LMFDB - Embedded newform 45.14.a.g.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,14,Mod(1,45)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("45.1"); S:= CuspForms(chi, 14); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 14, names="a")

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	45.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-15] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.2539180284$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 24774x^{2} - 86616x + 52534656 $ x^4 - x^3 - 24774x^2 - 86616x + 52534656
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{4}\cdot 5^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-146.991$ of defining polynomial
Character	$\chi$	$=$	45.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-150.991 q^{2} +14606.2 q^{4} +15625.0 q^{5} -242904. q^{7} -968489. q^{8} -2.35923e6 q^{10} +6.86310e6 q^{11} +2.01315e7 q^{13} +3.66762e7 q^{14} +2.65787e7 q^{16} +1.41199e8 q^{17} -2.08405e8 q^{19} +2.28222e8 q^{20} -1.03626e9 q^{22} +2.35142e8 q^{23} +2.44141e8 q^{25} -3.03967e9 q^{26} -3.54790e9 q^{28} -2.00980e9 q^{29} -3.36358e9 q^{31} +3.92072e9 q^{32} -2.13197e10 q^{34} -3.79537e9 q^{35} -9.62526e9 q^{37} +3.14672e10 q^{38} -1.51326e10 q^{40} -2.34132e10 q^{41} +8.82167e9 q^{43} +1.00244e11 q^{44} -3.55042e10 q^{46} +1.04323e11 q^{47} -3.78868e10 q^{49} -3.68630e10 q^{50} +2.94045e11 q^{52} -1.44937e11 q^{53} +1.07236e11 q^{55} +2.35249e11 q^{56} +3.03461e11 q^{58} -1.25803e11 q^{59} +7.31437e11 q^{61} +5.07869e11 q^{62} -8.09725e11 q^{64} +3.14554e11 q^{65} +1.13018e12 q^{67} +2.06238e12 q^{68} +5.73066e11 q^{70} +2.11613e12 q^{71} -2.39041e12 q^{73} +1.45333e12 q^{74} -3.04401e12 q^{76} -1.66707e12 q^{77} -1.88805e12 q^{79} +4.15292e11 q^{80} +3.53517e12 q^{82} -6.14850e11 q^{83} +2.20623e12 q^{85} -1.33199e12 q^{86} -6.64683e12 q^{88} -2.90766e12 q^{89} -4.89001e12 q^{91} +3.43453e12 q^{92} -1.57518e13 q^{94} -3.25633e12 q^{95} +1.60768e13 q^{97} +5.72056e12 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 15 q^{2} + 16837 q^{4} + 62500 q^{5} + 343040 q^{7} - 14865 q^{8} - 234375 q^{10} + 12697800 q^{11} + 34336040 q^{13} + 26944650 q^{14} + 66562801 q^{16} + 84377280 q^{17} - 131821144 q^{19} + 263078125 q^{20}+ \cdots - 17650752985395 q^{98}+O(q^{100}) $ 4 * q - 15 * q^2 + 16837 * q^4 + 62500 * q^5 + 343040 * q^7 - 14865 * q^8 - 234375 * q^10 + 12697800 * q^11 + 34336040 * q^13 + 26944650 * q^14 + 66562801 * q^16 + 84377280 * q^17 - 131821144 * q^19 + 263078125 * q^20 - 1016682450 * q^22 + 1046106360 * q^23 + 976562500 * q^25 - 2998448100 * q^26 - 2229409030 * q^28 - 1826812200 * q^29 + 3501222872 * q^31 - 1278404505 * q^32 + 2712309690 * q^34 + 5360000000 * q^35 + 8297642120 * q^37 + 50666879340 * q^38 - 232265625 * q^40 + 54459122400 * q^41 - 48173425360 * q^43 + 141205041150 * q^44 - 131025090720 * q^46 + 202496682840 * q^47 + 116605824372 * q^49 - 3662109375 * q^50 + 183603958220 * q^52 + 116147122920 * q^53 + 198403125000 * q^55 + 985319673150 * q^56 - 209673904950 * q^58 + 885100702200 * q^59 + 467836185368 * q^61 + 1845870142380 * q^62 - 1316914666607 * q^64 + 536500625000 * q^65 + 819037749440 * q^67 + 4568941043490 * q^68 + 421010156250 * q^70 + 2886468559200 * q^71 - 1235535876280 * q^73 + 5480818576200 * q^74 - 1007297478532 * q^76 + 3590015832000 * q^77 - 3553486866616 * q^79 + 1040043765625 * q^80 + 9494817638400 * q^82 + 3441261762480 * q^83 + 1318395000000 * q^85 + 1418346623400 * q^86 + 3182744059050 * q^88 - 2322663606000 * q^89 - 4966358282000 * q^91 - 3469668771120 * q^92 - 19865804246880 * q^94 - 2059705375000 * q^95 + 8462989649240 * q^97 - 17650752985395 * q^98

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 $

$2$	−150.991	−1.66823	−0.834114	−	0.551592i	$-0.814021\pi$
$2$	−150.991	−1.66823	−0.834114	+	0.551592i	$0.814021\pi$
$3$	0	0
$3$	0	0
$4$	14606.2	1.78299
$5$	15625.0	0.447214
$5$	15625.0	0.447214
$6$	0	0
$7$	−242904.	−0.780363	−0.390182	−	0.920738i	$-0.627588\pi$
$7$	−242904.	−0.780363	−0.390182	+	0.920738i	$0.627588\pi$
$8$	−968489.	−1.30620
$9$	0	0
$10$	−2.35923e6	−0.746054
$11$	6.86310e6	1.16807	0.584034	−	0.811730i	$-0.301474\pi$
$11$	6.86310e6	1.16807	0.584034	+	0.811730i	$0.301474\pi$
$12$	0	0
$13$	2.01315e7	1.15676	0.578380	−	0.815767i	$-0.303685\pi$
$13$	2.01315e7	1.15676	0.578380	+	0.815767i	$0.303685\pi$
$14$	3.66762e7	1.30182
$15$	0	0
$16$	2.65787e7	0.396054
$17$	1.41199e8	1.41877	0.709387	−	0.704819i	$-0.248972\pi$
$17$	1.41199e8	1.41877	0.709387	+	0.704819i	$0.248972\pi$
$18$	0	0
$19$	−2.08405e8	−1.01627	−0.508135	−	0.861277i	$-0.669665\pi$
$19$	−2.08405e8	−1.01627	−0.508135	+	0.861277i	$0.669665\pi$
$20$	2.28222e8	0.797376
$21$	0	0
$22$	−1.03626e9	−1.94860
$23$	2.35142e8	0.331206	0.165603	−	0.986192i	$-0.447043\pi$
$23$	2.35142e8	0.331206	0.165603	+	0.986192i	$0.447043\pi$
$24$	0	0
$25$	2.44141e8	0.200000
$26$	−3.03967e9	−1.92974
$27$	0	0
$28$	−3.54790e9	−1.39138
$29$	−2.00980e9	−0.627430	−0.313715	−	0.949517i	$-0.601574\pi$
$29$	−2.00980e9	−0.627430	−0.313715	+	0.949517i	$0.601574\pi$
$30$	0	0
$31$	−3.36358e9	−0.680692	−0.340346	−	0.940300i	$-0.610544\pi$
$31$	−3.36358e9	−0.680692	−0.340346	+	0.940300i	$0.610544\pi$
$32$	3.92072e9	0.645492
$33$	0	0
$34$	−2.13197e10	−2.36684
$35$	−3.79537e9	−0.348989
$36$	0	0
$37$	−9.62526e9	−0.616739	−0.308369	−	0.951267i	$-0.599783\pi$
$37$	−9.62526e9	−0.616739	−0.308369	+	0.951267i	$0.599783\pi$
$38$	3.14672e10	1.69537
$39$	0	0
$40$	−1.51326e10	−0.584150
$41$	−2.34132e10	−0.769777	−0.384888	−	0.922963i	$-0.625760\pi$
$41$	−2.34132e10	−0.769777	−0.384888	+	0.922963i	$0.625760\pi$
