LMFDB - Embedded newform 245.10.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,10,Mod(1,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("245.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	245.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:	$126.183779860$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - x^{12} - 5109 x^{11} + 3203 x^{10} + 9635922 x^{9} + 242128 x^{8} - 8405086048 x^{7} + \cdots - 96\!\cdots\!52 $ x^13 - x^12 - 5109x^11 + 3203x^10 + 9635922x^9 + 242128x^8 - 8405086048x^7 - 4775917264x^6 + 3475841466656x^5 + 1485626212224x^4 - 636895157285888x^3 + 323463590100992x^2 + 40992624146128896*x - 96856528223895552
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{3}\cdot 5^{3}\cdot 7^{7} $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$21.3241$ of defining polynomial
Character	$\chi$	$=$	245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-21.3241 q^{2} -190.064 q^{3} -57.2810 q^{4} -625.000 q^{5} +4052.95 q^{6} +12139.4 q^{8} +16441.3 q^{9} +13327.6 q^{10} -79888.7 q^{11} +10887.1 q^{12} +148270. q^{13} +118790. q^{15} -229535. q^{16} +196807. q^{17} -350597. q^{18} +743096. q^{19} +35800.6 q^{20} +1.70356e6 q^{22} +1.25821e6 q^{23} -2.30727e6 q^{24} +390625. q^{25} -3.16173e6 q^{26} +616127. q^{27} +4.19204e6 q^{29} -2.53309e6 q^{30} +6.78769e6 q^{31} -1.32075e6 q^{32} +1.51840e7 q^{33} -4.19674e6 q^{34} -941775. q^{36} +9.11742e6 q^{37} -1.58459e7 q^{38} -2.81808e7 q^{39} -7.58714e6 q^{40} -1.68499e7 q^{41} +1.19411e7 q^{43} +4.57610e6 q^{44} -1.02758e7 q^{45} -2.68303e7 q^{46} -5.48209e7 q^{47} +4.36263e7 q^{48} -8.32974e6 q^{50} -3.74059e7 q^{51} -8.49305e6 q^{52} +3.28424e7 q^{53} -1.31384e7 q^{54} +4.99304e7 q^{55} -1.41236e8 q^{57} -8.93918e7 q^{58} -7.27412e7 q^{59} -6.80441e6 q^{60} -1.26993e8 q^{61} -1.44742e8 q^{62} +1.45686e8 q^{64} -9.26687e7 q^{65} -3.23785e8 q^{66} +8.96858e7 q^{67} -1.12733e7 q^{68} -2.39141e8 q^{69} +1.94271e8 q^{71} +1.99588e8 q^{72} +2.22243e8 q^{73} -1.94421e8 q^{74} -7.42437e7 q^{75} -4.25653e7 q^{76} +6.00931e8 q^{78} +2.27969e8 q^{79} +1.43459e8 q^{80} -4.40718e8 q^{81} +3.59309e8 q^{82} +1.48052e7 q^{83} -1.23004e8 q^{85} -2.54634e8 q^{86} -7.96757e8 q^{87} -9.69803e8 q^{88} +4.47570e8 q^{89} +2.19123e8 q^{90} -7.20717e7 q^{92} -1.29010e9 q^{93} +1.16901e9 q^{94} -4.64435e8 q^{95} +2.51027e8 q^{96} -6.78338e8 q^{97} -1.31347e9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - q^{2} - 268 q^{3} + 3563 q^{4} - 8125 q^{5} - 3040 q^{6} - 4695 q^{8} + 82107 q^{9} + 625 q^{10} + 129087 q^{11} - 356068 q^{12} - 35889 q^{13} + 167500 q^{15} + 1379187 q^{16} - 251650 q^{17} + 391089 q^{18}+ \cdots + 5266142099 q^{99}+O(q^{100}) $ 13 * q - q^2 - 268 * q^3 + 3563 * q^4 - 8125 * q^5 - 3040 * q^6 - 4695 * q^8 + 82107 * q^9 + 625 * q^10 + 129087 * q^11 - 356068 * q^12 - 35889 * q^13 + 167500 * q^15 + 1379187 * q^16 - 251650 * q^17 + 391089 * q^18 + 22537 * q^19 - 2226875 * q^20 - 867693 * q^22 + 1640451 * q^23 - 3300104 * q^24 + 5078125 * q^25 + 215949 * q^26 - 4833346 * q^27 + 175192 * q^29 + 1900000 * q^30 - 3363236 * q^31 + 11618029 * q^32 - 1855346 * q^33 - 30084718 * q^34 + 2860133 * q^36 + 57134155 * q^37 - 28383363 * q^38 - 10216786 * q^39 + 2934375 * q^40 - 46043487 * q^41 - 23472776 * q^43 + 171806579 * q^44 - 51316875 * q^45 - 53075341 * q^46 + 1336787 * q^47 - 304545376 * q^48 - 390625 * q^50 + 234254040 * q^51 - 80069087 * q^52 + 122467755 * q^53 + 99983754 * q^54 - 80679375 * q^55 + 32652030 * q^57 + 37365152 * q^58 + 136056496 * q^59 + 222542500 * q^60 + 50448822 * q^61 + 109231036 * q^62 + 1109008839 * q^64 + 22430625 * q^65 + 500348610 * q^66 + 585418818 * q^67 - 480342846 * q^68 + 521416036 * q^69 + 497194080 * q^71 - 870751121 * q^72 + 346232592 * q^73 - 613439215 * q^74 - 104687500 * q^75 + 1436195213 * q^76 - 327612138 * q^78 + 136453862 * q^79 - 861991875 * q^80 - 344488883 * q^81 + 224863259 * q^82 + 523042314 * q^83 + 157281250 * q^85 - 2285275668 * q^86 - 2310323838 * q^87 - 1816859183 * q^88 + 843350780 * q^89 - 244430625 * q^90 - 172327913 * q^92 + 198172924 * q^93 - 2777934057 * q^94 - 14085625 * q^95 + 1583073484 * q^96 - 694973158 * q^97 + 5266142099 * q^99

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 $

$2$	−21.3241	−0.942403	−0.471201	−	0.882026i	$-0.656180\pi$
$2$	−21.3241	−0.942403	−0.471201	+	0.882026i	$0.656180\pi$
$3$	−190.064	−1.35473	−0.677367	−	0.735645i	$-0.736879\pi$
$3$	−190.064	−1.35473	−0.677367	+	0.735645i	$0.736879\pi$
$4$	−57.2810	−0.111877
$5$	−625.000	−0.447214
$5$	−625.000	−0.447214
$6$	4052.95	1.27671
$7$	0	0
$7$	0	0
$8$	12139.4	1.04784
$9$	16441.3	0.835306
$10$	13327.6	0.421455
$11$	−79888.7	−1.64520	−0.822599	−	0.568622i	$-0.807477\pi$
$11$	−79888.7	−1.64520	−0.822599	+	0.568622i	$0.807477\pi$
$12$	10887.1	0.151564
$13$	148270.	1.43982	0.719910	−	0.694067i	$-0.244183\pi$
$13$	148270.	1.43982	0.719910	+	0.694067i	$0.244183\pi$
$14$	0	0
$15$	118790.	0.605856
$16$	−229535.	−0.875607
$17$	196807.	0.571506	0.285753	−	0.958303i	$-0.407756\pi$
$17$	196807.	0.571506	0.285753	+	0.958303i	$0.407756\pi$
$18$	−350597.	−0.787194
$19$	743096.	1.30814	0.654069	−	0.756435i	$-0.273061\pi$
$19$	743096.	1.30814	0.654069	+	0.756435i	$0.273061\pi$
$20$	35800.6	0.0500329
$21$	0	0
$22$	1.70356e6	1.55044
$23$	1.25821e6	0.937517	0.468758	−	0.883326i	$-0.344702\pi$
$23$	1.25821e6	0.937517	0.468758	+	0.883326i	$0.344702\pi$
$24$	−2.30727e6	−1.41954
$25$	390625.	0.200000
$26$	−3.16173e6	−1.35689
$27$	616127.	0.223117
$28$	0	0
$29$	4.19204e6	1.10061	0.550307	−	0.834963i	$-0.314511\pi$
$29$	4.19204e6	1.10061	0.550307	+	0.834963i	$0.314511\pi$
$30$	−2.53309e6	−0.570960
$31$	6.78769e6	1.32006	0.660031	−	0.751238i	$-0.270543\pi$
$31$	6.78769e6	1.32006	0.660031	+	0.751238i	$0.270543\pi$
$32$	−1.32075e6	−0.222662
$33$	1.51840e7	2.22881
$34$	−4.19674e6	−0.538588
$35$	0	0
$36$	−941775.	−0.0934514
$37$	9.11742e6	0.799768	0.399884	−	0.916566i	$-0.369050\pi$
$37$	9.11742e6	0.799768	0.399884	+	0.916566i	$0.369050\pi$
$38$	−1.58459e7	−1.23279
$39$	−2.81808e7	−1.95057
$40$	−7.58714e6	−0.468606
$41$	−1.68499e7	−0.931255	−0.465628	−	0.884981i	$-0.654171\pi$
