LMFDB - Embedded newform 825.6.a.y.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.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:	$132.316651346$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400 $ x^13 - 306x^11 - 206x^10 + 34574x^9 + 39928x^8 - 1788312x^7 - 2591628x^6 + 42852537x^5 + 63733360x^4 - 448113518x^3 - 549984598x^2 + 1518551280*x + 522579400
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2}\cdot 5^{7} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.62206$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.62206 q^{2} +9.00000 q^{3} -18.8807 q^{4} -32.5985 q^{6} -168.040 q^{7} +184.293 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-3.62206 q^{2} +9.00000 q^{3} -18.8807 q^{4} -32.5985 q^{6} -168.040 q^{7} +184.293 q^{8} +81.0000 q^{9} +121.000 q^{11} -169.926 q^{12} +155.005 q^{13} +608.652 q^{14} -63.3377 q^{16} +426.797 q^{17} -293.387 q^{18} -1674.14 q^{19} -1512.36 q^{21} -438.269 q^{22} +11.2327 q^{23} +1658.64 q^{24} -561.437 q^{26} +729.000 q^{27} +3172.72 q^{28} -1107.81 q^{29} +7186.50 q^{31} -5667.96 q^{32} +1089.00 q^{33} -1545.88 q^{34} -1529.34 q^{36} +4576.30 q^{37} +6063.84 q^{38} +1395.05 q^{39} -14041.6 q^{41} +5477.87 q^{42} -20306.0 q^{43} -2284.56 q^{44} -40.6856 q^{46} -10551.8 q^{47} -570.039 q^{48} +11430.6 q^{49} +3841.17 q^{51} -2926.60 q^{52} -27069.4 q^{53} -2640.48 q^{54} -30968.6 q^{56} -15067.3 q^{57} +4012.57 q^{58} +22773.1 q^{59} -916.341 q^{61} -26029.9 q^{62} -13611.3 q^{63} +22556.5 q^{64} -3944.42 q^{66} +25328.4 q^{67} -8058.22 q^{68} +101.095 q^{69} -16788.6 q^{71} +14927.7 q^{72} +55976.9 q^{73} -16575.6 q^{74} +31608.9 q^{76} -20332.9 q^{77} -5052.94 q^{78} -47517.8 q^{79} +6561.00 q^{81} +50859.4 q^{82} +24415.3 q^{83} +28554.5 q^{84} +73549.4 q^{86} -9970.33 q^{87} +22299.4 q^{88} -22081.6 q^{89} -26047.1 q^{91} -212.082 q^{92} +64678.5 q^{93} +38219.1 q^{94} -51011.6 q^{96} +15706.3 q^{97} -41402.1 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10}) $ 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9	\( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9} + 1573 q^{11} + 1881 q^{12} + 986 q^{13} - 610 q^{14} + 3501 q^{16} + 1476 q^{17} + 1053 q^{18} + 270 q^{19} + 2736 q^{21} + 1573 q^{22} + 9084 q^{23} + 3591 q^{24} + 2652 q^{26} + 9477 q^{27} + 10920 q^{28} + 11952 q^{29} + 19096 q^{31} + 11661 q^{32} + 14157 q^{33} - 1302 q^{34} + 16929 q^{36} + 39964 q^{37} + 1574 q^{38} + 8874 q^{39} + 35184 q^{41} - 5490 q^{42} - 96 q^{43} + 25289 q^{44} - 4120 q^{46} + 34984 q^{47} + 31509 q^{48} + 14557 q^{49} + 13284 q^{51} + 39002 q^{52} + 22984 q^{53} + 9477 q^{54} + 59802 q^{56} + 2430 q^{57} + 18896 q^{58} - 9192 q^{59} + 5438 q^{61} + 272 q^{62} + 24624 q^{63} + 106557 q^{64} + 14157 q^{66} + 71508 q^{67} + 127948 q^{68} + 81756 q^{69} + 101700 q^{71} + 32319 q^{72} + 77390 q^{73} + 13676 q^{74} + 139966 q^{76} + 36784 q^{77} + 23868 q^{78} + 93954 q^{79} + 85293 q^{81} + 53284 q^{82} + 185918 q^{83} + 98280 q^{84} + 370930 q^{86} + 107568 q^{87} + 48279 q^{88} - 18418 q^{89} + 174536 q^{91} + 274264 q^{92} + 171864 q^{93} + 64520 q^{94} + 104949 q^{96} + 94312 q^{97} + 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9 + 1573 * q^11 + 1881 * q^12 + 986 * q^13 - 610 * q^14 + 3501 * q^16 + 1476 * q^17 + 1053 * q^18 + 270 * q^19 + 2736 * q^21 + 1573 * q^22 + 9084 * q^23 + 3591 * q^24 + 2652 * q^26 + 9477 * q^27 + 10920 * q^28 + 11952 * q^29 + 19096 * q^31 + 11661 * q^32 + 14157 * q^33 - 1302 * q^34 + 16929 * q^36 + 39964 * q^37 + 1574 * q^38 + 8874 * q^39 + 35184 * q^41 - 5490 * q^42 - 96 * q^43 + 25289 * q^44 - 4120 * q^46 + 34984 * q^47 + 31509 * q^48 + 14557 * q^49 + 13284 * q^51 + 39002 * q^52 + 22984 * q^53 + 9477 * q^54 + 59802 * q^56 + 2430 * q^57 + 18896 * q^58 - 9192 * q^59 + 5438 * q^61 + 272 * q^62 + 24624 * q^63 + 106557 * q^64 + 14157 * q^66 + 71508 * q^67 + 127948 * q^68 + 81756 * q^69 + 101700 * q^71 + 32319 * q^72 + 77390 * q^73 + 13676 * q^74 + 139966 * q^76 + 36784 * q^77 + 23868 * q^78 + 93954 * q^79 + 85293 * q^81 + 53284 * q^82 + 185918 * q^83 + 98280 * q^84 + 370930 * q^86 + 107568 * q^87 + 48279 * q^88 - 18418 * q^89 + 174536 * q^91 + 274264 * q^92 + 171864 * q^93 + 64520 * q^94 + 104949 * q^96 + 94312 * q^97 + 145677 * q^98 + 127413 * q^99

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 $

$2$	−3.62206	−0.640296	−0.320148	−	0.947368i	$-0.603733\pi$
$2$	−3.62206	−0.640296	−0.320148	+	0.947368i	$0.603733\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−18.8807	−0.590021
$5$	0	0
$5$	0	0
$6$	−32.5985	−0.369675
$7$	−168.040	−1.29619	−0.648095	−	0.761560i	$-0.724434\pi$
$7$	−168.040	−1.29619	−0.648095	+	0.761560i	$0.724434\pi$
$8$	184.293	1.01808
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−169.926	−0.340649
$13$	155.005	0.254383	0.127191	−	0.991878i	$-0.459404\pi$
$13$	155.005	0.254383	0.127191	+	0.991878i	$0.459404\pi$
$14$	608.652	0.829944
$15$	0	0
$16$	−63.3377	−0.0618532
$17$	426.797	0.358178	0.179089	−	0.983833i	$-0.442685\pi$
$17$	426.797	0.358178	0.179089	+	0.983833i	$0.442685\pi$
$18$	−293.387	−0.213432
$19$	−1674.14	−1.06392	−0.531959	−	0.846770i	$-0.678544\pi$
$19$	−1674.14	−1.06392	−0.531959	+	0.846770i	$0.678544\pi$
$20$	0	0
$21$	−1512.36	−0.748355
$22$	−438.269	−0.193056
$23$	11.2327	0.00442757	0.00221379	−	0.999998i	$-0.499295\pi$
$23$	11.2327	0.00442757	0.00221379	+	0.999998i	$0.499295\pi$
$24$	1658.64	0.587791
$25$	0	0
$26$	−561.437	−0.162880
$27$	729.000	0.192450
$28$	3172.72	0.764780
$29$	−1107.81	−0.244609	−0.122304	−	0.992493i	$-0.539028\pi$
$29$	−1107.81	−0.244609	−0.122304	+	0.992493i	$0.539028\pi$
$30$	0	0
$31$	7186.50	1.34312	0.671558	−	0.740952i	$-0.265625\pi$
$31$	7186.50	1.34312	0.671558	+	0.740952i	$0.265625\pi$
$32$	−5667.96	−0.978480
$33$	1089.00	0.174078
$34$	−1545.88	−0.229340
$35$	0	0
$36$	−1529.34	−0.196674
