0.560921
$51$		0			0
$52$	−1277.45			−0.472431
$53$			2079.78i			0.740397i	0.928953	+	0.370199i	$0.120710\pi$
$53$			2079.78i			0.740397i	−0.928953	+	0.370199i	$0.879290\pi$
$54$		0			0
$55$	−1808.28			−0.597780
$56$		−	445.248i		−	0.141980i
$57$		0			0
$58$	4595.01			1.36594
$59$	−8.57641			−0.00246378			−0.00123189	−	0.999999i	$-0.500392\pi$
$59$	−8.57641			−0.00246378			−0.00123189	+	0.999999i	$0.500392\pi$
$60$		0			0
$61$		−	1060.41i		−	0.284981i	−0.989796	−	0.142490i	$-0.954489\pi$
$61$		−	1060.41i		−	0.284981i	0.989796	−	0.142490i	$-0.0455110\pi$
$62$	−1844.11			−0.479737
$63$		0			0
$64$	512.000			0.125000
$65$			5345.85i			1.26529i
$66$		0			0
$67$		−	4746.67i		−	1.05740i	−0.848809	−	0.528700i	$-0.822680\pi$
$67$		−	4746.67i		−	1.05740i	0.848809	−	0.528700i	$-0.177320\pi$
$68$		−	617.840i		−	0.133616i
$69$		0			0
$70$	−1863.26			−0.380258
$71$	−1386.31			−0.275007			−0.137503	−	0.990501i	$-0.543908\pi$
$71$	−1386.31			−0.275007			−0.137503	+	0.990501i	$0.543908\pi$
$72$		0			0
$73$	−2617.74			−0.491225			−0.245613	−	0.969368i	$-0.578989\pi$
$73$	−2617.74			−0.491225			−0.245613	+	0.969368i	$0.578989\pi$
$74$			2757.45i			0.503553i
$75$		0			0
$76$			3037.26i			0.525841i
$77$	1062.85			0.179263
$78$		0			0
$79$			3921.65i			0.628368i	0.949362	+	0.314184i	$0.101731\pi$
$79$			3921.65i			0.628368i	−0.949362	+	0.314184i	$0.898269\pi$
$80$		−	2142.60i		−	0.334782i
$81$		0			0
$82$	−484.944			−0.0721214
$83$			10557.1i			1.53245i	0.642571	+	0.766226i	$0.277868\pi$
$83$			10557.1i			1.53245i	−0.642571	+	0.766226i	$0.722132\pi$
$84$		0			0
$85$	−2585.52			−0.357858
$86$		−	1503.68i		−	0.203311i
$87$		0			0
$88$			1222.19i			0.157825i
$89$			11318.8i			1.42896i	0.699655	+	0.714481i	$0.253337\pi$
$89$			11318.8i			1.42896i	−0.699655	+	0.714481i	$0.746663\pi$
$90$		0			0
$91$		−	3142.12i		−	0.379437i
$92$	−1138.07	+	4076.10i	−0.134460	+	0.481581i
$93$		0			0
$94$	−7276.22			−0.823474
$95$	12710.2			1.40834
$96$		0			0
$97$			2774.76i			0.294905i	0.989069	+	0.147453i	$0.0471074\pi$
$97$			2774.76i			0.294905i	−0.989069	+	0.147453i	$0.952893\pi$
$98$	−5695.89			−0.593075
$99$		0			0
$100$	−3966.31			−0.396631
$101$	3202.26			0.313916			0.156958	−	0.987605i	$-0.449831\pi$
$101$	3202.26			0.313916			0.156958	+	0.987605i	$0.449831\pi$
$102$		0			0
$103$			14013.2i			1.32087i	0.750881	+	0.660437i	$0.229629\pi$
$103$			14013.2i			1.32087i	−0.750881	+	0.660437i	$0.770371\pi$
$104$	3613.18			0.334059
$105$		0			0
$106$		−	5882.49i		−	0.523540i
$107$			19115.3i			1.66961i	0.550548	+	0.834804i	$0.314419\pi$
$107$			19115.3i			1.66961i	−0.550548	+	0.834804i	$0.685581\pi$
$108$		0			0
$109$			11486.1i			0.966764i	0.875409	+	0.483382i	$0.160592\pi$
$109$			11486.1i			0.966764i	−0.875409	+	0.483382i	$0.839408\pi$
$110$	5114.60			0.422694
$111$		0			0
$112$			1259.35i			0.100395i
$113$			16746.6i			1.31150i	0.754977	+	0.655751i	$0.227648\pi$
$113$			16746.6i			1.31150i	−0.754977	+	0.655751i	$0.772352\pi$
$114$		0			0
$115$	17057.6	+	4762.55i	1.28980	+	0.360117i
$116$	−12996.7			−0.965863
$117$		0			0
$118$	24.2578			0.00174215
$119$	1519.69			0.107315
$120$		0			0
$121$	11723.5			0.800731
$122$			2999.30i			0.201512i
$123$		0			0
$124$	5215.93			0.339225
$125$		−	4325.75i		−	0.276848i
$126$		0			0
$127$	6576.24			0.407728			0.203864	−	0.978999i	$-0.434650\pi$
$127$	6576.24			0.407728			0.203864	+	0.978999i	$0.434650\pi$
$128$	−1448.15			−0.0883883
$129$		0			0
$130$		−	15120.4i		−	0.894695i
$131$	7041.13			0.410298			0.205149	−	0.978731i	$-0.434232\pi$
$131$	7041.13			0.410298			0.205149	+	0.978731i	$0.434232\pi$
$132$		0			0
$133$	−7470.66			−0.422334
$134$			13425.6i			0.747695i
$135$		0			0
$136$			1747.52i			0.0944808i
$137$		−	26091.9i		−	1.39016i	−0.718932	−	0.695080i	$-0.755369\pi$
$137$		−	26091.9i		−	1.39016i	0.718932	−	0.695080i	$-0.244631\pi$
$138$		0			0
$139$	1401.19			0.0725216			0.0362608	−	0.999342i	$-0.488455\pi$
$139$	1401.19			0.0725216			0.0362608	+	0.999342i	$0.488455\pi$
$140$	5270.11			0.268883
$141$		0			0
$142$	3921.07			0.194459
$143$			8625.01i			0.421782i
$144$		0			0
$145$			54388.1i			2.58683i
$146$	7404.08			0.347349
$147$		0			0
$148$		−	7799.26i		−	0.356065i
$149$		−	16105.3i		−	0.725433i	−0.931899	−	0.362717i	$-0.881849\pi$
$149$		−	16105.3i		−	0.725433i	0.931899	−	0.362717i	$-0.118151\pi$
$150$		0			0
$151$	−5562.22			−0.243946			−0.121973	−	0.992533i	$-0.538922\pi$
$151$	−5562.22			−0.243946			−0.121973	+	0.992533i	$0.538922\pi$
$152$		−	8590.67i		−	0.371826i
$153$		0			0
$154$	−3006.20			−0.126758
$155$		−	21827.5i		−	0.908532i
$156$		0			0
$157$			41229.4i			1.67266i	0.548227	+	0.836330i	$0.315303\pi$
$157$			41229.4i			1.67266i	−0.548227	+	0.836330i	$0.684697\pi$
$158$		−	11092.1i		−	0.444323i
$159$		0			0
$160$			6060.20i			0.236727i
$161$	−10025.9	−	2799.27i	−0.386786	−	0.107992i
$162$		0			0
$163$	38706.4			1.45683			0.728413	−	0.685138i	$-0.240258\pi$
$163$	38706.4			1.45683			0.728413	+	0.685138i	$0.240258\pi$
$164$	1371.63			0.0509975
$165$		0			0
$166$		−	29859.9i		−	1.08361i
