LMFDB - Embedded newform 405.8.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,8,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("405.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	405.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:	$126.515935321$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 1221 x^{12} + 3034 x^{11} + 559330 x^{10} - 1662468 x^{9} - 119658132 x^{8} + \cdots + 11075674978368 $ x^14 - 2x^13 - 1221x^12 + 3034x^11 + 559330x^10 - 1662468x^9 - 119658132x^8 + 388985472x^7 + 12117415968x^6 - 35096050656x^5 - 526695898608x^4 + 532226461536x^3 + 9146880114336x^2 + 18387947059776*x + 11075674978368
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{28}\cdot 13 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-13.0124$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.2803 q^{2} -22.3144 q^{4} +125.000 q^{5} -799.488 q^{7} +1545.28 q^{8} +O(q^{10})$	\(q-10.2803 q^{2} -22.3144 q^{4} +125.000 q^{5} -799.488 q^{7} +1545.28 q^{8} -1285.04 q^{10} -2729.99 q^{11} +11007.2 q^{13} +8219.01 q^{14} -13029.8 q^{16} -17979.4 q^{17} +19039.3 q^{19} -2789.30 q^{20} +28065.3 q^{22} +17882.1 q^{23} +15625.0 q^{25} -113158. q^{26} +17840.1 q^{28} +11095.9 q^{29} +38835.1 q^{31} -63845.3 q^{32} +184834. q^{34} -99936.0 q^{35} -197036. q^{37} -195731. q^{38} +193161. q^{40} +44720.7 q^{41} +118881. q^{43} +60918.2 q^{44} -183834. q^{46} -278162. q^{47} -184363. q^{49} -160630. q^{50} -245619. q^{52} +1.34964e6 q^{53} -341249. q^{55} -1.23544e6 q^{56} -114070. q^{58} +2.90252e6 q^{59} -1.56452e6 q^{61} -399238. q^{62} +2.32417e6 q^{64} +1.37590e6 q^{65} -4.19081e6 q^{67} +401199. q^{68} +1.02738e6 q^{70} -2.05376e6 q^{71} -5.33839e6 q^{73} +2.02560e6 q^{74} -424850. q^{76} +2.18260e6 q^{77} +1.59709e6 q^{79} -1.62873e6 q^{80} -459745. q^{82} -2.80089e6 q^{83} -2.24742e6 q^{85} -1.22214e6 q^{86} -4.21862e6 q^{88} -7.86812e6 q^{89} -8.80012e6 q^{91} -399028. q^{92} +2.85961e6 q^{94} +2.37991e6 q^{95} +5.35120e6 q^{97} +1.89531e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 16 q^{2} + 714 q^{4} + 1750 q^{5} + 1538 q^{7} - 126 q^{8}+O(q^{10}) $ 14 * q + 16 * q^2 + 714 * q^4 + 1750 * q^5 + 1538 * q^7 - 126 * q^8	\( 14 q + 16 q^{2} + 714 q^{4} + 1750 q^{5} + 1538 q^{7} - 126 q^{8} + 2000 q^{10} + 10648 q^{11} + 17268 q^{13} + 9180 q^{14} + 34122 q^{16} + 3886 q^{17} + 22986 q^{19} + 89250 q^{20} - 49706 q^{22} + 155040 q^{23} + 218750 q^{25} + 7804 q^{26} + 256160 q^{28} + 195836 q^{29} + 92786 q^{31} + 157778 q^{32} + 787348 q^{34} + 192250 q^{35} + 876406 q^{37} + 329320 q^{38} - 15750 q^{40} + 795164 q^{41} + 730350 q^{43} + 2360876 q^{44} - 225654 q^{46} + 2687842 q^{47} + 1663586 q^{49} + 250000 q^{50} + 3875836 q^{52} + 2533750 q^{53} + 1331000 q^{55} + 2055276 q^{56} - 318934 q^{58} + 2283340 q^{59} + 2600400 q^{61} + 5702022 q^{62} + 474098 q^{64} + 2158500 q^{65} + 2422160 q^{67} + 1114364 q^{68} + 1147500 q^{70} + 6395324 q^{71} - 540774 q^{73} + 260516 q^{74} - 2417042 q^{76} - 1384890 q^{77} - 307384 q^{79} + 4265250 q^{80} - 14044738 q^{82} + 10991322 q^{83} + 485750 q^{85} + 2847712 q^{86} - 19226592 q^{88} + 2094000 q^{89} - 9496256 q^{91} + 39213378 q^{92} - 28132682 q^{94} + 2873250 q^{95} - 12859994 q^{97} + 28336478 q^{98}+O(q^{100}) \) 14 * q + 16 * q^2 + 714 * q^4 + 1750 * q^5 + 1538 * q^7 - 126 * q^8 + 2000 * q^10 + 10648 * q^11 + 17268 * q^13 + 9180 * q^14 + 34122 * q^16 + 3886 * q^17 + 22986 * q^19 + 89250 * q^20 - 49706 * q^22 + 155040 * q^23 + 218750 * q^25 + 7804 * q^26 + 256160 * q^28 + 195836 * q^29 + 92786 * q^31 + 157778 * q^32 + 787348 * q^34 + 192250 * q^35 + 876406 * q^37 + 329320 * q^38 - 15750 * q^40 + 795164 * q^41 + 730350 * q^43 + 2360876 * q^44 - 225654 * q^46 + 2687842 * q^47 + 1663586 * q^49 + 250000 * q^50 + 3875836 * q^52 + 2533750 * q^53 + 1331000 * q^55 + 2055276 * q^56 - 318934 * q^58 + 2283340 * q^59 + 2600400 * q^61 + 5702022 * q^62 + 474098 * q^64 + 2158500 * q^65 + 2422160 * q^67 + 1114364 * q^68 + 1147500 * q^70 + 6395324 * q^71 - 540774 * q^73 + 260516 * q^74 - 2417042 * q^76 - 1384890 * q^77 - 307384 * q^79 + 4265250 * q^80 - 14044738 * q^82 + 10991322 * q^83 + 485750 * q^85 + 2847712 * q^86 - 19226592 * q^88 + 2094000 * q^89 - 9496256 * q^91 + 39213378 * q^92 - 28132682 * q^94 + 2873250 * q^95 - 12859994 * q^97 + 28336478 * q^98

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 $

$2$	−10.2803	−0.908663	−0.454332	−	0.890833i	$-0.650122\pi$
$2$	−10.2803	−0.908663	−0.454332	+	0.890833i	$0.650122\pi$
$3$	0	0
$3$	0	0
$4$	−22.3144	−0.174331
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	−799.488	−0.880985	−0.440493	−	0.897756i	$-0.645196\pi$
$7$	−799.488	−0.880985	−0.440493	+	0.897756i	$0.645196\pi$
$8$	1545.28	1.06707
$9$	0	0
$10$	−1285.04	−0.406367
$11$	−2729.99	−0.618426	−0.309213	−	0.950993i	$-0.600066\pi$
$11$	−2729.99	−0.618426	−0.309213	+	0.950993i	$0.600066\pi$
$12$	0	0
$13$	11007.2	1.38955	0.694776	−	0.719226i	$-0.255503\pi$
$13$	11007.2	1.38955	0.694776	+	0.719226i	$0.255503\pi$
$14$	8219.01	0.800519
$15$	0	0
$16$	−13029.8	−0.795277
$17$	−17979.4	−0.887571	−0.443786	−	0.896133i	$-0.646365\pi$
$17$	−17979.4	−0.887571	−0.443786	+	0.896133i	$0.646365\pi$
$18$	0	0
$19$	19039.3	0.636815	0.318407	−	0.947954i	$-0.396852\pi$
$19$	19039.3	0.636815	0.318407	+	0.947954i	$0.396852\pi$
$20$	−2789.30	−0.0779633
$21$	0	0
$22$	28065.3	0.561940
$23$	17882.1	0.306458	0.153229	−	0.988191i	$-0.451033\pi$
$23$	17882.1	0.306458	0.153229	+	0.988191i	$0.451033\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	−113158.	−1.26264
$27$	0	0
$28$	17840.1	0.153583
$29$	11095.9	0.0844834	0.0422417	−	0.999107i	$-0.486550\pi$
