LMFDB - Embedded newform 825.6.a.d.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{313}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 78 $ x^2 - x - 78
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-8.34590$ 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)

$f(q)$	$=$	$q+8.34590 q^{2} +9.00000 q^{3} +37.6541 q^{4} +75.1131 q^{6} +26.6918 q^{7} +47.1885 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+8.34590 q^{2} +9.00000 q^{3} +37.6541 q^{4} +75.1131 q^{6} +26.6918 q^{7} +47.1885 q^{8} +81.0000 q^{9} +121.000 q^{11} +338.887 q^{12} -904.666 q^{13} +222.767 q^{14} -811.100 q^{16} +495.384 q^{17} +676.018 q^{18} -1501.38 q^{19} +240.226 q^{21} +1009.85 q^{22} -2393.01 q^{23} +424.697 q^{24} -7550.25 q^{26} +729.000 q^{27} +1005.06 q^{28} -5132.34 q^{29} -410.309 q^{31} -8279.40 q^{32} +1089.00 q^{33} +4134.42 q^{34} +3049.98 q^{36} -5828.69 q^{37} -12530.4 q^{38} -8141.99 q^{39} +18561.8 q^{41} +2004.91 q^{42} +788.920 q^{43} +4556.15 q^{44} -19971.8 q^{46} -3979.11 q^{47} -7299.90 q^{48} -16094.5 q^{49} +4458.45 q^{51} -34064.4 q^{52} -15130.5 q^{53} +6084.16 q^{54} +1259.55 q^{56} -13512.5 q^{57} -42834.0 q^{58} +41623.5 q^{59} -29989.8 q^{61} -3424.40 q^{62} +2162.04 q^{63} -43143.8 q^{64} +9088.69 q^{66} -25890.3 q^{67} +18653.2 q^{68} -21537.1 q^{69} +58163.0 q^{71} +3822.27 q^{72} -23875.3 q^{73} -48645.7 q^{74} -56533.2 q^{76} +3229.71 q^{77} -67952.3 q^{78} +27402.8 q^{79} +6561.00 q^{81} +154915. q^{82} -24056.3 q^{83} +9045.50 q^{84} +6584.25 q^{86} -46191.1 q^{87} +5709.81 q^{88} -55820.0 q^{89} -24147.2 q^{91} -90106.5 q^{92} -3692.78 q^{93} -33209.3 q^{94} -74514.6 q^{96} +101188. q^{97} -134324. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 18 q^{3} + 93 q^{4} - 9 q^{6} + 18 q^{7} - 171 q^{8} + 162 q^{9}+O(q^{10}) $ 2 * q - q^2 + 18 * q^3 + 93 * q^4 - 9 * q^6 + 18 * q^7 - 171 * q^8 + 162 * q^9	\( 2 q - q^{2} + 18 q^{3} + 93 q^{4} - 9 q^{6} + 18 q^{7} - 171 q^{8} + 162 q^{9} + 242 q^{11} + 837 q^{12} + 66 q^{13} + 304 q^{14} - 543 q^{16} + 920 q^{17} - 81 q^{18} - 2932 q^{19} + 162 q^{21} - 121 q^{22} - 5246 q^{23} - 1539 q^{24} - 16622 q^{26} + 1458 q^{27} + 524 q^{28} - 12600 q^{29} + 9936 q^{31} - 3803 q^{32} + 2178 q^{33} + 166 q^{34} + 7533 q^{36} - 5996 q^{37} + 840 q^{38} + 594 q^{39} + 24244 q^{41} + 2736 q^{42} - 20360 q^{43} + 11253 q^{44} + 6692 q^{46} + 5806 q^{47} - 4887 q^{48} - 32826 q^{49} + 8280 q^{51} + 19658 q^{52} - 40770 q^{53} - 729 q^{54} + 3156 q^{56} - 26388 q^{57} + 26958 q^{58} + 18212 q^{59} - 11398 q^{61} - 100120 q^{62} + 1458 q^{63} - 93559 q^{64} - 1089 q^{66} - 65368 q^{67} + 42154 q^{68} - 47214 q^{69} + 61446 q^{71} - 13851 q^{72} - 53412 q^{73} - 47082 q^{74} - 135712 q^{76} + 2178 q^{77} - 149598 q^{78} + 17122 q^{79} + 13122 q^{81} + 101810 q^{82} + 14304 q^{83} + 4716 q^{84} + 204240 q^{86} - 113400 q^{87} - 20691 q^{88} - 58140 q^{89} - 32584 q^{91} - 248008 q^{92} + 89424 q^{93} - 124660 q^{94} - 34227 q^{96} + 183056 q^{97} + 22047 q^{98} + 19602 q^{99}+O(q^{100}) \) 2 * q - q^2 + 18 * q^3 + 93 * q^4 - 9 * q^6 + 18 * q^7 - 171 * q^8 + 162 * q^9 + 242 * q^11 + 837 * q^12 + 66 * q^13 + 304 * q^14 - 543 * q^16 + 920 * q^17 - 81 * q^18 - 2932 * q^19 + 162 * q^21 - 121 * q^22 - 5246 * q^23 - 1539 * q^24 - 16622 * q^26 + 1458 * q^27 + 524 * q^28 - 12600 * q^29 + 9936 * q^31 - 3803 * q^32 + 2178 * q^33 + 166 * q^34 + 7533 * q^36 - 5996 * q^37 + 840 * q^38 + 594 * q^39 + 24244 * q^41 + 2736 * q^42 - 20360 * q^43 + 11253 * q^44 + 6692 * q^46 + 5806 * q^47 - 4887 * q^48 - 32826 * q^49 + 8280 * q^51 + 19658 * q^52 - 40770 * q^53 - 729 * q^54 + 3156 * q^56 - 26388 * q^57 + 26958 * q^58 + 18212 * q^59 - 11398 * q^61 - 100120 * q^62 + 1458 * q^63 - 93559 * q^64 - 1089 * q^66 - 65368 * q^67 + 42154 * q^68 - 47214 * q^69 + 61446 * q^71 - 13851 * q^72 - 53412 * q^73 - 47082 * q^74 - 135712 * q^76 + 2178 * q^77 - 149598 * q^78 + 17122 * q^79 + 13122 * q^81 + 101810 * q^82 + 14304 * q^83 + 4716 * q^84 + 204240 * q^86 - 113400 * q^87 - 20691 * q^88 - 58140 * q^89 - 32584 * q^91 - 248008 * q^92 + 89424 * q^93 - 124660 * q^94 - 34227 * q^96 + 183056 * q^97 + 22047 * q^98 + 19602 * 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 $

$2$	8.34590	1.47536	0.737681	−	0.675150i	$-0.235921\pi$
$2$	8.34590	1.47536	0.737681	+	0.675150i	$0.235921\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	37.6541	1.17669
$5$	0	0
$5$	0	0
$6$	75.1131	0.851800
$7$	26.6918	0.205889	0.102944	−	0.994687i	$-0.467174\pi$
$7$	26.6918	0.205889	0.102944	+	0.994687i	$0.467174\pi$
$8$	47.1885	0.260682
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	338.887	0.679363
$13$	−904.666	−1.48467	−0.742335	−	0.670029i	$-0.766282\pi$
$13$	−904.666	−1.48467	−0.742335	+	0.670029i	$0.766282\pi$
$14$	222.767	0.303760
$15$	0	0
$16$	−811.100	−0.792090
$17$	495.384	0.415738	0.207869	−	0.978157i	$-0.433347\pi$
$17$	495.384	0.415738	0.207869	+	0.978157i	$0.433347\pi$
$18$	676.018	0.491787
$19$	−1501.38	−0.954130	−0.477065	−	0.878868i	$-0.658299\pi$
$19$	−1501.38	−0.954130	−0.477065	+	0.878868i	$0.658299\pi$
$20$	0	0
$21$	240.226	0.118870
$22$	1009.85	0.444838
$23$	−2393.01	−0.943245	−0.471622	−	0.881801i	$-0.656331\pi$
$23$	−2393.01	−0.943245	−0.471622	+	0.881801i	$0.656331\pi$
$24$	424.697	0.150505
$25$	0	0
$26$	−7550.25	−2.19042
$27$	729.000	0.192450
$28$	1005.06	0.242267
$29$	−5132.34	−1.13324	−0.566618	−	0.823980i	$-0.691749\pi$
$29$	−5132.34	−1.13324	−0.566618	+	0.823980i	$0.691749\pi$
$30$	0	0
$31$	−410.309	−0.0766844	−0.0383422	−	0.999265i	$-0.512208\pi$
$31$	−410.309	−0.0766844	−0.0383422	+	0.999265i	$0.512208\pi$
$32$	−8279.40	−1.42930
$33$	1089.00	0.174078
$34$	4134.42	0.613363
$35$	0	0
$36$	3049.98	0.392230
$37$	−5828.69	−0.699949	−0.349975	−	0.936759i	$-0.613810\pi$
$37$	−5828.69	−0.699949	−0.349975	+	0.936759i	$0.613810\pi$
$38$	−12530.4	−1.40769
$39$	−8141.99	−0.857174
$40$	0	0
