LMFDB - Embedded newform 16.48.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,48,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 48, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 48);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 48 $
Character orbit:	$[\chi]$	$=$	16.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:	$223.852260248$
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} - x^{3} - 832803191366x^{2} + 3710135215485780x + 13175318942671469337000 $ x^4 - x^3 - 832803191366x^2 + 3710135215485780x + 13175318942671469337000
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{46}\cdot 3^{7}\cdot 5^{3}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-124721.$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.52034e11 q^{3} +4.23962e16 q^{5} +3.90714e19 q^{7} -3.47448e21 q^{9} +O(q^{10})$	$q-1.52034e11 q^{3} +4.23962e16 q^{5} +3.90714e19 q^{7} -3.47448e21 q^{9} -2.66558e24 q^{11} +2.26259e26 q^{13} -6.44566e27 q^{15} +8.41583e28 q^{17} -3.16735e29 q^{19} -5.94018e30 q^{21} -2.26985e31 q^{23} +1.08689e33 q^{25} +4.57064e33 q^{27} -3.39685e34 q^{29} -1.81646e35 q^{31} +4.05259e35 q^{33} +1.65648e36 q^{35} -7.13789e36 q^{37} -3.43991e37 q^{39} -2.78383e37 q^{41} +4.92777e37 q^{43} -1.47305e38 q^{45} -1.86825e39 q^{47} -3.71676e39 q^{49} -1.27949e40 q^{51} -2.48904e40 q^{53} -1.13011e41 q^{55} +4.81545e40 q^{57} -3.69239e40 q^{59} +3.52589e41 q^{61} -1.35753e41 q^{63} +9.59253e42 q^{65} +1.26183e42 q^{67} +3.45094e42 q^{69} +3.26023e43 q^{71} +3.59179e43 q^{73} -1.65245e44 q^{75} -1.04148e44 q^{77} +5.82596e44 q^{79} -6.02511e44 q^{81} +3.51619e44 q^{83} +3.56799e45 q^{85} +5.16437e45 q^{87} -2.92401e45 q^{89} +8.84027e45 q^{91} +2.76164e46 q^{93} -1.34284e46 q^{95} +2.17182e46 q^{97} +9.26152e45 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 38461494960 q^{3} - 31\!\cdots\!00 q^{5}+ \cdots - 17\!\cdots\!72 q^{9}+O(q^{10}) $ 4 * q - 38461494960 * q^3 - 31114680242272200 * q^5 + 39169218725888423200 * q^7 - 17071972417358142200172 * q^9	$ 4 q - 38461494960 q^{3} - 31\!\cdots\!00 q^{5}+ \cdots - 53\!\cdots\!16 q^{99}+O(q^{100}) $ 4 * q - 38461494960 * q^3 - 31114680242272200 * q^5 + 39169218725888423200 * q^7 - 17071972417358142200172 * q^9 + 1902165515717740404742512 * q^11 + 126959149195199610466858520 * q^13 - 12437812896841241272880733600 * q^15 + 210426285758836687099688769480 * q^17 + 1059119811802437301721159140240 * q^19 - 26054906182888686128940663165312 * q^21 - 137531418344493639994288859179680 * q^23 + 1009909900344396506188154274977500 * q^25 + 7345952425324536073931587214783520 * q^27 - 22743132490172744184500850941726760 * q^29 - 75786540814706434901630981303934848 * q^31 + 263381387470144952341649359926294720 * q^33 + 1313148822925052387484546800067979200 * q^35 - 11291457045120157869429951769632191560 * q^37 - 39294375338503805226442587829551714336 * q^39 + 130658126291089294119301677048020391528 * q^41 + 445357287738427943170332944488133090800 * q^43 + 86022747271219179966948746072020632600 * q^45 - 2031912465958423919194866366997063590720 * q^47 + 9153745744156628626499502447643951451172 * q^49 + 18534366050002170683962116343746386369952 * q^51 + 29171719896619533806321793123448457437560 * q^53 - 192172580604633896452606078657416872901600 * q^55 + 558743644452458098451223285784560964733760 * q^57 - 478242858216320686410790860443879577596880 * q^59 + 626683213833907852009669690320782900857688 * q^61 - 5811636950821577036406237472810763671724640 * q^63 + 12996735432890074297558542478104126872331600 * q^65 - 18701411419127220668869505853739529194216880 * q^67 - 2096956664577882726873494616728974107851904 * q^69 - 22196594617762006008028198712153461686540768 * q^71 + 104911201720506650033573412943840886878191080 * q^73 - 32388703324174028323482311940739905891330000 * q^75 - 263490744376507692433815778071300370399209600 * q^77 + 1327995310109394234162785254414226288730352960 * q^79 - 1061448205755511803861508510986654404170081116 * q^81 + 1439130627532364606968733068855111784504073360 * q^83 - 1001895286410663502447652611446390227458237200 * q^85 + 4326251093943206485496474339443178030108648160 * q^87 - 7960590522674945421108320896849737731421281880 * q^89 - 5325194479227227497182497798631911205603381568 * q^91 + 10728611999875738337226031496188630272675064320 * q^93 - 75421152415106939771624998522249518123547668000 * q^95 + 95066248961258904822565865801364599382905757320 * q^97 - 530900415919166351015781326273177630089784016 * 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
$3$	−1.52034e11	−0.932376	−0.466188	−	0.884686i	$-0.654373\pi$
$3$	−1.52034e11	−0.932376	−0.466188	+	0.884686i	$0.654373\pi$
$4$	0	0
$5$	4.23962e16	1.59049	0.795246	−	0.606286i	$-0.207341\pi$
$5$	4.23962e16	1.59049	0.795246	+	0.606286i	$0.207341\pi$
$6$	0	0
$7$	3.90714e19	0.539579	0.269790	−	0.962919i	$-0.413046\pi$
$7$	3.90714e19	0.539579	0.269790	+	0.962919i	$0.413046\pi$
$8$	0	0
$9$	−3.47448e21	−0.130675
$10$	0	0
$11$	−2.66558e24	−0.897561	−0.448781	−	0.893642i	$-0.648142\pi$
$11$	−2.66558e24	−0.897561	−0.448781	+	0.893642i	$0.648142\pi$
$12$	0	0
$13$	2.26259e26	1.50293	0.751463	−	0.659776i	$-0.229349\pi$
$13$	2.26259e26	1.50293	0.751463	+	0.659776i	$0.229349\pi$
$14$	0	0
$15$	−6.44566e27	−1.48294
$16$	0	0
$17$	8.41583e28	1.02223	0.511114	−	0.859513i	$-0.329233\pi$
$17$	8.41583e28	1.02223	0.511114	+	0.859513i	$0.329233\pi$
$18$	0	0
$19$	−3.16735e29	−0.281830	−0.140915	−	0.990022i	$-0.545004\pi$
