LMFDB - Embedded newform 32.9.d.b.15.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,9,Mod(15,32)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("32.15");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	32.d (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:	$13.0361155220$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 78x^{4} - 514x^{3} + 4237x^{2} - 18333x + 238980 $ x^6 - x^5 + 78x^4 - 514x^3 + 4237x^2 - 18333x + 238980
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{33} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			15.3
Root			$5.83239 + 4.16154i$ of defining polynomial
Character	$\chi$	$=$	32.15
Dual form			32.9.d.b.15.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+17.4864 q^{3} -796.785i q^{5} +1779.55i q^{7} -6255.23 q^{9} +O(q^{10})$	$q+17.4864 q^{3} -796.785i q^{5} +1779.55i q^{7} -6255.23 q^{9} -1458.80 q^{11} -30485.3i q^{13} -13932.9i q^{15} -82919.0 q^{17} -214604. q^{19} +31118.0i q^{21} -488296. i q^{23} -244241. q^{25} -224110. q^{27} -499553. i q^{29} +1.35347e6i q^{31} -25509.1 q^{33} +1.41792e6 q^{35} +334053. i q^{37} -533079. i q^{39} +3.11641e6 q^{41} +1.81268e6 q^{43} +4.98407e6i q^{45} +773477. i q^{47} +2.59799e6 q^{49} -1.44996e6 q^{51} -1.78592e6i q^{53} +1.16235e6i q^{55} -3.75265e6 q^{57} +1.59466e6 q^{59} -1.18438e7i q^{61} -1.11315e7i q^{63} -2.42902e7 q^{65} -2.22687e7 q^{67} -8.53854e6i q^{69} +2.06234e7i q^{71} -4.21796e6 q^{73} -4.27090e6 q^{75} -2.59601e6i q^{77} -4.96781e7i q^{79} +3.71217e7 q^{81} +1.13729e7 q^{83} +6.60686e7i q^{85} -8.73538e6i q^{87} +2.48076e7 q^{89} +5.42502e7 q^{91} +2.36672e7i q^{93} +1.70993e8i q^{95} +1.14484e7 q^{97} +9.12511e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 36 q^{3} + 16338 q^{9}+O(q^{10}) $ 6 * q + 36 * q^3 + 16338 * q^9	$ 6 q + 36 q^{3} + 16338 q^{9} - 46940 q^{11} - 201076 q^{17} + 95268 q^{19} - 447930 q^{25} - 419256 q^{27} + 1736856 q^{33} - 1989120 q^{35} + 3817100 q^{41} + 9881508 q^{43} - 12655482 q^{49} + 4303176 q^{51} + 7523736 q^{57} - 25243484 q^{59} + 27060480 q^{65} - 47850204 q^{67} - 55484916 q^{73} + 126075300 q^{75} - 79145442 q^{81} - 144646364 q^{83} + 142173452 q^{89} + 273971712 q^{91} + 125193612 q^{97} - 298320276 q^{99}+O(q^{100}) $ 6 * q + 36 * q^3 + 16338 * q^9 - 46940 * q^11 - 201076 * q^17 + 95268 * q^19 - 447930 * q^25 - 419256 * q^27 + 1736856 * q^33 - 1989120 * q^35 + 3817100 * q^41 + 9881508 * q^43 - 12655482 * q^49 + 4303176 * q^51 + 7523736 * q^57 - 25243484 * q^59 + 27060480 * q^65 - 47850204 * q^67 - 55484916 * q^73 + 126075300 * q^75 - 79145442 * q^81 - 144646364 * q^83 + 142173452 * q^89 + 273971712 * q^91 + 125193612 * q^97 - 298320276 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$.

$n$	$5$	$31$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	17.4864	0.215882	0.107941	−	0.994157i	$-0.465574\pi$
$3$	17.4864	0.215882	0.107941	+	0.994157i	$0.465574\pi$
$4$	0	0
$5$	− 796.785i	− 1.27486i	−0.770510	−	0.637428i	$-0.779998\pi$
$5$	− 796.785i	− 1.27486i	0.770510	−	0.637428i	$-0.220002\pi$
$6$	0	0
$7$	1779.55i	0.741171i	0.928798	+	0.370586i	$0.120843\pi$
$7$	1779.55i	0.741171i	−0.928798	+	0.370586i	$0.879157\pi$
$8$	0	0
$9$	−6255.23	−0.953395
$10$	0	0
$11$	−1458.80	−0.0996379	−0.0498189	−	0.998758i	$-0.515864\pi$
$11$	−1458.80	−0.0996379	−0.0498189	+	0.998758i	$0.515864\pi$
$12$	0	0
$13$	− 30485.3i	− 1.06738i	−0.845682	−	0.533688i	$-0.820806\pi$
$13$	− 30485.3i	− 1.06738i	0.845682	−	0.533688i	$-0.179194\pi$
$14$	0	0
$15$	− 13932.9i	− 0.275218i
$16$	0	0
$17$	−82919.0	−0.992792	−0.496396	−	0.868096i	$-0.665344\pi$
$17$	−82919.0	−0.992792	−0.496396	+	0.868096i	$0.665344\pi$
$18$	0	0
$19$	−214604.	−1.64673	−0.823366	−	0.567510i	$-0.807907\pi$
$19$	−214604.	−1.64673	−0.823366	+	0.567510i	$0.807907\pi$
$20$	0	0
$21$	31118.0i	0.160005i
$22$	0	0
$23$	− 488296.i	− 1.74490i	−0.488700	−	0.872452i	$-0.662529\pi$
$23$	− 488296.i	− 1.74490i	0.488700	−	0.872452i	$-0.337471\pi$
$24$	0	0
$25$	−244241.	−0.625257
$26$	0	0
$27$	−224110.	−0.421702
$28$	0	0
$29$	− 499553.i	− 0.706300i	−0.935567	−	0.353150i	$-0.885110\pi$
$29$	− 499553.i	− 0.706300i	0.935567	−	0.353150i	$-0.114890\pi$
$30$	0	0
$31$	1.35347e6i	1.46555i	0.680472	+	0.732774i	$0.261775\pi$
$31$	1.35347e6i	1.46555i	−0.680472	+	0.732774i	$0.738225\pi$
$32$	0	0
$33$	−25509.1	−0.0215100
$34$	0	0
$35$	1.41792e6	0.944887
$36$	0	0
$37$	334053.i	0.178241i	0.996021	+	0.0891206i	$0.0284057\pi$
$37$	334053.i	0.178241i	−0.996021	+	0.0891206i	$0.971594\pi$
$38$	0	0
$39$	− 533079.i	− 0.230427i
$40$	0	0
$41$	3.11641e6	1.10286	0.551428	−	0.834223i	$-0.314083\pi$
$41$	3.11641e6	1.10286	0.551428	+	0.834223i	$0.314083\pi$
$42$	0	0
$43$	1.81268e6	0.530208	0.265104	−	0.964220i	$-0.414594\pi$
$43$	1.81268e6	0.530208	0.265104	+	0.964220i	$0.414594\pi$
$44$	0	0
$45$	4.98407e6i	1.21544i
$46$	0	0
$47$	773477.i	0.158510i	0.996854	+	0.0792548i	$0.0252541\pi$
$47$	773477.i	0.158510i	−0.996854	+	0.0792548i	$0.974746\pi$
$48$	0	0
$49$	2.59799e6	0.450665
$50$	0	0
$51$	−1.44996e6	−0.214326
$52$	0	0
$53$	− 1.78592e6i	− 0.226339i	−0.993576	−	0.113169i	$-0.963900\pi$
$53$	− 1.78592e6i	− 0.226339i	0.993576	−	0.113169i	$-0.0361003\pi$
$54$	0	0
$55$	1.16235e6i	0.127024i
$56$	0	0
