LMFDB - Embedded newform 825.6.a.s.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 229 x^{7} + 267 x^{6} + 16434 x^{5} - 16568 x^{4} - 405504 x^{3} + 202288 x^{2} + \cdots + 1190400 $ x^9 - x^8 - 229x^7 + 267x^6 + 16434x^5 - 16568x^4 - 405504x^3 + 202288x^2 + 2184608*x + 1190400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$6.71211$ 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)

$f(q)$	$=$	$q+6.71211 q^{2} -9.00000 q^{3} +13.0525 q^{4} -60.4090 q^{6} -177.172 q^{7} -127.178 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+6.71211 q^{2} -9.00000 q^{3} +13.0525 q^{4} -60.4090 q^{6} -177.172 q^{7} -127.178 q^{8} +81.0000 q^{9} -121.000 q^{11} -117.472 q^{12} -783.433 q^{13} -1189.20 q^{14} -1271.31 q^{16} -1335.12 q^{17} +543.681 q^{18} -915.496 q^{19} +1594.55 q^{21} -812.166 q^{22} -1684.59 q^{23} +1144.60 q^{24} -5258.49 q^{26} -729.000 q^{27} -2312.53 q^{28} +1720.16 q^{29} +7741.92 q^{31} -4463.49 q^{32} +1089.00 q^{33} -8961.50 q^{34} +1057.25 q^{36} -9013.63 q^{37} -6144.91 q^{38} +7050.90 q^{39} +13835.8 q^{41} +10702.8 q^{42} +11885.9 q^{43} -1579.35 q^{44} -11307.2 q^{46} -20567.8 q^{47} +11441.8 q^{48} +14583.0 q^{49} +12016.1 q^{51} -10225.7 q^{52} -3483.63 q^{53} -4893.13 q^{54} +22532.4 q^{56} +8239.46 q^{57} +11545.9 q^{58} -22223.3 q^{59} -40834.7 q^{61} +51964.6 q^{62} -14350.9 q^{63} +10722.5 q^{64} +7309.49 q^{66} +2692.02 q^{67} -17426.6 q^{68} +15161.3 q^{69} +13211.7 q^{71} -10301.4 q^{72} -37232.0 q^{73} -60500.5 q^{74} -11949.5 q^{76} +21437.8 q^{77} +47326.4 q^{78} -46277.2 q^{79} +6561.00 q^{81} +92867.6 q^{82} +401.818 q^{83} +20812.8 q^{84} +79779.7 q^{86} -15481.5 q^{87} +15388.5 q^{88} -33472.8 q^{89} +138803. q^{91} -21988.1 q^{92} -69677.3 q^{93} -138053. q^{94} +40171.4 q^{96} +168527. q^{97} +97882.6 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) $ 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9	\( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9} - 1089 q^{11} - 1539 q^{12} + 723 q^{13} + 720 q^{14} + 4147 q^{16} - 2804 q^{17} + 81 q^{18} - 1601 q^{19} + 513 q^{21} - 121 q^{22} - 2392 q^{23} + 1593 q^{24} - 1987 q^{26} - 6561 q^{27} + 4436 q^{28} + 5966 q^{29} + 21575 q^{31} - 25493 q^{32} + 9801 q^{33} - 3098 q^{34} + 13851 q^{36} - 10228 q^{37} + 12765 q^{38} - 6507 q^{39} + 13304 q^{41} - 6480 q^{42} - 13829 q^{43} - 20691 q^{44} + 81283 q^{46} - 13998 q^{47} - 37323 q^{48} - 19468 q^{49} + 25236 q^{51} + 37131 q^{52} - 44166 q^{53} - 729 q^{54} + 79160 q^{56} + 14409 q^{57} - 7635 q^{58} + 11626 q^{59} + 49481 q^{61} - 94479 q^{62} - 4617 q^{63} + 109367 q^{64} + 1089 q^{66} + 26567 q^{67} - 119506 q^{68} + 21528 q^{69} + 78454 q^{71} - 14337 q^{72} - 100086 q^{73} + 162360 q^{74} + 194115 q^{76} + 6897 q^{77} + 17883 q^{78} - 8478 q^{79} + 59049 q^{81} - 52700 q^{82} - 157476 q^{83} - 39924 q^{84} + 251663 q^{86} - 53694 q^{87} + 21417 q^{88} - 65548 q^{89} - 106849 q^{91} - 350115 q^{92} - 194175 q^{93} - 35742 q^{94} + 229437 q^{96} + 116757 q^{97} - 14949 q^{98} - 88209 q^{99}+O(q^{100}) \) 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9 - 1089 * q^11 - 1539 * q^12 + 723 * q^13 + 720 * q^14 + 4147 * q^16 - 2804 * q^17 + 81 * q^18 - 1601 * q^19 + 513 * q^21 - 121 * q^22 - 2392 * q^23 + 1593 * q^24 - 1987 * q^26 - 6561 * q^27 + 4436 * q^28 + 5966 * q^29 + 21575 * q^31 - 25493 * q^32 + 9801 * q^33 - 3098 * q^34 + 13851 * q^36 - 10228 * q^37 + 12765 * q^38 - 6507 * q^39 + 13304 * q^41 - 6480 * q^42 - 13829 * q^43 - 20691 * q^44 + 81283 * q^46 - 13998 * q^47 - 37323 * q^48 - 19468 * q^49 + 25236 * q^51 + 37131 * q^52 - 44166 * q^53 - 729 * q^54 + 79160 * q^56 + 14409 * q^57 - 7635 * q^58 + 11626 * q^59 + 49481 * q^61 - 94479 * q^62 - 4617 * q^63 + 109367 * q^64 + 1089 * q^66 + 26567 * q^67 - 119506 * q^68 + 21528 * q^69 + 78454 * q^71 - 14337 * q^72 - 100086 * q^73 + 162360 * q^74 + 194115 * q^76 + 6897 * q^77 + 17883 * q^78 - 8478 * q^79 + 59049 * q^81 - 52700 * q^82 - 157476 * q^83 - 39924 * q^84 + 251663 * q^86 - 53694 * q^87 + 21417 * q^88 - 65548 * q^89 - 106849 * q^91 - 350115 * q^92 - 194175 * q^93 - 35742 * q^94 + 229437 * q^96 + 116757 * q^97 - 14949 * q^98 - 88209 * 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 $

$2$	6.71211	1.18655	0.593273	−	0.805002i	$-0.297836\pi$
$2$	6.71211	1.18655	0.593273	+	0.805002i	$0.297836\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	13.0525	0.407889
$5$	0	0
$5$	0	0
$6$	−60.4090	−0.685052
$7$	−177.172	−1.36663	−0.683314	−	0.730124i	$-0.739462\pi$
$7$	−177.172	−1.36663	−0.683314	+	0.730124i	$0.739462\pi$
$8$	−127.178	−0.702566
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−117.472	−0.235495
$13$	−783.433	−1.28571	−0.642856	−	0.765987i	$-0.722251\pi$
$13$	−783.433	−1.28571	−0.642856	+	0.765987i	$0.722251\pi$
$14$	−1189.20	−1.62157
$15$	0	0
$16$	−1271.31	−1.24152
$17$	−1335.12	−1.12047	−0.560233	−	0.828335i	$-0.689289\pi$
$17$	−1335.12	−1.12047	−0.560233	+	0.828335i	$0.689289\pi$
$18$	543.681	0.395515
$19$	−915.496	−0.581798	−0.290899	−	0.956754i	$-0.593954\pi$
$19$	−915.496	−0.581798	−0.290899	+	0.956754i	$0.593954\pi$
$20$	0	0
$21$	1594.55	0.789023
$22$	−812.166	−0.357757
$23$	−1684.59	−0.664011	−0.332006	−	0.943277i	$-0.607725\pi$
$23$	−1684.59	−0.664011	−0.332006	+	0.943277i	$0.607725\pi$
$24$	1144.60	0.405627
$25$	0	0
$26$	−5258.49	−1.52555
$27$	−729.000	−0.192450
$28$	−2312.53	−0.557433
$29$	1720.16	0.379817	0.189909	−	0.981802i	$-0.439181\pi$
$29$	1720.16	0.379817	0.189909	+	0.981802i	$0.439181\pi$
$30$	0	0
$31$	7741.92	1.44692	0.723460	−	0.690366i	$-0.242551\pi$
$31$	7741.92	1.44692	0.723460	+	0.690366i	$0.242551\pi$
$32$	−4463.49	−0.770548
$33$	1089.00	0.174078
$34$	−8961.50	−1.32948
$35$	0	0
$36$	1057.25	0.135963
$37$	−9013.63	−1.08242	−0.541209	−	0.840888i	$-0.682033\pi$
$37$	−9013.63	−1.08242	−0.541209	+	0.840888i	$0.682033\pi$
