LMFDB - Embedded newform 768.6.a.ba.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,6,Mod(1,768)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("768.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	768.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-45,0,-50,0,98,0,405,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$123.174773616$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 65x^{3} + 85x^{2} + 856x - 1692 $ x^5 - x^4 - 65x^3 + 85x^2 + 856*x - 1692
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.60811$ of defining polynomial
Character	$\chi$	$=$	768.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -66.8717 q^{5} +230.960 q^{7} +81.0000 q^{9} -360.698 q^{11} +674.528 q^{13} +601.845 q^{15} -765.210 q^{17} -2277.37 q^{19} -2078.64 q^{21} +465.702 q^{23} +1346.82 q^{25} -729.000 q^{27} +1430.14 q^{29} +3327.79 q^{31} +3246.28 q^{33} -15444.7 q^{35} -5736.07 q^{37} -6070.75 q^{39} +12666.7 q^{41} +2017.82 q^{43} -5416.61 q^{45} -1605.83 q^{47} +36535.6 q^{49} +6886.89 q^{51} +11998.3 q^{53} +24120.5 q^{55} +20496.4 q^{57} -24401.4 q^{59} +19670.5 q^{61} +18707.8 q^{63} -45106.8 q^{65} +50239.4 q^{67} -4191.32 q^{69} +16572.5 q^{71} -14263.2 q^{73} -12121.4 q^{75} -83306.8 q^{77} -47850.0 q^{79} +6561.00 q^{81} +60126.7 q^{83} +51170.9 q^{85} -12871.3 q^{87} -139147. q^{89} +155789. q^{91} -29950.1 q^{93} +152292. q^{95} +85226.4 q^{97} -29216.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 45 q^{3} - 50 q^{5} + 98 q^{7} + 405 q^{9} - 676 q^{13} + 450 q^{15} - 202 q^{17} - 716 q^{19} - 882 q^{21} - 836 q^{23} + 4683 q^{25} - 3645 q^{27} - 5046 q^{29} + 9346 q^{31} - 436 q^{35} - 10952 q^{37}+ \cdots + 29686 q^{97}+O(q^{100}) $ 5 * q - 45 * q^3 - 50 * q^5 + 98 * q^7 + 405 * q^9 - 676 * q^13 + 450 * q^15 - 202 * q^17 - 716 * q^19 - 882 * q^21 - 836 * q^23 + 4683 * q^25 - 3645 * q^27 - 5046 * q^29 + 9346 * q^31 - 436 * q^35 - 10952 * q^37 + 6084 * q^39 + 1238 * q^41 - 1676 * q^43 - 4050 * q^45 + 32628 * q^47 + 4689 * q^49 + 1818 * q^51 + 8490 * q^53 + 51712 * q^55 + 6444 * q^57 + 15884 * q^59 + 13824 * q^61 + 7938 * q^63 + 25768 * q^65 - 26652 * q^67 + 7524 * q^69 + 136116 * q^71 + 13694 * q^73 - 42147 * q^75 - 89856 * q^77 + 126154 * q^79 + 32805 * q^81 + 64720 * q^83 - 164636 * q^85 + 45414 * q^87 - 87790 * q^89 + 179992 * q^91 - 84114 * q^93 + 121592 * q^95 + 29686 * q^97

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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−66.8717	−1.19624	−0.598119	−	0.801408i	$-0.704085\pi$
$5$	−66.8717	−1.19624	−0.598119	+	0.801408i	$0.704085\pi$
$6$	0	0
$7$	230.960	1.78152	0.890762	−	0.454469i	$-0.150171\pi$
$7$	230.960	1.78152	0.890762	+	0.454469i	$0.150171\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−360.698	−0.898797	−0.449399	−	0.893331i	$-0.648362\pi$
$11$	−360.698	−0.898797	−0.449399	+	0.893331i	$0.648362\pi$
$12$	0	0
$13$	674.528	1.10698	0.553492	−	0.832854i	$-0.313295\pi$
$13$	674.528	1.10698	0.553492	+	0.832854i	$0.313295\pi$
$14$	0	0
$15$	601.845	0.690648
$16$	0	0
$17$	−765.210	−0.642182	−0.321091	−	0.947048i	$-0.604050\pi$
$17$	−765.210	−0.642182	−0.321091	+	0.947048i	$0.604050\pi$
$18$	0	0
$19$	−2277.37	−1.44727	−0.723636	−	0.690182i	$-0.757530\pi$
$19$	−2277.37	−1.44727	−0.723636	+	0.690182i	$0.757530\pi$
$20$	0	0
$21$	−2078.64	−1.02856
$22$	0	0
$23$	465.702	0.183565	0.0917823	−	0.995779i	$-0.470744\pi$
$23$	465.702	0.183565	0.0917823	+	0.995779i	$0.470744\pi$
$24$	0	0
$25$	1346.82	0.430984
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	1430.14	0.315779	0.157890	−	0.987457i	$-0.449531\pi$
$29$	1430.14	0.315779	0.157890	+	0.987457i	$0.449531\pi$
$30$	0	0
$31$	3327.79	0.621945	0.310972	−	0.950419i	$-0.399345\pi$
$31$	3327.79	0.621945	0.310972	+	0.950419i	$0.399345\pi$
$32$	0	0
$33$	3246.28	0.518921
$34$	0	0
$35$	−15444.7	−2.13113
$36$	0	0
$37$	−5736.07	−0.688827	−0.344414	−	0.938818i	$-0.611922\pi$
$37$	−5736.07	−0.688827	−0.344414	+	0.938818i	$0.611922\pi$
$38$	0	0
$39$	−6070.75	−0.639118
$40$	0	0
$41$	12666.7	1.17681	0.588403	−	0.808568i	$-0.299757\pi$
$41$	12666.7	1.17681	0.588403	+	0.808568i	$0.299757\pi$
$42$	0	0
$43$	2017.82	0.166422	0.0832112	−	0.996532i	$-0.473482\pi$
$43$	2017.82	0.166422	0.0832112	+	0.996532i	$0.473482\pi$
$44$	0	0
$45$	−5416.61	−0.398746
$46$	0	0
$47$	−1605.83	−0.106036	−0.0530182	−	0.998594i	$-0.516884\pi$
$47$	−1605.83	−0.106036	−0.0530182	+	0.998594i	$0.516884\pi$
$48$	0	0
$49$	36535.6	2.17383
$50$	0	0
$51$	6886.89	0.370764
$52$	0	0
$53$	11998.3	0.586718	0.293359	−	0.956002i	$-0.405227\pi$
$53$	11998.3	0.586718	0.293359	+	0.956002i	$0.405227\pi$
$54$	0	0
$55$	24120.5	1.07517
$56$	0	0
$57$	20496.4	0.835583
$58$	0	0
$59$	−24401.4	−0.912608	−0.456304	−	0.889824i	$-0.650827\pi$
$59$	−24401.4	−0.912608	−0.456304	+	0.889824i	$0.650827\pi$
$60$	0	0
$61$	19670.5	0.676848	0.338424	−	0.940994i	$-0.390106\pi$
