LMFDB - Embedded newform 40.8.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [40,8,Mod(1,40)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("40.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 40 = 2^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	40.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,76] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$12.4954010194$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{601}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 150 $ x^2 - x - 150
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-11.7577$ of defining polynomial
Character	$\chi$	$=$	40.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+87.0306 q^{3} +125.000 q^{5} +741.214 q^{7} +5387.33 q^{9} -6065.61 q^{11} -4234.04 q^{13} +10878.8 q^{15} +18488.3 q^{17} -28071.6 q^{19} +64508.3 q^{21} +104956. q^{23} +15625.0 q^{25} +278526. q^{27} -204768. q^{29} +5109.59 q^{31} -527894. q^{33} +92651.8 q^{35} +186181. q^{37} -368491. q^{39} -618450. q^{41} -329301. q^{43} +673416. q^{45} +56218.8 q^{47} -274144. q^{49} +1.60905e6 q^{51} +143301. q^{53} -758201. q^{55} -2.44309e6 q^{57} -3.09530e6 q^{59} -701736. q^{61} +3.99316e6 q^{63} -529255. q^{65} -1.15239e6 q^{67} +9.13438e6 q^{69} +5.03809e6 q^{71} -4.64023e6 q^{73} +1.35985e6 q^{75} -4.49592e6 q^{77} +2.67305e6 q^{79} +1.24582e7 q^{81} +2.64570e6 q^{83} +2.31104e6 q^{85} -1.78211e7 q^{87} +3.11153e6 q^{89} -3.13833e6 q^{91} +444690. q^{93} -3.50895e6 q^{95} -4.05990e6 q^{97} -3.26774e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 76 q^{3} + 250 q^{5} + 796 q^{7} + 3322 q^{9} - 560 q^{11} + 4476 q^{13} + 9500 q^{15} + 29524 q^{17} + 19560 q^{19} + 63904 q^{21} + 129796 q^{23} + 31250 q^{25} + 325432 q^{27} - 211060 q^{29} + 165352 q^{31}+ \cdots - 44048304 q^{99}+O(q^{100}) $ 2 * q + 76 * q^3 + 250 * q^5 + 796 * q^7 + 3322 * q^9 - 560 * q^11 + 4476 * q^13 + 9500 * q^15 + 29524 * q^17 + 19560 * q^19 + 63904 * q^21 + 129796 * q^23 + 31250 * q^25 + 325432 * q^27 - 211060 * q^29 + 165352 * q^31 - 588624 * q^33 + 99500 * q^35 - 292100 * q^37 - 464568 * q^39 - 851324 * q^41 - 197812 * q^43 + 415250 * q^45 - 34164 * q^47 - 1094686 * q^49 + 1487320 * q^51 - 876404 * q^53 - 70000 * q^55 - 2968496 * q^57 - 2088504 * q^59 + 1409708 * q^61 + 3880012 * q^63 + 559500 * q^65 - 2882076 * q^67 + 8860384 * q^69 + 9819448 * q^71 - 189404 * q^73 + 1187500 * q^75 - 4194288 * q^77 - 2028496 * q^79 + 16457698 * q^81 - 1393332 * q^83 + 3690500 * q^85 - 17751672 * q^87 + 5688820 * q^89 - 2661144 * q^91 - 1322880 * q^93 + 2445000 * q^95 - 20045996 * q^97 - 44048304 * 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
$3$	87.0306	1.86101	0.930503	−	0.366285i	$-0.119371\pi$
$3$	87.0306	1.86101	0.930503	+	0.366285i	$0.119371\pi$
$4$	0	0
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	741.214	0.816772	0.408386	−	0.912809i	$-0.366092\pi$
$7$	741.214	0.816772	0.408386	+	0.912809i	$0.366092\pi$
$8$	0	0
$9$	5387.33	2.46334
$10$	0	0
$11$	−6065.61	−1.37404	−0.687021	−	0.726637i	$-0.741082\pi$
$11$	−6065.61	−1.37404	−0.687021	+	0.726637i	$0.741082\pi$
$12$	0	0
$13$	−4234.04	−0.534507	−0.267253	−	0.963626i	$-0.586116\pi$
$13$	−4234.04	−0.534507	−0.267253	+	0.963626i	$0.586116\pi$
$14$	0	0
$15$	10878.8	0.832267
$16$	0	0
$17$	18488.3	0.912696	0.456348	−	0.889801i	$-0.349157\pi$
$17$	18488.3	0.912696	0.456348	+	0.889801i	$0.349157\pi$
$18$	0	0
$19$	−28071.6	−0.938923	−0.469461	−	0.882953i	$-0.655552\pi$
$19$	−28071.6	−0.938923	−0.469461	+	0.882953i	$0.655552\pi$
$20$	0	0
$21$	64508.3	1.52002
$22$	0	0
$23$	104956.	1.79870	0.899352	−	0.437225i	$-0.144039\pi$
$23$	104956.	1.79870	0.899352	+	0.437225i	$0.144039\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	278526.	2.72328
$28$	0	0
$29$	−204768.	−1.55908	−0.779541	−	0.626351i	$-0.784548\pi$
$29$	−204768.	−1.55908	−0.779541	+	0.626351i	$0.784548\pi$
$30$	0	0
$31$	5109.59	0.0308049	0.0154025	−	0.999881i	$-0.495097\pi$
$31$	5109.59	0.0308049	0.0154025	+	0.999881i	$0.495097\pi$
$32$	0	0
$33$	−527894.	−2.55710
$34$	0	0
$35$	92651.8	0.365271
$36$	0	0
$37$	186181.	0.604269	0.302134	−	0.953265i	$-0.402301\pi$
$37$	186181.	0.604269	0.302134	+	0.953265i	$0.402301\pi$
$38$	0	0
$39$	−368491.	−0.994720
$40$	0	0
$41$	−618450.	−1.40140	−0.700699	−	0.713457i	$-0.747128\pi$
$41$	−618450.	−1.40140	−0.700699	+	0.713457i	$0.747128\pi$
$42$	0	0
$43$	−329301.	−0.631616	−0.315808	−	0.948823i	$-0.602276\pi$
$43$	−329301.	−0.631616	−0.315808	+	0.948823i	$0.602276\pi$
$44$	0	0
$45$	673416.	1.10164
$46$	0	0
$47$	56218.8	0.0789839	0.0394920	−	0.999220i	$-0.487426\pi$
$47$	56218.8	0.0789839	0.0394920	+	0.999220i	$0.487426\pi$
$48$	0	0
$49$	−274144.	−0.332884
$50$	0	0
$51$	1.60905e6	1.69853
$52$	0	0
$53$	143301.	0.132216	0.0661079	−	0.997812i	$-0.478942\pi$
$53$	143301.	0.132216	0.0661079	+	0.997812i	$0.478942\pi$
$54$	0	0
$55$	−758201.	−0.614490
$56$	0	0
$57$	−2.44309e6	−1.74734
$58$	0	0
$59$	−3.09530e6	−1.96210	−0.981049	−	0.193759i	$-0.937932\pi$
