LMFDB - Embedded newform 825.4.c.r.199.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 83x^{8} + 2275x^{6} + 24517x^{4} + 87636x^{2} + 40000 $ x^10 + 83x^8 + 2275x^6 + 24517x^4 + 87636x^2 + 40000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.6
Root			$-0.729174i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.r.199.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.270826i q^{2} -3.00000i q^{3} +7.92665 q^{4} +0.812479 q^{6} -33.4133i q^{7} +4.31336i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q+0.270826i q^{2} -3.00000i q^{3} +7.92665 q^{4} +0.812479 q^{6} -33.4133i q^{7} +4.31336i q^{8} -9.00000 q^{9} +11.0000 q^{11} -23.7800i q^{12} +33.5174i q^{13} +9.04920 q^{14} +62.2451 q^{16} -71.1540i q^{17} -2.43744i q^{18} +48.9667 q^{19} -100.240 q^{21} +2.97909i q^{22} -66.7490i q^{23} +12.9401 q^{24} -9.07738 q^{26} +27.0000i q^{27} -264.856i q^{28} +66.8864 q^{29} +145.759 q^{31} +51.3644i q^{32} -33.0000i q^{33} +19.2704 q^{34} -71.3399 q^{36} -37.8491i q^{37} +13.2615i q^{38} +100.552 q^{39} -344.036 q^{41} -27.1476i q^{42} -34.4830i q^{43} +87.1932 q^{44} +18.0774 q^{46} -270.887i q^{47} -186.735i q^{48} -773.448 q^{49} -213.462 q^{51} +265.681i q^{52} -666.088i q^{53} -7.31231 q^{54} +144.123 q^{56} -146.900i q^{57} +18.1146i q^{58} -876.971 q^{59} +783.523 q^{61} +39.4753i q^{62} +300.720i q^{63} +484.050 q^{64} +8.93727 q^{66} +876.724i q^{67} -564.013i q^{68} -200.247 q^{69} -523.428 q^{71} -38.8202i q^{72} -91.0973i q^{73} +10.2505 q^{74} +388.142 q^{76} -367.546i q^{77} +27.2322i q^{78} +96.9287 q^{79} +81.0000 q^{81} -93.1741i q^{82} +1395.68i q^{83} -794.567 q^{84} +9.33889 q^{86} -200.659i q^{87} +47.4469i q^{88} -508.117 q^{89} +1119.93 q^{91} -529.096i q^{92} -437.276i q^{93} +73.3634 q^{94} +154.093 q^{96} -644.082i q^{97} -209.470i q^{98} -99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 92 q^{4} + 24 q^{6} - 90 q^{9}+O(q^{10}) $ 10 * q - 92 * q^4 + 24 * q^6 - 90 * q^9	$ 10 q - 92 q^{4} + 24 q^{6} - 90 q^{9} + 110 q^{11} - 110 q^{14} + 1268 q^{16} + 674 q^{19} - 228 q^{21} - 432 q^{24} + 718 q^{26} + 306 q^{29} + 526 q^{31} - 1034 q^{34} + 828 q^{36} - 258 q^{39} - 176 q^{41} - 1012 q^{44} + 872 q^{46} - 4762 q^{49} + 900 q^{51} - 216 q^{54} + 422 q^{56} - 820 q^{59} - 2260 q^{61} - 6340 q^{64} + 264 q^{66} + 1206 q^{69} + 2498 q^{71} + 5970 q^{74} - 5112 q^{76} - 4516 q^{79} + 810 q^{81} + 2646 q^{84} - 2870 q^{86} - 694 q^{89} - 1224 q^{91} - 1814 q^{94} + 1680 q^{96} - 990 q^{99}+O(q^{100}) $ 10 * q - 92 * q^4 + 24 * q^6 - 90 * q^9 + 110 * q^11 - 110 * q^14 + 1268 * q^16 + 674 * q^19 - 228 * q^21 - 432 * q^24 + 718 * q^26 + 306 * q^29 + 526 * q^31 - 1034 * q^34 + 828 * q^36 - 258 * q^39 - 176 * q^41 - 1012 * q^44 + 872 * q^46 - 4762 * q^49 + 900 * q^51 - 216 * q^54 + 422 * q^56 - 820 * q^59 - 2260 * q^61 - 6340 * q^64 + 264 * q^66 + 1206 * q^69 + 2498 * q^71 + 5970 * q^74 - 5112 * q^76 - 4516 * q^79 + 810 * q^81 + 2646 * q^84 - 2870 * q^86 - 694 * q^89 - 1224 * q^91 - 1814 * q^94 + 1680 * q^96 - 990 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	0.270826i	0.0957515i	0.998853	+	0.0478758i	$0.0152452\pi$
$2$	0.270826i	0.0957515i	−0.998853	+	0.0478758i	$0.984755\pi$
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	7.92665	0.990832
$5$	0	0
$5$	0	0
$6$	0.812479	0.0552822
$7$	− 33.4133i	− 1.80415i	−0.431581	−	0.902074i	$-0.642044\pi$
$7$	− 33.4133i	− 1.80415i	0.431581	−	0.902074i	$-0.357956\pi$
$8$	4.31336i	0.190625i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	− 23.7800i	− 0.572057i
$13$	33.5174i	0.715081i	0.933898	+	0.357540i	$0.116385\pi$
$13$	33.5174i	0.715081i	−0.933898	+	0.357540i	$0.883615\pi$
$14$	9.04920	0.172750
$15$	0	0
$16$	62.2451	0.972579
$17$	− 71.1540i	− 1.01514i	−0.861610	−	0.507570i	$-0.830544\pi$
$17$	− 71.1540i	− 1.01514i	0.861610	−	0.507570i	$-0.169456\pi$
$18$	− 2.43744i	− 0.0319172i
$19$	48.9667	0.591249	0.295625	−	0.955304i	$-0.404472\pi$
$19$	48.9667	0.591249	0.295625	+	0.955304i	$0.404472\pi$
$20$	0	0
$21$	−100.240	−1.04163
$22$	2.97909i	0.0288702i
$23$	− 66.7490i	− 0.605136i	−0.953128	−	0.302568i	$-0.902156\pi$
$23$	− 66.7490i	− 0.605136i	0.953128	−	0.302568i	$-0.0978439\pi$
$24$	12.9401	0.110058
$25$	0	0
$26$	−9.07738	−0.0684701
$27$	27.0000i	0.192450i
$28$	− 264.856i	− 1.78761i
$29$	66.8864	0.428293	0.214146	−	0.976802i	$-0.431303\pi$
$29$	66.8864	0.428293	0.214146	+	0.976802i	$0.431303\pi$
$30$	0	0
$31$	145.759	0.844486	0.422243	−	0.906483i	$-0.361243\pi$
$31$	145.759	0.844486	0.422243	+	0.906483i	$0.361243\pi$
$32$	51.3644i	0.283751i
$33$	− 33.0000i	− 0.174078i
$34$	19.2704	0.0972013
$35$	0	0
$36$	−71.3399	−0.330277
$37$	− 37.8491i	− 0.168172i	−0.996459	−	0.0840859i	$-0.973203\pi$
$37$	− 37.8491i	− 0.168172i	0.996459	−	0.0840859i	$-0.0267970\pi$
$38$	13.2615i	0.0566130i
$39$	100.552	0.412852
$40$	0	0
$41$	−344.036	−1.31047	−0.655237	−	0.755423i	$-0.727431\pi$
$41$	−344.036	−1.31047	−0.655237	+	0.755423i	$0.727431\pi$
$42$	− 27.1476i	− 0.0997373i
$43$	− 34.4830i	− 0.122293i	−0.998129	−	0.0611465i	$-0.980524\pi$
$43$	− 34.4830i	− 0.122293i	0.998129	−	0.0611465i	$-0.0194757\pi$
$44$	87.1932	0.298747
$45$	0	0
$46$	18.0774	0.0579427
$47$	− 270.887i	− 0.840701i	−0.907362	−	0.420351i	$-0.861907\pi$
$47$	− 270.887i	− 0.840701i	0.907362	−	0.420351i	$-0.138093\pi$
$48$	− 186.735i	− 0.561519i
$49$	−773.448	−2.25495
$50$	0	0
$51$	−213.462	−0.586092
$52$	265.681i	0.708524i
$53$	− 666.088i	− 1.72631i	−0.504941	−	0.863154i	$-0.668486\pi$
$53$	− 666.088i	− 1.72631i	0.504941	−	0.863154i	$-0.331514\pi$
$54$	−7.31231	−0.0184274
$55$	0	0
$56$	144.123	0.343916
$57$	− 146.900i	− 0.341358i
$58$	18.1146i	0.0410097i