$41$	−1.68499e7	−0.931255	−0.465628	+	0.884981i	$0.654171\pi$
$42$	0	0
$43$	1.19411e7	0.532644	0.266322	−	0.963884i	$-0.414192\pi$
$43$	1.19411e7	0.532644	0.266322	+	0.963884i	$0.414192\pi$
$44$	4.57610e6	0.184060
$45$	−1.02758e7	−0.373560
$46$	−2.68303e7	−0.883518
$47$	−5.48209e7	−1.63872	−0.819362	−	0.573276i	$-0.805672\pi$
$47$	−5.48209e7	−1.63872	−0.819362	+	0.573276i	$0.805672\pi$
$48$	4.36263e7	1.18621
$49$	0	0
$50$	−8.32974e6	−0.188481
$51$	−3.74059e7	−0.774238
$52$	−8.49305e6	−0.161083
$53$	3.28424e7	0.571734	0.285867	−	0.958269i	$-0.407718\pi$
$53$	3.28424e7	0.571734	0.285867	+	0.958269i	$0.407718\pi$
$54$	−1.31384e7	−0.210266
$55$	4.99304e7	0.735755
$56$	0	0
$57$	−1.41236e8	−1.77218
$58$	−8.93918e7	−1.03722
$59$	−7.27412e7	−0.781531	−0.390766	−	0.920490i	$-0.627790\pi$
$59$	−7.27412e7	−0.781531	−0.390766	+	0.920490i	$0.627790\pi$
$60$	−6.80441e6	−0.0677813
$61$	−1.26993e8	−1.17434	−0.587170	−	0.809463i	$-0.699758\pi$
$61$	−1.26993e8	−1.17434	−0.587170	+	0.809463i	$0.699758\pi$
$62$	−1.44742e8	−1.24403
$63$	0	0
$64$	1.45686e8	1.08544
$65$	−9.26687e7	−0.643907
$66$	−3.23785e8	−2.10043
$67$	8.96858e7	0.543735	0.271867	−	0.962335i	$-0.412359\pi$
$67$	8.96858e7	0.543735	0.271867	+	0.962335i	$0.412359\pi$
$68$	−1.12733e7	−0.0639383
$69$	−2.39141e8	−1.27009
$70$	0	0
$71$	1.94271e8	0.907289	0.453645	−	0.891183i	$-0.350124\pi$
$71$	1.94271e8	0.907289	0.453645	+	0.891183i	$0.350124\pi$
$72$	1.99588e8	0.875263
$73$	2.22243e8	0.915958	0.457979	−	0.888963i	$-0.348573\pi$
$73$	2.22243e8	0.915958	0.457979	+	0.888963i	$0.348573\pi$
$74$	−1.94421e8	−0.753704
$75$	−7.42437e7	−0.270947
$76$	−4.25653e7	−0.146350
$77$	0	0
$78$	6.00931e8	1.83823
$79$	2.27969e8	0.658497	0.329249	−	0.944243i	$-0.393205\pi$
$79$	2.27969e8	0.658497	0.329249	+	0.944243i	$0.393205\pi$
$80$	1.43459e8	0.391583
$81$	−4.40718e8	−1.13757
$82$	3.59309e8	0.877617
$83$	1.48052e7	0.0342423	0.0171212	−	0.999853i	$-0.494550\pi$
$83$	1.48052e7	0.0342423	0.0171212	+	0.999853i	$0.494550\pi$
$84$	0	0
$85$	−1.23004e8	−0.255585
$86$	−2.54634e8	−0.501965
$87$	−7.96757e8	−1.49104
$88$	−9.69803e8	−1.72390
$89$	4.47570e8	0.756147	0.378074	−	0.925776i	$-0.376587\pi$
$89$	4.47570e8	0.756147	0.378074	+	0.925776i	$0.376587\pi$
$90$	2.19123e8	0.352044
$91$	0	0
$92$	−7.20717e7	−0.104886
$93$	−1.29010e9	−1.78833
$94$	1.16901e9	1.54434
$95$	−4.64435e8	−0.585017
$96$	2.51027e8	0.301648
$97$	−6.78338e8	−0.777989	−0.388994	−	0.921240i	$-0.627177\pi$
$97$	−6.78338e8	−0.777989	−0.388994	+	0.921240i	$0.627177\pi$
$98$	0	0
$99$	−1.31347e9	−1.37424
$100$	−2.23754e7	−0.0223754
$101$	−1.87300e9	−1.79098	−0.895492	−	0.445079i	$-0.853176\pi$
$101$	−1.87300e9	−1.79098	−0.895492	+	0.445079i	$0.853176\pi$
$102$	7.97649e8	0.729644
$103$	1.87149e9	1.63840	0.819201	−	0.573507i	$-0.194417\pi$
$103$	1.87149e9	1.63840	0.819201	+	0.573507i	$0.194417\pi$
$104$	1.79991e9	1.50870
$105$	0	0
$106$	−7.00337e8	−0.538804
$107$	1.08424e9	0.799644	0.399822	−	0.916593i	$-0.369072\pi$
$107$	1.08424e9	0.799644	0.399822	+	0.916593i	$0.369072\pi$
$108$	−3.52923e7	−0.0249617
$109$	−1.13564e9	−0.770584	−0.385292	−	0.922795i	$-0.625899\pi$
$109$	−1.13564e9	−0.770584	−0.385292	+	0.922795i	$0.625899\pi$
$110$	−1.06472e9	−0.693377
$111$	−1.73289e9	−1.08347
$112$	0	0
$113$	−1.05120e9	−0.606502	−0.303251	−	0.952911i	$-0.598072\pi$
$113$	−1.05120e9	−0.606502	−0.303251	+	0.952911i	$0.598072\pi$
$114$	3.01173e9	1.67011
$115$	−7.86383e8	−0.419270
$116$	−2.40124e8	−0.123133
$117$	2.43775e9	1.20269
$118$	1.55114e9	0.736517
$119$	0	0
$120$	1.44204e9	0.634837
$121$	4.02425e9	1.70667
$122$	2.70801e9	1.10670
$123$	3.20255e9	1.26160
$124$	−3.88806e8	−0.147684
$125$	−2.44141e8	−0.0894427
$126$	0	0
$127$	−1.82342e9	−0.621969	−0.310985	−	0.950415i	$-0.600659\pi$
$127$	−1.82342e9	−0.621969	−0.310985	+	0.950415i	$0.600659\pi$
$128$	−2.43040e9	−0.800263
$129$	−2.26958e9	−0.721591
$130$	1.97608e9	0.606820
$131$	2.64606e9	0.785018	0.392509	−	0.919748i	$-0.371607\pi$
$131$	2.64606e9	0.785018	0.392509	+	0.919748i	$0.371607\pi$
$132$	−8.69752e8	−0.249352
$133$	0	0
$134$	−1.91247e9	−0.512417
$135$	−3.85079e8	−0.0997810
$136$	2.38912e9	0.598844
$137$	−6.37677e8	−0.154653	−0.0773265	−	0.997006i	$-0.524638\pi$
$137$	−6.37677e8	−0.154653	−0.0773265	+	0.997006i	$0.524638\pi$
$138$	5.09948e9	1.19693
$139$	2.75678e9	0.626376	0.313188	−	0.949691i	$-0.398603\pi$
$139$	2.75678e9	0.626376	0.313188	+	0.949691i	$0.398603\pi$
$140$	0	0
$141$	1.04195e10	2.22004
$142$	−4.14267e9	−0.855032
$143$	−1.18451e10	−2.36879
$144$	−3.77386e9	−0.731399
$145$	−2.62003e9	−0.492209
$146$	−4.73915e9	−0.863202
$147$	0	0
$148$	−5.22254e8	−0.0894756
$149$	7.86514e9	1.30728	0.653639	−	0.756806i	$-0.273241\pi$
$149$	7.86514e9	1.30728	0.653639	+	0.756806i	$0.273241\pi$
$150$	1.58318e9	0.255341
$151$	1.15679e10	1.81075	0.905373	−	0.424617i	$-0.139591\pi$
$151$	1.15679e10	1.81075	0.905373	+	0.424617i	$0.139591\pi$
$152$	9.02076e9	1.37071
$153$	3.23577e9	0.477382
$154$	0	0
$155$	−4.24231e9	−0.590350
$156$	1.61422e9	0.218224
$157$	1.55705e8	0.0204528	0.0102264	−	0.999948i	$-0.496745\pi$
$157$	1.55705e8	0.0204528	0.0102264	+	0.999948i	$0.496745\pi$
$158$	−4.86124e9	−0.620570
$159$	−6.24216e9	−0.774548
$160$	8.25469e8	0.0995774
$161$	0	0
$162$	9.39793e9	1.07205
$163$	−1.11616e10	−1.23846	−0.619229	−	0.785210i	$-0.712555\pi$
$163$	−1.11616e10	−1.23846	−0.619229	+	0.785210i	$0.712555\pi$
$164$	9.65176e8	0.104186
$165$	−9.48997e9	−0.996752
$166$	−3.15708e8	−0.0322700
$167$	−5.25167e9	−0.522485	−0.261242	−	0.965273i	$-0.584132\pi$
$167$	−5.25167e9	−0.522485	−0.261242	+	0.965273i	$0.584132\pi$
$168$	0	0
$169$	1.13795e10	1.07308
$170$	2.62296e9	0.240864
$171$	1.22175e10	1.09270
$172$	−6.83999e8	−0.0595906
$173$	−7.59307e9	−0.644481	−0.322240	−	0.946658i	$-0.604436\pi$
$173$	−7.59307e9	−0.644481	−0.322240	+	0.946658i	$0.604436\pi$
$174$	1.69902e10	1.40516
$175$	0	0
$176$	1.83372e10	1.44055
$177$	1.38255e10	1.05877
$178$	−9.54406e9	−0.712595
$179$	1.27063e10	0.925082	0.462541	−	0.886598i	$-0.346938\pi$
$179$	1.27063e10	0.925082	0.462541	+	0.886598i	$0.346938\pi$
$180$	5.88609e8	0.0417927
