LMFDB - Embedded newform 25.8.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,8,Mod(1,25)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("25.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 25 = 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	25.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:	$7.80962563710$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{29}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 7 $ x^2 - x - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.19258$ of defining polynomial
Character	$\chi$	$=$	25.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.7703 q^{2} +32.3110 q^{3} -12.0000 q^{4} -348.000 q^{6} +420.043 q^{7} +1507.85 q^{8} -1143.00 q^{9} +O(q^{10})$	\(q-10.7703 q^{2} +32.3110 q^{3} -12.0000 q^{4} -348.000 q^{6} +420.043 q^{7} +1507.85 q^{8} -1143.00 q^{9} -6828.00 q^{11} -387.732 q^{12} -10145.7 q^{13} -4524.00 q^{14} -14704.0 q^{16} -15681.6 q^{17} +12310.5 q^{18} -6860.00 q^{19} +13572.0 q^{21} +73539.8 q^{22} +29219.9 q^{23} +48720.0 q^{24} +109272. q^{26} -107596. q^{27} -5040.51 q^{28} -25590.0 q^{29} +82112.0 q^{31} -34637.4 q^{32} -220619. q^{33} +168896. q^{34} +13716.0 q^{36} +223527. q^{37} +73884.5 q^{38} -327816. q^{39} -533118. q^{41} -146175. q^{42} +708935. q^{43} +81936.0 q^{44} -314708. q^{46} -5826.75 q^{47} -475101. q^{48} -647107. q^{49} -506688. q^{51} +121748. q^{52} -589374. q^{53} +1.15884e6 q^{54} +633360. q^{56} -221653. q^{57} +275613. q^{58} -1.43898e6 q^{59} +1.38102e6 q^{61} -884373. q^{62} -480109. q^{63} +2.25517e6 q^{64} +2.37614e6 q^{66} +2.71487e6 q^{67} +188179. q^{68} +944124. q^{69} -481608. q^{71} -1.72347e6 q^{72} +1.48618e6 q^{73} -2.40746e6 q^{74} +82320.0 q^{76} -2.86805e6 q^{77} +3.53069e6 q^{78} +1.05976e6 q^{79} -976779. q^{81} +5.74186e6 q^{82} -2.60380e6 q^{83} -162864. q^{84} -7.63547e6 q^{86} -826838. q^{87} -1.02956e7 q^{88} -5.64417e6 q^{89} -4.26161e6 q^{91} -350639. q^{92} +2.65312e6 q^{93} +62756.0 q^{94} -1.11917e6 q^{96} -1.20091e7 q^{97} +6.96956e6 q^{98} +7.80440e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 24 q^{4} - 696 q^{6} - 2286 q^{9}+O(q^{10}) $ 2 * q - 24 * q^4 - 696 * q^6 - 2286 * q^9	$ 2 q - 24 q^{4} - 696 q^{6} - 2286 q^{9} - 13656 q^{11} - 9048 q^{14} - 29408 q^{16} - 13720 q^{19} + 27144 q^{21} + 97440 q^{24} + 218544 q^{26} - 51180 q^{29} + 164224 q^{31} + 337792 q^{34} + 27432 q^{36} - 655632 q^{39} - 1066236 q^{41} + 163872 q^{44} - 629416 q^{46} - 1294214 q^{49} - 1013376 q^{51} + 2317680 q^{54} + 1266720 q^{56} - 2877960 q^{59} + 2762044 q^{61} + 4510336 q^{64} + 4752288 q^{66} + 1888248 q^{69} - 963216 q^{71} - 4814928 q^{74} + 164640 q^{76} + 2119520 q^{79} - 1953558 q^{81} - 325728 q^{84} - 15270936 q^{86} - 11288340 q^{89} - 8523216 q^{91} + 125512 q^{94} - 2238336 q^{96} + 15608808 q^{99}+O(q^{100}) $ 2 * q - 24 * q^4 - 696 * q^6 - 2286 * q^9 - 13656 * q^11 - 9048 * q^14 - 29408 * q^16 - 13720 * q^19 + 27144 * q^21 + 97440 * q^24 + 218544 * q^26 - 51180 * q^29 + 164224 * q^31 + 337792 * q^34 + 27432 * q^36 - 655632 * q^39 - 1066236 * q^41 + 163872 * q^44 - 629416 * q^46 - 1294214 * q^49 - 1013376 * q^51 + 2317680 * q^54 + 1266720 * q^56 - 2877960 * q^59 + 2762044 * q^61 + 4510336 * q^64 + 4752288 * q^66 + 1888248 * q^69 - 963216 * q^71 - 4814928 * q^74 + 164640 * q^76 + 2119520 * q^79 - 1953558 * q^81 - 325728 * q^84 - 15270936 * q^86 - 11288340 * q^89 - 8523216 * q^91 + 125512 * q^94 - 2238336 * q^96 + 15608808 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−10.7703	−0.951972	−0.475986	−	0.879453i	$-0.657909\pi$
$2$	−10.7703	−0.951972	−0.475986	+	0.879453i	$0.657909\pi$
$3$	32.3110	0.690917	0.345458	−	0.938434i	$-0.387723\pi$
$3$	32.3110	0.690917	0.345458	+	0.938434i	$0.387723\pi$
$4$	−12.0000	−0.0937500
$5$	0	0
$5$	0	0
$6$	−348.000	−0.657733
$7$	420.043	0.462861	0.231430	−	0.972851i	$-0.425659\pi$
$7$	420.043	0.462861	0.231430	+	0.972851i	$0.425659\pi$
$8$	1507.85	1.04122
$9$	−1143.00	−0.522634
$10$	0	0
$11$	−6828.00	−1.54675	−0.773373	−	0.633951i	$-0.781432\pi$
$11$	−6828.00	−1.54675	−0.773373	+	0.633951i	$0.781432\pi$
$12$	−387.732	−0.0647735
$13$	−10145.7	−1.28079	−0.640395	−	0.768045i	$-0.721229\pi$
$13$	−10145.7	−1.28079	−0.640395	+	0.768045i	$0.721229\pi$
$14$	−4524.00	−0.440630
$15$	0	0
$16$	−14704.0	−0.897461
$17$	−15681.6	−0.774139	−0.387070	−	0.922050i	$-0.626513\pi$
$17$	−15681.6	−0.774139	−0.387070	+	0.922050i	$0.626513\pi$
$18$	12310.5	0.497533
$19$	−6860.00	−0.229449	−0.114725	−	0.993397i	$-0.536599\pi$
$19$	−6860.00	−0.229449	−0.114725	+	0.993397i	$0.536599\pi$
$20$	0	0
$21$	13572.0	0.319798
$22$	73539.8	1.47246
$23$	29219.9	0.500762	0.250381	−	0.968147i	$-0.419444\pi$
$23$	29219.9	0.500762	0.250381	+	0.968147i	$0.419444\pi$
$24$	48720.0	0.719396
$25$	0	0
$26$	109272.	1.21928
$27$	−107596.	−1.05201
$28$	−5040.51	−0.0433932
$29$	−25590.0	−0.194840	−0.0974198	−	0.995243i	$-0.531059\pi$
$29$	−25590.0	−0.194840	−0.0974198	+	0.995243i	$0.531059\pi$
$30$	0	0
$31$	82112.0	0.495040	0.247520	−	0.968883i	$-0.420384\pi$
$31$	82112.0	0.495040	0.247520	+	0.968883i	$0.420384\pi$
$32$	−34637.4	−0.186862
$33$	−220619.	−1.06867
$34$	168896.	0.736959
$35$	0	0
$36$	13716.0	0.0489969
$37$	223527.	0.725479	0.362739	−	0.931891i	$-0.381842\pi$
$37$	223527.	0.725479	0.362739	+	0.931891i	$0.381842\pi$
$38$	73884.5	0.218429
$39$	−327816.	−0.884920
$40$	0	0
$41$	−533118.	−1.20804	−0.604018	−	0.796971i	$-0.706434\pi$
$41$	−533118.	−1.20804	−0.604018	+	0.796971i	$0.706434\pi$
$42$	−146175.	−0.304439
$43$	708935.	1.35978	0.679888	−	0.733316i	$-0.262029\pi$
$43$	708935.	1.35978	0.679888	+	0.733316i	$0.262029\pi$
$44$	81936.0	0.145007
$45$	0	0
$46$	−314708.	−0.476711
$47$	−5826.75	−0.00818623	−0.00409311	−	0.999992i	$-0.501303\pi$
$47$	−5826.75	−0.00818623	−0.00409311	+	0.999992i	$0.501303\pi$
$48$	−475101.	−0.620071
$49$	−647107.	−0.785760