$19$	−3.16735e29	−0.281830	−0.140915	+	0.990022i	$0.545004\pi$
$20$	0	0
$21$	−5.94018e30	−0.503091
$22$	0	0
$23$	−2.26985e31	−0.226669	−0.113335	−	0.993557i	$-0.536153\pi$
$23$	−2.26985e31	−0.226669	−0.113335	+	0.993557i	$0.536153\pi$
$24$	0	0
$25$	1.08689e33	1.52967
$26$	0	0
$27$	4.57064e33	1.05421
$28$	0	0
$29$	−3.39685e34	−1.46125	−0.730624	−	0.682780i	$-0.760771\pi$
$29$	−3.39685e34	−1.46125	−0.730624	+	0.682780i	$0.760771\pi$
$30$	0	0
$31$	−1.81646e35	−1.63015	−0.815074	−	0.579357i	$-0.803304\pi$
$31$	−1.81646e35	−1.63015	−0.815074	+	0.579357i	$0.803304\pi$
$32$	0	0
$33$	4.05259e35	0.836865
$34$	0	0
$35$	1.65648e36	0.858197
$36$	0	0
$37$	−7.13789e36	−1.00191	−0.500956	−	0.865473i	$-0.667018\pi$
$37$	−7.13789e36	−1.00191	−0.500956	+	0.865473i	$0.667018\pi$
$38$	0	0
$39$	−3.43991e37	−1.40129
$40$	0	0
$41$	−2.78383e37	−0.350124	−0.175062	−	0.984557i	$-0.556013\pi$
$41$	−2.78383e37	−0.350124	−0.175062	+	0.984557i	$0.556013\pi$
$42$	0	0
$43$	4.92777e37	0.202368	0.101184	−	0.994868i	$-0.467737\pi$
$43$	4.92777e37	0.202368	0.101184	+	0.994868i	$0.467737\pi$
$44$	0	0
$45$	−1.47305e38	−0.207837
$46$	0	0
$47$	−1.86825e39	−0.948716	−0.474358	−	0.880332i	$-0.657320\pi$
$47$	−1.86825e39	−0.948716	−0.474358	+	0.880332i	$0.657320\pi$
$48$	0	0
$49$	−3.71676e39	−0.708854
$50$	0	0
$51$	−1.27949e40	−0.953100
$52$	0	0
$53$	−2.48904e40	−0.750843	−0.375421	−	0.926854i	$-0.622502\pi$
$53$	−2.48904e40	−0.750843	−0.375421	+	0.926854i	$0.622502\pi$
$54$	0	0
$55$	−1.13011e41	−1.42756
$56$	0	0
$57$	4.81545e40	0.262771
$58$	0	0
$59$	−3.69239e40	−0.0895953	−0.0447976	−	0.998996i	$-0.514264\pi$
$59$	−3.69239e40	−0.0895953	−0.0447976	+	0.998996i	$0.514264\pi$
$60$	0	0
$61$	3.52589e41	0.390857	0.195429	−	0.980718i	$-0.437390\pi$
$61$	3.52589e41	0.390857	0.195429	+	0.980718i	$0.437390\pi$
$62$	0	0
$63$	−1.35753e41	−0.0705093
$64$	0	0
$65$	9.59253e42	2.39039
$66$	0	0
$67$	1.26183e42	0.154257	0.0771283	−	0.997021i	$-0.475425\pi$
$67$	1.26183e42	0.154257	0.0771283	+	0.997021i	$0.475425\pi$
$68$	0	0
$69$	3.45094e42	0.211341
$70$	0	0
$71$	3.26023e43	1.02018	0.510090	−	0.860121i	$-0.329612\pi$
$71$	3.26023e43	1.02018	0.510090	+	0.860121i	$0.329612\pi$
$72$	0	0
$73$	3.59179e43	0.585092	0.292546	−	0.956251i	$-0.405498\pi$
$73$	3.59179e43	0.585092	0.292546	+	0.956251i	$0.405498\pi$
$74$	0	0
$75$	−1.65245e44	−1.42623
$76$	0	0
$77$	−1.04148e44	−0.484305
$78$	0	0
$79$	5.82596e44	1.48297	0.741483	−	0.670972i	$-0.234123\pi$
$79$	5.82596e44	1.48297	0.741483	+	0.670972i	$0.234123\pi$
$80$	0	0
$81$	−6.02511e44	−0.852250
$82$	0	0
$83$	3.51619e44	0.280373	0.140187	−	0.990125i	$-0.455230\pi$
$83$	3.51619e44	0.280373	0.140187	+	0.990125i	$0.455230\pi$
$84$	0	0
$85$	3.56799e45	1.62584
$86$	0	0
$87$	5.16437e45	1.36243
$88$	0	0
$89$	−2.92401e45	−0.452182	−0.226091	−	0.974106i	$-0.572595\pi$
$89$	−2.92401e45	−0.452182	−0.226091	+	0.974106i	$0.572595\pi$
$90$	0	0
$91$	8.84027e45	0.810947
$92$	0	0
$93$	2.76164e46	1.51991
$94$	0	0
$95$	−1.34284e46	−0.448248
$96$	0	0
$97$	2.17182e46	0.444311	0.222156	−	0.975011i	$-0.428691\pi$
$97$	2.17182e46	0.444311	0.222156	+	0.975011i	$0.428691\pi$
$98$	0	0
$99$	9.26152e45	0.117288
$100$	0	0
$101$	−5.11557e46	−0.404894	−0.202447	−	0.979293i	$-0.564889\pi$
$101$	−5.11557e46	−0.404894	−0.202447	+	0.979293i	$0.564889\pi$
$102$	0	0
$103$	−3.19569e47	−1.59547	−0.797736	−	0.603006i	$-0.793969\pi$
$103$	−3.19569e47	−1.59547	−0.797736	+	0.603006i	$0.793969\pi$
$104$	0	0
$105$	−2.51841e47	−0.800162
$106$	0	0
$107$	−2.00584e47	−0.409051	−0.204526	−	0.978861i	$-0.565565\pi$
$107$	−2.00584e47	−0.409051	−0.204526	+	0.978861i	$0.565565\pi$
$108$	0	0
$109$	−5.86410e47	−0.773888	−0.386944	−	0.922103i	$-0.626469\pi$
$109$	−5.86410e47	−0.773888	−0.386944	+	0.922103i	$0.626469\pi$
$110$	0	0
$111$	1.08520e48	0.934158
$112$	0	0
$113$	3.11593e47	0.176297	0.0881484	−	0.996107i	$-0.471905\pi$
$113$	3.11593e47	0.176297	0.0881484	+	0.996107i	$0.471905\pi$
$114$	0	0
$115$	−9.62329e47	−0.360516
$116$	0	0
$117$	−7.86134e47	−0.196394
$118$	0	0
$119$	3.28818e48	0.551572
$120$	0	0
$121$	−1.71442e48	−0.194384
$122$	0	0
$123$	4.23236e48	0.326447
$124$	0	0
$125$	1.59559e49	0.842432
$126$	0	0
$127$	−2.66165e49	−0.967749	−0.483875	−	0.875137i	$-0.660771\pi$
$127$	−2.66165e49	−0.967749	−0.483875	+	0.875137i	$0.660771\pi$
$128$	0	0
$129$	−7.49188e48	−0.188683
$130$	0	0
$131$	−6.34924e49	−1.11390	−0.556950	−	0.830546i	$-0.688028\pi$
$131$	−6.34924e49	−1.11390	−0.556950	+	0.830546i	$0.688028\pi$
$132$	0	0
$133$	−1.23753e49	−0.152069
$134$	0	0
$135$	1.93778e50	1.67672
$136$	0	0
$137$	1.99734e50	1.22326	0.611631	−	0.791143i	$-0.290514\pi$
$137$	1.99734e50	1.22326	0.611631	+	0.791143i	$0.290514\pi$
$138$	0	0
$139$	3.16608e50	1.37935	0.689674	−	0.724120i	$-0.257754\pi$
$139$	3.16608e50	1.37935	0.689674	+	0.724120i	$0.257754\pi$
$140$	0	0
$141$	2.84038e50	0.884560
$142$	0	0
$143$	−6.03113e50	−1.34897
$144$	0	0
$145$	−1.44014e51	−2.32410
$146$	0	0
$147$	5.65074e50	0.660919
$148$	0	0
$149$	1.00267e50	0.0853651	0.0426826	−	0.999089i	$-0.486410\pi$
$149$	1.00267e50	0.0853651	0.0426826	+	0.999089i	$0.486410\pi$
$150$	0	0
$151$	2.40479e49	0.0149664	0.00748321	−	0.999972i	$-0.497618\pi$
$151$	2.40479e49	0.0149664	0.00748321	+	0.999972i	$0.497618\pi$
$152$	0	0
$153$	−2.92407e50	−0.133579
$154$	0	0
$155$	−7.70111e51	−2.59274
$156$	0	0
$157$	−1.42274e51	−0.354391	−0.177196	−	0.984176i	$-0.556702\pi$
$157$	−1.42274e51	−0.354391	−0.177196	+	0.984176i	$0.556702\pi$
$158$	0	0
$159$	3.78419e51	0.700068
