LMFDB - Embedded newform 650.6.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,6,Mod(1,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("650.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	650.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:	$104.249482878$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1458804.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 361x - 1139 $ x^3 - x^2 - 361*x - 1139
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$20.8905$ of defining polynomial
Character	$\chi$	$=$	650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} +27.8905 q^{3} +16.0000 q^{4} +111.562 q^{6} +240.426 q^{7} +64.0000 q^{8} +534.880 q^{9} +O(q^{10})$	\(q+4.00000 q^{2} +27.8905 q^{3} +16.0000 q^{4} +111.562 q^{6} +240.426 q^{7} +64.0000 q^{8} +534.880 q^{9} +544.151 q^{11} +446.248 q^{12} -169.000 q^{13} +961.702 q^{14} +256.000 q^{16} -1629.30 q^{17} +2139.52 q^{18} -805.920 q^{19} +6705.59 q^{21} +2176.61 q^{22} -373.152 q^{23} +1784.99 q^{24} -676.000 q^{26} +8140.67 q^{27} +3846.81 q^{28} +1503.62 q^{29} -2200.08 q^{31} +1024.00 q^{32} +15176.7 q^{33} -6517.20 q^{34} +8558.08 q^{36} +13109.1 q^{37} -3223.68 q^{38} -4713.49 q^{39} -17099.9 q^{41} +26822.4 q^{42} +8935.58 q^{43} +8706.42 q^{44} -1492.61 q^{46} -15749.7 q^{47} +7139.97 q^{48} +40997.5 q^{49} -45442.0 q^{51} -2704.00 q^{52} -40379.7 q^{53} +32562.7 q^{54} +15387.2 q^{56} -22477.5 q^{57} +6014.46 q^{58} -47562.1 q^{59} -30280.0 q^{61} -8800.31 q^{62} +128599. q^{63} +4096.00 q^{64} +60706.6 q^{66} -38769.4 q^{67} -26068.8 q^{68} -10407.4 q^{69} -10519.7 q^{71} +34232.3 q^{72} -1582.12 q^{73} +52436.5 q^{74} -12894.7 q^{76} +130828. q^{77} -18854.0 q^{78} -6191.23 q^{79} +97071.7 q^{81} -68399.8 q^{82} -37849.2 q^{83} +107289. q^{84} +35742.3 q^{86} +41936.6 q^{87} +34825.7 q^{88} +49151.0 q^{89} -40631.9 q^{91} -5970.43 q^{92} -61361.3 q^{93} -62999.0 q^{94} +28559.9 q^{96} +15654.2 q^{97} +163990. q^{98} +291056. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 12 q^{2} + 22 q^{3} + 48 q^{4} + 88 q^{6} + 234 q^{7} + 192 q^{8} + 155 q^{9}+O(q^{10}) $ 3 * q + 12 * q^2 + 22 * q^3 + 48 * q^4 + 88 * q^6 + 234 * q^7 + 192 * q^8 + 155 * q^9	\( 3 q + 12 q^{2} + 22 q^{3} + 48 q^{4} + 88 q^{6} + 234 q^{7} + 192 q^{8} + 155 q^{9} + 48 q^{11} + 352 q^{12} - 507 q^{13} + 936 q^{14} + 768 q^{16} - 1506 q^{17} + 620 q^{18} - 360 q^{19} + 6904 q^{21} + 192 q^{22} + 2370 q^{23} + 1408 q^{24} - 2028 q^{26} + 10168 q^{27} + 3744 q^{28} - 3078 q^{29} - 5388 q^{31} + 3072 q^{32} + 22572 q^{33} - 6024 q^{34} + 2480 q^{36} + 25362 q^{37} - 1440 q^{38} - 3718 q^{39} - 15906 q^{41} + 27616 q^{42} + 39306 q^{43} + 768 q^{44} + 9480 q^{46} + 17778 q^{47} + 5632 q^{48} + 7767 q^{49} - 33972 q^{51} - 8112 q^{52} - 9246 q^{53} + 40672 q^{54} + 14976 q^{56} + 2996 q^{57} - 12312 q^{58} - 77760 q^{59} + 17982 q^{61} - 21552 q^{62} + 128762 q^{63} + 12288 q^{64} + 90288 q^{66} + 2922 q^{67} - 24096 q^{68} - 31860 q^{69} - 4944 q^{71} + 9920 q^{72} - 43278 q^{73} + 101448 q^{74} - 5760 q^{76} + 144420 q^{77} - 14872 q^{78} + 42120 q^{79} + 146519 q^{81} - 63624 q^{82} - 58098 q^{83} + 110464 q^{84} + 157224 q^{86} - 2016 q^{87} + 3072 q^{88} + 19614 q^{89} - 39546 q^{91} + 37920 q^{92} - 162796 q^{93} + 71112 q^{94} + 22528 q^{96} + 87078 q^{97} + 31068 q^{98} + 350340 q^{99}+O(q^{100}) \) 3 * q + 12 * q^2 + 22 * q^3 + 48 * q^4 + 88 * q^6 + 234 * q^7 + 192 * q^8 + 155 * q^9 + 48 * q^11 + 352 * q^12 - 507 * q^13 + 936 * q^14 + 768 * q^16 - 1506 * q^17 + 620 * q^18 - 360 * q^19 + 6904 * q^21 + 192 * q^22 + 2370 * q^23 + 1408 * q^24 - 2028 * q^26 + 10168 * q^27 + 3744 * q^28 - 3078 * q^29 - 5388 * q^31 + 3072 * q^32 + 22572 * q^33 - 6024 * q^34 + 2480 * q^36 + 25362 * q^37 - 1440 * q^38 - 3718 * q^39 - 15906 * q^41 + 27616 * q^42 + 39306 * q^43 + 768 * q^44 + 9480 * q^46 + 17778 * q^47 + 5632 * q^48 + 7767 * q^49 - 33972 * q^51 - 8112 * q^52 - 9246 * q^53 + 40672 * q^54 + 14976 * q^56 + 2996 * q^57 - 12312 * q^58 - 77760 * q^59 + 17982 * q^61 - 21552 * q^62 + 128762 * q^63 + 12288 * q^64 + 90288 * q^66 + 2922 * q^67 - 24096 * q^68 - 31860 * q^69 - 4944 * q^71 + 9920 * q^72 - 43278 * q^73 + 101448 * q^74 - 5760 * q^76 + 144420 * q^77 - 14872 * q^78 + 42120 * q^79 + 146519 * q^81 - 63624 * q^82 - 58098 * q^83 + 110464 * q^84 + 157224 * q^86 - 2016 * q^87 + 3072 * q^88 + 19614 * q^89 - 39546 * q^91 + 37920 * q^92 - 162796 * q^93 + 71112 * q^94 + 22528 * q^96 + 87078 * q^97 + 31068 * q^98 + 350340 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	27.8905	1.78918	0.894588	−	0.446892i	$-0.147469\pi$
$3$	27.8905	1.78918	0.894588	+	0.446892i	$0.147469\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	111.562	1.26514
$7$	240.426	1.85454	0.927269	−	0.374397i	$-0.122150\pi$
$7$	240.426	1.85454	0.927269	+	0.374397i	$0.122150\pi$
$8$	64.0000	0.353553
$9$	534.880	2.20115
$10$	0	0
$11$	544.151	1.35593	0.677966	−	0.735093i	$-0.262862\pi$
$11$	544.151	1.35593	0.677966	+	0.735093i	$0.262862\pi$
$12$	446.248	0.894588
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	961.702	1.31136
$15$	0	0
$16$	256.000	0.250000
$17$	−1629.30	−1.36735	−0.683673	−	0.729788i	$-0.739619\pi$
$17$	−1629.30	−1.36735	−0.683673	+	0.729788i	$0.739619\pi$
$18$	2139.52	1.55645
$19$	−805.920	−0.512162	−0.256081	−	0.966655i	$-0.582431\pi$
$19$	−805.920	−0.512162	−0.256081	+	0.966655i	$0.582431\pi$
$20$	0	0
$21$	6705.59	3.31809
$22$	2176.61	0.958789
$23$	−373.152	−0.147084	−0.0735421	−	0.997292i	$-0.523430\pi$
$23$	−373.152	−0.147084	−0.0735421	+	0.997292i	$0.523430\pi$
$24$	1784.99	0.632569
$25$	0	0
$26$	−676.000	−0.196116
$27$	8140.67	2.14907
$28$	3846.81	0.927269
$29$	1503.62	0.332003	0.166001	−	0.986126i	$-0.446914\pi$
$29$	1503.62	0.332003	0.166001	+	0.986126i	$0.446914\pi$
$30$	0	0
$31$	−2200.08	−0.411182	−0.205591	−	0.978638i	$-0.565912\pi$
$31$	−2200.08	−0.411182	−0.205591	+	0.978638i	$0.565912\pi$
