LMFDB - Embedded newform 405.8.a.d.1.5

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 660x^{5} + 510x^{4} + 118692x^{3} - 171216x^{2} - 4927392x + 8926848 $ x^7 - x^6 - 660x^5 + 510x^4 + 118692x^3 - 171216x^2 - 4927392*x + 8926848
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{6} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$7.61673$ 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)

$f(q)$	$=$	$q+9.61673 q^{2} -35.5185 q^{4} -125.000 q^{5} -1128.38 q^{7} -1572.51 q^{8} +O(q^{10})$	\(q+9.61673 q^{2} -35.5185 q^{4} -125.000 q^{5} -1128.38 q^{7} -1572.51 q^{8} -1202.09 q^{10} +7275.33 q^{11} -6319.54 q^{13} -10851.3 q^{14} -10576.1 q^{16} -36348.4 q^{17} +5248.19 q^{19} +4439.81 q^{20} +69964.9 q^{22} -77698.7 q^{23} +15625.0 q^{25} -60773.3 q^{26} +40078.4 q^{28} +19167.5 q^{29} -260576. q^{31} +99574.5 q^{32} -349553. q^{34} +141048. q^{35} +192332. q^{37} +50470.4 q^{38} +196564. q^{40} -371018. q^{41} -328752. q^{43} -258409. q^{44} -747208. q^{46} -970324. q^{47} +449702. q^{49} +150261. q^{50} +224461. q^{52} -536538. q^{53} -909417. q^{55} +1.77439e6 q^{56} +184329. q^{58} +3.06357e6 q^{59} +3.38455e6 q^{61} -2.50589e6 q^{62} +2.31132e6 q^{64} +789943. q^{65} +2.87590e6 q^{67} +1.29104e6 q^{68} +1.35642e6 q^{70} +1.89666e6 q^{71} +1.05291e6 q^{73} +1.84961e6 q^{74} -186408. q^{76} -8.20935e6 q^{77} +4.84825e6 q^{79} +1.32201e6 q^{80} -3.56798e6 q^{82} +7.98760e6 q^{83} +4.54355e6 q^{85} -3.16152e6 q^{86} -1.14406e7 q^{88} -895568. q^{89} +7.13085e6 q^{91} +2.75974e6 q^{92} -9.33134e6 q^{94} -656024. q^{95} -7.37266e6 q^{97} +4.32466e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 15 q^{2} + 457 q^{4} - 875 q^{5} + 614 q^{7} + 4605 q^{8}+O(q^{10}) $ 7 * q + 15 * q^2 + 457 * q^4 - 875 * q^5 + 614 * q^7 + 4605 * q^8	\( 7 q + 15 q^{2} + 457 q^{4} - 875 q^{5} + 614 q^{7} + 4605 q^{8} - 1875 q^{10} + 3753 q^{11} - 6562 q^{13} + 19653 q^{14} + 27625 q^{16} - 23862 q^{17} + 33251 q^{19} - 57125 q^{20} + 108060 q^{22} + 62682 q^{23} + 109375 q^{25} - 186789 q^{26} + 149303 q^{28} + 78939 q^{29} - 703165 q^{31} + 1266165 q^{32} - 589398 q^{34} - 76750 q^{35} + 336776 q^{37} + 498855 q^{38} - 575625 q^{40} + 480801 q^{41} - 442792 q^{43} - 9372 q^{44} + 213447 q^{46} + 1122744 q^{47} - 954843 q^{49} + 234375 q^{50} - 1324819 q^{52} + 692028 q^{53} - 469125 q^{55} + 6614175 q^{56} - 3971220 q^{58} + 1274487 q^{59} - 2485258 q^{61} - 2438700 q^{62} + 10358137 q^{64} + 820250 q^{65} - 892402 q^{67} + 926646 q^{68} - 2456625 q^{70} + 8286201 q^{71} + 788666 q^{73} - 537042 q^{74} + 16155485 q^{76} - 5060322 q^{77} - 1308352 q^{79} - 3453125 q^{80} + 716235 q^{82} + 9226704 q^{83} + 2982750 q^{85} + 15948900 q^{86} + 16624320 q^{88} + 4459791 q^{89} - 5701496 q^{91} + 8926029 q^{92} + 43736397 q^{94} - 4156375 q^{95} - 11086048 q^{97} + 21087450 q^{98}+O(q^{100}) \) 7 * q + 15 * q^2 + 457 * q^4 - 875 * q^5 + 614 * q^7 + 4605 * q^8 - 1875 * q^10 + 3753 * q^11 - 6562 * q^13 + 19653 * q^14 + 27625 * q^16 - 23862 * q^17 + 33251 * q^19 - 57125 * q^20 + 108060 * q^22 + 62682 * q^23 + 109375 * q^25 - 186789 * q^26 + 149303 * q^28 + 78939 * q^29 - 703165 * q^31 + 1266165 * q^32 - 589398 * q^34 - 76750 * q^35 + 336776 * q^37 + 498855 * q^38 - 575625 * q^40 + 480801 * q^41 - 442792 * q^43 - 9372 * q^44 + 213447 * q^46 + 1122744 * q^47 - 954843 * q^49 + 234375 * q^50 - 1324819 * q^52 + 692028 * q^53 - 469125 * q^55 + 6614175 * q^56 - 3971220 * q^58 + 1274487 * q^59 - 2485258 * q^61 - 2438700 * q^62 + 10358137 * q^64 + 820250 * q^65 - 892402 * q^67 + 926646 * q^68 - 2456625 * q^70 + 8286201 * q^71 + 788666 * q^73 - 537042 * q^74 + 16155485 * q^76 - 5060322 * q^77 - 1308352 * q^79 - 3453125 * q^80 + 716235 * q^82 + 9226704 * q^83 + 2982750 * q^85 + 15948900 * q^86 + 16624320 * q^88 + 4459791 * q^89 - 5701496 * q^91 + 8926029 * q^92 + 43736397 * q^94 - 4156375 * q^95 - 11086048 * q^97 + 21087450 * 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 $

$2$	9.61673	0.850007	0.425003	−	0.905192i	$-0.360273\pi$
$2$	9.61673	0.850007	0.425003	+	0.905192i	$0.360273\pi$
$3$	0	0
$3$	0	0
$4$	−35.5185	−0.277488
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	−1128.38	−1.24341	−0.621703	−	0.783253i	$-0.713559\pi$
$7$	−1128.38	−1.24341	−0.621703	+	0.783253i	$0.713559\pi$
$8$	−1572.51	−1.08587
$9$	0	0
$10$	−1202.09	−0.380135
$11$	7275.33	1.64808	0.824041	−	0.566531i	$-0.191715\pi$
$11$	7275.33	1.64808	0.824041	+	0.566531i	$0.191715\pi$
$12$	0	0
$13$	−6319.54	−0.797781	−0.398891	−	0.916999i	$-0.630605\pi$
$13$	−6319.54	−0.797781	−0.398891	+	0.916999i	$0.630605\pi$
$14$	−10851.3	−1.05690
$15$	0	0
$16$	−10576.1	−0.645512
$17$	−36348.4	−1.79438	−0.897190	−	0.441646i	$-0.854395\pi$
$17$	−36348.4	−1.79438	−0.897190	+	0.441646i	$0.854395\pi$
$18$	0	0
$19$	5248.19	0.175538	0.0877691	−	0.996141i	$-0.472026\pi$
$19$	5248.19	0.175538	0.0877691	+	0.996141i	$0.472026\pi$
$20$	4439.81	0.124097
$21$	0	0
$22$	69964.9	1.40088
$23$	−77698.7	−1.33158	−0.665789	−	0.746140i	$-0.731905\pi$
$23$	−77698.7	−1.33158	−0.665789	+	0.746140i	$0.731905\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	−60773.3	−0.678119
$27$	0	0
$28$	40078.4	0.345031
$29$	19167.5	0.145939	0.0729697	−	0.997334i	$-0.476752\pi$
$29$	19167.5	0.145939	0.0729697	+	0.997334i	$0.476752\pi$
$30$	0	0
$31$	−260576.	−1.57097	−0.785486	−	0.618879i	$-0.787587\pi$
$31$	−260576.	−1.57097	−0.785486	+	0.618879i	$0.787587\pi$
$32$	99574.5	0.537184
$33$	0	0
$34$	−349553.	−1.52523
$35$	141048.	0.556068
$36$	0	0
$37$	192332.	0.624233	0.312116	−	0.950044i	$-0.398962\pi$
$37$	192332.	0.624233	0.312116	+	0.950044i	$0.398962\pi$
