LMFDB - Embedded newform 1104.5.c.a.1057.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,5,Mod(1057,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1104.1057");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1104.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$114.120439245$
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			1057.7
Root			$0.707107 + 14.6559i$ of defining polynomial
Character	$\chi$	$=$	1104.1057
Dual form			1104.5.c.a.1057.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615 q^{3} +33.4782i q^{5} -19.6774i q^{7} +27.0000 q^{9} +O(q^{10})$	$q-5.19615 q^{3} +33.4782i q^{5} -19.6774i q^{7} +27.0000 q^{9} -54.0138i q^{11} -159.682 q^{13} -173.958i q^{15} +77.2301i q^{17} -379.657i q^{19} +102.247i q^{21} +(-142.258 + 509.513i) q^{23} -495.789 q^{25} -140.296 q^{27} +1624.58 q^{29} -651.991 q^{31} +280.664i q^{33} +658.763 q^{35} -974.907i q^{37} +829.730 q^{39} -171.454 q^{41} -531.633i q^{43} +903.911i q^{45} +2572.53 q^{47} +2013.80 q^{49} -401.299i q^{51} -2079.78i q^{53} +1808.28 q^{55} +1972.76i q^{57} -8.57641 q^{59} -1060.41i q^{61} -531.289i q^{63} -5345.85i q^{65} +4746.67i q^{67} +(739.195 - 2647.51i) q^{69} -1386.31 q^{71} -2617.74 q^{73} +2576.20 q^{75} -1062.85 q^{77} -3921.65i q^{79} +729.000 q^{81} +10557.1i q^{83} -2585.52 q^{85} -8441.58 q^{87} -11318.8i q^{89} +3142.12i q^{91} +3387.85 q^{93} +12710.2 q^{95} +2774.76i q^{97} -1458.37i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 432 q^{9}+O(q^{10}) $ 16 * q + 432 * q^9	$ 16 q + 432 q^{9} - 208 q^{13} - 840 q^{23} + 1056 q^{25} + 3600 q^{29} - 224 q^{31} + 3264 q^{35} + 2016 q^{39} - 6144 q^{41} - 8880 q^{47} - 13888 q^{49} - 832 q^{55} + 18240 q^{59} + 10584 q^{69} + 30048 q^{71} + 9536 q^{73} + 4176 q^{75} + 14160 q^{77} + 11664 q^{81} - 32496 q^{85} + 8064 q^{87} - 11952 q^{93} + 20064 q^{95}+O(q^{100}) $ 16 * q + 432 * q^9 - 208 * q^13 - 840 * q^23 + 1056 * q^25 + 3600 * q^29 - 224 * q^31 + 3264 * q^35 + 2016 * q^39 - 6144 * q^41 - 8880 * q^47 - 13888 * q^49 - 832 * q^55 + 18240 * q^59 + 10584 * q^69 + 30048 * q^71 + 9536 * q^73 + 4176 * q^75 + 14160 * q^77 + 11664 * q^81 - 32496 * q^85 + 8064 * q^87 - 11952 * q^93 + 20064 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times$.

$n$	$97$	$277$	$415$	$737$
$\chi(n)$	$-1$	$1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−5.19615			−0.577350
$3$	−5.19615			−0.577350
$4$		0			0
$5$			33.4782i			1.33913i	0.742755	+	0.669564i	$0.233519\pi$
$5$			33.4782i			1.33913i	−0.742755	+	0.669564i	$0.766481\pi$
$6$		0			0
$7$		−	19.6774i		−	0.401579i	−0.979634	−	0.200790i	$-0.935649\pi$
$7$		−	19.6774i		−	0.401579i	0.979634	−	0.200790i	$-0.0643508\pi$
$8$		0			0
$9$	27.0000			0.333333
$10$		0			0
$11$		−	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	$0.656623\pi$
$14$		0			0
$15$		−	173.958i		−	0.773146i
$16$		0			0
$17$			77.2301i			0.267232i	0.991033	+	0.133616i	$0.0426589\pi$
$17$			77.2301i			0.267232i	−0.991033	+	0.133616i	$0.957341\pi$
$18$		0			0
$19$		−	379.657i		−	1.05168i	−0.850583	−	0.525841i	$-0.823751\pi$
$19$		−	379.657i		−	1.05168i	0.850583	−	0.525841i	$-0.176249\pi$
$20$		0			0
$21$			102.247i			0.231852i
$22$		0			0
$23$	−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
$26$		0			0
$27$	−140.296			−0.192450
$28$		0			0
$29$	1624.58			1.93173			0.965863	−	0.259052i	$-0.0834100\pi$
$29$	1624.58			1.93173			0.965863	+	0.259052i	$0.0834100\pi$
$30$		0			0
$31$	−651.991			−0.678451			−0.339225	−	0.940705i	$-0.610165\pi$
$31$	−651.991			−0.678451			−0.339225	+	0.940705i	$0.610165\pi$
$32$		0			0
$33$			280.664i			0.257726i
$34$		0			0
$35$	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	$-0.115882\pi$
$38$		0			0
$39$	829.730			0.545516
$40$		0			0
$41$	−171.454			−0.101995			−0.0509975	−	0.998699i	$-0.516240\pi$
$41$	−171.454			−0.101995			−0.0509975	+	0.998699i	$0.516240\pi$
$42$		0			0
$43$		−	531.633i		−	0.287524i	−0.989612	−	0.143762i	$-0.954080\pi$
$43$		−	531.633i		−	0.287524i	0.989612	−	0.143762i	$-0.0459201\pi$
$44$		0			0
$45$			903.911i			0.446376i
$46$		0			0
$47$	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
$50$		0			0
$51$		−	401.299i		−	0.154286i
$52$		0			0
$53$		−	2079.78i		−	0.740397i	−0.928953	−	0.370199i	$-0.879290\pi$
$53$		−	2079.78i		−	0.740397i	0.928953	−	0.370199i	$-0.120710\pi$