$167$	38699.6			1.38763			0.693815	−	0.720153i	$-0.255929\pi$
$167$	38699.6			1.38763			0.693815	+	0.720153i	$0.255929\pi$
$168$		0			0
$169$	−3062.78			−0.107236
$170$	7312.96			0.253044
$171$		0			0
$172$			4253.06i			0.143762i
$173$	−30765.0			−1.02793			−0.513966	−	0.857811i	$-0.671824\pi$
$173$	−30765.0			−1.02793			−0.513966	+	0.857811i	$0.671824\pi$
$174$		0			0
$175$		−	9755.83i		−	0.318558i
$176$		−	3456.88i		−	0.111599i
$177$		0			0
$178$		−	32014.4i		−	1.01043i
$179$	−39384.6			−1.22920			−0.614598	−	0.788841i	$-0.710682\pi$
$179$	−39384.6			−1.22920			−0.614598	+	0.788841i	$0.710682\pi$
$180$		0			0
$181$			32728.1i			0.998997i	0.866315	+	0.499499i	$0.166482\pi$
$181$			32728.1i			0.998997i	−0.866315	+	0.499499i	$0.833518\pi$
$182$			8887.25i			0.268302i
$183$		0			0
$184$	3218.94	−	11529.0i	0.0950773	−	0.340529i
$185$	−32638.1			−0.953634
$186$		0			0
$187$	−4171.49			−0.119291
$188$	20580.3			0.582284
$189$		0			0
$190$	−35950.0			−0.995845
$191$		−	22931.5i		−	0.628588i	−0.949326	−	0.314294i	$-0.898232\pi$
$191$		−	22931.5i		−	0.628588i	0.949326	−	0.314294i	$-0.101768\pi$
$192$		0			0
$193$	−25528.5			−0.685347			−0.342674	−	0.939454i	$-0.611333\pi$
$193$	−25528.5			−0.685347			−0.342674	+	0.939454i	$0.611333\pi$
$194$		−	7848.21i		−	0.208529i
$195$		0			0
$196$	16110.4			0.419367
$197$	−47220.8			−1.21675			−0.608374	−	0.793651i	$-0.708178\pi$
$197$	−47220.8			−1.21675			−0.608374	+	0.793651i	$0.708178\pi$
$198$		0			0
$199$		−	43028.7i		−	1.08655i	−0.839553	−	0.543277i	$-0.817183\pi$
$199$		−	43028.7i		−	1.08655i	0.839553	−	0.543277i	$-0.182817\pi$
$200$	11218.4			0.280461
$201$		0			0
$202$	−9057.36			−0.221972
$203$		−	31967.5i		−	0.775742i
$204$		0			0
$205$		−	5739.96i		−	0.136584i
$206$		−	39635.2i		−	0.933999i
$207$		0			0
$208$	−10219.6			−0.236215
$209$	20506.7			0.469466
$210$		0			0
$211$	−37106.5			−0.833462			−0.416731	−	0.909030i	$-0.636824\pi$
$211$	−37106.5			−0.833462			−0.416731	+	0.909030i	$0.636824\pi$
$212$			16638.2i			0.370199i
$213$		0			0
$214$		−	54066.3i		−	1.18059i
$215$	17798.1			0.385032
$216$		0			0
$217$			12829.5i			0.272452i
$218$		−	32487.7i		−	0.683606i
$219$		0			0
$220$	−14466.3			−0.298890
$221$			12332.2i			0.252497i
$222$		0			0
$223$	−79480.0			−1.59826			−0.799131	−	0.601157i	$-0.794707\pi$
$223$	−79480.0			−1.59826			−0.799131	+	0.601157i	$0.794707\pi$
$224$		−	3561.99i		−	0.0709899i
$225$		0			0
$226$		−	47366.5i		−	0.927372i
$227$			74663.7i			1.44896i	0.689293	+	0.724482i	$0.257921\pi$
$227$			74663.7i			1.44896i	−0.689293	+	0.724482i	$0.742079\pi$
$228$		0			0
$229$		−	32658.3i		−	0.622763i	−0.950285	−	0.311381i	$-0.899208\pi$
$229$		−	32658.3i		−	0.622763i	0.950285	−	0.311381i	$-0.100792\pi$
$230$	−48246.1	−	13470.5i	−0.912025	−	0.254641i
$231$		0			0
$232$	36760.1			0.682969
$233$	−19403.8			−0.357417			−0.178708	−	0.983902i	$-0.557192\pi$
$233$	−19403.8			−0.357417			−0.178708	+	0.983902i	$0.557192\pi$
$234$		0			0
$235$		−	86123.7i		−	1.55951i
$236$	−68.6113			−0.00123189
$237$		0			0
$238$	−4298.32			−0.0758831
$239$	74488.3			1.30404			0.652022	−	0.758200i	$-0.273921\pi$
$239$	74488.3			1.30404			0.652022	+	0.758200i	$0.273921\pi$
$240$		0			0
$241$		−	30092.6i		−	0.518114i	−0.965862	−	0.259057i	$-0.916588\pi$
$241$		−	30092.6i		−	0.518114i	0.965862	−	0.259057i	$-0.0834117\pi$
$242$	−33159.1			−0.566203
$243$		0			0
$244$		−	8483.31i		−	0.142490i
$245$		−	67418.4i		−	1.12317i
$246$		0			0
$247$		−	60624.3i		−	0.993694i
$248$	−14752.9			−0.239869
$249$		0			0
$250$			12235.1i			0.195761i
$251$			63697.7i			1.01106i	0.862809	+	0.505530i	$0.168703\pi$
$251$			63697.7i			1.01106i	−0.862809	+	0.505530i	$0.831297\pi$
$252$		0			0
$253$	27520.7	+	7683.91i	0.429951	+	0.120044i
$254$	−18600.4			−0.288307
$255$		0			0
$256$	4096.00			0.0625000
$257$	70085.0			1.06111			0.530553	−	0.847652i	$-0.321984\pi$
$257$	70085.0			1.06111			0.530553	+	0.847652i	$0.321984\pi$
$258$		0			0
$259$	19183.6			0.285977
$260$			42766.8i			0.632645i
$261$		0			0
$262$	−19915.3			−0.290125
$263$			23606.9i			0.341293i	0.985332	+	0.170647i	$0.0545857\pi$
$263$			23606.9i			0.341293i	−0.985332	+	0.170647i	$0.945414\pi$
$264$		0			0
$265$	69627.1			0.991486
$266$	21130.2			0.298635
$267$		0			0
$268$		−	37973.3i		−	0.528700i
$269$	12740.0			0.176062			0.0880308	−	0.996118i	$-0.471943\pi$
$269$	12740.0			0.176062			0.0880308	+	0.996118i	$0.471943\pi$
$270$		0			0
$271$	455.015			0.00619565			0.00309783	−	0.999995i	$-0.499014\pi$
$271$	455.015			0.00619565			0.00309783	+	0.999995i	$0.499014\pi$
$272$		−	4942.72i		−	0.0668080i
$273$		0			0
$274$			73799.1i			0.982992i
$275$			26779.5i			0.354109i
$276$		0			0
$277$	46365.1			0.604271			0.302136	−	0.953265i	$-0.402300\pi$
$277$	46365.1			0.604271			0.302136	+	0.953265i	$0.402300\pi$
$278$	−3963.16			−0.0512805
$279$		0			0
$280$	−14906.1			−0.190129
$281$		−	117828.i		−	1.49223i	−0.665819	−	0.746113i	$-0.731918\pi$
$281$		−	117828.i		−	1.49223i	0.665819	−	0.746113i	$-0.268082\pi$
$282$		0			0
