LMFDB - Embedded newform 24.22.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [24,22,Mod(1,24)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(24, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("24.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 24 = 2^{3} \cdot 3 $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	24.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:	$67.0745626289$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 53560x - 70812 $ x^3 - 53560*x - 70812
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{4}\cdot 5\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$232.089$ of defining polynomial
Character	$\chi$	$=$	24.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-59049.0 q^{3} -2.28989e7 q^{5} +1.04414e9 q^{7} +3.48678e9 q^{9} +O(q^{10})$	$q-59049.0 q^{3} -2.28989e7 q^{5} +1.04414e9 q^{7} +3.48678e9 q^{9} +1.18125e11 q^{11} -6.38574e11 q^{13} +1.35216e12 q^{15} -8.74375e12 q^{17} -4.45645e13 q^{19} -6.16552e13 q^{21} +3.74324e14 q^{23} +4.75214e13 q^{25} -2.05891e14 q^{27} -4.06101e15 q^{29} +4.54408e15 q^{31} -6.97517e15 q^{33} -2.39095e16 q^{35} +3.39306e16 q^{37} +3.77072e16 q^{39} +3.32676e16 q^{41} +8.77348e16 q^{43} -7.98434e16 q^{45} -2.38472e17 q^{47} +5.31674e17 q^{49} +5.16310e17 q^{51} -3.19252e17 q^{53} -2.70493e18 q^{55} +2.63149e18 q^{57} -2.47374e18 q^{59} +3.20039e18 q^{61} +3.64068e18 q^{63} +1.46226e19 q^{65} +2.70310e19 q^{67} -2.21035e19 q^{69} +1.24185e19 q^{71} -6.78649e18 q^{73} -2.80609e18 q^{75} +1.23339e20 q^{77} +1.12229e20 q^{79} +1.21577e19 q^{81} +5.23835e19 q^{83} +2.00222e20 q^{85} +2.39798e20 q^{87} +1.16164e20 q^{89} -6.66758e20 q^{91} -2.68324e20 q^{93} +1.02048e21 q^{95} +1.21383e21 q^{97} +4.11877e20 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 177147 q^{3} - 4833126 q^{5} + 271431024 q^{7} + 10460353203 q^{9}+O(q^{10}) $ 3 * q - 177147 * q^3 - 4833126 * q^5 + 271431024 * q^7 + 10460353203 * q^9	$ 3 q - 177147 q^{3} - 4833126 q^{5} + 271431024 q^{7} + 10460353203 q^{9} + 61194658188 q^{11} - 594486202422 q^{13} + 285391257174 q^{15} + 1424819519334 q^{17} - 35094825681804 q^{19} - 16027730536176 q^{21} - 94911043073592 q^{23} - 613780025295699 q^{25} - 617673396283947 q^{27} - 32\!\cdots\!34 q^{29}+ \cdots + 21\!\cdots\!88 q^{99}+O(q^{100}) $ 3 * q - 177147 * q^3 - 4833126 * q^5 + 271431024 * q^7 + 10460353203 * q^9 + 61194658188 * q^11 - 594486202422 * q^13 + 285391257174 * q^15 + 1424819519334 * q^17 - 35094825681804 * q^19 - 16027730536176 * q^21 - 94911043073592 * q^23 - 613780025295699 * q^25 - 617673396283947 * q^27 - 3292734981070734 * q^29 - 1928658139449144 * q^31 - 3613483371343212 * q^33 - 18129892754606304 * q^35 + 22185275061682722 * q^37 + 35103815766816678 * q^39 + 167394191883971118 * q^41 + 69547615491362508 * q^43 - 16852068344867526 * q^45 - 49110618973894752 * q^47 + 973320764343712395 * q^49 - 84134167797153366 * q^51 + 488391341947078074 * q^53 - 3110835549535695000 * q^55 + 2072314361684844396 * q^57 - 1846144806145488852 * q^59 + 8183707455377473002 * q^61 + 946421460430656624 * q^63 + 28951429805381293356 * q^65 + 37438935583939030164 * q^67 + 5604402182452534008 * q^69 + 65865243803437791096 * q^71 + 88415465494171854846 * q^73 + 36243096713685730251 * q^75 + 156034186748470726080 * q^77 + 221381156829825400920 * q^79 + 36472996377170786403 * q^81 + 305200914127129955700 * q^83 + 184695330643817576052 * q^85 + 194432707897245771966 * q^87 + 327556458098783035470 * q^89 + 692083854852141366816 * q^91 + 113885334476332504056 * q^93 + 1486593794603974227096 * q^95 + 1980043166774365090758 * q^97 + 213372579594445325388 * 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
$3$	−59049.0	−0.577350
$3$	−59049.0	−0.577350
$4$	0	0
$5$	−2.28989e7	−1.04865	−0.524323	−	0.851519i	$-0.675682\pi$
$5$	−2.28989e7	−1.04865	−0.524323	+	0.851519i	$0.675682\pi$
$6$	0	0
$7$	1.04414e9	1.39710	0.698550	−	0.715561i	$-0.253829\pi$
$7$	1.04414e9	1.39710	0.698550	+	0.715561i	$0.253829\pi$
$8$	0	0
$9$	3.48678e9	0.333333
$10$	0	0
$11$	1.18125e11	1.37315	0.686576	−	0.727058i	$-0.259113\pi$
$11$	1.18125e11	1.37315	0.686576	+	0.727058i	$0.259113\pi$
$12$	0	0
$13$	−6.38574e11	−1.28471	−0.642356	−	0.766406i	$-0.722043\pi$
$13$	−6.38574e11	−1.28471	−0.642356	+	0.766406i	$0.722043\pi$
$14$	0	0
$15$	1.35216e12	0.605436
$16$	0	0
$17$	−8.74375e12	−1.05192	−0.525962	−	0.850508i	$-0.676295\pi$
$17$	−8.74375e12	−1.05192	−0.525962	+	0.850508i	$0.676295\pi$
$18$	0	0
$19$	−4.45645e13	−1.66754	−0.833769	−	0.552113i	$-0.813822\pi$
$19$	−4.45645e13	−1.66754	−0.833769	+	0.552113i	$0.813822\pi$
$20$	0	0
$21$	−6.16552e13	−0.806616
$22$	0	0
$23$	3.74324e14	1.88411	0.942054	−	0.335463i	$-0.108893\pi$
$23$	3.74324e14	1.88411	0.942054	+	0.335463i	$0.108893\pi$
$24$	0	0
$25$	4.75214e13	0.0996596
$26$	0	0
$27$	−2.05891e14	−0.192450
$28$	0	0
$29$	−4.06101e15	−1.79248	−0.896241	−	0.443567i	$-0.853713\pi$
$29$	−4.06101e15	−1.79248	−0.896241	+	0.443567i	$0.853713\pi$
$30$	0	0
$31$	4.54408e15	0.995746	0.497873	−	0.867250i	$-0.334115\pi$
$31$	4.54408e15	0.995746	0.497873	+	0.867250i	$0.334115\pi$
$32$	0	0
$33$	−6.97517e15	−0.792790
$34$	0	0
$35$	−2.39095e16	−1.46506
$36$	0	0
$37$	3.39306e16	1.16004	0.580020	−	0.814602i	$-0.303045\pi$