$41$	18561.8	1.72449	0.862245	−	0.506491i	$-0.169058\pi$
$41$	18561.8	1.72449	0.862245	+	0.506491i	$0.169058\pi$
$42$	2004.91	0.175376
$43$	788.920	0.0650671	0.0325336	−	0.999471i	$-0.489642\pi$
$43$	788.920	0.0650671	0.0325336	+	0.999471i	$0.489642\pi$
$44$	4556.15	0.354786
$45$	0	0
$46$	−19971.8	−1.39163
$47$	−3979.11	−0.262749	−0.131375	−	0.991333i	$-0.541939\pi$
$47$	−3979.11	−0.262749	−0.131375	+	0.991333i	$0.541939\pi$
$48$	−7299.90	−0.457313
$49$	−16094.5	−0.957610
$50$	0	0
$51$	4458.45	0.240026
$52$	−34064.4	−1.74700
$53$	−15130.5	−0.739886	−0.369943	−	0.929055i	$-0.620623\pi$
$53$	−15130.5	−0.739886	−0.369943	+	0.929055i	$0.620623\pi$
$54$	6084.16	0.283933
$55$	0	0
$56$	1259.55	0.0536716
$57$	−13512.5	−0.550867
$58$	−42834.0	−1.67193
$59$	41623.5	1.55671	0.778357	−	0.627822i	$-0.216053\pi$
$59$	41623.5	1.55671	0.778357	+	0.627822i	$0.216053\pi$
$60$	0	0
$61$	−29989.8	−1.03193	−0.515964	−	0.856610i	$-0.672566\pi$
$61$	−29989.8	−1.03193	−0.515964	+	0.856610i	$0.672566\pi$
$62$	−3424.40	−0.113137
$63$	2162.04	0.0686296
$64$	−43143.8	−1.31665
$65$	0	0
$66$	9088.69	0.256827
$67$	−25890.3	−0.704613	−0.352307	−	0.935885i	$-0.614603\pi$
$67$	−25890.3	−0.704613	−0.352307	+	0.935885i	$0.614603\pi$
$68$	18653.2	0.489195
$69$	−21537.1	−0.544582
$70$	0	0
$71$	58163.0	1.36931	0.684654	−	0.728869i	$-0.259954\pi$
$71$	58163.0	1.36931	0.684654	+	0.728869i	$0.259954\pi$
$72$	3822.27	0.0868941
$73$	−23875.3	−0.524375	−0.262187	−	0.965017i	$-0.584444\pi$
$73$	−23875.3	−0.524375	−0.262187	+	0.965017i	$0.584444\pi$
$74$	−48645.7	−1.03268
$75$	0	0
$76$	−56533.2	−1.12272
$77$	3229.71	0.0620778
$78$	−67952.3	−1.26464
$79$	27402.8	0.494000	0.247000	−	0.969016i	$-0.420555\pi$
$79$	27402.8	0.494000	0.247000	+	0.969016i	$0.420555\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	154915.	2.54425
$83$	−24056.3	−0.383296	−0.191648	−	0.981464i	$-0.561383\pi$
$83$	−24056.3	−0.383296	−0.191648	+	0.981464i	$0.561383\pi$
$84$	9045.50	0.139873
$85$	0	0
$86$	6584.25	0.0959975
$87$	−46191.1	−0.654274
$88$	5709.81	0.0785987
$89$	−55820.0	−0.746990	−0.373495	−	0.927632i	$-0.621841\pi$
$89$	−55820.0	−0.746990	−0.373495	+	0.927632i	$0.621841\pi$
$90$	0	0
$91$	−24147.2	−0.305677
$92$	−90106.5	−1.10991
$93$	−3692.78	−0.0442737
$94$	−33209.3	−0.387650
$95$	0	0
$96$	−74514.6	−0.825207
$97$	101188.	1.09194	0.545970	−	0.837805i	$-0.316161\pi$
$97$	101188.	1.09194	0.545970	+	0.837805i	$0.316161\pi$
$98$	−134324.	−1.41282
$99$	9801.00	0.100504
$100$	0	0
$101$	−75190.1	−0.733428	−0.366714	−	0.930334i	$-0.619517\pi$
$101$	−75190.1	−0.733428	−0.366714	+	0.930334i	$0.619517\pi$
$102$	37209.8	0.354125
$103$	−176059.	−1.63518	−0.817590	−	0.575801i	$-0.804690\pi$
$103$	−176059.	−1.63518	−0.817590	+	0.575801i	$0.804690\pi$
$104$	−42689.9	−0.387027
$105$	0	0
$106$	−126278.	−1.09160
$107$	124963.	1.05517	0.527586	−	0.849501i	$-0.323097\pi$
$107$	124963.	1.05517	0.527586	+	0.849501i	$0.323097\pi$
$108$	27449.8	0.226454
$109$	48001.5	0.386980	0.193490	−	0.981102i	$-0.438019\pi$
$109$	48001.5	0.386980	0.193490	+	0.981102i	$0.438019\pi$
$110$	0	0
$111$	−52458.2	−0.404116
$112$	−21649.7	−0.163082
$113$	−38955.6	−0.286995	−0.143497	−	0.989651i	$-0.545835\pi$
$113$	−38955.6	−0.286995	−0.143497	+	0.989651i	$0.545835\pi$
$114$	−112774.	−0.812728
$115$	0	0
$116$	−193254.	−1.33347
$117$	−73277.9	−0.494890
$118$	347386.	2.29672
$119$	13222.7	0.0855957
$120$	0	0
$121$	14641.0	0.0909091
$122$	−250292.	−1.52247
$123$	167056.	0.995635
$124$	−15449.8	−0.0902338
$125$	0	0
$126$	18044.1	0.101253
$127$	−186757.	−1.02746	−0.513732	−	0.857951i	$-0.671738\pi$
$127$	−186757.	−1.02746	−0.513732	+	0.857951i	$0.671738\pi$
$128$	−95133.6	−0.513226
$129$	7100.28	0.0375665
$130$	0	0
$131$	−363603.	−1.85118	−0.925592	−	0.378523i	$-0.876432\pi$
$131$	−363603.	−1.85118	−0.925592	+	0.378523i	$0.876432\pi$
$132$	41005.3	0.204836
$133$	−40074.6	−0.196445
$134$	−216078.	−1.03956
$135$	0	0
$136$	23376.4	0.108375
$137$	74002.6	0.336857	0.168429	−	0.985714i	$-0.446131\pi$
$137$	74002.6	0.336857	0.168429	+	0.985714i	$0.446131\pi$
$138$	−179746.	−0.803456
$139$	70042.3	0.307485	0.153742	−	0.988111i	$-0.450867\pi$
$139$	70042.3	0.307485	0.153742	+	0.988111i	$0.450867\pi$
$140$	0	0
$141$	−35812.0	−0.151698
$142$	485423.	2.02022
$143$	−109465.	−0.447645
$144$	−65699.1	−0.264030
$145$	0	0
$146$	−199261.	−0.773642
$147$	−144851.	−0.552876
$148$	−219474.	−0.823624
$149$	327767.	1.20948	0.604741	−	0.796422i	$-0.293276\pi$
$149$	327767.	1.20948	0.604741	+	0.796422i	$0.293276\pi$
$150$	0	0
$151$	519.828	0.00185531	0.000927656	−	1.00000i	$-0.499705\pi$
$151$	519.828	0.00185531	0.000927656		1.00000i	$0.499705\pi$
$152$	−70848.1	−0.248725
$153$	40126.1	0.138579
$154$	26954.8	0.0915872
$155$	0	0
$156$	−306579.	−1.00863
$157$	119470.	0.386821	0.193411	−	0.981118i	$-0.438045\pi$
$157$	119470.	0.386821	0.193411	+	0.981118i	$0.438045\pi$
$158$	228701.	0.728828
$159$	−136175.	−0.427173
$160$	0	0
$161$	−63873.7	−0.194204
$162$	54757.5	0.163929
$163$	−314012.	−0.925716	−0.462858	−	0.886432i	$-0.653176\pi$
$163$	−314012.	−0.925716	−0.462858	+	0.886432i	$0.653176\pi$
$164$	698928.	2.02919
$165$	0	0
$166$	−200772.	−0.565500
$167$	−281877.	−0.782111	−0.391056	−	0.920367i	$-0.627890\pi$
$167$	−281877.	−0.782111	−0.391056	+	0.920367i	$0.627890\pi$
$168$	11335.9	0.0309873
$169$	447127.	1.20424
$170$	0	0
$171$	−121612.	−0.318043
$172$	29706.1	0.0765639
$173$	−154131.	−0.391538	−0.195769	−	0.980650i	$-0.562720\pi$
$173$	−154131.	−0.391538	−0.195769	+	0.980650i	$0.562720\pi$
$174$	−385506.	−0.965291
$175$	0	0
$176$	−98143.1	−0.238824
$177$	374612.	0.898770
$178$	−465868.	−1.10208
$179$	603292.	1.40733	0.703664	−	0.710533i	$-0.251546\pi$
$179$	603292.	1.40733	0.703664	+	0.710533i	$0.251546\pi$
$180$	0	0
$181$	−560418.	−1.27150	−0.635749	−	0.771896i	$-0.719309\pi$
$181$	−560418.	−1.27150	−0.635749	+	0.771896i	$0.719309\pi$
$182$	−201530.	−0.450984