$283$		−	32448.1i		−	0.405151i	−0.979267	−	0.202575i	$-0.935069\pi$
$283$		−	32448.1i		−	0.405151i	0.979267	−	0.202575i	$-0.0649311\pi$
$284$	−11090.5			−0.137503
$285$		0			0
$286$		−	24395.2i		−	0.298245i
$287$			3373.76i			0.0409591i
$288$		0			0
$289$	77556.5			0.928587
$290$		−	153833.i		−	1.82916i
$291$		0			0
$292$	−20941.9			−0.245613
$293$		−	88859.5i		−	1.03507i	−0.855663	−	0.517534i	$-0.826850\pi$
$293$		−	88859.5i		−	1.03507i	0.855663	−	0.517534i	$-0.173150\pi$
$294$		0			0
$295$			287.123i			0.00329931i
$296$			22059.6i			0.251776i
$297$		0			0
$298$			45552.8i			0.512959i
$299$	22716.0	−	81359.9i	0.254091	−	0.910056i
$300$		0			0
$301$	−10461.1			−0.115464
$302$	15732.3			0.172496
$303$		0			0
$304$			24298.1i			0.262921i
$305$	−35500.7			−0.381626
$306$		0			0
$307$	−20919.2			−0.221957			−0.110978	−	0.993823i	$-0.535398\pi$
$307$	−20919.2			−0.221957			−0.110978	+	0.993823i	$0.535398\pi$
$308$	8502.80			0.0896315
$309$		0			0
$310$			61737.4i			0.642429i
$311$	−29482.6			−0.304821			−0.152411	−	0.988317i	$-0.548704\pi$
$311$	−29482.6			−0.304821			−0.152411	+	0.988317i	$0.548704\pi$
$312$		0			0
$313$			80312.8i			0.819777i	0.912136	+	0.409889i	$0.134432\pi$
$313$			80312.8i			0.819777i	−0.912136	+	0.409889i	$0.865568\pi$
$314$		−	116614.i		−	1.18275i
$315$		0			0
$316$			31373.2i			0.314184i
$317$	24797.0			0.246763			0.123381	−	0.992359i	$-0.460626\pi$
$317$	24797.0			0.246763			0.123381	+	0.992359i	$0.460626\pi$
$318$		0			0
$319$			87749.9i			0.862314i
$320$		−	17140.8i		−	0.167391i
$321$		0			0
$322$	28357.5	+	7917.53i	0.273499	+	0.0763621i
$323$	29321.0			0.281043
$324$		0			0
$325$	79168.4			0.749523
$326$	−109478.			−1.03013
$327$		0			0
$328$	−3879.55			−0.0360607
$329$			50620.7i			0.467667i
$330$		0			0
$331$	−127053.			−1.15966			−0.579830	−	0.814738i	$-0.696881\pi$
$331$	−127053.			−1.15966			−0.579830	+	0.814738i	$0.696881\pi$
$332$			84456.5i			0.766226i
$333$		0			0
$334$	−109459.			−0.981203
$335$	−158910.			−1.41599
$336$		0			0
$337$			70632.6i			0.621935i	0.950420	+	0.310968i	$0.100653\pi$
$337$			70632.6i			0.621935i	−0.950420	+	0.310968i	$0.899347\pi$
$338$	8662.85			0.0758276
$339$		0			0
$340$	−20684.2			−0.178929
$341$		−	35216.5i		−	0.302857i
$342$		0			0
$343$			86871.7i			0.738398i
$344$		−	12029.5i		−	0.101655i
$345$		0			0
$346$	87016.5			0.726857
$347$	−191595.			−1.59120			−0.795602	−	0.605819i	$-0.792845\pi$
$347$	−191595.			−1.59120			−0.795602	+	0.605819i	$0.792845\pi$
$348$		0			0
$349$	−145705.			−1.19625			−0.598126	−	0.801402i	$-0.704088\pi$
$349$	−145705.			−1.19625			−0.598126	+	0.801402i	$0.704088\pi$
$350$			27593.7i			0.225254i
$351$		0			0
$352$			9777.54i			0.0789123i
$353$	68772.2			0.551904			0.275952	−	0.961171i	$-0.411007\pi$
$353$	68772.2			0.551904			0.275952	+	0.961171i	$0.411007\pi$
$354$		0			0
$355$			46411.1i			0.368269i
$356$			90550.4i			0.714481i
$357$		0			0
$358$	111397.			0.869172
$359$			133581.i			1.03647i	0.855239	+	0.518234i	$0.173410\pi$
$359$			133581.i			1.03647i	−0.855239	+	0.518234i	$0.826590\pi$
$360$		0			0
$361$	−13818.7			−0.106036
$362$		−	92569.2i		−	0.706398i
$363$		0			0
$364$		−	25136.9i		−	0.189718i
$365$			87637.1i			0.657813i
$366$		0			0
$367$		−	194975.i		−	1.44760i	−0.690012	−	0.723798i	$-0.742395\pi$
$367$		−	194975.i		−	1.44760i	0.690012	−	0.723798i	$-0.257605\pi$
$368$	−9104.53	+	32608.8i	−0.0672298	+	0.240791i
$369$		0			0
$370$	92314.6			0.674321
$371$	−40924.5			−0.297328
$372$		0			0
$373$			245451.i			1.76419i	0.471067	+	0.882097i	$0.343869\pi$
$373$			245451.i			1.76419i	−0.471067	+	0.882097i	$0.656131\pi$
$374$	11798.8			0.0843515
$375$		0			0
$376$	−58209.7			−0.411737
$377$	259416.			1.82521
$378$		0			0
$379$		−	211323.i		−	1.47119i	−0.677422	−	0.735594i	$-0.736903\pi$
$379$		−	211323.i		−	1.47119i	0.677422	−	0.735594i	$-0.263097\pi$
$380$	101682.			0.704168
$381$		0			0
$382$			64860.1i			0.444479i
$383$		−	172464.i		−	1.17571i	−0.808965	−	0.587857i	$-0.799972\pi$
$383$		−	172464.i		−	1.17571i	0.808965	−	0.587857i	$-0.200028\pi$
$384$		0			0
$385$		−	35582.3i		−	0.240056i
$386$	72205.5			0.484614
$387$		0			0
$388$			22198.1i			0.147453i
$389$			140755.i			0.930177i	0.885264	+	0.465088i	$0.153977\pi$
$389$			140755.i			0.930177i	−0.885264	+	0.465088i	$0.846023\pi$
$390$		0			0
$391$	39349.7	+	10986.6i	0.257388	+	0.0718638i
$392$	−45567.1			−0.296537
$393$		0			0
$394$	133560.			0.860370
$395$	131290.			0.841465
$396$		0			0
$397$	−302137.			−1.91700			−0.958502	−	0.285085i	$-0.907978\pi$
$397$	−302137.			−1.91700			−0.958502	+	0.285085i	$0.907978\pi$
$398$			121703.i			0.768310i
$399$		0			0
$400$	−31730.5			−0.198316
$401$			251342.i			1.56306i	0.623865	+	0.781532i	$0.285561\pi$
$401$			251342.i			1.56306i	−0.623865	+	0.781532i	$0.714439\pi$
$402$		0			0
$403$	−104111.			−0.641042
$404$	25618.1			0.156958
$405$		0			0
$406$			90417.8i			0.548532i
$407$	−52658.4			−0.317892
$408$		0			0
$409$	118085.			0.705909			0.352954	−	0.935641i	$-0.385177\pi$