$38$	−6144.91	−0.690330
$39$	7050.90	0.742306
$40$	0	0
$41$	13835.8	1.28542	0.642710	−	0.766109i	$-0.277810\pi$
$41$	13835.8	1.28542	0.642710	+	0.766109i	$0.277810\pi$
$42$	10702.8	0.936212
$43$	11885.9	0.980307	0.490153	−	0.871636i	$-0.336941\pi$
$43$	11885.9	0.980307	0.490153	+	0.871636i	$0.336941\pi$
$44$	−1579.35	−0.122983
$45$	0	0
$46$	−11307.2	−0.787880
$47$	−20567.8	−1.35814	−0.679068	−	0.734075i	$-0.737616\pi$
$47$	−20567.8	−1.35814	−0.679068	+	0.734075i	$0.737616\pi$
$48$	11441.8	0.716789
$49$	14583.0	0.867673
$50$	0	0
$51$	12016.1	0.646902
$52$	−10225.7	−0.524428
$53$	−3483.63	−0.170350	−0.0851750	−	0.996366i	$-0.527145\pi$
$53$	−3483.63	−0.170350	−0.0851750	+	0.996366i	$0.527145\pi$
$54$	−4893.13	−0.228351
$55$	0	0
$56$	22532.4	0.960147
$57$	8239.46	0.335901
$58$	11545.9	0.450670
$59$	−22223.3	−0.831147	−0.415574	−	0.909560i	$-0.636419\pi$
$59$	−22223.3	−0.831147	−0.415574	+	0.909560i	$0.636419\pi$
$60$	0	0
$61$	−40834.7	−1.40509	−0.702546	−	0.711638i	$-0.747954\pi$
$61$	−40834.7	−1.40509	−0.702546	+	0.711638i	$0.747954\pi$
$62$	51964.6	1.71684
$63$	−14350.9	−0.455543
$64$	10722.5	0.327225
$65$	0	0
$66$	7309.49	0.206551
$67$	2692.02	0.0732640	0.0366320	−	0.999329i	$-0.488337\pi$
$67$	2692.02	0.0732640	0.0366320	+	0.999329i	$0.488337\pi$
$68$	−17426.6	−0.457026
$69$	15161.3	0.383367
$70$	0	0
$71$	13211.7	0.311039	0.155519	−	0.987833i	$-0.450295\pi$
$71$	13211.7	0.311039	0.155519	+	0.987833i	$0.450295\pi$
$72$	−10301.4	−0.234189
$73$	−37232.0	−0.817728	−0.408864	−	0.912595i	$-0.634075\pi$
$73$	−37232.0	−0.817728	−0.408864	+	0.912595i	$0.634075\pi$
$74$	−60500.5	−1.28434
$75$	0	0
$76$	−11949.5	−0.237309
$77$	21437.8	0.412054
$78$	47326.4	0.880779
$79$	−46277.2	−0.834257	−0.417128	−	0.908848i	$-0.636963\pi$
$79$	−46277.2	−0.834257	−0.417128	+	0.908848i	$0.636963\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	92867.6	1.52521
$83$	401.818	0.00640227	0.00320113	−	0.999995i	$-0.498981\pi$
$83$	401.818	0.00640227	0.00320113	+	0.999995i	$0.498981\pi$
$84$	20812.8	0.321834
$85$	0	0
$86$	79779.7	1.16318
$87$	−15481.5	−0.219288
$88$	15388.5	0.211832
$89$	−33472.8	−0.447937	−0.223969	−	0.974596i	$-0.571901\pi$
$89$	−33472.8	−0.447937	−0.223969	+	0.974596i	$0.571901\pi$
$90$	0	0
$91$	138803.	1.75709
$92$	−21988.1	−0.270843
$93$	−69677.3	−0.835380
$94$	−138053.	−1.61149
$95$	0	0
$96$	40171.4	0.444876
$97$	168527.	1.81861	0.909305	−	0.416130i	$-0.136614\pi$
$97$	168527.	1.81861	0.909305	+	0.416130i	$0.136614\pi$
$98$	97882.6	1.02953
$99$	−9801.00	−0.100504
$100$	0	0
$101$	176215.	1.71885	0.859425	−	0.511261i	$-0.170821\pi$
$101$	176215.	1.71885	0.859425	+	0.511261i	$0.170821\pi$
$102$	80653.5	0.767578
$103$	−66485.7	−0.617497	−0.308749	−	0.951144i	$-0.599910\pi$
$103$	−66485.7	−0.617497	−0.308749	+	0.951144i	$0.599910\pi$
$104$	99635.5	0.903297
$105$	0	0
$106$	−23382.5	−0.202128
$107$	87124.1	0.735663	0.367831	−	0.929892i	$-0.380100\pi$
$107$	87124.1	0.735663	0.367831	+	0.929892i	$0.380100\pi$
$108$	−9515.24	−0.0784983
$109$	17894.5	0.144263	0.0721314	−	0.997395i	$-0.477020\pi$
$109$	17894.5	0.144263	0.0721314	+	0.997395i	$0.477020\pi$
$110$	0	0
$111$	81122.7	0.624935
$112$	225241.	1.69669
$113$	−32973.4	−0.242922	−0.121461	−	0.992596i	$-0.538758\pi$
$113$	−32973.4	−0.242922	−0.121461	+	0.992596i	$0.538758\pi$
$114$	55304.2	0.398562
$115$	0	0
$116$	22452.4	0.154923
$117$	−63458.1	−0.428570
$118$	−149165.	−0.986194
$119$	236547.	1.53126
$120$	0	0
$121$	14641.0	0.0909091
$122$	−274087.	−1.66721
$123$	−124522.	−0.742138
$124$	101051.	0.590183
$125$	0	0
$126$	−96325.2	−0.540522
$127$	297307.	1.63567	0.817836	−	0.575451i	$-0.195174\pi$
$127$	297307.	1.63567	0.817836	+	0.575451i	$0.195174\pi$
$128$	214803.	1.15882
$129$	−106973.	−0.565980
$130$	0	0
$131$	44531.9	0.226722	0.113361	−	0.993554i	$-0.463838\pi$
$131$	44531.9	0.226722	0.113361	+	0.993554i	$0.463838\pi$
$132$	14214.1	0.0710044
$133$	162200.	0.795102
$134$	18069.1	0.0869310
$135$	0	0
$136$	169798.	0.787202
$137$	193093.	0.878953	0.439476	−	0.898254i	$-0.355164\pi$
$137$	193093.	0.878953	0.439476	+	0.898254i	$0.355164\pi$
$138$	101765.	0.454882
$139$	−45294.3	−0.198841	−0.0994206	−	0.995046i	$-0.531699\pi$
$139$	−45294.3	−0.198841	−0.0994206	+	0.995046i	$0.531699\pi$
$140$	0	0
$141$	185110.	0.784120
$142$	88678.7	0.369061
$143$	94795.4	0.387657
$144$	−102976.	−0.413839
$145$	0	0
$146$	−249905.	−0.970271
$147$	−131247.	−0.500951
$148$	−117650.	−0.441507
$149$	−461266.	−1.70210	−0.851051	−	0.525083i	$-0.824034\pi$
$149$	−461266.	−1.70210	−0.851051	+	0.525083i	$0.824034\pi$
$150$	0	0
$151$	−356514.	−1.27243	−0.636216	−	0.771511i	$-0.719501\pi$
$151$	−356514.	−1.27243	−0.636216	+	0.771511i	$0.719501\pi$
$152$	116431.	0.408752
$153$	−108145.	−0.373489
$154$	143893.	0.488921
$155$	0	0
$156$	92031.5	0.302779
$157$	−92075.8	−0.298124	−0.149062	−	0.988828i	$-0.547625\pi$
$157$	−92075.8	−0.298124	−0.149062	+	0.988828i	$0.547625\pi$
$158$	−310618.	−0.989883
$159$	31352.7	0.0983517
$160$	0	0
$161$	298463.	0.907457
$162$	44038.2	0.131838
$163$	−497333.	−1.46615	−0.733074	−	0.680149i	$-0.761915\pi$
$163$	−497333.	−1.46615	−0.733074	+	0.680149i	$0.761915\pi$
$164$	180591.	0.524309
$165$	0	0
$166$	2697.05	0.00759658
$167$	−292934.	−0.812792	−0.406396	−	0.913697i	$-0.633215\pi$
$167$	−292934.	−0.812792	−0.406396	+	0.913697i	$0.633215\pi$
$168$	−202792.	−0.554341
$169$	242474.	0.653054
$170$	0	0
$171$	−74155.1	−0.193933
$172$	155141.	0.399857
$173$	446012.	1.13300	0.566502	−	0.824060i	$-0.308296\pi$
$173$	446012.	1.13300	0.566502	+	0.824060i	$0.308296\pi$
$174$	−103913.	−0.260195
$175$	0	0
$176$	153829.	0.374331
$177$	200009.	0.479863
$178$	−224673.	−0.531498
$179$	−414071.	−0.965922	−0.482961	−	0.875642i	$-0.660439\pi$
$179$	−414071.	−0.965922	−0.482961	+	0.875642i	$0.660439\pi$
$180$	0	0
$181$	−560334.	−1.27131	−0.635654	−	0.771974i	$-0.719269\pi$