$160$	0	0
$161$	−8.86862e50	−0.122306
$162$	0	0
$163$	4.79256e51	0.494492	0.247246	−	0.968953i	$-0.420475\pi$
$163$	4.79256e51	0.494492	0.247246	+	0.968953i	$0.420475\pi$
$164$	0	0
$165$	1.71814e52	1.33103
$166$	0	0
$167$	9.89736e51	0.577673	0.288837	−	0.957378i	$-0.406732\pi$
$167$	9.89736e51	0.577673	0.288837	+	0.957378i	$0.406732\pi$
$168$	0	0
$169$	2.85292e52	1.25878
$170$	0	0
$171$	1.10049e51	0.0368280
$172$	0	0
$173$	−7.43991e52	−1.89446	−0.947228	−	0.320562i	$-0.896128\pi$
$173$	−7.43991e52	−1.89446	−0.947228	+	0.320562i	$0.896128\pi$
$174$	0	0
$175$	4.24665e52	0.825377
$176$	0	0
$177$	5.61369e51	0.0835365
$178$	0	0
$179$	−3.60632e52	−0.412115	−0.206057	−	0.978540i	$-0.566063\pi$
$179$	−3.60632e52	−0.412115	−0.206057	+	0.978540i	$0.566063\pi$
$180$	0	0
$181$	6.19557e52	0.545299	0.272650	−	0.962113i	$-0.412100\pi$
$181$	6.19557e52	0.545299	0.272650	+	0.962113i	$0.412100\pi$
$182$	0	0
$183$	−5.36056e52	−0.364426
$184$	0	0
$185$	−3.02619e53	−1.59353
$186$	0	0
$187$	−2.24331e53	−0.917511
$188$	0	0
$189$	1.78581e53	0.568832
$190$	0	0
$191$	3.09862e53	0.770699	0.385349	−	0.922771i	$-0.374081\pi$
$191$	3.09862e53	0.770699	0.385349	+	0.922771i	$0.374081\pi$
$192$	0	0
$193$	4.75475e53	0.925829	0.462914	−	0.886403i	$-0.346804\pi$
$193$	4.75475e53	0.925829	0.462914	+	0.886403i	$0.346804\pi$
$194$	0	0
$195$	−1.45839e54	−2.22874
$196$	0	0
$197$	−7.61644e53	−0.915790	−0.457895	−	0.889006i	$-0.651396\pi$
$197$	−7.61644e53	−0.915790	−0.457895	+	0.889006i	$0.651396\pi$
$198$	0	0
$199$	1.90631e54	1.80778	0.903890	−	0.427765i	$-0.140699\pi$
$199$	1.90631e54	1.80778	0.903890	+	0.427765i	$0.140699\pi$
$200$	0	0
$201$	−1.91841e53	−0.143825
$202$	0	0
$203$	−1.32720e54	−0.788459
$204$	0	0
$205$	−1.18024e54	−0.556869
$206$	0	0
$207$	7.88655e52	0.0296199
$208$	0	0
$209$	8.44284e53	0.252959
$210$	0	0
$211$	−1.25520e54	−0.300660	−0.150330	−	0.988636i	$-0.548034\pi$
$211$	−1.25520e54	−0.300660	−0.150330	+	0.988636i	$0.548034\pi$
$212$	0	0
$213$	−4.95665e54	−0.951191
$214$	0	0
$215$	2.08918e54	0.321864
$216$	0	0
$217$	−7.09718e54	−0.879594
$218$	0	0
$219$	−5.46075e54	−0.545526
$220$	0	0
$221$	1.90416e55	1.53633
$222$	0	0
$223$	1.43141e54	0.0934545	0.0467272	−	0.998908i	$-0.485121\pi$
$223$	1.43141e54	0.0934545	0.0467272	+	0.998908i	$0.485121\pi$
$224$	0	0
$225$	−3.77639e54	−0.199889
$226$	0	0
$227$	−3.38487e55	−1.45524	−0.727619	−	0.685982i	$-0.759373\pi$
$227$	−3.38487e55	−1.45524	−0.727619	+	0.685982i	$0.759373\pi$
$228$	0	0
$229$	−4.65218e54	−0.162750	−0.0813752	−	0.996684i	$-0.525931\pi$
$229$	−4.65218e54	−0.162750	−0.0813752	+	0.996684i	$0.525931\pi$
$230$	0	0
$231$	1.58341e55	0.451555
$232$	0	0
$233$	−6.87236e55	−1.60045	−0.800225	−	0.599700i	$-0.795287\pi$
$233$	−6.87236e55	−1.60045	−0.800225	+	0.599700i	$0.795287\pi$
$234$	0	0
$235$	−7.92068e55	−1.50893
$236$	0	0
$237$	−8.85744e55	−1.38268
$238$	0	0
$239$	−1.09219e56	−1.39942	−0.699710	−	0.714427i	$-0.746687\pi$
$239$	−1.09219e56	−1.39942	−0.699710	+	0.714427i	$0.746687\pi$
$240$	0	0
$241$	−4.82622e55	−0.508401	−0.254201	−	0.967152i	$-0.581812\pi$
$241$	−4.82622e55	−0.508401	−0.254201	+	0.967152i	$0.581812\pi$
$242$	0	0
$243$	−2.99259e55	−0.259597
$244$	0	0
$245$	−1.57577e56	−1.12743
$246$	0	0
$247$	−7.16642e55	−0.423569
$248$	0	0
$249$	−5.34580e55	−0.261414
$250$	0	0
$251$	4.06217e56	1.64599	0.822993	−	0.568052i	$-0.192303\pi$
$251$	4.06217e56	1.64599	0.822993	+	0.568052i	$0.192303\pi$
$252$	0	0
$253$	6.05047e55	0.203450
$254$	0	0
$255$	−5.42456e56	−1.51590
$256$	0	0
$257$	1.93698e56	0.450501	0.225251	−	0.974301i	$-0.427680\pi$
$257$	1.93698e56	0.450501	0.225251	+	0.974301i	$0.427680\pi$
$258$	0	0
$259$	−2.78887e56	−0.540611
$260$	0	0
$261$	1.18023e56	0.190948
$262$	0	0
$263$	−8.33659e56	−1.12727	−0.563636	−	0.826023i	$-0.690598\pi$
$263$	−8.33659e56	−1.12727	−0.563636	+	0.826023i	$0.690598\pi$
$264$	0	0
$265$	−1.05526e57	−1.19421
$266$	0	0
$267$	4.44548e56	0.421604
$268$	0	0
$269$	−1.88490e57	−1.50006	−0.750032	−	0.661402i	$-0.769962\pi$
$269$	−1.88490e57	−1.50006	−0.750032	+	0.661402i	$0.769962\pi$
$270$	0	0
$271$	−2.56612e57	−1.71593	−0.857963	−	0.513712i	$-0.828270\pi$
$271$	−2.56612e57	−1.71593	−0.857963	+	0.513712i	$0.828270\pi$
$272$	0	0
$273$	−1.34402e57	−0.756108
$274$	0	0
$275$	−2.89721e57	−1.37297
$276$	0	0
$277$	−8.36679e56	−0.334414	−0.167207	−	0.985922i	$-0.553475\pi$
$277$	−8.36679e56	−0.334414	−0.167207	+	0.985922i	$0.553475\pi$
$278$	0	0
$279$	6.31127e56	0.213019
$280$	0	0
$281$	−3.96974e57	−1.13283	−0.566414	−	0.824121i	$-0.691670\pi$
$281$	−3.96974e57	−1.13283	−0.566414	+	0.824121i	$0.691670\pi$
$282$	0	0
$283$	4.45361e57	1.07580	0.537900	−	0.843009i	$-0.319218\pi$
$283$	4.45361e57	1.07580	0.537900	+	0.843009i	$0.319218\pi$
$284$	0	0
$285$	2.04157e57	0.417936
$286$	0	0
$287$	−1.08768e57	−0.188919
$288$	0	0
$289$	3.04657e56	0.0449481
$290$	0	0
$291$	−3.30190e57	−0.414265
$292$	0	0
$293$	1.05167e58	1.12328	0.561641	−	0.827381i	$-0.310170\pi$
$293$	1.05167e58	1.12328	0.561641	+	0.827381i	$0.310170\pi$
$294$	0	0
$295$	−1.56543e57	−0.142501
$296$	0	0
$297$	−1.21834e58	−0.946222
$298$	0	0
$299$	−5.13574e57	−0.340667
$300$	0	0
$301$	1.92535e57	0.109193
$302$	0	0
$303$	7.77740e57	0.377514
$304$	0	0
$305$	1.49484e58	0.621656
$306$	0	0
$307$	−5.43218e57	−0.193741	−0.0968707	−	0.995297i	$-0.530883\pi$
$307$	−5.43218e57	−0.193741	−0.0968707	+	0.995297i	$0.530883\pi$
$308$	0	0
$309$	4.85853e58	1.48758
$310$	0	0
$311$	4.80030e58	1.26299	0.631493	−	0.775381i	$-0.282442\pi$
