LMFDB - Embedded newform 150.6.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,6,Mod(1,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, 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("150.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	150.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:	$24.0575729719$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1249}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 312 $ x^2 - x - 312
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 30)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-17.1706$ of defining polynomial
Character	$\chi$	$=$	150.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} +233.706 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} +233.706 q^{7} +64.0000 q^{8} +81.0000 q^{9} -89.7060 q^{11} +144.000 q^{12} -209.118 q^{13} +934.824 q^{14} +256.000 q^{16} +226.882 q^{17} +324.000 q^{18} -2180.47 q^{19} +2103.35 q^{21} -358.824 q^{22} +4296.47 q^{23} +576.000 q^{24} -836.472 q^{26} +729.000 q^{27} +3739.30 q^{28} +3556.06 q^{29} +6902.24 q^{31} +1024.00 q^{32} -807.354 q^{33} +907.528 q^{34} +1296.00 q^{36} -5435.59 q^{37} -8721.89 q^{38} -1882.06 q^{39} +6484.00 q^{41} +8413.41 q^{42} -21979.5 q^{43} -1435.30 q^{44} +17185.9 q^{46} -7718.24 q^{47} +2304.00 q^{48} +37811.5 q^{49} +2041.94 q^{51} -3345.89 q^{52} +12765.4 q^{53} +2916.00 q^{54} +14957.2 q^{56} -19624.2 q^{57} +14224.2 q^{58} -44555.6 q^{59} +11598.2 q^{61} +27608.9 q^{62} +18930.2 q^{63} +4096.00 q^{64} -3229.41 q^{66} -3961.29 q^{67} +3630.11 q^{68} +38668.2 q^{69} +46413.6 q^{71} +5184.00 q^{72} -61591.2 q^{73} -21742.3 q^{74} -34887.5 q^{76} -20964.8 q^{77} -7528.24 q^{78} -27877.8 q^{79} +6561.00 q^{81} +25936.0 q^{82} -50941.0 q^{83} +33653.7 q^{84} -87918.1 q^{86} +32004.6 q^{87} -5741.18 q^{88} +5131.19 q^{89} -48872.1 q^{91} +68743.5 q^{92} +62120.1 q^{93} -30872.9 q^{94} +9216.00 q^{96} -88435.3 q^{97} +151246. q^{98} -7266.18 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 18 q^{3} + 32 q^{4} + 72 q^{6} + 114 q^{7} + 128 q^{8} + 162 q^{9}+O(q^{10}) $ 2 * q + 8 * q^2 + 18 * q^3 + 32 * q^4 + 72 * q^6 + 114 * q^7 + 128 * q^8 + 162 * q^9	\( 2 q + 8 q^{2} + 18 q^{3} + 32 q^{4} + 72 q^{6} + 114 q^{7} + 128 q^{8} + 162 q^{9} + 174 q^{11} + 288 q^{12} + 642 q^{13} + 456 q^{14} + 512 q^{16} + 1514 q^{17} + 648 q^{18} - 120 q^{19} + 1026 q^{21} + 696 q^{22} + 4352 q^{23} + 1152 q^{24} + 2568 q^{26} + 1458 q^{27} + 1824 q^{28} - 2430 q^{29} + 11684 q^{31} + 2048 q^{32} + 1566 q^{33} + 6056 q^{34} + 2592 q^{36} - 17586 q^{37} - 480 q^{38} + 5778 q^{39} + 24984 q^{41} + 4104 q^{42} - 24168 q^{43} + 2784 q^{44} + 17408 q^{46} - 13316 q^{47} + 4608 q^{48} + 35334 q^{49} + 13626 q^{51} + 10272 q^{52} - 13698 q^{53} + 5832 q^{54} + 7296 q^{56} - 1080 q^{57} - 9720 q^{58} - 23730 q^{59} + 57124 q^{61} + 46736 q^{62} + 9234 q^{63} + 8192 q^{64} + 6264 q^{66} - 38316 q^{67} + 24224 q^{68} + 39168 q^{69} - 11076 q^{71} + 10368 q^{72} - 88548 q^{73} - 70344 q^{74} - 1920 q^{76} - 52532 q^{77} + 23112 q^{78} + 14220 q^{79} + 13122 q^{81} + 99936 q^{82} + 50792 q^{83} + 16416 q^{84} - 96672 q^{86} - 21870 q^{87} + 11136 q^{88} - 60420 q^{89} - 150756 q^{91} + 69632 q^{92} + 105156 q^{93} - 53264 q^{94} + 18432 q^{96} - 171216 q^{97} + 141336 q^{98} + 14094 q^{99}+O(q^{100}) \) 2 * q + 8 * q^2 + 18 * q^3 + 32 * q^4 + 72 * q^6 + 114 * q^7 + 128 * q^8 + 162 * q^9 + 174 * q^11 + 288 * q^12 + 642 * q^13 + 456 * q^14 + 512 * q^16 + 1514 * q^17 + 648 * q^18 - 120 * q^19 + 1026 * q^21 + 696 * q^22 + 4352 * q^23 + 1152 * q^24 + 2568 * q^26 + 1458 * q^27 + 1824 * q^28 - 2430 * q^29 + 11684 * q^31 + 2048 * q^32 + 1566 * q^33 + 6056 * q^34 + 2592 * q^36 - 17586 * q^37 - 480 * q^38 + 5778 * q^39 + 24984 * q^41 + 4104 * q^42 - 24168 * q^43 + 2784 * q^44 + 17408 * q^46 - 13316 * q^47 + 4608 * q^48 + 35334 * q^49 + 13626 * q^51 + 10272 * q^52 - 13698 * q^53 + 5832 * q^54 + 7296 * q^56 - 1080 * q^57 - 9720 * q^58 - 23730 * q^59 + 57124 * q^61 + 46736 * q^62 + 9234 * q^63 + 8192 * q^64 + 6264 * q^66 - 38316 * q^67 + 24224 * q^68 + 39168 * q^69 - 11076 * q^71 + 10368 * q^72 - 88548 * q^73 - 70344 * q^74 - 1920 * q^76 - 52532 * q^77 + 23112 * q^78 + 14220 * q^79 + 13122 * q^81 + 99936 * q^82 + 50792 * q^83 + 16416 * q^84 - 96672 * q^86 - 21870 * q^87 + 11136 * q^88 - 60420 * q^89 - 150756 * q^91 + 69632 * q^92 + 105156 * q^93 - 53264 * q^94 + 18432 * q^96 - 171216 * q^97 + 141336 * q^98 + 14094 * 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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	36.0000	0.408248
$7$	233.706	1.80271	0.901353	−	0.433086i	$-0.142575\pi$
$7$	233.706	1.80271	0.901353	+	0.433086i	$0.142575\pi$
$8$	64.0000	0.353553
$9$	81.0000	0.333333
$10$	0	0
$11$	−89.7060	−0.223532	−0.111766	−	0.993735i	$-0.535651\pi$
$11$	−89.7060	−0.223532	−0.111766	+	0.993735i	$0.535651\pi$
$12$	144.000	0.288675
$13$	−209.118	−0.343189	−0.171594	−	0.985168i	$-0.554892\pi$
$13$	−209.118	−0.343189	−0.171594	+	0.985168i	$0.554892\pi$
$14$	934.824	1.27471
$15$	0	0
$16$	256.000	0.250000
$17$	226.882	0.190405	0.0952024	−	0.995458i	$-0.469650\pi$
$17$	226.882	0.190405	0.0952024	+	0.995458i	$0.469650\pi$
$18$	324.000	0.235702
$19$	−2180.47	−1.38569	−0.692846	−	0.721086i	$-0.743643\pi$
$19$	−2180.47	−1.38569	−0.692846	+	0.721086i	$0.743643\pi$
$20$	0	0
$21$	2103.35	1.04079
$22$	−358.824	−0.158061
$23$	4296.47	1.69353	0.846764	−	0.531969i	$-0.178548\pi$
$23$	4296.47	1.69353	0.846764	+	0.531969i	$0.178548\pi$
$24$	576.000	0.204124
$25$	0	0
$26$	−836.472	−0.242671
$27$	729.000	0.192450
$28$	3739.30	0.901353
$29$	3556.06	0.785189	0.392595	−	0.919712i	$-0.371578\pi$
$29$	3556.06	0.785189	0.392595	+	0.919712i	$0.371578\pi$
$30$	0	0
$31$	6902.24	1.28999	0.644994	−	0.764188i	$-0.276860\pi$
$31$	6902.24	1.28999	0.644994	+	0.764188i	$0.276860\pi$
$32$	1024.00	0.176777
$33$	−807.354	−0.129056
$34$	907.528	0.134637
$35$	0	0
$36$	1296.00	0.166667