$181$	−560334.	−1.27131	−0.635654	+	0.771974i	$0.719269\pi$
$182$	931658.	2.08487
$183$	367512.	0.811230
$184$	214243.	0.466512
$185$	0	0
$186$	−467682.	−0.991216
$187$	161550.	0.337833
$188$	−268460.	−0.553969
$189$	129159.	0.263008
$190$	0	0
$191$	720559.	1.42918	0.714589	−	0.699545i	$-0.246614\pi$
$191$	720559.	1.42918	0.714589	+	0.699545i	$0.246614\pi$
$192$	−96502.7	−0.188924
$193$	−432118.	−0.835044	−0.417522	−	0.908667i	$-0.637101\pi$
$193$	−432118.	−0.835044	−0.417522	+	0.908667i	$0.637101\pi$
$194$	1.13117e6	2.15786
$195$	0	0
$196$	190344.	0.353915
$197$	786560.	1.44400	0.721998	−	0.691895i	$-0.243224\pi$
$197$	786560.	1.44400	0.721998	+	0.691895i	$0.243224\pi$
$198$	−65785.4	−0.119252
$199$	284214.	0.508761	0.254380	−	0.967104i	$-0.418129\pi$
$199$	284214.	0.508761	0.254380	+	0.967104i	$0.418129\pi$
$200$	0	0
$201$	−24228.1	−0.0422990
$202$	1.18277e6	2.03949
$203$	−304765.	−0.519069
$204$	156840.	0.263864
$205$	0	0
$206$	−446259.	−0.732688
$207$	−136452.	−0.221337
$208$	995988.	1.59623
$209$	110775.	0.175419
$210$	0	0
$211$	1.03608e6	1.60209	0.801045	−	0.598604i	$-0.204278\pi$
$211$	1.03608e6	1.60209	0.801045	+	0.598604i	$0.204278\pi$
$212$	−45469.9	−0.0694840
$213$	−118906.	−0.179578
$214$	584787.	0.872897
$215$	0	0
$216$	92712.8	0.135209
$217$	−1.37165e6	−1.97740
$218$	120110.	0.171174
$219$	335088.	0.472116
$220$	0	0
$221$	1.04598e6	1.44060
$222$	544504.	0.741513
$223$	1.13339e6	1.52622	0.763110	−	0.646269i	$-0.223672\pi$
$223$	1.13339e6	1.52622	0.763110	+	0.646269i	$0.223672\pi$
$224$	790807.	1.05305
$225$	0	0
$226$	−221321.	−0.288238
$227$	−681222.	−0.877453	−0.438727	−	0.898621i	$-0.644570\pi$
$227$	−681222.	−0.877453	−0.438727	+	0.898621i	$0.644570\pi$
$228$	107545.	0.137011
$229$	−620686.	−0.782137	−0.391069	−	0.920361i	$-0.627895\pi$
$229$	−620686.	−0.782137	−0.391069	+	0.920361i	$0.627895\pi$
$230$	0	0
$231$	−192941.	−0.237899
$232$	−218767.	−0.266847
$233$	−31388.4	−0.0378774	−0.0189387	−	0.999821i	$-0.506029\pi$
$233$	−31388.4	−0.0378774	−0.0189387	+	0.999821i	$0.506029\pi$
$234$	−425938.	−0.508518
$235$	0	0
$236$	−290068.	−0.339016
$237$	416495.	0.481658
$238$	1.58773e6	1.81691
$239$	−195859.	−0.221794	−0.110897	−	0.993832i	$-0.535372\pi$
$239$	−195859.	−0.221794	−0.110897	+	0.993832i	$0.535372\pi$
$240$	0	0
$241$	−1.33370e6	−1.47916	−0.739578	−	0.673071i	$-0.764975\pi$
$241$	−1.33370e6	−1.47916	−0.739578	+	0.673071i	$0.764975\pi$
$242$	98272.0	0.107868
$243$	−59049.0	−0.0641500
$244$	−532993.	−0.573122
$245$	0	0
$246$	−835808.	−0.880580
$247$	717229.	0.748024
$248$	−984602.	−1.01656
$249$	−3616.36	−0.00369635
$250$	0	0
$251$	−98131.5	−0.0983160	−0.0491580	−	0.998791i	$-0.515654\pi$
$251$	−98131.5	−0.0983160	−0.0491580	+	0.998791i	$0.515654\pi$
$252$	−187315.	−0.185811
$253$	203836.	0.200207
$254$	1.99556e6	1.94080
$255$	0	0
$256$	1.09866e6	1.04776
$257$	−717413.	−0.677543	−0.338771	−	0.940869i	$-0.610011\pi$
$257$	−717413.	−0.677543	−0.338771	+	0.940869i	$0.610011\pi$
$258$	−718017.	−0.671561
$259$	1.59696e6	1.47926
$260$	0	0
$261$	139333.	0.126606
$262$	298903.	0.269015
$263$	−1.90624e6	−1.69937	−0.849685	−	0.527290i	$-0.823208\pi$
$263$	−1.90624e6	−1.69937	−0.849685	+	0.527290i	$0.823208\pi$
$264$	−138497.	−0.122301
$265$	0	0
$266$	1.08871e6	0.943424
$267$	301255.	0.258617
$268$	35137.4	0.0298836
$269$	1.80790e6	1.52333	0.761665	−	0.647971i	$-0.224382\pi$
$269$	1.80790e6	1.52333	0.761665	+	0.647971i	$0.224382\pi$
$270$	0	0
$271$	1.27169e6	1.05186	0.525929	−	0.850529i	$-0.323718\pi$
$271$	1.27169e6	1.05186	0.525929	+	0.850529i	$0.323718\pi$
$272$	1.69736e6	1.39108
$273$	−1.24922e6	−1.01446
$274$	1.29606e6	1.04292
$275$	0	0
$276$	197893.	0.156371
$277$	1.80095e6	1.41027	0.705134	−	0.709074i	$-0.250887\pi$
$277$	1.80095e6	1.41027	0.705134	+	0.709074i	$0.250887\pi$
$278$	−304020.	−0.235934
$279$	627096.	0.482307
$280$	0	0
$281$	1.54030e6	1.16370	0.581849	−	0.813297i	$-0.302330\pi$
$281$	1.54030e6	1.16370	0.581849	+	0.813297i	$0.302330\pi$
$282$	1.24248e6	0.930394
$283$	−1.87910e6	−1.39471	−0.697354	−	0.716727i	$-0.745640\pi$
$283$	−1.87910e6	−1.39471	−0.697354	+	0.716727i	$0.745640\pi$
$284$	172446.	0.126869
$285$	0	0
$286$	636277.	0.459972
$287$	−2.45132e6	−1.75669
$288$	−361543.	−0.256849
$289$	362696.	0.255445
$290$	0	0
$291$	−1.51674e6	−1.04998
$292$	−485969.	−0.333543
$293$	−307140.	−0.209010	−0.104505	−	0.994524i	$-0.533326\pi$
$293$	−307140.	−0.209010	−0.104505	+	0.994524i	$0.533326\pi$
$294$	−880944.	−0.594401
$295$	0	0
$296$	1.14634e6	0.760471
$297$	88209.0	0.0580259
$298$	−3.09607e6	−2.01962
$299$	1.31977e6	0.853727
$300$	0	0
$301$	−2.10586e6	−1.33971
$302$	−2.39296e6	−1.50980
$303$	−1.58593e6	−0.992379
$304$	1.16388e6	0.722311
$305$	0	0
$306$	−725881.	−0.443161
$307$	−1.32237e6	−0.800771	−0.400386	−	0.916347i	$-0.631124\pi$
$307$	−1.32237e6	−0.800771	−0.400386	+	0.916347i	$0.631124\pi$
$308$	279816.	0.168072
$309$	598371.	0.356512
$310$	0	0
$311$	2.22049e6	1.30181	0.650904	−	0.759160i	$-0.274390\pi$
$311$	2.22049e6	1.30181	0.650904	+	0.759160i	$0.274390\pi$
$312$	−896719.	−0.521519
$313$	−978241.	−0.564397	−0.282199	−	0.959356i	$-0.591064\pi$
$313$	−978241.	−0.564397	−0.282199	+	0.959356i	$0.591064\pi$
$314$	−618023.	−0.353737
$315$	0	0
$316$	−604032.	−0.340284
$317$	−2.64915e6	−1.48067	−0.740335	−	0.672239i	$-0.765333\pi$
$317$	−2.64915e6	−1.48067	−0.740335	+	0.672239i	$0.765333\pi$
$318$	210443.	0.116699
$319$	−208140.	−0.114519
$320$	0	0
$321$	−784117.	−0.424735
$322$	2.00332e6	1.07674
$323$	1.22230e6	0.651885
$324$	85637.2	0.0453210
$325$	0	0
$326$	−3.33815e6	−1.73965
$327$	−161051.	−0.0832901
$328$	−1.75961e6	−0.903093
$329$	3.64404e6	1.85607
$330$	0	0
$331$	−555272.	−0.278571	−0.139286	−	0.990252i	$-0.544481\pi$
$331$	−555272.	−0.278571	−0.139286	+	0.990252i	$0.544481\pi$
$332$	5244.71	0.00261142
$333$	−730104.	−0.360806