$32$	1024.00	0.176777
$33$	15176.7	2.42600
$34$	−6517.20	−0.966860
$35$	0	0
$36$	8558.08	1.10058
$37$	13109.1	1.57423	0.787117	−	0.616804i	$-0.211573\pi$
$37$	13109.1	1.57423	0.787117	+	0.616804i	$0.211573\pi$
$38$	−3223.68	−0.362154
$39$	−4713.49	−0.496228
$40$	0	0
$41$	−17099.9	−1.58867	−0.794337	−	0.607477i	$-0.792182\pi$
$41$	−17099.9	−1.58867	−0.794337	+	0.607477i	$0.792182\pi$
$42$	26822.4	2.34625
$43$	8935.58	0.736973	0.368487	−	0.929633i	$-0.379876\pi$
$43$	8935.58	0.736973	0.368487	+	0.929633i	$0.379876\pi$
$44$	8706.42	0.677966
$45$	0	0
$46$	−1492.61	−0.104004
$47$	−15749.7	−1.03999	−0.519995	−	0.854170i	$-0.674066\pi$
$47$	−15749.7	−1.03999	−0.519995	+	0.854170i	$0.674066\pi$
$48$	7139.97	0.447294
$49$	40997.5	2.43931
$50$	0	0
$51$	−45442.0	−2.44642
$52$	−2704.00	−0.138675
$53$	−40379.7	−1.97457	−0.987286	−	0.158951i	$-0.949189\pi$
$53$	−40379.7	−1.97457	−0.987286	+	0.158951i	$0.949189\pi$
$54$	32562.7	1.51962
$55$	0	0
$56$	15387.2	0.655678
$57$	−22477.5	−0.916349
$58$	6014.46	0.234761
$59$	−47562.1	−1.77881	−0.889407	−	0.457116i	$-0.848882\pi$
$59$	−47562.1	−1.77881	−0.889407	+	0.457116i	$0.848882\pi$
$60$	0	0
$61$	−30280.0	−1.04191	−0.520956	−	0.853584i	$-0.674424\pi$
$61$	−30280.0	−1.04191	−0.520956	+	0.853584i	$0.674424\pi$
$62$	−8800.31	−0.290749
$63$	128599.	4.08212
$64$	4096.00	0.125000
$65$	0	0
$66$	60706.6	1.71544
$67$	−38769.4	−1.05512	−0.527561	−	0.849517i	$-0.676893\pi$
$67$	−38769.4	−1.05512	−0.527561	+	0.849517i	$0.676893\pi$
$68$	−26068.8	−0.683673
$69$	−10407.4	−0.263160
$70$	0	0
$71$	−10519.7	−0.247660	−0.123830	−	0.992303i	$-0.539518\pi$
$71$	−10519.7	−0.247660	−0.123830	+	0.992303i	$0.539518\pi$
$72$	34232.3	0.778225
$73$	−1582.12	−0.0347483	−0.0173742	−	0.999849i	$-0.505531\pi$
$73$	−1582.12	−0.0347483	−0.0173742	+	0.999849i	$0.505531\pi$
$74$	52436.5	1.11315
$75$	0	0
$76$	−12894.7	−0.256081
$77$	130828.	2.51463
$78$	−18854.0	−0.350886
$79$	−6191.23	−0.111612	−0.0558058	−	0.998442i	$-0.517773\pi$
$79$	−6191.23	−0.111612	−0.0558058	+	0.998442i	$0.517773\pi$
$80$	0	0
$81$	97071.7	1.64392
$82$	−68399.8	−1.12336
$83$	−37849.2	−0.603062	−0.301531	−	0.953456i	$-0.597498\pi$
$83$	−37849.2	−0.603062	−0.301531	+	0.953456i	$0.597498\pi$
$84$	107289.	1.65905
$85$	0	0
$86$	35742.3	0.521119
$87$	41936.6	0.594011
$88$	34825.7	0.479394
$89$	49151.0	0.657744	0.328872	−	0.944374i	$-0.393331\pi$
$89$	49151.0	0.657744	0.328872	+	0.944374i	$0.393331\pi$
$90$	0	0
$91$	−40631.9	−0.514356
$92$	−5970.43	−0.0735421
$93$	−61361.3	−0.735677
$94$	−62999.0	−0.735383
$95$	0	0
$96$	28559.9	0.316285
$97$	15654.2	0.168928	0.0844641	−	0.996427i	$-0.473082\pi$
$97$	15654.2	0.168928	0.0844641	+	0.996427i	$0.473082\pi$
$98$	163990.	1.72485
$99$	291056.	2.98461
$100$	0	0
$101$	−138108.	−1.34714	−0.673572	−	0.739122i	$-0.735241\pi$
$101$	−138108.	−1.34714	−0.673572	+	0.739122i	$0.735241\pi$
$102$	−181768.	−1.72988
$103$	47642.1	0.442484	0.221242	−	0.975219i	$-0.428989\pi$
$103$	47642.1	0.442484	0.221242	+	0.975219i	$0.428989\pi$
$104$	−10816.0	−0.0980581
$105$	0	0
$106$	−161519.	−1.39623
$107$	186527.	1.57500	0.787501	−	0.616313i	$-0.211374\pi$
$107$	186527.	1.57500	0.787501	+	0.616313i	$0.211374\pi$
$108$	130251.	1.07454
$109$	14407.1	0.116147	0.0580736	−	0.998312i	$-0.481504\pi$
$109$	14407.1	0.116147	0.0580736	+	0.998312i	$0.481504\pi$
$110$	0	0
$111$	365620.	2.81658
$112$	61548.9	0.463634
$113$	50802.0	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$113$	50802.0	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$114$	−89910.0	−0.647957
$115$	0	0
$116$	24057.8	0.166001
$117$	−90394.7	−0.610490
$118$	−190248.	−1.25781
$119$	−391725.	−2.53580
$120$	0	0
$121$	135050.	0.838552
$122$	−121120.	−0.736742
$123$	−476926.	−2.84242
$124$	−35201.3	−0.205591
$125$	0	0
$126$	514395.	2.88649
$127$	182278.	1.00282	0.501411	−	0.865209i	$-0.332814\pi$
$127$	182278.	1.00282	0.501411	+	0.865209i	$0.332814\pi$
$128$	16384.0	0.0883883
$129$	249218.	1.31858
$130$	0	0
$131$	199714.	1.01679	0.508394	−	0.861124i	$-0.330239\pi$
$131$	199714.	1.01679	0.508394	+	0.861124i	$0.330239\pi$
$132$	242826.	1.21300
$133$	−193764.	−0.949824
$134$	−155078.	−0.746083
$135$	0	0
$136$	−104275.	−0.483430
$137$	345418.	1.57233	0.786165	−	0.618017i	$-0.212064\pi$
$137$	345418.	1.57233	0.786165	+	0.618017i	$0.212064\pi$
$138$	−41629.6	−0.186082
$139$	183088.	0.803754	0.401877	−	0.915694i	$-0.368358\pi$
$139$	183088.	0.803754	0.401877	+	0.915694i	$0.368358\pi$
$140$	0	0
$141$	−439268.	−1.86072
$142$	−42078.7	−0.175122
$143$	−91961.6	−0.376068
$144$	136929.	0.550288
$145$	0	0
$146$	−6328.50	−0.0245708
$147$	1.14344e6	4.36435
$148$	209746.	0.787117
$149$	−187082.	−0.690344	−0.345172	−	0.938540i	$-0.612179\pi$
$149$	−187082.	−0.690344	−0.345172	+	0.938540i	$0.612179\pi$
$150$	0	0
$151$	−10616.3	−0.0378907	−0.0189454	−	0.999821i	$-0.506031\pi$
$151$	−10616.3	−0.0378907	−0.0189454	+	0.999821i	$0.506031\pi$
$152$	−51578.9	−0.181077
$153$	−871480.	−3.00974
$154$	523312.	1.77811
$155$	0	0
$156$	−75415.9	−0.248114
$157$	335884.	1.08753	0.543764	−	0.839238i	$-0.316999\pi$
$157$	335884.	1.08753	0.543764	+	0.839238i	$0.316999\pi$
$158$	−24764.9	−0.0789213
$159$	−1.12621e6	−3.53286
$160$	0	0
$161$	−89715.3	−0.272773
$162$	388287.	1.16242
$163$	−201881.	−0.595150	−0.297575	−	0.954698i	$-0.596178\pi$
$163$	−201881.	−0.595150	−0.297575	+	0.954698i	$0.596178\pi$
$164$	−273599.	−0.794337
$165$	0	0
$166$	−151397.	−0.426429
$167$	446050.	1.23764	0.618818	−	0.785535i	$-0.287612\pi$
$167$	446050.	1.23764	0.618818	+	0.785535i	$0.287612\pi$
$168$	429158.	1.17312
$169$	28561.0	0.0769231
$170$	0	0
$171$	−431070.	−1.12735
$172$	142969.	0.368487
$173$	−57408.8	−0.145835	−0.0729177	−	0.997338i	$-0.523231\pi$
$173$	−57408.8	−0.145835	−0.0729177	+	0.997338i	$0.523231\pi$
$174$	167746.	0.420030
$175$	0	0
$176$	139303.	0.338983
$177$	−1.32653e6	−3.18261
$178$	196604.	0.465095
$179$	87439.4	0.203974	0.101987	−	0.994786i	$-0.467480\pi$