$409$	118085.			0.705909			0.352954	+	0.935641i	$0.385177\pi$
$410$			16235.1i			0.0965797i
$411$		0			0
$412$			112105.i			0.660437i
$413$		−	168.761i		−	0.000989402i
$414$		0			0
$415$	353431.			2.05215
$416$	28905.5			0.167030
$417$		0			0
$418$	−58001.8			−0.331963
$419$			102383.i			0.583176i	0.956544	+	0.291588i	$0.0941837\pi$
$419$			102383.i			0.583176i	−0.956544	+	0.291588i	$0.905816\pi$
$420$		0			0
$421$			13174.0i			0.0743283i	0.999309	+	0.0371641i	$0.0118324\pi$
$421$			13174.0i			0.0743283i	−0.999309	+	0.0371641i	$0.988168\pi$
$422$	104953.			0.589346
$423$		0			0
$424$		−	47059.9i		−	0.261770i
$425$			38289.8i			0.211985i
$426$		0			0
$427$	20866.2			0.114442
$428$			152923.i			0.834804i
$429$		0			0
$430$	−50340.6			−0.272259
$431$		−	83944.7i		−	0.451896i	−0.974139	−	0.225948i	$-0.927452\pi$
$431$		−	83944.7i		−	0.451896i	0.974139	−	0.225948i	$-0.0725480\pi$
$432$		0			0
$433$			57692.2i			0.307710i	0.988093	+	0.153855i	$0.0491689\pi$
$433$			57692.2i			0.307710i	−0.988093	+	0.153855i	$0.950831\pi$
$434$		−	36287.3i		−	0.192652i
$435$		0			0
$436$			91889.0i			0.483382i
$437$	−193440.	−	54009.4i	−1.01294	−	0.282818i
$438$		0			0
$439$	−182534.			−0.947140			−0.473570	−	0.880756i	$-0.657035\pi$
$439$	−182534.			−0.947140			−0.473570	+	0.880756i	$0.657035\pi$
$440$	40916.8			0.211347
$441$		0			0
$442$		−	34880.8i		−	0.178543i
$443$	−216357.			−1.10246			−0.551230	−	0.834353i	$-0.685841\pi$
$443$	−216357.			−1.10246			−0.551230	+	0.834353i	$0.685841\pi$
$444$		0			0
$445$	378933.			1.91356
$446$	224803.			1.13014
$447$		0			0
$448$			10074.8i			0.0501974i
$449$	368789.			1.82930			0.914651	−	0.404244i	$-0.132466\pi$
$449$	368789.			1.82930			0.914651	+	0.404244i	$0.132466\pi$
$450$		0			0
$451$		−	9260.86i		−	0.0455301i
$452$			133973.i			0.655751i
$453$		0			0
$454$		−	211181.i		−	1.02457i
$455$	−105192.			−0.508114
$456$		0			0
$457$		−	154736.i		−	0.740897i	−0.928853	−	0.370448i	$-0.879204\pi$
$457$		−	154736.i		−	0.740897i	0.928853	−	0.370448i	$-0.120796\pi$
$458$			92371.6i			0.440360i
$459$		0			0
$460$	136461.	+	38100.4i	0.644899	+	0.180059i
$461$	−138326.			−0.650880			−0.325440	−	0.945563i	$-0.605512\pi$
$461$	−138326.			−0.650880			−0.325440	+	0.945563i	$0.605512\pi$
$462$		0			0
$463$	48659.5			0.226989			0.113495	−	0.993539i	$-0.463795\pi$
$463$	48659.5			0.226989			0.113495	+	0.993539i	$0.463795\pi$
$464$	−103973.			−0.482932
$465$		0			0
$466$	54882.2			0.252732
$467$			273257.i			1.25296i	0.779437	+	0.626480i	$0.215505\pi$
$467$			273257.i			1.25296i	−0.779437	+	0.626480i	$0.784495\pi$
$468$		0			0
$469$	93402.0			0.424630
$470$			243595.i			1.10274i
$471$		0			0
$472$	194.062			0.000871077
$473$	28715.5			0.128350
$474$		0			0
$475$		−	188230.i		−	0.834260i
$476$	12157.5			0.0536574
$477$		0			0
$478$	−210685.			−0.922098
$479$		−	206455.i		−	0.899816i	−0.893075	−	0.449908i	$-0.851457\pi$
$479$		−	206455.i		−	0.899816i	0.893075	−	0.449908i	$-0.148543\pi$
$480$		0			0
$481$			155675.i			0.672865i
$482$			85114.6i			0.366362i
$483$		0			0
$484$	93788.1			0.400366
$485$	92894.0			0.394916
$486$		0			0
$487$	−119940.			−0.505716			−0.252858	−	0.967503i	$-0.581371\pi$
$487$	−119940.			−0.505716			−0.252858	+	0.967503i	$0.581371\pi$
$488$			23994.4i			0.100756i
$489$		0			0
$490$			190688.i			0.794202i
$491$	103912.			0.431024			0.215512	−	0.976501i	$-0.430858\pi$
$491$	103912.			0.431024			0.215512	+	0.976501i	$0.430858\pi$
$492$		0			0
$493$			125467.i			0.516219i
$494$			171471.i			0.702648i
$495$		0			0
$496$	41727.4			0.169613
$497$		−	27278.9i		−	0.110437i
$498$		0			0
$499$	−454701.			−1.82610			−0.913050	−	0.407848i	$-0.866279\pi$
$499$	−454701.			−1.82610			−0.913050	+	0.407848i	$0.866279\pi$
$500$		−	34606.0i		−	0.138424i
$501$		0			0
$502$		−	180164.i		−	0.714927i
$503$			154319.i			0.609934i	0.952363	+	0.304967i	$0.0986454\pi$
$503$			154319.i			0.609934i	−0.952363	+	0.304967i	$0.901355\pi$
$504$		0			0
$505$		−	107206.i		−	0.420374i
$506$	−77840.4	−	21733.4i	−0.304021	−	0.0848841i
$507$		0			0
$508$	52610.0			0.203864
$509$	−384796.			−1.48523			−0.742617	−	0.669717i	$-0.766415\pi$
$509$	−384796.			−1.48523			−0.742617	+	0.669717i	$0.766415\pi$
$510$		0			0
$511$		−	51510.2i		−	0.197266i
$512$	−11585.2			−0.0441942
$513$		0			0
$514$	−198230.			−0.750315
$515$	469135.			1.76882
$516$		0			0
$517$		−	138952.i		−	0.519858i
$518$	−54259.5			−0.202216
$519$		0			0
$520$		−	120963.i		−	0.447348i
$521$		−	45060.9i		−	0.166006i	−0.996549	−	0.0830031i	$-0.973549\pi$
$521$		−	45060.9i		−	0.166006i	0.996549	−	0.0830031i	$-0.0264511\pi$
$522$		0			0
$523$		−	152841.i		−	0.558774i	−0.960179	−	0.279387i	$-0.909869\pi$
$523$		−	152841.i		−	0.558774i	0.960179	−	0.279387i	$-0.0901313\pi$
$524$	56329.0			0.205149
$525$		0			0
$526$		−	66770.5i		−	0.241331i
$527$		−	50353.3i		−	0.181304i
$528$		0			0
$529$	−239366.	−	144965.i	−0.855365	−	0.518026i
$530$	−196935.			−0.701086
$531$		0			0
$532$	−59765.3			−0.211167
$533$	−27378.0			−0.0963712
$534$		0			0
$535$	639947.			2.23582
$536$			107405.i			0.373847i
