LMFDB - Embedded newform 48.22.a.j.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [48,22,Mod(1,48)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("48.1"); S:= CuspForms(chi, 22); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(48, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 22, names="a")

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	48.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-177147,0,5280498] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$134.149125258$
Analytic rank:	$1$
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} - x^{2} - 12529199x - 17012391021 $ x^3 - x^2 - 12529199*x - 17012391021
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{4}\cdot 7 $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4086.16$ of defining polynomial
Character	$\chi$	$=$	48.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-59049.0 q^{3} +1.51338e7 q^{5} +8.40758e8 q^{7} +3.48678e9 q^{9} +3.34735e10 q^{11} -3.01154e11 q^{13} -8.93637e11 q^{15} +5.14303e12 q^{17} -4.56647e13 q^{19} -4.96459e13 q^{21} -2.41579e13 q^{23} -2.47804e14 q^{25} -2.05891e14 q^{27} +7.82910e14 q^{29} -8.00815e15 q^{31} -1.97657e15 q^{33} +1.27239e16 q^{35} +2.06600e16 q^{37} +1.77829e16 q^{39} -7.03094e16 q^{41} -6.95453e15 q^{43} +5.27684e16 q^{45} +2.12675e17 q^{47} +1.48329e17 q^{49} -3.03691e17 q^{51} +8.69454e17 q^{53} +5.06581e17 q^{55} +2.69646e18 q^{57} -9.80538e17 q^{59} +5.62594e18 q^{61} +2.93154e18 q^{63} -4.55762e18 q^{65} -2.36685e18 q^{67} +1.42650e18 q^{69} -4.99695e19 q^{71} +2.63228e19 q^{73} +1.46326e19 q^{75} +2.81431e19 q^{77} -1.05996e20 q^{79} +1.21577e19 q^{81} -8.96505e19 q^{83} +7.78337e19 q^{85} -4.62301e19 q^{87} +4.71064e20 q^{89} -2.53198e20 q^{91} +4.72873e20 q^{93} -6.91082e20 q^{95} -3.93900e20 q^{97} +1.16715e20 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 177147 q^{3} + 5280498 q^{5} - 852542376 q^{7} + 10460353203 q^{9} - 62490757668 q^{11} + 203765207802 q^{13} - 311808126402 q^{15} + 695827819926 q^{17} - 4955504123196 q^{19} + 50341774760424 q^{21}+ \cdots - 21\!\cdots\!68 q^{99}+O(q^{100}) $ 3 * q - 177147 * q^3 + 5280498 * q^5 - 852542376 * q^7 + 10460353203 * q^9 - 62490757668 * q^11 + 203765207802 * q^13 - 311808126402 * q^15 + 695827819926 * q^17 - 4955504123196 * q^19 + 50341774760424 * q^21 - 150867407938152 * q^23 + 678194854969869 * q^25 - 617673396283947 * q^27 + 3242224905226746 * q^29 - 652508601550896 * q^31 + 3690016749537732 * q^33 - 15443719285839600 * q^35 - 20693271836916222 * q^37 - 12032131755500298 * q^39 - 47608600312552002 * q^41 + 277110211230415548 * q^43 + 18411958055911698 * q^45 + 535942507326221328 * q^47 + 1192822454236615755 * q^49 - 41087936938810374 * q^51 + 2849297892348987426 * q^53 - 2255425539904757592 * q^55 + 292617562970600604 * q^57 - 881046377434726548 * q^59 + 10620890455615553322 * q^61 - 2972631457828276776 * q^63 + 24923333196516863004 * q^65 - 14840080728867715116 * q^67 + 8908569571339937448 * q^69 - 14608944838311941496 * q^71 + 13575086263949291454 * q^73 - 40046727991115794581 * q^75 + 173887987291330337760 * q^77 - 278096577517755548160 * q^79 + 36472996377170786403 * q^81 - 585716918283456830460 * q^83 + 442542794823708598596 * q^85 - 191450138428734124554 * q^87 + 477818701792662526830 * q^89 - 1318091663263676539056 * q^91 + 38529980412978857904 * q^93 - 698400977547585407208 * q^95 + 722411740013721590502 * q^97 - 217891799043453536868 * q^99

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
$5$	1.51338e7	0.693049	0.346524	−	0.938041i	$-0.387362\pi$
$5$	1.51338e7	0.693049	0.346524	+	0.938041i	$0.387362\pi$
$6$	0	0
$7$	8.40758e8	1.12497	0.562486	−	0.826807i	$-0.309845\pi$
$7$	8.40758e8	1.12497	0.562486	+	0.826807i	$0.309845\pi$
$8$	0	0
$9$	3.48678e9	0.333333
$10$	0	0
$11$	3.34735e10	0.389114	0.194557	−	0.980891i	$-0.437673\pi$
$11$	3.34735e10	0.389114	0.194557	+	0.980891i	$0.437673\pi$
$12$	0	0
$13$	−3.01154e11	−0.605876	−0.302938	−	0.953010i	$-0.597968\pi$
$13$	−3.01154e11	−0.605876	−0.302938	+	0.953010i	$0.597968\pi$
$14$	0	0
$15$	−8.93637e11	−0.400132
$16$	0	0
$17$	5.14303e12	0.618736	0.309368	−	0.950942i	$-0.399883\pi$
$17$	5.14303e12	0.618736	0.309368	+	0.950942i	$0.399883\pi$
$18$	0	0
$19$	−4.56647e13	−1.70871	−0.854355	−	0.519690i	$-0.826047\pi$
$19$	−4.56647e13	−1.70871	−0.854355	+	0.519690i	$0.826047\pi$
$20$	0	0
$21$	−4.96459e13	−0.649503
$22$	0	0
$23$	−2.41579e13	−0.121596	−0.0607978	−	0.998150i	$-0.519364\pi$
$23$	−2.41579e13	−0.121596	−0.0607978	+	0.998150i	$0.519364\pi$
$24$	0	0
$25$	−2.47804e14	−0.519684
$26$	0	0
$27$	−2.05891e14	−0.192450
$28$	0	0
$29$	7.82910e14	0.345568	0.172784	−	0.984960i	$-0.444724\pi$
$29$	7.82910e14	0.345568	0.172784	+	0.984960i	$0.444724\pi$
$30$	0	0
$31$	−8.00815e15	−1.75483	−0.877414	−	0.479734i	$-0.840733\pi$
$31$	−8.00815e15	−1.75483	−0.877414	+	0.479734i	$0.840733\pi$
$32$	0	0
$33$	−1.97657e15	−0.224655
$34$	0	0
$35$	1.27239e16	0.779660
$36$	0	0
$37$	2.06600e16	0.706339	0.353169	−	0.935559i	$-0.385104\pi$
$37$	2.06600e16	0.706339	0.353169	+	0.935559i	$0.385104\pi$
$38$	0	0
$39$	1.77829e16	0.349803
$40$	0	0
$41$	−7.03094e16	−0.818056	−0.409028	−	0.912522i	$-0.634132\pi$