$334$	−1.96621e6	−0.964414
$335$	0	0
$336$	−2.02717e6	−0.979585
$337$	−1.96065e6	−0.940427	−0.470213	−	0.882553i	$-0.655823\pi$
$337$	−1.96065e6	−0.940427	−0.470213	+	0.882553i	$0.655823\pi$
$338$	1.62751e6	0.774878
$339$	296761.	0.140251
$340$	0	0
$341$	−936772.	−0.436263
$342$	−497738.	−0.230110
$343$	394034.	0.180841
$344$	−1.51163e6	−0.688730
$345$	0	0
$346$	2.99368e6	1.34436
$347$	899300.	0.400942	0.200471	−	0.979700i	$-0.435753\pi$
$347$	899300.	0.400942	0.200471	+	0.979700i	$0.435753\pi$
$348$	−202071.	−0.0894451
$349$	852454.	0.374634	0.187317	−	0.982299i	$-0.440021\pi$
$349$	852454.	0.374634	0.187317	+	0.982299i	$0.440021\pi$
$350$	0	0
$351$	571123.	0.247435
$352$	540083.	0.232329
$353$	−58325.4	−0.0249127	−0.0124564	−	0.999922i	$-0.503965\pi$
$353$	−58325.4	−0.0249127	−0.0124564	+	0.999922i	$0.503965\pi$
$354$	1.34249e6	0.569379
$355$	0	0
$356$	−436903.	−0.182709
$357$	−2.12892e6	−0.884074
$358$	−2.77929e6	−1.14611
$359$	4.35963e6	1.78531	0.892654	−	0.450743i	$-0.148841\pi$
$359$	4.35963e6	1.78531	0.892654	+	0.450743i	$0.148841\pi$
$360$	0	0
$361$	−1.63797e6	−0.661511
$362$	−3.76103e6	−1.50846
$363$	−131769.	−0.0524864
$364$	1.81171e6	0.716698
$365$	0	0
$366$	2.46678e6	0.962562
$367$	−1.51064e6	−0.585459	−0.292730	−	0.956195i	$-0.594564\pi$
$367$	−1.51064e6	−0.585459	−0.292730	+	0.956195i	$0.594564\pi$
$368$	2.14164e6	0.824381
$369$	1.12070e6	0.428474
$370$	0	0
$371$	617202.	0.232805
$372$	−909460.	−0.340742
$373$	1.33021e6	0.495050	0.247525	−	0.968881i	$-0.420383\pi$
$373$	1.33021e6	0.495050	0.247525	+	0.968881i	$0.420383\pi$
$374$	1.08434e6	0.400855
$375$	0	0
$376$	2.61577e6	0.954180
$377$	−1.34763e6	−0.488335
$378$	866927.	0.312071
$379$	−4.17882e6	−1.49436	−0.747181	−	0.664621i	$-0.768593\pi$
$379$	−4.17882e6	−1.49436	−0.747181	+	0.664621i	$0.768593\pi$
$380$	0	0
$381$	−2.67577e6	−0.944356
$382$	4.83647e6	1.69578
$383$	−1.31585e6	−0.458362	−0.229181	−	0.973384i	$-0.573605\pi$
$383$	−1.31585e6	−0.458362	−0.229181	+	0.973384i	$0.573605\pi$
$384$	−1.93322e6	−0.669043
$385$	0	0
$386$	−2.90043e6	−0.990818
$387$	962760.	0.326769
$388$	2.19969e6	0.741792
$389$	−2.78879e6	−0.934420	−0.467210	−	0.884146i	$-0.654741\pi$
$389$	−2.78879e6	−0.934420	−0.467210	+	0.884146i	$0.654741\pi$
$390$	0	0
$391$	2.24914e6	0.744003
$392$	−1.85464e6	−0.609598
$393$	−400787.	−0.130898
$394$	5.27948e6	1.71337
$395$	0	0
$396$	−127927.	−0.0409944
$397$	−265601.	−0.0845772	−0.0422886	−	0.999105i	$-0.513465\pi$
$397$	−265601.	−0.0845772	−0.0422886	+	0.999105i	$0.513465\pi$
$398$	1.90768e6	0.603667
$399$	−1.45980e6	−0.459052
$400$	0	0
$401$	−69100.4	−0.0214595	−0.0107298	−	0.999942i	$-0.503415\pi$
$401$	−69100.4	−0.0214595	−0.0107298	+	0.999942i	$0.503415\pi$
$402$	−162622.	−0.0501896
$403$	−6.06528e6	−1.86032
$404$	2.30003e6	0.701101
$405$	0	0
$406$	−2.04562e6	−0.615899
$407$	1.09065e6	0.326362
$408$	−1.52818e6	−0.454491
$409$	−5.05776e6	−1.49503	−0.747515	−	0.664244i	$-0.768753\pi$
$409$	−5.05776e6	−1.49503	−0.747515	+	0.664244i	$0.768753\pi$
$410$	0	0
$411$	−1.73784e6	−0.507464
$412$	−867801.	−0.251870
$413$	3.93735e6	1.13587
$414$	−915882.	−0.262627
$415$	0	0
$416$	3.49685e6	0.990703
$417$	407648.	0.114801
$418$	743534.	0.208142
$419$	−4.26079e6	−1.18565	−0.592824	−	0.805332i	$-0.701987\pi$
$419$	−4.26079e6	−1.18565	−0.592824	+	0.805332i	$0.701987\pi$
$420$	0	0
$421$	−6.24929e6	−1.71840	−0.859202	−	0.511636i	$-0.829040\pi$
$421$	−6.24929e6	−1.71840	−0.859202	+	0.511636i	$0.829040\pi$
$422$	6.95428e6	1.90095
$423$	−1.66599e6	−0.452712
$424$	443041.	0.119682
$425$	0	0
$426$	−798108.	−0.213078
$427$	7.23477e6	1.92024
$428$	1.13718e6	0.300069
$429$	−853159.	−0.223814
$430$	0	0
$431$	1.70549e6	0.442238	0.221119	−	0.975247i	$-0.429029\pi$
$431$	1.70549e6	0.442238	0.221119	+	0.975247i	$0.429029\pi$
$432$	926786.	0.238930
$433$	−605782.	−0.155273	−0.0776367	−	0.996982i	$-0.524737\pi$
$433$	−605782.	−0.155273	−0.0776367	+	0.996982i	$0.524737\pi$
$434$	−9.20669e6	−2.34628
$435$	0	0
$436$	233568.	0.0588432
$437$	1.54224e6	0.386321
$438$	2.24915e6	0.560186
$439$	−2.77526e6	−0.687293	−0.343647	−	0.939099i	$-0.611662\pi$
$439$	−2.77526e6	−0.687293	−0.343647	+	0.939099i	$0.611662\pi$
$440$	0	0
$441$	1.18122e6	0.289224
$442$	7.02073e6	1.70933
$443$	−394928.	−0.0956111	−0.0478055	−	0.998857i	$-0.515223\pi$
$443$	−394928.	−0.0956111	−0.0478055	+	0.998857i	$0.515223\pi$
$444$	1.05885e6	0.254904
$445$	0	0
$446$	7.60744e6	1.81093
$447$	4.15139e6	0.982709
$448$	−1.89973e6	−0.447196
$449$	7.55318e6	1.76813	0.884064	−	0.467366i	$-0.154797\pi$
$449$	7.55318e6	1.76813	0.884064	+	0.467366i	$0.154797\pi$
$450$	0	0
$451$	−1.67413e6	−0.387569
$452$	−430384.	−0.0990854
$453$	3.20863e6	0.734639
$454$	−4.57244e6	−1.04114
$455$	0	0
$456$	−1.04788e6	−0.235993
$457$	−2.81470e6	−0.630436	−0.315218	−	0.949019i	$-0.602078\pi$
$457$	−2.81470e6	−0.630436	−0.315218	+	0.949019i	$0.602078\pi$
$458$	−4.16611e6	−0.928041
$459$	973305.	0.215634
$460$	0	0
$461$	1.15222e6	0.252512	0.126256	−	0.991998i	$-0.459704\pi$
$461$	1.15222e6	0.252512	0.126256	+	0.991998i	$0.459704\pi$
$462$	−1.29504e6	−0.282278
$463$	2.64702e6	0.573857	0.286929	−	0.957952i	$-0.407366\pi$
$463$	2.64702e6	0.573857	0.286929	+	0.957952i	$0.407366\pi$
$464$	−2.18686e6	−0.471549
$465$	0	0
$466$	−210683.	−0.0449432
$467$	−8.05020e6	−1.70810	−0.854052	−	0.520188i	$-0.825862\pi$
$467$	−8.05020e6	−1.70810	−0.854052	+	0.520188i	$0.825862\pi$
$468$	−828284.	−0.174809
$469$	−476950.	−0.100125
$470$	0	0
$471$	828682.	0.172122
$472$	2.82631e6	0.583936
$473$	−1.43820e6	−0.295574
$474$	2.79556e6	0.571509
$475$	0	0
$476$	3.08751e6	0.624585
$477$	−282174.	−0.0567834
$478$	−1.31463e6	−0.263168
$479$	2.33516e6	0.465026	0.232513	−	0.972593i	$-0.425305\pi$
$479$	2.33516e6	0.465026	0.232513	+	0.972593i	$0.425305\pi$
$480$	0	0
$481$	7.06157e6	1.39168
$482$	−8.95191e6	−1.75509