$50$	0	0
$51$	−506688.	−0.534866
$52$	121748.	0.120074
$53$	−589374.	−0.543783	−0.271891	−	0.962328i	$-0.587649\pi$
$53$	−589374.	−0.543783	−0.271891	+	0.962328i	$0.587649\pi$
$54$	1.15884e6	1.00149
$55$	0	0
$56$	633360.	0.481940
$57$	−221653.	−0.158530
$58$	275613.	0.185482
$59$	−1.43898e6	−0.912164	−0.456082	−	0.889938i	$-0.650748\pi$
$59$	−1.43898e6	−0.912164	−0.456082	+	0.889938i	$0.650748\pi$
$60$	0	0
$61$	1.38102e6	0.779016	0.389508	−	0.921023i	$-0.372645\pi$
$61$	1.38102e6	0.779016	0.389508	+	0.921023i	$0.372645\pi$
$62$	−884373.	−0.471264
$63$	−480109.	−0.241907
$64$	2.25517e6	1.07535
$65$	0	0
$66$	2.37614e6	1.01735
$67$	2.71487e6	1.10277	0.551387	−	0.834250i	$-0.314099\pi$
$67$	2.71487e6	1.10277	0.551387	+	0.834250i	$0.314099\pi$
$68$	188179.	0.0725756
$69$	944124.	0.345985
$70$	0	0
$71$	−481608.	−0.159694	−0.0798472	−	0.996807i	$-0.525443\pi$
$71$	−481608.	−0.159694	−0.0798472	+	0.996807i	$0.525443\pi$
$72$	−1.72347e6	−0.544176
$73$	1.48618e6	0.447137	0.223568	−	0.974688i	$-0.428229\pi$
$73$	1.48618e6	0.447137	0.223568	+	0.974688i	$0.428229\pi$
$74$	−2.40746e6	−0.690635
$75$	0	0
$76$	82320.0	0.0215109
$77$	−2.86805e6	−0.715928
$78$	3.53069e6	0.842419
$79$	1.05976e6	0.241831	0.120916	−	0.992663i	$-0.461417\pi$
$79$	1.05976e6	0.241831	0.120916	+	0.992663i	$0.461417\pi$
$80$	0	0
$81$	−976779.	−0.204220
$82$	5.74186e6	1.15002
$83$	−2.60380e6	−0.499844	−0.249922	−	0.968266i	$-0.580405\pi$
$83$	−2.60380e6	−0.499844	−0.249922	+	0.968266i	$0.580405\pi$
$84$	−162864.	−0.0299811
$85$	0	0
$86$	−7.63547e6	−1.29447
$87$	−826838.	−0.134618
$88$	−1.02956e7	−1.61050
$89$	−5.64417e6	−0.848663	−0.424331	−	0.905507i	$-0.639491\pi$
$89$	−5.64417e6	−0.848663	−0.424331	+	0.905507i	$0.639491\pi$
$90$	0	0
$91$	−4.26161e6	−0.592828
$92$	−350639.	−0.0469464
$93$	2.65312e6	0.342032
$94$	62756.0	0.00779306
$95$	0	0
$96$	−1.11917e6	−0.129106
$97$	−1.20091e7	−1.33601	−0.668004	−	0.744158i	$-0.732851\pi$
$97$	−1.20091e7	−1.33601	−0.668004	+	0.744158i	$0.732851\pi$
$98$	6.96956e6	0.748021
$99$	7.80440e6	0.808382
$100$	0	0
$101$	5.14270e6	0.496668	0.248334	−	0.968674i	$-0.420117\pi$
$101$	5.14270e6	0.496668	0.248334	+	0.968674i	$0.420117\pi$
$102$	5.45720e6	0.509177
$103$	3.48477e6	0.314227	0.157114	−	0.987581i	$-0.449781\pi$
$103$	3.48477e6	0.314227	0.157114	+	0.987581i	$0.449781\pi$
$104$	−1.52981e7	−1.33358
$105$	0	0
$106$	6.34775e6	0.517666
$107$	1.48640e7	1.17299	0.586493	−	0.809954i	$-0.300508\pi$
$107$	1.48640e7	1.17299	0.586493	+	0.809954i	$0.300508\pi$
$108$	1.29115e6	0.0986263
$109$	2.01124e7	1.48755	0.743773	−	0.668432i	$-0.233034\pi$
$109$	2.01124e7	1.48755	0.743773	+	0.668432i	$0.233034\pi$
$110$	0	0
$111$	7.22239e6	0.501246
$112$	−6.17631e6	−0.415400
$113$	−5.62633e6	−0.366818	−0.183409	−	0.983037i	$-0.558713\pi$
$113$	−5.62633e6	−0.366818	−0.183409	+	0.983037i	$0.558713\pi$
$114$	2.38728e6	0.150916
$115$	0	0
$116$	307080.	0.0182662
$117$	1.15965e7	0.669384
$118$	1.54983e7	0.868354
$119$	−6.58694e6	−0.358319
$120$	0	0
$121$	2.71344e7	1.39242
$122$	−1.48741e7	−0.741601
$123$	−1.72256e7	−0.834653
$124$	−985344.	−0.0464100
$125$	0	0
$126$	5.17093e6	0.230288
$127$	−2.85360e7	−1.23618	−0.618088	−	0.786109i	$-0.712092\pi$
$127$	−2.85360e7	−1.23618	−0.618088	+	0.786109i	$0.712092\pi$
$128$	−1.98553e7	−0.836839
$129$	2.29064e7	0.939492
$130$	0	0
$131$	−3.33132e7	−1.29469	−0.647346	−	0.762196i	$-0.724121\pi$
$131$	−3.33132e7	−1.29469	−0.647346	+	0.762196i	$0.724121\pi$
$132$	2.64743e6	0.100188
$133$	−2.88149e6	−0.106203
$134$	−2.92400e7	−1.04981
$135$	0	0
$136$	−2.36454e7	−0.806049
$137$	4.28099e7	1.42240	0.711202	−	0.702988i	$-0.248151\pi$
$137$	4.28099e7	1.42240	0.711202	+	0.702988i	$0.248151\pi$
$138$	−1.01685e7	−0.329368
$139$	−1.13808e7	−0.359436	−0.179718	−	0.983718i	$-0.557519\pi$
$139$	−1.13808e7	−0.359436	−0.179718	+	0.983718i	$0.557519\pi$
$140$	0	0
$141$	−188268.	−0.00565600
$142$	5.18708e6	0.152024
$143$	6.92745e7	1.98106
$144$	1.68067e7	0.469043
$145$	0	0
$146$	−1.60066e7	−0.425661
$147$	−2.09087e7	−0.542895
$148$	−2.68233e6	−0.0680136
$149$	−4.00070e7	−0.990794	−0.495397	−	0.868667i	$-0.664977\pi$
$149$	−4.00070e7	−0.990794	−0.495397	+	0.868667i	$0.664977\pi$
$150$	0	0
$151$	−2.86594e7	−0.677405	−0.338703	−	0.940893i	$-0.609988\pi$
$151$	−2.86594e7	−0.677405	−0.338703	+	0.940893i	$0.609988\pi$
$152$	−1.03438e7	−0.238907
$153$	1.79241e7	0.404591
$154$	3.08899e7	0.681544
$155$	0	0
$156$	3.93379e6	0.0829613
$157$	3.01958e7	0.622728	0.311364	−	0.950291i	$-0.399214\pi$
$157$	3.01958e7	0.622728	0.311364	+	0.950291i	$0.399214\pi$
$158$	−1.14140e7	−0.230217
$159$	−1.90433e7	−0.375709
$160$	0	0
$161$	1.22736e7	0.231783
$162$	1.05202e7	0.194412
$163$	−9.35416e7	−1.69180	−0.845898	−	0.533345i	$-0.820935\pi$
$163$	−9.35416e7	−1.69180	−0.845898	+	0.533345i	$0.820935\pi$
$164$	6.39742e6	0.113253
$165$	0	0
$166$	2.80438e7	0.475838
$167$	−5.73507e7	−0.952865	−0.476432	−	0.879211i	$-0.658070\pi$
$167$	−5.73507e7	−0.952865	−0.476432	+	0.879211i	$0.658070\pi$
$168$	2.04645e7	0.332980
$169$	4.01857e7	0.640425
$170$	0	0
$171$	7.84098e6	0.119918
$172$	−8.50722e6	−0.127479
$173$	−4.87192e7	−0.715383	−0.357691	−	0.933840i	$-0.616436\pi$
$173$	−4.87192e7	−0.715383	−0.357691	+	0.933840i	$0.616436\pi$
$174$	8.90532e6	0.128153
$175$	0	0
$176$	1.00399e8	1.38814
$177$	−4.64949e7	−0.630229
$178$	6.07896e7	0.807903
$179$	−1.93505e7	−0.252178	−0.126089	−	0.992019i	$-0.540243\pi$
$179$	−1.93505e7	−0.252178	−0.126089	+	0.992019i	$0.540243\pi$
$180$	0	0
$181$	7.82617e7	0.981011	0.490506	−	0.871438i	$-0.336812\pi$
$181$	7.82617e7	0.981011	0.490506	+	0.871438i	$0.336812\pi$
$182$	4.58989e7	0.564355
$183$	4.46222e7	0.538235
$184$	4.40591e7	0.521403
$185$	0	0
$186$	−2.85750e7	−0.325605
$187$	1.07074e8	1.19740
$188$	69921.0	0.000767459 0
$189$	−4.51948e7	−0.486936
$190$	0	0
$191$	−1.19454e8	−1.24046	−0.620229	−	0.784420i	$-0.712960\pi$