$42$	0	0
$43$	8.82167e9	0.212817	0.106408	−	0.994323i	$-0.466065\pi$
$43$	8.82167e9	0.212817	0.106408	+	0.994323i	$0.466065\pi$
$44$	1.00244e11	2.08265
$45$	0	0
$46$	−3.55042e10	−0.552527
$47$	1.04323e11	1.41170	0.705851	−	0.708360i	$-0.250565\pi$
$47$	1.04323e11	1.41170	0.705851	+	0.708360i	$0.250565\pi$
$48$	0	0
$49$	−3.78868e10	−0.391033
$50$	−3.68630e10	−0.333646
$51$	0	0
$52$	2.94045e11	2.06249
$53$	−1.44937e11	−0.898225	−0.449112	−	0.893475i	$-0.648260\pi$
$53$	−1.44937e11	−0.898225	−0.449112	+	0.893475i	$0.648260\pi$
$54$	0	0
$55$	1.07236e11	0.522375
$56$	2.35249e11	1.01931
$57$	0	0
$58$	3.03461e11	1.04670
$59$	−1.25803e11	−0.388286	−0.194143	−	0.980973i	$-0.562193\pi$
$59$	−1.25803e11	−0.388286	−0.194143	+	0.980973i	$0.562193\pi$
$60$	0	0
$61$	7.31437e11	1.81775	0.908873	−	0.417073i	$-0.136944\pi$
$61$	7.31437e11	1.81775	0.908873	+	0.417073i	$0.136944\pi$
$62$	5.07869e11	1.13555
$63$	0	0
$64$	−8.09725e11	−1.47288
$65$	3.14554e11	0.517319
$66$	0	0
$67$	1.13018e12	1.52637	0.763186	−	0.646179i	$-0.223634\pi$
$67$	1.13018e12	1.52637	0.763186	+	0.646179i	$0.223634\pi$
$68$	2.06238e12	2.52966
$69$	0	0
$70$	5.73066e11	0.582193
$71$	2.11613e12	1.96049	0.980244	−	0.197793i	$-0.0633774\pi$
$71$	2.11613e12	1.96049	0.980244	+	0.197793i	$0.0633774\pi$
$72$	0	0
$73$	−2.39041e12	−1.84873	−0.924365	−	0.381509i	$-0.875405\pi$
$73$	−2.39041e12	−1.84873	−0.924365	+	0.381509i	$0.875405\pi$
$74$	1.45333e12	1.02886
$75$	0	0
$76$	−3.04401e12	−1.81200
$77$	−1.66707e12	−0.911516
$78$	0	0
$79$	−1.88805e12	−0.873850	−0.436925	−	0.899498i	$-0.643932\pi$
$79$	−1.88805e12	−0.873850	−0.436925	+	0.899498i	$0.643932\pi$
$80$	4.15292e11	0.177121
$81$	0	0
$82$	3.53517e12	1.28416
$83$	−6.14850e11	−0.206425	−0.103212	−	0.994659i	$-0.532912\pi$
$83$	−6.14850e11	−0.206425	−0.103212	+	0.994659i	$0.532912\pi$
$84$	0	0
$85$	2.20623e12	0.634495
$86$	−1.33199e12	−0.355027
$87$	0	0
$88$	−6.64683e12	−1.52573
$89$	−2.90766e12	−0.620167	−0.310084	−	0.950709i	$-0.600357\pi$
$89$	−2.90766e12	−0.620167	−0.310084	+	0.950709i	$0.600357\pi$
$90$	0	0
$91$	−4.89001e12	−0.902693
$92$	3.43453e12	0.590536
$93$	0	0
$94$	−1.57518e13	−2.35504
$95$	−3.25633e12	−0.454490
$96$	0	0
$97$	1.60768e13	1.95967	0.979836	−	0.199804i	$-0.0640306\pi$
$97$	1.60768e13	1.95967	0.979836	+	0.199804i	$0.0640306\pi$
$98$	5.72056e12	0.652333
$99$	0	0
$100$	3.56597e12	0.356597
$101$	2.29304e12	0.214942	0.107471	−	0.994208i	$-0.465725\pi$
$101$	2.29304e12	0.214942	0.107471	+	0.994208i	$0.465725\pi$
$102$	0	0
$103$	7.44896e12	0.614687	0.307343	−	0.951599i	$-0.400560\pi$
$103$	7.44896e12	0.614687	0.307343	+	0.951599i	$0.400560\pi$
$104$	−1.94971e13	−1.51096
$105$	0	0
$106$	2.18841e13	1.49844
$107$	2.36993e13	1.52666	0.763329	−	0.646010i	$-0.223564\pi$
$107$	2.36993e13	1.52666	0.763329	+	0.646010i	$0.223564\pi$
$108$	0	0
$109$	2.66249e13	1.52061	0.760303	−	0.649569i	$-0.225051\pi$
$109$	2.66249e13	1.52061	0.760303	+	0.649569i	$0.225051\pi$
$110$	−1.61916e13	−0.871442
$111$	0	0
$112$	−6.45606e12	−0.309066
$113$	−1.05382e13	−0.476163	−0.238081	−	0.971245i	$-0.576518\pi$
$113$	−1.05382e13	−0.476163	−0.238081	+	0.971245i	$0.576518\pi$
$114$	0	0
$115$	3.67409e12	0.148120
$116$	−2.93556e13	−1.11870
$117$	0	0
$118$	1.89951e13	0.647750
$119$	−3.42977e13	−1.10716
$120$	0	0
$121$	1.25794e13	0.364381
$122$	−1.10440e14	−3.03241
$123$	0	0
$124$	−4.91292e13	−1.21366
$125$	3.81470e12	0.0894427
$126$	0	0
$127$	4.72856e12	0.100001	0.0500005	−	0.998749i	$-0.484078\pi$
$127$	4.72856e12	0.100001	0.0500005	+	0.998749i	$0.484078\pi$
$128$	9.01425e13	1.81161
$129$	0	0
$130$	−4.74948e13	−0.863006
$131$	2.10999e13	0.364769	0.182384	−	0.983227i	$-0.441618\pi$
$131$	2.10999e13	0.364769	0.182384	+	0.983227i	$0.441618\pi$
$132$	0	0
$133$	5.06223e13	0.793060
$134$	−1.70646e14	−2.54634
$135$	0	0
$136$	−1.36750e14	−1.85320
$137$	6.73125e13	0.869785	0.434893	−	0.900482i	$-0.356786\pi$
$137$	6.73125e13	0.869785	0.434893	+	0.900482i	$0.356786\pi$
$138$	0	0
$139$	1.24517e14	1.46431	0.732156	−	0.681137i	$-0.238514\pi$
$139$	1.24517e14	1.46431	0.732156	+	0.681137i	$0.238514\pi$
$140$	−5.54360e13	−0.622243
$141$	0	0
$142$	−3.19517e14	−3.27054
$143$	1.38164e14	1.35117
$144$	0	0
$145$	−3.14031e13	−0.280595
$146$	3.60930e14	3.08410
$147$	0	0
$148$	−1.40589e14	−1.09964
$149$	−2.13686e14	−1.59980	−0.799899	−	0.600135i	$-0.795114\pi$
$149$	−2.13686e14	−1.59980	−0.799899	+	0.600135i	$0.795114\pi$
$150$	0	0
$151$	1.33907e14	0.919291	0.459645	−	0.888103i	$-0.347977\pi$
$151$	1.33907e14	0.919291	0.459645	+	0.888103i	$0.347977\pi$
$152$	2.01838e14	1.32745
$153$	0	0
$154$	2.51712e14	1.52062
$155$	−5.25559e13	−0.304415
$156$	0	0
$157$	−1.12079e14	−0.597278	−0.298639	−	0.954366i	$-0.596533\pi$
$157$	−1.12079e14	−0.597278	−0.298639	+	0.954366i	$0.596533\pi$
$158$	2.85078e14	1.45778
$159$	0	0
$160$	6.12612e13	0.288673
$161$	−5.71167e13	−0.258461
$162$	0	0
$163$	3.15192e14	1.31630	0.658151	−	0.752886i	$-0.271339\pi$
$163$	3.15192e14	1.31630	0.658151	+	0.752886i	$0.271339\pi$
$164$	−3.41978e14	−1.37250
$165$	0	0
$166$	9.28366e13	0.344363
$167$	−2.75488e13	−0.0982755	−0.0491378	−	0.998792i	$-0.515647\pi$
$167$	−2.75488e13	−0.0982755	−0.0491378	+	0.998792i	$0.515647\pi$
$168$	0	0
$169$	1.02401e14	0.338095
$170$	−3.33121e14	−1.05848
$171$	0	0
$172$	1.28851e14	0.379449
$173$	6.39827e14	1.81452	0.907262	−	0.420566i	$-0.138168\pi$
$173$	6.39827e14	1.81452	0.907262	+	0.420566i	$0.138168\pi$
$174$	0	0
$175$	−5.93026e13	−0.156073
$176$	1.82412e14	0.462617
$177$	0	0
$178$	4.39030e14	1.03458
$179$	1.93924e14	0.440644	0.220322	−	0.975427i	$-0.429289\pi$