$37$	3.39306e16	1.16004	0.580020	+	0.814602i	$0.303045\pi$
$38$	0	0
$39$	3.77072e16	0.741729
$40$	0	0
$41$	3.32676e16	0.387071	0.193535	−	0.981093i	$-0.438005\pi$
$41$	3.32676e16	0.387071	0.193535	+	0.981093i	$0.438005\pi$
$42$	0	0
$43$	8.77348e16	0.619089	0.309545	−	0.950885i	$-0.399823\pi$
$43$	8.77348e16	0.619089	0.309545	+	0.950885i	$0.399823\pi$
$44$	0	0
$45$	−7.98434e16	−0.349549
$46$	0	0
$47$	−2.38472e17	−0.661318	−0.330659	−	0.943750i	$-0.607271\pi$
$47$	−2.38472e17	−0.661318	−0.330659	+	0.943750i	$0.607271\pi$
$48$	0	0
$49$	5.31674e17	0.951889
$50$	0	0
$51$	5.16310e17	0.607328
$52$	0	0
$53$	−3.19252e17	−0.250747	−0.125374	−	0.992110i	$-0.540013\pi$
$53$	−3.19252e17	−0.250747	−0.125374	+	0.992110i	$0.540013\pi$
$54$	0	0
$55$	−2.70493e18	−1.43995
$56$	0	0
$57$	2.63149e18	0.962754
$58$	0	0
$59$	−2.47374e18	−0.630098	−0.315049	−	0.949075i	$-0.602021\pi$
$59$	−2.47374e18	−0.630098	−0.315049	+	0.949075i	$0.602021\pi$
$60$	0	0
$61$	3.20039e18	0.574434	0.287217	−	0.957866i	$-0.407270\pi$
$61$	3.20039e18	0.574434	0.287217	+	0.957866i	$0.407270\pi$
$62$	0	0
$63$	3.64068e18	0.465700
$64$	0	0
$65$	1.46226e19	1.34721
$66$	0	0
$67$	2.70310e19	1.81166	0.905830	−	0.423641i	$-0.139248\pi$
$67$	2.70310e19	1.81166	0.905830	+	0.423641i	$0.139248\pi$
$68$	0	0
$69$	−2.21035e19	−1.08779
$70$	0	0
$71$	1.24185e19	0.452748	0.226374	−	0.974040i	$-0.427313\pi$
$71$	1.24185e19	0.452748	0.226374	+	0.974040i	$0.427313\pi$
$72$	0	0
$73$	−6.78649e18	−0.184823	−0.0924113	−	0.995721i	$-0.529457\pi$
$73$	−6.78649e18	−0.184823	−0.0924113	+	0.995721i	$0.529457\pi$
$74$	0	0
$75$	−2.80609e18	−0.0575385
$76$	0	0
$77$	1.23339e20	1.91843
$78$	0	0
$79$	1.12229e20	1.33359	0.666795	−	0.745241i	$-0.267666\pi$
$79$	1.12229e20	1.33359	0.666795	+	0.745241i	$0.267666\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	0	0
$83$	5.23835e19	0.370573	0.185287	−	0.982684i	$-0.440679\pi$
$83$	5.23835e19	0.370573	0.185287	+	0.982684i	$0.440679\pi$
$84$	0	0
$85$	2.00222e20	1.10310
$86$	0	0
$87$	2.39798e20	1.03489
$88$	0	0
$89$	1.16164e20	0.394891	0.197446	−	0.980314i	$-0.436735\pi$
$89$	1.16164e20	0.394891	0.197446	+	0.980314i	$0.436735\pi$
$90$	0	0
$91$	−6.66758e20	−1.79487
$92$	0	0
$93$	−2.68324e20	−0.574894
$94$	0	0
$95$	1.02048e21	1.74866
$96$	0	0
$97$	1.21383e21	1.67129	0.835647	−	0.549266i	$-0.185093\pi$
$97$	1.21383e21	1.67129	0.835647	+	0.549266i	$0.185093\pi$
$98$	0	0
$99$	4.11877e20	0.457718
$100$	0	0
$101$	−1.09039e21	−0.982220	−0.491110	−	0.871098i	$-0.663409\pi$
$101$	−1.09039e21	−0.982220	−0.491110	+	0.871098i	$0.663409\pi$
$102$	0	0
$103$	1.90244e21	1.39483	0.697414	−	0.716669i	$-0.254334\pi$
$103$	1.90244e21	1.39483	0.697414	+	0.716669i	$0.254334\pi$
$104$	0	0
$105$	1.41183e21	0.845855
$106$	0	0
$107$	−1.17732e20	−0.0578581	−0.0289290	−	0.999581i	$-0.509210\pi$
$107$	−1.17732e20	−0.0578581	−0.0289290	+	0.999581i	$0.509210\pi$
$108$	0	0
$109$	1.10264e21	0.446124	0.223062	−	0.974804i	$-0.428395\pi$
$109$	1.10264e21	0.446124	0.223062	+	0.974804i	$0.428395\pi$
$110$	0	0
$111$	−2.00357e21	−0.669749
$112$	0	0
$113$	−2.47288e21	−0.685298	−0.342649	−	0.939464i	$-0.611324\pi$
$113$	−2.47288e21	−0.685298	−0.342649	+	0.939464i	$0.611324\pi$
$114$	0	0
$115$	−8.57160e21	−1.97576
$116$	0	0
$117$	−2.22657e21	−0.428238
$118$	0	0
$119$	−9.12967e21	−1.46964
$120$	0	0
$121$	6.55328e21	0.885548
$122$	0	0
$123$	−1.96442e21	−0.223475
$124$	0	0
$125$	9.83085e21	0.944139
$126$	0	0
$127$	−1.43248e22	−1.16452	−0.582262	−	0.813001i	$-0.697832\pi$
$127$	−1.43248e22	−1.16452	−0.582262	+	0.813001i	$0.697832\pi$
$128$	0	0
$129$	−5.18065e21	−0.357431
$130$	0	0
$131$	1.20309e22	0.706236	0.353118	−	0.935579i	$-0.385121\pi$
$131$	1.20309e22	0.706236	0.353118	+	0.935579i	$0.385121\pi$
$132$	0	0
$133$	−4.65314e22	−2.32972
$134$	0	0
$135$	4.71468e21	0.201812
$136$	0	0
$137$	−3.72090e21	−0.136484	−0.0682420	−	0.997669i	$-0.521739\pi$
$137$	−3.72090e21	−0.136484	−0.0682420	+	0.997669i	$0.521739\pi$
$138$	0	0
$139$	−3.58816e21	−0.113036	−0.0565180	−	0.998402i	$-0.518000\pi$
$139$	−3.58816e21	−0.113036	−0.0565180	+	0.998402i	$0.518000\pi$
$140$	0	0
$141$	1.40816e22	0.381812
$142$	0	0
$143$	−7.54316e22	−1.76411
$144$	0	0
$145$	9.29925e22	1.87968
$146$	0	0
$147$	−3.13948e22	−0.549574
$148$	0	0
$149$	−2.91546e22	−0.442844	−0.221422	−	0.975178i	$-0.571070\pi$
$149$	−2.91546e22	−0.442844	−0.221422	+	0.975178i	$0.571070\pi$
$150$	0	0
$151$	5.30234e22	0.700180	0.350090	−	0.936716i	$-0.386151\pi$
$151$	5.30234e22	0.700180	0.350090	+	0.936716i	$0.386151\pi$
$152$	0	0
$153$	−3.04876e22	−0.350641
$154$	0	0
$155$	−1.04054e23	−1.04419
$156$	0	0
$157$	2.02430e22	0.177554	0.0887768	−	0.996052i	$-0.471704\pi$
$157$	2.02430e22	0.177554	0.0887768	+	0.996052i	$0.471704\pi$
$158$	0	0
$159$	1.88515e22	0.144769
$160$	0	0
$161$	3.90845e23	2.63229
$162$	0	0
$163$	5.48310e22	0.324382	0.162191	−	0.986759i	$-0.448144\pi$
$163$	5.48310e22	0.324382	0.162191	+	0.986759i	$0.448144\pi$
$164$	0	0
$165$	1.59723e23	0.831356
$166$	0	0
$167$	−2.05923e23	−0.944455	−0.472228	−	0.881477i	$-0.656550\pi$
$167$	−2.05923e23	−0.944455	−0.472228	+	0.881477i	$0.656550\pi$
$168$	0	0
$169$	1.60712e23	0.650487
$170$	0	0
$171$	−1.55387e23	−0.555846
$172$	0	0
$173$	3.39602e23	1.07519	0.537596	−	0.843203i	$-0.319333\pi$
$173$	3.39602e23	1.07519	0.537596	+	0.843203i	$0.319333\pi$