$57$	−3.75265e6	−0.355499
$58$	0	0
$59$	1.59466e6	0.131602	0.0658008	−	0.997833i	$-0.479040\pi$
$59$	1.59466e6	0.131602	0.0658008	+	0.997833i	$0.479040\pi$
$60$	0	0
$61$	− 1.18438e7i	− 0.855406i	−0.903919	−	0.427703i	$-0.859323\pi$
$61$	− 1.18438e7i	− 0.855406i	0.903919	−	0.427703i	$-0.140677\pi$
$62$	0	0
$63$	− 1.11315e7i	− 0.706629i
$64$	0	0
$65$	−2.42902e7	−1.36075
$66$	0	0
$67$	−2.22687e7	−1.10509	−0.552543	−	0.833485i	$-0.686342\pi$
$67$	−2.22687e7	−1.10509	−0.552543	+	0.833485i	$0.686342\pi$
$68$	0	0
$69$	− 8.53854e6i	− 0.376693i
$70$	0	0
$71$	2.06234e7i	0.811572i	0.913968	+	0.405786i	$0.133002\pi$
$71$	2.06234e7i	0.811572i	−0.913968	+	0.405786i	$0.866998\pi$
$72$	0	0
$73$	−4.21796e6	−0.148529	−0.0742644	−	0.997239i	$-0.523661\pi$
$73$	−4.21796e6	−0.148529	−0.0742644	+	0.997239i	$0.523661\pi$
$74$	0	0
$75$	−4.27090e6	−0.134981
$76$	0	0
$77$	− 2.59601e6i	− 0.0738487i
$78$	0	0
$79$	− 4.96781e7i	− 1.27543i	−0.770272	−	0.637715i	$-0.779880\pi$
$79$	− 4.96781e7i	− 1.27543i	0.770272	−	0.637715i	$-0.220120\pi$
$80$	0	0
$81$	3.71217e7	0.862357
$82$	0	0
$83$	1.13729e7	0.239640	0.119820	−	0.992796i	$-0.461768\pi$
$83$	1.13729e7	0.239640	0.119820	+	0.992796i	$0.461768\pi$
$84$	0	0
$85$	6.60686e7i	1.26567i
$86$	0	0
$87$	− 8.73538e6i	− 0.152477i
$88$	0	0
$89$	2.48076e7	0.395389	0.197695	−	0.980264i	$-0.436655\pi$
$89$	2.48076e7	0.395389	0.197695	+	0.980264i	$0.436655\pi$
$90$	0	0
$91$	5.42502e7	0.791108
$92$	0	0
$93$	2.36672e7i	0.316385i
$94$	0	0
$95$	1.70993e8i	2.09935i
$96$	0	0
$97$	1.14484e7	0.129318	0.0646589	−	0.997907i	$-0.479404\pi$
$97$	1.14484e7	0.129318	0.0646589	+	0.997907i	$0.479404\pi$
$98$	0	0
$99$	9.12511e6	0.0949943
$100$	0	0
$101$	− 1.47673e8i	− 1.41910i	−0.704653	−	0.709552i	$-0.748897\pi$
$101$	− 1.47673e8i	− 1.41910i	0.704653	−	0.709552i	$-0.251103\pi$
$102$	0	0
$103$	− 5.72198e7i	− 0.508390i	−0.967153	−	0.254195i	$-0.918189\pi$
$103$	− 5.72198e7i	− 0.508390i	0.967153	−	0.254195i	$-0.0818106\pi$
$104$	0	0
$105$	2.47943e7	0.203984
$106$	0	0
$107$	1.90189e8	1.45094	0.725471	−	0.688252i	$-0.241622\pi$
$107$	1.90189e8	1.45094	0.725471	+	0.688252i	$0.241622\pi$
$108$	0	0
$109$	5.09345e7i	0.360833i	0.983590	+	0.180416i	$0.0577445\pi$
$109$	5.09345e7i	0.360833i	−0.983590	+	0.180416i	$0.942256\pi$
$110$	0	0
$111$	5.84138e6i	0.0384790i
$112$	0	0
$113$	−7.76455e6	−0.0476214	−0.0238107	−	0.999716i	$-0.507580\pi$
$113$	−7.76455e6	−0.0476214	−0.0238107	+	0.999716i	$0.507580\pi$
$114$	0	0
$115$	−3.89067e8	−2.22450
$116$	0	0
$117$	1.90693e8i	1.01763i
$118$	0	0
$119$	− 1.47559e8i	− 0.735829i
$120$	0	0
$121$	−2.12231e8	−0.990072
$122$	0	0
$123$	5.44948e7	0.238086
$124$	0	0
$125$	− 1.16637e8i	− 0.477743i
$126$	0	0
$127$	− 2.54116e8i	− 0.976826i	−0.872612	−	0.488413i	$-0.837576\pi$
$127$	− 2.54116e8i	− 0.976826i	0.872612	−	0.488413i	$-0.162424\pi$
$128$	0	0
$129$	3.16972e7	0.114462
$130$	0	0
$131$	−2.72967e8	−0.926885	−0.463442	−	0.886127i	$-0.653386\pi$
$131$	−2.72967e8	−0.926885	−0.463442	+	0.886127i	$0.653386\pi$
$132$	0	0
$133$	− 3.81899e8i	− 1.22051i
$134$	0	0
$135$	1.78567e8i	0.537609i
$136$	0	0
$137$	−3.72346e7	−0.105698	−0.0528488	−	0.998603i	$-0.516830\pi$
$137$	−3.72346e7	−0.105698	−0.0528488	+	0.998603i	$0.516830\pi$
$138$	0	0
$139$	1.05417e8	0.282392	0.141196	−	0.989982i	$-0.454905\pi$
$139$	1.05417e8	0.282392	0.141196	+	0.989982i	$0.454905\pi$
$140$	0	0
$141$	1.35253e7i	0.0342193i
$142$	0	0
$143$	4.44719e7i	0.106351i
$144$	0	0
$145$	−3.98036e8	−0.900431
$146$	0	0
$147$	4.54296e7	0.0972903
$148$	0	0
$149$	6.08521e8i	1.23461i	0.786723	+	0.617306i	$0.211776\pi$
$149$	6.08521e8i	1.23461i	−0.786723	+	0.617306i	$0.788224\pi$
$150$	0	0
$151$	4.73729e8i	0.911219i	0.890180	+	0.455609i	$0.150579\pi$
$151$	4.73729e8i	0.911219i	−0.890180	+	0.455609i	$0.849421\pi$
$152$	0	0
$153$	5.18677e8	0.946523
$154$	0	0
$155$	1.07842e9	1.86836
$156$	0	0
$157$	− 4.88576e8i	− 0.804144i	−0.915608	−	0.402072i	$-0.868290\pi$
$157$	− 4.88576e8i	− 0.804144i	0.915608	−	0.402072i	$-0.131710\pi$
$158$	0	0
$159$	− 3.12294e7i	− 0.0488624i
$160$	0	0
$161$	8.68948e8	1.29327
$162$	0	0
$163$	−8.94547e8	−1.26722	−0.633611	−	0.773651i	$-0.718428\pi$
$163$	−8.94547e8	−1.26722	−0.633611	+	0.773651i	$0.718428\pi$
$164$	0	0
$165$	2.03253e7i	0.0274221i
$166$	0	0
$167$	7.04084e7i	0.0905229i	0.998975	+	0.0452615i	$0.0144121\pi$
$167$	7.04084e7i	0.0905229i	−0.998975	+	0.0452615i	$0.985588\pi$
$168$	0	0
$169$	−1.13624e8	−0.139291
$170$	0	0
$171$	1.34240e9	1.56999
$172$	0	0
$173$	1.52030e9i	1.69725i	0.528999	+	0.848623i	$0.322568\pi$
$173$	1.52030e9i	1.69725i	−0.528999	+	0.848623i	$0.677432\pi$
$174$	0	0
$175$	− 4.34640e8i	− 0.463423i
$176$	0	0
$177$	2.78850e7	0.0284104
$178$	0	0
$179$	−1.46877e9	−1.43068	−0.715341	−	0.698776i	$-0.753729\pi$
$179$	−1.46877e9	−1.43068	−0.715341	+	0.698776i	$0.753729\pi$
$180$	0	0
$181$	− 4.02425e8i	− 0.374948i	−0.982270	−	0.187474i	$-0.939970\pi$
$181$	− 4.02425e8i	− 0.374948i	0.982270	−	0.187474i	$-0.0600300\pi$
$182$	0	0
$183$	− 2.07106e8i	− 0.184667i
$184$	0	0
$185$	2.66168e8	0.227232
$186$	0	0
$187$	1.20962e8	0.0989197
$188$	0	0
$189$	− 3.98815e8i	− 0.312554i
$190$	0	0
$191$	3.03920e8i	0.228363i	0.993460	+	0.114182i	$0.0364246\pi$
$191$	3.03920e8i	0.228363i	−0.993460	+	0.114182i	$0.963575\pi$
