LMFDB - Embedded newform 1472.3.f.f.321.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,3,Mod(321,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1472.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1472.f (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:	$40.1090949138$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.613376.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 58x^{2} + 599 $ x^4 + 58*x^2 + 599
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 46)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.4
Root			$-6.67505i$ of defining polynomial
Character	$\chi$	$=$	1472.321
Dual form			1472.3.f.f.321.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{3} +9.43995i q^{5} -3.91016i q^{7} -3.17157 q^{9} +O(q^{10})$	$q+2.41421 q^{3} +9.43995i q^{5} -3.91016i q^{7} -3.17157 q^{9} -3.91016i q^{11} -10.3137 q^{13} +22.7901i q^{15} -17.2603i q^{17} -26.7002i q^{19} -9.43995i q^{21} +(3.10051 - 22.7901i) q^{23} -64.1127 q^{25} -29.3848 q^{27} -35.1421 q^{29} +33.2426 q^{31} -9.43995i q^{33} +36.9117 q^{35} +39.3794i q^{37} -24.8995 q^{39} +13.4853 q^{41} -29.9395i q^{43} -29.9395i q^{45} -31.0416 q^{47} +33.7107 q^{49} -41.6700i q^{51} -61.2207i q^{53} +36.9117 q^{55} -64.4600i q^{57} +26.2010 q^{59} +15.6406i q^{61} +12.4013i q^{63} -97.3609i q^{65} +68.3702i q^{67} +(7.48528 - 55.0201i) q^{69} -111.267 q^{71} +47.8528 q^{73} -154.782 q^{75} -15.2893 q^{77} -117.860i q^{79} -42.3970 q^{81} -74.8487i q^{83} +162.936 q^{85} -84.8406 q^{87} +39.1016i q^{89} +40.3282i q^{91} +80.2548 q^{93} +252.049 q^{95} -66.0797i q^{97} +12.4013i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 24 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 24 * q^9	$ 4 q + 4 q^{3} - 24 q^{9} + 4 q^{13} + 52 q^{23} - 132 q^{25} - 44 q^{27} - 84 q^{29} + 116 q^{31} - 56 q^{35} - 60 q^{39} + 20 q^{41} - 28 q^{47} - 148 q^{49} - 56 q^{55} + 184 q^{59} - 4 q^{69} - 100 q^{71} - 148 q^{73} - 308 q^{75} - 344 q^{77} + 68 q^{81} + 120 q^{85} - 164 q^{87} + 140 q^{93} + 352 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 - 24 * q^9 + 4 * q^13 + 52 * q^23 - 132 * q^25 - 44 * q^27 - 84 * q^29 + 116 * q^31 - 56 * q^35 - 60 * q^39 + 20 * q^41 - 28 * q^47 - 148 * q^49 - 56 * q^55 + 184 * q^59 - 4 * q^69 - 100 * q^71 - 148 * q^73 - 308 * q^75 - 344 * q^77 + 68 * q^81 + 120 * q^85 - 164 * q^87 + 140 * q^93 + 352 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1472\mathbb{Z}\right)^\times$.

$n$	$645$	$833$	$1151$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	2.41421			0.804738			0.402369	−	0.915478i	$-0.368187\pi$
$3$	2.41421			0.804738			0.402369	+	0.915478i	$0.368187\pi$
$4$		0			0
$5$			9.43995i			1.88799i	0.329959	+	0.943995i	$0.392965\pi$
$5$			9.43995i			1.88799i	−0.329959	+	0.943995i	$0.607035\pi$
$6$		0			0
$7$		−	3.91016i		−	0.558594i	−0.960205	−	0.279297i	$-0.909899\pi$
$7$		−	3.91016i		−	0.558594i	0.960205	−	0.279297i	$-0.0901014\pi$
$8$		0			0
$9$	−3.17157			−0.352397
$10$		0			0
$11$		−	3.91016i		−	0.355469i	−0.984078	−	0.177734i	$-0.943123\pi$
$11$		−	3.91016i		−	0.355469i	0.984078	−	0.177734i	$-0.0568768\pi$
$12$		0			0
$13$	−10.3137			−0.793362			−0.396681	−	0.917956i	$-0.629838\pi$
$13$	−10.3137			−0.793362			−0.396681	+	0.917956i	$0.629838\pi$
$14$		0			0
$15$			22.7901i			1.51934i
$16$		0			0
$17$		−	17.2603i		−	1.01531i	−0.861561	−	0.507655i	$-0.830513\pi$
$17$		−	17.2603i		−	1.01531i	0.861561	−	0.507655i	$-0.169487\pi$
$18$		0			0
$19$		−	26.7002i		−	1.40527i	−0.711548	−	0.702637i	$-0.752006\pi$
$19$		−	26.7002i		−	1.40527i	0.711548	−	0.702637i	$-0.247994\pi$
$20$		0			0
$21$		−	9.43995i		−	0.449522i
$22$		0			0
$23$	3.10051	−	22.7901i	0.134805	−	0.990872i
$23$	3.10051	−	22.7901i	0.134805	−	0.990872i
$24$		0			0
$25$	−64.1127			−2.56451
$26$		0			0
$27$	−29.3848			−1.08833
$28$		0			0
$29$	−35.1421			−1.21180			−0.605899	−	0.795542i	$-0.707186\pi$
$29$	−35.1421			−1.21180			−0.605899	+	0.795542i	$0.707186\pi$
$30$		0			0
$31$	33.2426			1.07234			0.536172	−	0.844109i	$-0.319870\pi$
$31$	33.2426			1.07234			0.536172	+	0.844109i	$0.319870\pi$
$32$		0			0
$33$		−	9.43995i		−	0.286059i
$34$		0			0
$35$	36.9117			1.05462
$36$		0			0
$37$			39.3794i			1.06431i	0.846647	+	0.532155i	$0.178618\pi$
$37$			39.3794i			1.06431i	−0.846647	+	0.532155i	$0.821382\pi$
$38$		0			0
$39$	−24.8995			−0.638449
$40$		0			0
$41$	13.4853			0.328909			0.164455	−	0.986385i	$-0.447414\pi$
$41$	13.4853			0.328909			0.164455	+	0.986385i	$0.447414\pi$
$42$		0			0
$43$		−	29.9395i		−	0.696267i	−0.937445	−	0.348134i	$-0.886816\pi$
$43$		−	29.9395i		−	0.696267i	0.937445	−	0.348134i	$-0.113184\pi$
$44$		0			0
$45$		−	29.9395i		−	0.665322i
$46$		0			0
$47$	−31.0416			−0.660460			−0.330230	−	0.943900i	$-0.607126\pi$
$47$	−31.0416			−0.660460			−0.330230	+	0.943900i	$0.607126\pi$
$48$		0			0
$49$	33.7107			0.687973
$50$		0			0
$51$		−	41.6700i		−	0.817058i
$52$		0			0
$53$		−	61.2207i		−	1.15511i	−0.816352	−	0.577554i	$-0.804007\pi$
$53$		−	61.2207i		−	1.15511i	0.816352	−	0.577554i	$-0.195993\pi$
$54$		0			0
$55$	36.9117			0.671122
$56$		0			0
$57$		−	64.4600i		−	1.13088i
$58$		0			0
$59$	26.2010			0.444085			0.222042	−	0.975037i	$-0.428728\pi$
$59$	26.2010			0.444085			0.222042	+	0.975037i	$0.428728\pi$
$60$		0			0
$61$			15.6406i			0.256404i	0.991748	+	0.128202i	$0.0409205\pi$
$61$			15.6406i			0.256404i	−0.991748	+	0.128202i	$0.959079\pi$
$62$		0			0
$63$			12.4013i			0.196847i
$64$		0			0