$54$		0			0
$55$	1808.28			0.597780
$56$		0			0
$57$			1972.76i			0.607189i
$58$		0			0
$59$	−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	$-0.0455110\pi$
$62$		0			0
$63$		−	531.289i		−	0.133860i
$64$		0			0
$65$		−	5345.85i		−	1.26529i
$66$		0			0
$67$			4746.67i			1.05740i	0.848809	+	0.528700i	$0.177320\pi$
$67$			4746.67i			1.05740i	−0.848809	+	0.528700i	$0.822680\pi$
$68$		0			0
$69$	739.195	−	2647.51i	0.155261	−	0.556082i
$70$		0			0
$71$	−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	$0.578989\pi$
$74$		0			0
$75$	2576.20			0.457990
$76$		0			0
$77$	−1062.85			−0.179263
$78$		0			0
$79$		−	3921.65i		−	0.628368i	−0.949362	−	0.314184i	$-0.898269\pi$
$79$		−	3921.65i		−	0.628368i	0.949362	−	0.314184i	$-0.101731\pi$
$80$		0			0
$81$	729.000			0.111111
$82$		0			0
$83$			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
$86$		0			0
$87$	−8441.58			−1.11528
$88$		0			0
$89$		−	11318.8i		−	1.42896i	−0.699655	−	0.714481i	$-0.746663\pi$
$89$		−	11318.8i		−	1.42896i	0.699655	−	0.714481i	$-0.253337\pi$
$90$		0			0
$91$			3142.12i			0.379437i
$92$		0			0
$93$	3387.85			0.391704
$94$		0			0
$95$	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	$0.952893\pi$
$98$		0			0
$99$		−	1458.37i		−	0.148798i
$100$		0			0
$101$	−3202.26			−0.313916			−0.156958	−	0.987605i	$-0.550169\pi$
$101$	−3202.26			−0.313916			−0.156958	+	0.987605i	$0.550169\pi$
$102$		0			0
$103$		−	14013.2i		−	1.32087i	−0.750881	−	0.660437i	$-0.770371\pi$
$103$		−	14013.2i		−	1.32087i	0.750881	−	0.660437i	$-0.229629\pi$
$104$		0			0
$105$	−3423.03			−0.310479
$106$		0			0
$107$			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	$0.839408\pi$
$110$		0			0
$111$			5065.77i			0.411149i
$112$		0			0
$113$		−	16746.6i		−	1.31150i	−0.754977	−	0.655751i	$-0.772352\pi$
$113$		−	16746.6i		−	1.31150i	0.754977	−	0.655751i	$-0.227648\pi$
$114$		0			0
$115$	−17057.6	−	4762.55i	−1.28980	−	0.360117i
$116$		0			0
$117$	−4311.40			−0.314954
$118$		0			0
$119$	1519.69			0.107315
$120$		0			0
$121$	11723.5			0.800731
$122$		0			0
$123$	890.899			0.0588869
$124$		0			0
$125$			4325.75i			0.276848i
$126$		0			0
$127$	−6576.24			−0.407728			−0.203864	−	0.978999i	$-0.565350\pi$
$127$	−6576.24			−0.407728			−0.203864	+	0.978999i	$0.565350\pi$
$128$		0			0
$129$			2762.44i			0.166002i
$130$		0			0
$131$	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
$134$		0			0
$135$		−	4696.86i		−	0.257715i
$136$		0			0
$137$			26091.9i			1.39016i	0.718932	+	0.695080i	$0.244631\pi$
$137$			26091.9i			1.39016i	−0.718932	+	0.695080i	$0.755369\pi$
$138$		0			0
$139$	−1401.19			−0.0725216			−0.0362608	−	0.999342i	$-0.511545\pi$
$139$	−1401.19			−0.0725216			−0.0362608	+	0.999342i	$0.511545\pi$
$140$		0			0
$141$	−13367.3			−0.672364
$142$		0			0
$143$			8625.01i			0.421782i
$144$		0			0
$145$			54388.1i			2.58683i
$146$		0			0
$147$	−10464.0			−0.484243
$148$		0			0
$149$			16105.3i			0.725433i	0.931899	+	0.362717i	$0.118151\pi$
$149$			16105.3i			0.725433i	−0.931899	+	0.362717i	$0.881849\pi$
$150$		0			0
$151$	5562.22			0.243946			0.121973	−	0.992533i	$-0.461078\pi$
$151$	5562.22			0.243946			0.121973	+	0.992533i	$0.461078\pi$
$152$		0			0
$153$			2085.21i			0.0890773i
$154$		0			0
$155$		−	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	$0.684697\pi$
$158$		0			0
$159$			10806.8i			0.427468i
$160$		0			0
$161$	10025.9	+	2799.27i	0.386786	+	0.107992i
$162$		0			0
$163$	−38706.4			−1.45683			−0.728413	−	0.685138i	$-0.759742\pi$
$163$	−38706.4			−1.45683			−0.728413	+	0.685138i	$0.759742\pi$
$164$		0			0
$165$	−9396.12			−0.345128
$166$		0			0
$167$	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
$170$		0			0
$171$		−	10250.7i		−	0.350561i
$172$		0			0
$173$	30765.0			1.02793			0.513966	−	0.857811i	$-0.328176\pi$
$173$	30765.0			1.02793			0.513966	+	0.857811i	$0.328176\pi$
$174$		0			0
$175$			9755.83i			0.318558i
$176$		0			0
$177$	44.5643			0.00142246
$178$		0			0
$179$	−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	$0.833518\pi$
$182$		0			0
$183$			5510.07i			0.164534i
$184$		0			0
$185$	32638.1			0.953634
$186$		0			0
$187$	4171.49			0.119291
$188$		0			0
$189$			2760.66i			0.0772840i
$190$		0			0
$191$		−	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	$0.611333\pi$
$194$		0			0
$195$			27777.9i			0.730516i
