LMFDB - Embedded newform 825.6.a.x.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 3 x^{12} - 318 x^{11} + 776 x^{10} + 37929 x^{9} - 75673 x^{8} - 2114192 x^{7} + \cdots + 1037920000 $ x^13 - 3x^12 - 318x^11 + 776x^10 + 37929x^9 - 75673x^8 - 2114192x^7 + 3527024x^6 + 55736416x^5 - 75128864x^4 - 615221248x^3 + 450708480x^2 + 2256236800x + 1037920000
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 5^{6} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-6.98295$ 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)

$f(q)$	$=$	$q-6.98295 q^{2} -9.00000 q^{3} +16.7616 q^{4} +62.8465 q^{6} +45.4926 q^{7} +106.409 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-6.98295 q^{2} -9.00000 q^{3} +16.7616 q^{4} +62.8465 q^{6} +45.4926 q^{7} +106.409 q^{8} +81.0000 q^{9} -121.000 q^{11} -150.854 q^{12} -398.031 q^{13} -317.673 q^{14} -1279.42 q^{16} -2129.16 q^{17} -565.619 q^{18} -1809.70 q^{19} -409.433 q^{21} +844.937 q^{22} +2152.64 q^{23} -957.682 q^{24} +2779.43 q^{26} -729.000 q^{27} +762.528 q^{28} +7073.02 q^{29} -6775.82 q^{31} +5529.03 q^{32} +1089.00 q^{33} +14867.8 q^{34} +1357.69 q^{36} +13514.9 q^{37} +12637.0 q^{38} +3582.28 q^{39} -6086.26 q^{41} +2859.05 q^{42} +10109.1 q^{43} -2028.15 q^{44} -15031.8 q^{46} -20147.5 q^{47} +11514.8 q^{48} -14737.4 q^{49} +19162.4 q^{51} -6671.63 q^{52} -11024.9 q^{53} +5090.57 q^{54} +4840.83 q^{56} +16287.3 q^{57} -49390.6 q^{58} +687.001 q^{59} -39571.5 q^{61} +47315.2 q^{62} +3684.90 q^{63} +2332.49 q^{64} -7604.43 q^{66} +35405.3 q^{67} -35688.0 q^{68} -19373.7 q^{69} -72900.0 q^{71} +8619.14 q^{72} +34875.0 q^{73} -94373.7 q^{74} -30333.4 q^{76} -5504.61 q^{77} -25014.9 q^{78} -14392.1 q^{79} +6561.00 q^{81} +42500.1 q^{82} -8705.46 q^{83} -6862.75 q^{84} -70591.2 q^{86} -63657.2 q^{87} -12875.5 q^{88} -113547. q^{89} -18107.5 q^{91} +36081.6 q^{92} +60982.3 q^{93} +140689. q^{94} -49761.3 q^{96} -104432. q^{97} +102911. q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 3 q^{2} - 117 q^{3} + 229 q^{4} - 27 q^{6} + 284 q^{7} + 369 q^{8} + 1053 q^{9}+O(q^{10}) $ 13 * q + 3 * q^2 - 117 * q^3 + 229 * q^4 - 27 * q^6 + 284 * q^7 + 369 * q^8 + 1053 * q^9	\( 13 q + 3 q^{2} - 117 q^{3} + 229 q^{4} - 27 q^{6} + 284 q^{7} + 369 q^{8} + 1053 q^{9} - 1573 q^{11} - 2061 q^{12} + 366 q^{13} - 2758 q^{14} + 4141 q^{16} + 2056 q^{17} + 243 q^{18} - 310 q^{19} - 2556 q^{21} - 363 q^{22} + 3612 q^{23} - 3321 q^{24} + 2280 q^{26} - 9477 q^{27} + 7896 q^{28} - 4848 q^{29} - 24 q^{31} + 38111 q^{32} + 14157 q^{33} + 5518 q^{34} + 18549 q^{36} - 8420 q^{37} + 474 q^{38} - 3294 q^{39} - 15120 q^{41} + 24822 q^{42} + 35492 q^{43} - 27709 q^{44} - 20280 q^{46} + 46544 q^{47} - 37269 q^{48} + 81837 q^{49} - 18504 q^{51} + 107194 q^{52} + 42256 q^{53} - 2187 q^{54} - 196602 q^{56} + 2790 q^{57} + 114160 q^{58} - 65592 q^{59} - 52042 q^{61} + 94972 q^{62} + 23004 q^{63} + 185977 q^{64} + 3267 q^{66} + 80580 q^{67} + 61108 q^{68} - 32508 q^{69} - 77820 q^{71} + 29889 q^{72} + 103050 q^{73} - 240028 q^{74} - 271174 q^{76} - 34364 q^{77} - 20520 q^{78} - 112258 q^{79} + 85293 q^{81} + 64060 q^{82} + 292150 q^{83} - 71064 q^{84} - 319250 q^{86} + 43632 q^{87} - 44649 q^{88} - 295810 q^{89} - 24200 q^{91} + 121328 q^{92} + 216 q^{93} - 358144 q^{94} - 342999 q^{96} + 49072 q^{97} + 101815 q^{98} - 127413 q^{99}+O(q^{100}) \) 13 * q + 3 * q^2 - 117 * q^3 + 229 * q^4 - 27 * q^6 + 284 * q^7 + 369 * q^8 + 1053 * q^9 - 1573 * q^11 - 2061 * q^12 + 366 * q^13 - 2758 * q^14 + 4141 * q^16 + 2056 * q^17 + 243 * q^18 - 310 * q^19 - 2556 * q^21 - 363 * q^22 + 3612 * q^23 - 3321 * q^24 + 2280 * q^26 - 9477 * q^27 + 7896 * q^28 - 4848 * q^29 - 24 * q^31 + 38111 * q^32 + 14157 * q^33 + 5518 * q^34 + 18549 * q^36 - 8420 * q^37 + 474 * q^38 - 3294 * q^39 - 15120 * q^41 + 24822 * q^42 + 35492 * q^43 - 27709 * q^44 - 20280 * q^46 + 46544 * q^47 - 37269 * q^48 + 81837 * q^49 - 18504 * q^51 + 107194 * q^52 + 42256 * q^53 - 2187 * q^54 - 196602 * q^56 + 2790 * q^57 + 114160 * q^58 - 65592 * q^59 - 52042 * q^61 + 94972 * q^62 + 23004 * q^63 + 185977 * q^64 + 3267 * q^66 + 80580 * q^67 + 61108 * q^68 - 32508 * q^69 - 77820 * q^71 + 29889 * q^72 + 103050 * q^73 - 240028 * q^74 - 271174 * q^76 - 34364 * q^77 - 20520 * q^78 - 112258 * q^79 + 85293 * q^81 + 64060 * q^82 + 292150 * q^83 - 71064 * q^84 - 319250 * q^86 + 43632 * q^87 - 44649 * q^88 - 295810 * q^89 - 24200 * q^91 + 121328 * q^92 + 216 * q^93 - 358144 * q^94 - 342999 * q^96 + 49072 * q^97 + 101815 * q^98 - 127413 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−6.98295	−1.23442	−0.617211	−	0.786797i	$-0.711738\pi$
$2$	−6.98295	−1.23442	−0.617211	+	0.786797i	$0.711738\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	16.7616	0.523799
$5$	0	0
$5$	0	0
$6$	62.8465	0.712694
$7$	45.4926	0.350910	0.175455	−	0.984487i	$-0.443860\pi$
$7$	45.4926	0.350910	0.175455	+	0.984487i	$0.443860\pi$
$8$	106.409	0.587833
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	−150.854	−0.302416
$13$	−398.031	−0.653219	−0.326609	−	0.945159i	$-0.605906\pi$
$13$	−398.031	−0.653219	−0.326609	+	0.945159i	$0.605906\pi$
$14$	−317.673	−0.433171
$15$	0	0
$16$	−1279.42	−1.24943
$17$	−2129.16	−1.78684	−0.893419	−	0.449224i	$-0.851701\pi$
$17$	−2129.16	−1.78684	−0.893419	+	0.449224i	$0.851701\pi$
$18$	−565.619	−0.411474
$19$	−1809.70	−1.15006	−0.575032	−	0.818131i	$-0.695010\pi$
$19$	−1809.70	−1.15006	−0.575032	+	0.818131i	$0.695010\pi$
$20$	0	0
$21$	−409.433	−0.202598
$22$	844.937	0.372192
$23$	2152.64	0.848499	0.424250	−	0.905545i	$-0.360538\pi$
$23$	2152.64	0.848499	0.424250	+	0.905545i	$0.360538\pi$
$24$	−957.682	−0.339386
$25$	0	0
$26$	2779.43	0.806348
$27$	−729.000	−0.192450
$28$	762.528	0.183806
$29$	7073.02	1.56174	0.780872	−	0.624690i	$-0.214775\pi$
$29$	7073.02	1.56174	0.780872	+	0.624690i	$0.214775\pi$
$30$	0	0
$31$	−6775.82	−1.26636	−0.633180	−	0.774004i	$-0.718251\pi$