$29$	11095.9	0.0844834	0.0422417	+	0.999107i	$0.486550\pi$
$30$	0	0
$31$	38835.1	0.234131	0.117065	−	0.993124i	$-0.462651\pi$
$31$	38835.1	0.234131	0.117065	+	0.993124i	$0.462651\pi$
$32$	−63845.3	−0.344432
$33$	0	0
$34$	184834.	0.806503
$35$	−99936.0	−0.393989
$36$	0	0
$37$	−197036.	−0.639498	−0.319749	−	0.947502i	$-0.603599\pi$
$37$	−197036.	−0.639498	−0.319749	+	0.947502i	$0.603599\pi$
$38$	−195731.	−0.578650
$39$	0	0
$40$	193161.	0.477209
$41$	44720.7	0.101336	0.0506682	−	0.998716i	$-0.483865\pi$
$41$	44720.7	0.101336	0.0506682	+	0.998716i	$0.483865\pi$
$42$	0	0
$43$	118881.	0.228020	0.114010	−	0.993480i	$-0.463630\pi$
$43$	118881.	0.228020	0.114010	+	0.993480i	$0.463630\pi$
$44$	60918.2	0.107811
$45$	0	0
$46$	−183834.	−0.278467
$47$	−278162.	−0.390801	−0.195401	−	0.980724i	$-0.562601\pi$
$47$	−278162.	−0.390801	−0.195401	+	0.980724i	$0.562601\pi$
$48$	0	0
$49$	−184363.	−0.223865
$50$	−160630.	−0.181733
$51$	0	0
$52$	−245619.	−0.242243
$53$	1.34964e6	1.24524	0.622620	−	0.782524i	$-0.286068\pi$
$53$	1.34964e6	1.24524	0.622620	+	0.782524i	$0.286068\pi$
$54$	0	0
$55$	−341249.	−0.276568
$56$	−1.23544e6	−0.940074
$57$	0	0
$58$	−114070.	−0.0767670
$59$	2.90252e6	1.83989	0.919947	−	0.392042i	$-0.128231\pi$
$59$	2.90252e6	1.83989	0.919947	+	0.392042i	$0.128231\pi$
$60$	0	0
$61$	−1.56452e6	−0.882525	−0.441263	−	0.897378i	$-0.645469\pi$
$61$	−1.56452e6	−0.882525	−0.441263	+	0.897378i	$0.645469\pi$
$62$	−399238.	−0.212746
$63$	0	0
$64$	2.32417e6	1.10825
$65$	1.37590e6	0.621427
$66$	0	0
$67$	−4.19081e6	−1.70230	−0.851151	−	0.524921i	$-0.824095\pi$
$67$	−4.19081e6	−1.70230	−0.851151	+	0.524921i	$0.824095\pi$
$68$	401199.	0.154731
$69$	0	0
$70$	1.02738e6	0.358003
$71$	−2.05376e6	−0.680996	−0.340498	−	0.940245i	$-0.610596\pi$
$71$	−2.05376e6	−0.680996	−0.340498	+	0.940245i	$0.610596\pi$
$72$	0	0
$73$	−5.33839e6	−1.60613	−0.803064	−	0.595893i	$-0.796798\pi$
$73$	−5.33839e6	−1.60613	−0.803064	+	0.595893i	$0.796798\pi$
$74$	2.02560e6	0.581089
$75$	0	0
$76$	−424850.	−0.111017
$77$	2.18260e6	0.544824
$78$	0	0
$79$	1.59709e6	0.364447	0.182223	−	0.983257i	$-0.441671\pi$
$79$	1.59709e6	0.364447	0.182223	+	0.983257i	$0.441671\pi$
$80$	−1.62873e6	−0.355659
$81$	0	0
$82$	−459745.	−0.0920806
$83$	−2.80089e6	−0.537678	−0.268839	−	0.963185i	$-0.586640\pi$
$83$	−2.80089e6	−0.537678	−0.268839	+	0.963185i	$0.586640\pi$
$84$	0	0
$85$	−2.24742e6	−0.396934
$86$	−1.22214e6	−0.207194
$87$	0	0
$88$	−4.21862e6	−0.659904
$89$	−7.86812e6	−1.18306	−0.591529	−	0.806284i	$-0.701475\pi$
$89$	−7.86812e6	−1.18306	−0.591529	+	0.806284i	$0.701475\pi$
$90$	0	0
$91$	−8.80012e6	−1.22418
$92$	−399028.	−0.0534252
$93$	0	0
$94$	2.85961e6	0.355107
$95$	2.37991e6	0.284792
$96$	0	0
$97$	5.35120e6	0.595319	0.297660	−	0.954672i	$-0.403794\pi$
$97$	5.35120e6	0.595319	0.297660	+	0.954672i	$0.403794\pi$
$98$	1.89531e6	0.203418
$99$	0	0
$100$	−348663.	−0.0348663
$101$	3.19582e6	0.308644	0.154322	−	0.988021i	$-0.450681\pi$
$101$	3.19582e6	0.308644	0.154322	+	0.988021i	$0.450681\pi$
$102$	0	0
$103$	9.36533e6	0.844487	0.422243	−	0.906483i	$-0.361243\pi$
$103$	9.36533e6	0.844487	0.422243	+	0.906483i	$0.361243\pi$
$104$	1.70093e7	1.48275
$105$	0	0
$106$	−1.38748e7	−1.13150
$107$	−6.41322e6	−0.506096	−0.253048	−	0.967454i	$-0.581433\pi$
$107$	−6.41322e6	−0.506096	−0.253048	+	0.967454i	$0.581433\pi$
$108$	0	0
$109$	2.84304e6	0.210277	0.105138	−	0.994458i	$-0.466471\pi$
$109$	2.84304e6	0.210277	0.105138	+	0.994458i	$0.466471\pi$
$110$	3.50816e6	0.251307
$111$	0	0
$112$	1.04172e7	0.700628
$113$	1.50045e7	0.978244	0.489122	−	0.872215i	$-0.337317\pi$
$113$	1.50045e7	0.978244	0.489122	+	0.872215i	$0.337317\pi$
$114$	0	0
$115$	2.23526e6	0.137052
$116$	−247599.	−0.0147281
$117$	0	0
$118$	−2.98389e7	−1.67184
$119$	1.43743e7	0.781937
$120$	0	0
$121$	−1.20343e7	−0.617550
$122$	1.60838e7	0.801918
$123$	0	0
$124$	−866581.	−0.0408163
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	−2.22950e7	−0.965817	−0.482909	−	0.875671i	$-0.660420\pi$
$127$	−2.22950e7	−0.965817	−0.482909	+	0.875671i	$0.660420\pi$
$128$	−1.57211e7	−0.662594
$129$	0	0
$130$	−1.41447e7	−0.564668
$131$	1.52425e6	0.0592389	0.0296195	−	0.999561i	$-0.490570\pi$
$131$	1.52425e6	0.0592389	0.0296195	+	0.999561i	$0.490570\pi$
$132$	0	0
$133$	−1.52217e7	−0.561024
$134$	4.30830e7	1.54682
$135$	0	0
$136$	−2.77832e7	−0.947102
$137$	−5.47844e7	−1.82027	−0.910133	−	0.414316i	$-0.864021\pi$
$137$	−5.47844e7	−1.82027	−0.910133	+	0.414316i	$0.864021\pi$
$138$	0	0
$139$	−4.79747e7	−1.51517	−0.757583	−	0.652739i	$-0.773620\pi$
$139$	−4.79747e7	−1.51517	−0.757583	+	0.652739i	$0.773620\pi$
$140$	2.23001e6	0.0686845
$141$	0	0
$142$	2.11133e7	0.618796
$143$	−3.00496e7	−0.859335
$144$	0	0
$145$	1.38699e6	0.0377821
$146$	5.48805e7	1.45943
$147$	0	0
$148$	4.39674e6	0.111485
$149$	4.26155e7	1.05540	0.527698	−	0.849432i	$-0.323055\pi$
$149$	4.26155e7	1.05540	0.527698	+	0.849432i	$0.323055\pi$
$150$	0	0
$151$	5.62032e7	1.32844	0.664220	−	0.747537i	$-0.268764\pi$
$151$	5.62032e7	1.32844	0.664220	+	0.747537i	$0.268764\pi$
$152$	2.94211e7	0.679527
$153$	0	0
$154$	−2.24379e7	−0.495061
$155$	4.85438e6	0.104706
$156$	0	0
$157$	7.24291e7	1.49370	0.746852	−	0.664991i	$-0.231565\pi$
$157$	7.24291e7	1.49370	0.746852	+	0.664991i	$0.231565\pi$
$158$	−1.64186e7	−0.331159
$159$	0	0
$160$	−7.98066e6	−0.154035
$161$	−1.42965e7	−0.269985
$162$	0	0
$163$	2.48294e7	0.449066	0.224533	−	0.974467i	$-0.427914\pi$
$163$	2.48294e7	0.449066	0.224533	+	0.974467i	$0.427914\pi$
$164$	−997916.	−0.0176661
$165$	0	0
$166$	2.87941e7	0.488568
$167$	7.58424e7	1.26010	0.630049	−	0.776555i	$-0.283035\pi$
$167$	7.58424e7	1.26010	0.630049	+	0.776555i	$0.283035\pi$
$168$	0	0
