LMFDB - Embedded newform 325.6.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, 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("325.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4 x^{8} - 181 x^{7} + 688 x^{6} + 10455 x^{5} - 37904 x^{4} - 197375 x^{3} + 702868 x^{2} + \cdots - 366960 $ x^9 - 4x^8 - 181x^7 + 688x^6 + 10455x^5 - 37904x^4 - 197375x^3 + 702868x^2 + 262812x - 366960
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-9.17271$ of defining polynomial
Character	$\chi$	$=$	325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.1727 q^{2} +23.0461 q^{3} +71.4841 q^{4} -234.441 q^{6} -51.0548 q^{7} -401.660 q^{8} +288.123 q^{9} +O(q^{10})$	\(q-10.1727 q^{2} +23.0461 q^{3} +71.4841 q^{4} -234.441 q^{6} -51.0548 q^{7} -401.660 q^{8} +288.123 q^{9} -512.445 q^{11} +1647.43 q^{12} +169.000 q^{13} +519.366 q^{14} +1798.48 q^{16} +978.964 q^{17} -2930.99 q^{18} +1353.56 q^{19} -1176.61 q^{21} +5212.95 q^{22} +11.3026 q^{23} -9256.71 q^{24} -1719.19 q^{26} +1039.91 q^{27} -3649.61 q^{28} -8050.51 q^{29} +759.478 q^{31} -5442.33 q^{32} -11809.9 q^{33} -9958.72 q^{34} +20596.2 q^{36} -12780.7 q^{37} -13769.3 q^{38} +3894.79 q^{39} -18129.3 q^{41} +11969.4 q^{42} +19533.0 q^{43} -36631.6 q^{44} -114.978 q^{46} +22049.0 q^{47} +41448.1 q^{48} -14200.4 q^{49} +22561.3 q^{51} +12080.8 q^{52} +13861.1 q^{53} -10578.7 q^{54} +20506.7 q^{56} +31194.2 q^{57} +81895.5 q^{58} -18164.6 q^{59} +7623.02 q^{61} -7725.95 q^{62} -14710.1 q^{63} -2188.19 q^{64} +120138. q^{66} +3764.44 q^{67} +69980.4 q^{68} +260.481 q^{69} -75797.0 q^{71} -115728. q^{72} -86605.1 q^{73} +130015. q^{74} +96757.7 q^{76} +26162.8 q^{77} -39620.6 q^{78} +61535.6 q^{79} -46048.0 q^{81} +184425. q^{82} -46849.0 q^{83} -84109.2 q^{84} -198704. q^{86} -185533. q^{87} +205829. q^{88} +1237.02 q^{89} -8628.26 q^{91} +807.955 q^{92} +17503.0 q^{93} -224298. q^{94} -125425. q^{96} +57565.4 q^{97} +144457. q^{98} -147647. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} - 11 q^{3} + 91 q^{4} - 83 q^{6} + 12 q^{7} - 639 q^{8} + 562 q^{9}+O(q^{10}) $ 9 * q - 5 * q^2 - 11 * q^3 + 91 * q^4 - 83 * q^6 + 12 * q^7 - 639 * q^8 + 562 * q^9	\( 9 q - 5 q^{2} - 11 q^{3} + 91 q^{4} - 83 q^{6} + 12 q^{7} - 639 q^{8} + 562 q^{9} - 1422 q^{11} + 1567 q^{12} + 1521 q^{13} - 342 q^{14} - 1061 q^{16} + 648 q^{17} + 418 q^{18} - 408 q^{19} - 3912 q^{21} + 4345 q^{22} + 1839 q^{23} - 8469 q^{24} - 845 q^{26} - 7649 q^{27} - 2836 q^{28} - 8737 q^{29} + 748 q^{31} - 423 q^{32} - 356 q^{33} - 17789 q^{34} + 512 q^{36} - 15486 q^{37} - 3425 q^{38} - 1859 q^{39} - 28676 q^{41} + 6876 q^{42} + 28665 q^{43} - 30599 q^{44} - 12056 q^{46} - 29452 q^{47} + 64759 q^{48} - 40907 q^{49} - 31006 q^{51} + 15379 q^{52} + 75977 q^{53} - 102761 q^{54} - 23002 q^{56} - 38038 q^{57} + 142384 q^{58} - 88142 q^{59} + 28165 q^{61} - 137308 q^{62} + 41492 q^{63} - 100845 q^{64} + 42577 q^{66} - 94754 q^{67} + 89267 q^{68} - 181747 q^{69} - 70562 q^{71} - 263778 q^{72} + 60602 q^{73} - 135676 q^{74} + 46373 q^{76} - 140292 q^{77} - 14027 q^{78} - 164073 q^{79} - 69935 q^{81} - 72887 q^{82} - 22458 q^{83} - 345656 q^{84} - 294920 q^{86} - 87031 q^{87} + 430607 q^{88} - 252698 q^{89} + 2028 q^{91} - 237824 q^{92} + 56556 q^{93} - 501606 q^{94} - 319181 q^{96} + 137986 q^{97} + 378699 q^{98} - 757776 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 - 11 * q^3 + 91 * q^4 - 83 * q^6 + 12 * q^7 - 639 * q^8 + 562 * q^9 - 1422 * q^11 + 1567 * q^12 + 1521 * q^13 - 342 * q^14 - 1061 * q^16 + 648 * q^17 + 418 * q^18 - 408 * q^19 - 3912 * q^21 + 4345 * q^22 + 1839 * q^23 - 8469 * q^24 - 845 * q^26 - 7649 * q^27 - 2836 * q^28 - 8737 * q^29 + 748 * q^31 - 423 * q^32 - 356 * q^33 - 17789 * q^34 + 512 * q^36 - 15486 * q^37 - 3425 * q^38 - 1859 * q^39 - 28676 * q^41 + 6876 * q^42 + 28665 * q^43 - 30599 * q^44 - 12056 * q^46 - 29452 * q^47 + 64759 * q^48 - 40907 * q^49 - 31006 * q^51 + 15379 * q^52 + 75977 * q^53 - 102761 * q^54 - 23002 * q^56 - 38038 * q^57 + 142384 * q^58 - 88142 * q^59 + 28165 * q^61 - 137308 * q^62 + 41492 * q^63 - 100845 * q^64 + 42577 * q^66 - 94754 * q^67 + 89267 * q^68 - 181747 * q^69 - 70562 * q^71 - 263778 * q^72 + 60602 * q^73 - 135676 * q^74 + 46373 * q^76 - 140292 * q^77 - 14027 * q^78 - 164073 * q^79 - 69935 * q^81 - 72887 * q^82 - 22458 * q^83 - 345656 * q^84 - 294920 * q^86 - 87031 * q^87 + 430607 * q^88 - 252698 * q^89 + 2028 * q^91 - 237824 * q^92 + 56556 * q^93 - 501606 * q^94 - 319181 * q^96 + 137986 * q^97 + 378699 * q^98 - 757776 * 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 $

$2$	−10.1727	−1.79830	−0.899149	−	0.437642i	$-0.855814\pi$
$2$	−10.1727	−1.79830	−0.899149	+	0.437642i	$0.855814\pi$
$3$	23.0461	1.47841	0.739204	−	0.673481i	$-0.235202\pi$
$3$	23.0461	1.47841	0.739204	+	0.673481i	$0.235202\pi$
$4$	71.4841	2.23388
$5$	0	0
$5$	0	0
$6$	−234.441	−2.65862
$7$	−51.0548	−0.393814	−0.196907	−	0.980422i	$-0.563090\pi$
$7$	−51.0548	−0.393814	−0.196907	+	0.980422i	$0.563090\pi$
$8$	−401.660	−2.21888
$9$	288.123	1.18569
$10$	0	0
$11$	−512.445	−1.27692	−0.638462	−	0.769653i	$-0.720429\pi$
$11$	−512.445	−1.27692	−0.638462	+	0.769653i	$0.720429\pi$
$12$	1647.43	3.30258
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	519.366	0.708196
$15$	0	0
$16$	1798.48	1.75633
$17$	978.964	0.821570	0.410785	−	0.911732i	$-0.365255\pi$
$17$	978.964	0.821570	0.410785	+	0.911732i	$0.365255\pi$
$18$	−2930.99	−2.13223
$19$	1353.56	0.860186	0.430093	−	0.902785i	$-0.358481\pi$
$19$	1353.56	0.860186	0.430093	+	0.902785i	$0.358481\pi$
$20$	0	0
$21$	−1176.61	−0.582218
$22$	5212.95	2.29629
$23$	11.3026	0.00445511	0.00222755	−	0.999998i	$-0.499291\pi$
$23$	11.3026	0.00445511	0.00222755	+	0.999998i	$0.499291\pi$
$24$	−9256.71	−3.28041
$25$	0	0
$26$	−1719.19	−0.498758
$27$	1039.91	0.274528
$28$	−3649.61	−0.879733
$29$	−8050.51	−1.77758	−0.888788	−	0.458318i	$-0.848452\pi$
$29$	−8050.51	−1.77758	−0.888788	+	0.458318i	$0.848452\pi$