$41$	−7.03094e16	−0.818056	−0.409028	+	0.912522i	$0.634132\pi$
$42$	0	0
$43$	−6.95453e15	−0.0490737	−0.0245369	−	0.999699i	$-0.507811\pi$
$43$	−6.95453e15	−0.0490737	−0.0245369	+	0.999699i	$0.507811\pi$
$44$	0	0
$45$	5.27684e16	0.231016
$46$	0	0
$47$	2.12675e17	0.589778	0.294889	−	0.955532i	$-0.404717\pi$
$47$	2.12675e17	0.589778	0.294889	+	0.955532i	$0.404717\pi$
$48$	0	0
$49$	1.48329e17	0.265562
$50$	0	0
$51$	−3.03691e17	−0.357227
$52$	0	0
$53$	8.69454e17	0.682889	0.341445	−	0.939902i	$-0.389084\pi$
$53$	8.69454e17	0.682889	0.341445	+	0.939902i	$0.389084\pi$
$54$	0	0
$55$	5.06581e17	0.269675
$56$	0	0
$57$	2.69646e18	0.986524
$58$	0	0
$59$	−9.80538e17	−0.249757	−0.124879	−	0.992172i	$-0.539854\pi$
$59$	−9.80538e17	−0.249757	−0.124879	+	0.992172i	$0.539854\pi$
$60$	0	0
$61$	5.62594e18	1.00979	0.504896	−	0.863180i	$-0.331531\pi$
$61$	5.62594e18	1.00979	0.504896	+	0.863180i	$0.331531\pi$
$62$	0	0
$63$	2.93154e18	0.374991
$64$	0	0
$65$	−4.55762e18	−0.419902
$66$	0	0
$67$	−2.36685e18	−0.158630	−0.0793152	−	0.996850i	$-0.525273\pi$
$67$	−2.36685e18	−0.158630	−0.0793152	+	0.996850i	$0.525273\pi$
$68$	0	0
$69$	1.42650e18	0.0702032
$70$	0	0
$71$	−4.99695e19	−1.82177	−0.910883	−	0.412666i	$-0.864598\pi$
$71$	−4.99695e19	−1.82177	−0.910883	+	0.412666i	$0.864598\pi$
$72$	0	0
$73$	2.63228e19	0.716873	0.358437	−	0.933554i	$-0.383310\pi$
$73$	2.63228e19	0.716873	0.358437	+	0.933554i	$0.383310\pi$
$74$	0	0
$75$	1.46326e19	0.300040
$76$	0	0
$77$	2.81431e19	0.437743
$78$	0	0
$79$	−1.05996e20	−1.25952	−0.629762	−	0.776788i	$-0.716848\pi$
$79$	−1.05996e20	−1.25952	−0.629762	+	0.776788i	$0.716848\pi$
$80$	0	0
$81$	1.21577e19	0.111111
$82$	0	0
$83$	−8.96505e19	−0.634209	−0.317105	−	0.948391i	$-0.602711\pi$
$83$	−8.96505e19	−0.634209	−0.317105	+	0.948391i	$0.602711\pi$
$84$	0	0
$85$	7.78337e19	0.428814
$86$	0	0
$87$	−4.62301e19	−0.199513
$88$	0	0
$89$	4.71064e20	1.60135	0.800673	−	0.599102i	$-0.204476\pi$
$89$	4.71064e20	1.60135	0.800673	+	0.599102i	$0.204476\pi$
$90$	0	0
$91$	−2.53198e20	−0.681594
$92$	0	0
$93$	4.72873e20	1.01315
$94$	0	0
$95$	−6.91082e20	−1.18422
$96$	0	0
$97$	−3.93900e20	−0.542354	−0.271177	−	0.962530i	$-0.587413\pi$
$97$	−3.93900e20	−0.542354	−0.271177	+	0.962530i	$0.587413\pi$
$98$	0	0
$99$	1.16715e20	0.129705
$100$	0	0
$101$	−1.27087e21	−1.14479	−0.572397	−	0.819976i	$-0.693986\pi$
$101$	−1.27087e21	−1.14479	−0.572397	+	0.819976i	$0.693986\pi$
$102$	0	0
$103$	−1.74236e21	−1.27746	−0.638731	−	0.769430i	$-0.720540\pi$
$103$	−1.74236e21	−1.27746	−0.638731	+	0.769430i	$0.720540\pi$
$104$	0	0
$105$	−7.51333e20	−0.450137
$106$	0	0
$107$	2.92887e21	1.43936	0.719681	−	0.694305i	$-0.244288\pi$
$107$	2.92887e21	1.43936	0.719681	+	0.694305i	$0.244288\pi$
$108$	0	0
$109$	8.72296e19	0.0352928	0.0176464	−	0.999844i	$-0.494383\pi$
$109$	8.72296e19	0.0352928	0.0176464	+	0.999844i	$0.494383\pi$
$110$	0	0
$111$	−1.21995e21	−0.407805
$112$	0	0
$113$	5.59051e21	1.54927	0.774636	−	0.632408i	$-0.217933\pi$
$113$	5.59051e21	1.54927	0.774636	+	0.632408i	$0.217933\pi$
$114$	0	0
$115$	−3.65602e20	−0.0842716
$116$	0	0
$117$	−1.05006e21	−0.201959
$118$	0	0
$119$	4.32404e21	0.696060
$120$	0	0
$121$	−6.27978e21	−0.848590
$122$	0	0
$123$	4.15170e21	0.472305
$124$	0	0
$125$	−1.09666e22	−1.05321
$126$	0	0
$127$	1.18421e22	0.962695	0.481348	−	0.876530i	$-0.340148\pi$
$127$	1.18421e22	0.962695	0.481348	+	0.876530i	$0.340148\pi$
$128$	0	0
$129$	4.10658e20	0.0283327
$130$	0	0
$131$	−1.85199e22	−1.08715	−0.543577	−	0.839360i	$-0.682930\pi$
$131$	−1.85199e22	−1.08715	−0.543577	+	0.839360i	$0.682930\pi$
$132$	0	0
$133$	−3.83930e22	−1.92225
$134$	0	0
$135$	−3.11592e21	−0.133377
$136$	0	0
$137$	−3.79206e22	−1.39094	−0.695472	−	0.718553i	$-0.744805\pi$
$137$	−3.79206e22	−1.39094	−0.695472	+	0.718553i	$0.744805\pi$
$138$	0	0
$139$	−3.59104e21	−0.113127	−0.0565634	−	0.998399i	$-0.518014\pi$
$139$	−3.59104e21	−0.113127	−0.0565634	+	0.998399i	$0.518014\pi$
$140$	0	0
$141$	−1.25582e22	−0.340508
$142$	0	0
$143$	−1.00807e22	−0.235755
$144$	0	0
$145$	1.18484e22	0.239495
$146$	0	0
$147$	−8.75865e21	−0.153322
$148$	0	0
$149$	−6.86199e22	−1.04230	−0.521152	−	0.853464i	$-0.674498\pi$
$149$	−6.86199e22	−1.04230	−0.521152	+	0.853464i	$0.674498\pi$
$150$	0	0
$151$	−5.56868e22	−0.735351	−0.367675	−	0.929954i	$-0.619846\pi$
$151$	−5.56868e22	−0.735351	−0.367675	+	0.929954i	$0.619846\pi$
$152$	0	0
$153$	1.79326e22	0.206245
$154$	0	0
$155$	−1.21194e23	−1.21618
$156$	0	0
$157$	5.36204e22	0.470310	0.235155	−	0.971958i	$-0.424440\pi$
$157$	5.36204e22	0.470310	0.235155	+	0.971958i	$0.424440\pi$
$158$	0	0
$159$	−5.13404e22	−0.394266
$160$	0	0
$161$	−2.03110e22	−0.136792
$162$	0	0
$163$	−5.37253e22	−0.317840	−0.158920	−	0.987291i	$-0.550801\pi$
$163$	−5.37253e22	−0.317840	−0.158920	+	0.987291i	$0.550801\pi$
$164$	0	0
$165$	−2.99131e22	−0.155697
$166$	0	0
$167$	−2.18267e23	−1.00107	−0.500537	−	0.865715i	$-0.666864\pi$
$167$	−2.18267e23	−1.00107	−0.500537	+	0.865715i	$0.666864\pi$
$168$	0	0
$169$	−1.56371e23	−0.632914
$170$	0	0
$171$	−1.59223e23	−0.569570
$172$	0	0
$173$	2.08038e23	0.658657	0.329328	−	0.944215i	$-0.393178\pi$
$173$	2.08038e23	0.658657	0.329328	+	0.944215i	$0.393178\pi$
$174$	0	0
$175$	−2.08344e23	−0.584630
