LMFDB - Embedded newform 207.6.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,6,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, 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("207.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	207.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:	$33.1994507013$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{29}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 7 $ x^2 - x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.19258$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.7703 q^{2} +84.0000 q^{4} -41.6148 q^{5} +0.236813 q^{7} +560.057 q^{8} +O(q^{10})$	\(q+10.7703 q^{2} +84.0000 q^{4} -41.6148 q^{5} +0.236813 q^{7} +560.057 q^{8} -448.205 q^{10} +421.598 q^{11} +254.502 q^{13} +2.55055 q^{14} +3344.00 q^{16} +975.007 q^{17} +2039.79 q^{19} -3495.65 q^{20} +4540.75 q^{22} -529.000 q^{23} -1393.21 q^{25} +2741.07 q^{26} +19.8923 q^{28} -2671.55 q^{29} +9039.14 q^{31} +18094.2 q^{32} +10501.1 q^{34} -9.85493 q^{35} -12665.3 q^{37} +21969.2 q^{38} -23306.7 q^{40} -10146.4 q^{41} +19523.2 q^{43} +35414.2 q^{44} -5697.50 q^{46} -27679.7 q^{47} -16806.9 q^{49} -15005.3 q^{50} +21378.2 q^{52} -10852.8 q^{53} -17544.7 q^{55} +132.629 q^{56} -28773.4 q^{58} -11907.7 q^{59} +39861.9 q^{61} +97354.5 q^{62} +87872.0 q^{64} -10591.1 q^{65} -28550.5 q^{67} +81900.6 q^{68} -106.141 q^{70} -52179.8 q^{71} +56918.0 q^{73} -136410. q^{74} +171342. q^{76} +99.8398 q^{77} +23178.3 q^{79} -139160. q^{80} -109281. q^{82} +18344.6 q^{83} -40574.8 q^{85} +210272. q^{86} +236119. q^{88} -47362.4 q^{89} +60.2694 q^{91} -44436.0 q^{92} -298120. q^{94} -84885.4 q^{95} -140379. q^{97} -181016. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 168 q^{4} - 94 q^{5} - 118 q^{7}+O(q^{10}) $ 2 * q + 168 * q^4 - 94 * q^5 - 118 * q^7	$ 2 q + 168 q^{4} - 94 q^{5} - 118 q^{7} + 116 q^{10} - 320 q^{11} - 288 q^{13} + 1276 q^{14} + 6688 q^{16} + 1810 q^{17} + 730 q^{19} - 7896 q^{20} + 12528 q^{22} - 1058 q^{23} - 1774 q^{25} + 8584 q^{26} - 9912 q^{28} - 8208 q^{29} + 1772 q^{31} + 1508 q^{34} + 6184 q^{35} - 23112 q^{37} + 36076 q^{38} + 6032 q^{40} - 5516 q^{41} + 10322 q^{43} - 26880 q^{44} - 42952 q^{47} - 19634 q^{49} - 10904 q^{50} - 24192 q^{52} + 25350 q^{53} + 21304 q^{55} + 66352 q^{56} + 30856 q^{58} - 18344 q^{59} + 37224 q^{61} + 175624 q^{62} + 175744 q^{64} + 17828 q^{65} - 7482 q^{67} + 152040 q^{68} - 66816 q^{70} - 126848 q^{71} + 137660 q^{73} - 23896 q^{74} + 61320 q^{76} + 87784 q^{77} + 62286 q^{79} - 314336 q^{80} - 159152 q^{82} - 83120 q^{83} - 84316 q^{85} + 309372 q^{86} + 651456 q^{88} - 69770 q^{89} + 64204 q^{91} - 88872 q^{92} - 133632 q^{94} - 16272 q^{95} - 170104 q^{97} - 150568 q^{98}+O(q^{100}) $ 2 * q + 168 * q^4 - 94 * q^5 - 118 * q^7 + 116 * q^10 - 320 * q^11 - 288 * q^13 + 1276 * q^14 + 6688 * q^16 + 1810 * q^17 + 730 * q^19 - 7896 * q^20 + 12528 * q^22 - 1058 * q^23 - 1774 * q^25 + 8584 * q^26 - 9912 * q^28 - 8208 * q^29 + 1772 * q^31 + 1508 * q^34 + 6184 * q^35 - 23112 * q^37 + 36076 * q^38 + 6032 * q^40 - 5516 * q^41 + 10322 * q^43 - 26880 * q^44 - 42952 * q^47 - 19634 * q^49 - 10904 * q^50 - 24192 * q^52 + 25350 * q^53 + 21304 * q^55 + 66352 * q^56 + 30856 * q^58 - 18344 * q^59 + 37224 * q^61 + 175624 * q^62 + 175744 * q^64 + 17828 * q^65 - 7482 * q^67 + 152040 * q^68 - 66816 * q^70 - 126848 * q^71 + 137660 * q^73 - 23896 * q^74 + 61320 * q^76 + 87784 * q^77 + 62286 * q^79 - 314336 * q^80 - 159152 * q^82 - 83120 * q^83 - 84316 * q^85 + 309372 * q^86 + 651456 * q^88 - 69770 * q^89 + 64204 * q^91 - 88872 * q^92 - 133632 * q^94 - 16272 * q^95 - 170104 * q^97 - 150568 * 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 $

$2$	10.7703	1.90394	0.951972	−	0.306186i	$-0.0990530\pi$
$2$	10.7703	1.90394	0.951972	+	0.306186i	$0.0990530\pi$
$3$	0	0
$3$	0	0
$4$	84.0000	2.62500
$5$	−41.6148	−0.744429	−0.372214	−	0.928147i	$-0.621401\pi$
$5$	−41.6148	−0.744429	−0.372214	+	0.928147i	$0.621401\pi$
$6$	0	0
$7$	0.236813	0.00182667	0.000913335	−	1.00000i	$-0.499709\pi$
$7$	0.236813	0.00182667	0.000913335		1.00000i	$0.499709\pi$
$8$	560.057	3.09391
$9$	0	0
$10$	−448.205	−1.41735
$11$	421.598	1.05055	0.525275	−	0.850933i	$-0.323963\pi$
$11$	421.598	1.05055	0.525275	+	0.850933i	$0.323963\pi$
$12$	0	0
$13$	254.502	0.417670	0.208835	−	0.977951i	$-0.433033\pi$
$13$	254.502	0.417670	0.208835	+	0.977951i	$0.433033\pi$
$14$	2.55055	0.00347788
$15$	0	0
$16$	3344.00	3.26562
$17$	975.007	0.818249	0.409125	−	0.912479i	$-0.365834\pi$
$17$	975.007	0.818249	0.409125	+	0.912479i	$0.365834\pi$
$18$	0	0
$19$	2039.79	1.29629	0.648143	−	0.761519i	$-0.275546\pi$
$19$	2039.79	1.29629	0.648143	+	0.761519i	$0.275546\pi$
$20$	−3495.65	−1.95413
$21$	0	0
$22$	4540.75	2.00019
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−1393.21	−0.445826
$26$	2741.07	0.795220
$27$	0	0
$28$	19.8923	0.00479501
$29$	−2671.55	−0.589885	−0.294943	−	0.955515i	$-0.595301\pi$
$29$	−2671.55	−0.589885	−0.294943	+	0.955515i	$0.595301\pi$
$30$	0	0
$31$	9039.14	1.68936	0.844681	−	0.535270i	$-0.179790\pi$
$31$	9039.14	1.68936	0.844681	+	0.535270i	$0.179790\pi$
$32$	18094.2	3.12366
$33$	0	0
$34$	10501.1	1.55790
$35$	−9.85493	−0.00135983
$36$	0	0
$37$	−12665.3	−1.52094	−0.760471	−	0.649372i	$-0.775032\pi$
$37$	−12665.3	−1.52094	−0.760471	+	0.649372i	$0.775032\pi$
$38$	21969.2	2.46805
$39$	0	0
$40$	−23306.7	−2.30319
$41$	−10146.4	−0.942658	−0.471329	−	0.881957i	$-0.656226\pi$