$191$	−1.19454e8	−1.24046	−0.620229	+	0.784420i	$0.712960\pi$
$192$	7.28667e7	0.742976
$193$	5.98469e7	0.599227	0.299613	−	0.954061i	$-0.403142\pi$
$193$	5.98469e7	0.599227	0.299613	+	0.954061i	$0.403142\pi$
$194$	1.29342e8	1.27184
$195$	0	0
$196$	7.76528e6	0.0736650
$197$	−1.22964e8	−1.14590	−0.572952	−	0.819589i	$-0.694202\pi$
$197$	−1.22964e8	−1.14590	−0.572952	+	0.819589i	$0.694202\pi$
$198$	−8.40560e7	−0.769557
$199$	−1.69053e8	−1.52067	−0.760337	−	0.649529i	$-0.774966\pi$
$199$	−1.69053e8	−1.52067	−0.760337	+	0.649529i	$0.774966\pi$
$200$	0	0
$201$	8.77200e7	0.761925
$202$	−5.53886e7	−0.472814
$203$	−1.07489e7	−0.0901836
$204$	6.08026e6	0.0501437
$205$	0	0
$206$	−3.75321e7	−0.299136
$207$	−3.33984e7	−0.261715
$208$	1.49182e8	1.14946
$209$	4.68401e7	0.354900
$210$	0	0
$211$	−2.67605e8	−1.96113	−0.980565	−	0.196195i	$-0.937141\pi$
$211$	−2.67605e8	−1.96113	−0.980565	+	0.196195i	$0.937141\pi$
$212$	7.07249e6	0.0509796
$213$	−1.55612e7	−0.110336
$214$	−1.60090e8	−1.11665
$215$	0	0
$216$	−1.62238e8	−1.09538
$217$	3.44906e7	0.229135
$218$	−2.16617e8	−1.41610
$219$	4.80198e7	0.308934
$220$	0	0
$221$	1.59100e8	0.991510
$222$	−7.77875e7	−0.477172
$223$	−1.49333e8	−0.901753	−0.450877	−	0.892586i	$-0.648889\pi$
$223$	−1.49333e8	−0.901753	−0.450877	+	0.892586i	$0.648889\pi$
$224$	−1.45492e7	−0.0864909
$225$	0	0
$226$	6.05975e7	0.349201
$227$	2.34185e8	1.32883	0.664414	−	0.747365i	$-0.268681\pi$
$227$	2.34185e8	1.32883	0.664414	+	0.747365i	$0.268681\pi$
$228$	2.65984e6	0.0148622
$229$	1.31882e8	0.725706	0.362853	−	0.931846i	$-0.381803\pi$
$229$	1.31882e8	0.725706	0.362853	+	0.931846i	$0.381803\pi$
$230$	0	0
$231$	−9.26696e7	−0.494647
$232$	−3.85858e7	−0.202871
$233$	1.83419e8	0.949948	0.474974	−	0.880000i	$-0.342458\pi$
$233$	1.83419e8	0.949948	0.474974	+	0.880000i	$0.342458\pi$
$234$	−1.24898e8	−0.637235
$235$	0	0
$236$	1.72678e7	0.0855153
$237$	3.42419e7	0.167085
$238$	7.09436e7	0.341109
$239$	1.05117e8	0.498058	0.249029	−	0.968496i	$-0.419889\pi$
$239$	1.05117e8	0.498058	0.249029	+	0.968496i	$0.419889\pi$
$240$	0	0
$241$	1.94216e8	0.893770	0.446885	−	0.894591i	$-0.352533\pi$
$241$	1.94216e8	0.893770	0.446885	+	0.894591i	$0.352533\pi$
$242$	−2.92247e8	−1.32555
$243$	2.03751e8	0.910914
$244$	−1.65723e7	−0.0730327
$245$	0	0
$246$	1.85525e8	0.794566
$247$	6.95992e7	0.293876
$248$	1.23812e8	0.515446
$249$	−8.41314e7	−0.345351
$250$	0	0
$251$	2.02689e8	0.809045	0.404523	−	0.914528i	$-0.367438\pi$
$251$	2.02689e8	0.809045	0.404523	+	0.914528i	$0.367438\pi$
$252$	5.76131e6	0.0226788
$253$	−1.99514e8	−0.774552
$254$	3.07342e8	1.17680
$255$	0	0
$256$	−7.48132e7	−0.278701
$257$	1.34593e7	0.0494603	0.0247301	−	0.999694i	$-0.492127\pi$
$257$	1.34593e7	0.0494603	0.0247301	+	0.999694i	$0.492127\pi$
$258$	−2.46710e8	−0.894369
$259$	9.38911e7	0.335796
$260$	0	0
$261$	2.92494e7	0.101830
$262$	3.58794e8	1.23251
$263$	1.42205e8	0.482026	0.241013	−	0.970522i	$-0.422520\pi$
$263$	1.42205e8	0.482026	0.241013	+	0.970522i	$0.422520\pi$
$264$	−3.32660e8	−1.11272
$265$	0	0
$266$	3.10346e7	0.101102
$267$	−1.82369e8	−0.586355
$268$	−3.25784e7	−0.103385
$269$	5.07548e8	1.58981	0.794903	−	0.606737i	$-0.207522\pi$
$269$	5.07548e8	1.58981	0.794903	+	0.606737i	$0.207522\pi$
$270$	0	0
$271$	1.12836e8	0.344393	0.172197	−	0.985063i	$-0.444914\pi$
$271$	1.12836e8	0.344393	0.172197	+	0.985063i	$0.444914\pi$
$272$	2.30582e8	0.694760
$273$	−1.37697e8	−0.409595
$274$	−4.61077e8	−1.35409
$275$	0	0
$276$	−1.13295e7	−0.0324361
$277$	−5.10728e8	−1.44381	−0.721906	−	0.691991i	$-0.756734\pi$
$277$	−5.10728e8	−1.44381	−0.721906	+	0.691991i	$0.756734\pi$
$278$	1.22575e8	0.342173
$279$	−9.38540e7	−0.258725
$280$	0	0
$281$	−1.70459e8	−0.458297	−0.229148	−	0.973391i	$-0.573594\pi$
$281$	−1.70459e8	−0.458297	−0.229148	+	0.973391i	$0.573594\pi$
$282$	2.02771e6	0.00538435
$283$	−1.62144e8	−0.425253	−0.212626	−	0.977134i	$-0.568202\pi$
$283$	−1.62144e8	−0.425253	−0.212626	+	0.977134i	$0.568202\pi$
$284$	5.77930e6	0.0149713
$285$	0	0
$286$	−7.46109e8	−1.88591
$287$	−2.23932e8	−0.559153
$288$	3.95905e7	0.0976602
$289$	−1.64426e8	−0.400708
$290$	0	0
$291$	−3.88026e8	−0.923071
$292$	−1.78341e7	−0.0419191
$293$	−3.85845e8	−0.896141	−0.448070	−	0.893998i	$-0.647889\pi$
$293$	−3.85845e8	−0.896141	−0.448070	+	0.893998i	$0.647889\pi$
$294$	2.25193e8	0.516820
$295$	0	0
$296$	3.37045e8	0.755382
$297$	7.34663e8	1.62720
$298$	4.30888e8	0.943208
$299$	−2.96455e8	−0.641371
$300$	0	0
$301$	2.97783e8	0.629387
$302$	3.08672e8	0.644871
$303$	1.66166e8	0.343157
$304$	1.00869e8	0.205922
$305$	0	0
$306$	−1.93048e8	−0.385159
$307$	−6.37817e8	−1.25809	−0.629045	−	0.777369i	$-0.716554\pi$
$307$	−6.37817e8	−1.25809	−0.629045	+	0.777369i	$0.716554\pi$
$308$	3.44166e7	0.0671183
$309$	1.12596e8	0.217105
$310$	0	0
$311$	−7.27817e8	−1.37202	−0.686010	−	0.727592i	$-0.740640\pi$
$311$	−7.27817e8	−1.37202	−0.686010	+	0.727592i	$0.740640\pi$
$312$	−4.94296e8	−0.921396
$313$	4.46869e8	0.823711	0.411855	−	0.911249i	$-0.364881\pi$
$313$	4.46869e8	0.823711	0.411855	+	0.911249i	$0.364881\pi$
$314$	−3.25219e8	−0.592819
$315$	0	0
$316$	−1.27171e7	−0.0226717
$317$	5.91083e8	1.04218	0.521088	−	0.853503i	$-0.325526\pi$
$317$	5.91083e8	1.04218	0.521088	+	0.853503i	$0.325526\pi$
$318$	2.05102e8	0.357664
$319$	1.74729e8	0.301367
$320$	0	0
$321$	4.80271e8	0.810436
$322$	−1.32191e8	−0.220651
$323$	1.07576e8	0.177626
$324$	1.17213e7	0.0191456
$325$	0	0
$326$	1.00747e9	1.61054
$327$	6.49851e8	1.02777
$328$	−8.03860e8	−1.25783
$329$	−2.44748e6	−0.00378908
$330$	0	0
$331$	5.84868e8	0.886462	0.443231	−	0.896407i	$-0.353832\pi$
$331$	5.84868e8	0.886462	0.443231	+	0.896407i	$0.353832\pi$
$332$	3.12456e7	0.0468604
$333$	−2.55492e8	−0.379160
$334$	6.17686e8	0.907100
$335$	0	0
$336$	−1.99563e8	−0.287007
$337$	−7.39373e8	−1.05235	−0.526174	−	0.850377i	$-0.676374\pi$
$337$	−7.39373e8	−1.05235	−0.526174	+	0.850377i	$0.676374\pi$