$174$	0	0
$175$	4.96188e22	0.139234
$176$	0	0
$177$	1.46072e23	0.363787
$178$	0	0
$179$	4.35144e23	0.963111	0.481556	−	0.876416i	$-0.340072\pi$
$179$	4.35144e23	0.963111	0.481556	+	0.876416i	$0.340072\pi$
$180$	0	0
$181$	8.73828e23	1.72108	0.860540	−	0.509384i	$-0.170127\pi$
$181$	8.73828e23	1.72108	0.860540	+	0.509384i	$0.170127\pi$
$182$	0	0
$183$	−1.88980e23	−0.331649
$184$	0	0
$185$	−7.76972e23	−1.21647
$186$	0	0
$187$	−1.03286e24	−1.44445
$188$	0	0
$189$	−2.14978e23	−0.268872
$190$	0	0
$191$	7.69847e23	0.862093	0.431046	−	0.902330i	$-0.358145\pi$
$191$	7.69847e23	0.862093	0.431046	+	0.902330i	$0.358145\pi$
$192$	0	0
$193$	1.43229e24	1.43774	0.718870	−	0.695145i	$-0.244660\pi$
$193$	1.43229e24	1.43774	0.718870	+	0.695145i	$0.244660\pi$
$194$	0	0
$195$	−8.63451e23	−0.777812
$196$	0	0
$197$	6.99558e23	0.566146	0.283073	−	0.959098i	$-0.408646\pi$
$197$	6.99558e23	0.566146	0.283073	+	0.959098i	$0.408646\pi$
$198$	0	0
$199$	1.42670e23	0.103842	0.0519211	−	0.998651i	$-0.483466\pi$
$199$	1.42670e23	0.103842	0.0519211	+	0.998651i	$0.483466\pi$
$200$	0	0
$201$	−1.59615e24	−1.04596
$202$	0	0
$203$	−4.24024e24	−2.50428
$204$	0	0
$205$	−7.61790e23	−0.405901
$206$	0	0
$207$	1.30519e24	0.628036
$208$	0	0
$209$	−5.26418e24	−2.28979
$210$	0	0
$211$	1.43060e24	0.563058	0.281529	−	0.959553i	$-0.409159\pi$
$211$	1.43060e24	0.563058	0.281529	+	0.959553i	$0.409159\pi$
$212$	0	0
$213$	−7.33300e23	−0.261394
$214$	0	0
$215$	−2.00903e24	−0.649206
$216$	0	0
$217$	4.74464e24	1.39116
$218$	0	0
$219$	4.00735e23	0.106707
$220$	0	0
$221$	5.58353e24	1.35142
$222$	0	0
$223$	1.94827e24	0.428990	0.214495	−	0.976725i	$-0.431189\pi$
$223$	1.94827e24	0.428990	0.214495	+	0.976725i	$0.431189\pi$
$224$	0	0
$225$	1.65697e23	0.0332199
$226$	0	0
$227$	−4.61102e24	−0.842414	−0.421207	−	0.906965i	$-0.638393\pi$
$227$	−4.61102e24	−0.842414	−0.421207	+	0.906965i	$0.638393\pi$
$228$	0	0
$229$	−7.86792e24	−1.31095	−0.655477	−	0.755215i	$-0.727533\pi$
$229$	−7.86792e24	−1.31095	−0.655477	+	0.755215i	$0.727533\pi$
$230$	0	0
$231$	−7.28302e24	−1.10761
$232$	0	0
$233$	−6.34306e24	−0.881173	−0.440587	−	0.897710i	$-0.645230\pi$
$233$	−6.34306e24	−0.881173	−0.440587	+	0.897710i	$0.645230\pi$
$234$	0	0
$235$	5.46075e24	0.693489
$236$	0	0
$237$	−6.62704e24	−0.769948
$238$	0	0
$239$	9.95341e24	1.05875	0.529376	−	0.848388i	$-0.322426\pi$
$239$	9.95341e24	1.05875	0.529376	+	0.848388i	$0.322426\pi$
$240$	0	0
$241$	1.85126e24	0.180421	0.0902107	−	0.995923i	$-0.471246\pi$
$241$	1.85126e24	0.180421	0.0902107	+	0.995923i	$0.471246\pi$
$242$	0	0
$243$	−7.17898e23	−0.0641500
$244$	0	0
$245$	−1.21747e25	−0.998195
$246$	0	0
$247$	2.84577e25	2.14231
$248$	0	0
$249$	−3.09319e24	−0.213951
$250$	0	0
$251$	1.30039e25	0.826992	0.413496	−	0.910506i	$-0.364308\pi$
$251$	1.30039e25	0.826992	0.413496	+	0.910506i	$0.364308\pi$
$252$	0	0
$253$	4.42171e25	2.58717
$254$	0	0
$255$	−1.18229e25	−0.636873
$256$	0	0
$257$	1.65071e24	0.0819168	0.0409584	−	0.999161i	$-0.486959\pi$
$257$	1.65071e24	0.0819168	0.0409584	+	0.999161i	$0.486959\pi$
$258$	0	0
$259$	3.54281e25	1.62069
$260$	0	0
$261$	−1.41599e25	−0.597494
$262$	0	0
$263$	−8.84343e23	−0.0344418	−0.0172209	−	0.999852i	$-0.505482\pi$
$263$	−8.84343e23	−0.0344418	−0.0172209	+	0.999852i	$0.505482\pi$
$264$	0	0
$265$	7.31050e24	0.262945
$266$	0	0
$267$	−6.85938e24	−0.227991
$268$	0	0
$269$	−2.31988e25	−0.712964	−0.356482	−	0.934302i	$-0.616024\pi$
$269$	−2.31988e25	−0.712964	−0.356482	+	0.934302i	$0.616024\pi$
$270$	0	0
$271$	1.37286e25	0.390345	0.195172	−	0.980769i	$-0.437473\pi$
$271$	1.37286e25	0.390345	0.195172	+	0.980769i	$0.437473\pi$
$272$	0	0
$273$	3.93714e25	1.03627
$274$	0	0
$275$	5.61347e24	0.136848
$276$	0	0
$277$	−5.66666e25	−1.28023	−0.640117	−	0.768277i	$-0.721114\pi$
$277$	−5.66666e25	−1.28023	−0.640117	+	0.768277i	$0.721114\pi$
$278$	0	0
$279$	1.58442e25	0.331915
$280$	0	0
$281$	−5.69168e25	−1.10617	−0.553087	−	0.833123i	$-0.686550\pi$
$281$	−5.69168e25	−1.10617	−0.553087	+	0.833123i	$0.686550\pi$
$282$	0	0
$283$	7.44994e25	1.34399	0.671994	−	0.740557i	$-0.265438\pi$
$283$	7.44994e25	1.34399	0.671994	+	0.740557i	$0.265438\pi$
$284$	0	0
$285$	−6.02581e25	−1.00959
$286$	0	0
$287$	3.47359e25	0.540777
$288$	0	0
$289$	7.36129e24	0.106543
$290$	0	0
$291$	−7.16752e25	−0.964922
$292$	0	0
$293$	−3.86412e25	−0.484106	−0.242053	−	0.970263i	$-0.577821\pi$
$293$	−3.86412e25	−0.484106	−0.242053	+	0.970263i	$0.577821\pi$
$294$	0	0
$295$	5.66459e25	0.660750
$296$	0	0
$297$	−2.43209e25	−0.264263
$298$	0	0
$299$	−2.39034e26	−2.42054
$300$	0	0
$301$	9.16071e25	0.864930
$302$	0	0
$303$	6.43866e25	0.567085
$304$	0	0
$305$	−7.32854e25	−0.602378
$306$	0	0
$307$	−2.14147e26	−1.64346	−0.821731	−	0.569875i	$-0.806992\pi$
$307$	−2.14147e26	−1.64346	−0.821731	+	0.569875i	$0.806992\pi$
$308$	0	0
$309$	−1.12337e26	−0.805304
$310$	0	0
$311$	−1.79488e25	−0.120240	−0.0601202	−	0.998191i	$-0.519148\pi$
$311$	−1.79488e25	−0.120240	−0.0601202	+	0.998191i	$0.519148\pi$
$312$	0	0
$313$	2.54702e26	1.59520	0.797602	−	0.603185i	$-0.206102\pi$
$313$	2.54702e26	1.59520	0.797602	+	0.603185i	$0.206102\pi$
$314$	0	0
$315$	−8.33674e25	−0.488355
$316$	0	0
$317$	2.88166e26	1.57950	0.789752	−	0.613427i	$-0.210209\pi$
$317$	2.88166e26	1.57950	0.789752	+	0.613427i	$0.210209\pi$