$181$	1.36557e10	0.945718	0.472859	−	0.881138i	$-0.343222\pi$
$181$	1.36557e10	0.945718	0.472859	+	0.881138i	$0.343222\pi$
$182$	0	0
$183$	2.41367e10	1.59092
$184$	1.52740e10	0.982364
$185$	−5.69838e9	−0.357667
$186$	2.75102e10	1.68533
$187$	−1.57226e10	−0.940239
$188$	3.14020e9	0.183335
$189$	0	0
$190$	9.90367e9	0.551322
$191$	5.55799e9	0.302181	0.151091	−	0.988520i	$-0.451721\pi$
$191$	5.55799e9	0.302181	0.151091	+	0.988520i	$0.451721\pi$
$192$	−2.76896e10	−1.47049
$193$	−1.23134e10	−0.638808	−0.319404	−	0.947619i	$-0.603483\pi$
$193$	−1.23134e10	−0.638808	−0.319404	+	0.947619i	$0.603483\pi$
$194$	1.44650e10	0.733179
$195$	1.76130e10	0.872323
$196$	0	0
$197$	6.37549e9	0.301589	0.150795	−	0.988565i	$-0.451817\pi$
$197$	6.37549e9	0.301589	0.150795	+	0.988565i	$0.451817\pi$
$198$	2.80087e10	1.29509
$199$	−4.28681e9	−0.193774	−0.0968869	−	0.995295i	$-0.530889\pi$
$199$	−4.28681e9	−0.193774	−0.0968869	+	0.995295i	$0.530889\pi$
$200$	4.74196e9	0.209567
$201$	−1.70460e10	−0.736617
$202$	3.99401e10	1.68783
$203$	0	0
$204$	2.14265e9	0.0866194
$205$	1.05312e10	0.416470
$206$	−3.99080e10	−1.54403
$207$	2.06867e10	0.783113
$208$	−3.40332e10	−1.26072
$209$	−5.93649e10	−2.15215
$210$	0	0
$211$	2.59322e10	0.900676	0.450338	−	0.892858i	$-0.351303\pi$
$211$	2.59322e10	0.900676	0.450338	+	0.892858i	$0.351303\pi$
$212$	−1.88125e9	−0.0639638
$213$	−3.69239e10	−1.22914
$214$	−2.31204e10	−0.753587
$215$	−7.46320e9	−0.238206
$216$	7.47942e9	0.233790
$217$	0	0
$218$	2.42165e10	0.726201
$219$	−4.22404e10	−1.24088
$220$	−2.86006e9	−0.0823140
$221$	2.91806e10	0.822865
$222$	3.69524e10	1.02107
$223$	−1.98676e10	−0.537990	−0.268995	−	0.963142i	$-0.586692\pi$
$223$	−1.98676e10	−0.537990	−0.268995	+	0.963142i	$0.586692\pi$
$224$	0	0
$225$	6.42239e9	0.167061
$226$	2.24159e10	0.571569
$227$	1.13329e10	0.283287	0.141643	−	0.989918i	$-0.454761\pi$
$227$	1.13329e10	0.283287	0.141643	+	0.989918i	$0.454761\pi$
$228$	8.09012e9	0.198266
$229$	−2.76440e10	−0.664265	−0.332132	−	0.943233i	$-0.607768\pi$
$229$	−2.76440e10	−0.664265	−0.332132	+	0.943233i	$0.607768\pi$
$230$	1.67689e10	0.395121
$231$	0	0
$232$	5.08890e10	1.15326
$233$	−3.57013e10	−0.793565	−0.396783	−	0.917913i	$-0.629873\pi$
$233$	−3.57013e10	−0.793565	−0.396783	+	0.917913i	$0.629873\pi$
$234$	−5.19830e10	−1.13342
$235$	3.42631e10	0.732860
$236$	4.16669e9	0.0874353
$237$	−4.33287e10	−0.892089
$238$	0	0
$239$	6.44509e10	1.27773	0.638864	−	0.769319i	$-0.279405\pi$
$239$	6.44509e10	1.27773	0.638864	+	0.769319i	$0.279405\pi$
$240$	−2.72665e10	−0.530491
$241$	5.79781e10	1.10710	0.553550	−	0.832816i	$-0.313273\pi$
$241$	5.79781e10	1.10710	0.553550	+	0.832816i	$0.313273\pi$
$242$	−8.58137e10	−1.60837
$243$	7.16374e10	1.31799
$244$	7.27426e9	0.131382
$245$	0	0
$246$	−6.82916e10	−1.18894
$247$	1.10179e11	1.88348
$248$	8.23987e10	1.38321
$249$	−2.81394e9	−0.0463892
$250$	5.20609e9	0.0842911
$251$	−1.40076e10	−0.222757	−0.111378	−	0.993778i	$-0.535527\pi$
$251$	−1.40076e10	−0.222757	−0.111378	+	0.993778i	$0.535527\pi$
$252$	0	0
$253$	−1.00517e11	−1.54240
$254$	3.88828e10	0.586146
$255$	2.33787e10	0.346250
$256$	−2.27649e10	−0.331273
$257$	6.94525e10	0.993091	0.496545	−	0.868011i	$-0.334602\pi$
$257$	6.94525e10	0.993091	0.496545	+	0.868011i	$0.334602\pi$
$258$	4.83968e10	0.680030
$259$	0	0
$260$	5.30816e9	0.0720383
$261$	6.89227e10	0.919349
$262$	−5.64250e10	−0.739803
$263$	1.12309e11	1.44748	0.723742	−	0.690071i	$-0.242421\pi$
$263$	1.12309e11	1.44748	0.723742	+	0.690071i	$0.242421\pi$
$264$	1.84325e11	2.33542
$265$	−2.05265e10	−0.255687
$266$	0	0
$267$	−8.50670e10	−1.02438
$268$	−5.13729e9	−0.0608314
$269$	−4.71714e9	−0.0549279	−0.0274640	−	0.999623i	$-0.508743\pi$
$269$	−4.71714e9	−0.0549279	−0.0274640	+	0.999623i	$0.508743\pi$
$270$	8.21148e9	0.0940339
$271$	−6.13139e10	−0.690554	−0.345277	−	0.938501i	$-0.612215\pi$
$271$	−6.13139e10	−0.690554	−0.345277	+	0.938501i	$0.612215\pi$
$272$	−4.51741e10	−0.500414
$273$	0	0
$274$	1.35979e10	0.145745
$275$	−3.12065e10	−0.329039
$276$	1.36982e10	0.142093
$277$	−1.06922e11	−1.09121	−0.545605	−	0.838043i	$-0.683700\pi$
$277$	−1.06922e11	−1.09121	−0.545605	+	0.838043i	$0.683700\pi$
$278$	−5.87859e10	−0.590298
$279$	1.11599e11	1.10266
$280$	0	0
$281$	9.68968e10	0.927110	0.463555	−	0.886068i	$-0.346574\pi$
$281$	9.68968e10	0.927110	0.463555	+	0.886068i	$0.346574\pi$
$282$	−2.22187e11	−2.09217
$283$	−2.94324e9	−0.0272763	−0.0136382	−	0.999907i	$-0.504341\pi$
$283$	−2.94324e9	−0.0272763	−0.0136382	+	0.999907i	$0.504341\pi$
$284$	−1.11280e10	−0.101505
$285$	8.82723e10	0.792543
$286$	2.52586e11	2.23235
$287$	0	0
$288$	−2.17149e10	−0.185991
$289$	−7.98549e10	−0.673381
$290$	5.58698e10	0.463860
$291$	1.28928e11	1.05397
$292$	−1.27303e10	−0.102475
$293$	−5.41297e10	−0.429073	−0.214536	−	0.976716i	$-0.568824\pi$
$293$	−5.41297e10	−0.429073	−0.214536	+	0.976716i	$0.568824\pi$
$294$	0	0
$295$	4.54632e10	0.349511
$296$	1.10680e11	0.838026
$297$	−4.92215e10	−0.367072
$298$	−1.67717e11	−1.23198
$299$	1.86555e11	1.34986
$300$	4.25275e9	0.0303127
$301$	0	0
$302$	−2.46675e11	−1.70645
$303$	3.55990e11	2.42631
$304$	−1.70567e11	−1.14541
$305$	7.93704e10	0.525181
$306$	−6.90000e10	−0.449886
$307$	−1.11130e11	−0.714014	−0.357007	−	0.934102i	$-0.616203\pi$
$307$	−1.11130e11	−0.714014	−0.357007	+	0.934102i	$0.616203\pi$
$308$	0	0
$309$	−3.55703e11	−2.21960
$310$	9.04635e10	0.556347
$311$	−3.88180e10	−0.235294	−0.117647	−	0.993055i	$-0.537535\pi$
$311$	−3.88180e10	−0.235294	−0.117647	+	0.993055i	$0.537535\pi$
$312$	−3.42099e11	−2.04388
$313$	−1.04187e11	−0.613568	−0.306784	−	0.951779i	$-0.599253\pi$
$313$	−1.04187e11	−0.613568	−0.306784	+	0.951779i	$0.599253\pi$
$314$	−3.32027e9	−0.0192748
$315$	0	0
$316$	−1.30583e10	−0.0736706
$317$	−2.80972e11	−1.56278	−0.781388	−	0.624045i	$-0.785488\pi$
$317$	−2.80972e11	−1.56278	−0.781388	+	0.624045i	$0.785488\pi$
$318$	1.33109e11	0.729936
$319$	−3.34897e11	−1.81073
$320$	−9.10536e10	−0.485425
$321$	−2.06074e11	−1.08331
$322$	0	0
$323$	1.46246e11	0.747608
$324$	2.52448e10	0.127268
$325$	5.79180e10	0.287964
$326$	2.38011e11	1.16713
$327$	2.15844e11	1.04394
$328$	−2.04548e11	−0.975803