$61$	19670.5	0.676848	0.338424	+	0.940994i	$0.390106\pi$
$62$	0	0
$63$	18707.8	0.593842
$64$	0	0
$65$	−45106.8	−1.32422
$66$	0	0
$67$	50239.4	1.36728	0.683640	−	0.729820i	$-0.260396\pi$
$67$	50239.4	1.36728	0.683640	+	0.729820i	$0.260396\pi$
$68$	0	0
$69$	−4191.32	−0.105981
$70$	0	0
$71$	16572.5	0.390160	0.195080	−	0.980787i	$-0.437503\pi$
$71$	16572.5	0.390160	0.195080	+	0.980787i	$0.437503\pi$
$72$	0	0
$73$	−14263.2	−0.313264	−0.156632	−	0.987657i	$-0.550064\pi$
$73$	−14263.2	−0.313264	−0.156632	+	0.987657i	$0.550064\pi$
$74$	0	0
$75$	−12121.4	−0.248828
$76$	0	0
$77$	−83306.8	−1.60123
$78$	0	0
$79$	−47850.0	−0.862609	−0.431305	−	0.902206i	$-0.641947\pi$
$79$	−47850.0	−0.862609	−0.431305	+	0.902206i	$0.641947\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	60126.7	0.958014	0.479007	−	0.877811i	$-0.340997\pi$
$83$	60126.7	0.958014	0.479007	+	0.877811i	$0.340997\pi$
$84$	0	0
$85$	51170.9	0.768203
$86$	0	0
$87$	−12871.3	−0.182315
$88$	0	0
$89$	−139147.	−1.86208	−0.931041	−	0.364914i	$-0.881098\pi$
$89$	−139147.	−1.86208	−0.931041	+	0.364914i	$0.881098\pi$
$90$	0	0
$91$	155789.	1.97212
$92$	0	0
$93$	−29950.1	−0.359080
$94$	0	0
$95$	152292.	1.73128
$96$	0	0
$97$	85226.4	0.919697	0.459849	−	0.887997i	$-0.347904\pi$
$97$	85226.4	0.919697	0.459849	+	0.887997i	$0.347904\pi$
$98$	0	0
$99$	−29216.5	−0.299599
$100$	0	0
$101$	−169928.	−1.65753	−0.828765	−	0.559597i	$-0.810956\pi$
$101$	−169928.	−1.65753	−0.828765	+	0.559597i	$0.810956\pi$
$102$	0	0
$103$	165056.	1.53299	0.766494	−	0.642252i	$-0.222000\pi$
$103$	165056.	1.53299	0.766494	+	0.642252i	$0.222000\pi$
$104$	0	0
$105$	139002.	1.23041
$106$	0	0
$107$	−4855.53	−0.0409994	−0.0204997	−	0.999790i	$-0.506526\pi$
$107$	−4855.53	−0.0409994	−0.0204997	+	0.999790i	$0.506526\pi$
$108$	0	0
$109$	−50783.0	−0.409404	−0.204702	−	0.978824i	$-0.565622\pi$
$109$	−50783.0	−0.409404	−0.204702	+	0.978824i	$0.565622\pi$
$110$	0	0
$111$	51624.7	0.397695
$112$	0	0
$113$	−250074.	−1.84235	−0.921177	−	0.389145i	$-0.872771\pi$
$113$	−250074.	−1.84235	−0.921177	+	0.389145i	$0.872771\pi$
$114$	0	0
$115$	−31142.3	−0.219587
$116$	0	0
$117$	54636.7	0.368995
$118$	0	0
$119$	−176733.	−1.14406
$120$	0	0
$121$	−30948.2	−0.192164
$122$	0	0
$123$	−114001.	−0.679429
$124$	0	0
$125$	118910.	0.680679
$126$	0	0
$127$	−75539.2	−0.415588	−0.207794	−	0.978173i	$-0.566628\pi$
$127$	−75539.2	−0.415588	−0.207794	+	0.978173i	$0.566628\pi$
$128$	0	0
$129$	−18160.4	−0.0960840
$130$	0	0
$131$	−215478.	−1.09704	−0.548522	−	0.836136i	$-0.684809\pi$
$131$	−215478.	−1.09704	−0.548522	+	0.836136i	$0.684809\pi$
$132$	0	0
$133$	−525982.	−2.57835
$134$	0	0
$135$	48749.5	0.230216
$136$	0	0
$137$	−140259.	−0.638455	−0.319228	−	0.947678i	$-0.603423\pi$
$137$	−140259.	−0.638455	−0.319228	+	0.947678i	$0.603423\pi$
$138$	0	0
$139$	−35488.8	−0.155795	−0.0778977	−	0.996961i	$-0.524821\pi$
$139$	−35488.8	−0.155795	−0.0778977	+	0.996961i	$0.524821\pi$
$140$	0	0
$141$	14452.5	0.0612202
$142$	0	0
$143$	−243301.	−0.994954
$144$	0	0
$145$	−95635.9	−0.377747
$146$	0	0
$147$	−328820.	−1.25506
$148$	0	0
$149$	92665.3	0.341941	0.170971	−	0.985276i	$-0.445310\pi$
$149$	92665.3	0.341941	0.170971	+	0.985276i	$0.445310\pi$
$150$	0	0
$151$	−335751.	−1.19833	−0.599163	−	0.800627i	$-0.704500\pi$
$151$	−335751.	−1.19833	−0.599163	+	0.800627i	$0.704500\pi$
$152$	0	0
$153$	−61982.0	−0.214061
$154$	0	0
$155$	−222535.	−0.743994
$156$	0	0
$157$	−411865.	−1.33354	−0.666770	−	0.745263i	$-0.732324\pi$
$157$	−411865.	−1.33354	−0.666770	+	0.745263i	$0.732324\pi$
$158$	0	0
$159$	−107985.	−0.338742
$160$	0	0
$161$	107559.	0.327025
$162$	0	0
$163$	201696.	0.594605	0.297302	−	0.954783i	$-0.403913\pi$
$163$	201696.	0.594605	0.297302	+	0.954783i	$0.403913\pi$
$164$	0	0
$165$	−217084.	−0.620752
$166$	0	0
$167$	232101.	0.644000	0.322000	−	0.946740i	$-0.395645\pi$
$167$	232101.	0.644000	0.322000	+	0.946740i	$0.395645\pi$
$168$	0	0
$169$	83694.6	0.225414
$170$	0	0
$171$	−184467.	−0.482424
$172$	0	0
$173$	−604310.	−1.53513	−0.767564	−	0.640972i	$-0.778531\pi$
$173$	−604310.	−1.53513	−0.767564	+	0.640972i	$0.778531\pi$
$174$	0	0
$175$	311063.	0.767808
$176$	0	0
$177$	219612.	0.526894
$178$	0	0
$179$	48331.8	0.112746	0.0563729	−	0.998410i	$-0.482046\pi$
$179$	48331.8	0.112746	0.0563729	+	0.998410i	$0.482046\pi$
$180$	0	0
$181$	157000.	0.356207	0.178104	−	0.984012i	$-0.443004\pi$
$181$	157000.	0.356207	0.178104	+	0.984012i	$0.443004\pi$
$182$	0	0
$183$	−177035.	−0.390778
$184$	0	0
$185$	383581.	0.824001
$186$	0	0
$187$	276010.	0.577192
$188$	0	0
$189$	−168370.	−0.342855
$190$	0	0
$191$	−271788.	−0.539073	−0.269536	−	0.962990i	$-0.586870\pi$
$191$	−271788.	−0.539073	−0.269536	+	0.962990i	$0.586870\pi$
$192$	0	0
$193$	211920.	0.409524	0.204762	−	0.978812i	$-0.434358\pi$
$193$	211920.	0.409524	0.204762	+	0.978812i	$0.434358\pi$
$194$	0	0
$195$	405961.	0.764536
$196$	0	0
$197$	−256636.	−0.471142	−0.235571	−	0.971857i	$-0.575696\pi$