$59$	−3.09530e6	−1.96210	−0.981049	+	0.193759i	$0.937932\pi$
$60$	0	0
$61$	−701736.	−0.395840	−0.197920	−	0.980218i	$-0.563419\pi$
$61$	−701736.	−0.395840	−0.197920	+	0.980218i	$0.563419\pi$
$62$	0	0
$63$	3.99316e6	2.01199
$64$	0	0
$65$	−529255.	−0.239039
$66$	0	0
$67$	−1.15239e6	−0.468101	−0.234050	−	0.972224i	$-0.575198\pi$
$67$	−1.15239e6	−0.468101	−0.234050	+	0.972224i	$0.575198\pi$
$68$	0	0
$69$	9.13438e6	3.34740
$70$	0	0
$71$	5.03809e6	1.67056	0.835279	−	0.549827i	$-0.185306\pi$
$71$	5.03809e6	1.67056	0.835279	+	0.549827i	$0.185306\pi$
$72$	0	0
$73$	−4.64023e6	−1.39608	−0.698039	−	0.716060i	$-0.745944\pi$
$73$	−4.64023e6	−1.39608	−0.698039	+	0.716060i	$0.745944\pi$
$74$	0	0
$75$	1.35985e6	0.372201
$76$	0	0
$77$	−4.49592e6	−1.12228
$78$	0	0
$79$	2.67305e6	0.609975	0.304988	−	0.952356i	$-0.401348\pi$
$79$	2.67305e6	0.609975	0.304988	+	0.952356i	$0.401348\pi$
$80$	0	0
$81$	1.24582e7	2.60471
$82$	0	0
$83$	2.64570e6	0.507887	0.253944	−	0.967219i	$-0.418272\pi$
$83$	2.64570e6	0.507887	0.253944	+	0.967219i	$0.418272\pi$
$84$	0	0
$85$	2.31104e6	0.408170
$86$	0	0
$87$	−1.78211e7	−2.90146
$88$	0	0
$89$	3.11153e6	0.467852	0.233926	−	0.972254i	$-0.424843\pi$
$89$	3.11153e6	0.467852	0.233926	+	0.972254i	$0.424843\pi$
$90$	0	0
$91$	−3.13833e6	−0.436570
$92$	0	0
$93$	444690.	0.0573281
$94$	0	0
$95$	−3.50895e6	−0.419899
$96$	0	0
$97$	−4.05990e6	−0.451662	−0.225831	−	0.974166i	$-0.572510\pi$
$97$	−4.05990e6	−0.451662	−0.225831	+	0.974166i	$0.572510\pi$
$98$	0	0
$99$	−3.26774e7	−3.38473
$100$	0	0
$101$	−550732.	−0.0531883	−0.0265941	−	0.999646i	$-0.508466\pi$
$101$	−550732.	−0.0531883	−0.0265941	+	0.999646i	$0.508466\pi$
$102$	0	0
$103$	−4.05931e6	−0.366034	−0.183017	−	0.983110i	$-0.558586\pi$
$103$	−4.05931e6	−0.366034	−0.183017	+	0.983110i	$0.558586\pi$
$104$	0	0
$105$	8.06354e6	0.679772
$106$	0	0
$107$	1.79134e7	1.41363	0.706814	−	0.707399i	$-0.250132\pi$
$107$	1.79134e7	1.41363	0.706814	+	0.707399i	$0.250132\pi$
$108$	0	0
$109$	−1.85504e7	−1.37202	−0.686011	−	0.727592i	$-0.740640\pi$
$109$	−1.85504e7	−1.37202	−0.686011	+	0.727592i	$0.740640\pi$
$110$	0	0
$111$	1.62035e7	1.12455
$112$	0	0
$113$	1.19073e7	0.776314	0.388157	−	0.921593i	$-0.373112\pi$
$113$	1.19073e7	0.776314	0.388157	+	0.921593i	$0.373112\pi$
$114$	0	0
$115$	1.31195e7	0.804405
$116$	0	0
$117$	−2.28102e7	−1.31667
$118$	0	0
$119$	1.37038e7	0.745464
$120$	0	0
$121$	1.73045e7	0.887993
$122$	0	0
$123$	−5.38241e7	−2.60801
$124$	0	0
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	3.43595e7	1.48845	0.744223	−	0.667931i	$-0.232820\pi$
$127$	3.43595e7	1.48845	0.744223	+	0.667931i	$0.232820\pi$
$128$	0	0
$129$	−2.86592e7	−1.17544
$130$	0	0
$131$	2.21781e7	0.861936	0.430968	−	0.902367i	$-0.358172\pi$
$131$	2.21781e7	0.861936	0.430968	+	0.902367i	$0.358172\pi$
$132$	0	0
$133$	−2.08071e7	−0.766886
$134$	0	0
$135$	3.48158e7	1.21789
$136$	0	0
$137$	−1.16559e7	−0.387279	−0.193639	−	0.981073i	$-0.562029\pi$
$137$	−1.16559e7	−0.387279	−0.193639	+	0.981073i	$0.562029\pi$
$138$	0	0
$139$	5.62296e6	0.177588	0.0887939	−	0.996050i	$-0.471699\pi$
$139$	5.62296e6	0.177588	0.0887939	+	0.996050i	$0.471699\pi$
$140$	0	0
$141$	4.89275e6	0.146989
$142$	0	0
$143$	2.56820e7	0.734435
$144$	0	0
$145$	−2.55960e7	−0.697243
$146$	0	0
$147$	−2.38590e7	−0.619499
$148$	0	0
$149$	2.74946e7	0.680918	0.340459	−	0.940259i	$-0.389418\pi$
$149$	2.74946e7	0.680918	0.340459	+	0.940259i	$0.389418\pi$
$150$	0	0
$151$	1.37682e7	0.325430	0.162715	−	0.986673i	$-0.447975\pi$
$151$	1.37682e7	0.325430	0.162715	+	0.986673i	$0.447975\pi$
$152$	0	0
$153$	9.96026e7	2.24828
$154$	0	0
$155$	638698.	0.0137764
$156$	0	0
$157$	2.97641e7	0.613825	0.306912	−	0.951738i	$-0.400704\pi$
$157$	2.97641e7	0.613825	0.306912	+	0.951738i	$0.400704\pi$
$158$	0	0
$159$	1.24716e7	0.246054
$160$	0	0
$161$	7.77949e7	1.46913
$162$	0	0
$163$	1.91229e7	0.345858	0.172929	−	0.984934i	$-0.444677\pi$
$163$	1.91229e7	0.345858	0.172929	+	0.984934i	$0.444677\pi$
$164$	0	0
$165$	−6.59867e7	−1.14357
$166$	0	0
$167$	1.55629e7	0.258574	0.129287	−	0.991607i	$-0.458731\pi$
$167$	1.55629e7	0.258574	0.129287	+	0.991607i	$0.458731\pi$
$168$	0	0
$169$	−4.48214e7	−0.714303
$170$	0	0
$171$	−1.51231e8	−2.31289
$172$	0	0
$173$	−7.98560e6	−0.117259	−0.0586295	−	0.998280i	$-0.518673\pi$
$173$	−7.98560e6	−0.117259	−0.0586295	+	0.998280i	$0.518673\pi$
$174$	0	0
$175$	1.15815e7	0.163354
$176$	0	0
$177$	−2.69386e8	−3.65147
$178$	0	0
$179$	9.75110e7	1.27077	0.635386	−	0.772194i	$-0.280841\pi$
$179$	9.75110e7	1.27077	0.635386	+	0.772194i	$0.280841\pi$
$180$	0	0
$181$	−9.94948e7	−1.24717	−0.623585	−	0.781756i	$-0.714324\pi$
$181$	−9.94948e7	−1.24717	−0.623585	+	0.781756i	$0.714324\pi$
$182$	0	0
$183$	−6.10725e7	−0.736660
$184$	0	0
$185$	2.32727e7	0.270237
$186$	0	0
$187$	−1.12143e8	−1.25408
$188$	0	0
$189$	2.06448e8	2.22430
$190$	0	0
$191$	6.12311e7	0.635851	0.317925	−	0.948116i	$-0.397014\pi$
$191$	6.12311e7	0.635851	0.317925	+	0.948116i	$0.397014\pi$
$192$	0	0
$193$	−8.89562e7	−0.890688	−0.445344	−	0.895360i	$-0.646919\pi$