$30$	0	0
$31$	759.478	0.141942	0.0709710	−	0.997478i	$-0.477390\pi$
$31$	759.478	0.141942	0.0709710	+	0.997478i	$0.477390\pi$
$32$	−5442.33	−0.939529
$33$	−11809.9	−1.88782
$34$	−9958.72	−1.47743
$35$	0	0
$36$	20596.2	2.64869
$37$	−12780.7	−1.53480	−0.767399	−	0.641170i	$-0.778449\pi$
$37$	−12780.7	−1.53480	−0.767399	+	0.641170i	$0.778449\pi$
$38$	−13769.3	−1.54687
$39$	3894.79	0.410037
$40$	0	0
$41$	−18129.3	−1.68431	−0.842156	−	0.539234i	$-0.818714\pi$
$41$	−18129.3	−1.68431	−0.842156	+	0.539234i	$0.818714\pi$
$42$	11969.4	1.04700
$43$	19533.0	1.61101	0.805505	−	0.592589i	$-0.201894\pi$
$43$	19533.0	1.61101	0.805505	+	0.592589i	$0.201894\pi$
$44$	−36631.6	−2.85249
$45$	0	0
$46$	−114.978	−0.00801161
$47$	22049.0	1.45594	0.727971	−	0.685608i	$-0.240464\pi$
$47$	22049.0	1.45594	0.727971	+	0.685608i	$0.240464\pi$
$48$	41448.1	2.59658
$49$	−14200.4	−0.844910
$50$	0	0
$51$	22561.3	1.21462
$52$	12080.8	0.619566
$53$	13861.1	0.677811	0.338905	−	0.940821i	$-0.389943\pi$
$53$	13861.1	0.677811	0.338905	+	0.940821i	$0.389943\pi$
$54$	−10578.7	−0.493683
$55$	0	0
$56$	20506.7	0.873827
$57$	31194.2	1.27171
$58$	81895.5	3.19661
$59$	−18164.6	−0.679352	−0.339676	−	0.940542i	$-0.610317\pi$
$59$	−18164.6	−0.679352	−0.339676	+	0.940542i	$0.610317\pi$
$60$	0	0
$61$	7623.02	0.262303	0.131151	−	0.991362i	$-0.458133\pi$
$61$	7623.02	0.262303	0.131151	+	0.991362i	$0.458133\pi$
$62$	−7725.95	−0.255254
$63$	−14710.1	−0.466942
$64$	−2188.19	−0.0667783
$65$	0	0
$66$	120138.	3.39486
$67$	3764.44	0.102450	0.0512252	−	0.998687i	$-0.483687\pi$
$67$	3764.44	0.102450	0.0512252	+	0.998687i	$0.483687\pi$
$68$	69980.4	1.83529
$69$	260.481	0.00658647
$70$	0	0
$71$	−75797.0	−1.78446	−0.892229	−	0.451583i	$-0.850859\pi$
$71$	−75797.0	−1.78446	−0.892229	+	0.451583i	$0.850859\pi$
$72$	−115728.	−2.63091
$73$	−86605.1	−1.90211	−0.951057	−	0.309015i	$-0.900001\pi$
$73$	−86605.1	−1.90211	−0.951057	+	0.309015i	$0.900001\pi$
$74$	130015.	2.76003
$75$	0	0
$76$	96757.7	1.92155
$77$	26162.8	0.502871
$78$	−39620.6	−0.737368
$79$	61535.6	1.10932	0.554662	−	0.832076i	$-0.312847\pi$
$79$	61535.6	1.10932	0.554662	+	0.832076i	$0.312847\pi$
$80$	0	0
$81$	−46048.0	−0.779827
$82$	184425.	3.02890
$83$	−46849.0	−0.746458	−0.373229	−	0.927739i	$-0.621749\pi$
$83$	−46849.0	−0.746458	−0.373229	+	0.927739i	$0.621749\pi$
$84$	−84109.2	−1.30060
$85$	0	0
$86$	−198704.	−2.89708
$87$	−185533.	−2.62798
$88$	205829.	2.83334
$89$	1237.02	0.0165540	0.00827698	−	0.999966i	$-0.497365\pi$
$89$	1237.02	0.0165540	0.00827698	+	0.999966i	$0.497365\pi$
$90$	0	0
$91$	−8628.26	−0.109224
$92$	807.955	0.00995217
$93$	17503.0	0.209848
$94$	−224298.	−2.61822
$95$	0	0
$96$	−125425.	−1.38901
$97$	57565.4	0.621201	0.310600	−	0.950541i	$-0.399470\pi$
$97$	57565.4	0.621201	0.310600	+	0.950541i	$0.399470\pi$
$98$	144457.	1.51940
$99$	−147647.	−1.51404
$100$	0	0
$101$	−13535.7	−0.132031	−0.0660155	−	0.997819i	$-0.521029\pi$
$101$	−13535.7	−0.132031	−0.0660155	+	0.997819i	$0.521029\pi$
$102$	−229510.	−2.18424
$103$	−99640.9	−0.925432	−0.462716	−	0.886507i	$-0.653125\pi$
$103$	−99640.9	−0.925432	−0.462716	+	0.886507i	$0.653125\pi$
$104$	−67880.6	−0.615407
$105$	0	0
$106$	−141005.	−1.21891
$107$	−3447.88	−0.0291134	−0.0145567	−	0.999894i	$-0.504634\pi$
$107$	−3447.88	−0.0291134	−0.0145567	+	0.999894i	$0.504634\pi$
$108$	74337.0	0.613261
$109$	54235.5	0.437238	0.218619	−	0.975810i	$-0.429845\pi$
$109$	54235.5	0.437238	0.218619	+	0.975810i	$0.429845\pi$
$110$	0	0
$111$	−294546.	−2.26906
$112$	−91821.3	−0.691669
$113$	−103423.	−0.761937	−0.380969	−	0.924588i	$-0.624409\pi$
$113$	−103423.	−0.761937	−0.380969	+	0.924588i	$0.624409\pi$
$114$	−317330.	−2.28691
$115$	0	0
$116$	−575483.	−3.97089
$117$	48692.8	0.328852
$118$	184783.	1.22168
$119$	−49980.8	−0.323546
$120$	0	0
$121$	101549.	0.630537
$122$	−77546.8	−0.471698
$123$	−417811.	−2.49010
$124$	54290.6	0.317081
$125$	0	0
$126$	149641.	0.839702
$127$	−86597.6	−0.476427	−0.238214	−	0.971213i	$-0.576562\pi$
$127$	−86597.6	−0.476427	−0.238214	+	0.971213i	$0.576562\pi$
$128$	196415.	1.05962
$129$	450160.	2.38173
$130$	0	0
$131$	212824.	1.08354	0.541768	−	0.840528i	$-0.317755\pi$
$131$	212824.	1.08354	0.541768	+	0.840528i	$0.317755\pi$
$132$	−844217.	−4.21715
$133$	−69105.6	−0.338754
$134$	−38294.6	−0.184236
$135$	0	0
$136$	−393211.	−1.82297
$137$	−392531.	−1.78678	−0.893392	−	0.449277i	$-0.851681\pi$
$137$	−392531.	−1.78678	−0.893392	+	0.449277i	$0.851681\pi$
$138$	−2649.79	−0.0118444
$139$	64771.0	0.284344	0.142172	−	0.989842i	$-0.454591\pi$
$139$	64771.0	0.284344	0.142172	+	0.989842i	$0.454591\pi$
$140$	0	0
$141$	508143.	2.15248
$142$	771061.	3.20899
$143$	−86603.2	−0.354155
$144$	518185.	2.08247
$145$	0	0
$146$	881009.	3.42057
$147$	−327264.	−1.24912
$148$	−913619.	−3.42855
$149$	−277263.	−1.02312	−0.511560	−	0.859248i	$-0.670932\pi$
$149$	−277263.	−1.02312	−0.511560	+	0.859248i	$0.670932\pi$
$150$	0	0
$151$	−56639.1	−0.202150	−0.101075	−	0.994879i	$-0.532228\pi$
$151$	−56639.1	−0.202150	−0.101075	+	0.994879i	$0.532228\pi$
$152$	−543670.	−1.90865
$153$	282062.	0.974128
$154$	−266146.	−0.904313
$155$	0	0
$156$	278416.	0.915972
$157$	−229864.	−0.744257	−0.372128	−	0.928181i	$-0.621372\pi$
$157$	−229864.	−0.744257	−0.372128	+	0.928181i	$0.621372\pi$
$158$	−625984.	−1.99490
$159$	319445.	1.00208
$160$	0	0
$161$	−577.051	−0.00175449
$162$	468433.	1.40236
$163$	400206.	1.17982	0.589908	−	0.807470i	$-0.299164\pi$
$163$	400206.	1.17982	0.589908	+	0.807470i	$0.299164\pi$
$164$	−1.29596e6	−3.76255
$165$	0	0
$166$	476582.	1.34235
$167$	276559.	0.767355	0.383677	−	0.923467i	$-0.374657\pi$
$167$	276559.	0.767355	0.383677	+	0.923467i	$0.374657\pi$
$168$	472599.	1.29187
$169$	28561.0	0.0769231
$170$	0	0
$171$	389991.	1.01991
$172$	1.39630e6	3.59880
$173$	−323519.	−0.821834	−0.410917	−	0.911673i	$-0.634791\pi$
$173$	−323519.	−0.821834	−0.410917	+	0.911673i	$0.634791\pi$
$174$	1.88737e6	4.72590
$175$	0	0
