LMFDB - Embedded newform 320.8.a.u.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,8,Mod(1,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	320.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,20,0,250,0,100,0,5554] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$99.9632081549$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{19}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 19 $ x^2 - 19
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.35890$ of defining polynomial
Character	$\chi$	$=$	320.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+79.7424 q^{3} +125.000 q^{5} +538.197 q^{7} +4171.85 q^{9} -1215.12 q^{11} -7070.42 q^{13} +9967.80 q^{15} -3348.13 q^{17} +22169.7 q^{19} +42917.1 q^{21} +58513.1 q^{23} +15625.0 q^{25} +158276. q^{27} +206301. q^{29} -177822. q^{31} -96896.5 q^{33} +67274.6 q^{35} +284128. q^{37} -563812. q^{39} +627353. q^{41} -164889. q^{43} +521481. q^{45} +449355. q^{47} -533887. q^{49} -266988. q^{51} +730190. q^{53} -151890. q^{55} +1.76786e6 q^{57} +1.42202e6 q^{59} +266326. q^{61} +2.24527e6 q^{63} -883803. q^{65} +2.95028e6 q^{67} +4.66598e6 q^{69} -921138. q^{71} +4.25657e6 q^{73} +1.24597e6 q^{75} -653973. q^{77} -6.28551e6 q^{79} +3.49751e6 q^{81} -9.17165e6 q^{83} -418516. q^{85} +1.64510e7 q^{87} +242643. q^{89} -3.80528e6 q^{91} -1.41799e7 q^{93} +2.77121e6 q^{95} -2.59198e6 q^{97} -5.06929e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 20 q^{3} + 250 q^{5} + 100 q^{7} + 5554 q^{9} + 4544 q^{11} - 3540 q^{13} + 2500 q^{15} - 27340 q^{17} + 38760 q^{19} + 69096 q^{21} + 124140 q^{23} + 31250 q^{25} + 206360 q^{27} + 72260 q^{29} - 306824 q^{31}+ \cdots + 2890688 q^{99}+O(q^{100}) $ 2 * q + 20 * q^3 + 250 * q^5 + 100 * q^7 + 5554 * q^9 + 4544 * q^11 - 3540 * q^13 + 2500 * q^15 - 27340 * q^17 + 38760 * q^19 + 69096 * q^21 + 124140 * q^23 + 31250 * q^25 + 206360 * q^27 + 72260 * q^29 - 306824 * q^31 - 440960 * q^33 + 12500 * q^35 + 123020 * q^37 - 774728 * q^39 + 264364 * q^41 + 423300 * q^43 + 694250 * q^45 + 105460 * q^47 - 1165414 * q^49 + 1166344 * q^51 + 2391580 * q^53 + 568000 * q^55 + 776720 * q^57 - 1120120 * q^59 - 2257044 * q^61 + 1639620 * q^63 - 442500 * q^65 + 4516460 * q^67 + 745272 * q^69 - 621784 * q^71 + 4569060 * q^73 + 312500 * q^75 - 3177600 * q^77 - 4333040 * q^79 - 2397878 * q^81 - 9793020 * q^83 - 3417500 * q^85 + 24458920 * q^87 + 6025620 * q^89 - 5352296 * q^91 - 6473040 * q^93 + 4845000 * q^95 + 4609540 * q^97 + 2890688 * 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
$3$	79.7424	1.70516	0.852579	−	0.522598i	$-0.175037\pi$
$3$	79.7424	1.70516	0.852579	+	0.522598i	$0.175037\pi$
$4$	0	0
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	538.197	0.593059	0.296529	−	0.955024i	$-0.404171\pi$
$7$	538.197	0.593059	0.296529	+	0.955024i	$0.404171\pi$
$8$	0	0
$9$	4171.85	1.90757
$10$	0	0
$11$	−1215.12	−0.275261	−0.137630	−	0.990484i	$-0.543949\pi$
$11$	−1215.12	−0.275261	−0.137630	+	0.990484i	$0.543949\pi$
$12$	0	0
$13$	−7070.42	−0.892573	−0.446286	−	0.894890i	$-0.647254\pi$
$13$	−7070.42	−0.892573	−0.446286	+	0.894890i	$0.647254\pi$
$14$	0	0
$15$	9967.80	0.762570
$16$	0	0
$17$	−3348.13	−0.165284	−0.0826420	−	0.996579i	$-0.526336\pi$
$17$	−3348.13	−0.165284	−0.0826420	+	0.996579i	$0.526336\pi$
$18$	0	0
$19$	22169.7	0.741519	0.370759	−	0.928729i	$-0.379097\pi$
$19$	22169.7	0.741519	0.370759	+	0.928729i	$0.379097\pi$
$20$	0	0
$21$	42917.1	1.01126
$22$	0	0
$23$	58513.1	1.00278	0.501390	−	0.865221i	$-0.332822\pi$
$23$	58513.1	1.00278	0.501390	+	0.865221i	$0.332822\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	0	0
$27$	158276.	1.54754
$28$	0	0
$29$	206301.	1.57076	0.785379	−	0.619015i	$-0.212468\pi$
$29$	206301.	1.57076	0.785379	+	0.619015i	$0.212468\pi$
$30$	0	0
$31$	−177822.	−1.07206	−0.536030	−	0.844199i	$-0.680077\pi$
$31$	−177822.	−1.07206	−0.536030	+	0.844199i	$0.680077\pi$
$32$	0	0
$33$	−96896.5	−0.469363
$34$	0	0
$35$	67274.6	0.265224
$36$	0	0
$37$	284128.	0.922163	0.461081	−	0.887358i	$-0.347462\pi$
$37$	284128.	0.922163	0.461081	+	0.887358i	$0.347462\pi$
$38$	0	0
$39$	−563812.	−1.52198
$40$	0	0
$41$	627353.	1.42157	0.710785	−	0.703409i	$-0.248340\pi$
$41$	627353.	1.42157	0.710785	+	0.703409i	$0.248340\pi$
$42$	0	0
$43$	−164889.	−0.316266	−0.158133	−	0.987418i	$-0.550547\pi$
$43$	−164889.	−0.316266	−0.158133	+	0.987418i	$0.550547\pi$
$44$	0	0
$45$	521481.	0.853090
$46$	0	0
$47$	449355.	0.631316	0.315658	−	0.948873i	$-0.397775\pi$
$47$	449355.	0.631316	0.315658	+	0.948873i	$0.397775\pi$
$48$	0	0
$49$	−533887.	−0.648281
$50$	0	0
$51$	−266988.	−0.281835
$52$	0	0
$53$	730190.	0.673706	0.336853	−	0.941557i	$-0.390638\pi$
$53$	730190.	0.673706	0.336853	+	0.941557i	$0.390638\pi$
$54$	0	0
$55$	−151890.	−0.123100
$56$	0	0
$57$	1.76786e6	1.26441
$58$	0	0
$59$	1.42202e6	0.901412	0.450706	−	0.892673i	$-0.351172\pi$
$59$	1.42202e6	0.901412	0.450706	+	0.892673i	$0.351172\pi$
$60$	0	0
$61$	266326.	0.150231	0.0751153	−	0.997175i	$-0.476068\pi$