$338$	−4.32813e8	−0.609666
$339$	−1.81792e8	−0.253441
$340$	0	0
$341$	−5.60661e8	−0.765702
$342$	−8.44499e7	−0.114158
$343$	−6.17736e8	−0.826558
$344$	1.06897e9	1.41582
$345$	0	0
$346$	5.24722e8	0.681024
$347$	−3.70870e8	−0.476506	−0.238253	−	0.971203i	$-0.576575\pi$
$347$	−3.70870e8	−0.476506	−0.238253	+	0.971203i	$0.576575\pi$
$348$	9.92206e6	0.0126204
$349$	−1.13274e9	−1.42640	−0.713199	−	0.700962i	$-0.752754\pi$
$349$	−1.13274e9	−1.42640	−0.713199	+	0.700962i	$0.752754\pi$
$350$	0	0
$351$	1.09163e9	1.34741
$352$	2.36504e8	0.289028
$353$	8.32858e8	1.00777	0.503883	−	0.863772i	$-0.331904\pi$
$353$	8.32858e8	1.00777	0.503883	+	0.863772i	$0.331904\pi$
$354$	5.00765e8	0.599960
$355$	0	0
$356$	6.77300e7	0.0795621
$357$	−2.12831e8	−0.247569
$358$	2.08412e8	0.240066
$359$	−6.75318e8	−0.770332	−0.385166	−	0.922847i	$-0.625856\pi$
$359$	−6.75318e8	−0.770332	−0.385166	+	0.922847i	$0.625856\pi$
$360$	0	0
$361$	−8.46812e8	−0.947353
$362$	−8.42904e8	−0.933895
$363$	8.76740e8	0.962050
$364$	5.11393e7	0.0555776
$365$	0	0
$366$	−4.80596e8	−0.512385
$367$	1.80237e9	1.90333	0.951664	−	0.307141i	$-0.0993723\pi$
$367$	1.80237e9	1.90333	0.951664	+	0.307141i	$0.0993723\pi$
$368$	−4.29649e8	−0.449414
$369$	6.09354e8	0.631360
$370$	0	0
$371$	−2.47562e8	−0.251696
$372$	−3.18374e7	−0.0320655
$373$	9.29928e8	0.927830	0.463915	−	0.885880i	$-0.346444\pi$
$373$	9.29928e8	0.927830	0.463915	+	0.885880i	$0.346444\pi$
$374$	−1.15322e9	−1.13989
$375$	0	0
$376$	−8.78584e6	−0.00852365
$377$	2.59627e8	0.249549
$378$	4.86762e8	0.463549
$379$	1.43545e9	1.35441	0.677206	−	0.735794i	$-0.263191\pi$
$379$	1.43545e9	1.35441	0.677206	+	0.735794i	$0.263191\pi$
$380$	0	0
$381$	−9.22026e8	−0.854094
$382$	1.28655e9	1.18088
$383$	−1.57707e9	−1.43435	−0.717174	−	0.696894i	$-0.754565\pi$
$383$	−1.57707e9	−1.43435	−0.717174	+	0.696894i	$0.754565\pi$
$384$	−6.41545e8	−0.578186
$385$	0	0
$386$	−6.44571e8	−0.570447
$387$	−8.10313e8	−0.710664
$388$	1.44109e8	0.125251
$389$	−2.24425e9	−1.93307	−0.966537	−	0.256528i	$-0.917421\pi$
$389$	−2.24425e9	−1.93307	−0.966537	+	0.256528i	$0.917421\pi$
$390$	0	0
$391$	−4.58215e8	−0.387660
$392$	−9.75738e8	−0.818148
$393$	−1.07638e9	−0.894525
$394$	1.32437e9	1.09087
$395$	0	0
$396$	−9.36528e7	−0.0757858
$397$	2.26641e8	0.181791	0.0908956	−	0.995860i	$-0.471027\pi$
$397$	2.26641e8	0.181791	0.0908956	+	0.995860i	$0.471027\pi$
$398$	1.82075e9	1.44764
$399$	−9.31039e7	−0.0733775
$400$	0	0
$401$	−9.11721e8	−0.706085	−0.353042	−	0.935607i	$-0.614853\pi$
$401$	−9.11721e8	−0.706085	−0.353042	+	0.935607i	$0.614853\pi$
$402$	−9.44773e8	−0.725331
$403$	−8.33080e8	−0.634043
$404$	−6.17124e7	−0.0465627
$405$	0	0
$406$	1.15769e8	0.0858523
$407$	−1.52625e9	−1.12213
$408$	−7.64008e8	−0.556913
$409$	2.55215e8	0.184449	0.0922243	−	0.995738i	$-0.470602\pi$
$409$	2.55215e8	0.184449	0.0922243	+	0.995738i	$0.470602\pi$
$410$	0	0
$411$	1.38323e9	0.982763
$412$	−4.18173e7	−0.0294588
$413$	−6.04433e8	−0.422205
$414$	3.59711e8	0.249145
$415$	0	0
$416$	3.51419e8	0.239331
$417$	−3.67726e8	−0.248341
$418$	−5.04483e8	−0.337854
$419$	−2.96316e8	−0.196791	−0.0983957	−	0.995147i	$-0.531371\pi$
$419$	−2.96316e8	−0.196791	−0.0983957	+	0.995147i	$0.531371\pi$
$420$	0	0
$421$	1.06676e9	0.696754	0.348377	−	0.937354i	$-0.386733\pi$
$421$	1.06676e9	0.696754	0.348377	+	0.937354i	$0.386733\pi$
$422$	2.88220e9	1.86694
$423$	6.65997e6	0.00427840
$424$	−8.88685e8	−0.566197
$425$	0	0
$426$	1.67600e8	0.105036
$427$	5.80088e8	0.360576
$428$	−1.78368e8	−0.109967
$429$	2.23833e9	1.36875
$430$	0	0
$431$	9.53169e7	0.0573455	0.0286728	−	0.999589i	$-0.490872\pi$
$431$	9.53169e7	0.0573455	0.0286728	+	0.999589i	$0.490872\pi$
$432$	1.58209e9	0.944141
$433$	1.89973e9	1.12456	0.562281	−	0.826946i	$-0.309924\pi$
$433$	1.89973e9	1.12456	0.562281	+	0.826946i	$0.309924\pi$
$434$	−3.71475e8	−0.218130
$435$	0	0
$436$	−2.41348e8	−0.139457
$437$	−2.00449e8	−0.114899
$438$	−5.17189e8	−0.294097
$439$	1.11226e9	0.627450	0.313725	−	0.949514i	$-0.398423\pi$
$439$	1.11226e9	0.627450	0.313725	+	0.949514i	$0.398423\pi$
$440$	0	0
$441$	7.39643e8	0.410665
$442$	−1.71356e9	−0.943890
$443$	3.22249e9	1.76108	0.880540	−	0.473972i	$-0.157180\pi$
$443$	3.22249e9	1.76108	0.880540	+	0.473972i	$0.157180\pi$
$444$	−8.66687e7	−0.0469918
$445$	0	0
$446$	1.60836e9	0.858443
$447$	−1.29266e9	−0.684557
$448$	9.47267e8	0.497736
$449$	7.29482e7	0.0380323	0.0190161	−	0.999819i	$-0.493947\pi$
$449$	7.29482e7	0.0380323	0.0190161	+	0.999819i	$0.493947\pi$
$450$	0	0
$451$	3.64013e9	1.86853
$452$	6.75160e7	0.0343892
$453$	−9.26015e8	−0.468031
$454$	−2.52225e9	−1.26501
$455$	0	0
$456$	−3.34219e8	−0.165065
$457$	−2.45286e8	−0.120217	−0.0601085	−	0.998192i	$-0.519145\pi$
$457$	−2.45286e8	−0.120217	−0.0601085	+	0.998192i	$0.519145\pi$
$458$	−1.42041e9	−0.690852
$459$	1.68727e9	0.814405
$460$	0	0
$461$	−3.25654e9	−1.54812	−0.774058	−	0.633115i	$-0.781776\pi$
$461$	−3.25654e9	−1.54812	−0.774058	+	0.633115i	$0.781776\pi$
$462$	9.98082e8	0.470890
$463$	5.48463e8	0.256811	0.128406	−	0.991722i	$-0.459014\pi$
$463$	5.48463e8	0.256811	0.128406	+	0.991722i	$0.459014\pi$
$464$	3.76275e8	0.174861
$465$	0	0
$466$	−1.97549e9	−0.904323
$467$	1.31891e9	0.599245	0.299623	−	0.954058i	$-0.403139\pi$
$467$	1.31891e9	0.599245	0.299623	+	0.954058i	$0.403139\pi$
$468$	−1.39158e8	−0.0627548
$469$	1.14036e9	0.510431
$470$	0	0
$471$	9.75658e8	0.430253
$472$	−2.16976e9	−0.949762
$473$	−4.84061e9	−2.10323
$474$	−3.68796e8	−0.159061
$475$	0	0
$476$	7.90433e7	0.0335924
$477$	6.73654e8	0.284199
$478$	−1.13214e9	−0.474137
$479$	−1.59989e9	−0.665144	−0.332572	−	0.943078i	$-0.607916\pi$
$479$	−1.59989e9	−0.665144	−0.332572	+	0.943078i	$0.607916\pi$
$480$	0	0
$481$	−2.26783e9	−0.929187
$482$	−2.09177e9	−0.850844
$483$	3.96573e8	0.160143
$484$	−3.25613e8	−0.130540
$485$	0	0
$486$	−2.19446e9	−0.867165