$193$	−8.89562e7	−0.890688	−0.445344	+	0.895360i	$0.646919\pi$
$194$	0	0
$195$	−4.60614e7	−0.444852
$196$	0	0
$197$	1.14248e8	1.06467	0.532335	−	0.846534i	$-0.321315\pi$
$197$	1.14248e8	1.06467	0.532335	+	0.846534i	$0.321315\pi$
$198$	0	0
$199$	−1.12801e8	−1.01468	−0.507338	−	0.861747i	$-0.669370\pi$
$199$	−1.12801e8	−1.01468	−0.507338	+	0.861747i	$0.669370\pi$
$200$	0	0
$201$	−1.00294e8	−0.871138
$202$	0	0
$203$	−1.51777e8	−1.27341
$204$	0	0
$205$	−7.73063e7	−0.626724
$206$	0	0
$207$	5.65432e8	4.43082
$208$	0	0
$209$	1.70272e8	1.29012
$210$	0	0
$211$	−4.35111e7	−0.318868	−0.159434	−	0.987209i	$-0.550967\pi$
$211$	−4.35111e7	−0.318868	−0.159434	+	0.987209i	$0.550967\pi$
$212$	0	0
$213$	4.38468e8	3.10892
$214$	0	0
$215$	−4.11626e7	−0.282467
$216$	0	0
$217$	3.78730e6	0.0251606
$218$	0	0
$219$	−4.03842e8	−2.59811
$220$	0	0
$221$	−7.82803e7	−0.487842
$222$	0	0
$223$	−4.72521e7	−0.285334	−0.142667	−	0.989771i	$-0.545568\pi$
$223$	−4.72521e7	−0.285334	−0.142667	+	0.989771i	$0.545568\pi$
$224$	0	0
$225$	8.41770e7	0.492668
$226$	0	0
$227$	1.84159e8	1.04497	0.522483	−	0.852650i	$-0.325006\pi$
$227$	1.84159e8	1.04497	0.522483	+	0.852650i	$0.325006\pi$
$228$	0	0
$229$	1.20722e8	0.664297	0.332149	−	0.943227i	$-0.392226\pi$
$229$	1.20722e8	0.664297	0.332149	+	0.943227i	$0.392226\pi$
$230$	0	0
$231$	−3.91282e8	−2.08857
$232$	0	0
$233$	−1.99119e8	−1.03126	−0.515629	−	0.856812i	$-0.672442\pi$
$233$	−1.99119e8	−1.03126	−0.515629	+	0.856812i	$0.672442\pi$
$234$	0	0
$235$	7.02734e6	0.0353227
$236$	0	0
$237$	2.32637e8	1.13517
$238$	0	0
$239$	−2.95160e8	−1.39851	−0.699255	−	0.714872i	$-0.746485\pi$
$239$	−2.95160e8	−1.39851	−0.699255	+	0.714872i	$0.746485\pi$
$240$	0	0
$241$	3.22537e8	1.48429	0.742146	−	0.670238i	$-0.233808\pi$
$241$	3.22537e8	1.48429	0.742146	+	0.670238i	$0.233808\pi$
$242$	0	0
$243$	4.75110e8	2.12409
$244$	0	0
$245$	−3.42681e7	−0.148870
$246$	0	0
$247$	1.18856e8	0.501861
$248$	0	0
$249$	2.30257e8	0.945181
$250$	0	0
$251$	1.24978e8	0.498856	0.249428	−	0.968393i	$-0.419757\pi$
$251$	1.24978e8	0.498856	0.249428	+	0.968393i	$0.419757\pi$
$252$	0	0
$253$	−6.36622e8	−2.47150
$254$	0	0
$255$	2.01131e8	0.759607
$256$	0	0
$257$	1.95980e8	0.720188	0.360094	−	0.932916i	$-0.382745\pi$
$257$	1.95980e8	0.720188	0.360094	+	0.932916i	$0.382745\pi$
$258$	0	0
$259$	1.38000e8	0.493550
$260$	0	0
$261$	−1.10315e9	−3.84055
$262$	0	0
$263$	2.03297e8	0.689105	0.344552	−	0.938767i	$-0.388031\pi$
$263$	2.03297e8	0.689105	0.344552	+	0.938767i	$0.388031\pi$
$264$	0	0
$265$	1.79126e7	0.0591287
$266$	0	0
$267$	2.70798e8	0.870676
$268$	0	0
$269$	2.29992e8	0.720409	0.360205	−	0.932873i	$-0.382707\pi$
$269$	2.29992e8	0.720409	0.360205	+	0.932873i	$0.382707\pi$
$270$	0	0
$271$	2.98723e8	0.911752	0.455876	−	0.890043i	$-0.349326\pi$
$271$	2.98723e8	0.911752	0.455876	+	0.890043i	$0.349326\pi$
$272$	0	0
$273$	−2.73131e8	−0.812459
$274$	0	0
$275$	−9.47752e7	−0.274808
$276$	0	0
$277$	−2.55584e8	−0.722528	−0.361264	−	0.932464i	$-0.617655\pi$
$277$	−2.55584e8	−0.722528	−0.361264	+	0.932464i	$0.617655\pi$
$278$	0	0
$279$	2.75270e7	0.0758830
$280$	0	0
$281$	4.50862e8	1.21219	0.606096	−	0.795392i	$-0.292735\pi$
$281$	4.50862e8	1.21219	0.606096	+	0.795392i	$0.292735\pi$
$282$	0	0
$283$	−3.23467e8	−0.848356	−0.424178	−	0.905579i	$-0.639437\pi$
$283$	−3.23467e8	−0.848356	−0.424178	+	0.905579i	$0.639437\pi$
$284$	0	0
$285$	−3.05386e8	−0.781434
$286$	0	0
$287$	−4.58404e8	−1.14462
$288$	0	0
$289$	−6.85205e7	−0.166985
$290$	0	0
$291$	−3.53335e8	−0.840546
$292$	0	0
$293$	5.52109e8	1.28230	0.641148	−	0.767418i	$-0.278459\pi$
$293$	5.52109e8	1.28230	0.641148	+	0.767418i	$0.278459\pi$
$294$	0	0
$295$	−3.86913e8	−0.877477
$296$	0	0
$297$	−1.68943e9	−3.74191
$298$	0	0
$299$	−4.44388e8	−0.961420
$300$	0	0
$301$	−2.44082e8	−0.515886
$302$	0	0
$303$	−4.79306e7	−0.0989836
$304$	0	0
$305$	−8.77170e7	−0.177025
$306$	0	0
$307$	5.38962e8	1.06310	0.531550	−	0.847027i	$-0.321610\pi$
$307$	5.38962e8	1.06310	0.531550	+	0.847027i	$0.321610\pi$
$308$	0	0
$309$	−3.53284e8	−0.681191
$310$	0	0
$311$	6.11434e8	1.15263	0.576313	−	0.817229i	$-0.304491\pi$
$311$	6.11434e8	1.15263	0.576313	+	0.817229i	$0.304491\pi$
$312$	0	0
$313$	1.01255e9	1.86642	0.933211	−	0.359328i	$-0.116994\pi$
$313$	1.01255e9	1.86642	0.933211	+	0.359328i	$0.116994\pi$
$314$	0	0
$315$	4.99145e8	0.899788
$316$	0	0
$317$	4.22278e8	0.744545	0.372273	−	0.928123i	$-0.378579\pi$
$317$	4.22278e8	0.744545	0.372273	+	0.928123i	$0.378579\pi$
$318$	0	0
$319$	1.24204e9	2.14224
$320$	0	0
$321$	1.55902e9	2.63077
$322$	0	0
$323$	−5.18997e8	−0.856952
$324$	0	0
$325$	−6.61569e7	−0.106901
$326$	0	0
$327$	−1.61445e9	−2.55334
$328$	0	0
$329$	4.16701e7	0.0645118
$330$	0	0
$331$	6.58640e8	0.998275	0.499138	−	0.866523i	$-0.333650\pi$
$331$	6.58640e8	0.998275	0.499138	+	0.866523i	$0.333650\pi$
$332$	0	0
$333$	1.00302e9	1.48852
$334$	0	0
$335$	−1.44049e8	−0.209341
$336$	0	0
$337$	−4.51883e8	−0.643164	−0.321582	−	0.946882i	$-0.604215\pi$