$183$	−269909.	−0.595784
$184$	−112922.	−0.245887
$185$	0	0
$186$	−30819.6	−0.0653198
$187$	59941.4	0.125350
$188$	−149830.	−0.309175
$189$	19458.3	0.0396233
$190$	0	0
$191$	337255.	0.668921	0.334461	−	0.942410i	$-0.391446\pi$
$191$	337255.	0.668921	0.334461	+	0.942410i	$0.391446\pi$
$192$	−388295.	−0.760166
$193$	−427002.	−0.825158	−0.412579	−	0.910922i	$-0.635372\pi$
$193$	−427002.	−0.825158	−0.412579	+	0.910922i	$0.635372\pi$
$194$	844503.	1.61100
$195$	0	0
$196$	−606026.	−1.12681
$197$	−387552.	−0.711483	−0.355741	−	0.934584i	$-0.615772\pi$
$197$	−387552.	−0.711483	−0.355741	+	0.934584i	$0.615772\pi$
$198$	81798.2	0.148279
$199$	564409.	1.01033	0.505163	−	0.863024i	$-0.331432\pi$
$199$	564409.	1.01033	0.505163	+	0.863024i	$0.331432\pi$
$200$	0	0
$201$	−233013.	−0.406809
$202$	−627529.	−1.08207
$203$	−136991.	−0.233321
$204$	167879.	0.282437
$205$	0	0
$206$	−1.46937e6	−2.41248
$207$	−193834.	−0.314415
$208$	733774.	1.17599
$209$	−181667.	−0.287681
$210$	0	0
$211$	495448.	0.766112	0.383056	−	0.923725i	$-0.374872\pi$
$211$	495448.	0.766112	0.383056	+	0.923725i	$0.374872\pi$
$212$	−569727.	−0.870616
$213$	523467.	0.790570
$214$	1.04293e6	1.55676
$215$	0	0
$216$	34400.4	0.0501683
$217$	−10951.9	−0.0157885
$218$	400616.	0.570935
$219$	−214878.	−0.302748
$220$	0	0
$221$	−448157.	−0.617233
$222$	−437811.	−0.596217
$223$	−1.23517e6	−1.66327	−0.831637	−	0.555320i	$-0.812596\pi$
$223$	−1.23517e6	−1.66327	−0.831637	+	0.555320i	$0.812596\pi$
$224$	−220992.	−0.294277
$225$	0	0
$226$	−325120.	−0.423421
$227$	802179.	1.03325	0.516627	−	0.856211i	$-0.327188\pi$
$227$	802179.	1.03325	0.516627	+	0.856211i	$0.327188\pi$
$228$	−508799.	−0.648200
$229$	−162526.	−0.204802	−0.102401	−	0.994743i	$-0.532652\pi$
$229$	−162526.	−0.204802	−0.102401	+	0.994743i	$0.532652\pi$
$230$	0	0
$231$	29067.4	0.0358407
$232$	−242188.	−0.295415
$233$	1.45802e6	1.75944	0.879718	−	0.475496i	$-0.157731\pi$
$233$	1.45802e6	1.75944	0.879718	+	0.475496i	$0.157731\pi$
$234$	−611570.	−0.730141
$235$	0	0
$236$	1.56730e6	1.83177
$237$	246625.	0.285211
$238$	110355.	0.126285
$239$	−582475.	−0.659603	−0.329802	−	0.944050i	$-0.606982\pi$
$239$	−582475.	−0.659603	−0.329802	+	0.944050i	$0.606982\pi$
$240$	0	0
$241$	434564.	0.481961	0.240980	−	0.970530i	$-0.422531\pi$
$241$	434564.	0.481961	0.240980	+	0.970530i	$0.422531\pi$
$242$	122192.	0.134124
$243$	59049.0	0.0641500
$244$	−1.12924e6	−1.21426
$245$	0	0
$246$	1.39424e6	1.46892
$247$	1.35825e6	1.41657
$248$	−19361.9	−0.0199903
$249$	−216507.	−0.221296
$250$	0	0
$251$	−948184.	−0.949967	−0.474983	−	0.879995i	$-0.657546\pi$
$251$	−948184.	−0.949967	−0.474983	+	0.879995i	$0.657546\pi$
$252$	81409.5	0.0807558
$253$	−289554.	−0.284399
$254$	−1.55865e6	−1.51588
$255$	0	0
$256$	586627.	0.559451
$257$	−1.14089e6	−1.07749	−0.538744	−	0.842470i	$-0.681101\pi$
$257$	−1.14089e6	−1.07749	−0.538744	+	0.842470i	$0.681101\pi$
$258$	59258.2	0.0554242
$259$	−155578.	−0.144112
$260$	0	0
$261$	−415720.	−0.377745
$262$	−3.03460e6	−2.73116
$263$	−280915.	−0.250430	−0.125215	−	0.992130i	$-0.539962\pi$
$263$	−280915.	−0.250430	−0.125215	+	0.992130i	$0.539962\pi$
$264$	51388.3	0.0453790
$265$	0	0
$266$	−334459.	−0.289827
$267$	−502380.	−0.431275
$268$	−974878.	−0.829112
$269$	1.08759e6	0.916397	0.458198	−	0.888850i	$-0.348495\pi$
$269$	1.08759e6	0.916397	0.458198	+	0.888850i	$0.348495\pi$
$270$	0	0
$271$	1.17929e6	0.975431	0.487716	−	0.873003i	$-0.337830\pi$
$271$	1.17929e6	0.975431	0.487716	+	0.873003i	$0.337830\pi$
$272$	−401806.	−0.329302
$273$	−217324.	−0.176483
$274$	617619.	0.496986
$275$	0	0
$276$	−810958.	−0.640805
$277$	621357.	0.486566	0.243283	−	0.969955i	$-0.421776\pi$
$277$	621357.	0.486566	0.243283	+	0.969955i	$0.421776\pi$
$278$	584566.	0.453651
$279$	−33235.0	−0.0255615
$280$	0	0
$281$	−243988.	−0.184333	−0.0921666	−	0.995744i	$-0.529379\pi$
$281$	−243988.	−0.184333	−0.0921666	+	0.995744i	$0.529379\pi$
$282$	−298884.	−0.223810
$283$	1.84579e6	1.36999	0.684993	−	0.728549i	$-0.259805\pi$
$283$	1.84579e6	1.36999	0.684993	+	0.728549i	$0.259805\pi$
$284$	2.19007e6	1.61125
$285$	0	0
$286$	−913581.	−0.660438
$287$	495448.	0.355053
$288$	−670631.	−0.476434
$289$	−1.17445e6	−0.827162
$290$	0	0
$291$	910690.	0.630431
$292$	−899003.	−0.617027
$293$	1.34899e6	0.917995	0.458998	−	0.888437i	$-0.348209\pi$
$293$	1.34899e6	0.917995	0.458998	+	0.888437i	$0.348209\pi$
$294$	−1.20891e6	−0.815692
$295$	0	0
$296$	−275047.	−0.182464
$297$	88209.0	0.0580259
$298$	2.73551e6	1.78442
$299$	2.16487e6	1.40041
$300$	0	0
$301$	21057.7	0.0133966
$302$	4338.43	0.00273726
$303$	−676711.	−0.423445
$304$	1.21777e6	0.755757
$305$	0	0
$306$	334888.	0.204454
$307$	1.20545e6	0.729970	0.364985	−	0.931013i	$-0.381074\pi$
$307$	1.20545e6	0.729970	0.364985	+	0.931013i	$0.381074\pi$
$308$	121612.	0.0730464
$309$	−1.58453e6	−0.944072
$310$	0	0
$311$	636875.	0.373382	0.186691	−	0.982419i	$-0.440224\pi$
$311$	636875.	0.373382	0.186691	+	0.982419i	$0.440224\pi$
$312$	−384209.	−0.223450
$313$	−1.86526e6	−1.07616	−0.538081	−	0.842893i	$-0.680850\pi$
$313$	−1.86526e6	−1.07616	−0.538081	+	0.842893i	$0.680850\pi$
$314$	997086.	0.570701
$315$	0	0
$316$	1.03183e6	0.581285
$317$	−2.35417e6	−1.31580	−0.657900	−	0.753105i	$-0.728555\pi$
$317$	−2.35417e6	−1.31580	−0.657900	+	0.753105i	$0.728555\pi$
$318$	−1.13650e6	−0.630235
$319$	−621013.	−0.341684
$320$	0	0
$321$	1.12467e6	0.609204
$322$	−533083.	−0.286520
$323$	−743761.	−0.396668
$324$	247049.	0.130743
$325$	0	0
$326$	−2.62072e6	−1.36577
$327$	432013.	0.223423
$328$	875905.	0.449544
$329$	−106210.	−0.0540972
$330$	0	0
$331$	1.97887e6	0.992768	0.496384	−	0.868103i	$-0.334661\pi$
$331$	1.97887e6	0.992768	0.496384	+	0.868103i	$0.334661\pi$
$332$	−905820.	−0.451021
$333$	−472124.	−0.233316
$334$	−2.35252e6	−1.15390
$335$	0	0
$336$	−194848.	−0.0941557
$337$	−2.86225e6	−1.37288	−0.686439	−	0.727187i	$-0.740827\pi$