$38$	50470.4	0.149209
$39$	0	0
$40$	196564.	0.485618
$41$	−371018.	−0.840720	−0.420360	−	0.907358i	$-0.638096\pi$
$41$	−371018.	−0.840720	−0.420360	+	0.907358i	$0.638096\pi$
$42$	0	0
$43$	−328752.	−0.630563	−0.315281	−	0.948998i	$-0.602099\pi$
$43$	−328752.	−0.630563	−0.315281	+	0.948998i	$0.602099\pi$
$44$	−258409.	−0.457323
$45$	0	0
$46$	−747208.	−1.13185
$47$	−970324.	−1.36325	−0.681623	−	0.731704i	$-0.738725\pi$
$47$	−970324.	−1.36325	−0.681623	+	0.731704i	$0.738725\pi$
$48$	0	0
$49$	449702.	0.546057
$50$	150261.	0.170001
$51$	0	0
$52$	224461.	0.221375
$53$	−536538.	−0.495034	−0.247517	−	0.968884i	$-0.579615\pi$
$53$	−536538.	−0.495034	−0.247517	+	0.968884i	$0.579615\pi$
$54$	0	0
$55$	−909417.	−0.737044
$56$	1.77439e6	1.35018
$57$	0	0
$58$	184329.	0.124049
$59$	3.06357e6	1.94198	0.970991	−	0.239115i	$-0.0768572\pi$
$59$	3.06357e6	1.94198	0.970991	+	0.239115i	$0.0768572\pi$
$60$	0	0
$61$	3.38455e6	1.90918	0.954590	−	0.297923i	$-0.0962940\pi$
$61$	3.38455e6	1.90918	0.954590	+	0.297923i	$0.0962940\pi$
$62$	−2.50589e6	−1.33534
$63$	0	0
$64$	2.31132e6	1.10212
$65$	789943.	0.356779
$66$	0	0
$67$	2.87590e6	1.16818	0.584092	−	0.811687i	$-0.301451\pi$
$67$	2.87590e6	1.16818	0.584092	+	0.811687i	$0.301451\pi$
$68$	1.29104e6	0.497919
$69$	0	0
$70$	1.35642e6	0.472661
$71$	1.89666e6	0.628907	0.314453	−	0.949273i	$-0.398179\pi$
$71$	1.89666e6	0.628907	0.314453	+	0.949273i	$0.398179\pi$
$72$	0	0
$73$	1.05291e6	0.316782	0.158391	−	0.987376i	$-0.449369\pi$
$73$	1.05291e6	0.316782	0.158391	+	0.987376i	$0.449369\pi$
$74$	1.84961e6	0.530602
$75$	0	0
$76$	−186408.	−0.0487098
$77$	−8.20935e6	−2.04923
$78$	0	0
$79$	4.84825e6	1.10634	0.553172	−	0.833067i	$-0.313417\pi$
$79$	4.84825e6	1.10634	0.553172	+	0.833067i	$0.313417\pi$
$80$	1.32201e6	0.288682
$81$	0	0
$82$	−3.56798e6	−0.714617
$83$	7.98760e6	1.53336	0.766678	−	0.642032i	$-0.221909\pi$
$83$	7.98760e6	1.53336	0.766678	+	0.642032i	$0.221909\pi$
$84$	0	0
$85$	4.54355e6	0.802471
$86$	−3.16152e6	−0.535983
$87$	0	0
$88$	−1.14406e7	−1.78961
$89$	−895568.	−0.134658	−0.0673292	−	0.997731i	$-0.521448\pi$
$89$	−895568.	−0.134658	−0.0673292	+	0.997731i	$0.521448\pi$
$90$	0	0
$91$	7.13085e6	0.991966
$92$	2.75974e6	0.369497
$93$	0	0
$94$	−9.33134e6	−1.15877
$95$	−656024.	−0.0785031
$96$	0	0
$97$	−7.37266e6	−0.820206	−0.410103	−	0.912039i	$-0.634507\pi$
$97$	−7.37266e6	−0.820206	−0.410103	+	0.912039i	$0.634507\pi$
$98$	4.32466e6	0.464152
$99$	0	0
$100$	−554977.	−0.0554977
$101$	−2.04305e6	−0.197312	−0.0986561	−	0.995122i	$-0.531454\pi$
$101$	−2.04305e6	−0.197312	−0.0986561	+	0.995122i	$0.531454\pi$
$102$	0	0
$103$	1.23366e7	1.11241	0.556205	−	0.831045i	$-0.312257\pi$
$103$	1.23366e7	1.11241	0.556205	+	0.831045i	$0.312257\pi$
$104$	9.93756e6	0.866290
$105$	0	0
$106$	−5.15974e6	−0.420782
$107$	−1.99439e7	−1.57386	−0.786930	−	0.617042i	$-0.788331\pi$
$107$	−1.99439e7	−1.57386	−0.786930	+	0.617042i	$0.788331\pi$
$108$	0	0
$109$	5.53754e6	0.409566	0.204783	−	0.978807i	$-0.434351\pi$
$109$	5.53754e6	0.409566	0.204783	+	0.978807i	$0.434351\pi$
$110$	−8.74562e6	−0.626493
$111$	0	0
$112$	1.19338e7	0.802633
$113$	−7.98558e6	−0.520633	−0.260317	−	0.965523i	$-0.583827\pi$
$113$	−7.98558e6	−0.520633	−0.260317	+	0.965523i	$0.583827\pi$
$114$	0	0
$115$	9.71234e6	0.595500
$116$	−680801.	−0.0404965
$117$	0	0
$118$	2.94615e7	1.65070
$119$	4.10149e7	2.23114
$120$	0	0
$121$	3.34433e7	1.71617
$122$	3.25483e7	1.62282
$123$	0	0
$124$	9.25527e6	0.435927
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	−3.95294e7	−1.71241	−0.856204	−	0.516638i	$-0.827183\pi$
$127$	−3.95294e7	−1.71241	−0.856204	+	0.516638i	$0.827183\pi$
$128$	9.48178e6	0.399627
$129$	0	0
$130$	7.59666e6	0.303264
$131$	−2.62801e7	−1.02136	−0.510679	−	0.859771i	$-0.670606\pi$
$131$	−2.62801e7	−1.02136	−0.510679	+	0.859771i	$0.670606\pi$
$132$	0	0
$133$	−5.92196e6	−0.218265
$134$	2.76567e7	0.992965
$135$	0	0
$136$	5.71584e7	1.94847
$137$	2.71472e7	0.901994	0.450997	−	0.892525i	$-0.351068\pi$
$137$	2.71472e7	0.901994	0.450997	+	0.892525i	$0.351068\pi$
$138$	0	0
$139$	1.39269e7	0.439849	0.219925	−	0.975517i	$-0.429419\pi$
$139$	1.39269e7	0.439849	0.219925	+	0.975517i	$0.429419\pi$
$140$	−5.00980e6	−0.154302
$141$	0	0
$142$	1.82397e7	0.534575
$143$	−4.59768e7	−1.31481
$144$	0	0
$145$	−2.39594e6	−0.0652661
$146$	1.01255e7	0.269267
$147$	0	0
$148$	−6.83136e6	−0.173217
$149$	5.94608e7	1.47258	0.736290	−	0.676666i	$-0.236576\pi$
$149$	5.94608e7	1.47258	0.736290	+	0.676666i	$0.236576\pi$
$150$	0	0
$151$	−4.37411e7	−1.03388	−0.516940	−	0.856022i	$-0.672929\pi$
$151$	−4.37411e7	−1.03388	−0.516940	+	0.856022i	$0.672929\pi$
$152$	−8.25285e6	−0.190612
$153$	0	0
$154$	−7.89471e7	−1.74186
$155$	3.25720e7	0.702560
$156$	0	0
$157$	−9.92229e6	−0.204627	−0.102313	−	0.994752i	$-0.532624\pi$
$157$	−9.92229e6	−0.204627	−0.102313	+	0.994752i	$0.532624\pi$
$158$	4.66243e7	0.940400
$159$	0	0
$160$	−1.24468e7	−0.240236
$161$	8.76738e7	1.65569
$162$	0	0
$163$	−2.53775e7	−0.458979	−0.229489	−	0.973311i	$-0.573706\pi$
$163$	−2.53775e7	−0.458979	−0.229489	+	0.973311i	$0.573706\pi$
$164$	1.31780e7	0.233290
$165$	0	0
$166$	7.68145e7	1.30336
$167$	8.98743e7	1.49323	0.746617	−	0.665254i	$-0.231677\pi$
$167$	8.98743e7	1.49323	0.746617	+	0.665254i	$0.231677\pi$
$168$	0	0
$169$	−2.28119e7	−0.363545
$170$	4.36941e7	0.682106
$171$	0	0
$172$	1.16768e7	0.174974
$173$	−1.03692e8	−1.52260	−0.761298	−	0.648402i	$-0.775437\pi$
$173$	−1.03692e8	−1.52260	−0.761298	+	0.648402i	$0.775437\pi$
$174$	0	0
$175$	−1.76310e7	−0.248681
$176$	−7.69444e7	−1.06386
$177$	0	0
$178$	−8.61244e6	−0.114461
$179$	7.70497e6	0.100412	0.0502060	−	0.998739i	$-0.484012\pi$