$337$	−4.51883e8	−0.643164	−0.321582	+	0.946882i	$0.604215\pi$
$338$	0	0
$339$	1.03630e9	1.44472
$340$	0	0
$341$	−3.09928e7	−0.0423272
$342$	0	0
$343$	−8.13622e8	−1.08866
$344$	0	0
$345$	1.14180e9	1.49700
$346$	0	0
$347$	2.77854e8	0.356996	0.178498	−	0.983940i	$-0.442876\pi$
$347$	2.77854e8	0.356996	0.178498	+	0.983940i	$0.442876\pi$
$348$	0	0
$349$	1.66487e8	0.209648	0.104824	−	0.994491i	$-0.466572\pi$
$349$	1.66487e8	0.209648	0.104824	+	0.994491i	$0.466572\pi$
$350$	0	0
$351$	−1.17929e9	−1.45561
$352$	0	0
$353$	−1.40229e9	−1.69678	−0.848392	−	0.529369i	$-0.822429\pi$
$353$	−1.40229e9	−1.69678	−0.848392	+	0.529369i	$0.822429\pi$
$354$	0	0
$355$	6.29761e8	0.747096
$356$	0	0
$357$	1.19265e9	1.38731
$358$	0	0
$359$	−6.43009e8	−0.733477	−0.366739	−	0.930324i	$-0.619526\pi$
$359$	−6.43009e8	−0.733477	−0.366739	+	0.930324i	$0.619526\pi$
$360$	0	0
$361$	−1.05856e8	−0.118424
$362$	0	0
$363$	1.50602e9	1.65256
$364$	0	0
$365$	−5.80029e8	−0.624345
$366$	0	0
$367$	−9.38196e7	−0.0990746	−0.0495373	−	0.998772i	$-0.515775\pi$
$367$	−9.38196e7	−0.0990746	−0.0495373	+	0.998772i	$0.515775\pi$
$368$	0	0
$369$	−3.33179e9	−3.45212
$370$	0	0
$371$	1.06217e8	0.107990
$372$	0	0
$373$	−1.01907e9	−1.01677	−0.508386	−	0.861129i	$-0.669758\pi$
$373$	−1.01907e9	−1.01677	−0.508386	+	0.861129i	$0.669758\pi$
$374$	0	0
$375$	1.69982e8	0.166453
$376$	0	0
$377$	8.66996e8	0.833340
$378$	0	0
$379$	−6.10359e7	−0.0575902	−0.0287951	−	0.999585i	$-0.509167\pi$
$379$	−6.10359e7	−0.0575902	−0.0287951	+	0.999585i	$0.509167\pi$
$380$	0	0
$381$	2.99032e9	2.77001
$382$	0	0
$383$	−1.85097e9	−1.68346	−0.841732	−	0.539895i	$-0.818464\pi$
$383$	−1.85097e9	−1.68346	−0.841732	+	0.539895i	$0.818464\pi$
$384$	0	0
$385$	−5.61990e8	−0.501898
$386$	0	0
$387$	−1.77405e9	−1.55589
$388$	0	0
$389$	−6.80195e8	−0.585882	−0.292941	−	0.956131i	$-0.594634\pi$
$389$	−6.80195e8	−0.585882	−0.292941	+	0.956131i	$0.594634\pi$
$390$	0	0
$391$	1.94046e9	1.64167
$392$	0	0
$393$	1.93017e9	1.60407
$394$	0	0
$395$	3.34131e8	0.272789
$396$	0	0
$397$	1.86725e9	1.49774	0.748868	−	0.662720i	$-0.230598\pi$
$397$	1.86725e9	1.49774	0.748868	+	0.662720i	$0.230598\pi$
$398$	0	0
$399$	−1.81085e9	−1.42718
$400$	0	0
$401$	7.41095e7	0.0573943	0.0286971	−	0.999588i	$-0.490864\pi$
$401$	7.41095e7	0.0573943	0.0286971	+	0.999588i	$0.490864\pi$
$402$	0	0
$403$	−2.16342e7	−0.0164654
$404$	0	0
$405$	1.55728e9	1.16486
$406$	0	0
$407$	−1.12930e9	−0.830291
$408$	0	0
$409$	−1.24701e9	−0.901237	−0.450618	−	0.892717i	$-0.648796\pi$
$409$	−1.24701e9	−0.901237	−0.450618	+	0.892717i	$0.648796\pi$
$410$	0	0
$411$	−1.01442e9	−0.720728
$412$	0	0
$413$	−2.29428e9	−1.60259
$414$	0	0
$415$	3.30713e8	0.227134
$416$	0	0
$417$	4.89370e8	0.330492
$418$	0	0
$419$	−5.87311e8	−0.390049	−0.195025	−	0.980798i	$-0.562479\pi$
$419$	−5.87311e8	−0.390049	−0.195025	+	0.980798i	$0.562479\pi$
$420$	0	0
$421$	−2.72214e9	−1.77796	−0.888982	−	0.457942i	$-0.848587\pi$
$421$	−2.72214e9	−1.77796	−0.888982	+	0.457942i	$0.848587\pi$
$422$	0	0
$423$	3.02869e8	0.194564
$424$	0	0
$425$	2.88880e8	0.182539
$426$	0	0
$427$	−5.20137e8	−0.323311
$428$	0	0
$429$	2.23512e9	1.36679
$430$	0	0
$431$	−1.33258e9	−0.801721	−0.400861	−	0.916139i	$-0.631289\pi$
$431$	−1.33258e9	−0.801721	−0.400861	+	0.916139i	$0.631289\pi$
$432$	0	0
$433$	−1.39398e9	−0.825179	−0.412590	−	0.910917i	$-0.635376\pi$
$433$	−1.39398e9	−0.825179	−0.412590	+	0.910917i	$0.635376\pi$
$434$	0	0
$435$	−2.22763e9	−1.29757
$436$	0	0
$437$	−2.94629e9	−1.68885
$438$	0	0
$439$	1.96180e9	1.10670	0.553348	−	0.832950i	$-0.313350\pi$
$439$	1.96180e9	1.10670	0.553348	+	0.832950i	$0.313350\pi$
$440$	0	0
$441$	−1.47691e9	−0.820007
$442$	0	0
$443$	1.36082e9	0.743684	0.371842	−	0.928296i	$-0.378726\pi$
$443$	1.36082e9	0.743684	0.371842	+	0.928296i	$0.378726\pi$
$444$	0	0
$445$	3.88941e8	0.209230
$446$	0	0
$447$	2.39287e9	1.26719
$448$	0	0
$449$	2.43167e9	1.26778	0.633889	−	0.773424i	$-0.281458\pi$
$449$	2.43167e9	1.26778	0.633889	+	0.773424i	$0.281458\pi$
$450$	0	0
$451$	3.75128e9	1.92558
$452$	0	0
$453$	1.19826e9	0.605628
$454$	0	0
$455$	−3.92291e8	−0.195240
$456$	0	0
$457$	−3.09420e8	−0.151650	−0.0758250	−	0.997121i	$-0.524159\pi$
$457$	−3.09420e8	−0.151650	−0.0758250	+	0.997121i	$0.524159\pi$
$458$	0	0
$459$	5.14948e9	2.48553
$460$	0	0
$461$	−2.43256e9	−1.15640	−0.578202	−	0.815894i	$-0.696245\pi$
$461$	−2.43256e9	−1.15640	−0.578202	+	0.815894i	$0.696245\pi$
$462$	0	0
$463$	1.49638e9	0.700664	0.350332	−	0.936626i	$-0.386069\pi$
$463$	1.49638e9	0.700664	0.350332	+	0.936626i	$0.386069\pi$
$464$	0	0
$465$	5.55863e7	0.0256379
$466$	0	0
$467$	−3.09165e9	−1.40469	−0.702346	−	0.711836i	$-0.747864\pi$
$467$	−3.09165e9	−1.40469	−0.702346	+	0.711836i	$0.747864\pi$
$468$	0	0
$469$	−8.54171e8	−0.382332
$470$	0	0
$471$	2.59039e9	1.14233
$472$	0	0
$473$	1.99741e9	0.867867
$474$	0	0
$475$	−4.38619e8	−0.187785
$476$	0	0
$477$	7.72009e8	0.325693
$478$	0	0
$479$	2.24028e8	0.0931384	0.0465692	−	0.998915i	$-0.485171\pi$
$479$	2.24028e8	0.0931384	0.0465692	+	0.998915i	$0.485171\pi$