$59$	−876.971	−1.93512	−0.967559	−	0.252645i	$-0.918699\pi$
$59$	−876.971	−1.93512	−0.967559	+	0.252645i	$0.918699\pi$
$60$	0	0
$61$	783.523	1.64459	0.822294	−	0.569063i	$-0.192694\pi$
$61$	783.523	1.64459	0.822294	+	0.569063i	$0.192694\pi$
$62$	39.4753i	0.0808608i
$63$	300.720i	0.601383i
$64$	484.050	0.945409
$65$	0	0
$66$	8.93727	0.0166682
$67$	876.724i	1.59864i	0.600906	+	0.799320i	$0.294807\pi$
$67$	876.724i	1.59864i	−0.600906	+	0.799320i	$0.705193\pi$
$68$	− 564.013i	− 1.00583i
$69$	−200.247	−0.349375
$70$	0	0
$71$	−523.428	−0.874922	−0.437461	−	0.899237i	$-0.644122\pi$
$71$	−523.428	−0.874922	−0.437461	+	0.899237i	$0.644122\pi$
$72$	− 38.8202i	− 0.0635417i
$73$	− 91.0973i	− 0.146057i	−0.997330	−	0.0730283i	$-0.976734\pi$
$73$	− 91.0973i	− 0.146057i	0.997330	−	0.0730283i	$-0.0232663\pi$
$74$	10.2505	0.0161027
$75$	0	0
$76$	388.142	0.585829
$77$	− 367.546i	− 0.543971i
$78$	27.2322i	0.0395312i
$79$	96.9287	0.138042	0.0690211	−	0.997615i	$-0.478012\pi$
$79$	96.9287	0.138042	0.0690211	+	0.997615i	$0.478012\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 93.1741i	− 0.125480i
$83$	1395.68i	1.84574i	0.385116	+	0.922868i	$0.374161\pi$
$83$	1395.68i	1.84574i	−0.385116	+	0.922868i	$0.625839\pi$
$84$	−794.567	−1.03208
$85$	0	0
$86$	9.33889	0.0117097
$87$	− 200.659i	− 0.247275i
$88$	47.4469i	0.0574757i
$89$	−508.117	−0.605172	−0.302586	−	0.953122i	$-0.597850\pi$
$89$	−508.117	−0.605172	−0.302586	+	0.953122i	$0.597850\pi$
$90$	0	0
$91$	1119.93	1.29011
$92$	− 529.096i	− 0.599588i
$93$	− 437.276i	− 0.487564i
$94$	73.3634	0.0804984
$95$	0	0
$96$	154.093	0.163824
$97$	− 644.082i	− 0.674192i	−0.941470	−	0.337096i	$-0.890555\pi$
$97$	− 644.082i	− 0.674192i	0.941470	−	0.337096i	$-0.109445\pi$
$98$	− 209.470i	− 0.215915i
$99$	−99.0000	−0.100504
$100$	0	0
$101$	1485.24	1.46324	0.731618	−	0.681715i	$-0.238766\pi$
$101$	1485.24	1.46324	0.731618	+	0.681715i	$0.238766\pi$
$102$	− 57.8111i	− 0.0561192i
$103$	− 605.952i	− 0.579672i	−0.957076	−	0.289836i	$-0.906399\pi$
$103$	− 605.952i	− 0.579672i	0.957076	−	0.289836i	$-0.0936008\pi$
$104$	−144.572	−0.136312
$105$	0	0
$106$	180.394	0.165297
$107$	− 352.908i	− 0.318850i	−0.987210	−	0.159425i	$-0.949036\pi$
$107$	− 352.908i	− 0.318850i	0.987210	−	0.159425i	$-0.0509640\pi$
$108$	214.020i	0.190686i
$109$	−1413.29	−1.24191	−0.620957	−	0.783844i	$-0.713256\pi$
$109$	−1413.29	−1.24191	−0.620957	+	0.783844i	$0.713256\pi$
$110$	0	0
$111$	−113.547	−0.0970940
$112$	− 2079.81i	− 1.75468i
$113$	− 103.753i	− 0.0863741i	−0.999067	−	0.0431870i	$-0.986249\pi$
$113$	− 103.753i	− 0.0863741i	0.999067	−	0.0431870i	$-0.0137511\pi$
$114$	39.7844	0.0326856
$115$	0	0
$116$	530.185	0.424366
$117$	− 301.656i	− 0.238360i
$118$	− 237.507i	− 0.185291i
$119$	−2377.49	−1.83146
$120$	0	0
$121$	121.000	0.0909091
$122$	212.199i	0.157472i
$123$	1032.11i	0.756603i
$124$	1155.38	0.836743
$125$	0	0
$126$	−81.4428	−0.0575833
$127$	− 1576.66i	− 1.10162i	−0.834631	−	0.550809i	$-0.814319\pi$
$127$	− 1576.66i	− 1.10162i	0.834631	−	0.550809i	$-0.185681\pi$
$128$	542.009i	0.374276i
$129$	−103.449	−0.0706059
$130$	0	0
$131$	−1085.16	−0.723747	−0.361873	−	0.932227i	$-0.617863\pi$
$131$	−1085.16	−0.723747	−0.361873	+	0.932227i	$0.617863\pi$
$132$	− 261.580i	− 0.172482i
$133$	− 1636.14i	− 1.06670i
$134$	−237.440	−0.153072
$135$	0	0
$136$	306.913	0.193511
$137$	1389.46i	0.866496i	0.901275	+	0.433248i	$0.142633\pi$
$137$	1389.46i	0.866496i	−0.901275	+	0.433248i	$0.857367\pi$
$138$	− 54.2321i	− 0.0334532i
$139$	−500.066	−0.305144	−0.152572	−	0.988292i	$-0.548756\pi$
$139$	−500.066	−0.305144	−0.152572	+	0.988292i	$0.548756\pi$
$140$	0	0
$141$	−812.662	−0.485379
$142$	− 141.758i	− 0.0837751i
$143$	368.691i	0.215605i
$144$	−560.205	−0.324193
$145$	0	0
$146$	24.6715	0.0139851
$147$	2320.35i	1.30190i
$148$	− 300.017i	− 0.166630i
$149$	−685.689	−0.377005	−0.188503	−	0.982073i	$-0.560363\pi$
$149$	−685.689	−0.377005	−0.188503	+	0.982073i	$0.560363\pi$
$150$	0	0
$151$	−1208.96	−0.651550	−0.325775	−	0.945447i	$-0.605625\pi$
$151$	−1208.96	−0.651550	−0.325775	+	0.945447i	$0.605625\pi$
$152$	211.211i	0.112707i
$153$	640.386i	0.338380i
$154$	99.5412	0.0520861
$155$	0	0
$156$	797.042	0.409067
$157$	− 3322.47i	− 1.68893i	−0.535610	−	0.844465i	$-0.679918\pi$
$157$	− 3322.47i	− 1.68893i	0.535610	−	0.844465i	$-0.320082\pi$
$158$	26.2509i	0.0132178i
$159$	−1998.27	−0.996684
$160$	0	0
$161$	−2230.30	−1.09176
$162$	21.9369i	0.0106391i
$163$	− 2448.31i	− 1.17648i	−0.808686	−	0.588240i	$-0.799821\pi$
$163$	− 2448.31i	− 1.17648i	0.808686	−	0.588240i	$-0.200179\pi$
$164$	−2727.06	−1.29846
$165$	0	0
$166$	−377.988	−0.176732
$167$	− 1518.71i	− 0.703720i	−0.936053	−	0.351860i	$-0.885549\pi$
$167$	− 1518.71i	− 0.703720i	0.936053	−	0.351860i	$-0.114451\pi$
$168$	− 432.370i	− 0.198560i
$169$	1073.59	0.488660
$170$	0	0
$171$	−440.701	−0.197083
$172$	− 273.334i	− 0.121172i
$173$	− 2636.80i	− 1.15880i	−0.815044	−	0.579399i	$-0.803287\pi$
$173$	− 2636.80i	− 1.15880i	0.815044	−	0.579399i	$-0.196713\pi$
$174$	54.3438	0.0236770
$175$	0	0
$176$	684.696	0.293244
$177$	2630.91i	1.11724i
$178$	− 137.611i	− 0.0579461i
$179$	1773.31	0.740466	0.370233	−	0.928939i	$-0.379278\pi$
$179$	1773.31	0.740466	0.370233	+	0.928939i	$0.379278\pi$
$180$	0	0
$181$	529.751	0.217547	0.108774	−	0.994067i	$-0.465308\pi$
$181$	529.751	0.217547	0.108774	+	0.994067i	$0.465308\pi$
$182$	303.305i	0.123530i
$183$	− 2350.57i	− 0.949503i
$184$	287.912	0.115354
$185$	0	0
$186$	118.426	0.0466850
$187$	− 782.694i	− 0.306076i
$188$	− 2147.23i	− 0.832993i
$189$	902.159	0.347209
$190$	0	0
$191$	2061.93	0.781130	0.390565	−	0.920575i	$-0.372280\pi$
$191$	2061.93	0.781130	0.390565	+	0.920575i	$0.372280\pi$
$192$	− 1452.15i	− 0.545832i