$179$	7.70497e6	0.100412	0.0502060	+	0.998739i	$0.484012\pi$
$180$	0	0
$181$	−1.17436e8	−1.47206	−0.736030	−	0.676949i	$-0.763302\pi$
$181$	−1.17436e8	−1.47206	−0.736030	+	0.676949i	$0.763302\pi$
$182$	6.85755e7	0.843178
$183$	0	0
$184$	1.22182e8	1.44593
$185$	−2.40416e7	−0.279165
$186$	0	0
$187$	−2.64447e8	−2.95728
$188$	3.44645e7	0.378285
$189$	0	0
$190$	−6.30880e6	−0.0667282
$191$	4.37944e7	0.454781	0.227390	−	0.973804i	$-0.426981\pi$
$191$	4.37944e7	0.454781	0.227390	+	0.973804i	$0.426981\pi$
$192$	0	0
$193$	9.30638e7	0.931816	0.465908	−	0.884833i	$-0.345728\pi$
$193$	9.30638e7	0.931816	0.465908	+	0.884833i	$0.345728\pi$
$194$	−7.09008e7	−0.697181
$195$	0	0
$196$	−1.59727e7	−0.151525
$197$	5.55205e7	0.517394	0.258697	−	0.965959i	$-0.416707\pi$
$197$	5.55205e7	0.517394	0.258697	+	0.965959i	$0.416707\pi$
$198$	0	0
$199$	−5.76060e7	−0.518182	−0.259091	−	0.965853i	$-0.583423\pi$
$199$	−5.76060e7	−0.518182	−0.259091	+	0.965853i	$0.583423\pi$
$200$	−2.45705e7	−0.217175
$201$	0	0
$202$	−1.96474e7	−0.167717
$203$	−2.16283e7	−0.181462
$204$	0	0
$205$	4.63772e7	0.375981
$206$	1.18638e8	0.945557
$207$	0	0
$208$	6.68359e7	0.514977
$209$	3.81823e7	0.289301
$210$	0	0
$211$	1.50749e8	1.10476	0.552378	−	0.833594i	$-0.313721\pi$
$211$	1.50749e8	1.10476	0.552378	+	0.833594i	$0.313721\pi$
$212$	1.90570e7	0.137366
$213$	0	0
$214$	−1.91795e8	−1.33779
$215$	4.10940e7	0.281996
$216$	0	0
$217$	2.94029e8	1.95336
$218$	5.32530e7	0.348134
$219$	0	0
$220$	3.23011e7	0.204521
$221$	2.29705e8	1.43152
$222$	0	0
$223$	−1.85283e8	−1.11884	−0.559421	−	0.828883i	$-0.688977\pi$
$223$	−1.85283e8	−1.11884	−0.559421	+	0.828883i	$0.688977\pi$
$224$	−1.12358e8	−0.667938
$225$	0	0
$226$	−7.67952e7	−0.442542
$227$	−1.04185e8	−0.591175	−0.295587	−	0.955316i	$-0.595515\pi$
$227$	−1.04185e8	−0.591175	−0.295587	+	0.955316i	$0.595515\pi$
$228$	0	0
$229$	−6.72459e7	−0.370034	−0.185017	−	0.982735i	$-0.559234\pi$
$229$	−6.72459e7	−0.370034	−0.185017	+	0.982735i	$0.559234\pi$
$230$	9.34010e7	0.506179
$231$	0	0
$232$	−3.01412e7	−0.158472
$233$	2.34137e8	1.21262	0.606308	−	0.795230i	$-0.292650\pi$
$233$	2.34137e8	1.21262	0.606308	+	0.795230i	$0.292650\pi$
$234$	0	0
$235$	1.21290e8	0.609662
$236$	−1.08813e8	−0.538878
$237$	0	0
$238$	3.94429e8	1.89648
$239$	4.80970e7	0.227890	0.113945	−	0.993487i	$-0.463651\pi$
$239$	4.80970e7	0.227890	0.113945	+	0.993487i	$0.463651\pi$
$240$	0	0
$241$	1.40335e8	0.645813	0.322906	−	0.946431i	$-0.395340\pi$
$241$	1.40335e8	0.645813	0.322906	+	0.946431i	$0.395340\pi$
$242$	3.21615e8	1.45876
$243$	0	0
$244$	−1.20214e8	−0.529775
$245$	−5.62127e7	−0.244204
$246$	0	0
$247$	−3.31661e7	−0.140041
$248$	4.09759e8	1.70588
$249$	0	0
$250$	−1.87827e7	−0.0760269
$251$	−6.41581e6	−0.0256091	−0.0128045	−	0.999918i	$-0.504076\pi$
$251$	−6.41581e6	−0.0256091	−0.0128045	+	0.999918i	$0.504076\pi$
$252$	0	0
$253$	−5.65284e8	−2.19455
$254$	−3.80144e8	−1.45556
$255$	0	0
$256$	−2.04665e8	−0.762437
$257$	9.17941e7	0.337325	0.168663	−	0.985674i	$-0.446055\pi$
$257$	9.17941e7	0.337325	0.168663	+	0.985674i	$0.446055\pi$
$258$	0	0
$259$	−2.17024e8	−0.776174
$260$	−2.80576e7	−0.0990019
$261$	0	0
$262$	−2.52729e8	−0.868162
$263$	2.52796e8	0.856892	0.428446	−	0.903567i	$-0.359061\pi$
$263$	2.52796e8	0.856892	0.428446	+	0.903567i	$0.359061\pi$
$264$	0	0
$265$	6.70672e7	0.221386
$266$	−5.69499e7	−0.185527
$267$	0	0
$268$	−1.02148e8	−0.324158
$269$	3.31448e8	1.03820	0.519101	−	0.854713i	$-0.326267\pi$
$269$	3.31448e8	1.03820	0.519101	+	0.854713i	$0.326267\pi$
$270$	0	0
$271$	−3.90459e8	−1.19175	−0.595873	−	0.803079i	$-0.703194\pi$
$271$	−3.90459e8	−1.19175	−0.595873	+	0.803079i	$0.703194\pi$
$272$	3.84423e8	1.15829
$273$	0	0
$274$	2.61068e8	0.766701
$275$	1.13677e8	0.329616
$276$	0	0
$277$	−3.23144e7	−0.0913518	−0.0456759	−	0.998956i	$-0.514544\pi$
$277$	−3.23144e7	−0.0913518	−0.0456759	+	0.998956i	$0.514544\pi$
$278$	1.33932e8	0.373875
$279$	0	0
$280$	−2.21799e8	−0.603820
$281$	2.76946e8	0.744601	0.372300	−	0.928112i	$-0.378569\pi$
$281$	2.76946e8	0.744601	0.372300	+	0.928112i	$0.378569\pi$
$282$	0	0
$283$	−5.14352e8	−1.34899	−0.674494	−	0.738281i	$-0.735638\pi$
$283$	−5.14352e8	−1.34899	−0.674494	+	0.738281i	$0.735638\pi$
$284$	−6.73667e7	−0.174514
$285$	0	0
$286$	−4.42146e8	−1.11760
$287$	4.18649e8	1.04536
$288$	0	0
$289$	9.10868e8	2.21980
$290$	−2.30411e7	−0.0554766
$291$	0	0
$292$	−3.73977e7	−0.0879032
$293$	6.89295e7	0.160092	0.0800458	−	0.996791i	$-0.474493\pi$
$293$	6.89295e7	0.160092	0.0800458	+	0.996791i	$0.474493\pi$
$294$	0	0
$295$	−3.82946e8	−0.868481
$296$	−3.02445e8	−0.677838
$297$	0	0
$298$	5.71818e8	1.25170
$299$	4.91020e8	1.06231
$300$	0	0
$301$	3.70957e8	0.784045
$302$	−4.20646e8	−0.878805
$303$	0	0
$304$	−5.55052e7	−0.113312
$305$	−4.23069e8	−0.853811
$306$	0	0
$307$	−3.73739e8	−0.737197	−0.368599	−	0.929589i	$-0.620162\pi$
$307$	−3.73739e8	−0.737197	−0.368599	+	0.929589i	$0.620162\pi$
$308$	2.91584e8	0.568638
$309$	0	0
$310$	3.13236e8	0.597181
$311$	−3.86615e8	−0.728816	−0.364408	−	0.931239i	$-0.618729\pi$
$311$	−3.86615e8	−0.728816	−0.364408	+	0.931239i	$0.618729\pi$
$312$	0	0
$313$	8.00164e8	1.47494	0.737469	−	0.675381i	$-0.236021\pi$
$313$	8.00164e8	1.47494	0.737469	+	0.675381i	$0.236021\pi$
$314$	−9.54199e7	−0.173934
$315$	0	0
$316$	−1.72203e8	−0.306998
$317$	−2.63720e7	−0.0464981	−0.0232490	−	0.999730i	$-0.507401\pi$
$317$	−2.63720e7	−0.0464981	−0.0232490	+	0.999730i	$0.507401\pi$
$318$	0	0
$319$	1.39450e8	0.240520
$320$	−2.88915e8	−0.492884
$321$	0	0
$322$	8.43135e8	1.40735
$323$	−1.90763e8	−0.314982
$324$	0	0
$325$	−9.87428e7	−0.159556