$480$	0	0
$481$	−7.88299e8	−0.322986
$482$	0	0
$483$	6.77054e9	2.73406
$484$	0	0
$485$	−5.07487e8	−0.201990
$486$	0	0
$487$	2.38421e9	0.935390	0.467695	−	0.883890i	$-0.345085\pi$
$487$	2.38421e9	0.935390	0.467695	+	0.883890i	$0.345085\pi$
$488$	0	0
$489$	1.66428e9	0.643644
$490$	0	0
$491$	1.28112e8	0.0488432	0.0244216	−	0.999702i	$-0.492226\pi$
$491$	1.28112e8	0.0488432	0.0244216	+	0.999702i	$0.492226\pi$
$492$	0	0
$493$	−3.78582e9	−1.42297
$494$	0	0
$495$	−4.08468e9	−1.51370
$496$	0	0
$497$	3.73430e9	1.36446
$498$	0	0
$499$	−2.42499e9	−0.873690	−0.436845	−	0.899537i	$-0.643904\pi$
$499$	−2.42499e9	−0.873690	−0.436845	+	0.899537i	$0.643904\pi$
$500$	0	0
$501$	1.35445e9	0.481207
$502$	0	0
$503$	2.22661e9	0.780110	0.390055	−	0.920792i	$-0.372456\pi$
$503$	2.22661e9	0.780110	0.390055	+	0.920792i	$0.372456\pi$
$504$	0	0
$505$	−6.88416e7	−0.0237865
$506$	0	0
$507$	−3.90084e9	−1.32932
$508$	0	0
$509$	−3.67616e9	−1.23561	−0.617807	−	0.786330i	$-0.711979\pi$
$509$	−3.67616e9	−1.23561	−0.617807	+	0.786330i	$0.711979\pi$
$510$	0	0
$511$	−3.43941e9	−1.14028
$512$	0	0
$513$	−7.81869e9	−2.55695
$514$	0	0
$515$	−5.07413e8	−0.163695
$516$	0	0
$517$	−3.41001e8	−0.108527
$518$	0	0
$519$	−6.94992e8	−0.218220
$520$	0	0
$521$	9.21625e8	0.285511	0.142755	−	0.989758i	$-0.454404\pi$
$521$	9.21625e8	0.285511	0.142755	+	0.989758i	$0.454404\pi$
$522$	0	0
$523$	−1.57199e8	−0.0480502	−0.0240251	−	0.999711i	$-0.507648\pi$
$523$	−1.57199e8	−0.0480502	−0.0240251	+	0.999711i	$0.507648\pi$
$524$	0	0
$525$	1.00794e9	0.304003
$526$	0	0
$527$	9.44677e7	0.0281155
$528$	0	0
$529$	7.61094e9	2.23534
$530$	0	0
$531$	−1.66754e10	−4.83332
$532$	0	0
$533$	2.61854e9	0.749056
$534$	0	0
$535$	2.23918e9	0.632194
$536$	0	0
$537$	8.48644e9	2.36491
$538$	0	0
$539$	1.66285e9	0.457397
$540$	0	0
$541$	−2.80816e8	−0.0762486	−0.0381243	−	0.999273i	$-0.512138\pi$
$541$	−2.80816e8	−0.0762486	−0.0381243	+	0.999273i	$0.512138\pi$
$542$	0	0
$543$	−8.65909e9	−2.32099
$544$	0	0
$545$	−2.31880e9	−0.613587
$546$	0	0
$547$	1.15456e9	0.301622	0.150811	−	0.988563i	$-0.451812\pi$
$547$	1.15456e9	0.301622	0.150811	+	0.988563i	$0.451812\pi$
$548$	0	0
$549$	−3.78048e9	−0.975088
$550$	0	0
$551$	5.74817e9	1.46386
$552$	0	0
$553$	1.98130e9	0.498210
$554$	0	0
$555$	2.02543e9	0.502913
$556$	0	0
$557$	2.36841e9	0.580715	0.290357	−	0.956918i	$-0.406226\pi$
$557$	2.36841e9	0.580715	0.290357	+	0.956918i	$0.406226\pi$
$558$	0	0
$559$	1.39427e9	0.337603
$560$	0	0
$561$	−9.75987e9	−2.33386
$562$	0	0
$563$	−2.50492e9	−0.591580	−0.295790	−	0.955253i	$-0.595583\pi$
$563$	−2.50492e9	−0.591580	−0.295790	+	0.955253i	$0.595583\pi$
$564$	0	0
$565$	1.48841e9	0.347178
$566$	0	0
$567$	9.23422e9	2.12745
$568$	0	0
$569$	−4.42619e9	−1.00725	−0.503625	−	0.863922i	$-0.668001\pi$
$569$	−4.42619e9	−1.00725	−0.503625	+	0.863922i	$0.668001\pi$
$570$	0	0
$571$	−1.32550e9	−0.297956	−0.148978	−	0.988841i	$-0.547598\pi$
$571$	−1.32550e9	−0.297956	−0.148978	+	0.988841i	$0.547598\pi$
$572$	0	0
$573$	5.32898e9	1.18332
$574$	0	0
$575$	1.63994e9	0.359741
$576$	0	0
$577$	2.76604e9	0.599436	0.299718	−	0.954028i	$-0.403107\pi$
$577$	2.76604e9	0.599436	0.299718	+	0.954028i	$0.403107\pi$
$578$	0	0
$579$	−7.74191e9	−1.65757
$580$	0	0
$581$	1.96103e9	0.414828
$582$	0	0
$583$	−8.69208e8	−0.181670
$584$	0	0
$585$	−2.85127e9	−0.588834
$586$	0	0
$587$	2.15575e9	0.439912	0.219956	−	0.975510i	$-0.429409\pi$
$587$	2.15575e9	0.439912	0.219956	+	0.975510i	$0.429409\pi$
$588$	0	0
$589$	−1.43434e8	−0.0289234
$590$	0	0
$591$	9.94303e9	1.98136
$592$	0	0
$593$	9.50086e9	1.87099	0.935495	−	0.353341i	$-0.114954\pi$
$593$	9.50086e9	1.87099	0.935495	+	0.353341i	$0.114954\pi$
$594$	0	0
$595$	1.71298e9	0.333382
$596$	0	0
$597$	−9.81715e9	−1.88832
$598$	0	0
$599$	−9.55306e8	−0.181614	−0.0908068	−	0.995869i	$-0.528945\pi$
$599$	−9.55306e8	−0.181614	−0.0908068	+	0.995869i	$0.528945\pi$
$600$	0	0
$601$	−8.41945e9	−1.58206	−0.791030	−	0.611777i	$-0.790455\pi$
$601$	−8.41945e9	−1.58206	−0.791030	+	0.611777i	$0.790455\pi$
$602$	0	0
$603$	−6.20833e9	−1.15309
$604$	0	0
$605$	2.16306e9	0.397122
$606$	0	0
$607$	−9.80243e9	−1.77899	−0.889495	−	0.456945i	$-0.848944\pi$
$607$	−9.80243e9	−1.77899	−0.889495	+	0.456945i	$0.848944\pi$
$608$	0	0
$609$	−1.32092e10	−2.36983
$610$	0	0
$611$	−2.38032e8	−0.0422174
$612$	0	0
$613$	−1.33970e9	−0.234907	−0.117454	−	0.993078i	$-0.537473\pi$
$613$	−1.33970e9	−0.234907	−0.117454	+	0.993078i	$0.537473\pi$
$614$	0	0
$615$	−6.72801e9	−1.16634
$616$	0	0
$617$	1.03037e10	1.76602	0.883012	−	0.469351i	$-0.155512\pi$
$617$	1.03037e10	1.76602	0.883012	+	0.469351i	$0.155512\pi$
$618$	0	0
$619$	2.87364e9	0.486984	0.243492	−	0.969903i	$-0.421707\pi$
$619$	2.87364e9	0.486984	0.243492	+	0.969903i	$0.421707\pi$
$620$	0	0
$621$	2.92330e10	4.89838
$622$	0	0
$623$	2.30631e9	0.382128
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	1.48188e10	2.40092
$628$	0	0
$629$	3.44218e9	0.551514
$630$	0	0
$631$	−6.17623e9	−0.978635	−0.489317	−	0.872106i	$-0.662754\pi$