$318$	0	0
$319$	−4.79707e26	−2.46135
$320$	0	0
$321$	6.95194e24	0.0334044
$322$	0	0
$323$	3.89661e26	1.75412
$324$	0	0
$325$	−3.03459e25	−0.128034
$326$	0	0
$327$	−6.51099e25	−0.257570
$328$	0	0
$329$	−2.48997e26	−0.923928
$330$	0	0
$331$	4.05229e26	1.41093	0.705467	−	0.708743i	$-0.250737\pi$
$331$	4.05229e26	1.41093	0.705467	+	0.708743i	$0.250737\pi$
$332$	0	0
$333$	1.18309e26	0.386680
$334$	0	0
$335$	−6.18980e26	−1.89979
$336$	0	0
$337$	−2.33842e26	−0.674231	−0.337116	−	0.941463i	$-0.609451\pi$
$337$	−2.33842e26	−0.674231	−0.337116	+	0.941463i	$0.609451\pi$
$338$	0	0
$339$	1.46021e26	0.395657
$340$	0	0
$341$	5.36770e26	1.36731
$342$	0	0
$343$	−2.80581e25	−0.0672156
$344$	0	0
$345$	5.06145e26	1.14071
$346$	0	0
$347$	4.99802e26	1.06008	0.530040	−	0.847973i	$-0.322177\pi$
$347$	4.99802e26	1.06008	0.530040	+	0.847973i	$0.322177\pi$
$348$	0	0
$349$	−1.02863e26	−0.205396	−0.102698	−	0.994713i	$-0.532748\pi$
$349$	−1.02863e26	−0.205396	−0.102698	+	0.994713i	$0.532748\pi$
$350$	0	0
$351$	1.31477e26	0.247243
$352$	0	0
$353$	−5.98988e26	−1.06117	−0.530583	−	0.847633i	$-0.678027\pi$
$353$	−5.98988e26	−1.06117	−0.530583	+	0.847633i	$0.678027\pi$
$354$	0	0
$355$	−2.84370e26	−0.474772
$356$	0	0
$357$	5.39098e26	0.848499
$358$	0	0
$359$	8.95419e26	1.32903	0.664515	−	0.747275i	$-0.268638\pi$
$359$	8.95419e26	1.32903	0.664515	+	0.747275i	$0.268638\pi$
$360$	0	0
$361$	1.27178e27	1.78069
$362$	0	0
$363$	−3.86964e26	−0.511271
$364$	0	0
$365$	1.55403e26	0.193814
$366$	0	0
$367$	−8.91935e26	−1.05036	−0.525181	−	0.850990i	$-0.676003\pi$
$367$	−8.91935e26	−1.05036	−0.525181	+	0.850990i	$0.676003\pi$
$368$	0	0
$369$	1.15997e26	0.129024
$370$	0	0
$371$	−3.33342e26	−0.350319
$372$	0	0
$373$	−2.25173e26	−0.223653	−0.111826	−	0.993728i	$-0.535670\pi$
$373$	−2.25173e26	−0.223653	−0.111826	+	0.993728i	$0.535670\pi$
$374$	0	0
$375$	−5.80502e26	−0.545099
$376$	0	0
$377$	2.59325e27	2.30282
$378$	0	0
$379$	1.12547e27	0.945417	0.472708	−	0.881219i	$-0.343276\pi$
$379$	1.12547e27	0.945417	0.472708	+	0.881219i	$0.343276\pi$
$380$	0	0
$381$	8.45863e26	0.672338
$382$	0	0
$383$	1.59881e27	1.20285	0.601423	−	0.798931i	$-0.294601\pi$
$383$	1.59881e27	1.20285	0.601423	+	0.798931i	$0.294601\pi$
$384$	0	0
$385$	−2.82432e27	−2.01176
$386$	0	0
$387$	3.05912e26	0.206363
$388$	0	0
$389$	−1.90159e27	−1.21519	−0.607596	−	0.794246i	$-0.707866\pi$
$389$	−1.90159e27	−1.21519	−0.607596	+	0.794246i	$0.707866\pi$
$390$	0	0
$391$	−3.27300e27	−1.98194
$392$	0	0
$393$	−7.10413e26	−0.407746
$394$	0	0
$395$	−2.56993e27	−1.39846
$396$	0	0
$397$	−2.23175e27	−1.15172	−0.575858	−	0.817550i	$-0.695332\pi$
$397$	−2.23175e27	−1.15172	−0.575858	+	0.817550i	$0.695332\pi$
$398$	0	0
$399$	2.74763e27	1.34506
$400$	0	0
$401$	−1.00234e27	−0.465587	−0.232793	−	0.972526i	$-0.574787\pi$
$401$	−1.00234e27	−0.465587	−0.232793	+	0.972526i	$0.574787\pi$
$402$	0	0
$403$	−2.90173e27	−1.27925
$404$	0	0
$405$	−2.78397e26	−0.116516
$406$	0	0
$407$	4.00805e27	1.59291
$408$	0	0
$409$	−1.44963e27	−0.547220	−0.273610	−	0.961841i	$-0.588218\pi$
$409$	−1.44963e27	−0.547220	−0.273610	+	0.961841i	$0.588218\pi$
$410$	0	0
$411$	2.19715e26	0.0787990
$412$	0	0
$413$	−2.58292e27	−0.880310
$414$	0	0
$415$	−1.19952e27	−0.388601
$416$	0	0
$417$	2.11877e26	0.0652613
$418$	0	0
$419$	4.01306e27	1.17552	0.587758	−	0.809037i	$-0.300011\pi$
$419$	4.01306e27	1.17552	0.587758	+	0.809037i	$0.300011\pi$
$420$	0	0
$421$	−1.87291e27	−0.521862	−0.260931	−	0.965358i	$-0.584029\pi$
$421$	−1.87291e27	−0.521862	−0.260931	+	0.965358i	$0.584029\pi$
$422$	0	0
$423$	−8.31502e26	−0.220439
$424$	0	0
$425$	−4.15515e26	−0.104834
$426$	0	0
$427$	3.34164e27	0.802541
$428$	0	0
$429$	4.45416e27	1.01851
$430$	0	0
$431$	1.48645e27	0.323698	0.161849	−	0.986816i	$-0.448254\pi$
$431$	1.48645e27	0.323698	0.161849	+	0.986816i	$0.448254\pi$
$432$	0	0
$433$	7.59546e27	1.57555	0.787773	−	0.615966i	$-0.211234\pi$
$433$	7.59546e27	1.57555	0.787773	+	0.615966i	$0.211234\pi$
$434$	0	0
$435$	−5.49112e27	−1.08523
$436$	0	0
$437$	−1.66816e28	−3.14182
$438$	0	0
$439$	−6.89476e26	−0.123777	−0.0618887	−	0.998083i	$-0.519712\pi$
$439$	−6.89476e26	−0.123777	−0.0618887	+	0.998083i	$0.519712\pi$
$440$	0	0
$441$	1.85383e27	0.317296
$442$	0	0
$443$	4.78907e27	0.781650	0.390825	−	0.920465i	$-0.372190\pi$
$443$	4.78907e27	0.781650	0.390825	+	0.920465i	$0.372190\pi$
$444$	0	0
$445$	−2.66003e27	−0.414101
$446$	0	0
$447$	1.72155e27	0.255676
$448$	0	0
$449$	−5.60799e27	−0.794732	−0.397366	−	0.917660i	$-0.630076\pi$
$449$	−5.60799e27	−0.794732	−0.397366	+	0.917660i	$0.630076\pi$
$450$	0	0
$451$	3.92973e27	0.531507
$452$	0	0
$453$	−3.13098e27	−0.404249
$454$	0	0
$455$	1.52680e28	1.88219
$456$	0	0
$457$	4.51711e26	0.0531791	0.0265895	−	0.999646i	$-0.491535\pi$
$457$	4.51711e26	0.0531791	0.0265895	+	0.999646i	$0.491535\pi$
$458$	0	0
$459$	1.80026e27	0.202443
$460$	0	0
$461$	1.45912e28	1.56758	0.783792	−	0.621023i	$-0.213283\pi$
$461$	1.45912e28	1.56758	0.783792	+	0.621023i	$0.213283\pi$
$462$	0	0
$463$	−7.56353e27	−0.776469	−0.388234	−	0.921561i	$-0.626915\pi$
$463$	−7.56353e27	−0.776469	−0.388234	+	0.921561i	$0.626915\pi$
$464$	0	0
$465$	6.14431e27	0.602861
$466$	0	0
$467$	4.22896e27	0.396649	0.198324	−	0.980136i	$-0.436450\pi$
$467$	4.22896e27	0.396649	0.198324	+	0.980136i	$0.436450\pi$