$61$	266326.	0.150231	0.0751153	+	0.997175i	$0.476068\pi$
$62$	0	0
$63$	2.24527e6	1.13130
$64$	0	0
$65$	−883803.	−0.399171
$66$	0	0
$67$	2.95028e6	1.19840	0.599200	−	0.800599i	$-0.295485\pi$
$67$	2.95028e6	1.19840	0.599200	+	0.800599i	$0.295485\pi$
$68$	0	0
$69$	4.66598e6	1.70990
$70$	0	0
$71$	−921138.	−0.305436	−0.152718	−	0.988270i	$-0.548803\pi$
$71$	−921138.	−0.305436	−0.152718	+	0.988270i	$0.548803\pi$
$72$	0	0
$73$	4.25657e6	1.28065	0.640323	−	0.768105i	$-0.278800\pi$
$73$	4.25657e6	1.28065	0.640323	+	0.768105i	$0.278800\pi$
$74$	0	0
$75$	1.24597e6	0.341032
$76$	0	0
$77$	−653973.	−0.163246
$78$	0	0
$79$	−6.28551e6	−1.43432	−0.717159	−	0.696910i	$-0.754558\pi$
$79$	−6.28551e6	−1.43432	−0.717159	+	0.696910i	$0.754558\pi$
$80$	0	0
$81$	3.49751e6	0.731243
$82$	0	0
$83$	−9.17165e6	−1.76065	−0.880327	−	0.474367i	$-0.842677\pi$
$83$	−9.17165e6	−1.76065	−0.880327	+	0.474367i	$0.842677\pi$
$84$	0	0
$85$	−418516.	−0.0739172
$86$	0	0
$87$	1.64510e7	2.67839
$88$	0	0
$89$	242643.	0.0364840	0.0182420	−	0.999834i	$-0.494193\pi$
$89$	242643.	0.0364840	0.0182420	+	0.999834i	$0.494193\pi$
$90$	0	0
$91$	−3.80528e6	−0.529348
$92$	0	0
$93$	−1.41799e7	−1.82803
$94$	0	0
$95$	2.77121e6	0.331617
$96$	0	0
$97$	−2.59198e6	−0.288357	−0.144179	−	0.989552i	$-0.546054\pi$
$97$	−2.59198e6	−0.288357	−0.144179	+	0.989552i	$0.546054\pi$
$98$	0	0
$99$	−5.06929e6	−0.525078
$100$	0	0
$101$	−3.69169e6	−0.356534	−0.178267	−	0.983982i	$-0.557049\pi$
$101$	−3.69169e6	−0.356534	−0.178267	+	0.983982i	$0.557049\pi$
$102$	0	0
$103$	−8.68203e6	−0.782873	−0.391436	−	0.920205i	$-0.628022\pi$
$103$	−8.68203e6	−0.782873	−0.391436	+	0.920205i	$0.628022\pi$
$104$	0	0
$105$	5.36464e6	0.452249
$106$	0	0
$107$	1.52118e7	1.20043	0.600216	−	0.799838i	$-0.295081\pi$
$107$	1.52118e7	1.20043	0.600216	+	0.799838i	$0.295081\pi$
$108$	0	0
$109$	1.60843e6	0.118963	0.0594813	−	0.998229i	$-0.481055\pi$
$109$	1.60843e6	0.118963	0.0594813	+	0.998229i	$0.481055\pi$
$110$	0	0
$111$	2.26570e7	1.57243
$112$	0	0
$113$	−2.62766e7	−1.71315	−0.856574	−	0.516024i	$-0.827411\pi$
$113$	−2.62766e7	−1.71315	−0.856574	+	0.516024i	$0.827411\pi$
$114$	0	0
$115$	7.31414e6	0.448457
$116$	0	0
$117$	−2.94967e7	−1.70264
$118$	0	0
$119$	−1.80195e6	−0.0980231
$120$	0	0
$121$	−1.80107e7	−0.924231
$122$	0	0
$123$	5.00266e7	2.42400
$124$	0	0
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	1.91864e7	0.831152	0.415576	−	0.909559i	$-0.363580\pi$
$127$	1.91864e7	0.831152	0.415576	+	0.909559i	$0.363580\pi$
$128$	0	0
$129$	−1.31487e7	−0.539284
$130$	0	0
$131$	3.07018e7	1.19320	0.596602	−	0.802537i	$-0.296517\pi$
$131$	3.07018e7	1.19320	0.596602	+	0.802537i	$0.296517\pi$
$132$	0	0
$133$	1.19317e7	0.439764
$134$	0	0
$135$	1.97846e7	0.692083
$136$	0	0
$137$	−3.16847e7	−1.05276	−0.526378	−	0.850251i	$-0.676450\pi$
$137$	−3.16847e7	−1.05276	−0.526378	+	0.850251i	$0.676450\pi$
$138$	0	0
$139$	2.07581e7	0.655596	0.327798	−	0.944748i	$-0.393693\pi$
$139$	2.07581e7	0.655596	0.327798	+	0.944748i	$0.393693\pi$
$140$	0	0
$141$	3.58326e7	1.07649
$142$	0	0
$143$	8.59140e6	0.245690
$144$	0	0
$145$	2.57877e7	0.702464
$146$	0	0
$147$	−4.25734e7	−1.10542
$148$	0	0
$149$	−2.94607e7	−0.729611	−0.364806	−	0.931084i	$-0.618865\pi$
$149$	−2.94607e7	−0.729611	−0.364806	+	0.931084i	$0.618865\pi$
$150$	0	0
$151$	−4.60002e7	−1.08728	−0.543638	−	0.839320i	$-0.682954\pi$
$151$	−4.60002e7	−1.08728	−0.543638	+	0.839320i	$0.682954\pi$
$152$	0	0
$153$	−1.39679e7	−0.315290
$154$	0	0
$155$	−2.22277e7	−0.479440
$156$	0	0
$157$	−1.92998e7	−0.398019	−0.199010	−	0.979998i	$-0.563773\pi$
$157$	−1.92998e7	−0.398019	−0.199010	+	0.979998i	$0.563773\pi$
$158$	0	0
$159$	5.82271e7	1.14878
$160$	0	0
$161$	3.14916e7	0.594708
$162$	0	0
$163$	2.35624e7	0.426151	0.213076	−	0.977036i	$-0.431652\pi$
$163$	2.35624e7	0.426151	0.213076	+	0.977036i	$0.431652\pi$
$164$	0	0
$165$	−1.21121e7	−0.209906
$166$	0	0
$167$	−2.18824e7	−0.363570	−0.181785	−	0.983338i	$-0.558187\pi$
$167$	−2.18824e7	−0.363570	−0.181785	+	0.983338i	$0.558187\pi$
$168$	0	0
$169$	−1.27577e7	−0.203314
$170$	0	0
$171$	9.24886e7	1.41450
$172$	0	0
$173$	9.62312e7	1.41304	0.706520	−	0.707693i	$-0.250264\pi$
$173$	9.62312e7	1.41304	0.706520	+	0.707693i	$0.250264\pi$
$174$	0	0
$175$	8.40932e6	0.118612
$176$	0	0
$177$	1.13395e8	1.53705
$178$	0	0
$179$	8.46776e7	1.10353	0.551763	−	0.834001i	$-0.313955\pi$
$179$	8.46776e7	1.10353	0.551763	+	0.834001i	$0.313955\pi$
$180$	0	0
$181$	9.65249e7	1.20994	0.604971	−	0.796248i	$-0.293185\pi$
$181$	9.65249e7	1.20994	0.604971	+	0.796248i	$0.293185\pi$
$182$	0	0
$183$	2.12374e7	0.256167
$184$	0	0
$185$	3.55160e7	0.412404
$186$	0	0
$187$	4.06837e6	0.0454962
$188$	0	0
$189$	8.51839e7	0.917785
$190$	0	0
$191$	1.46467e8	1.52098	0.760491	−	0.649348i	$-0.224958\pi$
$191$	1.46467e8	1.52098	0.760491	+	0.649348i	$0.224958\pi$
$192$	0	0
$193$	−8.53275e7	−0.854355	−0.427177	−	0.904168i	$-0.640492\pi$
$193$	−8.53275e7	−0.854355	−0.427177	+	0.904168i	$0.640492\pi$
$194$	0	0
$195$	−7.04765e7	−0.680649
$196$	0	0
$197$	−1.44802e8	−1.34941	−0.674705	−	0.738087i	$-0.735729\pi$