$483$	−2.68617e6	−0.523921
$484$	191101.	0.0370808
$485$	0	0
$486$	−396344.	−0.0761169
$487$	8.09784e6	1.54720	0.773600	−	0.633674i	$-0.218454\pi$
$487$	8.09784e6	1.54720	0.773600	+	0.633674i	$0.218454\pi$
$488$	5.19328e6	0.987170
$489$	4.47599e6	0.846481
$490$	0	0
$491$	5.53902e6	1.03688	0.518441	−	0.855114i	$-0.326513\pi$
$491$	5.53902e6	1.03688	0.518441	+	0.855114i	$0.326513\pi$
$492$	−1.62532e6	−0.302710
$493$	−2.29663e6	−0.425573
$494$	4.81413e6	0.887565
$495$	0	0
$496$	−9.84240e6	−1.79637
$497$	−2.34075e6	−0.425074
$498$	−24273.4	−0.00438589
$499$	−1.00373e7	−1.80454	−0.902271	−	0.431169i	$-0.858101\pi$
$499$	−1.00373e7	−1.80454	−0.902271	+	0.431169i	$0.858101\pi$
$500$	0	0
$501$	2.63641e6	0.469266
$502$	−658670.	−0.116656
$503$	2.38538e6	0.420376	0.210188	−	0.977661i	$-0.432592\pi$
$503$	2.38538e6	0.420376	0.210188	+	0.977661i	$0.432592\pi$
$504$	1.82513e6	0.320049
$505$	0	0
$506$	1.36817e6	0.237555
$507$	−2.18227e6	−0.377041
$508$	3.88059e6	0.667173
$509$	−8.86436e6	−1.51654	−0.758268	−	0.651943i	$-0.773954\pi$
$509$	−8.86436e6	−1.51654	−0.758268	+	0.651943i	$0.773954\pi$
$510$	0	0
$511$	6.59647e6	1.11753
$512$	500636.	0.0844009
$513$	667396.	0.111967
$514$	−4.81536e6	−0.803935
$515$	0	0
$516$	−1.39626e6	−0.230857
$517$	2.48870e6	0.409493
$518$	1.07190e7	1.75521
$519$	−4.01411e6	−0.654140
$520$	0	0
$521$	5.39366e6	0.870542	0.435271	−	0.900300i	$-0.356653\pi$
$521$	5.39366e6	0.870542	0.435271	+	0.900300i	$0.356653\pi$
$522$	935220.	0.150223
$523$	850386.	0.135945	0.0679723	−	0.997687i	$-0.478347\pi$
$523$	850386.	0.135945	0.0679723	+	0.997687i	$0.478347\pi$
$524$	581250.	0.0924773
$525$	0	0
$526$	−1.27949e7	−2.01638
$527$	−1.03364e7	−1.62123
$528$	−1.38446e6	−0.216120
$529$	−3.59849e6	−0.559089
$530$	0	0
$531$	−1.80008e6	−0.277049
$532$	2.11711e6	0.324313
$533$	−1.08394e7	−1.65268
$534$	2.02206e6	0.306860
$535$	0	0
$536$	−342365.	−0.0514728
$537$	3.72664e6	0.557675
$538$	1.21348e7	1.80750
$539$	−1.76454e6	−0.261613
$540$	0	0
$541$	7.13219e6	1.04768	0.523841	−	0.851816i	$-0.324499\pi$
$541$	7.13219e6	1.04768	0.523841	+	0.851816i	$0.324499\pi$
$542$	8.53570e6	1.24808
$543$	5.04301e6	0.733990
$544$	5.95931e6	0.863374
$545$	0	0
$546$	−8.38492e6	−1.20370
$547$	−1.39768e6	−0.199729	−0.0998644	−	0.995001i	$-0.531841\pi$
$547$	−1.39768e6	−0.199729	−0.0998644	+	0.995001i	$0.531841\pi$
$548$	2.52034e6	0.358515
$549$	−3.30761e6	−0.468364
$550$	0	0
$551$	−1.57480e6	−0.220977
$552$	−1.92819e6	−0.269341
$553$	8.19904e6	1.14012
$554$	1.20882e7	1.67335
$555$	0	0
$556$	−591201.	−0.0811052
$557$	1.91000e6	0.260852	0.130426	−	0.991458i	$-0.458365\pi$
$557$	1.91000e6	0.260852	0.130426	+	0.991458i	$0.458365\pi$
$558$	4.20914e6	0.572279
$559$	−9.31183e6	−1.26039
$560$	0	0
$561$	−1.45395e6	−0.195048
$562$	1.03387e7	1.38078
$563$	1.33043e7	1.76898	0.884488	−	0.466563i	$-0.154508\pi$
$563$	1.33043e7	1.76898	0.884488	+	0.466563i	$0.154508\pi$
$564$	2.41614e6	0.319834
$565$	0	0
$566$	−1.26127e7	−1.65488
$567$	−1.16243e6	−0.151848
$568$	−1.68024e6	−0.218525
$569$	692234.	0.0896339	0.0448170	−	0.998995i	$-0.485730\pi$
$569$	692234.	0.0896339	0.0448170	+	0.998995i	$0.485730\pi$
$570$	0	0
$571$	7.25834e6	0.931638	0.465819	−	0.884880i	$-0.345760\pi$
$571$	7.25834e6	0.931638	0.465819	+	0.884880i	$0.345760\pi$
$572$	1.23731e6	0.158121
$573$	−6.48503e6	−0.825136
$574$	−1.64536e7	−2.08439
$575$	0	0
$576$	868524.	0.109075
$577$	−1.41479e7	−1.76910	−0.884550	−	0.466446i	$-0.845534\pi$
$577$	−1.41479e7	−1.76910	−0.884550	+	0.466446i	$0.845534\pi$
$578$	2.43446e6	0.303098
$579$	3.88907e6	0.482113
$580$	0	0
$581$	−71190.9	−0.00874952
$582$	−1.01805e7	−1.24584
$583$	421519.	0.0513625
$584$	4.73509e6	0.574508
$585$	0	0
$586$	−2.06156e6	−0.248000
$587$	5.01437e6	0.600649	0.300325	−	0.953837i	$-0.402905\pi$
$587$	5.01437e6	0.600649	0.300325	+	0.953837i	$0.402905\pi$
$588$	−1.71309e6	−0.204333
$589$	−7.08769e6	−0.841815
$590$	0	0
$591$	−7.07904e6	−0.833692
$592$	1.14591e7	1.34384
$593$	−6.71458e6	−0.784119	−0.392060	−	0.919940i	$-0.628237\pi$
$593$	−6.71458e6	−0.784119	−0.392060	+	0.919940i	$0.628237\pi$
$594$	592069.	0.0688503
$595$	0	0
$596$	−6.02065e6	−0.694269
$597$	−2.55793e6	−0.293733
$598$	8.85842e6	1.01299
$599$	−1.12021e7	−1.27565	−0.637825	−	0.770182i	$-0.720166\pi$
$599$	−1.12021e7	−1.27565	−0.637825	+	0.770182i	$0.720166\pi$
$600$	0	0
$601$	1.05901e7	1.19596	0.597978	−	0.801512i	$-0.295971\pi$
$601$	1.05901e7	1.19596	0.597978	+	0.801512i	$0.295971\pi$
$602$	−1.41347e7	−1.58963
$603$	218053.	0.0244213
$604$	−4.65339e6	−0.519011
$605$	0	0
$606$	−1.06449e7	−1.17750
$607$	1.39277e7	1.53429	0.767147	−	0.641472i	$-0.221676\pi$
$607$	1.39277e7	1.53429	0.767147	+	0.641472i	$0.221676\pi$
$608$	4.08631e6	0.448303
$609$	2.74289e6	0.299685
$610$	0	0
$611$	1.61135e7	1.74617
$612$	−1.41156e6	−0.152342
$613$	1.91560e6	0.205898	0.102949	−	0.994687i	$-0.467172\pi$
$613$	1.91560e6	0.205898	0.102949	+	0.994687i	$0.467172\pi$
$614$	−8.87592e6	−0.950151
$615$	0	0
$616$	−2.72642e6	−0.289495
$617$	−524280.	−0.0554435	−0.0277217	−	0.999616i	$-0.508825\pi$
$617$	−524280.	−0.0554435	−0.0277217	+	0.999616i	$0.508825\pi$
$618$	4.01633e6	0.423018
$619$	9.20908e6	0.966028	0.483014	−	0.875613i	$-0.339542\pi$
$619$	9.20908e6	0.966028	0.483014	+	0.875613i	$0.339542\pi$
$620$	0	0
$621$	1.22807e6	0.127789
$622$	1.49042e7	1.54465
$623$	5.93045e6	0.612164
$624$	−8.96389e6	−0.921584
$625$	0	0
$626$	−6.56606e6	−0.669683
$627$	−996975.	−0.101278
$628$	−1.20182e6	−0.121601
$629$	1.20343e7	1.21281
$630$	0	0
$631$	1.98666e6	0.198633	0.0993164	−	0.995056i	$-0.468334\pi$
$631$	1.98666e6	0.198633	0.0993164	+	0.995056i	$0.468334\pi$
$632$	5.88545e6	0.586121
$633$	−9.32471e6	−0.924967
$634$	−1.77814e7	−1.75688
$635$	0	0
$636$	409229.	0.0401166
$637$	−1.14248e7	−1.11558
$638$	−1.39706e6	−0.135882
$639$	1.07015e6	0.103680
$640$	0	0