$193$	1637.24i	0.610627i	0.952252	+	0.305314i	$0.0987613\pi$
$193$	1637.24i	0.610627i	−0.952252	+	0.305314i	$0.901239\pi$
$194$	174.434	0.0645550
$195$	0	0
$196$	−6130.86	−2.23428
$197$	3414.86i	1.23502i	0.786563	+	0.617510i	$0.211858\pi$
$197$	3414.86i	1.23502i	−0.786563	+	0.617510i	$0.788142\pi$
$198$	− 26.8118i	− 0.00962339i
$199$	−3938.93	−1.40313	−0.701566	−	0.712605i	$-0.747515\pi$
$199$	−3938.93	−1.40313	−0.701566	+	0.712605i	$0.747515\pi$
$200$	0	0
$201$	2630.17	0.922975
$202$	402.242i	0.140107i
$203$	− 2234.90i	− 0.772704i
$204$	−1692.04	−0.580718
$205$	0	0
$206$	164.108	0.0555045
$207$	600.741i	0.201712i
$208$	2086.29i	0.695472i
$209$	538.634	0.178268
$210$	0	0
$211$	3543.14	1.15602	0.578009	−	0.816031i	$-0.303830\pi$
$211$	3543.14	1.15602	0.578009	+	0.816031i	$0.303830\pi$
$212$	− 5279.85i	− 1.71048i
$213$	1570.28i	0.505136i
$214$	95.5768	0.0305304
$215$	0	0
$216$	−116.461	−0.0366858
$217$	− 4870.28i	− 1.52358i
$218$	− 382.756i	− 0.118915i
$219$	−273.292	−0.0843258
$220$	0	0
$221$	2384.90	0.725907
$222$	− 30.7516i	− 0.00929690i
$223$	850.930i	0.255527i	0.991805	+	0.127763i	$0.0407798\pi$
$223$	850.930i	0.255527i	−0.991805	+	0.127763i	$0.959220\pi$
$224$	1716.26	0.511929
$225$	0	0
$226$	28.0991	0.00827045
$227$	5040.56i	1.47381i	0.675999	+	0.736903i	$0.263712\pi$
$227$	5040.56i	1.47381i	−0.675999	+	0.736903i	$0.736288\pi$
$228$	− 1164.43i	− 0.338228i
$229$	3624.34	1.04587	0.522933	−	0.852374i	$-0.324838\pi$
$229$	3624.34	1.04587	0.522933	+	0.852374i	$0.324838\pi$
$230$	0	0
$231$	−1102.64	−0.314062
$232$	288.505i	0.0816434i
$233$	2984.56i	0.839164i	0.907718	+	0.419582i	$0.137823\pi$
$233$	2984.56i	0.839164i	−0.907718	+	0.419582i	$0.862177\pi$
$234$	81.6965	0.0228234
$235$	0	0
$236$	−6951.45	−1.91738
$237$	− 290.786i	− 0.0796987i
$238$	− 643.887i	− 0.175366i
$239$	−1298.11	−0.351331	−0.175665	−	0.984450i	$-0.556208\pi$
$239$	−1298.11	−0.351331	−0.175665	+	0.984450i	$0.556208\pi$
$240$	0	0
$241$	3978.71	1.06345	0.531725	−	0.846917i	$-0.321544\pi$
$241$	3978.71	1.06345	0.531725	+	0.846917i	$0.321544\pi$
$242$	32.7700i	0.00870469i
$243$	− 243.000i	− 0.0641500i
$244$	6210.71	1.62951
$245$	0	0
$246$	−279.522	−0.0724459
$247$	1641.24i	0.422791i
$248$	628.710i	0.160980i
$249$	4187.05	1.06564
$250$	0	0
$251$	−1884.05	−0.473784	−0.236892	−	0.971536i	$-0.576129\pi$
$251$	−1884.05	−0.473784	−0.236892	+	0.971536i	$0.576129\pi$
$252$	2383.70i	0.595869i
$253$	− 734.239i	− 0.182455i
$254$	427.000	0.105482
$255$	0	0
$256$	3725.61	0.909572
$257$	− 4012.23i	− 0.973836i	−0.873448	−	0.486918i	$-0.838121\pi$
$257$	− 4012.23i	− 0.973836i	0.873448	−	0.486918i	$-0.161879\pi$
$258$	− 28.0167i	− 0.00676063i
$259$	−1264.66	−0.303407
$260$	0	0
$261$	−601.978	−0.142764
$262$	− 293.890i	− 0.0692999i
$263$	− 290.429i	− 0.0680935i	−0.999420	−	0.0340468i	$-0.989160\pi$
$263$	− 290.429i	− 0.0680935i	0.999420	−	0.0340468i	$-0.0108395\pi$
$264$	142.341	0.0331836
$265$	0	0
$266$	443.110	0.102138
$267$	1524.35i	0.349396i
$268$	6949.48i	1.58398i
$269$	6743.15	1.52839	0.764196	−	0.644985i	$-0.223136\pi$
$269$	6743.15	1.52839	0.764196	+	0.644985i	$0.223136\pi$
$270$	0	0
$271$	7182.11	1.60990	0.804949	−	0.593344i	$-0.202193\pi$
$271$	7182.11	1.60990	0.804949	+	0.593344i	$0.202193\pi$
$272$	− 4428.99i	− 0.987304i
$273$	− 3359.78i	− 0.744846i
$274$	−376.304	−0.0829683
$275$	0	0
$276$	−1587.29	−0.346172
$277$	− 5708.35i	− 1.23820i	−0.785312	−	0.619100i	$-0.787498\pi$
$277$	− 5708.35i	− 1.23820i	0.785312	−	0.619100i	$-0.212502\pi$
$278$	− 135.431i	− 0.0292180i
$279$	−1311.83	−0.281495
$280$	0	0
$281$	8740.94	1.85566	0.927831	−	0.373002i	$-0.121672\pi$
$281$	8740.94	1.85566	0.927831	+	0.373002i	$0.121672\pi$
$282$	− 220.090i	− 0.0464758i
$283$	7883.57i	1.65593i	0.560777	+	0.827967i	$0.310503\pi$
$283$	7883.57i	1.65593i	−0.560777	+	0.827967i	$0.689497\pi$
$284$	−4149.03	−0.866900
$285$	0	0
$286$	−99.8512	−0.0206445
$287$	11495.4i	2.36429i
$288$	− 462.280i	− 0.0945837i
$289$	−149.897	−0.0305102
$290$	0	0
$291$	−1932.25	−0.389245
$292$	− 722.097i	− 0.144718i
$293$	− 2328.54i	− 0.464282i	−0.972682	−	0.232141i	$-0.925427\pi$
$293$	− 2328.54i	− 0.464282i	0.972682	−	0.232141i	$-0.0745731\pi$
$294$	−628.410	−0.124659
$295$	0	0
$296$	163.257	0.0320578
$297$	297.000i	0.0580259i
$298$	− 185.703i	− 0.0360989i
$299$	2237.25	0.432721
$300$	0	0
$301$	−1152.19	−0.220635
$302$	− 327.419i	− 0.0623869i
$303$	− 4455.72i	− 0.844799i
$304$	3047.94	0.575037
$305$	0	0
$306$	−173.433	−0.0324004
$307$	2830.31i	0.526170i	0.964773	+	0.263085i	$0.0847399\pi$
$307$	2830.31i	0.526170i	−0.964773	+	0.263085i	$0.915260\pi$
$308$	− 2913.41i	− 0.538984i
$309$	−1817.86	−0.334674
$310$	0	0
$311$	−6580.67	−1.19986	−0.599929	−	0.800053i	$-0.704804\pi$
$311$	−6580.67	−1.19986	−0.599929	+	0.800053i	$0.704804\pi$
$312$	433.717i	0.0787000i
$313$	7837.96i	1.41542i	0.706501	+	0.707712i	$0.250272\pi$
$313$	7837.96i	1.41542i	−0.706501	+	0.707712i	$0.749728\pi$
$314$	899.813	0.161718
$315$	0	0
$316$	768.321	0.136777
$317$	5853.61i	1.03713i	0.855037	+	0.518567i	$0.173534\pi$
$317$	5853.61i	1.03713i	−0.855037	+	0.518567i	$0.826466\pi$
$318$	− 541.183i	− 0.0954340i
$319$	735.751	0.129135
$320$	0	0
$321$	−1058.72	−0.184088
$322$	− 604.025i	− 0.104537i
$323$	− 3484.18i	− 0.600201i
$324$	642.059	0.110092
$325$	0	0
$326$	663.066	0.112650
$327$	4239.87i	0.717020i
$328$	− 1483.95i	− 0.249809i
$329$	−9051.23	−1.51675
$330$	0	0
$331$	5347.61	0.888010	0.444005	−	0.896024i	$-0.353557\pi$
$331$	5347.61	0.888010	0.444005	+	0.896024i	$0.353557\pi$
$332$	11063.1i	1.82881i
$333$	340.642i	0.0560573i
$334$	411.306	0.0673823
$335$	0	0
$336$	−6239.44	−1.01306
$337$	2763.99i	0.446779i	0.974729	+	0.223389i	$0.0717121\pi$