$41$	−10146.4	−0.942658	−0.471329	+	0.881957i	$0.656226\pi$
$42$	0	0
$43$	19523.2	1.61020	0.805102	−	0.593137i	$-0.202111\pi$
$43$	19523.2	1.61020	0.805102	+	0.593137i	$0.202111\pi$
$44$	35414.2	2.75769
$45$	0	0
$46$	−5697.50	−0.397000
$47$	−27679.7	−1.82775	−0.913875	−	0.405995i	$-0.866925\pi$
$47$	−27679.7	−1.82775	−0.913875	+	0.405995i	$0.866925\pi$
$48$	0	0
$49$	−16806.9	−0.999997
$50$	−15005.3	−0.848827
$51$	0	0
$52$	21378.2	1.09638
$53$	−10852.8	−0.530703	−0.265351	−	0.964152i	$-0.585488\pi$
$53$	−10852.8	−0.530703	−0.265351	+	0.964152i	$0.585488\pi$
$54$	0	0
$55$	−17544.7	−0.782059
$56$	132.629	0.00565155
$57$	0	0
$58$	−28773.4	−1.12311
$59$	−11907.7	−0.445345	−0.222672	−	0.974893i	$-0.571478\pi$
$59$	−11907.7	−0.445345	−0.222672	+	0.974893i	$0.571478\pi$
$60$	0	0
$61$	39861.9	1.37162	0.685809	−	0.727782i	$-0.259449\pi$
$61$	39861.9	1.37162	0.685809	+	0.727782i	$0.259449\pi$
$62$	97354.5	3.21645
$63$	0	0
$64$	87872.0	2.68164
$65$	−10591.1	−0.310925
$66$	0	0
$67$	−28550.5	−0.777009	−0.388504	−	0.921447i	$-0.627008\pi$
$67$	−28550.5	−0.777009	−0.388504	+	0.921447i	$0.627008\pi$
$68$	81900.6	2.14790
$69$	0	0
$70$	−106.141	−0.00258903
$71$	−52179.8	−1.22845	−0.614223	−	0.789132i	$-0.710531\pi$
$71$	−52179.8	−1.22845	−0.614223	+	0.789132i	$0.710531\pi$
$72$	0	0
$73$	56918.0	1.25009	0.625047	−	0.780587i	$-0.285080\pi$
$73$	56918.0	1.25009	0.625047	+	0.780587i	$0.285080\pi$
$74$	−136410.	−2.89579
$75$	0	0
$76$	171342.	3.40275
$77$	99.8398	0.00191901
$78$	0	0
$79$	23178.3	0.417844	0.208922	−	0.977932i	$-0.433004\pi$
$79$	23178.3	0.417844	0.208922	+	0.977932i	$0.433004\pi$
$80$	−139160.	−2.43103
$81$	0	0
$82$	−109281.	−1.79477
$83$	18344.6	0.292289	0.146144	−	0.989263i	$-0.453314\pi$
$83$	18344.6	0.292289	0.146144	+	0.989263i	$0.453314\pi$
$84$	0	0
$85$	−40574.8	−0.609128
$86$	210272.	3.06574
$87$	0	0
$88$	236119.	3.25030
$89$	−47362.4	−0.633810	−0.316905	−	0.948457i	$-0.602644\pi$
$89$	−47362.4	−0.633810	−0.316905	+	0.948457i	$0.602644\pi$
$90$	0	0
$91$	60.2694	0.000762945 0
$92$	−44436.0	−0.547350
$93$	0	0
$94$	−298120.	−3.47993
$95$	−84885.4	−0.964992
$96$	0	0
$97$	−140379.	−1.51486	−0.757432	−	0.652915i	$-0.773546\pi$
$97$	−140379.	−1.51486	−0.757432	+	0.652915i	$0.773546\pi$
$98$	−181016.	−1.90394
$99$	0	0
$100$	−117029.	−1.17029
$101$	−87080.3	−0.849408	−0.424704	−	0.905332i	$-0.639622\pi$
$101$	−87080.3	−0.849408	−0.424704	+	0.905332i	$0.639622\pi$
$102$	0	0
$103$	17988.9	0.167075	0.0835375	−	0.996505i	$-0.473378\pi$
$103$	17988.9	0.167075	0.0835375	+	0.996505i	$0.473378\pi$
$104$	142536.	1.29223
$105$	0	0
$106$	−116888.	−1.01043
$107$	127485.	1.07646	0.538231	−	0.842797i	$-0.319093\pi$
$107$	127485.	1.07646	0.538231	+	0.842797i	$0.319093\pi$
$108$	0	0
$109$	−136418.	−1.09978	−0.549890	−	0.835237i	$-0.685330\pi$
$109$	−136418.	−1.09978	−0.549890	+	0.835237i	$0.685330\pi$
$110$	−188962.	−1.48900
$111$	0	0
$112$	791.902	0.00596522
$113$	104012.	0.766279	0.383139	−	0.923691i	$-0.374843\pi$
$113$	104012.	0.766279	0.383139	+	0.923691i	$0.374843\pi$
$114$	0	0
$115$	22014.2	0.155224
$116$	−224410.	−1.54845
$117$	0	0
$118$	−128249.	−0.847912
$119$	230.894	0.00149467
$120$	0	0
$121$	16693.7	0.103655
$122$	429325.	2.61148
$123$	0	0
$124$	759288.	4.43458
$125$	188024.	1.07631
$126$	0	0
$127$	203763.	1.12103	0.560513	−	0.828145i	$-0.310604\pi$
$127$	203763.	1.12103	0.560513	+	0.828145i	$0.310604\pi$
$128$	367397.	1.98203
$129$	0	0
$130$	−114069.	−0.591984
$131$	−60674.1	−0.308905	−0.154453	−	0.988000i	$-0.549361\pi$
$131$	−60674.1	−0.308905	−0.154453	+	0.988000i	$0.549361\pi$
$132$	0	0
$133$	483.048	0.00236789
$134$	−307498.	−1.47938
$135$	0	0
$136$	546060.	2.53159
$137$	−94446.5	−0.429917	−0.214958	−	0.976623i	$-0.568962\pi$
$137$	−94446.5	−0.429917	−0.214958	+	0.976623i	$0.568962\pi$
$138$	0	0
$139$	403508.	1.77139	0.885697	−	0.464263i	$-0.153681\pi$
$139$	403508.	1.77139	0.885697	+	0.464263i	$0.153681\pi$
$140$	−827.814	−0.00356954
$141$	0	0
$142$	−561993.	−2.33889
$143$	107298.	0.438783
$144$	0	0
$145$	111176.	0.439128
$146$	613026.	2.38011
$147$	0	0
$148$	−1.06389e6	−3.99247
$149$	−384528.	−1.41893	−0.709467	−	0.704738i	$-0.751064\pi$
$149$	−384528.	−1.41893	−0.709467	+	0.704738i	$0.751064\pi$
$150$	0	0
$151$	−442508.	−1.57935	−0.789675	−	0.613525i	$-0.789751\pi$
$151$	−442508.	−1.57935	−0.789675	+	0.613525i	$0.789751\pi$
$152$	1.14240e6	4.01059
$153$	0	0
$154$	1075.31	0.00365368
$155$	−376162.	−1.25761
$156$	0	0
$157$	240724.	0.779416	0.389708	−	0.920938i	$-0.372576\pi$
$157$	240724.	0.779416	0.389708	+	0.920938i	$0.372576\pi$
$158$	249638.	0.795552
$159$	0	0
$160$	−752985.	−2.32534
$161$	−125.274	−0.000380887 0
$162$	0	0
$163$	−178188.	−0.525301	−0.262651	−	0.964891i	$-0.584597\pi$
$163$	−178188.	−0.525301	−0.262651	+	0.964891i	$0.584597\pi$
$164$	−852301.	−2.47448
$165$	0	0
$166$	197577.	0.556502
$167$	−25525.2	−0.0708236	−0.0354118	−	0.999373i	$-0.511274\pi$
$167$	−25525.2	−0.0708236	−0.0354118	+	0.999373i	$0.511274\pi$
$168$	0	0
$169$	−306522.	−0.825552
$170$	−437004.	−1.15975
$171$	0	0
$172$	1.63995e6	4.22678
$173$	−701865.	−1.78295	−0.891474	−	0.453072i	$-0.850328\pi$
$173$	−701865.	−1.78295	−0.891474	+	0.453072i	$0.850328\pi$
$174$	0	0
$175$	−329.929	−0.000814377 0
$176$	1.40982e6	3.43070
$177$	0	0
$178$	−510109.	−1.20674
$179$	292917.	0.683302	0.341651	−	0.939827i	$-0.389014\pi$
$179$	292917.	0.683302	0.341651	+	0.939827i	$0.389014\pi$
$180$	0	0
$181$	−38932.7	−0.0883321	−0.0441660	−	0.999024i	$-0.514063\pi$