$468$	0	0
$469$	2.82240e28	2.53107
$470$	0	0
$471$	−1.19533e27	−0.102511
$472$	0	0
$473$	1.03637e28	0.850104
$474$	0	0
$475$	−2.11777e27	−0.166186
$476$	0	0
$477$	−1.11316e27	−0.0835825
$478$	0	0
$479$	−3.36732e27	−0.241970	−0.120985	−	0.992654i	$-0.538605\pi$
$479$	−3.36732e27	−0.241970	−0.120985	+	0.992654i	$0.538605\pi$
$480$	0	0
$481$	−2.16672e28	−1.49032
$482$	0	0
$483$	−2.30790e28	−1.51975
$484$	0	0
$485$	−2.77952e28	−1.75260
$486$	0	0
$487$	1.68799e28	1.01933	0.509665	−	0.860373i	$-0.329769\pi$
$487$	1.68799e28	1.01933	0.509665	+	0.860373i	$0.329769\pi$
$488$	0	0
$489$	−3.23772e27	−0.187282
$490$	0	0
$491$	6.31511e27	0.349966	0.174983	−	0.984571i	$-0.444013\pi$
$491$	6.31511e27	0.349966	0.174983	+	0.984571i	$0.444013\pi$
$492$	0	0
$493$	3.55085e28	1.88555
$494$	0	0
$495$	−9.43151e27	−0.479984
$496$	0	0
$497$	1.29666e28	0.632534
$498$	0	0
$499$	7.47715e27	0.349688	0.174844	−	0.984596i	$-0.444058\pi$
$499$	7.47715e27	0.349688	0.174844	+	0.984596i	$0.444058\pi$
$500$	0	0
$501$	1.21595e28	0.545281
$502$	0	0
$503$	−2.22587e28	−0.957273	−0.478637	−	0.878013i	$-0.658869\pi$
$503$	−2.22587e28	−0.957273	−0.478637	+	0.878013i	$0.658869\pi$
$504$	0	0
$505$	2.49688e28	1.03000
$506$	0	0
$507$	−9.48989e27	−0.375559
$508$	0	0
$509$	−4.18869e28	−1.59053	−0.795264	−	0.606264i	$-0.792668\pi$
$509$	−4.18869e28	−1.59053	−0.795264	+	0.606264i	$0.792668\pi$
$510$	0	0
$511$	−7.08601e27	−0.258216
$512$	0	0
$513$	9.17543e27	0.320918
$514$	0	0
$515$	−4.35638e28	−1.46268
$516$	0	0
$517$	−2.81696e28	−0.908091
$518$	0	0
$519$	−2.00531e28	−0.620762
$520$	0	0
$521$	−3.15615e28	−0.938343	−0.469171	−	0.883107i	$-0.655447\pi$
$521$	−3.15615e28	−0.938343	−0.469171	+	0.883107i	$0.655447\pi$
$522$	0	0
$523$	−1.28106e28	−0.365849	−0.182924	−	0.983127i	$-0.558556\pi$
$523$	−1.28106e28	−0.365849	−0.182924	+	0.983127i	$0.558556\pi$
$524$	0	0
$525$	−2.92994e27	−0.0803870
$526$	0	0
$527$	−3.97323e28	−1.04745
$528$	0	0
$529$	1.00647e29	2.54986
$530$	0	0
$531$	−8.62540e27	−0.210033
$532$	0	0
$533$	−2.12438e28	−0.497275
$534$	0	0
$535$	2.69592e27	0.0606726
$536$	0	0
$537$	−2.56948e28	−0.556053
$538$	0	0
$539$	6.28040e28	1.30709
$540$	0	0
$541$	−7.92358e28	−1.58617	−0.793086	−	0.609110i	$-0.791527\pi$
$541$	−7.92358e28	−1.58617	−0.793086	+	0.609110i	$0.791527\pi$
$542$	0	0
$543$	−5.15987e28	−0.993665
$544$	0	0
$545$	−2.52492e28	−0.467827
$546$	0	0
$547$	−9.53630e28	−1.70025	−0.850126	−	0.526580i	$-0.823474\pi$
$547$	−9.53630e28	−1.70025	−0.850126	+	0.526580i	$0.823474\pi$
$548$	0	0
$549$	1.11591e28	0.191478
$550$	0	0
$551$	1.80977e29	2.98903
$552$	0	0
$553$	1.17183e29	1.86316
$554$	0	0
$555$	4.58794e28	0.702330
$556$	0	0
$557$	−1.55708e28	−0.229526	−0.114763	−	0.993393i	$-0.536611\pi$
$557$	−1.55708e28	−0.229526	−0.114763	+	0.993393i	$0.536611\pi$
$558$	0	0
$559$	−5.60252e28	−0.795352
$560$	0	0
$561$	6.09891e28	0.833955
$562$	0	0
$563$	1.00662e28	0.132595	0.0662973	−	0.997800i	$-0.478881\pi$
$563$	1.00662e28	0.132595	0.0662973	+	0.997800i	$0.478881\pi$
$564$	0	0
$565$	5.66262e28	0.718635
$566$	0	0
$567$	1.26943e28	0.155233
$568$	0	0
$569$	−1.23826e29	−1.45926	−0.729628	−	0.683844i	$-0.760307\pi$
$569$	−1.23826e29	−1.45926	−0.729628	+	0.683844i	$0.760307\pi$
$570$	0	0
$571$	7.61732e28	0.865214	0.432607	−	0.901583i	$-0.357594\pi$
$571$	7.61732e28	0.865214	0.432607	+	0.901583i	$0.357594\pi$
$572$	0	0
$573$	−4.54587e28	−0.497729
$574$	0	0
$575$	1.77884e28	0.187769
$576$	0	0
$577$	1.56269e29	1.59047	0.795237	−	0.606299i	$-0.207347\pi$
$577$	1.56269e29	1.59047	0.795237	+	0.606299i	$0.207347\pi$
$578$	0	0
$579$	−8.45755e28	−0.830080
$580$	0	0
$581$	5.46955e28	0.517728
$582$	0	0
$583$	−3.77116e28	−0.344314
$584$	0	0
$585$	5.09859e28	0.449070
$586$	0	0
$587$	1.38917e29	1.18047	0.590237	−	0.807230i	$-0.299034\pi$
$587$	1.38917e29	1.18047	0.590237	+	0.807230i	$0.299034\pi$
$588$	0	0
$589$	−2.02505e29	−1.66044
$590$	0	0
$591$	−4.13082e28	−0.326864
$592$	0	0
$593$	9.99955e28	0.763671	0.381835	−	0.924230i	$-0.375292\pi$
$593$	9.99955e28	0.763671	0.381835	+	0.924230i	$0.375292\pi$
$594$	0	0
$595$	2.09059e29	1.54114
$596$	0	0
$597$	−8.42450e27	−0.0599534
$598$	0	0
$599$	−1.26122e29	−0.866583	−0.433292	−	0.901254i	$-0.642648\pi$
$599$	−1.26122e29	−0.866583	−0.433292	+	0.901254i	$0.642648\pi$
$600$	0	0
$601$	−6.37808e28	−0.423163	−0.211582	−	0.977360i	$-0.567861\pi$
$601$	−6.37808e28	−0.423163	−0.211582	+	0.977360i	$0.567861\pi$
$602$	0	0
$603$	9.42513e28	0.603887
$604$	0	0
$605$	−1.50063e29	−0.928627
$606$	0	0
$607$	5.82992e28	0.348483	0.174241	−	0.984703i	$-0.444253\pi$
$607$	5.82992e28	0.348483	0.174241	+	0.984703i	$0.444253\pi$
$608$	0	0
$609$	2.50382e29	1.44585
$610$	0	0
$611$	1.52282e29	0.849604
$612$	0	0
$613$	−2.59410e29	−1.39847	−0.699233	−	0.714894i	$-0.746475\pi$
$613$	−2.59410e29	−1.39847	−0.699233	+	0.714894i	$0.746475\pi$
$614$	0	0
$615$	4.49829e28	0.234347
$616$	0	0
$617$	1.85097e29	0.931975	0.465988	−	0.884791i	$-0.345699\pi$
$617$	1.85097e29	0.931975	0.465988	+	0.884791i	$0.345699\pi$
$618$	0	0
$619$	−2.73363e29	−1.33042	−0.665208	−	0.746658i	$-0.731657\pi$
$619$	−2.73363e29	−1.33042	−0.665208	+	0.746658i	$0.731657\pi$
$620$	0	0
$621$	−7.70700e28	−0.362597
$622$	0	0
$623$	1.21291e29	0.551703
$624$	0	0
$625$	−2.47775e29	−1.08973
$626$	0	0