$179$	1.93924e14	0.440644	0.220322	+	0.975427i	$0.429289\pi$
$180$	0	0
$181$	−2.09515e14	−0.442899	−0.221449	−	0.975172i	$-0.571079\pi$
$181$	−2.09515e14	−0.442899	−0.221449	+	0.975172i	$0.571079\pi$
$182$	7.38346e14	1.50590
$183$	0	0
$184$	−2.27732e14	−0.432621
$185$	−1.50395e14	−0.275814
$186$	0	0
$187$	9.69062e14	1.65722
$188$	1.52376e15	2.51705
$189$	0	0
$190$	4.91675e14	0.758193
$191$	7.24660e14	1.07998	0.539992	−	0.841670i	$-0.318427\pi$
$191$	7.24660e14	1.07998	0.539992	+	0.841670i	$0.318427\pi$
$192$	0	0
$193$	−6.49165e14	−0.904133	−0.452067	−	0.891984i	$-0.649313\pi$
$193$	−6.49165e14	−0.904133	−0.452067	+	0.891984i	$0.649313\pi$
$194$	−2.42745e15	−3.26918
$195$	0	0
$196$	−5.53384e14	−0.697207
$197$	−1.71994e14	−0.209644	−0.104822	−	0.994491i	$-0.533427\pi$
$197$	−1.71994e14	−0.209644	−0.104822	+	0.994491i	$0.533427\pi$
$198$	0	0
$199$	−1.24687e15	−1.42323	−0.711614	−	0.702571i	$-0.752036\pi$
$199$	−1.24687e15	−1.42323	−0.711614	+	0.702571i	$0.752036\pi$
$200$	−2.36447e14	−0.261240
$201$	0	0
$202$	−3.46228e14	−0.358573
$203$	4.88187e14	0.489624
$204$	0	0
$205$	−3.65831e14	−0.344255
$206$	−1.12472e15	−1.02544
$207$	0	0
$208$	5.35068e14	0.458139
$209$	−1.43030e15	−1.18707
$210$	0	0
$211$	−1.75030e15	−1.36545	−0.682727	−	0.730674i	$-0.739206\pi$
$211$	−1.75030e15	−1.36545	−0.682727	+	0.730674i	$0.739206\pi$
$212$	−2.11698e15	−1.60152
$213$	0	0
$214$	−3.57838e15	−2.54681
$215$	1.37839e14	0.0951746
$216$	0	0
$217$	8.17025e14	0.531187
$218$	−4.02012e15	−2.53672
$219$	0	0
$220$	1.56631e15	0.931388
$221$	2.84254e15	1.64118
$222$	0	0
$223$	−8.33292e14	−0.453749	−0.226874	−	0.973924i	$-0.572851\pi$
$223$	−8.33292e14	−0.453749	−0.226874	+	0.973924i	$0.572851\pi$
$224$	−9.52357e14	−0.503718
$225$	0	0
$226$	1.59117e15	0.794348
$227$	3.16284e14	0.153430	0.0767148	−	0.997053i	$-0.475557\pi$
$227$	3.16284e14	0.153430	0.0767148	+	0.997053i	$0.475557\pi$
$228$	0	0
$229$	−6.45320e14	−0.295696	−0.147848	−	0.989010i	$-0.547235\pi$
$229$	−6.45320e14	−0.295696	−0.147848	+	0.989010i	$0.547235\pi$
$230$	−5.54753e14	−0.247098
$231$	0	0
$232$	1.94647e15	0.819549
$233$	6.81387e14	0.278985	0.139492	−	0.990223i	$-0.455453\pi$
$233$	6.81387e14	0.278985	0.139492	+	0.990223i	$0.455453\pi$
$234$	0	0
$235$	1.63004e15	0.631333
$236$	−1.83750e15	−0.692309
$237$	0	0
$238$	5.17864e15	1.84699
$239$	5.12913e15	1.78015	0.890077	−	0.455810i	$-0.150650\pi$
$239$	5.12913e15	1.78015	0.890077	+	0.455810i	$0.150650\pi$
$240$	0	0
$241$	−2.86463e15	−0.941799	−0.470900	−	0.882187i	$-0.656071\pi$
$241$	−2.86463e15	−0.941799	−0.470900	+	0.882187i	$0.656071\pi$
$242$	−1.89938e15	−0.607870
$243$	0	0
$244$	1.06835e16	3.24102
$245$	−5.91982e14	−0.174875
$246$	0	0
$247$	−4.19549e15	−1.17558
$248$	3.25759e15	0.889120
$249$	0	0
$250$	−5.75984e14	−0.149211
$251$	−4.26181e15	−1.07576	−0.537880	−	0.843022i	$-0.680774\pi$
$251$	−4.26181e15	−1.07576	−0.537880	+	0.843022i	$0.680774\pi$
$252$	0	0
$253$	1.61380e15	0.386871
$254$	−7.13969e14	−0.166825
$255$	0	0
$256$	−6.97743e15	−1.54930
$257$	4.44912e15	0.963182	0.481591	−	0.876396i	$-0.340059\pi$
$257$	4.44912e15	0.963182	0.481591	+	0.876396i	$0.340059\pi$
$258$	0	0
$259$	2.33801e15	0.481280
$260$	4.59445e15	0.922373
$261$	0	0
$262$	−3.18590e15	−0.608518
$263$	−4.56745e15	−0.851062	−0.425531	−	0.904944i	$-0.639913\pi$
$263$	−4.56745e15	−0.851062	−0.425531	+	0.904944i	$0.639913\pi$
$264$	0	0
$265$	−2.26463e15	−0.401698
$266$	−7.64350e15	−1.32301
$267$	0	0
$268$	1.65076e16	2.72150
$269$	9.36290e15	1.50668	0.753340	−	0.657632i	$-0.228442\pi$
$269$	9.36290e15	1.50668	0.753340	+	0.657632i	$0.228442\pi$
$270$	0	0
$271$	5.53192e15	0.848352	0.424176	−	0.905580i	$-0.360564\pi$
$271$	5.53192e15	0.848352	0.424176	+	0.905580i	$0.360564\pi$
$272$	3.75289e15	0.561911
$273$	0	0
$274$	−1.01636e16	−1.45100
$275$	1.67556e15	0.233613
$276$	0	0
$277$	−1.34631e16	−1.79072	−0.895360	−	0.445344i	$-0.853082\pi$
$277$	−1.34631e16	−1.79072	−0.895360	+	0.445344i	$0.853082\pi$
$278$	−1.88010e16	−2.44281
$279$	0	0
$280$	3.67577e15	0.455849
$281$	3.79294e15	0.459605	0.229802	−	0.973237i	$-0.426192\pi$
$281$	3.79294e15	0.459605	0.229802	+	0.973237i	$0.426192\pi$
$282$	0	0
$283$	−8.08884e15	−0.935997	−0.467998	−	0.883729i	$-0.655025\pi$
$283$	−8.08884e15	−0.935997	−0.467998	+	0.883729i	$0.655025\pi$
$284$	3.09087e16	3.49552
$285$	0	0
$286$	−2.08615e16	−2.25407
$287$	5.68714e15	0.600706
$288$	0	0
$289$	1.00326e16	1.01292
$290$	4.74158e15	0.468097
$291$	0	0
$292$	−3.49148e16	−3.29626
$293$	1.17859e16	1.08824	0.544118	−	0.839009i	$-0.316864\pi$
$293$	1.17859e16	1.08824	0.544118	+	0.839009i	$0.316864\pi$
$294$	0	0
$295$	−1.96567e15	−0.173647
$296$	9.32196e15	0.805584
$297$	0	0
$298$	3.22646e16	2.66883
$299$	4.73374e15	0.383126
$300$	0	0
$301$	−2.14282e15	−0.166074
$302$	−2.02187e16	−1.53359
$303$	0	0
$304$	−5.53913e15	−0.402498
$305$	1.14287e16	0.812921
$306$	0	0
$307$	2.04992e16	1.39746	0.698728	−	0.715387i	$-0.253750\pi$
$307$	2.04992e16	1.39746	0.698728	+	0.715387i	$0.253750\pi$
$308$	−2.43496e16	−1.62522
$309$	0	0
$310$	7.93546e15	0.507833
$311$	1.56792e16	0.982613	0.491307	−	0.870987i	$-0.336519\pi$
$311$	1.56792e16	0.982613	0.491307	+	0.870987i	$0.336519\pi$
$312$	0	0
$313$	2.03960e16	1.22605	0.613023	−	0.790065i	$-0.289953\pi$
$313$	2.03960e16	1.22605	0.613023	+	0.790065i	$0.289953\pi$
$314$	1.69229e16	0.996396
$315$	0	0
$316$	−2.75772e16	−1.55806
$317$	−6.89350e15	−0.381553	−0.190776	−	0.981634i	$-0.561101\pi$
$317$	−6.89350e15	−0.381553	−0.190776	+	0.981634i	$0.561101\pi$
$318$	0	0
$319$	−1.37934e16	−0.732881
$320$	−1.26520e16	−0.658693
$321$	0	0