$65$		−	97.3609i		−	1.49786i
$66$		0			0
$67$			68.3702i			1.02045i	0.860041	+	0.510225i	$0.170438\pi$
$67$			68.3702i			1.02045i	−0.860041	+	0.510225i	$0.829562\pi$
$68$		0			0
$69$	7.48528	−	55.0201i	0.108482	−	0.797392i
$70$		0			0
$71$	−111.267			−1.56714			−0.783571	−	0.621303i	$-0.786604\pi$
$71$	−111.267			−1.56714			−0.783571	+	0.621303i	$0.786604\pi$
$72$		0			0
$73$	47.8528			0.655518			0.327759	−	0.944761i	$-0.393707\pi$
$73$	47.8528			0.655518			0.327759	+	0.944761i	$0.393707\pi$
$74$		0			0
$75$	−154.782			−2.06376
$76$		0			0
$77$	−15.2893			−0.198563
$78$		0			0
$79$		−	117.860i		−	1.49190i	−0.665999	−	0.745952i	$-0.731995\pi$
$79$		−	117.860i		−	1.49190i	0.665999	−	0.745952i	$-0.268005\pi$
$80$		0			0
$81$	−42.3970			−0.523419
$82$		0			0
$83$		−	74.8487i		−	0.901792i	−0.892576	−	0.450896i	$-0.851105\pi$
$83$		−	74.8487i		−	0.901792i	0.892576	−	0.450896i	$-0.148895\pi$
$84$		0			0
$85$	162.936			1.91689
$86$		0			0
$87$	−84.8406			−0.975180
$88$		0			0
$89$			39.1016i			0.439343i	0.975574	+	0.219672i	$0.0704986\pi$
$89$			39.1016i			0.439343i	−0.975574	+	0.219672i	$0.929501\pi$
$90$		0			0
$91$			40.3282i			0.443167i
$92$		0			0
$93$	80.2548			0.862955
$94$		0			0
$95$	252.049			2.65314
$96$		0			0
$97$		−	66.0797i		−	0.681234i	−0.940202	−	0.340617i	$-0.889364\pi$
$97$		−	66.0797i		−	0.681234i	0.940202	−	0.340617i	$-0.110636\pi$
$98$		0			0
$99$			12.4013i			0.125266i
$100$		0			0
$101$	−101.338			−1.00335			−0.501674	−	0.865057i	$-0.667282\pi$
$101$	−101.338			−1.00335			−0.501674	+	0.865057i	$0.667282\pi$
$102$		0			0
$103$			101.549i			0.985912i	0.870054	+	0.492956i	$0.164084\pi$
$103$			101.549i			0.985912i	−0.870054	+	0.492956i	$0.835916\pi$
$104$		0			0
$105$	89.1127			0.848692
$106$		0			0
$107$			75.5196i			0.705791i	0.935663	+	0.352895i	$0.114803\pi$
$107$			75.5196i			0.705791i	−0.935663	+	0.352895i	$0.885197\pi$
$108$		0			0
$109$		−	138.638i		−	1.27191i	−0.771727	−	0.635954i	$-0.780607\pi$
$109$		−	138.638i		−	1.27191i	0.771727	−	0.635954i	$-0.219393\pi$
$110$		0			0
$111$			95.0704i			0.856490i
$112$		0			0
$113$			180.701i			1.59912i	0.600585	+	0.799561i	$0.294935\pi$
$113$			180.701i			1.59912i	−0.600585	+	0.799561i	$0.705065\pi$
$114$		0			0
$115$	215.137	+	29.2686i	1.87076	+	0.254510i
$116$		0			0
$117$	32.7107			0.279578
$118$		0			0
$119$	−67.4903			−0.567146
$120$		0			0
$121$	105.711			0.873642
$122$		0			0
$123$	32.5563			0.264686
$124$		0			0
$125$		−	369.222i		−	2.95378i
$126$		0			0
$127$	36.3898			0.286534			0.143267	−	0.989684i	$-0.454239\pi$
$127$	36.3898			0.286534			0.143267	+	0.989684i	$0.454239\pi$
$128$		0			0
$129$		−	72.2803i		−	0.560313i
$130$		0			0
$131$	65.3848			0.499120			0.249560	−	0.968359i	$-0.419714\pi$
$131$	65.3848			0.499120			0.249560	+	0.968359i	$0.419714\pi$
$132$		0			0
$133$	−104.402			−0.784978
$134$		0			0
$135$		−	277.391i		−	2.05475i
$136$		0			0
$137$		−	65.8018i		−	0.480305i	−0.970735	−	0.240152i	$-0.922803\pi$
$137$		−	65.8018i		−	0.480305i	0.970735	−	0.240152i	$-0.0771974\pi$
$138$		0			0
$139$	−29.4092			−0.211577			−0.105788	−	0.994389i	$-0.533737\pi$
$139$	−29.4092			−0.211577			−0.105788	+	0.994389i	$0.533737\pi$
$140$		0			0
$141$	−74.9411			−0.531497
$142$		0			0
$143$			40.3282i			0.282015i
$144$		0			0
$145$		−	331.740i		−	2.28786i
$146$		0			0
$147$	81.3848			0.553638
$148$		0			0
$149$			3.23928i			0.0217401i	0.999941	+	0.0108701i	$0.00346012\pi$
$149$			3.23928i			0.0217401i	−0.999941	+	0.0108701i	$0.996540\pi$
$150$		0			0
$151$	−2.41421			−0.0159882			−0.00799408	−	0.999968i	$-0.502545\pi$
$151$	−2.41421			−0.0159882			−0.00799408	+	0.999968i	$0.502545\pi$
$152$		0			0
$153$			54.7422i			0.357792i
$154$		0			0
$155$			313.809i			2.02457i
$156$		0			0
$157$		−	73.6221i		−	0.468931i	−0.972124	−	0.234465i	$-0.924666\pi$
$157$		−	73.6221i		−	0.468931i	0.972124	−	0.234465i	$-0.0753339\pi$
$158$		0			0
$159$		−	147.800i		−	0.929559i
$160$		0			0
$161$	−89.1127	−	12.1235i	−0.553495	−	0.0753010i
$162$		0			0
$163$	−125.586			−0.770465			−0.385232	−	0.922820i	$-0.625879\pi$
$163$	−125.586			−0.770465			−0.385232	+	0.922820i	$0.625879\pi$
$164$		0			0
$165$	89.1127			0.540077
$166$		0			0
$167$	−92.8284			−0.555859			−0.277929	−	0.960601i	$-0.589648\pi$
$167$	−92.8284			−0.555859			−0.277929	+	0.960601i	$0.589648\pi$
$168$		0			0
$169$	−62.6274			−0.370576
$170$		0			0
$171$			84.6817i			0.495215i
$172$		0			0
$173$	6.71068			0.0387900			0.0193950	−	0.999812i	$-0.493826\pi$
$173$	6.71068			0.0387900			0.0193950	+	0.999812i	$0.493826\pi$
$174$		0			0
$175$			250.691i			1.43252i
$176$		0			0
$177$	63.2548			0.357372
$178$		0			0
$179$	−278.262			−1.55454			−0.777268	−	0.629170i	$-0.783395\pi$
$179$	−278.262			−1.55454			−0.777268	+	0.629170i	$0.783395\pi$
$180$		0			0
$181$		−	53.6783i		−	0.296565i	−0.988945	−	0.148283i	$-0.952625\pi$
$181$		−	53.6783i		−	0.296565i	0.988945	−	0.148283i	$-0.0473745\pi$
$182$		0			0
$183$			37.7598i			0.206338i
$184$		0			0
$185$	−371.740			−2.00941
$186$		0			0
$187$	−67.4903			−0.360911
$188$		0			0
$189$			114.899i			0.607932i
$190$		0			0
$191$			168.692i			0.883207i	0.897210	+	0.441603i	$0.145590\pi$
$191$			168.692i			0.883207i	−0.897210	+	0.441603i	$0.854410\pi$
$192$		0			0
$193$	−95.3188			−0.493880			−0.246940	−	0.969031i	$-0.579425\pi$
$193$	−95.3188			−0.493880			−0.246940	+	0.969031i	$0.579425\pi$
$194$		0			0
$195$		−	235.050i		−	1.20538i
$196$		0			0
$197$	−249.083			−1.26438			−0.632191	−	0.774813i	$-0.717844\pi$
