LMFDB - Embedded newform 100.18.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [100,18,Mod(1,100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 18, names="a")

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

chi := DirichletCharacter("100.1");

S:= CuspForms(chi, 18);

N := Newforms(S);

Level:	$ N $	$=$	$ 100 = 2^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 18 $
Character orbit:	$[\chi]$	$=$	100.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$183.222087345$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3 x^{7} - 88310852 x^{6} - 46424873565 x^{5} + \cdots + 10\!\cdots\!75 $ x^8 - 3x^7 - 88310852x^6 - 46424873565x^5 + 2019994729783288x^4 + 789267258558513075x^3 - 10607817373781103559800x^2 - 636492268853844788401875*x + 10491707875810205748413671875
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{35}\cdot 3^{4}\cdot 5^{14} $
Twist minimal:	no (minimal twist has level 20)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$4835.86$ of defining polynomial
Character	$\chi$	$=$	100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4988.24 q^{3} +8.73679e6 q^{7} -1.04258e8 q^{9} +6.61458e8 q^{11} -3.36520e7 q^{13} -2.31403e9 q^{17} -6.14052e10 q^{19} +4.35812e10 q^{21} +7.01987e9 q^{23} -1.16424e12 q^{27} +1.13007e12 q^{29} +3.64909e12 q^{31} +3.29951e12 q^{33} -3.38698e13 q^{37} -1.67864e11 q^{39} -2.80398e13 q^{41} +9.56968e13 q^{43} -2.08854e14 q^{47} -1.56299e14 q^{49} -1.15429e13 q^{51} +5.01273e14 q^{53} -3.06304e14 q^{57} +7.12609e14 q^{59} +1.49205e15 q^{61} -9.10877e14 q^{63} +5.73469e14 q^{67} +3.50168e13 q^{69} -6.67771e15 q^{71} -1.14580e16 q^{73} +5.77902e15 q^{77} -1.20012e16 q^{79} +7.65632e15 q^{81} -2.56333e16 q^{83} +5.63707e15 q^{87} +4.90595e16 q^{89} -2.94011e14 q^{91} +1.82025e16 q^{93} -2.20754e16 q^{97} -6.89620e16 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 376848248 q^{9} - 396200640 q^{11} - 11821646592 q^{19} - 420670059472 q^{21} - 1543712861232 q^{29} - 13722543013312 q^{31} + 79298226023808 q^{39} + 29824965017568 q^{41} + 535615080275736 q^{49}+ \cdots + 23\!\cdots\!60 q^{99}+O(q^{100}) $ 8 * q + 376848248 * q^9 - 396200640 * q^11 - 11821646592 * q^19 - 420670059472 * q^21 - 1543712861232 * q^29 - 13722543013312 * q^31 + 79298226023808 * q^39 + 29824965017568 * q^41 + 535615080275736 * q^49 - 764079320419264 * q^51 - 94539545151744 * q^59 - 3380827226969024 * q^61 - 5559752175554416 * q^69 + 20155606060098624 * q^71 + 3634057920126144 * q^79 + 15949137882627512 * q^81 - 103487949043655568 * q^89 - 69783870139121088 * q^91 + 239420571619653760 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	4988.24	0.438952	0.219476	−	0.975618i	$-0.429565\pi$
$3$	4988.24	0.438952	0.219476	+	0.975618i	$0.429565\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	8.73679e6	0.572820	0.286410	−	0.958107i	$-0.407538\pi$
$7$	8.73679e6	0.572820	0.286410	+	0.958107i	$0.407538\pi$
$8$	0	0
$9$	−1.04258e8	−0.807322
$10$	0	0
$11$	6.61458e8	0.930389	0.465194	−	0.885209i	$-0.345984\pi$
$11$	6.61458e8	0.930389	0.465194	+	0.885209i	$0.345984\pi$
$12$	0	0
$13$	−3.36520e7	−0.0114418	−0.00572088	−	0.999984i	$-0.501821\pi$
$13$	−3.36520e7	−0.0114418	−0.00572088	+	0.999984i	$0.501821\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.31403e9	−0.0804550	−0.0402275	−	0.999191i	$-0.512808\pi$
$17$	−2.31403e9	−0.0804550	−0.0402275	+	0.999191i	$0.512808\pi$
$18$	0	0
$19$	−6.14052e10	−0.829467	−0.414734	−	0.909943i	$-0.636125\pi$
$19$	−6.14052e10	−0.829467	−0.414734	+	0.909943i	$0.636125\pi$
$20$	0	0