$192$	0	0
$193$	−2.02929e9	−1.46257	−0.731284	−	0.682073i	$-0.761078\pi$
$193$	−2.02929e9	−1.46257	−0.731284	+	0.682073i	$0.761078\pi$
$194$	0	0
$195$	−4.24749e8	−0.293761
$196$	0	0
$197$	− 1.17091e9i	− 0.777422i	−0.921360	−	0.388711i	$-0.872920\pi$
$197$	− 1.17091e9i	− 0.777422i	0.921360	−	0.388711i	$-0.127080\pi$
$198$	0	0
$199$	− 1.40831e9i	− 0.898020i	−0.893527	−	0.449010i	$-0.851777\pi$
$199$	− 1.40831e9i	− 0.898020i	0.893527	−	0.449010i	$-0.148223\pi$
$200$	0	0
$201$	−3.89400e8	−0.238568
$202$	0	0
$203$	8.88980e8	0.523489
$204$	0	0
$205$	− 2.48311e9i	− 1.40598i
$206$	0	0
$207$	3.05440e9i	1.66358i
$208$	0	0
$209$	3.13064e8	0.164077
$210$	0	0
$211$	−2.76916e9	−1.39707	−0.698535	−	0.715575i	$-0.746165\pi$
$211$	−2.76916e9	−1.39707	−0.698535	+	0.715575i	$0.746165\pi$
$212$	0	0
$213$	3.60629e8i	0.175203i
$214$	0	0
$215$	− 1.44431e9i	− 0.675939i
$216$	0	0
$217$	−2.40856e9	−1.08622
$218$	0	0
$219$	−7.37569e7	−0.0320646
$220$	0	0
$221$	2.52781e9i	1.05968i
$222$	0	0
$223$	− 2.53904e9i	− 1.02672i	−0.858174	−	0.513358i	$-0.828401\pi$
$223$	− 2.53904e9i	− 1.02672i	0.858174	−	0.513358i	$-0.171599\pi$
$224$	0	0
$225$	1.52778e9	0.596117
$226$	0	0
$227$	1.37848e9	0.519153	0.259577	−	0.965723i	$-0.416417\pi$
$227$	1.37848e9	0.519153	0.259577	+	0.965723i	$0.416417\pi$
$228$	0	0
$229$	3.26124e9i	1.18588i	0.805246	+	0.592941i	$0.202033\pi$
$229$	3.26124e9i	1.18588i	−0.805246	+	0.592941i	$0.797967\pi$
$230$	0	0
$231$	− 4.53949e7i	− 0.0159426i
$232$	0	0
$233$	3.98642e9	1.35257	0.676285	−	0.736640i	$-0.263589\pi$
$233$	3.98642e9	1.35257	0.676285	+	0.736640i	$0.263589\pi$
$234$	0	0
$235$	6.16295e8	0.202077
$236$	0	0
$237$	− 8.68692e8i	− 0.275342i
$238$	0	0
$239$	− 9.59043e8i	− 0.293932i	−0.989142	−	0.146966i	$-0.953049\pi$
$239$	− 9.59043e8i	− 0.293932i	0.989142	−	0.146966i	$-0.0469507\pi$
$240$	0	0
$241$	1.65969e9	0.491993	0.245996	−	0.969271i	$-0.420885\pi$
$241$	1.65969e9	0.491993	0.245996	+	0.969271i	$0.420885\pi$
$242$	0	0
$243$	2.11951e9	0.607869
$244$	0	0
$245$	− 2.07004e9i	− 0.574533i
$246$	0	0
$247$	6.54227e9i	1.75768i
$248$	0	0
$249$	1.98871e8	0.0517338
$250$	0	0
$251$	2.93567e9	0.739627	0.369814	−	0.929106i	$-0.379422\pi$
$251$	2.93567e9	0.739627	0.369814	+	0.929106i	$0.379422\pi$
$252$	0	0
$253$	7.12325e8i	0.173859i
$254$	0	0
$255$	1.15530e9i	0.273234i
$256$	0	0
$257$	−9.42169e8	−0.215971	−0.107986	−	0.994152i	$-0.534440\pi$
$257$	−9.42169e8	−0.215971	−0.107986	+	0.994152i	$0.534440\pi$
$258$	0	0
$259$	−5.94464e8	−0.132107
$260$	0	0
$261$	3.12481e9i	0.673383i
$262$	0	0
$263$	4.35300e9i	0.909841i	0.890532	+	0.454920i	$0.150332\pi$
$263$	4.35300e9i	0.909841i	−0.890532	+	0.454920i	$0.849668\pi$
$264$	0	0
$265$	−1.42300e9	−0.288549
$266$	0	0
$267$	4.33796e8	0.0853573
$268$	0	0
$269$	− 1.63941e9i	− 0.313098i	−0.987670	−	0.156549i	$-0.949963\pi$
$269$	− 1.63941e9i	− 0.313098i	0.987670	−	0.156549i	$-0.0500368\pi$
$270$	0	0
$271$	− 6.53537e9i	− 1.21169i	−0.795581	−	0.605847i	$-0.792834\pi$
$271$	− 6.53537e9i	− 1.21169i	0.795581	−	0.605847i	$-0.207166\pi$
$272$	0	0
$273$	9.48642e8	0.170786
$274$	0	0
$275$	3.56298e8	0.0622993
$276$	0	0
$277$	− 8.55840e9i	− 1.45369i	−0.686799	−	0.726847i	$-0.740985\pi$
$277$	− 8.55840e9i	− 1.45369i	0.686799	−	0.726847i	$-0.259015\pi$
$278$	0	0
$279$	− 8.46623e9i	− 1.39725i
$280$	0	0
$281$	−5.19165e9	−0.832683	−0.416342	−	0.909208i	$-0.636688\pi$
$281$	−5.19165e9	−0.832683	−0.416342	+	0.909208i	$0.636688\pi$
$282$	0	0
$283$	−7.13116e9	−1.11177	−0.555885	−	0.831259i	$-0.687620\pi$
$283$	−7.13116e9	−1.11177	−0.555885	+	0.831259i	$0.687620\pi$
$284$	0	0
$285$	2.99006e9i	0.453210i
$286$	0	0
$287$	5.54581e9i	0.817405i
$288$	0	0
$289$	−1.00194e8	−0.0143632
$290$	0	0
$291$	2.00192e8	0.0279173
$292$	0	0
$293$	2.75986e9i	0.374470i	0.982315	+	0.187235i	$0.0599526\pi$
$293$	2.75986e9i	0.374470i	−0.982315	+	0.187235i	$0.940047\pi$
$294$	0	0
$295$	− 1.27060e9i	− 0.167773i
$296$	0	0
$297$	3.26931e8	0.0420175
$298$	0	0
$299$	−1.48858e10	−1.86247
$300$	0	0
$301$	3.22575e9i	0.392975i
$302$	0	0
$303$	− 2.58226e9i	− 0.306359i
$304$	0	0
$305$	−9.43698e9	−1.09052
$306$	0	0
$307$	1.03453e10	1.16463	0.582316	−	0.812963i	$-0.302147\pi$
$307$	1.03453e10	1.16463	0.582316	+	0.812963i	$0.302147\pi$
$308$	0	0
$309$	− 1.00057e9i	− 0.109752i
$310$	0	0
$311$	− 6.19845e9i	− 0.662585i	−0.943528	−	0.331292i	$-0.892515\pi$
$311$	− 6.19845e9i	− 0.662585i	0.943528	−	0.331292i	$-0.107485\pi$
$312$	0	0
$313$	7.26539e9	0.756975	0.378488	−	0.925606i	$-0.376444\pi$
$313$	7.26539e9	0.756975	0.378488	+	0.925606i	$0.376444\pi$
$314$	0	0
$315$	−8.86941e9	−0.900850
$316$	0	0
$317$	− 1.04449e10i	− 1.03435i	−0.855879	−	0.517175i	$-0.826983\pi$
$317$	− 1.04449e10i	− 1.03435i	0.855879	−	0.517175i	$-0.173017\pi$
$318$	0	0
$319$	7.28747e8i	0.0703743i
$320$	0	0
$321$	3.32572e9	0.313232
$322$	0	0
$323$	1.77947e10	1.63486
$324$	0	0
$325$	7.44576e9i	0.667384i
$326$	0	0
$327$	8.90661e8i	0.0778971i
$328$	0	0
$329$	−1.37644e9	−0.117483
$330$	0	0
$331$	4.99859e9	0.416424	0.208212	−	0.978084i	$-0.433236\pi$
$331$	4.99859e9	0.416424	0.208212	+	0.978084i	$0.433236\pi$
$332$	0	0
$333$	− 2.08958e9i	− 0.169934i
$334$	0	0
$335$	1.77434e10i	1.40882i
$336$	0	0
$337$	3.29299e9	0.255312	0.127656	−	0.991819i	$-0.459255\pi$
