LMFDB - Embedded newform 387.6.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,6,Mod(1,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("387.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	387.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$62.0685382676$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 $ x^8 - 4x^7 - 173x^6 + 462x^5 + 9118x^4 - 14192x^3 - 167688x^2 + 106368*x + 681984
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.65705$ of defining polynomial
Character	$\chi$	$=$	387.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.65705 q^{2} -18.6260 q^{4} +107.102 q^{5} -25.5214 q^{7} +185.142 q^{8} +O(q^{10})$	\(q-3.65705 q^{2} -18.6260 q^{4} +107.102 q^{5} -25.5214 q^{7} +185.142 q^{8} -391.677 q^{10} -512.073 q^{11} +862.516 q^{13} +93.3331 q^{14} -81.0406 q^{16} +1521.49 q^{17} -1543.11 q^{19} -1994.88 q^{20} +1872.67 q^{22} +3126.31 q^{23} +8345.85 q^{25} -3154.26 q^{26} +475.362 q^{28} +947.120 q^{29} +339.499 q^{31} -5628.17 q^{32} -5564.17 q^{34} -2733.40 q^{35} -7448.67 q^{37} +5643.25 q^{38} +19829.1 q^{40} -5116.45 q^{41} -1849.00 q^{43} +9537.86 q^{44} -11433.1 q^{46} +17159.9 q^{47} -16155.7 q^{49} -30521.2 q^{50} -16065.2 q^{52} -18090.4 q^{53} -54844.0 q^{55} -4725.08 q^{56} -3463.66 q^{58} -17031.6 q^{59} +10664.9 q^{61} -1241.56 q^{62} +23175.8 q^{64} +92377.3 q^{65} -8799.60 q^{67} -28339.3 q^{68} +9996.16 q^{70} +77057.3 q^{71} -7964.75 q^{73} +27240.1 q^{74} +28742.0 q^{76} +13068.8 q^{77} +68997.0 q^{79} -8679.62 q^{80} +18711.1 q^{82} -40813.4 q^{83} +162955. q^{85} +6761.88 q^{86} -94806.0 q^{88} +83692.1 q^{89} -22012.6 q^{91} -58230.7 q^{92} -62754.7 q^{94} -165271. q^{95} +159824. q^{97} +59082.0 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8}+O(q^{10}) $ 8 * q + 12 * q^2 + 122 * q^4 + 212 * q^5 - 136 * q^7 + 666 * q^8	$ 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8} - 617 q^{10} + 532 q^{11} - 2492 q^{13} + 4240 q^{14} + 1882 q^{16} + 2534 q^{17} - 1678 q^{19} + 2607 q^{20} + 11502 q^{22} + 2488 q^{23} + 4378 q^{25} - 4586 q^{26} + 18640 q^{28} + 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 30007 q^{34} - 5640 q^{35} - 3772 q^{37} + 6559 q^{38} + 14869 q^{40} + 10698 q^{41} - 14792 q^{43} + 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 7188 q^{49} - 26877 q^{50} - 60736 q^{52} + 62352 q^{53} - 49552 q^{55} + 144528 q^{56} + 52951 q^{58} + 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 153858 q^{64} + 5000 q^{65} + 27784 q^{67} - 40507 q^{68} + 185910 q^{70} + 9504 q^{71} + 14260 q^{73} + 15239 q^{74} + 1279 q^{76} + 218140 q^{77} + 160248 q^{79} + 1291 q^{80} - 47781 q^{82} + 77176 q^{83} + 141096 q^{85} - 22188 q^{86} + 129544 q^{88} + 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 248737 q^{94} - 135884 q^{95} + 144742 q^{97} + 292244 q^{98}+O(q^{100}) $ 8 * q + 12 * q^2 + 122 * q^4 + 212 * q^5 - 136 * q^7 + 666 * q^8 - 617 * q^10 + 532 * q^11 - 2492 * q^13 + 4240 * q^14 + 1882 * q^16 + 2534 * q^17 - 1678 * q^19 + 2607 * q^20 + 11502 * q^22 + 2488 * q^23 + 4378 * q^25 - 4586 * q^26 + 18640 * q^28 + 4360 * q^29 + 5704 * q^31 + 18294 * q^32 + 30007 * q^34 - 5640 * q^35 - 3772 * q^37 + 6559 * q^38 + 14869 * q^40 + 10698 * q^41 - 14792 * q^43 + 356 * q^44 - 19389 * q^46 + 77864 * q^47 + 7188 * q^49 - 26877 * q^50 - 60736 * q^52 + 62352 * q^53 - 49552 * q^55 + 144528 * q^56 + 52951 * q^58 + 26224 * q^59 - 82540 * q^61 + 9023 * q^62 + 153858 * q^64 + 5000 * q^65 + 27784 * q^67 - 40507 * q^68 + 185910 * q^70 + 9504 * q^71 + 14260 * q^73 + 15239 * q^74 + 1279 * q^76 + 218140 * q^77 + 160248 * q^79 + 1291 * q^80 - 47781 * q^82 + 77176 * q^83 + 141096 * q^85 - 22188 * q^86 + 129544 * q^88 + 265692 * q^89 + 401148 * q^91 - 190391 * q^92 + 248737 * q^94 - 135884 * q^95 + 144742 * q^97 + 292244 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−3.65705	−0.646481	−0.323241	−	0.946317i	$-0.604772\pi$
$2$	−3.65705	−0.646481	−0.323241	+	0.946317i	$0.604772\pi$
$3$	0	0
$3$	0	0
$4$	−18.6260	−0.582062
$5$	107.102	1.91590	0.957950	−	0.286936i	$-0.0926367\pi$
$5$	107.102	1.91590	0.957950	+	0.286936i	$0.0926367\pi$
$6$	0	0
$7$	−25.5214	−0.196861	−0.0984305	−	0.995144i	$-0.531382\pi$
$7$	−25.5214	−0.196861	−0.0984305	+	0.995144i	$0.531382\pi$
$8$	185.142	1.02277
$9$	0	0
$10$	−391.677	−1.23859
$11$	−512.073	−1.27600	−0.637999	−	0.770037i	$-0.720238\pi$
$11$	−512.073	−1.27600	−0.637999	+	0.770037i	$0.720238\pi$
$12$	0	0
$13$	862.516	1.41550	0.707749	−	0.706464i	$-0.249711\pi$
$13$	862.516	1.41550	0.707749	+	0.706464i	$0.249711\pi$
$14$	93.3331	0.127267
$15$	0	0
$16$	−81.0406	−0.0791413
$17$	1521.49	1.27687	0.638435	−	0.769675i	$-0.279582\pi$
$17$	1521.49	1.27687	0.638435	+	0.769675i	$0.279582\pi$
$18$	0	0
$19$	−1543.11	−0.980650	−0.490325	−	0.871540i	$-0.663122\pi$
$19$	−1543.11	−0.980650	−0.490325	+	0.871540i	$0.663122\pi$
$20$	−1994.88	−1.11517
$21$	0	0
$22$	1872.67	0.824908
$23$	3126.31	1.23229	0.616145	−	0.787633i	$-0.288694\pi$
$23$	3126.31	1.23229	0.616145	+	0.787633i	$0.288694\pi$
$24$	0	0
$25$	8345.85	2.67067
$26$	−3154.26	−0.915092
$27$	0	0
$28$	475.362	0.114585
$29$	947.120	0.209127	0.104563	−	0.994518i	$-0.466655\pi$
$29$	947.120	0.209127	0.104563	+	0.994518i	$0.466655\pi$
$30$	0	0
$31$	339.499	0.0634504	0.0317252	−	0.999497i	$-0.489900\pi$
$31$	339.499	0.0634504	0.0317252	+	0.999497i	$0.489900\pi$
$32$	−5628.17	−0.971610
$33$	0	0
$34$	−5564.17	−0.825473
$35$	−2733.40	−0.377166
$36$	0	0
$37$	−7448.67	−0.894488	−0.447244	−	0.894412i	$-0.647594\pi$
