LMFDB - Embedded newform 4.24.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4,24,Mod(1,4)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 24, names="a")

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

chi := DirichletCharacter("4.1");

S:= CuspForms(chi, 24);

N := Newforms(S);

Level:	$ N $	$=$	$ 4 = 2^{2} $
Weight:	$ k $	$=$	$ 24 $
Character orbit:	$[\chi]$	$=$	4.a (trivial)

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:	yes
Analytic conductor:	$13.4081614938$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\mathbb{Q}[x]/(x^{2} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 618312 $ x^2 - x - 618312
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-785.828$ of defining polynomial
Character	$\chi$	$=$	4.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+387210. q^{3} -2.09186e8 q^{5} +7.31949e8 q^{7} +5.57885e10 q^{9} +O(q^{10})$	$q+387210. q^{3} -2.09186e8 q^{5} +7.31949e8 q^{7} +5.57885e10 q^{9} -1.52441e12 q^{11} +3.20818e12 q^{13} -8.09989e13 q^{15} -1.42698e14 q^{17} -3.73906e14 q^{19} +2.83418e14 q^{21} -1.74397e14 q^{23} +3.18379e16 q^{25} -1.48513e16 q^{27} -6.76214e16 q^{29} +1.20141e17 q^{31} -5.90269e17 q^{33} -1.53113e17 q^{35} +1.03997e18 q^{37} +1.24224e18 q^{39} +4.44857e18 q^{41} -3.59241e18 q^{43} -1.16702e19 q^{45} +1.37780e19 q^{47} -2.68330e19 q^{49} -5.52541e19 q^{51} +1.46557e19 q^{53} +3.18886e20 q^{55} -1.44780e20 q^{57} +4.80044e19 q^{59} -1.58529e20 q^{61} +4.08343e19 q^{63} -6.71107e20 q^{65} -4.69567e20 q^{67} -6.75283e19 q^{69} +1.33886e21 q^{71} -2.99578e21 q^{73} +1.23279e22 q^{75} -1.11579e21 q^{77} -5.35321e21 q^{79} -1.10027e22 q^{81} -1.08227e22 q^{83} +2.98504e22 q^{85} -2.61837e22 q^{87} -1.80097e22 q^{89} +2.34822e21 q^{91} +4.65196e22 q^{93} +7.82158e22 q^{95} +2.49638e22 q^{97} -8.50448e22 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 170520 q^{3} - 92266020 q^{5} + 192083440 q^{7} + 8599879818 q^{9}+O(q^{10}) $ 2 * q + 170520 * q^3 - 92266020 * q^5 + 192083440 * q^7 + 8599879818 * q^9	$ 2 q + 170520 q^{3} - 92266020 q^{5} + 192083440 q^{7} + 8599879818 q^{9} - 1247174695800 q^{11} - 7460299980980 q^{13} - 106334360092080 q^{15} - 374897347903260 q^{17} - 840360279212552 q^{19} + 400401295079232 q^{21} + 64\!\cdots\!20 q^{23}+ \cdots - 98\!\cdots\!00 q^{99}+O(q^{100}) $ 2 * q + 170520 * q^3 - 92266020 * q^5 + 192083440 * q^7 + 8599879818 * q^9 - 1247174695800 * q^11 - 7460299980980 * q^13 - 106334360092080 * q^15 - 374897347903260 * q^17 - 840360279212552 * q^19 + 400401295079232 * q^21 + 6433357923072720 * q^23 + 33587241295679150 * q^25 + 15773861091124080 * q^27 - 68055499247434452 * q^29 - 145584514546845248 * q^31 - 650343867352613280 * q^33 - 216234488190163680 * q^35 + 1211894143551157660 * q^37 + 3553993994899379088 * q^39 + 5036778367134688692 * q^41 + 935180945919935560 * q^43 - 17187460645010169780 * q^45 - 8431168896277596000 * q^47 - 53910291562566659694 * q^49 - 4938798717170442576 * q^51 + 3763137197370204540 * q^53 + 351301189218989209200 * q^55 - 43703893486254219360 * q^57 + 348561606512901387816 * q^59 - 582966592720711355156 * q^61 + 66309764232647559600 * q^63 - 1918465858864426149720 * q^65 + 211615268845414654360 * q^67 - 1499363181860662472256 * q^69 + 3840047757133544010096 * q^71 - 680666251913594096780 * q^73 + 11948876435052098946600 * q^75 - 1265465342298122095680 * q^77 - 642991395306723640736 * q^79 - 13196380320218478599982 * q^81 - 18799502632508717771400 * q^83 + 2701670550661359899640 * q^85 - 26089604267151850210800 * q^87 + 5001398778883250139732 * q^89 + 8107764572821621082528 * q^91 + 104099589081258166744320 * q^93 + 23677928248035152895120 * q^95 + 122022794260559820614980 * q^97 - 98127319293495379027800 * q^99

Display 100 coefficients 10 coefficients

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
$3$	387210.	1.26198	0.630990	−	0.775791i	$-0.282649\pi$
$3$	387210.	1.26198	0.630990	+	0.775791i	$0.282649\pi$
$4$	0	0
$5$	−2.09186e8	−1.91592	−0.957961	−	0.286900i	$-0.907375\pi$
$5$	−2.09186e8	−1.91592	−0.957961	+	0.286900i	$0.907375\pi$
$6$	0	0
$7$	7.31949e8	0.139911	0.0699557	−	0.997550i	$-0.477714\pi$
$7$	7.31949e8	0.139911	0.0699557	+	0.997550i	$0.477714\pi$
$8$	0	0
$9$	5.57885e10	0.592592
$10$	0	0
$11$	−1.52441e12	−1.61097	−0.805485	−	0.592617i	$-0.798095\pi$
$11$	−1.52441e12	−1.61097	−0.805485	+	0.592617i	$0.798095\pi$
$12$	0	0
$13$	3.20818e12	0.496490	0.248245	−	0.968697i	$-0.420146\pi$
$13$	3.20818e12	0.496490	0.248245	+	0.968697i	$0.420146\pi$
$14$	0	0
$15$	−8.09989e13	−2.41785
$16$	0	0
$17$	−1.42698e14	−1.00985	−0.504923	−	0.863164i	$-0.668479\pi$
$17$	−1.42698e14	−1.00985	−0.504923	+	0.863164i	$0.668479\pi$
$18$	0	0
$19$	−3.73906e14	−0.736369	−0.368185	−	0.929753i	$-0.620021\pi$
$19$	−3.73906e14	−0.736369	−0.368185	+	0.929753i	$0.620021\pi$
$20$	0	0
$21$	2.83418e14	0.176565
$22$	0	0
$23$	−1.74397e14	−0.0381653	−0.0190827	−	0.999818i	$-0.506075\pi$
$23$	−1.74397e14	−0.0381653	−0.0190827	+	0.999818i	$0.506075\pi$
$24$	0	0
$25$	3.18379e16	2.67075
$26$	0	0
$27$	−1.48513e16	−0.514141
$28$	0	0
$29$	−6.76214e16	−1.02922	−0.514609	−	0.857425i	$-0.672063\pi$
$29$	−6.76214e16	−1.02922	−0.514609	+	0.857425i	$0.672063\pi$
$30$	0	0
$31$	1.20141e17	0.849239	0.424620	−	0.905372i	$-0.360408\pi$