$181$	−38932.7	−0.0883321	−0.0441660	+	0.999024i	$0.514063\pi$
$182$	649.121	0.00145260
$183$	0	0
$184$	−296270.	−0.645124
$185$	527066.	1.13223
$186$	0	0
$187$	411061.	0.859611
$188$	−2.32510e6	−4.79784
$189$	0	0
$190$	−914243.	−1.83729
$191$	−37244.8	−0.0738724	−0.0369362	−	0.999318i	$-0.511760\pi$
$191$	−37244.8	−0.0738724	−0.0369362	+	0.999318i	$0.511760\pi$
$192$	0	0
$193$	−42454.3	−0.0820406	−0.0410203	−	0.999158i	$-0.513061\pi$
$193$	−42454.3	−0.0820406	−0.0410203	+	0.999158i	$0.513061\pi$
$194$	−1.51193e6	−2.88421
$195$	0	0
$196$	−1.41178e6	−2.62499
$197$	−87335.1	−0.160333	−0.0801666	−	0.996781i	$-0.525545\pi$
$197$	−87335.1	−0.160333	−0.0801666	+	0.996781i	$0.525545\pi$
$198$	0	0
$199$	1.08779e6	1.94721	0.973604	−	0.228245i	$-0.0732989\pi$
$199$	1.08779e6	1.94721	0.973604	+	0.228245i	$0.0732989\pi$
$200$	−780275.	−1.37934
$201$	0	0
$202$	−937883.	−1.61722
$203$	−632.657	−0.00107753
$204$	0	0
$205$	422243.	0.701742
$206$	193746.	0.318101
$207$	0	0
$208$	851055.	1.36395
$209$	859969.	1.36181
$210$	0	0
$211$	−1.02777e6	−1.58923	−0.794617	−	0.607111i	$-0.792328\pi$
$211$	−1.02777e6	−1.58923	−0.794617	+	0.607111i	$0.792328\pi$
$212$	−911634.	−1.39310
$213$	0	0
$214$	1.37305e6	2.04952
$215$	−812456.	−1.19868
$216$	0	0
$217$	2140.58	0.00308591
$218$	−1.46927e6	−2.09392
$219$	0	0
$220$	−1.47376e6	−2.05291
$221$	248141.	0.341758
$222$	0	0
$223$	−209627.	−0.282284	−0.141142	−	0.989989i	$-0.545077\pi$
$223$	−209627.	−0.282284	−0.141142	+	0.989989i	$0.545077\pi$
$224$	4284.93	0.00570589
$225$	0	0
$226$	1.12024e6	1.45895
$227$	347103.	0.447088	0.223544	−	0.974694i	$-0.428237\pi$
$227$	347103.	0.447088	0.223544	+	0.974694i	$0.428237\pi$
$228$	0	0
$229$	−474094.	−0.597414	−0.298707	−	0.954345i	$-0.596555\pi$
$229$	−474094.	−0.597414	−0.298707	+	0.954345i	$0.596555\pi$
$230$	237101.	0.295538
$231$	0	0
$232$	−1.49622e6	−1.82505
$233$	−67426.2	−0.0813652	−0.0406826	−	0.999172i	$-0.512953\pi$
$233$	−67426.2	−0.0813652	−0.0406826	+	0.999172i	$0.512953\pi$
$234$	0	0
$235$	1.15189e6	1.36063
$236$	−1.00024e6	−1.16903
$237$	0	0
$238$	2486.81	0.00284577
$239$	631686.	0.715331	0.357665	−	0.933850i	$-0.383573\pi$
$239$	631686.	0.715331	0.357665	+	0.933850i	$0.383573\pi$
$240$	0	0
$241$	−582036.	−0.645516	−0.322758	−	0.946482i	$-0.604610\pi$
$241$	−582036.	−0.645516	−0.322758	+	0.946482i	$0.604610\pi$
$242$	179797.	0.197353
$243$	0	0
$244$	3.34840e6	3.60050
$245$	699418.	0.744426
$246$	0	0
$247$	519130.	0.541419
$248$	5.06243e6	5.22673
$249$	0	0
$250$	2.02508e6	2.04924
$251$	1.28835e6	1.29077	0.645386	−	0.763856i	$-0.276696\pi$
$251$	1.28835e6	1.29077	0.645386	+	0.763856i	$0.276696\pi$
$252$	0	0
$253$	−223025.	−0.219055
$254$	2.19459e6	2.13437
$255$	0	0
$256$	1.14509e6	1.09204
$257$	204908.	0.193520	0.0967601	−	0.995308i	$-0.469152\pi$
$257$	204908.	0.193520	0.0967601	+	0.995308i	$0.469152\pi$
$258$	0	0
$259$	−2999.32	−0.00277826
$260$	−889650.	−0.816179
$261$	0	0
$262$	−653480.	−0.588138
$263$	−418862.	−0.373407	−0.186703	−	0.982416i	$-0.559780\pi$
$263$	−418862.	−0.373407	−0.186703	+	0.982416i	$0.559780\pi$
$264$	0	0
$265$	451637.	0.395071
$266$	5202.58	0.00450832
$267$	0	0
$268$	−2.39824e6	−2.03965
$269$	1.48662e6	1.25262	0.626308	−	0.779576i	$-0.284565\pi$
$269$	1.48662e6	1.25262	0.626308	+	0.779576i	$0.284565\pi$
$270$	0	0
$271$	−658640.	−0.544785	−0.272392	−	0.962186i	$-0.587815\pi$
$271$	−658640.	−0.544785	−0.272392	+	0.962186i	$0.587815\pi$
$272$	3.26042e6	2.67209
$273$	0	0
$274$	−1.01722e6	−0.818537
$275$	−587372.	−0.468362
$276$	0	0
$277$	236526.	0.185216	0.0926082	−	0.995703i	$-0.470480\pi$
$277$	236526.	0.185216	0.0926082	+	0.995703i	$0.470480\pi$
$278$	4.34592e6	3.37263
$279$	0	0
$280$	−5519.32	−0.00420718
$281$	1.66253e6	1.25604	0.628020	−	0.778197i	$-0.283866\pi$
$281$	1.66253e6	1.25604	0.628020	+	0.778197i	$0.283866\pi$
$282$	0	0
$283$	966812.	0.717589	0.358795	−	0.933417i	$-0.383188\pi$
$283$	966812.	0.717589	0.358795	+	0.933417i	$0.383188\pi$
$284$	−4.38310e6	−3.22467
$285$	0	0
$286$	1.15563e6	0.835418
$287$	−2402.81	−0.00172193
$288$	0	0
$289$	−469218.	−0.330469
$290$	1.19740e6	0.836074
$291$	0	0
$292$	4.78111e6	3.28150
$293$	82436.0	0.0560981	0.0280490	−	0.999607i	$-0.491071\pi$
$293$	82436.0	0.0560981	0.0280490	+	0.999607i	$0.491071\pi$
$294$	0	0
$295$	495535.	0.331528
$296$	−7.09332e6	−4.70565
$297$	0	0
$298$	−4.14149e6	−2.70157
$299$	−134632.	−0.0870902
$300$	0	0
$301$	4623.35	0.00294131
$302$	−4.76595e6	−3.00699
$303$	0	0
$304$	6.82105e6	4.23318
$305$	−1.65884e6	−1.02107
$306$	0	0
$307$	471104.	0.285280	0.142640	−	0.989775i	$-0.454441\pi$
$307$	471104.	0.285280	0.142640	+	0.989775i	$0.454441\pi$
$308$	8386.54	0.00503740
$309$	0	0
$310$	−4.05139e6	−2.39442
$311$	−974141.	−0.571112	−0.285556	−	0.958362i	$-0.592178\pi$
$311$	−974141.	−0.571112	−0.285556	+	0.958362i	$0.592178\pi$
$312$	0	0
$313$	203897.	0.117639	0.0588194	−	0.998269i	$-0.481266\pi$
$313$	203897.	0.117639	0.0588194	+	0.998269i	$0.481266\pi$
$314$	2.59267e6	1.48396
$315$	0	0
$316$	1.94698e6	1.09684
$317$	1.11600e6	0.623757	0.311879	−	0.950122i	$-0.399042\pi$
$317$	1.11600e6	0.623757	0.311879	+	0.950122i	$0.399042\pi$
$318$	0	0
$319$	−1.12632e6	−0.619704
$320$	−3.65678e6	−1.99629
$321$	0	0
$322$	−1349.24	−0.000725187 0
$323$	1.98881e6	1.06068
$324$	0	0
$325$	−354574.	−0.186208
$326$	−1.91914e6	−1.00014
$327$	0	0
$328$	−5.68259e6	−2.91650
$329$	−6554.91	−0.00333870
$330$	0	0
$331$	2.08998e6	1.04851	0.524254	−	0.851562i	$-0.324344\pi$