$196$		0			0
$197$	47220.8			1.21675			0.608374	−	0.793651i	$-0.291822\pi$
$197$	47220.8			1.21675			0.608374	+	0.793651i	$0.291822\pi$
$198$		0			0
$199$			43028.7i			1.08655i	0.839553	+	0.543277i	$0.182817\pi$
$199$			43028.7i			1.08655i	−0.839553	+	0.543277i	$0.817183\pi$
$200$		0			0
$201$		−	24664.4i		−	0.610490i
$202$		0			0
$203$		−	31967.5i		−	0.775742i
$204$		0			0
$205$		−	5739.96i		−	0.136584i
$206$		0			0
$207$	−3840.97	+	13756.9i	−0.0896397	+	0.321054i
$208$		0			0
$209$	−20506.7			−0.469466
$210$		0			0
$211$	37106.5			0.833462			0.416731	−	0.909030i	$-0.363176\pi$
$211$	37106.5			0.833462			0.416731	+	0.909030i	$0.363176\pi$
$212$		0			0
$213$	7203.47			0.158775
$214$		0			0
$215$	17798.1			0.385032
$216$		0			0
$217$			12829.5i			0.272452i
$218$		0			0
$219$	13602.2			0.283609
$220$		0			0
$221$		−	12332.2i		−	0.252497i
$222$		0			0
$223$	79480.0			1.59826			0.799131	−	0.601157i	$-0.205293\pi$
$223$	79480.0			1.59826			0.799131	+	0.601157i	$0.205293\pi$
$224$		0			0
$225$	−13386.3			−0.264421
$226$		0			0
$227$			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	$-0.100792\pi$
$230$		0			0
$231$	5522.73			0.103498
$232$		0			0
$233$	19403.8			0.357417			0.178708	−	0.983902i	$-0.442808\pi$
$233$	19403.8			0.357417			0.178708	+	0.983902i	$0.442808\pi$
$234$		0			0
$235$			86123.7i			1.55951i
$236$		0			0
$237$			20377.5i			0.362789i
$238$		0			0
$239$	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	$-0.0834117\pi$
$242$		0			0
$243$	−3788.00			−0.0641500
$244$		0			0
$245$			67418.4i			1.12317i
$246$		0			0
$247$			60624.3i			0.993694i
$248$		0			0
$249$		−	54856.1i		−	0.884761i
$250$		0			0
$251$			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
$254$		0			0
$255$	13434.8			0.206609
$256$		0			0
$257$	−70085.0			−1.06111			−0.530553	−	0.847652i	$-0.678016\pi$
$257$	−70085.0			−1.06111			−0.530553	+	0.847652i	$0.678016\pi$
$258$		0			0
$259$	−19183.6			−0.285977
$260$		0			0
$261$	43863.7			0.643909
$262$		0			0
$263$			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
$266$		0			0
$267$			58814.2i			0.825011i
$268$		0			0
$269$	−12740.0			−0.176062			−0.0880308	−	0.996118i	$-0.528057\pi$
$269$	−12740.0			−0.176062			−0.0880308	+	0.996118i	$0.528057\pi$
$270$		0			0
$271$	−455.015			−0.00619565			−0.00309783	−	0.999995i	$-0.500986\pi$
$271$	−455.015			−0.00619565			−0.00309783	+	0.999995i	$0.500986\pi$
$272$		0			0
$273$		−	16326.9i		−	0.219068i
$274$		0			0
$275$			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	$0.402300\pi$
$278$		0			0
$279$	−17603.8			−0.226150
$280$		0			0
$281$			117828.i			1.49223i	0.665819	+	0.746113i	$0.268082\pi$
$281$			117828.i			1.49223i	−0.665819	+	0.746113i	$0.731918\pi$
$282$		0			0
$283$			32448.1i			0.405151i	0.979267	+	0.202575i	$0.0649311\pi$
$283$			32448.1i			0.405151i	−0.979267	+	0.202575i	$0.935069\pi$
$284$		0			0
$285$	−66044.3			−0.813104
$286$		0			0
$287$			3373.76i			0.0409591i
$288$		0			0
$289$	77556.5			0.928587
$290$		0			0
$291$		−	14418.1i		−	0.170264i
$292$		0			0
$293$			88859.5i			1.03507i	0.855663	+	0.517534i	$0.173150\pi$
$293$			88859.5i			1.03507i	−0.855663	+	0.517534i	$0.826850\pi$
$294$		0			0
$295$		−	287.123i		−	0.00329931i
$296$		0			0
$297$			7577.93i			0.0859088i
$298$		0			0
$299$	22716.0	−	81359.9i	0.254091	−	0.910056i
$300$		0			0
$301$	−10461.1			−0.115464
$302$		0			0
$303$	16639.4			0.181240
$304$		0			0
$305$	35500.7			0.381626
$306$		0			0
$307$	20919.2			0.221957			0.110978	−	0.993823i	$-0.464602\pi$
$307$	20919.2			0.221957			0.110978	+	0.993823i	$0.464602\pi$
$308$		0			0
$309$			72814.5i			0.762607i
$310$		0			0
$311$	−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	$0.865568\pi$
$314$		0			0
$315$	17786.6			0.179255
$316$		0			0
$317$	−24797.0			−0.246763			−0.123381	−	0.992359i	$-0.539374\pi$
$317$	−24797.0			−0.246763			−0.123381	+	0.992359i	$0.539374\pi$
$318$		0			0
$319$		−	87749.9i		−	0.862314i
$320$		0			0
$321$		−	99326.2i		−	0.963948i
$322$		0			0
$323$	29321.0			0.281043
$324$		0			0
$325$	79168.4			0.749523
$326$		0			0
$327$		−	59683.7i		−	0.558162i
$328$		0			0
$329$		−	50620.7i		−	0.467667i
$330$		0			0
$331$	127053.			1.15966			0.579830	−	0.814738i	$-0.303119\pi$
$331$	127053.			1.15966			0.579830	+	0.814738i	$0.303119\pi$
$332$		0			0
$333$		−	26322.5i		−	0.237377i