$176$	0	0
$177$	5.78998e22	0.144197
$178$	0	0
$179$	−2.73346e23	−0.605002	−0.302501	−	0.953149i	$-0.597821\pi$
$179$	−2.73346e23	−0.605002	−0.302501	+	0.953149i	$0.597821\pi$
$180$	0	0
$181$	−4.47620e23	−0.881627	−0.440813	−	0.897599i	$-0.645310\pi$
$181$	−4.47620e23	−0.881627	−0.440813	+	0.897599i	$0.645310\pi$
$182$	0	0
$183$	−3.32206e23	−0.583004
$184$	0	0
$185$	3.12665e23	0.489527
$186$	0	0
$187$	1.72155e23	0.240759
$188$	0	0
$189$	−1.73105e23	−0.216501
$190$	0	0
$191$	−7.33413e23	−0.821293	−0.410647	−	0.911795i	$-0.634697\pi$
$191$	−7.33413e23	−0.821293	−0.410647	+	0.911795i	$0.634697\pi$
$192$	0	0
$193$	−1.35102e24	−1.35616	−0.678079	−	0.734989i	$-0.737187\pi$
$193$	−1.35102e24	−1.35616	−0.678079	+	0.734989i	$0.737187\pi$
$194$	0	0
$195$	2.69123e23	0.242430
$196$	0	0
$197$	−9.02011e23	−0.729989	−0.364994	−	0.931010i	$-0.618929\pi$
$197$	−9.02011e23	−0.729989	−0.364994	+	0.931010i	$0.618929\pi$
$198$	0	0
$199$	−1.91315e24	−1.39249	−0.696244	−	0.717806i	$-0.745147\pi$
$199$	−1.91315e24	−1.39249	−0.696244	+	0.717806i	$0.745147\pi$
$200$	0	0
$201$	1.39760e23	0.0915853
$202$	0	0
$203$	6.58238e23	0.388754
$204$	0	0
$205$	−1.06405e24	−0.566953
$206$	0	0
$207$	−8.42336e22	−0.0405319
$208$	0	0
$209$	−1.52856e24	−0.664884
$210$	0	0
$211$	4.45736e24	1.75433	0.877167	−	0.480185i	$-0.159431\pi$
$211$	4.45736e24	1.75433	0.877167	+	0.480185i	$0.159431\pi$
$212$	0	0
$213$	2.95065e24	1.05180
$214$	0	0
$215$	−1.05249e23	−0.0340105
$216$	0	0
$217$	−6.73292e24	−1.97413
$218$	0	0
$219$	−1.55434e24	−0.413887
$220$	0	0
$221$	−1.54884e24	−0.374877
$222$	0	0
$223$	8.25290e24	1.81721	0.908606	−	0.417654i	$-0.137148\pi$
$223$	8.25290e24	1.81721	0.908606	+	0.417654i	$0.137148\pi$
$224$	0	0
$225$	−8.64041e23	−0.173228
$226$	0	0
$227$	2.17673e24	0.397680	0.198840	−	0.980032i	$-0.436283\pi$
$227$	2.17673e24	0.397680	0.198840	+	0.980032i	$0.436283\pi$
$228$	0	0
$229$	2.70002e24	0.449877	0.224939	−	0.974373i	$-0.427782\pi$
$229$	2.70002e24	0.449877	0.224939	+	0.974373i	$0.427782\pi$
$230$	0	0
$231$	−1.66182e24	−0.252731
$232$	0	0
$233$	1.29388e25	1.79745	0.898725	−	0.438512i	$-0.144494\pi$
$233$	1.29388e25	1.79745	0.898725	+	0.438512i	$0.144494\pi$
$234$	0	0
$235$	3.21858e24	0.408745
$236$	0	0
$237$	6.25898e24	0.727187
$238$	0	0
$239$	−1.36280e25	−1.44962	−0.724808	−	0.688951i	$-0.758071\pi$
$239$	−1.36280e25	−1.44962	−0.724808	+	0.688951i	$0.758071\pi$
$240$	0	0
$241$	1.58532e24	0.154503	0.0772516	−	0.997012i	$-0.475386\pi$
$241$	1.58532e24	0.154503	0.0772516	+	0.997012i	$0.475386\pi$
$242$	0	0
$243$	−7.17898e23	−0.0641500
$244$	0	0
$245$	2.24478e24	0.184047
$246$	0	0
$247$	1.37521e25	1.03527
$248$	0	0
$249$	5.29377e24	0.366161
$250$	0	0
$251$	−7.10015e23	−0.0451537	−0.0225768	−	0.999745i	$-0.507187\pi$
$251$	−7.10015e23	−0.0451537	−0.0225768	+	0.999745i	$0.507187\pi$
$252$	0	0
$253$	−8.08650e23	−0.0473146
$254$	0	0
$255$	−4.59600e24	−0.247576
$256$	0	0
$257$	3.27150e25	1.62349	0.811743	−	0.584015i	$-0.198519\pi$
$257$	3.27150e25	1.62349	0.811743	+	0.584015i	$0.198519\pi$
$258$	0	0
$259$	1.73701e25	0.794611
$260$	0	0
$261$	2.72984e24	0.115189
$262$	0	0
$263$	1.81651e25	0.707461	0.353730	−	0.935347i	$-0.384913\pi$
$263$	1.81651e25	0.707461	0.353730	+	0.935347i	$0.384913\pi$
$264$	0	0
$265$	1.31582e25	0.473275
$266$	0	0
$267$	−2.78159e25	−0.924537
$268$	0	0
$269$	4.75606e25	1.46167	0.730833	−	0.682556i	$-0.239132\pi$
$269$	4.75606e25	1.46167	0.730833	+	0.682556i	$0.239132\pi$
$270$	0	0
$271$	1.10127e25	0.313123	0.156561	−	0.987668i	$-0.449959\pi$
$271$	1.10127e25	0.313123	0.156561	+	0.987668i	$0.449959\pi$
$272$	0	0
$273$	1.49511e25	0.393518
$274$	0	0
$275$	−8.29487e24	−0.202216
$276$	0	0
$277$	−4.48744e25	−1.01382	−0.506911	−	0.861999i	$-0.669213\pi$
$277$	−4.48744e25	−1.01382	−0.506911	+	0.861999i	$0.669213\pi$
$278$	0	0
$279$	−2.79227e25	−0.584942
$280$	0	0
$281$	−8.48250e24	−0.164857	−0.0824285	−	0.996597i	$-0.526268\pi$
$281$	−8.48250e24	−0.164857	−0.0824285	+	0.996597i	$0.526268\pi$
$282$	0	0
$283$	9.02788e25	1.62865	0.814326	−	0.580407i	$-0.197107\pi$
$283$	9.02788e25	1.62865	0.814326	+	0.580407i	$0.197107\pi$
$284$	0	0
$285$	4.08077e25	0.683709
$286$	0	0
$287$	−5.91132e25	−0.920290
$288$	0	0
$289$	−4.26412e25	−0.617166
$290$	0	0
$291$	2.32594e25	0.313128
$292$	0	0
$293$	−1.20579e26	−1.51065	−0.755324	−	0.655352i	$-0.772520\pi$
$293$	−1.20579e26	−1.51065	−0.755324	+	0.655352i	$0.772520\pi$
$294$	0	0
$295$	−1.48393e25	−0.173094
$296$	0	0
$297$	−6.89189e24	−0.0748851
$298$	0	0
$299$	7.27527e24	0.0736719
$300$	0	0
$301$	−5.84708e24	−0.0552065
$302$	0	0
$303$	7.50437e25	0.660947
$304$	0	0
$305$	8.51420e25	0.699835
$306$	0	0
$307$	1.33188e26	1.02215	0.511074	−	0.859537i	$-0.329248\pi$
$307$	1.33188e26	1.02215	0.511074	+	0.859537i	$0.329248\pi$
$308$	0	0
$309$	1.02885e26	0.737543
$310$	0	0
$311$	1.32644e26	0.888593	0.444296	−	0.895880i	$-0.353454\pi$
$311$	1.32644e26	0.888593	0.444296	+	0.895880i	$0.353454\pi$
$312$	0	0
$313$	−1.86750e26	−1.16962	−0.584811	−	0.811169i	$-0.698831\pi$
$313$	−1.86750e26	−1.16962	−0.584811	+	0.811169i	$0.698831\pi$
$314$	0	0
$315$	4.43655e25	0.259887
$316$	0	0
$317$	−1.99163e26	−1.09166	−0.545829	−	0.837896i	$-0.683785\pi$