$337$	−2.86225e6	−1.37288	−0.686439	+	0.727187i	$0.740827\pi$
$338$	3.73168e6	1.77669
$339$	−350601.	−0.165697
$340$	0	0
$341$	−49647.4	−0.0231212
$342$	−1.01496e6	−0.469229
$343$	−878202.	−0.403050
$344$	37228.0	0.0169619
$345$	0	0
$346$	−1.28636e6	−0.577660
$347$	1.81514e6	0.809257	0.404629	−	0.914481i	$-0.367401\pi$
$347$	1.81514e6	0.809257	0.404629	+	0.914481i	$0.367401\pi$
$348$	−1.73928e6	−0.769878
$349$	−1.77907e6	−0.781861	−0.390930	−	0.920420i	$-0.627847\pi$
$349$	−1.77907e6	−0.781861	−0.390930	+	0.920420i	$0.627847\pi$
$350$	0	0
$351$	−659501.	−0.285725
$352$	−1.00181e6	−0.430950
$353$	4.18817e6	1.78890	0.894452	−	0.447164i	$-0.147566\pi$
$353$	4.18817e6	1.78890	0.894452	+	0.447164i	$0.147566\pi$
$354$	3.12647e6	1.32601
$355$	0	0
$356$	−2.10185e6	−0.878976
$357$	119004.	0.0494187
$358$	5.03502e6	2.07632
$359$	1.13143e6	0.463329	0.231665	−	0.972796i	$-0.425583\pi$
$359$	1.13143e6	0.463329	0.231665	+	0.972796i	$0.425583\pi$
$360$	0	0
$361$	−221946.	−0.0896355
$362$	−4.67720e6	−1.87592
$363$	131769.	0.0524864
$364$	−909240.	−0.359687
$365$	0	0
$366$	−2.25263e6	−0.878997
$367$	−1.31924e6	−0.511279	−0.255639	−	0.966772i	$-0.582286\pi$
$367$	−1.31924e6	−0.511279	−0.255639	+	0.966772i	$0.582286\pi$
$368$	1.94097e6	0.747134
$369$	1.50351e6	0.574830
$370$	0	0
$371$	−403861.	−0.152334
$372$	−139048.	−0.0520965
$373$	−741.945	−0.000276121 0	−0.000138061	−	1.00000i	$-0.500044\pi$
$373$	−741.945	−0.000276121 0	−0.000138061		1.00000i	$0.500044\pi$
$374$	500265.	0.184936
$375$	0	0
$376$	−187769.	−0.0684941
$377$	4.64305e6	1.68248
$378$	162397.	0.0584587
$379$	−567437.	−0.202917	−0.101459	−	0.994840i	$-0.532351\pi$
$379$	−567437.	−0.202917	−0.101459	+	0.994840i	$0.532351\pi$
$380$	0	0
$381$	−1.68081e6	−0.593207
$382$	2.81470e6	0.986900
$383$	−3.54829e6	−1.23601	−0.618006	−	0.786174i	$-0.712059\pi$
$383$	−3.54829e6	−1.23601	−0.618006	+	0.786174i	$0.712059\pi$
$384$	−856202.	−0.296311
$385$	0	0
$386$	−3.56372e6	−1.21741
$387$	63902.5	0.0216890
$388$	3.81013e6	1.28487
$389$	4.09335e6	1.37153	0.685764	−	0.727824i	$-0.259468\pi$
$389$	4.09335e6	1.37153	0.685764	+	0.727824i	$0.259468\pi$
$390$	0	0
$391$	−1.18546e6	−0.392142
$392$	−759478.	−0.249632
$393$	−3.27243e6	−1.06878
$394$	−3.23447e6	−1.04969
$395$	0	0
$396$	369048.	0.118262
$397$	−153511.	−0.0488835	−0.0244417	−	0.999701i	$-0.507781\pi$
$397$	−153511.	−0.0488835	−0.0244417	+	0.999701i	$0.507781\pi$
$398$	4.71050e6	1.49059
$399$	−360672.	−0.113417
$400$	0	0
$401$	474424.	0.147335	0.0736674	−	0.997283i	$-0.476530\pi$
$401$	474424.	0.147335	0.0736674	+	0.997283i	$0.476530\pi$
$402$	−1.94470e6	−0.600190
$403$	371193.	0.113851
$404$	−2.83122e6	−0.863017
$405$	0	0
$406$	−1.14332e6	−0.344232
$407$	−705271.	−0.211043
$408$	210388.	0.0625706
$409$	2.85246e6	0.843164	0.421582	−	0.906790i	$-0.361475\pi$
$409$	2.85246e6	0.843164	0.421582	+	0.906790i	$0.361475\pi$
$410$	0	0
$411$	666023.	0.194484
$412$	−6.62935e6	−1.92410
$413$	1.11101e6	0.320510
$414$	−1.61772e6	−0.463875
$415$	0	0
$416$	7.49009e6	2.12204
$417$	630381.	0.177526
$418$	−1.51618e6	−0.424434
$419$	6.61913e6	1.84190	0.920950	−	0.389681i	$-0.127415\pi$
$419$	6.61913e6	1.84190	0.920950	+	0.389681i	$0.127415\pi$
$420$	0	0
$421$	1.67579e6	0.460802	0.230401	−	0.973096i	$-0.425996\pi$
$421$	1.67579e6	0.460802	0.230401	+	0.973096i	$0.425996\pi$
$422$	4.13496e6	1.13029
$423$	−322308.	−0.0875831
$424$	−713988.	−0.192875
$425$	0	0
$426$	4.36880e6	1.16638
$427$	−800483.	−0.212463
$428$	4.70539e6	1.24161
$429$	−985181.	−0.258448
$430$	0	0
$431$	−4.53856e6	−1.17686	−0.588430	−	0.808548i	$-0.700254\pi$
$431$	−4.53856e6	−1.17686	−0.588430	+	0.808548i	$0.700254\pi$
$432$	−591292.	−0.152438
$433$	−379605.	−0.0972999	−0.0486500	−	0.998816i	$-0.515492\pi$
$433$	−379605.	−0.0972999	−0.0486500	+	0.998816i	$0.515492\pi$
$434$	−91403.4	−0.0232937
$435$	0	0
$436$	1.80745e6	0.455355
$437$	3.59282e6	0.899978
$438$	−1.79335e6	−0.446663
$439$	−1.97778e6	−0.489797	−0.244899	−	0.969549i	$-0.578755\pi$
$439$	−1.97778e6	−0.489797	−0.244899	+	0.969549i	$0.578755\pi$
$440$	0	0
$441$	−1.30366e6	−0.319203
$442$	−3.74027e6	−0.910641
$443$	−3.58373e6	−0.867613	−0.433807	−	0.901006i	$-0.642830\pi$
$443$	−3.58373e6	−0.867613	−0.433807	+	0.901006i	$0.642830\pi$
$444$	−1.97527e6	−0.475519
$445$	0	0
$446$	−1.03086e7	−2.45393
$447$	2.94990e6	0.698295
$448$	−1.15159e6	−0.271083
$449$	−1.05735e6	−0.247515	−0.123758	−	0.992312i	$-0.539495\pi$
$449$	−1.05735e6	−0.247515	−0.123758	+	0.992312i	$0.539495\pi$
$450$	0	0
$451$	2.24598e6	0.519954
$452$	−1.46684e6	−0.337704
$453$	4678.45	0.00107117
$454$	6.69491e6	1.52442
$455$	0	0
$456$	−637633.	−0.143601
$457$	−1.39016e6	−0.311368	−0.155684	−	0.987807i	$-0.549758\pi$
$457$	−1.39016e6	−0.311368	−0.155684	+	0.987807i	$0.549758\pi$
$458$	−1.35643e6	−0.302157
$459$	361135.	0.0800087
$460$	0	0
$461$	−1.09360e6	−0.239666	−0.119833	−	0.992794i	$-0.538236\pi$
$461$	−1.09360e6	−0.239666	−0.119833	+	0.992794i	$0.538236\pi$
$462$	242594.	0.0528779
$463$	−1.94891e6	−0.422513	−0.211256	−	0.977431i	$-0.567756\pi$
$463$	−1.94891e6	−0.422513	−0.211256	+	0.977431i	$0.567756\pi$
$464$	4.16284e6	0.897625
$465$	0	0
$466$	1.21685e7	2.59580
$467$	4.69017e6	0.995169	0.497584	−	0.867416i	$-0.334220\pi$
$467$	4.69017e6	0.995169	0.497584	+	0.867416i	$0.334220\pi$
$468$	−2.75921e6	−0.582332
$469$	−691060.	−0.145072
$470$	0	0
$471$	1.07523e6	0.223331
$472$	1.96415e6	0.405808
$473$	95459.3	0.0196185
$474$	2.05831e6	0.420789
$475$	0	0
$476$	497888.	0.100720
$477$	−1.22557e6	−0.246629
$478$	−4.86128e6	−0.973153
$479$	4.15576e6	0.827584	0.413792	−	0.910372i	$-0.364204\pi$
$479$	4.15576e6	0.827584	0.413792	+	0.910372i	$0.364204\pi$
$480$	0	0
$481$	5.27302e6	1.03919
$482$	3.62683e6	0.711066
$483$	−574863.	−0.112123
$484$	551294.	0.106972
$485$	0	0
$486$	492817.	0.0946445
$487$	2.37242e6	0.453282	0.226641	−	0.973978i	$-0.427226\pi$
$487$	2.37242e6	0.453282	0.226641	+	0.973978i	$0.427226\pi$