$537$		0			0
$538$	−36034.1			−0.124494
$539$		−	108773.i		−	0.374407i
$540$		0			0
$541$	−538523.			−1.83997			−0.919983	−	0.391958i	$-0.871798\pi$
$541$	−538523.			−1.83997			−0.919983	+	0.391958i	$0.871798\pi$
$542$	−1286.98			−0.00438099
$543$		0			0
$544$			13980.1i			0.0472404i
$545$	384535.			1.29462
$546$		0			0
$547$	−392350.			−1.31129			−0.655645	−	0.755069i	$-0.727603\pi$
$547$	−392350.			−1.31129			−0.655645	+	0.755069i	$0.727603\pi$
$548$		−	208735.i		−	0.695080i
$549$		0			0
$550$		−	75743.8i		−	0.250393i
$551$		−	616785.i		−	2.03156i
$552$		0			0
$553$	−77167.7			−0.252340
$554$	−131140.			−0.427284
$555$		0			0
$556$	11209.5			0.0362608
$557$			558524.i			1.80025i	0.435637	+	0.900123i	$0.356523\pi$
$557$			558524.i			1.80025i	−0.435637	+	0.900123i	$0.643477\pi$
$558$		0			0
$559$		−	84892.0i		−	0.271671i
$560$	42160.8			0.134441
$561$		0			0
$562$			333267.i			1.05516i
$563$		−	417048.i		−	1.31574i	−0.753132	−	0.657869i	$-0.771458\pi$
$563$		−	417048.i		−	1.31574i	0.753132	−	0.657869i	$-0.228542\pi$
$564$		0			0
$565$	560645.			1.75627
$566$			91777.1i			0.286485i
$567$		0			0
$568$	31368.6			0.0972296
$569$		−	56795.6i		−	0.175425i	−0.996146	−	0.0877123i	$-0.972044\pi$
$569$		−	56795.6i		−	0.175425i	0.996146	−	0.0877123i	$-0.0279556\pi$
$570$		0			0
$571$			281453.i			0.863243i	0.902055	+	0.431621i	$0.142058\pi$
$571$			281453.i			0.863243i	−0.902055	+	0.431621i	$0.857942\pi$
$572$			69000.1i			0.210891i
$573$		0			0
$574$		−	9542.43i		−	0.0289625i
$575$	70530.1	−	252611.i	0.213324	−	0.764041i
$576$		0			0
$577$	491209.			1.47542			0.737709	−	0.675119i	$-0.235908\pi$
$577$	491209.			1.47542			0.737709	+	0.675119i	$0.235908\pi$
$578$	−219363.			−0.656610
$579$		0			0
$580$			435105.i			1.29341i
$581$	−207735.			−0.615401
$582$		0			0
$583$	112337.			0.330510
$584$	59232.7			0.173674
$585$		0			0
$586$			251333.i			0.731903i
$587$	169969.			0.493281			0.246640	−	0.969107i	$-0.420673\pi$
$587$	169969.			0.493281			0.246640	+	0.969107i	$0.420673\pi$
$588$		0			0
$589$			247533.i			0.713515i
$590$		−	812.106i		−	0.00233297i
$591$		0			0
$592$		−	62394.1i		−	0.178033i
$593$	−401812.			−1.14265			−0.571325	−	0.820724i	$-0.693570\pi$
$593$	−401812.			−1.14265			−0.571325	+	0.820724i	$0.693570\pi$
$594$		0			0
$595$		−	50876.3i		−	0.143708i
$596$		−	128843.i		−	0.362717i
$597$		0			0
$598$	−64250.6	+	230120.i	−0.179670	+	0.643506i
$599$	−182719.			−0.509250			−0.254625	−	0.967040i	$-0.581952\pi$
$599$	−182719.			−0.509250			−0.254625	+	0.967040i	$0.581952\pi$
$600$		0			0
$601$	281509.			0.779369			0.389685	−	0.920948i	$-0.372584\pi$
$601$	281509.			0.779369			0.389685	+	0.920948i	$0.372584\pi$
$602$	29588.6			0.0816453
$603$		0			0
$604$	−44497.7			−0.121973
$605$		−	392482.i		−	1.07228i
$606$		0			0
$607$	−422711.			−1.14727			−0.573635	−	0.819111i	$-0.694467\pi$
$607$	−422711.			−1.14727			−0.573635	+	0.819111i	$0.694467\pi$
$608$		−	68725.3i		−	0.185913i
$609$		0			0
$610$	100411.			0.269850
$611$	−410786.			−1.10036
$612$		0			0
$613$		−	359369.i		−	0.956355i	−0.878263	−	0.478177i	$-0.841298\pi$
$613$		−	359369.i		−	0.956355i	0.878263	−	0.478177i	$-0.158702\pi$
$614$	59168.5			0.156947
$615$		0			0
$616$	−24049.6			−0.0633791
$617$			343800.i			0.903099i	0.892246	+	0.451550i	$0.149129\pi$
$617$			343800.i			0.903099i	−0.892246	+	0.451550i	$0.850871\pi$
$618$		0			0
$619$		−	625411.i		−	1.63224i	−0.577882	−	0.816120i	$-0.696121\pi$
$619$		−	625411.i		−	1.63224i	0.577882	−	0.816120i	$-0.303879\pi$
$620$		−	174620.i		−	0.454266i
$621$		0			0
$622$	83389.4			0.215541
$623$	−222724.			−0.573841
$624$		0			0
$625$	−454686.			−1.16400
$626$		−	227159.i		−	0.579670i
$627$		0			0
$628$			329835.i			0.836330i
$629$	−75292.1			−0.190304
$630$		0			0
$631$			384826.i			0.966508i	0.875480	+	0.483254i	$0.160545\pi$
$631$			384826.i			0.966508i	−0.875480	+	0.483254i	$0.839455\pi$
$632$		−	88736.7i		−	0.222162i
$633$		0			0
$634$	−70136.4			−0.174488
$635$		−	220161.i		−	0.546000i
$636$		0			0
$637$	−321567.			−0.792488
$638$		−	248194.i		−	0.609748i
$639$		0			0
$640$			48481.6i			0.118363i
$641$		−	284276.i		−	0.691870i	−0.938258	−	0.345935i	$-0.887562\pi$
$641$		−	284276.i		−	0.691870i	0.938258	−	0.345935i	$-0.112438\pi$
$642$		0			0
$643$			416229.i			1.00672i	0.864076	+	0.503361i	$0.167904\pi$
$643$			416229.i			1.00672i	−0.864076	+	0.503361i	$0.832096\pi$
$644$	−80207.1	−	22394.2i	−0.193393	−	0.0539962i
$645$		0			0
$646$	−82932.2			−0.198728
$647$	501651.			1.19838			0.599188	−	0.800609i	$-0.295490\pi$
$647$	501651.			1.19838			0.599188	+	0.800609i	$0.295490\pi$
$648$		0			0
$649$			463.245i			0.00109982i
$650$	−223922.			−0.529993
$651$		0			0
$652$	309651.			0.728413
$653$	243366.			0.570734			0.285367	−	0.958418i	$-0.407884\pi$
$653$	243366.			0.570734			0.285367	+	0.958418i	$0.407884\pi$
$654$		0			0
$655$		−	235724.i		−	0.549442i
$656$	10973.0			0.0254988
$657$		0			0
$658$		−	143177.i		−	0.330690i
$659$		−	91222.5i		−	0.210054i	−0.994469	−	0.105027i	$-0.966507\pi$
