LMFDB - Embedded newform 64.12.a.m.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [64,12,Mod(1,64)] 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("64.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(64, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	64.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,440,0,-770] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.1739635558$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1056820.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 331x - 1379 $ x^3 - x^2 - 331*x - 1379
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.50335$ of defining polynomial
Character	$\chi$	$=$	64.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-8.10720 q^{3} -9611.74 q^{5} -77473.2 q^{7} -177081. q^{9} -433944. q^{11} -1.20991e6 q^{13} +77924.3 q^{15} -3.77507e6 q^{17} +1.37832e7 q^{19} +628090. q^{21} -1.99143e7 q^{23} +4.35575e7 q^{25} +2.87180e6 q^{27} +1.15430e8 q^{29} -3.00417e8 q^{31} +3.51807e6 q^{33} +7.44652e8 q^{35} -4.52440e8 q^{37} +9.80901e6 q^{39} -3.74000e8 q^{41} -8.22566e7 q^{43} +1.70206e9 q^{45} -2.34368e9 q^{47} +4.02476e9 q^{49} +3.06052e7 q^{51} +2.08470e9 q^{53} +4.17096e9 q^{55} -1.11743e8 q^{57} -2.89241e9 q^{59} +7.03714e9 q^{61} +1.37190e10 q^{63} +1.16294e10 q^{65} -1.42883e10 q^{67} +1.61449e8 q^{69} -1.41180e10 q^{71} +5.07004e9 q^{73} -3.53129e8 q^{75} +3.36190e10 q^{77} +1.86354e10 q^{79} +3.13461e10 q^{81} -1.60715e9 q^{83} +3.62850e10 q^{85} -9.35814e8 q^{87} +8.15726e10 q^{89} +9.37358e10 q^{91} +2.43554e9 q^{93} -1.32480e11 q^{95} -1.21144e11 q^{97} +7.68433e10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 440 q^{3} - 770 q^{5} + 6032 q^{7} + 211663 q^{9} + 236136 q^{11} - 1435722 q^{13} - 222160 q^{15} - 756186 q^{17} + 23267992 q^{19} + 20368000 q^{21} - 42366288 q^{23} - 6913515 q^{25} + 298561328 q^{27}+ \cdots + 276379454984 q^{99}+O(q^{100}) $ 3 * q + 440 * q^3 - 770 * q^5 + 6032 * q^7 + 211663 * q^9 + 236136 * q^11 - 1435722 * q^13 - 222160 * q^15 - 756186 * q^17 + 23267992 * q^19 + 20368000 * q^21 - 42366288 * q^23 - 6913515 * q^25 + 298561328 * q^27 - 42667514 * q^29 - 555856064 * q^31 + 308282944 * q^33 + 1110160800 * q^35 + 462651966 * q^37 - 2309231888 * q^39 + 357860750 * q^41 + 1400057768 * q^43 + 2398467190 * q^45 - 1620045984 * q^47 + 3558319419 * q^49 - 1197100944 * q^51 + 5087465774 * q^53 + 6584414480 * q^55 + 15809413568 * q^57 - 13552245240 * q^59 + 18288485958 * q^61 + 30411798992 * q^63 + 18683192540 * q^65 - 17471404360 * q^67 + 37933678464 * q^69 + 23838346512 * q^71 + 35602893150 * q^73 - 31830320120 * q^75 + 61844462464 * q^77 - 4634961248 * q^79 + 104321985931 * q^81 + 49374840984 * q^83 + 56389897020 * q^85 - 146872315024 * q^87 + 58640505454 * q^89 + 80670986272 * q^91 + 10026258944 * q^93 - 139519303440 * q^95 - 66696660810 * q^97 + 276379454984 * 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
$3$	−8.10720	−0.0192621	−0.00963105	−	0.999954i	$-0.503066\pi$
$3$	−8.10720	−0.0192621	−0.00963105	+	0.999954i	$0.503066\pi$
$4$	0	0
$5$	−9611.74	−1.37552	−0.687760	−	0.725938i	$-0.741406\pi$
$5$	−9611.74	−1.37552	−0.687760	+	0.725938i	$0.741406\pi$
$6$	0	0
$7$	−77473.2	−1.74226	−0.871128	−	0.491056i	$-0.836611\pi$
$7$	−77473.2	−1.74226	−0.871128	+	0.491056i	$0.836611\pi$
$8$	0	0
$9$	−177081.	−0.999629
$10$	0	0
$11$	−433944.	−0.812407	−0.406204	−	0.913783i	$-0.633148\pi$
$11$	−433944.	−0.812407	−0.406204	+	0.913783i	$0.633148\pi$
$12$	0	0
$13$	−1.20991e6	−0.903787	−0.451893	−	0.892072i	$-0.649251\pi$
$13$	−1.20991e6	−0.903787	−0.451893	+	0.892072i	$0.649251\pi$
$14$	0	0
$15$	77924.3	0.0264954
$16$	0	0
$17$	−3.77507e6	−0.644846	−0.322423	−	0.946596i	$-0.604497\pi$
$17$	−3.77507e6	−0.644846	−0.322423	+	0.946596i	$0.604497\pi$
$18$	0	0
$19$	1.37832e7	1.27704	0.638519	−	0.769606i	$-0.279547\pi$
$19$	1.37832e7	1.27704	0.638519	+	0.769606i	$0.279547\pi$
$20$	0	0
$21$	628090.	0.0335595
$22$	0	0
$23$	−1.99143e7	−0.645152	−0.322576	−	0.946544i	$-0.604549\pi$
$23$	−1.99143e7	−0.645152	−0.322576	+	0.946544i	$0.604549\pi$
$24$	0	0
$25$	4.35575e7	0.892057
$26$	0	0
$27$	2.87180e6	0.0385171
$28$	0	0
$29$	1.15430e8	1.04503	0.522516	−	0.852629i	$-0.324993\pi$
$29$	1.15430e8	1.04503	0.522516	+	0.852629i	$0.324993\pi$
$30$	0	0
$31$	−3.00417e8	−1.88467	−0.942336	−	0.334669i	$-0.891376\pi$
$31$	−3.00417e8	−1.88467	−0.942336	+	0.334669i	$0.891376\pi$
$32$	0	0
$33$	3.51807e6	0.0156487
$34$	0	0
$35$	7.44652e8	2.39651
$36$	0	0
$37$	−4.52440e8	−1.07263	−0.536317	−	0.844017i	$-0.680185\pi$
$37$	−4.52440e8	−1.07263	−0.536317	+	0.844017i	$0.680185\pi$
$38$	0	0
$39$	9.80901e6	0.0174088
$40$	0	0
$41$	−3.74000e8	−0.504151	−0.252076	−	0.967708i	$-0.581113\pi$
$41$	−3.74000e8	−0.504151	−0.252076	+	0.967708i	$0.581113\pi$
$42$	0	0
$43$	−8.22566e7	−0.0853286	−0.0426643	−	0.999089i	$-0.513585\pi$
$43$	−8.22566e7	−0.0853286	−0.0426643	+	0.999089i	$0.513585\pi$
$44$	0	0
$45$	1.70206e9	1.37501
$46$	0	0
$47$	−2.34368e9	−1.49060	−0.745299	−	0.666730i	$-0.767693\pi$
$47$	−2.34368e9	−1.49060	−0.745299	+	0.666730i	$0.767693\pi$
$48$	0	0
$49$	4.02476e9	2.03546
$50$	0	0
$51$	3.06052e7	0.0124211
$52$	0	0