$488$	−1.41518e6	−0.269006
$489$	−2.82611e6	−0.534463
$490$	0	0
$491$	−1.18797e6	−0.222384	−0.111192	−	0.993799i	$-0.535467\pi$
$491$	−1.18797e6	−0.222384	−0.111192	+	0.993799i	$0.535467\pi$
$492$	6.29036e6	1.17155
$493$	−2.54248e6	−0.471129
$494$	1.13358e7	2.08995
$495$	0	0
$496$	332802.	0.0607409
$497$	1.55248e6	0.281925
$498$	−1.80695e6	−0.326492
$499$	710493.	0.127735	0.0638673	−	0.997958i	$-0.479657\pi$
$499$	710493.	0.127735	0.0638673	+	0.997958i	$0.479657\pi$
$500$	0	0
$501$	−2.53689e6	−0.451552
$502$	−7.91345e6	−1.40154
$503$	−4.75381e6	−0.837765	−0.418882	−	0.908041i	$-0.637578\pi$
$503$	−4.75381e6	−0.837765	−0.418882	+	0.908041i	$0.637578\pi$
$504$	102023.	0.0178905
$505$	0	0
$506$	−2.41659e6	−0.419591
$507$	4.02414e6	0.695270
$508$	−7.03215e6	−1.20901
$509$	−5.05479e6	−0.864787	−0.432393	−	0.901685i	$-0.642331\pi$
$509$	−5.05479e6	−0.864787	−0.432393	+	0.901685i	$0.642331\pi$
$510$	0	0
$511$	−637275.	−0.107963
$512$	7.94021e6	1.33862
$513$	−1.09451e6	−0.183622
$514$	−9.52179e6	−1.58968
$515$	0	0
$516$	267355.	0.0442042
$517$	−481473.	−0.0792219
$518$	−1.29844e6	−0.212617
$519$	−1.38718e6	−0.226055
$520$	0	0
$521$	−4.69775e6	−0.758220	−0.379110	−	0.925352i	$-0.623770\pi$
$521$	−4.69775e6	−0.758220	−0.379110	+	0.925352i	$0.623770\pi$
$522$	−3.46956e6	−0.557311
$523$	1.97083e6	0.315062	0.157531	−	0.987514i	$-0.449647\pi$
$523$	1.97083e6	0.315062	0.157531	+	0.987514i	$0.449647\pi$
$524$	−1.36912e7	−2.17827
$525$	0	0
$526$	−2.34449e6	−0.369474
$527$	−203260.	−0.0318806
$528$	−883288.	−0.137885
$529$	−709863.	−0.110290
$530$	0	0
$531$	3.37151e6	0.518905
$532$	−1.50897e6	−0.231155
$533$	−1.67922e7	−2.56030
$534$	−4.19282e6	−0.636286
$535$	0	0
$536$	−1.22173e6	−0.183680
$537$	5.42963e6	0.812521
$538$	9.07690e6	1.35202
$539$	−1.94744e6	−0.288730
$540$	0	0
$541$	−5.76437e6	−0.846757	−0.423378	−	0.905953i	$-0.639156\pi$
$541$	−5.76437e6	−0.846757	−0.423378	+	0.905953i	$0.639156\pi$
$542$	9.84222e6	1.43911
$543$	−5.04377e6	−0.734100
$544$	−4.10148e6	−0.594214
$545$	0	0
$546$	−1.81377e6	−0.260376
$547$	4.65202e6	0.664772	0.332386	−	0.943143i	$-0.392146\pi$
$547$	4.65202e6	0.664772	0.332386	+	0.943143i	$0.392146\pi$
$548$	2.78650e6	0.396376
$549$	−2.42918e6	−0.343976
$550$	0	0
$551$	7.70561e6	1.08126
$552$	−1.01630e6	−0.141963
$553$	731430.	0.101709
$554$	5.18578e6	0.717860
$555$	0	0
$556$	2.63738e6	0.361814
$557$	4.24834e6	0.580204	0.290102	−	0.956996i	$-0.406311\pi$
$557$	4.24834e6	0.580204	0.290102	+	0.956996i	$0.406311\pi$
$558$	−277376.	−0.0377124
$559$	−713709.	−0.0966032
$560$	0	0
$561$	539473.	0.0723706
$562$	−2.03630e6	−0.271958
$563$	3.09512e6	0.411534	0.205767	−	0.978601i	$-0.434031\pi$
$563$	3.09512e6	0.411534	0.205767	+	0.978601i	$0.434031\pi$
$564$	−1.34847e6	−0.178502
$565$	0	0
$566$	1.54048e7	2.02122
$567$	175125.	0.0228765
$568$	2.74463e6	0.356954
$569$	−7.60568e6	−0.984821	−0.492411	−	0.870363i	$-0.663884\pi$
$569$	−7.60568e6	−0.984821	−0.492411	+	0.870363i	$0.663884\pi$
$570$	0	0
$571$	−2.28582e6	−0.293395	−0.146697	−	0.989181i	$-0.546864\pi$
$571$	−2.28582e6	−0.293395	−0.146697	+	0.989181i	$0.546864\pi$
$572$	−4.12179e6	−0.526739
$573$	3.03529e6	0.386202
$574$	4.13496e6	0.523832
$575$	0	0
$576$	−3.49465e6	−0.438882
$577$	6.27770e6	0.784984	0.392492	−	0.919755i	$-0.371613\pi$
$577$	6.27770e6	0.784984	0.392492	+	0.919755i	$0.371613\pi$
$578$	−9.80186e6	−1.22036
$579$	−3.84302e6	−0.476405
$580$	0	0
$581$	−642107.	−0.0789164
$582$	7.60053e6	0.930114
$583$	−1.83079e6	−0.223084
$584$	−1.12664e6	−0.136695
$585$	0	0
$586$	1.12586e7	1.35437
$587$	1.48651e6	0.178063	0.0890314	−	0.996029i	$-0.471623\pi$
$587$	1.48651e6	0.178063	0.0890314	+	0.996029i	$0.471623\pi$
$588$	−5.45423e6	−0.650564
$589$	616031.	0.0731669
$590$	0	0
$591$	−3.48797e6	−0.410775
$592$	4.72765e6	0.554423
$593$	−4.72874e6	−0.552216	−0.276108	−	0.961127i	$-0.589045\pi$
$593$	−4.72874e6	−0.552216	−0.276108	+	0.961127i	$0.589045\pi$
$594$	736184.	0.0856091
$595$	0	0
$596$	1.23418e7	1.42319
$597$	5.07968e6	0.583312
$598$	1.80678e7	2.06611
$599$	−325927.	−0.0371153	−0.0185577	−	0.999828i	$-0.505907\pi$
$599$	−325927.	−0.0371153	−0.0185577	+	0.999828i	$0.505907\pi$
$600$	0	0
$601$	−1.30291e7	−1.47140	−0.735698	−	0.677310i	$-0.763146\pi$
$601$	−1.30291e7	−1.47140	−0.735698	+	0.677310i	$0.763146\pi$
$602$	175745.	0.0197648
$603$	−2.09712e6	−0.234871
$604$	19573.6	0.00218313
$605$	0	0
$606$	−5.64776e6	−0.624734
$607$	1.45956e6	0.160787	0.0803933	−	0.996763i	$-0.474382\pi$
$607$	1.45956e6	0.160787	0.0803933	+	0.996763i	$0.474382\pi$
$608$	1.24305e7	1.36374
$609$	−1.23292e6	−0.134708
$610$	0	0
$611$	3.59977e6	0.390096
$612$	1.51091e6	0.163065
$613$	−1.26574e7	−1.36048	−0.680241	−	0.732989i	$-0.738125\pi$
$613$	−1.26574e7	−1.36048	−0.680241	+	0.732989i	$0.738125\pi$
$614$	1.00606e7	1.07697
$615$	0	0
$616$	152405.	0.0161826
$617$	−2.57472e6	−0.272281	−0.136140	−	0.990690i	$-0.543470\pi$
$617$	−2.57472e6	−0.272281	−0.136140	+	0.990690i	$0.543470\pi$
$618$	−1.32244e7	−1.39285
$619$	−1.82556e7	−1.91500	−0.957500	−	0.288434i	$-0.906866\pi$
$619$	−1.82556e7	−1.91500	−0.957500	+	0.288434i	$0.906866\pi$
$620$	0	0
$621$	−1.74450e6	−0.181527
$622$	5.31530e6	0.550873
$623$	−1.48994e6	−0.153797
$624$	6.60397e6	0.678959
$625$	0	0
$626$	−1.55672e7	−1.58773
$627$	−1.63501e6	−0.166093
$628$	4.49854e6	0.455169
$629$	−2.88744e6	−0.290995
$630$	0	0
$631$	7.91613e6	0.791480	0.395740	−	0.918363i	$-0.370488\pi$
$631$	7.91613e6	0.791480	0.395740	+	0.918363i	$0.370488\pi$
$632$	1.29310e6	0.128777
$633$	4.45903e6	0.442315
$634$	−1.96477e7	−1.94128
$635$	0	0
$636$	−5.12754e6	−0.502651
$637$	1.45602e7	1.42173
$638$	−5.18292e6	−0.504107
$639$	4.71120e6	0.456436
$640$	0	0
$641$	−3.40205e6	−0.327036	−0.163518	−	0.986540i	$-0.552284\pi$
$641$	−3.40205e6	−0.327036	−0.163518	+	0.986540i	$0.552284\pi$
$642$	9.38640e6	0.898796
$643$	6.92901e6	0.660912	0.330456	−	0.943821i	$-0.392797\pi$