$197$	−249.083			−1.26438			−0.632191	+	0.774813i	$0.717844\pi$
$198$		0			0
$199$			76.1905i			0.382867i	0.981506	+	0.191433i	$0.0613136\pi$
$199$			76.1905i			0.382867i	−0.981506	+	0.191433i	$0.938686\pi$
$200$		0			0
$201$			165.060i			0.821195i
$202$		0			0
$203$			137.411i			0.676903i
$204$		0			0
$205$			127.300i			0.620978i
$206$		0			0
$207$	−9.83348	+	72.2803i	−0.0475047	+	0.349180i
$208$		0			0
$209$	−104.402			−0.499531
$210$		0			0
$211$	71.1716			0.337306			0.168653	−	0.985675i	$-0.446058\pi$
$211$	71.1716			0.337306			0.168653	+	0.985675i	$0.446058\pi$
$212$		0			0
$213$	−268.622			−1.26114
$214$		0			0
$215$	282.627			1.31455
$216$		0			0
$217$		−	129.984i		−	0.599004i
$218$		0			0
$219$	115.527			0.527520
$220$		0			0
$221$			178.017i			0.805508i
$222$		0			0
$223$	−84.0833			−0.377055			−0.188527	−	0.982068i	$-0.560371\pi$
$223$	−84.0833			−0.377055			−0.188527	+	0.982068i	$0.560371\pi$
$224$		0			0
$225$	203.338			0.903725
$226$		0			0
$227$		−	263.092i		−	1.15900i	−0.814974	−	0.579498i	$-0.803249\pi$
$227$		−	263.092i		−	1.15900i	0.814974	−	0.579498i	$-0.196751\pi$
$228$		0			0
$229$		−	141.321i		−	0.617124i	−0.951204	−	0.308562i	$-0.900152\pi$
$229$		−	141.321i		−	0.617124i	0.951204	−	0.308562i	$-0.0998477\pi$
$230$		0			0
$231$	−36.9117			−0.159791
$232$		0			0
$233$	−353.784			−1.51839			−0.759193	−	0.650866i	$-0.774406\pi$
$233$	−353.784			−1.51839			−0.759193	+	0.650866i	$0.774406\pi$
$234$		0			0
$235$		−	293.032i		−	1.24694i
$236$		0			0
$237$		−	284.540i		−	1.20059i
$238$		0			0
$239$	−272.286			−1.13927			−0.569637	−	0.821897i	$-0.692916\pi$
$239$	−272.286			−1.13927			−0.569637	+	0.821897i	$0.692916\pi$
$240$		0			0
$241$		−	443.678i		−	1.84099i	−0.390758	−	0.920493i	$-0.627787\pi$
$241$		−	443.678i		−	1.84099i	0.390758	−	0.920493i	$-0.372213\pi$
$242$		0			0
$243$	162.108			0.667110
$244$		0			0
$245$			318.227i			1.29889i
$246$		0			0
$247$			275.378i			1.11489i
$248$		0			0
$249$		−	180.701i		−	0.725706i
$250$		0			0
$251$		−	194.607i		−	0.775326i	−0.921801	−	0.387663i	$-0.873282\pi$
$251$		−	194.607i		−	0.775326i	0.921801	−	0.387663i	$-0.126718\pi$
$252$		0			0
$253$	−89.1127	−	12.1235i	−0.352224	−	0.0479188i
$254$		0			0
$255$	393.362			1.54260
$256$		0			0
$257$	−22.1371			−0.0861365			−0.0430683	−	0.999072i	$-0.513713\pi$
$257$	−22.1371			−0.0861365			−0.0430683	+	0.999072i	$0.513713\pi$
$258$		0			0
$259$	153.980			0.594517
$260$		0			0
$261$	111.456			0.427034
$262$		0			0
$263$		−	425.076i		−	1.61626i	−0.589006	−	0.808129i	$-0.700481\pi$
$263$		−	425.076i		−	1.61626i	0.589006	−	0.808129i	$-0.299519\pi$
$264$		0			0
$265$	577.921			2.18083
$266$		0			0
$267$			94.3995i			0.353556i
$268$		0			0
$269$	−411.215			−1.52868			−0.764341	−	0.644813i	$-0.776935\pi$
$269$	−411.215			−1.52868			−0.764341	+	0.644813i	$0.776935\pi$
$270$		0			0
$271$	−98.8183			−0.364643			−0.182322	−	0.983239i	$-0.558361\pi$
$271$	−98.8183			−0.364643			−0.182322	+	0.983239i	$0.558361\pi$
$272$		0			0
$273$			97.3609i			0.356633i
$274$		0			0
$275$			250.691i			0.911602i
$276$		0			0
$277$	207.319			0.748443			0.374222	−	0.927339i	$-0.377910\pi$
$277$	207.319			0.748443			0.374222	+	0.927339i	$0.377910\pi$
$278$		0			0
$279$	−105.431			−0.377891
$280$		0			0
$281$			31.0034i			0.110332i	0.998477	+	0.0551661i	$0.0175688\pi$
$281$			31.0034i			0.110332i	−0.998477	+	0.0551661i	$0.982431\pi$
$282$		0			0
$283$		−	225.888i		−	0.798191i	−0.916909	−	0.399095i	$-0.869324\pi$
$283$		−	225.888i		−	0.798191i	0.916909	−	0.399095i	$-0.130676\pi$
$284$		0			0
$285$	608.500			2.13509
$286$		0			0
$287$		−	52.7296i		−	0.183727i
$288$		0			0
$289$	−8.91674			−0.0308538
$290$		0			0
$291$		−	159.530i		−	0.548215i
$292$		0			0
$293$			162.099i			0.553238i	0.960980	+	0.276619i	$0.0892140\pi$
$293$			162.099i			0.553238i	−0.960980	+	0.276619i	$0.910786\pi$
$294$		0			0
$295$			247.336i			0.838428i
$296$		0			0
$297$			114.899i			0.386866i
$298$		0			0
$299$	−31.9777	+	235.050i	−0.106949	+	0.786121i
$300$		0			0
$301$	−117.068			−0.388931
$302$		0			0
$303$	−244.652			−0.807432
$304$		0			0
$305$	−147.647			−0.484088
$306$		0			0
$307$	−342.299			−1.11498			−0.557490	−	0.830184i	$-0.688235\pi$
$307$	−342.299			−1.11498			−0.557490	+	0.830184i	$0.688235\pi$
$308$		0			0
$309$			245.161i			0.793401i
$310$		0			0
$311$	464.654			1.49406			0.747032	−	0.664788i	$-0.231478\pi$
$311$	464.654			1.49406			0.747032	+	0.664788i	$0.231478\pi$
$312$		0			0
$313$		−	95.1855i		−	0.304107i	−0.988372	−	0.152054i	$-0.951411\pi$
$313$		−	95.1855i		−	0.304107i	0.988372	−	0.152054i	$-0.0485886\pi$
$314$		0			0
$315$	−117.068			−0.371645
$316$		0			0
$317$	210.142			0.662909			0.331454	−	0.943471i	$-0.392461\pi$
$317$	210.142			0.662909			0.331454	+	0.943471i	$0.392461\pi$
$318$		0			0
$319$			137.411i			0.430756i
$320$		0			0
$321$			182.320i			0.567977i
$322$		0			0
$323$	−460.853			−1.42679
$324$		0			0
$325$	661.240			2.03458
$326$		0			0
$327$		−	334.701i		−	1.02355i
$328$		0			0
$329$			121.378i			0.368929i
$330$		0			0
$331$	−66.2822			−0.200248			−0.100124	−	0.994975i	$-0.531924\pi$
$331$	−66.2822			−0.200248			−0.100124	+	0.994975i	$0.531924\pi$
$332$		0			0
$333$		−	124.895i		−	0.375059i
$334$		0			0
$335$	−645.411			−1.92660
$336$		0			0
$337$			453.348i			1.34525i	0.739985	+	0.672623i	$0.234832\pi$
$337$			453.348i			1.34525i	−0.739985	+	0.672623i	$0.765168\pi$
$338$		0			0
$339$			436.250i			1.28687i