$176$	−921624.	−2.24270
$177$	−418623.	−1.00436
$178$	−12583.9	−0.0297690
$179$	−757066.	−1.76604	−0.883021	−	0.469333i	$-0.844494\pi$
$179$	−757066.	−1.76604	−0.883021	+	0.469333i	$0.844494\pi$
$180$	0	0
$181$	640530.	1.45326	0.726629	−	0.687030i	$-0.241086\pi$
$181$	640530.	1.45326	0.726629	+	0.687030i	$0.241086\pi$
$182$	87772.8	0.196418
$183$	175681.	0.387790
$184$	−4539.80	−0.00988535
$185$	0	0
$186$	−178053.	−0.377370
$187$	−501665.	−1.04908
$188$	1.57615e6	3.25239
$189$	−53092.4	−0.108113
$190$	0	0
$191$	−611932.	−1.21372	−0.606862	−	0.794807i	$-0.707572\pi$
$191$	−611932.	−1.21372	−0.606862	+	0.794807i	$0.707572\pi$
$192$	−50429.3	−0.0987256
$193$	−134676.	−0.260254	−0.130127	−	0.991497i	$-0.541539\pi$
$193$	−134676.	−0.260254	−0.130127	+	0.991497i	$0.541539\pi$
$194$	−585596.	−1.11710
$195$	0	0
$196$	−1.01510e6	−1.88743
$197$	−435184.	−0.798927	−0.399463	−	0.916749i	$-0.630804\pi$
$197$	−435184.	−0.798927	−0.399463	+	0.916749i	$0.630804\pi$
$198$	1.50197e6	2.72269
$199$	179852.	0.321946	0.160973	−	0.986959i	$-0.448537\pi$
$199$	179852.	0.321946	0.160973	+	0.986959i	$0.448537\pi$
$200$	0	0
$201$	86755.8	0.151464
$202$	137694.	0.237431
$203$	411017.	0.700035
$204$	1.61277e6	2.71330
$205$	0	0
$206$	1.01362e6	1.66420
$207$	3256.54	0.00528238
$208$	303944.	0.487119
$209$	−693623.	−1.09839
$210$	0	0
$211$	−1.03041e6	−1.59333	−0.796663	−	0.604424i	$-0.793403\pi$
$211$	−1.03041e6	−1.59333	−0.796663	+	0.604424i	$0.793403\pi$
$212$	990849.	1.51415
$213$	−1.74683e6	−2.63816
$214$	35074.3	0.0523546
$215$	0	0
$216$	−417690.	−0.609144
$217$	−38775.0	−0.0558988
$218$	−551722.	−0.786284
$219$	−1.99591e6	−2.81210
$220$	0	0
$221$	165445.	0.227862
$222$	2.99633e6	4.08044
$223$	−1.04356e6	−1.40525	−0.702626	−	0.711559i	$-0.747989\pi$
$223$	−1.04356e6	−1.40525	−0.702626	+	0.711559i	$0.747989\pi$
$224$	277857.	0.370000
$225$	0	0
$226$	1.05209e6	1.37019
$227$	1.04811e6	1.35003	0.675016	−	0.737803i	$-0.264137\pi$
$227$	1.04811e6	1.35003	0.675016	+	0.737803i	$0.264137\pi$
$228$	2.22989e6	2.84084
$229$	718967.	0.905984	0.452992	−	0.891515i	$-0.350357\pi$
$229$	718967.	0.905984	0.452992	+	0.891515i	$0.350357\pi$
$230$	0	0
$231$	602950.	0.743449
$232$	3.23357e6	3.94423
$233$	−714611.	−0.862343	−0.431172	−	0.902270i	$-0.641900\pi$
$233$	−714611.	−0.862343	−0.431172	+	0.902270i	$0.641900\pi$
$234$	−495338.	−0.591373
$235$	0	0
$236$	−1.29848e6	−1.51759
$237$	1.41816e6	1.64003
$238$	508441.	0.581832
$239$	114365.	0.129508	0.0647541	−	0.997901i	$-0.479374\pi$
$239$	114365.	0.129508	0.0647541	+	0.997901i	$0.479374\pi$
$240$	0	0
$241$	1.01606e6	1.12688	0.563440	−	0.826157i	$-0.309478\pi$
$241$	1.01606e6	1.12688	0.563440	+	0.826157i	$0.309478\pi$
$242$	−1.03303e6	−1.13389
$243$	−1.31393e6	−1.42743
$244$	544925.	0.585952
$245$	0	0
$246$	4.25027e6	4.47794
$247$	228751.	0.238573
$248$	−305052.	−0.314952
$249$	−1.07969e6	−1.10357
$250$	0	0
$251$	−1.13208e6	−1.13421	−0.567105	−	0.823645i	$-0.691937\pi$
$251$	−1.13208e6	−1.13421	−0.567105	+	0.823645i	$0.691937\pi$
$252$	−1.05154e6	−1.04309
$253$	−5791.95	−0.00568884
$254$	880933.	0.856758
$255$	0	0
$256$	−1.92805e6	−1.83873
$257$	357876.	0.337987	0.168993	−	0.985617i	$-0.445948\pi$
$257$	357876.	0.337987	0.168993	+	0.985617i	$0.445948\pi$
$258$	−4.57935e6	−4.28306
$259$	652518.	0.604426
$260$	0	0
$261$	−2.31954e6	−2.10766
$262$	−2.16500e6	−1.94852
$263$	305594.	0.272431	0.136215	−	0.990679i	$-0.456506\pi$
$263$	305594.	0.272431	0.136215	+	0.990679i	$0.456506\pi$
$264$	4.74355e6	4.18884
$265$	0	0
$266$	702991.	0.609180
$267$	28508.5	0.0244735
$268$	269098.	0.228862
$269$	−667878.	−0.562751	−0.281375	−	0.959598i	$-0.590791\pi$
$269$	−667878.	−0.562751	−0.281375	+	0.959598i	$0.590791\pi$
$270$	0	0
$271$	−1.05893e6	−0.875875	−0.437937	−	0.899005i	$-0.644291\pi$
$271$	−1.05893e6	−0.875875	−0.437937	+	0.899005i	$0.644291\pi$
$272$	1.76065e6	1.44295
$273$	−198848.	−0.161478
$274$	3.99310e6	3.21317
$275$	0	0
$276$	18620.2	0.0147134
$277$	1.91406e6	1.49884	0.749420	−	0.662095i	$-0.230333\pi$
$277$	1.91406e6	1.49884	0.749420	+	0.662095i	$0.230333\pi$
$278$	−658896.	−0.511335
$279$	218823.	0.168299
$280$	0	0
$281$	1.08128e6	0.816908	0.408454	−	0.912779i	$-0.366068\pi$
$281$	1.08128e6	0.816908	0.408454	+	0.912779i	$0.366068\pi$
$282$	−5.16919e6	−3.87079
$283$	−859352.	−0.637830	−0.318915	−	0.947783i	$-0.603318\pi$
$283$	−859352.	−0.637830	−0.318915	+	0.947783i	$0.603318\pi$
$284$	−5.41828e6	−3.98626
$285$	0	0
$286$	880989.	0.636877
$287$	925590.	0.663306
$288$	−1.56806e6	−1.11399
$289$	−461486.	−0.325023
$290$	0	0
$291$	1.32666e6	0.918389
$292$	−6.19089e6	−4.24909
$293$	2.36378e6	1.60856	0.804280	−	0.594250i	$-0.202551\pi$
$293$	2.36378e6	1.60856	0.804280	+	0.594250i	$0.202551\pi$
$294$	3.32916e6	2.24630
$295$	0	0
$296$	5.13351e6	3.40553
$297$	−532896.	−0.350551
$298$	2.82052e6	1.83988
$299$	1910.14	0.00123562
$300$	0	0
$301$	−997255.	−0.634439
$302$	576173.	0.363526
$303$	−311944.	−0.195196
$304$	2.43435e6	1.51077
$305$	0	0
$306$	−2.86934e6	−1.75177
$307$	1.61376e6	0.977220	0.488610	−	0.872502i	$-0.337504\pi$
$307$	1.61376e6	0.977220	0.488610	+	0.872502i	$0.337504\pi$
$308$	1.87022e6	1.12335
$309$	−2.29633e6	−1.36817
$310$	0	0
$311$	776229.	0.455081	0.227541	−	0.973769i	$-0.426932\pi$
$311$	776229.	0.455081	0.227541	+	0.973769i	$0.426932\pi$
$312$	−1.56438e6	−0.909823
$313$	−656850.	−0.378970	−0.189485	−	0.981884i	$-0.560682\pi$
$313$	−656850.	−0.378970	−0.189485	+	0.981884i	$0.560682\pi$
$314$	2.33835e6	1.33840
$315$	0	0
$316$	4.39882e6	2.47810
$317$	−1.41713e6	−0.792069	−0.396034	−	0.918236i	$-0.629614\pi$
$317$	−1.41713e6	−0.792069	−0.396034	+	0.918236i	$0.629614\pi$
$318$	−3.24962e6	−1.80204
$319$	4.12544e6	2.26983
$320$	0	0
$321$	−79460.3	−0.0430415
$322$	5870.18	0.00315509
$323$	1.32508e6	0.706703
$324$	−3.29170e6	−1.74204
$325$	0	0
$326$	−4.07118e6	−2.12166
$327$	1.24992e6	0.646416
$328$	7.28184e6	3.73729
$329$	−1.12571e6	−0.573371
$330$	0	0