$311$	4.80030e58	1.26299	0.631493	+	0.775381i	$0.282442\pi$
$312$	0	0
$313$	−2.50321e58	−0.566507	−0.283253	−	0.959045i	$-0.591414\pi$
$313$	−2.50321e58	−0.566507	−0.283253	+	0.959045i	$0.591414\pi$
$314$	0	0
$315$	−5.75541e57	−0.112145
$316$	0	0
$317$	−6.65301e58	−1.11719	−0.558593	−	0.829442i	$-0.688659\pi$
$317$	−6.65301e58	−1.11719	−0.558593	+	0.829442i	$0.688659\pi$
$318$	0	0
$319$	9.05460e58	1.31156
$320$	0	0
$321$	3.04956e58	0.381390
$322$	0	0
$323$	−2.66559e58	−0.288094
$324$	0	0
$325$	2.45920e59	2.29898
$326$	0	0
$327$	8.91542e58	0.721555
$328$	0	0
$329$	−7.29953e58	−0.511907
$330$	0	0
$331$	1.52984e58	0.0930438	0.0465219	−	0.998917i	$-0.485186\pi$
$331$	1.52984e58	0.0930438	0.0465219	+	0.998917i	$0.485186\pi$
$332$	0	0
$333$	2.48005e58	0.130924
$334$	0	0
$335$	5.34969e58	0.245344
$336$	0	0
$337$	−1.19696e59	−0.477284	−0.238642	−	0.971108i	$-0.576702\pi$
$337$	−1.19696e59	−0.477284	−0.238642	+	0.971108i	$0.576702\pi$
$338$	0	0
$339$	−4.73727e58	−0.164375
$340$	0	0
$341$	4.84194e59	1.46316
$342$	0	0
$343$	−3.50084e59	−0.922062
$344$	0	0
$345$	1.46307e59	0.336136
$346$	0	0
$347$	6.60768e59	1.32528	0.662638	−	0.748940i	$-0.269437\pi$
$347$	6.60768e59	1.32528	0.662638	+	0.748940i	$0.269437\pi$
$348$	0	0
$349$	7.13810e59	1.25079	0.625396	−	0.780308i	$-0.284937\pi$
$349$	7.13810e59	1.25079	0.625396	+	0.780308i	$0.284937\pi$
$350$	0	0
$351$	1.03415e60	1.58441
$352$	0	0
$353$	4.98386e58	0.0668129	0.0334064	−	0.999442i	$-0.489364\pi$
$353$	4.98386e58	0.0668129	0.0334064	+	0.999442i	$0.489364\pi$
$354$	0	0
$355$	1.38221e60	1.62259
$356$	0	0
$357$	−4.99916e59	−0.514273
$358$	0	0
$359$	−2.08989e60	−1.88540	−0.942700	−	0.333641i	$-0.891723\pi$
$359$	−2.08989e60	−1.88540	−0.942700	+	0.333641i	$0.891723\pi$
$360$	0	0
$361$	−1.16273e60	−0.920572
$362$	0	0
$363$	2.60649e59	0.181239
$364$	0	0
$365$	1.52278e60	0.930585
$366$	0	0
$367$	1.22059e59	0.0656021	0.0328011	−	0.999462i	$-0.489557\pi$
$367$	1.22059e59	0.0656021	0.0328011	+	0.999462i	$0.489557\pi$
$368$	0	0
$369$	9.67236e58	0.0457523
$370$	0	0
$371$	−9.72503e59	−0.405139
$372$	0	0
$373$	−2.41910e60	−0.888172	−0.444086	−	0.895984i	$-0.646472\pi$
$373$	−2.41910e60	−0.888172	−0.444086	+	0.895984i	$0.646472\pi$
$374$	0	0
$375$	−2.42583e60	−0.785463
$376$	0	0
$377$	−7.68569e60	−2.19615
$378$	0	0
$379$	−6.29795e60	−1.58920	−0.794598	−	0.607135i	$-0.792319\pi$
$379$	−6.29795e60	−1.58920	−0.794598	+	0.607135i	$0.792319\pi$
$380$	0	0
$381$	4.04661e60	0.902306
$382$	0	0
$383$	−3.88964e60	−0.766898	−0.383449	−	0.923562i	$-0.625264\pi$
$383$	−3.88964e60	−0.766898	−0.383449	+	0.923562i	$0.625264\pi$
$384$	0	0
$385$	−4.41548e60	−0.770284
$386$	0	0
$387$	−1.71214e59	−0.0264443
$388$	0	0
$389$	5.15774e60	0.705739	0.352869	−	0.935673i	$-0.385206\pi$
$389$	5.15774e60	0.705739	0.352869	+	0.935673i	$0.385206\pi$
$390$	0	0
$391$	−1.91027e60	−0.231707
$392$	0	0
$393$	9.65300e60	1.03857
$394$	0	0
$395$	2.46999e61	2.35865
$396$	0	0
$397$	−1.92619e61	−1.63351	−0.816754	−	0.576986i	$-0.804229\pi$
$397$	−1.92619e61	−1.63351	−0.816754	+	0.576986i	$0.804229\pi$
$398$	0	0
$399$	1.88146e60	0.141786
$400$	0	0
$401$	5.06490e58	0.00339374	0.00169687	−	0.999999i	$-0.499460\pi$
$401$	5.06490e58	0.00339374	0.00169687	+	0.999999i	$0.499460\pi$
$402$	0	0
$403$	−4.10992e61	−2.44999
$404$	0	0
$405$	−2.55442e61	−1.35550
$406$	0	0
$407$	1.90266e61	0.899277
$408$	0	0
$409$	1.50217e61	0.632733	0.316367	−	0.948637i	$-0.397537\pi$
$409$	1.50217e61	0.632733	0.316367	+	0.948637i	$0.397537\pi$
$410$	0	0
$411$	−3.03664e61	−1.14054
$412$	0	0
$413$	−1.44267e60	−0.0483437
$414$	0	0
$415$	1.49073e61	0.445932
$416$	0	0
$417$	−4.81352e61	−1.28607
$418$	0	0
$419$	−6.95944e61	−1.66167	−0.830835	−	0.556518i	$-0.812137\pi$
$419$	−6.95944e61	−1.66167	−0.830835	+	0.556518i	$0.812137\pi$
$420$	0	0
$421$	−3.78966e61	−0.809042	−0.404521	−	0.914529i	$-0.632562\pi$
$421$	−3.78966e61	−0.809042	−0.404521	+	0.914529i	$0.632562\pi$
$422$	0	0
$423$	6.49121e60	0.123973
$424$	0	0
$425$	9.14712e61	1.56367
$426$	0	0
$427$	1.37762e61	0.210898
$428$	0	0
$429$	9.16936e61	1.25775
$430$	0	0
$431$	−4.06747e60	−0.0500160	−0.0250080	−	0.999687i	$-0.507961\pi$
$431$	−4.06747e60	−0.0500160	−0.0250080	+	0.999687i	$0.507961\pi$
$432$	0	0
$433$	−2.59125e61	−0.285788	−0.142894	−	0.989738i	$-0.545641\pi$
$433$	−2.59125e61	−0.285788	−0.142894	+	0.989738i	$0.545641\pi$
$434$	0	0
$435$	2.18950e62	2.16694
$436$	0	0
$437$	7.18941e60	0.0638821
$438$	0	0
$439$	−3.51058e61	−0.280196	−0.140098	−	0.990138i	$-0.544742\pi$
$439$	−3.51058e61	−0.280196	−0.140098	+	0.990138i	$0.544742\pi$
$440$	0	0
$441$	1.29138e61	0.0926293
$442$	0	0
$443$	4.43894e61	0.286281	0.143140	−	0.989702i	$-0.454280\pi$
$443$	4.43894e61	0.286281	0.143140	+	0.989702i	$0.454280\pi$
$444$	0	0
$445$	−1.23967e62	−0.719192
$446$	0	0
$447$	−1.52440e61	−0.0795924
$448$	0	0
$449$	3.43198e62	1.61345	0.806725	−	0.590927i	$-0.201238\pi$
$449$	3.43198e62	1.61345	0.806725	+	0.590927i	$0.201238\pi$
$450$	0	0
$451$	7.42052e61	0.314257
$452$	0	0
$453$	−3.65609e60	−0.0139543
$454$	0	0
$455$	3.74794e62	1.28981
$456$	0	0
$457$	−9.64466e60	−0.0299403	−0.0149701	−	0.999888i	$-0.504765\pi$
$457$	−9.64466e60	−0.0299403	−0.0149701	+	0.999888i	$0.504765\pi$
$458$	0	0
$459$	3.84658e62	1.07765
$460$	0	0
$461$	−6.32143e61	−0.159898	−0.0799491	−	0.996799i	$-0.525476\pi$
$461$	−6.32143e61	−0.159898	−0.0799491	+	0.996799i	$0.525476\pi$
$462$	0	0
$463$	−6.86213e62	−1.56785	−0.783927	−	0.620853i	$-0.786786\pi$
$463$	−6.86213e62	−1.56785	−0.783927	+	0.620853i	$0.786786\pi$