$179$	87439.4	0.203974	0.101987	+	0.994786i	$0.467480\pi$
$180$	0	0
$181$	21341.6	0.0484206	0.0242103	−	0.999707i	$-0.492293\pi$
$181$	21341.6	0.0484206	0.0242103	+	0.999707i	$0.492293\pi$
$182$	−162528.	−0.363705
$183$	−844523.	−1.86416
$184$	−23881.7	−0.0520021
$185$	0	0
$186$	−245445.	−0.520202
$187$	−886586.	−1.85403
$188$	−251996.	−0.519995
$189$	1.95723e6	3.98553
$190$	0	0
$191$	307517.	0.609938	0.304969	−	0.952362i	$-0.401354\pi$
$191$	307517.	0.609938	0.304969	+	0.952362i	$0.401354\pi$
$192$	114239.	0.223647
$193$	−182409.	−0.352496	−0.176248	−	0.984346i	$-0.556396\pi$
$193$	−182409.	−0.352496	−0.176248	+	0.984346i	$0.556396\pi$
$194$	62616.9	0.119450
$195$	0	0
$196$	655959.	1.21965
$197$	152508.	0.279980	0.139990	−	0.990153i	$-0.455293\pi$
$197$	152508.	0.279980	0.139990	+	0.990153i	$0.455293\pi$
$198$	1.16422e6	2.11044
$199$	−586606.	−1.05006	−0.525029	−	0.851084i	$-0.675946\pi$
$199$	−586606.	−1.05006	−0.525029	+	0.851084i	$0.675946\pi$
$200$	0	0
$201$	−1.08130e6	−1.88780
$202$	−552430.	−0.952574
$203$	361508.	0.615711
$204$	−727072.	−1.22321
$205$	0	0
$206$	190568.	0.312884
$207$	−199592.	−0.323755
$208$	−43264.0	−0.0693375
$209$	−438542.	−0.694458
$210$	0	0
$211$	−207041.	−0.320147	−0.160073	−	0.987105i	$-0.551173\pi$
$211$	−207041.	−0.320147	−0.160073	+	0.987105i	$0.551173\pi$
$212$	−646075.	−0.987286
$213$	−293399.	−0.443108
$214$	746106.	1.11369
$215$	0	0
$216$	521003.	0.759812
$217$	−528955.	−0.762552
$218$	57628.2	0.0821285
$219$	−44126.3	−0.0621708
$220$	0	0
$221$	275352.	0.379234
$222$	1.46248e6	1.99162
$223$	1.04889e6	1.41243	0.706214	−	0.707998i	$-0.250401\pi$
$223$	1.04889e6	1.41243	0.706214	+	0.707998i	$0.250401\pi$
$224$	246196.	0.327839
$225$	0	0
$226$	203208.	0.264649
$227$	100709.	0.129719	0.0648593	−	0.997894i	$-0.479340\pi$
$227$	100709.	0.129719	0.0648593	+	0.997894i	$0.479340\pi$
$228$	−359640.	−0.458174
$229$	487174.	0.613896	0.306948	−	0.951726i	$-0.400692\pi$
$229$	487174.	0.613896	0.306948	+	0.951726i	$0.400692\pi$
$230$	0	0
$231$	3.64886e6	4.49911
$232$	96231.4	0.117381
$233$	832844.	1.00502	0.502509	−	0.864572i	$-0.332410\pi$
$233$	832844.	1.00502	0.502509	+	0.864572i	$0.332410\pi$
$234$	−361579.	−0.431681
$235$	0	0
$236$	−760993.	−0.889407
$237$	−172677.	−0.199693
$238$	−1.56690e6	−1.79308
$239$	1.34919e6	1.52785	0.763923	−	0.645308i	$-0.223271\pi$
$239$	1.34919e6	1.52785	0.763923	+	0.645308i	$0.223271\pi$
$240$	0	0
$241$	−1.66283e6	−1.84419	−0.922096	−	0.386962i	$-0.873524\pi$
$241$	−1.66283e6	−1.84419	−0.922096	+	0.386962i	$0.873524\pi$
$242$	540199.	0.592946
$243$	729193.	0.792185
$244$	−484479.	−0.520956
$245$	0	0
$246$	−1.90770e6	−2.00989
$247$	136200.	0.142048
$248$	−140805.	−0.145375
$249$	−1.05563e6	−1.07898
$250$	0	0
$251$	−291273.	−0.291820	−0.145910	−	0.989298i	$-0.546611\pi$
$251$	−291273.	−0.291820	−0.145910	+	0.989298i	$0.546611\pi$
$252$	2.05758e6	2.04106
$253$	−203051.	−0.199436
$254$	729110.	0.709102
$255$	0	0
$256$	65536.0	0.0625000
$257$	548204.	0.517737	0.258868	−	0.965913i	$-0.416650\pi$
$257$	548204.	0.517737	0.258868	+	0.965913i	$0.416650\pi$
$258$	996872.	0.932374
$259$	3.15177e6	2.91947
$260$	0	0
$261$	804253.	0.730788
$262$	798857.	0.718978
$263$	−818412.	−0.729596	−0.364798	−	0.931087i	$-0.618862\pi$
$263$	−818412.	−0.729596	−0.364798	+	0.931087i	$0.618862\pi$
$264$	971306.	0.857721
$265$	0	0
$266$	−775055.	−0.671627
$267$	1.37085e6	1.17682
$268$	−620311.	−0.527561
$269$	−939455.	−0.791581	−0.395790	−	0.918341i	$-0.629529\pi$
$269$	−939455.	−0.791581	−0.395790	+	0.918341i	$0.629529\pi$
$270$	0	0
$271$	−1.52416e6	−1.26069	−0.630345	−	0.776315i	$-0.717087\pi$
$271$	−1.52416e6	−1.26069	−0.630345	+	0.776315i	$0.717087\pi$
$272$	−417101.	−0.341837
$273$	−1.13324e6	−0.920274
$274$	1.38167e6	1.11180
$275$	0	0
$276$	−166518.	−0.131580
$277$	439704.	0.344319	0.172159	−	0.985069i	$-0.444926\pi$
$277$	439704.	0.344319	0.172159	+	0.985069i	$0.444926\pi$
$278$	732352.	0.568340
$279$	−1.17678e6	−0.905074
$280$	0	0
$281$	−592033.	−0.447280	−0.223640	−	0.974672i	$-0.571794\pi$
$281$	−592033.	−0.447280	−0.223640	+	0.974672i	$0.571794\pi$
$282$	−1.75707e6	−1.31573
$283$	777226.	0.576875	0.288437	−	0.957499i	$-0.406864\pi$
$283$	777226.	0.576875	0.288437	+	0.957499i	$0.406864\pi$
$284$	−168315.	−0.123830
$285$	0	0
$286$	−367846.	−0.265920
$287$	−4.11126e6	−2.94626
$288$	547717.	0.389112
$289$	1.23476e6	0.869637
$290$	0	0
$291$	436604.	0.302242
$292$	−25314.0	−0.0173742
$293$	−609025.	−0.414444	−0.207222	−	0.978294i	$-0.566442\pi$
$293$	−609025.	−0.414444	−0.207222	+	0.978294i	$0.566442\pi$
$294$	4.57376e6	3.08606
$295$	0	0
$296$	838984.	0.556576
$297$	4.42976e6	2.91400
$298$	−748326.	−0.488147
$299$	63062.7	0.0407938
$300$	0	0
$301$	2.14834e6	1.36674
$302$	−42465.4	−0.0267928
$303$	−3.85189e6	−2.41028
$304$	−206315.	−0.128041
$305$	0	0
$306$	−3.48592e6	−2.12821
$307$	2.66761e6	1.61538	0.807692	−	0.589604i	$-0.200716\pi$
$307$	2.66761e6	1.61538	0.807692	+	0.589604i	$0.200716\pi$
$308$	2.09325e6	1.25731
$309$	1.32876e6	0.791682
$310$	0	0
$311$	−1.62320e6	−0.951634	−0.475817	−	0.879544i	$-0.657847\pi$
$311$	−1.62320e6	−0.951634	−0.475817	+	0.879544i	$0.657847\pi$
$312$	−301664.	−0.175443
$313$	−592115.	−0.341622	−0.170811	−	0.985304i	$-0.554639\pi$
$313$	−592115.	−0.341622	−0.170811	+	0.985304i	$0.554639\pi$
$314$	1.34354e6	0.768998
$315$	0	0
$316$	−99059.7	−0.0558058
$317$	2.61036e6	1.45899	0.729494	−	0.683987i	$-0.239756\pi$
$317$	2.61036e6	1.45899	0.729494	+	0.683987i	$0.239756\pi$
$318$	−4.50484e6	−2.49811
$319$	818194.	0.450173
$320$	0	0
$321$	5.20232e6	2.81796
$322$	−358861.	−0.192880
$323$	1.31308e6	0.700304
$324$	1.55315e6	0.821959
$325$	0	0
$326$	−807524.	−0.420834
$327$	401820.	0.207808
$328$	−1.09440e6	−0.561681
$329$	−3.78664e6	−1.92870
$330$	0	0
$331$	818785.	0.410771	0.205386	−	0.978681i	$-0.434155\pi$
$331$	818785.	0.410771	0.205386	+	0.978681i	$0.434155\pi$
$332$	−605588.	−0.301531