$37$	−7448.67	−0.894488	−0.447244	+	0.894412i	$0.647594\pi$
$38$	5643.25	0.633972
$39$	0	0
$40$	19829.1	1.95953
$41$	−5116.45	−0.475345	−0.237672	−	0.971345i	$-0.576384\pi$
$41$	−5116.45	−0.475345	−0.237672	+	0.971345i	$0.576384\pi$
$42$	0	0
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	9537.86	0.742710
$45$	0	0
$46$	−11433.1	−0.796652
$47$	17159.9	1.13311	0.566553	−	0.824025i	$-0.308276\pi$
$47$	17159.9	1.13311	0.566553	+	0.824025i	$0.308276\pi$
$48$	0	0
$49$	−16155.7	−0.961246
$50$	−30521.2	−1.72654
$51$	0	0
$52$	−16065.2	−0.823907
$53$	−18090.4	−0.884623	−0.442312	−	0.896861i	$-0.645841\pi$
$53$	−18090.4	−0.884623	−0.442312	+	0.896861i	$0.645841\pi$
$54$	0	0
$55$	−54844.0	−2.44468
$56$	−4725.08	−0.201344
$57$	0	0
$58$	−3463.66	−0.135197
$59$	−17031.6	−0.636980	−0.318490	−	0.947926i	$-0.603176\pi$
$59$	−17031.6	−0.636980	−0.318490	+	0.947926i	$0.603176\pi$
$60$	0	0
$61$	10664.9	0.366972	0.183486	−	0.983022i	$-0.441262\pi$
$61$	10664.9	0.366972	0.183486	+	0.983022i	$0.441262\pi$
$62$	−1241.56	−0.0410195
$63$	0	0
$64$	23175.8	0.707269
$65$	92377.3	2.71195
$66$	0	0
$67$	−8799.60	−0.239484	−0.119742	−	0.992805i	$-0.538207\pi$
$67$	−8799.60	−0.239484	−0.119742	+	0.992805i	$0.538207\pi$
$68$	−28339.3	−0.743218
$69$	0	0
$70$	9996.16	0.243831
$71$	77057.3	1.81413	0.907064	−	0.420992i	$-0.138318\pi$
$71$	77057.3	1.81413	0.907064	+	0.420992i	$0.138318\pi$
$72$	0	0
$73$	−7964.75	−0.174930	−0.0874652	−	0.996168i	$-0.527877\pi$
$73$	−7964.75	−0.174930	−0.0874652	+	0.996168i	$0.527877\pi$
$74$	27240.1	0.578269
$75$	0	0
$76$	28742.0	0.570800
$77$	13068.8	0.251194
$78$	0	0
$79$	68997.0	1.24383	0.621917	−	0.783083i	$-0.286354\pi$
$79$	68997.0	1.24383	0.621917	+	0.783083i	$0.286354\pi$
$80$	−8679.62	−0.151627
$81$	0	0
$82$	18711.1	0.307301
$83$	−40813.4	−0.650290	−0.325145	−	0.945664i	$-0.605413\pi$
$83$	−40813.4	−0.650290	−0.325145	+	0.945664i	$0.605413\pi$
$84$	0	0
$85$	162955.	2.44636
$86$	6761.88	0.0985874
$87$	0	0
$88$	−94806.0	−1.30506
$89$	83692.1	1.11998	0.559989	−	0.828500i	$-0.310805\pi$
$89$	83692.1	1.11998	0.559989	+	0.828500i	$0.310805\pi$
$90$	0	0
$91$	−22012.6	−0.278656
$92$	−58230.7	−0.717269
$93$	0	0
$94$	−62754.7	−0.732532
$95$	−165271.	−1.87883
$96$	0	0
$97$	159824.	1.72469	0.862346	−	0.506320i	$-0.168994\pi$
$97$	159824.	1.72469	0.862346	+	0.506320i	$0.168994\pi$
$98$	59082.0	0.621427
$99$	0	0
$100$	−155450.	−1.55450
$101$	168968.	1.64817	0.824085	−	0.566466i	$-0.191690\pi$
$101$	168968.	1.64817	0.824085	+	0.566466i	$0.191690\pi$
$102$	0	0
$103$	−145084.	−1.34749	−0.673747	−	0.738962i	$-0.735316\pi$
$103$	−145084.	−1.34749	−0.673747	+	0.738962i	$0.735316\pi$
$104$	159688.	1.44773
$105$	0	0
$106$	66157.5	0.571892
$107$	−72693.1	−0.613810	−0.306905	−	0.951740i	$-0.599293\pi$
$107$	−72693.1	−0.613810	−0.306905	+	0.951740i	$0.599293\pi$
$108$	0	0
$109$	−3254.39	−0.0262364	−0.0131182	−	0.999914i	$-0.504176\pi$
$109$	−3254.39	−0.0262364	−0.0131182	+	0.999914i	$0.504176\pi$
$110$	200567.	1.58044
$111$	0	0
$112$	2068.27	0.0155798
$113$	−116394.	−0.857503	−0.428752	−	0.903422i	$-0.641046\pi$
$113$	−116394.	−0.857503	−0.428752	+	0.903422i	$0.641046\pi$
$114$	0	0
$115$	334834.	2.36094
$116$	−17641.0	−0.121725
$117$	0	0
$118$	62285.4	0.411795
$119$	−38830.6	−0.251366
$120$	0	0
$121$	101167.	0.628170
$122$	−39002.1	−0.237240
$123$	0	0
$124$	−6323.50	−0.0369321
$125$	559163.	3.20084
$126$	0	0
$127$	−6451.01	−0.0354910	−0.0177455	−	0.999843i	$-0.505649\pi$
$127$	−6451.01	−0.0354910	−0.0177455	+	0.999843i	$0.505649\pi$
$128$	95346.3	0.514374
$129$	0	0
$130$	−337828.	−1.75322
$131$	48048.9	0.244628	0.122314	−	0.992491i	$-0.460969\pi$
$131$	48048.9	0.244628	0.122314	+	0.992491i	$0.460969\pi$
$132$	0	0
$133$	39382.5	0.193052
$134$	32180.6	0.154822
$135$	0	0
$136$	281691.	1.30595
$137$	103084.	0.469233	0.234616	−	0.972088i	$-0.424617\pi$
$137$	103084.	0.469233	0.234616	+	0.972088i	$0.424617\pi$
$138$	0	0
$139$	219626.	0.964156	0.482078	−	0.876128i	$-0.339882\pi$
$139$	219626.	0.964156	0.482078	+	0.876128i	$0.339882\pi$
$140$	50912.2	0.219534
$141$	0	0
$142$	−281802.	−1.17280
$143$	−441671.	−1.80617
$144$	0	0
$145$	101438.	0.400666
$146$	29127.5	0.113089
$147$	0	0
$148$	138739.	0.520648
$149$	335627.	1.23849	0.619243	−	0.785200i	$-0.287440\pi$
$149$	335627.	1.23849	0.619243	+	0.785200i	$0.287440\pi$
$150$	0	0
$151$	−84920.3	−0.303088	−0.151544	−	0.988450i	$-0.548425\pi$
$151$	−84920.3	−0.303088	−0.151544	+	0.988450i	$0.548425\pi$
$152$	−285695.	−1.00298
$153$	0	0
$154$	−47793.3	−0.162392
$155$	36361.0	0.121565
$156$	0	0
$157$	313767.	1.01592	0.507958	−	0.861382i	$-0.330401\pi$
$157$	313767.	1.01592	0.507958	+	0.861382i	$0.330401\pi$
$158$	−252325.	−0.804115
$159$	0	0
$160$	−602788.	−1.86151
$161$	−79787.9	−0.242590
$162$	0	0
$163$	239642.	0.706469	0.353235	−	0.935535i	$-0.385082\pi$
$163$	239642.	0.706469	0.353235	+	0.935535i	$0.385082\pi$
$164$	95298.9	0.276680
$165$	0	0
$166$	149257.	0.420400
$167$	506675.	1.40585	0.702923	−	0.711266i	$-0.251878\pi$
$167$	506675.	1.40585	0.702923	+	0.711266i	$0.251878\pi$
$168$	0	0
$169$	372641.	1.00363
$170$	−595933.	−1.58152
$171$	0	0
$172$	34439.5	0.0887637
$173$	427174.	1.08515	0.542575	−	0.840007i	$-0.317449\pi$
$173$	427174.	1.08515	0.542575	+	0.840007i	$0.317449\pi$
$174$	0	0
$175$	−212998.	−0.525751
$176$	41498.7	0.100984
$177$	0	0
$178$	−306066.	−0.724044
$179$	−11674.9	−0.0272345	−0.0136172	−	0.999907i	$-0.504335\pi$
$179$	−11674.9	−0.0272345	−0.0136172	+	0.999907i	$0.504335\pi$
$180$	0	0