$334$		0			0
$335$	−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	$0.899347\pi$
$338$		0			0
$339$			87017.7i			0.757196i
$340$		0			0
$341$			35216.5i			0.302857i
$342$		0			0
$343$		−	86871.7i		−	0.738398i
$344$		0			0
$345$	88633.8	+	24746.9i	0.744665	+	0.207914i
$346$		0			0
$347$	−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	$0.704088\pi$
$350$		0			0
$351$	22402.7			0.181839
$352$		0			0
$353$	−68772.2			−0.551904			−0.275952	−	0.961171i	$-0.588993\pi$
$353$	−68772.2			−0.551904			−0.275952	+	0.961171i	$0.588993\pi$
$354$		0			0
$355$		−	46411.1i		−	0.368269i
$356$		0			0
$357$	−7896.52			−0.0619583
$358$		0			0
$359$			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
$362$		0			0
$363$	−60917.1			−0.462302
$364$		0			0
$365$		−	87637.1i		−	0.657813i
$366$		0			0
$367$			194975.i			1.44760i	0.690012	+	0.723798i	$0.257605\pi$
$367$			194975.i			1.44760i	−0.690012	+	0.723798i	$0.742395\pi$
$368$		0			0
$369$	−4629.25			−0.0339983
$370$		0			0
$371$	−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	$0.656131\pi$
$374$		0			0
$375$		−	22477.2i		−	0.159838i
$376$		0			0
$377$	−259416.			−1.82521
$378$		0			0
$379$			211323.i			1.47119i	0.677422	+	0.735594i	$0.263097\pi$
$379$			211323.i			1.47119i	−0.677422	+	0.735594i	$0.736903\pi$
$380$		0			0
$381$	34171.2			0.235402
$382$		0			0
$383$		−	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
$386$		0			0
$387$		−	14354.1i		−	0.0958415i
$388$		0			0
$389$		−	140755.i		−	0.930177i	−0.885264	−	0.465088i	$-0.846023\pi$
$389$		−	140755.i		−	0.930177i	0.885264	−	0.465088i	$-0.153977\pi$
$390$		0			0
$391$	−39349.7	−	10986.6i	−0.257388	−	0.0718638i
$392$		0			0
$393$	−36586.8			−0.236886
$394$		0			0
$395$	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	$0.907978\pi$
$398$		0			0
$399$	38818.7			0.243835
$400$		0			0
$401$		−	251342.i		−	1.56306i	−0.623865	−	0.781532i	$-0.714439\pi$
$401$		−	251342.i		−	1.56306i	0.623865	−	0.781532i	$-0.285561\pi$
$402$		0			0
$403$	104111.			0.641042
$404$		0			0
$405$			24405.6i			0.148792i
$406$		0			0
$407$	−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	$0.385177\pi$
$410$		0			0
$411$		−	135578.i		−	0.802610i
$412$		0			0
$413$			168.761i			0.000989402i
$414$		0			0
$415$	−353431.			−2.05215
$416$		0			0
$417$	7280.79			0.0418703
$418$		0			0
$419$			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	$0.988168\pi$
$422$		0			0
$423$	69458.4			0.388189
$424$		0			0
$425$		−	38289.8i		−	0.211985i
$426$		0			0
$427$	−20866.2			−0.114442
$428$		0			0
$429$		−	44816.9i		−	0.243516i
$430$		0			0
$431$		−	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	$0.950831\pi$
$434$		0			0
$435$		−	282609.i		−	1.49351i
$436$		0			0
$437$	193440.	+	54009.4i	1.01294	+	0.282818i
$438$		0			0
$439$	182534.			0.947140			0.473570	−	0.880756i	$-0.342965\pi$
$439$	182534.			0.947140			0.473570	+	0.880756i	$0.342965\pi$
$440$		0			0
$441$	54372.6			0.279578
$442$		0			0
$443$	−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
$446$		0			0
$447$		−	83685.8i		−	0.418829i
$448$		0			0
$449$	−368789.			−1.82930			−0.914651	−	0.404244i	$-0.867534\pi$
$449$	−368789.			−1.82930			−0.914651	+	0.404244i	$0.867534\pi$
$450$		0			0
$451$			9260.86i			0.0455301i
$452$		0			0
$453$	−28902.1			−0.140842
$454$		0			0
$455$	−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	$-0.120796\pi$
$458$		0			0
$459$		−	10835.1i		−	0.0514288i
$460$		0			0
$461$	138326.			0.650880			0.325440	−	0.945563i	$-0.394488\pi$
$461$	138326.			0.650880			0.325440	+	0.945563i	$0.394488\pi$
$462$		0			0
$463$	−48659.5			−0.226989			−0.113495	−	0.993539i	$-0.536205\pi$
$463$	−48659.5			−0.226989			−0.113495	+	0.993539i	$0.536205\pi$
$464$		0			0
$465$			113419.i			0.524541i
$466$		0			0
$467$			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
$470$		0			0
$471$		−	214234.i		−	0.965710i
$472$		0			0
$473$	−28715.5			−0.128350
$474$		0			0
$475$			188230.i			0.834260i
$476$		0			0
$477$		−	56153.9i		−	0.246799i
$478$		0			0
$479$		−	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
$482$		0			0
$483$	−52096.0	−	14545.4i	−0.223311	−	0.0623494i
$484$		0			0
$485$	−92894.0			−0.394916
$486$		0			0
$487$	119940.			0.505716			0.252858	−	0.967503i	$-0.418629\pi$
$487$	119940.			0.505716			0.252858	+	0.967503i	$0.418629\pi$