$331$	−2.12709e6	−1.06713	−0.533564	−	0.845760i	$-0.679148\pi$
$331$	−2.12709e6	−1.06713	−0.533564	+	0.845760i	$0.679148\pi$
$332$	−3.34896e6	−1.66750
$333$	−3.68242e6	−1.81980
$334$	−2.81335e6	−1.37993
$335$	0	0
$336$	−2.11612e6	−1.02257
$337$	−483390.	−0.231859	−0.115929	−	0.993257i	$-0.536985\pi$
$337$	−483390.	−0.231859	−0.115929	+	0.993257i	$0.536985\pi$
$338$	−290543.	−0.138331
$339$	−2.38349e6	−1.12645
$340$	0	0
$341$	−389190.	−0.181249
$342$	−3.96726e6	−1.83411
$343$	1.58308e6	0.726552
$344$	−7.84564e6	−3.57464
$345$	0	0
$346$	3.29106e6	1.47790
$347$	−1.95372e6	−0.871041	−0.435521	−	0.900179i	$-0.643436\pi$
$347$	−1.95372e6	−0.871041	−0.435521	+	0.900179i	$0.643436\pi$
$348$	−1.32626e7	−5.87060
$349$	471784.	0.207338	0.103669	−	0.994612i	$-0.466942\pi$
$349$	471784.	0.207338	0.103669	+	0.994612i	$0.466942\pi$
$350$	0	0
$351$	175745.	0.0761403
$352$	2.78890e6	1.19971
$353$	−4.45914e6	−1.90465	−0.952324	−	0.305090i	$-0.901314\pi$
$353$	−4.45914e6	−1.90465	−0.952324	+	0.305090i	$0.901314\pi$
$354$	4.25853e6	1.80614
$355$	0	0
$356$	88427.3	0.0369795
$357$	−1.15186e6	−0.478333
$358$	7.70141e6	3.17587
$359$	−22798.5	−0.00933619	−0.00466810	−	0.999989i	$-0.501486\pi$
$359$	−22798.5	−0.00933619	−0.00466810	+	0.999989i	$0.501486\pi$
$360$	0	0
$361$	−643984.	−0.260080
$362$	−6.51593e6	−2.61339
$363$	2.34030e6	0.932192
$364$	−616784.	−0.243994
$365$	0	0
$366$	−1.78715e6	−0.697363
$367$	−1.45349e6	−0.563307	−0.281654	−	0.959516i	$-0.590883\pi$
$367$	−1.45349e6	−0.563307	−0.281654	+	0.959516i	$0.590883\pi$
$368$	20327.5	0.00782465
$369$	−5.22348e6	−1.99707
$370$	0	0
$371$	−707676.	−0.266932
$372$	1.25119e6	0.468775
$373$	−1.98547e6	−0.738910	−0.369455	−	0.929249i	$-0.620456\pi$
$373$	−1.98547e6	−0.738910	−0.369455	+	0.929249i	$0.620456\pi$
$374$	5.10329e6	1.88656
$375$	0	0
$376$	−8.85620e6	−3.23056
$377$	−1.36054e6	−0.493011
$378$	540094.	0.194419
$379$	1.09576e6	0.391847	0.195924	−	0.980619i	$-0.437230\pi$
$379$	1.09576e6	0.391847	0.195924	+	0.980619i	$0.437230\pi$
$380$	0	0
$381$	−1.99574e6	−0.704354
$382$	6.22501e6	2.18264
$383$	−2.86567e6	−0.998225	−0.499113	−	0.866537i	$-0.666341\pi$
$383$	−2.86567e6	−0.998225	−0.499113	+	0.866537i	$0.666341\pi$
$384$	4.52659e6	1.56655
$385$	0	0
$386$	1.37002e6	0.468015
$387$	5.62791e6	1.91016
$388$	4.11501e6	1.38769
$389$	1.42325e6	0.476878	0.238439	−	0.971158i	$-0.423364\pi$
$389$	1.42325e6	0.476878	0.238439	+	0.971158i	$0.423364\pi$
$390$	0	0
$391$	11064.8	0.00366018
$392$	5.70374e6	1.87476
$393$	4.90477e6	1.60191
$394$	4.42700e6	1.43671
$395$	0	0
$396$	−1.05544e7	−3.38218
$397$	6.03873e6	1.92296	0.961478	−	0.274882i	$-0.0886387\pi$
$397$	6.03873e6	1.92296	0.961478	+	0.274882i	$0.0886387\pi$
$398$	−1.82958e6	−0.578955
$399$	−1.59261e6	−0.500816
$400$	0	0
$401$	5.69168e6	1.76758	0.883791	−	0.467881i	$-0.154983\pi$
$401$	5.69168e6	1.76758	0.883791	+	0.467881i	$0.154983\pi$
$402$	−882541.	−0.272377
$403$	128352.	0.0393676
$404$	−967584.	−0.294941
$405$	0	0
$406$	−4.18116e6	−1.25887
$407$	6.54942e6	1.95982
$408$	−9.06198e6	−2.69509
$409$	−972884.	−0.287576	−0.143788	−	0.989609i	$-0.545928\pi$
$409$	−972884.	−0.287576	−0.143788	+	0.989609i	$0.545928\pi$
$410$	0	0
$411$	−9.04630e6	−2.64160
$412$	−7.12274e6	−2.06730
$413$	927388.	0.267539
$414$	−33127.8	−0.00949930
$415$	0	0
$416$	−919755.	−0.260579
$417$	1.49272e6	0.420376
$418$	7.05603e6	1.97524
$419$	3.58406e6	0.997334	0.498667	−	0.866794i	$-0.333823\pi$
$419$	3.58406e6	0.997334	0.498667	+	0.866794i	$0.333823\pi$
$420$	0	0
$421$	1.21460e6	0.333986	0.166993	−	0.985958i	$-0.446594\pi$
$421$	1.21460e6	0.333986	0.166993	+	0.985958i	$0.446594\pi$
$422$	1.04821e7	2.86528
$423$	6.35282e6	1.72630
$424$	−5.56746e6	−1.50398
$425$	0	0
$426$	1.77700e7	4.74419
$427$	−389192.	−0.103299
$428$	−246469.	−0.0650358
$429$	−1.99587e6	−0.523586
$430$	0	0
$431$	3.18545e6	0.825996	0.412998	−	0.910732i	$-0.364482\pi$
$431$	3.18545e6	0.825996	0.412998	+	0.910732i	$0.364482\pi$
$432$	1.87026e6	0.482162
$433$	5.64608e6	1.44720	0.723598	−	0.690221i	$-0.242487\pi$
$433$	5.64608e6	1.44720	0.723598	+	0.690221i	$0.242487\pi$
$434$	394447.	0.100523
$435$	0	0
$436$	3.87698e6	0.976735
$437$	15298.7	0.00383222
$438$	2.03038e7	5.05700
$439$	7.65369e6	1.89544	0.947719	−	0.319108i	$-0.103383\pi$
$439$	7.65369e6	1.89544	0.947719	+	0.319108i	$0.103383\pi$
$440$	0	0
$441$	−4.09146e6	−1.00180
$442$	−1.68302e6	−0.409765
$443$	5.65650e6	1.36943	0.684713	−	0.728813i	$-0.259927\pi$
$443$	5.65650e6	1.36943	0.684713	+	0.728813i	$0.259927\pi$
$444$	−2.10554e7	−5.06880
$445$	0	0
$446$	1.06158e7	2.52706
$447$	−6.38984e6	−1.51259
$448$	111718.	0.0262983
$449$	−1.93752e6	−0.453556	−0.226778	−	0.973947i	$-0.572819\pi$
$449$	−1.93752e6	−0.453556	−0.226778	+	0.973947i	$0.572819\pi$
$450$	0	0
$451$	9.29029e6	2.15074
$452$	−7.39307e6	−1.70208
$453$	−1.30531e6	−0.298860
$454$	−1.06622e7	−2.42776
$455$	0	0
$456$	−1.25295e7	−2.82176
$457$	3.22536e6	0.722417	0.361208	−	0.932485i	$-0.382364\pi$
$457$	3.22536e6	0.722417	0.361208	+	0.932485i	$0.382364\pi$
$458$	−7.31385e6	−1.62923
$459$	1.01803e6	0.225544
$460$	0	0
$461$	−4.53894e6	−0.994723	−0.497362	−	0.867543i	$-0.665698\pi$
$461$	−4.53894e6	−0.994723	−0.497362	+	0.867543i	$0.665698\pi$
$462$	−6.13364e6	−1.33694
$463$	4.40536e6	0.955056	0.477528	−	0.878617i	$-0.341533\pi$
$463$	4.40536e6	0.955056	0.477528	+	0.878617i	$0.341533\pi$
$464$	−1.44787e7	−3.12202
$465$	0	0
$466$	7.26954e6	1.55075
$467$	−7.89872e6	−1.67596	−0.837982	−	0.545698i	$-0.816265\pi$
$467$	−7.89872e6	−1.67596	−0.837982	+	0.545698i	$0.816265\pi$
$468$	3.48076e6	0.734614
$469$	−192193.	−0.0403464
$470$	0	0
$471$	−5.29748e6	−1.10032
$472$	7.29599e6	1.50740
$473$	−1.00096e7	−2.05714
$474$	−1.44265e7	−2.94927
$475$	0	0
$476$	−3.57283e6	−0.722762
$477$	3.99370e6	0.803674
$478$	−1.16340e6	−0.232895
$479$	1.62018e6	0.322644	0.161322	−	0.986902i	$-0.448424\pi$
$479$	1.62018e6	0.322644	0.161322	+	0.986902i	$0.448424\pi$
$480$	0	0