$322$	8.62410e15	0.431172
$323$	−2.94265e16	−1.44186
$324$	0	0
$325$	4.91491e15	0.231352
$326$	−4.75911e16	−2.19589
$327$	0	0
$328$	2.26754e16	1.00548
$329$	−2.53404e16	−1.10164
$330$	0	0
$331$	2.31221e16	0.966375	0.483188	−	0.875517i	$-0.339479\pi$
$331$	2.31221e16	0.966375	0.483188	+	0.875517i	$0.339479\pi$
$332$	−8.98063e15	−0.368052
$333$	0	0
$334$	4.15961e15	0.163946
$335$	1.76590e16	0.682614
$336$	0	0
$337$	3.60387e16	1.34021	0.670107	−	0.742264i	$-0.266248\pi$
$337$	3.60387e16	1.34021	0.670107	+	0.742264i	$0.266248\pi$
$338$	−1.54616e16	−0.564020
$339$	0	0
$340$	3.22247e16	1.13130
$341$	−2.30846e16	−0.795094
$342$	0	0
$343$	3.27375e16	1.08551
$344$	−8.54369e15	−0.277981
$345$	0	0
$346$	−9.66080e16	−3.02704
$347$	9.81147e15	0.301712	0.150856	−	0.988556i	$-0.451797\pi$
$347$	9.81147e15	0.301712	0.150856	+	0.988556i	$0.451797\pi$
$348$	0	0
$349$	−1.87719e15	−0.0556087	−0.0278044	−	0.999613i	$-0.508852\pi$
$349$	−1.87719e15	−0.0556087	−0.0278044	+	0.999613i	$0.508852\pi$
$350$	8.95415e15	0.260365
$351$	0	0
$352$	2.69083e16	0.753978
$353$	4.82934e16	1.32847	0.664236	−	0.747523i	$-0.268757\pi$
$353$	4.82934e16	1.32847	0.664236	+	0.747523i	$0.268757\pi$
$354$	0	0
$355$	3.30646e16	0.876757
$356$	−4.24700e16	−1.10575
$357$	0	0
$358$	−2.92808e16	−0.735095
$359$	6.98527e16	1.72214	0.861072	−	0.508483i	$-0.169794\pi$
$359$	6.98527e16	1.72214	0.861072	+	0.508483i	$0.169794\pi$
$360$	0	0
$361$	1.37958e15	0.0328059
$362$	3.16348e16	0.738856
$363$	0	0
$364$	−7.14245e16	−1.60949
$365$	−3.73501e16	−0.826777
$366$	0	0
$367$	−5.27475e16	−1.12687	−0.563433	−	0.826162i	$-0.690520\pi$
$367$	−5.27475e16	−1.12687	−0.563433	+	0.826162i	$0.690520\pi$
$368$	6.24976e15	0.131175
$369$	0	0
$370$	2.27082e16	0.460121
$371$	3.52056e16	0.700941
$372$	0	0
$373$	−2.41945e16	−0.465168	−0.232584	−	0.972576i	$-0.574718\pi$
$373$	−2.41945e16	−0.465168	−0.232584	+	0.972576i	$0.574718\pi$
$374$	−1.46319e17	−2.76463
$375$	0	0
$376$	−1.01035e17	−1.84397
$377$	−4.04602e16	−0.725787
$378$	0	0
$379$	−6.92419e16	−1.20009	−0.600045	−	0.799966i	$-0.704851\pi$
$379$	−6.92419e16	−1.20009	−0.600045	+	0.799966i	$0.704851\pi$
$380$	−4.75626e16	−0.810349
$381$	0	0
$382$	−1.09417e17	−1.80166
$383$	−1.96503e16	−0.318110	−0.159055	−	0.987270i	$-0.550845\pi$
$383$	−1.96503e16	−0.318110	−0.159055	+	0.987270i	$0.550845\pi$
$384$	0	0
$385$	−2.60480e16	−0.407643
$386$	9.80179e16	1.50830
$387$	0	0
$388$	2.34821e17	3.49407
$389$	−8.29966e16	−1.21447	−0.607236	−	0.794521i	$-0.707722\pi$
$389$	−8.29966e16	−1.21447	−0.607236	+	0.794521i	$0.707722\pi$
$390$	0	0
$391$	3.32017e16	0.469907
$392$	3.66930e16	0.510768
$393$	0	0
$394$	2.59696e16	0.349735
$395$	−2.95007e16	−0.390798
$396$	0	0
$397$	2.16288e16	0.277264	0.138632	−	0.990344i	$-0.455729\pi$
$397$	2.16288e16	0.277264	0.138632	+	0.990344i	$0.455729\pi$
$398$	1.88265e17	2.37427
$399$	0	0
$400$	6.48894e15	0.0792107
$401$	9.73010e16	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$401$	9.73010e16	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$402$	0	0
$403$	−6.77138e16	−0.787398
$404$	3.34926e16	0.383239
$405$	0	0
$406$	−7.37118e16	−0.816804
$407$	−6.60591e16	−0.720392
$408$	0	0
$409$	8.60330e16	0.908790	0.454395	−	0.890800i	$-0.349855\pi$
$409$	8.60330e16	0.908790	0.454395	+	0.890800i	$0.349855\pi$
$410$	5.52371e16	0.574296
$411$	0	0
$412$	1.08801e17	1.09598
$413$	3.05579e16	0.303004
$414$	0	0
$415$	−9.60703e15	−0.0923159
$416$	7.89298e16	0.746680
$417$	0	0
$418$	2.15963e17	1.98031
$419$	−1.65561e17	−1.49475	−0.747374	−	0.664403i	$-0.768686\pi$
$419$	−1.65561e17	−1.49475	−0.747374	+	0.664403i	$0.768686\pi$
$420$	0	0
$421$	−1.20937e17	−1.05859	−0.529293	−	0.848439i	$-0.677543\pi$
$421$	−1.20937e17	−1.05859	−0.529293	+	0.848439i	$0.677543\pi$
$422$	2.64279e17	2.27789
$423$	0	0
$424$	1.40369e17	1.17326
$425$	3.44724e16	0.283755
$426$	0	0
$427$	−1.77669e17	−1.41850
$428$	3.46158e17	2.72201
$429$	0	0
$430$	−2.08124e16	−0.158773
$431$	7.43099e16	0.558399	0.279199	−	0.960233i	$-0.409931\pi$
$431$	7.43099e16	0.558399	0.279199	+	0.960233i	$0.409931\pi$
$432$	0	0
$433$	8.82929e16	0.643805	0.321903	−	0.946773i	$-0.395678\pi$
$433$	8.82929e16	0.643805	0.321903	+	0.946773i	$0.395678\pi$
$434$	−1.23363e17	−0.886141
$435$	0	0
$436$	3.88890e17	2.71122
$437$	−4.90046e16	−0.336595
$438$	0	0
$439$	2.00274e17	1.33538	0.667690	−	0.744439i	$-0.267283\pi$
$439$	2.00274e17	1.33538	0.667690	+	0.744439i	$0.267283\pi$
$440$	−1.03857e17	−0.682327
$441$	0	0
$442$	−4.29198e17	−2.73787
$443$	−5.54083e16	−0.348298	−0.174149	−	0.984719i	$-0.555717\pi$
$443$	−5.54083e16	−0.348298	−0.174149	+	0.984719i	$0.555717\pi$
$444$	0	0
$445$	−4.54322e16	−0.277347
$446$	1.25819e17	0.756957
$447$	0	0
$448$	1.96685e17	1.14938
$449$	−8.54624e16	−0.492237	−0.246118	−	0.969240i	$-0.579155\pi$
$449$	−8.54624e16	−0.492237	−0.246118	+	0.969240i	$0.579155\pi$
$450$	0	0
$451$	−1.60687e17	−0.899151
$452$	−1.53923e17	−0.848992
$453$	0	0
$454$	−4.77560e16	−0.255956
$455$	−7.64063e16	−0.403697
$456$	0	0
$457$	7.96679e16	0.409099	0.204549	−	0.978856i	$-0.434427\pi$
$457$	7.96679e16	0.409099	0.204549	+	0.978856i	$0.434427\pi$
$458$	9.74374e16	0.493288
$459$	0	0
$460$	5.36645e16	0.264096
$461$	−2.95040e17	−1.43161	−0.715805	−	0.698300i	$-0.753940\pi$
$461$	−2.95040e17	−1.43161	−0.715805	+	0.698300i	$0.753940\pi$
$462$	0	0
$463$	−1.46334e17	−0.690350	−0.345175	−	0.938538i	$-0.612180\pi$
$463$	−1.46334e17	−0.690350	−0.345175	+	0.938538i	$0.612180\pi$
$464$	−5.34178e16	−0.248496
$465$	0	0
$466$	−1.02883e17	−0.465410
$467$	−9.18314e16	−0.409668	−0.204834	−	0.978797i	$-0.565665\pi$