$643$	6.92901e6	0.660912	0.330456	+	0.943821i	$0.392797\pi$
$644$	−2.40511e6	−0.228517
$645$	0	0
$646$	−6.20736e6	−0.585228
$647$	−1.11125e7	−1.04364	−0.521819	−	0.853056i	$-0.674746\pi$
$647$	−1.11125e7	−1.04364	−0.521819	+	0.853056i	$0.674746\pi$
$648$	309604.	0.0289647
$649$	5.03645e6	0.469367
$650$	0	0
$651$	−98567.0	−0.00911547
$652$	−1.18239e7	−1.08928
$653$	−6.61748e6	−0.607309	−0.303654	−	0.952782i	$-0.598207\pi$
$653$	−6.61748e6	−0.607309	−0.303654	+	0.952782i	$0.598207\pi$
$654$	3.60554e6	0.329629
$655$	0	0
$656$	−1.50555e7	−1.36595
$657$	−1.93390e6	−0.174792
$658$	−886416.	−0.0798128
$659$	1.88838e7	1.69385	0.846926	−	0.531711i	$-0.178451\pi$
$659$	1.88838e7	1.69385	0.846926	+	0.531711i	$0.178451\pi$
$660$	0	0
$661$	7.10304e6	0.632326	0.316163	−	0.948705i	$-0.397605\pi$
$661$	7.10304e6	0.632326	0.316163	+	0.948705i	$0.397605\pi$
$662$	1.65155e7	1.46469
$663$	−4.03341e6	−0.356360
$664$	−1.13518e6	−0.0999185
$665$	0	0
$666$	−3.94030e6	−0.344226
$667$	1.22817e7	1.06892
$668$	−1.06138e7	−0.920303
$669$	−1.11165e7	−0.960291
$670$	0	0
$671$	−3.62877e6	−0.311138
$672$	−1.98893e6	−0.169901
$673$	1.79739e7	1.52970	0.764849	−	0.644209i	$-0.222813\pi$
$673$	1.79739e7	1.52970	0.764849	+	0.644209i	$0.222813\pi$
$674$	−2.38880e7	−2.02549
$675$	0	0
$676$	1.68362e7	1.41702
$677$	617034.	0.0517413	0.0258706	−	0.999665i	$-0.491764\pi$
$677$	617034.	0.0517413	0.0258706	+	0.999665i	$0.491764\pi$
$678$	−2.92608e6	−0.244462
$679$	2.70088e6	0.224818
$680$	0	0
$681$	7.21961e6	0.596549
$682$	−414352.	−0.0341121
$683$	−924287.	−0.0758150	−0.0379075	−	0.999281i	$-0.512069\pi$
$683$	−924287.	−0.0758150	−0.0379075	+	0.999281i	$0.512069\pi$
$684$	−4.57919e6	−0.374239
$685$	0	0
$686$	−7.32939e6	−0.594644
$687$	−1.46273e6	−0.118242
$688$	−639893.	−0.0515390
$689$	1.36881e7	1.09849
$690$	0	0
$691$	2.33550e7	1.86074	0.930368	−	0.366627i	$-0.119487\pi$
$691$	2.33550e7	1.86074	0.930368	+	0.366627i	$0.119487\pi$
$692$	−5.80365e6	−0.460719
$693$	261606.	0.0206926
$694$	1.51490e7	1.19395
$695$	0	0
$696$	−2.17969e6	−0.170558
$697$	9.19522e6	0.716936
$698$	−1.48479e7	−1.15353
$699$	1.31222e7	1.01581
$700$	0	0
$701$	1.15300e7	0.886206	0.443103	−	0.896471i	$-0.353878\pi$
$701$	1.15300e7	0.886206	0.443103	+	0.896471i	$0.353878\pi$
$702$	−5.50413e6	−0.421547
$703$	8.75110e6	0.667843
$704$	−5.22040e6	−0.396984
$705$	0	0
$706$	3.49540e7	2.63928
$707$	−2.00696e6	−0.151005
$708$	1.41057e7	1.05757
$709$	−1.13035e7	−0.844497	−0.422248	−	0.906480i	$-0.638759\pi$
$709$	−1.13035e7	−0.844497	−0.422248	+	0.906480i	$0.638759\pi$
$710$	0	0
$711$	2.21962e6	0.164667
$712$	−2.63407e6	−0.194727
$713$	981872.	0.0723321
$714$	993197.	0.0729105
$715$	0	0
$716$	2.27164e7	1.65599
$717$	−5.24228e6	−0.380822
$718$	9.44276e6	0.683578
$719$	406909.	0.0293545	0.0146773	−	0.999892i	$-0.495328\pi$
$719$	406909.	0.0293545	0.0146773	+	0.999892i	$0.495328\pi$
$720$	0	0
$721$	−4.69934e6	−0.336665
$722$	−1.85234e6	−0.132245
$723$	3.91108e6	0.278260
$724$	−2.11020e7	−1.49616
$725$	0	0
$726$	1.09973e6	0.0774364
$727$	−1.93239e6	−0.135600	−0.0678000	−	0.997699i	$-0.521598\pi$
$727$	−1.93239e6	−0.135600	−0.0678000	+	0.997699i	$0.521598\pi$
$728$	−1.13947e6	−0.0796846
$729$	531441.	0.0370370
$730$	0	0
$731$	390818.	0.0270509
$732$	−1.01632e7	−0.701054
$733$	−357394.	−0.0245690	−0.0122845	−	0.999925i	$-0.503910\pi$
$733$	−357394.	−0.0245690	−0.0122845	+	0.999925i	$0.503910\pi$
$734$	−1.10102e7	−0.754320
$735$	0	0
$736$	1.98126e7	1.34818
$737$	−3.13273e6	−0.212449
$738$	1.25481e7	0.848082
$739$	2.22117e7	1.49613	0.748066	−	0.663624i	$-0.230983\pi$
$739$	2.22117e7	1.49613	0.748066	+	0.663624i	$0.230983\pi$
$740$	0	0
$741$	1.22243e7	0.817856
$742$	−3.37059e6	−0.224748
$743$	2.10894e7	1.40149	0.700747	−	0.713410i	$-0.252850\pi$
$743$	2.10894e7	1.40149	0.700747	+	0.713410i	$0.252850\pi$
$744$	−174257.	−0.0115414
$745$	0	0
$746$	−6192.20	−0.000407378 0
$747$	−1.94856e6	−0.127765
$748$	2.25704e6	0.147498
$749$	3.33550e6	0.217248
$750$	0	0
$751$	−3.08277e7	−1.99453	−0.997267	−	0.0738774i	$-0.976463\pi$
$751$	−3.08277e7	−1.99453	−0.997267	+	0.0738774i	$0.976463\pi$
$752$	3.22746e6	0.208121
$753$	−8.53366e6	−0.548464
$754$	3.87505e7	2.48227
$755$	0	0
$756$	732686.	0.0466244
$757$	−3.26795e6	−0.207270	−0.103635	−	0.994615i	$-0.533047\pi$
$757$	−3.26795e6	−0.207270	−0.103635	+	0.994615i	$0.533047\pi$
$758$	−4.73577e6	−0.299376
$759$	−2.60598e6	−0.164198
$760$	0	0
$761$	1.01120e7	0.632961	0.316480	−	0.948599i	$-0.397499\pi$
$761$	1.01120e7	0.632961	0.316480	+	0.948599i	$0.397499\pi$
$762$	−1.40279e7	−0.875194
$763$	1.28125e6	0.0796748
$764$	1.26990e7	0.787113
$765$	0	0
$766$	−2.96137e7	−1.82356
$767$	−3.76554e7	−2.31121
$768$	5.27964e6	0.322999
$769$	2.14392e7	1.30735	0.653676	−	0.756775i	$-0.273226\pi$
$769$	2.14392e7	1.30735	0.653676	+	0.756775i	$0.273226\pi$
$770$	0	0
$771$	−1.02680e7	−0.622088
$772$	−1.60784e7	−0.970955
$773$	1.89394e7	1.14003	0.570017	−	0.821633i	$-0.306937\pi$
$773$	1.89394e7	1.14003	0.570017	+	0.821633i	$0.306937\pi$
$774$	533324.	0.0319992
$775$	0	0
$776$	4.77490e6	0.284649
$777$	−1.40020e6	−0.0832030
$778$	3.41627e7	2.02350
$779$	−2.78684e7	−1.64539
$780$	0	0
$781$	7.03772e6	0.412862
$782$	−9.89370e6	−0.578551
$783$	−3.74148e6	−0.218091
$784$	1.30543e7	0.758513
$785$	0	0
$786$	−2.73114e7	−1.57684
$787$	1.10351e7	0.635096	0.317548	−	0.948242i	$-0.397141\pi$
$787$	1.10351e7	0.635096	0.317548	+	0.948242i	$0.397141\pi$
$788$	−1.45929e7	−0.837195
$789$	−2.52824e6	−0.144586
$790$	0	0
$791$	−1.03980e6	−0.0590890
$792$	462495.	0.0261996
$793$	2.71308e7	1.53207
$794$	−1.28118e6	−0.0721208
$795$	0	0
$796$	2.12523e7	1.18884
$797$	1.22483e7	0.683015	0.341507	−	0.939879i	$-0.389063\pi$
$797$	1.22483e7	0.683015	0.341507	+	0.939879i	$0.389063\pi$
$798$	−3.01013e6	−0.167332
$799$	−1.97119e6	−0.109235
$800$	0	0
$801$	−4.52142e6	−0.248997
$802$	3.95949e6	0.217372
$803$	−2.88891e6	−0.158105