$337$	2763.99i	0.446779i	−0.974729	+	0.223389i	$0.928288\pi$
$338$	290.755i	0.0467899i
$339$	−311.259	−0.0498681
$340$	0	0
$341$	1603.35	0.254622
$342$	− 119.353i	− 0.0188710i
$343$	14382.7i	2.26412i
$344$	148.737	0.0233121
$345$	0	0
$346$	714.114	0.110957
$347$	5038.66i	0.779509i	0.920919	+	0.389754i	$0.127440\pi$
$347$	5038.66i	0.779509i	−0.920919	+	0.389754i	$0.872560\pi$
$348$	− 1590.56i	− 0.245008i
$349$	7429.49	1.13952	0.569759	−	0.821812i	$-0.307037\pi$
$349$	7429.49	1.13952	0.569759	+	0.821812i	$0.307037\pi$
$350$	0	0
$351$	−904.969	−0.137617
$352$	565.009i	0.0855542i
$353$	3151.84i	0.475228i	0.971360	+	0.237614i	$0.0763653\pi$
$353$	3151.84i	0.475228i	−0.971360	+	0.237614i	$0.923635\pi$
$354$	−712.521	−0.106978
$355$	0	0
$356$	−4027.67	−0.599623
$357$	7132.47i	1.05740i
$358$	480.259i	0.0709008i
$359$	−903.954	−0.132894	−0.0664469	−	0.997790i	$-0.521166\pi$
$359$	−903.954	−0.132894	−0.0664469	+	0.997790i	$0.521166\pi$
$360$	0	0
$361$	−4461.26	−0.650424
$362$	143.470i	0.0208305i
$363$	− 363.000i	− 0.0524864i
$364$	8877.26	1.27828
$365$	0	0
$366$	636.596	0.0909164
$367$	3978.31i	0.565848i	0.959142	+	0.282924i	$0.0913044\pi$
$367$	3978.31i	0.565848i	−0.959142	+	0.282924i	$0.908696\pi$
$368$	− 4154.79i	− 0.588543i
$369$	3096.33	0.436825
$370$	0	0
$371$	−22256.2	−3.11451
$372$	− 3466.14i	− 0.483094i
$373$	5493.67i	0.762605i	0.924450	+	0.381302i	$0.124524\pi$
$373$	5493.67i	0.762605i	−0.924450	+	0.381302i	$0.875476\pi$
$374$	211.974	0.0293073
$375$	0	0
$376$	1168.43	0.160259
$377$	2241.86i	0.306264i
$378$	244.328i	0.0332458i
$379$	−3035.93	−0.411465	−0.205733	−	0.978608i	$-0.565958\pi$
$379$	−3035.93	−0.411465	−0.205733	+	0.978608i	$0.565958\pi$
$380$	0	0
$381$	−4729.97	−0.636020
$382$	558.424i	0.0747944i
$383$	3556.77i	0.474524i	0.971446	+	0.237262i	$0.0762500\pi$
$383$	3556.77i	0.474524i	−0.971446	+	0.237262i	$0.923750\pi$
$384$	1626.03	0.216088
$385$	0	0
$386$	−443.408	−0.0584685
$387$	310.347i	0.0407644i
$388$	− 5105.42i	− 0.668011i
$389$	5211.56	0.679272	0.339636	−	0.940557i	$-0.389696\pi$
$389$	5211.56	0.679272	0.339636	+	0.940557i	$0.389696\pi$
$390$	0	0
$391$	−4749.46	−0.614298
$392$	− 3336.16i	− 0.429851i
$393$	3255.48i	0.417855i
$394$	−924.835	−0.118255
$395$	0	0
$396$	−784.739	−0.0995823
$397$	− 6547.28i	− 0.827704i	−0.910344	−	0.413852i	$-0.864183\pi$
$397$	− 6547.28i	− 0.827704i	0.910344	−	0.413852i	$-0.135817\pi$
$398$	− 1066.76i	− 0.134352i
$399$	−4908.42	−0.615860
$400$	0	0
$401$	−1661.59	−0.206922	−0.103461	−	0.994634i	$-0.532992\pi$
$401$	−1661.59	−0.206922	−0.103461	+	0.994634i	$0.532992\pi$
$402$	712.319i	0.0883763i
$403$	4885.45i	0.603875i
$404$	11773.0	1.44982
$405$	0	0
$406$	605.268	0.0739876
$407$	− 416.340i	− 0.0507057i
$408$	− 920.738i	− 0.111724i
$409$	−7278.96	−0.880004	−0.440002	−	0.897997i	$-0.645022\pi$
$409$	−7278.96	−0.880004	−0.440002	+	0.897997i	$0.645022\pi$
$410$	0	0
$411$	4168.39	0.500272
$412$	− 4803.17i	− 0.574357i
$413$	29302.5i	3.49124i
$414$	−162.696	−0.0193142
$415$	0	0
$416$	−1721.60	−0.202905
$417$	1500.20i	0.176175i
$418$	145.876i	0.0170695i
$419$	5892.15	0.686994	0.343497	−	0.939154i	$-0.388389\pi$
$419$	5892.15	0.686994	0.343497	+	0.939154i	$0.388389\pi$
$420$	0	0
$421$	11086.3	1.28340	0.641700	−	0.766955i	$-0.278229\pi$
$421$	11086.3	1.28340	0.641700	+	0.766955i	$0.278229\pi$
$422$	959.575i	0.110690i
$423$	2437.98i	0.280234i
$424$	2873.08	0.329078
$425$	0	0
$426$	−425.274	−0.0483676
$427$	− 26180.1i	− 2.96708i
$428$	− 2797.38i	− 0.315926i
$429$	1106.07	0.124480
$430$	0	0
$431$	5658.06	0.632341	0.316170	−	0.948702i	$-0.397603\pi$
$431$	5658.06	0.632341	0.316170	+	0.948702i	$0.397603\pi$
$432$	1680.62i	0.187173i
$433$	− 3065.03i	− 0.340176i	−0.985429	−	0.170088i	$-0.945595\pi$
$433$	− 3065.03i	− 0.340176i	0.985429	−	0.170088i	$-0.0544052\pi$
$434$	1319.00	0.145885
$435$	0	0
$436$	−11202.7	−1.23053
$437$	− 3268.48i	− 0.357786i
$438$	− 74.0146i	− 0.00807433i
$439$	3017.50	0.328058	0.164029	−	0.986456i	$-0.447551\pi$
$439$	3017.50	0.328058	0.164029	+	0.986456i	$0.447551\pi$
$440$	0	0
$441$	6961.04	0.751650
$442$	645.893i	0.0695067i
$443$	− 17098.3i	− 1.83378i	−0.399144	−	0.916888i	$-0.630693\pi$
$443$	− 17098.3i	− 1.83378i	0.399144	−	0.916888i	$-0.369307\pi$
$444$	−900.050	−0.0962038
$445$	0	0
$446$	−230.454	−0.0244671
$447$	2057.07i	0.217664i
$448$	− 16173.7i	− 1.70566i
$449$	10823.4	1.13762	0.568808	−	0.822470i	$-0.307405\pi$
$449$	10823.4	1.13762	0.568808	+	0.822470i	$0.307405\pi$
$450$	0	0
$451$	−3784.40	−0.395123
$452$	− 822.415i	− 0.0855821i
$453$	3626.89i	0.376172i
$454$	−1365.12	−0.141119
$455$	0	0
$456$	633.633	0.0650714
$457$	− 6859.01i	− 0.702081i	−0.936360	−	0.351040i	$-0.885828\pi$
$457$	− 6859.01i	− 0.702081i	0.936360	−	0.351040i	$-0.114172\pi$
$458$	981.567i	0.100143i
$459$	1921.16	0.195364
$460$	0	0
$461$	9651.72	0.975109	0.487555	−	0.873093i	$-0.337889\pi$
$461$	9651.72	0.975109	0.487555	+	0.873093i	$0.337889\pi$
$462$	− 298.624i	− 0.0300719i
$463$	4744.49i	0.476232i	0.971237	+	0.238116i	$0.0765298\pi$
$463$	4744.49i	0.476232i	−0.971237	+	0.238116i	$0.923470\pi$
$464$	4163.35	0.416549
$465$	0	0
$466$	−808.298	−0.0803512
$467$	− 11038.1i	− 1.09376i	−0.837213	−	0.546878i	$-0.815816\pi$
$467$	− 11038.1i	− 1.09376i	0.837213	−	0.546878i	$-0.184184\pi$
$468$	− 2391.13i	− 0.236175i
$469$	29294.2	2.88418
$470$	0	0
$471$	−9967.42	−0.975105
$472$	− 3782.69i	− 0.368882i
$473$	− 379.313i	− 0.0368727i
$474$	78.7526	0.00763128
$475$	0	0
$476$	−18845.5	−1.81467
$477$	5994.80i	0.575436i
$478$	− 351.563i	− 0.0336404i
$479$	−17476.9	−1.66710	−0.833550	−	0.552444i	$-0.813695\pi$
$479$	−17476.9	−1.66710	−0.833550	+	0.552444i	$0.813695\pi$
$480$	0	0