$181$	71691.3	0.162656	0.0813280	−	0.996687i	$-0.474084\pi$
$181$	71691.3	0.162656	0.0813280	+	0.996687i	$0.474084\pi$
$182$	80501.3	0.180146
$183$	0	0
$184$	578811.	1.26035
$185$	−797768.	−1.71375
$186$	0	0
$187$	−779114.	−1.62928
$188$	−319621.	−0.659539
$189$	0	0
$190$	604403.	1.21463
$191$	−212639.	−0.421755	−0.210877	−	0.977513i	$-0.567632\pi$
$191$	−212639.	−0.421755	−0.210877	+	0.977513i	$0.567632\pi$
$192$	0	0
$193$	−137469.	−0.265651	−0.132825	−	0.991139i	$-0.542405\pi$
$193$	−137469.	−0.265651	−0.132825	+	0.991139i	$0.542405\pi$
$194$	−584483.	−1.11498
$195$	0	0
$196$	300915.	0.559505
$197$	28517.3	0.0523531	0.0261766	−	0.999657i	$-0.491667\pi$
$197$	28517.3	0.0523531	0.0261766	+	0.999657i	$0.491667\pi$
$198$	0	0
$199$	−353575.	−0.632921	−0.316460	−	0.948606i	$-0.602494\pi$
$199$	−353575.	−0.632921	−0.316460	+	0.948606i	$0.602494\pi$
$200$	1.54516e6	2.73149
$201$	0	0
$202$	−617926.	−1.06551
$203$	−24171.8	−0.0411689
$204$	0	0
$205$	−547982.	−0.910713
$206$	530580.	0.871130
$207$	0	0
$208$	−69898.9	−0.112024
$209$	790187.	1.25131
$210$	0	0
$211$	327840.	0.506939	0.253469	−	0.967343i	$-0.418428\pi$
$211$	327840.	0.506939	0.253469	+	0.967343i	$0.418428\pi$
$212$	336952.	0.514906
$213$	0	0
$214$	265842.	0.396816
$215$	−198032.	−0.292172
$216$	0	0
$217$	−8664.49	−0.0124909
$218$	11901.5	0.0169613
$219$	0	0
$220$	1.02152e6	1.42296
$221$	1.31231e6	1.80741
$222$	0	0
$223$	247951.	0.333890	0.166945	−	0.985966i	$-0.446610\pi$
$223$	247951.	0.333890	0.166945	+	0.985966i	$0.446610\pi$
$224$	143639.	0.191272
$225$	0	0
$226$	425660.	0.554359
$227$	868910.	1.11921	0.559603	−	0.828761i	$-0.310954\pi$
$227$	868910.	1.11921	0.559603	+	0.828761i	$0.310954\pi$
$228$	0	0
$229$	238721.	0.300817	0.150408	−	0.988624i	$-0.451941\pi$
$229$	238721.	0.300817	0.150408	+	0.988624i	$0.451941\pi$
$230$	−1.22451e6	−1.52631
$231$	0	0
$232$	175351.	0.213889
$233$	−507726.	−0.612688	−0.306344	−	0.951921i	$-0.599106\pi$
$233$	−507726.	−0.612688	−0.306344	+	0.951921i	$0.599106\pi$
$234$	0	0
$235$	1.83786e6	2.17092
$236$	317231.	0.370762
$237$	0	0
$238$	142005.	0.162503
$239$	−143880.	−0.162932	−0.0814660	−	0.996676i	$-0.525960\pi$
$239$	−143880.	−0.162932	−0.0814660	+	0.996676i	$0.525960\pi$
$240$	0	0
$241$	398237.	0.441671	0.220835	−	0.975311i	$-0.429122\pi$
$241$	398237.	0.441671	0.220835	+	0.975311i	$0.429122\pi$
$242$	−369974.	−0.406100
$243$	0	0
$244$	−198645.	−0.213600
$245$	−1.73030e6	−1.84165
$246$	0	0
$247$	−1.33096e6	−1.38811
$248$	62855.4	0.0648953
$249$	0	0
$250$	−2.04489e6	−2.06928
$251$	−1.65625e6	−1.65937	−0.829684	−	0.558233i	$-0.811480\pi$
$251$	−1.65625e6	−1.65937	−0.829684	+	0.558233i	$0.811480\pi$
$252$	0	0
$253$	−1.60090e6	−1.57240
$254$	23591.7	0.0229443
$255$	0	0
$256$	−1.09031e6	−1.03980
$257$	−32691.4	−0.0308746	−0.0154373	−	0.999881i	$-0.504914\pi$
$257$	−32691.4	−0.0308746	−0.0154373	+	0.999881i	$0.504914\pi$
$258$	0	0
$259$	190101.	0.176090
$260$	−1.72062e6	−1.57852
$261$	0	0
$262$	−175717.	−0.158147
$263$	−1.02727e6	−0.915791	−0.457895	−	0.889006i	$-0.651397\pi$
$263$	−1.02727e6	−0.915791	−0.457895	+	0.889006i	$0.651397\pi$
$264$	0	0
$265$	−1.93752e6	−1.69485
$266$	−144024.	−0.124804
$267$	0	0
$268$	163901.	0.139394
$269$	1.88917e6	1.59180	0.795902	−	0.605426i	$-0.206997\pi$
$269$	1.88917e6	1.59180	0.795902	+	0.605426i	$0.206997\pi$
$270$	0	0
$271$	605259.	0.500631	0.250316	−	0.968164i	$-0.419466\pi$
$271$	605259.	0.500631	0.250316	+	0.968164i	$0.419466\pi$
$272$	−123303.	−0.101053
$273$	0	0
$274$	−376982.	−0.303350
$275$	−4.27368e6	−3.40777
$276$	0	0
$277$	−1.16555e6	−0.912710	−0.456355	−	0.889798i	$-0.650845\pi$
$277$	−1.16555e6	−0.912710	−0.456355	+	0.889798i	$0.650845\pi$
$278$	−803184.	−0.623309
$279$	0	0
$280$	−506066.	−0.385755
$281$	−938873.	−0.709318	−0.354659	−	0.934996i	$-0.615403\pi$
$281$	−938873.	−0.709318	−0.354659	+	0.934996i	$0.615403\pi$
$282$	0	0
$283$	−2.54537e6	−1.88923	−0.944616	−	0.328179i	$-0.893565\pi$
$283$	−2.54537e6	−1.88923	−0.944616	+	0.328179i	$0.893565\pi$
$284$	−1.43527e6	−1.05594
$285$	0	0
$286$	1.61521e6	1.16766
$287$	130579.	0.0935768
$288$	0	0
$289$	895076.	0.630399
$290$	−370965.	−0.259023
$291$	0	0
$292$	148351.	0.101820
$293$	1.61560e6	1.09942	0.549712	−	0.835354i	$-0.314737\pi$
$293$	1.61560e6	1.09942	0.549712	+	0.835354i	$0.314737\pi$
$294$	0	0
$295$	−1.82412e6	−1.22039
$296$	−1.37906e6	−0.914858
$297$	0	0
$298$	−1.22740e6	−0.800657
$299$	2.69650e6	1.74430
$300$	0	0
$301$	47189.1	0.0300210
$302$	310558.	0.195941
$303$	0	0
$304$	125055.	0.0776099
$305$	1.14223e6	0.703081
$306$	0	0
$307$	−1.49055e6	−0.902613	−0.451307	−	0.892369i	$-0.649042\pi$
$307$	−1.49055e6	−0.902613	−0.451307	+	0.892369i	$0.649042\pi$
$308$	−243420.	−0.146211
$309$	0	0
$310$	−132974.	−0.0785892
$311$	754959.	0.442611	0.221305	−	0.975205i	$-0.428968\pi$
$311$	754959.	0.442611	0.221305	+	0.975205i	$0.428968\pi$
$312$	0	0
$313$	1.31681e6	0.759735	0.379867	−	0.925041i	$-0.375970\pi$
$313$	1.31681e6	0.759735	0.379867	+	0.925041i	$0.375970\pi$
$314$	−1.14746e6	−0.656770
$315$	0	0
$316$	−1.28514e6	−0.723988
$317$	2.19577e6	1.22727	0.613633	−	0.789591i	$-0.289707\pi$
$317$	2.19577e6	1.22727	0.613633	+	0.789591i	$0.289707\pi$
$318$	0	0
$319$	−484994.	−0.266845
$320$	2.48217e6	1.35506
$321$	0	0
$322$	291788.	0.156830
$323$	−2.34783e6	−1.25216
$324$	0	0
$325$	7.19843e6	3.78033
$326$	−876381.	−0.456719
$327$	0	0
$328$	−947268.	−0.486170
$329$	−437946.	−0.223065
$330$	0	0
$331$	727999.	0.365225	0.182613	−	0.983185i	$-0.441545\pi$