$631$	−6.17623e9	−0.978635	−0.489317	+	0.872106i	$0.662754\pi$
$632$	0	0
$633$	−3.78680e9	−0.593416
$634$	0	0
$635$	4.29493e9	0.665653
$636$	0	0
$637$	1.16074e9	0.177929
$638$	0	0
$639$	2.71418e10	4.11515
$640$	0	0
$641$	−9.14422e9	−1.37134	−0.685668	−	0.727915i	$-0.740490\pi$
$641$	−9.14422e9	−1.37134	−0.685668	+	0.727915i	$0.740490\pi$
$642$	0	0
$643$	−7.48915e8	−0.111095	−0.0555475	−	0.998456i	$-0.517690\pi$
$643$	−7.48915e8	−0.111095	−0.0555475	+	0.998456i	$0.517690\pi$
$644$	0	0
$645$	−3.58241e9	−0.525673
$646$	0	0
$647$	−6.13359e9	−0.890327	−0.445163	−	0.895449i	$-0.646854\pi$
$647$	−6.13359e9	−0.890327	−0.445163	+	0.895449i	$0.646854\pi$
$648$	0	0
$649$	1.87749e10	2.69601
$650$	0	0
$651$	3.29611e8	0.0468239
$652$	0	0
$653$	1.37749e10	1.93594	0.967972	−	0.251059i	$-0.0807787\pi$
$653$	1.37749e10	1.93594	0.967972	+	0.251059i	$0.0807787\pi$
$654$	0	0
$655$	2.77226e9	0.385469
$656$	0	0
$657$	−2.49984e10	−3.43901
$658$	0	0
$659$	−8.08420e9	−1.10037	−0.550184	−	0.835044i	$-0.685442\pi$
$659$	−8.08420e9	−1.10037	−0.550184	+	0.835044i	$0.685442\pi$
$660$	0	0
$661$	−2.44260e9	−0.328964	−0.164482	−	0.986380i	$-0.552595\pi$
$661$	−2.44260e9	−0.328964	−0.164482	+	0.986380i	$0.552595\pi$
$662$	0	0
$663$	−6.81278e9	−0.907877
$664$	0	0
$665$	−2.60089e9	−0.342962
$666$	0	0
$667$	−2.14916e10	−2.80433
$668$	0	0
$669$	−4.11238e9	−0.531009
$670$	0	0
$671$	4.25646e9	0.543900
$672$	0	0
$673$	−1.23905e8	−0.0156689	−0.00783443	−	0.999969i	$-0.502494\pi$
$673$	−1.23905e8	−0.0156689	−0.00783443	+	0.999969i	$0.502494\pi$
$674$	0	0
$675$	4.35197e9	0.544657
$676$	0	0
$677$	−1.14942e10	−1.42369	−0.711847	−	0.702335i	$-0.752141\pi$
$677$	−1.14942e10	−1.42369	−0.711847	+	0.702335i	$0.752141\pi$
$678$	0	0
$679$	−3.00925e9	−0.368905
$680$	0	0
$681$	1.60275e10	1.94469
$682$	0	0
$683$	8.92504e9	1.07186	0.535929	−	0.844263i	$-0.319961\pi$
$683$	8.92504e9	1.07186	0.535929	+	0.844263i	$0.319961\pi$
$684$	0	0
$685$	−1.45699e9	−0.173196
$686$	0	0
$687$	1.05065e10	1.23626
$688$	0	0
$689$	−6.06742e8	−0.0706703
$690$	0	0
$691$	−2.95336e9	−0.340521	−0.170260	−	0.985399i	$-0.554461\pi$
$691$	−2.95336e9	−0.340521	−0.170260	+	0.985399i	$0.554461\pi$
$692$	0	0
$693$	−2.42210e10	−2.76455
$694$	0	0
$695$	7.02870e8	0.0794197
$696$	0	0
$697$	−1.14341e10	−1.27905
$698$	0	0
$699$	−1.73295e10	−1.91918
$700$	0	0
$701$	−1.43095e9	−0.156896	−0.0784479	−	0.996918i	$-0.524996\pi$
$701$	−1.43095e9	−0.156896	−0.0784479	+	0.996918i	$0.524996\pi$
$702$	0	0
$703$	−5.22641e9	−0.567362
$704$	0	0
$705$	6.11594e8	0.0657357
$706$	0	0
$707$	−4.08211e8	−0.0434427
$708$	0	0
$709$	−1.25151e10	−1.31878	−0.659392	−	0.751799i	$-0.729186\pi$
$709$	−1.25151e10	−1.31878	−0.659392	+	0.751799i	$0.729186\pi$
$710$	0	0
$711$	1.44006e10	1.50258
$712$	0	0
$713$	5.36282e8	0.0554089
$714$	0	0
$715$	3.21025e9	0.328449
$716$	0	0
$717$	−2.56880e10	−2.60263
$718$	0	0
$719$	−8.89943e9	−0.892917	−0.446459	−	0.894804i	$-0.647315\pi$
$719$	−8.89943e9	−0.892917	−0.446459	+	0.894804i	$0.647315\pi$
$720$	0	0
$721$	−3.00881e9	−0.298966
$722$	0	0
$723$	2.80705e10	2.76227
$724$	0	0
$725$	−3.19950e9	−0.311816
$726$	0	0
$727$	3.56000e9	0.343621	0.171810	−	0.985130i	$-0.445038\pi$
$727$	3.56000e9	0.343621	0.171810	+	0.985130i	$0.445038\pi$
$728$	0	0
$729$	1.41030e10	1.34823
$730$	0	0
$731$	−6.08822e9	−0.576474
$732$	0	0
$733$	5.05266e9	0.473867	0.236934	−	0.971526i	$-0.423858\pi$
$733$	5.05266e9	0.473867	0.236934	+	0.971526i	$0.423858\pi$
$734$	0	0
$735$	−2.98237e9	−0.277049
$736$	0	0
$737$	6.98998e9	0.643191
$738$	0	0
$739$	3.15300e9	0.287387	0.143694	−	0.989622i	$-0.454102\pi$
$739$	3.15300e9	0.287387	0.143694	+	0.989622i	$0.454102\pi$
$740$	0	0
$741$	1.03441e10	0.933965
$742$	0	0
$743$	1.93745e9	0.173288	0.0866442	−	0.996239i	$-0.472386\pi$
$743$	1.93745e9	0.173288	0.0866442	+	0.996239i	$0.472386\pi$
$744$	0	0
$745$	3.43682e9	0.304516
$746$	0	0
$747$	1.42533e10	1.25110
$748$	0	0
$749$	1.32777e10	1.15461
$750$	0	0
$751$	1.58399e10	1.36462	0.682311	−	0.731062i	$-0.260975\pi$
$751$	1.58399e10	1.36462	0.682311	+	0.731062i	$0.260975\pi$
$752$	0	0
$753$	1.08769e10	0.928374
$754$	0	0
$755$	1.72103e9	0.145537
$756$	0	0
$757$	1.57162e10	1.31678	0.658389	−	0.752678i	$-0.271238\pi$
$757$	1.57162e10	1.31678	0.658389	+	0.752678i	$0.271238\pi$
$758$	0	0
$759$	−5.54056e10	−4.59947
$760$	0	0
$761$	−2.50123e9	−0.205735	−0.102867	−	0.994695i	$-0.532802\pi$
$761$	−2.50123e9	−0.205735	−0.102867	+	0.994695i	$0.532802\pi$
$762$	0	0
$763$	−1.37498e10	−1.12063
$764$	0	0
$765$	1.24503e10	1.00546
$766$	0	0
$767$	1.31056e10	1.04875
$768$	0	0
$769$	8.48850e9	0.673114	0.336557	−	0.941663i	$-0.390737\pi$
$769$	8.48850e9	0.673114	0.336557	+	0.941663i	$0.390737\pi$
$770$	0	0
$771$	1.70563e10	1.34027
$772$	0	0
$773$	−2.42832e10	−1.89094	−0.945471	−	0.325707i	$-0.894398\pi$
$773$	−2.42832e10	−1.89094	−0.945471	+	0.325707i	$0.894398\pi$
$774$	0	0
$775$	7.98373e7	0.00616098
$776$	0	0
$777$	1.20102e10	0.918498
$778$	0	0
$779$	1.73609e10	1.31580
$780$	0	0
$781$	−3.05591e10	−2.29542