$337$	3.29299e9	0.255312	0.127656	+	0.991819i	$0.459255\pi$
$338$	0	0
$339$	−1.35774e8	−0.0102806
$340$	0	0
$341$	− 1.97443e9i	− 0.146024i
$342$	0	0
$343$	1.48820e10i	1.07519i
$344$	0	0
$345$	−6.80338e9	−0.480229
$346$	0	0
$347$	−2.09239e9	−0.144319	−0.0721596	−	0.997393i	$-0.522989\pi$
$347$	−2.09239e9	−0.144319	−0.0721596	+	0.997393i	$0.522989\pi$
$348$	0	0
$349$	1.00832e10i	0.679665i	0.940486	+	0.339833i	$0.110370\pi$
$349$	1.00832e10i	0.679665i	−0.940486	+	0.339833i	$0.889630\pi$
$350$	0	0
$351$	6.83206e9i	0.450115i
$352$	0	0
$353$	1.75112e10	1.12776	0.563879	−	0.825857i	$-0.309308\pi$
$353$	1.75112e10	1.12776	0.563879	+	0.825857i	$0.309308\pi$
$354$	0	0
$355$	1.64324e10	1.03464
$356$	0	0
$357$	− 2.58027e9i	− 0.158852i
$358$	0	0
$359$	− 2.12923e10i	− 1.28187i	−0.767594	−	0.640937i	$-0.778546\pi$
$359$	− 2.12923e10i	− 1.28187i	0.767594	−	0.640937i	$-0.221454\pi$
$360$	0	0
$361$	2.90713e10	1.71173
$362$	0	0
$363$	−3.71115e9	−0.213738
$364$	0	0
$365$	3.36080e9i	0.189353i
$366$	0	0
$367$	− 1.81508e10i	− 1.00054i	−0.865871	−	0.500268i	$-0.833235\pi$
$367$	− 1.81508e10i	− 1.00054i	0.865871	−	0.500268i	$-0.166765\pi$
$368$	0	0
$369$	−1.94938e10	−1.05146
$370$	0	0
$371$	3.17814e9	0.167756
$372$	0	0
$373$	2.88943e10i	1.49271i	0.665547	+	0.746356i	$0.268198\pi$
$373$	2.88943e10i	1.49271i	−0.665547	+	0.746356i	$0.731802\pi$
$374$	0	0
$375$	− 2.03955e9i	− 0.103136i
$376$	0	0
$377$	−1.52290e10	−0.753888
$378$	0	0
$379$	−1.74756e10	−0.846983	−0.423492	−	0.905900i	$-0.639196\pi$
$379$	−1.74756e10	−0.846983	−0.423492	+	0.905900i	$0.639196\pi$
$380$	0	0
$381$	− 4.44358e9i	− 0.210879i
$382$	0	0
$383$	− 2.01489e10i	− 0.936389i	−0.883625	−	0.468195i	$-0.844905\pi$
$383$	− 2.01489e10i	− 0.936389i	0.883625	−	0.468195i	$-0.155095\pi$
$384$	0	0
$385$	−2.06846e9	−0.0941465
$386$	0	0
$387$	−1.13387e10	−0.505498
$388$	0	0
$389$	− 3.27161e10i	− 1.42877i	−0.699751	−	0.714387i	$-0.746706\pi$
$389$	− 3.27161e10i	− 1.42877i	0.699751	−	0.714387i	$-0.253294\pi$
$390$	0	0
$391$	4.04890e10i	1.73233i
$392$	0	0
$393$	−4.77322e9	−0.200097
$394$	0	0
$395$	−3.95828e10	−1.62599
$396$	0	0
$397$	− 2.27039e9i	− 0.0913985i	−0.998955	−	0.0456992i	$-0.985448\pi$
$397$	− 2.27039e9i	− 0.0913985i	0.998955	−	0.0456992i	$-0.0145516\pi$
$398$	0	0
$399$	− 6.67804e9i	− 0.263486i
$400$	0	0
$401$	−1.40935e10	−0.545055	−0.272528	−	0.962148i	$-0.587860\pi$
$401$	−1.40935e10	−0.545055	−0.272528	+	0.962148i	$0.587860\pi$
$402$	0	0
$403$	4.12608e10	1.56429
$404$	0	0
$405$	− 2.95780e10i	− 1.09938i
$406$	0	0
$407$	− 4.87316e8i	− 0.0177596i
$408$	0	0
$409$	−3.82460e10	−1.36676	−0.683380	−	0.730063i	$-0.739491\pi$
$409$	−3.82460e10	−1.36676	−0.683380	+	0.730063i	$0.739491\pi$
$410$	0	0
$411$	−6.51100e8	−0.0228182
$412$	0	0
$413$	2.83779e9i	0.0975394i
$414$	0	0
$415$	− 9.06175e9i	− 0.305506i
$416$	0	0
$417$	1.84337e9	0.0609632
$418$	0	0
$419$	−2.64682e10	−0.858754	−0.429377	−	0.903125i	$-0.641267\pi$
$419$	−2.64682e10	−0.858754	−0.429377	+	0.903125i	$0.641267\pi$
$420$	0	0
$421$	− 2.77128e10i	− 0.882170i	−0.897465	−	0.441085i	$-0.854594\pi$
$421$	− 2.77128e10i	− 0.882170i	0.897465	−	0.441085i	$-0.145406\pi$
$422$	0	0
$423$	− 4.83827e9i	− 0.151122i
$424$	0	0
$425$	2.02522e10	0.620750
$426$	0	0
$427$	2.10767e10	0.634003
$428$	0	0
$429$	7.77654e8i	0.0229592i
$430$	0	0
$431$	6.54522e9i	0.189677i	0.995493	+	0.0948386i	$0.0302335\pi$
$431$	6.54522e9i	0.189677i	−0.995493	+	0.0948386i	$0.969767\pi$
$432$	0	0
$433$	3.87616e10	1.10268	0.551341	−	0.834280i	$-0.314116\pi$
$433$	3.87616e10	1.10268	0.551341	+	0.834280i	$0.314116\pi$
$434$	0	0
$435$	−6.96022e9	−0.194386
$436$	0	0
$437$	1.04790e11i	2.87339i
$438$	0	0
$439$	− 5.32803e9i	− 0.143453i	−0.997424	−	0.0717264i	$-0.977149\pi$
$439$	− 5.32803e9i	− 0.143453i	0.997424	−	0.0717264i	$-0.0228508\pi$
$440$	0	0
$441$	−1.62510e10	−0.429662
$442$	0	0
$443$	2.40425e10	0.624258	0.312129	−	0.950040i	$-0.398958\pi$
$443$	2.40425e10	0.624258	0.312129	+	0.950040i	$0.398958\pi$
$444$	0	0
$445$	− 1.97663e10i	− 0.504065i
$446$	0	0
$447$	1.06408e10i	0.266530i
$448$	0	0
$449$	−1.39499e10	−0.343231	−0.171616	−	0.985164i	$-0.554899\pi$
$449$	−1.39499e10	−0.343231	−0.171616	+	0.985164i	$0.554899\pi$
$450$	0	0
$451$	−4.54621e9	−0.109886
$452$	0	0
$453$	8.28383e9i	0.196715i
$454$	0	0
$455$	− 4.32258e10i	− 1.00855i
$456$	0	0
$457$	7.15922e10	1.64135	0.820675	−	0.571396i	$-0.193598\pi$
$457$	7.15922e10	1.64135	0.820675	+	0.571396i	$0.193598\pi$
$458$	0	0
$459$	1.85830e10	0.418663
$460$	0	0
$461$	1.49659e10i	0.331360i	0.986180	+	0.165680i	$0.0529819\pi$
$461$	1.49659e10i	0.331360i	−0.986180	+	0.165680i	$0.947018\pi$
$462$	0	0
$463$	8.49544e10i	1.84868i	0.381568	+	0.924341i	$0.375384\pi$
$463$	8.49544e10i	1.84868i	−0.381568	+	0.924341i	$0.624616\pi$
$464$	0	0
$465$	1.88577e10	0.403345
$466$	0	0
$467$	−2.02009e10	−0.424721	−0.212361	−	0.977191i	$-0.568115\pi$
$467$	−2.02009e10	−0.424721	−0.212361	+	0.977191i	$0.568115\pi$
$468$	0	0
$469$	− 3.96283e10i	− 0.819058i
$470$	0	0
$471$	− 8.54344e9i	− 0.173600i
$472$	0	0
$473$	−2.64433e9	−0.0528288
$474$	0	0
$475$	5.24151e10	1.02963
$476$	0	0
$477$	1.11713e10i	0.215790i
$478$	0	0
$479$	− 9.48673e9i	− 0.180208i	−0.995932	−	0.0901041i	$-0.971280\pi$