$53$	2.08470e9	0.684739	0.342370	−	0.939565i	$-0.388770\pi$
$53$	2.08470e9	0.684739	0.342370	+	0.939565i	$0.388770\pi$
$54$	0	0
$55$	4.17096e9	1.11748
$56$	0	0
$57$	−1.11743e8	−0.0245984
$58$	0	0
$59$	−2.89241e9	−0.526713	−0.263357	−	0.964699i	$-0.584830\pi$
$59$	−2.89241e9	−0.526713	−0.263357	+	0.964699i	$0.584830\pi$
$60$	0	0
$61$	7.03714e9	1.06680	0.533399	−	0.845864i	$-0.320915\pi$
$61$	7.03714e9	1.06680	0.533399	+	0.845864i	$0.320915\pi$
$62$	0	0
$63$	1.37190e10	1.74161
$64$	0	0
$65$	1.16294e10	1.24318
$66$	0	0
$67$	−1.42883e10	−1.29292	−0.646458	−	0.762949i	$-0.723751\pi$
$67$	−1.42883e10	−1.29292	−0.646458	+	0.762949i	$0.723751\pi$
$68$	0	0
$69$	1.61449e8	0.0124270
$70$	0	0
$71$	−1.41180e10	−0.928650	−0.464325	−	0.885665i	$-0.653703\pi$
$71$	−1.41180e10	−0.928650	−0.464325	+	0.885665i	$0.653703\pi$
$72$	0	0
$73$	5.07004e9	0.286244	0.143122	−	0.989705i	$-0.454286\pi$
$73$	5.07004e9	0.286244	0.143122	+	0.989705i	$0.454286\pi$
$74$	0	0
$75$	−3.53129e8	−0.0171829
$76$	0	0
$77$	3.36190e10	1.41542
$78$	0	0
$79$	1.86354e10	0.681381	0.340691	−	0.940175i	$-0.389339\pi$
$79$	1.86354e10	0.681381	0.340691	+	0.940175i	$0.389339\pi$
$80$	0	0
$81$	3.13461e10	0.998887
$82$	0	0
$83$	−1.60715e9	−0.0447844	−0.0223922	−	0.999749i	$-0.507128\pi$
$83$	−1.60715e9	−0.0447844	−0.0223922	+	0.999749i	$0.507128\pi$
$84$	0	0
$85$	3.62850e10	0.886999
$86$	0	0
$87$	−9.35814e8	−0.0201295
$88$	0	0
$89$	8.15726e10	1.54846	0.774229	−	0.632905i	$-0.218138\pi$
$89$	8.15726e10	1.54846	0.774229	+	0.632905i	$0.218138\pi$
$90$	0	0
$91$	9.37358e10	1.57463
$92$	0	0
$93$	2.43554e9	0.0363028
$94$	0	0
$95$	−1.32480e11	−1.75659
$96$	0	0
$97$	−1.21144e11	−1.43238	−0.716188	−	0.697907i	$-0.754115\pi$
$97$	−1.21144e11	−1.43238	−0.716188	+	0.697907i	$0.754115\pi$
$98$	0	0
$99$	7.68433e10	0.812106
$100$	0	0
$101$	−3.41510e8	−0.00323323	−0.00161661	−	0.999999i	$-0.500515\pi$
$101$	−3.41510e8	−0.00323323	−0.00161661	+	0.999999i	$0.500515\pi$
$102$	0	0
$103$	−3.38650e10	−0.287837	−0.143919	−	0.989590i	$-0.545970\pi$
$103$	−3.38650e10	−0.287837	−0.143919	+	0.989590i	$0.545970\pi$
$104$	0	0
$105$	−6.03704e9	−0.0461618
$106$	0	0
$107$	−2.04508e11	−1.40961	−0.704806	−	0.709400i	$-0.748966\pi$
$107$	−2.04508e11	−1.40961	−0.704806	+	0.709400i	$0.748966\pi$
$108$	0	0
$109$	−2.96478e11	−1.84564	−0.922819	−	0.385234i	$-0.874121\pi$
$109$	−2.96478e11	−1.84564	−0.922819	+	0.385234i	$0.874121\pi$
$110$	0	0
$111$	3.66802e9	0.0206612
$112$	0	0
$113$	−8.68449e10	−0.443418	−0.221709	−	0.975113i	$-0.571163\pi$
$113$	−8.68449e10	−0.443418	−0.221709	+	0.975113i	$0.571163\pi$
$114$	0	0
$115$	1.91411e11	0.887420
$116$	0	0
$117$	2.14253e11	0.903451
$118$	0	0
$119$	2.92467e11	1.12349
$120$	0	0
$121$	−9.70045e10	−0.339995
$122$	0	0
$123$	3.03210e9	0.00971102
$124$	0	0
$125$	5.06600e10	0.148477
$126$	0	0
$127$	2.07215e11	0.556546	0.278273	−	0.960502i	$-0.410238\pi$
$127$	2.07215e11	0.556546	0.278273	+	0.960502i	$0.410238\pi$
$128$	0	0
$129$	6.66871e8	0.00164361
$130$	0	0
$131$	−7.06063e11	−1.59901	−0.799505	−	0.600659i	$-0.794905\pi$
$131$	−7.06063e11	−1.59901	−0.799505	+	0.600659i	$0.794905\pi$
$132$	0	0
$133$	−1.06782e12	−2.22493
$134$	0	0
$135$	−2.76030e10	−0.0529810
$136$	0	0
$137$	−6.91127e11	−1.22347	−0.611737	−	0.791061i	$-0.709529\pi$
$137$	−6.91127e11	−1.22347	−0.611737	+	0.791061i	$0.709529\pi$
$138$	0	0
$139$	−5.74440e11	−0.938996	−0.469498	−	0.882934i	$-0.655565\pi$
$139$	−5.74440e11	−0.938996	−0.469498	+	0.882934i	$0.655565\pi$
$140$	0	0
$141$	1.90007e10	0.0287121
$142$	0	0
$143$	5.25034e11	0.734243
$144$	0	0
$145$	−1.10948e12	−1.43746
$146$	0	0
$147$	−3.26296e10	−0.0392072
$148$	0	0
$149$	−7.50213e11	−0.836874	−0.418437	−	0.908246i	$-0.637422\pi$
$149$	−7.50213e11	−0.836874	−0.418437	+	0.908246i	$0.637422\pi$
$150$	0	0
$151$	−7.78919e11	−0.807456	−0.403728	−	0.914879i	$-0.632286\pi$
$151$	−7.78919e11	−0.807456	−0.403728	+	0.914879i	$0.632286\pi$
$152$	0	0
$153$	6.68494e11	0.644607
$154$	0	0
$155$	2.88753e12	2.59241
$156$	0	0
$157$	1.03515e12	0.866072	0.433036	−	0.901377i	$-0.357442\pi$
$157$	1.03515e12	0.866072	0.433036	+	0.901377i	$0.357442\pi$
$158$	0	0
$159$	−1.69010e10	−0.0131895
$160$	0	0
$161$	1.54283e12	1.12402
$162$	0	0
$163$	−1.04879e10	−0.00713929	−0.00356964	−	0.999994i	$-0.501136\pi$
$163$	−1.04879e10	−0.00713929	−0.00356964	+	0.999994i	$0.501136\pi$
$164$	0	0
$165$	−3.38148e10	−0.0215251
$166$	0	0
$167$	−6.17418e11	−0.367823	−0.183911	−	0.982943i	$-0.558876\pi$
$167$	−6.17418e11	−0.367823	−0.183911	+	0.982943i	$0.558876\pi$
$168$	0	0
$169$	−3.28269e11	−0.183170
$170$	0	0
$171$	−2.44074e12	−1.27656
$172$	0	0
$173$	2.20665e12	1.08263	0.541315	−	0.840820i	$-0.317927\pi$
$173$	2.20665e12	1.08263	0.541315	+	0.840820i	$0.317927\pi$
$174$	0	0
$175$	−3.37454e12	−1.55419
$176$	0	0
$177$	2.34494e10	0.0101456
$178$	0	0
$179$	2.94005e12	1.19581	0.597907	−	0.801565i	$-0.295999\pi$
$179$	2.94005e12	1.19581	0.597907	+	0.801565i	$0.295999\pi$
$180$	0	0
$181$	8.07676e11	0.309033	0.154517	−	0.987990i	$-0.450618\pi$
$181$	8.07676e11	0.309033	0.154517	+	0.987990i	$0.450618\pi$
$182$	0	0
$183$	−5.70515e10	−0.0205488
$184$	0	0
$185$	4.34874e12	1.47543
$186$	0	0