$329$	0	0
$330$	2.02366e11	0.939342
$331$	−3.90426e11	−1.78778	−0.893888	−	0.448290i	$-0.852033\pi$
$331$	−3.90426e11	−1.78778	−0.893888	+	0.448290i	$0.852033\pi$
$332$	−8.48057e8	−0.00383092
$333$	1.49902e11	0.668051
$334$	1.11987e11	0.492391
$335$	−5.60536e10	−0.243166
$336$	0	0
$337$	−1.89901e11	−0.802034	−0.401017	−	0.916071i	$-0.631343\pi$
$337$	−1.89901e11	−0.802034	−0.401017	+	0.916071i	$0.631343\pi$
$338$	−2.42658e11	−1.01128
$339$	1.99795e11	0.821649
$340$	7.04581e9	0.0285941
$341$	−5.42259e11	−2.17176
$342$	−2.60527e11	−1.02976
$343$	0	0
$344$	1.44958e11	0.558124
$345$	1.49463e11	0.568000
$346$	1.61916e11	0.607361
$347$	3.81954e11	1.41426	0.707129	−	0.707085i	$-0.249990\pi$
$347$	3.81954e11	1.41426	0.707129	+	0.707085i	$0.249990\pi$
$348$	4.56390e10	0.166813
$349$	−2.33327e10	−0.0841881	−0.0420940	−	0.999114i	$-0.513403\pi$
$349$	−2.33327e10	−0.0841881	−0.0420940	+	0.999114i	$0.513403\pi$
$350$	0	0
$351$	9.13531e10	0.321249
$352$	1.05513e11	0.366323
$353$	−1.84317e11	−0.631798	−0.315899	−	0.948793i	$-0.602306\pi$
$353$	−1.84317e11	−0.631798	−0.315899	+	0.948793i	$0.602306\pi$
$354$	−2.94817e11	−0.997785
$355$	−1.21419e11	−0.405752
$356$	−2.56373e10	−0.0845954
$357$	0	0
$358$	−2.70951e11	−0.871800
$359$	−1.13296e11	−0.359990	−0.179995	−	0.983668i	$-0.557608\pi$
$359$	−1.13296e11	−0.359990	−0.179995	+	0.983668i	$0.557608\pi$
$360$	−1.24743e11	−0.391430
$361$	2.29504e11	0.711225
$362$	−2.91197e11	−0.891247
$363$	−7.64865e11	−2.31209
$364$	0	0
$365$	−1.38902e11	−0.409629
$366$	−5.14695e11	−1.49929
$367$	−2.24258e10	−0.0645283	−0.0322642	−	0.999479i	$-0.510272\pi$
$367$	−2.24258e10	−0.0645283	−0.0322642	+	0.999479i	$0.510272\pi$
$368$	−2.88804e11	−0.820896
$369$	−2.77034e11	−0.777883
$370$	1.21513e11	0.337066
$371$	0	0
$372$	7.38979e10	0.200073
$373$	−1.03980e11	−0.278139	−0.139069	−	0.990283i	$-0.544411\pi$
$373$	−1.03980e11	−0.278139	−0.139069	+	0.990283i	$0.544411\pi$
$374$	3.35272e11	0.886084
$375$	4.64023e10	0.121171
$376$	−6.65495e11	−1.71711
$377$	6.21554e11	1.58469
$378$	0	0
$379$	2.82403e11	0.703060	0.351530	−	0.936177i	$-0.385662\pi$
$379$	2.82403e11	0.703060	0.351530	+	0.936177i	$0.385662\pi$
$380$	2.66033e10	0.0654499
$381$	3.46566e11	0.842603
$382$	−1.18519e11	−0.284777
$383$	−5.95330e11	−1.41372	−0.706860	−	0.707354i	$-0.749889\pi$
$383$	−5.95330e11	−1.41372	−0.706860	+	0.707354i	$0.749889\pi$
$384$	4.61932e11	1.08414
$385$	0	0
$386$	2.62573e11	0.602015
$387$	1.96328e11	0.444921
$388$	3.88559e10	0.0870390
$389$	−5.34860e11	−1.18431	−0.592157	−	0.805822i	$-0.701724\pi$
$389$	−5.34860e11	−1.18431	−0.592157	+	0.805822i	$0.701724\pi$
$390$	−3.75582e11	−0.822080
$391$	2.47625e11	0.535796
$392$	0	0
$393$	−5.02921e11	−1.06349
$394$	−1.35952e11	−0.284219
$395$	−1.42481e11	−0.294489
$396$	7.52371e10	0.153746
$397$	7.42008e11	1.49917	0.749586	−	0.661907i	$-0.230253\pi$
$397$	7.42008e11	1.49917	0.749586	+	0.661907i	$0.230253\pi$
$398$	9.14125e10	0.182613
$399$	0	0
$400$	−8.96621e10	−0.175121
$401$	2.20598e11	0.426041	0.213021	−	0.977048i	$-0.431670\pi$
$401$	2.20598e11	0.426041	0.213021	+	0.977048i	$0.431670\pi$
$402$	3.63492e11	0.694189
$403$	1.00641e12	1.90065
$404$	1.07287e11	0.200370
$405$	2.75449e11	0.508737
$406$	0	0
$407$	−7.28378e11	−1.31578
$408$	−4.54087e11	−0.811275
$409$	7.10410e11	1.25532	0.627660	−	0.778488i	$-0.284013\pi$
$409$	7.10410e11	1.25532	0.627660	+	0.778488i	$0.284013\pi$
$410$	−2.24568e11	−0.392482
$411$	1.21199e11	0.209514
$412$	−1.07201e11	−0.183299
$413$	0	0
$414$	−4.41126e11	−0.738008
$415$	−9.25325e9	−0.0153136
$416$	−1.95828e11	−0.320593
$417$	−5.23964e11	−0.848573
$418$	1.26591e12	2.02819
$419$	5.27046e11	0.835382	0.417691	−	0.908589i	$-0.362839\pi$
$419$	5.27046e11	0.835382	0.417691	+	0.908589i	$0.362839\pi$
$420$	0	0
$421$	7.36474e11	1.14258	0.571292	−	0.820747i	$-0.306442\pi$
$421$	7.36474e11	1.14258	0.571292	+	0.820747i	$0.306442\pi$
$422$	−5.52982e11	−0.848800
$423$	−9.01328e11	−1.36884
$424$	3.98688e11	0.599083
$425$	7.68777e10	0.114301
$426$	7.87372e11	1.15834
$427$	0	0
$428$	−6.21061e10	−0.0894617
$429$	2.25133e12	3.20908
$430$	1.59146e11	0.224486
$431$	7.00260e11	0.977488	0.488744	−	0.872427i	$-0.337455\pi$
$431$	7.00260e11	0.977488	0.488744	+	0.872427i	$0.337455\pi$
$432$	−1.41423e11	−0.195363
$433$	−7.26185e11	−0.992777	−0.496388	−	0.868101i	$-0.665341\pi$
$433$	−7.26185e11	−0.992777	−0.496388	+	0.868101i	$0.665341\pi$
$434$	0	0
$435$	4.97973e11	0.666813
$436$	6.50504e10	0.0862106
$437$	9.34973e11	1.22640
$438$	9.00741e11	1.16941
$439$	−3.28487e11	−0.422113	−0.211056	−	0.977474i	$-0.567690\pi$
$439$	−3.28487e11	−0.422113	−0.211056	+	0.977474i	$0.567690\pi$
$440$	6.06127e11	0.770950
$441$	0	0
$442$	−6.22251e11	−0.775470
$443$	−1.48530e12	−1.83231	−0.916154	−	0.400827i	$-0.868723\pi$
$443$	−1.48530e12	−1.83231	−0.916154	+	0.400827i	$0.868723\pi$
$444$	9.92618e10	0.121216
$445$	−2.79732e11	−0.338159
$446$	4.23660e11	0.507003
$447$	−1.49488e12	−1.77102
$448$	0	0
$449$	−6.21938e11	−0.722168	−0.361084	−	0.932533i	$-0.617593\pi$
$449$	−6.21938e11	−0.722168	−0.361084	+	0.932533i	$0.617593\pi$
$450$	−1.36952e11	−0.157439
$451$	1.34611e12	1.53210
$452$	6.02137e10	0.0678535
$453$	−2.19864e12	−2.45308
$454$	−2.41665e11	−0.266970
$455$	0	0
$456$	−1.71452e12	−1.85695
$457$	9.59245e11	1.02874	0.514371	−	0.857568i	$-0.328025\pi$
$457$	9.59245e11	1.02874	0.514371	+	0.857568i	$0.328025\pi$
$458$	5.89485e11	0.626005
$459$	1.21258e11	0.127513
$460$	4.50448e10	0.0469066
$461$	−1.83403e12	−1.89127	−0.945633	−	0.325236i	$-0.894556\pi$
$461$	−1.83403e12	−1.89127	−0.945633	+	0.325236i	$0.894556\pi$
$462$	0	0
$463$	1.43862e12	1.45489	0.727446	−	0.686165i	$-0.240707\pi$
$463$	1.43862e12	1.45489	0.727446	+	0.686165i	$0.240707\pi$
$464$	−9.62221e11	−0.963705
$465$	8.06310e11	0.799767
$466$	7.61300e11	0.747858
$467$	4.34144e11	0.422385	0.211192	−	0.977445i	$-0.432265\pi$
$467$	4.34144e11	0.422385	0.211192	+	0.977445i	$0.432265\pi$
$468$	−1.39637e11	−0.134553
$469$	0	0
$470$	−7.30631e11	−0.690649
$471$	−2.95939e10	−0.0277082
$472$	−8.83036e11	−0.818917
$473$	−9.53960e11	−0.876305
$474$	9.23947e11	0.840707