$37$	−5435.59	−0.652743	−0.326371	−	0.945242i	$-0.605826\pi$
$37$	−5435.59	−0.652743	−0.326371	+	0.945242i	$0.605826\pi$
$38$	−8721.89	−0.979832
$39$	−1882.06	−0.198140
$40$	0	0
$41$	6484.00	0.602398	0.301199	−	0.953561i	$-0.402613\pi$
$41$	6484.00	0.602398	0.301199	+	0.953561i	$0.402613\pi$
$42$	8413.41	0.735951
$43$	−21979.5	−1.81279	−0.906395	−	0.422432i	$-0.861177\pi$
$43$	−21979.5	−1.81279	−0.906395	+	0.422432i	$0.861177\pi$
$44$	−1435.30	−0.111766
$45$	0	0
$46$	17185.9	1.19751
$47$	−7718.24	−0.509652	−0.254826	−	0.966987i	$-0.582018\pi$
$47$	−7718.24	−0.509652	−0.254826	+	0.966987i	$0.582018\pi$
$48$	2304.00	0.144338
$49$	37811.5	2.24975
$50$	0	0
$51$	2041.94	0.109930
$52$	−3345.89	−0.171594
$53$	12765.4	0.624228	0.312114	−	0.950045i	$-0.398963\pi$
$53$	12765.4	0.624228	0.312114	+	0.950045i	$0.398963\pi$
$54$	2916.00	0.136083
$55$	0	0
$56$	14957.2	0.637353
$57$	−19624.2	−0.800029
$58$	14224.2	0.555212
$59$	−44555.6	−1.66637	−0.833187	−	0.552992i	$-0.813486\pi$
$59$	−44555.6	−1.66637	−0.833187	+	0.552992i	$0.813486\pi$
$60$	0	0
$61$	11598.2	0.399086	0.199543	−	0.979889i	$-0.436054\pi$
$61$	11598.2	0.399086	0.199543	+	0.979889i	$0.436054\pi$
$62$	27608.9	0.912159
$63$	18930.2	0.600902
$64$	4096.00	0.125000
$65$	0	0
$66$	−3229.41	−0.0912565
$67$	−3961.29	−0.107808	−0.0539038	−	0.998546i	$-0.517166\pi$
$67$	−3961.29	−0.107808	−0.0539038	+	0.998546i	$0.517166\pi$
$68$	3630.11	0.0952024
$69$	38668.2	0.977759
$70$	0	0
$71$	46413.6	1.09270	0.546348	−	0.837559i	$-0.316018\pi$
$71$	46413.6	1.09270	0.546348	+	0.837559i	$0.316018\pi$
$72$	5184.00	0.117851
$73$	−61591.2	−1.35273	−0.676365	−	0.736566i	$-0.736446\pi$
$73$	−61591.2	−1.35273	−0.676365	+	0.736566i	$0.736446\pi$
$74$	−21742.3	−0.461559
$75$	0	0
$76$	−34887.5	−0.692846
$77$	−20964.8	−0.402962
$78$	−7528.24	−0.140106
$79$	−27877.8	−0.502563	−0.251281	−	0.967914i	$-0.580852\pi$
$79$	−27877.8	−0.502563	−0.251281	+	0.967914i	$0.580852\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	25936.0	0.425959
$83$	−50941.0	−0.811656	−0.405828	−	0.913949i	$-0.633017\pi$
$83$	−50941.0	−0.811656	−0.405828	+	0.913949i	$0.633017\pi$
$84$	33653.7	0.520396
$85$	0	0
$86$	−87918.1	−1.28184
$87$	32004.6	0.453329
$88$	−5741.18	−0.0790305
$89$	5131.19	0.0686663	0.0343331	−	0.999410i	$-0.489069\pi$
$89$	5131.19	0.0686663	0.0343331	+	0.999410i	$0.489069\pi$
$90$	0	0
$91$	−48872.1	−0.618668
$92$	68743.5	0.846764
$93$	62120.1	0.744775
$94$	−30872.9	−0.360378
$95$	0	0
$96$	9216.00	0.102062
$97$	−88435.3	−0.954325	−0.477162	−	0.878815i	$-0.658335\pi$
$97$	−88435.3	−0.954325	−0.477162	+	0.878815i	$0.658335\pi$
$98$	151246.	1.59081
$99$	−7266.18	−0.0745107
$100$	0	0
$101$	−116471.	−1.13609	−0.568047	−	0.822996i	$-0.692301\pi$
$101$	−116471.	−1.13609	−0.568047	+	0.822996i	$0.692301\pi$
$102$	8167.76	0.0777324
$103$	22369.1	0.207756	0.103878	−	0.994590i	$-0.466875\pi$
$103$	22369.1	0.207756	0.103878	+	0.994590i	$0.466875\pi$
$104$	−13383.5	−0.121335
$105$	0	0
$106$	51061.5	0.441396
$107$	−130626.	−1.10299	−0.551495	−	0.834178i	$-0.685942\pi$
$107$	−130626.	−1.10299	−0.551495	+	0.834178i	$0.685942\pi$
$108$	11664.0	0.0962250
$109$	−51443.5	−0.414729	−0.207365	−	0.978264i	$-0.566489\pi$
$109$	−51443.5	−0.414729	−0.207365	+	0.978264i	$0.566489\pi$
$110$	0	0
$111$	−48920.3	−0.376861
$112$	59828.7	0.450676
$113$	138259.	1.01858	0.509291	−	0.860594i	$-0.329908\pi$
$113$	138259.	1.01858	0.509291	+	0.860594i	$0.329908\pi$
$114$	−78497.0	−0.565706
$115$	0	0
$116$	56897.0	0.392595
$117$	−16938.6	−0.114396
$118$	−178222.	−1.17830
$119$	53023.7	0.343244
$120$	0	0
$121$	−153004.	−0.950033
$122$	46392.9	0.282197
$123$	58356.0	0.347794
$124$	110436.	0.644994
$125$	0	0
$126$	75720.7	0.424902
$127$	−199662.	−1.09847	−0.549233	−	0.835669i	$-0.685080\pi$
$127$	−199662.	−1.09847	−0.549233	+	0.835669i	$0.685080\pi$
$128$	16384.0	0.0883883
$129$	−197816.	−1.04661
$130$	0	0
$131$	−314223.	−1.59978	−0.799889	−	0.600148i	$-0.795109\pi$
$131$	−314223.	−1.59978	−0.799889	+	0.600148i	$0.795109\pi$
$132$	−12917.7	−0.0645281
$133$	−509589.	−2.49799
$134$	−15845.1	−0.0762315
$135$	0	0
$136$	14520.5	0.0673183
$137$	256841.	1.16913	0.584564	−	0.811348i	$-0.301266\pi$
$137$	256841.	1.16913	0.584564	+	0.811348i	$0.301266\pi$
$138$	154673.	0.691380
$139$	−10777.2	−0.0473118	−0.0236559	−	0.999720i	$-0.507531\pi$
$139$	−10777.2	−0.0473118	−0.0236559	+	0.999720i	$0.507531\pi$
$140$	0	0
$141$	−69464.1	−0.294247
$142$	185654.	0.772652
$143$	18759.1	0.0767136
$144$	20736.0	0.0833333
$145$	0	0
$146$	−246365.	−0.956525
$147$	340303.	1.29889
$148$	−86969.4	−0.326371
$149$	4103.27	0.0151413	0.00757067	−	0.999971i	$-0.497590\pi$
$149$	4103.27	0.0151413	0.00757067	+	0.999971i	$0.497590\pi$
$150$	0	0
$151$	107449.	0.383495	0.191748	−	0.981444i	$-0.438585\pi$
$151$	107449.	0.383495	0.191748	+	0.981444i	$0.438585\pi$
$152$	−139550.	−0.489916
$153$	18377.4	0.0634683
$154$	−83859.3	−0.284937
$155$	0	0
$156$	−30113.0	−0.0990700
$157$	−345824.	−1.11971	−0.559855	−	0.828591i	$-0.689143\pi$
$157$	−345824.	−1.11971	−0.559855	+	0.828591i	$0.689143\pi$
$158$	−111511.	−0.355366
$159$	114888.	0.360398
$160$	0	0
$161$	1.00411e6	3.05293
$162$	26244.0	0.0785674
$163$	58776.3	0.173274	0.0866370	−	0.996240i	$-0.472388\pi$
$163$	58776.3	0.173274	0.0866370	+	0.996240i	$0.472388\pi$
$164$	103744.	0.301199
$165$	0	0
$166$	−203764.	−0.573928
$167$	154654.	0.429111	0.214556	−	0.976712i	$-0.431170\pi$
$167$	154654.	0.429111	0.214556	+	0.976712i	$0.431170\pi$
$168$	134615.	0.367976
$169$	−327563.	−0.882222
$170$	0	0
$171$	−176618.	−0.461897
$172$	−351673.	−0.906395
$173$	171600.	0.435916	0.217958	−	0.975958i	$-0.430060\pi$
$173$	171600.	0.435916	0.217958	+	0.975958i	$0.430060\pi$
$174$	128018.	0.320552
$175$	0	0
$176$	−22964.7	−0.0558830
$177$	−401000.	−0.962081
$178$	20524.8	0.0485544
$179$	595571.	1.38932	0.694658	−	0.719340i	$-0.255556\pi$
$179$	595571.	1.38932	0.694658	+	0.719340i	$0.255556\pi$
$180$	0	0