$187$	1.63817e12	0.523877
$188$	0	0
$189$	−2.22487e11	−0.0671066
$190$	0	0
$191$	2.07223e12	0.589866	0.294933	−	0.955518i	$-0.404703\pi$
$191$	2.07223e12	0.589866	0.294933	+	0.955518i	$0.404703\pi$
$192$	0	0
$193$	2.11276e11	0.0567918	0.0283959	−	0.999597i	$-0.490960\pi$
$193$	2.11276e11	0.0567918	0.0283959	+	0.999597i	$0.490960\pi$
$194$	0	0
$195$	−9.42817e10	−0.0239462
$196$	0	0
$197$	1.79929e12	0.432052	0.216026	−	0.976388i	$-0.430690\pi$
$197$	1.79929e12	0.432052	0.216026	+	0.976388i	$0.430690\pi$
$198$	0	0
$199$	5.44550e12	1.23693	0.618466	−	0.785812i	$-0.287755\pi$
$199$	5.44550e12	1.23693	0.618466	+	0.785812i	$0.287755\pi$
$200$	0	0
$201$	1.15838e11	0.0249043
$202$	0	0
$203$	−8.94273e12	−1.82071
$204$	0	0
$205$	3.59480e12	0.693471
$206$	0	0
$207$	3.52645e12	0.644913
$208$	0	0
$209$	−5.98111e12	−1.03747
$210$	0	0
$211$	−1.45420e12	−0.239371	−0.119685	−	0.992812i	$-0.538189\pi$
$211$	−1.45420e12	−0.239371	−0.119685	+	0.992812i	$0.538189\pi$
$212$	0	0
$213$	1.14457e11	0.0178878
$214$	0	0
$215$	7.90630e11	0.117371
$216$	0	0
$217$	2.32743e13	3.28358
$218$	0	0
$219$	−4.11038e10	−0.00551366
$220$	0	0
$221$	4.56751e12	0.582803
$222$	0	0
$223$	−6.20554e12	−0.753534	−0.376767	−	0.926308i	$-0.622964\pi$
$223$	−6.20554e12	−0.753534	−0.376767	+	0.926308i	$0.622964\pi$
$224$	0	0
$225$	−7.71322e12	−0.891726
$226$	0	0
$227$	−1.18576e13	−1.30574	−0.652868	−	0.757472i	$-0.726434\pi$
$227$	−1.18576e13	−1.30574	−0.652868	+	0.757472i	$0.726434\pi$
$228$	0	0
$229$	−2.83010e11	−0.0296966	−0.0148483	−	0.999890i	$-0.504727\pi$
$229$	−2.83010e11	−0.0296966	−0.0148483	+	0.999890i	$0.504727\pi$
$230$	0	0
$231$	−2.72556e11	−0.0272640
$232$	0	0
$233$	−9.70846e12	−0.926175	−0.463087	−	0.886313i	$-0.653258\pi$
$233$	−9.70846e12	−0.926175	−0.463087	+	0.886313i	$0.653258\pi$
$234$	0	0
$235$	2.25269e13	2.05035
$236$	0	0
$237$	−1.51081e11	−0.0131248
$238$	0	0
$239$	1.32519e13	1.09924	0.549618	−	0.835416i	$-0.314773\pi$
$239$	1.32519e13	1.09924	0.549618	+	0.835416i	$0.314773\pi$
$240$	0	0
$241$	−1.77216e13	−1.40413	−0.702066	−	0.712112i	$-0.747739\pi$
$241$	−1.77216e13	−1.40413	−0.702066	+	0.712112i	$0.747739\pi$
$242$	0	0
$243$	−7.62860e11	−0.0577577
$244$	0	0
$245$	−3.86850e13	−2.79981
$246$	0	0
$247$	−1.66764e13	−1.15417
$248$	0	0
$249$	1.30295e10	0.000862642 0
$250$	0	0
$251$	−4.29322e12	−0.272005	−0.136003	−	0.990708i	$-0.543426\pi$
$251$	−4.29322e12	−0.272005	−0.136003	+	0.990708i	$0.543426\pi$
$252$	0	0
$253$	8.64169e12	0.524126
$254$	0	0
$255$	−2.94170e11	−0.0170855
$256$	0	0
$257$	−4.51002e12	−0.250926	−0.125463	−	0.992098i	$-0.540042\pi$
$257$	−4.51002e12	−0.250926	−0.125463	+	0.992098i	$0.540042\pi$
$258$	0	0
$259$	3.50519e13	1.86880
$260$	0	0
$261$	−2.04405e13	−1.04464
$262$	0	0
$263$	−2.96086e13	−1.45098	−0.725489	−	0.688233i	$-0.758387\pi$
$263$	−2.96086e13	−1.45098	−0.725489	+	0.688233i	$0.758387\pi$
$264$	0	0
$265$	−2.00376e13	−0.941873
$266$	0	0
$267$	−6.61326e11	−0.0298266
$268$	0	0
$269$	2.87428e13	1.24420	0.622102	−	0.782936i	$-0.286279\pi$
$269$	2.87428e13	1.24420	0.622102	+	0.782936i	$0.286279\pi$
$270$	0	0
$271$	−3.78025e13	−1.57105	−0.785525	−	0.618830i	$-0.787607\pi$
$271$	−3.78025e13	−1.57105	−0.785525	+	0.618830i	$0.787607\pi$
$272$	0	0
$273$	−7.59935e11	−0.0303307
$274$	0	0
$275$	−1.89015e13	−0.724714
$276$	0	0
$277$	−1.61698e13	−0.595754	−0.297877	−	0.954604i	$-0.596278\pi$
$277$	−1.61698e13	−0.595754	−0.297877	+	0.954604i	$0.596278\pi$
$278$	0	0
$279$	5.31983e13	1.88397
$280$	0	0
$281$	3.10776e13	1.05819	0.529094	−	0.848563i	$-0.322532\pi$
$281$	3.10776e13	1.05819	0.529094	+	0.848563i	$0.322532\pi$
$282$	0	0
$283$	−2.16844e13	−0.710103	−0.355051	−	0.934847i	$-0.615537\pi$
$283$	−2.16844e13	−0.710103	−0.355051	+	0.934847i	$0.615537\pi$
$284$	0	0
$285$	1.07404e12	0.0338357
$286$	0	0
$287$	2.89750e13	0.878361
$288$	0	0
$289$	−2.00207e13	−0.584174
$290$	0	0
$291$	9.82138e11	0.0275906
$292$	0	0
$293$	−2.94763e13	−0.797445	−0.398722	−	0.917072i	$-0.630546\pi$
$293$	−2.94763e13	−0.797445	−0.398722	+	0.917072i	$0.630546\pi$
$294$	0	0
$295$	2.78011e13	0.724505
$296$	0	0
$297$	−1.24620e12	−0.0312915
$298$	0	0
$299$	2.40946e13	0.583080
$300$	0	0
$301$	6.37268e12	0.148664
$302$	0	0
$303$	2.76869e9	6.22788e−5 0
$304$	0	0
$305$	−6.76392e13	−1.46740
$306$	0	0
$307$	4.09035e12	0.0856051	0.0428025	−	0.999084i	$-0.486371\pi$
$307$	4.09035e12	0.0856051	0.0428025	+	0.999084i	$0.486371\pi$
$308$	0	0
$309$	2.74550e11	0.00554435
$310$	0	0
$311$	2.06575e13	0.402621	0.201310	−	0.979528i	$-0.435480\pi$
$311$	2.06575e13	0.402621	0.201310	+	0.979528i	$0.435480\pi$
$312$	0	0
$313$	7.58015e13	1.42621	0.713105	−	0.701057i	$-0.247288\pi$
$313$	7.58015e13	1.42621	0.713105	+	0.701057i	$0.247288\pi$
$314$	0	0
$315$	−1.31864e14	−2.39562
$316$	0	0
$317$	8.64969e13	1.51766	0.758831	−	0.651288i	$-0.225771\pi$
$317$	8.64969e13	1.51766	0.758831	+	0.651288i	$0.225771\pi$
$318$	0	0
$319$	−5.00901e13	−0.848992
$320$	0	0
$321$	1.65799e12	0.0271521
$322$	0	0
$323$	−5.20324e13	−0.823493
$324$	0	0
$325$	−5.27008e13	−0.806230
$326$	0	0
$327$	2.40360e12	0.0355509
$328$	0	0
$329$	1.81572e14	2.59700
$330$	0	0
$331$	9.59234e13	1.32700	0.663500	−	0.748177i	$-0.269070\pi$