$333$	7.01180e6	3.46513
$334$	1.78420e6	0.875140
$335$	0	0
$336$	1.71663e6	0.829523
$337$	−3.28910e6	−1.57762	−0.788809	−	0.614638i	$-0.789302\pi$
$337$	−3.28910e6	−1.57762	−0.788809	+	0.614638i	$0.789302\pi$
$338$	114244.	0.0543928
$339$	1.41689e6	0.669634
$340$	0	0
$341$	−1.19718e6	−0.557535
$342$	−1.72428e6	−0.797155
$343$	5.81600e6	2.66925
$344$	571877.	0.260559
$345$	0	0
$346$	−229635.	−0.103121
$347$	−139108.	−0.0620195	−0.0310098	−	0.999519i	$-0.509872\pi$
$347$	−139108.	−0.0620195	−0.0310098	+	0.999519i	$0.509872\pi$
$348$	670985.	0.297006
$349$	1.41781e6	0.623097	0.311549	−	0.950230i	$-0.399152\pi$
$349$	1.41781e6	0.623097	0.311549	+	0.950230i	$0.399152\pi$
$350$	0	0
$351$	−1.37577e6	−0.596045
$352$	557211.	0.239697
$353$	−2.45675e6	−1.04936	−0.524679	−	0.851300i	$-0.675815\pi$
$353$	−2.45675e6	−1.04936	−0.524679	+	0.851300i	$0.675815\pi$
$354$	−5.30612e6	−2.25045
$355$	0	0
$356$	786416.	0.328872
$357$	−1.09254e7	−4.53698
$358$	349758.	0.144231
$359$	−2.88601e6	−1.18185	−0.590924	−	0.806727i	$-0.701237\pi$
$359$	−2.88601e6	−1.18185	−0.590924	+	0.806727i	$0.701237\pi$
$360$	0	0
$361$	−1.82659e6	−0.737690
$362$	85366.4	0.0342386
$363$	3.76660e6	1.50032
$364$	−650111.	−0.257178
$365$	0	0
$366$	−3.37809e6	−1.31816
$367$	2.83587e6	1.09906	0.549530	−	0.835474i	$-0.314807\pi$
$367$	2.83587e6	1.09906	0.549530	+	0.835474i	$0.314807\pi$
$368$	−95526.9	−0.0367711
$369$	−9.14641e6	−3.49691
$370$	0	0
$371$	−9.70831e6	−3.66192
$372$	−981781.	−0.367838
$373$	−5.02691e6	−1.87081	−0.935403	−	0.353583i	$-0.884963\pi$
$373$	−5.02691e6	−1.87081	−0.935403	+	0.353583i	$0.884963\pi$
$374$	−3.54634e6	−1.31100
$375$	0	0
$376$	−1.00798e6	−0.367692
$377$	−254111.	−0.0920810
$378$	7.82891e6	2.81820
$379$	2.14892e6	0.768462	0.384231	−	0.923237i	$-0.374467\pi$
$379$	2.14892e6	0.768462	0.384231	+	0.923237i	$0.374467\pi$
$380$	0	0
$381$	5.08381e6	1.79423
$382$	1.23007e6	0.431291
$383$	−832971.	−0.290157	−0.145078	−	0.989420i	$-0.546343\pi$
$383$	−832971.	−0.290157	−0.145078	+	0.989420i	$0.546343\pi$
$384$	456958.	0.158142
$385$	0	0
$386$	−729637.	−0.249252
$387$	4.77946e6	1.62219
$388$	250468.	0.0844641
$389$	−725243.	−0.243002	−0.121501	−	0.992591i	$-0.538771\pi$
$389$	−725243.	−0.243002	−0.121501	+	0.992591i	$0.538771\pi$
$390$	0	0
$391$	607976.	0.201115
$392$	2.62384e6	0.862426
$393$	5.57013e6	1.81921
$394$	610032.	0.197976
$395$	0	0
$396$	4.65689e6	1.49231
$397$	−2.42654e6	−0.772701	−0.386351	−	0.922352i	$-0.626265\pi$
$397$	−2.42654e6	−0.772701	−0.386351	+	0.922352i	$0.626265\pi$
$398$	−2.34642e6	−0.742504
$399$	−5.40417e6	−1.69940
$400$	0	0
$401$	−2.14376e6	−0.665758	−0.332879	−	0.942970i	$-0.608020\pi$
$401$	−2.14376e6	−0.665758	−0.332879	+	0.942970i	$0.608020\pi$
$402$	−4.32519e6	−1.33487
$403$	371813.	0.114041
$404$	−2.20972e6	−0.673572
$405$	0	0
$406$	1.44603e6	0.435374
$407$	7.13335e6	2.13455
$408$	−2.90829e6	−0.864942
$409$	−3.66144e6	−1.08229	−0.541145	−	0.840929i	$-0.682009\pi$
$409$	−3.66144e6	−1.08229	−0.541145	+	0.840929i	$0.682009\pi$
$410$	0	0
$411$	9.63388e6	2.81317
$412$	762273.	0.221242
$413$	−1.14351e7	−3.29888
$414$	−798366.	−0.228929
$415$	0	0
$416$	−173056.	−0.0490290
$417$	5.10642e6	1.43806
$418$	−1.75417e6	−0.491056
$419$	3.03786e6	0.845341	0.422671	−	0.906283i	$-0.361093\pi$
$419$	3.03786e6	0.845341	0.422671	+	0.906283i	$0.361093\pi$
$420$	0	0
$421$	−1.88663e6	−0.518777	−0.259389	−	0.965773i	$-0.583521\pi$
$421$	−1.88663e6	−0.518777	−0.259389	+	0.965773i	$0.583521\pi$
$422$	−828162.	−0.226378
$423$	−8.42422e6	−2.28917
$424$	−2.58430e6	−0.698117
$425$	0	0
$426$	−1.17360e6	−0.313325
$427$	−7.28008e6	−1.93226
$428$	2.98442e6	0.787501
$429$	−2.56485e6	−0.672852
$430$	0	0
$431$	717193.	0.185970	0.0929850	−	0.995668i	$-0.470359\pi$
$431$	717193.	0.185970	0.0929850	+	0.995668i	$0.470359\pi$
$432$	2.08401e6	0.537268
$433$	−2.64569e6	−0.678140	−0.339070	−	0.940761i	$-0.610112\pi$
$433$	−2.64569e6	−0.678140	−0.339070	+	0.940761i	$0.610112\pi$
$434$	−2.11582e6	−0.539206
$435$	0	0
$436$	230513.	0.0580736
$437$	300731.	0.0753310
$438$	−176505.	−0.0439614
$439$	4.47973e6	1.10941	0.554704	−	0.832048i	$-0.312832\pi$
$439$	4.47973e6	1.10941	0.554704	+	0.832048i	$0.312832\pi$
$440$	0	0
$441$	2.19287e7	5.36929
$442$	1.10141e6	0.268159
$443$	−4.30731e6	−1.04279	−0.521395	−	0.853315i	$-0.674588\pi$
$443$	−4.30731e6	−1.04279	−0.521395	+	0.853315i	$0.674588\pi$
$444$	5.84992e6	1.40829
$445$	0	0
$446$	4.19555e6	0.998738
$447$	−5.21780e6	−1.23515
$448$	984783.	0.231817
$449$	−25272.9	−0.00591615	−0.00295808	−	0.999996i	$-0.500942\pi$
$449$	−25272.9	−0.00591615	−0.00295808	+	0.999996i	$0.500942\pi$
$450$	0	0
$451$	−9.30495e6	−2.15413
$452$	812832.	0.187135
$453$	−296095.	−0.0677932
$454$	402835.	0.0917250
$455$	0	0
$456$	−1.43856e6	−0.323978
$457$	−1.14360e6	−0.256143	−0.128072	−	0.991765i	$-0.540879\pi$
$457$	−1.14360e6	−0.256143	−0.128072	+	0.991765i	$0.540879\pi$
$458$	1.94869e6	0.434090
$459$	−1.32636e7	−2.93853
$460$	0	0
$461$	−8.34911e6	−1.82973	−0.914867	−	0.403755i	$-0.867705\pi$
$461$	−8.34911e6	−1.82973	−0.914867	+	0.403755i	$0.867705\pi$
$462$	1.45954e7	3.18135
$463$	3.41596e6	0.740561	0.370280	−	0.928920i	$-0.379262\pi$
$463$	3.41596e6	0.740561	0.370280	+	0.928920i	$0.379262\pi$
$464$	384925.	0.0830007
$465$	0	0
$466$	3.33138e6	0.710655
$467$	−5.67105e6	−1.20329	−0.601646	−	0.798763i	$-0.705488\pi$
$467$	−5.67105e6	−1.20329	−0.601646	+	0.798763i	$0.705488\pi$
$468$	−1.44632e6	−0.305245
$469$	−9.32116e6	−1.95676
$470$	0	0
$471$	9.36797e6	1.94578
$472$	−3.04397e6	−0.628906
$473$	4.86231e6	0.999286
$474$	−690706.	−0.141204
$475$	0	0
$476$	−6.26760e6	−1.26790
$477$	−2.15983e7	−4.34633
$478$	5.39677e6	1.08035
$479$	−6.16340e6	−1.22739	−0.613694	−	0.789544i	$-0.710317\pi$
$479$	−6.16340e6	−1.22739	−0.613694	+	0.789544i	$0.710317\pi$
$480$	0	0
$481$	−2.21544e6	−0.436614
$482$	−6.65133e6	−1.30404
$483$	−2.50220e6	−0.488039
$484$	2.16080e6	0.419276
$485$	0	0