$181$	144518.	0.327889	0.163944	−	0.986470i	$-0.447578\pi$
$181$	144518.	0.327889	0.163944	+	0.986470i	$0.447578\pi$
$182$	−195488.	−0.437464
$183$	104384.	0.230413
$184$	274974.	0.598753
$185$	0	0
$186$	248480.	0.526635
$187$	−20352.7	−0.0425616
$188$	−123492.	−0.254826
$189$	170372.	0.346931
$190$	0	0
$191$	815918.	1.61831	0.809157	−	0.587592i	$-0.199924\pi$
$191$	815918.	1.61831	0.809157	+	0.587592i	$0.199924\pi$
$192$	36864.0	0.0721688
$193$	−665030.	−1.28513	−0.642566	−	0.766230i	$-0.722130\pi$
$193$	−665030.	−1.28513	−0.642566	+	0.766230i	$0.722130\pi$
$194$	−353741.	−0.674810
$195$	0	0
$196$	604984.	1.12487
$197$	332972.	0.611283	0.305642	−	0.952147i	$-0.401129\pi$
$197$	332972.	0.611283	0.305642	+	0.952147i	$0.401129\pi$
$198$	−29064.7	−0.0526870
$199$	−19565.3	−0.0350231	−0.0175115	−	0.999847i	$-0.505574\pi$
$199$	−19565.3	−0.0350231	−0.0175115	+	0.999847i	$0.505574\pi$
$200$	0	0
$201$	−35651.6	−0.0622427
$202$	−465884.	−0.803339
$203$	831073.	1.41546
$204$	32671.0	0.0549651
$205$	0	0
$206$	89476.2	0.146906
$207$	348014.	0.564509
$208$	−53534.2	−0.0857971
$209$	195601.	0.309746
$210$	0	0
$211$	−669474.	−1.03521	−0.517604	−	0.855620i	$-0.673176\pi$
$211$	−669474.	−1.03521	−0.517604	+	0.855620i	$0.673176\pi$
$212$	204246.	0.312114
$213$	417722.	0.630868
$214$	−522506.	−0.779932
$215$	0	0
$216$	46656.0	0.0680414
$217$	1.61309e6	2.32547
$218$	−205774.	−0.293258
$219$	−554321.	−0.781000
$220$	0	0
$221$	−47445.1	−0.0653448
$222$	−195681.	−0.266481
$223$	841915.	1.13372	0.566860	−	0.823814i	$-0.308158\pi$
$223$	841915.	1.13372	0.566860	+	0.823814i	$0.308158\pi$
$224$	239315.	0.318676
$225$	0	0
$226$	553034.	0.720246
$227$	−106296.	−0.136916	−0.0684580	−	0.997654i	$-0.521808\pi$
$227$	−106296.	−0.136916	−0.0684580	+	0.997654i	$0.521808\pi$
$228$	−313988.	−0.400015
$229$	1.52931e6	1.92711	0.963553	−	0.267518i	$-0.0862033\pi$
$229$	1.52931e6	1.92711	0.963553	+	0.267518i	$0.0862033\pi$
$230$	0	0
$231$	−188683.	−0.232650
$232$	227588.	0.277606
$233$	−362144.	−0.437011	−0.218505	−	0.975836i	$-0.570118\pi$
$233$	−362144.	−0.437011	−0.218505	+	0.975836i	$0.570118\pi$
$234$	−67754.2	−0.0808903
$235$	0	0
$236$	−712890.	−0.833187
$237$	−250900.	−0.290155
$238$	212095.	0.242710
$239$	38495.6	0.0435929	0.0217965	−	0.999762i	$-0.493061\pi$
$239$	38495.6	0.0435929	0.0217965	+	0.999762i	$0.493061\pi$
$240$	0	0
$241$	567433.	0.629320	0.314660	−	0.949204i	$-0.398109\pi$
$241$	567433.	0.629320	0.314660	+	0.949204i	$0.398109\pi$
$242$	−612015.	−0.671775
$243$	59049.0	0.0641500
$244$	185572.	0.199543
$245$	0	0
$246$	233424.	0.245928
$247$	455976.	0.475553
$248$	441743.	0.456080
$249$	−458469.	−0.468610
$250$	0	0
$251$	19067.0	0.0191028	0.00955141	−	0.999954i	$-0.496960\pi$
$251$	19067.0	0.0191028	0.00955141	+	0.999954i	$0.496960\pi$
$252$	302883.	0.300451
$253$	−385419.	−0.378558
$254$	−798649.	−0.776733
$255$	0	0
$256$	65536.0	0.0625000
$257$	−854054.	−0.806589	−0.403295	−	0.915070i	$-0.632135\pi$
$257$	−854054.	−0.806589	−0.403295	+	0.915070i	$0.632135\pi$
$258$	−791263.	−0.740068
$259$	−1.27033e6	−1.17670
$260$	0	0
$261$	288041.	0.261730
$262$	−1.25689e6	−1.13121
$263$	−755220.	−0.673262	−0.336631	−	0.941637i	$-0.609288\pi$
$263$	−755220.	−0.673262	−0.336631	+	0.941637i	$0.609288\pi$
$264$	−51670.6	−0.0456283
$265$	0	0
$266$	−2.03836e6	−1.76635
$267$	46180.7	0.0396445
$268$	−63380.6	−0.0539038
$269$	−256615.	−0.216222	−0.108111	−	0.994139i	$-0.534480\pi$
$269$	−256615.	−0.216222	−0.108111	+	0.994139i	$0.534480\pi$
$270$	0	0
$271$	598518.	0.495056	0.247528	−	0.968881i	$-0.420382\pi$
$271$	598518.	0.495056	0.247528	+	0.968881i	$0.420382\pi$
$272$	58081.8	0.0476012
$273$	−439849.	−0.357188
$274$	1.02736e6	0.826698
$275$	0	0
$276$	618692.	0.488879
$277$	−425045.	−0.332840	−0.166420	−	0.986055i	$-0.553221\pi$
$277$	−425045.	−0.332840	−0.166420	+	0.986055i	$0.553221\pi$
$278$	−43108.9	−0.0334545
$279$	559081.	0.429996
$280$	0	0
$281$	−949963.	−0.717696	−0.358848	−	0.933396i	$-0.616830\pi$
$281$	−949963.	−0.717696	−0.358848	+	0.933396i	$0.616830\pi$
$282$	−277856.	−0.208064
$283$	2.12422e6	1.57664	0.788321	−	0.615264i	$-0.210950\pi$
$283$	2.12422e6	1.57664	0.788321	+	0.615264i	$0.210950\pi$
$284$	742617.	0.546348
$285$	0	0
$286$	75036.5	0.0542447
$287$	1.51535e6	1.08595
$288$	82944.0	0.0589256
$289$	−1.36838e6	−0.963746
$290$	0	0
$291$	−795918.	−0.550980
$292$	−985459.	−0.676365
$293$	−1.17556e6	−0.799977	−0.399988	−	0.916520i	$-0.630986\pi$
$293$	−1.17556e6	−0.799977	−0.399988	+	0.916520i	$0.630986\pi$
$294$	1.36121e6	0.918455
$295$	0	0
$296$	−347878.	−0.230779
$297$	−65395.7	−0.0430187
$298$	16413.1	0.0107065
$299$	−898469.	−0.581199
$300$	0	0
$301$	−5.13675e6	−3.26792
$302$	429796.	0.271172
$303$	−1.04824e6	−0.655924
$304$	−558201.	−0.346423
$305$	0	0
$306$	73509.8	0.0448788
$307$	588520.	0.356382	0.178191	−	0.983996i	$-0.442976\pi$
$307$	588520.	0.356382	0.178191	+	0.983996i	$0.442976\pi$
$308$	−335437.	−0.201481
$309$	201321.	0.119948
$310$	0	0
$311$	1.61863e6	0.948958	0.474479	−	0.880267i	$-0.342637\pi$
$311$	1.61863e6	0.948958	0.474479	+	0.880267i	$0.342637\pi$
$312$	−120452.	−0.0700531
$313$	−1.58901e6	−0.916783	−0.458392	−	0.888750i	$-0.651574\pi$
$313$	−1.58901e6	−0.916783	−0.458392	+	0.888750i	$0.651574\pi$
$314$	−1.38329e6	−0.791754
$315$	0	0
$316$	−446045.	−0.251281
$317$	−2.35137e6	−1.31423	−0.657117	−	0.753789i	$-0.728224\pi$
$317$	−2.35137e6	−1.31423	−0.657117	+	0.753789i	$0.728224\pi$
$318$	459553.	0.254840
$319$	−319000.	−0.175515
$320$	0	0
$321$	−1.17564e6	−0.636811
$322$	4.01644e6	2.15875
$323$	−494710.	−0.263842
$324$	104976.	0.0555556
$325$	0	0
$326$	235105.	0.122523
$327$	−462992.	−0.239444
$328$	414976.	0.212980
$329$	−1.80380e6	−0.918752
$330$	0	0
$331$	480922.	0.241271	0.120635	−	0.992697i	$-0.461507\pi$
$331$	480922.	0.241271	0.120635	+	0.992697i	$0.461507\pi$
$332$	−815056.	−0.405828
$333$	−440283.	−0.217581
$334$	618616.	0.303427
$335$	0	0
$336$	538459.	0.260198