$467$	−9.18314e16	−0.409668	−0.204834	+	0.978797i	$0.565665\pi$
$468$	0	0
$469$	−2.74524e17	−1.19112
$470$	−2.46122e17	−1.05321
$471$	0	0
$472$	1.21839e17	0.507179
$473$	6.05440e16	0.248584
$474$	0	0
$475$	−5.08801e16	−0.203254
$476$	−5.00960e17	−1.97405
$477$	0	0
$478$	−7.74452e17	−2.96970
$479$	−1.85480e17	−0.701644	−0.350822	−	0.936442i	$-0.614098\pi$
$479$	−1.85480e17	−0.701644	−0.350822	+	0.936442i	$0.614098\pi$
$480$	0	0
$481$	−1.93771e17	−0.713419
$482$	4.32533e17	1.57114
$483$	0	0
$484$	1.83738e17	0.649686
$485$	2.51200e17	0.876392
$486$	0	0
$487$	−5.22828e16	−0.177591	−0.0887954	−	0.996050i	$-0.528302\pi$
$487$	−5.22828e16	−0.177591	−0.0887954	+	0.996050i	$0.528302\pi$
$488$	−7.08389e17	−2.37434
$489$	0	0
$490$	8.93838e16	0.291732
$491$	5.38376e17	1.73403	0.867013	−	0.498286i	$-0.166037\pi$
$491$	5.38376e17	1.73403	0.867013	+	0.498286i	$0.166037\pi$
$492$	0	0
$493$	−2.83781e17	−0.890182
$494$	6.33481e17	1.96114
$495$	0	0
$496$	−8.93996e16	−0.269590
$497$	−5.14017e17	−1.52989
$498$	0	0
$499$	−3.09197e16	−0.0896565	−0.0448283	−	0.998995i	$-0.514274\pi$
$499$	−3.09197e16	−0.0896565	−0.0448283	+	0.998995i	$0.514274\pi$
$500$	5.57183e16	0.159475
$501$	0	0
$502$	6.43494e17	1.79461
$503$	−2.50927e17	−0.690804	−0.345402	−	0.938455i	$-0.612257\pi$
$503$	−2.50927e17	−0.690804	−0.345402	+	0.938455i	$0.612257\pi$
$504$	0	0
$505$	3.58287e16	0.0961252
$506$	−2.43669e17	−0.645389
$507$	0	0
$508$	6.90664e16	0.178301
$509$	−8.69163e16	−0.221532	−0.110766	−	0.993847i	$-0.535330\pi$
$509$	−8.69163e16	−0.221532	−0.110766	+	0.993847i	$0.535330\pi$
$510$	0	0
$511$	5.80639e17	1.44268
$512$	3.15079e17	0.772975
$513$	0	0
$514$	−6.71776e17	−1.60681
$515$	1.16390e17	0.274896
$516$	0	0
$517$	7.15978e17	1.64896
$518$	−3.53018e17	−0.802885
$519$	0	0
$520$	−3.04642e17	−0.675722
$521$	6.04613e16	0.132444	0.0662220	−	0.997805i	$-0.478905\pi$
$521$	6.04613e16	0.132444	0.0662220	+	0.997805i	$0.478905\pi$
$522$	0	0
$523$	6.29268e17	1.34454	0.672271	−	0.740305i	$-0.265319\pi$
$523$	6.29268e17	1.34454	0.672271	+	0.740305i	$0.265319\pi$
$524$	3.08190e17	0.650378
$525$	0	0
$526$	6.89643e17	1.41977
$527$	−4.74934e17	−0.965748
$528$	0	0
$529$	−4.48745e17	−0.890303
$530$	3.41939e17	0.670125
$531$	0	0
$532$	7.39400e17	1.41402
$533$	−4.71341e17	−0.890448
$534$	0	0
$535$	3.70302e17	0.682742
$536$	−1.09456e18	−1.99375
$537$	0	0
$538$	−1.41371e18	−2.51349
$539$	−2.60021e17	−0.456753
$540$	0	0
$541$	−5.09694e16	−0.0874033	−0.0437016	−	0.999045i	$-0.513915\pi$
$541$	−5.09694e16	−0.0874033	−0.0437016	+	0.999045i	$0.513915\pi$
$542$	−8.35270e17	−1.41524
$543$	0	0
$544$	5.53601e17	0.915808
$545$	4.16015e17	0.680035
$546$	0	0
$547$	4.30137e17	0.686576	0.343288	−	0.939230i	$-0.388459\pi$
$547$	4.30137e17	0.686576	0.343288	+	0.939230i	$0.388459\pi$
$548$	9.83181e17	1.55081
$549$	0	0
$550$	−2.52994e17	−0.389721
$551$	4.18852e17	0.637639
$552$	0	0
$553$	4.58614e17	0.681920
$554$	2.03281e18	2.98733
$555$	0	0
$556$	1.81873e18	2.61085
$557$	−3.17240e17	−0.450120	−0.225060	−	0.974345i	$-0.572258\pi$
$557$	−3.17240e17	−0.450120	−0.225060	+	0.974345i	$0.572258\pi$
$558$	0	0
$559$	1.77593e17	0.246178
$560$	−1.00876e17	−0.138218
$561$	0	0
$562$	−5.72698e17	−0.766726
$563$	−7.30740e17	−0.967071	−0.483536	−	0.875325i	$-0.660648\pi$
$563$	−7.30740e17	−0.967071	−0.483536	+	0.875325i	$0.660648\pi$
$564$	0	0
$565$	−1.64659e17	−0.212946
$566$	1.22134e18	1.56146
$567$	0	0
$568$	−2.04945e18	−2.56079
$569$	1.08027e18	1.33445	0.667225	−	0.744856i	$-0.267482\pi$
$569$	1.08027e18	1.33445	0.667225	+	0.744856i	$0.267482\pi$
$570$	0	0
$571$	−2.03921e16	−0.0246222	−0.0123111	−	0.999924i	$-0.503919\pi$
$571$	−2.03921e16	−0.0246222	−0.0123111	+	0.999924i	$0.503919\pi$
$572$	2.01806e18	2.40912
$573$	0	0
$574$	−8.58706e17	−1.00211
$575$	5.74076e16	0.0662412
$576$	0	0
$577$	−6.06626e17	−0.684350	−0.342175	−	0.939636i	$-0.611164\pi$
$577$	−6.06626e17	−0.684350	−0.342175	+	0.939636i	$0.611164\pi$
$578$	−1.51482e18	−1.68978
$579$	0	0
$580$	−4.58681e17	−0.500298
$581$	1.49349e17	0.161086
$582$	0	0
$583$	−9.94714e17	−1.04919
$584$	2.31508e18	2.41481
$585$	0	0
$586$	−1.77956e18	−1.81543
$587$	1.04007e18	1.04934	0.524670	−	0.851306i	$-0.324189\pi$
$587$	1.04007e18	1.04934	0.524670	+	0.851306i	$0.324189\pi$
$588$	0	0
$589$	7.00986e17	0.691767
$590$	2.96798e17	0.289683
$591$	0	0
$592$	−2.55827e17	−0.244262
$593$	1.12622e17	0.106357	0.0531785	−	0.998585i	$-0.483065\pi$
$593$	1.12622e17	0.106357	0.0531785	+	0.998585i	$0.483065\pi$
$594$	0	0
$595$	−5.35902e17	−0.495137
$596$	−3.12114e18	−2.85242
$597$	0	0
$598$	−7.14752e17	−0.639142
$599$	1.80328e17	0.159510	0.0797552	−	0.996814i	$-0.474586\pi$
$599$	1.80328e17	0.159510	0.0797552	+	0.996814i	$0.474586\pi$
$600$	0	0
$601$	1.46700e18	1.26983	0.634917	−	0.772580i	$-0.281034\pi$
$601$	1.46700e18	1.26983	0.634917	+	0.772580i	$0.281034\pi$
$602$	3.23546e17	0.277050
$603$	0	0
$604$	1.95587e18	1.63908
$605$	1.96553e17	0.162956
$606$	0	0
$607$	−1.92051e17	−0.155844	−0.0779220	−	0.996959i	$-0.524829\pi$
$607$	−1.92051e17	−0.155844	−0.0779220	+	0.996959i	$0.524829\pi$
$608$	−8.17097e17	−0.655994
$609$	0	0
$610$	−1.72563e18	−1.35614
$611$	2.10017e18	1.63300
$612$	0	0
$613$	1.28405e18	0.977434	0.488717	−	0.872442i	$-0.337465\pi$
$613$	1.28405e18	0.977434	0.488717	+	0.872442i	$0.337465\pi$
$614$	−3.09520e18	−2.33128
$615$	0	0
$616$	1.61454e18	1.19062
$617$	−2.18567e18	−1.59489	−0.797447	−	0.603390i	$-0.793816\pi$
$617$	−2.18567e18	−1.59489	−0.797447	+	0.603390i	$0.793816\pi$
$618$	0	0
$619$	2.16189e18	1.54470	0.772350	−	0.635197i	$-0.219081\pi$
$619$	2.16189e18	1.54470	0.772350	+	0.635197i	$0.219081\pi$