$331$	9.59234e13	1.32700	0.663500	+	0.748177i	$0.269070\pi$
$332$	0	0
$333$	8.01186e13	1.07224
$334$	0	0
$335$	1.37336e14	1.77843
$336$	0	0
$337$	−5.82689e13	−0.730251	−0.365126	−	0.930958i	$-0.618974\pi$
$337$	−5.82689e13	−0.730251	−0.365126	+	0.930958i	$0.618974\pi$
$338$	0	0
$339$	7.04069e11	0.00854116
$340$	0	0
$341$	1.30364e14	1.53112
$342$	0	0
$343$	−1.58622e14	−1.80403
$344$	0	0
$345$	−1.55181e12	−0.0170936
$346$	0	0
$347$	1.39079e14	1.48406	0.742029	−	0.670368i	$-0.233864\pi$
$347$	1.39079e14	1.48406	0.742029	+	0.670368i	$0.233864\pi$
$348$	0	0
$349$	−1.21040e12	−0.0125138	−0.00625691	−	0.999980i	$-0.501992\pi$
$349$	−1.21040e12	−0.0125138	−0.00625691	+	0.999980i	$0.501992\pi$
$350$	0	0
$351$	−3.47463e12	−0.0348112
$352$	0	0
$353$	4.46583e13	0.433652	0.216826	−	0.976210i	$-0.430430\pi$
$353$	4.46583e13	0.433652	0.216826	+	0.976210i	$0.430430\pi$
$354$	0	0
$355$	1.35699e14	1.27738
$356$	0	0
$357$	−2.37108e12	−0.0216407
$358$	0	0
$359$	−4.01912e13	−0.355722	−0.177861	−	0.984056i	$-0.556918\pi$
$359$	−4.01912e13	−0.355722	−0.177861	+	0.984056i	$0.556918\pi$
$360$	0	0
$361$	7.34850e13	0.630825
$362$	0	0
$363$	7.86434e11	0.00654902
$364$	0	0
$365$	−4.87320e13	−0.393734
$366$	0	0
$367$	−1.86874e14	−1.46516	−0.732582	−	0.680679i	$-0.761685\pi$
$367$	−1.86874e14	−1.46516	−0.732582	+	0.680679i	$0.761685\pi$
$368$	0	0
$369$	6.62285e13	0.503964
$370$	0	0
$371$	−1.61508e14	−1.19299
$372$	0	0
$373$	−4.10882e13	−0.294658	−0.147329	−	0.989088i	$-0.547068\pi$
$373$	−4.10882e13	−0.294658	−0.147329	+	0.989088i	$0.547068\pi$
$374$	0	0
$375$	−4.10710e11	−0.00285998
$376$	0	0
$377$	−1.39660e14	−0.944486
$378$	0	0
$379$	1.36292e13	0.0895270	0.0447635	−	0.998998i	$-0.485747\pi$
$379$	1.36292e13	0.0895270	0.0447635	+	0.998998i	$0.485747\pi$
$380$	0	0
$381$	−1.67994e12	−0.0107203
$382$	0	0
$383$	7.88646e13	0.488978	0.244489	−	0.969652i	$-0.421380\pi$
$383$	7.88646e13	0.488978	0.244489	+	0.969652i	$0.421380\pi$
$384$	0	0
$385$	−3.23137e14	−1.94694
$386$	0	0
$387$	1.45661e13	0.0852969
$388$	0	0
$389$	−7.25260e13	−0.412829	−0.206415	−	0.978465i	$-0.566180\pi$
$389$	−7.25260e13	−0.412829	−0.206415	+	0.978465i	$0.566180\pi$
$390$	0	0
$391$	7.51779e13	0.416024
$392$	0	0
$393$	5.72419e12	0.0308003
$394$	0	0
$395$	−1.79119e14	−0.937254
$396$	0	0
$397$	−1.22466e13	−0.0623260	−0.0311630	−	0.999514i	$-0.509921\pi$
$397$	−1.22466e13	−0.0623260	−0.0311630	+	0.999514i	$0.509921\pi$
$398$	0	0
$399$	8.65706e12	0.0428568
$400$	0	0
$401$	3.06634e14	1.47682	0.738408	−	0.674354i	$-0.235578\pi$
$401$	3.06634e14	1.47682	0.738408	+	0.674354i	$0.235578\pi$
$402$	0	0
$403$	3.63479e14	1.70334
$404$	0	0
$405$	−3.01291e14	−1.37399
$406$	0	0
$407$	1.96333e14	0.871415
$408$	0	0
$409$	−1.75007e14	−0.756097	−0.378049	−	0.925786i	$-0.623405\pi$
$409$	−1.75007e14	−0.756097	−0.378049	+	0.925786i	$0.623405\pi$
$410$	0	0
$411$	5.60310e12	0.0235667
$412$	0	0
$413$	2.24084e14	0.917670
$414$	0	0
$415$	1.54475e13	0.0616019
$416$	0	0
$417$	4.65710e12	0.0180870
$418$	0	0
$419$	−4.92094e14	−1.86153	−0.930767	−	0.365613i	$-0.880859\pi$
$419$	−4.92094e14	−1.86153	−0.930767	+	0.365613i	$0.880859\pi$
$420$	0	0
$421$	−7.15111e12	−0.0263525	−0.0131763	−	0.999913i	$-0.504194\pi$
$421$	−7.15111e12	−0.0263525	−0.0131763	+	0.999913i	$0.504194\pi$
$422$	0	0
$423$	4.15022e14	1.49004
$424$	0	0
$425$	−1.64433e14	−0.575240
$426$	0	0
$427$	−5.45189e14	−1.85863
$428$	0	0
$429$	−4.25656e12	−0.0141431
$430$	0	0
$431$	−9.30701e13	−0.301429	−0.150715	−	0.988577i	$-0.548157\pi$
$431$	−9.30701e13	−0.301429	−0.150715	+	0.988577i	$0.548157\pi$
$432$	0	0
$433$	−7.17950e13	−0.226679	−0.113339	−	0.993556i	$-0.536155\pi$
$433$	−7.17950e13	−0.226679	−0.113339	+	0.993556i	$0.536155\pi$
$434$	0	0
$435$	8.99480e12	0.0276886
$436$	0	0
$437$	−2.74482e14	−0.823884
$438$	0	0
$439$	−6.47772e13	−0.189613	−0.0948063	−	0.995496i	$-0.530223\pi$
$439$	−6.47772e13	−0.189613	−0.0948063	+	0.995496i	$0.530223\pi$
$440$	0	0
$441$	−7.12710e14	−2.03470
$442$	0	0
$443$	4.49312e14	1.25120	0.625601	−	0.780143i	$-0.284854\pi$
$443$	4.49312e14	1.25120	0.625601	+	0.780143i	$0.284854\pi$
$444$	0	0
$445$	−7.84055e14	−2.12994
$446$	0	0
$447$	6.08213e12	0.0161200
$448$	0	0
$449$	−1.01226e14	−0.261780	−0.130890	−	0.991397i	$-0.541784\pi$
$449$	−1.01226e14	−0.261780	−0.130890	+	0.991397i	$0.541784\pi$
$450$	0	0
$451$	1.62295e14	0.409576
$452$	0	0
$453$	6.31485e12	0.0155533
$454$	0	0
$455$	−9.00965e14	−2.16593
$456$	0	0
$457$	5.47245e14	1.28423	0.642115	−	0.766608i	$-0.278057\pi$
$457$	5.47245e14	1.28423	0.642115	+	0.766608i	$0.278057\pi$
$458$	0	0
$459$	−1.08412e13	−0.0248376
$460$	0	0
$461$	4.10181e13	0.0917531	0.0458765	−	0.998947i	$-0.485392\pi$
$461$	4.10181e13	0.0917531	0.0458765	+	0.998947i	$0.485392\pi$
$462$	0	0
$463$	4.31660e14	0.942859	0.471429	−	0.881904i	$-0.343738\pi$
$463$	4.31660e14	0.942859	0.471429	+	0.881904i	$0.343738\pi$
$464$	0	0
$465$	−2.34098e13	−0.0499352
$466$	0	0
$467$	3.19604e14	0.665839	0.332919	−	0.942955i	$-0.391966\pi$
$467$	3.19604e14	0.665839	0.332919	+	0.942955i	$0.391966\pi$
$468$	0	0
$469$	1.10696e15	2.25259
$470$	0	0
$471$	−8.39215e12	−0.0166824
$472$	0	0