$337$	−981449.	−0.470753	−0.235376	−	0.971904i	$-0.575632\pi$
$337$	−981449.	−0.470753	−0.235376	+	0.971904i	$0.575632\pi$
$338$	−1.31025e6	−0.623825
$339$	1.24433e6	0.588079
$340$	0	0
$341$	−619172.	−0.288353
$342$	−706473.	−0.326611
$343$	4.90887e6	2.25292
$344$	−1.40669e6	−0.640918
$345$	0	0
$346$	686400.	0.308239
$347$	1.91647e6	0.854432	0.427216	−	0.904150i	$-0.359494\pi$
$347$	1.91647e6	0.854432	0.427216	+	0.904150i	$0.359494\pi$
$348$	512073.	0.226665
$349$	1.22795e6	0.539658	0.269829	−	0.962908i	$-0.413033\pi$
$349$	1.22795e6	0.539658	0.269829	+	0.962908i	$0.413033\pi$
$350$	0	0
$351$	−152447.	−0.0660467
$352$	−91858.9	−0.0395152
$353$	−547823.	−0.233993	−0.116997	−	0.993132i	$-0.537327\pi$
$353$	−547823.	−0.233993	−0.116997	+	0.993132i	$0.537327\pi$
$354$	−1.60400e6	−0.680294
$355$	0	0
$356$	82099.1	0.0343331
$357$	477213.	0.198172
$358$	2.38228e6	0.982395
$359$	1.02314e6	0.418984	0.209492	−	0.977810i	$-0.432819\pi$
$359$	1.02314e6	0.418984	0.209492	+	0.977810i	$0.432819\pi$
$360$	0	0
$361$	2.27836e6	0.920140
$362$	578073.	0.231852
$363$	−1.37703e6	−0.548502
$364$	−781954.	−0.309334
$365$	0	0
$366$	417536.	0.162926
$367$	−3.05305e6	−1.18323	−0.591614	−	0.806221i	$-0.701509\pi$
$367$	−3.05305e6	−1.18323	−0.591614	+	0.806221i	$0.701509\pi$
$368$	1.09990e6	0.423382
$369$	525204.	0.200799
$370$	0	0
$371$	2.98334e6	1.12530
$372$	993922.	0.372387
$373$	3.69411e6	1.37479	0.687397	−	0.726282i	$-0.258753\pi$
$373$	3.69411e6	1.37479	0.687397	+	0.726282i	$0.258753\pi$
$374$	−81410.7	−0.0300956
$375$	0	0
$376$	−493967.	−0.180189
$377$	−743636.	−0.269468
$378$	681487.	0.245317
$379$	4.48524e6	1.60394	0.801969	−	0.597365i	$-0.203786\pi$
$379$	4.48524e6	1.60394	0.801969	+	0.597365i	$0.203786\pi$
$380$	0	0
$381$	−1.79696e6	−0.634200
$382$	3.26367e6	1.14432
$383$	4.53062e6	1.57819	0.789097	−	0.614269i	$-0.210549\pi$
$383$	4.53062e6	1.57819	0.789097	+	0.614269i	$0.210549\pi$
$384$	147456.	0.0510310
$385$	0	0
$386$	−2.66012e6	−0.908726
$387$	−1.78034e6	−0.604263
$388$	−1.41496e6	−0.477162
$389$	2.73557e6	0.916588	0.458294	−	0.888801i	$-0.348461\pi$
$389$	2.73557e6	0.916588	0.458294	+	0.888801i	$0.348461\pi$
$390$	0	0
$391$	974792.	0.322456
$392$	2.41993e6	0.795405
$393$	−2.82801e6	−0.923633
$394$	1.33189e6	0.432242
$395$	0	0
$396$	−116259.	−0.0372553
$397$	1.54948e6	0.493413	0.246707	−	0.969090i	$-0.420652\pi$
$397$	1.54948e6	0.493413	0.246707	+	0.969090i	$0.420652\pi$
$398$	−78261.3	−0.0247651
$399$	−4.58630e6	−1.44222
$400$	0	0
$401$	4.46136e6	1.38550	0.692749	−	0.721179i	$-0.256399\pi$
$401$	4.46136e6	1.38550	0.692749	+	0.721179i	$0.256399\pi$
$402$	−142606.	−0.0440123
$403$	−1.44338e6	−0.442709
$404$	−1.86353e6	−0.568047
$405$	0	0
$406$	3.32429e6	1.00088
$407$	487605.	0.145909
$408$	130684.	0.0388662
$409$	−2.36815e6	−0.700003	−0.350002	−	0.936749i	$-0.613819\pi$
$409$	−2.36815e6	−0.700003	−0.350002	+	0.936749i	$0.613819\pi$
$410$	0	0
$411$	2.31156e6	0.674996
$412$	357905.	0.103878
$413$	−1.04129e7	−3.00398
$414$	1.39206e6	0.399168
$415$	0	0
$416$	−214137.	−0.0606677
$417$	−96995.0	−0.0273155
$418$	782405.	0.219024
$419$	5.76363e6	1.60384	0.801920	−	0.597432i	$-0.203812\pi$
$419$	5.76363e6	1.60384	0.801920	+	0.597432i	$0.203812\pi$
$420$	0	0
$421$	3.26572e6	0.897995	0.448998	−	0.893533i	$-0.351781\pi$
$421$	3.26572e6	0.897995	0.448998	+	0.893533i	$0.351781\pi$
$422$	−2.67789e6	−0.732002
$423$	−625177.	−0.169884
$424$	816983.	0.220698
$425$	0	0
$426$	1.67089e6	0.446091
$427$	2.71057e6	0.719435
$428$	−2.09002e6	−0.551495
$429$	168832.	0.0442906
$430$	0	0
$431$	−5.62110e6	−1.45756	−0.728782	−	0.684745i	$-0.759913\pi$
$431$	−5.62110e6	−1.45756	−0.728782	+	0.684745i	$0.759913\pi$
$432$	186624.	0.0481125
$433$	−3.18656e6	−0.816774	−0.408387	−	0.912809i	$-0.633909\pi$
$433$	−3.18656e6	−0.816774	−0.408387	+	0.912809i	$0.633909\pi$
$434$	6.45237e6	1.64435
$435$	0	0
$436$	−823097.	−0.207365
$437$	−9.36833e6	−2.34671
$438$	−2.21728e6	−0.552250
$439$	−7.41102e6	−1.83534	−0.917671	−	0.397342i	$-0.869933\pi$
$439$	−7.41102e6	−1.83534	−0.917671	+	0.397342i	$0.869933\pi$
$440$	0	0
$441$	3.06273e6	0.749915
$442$	−189780.	−0.0462057
$443$	2.62862e6	0.636382	0.318191	−	0.948027i	$-0.396925\pi$
$443$	2.62862e6	0.636382	0.318191	+	0.948027i	$0.396925\pi$
$444$	−782724.	−0.188431
$445$	0	0
$446$	3.36766e6	0.801662
$447$	36929.4	0.00874185
$448$	957260.	0.225338
$449$	71764.4	0.0167994	0.00839969	−	0.999965i	$-0.497326\pi$
$449$	71764.4	0.0167994	0.00839969	+	0.999965i	$0.497326\pi$
$450$	0	0
$451$	−581653.	−0.134655
$452$	2.21214e6	0.509291
$453$	967040.	0.221411
$454$	−425186.	−0.0968142
$455$	0	0
$456$	−1.25595e6	−0.282853
$457$	−3.51471e6	−0.787225	−0.393613	−	0.919276i	$-0.628775\pi$
$457$	−3.51471e6	−0.787225	−0.393613	+	0.919276i	$0.628775\pi$
$458$	6.11722e6	1.36267
$459$	165397.	0.0366434
$460$	0	0
$461$	3.35569e6	0.735410	0.367705	−	0.929943i	$-0.380144\pi$
$461$	3.35569e6	0.735410	0.367705	+	0.929943i	$0.380144\pi$
$462$	−754734.	−0.164509
$463$	−3.47269e6	−0.752860	−0.376430	−	0.926445i	$-0.622848\pi$
$463$	−3.47269e6	−0.752860	−0.376430	+	0.926445i	$0.622848\pi$
$464$	910352.	0.196297
$465$	0	0
$466$	−1.44858e6	−0.309013
$467$	−2.35495e6	−0.499678	−0.249839	−	0.968287i	$-0.580378\pi$
$467$	−2.35495e6	−0.499678	−0.249839	+	0.968287i	$0.580378\pi$
$468$	−271017.	−0.0571981
$469$	−925776.	−0.194345
$470$	0	0
$471$	−3.11241e6	−0.646465
$472$	−2.85156e6	−0.589152
$473$	1.97170e6	0.405216
$474$	−1.00360e6	−0.205170
$475$	0	0
$476$	848379.	0.171622
$477$	1.03399e6	0.208076
$478$	153982.	0.0308249
$479$	−187771.	−0.0373929	−0.0186964	−	0.999825i	$-0.505952\pi$
$479$	−187771.	−0.0373929	−0.0186964	+	0.999825i	$0.505952\pi$
$480$	0	0
$481$	1.13668e6	0.224014
$482$	2.26973e6	0.444997
$483$	9.03700e6	1.76261
$484$	−2.44806e6	−0.475017
$485$	0	0
$486$	236196.	0.0453609
$487$	2.01505e6	0.385002	0.192501	−	0.981297i	$-0.438340\pi$
$487$	2.01505e6	0.385002	0.192501	+	0.981297i	$0.438340\pi$
$488$	742287.	0.141098
$489$	528987.	0.100040