$197$	−1.44802e8	−1.34941	−0.674705	+	0.738087i	$0.735729\pi$
$198$	0	0
$199$	−1.26870e7	−0.114123	−0.0570617	−	0.998371i	$-0.518173\pi$
$199$	−1.26870e7	−0.114123	−0.0570617	+	0.998371i	$0.518173\pi$
$200$	0	0
$201$	2.35263e8	2.04346
$202$	0	0
$203$	1.11031e8	0.931552
$204$	0	0
$205$	7.84191e7	0.635746
$206$	0	0
$207$	2.44108e8	1.91287
$208$	0	0
$209$	−2.69388e7	−0.204111
$210$	0	0
$211$	−9.47331e7	−0.694246	−0.347123	−	0.937820i	$-0.612841\pi$
$211$	−9.47331e7	−0.694246	−0.347123	+	0.937820i	$0.612841\pi$
$212$	0	0
$213$	−7.34537e7	−0.520817
$214$	0	0
$215$	−2.06111e7	−0.141438
$216$	0	0
$217$	−9.57031e7	−0.635795
$218$	0	0
$219$	3.39429e8	2.18371
$220$	0	0
$221$	2.36727e7	0.147528
$222$	0	0
$223$	2.28554e8	1.38014	0.690069	−	0.723744i	$-0.257580\pi$
$223$	2.28554e8	1.38014	0.690069	+	0.723744i	$0.257580\pi$
$224$	0	0
$225$	6.51851e7	0.381513
$226$	0	0
$227$	−3.00587e8	−1.70561	−0.852806	−	0.522229i	$-0.825101\pi$
$227$	−3.00587e8	−1.70561	−0.852806	+	0.522229i	$0.825101\pi$
$228$	0	0
$229$	1.05343e8	0.579669	0.289835	−	0.957077i	$-0.406400\pi$
$229$	1.05343e8	0.579669	0.289835	+	0.957077i	$0.406400\pi$
$230$	0	0
$231$	−5.21494e7	−0.278360
$232$	0	0
$233$	3.27196e8	1.69458	0.847291	−	0.531130i	$-0.178232\pi$
$233$	3.27196e8	1.69458	0.847291	+	0.531130i	$0.178232\pi$
$234$	0	0
$235$	5.61694e7	0.282333
$236$	0	0
$237$	−5.01221e8	−2.44574
$238$	0	0
$239$	5.72888e7	0.271442	0.135721	−	0.990747i	$-0.456665\pi$
$239$	5.72888e7	0.271442	0.135721	+	0.990747i	$0.456665\pi$
$240$	0	0
$241$	2.84287e8	1.30827	0.654134	−	0.756379i	$-0.273033\pi$
$241$	2.84287e8	1.30827	0.654134	+	0.756379i	$0.273033\pi$
$242$	0	0
$243$	−6.72506e7	−0.300659
$244$	0	0
$245$	−6.67359e7	−0.289920
$246$	0	0
$247$	−1.56749e8	−0.661859
$248$	0	0
$249$	−7.31369e8	−3.00219
$250$	0	0
$251$	−2.69178e7	−0.107444	−0.0537220	−	0.998556i	$-0.517108\pi$
$251$	−2.69178e7	−0.107444	−0.0537220	+	0.998556i	$0.517108\pi$
$252$	0	0
$253$	−7.11004e7	−0.276026
$254$	0	0
$255$	−3.33735e7	−0.126041
$256$	0	0
$257$	−3.28749e8	−1.20809	−0.604044	−	0.796951i	$-0.706445\pi$
$257$	−3.28749e8	−1.20809	−0.604044	+	0.796951i	$0.706445\pi$
$258$	0	0
$259$	1.52917e8	0.546897
$260$	0	0
$261$	8.60658e8	2.99632
$262$	0	0
$263$	−1.85590e8	−0.629084	−0.314542	−	0.949244i	$-0.601851\pi$
$263$	−1.85590e8	−0.629084	−0.314542	+	0.949244i	$0.601851\pi$
$264$	0	0
$265$	9.12737e7	0.301290
$266$	0	0
$267$	1.93489e7	0.0622109
$268$	0	0
$269$	−2.70876e8	−0.848471	−0.424236	−	0.905552i	$-0.639457\pi$
$269$	−2.70876e8	−0.848471	−0.424236	+	0.905552i	$0.639457\pi$
$270$	0	0
$271$	−2.30743e8	−0.704263	−0.352132	−	0.935950i	$-0.614543\pi$
$271$	−2.30743e8	−0.704263	−0.352132	+	0.935950i	$0.614543\pi$
$272$	0	0
$273$	−3.03442e8	−0.902623
$274$	0	0
$275$	−1.89862e7	−0.0550522
$276$	0	0
$277$	−1.42059e8	−0.401595	−0.200798	−	0.979633i	$-0.564353\pi$
$277$	−1.42059e8	−0.401595	−0.200798	+	0.979633i	$0.564353\pi$
$278$	0	0
$279$	−7.41846e8	−2.04503
$280$	0	0
$281$	−1.15362e8	−0.310164	−0.155082	−	0.987902i	$-0.549564\pi$
$281$	−1.15362e8	−0.310164	−0.155082	+	0.987902i	$0.549564\pi$
$282$	0	0
$283$	2.18080e8	0.571958	0.285979	−	0.958236i	$-0.407681\pi$
$283$	2.18080e8	0.571958	0.285979	+	0.958236i	$0.407681\pi$
$284$	0	0
$285$	2.20983e8	0.565460
$286$	0	0
$287$	3.37639e8	0.843075
$288$	0	0
$289$	−3.99129e8	−0.972681
$290$	0	0
$291$	−2.06691e8	−0.491695
$292$	0	0
$293$	−4.12384e8	−0.957778	−0.478889	−	0.877875i	$-0.658960\pi$
$293$	−4.12384e8	−0.957778	−0.478889	+	0.877875i	$0.658960\pi$
$294$	0	0
$295$	1.77752e8	0.403124
$296$	0	0
$297$	−1.92325e8	−0.425979
$298$	0	0
$299$	−4.13713e8	−0.895055
$300$	0	0
$301$	−8.87428e7	−0.187564
$302$	0	0
$303$	−2.94384e8	−0.607946
$304$	0	0
$305$	3.32907e7	0.0671852
$306$	0	0
$307$	1.57983e8	0.311621	0.155810	−	0.987787i	$-0.450201\pi$
$307$	1.57983e8	0.311621	0.155810	+	0.987787i	$0.450201\pi$
$308$	0	0
$309$	−6.92326e8	−1.33492
$310$	0	0
$311$	1.69169e8	0.318904	0.159452	−	0.987206i	$-0.449027\pi$
$311$	1.69169e8	0.318904	0.159452	+	0.987206i	$0.449027\pi$
$312$	0	0
$313$	−5.22117e8	−0.962415	−0.481208	−	0.876607i	$-0.659802\pi$
$313$	−5.22117e8	−0.962415	−0.481208	+	0.876607i	$0.659802\pi$
$314$	0	0
$315$	2.80659e8	0.505932
$316$	0	0
$317$	−4.73280e8	−0.834470	−0.417235	−	0.908799i	$-0.637001\pi$
$317$	−4.73280e8	−0.834470	−0.417235	+	0.908799i	$0.637001\pi$
$318$	0	0
$319$	−2.50681e8	−0.432368
$320$	0	0
$321$	1.21303e9	2.04693
$322$	0	0
$323$	−7.42270e7	−0.122561
$324$	0	0
$325$	−1.10475e8	−0.178515
$326$	0	0
$327$	1.28260e8	0.202850
$328$	0	0
$329$	2.41841e8	0.374408
$330$	0	0
$331$	−8.75892e7	−0.132756	−0.0663778	−	0.997795i	$-0.521144\pi$
$331$	−8.75892e7	−0.132756	−0.0663778	+	0.997795i	$0.521144\pi$
$332$	0	0
$333$	1.18534e9	1.75909
$334$	0	0
$335$	3.68785e8	0.535941
$336$	0	0
$337$	−3.04119e8	−0.432851	−0.216426	−	0.976299i	$-0.569440\pi$
$337$	−3.04119e8	−0.432851	−0.216426	+	0.976299i	$0.569440\pi$
$338$	0	0
$339$	−2.09536e9	−2.92119
$340$	0	0
$341$	2.16075e8	0.295096
$342$	0	0