$473$	3.56948e13	0.0693215
$474$	0	0
$475$	6.00360e14	1.13919
$476$	0	0
$477$	−3.69160e14	−0.684485
$478$	0	0
$479$	2.31754e13	0.0419935	0.0209968	−	0.999780i	$-0.493316\pi$
$479$	2.31754e13	0.0419935	0.0209968	+	0.999780i	$0.493316\pi$
$480$	0	0
$481$	5.47413e14	0.969432
$482$	0	0
$483$	−1.25080e13	−0.0216510
$484$	0	0
$485$	1.16440e15	1.97026
$486$	0	0
$487$	−9.87799e12	−0.0163403	−0.00817014	−	0.999967i	$-0.502601\pi$
$487$	−9.87799e12	−0.0163403	−0.00817014	+	0.999967i	$0.502601\pi$
$488$	0	0
$489$	8.50271e10	0.000137518 0
$490$	0	0
$491$	5.08043e14	0.803437	0.401719	−	0.915763i	$-0.368413\pi$
$491$	5.08043e14	0.803437	0.401719	+	0.915763i	$0.368413\pi$
$492$	0	0
$493$	−4.35756e14	−0.673885
$494$	0	0
$495$	−7.38598e14	−1.11707
$496$	0	0
$497$	1.09377e15	1.61795
$498$	0	0
$499$	−9.67307e14	−1.39962	−0.699812	−	0.714327i	$-0.746733\pi$
$499$	−9.67307e14	−1.39962	−0.699812	+	0.714327i	$0.746733\pi$
$500$	0	0
$501$	5.00553e12	0.00708504
$502$	0	0
$503$	−3.65928e14	−0.506724	−0.253362	−	0.967372i	$-0.581536\pi$
$503$	−3.65928e14	−0.506724	−0.253362	+	0.967372i	$0.581536\pi$
$504$	0	0
$505$	3.28251e12	0.00444737
$506$	0	0
$507$	2.66134e12	0.00352823
$508$	0	0
$509$	−2.27986e13	−0.0295775	−0.0147887	−	0.999891i	$-0.504708\pi$
$509$	−2.27986e13	−0.0295775	−0.0147887	+	0.999891i	$0.504708\pi$
$510$	0	0
$511$	−3.92792e14	−0.498710
$512$	0	0
$513$	3.95824e13	0.0491878
$514$	0	0
$515$	3.25502e14	0.395926
$516$	0	0
$517$	1.01703e15	1.21097
$518$	0	0
$519$	−1.78897e13	−0.0208537
$520$	0	0
$521$	−3.19507e14	−0.364647	−0.182324	−	0.983239i	$-0.558362\pi$
$521$	−3.19507e14	−0.364647	−0.182324	+	0.983239i	$0.558362\pi$
$522$	0	0
$523$	−4.76795e14	−0.532810	−0.266405	−	0.963861i	$-0.585836\pi$
$523$	−4.76795e14	−0.532810	−0.266405	+	0.963861i	$0.585836\pi$
$524$	0	0
$525$	2.73580e13	0.0299370
$526$	0	0
$527$	1.13410e15	1.21532
$528$	0	0
$529$	−5.56230e14	−0.583778
$530$	0	0
$531$	5.12192e14	0.526518
$532$	0	0
$533$	4.52508e14	0.455645
$534$	0	0
$535$	1.96568e15	1.93895
$536$	0	0
$537$	−2.38356e13	−0.0230339
$538$	0	0
$539$	−1.74652e15	−1.65362
$540$	0	0
$541$	−7.96966e14	−0.739359	−0.369679	−	0.929159i	$-0.620532\pi$
$541$	−7.96966e14	−0.739359	−0.369679	+	0.929159i	$0.620532\pi$
$542$	0	0
$543$	−6.54799e12	−0.00595263
$544$	0	0
$545$	2.84967e15	2.53871
$546$	0	0
$547$	−3.18708e14	−0.278268	−0.139134	−	0.990274i	$-0.544432\pi$
$547$	−3.18708e14	−0.278268	−0.139134	+	0.990274i	$0.544432\pi$
$548$	0	0
$549$	−1.24615e15	−1.06640
$550$	0	0
$551$	1.59099e15	1.33455
$552$	0	0
$553$	−1.44374e15	−1.18714
$554$	0	0
$555$	−3.52561e13	−0.0284199
$556$	0	0
$557$	1.81222e15	1.43221	0.716107	−	0.697990i	$-0.245922\pi$
$557$	1.81222e15	1.43221	0.716107	+	0.697990i	$0.245922\pi$
$558$	0	0
$559$	9.95234e13	0.0771188
$560$	0	0
$561$	−1.32810e13	−0.0100910
$562$	0	0
$563$	1.10209e15	0.821147	0.410573	−	0.911828i	$-0.365329\pi$
$563$	1.10209e15	0.821147	0.410573	+	0.911828i	$0.365329\pi$
$564$	0	0
$565$	8.34731e14	0.609930
$566$	0	0
$567$	−2.42848e15	−1.74032
$568$	0	0
$569$	5.11497e14	0.359522	0.179761	−	0.983710i	$-0.442467\pi$
$569$	5.11497e14	0.359522	0.179761	+	0.983710i	$0.442467\pi$
$570$	0	0
$571$	−2.06109e15	−1.42101	−0.710505	−	0.703692i	$-0.751534\pi$
$571$	−2.06109e15	−1.42101	−0.710505	+	0.703692i	$0.751534\pi$
$572$	0	0
$573$	−1.67999e13	−0.0113621
$574$	0	0
$575$	−8.67418e14	−0.575513
$576$	0	0
$577$	−1.20295e15	−0.783036	−0.391518	−	0.920171i	$-0.628050\pi$
$577$	−1.20295e15	−0.783036	−0.391518	+	0.920171i	$0.628050\pi$
$578$	0	0
$579$	−1.71286e12	−0.00109393
$580$	0	0
$581$	1.24511e14	0.0780259
$582$	0	0
$583$	−9.04640e14	−0.556287
$584$	0	0
$585$	−2.05935e15	−1.24272
$586$	0	0
$587$	2.33803e15	1.38465	0.692325	−	0.721586i	$-0.256587\pi$
$587$	2.33803e15	1.38465	0.692325	+	0.721586i	$0.256587\pi$
$588$	0	0
$589$	−4.14070e15	−2.40680
$590$	0	0
$591$	−1.45872e13	−0.00832223
$592$	0	0
$593$	1.65885e15	0.928981	0.464490	−	0.885578i	$-0.346238\pi$
$593$	1.65885e15	0.928981	0.464490	+	0.885578i	$0.346238\pi$
$594$	0	0
$595$	−2.81111e15	−1.54538
$596$	0	0
$597$	−4.41477e13	−0.0238259
$598$	0	0
$599$	1.01907e15	0.539951	0.269976	−	0.962867i	$-0.412984\pi$
$599$	1.01907e15	0.539951	0.269976	+	0.962867i	$0.412984\pi$
$600$	0	0
$601$	−9.47247e14	−0.492781	−0.246390	−	0.969171i	$-0.579245\pi$
$601$	−9.47247e14	−0.492781	−0.246390	+	0.969171i	$0.579245\pi$
$602$	0	0
$603$	2.53020e15	1.29244
$604$	0	0
$605$	9.32382e14	0.467670
$606$	0	0
$607$	2.47092e15	1.21708	0.608542	−	0.793521i	$-0.291755\pi$
$607$	2.47092e15	1.21708	0.608542	+	0.793521i	$0.291755\pi$
$608$	0	0
$609$	7.25005e13	0.0350708
$610$	0	0
$611$	2.83565e15	1.34718
$612$	0	0
$613$	−3.57299e15	−1.66725	−0.833623	−	0.552334i	$-0.813737\pi$
$613$	−3.57299e15	−1.66725	−0.833623	+	0.552334i	$0.813737\pi$
$614$	0	0
$615$	−2.91437e13	−0.0133577
$616$	0	0
$617$	−4.07316e15	−1.83385	−0.916923	−	0.399063i	$-0.869335\pi$
$617$	−4.07316e15	−1.83385	−0.916923	+	0.399063i	$0.869335\pi$
$618$	0	0
$619$	3.18768e15	1.40986	0.704931	−	0.709276i	$-0.250978\pi$
$619$	3.18768e15	1.40986	0.704931	+	0.709276i	$0.250978\pi$
$620$	0	0
$621$	−5.71899e13	−0.0248494
$622$	0	0
$623$	−6.31969e15	−2.69781