$169$	5.84099e7	0.930857
$170$	2.31043e7	0.360679
$171$	0	0
$172$	−2.65277e6	−0.0397511
$173$	6.09532e7	0.895025	0.447512	−	0.894278i	$-0.352310\pi$
$173$	6.09532e7	0.895025	0.447512	+	0.894278i	$0.352310\pi$
$174$	0	0
$175$	−1.24920e7	−0.176197
$176$	3.55713e7	0.491820
$177$	0	0
$178$	8.08871e7	1.07500
$179$	8.09700e7	1.05521	0.527605	−	0.849490i	$-0.323090\pi$
$179$	8.09700e7	1.05521	0.527605	+	0.849490i	$0.323090\pi$
$180$	0	0
$181$	−1.02786e8	−1.28843	−0.644213	−	0.764847i	$-0.722815\pi$
$181$	−1.02786e8	−1.28843	−0.644213	+	0.764847i	$0.722815\pi$
$182$	9.04683e7	1.11236
$183$	0	0
$184$	2.76329e7	0.327012
$185$	−2.46295e7	−0.285992
$186$	0	0
$187$	4.90836e7	0.548897
$188$	6.20703e6	0.0681289
$189$	0	0
$190$	−2.44663e7	−0.258780
$191$	9.44313e7	0.980616	0.490308	−	0.871549i	$-0.336884\pi$
$191$	9.44313e7	0.980616	0.490308	+	0.871549i	$0.336884\pi$
$192$	0	0
$193$	1.52729e8	1.52922	0.764611	−	0.644492i	$-0.222931\pi$
$193$	1.52729e8	1.52922	0.764611	+	0.644492i	$0.222931\pi$
$194$	−5.50122e7	−0.540945
$195$	0	0
$196$	4.11394e6	0.0390267
$197$	1.12659e8	1.04987	0.524935	−	0.851142i	$-0.324090\pi$
$197$	1.12659e8	1.04987	0.524935	+	0.851142i	$0.324090\pi$
$198$	0	0
$199$	−1.31546e8	−1.18329	−0.591646	−	0.806198i	$-0.701522\pi$
$199$	−1.31546e8	−1.18329	−0.591646	+	0.806198i	$0.701522\pi$
$200$	2.41451e7	0.213414
$201$	0	0
$202$	−3.28542e7	−0.280453
$203$	−8.87107e6	−0.0744286
$204$	0	0
$205$	5.59009e6	0.0453190
$206$	−9.62789e7	−0.767354
$207$	0	0
$208$	−1.43422e8	−1.10508
$209$	−5.19771e7	−0.393822
$210$	0	0
$211$	7.66919e7	0.562032	0.281016	−	0.959703i	$-0.409329\pi$
$211$	7.66919e7	0.562032	0.281016	+	0.959703i	$0.409329\pi$
$212$	−3.01165e7	−0.217084
$213$	0	0
$214$	6.59301e7	0.459871
$215$	1.48602e7	0.101974
$216$	0	0
$217$	−3.10482e7	−0.206266
$218$	−2.92275e7	−0.191071
$219$	0	0
$220$	7.61478e6	0.0482145
$221$	−1.97902e8	−1.23333
$222$	0	0
$223$	1.32370e8	0.799321	0.399660	−	0.916663i	$-0.369128\pi$
$223$	1.32370e8	0.799321	0.399660	+	0.916663i	$0.369128\pi$
$224$	5.10435e7	0.303440
$225$	0	0
$226$	−1.54252e8	−0.888895
$227$	2.23606e8	1.26880	0.634398	−	0.773006i	$-0.281248\pi$
$227$	2.23606e8	1.26880	0.634398	+	0.773006i	$0.281248\pi$
$228$	0	0
$229$	1.07490e8	0.591486	0.295743	−	0.955268i	$-0.404433\pi$
$229$	1.07490e8	0.591486	0.295743	+	0.955268i	$0.404433\pi$
$230$	−2.29792e7	−0.124534
$231$	0	0
$232$	1.71464e7	0.0901498
$233$	2.97585e8	1.54122	0.770612	−	0.637305i	$-0.219951\pi$
$233$	2.97585e8	1.54122	0.770612	+	0.637305i	$0.219951\pi$
$234$	0	0
$235$	−3.47703e7	−0.174772
$236$	−6.47680e7	−0.320751
$237$	0	0
$238$	−1.47773e8	−0.710517
$239$	5.70260e7	0.270197	0.135098	−	0.990832i	$-0.456865\pi$
$239$	5.70260e7	0.270197	0.135098	+	0.990832i	$0.456865\pi$
$240$	0	0
$241$	2.70084e8	1.24291	0.621455	−	0.783450i	$-0.286542\pi$
$241$	2.70084e8	1.24291	0.621455	+	0.783450i	$0.286542\pi$
$242$	1.23717e8	0.561145
$243$	0	0
$244$	3.49114e7	0.153852
$245$	−2.30453e7	−0.100116
$246$	0	0
$247$	2.09569e8	0.884888
$248$	6.00112e7	0.249834
$249$	0	0
$250$	−2.00788e7	−0.0812733
$251$	4.49250e8	1.79321	0.896603	−	0.442834i	$-0.146027\pi$
$251$	4.49250e8	1.79321	0.896603	+	0.442834i	$0.146027\pi$
$252$	0	0
$253$	−4.88180e7	−0.189521
$254$	2.29201e8	0.877602
$255$	0	0
$256$	−1.35876e8	−0.506176
$257$	1.20727e8	0.443648	0.221824	−	0.975087i	$-0.428799\pi$
$257$	1.20727e8	0.443648	0.221824	+	0.975087i	$0.428799\pi$
$258$	0	0
$259$	1.57528e8	0.563389
$260$	−3.07024e7	−0.108334
$261$	0	0
$262$	−1.56698e7	−0.0538282
$263$	2.86117e8	0.969836	0.484918	−	0.874560i	$-0.338849\pi$
$263$	2.86117e8	0.969836	0.484918	+	0.874560i	$0.338849\pi$
$264$	0	0
$265$	1.68705e8	0.556888
$266$	1.56484e8	0.509782
$267$	0	0
$268$	9.35155e7	0.296764
$269$	−1.27160e8	−0.398305	−0.199153	−	0.979968i	$-0.563819\pi$
$269$	−1.27160e8	−0.398305	−0.199153	+	0.979968i	$0.563819\pi$
$270$	0	0
$271$	3.50511e8	1.06982	0.534908	−	0.844910i	$-0.320346\pi$
$271$	3.50511e8	1.06982	0.534908	+	0.844910i	$0.320346\pi$
$272$	2.34268e8	0.705865
$273$	0	0
$274$	5.63203e8	1.65401
$275$	−4.26562e7	−0.123685
$276$	0	0
$277$	−1.92205e8	−0.543356	−0.271678	−	0.962388i	$-0.587579\pi$
$277$	−1.92205e8	−0.543356	−0.271678	+	0.962388i	$0.587579\pi$
$278$	4.93196e8	1.37678
$279$	0	0
$280$	−1.54430e8	−0.420414
$281$	−1.43831e8	−0.386704	−0.193352	−	0.981129i	$-0.561936\pi$
$281$	−1.43831e8	−0.386704	−0.193352	+	0.981129i	$0.561936\pi$
$282$	0	0
$283$	−3.57686e8	−0.938101	−0.469050	−	0.883171i	$-0.655404\pi$
$283$	−3.57686e8	−0.938101	−0.469050	+	0.883171i	$0.655404\pi$
$284$	4.58283e7	0.118719
$285$	0	0
$286$	3.08920e8	0.780846
$287$	−3.57537e7	−0.0892758
$288$	0	0
$289$	−8.70810e7	−0.212217
$290$	−1.42588e7	−0.0343312
$291$	0	0
$292$	1.19123e8	0.279998
$293$	5.33219e8	1.23842	0.619212	−	0.785224i	$-0.287452\pi$
$293$	5.33219e8	1.23842	0.619212	+	0.785224i	$0.287452\pi$
$294$	0	0
$295$	3.62815e8	0.822826
$296$	−3.04477e8	−0.682390
$297$	0	0
$298$	−4.38102e8	−0.958999
$299$	1.96832e8	0.425839
$300$	0	0
$301$	−9.50441e7	−0.200883
$302$	−5.77789e8	−1.20710
$303$	0	0
$304$	−2.48079e8	−0.506444
$305$	−1.95565e8	−0.394677
$306$	0	0
$307$	−2.72515e8	−0.537533	−0.268767	−	0.963205i	$-0.586616\pi$
$307$	−2.72515e8	−0.537533	−0.268767	+	0.963205i	$0.586616\pi$
$308$	−4.87034e7	−0.0949798
$309$	0	0
$310$	−4.99048e7	−0.0951428
$311$	3.49075e8	0.658048	0.329024	−	0.944322i	$-0.393280\pi$
$311$	3.49075e8	0.658048	0.329024	+	0.944322i	$0.393280\pi$
$312$	0	0
$313$	−3.56419e8	−0.656986	−0.328493	−	0.944506i	$-0.606541\pi$
$313$	−3.56419e8	−0.656986	−0.328493	+	0.944506i	$0.606541\pi$
$314$	−7.44596e8	−1.35727
$315$	0	0
$316$	−3.56381e7	−0.0635345
$317$	−4.53312e8	−0.799263	−0.399631	−	0.916676i	$-0.630862\pi$