$37$	4576.30	0.549554	0.274777	−	0.961508i	$-0.411396\pi$
$37$	4576.30	0.549554	0.274777	+	0.961508i	$0.411396\pi$
$38$	6063.84	0.681222
$39$	1395.05	0.146868
$40$	0	0
$41$	−14041.6	−1.30454	−0.652268	−	0.757988i	$-0.726182\pi$
$41$	−14041.6	−1.30454	−0.652268	+	0.757988i	$0.726182\pi$
$42$	5477.87	0.479169
$43$	−20306.0	−1.67476	−0.837380	−	0.546621i	$-0.815914\pi$
$43$	−20306.0	−1.67476	−0.837380	+	0.546621i	$0.815914\pi$
$44$	−2284.56	−0.177898
$45$	0	0
$46$	−40.6856	−0.00283495
$47$	−10551.8	−0.696756	−0.348378	−	0.937354i	$-0.613267\pi$
$47$	−10551.8	−0.696756	−0.348378	+	0.937354i	$0.613267\pi$
$48$	−570.039	−0.0357110
$49$	11430.6	0.680107
$50$	0	0
$51$	3841.17	0.206794
$52$	−2926.60	−0.150091
$53$	−27069.4	−1.32370	−0.661849	−	0.749637i	$-0.730228\pi$
$53$	−27069.4	−1.32370	−0.661849	+	0.749637i	$0.730228\pi$
$54$	−2640.48	−0.123225
$55$	0	0
$56$	−30968.6	−1.31963
$57$	−15067.3	−0.614253
$58$	4012.57	0.156622
$59$	22773.1	0.851709	0.425854	−	0.904792i	$-0.359974\pi$
$59$	22773.1	0.851709	0.425854	+	0.904792i	$0.359974\pi$
$60$	0	0
$61$	−916.341	−0.0315306	−0.0157653	−	0.999876i	$-0.505018\pi$
$61$	−916.341	−0.0315306	−0.0157653	+	0.999876i	$0.505018\pi$
$62$	−26029.9	−0.859991
$63$	−13611.3	−0.432063
$64$	22556.5	0.688369
$65$	0	0
$66$	−3944.42	−0.111461
$67$	25328.4	0.689321	0.344660	−	0.938727i	$-0.387994\pi$
$67$	25328.4	0.689321	0.344660	+	0.938727i	$0.387994\pi$
$68$	−8058.22	−0.211333
$69$	101.095	0.00255626
$70$	0	0
$71$	−16788.6	−0.395248	−0.197624	−	0.980278i	$-0.563323\pi$
$71$	−16788.6	−0.395248	−0.197624	+	0.980278i	$0.563323\pi$
$72$	14927.7	0.339361
$73$	55976.9	1.22942	0.614712	−	0.788752i	$-0.289272\pi$
$73$	55976.9	1.22942	0.614712	+	0.788752i	$0.289272\pi$
$74$	−16575.6	−0.351877
$75$	0	0
$76$	31608.9	0.627735
$77$	−20332.9	−0.390816
$78$	−5052.94	−0.0940389
$79$	−47517.8	−0.856621	−0.428310	−	0.903632i	$-0.640891\pi$
$79$	−47517.8	−0.856621	−0.428310	+	0.903632i	$0.640891\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	50859.4	0.835289
$83$	24415.3	0.389016	0.194508	−	0.980901i	$-0.437689\pi$
$83$	24415.3	0.389016	0.194508	+	0.980901i	$0.437689\pi$
$84$	28554.5	0.441546
$85$	0	0
$86$	73549.4	1.07234
$87$	−9970.33	−0.141225
$88$	22299.4	0.306964
$89$	−22081.6	−0.295499	−0.147749	−	0.989025i	$-0.547203\pi$
$89$	−22081.6	−0.295499	−0.147749	+	0.989025i	$0.547203\pi$
$90$	0	0
$91$	−26047.1	−0.329728
$92$	−212.082	−0.00261236
$93$	64678.5	0.775448
$94$	38219.1	0.446130
$95$	0	0
$96$	−51011.6	−0.564925
$97$	15706.3	0.169490	0.0847449	−	0.996403i	$-0.472992\pi$
$97$	15706.3	0.169490	0.0847449	+	0.996403i	$0.472992\pi$
$98$	−41402.1	−0.435469
$99$	9801.00	0.100504
$100$	0	0
$101$	10765.2	0.105007	0.0525033	−	0.998621i	$-0.483280\pi$
$101$	10765.2	0.105007	0.0525033	+	0.998621i	$0.483280\pi$
$102$	−13913.0	−0.132409
$103$	28684.5	0.266412	0.133206	−	0.991088i	$-0.457473\pi$
$103$	28684.5	0.266412	0.133206	+	0.991088i	$0.457473\pi$
$104$	28566.3	0.258983
$105$	0	0
$106$	98046.9	0.847558
$107$	−121918.	−1.02946	−0.514728	−	0.857354i	$-0.672107\pi$
$107$	−121918.	−1.02946	−0.514728	+	0.857354i	$0.672107\pi$
$108$	−13764.0	−0.113550
$109$	−169782.	−1.36876	−0.684378	−	0.729128i	$-0.739926\pi$
$109$	−169782.	−1.36876	−0.684378	+	0.729128i	$0.739926\pi$
$110$	0	0
$111$	41186.7	0.317285
$112$	10643.3	0.0801735
$113$	220268.	1.62276	0.811380	−	0.584519i	$-0.198717\pi$
$113$	220268.	1.62276	0.811380	+	0.584519i	$0.198717\pi$
$114$	54574.6	0.393304
$115$	0	0
$116$	20916.3	0.144324
$117$	12555.4	0.0847942
$118$	−82485.3	−0.545346
$119$	−71719.1	−0.464267
$120$	0	0
$121$	14641.0	0.0909091
$122$	3319.04	0.0201889
$123$	−126374.	−0.753174
$124$	−135686.	−0.792467
$125$	0	0
$126$	49300.8	0.276648
$127$	230988.	1.27081	0.635404	−	0.772180i	$-0.280834\pi$
$127$	230988.	1.27081	0.635404	+	0.772180i	$0.280834\pi$
$128$	99673.7	0.537720
$129$	−182754.	−0.966923
$130$	0	0
$131$	175894.	0.895513	0.447757	−	0.894155i	$-0.352223\pi$
$131$	175894.	0.895513	0.447757	+	0.894155i	$0.352223\pi$
$132$	−20561.1	−0.102710
$133$	281323.	1.37904
$134$	−91741.1	−0.441369
$135$	0	0
$136$	78655.7	0.364655
$137$	8437.54	0.0384073	0.0192037	−	0.999816i	$-0.493887\pi$
$137$	8437.54	0.0384073	0.0192037	+	0.999816i	$0.493887\pi$
$138$	−366.170	−0.00163676
$139$	−75054.6	−0.329489	−0.164744	−	0.986336i	$-0.552680\pi$
$139$	−75054.6	−0.329489	−0.164744	+	0.986336i	$0.552680\pi$
$140$	0	0
$141$	−94965.9	−0.402272
$142$	60809.5	0.253076
$143$	18755.6	0.0766992
$144$	−5130.35	−0.0206177
$145$	0	0
$146$	−202752.	−0.787194
$147$	102875.	0.392660
$148$	−86403.7	−0.324248
$149$	−119523.	−0.441049	−0.220524	−	0.975381i	$-0.570777\pi$
$149$	−119523.	−0.441049	−0.220524	+	0.975381i	$0.570777\pi$
$150$	0	0
$151$	372713.	1.33025	0.665124	−	0.746733i	$-0.268379\pi$
$151$	372713.	1.33025	0.665124	+	0.746733i	$0.268379\pi$
$152$	−308532.	−1.08316
$153$	34570.6	0.119393
$154$	73646.9	0.250238
$155$	0	0
$156$	−26339.4	−0.0866552
$157$	407799.	1.32038	0.660188	−	0.751101i	$-0.270477\pi$
$157$	407799.	1.32038	0.660188	+	0.751101i	$0.270477\pi$
$158$	172112.	0.548491
$159$	−243624.	−0.764237
$160$	0	0
$161$	−1887.55	−0.00573897
$162$	−23764.3	−0.0711440
$163$	360328.	1.06225	0.531127	−	0.847292i	$-0.321769\pi$
$163$	360328.	1.06225	0.531127	+	0.847292i	$0.321769\pi$
$164$	265115.	0.769704
$165$	0	0
$166$	−88433.8	−0.249085
$167$	69666.3	0.193300	0.0966500	−	0.995318i	$-0.469187\pi$
$167$	69666.3	0.193300	0.0966500	+	0.995318i	$0.469187\pi$
$168$	−278718.	−0.761888
$169$	−347266.	−0.935289
$170$	0	0
$171$	−135605.	−0.354639
$172$	383391.	0.988144
$173$	627131.	1.59310	0.796550	−	0.604572i	$-0.206656\pi$
$173$	627131.	1.59310	0.796550	+	0.604572i	$0.206656\pi$
$174$	36113.1	0.0904257
$175$	0	0
$176$	−7663.86	−0.0186494
$177$	204957.	0.491734
$178$	79980.8	0.189206
$179$	−662582.	−1.54564	−0.772818	−	0.634628i	$-0.781154\pi$
$179$	−662582.	−1.54564	−0.772818	+	0.634628i	$0.781154\pi$