$197$	−256636.	−0.471142	−0.235571	+	0.971857i	$0.575696\pi$
$198$	0	0
$199$	165826.	0.296839	0.148419	−	0.988925i	$-0.452581\pi$
$199$	165826.	0.296839	0.148419	+	0.988925i	$0.452581\pi$
$200$	0	0
$201$	−452154.	−0.789399
$202$	0	0
$203$	330305.	0.562568
$204$	0	0
$205$	−847046.	−1.40774
$206$	0	0
$207$	37721.9	0.0611882
$208$	0	0
$209$	821443.	1.30080
$210$	0	0
$211$	1.27854e6	1.97701	0.988505	−	0.151187i	$-0.0483097\pi$
$211$	1.27854e6	1.97701	0.988505	+	0.151187i	$0.0483097\pi$
$212$	0	0
$213$	−149153.	−0.225259
$214$	0	0
$215$	−134935.	−0.199081
$216$	0	0
$217$	768587.	1.10801
$218$	0	0
$219$	128369.	0.180863
$220$	0	0
$221$	−516155.	−0.710886
$222$	0	0
$223$	329203.	0.443304	0.221652	−	0.975126i	$-0.428855\pi$
$223$	329203.	0.443304	0.221652	+	0.975126i	$0.428855\pi$
$224$	0	0
$225$	109093.	0.143661
$226$	0	0
$227$	−242920.	−0.312896	−0.156448	−	0.987686i	$-0.550004\pi$
$227$	−242920.	−0.312896	−0.156448	+	0.987686i	$0.550004\pi$
$228$	0	0
$229$	−843699.	−1.06316	−0.531580	−	0.847008i	$-0.678402\pi$
$229$	−843699.	−1.06316	−0.531580	+	0.847008i	$0.678402\pi$
$230$	0	0
$231$	749761.	0.924470
$232$	0	0
$233$	980922.	1.18371	0.591854	−	0.806045i	$-0.298396\pi$
$233$	980922.	1.18371	0.591854	+	0.806045i	$0.298396\pi$
$234$	0	0
$235$	107385.	0.126845
$236$	0	0
$237$	430650.	0.498028
$238$	0	0
$239$	−1.48396e6	−1.68045	−0.840226	−	0.542237i	$-0.817578\pi$
$239$	−1.48396e6	−1.68045	−0.840226	+	0.542237i	$0.817578\pi$
$240$	0	0
$241$	−1.47497e6	−1.63584	−0.817919	−	0.575333i	$-0.804872\pi$
$241$	−1.47497e6	−1.63584	−0.817919	+	0.575333i	$0.804872\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−2.44320e6	−2.60042
$246$	0	0
$247$	−1.53615e6	−1.60211
$248$	0	0
$249$	−541140.	−0.553110
$250$	0	0
$251$	−384697.	−0.385421	−0.192710	−	0.981256i	$-0.561728\pi$
$251$	−384697.	−0.385421	−0.192710	+	0.981256i	$0.561728\pi$
$252$	0	0
$253$	−167978.	−0.164987
$254$	0	0
$255$	−460538.	−0.443522
$256$	0	0
$257$	1.47809e6	1.39594	0.697970	−	0.716127i	$-0.254087\pi$
$257$	1.47809e6	1.39594	0.697970	+	0.716127i	$0.254087\pi$
$258$	0	0
$259$	−1.32480e6	−1.22716
$260$	0	0
$261$	115841.	0.105260
$262$	0	0
$263$	−768376.	−0.684990	−0.342495	−	0.939520i	$-0.611272\pi$
$263$	−768376.	−0.684990	−0.342495	+	0.939520i	$0.611272\pi$
$264$	0	0
$265$	−802346.	−0.701854
$266$	0	0
$267$	1.25232e6	1.07507
$268$	0	0
$269$	−1.23415e6	−1.03989	−0.519943	−	0.854201i	$-0.674047\pi$
$269$	−1.23415e6	−1.03989	−0.519943	+	0.854201i	$0.674047\pi$
$270$	0	0
$271$	305284.	0.252511	0.126256	−	0.991998i	$-0.459704\pi$
$271$	305284.	0.252511	0.126256	+	0.991998i	$0.459704\pi$
$272$	0	0
$273$	−1.40210e6	−1.13860
$274$	0	0
$275$	−485796.	−0.387367
$276$	0	0
$277$	−773218.	−0.605483	−0.302742	−	0.953073i	$-0.597902\pi$
$277$	−773218.	−0.605483	−0.302742	+	0.953073i	$0.597902\pi$
$278$	0	0
$279$	269551.	0.207315
$280$	0	0
$281$	−815352.	−0.615998	−0.307999	−	0.951387i	$-0.599659\pi$
$281$	−815352.	−0.615998	−0.307999	+	0.951387i	$0.599659\pi$
$282$	0	0
$283$	−186289.	−0.138268	−0.0691338	−	0.997607i	$-0.522024\pi$
$283$	−186289.	−0.138268	−0.0691338	+	0.997607i	$0.522024\pi$
$284$	0	0
$285$	−1.37063e6	−0.999555
$286$	0	0
$287$	2.92551e6	2.09651
$288$	0	0
$289$	−834310.	−0.587602
$290$	0	0
$291$	−767038.	−0.530988
$292$	0	0
$293$	281361.	0.191467	0.0957337	−	0.995407i	$-0.469480\pi$
$293$	281361.	0.191467	0.0957337	+	0.995407i	$0.469480\pi$
$294$	0	0
$295$	1.63176e6	1.09170
$296$	0	0
$297$	262949.	0.172974
$298$	0	0
$299$	314129.	0.203203
$300$	0	0
$301$	466036.	0.296486
$302$	0	0
$303$	1.52935e6	0.956975
$304$	0	0
$305$	−1.31540e6	−0.809671
$306$	0	0
$307$	−2.34582e6	−1.42053	−0.710263	−	0.703936i	$-0.751424\pi$
$307$	−2.34582e6	−1.42053	−0.710263	+	0.703936i	$0.751424\pi$
$308$	0	0
$309$	−1.48551e6	−0.885071
$310$	0	0
$311$	−14836.6	−0.00869831	−0.00434915	−	0.999991i	$-0.501384\pi$
$311$	−14836.6	−0.00869831	−0.00434915	+	0.999991i	$0.501384\pi$
$312$	0	0
$313$	−2.23974e6	−1.29222	−0.646111	−	0.763243i	$-0.723606\pi$
$313$	−2.23974e6	−1.29222	−0.646111	+	0.763243i	$0.723606\pi$
$314$	0	0
$315$	−1.25102e6	−0.710375
$316$	0	0
$317$	198362.	0.110869	0.0554346	−	0.998462i	$-0.482346\pi$
$317$	198362.	0.110869	0.0554346	+	0.998462i	$0.482346\pi$
$318$	0	0
$319$	−515848.	−0.283821
$320$	0	0
$321$	43699.8	0.0236710
$322$	0	0
$323$	1.74267e6	0.929413
$324$	0	0
$325$	908470.	0.477092
$326$	0	0
$327$	457047.	0.236369
$328$	0	0
$329$	−370883.	−0.188907
$330$	0	0
$331$	−3.70558e6	−1.85903	−0.929515	−	0.368784i	$-0.879774\pi$
$331$	−3.70558e6	−1.85903	−0.929515	+	0.368784i	$0.879774\pi$
$332$	0	0
$333$	−464622.	−0.229609
$334$	0	0
$335$	−3.35959e6	−1.63559
$336$	0	0
$337$	815082.	0.390955	0.195477	−	0.980708i	$-0.437374\pi$
$337$	815082.	0.390955	0.195477	+	0.980708i	$0.437374\pi$
$338$	0	0
$339$	2.25067e6	1.06368
$340$	0	0
$341$	−1.20033e6	−0.559002
$342$	0	0
$343$	4.55651e6	2.09121
$344$	0	0