$31$	1.20141e17	0.849239	0.424620	+	0.905372i	$0.360408\pi$
$32$	0	0
$33$	−5.90269e17	−2.03301
$34$	0	0
$35$	−1.53113e17	−0.268059
$36$	0	0
$37$	1.03997e18	0.960952	0.480476	−	0.877008i	$-0.340464\pi$
$37$	1.03997e18	0.960952	0.480476	+	0.877008i	$0.340464\pi$
$38$	0	0
$39$	1.24224e18	0.626560
$40$	0	0
$41$	4.44857e18	1.26243	0.631214	−	0.775609i	$-0.282557\pi$
$41$	4.44857e18	1.26243	0.631214	+	0.775609i	$0.282557\pi$
$42$	0	0
$43$	−3.59241e18	−0.589520	−0.294760	−	0.955571i	$-0.595240\pi$
$43$	−3.59241e18	−0.589520	−0.294760	+	0.955571i	$0.595240\pi$
$44$	0	0
$45$	−1.16702e19	−1.13536
$46$	0	0
$47$	1.37780e19	0.812946	0.406473	−	0.913663i	$-0.366759\pi$
$47$	1.37780e19	0.812946	0.406473	+	0.913663i	$0.366759\pi$
$48$	0	0
$49$	−2.68330e19	−0.980425
$50$	0	0
$51$	−5.52541e19	−1.27440
$52$	0	0
$53$	1.46557e19	0.217187	0.108593	−	0.994086i	$-0.465365\pi$
$53$	1.46557e19	0.217187	0.108593	+	0.994086i	$0.465365\pi$
$54$	0	0
$55$	3.18886e20	3.08649
$56$	0	0
$57$	−1.44780e20	−0.929282
$58$	0	0
$59$	4.80044e19	0.207244	0.103622	−	0.994617i	$-0.466957\pi$
$59$	4.80044e19	0.207244	0.103622	+	0.994617i	$0.466957\pi$
$60$	0	0
$61$	−1.58529e20	−0.466462	−0.233231	−	0.972421i	$-0.574930\pi$
$61$	−1.58529e20	−0.466462	−0.233231	+	0.972421i	$0.574930\pi$
$62$	0	0
$63$	4.08343e19	0.0829103
$64$	0	0
$65$	−6.71107e20	−0.951235
$66$	0	0
$67$	−4.69567e20	−0.469718	−0.234859	−	0.972029i	$-0.575463\pi$
$67$	−4.69567e20	−0.469718	−0.234859	+	0.972029i	$0.575463\pi$
$68$	0	0
$69$	−6.75283e19	−0.0481638
$70$	0	0
$71$	1.33886e21	0.687487	0.343743	−	0.939064i	$-0.388305\pi$
$71$	1.33886e21	0.687487	0.343743	+	0.939064i	$0.388305\pi$
$72$	0	0
$73$	−2.99578e21	−1.11763	−0.558814	−	0.829293i	$-0.688743\pi$
$73$	−2.99578e21	−1.11763	−0.558814	+	0.829293i	$0.688743\pi$
$74$	0	0
$75$	1.23279e22	3.37044
$76$	0	0
$77$	−1.11579e21	−0.225393
$78$	0	0
$79$	−5.35321e21	−0.805198	−0.402599	−	0.915377i	$-0.631893\pi$
$79$	−5.35321e21	−0.805198	−0.402599	+	0.915377i	$0.631893\pi$
$80$	0	0
$81$	−1.10027e22	−1.24143
$82$	0	0
$83$	−1.08227e22	−0.922439	−0.461219	−	0.887286i	$-0.652588\pi$
$83$	−1.08227e22	−0.922439	−0.461219	+	0.887286i	$0.652588\pi$
$84$	0	0
$85$	2.98504e22	1.93479
$86$	0	0
$87$	−2.61837e22	−1.29885
$88$	0	0
$89$	−1.80097e22	−0.687894	−0.343947	−	0.938989i	$-0.611764\pi$
$89$	−1.80097e22	−0.687894	−0.343947	+	0.938989i	$0.611764\pi$
$90$	0	0
$91$	2.34822e21	0.0694646
$92$	0	0
$93$	4.65196e22	1.07172
$94$	0	0
$95$	7.82158e22	1.41083
$96$	0	0
$97$	2.49638e22	0.354352	0.177176	−	0.984179i	$-0.443304\pi$
$97$	2.49638e22	0.354352	0.177176	+	0.984179i	$0.443304\pi$
$98$	0	0
$99$	−8.50448e22	−0.954647
$100$	0	0
$101$	4.15528e22	0.370599	0.185300	−	0.982682i	$-0.440674\pi$
$101$	4.15528e22	0.370599	0.185300	+	0.982682i	$0.440674\pi$
$102$	0	0
$103$	−1.58995e23	−1.13176	−0.565881	−	0.824487i	$-0.691464\pi$
$103$	−1.58995e23	−1.13176	−0.565881	+	0.824487i	$0.691464\pi$
$104$	0	0
$105$	−5.92871e22	−0.338285
$106$	0	0
$107$	−3.18100e23	−1.46100	−0.730499	−	0.682914i	$-0.760712\pi$
$107$	−3.18100e23	−1.46100	−0.730499	+	0.682914i	$0.760712\pi$
$108$	0	0
$109$	2.49191e23	0.924968	0.462484	−	0.886628i	$-0.346958\pi$
$109$	2.49191e23	0.924968	0.462484	+	0.886628i	$0.346958\pi$
$110$	0	0
$111$	4.02688e23	1.21270
$112$	0	0
$113$	−1.93004e23	−0.473330	−0.236665	−	0.971591i	$-0.576054\pi$
$113$	−1.93004e23	−0.473330	−0.236665	+	0.971591i	$0.576054\pi$
$114$	0	0
$115$	3.64814e22	0.0731217
$116$	0	0
$117$	1.78979e23	0.294216
$118$	0	0
$119$	−1.04448e23	−0.141289
$120$	0	0
$121$	1.42841e24	1.59522
$122$	0	0
$123$	1.72253e24	1.59316
$124$	0	0
$125$	−4.16635e24	−3.20103
$126$	0	0
$127$	−1.65775e24	−1.06115	−0.530575	−	0.847638i	$-0.678024\pi$
$127$	−1.65775e24	−1.06115	−0.530575	+	0.847638i	$0.678024\pi$
$128$	0	0
$129$	−1.39102e24	−0.743962
$130$	0	0
$131$	2.16608e24	0.970630	0.485315	−	0.874339i	$-0.338705\pi$
$131$	2.16608e24	0.970630	0.485315	+	0.874339i	$0.338705\pi$
$132$	0	0
$133$	−2.73680e23	−0.103026
$134$	0	0
$135$	3.10669e24	0.985053
$136$	0	0
$137$	−1.36579e24	−0.365678	−0.182839	−	0.983143i	$-0.558529\pi$
$137$	−1.36579e24	−0.365678	−0.182839	+	0.983143i	$0.558529\pi$
$138$	0	0
$139$	1.01914e24	0.230975	0.115488	−	0.993309i	$-0.463157\pi$
$139$	1.01914e24	0.230975	0.115488	+	0.993309i	$0.463157\pi$
$140$	0	0
$141$	5.33499e24	1.02592
$142$	0	0
$143$	−4.89060e24	−0.799830
$144$	0	0
$145$	1.41454e25	1.97190
$146$	0	0
$147$	−1.03900e25	−1.23728
$148$	0	0
$149$	−1.32992e25	−1.35576	−0.677881	−	0.735171i	$-0.737102\pi$
$149$	−1.32992e25	−1.35576	−0.677881	+	0.735171i	$0.737102\pi$
$150$	0	0
$151$	1.75843e25	1.53776	0.768882	−	0.639391i	$-0.220814\pi$
$151$	1.75843e25	1.53776	0.768882	+	0.639391i	$0.220814\pi$
$152$	0	0
$153$	−7.96090e24	−0.598426
$154$	0	0
$155$	−2.51317e25	−1.62708
$156$	0	0
$157$	−1.12704e25	−0.629640	−0.314820	−	0.949151i	$-0.601944\pi$
$157$	−1.12704e25	−0.629640	−0.314820	+	0.949151i	$0.601944\pi$
$158$	0	0
$159$	5.67483e24	0.274085
$160$	0	0
$161$	−1.27650e23	−0.00533976
$162$	0	0
$163$	1.39967e25	0.508006	0.254003	−	0.967203i	$-0.418253\pi$
$163$	1.39967e25	0.508006	0.254003	+	0.967203i	$0.418253\pi$
$164$	0	0
$165$	1.23476e26	3.89509
$166$	0	0
$167$	−5.01925e25	−1.37848	−0.689238	−	0.724535i	$-0.742054\pi$
$167$	−5.01925e25	−1.37848	−0.689238	+	0.724535i	$0.742054\pi$
$168$	0	0
$169$	−3.14615e25	−0.753498