$180$	0	0
$181$	−525680.	−1.19268	−0.596341	−	0.802731i	$-0.703379\pi$
$181$	−525680.	−1.19268	−0.596341	+	0.802731i	$0.703379\pi$
$182$	94344.1	0.211123
$183$	−8247.07	−0.0182042
$184$	2070.11	0.00450764
$185$	0	0
$186$	−234269.	−0.496516
$187$	51642.5	0.107995
$188$	199225.	0.411101
$189$	−122501.	−0.249452
$190$	0	0
$191$	−824338.	−1.63502	−0.817508	−	0.575918i	$-0.804645\pi$
$191$	−824338.	−1.63502	−0.817508	+	0.575918i	$0.804645\pi$
$192$	203008.	0.397430
$193$	−783372.	−1.51382	−0.756911	−	0.653518i	$-0.773293\pi$
$193$	−783372.	−1.51382	−0.756911	+	0.653518i	$0.773293\pi$
$194$	−56889.0	−0.108524
$195$	0	0
$196$	−215817.	−0.401278
$197$	−272361.	−0.500010	−0.250005	−	0.968245i	$-0.580432\pi$
$197$	−272361.	−0.500010	−0.250005	+	0.968245i	$0.580432\pi$
$198$	−35499.8	−0.0643521
$199$	982522.	1.75877	0.879386	−	0.476109i	$-0.157953\pi$
$199$	982522.	1.75877	0.879386	+	0.476109i	$0.157953\pi$
$200$	0	0
$201$	227956.	0.397979
$202$	−38992.0	−0.0672353
$203$	186157.	0.317059
$204$	−72524.0	−0.122013
$205$	0	0
$206$	−103897.	−0.170583
$207$	909.851	0.00147586
$208$	−9817.66	−0.0157344
$209$	−202571.	−0.320783
$210$	0	0
$211$	−430400.	−0.665527	−0.332764	−	0.943010i	$-0.607981\pi$
$211$	−430400.	−0.665527	−0.332764	+	0.943010i	$0.607981\pi$
$212$	511089.	0.781010
$213$	−151098.	−0.228197
$214$	441593.	0.659156
$215$	0	0
$216$	134349.	0.195930
$217$	−1.20762e6	−1.74093
$218$	614961.	0.876408
$219$	503792.	0.709808
$220$	0	0
$221$	66155.7	0.0911143
$222$	−149181.	−0.203156
$223$	532033.	0.716434	0.358217	−	0.933638i	$-0.383385\pi$
$223$	532033.	0.716434	0.358217	+	0.933638i	$0.383385\pi$
$224$	952446.	1.26829
$225$	0	0
$226$	−797822.	−1.03905
$227$	792684.	1.02102	0.510512	−	0.859871i	$-0.329456\pi$
$227$	792684.	1.02102	0.510512	+	0.859871i	$0.329456\pi$
$228$	284481.	0.362423
$229$	286558.	0.361097	0.180549	−	0.983566i	$-0.442213\pi$
$229$	286558.	0.361097	0.180549	+	0.983566i	$0.442213\pi$
$230$	0	0
$231$	−182996.	−0.225638
$232$	−204162.	−0.249032
$233$	1.41434e6	1.70673	0.853366	−	0.521312i	$-0.174557\pi$
$233$	1.41434e6	1.70673	0.853366	+	0.521312i	$0.174557\pi$
$234$	−45476.4	−0.0542934
$235$	0	0
$236$	−429971.	−0.502527
$237$	−427660.	−0.494570
$238$	259771.	0.297268
$239$	1.00694e6	1.14028	0.570138	−	0.821549i	$-0.306890\pi$
$239$	1.00694e6	1.14028	0.570138	+	0.821549i	$0.306890\pi$
$240$	0	0
$241$	1.08716e6	1.20573	0.602866	−	0.797843i	$-0.294025\pi$
$241$	1.08716e6	1.20573	0.602866	+	0.797843i	$0.294025\pi$
$242$	−53030.6	−0.0582087
$243$	59049.0	0.0641500
$244$	17301.1	0.0186037
$245$	0	0
$246$	457735.	0.482254
$247$	−259500.	−0.270642
$248$	1.32442e6	1.36740
$249$	219738.	0.224598
$250$	0	0
$251$	312788.	0.313376	0.156688	−	0.987648i	$-0.449918\pi$
$251$	312788.	0.313376	0.156688	+	0.987648i	$0.449918\pi$
$252$	256990.	0.254927
$253$	1359.16	0.00133496
$254$	−836652.	−0.813693
$255$	0	0
$256$	−1.08283e6	−1.03267
$257$	1.50645e6	1.42273	0.711366	−	0.702822i	$-0.248077\pi$
$257$	1.50645e6	1.42273	0.711366	+	0.702822i	$0.248077\pi$
$258$	661945.	0.619117
$259$	−769003.	−0.712326
$260$	0	0
$261$	−89732.9	−0.0815362
$262$	−637097.	−0.573393
$263$	189662.	0.169079	0.0845397	−	0.996420i	$-0.473058\pi$
$263$	189662.	0.169079	0.0845397	+	0.996420i	$0.473058\pi$
$264$	200695.	0.177226
$265$	0	0
$266$	−1.01897e6	−0.882993
$267$	−198734.	−0.170606
$268$	−478218.	−0.406714
$269$	1.60463e6	1.35206	0.676029	−	0.736875i	$-0.263699\pi$
$269$	1.60463e6	1.35206	0.676029	+	0.736875i	$0.263699\pi$
$270$	0	0
$271$	1.71899e6	1.42184	0.710921	−	0.703272i	$-0.248278\pi$
$271$	1.71899e6	1.42184	0.710921	+	0.703272i	$0.248278\pi$
$272$	−27032.3	−0.0221545
$273$	−234424.	−0.190369
$274$	−30561.3	−0.0245921
$275$	0	0
$276$	−1908.73	−0.00150825
$277$	756133.	0.592105	0.296053	−	0.955172i	$-0.404330\pi$
$277$	756133.	0.592105	0.296053	+	0.955172i	$0.404330\pi$
$278$	271852.	0.210970
$279$	582107.	0.447705
$280$	0	0
$281$	772332.	0.583497	0.291748	−	0.956495i	$-0.405763\pi$
$281$	772332.	0.583497	0.291748	+	0.956495i	$0.405763\pi$
$282$	343972.	0.257573
$283$	1.93103e6	1.43325	0.716625	−	0.697458i	$-0.245686\pi$
$283$	1.93103e6	1.43325	0.716625	+	0.697458i	$0.245686\pi$
$284$	316981.	0.233205
$285$	0	0
$286$	−67933.9	−0.0491102
$287$	2.35955e6	1.69093
$288$	−459105.	−0.326160
$289$	−1.23770e6	−0.871708
$290$	0	0
$291$	141356.	0.0978550
$292$	−1.05688e6	−0.725386
$293$	−1.70779e6	−1.16216	−0.581080	−	0.813847i	$-0.697370\pi$
$293$	−1.70779e6	−1.16216	−0.581080	+	0.813847i	$0.697370\pi$
$294$	−372619.	−0.251418
$295$	0	0
$296$	843379.	0.559492
$297$	88209.0	0.0580259
$298$	432920.	0.282402
$299$	1741.13	0.00112630
$300$	0	0
$301$	3.41222e6	2.17081
$302$	−1.34999e6	−0.851752
$303$	96886.4	0.0606256
$304$	106036.	0.0658068
$305$	0	0
$306$	−125217.	−0.0764467
$307$	−256601.	−0.155386	−0.0776931	−	0.996977i	$-0.524755\pi$
$307$	−256601.	−0.155386	−0.0776931	+	0.996977i	$0.524755\pi$
$308$	383899.	0.230590
$309$	258160.	0.153813
$310$	0	0
$311$	2.08034e6	1.21964	0.609822	−	0.792539i	$-0.291241\pi$
$311$	2.08034e6	1.21964	0.609822	+	0.792539i	$0.291241\pi$
$312$	257097.	0.149524
$313$	1.05439e6	0.608331	0.304166	−	0.952619i	$-0.401622\pi$
$313$	1.05439e6	0.608331	0.304166	+	0.952619i	$0.401622\pi$
$314$	−1.47707e6	−0.845431
$315$	0	0
$316$	897169.	0.505425
$317$	2.35341e6	1.31538	0.657688	−	0.753291i	$-0.271535\pi$
$317$	2.35341e6	1.31538	0.657688	+	0.753291i	$0.271535\pi$
$318$	882422.	0.489338
$319$	−134045.	−0.0737523
$320$	0	0
$321$	−1.09726e6	−0.594356
$322$	6836.82	0.00367464
$323$	−714519.	−0.381072
$324$	−123876.	−0.0655579
$325$	0	0
$326$	−1.30513e6	−0.680157
$327$	−1.52804e6	−0.790251
$328$	−2.58776e6	−1.32813
$329$	1.77312e6	0.903127
$330$	0	0
$331$	−376896.	−0.189083	−0.0945414	−	0.995521i	$-0.530138\pi$
$331$	−376896.	−0.189083	−0.0945414	+	0.995521i	$0.530138\pi$
$332$	−460978.	−0.229528
$333$	370680.	0.183185
$334$	−252336.	−0.123769