$31$	−6775.82	−1.26636	−0.633180	+	0.774004i	$0.718251\pi$
$32$	5529.03	0.954496
$33$	1089.00	0.174078
$34$	14867.8	2.20571
$35$	0	0
$36$	1357.69	0.174600
$37$	13514.9	1.62296	0.811480	−	0.584380i	$-0.198662\pi$
$37$	13514.9	1.62296	0.811480	+	0.584380i	$0.198662\pi$
$38$	12637.0	1.41966
$39$	3582.28	0.377136
$40$	0	0
$41$	−6086.26	−0.565446	−0.282723	−	0.959202i	$-0.591238\pi$
$41$	−6086.26	−0.565446	−0.282723	+	0.959202i	$0.591238\pi$
$42$	2859.05	0.250092
$43$	10109.1	0.833759	0.416880	−	0.908962i	$-0.363124\pi$
$43$	10109.1	0.833759	0.416880	+	0.908962i	$0.363124\pi$
$44$	−2028.15	−0.157931
$45$	0	0
$46$	−15031.8	−1.04741
$47$	−20147.5	−1.33038	−0.665192	−	0.746672i	$-0.731650\pi$
$47$	−20147.5	−1.33038	−0.665192	+	0.746672i	$0.731650\pi$
$48$	11514.8	0.721361
$49$	−14737.4	−0.876862
$50$	0	0
$51$	19162.4	1.03163
$52$	−6671.63	−0.342156
$53$	−11024.9	−0.539118	−0.269559	−	0.962984i	$-0.586878\pi$
$53$	−11024.9	−0.539118	−0.269559	+	0.962984i	$0.586878\pi$
$54$	5090.57	0.237565
$55$	0	0
$56$	4840.83	0.206276
$57$	16287.3	0.663989
$58$	−49390.6	−1.92785
$59$	687.001	0.0256937	0.0128469	−	0.999917i	$-0.495911\pi$
$59$	687.001	0.0256937	0.0128469	+	0.999917i	$0.495911\pi$
$60$	0	0
$61$	−39571.5	−1.36163	−0.680813	−	0.732457i	$-0.738373\pi$
$61$	−39571.5	−1.36163	−0.680813	+	0.732457i	$0.738373\pi$
$62$	47315.2	1.56322
$63$	3684.90	0.116970
$64$	2332.49	0.0711821
$65$	0	0
$66$	−7604.43	−0.214885
$67$	35405.3	0.963565	0.481783	−	0.876291i	$-0.339989\pi$
$67$	35405.3	0.963565	0.481783	+	0.876291i	$0.339989\pi$
$68$	−35688.0	−0.935944
$69$	−19373.7	−0.489881
$70$	0	0
$71$	−72900.0	−1.71626	−0.858128	−	0.513436i	$-0.828372\pi$
$71$	−72900.0	−1.71626	−0.858128	+	0.513436i	$0.828372\pi$
$72$	8619.14	0.195944
$73$	34875.0	0.765961	0.382981	−	0.923756i	$-0.374898\pi$
$73$	34875.0	0.765961	0.382981	+	0.923756i	$0.374898\pi$
$74$	−94373.7	−2.00342
$75$	0	0
$76$	−30333.4	−0.602402
$77$	−5504.61	−0.105803
$78$	−25014.9	−0.465545
$79$	−14392.1	−0.259452	−0.129726	−	0.991550i	$-0.541410\pi$
$79$	−14392.1	−0.259452	−0.129726	+	0.991550i	$0.541410\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	42500.1	0.697999
$83$	−8705.46	−0.138706	−0.0693532	−	0.997592i	$-0.522094\pi$
$83$	−8705.46	−0.138706	−0.0693532	+	0.997592i	$0.522094\pi$
$84$	−6862.75	−0.106121
$85$	0	0
$86$	−70591.2	−1.02921
$87$	−63657.2	−0.901674
$88$	−12875.5	−0.177238
$89$	−113547.	−1.51950	−0.759748	−	0.650217i	$-0.774678\pi$
$89$	−113547.	−1.51950	−0.759748	+	0.650217i	$0.774678\pi$
$90$	0	0
$91$	−18107.5	−0.229221
$92$	36081.6	0.444443
$93$	60982.3	0.731134
$94$	140689.	1.64226
$95$	0	0
$96$	−49761.3	−0.551079
$97$	−104432.	−1.12695	−0.563476	−	0.826132i	$-0.690536\pi$
$97$	−104432.	−1.12695	−0.563476	+	0.826132i	$0.690536\pi$
$98$	102911.	1.08242
$99$	−9801.00	−0.100504
$100$	0	0
$101$	108911.	1.06235	0.531175	−	0.847262i	$-0.321751\pi$
$101$	108911.	1.06235	0.531175	+	0.847262i	$0.321751\pi$
$102$	−133810.	−1.27347
$103$	163858.	1.52186	0.760928	−	0.648836i	$-0.224744\pi$
$103$	163858.	1.52186	0.760928	+	0.648836i	$0.224744\pi$
$104$	−42354.2	−0.383984
$105$	0	0
$106$	76986.1	0.665499
$107$	86510.1	0.730479	0.365239	−	0.930914i	$-0.380987\pi$
$107$	86510.1	0.730479	0.365239	+	0.930914i	$0.380987\pi$
$108$	−12219.2	−0.100805
$109$	−148364.	−1.19609	−0.598044	−	0.801463i	$-0.704055\pi$
$109$	−148364.	−1.19609	−0.598044	+	0.801463i	$0.704055\pi$
$110$	0	0
$111$	−121634.	−0.937017
$112$	−58204.2	−0.438439
$113$	54533.7	0.401762	0.200881	−	0.979616i	$-0.435620\pi$
$113$	54533.7	0.401762	0.200881	+	0.979616i	$0.435620\pi$
$114$	−113733.	−0.819644
$115$	0	0
$116$	118555.	0.818041
$117$	−32240.5	−0.217740
$118$	−4797.29	−0.0317169
$119$	−96860.9	−0.627019
$120$	0	0
$121$	14641.0	0.0909091
$122$	276326.	1.68082
$123$	54776.4	0.326460
$124$	−113573.	−0.663319
$125$	0	0
$126$	−25731.5	−0.144390
$127$	−314454.	−1.73001	−0.865003	−	0.501767i	$-0.832684\pi$
$127$	−314454.	−1.73001	−0.865003	+	0.501767i	$0.832684\pi$
$128$	−193217.	−1.04236
$129$	−90981.8	−0.481371
$130$	0	0
$131$	104647.	0.532780	0.266390	−	0.963865i	$-0.414169\pi$
$131$	104647.	0.532780	0.266390	+	0.963865i	$0.414169\pi$
$132$	18253.4	0.0911817
$133$	−82327.8	−0.403569
$134$	−247233.	−1.18945
$135$	0	0
$136$	−226562.	−1.05036
$137$	−307469.	−1.39959	−0.699793	−	0.714345i	$-0.746725\pi$
$137$	−307469.	−1.39959	−0.699793	+	0.714345i	$0.746725\pi$
$138$	135286.	0.604720
$139$	−204046.	−0.895759	−0.447880	−	0.894094i	$-0.647821\pi$
$139$	−204046.	−0.895759	−0.447880	+	0.894094i	$0.647821\pi$
$140$	0	0
$141$	181328.	0.768098
$142$	509057.	2.11858
$143$	48161.8	0.196953
$144$	−103633.	−0.416478
$145$	0	0
$146$	−243530.	−0.945520
$147$	132637.	0.506257
$148$	226531.	0.850105
$149$	105309.	0.388598	0.194299	−	0.980942i	$-0.437757\pi$
$149$	105309.	0.388598	0.194299	+	0.980942i	$0.437757\pi$
$150$	0	0
$151$	200814.	0.716725	0.358362	−	0.933583i	$-0.383335\pi$
$151$	200814.	0.716725	0.358362	+	0.933583i	$0.383335\pi$
$152$	−192568.	−0.676045
$153$	−172462.	−0.595613
$154$	38438.4	0.130606
$155$	0	0
$156$	60044.7	0.197544
$157$	103777.	0.336009	0.168004	−	0.985786i	$-0.446268\pi$
$157$	103777.	0.336009	0.168004	+	0.985786i	$0.446268\pi$
$158$	100499.	0.320274
$159$	99223.8	0.311260
$160$	0	0
$161$	97929.1	0.297747
$162$	−45815.1	−0.137158
$163$	449193.	1.32423	0.662116	−	0.749402i	$-0.269659\pi$
$163$	449193.	1.32423	0.662116	+	0.749402i	$0.269659\pi$
$164$	−102015.	−0.296180
$165$	0	0
$166$	60789.8	0.171222
$167$	217540.	0.603599	0.301799	−	0.953371i	$-0.402413\pi$
$167$	217540.	0.603599	0.301799	+	0.953371i	$0.402413\pi$
$168$	−43567.5	−0.119094
$169$	−212864.	−0.573305
$170$	0	0
$171$	−146585.	−0.383354
$172$	169444.	0.436723
$173$	−488266.	−1.24034	−0.620170	−	0.784467i	$-0.712936\pi$
$173$	−488266.	−1.24034	−0.620170	+	0.784467i	$0.712936\pi$
$174$	444515.	1.11305
$175$	0	0
$176$	154810.	0.376718
$177$	−6183.01	−0.0148343
$178$	792891.	1.87570