$490$	0	0
$491$	5.80814e6	1.08726	0.543630	−	0.839325i	$-0.317049\pi$
$491$	5.80814e6	1.08726	0.543630	+	0.839325i	$0.317049\pi$
$492$	933696.	0.173897
$493$	806807.	0.149504
$494$	1.82390e6	0.336267
$495$	0	0
$496$	1.76697e6	0.322497
$497$	1.08471e7	1.96981
$498$	−1.83388e6	−0.331357
$499$	580611.	0.104384	0.0521920	−	0.998637i	$-0.483379\pi$
$499$	580611.	0.104384	0.0521920	+	0.998637i	$0.483379\pi$
$500$	0	0
$501$	1.39188e6	0.247747
$502$	76267.9	0.0135077
$503$	557751.	0.0982926	0.0491463	−	0.998792i	$-0.484350\pi$
$503$	557751.	0.0982926	0.0491463	+	0.998792i	$0.484350\pi$
$504$	1.21153e6	0.212451
$505$	0	0
$506$	−1.54168e6	−0.267681
$507$	−2.94806e6	−0.509351
$508$	−3.19460e6	−0.549233
$509$	−2.63318e6	−0.450491	−0.225246	−	0.974302i	$-0.572318\pi$
$509$	−2.63318e6	−0.450491	−0.225246	+	0.974302i	$0.572318\pi$
$510$	0	0
$511$	−1.43942e7	−2.43858
$512$	262144.	0.0441942
$513$	−1.58956e6	−0.266676
$514$	−3.41621e6	−0.570345
$515$	0	0
$516$	−3.16505e6	−0.523307
$517$	692372.	0.113923
$518$	−5.08132e6	−0.832055
$519$	1.54440e6	0.251676
$520$	0	0
$521$	−6.78129e6	−1.09451	−0.547253	−	0.836967i	$-0.684326\pi$
$521$	−6.78129e6	−1.09451	−0.547253	+	0.836967i	$0.684326\pi$
$522$	1.15216e6	0.185071
$523$	−1.55863e6	−0.249166	−0.124583	−	0.992209i	$-0.539759\pi$
$523$	−1.55863e6	−0.249166	−0.124583	+	0.992209i	$0.539759\pi$
$524$	−5.02757e6	−0.799889
$525$	0	0
$526$	−3.02088e6	−0.476068
$527$	1.56599e6	0.245620
$528$	−206683.	−0.0322641
$529$	1.20233e7	1.86804
$530$	0	0
$531$	−3.60900e6	−0.555458
$532$	−8.15343e6	−1.24900
$533$	−1.35592e6	−0.206736
$534$	184723.	0.0280329
$535$	0	0
$536$	−253522.	−0.0381157
$537$	5.36014e6	0.802122
$538$	−1.02646e6	−0.152892
$539$	−3.39192e6	−0.502890
$540$	0	0
$541$	7.29465e6	1.07155	0.535774	−	0.844362i	$-0.320020\pi$
$541$	7.29465e6	1.07155	0.535774	+	0.844362i	$0.320020\pi$
$542$	2.39407e6	0.350057
$543$	1.30066e6	0.189307
$544$	232327.	0.0336591
$545$	0	0
$546$	−1.75940e6	−0.252570
$547$	5.15944e6	0.737283	0.368642	−	0.929572i	$-0.379823\pi$
$547$	5.15944e6	0.737283	0.368642	+	0.929572i	$0.379823\pi$
$548$	4.10945e6	0.584564
$549$	939456.	0.133029
$550$	0	0
$551$	−7.75389e6	−1.08803
$552$	2.47477e6	0.345690
$553$	−6.51520e6	−0.905973
$554$	−1.70018e6	−0.235353
$555$	0	0
$556$	−172436.	−0.0236559
$557$	2.18378e6	0.298244	0.149122	−	0.988819i	$-0.452355\pi$
$557$	2.18378e6	0.298244	0.149122	+	0.988819i	$0.452355\pi$
$558$	2.23632e6	0.304053
$559$	4.59631e6	0.622129
$560$	0	0
$561$	−183174.	−0.0245729
$562$	−3.79985e6	−0.507488
$563$	1.13003e7	1.50252	0.751259	−	0.660007i	$-0.229447\pi$
$563$	1.13003e7	1.50252	0.751259	+	0.660007i	$0.229447\pi$
$564$	−1.11143e6	−0.147124
$565$	0	0
$566$	8.49688e6	1.11485
$567$	1.53334e6	0.200301
$568$	2.97047e6	0.386326
$569$	−1.01686e7	−1.31669	−0.658343	−	0.752718i	$-0.728742\pi$
$569$	−1.01686e7	−1.31669	−0.658343	+	0.752718i	$0.728742\pi$
$570$	0	0
$571$	−284871.	−0.0365644	−0.0182822	−	0.999833i	$-0.505820\pi$
$571$	−284871.	−0.0365644	−0.0182822	+	0.999833i	$0.505820\pi$
$572$	300146.	0.0383568
$573$	7.34326e6	0.934334
$574$	6.06140e6	0.767879
$575$	0	0
$576$	331776.	0.0416667
$577$	−9.14119e6	−1.14304	−0.571522	−	0.820587i	$-0.693647\pi$
$577$	−9.14119e6	−1.14304	−0.571522	+	0.820587i	$0.693647\pi$
$578$	−5.47353e6	−0.681471
$579$	−5.98527e6	−0.741971
$580$	0	0
$581$	−1.19052e7	−1.46318
$582$	−3.18367e6	−0.389602
$583$	−1.14513e6	−0.139535
$584$	−3.94184e6	−0.478263
$585$	0	0
$586$	−4.70226e6	−0.565669
$587$	1.85347e6	0.222019	0.111010	−	0.993819i	$-0.464592\pi$
$587$	1.85347e6	0.222019	0.111010	+	0.993819i	$0.464592\pi$
$588$	5.44485e6	0.649446
$589$	−1.50501e7	−1.78752
$590$	0	0
$591$	2.99675e6	0.352924
$592$	−1.39151e6	−0.163186
$593$	1.30967e7	1.52941	0.764705	−	0.644380i	$-0.222884\pi$
$593$	1.30967e7	1.52941	0.764705	+	0.644380i	$0.222884\pi$
$594$	−261583.	−0.0304188
$595$	0	0
$596$	65652.3	0.00757067
$597$	−176088.	−0.0202206
$598$	−3.59388e6	−0.410970
$599$	−7.13874e6	−0.812933	−0.406466	−	0.913666i	$-0.633239\pi$
$599$	−7.13874e6	−0.812933	−0.406466	+	0.913666i	$0.633239\pi$
$600$	0	0
$601$	−7.27856e6	−0.821976	−0.410988	−	0.911641i	$-0.634816\pi$
$601$	−7.27856e6	−0.821976	−0.410988	+	0.911641i	$0.634816\pi$
$602$	−2.05470e7	−2.31077
$603$	−320864.	−0.0359359
$604$	1.71918e6	0.191748
$605$	0	0
$606$	−4.19295e6	−0.463808
$607$	466696.	0.0514117	0.0257058	−	0.999670i	$-0.491817\pi$
$607$	466696.	0.0514117	0.0257058	+	0.999670i	$0.491817\pi$
$608$	−2.23280e6	−0.244958
$609$	7.47965e6	0.817219
$610$	0	0
$611$	1.61402e6	0.174907
$612$	294039.	0.0317341
$613$	−9.21342e6	−0.990307	−0.495153	−	0.868806i	$-0.664888\pi$
$613$	−9.21342e6	−0.990307	−0.495153	+	0.868806i	$0.664888\pi$
$614$	2.35408e6	0.252000
$615$	0	0
$616$	−1.34175e6	−0.142469
$617$	8.53638e6	0.902736	0.451368	−	0.892338i	$-0.350936\pi$
$617$	8.53638e6	0.902736	0.451368	+	0.892338i	$0.350936\pi$
$618$	805286.	0.0848162
$619$	7.72476e6	0.810323	0.405162	−	0.914245i	$-0.367215\pi$
$619$	7.72476e6	0.810323	0.405162	+	0.914245i	$0.367215\pi$
$620$	0	0
$621$	3.13213e6	0.325920
$622$	6.47452e6	0.671014
$623$	1.19919e6	0.123785
$624$	−481808.	−0.0495350
$625$	0	0
$626$	−6.35605e6	−0.648264
$627$	1.76041e6	0.178832
$628$	−5.53318e6	−0.559855
$629$	−1.23324e6	−0.124285
$630$	0	0
$631$	1.50119e7	1.50094	0.750470	−	0.660904i	$-0.229827\pi$
$631$	1.50119e7	1.50094	0.750470	+	0.660904i	$0.229827\pi$
$632$	−1.78418e6	−0.177683
$633$	−6.02526e6	−0.597677
$634$	−9.40547e6	−0.929303
$635$	0	0
$636$	1.83821e6	0.180199
$637$	−7.90706e6	−0.772087
$638$	−1.27600e6	−0.124108
$639$	3.75950e6	0.364232
$640$	0	0
$641$	−8.60718e6	−0.827401	−0.413700	−	0.910413i	$-0.635764\pi$
$641$	−8.60718e6	−0.827401	−0.413700	+	0.910413i	$0.635764\pi$
$642$	−4.70255e6	−0.450294
$643$	−1.97196e7	−1.88092	−0.940460	−	0.339904i	$-0.889605\pi$
$643$	−1.97196e7	−1.88092	−0.940460	+	0.339904i	$0.889605\pi$
$644$	1.60658e7	1.52647
$645$	0	0
$646$	−1.97884e6	−0.186565