$475$	2.90272e11	0.261628
$476$	0	0
$477$	5.39973e11	0.477573
$478$	−1.37436e12	−1.20414
$479$	1.45935e11	0.126663	0.0633314	−	0.997993i	$-0.479828\pi$
$479$	1.45935e11	0.126663	0.0633314	+	0.997993i	$0.479828\pi$
$480$	−1.56892e11	−0.134901
$481$	1.35184e12	1.15152
$482$	−1.23633e12	−1.04333
$483$	0	0
$484$	−2.30513e11	−0.190937
$485$	4.23961e11	0.347927
$486$	−1.52761e12	−1.24208
$487$	8.41559e11	0.677961	0.338980	−	0.940793i	$-0.389918\pi$
$487$	8.41559e11	0.677961	0.338980	+	0.940793i	$0.389918\pi$
$488$	−1.54162e12	−1.23052
$489$	2.12141e12	1.67778
$490$	0	0
$491$	−9.35646e11	−0.726515	−0.363258	−	0.931689i	$-0.618336\pi$
$491$	−9.35646e11	−0.726515	−0.363258	+	0.931689i	$0.618336\pi$
$492$	−1.83445e11	−0.141144
$493$	8.25024e11	0.629007
$494$	−2.34947e12	−1.77500
$495$	8.20922e11	0.614580
$496$	−1.55801e12	−1.15586
$497$	0	0
$498$	6.00048e10	0.0437173
$499$	−2.90555e11	−0.209786	−0.104893	−	0.994484i	$-0.533450\pi$
$499$	−2.90555e11	−0.209786	−0.104893	+	0.994484i	$0.533450\pi$
$500$	1.39846e10	0.0100066
$501$	9.98154e11	0.707828
$502$	2.98699e11	0.209926
$503$	1.11722e12	0.778185	0.389092	−	0.921199i	$-0.372789\pi$
$503$	1.11722e12	0.778185	0.389092	+	0.921199i	$0.372789\pi$
$504$	0	0
$505$	1.17062e12	0.800952
$506$	2.14344e12	1.45356
$507$	−2.16283e12	−1.45374
$508$	1.04447e11	0.0695840
$509$	1.47628e12	0.974854	0.487427	−	0.873164i	$-0.337936\pi$
$509$	1.47628e12	0.974854	0.487427	+	0.873164i	$0.337936\pi$
$510$	−4.98531e11	−0.326307
$511$	0	0
$512$	1.72981e12	1.11246
$513$	4.57841e11	0.291868
$514$	−1.48102e12	−0.935892
$515$	−1.16968e12	−0.732716
$516$	1.30004e11	0.0807294
$517$	4.37957e12	2.69603
$518$	0	0
$519$	1.44317e12	0.873101
$520$	−1.12495e12	−0.674709
$521$	5.48457e11	0.326116	0.163058	−	0.986616i	$-0.447864\pi$
$521$	5.48457e11	0.326116	0.163058	+	0.986616i	$0.447864\pi$
$522$	−1.46972e12	−0.866397
$523$	1.49093e12	0.871364	0.435682	−	0.900101i	$-0.356507\pi$
$523$	1.49093e12	0.871364	0.435682	+	0.900101i	$0.356507\pi$
$524$	−1.51569e11	−0.0878254
$525$	0	0
$526$	−2.39489e12	−1.36411
$527$	1.33587e12	0.754423
$528$	−3.48525e12	−1.95156
$529$	−2.18053e11	−0.121063
$530$	4.37710e11	0.240960
$531$	−1.19596e12	−0.652817
$532$	0	0
$533$	−2.49833e12	−1.34084
$534$	1.81398e12	0.965377
$535$	−6.77648e11	−0.357612
$536$	1.08873e12	0.569745
$537$	−2.41501e12	−1.25324
$538$	1.00589e11	0.0517642
$539$	0	0
$540$	2.20577e10	0.0111632
$541$	2.09478e12	1.05136	0.525679	−	0.850683i	$-0.323811\pi$
$541$	2.09478e12	1.05136	0.525679	+	0.850683i	$0.323811\pi$
$542$	1.30747e12	0.650780
$543$	−2.59546e12	−1.28120
$544$	−2.59933e11	−0.127252
$545$	7.09773e11	0.344616
$546$	0	0
$547$	3.14192e12	1.50056	0.750279	−	0.661121i	$-0.229919\pi$
$547$	3.14192e12	1.50056	0.750279	+	0.661121i	$0.229919\pi$
$548$	3.65268e10	0.0173021
$549$	−2.08793e12	−0.980934
$550$	6.65452e11	0.310088
$551$	3.11509e12	1.43975
$552$	−2.90303e12	−1.33084
$553$	0	0
$554$	2.28002e12	1.02836
$555$	1.08306e12	0.484544
$556$	−1.57911e11	−0.0700770
$557$	2.80927e12	1.23664	0.618322	−	0.785925i	$-0.287813\pi$
$557$	2.80927e12	1.23664	0.618322	+	0.785925i	$0.287813\pi$
$558$	−2.37974e12	−1.03915
$559$	1.77051e12	0.766912
$560$	0	0
$561$	2.98831e12	1.27377
$562$	−2.06624e12	−0.873711
$563$	1.39940e12	0.587022	0.293511	−	0.955956i	$-0.405176\pi$
$563$	1.39940e12	0.587022	0.293511	+	0.955956i	$0.405176\pi$
$564$	−5.96838e11	−0.248371
$565$	6.56999e11	0.271236
$566$	6.27620e10	0.0257053
$567$	0	0
$568$	2.35834e12	0.950690
$569$	−2.48494e12	−0.993828	−0.496914	−	0.867800i	$-0.665534\pi$
$569$	−2.48494e12	−0.993828	−0.496914	+	0.867800i	$0.665534\pi$
$570$	−1.88233e12	−0.746895
$571$	−3.24391e12	−1.27705	−0.638523	−	0.769602i	$-0.720454\pi$
$571$	−3.24391e12	−1.27705	−0.638523	+	0.769602i	$0.720454\pi$
$572$	6.78498e11	0.265013
$573$	−1.05637e12	−0.409376
$574$	0	0
$575$	4.91489e11	0.187503
$576$	2.39527e12	0.906677
$577$	−7.31430e11	−0.274715	−0.137357	−	0.990522i	$-0.543861\pi$
$577$	−7.31430e11	−0.274715	−0.137357	+	0.990522i	$0.543861\pi$
$578$	1.70284e12	0.634597
$579$	2.34034e12	0.865415
$580$	1.50078e11	0.0550669
$581$	0	0
$582$	−2.74927e12	−0.993263
$583$	−2.62374e12	−0.940615
$584$	2.69791e12	0.959774
$585$	−1.52360e12	−0.537859
$586$	1.15427e12	0.404360
$587$	1.33193e12	0.463032	0.231516	−	0.972831i	$-0.425631\pi$
$587$	1.33193e12	0.463032	0.231516	+	0.972831i	$0.425631\pi$
$588$	0	0
$589$	5.04390e12	1.72682
$590$	−9.69465e11	−0.329381
$591$	−1.21175e12	−0.408573
$592$	−2.09277e12	−0.700282
$593$	−5.41331e12	−1.79770	−0.898850	−	0.438257i	$-0.855596\pi$
$593$	−5.41331e12	−1.79770	−0.898850	+	0.438257i	$0.855596\pi$
$594$	1.04961e12	0.345929
$595$	0	0
$596$	−4.50523e11	−0.146254
$597$	8.14768e11	0.262512
$598$	−3.97813e12	−1.27211
$599$	−3.27443e11	−0.103924	−0.0519619	−	0.998649i	$-0.516547\pi$
$599$	−3.27443e11	−0.103924	−0.0519619	+	0.998649i	$0.516547\pi$
$600$	−9.01277e11	−0.283908
$601$	7.56209e11	0.236432	0.118216	−	0.992988i	$-0.462282\pi$
$601$	7.56209e11	0.236432	0.118216	+	0.992988i	$0.462282\pi$
$602$	0	0
$603$	1.47455e12	0.454185
$604$	−6.62620e11	−0.202581
$605$	−2.51516e12	−0.763248
$606$	−7.59118e12	−2.28656
$607$	−4.79752e12	−1.43439	−0.717195	−	0.696872i	$-0.754574\pi$
$607$	−4.79752e12	−1.43439	−0.717195	+	0.696872i	$0.754574\pi$
$608$	−9.81444e11	−0.291272
$609$	0	0
$610$	−1.69251e12	−0.494932
$611$	−8.12830e12	−2.35947
$612$	−1.85348e11	−0.0534080
$613$	4.44786e12	1.27227	0.636134	−	0.771579i	$-0.280533\pi$
$613$	4.44786e12	1.27227	0.636134	+	0.771579i	$0.280533\pi$
$614$	2.36974e12	0.672889
$615$	−2.00159e12	−0.564206
$616$	0	0
$617$	−1.63327e12	−0.453707	−0.226854	−	0.973929i	$-0.572844\pi$
$617$	−1.63327e12	−0.453707	−0.226854	+	0.973929i	$0.572844\pi$
$618$	7.58507e12	2.09176
$619$	1.77545e12	0.486071	0.243035	−	0.970017i	$-0.421857\pi$
$619$	1.77545e12	0.486071	0.243035	+	0.970017i	$0.421857\pi$
$620$	2.43003e11	0.0660465
$621$	7.75218e11	0.209176
$622$	8.27761e11	0.221742
$623$	0	0
$624$	6.46848e12	1.70794
$625$	1.52588e11	0.0400000
$626$	2.22169e12	0.578228
$627$	1.12831e13	2.91559
$628$	−8.91892e9	−0.00228820
$629$	1.79437e12	0.457072
$630$	0	0