$179$	−515802.	−1.20323	−0.601617	−	0.798784i	$-0.705477\pi$
$179$	−515802.	−1.20323	−0.601617	+	0.798784i	$0.705477\pi$
$180$	0	0
$181$	807513.	1.83212	0.916059	−	0.401044i	$-0.131353\pi$
$181$	807513.	1.83212	0.916059	+	0.401044i	$0.131353\pi$
$182$	126444.	0.282956
$183$	356143.	0.786135
$184$	229060.	0.498776
$185$	0	0
$186$	−425837.	−0.902528
$187$	257628.	0.538752
$188$	−337704.	−0.696854
$189$	−33164.1	−0.0675327
$190$	0	0
$191$	−416366.	−0.825832	−0.412916	−	0.910769i	$-0.635490\pi$
$191$	−416366.	−0.825832	−0.412916	+	0.910769i	$0.635490\pi$
$192$	−20992.4	−0.0410970
$193$	−617199.	−1.19270	−0.596351	−	0.802724i	$-0.703383\pi$
$193$	−617199.	−1.19270	−0.596351	+	0.802724i	$0.703383\pi$
$194$	729246.	1.39114
$195$	0	0
$196$	−247022.	−0.459300
$197$	332535.	0.610481	0.305240	−	0.952275i	$-0.401263\pi$
$197$	332535.	0.610481	0.305240	+	0.952275i	$0.401263\pi$
$198$	68439.9	0.124064
$199$	−294341.	−0.526887	−0.263444	−	0.964675i	$-0.584858\pi$
$199$	−294341.	−0.526887	−0.263444	+	0.964675i	$0.584858\pi$
$200$	0	0
$201$	−318648.	−0.556315
$202$	−760518.	−1.31139
$203$	321770.	0.548032
$204$	321192.	0.540368
$205$	0	0
$206$	−1.14421e6	−1.87861
$207$	174364.	0.282833
$208$	509249.	0.816154
$209$	218973.	0.346757
$210$	0	0
$211$	255300.	0.394770	0.197385	−	0.980326i	$-0.436755\pi$
$211$	255300.	0.394770	0.197385	+	0.980326i	$0.436755\pi$
$212$	−184794.	−0.282390
$213$	656100.	0.990881
$214$	−604096.	−0.901719
$215$	0	0
$216$	−77572.3	−0.113129
$217$	−308250.	−0.444379
$218$	1.03602e6	1.47648
$219$	−313875.	−0.442228
$220$	0	0
$221$	847471.	1.16720
$222$	849363.	1.15667
$223$	1.27785e6	1.72074	0.860372	−	0.509667i	$-0.170231\pi$
$223$	1.27785e6	1.72074	0.860372	+	0.509667i	$0.170231\pi$
$224$	251530.	0.334942
$225$	0	0
$226$	−380806.	−0.495944
$227$	−739980.	−0.953137	−0.476568	−	0.879137i	$-0.658119\pi$
$227$	−739980.	−0.953137	−0.476568	+	0.879137i	$0.658119\pi$
$228$	273000.	0.347797
$229$	1.10548e6	1.39304	0.696520	−	0.717538i	$-0.254731\pi$
$229$	1.10548e6	1.39304	0.696520	+	0.717538i	$0.254731\pi$
$230$	0	0
$231$	49541.5	0.0610856
$232$	752634.	0.918045
$233$	−686353.	−0.828242	−0.414121	−	0.910222i	$-0.635911\pi$
$233$	−686353.	−0.828242	−0.414121	+	0.910222i	$0.635911\pi$
$234$	225134.	0.268783
$235$	0	0
$236$	11515.2	0.0134584
$237$	129529.	0.149795
$238$	676375.	0.774007
$239$	−267116.	−0.302486	−0.151243	−	0.988497i	$-0.548328\pi$
$239$	−267116.	−0.302486	−0.151243	+	0.988497i	$0.548328\pi$
$240$	0	0
$241$	−98881.4	−0.109666	−0.0548330	−	0.998496i	$-0.517463\pi$
$241$	−98881.4	−0.109666	−0.0548330	+	0.998496i	$0.517463\pi$
$242$	−102237.	−0.112220
$243$	−59049.0	−0.0641500
$244$	−663280.	−0.713218
$245$	0	0
$246$	−382501.	−0.402990
$247$	720316.	0.751243
$248$	−721009.	−0.744409
$249$	78349.1	0.0800822
$250$	0	0
$251$	−1.06783e6	−1.06984	−0.534919	−	0.844903i	$-0.679658\pi$
$251$	−1.06783e6	−1.06984	−0.534919	+	0.844903i	$0.679658\pi$
$252$	61764.8	0.0612688
$253$	−260469.	−0.255832
$254$	2.19582e6	2.13556
$255$	0	0
$256$	1.27458e6	1.21554
$257$	−876600.	−0.827883	−0.413941	−	0.910304i	$-0.635848\pi$
$257$	−876600.	−0.827883	−0.413941	+	0.910304i	$0.635848\pi$
$258$	635321.	0.594216
$259$	614827.	0.569513
$260$	0	0
$261$	572915.	0.520582
$262$	−730744.	−0.657676
$263$	−572993.	−0.510811	−0.255405	−	0.966834i	$-0.582209\pi$
$263$	−572993.	−0.510811	−0.255405	+	0.966834i	$0.582209\pi$
$264$	115880.	0.102329
$265$	0	0
$266$	574891.	0.498174
$267$	1.02192e6	0.877282
$268$	593448.	0.504715
$269$	−703942.	−0.593139	−0.296569	−	0.955011i	$-0.595843\pi$
$269$	−703942.	−0.593139	−0.296569	+	0.955011i	$0.595843\pi$
$270$	0	0
$271$	144761.	0.119737	0.0598684	−	0.998206i	$-0.480932\pi$
$271$	144761.	0.119737	0.0598684	+	0.998206i	$0.480932\pi$
$272$	2.72409e6	2.23254
$273$	162967.	0.132341
$274$	2.14704e6	1.72768
$275$	0	0
$276$	−324734.	−0.256599
$277$	1.66352e6	1.30265	0.651326	−	0.758798i	$-0.274213\pi$
$277$	1.66352e6	1.30265	0.651326	+	0.758798i	$0.274213\pi$
$278$	1.42484e6	1.10575
$279$	−548841.	−0.422120
$280$	0	0
$281$	2.08675e6	1.57654	0.788269	−	0.615330i	$-0.210977\pi$
$281$	2.08675e6	1.57654	0.788269	+	0.615330i	$0.210977\pi$
$282$	−1.26620e6	−0.948157
$283$	−1.90741e6	−1.41572	−0.707861	−	0.706351i	$-0.750340\pi$
$283$	−1.90741e6	−1.41572	−0.707861	+	0.706351i	$0.750340\pi$
$284$	−1.22192e6	−0.898973
$285$	0	0
$286$	−336311.	−0.243123
$287$	−276880.	−0.198421
$288$	447852.	0.318165
$289$	3.11345e6	2.19279
$290$	0	0
$291$	939891.	0.650646
$292$	584560.	0.401210
$293$	1.79926e6	1.22441	0.612204	−	0.790700i	$-0.290283\pi$
$293$	1.79926e6	1.22441	0.612204	+	0.790700i	$0.290283\pi$
$294$	−926196.	−0.624935
$295$	0	0
$296$	1.43811e6	0.954030
$297$	88209.0	0.0580259
$298$	−735368.	−0.479694
$299$	−856817.	−0.554256
$300$	0	0
$301$	459889.	0.292575
$302$	−1.40228e6	−0.884741
$303$	−980197.	−0.613348
$304$	2.31536e6	1.43693
$305$	0	0
$306$	1.20429e6	0.735238
$307$	1.85153e6	1.12120	0.560601	−	0.828086i	$-0.310570\pi$
$307$	1.85153e6	1.12120	0.560601	+	0.828086i	$0.310570\pi$
$308$	−92265.9	−0.0554197
$309$	−1.47472e6	−0.878644
$310$	0	0
$311$	−18208.0	−0.0106748	−0.00533742	−	0.999986i	$-0.501699\pi$
$311$	−18208.0	−0.0106748	−0.00533742	+	0.999986i	$0.501699\pi$
$312$	381187.	0.221693
$313$	2.69863e6	1.55698	0.778489	−	0.627659i	$-0.215987\pi$
$313$	2.69863e6	1.55698	0.778489	+	0.627659i	$0.215987\pi$
$314$	−724667.	−0.414777
$315$	0	0
$316$	−241235.	−0.135901
$317$	−1.81376e6	−1.01375	−0.506875	−	0.862020i	$-0.669199\pi$
$317$	−1.81376e6	−1.01375	−0.506875	+	0.862020i	$0.669199\pi$
$318$	−692875.	−0.384226
$319$	−855836.	−0.470884
$320$	0	0
$321$	−778591.	−0.421742
$322$	−683834.	−0.367545
$323$	3.85313e6	2.05498
$324$	109973.	0.0581999
$325$	0	0
$326$	−3.13669e6	−1.63466
$327$	1.33528e6	0.690562
$328$	−647634.	−0.332388
$329$	−916563.	−0.466845
$330$	0	0
$331$	−399962.	−0.200654	−0.100327	−	0.994955i	$-0.531989\pi$
$331$	−399962.	−0.200654	−0.100327	+	0.994955i	$0.531989\pi$
