LMFDB - Embedded newform 384.10.a.a.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,10,Mod(1,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("384.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	384.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:	$197.773761087$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 14124x^{2} - 170336x + 18391464 $ x^4 - 14124x^2 - 170336x + 18391464
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-47.1038$ of defining polynomial
Character	$\chi$	$=$	384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-81.0000 q^{3} +1350.54 q^{5} +4629.00 q^{7} +6561.00 q^{9} +O(q^{10})$	$q-81.0000 q^{3} +1350.54 q^{5} +4629.00 q^{7} +6561.00 q^{9} -35546.7 q^{11} +48411.4 q^{13} -109394. q^{15} -130374. q^{17} +561717. q^{19} -374949. q^{21} +400621. q^{23} -129157. q^{25} -531441. q^{27} -3.25076e6 q^{29} +379535. q^{31} +2.87928e6 q^{33} +6.25167e6 q^{35} -1.76862e7 q^{37} -3.92133e6 q^{39} +4.50782e6 q^{41} -2.91165e7 q^{43} +8.86092e6 q^{45} -3.36056e7 q^{47} -1.89260e7 q^{49} +1.05603e7 q^{51} -5.68080e6 q^{53} -4.80074e7 q^{55} -4.54991e7 q^{57} +8.79057e7 q^{59} -1.07388e8 q^{61} +3.03709e7 q^{63} +6.53817e7 q^{65} +1.95661e8 q^{67} -3.24503e7 q^{69} -4.89316e7 q^{71} -1.62014e8 q^{73} +1.04617e7 q^{75} -1.64546e8 q^{77} +1.10329e8 q^{79} +4.30467e7 q^{81} +2.60270e8 q^{83} -1.76076e8 q^{85} +2.63311e8 q^{87} +1.44221e8 q^{89} +2.24096e8 q^{91} -3.07423e7 q^{93} +7.58623e8 q^{95} +5.59756e8 q^{97} -2.33222e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 324 q^{3} - 1728 q^{5} + 4840 q^{7} + 26244 q^{9}+O(q^{10}) $ 4 * q - 324 * q^3 - 1728 * q^5 + 4840 * q^7 + 26244 * q^9	$ 4 q - 324 q^{3} - 1728 q^{5} + 4840 q^{7} + 26244 q^{9} - 15824 q^{11} - 82440 q^{13} + 139968 q^{15} + 165912 q^{17} - 539904 q^{19} - 392040 q^{21} + 729680 q^{23} + 224812 q^{25} - 2125764 q^{27} - 1850864 q^{29} + 3197960 q^{31} + 1281744 q^{33} - 8574912 q^{35} - 4187992 q^{37} + 6677640 q^{39} + 227704 q^{41} - 1600352 q^{43} - 11337408 q^{45} + 18053904 q^{47} + 57728820 q^{49} - 13438872 q^{51} - 29418288 q^{53} + 45906816 q^{55} + 43732224 q^{57} - 38300048 q^{59} + 99764648 q^{61} + 31755240 q^{63} + 314120832 q^{65} + 183717008 q^{67} - 59104080 q^{69} + 181868080 q^{71} + 254539160 q^{73} - 18209772 q^{75} + 230564704 q^{77} + 831578184 q^{79} + 172186884 q^{81} + 687923952 q^{83} - 1041391104 q^{85} + 149919984 q^{87} + 627627272 q^{89} + 1018147632 q^{91} - 259034760 q^{93} + 2167118208 q^{95} - 889385880 q^{97} - 103821264 q^{99}+O(q^{100}) $ 4 * q - 324 * q^3 - 1728 * q^5 + 4840 * q^7 + 26244 * q^9 - 15824 * q^11 - 82440 * q^13 + 139968 * q^15 + 165912 * q^17 - 539904 * q^19 - 392040 * q^21 + 729680 * q^23 + 224812 * q^25 - 2125764 * q^27 - 1850864 * q^29 + 3197960 * q^31 + 1281744 * q^33 - 8574912 * q^35 - 4187992 * q^37 + 6677640 * q^39 + 227704 * q^41 - 1600352 * q^43 - 11337408 * q^45 + 18053904 * q^47 + 57728820 * q^49 - 13438872 * q^51 - 29418288 * q^53 + 45906816 * q^55 + 43732224 * q^57 - 38300048 * q^59 + 99764648 * q^61 + 31755240 * q^63 + 314120832 * q^65 + 183717008 * q^67 - 59104080 * q^69 + 181868080 * q^71 + 254539160 * q^73 - 18209772 * q^75 + 230564704 * q^77 + 831578184 * q^79 + 172186884 * q^81 + 687923952 * q^83 - 1041391104 * q^85 + 149919984 * q^87 + 627627272 * q^89 + 1018147632 * q^91 - 259034760 * q^93 + 2167118208 * q^95 - 889385880 * q^97 - 103821264 * 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
$3$	−81.0000	−0.577350
$3$	−81.0000	−0.577350
$4$	0	0
$5$	1350.54	0.966370	0.483185	−	0.875518i	$-0.339480\pi$
$5$	1350.54	0.966370	0.483185	+	0.875518i	$0.339480\pi$
$6$	0	0
$7$	4629.00	0.728695	0.364348	−	0.931263i	$-0.381292\pi$
$7$	4629.00	0.728695	0.364348	+	0.931263i	$0.381292\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	−35546.7	−0.732036	−0.366018	−	0.930608i	$-0.619279\pi$
$11$	−35546.7	−0.732036	−0.366018	+	0.930608i	$0.619279\pi$
$12$	0	0
$13$	48411.4	0.470114	0.235057	−	0.971982i	$-0.424472\pi$
$13$	48411.4	0.470114	0.235057	+	0.971982i	$0.424472\pi$
$14$	0	0
$15$	−109394.	−0.557934
$16$	0	0
$17$	−130374.	−0.378591	−0.189296	−	0.981920i	$-0.560620\pi$
$17$	−130374.	−0.378591	−0.189296	+	0.981920i	$0.560620\pi$
$18$	0	0
$19$	561717.	0.988841	0.494420	−	0.869223i	$-0.335380\pi$
$19$	561717.	0.988841	0.494420	+	0.869223i	$0.335380\pi$
$20$	0	0
$21$	−374949.	−0.420712
$22$	0	0
$23$	400621.	0.298510	0.149255	−	0.988799i	$-0.452313\pi$
$23$	400621.	0.298510	0.149255	+	0.988799i	$0.452313\pi$
$24$	0	0
$25$	−129157.	−0.0661285
$26$	0	0
$27$	−531441.	−0.192450
$28$	0	0
$29$	−3.25076e6	−0.853481	−0.426740	−	0.904374i	$-0.640338\pi$
$29$	−3.25076e6	−0.853481	−0.426740	+	0.904374i	$0.640338\pi$
$30$	0	0
$31$	379535.	0.0738115	0.0369057	−	0.999319i	$-0.488250\pi$
$31$	379535.	0.0738115	0.0369057	+	0.999319i	$0.488250\pi$
$32$	0	0
$33$	2.87928e6	0.422641
$34$	0	0
$35$	6.25167e6	0.704190
$36$	0	0
$37$	−1.76862e7	−1.55141	−0.775707	−	0.631093i	$-0.782606\pi$
$37$	−1.76862e7	−1.55141	−0.775707	+	0.631093i	$0.782606\pi$
$38$	0	0
$39$	−3.92133e6	−0.271420
$40$	0	0
$41$	4.50782e6	0.249138	0.124569	−	0.992211i	$-0.460245\pi$
$41$	4.50782e6	0.249138	0.124569	+	0.992211i	$0.460245\pi$
$42$	0	0