$782$	0	0
$783$	−5.70333e10	−4.24582
$784$	0	0
$785$	3.72052e9	0.274511
$786$	0	0
$787$	−6.20737e9	−0.453938	−0.226969	−	0.973902i	$-0.572882\pi$
$787$	−6.20737e9	−0.453938	−0.226969	+	0.973902i	$0.572882\pi$
$788$	0	0
$789$	1.76930e10	1.28243
$790$	0	0
$791$	8.82583e9	0.634071
$792$	0	0
$793$	2.97118e9	0.211579
$794$	0	0
$795$	1.55895e9	0.110039
$796$	0	0
$797$	1.88550e7	0.00131924	0.000659619	−	1.00000i	$-0.499790\pi$
$797$	1.88550e7	0.00131924	0.000659619		1.00000i	$0.499790\pi$
$798$	0	0
$799$	1.03939e9	0.0720883
$800$	0	0
$801$	1.67628e10	1.15248
$802$	0	0
$803$	2.81458e10	1.91827
$804$	0	0
$805$	9.72436e9	0.657015
$806$	0	0
$807$	2.00163e10	1.34069
$808$	0	0
$809$	−1.59140e10	−1.05672	−0.528360	−	0.849021i	$-0.677193\pi$
$809$	−1.59140e10	−1.05672	−0.528360	+	0.849021i	$0.677193\pi$
$810$	0	0
$811$	−2.62774e10	−1.72986	−0.864928	−	0.501896i	$-0.832636\pi$
$811$	−2.62774e10	−1.72986	−0.864928	+	0.501896i	$0.832636\pi$
$812$	0	0
$813$	2.59981e10	1.69678
$814$	0	0
$815$	2.39037e9	0.154673
$816$	0	0
$817$	9.24401e9	0.593039
$818$	0	0
$819$	−1.69072e10	−1.07542
$820$	0	0
$821$	9.37571e9	0.591293	0.295647	−	0.955297i	$-0.404465\pi$
$821$	9.37571e9	0.591293	0.295647	+	0.955297i	$0.404465\pi$
$822$	0	0
$823$	−1.03455e10	−0.646920	−0.323460	−	0.946242i	$-0.604846\pi$
$823$	−1.03455e10	−0.646920	−0.323460	+	0.946242i	$0.604846\pi$
$824$	0	0
$825$	−8.24834e9	−0.511420
$826$	0	0
$827$	−2.19356e10	−1.34859	−0.674296	−	0.738461i	$-0.735553\pi$
$827$	−2.19356e10	−1.34859	−0.674296	+	0.738461i	$0.735553\pi$
$828$	0	0
$829$	3.05413e10	1.86186	0.930929	−	0.365199i	$-0.118999\pi$
$829$	3.05413e10	1.86186	0.930929	+	0.365199i	$0.118999\pi$
$830$	0	0
$831$	−2.22436e10	−1.34463
$832$	0	0
$833$	−5.06847e9	−0.303822
$834$	0	0
$835$	1.94537e9	0.115638
$836$	0	0
$837$	1.42315e9	0.0838905
$838$	0	0
$839$	−2.52012e10	−1.47318	−0.736588	−	0.676342i	$-0.763564\pi$
$839$	−2.52012e10	−1.47318	−0.736588	+	0.676342i	$0.763564\pi$
$840$	0	0
$841$	2.46800e10	1.43074
$842$	0	0
$843$	3.92388e10	2.25589
$844$	0	0
$845$	−5.60268e9	−0.319446
$846$	0	0
$847$	1.28263e10	0.725287
$848$	0	0
$849$	−2.81516e10	−1.57879
$850$	0	0
$851$	1.95409e10	1.08690
$852$	0	0
$853$	3.04039e9	0.167729	0.0838644	−	0.996477i	$-0.473274\pi$
$853$	3.04039e9	0.167729	0.0838644	+	0.996477i	$0.473274\pi$
$854$	0	0
$855$	−1.89039e10	−1.03435
$856$	0	0
$857$	−7.30094e9	−0.396229	−0.198114	−	0.980179i	$-0.563482\pi$
$857$	−7.30094e9	−0.396229	−0.198114	+	0.980179i	$0.563482\pi$
$858$	0	0
$859$	2.16215e10	1.16388	0.581941	−	0.813231i	$-0.302293\pi$
$859$	2.16215e10	1.16388	0.581941	+	0.813231i	$0.302293\pi$
$860$	0	0
$861$	−3.98952e10	−2.13015
$862$	0	0
$863$	3.05503e10	1.61800	0.808999	−	0.587810i	$-0.200010\pi$
$863$	3.05503e10	1.61800	0.808999	+	0.587810i	$0.200010\pi$
$864$	0	0
$865$	−9.98200e8	−0.0524398
$866$	0	0
$867$	−5.96338e9	−0.310760
$868$	0	0
$869$	−1.62137e10	−0.838132
$870$	0	0
$871$	4.87929e9	0.250203
$872$	0	0
$873$	−2.18720e10	−1.11260
$874$	0	0
$875$	1.44768e9	0.0730543
$876$	0	0
$877$	2.48353e8	0.0124329	0.00621644	−	0.999981i	$-0.498021\pi$
$877$	2.48353e8	0.0124329	0.00621644	+	0.999981i	$0.498021\pi$
$878$	0	0
$879$	4.80504e10	2.38636
$880$	0	0
$881$	−2.96667e10	−1.46169	−0.730843	−	0.682546i	$-0.760873\pi$
$881$	−2.96667e10	−1.46169	−0.730843	+	0.682546i	$0.760873\pi$
$882$	0	0
$883$	−7.57977e9	−0.370505	−0.185252	−	0.982691i	$-0.559310\pi$
$883$	−7.57977e9	−0.370505	−0.185252	+	0.982691i	$0.559310\pi$
$884$	0	0
$885$	−3.36732e10	−1.63299
$886$	0	0
$887$	−1.01060e10	−0.486235	−0.243117	−	0.969997i	$-0.578170\pi$
$887$	−1.01060e10	−0.486235	−0.243117	+	0.969997i	$0.578170\pi$
$888$	0	0
$889$	2.54677e10	1.21572
$890$	0	0
$891$	−7.55668e10	−3.57898
$892$	0	0
$893$	−1.57815e9	−0.0741598
$894$	0	0
$895$	1.21889e10	0.568307
$896$	0	0
$897$	−3.86753e10	−1.78921
$898$	0	0
$899$	−1.04628e9	−0.0480274
$900$	0	0
$901$	2.64939e9	0.120673
$902$	0	0
$903$	−2.12426e10	−0.960067
$904$	0	0
$905$	−1.24369e10	−0.557751
$906$	0	0
$907$	1.98649e10	0.884016	0.442008	−	0.897011i	$-0.354266\pi$
$907$	1.98649e10	0.884016	0.442008	+	0.897011i	$0.354266\pi$
$908$	0	0
$909$	−2.96698e9	−0.131021
$910$	0	0
$911$	−2.90396e10	−1.27256	−0.636278	−	0.771460i	$-0.719527\pi$
$911$	−2.90396e10	−1.27256	−0.636278	+	0.771460i	$0.719527\pi$
$912$	0	0
$913$	−1.60478e10	−0.697859
$914$	0	0
$915$	−7.63406e9	−0.329444
$916$	0	0
$917$	1.64387e10	0.704005
$918$	0	0
$919$	3.06973e10	1.30465	0.652327	−	0.757938i	$-0.273793\pi$
$919$	3.06973e10	1.30465	0.652327	+	0.757938i	$0.273793\pi$
$920$	0	0
$921$	4.69062e10	1.97843
$922$	0	0
$923$	−2.13315e10	−0.892924
$924$	0	0
$925$	2.90908e9	0.120854
$926$	0	0
$927$	−2.18688e10	−0.901666
$928$	0	0
$929$	−1.29984e10	−0.531904	−0.265952	−	0.963986i	$-0.585686\pi$
$929$	−1.29984e10	−0.531904	−0.265952	+	0.963986i	$0.585686\pi$
$930$	0	0
$931$	7.69568e9	0.312553
$932$	0	0
$933$	5.32135e10	2.14504
$934$	0	0
$935$	−1.40179e10	−0.560843
$936$	0	0
$937$	−3.51243e10	−1.39482	−0.697412	−	0.716670i	$-0.745665\pi$