$332$	−145917.	−0.0726543
$333$	1.09471e6	0.540987
$334$	−1.51907e6	−0.745096
$335$	0	0
$336$	523837.	0.253133
$337$	2.82050e6	1.35285	0.676427	−	0.736509i	$-0.263527\pi$
$337$	2.82050e6	1.35285	0.676427	+	0.736509i	$0.263527\pi$
$338$	1.48642e6	0.707701
$339$	−490803.	−0.231957
$340$	0	0
$341$	819874.	0.381822
$342$	1.02360e6	0.473221
$343$	−1.43504e6	−0.658610
$344$	1.07570e6	0.490111
$345$	0	0
$346$	3.40953e6	1.53110
$347$	−1.89890e6	−0.846602	−0.423301	−	0.905989i	$-0.639129\pi$
$347$	−1.89890e6	−0.846602	−0.423301	+	0.905989i	$0.639129\pi$
$348$	−1.06700e6	−0.472296
$349$	2.08077e6	0.914452	0.457226	−	0.889351i	$-0.348843\pi$
$349$	2.08077e6	0.914452	0.457226	+	0.889351i	$0.348843\pi$
$350$	0	0
$351$	290165.	0.125712
$352$	−669013.	−0.287791
$353$	288540.	0.123245	0.0616224	−	0.998100i	$-0.480373\pi$
$353$	288540.	0.123245	0.0616224	+	0.998100i	$0.480373\pi$
$354$	43175.6	0.0183118
$355$	0	0
$356$	−1.90322e6	−0.795911
$357$	871748.	0.362010
$358$	3.60182e6	1.48530
$359$	691161.	0.283037	0.141518	−	0.989936i	$-0.454802\pi$
$359$	691161.	0.283037	0.141518	+	0.989936i	$0.454802\pi$
$360$	0	0
$361$	798903.	0.322646
$362$	−5.63883e6	−2.26161
$363$	−131769.	−0.0524864
$364$	−303510.	−0.120066
$365$	0	0
$366$	−2.48693e6	−0.970423
$367$	−2.38878e6	−0.925788	−0.462894	−	0.886414i	$-0.653189\pi$
$367$	−2.38878e6	−0.925788	−0.462894	+	0.886414i	$0.653189\pi$
$368$	−2.75413e6	−1.06014
$369$	−492987.	−0.188482
$370$	0	0
$371$	−501550.	−0.189182
$372$	1.02216e6	0.382967
$373$	1.60656e6	0.597897	0.298948	−	0.954269i	$-0.403364\pi$
$373$	1.60656e6	0.597897	0.298948	+	0.954269i	$0.403364\pi$
$374$	−1.79900e6	−0.665048
$375$	0	0
$376$	−2.14388e6	−0.782044
$377$	−2.81528e6	−1.02016
$378$	231583.	0.0833638
$379$	2.53522e6	0.906605	0.453303	−	0.891357i	$-0.350246\pi$
$379$	2.53522e6	0.906605	0.453303	+	0.891357i	$0.350246\pi$
$380$	0	0
$381$	2.83008e6	0.998819
$382$	2.90746e6	1.01943
$383$	−1.42117e6	−0.495052	−0.247526	−	0.968881i	$-0.579618\pi$
$383$	−1.42117e6	−0.495052	−0.247526	+	0.968881i	$0.579618\pi$
$384$	1.73895e6	0.601810
$385$	0	0
$386$	4.30987e6	1.47230
$387$	818836.	0.277920
$388$	−1.75045e6	−0.590297
$389$	2.90780e6	0.974296	0.487148	−	0.873319i	$-0.338037\pi$
$389$	2.90780e6	0.974296	0.487148	+	0.873319i	$0.338037\pi$
$390$	0	0
$391$	−4.58330e6	−1.51613
$392$	−1.56820e6	−0.515449
$393$	−941822.	−0.307601
$394$	−2.32208e6	−0.753591
$395$	0	0
$396$	−164280.	−0.0526438
$397$	4.63823e6	1.47698	0.738492	−	0.674263i	$-0.235538\pi$
$397$	4.63823e6	1.47698	0.738492	+	0.674263i	$0.235538\pi$
$398$	2.05537e6	0.650402
$399$	740950.	0.233001
$400$	0	0
$401$	1.29004e6	0.400629	0.200315	−	0.979732i	$-0.435804\pi$
$401$	1.29004e6	0.400629	0.200315	+	0.979732i	$0.435804\pi$
$402$	2.22510e6	0.686727
$403$	2.69699e6	0.827211
$404$	1.82552e6	0.556458
$405$	0	0
$406$	−2.24691e6	−0.676503
$407$	−1.63530e6	−0.489341
$408$	2.03906e6	0.606427
$409$	−3.39282e6	−1.00289	−0.501444	−	0.865190i	$-0.667198\pi$
$409$	−3.39282e6	−1.00289	−0.501444	+	0.865190i	$0.667198\pi$
$410$	0	0
$411$	2.76722e6	0.808052
$412$	2.74651e6	0.797147
$413$	31253.5	0.00901619
$414$	−1.21757e6	−0.349136
$415$	0	0
$416$	−2.20073e6	−0.623495
$417$	1.83641e6	0.517167
$418$	−1.52908e6	−0.428045
$419$	2.99840e6	0.834363	0.417181	−	0.908823i	$-0.363018\pi$
$419$	2.99840e6	0.834363	0.417181	+	0.908823i	$0.363018\pi$
$420$	0	0
$421$	336838.	0.0926224	0.0463112	−	0.998927i	$-0.485253\pi$
$421$	336838.	0.0926224	0.0463112	+	0.998927i	$0.485253\pi$
$422$	−1.78275e6	−0.487314
$423$	−1.63195e6	−0.443461
$424$	−1.17315e6	−0.316911
$425$	0	0
$426$	−4.58152e6	−1.22317
$427$	−1.80021e6	−0.477808
$428$	1.45005e6	0.382624
$429$	−433456.	−0.113711
$430$	0	0
$431$	5.94562e6	1.54171	0.770857	−	0.637008i	$-0.219828\pi$
$431$	5.94562e6	1.54171	0.770857	+	0.637008i	$0.219828\pi$
$432$	932697.	0.240454
$433$	1.83495e6	0.470331	0.235166	−	0.971955i	$-0.424437\pi$
$433$	1.83495e6	0.470331	0.235166	+	0.971955i	$0.424437\pi$
$434$	2.15249e6	0.548551
$435$	0	0
$436$	−2.48682e6	−0.626510
$437$	−3.89562e6	−0.975828
$438$	2.19177e6	0.545896
$439$	−408906.	−0.101266	−0.0506329	−	0.998717i	$-0.516124\pi$
$439$	−408906.	−0.101266	−0.0506329	+	0.998717i	$0.516124\pi$
$440$	0	0
$441$	−1.19373e6	−0.292287
$442$	−5.91784e6	−1.44081
$443$	3.91300e6	0.947328	0.473664	−	0.880706i	$-0.342931\pi$
$443$	3.91300e6	0.947328	0.473664	+	0.880706i	$0.342931\pi$
$444$	−2.03878e6	−0.490809
$445$	0	0
$446$	−8.92313e6	−2.12412
$447$	−947782.	−0.224357
$448$	106111.	0.0249785
$449$	131666.	0.0308218	0.0154109	−	0.999881i	$-0.495094\pi$
$449$	131666.	0.0308218	0.0154109	+	0.999881i	$0.495094\pi$
$450$	0	0
$451$	736438.	0.170488
$452$	914071.	0.210443
$453$	−1.80733e6	−0.413801
$454$	5.16724e6	1.17657
$455$	0	0
$456$	1.73311e6	0.390315
$457$	1.00378e6	0.224827	0.112414	−	0.993662i	$-0.464142\pi$
$457$	1.00378e6	0.224827	0.112414	+	0.993662i	$0.464142\pi$
$458$	−7.71953e6	−1.71960
$459$	1.55215e6	0.343877
$460$	0	0
$461$	−6.37043e6	−1.39610	−0.698050	−	0.716049i	$-0.745949\pi$
$461$	−6.37043e6	−1.39610	−0.698050	+	0.716049i	$0.745949\pi$
$462$	−345945.	−0.0754054
$463$	−1.08365e6	−0.234930	−0.117465	−	0.993077i	$-0.537477\pi$
$463$	−1.08365e6	−0.234930	−0.117465	+	0.993077i	$0.537477\pi$
$464$	−9.04937e6	−1.95130
$465$	0	0
$466$	4.79277e6	1.02240
$467$	−1.98318e6	−0.420794	−0.210397	−	0.977616i	$-0.567476\pi$
$467$	−1.98318e6	−0.420794	−0.210397	+	0.977616i	$0.567476\pi$
$468$	−540402.	−0.114052
$469$	1.61068e6	0.338125
$470$	0	0
$471$	−933990.	−0.193995
$472$	73103.2	0.0151036
$473$	−1.22320e6	−0.251388
$474$	−904495.	−0.184910
$475$	0	0
$476$	−1.62354e6	−0.328432
$477$	−893014.	−0.179706
$478$	1.86526e6	0.373395
$479$	3.85481e6	0.767652	0.383826	−	0.923405i	$-0.374606\pi$
$479$	3.85481e6	0.767652	0.383826	+	0.923405i	$0.374606\pi$
$480$	0	0
$481$	−5.37934e6	−1.06015
$482$	690484.	0.135374
$483$	−881362.	−0.171904
$484$	245406.	0.0476181