$340$		0			0
$341$		−	129.984i		−	0.381185i
$342$		0			0
$343$		−	323.412i		−	0.942891i
$344$		0			0
$345$	519.387	+	70.6607i	1.50547	+	0.204814i
$346$		0			0
$347$	291.750			0.840779			0.420389	−	0.907344i	$-0.361893\pi$
$347$	291.750			0.840779			0.420389	+	0.907344i	$0.361893\pi$
$348$		0			0
$349$	−150.951			−0.432525			−0.216263	−	0.976335i	$-0.569387\pi$
$349$	−150.951			−0.432525			−0.216263	+	0.976335i	$0.569387\pi$
$350$		0			0
$351$	303.066			0.863436
$352$		0			0
$353$	377.059			1.06816			0.534078	−	0.845435i	$-0.320659\pi$
$353$	377.059			1.06816			0.534078	+	0.845435i	$0.320659\pi$
$354$		0			0
$355$		−	1050.36i		−	2.95875i
$356$		0			0
$357$	−162.936			−0.456404
$358$		0			0
$359$		−	352.240i		−	0.981169i	−0.871394	−	0.490584i	$-0.836783\pi$
$359$		−	352.240i		−	0.981169i	0.871394	−	0.490584i	$-0.163217\pi$
$360$		0			0
$361$	−351.902			−0.974797
$362$		0			0
$363$	255.208			0.703053
$364$		0			0
$365$			451.728i			1.23761i
$366$		0			0
$367$			320.843i			0.874232i	0.899405	+	0.437116i	$0.144000\pi$
$367$			320.843i			0.874232i	−0.899405	+	0.437116i	$0.856000\pi$
$368$		0			0
$369$	−42.7696			−0.115907
$370$		0			0
$371$	−239.383			−0.645236
$372$		0			0
$373$			156.128i			0.418575i	0.977854	+	0.209287i	$0.0671144\pi$
$373$			156.128i			0.418575i	−0.977854	+	0.209287i	$0.932886\pi$
$374$		0			0
$375$		−	891.381i		−	2.37702i
$376$		0			0
$377$	362.446			0.961395
$378$		0			0
$379$			451.220i			1.19055i	0.803520	+	0.595277i	$0.202958\pi$
$379$			451.220i			1.19055i	−0.803520	+	0.595277i	$0.797042\pi$
$380$		0			0
$381$	87.8528			0.230585
$382$		0			0
$383$			423.178i			1.10490i	0.833545	+	0.552452i	$0.186308\pi$
$383$			423.178i			1.10490i	−0.833545	+	0.552452i	$0.813692\pi$
$384$		0			0
$385$		−	144.330i		−	0.374884i
$386$		0			0
$387$			94.9553i			0.245363i
$388$		0			0
$389$			11.0596i			0.0284308i	0.999899	+	0.0142154i	$0.00452506\pi$
$389$			11.0596i			0.0284308i	−0.999899	+	0.0142154i	$0.995475\pi$
$390$		0			0
$391$	−393.362	−	53.5155i	−1.00604	−	0.136868i
$392$		0			0
$393$	157.853			0.401661
$394$		0			0
$395$	1112.60			2.81670
$396$		0			0
$397$	332.578			0.837727			0.418864	−	0.908049i	$-0.362429\pi$
$397$	332.578			0.837727			0.418864	+	0.908049i	$0.362429\pi$
$398$		0			0
$399$	−252.049			−0.631701
$400$		0			0
$401$		−	51.7808i		−	0.129129i	−0.997914	−	0.0645646i	$-0.979434\pi$
$401$		−	51.7808i		−	0.129129i	0.997914	−	0.0645646i	$-0.0205658\pi$
$402$		0			0
$403$	−342.855			−0.850757
$404$		0			0
$405$		−	400.225i		−	0.988211i
$406$		0			0
$407$	153.980			0.378329
$408$		0			0
$409$	256.088			0.626133			0.313066	−	0.949731i	$-0.398644\pi$
$409$	256.088			0.626133			0.313066	+	0.949731i	$0.398644\pi$
$410$		0			0
$411$		−	158.860i		−	0.386520i
$412$		0			0
$413$		−	102.450i		−	0.248063i
$414$		0			0
$415$	706.569			1.70257
$416$		0			0
$417$	−71.0000			−0.170264
$418$		0			0
$419$			241.644i			0.576715i	0.957523	+	0.288358i	$0.0931092\pi$
$419$			241.644i			0.576715i	−0.957523	+	0.288358i	$0.906891\pi$
$420$		0			0
$421$			312.860i			0.743136i	0.928406	+	0.371568i	$0.121180\pi$
$421$			312.860i			0.743136i	−0.928406	+	0.371568i	$0.878820\pi$
$422$		0			0
$423$	98.4508			0.232744
$424$		0			0
$425$			1106.60i			2.60377i
$426$		0			0
$427$	61.1573			0.143225
$428$		0			0
$429$			97.3609i			0.226949i
$430$		0			0
$431$			672.412i			1.56012i	0.625704	+	0.780060i	$0.284812\pi$
$431$			672.412i			1.56012i	−0.625704	+	0.780060i	$0.715188\pi$
$432$		0			0
$433$			395.414i			0.913197i	0.889673	+	0.456598i	$0.150932\pi$
$433$			395.414i			0.913197i	−0.889673	+	0.456598i	$0.849068\pi$
$434$		0			0
$435$		−	800.891i		−	1.84113i
$436$		0			0
$437$	−608.500	−	82.7842i	−1.39245	−	0.189437i
$438$		0			0
$439$	836.997			1.90660			0.953300	−	0.302026i	$-0.0976630\pi$
$439$	836.997			1.90660			0.953300	+	0.302026i	$0.0976630\pi$
$440$		0			0
$441$	−106.916			−0.242440
$442$		0			0
$443$	227.905			0.514457			0.257229	−	0.966351i	$-0.417191\pi$
$443$	227.905			0.514457			0.257229	+	0.966351i	$0.417191\pi$
$444$		0			0
$445$	−369.117			−0.829476
$446$		0			0
$447$			7.82031i			0.0174951i
$448$		0			0
$449$	−439.019			−0.977771			−0.488886	−	0.872348i	$-0.662596\pi$
$449$	−439.019			−0.977771			−0.488886	+	0.872348i	$0.662596\pi$
$450$		0			0
$451$		−	52.7296i		−	0.116917i
$452$		0			0
$453$	−5.82843			−0.0128663
$454$		0			0
$455$	−380.696			−0.836695
$456$		0			0
$457$			40.1654i			0.0878893i	0.999034	+	0.0439447i	$0.0139925\pi$
$457$			40.1654i			0.0878893i	−0.999034	+	0.0439447i	$0.986007\pi$
$458$		0			0
$459$			507.189i			1.10499i
$460$		0			0
$461$	−65.6762			−0.142465			−0.0712323	−	0.997460i	$-0.522693\pi$
$461$	−65.6762			−0.142465			−0.0712323	+	0.997460i	$0.522693\pi$
$462$		0			0
$463$	−673.436			−1.45450			−0.727252	−	0.686370i	$-0.759203\pi$
$463$	−673.436			−1.45450			−0.727252	+	0.686370i	$0.759203\pi$
$464$		0			0
$465$			757.602i			1.62925i
$466$		0			0
$467$		−	310.685i		−	0.665278i	−0.943054	−	0.332639i	$-0.892061\pi$
$467$		−	310.685i		−	0.665278i	0.943054	−	0.332639i	$-0.107939\pi$
$468$		0			0
$469$	267.338			0.570017
$470$		0			0
$471$		−	177.739i		−	0.377366i
$472$		0			0
$473$	−117.068			−0.247501
$474$		0			0
$475$			1711.82i			3.60384i
$476$		0			0
$477$			194.166i			0.407057i
$478$		0			0
$479$		−	230.469i		−	0.481146i	−0.970631	−	0.240573i	$-0.922665\pi$
$479$		−	230.469i		−	0.481146i	0.970631	−	0.240573i	$-0.0773354\pi$
$480$		0			0
$481$		−	406.148i		−	0.844383i