$647$	−541311.	−0.0508378	−0.0254189	−	0.999677i	$-0.508092\pi$
$647$	−541311.	−0.0508378	−0.0254189	+	0.999677i	$0.508092\pi$
$648$	419904.	0.0392837
$649$	3.99690e6	0.372488
$650$	0	0
$651$	1.45178e7	1.34261
$652$	940421.	0.0866370
$653$	−9.28231e6	−0.851870	−0.425935	−	0.904754i	$-0.640055\pi$
$653$	−9.28231e6	−0.851870	−0.425935	+	0.904754i	$0.640055\pi$
$654$	−1.85197e6	−0.169312
$655$	0	0
$656$	1.65990e6	0.150599
$657$	−4.98889e6	−0.450910
$658$	−7.21519e6	−0.649655
$659$	1.79964e7	1.61425	0.807127	−	0.590378i	$-0.201021\pi$
$659$	1.79964e7	1.61425	0.807127	+	0.590378i	$0.201021\pi$
$660$	0	0
$661$	7.78607e6	0.693130	0.346565	−	0.938026i	$-0.387348\pi$
$661$	7.78607e6	0.693130	0.346565	+	0.938026i	$0.387348\pi$
$662$	1.92369e6	0.170604
$663$	−427006.	−0.0377268
$664$	−3.26022e6	−0.286964
$665$	0	0
$666$	−1.76113e6	−0.153853
$667$	1.52785e7	1.32974
$668$	2.47446e6	0.214556
$669$	7.57724e6	0.654554
$670$	0	0
$671$	−1.04043e6	−0.0892086
$672$	2.15383e6	0.183988
$673$	1.35540e7	1.15353	0.576766	−	0.816910i	$-0.304315\pi$
$673$	1.35540e7	1.15353	0.576766	+	0.816910i	$0.304315\pi$
$674$	−3.92579e6	−0.332872
$675$	0	0
$676$	−5.24100e6	−0.441111
$677$	1.46008e7	1.22435	0.612173	−	0.790724i	$-0.290296\pi$
$677$	1.46008e7	1.22435	0.612173	+	0.790724i	$0.290296\pi$
$678$	4.97731e6	0.415834
$679$	−2.06679e7	−1.72037
$680$	0	0
$681$	−956668.	−0.0790485
$682$	−2.47669e6	−0.203897
$683$	−9.71085e6	−0.796536	−0.398268	−	0.917269i	$-0.630389\pi$
$683$	−9.71085e6	−0.796536	−0.398268	+	0.917269i	$0.630389\pi$
$684$	−2.82589e6	−0.230949
$685$	0	0
$686$	1.96355e7	1.59306
$687$	1.37638e7	1.11262
$688$	−5.62676e6	−0.453197
$689$	−2.66947e6	−0.214228
$690$	0	0
$691$	1.30382e7	1.03878	0.519389	−	0.854538i	$-0.326159\pi$
$691$	1.30382e7	1.03878	0.519389	+	0.854538i	$0.326159\pi$
$692$	2.74560e6	0.217958
$693$	−1.69815e6	−0.134321
$694$	7.66587e6	0.604175
$695$	0	0
$696$	2.04829e6	0.160276
$697$	1.47110e6	0.114699
$698$	4.91182e6	0.381596
$699$	−3.25930e6	−0.252308
$700$	0	0
$701$	4.65952e6	0.358134	0.179067	−	0.983837i	$-0.442692\pi$
$701$	4.65952e6	0.358134	0.179067	+	0.983837i	$0.442692\pi$
$702$	−609788.	−0.0467021
$703$	1.18521e7	0.904500
$704$	−367436.	−0.0279415
$705$	0	0
$706$	−2.19129e6	−0.165458
$707$	−2.72199e7	−2.04804
$708$	−6.41601e6	−0.481041
$709$	1.94059e7	1.44983	0.724916	−	0.688837i	$-0.241878\pi$
$709$	1.94059e7	1.44983	0.724916	+	0.688837i	$0.241878\pi$
$710$	0	0
$711$	−2.25810e6	−0.167521
$712$	328396.	0.0242772
$713$	2.96553e7	2.18463
$714$	1.90885e6	0.140129
$715$	0	0
$716$	9.52914e6	0.694658
$717$	346460.	0.0251684
$718$	4.09254e6	0.296266
$719$	1.20100e7	0.866405	0.433202	−	0.901297i	$-0.357384\pi$
$719$	1.20100e7	0.866405	0.433202	+	0.901297i	$0.357384\pi$
$720$	0	0
$721$	5.22778e6	0.374524
$722$	9.11343e6	0.650637
$723$	5.10689e6	0.363338
$724$	2.31229e6	0.163944
$725$	0	0
$726$	−5.50814e6	−0.387850
$727$	−1.14993e7	−0.806927	−0.403463	−	0.914996i	$-0.632194\pi$
$727$	−1.14993e7	−0.806927	−0.403463	+	0.914996i	$0.632194\pi$
$728$	−3.12781e6	−0.218732
$729$	531441.	0.0370370
$730$	0	0
$731$	−4.98676e6	−0.345164
$732$	1.67014e6	0.115206
$733$	2.04473e7	1.40565	0.702823	−	0.711364i	$-0.251922\pi$
$733$	2.04473e7	1.40565	0.702823	+	0.711364i	$0.251922\pi$
$734$	−1.22122e7	−0.836668
$735$	0	0
$736$	4.39959e6	0.299376
$737$	355351.	0.0240984
$738$	2.10082e6	0.141986
$739$	220621.	0.0148606	0.00743030	−	0.999972i	$-0.497635\pi$
$739$	220621.	0.0148606	0.00743030	+	0.999972i	$0.497635\pi$
$740$	0	0
$741$	4.10378e6	0.274561
$742$	1.19334e7	0.795707
$743$	−2.51435e7	−1.67091	−0.835456	−	0.549558i	$-0.814796\pi$
$743$	−2.51435e7	−1.67091	−0.835456	+	0.549558i	$0.814796\pi$
$744$	3.97569e6	0.263318
$745$	0	0
$746$	1.47764e7	0.972126
$747$	−4.12622e6	−0.270552
$748$	−325643.	−0.0212808
$749$	−3.05282e7	−1.98837
$750$	0	0
$751$	−667547.	−0.0431899	−0.0215950	−	0.999767i	$-0.506874\pi$
$751$	−667547.	−0.0431899	−0.0215950	+	0.999767i	$0.506874\pi$
$752$	−1.97587e6	−0.127413
$753$	171603.	0.0110290
$754$	−2.97454e6	−0.190543
$755$	0	0
$756$	2.72595e6	0.173465
$757$	−1.62490e7	−1.03059	−0.515297	−	0.857012i	$-0.672318\pi$
$757$	−1.62490e7	−1.03059	−0.515297	+	0.857012i	$0.672318\pi$
$758$	1.79410e7	1.13416
$759$	−3.46877e6	−0.218560
$760$	0	0
$761$	−1.43662e7	−0.899253	−0.449626	−	0.893217i	$-0.648443\pi$
$761$	−1.43662e7	−0.899253	−0.449626	+	0.893217i	$0.648443\pi$
$762$	−7.18784e6	−0.448447
$763$	−1.20227e7	−0.747634
$764$	1.30547e7	0.809157
$765$	0	0
$766$	1.81225e7	1.11595
$767$	9.31737e6	0.571880
$768$	589824.	0.0360844
$769$	1.92705e7	1.17511	0.587553	−	0.809185i	$-0.300091\pi$
$769$	1.92705e7	1.17511	0.587553	+	0.809185i	$0.300091\pi$
$770$	0	0
$771$	−7.68648e6	−0.465684
$772$	−1.06405e7	−0.642566
$773$	−2.40693e7	−1.44882	−0.724410	−	0.689370i	$-0.757888\pi$
$773$	−2.40693e7	−1.44882	−0.724410	+	0.689370i	$0.757888\pi$
$774$	−7.12137e6	−0.427279
$775$	0	0
$776$	−5.65986e6	−0.337405
$777$	−1.14330e7	−0.679370
$778$	1.09423e7	0.648125
$779$	−1.41382e7	−0.834737
$780$	0	0
$781$	−4.16357e6	−0.244252
$782$	3.89917e6	0.228011
$783$	2.59237e6	0.151110
$784$	9.67974e6	0.562436
$785$	0	0
$786$	−1.13120e7	−0.653107
$787$	−3.76223e6	−0.216525	−0.108263	−	0.994122i	$-0.534529\pi$
$787$	−3.76223e6	−0.216525	−0.108263	+	0.994122i	$0.534529\pi$
$788$	5.32756e6	0.305642
$789$	−6.79698e6	−0.388708
$790$	0	0
$791$	3.23119e7	1.83620
$792$	−465036.	−0.0263435
$793$	−2.42540e6	−0.136962
$794$	6.19794e6	0.348896
$795$	0	0
$796$	−313045.	−0.0175115
$797$	−9.10423e6	−0.507689	−0.253844	−	0.967245i	$-0.581695\pi$
$797$	−9.10423e6	−0.507689	−0.253844	+	0.967245i	$0.581695\pi$
$798$	−1.83452e7	−1.01980
$799$	−1.75113e6	−0.0970401
$800$	0	0
$801$	415627.	0.0228888
$802$	1.78454e7	0.979695
$803$	5.52510e6	0.302379
$804$	−570425.	−0.0311214
$805$	0	0
$806$	−5.77352e6	−0.313043
$807$	−2.30953e6	−0.124836
$808$	−7.45414e6	−0.401670
$809$	1.20232e7	0.645877	0.322938	−	0.946420i	$-0.395329\pi$