$317$	−4.53312e8	−0.799263	−0.399631	+	0.916676i	$0.630862\pi$
$318$	0	0
$319$	−3.02919e7	−0.0522467
$320$	2.90521e8	0.495625
$321$	0	0
$322$	1.46973e8	0.245325
$323$	−3.42314e8	−0.565218
$324$	0	0
$325$	1.71987e8	0.277911
$326$	−2.55255e8	−0.408050
$327$	0	0
$328$	6.91062e7	0.108133
$329$	2.22387e8	0.344290
$330$	0	0
$331$	7.55690e8	1.14537	0.572685	−	0.819776i	$-0.305902\pi$
$331$	7.55690e8	1.14537	0.572685	+	0.819776i	$0.305902\pi$
$332$	6.25001e7	0.0937341
$333$	0	0
$334$	−7.79687e8	−1.14501
$335$	−5.23852e8	−0.761292
$336$	0	0
$337$	8.13087e7	0.115726	0.0578632	−	0.998325i	$-0.481571\pi$
$337$	8.13087e7	0.115726	0.0578632	+	0.998325i	$0.481571\pi$
$338$	−6.00474e8	−0.845835
$339$	0	0
$340$	5.01499e7	0.0691980
$341$	−1.06020e8	−0.144792
$342$	0	0
$343$	8.05808e8	1.07821
$344$	1.83705e8	0.243314
$345$	0	0
$346$	−6.26620e8	−0.813276
$347$	9.70381e8	1.24678	0.623389	−	0.781912i	$-0.285755\pi$
$347$	9.70381e8	1.24678	0.623389	+	0.781912i	$0.285755\pi$
$348$	0	0
$349$	−1.38311e9	−1.74167	−0.870836	−	0.491573i	$-0.836422\pi$
$349$	−1.38311e9	−1.74167	−0.870836	+	0.491573i	$0.836422\pi$
$350$	1.28422e8	0.160104
$351$	0	0
$352$	1.74297e8	0.213006
$353$	−7.88869e8	−0.954539	−0.477269	−	0.878757i	$-0.658373\pi$
$353$	−7.88869e8	−0.954539	−0.477269	+	0.878757i	$0.658373\pi$
$354$	0	0
$355$	−2.56719e8	−0.304551
$356$	1.75572e8	0.206244
$357$	0	0
$358$	−8.32400e8	−0.958830
$359$	6.16986e8	0.703792	0.351896	−	0.936039i	$-0.385537\pi$
$359$	6.16986e8	0.703792	0.351896	+	0.936039i	$0.385537\pi$
$360$	0	0
$361$	−5.31377e8	−0.594467
$362$	1.05668e9	1.17074
$363$	0	0
$364$	1.96369e8	0.213412
$365$	−6.67298e8	−0.718282
$366$	0	0
$367$	−1.41429e9	−1.49350	−0.746752	−	0.665102i	$-0.768388\pi$
$367$	−1.41429e9	−1.49350	−0.746752	+	0.665102i	$0.768388\pi$
$368$	−2.33000e8	−0.243719
$369$	0	0
$370$	2.53200e8	0.259871
$371$	−1.07902e9	−1.09704
$372$	0	0
$373$	−3.86163e8	−0.385291	−0.192646	−	0.981268i	$-0.561707\pi$
$373$	−3.86163e8	−0.385291	−0.192646	+	0.981268i	$0.561707\pi$
$374$	−5.04596e8	−0.498762
$375$	0	0
$376$	−4.29840e8	−0.417013
$377$	1.22135e8	0.117394
$378$	0	0
$379$	1.36566e8	0.128856	0.0644281	−	0.997922i	$-0.479478\pi$
$379$	1.36566e8	0.128856	0.0644281	+	0.997922i	$0.479478\pi$
$380$	−5.31063e7	−0.0496482
$381$	0	0
$382$	−9.70787e8	−0.891050
$383$	3.37957e8	0.307373	0.153686	−	0.988120i	$-0.450885\pi$
$383$	3.37957e8	0.307373	0.153686	+	0.988120i	$0.450885\pi$
$384$	0	0
$385$	2.72825e8	0.243653
$386$	−1.57011e9	−1.38955
$387$	0	0
$388$	−1.19409e8	−0.103783
$389$	−1.06230e9	−0.915007	−0.457503	−	0.889208i	$-0.651256\pi$
$389$	−1.06230e9	−0.915007	−0.457503	+	0.889208i	$0.651256\pi$
$390$	0	0
$391$	−3.21508e8	−0.272003
$392$	−2.84893e8	−0.238880
$393$	0	0
$394$	−1.15818e9	−0.953978
$395$	1.99636e8	0.162986
$396$	0	0
$397$	8.05226e8	0.645879	0.322939	−	0.946420i	$-0.395329\pi$
$397$	8.05226e8	0.645879	0.322939	+	0.946420i	$0.395329\pi$
$398$	1.35234e9	1.07521
$399$	0	0
$400$	−2.03591e8	−0.159055
$401$	−5.83808e8	−0.452131	−0.226066	−	0.974112i	$-0.572586\pi$
$401$	−5.83808e8	−0.452131	−0.226066	+	0.974112i	$0.572586\pi$
$402$	0	0
$403$	4.27465e8	0.325337
$404$	−7.13128e7	−0.0538063
$405$	0	0
$406$	9.11977e7	0.0676306
$407$	5.37907e8	0.395482
$408$	0	0
$409$	−2.34604e9	−1.69552	−0.847762	−	0.530377i	$-0.822050\pi$
$409$	−2.34604e9	−1.69552	−0.847762	+	0.530377i	$0.822050\pi$
$410$	−5.74681e7	−0.0411797
$411$	0	0
$412$	−2.08982e8	−0.147220
$413$	−2.32053e9	−1.62092
$414$	0	0
$415$	−3.50111e8	−0.240457
$416$	−7.02758e8	−0.478607
$417$	0	0
$418$	5.34343e8	0.357852
$419$	2.45737e9	1.63201	0.816003	−	0.578048i	$-0.196185\pi$
$419$	2.45737e9	1.63201	0.816003	+	0.578048i	$0.196185\pi$
$420$	0	0
$421$	2.72181e9	1.77775	0.888875	−	0.458151i	$-0.151488\pi$
$421$	2.72181e9	1.77775	0.888875	+	0.458151i	$0.151488\pi$
$422$	−7.88420e8	−0.510698
$423$	0	0
$424$	2.08558e9	1.32876
$425$	−2.80928e8	−0.177514
$426$	0	0
$427$	1.25082e9	0.777492
$428$	1.43107e8	0.0882284
$429$	0	0
$430$	−1.52768e8	−0.0926599
$431$	−1.54477e9	−0.929380	−0.464690	−	0.885474i	$-0.653834\pi$
$431$	−1.54477e9	−0.929380	−0.464690	+	0.885474i	$0.653834\pi$
$432$	0	0
$433$	1.97556e9	1.16945	0.584725	−	0.811232i	$-0.301202\pi$
$433$	1.97556e9	1.16945	0.584725	+	0.811232i	$0.301202\pi$
$434$	3.19186e8	0.187426
$435$	0	0
$436$	−6.34408e7	−0.0366578
$437$	3.40462e8	0.195157
$438$	0	0
$439$	4.13297e8	0.233150	0.116575	−	0.993182i	$-0.462808\pi$
$439$	4.13297e8	0.233150	0.116575	+	0.993182i	$0.462808\pi$
$440$	−5.27327e8	−0.295118
$441$	0	0
$442$	2.03451e9	1.12068
$443$	−1.67006e9	−0.912680	−0.456340	−	0.889806i	$-0.650840\pi$
$443$	−1.67006e9	−0.912680	−0.456340	+	0.889806i	$0.650840\pi$
$444$	0	0
$445$	−9.83515e8	−0.529080
$446$	−1.36080e9	−0.726313
$447$	0	0
$448$	−1.85814e9	−0.976352
$449$	1.92534e9	1.00379	0.501897	−	0.864927i	$-0.332636\pi$
$449$	1.92534e9	1.00379	0.501897	+	0.864927i	$0.332636\pi$
$450$	0	0
$451$	−1.22087e8	−0.0626690
$452$	−3.34817e8	−0.170539
$453$	0	0
$454$	−2.29874e9	−1.15291
$455$	−1.10001e9	−0.547468
$456$	0	0
$457$	−2.45414e9	−1.20280	−0.601398	−	0.798950i	$-0.705389\pi$
$457$	−2.45414e9	−1.20280	−0.601398	+	0.798950i	$0.705389\pi$
$458$	−1.10504e9	−0.537462
$459$	0	0
$460$	−4.98785e7	−0.0238925
$461$	−2.03894e9	−0.969287	−0.484643	−	0.874712i	$-0.661051\pi$
$461$	−2.03894e9	−0.969287	−0.484643	+	0.874712i	$0.661051\pi$
$462$	0	0
$463$	1.42445e9	0.666980	0.333490	−	0.942754i	$-0.391774\pi$
$463$	1.42445e9	0.666980	0.333490	+	0.942754i	$0.391774\pi$
$464$	−1.44578e8	−0.0671877
$465$	0	0
$466$	−3.05928e9	−1.40045
$467$	2.41691e9	1.09812	0.549062	−	0.835782i	$-0.314985\pi$