$331$	2.08998e6	1.04851	0.524254	+	0.851562i	$0.324344\pi$
$332$	1.54094e6	0.767258
$333$	0	0
$334$	−274915.	−0.134844
$335$	1.18812e6	0.578428
$336$	0	0
$337$	901699.	0.432501	0.216250	−	0.976338i	$-0.430617\pi$
$337$	901699.	0.432501	0.216250	+	0.976338i	$0.430617\pi$
$338$	−3.30134e6	−1.57180
$339$	0	0
$340$	−3.40828e6	−1.59896
$341$	3.81088e6	1.77476
$342$	0	0
$343$	−7960.21	−0.00365333
$344$	1.09341e7	4.98182
$345$	0	0
$346$	−7.55932e6	−3.39463
$347$	1.53207e6	0.683053	0.341526	−	0.939872i	$-0.389056\pi$
$347$	1.53207e6	0.683053	0.341526	+	0.939872i	$0.389056\pi$
$348$	0	0
$349$	1.28353e6	0.564084	0.282042	−	0.959402i	$-0.408988\pi$
$349$	1.28353e6	0.564084	0.282042	+	0.959402i	$0.408988\pi$
$350$	−3553.44	−0.00155053
$351$	0	0
$352$	7.62846e6	3.28156
$353$	4.45978e6	1.90492	0.952460	−	0.304663i	$-0.0985440\pi$
$353$	4.45978e6	1.90492	0.952460	+	0.304663i	$0.0985440\pi$
$354$	0	0
$355$	2.17145e6	0.914491
$356$	−3.97844e6	−1.66375
$357$	0	0
$358$	3.15482e6	1.30097
$359$	−4.67506e6	−1.91448	−0.957240	−	0.289296i	$-0.906579\pi$
$359$	−4.67506e6	−1.91448	−0.957240	+	0.289296i	$0.906579\pi$
$360$	0	0
$361$	1.68463e6	0.680356
$362$	−419318.	−0.168179
$363$	0	0
$364$	5062.63	0.00200273
$365$	−2.36863e6	−0.930606
$366$	0	0
$367$	2.43229e6	0.942650	0.471325	−	0.881960i	$-0.343776\pi$
$367$	2.43229e6	0.942650	0.471325	+	0.881960i	$0.343776\pi$
$368$	−1.76898e6	−0.680930
$369$	0	0
$370$	5.67668e6	2.15571
$371$	−2570.08	−0.000969419 0
$372$	0	0
$373$	1.37458e6	0.511561	0.255780	−	0.966735i	$-0.417668\pi$
$373$	1.37458e6	0.511561	0.255780	+	0.966735i	$0.417668\pi$
$374$	4.42726e6	1.63665
$375$	0	0
$376$	−1.55022e7	−5.65489
$377$	−679914.	−0.246377
$378$	0	0
$379$	2.01997e6	0.722350	0.361175	−	0.932498i	$-0.382376\pi$
$379$	2.01997e6	0.722350	0.361175	+	0.932498i	$0.382376\pi$
$380$	−7.13037e6	−2.53310
$381$	0	0
$382$	−401139.	−0.140649
$383$	−2.00834e6	−0.699584	−0.349792	−	0.936827i	$-0.613748\pi$
$383$	−2.00834e6	−0.699584	−0.349792	+	0.936827i	$0.613748\pi$
$384$	0	0
$385$	−4154.82	−0.00142857
$386$	−457247.	−0.156201
$387$	0	0
$388$	−1.17919e7	−3.97652
$389$	−3.25055e6	−1.08914	−0.544569	−	0.838716i	$-0.683307\pi$
$389$	−3.25055e6	−1.08914	−0.544569	+	0.838716i	$0.683307\pi$
$390$	0	0
$391$	−515779.	−0.170617
$392$	−9.41285e6	−3.09390
$393$	0	0
$394$	−940628.	−0.305265
$395$	−964563.	−0.311055
$396$	0	0
$397$	5.43707e6	1.73137	0.865683	−	0.500592i	$-0.166884\pi$
$397$	5.43707e6	1.73137	0.865683	+	0.500592i	$0.166884\pi$
$398$	1.17159e7	3.70737
$399$	0	0
$400$	−4.65888e6	−1.45590
$401$	−3.27208e6	−1.01616	−0.508081	−	0.861309i	$-0.669645\pi$
$401$	−3.27208e6	−1.01616	−0.508081	+	0.861309i	$0.669645\pi$
$402$	0	0
$403$	2.30048e6	0.705596
$404$	−7.31474e6	−2.22970
$405$	0	0
$406$	−6813.92	−0.00205155
$407$	−5.33968e6	−1.59783
$408$	0	0
$409$	−1.56299e6	−0.462007	−0.231003	−	0.972953i	$-0.574201\pi$
$409$	−1.56299e6	−0.462007	−0.231003	+	0.972953i	$0.574201\pi$
$410$	4.54769e6	1.33608
$411$	0	0
$412$	1.51107e6	0.438572
$413$	−2819.89	−0.000813498 0
$414$	0	0
$415$	−763406.	−0.217588
$416$	4.60500e6	1.30466
$417$	0	0
$418$	9.26215e6	2.59281
$419$	4.76111e6	1.32487	0.662434	−	0.749120i	$-0.269523\pi$
$419$	4.76111e6	1.32487	0.662434	+	0.749120i	$0.269523\pi$
$420$	0	0
$421$	−4.56989e6	−1.25661	−0.628305	−	0.777967i	$-0.716251\pi$
$421$	−4.56989e6	−1.25661	−0.628305	+	0.777967i	$0.716251\pi$
$422$	−1.10694e7	−3.02581
$423$	0	0
$424$	−6.07818e6	−1.64195
$425$	−1.35839e6	−0.364796
$426$	0	0
$427$	9439.80	0.00250549
$428$	1.07087e7	2.82571
$429$	0	0
$430$	−8.75042e6	−2.28222
$431$	3.28891e6	0.852824	0.426412	−	0.904529i	$-0.359777\pi$
$431$	3.28891e6	0.852824	0.426412	+	0.904529i	$0.359777\pi$
$432$	0	0
$433$	2.11805e6	0.542895	0.271447	−	0.962453i	$-0.412498\pi$
$433$	2.11805e6	0.542895	0.271447	+	0.962453i	$0.412498\pi$
$434$	23054.8	0.00587540
$435$	0	0
$436$	−1.14591e7	−2.88692
$437$	−1.07905e6	−0.270294
$438$	0	0
$439$	758443.	0.187829	0.0939143	−	0.995580i	$-0.470062\pi$
$439$	758443.	0.187829	0.0939143	+	0.995580i	$0.470062\pi$
$440$	−9.82605e6	−2.41962
$441$	0	0
$442$	2.67257e6	0.650688
$443$	−650377.	−0.157455	−0.0787274	−	0.996896i	$-0.525086\pi$
$443$	−650377.	−0.157455	−0.0787274	+	0.996896i	$0.525086\pi$
$444$	0	0
$445$	1.97098e6	0.471826
$446$	−2.25776e6	−0.537452
$447$	0	0
$448$	20809.2	0.00489847
$449$	7.58111e6	1.77467	0.887333	−	0.461129i	$-0.152555\pi$
$449$	7.58111e6	1.77467	0.887333	+	0.461129i	$0.152555\pi$
$450$	0	0
$451$	−4.27772e6	−0.990309
$452$	8.73699e6	2.01148
$453$	0	0
$454$	3.73841e6	0.851231
$455$	−2508.10	−0.000567958 0
$456$	0	0
$457$	−8.04537e6	−1.80200	−0.901002	−	0.433815i	$-0.857167\pi$
$457$	−8.04537e6	−1.80200	−0.901002	+	0.433815i	$0.857167\pi$
$458$	−5.10614e6	−1.13744
$459$	0	0
$460$	1.84920e6	0.407463
$461$	−5.63578e6	−1.23510	−0.617550	−	0.786532i	$-0.711875\pi$
$461$	−5.63578e6	−1.23510	−0.617550	+	0.786532i	$0.711875\pi$
$462$	0	0
$463$	1.39156e6	0.301682	0.150841	−	0.988558i	$-0.451802\pi$
$463$	1.39156e6	0.301682	0.150841	+	0.988558i	$0.451802\pi$
$464$	−8.93365e6	−1.92634
$465$	0	0
$466$	−726202.	−0.154915
$467$	−4.05063e6	−0.859469	−0.429734	−	0.902955i	$-0.641393\pi$
$467$	−4.05063e6	−0.859469	−0.429734	+	0.902955i	$0.641393\pi$
$468$	0	0
$469$	−6761.12	−0.00141934
$470$	1.24062e7	2.59056
$471$	0	0
$472$	−6.66897e6	−1.37786
$473$	8.23095e6	1.69160
$474$	0	0
$475$	−2.84184e6	−0.577917
$476$	19395.1	0.00392351
$477$	0	0
$478$	6.80347e6	1.36195