$627$	3.10845e29	1.32201
$628$	0	0
$629$	−2.96680e29	−1.22027
$630$	0	0
$631$	2.58948e29	1.03016	0.515080	−	0.857142i	$-0.327762\pi$
$631$	2.58948e29	1.03016	0.515080	+	0.857142i	$0.327762\pi$
$632$	0	0
$633$	−8.44756e28	−0.325081
$634$	0	0
$635$	3.28021e29	1.22117
$636$	0	0
$637$	−3.39513e29	−1.22290
$638$	0	0
$639$	4.33006e28	0.150916
$640$	0	0
$641$	−2.88852e29	−0.974240	−0.487120	−	0.873335i	$-0.661953\pi$
$641$	−2.88852e29	−0.974240	−0.487120	+	0.873335i	$0.661953\pi$
$642$	0	0
$643$	−3.13854e29	−1.02450	−0.512250	−	0.858836i	$-0.671188\pi$
$643$	−3.13854e29	−1.02450	−0.512250	+	0.858836i	$0.671188\pi$
$644$	0	0
$645$	1.18631e29	0.374819
$646$	0	0
$647$	−3.63788e29	−1.11264	−0.556318	−	0.830969i	$-0.687786\pi$
$647$	−3.63788e29	−1.11264	−0.556318	+	0.830969i	$0.687786\pi$
$648$	0	0
$649$	−2.92211e29	−0.865220
$650$	0	0
$651$	−2.80166e29	−0.803185
$652$	0	0
$653$	2.24989e29	0.624557	0.312279	−	0.949991i	$-0.398908\pi$
$653$	2.24989e29	0.624557	0.312279	+	0.949991i	$0.398908\pi$
$654$	0	0
$655$	−2.75494e29	−0.740592
$656$	0	0
$657$	−2.36630e28	−0.0616075
$658$	0	0
$659$	−4.87114e29	−1.22838	−0.614191	−	0.789157i	$-0.710518\pi$
$659$	−4.87114e29	−1.22838	−0.614191	+	0.789157i	$0.710518\pi$
$660$	0	0
$661$	6.65628e28	0.162599	0.0812993	−	0.996690i	$-0.474093\pi$
$661$	6.65628e28	0.162599	0.0812993	+	0.996690i	$0.474093\pi$
$662$	0	0
$663$	−3.29702e29	−0.780243
$664$	0	0
$665$	1.06552e30	2.44305
$666$	0	0
$667$	−1.52013e30	−3.37723
$668$	0	0
$669$	−1.15043e29	−0.247677
$670$	0	0
$671$	3.78046e29	0.788785
$672$	0	0
$673$	7.39137e29	1.49474	0.747372	−	0.664406i	$-0.231316\pi$
$673$	7.39137e29	1.49474	0.747372	+	0.664406i	$0.231316\pi$
$674$	0	0
$675$	−9.78423e27	−0.0191795
$676$	0	0
$677$	−8.52490e29	−1.61997	−0.809987	−	0.586448i	$-0.800526\pi$
$677$	−8.52490e29	−1.61997	−0.809987	+	0.586448i	$0.800526\pi$
$678$	0	0
$679$	1.26740e30	2.33497
$680$	0	0
$681$	2.72276e29	0.486368
$682$	0	0
$683$	4.31556e29	0.747515	0.373757	−	0.927527i	$-0.378069\pi$
$683$	4.31556e29	0.747515	0.373757	+	0.927527i	$0.378069\pi$
$684$	0	0
$685$	8.52043e28	0.143123
$686$	0	0
$687$	4.64593e29	0.756880
$688$	0	0
$689$	2.03866e29	0.322138
$690$	0	0
$691$	2.76157e29	0.423288	0.211644	−	0.977347i	$-0.432118\pi$
$691$	2.76157e29	0.423288	0.211644	+	0.977347i	$0.432118\pi$
$692$	0	0
$693$	4.30055e29	0.639477
$694$	0	0
$695$	8.21648e28	0.118535
$696$	0	0
$697$	−2.90883e29	−0.407169
$698$	0	0
$699$	3.74551e29	0.508746
$700$	0	0
$701$	−1.58719e28	−0.0209214	−0.0104607	−	0.999945i	$-0.503330\pi$
$701$	−1.58719e28	−0.0209214	−0.0104607	+	0.999945i	$0.503330\pi$
$702$	0	0
$703$	−1.51210e30	−1.93441
$704$	0	0
$705$	−3.22452e29	−0.400386
$706$	0	0
$707$	−1.13852e30	−1.37226
$708$	0	0
$709$	1.37623e29	0.161029	0.0805146	−	0.996753i	$-0.474344\pi$
$709$	1.37623e29	0.161029	0.0805146	+	0.996753i	$0.474344\pi$
$710$	0	0
$711$	3.91320e29	0.444530
$712$	0	0
$713$	1.70096e30	1.87609
$714$	0	0
$715$	1.72730e30	1.84992
$716$	0	0
$717$	−5.87739e29	−0.611270
$718$	0	0
$719$	−6.55225e29	−0.661816	−0.330908	−	0.943663i	$-0.607355\pi$
$719$	−6.55225e29	−0.661816	−0.330908	+	0.943663i	$0.607355\pi$
$720$	0	0
$721$	1.98641e30	1.94871
$722$	0	0
$723$	−1.09315e29	−0.104166
$724$	0	0
$725$	−1.92985e29	−0.178638
$726$	0	0
$727$	2.29870e29	0.206714	0.103357	−	0.994644i	$-0.467042\pi$
$727$	2.29870e29	0.206714	0.103357	+	0.994644i	$0.467042\pi$
$728$	0	0
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−7.67132e29	−0.651235
$732$	0	0
$733$	−4.14506e29	−0.341931	−0.170966	−	0.985277i	$-0.554689\pi$
$733$	−4.14506e29	−0.341931	−0.170966	+	0.985277i	$0.554689\pi$
$734$	0	0
$735$	7.18906e29	0.576308
$736$	0	0
$737$	3.19304e30	2.48769
$738$	0	0
$739$	1.35190e30	1.02371	0.511855	−	0.859072i	$-0.328958\pi$
$739$	1.35190e30	1.02371	0.511855	+	0.859072i	$0.328958\pi$
$740$	0	0
$741$	−1.68040e30	−1.23686
$742$	0	0
$743$	−1.85381e30	−1.32643	−0.663214	−	0.748430i	$-0.730808\pi$
$743$	−1.85381e30	−1.32643	−0.663214	+	0.748430i	$0.730808\pi$
$744$	0	0
$745$	6.67607e29	0.464387
$746$	0	0
$747$	1.82650e29	0.123524
$748$	0	0
$749$	−1.22928e29	−0.0808335
$750$	0	0
$751$	1.14505e29	0.0732159	0.0366080	−	0.999330i	$-0.488345\pi$
$751$	1.14505e29	0.0732159	0.0366080	+	0.999330i	$0.488345\pi$
$752$	0	0
$753$	−7.67870e29	−0.477464
$754$	0	0
$755$	−1.21418e30	−0.734241
$756$	0	0
$757$	−2.80394e30	−1.64916	−0.824579	−	0.565746i	$-0.808588\pi$
$757$	−2.80394e30	−1.64916	−0.824579	+	0.565746i	$0.808588\pi$
$758$	0	0
$759$	−2.61097e30	−1.49370
$760$	0	0
$761$	−6.27406e29	−0.349148	−0.174574	−	0.984644i	$-0.555855\pi$
$761$	−6.27406e29	−0.349148	−0.174574	+	0.984644i	$0.555855\pi$
$762$	0	0
$763$	1.15131e30	0.623280
$764$	0	0
$765$	6.98131e29	0.367699
$766$	0	0
$767$	1.57967e30	0.809495
$768$	0	0
$769$	2.09742e30	1.04582	0.522911	−	0.852387i	$-0.324846\pi$
$769$	2.09742e30	1.04582	0.522911	+	0.852387i	$0.324846\pi$
$770$	0	0
$771$	−9.74728e28	−0.0472947
$772$	0	0
$773$	−3.96887e30	−1.87406	−0.937028	−	0.349255i	$-0.886435\pi$
$773$	−3.96887e30	−1.87406	−0.937028	+	0.349255i	$0.886435\pi$
$774$	0	0
$775$	2.15941e29	0.0992356
$776$	0	0
$777$	−2.09199e30	−0.935707
$778$	0	0
$779$	−1.48255e30	−0.645456
$780$	0	0
$781$	1.46694e30	0.621692
$782$	0	0
$783$	8.36126e29	0.344963
$784$	0	0
$785$	−4.63543e29	−0.186191