$345$	280281.	0.126778
$346$	0	0
$347$	−159705.	−0.0712024	−0.0356012	−	0.999366i	$-0.511335\pi$
$347$	−159705.	−0.0712024	−0.0356012	+	0.999366i	$0.511335\pi$
$348$	0	0
$349$	−3.05102e6	−1.34086	−0.670428	−	0.741975i	$-0.733889\pi$
$349$	−3.05102e6	−1.34086	−0.670428	+	0.741975i	$0.733889\pi$
$350$	0	0
$351$	−491731.	−0.213039
$352$	0	0
$353$	−529374.	−0.226113	−0.113057	−	0.993589i	$-0.536064\pi$
$353$	−529374.	−0.226113	−0.113057	+	0.993589i	$0.536064\pi$
$354$	0	0
$355$	−1.10823e6	−0.466724
$356$	0	0
$357$	1.59060e6	0.660526
$358$	0	0
$359$	−2.74711e6	−1.12497	−0.562484	−	0.826808i	$-0.690154\pi$
$359$	−2.74711e6	−1.12497	−0.562484	+	0.826808i	$0.690154\pi$
$360$	0	0
$361$	2.71033e6	1.09460
$362$	0	0
$363$	278534.	0.110946
$364$	0	0
$365$	953805.	0.374738
$366$	0	0
$367$	2.09523e6	0.812020	0.406010	−	0.913869i	$-0.366920\pi$
$367$	2.09523e6	0.812020	0.406010	+	0.913869i	$0.366920\pi$
$368$	0	0
$369$	1.02601e6	0.392269
$370$	0	0
$371$	2.77113e6	1.04525
$372$	0	0
$373$	−1.85054e6	−0.688693	−0.344347	−	0.938843i	$-0.611900\pi$
$373$	−1.85054e6	−0.688693	−0.344347	+	0.938843i	$0.611900\pi$
$374$	0	0
$375$	−1.07019e6	−0.392990
$376$	0	0
$377$	964669.	0.349562
$378$	0	0
$379$	−250828.	−0.0896972	−0.0448486	−	0.998994i	$-0.514281\pi$
$379$	−250828.	−0.0896972	−0.0448486	+	0.998994i	$0.514281\pi$
$380$	0	0
$381$	679853.	0.239940
$382$	0	0
$383$	1.25720e6	0.437932	0.218966	−	0.975733i	$-0.429732\pi$
$383$	1.25720e6	0.437932	0.218966	+	0.975733i	$0.429732\pi$
$384$	0	0
$385$	5.57087e6	1.91545
$386$	0	0
$387$	163444.	0.0554741
$388$	0	0
$389$	5.48629e6	1.83825	0.919126	−	0.393964i	$-0.128897\pi$
$389$	5.48629e6	1.83825	0.919126	+	0.393964i	$0.128897\pi$
$390$	0	0
$391$	−356360.	−0.117882
$392$	0	0
$393$	1.93930e6	0.633379
$394$	0	0
$395$	3.19981e6	1.03189
$396$	0	0
$397$	−926145.	−0.294919	−0.147459	−	0.989068i	$-0.547110\pi$
$397$	−926145.	−0.294919	−0.147459	+	0.989068i	$0.547110\pi$
$398$	0	0
$399$	4.73384e6	1.48861
$400$	0	0
$401$	−4.52840e6	−1.40632	−0.703159	−	0.711032i	$-0.748228\pi$
$401$	−4.52840e6	−1.40632	−0.703159	+	0.711032i	$0.748228\pi$
$402$	0	0
$403$	2.24469e6	0.688483
$404$	0	0
$405$	−438745.	−0.132915
$406$	0	0
$407$	2.06899e6	0.619116
$408$	0	0
$409$	−4.09167e6	−1.20946	−0.604731	−	0.796429i	$-0.706720\pi$
$409$	−4.09167e6	−1.20946	−0.604731	+	0.796429i	$0.706720\pi$
$410$	0	0
$411$	1.26233e6	0.368612
$412$	0	0
$413$	−5.63574e6	−1.62583
$414$	0	0
$415$	−4.02077e6	−1.14601
$416$	0	0
$417$	319399.	0.0899485
$418$	0	0
$419$	−121773.	−0.0338856	−0.0169428	−	0.999856i	$-0.505393\pi$
$419$	−121773.	−0.0338856	−0.0169428	+	0.999856i	$0.505393\pi$
$420$	0	0
$421$	−6.22499e6	−1.71172	−0.855861	−	0.517206i	$-0.826972\pi$
$421$	−6.22499e6	−1.71172	−0.855861	+	0.517206i	$0.826972\pi$
$422$	0	0
$423$	−130072.	−0.0353455
$424$	0	0
$425$	−1.03060e6	−0.276770
$426$	0	0
$427$	4.54310e6	1.20582
$428$	0	0
$429$	2.18971e6	0.574437
$430$	0	0
$431$	−5.69823e6	−1.47757	−0.738783	−	0.673943i	$-0.764599\pi$
$431$	−5.69823e6	−1.47757	−0.738783	+	0.673943i	$0.764599\pi$
$432$	0	0
$433$	3.78582e6	0.970377	0.485188	−	0.874410i	$-0.338751\pi$
$433$	3.78582e6	0.970377	0.485188	+	0.874410i	$0.338751\pi$
$434$	0	0
$435$	860723.	0.218092
$436$	0	0
$437$	−1.06058e6	−0.265668
$438$	0	0
$439$	212373.	0.0525943	0.0262971	−	0.999654i	$-0.491628\pi$
$439$	212373.	0.0525943	0.0262971	+	0.999654i	$0.491628\pi$
$440$	0	0
$441$	2.95938e6	0.724610
$442$	0	0
$443$	4.53614e6	1.09819	0.549094	−	0.835760i	$-0.314973\pi$
$443$	4.53614e6	1.09819	0.549094	+	0.835760i	$0.314973\pi$
$444$	0	0
$445$	9.30500e6	2.22749
$446$	0	0
$447$	−833988.	−0.197420
$448$	0	0
$449$	−2.61235e6	−0.611526	−0.305763	−	0.952108i	$-0.598912\pi$
$449$	−2.61235e6	−0.611526	−0.305763	+	0.952108i	$0.598912\pi$
$450$	0	0
$451$	−4.56886e6	−1.05771
$452$	0	0
$453$	3.02176e6	0.691854
$454$	0	0
$455$	−1.04179e7	−2.35912
$456$	0	0
$457$	3.79986e6	0.851093	0.425546	−	0.904937i	$-0.360082\pi$
$457$	3.79986e6	0.851093	0.425546	+	0.904937i	$0.360082\pi$
$458$	0	0
$459$	557838.	0.123588
$460$	0	0
$461$	7.43429e6	1.62925	0.814624	−	0.579989i	$-0.196943\pi$
$461$	7.43429e6	1.62925	0.814624	+	0.579989i	$0.196943\pi$
$462$	0	0
$463$	7.91454e6	1.71583	0.857913	−	0.513795i	$-0.171761\pi$
$463$	7.91454e6	1.71583	0.857913	+	0.513795i	$0.171761\pi$
$464$	0	0
$465$	2.00282e6	0.429545
$466$	0	0
$467$	−2.39528e6	−0.508235	−0.254117	−	0.967173i	$-0.581785\pi$
$467$	−2.39528e6	−0.508235	−0.254117	+	0.967173i	$0.581785\pi$
$468$	0	0
$469$	1.16033e7	2.43584
$470$	0	0
$471$	3.70679e6	0.769920
$472$	0	0
$473$	−727824.	−0.149580
$474$	0	0
$475$	−3.06722e6	−0.623750
$476$	0	0
$477$	971861.	0.195573
$478$	0	0
$479$	−4.94106e6	−0.983968	−0.491984	−	0.870604i	$-0.663728\pi$
$479$	−4.94106e6	−0.983968	−0.491984	+	0.870604i	$0.663728\pi$
$480$	0	0
$481$	−3.86914e6	−0.762521
$482$	0	0
$483$	−968028.	−0.188808
$484$	0	0
$485$	−5.69924e6	−1.10018
$486$	0	0