$343$	−7.30565e8	−0.977528
$344$	0	0
$345$	5.83247e8	0.764691
$346$	0	0
$347$	3.15347e8	0.405169	0.202584	−	0.979265i	$-0.435066\pi$
$347$	3.15347e8	0.405169	0.202584	+	0.979265i	$0.435066\pi$
$348$	0	0
$349$	5.23418e8	0.659112	0.329556	−	0.944136i	$-0.393101\pi$
$349$	5.23418e8	0.659112	0.329556	+	0.944136i	$0.393101\pi$
$350$	0	0
$351$	−1.11908e9	−1.38130
$352$	0	0
$353$	−4.13964e8	−0.500900	−0.250450	−	0.968130i	$-0.580579\pi$
$353$	−4.13964e8	−0.500900	−0.250450	+	0.968130i	$0.580579\pi$
$354$	0	0
$355$	−1.15142e8	−0.136595
$356$	0	0
$357$	−1.43692e8	−0.167145
$358$	0	0
$359$	2.48768e8	0.283768	0.141884	−	0.989883i	$-0.454684\pi$
$359$	2.48768e8	0.283768	0.141884	+	0.989883i	$0.454684\pi$
$360$	0	0
$361$	−4.02376e8	−0.450150
$362$	0	0
$363$	−1.43621e9	−1.57596
$364$	0	0
$365$	5.32071e8	0.572723
$366$	0	0
$367$	−1.45884e9	−1.54056	−0.770278	−	0.637708i	$-0.779883\pi$
$367$	−1.45884e9	−1.54056	−0.770278	+	0.637708i	$0.779883\pi$
$368$	0	0
$369$	2.61722e9	2.71174
$370$	0	0
$371$	3.92986e8	0.399547
$372$	0	0
$373$	−6.29125e8	−0.627706	−0.313853	−	0.949472i	$-0.601620\pi$
$373$	−6.29125e8	−0.627706	−0.313853	+	0.949472i	$0.601620\pi$
$374$	0	0
$375$	1.55747e8	0.152514
$376$	0	0
$377$	−1.45864e9	−1.40202
$378$	0	0
$379$	8.54658e7	0.0806409	0.0403204	−	0.999187i	$-0.487162\pi$
$379$	8.54658e7	0.0806409	0.0403204	+	0.999187i	$0.487162\pi$
$380$	0	0
$381$	1.52997e9	1.41725
$382$	0	0
$383$	7.56340e8	0.687893	0.343947	−	0.938989i	$-0.388236\pi$
$383$	7.56340e8	0.687893	0.343947	+	0.938989i	$0.388236\pi$
$384$	0	0
$385$	−8.17466e7	−0.0730058
$386$	0	0
$387$	−6.87892e8	−0.603298
$388$	0	0
$389$	−1.42837e9	−1.23032	−0.615160	−	0.788402i	$-0.710908\pi$
$389$	−1.42837e9	−1.23032	−0.615160	+	0.788402i	$0.710908\pi$
$390$	0	0
$391$	−1.95909e8	−0.165744
$392$	0	0
$393$	2.44824e9	2.03460
$394$	0	0
$395$	−7.85688e8	−0.641446
$396$	0	0
$397$	1.67152e8	0.134074	0.0670372	−	0.997750i	$-0.478645\pi$
$397$	1.67152e8	0.134074	0.0670372	+	0.997750i	$0.478645\pi$
$398$	0	0
$399$	9.51459e8	0.749868
$400$	0	0
$401$	2.32051e9	1.79712	0.898561	−	0.438848i	$-0.144613\pi$
$401$	2.32051e9	1.79712	0.898561	+	0.438848i	$0.144613\pi$
$402$	0	0
$403$	1.25728e9	0.956892
$404$	0	0
$405$	4.37189e8	0.327022
$406$	0	0
$407$	−3.45249e8	−0.253835
$408$	0	0
$409$	−8.24735e8	−0.596050	−0.298025	−	0.954558i	$-0.596328\pi$
$409$	−8.24735e8	−0.596050	−0.298025	+	0.954558i	$0.596328\pi$
$410$	0	0
$411$	−2.52661e9	−1.79512
$412$	0	0
$413$	7.65326e8	0.534590
$414$	0	0
$415$	−1.14646e9	−0.787389
$416$	0	0
$417$	1.65530e9	1.11790
$418$	0	0
$419$	8.23968e8	0.547219	0.273610	−	0.961841i	$-0.411782\pi$
$419$	8.23968e8	0.547219	0.273610	+	0.961841i	$0.411782\pi$
$420$	0	0
$421$	2.12046e8	0.138498	0.0692488	−	0.997599i	$-0.477940\pi$
$421$	2.12046e8	0.138498	0.0692488	+	0.997599i	$0.477940\pi$
$422$	0	0
$423$	1.87464e9	1.20428
$424$	0	0
$425$	−5.23145e7	−0.0330568
$426$	0	0
$427$	1.43336e8	0.0890956
$428$	0	0
$429$	6.85099e8	0.418941
$430$	0	0
$431$	−7.44023e8	−0.447627	−0.223813	−	0.974632i	$-0.571851\pi$
$431$	−7.44023e8	−0.447627	−0.223813	+	0.974632i	$0.571851\pi$
$432$	0	0
$433$	−1.93573e9	−1.14588	−0.572938	−	0.819598i	$-0.694197\pi$
$433$	−1.93573e9	−1.14588	−0.572938	+	0.819598i	$0.694197\pi$
$434$	0	0
$435$	2.05637e9	1.19781
$436$	0	0
$437$	1.29722e9	0.743581
$438$	0	0
$439$	1.35485e9	0.764300	0.382150	−	0.924100i	$-0.375184\pi$
$439$	1.35485e9	0.764300	0.382150	+	0.924100i	$0.375184\pi$
$440$	0	0
$441$	−2.22730e9	−1.23664
$442$	0	0
$443$	1.05985e9	0.579203	0.289601	−	0.957147i	$-0.406477\pi$
$443$	1.05985e9	0.579203	0.289601	+	0.957147i	$0.406477\pi$
$444$	0	0
$445$	3.03303e7	0.0163161
$446$	0	0
$447$	−2.34927e9	−1.24410
$448$	0	0
$449$	−1.70272e9	−0.887730	−0.443865	−	0.896094i	$-0.646393\pi$
$449$	−1.70272e9	−0.887730	−0.443865	+	0.896094i	$0.646393\pi$
$450$	0	0
$451$	−7.62309e8	−0.391303
$452$	0	0
$453$	−3.66816e9	−1.85398
$454$	0	0
$455$	−4.75660e8	−0.236732
$456$	0	0
$457$	−3.76431e9	−1.84493	−0.922463	−	0.386085i	$-0.873827\pi$
$457$	−3.76431e9	−1.84493	−0.922463	+	0.386085i	$0.873827\pi$
$458$	0	0
$459$	−5.29930e8	−0.255784
$460$	0	0
$461$	−3.59084e9	−1.70704	−0.853519	−	0.521062i	$-0.825536\pi$
$461$	−3.59084e9	−1.70704	−0.853519	+	0.521062i	$0.825536\pi$
$462$	0	0
$463$	2.45649e8	0.115022	0.0575111	−	0.998345i	$-0.481684\pi$
$463$	2.45649e8	0.115022	0.0575111	+	0.998345i	$0.481684\pi$
$464$	0	0
$465$	−1.77249e9	−0.817521
$466$	0	0
$467$	−4.04985e8	−0.184005	−0.0920025	−	0.995759i	$-0.529327\pi$
$467$	−4.04985e8	−0.184005	−0.0920025	+	0.995759i	$0.529327\pi$
$468$	0	0
$469$	1.58783e9	0.710722
$470$	0	0
$471$	−1.53901e9	−0.678686
$472$	0	0
$473$	2.00360e8	0.0870556
$474$	0	0
$475$	3.46401e8	0.148304
$476$	0	0
$477$	3.04624e9	1.28514
$478$	0	0
$479$	−4.18334e9	−1.73920	−0.869598	−	0.493760i	$-0.835622\pi$
$479$	−4.18334e9	−1.73920	−0.869598	+	0.493760i	$0.835622\pi$
$480$	0	0
$481$	−2.00890e9	−0.823097
$482$	0	0
$483$	2.51121e9	1.01407
$484$	0	0
$485$	−3.23998e8	−0.128957
$486$	0	0