$331$	727999.	0.365225	0.182613	+	0.983185i	$0.441545\pi$
$332$	760190.	0.378509
$333$	0	0
$334$	−1.85293e6	−0.908853
$335$	−942455.	−0.458827
$336$	0	0
$337$	626937.	0.300711	0.150355	−	0.988632i	$-0.451958\pi$
$337$	626937.	0.300711	0.150355	+	0.988632i	$0.451958\pi$
$338$	−1.36277e6	−0.648829
$339$	0	0
$340$	−3.03519e6	−1.42393
$341$	−173848.	−0.0809625
$342$	0	0
$343$	841254.	0.386093
$344$	−342327.	−0.155971
$345$	0	0
$346$	−1.56220e6	−0.701529
$347$	3.30393e6	1.47302	0.736508	−	0.676428i	$-0.236473\pi$
$347$	3.30393e6	1.47302	0.736508	+	0.676428i	$0.236473\pi$
$348$	0	0
$349$	−2.93486e6	−1.28980	−0.644902	−	0.764265i	$-0.723102\pi$
$349$	−2.93486e6	−1.28980	−0.644902	+	0.764265i	$0.723102\pi$
$350$	778943.	0.339888
$351$	0	0
$352$	2.88203e6	1.23977
$353$	−633691.	−0.270670	−0.135335	−	0.990800i	$-0.543211\pi$
$353$	−633691.	−0.270670	−0.135335	+	0.990800i	$0.543211\pi$
$354$	0	0
$355$	8.25300e6	3.47569
$356$	−1.55885e6	−0.651897
$357$	0	0
$358$	42695.6	0.0176066
$359$	−3.08062e6	−1.26154	−0.630771	−	0.775969i	$-0.717261\pi$
$359$	−3.08062e6	−1.26154	−0.630771	+	0.775969i	$0.717261\pi$
$360$	0	0
$361$	−94896.2	−0.0383249
$362$	−262179.	−0.105154
$363$	0	0
$364$	410007.	0.162195
$365$	−853042.	−0.335149
$366$	0	0
$367$	−1.32881e6	−0.514988	−0.257494	−	0.966280i	$-0.582897\pi$
$367$	−1.32881e6	−0.514988	−0.257494	+	0.966280i	$0.582897\pi$
$368$	−253358.	−0.0975250
$369$	0	0
$370$	2.91748e6	1.10791
$371$	461692.	0.174148
$372$	0	0
$373$	−1.91319e6	−0.712011	−0.356006	−	0.934484i	$-0.615862\pi$
$373$	−1.91319e6	−0.712011	−0.356006	+	0.934484i	$0.615862\pi$
$374$	2.84926e6	1.05330
$375$	0	0
$376$	3.17702e6	1.15891
$377$	816906.	0.296019
$378$	0	0
$379$	703691.	0.251642	0.125821	−	0.992053i	$-0.459843\pi$
$379$	703691.	0.251642	0.125821	+	0.992053i	$0.459843\pi$
$380$	3.07833e6	1.09359
$381$	0	0
$382$	777632.	0.272656
$383$	−1.73748e6	−0.605234	−0.302617	−	0.953112i	$-0.597860\pi$
$383$	−1.73748e6	−0.605234	−0.302617	+	0.953112i	$0.597860\pi$
$384$	0	0
$385$	1.39970e6	0.481263
$386$	502731.	0.171738
$387$	0	0
$388$	−2.97687e6	−1.00388
$389$	−3.93096e6	−1.31712	−0.658559	−	0.752529i	$-0.728834\pi$
$389$	−3.93096e6	−1.31712	−0.658559	+	0.752529i	$0.728834\pi$
$390$	0	0
$391$	4.75666e6	1.57347
$392$	−2.99109e6	−0.983136
$393$	0	0
$394$	−104289.	−0.0338453
$395$	7.38971e6	2.38306
$396$	0	0
$397$	3.99784e6	1.27306	0.636530	−	0.771252i	$-0.280369\pi$
$397$	3.99784e6	1.27306	0.636530	+	0.771252i	$0.280369\pi$
$398$	1.29304e6	0.409171
$399$	0	0
$400$	−676353.	−0.211360
$401$	−4.39985e6	−1.36640	−0.683198	−	0.730233i	$-0.739412\pi$
$401$	−4.39985e6	−1.36640	−0.683198	+	0.730233i	$0.739412\pi$
$402$	0	0
$403$	292823.	0.0898138
$404$	−3.14721e6	−0.959338
$405$	0	0
$406$	88397.6	0.0266149
$407$	3.81426e6	1.14136
$408$	0	0
$409$	−1.64907e6	−0.487451	−0.243726	−	0.969844i	$-0.578370\pi$
$409$	−1.64907e6	−0.487451	−0.243726	+	0.969844i	$0.578370\pi$
$410$	2.00400e6	0.588759
$411$	0	0
$412$	2.70234e6	0.784326
$413$	434671.	0.125396
$414$	0	0
$415$	−4.37120e6	−1.24589
$416$	−4.85439e6	−1.37531
$417$	0	0
$418$	−2.88975e6	−0.808947
$419$	5.93011e6	1.65017	0.825083	−	0.565012i	$-0.191129\pi$
$419$	5.93011e6	1.65017	0.825083	+	0.565012i	$0.191129\pi$
$420$	0	0
$421$	4.85544e6	1.33513	0.667565	−	0.744552i	$-0.267337\pi$
$421$	4.85544e6	1.33513	0.667565	+	0.744552i	$0.267337\pi$
$422$	−1.19893e6	−0.327726
$423$	0	0
$424$	−3.34929e6	−0.904769
$425$	1.26981e7	3.41010
$426$	0	0
$427$	−272184.	−0.0722424
$428$	1.35398e6	0.357275
$429$	0	0
$430$	724212.	0.188884
$431$	−2.59168e6	−0.672029	−0.336015	−	0.941857i	$-0.609079\pi$
$431$	−2.59168e6	−0.672029	−0.336015	+	0.941857i	$0.609079\pi$
$432$	0	0
$433$	−5.08691e6	−1.30387	−0.651934	−	0.758275i	$-0.726042\pi$
$433$	−5.08691e6	−1.30387	−0.651934	+	0.758275i	$0.726042\pi$
$434$	31686.5	0.00807513
$435$	0	0
$436$	60616.3	0.0152712
$437$	−4.82426e6	−1.20845
$438$	0	0
$439$	−5.71552e6	−1.41545	−0.707725	−	0.706488i	$-0.750278\pi$
$439$	−5.71552e6	−1.41545	−0.707725	+	0.706488i	$0.750278\pi$
$440$	−1.01539e7	−2.50036
$441$	0	0
$442$	−4.79918e6	−1.16845
$443$	4.81790e6	1.16640	0.583201	−	0.812328i	$-0.301800\pi$
$443$	4.81790e6	1.16640	0.583201	+	0.812328i	$0.301800\pi$
$444$	0	0
$445$	8.96359e6	2.14576
$446$	−906769.	−0.215854
$447$	0	0
$448$	−591479.	−0.139234
$449$	2.36680e6	0.554046	0.277023	−	0.960863i	$-0.410652\pi$
$449$	2.36680e6	0.554046	0.277023	+	0.960863i	$0.410652\pi$
$450$	0	0
$451$	2.61999e6	0.606539
$452$	2.16796e6	0.499120
$453$	0	0
$454$	−3.17765e6	−0.723546
$455$	−2.35760e6	−0.533877
$456$	0	0
$457$	1.78378e6	0.399531	0.199766	−	0.979844i	$-0.435982\pi$
$457$	1.78378e6	0.399531	0.199766	+	0.979844i	$0.435982\pi$
$458$	−873015.	−0.194472
$459$	0	0
$460$	−6.23662e6	−1.37422
$461$	4.26575e6	0.934852	0.467426	−	0.884032i	$-0.345181\pi$
$461$	4.26575e6	0.934852	0.467426	+	0.884032i	$0.345181\pi$
$462$	0	0
$463$	−2.33897e6	−0.507074	−0.253537	−	0.967326i	$-0.581594\pi$
$463$	−2.33897e6	−0.507074	−0.253537	+	0.967326i	$0.581594\pi$
$464$	−76755.2	−0.0165506
$465$	0	0
$466$	1.85678e6	0.396091
$467$	6.84641e6	1.45268	0.726342	−	0.687334i	$-0.241219\pi$
$467$	6.84641e6	1.45268	0.726342	+	0.687334i	$0.241219\pi$
$468$	0	0
$469$	224578.	0.0471450
$470$	−6.72116e6	−1.40346
$471$	0	0
$472$	−3.15326e6	−0.651486
$473$	946822.	0.194588
$474$	0	0
$475$	−1.28786e7	−2.61899
$476$	723258.	0.146311
$477$	0	0
$478$	526177.	0.105332
$479$	−9.12451e6	−1.81707	−0.908533	−	0.417814i	$-0.862797\pi$
$479$	−9.12451e6	−1.81707	−0.908533	+	0.417814i	$0.862797\pi$