$326$	−2.44049e8	−0.390135
$327$	0	0
$328$	5.83430e8	0.912915
$329$	1.09490e9	1.69507
$330$	0	0
$331$	−9.04550e8	−1.37099	−0.685495	−	0.728077i	$-0.740414\pi$
$331$	−9.04550e8	−1.37099	−0.685495	+	0.728077i	$0.740414\pi$
$332$	−2.83708e8	−0.425488
$333$	0	0
$334$	8.64297e8	1.26926
$335$	−3.59487e8	−0.522428
$336$	0	0
$337$	5.52918e8	0.786966	0.393483	−	0.919332i	$-0.371270\pi$
$337$	5.52918e8	0.786966	0.393483	+	0.919332i	$0.371270\pi$
$338$	−2.19376e8	−0.309016
$339$	0	0
$340$	−1.61380e8	−0.222676
$341$	−1.89578e9	−2.58909
$342$	0	0
$343$	4.21836e8	0.564435
$344$	5.16966e8	0.684712
$345$	0	0
$346$	−9.97179e8	−1.29422
$347$	4.17834e8	0.536847	0.268423	−	0.963301i	$-0.413497\pi$
$347$	4.17834e8	0.536847	0.268423	+	0.963301i	$0.413497\pi$
$348$	0	0
$349$	3.90455e8	0.491679	0.245840	−	0.969311i	$-0.420936\pi$
$349$	3.90455e8	0.491679	0.245840	+	0.969311i	$0.420936\pi$
$350$	−1.69552e8	−0.211381
$351$	0	0
$352$	7.24438e8	0.885323
$353$	−3.62310e8	−0.438398	−0.219199	−	0.975680i	$-0.570344\pi$
$353$	−3.62310e8	−0.438398	−0.219199	+	0.975680i	$0.570344\pi$
$354$	0	0
$355$	−2.37083e8	−0.281256
$356$	3.18093e7	0.0373662
$357$	0	0
$358$	7.40966e7	0.0853508
$359$	4.52892e8	0.516611	0.258306	−	0.966063i	$-0.416836\pi$
$359$	4.52892e8	0.516611	0.258306	+	0.966063i	$0.416836\pi$
$360$	0	0
$361$	−8.66328e8	−0.969186
$362$	−1.12935e9	−1.25126
$363$	0	0
$364$	−2.53277e8	−0.275259
$365$	−1.31613e8	−0.141669
$366$	0	0
$367$	−2.56397e8	−0.270759	−0.135379	−	0.990794i	$-0.543225\pi$
$367$	−2.56397e8	−0.270759	−0.135379	+	0.990794i	$0.543225\pi$
$368$	8.21747e8	0.859549
$369$	0	0
$370$	−2.31201e8	−0.237292
$371$	6.05419e8	0.615528
$372$	0	0
$373$	−1.21192e9	−1.20919	−0.604593	−	0.796535i	$-0.706664\pi$
$373$	−1.21192e9	−1.20919	−0.604593	+	0.796535i	$0.706664\pi$
$374$	−2.54311e9	−2.51371
$375$	0	0
$376$	1.52585e9	1.48031
$377$	−1.21130e8	−0.116428
$378$	0	0
$379$	−3.05842e8	−0.288576	−0.144288	−	0.989536i	$-0.546089\pi$
$379$	−3.05842e8	−0.288576	−0.144288	+	0.989536i	$0.546089\pi$
$380$	2.33010e7	0.0217837
$381$	0	0
$382$	4.21159e8	0.386567
$383$	−4.39109e8	−0.399371	−0.199686	−	0.979860i	$-0.563992\pi$
$383$	−4.39109e8	−0.399371	−0.199686	+	0.979860i	$0.563992\pi$
$384$	0	0
$385$	1.02617e9	0.916445
$386$	8.94969e8	0.792050
$387$	0	0
$388$	2.61866e8	0.227598
$389$	6.48774e7	0.0558818	0.0279409	−	0.999610i	$-0.491105\pi$
$389$	6.48774e7	0.0558818	0.0279409	+	0.999610i	$0.491105\pi$
$390$	0	0
$391$	2.82423e9	2.38936
$392$	−7.07162e8	−0.592949
$393$	0	0
$394$	5.33925e8	0.439788
$395$	−6.06031e8	−0.494772
$396$	0	0
$397$	9.24611e8	0.741639	0.370819	−	0.928705i	$-0.379077\pi$
$397$	9.24611e8	0.741639	0.370819	+	0.928705i	$0.379077\pi$
$398$	−5.53982e8	−0.440458
$399$	0	0
$400$	−1.65251e8	−0.129102
$401$	7.34813e8	0.569078	0.284539	−	0.958665i	$-0.408160\pi$
$401$	7.34813e8	0.569078	0.284539	+	0.958665i	$0.408160\pi$
$402$	0	0
$403$	1.64672e9	1.25329
$404$	7.25660e7	0.0547518
$405$	0	0
$406$	−2.07993e8	−0.154244
$407$	1.39928e9	1.02879
$408$	0	0
$409$	−2.51950e9	−1.82089	−0.910444	−	0.413632i	$-0.864260\pi$
$409$	−2.51950e9	−1.82089	−0.910444	+	0.413632i	$0.864260\pi$
$410$	4.45997e8	0.319587
$411$	0	0
$412$	−4.38178e8	−0.308681
$413$	−3.45687e9	−2.41467
$414$	0	0
$415$	−9.98449e8	−0.685737
$416$	−6.29265e8	−0.428556
$417$	0	0
$418$	3.67189e8	0.245908
$419$	7.23835e8	0.480718	0.240359	−	0.970684i	$-0.422735\pi$
$419$	7.23835e8	0.480718	0.240359	+	0.970684i	$0.422735\pi$
$420$	0	0
$421$	1.42856e9	0.933066	0.466533	−	0.884504i	$-0.345503\pi$
$421$	1.42856e9	0.933066	0.466533	+	0.884504i	$0.345503\pi$
$422$	1.44971e9	0.939050
$423$	0	0
$424$	8.43713e8	0.537544
$425$	−5.67944e8	−0.358876
$426$	0	0
$427$	−3.81907e9	−2.37388
$428$	7.08376e8	0.436728
$429$	0	0
$430$	3.95189e8	0.239699
$431$	−1.29599e9	−0.779708	−0.389854	−	0.920877i	$-0.627475\pi$
$431$	−1.29599e9	−0.779708	−0.389854	+	0.920877i	$0.627475\pi$
$432$	0	0
$433$	1.13189e9	0.670031	0.335015	−	0.942213i	$-0.391258\pi$
$433$	1.13189e9	0.670031	0.335015	+	0.942213i	$0.391258\pi$
$434$	2.82760e9	1.66037
$435$	0	0
$436$	−1.96685e8	−0.113650
$437$	−4.07778e8	−0.233743
$438$	0	0
$439$	2.54714e9	1.43690	0.718452	−	0.695577i	$-0.244851\pi$
$439$	2.54714e9	1.43690	0.718452	+	0.695577i	$0.244851\pi$
$440$	1.43007e9	0.800337
$441$	0	0
$442$	2.20901e9	1.21680
$443$	1.54563e9	0.844682	0.422341	−	0.906437i	$-0.361208\pi$
$443$	1.54563e9	0.844682	0.422341	+	0.906437i	$0.361208\pi$
$444$	0	0
$445$	1.11946e8	0.0602211
$446$	−1.78182e9	−0.951024
$447$	0	0
$448$	−2.60805e9	−1.37038
$449$	−3.45903e9	−1.80340	−0.901701	−	0.432361i	$-0.857681\pi$
$449$	−3.45903e9	−1.80340	−0.901701	+	0.432361i	$0.857681\pi$
$450$	0	0
$451$	−2.69928e9	−1.38557
$452$	2.83636e8	0.144470
$453$	0	0
$454$	−1.00192e9	−0.502502
$455$	−8.91357e8	−0.443620
$456$	0	0
$457$	−1.11696e9	−0.547432	−0.273716	−	0.961811i	$-0.588253\pi$
$457$	−1.11696e9	−0.547432	−0.273716	+	0.961811i	$0.588253\pi$
$458$	−6.46685e8	−0.314531
$459$	0	0
$460$	−3.44968e8	−0.165244
$461$	−3.08047e9	−1.46442	−0.732208	−	0.681081i	$-0.761510\pi$
$461$	−3.08047e9	−1.46442	−0.732208	+	0.681081i	$0.761510\pi$
$462$	0	0
$463$	−1.39607e9	−0.653694	−0.326847	−	0.945077i	$-0.605986\pi$
$463$	−1.39607e9	−0.653694	−0.326847	+	0.945077i	$0.605986\pi$
$464$	−2.02717e8	−0.0942056
$465$	0	0
$466$	2.25163e9	1.03073
$467$	1.65734e9	0.753014	0.376507	−	0.926414i	$-0.377125\pi$
$467$	1.65734e9	0.753014	0.376507	+	0.926414i	$0.377125\pi$
$468$	0	0
$469$	−3.24511e9	−1.45253
$470$	1.16642e9	0.518217
$471$	0	0
$472$	−4.81750e9	−2.10875
$473$	−2.39178e9	−1.03922
$474$	0	0
$475$	8.20029e7	0.0351077
$476$	−1.45679e9	−0.619116
$477$	0	0