$485$	0	0
$486$	412336.	0.0791882
$487$	7.19241e6	1.37421	0.687103	−	0.726560i	$-0.258882\pi$
$487$	7.19241e6	1.37421	0.687103	+	0.726560i	$0.258882\pi$
$488$	−4.21077e6	−0.800408
$489$	−4.04274e6	−0.764545
$490$	0	0
$491$	−2.67516e6	−0.500780	−0.250390	−	0.968145i	$-0.580559\pi$
$491$	−2.67516e6	−0.500780	−0.250390	+	0.968145i	$0.580559\pi$
$492$	918138.	0.171000
$493$	−1.50596e7	−2.79059
$494$	−5.02993e6	−0.927351
$495$	0	0
$496$	8.66912e6	1.58223
$497$	−3.31641e6	−0.602251
$498$	−547108.	−0.0988552
$499$	232922.	0.0418753	0.0209377	−	0.999781i	$-0.493335\pi$
$499$	232922.	0.0418753	0.0209377	+	0.999781i	$0.493335\pi$
$500$	0	0
$501$	−1.95786e6	−0.348488
$502$	7.45661e6	1.32063
$503$	−3.05908e6	−0.539102	−0.269551	−	0.962986i	$-0.586875\pi$
$503$	−3.05908e6	−0.539102	−0.269551	+	0.962986i	$0.586875\pi$
$504$	392107.	0.0687588
$505$	0	0
$506$	1.81884e6	0.315805
$507$	1.91578e6	0.330998
$508$	−5.27074e6	−0.906176
$509$	−8.37388e6	−1.43262	−0.716312	−	0.697780i	$-0.754171\pi$
$509$	−8.37388e6	−1.43262	−0.716312	+	0.697780i	$0.754171\pi$
$510$	0	0
$511$	1.58655e6	0.268784
$512$	−2.71741e6	−0.458121
$513$	1.31927e6	0.221330
$514$	6.12126e6	1.02196
$515$	0	0
$516$	−1.52500e6	−0.252142
$517$	2.43785e6	0.401126
$518$	−4.29331e6	−0.703020
$519$	4.39439e6	0.716111
$520$	0	0
$521$	978161.	0.157876	0.0789380	−	0.996880i	$-0.474847\pi$
$521$	978161.	0.157876	0.0789380	+	0.996880i	$0.474847\pi$
$522$	−4.00064e6	−0.642618
$523$	2.05926e6	0.329199	0.164599	−	0.986361i	$-0.447367\pi$
$523$	2.05926e6	0.329199	0.164599	+	0.986361i	$0.447367\pi$
$524$	1.75405e6	0.279070
$525$	0	0
$526$	4.00118e6	0.630557
$527$	1.44268e7	2.26278
$528$	−1.39329e6	−0.217498
$529$	−1.80249e6	−0.280049
$530$	0	0
$531$	55647.1	0.00856458
$532$	−1.37994e6	−0.211389
$533$	2.42252e6	0.369360
$534$	−7.13602e6	−1.08294
$535$	0	0
$536$	3.76745e6	0.566415
$537$	4.64222e6	0.694688
$538$	4.91559e6	0.732184
$539$	1.78323e6	0.264384
$540$	0	0
$541$	1.00838e7	1.48127	0.740633	−	0.671910i	$-0.234526\pi$
$541$	1.00838e7	1.48127	0.740633	+	0.671910i	$0.234526\pi$
$542$	−1.01086e6	−0.147806
$543$	−7.26762e6	−1.05777
$544$	−1.17722e7	−1.70553
$545$	0	0
$546$	−1.13799e6	−0.163365
$547$	−4.34085e6	−0.620307	−0.310154	−	0.950686i	$-0.600380\pi$
$547$	−4.34085e6	−0.620307	−0.310154	+	0.950686i	$0.600380\pi$
$548$	−5.15366e6	−0.733102
$549$	−3.20529e6	−0.453875
$550$	0	0
$551$	−1.28000e7	−1.79611
$552$	−2.06154e6	−0.287968
$553$	−654735.	−0.0910444
$554$	−1.16163e7	−1.60802
$555$	0	0
$556$	−3.42013e6	−0.469198
$557$	−7.27272e6	−0.993251	−0.496625	−	0.867965i	$-0.665428\pi$
$557$	−7.27272e6	−0.993251	−0.496625	+	0.867965i	$0.665428\pi$
$558$	3.83253e6	0.521075
$559$	−4.02373e6	−0.544627
$560$	0	0
$561$	−2.31865e6	−0.311049
$562$	−1.45717e7	−1.94611
$563$	8.99392e6	1.19585	0.597927	−	0.801551i	$-0.295991\pi$
$563$	8.99392e6	1.19585	0.597927	+	0.801551i	$0.295991\pi$
$564$	3.03934e6	0.402329
$565$	0	0
$566$	1.33194e7	1.74760
$567$	298477.	0.0389900
$568$	−7.75723e6	−1.00887
$569$	5.76031e6	0.745873	0.372937	−	0.927857i	$-0.378351\pi$
$569$	5.76031e6	0.745873	0.372937	+	0.927857i	$0.378351\pi$
$570$	0	0
$571$	9.58090e6	1.22975	0.614874	−	0.788626i	$-0.289207\pi$
$571$	9.58090e6	1.22975	0.614874	+	0.788626i	$0.289207\pi$
$572$	807267.	0.103164
$573$	3.74729e6	0.476794
$574$	1.93344e6	0.244935
$575$	0	0
$576$	188932.	0.0237274
$577$	−8.22427e6	−1.02839	−0.514195	−	0.857673i	$-0.671909\pi$
$577$	−8.22427e6	−1.02839	−0.514195	+	0.857673i	$0.671909\pi$
$578$	−2.17411e7	−2.70683
$579$	5.55479e6	0.688607
$580$	0	0
$581$	−396034.	−0.0486735
$582$	−6.56321e6	−0.803173
$583$	1.33401e6	0.162550
$584$	3.71102e6	0.450257
$585$	0	0
$586$	−1.25642e7	−1.51144
$587$	9.59481e6	1.14932	0.574660	−	0.818392i	$-0.305134\pi$
$587$	9.59481e6	1.14932	0.574660	+	0.818392i	$0.305134\pi$
$588$	2.22320e6	0.265177
$589$	1.22622e7	1.45640
$590$	0	0
$591$	−2.99282e6	−0.352461
$592$	−1.72912e7	−2.02778
$593$	7.08140e6	0.826956	0.413478	−	0.910514i	$-0.364314\pi$
$593$	7.08140e6	0.826956	0.413478	+	0.910514i	$0.364314\pi$
$594$	−615959.	−0.0716285
$595$	0	0
$596$	1.76515e6	0.203547
$597$	2.64907e6	0.304199
$598$	5.98311e6	0.684186
$599$	−2.77244e6	−0.315715	−0.157858	−	0.987462i	$-0.550459\pi$
$599$	−2.77244e6	−0.315715	−0.157858	+	0.987462i	$0.550459\pi$
$600$	0	0
$601$	1.01531e7	1.14661	0.573303	−	0.819344i	$-0.305662\pi$
$601$	1.01531e7	1.14661	0.573303	+	0.819344i	$0.305662\pi$
$602$	−3.21138e6	−0.361161
$603$	2.86783e6	0.321188
$604$	3.36596e6	0.375420
$605$	0	0
$606$	6.84466e6	0.757130
$607$	6.91952e6	0.762262	0.381131	−	0.924521i	$-0.375535\pi$
$607$	6.91952e6	0.762262	0.381131	+	0.924521i	$0.375535\pi$
$608$	−1.00059e7	−1.09773
$609$	−2.89593e6	−0.316406
$610$	0	0
$611$	8.01934e6	0.869032
$612$	−2.89073e6	−0.311981
$613$	5.34824e6	0.574857	0.287428	−	0.957802i	$-0.407200\pi$
$613$	5.34824e6	0.574857	0.287428	+	0.957802i	$0.407200\pi$
$614$	−1.29291e7	−1.38404
$615$	0	0
$616$	−585740.	−0.0621947
$617$	−6.58637e6	−0.696519	−0.348260	−	0.937398i	$-0.613227\pi$
$617$	−6.58637e6	−0.696519	−0.348260	+	0.937398i	$0.613227\pi$
$618$	1.02979e7	1.08462
$619$	3.16144e6	0.331633	0.165816	−	0.986157i	$-0.446974\pi$
$619$	3.16144e6	0.331633	0.165816	+	0.986157i	$0.446974\pi$
$620$	0	0
$621$	−1.56927e6	−0.163294
$622$	127146.	0.0131773
$623$	−5.16554e6	−0.533207
$624$	−4.58324e6	−0.471207
$625$	0	0
$626$	−1.88444e7	−1.92197
$627$	−1.97076e6	−0.200200
$628$	1.73946e6	0.176001
$629$	−2.87753e7	−2.89997
$630$	0	0
$631$	−5.41296e6	−0.541204	−0.270602	−	0.962691i	$-0.587223\pi$
$631$	−5.41296e6	−0.541204	−0.270602	+	0.962691i	$0.587223\pi$
$632$	−1.53145e6	−0.152515
$633$	−2.29770e6	−0.227921
$634$	1.26654e7	1.25140
$635$	0	0
$636$	1.66315e6	0.163038
$637$	5.86595e6	0.572783
$638$	5.97626e6	0.581270
$639$	−5.90490e6	−0.572085
$640$	0	0
$641$	1.44407e7	1.38817	0.694085	−	0.719893i	$-0.255809\pi$
$641$	1.44407e7	1.38817	0.694085	+	0.719893i	$0.255809\pi$