$937$	−3.51243e10	−1.39482	−0.697412	+	0.716670i	$0.745665\pi$
$938$	0	0
$939$	8.81226e10	3.47342
$940$	0	0
$941$	4.51801e9	0.176760	0.0883800	−	0.996087i	$-0.471831\pi$
$941$	4.51801e9	0.176760	0.0883800	+	0.996087i	$0.471831\pi$
$942$	0	0
$943$	−6.49101e10	−2.52070
$944$	0	0
$945$	2.58060e10	0.994738
$946$	0	0
$947$	7.07244e9	0.270610	0.135305	−	0.990804i	$-0.456799\pi$
$947$	7.07244e9	0.270610	0.135305	+	0.990804i	$0.456799\pi$
$948$	0	0
$949$	1.96469e10	0.746213
$950$	0	0
$951$	3.67511e10	1.38560
$952$	0	0
$953$	7.75183e9	0.290121	0.145060	−	0.989423i	$-0.453662\pi$
$953$	7.75183e9	0.290121	0.145060	+	0.989423i	$0.453662\pi$
$954$	0	0
$955$	7.65388e9	0.284361
$956$	0	0
$957$	1.08096e11	3.98673
$958$	0	0
$959$	−8.63951e9	−0.316318
$960$	0	0
$961$	−2.74865e10	−0.999051
$962$	0	0
$963$	9.65054e10	3.48225
$964$	0	0
$965$	−1.11195e10	−0.398328
$966$	0	0
$967$	−2.07254e9	−0.0737071	−0.0368536	−	0.999321i	$-0.511734\pi$
$967$	−2.07254e9	−0.0737071	−0.0368536	+	0.999321i	$0.511734\pi$
$968$	0	0
$969$	−4.51687e10	−1.59479
$970$	0	0
$971$	3.00236e10	1.05244	0.526218	−	0.850350i	$-0.323610\pi$
$971$	3.00236e10	1.05244	0.526218	+	0.850350i	$0.323610\pi$
$972$	0	0
$973$	4.16782e9	0.145049
$974$	0	0
$975$	−5.75767e9	−0.198944
$976$	0	0
$977$	−2.62484e9	−0.0900476	−0.0450238	−	0.998986i	$-0.514336\pi$
$977$	−2.62484e9	−0.0900476	−0.0450238	+	0.998986i	$0.514336\pi$
$978$	0	0
$979$	−1.88733e10	−0.642849
$980$	0	0
$981$	−9.99371e10	−3.37976
$982$	0	0
$983$	3.23601e9	0.108661	0.0543304	−	0.998523i	$-0.482698\pi$
$983$	3.23601e9	0.108661	0.0543304	+	0.998523i	$0.482698\pi$
$984$	0	0
$985$	1.42809e10	0.476135
$986$	0	0
$987$	3.62658e9	0.120057
$988$	0	0
$989$	−3.45621e10	−1.13609
$990$	0	0
$991$	3.91654e10	1.27834	0.639168	−	0.769067i	$-0.279279\pi$
$991$	3.91654e10	1.27834	0.639168	+	0.769067i	$0.279279\pi$
$992$	0	0
$993$	5.73219e10	1.85780
$994$	0	0
$995$	−1.41001e10	−0.453777
$996$	0	0
$997$	4.65367e10	1.48718	0.743588	−	0.668638i	$-0.233123\pi$
$997$	4.65367e10	1.48718	0.743588	+	0.668638i	$0.233123\pi$
$998$	0	0
$999$	5.18564e10	1.64560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	40.8.a.d.1.2	✓	2
3.2	odd	2		360.8.a.f.1.2		2
4.3	odd	2		80.8.a.e.1.1		2
5.2	odd	4		200.8.c.g.49.1		4
5.3	odd	4		200.8.c.g.49.4		4
5.4	even	2		200.8.a.j.1.1		2
8.3	odd	2		320.8.a.w.1.2		2
8.5	even	2		320.8.a.i.1.1		2
20.3	even	4		400.8.c.p.49.1		4
20.7	even	4		400.8.c.p.49.4		4
20.19	odd	2		400.8.a.bg.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.8.a.d.1.2	✓	2	1.1	even	1	trivial
80.8.a.e.1.1		2	4.3	odd	2
200.8.a.j.1.1		2	5.4	even	2
200.8.c.g.49.1		4	5.2	odd	4
200.8.c.g.49.4		4	5.3	odd	4
320.8.a.i.1.1		2	8.5	even	2
320.8.a.w.1.2		2	8.3	odd	2
360.8.a.f.1.2		2	3.2	odd	2
400.8.a.bg.1.2		2	20.19	odd	2
400.8.c.p.49.1		4	20.3	even	4
400.8.c.p.49.4		4	20.7	even	4

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:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	40.a (trivial)

\(f(q)\)	\(=\)	\(q+87.0306 q^{3} +125.000 q^{5} +741.214 q^{7} +5387.33 q^{9} -6065.61 q^{11} -4234.04 q^{13} +10878.8 q^{15} +18488.3 q^{17} -28071.6 q^{19} +64508.3 q^{21} +104956. q^{23} +15625.0 q^{25} +278526. q^{27} -204768. q^{29} +5109.59 q^{31} -527894. q^{33} +92651.8 q^{35} +186181. q^{37} -368491. q^{39} -618450. q^{41} -329301. q^{43} +673416. q^{45} +56218.8 q^{47} -274144. q^{49} +1.60905e6 q^{51} +143301. q^{53} -758201. q^{55} -2.44309e6 q^{57} -3.09530e6 q^{59} -701736. q^{61} +3.99316e6 q^{63} -529255. q^{65} -1.15239e6 q^{67} +9.13438e6 q^{69} +5.03809e6 q^{71} -4.64023e6 q^{73} +1.35985e6 q^{75} -4.49592e6 q^{77} +2.67305e6 q^{79} +1.24582e7 q^{81} +2.64570e6 q^{83} +2.31104e6 q^{85} -1.78211e7 q^{87} +3.11153e6 q^{89} -3.13833e6 q^{91} +444690. q^{93} -3.50895e6 q^{95} -4.05990e6 q^{97} -3.26774e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 76 q^{3} + 250 q^{5} + 796 q^{7} + 3322 q^{9} - 560 q^{11} + 4476 q^{13} + 9500 q^{15} + 29524 q^{17} + 19560 q^{19} + 63904 q^{21} + 129796 q^{23} + 31250 q^{25} + 325432 q^{27} - 211060 q^{29} + 165352 q^{31}+ \cdots - 44048304 q^{99}+O(q^{100}) \) 2 * q + 76 * q^3 + 250 * q^5 + 796 * q^7 + 3322 * q^9 - 560 * q^11 + 4476 * q^13 + 9500 * q^15 + 29524 * q^17 + 19560 * q^19 + 63904 * q^21 + 129796 * q^23 + 31250 * q^25 + 325432 * q^27 - 211060 * q^29 + 165352 * q^31 - 588624 * q^33 + 99500 * q^35 - 292100 * q^37 - 464568 * q^39 - 851324 * q^41 - 197812 * q^43 + 415250 * q^45 - 34164 * q^47 - 1094686 * q^49 + 1487320 * q^51 - 876404 * q^53 - 70000 * q^55 - 2968496 * q^57 - 2088504 * q^59 + 1409708 * q^61 + 3880012 * q^63 + 559500 * q^65 - 2882076 * q^67 + 8860384 * q^69 + 9819448 * q^71 - 189404 * q^73 + 1187500 * q^75 - 4194288 * q^77 - 2028496 * q^79 + 16457698 * q^81 - 1393332 * q^83 + 3690500 * q^85 - 17751672 * q^87 + 5688820 * q^89 - 2661144 * q^91 - 1322880 * q^93 + 2445000 * q^95 - 20045996 * q^97 - 44048304 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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