$43$	−2.91165e7	−1.29877	−0.649384	−	0.760460i	$-0.724973\pi$
$43$	−2.91165e7	−1.29877	−0.649384	+	0.760460i	$0.724973\pi$
$44$	0	0
$45$	8.86092e6	0.322123
$46$	0	0
$47$	−3.36056e7	−1.00455	−0.502275	−	0.864708i	$-0.667504\pi$
$47$	−3.36056e7	−1.00455	−0.502275	+	0.864708i	$0.667504\pi$
$48$	0	0
$49$	−1.89260e7	−0.469003
$50$	0	0
$51$	1.05603e7	0.218580
$52$	0	0
$53$	−5.68080e6	−0.0988935	−0.0494468	−	0.998777i	$-0.515746\pi$
$53$	−5.68080e6	−0.0988935	−0.0494468	+	0.998777i	$0.515746\pi$
$54$	0	0
$55$	−4.80074e7	−0.707418
$56$	0	0
$57$	−4.54991e7	−0.570907
$58$	0	0
$59$	8.79057e7	0.944459	0.472229	−	0.881476i	$-0.343449\pi$
$59$	8.79057e7	0.944459	0.472229	+	0.881476i	$0.343449\pi$
$60$	0	0
$61$	−1.07388e8	−0.993051	−0.496526	−	0.868022i	$-0.665391\pi$
$61$	−1.07388e8	−0.993051	−0.496526	+	0.868022i	$0.665391\pi$
$62$	0	0
$63$	3.03709e7	0.242898
$64$	0	0
$65$	6.53817e7	0.454304
$66$	0	0
$67$	1.95661e8	1.18622	0.593112	−	0.805120i	$-0.297899\pi$
$67$	1.95661e8	1.18622	0.593112	+	0.805120i	$0.297899\pi$
$68$	0	0
$69$	−3.24503e7	−0.172345
$70$	0	0
$71$	−4.89316e7	−0.228521	−0.114261	−	0.993451i	$-0.536450\pi$
$71$	−4.89316e7	−0.228521	−0.114261	+	0.993451i	$0.536450\pi$
$72$	0	0
$73$	−1.62014e8	−0.667726	−0.333863	−	0.942622i	$-0.608352\pi$
$73$	−1.62014e8	−0.667726	−0.333863	+	0.942622i	$0.608352\pi$
$74$	0	0
$75$	1.04617e7	0.0381793
$76$	0	0
$77$	−1.64546e8	−0.533431
$78$	0	0
$79$	1.10329e8	0.318689	0.159344	−	0.987223i	$-0.449062\pi$
$79$	1.10329e8	0.318689	0.159344	+	0.987223i	$0.449062\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	2.60270e8	0.601967	0.300983	−	0.953629i	$-0.402685\pi$
$83$	2.60270e8	0.601967	0.300983	+	0.953629i	$0.402685\pi$
$84$	0	0
$85$	−1.76076e8	−0.365859
$86$	0	0
$87$	2.63311e8	0.492757
$88$	0	0
$89$	1.44221e8	0.243654	0.121827	−	0.992551i	$-0.461125\pi$
$89$	1.44221e8	0.243654	0.121827	+	0.992551i	$0.461125\pi$
$90$	0	0
$91$	2.24096e8	0.342570
$92$	0	0
$93$	−3.07423e7	−0.0426151
$94$	0	0
$95$	7.58623e8	0.955586
$96$	0	0
$97$	5.59756e8	0.641987	0.320993	−	0.947081i	$-0.395983\pi$
$97$	5.59756e8	0.641987	0.320993	+	0.947081i	$0.395983\pi$
$98$	0	0
$99$	−2.33222e8	−0.244012
$100$	0	0
$101$	9.19152e8	0.878904	0.439452	−	0.898266i	$-0.355173\pi$
$101$	9.19152e8	0.878904	0.439452	+	0.898266i	$0.355173\pi$
$102$	0	0
$103$	1.21707e8	0.106549	0.0532744	−	0.998580i	$-0.483034\pi$
$103$	1.21707e8	0.106549	0.0532744	+	0.998580i	$0.483034\pi$
$104$	0	0
$105$	−5.06385e8	−0.406564
$106$	0	0
$107$	−5.45508e8	−0.402322	−0.201161	−	0.979558i	$-0.564471\pi$
$107$	−5.45508e8	−0.402322	−0.201161	+	0.979558i	$0.564471\pi$
$108$	0	0
$109$	−1.03012e9	−0.698988	−0.349494	−	0.936939i	$-0.613646\pi$
$109$	−1.03012e9	−0.698988	−0.349494	+	0.936939i	$0.613646\pi$
$110$	0	0
$111$	1.43258e9	0.895709
$112$	0	0
$113$	1.78088e9	1.02750	0.513750	−	0.857940i	$-0.328256\pi$
$113$	1.78088e9	1.02750	0.513750	+	0.857940i	$0.328256\pi$
$114$	0	0
$115$	5.41056e8	0.288471
$116$	0	0
$117$	3.17627e8	0.156705
$118$	0	0
$119$	−6.03501e8	−0.275878
$120$	0	0
$121$	−1.09438e9	−0.464124
$122$	0	0
$123$	−3.65134e8	−0.143840
$124$	0	0
$125$	−2.81221e9	−1.03027
$126$	0	0
$127$	2.25709e9	0.769897	0.384949	−	0.922938i	$-0.374219\pi$
$127$	2.25709e9	0.769897	0.384949	+	0.922938i	$0.374219\pi$
$128$	0	0
$129$	2.35844e9	0.749844
$130$	0	0
$131$	−5.72321e9	−1.69793	−0.848964	−	0.528451i	$-0.822773\pi$
$131$	−5.72321e9	−1.69793	−0.848964	+	0.528451i	$0.822773\pi$
$132$	0	0
$133$	2.60019e9	0.720563
$134$	0	0
$135$	−7.17734e8	−0.185978
$136$	0	0
$137$	3.64308e9	0.883539	0.441769	−	0.897129i	$-0.354351\pi$
$137$	3.64308e9	0.883539	0.441769	+	0.897129i	$0.354351\pi$
$138$	0	0
$139$	1.57294e9	0.357394	0.178697	−	0.983904i	$-0.442812\pi$
$139$	1.57294e9	0.357394	0.178697	+	0.983904i	$0.442812\pi$
$140$	0	0
$141$	2.72206e9	0.579978
$142$	0	0
$143$	−1.72087e9	−0.344140
$144$	0	0
$145$	−4.39029e9	−0.824778
$146$	0	0
$147$	1.53300e9	0.270779
$148$	0	0
$149$	−6.20779e9	−1.03181	−0.515904	−	0.856646i	$-0.672544\pi$
$149$	−6.20779e9	−1.03181	−0.515904	+	0.856646i	$0.672544\pi$
$150$	0	0
$151$	1.61701e8	0.0253114	0.0126557	−	0.999920i	$-0.495971\pi$
$151$	1.61701e8	0.0253114	0.0126557	+	0.999920i	$0.495971\pi$
$152$	0	0
$153$	−8.55383e8	−0.126197
$154$	0	0
$155$	5.12578e8	0.0713292
$156$	0	0
$157$	−9.93712e9	−1.30530	−0.652652	−	0.757657i	$-0.726344\pi$
$157$	−9.93712e9	−1.30530	−0.652652	+	0.757657i	$0.726344\pi$
$158$	0	0
$159$	4.60145e8	0.0570962
$160$	0	0
$161$	1.85448e9	0.217523
$162$	0	0
$163$	1.76080e9	0.195373	0.0976866	−	0.995217i	$-0.468856\pi$
$163$	1.76080e9	0.195373	0.0976866	+	0.995217i	$0.468856\pi$
$164$	0	0
$165$	3.88860e9	0.408428
$166$	0	0
$167$	7.70998e9	0.767060	0.383530	−	0.923528i	$-0.374708\pi$
$167$	7.70998e9	0.767060	0.383530	+	0.923528i	$0.374708\pi$
$168$	0	0
$169$	−8.26083e9	−0.778993
$170$	0	0
$171$	3.68542e9	0.329614
$172$	0	0
$173$	2.06841e9	0.175562	0.0877808	−	0.996140i	$-0.472023\pi$
$173$	2.06841e9	0.175562	0.0877808	+	0.996140i	$0.472023\pi$
$174$	0	0
$175$	−5.97869e8	−0.0481875
$176$	0	0
$177$	−7.12036e9	−0.545283
$178$	0	0
$179$	−1.60048e10	−1.16523	−0.582616	−	0.812747i	$-0.697971\pi$
$179$	−1.60048e10	−1.16523	−0.582616	+	0.812747i	$0.697971\pi$
$180$	0	0
$181$	1.01069e10	0.699945	0.349973	−	0.936760i	$-0.386191\pi$