$481$	−2.15994e6	−0.425676
$482$	−1.03361e7	−2.02647
$483$	−13298.8	−0.00259385
$484$	7.25911e6	1.40854
$485$	0	0
$486$	1.33662e7	2.56695
$487$	4.09572e6	0.782543	0.391272	−	0.920275i	$-0.372035\pi$
$487$	4.09572e6	0.782543	0.391272	+	0.920275i	$0.372035\pi$
$488$	−3.06187e6	−0.582018
$489$	9.22319e6	1.74425
$490$	0	0
$491$	−7.09337e6	−1.32785	−0.663925	−	0.747799i	$-0.731111\pi$
$491$	−7.09337e6	−1.32785	−0.663925	+	0.747799i	$0.731111\pi$
$492$	−2.98668e7	−5.56258
$493$	−7.88116e6	−1.46040
$494$	−2.32702e6	−0.429025
$495$	0	0
$496$	1.36591e6	0.249297
$497$	3.86980e6	0.702745
$498$	1.09834e7	1.98455
$499$	−1.60514e6	−0.288577	−0.144289	−	0.989536i	$-0.546089\pi$
$499$	−1.60514e6	−0.288577	−0.144289	+	0.989536i	$0.546089\pi$
$500$	0	0
$501$	6.37360e6	1.13446
$502$	1.15164e7	2.03965
$503$	5.32213e6	0.937920	0.468960	−	0.883219i	$-0.344629\pi$
$503$	5.32213e6	0.937920	0.468960	+	0.883219i	$0.344629\pi$
$504$	5.90845e6	1.03609
$505$	0	0
$506$	58919.9	0.0102302
$507$	658220.	0.113724
$508$	−6.19035e6	−1.06428
$509$	2.38983e6	0.408857	0.204429	−	0.978881i	$-0.434466\pi$
$509$	2.38983e6	0.408857	0.204429	+	0.978881i	$0.434466\pi$
$510$	0	0
$511$	4.42161e6	0.749080
$512$	1.33282e7	2.24697
$513$	1.40758e6	0.236145
$514$	−3.64057e6	−0.607801
$515$	0	0
$516$	3.21793e7	5.32050
$517$	−1.12989e7	−1.85913
$518$	−6.63787e6	−1.08694
$519$	−7.45584e6	−1.21501
$520$	0	0
$521$	−970922.	−0.156708	−0.0783538	−	0.996926i	$-0.524966\pi$
$521$	−970922.	−0.156708	−0.0783538	+	0.996926i	$0.524966\pi$
$522$	2.35960e7	3.79020
$523$	7.93187e6	1.26801	0.634003	−	0.773330i	$-0.281410\pi$
$523$	7.93187e6	1.26801	0.634003	+	0.773330i	$0.281410\pi$
$524$	1.52136e7	2.42049
$525$	0	0
$526$	−3.10872e6	−0.489912
$527$	743501.	0.116615
$528$	−2.12398e7	−3.31563
$529$	−6.43622e6	−0.999980
$530$	0	0
$531$	−5.23363e6	−0.805502
$532$	−4.93995e6	−0.756734
$533$	−3.06386e6	−0.467144
$534$	−290009.	−0.0440107
$535$	0	0
$536$	−1.51203e6	−0.227325
$537$	−1.74474e7	−2.61093
$538$	6.79413e6	1.01199
$539$	7.27692e6	1.07889
$540$	0	0
$541$	−6.96364e6	−1.02292	−0.511462	−	0.859306i	$-0.670896\pi$
$541$	−6.96364e6	−1.02292	−0.511462	+	0.859306i	$0.670896\pi$
$542$	1.07721e7	1.57508
$543$	1.47617e7	2.14851
$544$	−5.32785e6	−0.771889
$545$	0	0
$546$	2.02282e6	0.290386
$547$	1.01847e7	1.45539	0.727693	−	0.685903i	$-0.240593\pi$
$547$	1.01847e7	1.45539	0.727693	+	0.685903i	$0.240593\pi$
$548$	−2.80597e7	−3.99146
$549$	2.19637e6	0.311010
$550$	0	0
$551$	−1.08968e7	−1.52905
$552$	−104625.	−0.0146146
$553$	−3.14169e6	−0.436868
$554$	−1.94711e7	−2.69536
$555$	0	0
$556$	4.63009e6	0.635189
$557$	1.17433e7	1.60381	0.801903	−	0.597454i	$-0.203821\pi$
$557$	1.17433e7	1.60381	0.801903	+	0.597454i	$0.203821\pi$
$558$	−2.22602e6	−0.302652
$559$	3.30108e6	0.446814
$560$	0	0
$561$	−1.15614e7	−1.55097
$562$	−1.09996e7	−1.46904
$563$	−1.14861e7	−1.52722	−0.763611	−	0.645677i	$-0.776575\pi$
$563$	−1.14861e7	−1.52722	−0.763611	+	0.645677i	$0.776575\pi$
$564$	3.63241e7	4.80837
$565$	0	0
$566$	8.74194e6	1.14701
$567$	2.35097e6	0.307107
$568$	3.04447e7	3.95950
$569$	1.51914e7	1.96706	0.983532	−	0.180734i	$-0.0578473\pi$
$569$	1.51914e7	1.96706	0.983532	+	0.180734i	$0.0578473\pi$
$570$	0	0
$571$	−7.26504e6	−0.932498	−0.466249	−	0.884654i	$-0.654395\pi$
$571$	−7.26504e6	−0.932498	−0.466249	+	0.884654i	$0.654395\pi$
$572$	−6.19075e6	−0.791140
$573$	−1.41027e7	−1.79438
$574$	−9.41576e6	−1.19282
$575$	0	0
$576$	−630468.	−0.0791784
$577$	5.66789e6	0.708731	0.354366	−	0.935107i	$-0.384697\pi$
$577$	5.66789e6	0.708731	0.354366	+	0.935107i	$0.384697\pi$
$578$	4.69457e6	0.584489
$579$	−3.10377e6	−0.384762
$580$	0	0
$581$	2.39187e6	0.293966
$582$	−1.34957e7	−1.65154
$583$	−7.10305e6	−0.865513
$584$	3.47859e7	4.22056
$585$	0	0
$586$	−2.40460e7	−2.89267
$587$	1.09050e6	0.130627	0.0653133	−	0.997865i	$-0.479195\pi$
$587$	1.09050e6	0.130627	0.0653133	+	0.997865i	$0.479195\pi$
$588$	−2.33942e7	−2.79039
$589$	1.02800e6	0.122096
$590$	0	0
$591$	−1.00293e7	−1.18114
$592$	−2.29859e7	−2.69562
$593$	4.95413e6	0.578536	0.289268	−	0.957248i	$-0.406588\pi$
$593$	4.95413e6	0.578536	0.289268	+	0.957248i	$0.406588\pi$
$594$	5.42100e6	0.630396
$595$	0	0
$596$	−1.98199e7	−2.28553
$597$	4.14489e6	0.475968
$598$	−19431.3	−0.00222202
$599$	6.54475e6	0.745292	0.372646	−	0.927974i	$-0.378451\pi$
$599$	6.54475e6	0.745292	0.372646	+	0.927974i	$0.378451\pi$
$600$	0	0
$601$	−2.16617e6	−0.244628	−0.122314	−	0.992491i	$-0.539031\pi$
$601$	−2.16617e6	−0.244628	−0.122314	+	0.992491i	$0.539031\pi$
$602$	1.01448e7	1.14091
$603$	1.08462e6	0.121475
$604$	−4.04879e6	−0.451578
$605$	0	0
$606$	3.17332e6	0.351020
$607$	−986604.	−0.108685	−0.0543427	−	0.998522i	$-0.517306\pi$
$607$	−986604.	−0.108685	−0.0543427	+	0.998522i	$0.517306\pi$
$608$	−7.36651e6	−0.808170
$609$	9.47235e6	1.03494
$610$	0	0
$611$	3.72628e6	0.403805
$612$	2.01630e7	2.17608
$613$	1.50041e7	1.61272	0.806360	−	0.591425i	$-0.201435\pi$
$613$	1.50041e7	1.61272	0.806360	+	0.591425i	$0.201435\pi$
$614$	−1.64163e7	−1.75733
$615$	0	0
$616$	−1.05085e7	−1.11581
$617$	1.27376e6	0.134703	0.0673514	−	0.997729i	$-0.478545\pi$
$617$	1.27376e6	0.134703	0.0673514	+	0.997729i	$0.478545\pi$
$618$	2.33599e7	2.46037
$619$	−2.59865e6	−0.272597	−0.136299	−	0.990668i	$-0.543521\pi$
$619$	−2.59865e6	−0.272597	−0.136299	+	0.990668i	$0.543521\pi$
$620$	0	0
$621$	11753.7	0.00122305
$622$	−7.89635e6	−0.818372
$623$	−63155.8	−0.00651919
$624$	7.00472e6	0.720161
$625$	0	0
$626$	6.68195e6	0.681502
$627$	−1.59853e7	−1.62387
$628$	−1.64317e7	−1.66258
$629$	−1.25119e7	−1.26094
$630$	0	0
$631$	−1.92424e7	−1.92392	−0.961959	−	0.273194i	$-0.911920\pi$
$631$	−1.92424e7	−1.92392	−0.961959	+	0.273194i	$0.911920\pi$
$632$	−2.47164e7	−2.46146
$633$	−2.37470e7	−2.35559
$634$	1.44161e7	1.42438
$635$	0	0
$636$	2.28352e7	2.23853
$637$	−2.39987e6	−0.234336
$638$	−4.19669e7	−4.08184
$639$	−2.18389e7	−2.11582
$640$	0	0
$641$	1.83373e7	1.76275	0.881375	−	0.472417i	$-0.156618\pi$