$482$		0			0
$483$	−215.137	−	29.2686i	−0.445418	−	0.0605976i
$484$		0			0
$485$	623.789			1.28616
$486$		0			0
$487$	372.360			0.764599			0.382299	−	0.924039i	$-0.375132\pi$
$487$	372.360			0.764599			0.382299	+	0.924039i	$0.375132\pi$
$488$		0			0
$489$	−303.191			−0.620022
$490$		0			0
$491$	−258.026			−0.525512			−0.262756	−	0.964862i	$-0.584631\pi$
$491$	−258.026			−0.525512			−0.262756	+	0.964862i	$0.584631\pi$
$492$		0			0
$493$			606.563i			1.23035i
$494$		0			0
$495$	−117.068			−0.236501
$496$		0			0
$497$			435.071i			0.875395i
$498$		0			0
$499$	372.747			0.746988			0.373494	−	0.927632i	$-0.378160\pi$
$499$	372.747			0.746988			0.373494	+	0.927632i	$0.378160\pi$
$500$		0			0
$501$	−224.108			−0.447321
$502$		0			0
$503$		−	611.191i		−	1.21509i	−0.794284	−	0.607546i	$-0.792154\pi$
$503$		−	611.191i		−	1.21509i	0.794284	−	0.607546i	$-0.207846\pi$
$504$		0			0
$505$		−	956.627i		−	1.89431i
$506$		0			0
$507$	−151.196			−0.298217
$508$		0			0
$509$	−357.985			−0.703310			−0.351655	−	0.936130i	$-0.614381\pi$
$509$	−357.985			−0.703310			−0.351655	+	0.936130i	$0.614381\pi$
$510$		0			0
$511$		−	187.112i		−	0.366168i
$512$		0			0
$513$			784.580i			1.52940i
$514$		0			0
$515$	−958.617			−1.86139
$516$		0			0
$517$			121.378i			0.234773i
$518$		0			0
$519$	16.2010			0.0312158
$520$		0			0
$521$			123.227i			0.236521i	0.992983	+	0.118261i	$0.0377318\pi$
$521$			123.227i			0.236521i	−0.992983	+	0.118261i	$0.962268\pi$
$522$		0			0
$523$		−	246.110i		−	0.470573i	−0.971926	−	0.235286i	$-0.924397\pi$
$523$		−	246.110i		−	0.470573i	0.971926	−	0.235286i	$-0.0756028\pi$
$524$		0			0
$525$			605.221i			1.15280i
$526$		0			0
$527$		−	573.777i		−	1.08876i
$528$		0			0
$529$	−509.774	−	141.321i	−0.963655	−	0.267148i
$530$		0			0
$531$	−83.0984			−0.156494
$532$		0			0
$533$	−139.083			−0.260944
$534$		0			0
$535$	−712.902			−1.33253
$536$		0			0
$537$	−671.784			−1.25099
$538$		0			0
$539$		−	131.814i		−	0.244553i
$540$		0			0
$541$	425.922			0.787286			0.393643	−	0.919263i	$-0.371215\pi$
$541$	425.922			0.787286			0.393643	+	0.919263i	$0.371215\pi$
$542$		0			0
$543$		−	129.591i		−	0.238657i
$544$		0			0
$545$	1308.74			2.40135
$546$		0			0
$547$	715.070			1.30726			0.653629	−	0.756815i	$-0.273246\pi$
$547$	715.070			1.30726			0.653629	+	0.756815i	$0.273246\pi$
$548$		0			0
$549$		−	49.6054i		−	0.0903559i
$550$		0			0
$551$			938.303i			1.70291i
$552$		0			0
$553$	−460.853			−0.833369
$554$		0			0
$555$	−897.460			−1.61705
$556$		0			0
$557$		−	325.539i		−	0.584451i	−0.956349	−	0.292226i	$-0.905604\pi$
$557$		−	325.539i		−	0.584451i	0.956349	−	0.292226i	$-0.0943958\pi$
$558$		0			0
$559$			308.787i			0.552392i
$560$		0			0
$561$	−162.936			−0.290439
$562$		0			0
$563$			636.780i			1.13105i	0.824732	+	0.565524i	$0.191326\pi$
$563$			636.780i			1.13105i	−0.824732	+	0.565524i	$0.808674\pi$
$564$		0			0
$565$	−1705.81			−3.01913
$566$		0			0
$567$			165.779i			0.292379i
$568$		0			0
$569$			359.226i			0.631329i	0.948871	+	0.315665i	$0.102227\pi$
$569$			359.226i			0.631329i	−0.948871	+	0.315665i	$0.897773\pi$
$570$		0			0
$571$			766.371i			1.34216i	0.741387	+	0.671078i	$0.234168\pi$
$571$			766.371i			1.34216i	−0.741387	+	0.671078i	$0.765832\pi$
$572$		0			0
$573$			407.260i			0.710750i
$574$		0			0
$575$	−198.782	+	1461.13i	−0.345707	+	2.54110i
$576$		0			0
$577$	−966.661			−1.67532			−0.837661	−	0.546190i	$-0.816078\pi$
$577$	−966.661			−1.67532			−0.837661	+	0.546190i	$0.816078\pi$
$578$		0			0
$579$	−230.120			−0.397444
$580$		0			0
$581$	−292.670			−0.503735
$582$		0			0
$583$	−239.383			−0.410605
$584$		0			0
$585$			308.787i			0.527841i
$586$		0			0
$587$	855.933			1.45815			0.729074	−	0.684435i	$-0.239951\pi$
$587$	855.933			1.45815			0.729074	+	0.684435i	$0.239951\pi$
$588$		0			0
$589$		−	887.586i		−	1.50694i
$590$		0			0
$591$	−601.340			−1.01750
$592$		0			0
$593$	159.563			0.269078			0.134539	−	0.990908i	$-0.457045\pi$
$593$	159.563			0.269078			0.134539	+	0.990908i	$0.457045\pi$
$594$		0			0
$595$		−	637.106i		−	1.07077i
$596$		0			0
$597$			183.940i			0.308107i
$598$		0			0
$599$	−46.1320			−0.0770151			−0.0385075	−	0.999258i	$-0.512260\pi$
$599$	−46.1320			−0.0770151			−0.0385075	+	0.999258i	$0.512260\pi$
$600$		0			0
$601$	−283.010			−0.470899			−0.235449	−	0.971887i	$-0.575656\pi$
$601$	−283.010			−0.470899			−0.235449	+	0.971887i	$0.575656\pi$
$602$		0			0
$603$		−	216.841i		−	0.359604i
$604$		0			0
$605$			997.904i			1.64943i
$606$		0			0
$607$	690.250			1.13715			0.568575	−	0.822632i	$-0.307495\pi$
$607$	690.250			1.13715			0.568575	+	0.822632i	$0.307495\pi$
$608$		0			0
$609$			331.740i			0.544729i
$610$		0			0
$611$	320.154			0.523984
$612$		0			0
$613$		−	247.058i		−	0.403032i	−0.979485	−	0.201516i	$-0.935413\pi$
$613$		−	247.058i		−	0.403032i	0.979485	−	0.201516i	$-0.0645867\pi$
$614$		0			0
$615$			307.330i			0.499724i
$616$		0			0
$617$			548.303i			0.888660i	0.895863	+	0.444330i	$0.146558\pi$
$617$			548.303i			0.888660i	−0.895863	+	0.444330i	$0.853442\pi$
$618$		0			0
$619$		−	1090.68i		−	1.76201i	−0.473108	−	0.881005i	$-0.656868\pi$
$619$		−	1090.68i		−	1.76201i	0.473108	−	0.881005i	$-0.343132\pi$
$620$		0			0
$621$	−91.1076	+	669.681i	−0.146711	+	1.07839i
$622$		0			0
$623$	152.893			0.245414
$624$		0			0
$625$	1882.62			3.01219
$626$		0			0
$627$	−252.049			−0.401992
$628$		0			0
$629$	679.700			1.08060
$630$		0			0
$631$		−	132.159i		−	0.209444i	−0.994502	−	0.104722i	$-0.966605\pi$