$478$	4.62536e8	0.193708
$479$	−4.23641e9	−1.76126	−0.880630	−	0.473804i	$-0.842880\pi$
$479$	−4.23641e9	−1.76126	−0.880630	+	0.473804i	$0.842880\pi$
$480$	0	0
$481$	−1.21545e9	−0.498001
$482$	1.34956e9	0.548945
$483$	0	0
$484$	−1.18786e9	−0.476218
$485$	9.21582e8	0.366807
$486$	0	0
$487$	2.39585e9	0.939959	0.469979	−	0.882677i	$-0.344261\pi$
$487$	2.39585e9	0.939959	0.469979	+	0.882677i	$0.344261\pi$
$488$	−5.32225e9	−2.07313
$489$	0	0
$490$	−5.40582e8	−0.207575
$491$	−3.93760e9	−1.50123	−0.750613	−	0.660742i	$-0.770242\pi$
$491$	−3.93760e9	−1.50123	−0.750613	+	0.660742i	$0.770242\pi$
$492$	0	0
$493$	−6.96708e8	−0.261871
$494$	−3.18950e8	−0.119036
$495$	0	0
$496$	2.75587e9	1.01408
$497$	−2.14016e9	−0.781986
$498$	0	0
$499$	3.63697e9	1.31035	0.655176	−	0.755476i	$-0.272595\pi$
$499$	3.63697e9	1.31035	0.655176	+	0.755476i	$0.272595\pi$
$500$	6.93721e7	0.0248193
$501$	0	0
$502$	−6.16991e7	−0.0217679
$503$	1.93139e9	0.676678	0.338339	−	0.941024i	$-0.390135\pi$
$503$	1.93139e9	0.676678	0.338339	+	0.941024i	$0.390135\pi$
$504$	0	0
$505$	2.55381e8	0.0882407
$506$	−5.43619e9	−1.86538
$507$	0	0
$508$	1.40403e9	0.475173
$509$	4.12567e9	1.38670	0.693350	−	0.720601i	$-0.256134\pi$
$509$	4.12567e9	1.38670	0.693350	+	0.720601i	$0.256134\pi$
$510$	0	0
$511$	−1.18808e9	−0.393888
$512$	−3.18188e9	−1.04770
$513$	0	0
$514$	8.82759e8	0.286729
$515$	−1.54208e9	−0.497485
$516$	0	0
$517$	−7.05943e9	−2.24674
$518$	−2.08706e9	−0.659754
$519$	0	0
$520$	−1.24220e9	−0.387417
$521$	1.88529e9	0.584044	0.292022	−	0.956412i	$-0.405672\pi$
$521$	1.88529e9	0.584044	0.292022	+	0.956412i	$0.405672\pi$
$522$	0	0
$523$	−7.09365e8	−0.216827	−0.108414	−	0.994106i	$-0.534577\pi$
$523$	−7.09365e8	−0.216827	−0.108414	+	0.994106i	$0.534577\pi$
$524$	9.33432e8	0.283415
$525$	0	0
$526$	2.43108e9	0.728364
$527$	9.47152e9	2.81892
$528$	0	0
$529$	2.63227e9	0.773100
$530$	6.44968e8	0.188180
$531$	0	0
$532$	2.10339e8	0.0605661
$533$	2.34466e9	0.670710
$534$	0	0
$535$	2.49298e9	0.703852
$536$	−4.52239e9	−1.26850
$537$	0	0
$538$	3.18744e9	0.882479
$539$	3.27173e9	0.899947
$540$	0	0
$541$	−3.77819e9	−1.02587	−0.512936	−	0.858427i	$-0.671442\pi$
$541$	−3.77819e9	−1.02587	−0.512936	+	0.858427i	$0.671442\pi$
$542$	−3.75494e9	−1.01299
$543$	0	0
$544$	−3.61938e9	−0.963912
$545$	−6.92192e8	−0.183163
$546$	0	0
$547$	−1.72988e8	−0.0451920	−0.0225960	−	0.999745i	$-0.507193\pi$
$547$	−1.72988e8	−0.0451920	−0.0225960	+	0.999745i	$0.507193\pi$
$548$	−9.64230e8	−0.250293
$549$	0	0
$550$	1.09320e9	0.280176
$551$	1.00595e8	0.0256180
$552$	0	0
$553$	−5.47067e9	−1.37563
$554$	−3.10759e8	−0.0776496
$555$	0	0
$556$	−4.94664e8	−0.122053
$557$	6.54188e9	1.60402	0.802009	−	0.597312i	$-0.203764\pi$
$557$	6.54188e9	1.60402	0.802009	+	0.597312i	$0.203764\pi$
$558$	0	0
$559$	2.07756e9	0.503051
$560$	−1.49173e9	−0.358948
$561$	0	0
$562$	2.66332e9	0.632916
$563$	1.23750e9	0.292257	0.146129	−	0.989266i	$-0.453319\pi$
$563$	1.23750e9	0.292257	0.146129	+	0.989266i	$0.453319\pi$
$564$	0	0
$565$	9.98197e8	0.232834
$566$	−4.94638e9	−1.14665
$567$	0	0
$568$	−2.98253e9	−0.682913
$569$	6.92969e9	1.57696	0.788481	−	0.615059i	$-0.210868\pi$
$569$	6.92969e9	1.57696	0.788481	+	0.615059i	$0.210868\pi$
$570$	0	0
$571$	3.49878e8	0.0786486	0.0393243	−	0.999227i	$-0.487479\pi$
$571$	3.49878e8	0.0786486	0.0393243	+	0.999227i	$0.487479\pi$
$572$	1.63303e9	0.364844
$573$	0	0
$574$	4.02604e9	0.888559
$575$	−1.21404e9	−0.266316
$576$	0	0
$577$	−4.18307e9	−0.906525	−0.453262	−	0.891377i	$-0.649740\pi$
$577$	−4.18307e9	−0.906525	−0.453262	+	0.891377i	$0.649740\pi$
$578$	8.75957e9	1.88684
$579$	0	0
$580$	8.51002e7	0.0181106
$581$	−9.01305e9	−1.90658
$582$	0	0
$583$	−3.90349e9	−0.815856
$584$	−1.65571e9	−0.343985
$585$	0	0
$586$	6.62876e8	0.136079
$587$	3.65847e9	0.746563	0.373281	−	0.927718i	$-0.378233\pi$
$587$	3.65847e9	0.746563	0.373281	+	0.927718i	$0.378233\pi$
$588$	0	0
$589$	−1.36755e9	−0.275766
$590$	−3.68269e9	−0.738215
$591$	0	0
$592$	−2.03412e9	−0.402950
$593$	2.06095e9	0.405860	0.202930	−	0.979193i	$-0.434954\pi$
$593$	2.06095e9	0.405860	0.202930	+	0.979193i	$0.434954\pi$
$594$	0	0
$595$	−5.12686e9	−0.997796
$596$	−2.11196e9	−0.408624
$597$	0	0
$598$	4.72201e9	0.902969
$599$	−5.41224e8	−0.102892	−0.0514462	−	0.998676i	$-0.516383\pi$
$599$	−5.41224e8	−0.102892	−0.0514462	+	0.998676i	$0.516383\pi$
$600$	0	0
$601$	5.57372e9	1.04733	0.523666	−	0.851923i	$-0.324564\pi$
$601$	5.57372e9	1.04733	0.523666	+	0.851923i	$0.324564\pi$
$602$	3.56740e9	0.666444
$603$	0	0
$604$	1.55362e9	0.286890
$605$	−4.18042e9	−0.767495
$606$	0	0
$607$	−5.57346e9	−1.01150	−0.505748	−	0.862681i	$-0.668783\pi$
$607$	−5.57346e9	−1.01150	−0.505748	+	0.862681i	$0.668783\pi$
$608$	5.22586e8	0.0942964
$609$	0	0
$610$	−4.06854e9	−0.725745
$611$	6.13200e9	1.08757
$612$	0	0
$613$	−2.83033e9	−0.496279	−0.248140	−	0.968724i	$-0.579819\pi$
$613$	−2.83033e9	−0.496279	−0.248140	+	0.968724i	$0.579819\pi$
$614$	−3.59414e9	−0.626623
$615$	0	0
$616$	1.29093e10	2.22521
$617$	−8.56656e8	−0.146828	−0.0734139	−	0.997302i	$-0.523389\pi$
$617$	−8.56656e8	−0.146828	−0.0734139	+	0.997302i	$0.523389\pi$
$618$	0	0
$619$	−2.44610e9	−0.414530	−0.207265	−	0.978285i	$-0.566456\pi$
$619$	−2.44610e9	−0.414530	−0.207265	+	0.978285i	$0.566456\pi$
$620$	−1.15691e9	−0.194952
$621$	0	0
$622$	−3.71798e9	−0.619498
$623$	1.01054e9	0.167435
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	7.69496e9	1.25371
$627$	0	0
$628$	3.52425e8	0.0567816
$629$	−6.99098e9	−1.12011
$630$	0	0
$631$	−8.44592e9	−1.33827	−0.669136	−	0.743140i	$-0.733336\pi$
$631$	−8.44592e9	−1.33827	−0.669136	+	0.743140i	$0.733336\pi$