$487$	−3.25089e9	−1.27541	−0.637706	−	0.770280i	$-0.720117\pi$
$487$	−3.25089e9	−1.27541	−0.637706	+	0.770280i	$0.720117\pi$
$488$	0	0
$489$	1.87893e9	0.726656
$490$	0	0
$491$	−3.37360e9	−1.28620	−0.643100	−	0.765782i	$-0.722352\pi$
$491$	−3.37360e9	−1.28620	−0.643100	+	0.765782i	$0.722352\pi$
$492$	0	0
$493$	−6.90723e8	−0.259621
$494$	0	0
$495$	−6.33662e8	−0.234822
$496$	0	0
$497$	−4.95753e8	−0.181142
$498$	0	0
$499$	3.74951e9	1.35090	0.675449	−	0.737406i	$-0.263950\pi$
$499$	3.74951e9	1.35090	0.675449	+	0.737406i	$0.263950\pi$
$500$	0	0
$501$	−1.74496e9	−0.619945
$502$	0	0
$503$	3.09301e9	1.08366	0.541830	−	0.840488i	$-0.317732\pi$
$503$	3.09301e9	1.08366	0.541830	+	0.840488i	$0.317732\pi$
$504$	0	0
$505$	−4.61461e8	−0.159447
$506$	0	0
$507$	−1.01733e9	−0.346683
$508$	0	0
$509$	−1.36862e9	−0.460015	−0.230007	−	0.973189i	$-0.573875\pi$
$509$	−1.36862e9	−0.460015	−0.230007	+	0.973189i	$0.573875\pi$
$510$	0	0
$511$	2.29087e9	0.759499
$512$	0	0
$513$	3.50894e9	1.14753
$514$	0	0
$515$	−1.08525e9	−0.350111
$516$	0	0
$517$	−5.46020e8	−0.173777
$518$	0	0
$519$	7.67371e9	2.40946
$520$	0	0
$521$	−4.47414e9	−1.38604	−0.693022	−	0.720916i	$-0.743721\pi$
$521$	−4.47414e9	−1.38604	−0.693022	+	0.720916i	$0.743721\pi$
$522$	0	0
$523$	−3.01993e9	−0.923085	−0.461542	−	0.887118i	$-0.652704\pi$
$523$	−3.01993e9	−0.923085	−0.461542	+	0.887118i	$0.652704\pi$
$524$	0	0
$525$	6.70579e8	0.202252
$526$	0	0
$527$	5.95370e8	0.177194
$528$	0	0
$529$	1.89619e7	0.00556913
$530$	0	0
$531$	5.93244e9	1.71950
$532$	0	0
$533$	−4.43565e9	−1.26885
$534$	0	0
$535$	1.90148e9	0.536850
$536$	0	0
$537$	6.75239e9	1.88169
$538$	0	0
$539$	6.48737e8	0.178446
$540$	0	0
$541$	−3.23946e9	−0.879593	−0.439796	−	0.898097i	$-0.644949\pi$
$541$	−3.23946e9	−0.879593	−0.439796	+	0.898097i	$0.644949\pi$
$542$	0	0
$543$	7.69713e9	2.06314
$544$	0	0
$545$	2.01054e8	0.0532017
$546$	0	0
$547$	4.25742e9	1.11222	0.556111	−	0.831108i	$-0.312293\pi$
$547$	4.25742e9	1.11222	0.556111	+	0.831108i	$0.312293\pi$
$548$	0	0
$549$	1.11107e9	0.286575
$550$	0	0
$551$	4.57364e9	1.16475
$552$	0	0
$553$	−3.38284e9	−0.850635
$554$	0	0
$555$	2.83213e9	0.703214
$556$	0	0
$557$	5.65828e9	1.38737	0.693684	−	0.720280i	$-0.255987\pi$
$557$	5.65828e9	1.38737	0.693684	+	0.720280i	$0.255987\pi$
$558$	0	0
$559$	1.16584e9	0.282290
$560$	0	0
$561$	3.24422e8	0.0775783
$562$	0	0
$563$	4.37511e9	1.03326	0.516630	−	0.856209i	$-0.327186\pi$
$563$	4.37511e9	1.03326	0.516630	+	0.856209i	$0.327186\pi$
$564$	0	0
$565$	−3.28458e9	−0.766143
$566$	0	0
$567$	1.88235e9	0.433670
$568$	0	0
$569$	1.67226e9	0.380550	0.190275	−	0.981731i	$-0.439062\pi$
$569$	1.67226e9	0.380550	0.190275	+	0.981731i	$0.439062\pi$
$570$	0	0
$571$	6.39802e8	0.143820	0.0719100	−	0.997411i	$-0.477091\pi$
$571$	6.39802e8	0.143820	0.0719100	+	0.997411i	$0.477091\pi$
$572$	0	0
$573$	1.16797e10	2.59352
$574$	0	0
$575$	9.14268e8	0.200556
$576$	0	0
$577$	4.87101e9	1.05561	0.527805	−	0.849366i	$-0.323015\pi$
$577$	4.87101e9	1.05561	0.527805	+	0.849366i	$0.323015\pi$
$578$	0	0
$579$	−6.80422e9	−1.45681
$580$	0	0
$581$	−4.93615e9	−1.04417
$582$	0	0
$583$	−8.87268e8	−0.185445
$584$	0	0
$585$	−3.68709e9	−0.761444
$586$	0	0
$587$	−2.32406e9	−0.474258	−0.237129	−	0.971478i	$-0.576206\pi$
$587$	−2.32406e9	−0.474258	−0.237129	+	0.971478i	$0.576206\pi$
$588$	0	0
$589$	−3.94226e9	−0.794953
$590$	0	0
$591$	−1.15469e10	−2.30096
$592$	0	0
$593$	4.27080e9	0.841043	0.420522	−	0.907283i	$-0.361847\pi$
$593$	4.27080e9	0.841043	0.420522	+	0.907283i	$0.361847\pi$
$594$	0	0
$595$	−2.25244e8	−0.0438373
$596$	0	0
$597$	−1.01169e9	−0.194598
$598$	0	0
$599$	−3.50335e9	−0.666023	−0.333012	−	0.942923i	$-0.608065\pi$
$599$	−3.50335e9	−0.666023	−0.333012	+	0.942923i	$0.608065\pi$
$600$	0	0
$601$	2.98866e9	0.561585	0.280792	−	0.959769i	$-0.409403\pi$
$601$	2.98866e9	0.561585	0.280792	+	0.959769i	$0.409403\pi$
$602$	0	0
$603$	1.23081e10	2.28603
$604$	0	0
$605$	−2.25133e9	−0.413329
$606$	0	0
$607$	7.31901e9	1.32829	0.664143	−	0.747605i	$-0.268797\pi$
$607$	7.31901e9	1.32829	0.664143	+	0.747605i	$0.268797\pi$
$608$	0	0
$609$	8.85386e9	1.58844
$610$	0	0
$611$	−3.17713e9	−0.563496
$612$	0	0
$613$	8.32799e9	1.46026	0.730128	−	0.683311i	$-0.239461\pi$
$613$	8.32799e9	1.46026	0.730128	+	0.683311i	$0.239461\pi$
$614$	0	0
$615$	6.25333e9	1.08405
$616$	0	0
$617$	−2.88456e9	−0.494403	−0.247201	−	0.968964i	$-0.579511\pi$
$617$	−2.88456e9	−0.494403	−0.247201	+	0.968964i	$0.579511\pi$
$618$	0	0
$619$	−2.27213e9	−0.385049	−0.192525	−	0.981292i	$-0.561668\pi$
$619$	−2.27213e9	−0.385049	−0.192525	+	0.981292i	$0.561668\pi$
$620$	0	0
$621$	9.26125e9	1.55185
$622$	0	0
$623$	1.30589e8	0.0216371
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	0	0
$627$	−2.14817e9	−0.348042
$628$	0	0
$629$	−9.51296e8	−0.152419
$630$	0	0
$631$	2.90856e7	0.00460867	0.00230434	−	0.999997i	$-0.499267\pi$
$631$	2.90856e7	0.00460867	0.00230434	+	0.999997i	$0.499267\pi$
$632$	0	0
$633$	−7.55424e9	−1.18380
$634$	0	0
$635$	2.39830e9	0.371702
$636$	0	0
$637$	3.77481e9	0.578638
$638$	0	0