$642$	5.43686e6	0.520608
$643$	3.49700e6	0.333556	0.166778	−	0.985994i	$-0.446664\pi$
$643$	3.49700e6	0.333556	0.166778	+	0.985994i	$0.446664\pi$
$644$	1.64145e6	0.155960
$645$	0	0
$646$	−2.69062e7	−2.53671
$647$	−106840.	−0.0100340	−0.00501701	−	0.999987i	$-0.501597\pi$
$647$	−106840.	−0.0100340	−0.00501701	+	0.999987i	$0.501597\pi$
$648$	698150.	0.0653148
$649$	−83127.1	−0.00774695
$650$	0	0
$651$	2.77425e6	0.256562
$652$	7.52918e6	0.693631
$653$	−9.04446e6	−0.830042	−0.415021	−	0.909812i	$-0.636226\pi$
$653$	−9.04446e6	−0.830042	−0.415021	+	0.909812i	$0.636226\pi$
$654$	−9.32418e6	−0.852445
$655$	0	0
$656$	7.78689e6	0.706487
$657$	2.82487e6	0.255320
$658$	6.40032e6	0.576284
$659$	1.26896e7	1.13824	0.569119	−	0.822255i	$-0.307284\pi$
$659$	1.26896e7	1.13824	0.569119	+	0.822255i	$0.307284\pi$
$660$	0	0
$661$	−2.23689e6	−0.199132	−0.0995659	−	0.995031i	$-0.531745\pi$
$661$	−2.23689e6	−0.199132	−0.0995659	+	0.995031i	$0.531745\pi$
$662$	2.79291e6	0.247692
$663$	−7.62724e6	−0.673881
$664$	−926340.	−0.0815362
$665$	0	0
$666$	−7.64427e6	−0.667806
$667$	1.52257e7	1.32514
$668$	3.64632e6	0.316165
$669$	−1.15006e7	−0.993472
$670$	0	0
$671$	4.78815e6	0.410546
$672$	−2.26377e6	−0.193379
$673$	7.14632e6	0.608198	0.304099	−	0.952640i	$-0.401645\pi$
$673$	7.14632e6	0.608198	0.304099	+	0.952640i	$0.401645\pi$
$674$	−1.96954e7	−1.66999
$675$	0	0
$676$	−3.56794e6	−0.300297
$677$	6.71552e6	0.563129	0.281564	−	0.959542i	$-0.409147\pi$
$677$	6.71552e6	0.563129	0.281564	+	0.959542i	$0.409147\pi$
$678$	3.42725e6	0.286333
$679$	−4.75090e6	−0.395459
$680$	0	0
$681$	6.65982e6	0.550294
$682$	−5.72514e6	−0.471330
$683$	−4.04188e6	−0.331537	−0.165768	−	0.986165i	$-0.553010\pi$
$683$	−4.04188e6	−0.331537	−0.165768	+	0.986165i	$0.553010\pi$
$684$	−2.45700e6	−0.200801
$685$	0	0
$686$	1.00208e7	0.813003
$687$	−9.94935e6	−0.804272
$688$	−1.29338e7	−1.04173
$689$	4.38824e6	0.352162
$690$	0	0
$691$	1.09737e7	0.874295	0.437148	−	0.899390i	$-0.355989\pi$
$691$	1.09737e7	0.874295	0.437148	+	0.899390i	$0.355989\pi$
$692$	−8.18410e6	−0.649689
$693$	−445873.	−0.0352678
$694$	1.32600e7	1.04507
$695$	0	0
$696$	−6.77371e6	−0.530034
$697$	1.29586e7	1.01036
$698$	−1.45299e7	−1.12882
$699$	6.17717e6	0.478186
$700$	0	0
$701$	2.54758e6	0.195809	0.0979046	−	0.995196i	$-0.468786\pi$
$701$	2.54758e6	0.195809	0.0979046	+	0.995196i	$0.468786\pi$
$702$	−2.02621e6	−0.155182
$703$	−2.44578e7	−1.86651
$704$	−282232.	−0.0214622
$705$	0	0
$706$	−2.01486e6	−0.152136
$707$	4.95463e6	0.372789
$708$	−103637.	−0.00777019
$709$	3.37269e6	0.251977	0.125989	−	0.992032i	$-0.459790\pi$
$709$	3.37269e6	0.251977	0.125989	+	0.992032i	$0.459790\pi$
$710$	0	0
$711$	−1.16576e6	−0.0864841
$712$	−1.20824e7	−0.893210
$713$	−1.45859e7	−1.07451
$714$	−6.08737e6	−0.446873
$715$	0	0
$716$	−8.64565e6	−0.630253
$717$	2.40404e6	0.174640
$718$	−4.82634e6	−0.349387
$719$	−6.72012e6	−0.484791	−0.242396	−	0.970177i	$-0.577933\pi$
$719$	−6.72012e6	−0.484791	−0.242396	+	0.970177i	$0.577933\pi$
$720$	0	0
$721$	7.45431e6	0.534035
$722$	−5.57870e6	−0.398281
$723$	889933.	0.0633157
$724$	1.35352e7	0.959662
$725$	0	0
$726$	920136.	0.0647904
$727$	−2.18623e7	−1.53412	−0.767060	−	0.641576i	$-0.778281\pi$
$727$	−2.18623e7	−1.53412	−0.767060	+	0.641576i	$0.778281\pi$
$728$	−1.92680e6	−0.134744
$729$	531441.	0.0370370
$730$	0	0
$731$	−2.15238e7	−1.48979
$732$	5.96952e6	0.411777
$733$	−9.06206e6	−0.622970	−0.311485	−	0.950251i	$-0.600826\pi$
$733$	−9.06206e6	−0.622970	−0.311485	+	0.950251i	$0.600826\pi$
$734$	1.66807e7	1.14281
$735$	0	0
$736$	1.19020e7	0.809889
$737$	−4.28404e6	−0.290526
$738$	3.44251e6	0.232666
$739$	2.46580e7	1.66092	0.830458	−	0.557082i	$-0.188079\pi$
$739$	2.46580e7	1.66092	0.830458	+	0.557082i	$0.188079\pi$
$740$	0	0
$741$	−6.48284e6	−0.433730
$742$	3.50230e6	0.233530
$743$	1.12989e7	0.750867	0.375434	−	0.926849i	$-0.377494\pi$
$743$	1.12989e7	0.750867	0.375434	+	0.926849i	$0.377494\pi$
$744$	6.48908e6	0.429785
$745$	0	0
$746$	−1.12186e7	−0.738057
$747$	−705142.	−0.0462355
$748$	4.31825e6	0.282198
$749$	3.93557e6	0.256332
$750$	0	0
$751$	6.98434e6	0.451882	0.225941	−	0.974141i	$-0.427454\pi$
$751$	6.98434e6	0.451882	0.225941	+	0.974141i	$0.427454\pi$
$752$	2.57771e7	1.66223
$753$	9.61048e6	0.617672
$754$	1.96590e7	1.25931
$755$	0	0
$756$	−555883.	−0.0353736
$757$	−4.85229e6	−0.307757	−0.153878	−	0.988090i	$-0.549176\pi$
$757$	−4.85229e6	−0.307757	−0.153878	+	0.988090i	$0.549176\pi$
$758$	−1.77033e7	−1.11913
$759$	2.34422e6	0.147705
$760$	0	0
$761$	1.57481e7	0.985747	0.492873	−	0.870101i	$-0.335947\pi$
$761$	1.57481e7	0.985747	0.492873	+	0.870101i	$0.335947\pi$
$762$	−1.97623e7	−1.23297
$763$	−6.74948e6	−0.419719
$764$	−6.97895e6	−0.432570
$765$	0	0
$766$	9.92399e6	0.611103
$767$	−273448.	−0.0167836
$768$	−1.14712e7	−0.701790
$769$	−2.63643e6	−0.160768	−0.0803842	−	0.996764i	$-0.525615\pi$
$769$	−2.63643e6	−0.160768	−0.0803842	+	0.996764i	$0.525615\pi$
$770$	0	0
$771$	7.88940e6	0.477978
$772$	−1.03452e7	−0.624736
$773$	−2.00553e7	−1.20720	−0.603602	−	0.797286i	$-0.706268\pi$
$773$	−2.00553e7	−1.20720	−0.603602	+	0.797286i	$0.706268\pi$
$774$	−5.71789e6	−0.343071
$775$	0	0
$776$	−1.11126e7	−0.662460
$777$	−5.53344e6	−0.328808
$778$	−2.03050e7	−1.20269
$779$	1.10143e7	0.650299
$780$	0	0
$781$	8.82091e6	0.517471
$782$	3.20050e7	1.87155
$783$	−5.15623e6	−0.300558
$784$	1.88554e7	1.09558
$785$	0	0
$786$	6.57670e6	0.379710
$787$	8.29630e6	0.477472	0.238736	−	0.971085i	$-0.423267\pi$
$787$	8.29630e6	0.477472	0.238736	+	0.971085i	$0.423267\pi$
$788$	5.57381e6	0.319769
$789$	5.15694e6	0.294917
$790$	0	0
$791$	2.48088e6	0.140982
$792$	−1.04292e6	−0.0590794
$793$	1.57507e7	0.889440
$794$	−3.23885e7	−1.82322
$795$	0	0
$796$	−4.93361e6	−0.275983
$797$	−3.63220e6	−0.202546	−0.101273	−	0.994859i	$-0.532292\pi$
$797$	−3.63220e6	−0.202546	−0.101273	+	0.994859i	$0.532292\pi$
$798$	−5.17402e6	−0.287621
$799$	4.28972e7	2.37718
$800$	0	0