$631$	−5.81058e12	−1.45911	−0.729554	−	0.683923i	$-0.760272\pi$
$631$	−5.81058e12	−1.45911	−0.729554	+	0.683923i	$0.760272\pi$
$632$	2.76741e12	0.689997
$633$	−4.92878e12	−1.22018
$634$	5.99149e12	1.47277
$635$	1.13963e12	0.278153
$636$	3.57557e11	0.0866540
$637$	0	0
$638$	7.14139e12	1.70643
$639$	3.19407e12	0.757864
$640$	1.51900e12	0.357889
$641$	1.81559e12	0.424772	0.212386	−	0.977186i	$-0.431877\pi$
$641$	1.81559e12	0.424772	0.212386	+	0.977186i	$0.431877\pi$
$642$	4.39436e12	1.02091
$643$	−8.51767e12	−1.96504	−0.982521	−	0.186154i	$-0.940398\pi$
$643$	−8.51767e12	−1.96504	−0.982521	+	0.186154i	$0.940398\pi$
$644$	0	0
$645$	1.41849e12	0.322705
$646$	−3.11858e12	−0.704548
$647$	4.28899e12	0.962245	0.481123	−	0.876653i	$-0.340229\pi$
$647$	4.28899e12	0.962245	0.481123	+	0.876653i	$0.340229\pi$
$648$	−5.35006e12	−1.19199
$649$	5.81120e12	1.28577
$650$	−1.23505e12	−0.271378
$651$	0	0
$652$	6.39346e11	0.138555
$653$	7.96742e12	1.71478	0.857390	−	0.514667i	$-0.172084\pi$
$653$	7.96742e12	1.71478	0.857390	+	0.514667i	$0.172084\pi$
$654$	−4.60268e12	−0.983809
$655$	−1.65379e12	−0.351071
$656$	3.86763e12	0.815413
$657$	3.65397e12	0.765105
$658$	0	0
$659$	−5.80696e12	−1.19940	−0.599701	−	0.800224i	$-0.704714\pi$
$659$	−5.80696e12	−1.19940	−0.599701	+	0.800224i	$0.704714\pi$
$660$	5.43595e11	0.111514
$661$	−4.58242e11	−0.0933660	−0.0466830	−	0.998910i	$-0.514865\pi$
$661$	−4.58242e11	−0.0933660	−0.0466830	+	0.998910i	$0.514865\pi$
$662$	8.32551e12	1.68481
$663$	−5.54618e12	−1.11476
$664$	1.79727e11	0.0358803
$665$	0	0
$666$	−3.19654e12	−0.629573
$667$	5.27449e12	1.03184
$668$	3.00821e11	0.0584540
$669$	3.77612e12	0.728834
$670$	1.19530e12	0.229160
$671$	1.01453e13	1.93202
$672$	0	0
$673$	−1.20278e12	−0.226005	−0.113002	−	0.993595i	$-0.536047\pi$
$673$	−1.20278e12	−0.226005	−0.113002	+	0.993595i	$0.536047\pi$
$674$	4.04948e12	0.755839
$675$	2.40674e11	0.0446234
$676$	−6.51829e11	−0.120053
$677$	−9.07750e11	−0.166080	−0.0830400	−	0.996546i	$-0.526463\pi$
$677$	−9.07750e11	−0.166080	−0.0830400	+	0.996546i	$0.526463\pi$
$678$	−4.26046e12	−0.774324
$679$	0	0
$680$	−1.49320e12	−0.267811
$681$	−2.15398e12	−0.383778
$682$	1.15632e13	2.04668
$683$	4.70127e12	0.826651	0.413325	−	0.910583i	$-0.364367\pi$
$683$	4.70127e12	0.826651	0.413325	+	0.910583i	$0.364367\pi$
$684$	−6.99829e11	−0.122247
$685$	3.98548e11	0.0691629
$686$	0	0
$687$	5.25413e12	0.899903
$688$	−2.74091e12	−0.466387
$689$	4.86955e12	0.823194
$690$	−3.18717e12	−0.535285
$691$	−2.88818e12	−0.481917	−0.240959	−	0.970535i	$-0.577462\pi$
$691$	−2.88818e12	−0.481917	−0.240959	+	0.970535i	$0.577462\pi$
$692$	4.34939e11	0.0721025
$693$	0	0
$694$	−8.14484e12	−1.33280
$695$	−1.72299e12	−0.280124
$696$	−9.67217e12	−1.56236
$697$	−3.31617e12	−0.532218
$698$	4.97550e11	0.0793391
$699$	6.78553e12	1.07507
$700$	0	0
$701$	4.88547e11	0.0764145	0.0382072	−	0.999270i	$-0.487835\pi$
$701$	4.88547e11	0.0764145	0.0382072	+	0.999270i	$0.487835\pi$
$702$	−1.94803e12	−0.302746
$703$	6.77511e12	1.04621
$704$	−1.16386e13	−1.78577
$705$	−6.51218e12	−0.992831
$706$	3.93040e12	0.595409
$707$	0	0
$708$	−7.91937e11	−0.118452
$709$	1.22900e13	1.82660	0.913302	−	0.407283i	$-0.133524\pi$
$709$	1.22900e13	1.82660	0.913302	+	0.407283i	$0.133524\pi$
$710$	2.58917e12	0.382382
$711$	3.74811e12	0.550046
$712$	5.43325e12	0.792318
$713$	8.54036e12	1.23758
$714$	0	0
$715$	7.40318e12	1.05935
$716$	−7.27829e11	−0.103495
$717$	−1.22498e13	−1.73098
$718$	2.41595e12	0.339256
$719$	−1.07824e13	−1.50465	−0.752326	−	0.658791i	$-0.771068\pi$
$719$	−1.07824e13	−1.50465	−0.752326	+	0.658791i	$0.771068\pi$
$720$	2.35866e12	0.327092
$721$	0	0
$722$	−4.89397e12	−0.670261
$723$	−1.10195e13	−1.49983
$724$	−7.82214e11	−0.105804
$725$	1.63752e12	0.220123
$726$	1.63101e13	2.17892
$727$	1.23298e13	1.63701	0.818503	−	0.574502i	$-0.194804\pi$
$727$	1.23298e13	1.63701	0.818503	+	0.574502i	$0.194804\pi$
$728$	0	0
$729$	−4.94104e12	−0.647954
$730$	2.96197e12	0.386035
$731$	2.35010e12	0.304409
$732$	−1.38258e12	−0.177987
$733$	−6.25325e10	−0.00800088	−0.00400044	−	0.999992i	$-0.501273\pi$
$733$	−6.25325e10	−0.00800088	−0.00400044	+	0.999992i	$0.501273\pi$
$734$	4.78211e11	0.0608117
$735$	0	0
$736$	−1.66178e12	−0.208749
$737$	−7.16488e12	−0.894551
$738$	5.90751e12	0.733079
$739$	1.37529e13	1.69627	0.848133	−	0.529783i	$-0.177727\pi$
$739$	1.37529e13	1.69627	0.848133	+	0.529783i	$0.177727\pi$
$740$	3.26409e11	0.0400147
$741$	−2.09410e13	−2.55162
$742$	0	0
$743$	5.86908e12	0.706514	0.353257	−	0.935526i	$-0.385074\pi$
$743$	5.86908e12	0.706514	0.353257	+	0.935526i	$0.385074\pi$
$744$	−1.56610e13	−1.87388
$745$	−4.91571e12	−0.584633
$746$	2.21729e12	0.262119
$747$	2.43417e11	0.0286028
$748$	9.00609e11	0.105191
$749$	0	0
$750$	−9.89490e11	−0.114192
$751$	9.01282e12	1.03391	0.516953	−	0.856014i	$-0.327066\pi$
$751$	9.01282e12	1.03391	0.516953	+	0.856014i	$0.327066\pi$
$752$	1.25833e13	1.43488
$753$	2.66233e12	0.301776
$754$	−1.32541e13	−1.49341
$755$	−7.22993e12	−0.809790
$756$	0	0
$757$	1.10513e13	1.22316	0.611580	−	0.791183i	$-0.290534\pi$
$757$	1.10513e13	1.22316	0.611580	+	0.791183i	$0.290534\pi$
$758$	−6.02199e12	−0.662565
$759$	1.91047e13	2.08954
$760$	−5.63797e12	−0.613002
$761$	−2.58275e12	−0.279159	−0.139579	−	0.990211i	$-0.544575\pi$
$761$	−2.58275e12	−0.279159	−0.139579	+	0.990211i	$0.544575\pi$
$762$	−7.39022e12	−0.794072
$763$	0	0
$764$	−3.18367e11	−0.0338071
$765$	−2.02235e12	−0.213492
$766$	1.26949e13	1.33229
$767$	−1.07853e13	−1.12526
$768$	4.32679e12	0.448787
$769$	−1.38646e13	−1.42968	−0.714838	−	0.699290i	$-0.753500\pi$
$769$	−1.38646e13	−1.42968	−0.714838	+	0.699290i	$0.753500\pi$
$770$	0	0
$771$	−1.32004e13	−1.34537
$772$	7.05324e11	0.0714679
$773$	−3.61445e12	−0.364111	−0.182056	−	0.983288i	$-0.558275\pi$
$773$	−3.61445e12	−0.364111	−0.182056	+	0.983288i	$0.558275\pi$
$774$	−4.18652e12	−0.419294
$775$	2.65144e12	0.264012
$776$	−8.23464e12	−0.815205
$777$	0	0
$778$	1.14054e13	1.11610
$779$	−1.25211e13	−1.21821
$780$	−1.00889e12	−0.0975928
$781$	−1.55201e13	−1.49267
$782$	−5.28039e12	−0.504936
$783$	2.58283e12	0.245566
$784$	0	0
$785$	−9.73155e10	−0.00914679