$464$	0	0
$465$	1.17083e63	2.41741
$466$	0	0
$467$	2.91371e62	0.543876	0.271938	−	0.962315i	$-0.412335\pi$
$467$	2.91371e62	0.543876	0.271938	+	0.962315i	$0.412335\pi$
$468$	0	0
$469$	4.93016e61	0.0832337
$470$	0	0
$471$	2.16304e62	0.330426
$472$	0	0
$473$	−1.31354e62	−0.181637
$474$	0	0
$475$	−3.44258e62	−0.431106
$476$	0	0
$477$	8.64813e61	0.0981161
$478$	0	0
$479$	9.39830e62	0.966417	0.483209	−	0.875505i	$-0.339471\pi$
$479$	9.39830e62	0.966417	0.483209	+	0.875505i	$0.339471\pi$
$480$	0	0
$481$	−1.61501e63	−1.50580
$482$	0	0
$483$	1.34833e62	0.114035
$484$	0	0
$485$	9.20767e62	0.706674
$486$	0	0
$487$	−2.39350e63	−1.66764	−0.833820	−	0.552037i	$-0.813851\pi$
$487$	−2.39350e63	−1.66764	−0.833820	+	0.552037i	$0.813851\pi$
$488$	0	0
$489$	−7.28632e62	−0.461052
$490$	0	0
$491$	−1.23586e63	−0.710481	−0.355241	−	0.934775i	$-0.615601\pi$
$491$	−1.23586e63	−0.710481	−0.355241	+	0.934775i	$0.615601\pi$
$492$	0	0
$493$	−2.85874e63	−1.49373
$494$	0	0
$495$	3.92653e62	0.186546
$496$	0	0
$497$	1.27382e63	0.550468
$498$	0	0
$499$	2.94657e63	1.15865	0.579327	−	0.815095i	$-0.303316\pi$
$499$	2.94657e63	1.15865	0.579327	+	0.815095i	$0.303316\pi$
$500$	0	0
$501$	−1.50473e63	−0.538609
$502$	0	0
$503$	1.27808e63	0.416591	0.208296	−	0.978066i	$-0.433208\pi$
$503$	1.27808e63	0.416591	0.208296	+	0.978066i	$0.433208\pi$
$504$	0	0
$505$	−2.16880e63	−0.643981
$506$	0	0
$507$	−4.33740e63	−1.17366
$508$	0	0
$509$	7.25177e63	1.78886	0.894429	−	0.447209i	$-0.147582\pi$
$509$	7.25177e63	1.78886	0.894429	+	0.447209i	$0.147582\pi$
$510$	0	0
$511$	1.40336e63	0.315704
$512$	0	0
$513$	−1.44768e63	−0.297109
$514$	0	0
$515$	−1.35485e64	−2.53759
$516$	0	0
$517$	4.97998e63	0.851531
$518$	0	0
$519$	1.13112e64	1.76634
$520$	0	0
$521$	−9.84138e63	−1.40401	−0.702004	−	0.712173i	$-0.747711\pi$
$521$	−9.84138e63	−1.40401	−0.702004	+	0.712173i	$0.747711\pi$
$522$	0	0
$523$	−4.91409e62	−0.0640698	−0.0320349	−	0.999487i	$-0.510199\pi$
$523$	−4.91409e62	−0.0640698	−0.0320349	+	0.999487i	$0.510199\pi$
$524$	0	0
$525$	−6.45635e63	−0.769561
$526$	0	0
$527$	−1.52871e64	−1.66638
$528$	0	0
$529$	−9.51264e63	−0.948621
$530$	0	0
$531$	1.28291e62	0.0117078
$532$	0	0
$533$	−6.29866e63	−0.526210
$534$	0	0
$535$	−8.50400e63	−0.650593
$536$	0	0
$537$	5.48283e63	0.384246
$538$	0	0
$539$	9.90734e63	0.636240
$540$	0	0
$541$	6.71742e61	0.00395428	0.00197714	−	0.999998i	$-0.499371\pi$
$541$	6.71742e61	0.00395428	0.00197714	+	0.999998i	$0.499371\pi$
$542$	0	0
$543$	−9.41938e63	−0.508424
$544$	0	0
$545$	−2.48615e64	−1.23086
$546$	0	0
$547$	−4.01645e62	−0.0182449	−0.00912243	−	0.999958i	$-0.502904\pi$
$547$	−4.01645e62	−0.0182449	−0.00912243	+	0.999958i	$0.502904\pi$
$548$	0	0
$549$	−1.22507e63	−0.0510751
$550$	0	0
$551$	1.07590e64	0.411823
$552$	0	0
$553$	2.27629e64	0.800177
$554$	0	0
$555$	4.60084e64	1.48577
$556$	0	0
$557$	6.03614e64	1.79127	0.895637	−	0.444785i	$-0.146720\pi$
$557$	6.03614e64	1.79127	0.895637	+	0.444785i	$0.146720\pi$
$558$	0	0
$559$	1.11495e64	0.304144
$560$	0	0
$561$	3.41059e64	0.855466
$562$	0	0
$563$	3.47404e63	0.0801470	0.0400735	−	0.999197i	$-0.487241\pi$
$563$	3.47404e63	0.0801470	0.0400735	+	0.999197i	$0.487241\pi$
$564$	0	0
$565$	1.32104e64	0.280399
$566$	0	0
$567$	−2.35409e64	−0.459856
$568$	0	0
$569$	9.25178e64	1.66375	0.831873	−	0.554966i	$-0.187269\pi$
$569$	9.25178e64	1.66375	0.831873	+	0.554966i	$0.187269\pi$
$570$	0	0
$571$	1.04188e65	1.72532	0.862658	−	0.505787i	$-0.168798\pi$
$571$	1.04188e65	1.72532	0.862658	+	0.505787i	$0.168798\pi$
$572$	0	0
$573$	−4.71096e64	−0.718581
$574$	0	0
$575$	−2.46708e64	−0.346728
$576$	0	0
$577$	−8.68105e64	−1.12445	−0.562225	−	0.826984i	$-0.690055\pi$
$577$	−8.68105e64	−1.12445	−0.562225	+	0.826984i	$0.690055\pi$
$578$	0	0
$579$	−7.22883e64	−0.863221
$580$	0	0
$581$	1.37382e64	0.151284
$582$	0	0
$583$	6.63474e64	0.673928
$584$	0	0
$585$	−3.33291e64	−0.312364
$586$	0	0
$587$	−5.04453e64	−0.436340	−0.218170	−	0.975911i	$-0.570009\pi$
$587$	−5.04453e64	−0.436340	−0.218170	+	0.975911i	$0.570009\pi$
$588$	0	0
$589$	5.75338e64	0.459424
$590$	0	0
$591$	1.15796e65	0.853861
$592$	0	0
$593$	1.82757e65	1.24477	0.622385	−	0.782712i	$-0.286164\pi$
$593$	1.82757e65	1.24477	0.622385	+	0.782712i	$0.286164\pi$
$594$	0	0
$595$	1.39406e65	0.877272
$596$	0	0
$597$	−2.89823e65	−1.68553
$598$	0	0
$599$	−6.30818e64	−0.339136	−0.169568	−	0.985519i	$-0.554237\pi$
$599$	−6.30818e64	−0.339136	−0.169568	+	0.985519i	$0.554237\pi$
$600$	0	0
$601$	1.16613e65	0.579693	0.289847	−	0.957073i	$-0.406396\pi$
$601$	1.16613e65	0.579693	0.289847	+	0.957073i	$0.406396\pi$
$602$	0	0
$603$	−4.38422e63	−0.0201574
$604$	0	0
$605$	−7.26847e64	−0.309166
$606$	0	0
$607$	8.69736e64	0.342337	0.171168	−	0.985242i	$-0.445246\pi$
$607$	8.69736e64	0.342337	0.171168	+	0.985242i	$0.445246\pi$
$608$	0	0
$609$	2.01779e65	0.735141
$610$	0	0
$611$	−4.22709e65	−1.42585
$612$	0	0
$613$	3.95467e65	1.23535	0.617674	−	0.786435i	$-0.288075\pi$
$613$	3.95467e65	1.23535	0.617674	+	0.786435i	$0.288075\pi$
$614$	0	0
$615$	1.79436e65	0.519212
$616$	0	0
$617$	−4.10300e65	−1.10002	−0.550009	−	0.835159i	$-0.685376\pi$
$617$	−4.10300e65	−1.10002	−0.550009	+	0.835159i	$0.685376\pi$
$618$	0	0
$619$	−1.39273e65	−0.346047	−0.173024	−	0.984918i	$-0.555354\pi$
$619$	−1.39273e65	−0.346047	−0.173024	+	0.984918i	$0.555354\pi$
$620$	0	0
$621$	−1.03747e65	−0.238958
$622$	0	0
$623$	−1.14245e65	−0.243988
$624$	0	0
$625$	−9.58171e64	−0.189785
$626$	0	0
$627$	−1.28360e65	−0.235853
$628$	0	0
$629$	−6.00713e65	−1.02418