$317$	−1.99163e26	−1.09166	−0.545829	+	0.837896i	$0.683785\pi$
$318$	0	0
$319$	2.62067e25	0.134465
$320$	0	0
$321$	−1.72947e26	−0.831016
$322$	0	0
$323$	−2.34855e26	−1.05724
$324$	0	0
$325$	7.46274e25	0.314864
$326$	0	0
$327$	−5.15082e24	−0.0203763
$328$	0	0
$329$	1.78808e26	0.663483
$330$	0	0
$331$	−3.61887e26	−1.26002	−0.630012	−	0.776586i	$-0.716950\pi$
$331$	−3.61887e26	−1.26002	−0.630012	+	0.776586i	$0.716950\pi$
$332$	0	0
$333$	7.20371e25	0.235446
$334$	0	0
$335$	−3.58196e25	−0.109939
$336$	0	0
$337$	−4.20174e26	−1.21148	−0.605739	−	0.795664i	$-0.707122\pi$
$337$	−4.20174e26	−1.21148	−0.605739	+	0.795664i	$0.707122\pi$
$338$	0	0
$339$	−3.30114e26	−0.894472
$340$	0	0
$341$	−2.68060e26	−0.682829
$342$	0	0
$343$	−3.44894e26	−0.826222
$344$	0	0
$345$	2.15884e25	0.0486543
$346$	0	0
$347$	−7.87467e26	−1.67022	−0.835108	−	0.550085i	$-0.814595\pi$
$347$	−7.87467e26	−1.67022	−0.835108	+	0.550085i	$0.814595\pi$
$348$	0	0
$349$	4.94479e26	0.987372	0.493686	−	0.869640i	$-0.335649\pi$
$349$	4.94479e26	0.987372	0.493686	+	0.869640i	$0.335649\pi$
$350$	0	0
$351$	6.20050e25	0.116601
$352$	0	0
$353$	−9.34606e26	−1.65575	−0.827874	−	0.560914i	$-0.810450\pi$
$353$	−9.34606e26	−1.65575	−0.827874	+	0.560914i	$0.810450\pi$
$354$	0	0
$355$	−7.56230e26	−1.26257
$356$	0	0
$357$	−2.55330e26	−0.401871
$358$	0	0
$359$	−3.08467e25	−0.0457844	−0.0228922	−	0.999738i	$-0.507287\pi$
$359$	−3.08467e25	−0.0457844	−0.0228922	+	0.999738i	$0.507287\pi$
$360$	0	0
$361$	1.37106e27	1.91969
$362$	0	0
$363$	3.70815e26	0.489934
$364$	0	0
$365$	3.98365e26	0.496828
$366$	0	0
$367$	−3.96982e26	−0.467495	−0.233747	−	0.972297i	$-0.575099\pi$
$367$	−3.96982e26	−0.467495	−0.233747	+	0.972297i	$0.575099\pi$
$368$	0	0
$369$	−2.45154e26	−0.272685
$370$	0	0
$371$	7.31001e26	0.768231
$372$	0	0
$373$	2.66510e26	0.264711	0.132355	−	0.991202i	$-0.457746\pi$
$373$	2.66510e26	0.264711	0.132355	+	0.991202i	$0.457746\pi$
$374$	0	0
$375$	6.47567e26	0.608074
$376$	0	0
$377$	−2.35777e26	−0.209371
$378$	0	0
$379$	4.17651e26	0.350834	0.175417	−	0.984494i	$-0.443873\pi$
$379$	4.17651e26	0.350834	0.175417	+	0.984494i	$0.443873\pi$
$380$	0	0
$381$	−6.99263e26	−0.555812
$382$	0	0
$383$	−2.09590e26	−0.157683	−0.0788414	−	0.996887i	$-0.525122\pi$
$383$	−2.09590e26	−0.157683	−0.0788414	+	0.996887i	$0.525122\pi$
$384$	0	0
$385$	4.25912e26	0.303377
$386$	0	0
$387$	−2.42489e25	−0.0163579
$388$	0	0
$389$	−1.25020e27	−0.798929	−0.399465	−	0.916749i	$-0.630804\pi$
$389$	−1.25020e27	−0.798929	−0.399465	+	0.916749i	$0.630804\pi$
$390$	0	0
$391$	−1.24245e26	−0.0752355
$392$	0	0
$393$	1.09358e27	0.627668
$394$	0	0
$395$	−1.60413e27	−0.872911
$396$	0	0
$397$	2.25158e27	1.16195	0.580974	−	0.813922i	$-0.302672\pi$
$397$	2.25158e27	1.16195	0.580974	+	0.813922i	$0.302672\pi$
$398$	0	0
$399$	2.26707e27	1.10981
$400$	0	0
$401$	−2.49719e27	−1.15994	−0.579971	−	0.814637i	$-0.696936\pi$
$401$	−2.49719e27	−1.15994	−0.579971	+	0.814637i	$0.696936\pi$
$402$	0	0
$403$	2.41169e27	1.06321
$404$	0	0
$405$	1.83992e26	0.0770054
$406$	0	0
$407$	6.91563e26	0.274847
$408$	0	0
$409$	7.98541e26	0.301441	0.150720	−	0.988576i	$-0.451841\pi$
$409$	7.98541e26	0.301441	0.150720	+	0.988576i	$0.451841\pi$
$410$	0	0
$411$	2.23917e27	0.803062
$412$	0	0
$413$	−8.24395e26	−0.280970
$414$	0	0
$415$	−1.35675e27	−0.439538
$416$	0	0
$417$	2.12048e26	0.0653138
$418$	0	0
$419$	6.23317e27	1.82584	0.912918	−	0.408144i	$-0.133824\pi$
$419$	6.23317e27	1.82584	0.912918	+	0.408144i	$0.133824\pi$
$420$	0	0
$421$	−1.93368e27	−0.538795	−0.269398	−	0.963029i	$-0.586825\pi$
$421$	−1.93368e27	−0.538795	−0.269398	+	0.963029i	$0.586825\pi$
$422$	0	0
$423$	7.41551e26	0.196593
$424$	0	0
$425$	−1.27447e27	−0.321547
$426$	0	0
$427$	4.73006e27	1.13599
$428$	0	0
$429$	5.95254e26	0.136113
$430$	0	0
$431$	1.38099e27	0.300732	0.150366	−	0.988630i	$-0.451955\pi$
$431$	1.38099e27	0.300732	0.150366	+	0.988630i	$0.451955\pi$
$432$	0	0
$433$	6.39470e27	1.32647	0.663235	−	0.748411i	$-0.269183\pi$
$433$	6.39470e27	1.32647	0.663235	+	0.748411i	$0.269183\pi$
$434$	0	0
$435$	−6.99638e26	−0.138273
$436$	0	0
$437$	1.10317e27	0.207772
$438$	0	0
$439$	−6.94609e27	−1.24699	−0.623495	−	0.781827i	$-0.714288\pi$
$439$	−6.94609e27	−1.24699	−0.623495	+	0.781827i	$0.714288\pi$
$440$	0	0
$441$	5.17190e26	0.0885206
$442$	0	0
$443$	−8.47333e27	−1.38298	−0.691488	−	0.722387i	$-0.743045\pi$
$443$	−8.47333e27	−1.38298	−0.691488	+	0.722387i	$0.743045\pi$
$444$	0	0
$445$	7.12900e27	1.10981
$446$	0	0
$447$	4.05194e27	0.601774
$448$	0	0
$449$	−5.62902e27	−0.797712	−0.398856	−	0.917014i	$-0.630593\pi$
$449$	−5.62902e27	−0.797712	−0.398856	+	0.917014i	$0.630593\pi$
$450$	0	0
$451$	−2.35350e27	−0.318317
$452$	0	0
$453$	3.28825e27	0.424555
$454$	0	0
$455$	−3.83185e27	−0.472378
$456$	0	0
$457$	−2.91070e26	−0.0342671	−0.0171335	−	0.999853i	$-0.505454\pi$
$457$	−2.91070e26	−0.0342671	−0.0171335	+	0.999853i	$0.505454\pi$
$458$	0	0
$459$	−1.05890e27	−0.119076
$460$	0	0
$461$	1.31214e28	1.40968	0.704840	−	0.709367i	$-0.251019\pi$
$461$	1.31214e28	1.40968	0.704840	+	0.709367i	$0.251019\pi$
$462$	0	0
$463$	1.50940e28	1.54955	0.774773	−	0.632239i	$-0.217864\pi$
$463$	1.50940e28	1.54955	0.774773	+	0.632239i	$0.217864\pi$
$464$	0	0
$465$	7.15638e27	0.702162