$639$	−3.84285e9	−0.582640
$640$	0	0
$641$	−3.12634e9	−0.468849	−0.234424	−	0.972134i	$-0.575320\pi$
$641$	−3.12634e9	−0.468849	−0.234424	+	0.972134i	$0.575320\pi$
$642$	0	0
$643$	2.48120e9	0.368064	0.184032	−	0.982920i	$-0.441085\pi$
$643$	2.48120e9	0.368064	0.184032	+	0.982920i	$0.441085\pi$
$644$	0	0
$645$	−1.64358e9	−0.241175
$646$	0	0
$647$	−3.10080e9	−0.450100	−0.225050	−	0.974347i	$-0.572254\pi$
$647$	−3.10080e9	−0.450100	−0.225050	+	0.974347i	$0.572254\pi$
$648$	0	0
$649$	−1.72792e9	−0.248123
$650$	0	0
$651$	−7.63159e9	−1.08413
$652$	0	0
$653$	−9.64911e8	−0.135610	−0.0678049	−	0.997699i	$-0.521600\pi$
$653$	−9.64911e8	−0.135610	−0.0678049	+	0.997699i	$0.521600\pi$
$654$	0	0
$655$	3.83773e9	0.533617
$656$	0	0
$657$	1.77577e10	2.44292
$658$	0	0
$659$	−7.70039e9	−1.04813	−0.524063	−	0.851679i	$-0.675584\pi$
$659$	−7.70039e9	−1.04813	−0.524063	+	0.851679i	$0.675584\pi$
$660$	0	0
$661$	−1.30650e10	−1.75956	−0.879779	−	0.475382i	$-0.842310\pi$
$661$	−1.30650e10	−1.75956	−0.879779	+	0.475382i	$0.842310\pi$
$662$	0	0
$663$	1.88772e9	0.251559
$664$	0	0
$665$	1.49146e9	0.196669
$666$	0	0
$667$	1.20713e10	1.57513
$668$	0	0
$669$	1.82255e10	2.35335
$670$	0	0
$671$	−3.23617e8	−0.0413526
$672$	0	0
$673$	1.31771e10	1.66636	0.833179	−	0.553003i	$-0.186518\pi$
$673$	1.31771e10	1.66636	0.833179	+	0.553003i	$0.186518\pi$
$674$	0	0
$675$	2.47307e9	0.309509
$676$	0	0
$677$	3.18151e9	0.394069	0.197035	−	0.980397i	$-0.436869\pi$
$677$	3.18151e9	0.394069	0.197035	+	0.980397i	$0.436869\pi$
$678$	0	0
$679$	−1.39500e9	−0.171013
$680$	0	0
$681$	−2.39695e10	−2.90834
$682$	0	0
$683$	−3.33139e8	−0.0400086	−0.0200043	−	0.999800i	$-0.506368\pi$
$683$	−3.33139e8	−0.0400086	−0.0200043	+	0.999800i	$0.506368\pi$
$684$	0	0
$685$	−3.96059e9	−0.470807
$686$	0	0
$687$	8.40028e9	0.988428
$688$	0	0
$689$	−5.16275e9	−0.601331
$690$	0	0
$691$	−4.45248e9	−0.513368	−0.256684	−	0.966495i	$-0.582630\pi$
$691$	−4.45248e9	−0.513368	−0.256684	+	0.966495i	$0.582630\pi$
$692$	0	0
$693$	−2.72828e9	−0.311402
$694$	0	0
$695$	2.59476e9	0.293191
$696$	0	0
$697$	−2.10046e9	−0.234963
$698$	0	0
$699$	2.60914e10	2.88953
$700$	0	0
$701$	1.91380e9	0.209838	0.104919	−	0.994481i	$-0.466542\pi$
$701$	1.91380e9	0.209838	0.104919	+	0.994481i	$0.466542\pi$
$702$	0	0
$703$	6.29902e9	0.683801
$704$	0	0
$705$	4.47908e9	0.481423
$706$	0	0
$707$	−1.98686e9	−0.211445
$708$	0	0
$709$	−1.38704e9	−0.146160	−0.0730799	−	0.997326i	$-0.523283\pi$
$709$	−1.38704e9	−0.146160	−0.0730799	+	0.997326i	$0.523283\pi$
$710$	0	0
$711$	−2.62222e10	−2.73606
$712$	0	0
$713$	−1.04049e10	−1.07504
$714$	0	0
$715$	1.07393e9	0.109876
$716$	0	0
$717$	4.56835e9	0.462852
$718$	0	0
$719$	−9.42958e9	−0.946109	−0.473055	−	0.881033i	$-0.656849\pi$
$719$	−9.42958e9	−0.946109	−0.473055	+	0.881033i	$0.656849\pi$
$720$	0	0
$721$	−4.67264e9	−0.464290
$722$	0	0
$723$	2.26697e10	2.23081
$724$	0	0
$725$	3.22346e9	0.314152
$726$	0	0
$727$	2.99387e9	0.288976	0.144488	−	0.989507i	$-0.453846\pi$
$727$	2.99387e9	0.288976	0.144488	+	0.989507i	$0.453846\pi$
$728$	0	0
$729$	−1.30118e10	−1.24391
$730$	0	0
$731$	5.52070e8	0.0522737
$732$	0	0
$733$	−1.11711e10	−1.04768	−0.523842	−	0.851815i	$-0.675502\pi$
$733$	−1.11711e10	−1.04768	−0.523842	+	0.851815i	$0.675502\pi$
$734$	0	0
$735$	−5.32168e9	−0.494360
$736$	0	0
$737$	−3.58495e9	−0.329873
$738$	0	0
$739$	−6.96449e9	−0.634796	−0.317398	−	0.948292i	$-0.602809\pi$
$739$	−6.96449e9	−0.634796	−0.317398	+	0.948292i	$0.602809\pi$
$740$	0	0
$741$	−1.24995e10	−1.12858
$742$	0	0
$743$	−1.27161e9	−0.113734	−0.0568672	−	0.998382i	$-0.518111\pi$
$743$	−1.27161e9	−0.113734	−0.0568672	+	0.998382i	$0.518111\pi$
$744$	0	0
$745$	−3.68259e9	−0.326292
$746$	0	0
$747$	−3.82627e10	−3.35856
$748$	0	0
$749$	8.18695e9	0.711927
$750$	0	0
$751$	1.70771e10	1.47121	0.735603	−	0.677413i	$-0.236899\pi$
$751$	1.70771e10	1.47121	0.735603	+	0.677413i	$0.236899\pi$
$752$	0	0
$753$	−2.14649e9	−0.183209
$754$	0	0
$755$	−5.75002e9	−0.486245
$756$	0	0
$757$	−1.37933e10	−1.15567	−0.577833	−	0.816155i	$-0.696101\pi$
$757$	−1.37933e10	−1.15567	−0.577833	+	0.816155i	$0.696101\pi$
$758$	0	0
$759$	−5.66972e9	−0.470669
$760$	0	0
$761$	1.71364e10	1.40953	0.704765	−	0.709441i	$-0.251053\pi$
$761$	1.71364e10	1.40953	0.704765	+	0.709441i	$0.251053\pi$
$762$	0	0
$763$	8.65654e8	0.0705518
$764$	0	0
$765$	−1.74598e9	−0.141002
$766$	0	0
$767$	−1.00543e10	−0.804575
$768$	0	0
$769$	−4.41529e9	−0.350120	−0.175060	−	0.984558i	$-0.556012\pi$
$769$	−4.41529e9	−0.350120	−0.175060	+	0.984558i	$0.556012\pi$
$770$	0	0
$771$	−2.62152e10	−2.05998
$772$	0	0
$773$	5.87300e8	0.0457332	0.0228666	−	0.999739i	$-0.492721\pi$
$773$	5.87300e8	0.0457332	0.0228666	+	0.999739i	$0.492721\pi$
$774$	0	0
$775$	−2.77847e9	−0.214412
$776$	0	0
$777$	1.21939e10	0.932546
$778$	0	0
$779$	1.39082e10	1.05412
$780$	0	0
$781$	1.11929e9	0.0840746
$782$	0	0
$783$	3.26527e10	2.43082
$784$	0	0
$785$	−2.41248e9	−0.178000
$786$	0	0
$787$	−2.30069e10	−1.68247	−0.841235	−	0.540670i	$-0.818171\pi$
$787$	−2.30069e10	−1.68247	−0.841235	+	0.540670i	$0.818171\pi$