$487$	846936.	0.161818	0.0809092	−	0.996721i	$-0.474218\pi$
$487$	846936.	0.161818	0.0809092	+	0.996721i	$0.474218\pi$
$488$	0	0
$489$	−1.81526e6	−0.343295
$490$	0	0
$491$	−8.97769e6	−1.68059	−0.840294	−	0.542132i	$-0.817617\pi$
$491$	−8.97769e6	−1.68059	−0.840294	+	0.542132i	$0.817617\pi$
$492$	0	0
$493$	−1.09436e6	−0.202788
$494$	0	0
$495$	1.95376e6	0.358391
$496$	0	0
$497$	3.82759e6	0.695079
$498$	0	0
$499$	7.29948e6	1.31232	0.656161	−	0.754621i	$-0.272179\pi$
$499$	7.29948e6	1.31232	0.656161	+	0.754621i	$0.272179\pi$
$500$	0	0
$501$	−2.08891e6	−0.371813
$502$	0	0
$503$	−2.35478e6	−0.414983	−0.207492	−	0.978237i	$-0.566530\pi$
$503$	−2.35478e6	−0.414983	−0.207492	+	0.978237i	$0.566530\pi$
$504$	0	0
$505$	1.13634e7	1.98280
$506$	0	0
$507$	−753251.	−0.130143
$508$	0	0
$509$	7.31469e6	1.25142	0.625708	−	0.780058i	$-0.284810\pi$
$509$	7.31469e6	1.25142	0.625708	+	0.780058i	$0.284810\pi$
$510$	0	0
$511$	−3.29423e6	−0.558087
$512$	0	0
$513$	1.66020e6	0.278528
$514$	0	0
$515$	−1.10376e7	−1.83382
$516$	0	0
$517$	579220.	0.0953053
$518$	0	0
$519$	5.43879e6	0.886306
$520$	0	0
$521$	−4.05729e6	−0.654850	−0.327425	−	0.944877i	$-0.606181\pi$
$521$	−4.05729e6	−0.654850	−0.327425	+	0.944877i	$0.606181\pi$
$522$	0	0
$523$	4.06432e6	0.649731	0.324865	−	0.945760i	$-0.394681\pi$
$523$	4.06432e6	0.649731	0.324865	+	0.945760i	$0.394681\pi$
$524$	0	0
$525$	−2.79956e6	−0.443294
$526$	0	0
$527$	−2.54646e6	−0.399402
$528$	0	0
$529$	−6.21946e6	−0.966304
$530$	0	0
$531$	−1.97651e6	−0.304203
$532$	0	0
$533$	8.54406e6	1.30271
$534$	0	0
$535$	324698.	0.0490450
$536$	0	0
$537$	−434986.	−0.0650938
$538$	0	0
$539$	−1.31783e7	−1.95383
$540$	0	0
$541$	2.19907e6	0.323033	0.161516	−	0.986870i	$-0.448362\pi$
$541$	2.19907e6	0.323033	0.161516	+	0.986870i	$0.448362\pi$
$542$	0	0
$543$	−1.41300e6	−0.205656
$544$	0	0
$545$	3.39594e6	0.489744
$546$	0	0
$547$	−5.46860e6	−0.781463	−0.390731	−	0.920505i	$-0.627778\pi$
$547$	−5.46860e6	−0.781463	−0.390731	+	0.920505i	$0.627778\pi$
$548$	0	0
$549$	1.59331e6	0.225616
$550$	0	0
$551$	−3.25696e6	−0.457018
$552$	0	0
$553$	−1.10514e7	−1.53676
$554$	0	0
$555$	−3.45223e6	−0.475737
$556$	0	0
$557$	1.14168e6	0.155921	0.0779607	−	0.996956i	$-0.475159\pi$
$557$	1.14168e6	0.155921	0.0779607	+	0.996956i	$0.475159\pi$
$558$	0	0
$559$	1.36108e6	0.184227
$560$	0	0
$561$	−2.48409e6	−0.333242
$562$	0	0
$563$	8.79987e6	1.17005	0.585026	−	0.811015i	$-0.301084\pi$
$563$	8.79987e6	1.17005	0.585026	+	0.811015i	$0.301084\pi$
$564$	0	0
$565$	1.67229e7	2.20389
$566$	0	0
$567$	1.51533e6	0.197947
$568$	0	0
$569$	−1.09177e7	−1.41367	−0.706836	−	0.707377i	$-0.749878\pi$
$569$	−1.09177e7	−1.41367	−0.706836	+	0.707377i	$0.749878\pi$
$570$	0	0
$571$	−3.13621e6	−0.402545	−0.201273	−	0.979535i	$-0.564508\pi$
$571$	−3.13621e6	−0.402545	−0.201273	+	0.979535i	$0.564508\pi$
$572$	0	0
$573$	2.44609e6	0.311234
$574$	0	0
$575$	627219.	0.0791133
$576$	0	0
$577$	−3.75626e6	−0.469695	−0.234848	−	0.972032i	$-0.575459\pi$
$577$	−3.75626e6	−0.469695	−0.234848	+	0.972032i	$0.575459\pi$
$578$	0	0
$579$	−1.90728e6	−0.236439
$580$	0	0
$581$	1.38869e7	1.70673
$582$	0	0
$583$	−4.32775e6	−0.527340
$584$	0	0
$585$	−3.65365e6	−0.441405
$586$	0	0
$587$	8.59479e6	1.02953	0.514766	−	0.857331i	$-0.327879\pi$
$587$	8.59479e6	1.02953	0.514766	+	0.857331i	$0.327879\pi$
$588$	0	0
$589$	−7.57862e6	−0.900123
$590$	0	0
$591$	2.30972e6	0.272014
$592$	0	0
$593$	−1.28180e6	−0.149687	−0.0748434	−	0.997195i	$-0.523846\pi$
$593$	−1.28180e6	−0.149687	−0.0748434	+	0.997195i	$0.523846\pi$
$594$	0	0
$595$	1.18184e7	1.36857
$596$	0	0
$597$	−1.49244e6	−0.171380
$598$	0	0
$599$	−3.91371e6	−0.445678	−0.222839	−	0.974855i	$-0.571533\pi$
$599$	−3.91371e6	−0.445678	−0.222839	+	0.974855i	$0.571533\pi$
$600$	0	0
$601$	−3.63280e6	−0.410257	−0.205128	−	0.978735i	$-0.565761\pi$
$601$	−3.63280e6	−0.410257	−0.205128	+	0.978735i	$0.565761\pi$
$602$	0	0
$603$	4.06939e6	0.455760
$604$	0	0
$605$	2.06956e6	0.229874
$606$	0	0
$607$	1.23312e7	1.35842	0.679210	−	0.733944i	$-0.262323\pi$
$607$	1.23312e7	1.35842	0.679210	+	0.733944i	$0.262323\pi$
$608$	0	0
$609$	−2.97275e6	−0.324799
$610$	0	0
$611$	−1.08318e6	−0.117381
$612$	0	0
$613$	−8.39494e6	−0.902332	−0.451166	−	0.892440i	$-0.648992\pi$
$613$	−8.39494e6	−0.902332	−0.451166	+	0.892440i	$0.648992\pi$
$614$	0	0
$615$	7.62341e6	0.812759
$616$	0	0
$617$	5.02025e6	0.530899	0.265450	−	0.964125i	$-0.414480\pi$
$617$	5.02025e6	0.530899	0.265450	+	0.964125i	$0.414480\pi$
$618$	0	0
$619$	8.02789e6	0.842122	0.421061	−	0.907032i	$-0.361658\pi$
$619$	8.02789e6	0.842122	0.421061	+	0.907032i	$0.361658\pi$
$620$	0	0
$621$	−339497.	−0.0353270
$622$	0	0
$623$	−3.21374e7	−3.31735
$624$	0	0
$625$	−1.21605e7	−1.24524
$626$	0	0
$627$	−7.39299e6	−0.751019
$628$	0	0
$629$	4.38930e6	0.442353
$630$	0	0
$631$	−118997.	−0.0118977	−0.00594885	−	0.999982i	$-0.501894\pi$
$631$	−118997.	−0.0118977	−0.00594885	+	0.999982i	$0.501894\pi$