$466$	0	0
$467$	−8.92048e27	−0.836683	−0.418341	−	0.908290i	$-0.637388\pi$
$467$	−8.92048e27	−0.836683	−0.418341	+	0.908290i	$0.637388\pi$
$468$	0	0
$469$	−1.98995e27	−0.178455
$470$	0	0
$471$	−3.16623e27	−0.271533
$472$	0	0
$473$	−2.32792e26	−0.0190953
$474$	0	0
$475$	1.13159e28	0.887989
$476$	0	0
$477$	3.03160e27	0.227630
$478$	0	0
$479$	−1.87487e28	−1.34725	−0.673625	−	0.739073i	$-0.735264\pi$
$479$	−1.87487e28	−1.34725	−0.673625	+	0.739073i	$0.735264\pi$
$480$	0	0
$481$	−6.22186e27	−0.427954
$482$	0	0
$483$	1.19934e27	0.0789767
$484$	0	0
$485$	−5.96121e27	−0.375878
$486$	0	0
$487$	−9.66602e27	−0.583705	−0.291853	−	0.956463i	$-0.594272\pi$
$487$	−9.66602e27	−0.583705	−0.291853	+	0.956463i	$0.594272\pi$
$488$	0	0
$489$	3.17242e27	0.183505
$490$	0	0
$491$	−2.43841e28	−1.35130	−0.675648	−	0.737224i	$-0.736136\pi$
$491$	−2.43841e28	−1.35130	−0.675648	+	0.737224i	$0.736136\pi$
$492$	0	0
$493$	4.02653e27	0.213815
$494$	0	0
$495$	1.76634e27	0.0898917
$496$	0	0
$497$	−4.20123e28	−2.04943
$498$	0	0
$499$	3.32191e28	1.55357	0.776787	−	0.629764i	$-0.216848\pi$
$499$	3.32191e28	1.55357	0.776787	+	0.629764i	$0.216848\pi$
$500$	0	0
$501$	1.28885e28	0.577970
$502$	0	0
$503$	−1.02085e28	−0.439036	−0.219518	−	0.975608i	$-0.570448\pi$
$503$	−1.02085e28	−0.439036	−0.219518	+	0.975608i	$0.570448\pi$
$504$	0	0
$505$	−1.92332e28	−0.793398
$506$	0	0
$507$	9.23353e27	0.365413
$508$	0	0
$509$	−2.97174e28	−1.12843	−0.564215	−	0.825628i	$-0.690821\pi$
$509$	−2.97174e28	−1.12843	−0.564215	+	0.825628i	$0.690821\pi$
$510$	0	0
$511$	2.21311e28	0.806462
$512$	0	0
$513$	9.40197e27	0.328841
$514$	0	0
$515$	−2.63686e28	−0.885344
$516$	0	0
$517$	7.11896e27	0.229491
$518$	0	0
$519$	−1.22845e28	−0.380276
$520$	0	0
$521$	−6.31442e28	−1.87732	−0.938658	−	0.344849i	$-0.887930\pi$
$521$	−6.31442e28	−1.87732	−0.938658	+	0.344849i	$0.887930\pi$
$522$	0	0
$523$	−6.21727e27	−0.177554	−0.0887772	−	0.996052i	$-0.528296\pi$
$523$	−6.21727e27	−0.177554	−0.0887772	+	0.996052i	$0.528296\pi$
$524$	0	0
$525$	1.23025e28	0.337536
$526$	0	0
$527$	−4.11861e28	−1.08577
$528$	0	0
$529$	−3.88880e28	−0.985215
$530$	0	0
$531$	−3.41892e27	−0.0832524
$532$	0	0
$533$	2.11740e28	0.495641
$534$	0	0
$535$	4.43250e28	0.997548
$536$	0	0
$537$	1.61408e28	0.349298
$538$	0	0
$539$	4.96507e27	0.103334
$540$	0	0
$541$	−5.85756e28	−1.17259	−0.586294	−	0.810099i	$-0.699414\pi$
$541$	−5.85756e28	−1.17259	−0.586294	+	0.810099i	$0.699414\pi$
$542$	0	0
$543$	2.64315e28	0.509008
$544$	0	0
$545$	1.32012e27	0.0244596
$546$	0	0
$547$	−2.21556e27	−0.0395017	−0.0197509	−	0.999805i	$-0.506287\pi$
$547$	−2.21556e27	−0.0395017	−0.0197509	+	0.999805i	$0.506287\pi$
$548$	0	0
$549$	1.96164e28	0.336597
$550$	0	0
$551$	−3.57514e28	−0.590475
$552$	0	0
$553$	−8.91173e28	−1.41693
$554$	0	0
$555$	−1.84626e28	−0.282629
$556$	0	0
$557$	5.45401e27	0.0803964	0.0401982	−	0.999192i	$-0.487201\pi$
$557$	5.45401e27	0.0803964	0.0401982	+	0.999192i	$0.487201\pi$
$558$	0	0
$559$	2.09439e27	0.0297326
$560$	0	0
$561$	−1.01656e28	−0.139002
$562$	0	0
$563$	−8.27033e27	−0.108939	−0.0544696	−	0.998515i	$-0.517347\pi$
$563$	−8.27033e27	−0.108939	−0.0544696	+	0.998515i	$0.517347\pi$
$564$	0	0
$565$	8.46057e28	1.07372
$566$	0	0
$567$	1.02217e28	0.124997
$568$	0	0
$569$	8.21967e28	0.968669	0.484335	−	0.874883i	$-0.339062\pi$
$569$	8.21967e28	0.968669	0.484335	+	0.874883i	$0.339062\pi$
$570$	0	0
$571$	−6.72222e28	−0.763544	−0.381772	−	0.924257i	$-0.624686\pi$
$571$	−6.72222e28	−0.763544	−0.381772	+	0.924257i	$0.624686\pi$
$572$	0	0
$573$	4.33073e28	0.474174
$574$	0	0
$575$	5.98645e27	0.0631912
$576$	0	0
$577$	1.51324e29	1.54015	0.770075	−	0.637953i	$-0.220219\pi$
$577$	1.51324e29	1.54015	0.770075	+	0.637953i	$0.220219\pi$
$578$	0	0
$579$	7.97764e28	0.782978
$580$	0	0
$581$	−7.53744e28	−0.713468
$582$	0	0
$583$	2.91036e28	0.265722
$584$	0	0
$585$	−1.58914e28	−0.139967
$586$	0	0
$587$	−7.61361e28	−0.646980	−0.323490	−	0.946232i	$-0.604856\pi$
$587$	−7.61361e28	−0.646980	−0.323490	+	0.946232i	$0.604856\pi$
$588$	0	0
$589$	3.65690e29	2.99849
$590$	0	0
$591$	5.32628e28	0.421459
$592$	0	0
$593$	6.44095e28	0.491899	0.245950	−	0.969283i	$-0.420900\pi$
$593$	6.44095e28	0.491899	0.245950	+	0.969283i	$0.420900\pi$
$594$	0	0
$595$	6.54393e28	0.482404
$596$	0	0
$597$	1.12969e29	0.803953
$598$	0	0
$599$	−5.13552e28	−0.352860	−0.176430	−	0.984313i	$-0.556455\pi$
$599$	−5.13552e28	−0.352860	−0.176430	+	0.984313i	$0.556455\pi$
$600$	0	0
$601$	−1.12674e29	−0.747554	−0.373777	−	0.927519i	$-0.621938\pi$
$601$	−1.12674e29	−0.747554	−0.373777	+	0.927519i	$0.621938\pi$
$602$	0	0
$603$	−8.25271e27	−0.0528768
$604$	0	0
$605$	−9.50371e28	−0.588114
$606$	0	0
$607$	1.15513e29	0.690475	0.345238	−	0.938515i	$-0.387798\pi$
$607$	1.15513e29	0.690475	0.345238	+	0.938515i	$0.387798\pi$
$608$	0	0
$609$	−3.88683e28	−0.224447
$610$	0	0
$611$	−6.40479e28	−0.357332
$612$	0	0
$613$	2.36968e29	1.27748	0.638742	−	0.769421i	$-0.279455\pi$
$613$	2.36968e29	1.27748	0.638742	+	0.769421i	$0.279455\pi$
$614$	0	0
$615$	6.28311e28	0.327330
$616$	0	0
$617$	−2.69051e29	−1.35469	−0.677346	−	0.735664i	$-0.736870\pi$
$617$	−2.69051e29	−1.35469	−0.677346	+	0.735664i	$0.736870\pi$
$618$	0	0
$619$	−2.88261e29	−1.40292	−0.701461	−	0.712708i	$-0.747468\pi$