$181$	1.01069e10	0.699945	0.349973	+	0.936760i	$0.386191\pi$
$182$	0	0
$183$	8.69843e9	0.573338
$184$	0	0
$185$	−2.38860e10	−1.49924
$186$	0	0
$187$	4.63436e9	0.277142
$188$	0	0
$189$	−2.46004e9	−0.140237
$190$	0	0
$191$	−7.60249e9	−0.413338	−0.206669	−	0.978411i	$-0.566262\pi$
$191$	−7.60249e9	−0.413338	−0.206669	+	0.978411i	$0.566262\pi$
$192$	0	0
$193$	−2.40493e10	−1.24765	−0.623827	−	0.781563i	$-0.714423\pi$
$193$	−2.40493e10	−1.24765	−0.623827	+	0.781563i	$0.714423\pi$
$194$	0	0
$195$	−5.29592e9	−0.262292
$196$	0	0
$197$	−3.54192e10	−1.67549	−0.837744	−	0.546064i	$-0.816126\pi$
$197$	−3.54192e10	−1.67549	−0.837744	+	0.546064i	$0.816126\pi$
$198$	0	0
$199$	−1.45012e10	−0.655487	−0.327743	−	0.944767i	$-0.606288\pi$
$199$	−1.45012e10	−0.655487	−0.327743	+	0.944767i	$0.606288\pi$
$200$	0	0
$201$	−1.58485e10	−0.684867
$202$	0	0
$203$	−1.50478e10	−0.621927
$204$	0	0
$205$	6.08801e9	0.240759
$206$	0	0
$207$	2.62848e9	0.0995033
$208$	0	0
$209$	−1.99672e10	−0.723867
$210$	0	0
$211$	−2.22563e10	−0.773004	−0.386502	−	0.922289i	$-0.626317\pi$
$211$	−2.22563e10	−0.773004	−0.386502	+	0.922289i	$0.626317\pi$
$212$	0	0
$213$	3.96346e9	0.131937
$214$	0	0
$215$	−3.93232e10	−1.25509
$216$	0	0
$217$	1.75687e9	0.0537861
$218$	0	0
$219$	1.31231e10	0.385512
$220$	0	0
$221$	−6.31159e9	−0.177981
$222$	0	0
$223$	6.93618e10	1.87823	0.939114	−	0.343606i	$-0.111648\pi$
$223$	6.93618e10	1.87823	0.939114	+	0.343606i	$0.111648\pi$
$224$	0	0
$225$	−8.47400e8	−0.0220428
$226$	0	0
$227$	−4.24558e10	−1.06126	−0.530629	−	0.847604i	$-0.678044\pi$
$227$	−4.24558e10	−1.06126	−0.530629	+	0.847604i	$0.678044\pi$
$228$	0	0
$229$	−2.55723e10	−0.614483	−0.307241	−	0.951632i	$-0.599406\pi$
$229$	−2.55723e10	−0.614483	−0.307241	+	0.951632i	$0.599406\pi$
$230$	0	0
$231$	1.33282e10	0.307977
$232$	0	0
$233$	7.42856e10	1.65121	0.825606	−	0.564247i	$-0.190833\pi$
$233$	7.42856e10	1.65121	0.825606	+	0.564247i	$0.190833\pi$
$234$	0	0
$235$	−4.53859e10	−0.970768
$236$	0	0
$237$	−8.93662e9	−0.183995
$238$	0	0
$239$	5.00750e10	0.992729	0.496365	−	0.868114i	$-0.334668\pi$
$239$	5.00750e10	0.992729	0.496365	+	0.868114i	$0.334668\pi$
$240$	0	0
$241$	−2.67268e9	−0.0510353	−0.0255176	−	0.999674i	$-0.508123\pi$
$241$	−2.67268e9	−0.0510353	−0.0255176	+	0.999674i	$0.508123\pi$
$242$	0	0
$243$	−3.48678e9	−0.0641500
$244$	0	0
$245$	−2.55603e10	−0.453231
$246$	0	0
$247$	2.71935e10	0.464867
$248$	0	0
$249$	−2.10819e10	−0.347546
$250$	0	0
$251$	−1.97889e10	−0.314696	−0.157348	−	0.987543i	$-0.550294\pi$
$251$	−1.97889e10	−0.314696	−0.157348	+	0.987543i	$0.550294\pi$
$252$	0	0
$253$	−1.42408e10	−0.218520
$254$	0	0
$255$	1.42621e10	0.211229
$256$	0	0
$257$	−9.15704e10	−1.30935	−0.654676	−	0.755910i	$-0.727195\pi$
$257$	−9.15704e10	−1.30935	−0.654676	+	0.755910i	$0.727195\pi$
$258$	0	0
$259$	−8.18696e10	−1.13051
$260$	0	0
$261$	−2.13282e10	−0.284494
$262$	0	0
$263$	5.43794e10	0.700863	0.350432	−	0.936588i	$-0.386035\pi$
$263$	5.43794e10	0.700863	0.350432	+	0.936588i	$0.386035\pi$
$264$	0	0
$265$	−7.67217e9	−0.0955678
$266$	0	0
$267$	−1.16819e10	−0.140674
$268$	0	0
$269$	3.46224e8	0.00403155	0.00201578	−	0.999998i	$-0.499358\pi$
$269$	3.46224e8	0.00403155	0.00201578	+	0.999998i	$0.499358\pi$
$270$	0	0
$271$	1.03029e11	1.16037	0.580184	−	0.814485i	$-0.302980\pi$
$271$	1.03029e11	1.16037	0.580184	+	0.814485i	$0.302980\pi$
$272$	0	0
$273$	−1.81518e10	−0.197783
$274$	0	0
$275$	4.59111e9	0.0484084
$276$	0	0
$277$	−3.01924e10	−0.308133	−0.154066	−	0.988060i	$-0.549237\pi$
$277$	−3.01924e10	−0.308133	−0.154066	+	0.988060i	$0.549237\pi$
$278$	0	0
$279$	2.49013e9	0.0246038
$280$	0	0
$281$	2.70465e9	0.0258781	0.0129391	−	0.999916i	$-0.495881\pi$
$281$	2.70465e9	0.0258781	0.0129391	+	0.999916i	$0.495881\pi$
$282$	0	0
$283$	−1.69965e11	−1.57514	−0.787572	−	0.616222i	$-0.788662\pi$
$283$	−1.69965e11	−1.57514	−0.787572	+	0.616222i	$0.788662\pi$
$284$	0	0
$285$	−6.14485e10	−0.551708
$286$	0	0
$287$	2.08667e10	0.181546
$288$	0	0
$289$	−1.01591e11	−0.856669
$290$	0	0
$291$	−4.53403e10	−0.370651
$292$	0	0
$293$	−7.58823e10	−0.601500	−0.300750	−	0.953703i	$-0.597237\pi$
$293$	−7.58823e10	−0.601500	−0.300750	+	0.953703i	$0.597237\pi$
$294$	0	0
$295$	1.18720e11	0.912697
$296$	0	0
$297$	1.88910e10	0.140880
$298$	0	0
$299$	1.93946e10	0.140333
$300$	0	0
$301$	−1.34780e11	−0.946407
$302$	0	0
$303$	−7.44513e10	−0.507435
$304$	0	0
$305$	−1.45032e11	−0.959655
$306$	0	0
$307$	1.64274e11	1.05547	0.527735	−	0.849409i	$-0.323042\pi$
$307$	1.64274e11	1.05547	0.527735	+	0.849409i	$0.323042\pi$
$308$	0	0
$309$	−9.85828e9	−0.0615160
$310$	0	0
$311$	2.36639e11	1.43438	0.717189	−	0.696878i	$-0.245428\pi$
$311$	2.36639e11	1.43438	0.717189	+	0.696878i	$0.245428\pi$
$312$	0	0
$313$	−2.03140e11	−1.19632	−0.598158	−	0.801378i	$-0.704100\pi$
$313$	−2.03140e11	−1.19632	−0.598158	+	0.801378i	$0.704100\pi$
$314$	0	0
$315$	4.10172e10	0.234730
$316$	0	0
$317$	−2.78065e11	−1.54661	−0.773304	−	0.634036i	$-0.781397\pi$
$317$	−2.78065e11	−1.54661	−0.773304	+	0.634036i	$0.781397\pi$
$318$	0	0
$319$	1.15554e11	0.624778
$320$	0	0
$321$	4.41861e10	0.232281
$322$	0	0
$323$	−7.32332e10	−0.374366
$324$	0	0
$325$	−6.25268e9	−0.0310879
$326$	0	0
$327$	8.34399e10	0.403561
$328$	0	0