$481$	1268.60	0.120256
$482$	1077.54i	0.101827i
$483$	6690.91i	0.630325i
$484$	959.125	0.0900756
$485$	0	0
$486$	65.8108	0.00614246
$487$	17014.6i	1.58317i	0.611058	+	0.791586i	$0.290744\pi$
$487$	17014.6i	1.58317i	−0.611058	+	0.791586i	$0.709256\pi$
$488$	3379.61i	0.313500i
$489$	−7344.92	−0.679241
$490$	0	0
$491$	19673.8	1.80828	0.904142	−	0.427233i	$-0.140512\pi$
$491$	19673.8	1.80828	0.904142	+	0.427233i	$0.140512\pi$
$492$	8181.17i	0.749666i
$493$	− 4759.24i	− 0.434778i
$494$	−444.490	−0.0404829
$495$	0	0
$496$	9072.76	0.821329
$497$	17489.4i	1.57849i
$498$	1133.96i	0.102036i
$499$	15754.6	1.41337	0.706687	−	0.707526i	$-0.250189\pi$
$499$	15754.6	1.41337	0.706687	+	0.707526i	$0.250189\pi$
$500$	0	0
$501$	−4556.13	−0.406293
$502$	− 510.249i	− 0.0453656i
$503$	16457.2i	1.45883i	0.684073	+	0.729414i	$0.260207\pi$
$503$	16457.2i	1.45883i	−0.684073	+	0.729414i	$0.739793\pi$
$504$	−1297.11	−0.114639
$505$	0	0
$506$	198.851	0.0174704
$507$	− 3220.76i	− 0.282128i
$508$	− 12497.6i	− 1.09152i
$509$	−13852.0	−1.20625	−0.603124	−	0.797647i	$-0.706078\pi$
$509$	−13852.0	−1.20625	−0.603124	+	0.797647i	$0.706078\pi$
$510$	0	0
$511$	−3043.86	−0.263508
$512$	5345.06i	0.461368i
$513$	1322.10i	0.113786i
$514$	1086.62	0.0932463
$515$	0	0
$516$	−820.003	−0.0699586
$517$	− 2979.76i	− 0.253481i
$518$	− 342.504i	− 0.0290517i
$519$	−7910.39	−0.669032
$520$	0	0
$521$	−12295.8	−1.03395	−0.516976	−	0.856000i	$-0.672942\pi$
$521$	−12295.8	−1.03395	−0.516976	+	0.856000i	$0.672942\pi$
$522$	− 163.031i	− 0.0136699i
$523$	− 1971.83i	− 0.164861i	−0.996597	−	0.0824305i	$-0.973732\pi$
$523$	− 1971.83i	− 0.164861i	0.996597	−	0.0824305i	$-0.0262682\pi$
$524$	−8601.68	−0.717111
$525$	0	0
$526$	78.6557	0.00652006
$527$	− 10371.3i	− 0.857271i
$528$	− 2054.09i	− 0.169304i
$529$	7711.57	0.633810
$530$	0	0
$531$	7892.74	0.645039
$532$	− 12969.1i	− 1.05692i
$533$	− 11531.2i	− 0.937095i
$534$	−412.834	−0.0334552
$535$	0	0
$536$	−3781.62	−0.304741
$537$	− 5319.93i	− 0.427508i
$538$	1826.22i	0.146346i
$539$	−8507.93	−0.679893
$540$	0	0
$541$	−8849.45	−0.703267	−0.351634	−	0.936138i	$-0.614374\pi$
$541$	−8849.45	−0.703267	−0.351634	+	0.936138i	$0.614374\pi$
$542$	1945.10i	0.154150i
$543$	− 1589.25i	− 0.125601i
$544$	3654.79	0.288047
$545$	0	0
$546$	909.916	0.0713202
$547$	10687.9i	0.835435i	0.908577	+	0.417717i	$0.137170\pi$
$547$	10687.9i	0.835435i	−0.908577	+	0.417717i	$0.862830\pi$
$548$	11013.8i	0.858552i
$549$	−7051.71	−0.548196
$550$	0	0
$551$	3275.21	0.253228
$552$	− 863.737i	− 0.0665998i
$553$	− 3238.71i	− 0.249049i
$554$	1545.97	0.118560
$555$	0	0
$556$	−3963.85	−0.302346
$557$	23864.2i	1.81537i	0.419655	+	0.907684i	$0.362151\pi$
$557$	23864.2i	1.81537i	−0.419655	+	0.907684i	$0.637849\pi$
$558$	− 355.278i	− 0.0269536i
$559$	1155.78	0.0874494
$560$	0	0
$561$	−2348.08	−0.176713
$562$	2367.28i	0.177682i
$563$	− 13610.5i	− 1.01886i	−0.860513	−	0.509428i	$-0.829857\pi$
$563$	− 13610.5i	− 1.01886i	0.860513	−	0.509428i	$-0.170143\pi$
$564$	−6441.69	−0.480929
$565$	0	0
$566$	−2135.08	−0.158558
$567$	− 2706.48i	− 0.200461i
$568$	− 2257.73i	− 0.166782i
$569$	−8564.56	−0.631011	−0.315505	−	0.948924i	$-0.602174\pi$
$569$	−8564.56	−0.631011	−0.315505	+	0.948924i	$0.602174\pi$
$570$	0	0
$571$	15329.9	1.12353	0.561764	−	0.827298i	$-0.310123\pi$
$571$	15329.9	1.12353	0.561764	+	0.827298i	$0.310123\pi$
$572$	2922.49i	0.213628i
$573$	− 6185.78i	− 0.450985i
$574$	−3113.25	−0.226384
$575$	0	0
$576$	−4356.45	−0.315136
$577$	− 9594.47i	− 0.692241i	−0.938190	−	0.346120i	$-0.887499\pi$
$577$	− 9594.47i	− 0.692241i	0.938190	−	0.346120i	$-0.112501\pi$
$578$	− 40.5959i	− 0.00292140i
$579$	4911.72	0.352546
$580$	0	0
$581$	46634.4	3.32998
$582$	− 523.303i	− 0.0372708i
$583$	− 7326.97i	− 0.520501i
$584$	392.935	0.0278421
$585$	0	0
$586$	630.629	0.0444557
$587$	16659.5i	1.17140i	0.810530	+	0.585698i	$0.199179\pi$
$587$	16659.5i	1.17140i	−0.810530	+	0.585698i	$0.800821\pi$
$588$	18392.6i	1.28996i
$589$	7137.33	0.499302
$590$	0	0
$591$	10244.6	0.713039
$592$	− 2355.92i	− 0.163560i
$593$	− 459.030i	− 0.0317877i	−0.999874	−	0.0158939i	$-0.994941\pi$
$593$	− 459.030i	− 0.0317877i	0.999874	−	0.0158939i	$-0.00505938\pi$
$594$	−80.4354	−0.00555607
$595$	0	0
$596$	−5435.22	−0.373549
$597$	11816.8i	0.810098i
$598$	605.906i	0.0414337i
$599$	−869.305	−0.0592969	−0.0296485	−	0.999560i	$-0.509439\pi$
$599$	−869.305	−0.0592969	−0.0296485	+	0.999560i	$0.509439\pi$
$600$	0	0
$601$	−5455.29	−0.370259	−0.185130	−	0.982714i	$-0.559271\pi$
$601$	−5455.29	−0.370259	−0.185130	+	0.982714i	$0.559271\pi$
$602$	− 312.043i	− 0.0211261i
$603$	− 7890.51i	− 0.532880i
$604$	−9583.03	−0.645576
$605$	0	0
$606$	1206.72	0.0808908
$607$	27152.9i	1.81565i	0.419347	+	0.907826i	$0.362259\pi$
$607$	27152.9i	1.81565i	−0.419347	+	0.907826i	$0.637741\pi$
$608$	2515.15i	0.167768i
$609$	−6704.69	−0.446121
$610$	0	0
$611$	9079.43	0.601169
$612$	5076.12i	0.335278i
$613$	10313.6i	0.679546i	0.940507	+	0.339773i	$0.110350\pi$
$613$	10313.6i	0.679546i	−0.940507	+	0.339773i	$0.889650\pi$
$614$	−766.521	−0.0503816
$615$	0	0
$616$	1585.36	0.103695
$617$	− 27148.4i	− 1.77140i	−0.464259	−	0.885699i	$-0.653679\pi$
$617$	− 27148.4i	− 1.77140i	0.464259	−	0.885699i	$-0.346321\pi$
$618$	− 492.323i	− 0.0320455i
$619$	−13110.4	−0.851294	−0.425647	−	0.904889i	$-0.639953\pi$
$619$	−13110.4	−0.851294	−0.425647	+	0.904889i	$0.639953\pi$
$620$	0	0
$621$	1802.22	0.116458
$622$	− 1782.22i	− 0.114888i
$623$	16977.9i	1.09182i
$624$	6258.87	0.401531
$625$	0	0
$626$	−2122.72	−0.135529
$627$	− 1615.90i	− 0.102923i
$628$	− 26336.1i	− 1.67345i
$629$	−2693.12	−0.170718
$630$	0	0
$631$	30199.9	1.90529	0.952647	−	0.304078i	$-0.0983482\pi$