$486$	2.91677e6	0.560160
$487$	7.32701e6	1.39992	0.699962	−	0.714180i	$-0.253200\pi$
$487$	7.32701e6	1.39992	0.699962	+	0.714180i	$0.253200\pi$
$488$	−1.93792e6	−0.368371
$489$	−5.63056e6	−1.06483
$490$	0	0
$491$	8.89739e6	1.66555	0.832777	−	0.553608i	$-0.186750\pi$
$491$	8.89739e6	1.66555	0.832777	+	0.553608i	$0.186750\pi$
$492$	−7.63081e6	−1.42121
$493$	−2.44984e6	−0.453963
$494$	544802.	0.100443
$495$	0	0
$496$	−563220.	−0.102795
$497$	−2.52920e6	−0.459295
$498$	−4.22254e6	−0.762957
$499$	3.02237e6	0.543371	0.271686	−	0.962386i	$-0.412419\pi$
$499$	3.02237e6	0.543371	0.271686	+	0.962386i	$0.412419\pi$
$500$	0	0
$501$	1.24406e7	2.21435
$502$	−1.16509e6	−0.206348
$503$	7.48799e6	1.31961	0.659805	−	0.751437i	$-0.270639\pi$
$503$	7.48799e6	1.31961	0.659805	+	0.751437i	$0.270639\pi$
$504$	8.23032e6	1.44325
$505$	0	0
$506$	−812205.	−0.141023
$507$	796581.	0.137629
$508$	2.91644e6	0.501411
$509$	4.87997e6	0.834878	0.417439	−	0.908705i	$-0.362928\pi$
$509$	4.87997e6	0.834878	0.417439	+	0.908705i	$0.362928\pi$
$510$	0	0
$511$	−380383.	−0.0644420
$512$	262144.	0.0441942
$513$	−6.56073e6	−1.10067
$514$	2.19281e6	0.366095
$515$	0	0
$516$	3.98749e6	0.659288
$517$	−8.57024e6	−1.41015
$518$	1.26071e7	2.06438
$519$	−1.60116e6	−0.260925
$520$	0	0
$521$	−3.79679e6	−0.612805	−0.306403	−	0.951902i	$-0.599125\pi$
$521$	−3.79679e6	−0.612805	−0.306403	+	0.951902i	$0.599125\pi$
$522$	3.21701e6	0.516745
$523$	5.66643e6	0.905848	0.452924	−	0.891549i	$-0.350381\pi$
$523$	5.66643e6	0.905848	0.452924	+	0.891549i	$0.350381\pi$
$524$	3.19543e6	0.508394
$525$	0	0
$526$	−3.27365e6	−0.515902
$527$	3.58459e6	0.562228
$528$	3.88522e6	0.606500
$529$	−6.29710e6	−0.978366
$530$	0	0
$531$	−2.54400e7	−3.91544
$532$	−3.10022e6	−0.474912
$533$	2.88989e6	0.440619
$534$	5.48338e6	0.832138
$535$	0	0
$536$	−2.48124e6	−0.373042
$537$	2.43873e6	0.364945
$538$	−3.75782e6	−0.559732
$539$	2.23088e7	3.30754
$540$	0	0
$541$	1.15648e7	1.69881	0.849403	−	0.527745i	$-0.176962\pi$
$541$	1.15648e7	1.69881	0.849403	+	0.527745i	$0.176962\pi$
$542$	−6.09665e6	−0.891442
$543$	595228.	0.0866330
$544$	−1.66840e6	−0.241715
$545$	0	0
$546$	−4.53298e6	−0.650732
$547$	1.28104e7	1.83060	0.915299	−	0.402776i	$-0.131955\pi$
$547$	1.28104e7	1.83060	0.915299	+	0.402776i	$0.131955\pi$
$548$	5.52669e6	0.786165
$549$	−1.61961e7	−2.29340
$550$	0	0
$551$	−1.21179e6	−0.170039
$552$	−666073.	−0.0930410
$553$	−1.48853e6	−0.206988
$554$	1.75881e6	0.243470
$555$	0	0
$556$	2.92941e6	0.401877
$557$	−3.51737e6	−0.480374	−0.240187	−	0.970727i	$-0.577209\pi$
$557$	−3.51737e6	−0.480374	−0.240187	+	0.970727i	$0.577209\pi$
$558$	−4.70711e6	−0.639984
$559$	−1.51011e6	−0.204400
$560$	0	0
$561$	−2.47273e7	−3.31719
$562$	−2.36813e6	−0.316275
$563$	−9.45258e6	−1.25684	−0.628419	−	0.777875i	$-0.716298\pi$
$563$	−9.45258e6	−1.25684	−0.628419	+	0.777875i	$0.716298\pi$
$564$	−7.02829e6	−0.930362
$565$	0	0
$566$	3.10891e6	0.407912
$567$	2.33385e7	3.04871
$568$	−673259.	−0.0875611
$569$	1.11315e7	1.44136	0.720679	−	0.693269i	$-0.243830\pi$
$569$	1.11315e7	1.44136	0.720679	+	0.693269i	$0.243830\pi$
$570$	0	0
$571$	4.92935e6	0.632702	0.316351	−	0.948642i	$-0.397542\pi$
$571$	4.92935e6	0.632702	0.316351	+	0.948642i	$0.397542\pi$
$572$	−1.47139e6	−0.188034
$573$	8.57680e6	1.09129
$574$	−1.64450e7	−2.08332
$575$	0	0
$576$	2.19087e6	0.275144
$577$	−1.09437e7	−1.36843	−0.684217	−	0.729279i	$-0.739856\pi$
$577$	−1.09437e7	−1.36843	−0.684217	+	0.729279i	$0.739856\pi$
$578$	4.93904e6	0.614926
$579$	−5.08749e6	−0.630677
$580$	0	0
$581$	−9.09993e6	−1.11840
$582$	1.74642e6	0.213718
$583$	−2.19727e7	−2.67739
$584$	−101256.	−0.0122854
$585$	0	0
$586$	−2.43610e6	−0.293056
$587$	−7.96444e6	−0.954025	−0.477013	−	0.878896i	$-0.658280\pi$
$587$	−7.96444e6	−0.954025	−0.477013	+	0.878896i	$0.658280\pi$
$588$	1.82950e7	2.18218
$589$	1.77309e6	0.210592
$590$	0	0
$591$	4.25352e6	0.500934
$592$	3.35593e6	0.393558
$593$	1.19793e7	1.39892	0.699460	−	0.714671i	$-0.253424\pi$
$593$	1.19793e7	1.39892	0.699460	+	0.714671i	$0.253424\pi$
$594$	1.77190e7	2.06051
$595$	0	0
$596$	−2.99330e6	−0.345172
$597$	−1.63607e7	−1.87874
$598$	252251.	0.0288456
$599$	7.45663e6	0.849132	0.424566	−	0.905397i	$-0.360427\pi$
$599$	7.45663e6	0.849132	0.424566	+	0.905397i	$0.360427\pi$
$600$	0	0
$601$	−1.04196e7	−1.17670	−0.588351	−	0.808606i	$-0.700223\pi$
$601$	−1.04196e7	−1.17670	−0.588351	+	0.808606i	$0.700223\pi$
$602$	8.59337e6	0.966434
$603$	−2.07370e7	−2.32248
$604$	−169862.	−0.0189454
$605$	0	0
$606$	−1.54075e7	−1.70432
$607$	−571939.	−0.0630054	−0.0315027	−	0.999504i	$-0.510029\pi$
$607$	−571939.	−0.0630054	−0.0315027	+	0.999504i	$0.510029\pi$
$608$	−825262.	−0.0905384
$609$	1.00826e7	1.10162
$610$	0	0
$611$	2.66171e6	0.288441
$612$	−1.39437e7	−1.50487
$613$	1.61323e7	1.73398	0.866991	−	0.498323i	$-0.166051\pi$
$613$	1.61323e7	1.73398	0.866991	+	0.498323i	$0.166051\pi$
$614$	1.06704e7	1.14225
$615$	0	0
$616$	8.37299e6	0.889055
$617$	6.39133e6	0.675894	0.337947	−	0.941165i	$-0.390268\pi$
$617$	6.39133e6	0.675894	0.337947	+	0.941165i	$0.390268\pi$
$618$	5.31505e6	0.559804
$619$	1.27062e7	1.33287	0.666435	−	0.745563i	$-0.267819\pi$
$619$	1.27062e7	1.33287	0.666435	+	0.745563i	$0.267819\pi$
$620$	0	0
$621$	−3.03771e6	−0.316095
$622$	−6.49278e6	−0.672907
$623$	1.18172e7	1.21981
$624$	−1.20665e6	−0.124057
$625$	0	0
$626$	−2.36846e6	−0.241563
$627$	−1.22312e7	−1.24251
$628$	5.37414e6	0.543764
$629$	−2.13587e7	−2.15252
$630$	0	0
$631$	−694354.	−0.0694237	−0.0347118	−	0.999397i	$-0.511051\pi$
$631$	−694354.	−0.0694237	−0.0347118	+	0.999397i	$0.511051\pi$
$632$	−396239.	−0.0394607
$633$	−5.77447e6	−0.572799
$634$	1.04414e7	1.03166
$635$	0	0
$636$	−1.80193e7	−1.76643
$637$	−6.92857e6	−0.676542
$638$	3.27278e6	0.318321
$639$	−5.62676e6	−0.545138
$640$	0	0
$641$	−8.71863e6	−0.838114	−0.419057	−	0.907960i	$-0.637639\pi$
$641$	−8.71863e6	−0.838114	−0.419057	+	0.907960i	$0.637639\pi$
$642$	2.08093e7	1.99260