$487$	1.95948e9	0.768759	0.384380	−	0.923175i	$-0.374415\pi$
$487$	1.95948e9	0.768759	0.384380	+	0.923175i	$0.374415\pi$
$488$	2.08237e9	0.811126
$489$	−3.02242e9	−1.16889
$490$	0	0
$491$	2.38785e8	0.0910376	0.0455188	−	0.998963i	$-0.485506\pi$
$491$	2.38785e8	0.0910376	0.0455188	+	0.998963i	$0.485506\pi$
$492$	2.06707e8	0.0782487
$493$	4.01292e8	0.150833
$494$	−7.49606e8	−0.279762
$495$	0	0
$496$	−1.20737e9	−0.444279
$497$	−2.02296e8	−0.0739163
$498$	9.06123e8	0.328764
$499$	3.06642e9	1.10479	0.552394	−	0.833583i	$-0.313714\pi$
$499$	3.06642e9	1.10479	0.552394	+	0.833583i	$0.313714\pi$
$500$	0	0
$501$	−1.85306e9	−0.658350
$502$	−2.18303e9	−0.770188
$503$	1.60348e9	0.561793	0.280897	−	0.959738i	$-0.409368\pi$
$503$	1.60348e9	0.561793	0.280897	+	0.959738i	$0.409368\pi$
$504$	−7.23930e8	−0.251878
$505$	0	0
$506$	2.14883e9	0.737351
$507$	1.29844e9	0.442480
$508$	3.42432e8	0.115891
$509$	1.21742e9	0.409192	0.204596	−	0.978847i	$-0.434412\pi$
$509$	1.21742e9	0.409192	0.204596	+	0.978847i	$0.434412\pi$
$510$	0	0
$511$	6.24258e8	0.206962
$512$	3.34724e9	1.10215
$513$	7.38106e8	0.241384
$514$	−1.44961e8	−0.0470848
$515$	0	0
$516$	−2.74877e8	−0.0880773
$517$	3.97850e7	0.0126620
$518$	−1.01124e9	−0.319668
$519$	−1.57416e9	−0.494270
$520$	0	0
$521$	−2.08635e9	−0.646331	−0.323166	−	0.946342i	$-0.604747\pi$
$521$	−2.08635e9	−0.646331	−0.323166	+	0.946342i	$0.604747\pi$
$522$	−3.15025e8	−0.0969390
$523$	−4.28922e9	−1.31106	−0.655531	−	0.755169i	$-0.727555\pi$
$523$	−4.28922e9	−1.31106	−0.655531	+	0.755169i	$0.727555\pi$
$524$	3.99758e8	0.121377
$525$	0	0
$526$	−1.53160e9	−0.458875
$527$	−1.28765e9	−0.383230
$528$	3.24399e9	0.959093
$529$	−2.55102e9	−0.749237
$530$	0	0
$531$	1.64475e9	0.476727
$532$	3.45779e7	0.00995654
$533$	5.40883e9	1.54724
$534$	1.96417e9	0.558194
$535$	0	0
$536$	4.09360e9	1.14823
$537$	−6.25235e8	−0.174234
$538$	−5.46646e9	−1.51345
$539$	4.41845e9	1.21537
$540$	0	0
$541$	4.91116e9	1.33350	0.666751	−	0.745280i	$-0.267684\pi$
$541$	4.91116e9	1.33350	0.666751	+	0.745280i	$0.267684\pi$
$542$	−1.21528e9	−0.327852
$543$	2.52871e9	0.677797
$544$	5.43170e8	0.144657
$545$	0	0
$546$	1.48304e9	0.389923
$547$	−1.76451e9	−0.460965	−0.230482	−	0.973077i	$-0.574030\pi$
$547$	−1.76451e9	−0.460965	−0.230482	+	0.973077i	$0.574030\pi$
$548$	−5.13719e8	−0.133350
$549$	−1.57851e9	−0.407140
$550$	0	0
$551$	1.75547e8	0.0447058
$552$	1.42359e9	0.360246
$553$	4.45145e8	0.111934
$554$	5.50071e9	1.37447
$555$	0	0
$556$	1.36570e8	0.0336971
$557$	4.13406e8	0.101364	0.0506820	−	0.998715i	$-0.483860\pi$
$557$	4.13406e8	0.101364	0.0506820	+	0.998715i	$0.483860\pi$
$558$	1.01084e9	0.246299
$559$	−7.19261e9	−1.74159
$560$	0	0
$561$	3.45967e9	0.827302
$562$	1.83590e9	0.436286
$563$	−7.57073e9	−1.78796	−0.893982	−	0.448104i	$-0.852100\pi$
$563$	−7.57073e9	−1.78796	−0.893982	+	0.448104i	$0.852100\pi$
$564$	2.25922e6	0.000530250 0
$565$	0	0
$566$	1.74634e9	0.404829
$567$	−4.10289e8	−0.0945255
$568$	−7.26191e8	−0.166277
$569$	8.90287e8	0.202599	0.101299	−	0.994856i	$-0.467700\pi$
$569$	8.90287e8	0.202599	0.101299	+	0.994856i	$0.467700\pi$
$570$	0	0
$571$	−4.96089e9	−1.11515	−0.557575	−	0.830126i	$-0.688268\pi$
$571$	−4.96089e9	−1.11515	−0.557575	+	0.830126i	$0.688268\pi$
$572$	−8.31294e8	−0.185724
$573$	−3.85966e9	−0.857054
$574$	2.41183e9	0.532297
$575$	0	0
$576$	−2.57766e9	−0.562013
$577$	−1.53066e9	−0.331713	−0.165856	−	0.986150i	$-0.553039\pi$
$577$	−1.53066e9	−0.331713	−0.165856	+	0.986150i	$0.553039\pi$
$578$	1.77092e9	0.381463
$579$	1.93371e9	0.414016
$580$	0	0
$581$	−1.09371e9	−0.231358
$582$	4.17916e9	0.878737
$583$	4.02425e9	0.841094
$584$	2.24093e9	0.465567
$585$	0	0
$586$	4.15568e9	0.853100
$587$	4.39564e9	0.896992	0.448496	−	0.893785i	$-0.351960\pi$
$587$	4.39564e9	0.896992	0.448496	+	0.893785i	$0.351960\pi$
$588$	2.50904e8	0.0508964
$589$	−5.63288e8	−0.113587
$590$	0	0
$591$	−3.97310e9	−0.791724
$592$	−3.28675e9	−0.651089
$593$	3.32990e9	0.655753	0.327877	−	0.944721i	$-0.393667\pi$
$593$	3.32990e9	0.655753	0.327877	+	0.944721i	$0.393667\pi$
$594$	−7.91256e9	−1.54905
$595$	0	0
$596$	4.80083e8	0.0928870
$597$	−5.46226e9	−1.05066
$598$	3.19292e9	0.610567
$599$	−4.53030e9	−0.861258	−0.430629	−	0.902529i	$-0.641708\pi$
$599$	−4.53030e9	−0.861258	−0.430629	+	0.902529i	$0.641708\pi$
$600$	0	0
$601$	−4.70479e9	−0.884056	−0.442028	−	0.897001i	$-0.645741\pi$
$601$	−4.70479e9	−0.884056	−0.442028	+	0.897001i	$0.645741\pi$
$602$	−3.20722e9	−0.599158
$603$	−3.10309e9	−0.576347
$604$	3.43913e8	0.0635067
$605$	0	0
$606$	−1.78966e9	−0.326675
$607$	−2.24429e9	−0.407303	−0.203652	−	0.979043i	$-0.565281\pi$
$607$	−2.24429e9	−0.407303	−0.203652	+	0.979043i	$0.565281\pi$
$608$	2.37612e8	0.0428752
$609$	−3.47307e8	−0.0623094
$610$	0	0
$611$	5.91162e7	0.0104848
$612$	−2.15089e8	−0.0379304
$613$	8.74415e9	1.53323	0.766613	−	0.642109i	$-0.221941\pi$
$613$	8.74415e9	1.53323	0.766613	+	0.642109i	$0.221941\pi$
$614$	6.86950e9	1.19767
$615$	0	0
$616$	−4.32458e9	−0.745438
$617$	4.49031e9	0.769623	0.384812	−	0.922995i	$-0.374266\pi$
$617$	4.49031e9	0.769623	0.384812	+	0.922995i	$0.374266\pi$
$618$	−1.21270e9	−0.206678
$619$	−3.74101e9	−0.633974	−0.316987	−	0.948430i	$-0.602671\pi$
$619$	−3.74101e9	−0.633974	−0.316987	+	0.948430i	$0.602671\pi$
$620$	0	0
$621$	−3.14393e9	−0.526808
$622$	7.83883e9	1.30612
$623$	−2.37079e9	−0.392813
$624$	4.82021e9	0.794181
$625$	0	0
$626$	−4.81292e9	−0.784149
$627$	1.51345e9	0.245206
$628$	−3.62350e8	−0.0583808
$629$	−3.50527e9	−0.561622
$630$	0	0
$631$	1.93545e9	0.306675	0.153337	−	0.988174i	$-0.450998\pi$
$631$	1.93545e9	0.306675	0.153337	+	0.988174i	$0.450998\pi$
$632$	1.59796e9	0.251799
$633$	−8.64660e9	−1.35498
$634$	−6.36616e9	−0.992122
$635$	0	0
$636$	2.28519e8	0.0352227
$637$	6.56532e9	1.00639
$638$	−1.88188e9	−0.286893
$639$	5.50478e8	0.0834616
$640$	0	0
$641$	5.89076e9	0.883422	0.441711	−	0.897157i	$-0.354372\pi$