$479$	−6.55529e6	−1.30543	−0.652714	−	0.757605i	$-0.726370\pi$
$479$	−6.55529e6	−1.30543	−0.652714	+	0.757605i	$0.726370\pi$
$480$	0	0
$481$	−3.22336e6	−0.635252
$482$	−6.26872e6	−1.22903
$483$	0	0
$484$	1.40227e6	0.272094
$485$	5.84186e6	1.12771
$486$	0	0
$487$	−1.98924e6	−0.380071	−0.190035	−	0.981777i	$-0.560860\pi$
$487$	−1.98924e6	−0.380071	−0.190035	+	0.981777i	$0.560860\pi$
$488$	2.23249e7	4.24366
$489$	0	0
$490$	7.53296e6	1.41735
$491$	−7.99205e6	−1.49608	−0.748040	−	0.663654i	$-0.769005\pi$
$491$	−7.99205e6	−1.49608	−0.748040	+	0.663654i	$0.769005\pi$
$492$	0	0
$493$	−2.60478e6	−0.482673
$494$	5.59120e6	1.03083
$495$	0	0
$496$	3.02269e7	5.51682
$497$	−12356.8	−0.00224397
$498$	0	0
$499$	−1.00308e7	−1.80338	−0.901688	−	0.432387i	$-0.857671\pi$
$499$	−1.00308e7	−1.80338	−0.901688	+	0.432387i	$0.857671\pi$
$500$	1.57940e7	2.82533
$501$	0	0
$502$	1.38760e7	2.45756
$503$	4.82601e6	0.850488	0.425244	−	0.905079i	$-0.360188\pi$
$503$	4.82601e6	0.850488	0.425244	+	0.905079i	$0.360188\pi$
$504$	0	0
$505$	3.62383e6	0.632324
$506$	−2.40206e6	−0.417068
$507$	0	0
$508$	1.71161e7	2.94270
$509$	−138289.	−0.0236589	−0.0118294	−	0.999930i	$-0.503766\pi$
$509$	−138289.	−0.0236589	−0.0118294	+	0.999930i	$0.503766\pi$
$510$	0	0
$511$	13478.9	0.00228351
$512$	576256.	0.0971494
$513$	0	0
$514$	2.20693e6	0.368451
$515$	−748605.	−0.124375
$516$	0	0
$517$	−1.16697e7	−1.92014
$518$	−32303.6	−0.00528965
$519$	0	0
$520$	−5.93160e6	−0.961975
$521$	−4.62831e6	−0.747012	−0.373506	−	0.927628i	$-0.621845\pi$
$521$	−4.62831e6	−0.747012	−0.373506	+	0.927628i	$0.621845\pi$
$522$	0	0
$523$	−1.54914e6	−0.247649	−0.123825	−	0.992304i	$-0.539516\pi$
$523$	−1.54914e6	−0.247649	−0.123825	+	0.992304i	$0.539516\pi$
$524$	−5.09662e6	−0.810876
$525$	0	0
$526$	−4.51129e6	−0.710945
$527$	8.81323e6	1.38232
$528$	0	0
$529$	279841.	0.0434783
$530$	4.86428e6	0.752192
$531$	0	0
$532$	40576.0	0.00621570
$533$	−2.58229e6	−0.393720
$534$	0	0
$535$	−5.30526e6	−0.801349
$536$	−1.59899e7	−2.40399
$537$	0	0
$538$	1.60113e7	2.38491
$539$	−7.08577e6	−1.05055
$540$	0	0
$541$	−6.52702e6	−0.958787	−0.479393	−	0.877600i	$-0.659143\pi$
$541$	−6.52702e6	−0.958787	−0.479393	+	0.877600i	$0.659143\pi$
$542$	−7.09377e6	−1.03724
$543$	0	0
$544$	1.76419e7	2.55593
$545$	5.67701e6	0.818707
$546$	0	0
$547$	1.06849e7	1.52686	0.763432	−	0.645888i	$-0.223513\pi$
$547$	1.06849e7	1.52686	0.763432	+	0.645888i	$0.223513\pi$
$548$	−7.93351e6	−1.12853
$549$	0	0
$550$	−6.32619e6	−0.891735
$551$	−5.44938e6	−0.764660
$552$	0	0
$553$	5488.93	0.000763264 0
$554$	2.54746e6	0.352642
$555$	0	0
$556$	3.38947e7	4.64991
$557$	3.07804e6	0.420374	0.210187	−	0.977661i	$-0.432593\pi$
$557$	3.07804e6	0.420374	0.210187	+	0.977661i	$0.432593\pi$
$558$	0	0
$559$	4.96871e6	0.672533
$560$	−32954.9	−0.00444068
$561$	0	0
$562$	1.79060e7	2.39143
$563$	−1.10366e7	−1.46745	−0.733724	−	0.679448i	$-0.762219\pi$
$563$	−1.10366e7	−1.46745	−0.733724	+	0.679448i	$0.762219\pi$
$564$	0	0
$565$	−4.32844e6	−0.570440
$566$	1.04129e7	1.36625
$567$	0	0
$568$	−2.92237e7	−3.80070
$569$	9.26308e6	1.19943	0.599715	−	0.800214i	$-0.295281\pi$
$569$	9.26308e6	1.19943	0.599715	+	0.800214i	$0.295281\pi$
$570$	0	0
$571$	−1.31913e7	−1.69315	−0.846576	−	0.532267i	$-0.821340\pi$
$571$	−1.31913e7	−1.69315	−0.846576	+	0.532267i	$0.821340\pi$
$572$	9.01300e6	1.15181
$573$	0	0
$574$	−25879.0	−0.00327845
$575$	737006.	0.0929611
$576$	0	0
$577$	−1.29741e6	−0.162232	−0.0811160	−	0.996705i	$-0.525848\pi$
$577$	−1.29741e6	−0.162232	−0.0811160	+	0.996705i	$0.525848\pi$
$578$	−5.05363e6	−0.629193
$579$	0	0
$580$	9.33878e6	1.15271
$581$	4344.23	0.000533916 0
$582$	0	0
$583$	−4.57551e6	−0.557530
$584$	3.18773e7	3.86767
$585$	0	0
$586$	887863.	0.106808
$587$	−4.51203e6	−0.540477	−0.270238	−	0.962793i	$-0.587103\pi$
$587$	−4.51203e6	−0.540477	−0.270238	+	0.962793i	$0.587103\pi$
$588$	0	0
$589$	1.84379e7	2.18990
$590$	5.33708e6	0.631210
$591$	0	0
$592$	−4.23529e7	−4.96683
$593$	−9.27288e6	−1.08287	−0.541437	−	0.840741i	$-0.682120\pi$
$593$	−9.27288e6	−1.08287	−0.541437	+	0.840741i	$0.682120\pi$
$594$	0	0
$595$	−9608.63	−0.00111268
$596$	−3.23004e7	−3.72470
$597$	0	0
$598$	−1.45003e6	−0.165815
$599$	6.32225e6	0.719954	0.359977	−	0.932961i	$-0.382785\pi$
$599$	6.32225e6	0.719954	0.359977	+	0.932961i	$0.382785\pi$
$600$	0	0
$601$	3.18785e6	0.360008	0.180004	−	0.983666i	$-0.442389\pi$
$601$	3.18785e6	0.360008	0.180004	+	0.983666i	$0.442389\pi$
$602$	49795.0	0.00560009
$603$	0	0
$604$	−3.71706e7	−4.14579
$605$	−694706.	−0.0771636
$606$	0	0
$607$	−5.70410e6	−0.628369	−0.314185	−	0.949362i	$-0.601731\pi$
$607$	−5.70410e6	−0.628369	−0.314185	+	0.949362i	$0.601731\pi$
$608$	3.69082e7	4.04915
$609$	0	0
$610$	−1.78663e7	−1.94406
$611$	−7.04455e6	−0.763396
$612$	0	0
$613$	1.17835e7	1.26655	0.633276	−	0.773926i	$-0.281710\pi$
$613$	1.17835e7	1.26655	0.633276	+	0.773926i	$0.281710\pi$
$614$	5.07394e6	0.543156
$615$	0	0
$616$	55916.0	0.00593723
$617$	−1.41628e6	−0.149774	−0.0748868	−	0.997192i	$-0.523860\pi$
$617$	−1.41628e6	−0.149774	−0.0748868	+	0.997192i	$0.523860\pi$
$618$	0	0
$619$	−4.37752e6	−0.459200	−0.229600	−	0.973285i	$-0.573742\pi$
$619$	−4.37752e6	−0.459200	−0.229600	+	0.973285i	$0.573742\pi$
$620$	−3.15976e7	−3.30123
$621$	0	0
$622$	−1.04918e7	−1.08736
$623$	−11216.0	−0.00115776
$624$	0	0
$625$	−3.47084e6	−0.355414
$626$	2.19604e6	0.223978
$627$	0	0
$628$	2.02208e7	2.04597
$629$	−1.23488e7	−1.24451
$630$	0	0
$631$	−1.61975e7	−1.61948	−0.809739	−	0.586790i	$-0.800391\pi$