$335$	0	0
$336$	95789.6	0.0462882
$337$	−3.13928e6	−1.50576	−0.752879	−	0.658159i	$-0.771335\pi$
$337$	−3.13928e6	−1.50576	−0.752879	+	0.658159i	$0.771335\pi$
$338$	1.25782e6	0.598862
$339$	1.98241e6	0.936901
$340$	0	0
$341$	869567.	0.404965
$342$	491171.	0.227074
$343$	903460.	0.414642
$344$	−3.74225e6	−1.70505
$345$	0	0
$346$	−2.27151e6	−1.02006
$347$	1.74141e6	0.776384	0.388192	−	0.921579i	$-0.373100\pi$
$347$	1.74141e6	0.776384	0.388192	+	0.921579i	$0.373100\pi$
$348$	188247.	0.0833257
$349$	−4.40916e6	−1.93772	−0.968862	−	0.247602i	$-0.920357\pi$
$349$	−4.40916e6	−1.93772	−0.968862	+	0.247602i	$0.920357\pi$
$350$	0	0
$351$	112999.	0.0489560
$352$	−685823.	−0.295023
$353$	1.97624e6	0.844116	0.422058	−	0.906569i	$-0.361308\pi$
$353$	1.97624e6	0.844116	0.422058	+	0.906569i	$0.361308\pi$
$354$	−742368.	−0.314855
$355$	0	0
$356$	416916.	0.174350
$357$	−645472.	−0.268045
$358$	2.39991e6	0.989664
$359$	340087.	0.139269	0.0696343	−	0.997573i	$-0.477817\pi$
$359$	340087.	0.139269	0.0696343	+	0.997573i	$0.477817\pi$
$360$	0	0
$361$	326652.	0.131922
$362$	1.90404e6	0.763669
$363$	131769.	0.0524864
$364$	491787.	0.194547
$365$	0	0
$366$	29871.4	0.0116561
$367$	3.60337e6	1.39651	0.698254	−	0.715850i	$-0.253961\pi$
$367$	3.60337e6	1.39651	0.698254	+	0.715850i	$0.253961\pi$
$368$	−711.455	−0.000273859 0
$369$	−1.13737e6	−0.434845
$370$	0	0
$371$	4.54875e6	1.71576
$372$	−1.22118e6	−0.457531
$373$	1.94994e6	0.725687	0.362844	−	0.931850i	$-0.381806\pi$
$373$	1.94994e6	0.725687	0.362844	+	0.931850i	$0.381806\pi$
$374$	−187052.	−0.0691486
$375$	0	0
$376$	−1.94462e6	−0.709356
$377$	−171717.	−0.0622242
$378$	443707.	0.159723
$379$	2.15699e6	0.771349	0.385674	−	0.922635i	$-0.373969\pi$
$379$	2.15699e6	0.771349	0.385674	+	0.922635i	$0.373969\pi$
$380$	0	0
$381$	2.07889e6	0.733701
$382$	2.98580e6	1.04689
$383$	−1.66805e6	−0.581049	−0.290525	−	0.956868i	$-0.593830\pi$
$383$	−1.66805e6	−0.581049	−0.290525	+	0.956868i	$0.593830\pi$
$384$	897064.	0.310453
$385$	0	0
$386$	2.83742e6	0.969294
$387$	−1.64478e6	−0.558253
$388$	−296545.	−0.100003
$389$	1.58917e6	0.532473	0.266236	−	0.963908i	$-0.414220\pi$
$389$	1.58917e6	0.532473	0.266236	+	0.963908i	$0.414220\pi$
$390$	0	0
$391$	4794.09	0.00158586
$392$	2.10657e6	0.692406
$393$	1.58304e6	0.517025
$394$	986506.	0.320154
$395$	0	0
$396$	−185050.	−0.0592994
$397$	−4.07780e6	−1.29852	−0.649261	−	0.760566i	$-0.724922\pi$
$397$	−4.07780e6	−1.29852	−0.649261	+	0.760566i	$0.724922\pi$
$398$	−3.55875e6	−1.12613
$399$	2.53191e6	0.796189
$400$	0	0
$401$	−4.89367e6	−1.51976	−0.759878	−	0.650066i	$-0.774741\pi$
$401$	−4.89367e6	−1.51976	−0.759878	+	0.650066i	$0.774741\pi$
$402$	−825670.	−0.254824
$403$	1.11394e6	0.341665
$404$	−203254.	−0.0619562
$405$	0	0
$406$	−674273.	−0.203012
$407$	553732.	0.165697
$408$	707901.	0.210534
$409$	508943.	0.150439	0.0752196	−	0.997167i	$-0.476034\pi$
$409$	508943.	0.150439	0.0752196	+	0.997167i	$0.476034\pi$
$410$	0	0
$411$	75937.8	0.0221745
$412$	−541583.	−0.157189
$413$	−3.82679e6	−1.10398
$414$	−3295.53	−0.000944985 0
$415$	0	0
$416$	−878562.	−0.248908
$417$	−675492.	−0.190230
$418$	733725.	0.205396
$419$	6.78930e6	1.88925	0.944626	−	0.328148i	$-0.106425\pi$
$419$	6.78930e6	1.88925	0.944626	+	0.328148i	$0.106425\pi$
$420$	0	0
$421$	−3.93204e6	−1.08122	−0.540608	−	0.841275i	$-0.681806\pi$
$421$	−3.93204e6	−1.08122	−0.540608	+	0.841275i	$0.681806\pi$
$422$	1.55893e6	0.426134
$423$	−854693.	−0.232252
$424$	−4.98869e6	−1.34764
$425$	0	0
$426$	547285.	0.146113
$427$	153982.	0.0408697
$428$	2.30189e6	0.607401
$429$	168800.	0.0442823
$430$	0	0
$431$	−3.24173e6	−0.840589	−0.420295	−	0.907388i	$-0.638073\pi$
$431$	−3.24173e6	−0.840589	−0.420295	+	0.907388i	$0.638073\pi$
$432$	−46173.2	−0.0119037
$433$	5.96730e6	1.52953	0.764765	−	0.644309i	$-0.222855\pi$
$433$	5.96730e6	1.52953	0.764765	+	0.644309i	$0.222855\pi$
$434$	4.37408e6	1.11471
$435$	0	0
$436$	3.20560e6	0.807595
$437$	−18805.2	−0.00471057
$438$	−1.82476e6	−0.454487
$439$	2.85969e6	0.708203	0.354101	−	0.935207i	$-0.384787\pi$
$439$	2.85969e6	0.708203	0.354101	+	0.935207i	$0.384787\pi$
$440$	0	0
$441$	925875.	0.226702
$442$	−239620.	−0.0583401
$443$	−2.52822e6	−0.612077	−0.306038	−	0.952019i	$-0.599004\pi$
$443$	−2.52822e6	−0.612077	−0.306038	+	0.952019i	$0.599004\pi$
$444$	−777633.	−0.187205
$445$	0	0
$446$	−1.92705e6	−0.458730
$447$	−1.07571e6	−0.254640
$448$	−3.79040e6	−0.892257
$449$	5.72753e6	1.34076	0.670380	−	0.742018i	$-0.266131\pi$
$449$	5.72753e6	1.34076	0.670380	+	0.742018i	$0.266131\pi$
$450$	0	0
$451$	−1.69903e6	−0.393333
$452$	−4.15880e6	−0.957464
$453$	3.35442e6	0.768019
$454$	−2.87115e6	−0.653757
$455$	0	0
$456$	−2.77679e6	−0.625362
$457$	4.94045e6	1.10656	0.553281	−	0.832994i	$-0.313375\pi$
$457$	4.94045e6	1.10656	0.553281	+	0.832994i	$0.313375\pi$
$458$	−1.03793e6	−0.231209
$459$	311135.	0.0689314
$460$	0	0
$461$	6.18580e6	1.35564	0.677818	−	0.735229i	$-0.262926\pi$
$461$	6.18580e6	1.35564	0.677818	+	0.735229i	$0.262926\pi$
$462$	662822.	0.144475
$463$	1.23230e6	0.267155	0.133577	−	0.991038i	$-0.457354\pi$
$463$	1.23230e6	0.267155	0.133577	+	0.991038i	$0.457354\pi$
$464$	70166.4	0.0151298
$465$	0	0
$466$	−5.12284e6	−1.09281
$467$	7.74380e6	1.64309	0.821546	−	0.570142i	$-0.193112\pi$
$467$	7.74380e6	1.64309	0.821546	+	0.570142i	$0.193112\pi$
$468$	−237055.	−0.0500304
$469$	−4.25620e6	−0.893490
$470$	0	0
$471$	3.67019e6	0.762319
$472$	4.19691e6	0.867111
$473$	−2.45702e6	−0.504959
$474$	1.54901e6	0.316671
$475$	0	0
$476$	1.35411e6	0.273927
$477$	−2.19262e6	−0.441233
$478$	−3.64721e6	−0.730114
$479$	−3.36135e6	−0.669384	−0.334692	−	0.942328i	$-0.608632\pi$
$479$	−3.36135e6	−0.669384	−0.334692	+	0.942328i	$0.608632\pi$
$480$	0	0
$481$	709349.	0.139797
$482$	−3.93776e6	−0.772025
$483$	−16988.0	−0.00331340
$484$	−276432.	−0.0536383
$485$	0	0
$486$	−213879.	−0.0410750