$488$		0			0
$489$	201124.			0.841099
$490$		0			0
$491$	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
$494$		0			0
$495$	48823.7			0.199260
$496$		0			0
$497$			27278.9i			0.110437i
$498$		0			0
$499$	454701.			1.82610			0.913050	−	0.407848i	$-0.133721\pi$
$499$	454701.			1.82610			0.913050	+	0.407848i	$0.133721\pi$
$500$		0			0
$501$	−201089.			−0.801149
$502$		0			0
$503$			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
$506$		0			0
$507$	15914.7			0.0619130
$508$		0			0
$509$	384796.			1.48523			0.742617	−	0.669717i	$-0.233585\pi$
$509$	384796.			1.48523			0.742617	+	0.669717i	$0.233585\pi$
$510$		0			0
$511$			51510.2i			0.197266i
$512$		0			0
$513$			53264.5i			0.202396i
$514$		0			0
$515$	469135.			1.76882
$516$		0			0
$517$		−	138952.i		−	0.519858i
$518$		0			0
$519$	−159859.			−0.593477
$520$		0			0
$521$			45060.9i			0.166006i	0.996549	+	0.0830031i	$0.0264511\pi$
$521$			45060.9i			0.166006i	−0.996549	+	0.0830031i	$0.973549\pi$
$522$		0			0
$523$			152841.i			0.558774i	0.960179	+	0.279387i	$0.0901313\pi$
$523$			152841.i			0.558774i	−0.960179	+	0.279387i	$0.909869\pi$
$524$		0			0
$525$		−	50692.8i		−	0.183919i
$526$		0			0
$527$		−	50353.3i		−	0.181304i
$528$		0			0
$529$	−239366.	−	144965.i	−0.855365	−	0.518026i
$530$		0			0
$531$	−231.563			−0.000821259
$532$		0			0
$533$	27378.0			0.0963712
$534$		0			0
$535$	−639947.			−2.23582
$536$		0			0
$537$	204649.			0.709676
$538$		0			0
$539$		−	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	$0.871798\pi$
$542$		0			0
$543$		−	170060.i		−	0.576771i
$544$		0			0
$545$	−384535.			−1.29462
$546$		0			0
$547$	392350.			1.31129			0.655645	−	0.755069i	$-0.272397\pi$
$547$	392350.			1.31129			0.655645	+	0.755069i	$0.272397\pi$
$548$		0			0
$549$		−	28631.2i		−	0.0949936i
$550$		0			0
$551$		−	616785.i		−	2.03156i
$552$		0			0
$553$	−77167.7			−0.252340
$554$		0			0
$555$	−169593.			−0.550581
$556$		0			0
$557$		−	558524.i		−	1.80025i	−0.435637	−	0.900123i	$-0.643477\pi$
$557$		−	558524.i		−	1.80025i	0.435637	−	0.900123i	$-0.356523\pi$
$558$		0			0
$559$			84892.0i			0.271671i
$560$		0			0
$561$	−21675.7			−0.0688727
$562$		0			0
$563$		−	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
$566$		0			0
$567$		−	14344.8i		−	0.0446199i
$568$		0			0
$569$			56795.6i			0.175425i	0.996146	+	0.0877123i	$0.0279556\pi$
$569$			56795.6i			0.175425i	−0.996146	+	0.0877123i	$0.972044\pi$
$570$		0			0
$571$		−	281453.i		−	0.863243i	−0.902055	−	0.431621i	$-0.857942\pi$
$571$		−	281453.i		−	0.863243i	0.902055	−	0.431621i	$-0.142058\pi$
$572$		0			0
$573$			119156.i			0.362915i
$574$		0			0
$575$	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	$0.235908\pi$
$578$		0			0
$579$	132650.			0.395685
$580$		0			0
$581$	207735.			0.615401
$582$		0			0
$583$	−112337.			−0.330510
$584$		0			0
$585$		−	144338.i		−	0.421763i
$586$		0			0
$587$	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
$590$		0			0
$591$	−245366.			−0.702490
$592$		0			0
$593$	401812.			1.14265			0.571325	−	0.820724i	$-0.306430\pi$
$593$	401812.			1.14265			0.571325	+	0.820724i	$0.306430\pi$
$594$		0			0
$595$			50876.3i			0.143708i
$596$		0			0
$597$		−	223583.i		−	0.627323i
$598$		0			0
$599$	−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	$0.372584\pi$
$602$		0			0
$603$			128160.i			0.352467i
$604$		0			0
$605$			392482.i			1.07228i
$606$		0			0
$607$	422711.			1.14727			0.573635	−	0.819111i	$-0.305533\pi$
$607$	422711.			1.14727			0.573635	+	0.819111i	$0.305533\pi$
$608$		0			0
$609$			166108.i			0.447875i
$610$		0			0
$611$	−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	$-0.158702\pi$
$614$		0			0
$615$			29825.7i			0.0788570i
$616$		0			0
$617$		−	343800.i		−	0.903099i	−0.892246	−	0.451550i	$-0.850871\pi$
$617$		−	343800.i		−	0.903099i	0.892246	−	0.451550i	$-0.149129\pi$
$618$		0			0
$619$			625411.i			1.63224i	0.577882	+	0.816120i	$0.303879\pi$
$619$			625411.i			1.63224i	−0.577882	+	0.816120i	$0.696121\pi$
$620$		0			0
$621$	19958.3	−	71482.7i	0.0517535	−	0.185361i
$622$		0			0
$623$	−222724.			−0.573841
$624$		0			0
$625$	−454686.			−1.16400
$626$		0			0
$627$	106556.			0.271046
$628$		0			0
$629$	75292.1			0.190304
$630$		0			0
$631$		−	384826.i		−	0.966508i	−0.875480	−	0.483254i	$-0.839455\pi$
$631$		−	384826.i		−	0.966508i	0.875480	−	0.483254i	$-0.160545\pi$