$619$	−2.88261e29	−1.40292	−0.701461	+	0.712708i	$0.747468\pi$
$620$	0	0
$621$	4.97391e27	0.0234011
$622$	0	0
$623$	3.96051e29	1.80147
$624$	0	0
$625$	−4.78042e28	−0.210245
$626$	0	0
$627$	9.02598e28	0.383871
$628$	0	0
$629$	1.06255e29	0.437037
$630$	0	0
$631$	1.60612e29	0.638953	0.319477	−	0.947594i	$-0.396493\pi$
$631$	1.60612e29	0.638953	0.319477	+	0.947594i	$0.396493\pi$
$632$	0	0
$633$	−2.63203e29	−1.01287
$634$	0	0
$635$	1.79216e29	0.667194
$636$	0	0
$637$	−4.46698e28	−0.160898
$638$	0	0
$639$	−1.74233e29	−0.607255
$640$	0	0
$641$	−5.00919e29	−1.68950	−0.844749	−	0.535162i	$-0.820250\pi$
$641$	−5.00919e29	−1.68950	−0.844749	+	0.535162i	$0.820250\pi$
$642$	0	0
$643$	2.11297e29	0.689728	0.344864	−	0.938653i	$-0.387925\pi$
$643$	2.11297e29	0.689728	0.344864	+	0.938653i	$0.387925\pi$
$644$	0	0
$645$	6.21483e27	0.0196359
$646$	0	0
$647$	−4.66535e29	−1.42689	−0.713443	−	0.700713i	$-0.752865\pi$
$647$	−4.66535e29	−1.42689	−0.713443	+	0.700713i	$0.752865\pi$
$648$	0	0
$649$	−3.28220e28	−0.0971842
$650$	0	0
$651$	3.97572e29	1.13977
$652$	0	0
$653$	6.87022e29	1.90714	0.953569	−	0.301175i	$-0.0973787\pi$
$653$	6.87022e29	1.90714	0.953569	+	0.301175i	$0.0973787\pi$
$654$	0	0
$655$	−2.80277e29	−0.753450
$656$	0	0
$657$	9.17820e28	0.238958
$658$	0	0
$659$	5.83008e29	1.47021	0.735103	−	0.677956i	$-0.237134\pi$
$659$	5.83008e29	1.47021	0.735103	+	0.677956i	$0.237134\pi$
$660$	0	0
$661$	7.08881e29	1.73164	0.865820	−	0.500355i	$-0.166797\pi$
$661$	7.08881e29	1.73164	0.865820	+	0.500355i	$0.166797\pi$
$662$	0	0
$663$	9.14577e28	0.216435
$664$	0	0
$665$	−5.81033e29	−1.33221
$666$	0	0
$667$	−1.89135e28	−0.0420195
$668$	0	0
$669$	−4.87326e29	−1.04917
$670$	0	0
$671$	1.88320e29	0.392925
$672$	0	0
$673$	−2.18749e29	−0.442373	−0.221186	−	0.975232i	$-0.570993\pi$
$673$	−2.18749e29	−0.442373	−0.221186	+	0.975232i	$0.570993\pi$
$674$	0	0
$675$	5.10207e28	0.100013
$676$	0	0
$677$	4.19570e29	0.797302	0.398651	−	0.917103i	$-0.369478\pi$
$677$	4.19570e29	0.797302	0.398651	+	0.917103i	$0.369478\pi$
$678$	0	0
$679$	−3.31175e29	−0.610133
$680$	0	0
$681$	−1.28534e29	−0.229601
$682$	0	0
$683$	4.62694e29	0.801450	0.400725	−	0.916198i	$-0.368758\pi$
$683$	4.62694e29	0.801450	0.400725	+	0.916198i	$0.368758\pi$
$684$	0	0
$685$	−5.73884e29	−0.963991
$686$	0	0
$687$	−1.59433e29	−0.259737
$688$	0	0
$689$	−2.61840e29	−0.413746
$690$	0	0
$691$	−7.05176e29	−1.08088	−0.540441	−	0.841382i	$-0.681743\pi$
$691$	−7.05176e29	−1.08088	−0.540441	+	0.841382i	$0.681743\pi$
$692$	0	0
$693$	9.81289e28	0.145914
$694$	0	0
$695$	−5.43462e28	−0.0784023
$696$	0	0
$697$	−3.61603e29	−0.506160
$698$	0	0
$699$	−7.64024e29	−1.03776
$700$	0	0
$701$	7.25728e29	0.956609	0.478305	−	0.878194i	$-0.341252\pi$
$701$	7.25728e29	0.956609	0.478305	+	0.878194i	$0.341252\pi$
$702$	0	0
$703$	−9.43435e29	−1.20693
$704$	0	0
$705$	−1.90054e29	−0.235989
$706$	0	0
$707$	−1.06850e30	−1.28786
$708$	0	0
$709$	−4.71108e29	−0.551233	−0.275616	−	0.961268i	$-0.588882\pi$
$709$	−4.71108e29	−0.551233	−0.275616	+	0.961268i	$0.588882\pi$
$710$	0	0
$711$	−3.69586e29	−0.419841
$712$	0	0
$713$	1.93460e29	0.213379
$714$	0	0
$715$	−1.52559e29	−0.163390
$716$	0	0
$717$	8.04717e29	0.836936
$718$	0	0
$719$	−5.11299e29	−0.516442	−0.258221	−	0.966086i	$-0.583136\pi$
$719$	−5.11299e29	−0.516442	−0.258221	+	0.966086i	$0.583136\pi$
$720$	0	0
$721$	−1.46491e30	−1.43711
$722$	0	0
$723$	−9.36115e28	−0.0892025
$724$	0	0
$725$	−1.94009e29	−0.179586
$726$	0	0
$727$	−2.09585e29	−0.188473	−0.0942366	−	0.995550i	$-0.530041\pi$
$727$	−2.09585e29	−0.188473	−0.0942366	+	0.995550i	$0.530041\pi$
$728$	0	0
$729$	4.23912e28	0.0370370
$730$	0	0
$731$	−3.57673e28	−0.0303637
$732$	0	0
$733$	2.06298e30	1.70178	0.850890	−	0.525345i	$-0.176064\pi$
$733$	2.06298e30	1.70178	0.850890	+	0.525345i	$0.176064\pi$
$734$	0	0
$735$	−1.32552e29	−0.106260
$736$	0	0
$737$	−7.92268e28	−0.0617253
$738$	0	0
$739$	8.76008e28	0.0663348	0.0331674	−	0.999450i	$-0.489441\pi$
$739$	8.76008e28	0.0663348	0.0331674	+	0.999450i	$0.489441\pi$
$740$	0	0
$741$	−8.12050e29	−0.597711
$742$	0	0
$743$	5.61196e29	0.401543	0.200771	−	0.979638i	$-0.435655\pi$
$743$	5.61196e29	0.401543	0.200771	+	0.979638i	$0.435655\pi$
$744$	0	0
$745$	−1.03848e30	−0.722367
$746$	0	0
$747$	−3.12592e29	−0.211403
$748$	0	0
$749$	2.46247e30	1.61924
$750$	0	0
$751$	1.05895e30	0.677106	0.338553	−	0.940947i	$-0.390063\pi$
$751$	1.05895e30	0.677106	0.338553	+	0.940947i	$0.390063\pi$
$752$	0	0
$753$	4.19256e28	0.0260695
$754$	0	0
$755$	−8.42755e29	−0.509634
$756$	0	0
$757$	−1.84921e30	−1.08763	−0.543813	−	0.839206i	$-0.683020\pi$
$757$	−1.84921e30	−1.08763	−0.543813	+	0.839206i	$0.683020\pi$
$758$	0	0
$759$	4.77500e28	0.0273171
$760$	0	0
$761$	−4.53605e29	−0.252429	−0.126214	−	0.992003i	$-0.540283\pi$
$761$	−4.53605e29	−0.252429	−0.126214	+	0.992003i	$0.540283\pi$
$762$	0	0
$763$	7.33390e28	0.0397034
$764$	0	0
$765$	2.71389e29	0.142938
$766$	0	0
$767$	2.95293e29	0.151322
$768$	0	0
$769$	−1.50798e30	−0.751916	−0.375958	−	0.926637i	$-0.622686\pi$
$769$	−1.50798e30	−0.751916	−0.375958	+	0.926637i	$0.622686\pi$
$770$	0	0
$771$	−1.93179e30	−0.937320
$772$	0	0
$773$	5.10001e29	0.240817	0.120408	−	0.992724i	$-0.461580\pi$