$170$	0	0
$171$	−2.08596e25	−0.436366
$172$	0	0
$173$	−1.29563e25	−0.237110	−0.118555	−	0.992948i	$-0.537826\pi$
$173$	−1.29563e25	−0.237110	−0.118555	+	0.992948i	$0.537826\pi$
$174$	0	0
$175$	2.33037e25	0.373669
$176$	0	0
$177$	1.85878e25	0.261538
$178$	0	0
$179$	2.22455e25	0.275063	0.137531	−	0.990497i	$-0.456083\pi$
$179$	2.22455e25	0.275063	0.137531	+	0.990497i	$0.456083\pi$
$180$	0	0
$181$	−1.15496e26	−1.25679	−0.628395	−	0.777895i	$-0.716288\pi$
$181$	−1.15496e26	−1.25679	−0.628395	+	0.777895i	$0.716288\pi$
$182$	0	0
$183$	−6.13841e25	−0.588665
$184$	0	0
$185$	−2.17548e26	−1.84111
$186$	0	0
$187$	2.17531e26	1.62683
$188$	0	0
$189$	−1.08704e25	−0.0719342
$190$	0	0
$191$	2.24365e26	1.31544	0.657720	−	0.753262i	$-0.271521\pi$
$191$	2.24365e26	1.31544	0.657720	+	0.753262i	$0.271521\pi$
$192$	0	0
$193$	1.97440e26	1.02689	0.513447	−	0.858121i	$-0.328368\pi$
$193$	1.97440e26	1.02689	0.513447	+	0.858121i	$0.328368\pi$
$194$	0	0
$195$	−2.59859e26	−1.20044
$196$	0	0
$197$	2.41560e26	0.992347	0.496174	−	0.868223i	$-0.334738\pi$
$197$	2.41560e26	0.992347	0.496174	+	0.868223i	$0.334738\pi$
$198$	0	0
$199$	3.29101e26	1.20370	0.601850	−	0.798609i	$-0.294431\pi$
$199$	3.29101e26	1.20370	0.601850	+	0.798609i	$0.294431\pi$
$200$	0	0
$201$	−1.81821e26	−0.592775
$202$	0	0
$203$	−4.94954e25	−0.143999
$204$	0	0
$205$	−9.30580e26	−2.41871
$206$	0	0
$207$	−9.72934e24	−0.0226164
$208$	0	0
$209$	5.69987e26	1.18627
$210$	0	0
$211$	−1.06204e27	−1.98104	−0.990518	−	0.137387i	$-0.956130\pi$
$211$	−1.06204e27	−1.98104	−0.990518	+	0.137387i	$0.956130\pi$
$212$	0	0
$213$	5.18420e26	0.867594
$214$	0	0
$215$	7.51482e26	1.12947
$216$	0	0
$217$	8.79367e25	0.118818
$218$	0	0
$219$	−1.16000e27	−1.41042
$220$	0	0
$221$	−4.57801e26	−0.501378
$222$	0	0
$223$	4.37454e26	0.431943	0.215971	−	0.976400i	$-0.430708\pi$
$223$	4.37454e26	0.431943	0.215971	+	0.976400i	$0.430708\pi$
$224$	0	0
$225$	1.77619e27	1.58267
$226$	0	0
$227$	1.76894e27	1.42369	0.711846	−	0.702335i	$-0.247859\pi$
$227$	1.76894e27	1.42369	0.711846	+	0.702335i	$0.247859\pi$
$228$	0	0
$229$	1.02884e27	0.748581	0.374291	−	0.927311i	$-0.377886\pi$
$229$	1.02884e27	0.748581	0.374291	+	0.927311i	$0.377886\pi$
$230$	0	0
$231$	−4.32046e26	−0.284441
$232$	0	0
$233$	−1.28652e27	−0.767050	−0.383525	−	0.923530i	$-0.625290\pi$
$233$	−1.28652e27	−0.767050	−0.383525	+	0.923530i	$0.625290\pi$
$234$	0	0
$235$	−2.88217e27	−1.55754
$236$	0	0
$237$	−2.07282e27	−1.01614
$238$	0	0
$239$	−1.39893e26	−0.0622614	−0.0311307	−	0.999515i	$-0.509911\pi$
$239$	−1.39893e26	−0.0622614	−0.0311307	+	0.999515i	$0.509911\pi$
$240$	0	0
$241$	8.54422e26	0.345522	0.172761	−	0.984964i	$-0.444731\pi$
$241$	8.54422e26	0.345522	0.172761	+	0.984964i	$0.444731\pi$
$242$	0	0
$243$	−2.86220e27	−1.05251
$244$	0	0
$245$	5.61309e27	1.87842
$246$	0	0
$247$	−1.19956e27	−0.365600
$248$	0	0
$249$	−4.19066e27	−1.16410
$250$	0	0
$251$	2.52129e27	0.638814	0.319407	−	0.947618i	$-0.396516\pi$
$251$	2.52129e27	0.638814	0.319407	+	0.947618i	$0.396516\pi$
$252$	0	0
$253$	2.65853e26	0.0614831
$254$	0	0
$255$	1.15584e28	2.44166
$256$	0	0
$257$	−5.52365e27	−1.06658	−0.533292	−	0.845931i	$-0.679045\pi$
$257$	−5.52365e27	−1.06658	−0.533292	+	0.845931i	$0.679045\pi$
$258$	0	0
$259$	7.61206e26	0.134448
$260$	0	0
$261$	−3.77249e27	−0.609906
$262$	0	0
$263$	1.10056e28	1.62976	0.814880	−	0.579630i	$-0.196803\pi$
$263$	1.10056e28	1.62976	0.814880	+	0.579630i	$0.196803\pi$
$264$	0	0
$265$	−3.06576e27	−0.416113
$266$	0	0
$267$	−6.97354e27	−0.868108
$268$	0	0
$269$	−3.38900e27	−0.387187	−0.193593	−	0.981082i	$-0.562014\pi$
$269$	−3.38900e27	−0.387187	−0.193593	+	0.981082i	$0.562014\pi$
$270$	0	0
$271$	1.30721e28	1.37151	0.685753	−	0.727834i	$-0.259473\pi$
$271$	1.30721e28	1.37151	0.685753	+	0.727834i	$0.259473\pi$
$272$	0	0
$273$	9.09256e26	0.0876629
$274$	0	0
$275$	−4.85341e28	−4.30250
$276$	0	0
$277$	−1.72270e27	−0.140505	−0.0702526	−	0.997529i	$-0.522381\pi$
$277$	−1.72270e27	−0.140505	−0.0702526	+	0.997529i	$0.522381\pi$
$278$	0	0
$279$	6.70245e27	0.503252
$280$	0	0
$281$	8.49530e27	0.587565	0.293783	−	0.955872i	$-0.405086\pi$
$281$	8.49530e27	0.587565	0.293783	+	0.955872i	$0.405086\pi$
$282$	0	0
$283$	1.48272e28	0.945178	0.472589	−	0.881283i	$-0.343319\pi$
$283$	1.48272e28	0.945178	0.472589	+	0.881283i	$0.343319\pi$
$284$	0	0
$285$	3.02860e28	1.78043
$286$	0	0
$287$	3.25613e27	0.176628
$288$	0	0
$289$	3.95146e26	0.0197894
$290$	0	0
$291$	9.66622e27	0.447185
$292$	0	0
$293$	−3.04503e28	−1.30201	−0.651004	−	0.759074i	$-0.725652\pi$
$293$	−3.04503e28	−1.30201	−0.651004	+	0.759074i	$0.725652\pi$
$294$	0	0
$295$	−1.00418e28	−0.397064
$296$	0	0
$297$	2.26396e28	0.828265
$298$	0	0
$299$	−5.59497e26	−0.0189487
$300$	0	0
$301$	−2.62946e27	−0.0824806
$302$	0	0
$303$	1.60897e28	0.467689
$304$	0	0
$305$	3.31621e28	0.893704
$306$	0	0
$307$	−6.06055e28	−1.51503	−0.757515	−	0.652818i	$-0.773587\pi$
$307$	−6.06055e28	−1.51503	−0.757515	+	0.652818i	$0.773587\pi$
$308$	0	0
$309$	−6.15644e28	−1.42826
$310$	0	0
$311$	−8.87533e28	−1.91179	−0.955894	−	0.293711i	$-0.905110\pi$
$311$	−8.87533e28	−1.91179	−0.955894	+	0.293711i	$0.905110\pi$
$312$	0	0
$313$	7.40764e28	1.48225	0.741123	−	0.671369i	$-0.234294\pi$
$313$	7.40764e28	1.48225	0.741123	+	0.671369i	$0.234294\pi$
$314$	0	0
$315$	−8.54196e27	−0.158850