$479$	− 9.48673e9i	− 0.180208i	0.995932	−	0.0901041i	$-0.0287200\pi$
$480$	0	0
$481$	1.01837e10	0.190250
$482$	0	0
$483$	1.51948e10	0.279194
$484$	0	0
$485$	− 9.12192e9i	− 0.164862i
$486$	0	0
$487$	− 2.60799e10i	− 0.463650i	−0.972758	−	0.231825i	$-0.925530\pi$
$487$	− 2.60799e10i	− 0.463650i	0.972758	−	0.231825i	$-0.0744697\pi$
$488$	0	0
$489$	−1.56424e10	−0.273570
$490$	0	0
$491$	5.60323e10	0.964079	0.482039	−	0.876150i	$-0.339896\pi$
$491$	5.60323e10	0.964079	0.482039	+	0.876150i	$0.339896\pi$
$492$	0	0
$493$	4.14224e10i	0.701210i
$494$	0	0
$495$	− 7.27075e9i	− 0.121104i
$496$	0	0
$497$	−3.67004e10	−0.601514
$498$	0	0
$499$	−1.12235e11	−1.81021	−0.905103	−	0.425192i	$-0.860206\pi$
$499$	−1.12235e11	−1.81021	−0.905103	+	0.425192i	$0.860206\pi$
$500$	0	0
$501$	1.23119e9i	0.0195422i
$502$	0	0
$503$	8.45980e10i	1.32156i	0.750578	+	0.660782i	$0.229775\pi$
$503$	8.45980e10i	1.32156i	−0.750578	+	0.660782i	$0.770225\pi$
$504$	0	0
$505$	−1.17663e11	−1.80915
$506$	0	0
$507$	−1.98687e9	−0.0300703
$508$	0	0
$509$	− 1.13293e11i	− 1.68784i	−0.536471	−	0.843919i	$-0.680243\pi$
$509$	− 1.13293e11i	− 1.68784i	0.536471	−	0.843919i	$-0.319757\pi$
$510$	0	0
$511$	− 7.50607e9i	− 0.110085i
$512$	0	0
$513$	4.80948e10	0.694431
$514$	0	0
$515$	−4.55919e10	−0.648124
$516$	0	0
$517$	− 1.12835e9i	− 0.0157936i
$518$	0	0
$519$	2.65846e10i	0.366404i
$520$	0	0
$521$	1.03318e11	1.40225	0.701125	−	0.713038i	$-0.252681\pi$
$521$	1.03318e11	1.40225	0.701125	+	0.713038i	$0.252681\pi$
$522$	0	0
$523$	3.55661e8	0.00475367	0.00237684	−	0.999997i	$-0.499243\pi$
$523$	3.55661e8	0.00475367	0.00237684	+	0.999997i	$0.499243\pi$
$524$	0	0
$525$	− 7.60029e9i	− 0.100044i
$526$	0	0
$527$	− 1.12228e11i	− 1.45499i
$528$	0	0
$529$	−1.60122e11	−2.04469
$530$	0	0
$531$	−9.97498e9	−0.125468
$532$	0	0
$533$	− 9.50047e10i	− 1.17716i
$534$	0	0
$535$	− 1.51540e11i	− 1.84974i
$536$	0	0
$537$	−2.56836e10	−0.308858
$538$	0	0
$539$	−3.78995e9	−0.0449033
$540$	0	0
$541$	1.30241e11i	1.52041i	0.649685	+	0.760204i	$0.274901\pi$
$541$	1.30241e11i	1.52041i	−0.649685	+	0.760204i	$0.725099\pi$
$542$	0	0
$543$	− 7.03697e9i	− 0.0809444i
$544$	0	0
$545$	4.05838e10	0.460009
$546$	0	0
$547$	−1.43558e11	−1.60353	−0.801766	−	0.597638i	$-0.796106\pi$
$547$	−1.43558e11	−1.60353	−0.801766	+	0.597638i	$0.796106\pi$
$548$	0	0
$549$	7.40858e10i	0.815540i
$550$	0	0
$551$	1.07206e11i	1.16309i
$552$	0	0
$553$	8.84048e10	0.945313
$554$	0	0
$555$	4.65433e9	0.0490552
$556$	0	0
$557$	2.53322e10i	0.263180i	0.991304	+	0.131590i	$0.0420082\pi$
$557$	2.53322e10i	0.263180i	−0.991304	+	0.131590i	$0.957992\pi$
$558$	0	0
$559$	− 5.52600e10i	− 0.565932i
$560$	0	0
$561$	2.11519e9	0.0213550
$562$	0	0
$563$	9.04001e10	0.899778	0.449889	−	0.893084i	$-0.351464\pi$
$563$	9.04001e10	0.899778	0.449889	+	0.893084i	$0.351464\pi$
$564$	0	0
$565$	6.18668e9i	0.0607105i
$566$	0	0
$567$	6.60599e10i	0.639155i
$568$	0	0
$569$	−1.46686e11	−1.39939	−0.699696	−	0.714441i	$-0.746681\pi$
$569$	−1.46686e11	−1.39939	−0.699696	+	0.714441i	$0.746681\pi$
$570$	0	0
$571$	1.32868e11	1.24990	0.624949	−	0.780666i	$-0.285120\pi$
$571$	1.32868e11	1.24990	0.624949	+	0.780666i	$0.285120\pi$
$572$	0	0
$573$	5.31447e9i	0.0492994i
$574$	0	0
$575$	1.19262e11i	1.09101i
$576$	0	0
$577$	1.26715e11	1.14321	0.571604	−	0.820529i	$-0.306321\pi$
$577$	1.26715e11	1.14321	0.571604	+	0.820529i	$0.306321\pi$
$578$	0	0
$579$	−3.54851e10	−0.315741
$580$	0	0
$581$	2.02387e10i	0.177614i
$582$	0	0
$583$	2.60530e9i	0.0225519i
$584$	0	0
$585$	1.51941e11	1.29733
$586$	0	0
$587$	2.10292e11	1.77121	0.885606	−	0.464438i	$-0.153744\pi$
$587$	2.10292e11	1.77121	0.885606	+	0.464438i	$0.153744\pi$
$588$	0	0
$589$	− 2.90459e11i	− 2.41337i
$590$	0	0
$591$	− 2.04749e10i	− 0.167831i
$592$	0	0
$593$	1.03227e10	0.0834789	0.0417394	−	0.999129i	$-0.486710\pi$
$593$	1.03227e10	0.0834789	0.0417394	+	0.999129i	$0.486710\pi$
$594$	0	0
$595$	−1.17573e11	−0.938076
$596$	0	0
$597$	− 2.46263e10i	− 0.193866i
$598$	0	0
$599$	1.38904e11i	1.07897i	0.841996	+	0.539483i	$0.181380\pi$
$599$	1.38904e11i	1.07897i	−0.841996	+	0.539483i	$0.818620\pi$
$600$	0	0
$601$	−6.88691e10	−0.527870	−0.263935	−	0.964540i	$-0.585020\pi$
$601$	−6.88691e10	−0.527870	−0.263935	+	0.964540i	$0.585020\pi$
$602$	0	0
$603$	1.39296e11	1.05358
$604$	0	0
$605$	1.69102e11i	1.26220i
$606$	0	0
$607$	4.98379e10i	0.367117i	0.983009	+	0.183559i	$0.0587617\pi$
$607$	4.98379e10i	0.367117i	−0.983009	+	0.183559i	$0.941238\pi$
$608$	0	0
$609$	1.55451e10	0.113012
$610$	0	0
$611$	2.35797e10	0.169189
$612$	0	0
$613$	1.64085e11i	1.16205i	0.813884	+	0.581027i	$0.197349\pi$
$613$	1.64085e11i	1.16205i	−0.813884	+	0.581027i	$0.802651\pi$
$614$	0	0
$615$	− 4.34206e10i	− 0.303526i
$616$	0	0
$617$	1.16747e11	0.805576	0.402788	−	0.915293i	$-0.368041\pi$
$617$	1.16747e11	0.805576	0.402788	+	0.915293i	$0.368041\pi$
$618$	0	0
$619$	−1.51634e11	−1.03285	−0.516423	−	0.856334i	$-0.672737\pi$
$619$	−1.51634e11	−1.03285	−0.516423	+	0.856334i	$0.672737\pi$
$620$	0	0
$621$	1.09432e11i	0.735830i
$622$	0	0
$623$	4.41465e10i	0.293051i
$624$	0	0
$625$	−1.88341e11	−1.23431
$626$	0	0
$627$	5.47436e9	0.0354212
$628$	0	0
$629$	− 2.76993e10i	− 0.176957i
$630$	0	0
$631$	− 2.66911e11i	− 1.68364i	−0.539757	−	0.841821i	$-0.681484\pi$