$480$	0	0
$481$	−6.42460e6	−1.26614
$482$	−1.45637e6	−0.285532
$483$	0	0
$484$	−1.88434e6	−0.365634
$485$	1.71174e7	3.30434
$486$	0	0
$487$	2.68151e6	0.512338	0.256169	−	0.966632i	$-0.417540\pi$
$487$	2.68151e6	0.512338	0.256169	+	0.966632i	$0.417540\pi$
$488$	1.97452e6	0.375329
$489$	0	0
$490$	6.32781e6	1.19059
$491$	−4.83155e6	−0.904446	−0.452223	−	0.891905i	$-0.649369\pi$
$491$	−4.83155e6	−0.904446	−0.452223	+	0.891905i	$0.649369\pi$
$492$	0	0
$493$	1.44103e6	0.267028
$494$	4.86739e6	0.897385
$495$	0	0
$496$	−27513.2	−0.00502154
$497$	−1.96661e6	−0.357131
$498$	0	0
$499$	8.26537e6	1.48597	0.742987	−	0.669306i	$-0.233409\pi$
$499$	8.26537e6	1.48597	0.742987	+	0.669306i	$0.233409\pi$
$500$	−1.04150e7	−1.86309
$501$	0	0
$502$	6.05701e6	1.07275
$503$	1.06078e7	1.86941	0.934703	−	0.355429i	$-0.115665\pi$
$503$	1.06078e7	1.86941	0.934703	+	0.355429i	$0.115665\pi$
$504$	0	0
$505$	1.80969e7	3.15773
$506$	5.85457e6	1.01653
$507$	0	0
$508$	120157.	0.0206580
$509$	1.89756e6	0.324639	0.162320	−	0.986738i	$-0.448102\pi$
$509$	1.89756e6	0.324639	0.162320	+	0.986738i	$0.448102\pi$
$510$	0	0
$511$	203272.	0.0344370
$512$	936238.	0.157838
$513$	0	0
$514$	119554.	0.0199598
$515$	−1.55388e7	−2.58166
$516$	0	0
$517$	−8.78713e6	−1.44584
$518$	−695207.	−0.113839
$519$	0	0
$520$	1.71029e7	2.77371
$521$	1.76155e6	0.284316	0.142158	−	0.989844i	$-0.454596\pi$
$521$	1.76155e6	0.284316	0.142158	+	0.989844i	$0.454596\pi$
$522$	0	0
$523$	−7.12167e6	−1.13849	−0.569243	−	0.822170i	$-0.692764\pi$
$523$	−7.12167e6	−1.13849	−0.569243	+	0.822170i	$0.692764\pi$
$524$	−894959.	−0.142388
$525$	0	0
$526$	3.75679e6	0.592041
$527$	516544.	0.0810179
$528$	0	0
$529$	3.33749e6	0.518538
$530$	7.08560e6	1.09569
$531$	0	0
$532$	−733538.	−0.112368
$533$	−4.41302e6	−0.672849
$534$	0	0
$535$	−7.78558e6	−1.17600
$536$	−1.62917e6	−0.244937
$537$	0	0
$538$	−6.90877e6	−1.02907
$539$	8.27287e6	1.22655
$540$	0	0
$541$	6.26443e6	0.920214	0.460107	−	0.887864i	$-0.347811\pi$
$541$	6.26443e6	0.920214	0.460107	+	0.887864i	$0.347811\pi$
$542$	−2.21346e6	−0.323649
$543$	0	0
$544$	−8.56320e6	−1.24062
$545$	−348552.	−0.0502663
$546$	0	0
$547$	−3.19734e6	−0.456899	−0.228449	−	0.973556i	$-0.573366\pi$
$547$	−3.19734e6	−0.456899	−0.228449	+	0.973556i	$0.573366\pi$
$548$	−1.92003e6	−0.273123
$549$	0	0
$550$	1.56291e7	2.20306
$551$	−1.46151e6	−0.205080
$552$	0	0
$553$	−1.76090e6	−0.244862
$554$	4.26248e6	0.590049
$555$	0	0
$556$	−4.09076e6	−0.561199
$557$	−1.13299e7	−1.54735	−0.773677	−	0.633581i	$-0.781584\pi$
$557$	−1.13299e7	−1.54735	−0.773677	+	0.633581i	$0.781584\pi$
$558$	0	0
$559$	−1.59479e6	−0.215861
$560$	221516.	0.0298494
$561$	0	0
$562$	3.43350e6	0.458561
$563$	4.20790e6	0.559492	0.279746	−	0.960074i	$-0.409750\pi$
$563$	4.20790e6	0.559492	0.279746	+	0.960074i	$0.409750\pi$
$564$	0	0
$565$	−1.24661e7	−1.64289
$566$	9.30855e6	1.22135
$567$	0	0
$568$	1.42665e7	1.85544
$569$	3.28836e6	0.425794	0.212897	−	0.977075i	$-0.431710\pi$
$569$	3.28836e6	0.425794	0.212897	+	0.977075i	$0.431710\pi$
$570$	0	0
$571$	−2.54135e6	−0.326192	−0.163096	−	0.986610i	$-0.552148\pi$
$571$	−2.54135e6	−0.326192	−0.163096	+	0.986610i	$0.552148\pi$
$572$	8.22656e6	1.05130
$573$	0	0
$574$	−477534.	−0.0604956
$575$	2.60917e7	3.29104
$576$	0	0
$577$	−1.69533e6	−0.211990	−0.105995	−	0.994367i	$-0.533803\pi$
$577$	−1.69533e6	−0.211990	−0.105995	+	0.994367i	$0.533803\pi$
$578$	−3.27334e6	−0.407541
$579$	0	0
$580$	−1.88939e6	−0.233213
$581$	1.04161e6	0.128017
$582$	0	0
$583$	9.26360e6	1.12878
$584$	−1.47461e6	−0.178914
$585$	0	0
$586$	−5.90834e6	−0.710757
$587$	−6.20781e6	−0.743607	−0.371803	−	0.928311i	$-0.621260\pi$
$587$	−6.20781e6	−0.743607	−0.371803	+	0.928311i	$0.621260\pi$
$588$	0	0
$589$	−523886.	−0.0622226
$590$	6.67090e6	0.788959
$591$	0	0
$592$	603645.	0.0707909
$593$	−3.01403e6	−0.351974	−0.175987	−	0.984393i	$-0.556312\pi$
$593$	−3.01403e6	−0.351974	−0.175987	+	0.984393i	$0.556312\pi$
$594$	0	0
$595$	−4.15884e6	−0.481592
$596$	−6.25138e6	−0.720876
$597$	0	0
$598$	−9.86122e6	−1.12766
$599$	−1.21463e7	−1.38317	−0.691586	−	0.722294i	$-0.743088\pi$
$599$	−1.21463e7	−1.38317	−0.691586	+	0.722294i	$0.743088\pi$
$600$	0	0
$601$	1.68363e6	0.190134	0.0950671	−	0.995471i	$-0.469693\pi$
$601$	1.68363e6	0.190134	0.0950671	+	0.995471i	$0.469693\pi$
$602$	−172573.	−0.0194080
$603$	0	0
$604$	1.58173e6	0.176416
$605$	1.08352e7	1.20351
$606$	0	0
$607$	−5.11718e6	−0.563714	−0.281857	−	0.959456i	$-0.590950\pi$
$607$	−5.11718e6	−0.563714	−0.281857	+	0.959456i	$0.590950\pi$
$608$	8.68491e6	0.952810
$609$	0	0
$610$	−4.17720e6	−0.454528
$611$	1.48007e7	1.60391
$612$	0	0
$613$	4.10152e6	0.440853	0.220426	−	0.975404i	$-0.429255\pi$
$613$	4.10152e6	0.440853	0.220426	+	0.975404i	$0.429255\pi$
$614$	5.45103e6	0.583522
$615$	0	0
$616$	2.41958e6	0.256915
$617$	−1.52311e7	−1.61071	−0.805355	−	0.592792i	$-0.798025\pi$
$617$	−1.52311e7	−1.61071	−0.805355	+	0.592792i	$0.798025\pi$
$618$	0	0
$619$	5.27961e6	0.553829	0.276914	−	0.960895i	$-0.410688\pi$
$619$	5.27961e6	0.553829	0.276914	+	0.960895i	$0.410688\pi$
$620$	−677260.	−0.0707581
$621$	0	0
$622$	−2.76092e6	−0.286140
$623$	−2.13594e6	−0.220480
$624$	0	0
$625$	3.38068e7	3.46181
$626$	−4.81564e6	−0.491154
$627$	0	0
$628$	−5.84421e6	−0.591326
$629$	−1.13331e7	−1.14215
$630$	0	0
$631$	−1.84888e7	−1.84857	−0.924283	−	0.381709i	$-0.875336\pi$
$631$	−1.84888e7	−1.84857	−0.924283	+	0.381709i	$0.875336\pi$
$632$	1.27742e7	1.27216
$633$	0	0
$634$	−8.03004e6	−0.793404
$635$	−690917.	−0.0679973
$636$	0	0