$641$	5.89076e9	0.883422	0.441711	+	0.897157i	$0.354372\pi$
$642$	−5.17268e9	−0.771512
$643$	3.16008e9	0.468770	0.234385	−	0.972144i	$-0.424692\pi$
$643$	3.16008e9	0.468770	0.234385	+	0.972144i	$0.424692\pi$
$644$	−1.47283e8	−0.0217297
$645$	0	0
$646$	−1.15863e9	−0.169095
$647$	−1.27557e10	−1.85157	−0.925783	−	0.378054i	$-0.876593\pi$
$647$	−1.27557e10	−1.85157	−0.925783	+	0.378054i	$0.876593\pi$
$648$	−1.47283e9	−0.212638
$649$	9.82536e9	1.41089
$650$	0	0
$651$	1.11442e9	0.158313
$652$	1.12250e9	0.158606
$653$	−4.43892e9	−0.623852	−0.311926	−	0.950106i	$-0.600974\pi$
$653$	−4.43892e9	−0.623852	−0.311926	+	0.950106i	$0.600974\pi$
$654$	−6.99910e9	−0.978409
$655$	0	0
$656$	7.83897e9	1.08417
$657$	−1.69870e9	−0.233689
$658$	2.63602e7	0.00360710
$659$	1.08526e10	1.47719	0.738595	−	0.674149i	$-0.235489\pi$
$659$	1.08526e10	1.47719	0.738595	+	0.674149i	$0.235489\pi$
$660$	0	0
$661$	8.49307e9	1.14382	0.571912	−	0.820315i	$-0.306202\pi$
$661$	8.49307e9	1.14382	0.571912	+	0.820315i	$0.306202\pi$
$662$	−6.29922e9	−0.843887
$663$	5.14068e9	0.685051
$664$	−3.92613e9	−0.520447
$665$	0	0
$666$	2.75173e9	0.360949
$667$	−7.47737e8	−0.0975683
$668$	6.88209e8	0.0893311
$669$	−4.82509e9	−0.623037
$670$	0	0
$671$	−9.42962e9	−1.20494
$672$	−4.70099e8	−0.0597581
$673$	−4.20188e9	−0.531362	−0.265681	−	0.964061i	$-0.585597\pi$
$673$	−4.20188e9	−0.531362	−0.265681	+	0.964061i	$0.585597\pi$
$674$	7.96330e9	1.00180
$675$	0	0
$676$	−4.82228e8	−0.0600398
$677$	6.56755e9	0.813472	0.406736	−	0.913546i	$-0.366667\pi$
$677$	6.56755e9	0.813472	0.406736	+	0.913546i	$0.366667\pi$
$678$	1.95796e9	0.241269
$679$	−5.04433e9	−0.618386
$680$	0	0
$681$	7.56676e9	0.918110
$682$	6.03850e9	0.728927
$683$	6.31484e9	0.758386	0.379193	−	0.925318i	$-0.376202\pi$
$683$	6.31484e9	0.758386	0.379193	+	0.925318i	$0.376202\pi$
$684$	−9.40918e7	−0.0112423
$685$	0	0
$686$	6.65322e9	0.786860
$687$	4.26123e9	0.501403
$688$	−1.04242e10	−1.22035
$689$	5.97958e9	0.696472
$690$	0	0
$691$	3.76447e9	0.434041	0.217020	−	0.976167i	$-0.430366\pi$
$691$	3.76447e9	0.434041	0.217020	+	0.976167i	$0.430366\pi$
$692$	5.84630e8	0.0670672
$693$	3.27818e9	0.374168
$694$	3.99439e9	0.453620
$695$	0	0
$696$	−1.24674e9	−0.140167
$697$	8.36014e9	0.935188
$698$	1.22000e10	1.35789
$699$	5.92646e9	0.656335
$700$	0	0
$701$	1.97083e9	0.216090	0.108045	−	0.994146i	$-0.465541\pi$
$701$	1.97083e9	0.216090	0.108045	+	0.994146i	$0.465541\pi$
$702$	−1.17572e10	−1.28270
$703$	−1.53340e9	−0.166461
$704$	−1.53983e10	−1.66329
$705$	0	0
$706$	−8.97016e9	−0.959364
$707$	2.16016e9	0.229888
$708$	5.57938e8	0.0590840
$709$	9.62853e9	1.01461	0.507304	−	0.861767i	$-0.330642\pi$
$709$	9.62853e9	1.01461	0.507304	+	0.861767i	$0.330642\pi$
$710$	0	0
$711$	−1.21131e9	−0.126389
$712$	−8.51054e9	−0.883644
$713$	2.39930e9	0.247897
$714$	2.29226e9	0.235678
$715$	0	0
$716$	2.32206e8	0.0236417
$717$	3.39643e9	0.344117
$718$	7.27340e9	0.733334
$719$	−1.89490e10	−1.90123	−0.950614	−	0.310376i	$-0.899545\pi$
$719$	−1.89490e10	−1.90123	−0.950614	+	0.310376i	$0.899545\pi$
$720$	0	0
$721$	1.46375e9	0.145444
$722$	9.12045e9	0.901853
$723$	6.27532e9	0.617521
$724$	−9.39140e8	−0.0919698
$725$	0	0
$726$	−9.44278e9	−0.915844
$727$	1.44446e9	0.139423	0.0697116	−	0.997567i	$-0.477792\pi$
$727$	1.44446e9	0.139423	0.0697116	+	0.997567i	$0.477792\pi$
$728$	−6.42585e9	−0.617264
$729$	8.71961e9	0.833586
$730$	0	0
$731$	−1.11172e10	−1.05266
$732$	−5.35466e8	−0.0504595
$733$	−1.38939e10	−1.30305	−0.651525	−	0.758627i	$-0.725871\pi$
$733$	−1.38939e10	−1.30305	−0.651525	+	0.758627i	$0.725871\pi$
$734$	−1.94122e10	−1.81191
$735$	0	0
$736$	−1.01210e9	−0.0935732
$737$	−1.85371e10	−1.70571
$738$	−6.56294e9	−0.601037
$739$	1.19008e10	1.08473	0.542366	−	0.840142i	$-0.317529\pi$
$739$	1.19008e10	1.08473	0.542366	+	0.840142i	$0.317529\pi$
$740$	0	0
$741$	2.24882e9	0.203044
$742$	2.66633e9	0.239607
$743$	−1.57512e10	−1.40882	−0.704408	−	0.709796i	$-0.748787\pi$
$743$	−1.57512e10	−1.40882	−0.704408	+	0.709796i	$0.748787\pi$
$744$	4.00050e9	0.356130
$745$	0	0
$746$	−1.00156e10	−0.883268
$747$	2.97615e9	0.261235
$748$	−1.28489e9	−0.112256
$749$	6.24352e9	0.542929
$750$	0	0
$751$	−1.60645e10	−1.38397	−0.691984	−	0.721912i	$-0.743263\pi$
$751$	−1.60645e10	−1.38397	−0.691984	+	0.721912i	$0.743263\pi$
$752$	8.56765e7	0.00734682
$753$	6.54909e9	0.558983
$754$	−2.79627e9	−0.237563
$755$	0	0
$756$	5.42337e8	0.0456502
$757$	−1.88969e10	−1.58327	−0.791636	−	0.610993i	$-0.790771\pi$
$757$	−1.88969e10	−1.58327	−0.791636	+	0.610993i	$0.790771\pi$
$758$	−1.54603e10	−1.28936
$759$	−6.44648e9	−0.535151
$760$	0	0
$761$	1.01100e10	0.831584	0.415792	−	0.909460i	$-0.363504\pi$
$761$	1.01100e10	0.831584	0.415792	+	0.909460i	$0.363504\pi$
$762$	9.93053e9	0.813074
$763$	8.44806e9	0.688527
$764$	1.43344e9	0.116293
$765$	0	0
$766$	1.69855e10	1.36546
$767$	1.45994e10	1.16829
$768$	−2.41729e9	−0.192559
$769$	−2.00677e10	−1.59132	−0.795658	−	0.605746i	$-0.792875\pi$
$769$	−2.00677e10	−1.59132	−0.795658	+	0.605746i	$0.792875\pi$
$770$	0	0
$771$	4.34883e8	0.0341729
$772$	−7.18163e8	−0.0561775
$773$	2.15770e10	1.68020	0.840102	−	0.542428i	$-0.182495\pi$
$773$	2.15770e10	1.68020	0.840102	+	0.542428i	$0.182495\pi$
$774$	8.72734e9	0.676532
$775$	0	0
$776$	−1.81079e10	−1.39108
$777$	3.03371e9	0.232007
$778$	2.41714e10	1.84023
$779$	3.65719e9	0.277183
$780$	0	0
$781$	3.28842e9	0.247007
$782$	4.93512e9	0.369041
$783$	2.75337e9	0.204974
$784$	9.51506e9	0.705189
$785$	0	0
$786$	1.15930e10	0.851563
$787$	−2.32055e10	−1.69699	−0.848494	−	0.529205i	$-0.822490\pi$
$787$	−2.32055e10	−1.69699	−0.848494	+	0.529205i	$0.822490\pi$
$788$	1.47557e9	0.107428
$789$	4.59479e9	0.333040
$790$	0	0
$791$	−2.36330e9	−0.169786
$792$	1.17678e10	0.841703
$793$	−1.40114e10	−0.997756
$794$	−2.44100e9	−0.173060
$795$	0	0
$796$	2.02863e9	0.142563
$797$	6.77123e8	0.0473766	0.0236883	−	0.999719i	$-0.492459\pi$