$786$	1.07244e13	1.00224
$787$	−4.22165e12	−0.392279	−0.196140	−	0.980576i	$-0.562841\pi$
$787$	−4.22165e12	−0.392279	−0.196140	+	0.980576i	$0.562841\pi$
$788$	−3.65195e11	−0.0337409
$789$	−2.13459e13	−1.96096
$790$	3.03828e12	0.277527
$791$	0	0
$792$	−1.59448e13	−1.43998
$793$	−1.88292e13	−1.69084
$794$	−1.58227e13	−1.41282
$795$	3.90135e12	0.346388
$796$	2.45552e11	0.0216788
$797$	−2.23662e13	−1.96349	−0.981747	−	0.190193i	$-0.939089\pi$
$797$	−2.23662e13	−1.96349	−0.981747	+	0.190193i	$0.939089\pi$
$798$	0	0
$799$	−1.07891e13	−0.936540
$800$	−5.15918e11	−0.0445324
$801$	7.35865e12	0.631614
$802$	−4.70406e12	−0.401503
$803$	−1.77547e13	−1.50693
$804$	9.76414e11	0.0824104
$805$	0	0
$806$	−2.14608e13	−1.79118
$807$	8.96558e11	0.0744128
$808$	−2.27371e13	−1.87666
$809$	−5.09731e12	−0.418382	−0.209191	−	0.977875i	$-0.567083\pi$
$809$	−5.09731e12	−0.418382	−0.209191	+	0.977875i	$0.567083\pi$
$810$	−5.87371e12	−0.479435
$811$	−7.10602e12	−0.576809	−0.288405	−	0.957509i	$-0.593125\pi$
$811$	−7.10602e12	−0.576809	−0.288405	+	0.957509i	$0.593125\pi$
$812$	0	0
$813$	1.16536e13	0.935517
$814$	1.55320e13	1.23999
$815$	6.97599e12	0.553856
$816$	8.58597e12	0.677928
$817$	8.87340e12	0.696772
$818$	−1.51489e13	−1.18302
$819$	0	0
$820$	−6.03235e11	−0.0465934
$821$	−1.43864e13	−1.10512	−0.552559	−	0.833474i	$-0.686349\pi$
$821$	−1.43864e13	−1.10512	−0.552559	+	0.833474i	$0.686349\pi$
$822$	−2.58447e12	−0.197446
$823$	−1.10546e13	−0.839932	−0.419966	−	0.907540i	$-0.637958\pi$
$823$	−1.10546e13	−0.839932	−0.419966	+	0.907540i	$0.637958\pi$
$824$	2.27188e13	1.71678
$825$	5.93123e12	0.445761
$826$	0	0
$827$	−6.53464e12	−0.485788	−0.242894	−	0.970053i	$-0.578097\pi$
$827$	−6.53464e12	−0.485788	−0.242894	+	0.970053i	$0.578097\pi$
$828$	−1.18495e12	−0.0876122
$829$	−6.58701e12	−0.484388	−0.242194	−	0.970228i	$-0.577867\pi$
$829$	−6.58701e12	−0.484388	−0.242194	+	0.970228i	$0.577867\pi$
$830$	1.97318e11	0.0144316
$831$	2.03220e13	1.47830
$832$	2.16008e13	1.56284
$833$	0	0
$834$	1.11731e13	0.799697
$835$	3.28230e12	0.233662
$836$	3.40048e12	0.240775
$837$	4.18208e12	0.294529
$838$	−1.12388e13	−0.787266
$839$	−4.76554e12	−0.332035	−0.166017	−	0.986123i	$-0.553091\pi$
$839$	−4.76554e12	−0.332035	−0.166017	+	0.986123i	$0.553091\pi$
$840$	0	0
$841$	3.06609e12	0.211350
$842$	−1.57047e13	−1.07677
$843$	−1.84166e13	−1.25599
$844$	−1.48542e12	−0.100765
$845$	−7.11218e12	−0.479897
$846$	1.92201e13	1.28999
$847$	0	0
$848$	−7.53849e12	−0.500614
$849$	5.59403e11	0.0369522
$850$	−1.63935e12	−0.107718
$851$	1.14717e13	0.749796
$852$	2.11504e12	0.137512
$853$	1.95984e13	1.26751	0.633754	−	0.773535i	$-0.281513\pi$
$853$	1.95984e13	1.26751	0.633754	+	0.773535i	$0.281513\pi$
$854$	0	0
$855$	−7.63592e12	−0.488668
$856$	1.31620e13	0.837896
$857$	2.63348e12	0.166769	0.0833847	−	0.996517i	$-0.473427\pi$
$857$	2.63348e12	0.166769	0.0833847	+	0.996517i	$0.473427\pi$
$858$	−4.80076e13	−3.02425
$859$	−1.98864e13	−1.24620	−0.623100	−	0.782142i	$-0.714127\pi$
$859$	−1.98864e13	−1.24620	−0.623100	+	0.782142i	$0.714127\pi$
$860$	4.27500e11	0.0266497
$861$	0	0
$862$	−1.49324e13	−0.921188
$863$	−2.16547e13	−1.32893	−0.664467	−	0.747318i	$-0.731341\pi$
$863$	−2.16547e13	−1.32893	−0.664467	+	0.747318i	$0.731341\pi$
$864$	−8.13749e11	−0.0496797
$865$	4.74567e12	0.288221
$866$	1.54853e13	0.935596
$867$	1.51775e13	0.912253
$868$	0	0
$869$	−1.82121e13	−1.08336
$870$	−1.06188e13	−0.628407
$871$	1.32977e13	0.782880
$872$	−1.37860e13	−0.807446
$873$	−1.11528e13	−0.649858
$874$	−1.99375e13	−1.15576
$875$	0	0
$876$	2.41957e12	0.138826
$877$	2.83238e12	0.161679	0.0808395	−	0.996727i	$-0.474240\pi$
$877$	2.83238e12	0.161679	0.0808395	+	0.996727i	$0.474240\pi$
$878$	7.00471e12	0.397800
$879$	1.02881e13	0.581280
$880$	−1.14608e13	−0.644232
$881$	2.62782e13	1.46962	0.734809	−	0.678274i	$-0.237272\pi$
$881$	2.62782e13	1.46962	0.734809	+	0.678274i	$0.237272\pi$
$882$	0	0
$883$	1.46614e13	0.811620	0.405810	−	0.913957i	$-0.366989\pi$
$883$	1.46614e13	0.811620	0.405810	+	0.913957i	$0.366989\pi$
$884$	−1.67149e12	−0.0920596
$885$	−8.64093e12	−0.473495
$886$	3.16728e13	1.72677
$887$	−2.09081e13	−1.13412	−0.567059	−	0.823677i	$-0.691919\pi$
$887$	−2.09081e13	−1.13412	−0.567059	+	0.823677i	$0.691919\pi$
$888$	−2.10363e13	−1.13530
$889$	0	0
$890$	5.96503e12	0.318682
$891$	3.52084e13	1.87153
$892$	1.13804e12	0.0601887
$893$	−4.07372e13	−2.14368
$894$	3.18770e13	1.66901
$895$	−7.94144e12	−0.413709
$896$	0	0
$897$	−3.54574e13	−1.82870
$898$	1.32623e13	0.680573
$899$	2.84543e13	1.45288
$900$	−3.67881e11	−0.0186903
$901$	6.46362e12	0.326749
$902$	−2.87047e13	−1.44385
$903$	0	0
$904$	−1.27610e13	−0.635514
$905$	−8.53483e12	−0.422938
$906$	4.68841e13	2.31179
$907$	1.89149e13	0.928049	0.464025	−	0.885822i	$-0.346405\pi$
$907$	1.89149e13	0.928049	0.464025	+	0.885822i	$0.346405\pi$
$908$	−6.49162e11	−0.0316932
$909$	−3.07946e13	−1.49602
$910$	0	0
$911$	−7.67112e12	−0.369000	−0.184500	−	0.982833i	$-0.559067\pi$
$911$	−7.67112e12	−0.369000	−0.184500	+	0.982833i	$0.559067\pi$
$912$	3.24186e13	1.55173
$913$	−1.18277e12	−0.0563353
$914$	−2.04551e13	−0.969490
$915$	−1.50855e13	−0.711481
$916$	1.58348e12	0.0743159
$917$	0	0
$918$	−2.58572e12	−0.120168
$919$	−3.52484e13	−1.63012	−0.815060	−	0.579377i	$-0.803296\pi$
$919$	−3.52484e13	−1.63012	−0.815060	+	0.579377i	$0.803296\pi$
$920$	−9.54624e12	−0.439326
$921$	2.11217e13	0.967300
$922$	3.91091e13	1.78233
$923$	2.88046e13	1.30633
$924$	0	0
$925$	3.56149e12	0.159954
$926$	−3.06773e13	−1.37109
$927$	3.07698e13	1.36857
$928$	−5.53664e12	−0.245065
$929$	3.48477e13	1.53498	0.767492	−	0.641059i	$-0.221505\pi$
$929$	3.48477e13	1.53498	0.767492	+	0.641059i	$0.221505\pi$
$930$	−1.71939e13	−0.753703
$931$	0	0
$932$	2.04501e12	0.0887816
$933$	7.37790e12	0.318761
$934$	−9.25776e12	−0.398057
$935$	9.82666e12	0.420488
$936$	2.95929e13	1.26022
$937$	−2.95277e13	−1.25142	−0.625709	−	0.780057i	$-0.715190\pi$
$937$	−2.95277e13	−1.25142	−0.625709	+	0.780057i	$0.715190\pi$
$938$	0	0
$939$	1.98021e13	0.831222
$940$	−1.96262e12	−0.0819901
$941$	2.62047e12	0.108950	0.0544748	−	0.998515i	$-0.482652\pi$