$786$	0	0
$787$	3.10916e30	1.21593	0.607964	−	0.793965i	$-0.291987\pi$
$787$	3.10916e30	1.21593	0.607964	+	0.793965i	$0.291987\pi$
$788$	0	0
$789$	5.22196e28	0.0198850
$790$	0	0
$791$	−2.58202e30	−0.957430
$792$	0	0
$793$	−2.04369e30	−0.737982
$794$	0	0
$795$	−4.31678e29	−0.151812
$796$	0	0
$797$	3.20026e30	1.09616	0.548078	−	0.836427i	$-0.315360\pi$
$797$	3.20026e30	1.09616	0.548078	+	0.836427i	$0.315360\pi$
$798$	0	0
$799$	2.08514e30	0.695656
$800$	0	0
$801$	4.05040e29	0.131630
$802$	0	0
$803$	−8.01654e29	−0.253790
$804$	0	0
$805$	−8.94992e30	−2.76034
$806$	0	0
$807$	1.36987e30	0.411630
$808$	0	0
$809$	1.18061e30	0.345657	0.172829	−	0.984952i	$-0.444709\pi$
$809$	1.18061e30	0.345657	0.172829	+	0.984952i	$0.444709\pi$
$810$	0	0
$811$	−4.04592e29	−0.115425	−0.0577123	−	0.998333i	$-0.518381\pi$
$811$	−4.04592e29	−0.115425	−0.0577123	+	0.998333i	$0.518381\pi$
$812$	0	0
$813$	−8.10660e29	−0.225366
$814$	0	0
$815$	−1.25557e30	−0.340162
$816$	0	0
$817$	−3.90986e30	−1.03236
$818$	0	0
$819$	−2.32484e30	−0.598291
$820$	0	0
$821$	−4.67234e29	−0.117201	−0.0586006	−	0.998282i	$-0.518664\pi$
$821$	−4.67234e29	−0.117201	−0.0586006	+	0.998282i	$0.518664\pi$
$822$	0	0
$823$	3.03065e29	0.0741033	0.0370516	−	0.999313i	$-0.488203\pi$
$823$	3.03065e29	0.0741033	0.0370516	+	0.999313i	$0.488203\pi$
$824$	0	0
$825$	−3.31470e29	−0.0790091
$826$	0	0
$827$	−5.56427e30	−1.29301	−0.646503	−	0.762912i	$-0.723769\pi$
$827$	−5.56427e30	−1.29301	−0.646503	+	0.762912i	$0.723769\pi$
$828$	0	0
$829$	6.34074e30	1.43654	0.718270	−	0.695765i	$-0.244934\pi$
$829$	6.34074e30	1.43654	0.718270	+	0.695765i	$0.244934\pi$
$830$	0	0
$831$	3.34611e30	0.739144
$832$	0	0
$833$	−4.64882e30	−1.00131
$834$	0	0
$835$	4.71540e30	0.990400
$836$	0	0
$837$	−9.35586e29	−0.191631
$838$	0	0
$839$	5.21434e30	1.04159	0.520797	−	0.853681i	$-0.325635\pi$
$839$	5.21434e30	1.04159	0.520797	+	0.853681i	$0.325635\pi$
$840$	0	0
$841$	1.13589e31	2.21299
$842$	0	0
$843$	3.36088e30	0.638650
$844$	0	0
$845$	−3.68013e30	−0.682131
$846$	0	0
$847$	6.84251e30	1.23720
$848$	0	0
$849$	−4.39912e30	−0.775952
$850$	0	0
$851$	1.27010e31	2.18564
$852$	0	0
$853$	6.32826e30	1.06248	0.531239	−	0.847222i	$-0.321727\pi$
$853$	6.32826e30	1.06248	0.531239	+	0.847222i	$0.321727\pi$
$854$	0	0
$855$	3.55818e30	0.582886
$856$	0	0
$857$	1.08234e30	0.173007	0.0865036	−	0.996252i	$-0.472431\pi$
$857$	1.08234e30	0.173007	0.0865036	+	0.996252i	$0.472431\pi$
$858$	0	0
$859$	−5.23064e30	−0.815880	−0.407940	−	0.913009i	$-0.633753\pi$
$859$	−5.23064e30	−0.815880	−0.407940	+	0.913009i	$0.633753\pi$
$860$	0	0
$861$	−2.05112e30	−0.312218
$862$	0	0
$863$	−4.34348e30	−0.645245	−0.322623	−	0.946528i	$-0.604564\pi$
$863$	−4.34348e30	−0.645245	−0.322623	+	0.946528i	$0.604564\pi$
$864$	0	0
$865$	−7.77650e30	−1.12750
$866$	0	0
$867$	−4.34677e29	−0.0615129
$868$	0	0
$869$	1.32571e31	1.83122
$870$	0	0
$871$	−1.72613e31	−2.32746
$872$	0	0
$873$	4.23235e30	0.557098
$874$	0	0
$875$	1.02647e31	1.31906
$876$	0	0
$877$	−1.09787e31	−1.37738	−0.688690	−	0.725056i	$-0.741814\pi$
$877$	−1.09787e31	−1.37738	−0.688690	+	0.725056i	$0.741814\pi$
$878$	0	0
$879$	2.28173e30	0.279499
$880$	0	0
$881$	1.59621e31	1.90917	0.954584	−	0.297943i	$-0.0963004\pi$
$881$	1.59621e31	1.90917	0.954584	+	0.297943i	$0.0963004\pi$
$882$	0	0
$883$	1.59335e31	1.86090	0.930452	−	0.366415i	$-0.119415\pi$
$883$	1.59335e31	1.86090	0.930452	+	0.366415i	$0.119415\pi$
$884$	0	0
$885$	−3.34488e30	−0.381484
$886$	0	0
$887$	−1.63044e30	−0.181597	−0.0907983	−	0.995869i	$-0.528942\pi$
$887$	−1.63044e30	−0.181597	−0.0907983	+	0.995869i	$0.528942\pi$
$888$	0	0
$889$	−1.49570e31	−1.62696
$890$	0	0
$891$	1.43612e30	0.152573
$892$	0	0
$893$	1.06274e31	1.10277
$894$	0	0
$895$	−9.96432e30	−1.00996
$896$	0	0
$897$	1.41147e31	1.39750
$898$	0	0
$899$	−1.84536e31	−1.78486
$900$	0	0
$901$	2.79146e30	0.263767
$902$	0	0
$903$	−5.40931e30	−0.499368
$904$	0	0
$905$	−2.00097e31	−1.80480
$906$	0	0
$907$	−2.06654e31	−1.82124	−0.910622	−	0.413240i	$-0.864397\pi$
$907$	−2.06654e31	−1.82124	−0.910622	+	0.413240i	$0.864397\pi$
$908$	0	0
$909$	−3.80196e30	−0.327407
$910$	0	0
$911$	−7.55745e30	−0.635964	−0.317982	−	0.948097i	$-0.603005\pi$
$911$	−7.55745e30	−0.635964	−0.317982	+	0.948097i	$0.603005\pi$
$912$	0	0
$913$	6.18780e30	0.508854
$914$	0	0
$915$	4.32743e30	0.347783
$916$	0	0
$917$	1.25619e31	0.986683
$918$	0	0
$919$	2.05403e30	0.157686	0.0788432	−	0.996887i	$-0.474877\pi$
$919$	2.05403e30	0.157686	0.0788432	+	0.996887i	$0.474877\pi$
$920$	0	0
$921$	1.26452e31	0.948854
$922$	0	0
$923$	−7.93013e30	−0.581651
$924$	0	0
$925$	1.61243e30	0.115609
$926$	0	0
$927$	6.63340e30	0.464943
$928$	0	0
$929$	−1.05220e31	−0.720996	−0.360498	−	0.932760i	$-0.617393\pi$
$929$	−1.05220e31	−0.720996	−0.360498	+	0.932760i	$0.617393\pi$
$930$	0	0
$931$	−2.36938e31	−1.58731
$932$	0	0
$933$	1.05986e30	0.0694208
$934$	0	0
$935$	2.36512e31	1.51472
$936$	0	0
$937$	−3.62945e29	−0.0227287	−0.0113643	−	0.999935i	$-0.503617\pi$
$937$	−3.62945e29	−0.0227287	−0.0113643	+	0.999935i	$0.503617\pi$
$938$	0	0
$939$	−1.50399e31	−0.920991
$940$	0	0
$941$	−1.37977e31	−0.826257	−0.413129	−	0.910673i	$-0.635564\pi$
$941$	−1.37977e31	−0.826257	−0.413129	+	0.910673i	$0.635564\pi$
$942$	0	0
$943$	1.24529e31	0.729283
$944$	0	0