$797$	6.77123e8	0.0473766	0.0236883	+	0.999719i	$0.492459\pi$
$798$	1.00276e9	0.0698533
$799$	9.13727e7	0.00633728
$800$	0	0
$801$	6.45129e9	0.443540
$802$	9.81954e9	0.672173
$803$	−1.01476e10	−0.691607
$804$	−1.05264e9	−0.0714305
$805$	0	0
$806$	8.97254e9	0.603591
$807$	1.63994e10	1.09842
$808$	7.75440e9	0.517141
$809$	−5.84504e9	−0.388122	−0.194061	−	0.980990i	$-0.562166\pi$
$809$	−5.84504e9	−0.388122	−0.194061	+	0.980990i	$0.562166\pi$
$810$	0	0
$811$	1.91491e10	1.26060	0.630299	−	0.776353i	$-0.282932\pi$
$811$	1.91491e10	1.26060	0.630299	+	0.776353i	$0.282932\pi$
$812$	1.28987e8	0.00845472
$813$	3.64584e9	0.237947
$814$	1.64382e10	1.06824
$815$	0	0
$816$	7.45034e9	0.480021
$817$	−4.86330e9	−0.311999
$818$	−2.74875e9	−0.175590
$819$	4.87102e9	0.309832
$820$	0	0
$821$	6.17006e9	0.389124	0.194562	−	0.980890i	$-0.437671\pi$
$821$	6.17006e9	0.389124	0.194562	+	0.980890i	$0.437671\pi$
$822$	−1.48979e10	−0.935562
$823$	−2.25285e10	−1.40875	−0.704373	−	0.709830i	$-0.748772\pi$
$823$	−2.25285e10	−1.40875	−0.704373	+	0.709830i	$0.748772\pi$
$824$	5.25450e9	0.327180
$825$	0	0
$826$	6.50995e9	0.401927
$827$	6.08545e9	0.374131	0.187065	−	0.982347i	$-0.440102\pi$
$827$	6.08545e9	0.374131	0.187065	+	0.982347i	$0.440102\pi$
$828$	4.00780e8	0.0245358
$829$	4.81588e9	0.293586	0.146793	−	0.989167i	$-0.453105\pi$
$829$	4.81588e9	0.293586	0.146793	+	0.989167i	$0.453105\pi$
$830$	0	0
$831$	−1.65021e10	−0.997554
$832$	−2.28801e10	−1.37730
$833$	1.01477e10	0.608288
$834$	3.96053e9	0.236413
$835$	0	0
$836$	−5.62081e8	−0.0332718
$837$	−8.83489e9	−0.520789
$838$	3.19142e9	0.187340
$839$	−2.92635e10	−1.71064	−0.855320	−	0.518100i	$-0.826640\pi$
$839$	−2.92635e10	−1.71064	−0.855320	+	0.518100i	$0.826640\pi$
$840$	0	0
$841$	−1.65950e10	−0.962038
$842$	−1.14894e10	−0.663290
$843$	−5.50769e9	−0.316645
$844$	3.21127e9	0.183856
$845$	0	0
$846$	−7.17301e7	−0.00407291
$847$	1.13976e10	0.644499
$848$	8.66615e9	0.488024
$849$	−5.23902e9	−0.293814
$850$	0	0
$851$	6.53145e9	0.363292
$852$	1.86735e8	0.0103440
$853$	−2.17155e10	−1.19797	−0.598987	−	0.800759i	$-0.704430\pi$
$853$	−2.17155e10	−1.19797	−0.598987	+	0.800759i	$0.704430\pi$
$854$	−6.24774e9	−0.343258
$855$	0	0
$856$	2.24126e10	1.22134
$857$	4.01757e9	0.218037	0.109019	−	0.994040i	$-0.465229\pi$
$857$	4.01757e9	0.218037	0.109019	+	0.994040i	$0.465229\pi$
$858$	−2.41075e10	−1.30301
$859$	1.62487e10	0.874666	0.437333	−	0.899300i	$-0.355923\pi$
$859$	1.62487e10	0.874666	0.437333	+	0.899300i	$0.355923\pi$
$860$	0	0
$861$	−7.23548e9	−0.386328
$862$	−1.02659e9	−0.0545913
$863$	1.95958e10	1.03783	0.518914	−	0.854826i	$-0.326336\pi$
$863$	1.95958e10	1.03783	0.518914	+	0.854826i	$0.326336\pi$
$864$	3.72683e9	0.196581
$865$	0	0
$866$	−2.04607e10	−1.07055
$867$	−5.31277e9	−0.276856
$868$	−4.13887e8	−0.0214814
$869$	−7.23604e9	−0.374052
$870$	0	0
$871$	−2.75441e10	−1.41242
$872$	3.03264e10	1.54886
$873$	1.37264e10	0.698243
$874$	2.15890e9	0.109381
$875$	0	0
$876$	−5.76238e8	−0.0289626
$877$	3.67840e10	1.84145	0.920727	−	0.390207i	$-0.127597\pi$
$877$	3.67840e10	1.84145	0.920727	+	0.390207i	$0.127597\pi$
$878$	−1.19794e10	−0.597314
$879$	−1.24670e10	−0.619159
$880$	0	0
$881$	1.48378e9	0.0731062	0.0365531	−	0.999332i	$-0.488362\pi$
$881$	1.48378e9	0.0731062	0.0365531	+	0.999332i	$0.488362\pi$
$882$	−7.96620e9	−0.390941
$883$	2.36597e10	1.15650	0.578250	−	0.815859i	$-0.303736\pi$
$883$	2.36597e10	1.15650	0.578250	+	0.815859i	$0.303736\pi$
$884$	−1.90920e9	−0.0929541
$885$	0	0
$886$	−3.47073e10	−1.67650
$887$	−4.21269e9	−0.202687	−0.101344	−	0.994851i	$-0.532314\pi$
$887$	−4.21269e9	−0.202687	−0.101344	+	0.994851i	$0.532314\pi$
$888$	1.08903e10	0.521907
$889$	−1.19863e10	−0.572177
$890$	0	0
$891$	6.66945e9	0.315877
$892$	1.79199e9	0.0845394
$893$	3.99715e7	0.00187832
$894$	1.39224e10	0.651678
$895$	0	0
$896$	−8.34008e9	−0.387340
$897$	−9.57875e9	−0.443134
$898$	−7.85676e8	−0.0362056
$899$	−2.10125e9	−0.0964535
$900$	0	0
$901$	9.24233e9	0.420964
$902$	−3.92054e10	−1.77878
$903$	9.62167e9	0.434854
$904$	−8.48365e9	−0.381938
$905$	0	0
$906$	9.97349e9	0.445552
$907$	−1.32662e10	−0.590367	−0.295184	−	0.955441i	$-0.595381\pi$
$907$	−1.32662e10	−0.590367	−0.295184	+	0.955441i	$0.595381\pi$
$908$	−2.81022e9	−0.124578
$909$	−5.87811e9	−0.259576
$910$	0	0
$911$	7.15727e9	0.313641	0.156821	−	0.987627i	$-0.449876\pi$
$911$	7.15727e9	0.313641	0.156821	+	0.987627i	$0.449876\pi$
$912$	3.25919e9	0.142275
$913$	1.77788e10	0.773132
$914$	2.64181e9	0.114443
$915$	0	0
$916$	−1.58258e9	−0.0680349
$917$	−1.39930e10	−0.599263
$918$	−1.81725e10	−0.775291
$919$	−9.78152e9	−0.415721	−0.207860	−	0.978158i	$-0.566650\pi$
$919$	−9.78152e9	−0.415721	−0.207860	+	0.978158i	$0.566650\pi$
$920$	0	0
$921$	−2.06085e10	−0.869236
$922$	3.50740e10	1.47376
$923$	4.88623e9	0.204535
$924$	1.11204e9	0.0463732
$925$	0	0
$926$	−5.90713e9	−0.244477
$927$	−3.98309e9	−0.164226
$928$	8.86371e8	0.0364080
$929$	2.92073e10	1.19519	0.597594	−	0.801799i	$-0.296124\pi$
$929$	2.92073e10	1.19519	0.597594	+	0.801799i	$0.296124\pi$
$930$	0	0
$931$	4.43915e9	0.180292
$932$	−2.20103e9	−0.0890576
$933$	−2.35165e10	−0.947952
$934$	−1.42050e10	−0.570464
$935$	0	0
$936$	1.74857e10	0.696976
$937$	3.72053e10	1.47746	0.738731	−	0.674000i	$-0.235425\pi$
$937$	3.72053e10	1.47746	0.738731	+	0.674000i	$0.235425\pi$
$938$	−1.22821e10	−0.485916
$939$	1.44388e10	0.569116
$940$	0	0
$941$	6.20016e9	0.242571	0.121286	−	0.992618i	$-0.461298\pi$
$941$	6.20016e9	0.242571	0.121286	+	0.992618i	$0.461298\pi$
$942$	−1.05082e10	−0.409589
$943$	−1.55777e10	−0.604938
$944$	2.11588e10	0.818631
$945$	0	0
$946$	5.21350e10	2.00221
$947$	−1.27543e10	−0.488015	−0.244008	−	0.969773i	$-0.578462\pi$
$947$	−1.27543e10	−0.488015	−0.244008	+	0.969773i	$0.578462\pi$
$948$	−4.10903e8	−0.0156643
$949$	−1.50782e10	−0.572689
$950$	0	0
$951$	1.90985e10	0.720057
$952$	−9.93210e9	−0.373088