$632$	0	0
$633$	−1.15069e7	−1.14143
$634$	0	0
$635$	5.05143e6	0.497142
$636$	0	0
$637$	2.46443e7	2.40640
$638$	0	0
$639$	1.34237e6	0.130053
$640$	0	0
$641$	1.30672e7	1.25614	0.628069	−	0.778158i	$-0.283846\pi$
$641$	1.30672e7	1.25614	0.628069	+	0.778158i	$0.283846\pi$
$642$	0	0
$643$	1.86984e7	1.78351	0.891757	−	0.452514i	$-0.149473\pi$
$643$	1.86984e7	1.78351	0.891757	+	0.452514i	$0.149473\pi$
$644$	0	0
$645$	1.21442e6	0.114939
$646$	0	0
$647$	−1.57250e7	−1.47683	−0.738414	−	0.674348i	$-0.764425\pi$
$647$	−1.57250e7	−1.47683	−0.738414	+	0.674348i	$0.764425\pi$
$648$	0	0
$649$	8.80151e6	0.820249
$650$	0	0
$651$	−6.91728e6	−0.639710
$652$	0	0
$653$	1.04382e7	0.957953	0.478977	−	0.877828i	$-0.341008\pi$
$653$	1.04382e7	0.957953	0.478977	+	0.877828i	$0.341008\pi$
$654$	0	0
$655$	1.44094e7	1.31233
$656$	0	0
$657$	−1.15532e6	−0.104421
$658$	0	0
$659$	1.56129e7	1.40045	0.700227	−	0.713920i	$-0.253082\pi$
$659$	1.56129e7	1.40045	0.700227	+	0.713920i	$0.253082\pi$
$660$	0	0
$661$	1.34894e7	1.20085	0.600425	−	0.799681i	$-0.294998\pi$
$661$	1.34894e7	1.20085	0.600425	+	0.799681i	$0.294998\pi$
$662$	0	0
$663$	4.64540e6	0.410430
$664$	0	0
$665$	3.51733e7	3.08432
$666$	0	0
$667$	666019.	0.0579659
$668$	0	0
$669$	−2.96283e6	−0.255942
$670$	0	0
$671$	−7.09511e6	−0.608349
$672$	0	0
$673$	−4.64517e6	−0.395334	−0.197667	−	0.980269i	$-0.563336\pi$
$673$	−4.64517e6	−0.395334	−0.197667	+	0.980269i	$0.563336\pi$
$674$	0	0
$675$	−981834.	−0.0829428
$676$	0	0
$677$	−1.80937e7	−1.51725	−0.758623	−	0.651530i	$-0.774128\pi$
$677$	−1.80937e7	−1.51725	−0.758623	+	0.651530i	$0.774128\pi$
$678$	0	0
$679$	1.96839e7	1.63846
$680$	0	0
$681$	2.18628e6	0.180650
$682$	0	0
$683$	1.75881e7	1.44267	0.721333	−	0.692588i	$-0.243530\pi$
$683$	1.75881e7	1.44267	0.721333	+	0.692588i	$0.243530\pi$
$684$	0	0
$685$	9.37938e6	0.763744
$686$	0	0
$687$	7.59329e6	0.613816
$688$	0	0
$689$	8.09318e6	0.649488
$690$	0	0
$691$	−8.24840e6	−0.657165	−0.328582	−	0.944475i	$-0.606571\pi$
$691$	−8.24840e6	−0.657165	−0.328582	+	0.944475i	$0.606571\pi$
$692$	0	0
$693$	−6.74785e6	−0.533743
$694$	0	0
$695$	2.37320e6	0.186368
$696$	0	0
$697$	−9.69271e6	−0.755724
$698$	0	0
$699$	−8.82830e6	−0.683414
$700$	0	0
$701$	−6.82859e6	−0.524851	−0.262426	−	0.964952i	$-0.584522\pi$
$701$	−6.82859e6	−0.524851	−0.262426	+	0.964952i	$0.584522\pi$
$702$	0	0
$703$	1.30632e7	0.996920
$704$	0	0
$705$	−966462.	−0.0732339
$706$	0	0
$707$	−3.92466e7	−2.95293
$708$	0	0
$709$	−781645.	−0.0583974	−0.0291987	−	0.999574i	$-0.509296\pi$
$709$	−781645.	−0.0583974	−0.0291987	+	0.999574i	$0.509296\pi$
$710$	0	0
$711$	−3.87585e6	−0.287536
$712$	0	0
$713$	1.54976e6	0.114167
$714$	0	0
$715$	1.62699e7	1.19020
$716$	0	0
$717$	1.33556e7	0.970209
$718$	0	0
$719$	2.25941e7	1.62995	0.814973	−	0.579499i	$-0.196752\pi$
$719$	2.25941e7	1.62995	0.814973	+	0.579499i	$0.196752\pi$
$720$	0	0
$721$	3.81214e7	2.73105
$722$	0	0
$723$	1.32747e7	0.944451
$724$	0	0
$725$	1.92615e6	0.136096
$726$	0	0
$727$	1.31915e7	0.925672	0.462836	−	0.886444i	$-0.346832\pi$
$727$	1.31915e7	0.925672	0.462836	+	0.886444i	$0.346832\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.54406e6	−0.106874
$732$	0	0
$733$	−9.78681e6	−0.672793	−0.336396	−	0.941721i	$-0.609208\pi$
$733$	−9.78681e6	−0.672793	−0.336396	+	0.941721i	$0.609208\pi$
$734$	0	0
$735$	2.19888e7	1.50135
$736$	0	0
$737$	−1.81212e7	−1.22891
$738$	0	0
$739$	−2.14877e7	−1.44737	−0.723685	−	0.690130i	$-0.757553\pi$
$739$	−2.14877e7	−1.44737	−0.723685	+	0.690130i	$0.757553\pi$
$740$	0	0
$741$	1.38254e7	0.924977
$742$	0	0
$743$	8.99108e6	0.597503	0.298751	−	0.954331i	$-0.403430\pi$
$743$	8.99108e6	0.597503	0.298751	+	0.954331i	$0.403430\pi$
$744$	0	0
$745$	−6.19668e6	−0.409043
$746$	0	0
$747$	4.87026e6	0.319338
$748$	0	0
$749$	−1.12143e6	−0.0730414
$750$	0	0
$751$	−2.19356e7	−1.41922	−0.709611	−	0.704594i	$-0.751129\pi$
$751$	−2.19356e7	−1.41922	−0.709611	+	0.704594i	$0.751129\pi$
$752$	0	0
$753$	3.46228e6	0.222523
$754$	0	0
$755$	2.24523e7	1.43348
$756$	0	0
$757$	−1.43342e7	−0.909144	−0.454572	−	0.890710i	$-0.650208\pi$
$757$	−1.43342e7	−0.909144	−0.454572	+	0.890710i	$0.650208\pi$
$758$	0	0
$759$	1.51180e6	0.0952554
$760$	0	0
$761$	−306000.	−0.0191540	−0.00957702	−	0.999954i	$-0.503049\pi$
$761$	−306000.	−0.0191540	−0.00957702	+	0.999954i	$0.503049\pi$
$762$	0	0
$763$	−1.17288e7	−0.729363
$764$	0	0
$765$	4.14484e6	0.256068
$766$	0	0
$767$	−1.64594e7	−1.01024
$768$	0	0
$769$	−1.22052e7	−0.744266	−0.372133	−	0.928179i	$-0.621373\pi$
$769$	−1.22052e7	−0.744266	−0.372133	+	0.928179i	$0.621373\pi$
$770$	0	0
$771$	−1.33028e7	−0.805947
$772$	0	0
$773$	3.18344e7	1.91623	0.958116	−	0.286380i	$-0.0924521\pi$
$773$	3.18344e7	1.91623	0.958116	+	0.286380i	$0.0924521\pi$
$774$	0	0
$775$	4.48195e6	0.268048
$776$	0	0
$777$	1.19232e7	0.708503
$778$	0	0
$779$	−2.88469e7	−1.70316
$780$	0	0
$781$	−5.97767e6	−0.350675
$782$	0	0