$773$	5.10001e29	0.240817	0.120408	+	0.992724i	$0.461580\pi$
$774$	0	0
$775$	1.98446e30	0.911955
$776$	0	0
$777$	−1.02569e30	−0.458769
$778$	0	0
$779$	3.21066e30	1.39782
$780$	0	0
$781$	−1.67265e30	−0.708875
$782$	0	0
$783$	−1.61194e29	−0.0665045
$784$	0	0
$785$	8.11482e29	0.325947
$786$	0	0
$787$	3.76408e30	1.47205	0.736027	−	0.676952i	$-0.236700\pi$
$787$	3.76408e30	1.47205	0.736027	+	0.676952i	$0.236700\pi$
$788$	0	0
$789$	−1.07263e30	−0.408453
$790$	0	0
$791$	4.70026e30	1.74289
$792$	0	0
$793$	−1.69428e30	−0.611809
$794$	0	0
$795$	−7.76977e29	−0.273246
$796$	0	0
$797$	5.68916e30	1.94866	0.974329	−	0.225129i	$-0.0722805\pi$
$797$	5.68916e30	1.94866	0.974329	+	0.225129i	$0.0722805\pi$
$798$	0	0
$799$	1.09379e30	0.364916
$800$	0	0
$801$	1.64250e30	0.533782
$802$	0	0
$803$	8.81116e29	0.278946
$804$	0	0
$805$	−3.07383e29	−0.0948032
$806$	0	0
$807$	−2.80840e30	−0.843893
$808$	0	0
$809$	−3.33803e29	−0.0977306	−0.0488653	−	0.998805i	$-0.515561\pi$
$809$	−3.33803e29	−0.0977306	−0.0488653	+	0.998805i	$0.515561\pi$
$810$	0	0
$811$	2.91268e30	0.830949	0.415474	−	0.909605i	$-0.363616\pi$
$811$	2.91268e30	0.830949	0.415474	+	0.909605i	$0.363616\pi$
$812$	0	0
$813$	−6.50287e29	−0.180782
$814$	0	0
$815$	−8.13069e29	−0.220279
$816$	0	0
$817$	3.17577e29	0.0838527
$818$	0	0
$819$	−8.82847e29	−0.227198
$820$	0	0
$821$	4.82227e30	1.20962	0.604809	−	0.796370i	$-0.293249\pi$
$821$	4.82227e30	1.20962	0.604809	+	0.796370i	$0.293249\pi$
$822$	0	0
$823$	−2.55917e30	−0.625751	−0.312875	−	0.949794i	$-0.601292\pi$
$823$	−2.55917e30	−0.625751	−0.312875	+	0.949794i	$0.601292\pi$
$824$	0	0
$825$	4.89804e29	0.116750
$826$	0	0
$827$	1.93866e30	0.450498	0.225249	−	0.974301i	$-0.427680\pi$
$827$	1.93866e30	0.450498	0.225249	+	0.974301i	$0.427680\pi$
$828$	0	0
$829$	7.77902e30	1.76239	0.881195	−	0.472753i	$-0.156740\pi$
$829$	7.77902e30	1.76239	0.881195	+	0.472753i	$0.156740\pi$
$830$	0	0
$831$	2.64979e30	0.585330
$832$	0	0
$833$	7.62858e29	0.164313
$834$	0	0
$835$	−3.30322e30	−0.693792
$836$	0	0
$837$	1.64881e30	0.337717
$838$	0	0
$839$	6.16703e30	1.23190	0.615950	−	0.787785i	$-0.288772\pi$
$839$	6.16703e30	1.23190	0.615950	+	0.787785i	$0.288772\pi$
$840$	0	0
$841$	−4.51989e30	−0.880583
$842$	0	0
$843$	5.00883e29	0.0951803
$844$	0	0
$845$	−2.36649e30	−0.438640
$846$	0	0
$847$	−5.27977e30	−0.954640
$848$	0	0
$849$	−5.33088e30	−0.940303
$850$	0	0
$851$	−4.99104e29	−0.0858876
$852$	0	0
$853$	−1.15935e30	−0.194648	−0.0973239	−	0.995253i	$-0.531028\pi$
$853$	−1.15935e30	−0.194648	−0.0973239	+	0.995253i	$0.531028\pi$
$854$	0	0
$855$	−2.40966e30	−0.394740
$856$	0	0
$857$	6.59419e30	1.05405	0.527027	−	0.849848i	$-0.323307\pi$
$857$	6.59419e30	1.05405	0.527027	+	0.849848i	$0.323307\pi$
$858$	0	0
$859$	−6.07395e30	−0.947421	−0.473710	−	0.880681i	$-0.657086\pi$
$859$	−6.07395e30	−0.947421	−0.473710	+	0.880681i	$0.657086\pi$
$860$	0	0
$861$	3.49058e30	0.531330
$862$	0	0
$863$	−4.55774e30	−0.677075	−0.338537	−	0.940953i	$-0.609932\pi$
$863$	−4.55774e30	−0.677075	−0.338537	+	0.940953i	$0.609932\pi$
$864$	0	0
$865$	3.14842e30	0.456481
$866$	0	0
$867$	2.51792e30	0.356321
$868$	0	0
$869$	−3.54806e30	−0.490099
$870$	0	0
$871$	7.12788e29	0.0961103
$872$	0	0
$873$	−1.37344e30	−0.180785
$874$	0	0
$875$	−9.22026e30	−1.18484
$876$	0	0
$877$	−6.30080e29	−0.0790496	−0.0395248	−	0.999219i	$-0.512584\pi$
$877$	−6.30080e29	−0.0790496	−0.0395248	+	0.999219i	$0.512584\pi$
$878$	0	0
$879$	7.12010e30	0.872173
$880$	0	0
$881$	4.36026e30	0.521513	0.260757	−	0.965405i	$-0.416028\pi$
$881$	4.36026e30	0.521513	0.260757	+	0.965405i	$0.416028\pi$
$882$	0	0
$883$	−3.58988e30	−0.419269	−0.209635	−	0.977780i	$-0.567227\pi$
$883$	−3.58988e30	−0.419269	−0.209635	+	0.977780i	$0.567227\pi$
$884$	0	0
$885$	8.76245e29	0.0999358
$886$	0	0
$887$	1.18282e30	0.131741	0.0658707	−	0.997828i	$-0.479018\pi$
$887$	1.18282e30	0.131741	0.0658707	+	0.997828i	$0.479018\pi$
$888$	0	0
$889$	9.95632e30	1.08300
$890$	0	0
$891$	4.06959e29	0.0432349
$892$	0	0
$893$	−9.71174e30	−1.00776
$894$	0	0
$895$	−4.13678e30	−0.419295
$896$	0	0
$897$	−4.29597e29	−0.0425345
$898$	0	0
$899$	−6.26966e30	−0.606411
$900$	0	0
$901$	4.47163e30	0.422528
$902$	0	0
$903$	3.45264e29	0.0318735
$904$	0	0
$905$	−6.77421e30	−0.611010
$906$	0	0
$907$	8.29044e30	0.730636	0.365318	−	0.930883i	$-0.380960\pi$
$907$	8.29044e30	0.730636	0.365318	+	0.930883i	$0.380960\pi$
$908$	0	0
$909$	−4.43126e30	−0.381598
$910$	0	0
$911$	6.67910e30	0.562050	0.281025	−	0.959700i	$-0.409326\pi$
$911$	6.67910e30	0.562050	0.281025	+	0.959700i	$0.409326\pi$
$912$	0	0
$913$	−3.00091e30	−0.246780
$914$	0	0
$915$	−5.02755e30	−0.404050
$916$	0	0
$917$	−1.55708e31	−1.22302
$918$	0	0
$919$	−1.36155e31	−1.04525	−0.522626	−	0.852562i	$-0.675048\pi$
$919$	−1.36155e31	−1.04525	−0.522626	+	0.852562i	$0.675048\pi$
$920$	0	0
$921$	−7.86464e30	−0.590137
$922$	0	0
$923$	1.50485e31	1.10376
$924$	0	0
$925$	−5.11965e30	−0.367073
$926$	0	0
$927$	−6.07525e30	−0.425821
$928$	0	0
$929$	6.88409e30	0.471717	0.235858	−	0.971787i	$-0.424210\pi$
$929$	6.88409e30	0.471717	0.235858	+	0.971787i	$0.424210\pi$
$930$	0	0
$931$	−6.77338e30	−0.453768
$932$	0	0
$933$	−7.83248e30	−0.513029
$934$	0	0
$935$	2.60536e30	0.166858