$630$	0	0
$631$	−1.54026e65	−0.243728	−0.121864	−	0.992547i	$-0.538887\pi$
$631$	−1.54026e65	−0.243728	−0.121864	+	0.992547i	$0.538887\pi$
$632$	0	0
$633$	1.90833e65	0.280328
$634$	0	0
$635$	−1.12844e66	−1.53920
$636$	0	0
$637$	−8.40952e65	−1.06536
$638$	0	0
$639$	−1.13276e65	−0.133312
$640$	0	0
$641$	−1.65837e65	−0.181350	−0.0906752	−	0.995881i	$-0.528903\pi$
$641$	−1.65837e65	−0.181350	−0.0906752	+	0.995881i	$0.528903\pi$
$642$	0	0
$643$	1.22213e66	1.24211	0.621053	−	0.783768i	$-0.286705\pi$
$643$	1.22213e66	1.24211	0.621053	+	0.783768i	$0.286705\pi$
$644$	0	0
$645$	−3.17627e65	−0.300099
$646$	0	0
$647$	9.62351e65	0.845440	0.422720	−	0.906260i	$-0.361075\pi$
$647$	9.62351e65	0.845440	0.422720	+	0.906260i	$0.361075\pi$
$648$	0	0
$649$	9.84237e64	0.0804173
$650$	0	0
$651$	1.07901e66	0.820112
$652$	0	0
$653$	3.91841e65	0.277109	0.138554	−	0.990355i	$-0.455754\pi$
$653$	3.91841e65	0.277109	0.138554	+	0.990355i	$0.455754\pi$
$654$	0	0
$655$	−2.69183e66	−1.77165
$656$	0	0
$657$	−1.24796e65	−0.0764567
$658$	0	0
$659$	−2.16961e66	−1.23758	−0.618792	−	0.785555i	$-0.712378\pi$
$659$	−2.16961e66	−1.23758	−0.618792	+	0.785555i	$0.712378\pi$
$660$	0	0
$661$	−9.87453e65	−0.524544	−0.262272	−	0.964994i	$-0.584472\pi$
$661$	−9.87453e65	−0.524544	−0.262272	+	0.964994i	$0.584472\pi$
$662$	0	0
$663$	−2.89497e66	−1.43244
$664$	0	0
$665$	−5.24665e65	−0.241865
$666$	0	0
$667$	7.71034e65	0.331220
$668$	0	0
$669$	−2.17623e65	−0.0871347
$670$	0	0
$671$	−9.39857e65	−0.350818
$672$	0	0
$673$	−5.18176e66	−1.80353	−0.901765	−	0.432227i	$-0.857728\pi$
$673$	−5.18176e66	−1.80353	−0.901765	+	0.432227i	$0.857728\pi$
$674$	0	0
$675$	4.96780e66	1.61260
$676$	0	0
$677$	5.62905e66	1.70452	0.852259	−	0.523120i	$-0.175232\pi$
$677$	5.62905e66	1.70452	0.852259	+	0.523120i	$0.175232\pi$
$678$	0	0
$679$	8.48559e65	0.239741
$680$	0	0
$681$	5.14615e66	1.35683
$682$	0	0
$683$	2.17309e66	0.534800	0.267400	−	0.963586i	$-0.413835\pi$
$683$	2.17309e66	0.534800	0.267400	+	0.963586i	$0.413835\pi$
$684$	0	0
$685$	8.46798e66	1.94559
$686$	0	0
$687$	7.07289e65	0.151745
$688$	0	0
$689$	−5.63168e66	−1.12846
$690$	0	0
$691$	−3.80507e66	−0.712244	−0.356122	−	0.934439i	$-0.615901\pi$
$691$	−3.80507e66	−0.712244	−0.356122	+	0.934439i	$0.615901\pi$
$692$	0	0
$693$	3.61861e65	0.0632864
$694$	0	0
$695$	1.34230e67	2.19384
$696$	0	0
$697$	−2.34282e66	−0.357906
$698$	0	0
$699$	1.04483e67	1.49222
$700$	0	0
$701$	4.48575e66	0.599048	0.299524	−	0.954089i	$-0.403172\pi$
$701$	4.48575e66	0.599048	0.299524	+	0.954089i	$0.403172\pi$
$702$	0	0
$703$	2.26082e66	0.282368
$704$	0	0
$705$	1.20421e67	1.40689
$706$	0	0
$707$	−1.99872e66	−0.218472
$708$	0	0
$709$	−1.23589e67	−1.26414	−0.632068	−	0.774913i	$-0.717794\pi$
$709$	−1.23589e67	−1.26414	−0.632068	+	0.774913i	$0.717794\pi$
$710$	0	0
$711$	−2.02422e66	−0.193786
$712$	0	0
$713$	4.12310e66	0.369504
$714$	0	0
$715$	−2.55697e67	−2.14552
$716$	0	0
$717$	1.66050e67	1.30479
$718$	0	0
$719$	5.05205e66	0.371825	0.185913	−	0.982566i	$-0.440476\pi$
$719$	5.05205e66	0.371825	0.185913	+	0.982566i	$0.440476\pi$
$720$	0	0
$721$	−1.24860e67	−0.860884
$722$	0	0
$723$	7.33749e66	0.474021
$724$	0	0
$725$	−3.69202e67	−2.23522
$726$	0	0
$727$	−2.10795e67	−1.19619	−0.598097	−	0.801424i	$-0.704076\pi$
$727$	−2.10795e67	−1.19619	−0.598097	+	0.801424i	$0.704076\pi$
$728$	0	0
$729$	2.05698e67	1.09429
$730$	0	0
$731$	4.14712e66	0.206866
$732$	0	0
$733$	2.34941e67	1.09905	0.549523	−	0.835478i	$-0.314809\pi$
$733$	2.34941e67	1.09905	0.549523	+	0.835478i	$0.314809\pi$
$734$	0	0
$735$	2.39570e67	1.05119
$736$	0	0
$737$	−3.36352e66	−0.138455
$738$	0	0
$739$	8.69448e66	0.335814	0.167907	−	0.985803i	$-0.446299\pi$
$739$	8.69448e66	0.335814	0.167907	+	0.985803i	$0.446299\pi$
$740$	0	0
$741$	1.08954e67	0.394926
$742$	0	0
$743$	−1.83903e67	−0.625679	−0.312840	−	0.949806i	$-0.601280\pi$
$743$	−1.83903e67	−0.625679	−0.312840	+	0.949806i	$0.601280\pi$
$744$	0	0
$745$	4.25094e66	0.135773
$746$	0	0
$747$	−1.22169e66	−0.0366377
$748$	0	0
$749$	−7.83710e66	−0.220715
$750$	0	0
$751$	2.92872e67	0.774709	0.387355	−	0.921931i	$-0.373389\pi$
$751$	2.92872e67	0.774709	0.387355	+	0.921931i	$0.373389\pi$
$752$	0	0
$753$	−6.17588e67	−1.53468
$754$	0	0
$755$	1.01954e66	0.0238040
$756$	0	0
$757$	−3.60671e67	−0.791330	−0.395665	−	0.918395i	$-0.629486\pi$
$757$	−3.60671e67	−0.791330	−0.395665	+	0.918395i	$0.629486\pi$
$758$	0	0
$759$	−9.19877e66	−0.189691
$760$	0	0
$761$	6.36047e67	1.23296	0.616481	−	0.787370i	$-0.288558\pi$
$761$	6.36047e67	1.23296	0.616481	+	0.787370i	$0.288558\pi$
$762$	0	0
$763$	−2.29118e67	−0.417574
$764$	0	0
$765$	−1.23969e67	−0.212457
$766$	0	0
$767$	−8.35437e66	−0.134655
$768$	0	0
$769$	−1.58030e67	−0.239591	−0.119795	−	0.992799i	$-0.538224\pi$
$769$	−1.58030e67	−0.239591	−0.119795	+	0.992799i	$0.538224\pi$
$770$	0	0
$771$	−2.94486e67	−0.420037
$772$	0	0
$773$	−1.20430e68	−1.61628	−0.808142	−	0.588988i	$-0.799526\pi$
$773$	−1.20430e68	−1.61628	−0.808142	+	0.588988i	$0.799526\pi$
$774$	0	0
$775$	−1.97430e68	−2.49358
$776$	0	0
$777$	4.24004e67	0.504052
$778$	0	0
$779$	8.81736e66	0.0986752
$780$	0	0
$781$	−8.69040e67	−0.915674
$782$	0	0
$783$	−1.55258e68	−1.54047
$784$	0	0
$785$	−6.03186e67	−0.563656
$786$	0	0
$787$	1.72836e68	1.52134	0.760671	−	0.649137i	$-0.224870\pi$
$787$	1.72836e68	1.52134	0.760671	+	0.649137i	$0.224870\pi$
$788$	0	0
$789$	1.26745e68	1.05104
$790$	0	0
$791$	1.21744e67	0.0951261
$792$	0	0
$793$	7.97766e67	0.587429
$794$	0	0
$795$	1.60435e68	1.11345
$796$	0	0