$316$	0	0
$317$	1.50150e28	0.259623	0.129812	−	0.991539i	$-0.458563\pi$
$317$	1.50150e28	0.259623	0.129812	+	0.991539i	$0.458563\pi$
$318$	0	0
$319$	1.03083e29	1.65804
$320$	0	0
$321$	−1.23171e29	−1.84375
$322$	0	0
$323$	5.33556e28	0.743619
$324$	0	0
$325$	1.02142e29	1.32600
$326$	0	0
$327$	9.64891e28	1.16729
$328$	0	0
$329$	1.00848e28	0.113740
$330$	0	0
$331$	−7.74199e28	−0.814386	−0.407193	−	0.913342i	$-0.633492\pi$
$331$	−7.74199e28	−0.814386	−0.407193	+	0.913342i	$0.633492\pi$
$332$	0	0
$333$	5.80185e28	0.569452
$334$	0	0
$335$	9.82269e28	0.899943
$336$	0	0
$337$	−2.10595e29	−1.80179	−0.900896	−	0.434035i	$-0.857089\pi$
$337$	−2.10595e29	−1.80179	−0.900896	+	0.434035i	$0.857089\pi$
$338$	0	0
$339$	−7.47330e28	−0.597332
$340$	0	0
$341$	−1.83144e29	−1.36810
$342$	0	0
$343$	−3.96729e28	−0.277084
$344$	0	0
$345$	1.41260e28	0.0922781
$346$	0	0
$347$	2.83300e29	1.73164	0.865821	−	0.500354i	$-0.166797\pi$
$347$	2.83300e29	1.73164	0.865821	+	0.500354i	$0.166797\pi$
$348$	0	0
$349$	−5.13553e28	−0.293828	−0.146914	−	0.989149i	$-0.546934\pi$
$349$	−5.13553e28	−0.293828	−0.146914	+	0.989149i	$0.546934\pi$
$350$	0	0
$351$	−4.76458e28	−0.255266
$352$	0	0
$353$	4.28041e28	0.214821	0.107410	−	0.994215i	$-0.465744\pi$
$353$	4.28041e28	0.214821	0.107410	+	0.994215i	$0.465744\pi$
$354$	0	0
$355$	−2.80071e29	−1.31717
$356$	0	0
$357$	−4.04432e28	−0.178304
$358$	0	0
$359$	−3.32369e28	−0.137415	−0.0687076	−	0.997637i	$-0.521888\pi$
$359$	−3.32369e28	−0.137415	−0.0687076	+	0.997637i	$0.521888\pi$
$360$	0	0
$361$	−1.18024e29	−0.457761
$362$	0	0
$363$	5.53095e29	2.01314
$364$	0	0
$365$	6.26675e29	2.14129
$366$	0	0
$367$	−1.37619e29	−0.441590	−0.220795	−	0.975320i	$-0.570865\pi$
$367$	−1.37619e29	−0.441590	−0.220795	+	0.975320i	$0.570865\pi$
$368$	0	0
$369$	2.48179e29	0.748104
$370$	0	0
$371$	1.07272e28	0.0303869
$372$	0	0
$373$	5.64990e29	1.50449	0.752244	−	0.658884i	$-0.228971\pi$
$373$	5.64990e29	1.50449	0.752244	+	0.658884i	$0.228971\pi$
$374$	0	0
$375$	−1.61325e30	−4.03964
$376$	0	0
$377$	−2.16942e29	−0.510996
$378$	0	0
$379$	3.18752e29	0.706484	0.353242	−	0.935532i	$-0.385079\pi$
$379$	3.18752e29	0.706484	0.353242	+	0.935532i	$0.385079\pi$
$380$	0	0
$381$	−6.41899e29	−1.33915
$382$	0	0
$383$	3.44375e29	0.676465	0.338233	−	0.941063i	$-0.390171\pi$
$383$	3.44375e29	0.676465	0.338233	+	0.941063i	$0.390171\pi$
$384$	0	0
$385$	2.33408e29	0.431835
$386$	0	0
$387$	−2.00415e29	−0.349345
$388$	0	0
$389$	−8.77881e29	−1.44217	−0.721083	−	0.692849i	$-0.756355\pi$
$389$	−8.77881e29	−1.44217	−0.721083	+	0.692849i	$0.756355\pi$
$390$	0	0
$391$	2.48861e28	0.0385411
$392$	0	0
$393$	8.38726e29	1.22491
$394$	0	0
$395$	1.11982e30	1.54270
$396$	0	0
$397$	−1.44903e30	−1.88359	−0.941793	−	0.336192i	$-0.890861\pi$
$397$	−1.44903e30	−1.88359	−0.941793	+	0.336192i	$0.890861\pi$
$398$	0	0
$399$	−1.05972e29	−0.130017
$400$	0	0
$401$	8.88431e29	1.02911	0.514557	−	0.857456i	$-0.327956\pi$
$401$	8.88431e29	1.02911	0.514557	+	0.857456i	$0.327956\pi$
$402$	0	0
$403$	3.85432e29	0.421639
$404$	0	0
$405$	2.30161e30	2.37848
$406$	0	0
$407$	−1.58535e30	−1.54806
$408$	0	0
$409$	−6.16570e29	−0.569068	−0.284534	−	0.958666i	$-0.591839\pi$
$409$	−6.16570e29	−0.569068	−0.284534	+	0.958666i	$0.591839\pi$
$410$	0	0
$411$	−5.28849e29	−0.461478
$412$	0	0
$413$	3.51367e28	0.0289959
$414$	0	0
$415$	2.26396e30	1.76732
$416$	0	0
$417$	3.94623e29	0.291486
$418$	0	0
$419$	−3.89006e29	−0.271954	−0.135977	−	0.990712i	$-0.543417\pi$
$419$	−3.89006e29	−0.271954	−0.135977	+	0.990712i	$0.543417\pi$
$420$	0	0
$421$	−1.64655e30	−1.08976	−0.544879	−	0.838515i	$-0.683424\pi$
$421$	−1.64655e30	−1.08976	−0.544879	+	0.838515i	$0.683424\pi$
$422$	0	0
$423$	7.68655e29	0.481745
$424$	0	0
$425$	−4.54320e30	−2.69705
$426$	0	0
$427$	−1.16035e29	−0.0652633
$428$	0	0
$429$	−1.89369e30	−1.00937
$430$	0	0
$431$	−2.94618e30	−1.48857	−0.744287	−	0.667859i	$-0.767211\pi$
$431$	−2.94618e30	−1.48857	−0.744287	+	0.667859i	$0.767211\pi$
$432$	0	0
$433$	2.55571e30	1.22434	0.612168	−	0.790728i	$-0.290298\pi$
$433$	2.55571e30	1.22434	0.612168	+	0.790728i	$0.290298\pi$
$434$	0	0
$435$	5.47726e30	2.48850
$436$	0	0
$437$	6.52080e28	0.0281037
$438$	0	0
$439$	1.96935e30	0.805341	0.402671	−	0.915345i	$-0.368082\pi$
$439$	1.96935e30	0.805341	0.402671	+	0.915345i	$0.368082\pi$
$440$	0	0
$441$	−1.49697e30	−0.580992
$442$	0	0
$443$	3.63340e30	1.33866	0.669330	−	0.742966i	$-0.266581\pi$
$443$	3.63340e30	1.33866	0.669330	+	0.742966i	$0.266581\pi$
$444$	0	0
$445$	3.76738e30	1.31795
$446$	0	0
$447$	−5.14959e30	−1.71094
$448$	0	0
$449$	−9.30596e29	−0.293717	−0.146858	−	0.989158i	$-0.546916\pi$
$449$	−9.30596e29	−0.293717	−0.146858	+	0.989158i	$0.546916\pi$
$450$	0	0
$451$	−6.78147e30	−2.03373
$452$	0	0
$453$	6.80881e30	1.94063
$454$	0	0
$455$	−4.91216e29	−0.133089
$456$	0	0
$457$	5.15005e30	1.32671	0.663354	−	0.748306i	$-0.269132\pi$
$457$	5.15005e30	1.32671	0.663354	+	0.748306i	$0.269132\pi$
$458$	0	0
$459$	2.11925e30	0.519203
$460$	0	0
$461$	−1.91029e30	−0.445183	−0.222592	−	0.974912i	$-0.571452\pi$
$461$	−1.91029e30	−0.445183	−0.222592	+	0.974912i	$0.571452\pi$
$462$	0	0
$463$	−2.39782e30	−0.531661	−0.265831	−	0.964020i	$-0.585646\pi$
$463$	−2.39782e30	−0.531661	−0.265831	+	0.964020i	$0.585646\pi$
$464$	0	0
$465$	−9.73125e30	−2.05334
$466$	0	0