$637$	−1.39345e7	−1.36064
$638$	1.77365e6	0.172511
$639$	0	0
$640$	1.02118e7	0.985489
$641$	3.87212e6	0.372223	0.186112	−	0.982529i	$-0.440411\pi$
$641$	3.87212e6	0.372223	0.186112	+	0.982529i	$0.440411\pi$
$642$	0	0
$643$	3.24879e6	0.309880	0.154940	−	0.987924i	$-0.450482\pi$
$643$	3.24879e6	0.309880	0.154940	+	0.987924i	$0.450482\pi$
$644$	1.48613e6	0.141202
$645$	0	0
$646$	8.58615e6	0.809500
$647$	−1.85942e7	−1.74629	−0.873147	−	0.487456i	$-0.837925\pi$
$647$	−1.85942e7	−1.74629	−0.873147	+	0.487456i	$0.837925\pi$
$648$	0	0
$649$	8.72142e6	0.812785
$650$	−2.63250e7	−2.44391
$651$	0	0
$652$	−4.46356e6	−0.411209
$653$	−9.77909e6	−0.897461	−0.448730	−	0.893667i	$-0.648124\pi$
$653$	−9.77909e6	−0.897461	−0.448730	+	0.893667i	$0.648124\pi$
$654$	0	0
$655$	5.14614e6	0.468682
$656$	414640.	0.0376194
$657$	0	0
$658$	1.60159e6	0.144207
$659$	−1.92292e7	−1.72484	−0.862418	−	0.506196i	$-0.831051\pi$
$659$	−1.92292e7	−1.72484	−0.862418	+	0.506196i	$0.831051\pi$
$660$	0	0
$661$	−1.39614e7	−1.24287	−0.621434	−	0.783467i	$-0.713449\pi$
$661$	−1.39614e7	−1.24287	−0.621434	+	0.783467i	$0.713449\pi$
$662$	−2.66233e6	−0.236111
$663$	0	0
$664$	−7.55626e6	−0.665099
$665$	4.21794e6	0.369868
$666$	0	0
$667$	2.96099e6	0.257705
$668$	−9.43732e6	−0.818290
$669$	0	0
$670$	3.44660e6	0.296623
$671$	−5.46121e6	−0.468255
$672$	0	0
$673$	1.13333e7	0.964537	0.482269	−	0.876023i	$-0.339813\pi$
$673$	1.13333e7	0.964537	0.482269	+	0.876023i	$0.339813\pi$
$674$	−2.29274e6	−0.194404
$675$	0	0
$676$	−6.94082e6	−0.584176
$677$	9.84009e6	0.825140	0.412570	−	0.910926i	$-0.364631\pi$
$677$	9.84009e6	0.825140	0.412570	+	0.910926i	$0.364631\pi$
$678$	0	0
$679$	−4.07892e6	−0.339524
$680$	3.01697e7	2.50207
$681$	0	0
$682$	635771.	0.0523407
$683$	−4282.68	−0.000351288 0	−0.000175644	−	1.00000i	$-0.500056\pi$
$683$	−4282.68	−0.000351288 0	−0.000175644		1.00000i	$0.500056\pi$
$684$	0	0
$685$	1.10405e7	0.899003
$686$	−3.07651e6	−0.249602
$687$	0	0
$688$	149844.	0.0120689
$689$	−1.56033e7	−1.25218
$690$	0	0
$691$	5.24701e6	0.418039	0.209019	−	0.977911i	$-0.432973\pi$
$691$	5.24701e6	0.418039	0.209019	+	0.977911i	$0.432973\pi$
$692$	−7.95655e6	−0.631625
$693$	0	0
$694$	−1.20826e7	−0.952277
$695$	2.35224e7	1.84723
$696$	0	0
$697$	−7.78462e6	−0.606954
$698$	1.07329e7	0.833834
$699$	0	0
$700$	3.96730e6	0.306020
$701$	3.73379e6	0.286982	0.143491	−	0.989652i	$-0.454167\pi$
$701$	3.73379e6	0.286982	0.143491	+	0.989652i	$0.454167\pi$
$702$	0	0
$703$	1.14942e7	0.877180
$704$	−1.18677e7	−0.902473
$705$	0	0
$706$	2.31744e6	0.174983
$707$	−4.31231e6	−0.324461
$708$	0	0
$709$	−1.21242e7	−0.905810	−0.452905	−	0.891559i	$-0.649612\pi$
$709$	−1.21242e7	−0.905810	−0.452905	+	0.891559i	$0.649612\pi$
$710$	−3.01816e7	−2.24697
$711$	0	0
$712$	1.54949e7	1.14548
$713$	1.06138e6	0.0781892
$714$	0	0
$715$	−4.73039e7	−3.46044
$716$	217456.	0.0158522
$717$	0	0
$718$	1.12660e7	0.815563
$719$	−1.59176e7	−1.14830	−0.574149	−	0.818751i	$-0.694667\pi$
$719$	−1.59176e7	−1.14830	−0.574149	+	0.818751i	$0.694667\pi$
$720$	0	0
$721$	3.70275e6	0.265269
$722$	347040.	0.0247763
$723$	0	0
$724$	−1.33532e6	−0.0946759
$725$	7.90452e6	0.558509
$726$	0	0
$727$	−633198.	−0.0444328	−0.0222164	−	0.999753i	$-0.507072\pi$
$727$	−633198.	−0.0444328	−0.0222164	+	0.999753i	$0.507072\pi$
$728$	−4.07546e6	−0.285002
$729$	0	0
$730$	3.11961e6	0.216668
$731$	−2.81324e6	−0.194721
$732$	0	0
$733$	−2.64288e7	−1.81684	−0.908421	−	0.418057i	$-0.862711\pi$
$733$	−2.64288e7	−1.81684	−0.908421	+	0.418057i	$0.862711\pi$
$734$	4.85952e6	0.332930
$735$	0	0
$736$	−1.75954e7	−1.19730
$737$	4.50603e6	0.305581
$738$	0	0
$739$	2.15407e6	0.145094	0.0725470	−	0.997365i	$-0.476887\pi$
$739$	2.15407e6	0.145094	0.0725470	+	0.997365i	$0.476887\pi$
$740$	1.48592e7	0.997508
$741$	0	0
$742$	−1.68843e6	−0.112583
$743$	−1.76951e7	−1.17593	−0.587964	−	0.808887i	$-0.700070\pi$
$743$	−1.76951e7	−1.17593	−0.587964	+	0.808887i	$0.700070\pi$
$744$	0	0
$745$	3.59463e7	2.37281
$746$	6.99664e6	0.460302
$747$	0	0
$748$	1.45118e7	0.948345
$749$	1.85523e6	0.120835
$750$	0	0
$751$	2.58345e7	1.67148	0.835739	−	0.549127i	$-0.185040\pi$
$751$	2.58345e7	1.67148	0.835739	+	0.549127i	$0.185040\pi$
$752$	−1.39065e6	−0.0896755
$753$	0	0
$754$	−2.98747e6	−0.191370
$755$	−9.09514e6	−0.580687
$756$	0	0
$757$	−2.47558e7	−1.57013	−0.785067	−	0.619411i	$-0.787371\pi$
$757$	−2.47558e7	−1.57013	−0.785067	+	0.619411i	$0.787371\pi$
$758$	−2.57343e6	−0.162682
$759$	0	0
$760$	−3.05985e7	−1.92161
$761$	1.12818e7	0.706183	0.353092	−	0.935589i	$-0.385130\pi$
$761$	1.12818e7	0.706183	0.353092	+	0.935589i	$0.385130\pi$
$762$	0	0
$763$	83056.7	0.00516492
$764$	3.96062e6	0.245488
$765$	0	0
$766$	6.35406e6	0.391272
$767$	−1.46900e7	−0.901643
$768$	0	0
$769$	−1.45315e7	−0.886124	−0.443062	−	0.896491i	$-0.646108\pi$
$769$	−1.45315e7	−0.886124	−0.443062	+	0.896491i	$0.646108\pi$
$770$	−5.11876e6	−0.311127
$771$	0	0
$772$	2.56050e6	0.154625
$773$	5.49667e6	0.330865	0.165433	−	0.986221i	$-0.447098\pi$
$773$	5.49667e6	0.330865	0.165433	+	0.986221i	$0.447098\pi$
$774$	0	0
$775$	2.83341e6	0.169455
$776$	2.95900e7	1.76397
$777$	0	0
$778$	1.43757e7	0.851492
$779$	7.89526e6	0.466147
$780$	0	0
$781$	−3.94590e7	−2.31482
$782$	−1.73953e7	−1.01722
$783$	0	0
$784$	1.30926e6	0.0760742
$785$	3.36050e7	1.94639
$786$	0	0
$787$	1.62860e7	0.937297	0.468649	−	0.883385i	$-0.344741\pi$
$787$	1.62860e7	0.937297	0.468649	+	0.883385i	$0.344741\pi$
$788$	−531163.	−0.0304728
$789$	0	0
$790$	−2.70245e7	−1.54060
$791$	2.97055e6	0.168809
$792$	0	0