$783$	−1.04257e6	−0.0607717
$784$	0	0
$785$	2.75421e7	1.59523
$786$	0	0
$787$	−1.58777e6	−0.0913799	−0.0456899	−	0.998956i	$-0.514549\pi$
$787$	−1.58777e6	−0.0913799	−0.0456899	+	0.998956i	$0.514549\pi$
$788$	0	0
$789$	6.91538e6	0.395479
$790$	0	0
$791$	−5.77572e7	−3.28220
$792$	0	0
$793$	1.32683e7	0.749260
$794$	0	0
$795$	7.22111e6	0.405216
$796$	0	0
$797$	−602565.	−0.0336014	−0.0168007	−	0.999859i	$-0.505348\pi$
$797$	−602565.	−0.0336014	−0.0168007	+	0.999859i	$0.505348\pi$
$798$	0	0
$799$	1.22880e6	0.0680948
$800$	0	0
$801$	−1.12709e7	−0.620694
$802$	0	0
$803$	5.14471e6	0.281561
$804$	0	0
$805$	−7.19263e6	−0.391199
$806$	0	0
$807$	1.11073e7	0.600379
$808$	0	0
$809$	−4.61727e6	−0.248036	−0.124018	−	0.992280i	$-0.539578\pi$
$809$	−4.61727e6	−0.248036	−0.124018	+	0.992280i	$0.539578\pi$
$810$	0	0
$811$	−1.46562e7	−0.782473	−0.391236	−	0.920290i	$-0.627953\pi$
$811$	−1.46562e7	−0.782473	−0.391236	+	0.920290i	$0.627953\pi$
$812$	0	0
$813$	−2.74755e6	−0.145787
$814$	0	0
$815$	−1.34878e7	−0.711288
$816$	0	0
$817$	−4.59533e6	−0.240858
$818$	0	0
$819$	1.26189e7	0.657373
$820$	0	0
$821$	−2.60620e7	−1.34943	−0.674714	−	0.738079i	$-0.735733\pi$
$821$	−2.60620e7	−1.34943	−0.674714	+	0.738079i	$0.735733\pi$
$822$	0	0
$823$	−661397.	−0.0340379	−0.0170190	−	0.999855i	$-0.505418\pi$
$823$	−661397.	−0.0340379	−0.0170190	+	0.999855i	$0.505418\pi$
$824$	0	0
$825$	4.37217e6	0.223646
$826$	0	0
$827$	−2.73064e7	−1.38835	−0.694176	−	0.719805i	$-0.744231\pi$
$827$	−2.73064e7	−1.38835	−0.694176	+	0.719805i	$0.744231\pi$
$828$	0	0
$829$	−2.82299e7	−1.42667	−0.713334	−	0.700825i	$-0.752816\pi$
$829$	−2.82299e7	−1.42667	−0.713334	+	0.700825i	$0.752816\pi$
$830$	0	0
$831$	6.95896e6	0.349576
$832$	0	0
$833$	−2.79574e7	−1.39600
$834$	0	0
$835$	−1.55210e7	−0.770377
$836$	0	0
$837$	−2.42596e6	−0.119693
$838$	0	0
$839$	1.82539e7	0.895266	0.447633	−	0.894217i	$-0.352267\pi$
$839$	1.82539e7	0.895266	0.447633	+	0.894217i	$0.352267\pi$
$840$	0	0
$841$	−1.84659e7	−0.900284
$842$	0	0
$843$	7.33816e6	0.355646
$844$	0	0
$845$	−5.59680e6	−0.269648
$846$	0	0
$847$	−7.14780e6	−0.342345
$848$	0	0
$849$	1.67660e6	0.0798288
$850$	0	0
$851$	−2.67130e6	−0.126444
$852$	0	0
$853$	−9.21479e6	−0.433624	−0.216812	−	0.976213i	$-0.569566\pi$
$853$	−9.21479e6	−0.433624	−0.216812	+	0.976213i	$0.569566\pi$
$854$	0	0
$855$	1.23356e7	0.577093
$856$	0	0
$857$	−3.64148e7	−1.69366	−0.846830	−	0.531864i	$-0.821492\pi$
$857$	−3.64148e7	−1.69366	−0.846830	+	0.531864i	$0.821492\pi$
$858$	0	0
$859$	−2.32507e7	−1.07511	−0.537555	−	0.843229i	$-0.680652\pi$
$859$	−2.32507e7	−1.07511	−0.537555	+	0.843229i	$0.680652\pi$
$860$	0	0
$861$	−2.63296e7	−1.21042
$862$	0	0
$863$	−2.69228e7	−1.23053	−0.615266	−	0.788319i	$-0.710952\pi$
$863$	−2.69228e7	−1.23053	−0.615266	+	0.788319i	$0.710952\pi$
$864$	0	0
$865$	4.04112e7	1.83638
$866$	0	0
$867$	7.50879e6	0.339252
$868$	0	0
$869$	1.72594e7	0.775311
$870$	0	0
$871$	3.38878e7	1.51356
$872$	0	0
$873$	6.90334e6	0.306566
$874$	0	0
$875$	2.74634e7	1.21265
$876$	0	0
$877$	8.06833e6	0.354229	0.177115	−	0.984190i	$-0.443324\pi$
$877$	8.06833e6	0.354229	0.177115	+	0.984190i	$0.443324\pi$
$878$	0	0
$879$	−2.53225e6	−0.110544
$880$	0	0
$881$	−3.51408e7	−1.52536	−0.762680	−	0.646776i	$-0.776117\pi$
$881$	−3.51408e7	−1.52536	−0.762680	+	0.646776i	$0.776117\pi$
$882$	0	0
$883$	4.42971e7	1.91194	0.955968	−	0.293472i	$-0.0948107\pi$
$883$	4.42971e7	1.91194	0.955968	+	0.293472i	$0.0948107\pi$
$884$	0	0
$885$	−1.46858e7	−0.630290
$886$	0	0
$887$	−4.24507e7	−1.81166	−0.905828	−	0.423645i	$-0.860750\pi$
$887$	−4.24507e7	−1.81166	−0.905828	+	0.423645i	$0.860750\pi$
$888$	0	0
$889$	−1.74465e7	−0.740380
$890$	0	0
$891$	−2.36654e6	−0.0998663
$892$	0	0
$893$	3.65708e6	0.153464
$894$	0	0
$895$	−3.23203e6	−0.134871
$896$	0	0
$897$	−2.82716e6	−0.117319
$898$	0	0
$899$	4.75921e6	0.196397
$900$	0	0
$901$	−9.18121e6	−0.376780
$902$	0	0
$903$	−4.19433e6	−0.171176
$904$	0	0
$905$	−1.04988e7	−0.426109
$906$	0	0
$907$	8.87749e6	0.358321	0.179160	−	0.983820i	$-0.442662\pi$
$907$	8.87749e6	0.358321	0.179160	+	0.983820i	$0.442662\pi$
$908$	0	0
$909$	−1.37642e7	−0.552510
$910$	0	0
$911$	3.01023e7	1.20172	0.600860	−	0.799354i	$-0.294825\pi$
$911$	3.01023e7	1.20172	0.600860	+	0.799354i	$0.294825\pi$
$912$	0	0
$913$	−2.16876e7	−0.861060
$914$	0	0
$915$	1.18386e7	0.467464
$916$	0	0
$917$	−4.97668e7	−1.95441
$918$	0	0
$919$	2.68865e6	0.105014	0.0525068	−	0.998621i	$-0.483279\pi$
$919$	2.68865e6	0.105014	0.0525068	+	0.998621i	$0.483279\pi$
$920$	0	0
$921$	2.11124e7	0.820141
$922$	0	0
$923$	1.11786e7	0.431901
$924$	0	0
$925$	−7.72548e6	−0.296873
$926$	0	0
$927$	1.33695e7	0.510996
$928$	0	0
$929$	5.37353e6	0.204277	0.102139	−	0.994770i	$-0.467431\pi$
$929$	5.37353e6	0.204277	0.102139	+	0.994770i	$0.467431\pi$
$930$	0	0
$931$	−8.32051e7	−3.14612
$932$	0	0
$933$	133530.	0.00502197
$934$	0	0
$935$	−1.84572e7	−0.690458
$936$	0	0