$945$	4.92276e30	0.281952
$946$	0	0
$947$	−6.03295e30	−0.337952	−0.168976	−	0.985620i	$-0.554046\pi$
$947$	−6.03295e30	−0.337952	−0.168976	+	0.985620i	$0.554046\pi$
$948$	0	0
$949$	4.33367e30	0.237444
$950$	0	0
$951$	−1.70159e31	−0.911927
$952$	0	0
$953$	−4.31703e30	−0.226313	−0.113156	−	0.993577i	$-0.536096\pi$
$953$	−4.31703e30	−0.226313	−0.113156	+	0.993577i	$0.536096\pi$
$954$	0	0
$955$	−1.76286e31	−0.904030
$956$	0	0
$957$	2.83262e31	1.42106
$958$	0	0
$959$	−3.88512e30	−0.190682
$960$	0	0
$961$	−1.76814e29	−0.00849024
$962$	0	0
$963$	−4.10505e29	−0.0192860
$964$	0	0
$965$	−3.27979e31	−1.50768
$966$	0	0
$967$	−3.09428e31	−1.39181	−0.695907	−	0.718132i	$-0.744998\pi$
$967$	−3.09428e31	−1.39181	−0.695907	+	0.718132i	$0.744998\pi$
$968$	0	0
$969$	−2.30091e31	−1.01274
$970$	0	0
$971$	2.67948e31	1.15411	0.577057	−	0.816704i	$-0.304201\pi$
$971$	2.67948e31	1.15411	0.577057	+	0.816704i	$0.304201\pi$
$972$	0	0
$973$	−3.74653e30	−0.157923
$974$	0	0
$975$	1.79190e30	0.0739204
$976$	0	0
$977$	2.92566e31	1.18122	0.590611	−	0.806957i	$-0.298887\pi$
$977$	2.92566e31	1.18122	0.590611	+	0.806957i	$0.298887\pi$
$978$	0	0
$979$	1.37219e31	0.542246
$980$	0	0
$981$	3.84467e30	0.148708
$982$	0	0
$983$	3.05616e31	1.15708	0.578540	−	0.815654i	$-0.303623\pi$
$983$	3.05616e31	1.15708	0.578540	+	0.815654i	$0.303623\pi$
$984$	0	0
$985$	−1.60191e31	−0.593687
$986$	0	0
$987$	1.47031e31	0.533430
$988$	0	0
$989$	3.28413e31	1.16643
$990$	0	0
$991$	3.66516e30	0.127444	0.0637220	−	0.997968i	$-0.479703\pi$
$991$	3.66516e30	0.127444	0.0637220	+	0.997968i	$0.479703\pi$
$992$	0	0
$993$	−2.39284e31	−0.814603
$994$	0	0
$995$	−3.26697e30	−0.108894
$996$	0	0
$997$	4.60718e30	0.150361	0.0751807	−	0.997170i	$-0.476047\pi$
$997$	4.60718e30	0.150361	0.0751807	+	0.997170i	$0.476047\pi$
$998$	0	0
$999$	−6.98600e30	−0.223250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	24.22.a.b.1.1	✓	3
3.2	odd	2		72.22.a.e.1.3		3
4.3	odd	2		48.22.a.k.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.22.a.b.1.1	✓	3	1.1	even	1	trivial
48.22.a.k.1.1		3	4.3	odd	2
72.22.a.e.1.3		3	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:	\( N \)	\(=\)	\( 24 = 2^{3} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	24.a (trivial)

\(f(q)\)	\(=\)	\(q-59049.0 q^{3} -2.28989e7 q^{5} +1.04414e9 q^{7} +3.48678e9 q^{9} +O(q^{10})\)	\(q-59049.0 q^{3} -2.28989e7 q^{5} +1.04414e9 q^{7} +3.48678e9 q^{9} +1.18125e11 q^{11} -6.38574e11 q^{13} +1.35216e12 q^{15} -8.74375e12 q^{17} -4.45645e13 q^{19} -6.16552e13 q^{21} +3.74324e14 q^{23} +4.75214e13 q^{25} -2.05891e14 q^{27} -4.06101e15 q^{29} +4.54408e15 q^{31} -6.97517e15 q^{33} -2.39095e16 q^{35} +3.39306e16 q^{37} +3.77072e16 q^{39} +3.32676e16 q^{41} +8.77348e16 q^{43} -7.98434e16 q^{45} -2.38472e17 q^{47} +5.31674e17 q^{49} +5.16310e17 q^{51} -3.19252e17 q^{53} -2.70493e18 q^{55} +2.63149e18 q^{57} -2.47374e18 q^{59} +3.20039e18 q^{61} +3.64068e18 q^{63} +1.46226e19 q^{65} +2.70310e19 q^{67} -2.21035e19 q^{69} +1.24185e19 q^{71} -6.78649e18 q^{73} -2.80609e18 q^{75} +1.23339e20 q^{77} +1.12229e20 q^{79} +1.21577e19 q^{81} +5.23835e19 q^{83} +2.00222e20 q^{85} +2.39798e20 q^{87} +1.16164e20 q^{89} -6.66758e20 q^{91} -2.68324e20 q^{93} +1.02048e21 q^{95} +1.21383e21 q^{97} +4.11877e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 177147 q^{3} - 4833126 q^{5} + 271431024 q^{7} + 10460353203 q^{9}+O(q^{10}) \) 3 * q - 177147 * q^3 - 4833126 * q^5 + 271431024 * q^7 + 10460353203 * q^9	\( 3 q - 177147 q^{3} - 4833126 q^{5} + 271431024 q^{7} + 10460353203 q^{9} + 61194658188 q^{11} - 594486202422 q^{13} + 285391257174 q^{15} + 1424819519334 q^{17} - 35094825681804 q^{19} - 16027730536176 q^{21} - 94911043073592 q^{23} - 613780025295699 q^{25} - 617673396283947 q^{27} - 32\!\cdots\!34 q^{29}+ \cdots + 21\!\cdots\!88 q^{99}+O(q^{100}) \) 3 * q - 177147 * q^3 - 4833126 * q^5 + 271431024 * q^7 + 10460353203 * q^9 + 61194658188 * q^11 - 594486202422 * q^13 + 285391257174 * q^15 + 1424819519334 * q^17 - 35094825681804 * q^19 - 16027730536176 * q^21 - 94911043073592 * q^23 - 613780025295699 * q^25 - 617673396283947 * q^27 - 3292734981070734 * q^29 - 1928658139449144 * q^31 - 3613483371343212 * q^33 - 18129892754606304 * q^35 + 22185275061682722 * q^37 + 35103815766816678 * q^39 + 167394191883971118 * q^41 + 69547615491362508 * q^43 - 16852068344867526 * q^45 - 49110618973894752 * q^47 + 973320764343712395 * q^49 - 84134167797153366 * q^51 + 488391341947078074 * q^53 - 3110835549535695000 * q^55 + 2072314361684844396 * q^57 - 1846144806145488852 * q^59 + 8183707455377473002 * q^61 + 946421460430656624 * q^63 + 28951429805381293356 * q^65 + 37438935583939030164 * q^67 + 5604402182452534008 * q^69 + 65865243803437791096 * q^71 + 88415465494171854846 * q^73 + 36243096713685730251 * q^75 + 156034186748470726080 * q^77 + 221381156829825400920 * q^79 + 36472996377170786403 * q^81 + 305200914127129955700 * q^83 + 184695330643817576052 * q^85 + 194432707897245771966 * q^87 + 327556458098783035470 * q^89 + 692083854852141366816 * q^91 + 113885334476332504056 * q^93 + 1486593794603974227096 * q^95 + 1980043166774365090758 * q^97 + 213372579594445325388 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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