$797$	−5.17743e67	−0.338722	−0.169361	−	0.985554i	$-0.554170\pi$
$797$	−5.17743e67	−0.338722	−0.169361	+	0.985554i	$0.554170\pi$
$798$	0	0
$799$	−1.57229e68	−0.969803
$800$	0	0
$801$	1.01594e67	0.0590887
$802$	0	0
$803$	−9.57423e67	−0.525156
$804$	0	0
$805$	−3.75995e67	−0.194527
$806$	0	0
$807$	2.86569e68	1.39862
$808$	0	0
$809$	−1.34819e68	−0.620812	−0.310406	−	0.950604i	$-0.600465\pi$
$809$	−1.34819e68	−0.620812	−0.310406	+	0.950604i	$0.600465\pi$
$810$	0	0
$811$	1.29768e68	0.563869	0.281934	−	0.959434i	$-0.409024\pi$
$811$	1.29768e68	0.563869	0.281934	+	0.959434i	$0.409024\pi$
$812$	0	0
$813$	3.90137e68	1.59989
$814$	0	0
$815$	2.03186e68	0.786485
$816$	0	0
$817$	−1.56080e67	−0.0570332
$818$	0	0
$819$	−3.07154e67	−0.105970
$820$	0	0
$821$	−3.05518e68	−0.995342	−0.497671	−	0.867366i	$-0.665811\pi$
$821$	−3.05518e68	−0.995342	−0.497671	+	0.867366i	$0.665811\pi$
$822$	0	0
$823$	3.54128e68	1.08959	0.544796	−	0.838568i	$-0.316607\pi$
$823$	3.54128e68	1.08959	0.544796	+	0.838568i	$0.316607\pi$
$824$	0	0
$825$	4.40474e68	1.28012
$826$	0	0
$827$	−5.88899e68	−1.61682	−0.808410	−	0.588620i	$-0.799672\pi$
$827$	−5.88899e68	−1.61682	−0.808410	+	0.588620i	$0.799672\pi$
$828$	0	0
$829$	7.14204e68	1.85264	0.926319	−	0.376740i	$-0.122955\pi$
$829$	7.14204e68	1.85264	0.926319	+	0.376740i	$0.122955\pi$
$830$	0	0
$831$	1.27204e68	0.311799
$832$	0	0
$833$	−3.12797e68	−0.724610
$834$	0	0
$835$	4.19610e68	0.918785
$836$	0	0
$837$	−8.30241e68	−1.71852
$838$	0	0
$839$	−4.08251e68	−0.798953	−0.399477	−	0.916743i	$-0.630808\pi$
$839$	−4.08251e68	−0.798953	−0.399477	+	0.916743i	$0.630808\pi$
$840$	0	0
$841$	6.13474e68	1.13525
$842$	0	0
$843$	6.03536e68	1.05622
$844$	0	0
$845$	1.20953e69	2.00209
$846$	0	0
$847$	−6.69846e67	−0.104885
$848$	0	0
$849$	−6.77100e68	−1.00305
$850$	0	0
$851$	1.62019e68	0.227102
$852$	0	0
$853$	6.69695e68	0.888331	0.444166	−	0.895945i	$-0.353500\pi$
$853$	6.69695e68	0.888331	0.444166	+	0.895945i	$0.353500\pi$
$854$	0	0
$855$	4.66566e67	0.0585746
$856$	0	0
$857$	1.01115e69	1.20162	0.600809	−	0.799393i	$-0.294845\pi$
$857$	1.01115e69	1.20162	0.600809	+	0.799393i	$0.294845\pi$
$858$	0	0
$859$	4.13647e68	0.465361	0.232681	−	0.972553i	$-0.425250\pi$
$859$	4.13647e68	0.465361	0.232681	+	0.972553i	$0.425250\pi$
$860$	0	0
$861$	1.65364e68	0.176144
$862$	0	0
$863$	−6.85332e68	−0.691270	−0.345635	−	0.938369i	$-0.612336\pi$
$863$	−6.85332e68	−0.691270	−0.345635	+	0.938369i	$0.612336\pi$
$864$	0	0
$865$	−3.15424e69	−3.01312
$866$	0	0
$867$	−4.63182e67	−0.0419086
$868$	0	0
$869$	−1.55296e69	−1.33105
$870$	0	0
$871$	2.85501e68	0.231836
$872$	0	0
$873$	−7.54594e67	−0.0580602
$874$	0	0
$875$	6.23418e68	0.454559
$876$	0	0
$877$	6.43840e68	0.444925	0.222463	−	0.974941i	$-0.428590\pi$
$877$	6.43840e68	0.444925	0.222463	+	0.974941i	$0.428590\pi$
$878$	0	0
$879$	−1.59889e69	−1.04732
$880$	0	0
$881$	8.86127e68	0.550253	0.275126	−	0.961408i	$-0.411280\pi$
$881$	8.86127e68	0.550253	0.275126	+	0.961408i	$0.411280\pi$
$882$	0	0
$883$	−4.16643e68	−0.245294	−0.122647	−	0.992450i	$-0.539138\pi$
$883$	−4.16643e68	−0.245294	−0.122647	+	0.992450i	$0.539138\pi$
$884$	0	0
$885$	2.37999e68	0.132864
$886$	0	0
$887$	−2.94393e69	−1.55856	−0.779278	−	0.626679i	$-0.784414\pi$
$887$	−2.94393e69	−1.55856	−0.779278	+	0.626679i	$0.784414\pi$
$888$	0	0
$889$	−1.03994e69	−0.522177
$890$	0	0
$891$	1.60604e69	0.764946
$892$	0	0
$893$	5.91742e68	0.267376
$894$	0	0
$895$	−1.52894e69	−0.655466
$896$	0	0
$897$	7.80807e68	0.317630
$898$	0	0
$899$	6.17026e69	2.38205
$900$	0	0
$901$	−2.09473e69	−0.767532
$902$	0	0
$903$	−2.92718e68	−0.101809
$904$	0	0
$905$	2.62669e69	0.867295
$906$	0	0
$907$	−3.36233e69	−1.05407	−0.527034	−	0.849844i	$-0.676696\pi$
$907$	−3.36233e69	−1.05407	−0.527034	+	0.849844i	$0.676696\pi$
$908$	0	0
$909$	1.77739e68	0.0529094
$910$	0	0
$911$	−4.29823e69	−1.21509	−0.607544	−	0.794286i	$-0.707845\pi$
$911$	−4.29823e69	−1.21509	−0.607544	+	0.794286i	$0.707845\pi$
$912$	0	0
$913$	−9.37269e68	−0.251652
$914$	0	0
$915$	−2.27267e69	−0.579617
$916$	0	0
$917$	−2.48074e69	−0.601037
$918$	0	0
$919$	−5.89124e69	−1.35610	−0.678050	−	0.735015i	$-0.737175\pi$
$919$	−5.89124e69	−1.35610	−0.678050	+	0.735015i	$0.737175\pi$
$920$	0	0
$921$	8.25876e68	0.180640
$922$	0	0
$923$	7.37656e69	1.53325
$924$	0	0
$925$	−7.75813e69	−1.53259
$926$	0	0
$927$	1.11034e69	0.208488
$928$	0	0
$929$	−3.81127e69	−0.680301	−0.340150	−	0.940371i	$-0.610478\pi$
$929$	−3.81127e69	−0.680301	−0.340150	+	0.940371i	$0.610478\pi$
$930$	0	0
$931$	1.17723e69	0.199776
$932$	0	0
$933$	−7.29808e69	−1.17758
$934$	0	0
$935$	−9.51078e69	−1.45930
$936$	0	0
$937$	−3.18868e69	−0.465298	−0.232649	−	0.972561i	$-0.574739\pi$
$937$	−3.18868e69	−0.465298	−0.232649	+	0.972561i	$0.574739\pi$
$938$	0	0
$939$	3.80573e69	0.528197
$940$	0	0
$941$	7.20176e69	0.950785	0.475393	−	0.879774i	$-0.342306\pi$
$941$	7.20176e69	0.950785	0.475393	+	0.879774i	$0.342306\pi$
$942$	0	0
$943$	6.31886e68	0.0793623
$944$	0	0
$945$	7.57117e69	0.904723
$946$	0	0
$947$	−7.19607e69	−0.818222	−0.409111	−	0.912485i	$-0.634161\pi$
$947$	−7.19607e69	−0.818222	−0.409111	+	0.912485i	$0.634161\pi$
$948$	0	0
$949$	8.12676e69	0.879350
$950$	0	0
$951$	1.01148e70	1.04164
$952$	0	0
$953$	1.03533e70	1.01483	0.507417	−	0.861700i	$-0.330600\pi$
$953$	1.03533e70	1.01483	0.507417	+	0.861700i	$0.330600\pi$
$954$	0	0
$955$	1.31370e70	1.22579
$956$	0	0
$957$	−1.37661e70	−1.22287
$958$	0	0
$959$	7.80391e69	0.660047
$960$	0	0
$961$	2.05789e70	1.65738
$962$	0	0