$641$	1.83373e7	1.76275	0.881375	+	0.472417i	$0.156618\pi$
$642$	808326.	0.0774015
$643$	−1.43215e6	−0.136603	−0.0683015	−	0.997665i	$-0.521758\pi$
$643$	−1.43215e6	−0.136603	−0.0683015	+	0.997665i	$0.521758\pi$
$644$	−41250.0	−0.00391931
$645$	0	0
$646$	−1.34797e7	−1.27086
$647$	1.04396e7	0.980440	0.490220	−	0.871599i	$-0.336916\pi$
$647$	1.04396e7	0.980440	0.490220	+	0.871599i	$0.336916\pi$
$648$	1.84957e7	1.73034
$649$	9.30834e6	0.867482
$650$	0	0
$651$	−893612.	−0.0826412
$652$	2.86084e7	2.63557
$653$	−1.32094e7	−1.21227	−0.606137	−	0.795360i	$-0.707282\pi$
$653$	−1.32094e7	−1.21227	−0.606137	+	0.795360i	$0.707282\pi$
$654$	−1.27151e7	−1.16245
$655$	0	0
$656$	−3.26053e7	−2.95821
$657$	−2.49529e7	−2.25532
$658$	1.14515e7	1.03109
$659$	3.20415e6	0.287408	0.143704	−	0.989621i	$-0.454099\pi$
$659$	3.20415e6	0.287408	0.143704	+	0.989621i	$0.454099\pi$
$660$	0	0
$661$	−2.22430e7	−1.98011	−0.990057	−	0.140667i	$-0.955075\pi$
$661$	−2.22430e7	−1.98011	−0.990057	+	0.140667i	$0.955075\pi$
$662$	2.16383e7	1.91901
$663$	3.81286e6	0.336874
$664$	1.88174e7	1.65630
$665$	0	0
$666$	3.74602e7	3.27254
$667$	−90991.6	−0.00791930
$668$	1.97696e7	1.71418
$669$	−2.40500e7	−2.07754
$670$	0	0
$671$	−3.90638e6	−0.334941
$672$	6.40353e6	0.547011
$673$	1.08386e7	0.922434	0.461217	−	0.887287i	$-0.347413\pi$
$673$	1.08386e7	0.922434	0.461217	+	0.887287i	$0.347413\pi$
$674$	4.91739e6	0.416951
$675$	0	0
$676$	2.04166e6	0.171837
$677$	1.21301e7	1.01717	0.508585	−	0.861012i	$-0.330169\pi$
$677$	1.21301e7	1.01717	0.508585	+	0.861012i	$0.330169\pi$
$678$	2.42465e7	2.02570
$679$	−2.93899e6	−0.244638
$680$	0	0
$681$	2.41550e7	1.99590
$682$	3.95912e6	0.325940
$683$	1.86818e7	1.53238	0.766191	−	0.642613i	$-0.222149\pi$
$683$	1.86818e7	1.53238	0.766191	+	0.642613i	$0.222149\pi$
$684$	2.78781e7	2.27837
$685$	0	0
$686$	−1.61042e7	−1.30656
$687$	1.65694e7	1.33941
$688$	3.51298e7	2.82947
$689$	2.34253e6	0.187991
$690$	0	0
$691$	−1.25804e7	−1.00230	−0.501151	−	0.865360i	$-0.667090\pi$
$691$	−1.25804e7	−1.00230	−0.501151	+	0.865360i	$0.667090\pi$
$692$	−2.31264e7	−1.83588
$693$	7.53810e6	0.596250
$694$	1.98746e7	1.56639
$695$	0	0
$696$	7.45212e7	5.83118
$697$	−1.77480e7	−1.38378
$698$	−4.79933e6	−0.372856
$699$	−1.64690e7	−1.27490
$700$	0	0
$701$	8.22912e6	0.632497	0.316249	−	0.948676i	$-0.397577\pi$
$701$	8.22912e6	0.632497	0.316249	+	0.948676i	$0.397577\pi$
$702$	−1.78780e6	−0.136923
$703$	−1.72994e7	−1.32021
$704$	1.12133e6	0.0852709
$705$	0	0
$706$	4.53616e7	3.42512
$707$	691060.	0.0519957
$708$	−2.99248e7	−2.24362
$709$	2.08389e7	1.55690	0.778449	−	0.627708i	$-0.216007\pi$
$709$	2.08389e7	1.55690	0.778449	+	0.627708i	$0.216007\pi$
$710$	0	0
$711$	1.77298e7	1.31532
$712$	−496862.	−0.0367313
$713$	8584.06	0.000632367 0
$714$	1.17176e7	0.860186
$715$	0	0
$716$	−5.41182e7	−3.94512
$717$	2.63566e6	0.191466
$718$	231922.	0.0167893
$719$	1.98244e7	1.43014	0.715068	−	0.699055i	$-0.246396\pi$
$719$	1.98244e7	1.43014	0.715068	+	0.699055i	$0.246396\pi$
$720$	0	0
$721$	5.08715e6	0.364448
$722$	6.55107e6	0.467702
$723$	2.34163e7	1.66599
$724$	4.57877e7	3.24640
$725$	0	0
$726$	−2.38072e7	−1.67636
$727$	−9.64612e6	−0.676888	−0.338444	−	0.940987i	$-0.609901\pi$
$727$	−9.64612e6	−0.676888	−0.338444	+	0.940987i	$0.609901\pi$
$728$	3.46563e6	0.242356
$729$	−1.90912e7	−1.33050
$730$	0	0
$731$	1.91221e7	1.32356
$732$	1.25584e7	0.866276
$733$	−1.74331e7	−1.19844	−0.599219	−	0.800585i	$-0.704522\pi$
$733$	−1.74331e7	−1.19844	−0.599219	+	0.800585i	$0.704522\pi$
$734$	1.47859e7	1.01299
$735$	0	0
$736$	−61512.5	−0.00418570
$737$	−1.92907e6	−0.130821
$738$	5.31370e7	3.59133
$739$	−3.56942e6	−0.240429	−0.120214	−	0.992748i	$-0.538358\pi$
$739$	−3.56942e6	−0.240429	−0.120214	+	0.992748i	$0.538358\pi$
$740$	0	0
$741$	5.27182e6	0.352708
$742$	7.19899e6	0.480023
$743$	−9.04400e6	−0.601019	−0.300510	−	0.953779i	$-0.597157\pi$
$743$	−9.04400e6	−0.601019	−0.300510	+	0.953779i	$0.597157\pi$
$744$	−7.03026e6	−0.465628
$745$	0	0
$746$	2.01976e7	1.32878
$747$	−1.34983e7	−0.885069
$748$	−3.58611e7	−2.34352
$749$	176031.	0.0114653
$750$	0	0
$751$	−2.27403e7	−1.47128	−0.735642	−	0.677371i	$-0.763119\pi$
$751$	−2.27403e7	−1.47128	−0.735642	+	0.677371i	$0.763119\pi$
$752$	3.96547e7	2.55712
$753$	−2.60901e7	−1.67683
$754$	1.38403e7	0.886581
$755$	0	0
$756$	−3.79526e6	−0.241511
$757$	−1.51736e7	−0.962385	−0.481192	−	0.876615i	$-0.659796\pi$
$757$	−1.51736e7	−0.962385	−0.481192	+	0.876615i	$0.659796\pi$
$758$	−1.11468e7	−0.704659
$759$	−133482.	−0.00841043
$760$	0	0
$761$	1.74364e7	1.09143	0.545714	−	0.837971i	$-0.316258\pi$
$761$	1.74364e7	1.09143	0.545714	+	0.837971i	$0.316258\pi$
$762$	2.03021e7	1.26664
$763$	−2.76898e6	−0.172190
$764$	−4.37434e7	−2.71131
$765$	0	0
$766$	2.91516e7	1.79511
$767$	−3.06981e6	−0.188418
$768$	−4.44340e7	−2.71839
$769$	1.64867e7	1.00535	0.502675	−	0.864476i	$-0.332349\pi$
$769$	1.64867e7	1.00535	0.502675	+	0.864476i	$0.332349\pi$
$770$	0	0
$771$	8.24764e6	0.499682
$772$	−9.62722e6	−0.581377
$773$	1.70607e7	1.02695	0.513474	−	0.858105i	$-0.328358\pi$
$773$	1.70607e7	1.02695	0.513474	+	0.858105i	$0.328358\pi$
$774$	−5.72512e7	−3.43504
$775$	0	0
$776$	−2.31217e7	−1.37837
$777$	1.50380e7	0.893588
$778$	−1.44783e7	−0.857569
$779$	−2.45391e7	−1.44882
$780$	0	0
$781$	3.88418e7	2.27862
$782$	−112559.	−0.00658210
$783$	−8.37180e6	−0.487994
$784$	−2.55392e7	−1.48394
$785$	0	0
$786$	−4.98949e7	−2.88071
$787$	2.38884e6	0.137483	0.0687417	−	0.997634i	$-0.478102\pi$
$787$	2.38884e6	0.137483	0.0687417	+	0.997634i	$0.478102\pi$
$788$	−3.11087e7	−1.78471
$789$	7.04276e6	0.402764
$790$	0	0
$791$	5.28022e6	0.300062
$792$	5.93040e7	3.35947
$793$	1.28829e6	0.0727496
$794$	−6.14303e7	−3.45805
$795$	0	0
$796$	1.28566e7	0.719188
$797$	7.46487e6	0.416272	0.208136	−	0.978100i	$-0.433260\pi$
$797$	7.46487e6	0.416272	0.208136	+	0.978100i	$0.433260\pi$
$798$	1.62012e7	0.900617
$799$	2.15852e7	1.19616
$800$	0	0
$801$	356414.	0.0196279