$632$		0			0
$633$	−192811.			−0.481199
$634$		0			0
$635$		−	220161.i		−	0.546000i
$636$		0			0
$637$	−321567.			−0.792488
$638$		0			0
$639$	−37430.3			−0.0916689
$640$		0			0
$641$			284276.i			0.691870i	0.938258	+	0.345935i	$0.112438\pi$
$641$			284276.i			0.691870i	−0.938258	+	0.345935i	$0.887562\pi$
$642$		0			0
$643$		−	416229.i		−	1.00672i	−0.864076	−	0.503361i	$-0.832096\pi$
$643$		−	416229.i		−	1.00672i	0.864076	−	0.503361i	$-0.167904\pi$
$644$		0			0
$645$	−92481.7			−0.222298
$646$		0			0
$647$	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
$650$		0			0
$651$		−	66663.9i		−	0.157300i
$652$		0			0
$653$	−243366.			−0.570734			−0.285367	−	0.958418i	$-0.592116\pi$
$653$	−243366.			−0.570734			−0.285367	+	0.958418i	$0.592116\pi$
$654$		0			0
$655$			235724.i			0.549442i
$656$		0			0
$657$	−70678.9			−0.163742
$658$		0			0
$659$		−	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	$0.672427\pi$
$662$		0			0
$663$			64080.1i			0.145779i
$664$		0			0
$665$		−	250104.i		−	0.565559i
$666$		0			0
$667$	−231110.	+	827746.i	−0.519478	+	1.86057i
$668$		0			0
$669$	−412990.			−0.922757
$670$		0			0
$671$	−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	$0.261675\pi$
$674$		0			0
$675$	69557.3			0.152663
$676$		0			0
$677$		−	823346.i		−	1.79641i	−0.439578	−	0.898205i	$-0.644872\pi$
$677$		−	823346.i		−	1.79641i	0.439578	−	0.898205i	$-0.355128\pi$
$678$		0			0
$679$	54600.1			0.118428
$680$		0			0
$681$		−	387964.i		−	0.836560i
$682$		0			0
$683$	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
$686$		0			0
$687$			169697.i			0.359552i
$688$		0			0
$689$			332102.i			0.699573i
$690$		0			0
$691$	−567858.			−1.18928			−0.594639	−	0.803993i	$-0.702705\pi$
$691$	−567858.			−1.18928			−0.594639	+	0.803993i	$0.702705\pi$
$692$		0			0
$693$	−28697.0			−0.0597543
$694$		0			0
$695$		−	46909.3i		−	0.0971156i
$696$		0			0
$697$		−	13241.4i		−	0.0272563i
$698$		0			0
$699$	−100825.			−0.206355
$700$		0			0
$701$			190169.i			0.386994i	0.981101	+	0.193497i	$0.0619830\pi$
$701$			190169.i			0.386994i	−0.981101	+	0.193497i	$0.938017\pi$
$702$		0			0
$703$	−370131.			−0.748935
$704$		0			0
$705$		−	447512.i		−	0.900381i
$706$		0			0
$707$			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	$-0.300436\pi$
$710$		0			0
$711$		−	105884.i		−	0.209456i
$712$		0			0
$713$	92751.1	−	332198.i	0.182448	−	0.653458i
$714$		0			0
$715$	−288750.			−0.564819
$716$		0			0
$717$	−387052.			−0.752890
$718$		0			0
$719$	−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
$722$		0			0
$723$			156366.i			0.299133i
$724$		0			0
$725$	−805450.			−1.53237
$726$		0			0
$727$			364607.i			0.689853i	0.938630	+	0.344926i	$0.112096\pi$
$727$			364607.i			0.689853i	−0.938630	+	0.344926i	$0.887904\pi$
$728$		0			0
$729$	19683.0			0.0370370
$730$		0			0
$731$	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	$0.732234\pi$
$734$		0			0
$735$		−	350316.i		−	0.648464i
$736$		0			0
$737$	256386.			0.472018
$738$		0			0
$739$	197657.			0.361929			0.180965	−	0.983490i	$-0.442078\pi$
$739$	197657.			0.361929			0.180965	+	0.983490i	$0.442078\pi$
$740$		0			0
$741$		−	315013.i		−	0.573710i
$742$		0			0
$743$		−	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
$746$		0			0
$747$			285041.i			0.510817i
$748$		0			0
$749$	376140.			0.670480
$750$		0			0
$751$		−	454544.i		−	0.805928i	−0.915216	−	0.402964i	$-0.867980\pi$
$751$		−	454544.i		−	0.805928i	0.915216	−	0.402964i	$-0.132020\pi$
$752$		0			0
$753$		−	330983.i		−	0.583735i
$754$		0			0
$755$			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	$-0.131831\pi$
$758$		0			0
$759$	−143002.	−	39926.8i	−0.248232	−	0.0693076i
$760$		0			0
$761$	408299.			0.705032			0.352516	−	0.935806i	$-0.385326\pi$
$761$	408299.			0.705032			0.352516	+	0.935806i	$0.385326\pi$
$762$		0			0
$763$	226017.			0.388233
$764$		0			0
$765$	−69809.1			−0.119286
$766$		0			0
$767$	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	$-0.0152324\pi$
$770$		0			0
$771$	364172.			0.612630
$772$		0			0
$773$		−	714995.i		−	1.19659i	−0.801277	−	0.598293i	$-0.795846\pi$
$773$		−	714995.i		−	1.19659i	0.801277	−	0.598293i	$-0.204154\pi$
$774$		0			0
$775$	323250.			0.538190
$776$		0			0
$777$	99681.0			0.165109
$778$		0			0
$779$			65093.6i			0.107266i
$780$		0			0