$467$	8.20935e29	0.164879	0.0824394	−	0.996596i	$-0.473729\pi$
$467$	8.20935e29	0.164879	0.0824394	+	0.996596i	$0.473729\pi$
$468$	0	0
$469$	−3.43699e29	−0.0657189
$470$	0	0
$471$	−4.36400e30	−0.794592
$472$	0	0
$473$	5.47632e30	0.949699
$474$	0	0
$475$	−1.19044e31	−1.96666
$476$	0	0
$477$	8.17618e29	0.128703
$478$	0	0
$479$	9.01484e30	1.35238	0.676191	−	0.736726i	$-0.263629\pi$
$479$	9.01484e30	1.35238	0.676191	+	0.736726i	$0.263629\pi$
$480$	0	0
$481$	3.33642e30	0.477103
$482$	0	0
$483$	−4.94272e28	−0.00673867
$484$	0	0
$485$	−5.22207e30	−0.678911
$486$	0	0
$487$	1.18875e31	1.47403	0.737014	−	0.675877i	$-0.236235\pi$
$487$	1.18875e31	1.47403	0.737014	+	0.675877i	$0.236235\pi$
$488$	0	0
$489$	5.41967e30	0.641093
$490$	0	0
$491$	−4.78471e30	−0.540031	−0.270015	−	0.962856i	$-0.587029\pi$
$491$	−4.78471e30	−0.540031	−0.270015	+	0.962856i	$0.587029\pi$
$492$	0	0
$493$	9.64943e30	1.03935
$494$	0	0
$495$	1.77902e31	1.82903
$496$	0	0
$497$	9.79977e29	0.0961872
$498$	0	0
$499$	−1.40022e31	−1.31232	−0.656160	−	0.754622i	$-0.727820\pi$
$499$	−1.40022e31	−1.31232	−0.656160	+	0.754622i	$0.727820\pi$
$500$	0	0
$501$	−1.94350e31	−1.73961
$502$	0	0
$503$	7.41798e30	0.634240	0.317120	−	0.948385i	$-0.397284\pi$
$503$	7.41798e30	0.634240	0.317120	+	0.948385i	$0.397284\pi$
$504$	0	0
$505$	−8.69227e30	−0.710039
$506$	0	0
$507$	−1.21822e31	−0.950899
$508$	0	0
$509$	−8.56637e30	−0.639061	−0.319530	−	0.947576i	$-0.603525\pi$
$509$	−8.56637e30	−0.639061	−0.319530	+	0.947576i	$0.603525\pi$
$510$	0	0
$511$	−2.19276e30	−0.156369
$512$	0	0
$513$	5.55300e30	0.378597
$514$	0	0
$515$	3.32595e31	2.16837
$516$	0	0
$517$	−2.10034e31	−1.30963
$518$	0	0
$519$	−5.01679e30	−0.299227
$520$	0	0
$521$	1.85808e31	1.06030	0.530151	−	0.847903i	$-0.322135\pi$
$521$	1.85808e31	1.06030	0.530151	+	0.847903i	$0.322135\pi$
$522$	0	0
$523$	−2.92489e31	−1.59713	−0.798564	−	0.601910i	$-0.794407\pi$
$523$	−2.92489e31	−1.59713	−0.798564	+	0.601910i	$0.794407\pi$
$524$	0	0
$525$	9.02342e30	0.471563
$526$	0	0
$527$	−1.71438e31	−0.857601
$528$	0	0
$529$	−2.08501e31	−0.998543
$530$	0	0
$531$	2.67809e30	0.122811
$532$	0	0
$533$	1.42718e31	0.626782
$534$	0	0
$535$	6.65420e31	2.79916
$536$	0	0
$537$	8.61367e30	0.347123
$538$	0	0
$539$	4.09046e31	1.57943
$540$	0	0
$541$	3.50213e31	1.29588	0.647938	−	0.761693i	$-0.275632\pi$
$541$	3.50213e31	1.29588	0.647938	+	0.761693i	$0.275632\pi$
$542$	0	0
$543$	−4.47211e31	−1.58604
$544$	0	0
$545$	−5.21272e31	−1.77217
$546$	0	0
$547$	1.82542e30	0.0594990	0.0297495	−	0.999557i	$-0.490529\pi$
$547$	1.82542e30	0.0594990	0.0297495	+	0.999557i	$0.490529\pi$
$548$	0	0
$549$	−8.84410e30	−0.276421
$550$	0	0
$551$	2.52840e31	0.757884
$552$	0	0
$553$	−3.91827e30	−0.112656
$554$	0	0
$555$	−8.42367e31	−2.32344
$556$	0	0
$557$	−2.56099e31	−0.677756	−0.338878	−	0.940830i	$-0.610047\pi$
$557$	−2.56099e31	−0.677756	−0.338878	+	0.940830i	$0.610047\pi$
$558$	0	0
$559$	−1.15251e31	−0.292691
$560$	0	0
$561$	8.42302e31	2.05303
$562$	0	0
$563$	−6.17127e31	−1.44387	−0.721935	−	0.691961i	$-0.756747\pi$
$563$	−6.17127e31	−1.44387	−0.721935	+	0.691961i	$0.756747\pi$
$564$	0	0
$565$	4.03737e31	0.906862
$566$	0	0
$567$	−8.05340e30	−0.173690
$568$	0	0
$569$	−4.42387e31	−0.916243	−0.458122	−	0.888889i	$-0.651478\pi$
$569$	−4.42387e31	−0.916243	−0.458122	+	0.888889i	$0.651478\pi$
$570$	0	0
$571$	−4.85052e31	−0.964880	−0.482440	−	0.875929i	$-0.660249\pi$
$571$	−4.85052e31	−0.964880	−0.482440	+	0.875929i	$0.660249\pi$
$572$	0	0
$573$	8.68764e31	1.66006
$574$	0	0
$575$	−5.55243e30	−0.101930
$576$	0	0
$577$	4.95368e31	0.873789	0.436894	−	0.899513i	$-0.356078\pi$
$577$	4.95368e31	0.873789	0.436894	+	0.899513i	$0.356078\pi$
$578$	0	0
$579$	7.64507e31	1.29592
$580$	0	0
$581$	−7.92166e30	−0.129060
$582$	0	0
$583$	−2.23413e31	−0.349881
$584$	0	0
$585$	−3.74400e31	−0.563694
$586$	0	0
$587$	5.70339e31	0.825648	0.412824	−	0.910811i	$-0.364542\pi$
$587$	5.70339e31	0.825648	0.412824	+	0.910811i	$0.364542\pi$
$588$	0	0
$589$	−4.49212e31	−0.625353
$590$	0	0
$591$	9.35345e31	1.25232
$592$	0	0
$593$	−1.33571e32	−1.72021	−0.860107	−	0.510113i	$-0.829604\pi$
$593$	−1.33571e32	−1.72021	−0.860107	+	0.510113i	$0.829604\pi$
$594$	0	0
$595$	2.18490e31	0.270699
$596$	0	0
$597$	1.27431e32	1.51904
$598$	0	0
$599$	9.93052e31	1.13910	0.569552	−	0.821955i	$-0.307117\pi$
$599$	9.93052e31	1.13910	0.569552	+	0.821955i	$0.307117\pi$
$600$	0	0
$601$	1.44615e31	0.159645	0.0798227	−	0.996809i	$-0.474565\pi$
$601$	1.44615e31	0.159645	0.0798227	+	0.996809i	$0.474565\pi$
$602$	0	0
$603$	−2.61964e31	−0.278351
$604$	0	0
$605$	−2.98803e32	−3.05632
$606$	0	0
$607$	−7.96312e31	−0.784175	−0.392088	−	0.919928i	$-0.628247\pi$
$607$	−7.96312e31	−0.784175	−0.392088	+	0.919928i	$0.628247\pi$
$608$	0	0
$609$	−1.91651e31	−0.181724
$610$	0	0
$611$	4.42024e31	0.403619
$612$	0	0
$613$	−4.28683e31	−0.376999	−0.188500	−	0.982073i	$-0.560362\pi$
$613$	−4.28683e31	−0.376999	−0.188500	+	0.982073i	$0.560362\pi$
$614$	0	0
$615$	−3.60330e32	−3.05236
$616$	0	0
$617$	1.17493e32	0.958810	0.479405	−	0.877594i	$-0.340853\pi$
$617$	1.17493e32	0.958810	0.479405	+	0.877594i	$0.340853\pi$
$618$	0	0
$619$	1.82548e32	1.43527	0.717637	−	0.696417i	$-0.245224\pi$
$619$	1.82548e32	1.43527	0.717637	+	0.696417i	$0.245224\pi$
$620$	0	0
$621$	2.59003e30	0.0196223
$622$	0	0
$623$	−1.31822e31	−0.0962442
$624$	0	0