$487$	−397760.	−0.0759973	−0.0379987	−	0.999278i	$-0.512098\pi$
$487$	−397760.	−0.0759973	−0.0379987	+	0.999278i	$0.512098\pi$
$488$	−168875.	−0.0321008
$489$	3.24295e6	0.613293
$490$	0	0
$491$	−585823.	−0.109664	−0.0548318	−	0.998496i	$-0.517462\pi$
$491$	−585823.	−0.109664	−0.0548318	+	0.998496i	$0.517462\pi$
$492$	2.38603e6	0.444389
$493$	−472812.	−0.0876135
$494$	939926.	0.173291
$495$	0	0
$496$	−455177.	−0.0830760
$497$	2.82117e6	0.512316
$498$	−795904.	−0.143809
$499$	7.82230e6	1.40632	0.703158	−	0.711034i	$-0.251773\pi$
$499$	7.82230e6	1.40632	0.703158	+	0.711034i	$0.251773\pi$
$500$	0	0
$501$	626997.	0.111602
$502$	−1.13294e6	−0.200653
$503$	−4.04320e6	−0.712534	−0.356267	−	0.934384i	$-0.615951\pi$
$503$	−4.04320e6	−0.712534	−0.356267	+	0.934384i	$0.615951\pi$
$504$	−2.50846e6	−0.439876
$505$	0	0
$506$	−4922.96	−0.000854771 0
$507$	−3.12540e6	−0.539990
$508$	−4.36121e6	−0.749804
$509$	−1.07585e7	−1.84060	−0.920299	−	0.391216i	$-0.872054\pi$
$509$	−1.07585e7	−1.84060	−0.920299	+	0.391216i	$0.872054\pi$
$510$	0	0
$511$	−9.40637e6	−1.59357
$512$	732521.	0.123494
$513$	−1.22045e6	−0.204751
$514$	−5.45646e6	−0.910969
$515$	0	0
$516$	3.45052e6	0.570505
$517$	−1.27676e6	−0.210080
$518$	2.78537e6	0.456099
$519$	5.64418e6	0.919777
$520$	0	0
$521$	−3.48830e6	−0.563014	−0.281507	−	0.959559i	$-0.590834\pi$
$521$	−3.48830e6	−0.563014	−0.281507	+	0.959559i	$0.590834\pi$
$522$	325018.	0.0522073
$523$	−5.47692e6	−0.875552	−0.437776	−	0.899084i	$-0.644234\pi$
$523$	−5.47692e6	−0.875552	−0.437776	+	0.899084i	$0.644234\pi$
$524$	−3.32099e6	−0.528372
$525$	0	0
$526$	−686967.	−0.108261
$527$	3.06718e6	0.481075
$528$	−68974.7	−0.0107673
$529$	−6.43622e6	−0.999980
$530$	0	0
$531$	1.84462e6	0.283903
$532$	−5.31158e6	−0.813663
$533$	−2.17651e6	−0.331851
$534$	719828.	0.109238
$535$	0	0
$536$	4.66785e6	0.701786
$537$	−5.96324e6	−0.892374
$538$	−5.81208e6	−0.865717
$539$	1.38310e6	0.205060
$540$	0	0
$541$	−7.78523e6	−1.14361	−0.571806	−	0.820389i	$-0.693757\pi$
$541$	−7.78523e6	−1.14361	−0.571806	+	0.820389i	$0.693757\pi$
$542$	−6.22630e6	−0.910399
$543$	−4.73112e6	−0.688596
$544$	−2.41907e6	−0.350470
$545$	0	0
$546$	849097.	0.121892
$547$	−3.92034e6	−0.560215	−0.280108	−	0.959969i	$-0.590370\pi$
$547$	−3.92034e6	−0.560215	−0.280108	+	0.959969i	$0.590370\pi$
$548$	−159307.	−0.0226612
$549$	−74223.6	−0.0105102
$550$	0	0
$551$	1.85464e6	0.260244
$552$	18631.0	0.00260249
$553$	7.98491e6	1.11034
$554$	−2.73876e6	−0.379122
$555$	0	0
$556$	1.41708e6	0.194405
$557$	−5.56520e6	−0.760051	−0.380026	−	0.924976i	$-0.624085\pi$
$557$	−5.56520e6	−0.760051	−0.380026	+	0.924976i	$0.624085\pi$
$558$	−2.10843e6	−0.286664
$559$	−3.14753e6	−0.426030
$560$	0	0
$561$	464782.	0.0623508
$562$	−2.79743e6	−0.373610
$563$	6.49216e6	0.863213	0.431607	−	0.902062i	$-0.357947\pi$
$563$	6.49216e6	0.863213	0.431607	+	0.902062i	$0.357947\pi$
$564$	1.79302e6	0.237349
$565$	0	0
$566$	−6.99429e6	−0.917704
$567$	−1.10251e6	−0.144021
$568$	−3.09403e6	−0.402396
$569$	−1.02437e7	−1.32641	−0.663204	−	0.748439i	$-0.730804\pi$
$569$	−1.02437e7	−1.32641	−0.663204	+	0.748439i	$0.730804\pi$
$570$	0	0
$571$	−1.42962e7	−1.83498	−0.917488	−	0.397764i	$-0.869786\pi$
$571$	−1.42962e7	−1.83498	−0.917488	+	0.397764i	$0.869786\pi$
$572$	−354119.	−0.0452542
$573$	−7.41904e6	−0.943976
$574$	−8.54643e6	−1.08269
$575$	0	0
$576$	1.82708e6	0.229456
$577$	5.37436e6	0.672027	0.336014	−	0.941857i	$-0.390921\pi$
$577$	5.37436e6	0.672027	0.336014	+	0.941857i	$0.390921\pi$
$578$	4.48303e6	0.558151
$579$	−7.05035e6	−0.874006
$580$	0	0
$581$	−4.10276e6	−0.504238
$582$	−512001.	−0.0626561
$583$	−3.27540e6	−0.399110
$584$	1.03161e7	1.25166
$585$	0	0
$586$	6.18572e6	0.744126
$587$	−1.17715e7	−1.41006	−0.705030	−	0.709178i	$-0.749066\pi$
$587$	−1.17715e7	−1.41006	−0.705030	+	0.709178i	$0.749066\pi$
$588$	−1.94235e6	−0.231678
$589$	−1.20312e7	−1.42897
$590$	0	0
$591$	−2.45125e6	−0.288681
$592$	−289852.	−0.0339917
$593$	3.16466e6	0.369564	0.184782	−	0.982780i	$-0.440842\pi$
$593$	3.16466e6	0.369564	0.184782	+	0.982780i	$0.440842\pi$
$594$	−319498.	−0.0371537
$595$	0	0
$596$	2.25668e6	0.260228
$597$	8.84270e6	1.01543
$598$	−6306.47	−0.000721163 0
$599$	4.34303e6	0.494567	0.247284	−	0.968943i	$-0.420462\pi$
$599$	4.34303e6	0.494567	0.247284	+	0.968943i	$0.420462\pi$
$600$	0	0
$601$	8.05946e6	0.910165	0.455082	−	0.890449i	$-0.349610\pi$
$601$	8.05946e6	0.910165	0.455082	+	0.890449i	$0.349610\pi$
$602$	−1.23593e7	−1.38996
$603$	2.05160e6	0.229774
$604$	−7.03708e6	−0.784874
$605$	0	0
$606$	−350928.	−0.0388183
$607$	284560.	0.0313475	0.0156737	−	0.999877i	$-0.495011\pi$
$607$	284560.	0.0313475	0.0156737	+	0.999877i	$0.495011\pi$
$608$	9.48897e6	1.04102
$609$	1.67542e6	0.183054
$610$	0	0
$611$	−1.63558e6	−0.177243
$612$	−652716.	−0.0704443
$613$	−1.43183e7	−1.53901	−0.769503	−	0.638644i	$-0.779496\pi$
$613$	−1.43183e7	−1.53901	−0.769503	+	0.638644i	$0.779496\pi$
$614$	929424.	0.0994931
$615$	0	0
$616$	−3.74720e6	−0.397883
$617$	1.35489e7	1.43282	0.716411	−	0.697679i	$-0.245784\pi$
$617$	1.35489e7	1.43282	0.716411	+	0.697679i	$0.245784\pi$
$618$	−935072.	−0.0984859
$619$	−1.25027e7	−1.31152	−0.655762	−	0.754968i	$-0.727652\pi$
$619$	−1.25027e7	−1.31152	−0.655762	+	0.754968i	$0.727652\pi$
$620$	0	0
$621$	8188.65	0.000852086 0
$622$	−7.53511e6	−0.780932
$623$	3.71060e6	0.383022
$624$	−88358.9	−0.00908425
$625$	0	0
$626$	−3.81906e6	−0.389512
$627$	−1.82314e6	−0.185204
$628$	−7.69953e6	−0.779050
$629$	1.95315e6	0.196838
$630$	0	0
$631$	272233.	0.0272187	0.0136093	−	0.999907i	$-0.495668\pi$
$631$	272233.	0.0272187	0.0136093	+	0.999907i	$0.495668\pi$
$632$	−8.75719e6	−0.872112
$633$	−3.87360e6	−0.384242
$634$	−8.52419e6	−0.842229
$635$	0	0
$636$	4.59980e6	0.450916
$637$	1.77179e6	0.173007
$638$	485521.	0.0472233
$639$	−1.35988e6	−0.131749
$640$	0	0
$641$	1.57242e7	1.51156	0.755779	−	0.654827i	$-0.227258\pi$
$641$	1.57242e7	1.51156	0.755779	+	0.654827i	$0.227258\pi$