$804$	−8.77390e6	−0.478688
$805$	0	0
$806$	3.09794e6	0.167971
$807$	9.78829e6	0.529082
$808$	−3.54811e6	−0.191192
$809$	−2.42528e7	−1.30284	−0.651421	−	0.758717i	$-0.725827\pi$
$809$	−2.42528e7	−1.30284	−0.651421	+	0.758717i	$0.725827\pi$
$810$	0	0
$811$	3.64893e7	1.94811	0.974055	−	0.226311i	$-0.0726664\pi$
$811$	3.64893e7	1.94811	0.974055	+	0.226311i	$0.0726664\pi$
$812$	−5.15829e6	−0.274546
$813$	1.06136e7	0.563166
$814$	−5.88613e6	−0.311364
$815$	0	0
$816$	−3.61625e6	−0.190122
$817$	−1.18447e6	−0.0620825
$818$	2.38064e7	1.24397
$819$	−1.95592e6	−0.101892
$820$	0	0
$821$	2.40297e7	1.24420	0.622100	−	0.782938i	$-0.286280\pi$
$821$	2.40297e7	1.24420	0.622100	+	0.782938i	$0.286280\pi$
$822$	5.55857e6	0.286935
$823$	1.16283e7	0.598432	0.299216	−	0.954185i	$-0.403275\pi$
$823$	1.16283e7	0.598432	0.299216	+	0.954185i	$0.403275\pi$
$824$	−8.30798e6	−0.426263
$825$	0	0
$826$	9.27236e6	0.472868
$827$	−1.01425e7	−0.515680	−0.257840	−	0.966188i	$-0.583011\pi$
$827$	−1.01425e7	−0.515680	−0.257840	+	0.966188i	$0.583011\pi$
$828$	−7.29863e6	−0.369969
$829$	1.51928e7	0.767804	0.383902	−	0.923374i	$-0.374580\pi$
$829$	1.51928e7	0.767804	0.383902	+	0.923374i	$0.374580\pi$
$830$	0	0
$831$	5.59221e6	0.280919
$832$	3.90307e7	1.95478
$833$	−7.97298e6	−0.398114
$834$	5.26110e6	0.261916
$835$	0	0
$836$	−6.84052e6	−0.338512
$837$	−299115.	−0.0147579
$838$	5.52426e7	2.71747
$839$	−9.62729e6	−0.472171	−0.236085	−	0.971732i	$-0.575864\pi$
$839$	−9.62729e6	−0.472171	−0.236085	+	0.971732i	$0.575864\pi$
$840$	0	0
$841$	5.82977e6	0.284225
$842$	1.39860e7	0.679849
$843$	−2.19590e6	−0.106425
$844$	1.86557e7	0.901477
$845$	0	0
$846$	−2.68995e6	−0.129217
$847$	390795.	0.0187172
$848$	1.22724e7	0.586056
$849$	1.66121e7	0.790962
$850$	0	0
$851$	1.39481e7	0.660223
$852$	1.97107e7	0.930256
$853$	−7.49543e6	−0.352715	−0.176358	−	0.984326i	$-0.556432\pi$
$853$	−7.49543e6	−0.352715	−0.176358	+	0.984326i	$0.556432\pi$
$854$	−6.68076e6	−0.313459
$855$	0	0
$856$	5.89684e6	0.275065
$857$	2.69493e6	0.125342	0.0626708	−	0.998034i	$-0.480038\pi$
$857$	2.69493e6	0.125342	0.0626708	+	0.998034i	$0.480038\pi$
$858$	−8.22222e6	−0.381304
$859$	−3.28791e7	−1.52033	−0.760164	−	0.649731i	$-0.774882\pi$
$859$	−3.28791e7	−1.52033	−0.760164	+	0.649731i	$0.774882\pi$
$860$	0	0
$861$	4.45904e6	0.204990
$862$	−3.78784e7	−1.73629
$863$	−3.03331e7	−1.38640	−0.693202	−	0.720744i	$-0.743801\pi$
$863$	−3.03331e7	−1.38640	−0.693202	+	0.720744i	$0.743801\pi$
$864$	−6.03568e6	−0.275069
$865$	0	0
$866$	−3.16815e6	−0.143553
$867$	−1.05701e7	−0.477562
$868$	−412383.	−0.0185781
$869$	3.31574e6	0.148947
$870$	0	0
$871$	2.34221e7	1.04612
$872$	2.26512e6	0.100879
$873$	8.19621e6	0.363980
$874$	2.99853e7	1.32779
$875$	0	0
$876$	−8.09103e6	−0.356241
$877$	3.81826e7	1.67636	0.838178	−	0.545396i	$-0.183621\pi$
$877$	3.81826e7	1.67636	0.838178	+	0.545396i	$0.183621\pi$
$878$	−1.65064e7	−0.722628
$879$	1.21409e7	0.530005
$880$	0	0
$881$	−8.09402e6	−0.351338	−0.175669	−	0.984449i	$-0.556209\pi$
$881$	−8.09402e6	−0.351338	−0.175669	+	0.984449i	$0.556209\pi$
$882$	−1.08802e7	−0.470940
$883$	−1.90914e7	−0.824018	−0.412009	−	0.911180i	$-0.635173\pi$
$883$	−1.90914e7	−0.824018	−0.412009	+	0.911180i	$0.635173\pi$
$884$	−1.68749e7	−0.726292
$885$	0	0
$886$	−2.99095e7	−1.28004
$887$	−1.40640e7	−0.600205	−0.300102	−	0.953907i	$-0.597021\pi$
$887$	−1.40640e7	−0.600205	−0.300102	+	0.953907i	$0.597021\pi$
$888$	−2.47543e6	−0.105346
$889$	−4.98487e6	−0.211543
$890$	0	0
$891$	793881.	0.0335013
$892$	−4.65091e7	−1.95716
$893$	5.97417e6	0.250697
$894$	2.46196e7	1.03024
$895$	0	0
$896$	−2.53929e6	−0.105668
$897$	1.94838e7	0.808525
$898$	−8.82452e6	−0.365174
$899$	2.10585e6	0.0869015
$900$	0	0
$901$	−7.49542e6	−0.307598
$902$	1.87447e7	0.767119
$903$	189519.	0.00773453
$904$	−1.83826e6	−0.0748145
$905$	0	0
$906$	39045.9	0.00158036
$907$	−1.37464e7	−0.554845	−0.277423	−	0.960748i	$-0.589480\pi$
$907$	−1.37464e7	−0.554845	−0.277423	+	0.960748i	$0.589480\pi$
$908$	3.02053e7	1.21582
$909$	−6.09040e6	−0.244476
$910$	0	0
$911$	3.83931e7	1.53270	0.766350	−	0.642423i	$-0.222071\pi$
$911$	3.83931e7	1.53270	0.766350	+	0.642423i	$0.222071\pi$
$912$	1.09600e7	0.436336
$913$	−2.91082e6	−0.115568
$914$	−1.16021e7	−0.459381
$915$	0	0
$916$	−6.11977e6	−0.240988
$917$	−9.70523e6	−0.381138
$918$	3.01399e6	0.118042
$919$	1.85935e7	0.726227	0.363114	−	0.931745i	$-0.381714\pi$
$919$	1.85935e7	0.726227	0.363114	+	0.931745i	$0.381714\pi$
$920$	0	0
$921$	1.08491e7	0.421448
$922$	−9.12710e6	−0.353595
$923$	−5.26181e7	−2.03297
$924$	1.09451e6	0.0421734
$925$	0	0
$926$	−1.62654e7	−0.623359
$927$	−1.42608e7	−0.545060
$928$	4.24927e7	1.61974
$929$	−2.77987e7	−1.05678	−0.528391	−	0.849001i	$-0.677204\pi$
$929$	−2.77987e7	−1.05678	−0.528391	+	0.849001i	$0.677204\pi$
$930$	0	0
$931$	2.41641e7	0.913684
$932$	5.49004e7	2.07031
$933$	5.73187e6	0.215572
$934$	3.91437e7	1.46823
$935$	0	0
$936$	−3.45788e6	−0.129009
$937$	−4.55794e7	−1.69598	−0.847988	−	0.530015i	$-0.822186\pi$
$937$	−4.55794e7	−1.69598	−0.847988	+	0.530015i	$0.822186\pi$
$938$	−5.76752e6	−0.214034
$939$	−1.67873e7	−0.621322
$940$	0	0
$941$	−2.38944e7	−0.879676	−0.439838	−	0.898077i	$-0.644964\pi$
$941$	−2.38944e7	−0.879676	−0.439838	+	0.898077i	$0.644964\pi$
$942$	8.97377e6	0.329494
$943$	−4.44186e7	−1.62662
$944$	−3.37609e7	−1.23306
$945$	0	0
$946$	796694.	0.0289443
$947$	−3.52637e7	−1.27777	−0.638885	−	0.769302i	$-0.720604\pi$
$947$	−3.52637e7	−1.27777	−0.638885	+	0.769302i	$0.720604\pi$
$948$	9.28644e6	0.335605
$949$	2.15992e7	0.778523
$950$	0	0
$951$	−2.11875e7	−0.759678
$952$	623959.	0.0223133
$953$	−5.08550e7	−1.81385	−0.906924	−	0.421293i	$-0.861576\pi$
$953$	−5.08550e7	−1.81385	−0.906924	+	0.421293i	$0.861576\pi$
$954$	−1.02285e7	−0.363866
$955$	0	0
$956$	−2.19326e7	−0.776149
$957$	−5.58912e6	−0.197271
$958$	3.46836e7	1.22098
$959$	1.97526e6	0.0693551
$960$	0	0
$961$	−2.84608e7	−0.994120