$625$	4.92005e32	3.46218
$626$	0	0
$627$	2.20705e32	1.49705
$628$	0	0
$629$	−1.48402e32	−0.970414
$630$	0	0
$631$	−8.49211e31	−0.535400	−0.267700	−	0.963502i	$-0.586264\pi$
$631$	−8.49211e31	−0.535400	−0.267700	+	0.963502i	$0.586264\pi$
$632$	0	0
$633$	−4.11232e32	−2.50003
$634$	0	0
$635$	3.46779e32	2.03308
$636$	0	0
$637$	−8.60851e31	−0.486771
$638$	0	0
$639$	7.46930e31	0.407399
$640$	0	0
$641$	4.48524e31	0.236004	0.118002	−	0.993013i	$-0.462351\pi$
$641$	4.48524e31	0.236004	0.118002	+	0.993013i	$0.462351\pi$
$642$	0	0
$643$	−2.69068e32	−1.36595	−0.682976	−	0.730441i	$-0.739314\pi$
$643$	−2.69068e32	−1.36595	−0.682976	+	0.730441i	$0.739314\pi$
$644$	0	0
$645$	2.90982e32	1.42537
$646$	0	0
$647$	8.10364e31	0.383072	0.191536	−	0.981486i	$-0.438653\pi$
$647$	8.10364e31	0.383072	0.191536	+	0.981486i	$0.438653\pi$
$648$	0	0
$649$	−7.31786e31	−0.333864
$650$	0	0
$651$	3.40500e31	0.149946
$652$	0	0
$653$	2.49897e32	1.06233	0.531166	−	0.847268i	$-0.321754\pi$
$653$	2.49897e32	1.06233	0.531166	+	0.847268i	$0.321754\pi$
$654$	0	0
$655$	−4.53113e32	−1.85965
$656$	0	0
$657$	−1.67130e32	−0.662297
$658$	0	0
$659$	−1.39914e32	−0.535400	−0.267700	−	0.963502i	$-0.586264\pi$
$659$	−1.39914e32	−0.535400	−0.267700	+	0.963502i	$0.586264\pi$
$660$	0	0
$661$	−2.05101e32	−0.757967	−0.378984	−	0.925403i	$-0.623726\pi$
$661$	−2.05101e32	−0.757967	−0.378984	+	0.925403i	$0.623726\pi$
$662$	0	0
$663$	−1.77265e32	−0.632729
$664$	0	0
$665$	5.72500e31	0.197391
$666$	0	0
$667$	1.17930e31	0.0392804
$668$	0	0
$669$	1.69386e32	0.545103
$670$	0	0
$671$	2.41664e32	0.751455
$672$	0	0
$673$	−5.12434e31	−0.153980	−0.0769900	−	0.997032i	$-0.524531\pi$
$673$	−5.12434e31	−0.153980	−0.0769900	+	0.997032i	$0.524531\pi$
$674$	0	0
$675$	−4.72835e32	−1.37314
$676$	0	0
$677$	7.07886e31	0.198698	0.0993490	−	0.995053i	$-0.468324\pi$
$677$	7.07886e31	0.198698	0.0993490	+	0.995053i	$0.468324\pi$
$678$	0	0
$679$	1.82722e31	0.0495779
$680$	0	0
$681$	6.84952e32	1.79667
$682$	0	0
$683$	−1.04620e31	−0.0265324	−0.0132662	−	0.999912i	$-0.504223\pi$
$683$	−1.04620e31	−0.0265324	−0.0132662	+	0.999912i	$0.504223\pi$
$684$	0	0
$685$	2.85705e32	0.700610
$686$	0	0
$687$	3.98377e32	0.944694
$688$	0	0
$689$	4.70181e31	0.107831
$690$	0	0
$691$	6.91309e32	1.53347	0.766734	−	0.641965i	$-0.221880\pi$
$691$	6.91309e32	1.53347	0.766734	+	0.641965i	$0.221880\pi$
$692$	0	0
$693$	−6.22484e31	−0.133566
$694$	0	0
$695$	−2.13191e32	−0.442531
$696$	0	0
$697$	−6.34803e32	−1.27486
$698$	0	0
$699$	−4.98154e32	−0.968002
$700$	0	0
$701$	4.87851e32	0.917339	0.458669	−	0.888607i	$-0.348326\pi$
$701$	4.87851e32	0.917339	0.458669	+	0.888607i	$0.348326\pi$
$702$	0	0
$703$	−3.88851e32	−0.707616
$704$	0	0
$705$	−1.11601e33	−1.96558
$706$	0	0
$707$	3.04145e31	0.0518511
$708$	0	0
$709$	2.32445e31	0.0383609	0.0191805	−	0.999816i	$-0.493894\pi$
$709$	2.32445e31	0.0383609	0.0191805	+	0.999816i	$0.493894\pi$
$710$	0	0
$711$	−2.98647e32	−0.477153
$712$	0	0
$713$	−2.09521e31	−0.0324115
$714$	0	0
$715$	1.02304e33	1.53241
$716$	0	0
$717$	−5.41678e31	−0.0785725
$718$	0	0
$719$	8.46511e32	1.18919	0.594594	−	0.804026i	$-0.297313\pi$
$719$	8.46511e32	1.18919	0.594594	+	0.804026i	$0.297313\pi$
$720$	0	0
$721$	−1.16376e32	−0.158346
$722$	0	0
$723$	3.30841e32	0.436042
$724$	0	0
$725$	−2.15292e33	−2.74879
$726$	0	0
$727$	−1.29417e33	−1.60083	−0.800415	−	0.599446i	$-0.795388\pi$
$727$	−1.29417e33	−1.60083	−0.800415	+	0.599446i	$0.795388\pi$
$728$	0	0
$729$	−7.24448e31	−0.0868242
$730$	0	0
$731$	5.12630e32	0.595325
$732$	0	0
$733$	5.02612e32	0.565636	0.282818	−	0.959174i	$-0.408731\pi$
$733$	5.02612e32	0.565636	0.282818	+	0.959174i	$0.408731\pi$
$734$	0	0
$735$	2.17344e33	2.37052
$736$	0	0
$737$	7.15815e32	0.756702
$738$	0	0
$739$	−6.75747e32	−0.692426	−0.346213	−	0.938156i	$-0.612533\pi$
$739$	−6.75747e32	−0.692426	−0.346213	+	0.938156i	$0.612533\pi$
$740$	0	0
$741$	−4.64480e32	−0.461379
$742$	0	0
$743$	1.22735e33	1.18194	0.590971	−	0.806693i	$-0.298745\pi$
$743$	1.22735e33	1.18194	0.590971	+	0.806693i	$0.298745\pi$
$744$	0	0
$745$	2.78201e33	2.59753
$746$	0	0
$747$	−6.03782e32	−0.546630
$748$	0	0
$749$	−2.32832e32	−0.204410
$750$	0	0
$751$	−8.58023e32	−0.730533	−0.365267	−	0.930903i	$-0.619022\pi$
$751$	−8.58023e32	−0.730533	−0.365267	+	0.930903i	$0.619022\pi$
$752$	0	0
$753$	9.76268e32	0.806171
$754$	0	0
$755$	−3.67838e33	−2.94623
$756$	0	0
$757$	3.90201e32	0.303170	0.151585	−	0.988444i	$-0.451562\pi$
$757$	3.90201e32	0.303170	0.151585	+	0.988444i	$0.451562\pi$
$758$	0	0
$759$	1.02941e32	0.0775904
$760$	0	0
$761$	−2.45216e33	−1.79319	−0.896593	−	0.442855i	$-0.853965\pi$
$761$	−2.45216e33	−1.79319	−0.896593	+	0.442855i	$0.853965\pi$
$762$	0	0
$763$	1.82395e32	0.129414
$764$	0	0
$765$	1.66531e33	1.14654
$766$	0	0
$767$	1.54007e32	0.102895
$768$	0	0
$769$	−9.56496e32	−0.620198	−0.310099	−	0.950704i	$-0.600362\pi$
$769$	−9.56496e32	−0.620198	−0.310099	+	0.950704i	$0.600362\pi$
$770$	0	0
$771$	−2.13881e33	−1.34601
$772$	0	0
$773$	−1.64837e33	−1.00691	−0.503454	−	0.864022i	$-0.667938\pi$
$773$	−1.64837e33	−1.00691	−0.503454	+	0.864022i	$0.667938\pi$
$774$	0	0
$775$	3.82502e33	2.26811
$776$	0	0
$777$	2.94747e32	0.169671
$778$	0	0
$779$	−1.66335e33	−0.929613
$780$	0	0
$781$	−2.04098e33	−1.10752
$782$	0	0
$783$	1.00427e33	0.529163
$784$	0	0
$785$	2.35760e33	1.20634
$786$	0	0