$801$	−9.19729e6	−0.506499
$802$	−9.00830e6	−0.494546
$803$	−4.21987e6	−0.230946
$804$	−5.34104e6	−0.291397
$805$	0	0
$806$	−1.88329e7	−1.02113
$807$	6.33548e6	0.342449
$808$	1.15891e7	0.624484
$809$	9.97936e6	0.536082	0.268041	−	0.963407i	$-0.413624\pi$
$809$	9.97936e6	0.536082	0.268041	+	0.963407i	$0.413624\pi$
$810$	0	0
$811$	−1.06832e7	−0.570360	−0.285180	−	0.958474i	$-0.592053\pi$
$811$	−1.06832e7	−0.570360	−0.285180	+	0.958474i	$0.592053\pi$
$812$	5.39338e6	0.287059
$813$	−1.30285e6	−0.0691300
$814$	1.14192e7	0.604054
$815$	0	0
$816$	−2.45168e7	−1.28896
$817$	−1.82944e7	−0.958876
$818$	2.36919e7	1.23799
$819$	−1.46671e6	−0.0764070
$820$	0	0
$821$	−2.04749e7	−1.06014	−0.530071	−	0.847953i	$-0.677835\pi$
$821$	−2.04749e7	−1.06014	−0.530071	+	0.847953i	$0.677835\pi$
$822$	−1.93234e7	−0.997477
$823$	6.99413e6	0.359943	0.179972	−	0.983672i	$-0.442399\pi$
$823$	6.99413e6	0.359943	0.179972	+	0.983672i	$0.442399\pi$
$824$	1.74360e7	0.894597
$825$	0	0
$826$	−218241.	−0.0111298
$827$	−1.29660e7	−0.659238	−0.329619	−	0.944114i	$-0.606920\pi$
$827$	−1.29660e7	−0.659238	−0.329619	+	0.944114i	$0.606920\pi$
$828$	2.92261e6	0.148148
$829$	−2.13081e7	−1.07686	−0.538428	−	0.842671i	$-0.680982\pi$
$829$	−2.13081e7	−1.07686	−0.538428	+	0.842671i	$0.680982\pi$
$830$	0	0
$831$	−1.49717e7	−0.752086
$832$	−928405.	−0.0464975
$833$	3.13783e7	1.56681
$834$	−1.28236e7	−0.638402
$835$	0	0
$836$	3.67034e6	0.181631
$837$	4.93957e6	0.243711
$838$	−2.09377e7	−1.02996
$839$	3.73909e7	1.83384	0.916920	−	0.399071i	$-0.130667\pi$
$839$	3.73909e7	1.83384	0.916920	+	0.399071i	$0.130667\pi$
$840$	0	0
$841$	2.95165e7	1.43905
$842$	−2.35212e6	−0.114335
$843$	−1.87807e7	−0.910215
$844$	4.27923e6	0.206780
$845$	0	0
$846$	1.13958e7	0.547419
$847$	666057.	0.0319009
$848$	1.41054e7	0.673592
$849$	1.71667e7	0.817368
$850$	0	0
$851$	2.90926e7	1.37708
$852$	1.09973e7	0.519022
$853$	3.77035e6	0.177422	0.0887112	−	0.996057i	$-0.471725\pi$
$853$	3.77035e6	0.177422	0.0887112	+	0.996057i	$0.471725\pi$
$854$	1.25708e7	0.589817
$855$	0	0
$856$	9.20547e6	0.429400
$857$	2.56412e7	1.19258	0.596288	−	0.802771i	$-0.296642\pi$
$857$	2.56412e7	1.19258	0.596288	+	0.802771i	$0.296642\pi$
$858$	3.02680e6	0.140367
$859$	−1.13736e7	−0.525916	−0.262958	−	0.964807i	$-0.584698\pi$
$859$	−1.13736e7	−0.525916	−0.262958	+	0.964807i	$0.584698\pi$
$860$	0	0
$861$	2.49192e6	0.114558
$862$	−4.15180e7	−1.90313
$863$	3.01515e7	1.37810	0.689051	−	0.724713i	$-0.258028\pi$
$863$	3.01515e7	1.37810	0.689051	+	0.724713i	$0.258028\pi$
$864$	−4.03066e6	−0.183693
$865$	0	0
$866$	−1.28133e7	−0.580587
$867$	−2.80210e7	−1.26601
$868$	−5.16675e6	−0.232765
$869$	1.74145e6	0.0782278
$870$	0	0
$871$	−1.40924e7	−0.629419
$872$	−1.57873e7	−0.703100
$873$	−8.45902e6	−0.375651
$874$	2.72029e7	1.20458
$875$	0	0
$876$	−5.26104e6	−0.231639
$877$	−2.44343e7	−1.07275	−0.536377	−	0.843979i	$-0.680207\pi$
$877$	−2.44343e7	−1.07275	−0.536377	+	0.843979i	$0.680207\pi$
$878$	2.85537e6	0.125005
$879$	−1.61934e7	−0.706912
$880$	0	0
$881$	−8.53133e6	−0.370320	−0.185160	−	0.982708i	$-0.559280\pi$
$881$	−8.53133e6	−0.370320	−0.185160	+	0.982708i	$0.559280\pi$
$882$	8.33576e6	0.360806
$883$	−3.67915e6	−0.158798	−0.0793991	−	0.996843i	$-0.525300\pi$
$883$	−3.67915e6	−0.158798	−0.0793991	+	0.996843i	$0.525300\pi$
$884$	1.42049e7	0.611377
$885$	0	0
$886$	−2.73243e7	−1.16940
$887$	3.57756e7	1.52679	0.763393	−	0.645934i	$-0.223532\pi$
$887$	3.57756e7	1.52679	0.763393	+	0.645934i	$0.223532\pi$
$888$	−1.29430e7	−0.550809
$889$	−1.43053e7	−0.607076
$890$	0	0
$891$	−793881.	−0.0335013
$892$	2.14187e7	0.901324
$893$	3.64609e7	1.53003
$894$	6.61831e6	0.276951
$895$	0	0
$896$	−8.78993e6	−0.365776
$897$	7.71135e6	0.320000
$898$	−919417.	−0.0380471
$899$	−4.79255e7	−1.97773
$900$	0	0
$901$	2.34737e7	0.963316
$902$	−5.14251e6	−0.210455
$903$	−4.13900e6	−0.168918
$904$	5.80288e6	0.236169
$905$	0	0
$906$	1.26205e7	0.510806
$907$	−7.77459e6	−0.313804	−0.156902	−	0.987614i	$-0.550151\pi$
$907$	−7.77459e6	−0.313804	−0.156902	+	0.987614i	$0.550151\pi$
$908$	−1.24032e7	−0.499252
$909$	8.82177e6	0.354116
$910$	0	0
$911$	−4.90093e7	−1.95651	−0.978256	−	0.207403i	$-0.933499\pi$
$911$	−4.90093e7	−1.95651	−0.978256	+	0.207403i	$0.933499\pi$
$912$	−2.08383e7	−0.829611
$913$	1.05336e6	0.0418215
$914$	−7.00936e6	−0.277532
$915$	0	0
$916$	1.85296e7	0.729673
$917$	4.76066e6	0.186958
$918$	−1.08386e7	−0.424490
$919$	−2.03739e7	−0.795765	−0.397882	−	0.917436i	$-0.630255\pi$
$919$	−2.03739e7	−0.795765	−0.397882	+	0.917436i	$0.630255\pi$
$920$	0	0
$921$	−1.66637e7	−0.647326
$922$	4.44844e7	1.72338
$923$	2.90165e7	1.12109
$924$	830393.	0.0319966
$925$	0	0
$926$	7.56710e6	0.290003
$927$	1.32725e7	0.507285
$928$	3.91070e7	1.49068
$929$	3.04946e7	1.15927	0.579634	−	0.814877i	$-0.303195\pi$
$929$	3.04946e7	1.15927	0.579634	+	0.814877i	$0.303195\pi$
$930$	0	0
$931$	2.66703e7	1.00845
$932$	−1.15044e7	−0.433833
$933$	163872.	0.00616312
$934$	1.38484e7	0.519437
$935$	0	0
$936$	−3.43069e6	−0.127995
$937$	3.22130e7	1.19862	0.599311	−	0.800517i	$-0.295441\pi$
$937$	3.22130e7	1.19862	0.599311	+	0.800517i	$0.295441\pi$
$938$	−1.12473e7	−0.417389
$939$	−2.42877e7	−0.898921
$940$	0	0
$941$	1.62640e7	0.598761	0.299381	−	0.954134i	$-0.403220\pi$
$941$	1.62640e7	0.598761	0.299381	+	0.954134i	$0.403220\pi$
$942$	6.52201e6	0.239472
$943$	−1.31015e7	−0.479780
$944$	−878963.	−0.0321026
$945$	0	0
$946$	8.54154e6	0.310319
$947$	3.15271e7	1.14238	0.571188	−	0.820819i	$-0.306483\pi$
$947$	3.15271e7	1.14238	0.571188	+	0.820819i	$0.306483\pi$
$948$	2.17111e6	0.0784624
$949$	−1.38813e7	−0.500340
$950$	0	0
$951$	1.63238e7	0.585289
$952$	−1.03069e7	−0.368583
$953$	4.20459e7	1.49966	0.749828	−	0.661633i	$-0.230136\pi$
$953$	4.20459e7	1.49966	0.749828	+	0.661633i	$0.230136\pi$
$954$	6.23587e6	0.221833
$955$	0	0
$956$	−4.47728e6	−0.158442
$957$	7.70252e6	0.271865
$958$	−2.69179e7	−0.947607
$959$	−1.39876e7	−0.491129
$960$	0	0