$631$	− 2.66911e11i	− 1.68364i	0.539757	−	0.841821i	$-0.318516\pi$
$632$	0	0
$633$	−4.84227e10	−0.301602
$634$	0	0
$635$	−2.02476e11	−1.24531
$636$	0	0
$637$	− 7.92007e10i	− 0.481029i
$638$	0	0
$639$	− 1.29004e11i	− 0.773748i
$640$	0	0
$641$	1.55457e11	0.920827	0.460414	−	0.887705i	$-0.347701\pi$
$641$	1.55457e11	0.920827	0.460414	+	0.887705i	$0.347701\pi$
$642$	0	0
$643$	1.27487e11	0.745800	0.372900	−	0.927871i	$-0.378363\pi$
$643$	1.27487e11	0.745800	0.372900	+	0.927871i	$0.378363\pi$
$644$	0	0
$645$	− 2.52559e10i	− 0.145923i
$646$	0	0
$647$	− 1.80017e11i	− 1.02730i	−0.858000	−	0.513649i	$-0.828293\pi$
$647$	− 1.80017e11i	− 1.02730i	0.858000	−	0.513649i	$-0.171707\pi$
$648$	0	0
$649$	−2.32629e9	−0.0131125
$650$	0	0
$651$	−4.21171e10	−0.234496
$652$	0	0
$653$	9.62574e9i	0.0529397i	0.999650	+	0.0264699i	$0.00842660\pi$
$653$	9.62574e9i	0.0529397i	−0.999650	+	0.0264699i	$0.991573\pi$
$654$	0	0
$655$	2.17496e11i	1.18164i
$656$	0	0
$657$	2.63843e10	0.141607
$658$	0	0
$659$	−4.10033e10	−0.217409	−0.108704	−	0.994074i	$-0.534670\pi$
$659$	−4.10033e10	−0.217409	−0.108704	+	0.994074i	$0.534670\pi$
$660$	0	0
$661$	− 2.41101e11i	− 1.26297i	−0.775387	−	0.631486i	$-0.782445\pi$
$661$	− 2.41101e11i	− 1.26297i	0.775387	−	0.631486i	$-0.217555\pi$
$662$	0	0
$663$	4.42024e10i	0.228766i
$664$	0	0
$665$	−3.04291e11	−1.55598
$666$	0	0
$667$	−2.43929e11	−1.23243
$668$	0	0
$669$	− 4.43987e10i	− 0.221649i
$670$	0	0
$671$	1.72777e10i	0.0852309i
$672$	0	0
$673$	−1.07571e11	−0.524367	−0.262183	−	0.965018i	$-0.584443\pi$
$673$	−1.07571e11	−0.524367	−0.262183	+	0.965018i	$0.584443\pi$
$674$	0	0
$675$	5.47368e10	0.263672
$676$	0	0
$677$	− 2.01309e9i	− 0.00958316i	−0.999989	−	0.00479158i	$-0.998475\pi$
$677$	− 2.01309e9i	− 0.00958316i	0.999989	−	0.00479158i	$-0.00152521\pi$
$678$	0	0
$679$	2.03730e10i	0.0958466i
$680$	0	0
$681$	2.41046e10	0.112076
$682$	0	0
$683$	−7.91693e10	−0.363810	−0.181905	−	0.983316i	$-0.558226\pi$
$683$	−7.91693e10	−0.363810	−0.181905	+	0.983316i	$0.558226\pi$
$684$	0	0
$685$	2.96680e10i	0.134749i
$686$	0	0
$687$	5.70275e10i	0.256010i
$688$	0	0
$689$	−5.44444e10	−0.241589
$690$	0	0
$691$	4.38242e10	0.192222	0.0961108	−	0.995371i	$-0.469360\pi$
$691$	4.38242e10	0.192222	0.0961108	+	0.995371i	$0.469360\pi$
$692$	0	0
$693$	1.62386e10i	0.0704070i
$694$	0	0
$695$	− 8.39948e10i	− 0.360009i
$696$	0	0
$697$	−2.58409e11	−1.09491
$698$	0	0
$699$	6.97083e10	0.291995
$700$	0	0
$701$	− 1.77030e10i	− 0.0733118i	−0.999328	−	0.0366559i	$-0.988329\pi$
$701$	− 1.77030e10i	− 0.0733118i	0.999328	−	0.0366559i	$-0.0116705\pi$
$702$	0	0
$703$	− 7.16890e10i	− 0.293516i
$704$	0	0
$705$	1.07768e10	0.0436247
$706$	0	0
$707$	2.62791e11	1.05180
$708$	0	0
$709$	− 6.68754e10i	− 0.264656i	−0.991206	−	0.132328i	$-0.957755\pi$
$709$	− 6.68754e10i	− 0.264656i	0.991206	−	0.132328i	$-0.0422452\pi$
$710$	0	0
$711$	3.10748e11i	1.21599i
$712$	0	0
$713$	6.60891e11	2.55724
$714$	0	0
$715$	3.54346e10	0.135582
$716$	0	0
$717$	− 1.67702e10i	− 0.0634544i
$718$	0	0
$719$	− 4.12968e11i	− 1.54526i	−0.634857	−	0.772630i	$-0.718941\pi$
$719$	− 4.12968e11i	− 1.54526i	0.634857	−	0.772630i	$-0.281059\pi$
$720$	0	0
$721$	1.01826e11	0.376804
$722$	0	0
$723$	2.90220e10	0.106212
$724$	0	0
$725$	1.22011e11i	0.441619i
$726$	0	0
$727$	4.23681e11i	1.51671i	0.651843	+	0.758354i	$0.273996\pi$
$727$	4.23681e11i	1.51671i	−0.651843	+	0.758354i	$0.726004\pi$
$728$	0	0
$729$	−2.06493e11	−0.731130
$730$	0	0
$731$	−1.50305e11	−0.526387
$732$	0	0
$733$	9.12513e10i	0.316099i	0.987431	+	0.158050i	$0.0505206\pi$
$733$	9.12513e10i	0.316099i	−0.987431	+	0.158050i	$0.949479\pi$
$734$	0	0
$735$	− 3.61976e10i	− 0.124031i
$736$	0	0
$737$	3.24855e10	0.110108
$738$	0	0
$739$	2.27721e11	0.763527	0.381764	−	0.924260i	$-0.375317\pi$
$739$	2.27721e11	0.763527	0.381764	+	0.924260i	$0.375317\pi$
$740$	0	0
$741$	1.14401e11i	0.379451i
$742$	0	0
$743$	− 2.30762e10i	− 0.0757196i	−0.999283	−	0.0378598i	$-0.987946\pi$
$743$	− 2.30762e10i	− 0.0757196i	0.999283	−	0.0378598i	$-0.0120540\pi$
$744$	0	0
$745$	4.84860e11	1.57395
$746$	0	0
$747$	−7.11401e10	−0.228471
$748$	0	0
$749$	3.38451e11i	1.07540i
$750$	0	0
$751$	8.42681e10i	0.264913i	0.991189	+	0.132457i	$0.0422865\pi$
$751$	8.42681e10i	0.264913i	−0.991189	+	0.132457i	$0.957713\pi$
$752$	0	0
$753$	5.13344e10	0.159672
$754$	0	0
$755$	3.77460e11	1.16167
$756$	0	0
$757$	4.53458e11i	1.38087i	0.723393	+	0.690436i	$0.242581\pi$
$757$	4.53458e11i	1.38087i	−0.723393	+	0.690436i	$0.757419\pi$
$758$	0	0
$759$	1.24560e10i	0.0375329i
$760$	0	0
$761$	−3.05406e11	−0.910623	−0.455311	−	0.890332i	$-0.650472\pi$
$761$	−3.05406e11	−0.910623	−0.455311	+	0.890332i	$0.650472\pi$
$762$	0	0
$763$	−9.06406e10	−0.267439
$764$	0	0
$765$	− 4.13274e11i	− 1.20668i
$766$	0	0
$767$	− 4.86138e10i	− 0.140468i
$768$	0	0
$769$	3.54824e11	1.01463	0.507316	−	0.861760i	$-0.330638\pi$
$769$	3.54824e11	1.01463	0.507316	+	0.861760i	$0.330638\pi$
$770$	0	0
$771$	−1.64751e10	−0.0466242
$772$	0	0
$773$	− 1.47448e11i	− 0.412971i	−0.978450	−	0.206485i	$-0.933797\pi$
$773$	− 1.47448e11i	− 0.412971i	0.978450	−	0.206485i	$-0.0662026\pi$
$774$	0	0
$775$	− 3.30572e11i	− 0.916345i
$776$	0	0
$777$	−1.03950e10	−0.0285195
$778$	0	0
$779$	−6.68793e11	−1.81611
$780$	0	0