$467$	2.41691e9	1.09812	0.549062	+	0.835782i	$0.314985\pi$
$468$	0	0
$469$	3.35050e9	1.49970
$470$	3.57451e8	0.158809
$471$	0	0
$472$	4.48522e9	1.96330
$473$	−3.24545e8	−0.141014
$474$	0	0
$475$	2.97489e8	0.127363
$476$	−3.20754e8	−0.136316
$477$	0	0
$478$	−5.86247e8	−0.245518
$479$	−1.17110e9	−0.486877	−0.243438	−	0.969916i	$-0.578275\pi$
$479$	−1.17110e9	−0.486877	−0.243438	+	0.969916i	$0.578275\pi$
$480$	0	0
$481$	−2.16881e9	−0.888617
$482$	−2.77656e9	−1.12939
$483$	0	0
$484$	2.68538e8	0.107658
$485$	6.68900e8	0.266235
$486$	0	0
$487$	3.96670e9	1.55625	0.778123	−	0.628112i	$-0.216172\pi$
$487$	3.96670e9	1.55625	0.778123	+	0.628112i	$0.216172\pi$
$488$	−2.41763e9	−0.941717
$489$	0	0
$490$	2.36914e8	0.0909713
$491$	−4.78800e9	−1.82544	−0.912722	−	0.408581i	$-0.866024\pi$
$491$	−4.78800e9	−1.82544	−0.912722	+	0.408581i	$0.866024\pi$
$492$	0	0
$493$	−1.99498e8	−0.0749850
$494$	−2.15444e9	−0.804065
$495$	0	0
$496$	−5.06014e8	−0.186199
$497$	1.64195e9	0.599947
$498$	0	0
$499$	2.98945e9	1.07706	0.538530	−	0.842606i	$-0.318980\pi$
$499$	2.98945e9	1.07706	0.538530	+	0.842606i	$0.318980\pi$
$500$	−4.35828e7	−0.0155927
$501$	0	0
$502$	−4.61845e9	−1.62942
$503$	2.07517e9	0.727051	0.363526	−	0.931584i	$-0.381573\pi$
$503$	2.07517e9	0.727051	0.363526	+	0.931584i	$0.381573\pi$
$504$	0	0
$505$	3.99478e8	0.138030
$506$	5.01866e8	0.172211
$507$	0	0
$508$	4.97500e8	0.168372
$509$	−2.35244e8	−0.0790689	−0.0395344	−	0.999218i	$-0.512587\pi$
$509$	−2.35244e8	−0.0790689	−0.0395344	+	0.999218i	$0.512587\pi$
$510$	0	0
$511$	4.26797e9	1.41497
$512$	3.40915e9	1.12254
$513$	0	0
$514$	−1.24112e9	−0.403126
$515$	1.17067e9	0.377666
$516$	0	0
$517$	7.59382e8	0.241681
$518$	−1.61944e9	−0.511930
$519$	0	0
$520$	2.12616e9	0.663107
$521$	−3.97644e9	−1.23186	−0.615931	−	0.787800i	$-0.711220\pi$
$521$	−3.97644e9	−1.23186	−0.615931	+	0.787800i	$0.711220\pi$
$522$	0	0
$523$	−8.58612e8	−0.262447	−0.131223	−	0.991353i	$-0.541891\pi$
$523$	−8.58612e8	−0.262447	−0.131223	+	0.991353i	$0.541891\pi$
$524$	−3.40128e7	−0.0103272
$525$	0	0
$526$	−2.94138e9	−0.881254
$527$	−6.98230e8	−0.207808
$528$	0	0
$529$	−3.08506e9	−0.906084
$530$	−1.73435e9	−0.506024
$531$	0	0
$532$	3.39663e8	0.0978041
$533$	4.92250e8	0.140812
$534$	0	0
$535$	−8.01653e8	−0.226333
$536$	−6.47600e9	−1.81648
$537$	0	0
$538$	1.30724e9	0.361925
$539$	5.03309e8	0.138444
$540$	0	0
$541$	3.90182e9	1.05944	0.529721	−	0.848172i	$-0.322297\pi$
$541$	3.90182e9	1.05944	0.529721	+	0.848172i	$0.322297\pi$
$542$	−3.60338e9	−0.972103
$543$	0	0
$544$	1.14790e9	0.305708
$545$	3.55381e8	0.0940385
$546$	0	0
$547$	2.98232e9	0.779109	0.389555	−	0.921003i	$-0.372629\pi$
$547$	2.98232e9	0.779109	0.389555	+	0.921003i	$0.372629\pi$
$548$	1.22248e9	0.317329
$549$	0	0
$550$	4.38520e8	0.112388
$551$	2.11259e8	0.0538003
$552$	0	0
$553$	−1.27685e9	−0.321072
$554$	1.97593e9	0.493728
$555$	0	0
$556$	1.07053e9	0.264141
$557$	4.41667e9	1.08293	0.541467	−	0.840722i	$-0.317869\pi$
$557$	4.41667e9	1.08293	0.541467	+	0.840722i	$0.317869\pi$
$558$	0	0
$559$	1.30855e9	0.316846
$560$	1.30215e9	0.313330
$561$	0	0
$562$	1.47863e9	0.351384
$563$	6.58887e8	0.155608	0.0778039	−	0.996969i	$-0.475209\pi$
$563$	6.58887e8	0.155608	0.0778039	+	0.996969i	$0.475209\pi$
$564$	0	0
$565$	1.87556e9	0.437484
$566$	3.67714e9	0.852418
$567$	0	0
$568$	−3.17364e9	−0.726671
$569$	2.82336e9	0.642500	0.321250	−	0.946994i	$-0.395897\pi$
$569$	2.82336e9	0.642500	0.321250	+	0.946994i	$0.395897\pi$
$570$	0	0
$571$	−8.23731e9	−1.85165	−0.925825	−	0.377953i	$-0.876628\pi$
$571$	−8.23731e9	−1.85165	−0.925825	+	0.377953i	$0.876628\pi$
$572$	6.70539e8	0.149809
$573$	0	0
$574$	3.67560e8	0.0811217
$575$	2.79407e8	0.0612915
$576$	0	0
$577$	7.64128e9	1.65596	0.827982	−	0.560755i	$-0.189489\pi$
$577$	7.64128e9	1.65596	0.827982	+	0.560755i	$0.189489\pi$
$578$	8.95223e8	0.192834
$579$	0	0
$580$	−3.09499e7	−0.00658661
$581$	2.23927e9	0.473686
$582$	0	0
$583$	−3.68452e9	−0.770088
$584$	−8.24933e9	−1.71385
$585$	0	0
$586$	−5.48168e9	−1.12531
$587$	3.82718e9	0.780991	0.390495	−	0.920605i	$-0.372304\pi$
$587$	3.82718e9	0.780991	0.390495	+	0.920605i	$0.372304\pi$
$588$	0	0
$589$	7.39392e8	0.149098
$590$	−3.72986e9	−0.747672
$591$	0	0
$592$	2.56734e9	0.508578
$593$	−7.93322e8	−0.156228	−0.0781138	−	0.996944i	$-0.524890\pi$
$593$	−7.93322e8	−0.156228	−0.0781138	+	0.996944i	$0.524890\pi$
$594$	0	0
$595$	1.79679e9	0.349693
$596$	−9.50939e8	−0.183989
$597$	0	0
$598$	−2.02350e9	−0.386944
$599$	3.64109e9	0.692210	0.346105	−	0.938196i	$-0.387504\pi$
$599$	3.64109e9	0.692210	0.346105	+	0.938196i	$0.387504\pi$
$600$	0	0
$601$	3.90961e9	0.734636	0.367318	−	0.930095i	$-0.380276\pi$
$601$	3.90961e9	0.734636	0.367318	+	0.930095i	$0.380276\pi$
$602$	9.77087e8	0.182535
$603$	0	0
$604$	−1.25414e9	−0.231589
$605$	−1.50429e9	−0.276177
$606$	0	0
$607$	8.98606e9	1.63083	0.815415	−	0.578877i	$-0.196509\pi$
$607$	8.98606e9	1.63083	0.815415	+	0.578877i	$0.196509\pi$
$608$	−1.21557e9	−0.219340
$609$	0	0
$610$	2.01048e9	0.358629
$611$	−3.06179e9	−0.543039
$612$	0	0
$613$	7.52987e9	1.32031	0.660155	−	0.751130i	$-0.270491\pi$
$613$	7.52987e9	1.32031	0.660155	+	0.751130i	$0.270491\pi$
$614$	2.80154e9	0.488437
$615$	0	0
$616$	3.37273e9	0.581366
$617$	1.27221e9	0.218052	0.109026	−	0.994039i	$-0.465227\pi$
$617$	1.27221e9	0.218052	0.109026	+	0.994039i	$0.465227\pi$
$618$	0	0
$619$	−5.08374e9	−0.861521	−0.430760	−	0.902466i	$-0.641755\pi$
$619$	−5.08374e9	−0.861521	−0.430760	+	0.902466i	$0.641755\pi$
$620$	−1.08323e8	−0.0182536
$621$	0	0
$622$	−3.58861e9	−0.597944
$623$	6.29047e9	1.04226
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	3.66411e9	0.596979
$627$	0	0
$628$	−1.61621e9	−0.260399
$629$	3.54258e9	0.567600