$641$	−1.31651e7	−1.26555	−0.632776	−	0.774335i	$-0.718085\pi$
$641$	−1.31651e7	−1.26555	−0.632776	+	0.774335i	$0.718085\pi$
$642$	−5.26308e6	−0.503967
$643$	7.46182e6	0.711733	0.355867	−	0.934537i	$-0.384186\pi$
$643$	7.46182e6	0.711733	0.355867	+	0.934537i	$0.384186\pi$
$644$	3.89568e6	0.370142
$645$	0	0
$646$	8.20421e6	0.773491
$647$	1.12027e7	1.05212	0.526058	−	0.850449i	$-0.323669\pi$
$647$	1.12027e7	1.05212	0.526058	+	0.850449i	$0.323669\pi$
$648$	−834415.	−0.0780629
$649$	2.68902e6	0.250600
$650$	0	0
$651$	1.23449e7	1.14165
$652$	−6.49141e6	−0.598026
$653$	−5.01727e6	−0.460452	−0.230226	−	0.973137i	$-0.573947\pi$
$653$	−5.01727e6	−0.460452	−0.230226	+	0.973137i	$0.573947\pi$
$654$	−1.08099e6	−0.0988275
$655$	0	0
$656$	−1.75896e7	−1.59587
$657$	−3.01579e6	−0.272576
$658$	2.44592e7	2.20231
$659$	1.59087e6	0.142699	0.0713495	−	0.997451i	$-0.477269\pi$
$659$	1.59087e6	0.142699	0.0713495	+	0.997451i	$0.477269\pi$
$660$	0	0
$661$	1.38029e7	1.22876	0.614378	−	0.789012i	$-0.289407\pi$
$661$	1.38029e7	1.22876	0.614378	+	0.789012i	$0.289407\pi$
$662$	−3.72705e6	−0.330537
$663$	−9.41381e6	−0.831729
$664$	−51102.4	−0.00449802
$665$	0	0
$666$	−4.90054e6	−0.428113
$667$	−2.89778e6	−0.252203
$668$	−3.82351e6	−0.331529
$669$	−1.02005e7	−0.881163
$670$	0	0
$671$	4.94100e6	0.423651
$672$	−7.11726e6	−0.607980
$673$	−1.14669e7	−0.975911	−0.487955	−	0.872869i	$-0.662257\pi$
$673$	−1.14669e7	−0.975911	−0.487955	+	0.872869i	$0.662257\pi$
$674$	−1.31601e7	−1.11586
$675$	0	0
$676$	3.16488e6	0.266374
$677$	−1.82188e7	−1.52774	−0.763869	−	0.645371i	$-0.776703\pi$
$677$	−1.82188e7	−1.52774	−0.763869	+	0.645371i	$0.776703\pi$
$678$	1.99189e6	0.166414
$679$	−2.98583e7	−2.48536
$680$	0	0
$681$	6.13100e6	0.506598
$682$	−6.28772e6	−0.517645
$683$	1.75425e7	1.43893	0.719464	−	0.694530i	$-0.244388\pi$
$683$	1.75425e7	1.43893	0.719464	+	0.694530i	$0.244388\pi$
$684$	−967907.	−0.0791031
$685$	0	0
$686$	2.64480e6	0.214577
$687$	5.58617e6	0.451567
$688$	−1.51107e7	−1.21707
$689$	2.72919e6	0.219021
$690$	0	0
$691$	7.62835e6	0.607765	0.303882	−	0.952710i	$-0.401717\pi$
$691$	7.62835e6	0.607765	0.303882	+	0.952710i	$0.401717\pi$
$692$	5.82155e6	0.462140
$693$	1.73646e6	0.137351
$694$	6.03621e6	0.475735
$695$	0	0
$696$	1.96890e6	0.154064
$697$	−1.84725e7	−1.44027
$698$	5.72177e6	0.444520
$699$	282496.	0.0218685
$700$	0	0
$701$	−1.99597e7	−1.53412	−0.767059	−	0.641577i	$-0.778281\pi$
$701$	−1.99597e7	−1.53412	−0.767059	+	0.641577i	$0.778281\pi$
$702$	3.83344e6	0.293593
$703$	8.25194e6	0.629749
$704$	−1.29743e6	−0.0986622
$705$	0	0
$706$	−391487.	−0.0295601
$707$	−3.12203e7	−2.34903
$708$	2.61061e6	0.195731
$709$	8.79537e6	0.657110	0.328555	−	0.944485i	$-0.393438\pi$
$709$	8.79537e6	0.657110	0.328555	+	0.944485i	$0.393438\pi$
$710$	0	0
$711$	−3.74846e6	−0.278086
$712$	4.25701e6	0.314706
$713$	−1.30420e7	−0.960771
$714$	−1.42895e7	−1.04899
$715$	0	0
$716$	−5.40464e6	−0.393989
$717$	1.76273e6	0.128053
$718$	2.92623e7	2.11835
$719$	−1.04268e7	−0.752190	−0.376095	−	0.926581i	$-0.622733\pi$
$719$	−1.04268e7	−0.752190	−0.376095	+	0.926581i	$0.622733\pi$
$720$	0	0
$721$	1.17794e7	0.843889
$722$	−1.09942e7	−0.784913
$723$	1.20033e7	0.853991
$724$	−7.31374e6	−0.518553
$725$	0	0
$726$	−884448.	−0.0622775
$727$	3.28724e6	0.230672	0.115336	−	0.993327i	$-0.463205\pi$
$727$	3.28724e6	0.230672	0.115336	+	0.993327i	$0.463205\pi$
$728$	−1.76526e7	−1.23447
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.58692e7	−1.09840
$732$	4.79694e6	0.330892
$733$	−5.45350e6	−0.374900	−0.187450	−	0.982274i	$-0.560022\pi$
$733$	−5.45350e6	−0.374900	−0.187450	+	0.982274i	$0.560022\pi$
$734$	−1.01396e7	−0.694674
$735$	0	0
$736$	7.51917e6	0.511653
$737$	−325734.	−0.0220899
$738$	7.52227e6	0.508403
$739$	2.58376e7	1.74037	0.870183	−	0.492729i	$-0.164001\pi$
$739$	2.58376e7	1.74037	0.870183	+	0.492729i	$0.164001\pi$
$740$	0	0
$741$	−6.45507e6	−0.431872
$742$	4.14273e6	0.276234
$743$	7.94543e6	0.528014	0.264007	−	0.964521i	$-0.414956\pi$
$743$	7.94543e6	0.528014	0.264007	+	0.964521i	$0.414956\pi$
$744$	8.86142e6	0.586909
$745$	0	0
$746$	8.92854e6	0.587399
$747$	32547.2	0.00213409
$748$	2.10862e6	0.137799
$749$	−1.54360e7	−1.00538
$750$	0	0
$751$	1.16319e7	0.752574	0.376287	−	0.926503i	$-0.377201\pi$
$751$	1.16319e7	0.752574	0.376287	+	0.926503i	$0.377201\pi$
$752$	2.61481e7	1.68615
$753$	883184.	0.0567628
$754$	−9.04546e6	−0.579432
$755$	0	0
$756$	1.68584e6	0.107278
$757$	1.46923e7	0.931856	0.465928	−	0.884823i	$-0.345721\pi$
$757$	1.46923e7	0.931856	0.465928	+	0.884823i	$0.345721\pi$
$758$	−2.80487e7	−1.77313
$759$	−1.83452e6	−0.115590
$760$	0	0
$761$	−1.52759e6	−0.0956194	−0.0478097	−	0.998856i	$-0.515224\pi$
$761$	−1.52759e6	−0.0956194	−0.0478097	+	0.998856i	$0.515224\pi$
$762$	−1.79600e7	−1.12052
$763$	−3.17041e6	−0.197154
$764$	9.40507e6	0.582946
$765$	0	0
$766$	−8.83212e6	−0.543867
$767$	1.74104e7	1.06862
$768$	−9.88792e6	−0.604926
$769$	1.53066e7	0.933389	0.466694	−	0.884419i	$-0.345445\pi$
$769$	1.53066e7	0.933389	0.466694	+	0.884419i	$0.345445\pi$
$770$	0	0
$771$	6.45672e6	0.391180
$772$	−5.64021e6	−0.340606
$773$	−2.13061e7	−1.28249	−0.641245	−	0.767336i	$-0.721582\pi$
$773$	−2.13061e7	−1.28249	−0.641245	+	0.767336i	$0.721582\pi$
$774$	6.46215e6	0.387726
$775$	0	0
$776$	−2.14329e7	−1.27769
$777$	−1.43727e7	−0.854054
$778$	−1.87187e7	−1.10873
$779$	−1.26666e7	−0.747855
$780$	0	0
$781$	−1.59862e6	−0.0937817
$782$	1.50965e7	0.882793
$783$	−1.25400e6	−0.0730959
$784$	−1.85395e7	−1.07723
$785$	0	0
$786$	−2.69013e6	−0.155316
$787$	1.60779e7	0.925319	0.462659	−	0.886536i	$-0.346895\pi$
$787$	1.60779e7	0.925319	0.462659	+	0.886536i	$0.346895\pi$
$788$	1.02665e7	0.588991
$789$	1.71562e7	0.981132
$790$	0	0
$791$	5.84197e6	0.331985
$792$	1.24647e6	0.0706105
$793$	3.19913e7	1.80654
$794$	−1.78274e6	−0.100355
$795$	0	0
$796$	3.70970e6	0.207518