$659$		−	91222.5i		−	0.210054i	0.994469	−	0.105027i	$-0.0334929\pi$
$660$		0			0
$661$			748739.i			1.71367i	0.515589	+	0.856836i	$0.327573\pi$
$661$			748739.i			1.71367i	−0.515589	+	0.856836i	$0.672427\pi$
$662$	359362.			0.820003
$663$		0			0
$664$		−	238879.i		−	0.541803i
$665$			250104.i			0.565559i
$666$		0			0
$667$	231110.	−	827746.i	0.519478	−	1.86057i
$668$	309597.			0.693815
$669$		0			0
$670$	449465.			1.00126
$671$	−57277.0			−0.127214
$672$		0			0
$673$	616620.			1.36141			0.680703	−	0.732560i	$-0.261675\pi$
$673$	616620.			1.36141			0.680703	+	0.732560i	$0.261675\pi$
$674$		−	199779.i		−	0.439775i
$675$		0			0
$676$	−24502.2			−0.0536182
$677$			823346.i			1.79641i	0.439578	+	0.898205i	$0.355128\pi$
$677$			823346.i			1.79641i	−0.439578	+	0.898205i	$0.644872\pi$
$678$		0			0
$679$	−54600.1			−0.118428
$680$	58503.7			0.126522
$681$		0			0
$682$			99607.4i			0.214152i
$683$	354815.			0.760606			0.380303	−	0.924862i	$-0.375820\pi$
$683$	354815.			0.760606			0.380303	+	0.924862i	$0.375820\pi$
$684$		0			0
$685$	−873511.			−1.86160
$686$		−	245710.i		−	0.522126i
$687$		0			0
$688$			34024.5i			0.0718811i
$689$		−	332102.i		−	0.699573i
$690$		0			0
$691$	567858.			1.18928			0.594639	−	0.803993i	$-0.297295\pi$
$691$	567858.			1.18928			0.594639	+	0.803993i	$0.297295\pi$
$692$	−246120.			−0.513966
$693$		0			0
$694$	541913.			1.12515
$695$		−	46909.3i		−	0.0971156i
$696$		0			0
$697$		−	13241.4i		−	0.0272563i
$698$	412115.			0.845878
$699$		0			0
$700$		−	78046.7i		−	0.159279i
$701$		−	190169.i		−	0.386994i	−0.981101	−	0.193497i	$-0.938017\pi$
$701$		−	190169.i		−	0.386994i	0.981101	−	0.193497i	$-0.0619830\pi$
$702$		0			0
$703$	370131.			0.748935
$704$		−	27655.1i		−	0.0557994i
$705$		0			0
$706$	−194517.			−0.390255
$707$			63012.1i			0.126062i
$708$		0			0
$709$		−	814163.i		−	1.61964i	−0.586677	−	0.809821i	$-0.699564\pi$
$709$		−	814163.i		−	1.61964i	0.586677	−	0.809821i	$-0.300436\pi$
$710$		−	131270.i		−	0.260406i
$711$		0			0
$712$		−	256115.i		−	0.505214i
$713$	−92751.1	+	332198.i	−0.182448	+	0.653458i
$714$		0			0
$715$	288750.			0.564819
$716$	−315077.			−0.614598
$717$		0			0
$718$		−	377824.i		−	0.732894i
$719$	−976490.			−1.88890			−0.944452	−	0.328649i	$-0.893407\pi$
$719$	−976490.			−1.88890			−0.944452	+	0.328649i	$0.893407\pi$
$720$		0			0
$721$	−275742.			−0.530436
$722$	39085.2			0.0749787
$723$		0			0
$724$			261825.i			0.499499i
$725$	805450.			1.53237
$726$		0			0
$727$		−	364607.i		−	0.689853i	−0.938630	−	0.344926i	$-0.887904\pi$
$727$		−	364607.i		−	0.689853i	0.938630	−	0.344926i	$-0.112096\pi$
$728$			71098.0i			0.134151i
$729$		0			0
$730$		−	247875.i		−	0.465144i
$731$	41058.0			0.0768357
$732$		0			0
$733$			801046.i			1.49090i	0.666560	+	0.745451i	$0.267766\pi$
$733$			801046.i			1.49090i	−0.666560	+	0.745451i	$0.732234\pi$
$734$			551473.i			1.02360i
$735$		0			0
$736$	25751.5	−	92231.7i	0.0475386	−	0.170265i
$737$	−256386.			−0.472018
$738$		0			0
$739$	−197657.			−0.361929			−0.180965	−	0.983490i	$-0.557922\pi$
$739$	−197657.			−0.361929			−0.180965	+	0.983490i	$0.557922\pi$
$740$	−261105.			−0.476817
$741$		0			0
$742$	115752.			0.210243
$743$		−	179668.i		−	0.325456i	−0.986671	−	0.162728i	$-0.947971\pi$
$743$		−	179668.i		−	0.325456i	0.986671	−	0.162728i	$-0.0520293\pi$
$744$		0			0
$745$	−539178.			−0.971448
$746$		−	694239.i		−	1.24747i
$747$		0			0
$748$	−33371.9			−0.0596455
$749$	−376140.			−0.670480
$750$		0			0
$751$			454544.i			0.805928i	0.915216	+	0.402964i	$0.132020\pi$
$751$			454544.i			0.805928i	−0.915216	+	0.402964i	$0.867980\pi$
$752$	164642.			0.291142
$753$		0			0
$754$	−733739.			−1.29062
$755$			186213.i			0.326675i
$756$		0			0
$757$		−	461212.i		−	0.804838i	−0.915456	−	0.402419i	$-0.868169\pi$
$757$		−	461212.i		−	0.804838i	0.915456	−	0.402419i	$-0.131831\pi$
$758$			597712.i			1.04029i
$759$		0			0
$760$	−287600.			−0.497922
$761$	−408299.			−0.705032			−0.352516	−	0.935806i	$-0.614674\pi$
$761$	−408299.			−0.705032			−0.352516	+	0.935806i	$0.614674\pi$
$762$		0			0
$763$	−226017.			−0.388233
$764$		−	183452.i		−	0.314294i
$765$		0			0
$766$			487803.i			0.831355i
$767$	1369.50			0.00232793
$768$		0			0
$769$		−	56576.3i		−	0.0956714i	−0.998855	−	0.0478357i	$-0.984768\pi$
$769$		−	56576.3i		−	0.0956714i	0.998855	−	0.0478357i	$-0.0152324\pi$
$770$			100642.i			0.169745i
$771$		0			0
$772$	−204228.			−0.342674
$773$			714995.i			1.19659i	0.801277	+	0.598293i	$0.204154\pi$
$773$			714995.i			1.19659i	−0.801277	+	0.598293i	$0.795846\pi$
$774$		0			0
$775$	−323250.			−0.538190
$776$		−	62785.7i		−	0.104265i
$777$		0			0
$778$		−	398116.i		−	0.657734i
$779$			65093.6i			0.107266i
$780$		0			0
$781$			74879.8i			0.122762i
$782$	−111298.	−	31074.8i	−0.182001	−	0.0508154i
$783$		0			0
$784$	128883.			0.209684
$785$	1.38028e6			2.23990
$786$		0			0
$787$		−	361359.i		−	0.583431i	−0.956505	−	0.291715i	$-0.905774\pi$
$787$		−	361359.i		−	0.583431i	0.956505	−	0.291715i	$-0.0942260\pi$
$788$	−377766.			−0.608374
$789$		0			0
$790$	−371343.			−0.595006