$624$	0	0
$625$	−2.61376e15	−1.09629
$626$	0	0
$627$	4.84901e13	0.0199839
$628$	0	0
$629$	1.70799e15	0.691683
$630$	0	0
$631$	−2.44443e15	−0.972784	−0.486392	−	0.873741i	$-0.661687\pi$
$631$	−2.44443e15	−0.972784	−0.486392	+	0.873741i	$0.661687\pi$
$632$	0	0
$633$	1.17895e13	0.00461078
$634$	0	0
$635$	−1.99170e15	−0.765541
$636$	0	0
$637$	−4.86962e15	−1.83962
$638$	0	0
$639$	2.50003e15	0.928306
$640$	0	0
$641$	1.14910e15	0.419411	0.209705	−	0.977765i	$-0.432749\pi$
$641$	1.14910e15	0.419411	0.209705	+	0.977765i	$0.432749\pi$
$642$	0	0
$643$	−1.51489e15	−0.543528	−0.271764	−	0.962364i	$-0.587607\pi$
$643$	−1.51489e15	−0.543528	−0.271764	+	0.962364i	$0.587607\pi$
$644$	0	0
$645$	−6.40979e12	−0.00226082
$646$	0	0
$647$	−3.71991e15	−1.28991	−0.644954	−	0.764221i	$-0.723124\pi$
$647$	−3.71991e15	−1.28991	−0.644954	+	0.764221i	$0.723124\pi$
$648$	0	0
$649$	1.25514e15	0.427906
$650$	0	0
$651$	−1.88689e14	−0.0632487
$652$	0	0
$653$	3.19693e15	1.05368	0.526842	−	0.849963i	$-0.323376\pi$
$653$	3.19693e15	1.05368	0.526842	+	0.849963i	$0.323376\pi$
$654$	0	0
$655$	6.78650e15	2.19947
$656$	0	0
$657$	−8.97810e14	−0.286137
$658$	0	0
$659$	−1.68897e15	−0.529361	−0.264681	−	0.964336i	$-0.585267\pi$
$659$	−1.68897e15	−0.529361	−0.264681	+	0.964336i	$0.585267\pi$
$660$	0	0
$661$	2.27989e15	0.702758	0.351379	−	0.936233i	$-0.385713\pi$
$661$	2.27989e15	0.702758	0.351379	+	0.936233i	$0.385713\pi$
$662$	0	0
$663$	−3.70297e13	−0.0112260
$664$	0	0
$665$	1.02637e16	3.06043
$666$	0	0
$667$	−2.29871e15	−0.674205
$668$	0	0
$669$	5.03095e13	0.0145146
$670$	0	0
$671$	−3.05372e15	−0.866674
$672$	0	0
$673$	−5.24583e15	−1.46464	−0.732321	−	0.680959i	$-0.761563\pi$
$673$	−5.24583e15	−1.46464	−0.732321	+	0.680959i	$0.761563\pi$
$674$	0	0
$675$	1.25088e14	0.0343594
$676$	0	0
$677$	1.96641e15	0.531419	0.265709	−	0.964053i	$-0.414394\pi$
$677$	1.96641e15	0.531419	0.265709	+	0.964053i	$0.414394\pi$
$678$	0	0
$679$	9.38541e15	2.49557
$680$	0	0
$681$	9.61321e13	0.0251512
$682$	0	0
$683$	−3.08409e15	−0.793988	−0.396994	−	0.917821i	$-0.629947\pi$
$683$	−3.08409e15	−0.793988	−0.396994	+	0.917821i	$0.629947\pi$
$684$	0	0
$685$	6.64294e15	1.68291
$686$	0	0
$687$	2.29442e12	0.000572020 0
$688$	0	0
$689$	−2.52230e15	−0.618858
$690$	0	0
$691$	3.36803e15	0.813292	0.406646	−	0.913586i	$-0.366698\pi$
$691$	3.36803e15	0.813292	0.406646	+	0.913586i	$0.366698\pi$
$692$	0	0
$693$	−5.95329e15	−1.41490
$694$	0	0
$695$	5.52137e15	1.29161
$696$	0	0
$697$	1.41188e15	0.325100
$698$	0	0
$699$	7.87084e13	0.0178401
$700$	0	0
$701$	−3.42925e15	−0.765157	−0.382579	−	0.923923i	$-0.624964\pi$
$701$	−3.42925e15	−0.765157	−0.382579	+	0.923923i	$0.624964\pi$
$702$	0	0
$703$	−6.23605e15	−1.36979
$704$	0	0
$705$	−1.82630e14	−0.0394940
$706$	0	0
$707$	2.64579e13	0.00563311
$708$	0	0
$709$	−1.25116e14	−0.0262275	−0.0131138	−	0.999914i	$-0.504174\pi$
$709$	−1.25116e14	−0.0262275	−0.0131138	+	0.999914i	$0.504174\pi$
$710$	0	0
$711$	−3.29998e15	−0.681128
$712$	0	0
$713$	5.98261e15	1.21590
$714$	0	0
$715$	−5.04650e15	−1.00997
$716$	0	0
$717$	−1.07436e14	−0.0211736
$718$	0	0
$719$	6.08027e15	1.18009	0.590044	−	0.807371i	$-0.299111\pi$
$719$	6.08027e15	1.18009	0.590044	+	0.807371i	$0.299111\pi$
$720$	0	0
$721$	2.62363e15	0.501486
$722$	0	0
$723$	1.43672e14	0.0270465
$724$	0	0
$725$	5.02784e15	0.932229
$726$	0	0
$727$	−2.84854e15	−0.520216	−0.260108	−	0.965580i	$-0.583758\pi$
$727$	−2.84854e15	−0.520216	−0.260108	+	0.965580i	$0.583758\pi$
$728$	0	0
$729$	−5.54669e15	−0.997775
$730$	0	0
$731$	3.10525e14	0.0550238
$732$	0	0
$733$	9.54311e15	1.66578	0.832890	−	0.553438i	$-0.186684\pi$
$733$	9.54311e15	1.66578	0.832890	+	0.553438i	$0.186684\pi$
$734$	0	0
$735$	3.13627e14	0.0539303
$736$	0	0
$737$	6.20034e15	1.05037
$738$	0	0
$739$	−5.08100e15	−0.848018	−0.424009	−	0.905658i	$-0.639378\pi$
$739$	−5.08100e15	−0.848018	−0.424009	+	0.905658i	$0.639378\pi$
$740$	0	0
$741$	1.35199e14	0.0222317
$742$	0	0
$743$	−1.30834e15	−0.211973	−0.105987	−	0.994368i	$-0.533800\pi$
$743$	−1.30834e15	−0.211973	−0.105987	+	0.994368i	$0.533800\pi$
$744$	0	0
$745$	7.21086e15	1.15114
$746$	0	0
$747$	2.84596e14	0.0447678
$748$	0	0
$749$	1.58439e16	2.45591
$750$	0	0
$751$	−1.02246e16	−1.56181	−0.780903	−	0.624652i	$-0.785241\pi$
$751$	−1.02246e16	−1.56181	−0.780903	+	0.624652i	$0.785241\pi$
$752$	0	0
$753$	3.48060e13	0.00523940
$754$	0	0
$755$	7.48677e15	1.11067
$756$	0	0
$757$	−1.16093e16	−1.69737	−0.848686	−	0.528897i	$-0.822606\pi$
$757$	−1.16093e16	−1.69737	−0.848686	+	0.528897i	$0.822606\pi$
$758$	0	0
$759$	−7.00599e13	−0.0100958
$760$	0	0
$761$	4.02600e14	0.0571818	0.0285909	−	0.999591i	$-0.490898\pi$
$761$	4.02600e14	0.0571818	0.0285909	+	0.999591i	$0.490898\pi$
$762$	0	0
$763$	2.29691e16	3.21557
$764$	0	0
$765$	−6.42540e15	−0.886670
$766$	0	0
$767$	3.49957e15	0.476036
$768$	0	0
$769$	3.45914e15	0.463845	0.231923	−	0.972734i	$-0.425498\pi$
$769$	3.45914e15	0.463845	0.231923	+	0.972734i	$0.425498\pi$
$770$	0	0
$771$	3.65636e13	0.00483337
$772$	0	0
$773$	−6.12705e15	−0.798480	−0.399240	−	0.916847i	$-0.630726\pi$
$773$	−6.12705e15	−0.798480	−0.399240	+	0.916847i	$0.630726\pi$
$774$	0	0