$953$	−3.08638e10	−1.15511	−0.577556	−	0.816351i	$-0.695994\pi$
$953$	−3.08638e10	−1.15511	−0.577556	+	0.816351i	$0.695994\pi$
$954$	−7.25548e9	−0.270550
$955$	0	0
$956$	−1.26140e9	−0.0466929
$957$	5.64565e9	0.208220
$958$	1.72313e10	0.633198
$959$	1.79820e10	0.658375
$960$	0	0
$961$	−2.07702e10	−0.754935
$962$	2.44253e10	0.884559
$963$	−1.69896e10	−0.613042
$964$	−2.33060e9	−0.0837910
$965$	0	0
$966$	−4.27122e9	−0.152451
$967$	−3.32539e10	−1.18263	−0.591317	−	0.806439i	$-0.701392\pi$
$967$	−3.32539e10	−1.18263	−0.591317	+	0.806439i	$0.701392\pi$
$968$	4.09145e10	1.44982
$969$	3.47588e9	0.122725
$970$	0	0
$971$	4.45095e10	1.56022	0.780108	−	0.625644i	$-0.215164\pi$
$971$	4.45095e10	1.56022	0.780108	+	0.625644i	$0.215164\pi$
$972$	−2.44501e9	−0.0853982
$973$	−4.78043e9	−0.166369
$974$	−2.11043e10	−0.731837
$975$	0	0
$976$	−2.03065e10	−0.699136
$977$	6.80736e9	0.233533	0.116766	−	0.993159i	$-0.462747\pi$
$977$	6.80736e9	0.233533	0.116766	+	0.993159i	$0.462747\pi$
$978$	3.25525e10	1.11275
$979$	3.85384e10	1.31267
$980$	0	0
$981$	−2.29884e10	−0.777442
$982$	−2.57179e9	−0.0866652
$983$	6.37498e8	0.0214063	0.0107031	−	0.999943i	$-0.496593\pi$
$983$	6.37498e8	0.0214063	0.0107031	+	0.999943i	$0.496593\pi$
$984$	−2.59735e10	−0.869056
$985$	0	0
$986$	−4.32205e9	−0.143589
$987$	−7.90806e7	−0.00261794
$988$	−8.35190e8	−0.0275509
$989$	2.07150e10	0.680924
$990$	0	0
$991$	5.60147e10	1.82828	0.914142	−	0.405393i	$-0.132865\pi$
$991$	5.60147e10	1.82828	0.914142	+	0.405393i	$0.132865\pi$
$992$	−2.84414e9	−0.0925041
$993$	1.88977e10	0.612472
$994$	2.17879e9	0.0703662
$995$	0	0
$996$	1.00958e9	0.0323766
$997$	−8.97443e9	−0.286796	−0.143398	−	0.989665i	$-0.545803\pi$
$997$	−8.97443e9	−0.286796	−0.143398	+	0.989665i	$0.545803\pi$
$998$	−3.30263e10	−1.05173
$999$	−2.40506e10	−0.763214

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.8.a.d.1.1		2
3.2	odd	2		225.8.a.n.1.2		2
4.3	odd	2		400.8.a.y.1.1		2
5.2	odd	4		5.8.b.a.4.1	✓	2
5.3	odd	4		5.8.b.a.4.2	yes	2
5.4	even	2	inner	25.8.a.d.1.2		2
15.2	even	4		45.8.b.a.19.2		2
15.8	even	4		45.8.b.a.19.1		2
15.14	odd	2		225.8.a.n.1.1		2
20.3	even	4		80.8.c.a.49.1		2
20.7	even	4		80.8.c.a.49.2		2
20.19	odd	2		400.8.a.y.1.2		2
40.3	even	4		320.8.c.c.129.2		2
40.13	odd	4		320.8.c.d.129.1		2
40.27	even	4		320.8.c.c.129.1		2
40.37	odd	4		320.8.c.d.129.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.8.b.a.4.1	✓	2	5.2	odd	4
5.8.b.a.4.2	yes	2	5.3	odd	4
25.8.a.d.1.1		2	1.1	even	1	trivial
25.8.a.d.1.2		2	5.4	even	2	inner
45.8.b.a.19.1		2	15.8	even	4
45.8.b.a.19.2		2	15.2	even	4
80.8.c.a.49.1		2	20.3	even	4
80.8.c.a.49.2		2	20.7	even	4
225.8.a.n.1.1		2	15.14	odd	2
225.8.a.n.1.2		2	3.2	odd	2
320.8.c.c.129.1		2	40.27	even	4
320.8.c.c.129.2		2	40.3	even	4
320.8.c.d.129.1		2	40.13	odd	4
320.8.c.d.129.2		2	40.37	odd	4
400.8.a.y.1.1		2	4.3	odd	2
400.8.a.y.1.2		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	25.a (trivial)

\(f(q)\)	\(=\)	\(q-10.7703 q^{2} +32.3110 q^{3} -12.0000 q^{4} -348.000 q^{6} +420.043 q^{7} +1507.85 q^{8} -1143.00 q^{9} +O(q^{10})\)	\(q-10.7703 q^{2} +32.3110 q^{3} -12.0000 q^{4} -348.000 q^{6} +420.043 q^{7} +1507.85 q^{8} -1143.00 q^{9} -6828.00 q^{11} -387.732 q^{12} -10145.7 q^{13} -4524.00 q^{14} -14704.0 q^{16} -15681.6 q^{17} +12310.5 q^{18} -6860.00 q^{19} +13572.0 q^{21} +73539.8 q^{22} +29219.9 q^{23} +48720.0 q^{24} +109272. q^{26} -107596. q^{27} -5040.51 q^{28} -25590.0 q^{29} +82112.0 q^{31} -34637.4 q^{32} -220619. q^{33} +168896. q^{34} +13716.0 q^{36} +223527. q^{37} +73884.5 q^{38} -327816. q^{39} -533118. q^{41} -146175. q^{42} +708935. q^{43} +81936.0 q^{44} -314708. q^{46} -5826.75 q^{47} -475101. q^{48} -647107. q^{49} -506688. q^{51} +121748. q^{52} -589374. q^{53} +1.15884e6 q^{54} +633360. q^{56} -221653. q^{57} +275613. q^{58} -1.43898e6 q^{59} +1.38102e6 q^{61} -884373. q^{62} -480109. q^{63} +2.25517e6 q^{64} +2.37614e6 q^{66} +2.71487e6 q^{67} +188179. q^{68} +944124. q^{69} -481608. q^{71} -1.72347e6 q^{72} +1.48618e6 q^{73} -2.40746e6 q^{74} +82320.0 q^{76} -2.86805e6 q^{77} +3.53069e6 q^{78} +1.05976e6 q^{79} -976779. q^{81} +5.74186e6 q^{82} -2.60380e6 q^{83} -162864. q^{84} -7.63547e6 q^{86} -826838. q^{87} -1.02956e7 q^{88} -5.64417e6 q^{89} -4.26161e6 q^{91} -350639. q^{92} +2.65312e6 q^{93} +62756.0 q^{94} -1.11917e6 q^{96} -1.20091e7 q^{97} +6.96956e6 q^{98} +7.80440e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 24 q^{4} - 696 q^{6} - 2286 q^{9}+O(q^{10}) \) 2 * q - 24 * q^4 - 696 * q^6 - 2286 * q^9	\( 2 q - 24 q^{4} - 696 q^{6} - 2286 q^{9} - 13656 q^{11} - 9048 q^{14} - 29408 q^{16} - 13720 q^{19} + 27144 q^{21} + 97440 q^{24} + 218544 q^{26} - 51180 q^{29} + 164224 q^{31} + 337792 q^{34} + 27432 q^{36} - 655632 q^{39} - 1066236 q^{41} + 163872 q^{44} - 629416 q^{46} - 1294214 q^{49} - 1013376 q^{51} + 2317680 q^{54} + 1266720 q^{56} - 2877960 q^{59} + 2762044 q^{61} + 4510336 q^{64} + 4752288 q^{66} + 1888248 q^{69} - 963216 q^{71} - 4814928 q^{74} + 164640 q^{76} + 2119520 q^{79} - 1953558 q^{81} - 325728 q^{84} - 15270936 q^{86} - 11288340 q^{89} - 8523216 q^{91} + 125512 q^{94} - 2238336 q^{96} + 15608808 q^{99}+O(q^{100}) \) 2 * q - 24 * q^4 - 696 * q^6 - 2286 * q^9 - 13656 * q^11 - 9048 * q^14 - 29408 * q^16 - 13720 * q^19 + 27144 * q^21 + 97440 * q^24 + 218544 * q^26 - 51180 * q^29 + 164224 * q^31 + 337792 * q^34 + 27432 * q^36 - 655632 * q^39 - 1066236 * q^41 + 163872 * q^44 - 629416 * q^46 - 1294214 * q^49 - 1013376 * q^51 + 2317680 * q^54 + 1266720 * q^56 - 2877960 * q^59 + 2762044 * q^61 + 4510336 * q^64 + 4752288 * q^66 + 1888248 * q^69 - 963216 * q^71 - 4814928 * q^74 + 164640 * q^76 + 2119520 * q^79 - 1953558 * q^81 - 325728 * q^84 - 15270936 * q^86 - 11288340 * q^89 - 8523216 * q^91 + 125512 * q^94 - 2238336 * q^96 + 15608808 * q^99

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