$329$	−1.55561e11	−0.732012
$330$	0	0
$331$	1.08372e10	0.0496240	0.0248120	−	0.999692i	$-0.492101\pi$
$331$	1.08372e10	0.0496240	0.0248120	+	0.999692i	$0.492101\pi$
$332$	0	0
$333$	−1.16039e11	−0.517138
$334$	0	0
$335$	2.64248e11	1.14633
$336$	0	0
$337$	1.39972e11	0.591161	0.295581	−	0.955318i	$-0.404487\pi$
$337$	1.39972e11	0.591161	0.295581	+	0.955318i	$0.404487\pi$
$338$	0	0
$339$	−1.44251e11	−0.593228
$340$	0	0
$341$	−1.34912e10	−0.0540326
$342$	0	0
$343$	−2.74405e11	−1.07046
$344$	0	0
$345$	−4.38256e10	−0.166549
$346$	0	0
$347$	−5.60241e10	−0.207440	−0.103720	−	0.994607i	$-0.533075\pi$
$347$	−5.60241e10	−0.207440	−0.103720	+	0.994607i	$0.533075\pi$
$348$	0	0
$349$	−2.40745e11	−0.868647	−0.434324	−	0.900757i	$-0.643013\pi$
$349$	−2.40745e11	−0.868647	−0.434324	+	0.900757i	$0.643013\pi$
$350$	0	0
$351$	−2.57278e10	−0.0904734
$352$	0	0
$353$	−4.30557e11	−1.47586	−0.737928	−	0.674879i	$-0.764196\pi$
$353$	−4.30557e11	−1.47586	−0.737928	+	0.674879i	$0.764196\pi$
$354$	0	0
$355$	−6.60843e10	−0.220836
$356$	0	0
$357$	4.88836e10	0.159278
$358$	0	0
$359$	2.82766e11	0.898467	0.449234	−	0.893414i	$-0.351697\pi$
$359$	2.82766e11	0.898467	0.449234	+	0.893414i	$0.351697\pi$
$360$	0	0
$361$	−7.16187e9	−0.0221944
$362$	0	0
$363$	8.86448e10	0.267962
$364$	0	0
$365$	−2.18806e11	−0.645271
$366$	0	0
$367$	−2.58503e11	−0.743821	−0.371910	−	0.928269i	$-0.621297\pi$
$367$	−2.58503e11	−0.743821	−0.371910	+	0.928269i	$0.621297\pi$
$368$	0	0
$369$	2.95758e10	0.0830459
$370$	0	0
$371$	−2.62964e10	−0.0720633
$372$	0	0
$373$	−3.13543e11	−0.838701	−0.419351	−	0.907824i	$-0.637742\pi$
$373$	−3.13543e11	−0.838701	−0.419351	+	0.907824i	$0.637742\pi$
$374$	0	0
$375$	2.27789e11	0.594829
$376$	0	0
$377$	−1.57374e11	−0.401233
$378$	0	0
$379$	−7.27877e11	−1.81210	−0.906049	−	0.423172i	$-0.860916\pi$
$379$	−7.27877e11	−1.81210	−0.906049	+	0.423172i	$0.860916\pi$
$380$	0	0
$381$	−1.82825e11	−0.444500
$382$	0	0
$383$	−2.35565e11	−0.559393	−0.279696	−	0.960089i	$-0.590234\pi$
$383$	−2.35565e11	−0.559393	−0.279696	+	0.960089i	$0.590234\pi$
$384$	0	0
$385$	−2.22226e11	−0.515492
$386$	0	0
$387$	−1.91034e11	−0.432923
$388$	0	0
$389$	4.88725e11	1.08216	0.541080	−	0.840971i	$-0.318016\pi$
$389$	4.88725e11	1.08216	0.541080	+	0.840971i	$0.318016\pi$
$390$	0	0
$391$	−5.22305e10	−0.113013
$392$	0	0
$393$	4.63580e11	0.980299
$394$	0	0
$395$	1.49004e11	0.307971
$396$	0	0
$397$	−5.70044e11	−1.15173	−0.575865	−	0.817545i	$-0.695335\pi$
$397$	−5.70044e11	−1.15173	−0.575865	+	0.817545i	$0.695335\pi$
$398$	0	0
$399$	−2.10615e11	−0.416018
$400$	0	0
$401$	1.61868e11	0.312617	0.156308	−	0.987708i	$-0.450041\pi$
$401$	1.61868e11	0.312617	0.156308	+	0.987708i	$0.450041\pi$
$402$	0	0
$403$	1.83738e10	0.0346998
$404$	0	0
$405$	5.81365e10	0.107374
$406$	0	0
$407$	6.28687e11	1.13569
$408$	0	0
$409$	−3.26776e11	−0.577424	−0.288712	−	0.957416i	$-0.593227\pi$
$409$	−3.26776e11	−0.577424	−0.288712	+	0.957416i	$0.593227\pi$
$410$	0	0
$411$	−2.95089e11	−0.510111
$412$	0	0
$413$	4.06915e11	0.688223
$414$	0	0
$415$	3.51506e11	0.581723
$416$	0	0
$417$	−1.27408e11	−0.206341
$418$	0	0
$419$	−9.19459e11	−1.45737	−0.728684	−	0.684850i	$-0.759868\pi$
$419$	−9.19459e11	−1.45737	−0.728684	+	0.684850i	$0.759868\pi$
$420$	0	0
$421$	8.51831e11	1.32155	0.660776	−	0.750583i	$-0.270227\pi$
$421$	8.51831e11	1.32155	0.660776	+	0.750583i	$0.270227\pi$
$422$	0	0
$423$	−2.20487e11	−0.334850
$424$	0	0
$425$	1.68387e10	0.0250357
$426$	0	0
$427$	−4.97099e11	−0.723632
$428$	0	0
$429$	1.39390e11	0.198689
$430$	0	0
$431$	−2.24830e10	−0.0313839	−0.0156920	−	0.999877i	$-0.504995\pi$
$431$	−2.24830e10	−0.0313839	−0.0156920	+	0.999877i	$0.504995\pi$
$432$	0	0
$433$	−1.37987e12	−1.88644	−0.943218	−	0.332173i	$-0.892218\pi$
$433$	−1.37987e12	−1.88644	−0.943218	+	0.332173i	$0.892218\pi$
$434$	0	0
$435$	3.55613e11	0.476186
$436$	0	0
$437$	2.25036e11	0.295179
$438$	0	0
$439$	−1.97737e11	−0.254096	−0.127048	−	0.991897i	$-0.540550\pi$
$439$	−1.97737e11	−0.254096	−0.127048	+	0.991897i	$0.540550\pi$
$440$	0	0
$441$	−1.24173e11	−0.156334
$442$	0	0
$443$	−4.21012e10	−0.0519371	−0.0259685	−	0.999663i	$-0.508267\pi$
$443$	−4.21012e10	−0.0519371	−0.0259685	+	0.999663i	$0.508267\pi$
$444$	0	0
$445$	1.94777e11	0.235460
$446$	0	0
$447$	5.02831e11	0.595715
$448$	0	0
$449$	2.04126e11	0.237023	0.118511	−	0.992953i	$-0.462188\pi$
$449$	2.04126e11	0.237023	0.118511	+	0.992953i	$0.462188\pi$
$450$	0	0
$451$	−1.60238e11	−0.182378
$452$	0	0
$453$	−1.30978e10	−0.0146135
$454$	0	0
$455$	3.02652e11	0.331049
$456$	0	0
$457$	1.44309e12	1.54764	0.773820	−	0.633405i	$-0.218343\pi$
$457$	1.44309e12	1.54764	0.773820	+	0.633405i	$0.218343\pi$
$458$	0	0
$459$	6.92860e10	0.0728599
$460$	0	0
$461$	−5.27006e11	−0.543452	−0.271726	−	0.962375i	$-0.587594\pi$
$461$	−5.27006e11	−0.543452	−0.271726	+	0.962375i	$0.587594\pi$
$462$	0	0
$463$	2.42042e11	0.244780	0.122390	−	0.992482i	$-0.460944\pi$
$463$	2.42042e11	0.244780	0.122390	+	0.992482i	$0.460944\pi$
$464$	0	0
$465$	−4.15188e10	−0.0411819
$466$	0	0
$467$	1.73785e12	1.69078	0.845390	−	0.534150i	$-0.179368\pi$
$467$	1.73785e12	1.69078	0.845390	+	0.534150i	$0.179368\pi$
$468$	0	0
$469$	9.05713e11	0.864396
$470$	0	0
$471$	8.04906e11	0.753618
$472$	0	0
$473$	1.03500e12	0.950745