$642$	3.97434e6	0.380564
$643$	−5.47692e6	−0.522407	−0.261203	−	0.965284i	$-0.584119\pi$
$643$	−5.47692e6	−0.522407	−0.261203	+	0.965284i	$0.584119\pi$
$644$	35638.2	0.00338611
$645$	0	0
$646$	2.58803e6	0.243999
$647$	−3.91671e6	−0.367841	−0.183921	−	0.982941i	$-0.558879\pi$
$647$	−3.91671e6	−0.367841	−0.183921	+	0.982941i	$0.558879\pi$
$648$	1.20915e6	0.113120
$649$	2.75554e6	0.256800
$650$	0	0
$651$	−1.08686e7	−1.00513
$652$	−6.80323e6	−0.626753
$653$	−1.51457e7	−1.38997	−0.694985	−	0.719024i	$-0.744589\pi$
$653$	−1.51457e7	−1.38997	−0.694985	+	0.719024i	$0.744589\pi$
$654$	5.53465e6	0.505995
$655$	0	0
$656$	889361.	0.0806898
$657$	4.53413e6	0.409808
$658$	−6.42235e6	−0.578269
$659$	1.56274e6	0.140176	0.0700880	−	0.997541i	$-0.477672\pi$
$659$	1.56274e6	0.140176	0.0700880	+	0.997541i	$0.477672\pi$
$660$	0	0
$661$	1.33712e7	1.19033	0.595163	−	0.803605i	$-0.297088\pi$
$661$	1.33712e7	1.19033	0.595163	+	0.803605i	$0.297088\pi$
$662$	1.36514e6	0.121069
$663$	595401.	0.0526049
$664$	4.49957e6	0.396051
$665$	0	0
$666$	−1.34263e6	−0.117292
$667$	−12443.8	−0.00108302
$668$	−1.31535e6	−0.114051
$669$	4.78829e6	0.413633
$670$	0	0
$671$	−110877.	−0.00950684
$672$	8.57201e6	0.732250
$673$	−1.30303e7	−1.10896	−0.554481	−	0.832196i	$-0.687083\pi$
$673$	−1.30303e7	−1.10896	−0.554481	+	0.832196i	$0.687083\pi$
$674$	1.13707e7	0.964130
$675$	0	0
$676$	6.55663e6	0.551841
$677$	1.20799e7	1.01296	0.506478	−	0.862253i	$-0.330947\pi$
$677$	1.20799e7	1.01296	0.506478	+	0.862253i	$0.330947\pi$
$678$	−7.18040e6	−0.599894
$679$	−2.63929e6	−0.219691
$680$	0	0
$681$	7.13416e6	0.589488
$682$	−3.14962e6	−0.259297
$683$	−1.84029e7	−1.50950	−0.754752	−	0.656011i	$-0.772243\pi$
$683$	−1.84029e7	−1.50950	−0.754752	+	0.656011i	$0.772243\pi$
$684$	2.56032e6	0.209245
$685$	0	0
$686$	−3.27238e6	−0.265494
$687$	2.57903e6	0.208480
$688$	1.28613e6	0.103589
$689$	−4.19589e6	−0.336726
$690$	0	0
$691$	5.23001e6	0.416684	0.208342	−	0.978056i	$-0.433193\pi$
$691$	5.23001e6	0.416684	0.208342	+	0.978056i	$0.433193\pi$
$692$	−1.18407e7	−0.939963
$693$	−1.64696e6	−0.130272
$694$	−6.30748e6	−0.497115
$695$	0	0
$696$	−1.83746e6	−0.143779
$697$	−5.99290e6	−0.467257
$698$	1.59702e7	1.24072
$699$	1.27291e7	0.985382
$700$	0	0
$701$	4.83055e6	0.371280	0.185640	−	0.982618i	$-0.440564\pi$
$701$	4.83055e6	0.371280	0.185640	+	0.982618i	$0.440564\pi$
$702$	−409288.	−0.0313463
$703$	−7.66137e6	−0.584680
$704$	2.72934e6	0.207551
$705$	0	0
$706$	−7.15804e6	−0.540484
$707$	−1.80898e6	−0.136109
$708$	−3.86974e6	−0.290134
$709$	−6.13980e6	−0.458711	−0.229355	−	0.973343i	$-0.573662\pi$
$709$	−6.13980e6	−0.458711	−0.229355	+	0.973343i	$0.573662\pi$
$710$	0	0
$711$	−3.84894e6	−0.285540
$712$	−4.06948e6	−0.300842
$713$	80724.0	0.00594674
$714$	2.33794e6	0.171628
$715$	0	0
$716$	1.25100e7	0.911959
$717$	9.06249e6	0.658339
$718$	−1.23181e6	−0.0891731
$719$	1.31130e7	0.945974	0.472987	−	0.881069i	$-0.343176\pi$
$719$	1.31130e7	0.945974	0.472987	+	0.881069i	$0.343176\pi$
$720$	0	0
$721$	−4.82015e6	−0.345321
$722$	−1.18315e6	−0.0844690
$723$	9.78444e6	0.696130
$724$	9.92520e6	0.703708
$725$	0	0
$726$	−477275.	−0.0336068
$727$	2.62296e7	1.84059	0.920293	−	0.391229i	$-0.127950\pi$
$727$	2.62296e7	1.84059	0.920293	+	0.391229i	$0.127950\pi$
$728$	−4.80029e6	−0.335691
$729$	531441.	0.0370370
$730$	0	0
$731$	−8.66653e6	−0.599863
$732$	155710.	0.0107409
$733$	−261993.	−0.0180106	−0.00900532	−	0.999959i	$-0.502867\pi$
$733$	−261993.	−0.0180106	−0.00900532	+	0.999959i	$0.502867\pi$
$734$	−1.30516e7	−0.894178
$735$	0	0
$736$	−63666.6	−0.00433229
$737$	3.06474e6	0.207838
$738$	4.11961e6	0.278430
$739$	2.14527e7	1.44501	0.722506	−	0.691365i	$-0.242990\pi$
$739$	2.14527e7	1.44501	0.722506	+	0.691365i	$0.242990\pi$
$740$	0	0
$741$	−2.33550e6	−0.156255
$742$	−1.64758e7	−1.09860
$743$	1.11848e7	0.743283	0.371642	−	0.928376i	$-0.378795\pi$
$743$	1.11848e7	0.743283	0.371642	+	0.928376i	$0.378795\pi$
$744$	1.19198e7	0.789471
$745$	0	0
$746$	−7.06280e6	−0.464654
$747$	1.97764e6	0.129672
$748$	−975045.	−0.0637192
$749$	2.04871e7	1.33437
$750$	0	0
$751$	5.63342e6	0.364479	0.182239	−	0.983254i	$-0.441665\pi$
$751$	5.63342e6	0.364479	0.182239	+	0.983254i	$0.441665\pi$
$752$	668325.	0.0430966
$753$	2.81509e6	0.180928
$754$	621968.	0.0398419
$755$	0	0
$756$	2.31291e6	0.147182
$757$	−7.69867e6	−0.488288	−0.244144	−	0.969739i	$-0.578507\pi$
$757$	−7.69867e6	−0.488288	−0.244144	+	0.969739i	$0.578507\pi$
$758$	−7.81276e6	−0.493891
$759$	12232.4	0.000770741 0
$760$	0	0
$761$	2.39899e7	1.50164	0.750822	−	0.660505i	$-0.229658\pi$
$761$	2.39899e7	1.50164	0.750822	+	0.660505i	$0.229658\pi$
$762$	−7.52987e6	−0.469786
$763$	2.85303e7	1.77417
$764$	1.55641e7	0.964694
$765$	0	0
$766$	6.04179e6	0.372043
$767$	3.52994e6	0.216660
$768$	−9.74549e6	−0.596212
$769$	−2.03436e7	−1.24054	−0.620272	−	0.784387i	$-0.712978\pi$
$769$	−2.03436e7	−1.24054	−0.620272	+	0.784387i	$0.712978\pi$
$770$	0	0
$771$	1.35581e7	0.821415
$772$	1.47906e7	0.893188
$773$	−1.86605e7	−1.12324	−0.561622	−	0.827394i	$-0.689822\pi$
$773$	−1.86605e7	−1.12324	−0.561622	+	0.827394i	$0.689822\pi$
$774$	5.95750e6	0.357447
$775$	0	0
$776$	2.89455e6	0.172555
$777$	−6.92102e6	−0.411261
$778$	−5.75608e6	−0.340940
$779$	2.35076e7	1.38792
$780$	0	0
$781$	−2.03143e6	−0.119172
$782$	−17364.5	−0.00101542
$783$	−807596.	−0.0470750
$784$	−723985.	−0.0420668
$785$	0	0
$786$	−5.73388e6	−0.331049
$787$	−2.98644e7	−1.71877	−0.859383	−	0.511333i	$-0.829152\pi$
$787$	−2.98644e7	−1.71877	−0.859383	+	0.511333i	$0.829152\pi$
$788$	5.14235e6	0.295017
$789$	1.70696e6	0.0976180
$790$	0	0
$791$	−3.70138e7	−2.10341
$792$	1.80625e6	0.102321
$793$	−142037.	−0.00802084
$794$	1.47700e7	0.831438
$795$	0	0
$796$	−1.85507e7	−1.03771
$797$	−2.60350e7	−1.45182	−0.725909	−	0.687791i	$-0.758581\pi$
$797$	−2.60350e7	−1.45182	−0.725909	+	0.687791i	$0.758581\pi$
$798$	−9.17073e6	−0.509796