$643$	−5.98737e6	−0.571096	−0.285548	−	0.958364i	$-0.592176\pi$
$643$	−5.98737e6	−0.571096	−0.285548	+	0.958364i	$0.592176\pi$
$644$	−1.43544e6	−0.136387
$645$	0	0
$646$	5.25234e6	0.495189
$647$	1.09581e7	1.02914	0.514571	−	0.857448i	$-0.327951\pi$
$647$	1.09581e7	1.02914	0.514571	+	0.857448i	$0.327951\pi$
$648$	6.21259e6	0.581212
$649$	−2.58810e7	−2.41195
$650$	0	0
$651$	−1.47528e7	−1.36434
$652$	−3.23009e6	−0.297575
$653$	−6.47503e6	−0.594236	−0.297118	−	0.954841i	$-0.596025\pi$
$653$	−6.47503e6	−0.594236	−0.297118	+	0.954841i	$0.596025\pi$
$654$	1.60728e6	0.146942
$655$	0	0
$656$	−4.37758e6	−0.397169
$657$	−846247.	−0.0764863
$658$	−1.51466e7	−1.36380
$659$	501723.	0.0450039	0.0225020	−	0.999747i	$-0.492837\pi$
$659$	501723.	0.0450039	0.0225020	+	0.999747i	$0.492837\pi$
$660$	0	0
$661$	4.00659e6	0.356674	0.178337	−	0.983969i	$-0.442928\pi$
$661$	4.00659e6	0.356674	0.178337	+	0.983969i	$0.442928\pi$
$662$	3.27514e6	0.290459
$663$	7.67969e6	0.678516
$664$	−2.42235e6	−0.213215
$665$	0	0
$666$	2.80472e7	2.45021
$667$	−561077.	−0.0488324
$668$	7.13681e6	0.618818
$669$	2.92540e7	2.52708
$670$	0	0
$671$	−1.64769e7	−1.41276
$672$	6.86652e6	0.586562
$673$	−8.17473e6	−0.695722	−0.347861	−	0.937546i	$-0.613092\pi$
$673$	−8.17473e6	−0.695722	−0.347861	+	0.937546i	$0.613092\pi$
$674$	−1.31564e7	−1.11554
$675$	0	0
$676$	456976.	0.0384615
$677$	−1.00935e6	−0.0846392	−0.0423196	−	0.999104i	$-0.513475\pi$
$677$	−1.00935e6	−0.0846392	−0.0423196	+	0.999104i	$0.513475\pi$
$678$	5.66757e6	0.473503
$679$	3.76368e6	0.313284
$680$	0	0
$681$	2.80882e6	0.232090
$682$	−4.78870e6	−0.394237
$683$	−1.99189e7	−1.63386	−0.816928	−	0.576739i	$-0.804325\pi$
$683$	−1.99189e7	−1.63386	−0.816928	+	0.576739i	$0.804325\pi$
$684$	−6.89712e6	−0.563674
$685$	0	0
$686$	2.32640e7	1.88745
$687$	1.35875e7	1.09837
$688$	2.28751e6	0.184243
$689$	6.82417e6	0.547648
$690$	0	0
$691$	−677407.	−0.0539703	−0.0269851	−	0.999636i	$-0.508591\pi$
$691$	−677407.	−0.0539703	−0.0269851	+	0.999636i	$0.508591\pi$
$692$	−918541.	−0.0729177
$693$	6.99772e7	5.53508
$694$	−556432.	−0.0438544
$695$	0	0
$696$	2.68394e6	0.210015
$697$	2.78609e7	2.17227
$698$	5.67126e6	0.440596
$699$	2.32284e7	1.79815
$700$	0	0
$701$	1.04542e7	0.803518	0.401759	−	0.915745i	$-0.368399\pi$
$701$	1.04542e7	0.803518	0.401759	+	0.915745i	$0.368399\pi$
$702$	−5.50310e6	−0.421468
$703$	−1.05649e7	−0.806263
$704$	2.22884e6	0.169492
$705$	0	0
$706$	−9.82700e6	−0.742009
$707$	−3.32046e7	−2.49833
$708$	−2.12245e7	−1.59131
$709$	−2.25125e7	−1.68193	−0.840964	−	0.541091i	$-0.818011\pi$
$709$	−2.25125e7	−1.68193	−0.840964	+	0.541091i	$0.818011\pi$
$710$	0	0
$711$	−3.31157e6	−0.245674
$712$	3.14566e6	0.232548
$713$	820964.	0.0604784
$714$	−4.37017e7	−3.20813
$715$	0	0
$716$	1.39903e6	0.101987
$717$	3.76297e7	2.73358
$718$	−1.15440e7	−0.835693
$719$	−2.77678e6	−0.200317	−0.100159	−	0.994971i	$-0.531935\pi$
$719$	−2.77678e6	−0.200317	−0.100159	+	0.994971i	$0.531935\pi$
$720$	0	0
$721$	1.14544e7	0.820603
$722$	−7.30637e6	−0.521625
$723$	−4.63772e7	−3.29958
$724$	341466.	0.0242103
$725$	0	0
$726$	1.50664e7	1.06089
$727$	−2.76872e6	−0.194287	−0.0971434	−	0.995270i	$-0.530971\pi$
$727$	−2.76872e6	−0.194287	−0.0971434	+	0.995270i	$0.530971\pi$
$728$	−2.60044e6	−0.181852
$729$	−3.25086e6	−0.226558
$730$	0	0
$731$	−1.45587e7	−1.00770
$732$	−1.35124e7	−0.932081
$733$	1.80998e7	1.24427	0.622133	−	0.782911i	$-0.286266\pi$
$733$	1.80998e7	1.24427	0.622133	+	0.782911i	$0.286266\pi$
$734$	1.13435e7	0.777152
$735$	0	0
$736$	−382108.	−0.0260011
$737$	−2.10964e7	−1.43067
$738$	−3.65857e7	−2.47269
$739$	−3.57016e6	−0.240479	−0.120239	−	0.992745i	$-0.538366\pi$
$739$	−3.57016e6	−0.240479	−0.120239	+	0.992745i	$0.538366\pi$
$740$	0	0
$741$	3.79870e6	0.254149
$742$	−3.88332e7	−2.58937
$743$	1.73285e7	1.15157	0.575783	−	0.817603i	$-0.304697\pi$
$743$	1.73285e7	1.15157	0.575783	+	0.817603i	$0.304697\pi$
$744$	−3.92712e6	−0.260101
$745$	0	0
$746$	−2.01076e7	−1.32286
$747$	−2.02448e7	−1.32743
$748$	−1.41854e7	−0.927015
$749$	4.48457e7	2.92090
$750$	0	0
$751$	3.04634e6	0.197096	0.0985480	−	0.995132i	$-0.468580\pi$
$751$	3.04634e6	0.197096	0.0985480	+	0.995132i	$0.468580\pi$
$752$	−4.03193e6	−0.259997
$753$	−8.12374e6	−0.522118
$754$	−1.01644e6	−0.0651111
$755$	0	0
$756$	3.13156e7	1.99277
$757$	8.98398e6	0.569808	0.284904	−	0.958556i	$-0.408038\pi$
$757$	8.98398e6	0.569808	0.284904	+	0.958556i	$0.408038\pi$
$758$	8.59568e6	0.543384
$759$	−5.66320e6	−0.356827
$760$	0	0
$761$	1.31757e7	0.824731	0.412366	−	0.911018i	$-0.364703\pi$
$761$	1.31757e7	0.824731	0.412366	+	0.911018i	$0.364703\pi$
$762$	2.03352e7	1.26871
$763$	3.46382e6	0.215399
$764$	4.92027e6	0.304969
$765$	0	0
$766$	−3.33188e6	−0.205172
$767$	8.03799e6	0.493354
$768$	1.82783e6	0.111824
$769$	3.53661e6	0.215661	0.107830	−	0.994169i	$-0.465610\pi$
$769$	3.53661e6	0.215661	0.107830	+	0.994169i	$0.465610\pi$
$770$	0	0
$771$	1.52897e7	0.926322
$772$	−2.91855e6	−0.176248
$773$	3.14397e7	1.89247	0.946235	−	0.323479i	$-0.104853\pi$
$773$	3.14397e7	1.89247	0.946235	+	0.323479i	$0.104853\pi$
$774$	1.91179e7	1.14706
$775$	0	0
$776$	1.00187e6	0.0597251
$777$	8.79044e7	5.22346
$778$	−2.90097e6	−0.171828
$779$	1.37812e7	0.813659
$780$	0	0
$781$	−5.72429e6	−0.335810
$782$	2.43191e6	0.142210
$783$	1.22404e7	0.713498
$784$	1.04953e7	0.609827
$785$	0	0
$786$	2.22805e7	1.28638
$787$	−1.27213e7	−0.732139	−0.366069	−	0.930588i	$-0.619297\pi$
$787$	−1.27213e7	−0.732139	−0.366069	+	0.930588i	$0.619297\pi$
$788$	2.44013e6	0.139990
$789$	−2.28259e7	−1.30538
$790$	0	0
$791$	1.22141e7	0.694097
$792$	1.86276e7	1.05522
$793$	5.11731e6	0.288974
$794$	−9.70617e6	−0.546382
$795$	0	0
$796$	−9.38569e6	−0.525029
$797$	−1.84237e7	−1.02738	−0.513689	−	0.857977i	$-0.671721\pi$
$797$	−1.84237e7	−1.02738	−0.513689	+	0.857977i	$0.671721\pi$
$798$	−2.16167e7	−1.20166
$799$	2.56610e7	1.42203
$800$	0	0
$801$	2.62899e7	1.44780
$802$	−8.57506e6	−0.470762