$788$	0	0
$789$	−1.47994e10	−1.07269
$790$	0	0
$791$	−1.41420e10	−1.01600
$792$	0	0
$793$	−1.88303e9	−0.134092
$794$	0	0
$795$	7.27838e9	0.513748
$796$	0	0
$797$	−2.49526e10	−1.74587	−0.872935	−	0.487836i	$-0.837787\pi$
$797$	−2.49526e10	−1.74587	−0.872935	+	0.487836i	$0.837787\pi$
$798$	0	0
$799$	−1.50450e9	−0.104346
$800$	0	0
$801$	1.01227e9	0.0695956
$802$	0	0
$803$	−5.17223e9	−0.352512
$804$	0	0
$805$	3.93645e9	0.265962
$806$	0	0
$807$	−2.16003e10	−1.44678
$808$	0	0
$809$	−1.29758e10	−0.861616	−0.430808	−	0.902443i	$-0.641771\pi$
$809$	−1.29758e10	−0.861616	−0.430808	+	0.902443i	$0.641771\pi$
$810$	0	0
$811$	−2.04887e10	−1.34878	−0.674391	−	0.738374i	$-0.735594\pi$
$811$	−2.04887e10	−1.34878	−0.674391	+	0.738374i	$0.735594\pi$
$812$	0	0
$813$	−1.84000e10	−1.20088
$814$	0	0
$815$	2.94531e9	0.190581
$816$	0	0
$817$	−3.65554e9	−0.234517
$818$	0	0
$819$	−1.58750e10	−1.00977
$820$	0	0
$821$	7.97556e9	0.502991	0.251495	−	0.967858i	$-0.419078\pi$
$821$	7.97556e9	0.502991	0.251495	+	0.967858i	$0.419078\pi$
$822$	0	0
$823$	−5.64462e9	−0.352968	−0.176484	−	0.984304i	$-0.556472\pi$
$823$	−5.64462e9	−0.352968	−0.176484	+	0.984304i	$0.556472\pi$
$824$	0	0
$825$	−1.51401e9	−0.0938727
$826$	0	0
$827$	1.22946e10	0.755866	0.377933	−	0.925833i	$-0.376635\pi$
$827$	1.22946e10	0.755866	0.377933	+	0.925833i	$0.376635\pi$
$828$	0	0
$829$	1.72844e10	1.05369	0.526845	−	0.849962i	$-0.323375\pi$
$829$	1.72844e10	1.05369	0.526845	+	0.849962i	$0.323375\pi$
$830$	0	0
$831$	−1.13281e10	−0.684784
$832$	0	0
$833$	1.78752e9	0.107150
$834$	0	0
$835$	−2.73530e9	−0.162593
$836$	0	0
$837$	−2.81450e10	−1.65906
$838$	0	0
$839$	4.87580e9	0.285023	0.142511	−	0.989793i	$-0.454482\pi$
$839$	4.87580e9	0.285023	0.142511	+	0.989793i	$0.454482\pi$
$840$	0	0
$841$	2.53104e10	1.46728
$842$	0	0
$843$	−9.19927e9	−0.528879
$844$	0	0
$845$	−1.59471e9	−0.0909249
$846$	0	0
$847$	−9.69328e9	−0.548124
$848$	0	0
$849$	1.73903e10	0.975279
$850$	0	0
$851$	1.66252e10	0.924727
$852$	0	0
$853$	1.67542e10	0.924277	0.462138	−	0.886808i	$-0.347082\pi$
$853$	1.67542e10	0.924277	0.462138	+	0.886808i	$0.347082\pi$
$854$	0	0
$855$	1.15611e10	0.632582
$856$	0	0
$857$	−1.45198e10	−0.788001	−0.394001	−	0.919110i	$-0.628909\pi$
$857$	−1.45198e10	−0.788001	−0.394001	+	0.919110i	$0.628909\pi$
$858$	0	0
$859$	2.44108e10	1.31403	0.657016	−	0.753877i	$-0.271819\pi$
$859$	2.44108e10	1.31403	0.657016	+	0.753877i	$0.271819\pi$
$860$	0	0
$861$	2.69242e10	1.43758
$862$	0	0
$863$	2.03914e10	1.07996	0.539981	−	0.841677i	$-0.318431\pi$
$863$	2.03914e10	1.07996	0.539981	+	0.841677i	$0.318431\pi$
$864$	0	0
$865$	1.20289e10	0.631931
$866$	0	0
$867$	−3.18275e10	−1.65858
$868$	0	0
$869$	7.63764e9	0.394812
$870$	0	0
$871$	−2.08597e10	−1.06966
$872$	0	0
$873$	−1.08134e10	−0.550061
$874$	0	0
$875$	1.05117e9	0.0530448
$876$	0	0
$877$	3.75418e10	1.87939	0.939693	−	0.342018i	$-0.111110\pi$
$877$	3.75418e10	1.87939	0.939693	+	0.342018i	$0.111110\pi$
$878$	0	0
$879$	−3.28845e10	−1.63316
$880$	0	0
$881$	1.05852e10	0.521533	0.260767	−	0.965402i	$-0.416025\pi$
$881$	1.05852e10	0.521533	0.260767	+	0.965402i	$0.416025\pi$
$882$	0	0
$883$	3.45129e10	1.68702	0.843509	−	0.537116i	$-0.180486\pi$
$883$	3.45129e10	1.68702	0.843509	+	0.537116i	$0.180486\pi$
$884$	0	0
$885$	1.41744e10	0.687390
$886$	0	0
$887$	−1.68685e10	−0.811602	−0.405801	−	0.913961i	$-0.633007\pi$
$887$	−1.68685e10	−0.811602	−0.405801	+	0.913961i	$0.633007\pi$
$888$	0	0
$889$	1.03261e10	0.492922
$890$	0	0
$891$	−4.24990e9	−0.201283
$892$	0	0
$893$	9.96206e9	0.468133
$894$	0	0
$895$	1.05847e10	0.493512
$896$	0	0
$897$	−3.29904e10	−1.52621
$898$	0	0
$899$	−3.66849e10	−1.68395
$900$	0	0
$901$	−2.44477e9	−0.111353
$902$	0	0
$903$	−7.07656e9	−0.319827
$904$	0	0
$905$	1.20656e10	0.541102
$906$	0	0
$907$	−1.14713e10	−0.510492	−0.255246	−	0.966876i	$-0.582156\pi$
$907$	−1.14713e10	−0.510492	−0.255246	+	0.966876i	$0.582156\pi$
$908$	0	0
$909$	−1.54012e10	−0.680112
$910$	0	0
$911$	−1.62306e10	−0.711248	−0.355624	−	0.934629i	$-0.615732\pi$
$911$	−1.62306e10	−0.711248	−0.355624	+	0.934629i	$0.615732\pi$
$912$	0	0
$913$	1.11446e10	0.484639
$914$	0	0
$915$	2.65468e9	0.114561
$916$	0	0
$917$	1.65236e10	0.707641
$918$	0	0
$919$	4.36259e10	1.85413	0.927065	−	0.374900i	$-0.122323\pi$
$919$	4.36259e10	1.85413	0.927065	+	0.374900i	$0.122323\pi$
$920$	0	0
$921$	1.25979e10	0.531363
$922$	0	0
$923$	6.51283e9	0.272624
$924$	0	0
$925$	4.43950e9	0.184433
$926$	0	0
$927$	−3.62201e10	−1.49338
$928$	0	0
$929$	1.67801e10	0.686656	0.343328	−	0.939216i	$-0.388446\pi$
$929$	1.67801e10	0.686656	0.343328	+	0.939216i	$0.388446\pi$
$930$	0	0
$931$	−1.18361e10	−0.480713
$932$	0	0
$933$	1.34900e10	0.543782
$934$	0	0
$935$	5.08547e8	0.0203465
$936$	0	0
$937$	1.94934e10	0.774105	0.387053	−	0.922058i	$-0.373493\pi$
$937$	1.94934e10	0.774105	0.387053	+	0.922058i	$0.373493\pi$
$938$	0	0
$939$	−4.16348e10	−1.64107
$940$	0	0
$941$	4.91523e10	1.92300	0.961502	−	0.274797i	$-0.0886108\pi$
$941$	4.91523e10	1.92300	0.961502	+	0.274797i	$0.0886108\pi$