$799$	−4.50346e6	−0.249563
$800$	0	0
$801$	−1.78861e6	−0.0984995
$802$	1.77252e7	0.973093
$803$	6.77320e6	0.370685
$804$	−4.30396e6	−0.234816
$805$	0	0
$806$	−4.03477e6	−0.218767
$807$	1.44417e7	0.780611
$808$	1.98394e6	0.106906
$809$	−1.97670e7	−1.06186	−0.530932	−	0.847414i	$-0.678158\pi$
$809$	−1.97670e7	−1.06186	−0.530932	+	0.847414i	$0.678158\pi$
$810$	0	0
$811$	7.26108e6	0.387658	0.193829	−	0.981035i	$-0.437909\pi$
$811$	7.26108e6	0.387658	0.193829	+	0.981035i	$0.437909\pi$
$812$	−3.51478e6	−0.187072
$813$	1.54710e7	0.820901
$814$	−2.00565e6	−0.106095
$815$	0	0
$816$	−243291.	−0.0127909
$817$	3.39951e7	1.78181
$818$	−1.84342e6	−0.0963256
$819$	−2.10981e6	−0.109909
$820$	0	0
$821$	3.10030e7	1.60526	0.802631	−	0.596476i	$-0.203433\pi$
$821$	3.10030e7	1.60526	0.802631	+	0.596476i	$0.203433\pi$
$822$	−275051.	−0.0141982
$823$	2.56767e7	1.32142	0.660709	−	0.750643i	$-0.270256\pi$
$823$	2.56767e7	1.32142	0.660709	+	0.750643i	$0.270256\pi$
$824$	5.28635e6	0.271230
$825$	0	0
$826$	1.38609e7	0.706871
$827$	1.60345e7	0.815254	0.407627	−	0.913149i	$-0.366356\pi$
$827$	1.60345e7	0.815254	0.407627	+	0.913149i	$0.366356\pi$
$828$	−17178.6	−0.000870787 0
$829$	−1.90352e7	−0.961992	−0.480996	−	0.876723i	$-0.659725\pi$
$829$	−1.90352e7	−0.961992	−0.480996	+	0.876723i	$0.659725\pi$
$830$	0	0
$831$	6.80520e6	0.341852
$832$	3.49637e6	0.175109
$833$	4.87853e6	0.243599
$834$	2.44667e6	0.121804
$835$	0	0
$836$	3.82468e6	0.189269
$837$	5.23896e6	0.258483
$838$	−2.45912e7	−1.20968
$839$	2.86604e7	1.40565	0.702826	−	0.711362i	$-0.251921\pi$
$839$	2.86604e7	1.40565	0.702826	+	0.711362i	$0.251921\pi$
$840$	0	0
$841$	−1.92839e7	−0.940167
$842$	1.42421e7	0.692297
$843$	6.95099e6	0.336882
$844$	8.12625e6	0.392676
$845$	0	0
$846$	3.09575e6	0.148710
$847$	−2.46028e6	−0.117835
$848$	1.71451e6	0.0818749
$849$	1.73792e7	0.827488
$850$	0	0
$851$	51404.3	0.00243319
$852$	2.85283e6	0.134641
$853$	5.22750e6	0.245992	0.122996	−	0.992407i	$-0.460750\pi$
$853$	5.22750e6	0.245992	0.122996	+	0.992407i	$0.460750\pi$
$854$	−557733.	−0.0261687
$855$	0	0
$856$	−2.24686e7	−1.04807
$857$	9.88397e6	0.459705	0.229853	−	0.973225i	$-0.426176\pi$
$857$	9.88397e6	0.459705	0.229853	+	0.973225i	$0.426176\pi$
$858$	−611405.	−0.0283538
$859$	−1.37127e7	−0.634076	−0.317038	−	0.948413i	$-0.602688\pi$
$859$	−1.37127e7	−0.634076	−0.317038	+	0.948413i	$0.602688\pi$
$860$	0	0
$861$	2.12360e7	0.976257
$862$	1.17417e7	0.538226
$863$	−3.39850e7	−1.55332	−0.776659	−	0.629921i	$-0.783087\pi$
$863$	−3.39850e7	−1.55332	−0.776659	+	0.629921i	$0.783087\pi$
$864$	−4.13194e6	−0.188308
$865$	0	0
$866$	−2.16139e7	−0.979352
$867$	−1.11393e7	−0.503281
$868$	2.28007e7	1.02719
$869$	−5.74965e6	−0.258281
$870$	0	0
$871$	3.92603e6	0.175351
$872$	−3.12896e7	−1.39351
$873$	1.27221e6	0.0564966
$874$	68113.4	0.00301616
$875$	0	0
$876$	−9.51193e6	−0.418802
$877$	−1.66649e7	−0.731653	−0.365826	−	0.930683i	$-0.619214\pi$
$877$	−1.66649e7	−0.731653	−0.365826	+	0.930683i	$0.619214\pi$
$878$	−1.03580e7	−0.453459
$879$	−1.53701e7	−0.670973
$880$	0	0
$881$	−1.85938e7	−0.807103	−0.403551	−	0.914957i	$-0.632224\pi$
$881$	−1.85938e7	−0.807103	−0.403551	+	0.914957i	$0.632224\pi$
$882$	−3.35357e6	−0.145156
$883$	1.41760e7	0.611859	0.305929	−	0.952054i	$-0.401033\pi$
$883$	1.41760e7	0.611859	0.305929	+	0.952054i	$0.401033\pi$
$884$	−1.24907e6	−0.0537594
$885$	0	0
$886$	9.15737e6	0.391910
$887$	1.10264e7	0.470570	0.235285	−	0.971926i	$-0.424398\pi$
$887$	1.10264e7	0.470570	0.235285	+	0.971926i	$0.424398\pi$
$888$	7.59041e6	0.323023
$889$	−3.88153e7	−1.64721
$890$	0	0
$891$	793881.	0.0335013
$892$	−1.00451e7	−0.422711
$893$	1.76652e7	0.741291
$894$	3.89628e6	0.163045
$895$	0	0
$896$	−1.67492e7	−0.696986
$897$	15670.2	0.000650268 0
$898$	−2.07454e7	−0.858483
$899$	−7.96131e6	−0.328538
$900$	0	0
$901$	−1.15531e7	−0.474120
$902$	6.15399e6	0.251849
$903$	3.07100e7	1.25332
$904$	4.05937e7	1.65211
$905$	0	0
$906$	−1.21499e7	−0.491759
$907$	1.14062e7	0.460388	0.230194	−	0.973145i	$-0.426064\pi$
$907$	1.14062e7	0.460388	0.230194	+	0.973145i	$0.426064\pi$
$908$	−1.49664e7	−0.602426
$909$	871978.	0.0350022
$910$	0	0
$911$	2.17910e7	0.869922	0.434961	−	0.900449i	$-0.356762\pi$
$911$	2.17910e7	0.869922	0.434961	+	0.900449i	$0.356762\pi$
$912$	954326.	0.0379935
$913$	2.95425e6	0.117293
$914$	−1.78946e7	−0.708527
$915$	0	0
$916$	−5.41042e6	−0.213055
$917$	−2.95572e7	−1.16075
$918$	−1.12695e6	−0.0441365
$919$	3.12982e7	1.22245	0.611225	−	0.791457i	$-0.290677\pi$
$919$	3.12982e7	1.22245	0.611225	+	0.791457i	$0.290677\pi$
$920$	0	0
$921$	−2.30941e6	−0.0897123
$922$	−2.24053e7	−0.868008
$923$	−2.60232e6	−0.100544
$924$	3.45509e6	0.133131
$925$	0	0
$926$	−4.46345e6	−0.171058
$927$	2.32344e6	0.0888040
$928$	6.27904e6	0.239345
$929$	−3.00058e7	−1.14069	−0.570343	−	0.821407i	$-0.693190\pi$
$929$	−3.00058e7	−1.14069	−0.570343	+	0.821407i	$0.693190\pi$
$930$	0	0
$931$	−1.91364e7	−0.723578
$932$	−2.67038e7	−1.00701
$933$	1.87230e7	0.704161
$934$	−2.80485e7	−1.05206
$935$	0	0
$936$	2.31387e6	0.0863276
$937$	2.12772e7	0.791711	0.395855	−	0.918313i	$-0.370448\pi$
$937$	2.12772e7	0.791711	0.395855	+	0.918313i	$0.370448\pi$
$938$	1.54162e7	0.572098
$939$	9.48950e6	0.351220
$940$	0	0
$941$	4.12864e7	1.51996	0.759982	−	0.649944i	$-0.225208\pi$
$941$	4.12864e7	1.51996	0.759982	+	0.649944i	$0.225208\pi$
$942$	−1.32937e7	−0.488110
$943$	−157725.	−0.00577593
$944$	−1.44239e6	−0.0526809
$945$	0	0
$946$	8.89948e6	0.323323
$947$	1.15296e7	0.417771	0.208886	−	0.977940i	$-0.433016\pi$
$947$	1.15296e7	0.417771	0.208886	+	0.977940i	$0.433016\pi$
$948$	8.07452e6	0.291807
$949$	8.67670e6	0.312744
$950$	0	0
$951$	2.11807e7	0.759432
$952$	−1.32173e7	−0.472663
$953$	−9.67798e6	−0.345185	−0.172593	−	0.984993i	$-0.555214\pi$
$953$	−9.67798e6	−0.345185	−0.172593	+	0.984993i	$0.555214\pi$
$954$	7.94180e6	0.282519
$955$	0	0
$956$	−1.90118e7	−0.672788