$793$	9.19866e6	0.519447
$794$	−1.46203e7	−0.823009
$795$	0	0
$796$	6.58569e6	0.368399
$797$	−1.65610e7	−0.923506	−0.461753	−	0.887009i	$-0.652779\pi$
$797$	−1.65610e7	−0.923506	−0.461753	+	0.887009i	$0.652779\pi$
$798$	0	0
$799$	2.61087e7	1.44683
$800$	−4.69718e7	−2.59485
$801$	0	0
$802$	1.60905e7	0.883350
$803$	4.07853e6	0.223211
$804$	0	0
$805$	−8.54545e6	−0.464778
$806$	−1.07087e6	−0.0580629
$807$	0	0
$808$	3.12831e7	1.68571
$809$	1.93881e7	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$809$	1.93881e7	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$810$	0	0
$811$	1.99327e7	1.06418	0.532089	−	0.846688i	$-0.321407\pi$
$811$	1.99327e7	1.06418	0.532089	+	0.846688i	$0.321407\pi$
$812$	450224.	0.0239629
$813$	0	0
$814$	−1.39489e7	−0.737870
$815$	2.56661e7	1.35352
$816$	0	0
$817$	2.85322e6	0.149548
$818$	6.03074e6	0.315128
$819$	0	0
$820$	1.02067e7	0.530092
$821$	2.97106e7	1.53834	0.769172	−	0.639041i	$-0.220669\pi$
$821$	2.97106e7	1.53834	0.769172	+	0.639041i	$0.220669\pi$
$822$	0	0
$823$	−3.70742e6	−0.190797	−0.0953987	−	0.995439i	$-0.530413\pi$
$823$	−3.70742e6	−0.190797	−0.0953987	+	0.995439i	$0.530413\pi$
$824$	−2.68611e7	−1.37818
$825$	0	0
$826$	−1.58961e6	−0.0810664
$827$	5.42093e6	0.275620	0.137810	−	0.990459i	$-0.455994\pi$
$827$	5.42093e6	0.275620	0.137810	+	0.990459i	$0.455994\pi$
$828$	0	0
$829$	−3.24596e7	−1.64043	−0.820214	−	0.572056i	$-0.806146\pi$
$829$	−3.24596e7	−1.64043	−0.820214	+	0.572056i	$0.806146\pi$
$830$	1.59857e7	0.805445
$831$	0	0
$832$	1.99895e7	1.00114
$833$	−2.45807e7	−1.22739
$834$	0	0
$835$	5.42659e7	2.69346
$836$	−1.47180e7	−0.728339
$837$	0	0
$838$	−2.16867e7	−1.06680
$839$	6.99699e6	0.343168	0.171584	−	0.985170i	$-0.445112\pi$
$839$	6.99699e6	0.343168	0.171584	+	0.985170i	$0.445112\pi$
$840$	0	0
$841$	−1.96141e7	−0.956266
$842$	−1.77566e7	−0.863136
$843$	0	0
$844$	−6.10634e6	−0.295070
$845$	3.99107e7	1.92286
$846$	0	0
$847$	−2.58194e6	−0.123662
$848$	1.46606e6	0.0700102
$849$	0	0
$850$	−4.64377e7	−2.20457
$851$	−2.32869e7	−1.10227
$852$	0	0
$853$	−5.68399e6	−0.267474	−0.133737	−	0.991017i	$-0.542698\pi$
$853$	−5.68399e6	−0.267474	−0.133737	+	0.991017i	$0.542698\pi$
$854$	995389.	0.0467033
$855$	0	0
$856$	−1.34585e7	−0.627788
$857$	8.71203e6	0.405198	0.202599	−	0.979262i	$-0.435061\pi$
$857$	8.71203e6	0.405198	0.202599	+	0.979262i	$0.435061\pi$
$858$	0	0
$859$	1.41568e7	0.654609	0.327304	−	0.944919i	$-0.393860\pi$
$859$	1.41568e7	0.654609	0.327304	+	0.944919i	$0.393860\pi$
$860$	3.68854e6	0.170062
$861$	0	0
$862$	9.47790e6	0.434454
$863$	−9.82195e6	−0.448922	−0.224461	−	0.974483i	$-0.572062\pi$
$863$	−9.82195e6	−0.448922	−0.224461	+	0.974483i	$0.572062\pi$
$864$	0	0
$865$	4.57513e7	2.07904
$866$	1.86031e7	0.842927
$867$	0	0
$868$	161385.	0.00727048
$869$	−3.53315e7	−1.58713
$870$	0	0
$871$	−7.58980e6	−0.338988
$872$	−602524.	−0.0268339
$873$	0	0
$874$	1.76426e7	0.781237
$875$	−1.42706e7	−0.630120
$876$	0	0
$877$	2.13034e7	0.935298	0.467649	−	0.883914i	$-0.345101\pi$
$877$	2.13034e7	0.935298	0.467649	+	0.883914i	$0.345101\pi$
$878$	2.09019e7	0.915061
$879$	0	0
$880$	4.44460e6	0.193475
$881$	−3.10088e7	−1.34600	−0.673000	−	0.739642i	$-0.734995\pi$
$881$	−3.10088e7	−1.34600	−0.673000	+	0.739642i	$0.734995\pi$
$882$	0	0
$883$	−2.21207e7	−0.954766	−0.477383	−	0.878695i	$-0.658415\pi$
$883$	−2.21207e7	−0.954766	−0.477383	+	0.878695i	$0.658415\pi$
$884$	−2.44431e7	−1.05202
$885$	0	0
$886$	−1.76193e7	−0.754057
$887$	−4.29991e7	−1.83506	−0.917530	−	0.397666i	$-0.869820\pi$
$887$	−4.29991e7	−1.83506	−0.917530	+	0.397666i	$0.869820\pi$
$888$	0	0
$889$	164639.	0.00698680
$890$	−3.27803e7	−1.38720
$891$	0	0
$892$	−4.61833e6	−0.194345
$893$	−2.64797e7	−1.11118
$894$	0	0
$895$	−1.25040e6	−0.0521786
$896$	−2.43337e6	−0.101260
$897$	0	0
$898$	−8.65551e6	−0.358181
$899$	321546.	0.0132692
$900$	0	0
$901$	−2.75244e7	−1.12955
$902$	−9.58144e6	−0.392116
$903$	0	0
$904$	−2.15494e7	−0.877031
$905$	7.67828e6	0.311632
$906$	0	0
$907$	3.34186e7	1.34887	0.674435	−	0.738334i	$-0.264387\pi$
$907$	3.34186e7	1.34887	0.674435	+	0.738334i	$0.264387\pi$
$908$	−1.61843e7	−0.651448
$909$	0	0
$910$	8.62185e6	0.345141
$911$	3.77821e7	1.50831	0.754155	−	0.656697i	$-0.228047\pi$
$911$	3.77821e7	1.50831	0.754155	+	0.656697i	$0.228047\pi$
$912$	0	0
$913$	2.08994e7	0.829769
$914$	−6.52337e6	−0.258289
$915$	0	0
$916$	−4.44642e6	−0.175094
$917$	−1.22628e6	−0.0481576
$918$	0	0
$919$	3.71228e7	1.44995	0.724973	−	0.688777i	$-0.241852\pi$
$919$	3.71228e7	1.44995	0.724973	+	0.688777i	$0.241852\pi$
$920$	6.19918e7	2.41471
$921$	0	0
$922$	−1.56000e7	−0.604364
$923$	6.64632e7	2.56789
$924$	0	0
$925$	−6.21655e7	−2.38888
$926$	8.55371e6	0.327814
$927$	0	0
$928$	−5.33055e6	−0.203190
$929$	−4.73143e6	−0.179868	−0.0899339	−	0.995948i	$-0.528666\pi$
$929$	−4.73143e6	−0.179868	−0.0899339	+	0.995948i	$0.528666\pi$
$930$	0	0
$931$	2.49300e7	0.942646
$932$	9.45690e6	0.356623
$933$	0	0
$934$	−2.50377e7	−0.939132
$935$	−8.34447e7	−3.12154
$936$	0	0
$937$	−1.74328e7	−0.648660	−0.324330	−	0.945944i	$-0.605139\pi$
$937$	−1.74328e7	−0.648660	−0.324330	+	0.945944i	$0.605139\pi$
$938$	−821293.	−0.0304783
$939$	0	0
$940$	−3.42320e7	−1.26361
$941$	−3.82735e7	−1.40904	−0.704521	−	0.709683i	$-0.748838\pi$
$941$	−3.82735e7	−1.40904	−0.704521	+	0.709683i	$0.748838\pi$
$942$	0	0
$943$	−1.59956e7	−0.585762
$944$	1.38025e6	0.0504114
$945$	0	0
$946$	−3.46258e6	−0.125797
$947$	1.65041e7	0.598021	0.299011	−	0.954250i	$-0.403343\pi$
$947$	1.65041e7	0.598021	0.299011	+	0.954250i	$0.403343\pi$