$631$	30199.9	1.90529	0.952647	+	0.304078i	$0.0983482\pi$
$632$	418.088i	0.0263143i
$633$	− 10629.4i	− 0.667427i
$634$	−1585.31	−0.0993072
$635$	0	0
$636$	−15839.6	−0.987546
$637$	− 25924.0i	− 1.61247i
$638$	199.261i	0.0123649i
$639$	4710.85	0.291641
$640$	0	0
$641$	−8889.43	−0.547756	−0.273878	−	0.961764i	$-0.588306\pi$
$641$	−8889.43	−0.547756	−0.273878	+	0.961764i	$0.588306\pi$
$642$	− 286.730i	− 0.0176267i
$643$	16570.2i	1.01627i	0.861276	+	0.508137i	$0.169666\pi$
$643$	16570.2i	1.01627i	−0.861276	+	0.508137i	$0.830334\pi$
$644$	−17678.8	−1.08175
$645$	0	0
$646$	943.608	0.0574702
$647$	− 13356.3i	− 0.811576i	−0.913967	−	0.405788i	$-0.866997\pi$
$647$	− 13356.3i	− 0.811576i	0.913967	−	0.405788i	$-0.133003\pi$
$648$	349.382i	0.0211806i
$649$	−9646.69	−0.583460
$650$	0	0
$651$	−14610.8	−0.879638
$652$	− 19406.9i	− 1.16569i
$653$	− 32689.5i	− 1.95902i	−0.201394	−	0.979510i	$-0.564547\pi$
$653$	− 32689.5i	− 1.95902i	0.201394	−	0.979510i	$-0.435453\pi$
$654$	−1148.27	−0.0686558
$655$	0	0
$656$	−21414.6	−1.27454
$657$	819.876i	0.0486855i
$658$	− 2451.31i	− 0.145231i
$659$	−16123.3	−0.953071	−0.476536	−	0.879155i	$-0.658108\pi$
$659$	−16123.3	−0.953071	−0.476536	+	0.879155i	$0.658108\pi$
$660$	0	0
$661$	22340.9	1.31461	0.657306	−	0.753624i	$-0.271696\pi$
$661$	22340.9	1.31461	0.657306	+	0.753624i	$0.271696\pi$
$662$	1448.27i	0.0850283i
$663$	− 7154.69i	− 0.419103i
$664$	−6020.08	−0.351844
$665$	0	0
$666$	−92.2548	−0.00536757
$667$	− 4464.60i	− 0.259175i
$668$	− 12038.3i	− 0.697268i
$669$	2552.79	0.147528
$670$	0	0
$671$	8618.75	0.495862
$672$	− 5148.77i	− 0.295562i
$673$	3415.65i	0.195637i	0.995204	+	0.0978184i	$0.0311864\pi$
$673$	3415.65i	0.195637i	−0.995204	+	0.0978184i	$0.968814\pi$
$674$	−748.562	−0.0427797
$675$	0	0
$676$	8509.94	0.484180
$677$	32841.9i	1.86442i	0.361912	+	0.932212i	$0.382124\pi$
$677$	32841.9i	1.86442i	−0.361912	+	0.932212i	$0.617876\pi$
$678$	− 84.2972i	− 0.00477495i
$679$	−21520.9	−1.21634
$680$	0	0
$681$	15121.7	0.850902
$682$	434.228i	0.0243804i
$683$	7227.43i	0.404905i	0.979292	+	0.202452i	$0.0648911\pi$
$683$	7227.43i	0.404905i	−0.979292	+	0.202452i	$0.935109\pi$
$684$	−3493.28	−0.195276
$685$	0	0
$686$	−3895.21	−0.216793
$687$	− 10873.0i	− 0.603831i
$688$	− 2146.39i	− 0.118940i
$689$	22325.5	1.23445
$690$	0	0
$691$	15397.5	0.847682	0.423841	−	0.905737i	$-0.360681\pi$
$691$	15397.5	0.847682	0.423841	+	0.905737i	$0.360681\pi$
$692$	− 20901.0i	− 1.14817i
$693$	3307.92i	0.181324i
$694$	−1364.60	−0.0746392
$695$	0	0
$696$	865.515	0.0471369
$697$	24479.6i	1.33032i
$698$	2012.10i	0.109111i
$699$	8953.68	0.484491
$700$	0	0
$701$	−22130.1	−1.19236	−0.596180	−	0.802851i	$-0.703315\pi$
$701$	−22130.1	−1.19236	−0.596180	+	0.802851i	$0.703315\pi$
$702$	− 245.089i	− 0.0131771i
$703$	− 1853.35i	− 0.0994315i
$704$	5324.55	0.285052
$705$	0	0
$706$	−853.601	−0.0455038
$707$	− 49626.7i	− 2.63989i
$708$	20854.3i	1.10700i
$709$	−10062.6	−0.533018	−0.266509	−	0.963832i	$-0.585870\pi$
$709$	−10062.6	−0.533018	−0.266509	+	0.963832i	$0.585870\pi$
$710$	0	0
$711$	−872.359	−0.0460141
$712$	− 2191.69i	− 0.115361i
$713$	− 9729.25i	− 0.511029i
$714$	−1931.66	−0.101247
$715$	0	0
$716$	14056.4	0.733677
$717$	3894.34i	0.202841i
$718$	− 244.814i	− 0.0127248i
$719$	18629.7	0.966303	0.483151	−	0.875537i	$-0.339492\pi$
$719$	18629.7	0.966303	0.483151	+	0.875537i	$0.339492\pi$
$720$	0	0
$721$	−20246.9	−1.04581
$722$	− 1208.23i	− 0.0622791i
$723$	− 11936.1i	− 0.613983i
$724$	4199.15	0.215553
$725$	0	0
$726$	98.3099	0.00502565
$727$	− 21605.8i	− 1.10222i	−0.834431	−	0.551112i	$-0.814204\pi$
$727$	− 21605.8i	− 1.10222i	0.834431	−	0.551112i	$-0.185796\pi$
$728$	4830.64i	0.245928i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−2453.60	−0.124145
$732$	− 18632.1i	− 0.940798i
$733$	− 23668.1i	− 1.19264i	−0.802749	−	0.596318i	$-0.796630\pi$
$733$	− 23668.1i	− 1.19264i	0.802749	−	0.596318i	$-0.203370\pi$
$734$	−1077.43	−0.0541808
$735$	0	0
$736$	3428.52	0.171708
$737$	9643.96i	0.482008i
$738$	838.567i	0.0418267i
$739$	8475.82	0.421906	0.210953	−	0.977496i	$-0.432343\pi$
$739$	8475.82	0.421906	0.210953	+	0.977496i	$0.432343\pi$
$740$	0	0
$741$	4923.71	0.244098
$742$	− 6027.57i	− 0.298220i
$743$	− 14491.2i	− 0.715520i	−0.933813	−	0.357760i	$-0.883541\pi$
$743$	− 14491.2i	− 0.715520i	0.933813	−	0.357760i	$-0.116459\pi$
$744$	1886.13	0.0929420
$745$	0	0
$746$	−1487.83	−0.0730206
$747$	− 12561.1i	− 0.615245i
$748$	− 6204.15i	− 0.303270i
$749$	−11791.8	−0.575252
$750$	0	0
$751$	11780.5	0.572408	0.286204	−	0.958169i	$-0.407607\pi$
$751$	11780.5	0.572408	0.286204	+	0.958169i	$0.407607\pi$
$752$	− 16861.4i	− 0.817648i
$753$	5652.14i	0.273540i
$754$	−607.154	−0.0293252
$755$	0	0
$756$	7151.10	0.344025
$757$	22616.7i	1.08589i	0.839768	+	0.542945i	$0.182691\pi$
$757$	22616.7i	1.08589i	−0.839768	+	0.542945i	$0.817309\pi$
$758$	− 822.210i	− 0.0393984i
$759$	−2202.72	−0.105341
$760$	0	0
$761$	2037.48	0.0970547	0.0485273	−	0.998822i	$-0.484547\pi$
$761$	2037.48	0.0970547	0.0485273	+	0.998822i	$0.484547\pi$
$762$	− 1281.00i	− 0.0608999i
$763$	47222.7i	2.24060i
$764$	16344.2	0.773968
$765$	0	0
$766$	−963.268	−0.0454364
$767$	− 29393.8i	− 1.38377i
$768$	− 11176.8i	− 0.525142i
$769$	−16192.1	−0.759298	−0.379649	−	0.925131i	$-0.623955\pi$
$769$	−16192.1	−0.759298	−0.379649	+	0.925131i	$0.623955\pi$
$770$	0	0
$771$	−12036.7	−0.562245
$772$	12977.8i	0.605029i
$773$	− 39726.3i	− 1.84845i	−0.381844	−	0.924227i	$-0.624711\pi$
$773$	− 39726.3i	− 1.84845i	0.381844	−	0.924227i	$-0.375289\pi$
$774$	−84.0500	−0.00390325
$775$	0	0
$776$	2778.16	0.128518
$777$	3793.99i	0.175172i
$778$	1411.43i	0.0650413i
$779$	−16846.3	−0.774817
$780$	0	0
$781$	−5757.71	−0.263799