$787$	7.91155e32	0.393145	0.196572	−	0.980489i	$-0.437019\pi$
$787$	7.91155e32	0.393145	0.196572	+	0.980489i	$0.437019\pi$
$788$	0	0
$789$	4.26149e33	2.05672
$790$	0	0
$791$	−1.41269e32	−0.0662242
$792$	0	0
$793$	−5.08590e32	−0.231593
$794$	0	0
$795$	−1.18709e33	−0.525126
$796$	0	0
$797$	−4.89061e32	−0.210181	−0.105090	−	0.994463i	$-0.533513\pi$
$797$	−4.89061e32	−0.210181	−0.105090	+	0.994463i	$0.533513\pi$
$798$	0	0
$799$	−1.96610e33	−0.820950
$800$	0	0
$801$	−1.00473e33	−0.407640
$802$	0	0
$803$	4.56681e33	1.80046
$804$	0	0
$805$	2.67025e31	0.0102306
$806$	0	0
$807$	−1.31226e33	−0.488622
$808$	0	0
$809$	3.88048e33	1.40436	0.702179	−	0.712000i	$-0.252211\pi$
$809$	3.88048e33	1.40436	0.702179	+	0.712000i	$0.252211\pi$
$810$	0	0
$811$	−4.12171e33	−1.44990	−0.724949	−	0.688803i	$-0.758137\pi$
$811$	−4.12171e33	−1.44990	−0.724949	+	0.688803i	$0.758137\pi$
$812$	0	0
$813$	5.06164e33	1.73081
$814$	0	0
$815$	−2.92792e33	−0.973299
$816$	0	0
$817$	1.34322e33	0.434105
$818$	0	0
$819$	1.31004e32	0.0411641
$820$	0	0
$821$	−4.87333e32	−0.148895	−0.0744474	−	0.997225i	$-0.523719\pi$
$821$	−4.87333e32	−0.148895	−0.0744474	+	0.997225i	$0.523719\pi$
$822$	0	0
$823$	3.29803e33	0.979842	0.489921	−	0.871767i	$-0.337026\pi$
$823$	3.29803e33	0.979842	0.489921	+	0.871767i	$0.337026\pi$
$824$	0	0
$825$	−1.87929e34	−5.42967
$826$	0	0
$827$	−5.54431e32	−0.155788	−0.0778940	−	0.996962i	$-0.524820\pi$
$827$	−5.54431e32	−0.155788	−0.0778940	+	0.996962i	$0.524820\pi$
$828$	0	0
$829$	−5.38218e33	−1.47089	−0.735446	−	0.677583i	$-0.763027\pi$
$829$	−5.38218e33	−1.47089	−0.735446	+	0.677583i	$0.763027\pi$
$830$	0	0
$831$	−6.67047e32	−0.177315
$832$	0	0
$833$	3.82901e33	0.990078
$834$	0	0
$835$	1.04996e34	2.64105
$836$	0	0
$837$	−1.78425e33	−0.436628
$838$	0	0
$839$	−4.71028e32	−0.112146	−0.0560730	−	0.998427i	$-0.517858\pi$
$839$	−4.71028e32	−0.112146	−0.0560730	+	0.998427i	$0.517858\pi$
$840$	0	0
$841$	2.55929e32	0.0592878
$842$	0	0
$843$	3.28947e33	0.741495
$844$	0	0
$845$	6.58130e33	1.44364
$846$	0	0
$847$	1.04552e33	0.223190
$848$	0	0
$849$	5.74122e33	1.19280
$850$	0	0
$851$	−1.81368e32	−0.0366750
$852$	0	0
$853$	−7.73307e33	−1.52208	−0.761041	−	0.648704i	$-0.775311\pi$
$853$	−7.73307e33	−1.52208	−0.761041	+	0.648704i	$0.775311\pi$
$854$	0	0
$855$	4.36354e33	0.836043
$856$	0	0
$857$	−3.06846e33	−0.572322	−0.286161	−	0.958182i	$-0.592379\pi$
$857$	−3.06846e33	−0.572322	−0.286161	+	0.958182i	$0.592379\pi$
$858$	0	0
$859$	−6.27658e33	−1.13973	−0.569864	−	0.821739i	$-0.693004\pi$
$859$	−6.27658e33	−1.13973	−0.569864	+	0.821739i	$0.693004\pi$
$860$	0	0
$861$	1.26081e33	0.222901
$862$	0	0
$863$	2.01784e33	0.347347	0.173674	−	0.984803i	$-0.444436\pi$
$863$	2.01784e33	0.347347	0.173674	+	0.984803i	$0.444436\pi$
$864$	0	0
$865$	2.71027e33	0.454283
$866$	0	0
$867$	1.53004e32	0.0249738
$868$	0	0
$869$	8.16051e33	1.29715
$870$	0	0
$871$	−1.50646e33	−0.233210
$872$	0	0
$873$	1.39269e33	0.209986
$874$	0	0
$875$	−3.04955e33	−0.447861
$876$	0	0
$877$	−7.10079e33	−1.01581	−0.507903	−	0.861414i	$-0.669579\pi$
$877$	−7.10079e33	−1.01581	−0.507903	+	0.861414i	$0.669579\pi$
$878$	0	0
$879$	−1.17907e34	−1.64311
$880$	0	0
$881$	6.35471e33	0.862726	0.431363	−	0.902178i	$-0.358033\pi$
$881$	6.35471e33	0.862726	0.431363	+	0.902178i	$0.358033\pi$
$882$	0	0
$883$	−4.03451e33	−0.533633	−0.266817	−	0.963747i	$-0.585972\pi$
$883$	−4.03451e33	−0.533633	−0.266817	+	0.963747i	$0.585972\pi$
$884$	0	0
$885$	−3.88830e33	−0.501086
$886$	0	0
$887$	−6.96616e33	−0.874726	−0.437363	−	0.899285i	$-0.644087\pi$
$887$	−6.96616e33	−0.874726	−0.437363	+	0.899285i	$0.644087\pi$
$888$	0	0
$889$	−1.21339e33	−0.148467
$890$	0	0
$891$	1.67727e34	1.99990
$892$	0	0
$893$	−5.15168e33	−0.598628
$894$	0	0
$895$	−4.65344e33	−0.526998
$896$	0	0
$897$	−2.16643e32	−0.0239128
$898$	0	0
$899$	−8.12407e33	−0.874051
$900$	0	0
$901$	−2.09134e33	−0.219325
$902$	0	0
$903$	−1.01815e33	−0.104089
$904$	0	0
$905$	2.41601e34	2.40791
$906$	0	0
$907$	−2.14289e33	−0.208217	−0.104109	−	0.994566i	$-0.533199\pi$
$907$	−2.14289e33	−0.208217	−0.104109	+	0.994566i	$0.533199\pi$
$908$	0	0
$909$	2.31817e33	0.219614
$910$	0	0
$911$	−6.44743e33	−0.595559	−0.297780	−	0.954635i	$-0.596246\pi$
$911$	−6.44743e33	−0.595559	−0.297780	+	0.954635i	$0.596246\pi$
$912$	0	0
$913$	1.64983e34	1.48602
$914$	0	0
$915$	1.28407e34	1.12784
$916$	0	0
$917$	1.58546e33	0.135802
$918$	0	0
$919$	1.53028e34	1.27833	0.639164	−	0.769071i	$-0.279281\pi$
$919$	1.53028e34	1.27833	0.639164	+	0.769071i	$0.279281\pi$
$920$	0	0
$921$	−2.34671e34	−1.91194
$922$	0	0
$923$	4.29531e33	0.341330
$924$	0	0
$925$	3.31105e34	2.56647
$926$	0	0
$927$	−8.87008e33	−0.670673
$928$	0	0
$929$	−2.50103e34	−1.84475	−0.922376	−	0.386294i	$-0.873755\pi$
$929$	−2.50103e34	−1.84475	−0.922376	+	0.386294i	$0.873755\pi$
$930$	0	0
$931$	1.00330e34	0.721954
$932$	0	0
$933$	−3.43662e34	−2.41264
$934$	0	0
$935$	−4.55044e34	−3.11688
$936$	0	0
$937$	−8.75099e33	−0.584860	−0.292430	−	0.956287i	$-0.594464\pi$
$937$	−8.75099e33	−0.584860	−0.292430	+	0.956287i	$0.594464\pi$
$938$	0	0
$939$	2.86831e34	1.87056
$940$	0	0
$941$	1.18186e34	0.752115	0.376057	−	0.926596i	$-0.377280\pi$
$941$	1.18186e34	0.752115	0.376057	+	0.926596i	$0.377280\pi$
$942$	0	0
$943$	−7.75818e32	−0.0481809
$944$	0	0
$945$	2.27394e33	0.137820
$946$	0	0