$797$	−3.34929e7	−1.86770	−0.933849	−	0.357668i	$-0.883572\pi$
$797$	−3.34929e7	−1.86770	−0.933849	+	0.357668i	$0.883572\pi$
$798$	−9.79836e6	−0.544686
$799$	2.74605e7	1.52175
$800$	0	0
$801$	−2.71130e6	−0.149312
$802$	−463810.	−0.0254627
$803$	4.50507e6	0.246554
$804$	−316237.	−0.0172533
$805$	0	0
$806$	−4.07108e7	−2.20736
$807$	−1.62711e7	−0.879495
$808$	−2.24106e7	−1.20761
$809$	−3.65008e7	−1.96079	−0.980396	−	0.197037i	$-0.936868\pi$
$809$	−3.65008e7	−1.96079	−0.980396	+	0.197037i	$0.936868\pi$
$810$	0	0
$811$	−1.57753e7	−0.842222	−0.421111	−	0.907009i	$-0.638360\pi$
$811$	−1.57753e7	−0.842222	−0.421111	+	0.907009i	$0.638360\pi$
$812$	−3.97793e6	−0.211723
$813$	−1.14452e7	−0.607290
$814$	7.32056e6	0.387243
$815$	0	0
$816$	−1.52762e7	−0.803139
$817$	−1.08815e7	−0.570340
$818$	−3.39483e7	−1.77392
$819$	1.12430e7	0.585697
$820$	0	0
$821$	−1.08891e7	−0.563811	−0.281906	−	0.959442i	$-0.590966\pi$
$821$	−1.08891e7	−0.563811	−0.281906	+	0.959442i	$0.590966\pi$
$822$	−1.16646e7	−0.602129
$823$	−4.77752e6	−0.245869	−0.122934	−	0.992415i	$-0.539230\pi$
$823$	−4.77752e6	−0.245869	−0.122934	+	0.992415i	$0.539230\pi$
$824$	8.45552e6	0.433832
$825$	0	0
$826$	2.64279e7	1.34776
$827$	1.24682e7	0.633927	0.316964	−	0.948438i	$-0.397337\pi$
$827$	1.24682e7	0.633927	0.316964	+	0.948438i	$0.397337\pi$
$828$	−1.78104e6	−0.0902811
$829$	−2.92619e7	−1.47882	−0.739412	−	0.673253i	$-0.764897\pi$
$829$	−2.92619e7	−1.47882	−0.739412	+	0.673253i	$0.764897\pi$
$830$	0	0
$831$	−1.62085e7	−0.814219
$832$	−8.40038e6	−0.420717
$833$	−1.94701e7	−0.972199
$834$	2.73618e6	0.136217
$835$	0	0
$836$	1.44589e6	0.0715514
$837$	−5.64386e6	−0.278460
$838$	−2.85989e7	−1.40682
$839$	1.47124e7	0.721571	0.360786	−	0.932649i	$-0.382509\pi$
$839$	1.47124e7	0.721571	0.360786	+	0.932649i	$0.382509\pi$
$840$	0	0
$841$	−1.75522e7	−0.855739
$842$	−4.19459e7	−2.03896
$843$	−1.38627e7	−0.671861
$844$	1.35234e7	0.653475
$845$	0	0
$846$	−1.11823e7	−0.537163
$847$	−2.59398e6	−0.124239
$848$	4.42878e6	0.211492
$849$	1.69119e7	0.805235
$850$	0	0
$851$	1.51843e7	0.718739
$852$	−1.55201e6	−0.0732480
$853$	4.14062e7	1.94846	0.974232	−	0.225548i	$-0.0724171\pi$
$853$	4.14062e7	1.94846	0.974232	+	0.225548i	$0.0724171\pi$
$854$	4.85606e7	2.27845
$855$	0	0
$856$	−1.10803e7	−0.516852
$857$	−1.96593e7	−0.914359	−0.457179	−	0.889375i	$-0.651140\pi$
$857$	−1.96593e7	−0.914359	−0.457179	+	0.889375i	$0.651140\pi$
$858$	−5.72650e6	−0.265565
$859$	3.89269e7	1.79997	0.899987	−	0.435916i	$-0.143575\pi$
$859$	3.89269e7	1.79997	0.899987	+	0.435916i	$0.143575\pi$
$860$	0	0
$861$	2.20619e7	1.01423
$862$	1.14475e7	0.524736
$863$	−2.21492e7	−1.01235	−0.506176	−	0.862430i	$-0.668941\pi$
$863$	−2.21492e7	−1.01235	−0.506176	+	0.862430i	$0.668941\pi$
$864$	3.25389e6	0.148292
$865$	0	0
$866$	−4.06608e6	−0.184239
$867$	−3.26426e6	−0.147482
$868$	−1.79034e7	−0.806561
$869$	5.59955e6	0.251538
$870$	0	0
$871$	−2.10901e6	−0.0941963
$872$	−2.27579e6	−0.101354
$873$	1.36507e7	0.606203
$874$	1.03517e7	0.458387
$875$	0	0
$876$	4.37372e6	0.192571
$877$	1.54661e7	0.679017	0.339509	−	0.940603i	$-0.389739\pi$
$877$	1.54661e7	0.679017	0.339509	+	0.940603i	$0.389739\pi$
$878$	−1.86278e7	−0.815505
$879$	2.76426e6	0.120672
$880$	0	0
$881$	2.87591e7	1.24835	0.624174	−	0.781286i	$-0.285436\pi$
$881$	2.87591e7	1.24835	0.624174	+	0.781286i	$0.285436\pi$
$882$	7.92849e6	0.343178
$883$	2.15386e7	0.929642	0.464821	−	0.885405i	$-0.346119\pi$
$883$	2.15386e7	0.929642	0.464821	+	0.885405i	$0.346119\pi$
$884$	1.36526e7	0.587604
$885$	0	0
$886$	−2.65080e6	−0.113447
$887$	−2.32877e7	−0.993844	−0.496922	−	0.867795i	$-0.665537\pi$
$887$	−2.32877e7	−0.993844	−0.496922	+	0.867795i	$0.665537\pi$
$888$	−1.03170e7	−0.439058
$889$	−5.26746e7	−2.23536
$890$	0	0
$891$	−793881.	−0.0335013
$892$	1.47935e7	0.622529
$893$	1.88297e7	0.790161
$894$	2.78646e7	1.16603
$895$	0	0
$896$	−3.80570e7	−1.58367
$897$	−1.18779e7	−0.492900
$898$	5.06978e7	2.09796
$899$	1.33174e7	0.549565
$900$	0	0
$901$	4.65107e6	0.190872
$902$	−1.12370e7	−0.459868
$903$	1.89527e7	0.773485
$904$	4.19349e6	0.170669
$905$	0	0
$906$	2.15367e7	0.871682
$907$	6.21299e6	0.250774	0.125387	−	0.992108i	$-0.459983\pi$
$907$	6.21299e6	0.250774	0.125387	+	0.992108i	$0.459983\pi$
$908$	−8.89162e6	−0.357904
$909$	1.42734e7	0.572950
$910$	0	0
$911$	2.40141e7	0.958674	0.479337	−	0.877631i	$-0.340877\pi$
$911$	2.40141e7	0.958674	0.479337	+	0.877631i	$0.340877\pi$
$912$	−1.04749e7	−0.417027
$913$	−48620.0	−0.00193036
$914$	−1.88926e7	−0.748041
$915$	0	0
$916$	−8.10147e6	−0.319025
$917$	−7.88981e6	−0.309844
$918$	6.53293e6	0.255859
$919$	1.20620e7	0.471117	0.235559	−	0.971860i	$-0.424308\pi$
$919$	1.20620e7	0.471117	0.235559	+	0.971860i	$0.424308\pi$
$920$	0	0
$921$	1.19014e7	0.462325
$922$	7.73380e6	0.299616
$923$	−1.03505e7	−0.399906
$924$	−2.51835e6	−0.0970366
$925$	0	0
$926$	1.77671e7	0.680908
$927$	−5.38534e6	−0.205832
$928$	−7.67794e6	−0.292668
$929$	5.56585e6	0.211589	0.105794	−	0.994388i	$-0.466262\pi$
$929$	5.56585e6	0.211589	0.105794	+	0.994388i	$0.466262\pi$
$930$	0	0
$931$	−1.33507e7	−0.504811
$932$	−409696.	−0.0154498
$933$	−1.99844e7	−0.751600
$934$	−5.40338e7	−2.02674
$935$	0	0
$936$	8.07047e6	0.301099
$937$	−1.39429e7	−0.518806	−0.259403	−	0.965769i	$-0.583526\pi$
$937$	−1.39429e7	−0.518806	−0.259403	+	0.965769i	$0.583526\pi$
$938$	−3.20134e6	−0.118802
$939$	8.80417e6	0.325855
$940$	0	0
$941$	−1.76975e7	−0.651536	−0.325768	−	0.945450i	$-0.605623\pi$
$941$	−1.76975e7	−0.651536	−0.325768	+	0.945450i	$0.605623\pi$
$942$	5.56221e6	0.204230
$943$	−2.33077e7	−0.853534
$944$	2.82527e7	1.03188
$945$	0	0
$946$	−9.65334e6	−0.350711
$947$	1.08090e7	0.391659	0.195830	−	0.980638i	$-0.437260\pi$
$947$	1.08090e7	0.391659	0.195830	+	0.980638i	$0.437260\pi$
$948$	5.43629e6	0.196463
$949$	2.91688e7	1.05136
$950$	0	0
$951$	2.38423e7	0.854865
$952$	−3.00835e7	−1.07581