$630$	0	0
$631$	−8.66412e9	−1.37285	−0.686423	−	0.727203i	$-0.740820\pi$
$631$	−8.66412e9	−1.37285	−0.686423	+	0.727203i	$0.740820\pi$
$632$	2.46796e9	0.388891
$633$	0	0
$634$	4.66020e9	0.726261
$635$	−2.78688e9	−0.431927
$636$	0	0
$637$	−2.02931e9	−0.311072
$638$	3.11411e8	0.0474746
$639$	0	0
$640$	−1.96513e9	−0.296321
$641$	−4.70457e8	−0.0705532	−0.0352766	−	0.999378i	$-0.511231\pi$
$641$	−4.70457e8	−0.0705532	−0.0352766	+	0.999378i	$0.511231\pi$
$642$	0	0
$643$	−1.22547e9	−0.181788	−0.0908939	−	0.995861i	$-0.528972\pi$
$643$	−1.22547e9	−0.181788	−0.0908939	+	0.995861i	$0.528972\pi$
$644$	3.19018e8	0.0470668
$645$	0	0
$646$	3.51911e9	0.513593
$647$	7.49800e9	1.08838	0.544190	−	0.838962i	$-0.316837\pi$
$647$	7.49800e9	1.08838	0.544190	+	0.838962i	$0.316837\pi$
$648$	0	0
$649$	−7.92386e9	−1.13784
$650$	−1.76809e9	−0.252527
$651$	0	0
$652$	−5.54054e8	−0.0782862
$653$	9.09314e9	1.27796	0.638981	−	0.769223i	$-0.279356\pi$
$653$	9.09314e9	1.27796	0.638981	+	0.769223i	$0.279356\pi$
$654$	0	0
$655$	1.90531e8	0.0264924
$656$	−5.82703e8	−0.0805905
$657$	0	0
$658$	−2.28622e9	−0.312844
$659$	−6.75965e9	−0.920079	−0.460040	−	0.887898i	$-0.652165\pi$
$659$	−6.75965e9	−0.920079	−0.460040	+	0.887898i	$0.652165\pi$
$660$	0	0
$661$	−7.08950e9	−0.954796	−0.477398	−	0.878687i	$-0.658420\pi$
$661$	−7.08950e9	−0.954796	−0.477398	+	0.878687i	$0.658420\pi$
$662$	−7.76875e9	−1.04075
$663$	0	0
$664$	−4.32817e9	−0.573741
$665$	−1.90271e9	−0.250898
$666$	0	0
$667$	1.98419e8	0.0258906
$668$	−1.69238e9	−0.219675
$669$	0	0
$670$	5.38538e9	0.691758
$671$	4.27113e9	0.545776
$672$	0	0
$673$	9.68405e9	1.22463	0.612314	−	0.790614i	$-0.290239\pi$
$673$	9.68405e9	1.22463	0.612314	+	0.790614i	$0.290239\pi$
$674$	−8.35882e8	−0.105156
$675$	0	0
$676$	−1.30338e9	−0.162278
$677$	7.23809e9	0.896528	0.448264	−	0.893901i	$-0.352042\pi$
$677$	7.23809e9	0.896528	0.448264	+	0.893901i	$0.352042\pi$
$678$	0	0
$679$	−4.27822e9	−0.524468
$680$	−3.47291e9	−0.423557
$681$	0	0
$682$	1.08992e9	0.131567
$683$	3.21828e9	0.386502	0.193251	−	0.981149i	$-0.438097\pi$
$683$	3.21828e9	0.386502	0.193251	+	0.981149i	$0.438097\pi$
$684$	0	0
$685$	−6.84805e9	−0.814048
$686$	−8.28399e9	−0.979727
$687$	0	0
$688$	−1.54900e9	−0.181340
$689$	1.48558e10	1.73033
$690$	0	0
$691$	1.18734e10	1.36900	0.684498	−	0.729015i	$-0.260022\pi$
$691$	1.18734e10	1.36900	0.684498	+	0.729015i	$0.260022\pi$
$692$	−1.36013e9	−0.156031
$693$	0	0
$694$	−9.97585e9	−1.13290
$695$	−5.99683e9	−0.677603
$696$	0	0
$697$	−8.04050e8	−0.0899432
$698$	1.42188e10	1.58259
$699$	0	0
$700$	2.78751e8	0.0307167
$701$	−6.78566e8	−0.0744011	−0.0372005	−	0.999308i	$-0.511844\pi$
$701$	−6.78566e8	−0.0744011	−0.0372005	+	0.999308i	$0.511844\pi$
$702$	0	0
$703$	−3.75142e9	−0.407242
$704$	−6.34497e9	−0.685370
$705$	0	0
$706$	8.10985e9	0.867354
$707$	−2.55502e9	−0.271911
$708$	0	0
$709$	−1.12904e10	−1.18973	−0.594863	−	0.803827i	$-0.702794\pi$
$709$	−1.12904e10	−1.18973	−0.594863	+	0.803827i	$0.702794\pi$
$710$	2.63917e9	0.276734
$711$	0	0
$712$	−1.21585e10	−1.26241
$713$	6.94451e8	0.0717511
$714$	0	0
$715$	−3.75620e9	−0.384306
$716$	−1.80680e9	−0.183956
$717$	0	0
$718$	−6.34283e9	−0.639510
$719$	6.82206e9	0.684486	0.342243	−	0.939612i	$-0.388813\pi$
$719$	6.82206e9	0.684486	0.342243	+	0.939612i	$0.388813\pi$
$720$	0	0
$721$	−7.48747e9	−0.743980
$722$	5.46275e9	0.540170
$723$	0	0
$724$	2.29361e9	0.224613
$725$	1.73374e8	0.0168967
$726$	0	0
$727$	8.24394e9	0.795727	0.397864	−	0.917445i	$-0.369752\pi$
$727$	8.24394e9	0.795727	0.397864	+	0.917445i	$0.369752\pi$
$728$	−1.35987e10	−1.30628
$729$	0	0
$730$	6.86006e9	0.652676
$731$	−2.13741e9	−0.202384
$732$	0	0
$733$	−6.52263e9	−0.611728	−0.305864	−	0.952075i	$-0.598945\pi$
$733$	−6.52263e9	−0.611728	−0.305864	+	0.952075i	$0.598945\pi$
$734$	1.45394e10	1.35709
$735$	0	0
$736$	−1.14169e9	−0.105554
$737$	1.14409e10	1.05275
$738$	0	0
$739$	−2.02753e10	−1.84804	−0.924020	−	0.382345i	$-0.875116\pi$
$739$	−2.02753e10	−1.84804	−0.924020	+	0.382345i	$0.875116\pi$
$740$	5.49592e8	0.0498574
$741$	0	0
$742$	1.10927e10	0.996838
$743$	1.23416e10	1.10385	0.551926	−	0.833893i	$-0.313893\pi$
$743$	1.23416e10	1.10385	0.551926	+	0.833893i	$0.313893\pi$
$744$	0	0
$745$	5.32693e9	0.471987
$746$	3.96989e9	0.350100
$747$	0	0
$748$	−1.09527e9	−0.0956899
$749$	5.12729e9	0.445863
$750$	0	0
$751$	3.52449e9	0.303638	0.151819	−	0.988408i	$-0.451487\pi$
$751$	3.52449e9	0.303638	0.151819	+	0.988408i	$0.451487\pi$
$752$	3.62441e9	0.310795
$753$	0	0
$754$	−1.25559e9	−0.106672
$755$	7.02541e9	0.594097
$756$	0	0
$757$	7.69064e9	0.644357	0.322179	−	0.946679i	$-0.395585\pi$
$757$	7.69064e9	0.644357	0.322179	+	0.946679i	$0.395585\pi$
$758$	−1.40395e9	−0.117087
$759$	0	0
$760$	3.67764e9	0.303894
$761$	2.22283e10	1.82835	0.914176	−	0.405318i	$-0.132839\pi$
$761$	2.22283e10	1.82835	0.914176	+	0.405318i	$0.132839\pi$
$762$	0	0
$763$	−2.27298e9	−0.185251
$764$	−2.10718e9	−0.170952
$765$	0	0
$766$	−3.47431e9	−0.279298
$767$	3.19486e10	2.55663
$768$	0	0
$769$	−8.14868e9	−0.646167	−0.323084	−	0.946370i	$-0.604720\pi$
$769$	−8.14868e9	−0.646167	−0.323084	+	0.946370i	$0.604720\pi$
$770$	−2.80473e9	−0.221398
$771$	0	0
$772$	−3.40806e9	−0.266591
$773$	−1.32756e10	−1.03378	−0.516888	−	0.856053i	$-0.672909\pi$
$773$	−1.32756e10	−1.03378	−0.516888	+	0.856053i	$0.672909\pi$
$774$	0	0
$775$	6.06798e8	0.0468261
$776$	8.26913e9	0.635248
$777$	0	0
$778$	1.09208e10	0.831433
$779$	8.51451e8	0.0645325
$780$	0	0
$781$	5.60674e9	0.421145
$782$	3.30522e9	0.247159
$783$	0	0
$784$	2.40221e9	0.178035
$785$	9.05364e9	0.668004
$786$	0	0
$787$	−1.52249e10	−1.11338	−0.556690	−	0.830720i	$-0.687929\pi$
$787$	−1.52249e10	−1.11338	−0.556690	+	0.830720i	$0.687929\pi$