$947$	2.79192e34	1.65150	0.825750	−	0.564037i	$-0.190752\pi$
$947$	2.79192e34	1.65150	0.825750	+	0.564037i	$0.190752\pi$
$948$	0	0
$949$	−9.61100e33	−0.554891
$950$	0	0
$951$	5.81396e33	0.327639
$952$	0	0
$953$	−8.61770e33	−0.474049	−0.237024	−	0.971504i	$-0.576172\pi$
$953$	−8.61770e33	−0.474049	−0.237024	+	0.971504i	$0.576172\pi$
$954$	0	0
$955$	−4.69340e34	−2.52028
$956$	0	0
$957$	3.99148e34	2.09241
$958$	0	0
$959$	−9.99691e32	−0.0511625
$960$	0	0
$961$	−5.57957e33	−0.278793
$962$	0	0
$963$	−1.77463e34	−0.865775
$964$	0	0
$965$	−4.13017e34	−1.96745
$966$	0	0
$967$	2.60088e34	1.20981	0.604903	−	0.796299i	$-0.293212\pi$
$967$	2.60088e34	1.20981	0.604903	+	0.796299i	$0.293212\pi$
$968$	0	0
$969$	2.06598e34	0.938432
$970$	0	0
$971$	8.04930e33	0.357057	0.178528	−	0.983935i	$-0.442866\pi$
$971$	8.04930e33	0.357057	0.178528	+	0.983935i	$0.442866\pi$
$972$	0	0
$973$	7.45961e32	0.0323161
$974$	0	0
$975$	3.95503e34	1.67339
$976$	0	0
$977$	2.25035e34	0.929958	0.464979	−	0.885322i	$-0.346062\pi$
$977$	2.25035e34	0.929958	0.464979	+	0.885322i	$0.346062\pi$
$978$	0	0
$979$	2.74543e34	1.10818
$980$	0	0
$981$	1.39020e34	0.548129
$982$	0	0
$983$	5.27108e33	0.203018	0.101509	−	0.994835i	$-0.467633\pi$
$983$	5.27108e33	0.203018	0.101509	+	0.994835i	$0.467633\pi$
$984$	0	0
$985$	−5.05310e34	−1.90126
$986$	0	0
$987$	3.90494e33	0.143538
$988$	0	0
$989$	6.26505e32	0.0224992
$990$	0	0
$991$	5.12466e34	1.79812	0.899058	−	0.437829i	$-0.144252\pi$
$991$	5.12466e34	1.79812	0.899058	+	0.437829i	$0.144252\pi$
$992$	0	0
$993$	−2.99778e34	−1.02774
$994$	0	0
$995$	−6.88432e34	−2.30619
$996$	0	0
$997$	−1.33679e34	−0.437591	−0.218796	−	0.975771i	$-0.570213\pi$
$997$	−1.33679e34	−0.437591	−0.218796	+	0.975771i	$0.570213\pi$
$998$	0	0
$999$	−1.54450e34	−0.494065

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.24.a.a.1.2	✓	2
3.2	odd	2		36.24.a.e.1.2		2
4.3	odd	2		16.24.a.c.1.1		2
8.3	odd	2		64.24.a.f.1.2		2
8.5	even	2		64.24.a.e.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.24.a.a.1.2	✓	2	1.1	even	1	trivial
16.24.a.c.1.1		2	4.3	odd	2
36.24.a.e.1.2		2	3.2	odd	2
64.24.a.e.1.1		2	8.5	even	2
64.24.a.f.1.2		2	8.3	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:	\( N \)	\(=\)	\( 4 = 2^{2} \)
Weight:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	4.a (trivial)

\(f(q)\)	\(=\)	\(q+387210. q^{3} -2.09186e8 q^{5} +7.31949e8 q^{7} +5.57885e10 q^{9} +O(q^{10})\)	\(q+387210. q^{3} -2.09186e8 q^{5} +7.31949e8 q^{7} +5.57885e10 q^{9} -1.52441e12 q^{11} +3.20818e12 q^{13} -8.09989e13 q^{15} -1.42698e14 q^{17} -3.73906e14 q^{19} +2.83418e14 q^{21} -1.74397e14 q^{23} +3.18379e16 q^{25} -1.48513e16 q^{27} -6.76214e16 q^{29} +1.20141e17 q^{31} -5.90269e17 q^{33} -1.53113e17 q^{35} +1.03997e18 q^{37} +1.24224e18 q^{39} +4.44857e18 q^{41} -3.59241e18 q^{43} -1.16702e19 q^{45} +1.37780e19 q^{47} -2.68330e19 q^{49} -5.52541e19 q^{51} +1.46557e19 q^{53} +3.18886e20 q^{55} -1.44780e20 q^{57} +4.80044e19 q^{59} -1.58529e20 q^{61} +4.08343e19 q^{63} -6.71107e20 q^{65} -4.69567e20 q^{67} -6.75283e19 q^{69} +1.33886e21 q^{71} -2.99578e21 q^{73} +1.23279e22 q^{75} -1.11579e21 q^{77} -5.35321e21 q^{79} -1.10027e22 q^{81} -1.08227e22 q^{83} +2.98504e22 q^{85} -2.61837e22 q^{87} -1.80097e22 q^{89} +2.34822e21 q^{91} +4.65196e22 q^{93} +7.82158e22 q^{95} +2.49638e22 q^{97} -8.50448e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 170520 q^{3} - 92266020 q^{5} + 192083440 q^{7} + 8599879818 q^{9}+O(q^{10}) \) 2 * q + 170520 * q^3 - 92266020 * q^5 + 192083440 * q^7 + 8599879818 * q^9	\( 2 q + 170520 q^{3} - 92266020 q^{5} + 192083440 q^{7} + 8599879818 q^{9} - 1247174695800 q^{11} - 7460299980980 q^{13} - 106334360092080 q^{15} - 374897347903260 q^{17} - 840360279212552 q^{19} + 400401295079232 q^{21} + 64\!\cdots\!20 q^{23}+ \cdots - 98\!\cdots\!00 q^{99}+O(q^{100}) \) 2 * q + 170520 * q^3 - 92266020 * q^5 + 192083440 * q^7 + 8599879818 * q^9 - 1247174695800 * q^11 - 7460299980980 * q^13 - 106334360092080 * q^15 - 374897347903260 * q^17 - 840360279212552 * q^19 + 400401295079232 * q^21 + 6433357923072720 * q^23 + 33587241295679150 * q^25 + 15773861091124080 * q^27 - 68055499247434452 * q^29 - 145584514546845248 * q^31 - 650343867352613280 * q^33 - 216234488190163680 * q^35 + 1211894143551157660 * q^37 + 3553993994899379088 * q^39 + 5036778367134688692 * q^41 + 935180945919935560 * q^43 - 17187460645010169780 * q^45 - 8431168896277596000 * q^47 - 53910291562566659694 * q^49 - 4938798717170442576 * q^51 + 3763137197370204540 * q^53 + 351301189218989209200 * q^55 - 43703893486254219360 * q^57 + 348561606512901387816 * q^59 - 582966592720711355156 * q^61 + 66309764232647559600 * q^63 - 1918465858864426149720 * q^65 + 211615268845414654360 * q^67 - 1499363181860662472256 * q^69 + 3840047757133544010096 * q^71 - 680666251913594096780 * q^73 + 11948876435052098946600 * q^75 - 1265465342298122095680 * q^77 - 642991395306723640736 * q^79 - 13196380320218478599982 * q^81 - 18799502632508717771400 * q^83 + 2701670550661359899640 * q^85 - 26089604267151850210800 * q^87 + 5001398778883250139732 * q^89 + 8107764572821621082528 * q^91 + 104099589081258166744320 * q^93 + 23677928248035152895120 * q^95 + 122022794260559820614980 * q^97 - 98127319293495379027800 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more