$961$	1.72825e7	0.603669
$962$	3.75637e7	1.30867
$963$	7.00732e6	0.243493
$964$	−1.65741e6	−0.0574430
$965$	0	0
$966$	6.15451e6	0.212202
$967$	3.72088e7	1.27962	0.639808	−	0.768535i	$-0.279014\pi$
$967$	3.72088e7	1.27962	0.639808	+	0.768535i	$0.279014\pi$
$968$	1.55794e6	0.0534394
$969$	−3.46781e7	−1.18644
$970$	0	0
$971$	2.66779e7	0.908036	0.454018	−	0.890992i	$-0.349990\pi$
$971$	2.66779e7	0.908036	0.454018	+	0.890992i	$0.349990\pi$
$972$	−989754.	−0.0336017
$973$	−9.28259e6	−0.314331
$974$	−5.02242e7	−1.69635
$975$	0	0
$976$	5.06285e7	1.70126
$977$	−3.55924e7	−1.19295	−0.596473	−	0.802633i	$-0.703432\pi$
$977$	−3.55924e7	−1.19295	−0.596473	+	0.802633i	$0.703432\pi$
$978$	2.82302e7	0.943772
$979$	1.37392e7	0.458145
$980$	0	0
$981$	−1.20175e7	−0.398696
$982$	1.86805e7	0.618174
$983$	−3.09648e6	−0.102208	−0.0511040	−	0.998693i	$-0.516274\pi$
$983$	−3.09648e6	−0.102208	−0.0511040	+	0.998693i	$0.516274\pi$
$984$	5.82871e6	0.191904
$985$	0	0
$986$	1.05160e8	3.44476
$987$	8.24907e6	0.269533
$988$	1.20736e7	0.393501
$989$	2.17612e7	0.707444
$990$	0	0
$991$	3.93685e7	1.27340	0.636699	−	0.771112i	$-0.280299\pi$
$991$	3.93685e7	1.27340	0.636699	+	0.771112i	$0.280299\pi$
$992$	−3.74637e7	−1.20874
$993$	3.59966e6	0.115848
$994$	2.31583e7	0.743433
$995$	0	0
$996$	1.31325e6	0.0419470
$997$	−2.54921e7	−0.812210	−0.406105	−	0.913826i	$-0.633113\pi$
$997$	−2.54921e7	−0.812210	−0.406105	+	0.913826i	$0.633113\pi$
$998$	−1.62648e6	−0.0516919
$999$	−9.85235e6	−0.312339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.x.1.4		13
5.2	odd	4		165.6.c.a.34.7	✓	26
5.3	odd	4		165.6.c.a.34.20	yes	26
5.4	even	2		825.6.a.w.1.10		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.a.34.7	✓	26	5.2	odd	4
165.6.c.a.34.20	yes	26	5.3	odd	4
825.6.a.w.1.10		13	5.4	even	2
825.6.a.x.1.4		13	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-6.98295 q^{2} -9.00000 q^{3} +16.7616 q^{4} +62.8465 q^{6} +45.4926 q^{7} +106.409 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-6.98295 q^{2} -9.00000 q^{3} +16.7616 q^{4} +62.8465 q^{6} +45.4926 q^{7} +106.409 q^{8} +81.0000 q^{9} -121.000 q^{11} -150.854 q^{12} -398.031 q^{13} -317.673 q^{14} -1279.42 q^{16} -2129.16 q^{17} -565.619 q^{18} -1809.70 q^{19} -409.433 q^{21} +844.937 q^{22} +2152.64 q^{23} -957.682 q^{24} +2779.43 q^{26} -729.000 q^{27} +762.528 q^{28} +7073.02 q^{29} -6775.82 q^{31} +5529.03 q^{32} +1089.00 q^{33} +14867.8 q^{34} +1357.69 q^{36} +13514.9 q^{37} +12637.0 q^{38} +3582.28 q^{39} -6086.26 q^{41} +2859.05 q^{42} +10109.1 q^{43} -2028.15 q^{44} -15031.8 q^{46} -20147.5 q^{47} +11514.8 q^{48} -14737.4 q^{49} +19162.4 q^{51} -6671.63 q^{52} -11024.9 q^{53} +5090.57 q^{54} +4840.83 q^{56} +16287.3 q^{57} -49390.6 q^{58} +687.001 q^{59} -39571.5 q^{61} +47315.2 q^{62} +3684.90 q^{63} +2332.49 q^{64} -7604.43 q^{66} +35405.3 q^{67} -35688.0 q^{68} -19373.7 q^{69} -72900.0 q^{71} +8619.14 q^{72} +34875.0 q^{73} -94373.7 q^{74} -30333.4 q^{76} -5504.61 q^{77} -25014.9 q^{78} -14392.1 q^{79} +6561.00 q^{81} +42500.1 q^{82} -8705.46 q^{83} -6862.75 q^{84} -70591.2 q^{86} -63657.2 q^{87} -12875.5 q^{88} -113547. q^{89} -18107.5 q^{91} +36081.6 q^{92} +60982.3 q^{93} +140689. q^{94} -49761.3 q^{96} -104432. q^{97} +102911. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 3 q^{2} - 117 q^{3} + 229 q^{4} - 27 q^{6} + 284 q^{7} + 369 q^{8} + 1053 q^{9}+O(q^{10}) \) 13 * q + 3 * q^2 - 117 * q^3 + 229 * q^4 - 27 * q^6 + 284 * q^7 + 369 * q^8 + 1053 * q^9	\( 13 q + 3 q^{2} - 117 q^{3} + 229 q^{4} - 27 q^{6} + 284 q^{7} + 369 q^{8} + 1053 q^{9} - 1573 q^{11} - 2061 q^{12} + 366 q^{13} - 2758 q^{14} + 4141 q^{16} + 2056 q^{17} + 243 q^{18} - 310 q^{19} - 2556 q^{21} - 363 q^{22} + 3612 q^{23} - 3321 q^{24} + 2280 q^{26} - 9477 q^{27} + 7896 q^{28} - 4848 q^{29} - 24 q^{31} + 38111 q^{32} + 14157 q^{33} + 5518 q^{34} + 18549 q^{36} - 8420 q^{37} + 474 q^{38} - 3294 q^{39} - 15120 q^{41} + 24822 q^{42} + 35492 q^{43} - 27709 q^{44} - 20280 q^{46} + 46544 q^{47} - 37269 q^{48} + 81837 q^{49} - 18504 q^{51} + 107194 q^{52} + 42256 q^{53} - 2187 q^{54} - 196602 q^{56} + 2790 q^{57} + 114160 q^{58} - 65592 q^{59} - 52042 q^{61} + 94972 q^{62} + 23004 q^{63} + 185977 q^{64} + 3267 q^{66} + 80580 q^{67} + 61108 q^{68} - 32508 q^{69} - 77820 q^{71} + 29889 q^{72} + 103050 q^{73} - 240028 q^{74} - 271174 q^{76} - 34364 q^{77} - 20520 q^{78} - 112258 q^{79} + 85293 q^{81} + 64060 q^{82} + 292150 q^{83} - 71064 q^{84} - 319250 q^{86} + 43632 q^{87} - 44649 q^{88} - 295810 q^{89} - 24200 q^{91} + 121328 q^{92} + 216 q^{93} - 358144 q^{94} - 342999 q^{96} + 49072 q^{97} + 101815 q^{98} - 127413 q^{99}+O(q^{100}) \) 13 * q + 3 * q^2 - 117 * q^3 + 229 * q^4 - 27 * q^6 + 284 * q^7 + 369 * q^8 + 1053 * q^9 - 1573 * q^11 - 2061 * q^12 + 366 * q^13 - 2758 * q^14 + 4141 * q^16 + 2056 * q^17 + 243 * q^18 - 310 * q^19 - 2556 * q^21 - 363 * q^22 + 3612 * q^23 - 3321 * q^24 + 2280 * q^26 - 9477 * q^27 + 7896 * q^28 - 4848 * q^29 - 24 * q^31 + 38111 * q^32 + 14157 * q^33 + 5518 * q^34 + 18549 * q^36 - 8420 * q^37 + 474 * q^38 - 3294 * q^39 - 15120 * q^41 + 24822 * q^42 + 35492 * q^43 - 27709 * q^44 - 20280 * q^46 + 46544 * q^47 - 37269 * q^48 + 81837 * q^49 - 18504 * q^51 + 107194 * q^52 + 42256 * q^53 - 2187 * q^54 - 196602 * q^56 + 2790 * q^57 + 114160 * q^58 - 65592 * q^59 - 52042 * q^61 + 94972 * q^62 + 23004 * q^63 + 185977 * q^64 + 3267 * q^66 + 80580 * q^67 + 61108 * q^68 - 32508 * q^69 - 77820 * q^71 + 29889 * q^72 + 103050 * q^73 - 240028 * q^74 - 271174 * q^76 - 34364 * q^77 - 20520 * q^78 - 112258 * q^79 + 85293 * q^81 + 64060 * q^82 + 292150 * q^83 - 71064 * q^84 - 319250 * q^86 + 43632 * q^87 - 44649 * q^88 - 295810 * q^89 - 24200 * q^91 + 121328 * q^92 + 216 * q^93 - 358144 * q^94 - 342999 * q^96 + 49072 * q^97 + 101815 * q^98 - 127413 * q^99

Introduction

L-functions

Modular forms

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