$781$	− 3.00854e10i	− 0.0808633i
$782$	0	0
$783$	1.11955e11i	0.297848i
$784$	0	0
$785$	−3.89290e11	−1.02517
$786$	0	0
$787$	5.12163e11	1.33509	0.667543	−	0.744572i	$-0.267346\pi$
$787$	5.12163e11	1.33509	0.667543	+	0.744572i	$0.267346\pi$
$788$	0	0
$789$	7.61183e10i	0.196418i
$790$	0	0
$791$	− 1.38174e10i	− 0.0352956i
$792$	0	0
$793$	−3.61063e11	−0.913040
$794$	0	0
$795$	−2.48831e10	−0.0622925
$796$	0	0
$797$	− 3.39499e11i	− 0.841404i	−0.907199	−	0.420702i	$-0.861784\pi$
$797$	− 3.39499e11i	− 0.841404i	0.907199	−	0.420702i	$-0.138216\pi$
$798$	0	0
$799$	− 6.41359e10i	− 0.157367i
$800$	0	0
$801$	−1.55177e11	−0.376962
$802$	0	0
$803$	6.15315e9	0.0147991
$804$	0	0
$805$	− 6.92364e11i	− 1.64874i
$806$	0	0
$807$	− 2.86675e10i	− 0.0675920i
$808$	0	0
$809$	1.13991e11	0.266120	0.133060	−	0.991108i	$-0.457520\pi$
$809$	1.13991e11	0.266120	0.133060	+	0.991108i	$0.457520\pi$
$810$	0	0
$811$	5.19425e10	0.120071	0.0600357	−	0.998196i	$-0.480879\pi$
$811$	5.19425e10	0.120071	0.0600357	+	0.998196i	$0.480879\pi$
$812$	0	0
$813$	− 1.14280e11i	− 0.261582i
$814$	0	0
$815$	7.12762e11i	1.61553i
$816$	0	0
$817$	−3.89007e11	−0.873112
$818$	0	0
$819$	−3.39347e11	−0.754239
$820$	0	0
$821$	− 1.77079e11i	− 0.389757i	−0.980827	−	0.194878i	$-0.937569\pi$
$821$	− 1.77079e11i	− 0.389757i	0.980827	−	0.194878i	$-0.0624312\pi$
$822$	0	0
$823$	− 6.99908e9i	− 0.0152560i	−0.999971	−	0.00762802i	$-0.997572\pi$
$823$	− 6.99908e9i	− 0.0152560i	0.999971	−	0.00762802i	$-0.00242810\pi$
$824$	0	0
$825$	6.23038e9	0.0134493
$826$	0	0
$827$	−7.66261e11	−1.63815	−0.819077	−	0.573683i	$-0.805514\pi$
$827$	−7.66261e11	−1.63815	−0.819077	+	0.573683i	$0.805514\pi$
$828$	0	0
$829$	6.81365e11i	1.44265i	0.692595	+	0.721327i	$0.256467\pi$
$829$	6.81365e11i	1.44265i	−0.692595	+	0.721327i	$0.743533\pi$
$830$	0	0
$831$	− 1.49656e11i	− 0.313826i
$832$	0	0
$833$	−2.15423e11	−0.447417
$834$	0	0
$835$	5.61003e10	0.115404
$836$	0	0
$837$	− 3.03325e11i	− 0.618025i
$838$	0	0
$839$	− 2.98466e11i	− 0.602348i	−0.953569	−	0.301174i	$-0.902622\pi$
$839$	− 2.98466e11i	− 0.602348i	0.953569	−	0.301174i	$-0.0973785\pi$
$840$	0	0
$841$	2.50693e11	0.501140
$842$	0	0
$843$	−9.07833e10	−0.179761
$844$	0	0
$845$	9.05337e10i	0.177576i
$846$	0	0
$847$	− 3.77676e11i	− 0.733813i
$848$	0	0
$849$	−1.24698e11	−0.240011
$850$	0	0
$851$	1.63117e11	0.311014
$852$	0	0
$853$	8.28052e11i	1.56409i	0.623223	+	0.782045i	$0.285823\pi$
$853$	8.28052e11i	1.56409i	−0.623223	+	0.782045i	$0.714177\pi$
$854$	0	0
$855$	− 1.06960e12i	− 2.00151i
$856$	0	0
$857$	−3.13054e11	−0.580357	−0.290179	−	0.956973i	$-0.593715\pi$
$857$	−3.13054e11	−0.580357	−0.290179	+	0.956973i	$0.593715\pi$
$858$	0	0
$859$	5.33113e11	0.979145	0.489572	−	0.871963i	$-0.337153\pi$
$859$	5.33113e11	0.979145	0.489572	+	0.871963i	$0.337153\pi$
$860$	0	0
$861$	9.69763e10i	0.176463i
$862$	0	0
$863$	− 1.76376e11i	− 0.317977i	−0.987280	−	0.158989i	$-0.949177\pi$
$863$	− 1.76376e11i	− 0.317977i	0.987280	−	0.158989i	$-0.0508233\pi$
$864$	0	0
$865$	1.21135e12	2.16374
$866$	0	0
$867$	−1.75203e9	−0.00310075
$868$	0	0
$869$	7.24703e10i	0.127081i
$870$	0	0
$871$	6.78869e11i	1.17954i
$872$	0	0
$873$	−7.16124e10	−0.123291
$874$	0	0
$875$	2.07561e11	0.354090
$876$	0	0
$877$	− 4.62552e11i	− 0.781920i	−0.920408	−	0.390960i	$-0.872143\pi$
$877$	− 4.62552e11i	− 0.781920i	0.920408	−	0.390960i	$-0.127857\pi$
$878$	0	0
$879$	4.82601e10i	0.0808412i
$880$	0	0
$881$	1.48349e10	0.0246253	0.0123127	−	0.999924i	$-0.496081\pi$
$881$	1.48349e10	0.0246253	0.0123127	+	0.999924i	$0.496081\pi$
$882$	0	0
$883$	−6.64762e11	−1.09351	−0.546756	−	0.837292i	$-0.684137\pi$
$883$	−6.64762e11	−1.09351	−0.546756	+	0.837292i	$0.684137\pi$
$884$	0	0
$885$	− 2.22183e10i	− 0.0362191i
$886$	0	0
$887$	5.25943e11i	0.849658i	0.905274	+	0.424829i	$0.139666\pi$
$887$	5.25943e11i	0.849658i	−0.905274	+	0.424829i	$0.860334\pi$
$888$	0	0
$889$	4.52213e11	0.723996
$890$	0	0
$891$	−5.41530e10	−0.0859235
$892$	0	0
$893$	− 1.65991e11i	− 0.261023i
$894$	0	0
$895$	1.17030e12i	1.82391i
$896$	0	0
$897$	−2.60300e11	−0.402073
$898$	0	0
$899$	6.76127e11	1.03512
$900$	0	0
$901$	1.48087e11i	0.224707i
$902$	0	0
$903$	5.64069e10i	0.0848361i
$904$	0	0
$905$	−3.20646e11	−0.478004
$906$	0	0
$907$	1.11825e12	1.65239	0.826193	−	0.563388i	$-0.190502\pi$
$907$	1.11825e12	1.65239	0.826193	+	0.563388i	$0.190502\pi$
$908$	0	0
$909$	9.23725e11i	1.35297i
$910$	0	0
$911$	− 3.98496e11i	− 0.578563i	−0.957244	−	0.289282i	$-0.906584\pi$
$911$	− 3.98496e11i	− 0.578563i	0.957244	−	0.289282i	$-0.0934164\pi$
$912$	0	0
$913$	−1.65908e10	−0.0238772
$914$	0	0
$915$	−1.65019e11	−0.235423
$916$	0	0
$917$	− 4.85760e11i	− 0.686980i
$918$	0	0
$919$	6.28911e11i	0.881713i	0.897578	+	0.440857i	$0.145325\pi$
$919$	6.28911e11i	0.881713i	−0.897578	+	0.440857i	$0.854675\pi$
$920$	0	0
$921$	1.80902e11	0.251423
$922$	0	0
$923$	6.28711e11	0.866252
$924$	0	0
$925$	− 8.15894e10i	− 0.111447i
$926$	0	0
$927$	3.57923e11i	0.484697i
$928$	0	0
$929$	7.31045e11	0.981480	0.490740	−	0.871306i	$-0.336727\pi$
$929$	7.31045e11	0.981480	0.490740	+	0.871306i	$0.336727\pi$
$930$	0	0
$931$	−5.57540e11	−0.742125
$932$	0	0
$933$	− 1.08389e11i	− 0.143040i
$934$	0	0
$935$	− 9.63808e10i	− 0.126108i
$936$	0	0
$937$	7.48792e11	0.971411	0.485705	−	0.874123i	$-0.338563\pi$