$802$	−5.78999e7	−3.17864
$803$	4.43804e7	2.42886
$804$	6.20166e6	0.338351
$805$	0	0
$806$	−1.30569e6	−0.0707947
$807$	−1.53920e7	−0.831976
$808$	5.43674e6	0.292961
$809$	−1.91578e7	−1.02914	−0.514569	−	0.857449i	$-0.672048\pi$
$809$	−1.91578e7	−1.02914	−0.514569	+	0.857449i	$0.672048\pi$
$810$	0	0
$811$	2.86526e6	0.152972	0.0764859	−	0.997071i	$-0.475630\pi$
$811$	2.86526e6	0.152972	0.0764859	+	0.997071i	$0.475630\pi$
$812$	2.93812e7	1.56379
$813$	−2.44041e7	−1.29490
$814$	−6.66253e7	−3.52435
$815$	0	0
$816$	4.05762e7	2.13327
$817$	2.64391e7	1.38577
$818$	9.89687e6	0.517147
$819$	−2.48600e6	−0.129507
$820$	0	0
$821$	−1.42195e7	−0.736254	−0.368127	−	0.929775i	$-0.620001\pi$
$821$	−1.42195e7	−0.736254	−0.368127	+	0.929775i	$0.620001\pi$
$822$	9.20255e7	4.75038
$823$	8.97879e6	0.462081	0.231041	−	0.972944i	$-0.425787\pi$
$823$	8.97879e6	0.462081	0.231041	+	0.972944i	$0.425787\pi$
$824$	4.00218e7	2.05342
$825$	0	0
$826$	−9.43406e6	−0.481114
$827$	1.46294e7	0.743813	0.371907	−	0.928270i	$-0.378704\pi$
$827$	1.46294e7	0.743813	0.371907	+	0.928270i	$0.378704\pi$
$828$	232790.	0.0118002
$829$	6.70195e6	0.338700	0.169350	−	0.985556i	$-0.445833\pi$
$829$	6.70195e6	0.338700	0.169350	+	0.985556i	$0.445833\pi$
$830$	0	0
$831$	4.41115e7	2.21590
$832$	−369804.	−0.0185210
$833$	−1.39017e7	−0.694153
$834$	−1.51850e7	−0.755961
$835$	0	0
$836$	−4.95830e7	−2.45368
$837$	789788.	0.0389670
$838$	−3.64597e7	−1.79350
$839$	−1.00902e7	−0.494876	−0.247438	−	0.968904i	$-0.579589\pi$
$839$	−1.00902e7	−0.494876	−0.247438	+	0.968904i	$0.579589\pi$
$840$	0	0
$841$	4.42995e7	2.15978
$842$	−1.23558e7	−0.600607
$843$	2.49193e7	1.20772
$844$	−7.36580e7	−3.55930
$845$	0	0
$846$	−6.46254e7	−3.10440
$847$	−5.18455e6	−0.248315
$848$	2.49290e7	1.19046
$849$	−1.98047e7	−0.942973
$850$	0	0
$851$	−144455.	−0.00683769
$852$	−1.24870e8	−5.89332
$853$	8.26689e6	0.389018	0.194509	−	0.980901i	$-0.437689\pi$
$853$	8.26689e6	0.389018	0.194509	+	0.980901i	$0.437689\pi$
$854$	3.95914e6	0.185762
$855$	0	0
$856$	1.38488e6	0.0645992
$857$	−1.34533e6	−0.0625714	−0.0312857	−	0.999510i	$-0.509960\pi$
$857$	−1.34533e6	−0.0625714	−0.0312857	+	0.999510i	$0.509960\pi$
$858$	2.03034e7	0.941564
$859$	−2.57272e7	−1.18962	−0.594812	−	0.803865i	$-0.702774\pi$
$859$	−2.57272e7	−1.18962	−0.594812	+	0.803865i	$0.702774\pi$
$860$	0	0
$861$	2.13313e7	0.980637
$862$	−3.24047e7	−1.48539
$863$	−9.60150e6	−0.438846	−0.219423	−	0.975630i	$-0.570417\pi$
$863$	−9.60150e6	−0.438846	−0.219423	+	0.975630i	$0.570417\pi$
$864$	−5.65954e6	−0.257927
$865$	0	0
$866$	−5.74360e7	−2.60249
$867$	−1.06355e7	−0.480517
$868$	−2.77179e6	−0.124871
$869$	−3.15336e7	−1.41652
$870$	0	0
$871$	636191.	0.0284146
$872$	−2.17843e7	−0.970178
$873$	1.65859e7	0.736552
$874$	−155629.	−0.00689148
$875$	0	0
$876$	−1.42676e8	−6.28189
$877$	−3.88143e6	−0.170409	−0.0852046	−	0.996363i	$-0.527154\pi$
$877$	−3.88143e6	−0.170409	−0.0852046	+	0.996363i	$0.527154\pi$
$878$	−7.78588e7	−3.40856
$879$	5.44759e7	2.37811
$880$	0	0
$881$	−1.47028e7	−0.638204	−0.319102	−	0.947720i	$-0.603381\pi$
$881$	−1.47028e7	−0.638204	−0.319102	+	0.947720i	$0.603381\pi$
$882$	4.16213e7	1.80154
$883$	−4.37837e7	−1.88978	−0.944889	−	0.327392i	$-0.893830\pi$
$883$	−4.37837e7	−1.88978	−0.944889	+	0.327392i	$0.893830\pi$
$884$	1.18267e7	0.509017
$885$	0	0
$886$	−5.75420e7	−2.46264
$887$	9.31739e6	0.397636	0.198818	−	0.980036i	$-0.436290\pi$
$887$	9.31739e6	0.397636	0.198818	+	0.980036i	$0.436290\pi$
$888$	1.18307e8	5.03477
$889$	4.42122e6	0.187624
$890$	0	0
$891$	2.35971e7	0.995781
$892$	−7.45978e7	−3.13916
$893$	2.98445e7	1.25238
$894$	6.50020e7	2.72009
$895$	0	0
$896$	−1.00279e7	−0.417292
$897$	44021.2	0.00182676
$898$	1.97098e7	0.815628
$899$	−6.11418e6	−0.252313
$900$	0	0
$901$	1.35695e7	0.556869
$902$	−9.45074e7	−3.86767
$903$	−2.29828e7	−0.937960
$904$	4.15407e7	1.69065
$905$	0	0
$906$	1.32785e7	0.537440
$907$	1.32431e7	0.534529	0.267265	−	0.963623i	$-0.413880\pi$
$907$	1.32431e7	0.534529	0.267265	+	0.963623i	$0.413880\pi$
$908$	7.49235e7	3.01581
$909$	−3.89993e6	−0.156548
$910$	0	0
$911$	2.78672e7	1.11249	0.556246	−	0.831018i	$-0.312241\pi$
$911$	2.78672e7	1.11249	0.556246	+	0.831018i	$0.312241\pi$
$912$	5.61023e7	2.23354
$913$	2.40075e7	0.953171
$914$	−3.28107e7	−1.29912
$915$	0	0
$916$	5.13947e7	2.02386
$917$	−1.08657e7	−0.426712
$918$	−1.03562e7	−0.405595
$919$	−1.81646e7	−0.709474	−0.354737	−	0.934966i	$-0.615430\pi$
$919$	−1.81646e7	−0.709474	−0.354737	+	0.934966i	$0.615430\pi$
$920$	0	0
$921$	3.71908e7	1.44473
$922$	4.61733e7	1.78881
$923$	−1.28097e7	−0.494920
$924$	4.31013e7	1.66077
$925$	0	0
$926$	−4.48145e7	−1.71748
$927$	−2.87088e7	−1.09728
$928$	4.38136e7	1.67009
$929$	3.97985e7	1.51296	0.756480	−	0.654017i	$-0.226917\pi$
$929$	3.97985e7	1.51296	0.756480	+	0.654017i	$0.226917\pi$
$930$	0	0
$931$	−1.92210e7	−0.726780
$932$	−5.10833e7	−1.92637
$933$	1.78891e7	0.672796
$934$	8.03515e7	3.01388
$935$	0	0
$936$	−1.95580e7	−0.729683
$937$	−3.23323e7	−1.20306	−0.601531	−	0.798850i	$-0.705442\pi$
$937$	−3.23323e7	−1.20306	−0.601531	+	0.798850i	$0.705442\pi$
$938$	1.95512e6	0.0725550
$939$	−1.51378e7	−0.560273
$940$	0	0
$941$	−1.24433e7	−0.458102	−0.229051	−	0.973414i	$-0.573562\pi$
$941$	−1.24433e7	−0.458102	−0.229051	+	0.973414i	$0.573562\pi$
$942$	5.38898e7	1.97870
$943$	−204908.	−0.00750379
$944$	−3.26687e7	−1.19317
$945$	0	0
$946$	1.01825e8	3.69935
$947$	−3.32452e7	−1.20463	−0.602316	−	0.798258i	$-0.705755\pi$
$947$	−3.32452e7	−1.20463	−0.602316	+	0.798258i	$0.705755\pi$
$948$	1.01376e8	3.66364
$949$	−1.46363e7	−0.527552
$950$	0	0
$951$	−3.26594e7	−1.17100
$952$	2.00753e7	0.717910
$953$	−4.55359e7	−1.62413	−0.812066	−	0.583566i	$-0.801657\pi$
$953$	−4.55359e7	−1.62413	−0.812066	+	0.583566i	$0.801657\pi$
$954$	−4.06268e7	−1.44525
$955$	0	0
$956$	8.17526e6	0.289306
$957$	9.50754e7	3.35574
$958$	−1.64816e7	−0.580210
$959$	2.00406e7	0.703661
$960$	0	0
$961$	−2.80523e7	−0.979852