$963$	6.96926e68	0.0534526
$964$	0	0
$965$	2.01583e70	1.47252
$966$	0	0
$967$	1.88655e70	1.31264	0.656321	−	0.754482i	$-0.272112\pi$
$967$	1.88655e70	1.31264	0.656321	+	0.754482i	$0.272112\pi$
$968$	0	0
$969$	4.05260e69	0.268612
$970$	0	0
$971$	1.54065e70	0.972862	0.486431	−	0.873719i	$-0.338299\pi$
$971$	1.54065e70	0.972862	0.486431	+	0.873719i	$0.338299\pi$
$972$	0	0
$973$	1.23703e70	0.744268
$974$	0	0
$975$	−3.73882e70	−2.14351
$976$	0	0
$977$	−1.28077e70	−0.699760	−0.349880	−	0.936795i	$-0.613778\pi$
$977$	−1.28077e70	−0.699760	−0.349880	+	0.936795i	$0.613778\pi$
$978$	0	0
$979$	7.79418e69	0.405861
$980$	0	0
$981$	2.03747e69	0.101128
$982$	0	0
$983$	−1.22658e70	−0.580349	−0.290174	−	0.956974i	$-0.593713\pi$
$983$	−1.22658e70	−0.580349	−0.290174	+	0.956974i	$0.593713\pi$
$984$	0	0
$985$	−3.22908e70	−1.45656
$986$	0	0
$987$	1.10978e70	0.477290
$988$	0	0
$989$	−1.11853e69	−0.0458705
$990$	0	0
$991$	−2.78865e70	−1.09059	−0.545296	−	0.838244i	$-0.683583\pi$
$991$	−2.78865e70	−1.09059	−0.545296	+	0.838244i	$0.683583\pi$
$992$	0	0
$993$	−2.32587e69	−0.0867519
$994$	0	0
$995$	8.08202e70	2.87526
$996$	0	0
$997$	2.05189e70	0.696336	0.348168	−	0.937432i	$-0.386804\pi$
$997$	2.05189e70	0.696336	0.348168	+	0.937432i	$0.386804\pi$
$998$	0	0
$999$	−3.26247e70	−1.05623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.48.a.d.1.1		4
4.3	odd	2		1.48.a.a.1.2	✓	4
12.11	even	2		9.48.a.c.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.48.a.a.1.2	✓	4	4.3	odd	2
9.48.a.c.1.3		4	12.11	even	2
16.48.a.d.1.1		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 48 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-1.52034e11 q^{3} +4.23962e16 q^{5} +3.90714e19 q^{7} -3.47448e21 q^{9} +O(q^{10})\)	\(q-1.52034e11 q^{3} +4.23962e16 q^{5} +3.90714e19 q^{7} -3.47448e21 q^{9} -2.66558e24 q^{11} +2.26259e26 q^{13} -6.44566e27 q^{15} +8.41583e28 q^{17} -3.16735e29 q^{19} -5.94018e30 q^{21} -2.26985e31 q^{23} +1.08689e33 q^{25} +4.57064e33 q^{27} -3.39685e34 q^{29} -1.81646e35 q^{31} +4.05259e35 q^{33} +1.65648e36 q^{35} -7.13789e36 q^{37} -3.43991e37 q^{39} -2.78383e37 q^{41} +4.92777e37 q^{43} -1.47305e38 q^{45} -1.86825e39 q^{47} -3.71676e39 q^{49} -1.27949e40 q^{51} -2.48904e40 q^{53} -1.13011e41 q^{55} +4.81545e40 q^{57} -3.69239e40 q^{59} +3.52589e41 q^{61} -1.35753e41 q^{63} +9.59253e42 q^{65} +1.26183e42 q^{67} +3.45094e42 q^{69} +3.26023e43 q^{71} +3.59179e43 q^{73} -1.65245e44 q^{75} -1.04148e44 q^{77} +5.82596e44 q^{79} -6.02511e44 q^{81} +3.51619e44 q^{83} +3.56799e45 q^{85} +5.16437e45 q^{87} -2.92401e45 q^{89} +8.84027e45 q^{91} +2.76164e46 q^{93} -1.34284e46 q^{95} +2.17182e46 q^{97} +9.26152e45 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 38461494960 q^{3} - 31\!\cdots\!00 q^{5}+ \cdots - 17\!\cdots\!72 q^{9}+O(q^{10}) \) 4 * q - 38461494960 * q^3 - 31114680242272200 * q^5 + 39169218725888423200 * q^7 - 17071972417358142200172 * q^9	\( 4 q - 38461494960 q^{3} - 31\!\cdots\!00 q^{5}+ \cdots - 53\!\cdots\!16 q^{99}+O(q^{100}) \) 4 * q - 38461494960 * q^3 - 31114680242272200 * q^5 + 39169218725888423200 * q^7 - 17071972417358142200172 * q^9 + 1902165515717740404742512 * q^11 + 126959149195199610466858520 * q^13 - 12437812896841241272880733600 * q^15 + 210426285758836687099688769480 * q^17 + 1059119811802437301721159140240 * q^19 - 26054906182888686128940663165312 * q^21 - 137531418344493639994288859179680 * q^23 + 1009909900344396506188154274977500 * q^25 + 7345952425324536073931587214783520 * q^27 - 22743132490172744184500850941726760 * q^29 - 75786540814706434901630981303934848 * q^31 + 263381387470144952341649359926294720 * q^33 + 1313148822925052387484546800067979200 * q^35 - 11291457045120157869429951769632191560 * q^37 - 39294375338503805226442587829551714336 * q^39 + 130658126291089294119301677048020391528 * q^41 + 445357287738427943170332944488133090800 * q^43 + 86022747271219179966948746072020632600 * q^45 - 2031912465958423919194866366997063590720 * q^47 + 9153745744156628626499502447643951451172 * q^49 + 18534366050002170683962116343746386369952 * q^51 + 29171719896619533806321793123448457437560 * q^53 - 192172580604633896452606078657416872901600 * q^55 + 558743644452458098451223285784560964733760 * q^57 - 478242858216320686410790860443879577596880 * q^59 + 626683213833907852009669690320782900857688 * q^61 - 5811636950821577036406237472810763671724640 * q^63 + 12996735432890074297558542478104126872331600 * q^65 - 18701411419127220668869505853739529194216880 * q^67 - 2096956664577882726873494616728974107851904 * q^69 - 22196594617762006008028198712153461686540768 * q^71 + 104911201720506650033573412943840886878191080 * q^73 - 32388703324174028323482311940739905891330000 * q^75 - 263490744376507692433815778071300370399209600 * q^77 + 1327995310109394234162785254414226288730352960 * q^79 - 1061448205755511803861508510986654404170081116 * q^81 + 1439130627532364606968733068855111784504073360 * q^83 - 1001895286410663502447652611446390227458237200 * q^85 + 4326251093943206485496474339443178030108648160 * q^87 - 7960590522674945421108320896849737731421281880 * q^89 - 5325194479227227497182497798631911205603381568 * q^91 + 10728611999875738337226031496188630272675064320 * q^93 - 75421152415106939771624998522249518123547668000 * q^95 + 95066248961258904822565865801364599382905757320 * q^97 - 530900415919166351015781326273177630089784016 * q^99

Introduction

L-functions

Modular forms

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