$781$			74879.8i			0.122762i
$782$		0			0
$783$	−227923.			−0.371761
$784$		0			0
$785$	−1.38028e6			−2.23990
$786$		0			0
$787$			361359.i			0.583431i	0.956505	+	0.291715i	$0.0942260\pi$
$787$			361359.i			0.583431i	−0.956505	+	0.291715i	$0.905774\pi$
$788$		0			0
$789$		−	122665.i		−	0.197046i
$790$		0			0
$791$	−329529.			−0.526672
$792$		0			0
$793$			169329.i			0.269267i
$794$		0			0
$795$	−361793.			−0.572435
$796$		0			0
$797$		−	388399.i		−	0.611451i	−0.952120	−	0.305726i	$-0.901101\pi$
$797$		−	388399.i		−	0.611451i	0.952120	−	0.305726i	$-0.0988990\pi$
$798$		0			0
$799$			198677.i			0.311210i
$800$		0			0
$801$		−	305608.i		−	0.476320i
$802$		0			0
$803$			141394.i			0.219280i
$804$		0			0
$805$	−93714.5	+	335648.i	−0.144616	+	0.517956i
$806$		0			0
$807$	66198.9			0.101649
$808$		0			0
$809$	170967.			0.261225			0.130612	−	0.991434i	$-0.458306\pi$
$809$	170967.			0.261225			0.130612	+	0.991434i	$0.458306\pi$
$810$		0			0
$811$	476324.			0.724203			0.362102	−	0.932139i	$-0.382059\pi$
$811$	476324.			0.724203			0.362102	+	0.932139i	$0.382059\pi$
$812$		0			0
$813$	2364.33			0.00357706
$814$		0			0
$815$		−	1.29582e6i		−	1.95088i
$816$		0			0
$817$	−201838.			−0.302384
$818$		0			0
$819$			84837.2i			0.126479i
$820$		0			0
$821$	1.06850e6			1.58521			0.792605	−	0.609736i	$-0.208724\pi$
$821$	1.06850e6			1.58521			0.792605	+	0.609736i	$0.208724\pi$
$822$		0			0
$823$	663222.			0.979172			0.489586	−	0.871955i	$-0.337148\pi$
$823$	663222.			0.979172			0.489586	+	0.871955i	$0.337148\pi$
$824$		0			0
$825$		−	139150.i		−	0.204445i
$826$		0			0
$827$			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	$0.500819\pi$
$830$		0			0
$831$	−240920.			−0.348876
$832$		0			0
$833$			155526.i			0.224137i
$834$		0			0
$835$			1.29559e6i			1.85821i
$836$		0			0
$837$	91471.8			0.130568
$838$		0			0
$839$		−	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
$842$		0			0
$843$		−	612251.i		−	0.861537i
$844$		0			0
$845$		−	102536.i		−	0.143603i
$846$		0			0
$847$		−	230688.i		−	0.321557i
$848$		0			0
$849$		−	168605.i		−	0.233914i
$850$		0			0
$851$	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	$0.508726\pi$
$854$		0			0
$855$	343176.			0.469446
$856$		0			0
$857$	1.06525e6			1.45040			0.725202	−	0.688536i	$-0.241746\pi$
$857$	1.06525e6			1.45040			0.725202	+	0.688536i	$0.241746\pi$
$858$		0			0
$859$	−292913.			−0.396965			−0.198483	−	0.980104i	$-0.563601\pi$
$859$	−292913.			−0.396965			−0.198483	+	0.980104i	$0.563601\pi$
$860$		0			0
$861$		−	17530.6i		−	0.0236477i
$862$		0			0
$863$	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
$866$		0			0
$867$	−402995.			−0.536120
$868$		0			0
$869$	−211823.			−0.280501
$870$		0			0
$871$		−	757956.i		−	0.999097i
$872$		0			0
$873$			74918.6i			0.0983017i
$874$		0			0
$875$	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	$0.469808\pi$
$878$		0			0
$879$		−	461728.i		−	0.597597i
$880$		0			0
$881$			781737.i			1.00718i	0.863941	+	0.503592i	$0.167989\pi$
$881$			781737.i			1.00718i	−0.863941	+	0.503592i	$0.832011\pi$
$882$		0			0
$883$	−1.11076e6			−1.42462			−0.712312	−	0.701863i	$-0.752352\pi$
$883$	−1.11076e6			−1.42462			−0.712312	+	0.701863i	$0.752352\pi$
$884$		0			0
$885$			1491.93i			0.00190486i
$886$		0			0
$887$	−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
$890$		0			0
$891$		−	39376.1i		−	0.0495995i
$892$		0			0
$893$		−	976681.i		−	1.22476i
$894$		0			0
$895$		−	1.31853e6i		−	1.64605i
$896$		0			0
$897$	−118036.	+	422758.i	−0.146700	+	0.525421i
$898$		0			0
$899$	−1.05921e6			−1.31058
$900$		0			0
$901$	160621.			0.197858
$902$		0			0
$903$	54357.7			0.0666631
$904$		0			0
$905$	−1.09568e6			−1.33778
$906$		0			0
$907$			772272.i			0.938762i	0.882996	+	0.469381i	$0.155523\pi$
$907$			772272.i			0.938762i	−0.882996	+	0.469381i	$0.844477\pi$
$908$		0			0
$909$	−86461.0			−0.104639
$910$		0			0
$911$		−	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
$914$		0			0
$915$	−184467.			−0.220332
$916$		0			0
$917$		−	138551.i		−	0.164767i
$918$		0			0
$919$		−	369970.i		−	0.438062i	−0.975718	−	0.219031i	$-0.929710\pi$
$919$		−	369970.i		−	0.438062i	0.975718	−	0.219031i	$-0.0702896\pi$
$920$		0			0
$921$	−108699.			−0.128147
$922$		0			0
$923$	221368.			0.259843
$924$		0			0
$925$			483348.i			0.564907i
$926$		0			0
$927$		−	378355.i		−	0.440292i