$809$	1.20232e7	0.645877	0.322938	+	0.946420i	$0.395329\pi$
$810$	0	0
$811$	−5.50728e6	−0.294026	−0.147013	−	0.989135i	$-0.546966\pi$
$811$	−5.50728e6	−0.294026	−0.147013	+	0.989135i	$0.546966\pi$
$812$	1.32972e7	0.707732
$813$	5.38666e6	0.285821
$814$	1.95042e6	0.103173
$815$	0	0
$816$	522736.	0.0274826
$817$	4.79258e7	2.51197
$818$	−9.47258e6	−0.494977
$819$	−3.95864e6	−0.206223
$820$	0	0
$821$	1.33971e7	0.693672	0.346836	−	0.937926i	$-0.387256\pi$
$821$	1.33971e7	0.693672	0.346836	+	0.937926i	$0.387256\pi$
$822$	9.24626e6	0.477295
$823$	1.41005e7	0.725664	0.362832	−	0.931855i	$-0.381810\pi$
$823$	1.41005e7	0.725664	0.362832	+	0.931855i	$0.381810\pi$
$824$	1.43162e6	0.0734530
$825$	0	0
$826$	−4.16516e7	−2.12413
$827$	−1.28958e7	−0.655669	−0.327834	−	0.944735i	$-0.606319\pi$
$827$	−1.28958e7	−0.655669	−0.327834	+	0.944735i	$0.606319\pi$
$828$	5.56823e6	0.282255
$829$	−372726.	−0.0188366	−0.00941831	−	0.999956i	$-0.502998\pi$
$829$	−372726.	−0.0188366	−0.00941831	+	0.999956i	$0.502998\pi$
$830$	0	0
$831$	−3.82541e6	−0.192165
$832$	−856547.	−0.0428986
$833$	8.57875e6	0.428362
$834$	−387980.	−0.0193150
$835$	0	0
$836$	3.12962e6	0.154873
$837$	5.03173e6	0.248258
$838$	2.30545e7	1.13409
$839$	1.57724e7	0.773560	0.386780	−	0.922172i	$-0.373587\pi$
$839$	1.57724e7	0.773560	0.386780	+	0.922172i	$0.373587\pi$
$840$	0	0
$841$	−7.86558e6	−0.383478
$842$	1.30629e7	0.634978
$843$	−8.54966e6	−0.414362
$844$	−1.07116e7	−0.517604
$845$	0	0
$846$	−2.50071e6	−0.120126
$847$	−3.57579e7	−1.71263
$848$	3.26793e6	0.156057
$849$	1.91180e7	0.910275
$850$	0	0
$851$	−2.33538e7	−1.10544
$852$	6.68355e6	0.315434
$853$	−3.39555e6	−0.159786	−0.0798928	−	0.996803i	$-0.525458\pi$
$853$	−3.39555e6	−0.159786	−0.0798928	+	0.996803i	$0.525458\pi$
$854$	1.08423e7	0.508718
$855$	0	0
$856$	−8.36009e6	−0.389966
$857$	4.62981e6	0.215333	0.107667	−	0.994187i	$-0.465662\pi$
$857$	4.62981e6	0.215333	0.107667	+	0.994187i	$0.465662\pi$
$858$	675329.	0.0313182
$859$	−1.57703e7	−0.729218	−0.364609	−	0.931161i	$-0.618797\pi$
$859$	−1.57703e7	−0.729218	−0.364609	+	0.931161i	$0.618797\pi$
$860$	0	0
$861$	1.36381e7	0.626971
$862$	−2.24844e7	−1.03065
$863$	−3.26217e7	−1.49101	−0.745503	−	0.666502i	$-0.767791\pi$
$863$	−3.26217e7	−1.49101	−0.745503	+	0.666502i	$0.767791\pi$
$864$	746496.	0.0340207
$865$	0	0
$866$	−1.27462e7	−0.577546
$867$	−1.23154e7	−0.556419
$868$	2.58095e7	1.16273
$869$	2.50080e6	0.112339
$870$	0	0
$871$	828376.	0.0369983
$872$	−3.29239e6	−0.146629
$873$	−7.16326e6	−0.318108
$874$	−3.74733e7	−1.65937
$875$	0	0
$876$	−8.86913e6	−0.390500
$877$	3.88139e6	0.170407	0.0852037	−	0.996364i	$-0.472846\pi$
$877$	3.88139e6	0.170407	0.0852037	+	0.996364i	$0.472846\pi$
$878$	−2.96441e7	−1.29778
$879$	−1.05801e7	−0.461867
$880$	0	0
$881$	−2.11909e7	−0.919834	−0.459917	−	0.887962i	$-0.652121\pi$
$881$	−2.11909e7	−0.919834	−0.459917	+	0.887962i	$0.652121\pi$
$882$	1.22509e7	0.530270
$883$	257162.	0.0110995	0.00554976	−	0.999985i	$-0.498233\pi$
$883$	257162.	0.0110995	0.00554976	+	0.999985i	$0.498233\pi$
$884$	−759122.	−0.0326724
$885$	0	0
$886$	1.05145e7	0.449990
$887$	−1.02364e7	−0.436856	−0.218428	−	0.975853i	$-0.570093\pi$
$887$	−1.02364e7	−0.436856	−0.218428	+	0.975853i	$0.570093\pi$
$888$	−3.13090e6	−0.133241
$889$	−4.66623e7	−1.98021
$890$	0	0
$891$	−588561.	−0.0248369
$892$	1.34706e7	0.566860
$893$	1.68294e7	0.706220
$894$	147718.	0.00618142
$895$	0	0
$896$	3.82904e6	0.159338
$897$	−8.08622e6	−0.335556
$898$	287058.	0.0118790
$899$	2.45448e7	1.01288
$900$	0	0
$901$	2.89623e6	0.118856
$902$	−2.32661e6	−0.0952155
$903$	−4.62307e7	−1.88674
$904$	8.84855e6	0.360123
$905$	0	0
$906$	3.86816e6	0.156561
$907$	1.30652e7	0.527350	0.263675	−	0.964612i	$-0.415065\pi$
$907$	1.30652e7	0.527350	0.263675	+	0.964612i	$0.415065\pi$
$908$	−1.70074e6	−0.0684580
$909$	−9.43414e6	−0.378698
$910$	0	0
$911$	−9.81608e6	−0.391870	−0.195935	−	0.980617i	$-0.562774\pi$
$911$	−9.81608e6	−0.391870	−0.195935	+	0.980617i	$0.562774\pi$
$912$	−5.02381e6	−0.200007
$913$	4.56971e6	0.181431
$914$	−1.40588e7	−0.556652
$915$	0	0
$916$	2.44689e7	0.963553
$917$	−7.34358e7	−2.88393
$918$	661588.	0.0259108
$919$	2.71414e6	0.106009	0.0530045	−	0.998594i	$-0.483120\pi$
$919$	2.71414e6	0.106009	0.0530045	+	0.998594i	$0.483120\pi$
$920$	0	0
$921$	5.29668e6	0.205757
$922$	1.34228e7	0.520013
$923$	−9.70591e6	−0.375000
$924$	−3.01893e6	−0.116325
$925$	0	0
$926$	−1.38908e7	−0.532352
$927$	1.81189e6	0.0692522
$928$	3.64141e6	0.138803
$929$	−4.31963e6	−0.164213	−0.0821064	−	0.996624i	$-0.526165\pi$
$929$	−4.31963e6	−0.164213	−0.0821064	+	0.996624i	$0.526165\pi$
$930$	0	0
$931$	−8.24469e7	−3.11745
$932$	−5.79431e6	−0.218505
$933$	1.45677e7	0.547881
$934$	−9.41981e6	−0.353326
$935$	0	0
$936$	−1.08407e6	−0.0404452
$937$	2.17380e7	0.808854	0.404427	−	0.914570i	$-0.367471\pi$
$937$	2.17380e7	0.808854	0.404427	+	0.914570i	$0.367471\pi$
$938$	−3.70311e6	−0.137423
$939$	−1.43011e7	−0.529305
$940$	0	0
$941$	−2.17794e7	−0.801811	−0.400905	−	0.916119i	$-0.631304\pi$
$941$	−2.17794e7	−0.801811	−0.400905	+	0.916119i	$0.631304\pi$
$942$	−1.24496e7	−0.457120
$943$	2.78583e7	1.02018
$944$	−1.14062e7	−0.416593
$945$	0	0
$946$	7.88678e6	0.286531
$947$	4.93262e7	1.78732	0.893660	−	0.448745i	$-0.148129\pi$
$947$	4.93262e7	1.78732	0.893660	+	0.448745i	$0.148129\pi$
$948$	−4.01440e6	−0.145077
$949$	1.28798e7	0.464242
$950$	0	0
$951$	−2.11623e7	−0.758773
$952$	3.39352e6	0.121355
$953$	3.37247e7	1.20286	0.601432	−	0.798924i	$-0.294597\pi$
$953$	3.37247e7	1.20286	0.601432	+	0.798924i	$0.294597\pi$
$954$	4.13598e6	0.147132
$955$	0	0
$956$	615929.	0.0217965
$957$	−2.87100e6	−0.101334
$958$	−751082.	−0.0264407
$959$	6.00252e7	2.10759
$960$	0	0
$961$	1.90117e7	0.664068
$962$	4.54671e6	0.158402
$963$	−1.05807e7	−0.367663
$964$	9.07892e6	0.314660
$965$	0	0
$966$	3.61480e7	1.24635
$967$	−2.77310e7	−0.953673	−0.476836	−	0.878992i	$-0.658217\pi$
$967$	−2.77310e7	−0.953673	−0.476836	+	0.878992i	$0.658217\pi$
$968$	−9.79225e6	−0.335888