$775$	−1.30854e16	−1.68124
$776$	0	0
$777$	−2.84173e14	−0.0359971
$778$	0	0
$779$	−5.15490e15	−0.643820
$780$	0	0
$781$	6.12642e15	0.754442
$782$	0	0
$783$	3.31492e14	0.0402516
$784$	0	0
$785$	−9.94957e15	−1.19130
$786$	0	0
$787$	1.63599e14	0.0193160	0.00965802	−	0.999953i	$-0.496926\pi$
$787$	1.63599e14	0.0193160	0.00965802	+	0.999953i	$0.496926\pi$
$788$	0	0
$789$	2.40043e14	0.0279489
$790$	0	0
$791$	6.72815e15	0.772547
$792$	0	0
$793$	−8.51433e15	−0.964157
$794$	0	0
$795$	1.62448e14	0.0181425
$796$	0	0
$797$	3.91386e14	0.0431107	0.0215553	−	0.999768i	$-0.493138\pi$
$797$	3.91386e14	0.0431107	0.0215553	+	0.999768i	$0.493138\pi$
$798$	0	0
$799$	8.84757e15	0.961206
$800$	0	0
$801$	−1.44450e16	−1.54788
$802$	0	0
$803$	−2.20011e15	−0.232546
$804$	0	0
$805$	−1.48292e16	−1.54611
$806$	0	0
$807$	−2.33024e14	−0.0239660
$808$	0	0
$809$	1.10789e16	1.12404	0.562020	−	0.827124i	$-0.310025\pi$
$809$	1.10789e16	1.12404	0.562020	+	0.827124i	$0.310025\pi$
$810$	0	0
$811$	−7.03872e15	−0.704497	−0.352248	−	0.935907i	$-0.614583\pi$
$811$	−7.03872e15	−0.704497	−0.352248	+	0.935907i	$0.614583\pi$
$812$	0	0
$813$	3.06473e14	0.0302617
$814$	0	0
$815$	1.00807e14	0.00982024
$816$	0	0
$817$	−1.13376e15	−0.108968
$818$	0	0
$819$	−1.65989e16	−1.57404
$820$	0	0
$821$	−5.60310e15	−0.524253	−0.262127	−	0.965033i	$-0.584424\pi$
$821$	−5.60310e15	−0.524253	−0.262127	+	0.965033i	$0.584424\pi$
$822$	0	0
$823$	−1.66745e16	−1.53940	−0.769702	−	0.638403i	$-0.779595\pi$
$823$	−1.66745e16	−1.53940	−0.769702	+	0.638403i	$0.779595\pi$
$824$	0	0
$825$	1.53238e14	0.0139595
$826$	0	0
$827$	−1.13939e15	−0.102422	−0.0512108	−	0.998688i	$-0.516308\pi$
$827$	−1.13939e15	−0.102422	−0.0512108	+	0.998688i	$0.516308\pi$
$828$	0	0
$829$	−1.44203e16	−1.27916	−0.639579	−	0.768726i	$-0.720891\pi$
$829$	−1.44203e16	−1.27916	−0.639579	+	0.768726i	$0.720891\pi$
$830$	0	0
$831$	1.31092e14	0.0114755
$832$	0	0
$833$	−1.51938e16	−1.31256
$834$	0	0
$835$	5.93446e15	0.505948
$836$	0	0
$837$	−8.62738e14	−0.0725920
$838$	0	0
$839$	1.33530e16	1.10889	0.554444	−	0.832221i	$-0.312931\pi$
$839$	1.33530e16	1.10889	0.554444	+	0.832221i	$0.312931\pi$
$840$	0	0
$841$	1.12358e15	0.0920925
$842$	0	0
$843$	−2.51952e14	−0.0203829
$844$	0	0
$845$	3.15524e15	0.251954
$846$	0	0
$847$	7.51524e15	0.592358
$848$	0	0
$849$	1.75799e14	0.0136781
$850$	0	0
$851$	9.01003e15	0.692012
$852$	0	0
$853$	9.20764e15	0.698118	0.349059	−	0.937101i	$-0.386501\pi$
$853$	9.20764e15	0.698118	0.349059	+	0.937101i	$0.386501\pi$
$854$	0	0
$855$	2.34597e16	1.75594
$856$	0	0
$857$	5.20571e14	0.0384668	0.0192334	−	0.999815i	$-0.493877\pi$
$857$	5.20571e14	0.0384668	0.0192334	+	0.999815i	$0.493877\pi$
$858$	0	0
$859$	−1.86500e16	−1.36056	−0.680280	−	0.732953i	$-0.738142\pi$
$859$	−1.86500e16	−1.36056	−0.680280	+	0.732953i	$0.738142\pi$
$860$	0	0
$861$	−2.34906e14	−0.0169191
$862$	0	0
$863$	−7.13279e14	−0.0507224	−0.0253612	−	0.999678i	$-0.508074\pi$
$863$	−7.13279e14	−0.0507224	−0.0253612	+	0.999678i	$0.508074\pi$
$864$	0	0
$865$	−2.12098e16	−1.48918
$866$	0	0
$867$	1.62312e14	0.0112524
$868$	0	0
$869$	−8.08672e15	−0.553559
$870$	0	0
$871$	1.72877e16	1.16852
$872$	0	0
$873$	2.14523e16	1.43185
$874$	0	0
$875$	−3.92479e15	−0.258685
$876$	0	0
$877$	1.70756e16	1.11142	0.555709	−	0.831377i	$-0.312447\pi$
$877$	1.70756e16	1.11142	0.555709	+	0.831377i	$0.312447\pi$
$878$	0	0
$879$	2.38970e14	0.0153605
$880$	0	0
$881$	1.65770e16	1.05229	0.526147	−	0.850393i	$-0.323636\pi$
$881$	1.65770e16	1.05229	0.526147	+	0.850393i	$0.323636\pi$
$882$	0	0
$883$	1.71471e16	1.07500	0.537499	−	0.843264i	$-0.319369\pi$
$883$	1.71471e16	1.07500	0.537499	+	0.843264i	$0.319369\pi$
$884$	0	0
$885$	−2.25389e14	−0.0139555
$886$	0	0
$887$	−1.51925e16	−0.929069	−0.464535	−	0.885555i	$-0.653778\pi$
$887$	−1.51925e16	−0.929069	−0.464535	+	0.885555i	$0.653778\pi$
$888$	0	0
$889$	−1.60536e16	−0.969646
$890$	0	0
$891$	−1.36025e16	−0.811503
$892$	0	0
$893$	−3.23033e16	−1.90355
$894$	0	0
$895$	−2.82591e16	−1.64487
$896$	0	0
$897$	−1.95340e14	−0.0112314
$898$	0	0
$899$	−3.46772e16	−1.96954
$900$	0	0
$901$	−7.86987e15	−0.441551
$902$	0	0
$903$	−5.16646e13	−0.00286359
$904$	0	0
$905$	−7.76318e15	−0.425082
$906$	0	0
$907$	3.52570e16	1.90724	0.953621	−	0.301011i	$-0.0973241\pi$
$907$	3.52570e16	1.90724	0.953621	+	0.301011i	$0.0973241\pi$
$908$	0	0
$909$	6.04751e13	0.00323203
$910$	0	0
$911$	−1.80307e16	−0.952052	−0.476026	−	0.879431i	$-0.657923\pi$
$911$	−1.80307e16	−0.952052	−0.476026	+	0.879431i	$0.657923\pi$
$912$	0	0
$913$	6.97412e14	0.0363832
$914$	0	0
$915$	5.48364e14	0.0282653
$916$	0	0
$917$	5.47009e16	2.78589
$918$	0	0
$919$	−1.18274e16	−0.595189	−0.297595	−	0.954692i	$-0.596184\pi$
$919$	−1.18274e16	−0.595189	−0.297595	+	0.954692i	$0.596184\pi$
$920$	0	0
$921$	−3.31613e13	−0.00164893
$922$	0	0
$923$	1.70816e16	0.839302
$924$	0	0
$925$	−1.97071e16	−0.956851
$926$	0	0
$927$	5.99686e15	0.287730
$928$	0	0
$929$	1.51253e16	0.717163	0.358582	−	0.933498i	$-0.383261\pi$
$929$	1.51253e16	0.717163	0.358582	+	0.933498i	$0.383261\pi$
$930$	0	0
$931$	5.54739e16	2.59936
$932$	0	0
$933$	−1.67475e14	−0.00775532
$934$	0	0
$935$	−1.57457e16	−0.720604
$936$	0	0