$620$	−7.67643e17	−0.542767
$621$	0	0
$622$	−2.36742e18	−1.63922
$623$	7.06282e17	0.483956
$624$	0	0
$625$	5.96046e16	0.0400000
$626$	−3.07961e18	−2.04532
$627$	0	0
$628$	−1.63705e18	−1.06494
$629$	−1.35908e18	−0.875014
$630$	0	0
$631$	−9.40530e17	−0.593174	−0.296587	−	0.955006i	$-0.595848\pi$
$631$	−9.40530e17	−0.593174	−0.296587	+	0.955006i	$0.595848\pi$
$632$	1.82855e18	1.14142
$633$	0	0
$634$	1.04086e18	0.636517
$635$	7.38837e16	0.0447218
$636$	0	0
$637$	−7.62717e17	−0.452332
$638$	2.08268e18	1.22261
$639$	0	0
$640$	1.40848e18	0.810177
$641$	2.94832e18	1.67879	0.839396	−	0.543520i	$-0.182909\pi$
$641$	2.94832e18	1.67879	0.839396	+	0.543520i	$0.182909\pi$
$642$	0	0
$643$	−2.30935e18	−1.28860	−0.644301	−	0.764772i	$-0.722851\pi$
$643$	−2.30935e18	−1.28860	−0.644301	+	0.764772i	$0.722851\pi$
$644$	−8.34260e17	−0.460832
$645$	0	0
$646$	4.44314e18	2.40535
$647$	−1.97636e18	−1.05923	−0.529613	−	0.848239i	$-0.677663\pi$
$647$	−1.97636e18	−1.05923	−0.529613	+	0.848239i	$0.677663\pi$
$648$	0	0
$649$	−8.63397e17	−0.453544
$650$	−7.42106e17	−0.385948
$651$	0	0
$652$	4.60376e18	2.34695
$653$	6.64081e17	0.335186	0.167593	−	0.985856i	$-0.446401\pi$
$653$	6.64081e17	0.335186	0.167593	+	0.985856i	$0.446401\pi$
$654$	0	0
$655$	3.29687e17	0.163130
$656$	−6.22291e17	−0.304873
$657$	0	0
$658$	3.82617e18	1.83779
$659$	9.44263e17	0.449094	0.224547	−	0.974463i	$-0.427910\pi$
$659$	9.44263e17	0.449094	0.224547	+	0.974463i	$0.427910\pi$
$660$	0	0
$661$	9.01620e17	0.420450	0.210225	−	0.977653i	$-0.432580\pi$
$661$	9.01620e17	0.420450	0.210225	+	0.977653i	$0.432580\pi$
$662$	−3.49123e18	−1.61213
$663$	0	0
$664$	5.95475e17	0.269632
$665$	7.90973e17	0.354667
$666$	0	0
$667$	−4.72587e17	−0.207809
$668$	−4.02384e17	−0.175224
$669$	0	0
$670$	−2.66635e18	−1.13876
$671$	5.01993e18	2.12325
$672$	0	0
$673$	−1.79416e18	−0.744327	−0.372164	−	0.928167i	$-0.621384\pi$
$673$	−1.79416e18	−0.744327	−0.372164	+	0.928167i	$0.621384\pi$
$674$	−5.44151e18	−2.23578
$675$	0	0
$676$	1.49569e18	0.602819
$677$	−4.18204e18	−1.66941	−0.834703	−	0.550700i	$-0.814361\pi$
$677$	−4.18204e18	−1.66941	−0.834703	+	0.550700i	$0.814361\pi$
$678$	0	0
$679$	−3.90511e18	−1.52926
$680$	−2.13671e18	−0.828778
$681$	0	0
$682$	3.48556e18	1.32640
$683$	3.12297e18	1.17715	0.588576	−	0.808442i	$-0.299689\pi$
$683$	3.12297e18	1.17715	0.588576	+	0.808442i	$0.299689\pi$
$684$	0	0
$685$	1.05176e18	0.388980
$686$	−4.94307e18	−1.81088
$687$	0	0
$688$	2.34469e17	0.0842869
$689$	−2.91779e18	−1.03903
$690$	0	0
$691$	−1.98112e18	−0.692313	−0.346157	−	0.938177i	$-0.612513\pi$
$691$	−1.98112e18	−0.692313	−0.346157	+	0.938177i	$0.612513\pi$
$692$	9.34545e18	3.23527
$693$	0	0
$694$	−1.48144e18	−0.503324
$695$	1.94558e18	0.654860
$696$	0	0
$697$	−3.30591e18	−1.09214
$698$	2.83438e17	0.0927681
$699$	0	0
$700$	−8.66188e17	−0.278275
$701$	2.35827e18	0.750631	0.375316	−	0.926897i	$-0.377534\pi$
$701$	2.35827e18	0.750631	0.375316	+	0.926897i	$0.377534\pi$
$702$	0	0
$703$	2.00595e18	0.626774
$704$	−5.55722e18	−1.72042
$705$	0	0
$706$	−7.29186e18	−2.21619
$707$	−5.56987e17	−0.167733
$708$	0	0
$709$	−5.38933e18	−1.59344	−0.796718	−	0.604352i	$-0.793432\pi$
$709$	−5.38933e18	−1.59344	−0.796718	+	0.604352i	$0.793432\pi$
$710$	−4.99245e18	−1.46263
$711$	0	0
$712$	2.81604e18	0.810062
$713$	−7.90917e17	−0.225449
$714$	0	0
$715$	2.15882e18	0.604263
$716$	2.83250e18	0.785662
$717$	0	0
$718$	−1.05471e19	−2.87293
$719$	−4.61827e18	−1.24664	−0.623321	−	0.781966i	$-0.714217\pi$
$719$	−4.61827e18	−1.24664	−0.623321	+	0.781966i	$0.714217\pi$
$720$	0	0
$721$	−1.80938e18	−0.479679
$722$	−2.08305e17	−0.0547277
$723$	0	0
$724$	−3.06022e18	−0.789682
$725$	−4.90673e17	−0.125486
$726$	0	0
$727$	6.89875e18	1.73299	0.866496	−	0.499184i	$-0.166367\pi$
$727$	6.89875e18	1.73299	0.866496	+	0.499184i	$0.166367\pi$
$728$	4.73591e18	1.17910
$729$	0	0
$730$	5.63953e18	1.37925
$731$	1.24561e18	0.301939
$732$	0	0
$733$	−1.03644e18	−0.246814	−0.123407	−	0.992356i	$-0.539382\pi$
$733$	−1.03644e18	−0.246814	−0.123407	+	0.992356i	$0.539382\pi$
$734$	7.96438e18	1.87987
$735$	0	0
$736$	9.21924e17	0.213791
$737$	7.75652e18	1.78290
$738$	0	0
$739$	6.31887e17	0.142709	0.0713543	−	0.997451i	$-0.477268\pi$
$739$	6.31887e17	0.142709	0.0713543	+	0.997451i	$0.477268\pi$
$740$	−2.19670e18	−0.491773
$741$	0	0
$742$	−5.31573e18	−1.16933
$743$	6.72185e18	1.46575	0.732877	−	0.680361i	$-0.238177\pi$
$743$	6.72185e18	1.46575	0.732877	+	0.680361i	$0.238177\pi$
$744$	0	0
$745$	−3.33884e18	−0.715451
$746$	3.65315e18	0.776007
$747$	0	0
$748$	1.41543e19	2.95481
$749$	−5.75665e18	−1.19135
$750$	0	0
$751$	−5.10562e18	−1.03846	−0.519229	−	0.854635i	$-0.673781\pi$
$751$	−5.10562e18	−1.03846	−0.519229	+	0.854635i	$0.673781\pi$
$752$	2.77277e18	0.559110
$753$	0	0
$754$	6.10912e18	1.21078
$755$	2.09230e18	0.411119
$756$	0	0
$757$	5.73702e18	1.10806	0.554030	−	0.832497i	$-0.313089\pi$
$757$	5.73702e18	1.10806	0.554030	+	0.832497i	$0.313089\pi$
$758$	1.04549e19	2.00203
$759$	0	0
$760$	3.15371e18	0.593655
$761$	2.23151e18	0.416485	0.208242	−	0.978077i	$-0.433226\pi$
$761$	2.23151e18	0.416485	0.208242	+	0.978077i	$0.433226\pi$
$762$	0	0
$763$	−6.46729e18	−1.18662
$764$	1.05845e19	1.92560
$765$	0	0
$766$	2.96701e18	0.530680
$767$	−2.53259e18	−0.449154
$768$	0	0
$769$	−1.49297e18	−0.260334	−0.130167	−	0.991492i	$-0.541551\pi$
$769$	−1.49297e18	−0.260334	−0.130167	+	0.991492i	$0.541551\pi$
$770$	3.93301e18	0.680041
$771$	0	0
$772$	−9.48185e18	−1.61206
$773$	−3.08330e18	−0.519815	−0.259908	−	0.965634i	$-0.583692\pi$
$773$	−3.08330e18	−0.519815	−0.259908	+	0.965634i	$0.583692\pi$
$774$	0	0