$632$	−7.62394e9	−1.20135
$633$	0	0
$634$	−2.53612e8	−0.0395237
$635$	4.94118e9	0.765812
$636$	0	0
$637$	−2.84191e9	−0.435634
$638$	1.34105e9	0.204444
$639$	0	0
$640$	−1.18522e9	−0.178719
$641$	2.33418e9	0.350051	0.175025	−	0.984564i	$-0.443999\pi$
$641$	2.33418e9	0.350051	0.175025	+	0.984564i	$0.443999\pi$
$642$	0	0
$643$	−3.12129e9	−0.463015	−0.231508	−	0.972833i	$-0.574366\pi$
$643$	−3.12129e9	−0.463015	−0.231508	+	0.972833i	$0.574366\pi$
$644$	−3.11404e9	−0.459435
$645$	0	0
$646$	−1.83452e9	−0.267737
$647$	−7.79684e8	−0.113176	−0.0565879	−	0.998398i	$-0.518022\pi$
$647$	−7.79684e8	−0.113176	−0.0565879	+	0.998398i	$0.518022\pi$
$648$	0	0
$649$	2.22885e10	3.20054
$650$	−9.49583e8	−0.135624
$651$	0	0
$652$	9.01372e8	0.127361
$653$	1.35874e9	0.190959	0.0954797	−	0.995431i	$-0.469562\pi$
$653$	1.35874e9	0.190959	0.0954797	+	0.995431i	$0.469562\pi$
$654$	0	0
$655$	3.28502e9	0.456765
$656$	3.92391e9	0.542694
$657$	0	0
$658$	1.05293e10	1.44082
$659$	5.06175e9	0.688972	0.344486	−	0.938791i	$-0.388053\pi$
$659$	5.06175e9	0.688972	0.344486	+	0.938791i	$0.388053\pi$
$660$	0	0
$661$	7.91589e9	1.06609	0.533046	−	0.846086i	$-0.321047\pi$
$661$	7.91589e9	1.06609	0.533046	+	0.846086i	$0.321047\pi$
$662$	−8.69881e9	−1.16535
$663$	0	0
$664$	−1.25606e10	−1.66503
$665$	7.40245e8	0.0976112
$666$	0	0
$667$	−1.48929e9	−0.194330
$668$	−3.19220e9	−0.414355
$669$	0	0
$670$	−3.45709e9	−0.444067
$671$	2.46237e10	3.14648
$672$	0	0
$673$	9.03515e9	1.14257	0.571285	−	0.820752i	$-0.306445\pi$
$673$	9.03515e9	1.14257	0.571285	+	0.820752i	$0.306445\pi$
$674$	5.31726e9	0.668927
$675$	0	0
$676$	8.10245e8	0.100880
$677$	5.57586e9	0.690640	0.345320	−	0.938485i	$-0.387770\pi$
$677$	5.57586e9	0.690640	0.345320	+	0.938485i	$0.387770\pi$
$678$	0	0
$679$	8.31917e9	1.01985
$680$	−7.14479e9	−0.871382
$681$	0	0
$682$	−1.82312e10	−2.20074
$683$	−2.19921e7	−0.00264116	−0.00132058	−	0.999999i	$-0.500420\pi$
$683$	−2.19921e7	−0.00264116	−0.00132058	+	0.999999i	$0.500420\pi$
$684$	0	0
$685$	−3.39341e9	−0.403384
$686$	4.05668e9	0.479773
$687$	0	0
$688$	3.47690e9	0.407036
$689$	3.39067e9	0.394929
$690$	0	0
$691$	8.12411e8	0.0936705	0.0468352	−	0.998903i	$-0.485086\pi$
$691$	8.12411e8	0.0936705	0.0468352	+	0.998903i	$0.485086\pi$
$692$	3.68299e9	0.422503
$693$	0	0
$694$	4.01819e9	0.456323
$695$	−1.74087e9	−0.196707
$696$	0	0
$697$	1.34859e10	1.50857
$698$	3.75490e9	0.417931
$699$	0	0
$700$	6.26226e8	0.0690061
$701$	−1.78499e10	−1.95714	−0.978570	−	0.205913i	$-0.933984\pi$
$701$	−1.78499e10	−1.95714	−0.978570	+	0.205913i	$0.933984\pi$
$702$	0	0
$703$	1.00940e9	0.109577
$704$	1.68156e10	1.81639
$705$	0	0
$706$	−3.48424e9	−0.372641
$707$	2.30534e9	0.245339
$708$	0	0
$709$	−6.55336e9	−0.690561	−0.345281	−	0.938499i	$-0.612216\pi$
$709$	−6.55336e9	−0.690561	−0.345281	+	0.938499i	$0.612216\pi$
$710$	−2.27996e9	−0.239069
$711$	0	0
$712$	1.40829e9	0.146222
$713$	2.02464e10	2.09187
$714$	0	0
$715$	5.74710e9	0.588000
$716$	−2.73669e8	−0.0278631
$717$	0	0
$718$	4.35534e9	0.439123
$719$	−1.36705e10	−1.37162	−0.685808	−	0.727783i	$-0.740551\pi$
$719$	−1.36705e10	−1.37162	−0.685808	+	0.727783i	$0.740551\pi$
$720$	0	0
$721$	−1.39204e10	−1.38318
$722$	−8.33124e9	−0.823815
$723$	0	0
$724$	4.17115e9	0.408480
$725$	2.99492e8	0.0291879
$726$	0	0
$727$	8.26629e9	0.797884	0.398942	−	0.916976i	$-0.369377\pi$
$727$	8.26629e9	0.797884	0.398942	+	0.916976i	$0.369377\pi$
$728$	−1.12134e10	−1.07715
$729$	0	0
$730$	−1.26569e9	−0.120420
$731$	1.19496e10	1.13147
$732$	0	0
$733$	9.06130e9	0.849819	0.424910	−	0.905236i	$-0.360306\pi$
$733$	9.06130e9	0.849819	0.424910	+	0.905236i	$0.360306\pi$
$734$	−2.46570e9	−0.230147
$735$	0	0
$736$	−7.73682e9	−0.715303
$737$	2.09231e10	1.92526
$738$	0	0
$739$	3.14272e9	0.286451	0.143225	−	0.989690i	$-0.454253\pi$
$739$	3.14272e9	0.286451	0.143225	+	0.989690i	$0.454253\pi$
$740$	8.53920e8	0.0774652
$741$	0	0
$742$	5.82216e9	0.523203
$743$	7.44283e9	0.665698	0.332849	−	0.942980i	$-0.391990\pi$
$743$	7.44283e9	0.665698	0.332849	+	0.942980i	$0.391990\pi$
$744$	0	0
$745$	−7.43260e9	−0.658558
$746$	−1.16547e10	−1.02782
$747$	0	0
$748$	9.39276e9	0.820611
$749$	2.25043e10	1.95695
$750$	0	0
$751$	2.05526e9	0.177063	0.0885315	−	0.996073i	$-0.471783\pi$
$751$	2.05526e9	0.177063	0.0885315	+	0.996073i	$0.471783\pi$
$752$	1.02622e10	0.879991
$753$	0	0
$754$	−1.16487e9	−0.0989644
$755$	5.46763e9	0.462365
$756$	0	0
$757$	1.36734e10	1.14562	0.572811	−	0.819688i	$-0.305853\pi$
$757$	1.36734e10	1.14562	0.572811	+	0.819688i	$0.305853\pi$
$758$	−2.94120e9	−0.245292
$759$	0	0
$760$	1.03161e9	0.0852445
$761$	−2.15764e10	−1.77473	−0.887364	−	0.461070i	$-0.847466\pi$
$761$	−2.15764e10	−1.77473	−0.887364	+	0.461070i	$0.847466\pi$
$762$	0	0
$763$	−6.24845e9	−0.509257
$764$	−1.55551e9	−0.126196
$765$	0	0
$766$	−4.22279e9	−0.339468
$767$	−1.93603e10	−1.54928
$768$	0	0
$769$	1.82302e10	1.44560	0.722801	−	0.691056i	$-0.242854\pi$
$769$	1.82302e10	1.44560	0.722801	+	0.691056i	$0.242854\pi$
$770$	9.86839e9	0.778984
$771$	0	0
$772$	−3.30549e9	−0.258568
$773$	1.37669e10	1.07203	0.536016	−	0.844208i	$-0.319929\pi$
$773$	1.37669e10	1.07203	0.536016	+	0.844208i	$0.319929\pi$
$774$	0	0
$775$	−4.07150e9	−0.314194
$776$	1.15936e10	0.890640
$777$	0	0
$778$	6.23909e8	0.0474999
$779$	−1.94717e9	−0.147578
$780$	0	0
$781$	1.37989e10	1.03649
$782$	2.71598e10	2.03097
$783$	0	0
$784$	−4.75607e9	−0.352486
$785$	1.24029e9	0.0915120
$786$	0	0
$787$	−1.48460e10	−1.08567	−0.542833	−	0.839840i	$-0.682648\pi$
$787$	−1.48460e10	−1.08567	−0.542833	+	0.839840i	$0.682648\pi$
$788$	−1.97200e9	−0.143571
$789$	0	0
$790$	−5.82804e9	−0.420560
$791$	9.01078e9	0.647358