$928$		0			0
$929$	−1.25590e6			−1.45520			−0.727599	−	0.686002i	$-0.759364\pi$
$929$	−1.25590e6			−1.45520			−0.727599	+	0.686002i	$0.759364\pi$
$930$		0			0
$931$		−	764554.i		−	0.882082i
$932$		0			0
$933$	153196.			0.175989
$934$		0			0
$935$			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	$0.708211\pi$
$938$		0			0
$939$		−	417317.i		−	0.473299i
$940$		0			0
$941$			1.37875e6i			1.55706i	0.627605	+	0.778532i	$0.284035\pi$
$941$			1.37875e6i			1.55706i	−0.627605	+	0.778532i	$0.715965\pi$
$942$		0			0
$943$	24390.7	−	87357.9i	0.0274284	−	0.0982378i
$944$		0			0
$945$	−92421.9			−0.103493
$946$		0			0
$947$	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
$950$		0			0
$951$	128849.			0.142469
$952$		0			0
$953$		−	161841.i		−	0.178198i	−0.996023	−	0.0890991i	$-0.971601\pi$
$953$		−	161841.i		−	0.178198i	0.996023	−	0.0890991i	$-0.0283988\pi$
$954$		0			0
$955$	767705.			0.841759
$956$		0			0
$957$			455962.i			0.497857i
$958$		0			0
$959$	513421.			0.558260
$960$		0			0
$961$	−498429.			−0.539705
$962$		0			0
$963$			516114.i			0.556536i
$964$		0			0
$965$		−	854648.i		−	0.917768i
$966$		0			0
$967$	234245.			0.250506			0.125253	−	0.992125i	$-0.460026\pi$
$967$	234245.			0.250506			0.125253	+	0.992125i	$0.460026\pi$
$968$		0			0
$969$	−152356.			−0.162260
$970$		0			0
$971$		−	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
$974$		0			0
$975$	−411371.			−0.432738
$976$		0			0
$977$		−	1.49726e6i		−	1.56858i	−0.620392	−	0.784292i	$-0.713027\pi$
$977$		−	1.49726e6i		−	1.56858i	0.620392	−	0.784292i	$-0.286973\pi$
$978$		0			0
$979$	−611372.			−0.637881
$980$		0			0
$981$			310125.i			0.322255i
$982$		0			0
$983$			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
$986$		0			0
$987$			263033.i			0.270007i
$988$		0			0
$989$	270874.	+	75629.1i	0.276933	+	0.0773208i
$990$		0			0
$991$	−362117.			−0.368724			−0.184362	−	0.982858i	$-0.559022\pi$
$991$	−362117.			−0.368724			−0.184362	+	0.982858i	$0.559022\pi$
$992$		0			0
$993$	−660189.			−0.669530
$994$		0			0
$995$	−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	$0.251694\pi$
$998$		0			0
$999$			136776.i			0.137050i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.b.a.91.13	✓	16	92.91	even	2
138.5.b.a.91.16	yes	16	4.3	odd	2
414.5.b.b.91.1		16	12.11	even	2
414.5.b.b.91.8		16	276.275	odd	2
1104.5.c.a.1057.2		16	23.22	odd	2	inner
1104.5.c.a.1057.7		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1104.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.19615 q^{3} +33.4782i q^{5} -19.6774i q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q-5.19615 q^{3} +33.4782i q^{5} -19.6774i q^{7} +27.0000 q^{9} -54.0138i q^{11} -159.682 q^{13} -173.958i q^{15} +77.2301i q^{17} -379.657i q^{19} +102.247i q^{21} +(-142.258 + 509.513i) q^{23} -495.789 q^{25} -140.296 q^{27} +1624.58 q^{29} -651.991 q^{31} +280.664i q^{33} +658.763 q^{35} -974.907i q^{37} +829.730 q^{39} -171.454 q^{41} -531.633i q^{43} +903.911i q^{45} +2572.53 q^{47} +2013.80 q^{49} -401.299i q^{51} -2079.78i q^{53} +1808.28 q^{55} +1972.76i q^{57} -8.57641 q^{59} -1060.41i q^{61} -531.289i q^{63} -5345.85i q^{65} +4746.67i q^{67} +(739.195 - 2647.51i) q^{69} -1386.31 q^{71} -2617.74 q^{73} +2576.20 q^{75} -1062.85 q^{77} -3921.65i q^{79} +729.000 q^{81} +10557.1i q^{83} -2585.52 q^{85} -8441.58 q^{87} -11318.8i q^{89} +3142.12i q^{91} +3387.85 q^{93} +12710.2 q^{95} +2774.76i q^{97} -1458.37i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 432 q^{9}+O(q^{10}) \) 16 * q + 432 * q^9	\( 16 q + 432 q^{9} - 208 q^{13} - 840 q^{23} + 1056 q^{25} + 3600 q^{29} - 224 q^{31} + 3264 q^{35} + 2016 q^{39} - 6144 q^{41} - 8880 q^{47} - 13888 q^{49} - 832 q^{55} + 18240 q^{59} + 10584 q^{69} + 30048 q^{71} + 9536 q^{73} + 4176 q^{75} + 14160 q^{77} + 11664 q^{81} - 32496 q^{85} + 8064 q^{87} - 11952 q^{93} + 20064 q^{95}+O(q^{100}) \) 16 * q + 432 * q^9 - 208 * q^13 - 840 * q^23 + 1056 * q^25 + 3600 * q^29 - 224 * q^31 + 3264 * q^35 + 2016 * q^39 - 6144 * q^41 - 8880 * q^47 - 13888 * q^49 - 832 * q^55 + 18240 * q^59 + 10584 * q^69 + 30048 * q^71 + 9536 * q^73 + 4176 * q^75 + 14160 * q^77 + 11664 * q^81 - 32496 * q^85 + 8064 * q^87 - 11952 * q^93 + 20064 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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