$775$	−8.21186e17	−0.136138
$776$	−1.55702e19	−2.55972
$777$	0	0
$778$	1.25317e19	2.02602
$779$	4.87941e18	0.782302
$780$	0	0
$781$	1.45232e19	2.28998
$782$	−5.01316e18	−0.783912
$783$	0	0
$784$	−1.00698e18	−0.154870
$785$	−1.75123e18	−0.267111
$786$	0	0
$787$	1.53244e18	0.229904	0.114952	−	0.993371i	$-0.463329\pi$
$787$	1.53244e18	0.229904	0.114952	+	0.993371i	$0.463329\pi$
$788$	−2.51219e18	−0.373793
$789$	0	0
$790$	4.45434e18	0.651940
$791$	2.55976e18	0.371580
$792$	0	0
$793$	1.47249e19	2.10270
$794$	−3.26574e18	−0.462540
$795$	0	0
$796$	−1.82120e19	−2.53760
$797$	−1.11316e19	−1.53843	−0.769216	−	0.638988i	$-0.779353\pi$
$797$	−1.11316e19	−1.53843	−0.769216	+	0.638988i	$0.779353\pi$
$798$	0	0
$799$	1.47303e19	2.00289
$800$	9.57207e17	0.129098
$801$	0	0
$802$	−1.46916e19	−1.94955
$803$	−1.64056e19	−2.15944
$804$	0	0
$805$	−8.92449e17	−0.115587
$806$	1.02242e19	1.31356
$807$	0	0
$808$	−2.22078e18	−0.280758
$809$	−3.24602e18	−0.407086	−0.203543	−	0.979066i	$-0.565246\pi$
$809$	−3.24602e18	−0.407086	−0.203543	+	0.979066i	$0.565246\pi$
$810$	0	0
$811$	−1.11593e19	−1.37721	−0.688606	−	0.725136i	$-0.741777\pi$
$811$	−1.11593e19	−1.37721	−0.688606	+	0.725136i	$0.741777\pi$
$812$	7.13057e18	0.872992
$813$	0	0
$814$	9.97432e18	1.20178
$815$	4.92487e18	0.588668
$816$	0	0
$817$	−1.83848e18	−0.216279
$818$	−1.29902e19	−1.51607
$819$	0	0
$820$	−5.34340e18	−0.613801
$821$	1.17292e19	1.33672	0.668358	−	0.743840i	$-0.266997\pi$
$821$	1.17292e19	1.33672	0.668358	+	0.743840i	$0.266997\pi$
$822$	0	0
$823$	3.13153e18	0.351284	0.175642	−	0.984454i	$-0.443800\pi$
$823$	3.13153e18	0.351284	0.175642	+	0.984454i	$0.443800\pi$
$824$	−7.21424e18	−0.802904
$825$	0	0
$826$	−4.61397e18	−0.505480
$827$	−8.27527e18	−0.899490	−0.449745	−	0.893157i	$-0.648485\pi$
$827$	−8.27527e18	−0.899490	−0.449745	+	0.893157i	$0.648485\pi$
$828$	0	0
$829$	−5.39923e17	−0.0577733	−0.0288867	−	0.999583i	$-0.509196\pi$
$829$	−5.39923e17	−0.0577733	−0.0288867	+	0.999583i	$0.509196\pi$
$830$	1.45057e18	0.154004
$831$	0	0
$832$	−1.63010e19	−1.70377
$833$	−5.34958e18	−0.554788
$834$	0	0
$835$	−4.30450e17	−0.0439501
$836$	−2.08913e19	−2.11653
$837$	0	0
$838$	2.49983e19	2.49358
$839$	1.48290e19	1.46777	0.733886	−	0.679273i	$-0.237705\pi$
$839$	1.48290e19	1.46777	0.733886	+	0.679273i	$0.237705\pi$
$840$	0	0
$841$	−6.22134e18	−0.606331
$842$	1.82604e19	1.76596
$843$	0	0
$844$	−2.55653e19	−2.43458
$845$	1.60001e18	0.151201
$846$	0	0
$847$	−3.05558e18	−0.284349
$848$	−3.85223e18	−0.355745
$849$	0	0
$850$	−5.20502e18	−0.473368
$851$	−2.26330e18	−0.204268
$852$	0	0
$853$	−3.49218e18	−0.310404	−0.155202	−	0.987883i	$-0.549603\pi$
$853$	−3.49218e18	−0.310404	−0.155202	+	0.987883i	$0.549603\pi$
$854$	2.68263e19	2.36638
$855$	0	0
$856$	−2.29525e19	−1.99412
$857$	−1.10408e19	−0.951977	−0.475988	−	0.879452i	$-0.657910\pi$
$857$	−1.10408e19	−0.951977	−0.475988	+	0.879452i	$0.657910\pi$
$858$	0	0
$859$	−5.84161e18	−0.496109	−0.248054	−	0.968746i	$-0.579791\pi$
$859$	−5.84161e18	−0.496109	−0.248054	+	0.968746i	$0.579791\pi$
$860$	2.01330e18	0.169695
$861$	0	0
$862$	−1.12201e19	−0.931537
$863$	2.70899e17	0.0223222	0.0111611	−	0.999938i	$-0.496447\pi$
$863$	2.70899e17	0.0223222	0.0111611	+	0.999938i	$0.496447\pi$
$864$	0	0
$865$	9.99729e18	0.811480
$866$	−1.33314e19	−1.07401
$867$	0	0
$868$	1.19337e19	0.947099
$869$	−1.29579e19	−1.02072
$870$	0	0
$871$	2.27521e19	1.76565
$872$	−2.57860e19	−1.98621
$873$	0	0
$874$	7.39925e18	0.561517
$875$	−9.26604e17	−0.0697978
$876$	0	0
$877$	1.18101e18	0.0876506	0.0438253	−	0.999039i	$-0.486045\pi$
$877$	1.18101e18	0.0876506	0.0438253	+	0.999039i	$0.486045\pi$
$878$	−3.02395e19	−2.22772
$879$	0	0
$880$	2.85019e18	0.206889
$881$	−1.72152e19	−1.24042	−0.620210	−	0.784436i	$-0.712952\pi$
$881$	−1.72152e19	−1.24042	−0.620210	+	0.784436i	$0.712952\pi$
$882$	0	0
$883$	−5.09224e18	−0.361547	−0.180773	−	0.983525i	$-0.557860\pi$
$883$	−5.09224e18	−0.361547	−0.180773	+	0.983525i	$0.557860\pi$
$884$	4.15188e19	2.92621
$885$	0	0
$886$	8.36614e18	0.581040
$887$	−1.84087e18	−0.126917	−0.0634584	−	0.997984i	$-0.520213\pi$
$887$	−1.84087e18	−0.126917	−0.0634584	+	0.997984i	$0.520213\pi$
$888$	0	0
$889$	−1.14858e18	−0.0780372
$890$	6.85985e18	0.462679
$891$	0	0
$892$	−1.21712e19	−0.809028
$893$	−2.17414e19	−1.43467
$894$	0	0
$895$	3.03007e18	0.197062
$896$	−2.18959e19	−1.41371
$897$	0	0
$898$	1.29040e19	0.821164
$899$	6.76012e18	0.427087
$900$	0	0
$901$	−2.04649e19	−1.27438
$902$	2.42622e19	1.49999
$903$	0	0
$904$	1.02061e19	0.621964
$905$	−3.27367e18	−0.198070
$906$	0	0
$907$	1.23998e19	0.739552	0.369776	−	0.929121i	$-0.379434\pi$
$907$	1.23998e19	0.739552	0.369776	+	0.929121i	$0.379434\pi$
$908$	4.61971e18	0.273563
$909$	0	0
$910$	1.15367e19	0.673458
$911$	−1.37704e18	−0.0798137	−0.0399068	−	0.999203i	$-0.512706\pi$
$911$	−1.37704e18	−0.0798137	−0.0399068	+	0.999203i	$0.512706\pi$
$912$	0	0
$913$	−4.21977e18	−0.241118
$914$	−1.20291e19	−0.682470
$915$	0	0
$916$	−9.42569e18	−0.527221
$917$	−5.12525e18	−0.284652
$918$	0	0
$919$	5.33293e18	0.292022	0.146011	−	0.989283i	$-0.453357\pi$
$919$	5.33293e18	0.292022	0.146011	+	0.989283i	$0.453357\pi$
$920$	−3.55831e18	−0.193474
$921$	0	0
$922$	4.45483e19	2.38825
$923$	4.26009e19	2.26781
$924$	0	0
$925$	−2.34992e18	−0.123348
$926$	2.20951e19	1.15166
$927$	0	0
$928$	−7.87985e18	−0.405001
$929$	3.63320e18	0.185433	0.0927164	−	0.995693i	$-0.470445\pi$
$929$	3.63320e18	0.185433	0.0927164	+	0.995693i	$0.470445\pi$
$930$	0	0
$931$	7.89580e18	0.397396
$932$	9.95249e18	0.497426
$933$	0	0
$934$	1.38657e19	0.683419
$935$	1.51416e19	0.741133
$936$	0	0