$792$	0	0
$793$	−2.13888e10	−1.52311
$794$	8.89174e9	0.630398
$795$	0	0
$796$	2.04608e9	0.143789
$797$	2.21766e9	0.155164	0.0775819	−	0.996986i	$-0.475280\pi$
$797$	2.21766e9	0.155164	0.0775819	+	0.996986i	$0.475280\pi$
$798$	0	0
$799$	3.52697e10	2.44618
$800$	1.55585e9	0.107437
$801$	0	0
$802$	7.06650e9	0.483720
$803$	7.66025e9	0.522082
$804$	0	0
$805$	−1.09592e10	−0.740448
$806$	1.58361e10	1.06531
$807$	0	0
$808$	3.21272e9	0.214256
$809$	2.10843e10	1.40004	0.700019	−	0.714124i	$-0.253175\pi$
$809$	2.10843e10	1.40004	0.700019	+	0.714124i	$0.253175\pi$
$810$	0	0
$811$	−4.32086e9	−0.284444	−0.142222	−	0.989835i	$-0.545425\pi$
$811$	−4.32086e9	−0.284444	−0.142222	+	0.989835i	$0.545425\pi$
$812$	7.68204e8	0.0503536
$813$	0	0
$814$	1.34565e10	0.874475
$815$	3.17219e9	0.205262
$816$	0	0
$817$	−1.72535e9	−0.110688
$818$	−2.42294e10	−1.54777
$819$	0	0
$820$	−1.64725e9	−0.104330
$821$	−2.04722e10	−1.29111	−0.645554	−	0.763714i	$-0.723374\pi$
$821$	−2.04722e10	−1.29111	−0.645554	+	0.763714i	$0.723374\pi$
$822$	0	0
$823$	3.10590e10	1.94217	0.971086	−	0.238729i	$-0.0767307\pi$
$823$	3.10590e10	1.94217	0.971086	+	0.238729i	$0.0767307\pi$
$824$	−1.93995e10	−1.20794
$825$	0	0
$826$	−3.32438e10	−2.05249
$827$	2.02954e10	1.24775	0.623876	−	0.781523i	$-0.285557\pi$
$827$	2.02954e10	1.24775	0.623876	+	0.781523i	$0.285557\pi$
$828$	0	0
$829$	−2.07480e10	−1.26484	−0.632421	−	0.774625i	$-0.717939\pi$
$829$	−2.07480e10	−1.26484	−0.632421	+	0.774625i	$0.717939\pi$
$830$	−9.60182e9	−0.582881
$831$	0	0
$832$	−1.46065e10	−0.879252
$833$	−1.63459e10	−0.979834
$834$	0	0
$835$	−1.12343e10	−0.667795
$836$	−1.35618e9	−0.0802778
$837$	0	0
$838$	6.96093e9	0.408614
$839$	−7.97235e9	−0.466036	−0.233018	−	0.972472i	$-0.574860\pi$
$839$	−7.97235e9	−0.466036	−0.233018	+	0.972472i	$0.574860\pi$
$840$	0	0
$841$	−1.68825e10	−0.978702
$842$	1.37381e10	0.793112
$843$	0	0
$844$	−5.35439e9	−0.306557
$845$	2.85149e9	0.162582
$846$	0	0
$847$	−3.77368e10	−2.13390
$848$	5.67446e9	0.319550
$849$	0	0
$850$	−5.46176e9	−0.305047
$851$	−1.49440e10	−0.831215
$852$	0	0
$853$	−8.83791e9	−0.487560	−0.243780	−	0.969831i	$-0.578387\pi$
$853$	−8.83791e9	−0.487560	−0.243780	+	0.969831i	$0.578387\pi$
$854$	−3.67269e10	−2.01782
$855$	0	0
$856$	3.13620e10	1.70901
$857$	2.22532e9	0.120770	0.0603852	−	0.998175i	$-0.480767\pi$
$857$	2.22532e9	0.120770	0.0603852	+	0.998175i	$0.480767\pi$
$858$	0	0
$859$	6.26760e9	0.337384	0.168692	−	0.985669i	$-0.446046\pi$
$859$	6.26760e9	0.337384	0.168692	+	0.985669i	$0.446046\pi$
$860$	−1.45960e9	−0.0782507
$861$	0	0
$862$	−1.24632e10	−0.662757
$863$	1.42420e10	0.754280	0.377140	−	0.926156i	$-0.376908\pi$
$863$	1.42420e10	0.754280	0.377140	+	0.926156i	$0.376908\pi$
$864$	0	0
$865$	1.29615e10	0.680925
$866$	1.08850e10	0.569531
$867$	0	0
$868$	−1.04435e10	−0.542034
$869$	3.52726e10	1.82334
$870$	0	0
$871$	−1.81743e10	−0.931956
$872$	−8.70785e9	−0.444737
$873$	0	0
$874$	−3.92149e9	−0.198683
$875$	2.20387e9	0.111214
$876$	0	0
$877$	−1.50667e10	−0.754255	−0.377128	−	0.926161i	$-0.623088\pi$
$877$	−1.50667e10	−0.754255	−0.377128	+	0.926161i	$0.623088\pi$
$878$	2.44952e10	1.22138
$879$	0	0
$880$	9.61805e9	0.475771
$881$	−1.20809e10	−0.595230	−0.297615	−	0.954686i	$-0.596191\pi$
$881$	−1.20809e10	−0.595230	−0.297615	+	0.954686i	$0.596191\pi$
$882$	0	0
$883$	2.22376e10	1.08699	0.543495	−	0.839412i	$-0.317101\pi$
$883$	2.22376e10	1.08699	0.543495	+	0.839412i	$0.317101\pi$
$884$	−8.15879e9	−0.397231
$885$	0	0
$886$	1.48639e10	0.717986
$887$	−1.47398e10	−0.709185	−0.354593	−	0.935021i	$-0.615380\pi$
$887$	−1.47398e10	−0.709185	−0.354593	+	0.935021i	$0.615380\pi$
$888$	0	0
$889$	4.46043e10	2.12922
$890$	1.07655e9	0.0511884
$891$	0	0
$892$	6.58098e9	0.310466
$893$	−5.09244e9	−0.239302
$894$	0	0
$895$	−9.63121e8	−0.0449056
$896$	−1.06991e10	−0.496898
$897$	0	0
$898$	−3.32646e10	−1.53290
$899$	−4.99459e9	−0.229267
$900$	0	0
$901$	1.95023e10	0.888278
$902$	−2.59582e10	−1.17775
$903$	0	0
$904$	1.25574e10	0.565342
$905$	1.46795e10	0.658325
$906$	0	0
$907$	4.24934e9	0.189102	0.0945509	−	0.995520i	$-0.469858\pi$
$907$	4.24934e9	0.189102	0.0945509	+	0.995520i	$0.469858\pi$
$908$	3.70051e9	0.164044
$909$	0	0
$910$	−8.57194e9	−0.377080
$911$	−2.25340e10	−0.987468	−0.493734	−	0.869613i	$-0.664368\pi$
$911$	−2.25340e10	−0.987468	−0.493734	+	0.869613i	$0.664368\pi$
$912$	0	0
$913$	5.81124e10	2.52709
$914$	−1.07415e10	−0.465321
$915$	0	0
$916$	2.38847e9	0.102680
$917$	2.96540e10	1.26996
$918$	0	0
$919$	3.03997e10	1.29201	0.646004	−	0.763334i	$-0.276439\pi$
$919$	3.03997e10	1.29201	0.646004	+	0.763334i	$0.276439\pi$
$920$	−1.52728e10	−0.646638
$921$	0	0
$922$	−2.96241e10	−1.24476
$923$	−1.19860e10	−0.501730
$924$	0	0
$925$	3.00519e9	0.124847
$926$	−1.34257e10	−0.555645
$927$	0	0
$928$	1.90860e9	0.0783964
$929$	1.39057e10	0.569034	0.284517	−	0.958671i	$-0.408167\pi$
$929$	1.39057e10	0.569034	0.284517	+	0.958671i	$0.408167\pi$
$930$	0	0
$931$	2.36012e9	0.0958540
$932$	−8.31618e9	−0.336487
$933$	0	0
$934$	1.59382e10	0.640067
$935$	3.30559e10	1.32254
$936$	0	0
$937$	−4.48169e10	−1.77973	−0.889864	−	0.456226i	$-0.849201\pi$
$937$	−4.48169e10	−1.77973	−0.889864	+	0.456226i	$0.849201\pi$
$938$	−3.12073e10	−1.23466
$939$	0	0
$940$	−4.30806e9	−0.169174
$941$	2.23581e10	0.874726	0.437363	−	0.899285i	$-0.355913\pi$
$941$	2.23581e10	0.874726	0.437363	+	0.899285i	$0.355913\pi$
$942$	0	0
$943$	2.88276e10	1.11948
$944$	−3.24005e10	−1.25357
$945$	0	0
$946$	−2.30011e10	−0.883343
$947$	−4.03196e10	−1.54273	−0.771367	−	0.636391i	$-0.780427\pi$
$947$	−4.03196e10	−1.54273	−0.771367	+	0.636391i	$0.780427\pi$
$948$	0	0