$782$	− 1286.28i	− 0.0588200i
$783$	1805.93i	0.0824250i
$784$	−48143.3	−2.19312
$785$	0	0
$786$	−881.669	−0.0400103
$787$	− 5160.53i	− 0.233739i	−0.993147	−	0.116870i	$-0.962714\pi$
$787$	− 5160.53i	− 0.233739i	0.993147	−	0.116870i	$-0.0372860\pi$
$788$	27068.4i	1.22370i
$789$	−871.286	−0.0393138
$790$	0	0
$791$	−3466.73	−0.155832
$792$	− 427.022i	− 0.0191586i
$793$	26261.6i	1.17601i
$794$	1773.17	0.0792539
$795$	0	0
$796$	−31222.5	−1.39027
$797$	9247.76i	0.411007i	0.978656	+	0.205504i	$0.0658832\pi$
$797$	9247.76i	0.411007i	−0.978656	+	0.205504i	$0.934117\pi$
$798$	− 1329.33i	− 0.0589696i
$799$	−19274.7	−0.853430
$800$	0	0
$801$	4573.05	0.201724
$802$	− 450.002i	− 0.0198131i
$803$	− 1002.07i	− 0.0440377i
$804$	20848.5	0.914513
$805$	0	0
$806$	−1323.11	−0.0578220
$807$	− 20229.5i	− 0.882417i
$808$	6406.36i	0.278930i
$809$	−29359.7	−1.27594	−0.637968	−	0.770063i	$-0.720225\pi$
$809$	−29359.7	−1.27594	−0.637968	+	0.770063i	$0.720225\pi$
$810$	0	0
$811$	13935.0	0.603358	0.301679	−	0.953410i	$-0.402453\pi$
$811$	13935.0	0.603358	0.301679	+	0.953410i	$0.402453\pi$
$812$	− 17715.2i	− 0.765620i
$813$	− 21546.3i	− 0.929475i
$814$	112.756	0.00485515
$815$	0	0
$816$	−13287.0	−0.570020
$817$	− 1688.52i	− 0.0723057i
$818$	− 1971.33i	− 0.0842617i
$819$	−10079.3	−0.430037
$820$	0	0
$821$	9965.28	0.423618	0.211809	−	0.977311i	$-0.432064\pi$
$821$	9965.28	0.423618	0.211809	+	0.977311i	$0.432064\pi$
$822$	1128.91i	0.0479018i
$823$	2339.58i	0.0990920i	0.998772	+	0.0495460i	$0.0157774\pi$
$823$	2339.58i	0.0990920i	−0.998772	+	0.0495460i	$0.984223\pi$
$824$	2613.69	0.110500
$825$	0	0
$826$	−7935.89	−0.334292
$827$	− 15883.4i	− 0.667861i	−0.942598	−	0.333931i	$-0.891625\pi$
$827$	− 15883.4i	− 0.667861i	0.942598	−	0.333931i	$-0.108375\pi$
$828$	4761.87i	0.199863i
$829$	−4388.04	−0.183839	−0.0919196	−	0.995766i	$-0.529300\pi$
$829$	−4388.04	−0.183839	−0.0919196	+	0.995766i	$0.529300\pi$
$830$	0	0
$831$	−17125.0	−0.714875
$832$	16224.1i	0.676044i
$833$	55034.0i	2.28909i
$834$	−406.293	−0.0168690
$835$	0	0
$836$	4269.57	0.176634
$837$	3935.49i	0.162521i
$838$	1595.75i	0.0657807i
$839$	−33830.2	−1.39207	−0.696036	−	0.718007i	$-0.745055\pi$
$839$	−33830.2	−1.39207	−0.696036	+	0.718007i	$0.745055\pi$
$840$	0	0
$841$	−19915.2	−0.816565
$842$	3002.45i	0.122888i
$843$	− 26222.8i	− 1.07137i
$844$	28085.2	1.14542
$845$	0	0
$846$	−660.270	−0.0268328
$847$	− 4043.01i	− 0.164013i
$848$	− 41460.7i	− 1.67897i
$849$	23650.7	0.956054
$850$	0	0
$851$	−2526.39	−0.101767
$852$	12447.1i	0.500505i
$853$	− 26047.8i	− 1.04556i	−0.852469	−	0.522778i	$-0.824896\pi$
$853$	− 26047.8i	− 1.04556i	0.852469	−	0.522778i	$-0.175104\pi$
$854$	7090.25	0.284102
$855$	0	0
$856$	1522.22	0.0607808
$857$	17210.7i	0.686006i	0.939334	+	0.343003i	$0.111444\pi$
$857$	17210.7i	0.686006i	−0.939334	+	0.343003i	$0.888556\pi$
$858$	299.554i	0.0119191i
$859$	−27367.9	−1.08706	−0.543528	−	0.839391i	$-0.682912\pi$
$859$	−27367.9	−1.08706	−0.543528	+	0.839391i	$0.682912\pi$
$860$	0	0
$861$	34486.2	1.36502
$862$	1532.35i	0.0605476i
$863$	− 34684.4i	− 1.36810i	−0.729436	−	0.684049i	$-0.760217\pi$
$863$	− 34684.4i	− 1.36810i	0.729436	−	0.684049i	$-0.239783\pi$
$864$	−1386.84	−0.0546079
$865$	0	0
$866$	830.092	0.0325724
$867$	449.690i	0.0176151i
$868$	− 38605.0i	− 1.50961i
$869$	1066.22	0.0416213
$870$	0	0
$871$	−29385.5	−1.14316
$872$	− 6096.03i	− 0.236740i
$873$	5796.74i	0.224731i
$874$	885.190	0.0342586
$875$	0	0
$876$	−2166.29	−0.0835527
$877$	45816.6i	1.76410i	0.471153	+	0.882051i	$0.343838\pi$
$877$	45816.6i	1.76410i	−0.471153	+	0.882051i	$0.656162\pi$
$878$	817.217i	0.0314120i
$879$	−6985.61	−0.268053
$880$	0	0
$881$	−26404.8	−1.00976	−0.504881	−	0.863189i	$-0.668464\pi$
$881$	−26404.8	−1.00976	−0.504881	+	0.863189i	$0.668464\pi$
$882$	1885.23i	0.0719717i
$883$	− 33977.1i	− 1.29493i	−0.762097	−	0.647463i	$-0.775830\pi$
$883$	− 33977.1i	− 1.29493i	0.762097	−	0.647463i	$-0.224170\pi$
$884$	18904.2	0.719252
$885$	0	0
$886$	4630.66	0.175587
$887$	− 3023.60i	− 0.114456i	−0.998361	−	0.0572280i	$-0.981774\pi$
$887$	− 3023.60i	− 0.114456i	0.998361	−	0.0572280i	$-0.0182262\pi$
$888$	− 489.770i	− 0.0185086i
$889$	−52681.2	−1.98748
$890$	0	0
$891$	891.000	0.0335013
$892$	6745.03i	0.253184i
$893$	− 13264.5i	− 0.497064i
$894$	−557.108	−0.0208417
$895$	0	0
$896$	18110.3	0.675249
$897$	− 6711.75i	− 0.249832i
$898$	2931.27i	0.108929i
$899$	9749.28	0.361687
$900$	0	0
$901$	−47394.9	−1.75244
$902$	− 1024.92i	− 0.0378336i
$903$	3456.57i	0.127384i
$904$	447.524	0.0164651
$905$	0	0
$906$	−982.257	−0.0360191
$907$	23951.0i	0.876825i	0.898774	+	0.438412i	$0.144459\pi$
$907$	23951.0i	0.876825i	−0.898774	+	0.438412i	$0.855541\pi$
$908$	39954.8i	1.46029i
$909$	−13367.1	−0.487745
$910$	0	0
$911$	24198.9	0.880071	0.440035	−	0.897980i	$-0.354966\pi$
$911$	24198.9	0.880071	0.440035	+	0.897980i	$0.354966\pi$
$912$	− 9143.81i	− 0.331998i
$913$	15352.5i	0.556510i
$914$	1857.60	0.0672253
$915$	0	0
$916$	28728.9	1.03628
$917$	36258.8i	1.30575i
$918$	520.300i	0.0187064i
$919$	−11056.9	−0.396880	−0.198440	−	0.980113i	$-0.563587\pi$
$919$	−11056.9	−0.396880	−0.198440	+	0.980113i	$0.563587\pi$
$920$	0	0
$921$	8490.92	0.303784
$922$	2613.94i	0.0933682i
$923$	− 17543.9i	− 0.625639i
$924$	−8740.24	−0.311182
$925$	0	0
$926$	−1284.93	−0.0455999
$927$	5453.57i	0.193224i
$928$	3435.58i	0.121529i
$929$	48242.5	1.70375	0.851876	−	0.523744i	$-0.175465\pi$
$929$	48242.5	1.70375	0.851876	+	0.523744i	$0.175465\pi$
$930$	0	0
$931$	−37873.2	−1.33324
$932$	23657.6i	0.831470i
$933$	19742.0i	0.692738i
$934$	2989.42	0.104729
$935$	0	0
$936$	1301.15	0.0454375
$937$	− 48721.9i	− 1.69869i	−0.527836	−	0.849346i	$-0.676996\pi$