$631$	−1.61975e7	−1.61948	−0.809739	+	0.586790i	$0.800391\pi$
$632$	1.29812e7	1.29277
$633$	0	0
$634$	1.20197e7	1.18760
$635$	−8.47956e6	−0.834525
$636$	0	0
$637$	−4.27740e6	−0.417668
$638$	−1.21308e7	−1.17988
$639$	0	0
$640$	−1.52892e7	−1.47548
$641$	1.56439e7	1.50383	0.751915	−	0.659260i	$-0.229130\pi$
$641$	1.56439e7	1.50383	0.751915	+	0.659260i	$0.229130\pi$
$642$	0	0
$643$	−1.57570e7	−1.50295	−0.751476	−	0.659760i	$-0.770658\pi$
$643$	−1.57570e7	−1.50295	−0.751476	+	0.659760i	$0.770658\pi$
$644$	−10523.0	−0.000999829 0
$645$	0	0
$646$	2.14201e7	2.01948
$647$	660291.	0.0620119	0.0310059	−	0.999519i	$-0.490129\pi$
$647$	660291.	0.0620119	0.0310059	+	0.999519i	$0.490129\pi$
$648$	0	0
$649$	−5.02024e6	−0.467857
$650$	−3.81888e6	−0.354529
$651$	0	0
$652$	−1.49678e7	−1.37892
$653$	−1.24310e7	−1.14084	−0.570419	−	0.821354i	$-0.693219\pi$
$653$	−1.24310e7	−1.14084	−0.570419	+	0.821354i	$0.693219\pi$
$654$	0	0
$655$	2.52494e6	0.229958
$656$	−3.39297e7	−3.07837
$657$	0	0
$658$	−70598.6	−0.00635669
$659$	13723.9	0.00123101	0.000615507	−	1.00000i	$-0.499804\pi$
$659$	13723.9	0.00123101	0.000615507		1.00000i	$0.499804\pi$
$660$	0	0
$661$	6.54194e6	0.582375	0.291187	−	0.956666i	$-0.405950\pi$
$661$	6.54194e6	0.582375	0.291187	+	0.956666i	$0.405950\pi$
$662$	2.25098e7	1.99630
$663$	0	0
$664$	1.02740e7	0.904315
$665$	−20101.9	−0.00176272
$666$	0	0
$667$	1.41325e6	0.123000
$668$	−2.14412e6	−0.185912
$669$	0	0
$670$	1.27965e7	1.10129
$671$	1.68057e7	1.44095
$672$	0	0
$673$	1.54266e7	1.31290	0.656451	−	0.754369i	$-0.272057\pi$
$673$	1.54266e7	1.31290	0.656451	+	0.754369i	$0.272057\pi$
$674$	9.71159e6	0.823456
$675$	0	0
$676$	−2.57478e7	−2.16707
$677$	−5.30462e6	−0.444818	−0.222409	−	0.974953i	$-0.571392\pi$
$677$	−5.30462e6	−0.444818	−0.222409	+	0.974953i	$0.571392\pi$
$678$	0	0
$679$	−33243.6	−0.00276716
$680$	−2.27242e7	−1.88459
$681$	0	0
$682$	4.10444e7	3.37904
$683$	−1.52230e7	−1.24867	−0.624337	−	0.781155i	$-0.714631\pi$
$683$	−1.52230e7	−1.24867	−0.624337	+	0.781155i	$0.714631\pi$
$684$	0	0
$685$	3.93038e6	0.320043
$686$	−85734.1	−0.00695574
$687$	0	0
$688$	6.52857e7	5.25832
$689$	−2.76206e6	−0.221659
$690$	0	0
$691$	1.04711e7	0.834248	0.417124	−	0.908850i	$-0.363038\pi$
$691$	1.04711e7	0.834248	0.417124	+	0.908850i	$0.363038\pi$
$692$	−5.89567e7	−4.68024
$693$	0	0
$694$	1.65009e7	1.30049
$695$	−1.67919e7	−1.31868
$696$	0	0
$697$	−9.89286e6	−0.771329
$698$	1.38241e7	1.07398
$699$	0	0
$700$	−27714.0	−0.00213774
$701$	2.28874e7	1.75914	0.879572	−	0.475765i	$-0.157829\pi$
$701$	2.28874e7	1.75914	0.879572	+	0.475765i	$0.157829\pi$
$702$	0	0
$703$	−2.58346e7	−1.97158
$704$	3.70466e7	2.81720
$705$	0	0
$706$	4.80333e7	3.62686
$707$	−20621.7	−0.00155159
$708$	0	0
$709$	1.85127e7	1.38310	0.691550	−	0.722329i	$-0.256928\pi$
$709$	1.85127e7	1.38310	0.691550	+	0.722329i	$0.256928\pi$
$710$	2.33873e7	1.74114
$711$	0	0
$712$	−2.65257e7	−1.96095
$713$	−4.78170e6	−0.352256
$714$	0	0
$715$	−4.46517e6	−0.326643
$716$	2.46051e7	1.79367
$717$	0	0
$718$	−5.03519e7	−3.64506
$719$	−1.25679e7	−0.906652	−0.453326	−	0.891345i	$-0.649763\pi$
$719$	−1.25679e7	−0.906652	−0.453326	+	0.891345i	$0.649763\pi$
$720$	0	0
$721$	4260.00	0.000305191 0
$722$	1.81440e7	1.29536
$723$	0	0
$724$	−3.27035e6	−0.231872
$725$	3.72201e6	0.262986
$726$	0	0
$727$	3.64050e6	0.255462	0.127731	−	0.991809i	$-0.459231\pi$
$727$	3.64050e6	0.255462	0.127731	+	0.991809i	$0.459231\pi$
$728$	33754.3	0.00236048
$729$	0	0
$730$	−2.55110e7	−1.77182
$731$	1.90353e7	1.31755
$732$	0	0
$733$	−4.38735e6	−0.301608	−0.150804	−	0.988564i	$-0.548186\pi$
$733$	−4.38735e6	−0.301608	−0.150804	+	0.988564i	$0.548186\pi$
$734$	2.61966e7	1.79475
$735$	0	0
$736$	−9.57181e6	−0.651327
$737$	−1.20368e7	−0.816287
$738$	0	0
$739$	1.95857e7	1.31925	0.659625	−	0.751595i	$-0.270715\pi$
$739$	1.95857e7	1.31925	0.659625	+	0.751595i	$0.270715\pi$
$740$	4.42736e7	2.97211
$741$	0	0
$742$	−27680.6	−0.00184572
$743$	2.94833e7	1.95931	0.979657	−	0.200678i	$-0.0643146\pi$
$743$	2.94833e7	1.95931	0.979657	+	0.200678i	$0.0643146\pi$
$744$	0	0
$745$	1.60021e7	1.05630
$746$	1.48047e7	0.973983
$747$	0	0
$748$	3.45291e7	2.25648
$749$	30190.0	0.00196634
$750$	0	0
$751$	1.27147e7	0.822634	0.411317	−	0.911492i	$-0.365069\pi$
$751$	1.27147e7	0.822634	0.411317	+	0.911492i	$0.365069\pi$
$752$	−9.25609e7	−5.96875
$753$	0	0
$754$	−7.32290e6	−0.469089
$755$	1.84149e7	1.17571
$756$	0	0
$757$	1.63980e7	1.04004	0.520022	−	0.854153i	$-0.325924\pi$
$757$	1.63980e7	1.04004	0.520022	+	0.854153i	$0.325924\pi$
$758$	2.17558e7	1.37531
$759$	0	0
$760$	−4.75407e7	−2.98560
$761$	−8.50863e6	−0.532596	−0.266298	−	0.963891i	$-0.585801\pi$
$761$	−8.50863e6	−0.532596	−0.266298	+	0.963891i	$0.585801\pi$
$762$	0	0
$763$	−32305.5	−0.00200893
$764$	−3.12856e6	−0.193915
$765$	0	0
$766$	−2.16305e7	−1.33197
$767$	−3.03053e6	−0.186007
$768$	0	0
$769$	1.70141e7	1.03751	0.518755	−	0.854923i	$-0.326396\pi$
$769$	1.70141e7	1.03751	0.518755	+	0.854923i	$0.326396\pi$
$770$	−44748.7	−0.00271991
$771$	0	0
$772$	−3.56616e6	−0.215356
$773$	−3.63916e6	−0.219055	−0.109527	−	0.993984i	$-0.534934\pi$
$773$	−3.63916e6	−0.219055	−0.109527	+	0.993984i	$0.534934\pi$
$774$	0	0
$775$	−1.25934e7	−0.753161
$776$	−7.86204e7	−4.68685
$777$	0	0
$778$	−3.50095e7	−2.07366
$779$	−2.06966e7	−1.22195
$780$	0	0
$781$	−2.19989e7	−1.29054
$782$	−5.55511e6	−0.324845
$783$	0	0
$784$	−5.62024e7	−3.26561
$785$	−1.00177e7	−0.580220
$786$	0	0