$791$	−329529.			−0.526672
$792$		0			0
$793$			169329.i			0.269267i
$794$	854573.			1.35553
$795$		0			0
$796$		−	344229.i		−	0.543277i
$797$			388399.i			0.611451i	0.952120	+	0.305726i	$0.0988990\pi$
$797$			388399.i			0.611451i	−0.952120	+	0.305726i	$0.901101\pi$
$798$		0			0
$799$		−	198677.i		−	0.311210i
$800$	89747.4			0.140230
$801$		0			0
$802$		−	710903.i		−	1.10525i
$803$			141394.i			0.219280i
$804$		0			0
$805$	−93714.5	+	335648.i	−0.144616	+	0.517956i
$806$	294470.			0.453285
$807$		0			0
$808$	−72458.9			−0.110986
$809$	−170967.			−0.261225			−0.130612	−	0.991434i	$-0.541694\pi$
$809$	−170967.			−0.261225			−0.130612	+	0.991434i	$0.541694\pi$
$810$		0			0
$811$	−476324.			−0.724203			−0.362102	−	0.932139i	$-0.617941\pi$
$811$	−476324.			−0.724203			−0.362102	+	0.932139i	$0.617941\pi$
$812$		−	255740.i		−	0.387871i
$813$		0			0
$814$	148941.			0.224783
$815$		−	1.29582e6i		−	1.95088i
$816$		0			0
$817$	−201838.			−0.302384
$818$	−333995.			−0.499153
$819$		0			0
$820$		−	45919.7i		−	0.0682922i
$821$	−1.06850e6			−1.58521			−0.792605	−	0.609736i	$-0.791276\pi$
$821$	−1.06850e6			−1.58521			−0.792605	+	0.609736i	$0.791276\pi$
$822$		0			0
$823$	−663222.			−0.979172			−0.489586	−	0.871955i	$-0.662852\pi$
$823$	−663222.			−0.979172			−0.489586	+	0.871955i	$0.662852\pi$
$824$		−	317082.i		−	0.467000i
$825$		0			0
$826$			477.329i			0.000699613i
$827$			285779.i			0.417850i	0.977932	+	0.208925i	$0.0669964\pi$
$827$			285779.i			0.417850i	−0.977932	+	0.208925i	$0.933004\pi$
$828$		0			0
$829$	−3536.66			−0.00514618			−0.00257309	−	0.999997i	$-0.500819\pi$
$829$	−3536.66			−0.00514618			−0.00257309	+	0.999997i	$0.500819\pi$
$830$	−999654.			−1.45109
$831$		0			0
$832$	−81757.0			−0.118108
$833$		−	155526.i		−	0.224137i
$834$		0			0
$835$		−	1.29559e6i		−	1.85821i
$836$	164054.			0.234733
$837$		0			0
$838$		−	289583.i		−	0.412368i
$839$		−	1.26209e6i		−	1.79294i	−0.443104	−	0.896470i	$-0.646123\pi$
$839$		−	1.26209e6i		−	1.79294i	0.443104	−	0.896470i	$-0.353877\pi$
$840$		0			0
$841$	1.93199e6			2.73157
$842$		−	37261.7i		−	0.0525580i
$843$		0			0
$844$	−296852.			−0.416731
$845$			102536.i			0.143603i
$846$		0			0
$847$			230688.i			0.321557i
$848$			133106.i			0.185099i
$849$		0			0
$850$		−	108300.i		−	0.149896i
$851$	496728.	+	138689.i	0.685898	+	0.191506i
$852$		0			0
$853$	−39888.6			−0.0548214			−0.0274107	−	0.999624i	$-0.508726\pi$
$853$	−39888.6			−0.0548214			−0.0274107	+	0.999624i	$0.508726\pi$
$854$	−59018.4			−0.0809230
$855$		0			0
$856$		−	432531.i		−	0.590295i
$857$	−1.06525e6			−1.45040			−0.725202	−	0.688536i	$-0.758254\pi$
$857$	−1.06525e6			−1.45040			−0.725202	+	0.688536i	$0.758254\pi$
$858$		0			0
$859$	292913.			0.396965			0.198483	−	0.980104i	$-0.436399\pi$
$859$	292913.			0.396965			0.198483	+	0.980104i	$0.436399\pi$
$860$	142385.			0.192516
$861$		0			0
$862$			237432.i			0.319539i
$863$	1.27218e6			1.70816			0.854080	−	0.520141i	$-0.174121\pi$
$863$	1.27218e6			1.70816			0.854080	+	0.520141i	$0.174121\pi$
$864$		0			0
$865$			1.02996e6i			1.37653i
$866$		−	163178.i		−	0.217584i
$867$		0			0
$868$			102636.i			0.136226i
$869$	211823.			0.280501
$870$		0			0
$871$			757956.i			0.999097i
$872$		−	259901.i		−	0.341803i
$873$		0			0
$874$	547132.	+	152762.i	0.716258	+	0.199982i
$875$	85119.4			0.111176
$876$		0			0
$877$	145686.			0.189417			0.0947083	−	0.995505i	$-0.469808\pi$
$877$	145686.			0.189417			0.0947083	+	0.995505i	$0.469808\pi$
$878$	516284.			0.669729
$879$		0			0
$880$	−115730.			−0.149445
$881$		−	781737.i		−	1.00718i	−0.863941	−	0.503592i	$-0.832011\pi$
$881$		−	781737.i		−	1.00718i	0.863941	−	0.503592i	$-0.167989\pi$
$882$		0			0
$883$	1.11076e6			1.42462			0.712312	−	0.701863i	$-0.247648\pi$
$883$	1.11076e6			1.42462			0.712312	+	0.701863i	$0.247648\pi$
$884$			98657.8i			0.126249i
$885$		0			0
$886$	611949.			0.779557
$887$	−275114.			−0.349676			−0.174838	−	0.984597i	$-0.555940\pi$
$887$	−275114.			−0.349676			−0.174838	+	0.984597i	$0.555940\pi$
$888$		0			0
$889$			129403.i			0.163735i
$890$	−1.07178e6			−1.35309
$891$		0			0
$892$	−635840.			−0.799131
$893$			976681.i			1.22476i
$894$		0			0
$895$			1.31853e6i			1.64605i
$896$		−	28495.9i		−	0.0354949i
$897$		0			0
$898$	−1.04309e6			−1.29351
$899$	−1.05921e6			−1.31058
$900$		0			0
$901$	160621.			0.197858
$902$			26193.7i			0.0321946i
$903$		0			0
$904$		−	378932.i		−	0.463686i
$905$	1.09568e6			1.33778
$906$		0			0
$907$		−	772272.i		−	0.938762i	−0.882996	−	0.469381i	$-0.844477\pi$
$907$		−	772272.i		−	0.938762i	0.882996	−	0.469381i	$-0.155523\pi$
$908$			597310.i			0.724482i
$909$		0			0
$910$	297529.			0.359291
$911$		−	1.42645e6i		−	1.71878i	−0.511321	−	0.859390i	$-0.670844\pi$
$911$		−	1.42645e6i		−	1.71878i	0.511321	−	0.859390i	$-0.329156\pi$
$912$		0			0
$913$	570227.			0.684079
$914$			437658.i			0.523893i
$915$		0			0
$916$		−	261266.i		−	0.311381i
$917$			138551.i			0.164767i
$918$		0			0
$919$			369970.i			0.438062i	0.975718	+	0.219031i	$0.0702896\pi$