$962$	4.40081e7	1.53319
$963$	1.01220e7	0.351724
$964$	1.63631e7	0.567119
$965$	0	0
$966$	−4.79775e6	−0.165423
$967$	−2.33318e7	−0.802384	−0.401192	−	0.915994i	$-0.631404\pi$
$967$	−2.33318e7	−0.802384	−0.401192	+	0.915994i	$0.631404\pi$
$968$	690887.	0.0236984
$969$	−6.69385e6	−0.229016
$970$	0	0
$971$	−2.99725e7	−1.02017	−0.510087	−	0.860123i	$-0.670387\pi$
$971$	−2.99725e7	−1.02017	−0.510087	+	0.860123i	$0.670387\pi$
$972$	2.22344e6	0.0754847
$973$	1.86956e6	0.0633077
$974$	1.97999e7	0.668754
$975$	0	0
$976$	2.43248e7	0.817380
$977$	−3.06238e7	−1.02642	−0.513208	−	0.858264i	$-0.671543\pi$
$977$	−3.06238e7	−1.02642	−0.513208	+	0.858264i	$0.671543\pi$
$978$	−2.35865e7	−0.788525
$979$	−6.75422e6	−0.225226
$980$	0	0
$981$	3.88812e6	0.128993
$982$	−9.91471e6	−0.328096
$983$	−2.10228e7	−0.693915	−0.346958	−	0.937881i	$-0.612785\pi$
$983$	−2.10228e7	−0.693915	−0.346958	+	0.937881i	$0.612785\pi$
$984$	7.88315e6	0.259545
$985$	0	0
$986$	−2.12193e7	−0.695085
$987$	−955887.	−0.0312330
$988$	5.11437e7	1.66686
$989$	−1.88789e6	−0.0613742
$990$	0	0
$991$	−2.43651e7	−0.788103	−0.394052	−	0.919088i	$-0.628927\pi$
$991$	−2.43651e7	−0.788103	−0.394052	+	0.919088i	$0.628927\pi$
$992$	3.39711e6	0.109605
$993$	1.78098e7	0.573175
$994$	1.29568e7	0.415941
$995$	0	0
$996$	−8.15238e6	−0.260397
$997$	1.04094e7	0.331655	0.165827	−	0.986155i	$-0.446971\pi$
$997$	1.04094e7	0.331655	0.165827	+	0.986155i	$0.446971\pi$
$998$	5.92971e6	0.188455
$999$	−4.24911e6	−0.134705

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.d.1.2		2
5.4	even	2		33.6.a.d.1.1	✓	2
15.14	odd	2		99.6.a.e.1.2		2
20.19	odd	2		528.6.a.q.1.1		2
55.54	odd	2		363.6.a.g.1.2		2
165.164	even	2		1089.6.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.d.1.1	✓	2	5.4	even	2
99.6.a.e.1.2		2	15.14	odd	2
363.6.a.g.1.2		2	55.54	odd	2
528.6.a.q.1.1		2	20.19	odd	2
825.6.a.d.1.2		2	1.1	even	1	trivial
1089.6.a.o.1.1		2	165.164	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+8.34590 q^{2} +9.00000 q^{3} +37.6541 q^{4} +75.1131 q^{6} +26.6918 q^{7} +47.1885 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+8.34590 q^{2} +9.00000 q^{3} +37.6541 q^{4} +75.1131 q^{6} +26.6918 q^{7} +47.1885 q^{8} +81.0000 q^{9} +121.000 q^{11} +338.887 q^{12} -904.666 q^{13} +222.767 q^{14} -811.100 q^{16} +495.384 q^{17} +676.018 q^{18} -1501.38 q^{19} +240.226 q^{21} +1009.85 q^{22} -2393.01 q^{23} +424.697 q^{24} -7550.25 q^{26} +729.000 q^{27} +1005.06 q^{28} -5132.34 q^{29} -410.309 q^{31} -8279.40 q^{32} +1089.00 q^{33} +4134.42 q^{34} +3049.98 q^{36} -5828.69 q^{37} -12530.4 q^{38} -8141.99 q^{39} +18561.8 q^{41} +2004.91 q^{42} +788.920 q^{43} +4556.15 q^{44} -19971.8 q^{46} -3979.11 q^{47} -7299.90 q^{48} -16094.5 q^{49} +4458.45 q^{51} -34064.4 q^{52} -15130.5 q^{53} +6084.16 q^{54} +1259.55 q^{56} -13512.5 q^{57} -42834.0 q^{58} +41623.5 q^{59} -29989.8 q^{61} -3424.40 q^{62} +2162.04 q^{63} -43143.8 q^{64} +9088.69 q^{66} -25890.3 q^{67} +18653.2 q^{68} -21537.1 q^{69} +58163.0 q^{71} +3822.27 q^{72} -23875.3 q^{73} -48645.7 q^{74} -56533.2 q^{76} +3229.71 q^{77} -67952.3 q^{78} +27402.8 q^{79} +6561.00 q^{81} +154915. q^{82} -24056.3 q^{83} +9045.50 q^{84} +6584.25 q^{86} -46191.1 q^{87} +5709.81 q^{88} -55820.0 q^{89} -24147.2 q^{91} -90106.5 q^{92} -3692.78 q^{93} -33209.3 q^{94} -74514.6 q^{96} +101188. q^{97} -134324. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 18 q^{3} + 93 q^{4} - 9 q^{6} + 18 q^{7} - 171 q^{8} + 162 q^{9}+O(q^{10}) \) 2 * q - q^2 + 18 * q^3 + 93 * q^4 - 9 * q^6 + 18 * q^7 - 171 * q^8 + 162 * q^9	\( 2 q - q^{2} + 18 q^{3} + 93 q^{4} - 9 q^{6} + 18 q^{7} - 171 q^{8} + 162 q^{9} + 242 q^{11} + 837 q^{12} + 66 q^{13} + 304 q^{14} - 543 q^{16} + 920 q^{17} - 81 q^{18} - 2932 q^{19} + 162 q^{21} - 121 q^{22} - 5246 q^{23} - 1539 q^{24} - 16622 q^{26} + 1458 q^{27} + 524 q^{28} - 12600 q^{29} + 9936 q^{31} - 3803 q^{32} + 2178 q^{33} + 166 q^{34} + 7533 q^{36} - 5996 q^{37} + 840 q^{38} + 594 q^{39} + 24244 q^{41} + 2736 q^{42} - 20360 q^{43} + 11253 q^{44} + 6692 q^{46} + 5806 q^{47} - 4887 q^{48} - 32826 q^{49} + 8280 q^{51} + 19658 q^{52} - 40770 q^{53} - 729 q^{54} + 3156 q^{56} - 26388 q^{57} + 26958 q^{58} + 18212 q^{59} - 11398 q^{61} - 100120 q^{62} + 1458 q^{63} - 93559 q^{64} - 1089 q^{66} - 65368 q^{67} + 42154 q^{68} - 47214 q^{69} + 61446 q^{71} - 13851 q^{72} - 53412 q^{73} - 47082 q^{74} - 135712 q^{76} + 2178 q^{77} - 149598 q^{78} + 17122 q^{79} + 13122 q^{81} + 101810 q^{82} + 14304 q^{83} + 4716 q^{84} + 204240 q^{86} - 113400 q^{87} - 20691 q^{88} - 58140 q^{89} - 32584 q^{91} - 248008 q^{92} + 89424 q^{93} - 124660 q^{94} - 34227 q^{96} + 183056 q^{97} + 22047 q^{98} + 19602 q^{99}+O(q^{100}) \) 2 * q - q^2 + 18 * q^3 + 93 * q^4 - 9 * q^6 + 18 * q^7 - 171 * q^8 + 162 * q^9 + 242 * q^11 + 837 * q^12 + 66 * q^13 + 304 * q^14 - 543 * q^16 + 920 * q^17 - 81 * q^18 - 2932 * q^19 + 162 * q^21 - 121 * q^22 - 5246 * q^23 - 1539 * q^24 - 16622 * q^26 + 1458 * q^27 + 524 * q^28 - 12600 * q^29 + 9936 * q^31 - 3803 * q^32 + 2178 * q^33 + 166 * q^34 + 7533 * q^36 - 5996 * q^37 + 840 * q^38 + 594 * q^39 + 24244 * q^41 + 2736 * q^42 - 20360 * q^43 + 11253 * q^44 + 6692 * q^46 + 5806 * q^47 - 4887 * q^48 - 32826 * q^49 + 8280 * q^51 + 19658 * q^52 - 40770 * q^53 - 729 * q^54 + 3156 * q^56 - 26388 * q^57 + 26958 * q^58 + 18212 * q^59 - 11398 * q^61 - 100120 * q^62 + 1458 * q^63 - 93559 * q^64 - 1089 * q^66 - 65368 * q^67 + 42154 * q^68 - 47214 * q^69 + 61446 * q^71 - 13851 * q^72 - 53412 * q^73 - 47082 * q^74 - 135712 * q^76 + 2178 * q^77 - 149598 * q^78 + 17122 * q^79 + 13122 * q^81 + 101810 * q^82 + 14304 * q^83 + 4716 * q^84 + 204240 * q^86 - 113400 * q^87 - 20691 * q^88 - 58140 * q^89 - 32584 * q^91 - 248008 * q^92 + 89424 * q^93 - 124660 * q^94 - 34227 * q^96 + 183056 * q^97 + 22047 * q^98 + 19602 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more