$937$	3.12251e7	1.16186	0.580931	−	0.813953i	$-0.302688\pi$
$937$	3.12251e7	1.16186	0.580931	+	0.813953i	$0.302688\pi$
$938$	0	0
$939$	2.01577e7	0.746065
$940$	0	0
$941$	−3.47494e7	−1.27930	−0.639651	−	0.768665i	$-0.720921\pi$
$941$	−3.47494e7	−1.27930	−0.639651	+	0.768665i	$0.720921\pi$
$942$	0	0
$943$	5.89893e6	0.216020
$944$	0	0
$945$	1.12592e7	0.410135
$946$	0	0
$947$	2.11046e7	0.764720	0.382360	−	0.924014i	$-0.375111\pi$
$947$	2.11046e7	0.764720	0.382360	+	0.924014i	$0.375111\pi$
$948$	0	0
$949$	−9.62093e6	−0.346778
$950$	0	0
$951$	−1.78526e6	−0.0640103
$952$	0	0
$953$	−1.22707e6	−0.0437661	−0.0218830	−	0.999761i	$-0.506966\pi$
$953$	−1.22707e6	−0.0437661	−0.0218830	+	0.999761i	$0.506966\pi$
$954$	0	0
$955$	1.81749e7	0.644859
$956$	0	0
$957$	4.64263e6	0.163864
$958$	0	0
$959$	−3.23943e7	−1.13742
$960$	0	0
$961$	−1.75550e7	−0.613185
$962$	0	0
$963$	−393298.	−0.0136665
$964$	0	0
$965$	−1.41715e7	−0.489888
$966$	0	0
$967$	−2.09032e7	−0.718865	−0.359433	−	0.933171i	$-0.617030\pi$
$967$	−2.09032e7	−0.718865	−0.359433	+	0.933171i	$0.617030\pi$
$968$	0	0
$969$	−1.56840e7	−0.536597
$970$	0	0
$971$	3.51130e7	1.19514	0.597572	−	0.801815i	$-0.296132\pi$
$971$	3.51130e7	1.19514	0.597572	+	0.801815i	$0.296132\pi$
$972$	0	0
$973$	−8.19650e6	−0.277553
$974$	0	0
$975$	−8.17623e6	−0.275449
$976$	0	0
$977$	−2.99334e7	−1.00327	−0.501637	−	0.865078i	$-0.667269\pi$
$977$	−2.99334e7	−1.00327	−0.501637	+	0.865078i	$0.667269\pi$
$978$	0	0
$979$	5.01900e7	1.67363
$980$	0	0
$981$	−4.11342e6	−0.136468
$982$	0	0
$983$	1.16148e7	0.383379	0.191689	−	0.981456i	$-0.438603\pi$
$983$	1.16148e7	0.383379	0.191689	+	0.981456i	$0.438603\pi$
$984$	0	0
$985$	1.71617e7	0.563598
$986$	0	0
$987$	3.33795e6	0.109065
$988$	0	0
$989$	939705.	0.0305493
$990$	0	0
$991$	5.48219e7	1.77325	0.886625	−	0.462488i	$-0.153043\pi$
$991$	5.48219e7	1.77325	0.886625	+	0.462488i	$0.153043\pi$
$992$	0	0
$993$	3.33502e7	1.07331
$994$	0	0
$995$	−1.10891e7	−0.355090
$996$	0	0
$997$	1.86399e7	0.593888	0.296944	−	0.954895i	$-0.404032\pi$
$997$	1.86399e7	0.593888	0.296944	+	0.954895i	$0.404032\pi$
$998$	0	0
$999$	4.18160e6	0.132565

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.6.a.ba.1.2		5
4.3	odd	2		768.6.a.bc.1.2		5
8.3	odd	2		768.6.a.bb.1.4		5
8.5	even	2		768.6.a.bd.1.4		5
16.3	odd	4		96.6.d.a.49.9		10
16.5	even	4		24.6.d.a.13.1	✓	10
16.11	odd	4		96.6.d.a.49.2		10
16.13	even	4		24.6.d.a.13.2	yes	10
48.5	odd	4		72.6.d.d.37.10		10
48.11	even	4		288.6.d.d.145.8		10
48.29	odd	4		72.6.d.d.37.9		10
48.35	even	4		288.6.d.d.145.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.6.d.a.13.1	✓	10	16.5	even	4
24.6.d.a.13.2	yes	10	16.13	even	4
72.6.d.d.37.9		10	48.29	odd	4
72.6.d.d.37.10		10	48.5	odd	4
96.6.d.a.49.2		10	16.11	odd	4
96.6.d.a.49.9		10	16.3	odd	4
288.6.d.d.145.3		10	48.35	even	4
288.6.d.d.145.8		10	48.11	even	4
768.6.a.ba.1.2		5	1.1	even	1	trivial
768.6.a.bb.1.4		5	8.3	odd	2
768.6.a.bc.1.2		5	4.3	odd	2
768.6.a.bd.1.4		5	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -66.8717 q^{5} +230.960 q^{7} +81.0000 q^{9} -360.698 q^{11} +674.528 q^{13} +601.845 q^{15} -765.210 q^{17} -2277.37 q^{19} -2078.64 q^{21} +465.702 q^{23} +1346.82 q^{25} -729.000 q^{27} +1430.14 q^{29} +3327.79 q^{31} +3246.28 q^{33} -15444.7 q^{35} -5736.07 q^{37} -6070.75 q^{39} +12666.7 q^{41} +2017.82 q^{43} -5416.61 q^{45} -1605.83 q^{47} +36535.6 q^{49} +6886.89 q^{51} +11998.3 q^{53} +24120.5 q^{55} +20496.4 q^{57} -24401.4 q^{59} +19670.5 q^{61} +18707.8 q^{63} -45106.8 q^{65} +50239.4 q^{67} -4191.32 q^{69} +16572.5 q^{71} -14263.2 q^{73} -12121.4 q^{75} -83306.8 q^{77} -47850.0 q^{79} +6561.00 q^{81} +60126.7 q^{83} +51170.9 q^{85} -12871.3 q^{87} -139147. q^{89} +155789. q^{91} -29950.1 q^{93} +152292. q^{95} +85226.4 q^{97} -29216.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 45 q^{3} - 50 q^{5} + 98 q^{7} + 405 q^{9} - 676 q^{13} + 450 q^{15} - 202 q^{17} - 716 q^{19} - 882 q^{21} - 836 q^{23} + 4683 q^{25} - 3645 q^{27} - 5046 q^{29} + 9346 q^{31} - 436 q^{35} - 10952 q^{37}+ \cdots + 29686 q^{97}+O(q^{100}) \) 5 * q - 45 * q^3 - 50 * q^5 + 98 * q^7 + 405 * q^9 - 676 * q^13 + 450 * q^15 - 202 * q^17 - 716 * q^19 - 882 * q^21 - 836 * q^23 + 4683 * q^25 - 3645 * q^27 - 5046 * q^29 + 9346 * q^31 - 436 * q^35 - 10952 * q^37 + 6084 * q^39 + 1238 * q^41 - 1676 * q^43 - 4050 * q^45 + 32628 * q^47 + 4689 * q^49 + 1818 * q^51 + 8490 * q^53 + 51712 * q^55 + 6444 * q^57 + 15884 * q^59 + 13824 * q^61 + 7938 * q^63 + 25768 * q^65 - 26652 * q^67 + 7524 * q^69 + 136116 * q^71 + 13694 * q^73 - 42147 * q^75 - 89856 * q^77 + 126154 * q^79 + 32805 * q^81 + 64720 * q^83 - 164636 * q^85 + 45414 * q^87 - 87790 * q^89 + 179992 * q^91 - 84114 * q^93 + 121592 * q^95 + 29686 * q^97

Introduction

L-functions

Modular forms

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Database

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