$631$		−	132.159i		−	0.209444i	0.994502	−	0.104722i	$-0.0333953\pi$
$632$		0			0
$633$	171.823			0.271443
$634$		0			0
$635$			343.518i			0.540974i
$636$		0			0
$637$	−347.682			−0.545812
$638$		0			0
$639$	352.891			0.552256
$640$		0			0
$641$		−	349.786i		−	0.545688i	−0.962058	−	0.272844i	$-0.912036\pi$
$641$		−	349.786i		−	0.545688i	0.962058	−	0.272844i	$-0.0879644\pi$
$642$		0			0
$643$			147.129i			0.228817i	0.993434	+	0.114408i	$0.0364972\pi$
$643$			147.129i			0.228817i	−0.993434	+	0.114408i	$0.963503\pi$
$644$		0			0
$645$	682.323			1.05787
$646$		0			0
$647$	966.252			1.49343			0.746717	−	0.665142i	$-0.231629\pi$
$647$	966.252			1.49343			0.746717	+	0.665142i	$0.231629\pi$
$648$		0			0
$649$		−	102.450i		−	0.157858i
$650$		0			0
$651$		−	313.809i		−	0.482041i
$652$		0			0
$653$	817.378			1.25173			0.625863	−	0.779933i	$-0.284747\pi$
$653$	817.378			1.25173			0.625863	+	0.779933i	$0.284747\pi$
$654$		0			0
$655$			617.229i			0.942335i
$656$		0			0
$657$	−151.769			−0.231003
$658$		0			0
$659$		−	1053.82i		−	1.59913i	−0.600581	−	0.799564i	$-0.705064\pi$
$659$		−	1053.82i		−	1.59913i	0.600581	−	0.799564i	$-0.294936\pi$
$660$		0			0
$661$			1162.41i			1.75856i	0.476305	+	0.879280i	$0.341976\pi$
$661$			1162.41i			1.75856i	−0.476305	+	0.879280i	$0.658024\pi$
$662$		0			0
$663$			429.772i			0.648223i
$664$		0			0
$665$		−	985.550i		−	1.48203i
$666$		0			0
$667$	−108.958	+	800.891i	−0.163356	+	1.20074i
$668$		0			0
$669$	−202.995			−0.303430
$670$		0			0
$671$	61.1573			0.0911435
$672$		0			0
$673$	−647.461			−0.962052			−0.481026	−	0.876706i	$-0.659736\pi$
$673$	−647.461			−0.962052			−0.481026	+	0.876706i	$0.659736\pi$
$674$		0			0
$675$	1883.94			2.79102
$676$		0			0
$677$			544.278i			0.803956i	0.915649	+	0.401978i	$0.131677\pi$
$677$			544.278i			0.803956i	−0.915649	+	0.401978i	$0.868323\pi$
$678$		0			0
$679$	−258.382			−0.380533
$680$		0			0
$681$		−	635.160i		−	0.932688i
$682$		0			0
$683$	−77.8314			−0.113955			−0.0569776	−	0.998375i	$-0.518146\pi$
$683$	−77.8314			−0.113955			−0.0569776	+	0.998375i	$0.518146\pi$
$684$		0			0
$685$	621.166			0.906811
$686$		0			0
$687$		−	341.180i		−	0.496623i
$688$		0			0
$689$			631.413i			0.916419i
$690$		0			0
$691$	223.769			0.323833			0.161917	−	0.986804i	$-0.448232\pi$
$691$	223.769			0.323833			0.161917	+	0.986804i	$0.448232\pi$
$692$		0			0
$693$	48.4912			0.0699729
$694$		0			0
$695$		−	277.621i		−	0.399455i
$696$		0			0
$697$		−	232.760i		−	0.333945i
$698$		0			0
$699$	−854.110			−1.22190
$700$		0			0
$701$			376.995i			0.537795i	0.963169	+	0.268898i	$0.0866594\pi$
$701$			376.995i			0.537795i	−0.963169	+	0.268898i	$0.913341\pi$
$702$		0			0
$703$	1051.44			1.49565
$704$		0			0
$705$		−	707.441i		−	1.00346i
$706$		0			0
$707$			396.248i			0.560464i
$708$		0			0
$709$		−	1182.12i		−	1.66731i	−0.552287	−	0.833654i	$-0.686245\pi$
$709$		−	1182.12i		−	1.66731i	0.552287	−	0.833654i	$-0.313755\pi$
$710$		0			0
$711$			373.803i			0.525743i
$712$		0			0
$713$	103.069	−	757.602i	0.144557	−	1.06256i
$714$		0			0
$715$	−380.696			−0.532443
$716$		0			0
$717$	−657.357			−0.916817
$718$		0			0
$719$	93.9941			0.130729			0.0653645	−	0.997861i	$-0.479179\pi$
$719$	93.9941			0.130729			0.0653645	+	0.997861i	$0.479179\pi$
$720$		0			0
$721$	397.072			0.550724
$722$		0			0
$723$		−	1071.13i		−	1.48151i
$724$		0			0
$725$	2253.06			3.10767
$726$		0			0
$727$		−	1303.61i		−	1.79314i	−0.442900	−	0.896571i	$-0.646050\pi$
$727$		−	1303.61i		−	1.79314i	0.442900	−	0.896571i	$-0.353950\pi$
$728$		0			0
$729$	772.935			1.06027
$730$		0			0
$731$	−516.764			−0.706927
$732$		0			0
$733$		−	635.946i		−	0.867594i	−0.901011	−	0.433797i	$-0.857174\pi$
$733$		−	635.946i		−	0.867594i	0.901011	−	0.433797i	$-0.142826\pi$
$734$		0			0
$735$			768.268i			1.04526i
$736$		0			0
$737$	267.338			0.362738
$738$		0			0
$739$	−200.380			−0.271150			−0.135575	−	0.990767i	$-0.543288\pi$
$739$	−200.380			−0.271150			−0.135575	+	0.990767i	$0.543288\pi$
$740$		0			0
$741$			664.822i			0.897196i
$742$		0			0
$743$		−	1210.79i		−	1.62959i	−0.579748	−	0.814796i	$-0.696849\pi$
$743$		−	1210.79i		−	1.62959i	0.579748	−	0.814796i	$-0.303151\pi$
$744$		0			0
$745$	−30.5786			−0.0410452
$746$		0			0
$747$			237.388i			0.317789i
$748$		0			0
$749$	295.294			0.394250
$750$		0			0
$751$			983.883i			1.31010i	0.755587	+	0.655048i	$0.227352\pi$
$751$			983.883i			1.31010i	−0.755587	+	0.655048i	$0.772648\pi$
$752$		0			0
$753$		−	469.822i		−	0.623934i
$754$		0			0
$755$		−	22.7901i		−	0.0301855i
$756$		0			0
$757$		−	1494.15i		−	1.97378i	−0.161407	−	0.986888i	$-0.551603\pi$
$757$		−	1494.15i		−	1.97378i	0.161407	−	0.986888i	$-0.448397\pi$
$758$		0			0
$759$	−215.137	−	29.2686i	−0.283448	−	0.0385621i
$760$		0			0
$761$	−779.132			−1.02383			−0.511913	−	0.859037i	$-0.671063\pi$
$761$	−779.132			−1.02383			−0.511913	+	0.859037i	$0.671063\pi$
$762$		0			0
$763$	−542.096			−0.710479
$764$		0			0
$765$	−516.764			−0.675508
$766$		0			0
$767$	−270.230			−0.352320
$768$		0			0
$769$			155.017i			0.201582i	0.994908	+	0.100791i	$0.0321374\pi$
$769$			155.017i			0.201582i	−0.994908	+	0.100791i	$0.967863\pi$
$770$		0			0
$771$	−53.4437			−0.0693173
$772$		0			0
$773$			97.8690i			0.126609i	0.997994	+	0.0633047i	$0.0201640\pi$
$773$			97.8690i			0.126609i	−0.997994	+	0.0633047i	$0.979836\pi$
$774$		0			0
$775$	−2131.28			−2.75003
$776$		0			0
$777$	371.740			0.478430
$778$		0			0
$779$		−	360.060i		−	0.462208i
$780$		0			0