$788$	−2.51393e9	−0.183025
$789$	0	0
$790$	−2.05233e9	−0.148099
$791$	−1.19959e10	−0.861819
$792$	0	0
$793$	−1.72210e10	−1.22632
$794$	−8.27801e9	−0.586886
$795$	0	0
$796$	2.93537e9	0.206285
$797$	−1.84022e10	−1.28755	−0.643776	−	0.765214i	$-0.722633\pi$
$797$	−1.84022e10	−1.28755	−0.643776	+	0.765214i	$0.722633\pi$
$798$	0	0
$799$	5.00118e9	0.346864
$800$	−9.97583e8	−0.0688865
$801$	0	0
$802$	6.00175e9	0.410835
$803$	1.45738e10	0.993270
$804$	0	0
$805$	−1.78706e9	−0.120741
$806$	−4.39449e9	−0.295622
$807$	0	0
$808$	4.93845e9	0.329345
$809$	−2.07148e10	−1.37550	−0.687751	−	0.725947i	$-0.741402\pi$
$809$	−2.07148e10	−1.37550	−0.687751	+	0.725947i	$0.741402\pi$
$810$	0	0
$811$	2.62798e10	1.73001	0.865006	−	0.501761i	$-0.167314\pi$
$811$	2.62798e10	1.73001	0.865006	+	0.501761i	$0.167314\pi$
$812$	1.97953e8	0.0129752
$813$	0	0
$814$	−5.52987e9	−0.359360
$815$	3.10368e9	0.200828
$816$	0	0
$817$	2.26341e9	0.145207
$818$	2.41181e10	1.54066
$819$	0	0
$820$	−1.24740e8	−0.00790052
$821$	−4.99659e9	−0.315117	−0.157559	−	0.987510i	$-0.550362\pi$
$821$	−4.99659e9	−0.315117	−0.157559	+	0.987510i	$0.550362\pi$
$822$	0	0
$823$	−8.60951e9	−0.538368	−0.269184	−	0.963089i	$-0.586754\pi$
$823$	−8.60951e9	−0.538368	−0.269184	+	0.963089i	$0.586754\pi$
$824$	1.44721e10	0.901128
$825$	0	0
$826$	2.38558e10	1.47287
$827$	2.07729e10	1.27711	0.638555	−	0.769576i	$-0.279533\pi$
$827$	2.07729e10	1.27711	0.638555	+	0.769576i	$0.279533\pi$
$828$	0	0
$829$	−1.30674e10	−0.796617	−0.398308	−	0.917252i	$-0.630403\pi$
$829$	−1.30674e10	−0.796617	−0.398308	+	0.917252i	$0.630403\pi$
$830$	3.59926e9	0.218494
$831$	0	0
$832$	2.55826e10	1.53997
$833$	3.31472e9	0.198696
$834$	0	0
$835$	9.48030e9	0.563533
$836$	1.15984e9	0.0686556
$837$	0	0
$838$	−2.52626e10	−1.48294
$839$	6.75156e9	0.394673	0.197336	−	0.980336i	$-0.436771\pi$
$839$	6.75156e9	0.394673	0.197336	+	0.980336i	$0.436771\pi$
$840$	0	0
$841$	−1.71268e10	−0.992863
$842$	−2.79812e10	−1.61538
$843$	0	0
$844$	−1.71133e9	−0.0979798
$845$	7.30124e9	0.416292
$846$	0	0
$847$	9.62127e9	0.544052
$848$	−1.75856e10	−0.990311
$849$	0	0
$850$	2.88803e9	0.161301
$851$	−3.52341e9	−0.195979
$852$	0	0
$853$	1.92149e9	0.106003	0.0530013	−	0.998594i	$-0.483121\pi$
$853$	1.92149e9	0.106003	0.0530013	+	0.998594i	$0.483121\pi$
$854$	−1.28588e10	−0.706478
$855$	0	0
$856$	−9.91025e9	−0.540041
$857$	−1.66262e10	−0.902318	−0.451159	−	0.892444i	$-0.648989\pi$
$857$	−1.66262e10	−0.902318	−0.451159	+	0.892444i	$0.648989\pi$
$858$	0	0
$859$	−3.14620e10	−1.69360	−0.846798	−	0.531915i	$-0.821473\pi$
$859$	−3.14620e10	−1.69360	−0.846798	+	0.531915i	$0.821473\pi$
$860$	−3.31596e8	−0.0177772
$861$	0	0
$862$	1.58808e10	0.844493
$863$	−1.64912e9	−0.0873402	−0.0436701	−	0.999046i	$-0.513905\pi$
$863$	−1.64912e9	−0.0873402	−0.0436701	+	0.999046i	$0.513905\pi$
$864$	0	0
$865$	7.61915e9	0.400267
$866$	−2.03094e10	−1.06264
$867$	0	0
$868$	6.92821e8	0.0359585
$869$	−4.36004e9	−0.225383
$870$	0	0
$871$	−4.61291e10	−2.36544
$872$	4.39331e9	0.224380
$873$	0	0
$874$	−3.50007e9	−0.177332
$875$	−1.56150e9	−0.0787977
$876$	0	0
$877$	2.70603e10	1.35467	0.677335	−	0.735675i	$-0.263135\pi$
$877$	2.70603e10	1.35467	0.677335	+	0.735675i	$0.263135\pi$
$878$	−4.24883e9	−0.211855
$879$	0	0
$880$	4.44642e9	0.219948
$881$	2.55275e10	1.25774	0.628872	−	0.777509i	$-0.283517\pi$
$881$	2.55275e10	1.25774	0.628872	+	0.777509i	$0.283517\pi$
$882$	0	0
$883$	−7.90645e9	−0.386473	−0.193236	−	0.981152i	$-0.561898\pi$
$883$	−7.90645e9	−0.386473	−0.193236	+	0.981152i	$0.561898\pi$
$884$	4.41608e9	0.215008
$885$	0	0
$886$	1.71688e10	0.829318
$887$	−2.70207e10	−1.30006	−0.650030	−	0.759909i	$-0.725244\pi$
$887$	−2.70207e10	−1.30006	−0.650030	+	0.759909i	$0.725244\pi$
$888$	0	0
$889$	1.78246e10	0.850871
$890$	1.01109e10	0.480755
$891$	0	0
$892$	−2.95375e9	−0.139347
$893$	−5.29601e9	−0.248868
$894$	0	0
$895$	1.01213e10	0.471904
$896$	1.25688e10	0.583735
$897$	0	0
$898$	−1.97931e10	−0.912111
$899$	4.30912e8	0.0197801
$900$	0	0
$901$	−2.42657e10	−1.10524
$902$	1.25510e9	0.0569450
$903$	0	0
$904$	2.31862e10	1.04386
$905$	−1.28483e10	−0.576201
$906$	0	0
$907$	1.19824e10	0.533237	0.266618	−	0.963802i	$-0.414094\pi$
$907$	1.19824e10	0.533237	0.266618	+	0.963802i	$0.414094\pi$
$908$	−4.98963e9	−0.221191
$909$	0	0
$910$	1.13085e10	0.497464
$911$	2.87625e10	1.26041	0.630207	−	0.776427i	$-0.282970\pi$
$911$	2.87625e10	1.26041	0.630207	+	0.776427i	$0.282970\pi$
$912$	0	0
$913$	7.64640e9	0.332514
$914$	2.52294e10	1.09294
$915$	0	0
$916$	−2.39858e9	−0.103115
$917$	−1.21862e9	−0.0521886
$918$	0	0
$919$	4.45392e10	1.89294	0.946472	−	0.322786i	$-0.104619\pi$
$919$	4.45392e10	1.89294	0.946472	+	0.322786i	$0.104619\pi$
$920$	3.45411e9	0.146244
$921$	0	0
$922$	2.09611e10	0.880755
$923$	−2.26061e10	−0.946280
$924$	0	0
$925$	−3.07869e9	−0.127900
$926$	−1.46438e10	−0.606060
$927$	0	0
$928$	−7.08424e8	−0.0290988
$929$	2.94819e10	1.20642	0.603212	−	0.797581i	$-0.293887\pi$
$929$	2.94819e10	1.20642	0.603212	+	0.797581i	$0.293887\pi$
$930$	0	0
$931$	−3.51013e9	−0.142561
$932$	−6.64044e9	−0.268684
$933$	0	0
$934$	−2.48467e10	−0.997824
$935$	6.13545e9	0.245474
$936$	0	0
$937$	2.28976e10	0.909286	0.454643	−	0.890674i	$-0.349767\pi$
$937$	2.28976e10	0.909286	0.454643	+	0.890674i	$0.349767\pi$
$938$	−3.44444e10	−1.36272
$939$	0	0
$940$	7.75879e8	0.0304682
$941$	−3.52084e10	−1.37747	−0.688736	−	0.725012i	$-0.741834\pi$
$941$	−3.52084e10	−1.37747	−0.688736	+	0.725012i	$0.741834\pi$
$942$	0	0
$943$	7.99699e8	0.0310553
$944$	−3.78193e10	−1.46323
$945$	0	0
$946$	3.33644e9	0.128134
$947$	−1.01547e10	−0.388546	−0.194273	−	0.980947i	$-0.562235\pi$