$919$			369970.i			0.438062i	−0.975718	+	0.219031i	$0.929710\pi$
$920$	−385969.	−	107764.i	−0.456012	−	0.127321i
$921$		0			0
$922$	391244.			0.460241
$923$	221368.			0.259843
$924$		0			0
$925$			483348.i			0.564907i
$926$	−137630.			−0.160506
$927$		0			0
$928$	294081.			0.341484
$929$	1.25590e6			1.45520			0.727599	−	0.686002i	$-0.240636\pi$
$929$	1.25590e6			1.45520			0.727599	+	0.686002i	$0.240636\pi$
$930$		0			0
$931$			764554.i			0.882082i
$932$	−155230.			−0.178708
$933$		0			0
$934$		−	772887.i		−	0.885977i
$935$			139654.i			0.159746i
$936$		0			0
$937$			1.39349e6i			1.58717i	0.608457	+	0.793587i	$0.291789\pi$
$937$			1.39349e6i			1.58717i	−0.608457	+	0.793587i	$0.708211\pi$
$938$	−264181.			−0.300259
$939$		0			0
$940$		−	688990.i		−	0.779753i
$941$		−	1.37875e6i		−	1.55706i	−0.627605	−	0.778532i	$-0.715965\pi$
$941$		−	1.37875e6i		−	1.55706i	0.627605	−	0.778532i	$-0.284035\pi$
$942$		0			0
$943$	−24390.7	+	87357.9i	−0.0274284	+	0.0982378i
$944$	−548.890			−0.000615945
$945$		0			0
$946$	−81219.7			−0.0907568
$947$	1.30227e6			1.45212			0.726059	−	0.687632i	$-0.241350\pi$
$947$	1.30227e6			1.45212			0.726059	+	0.687632i	$0.241350\pi$
$948$		0			0
$949$	418005.			0.464140
$950$			532395.i			0.589911i
$951$		0			0
$952$	−34386.6			−0.0379415
$953$			161841.i			0.178198i	0.996023	+	0.0890991i	$0.0283988\pi$
$953$			161841.i			0.178198i	−0.996023	+	0.0890991i	$0.971601\pi$
$954$		0			0
$955$	−767705.			−0.841759
$956$	595906.			0.652022
$957$		0			0
$958$			583942.i			0.636266i
$959$	513421.			0.558260
$960$		0			0
$961$	−498429.			−0.539705
$962$		−	440315.i		−	0.475787i
$963$		0			0
$964$		−	240740.i		−	0.259057i
$965$			854648.i			0.917768i
$966$		0			0
$967$	−234245.			−0.250506			−0.125253	−	0.992125i	$-0.539974\pi$
$967$	−234245.			−0.250506			−0.125253	+	0.992125i	$0.539974\pi$
$968$	−265273.			−0.283101
$969$		0			0
$970$	−262744.			−0.279248
$971$		−	27276.6i		−	0.0289302i	−0.999895	−	0.0144651i	$-0.995395\pi$
$971$		−	27276.6i		−	0.0289302i	0.999895	−	0.0144651i	$-0.00460454\pi$
$972$		0			0
$973$			27571.7i			0.0291232i
$974$	339242.			0.357595
$975$		0			0
$976$		−	67866.5i		−	0.0712452i
$977$			1.49726e6i			1.56858i	0.620392	+	0.784292i	$0.286973\pi$
$977$			1.49726e6i			1.56858i	−0.620392	+	0.784292i	$0.713027\pi$
$978$		0			0
$979$	611372.			0.637881
$980$		−	539347.i		−	0.561586i
$981$		0			0
$982$	−293907.			−0.304780
$983$			1.82952e6i			1.89335i	0.322190	+	0.946675i	$0.395581\pi$
$983$			1.82952e6i			1.89335i	−0.322190	+	0.946675i	$0.604419\pi$
$984$		0			0
$985$			1.58087e6i			1.62938i
$986$		−	354873.i		−	0.365022i
$987$		0			0
$988$		−	484994.i		−	0.496847i
$989$	−270874.	−	75629.1i	−0.276933	−	0.0773208i
$990$		0			0
$991$	362117.			0.368724			0.184362	−	0.982858i	$-0.440978\pi$
$991$	362117.			0.368724			0.184362	+	0.982858i	$0.440978\pi$
$992$	−118023.			−0.119934
$993$		0			0
$994$			77156.5i			0.0780908i
$995$	−1.44052e6			−1.45504
$996$		0			0
$997$	1.39824e6			1.40667			0.703335	−	0.710859i	$-0.251694\pi$
$997$	1.39824e6			1.40667			0.703335	+	0.710859i	$0.251694\pi$
$998$	1.28609e6			1.29125
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.5.b.b.91.1		16
3.2	odd	2		138.5.b.a.91.16	yes	16
12.11	even	2		1104.5.c.a.1057.7		16
23.22	odd	2	inner	414.5.b.b.91.8		16
69.68	even	2		138.5.b.a.91.13	✓	16
276.275	odd	2		1104.5.c.a.1057.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.b.a.91.13	✓	16	69.68	even	2
138.5.b.a.91.16	yes	16	3.2	odd	2
414.5.b.b.91.1		16	1.1	even	1	trivial
414.5.b.b.91.8		16	23.22	odd	2	inner
1104.5.c.a.1057.2		16	276.275	odd	2
1104.5.c.a.1057.7		16	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.82843 q^{2} +8.00000 q^{4} -33.4782i q^{5} +19.6774i q^{7} -22.6274 q^{8} +O(q^{10})\)	\(q-2.82843 q^{2} +8.00000 q^{4} -33.4782i q^{5} +19.6774i q^{7} -22.6274 q^{8} +94.6906i q^{10} -54.0138i q^{11} -159.682 q^{13} -55.6561i q^{14} +64.0000 q^{16} -77.2301i q^{17} +379.657i q^{19} -267.826i q^{20} +152.774i q^{22} +(-142.258 + 509.513i) q^{23} -495.789 q^{25} +451.648 q^{26} +157.419i q^{28} -1624.58 q^{29} +651.991 q^{31} -181.019 q^{32} +218.440i q^{34} +658.763 q^{35} -974.907i q^{37} -1073.83i q^{38} +757.525i q^{40} +171.454 q^{41} +531.633i q^{43} -432.110i q^{44} +(402.367 - 1441.12i) q^{46} +2572.53 q^{47} +2013.80 q^{49} +1402.30 q^{50} -1277.45 q^{52} +2079.78i q^{53} -1808.28 q^{55} -445.248i q^{56} +4595.01 q^{58} -8.57641 q^{59} -1060.41i q^{61} -1844.11 q^{62} +512.000 q^{64} +5345.85i q^{65} -4746.67i q^{67} -617.840i q^{68} -1863.26 q^{70} -1386.31 q^{71} -2617.74 q^{73} +2757.45i q^{74} +3037.26i q^{76} +1062.85 q^{77} +3921.65i q^{79} -2142.60i q^{80} -484.944 q^{82} +10557.1i q^{83} -2585.52 q^{85} -1503.68i q^{86} +1222.19i q^{88} +11318.8i q^{89} -3142.12i q^{91} +(-1138.07 + 4076.10i) q^{92} -7276.22 q^{94} +12710.2 q^{95} +2774.76i q^{97} -5695.89 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 128 q^{4}+O(q^{10}) \) 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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