$781$			435.071i			0.557070i
$782$		0			0
$783$	1032.64			1.31883
$784$		0			0
$785$	694.989			0.885336
$786$		0			0
$787$			728.726i			0.925954i	0.886370	+	0.462977i	$0.153219\pi$
$787$			728.726i			0.925954i	−0.886370	+	0.462977i	$0.846781\pi$
$788$		0			0
$789$		−	1026.22i		−	1.30066i
$790$		0			0
$791$	706.569			0.893260
$792$		0			0
$793$		−	161.313i		−	0.203421i
$794$		0			0
$795$	1395.22			1.75500
$796$		0			0
$797$		−	279.729i		−	0.350978i	−0.984481	−	0.175489i	$-0.943849\pi$
$797$		−	279.729i		−	0.350978i	0.984481	−	0.175489i	$-0.0561506\pi$
$798$		0			0
$799$			535.787i			0.670572i
$800$		0			0
$801$		−	124.013i		−	0.154823i
$802$		0			0
$803$		−	187.112i		−	0.233016i
$804$		0			0
$805$	114.445	−	841.220i	0.142168	−	1.04499i
$806$		0			0
$807$	−992.762			−1.23019
$808$		0			0
$809$	1156.82			1.42994			0.714968	−	0.699157i	$-0.246441\pi$
$809$	1156.82			1.42994			0.714968	+	0.699157i	$0.246441\pi$
$810$		0			0
$811$	−1178.54			−1.45319			−0.726594	−	0.687067i	$-0.758898\pi$
$811$	−1178.54			−1.45319			−0.726594	+	0.687067i	$0.758898\pi$
$812$		0			0
$813$	−238.569			−0.293442
$814$		0			0
$815$		−	1185.52i		−	1.45463i
$816$		0			0
$817$	−799.391			−0.978447
$818$		0			0
$819$		−	127.904i		−	0.156171i
$820$		0			0
$821$	732.347			0.892019			0.446009	−	0.895028i	$-0.352845\pi$
$821$	732.347			0.892019			0.446009	+	0.895028i	$0.352845\pi$
$822$		0			0
$823$	−882.012			−1.07170			−0.535852	−	0.844312i	$-0.680009\pi$
$823$	−882.012			−1.07170			−0.535852	+	0.844312i	$0.680009\pi$
$824$		0			0
$825$			605.221i			0.733601i
$826$		0			0
$827$			1039.74i			1.25724i	0.777713	+	0.628619i	$0.216380\pi$
$827$			1039.74i			1.25724i	−0.777713	+	0.628619i	$0.783620\pi$
$828$		0			0
$829$	−413.505			−0.498799			−0.249400	−	0.968401i	$-0.580233\pi$
$829$	−413.505			−0.498799			−0.249400	+	0.968401i	$0.580233\pi$
$830$		0			0
$831$	500.512			0.602301
$832$		0			0
$833$		−	581.855i		−	0.698506i
$834$		0			0
$835$		−	876.296i		−	1.04946i
$836$		0			0
$837$	−976.828			−1.16706
$838$		0			0
$839$		−	254.486i		−	0.303320i	−0.988433	−	0.151660i	$-0.951538\pi$
$839$		−	254.486i		−	0.303320i	0.988433	−	0.151660i	$-0.0484619\pi$
$840$		0			0
$841$	393.970			0.468454
$842$		0			0
$843$			74.8487i			0.0887885i
$844$		0			0
$845$		−	591.200i		−	0.699645i
$846$		0			0
$847$		−	413.345i		−	0.488011i
$848$		0			0
$849$		−	545.342i		−	0.642334i
$850$		0			0
$851$	897.460	+	122.096i	1.05459	+	0.143474i
$852$		0			0
$853$	−744.132			−0.872370			−0.436185	−	0.899857i	$-0.643671\pi$
$853$	−744.132			−0.872370			−0.436185	+	0.899857i	$0.643671\pi$
$854$		0			0
$855$	−799.391			−0.934960
$856$		0			0
$857$	−578.921			−0.675520			−0.337760	−	0.941232i	$-0.609669\pi$
$857$	−578.921			−0.675520			−0.337760	+	0.941232i	$0.609669\pi$
$858$		0			0
$859$	1395.20			1.62421			0.812106	−	0.583510i	$-0.198321\pi$
$859$	1395.20			1.62421			0.812106	+	0.583510i	$0.198321\pi$
$860$		0			0
$861$		−	127.300i		−	0.147852i
$862$		0			0
$863$	565.704			0.655508			0.327754	−	0.944763i	$-0.393708\pi$
$863$	565.704			0.655508			0.327754	+	0.944763i	$0.393708\pi$
$864$		0			0
$865$			63.3485i			0.0732352i
$866$		0			0
$867$	−21.5269			−0.0248292
$868$		0			0
$869$	−460.853			−0.530325
$870$		0			0
$871$		−	705.150i		−	0.809587i
$872$		0			0
$873$			209.576i			0.240065i
$874$		0			0
$875$	−1443.72			−1.64996
$876$		0			0
$877$	468.112			0.533765			0.266882	−	0.963729i	$-0.414006\pi$
$877$	468.112			0.533765			0.266882	+	0.963729i	$0.414006\pi$
$878$		0			0
$879$			391.341i			0.445212i
$880$		0			0
$881$			1317.52i			1.49548i	0.663990	+	0.747741i	$0.268862\pi$
$881$			1317.52i			1.49548i	−0.663990	+	0.747741i	$0.731138\pi$
$882$		0			0
$883$	257.377			0.291480			0.145740	−	0.989323i	$-0.453444\pi$
$883$	257.377			0.291480			0.145740	+	0.989323i	$0.453444\pi$
$884$		0			0
$885$			597.123i			0.674715i
$886$		0			0
$887$	331.722			0.373982			0.186991	−	0.982362i	$-0.440126\pi$
$887$	331.722			0.373982			0.186991	+	0.982362i	$0.440126\pi$
$888$		0			0
$889$		−	142.290i		−	0.160056i
$890$		0			0
$891$			165.779i			0.186059i
$892$		0			0
$893$			828.818i			0.928128i
$894$		0			0
$895$		−	2626.78i		−	2.93495i
$896$		0			0
$897$	−77.2010	+	567.461i	−0.0860658	+	0.632621i
$898$		0			0
$899$	−1168.22			−1.29946
$900$		0			0
$901$	−1056.69			−1.17279
$902$		0			0
$903$	−282.627			−0.312987
$904$		0			0
$905$	506.721			0.559912
$906$		0			0
$907$			436.596i			0.481362i	0.970604	+	0.240681i	$0.0773708\pi$
$907$			436.596i			0.481362i	−0.970604	+	0.240681i	$0.922629\pi$
$908$		0			0
$909$	321.401			0.353577
$910$		0			0
$911$			1530.29i			1.67979i	0.542749	+	0.839895i	$0.317383\pi$
$911$			1530.29i			1.67979i	−0.542749	+	0.839895i	$0.682617\pi$
$912$		0			0
$913$	−292.670			−0.320559
$914$		0			0
$915$	−356.451			−0.389564
$916$		0			0
$917$		−	255.665i		−	0.278806i
$918$		0			0
$919$			676.783i			0.736434i	0.929740	+	0.368217i	$0.120032\pi$
$919$			676.783i			0.736434i	−0.929740	+	0.368217i	$0.879968\pi$
$920$		0			0
$921$	−826.382			−0.897266
$922$		0			0
$923$	1147.58			1.24331
$924$		0			0
$925$		−	2524.72i		−	2.72943i
$926$		0			0
$927$		−	322.070i		−	0.347432i
$928$		0			0
$929$	−686.535			−0.739004			−0.369502	−	0.929230i	$-0.620472\pi$
$929$	−686.535			−0.739004			−0.369502	+	0.929230i	$0.620472\pi$
$930$		0			0
$931$		−	900.082i		−	0.966791i
$932$		0			0
$933$	1121.77			1.20233
$934$		0			0
$935$		−	637.106i		−	0.681396i
$936$		0			0