$937$	7.48792e11	0.971411	0.485705	+	0.874123i	$0.338563\pi$
$938$	0	0
$939$	1.27046e11	0.163417
$940$	0	0
$941$	− 9.67709e11i	− 1.23420i	−0.786884	−	0.617101i	$-0.788307\pi$
$941$	− 9.67709e11i	− 1.23420i	0.786884	−	0.617101i	$-0.211693\pi$
$942$	0	0
$943$	− 1.52173e12i	− 1.92438i
$944$	0	0
$945$	−3.17770e11	−0.398461
$946$	0	0
$947$	1.41847e11	0.176368	0.0881839	−	0.996104i	$-0.471894\pi$
$947$	1.41847e11	0.176368	0.0881839	+	0.996104i	$0.471894\pi$
$948$	0	0
$949$	1.28586e11i	0.158536i
$950$	0	0
$951$	− 1.82644e11i	− 0.223297i
$952$	0	0
$953$	−1.01418e12	−1.22954	−0.614771	−	0.788706i	$-0.710751\pi$
$953$	−1.01418e12	−1.22954	−0.614771	+	0.788706i	$0.710751\pi$
$954$	0	0
$955$	2.42159e11	0.291130
$956$	0	0
$957$	1.27432e10i	0.0151925i
$958$	0	0
$959$	− 6.62610e10i	− 0.0783400i
$960$	0	0
$961$	−9.78977e11	−1.14783
$962$	0	0
$963$	−1.18968e12	−1.38332
$964$	0	0
$965$	1.61691e12i	1.86456i
$966$	0	0
$967$	4.83390e11i	0.552830i	0.961038	+	0.276415i	$0.0891464\pi$
$967$	4.83390e11i	0.552830i	−0.961038	+	0.276415i	$0.910854\pi$
$968$	0	0
$969$	3.11166e11	0.352937
$970$	0	0
$971$	−1.16310e12	−1.30840	−0.654198	−	0.756323i	$-0.726994\pi$
$971$	−1.16310e12	−1.30840	−0.654198	+	0.756323i	$0.726994\pi$
$972$	0	0
$973$	1.87595e11i	0.209301i
$974$	0	0
$975$	1.30200e11i	0.144076i
$976$	0	0
$977$	4.01473e11	0.440634	0.220317	−	0.975428i	$-0.429291\pi$
$977$	4.01473e11	0.440634	0.220317	+	0.975428i	$0.429291\pi$
$978$	0	0
$979$	−3.61893e10	−0.0393958
$980$	0	0
$981$	− 3.18607e11i	− 0.344016i
$982$	0	0
$983$	1.72803e11i	0.185071i	0.995709	+	0.0925353i	$0.0294971\pi$
$983$	1.72803e11i	0.185071i	−0.995709	+	0.0925353i	$0.970503\pi$
$984$	0	0
$985$	−9.32960e11	−0.991101
$986$	0	0
$987$	−2.40690e10	−0.0253624
$988$	0	0
$989$	− 8.85122e11i	− 0.925163i
$990$	0	0
$991$	− 1.68640e12i	− 1.74850i	−0.485472	−	0.874252i	$-0.661352\pi$
$991$	− 1.68640e12i	− 1.74850i	0.485472	−	0.874252i	$-0.338648\pi$
$992$	0	0
$993$	8.74075e10	0.0898983
$994$	0	0
$995$	−1.12212e12	−1.14485
$996$	0	0
$997$	− 5.49409e11i	− 0.556052i	−0.960574	−	0.278026i	$-0.910320\pi$
$997$	− 5.49409e11i	− 0.556052i	0.960574	−	0.278026i	$-0.0896801\pi$
$998$	0	0
$999$	− 7.48645e10i	− 0.0751647i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.9.d.b.15.3		6
3.2	odd	2		288.9.b.b.271.5		6
4.3	odd	2		8.9.d.b.3.2	yes	6
8.3	odd	2	inner	32.9.d.b.15.4		6
8.5	even	2		8.9.d.b.3.1	✓	6
12.11	even	2		72.9.b.b.19.5		6
16.3	odd	4		256.9.c.n.255.5		12
16.5	even	4		256.9.c.n.255.6		12
16.11	odd	4		256.9.c.n.255.8		12
16.13	even	4		256.9.c.n.255.7		12
24.5	odd	2		72.9.b.b.19.6		6
24.11	even	2		288.9.b.b.271.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.9.d.b.3.1	✓	6	8.5	even	2
8.9.d.b.3.2	yes	6	4.3	odd	2
32.9.d.b.15.3		6	1.1	even	1	trivial
32.9.d.b.15.4		6	8.3	odd	2	inner
72.9.b.b.19.5		6	12.11	even	2
72.9.b.b.19.6		6	24.5	odd	2
256.9.c.n.255.5		12	16.3	odd	4
256.9.c.n.255.6		12	16.5	even	4
256.9.c.n.255.7		12	16.13	even	4
256.9.c.n.255.8		12	16.11	odd	4
288.9.b.b.271.2		6	24.11	even	2
288.9.b.b.271.5		6	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	32.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+17.4864 q^{3} -796.785i q^{5} +1779.55i q^{7} -6255.23 q^{9} +O(q^{10})\)	\(q+17.4864 q^{3} -796.785i q^{5} +1779.55i q^{7} -6255.23 q^{9} -1458.80 q^{11} -30485.3i q^{13} -13932.9i q^{15} -82919.0 q^{17} -214604. q^{19} +31118.0i q^{21} -488296. i q^{23} -244241. q^{25} -224110. q^{27} -499553. i q^{29} +1.35347e6i q^{31} -25509.1 q^{33} +1.41792e6 q^{35} +334053. i q^{37} -533079. i q^{39} +3.11641e6 q^{41} +1.81268e6 q^{43} +4.98407e6i q^{45} +773477. i q^{47} +2.59799e6 q^{49} -1.44996e6 q^{51} -1.78592e6i q^{53} +1.16235e6i q^{55} -3.75265e6 q^{57} +1.59466e6 q^{59} -1.18438e7i q^{61} -1.11315e7i q^{63} -2.42902e7 q^{65} -2.22687e7 q^{67} -8.53854e6i q^{69} +2.06234e7i q^{71} -4.21796e6 q^{73} -4.27090e6 q^{75} -2.59601e6i q^{77} -4.96781e7i q^{79} +3.71217e7 q^{81} +1.13729e7 q^{83} +6.60686e7i q^{85} -8.73538e6i q^{87} +2.48076e7 q^{89} +5.42502e7 q^{91} +2.36672e7i q^{93} +1.70993e8i q^{95} +1.14484e7 q^{97} +9.12511e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 36 q^{3} + 16338 q^{9}+O(q^{10}) \) 6 * q + 36 * q^3 + 16338 * q^9	\( 6 q + 36 q^{3} + 16338 q^{9} - 46940 q^{11} - 201076 q^{17} + 95268 q^{19} - 447930 q^{25} - 419256 q^{27} + 1736856 q^{33} - 1989120 q^{35} + 3817100 q^{41} + 9881508 q^{43} - 12655482 q^{49} + 4303176 q^{51} + 7523736 q^{57} - 25243484 q^{59} + 27060480 q^{65} - 47850204 q^{67} - 55484916 q^{73} + 126075300 q^{75} - 79145442 q^{81} - 144646364 q^{83} + 142173452 q^{89} + 273971712 q^{91} + 125193612 q^{97} - 298320276 q^{99}+O(q^{100}) \) 6 * q + 36 * q^3 + 16338 * q^9 - 46940 * q^11 - 201076 * q^17 + 95268 * q^19 - 447930 * q^25 - 419256 * q^27 + 1736856 * q^33 - 1989120 * q^35 + 3817100 * q^41 + 9881508 * q^43 - 12655482 * q^49 + 4303176 * q^51 + 7523736 * q^57 - 25243484 * q^59 + 27060480 * q^65 - 47850204 * q^67 - 55484916 * q^73 + 126075300 * q^75 - 79145442 * q^81 - 144646364 * q^83 + 142173452 * q^89 + 273971712 * q^91 + 125193612 * q^97 - 298320276 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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