$936$	0	0
$937$	−5.51478e30	−0.345352	−0.172676	−	0.984979i	$-0.555241\pi$
$937$	−5.51478e30	−0.345352	−0.172676	+	0.984979i	$0.555241\pi$
$938$	0	0
$939$	1.10274e31	0.675282
$940$	0	0
$941$	−2.31900e31	−1.38870	−0.694351	−	0.719636i	$-0.744309\pi$
$941$	−2.31900e31	−1.38870	−0.694351	+	0.719636i	$0.744309\pi$
$942$	0	0
$943$	1.69853e30	0.0994720
$944$	0	0
$945$	−2.61974e30	−0.150046
$946$	0	0
$947$	2.20661e31	1.23609	0.618047	−	0.786141i	$-0.287924\pi$
$947$	2.20661e31	1.23609	0.618047	+	0.786141i	$0.287924\pi$
$948$	0	0
$949$	−7.92723e30	−0.434336
$950$	0	0
$951$	1.17604e31	0.630269
$952$	0	0
$953$	−2.49350e31	−1.30718	−0.653588	−	0.756850i	$-0.726737\pi$
$953$	−2.49350e31	−1.30718	−0.653588	+	0.756850i	$0.726737\pi$
$954$	0	0
$955$	−1.10993e31	−0.569196
$956$	0	0
$957$	−1.54748e30	−0.0776336
$958$	0	0
$959$	−3.18821e31	−1.56477
$960$	0	0
$961$	4.33050e31	2.07942
$962$	0	0
$963$	1.02123e31	0.479787
$964$	0	0
$965$	−2.04461e31	−0.939883
$966$	0	0
$967$	4.48307e30	0.201650	0.100825	−	0.994904i	$-0.467852\pi$
$967$	4.48307e30	0.201650	0.100825	+	0.994904i	$0.467852\pi$
$968$	0	0
$969$	1.38680e31	0.610398
$970$	0	0
$971$	−2.61136e31	−1.12477	−0.562385	−	0.826875i	$-0.690116\pi$
$971$	−2.61136e31	−1.12477	−0.562385	+	0.826875i	$0.690116\pi$
$972$	0	0
$973$	−3.01920e30	−0.127264
$974$	0	0
$975$	−4.40667e30	−0.181787
$976$	0	0
$977$	2.42220e31	0.977949	0.488975	−	0.872298i	$-0.337371\pi$
$977$	2.42220e31	0.977949	0.488975	+	0.872298i	$0.337371\pi$
$978$	0	0
$979$	1.57681e31	0.623107
$980$	0	0
$981$	3.04151e29	0.0117643
$982$	0	0
$983$	−2.19828e31	−0.832282	−0.416141	−	0.909300i	$-0.636618\pi$
$983$	−2.19828e31	−0.832282	−0.416141	+	0.909300i	$0.636618\pi$
$984$	0	0
$985$	−1.36509e31	−0.505918
$986$	0	0
$987$	−1.05584e31	−0.383062
$988$	0	0
$989$	1.68007e29	0.00596715
$990$	0	0
$991$	1.58925e31	0.552609	0.276304	−	0.961070i	$-0.410890\pi$
$991$	1.58925e31	0.552609	0.276304	+	0.961070i	$0.410890\pi$
$992$	0	0
$993$	2.13690e31	0.727475
$994$	0	0
$995$	−2.89532e31	−0.965061
$996$	0	0
$997$	−3.74264e31	−1.22146	−0.610729	−	0.791840i	$-0.709123\pi$
$997$	−3.74264e31	−1.22146	−0.610729	+	0.791840i	$0.709123\pi$
$998$	0	0
$999$	−4.25372e30	−0.135935

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.22.a.j.1.2		3
4.3	odd	2		24.22.a.d.1.2	✓	3
12.11	even	2		72.22.a.c.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.22.a.d.1.2	✓	3	4.3	odd	2
48.22.a.j.1.2		3	1.1	even	1	trivial
72.22.a.c.1.2		3	12.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

\(f(q)\)	\(=\)	\(q-59049.0 q^{3} +1.51338e7 q^{5} +8.40758e8 q^{7} +3.48678e9 q^{9} +3.34735e10 q^{11} -3.01154e11 q^{13} -8.93637e11 q^{15} +5.14303e12 q^{17} -4.56647e13 q^{19} -4.96459e13 q^{21} -2.41579e13 q^{23} -2.47804e14 q^{25} -2.05891e14 q^{27} +7.82910e14 q^{29} -8.00815e15 q^{31} -1.97657e15 q^{33} +1.27239e16 q^{35} +2.06600e16 q^{37} +1.77829e16 q^{39} -7.03094e16 q^{41} -6.95453e15 q^{43} +5.27684e16 q^{45} +2.12675e17 q^{47} +1.48329e17 q^{49} -3.03691e17 q^{51} +8.69454e17 q^{53} +5.06581e17 q^{55} +2.69646e18 q^{57} -9.80538e17 q^{59} +5.62594e18 q^{61} +2.93154e18 q^{63} -4.55762e18 q^{65} -2.36685e18 q^{67} +1.42650e18 q^{69} -4.99695e19 q^{71} +2.63228e19 q^{73} +1.46326e19 q^{75} +2.81431e19 q^{77} -1.05996e20 q^{79} +1.21577e19 q^{81} -8.96505e19 q^{83} +7.78337e19 q^{85} -4.62301e19 q^{87} +4.71064e20 q^{89} -2.53198e20 q^{91} +4.72873e20 q^{93} -6.91082e20 q^{95} -3.93900e20 q^{97} +1.16715e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 177147 q^{3} + 5280498 q^{5} - 852542376 q^{7} + 10460353203 q^{9} - 62490757668 q^{11} + 203765207802 q^{13} - 311808126402 q^{15} + 695827819926 q^{17} - 4955504123196 q^{19} + 50341774760424 q^{21}+ \cdots - 21\!\cdots\!68 q^{99}+O(q^{100}) \) 3 * q - 177147 * q^3 + 5280498 * q^5 - 852542376 * q^7 + 10460353203 * q^9 - 62490757668 * q^11 + 203765207802 * q^13 - 311808126402 * q^15 + 695827819926 * q^17 - 4955504123196 * q^19 + 50341774760424 * q^21 - 150867407938152 * q^23 + 678194854969869 * q^25 - 617673396283947 * q^27 + 3242224905226746 * q^29 - 652508601550896 * q^31 + 3690016749537732 * q^33 - 15443719285839600 * q^35 - 20693271836916222 * q^37 - 12032131755500298 * q^39 - 47608600312552002 * q^41 + 277110211230415548 * q^43 + 18411958055911698 * q^45 + 535942507326221328 * q^47 + 1192822454236615755 * q^49 - 41087936938810374 * q^51 + 2849297892348987426 * q^53 - 2255425539904757592 * q^55 + 292617562970600604 * q^57 - 881046377434726548 * q^59 + 10620890455615553322 * q^61 - 2972631457828276776 * q^63 + 24923333196516863004 * q^65 - 14840080728867715116 * q^67 + 8908569571339937448 * q^69 - 14608944838311941496 * q^71 + 13575086263949291454 * q^73 - 40046727991115794581 * q^75 + 173887987291330337760 * q^77 - 278096577517755548160 * q^79 + 36472996377170786403 * q^81 - 585716918283456830460 * q^83 + 442542794823708598596 * q^85 - 191450138428734124554 * q^87 + 477818701792662526830 * q^89 - 1318091663263676539056 * q^91 + 38529980412978857904 * q^93 - 698400977547585407208 * q^95 + 722411740013721590502 * q^97 - 217891799043453536868 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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