$937$	−3.45437e15	−0.156243	−0.0781217	−	0.996944i	$-0.524892\pi$
$937$	−3.45437e15	−0.156243	−0.0781217	+	0.996944i	$0.524892\pi$
$938$	0	0
$939$	−6.14537e14	−0.0274718
$940$	0	0
$941$	2.80566e16	1.23963	0.619816	−	0.784747i	$-0.287207\pi$
$941$	2.80566e16	1.23963	0.619816	+	0.784747i	$0.287207\pi$
$942$	0	0
$943$	7.44796e15	0.325254
$944$	0	0
$945$	2.13849e15	0.0923065
$946$	0	0
$947$	−3.04374e16	−1.29862	−0.649310	−	0.760524i	$-0.724942\pi$
$947$	−3.04374e16	−1.29862	−0.649310	+	0.760524i	$0.724942\pi$
$948$	0	0
$949$	−6.13431e15	−0.258703
$950$	0	0
$951$	−7.01248e14	−0.0292334
$952$	0	0
$953$	1.86377e16	0.768036	0.384018	−	0.923326i	$-0.374540\pi$
$953$	1.86377e16	0.768036	0.384018	+	0.923326i	$0.374540\pi$
$954$	0	0
$955$	−1.99177e16	−0.811374
$956$	0	0
$957$	4.06091e14	0.0163534
$958$	0	0
$959$	5.35438e16	2.13161
$960$	0	0
$961$	6.48421e16	2.55199
$962$	0	0
$963$	3.62145e16	1.40909
$964$	0	0
$965$	−2.03073e15	−0.0781182
$966$	0	0
$967$	−1.77292e16	−0.674286	−0.337143	−	0.941454i	$-0.609461\pi$
$967$	−1.77292e16	−0.674286	−0.337143	+	0.941454i	$0.609461\pi$
$968$	0	0
$969$	4.21837e14	0.0158622
$970$	0	0
$971$	2.64145e16	0.982056	0.491028	−	0.871144i	$-0.336621\pi$
$971$	2.64145e16	0.982056	0.491028	+	0.871144i	$0.336621\pi$
$972$	0	0
$973$	4.45037e16	1.63597
$974$	0	0
$975$	4.27256e14	0.0155297
$976$	0	0
$977$	1.41752e16	0.509458	0.254729	−	0.967013i	$-0.418014\pi$
$977$	1.41752e16	0.509458	0.254729	+	0.967013i	$0.418014\pi$
$978$	0	0
$979$	−3.53979e16	−1.25798
$980$	0	0
$981$	5.25007e16	1.84495
$982$	0	0
$983$	−3.39879e16	−1.18108	−0.590541	−	0.807008i	$-0.701086\pi$
$983$	−3.39879e16	−1.18108	−0.590541	+	0.807008i	$0.701086\pi$
$984$	0	0
$985$	−1.72943e16	−0.594296
$986$	0	0
$987$	−1.47204e15	−0.0500238
$988$	0	0
$989$	1.63808e15	0.0550499
$990$	0	0
$991$	−4.42289e16	−1.46994	−0.734972	−	0.678097i	$-0.762805\pi$
$991$	−4.42289e16	−1.46994	−0.734972	+	0.678097i	$0.762805\pi$
$992$	0	0
$993$	−7.77670e14	−0.0255608
$994$	0	0
$995$	−5.23407e16	−1.70143
$996$	0	0
$997$	−2.00620e16	−0.644986	−0.322493	−	0.946572i	$-0.604521\pi$
$997$	−2.00620e16	−0.644986	−0.322493	+	0.946572i	$0.604521\pi$
$998$	0	0
$999$	−1.29932e15	−0.0413147

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.12.a.m.1.2		3
4.3	odd	2		64.12.a.l.1.2		3
8.3	odd	2		32.12.a.e.1.2	yes	3
8.5	even	2		32.12.a.d.1.2	✓	3
16.3	odd	4		256.12.b.m.129.4		6
16.5	even	4		256.12.b.l.129.4		6
16.11	odd	4		256.12.b.m.129.3		6
16.13	even	4		256.12.b.l.129.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.12.a.d.1.2	✓	3	8.5	even	2
32.12.a.e.1.2	yes	3	8.3	odd	2
64.12.a.l.1.2		3	4.3	odd	2
64.12.a.m.1.2		3	1.1	even	1	trivial
256.12.b.l.129.3		6	16.13	even	4
256.12.b.l.129.4		6	16.5	even	4
256.12.b.m.129.3		6	16.11	odd	4
256.12.b.m.129.4		6	16.3	odd	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:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

\(f(q)\)	\(=\)	\(q-8.10720 q^{3} -9611.74 q^{5} -77473.2 q^{7} -177081. q^{9} -433944. q^{11} -1.20991e6 q^{13} +77924.3 q^{15} -3.77507e6 q^{17} +1.37832e7 q^{19} +628090. q^{21} -1.99143e7 q^{23} +4.35575e7 q^{25} +2.87180e6 q^{27} +1.15430e8 q^{29} -3.00417e8 q^{31} +3.51807e6 q^{33} +7.44652e8 q^{35} -4.52440e8 q^{37} +9.80901e6 q^{39} -3.74000e8 q^{41} -8.22566e7 q^{43} +1.70206e9 q^{45} -2.34368e9 q^{47} +4.02476e9 q^{49} +3.06052e7 q^{51} +2.08470e9 q^{53} +4.17096e9 q^{55} -1.11743e8 q^{57} -2.89241e9 q^{59} +7.03714e9 q^{61} +1.37190e10 q^{63} +1.16294e10 q^{65} -1.42883e10 q^{67} +1.61449e8 q^{69} -1.41180e10 q^{71} +5.07004e9 q^{73} -3.53129e8 q^{75} +3.36190e10 q^{77} +1.86354e10 q^{79} +3.13461e10 q^{81} -1.60715e9 q^{83} +3.62850e10 q^{85} -9.35814e8 q^{87} +8.15726e10 q^{89} +9.37358e10 q^{91} +2.43554e9 q^{93} -1.32480e11 q^{95} -1.21144e11 q^{97} +7.68433e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 440 q^{3} - 770 q^{5} + 6032 q^{7} + 211663 q^{9} + 236136 q^{11} - 1435722 q^{13} - 222160 q^{15} - 756186 q^{17} + 23267992 q^{19} + 20368000 q^{21} - 42366288 q^{23} - 6913515 q^{25} + 298561328 q^{27}+ \cdots + 276379454984 q^{99}+O(q^{100}) \) 3 * q + 440 * q^3 - 770 * q^5 + 6032 * q^7 + 211663 * q^9 + 236136 * q^11 - 1435722 * q^13 - 222160 * q^15 - 756186 * q^17 + 23267992 * q^19 + 20368000 * q^21 - 42366288 * q^23 - 6913515 * q^25 + 298561328 * q^27 - 42667514 * q^29 - 555856064 * q^31 + 308282944 * q^33 + 1110160800 * q^35 + 462651966 * q^37 - 2309231888 * q^39 + 357860750 * q^41 + 1400057768 * q^43 + 2398467190 * q^45 - 1620045984 * q^47 + 3558319419 * q^49 - 1197100944 * q^51 + 5087465774 * q^53 + 6584414480 * q^55 + 15809413568 * q^57 - 13552245240 * q^59 + 18288485958 * q^61 + 30411798992 * q^63 + 18683192540 * q^65 - 17471404360 * q^67 + 37933678464 * q^69 + 23838346512 * q^71 + 35602893150 * q^73 - 31830320120 * q^75 + 61844462464 * q^77 - 4634961248 * q^79 + 104321985931 * q^81 + 49374840984 * q^83 + 56389897020 * q^85 - 146872315024 * q^87 + 58640505454 * q^89 + 80670986272 * q^91 + 10026258944 * q^93 - 139519303440 * q^95 - 66696660810 * q^97 + 276379454984 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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