$937$	3.67066e19	1.77189	0.885946	−	0.463789i	$-0.153510\pi$
$937$	3.67066e19	1.77189	0.885946	+	0.463789i	$0.153510\pi$
$938$	4.14506e19	1.98707
$939$	0	0
$940$	2.38088e19	1.12566
$941$	−3.60825e18	−0.169420	−0.0847099	−	0.996406i	$-0.526996\pi$
$941$	−3.60825e18	−0.169420	−0.0847099	+	0.996406i	$0.526996\pi$
$942$	0	0
$943$	−5.50541e18	−0.254955
$944$	−3.34367e18	−0.153782
$945$	0	0
$946$	−9.14159e18	−0.414695
$947$	2.38079e19	1.07262	0.536311	−	0.844021i	$-0.319818\pi$
$947$	2.38079e19	1.07262	0.536311	+	0.844021i	$0.319818\pi$
$948$	0	0
$949$	−4.81224e19	−2.13854
$950$	7.68242e18	0.339074
$951$	0	0
$952$	3.32170e19	1.44617
$953$	−4.14504e19	−1.79236	−0.896180	−	0.443691i	$-0.853669\pi$
$953$	−4.14504e19	−1.79236	−0.896180	+	0.443691i	$0.853669\pi$
$954$	0	0
$955$	1.13228e19	0.482984
$956$	7.49173e19	3.17399
$957$	0	0
$958$	2.80058e19	1.17050
$959$	−1.63504e19	−0.678748
$960$	0	0
$961$	−1.31039e19	−0.536659
$962$	2.92576e19	1.19015
$963$	0	0
$964$	−4.18415e19	−1.67921
$965$	−1.01432e19	−0.404341
$966$	0	0
$967$	−1.57592e19	−0.619814	−0.309907	−	0.950767i	$-0.600298\pi$
$967$	−1.57592e19	−0.619814	−0.309907	+	0.950767i	$0.600298\pi$
$968$	−1.21830e19	−0.475954
$969$	0	0
$970$	−3.79289e19	−1.46202
$971$	−5.15312e18	−0.197308	−0.0986542	−	0.995122i	$-0.531454\pi$
$971$	−5.15312e18	−0.197308	−0.0986542	+	0.995122i	$0.531454\pi$
$972$	0	0
$973$	−3.02457e19	−1.14269
$974$	7.89422e18	0.296262
$975$	0	0
$976$	1.94407e19	0.719925
$977$	−5.38800e19	−1.98204	−0.991022	−	0.133699i	$-0.957314\pi$
$977$	−5.38800e19	−1.98204	−0.991022	+	0.133699i	$0.957314\pi$
$978$	0	0
$979$	−1.99556e19	−0.724397
$980$	−8.64662e18	−0.311801
$981$	0	0
$982$	−8.12898e19	−2.89275
$983$	3.40528e19	1.20380	0.601902	−	0.798570i	$-0.294410\pi$
$983$	3.40528e19	1.20380	0.601902	+	0.798570i	$0.294410\pi$
$984$	0	0
$985$	−2.68741e18	−0.0937559
$986$	4.28484e19	1.48503
$987$	0	0
$988$	−6.12803e19	−2.09605
$989$	2.07434e18	0.0704862
$990$	0	0
$991$	−2.90886e19	−0.975537	−0.487768	−	0.872973i	$-0.662189\pi$
$991$	−2.90886e19	−0.975537	−0.487768	+	0.872973i	$0.662189\pi$
$992$	−1.31876e19	−0.439381
$993$	0	0
$994$	7.76118e19	2.55221
$995$	−1.94823e19	−0.636487
$996$	0	0
$997$	2.13264e19	0.687700	0.343850	−	0.939025i	$-0.388269\pi$
$997$	2.13264e19	0.687700	0.343850	+	0.939025i	$0.388269\pi$
$998$	4.66859e18	0.149568
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.14.a.g.1.1	✓	4
3.2	odd	2		45.14.a.h.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.14.a.g.1.1	✓	4	1.1	even	1	trivial
45.14.a.h.1.4	yes	4	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

\(f(q)\)	\(=\)	\(q-150.991 q^{2} +14606.2 q^{4} +15625.0 q^{5} -242904. q^{7} -968489. q^{8} -2.35923e6 q^{10} +6.86310e6 q^{11} +2.01315e7 q^{13} +3.66762e7 q^{14} +2.65787e7 q^{16} +1.41199e8 q^{17} -2.08405e8 q^{19} +2.28222e8 q^{20} -1.03626e9 q^{22} +2.35142e8 q^{23} +2.44141e8 q^{25} -3.03967e9 q^{26} -3.54790e9 q^{28} -2.00980e9 q^{29} -3.36358e9 q^{31} +3.92072e9 q^{32} -2.13197e10 q^{34} -3.79537e9 q^{35} -9.62526e9 q^{37} +3.14672e10 q^{38} -1.51326e10 q^{40} -2.34132e10 q^{41} +8.82167e9 q^{43} +1.00244e11 q^{44} -3.55042e10 q^{46} +1.04323e11 q^{47} -3.78868e10 q^{49} -3.68630e10 q^{50} +2.94045e11 q^{52} -1.44937e11 q^{53} +1.07236e11 q^{55} +2.35249e11 q^{56} +3.03461e11 q^{58} -1.25803e11 q^{59} +7.31437e11 q^{61} +5.07869e11 q^{62} -8.09725e11 q^{64} +3.14554e11 q^{65} +1.13018e12 q^{67} +2.06238e12 q^{68} +5.73066e11 q^{70} +2.11613e12 q^{71} -2.39041e12 q^{73} +1.45333e12 q^{74} -3.04401e12 q^{76} -1.66707e12 q^{77} -1.88805e12 q^{79} +4.15292e11 q^{80} +3.53517e12 q^{82} -6.14850e11 q^{83} +2.20623e12 q^{85} -1.33199e12 q^{86} -6.64683e12 q^{88} -2.90766e12 q^{89} -4.89001e12 q^{91} +3.43453e12 q^{92} -1.57518e13 q^{94} -3.25633e12 q^{95} +1.60768e13 q^{97} +5.72056e12 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 15 q^{2} + 16837 q^{4} + 62500 q^{5} + 343040 q^{7} - 14865 q^{8} - 234375 q^{10} + 12697800 q^{11} + 34336040 q^{13} + 26944650 q^{14} + 66562801 q^{16} + 84377280 q^{17} - 131821144 q^{19} + 263078125 q^{20}+ \cdots - 17650752985395 q^{98}+O(q^{100}) \) 4 * q - 15 * q^2 + 16837 * q^4 + 62500 * q^5 + 343040 * q^7 - 14865 * q^8 - 234375 * q^10 + 12697800 * q^11 + 34336040 * q^13 + 26944650 * q^14 + 66562801 * q^16 + 84377280 * q^17 - 131821144 * q^19 + 263078125 * q^20 - 1016682450 * q^22 + 1046106360 * q^23 + 976562500 * q^25 - 2998448100 * q^26 - 2229409030 * q^28 - 1826812200 * q^29 + 3501222872 * q^31 - 1278404505 * q^32 + 2712309690 * q^34 + 5360000000 * q^35 + 8297642120 * q^37 + 50666879340 * q^38 - 232265625 * q^40 + 54459122400 * q^41 - 48173425360 * q^43 + 141205041150 * q^44 - 131025090720 * q^46 + 202496682840 * q^47 + 116605824372 * q^49 - 3662109375 * q^50 + 183603958220 * q^52 + 116147122920 * q^53 + 198403125000 * q^55 + 985319673150 * q^56 - 209673904950 * q^58 + 885100702200 * q^59 + 467836185368 * q^61 + 1845870142380 * q^62 - 1316914666607 * q^64 + 536500625000 * q^65 + 819037749440 * q^67 + 4568941043490 * q^68 + 421010156250 * q^70 + 2886468559200 * q^71 - 1235535876280 * q^73 + 5480818576200 * q^74 - 1007297478532 * q^76 + 3590015832000 * q^77 - 3553486866616 * q^79 + 1040043765625 * q^80 + 9494817638400 * q^82 + 3441261762480 * q^83 + 1318395000000 * q^85 + 1418346623400 * q^86 + 3182744059050 * q^88 - 2322663606000 * q^89 - 4966358282000 * q^91 - 3469668771120 * q^92 - 19865804246880 * q^94 - 2059705375000 * q^95 + 8462989649240 * q^97 - 17650752985395 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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