$803$	−860915.	−0.0471164
$804$	−1.73008e7	−0.943899
$805$	0	0
$806$	1.48725e6	0.0806394
$807$	−2.62019e7	−1.41628
$808$	−8.83888e6	−0.476287
$809$	2.10360e7	1.13004	0.565018	−	0.825079i	$-0.308869\pi$
$809$	2.10360e7	1.13004	0.565018	+	0.825079i	$0.308869\pi$
$810$	0	0
$811$	3.53571e7	1.88767	0.943833	−	0.330424i	$-0.107192\pi$
$811$	3.53571e7	1.88767	0.943833	+	0.330424i	$0.107192\pi$
$812$	5.78412e6	0.307856
$813$	−4.25097e7	−2.25560
$814$	2.85334e7	1.50936
$815$	0	0
$816$	−1.16331e7	−0.611606
$817$	−7.20136e6	−0.377450
$818$	−1.46458e7	−0.765295
$819$	−2.17332e7	−1.13218
$820$	0	0
$821$	1.72038e7	0.890770	0.445385	−	0.895339i	$-0.353067\pi$
$821$	1.72038e7	0.890770	0.445385	+	0.895339i	$0.353067\pi$
$822$	3.85355e7	1.98921
$823$	947098.	0.0487411	0.0243705	−	0.999703i	$-0.492242\pi$
$823$	947098.	0.0487411	0.0243705	+	0.999703i	$0.492242\pi$
$824$	3.04909e6	0.156442
$825$	0	0
$826$	−4.57405e7	−2.33266
$827$	3.05385e7	1.55269	0.776345	−	0.630308i	$-0.217072\pi$
$827$	3.05385e7	1.55269	0.776345	+	0.630308i	$0.217072\pi$
$828$	−3.19346e6	−0.161877
$829$	−1.57415e7	−0.795538	−0.397769	−	0.917486i	$-0.630215\pi$
$829$	−1.57415e7	−0.795538	−0.397769	+	0.917486i	$0.630215\pi$
$830$	0	0
$831$	1.22636e7	0.616047
$832$	−692224.	−0.0346688
$833$	−6.67971e7	−3.33538
$834$	2.04257e7	1.01686
$835$	0	0
$836$	−7.01668e6	−0.347229
$837$	−1.79101e7	−0.883659
$838$	1.21514e7	0.597746
$839$	2.07700e7	1.01867	0.509333	−	0.860570i	$-0.329892\pi$
$839$	2.07700e7	1.01867	0.509333	+	0.860570i	$0.329892\pi$
$840$	0	0
$841$	−1.82503e7	−0.889774
$842$	−7.54651e6	−0.366831
$843$	−1.65121e7	−0.800263
$844$	−3.31265e6	−0.160073
$845$	0	0
$846$	−3.36969e7	−1.61869
$847$	3.24694e7	1.55513
$848$	−1.03372e7	−0.493643
$849$	2.16772e7	1.03213
$850$	0	0
$851$	−4.89169e6	−0.231545
$852$	−4.69438e6	−0.221554
$853$	1.91838e7	0.902740	0.451370	−	0.892337i	$-0.350936\pi$
$853$	1.91838e7	0.902740	0.451370	+	0.892337i	$0.350936\pi$
$854$	−2.91203e7	−1.36632
$855$	0	0
$856$	1.19377e7	0.556847
$857$	−2.64211e7	−1.22885	−0.614425	−	0.788975i	$-0.710612\pi$
$857$	−2.64211e7	−1.22885	−0.614425	+	0.788975i	$0.710612\pi$
$858$	−1.02594e7	−0.475778
$859$	−3.30576e7	−1.52858	−0.764289	−	0.644873i	$-0.776910\pi$
$859$	−3.30576e7	−1.52858	−0.764289	+	0.644873i	$0.776910\pi$
$860$	0	0
$861$	−1.14665e8	−5.27137
$862$	2.86877e6	0.131501
$863$	−1.19324e7	−0.545380	−0.272690	−	0.962102i	$-0.587913\pi$
$863$	−1.19324e7	−0.545380	−0.272690	+	0.962102i	$0.587913\pi$
$864$	8.33605e6	0.379906
$865$	0	0
$866$	−1.05828e7	−0.479518
$867$	3.44381e7	1.55593
$868$	−8.46328e6	−0.381276
$869$	−3.36897e6	−0.151338
$870$	0	0
$871$	6.55203e6	0.292638
$872$	922051.	0.0410643
$873$	8.37313e6	0.371837
$874$	1.20292e6	0.0532671
$875$	0	0
$876$	−706020.	−0.0310854
$877$	−3.88434e6	−0.170537	−0.0852684	−	0.996358i	$-0.527175\pi$
$877$	−3.88434e6	−0.170537	−0.0852684	+	0.996358i	$0.527175\pi$
$878$	1.79189e7	0.784469
$879$	−1.69860e7	−0.741513
$880$	0	0
$881$	−8.93397e6	−0.387797	−0.193899	−	0.981022i	$-0.562113\pi$
$881$	−8.93397e6	−0.387797	−0.193899	+	0.981022i	$0.562113\pi$
$882$	8.77148e7	3.79666
$883$	4.32552e6	0.186697	0.0933484	−	0.995634i	$-0.470243\pi$
$883$	4.32552e6	0.186697	0.0933484	+	0.995634i	$0.470243\pi$
$884$	4.40563e6	0.189617
$885$	0	0
$886$	−1.72292e7	−0.737364
$887$	−3.55552e7	−1.51738	−0.758689	−	0.651453i	$-0.774160\pi$
$887$	−3.55552e7	−1.51738	−0.758689	+	0.651453i	$0.774160\pi$
$888$	2.33997e7	0.995812
$889$	4.38242e7	1.85977
$890$	0	0
$891$	5.28217e7	2.22904
$892$	1.67822e7	0.706214
$893$	1.26930e7	0.532643
$894$	−2.08712e7	−0.873380
$895$	0	0
$896$	3.93913e6	0.163919
$897$	1.75885e6	0.0729874
$898$	−101092.	−0.00418335
$899$	−3.30807e6	−0.136514
$900$	0	0
$901$	6.57906e7	2.69993
$902$	−3.72198e7	−1.52320
$903$	5.99184e7	2.44535
$904$	3.25133e6	0.132324
$905$	0	0
$906$	−1.18438e6	−0.0479370
$907$	−2.44157e7	−0.985486	−0.492743	−	0.870175i	$-0.664006\pi$
$907$	−2.44157e7	−0.985486	−0.492743	+	0.870175i	$0.664006\pi$
$908$	1.61134e6	0.0648593
$909$	−7.38709e7	−2.96527
$910$	0	0
$911$	3.00267e7	1.19870	0.599352	−	0.800486i	$-0.295425\pi$
$911$	3.00267e7	1.19870	0.599352	+	0.800486i	$0.295425\pi$
$912$	−5.75424e6	−0.229087
$913$	−2.05957e7	−0.817711
$914$	−4.57439e6	−0.181120
$915$	0	0
$916$	7.79478e6	0.306948
$917$	4.80164e7	1.88567
$918$	−5.30544e7	−2.07785
$919$	−1.30118e7	−0.508216	−0.254108	−	0.967176i	$-0.581782\pi$
$919$	−1.30118e7	−0.508216	−0.254108	+	0.967176i	$0.581782\pi$
$920$	0	0
$921$	7.44009e7	2.89021
$922$	−3.33964e7	−1.29382
$923$	1.77782e6	0.0686886
$924$	5.83817e7	2.24956
$925$	0	0
$926$	1.36639e7	0.523656
$927$	2.54828e7	0.973975
$928$	1.53970e6	0.0586904
$929$	−5.16678e7	−1.96418	−0.982089	−	0.188419i	$-0.939664\pi$
$929$	−5.16678e7	−1.96418	−0.982089	+	0.188419i	$0.939664\pi$
$930$	0	0
$931$	−3.30407e7	−1.24932
$932$	1.33255e7	0.502509
$933$	−4.52717e7	−1.70264
$934$	−2.26842e7	−0.850857
$935$	0	0
$936$	−5.78526e6	−0.215841
$937$	4.04703e6	0.150587	0.0752936	−	0.997161i	$-0.476011\pi$
$937$	4.04703e6	0.150587	0.0752936	+	0.997161i	$0.476011\pi$
$938$	−3.72846e7	−1.38364
$939$	−1.65144e7	−0.611221
$940$	0	0
$941$	−4.43008e7	−1.63094	−0.815470	−	0.578799i	$-0.803521\pi$
$941$	−4.43008e7	−1.63094	−0.815470	+	0.578799i	$0.803521\pi$
$942$	3.74719e7	1.37587
$943$	6.38088e6	0.233669
$944$	−1.21759e7	−0.444704
$945$	0	0
$946$	1.94492e7	0.706602
$947$	3.60463e7	1.30613	0.653064	−	0.757303i	$-0.273483\pi$
$947$	3.60463e7	1.30613	0.653064	+	0.757303i	$0.273483\pi$
$948$	−2.76283e6	−0.0998464
$949$	267379.	0.00963745
$950$	0	0
$951$	7.28042e7	2.61039
$952$	−2.50704e7	−0.896539
$953$	1.31248e7	0.468123	0.234061	−	0.972222i	$-0.424798\pi$
$953$	1.31248e7	0.468123	0.234061	+	0.972222i	$0.424798\pi$
$954$	−8.63931e7	−3.07332
$955$	0	0
$956$	2.15871e7	0.763923
$957$	2.28198e7	0.805439
$958$	−2.46536e7	−0.867894
$959$	8.30473e7	2.91594
$960$	0	0
$961$	−2.37888e7	−0.830929
$962$	−8.86176e6	−0.308733