$962$	2.19725e7	0.765493
$963$	−993414.	−0.0345195
$964$	7.26323e7	2.51731
$965$	0	0
$966$	135285.	0.00466451
$967$	−1.52863e7	−0.525697	−0.262849	−	0.964837i	$-0.584662\pi$
$967$	−1.52863e7	−0.525697	−0.262849	+	0.964837i	$0.584662\pi$
$968$	−4.07881e7	−1.39909
$969$	3.05380e7	1.04480
$970$	0	0
$971$	−1.92581e7	−0.655490	−0.327745	−	0.944766i	$-0.606289\pi$
$971$	−1.92581e7	−0.655490	−0.327745	+	0.944766i	$0.606289\pi$
$972$	−9.39248e7	−3.18871
$973$	−3.30687e6	−0.111979
$974$	−4.16646e7	−1.40725
$975$	0	0
$976$	1.37099e7	0.460690
$977$	−795487.	−0.0266622	−0.0133311	−	0.999911i	$-0.504244\pi$
$977$	−795487.	−0.0266622	−0.0133311	+	0.999911i	$0.504244\pi$
$978$	−9.38248e7	−3.13668
$979$	−633905.	−0.0211382
$980$	0	0
$981$	1.56265e7	0.518429
$982$	7.21588e7	2.38787
$983$	−7.51037e6	−0.247901	−0.123950	−	0.992288i	$-0.539556\pi$
$983$	−7.51037e6	−0.247901	−0.123950	+	0.992288i	$0.539556\pi$
$984$	1.67818e8	5.52524
$985$	0	0
$986$	8.01728e7	2.62624
$987$	−2.59432e7	−0.847676
$988$	1.63521e7	0.532942
$989$	220774.	0.00717723
$990$	0	0
$991$	5.90076e7	1.90864	0.954319	−	0.298788i	$-0.0965824\pi$
$991$	5.90076e7	1.90864	0.954319	+	0.298788i	$0.0965824\pi$
$992$	−4.13333e6	−0.133359
$993$	−4.90212e7	−1.57765
$994$	−3.93664e7	−1.26375
$995$	0	0
$996$	−7.71805e7	−2.46524
$997$	−4.32622e7	−1.37839	−0.689193	−	0.724577i	$-0.742035\pi$
$997$	−4.32622e7	−1.37839	−0.689193	+	0.724577i	$0.742035\pi$
$998$	1.63287e7	0.518948
$999$	−1.32908e7	−0.421345

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.h.1.1	✓	9
5.2	odd	4		325.6.b.h.274.1		18
5.3	odd	4		325.6.b.h.274.18		18
5.4	even	2		325.6.a.i.1.9	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.1	✓	9	1.1	even	1	trivial
325.6.a.i.1.9	yes	9	5.4	even	2
325.6.b.h.274.1		18	5.2	odd	4
325.6.b.h.274.18		18	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

\(f(q)\)	\(=\)	\(q-10.1727 q^{2} +23.0461 q^{3} +71.4841 q^{4} -234.441 q^{6} -51.0548 q^{7} -401.660 q^{8} +288.123 q^{9} +O(q^{10})\)	\(q-10.1727 q^{2} +23.0461 q^{3} +71.4841 q^{4} -234.441 q^{6} -51.0548 q^{7} -401.660 q^{8} +288.123 q^{9} -512.445 q^{11} +1647.43 q^{12} +169.000 q^{13} +519.366 q^{14} +1798.48 q^{16} +978.964 q^{17} -2930.99 q^{18} +1353.56 q^{19} -1176.61 q^{21} +5212.95 q^{22} +11.3026 q^{23} -9256.71 q^{24} -1719.19 q^{26} +1039.91 q^{27} -3649.61 q^{28} -8050.51 q^{29} +759.478 q^{31} -5442.33 q^{32} -11809.9 q^{33} -9958.72 q^{34} +20596.2 q^{36} -12780.7 q^{37} -13769.3 q^{38} +3894.79 q^{39} -18129.3 q^{41} +11969.4 q^{42} +19533.0 q^{43} -36631.6 q^{44} -114.978 q^{46} +22049.0 q^{47} +41448.1 q^{48} -14200.4 q^{49} +22561.3 q^{51} +12080.8 q^{52} +13861.1 q^{53} -10578.7 q^{54} +20506.7 q^{56} +31194.2 q^{57} +81895.5 q^{58} -18164.6 q^{59} +7623.02 q^{61} -7725.95 q^{62} -14710.1 q^{63} -2188.19 q^{64} +120138. q^{66} +3764.44 q^{67} +69980.4 q^{68} +260.481 q^{69} -75797.0 q^{71} -115728. q^{72} -86605.1 q^{73} +130015. q^{74} +96757.7 q^{76} +26162.8 q^{77} -39620.6 q^{78} +61535.6 q^{79} -46048.0 q^{81} +184425. q^{82} -46849.0 q^{83} -84109.2 q^{84} -198704. q^{86} -185533. q^{87} +205829. q^{88} +1237.02 q^{89} -8628.26 q^{91} +807.955 q^{92} +17503.0 q^{93} -224298. q^{94} -125425. q^{96} +57565.4 q^{97} +144457. q^{98} -147647. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} - 11 q^{3} + 91 q^{4} - 83 q^{6} + 12 q^{7} - 639 q^{8} + 562 q^{9}+O(q^{10}) \) 9 * q - 5 * q^2 - 11 * q^3 + 91 * q^4 - 83 * q^6 + 12 * q^7 - 639 * q^8 + 562 * q^9	\( 9 q - 5 q^{2} - 11 q^{3} + 91 q^{4} - 83 q^{6} + 12 q^{7} - 639 q^{8} + 562 q^{9} - 1422 q^{11} + 1567 q^{12} + 1521 q^{13} - 342 q^{14} - 1061 q^{16} + 648 q^{17} + 418 q^{18} - 408 q^{19} - 3912 q^{21} + 4345 q^{22} + 1839 q^{23} - 8469 q^{24} - 845 q^{26} - 7649 q^{27} - 2836 q^{28} - 8737 q^{29} + 748 q^{31} - 423 q^{32} - 356 q^{33} - 17789 q^{34} + 512 q^{36} - 15486 q^{37} - 3425 q^{38} - 1859 q^{39} - 28676 q^{41} + 6876 q^{42} + 28665 q^{43} - 30599 q^{44} - 12056 q^{46} - 29452 q^{47} + 64759 q^{48} - 40907 q^{49} - 31006 q^{51} + 15379 q^{52} + 75977 q^{53} - 102761 q^{54} - 23002 q^{56} - 38038 q^{57} + 142384 q^{58} - 88142 q^{59} + 28165 q^{61} - 137308 q^{62} + 41492 q^{63} - 100845 q^{64} + 42577 q^{66} - 94754 q^{67} + 89267 q^{68} - 181747 q^{69} - 70562 q^{71} - 263778 q^{72} + 60602 q^{73} - 135676 q^{74} + 46373 q^{76} - 140292 q^{77} - 14027 q^{78} - 164073 q^{79} - 69935 q^{81} - 72887 q^{82} - 22458 q^{83} - 345656 q^{84} - 294920 q^{86} - 87031 q^{87} + 430607 q^{88} - 252698 q^{89} + 2028 q^{91} - 237824 q^{92} + 56556 q^{93} - 501606 q^{94} - 319181 q^{96} + 137986 q^{97} + 378699 q^{98} - 757776 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 - 11 * q^3 + 91 * q^4 - 83 * q^6 + 12 * q^7 - 639 * q^8 + 562 * q^9 - 1422 * q^11 + 1567 * q^12 + 1521 * q^13 - 342 * q^14 - 1061 * q^16 + 648 * q^17 + 418 * q^18 - 408 * q^19 - 3912 * q^21 + 4345 * q^22 + 1839 * q^23 - 8469 * q^24 - 845 * q^26 - 7649 * q^27 - 2836 * q^28 - 8737 * q^29 + 748 * q^31 - 423 * q^32 - 356 * q^33 - 17789 * q^34 + 512 * q^36 - 15486 * q^37 - 3425 * q^38 - 1859 * q^39 - 28676 * q^41 + 6876 * q^42 + 28665 * q^43 - 30599 * q^44 - 12056 * q^46 - 29452 * q^47 + 64759 * q^48 - 40907 * q^49 - 31006 * q^51 + 15379 * q^52 + 75977 * q^53 - 102761 * q^54 - 23002 * q^56 - 38038 * q^57 + 142384 * q^58 - 88142 * q^59 + 28165 * q^61 - 137308 * q^62 + 41492 * q^63 - 100845 * q^64 + 42577 * q^66 - 94754 * q^67 + 89267 * q^68 - 181747 * q^69 - 70562 * q^71 - 263778 * q^72 + 60602 * q^73 - 135676 * q^74 + 46373 * q^76 - 140292 * q^77 - 14027 * q^78 - 164073 * q^79 - 69935 * q^81 - 72887 * q^82 - 22458 * q^83 - 345656 * q^84 - 294920 * q^86 - 87031 * q^87 + 430607 * q^88 - 252698 * q^89 + 2028 * q^91 - 237824 * q^92 + 56556 * q^93 - 501606 * q^94 - 319181 * q^96 + 137986 * q^97 + 378699 * q^98 - 757776 * q^99

Introduction

L-functions

Modular forms

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