$474$	0	0
$475$	−7.25498e10	−0.0653905
$476$	0	0
$477$	−3.72717e10	−0.0329645
$478$	0	0
$479$	−7.67989e11	−0.666569	−0.333284	−	0.942826i	$-0.608157\pi$
$479$	−7.67989e11	−0.666569	−0.333284	+	0.942826i	$0.608157\pi$
$480$	0	0
$481$	−8.56216e11	−0.729341
$482$	0	0
$483$	−1.50212e11	−0.125587
$484$	0	0
$485$	7.55975e11	0.620397
$486$	0	0
$487$	−9.42997e11	−0.759678	−0.379839	−	0.925053i	$-0.624021\pi$
$487$	−9.42997e11	−0.759678	−0.379839	+	0.925053i	$0.624021\pi$
$488$	0	0
$489$	−1.42625e11	−0.112799
$490$	0	0
$491$	2.21900e10	0.0172302	0.00861511	−	0.999963i	$-0.497258\pi$
$491$	2.21900e10	0.0172302	0.00861511	+	0.999963i	$0.497258\pi$
$492$	0	0
$493$	4.23814e11	0.323120
$494$	0	0
$495$	−3.14976e11	−0.235806
$496$	0	0
$497$	−2.26504e11	−0.166523
$498$	0	0
$499$	1.67958e12	1.21269	0.606344	−	0.795202i	$-0.292635\pi$
$499$	1.67958e12	1.21269	0.606344	+	0.795202i	$0.292635\pi$
$500$	0	0
$501$	−6.24509e11	−0.442862
$502$	0	0
$503$	−2.12521e12	−1.48028	−0.740142	−	0.672451i	$-0.765242\pi$
$503$	−2.12521e12	−1.48028	−0.740142	+	0.672451i	$0.765242\pi$
$504$	0	0
$505$	1.24136e12	0.849347
$506$	0	0
$507$	6.69127e11	0.449752
$508$	0	0
$509$	−2.37137e12	−1.56592	−0.782958	−	0.622074i	$-0.786290\pi$
$509$	−2.37137e12	−1.56592	−0.782958	+	0.622074i	$0.786290\pi$
$510$	0	0
$511$	−7.49961e11	−0.486569
$512$	0	0
$513$	−2.98519e11	−0.190302
$514$	0	0
$515$	1.64371e11	0.102966
$516$	0	0
$517$	1.19457e12	0.735367
$518$	0	0
$519$	−1.67541e11	−0.101360
$520$	0	0
$521$	2.63954e12	1.56949	0.784744	−	0.619820i	$-0.212794\pi$
$521$	2.63954e12	1.56949	0.784744	+	0.619820i	$0.212794\pi$
$522$	0	0
$523$	4.55831e11	0.266407	0.133204	−	0.991089i	$-0.457474\pi$
$523$	4.55831e11	0.266407	0.133204	+	0.991089i	$0.457474\pi$
$524$	0	0
$525$	4.84274e10	0.0278211
$526$	0	0
$527$	−4.94814e10	−0.0279444
$528$	0	0
$529$	−1.64066e12	−0.910892
$530$	0	0
$531$	5.76749e11	0.314820
$532$	0	0
$533$	2.18230e11	0.117123
$534$	0	0
$535$	−7.36732e11	−0.388792
$536$	0	0
$537$	1.29639e12	0.672747
$538$	0	0
$539$	6.72756e11	0.343327
$540$	0	0
$541$	1.14393e12	0.574133	0.287067	−	0.957911i	$-0.407320\pi$
$541$	1.14393e12	0.574133	0.287067	+	0.957911i	$0.407320\pi$
$542$	0	0
$543$	−8.18658e11	−0.404114
$544$	0	0
$545$	−1.39122e12	−0.675481
$546$	0	0
$547$	−8.54220e11	−0.407969	−0.203984	−	0.978974i	$-0.565389\pi$
$547$	−8.54220e11	−0.407969	−0.203984	+	0.978974i	$0.565389\pi$
$548$	0	0
$549$	−7.04573e11	−0.331017
$550$	0	0
$551$	−1.82601e12	−0.843956
$552$	0	0
$553$	5.10711e11	0.232227
$554$	0	0
$555$	1.93477e12	0.865587
$556$	0	0
$557$	−1.98459e12	−0.873618	−0.436809	−	0.899554i	$-0.643891\pi$
$557$	−1.98459e12	−0.873618	−0.436809	+	0.899554i	$0.643891\pi$
$558$	0	0
$559$	−1.40957e12	−0.610569
$560$	0	0
$561$	−3.75383e11	−0.160008
$562$	0	0
$563$	−1.38422e11	−0.0580655	−0.0290327	−	0.999578i	$-0.509243\pi$
$563$	−1.38422e11	−0.0580655	−0.0290327	+	0.999578i	$0.509243\pi$
$564$	0	0
$565$	2.40516e12	0.992946
$566$	0	0
$567$	1.99263e11	0.0809662
$568$	0	0
$569$	2.40884e12	0.963391	0.481696	−	0.876339i	$-0.340021\pi$
$569$	2.40884e12	0.963391	0.481696	+	0.876339i	$0.340021\pi$
$570$	0	0
$571$	1.02972e12	0.405374	0.202687	−	0.979244i	$-0.435033\pi$
$571$	1.02972e12	0.405374	0.202687	+	0.979244i	$0.435033\pi$
$572$	0	0
$573$	6.15802e11	0.238641
$574$	0	0
$575$	−5.17431e10	−0.0197400
$576$	0	0
$577$	1.57101e12	0.590047	0.295023	−	0.955490i	$-0.404673\pi$
$577$	1.57101e12	0.590047	0.295023	+	0.955490i	$0.404673\pi$
$578$	0	0
$579$	1.94799e12	0.720333
$580$	0	0
$581$	1.20479e12	0.438650
$582$	0	0
$583$	2.01934e11	0.0723936
$584$	0	0
$585$	4.28970e11	0.151435
$586$	0	0
$587$	3.36810e12	1.17088	0.585442	−	0.810714i	$-0.300921\pi$
$587$	3.36810e12	1.17088	0.585442	+	0.810714i	$0.300921\pi$
$588$	0	0
$589$	2.13191e11	0.0729878
$590$	0	0
$591$	2.86896e12	0.967343
$592$	0	0
$593$	3.52259e12	1.16981	0.584906	−	0.811101i	$-0.301131\pi$
$593$	3.52259e12	1.16981	0.584906	+	0.811101i	$0.301131\pi$
$594$	0	0
$595$	−8.15054e11	−0.266600
$596$	0	0
$597$	1.17459e12	0.378445
$598$	0	0
$599$	1.57104e12	0.498617	0.249308	−	0.968424i	$-0.419797\pi$
$599$	1.57104e12	0.498617	0.249308	+	0.968424i	$0.419797\pi$
$600$	0	0
$601$	5.03628e12	1.57462	0.787309	−	0.616559i	$-0.211474\pi$
$601$	5.03628e12	1.57462	0.787309	+	0.616559i	$0.211474\pi$
$602$	0	0
$603$	1.28373e12	0.395408
$604$	0	0
$605$	−1.47801e12	−0.448515
$606$	0	0
$607$	−1.56139e11	−0.0466834	−0.0233417	−	0.999728i	$-0.507431\pi$
$607$	−1.56139e11	−0.0466834	−0.0233417	+	0.999728i	$0.507431\pi$
$608$	0	0
$609$	1.21887e12	0.359070
$610$	0	0
$611$	−1.62690e12	−0.472253
$612$	0	0
$613$	5.67864e12	1.62432	0.812160	−	0.583434i	$-0.198291\pi$
$613$	5.67864e12	1.62432	0.812160	+	0.583434i	$0.198291\pi$
$614$	0	0
$615$	−4.93129e11	−0.139003
$616$	0	0
$617$	2.84876e12	0.791357	0.395678	−	0.918389i	$-0.370510\pi$
$617$	2.84876e12	0.791357	0.395678	+	0.918389i	$0.370510\pi$
$618$	0	0
$619$	3.40171e12	0.931299	0.465650	−	0.884969i	$-0.345821\pi$
$619$	3.40171e12	0.931299	0.465650	+	0.884969i	$0.345821\pi$
$620$	0	0
$621$	−2.12906e11	−0.0574482
$622$	0	0
$623$	6.67598e11	0.177549
$624$	0	0
$625$	−3.54576e12	−0.929499
$626$	0	0
$627$	1.61734e12	0.417925
$628$	0	0
$629$	2.30582e12	0.587352
$630$	0	0