$963$	9.97693e7	3.46682
$964$	−2.66053e7	−0.922096
$965$	0	0
$966$	−1.00088e7	−0.345096
$967$	−2.48150e7	−0.853390	−0.426695	−	0.904396i	$-0.640322\pi$
$967$	−2.48150e7	−0.853390	−0.426695	+	0.904396i	$0.640322\pi$
$968$	8.64318e6	0.296473
$969$	3.66226e7	1.25297
$970$	0	0
$971$	−3.54227e6	−0.120569	−0.0602843	−	0.998181i	$-0.519201\pi$
$971$	−3.54227e6	−0.120569	−0.0602843	+	0.998181i	$0.519201\pi$
$972$	1.16671e7	0.396093
$973$	4.40191e7	1.49059
$974$	2.93080e7	0.989896
$975$	0	0
$976$	−7.75167e6	−0.260478
$977$	−9.16916e6	−0.307322	−0.153661	−	0.988124i	$-0.549106\pi$
$977$	−9.16916e6	−0.307322	−0.153661	+	0.988124i	$0.549106\pi$
$978$	−2.25222e7	−0.752947
$979$	2.67456e7	0.891857
$980$	0	0
$981$	7.70604e6	0.255658
$982$	3.55895e7	1.17772
$983$	3.05912e7	1.00975	0.504874	−	0.863193i	$-0.331539\pi$
$983$	3.05912e7	1.00975	0.504874	+	0.863193i	$0.331539\pi$
$984$	−3.05233e7	−1.00495
$985$	0	0
$986$	−9.79936e6	−0.321000
$987$	−1.05611e8	−3.45078
$988$	2.17921e6	0.0710242
$989$	−3.33433e6	−0.108397
$990$	0	0
$991$	3.47852e7	1.12515	0.562575	−	0.826746i	$-0.309811\pi$
$991$	3.47852e7	1.12515	0.562575	+	0.826746i	$0.309811\pi$
$992$	−2.25288e6	−0.0726874
$993$	2.28363e7	0.734942
$994$	−1.01168e7	−0.324771
$995$	0	0
$996$	−1.68902e7	−0.539492
$997$	4.21812e7	1.34394	0.671972	−	0.740577i	$-0.265448\pi$
$997$	4.21812e7	1.34394	0.671972	+	0.740577i	$0.265448\pi$
$998$	1.20895e7	0.384222
$999$	1.06717e8	3.38314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.6.a.j.1.3		3
5.2	odd	4		650.6.b.j.599.4		6
5.3	odd	4		650.6.b.j.599.3		6
5.4	even	2		130.6.a.f.1.1	✓	3
20.19	odd	2		1040.6.a.l.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.6.a.f.1.1	✓	3	5.4	even	2
650.6.a.j.1.3		3	1.1	even	1	trivial
650.6.b.j.599.3		6	5.3	odd	4
650.6.b.j.599.4		6	5.2	odd	4
1040.6.a.l.1.3		3	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	650.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +27.8905 q^{3} +16.0000 q^{4} +111.562 q^{6} +240.426 q^{7} +64.0000 q^{8} +534.880 q^{9} +O(q^{10})\)	\(q+4.00000 q^{2} +27.8905 q^{3} +16.0000 q^{4} +111.562 q^{6} +240.426 q^{7} +64.0000 q^{8} +534.880 q^{9} +544.151 q^{11} +446.248 q^{12} -169.000 q^{13} +961.702 q^{14} +256.000 q^{16} -1629.30 q^{17} +2139.52 q^{18} -805.920 q^{19} +6705.59 q^{21} +2176.61 q^{22} -373.152 q^{23} +1784.99 q^{24} -676.000 q^{26} +8140.67 q^{27} +3846.81 q^{28} +1503.62 q^{29} -2200.08 q^{31} +1024.00 q^{32} +15176.7 q^{33} -6517.20 q^{34} +8558.08 q^{36} +13109.1 q^{37} -3223.68 q^{38} -4713.49 q^{39} -17099.9 q^{41} +26822.4 q^{42} +8935.58 q^{43} +8706.42 q^{44} -1492.61 q^{46} -15749.7 q^{47} +7139.97 q^{48} +40997.5 q^{49} -45442.0 q^{51} -2704.00 q^{52} -40379.7 q^{53} +32562.7 q^{54} +15387.2 q^{56} -22477.5 q^{57} +6014.46 q^{58} -47562.1 q^{59} -30280.0 q^{61} -8800.31 q^{62} +128599. q^{63} +4096.00 q^{64} +60706.6 q^{66} -38769.4 q^{67} -26068.8 q^{68} -10407.4 q^{69} -10519.7 q^{71} +34232.3 q^{72} -1582.12 q^{73} +52436.5 q^{74} -12894.7 q^{76} +130828. q^{77} -18854.0 q^{78} -6191.23 q^{79} +97071.7 q^{81} -68399.8 q^{82} -37849.2 q^{83} +107289. q^{84} +35742.3 q^{86} +41936.6 q^{87} +34825.7 q^{88} +49151.0 q^{89} -40631.9 q^{91} -5970.43 q^{92} -61361.3 q^{93} -62999.0 q^{94} +28559.9 q^{96} +15654.2 q^{97} +163990. q^{98} +291056. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 12 q^{2} + 22 q^{3} + 48 q^{4} + 88 q^{6} + 234 q^{7} + 192 q^{8} + 155 q^{9}+O(q^{10}) \) 3 * q + 12 * q^2 + 22 * q^3 + 48 * q^4 + 88 * q^6 + 234 * q^7 + 192 * q^8 + 155 * q^9	\( 3 q + 12 q^{2} + 22 q^{3} + 48 q^{4} + 88 q^{6} + 234 q^{7} + 192 q^{8} + 155 q^{9} + 48 q^{11} + 352 q^{12} - 507 q^{13} + 936 q^{14} + 768 q^{16} - 1506 q^{17} + 620 q^{18} - 360 q^{19} + 6904 q^{21} + 192 q^{22} + 2370 q^{23} + 1408 q^{24} - 2028 q^{26} + 10168 q^{27} + 3744 q^{28} - 3078 q^{29} - 5388 q^{31} + 3072 q^{32} + 22572 q^{33} - 6024 q^{34} + 2480 q^{36} + 25362 q^{37} - 1440 q^{38} - 3718 q^{39} - 15906 q^{41} + 27616 q^{42} + 39306 q^{43} + 768 q^{44} + 9480 q^{46} + 17778 q^{47} + 5632 q^{48} + 7767 q^{49} - 33972 q^{51} - 8112 q^{52} - 9246 q^{53} + 40672 q^{54} + 14976 q^{56} + 2996 q^{57} - 12312 q^{58} - 77760 q^{59} + 17982 q^{61} - 21552 q^{62} + 128762 q^{63} + 12288 q^{64} + 90288 q^{66} + 2922 q^{67} - 24096 q^{68} - 31860 q^{69} - 4944 q^{71} + 9920 q^{72} - 43278 q^{73} + 101448 q^{74} - 5760 q^{76} + 144420 q^{77} - 14872 q^{78} + 42120 q^{79} + 146519 q^{81} - 63624 q^{82} - 58098 q^{83} + 110464 q^{84} + 157224 q^{86} - 2016 q^{87} + 3072 q^{88} + 19614 q^{89} - 39546 q^{91} + 37920 q^{92} - 162796 q^{93} + 71112 q^{94} + 22528 q^{96} + 87078 q^{97} + 31068 q^{98} + 350340 q^{99}+O(q^{100}) \) 3 * q + 12 * q^2 + 22 * q^3 + 48 * q^4 + 88 * q^6 + 234 * q^7 + 192 * q^8 + 155 * q^9 + 48 * q^11 + 352 * q^12 - 507 * q^13 + 936 * q^14 + 768 * q^16 - 1506 * q^17 + 620 * q^18 - 360 * q^19 + 6904 * q^21 + 192 * q^22 + 2370 * q^23 + 1408 * q^24 - 2028 * q^26 + 10168 * q^27 + 3744 * q^28 - 3078 * q^29 - 5388 * q^31 + 3072 * q^32 + 22572 * q^33 - 6024 * q^34 + 2480 * q^36 + 25362 * q^37 - 1440 * q^38 - 3718 * q^39 - 15906 * q^41 + 27616 * q^42 + 39306 * q^43 + 768 * q^44 + 9480 * q^46 + 17778 * q^47 + 5632 * q^48 + 7767 * q^49 - 33972 * q^51 - 8112 * q^52 - 9246 * q^53 + 40672 * q^54 + 14976 * q^56 + 2996 * q^57 - 12312 * q^58 - 77760 * q^59 + 17982 * q^61 - 21552 * q^62 + 128762 * q^63 + 12288 * q^64 + 90288 * q^66 + 2922 * q^67 - 24096 * q^68 - 31860 * q^69 - 4944 * q^71 + 9920 * q^72 - 43278 * q^73 + 101448 * q^74 - 5760 * q^76 + 144420 * q^77 - 14872 * q^78 + 42120 * q^79 + 146519 * q^81 - 63624 * q^82 - 58098 * q^83 + 110464 * q^84 + 157224 * q^86 - 2016 * q^87 + 3072 * q^88 + 19614 * q^89 - 39546 * q^91 + 37920 * q^92 - 162796 * q^93 + 71112 * q^94 + 22528 * q^96 + 87078 * q^97 + 31068 * q^98 + 350340 * q^99

Introduction

L-functions

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