$969$	−4.45239e6	−0.152329
$970$	0	0
$971$	−6.67841e6	−0.227314	−0.113657	−	0.993520i	$-0.536256\pi$
$971$	−6.67841e6	−0.227314	−0.113657	+	0.993520i	$0.536256\pi$
$972$	944784.	0.0320750
$973$	−2.51870e6	−0.0852893
$974$	8.06018e6	0.272237
$975$	0	0
$976$	2.96915e6	0.0997716
$977$	4.34946e7	1.45780	0.728902	−	0.684618i	$-0.240031\pi$
$977$	4.34946e7	1.45780	0.728902	+	0.684618i	$0.240031\pi$
$978$	2.11595e6	0.0707388
$979$	−460299.	−0.0153491
$980$	0	0
$981$	−4.16693e6	−0.138243
$982$	2.32326e7	0.768809
$983$	−1.24213e7	−0.410000	−0.205000	−	0.978762i	$-0.565719\pi$
$983$	−1.24213e7	−0.410000	−0.205000	+	0.978762i	$0.565719\pi$
$984$	3.73478e6	0.122964
$985$	0	0
$986$	3.22723e6	0.105715
$987$	−1.62342e7	−0.530441
$988$	7.29561e6	0.237777
$989$	−9.44344e7	−3.07001
$990$	0	0
$991$	5.93479e7	1.91965	0.959824	−	0.280604i	$-0.0905348\pi$
$991$	5.93479e7	1.91965	0.959824	+	0.280604i	$0.0905348\pi$
$992$	7.06789e6	0.228040
$993$	4.32830e6	0.139298
$994$	4.33885e7	1.39286
$995$	0	0
$996$	−7.33550e6	−0.234305
$997$	2.44456e7	0.778867	0.389434	−	0.921055i	$-0.372671\pi$
$997$	2.44456e7	0.778867	0.389434	+	0.921055i	$0.372671\pi$
$998$	2.32244e6	0.0738107
$999$	−3.96254e6	−0.125620

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.6.a.o.1.2		2
3.2	odd	2		450.6.a.bb.1.2		2
5.2	odd	4		30.6.c.b.19.4	yes	4
5.3	odd	4		30.6.c.b.19.2	✓	4
5.4	even	2		150.6.a.n.1.1		2
15.2	even	4		90.6.c.c.19.1		4
15.8	even	4		90.6.c.c.19.3		4
15.14	odd	2		450.6.a.bc.1.1		2
20.3	even	4		240.6.f.b.49.2		4
20.7	even	4		240.6.f.b.49.4		4
60.23	odd	4		720.6.f.i.289.2		4
60.47	odd	4		720.6.f.i.289.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.6.c.b.19.2	✓	4	5.3	odd	4
30.6.c.b.19.4	yes	4	5.2	odd	4
90.6.c.c.19.1		4	15.2	even	4
90.6.c.c.19.3		4	15.8	even	4
150.6.a.n.1.1		2	5.4	even	2
150.6.a.o.1.2		2	1.1	even	1	trivial
240.6.f.b.49.2		4	20.3	even	4
240.6.f.b.49.4		4	20.7	even	4
450.6.a.bb.1.2		2	3.2	odd	2
450.6.a.bc.1.1		2	15.14	odd	2
720.6.f.i.289.1		4	60.47	odd	4
720.6.f.i.289.2		4	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} +233.706 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} +233.706 q^{7} +64.0000 q^{8} +81.0000 q^{9} -89.7060 q^{11} +144.000 q^{12} -209.118 q^{13} +934.824 q^{14} +256.000 q^{16} +226.882 q^{17} +324.000 q^{18} -2180.47 q^{19} +2103.35 q^{21} -358.824 q^{22} +4296.47 q^{23} +576.000 q^{24} -836.472 q^{26} +729.000 q^{27} +3739.30 q^{28} +3556.06 q^{29} +6902.24 q^{31} +1024.00 q^{32} -807.354 q^{33} +907.528 q^{34} +1296.00 q^{36} -5435.59 q^{37} -8721.89 q^{38} -1882.06 q^{39} +6484.00 q^{41} +8413.41 q^{42} -21979.5 q^{43} -1435.30 q^{44} +17185.9 q^{46} -7718.24 q^{47} +2304.00 q^{48} +37811.5 q^{49} +2041.94 q^{51} -3345.89 q^{52} +12765.4 q^{53} +2916.00 q^{54} +14957.2 q^{56} -19624.2 q^{57} +14224.2 q^{58} -44555.6 q^{59} +11598.2 q^{61} +27608.9 q^{62} +18930.2 q^{63} +4096.00 q^{64} -3229.41 q^{66} -3961.29 q^{67} +3630.11 q^{68} +38668.2 q^{69} +46413.6 q^{71} +5184.00 q^{72} -61591.2 q^{73} -21742.3 q^{74} -34887.5 q^{76} -20964.8 q^{77} -7528.24 q^{78} -27877.8 q^{79} +6561.00 q^{81} +25936.0 q^{82} -50941.0 q^{83} +33653.7 q^{84} -87918.1 q^{86} +32004.6 q^{87} -5741.18 q^{88} +5131.19 q^{89} -48872.1 q^{91} +68743.5 q^{92} +62120.1 q^{93} -30872.9 q^{94} +9216.00 q^{96} -88435.3 q^{97} +151246. q^{98} -7266.18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 18 q^{3} + 32 q^{4} + 72 q^{6} + 114 q^{7} + 128 q^{8} + 162 q^{9}+O(q^{10}) \) 2 * q + 8 * q^2 + 18 * q^3 + 32 * q^4 + 72 * q^6 + 114 * q^7 + 128 * q^8 + 162 * q^9	\( 2 q + 8 q^{2} + 18 q^{3} + 32 q^{4} + 72 q^{6} + 114 q^{7} + 128 q^{8} + 162 q^{9} + 174 q^{11} + 288 q^{12} + 642 q^{13} + 456 q^{14} + 512 q^{16} + 1514 q^{17} + 648 q^{18} - 120 q^{19} + 1026 q^{21} + 696 q^{22} + 4352 q^{23} + 1152 q^{24} + 2568 q^{26} + 1458 q^{27} + 1824 q^{28} - 2430 q^{29} + 11684 q^{31} + 2048 q^{32} + 1566 q^{33} + 6056 q^{34} + 2592 q^{36} - 17586 q^{37} - 480 q^{38} + 5778 q^{39} + 24984 q^{41} + 4104 q^{42} - 24168 q^{43} + 2784 q^{44} + 17408 q^{46} - 13316 q^{47} + 4608 q^{48} + 35334 q^{49} + 13626 q^{51} + 10272 q^{52} - 13698 q^{53} + 5832 q^{54} + 7296 q^{56} - 1080 q^{57} - 9720 q^{58} - 23730 q^{59} + 57124 q^{61} + 46736 q^{62} + 9234 q^{63} + 8192 q^{64} + 6264 q^{66} - 38316 q^{67} + 24224 q^{68} + 39168 q^{69} - 11076 q^{71} + 10368 q^{72} - 88548 q^{73} - 70344 q^{74} - 1920 q^{76} - 52532 q^{77} + 23112 q^{78} + 14220 q^{79} + 13122 q^{81} + 99936 q^{82} + 50792 q^{83} + 16416 q^{84} - 96672 q^{86} - 21870 q^{87} + 11136 q^{88} - 60420 q^{89} - 150756 q^{91} + 69632 q^{92} + 105156 q^{93} - 53264 q^{94} + 18432 q^{96} - 171216 q^{97} + 141336 q^{98} + 14094 q^{99}+O(q^{100}) \) 2 * q + 8 * q^2 + 18 * q^3 + 32 * q^4 + 72 * q^6 + 114 * q^7 + 128 * q^8 + 162 * q^9 + 174 * q^11 + 288 * q^12 + 642 * q^13 + 456 * q^14 + 512 * q^16 + 1514 * q^17 + 648 * q^18 - 120 * q^19 + 1026 * q^21 + 696 * q^22 + 4352 * q^23 + 1152 * q^24 + 2568 * q^26 + 1458 * q^27 + 1824 * q^28 - 2430 * q^29 + 11684 * q^31 + 2048 * q^32 + 1566 * q^33 + 6056 * q^34 + 2592 * q^36 - 17586 * q^37 - 480 * q^38 + 5778 * q^39 + 24984 * q^41 + 4104 * q^42 - 24168 * q^43 + 2784 * q^44 + 17408 * q^46 - 13316 * q^47 + 4608 * q^48 + 35334 * q^49 + 13626 * q^51 + 10272 * q^52 - 13698 * q^53 + 5832 * q^54 + 7296 * q^56 - 1080 * q^57 - 9720 * q^58 - 23730 * q^59 + 57124 * q^61 + 46736 * q^62 + 9234 * q^63 + 8192 * q^64 + 6264 * q^66 - 38316 * q^67 + 24224 * q^68 + 39168 * q^69 - 11076 * q^71 + 10368 * q^72 - 88548 * q^73 - 70344 * q^74 - 1920 * q^76 - 52532 * q^77 + 23112 * q^78 + 14220 * q^79 + 13122 * q^81 + 99936 * q^82 + 50792 * q^83 + 16416 * q^84 - 96672 * q^86 - 21870 * q^87 + 11136 * q^88 - 60420 * q^89 - 150756 * q^91 + 69632 * q^92 + 105156 * q^93 - 53264 * q^94 + 18432 * q^96 - 171216 * q^97 + 141336 * q^98 + 14094 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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