$937$			774.354i			0.826418i	0.910636	+	0.413209i	$0.135592\pi$
$937$			774.354i			0.826418i	−0.910636	+	0.413209i	$0.864408\pi$
$938$		0			0
$939$		−	229.798i		−	0.244726i
$940$		0			0
$941$		−	774.814i		−	0.823395i	−0.911321	−	0.411697i	$-0.864936\pi$
$941$		−	774.814i		−	0.823395i	0.911321	−	0.411697i	$-0.135064\pi$
$942$		0			0
$943$	41.8112	−	307.330i	0.0443385	−	0.325907i
$944$		0			0
$945$	−1084.64			−1.14777
$946$		0			0
$947$	1090.69			1.15173			0.575865	−	0.817545i	$-0.304665\pi$
$947$	1090.69			1.15173			0.575865	+	0.817545i	$0.304665\pi$
$948$		0			0
$949$	−493.540			−0.520063
$950$		0			0
$951$	507.328			0.533468
$952$		0			0
$953$			60.7603i			0.0637569i	0.999492	+	0.0318785i	$0.0101489\pi$
$953$			60.7603i			0.0637569i	−0.999492	+	0.0318785i	$0.989851\pi$
$954$		0			0
$955$	−1592.45			−1.66749
$956$		0			0
$957$			331.740i			0.346646i
$958$		0			0
$959$	−257.295			−0.268295
$960$		0			0
$961$	144.073			0.149920
$962$		0			0
$963$		−	239.516i		−	0.248719i
$964$		0			0
$965$		−	899.805i		−	0.932440i
$966$		0			0
$967$	1715.83			1.77438			0.887192	−	0.461400i	$-0.152653\pi$
$967$	1715.83			1.77438			0.887192	+	0.461400i	$0.152653\pi$
$968$		0			0
$969$	−1112.60			−1.14819
$970$		0			0
$971$		−	1071.57i		−	1.10358i	−0.833984	−	0.551789i	$-0.813946\pi$
$971$		−	1071.57i		−	1.10358i	0.833984	−	0.551789i	$-0.186054\pi$
$972$		0			0
$973$			114.994i			0.118185i
$974$		0			0
$975$	1596.37			1.63731
$976$		0			0
$977$		−	456.079i		−	0.466816i	−0.972379	−	0.233408i	$-0.925012\pi$
$977$		−	456.079i		−	0.466816i	0.972379	−	0.233408i	$-0.0749877\pi$
$978$		0			0
$979$	152.893			0.156173
$980$		0			0
$981$			439.700i			0.448216i
$982$		0			0
$983$			216.055i			0.219791i	0.993943	+	0.109896i	$0.0350517\pi$
$983$			216.055i			0.219791i	−0.993943	+	0.109896i	$0.964948\pi$
$984$		0			0
$985$		−	2351.33i		−	2.38714i
$986$		0			0
$987$			293.032i			0.296891i
$988$		0			0
$989$	−682.323	−	92.8276i	−0.689912	−	0.0938600i
$990$		0			0
$991$	−520.142			−0.524866			−0.262433	−	0.964950i	$-0.584525\pi$
$991$	−520.142			−0.524866			−0.262433	+	0.964950i	$0.584525\pi$
$992$		0			0
$993$	−160.019			−0.161147
$994$		0			0
$995$	−719.235			−0.722849
$996$		0			0
$997$	−466.877			−0.468282			−0.234141	−	0.972203i	$-0.575228\pi$
$997$	−466.877			−0.468282			−0.234141	+	0.972203i	$0.575228\pi$
$998$		0			0
$999$		−	1157.16i		−	1.15831i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.3.f.f.321.4		4
4.3	odd	2		1472.3.f.c.321.2		4
8.3	odd	2		368.3.f.c.321.3		4
8.5	even	2		46.3.b.a.45.1	✓	4
23.22	odd	2	inner	1472.3.f.f.321.3		4
24.5	odd	2		414.3.b.a.91.4		4
40.13	odd	4		1150.3.c.a.1149.5		8
40.29	even	2		1150.3.d.a.551.4		4
40.37	odd	4		1150.3.c.a.1149.4		8
92.91	even	2		1472.3.f.c.321.1		4
184.45	odd	2		46.3.b.a.45.2	yes	4
184.91	even	2		368.3.f.c.321.4		4
552.413	even	2		414.3.b.a.91.3		4
920.229	odd	2		1150.3.d.a.551.3		4
920.413	even	4		1150.3.c.a.1149.6		8
920.597	even	4		1150.3.c.a.1149.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.3.b.a.45.1	✓	4	8.5	even	2
46.3.b.a.45.2	yes	4	184.45	odd	2
368.3.f.c.321.3		4	8.3	odd	2
368.3.f.c.321.4		4	184.91	even	2
414.3.b.a.91.3		4	552.413	even	2
414.3.b.a.91.4		4	24.5	odd	2
1150.3.c.a.1149.3		8	920.597	even	4
1150.3.c.a.1149.4		8	40.37	odd	4
1150.3.c.a.1149.5		8	40.13	odd	4
1150.3.c.a.1149.6		8	920.413	even	4
1150.3.d.a.551.3		4	920.229	odd	2
1150.3.d.a.551.4		4	40.29	even	2
1472.3.f.c.321.1		4	92.91	even	2
1472.3.f.c.321.2		4	4.3	odd	2
1472.3.f.f.321.3		4	23.22	odd	2	inner
1472.3.f.f.321.4		4	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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1472.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.41421 q^{3} +9.43995i q^{5} -3.91016i q^{7} -3.17157 q^{9} +O(q^{10})\)	\(q+2.41421 q^{3} +9.43995i q^{5} -3.91016i q^{7} -3.17157 q^{9} -3.91016i q^{11} -10.3137 q^{13} +22.7901i q^{15} -17.2603i q^{17} -26.7002i q^{19} -9.43995i q^{21} +(3.10051 - 22.7901i) q^{23} -64.1127 q^{25} -29.3848 q^{27} -35.1421 q^{29} +33.2426 q^{31} -9.43995i q^{33} +36.9117 q^{35} +39.3794i q^{37} -24.8995 q^{39} +13.4853 q^{41} -29.9395i q^{43} -29.9395i q^{45} -31.0416 q^{47} +33.7107 q^{49} -41.6700i q^{51} -61.2207i q^{53} +36.9117 q^{55} -64.4600i q^{57} +26.2010 q^{59} +15.6406i q^{61} +12.4013i q^{63} -97.3609i q^{65} +68.3702i q^{67} +(7.48528 - 55.0201i) q^{69} -111.267 q^{71} +47.8528 q^{73} -154.782 q^{75} -15.2893 q^{77} -117.860i q^{79} -42.3970 q^{81} -74.8487i q^{83} +162.936 q^{85} -84.8406 q^{87} +39.1016i q^{89} +40.3282i q^{91} +80.2548 q^{93} +252.049 q^{95} -66.0797i q^{97} +12.4013i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 24 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 24 * q^9	\( 4 q + 4 q^{3} - 24 q^{9} + 4 q^{13} + 52 q^{23} - 132 q^{25} - 44 q^{27} - 84 q^{29} + 116 q^{31} - 56 q^{35} - 60 q^{39} + 20 q^{41} - 28 q^{47} - 148 q^{49} - 56 q^{55} + 184 q^{59} - 4 q^{69} - 100 q^{71} - 148 q^{73} - 308 q^{75} - 344 q^{77} + 68 q^{81} + 120 q^{85} - 164 q^{87} + 140 q^{93} + 352 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 - 24 * q^9 + 4 * q^13 + 52 * q^23 - 132 * q^25 - 44 * q^27 - 84 * q^29 + 116 * q^31 - 56 * q^35 - 60 * q^39 + 20 * q^41 - 28 * q^47 - 148 * q^49 - 56 * q^55 + 184 * q^59 - 4 * q^69 - 100 * q^71 - 148 * q^73 - 308 * q^75 - 344 * q^77 + 68 * q^81 + 120 * q^85 - 164 * q^87 + 140 * q^93 + 352 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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