$947$	−1.01547e10	−0.388546	−0.194273	+	0.980947i	$0.562235\pi$
$948$	0	0
$949$	−5.87607e10	−2.23180
$950$	−3.05829e9	−0.115730
$951$	0	0
$952$	2.22124e10	0.834383
$953$	1.54884e10	0.579672	0.289836	−	0.957076i	$-0.406399\pi$
$953$	1.54884e10	0.579672	0.289836	+	0.957076i	$0.406399\pi$
$954$	0	0
$955$	1.18039e10	0.438545
$956$	−1.27250e9	−0.0471037
$957$	0	0
$958$	1.20393e10	0.442407
$959$	4.37994e10	1.60363
$960$	0	0
$961$	−2.60045e10	−0.945183
$962$	2.22962e10	0.807453
$963$	0	0
$964$	−6.02677e9	−0.216678
$965$	1.90911e10	0.683889
$966$	0	0
$967$	−9.34083e9	−0.332195	−0.166097	−	0.986109i	$-0.553117\pi$
$967$	−9.34083e9	−0.332195	−0.166097	+	0.986109i	$0.553117\pi$
$968$	−1.85964e10	−0.658970
$969$	0	0
$970$	−6.87652e9	−0.241918
$971$	1.35845e10	0.476186	0.238093	−	0.971242i	$-0.423478\pi$
$971$	1.35845e10	0.476186	0.238093	+	0.971242i	$0.423478\pi$
$972$	0	0
$973$	3.83552e10	1.33484
$974$	−4.07791e10	−1.41410
$975$	0	0
$976$	2.03854e10	0.701852
$977$	−5.09207e10	−1.74688	−0.873441	−	0.486930i	$-0.838117\pi$
$977$	−5.09207e10	−1.74688	−0.873441	+	0.486930i	$0.838117\pi$
$978$	0	0
$979$	2.14799e10	0.731633
$980$	5.14243e8	0.0174533
$981$	0	0
$982$	4.92223e10	1.65871
$983$	−1.61868e8	−0.00543530	−0.00271765	−	0.999996i	$-0.500865\pi$
$983$	−1.61868e8	−0.00543530	−0.00271765	+	0.999996i	$0.500865\pi$
$984$	0	0
$985$	1.40824e10	0.469516
$986$	2.05091e9	0.0681361
$987$	0	0
$988$	−4.67641e9	−0.154264
$989$	2.12584e9	0.0698786
$990$	0	0
$991$	5.36982e10	1.75268	0.876339	−	0.481695i	$-0.159979\pi$
$991$	5.36982e10	1.75268	0.876339	+	0.481695i	$0.159979\pi$
$992$	−2.47944e9	−0.0806421
$993$	0	0
$994$	−1.68798e10	−0.545150
$995$	−1.64433e10	−0.529185
$996$	0	0
$997$	5.76569e10	1.84255	0.921273	−	0.388917i	$-0.127151\pi$
$997$	5.76569e10	1.84255	0.921273	+	0.388917i	$0.127151\pi$
$998$	−3.07326e10	−0.978685
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.8.a.h.1.4	yes	14
3.2	odd	2		405.8.a.g.1.11	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.8.a.g.1.11	✓	14	3.2	odd	2
405.8.a.h.1.4	yes	14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q-10.2803 q^{2} -22.3144 q^{4} +125.000 q^{5} -799.488 q^{7} +1545.28 q^{8} +O(q^{10})\)	\(q-10.2803 q^{2} -22.3144 q^{4} +125.000 q^{5} -799.488 q^{7} +1545.28 q^{8} -1285.04 q^{10} -2729.99 q^{11} +11007.2 q^{13} +8219.01 q^{14} -13029.8 q^{16} -17979.4 q^{17} +19039.3 q^{19} -2789.30 q^{20} +28065.3 q^{22} +17882.1 q^{23} +15625.0 q^{25} -113158. q^{26} +17840.1 q^{28} +11095.9 q^{29} +38835.1 q^{31} -63845.3 q^{32} +184834. q^{34} -99936.0 q^{35} -197036. q^{37} -195731. q^{38} +193161. q^{40} +44720.7 q^{41} +118881. q^{43} +60918.2 q^{44} -183834. q^{46} -278162. q^{47} -184363. q^{49} -160630. q^{50} -245619. q^{52} +1.34964e6 q^{53} -341249. q^{55} -1.23544e6 q^{56} -114070. q^{58} +2.90252e6 q^{59} -1.56452e6 q^{61} -399238. q^{62} +2.32417e6 q^{64} +1.37590e6 q^{65} -4.19081e6 q^{67} +401199. q^{68} +1.02738e6 q^{70} -2.05376e6 q^{71} -5.33839e6 q^{73} +2.02560e6 q^{74} -424850. q^{76} +2.18260e6 q^{77} +1.59709e6 q^{79} -1.62873e6 q^{80} -459745. q^{82} -2.80089e6 q^{83} -2.24742e6 q^{85} -1.22214e6 q^{86} -4.21862e6 q^{88} -7.86812e6 q^{89} -8.80012e6 q^{91} -399028. q^{92} +2.85961e6 q^{94} +2.37991e6 q^{95} +5.35120e6 q^{97} +1.89531e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 16 q^{2} + 714 q^{4} + 1750 q^{5} + 1538 q^{7} - 126 q^{8}+O(q^{10}) \) 14 * q + 16 * q^2 + 714 * q^4 + 1750 * q^5 + 1538 * q^7 - 126 * q^8	\( 14 q + 16 q^{2} + 714 q^{4} + 1750 q^{5} + 1538 q^{7} - 126 q^{8} + 2000 q^{10} + 10648 q^{11} + 17268 q^{13} + 9180 q^{14} + 34122 q^{16} + 3886 q^{17} + 22986 q^{19} + 89250 q^{20} - 49706 q^{22} + 155040 q^{23} + 218750 q^{25} + 7804 q^{26} + 256160 q^{28} + 195836 q^{29} + 92786 q^{31} + 157778 q^{32} + 787348 q^{34} + 192250 q^{35} + 876406 q^{37} + 329320 q^{38} - 15750 q^{40} + 795164 q^{41} + 730350 q^{43} + 2360876 q^{44} - 225654 q^{46} + 2687842 q^{47} + 1663586 q^{49} + 250000 q^{50} + 3875836 q^{52} + 2533750 q^{53} + 1331000 q^{55} + 2055276 q^{56} - 318934 q^{58} + 2283340 q^{59} + 2600400 q^{61} + 5702022 q^{62} + 474098 q^{64} + 2158500 q^{65} + 2422160 q^{67} + 1114364 q^{68} + 1147500 q^{70} + 6395324 q^{71} - 540774 q^{73} + 260516 q^{74} - 2417042 q^{76} - 1384890 q^{77} - 307384 q^{79} + 4265250 q^{80} - 14044738 q^{82} + 10991322 q^{83} + 485750 q^{85} + 2847712 q^{86} - 19226592 q^{88} + 2094000 q^{89} - 9496256 q^{91} + 39213378 q^{92} - 28132682 q^{94} + 2873250 q^{95} - 12859994 q^{97} + 28336478 q^{98}+O(q^{100}) \) 14 * q + 16 * q^2 + 714 * q^4 + 1750 * q^5 + 1538 * q^7 - 126 * q^8 + 2000 * q^10 + 10648 * q^11 + 17268 * q^13 + 9180 * q^14 + 34122 * q^16 + 3886 * q^17 + 22986 * q^19 + 89250 * q^20 - 49706 * q^22 + 155040 * q^23 + 218750 * q^25 + 7804 * q^26 + 256160 * q^28 + 195836 * q^29 + 92786 * q^31 + 157778 * q^32 + 787348 * q^34 + 192250 * q^35 + 876406 * q^37 + 329320 * q^38 - 15750 * q^40 + 795164 * q^41 + 730350 * q^43 + 2360876 * q^44 - 225654 * q^46 + 2687842 * q^47 + 1663586 * q^49 + 250000 * q^50 + 3875836 * q^52 + 2533750 * q^53 + 1331000 * q^55 + 2055276 * q^56 - 318934 * q^58 + 2283340 * q^59 + 2600400 * q^61 + 5702022 * q^62 + 474098 * q^64 + 2158500 * q^65 + 2422160 * q^67 + 1114364 * q^68 + 1147500 * q^70 + 6395324 * q^71 - 540774 * q^73 + 260516 * q^74 - 2417042 * q^76 - 1384890 * q^77 - 307384 * q^79 + 4265250 * q^80 - 14044738 * q^82 + 10991322 * q^83 + 485750 * q^85 + 2847712 * q^86 - 19226592 * q^88 + 2094000 * q^89 - 9496256 * q^91 + 39213378 * q^92 - 28132682 * q^94 + 2873250 * q^95 - 12859994 * q^97 + 28336478 * q^98

Introduction

L-functions

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