$631$	−3.11154e12	−0.781347	−0.390673	−	0.920529i	$-0.627758\pi$
$631$	−3.11154e12	−0.781347	−0.390673	+	0.920529i	$0.627758\pi$
$632$	0	0
$633$	1.80276e12	0.446294
$634$	0	0
$635$	3.04830e12	0.744006
$636$	0	0
$637$	−9.16233e11	−0.220485
$638$	0	0
$639$	−3.21040e11	−0.0761738
$640$	0	0
$641$	5.65123e11	0.132215	0.0661077	−	0.997812i	$-0.478942\pi$
$641$	5.65123e11	0.132215	0.0661077	+	0.997812i	$0.478942\pi$
$642$	0	0
$643$	−6.39312e12	−1.47490	−0.737451	−	0.675401i	$-0.763971\pi$
$643$	−6.39312e12	−1.47490	−0.737451	+	0.675401i	$0.763971\pi$
$644$	0	0
$645$	3.18518e12	0.724627
$646$	0	0
$647$	−4.53481e12	−1.01740	−0.508698	−	0.860945i	$-0.669873\pi$
$647$	−4.53481e12	−1.01740	−0.508698	+	0.860945i	$0.669873\pi$
$648$	0	0
$649$	−3.12476e12	−0.691377
$650$	0	0
$651$	−1.42306e11	−0.0310534
$652$	0	0
$653$	3.45675e12	0.743975	0.371987	−	0.928238i	$-0.378676\pi$
$653$	3.45675e12	0.743975	0.371987	+	0.928238i	$0.378676\pi$
$654$	0	0
$655$	−7.72945e12	−1.64083
$656$	0	0
$657$	−1.06297e12	−0.222575
$658$	0	0
$659$	6.26729e12	1.29448	0.647240	−	0.762286i	$-0.275923\pi$
$659$	6.26729e12	1.29448	0.647240	+	0.762286i	$0.275923\pi$
$660$	0	0
$661$	5.54653e12	1.13010	0.565048	−	0.825058i	$-0.308858\pi$
$661$	5.54653e12	1.13010	0.565048	+	0.825058i	$0.308858\pi$
$662$	0	0
$663$	5.11239e11	0.102757
$664$	0	0
$665$	3.51167e12	0.696331
$666$	0	0
$667$	−1.30232e12	−0.254772
$668$	0	0
$669$	−5.61830e12	−1.08440
$670$	0	0
$671$	3.81729e12	0.726949
$672$	0	0
$673$	−3.74900e12	−0.704445	−0.352223	−	0.935916i	$-0.614574\pi$
$673$	−3.74900e12	−0.704445	−0.352223	+	0.935916i	$0.614574\pi$
$674$	0	0
$675$	6.86394e10	0.0127264
$676$	0	0
$677$	9.07618e12	1.66056	0.830279	−	0.557348i	$-0.188181\pi$
$677$	9.07618e12	1.66056	0.830279	+	0.557348i	$0.188181\pi$
$678$	0	0
$679$	2.59111e12	0.467813
$680$	0	0
$681$	3.43892e12	0.612718
$682$	0	0
$683$	−4.23293e12	−0.744300	−0.372150	−	0.928173i	$-0.621379\pi$
$683$	−4.23293e12	−0.744300	−0.372150	+	0.928173i	$0.621379\pi$
$684$	0	0
$685$	4.92013e12	0.853826
$686$	0	0
$687$	2.07136e12	0.354772
$688$	0	0
$689$	−2.75016e11	−0.0464912
$690$	0	0
$691$	1.16308e12	0.194071	0.0970354	−	0.995281i	$-0.469064\pi$
$691$	1.16308e12	0.194071	0.0970354	+	0.995281i	$0.469064\pi$
$692$	0	0
$693$	−1.07958e12	−0.177810
$694$	0	0
$695$	2.12433e12	0.345375
$696$	0	0
$697$	−5.87703e11	−0.0943214
$698$	0	0
$699$	−6.01713e12	−0.953328
$700$	0	0
$701$	−9.11768e12	−1.42611	−0.713055	−	0.701108i	$-0.752689\pi$
$701$	−9.11768e12	−1.42611	−0.713055	+	0.701108i	$0.752689\pi$
$702$	0	0
$703$	−9.93466e12	−1.53410
$704$	0	0
$705$	3.67626e12	0.560473
$706$	0	0
$707$	4.25476e12	0.640453
$708$	0	0
$709$	1.13764e13	1.69082	0.845411	−	0.534117i	$-0.179356\pi$
$709$	1.13764e13	1.69082	0.845411	+	0.534117i	$0.179356\pi$
$710$	0	0
$711$	7.23866e11	0.106230
$712$	0	0
$713$	1.52050e11	0.0220334
$714$	0	0
$715$	−2.32411e12	−0.332567
$716$	0	0
$717$	−4.05608e12	−0.573152
$718$	0	0
$719$	2.03795e12	0.284389	0.142194	−	0.989839i	$-0.454584\pi$
$719$	2.03795e12	0.284389	0.142194	+	0.989839i	$0.454584\pi$
$720$	0	0
$721$	5.63382e11	0.0776416
$722$	0	0
$723$	2.16487e11	0.0294652
$724$	0	0
$725$	4.19859e11	0.0564394
$726$	0	0
$727$	−1.32610e13	−1.76064	−0.880320	−	0.474381i	$-0.842672\pi$
$727$	−1.32610e13	−1.76064	−0.880320	+	0.474381i	$0.842672\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	3.79604e12	0.491702
$732$	0	0
$733$	−1.06385e13	−1.36117	−0.680587	−	0.732667i	$-0.738275\pi$
$733$	−1.06385e13	−1.36117	−0.680587	+	0.732667i	$0.738275\pi$
$734$	0	0
$735$	2.07039e12	0.261673
$736$	0	0
$737$	−6.95509e12	−0.868359
$738$	0	0
$739$	−1.85609e12	−0.228928	−0.114464	−	0.993427i	$-0.536515\pi$
$739$	−1.85609e12	−0.228928	−0.114464	+	0.993427i	$0.536515\pi$
$740$	0	0
$741$	−2.20267e12	−0.268391
$742$	0	0
$743$	−3.97256e12	−0.478212	−0.239106	−	0.970993i	$-0.576854\pi$
$743$	−3.97256e12	−0.478212	−0.239106	+	0.970993i	$0.576854\pi$
$744$	0	0
$745$	−8.38389e12	−0.997109
$746$	0	0
$747$	1.70763e12	0.200656
$748$	0	0
$749$	−2.52516e12	−0.293170
$750$	0	0
$751$	−6.16552e12	−0.707278	−0.353639	−	0.935382i	$-0.615056\pi$
$751$	−6.16552e12	−0.707278	−0.353639	+	0.935382i	$0.615056\pi$
$752$	0	0
$753$	1.60290e12	0.181690
$754$	0	0
$755$	2.18384e11	0.0244602
$756$	0	0
$757$	−1.04042e13	−1.15154	−0.575768	−	0.817613i	$-0.695297\pi$
$757$	−1.04042e13	−1.15154	−0.575768	+	0.817613i	$0.695297\pi$
$758$	0	0
$759$	1.15350e12	0.126162
$760$	0	0
$761$	1.30033e13	1.40547	0.702737	−	0.711450i	$-0.251961\pi$
$761$	1.30033e13	1.40547	0.702737	+	0.711450i	$0.251961\pi$
$762$	0	0
$763$	−4.76844e12	−0.509349
$764$	0	0
$765$	−1.15523e12	−0.121953
$766$	0	0
$767$	4.25564e12	0.444003
$768$	0	0
$769$	−1.32065e13	−1.36182	−0.680909	−	0.732368i	$-0.738415\pi$
$769$	−1.32065e13	−1.36182	−0.680909	+	0.732368i	$0.738415\pi$
$770$	0	0
$771$	7.41721e12	0.755954
$772$	0	0
$773$	−5.64488e11	−0.0568653	−0.0284326	−	0.999596i	$-0.509052\pi$
$773$	−5.64488e11	−0.0568653	−0.0284326	+	0.999596i	$0.509052\pi$
$774$	0	0
$775$	−4.90196e10	−0.00488104
$776$	0	0
$777$	6.63144e12	0.652699
$778$	0	0
$779$	2.53212e12	0.246358
$780$	0	0
$781$	1.73936e12	0.167286
$782$	0	0
$783$	1.72759e12	0.164252
$784$	0	0
$785$	−1.34205e13	−1.26141
$786$	0	0