$941$	2.62047e12	0.108950	0.0544748	+	0.998515i	$0.482652\pi$
$942$	6.31064e11	0.0261122
$943$	−2.12007e13	−0.873067
$944$	1.66967e13	0.684314
$945$	0	0
$946$	2.03424e13	0.825832
$947$	−1.38273e12	−0.0558680	−0.0279340	−	0.999610i	$-0.508893\pi$
$947$	−1.38273e12	−0.0558680	−0.0279340	+	0.999610i	$0.508893\pi$
$948$	2.48191e12	0.0998041
$949$	3.29520e13	1.31881
$950$	−6.18980e12	−0.246559
$951$	5.34027e13	2.11715
$952$	0	0
$953$	1.17712e12	0.0462278	0.0231139	−	0.999733i	$-0.492642\pi$
$953$	1.17712e12	0.0462278	0.0231139	+	0.999733i	$0.492642\pi$
$954$	−1.15145e13	−0.450066
$955$	−3.47374e12	−0.135140
$956$	−3.69181e12	−0.142948
$957$	6.36518e13	2.45305
$958$	−3.11193e12	−0.119367
$959$	0	0
$960$	1.73060e13	0.657622
$961$	1.96331e13	0.742564
$962$	−2.88268e13	−1.08520
$963$	1.78263e13	0.667947
$964$	−3.32104e12	−0.123859
$965$	7.69588e12	0.285684
$966$	0	0
$967$	4.43188e13	1.62993	0.814965	−	0.579510i	$-0.196756\pi$
$967$	4.43188e13	1.62993	0.814965	+	0.579510i	$0.196756\pi$
$968$	4.88521e13	1.78831
$969$	−2.77962e13	−1.01281
$970$	−9.04061e12	−0.327888
$971$	−1.26267e13	−0.455832	−0.227916	−	0.973681i	$-0.573191\pi$
$971$	−1.26267e13	−0.455832	−0.227916	+	0.973681i	$0.573191\pi$
$972$	−4.10346e12	−0.147452
$973$	0	0
$974$	−1.79455e13	−0.638912
$975$	−1.10081e13	−0.390115
$976$	2.91493e13	1.02826
$977$	−1.82582e13	−0.641110	−0.320555	−	0.947230i	$-0.603869\pi$
$977$	−1.82582e13	−0.641110	−0.320555	+	0.947230i	$0.603869\pi$
$978$	−4.52373e13	−1.58115
$979$	−3.57558e13	−1.24401
$980$	0	0
$981$	−1.86714e13	−0.643674
$982$	1.99518e13	0.684670
$983$	−2.13278e13	−0.728542	−0.364271	−	0.931293i	$-0.618682\pi$
$983$	−2.13278e13	−0.728542	−0.364271	+	0.931293i	$0.618682\pi$
$984$	3.88771e13	1.32195
$985$	−3.98468e12	−0.134875
$986$	−1.75929e13	−0.592778
$987$	0	0
$988$	−6.31115e12	−0.210718
$989$	1.50245e13	0.499363
$990$	−1.75055e13	−0.579182
$991$	3.39103e13	1.11686	0.558431	−	0.829551i	$-0.311404\pi$
$991$	3.39103e13	1.11686	0.558431	+	0.829551i	$0.311404\pi$
$992$	−8.96484e12	−0.293927
$993$	7.42060e13	2.42196
$994$	0	0
$995$	2.67925e12	0.0866583
$996$	1.61185e11	0.00518988
$997$	2.14211e13	0.686616	0.343308	−	0.939223i	$-0.388453\pi$
$997$	2.14211e13	0.686616	0.343308	+	0.939223i	$0.388453\pi$
$998$	6.19584e12	0.197703
$999$	5.61748e12	0.178442

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.10.a.l.1.4		13
7.3	odd	6		35.10.e.b.16.10	yes	26
7.5	odd	6		35.10.e.b.11.10	✓	26
7.6	odd	2		245.10.a.m.1.4		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.10.e.b.11.10	✓	26	7.5	odd	6
35.10.e.b.16.10	yes	26	7.3	odd	6
245.10.a.l.1.4		13	1.1	even	1	trivial
245.10.a.m.1.4		13	7.6	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:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)

\(f(q)\)	\(=\)	\(q-21.3241 q^{2} -190.064 q^{3} -57.2810 q^{4} -625.000 q^{5} +4052.95 q^{6} +12139.4 q^{8} +16441.3 q^{9} +13327.6 q^{10} -79888.7 q^{11} +10887.1 q^{12} +148270. q^{13} +118790. q^{15} -229535. q^{16} +196807. q^{17} -350597. q^{18} +743096. q^{19} +35800.6 q^{20} +1.70356e6 q^{22} +1.25821e6 q^{23} -2.30727e6 q^{24} +390625. q^{25} -3.16173e6 q^{26} +616127. q^{27} +4.19204e6 q^{29} -2.53309e6 q^{30} +6.78769e6 q^{31} -1.32075e6 q^{32} +1.51840e7 q^{33} -4.19674e6 q^{34} -941775. q^{36} +9.11742e6 q^{37} -1.58459e7 q^{38} -2.81808e7 q^{39} -7.58714e6 q^{40} -1.68499e7 q^{41} +1.19411e7 q^{43} +4.57610e6 q^{44} -1.02758e7 q^{45} -2.68303e7 q^{46} -5.48209e7 q^{47} +4.36263e7 q^{48} -8.32974e6 q^{50} -3.74059e7 q^{51} -8.49305e6 q^{52} +3.28424e7 q^{53} -1.31384e7 q^{54} +4.99304e7 q^{55} -1.41236e8 q^{57} -8.93918e7 q^{58} -7.27412e7 q^{59} -6.80441e6 q^{60} -1.26993e8 q^{61} -1.44742e8 q^{62} +1.45686e8 q^{64} -9.26687e7 q^{65} -3.23785e8 q^{66} +8.96858e7 q^{67} -1.12733e7 q^{68} -2.39141e8 q^{69} +1.94271e8 q^{71} +1.99588e8 q^{72} +2.22243e8 q^{73} -1.94421e8 q^{74} -7.42437e7 q^{75} -4.25653e7 q^{76} +6.00931e8 q^{78} +2.27969e8 q^{79} +1.43459e8 q^{80} -4.40718e8 q^{81} +3.59309e8 q^{82} +1.48052e7 q^{83} -1.23004e8 q^{85} -2.54634e8 q^{86} -7.96757e8 q^{87} -9.69803e8 q^{88} +4.47570e8 q^{89} +2.19123e8 q^{90} -7.20717e7 q^{92} -1.29010e9 q^{93} +1.16901e9 q^{94} -4.64435e8 q^{95} +2.51027e8 q^{96} -6.78338e8 q^{97} -1.31347e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - q^{2} - 268 q^{3} + 3563 q^{4} - 8125 q^{5} - 3040 q^{6} - 4695 q^{8} + 82107 q^{9} + 625 q^{10} + 129087 q^{11} - 356068 q^{12} - 35889 q^{13} + 167500 q^{15} + 1379187 q^{16} - 251650 q^{17} + 391089 q^{18}+ \cdots + 5266142099 q^{99}+O(q^{100}) \) 13 * q - q^2 - 268 * q^3 + 3563 * q^4 - 8125 * q^5 - 3040 * q^6 - 4695 * q^8 + 82107 * q^9 + 625 * q^10 + 129087 * q^11 - 356068 * q^12 - 35889 * q^13 + 167500 * q^15 + 1379187 * q^16 - 251650 * q^17 + 391089 * q^18 + 22537 * q^19 - 2226875 * q^20 - 867693 * q^22 + 1640451 * q^23 - 3300104 * q^24 + 5078125 * q^25 + 215949 * q^26 - 4833346 * q^27 + 175192 * q^29 + 1900000 * q^30 - 3363236 * q^31 + 11618029 * q^32 - 1855346 * q^33 - 30084718 * q^34 + 2860133 * q^36 + 57134155 * q^37 - 28383363 * q^38 - 10216786 * q^39 + 2934375 * q^40 - 46043487 * q^41 - 23472776 * q^43 + 171806579 * q^44 - 51316875 * q^45 - 53075341 * q^46 + 1336787 * q^47 - 304545376 * q^48 - 390625 * q^50 + 234254040 * q^51 - 80069087 * q^52 + 122467755 * q^53 + 99983754 * q^54 - 80679375 * q^55 + 32652030 * q^57 + 37365152 * q^58 + 136056496 * q^59 + 222542500 * q^60 + 50448822 * q^61 + 109231036 * q^62 + 1109008839 * q^64 + 22430625 * q^65 + 500348610 * q^66 + 585418818 * q^67 - 480342846 * q^68 + 521416036 * q^69 + 497194080 * q^71 - 870751121 * q^72 + 346232592 * q^73 - 613439215 * q^74 - 104687500 * q^75 + 1436195213 * q^76 - 327612138 * q^78 + 136453862 * q^79 - 861991875 * q^80 - 344488883 * q^81 + 224863259 * q^82 + 523042314 * q^83 + 157281250 * q^85 - 2285275668 * q^86 - 2310323838 * q^87 - 1816859183 * q^88 + 843350780 * q^89 - 244430625 * q^90 - 172327913 * q^92 + 198172924 * q^93 - 2777934057 * q^94 - 14085625 * q^95 + 1583073484 * q^96 - 694973158 * q^97 + 5266142099 * q^99

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