$948$	0	0
$949$	−6.86973e6	−0.247613
$950$	4.70977e7	1.69313
$951$	0	0
$952$	−7.18916e6	−0.257090
$953$	2.15575e7	0.768894	0.384447	−	0.923147i	$-0.374392\pi$
$953$	2.15575e7	0.768894	0.384447	+	0.923147i	$0.374392\pi$
$954$	0	0
$955$	−2.27741e7	−0.808040
$956$	2.67991e6	0.0948365
$957$	0	0
$958$	3.33688e7	1.17470
$959$	−2.63084e6	−0.0923736
$960$	0	0
$961$	−2.85139e7	−0.995974
$962$	2.34951e7	0.818539
$963$	0	0
$964$	−7.41755e6	−0.257080
$965$	−1.47232e7	−0.508960
$966$	0	0
$967$	−3.38528e7	−1.16420	−0.582101	−	0.813116i	$-0.697769\pi$
$967$	−3.38528e7	−1.16420	−0.582101	+	0.813116i	$0.697769\pi$
$968$	1.87303e7	0.642476
$969$	0	0
$970$	−6.25993e7	−2.13619
$971$	−1.12966e7	−0.384502	−0.192251	−	0.981346i	$-0.561579\pi$
$971$	−1.12966e7	−0.384502	−0.192251	+	0.981346i	$0.561579\pi$
$972$	0	0
$973$	−5.60517e6	−0.189805
$974$	−9.80641e6	−0.331217
$975$	0	0
$976$	−864291.	−0.0290426
$977$	2.28369e7	0.765423	0.382711	−	0.923868i	$-0.374990\pi$
$977$	2.28369e7	0.765423	0.382711	+	0.923868i	$0.374990\pi$
$978$	0	0
$979$	−4.28564e7	−1.42909
$980$	3.22286e7	1.07196
$981$	0	0
$982$	1.76692e7	0.584707
$983$	−1.60143e7	−0.528598	−0.264299	−	0.964441i	$-0.585140\pi$
$983$	−1.60143e7	−0.528598	−0.264299	+	0.964441i	$0.585140\pi$
$984$	0	0
$985$	3.05426e6	0.100303
$986$	−5.26993e6	−0.172629
$987$	0	0
$988$	2.47905e7	0.807965
$989$	−5.78055e6	−0.187922
$990$	0	0
$991$	1.70581e7	0.551755	0.275878	−	0.961193i	$-0.411032\pi$
$991$	1.70581e7	0.551755	0.275878	+	0.961193i	$0.411032\pi$
$992$	−1.91076e6	−0.0616490
$993$	0	0
$994$	7.19200e6	0.230879
$995$	−3.78687e7	−1.21261
$996$	0	0
$997$	−2.65006e7	−0.844341	−0.422171	−	0.906516i	$-0.638732\pi$
$997$	−2.65006e7	−0.844341	−0.422171	+	0.906516i	$0.638732\pi$
$998$	−3.02269e7	−0.960653
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.6.a.c.1.3		8
3.2	odd	2		43.6.a.a.1.6	✓	8
12.11	even	2		688.6.a.e.1.3		8
15.14	odd	2		1075.6.a.a.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.a.1.6	✓	8	3.2	odd	2
387.6.a.c.1.3		8	1.1	even	1	trivial
688.6.a.e.1.3		8	12.11	even	2
1075.6.a.a.1.3		8	15.14	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q-3.65705 q^{2} -18.6260 q^{4} +107.102 q^{5} -25.5214 q^{7} +185.142 q^{8} +O(q^{10})\)	\(q-3.65705 q^{2} -18.6260 q^{4} +107.102 q^{5} -25.5214 q^{7} +185.142 q^{8} -391.677 q^{10} -512.073 q^{11} +862.516 q^{13} +93.3331 q^{14} -81.0406 q^{16} +1521.49 q^{17} -1543.11 q^{19} -1994.88 q^{20} +1872.67 q^{22} +3126.31 q^{23} +8345.85 q^{25} -3154.26 q^{26} +475.362 q^{28} +947.120 q^{29} +339.499 q^{31} -5628.17 q^{32} -5564.17 q^{34} -2733.40 q^{35} -7448.67 q^{37} +5643.25 q^{38} +19829.1 q^{40} -5116.45 q^{41} -1849.00 q^{43} +9537.86 q^{44} -11433.1 q^{46} +17159.9 q^{47} -16155.7 q^{49} -30521.2 q^{50} -16065.2 q^{52} -18090.4 q^{53} -54844.0 q^{55} -4725.08 q^{56} -3463.66 q^{58} -17031.6 q^{59} +10664.9 q^{61} -1241.56 q^{62} +23175.8 q^{64} +92377.3 q^{65} -8799.60 q^{67} -28339.3 q^{68} +9996.16 q^{70} +77057.3 q^{71} -7964.75 q^{73} +27240.1 q^{74} +28742.0 q^{76} +13068.8 q^{77} +68997.0 q^{79} -8679.62 q^{80} +18711.1 q^{82} -40813.4 q^{83} +162955. q^{85} +6761.88 q^{86} -94806.0 q^{88} +83692.1 q^{89} -22012.6 q^{91} -58230.7 q^{92} -62754.7 q^{94} -165271. q^{95} +159824. q^{97} +59082.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8}+O(q^{10}) \) 8 * q + 12 * q^2 + 122 * q^4 + 212 * q^5 - 136 * q^7 + 666 * q^8	\( 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8} - 617 q^{10} + 532 q^{11} - 2492 q^{13} + 4240 q^{14} + 1882 q^{16} + 2534 q^{17} - 1678 q^{19} + 2607 q^{20} + 11502 q^{22} + 2488 q^{23} + 4378 q^{25} - 4586 q^{26} + 18640 q^{28} + 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 30007 q^{34} - 5640 q^{35} - 3772 q^{37} + 6559 q^{38} + 14869 q^{40} + 10698 q^{41} - 14792 q^{43} + 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 7188 q^{49} - 26877 q^{50} - 60736 q^{52} + 62352 q^{53} - 49552 q^{55} + 144528 q^{56} + 52951 q^{58} + 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 153858 q^{64} + 5000 q^{65} + 27784 q^{67} - 40507 q^{68} + 185910 q^{70} + 9504 q^{71} + 14260 q^{73} + 15239 q^{74} + 1279 q^{76} + 218140 q^{77} + 160248 q^{79} + 1291 q^{80} - 47781 q^{82} + 77176 q^{83} + 141096 q^{85} - 22188 q^{86} + 129544 q^{88} + 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 248737 q^{94} - 135884 q^{95} + 144742 q^{97} + 292244 q^{98}+O(q^{100}) \) 8 * q + 12 * q^2 + 122 * q^4 + 212 * q^5 - 136 * q^7 + 666 * q^8 - 617 * q^10 + 532 * q^11 - 2492 * q^13 + 4240 * q^14 + 1882 * q^16 + 2534 * q^17 - 1678 * q^19 + 2607 * q^20 + 11502 * q^22 + 2488 * q^23 + 4378 * q^25 - 4586 * q^26 + 18640 * q^28 + 4360 * q^29 + 5704 * q^31 + 18294 * q^32 + 30007 * q^34 - 5640 * q^35 - 3772 * q^37 + 6559 * q^38 + 14869 * q^40 + 10698 * q^41 - 14792 * q^43 + 356 * q^44 - 19389 * q^46 + 77864 * q^47 + 7188 * q^49 - 26877 * q^50 - 60736 * q^52 + 62352 * q^53 - 49552 * q^55 + 144528 * q^56 + 52951 * q^58 + 26224 * q^59 - 82540 * q^61 + 9023 * q^62 + 153858 * q^64 + 5000 * q^65 + 27784 * q^67 - 40507 * q^68 + 185910 * q^70 + 9504 * q^71 + 14260 * q^73 + 15239 * q^74 + 1279 * q^76 + 218140 * q^77 + 160248 * q^79 + 1291 * q^80 - 47781 * q^82 + 77176 * q^83 + 141096 * q^85 - 22188 * q^86 + 129544 * q^88 + 265692 * q^89 + 401148 * q^91 - 190391 * q^92 + 248737 * q^94 - 135884 * q^95 + 144742 * q^97 + 292244 * q^98

Introduction

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Database

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