$787$	−1.90473e13	−1.76989	−0.884946	−	0.465694i	$-0.845805\pi$
$787$	−1.90473e13	−1.76989	−0.884946	+	0.465694i	$0.845805\pi$
$788$	0	0
$789$	−4.40473e12	−0.404644
$790$	0	0
$791$	8.24370e12	0.748735
$792$	0	0
$793$	−5.19881e12	−0.466847
$794$	0	0
$795$	6.21445e11	0.0551761
$796$	0	0
$797$	1.11890e13	0.982270	0.491135	−	0.871084i	$-0.336582\pi$
$797$	1.11890e13	0.982270	0.491135	+	0.871084i	$0.336582\pi$
$798$	0	0
$799$	4.38130e12	0.380314
$800$	0	0
$801$	9.46233e11	0.0812179
$802$	0	0
$803$	5.75905e12	0.488799
$804$	0	0
$805$	2.50455e12	0.210207
$806$	0	0
$807$	−2.80442e10	−0.00232762
$808$	0	0
$809$	1.37095e12	0.112526	0.0562629	−	0.998416i	$-0.482082\pi$
$809$	1.37095e12	0.112526	0.0562629	+	0.998416i	$0.482082\pi$
$810$	0	0
$811$	−6.16928e12	−0.500772	−0.250386	−	0.968146i	$-0.580558\pi$
$811$	−6.16928e12	−0.500772	−0.250386	+	0.968146i	$0.580558\pi$
$812$	0	0
$813$	−8.34532e12	−0.669939
$814$	0	0
$815$	2.37803e12	0.188803
$816$	0	0
$817$	−1.63553e13	−1.28428
$818$	0	0
$819$	1.47030e12	0.114190
$820$	0	0
$821$	−1.77345e13	−1.36231	−0.681153	−	0.732141i	$-0.738521\pi$
$821$	−1.77345e13	−1.36231	−0.681153	+	0.732141i	$0.738521\pi$
$822$	0	0
$823$	3.20099e12	0.243212	0.121606	−	0.992578i	$-0.461196\pi$
$823$	3.20099e12	0.243212	0.121606	+	0.992578i	$0.461196\pi$
$824$	0	0
$825$	−3.71880e11	−0.0279486
$826$	0	0
$827$	1.27493e13	0.947787	0.473894	−	0.880582i	$-0.342848\pi$
$827$	1.27493e13	0.947787	0.473894	+	0.880582i	$0.342848\pi$
$828$	0	0
$829$	8.80649e12	0.647601	0.323801	−	0.946125i	$-0.395039\pi$
$829$	8.80649e12	0.647601	0.323801	+	0.946125i	$0.395039\pi$
$830$	0	0
$831$	2.44558e12	0.177901
$832$	0	0
$833$	2.46745e12	0.177560
$834$	0	0
$835$	1.04127e13	0.741264
$836$	0	0
$837$	−2.01700e11	−0.0142050
$838$	0	0
$839$	1.84537e13	1.28574	0.642872	−	0.765974i	$-0.277743\pi$
$839$	1.84537e13	1.28574	0.642872	+	0.765974i	$0.277743\pi$
$840$	0	0
$841$	−3.93972e12	−0.271571
$842$	0	0
$843$	−2.19077e11	−0.0149408
$844$	0	0
$845$	−1.11566e13	−0.752796
$846$	0	0
$847$	−5.06588e12	−0.338205
$848$	0	0
$849$	1.37672e13	0.909410
$850$	0	0
$851$	−7.08548e12	−0.463112
$852$	0	0
$853$	1.83968e13	1.18980	0.594898	−	0.803801i	$-0.297192\pi$
$853$	1.83968e13	1.18980	0.594898	+	0.803801i	$0.297192\pi$
$854$	0	0
$855$	4.97733e12	0.318529
$856$	0	0
$857$	−6.78324e12	−0.429560	−0.214780	−	0.976662i	$-0.568903\pi$
$857$	−6.78324e12	−0.429560	−0.214780	+	0.976662i	$0.568903\pi$
$858$	0	0
$859$	−2.47177e13	−1.54895	−0.774477	−	0.632602i	$-0.781987\pi$
$859$	−2.47177e13	−1.54895	−0.774477	+	0.632602i	$0.781987\pi$
$860$	0	0
$861$	−1.69020e12	−0.104815
$862$	0	0
$863$	2.85222e13	1.75039	0.875193	−	0.483773i	$-0.160734\pi$
$863$	2.85222e13	1.75039	0.875193	+	0.483773i	$0.160734\pi$
$864$	0	0
$865$	2.79348e12	0.169657
$866$	0	0
$867$	8.22883e12	0.494598
$868$	0	0
$869$	−3.92182e12	−0.233291
$870$	0	0
$871$	9.47221e12	0.557660
$872$	0	0
$873$	3.67256e12	0.213996
$874$	0	0
$875$	−1.30177e13	−0.750757
$876$	0	0
$877$	4.00775e12	0.228772	0.114386	−	0.993436i	$-0.463510\pi$
$877$	4.00775e12	0.228772	0.114386	+	0.993436i	$0.463510\pi$
$878$	0	0
$879$	6.14646e12	0.347276
$880$	0	0
$881$	−1.66457e13	−0.930914	−0.465457	−	0.885070i	$-0.654110\pi$
$881$	−1.66457e13	−0.930914	−0.465457	+	0.885070i	$0.654110\pi$
$882$	0	0
$883$	−3.02477e13	−1.67444	−0.837218	−	0.546869i	$-0.815819\pi$
$883$	−3.02477e13	−1.67444	−0.837218	+	0.546869i	$0.815819\pi$
$884$	0	0
$885$	−9.61636e12	−0.526946
$886$	0	0
$887$	1.79172e13	0.971883	0.485941	−	0.873991i	$-0.338477\pi$
$887$	1.79172e13	0.971883	0.485941	+	0.873991i	$0.338477\pi$
$888$	0	0
$889$	1.04481e13	0.561021
$890$	0	0
$891$	−1.53017e12	−0.0813373
$892$	0	0
$893$	−1.88769e13	−0.993341
$894$	0	0
$895$	−2.16152e13	−1.12605
$896$	0	0
$897$	−1.57097e12	−0.0810216
$898$	0	0
$899$	−1.23378e12	−0.0629967
$900$	0	0
$901$	7.40628e11	0.0374402
$902$	0	0
$903$	1.09172e13	0.546408
$904$	0	0
$905$	1.36498e13	0.676406
$906$	0	0
$907$	−2.41416e13	−1.18450	−0.592248	−	0.805756i	$-0.701759\pi$
$907$	−2.41416e13	−1.18450	−0.592248	+	0.805756i	$0.701759\pi$
$908$	0	0
$909$	6.03056e12	0.292968
$910$	0	0
$911$	1.75577e12	0.0844568	0.0422284	−	0.999108i	$-0.486554\pi$
$911$	1.75577e12	0.0844568	0.0422284	+	0.999108i	$0.486554\pi$
$912$	0	0
$913$	−9.25174e12	−0.440661
$914$	0	0
$915$	1.17476e13	0.554057
$916$	0	0
$917$	−2.64928e13	−1.23727
$918$	0	0
$919$	1.49135e13	0.689701	0.344851	−	0.938658i	$-0.387930\pi$
$919$	1.49135e13	0.689701	0.344851	+	0.938658i	$0.387930\pi$
$920$	0	0
$921$	−1.33062e13	−0.609376
$922$	0	0
$923$	−2.36885e12	−0.107431
$924$	0	0
$925$	2.28430e12	0.102593
$926$	0	0
$927$	7.98520e11	0.0355163
$928$	0	0
$929$	1.18835e13	0.523450	0.261725	−	0.965143i	$-0.415709\pi$
$929$	1.18835e13	0.523450	0.261725	+	0.965143i	$0.415709\pi$
$930$	0	0
$931$	−1.06310e13	−0.463769
$932$	0	0
$933$	−1.91677e13	−0.828139
$934$	0	0
$935$	6.25891e12	0.267822
$936$	0	0
$937$	−1.47326e13	−0.624384	−0.312192	−	0.950019i	$-0.601063\pi$
$937$	−1.47326e13	−0.624384	−0.312192	+	0.950019i	$0.601063\pi$
$938$	0	0
$939$	1.64543e13	0.690693
$940$	0	0
$941$	−1.80614e13	−0.750927	−0.375463	−	0.926837i	$-0.622516\pi$
$941$	−1.80614e13	−0.750927	−0.375463	+	0.926837i	$0.622516\pi$
$942$	0	0
$943$	1.80593e12	0.0743701
$944$	0	0