$949$	−6.65389e9	−0.252722
$950$	7.88600e8	0.0298417
$951$	0	0
$952$	−6.44964e10	−2.42274
$953$	−5.06709e10	−1.89641	−0.948207	−	0.317653i	$-0.897105\pi$
$953$	−5.06709e10	−1.89641	−0.948207	+	0.317653i	$0.897105\pi$
$954$	0	0
$955$	−5.47430e9	−0.203384
$956$	−1.70833e9	−0.0632368
$957$	0	0
$958$	−4.07404e10	−1.49708
$959$	−3.06324e10	−1.12154
$960$	0	0
$961$	4.03873e10	1.46795
$962$	−1.16887e10	−0.423304
$963$	0	0
$964$	−4.98449e9	−0.179205
$965$	−1.16330e10	−0.416721
$966$	0	0
$967$	−3.88042e10	−1.38002	−0.690012	−	0.723798i	$-0.742395\pi$
$967$	−3.88042e10	−1.38002	−0.690012	+	0.723798i	$0.742395\pi$
$968$	−5.25901e10	−1.86355
$969$	0	0
$970$	8.86260e9	0.311789
$971$	3.82427e10	1.34054	0.670272	−	0.742115i	$-0.266177\pi$
$971$	3.82427e10	1.34054	0.670272	+	0.742115i	$0.266177\pi$
$972$	0	0
$973$	−1.57149e10	−0.546911
$974$	2.30403e10	0.798971
$975$	0	0
$976$	−3.57952e10	−1.23240
$977$	3.75539e10	1.28832	0.644160	−	0.764891i	$-0.277207\pi$
$977$	3.75539e10	1.28832	0.644160	+	0.764891i	$0.277207\pi$
$978$	0	0
$979$	−6.51556e9	−0.221928
$980$	1.99659e9	0.0677638
$981$	0	0
$982$	−3.78668e10	−1.27605
$983$	2.22662e10	0.747668	0.373834	−	0.927496i	$-0.378043\pi$
$983$	2.22662e10	0.747668	0.373834	+	0.927496i	$0.378043\pi$
$984$	0	0
$985$	−6.94006e9	−0.231386
$986$	−6.70005e9	−0.222592
$987$	0	0
$988$	1.17801e9	0.0388598
$989$	2.55436e10	0.839643
$990$	0	0
$991$	2.43258e10	0.793981	0.396990	−	0.917823i	$-0.370055\pi$
$991$	2.43258e10	0.793981	0.396990	+	0.917823i	$0.370055\pi$
$992$	−2.59467e10	−0.843902
$993$	0	0
$994$	−2.05813e10	−0.664694
$995$	7.20075e9	0.231738
$996$	0	0
$997$	−3.31780e10	−1.06027	−0.530135	−	0.847913i	$-0.677859\pi$
$997$	−3.31780e10	−1.06027	−0.530135	+	0.847913i	$0.677859\pi$
$998$	3.49758e10	1.11381
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.8.a.d.1.5	yes	7
3.2	odd	2		405.8.a.a.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.8.a.a.1.3	✓	7	3.2	odd	2
405.8.a.d.1.5	yes	7	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)

\(f(q)\)	\(=\)	\(q+9.61673 q^{2} -35.5185 q^{4} -125.000 q^{5} -1128.38 q^{7} -1572.51 q^{8} +O(q^{10})\)	\(q+9.61673 q^{2} -35.5185 q^{4} -125.000 q^{5} -1128.38 q^{7} -1572.51 q^{8} -1202.09 q^{10} +7275.33 q^{11} -6319.54 q^{13} -10851.3 q^{14} -10576.1 q^{16} -36348.4 q^{17} +5248.19 q^{19} +4439.81 q^{20} +69964.9 q^{22} -77698.7 q^{23} +15625.0 q^{25} -60773.3 q^{26} +40078.4 q^{28} +19167.5 q^{29} -260576. q^{31} +99574.5 q^{32} -349553. q^{34} +141048. q^{35} +192332. q^{37} +50470.4 q^{38} +196564. q^{40} -371018. q^{41} -328752. q^{43} -258409. q^{44} -747208. q^{46} -970324. q^{47} +449702. q^{49} +150261. q^{50} +224461. q^{52} -536538. q^{53} -909417. q^{55} +1.77439e6 q^{56} +184329. q^{58} +3.06357e6 q^{59} +3.38455e6 q^{61} -2.50589e6 q^{62} +2.31132e6 q^{64} +789943. q^{65} +2.87590e6 q^{67} +1.29104e6 q^{68} +1.35642e6 q^{70} +1.89666e6 q^{71} +1.05291e6 q^{73} +1.84961e6 q^{74} -186408. q^{76} -8.20935e6 q^{77} +4.84825e6 q^{79} +1.32201e6 q^{80} -3.56798e6 q^{82} +7.98760e6 q^{83} +4.54355e6 q^{85} -3.16152e6 q^{86} -1.14406e7 q^{88} -895568. q^{89} +7.13085e6 q^{91} +2.75974e6 q^{92} -9.33134e6 q^{94} -656024. q^{95} -7.37266e6 q^{97} +4.32466e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 15 q^{2} + 457 q^{4} - 875 q^{5} + 614 q^{7} + 4605 q^{8}+O(q^{10}) \) 7 * q + 15 * q^2 + 457 * q^4 - 875 * q^5 + 614 * q^7 + 4605 * q^8	\( 7 q + 15 q^{2} + 457 q^{4} - 875 q^{5} + 614 q^{7} + 4605 q^{8} - 1875 q^{10} + 3753 q^{11} - 6562 q^{13} + 19653 q^{14} + 27625 q^{16} - 23862 q^{17} + 33251 q^{19} - 57125 q^{20} + 108060 q^{22} + 62682 q^{23} + 109375 q^{25} - 186789 q^{26} + 149303 q^{28} + 78939 q^{29} - 703165 q^{31} + 1266165 q^{32} - 589398 q^{34} - 76750 q^{35} + 336776 q^{37} + 498855 q^{38} - 575625 q^{40} + 480801 q^{41} - 442792 q^{43} - 9372 q^{44} + 213447 q^{46} + 1122744 q^{47} - 954843 q^{49} + 234375 q^{50} - 1324819 q^{52} + 692028 q^{53} - 469125 q^{55} + 6614175 q^{56} - 3971220 q^{58} + 1274487 q^{59} - 2485258 q^{61} - 2438700 q^{62} + 10358137 q^{64} + 820250 q^{65} - 892402 q^{67} + 926646 q^{68} - 2456625 q^{70} + 8286201 q^{71} + 788666 q^{73} - 537042 q^{74} + 16155485 q^{76} - 5060322 q^{77} - 1308352 q^{79} - 3453125 q^{80} + 716235 q^{82} + 9226704 q^{83} + 2982750 q^{85} + 15948900 q^{86} + 16624320 q^{88} + 4459791 q^{89} - 5701496 q^{91} + 8926029 q^{92} + 43736397 q^{94} - 4156375 q^{95} - 11086048 q^{97} + 21087450 q^{98}+O(q^{100}) \) 7 * q + 15 * q^2 + 457 * q^4 - 875 * q^5 + 614 * q^7 + 4605 * q^8 - 1875 * q^10 + 3753 * q^11 - 6562 * q^13 + 19653 * q^14 + 27625 * q^16 - 23862 * q^17 + 33251 * q^19 - 57125 * q^20 + 108060 * q^22 + 62682 * q^23 + 109375 * q^25 - 186789 * q^26 + 149303 * q^28 + 78939 * q^29 - 703165 * q^31 + 1266165 * q^32 - 589398 * q^34 - 76750 * q^35 + 336776 * q^37 + 498855 * q^38 - 575625 * q^40 + 480801 * q^41 - 442792 * q^43 - 9372 * q^44 + 213447 * q^46 + 1122744 * q^47 - 954843 * q^49 + 234375 * q^50 - 1324819 * q^52 + 692028 * q^53 - 469125 * q^55 + 6614175 * q^56 - 3971220 * q^58 + 1274487 * q^59 - 2485258 * q^61 - 2438700 * q^62 + 10358137 * q^64 + 820250 * q^65 - 892402 * q^67 + 926646 * q^68 - 2456625 * q^70 + 8286201 * q^71 + 788666 * q^73 - 537042 * q^74 + 16155485 * q^76 - 5060322 * q^77 - 1308352 * q^79 - 3453125 * q^80 + 716235 * q^82 + 9226704 * q^83 + 2982750 * q^85 + 15948900 * q^86 + 16624320 * q^88 + 4459791 * q^89 - 5701496 * q^91 + 8926029 * q^92 + 43736397 * q^94 - 4156375 * q^95 - 11086048 * q^97 + 21087450 * q^98

Introduction

L-functions

Modular forms

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