$953$	2.79930e7	0.998428	0.499214	−	0.866479i	$-0.333622\pi$
$953$	2.79930e7	0.998428	0.499214	+	0.866479i	$0.333622\pi$
$954$	−1.89398e6	−0.0673760
$955$	0	0
$956$	−2.55644e6	−0.0904673
$957$	1.87326e6	0.0661177
$958$	1.56738e7	0.551775
$959$	−3.42108e7	−1.20120
$960$	0	0
$961$	3.13082e7	1.09358
$962$	4.73981e7	1.65129
$963$	7.05705e6	0.245221
$964$	−1.74080e7	−0.603332
$965$	0	0
$966$	−1.80299e7	−0.621655
$967$	3.30769e7	1.13752	0.568760	−	0.822504i	$-0.307423\pi$
$967$	3.30769e7	1.13752	0.568760	+	0.822504i	$0.307423\pi$
$968$	−1.86201e6	−0.0638696
$969$	−1.10007e7	−0.376366
$970$	0	0
$971$	−3.56721e7	−1.21417	−0.607086	−	0.794636i	$-0.707662\pi$
$971$	−3.56721e7	−1.21417	−0.607086	+	0.794636i	$0.707662\pi$
$972$	−770735.	−0.0261661
$973$	8.02488e6	0.271742
$974$	5.43536e7	1.83582
$975$	0	0
$976$	5.19137e7	1.74444
$977$	4.13863e7	1.38714	0.693571	−	0.720389i	$-0.256037\pi$
$977$	4.13863e7	1.38714	0.693571	+	0.720389i	$0.256037\pi$
$978$	3.00434e7	1.00439
$979$	4.05021e6	0.135058
$980$	0	0
$981$	1.44946e6	0.0480876
$982$	3.71785e7	1.23031
$983$	2.54427e7	0.839805	0.419903	−	0.907569i	$-0.362064\pi$
$983$	2.54427e7	0.839805	0.419903	+	0.907569i	$0.362064\pi$
$984$	1.58365e7	0.521401
$985$	0	0
$986$	−1.54152e7	−0.504961
$987$	−3.27964e7	−1.07160
$988$	9.36161e6	0.305111
$989$	−2.00230e7	−0.650935
$990$	0	0
$991$	2.30502e7	0.745573	0.372786	−	0.927917i	$-0.378402\pi$
$991$	2.30502e7	0.745573	0.372786	+	0.927917i	$0.378402\pi$
$992$	−3.45560e7	−1.11492
$993$	4.99745e6	0.160833
$994$	−1.57114e7	−0.504370
$995$	0	0
$996$	−47202.4	−0.00150770
$997$	1.12684e7	0.359026	0.179513	−	0.983756i	$-0.442548\pi$
$997$	1.12684e7	0.359026	0.179513	+	0.983756i	$0.442548\pi$
$998$	−6.73717e7	−2.14117
$999$	6.57093e6	0.208312

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.s.1.7	yes	9
5.4	even	2		825.6.a.r.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.r.1.3	✓	9	5.4	even	2
825.6.a.s.1.7	yes	9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+6.71211 q^{2} -9.00000 q^{3} +13.0525 q^{4} -60.4090 q^{6} -177.172 q^{7} -127.178 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+6.71211 q^{2} -9.00000 q^{3} +13.0525 q^{4} -60.4090 q^{6} -177.172 q^{7} -127.178 q^{8} +81.0000 q^{9} -121.000 q^{11} -117.472 q^{12} -783.433 q^{13} -1189.20 q^{14} -1271.31 q^{16} -1335.12 q^{17} +543.681 q^{18} -915.496 q^{19} +1594.55 q^{21} -812.166 q^{22} -1684.59 q^{23} +1144.60 q^{24} -5258.49 q^{26} -729.000 q^{27} -2312.53 q^{28} +1720.16 q^{29} +7741.92 q^{31} -4463.49 q^{32} +1089.00 q^{33} -8961.50 q^{34} +1057.25 q^{36} -9013.63 q^{37} -6144.91 q^{38} +7050.90 q^{39} +13835.8 q^{41} +10702.8 q^{42} +11885.9 q^{43} -1579.35 q^{44} -11307.2 q^{46} -20567.8 q^{47} +11441.8 q^{48} +14583.0 q^{49} +12016.1 q^{51} -10225.7 q^{52} -3483.63 q^{53} -4893.13 q^{54} +22532.4 q^{56} +8239.46 q^{57} +11545.9 q^{58} -22223.3 q^{59} -40834.7 q^{61} +51964.6 q^{62} -14350.9 q^{63} +10722.5 q^{64} +7309.49 q^{66} +2692.02 q^{67} -17426.6 q^{68} +15161.3 q^{69} +13211.7 q^{71} -10301.4 q^{72} -37232.0 q^{73} -60500.5 q^{74} -11949.5 q^{76} +21437.8 q^{77} +47326.4 q^{78} -46277.2 q^{79} +6561.00 q^{81} +92867.6 q^{82} +401.818 q^{83} +20812.8 q^{84} +79779.7 q^{86} -15481.5 q^{87} +15388.5 q^{88} -33472.8 q^{89} +138803. q^{91} -21988.1 q^{92} -69677.3 q^{93} -138053. q^{94} +40171.4 q^{96} +168527. q^{97} +97882.6 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10}) \) 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9	\( 9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9} - 1089 q^{11} - 1539 q^{12} + 723 q^{13} + 720 q^{14} + 4147 q^{16} - 2804 q^{17} + 81 q^{18} - 1601 q^{19} + 513 q^{21} - 121 q^{22} - 2392 q^{23} + 1593 q^{24} - 1987 q^{26} - 6561 q^{27} + 4436 q^{28} + 5966 q^{29} + 21575 q^{31} - 25493 q^{32} + 9801 q^{33} - 3098 q^{34} + 13851 q^{36} - 10228 q^{37} + 12765 q^{38} - 6507 q^{39} + 13304 q^{41} - 6480 q^{42} - 13829 q^{43} - 20691 q^{44} + 81283 q^{46} - 13998 q^{47} - 37323 q^{48} - 19468 q^{49} + 25236 q^{51} + 37131 q^{52} - 44166 q^{53} - 729 q^{54} + 79160 q^{56} + 14409 q^{57} - 7635 q^{58} + 11626 q^{59} + 49481 q^{61} - 94479 q^{62} - 4617 q^{63} + 109367 q^{64} + 1089 q^{66} + 26567 q^{67} - 119506 q^{68} + 21528 q^{69} + 78454 q^{71} - 14337 q^{72} - 100086 q^{73} + 162360 q^{74} + 194115 q^{76} + 6897 q^{77} + 17883 q^{78} - 8478 q^{79} + 59049 q^{81} - 52700 q^{82} - 157476 q^{83} - 39924 q^{84} + 251663 q^{86} - 53694 q^{87} + 21417 q^{88} - 65548 q^{89} - 106849 q^{91} - 350115 q^{92} - 194175 q^{93} - 35742 q^{94} + 229437 q^{96} + 116757 q^{97} - 14949 q^{98} - 88209 q^{99}+O(q^{100}) \) 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9 - 1089 * q^11 - 1539 * q^12 + 723 * q^13 + 720 * q^14 + 4147 * q^16 - 2804 * q^17 + 81 * q^18 - 1601 * q^19 + 513 * q^21 - 121 * q^22 - 2392 * q^23 + 1593 * q^24 - 1987 * q^26 - 6561 * q^27 + 4436 * q^28 + 5966 * q^29 + 21575 * q^31 - 25493 * q^32 + 9801 * q^33 - 3098 * q^34 + 13851 * q^36 - 10228 * q^37 + 12765 * q^38 - 6507 * q^39 + 13304 * q^41 - 6480 * q^42 - 13829 * q^43 - 20691 * q^44 + 81283 * q^46 - 13998 * q^47 - 37323 * q^48 - 19468 * q^49 + 25236 * q^51 + 37131 * q^52 - 44166 * q^53 - 729 * q^54 + 79160 * q^56 + 14409 * q^57 - 7635 * q^58 + 11626 * q^59 + 49481 * q^61 - 94479 * q^62 - 4617 * q^63 + 109367 * q^64 + 1089 * q^66 + 26567 * q^67 - 119506 * q^68 + 21528 * q^69 + 78454 * q^71 - 14337 * q^72 - 100086 * q^73 + 162360 * q^74 + 194115 * q^76 + 6897 * q^77 + 17883 * q^78 - 8478 * q^79 + 59049 * q^81 - 52700 * q^82 - 157476 * q^83 - 39924 * q^84 + 251663 * q^86 - 53694 * q^87 + 21417 * q^88 - 65548 * q^89 - 106849 * q^91 - 350115 * q^92 - 194175 * q^93 - 35742 * q^94 + 229437 * q^96 + 116757 * q^97 - 14949 * q^98 - 88209 * q^99

Introduction

L-functions

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