$945$	−3.32239e12	−0.135521
$946$	0	0
$947$	−4.85911e12	−0.196328	−0.0981639	−	0.995170i	$-0.531297\pi$
$947$	−4.85911e12	−0.196328	−0.0981639	+	0.995170i	$0.531297\pi$
$948$	0	0
$949$	−7.84330e12	−0.313907
$950$	0	0
$951$	2.25233e13	0.892934
$952$	0	0
$953$	−9.86205e12	−0.387301	−0.193651	−	0.981071i	$-0.562033\pi$
$953$	−9.86205e12	−0.387301	−0.193651	+	0.981071i	$0.562033\pi$
$954$	0	0
$955$	−1.02675e13	−0.399438
$956$	0	0
$957$	−9.35985e12	−0.360716
$958$	0	0
$959$	1.68638e13	0.643831
$960$	0	0
$961$	−2.62956e13	−0.994552
$962$	0	0
$963$	−3.57908e12	−0.134107
$964$	0	0
$965$	−3.24796e13	−1.20569
$966$	0	0
$967$	1.32058e12	0.0485676	0.0242838	−	0.999705i	$-0.492269\pi$
$967$	1.32058e12	0.0485676	0.0242838	+	0.999705i	$0.492269\pi$
$968$	0	0
$969$	5.93189e12	0.216141
$970$	0	0
$971$	−1.01689e13	−0.367103	−0.183551	−	0.983010i	$-0.558759\pi$
$971$	−1.01689e13	−0.367103	−0.183551	+	0.983010i	$0.558759\pi$
$972$	0	0
$973$	7.28116e12	0.260431
$974$	0	0
$975$	5.06467e11	0.0179486
$976$	0	0
$977$	5.13095e13	1.80166	0.900829	−	0.434173i	$-0.142959\pi$
$977$	5.13095e13	1.80166	0.900829	+	0.434173i	$0.142959\pi$
$978$	0	0
$979$	−5.12658e12	−0.178363
$980$	0	0
$981$	−6.75863e12	−0.232996
$982$	0	0
$983$	−3.90789e13	−1.33491	−0.667455	−	0.744650i	$-0.732616\pi$
$983$	−3.90789e13	−1.33491	−0.667455	+	0.744650i	$0.732616\pi$
$984$	0	0
$985$	−4.78352e13	−1.61914
$986$	0	0
$987$	1.26004e13	0.422627
$988$	0	0
$989$	−1.16647e13	−0.387695
$990$	0	0
$991$	7.00140e11	0.0230597	0.0115298	−	0.999934i	$-0.496330\pi$
$991$	7.00140e11	0.0230597	0.0115298	+	0.999934i	$0.496330\pi$
$992$	0	0
$993$	−8.77814e11	−0.0286504
$994$	0	0
$995$	−1.95844e13	−0.633443
$996$	0	0
$997$	4.06223e13	1.30208	0.651038	−	0.759045i	$-0.274334\pi$
$997$	4.06223e13	1.30208	0.651038	+	0.759045i	$0.274334\pi$
$998$	0	0
$999$	9.39919e12	0.298570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.a.a.1.4	✓	4
4.3	odd	2		384.10.a.e.1.4	yes	4
8.3	odd	2		384.10.a.d.1.1	yes	4
8.5	even	2		384.10.a.h.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.a.a.1.4	✓	4	1.1	even	1	trivial
384.10.a.d.1.1	yes	4	8.3	odd	2
384.10.a.e.1.4	yes	4	4.3	odd	2
384.10.a.h.1.1	yes	4	8.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q-81.0000 q^{3} +1350.54 q^{5} +4629.00 q^{7} +6561.00 q^{9} +O(q^{10})\)	\(q-81.0000 q^{3} +1350.54 q^{5} +4629.00 q^{7} +6561.00 q^{9} -35546.7 q^{11} +48411.4 q^{13} -109394. q^{15} -130374. q^{17} +561717. q^{19} -374949. q^{21} +400621. q^{23} -129157. q^{25} -531441. q^{27} -3.25076e6 q^{29} +379535. q^{31} +2.87928e6 q^{33} +6.25167e6 q^{35} -1.76862e7 q^{37} -3.92133e6 q^{39} +4.50782e6 q^{41} -2.91165e7 q^{43} +8.86092e6 q^{45} -3.36056e7 q^{47} -1.89260e7 q^{49} +1.05603e7 q^{51} -5.68080e6 q^{53} -4.80074e7 q^{55} -4.54991e7 q^{57} +8.79057e7 q^{59} -1.07388e8 q^{61} +3.03709e7 q^{63} +6.53817e7 q^{65} +1.95661e8 q^{67} -3.24503e7 q^{69} -4.89316e7 q^{71} -1.62014e8 q^{73} +1.04617e7 q^{75} -1.64546e8 q^{77} +1.10329e8 q^{79} +4.30467e7 q^{81} +2.60270e8 q^{83} -1.76076e8 q^{85} +2.63311e8 q^{87} +1.44221e8 q^{89} +2.24096e8 q^{91} -3.07423e7 q^{93} +7.58623e8 q^{95} +5.59756e8 q^{97} -2.33222e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 324 q^{3} - 1728 q^{5} + 4840 q^{7} + 26244 q^{9}+O(q^{10}) \) 4 * q - 324 * q^3 - 1728 * q^5 + 4840 * q^7 + 26244 * q^9	\( 4 q - 324 q^{3} - 1728 q^{5} + 4840 q^{7} + 26244 q^{9} - 15824 q^{11} - 82440 q^{13} + 139968 q^{15} + 165912 q^{17} - 539904 q^{19} - 392040 q^{21} + 729680 q^{23} + 224812 q^{25} - 2125764 q^{27} - 1850864 q^{29} + 3197960 q^{31} + 1281744 q^{33} - 8574912 q^{35} - 4187992 q^{37} + 6677640 q^{39} + 227704 q^{41} - 1600352 q^{43} - 11337408 q^{45} + 18053904 q^{47} + 57728820 q^{49} - 13438872 q^{51} - 29418288 q^{53} + 45906816 q^{55} + 43732224 q^{57} - 38300048 q^{59} + 99764648 q^{61} + 31755240 q^{63} + 314120832 q^{65} + 183717008 q^{67} - 59104080 q^{69} + 181868080 q^{71} + 254539160 q^{73} - 18209772 q^{75} + 230564704 q^{77} + 831578184 q^{79} + 172186884 q^{81} + 687923952 q^{83} - 1041391104 q^{85} + 149919984 q^{87} + 627627272 q^{89} + 1018147632 q^{91} - 259034760 q^{93} + 2167118208 q^{95} - 889385880 q^{97} - 103821264 q^{99}+O(q^{100}) \) 4 * q - 324 * q^3 - 1728 * q^5 + 4840 * q^7 + 26244 * q^9 - 15824 * q^11 - 82440 * q^13 + 139968 * q^15 + 165912 * q^17 - 539904 * q^19 - 392040 * q^21 + 729680 * q^23 + 224812 * q^25 - 2125764 * q^27 - 1850864 * q^29 + 3197960 * q^31 + 1281744 * q^33 - 8574912 * q^35 - 4187992 * q^37 + 6677640 * q^39 + 227704 * q^41 - 1600352 * q^43 - 11337408 * q^45 + 18053904 * q^47 + 57728820 * q^49 - 13438872 * q^51 - 29418288 * q^53 + 45906816 * q^55 + 43732224 * q^57 - 38300048 * q^59 + 99764648 * q^61 + 31755240 * q^63 + 314120832 * q^65 + 183717008 * q^67 - 59104080 * q^69 + 181868080 * q^71 + 254539160 * q^73 - 18209772 * q^75 + 230564704 * q^77 + 831578184 * q^79 + 172186884 * q^81 + 687923952 * q^83 - 1041391104 * q^85 + 149919984 * q^87 + 627627272 * q^89 + 1018147632 * q^91 - 259034760 * q^93 + 2167118208 * q^95 - 889385880 * q^97 - 103821264 * q^99

Introduction

L-functions

Modular forms

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Database

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