$787$	−5.30127e6	−0.305101	−0.152550	−	0.988296i	$-0.548749\pi$
$787$	−5.30127e6	−0.305101	−0.152550	+	0.988296i	$0.548749\pi$
$788$	−7.33615e6	−0.420875
$789$	0	0
$790$	−1.03887e7	−0.592232
$791$	24631.3	0.00139974
$792$	0	0
$793$	1.01449e7	0.572883
$794$	5.85591e7	3.29642
$795$	0	0
$796$	9.13743e7	5.11142
$797$	−1.41955e7	−0.791598	−0.395799	−	0.918337i	$-0.629532\pi$
$797$	−1.41955e7	−0.791598	−0.395799	+	0.918337i	$0.629532\pi$
$798$	0	0
$799$	−2.69879e7	−1.49555
$800$	−2.52089e7	−1.39261
$801$	0	0
$802$	−3.52414e7	−1.93472
$803$	2.39965e7	1.31329
$804$	0	0
$805$	5213.26	0.000283543 0
$806$	2.47769e7	1.34341
$807$	0	0
$808$	−4.87699e7	−2.62799
$809$	1.58542e7	0.851675	0.425838	−	0.904800i	$-0.359979\pi$
$809$	1.58542e7	0.851675	0.425838	+	0.904800i	$0.359979\pi$
$810$	0	0
$811$	−1.40129e7	−0.748127	−0.374063	−	0.927403i	$-0.622036\pi$
$811$	−1.40129e7	−0.748127	−0.374063	+	0.927403i	$0.622036\pi$
$812$	−53143.1	−0.00282851
$813$	0	0
$814$	−5.75101e7	−3.04217
$815$	7.41524e6	0.391049
$816$	0	0
$817$	3.98232e7	2.08728
$818$	−1.68339e7	−0.879634
$819$	0	0
$820$	3.54684e7	1.84207
$821$	−1.16279e7	−0.602065	−0.301033	−	0.953614i	$-0.597331\pi$
$821$	−1.16279e7	−0.602065	−0.301033	+	0.953614i	$0.597331\pi$
$822$	0	0
$823$	−2.76237e7	−1.42161	−0.710807	−	0.703387i	$-0.751670\pi$
$823$	−2.76237e7	−1.42161	−0.710807	+	0.703387i	$0.751670\pi$
$824$	1.00748e7	0.516914
$825$	0	0
$826$	−30371.1	−0.00154885
$827$	2.78146e7	1.41420	0.707098	−	0.707116i	$-0.250004\pi$
$827$	2.78146e7	1.41420	0.707098	+	0.707116i	$0.250004\pi$
$828$	0	0
$829$	−1.23659e7	−0.624943	−0.312471	−	0.949927i	$-0.601157\pi$
$829$	−1.23659e7	−0.624943	−0.312471	+	0.949927i	$0.601157\pi$
$830$	−8.22214e6	−0.414276
$831$	0	0
$832$	2.23636e7	1.12004
$833$	−1.63869e7	−0.818246
$834$	0	0
$835$	1.06223e6	0.0527231
$836$	7.22374e7	3.57476
$837$	0	0
$838$	5.12787e7	2.52247
$839$	3.49502e7	1.71413	0.857067	−	0.515205i	$-0.172284\pi$
$839$	3.49502e7	1.71413	0.857067	+	0.515205i	$0.172284\pi$
$840$	0	0
$841$	−1.33740e7	−0.652035
$842$	−4.92192e7	−2.39251
$843$	0	0
$844$	−8.63323e7	−4.17174
$845$	1.27558e7	0.614565
$846$	0	0
$847$	3953.28	0.000189343 0
$848$	−3.62917e7	−1.73308
$849$	0	0
$850$	−1.46303e7	−0.694552
$851$	6.69997e6	0.317138
$852$	0	0
$853$	−9.99950e6	−0.470550	−0.235275	−	0.971929i	$-0.575599\pi$
$853$	−9.99950e6	−0.470550	−0.235275	+	0.971929i	$0.575599\pi$
$854$	101670.	0.00477032
$855$	0	0
$856$	7.13987e7	3.33047
$857$	−9.71240e6	−0.451725	−0.225863	−	0.974159i	$-0.572520\pi$
$857$	−9.71240e6	−0.451725	−0.225863	+	0.974159i	$0.572520\pi$
$858$	0	0
$859$	3.69393e7	1.70807	0.854035	−	0.520216i	$-0.174149\pi$
$859$	3.69393e7	1.70807	0.854035	+	0.520216i	$0.174149\pi$
$860$	−6.82463e7	−3.14654
$861$	0	0
$862$	3.54227e7	1.62373
$863$	3.31284e7	1.51416	0.757082	−	0.653319i	$-0.226624\pi$
$863$	3.31284e7	1.51416	0.757082	+	0.653319i	$0.226624\pi$
$864$	0	0
$865$	2.92080e7	1.32728
$866$	2.28121e7	1.03364
$867$	0	0
$868$	179809.	0.00810051
$869$	9.77194e6	0.438966
$870$	0	0
$871$	−7.26615e6	−0.324533
$872$	−7.64019e7	−3.40262
$873$	0	0
$874$	−1.16217e7	−0.514625
$875$	44526.6	0.00196607
$876$	0	0
$877$	1.38007e7	0.605902	0.302951	−	0.953006i	$-0.402028\pi$
$877$	1.38007e7	0.605902	0.302951	+	0.953006i	$0.402028\pi$
$878$	8.16869e6	0.357615
$879$	0	0
$880$	−5.86696e7	−2.55391
$881$	7.74868e6	0.336347	0.168174	−	0.985757i	$-0.446213\pi$
$881$	7.74868e6	0.336347	0.168174	+	0.985757i	$0.446213\pi$
$882$	0	0
$883$	3.79838e7	1.63944	0.819722	−	0.572761i	$-0.194128\pi$
$883$	3.79838e7	1.63944	0.819722	+	0.572761i	$0.194128\pi$
$884$	2.08439e7	0.897115
$885$	0	0
$886$	−7.00477e6	−0.299785
$887$	1.51721e7	0.647495	0.323747	−	0.946144i	$-0.395057\pi$
$887$	1.51721e7	0.647495	0.323747	+	0.946144i	$0.395057\pi$
$888$	0	0
$889$	48253.7	0.00204775
$890$	2.12281e7	0.898330
$891$	0	0
$892$	−1.76087e7	−0.740995
$893$	−5.64607e7	−2.36929
$894$	0	0
$895$	−1.21897e7	−0.508670
$896$	87004.5	0.00362052
$897$	0	0
$898$	8.16510e7	3.37886
$899$	−2.41485e7	−0.996530
$900$	0	0
$901$	−1.05815e7	−0.434247
$902$	−4.60724e7	−1.88549
$903$	0	0
$904$	5.82526e7	2.37080
$905$	1.62018e6	0.0657569
$906$	0	0
$907$	8.48315e6	0.342404	0.171202	−	0.985236i	$-0.445235\pi$
$907$	8.48315e6	0.342404	0.171202	+	0.985236i	$0.445235\pi$
$908$	2.91566e7	1.17361
$909$	0	0
$910$	−27013.1	−0.00108136
$911$	−2.96302e7	−1.18288	−0.591438	−	0.806351i	$-0.701439\pi$
$911$	−2.96302e7	−1.18288	−0.591438	+	0.806351i	$0.701439\pi$
$912$	0	0
$913$	7.73403e6	0.307064
$914$	−8.66513e7	−3.43091
$915$	0	0
$916$	−3.98239e7	−1.56821
$917$	−14368.4	−0.000564268 0
$918$	0	0
$919$	−1.27080e7	−0.496352	−0.248176	−	0.968715i	$-0.579831\pi$
$919$	−1.27080e7	−0.496352	−0.248176	+	0.968715i	$0.579831\pi$
$920$	1.23292e7	0.480249
$921$	0	0
$922$	−6.06992e7	−2.35156
$923$	−1.32799e7	−0.513085
$924$	0	0
$925$	1.76454e7	0.678075
$926$	1.49875e7	0.574385
$927$	0	0
$928$	−4.83394e7	−1.84260
$929$	4.50533e7	1.71272	0.856362	−	0.516376i	$-0.172719\pi$
$929$	4.50533e7	1.71272	0.856362	+	0.516376i	$0.172719\pi$
$930$	0	0
$931$	−3.42826e7	−1.29628
$932$	−5.66380e6	−0.213584
$933$	0	0
$934$	−4.36266e7	−1.63638
$935$	−1.71062e7	−0.639919
$936$	0	0
$937$	−2.49632e7	−0.928864	−0.464432	−	0.885609i	$-0.653741\pi$
$937$	−2.49632e7	−0.928864	−0.464432	+	0.885609i	$0.653741\pi$
$938$	−72819.4	−0.00270234
$939$	0	0
$940$	9.67585e7	3.57165
$941$	3.36141e7	1.23751	0.618754	−	0.785585i	$-0.287638\pi$
$941$	3.36141e7	1.23751	0.618754	+	0.785585i	$0.287638\pi$
$942$	0	0