$937$	− 48721.9i	− 1.69869i	0.527836	−	0.849346i	$-0.323004\pi$
$938$	7933.65i	0.276165i
$939$	23513.9	0.817195
$940$	0	0
$941$	−4978.66	−0.172476	−0.0862379	−	0.996275i	$-0.527485\pi$
$941$	−4978.66	−0.172476	−0.0862379	+	0.996275i	$0.527485\pi$
$942$	− 2699.44i	− 0.0933678i
$943$	22964.1i	0.793015i
$944$	−54587.1	−1.88205
$945$	0	0
$946$	102.728	0.00353062
$947$	27711.4i	0.950896i	0.879744	+	0.475448i	$0.157714\pi$
$947$	27711.4i	0.950896i	−0.879744	+	0.475448i	$0.842286\pi$
$948$	− 2304.96i	− 0.0789680i
$949$	3053.34	0.104442
$950$	0	0
$951$	17560.8	0.598790
$952$	− 10255.0i	− 0.349123i
$953$	49175.6i	1.67152i	0.549098	+	0.835758i	$0.314971\pi$
$953$	49175.6i	1.67152i	−0.549098	+	0.835758i	$0.685029\pi$
$954$	−1623.55	−0.0550989
$955$	0	0
$956$	−10289.7	−0.348109
$957$	− 2207.25i	− 0.0745562i
$958$	− 4733.21i	− 0.159627i
$959$	46426.6	1.56329
$960$	0	0
$961$	−8545.37	−0.286844
$962$	343.571i	0.0115147i
$963$	3176.17i	0.106283i
$964$	31537.9	1.05370
$965$	0	0
$966$	−1812.07	−0.0603546
$967$	− 52678.5i	− 1.75184i	−0.482461	−	0.875918i	$-0.660257\pi$
$967$	− 52678.5i	− 1.75184i	0.482461	−	0.875918i	$-0.339743\pi$
$968$	521.916i	0.0173296i
$969$	−10452.5	−0.346526
$970$	0	0
$971$	46877.5	1.54930	0.774651	−	0.632390i	$-0.217926\pi$
$971$	46877.5	1.54930	0.774651	+	0.632390i	$0.217926\pi$
$972$	− 1926.18i	− 0.0635619i
$973$	16708.8i	0.550525i
$974$	−4608.00	−0.151591
$975$	0	0
$976$	48770.4	1.59949
$977$	1335.33i	0.0437267i	0.999761	+	0.0218633i	$0.00695987\pi$
$977$	1335.33i	0.0437267i	−0.999761	+	0.0218633i	$0.993040\pi$
$978$	− 1989.20i	− 0.0650384i
$979$	−5589.29	−0.182466
$980$	0	0
$981$	12719.6	0.413972
$982$	5328.19i	0.173146i
$983$	− 15102.0i	− 0.490011i	−0.969522	−	0.245005i	$-0.921210\pi$
$983$	− 15102.0i	− 0.490011i	0.969522	−	0.245005i	$-0.0787897\pi$
$984$	−4451.85	−0.144228
$985$	0	0
$986$	1288.93	0.0416306
$987$	27153.7i	0.875696i
$988$	13009.5i	0.418915i
$989$	−2301.70	−0.0740039
$990$	0	0
$991$	−45849.8	−1.46969	−0.734847	−	0.678233i	$-0.762746\pi$
$991$	−45849.8	−1.46969	−0.734847	+	0.678233i	$0.762746\pi$
$992$	7486.82i	0.239624i
$993$	− 16042.8i	− 0.512693i
$994$	−4736.60	−0.151143
$995$	0	0
$996$	33189.3	1.05587
$997$	49662.4i	1.57756i	0.614678	+	0.788779i	$0.289286\pi$
$997$	49662.4i	1.57756i	−0.614678	+	0.788779i	$0.710714\pi$
$998$	4266.77i	0.135333i
$999$	1021.93	0.0323647

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.r.199.6		10
5.2	odd	4		825.4.a.w.1.3	✓	5
5.3	odd	4		825.4.a.z.1.3	yes	5
5.4	even	2	inner	825.4.c.r.199.5		10
15.2	even	4		2475.4.a.bm.1.3		5
15.8	even	4		2475.4.a.bf.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.w.1.3	✓	5	5.2	odd	4
825.4.a.z.1.3	yes	5	5.3	odd	4
825.4.c.r.199.5		10	5.4	even	2	inner
825.4.c.r.199.6		10	1.1	even	1	trivial
2475.4.a.bf.1.3		5	15.8	even	4
2475.4.a.bm.1.3		5	15.2	even	4

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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.270826i q^{2} -3.00000i q^{3} +7.92665 q^{4} +0.812479 q^{6} -33.4133i q^{7} +4.31336i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q+0.270826i q^{2} -3.00000i q^{3} +7.92665 q^{4} +0.812479 q^{6} -33.4133i q^{7} +4.31336i q^{8} -9.00000 q^{9} +11.0000 q^{11} -23.7800i q^{12} +33.5174i q^{13} +9.04920 q^{14} +62.2451 q^{16} -71.1540i q^{17} -2.43744i q^{18} +48.9667 q^{19} -100.240 q^{21} +2.97909i q^{22} -66.7490i q^{23} +12.9401 q^{24} -9.07738 q^{26} +27.0000i q^{27} -264.856i q^{28} +66.8864 q^{29} +145.759 q^{31} +51.3644i q^{32} -33.0000i q^{33} +19.2704 q^{34} -71.3399 q^{36} -37.8491i q^{37} +13.2615i q^{38} +100.552 q^{39} -344.036 q^{41} -27.1476i q^{42} -34.4830i q^{43} +87.1932 q^{44} +18.0774 q^{46} -270.887i q^{47} -186.735i q^{48} -773.448 q^{49} -213.462 q^{51} +265.681i q^{52} -666.088i q^{53} -7.31231 q^{54} +144.123 q^{56} -146.900i q^{57} +18.1146i q^{58} -876.971 q^{59} +783.523 q^{61} +39.4753i q^{62} +300.720i q^{63} +484.050 q^{64} +8.93727 q^{66} +876.724i q^{67} -564.013i q^{68} -200.247 q^{69} -523.428 q^{71} -38.8202i q^{72} -91.0973i q^{73} +10.2505 q^{74} +388.142 q^{76} -367.546i q^{77} +27.2322i q^{78} +96.9287 q^{79} +81.0000 q^{81} -93.1741i q^{82} +1395.68i q^{83} -794.567 q^{84} +9.33889 q^{86} -200.659i q^{87} +47.4469i q^{88} -508.117 q^{89} +1119.93 q^{91} -529.096i q^{92} -437.276i q^{93} +73.3634 q^{94} +154.093 q^{96} -644.082i q^{97} -209.470i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 92 q^{4} + 24 q^{6} - 90 q^{9}+O(q^{10}) \) 10 * q - 92 * q^4 + 24 * q^6 - 90 * q^9	\( 10 q - 92 q^{4} + 24 q^{6} - 90 q^{9} + 110 q^{11} - 110 q^{14} + 1268 q^{16} + 674 q^{19} - 228 q^{21} - 432 q^{24} + 718 q^{26} + 306 q^{29} + 526 q^{31} - 1034 q^{34} + 828 q^{36} - 258 q^{39} - 176 q^{41} - 1012 q^{44} + 872 q^{46} - 4762 q^{49} + 900 q^{51} - 216 q^{54} + 422 q^{56} - 820 q^{59} - 2260 q^{61} - 6340 q^{64} + 264 q^{66} + 1206 q^{69} + 2498 q^{71} + 5970 q^{74} - 5112 q^{76} - 4516 q^{79} + 810 q^{81} + 2646 q^{84} - 2870 q^{86} - 694 q^{89} - 1224 q^{91} - 1814 q^{94} + 1680 q^{96} - 990 q^{99}+O(q^{100}) \) 10 * q - 92 * q^4 + 24 * q^6 - 90 * q^9 + 110 * q^11 - 110 * q^14 + 1268 * q^16 + 674 * q^19 - 228 * q^21 - 432 * q^24 + 718 * q^26 + 306 * q^29 + 526 * q^31 - 1034 * q^34 + 828 * q^36 - 258 * q^39 - 176 * q^41 - 1012 * q^44 + 872 * q^46 - 4762 * q^49 + 900 * q^51 - 216 * q^54 + 422 * q^56 - 820 * q^59 - 2260 * q^61 - 6340 * q^64 + 264 * q^66 + 1206 * q^69 + 2498 * q^71 + 5970 * q^74 - 5112 * q^76 - 4516 * q^79 + 810 * q^81 + 2646 * q^84 - 2870 * q^86 - 694 * q^89 - 1224 * q^91 - 1814 * q^94 + 1680 * q^96 - 990 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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