$957$	−1.20641e6	−0.0425809
$958$	1.21750e7	0.428604
$959$	−1.41785e6	−0.0497832
$960$	0	0
$961$	2.30167e7	0.803959
$962$	−2.56931e6	−0.0895113
$963$	−9.87534e6	−0.343152
$964$	−2.05263e7	−0.711408
$965$	0	0
$966$	61531.4	0.00212155
$967$	3.88946e7	1.33759	0.668795	−	0.743447i	$-0.266811\pi$
$967$	3.88946e7	1.33759	0.668795	+	0.743447i	$0.266811\pi$
$968$	2.69823e6	0.0925531
$969$	−6.43067e6	−0.220012
$970$	0	0
$971$	2.79477e7	0.951259	0.475629	−	0.879646i	$-0.342220\pi$
$971$	2.79477e7	0.951259	0.475629	+	0.879646i	$0.342220\pi$
$972$	−1.11489e6	−0.0378499
$973$	1.26122e7	0.427080
$974$	1.44071e6	0.0486608
$975$	0	0
$976$	58038.9	0.00195027
$977$	7.01873e6	0.235246	0.117623	−	0.993058i	$-0.462473\pi$
$977$	7.01873e6	0.235246	0.117623	+	0.993058i	$0.462473\pi$
$978$	−1.17462e7	−0.392689
$979$	−2.67187e6	−0.0890962
$980$	0	0
$981$	−1.37524e7	−0.456252
$982$	2.12188e6	0.0702171
$983$	−4.12903e7	−1.36290	−0.681450	−	0.731864i	$-0.738650\pi$
$983$	−4.12903e7	−1.36290	−0.681450	+	0.731864i	$0.738650\pi$
$984$	−2.32899e7	−0.766795
$985$	0	0
$986$	1.71255e6	0.0560985
$987$	1.59581e7	0.521421
$988$	4.89955e6	0.159685
$989$	−228091.	−0.00741512
$990$	0	0
$991$	2.63627e6	0.0852717	0.0426359	−	0.999091i	$-0.486424\pi$
$991$	2.63627e6	0.0852717	0.0426359	+	0.999091i	$0.486424\pi$
$992$	−4.07328e7	−1.31421
$993$	−3.39207e6	−0.109167
$994$	−1.02184e7	−0.328034
$995$	0	0
$996$	−4.14880e6	−0.132518
$997$	4.09967e7	1.30620	0.653102	−	0.757270i	$-0.273467\pi$
$997$	4.09967e7	1.30620	0.653102	+	0.757270i	$0.273467\pi$
$998$	−2.83328e7	−0.900458
$999$	3.33612e6	0.105762

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.y.1.4		13
5.2	odd	4		165.6.c.b.34.10	✓	26
5.3	odd	4		165.6.c.b.34.17	yes	26
5.4	even	2		825.6.a.v.1.10		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.b.34.10	✓	26	5.2	odd	4
165.6.c.b.34.17	yes	26	5.3	odd	4
825.6.a.v.1.10		13	5.4	even	2
825.6.a.y.1.4		13	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q-3.62206 q^{2} +9.00000 q^{3} -18.8807 q^{4} -32.5985 q^{6} -168.040 q^{7} +184.293 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-3.62206 q^{2} +9.00000 q^{3} -18.8807 q^{4} -32.5985 q^{6} -168.040 q^{7} +184.293 q^{8} +81.0000 q^{9} +121.000 q^{11} -169.926 q^{12} +155.005 q^{13} +608.652 q^{14} -63.3377 q^{16} +426.797 q^{17} -293.387 q^{18} -1674.14 q^{19} -1512.36 q^{21} -438.269 q^{22} +11.2327 q^{23} +1658.64 q^{24} -561.437 q^{26} +729.000 q^{27} +3172.72 q^{28} -1107.81 q^{29} +7186.50 q^{31} -5667.96 q^{32} +1089.00 q^{33} -1545.88 q^{34} -1529.34 q^{36} +4576.30 q^{37} +6063.84 q^{38} +1395.05 q^{39} -14041.6 q^{41} +5477.87 q^{42} -20306.0 q^{43} -2284.56 q^{44} -40.6856 q^{46} -10551.8 q^{47} -570.039 q^{48} +11430.6 q^{49} +3841.17 q^{51} -2926.60 q^{52} -27069.4 q^{53} -2640.48 q^{54} -30968.6 q^{56} -15067.3 q^{57} +4012.57 q^{58} +22773.1 q^{59} -916.341 q^{61} -26029.9 q^{62} -13611.3 q^{63} +22556.5 q^{64} -3944.42 q^{66} +25328.4 q^{67} -8058.22 q^{68} +101.095 q^{69} -16788.6 q^{71} +14927.7 q^{72} +55976.9 q^{73} -16575.6 q^{74} +31608.9 q^{76} -20332.9 q^{77} -5052.94 q^{78} -47517.8 q^{79} +6561.00 q^{81} +50859.4 q^{82} +24415.3 q^{83} +28554.5 q^{84} +73549.4 q^{86} -9970.33 q^{87} +22299.4 q^{88} -22081.6 q^{89} -26047.1 q^{91} -212.082 q^{92} +64678.5 q^{93} +38219.1 q^{94} -51011.6 q^{96} +15706.3 q^{97} -41402.1 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9}+O(q^{10}) \) 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9	\( 13 q + 13 q^{2} + 117 q^{3} + 209 q^{4} + 117 q^{6} + 304 q^{7} + 399 q^{8} + 1053 q^{9} + 1573 q^{11} + 1881 q^{12} + 986 q^{13} - 610 q^{14} + 3501 q^{16} + 1476 q^{17} + 1053 q^{18} + 270 q^{19} + 2736 q^{21} + 1573 q^{22} + 9084 q^{23} + 3591 q^{24} + 2652 q^{26} + 9477 q^{27} + 10920 q^{28} + 11952 q^{29} + 19096 q^{31} + 11661 q^{32} + 14157 q^{33} - 1302 q^{34} + 16929 q^{36} + 39964 q^{37} + 1574 q^{38} + 8874 q^{39} + 35184 q^{41} - 5490 q^{42} - 96 q^{43} + 25289 q^{44} - 4120 q^{46} + 34984 q^{47} + 31509 q^{48} + 14557 q^{49} + 13284 q^{51} + 39002 q^{52} + 22984 q^{53} + 9477 q^{54} + 59802 q^{56} + 2430 q^{57} + 18896 q^{58} - 9192 q^{59} + 5438 q^{61} + 272 q^{62} + 24624 q^{63} + 106557 q^{64} + 14157 q^{66} + 71508 q^{67} + 127948 q^{68} + 81756 q^{69} + 101700 q^{71} + 32319 q^{72} + 77390 q^{73} + 13676 q^{74} + 139966 q^{76} + 36784 q^{77} + 23868 q^{78} + 93954 q^{79} + 85293 q^{81} + 53284 q^{82} + 185918 q^{83} + 98280 q^{84} + 370930 q^{86} + 107568 q^{87} + 48279 q^{88} - 18418 q^{89} + 174536 q^{91} + 274264 q^{92} + 171864 q^{93} + 64520 q^{94} + 104949 q^{96} + 94312 q^{97} + 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 13 * q + 13 * q^2 + 117 * q^3 + 209 * q^4 + 117 * q^6 + 304 * q^7 + 399 * q^8 + 1053 * q^9 + 1573 * q^11 + 1881 * q^12 + 986 * q^13 - 610 * q^14 + 3501 * q^16 + 1476 * q^17 + 1053 * q^18 + 270 * q^19 + 2736 * q^21 + 1573 * q^22 + 9084 * q^23 + 3591 * q^24 + 2652 * q^26 + 9477 * q^27 + 10920 * q^28 + 11952 * q^29 + 19096 * q^31 + 11661 * q^32 + 14157 * q^33 - 1302 * q^34 + 16929 * q^36 + 39964 * q^37 + 1574 * q^38 + 8874 * q^39 + 35184 * q^41 - 5490 * q^42 - 96 * q^43 + 25289 * q^44 - 4120 * q^46 + 34984 * q^47 + 31509 * q^48 + 14557 * q^49 + 13284 * q^51 + 39002 * q^52 + 22984 * q^53 + 9477 * q^54 + 59802 * q^56 + 2430 * q^57 + 18896 * q^58 - 9192 * q^59 + 5438 * q^61 + 272 * q^62 + 24624 * q^63 + 106557 * q^64 + 14157 * q^66 + 71508 * q^67 + 127948 * q^68 + 81756 * q^69 + 101700 * q^71 + 32319 * q^72 + 77390 * q^73 + 13676 * q^74 + 139966 * q^76 + 36784 * q^77 + 23868 * q^78 + 93954 * q^79 + 85293 * q^81 + 53284 * q^82 + 185918 * q^83 + 98280 * q^84 + 370930 * q^86 + 107568 * q^87 + 48279 * q^88 - 18418 * q^89 + 174536 * q^91 + 274264 * q^92 + 171864 * q^93 + 64520 * q^94 + 104949 * q^96 + 94312 * q^97 + 145677 * q^98 + 127413 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more