$942$	0	0
$943$	3.67084e10	1.42552
$944$	0	0
$945$	1.06480e10	0.410446
$946$	0	0
$947$	3.01570e10	1.15389	0.576944	−	0.816784i	$-0.304245\pi$
$947$	3.01570e10	1.15389	0.576944	+	0.816784i	$0.304245\pi$
$948$	0	0
$949$	−3.00957e10	−1.14307
$950$	0	0
$951$	−3.77405e10	−1.42290
$952$	0	0
$953$	−1.86916e10	−0.699554	−0.349777	−	0.936833i	$-0.613743\pi$
$953$	−1.86916e10	−0.699554	−0.349777	+	0.936833i	$0.613743\pi$
$954$	0	0
$955$	1.83084e10	0.680204
$956$	0	0
$957$	−1.99899e10	−0.737256
$958$	0	0
$959$	−1.70526e10	−0.624346
$960$	0	0
$961$	4.10799e9	0.149313
$962$	0	0
$963$	6.34614e10	2.28990
$964$	0	0
$965$	−1.06659e10	−0.382079
$966$	0	0
$967$	7.61328e9	0.270757	0.135378	−	0.990794i	$-0.456775\pi$
$967$	7.61328e9	0.270757	0.135378	+	0.990794i	$0.456775\pi$
$968$	0	0
$969$	−5.91903e9	−0.208986
$970$	0	0
$971$	−4.83242e10	−1.69394	−0.846969	−	0.531643i	$-0.821575\pi$
$971$	−4.83242e10	−1.69394	−0.846969	+	0.531643i	$0.821575\pi$
$972$	0	0
$973$	1.11720e10	0.388807
$974$	0	0
$975$	−8.80957e9	−0.304396
$976$	0	0
$977$	1.42491e9	0.0488829	0.0244414	−	0.999701i	$-0.492219\pi$
$977$	1.42491e9	0.0488829	0.0244414	+	0.999701i	$0.492219\pi$
$978$	0	0
$979$	−2.94840e8	−0.0100426
$980$	0	0
$981$	6.71014e9	0.226929
$982$	0	0
$983$	−4.85675e10	−1.63083	−0.815415	−	0.578877i	$-0.803491\pi$
$983$	−4.85675e10	−1.63083	−0.815415	+	0.578877i	$0.803491\pi$
$984$	0	0
$985$	−1.81003e10	−0.603475
$986$	0	0
$987$	1.92850e10	0.638425
$988$	0	0
$989$	−9.64818e9	−0.317145
$990$	0	0
$991$	1.69341e10	0.552719	0.276360	−	0.961054i	$-0.410872\pi$
$991$	1.69341e10	0.552719	0.276360	+	0.961054i	$0.410872\pi$
$992$	0	0
$993$	−6.98457e9	−0.226369
$994$	0	0
$995$	−1.58588e9	−0.0510375
$996$	0	0
$997$	−2.36849e10	−0.756900	−0.378450	−	0.925622i	$-0.623543\pi$
$997$	−2.36849e10	−0.756900	−0.378450	+	0.925622i	$0.623543\pi$
$998$	0	0
$999$	4.49707e10	1.42709

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.8.a.u.1.2		2
4.3	odd	2		320.8.a.l.1.1		2
8.3	odd	2		5.8.a.b.1.1	✓	2
8.5	even	2		80.8.a.g.1.1		2
24.11	even	2		45.8.a.h.1.2		2
40.3	even	4		25.8.b.c.24.2		4
40.13	odd	4		400.8.c.m.49.1		4
40.19	odd	2		25.8.a.b.1.2		2
40.27	even	4		25.8.b.c.24.3		4
40.29	even	2		400.8.a.bb.1.2		2
40.37	odd	4		400.8.c.m.49.4		4
56.27	even	2		245.8.a.c.1.1		2
88.43	even	2		605.8.a.d.1.2		2
120.59	even	2		225.8.a.w.1.1		2
120.83	odd	4		225.8.b.m.199.3		4
120.107	odd	4		225.8.b.m.199.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.8.a.b.1.1	✓	2	8.3	odd	2
25.8.a.b.1.2		2	40.19	odd	2
25.8.b.c.24.2		4	40.3	even	4
25.8.b.c.24.3		4	40.27	even	4
45.8.a.h.1.2		2	24.11	even	2
80.8.a.g.1.1		2	8.5	even	2
225.8.a.w.1.1		2	120.59	even	2
225.8.b.m.199.2		4	120.107	odd	4
225.8.b.m.199.3		4	120.83	odd	4
245.8.a.c.1.1		2	56.27	even	2
320.8.a.l.1.1		2	4.3	odd	2
320.8.a.u.1.2		2	1.1	even	1	trivial
400.8.a.bb.1.2		2	40.29	even	2
400.8.c.m.49.1		4	40.13	odd	4
400.8.c.m.49.4		4	40.37	odd	4
605.8.a.d.1.2		2	88.43	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)

\(f(q)\)	\(=\)	\(q+79.7424 q^{3} +125.000 q^{5} +538.197 q^{7} +4171.85 q^{9} -1215.12 q^{11} -7070.42 q^{13} +9967.80 q^{15} -3348.13 q^{17} +22169.7 q^{19} +42917.1 q^{21} +58513.1 q^{23} +15625.0 q^{25} +158276. q^{27} +206301. q^{29} -177822. q^{31} -96896.5 q^{33} +67274.6 q^{35} +284128. q^{37} -563812. q^{39} +627353. q^{41} -164889. q^{43} +521481. q^{45} +449355. q^{47} -533887. q^{49} -266988. q^{51} +730190. q^{53} -151890. q^{55} +1.76786e6 q^{57} +1.42202e6 q^{59} +266326. q^{61} +2.24527e6 q^{63} -883803. q^{65} +2.95028e6 q^{67} +4.66598e6 q^{69} -921138. q^{71} +4.25657e6 q^{73} +1.24597e6 q^{75} -653973. q^{77} -6.28551e6 q^{79} +3.49751e6 q^{81} -9.17165e6 q^{83} -418516. q^{85} +1.64510e7 q^{87} +242643. q^{89} -3.80528e6 q^{91} -1.41799e7 q^{93} +2.77121e6 q^{95} -2.59198e6 q^{97} -5.06929e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 20 q^{3} + 250 q^{5} + 100 q^{7} + 5554 q^{9} + 4544 q^{11} - 3540 q^{13} + 2500 q^{15} - 27340 q^{17} + 38760 q^{19} + 69096 q^{21} + 124140 q^{23} + 31250 q^{25} + 206360 q^{27} + 72260 q^{29} - 306824 q^{31}+ \cdots + 2890688 q^{99}+O(q^{100}) \) 2 * q + 20 * q^3 + 250 * q^5 + 100 * q^7 + 5554 * q^9 + 4544 * q^11 - 3540 * q^13 + 2500 * q^15 - 27340 * q^17 + 38760 * q^19 + 69096 * q^21 + 124140 * q^23 + 31250 * q^25 + 206360 * q^27 + 72260 * q^29 - 306824 * q^31 - 440960 * q^33 + 12500 * q^35 + 123020 * q^37 - 774728 * q^39 + 264364 * q^41 + 423300 * q^43 + 694250 * q^45 + 105460 * q^47 - 1165414 * q^49 + 1166344 * q^51 + 2391580 * q^53 + 568000 * q^55 + 776720 * q^57 - 1120120 * q^59 - 2257044 * q^61 + 1639620 * q^63 - 442500 * q^65 + 4516460 * q^67 + 745272 * q^69 - 621784 * q^71 + 4569060 * q^73 + 312500 * q^75 - 3177600 * q^77 - 4333040 * q^79 - 2397878 * q^81 - 9793020 * q^83 - 3417500 * q^85 + 24458920 * q^87 + 6025620 * q^89 - 5352296 * q^91 - 6473040 * q^93 + 4845000 * q^95 + 4609540 * q^97 + 2890688 * q^99

Introduction

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