$943$	5.36747e6	0.196558
$944$	−3.98192e7	−1.45433
$945$	0	0
$946$	8.86501e7	3.22071
$947$	5.33926e6	0.193467	0.0967333	−	0.995310i	$-0.469161\pi$
$947$	5.33926e6	0.193467	0.0967333	+	0.995310i	$0.469161\pi$
$948$	0	0
$949$	1.44858e7	0.522127
$950$	−3.06076e7	−1.10032
$951$	0	0
$952$	129314.	0.00462438
$953$	3.69391e7	1.31751	0.658755	−	0.752358i	$-0.271084\pi$
$953$	3.69391e7	1.31751	0.658755	+	0.752358i	$0.271084\pi$
$954$	0	0
$955$	1.54994e6	0.0549928
$956$	5.30617e7	1.87774
$957$	0	0
$958$	−7.06026e7	−2.48546
$959$	−22366.2	−0.000785317 0
$960$	0	0
$961$	5.30769e7	1.85395
$962$	−3.47166e7	−1.20948
$963$	0	0
$964$	−4.88910e7	−1.69448
$965$	1.76673e6	0.0610734
$966$	0	0
$967$	−4.47688e7	−1.53960	−0.769802	−	0.638283i	$-0.779645\pi$
$967$	−4.47688e7	−1.53960	−0.769802	+	0.638283i	$0.779645\pi$
$968$	9.34943e6	0.320698
$969$	0	0
$970$	6.29187e7	2.14709
$971$	4.07416e7	1.38672	0.693361	−	0.720590i	$-0.256129\pi$
$971$	4.07416e7	1.38672	0.693361	+	0.720590i	$0.256129\pi$
$972$	0	0
$973$	95555.9	0.00323575
$974$	−2.14247e7	−0.723633
$975$	0	0
$976$	1.33298e8	4.47919
$977$	−1.07576e7	−0.360561	−0.180281	−	0.983615i	$-0.557701\pi$
$977$	−1.07576e7	−0.360561	−0.180281	+	0.983615i	$0.557701\pi$
$978$	0	0
$979$	−1.99679e7	−0.665849
$980$	5.87511e7	1.95412
$981$	0	0
$982$	−8.60770e7	−2.84845
$983$	2.44766e7	0.807917	0.403958	−	0.914777i	$-0.367634\pi$
$983$	2.44766e7	0.807917	0.403958	+	0.914777i	$0.367634\pi$
$984$	0	0
$985$	3.63444e6	0.119357
$986$	−2.80543e7	−0.918982
$987$	0	0
$988$	4.36069e7	1.42123
$989$	−1.03278e7	−0.335751
$990$	0	0
$991$	−2.09086e6	−0.0676301	−0.0338151	−	0.999428i	$-0.510766\pi$
$991$	−2.09086e6	−0.0676301	−0.0338151	+	0.999428i	$0.510766\pi$
$992$	1.63556e8	5.27699
$993$	0	0
$994$	−133087.	−0.00427239
$995$	−4.52682e7	−1.44956
$996$	0	0
$997$	1.49624e7	0.476721	0.238360	−	0.971177i	$-0.423390\pi$
$997$	1.49624e7	0.476721	0.238360	+	0.971177i	$0.423390\pi$
$998$	−1.08036e8	−3.43353
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.a.1.2		2
3.2	odd	2		69.6.a.a.1.1	✓	2
12.11	even	2		1104.6.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.a.1.1	✓	2	3.2	odd	2
207.6.a.a.1.2		2	1.1	even	1	trivial
1104.6.a.h.1.1		2	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q+10.7703 q^{2} +84.0000 q^{4} -41.6148 q^{5} +0.236813 q^{7} +560.057 q^{8} +O(q^{10})\)	\(q+10.7703 q^{2} +84.0000 q^{4} -41.6148 q^{5} +0.236813 q^{7} +560.057 q^{8} -448.205 q^{10} +421.598 q^{11} +254.502 q^{13} +2.55055 q^{14} +3344.00 q^{16} +975.007 q^{17} +2039.79 q^{19} -3495.65 q^{20} +4540.75 q^{22} -529.000 q^{23} -1393.21 q^{25} +2741.07 q^{26} +19.8923 q^{28} -2671.55 q^{29} +9039.14 q^{31} +18094.2 q^{32} +10501.1 q^{34} -9.85493 q^{35} -12665.3 q^{37} +21969.2 q^{38} -23306.7 q^{40} -10146.4 q^{41} +19523.2 q^{43} +35414.2 q^{44} -5697.50 q^{46} -27679.7 q^{47} -16806.9 q^{49} -15005.3 q^{50} +21378.2 q^{52} -10852.8 q^{53} -17544.7 q^{55} +132.629 q^{56} -28773.4 q^{58} -11907.7 q^{59} +39861.9 q^{61} +97354.5 q^{62} +87872.0 q^{64} -10591.1 q^{65} -28550.5 q^{67} +81900.6 q^{68} -106.141 q^{70} -52179.8 q^{71} +56918.0 q^{73} -136410. q^{74} +171342. q^{76} +99.8398 q^{77} +23178.3 q^{79} -139160. q^{80} -109281. q^{82} +18344.6 q^{83} -40574.8 q^{85} +210272. q^{86} +236119. q^{88} -47362.4 q^{89} +60.2694 q^{91} -44436.0 q^{92} -298120. q^{94} -84885.4 q^{95} -140379. q^{97} -181016. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 168 q^{4} - 94 q^{5} - 118 q^{7}+O(q^{10}) \) 2 * q + 168 * q^4 - 94 * q^5 - 118 * q^7	\( 2 q + 168 q^{4} - 94 q^{5} - 118 q^{7} + 116 q^{10} - 320 q^{11} - 288 q^{13} + 1276 q^{14} + 6688 q^{16} + 1810 q^{17} + 730 q^{19} - 7896 q^{20} + 12528 q^{22} - 1058 q^{23} - 1774 q^{25} + 8584 q^{26} - 9912 q^{28} - 8208 q^{29} + 1772 q^{31} + 1508 q^{34} + 6184 q^{35} - 23112 q^{37} + 36076 q^{38} + 6032 q^{40} - 5516 q^{41} + 10322 q^{43} - 26880 q^{44} - 42952 q^{47} - 19634 q^{49} - 10904 q^{50} - 24192 q^{52} + 25350 q^{53} + 21304 q^{55} + 66352 q^{56} + 30856 q^{58} - 18344 q^{59} + 37224 q^{61} + 175624 q^{62} + 175744 q^{64} + 17828 q^{65} - 7482 q^{67} + 152040 q^{68} - 66816 q^{70} - 126848 q^{71} + 137660 q^{73} - 23896 q^{74} + 61320 q^{76} + 87784 q^{77} + 62286 q^{79} - 314336 q^{80} - 159152 q^{82} - 83120 q^{83} - 84316 q^{85} + 309372 q^{86} + 651456 q^{88} - 69770 q^{89} + 64204 q^{91} - 88872 q^{92} - 133632 q^{94} - 16272 q^{95} - 170104 q^{97} - 150568 q^{98}+O(q^{100}) \) 2 * q + 168 * q^4 - 94 * q^5 - 118 * q^7 + 116 * q^10 - 320 * q^11 - 288 * q^13 + 1276 * q^14 + 6688 * q^16 + 1810 * q^17 + 730 * q^19 - 7896 * q^20 + 12528 * q^22 - 1058 * q^23 - 1774 * q^25 + 8584 * q^26 - 9912 * q^28 - 8208 * q^29 + 1772 * q^31 + 1508 * q^34 + 6184 * q^35 - 23112 * q^37 + 36076 * q^38 + 6032 * q^40 - 5516 * q^41 + 10322 * q^43 - 26880 * q^44 - 42952 * q^47 - 19634 * q^49 - 10904 * q^50 - 24192 * q^52 + 25350 * q^53 + 21304 * q^55 + 66352 * q^56 + 30856 * q^58 - 18344 * q^59 + 37224 * q^61 + 175624 * q^62 + 175744 * q^64 + 17828 * q^65 - 7482 * q^67 + 152040 * q^68 - 66816 * q^70 - 126848 * q^71 + 137660 * q^73 - 23896 * q^74 + 61320 * q^76 + 87784 * q^77 + 62286 * q^79 - 314336 * q^80 - 159152 * q^82 - 83120 * q^83 - 84316 * q^85 + 309372 * q^86 + 651456 * q^88 - 69770 * q^89 + 64204 * q^91 - 88872 * q^92 - 133632 * q^94 - 16272 * q^95 - 170104 * q^97 - 150568 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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