$21$	4.35812e10	0.251440
$22$	0	0
$23$	7.01987e9	0.0186914	0.00934571	−	0.999956i	$-0.497025\pi$
$23$	7.01987e9	0.0186914	0.00934571	+	0.999956i	$0.497025\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.16424e12	−0.793327
$28$	0	0
$29$	1.13007e12	0.419491	0.209746	−	0.977756i	$-0.432736\pi$
$29$	1.13007e12	0.419491	0.209746	+	0.977756i	$0.432736\pi$
$30$	0	0
$31$	3.64909e12	0.768440	0.384220	−	0.923242i	$-0.374470\pi$
$31$	3.64909e12	0.768440	0.384220	+	0.923242i	$0.374470\pi$
$32$	0	0
$33$	3.29951e12	0.408396
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.38698e13	−1.58525	−0.792625	−	0.609709i	$-0.791286\pi$
$37$	−3.38698e13	−1.58525	−0.792625	+	0.609709i	$0.791286\pi$
$38$	0	0
$39$	−1.67864e11	−0.00502238
$40$	0	0
$41$	−2.80398e13	−0.548419	−0.274209	−	0.961670i	$-0.588416\pi$
$41$	−2.80398e13	−0.548419	−0.274209	+	0.961670i	$0.588416\pi$
$42$	0	0
$43$	9.56968e13	1.24858	0.624288	−	0.781194i	$-0.285389\pi$
$43$	9.56968e13	1.24858	0.624288	+	0.781194i	$0.285389\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.08854e14	−1.27941	−0.639707	−	0.768619i	$-0.720944\pi$
$47$	−2.08854e14	−1.27941	−0.639707	+	0.768619i	$0.720944\pi$
$48$	0	0
$49$	−1.56299e14	−0.671877
$50$	0	0
$51$	−1.15429e13	−0.0353158
$52$	0	0
$53$	5.01273e14	1.10593	0.552967	−	0.833203i	$-0.313495\pi$
$53$	5.01273e14	1.10593	0.552967	+	0.833203i	$0.313495\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.06304e14	−0.364096
$58$	0	0
$59$	7.12609e14	0.631843	0.315922	−	0.948785i	$-0.397686\pi$
$59$	7.12609e14	0.631843	0.315922	+	0.948785i	$0.397686\pi$
$60$	0	0
$61$	1.49205e15	0.996507	0.498254	−	0.867031i	$-0.333975\pi$
$61$	1.49205e15	0.996507	0.498254	+	0.867031i	$0.333975\pi$
$62$	0	0
$63$	−9.10877e14	−0.462450
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.73469e14	0.172534	0.0862669	−	0.996272i	$-0.472506\pi$
$67$	5.73469e14	0.172534	0.0862669	+	0.996272i	$0.472506\pi$
$68$	0	0
$69$	3.50168e13	0.00820463
$70$	0	0
$71$	−6.67771e15	−1.22725	−0.613623	−	0.789599i	$-0.710288\pi$
$71$	−6.67771e15	−1.22725	−0.613623	+	0.789599i	$0.710288\pi$
$72$	0	0
$73$	−1.14580e16	−1.66290	−0.831450	−	0.555600i	$-0.812489\pi$
$73$	−1.14580e16	−1.66290	−0.831450	+	0.555600i	$0.812489\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.77902e15	0.532946
$78$	0	0
$79$	−1.20012e16	−0.890005	−0.445003	−	0.895529i	$-0.646797\pi$
$79$	−1.20012e16	−0.890005	−0.445003	+	0.895529i	$0.646797\pi$
$80$	0	0
$81$	7.65632e15	0.459090
$82$	0	0
$83$	−2.56333e16	−1.24923	−0.624614	−	0.780934i	$-0.714744\pi$
$83$	−2.56333e16	−1.24923	−0.624614	+	0.780934i	$0.714744\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.63707e15	0.184136
$88$	0	0
$89$	4.90595e16	1.32102	0.660508	−	0.750819i	$-0.270341\pi$
$89$	4.90595e16	1.32102	0.660508	+	0.750819i	$0.270341\pi$
$90$	0	0
$91$	−2.94011e14	−0.00655407
$92$	0	0
$93$	1.82025e16	0.337308
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.20754e16	−0.285988	−0.142994	−	0.989724i	$-0.545673\pi$
$97$	−2.20754e16	−0.285988	−0.142994	+	0.989724i	$0.545673\pi$
$98$	0	0
$99$	−6.89620e16	−0.751123
$100$	0	0
$101$	4.35640e16	0.400310	0.200155	−	0.979764i	$-0.435855\pi$
$101$	4.35640e16	0.400310	0.200155	+	0.979764i	$0.435855\pi$
$102$	0	0
$103$	1.09433e17	0.851198	0.425599	−	0.904912i	$-0.360063\pi$
$103$	1.09433e17	0.851198	0.425599	+	0.904912i	$0.360063\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.96661e17	−1.10651	−0.553256	−	0.833011i	$-0.686615\pi$
$107$	−1.96661e17	−1.10651	−0.553256	+	0.833011i	$0.686615\pi$
$108$	0	0
$109$	−7.27689e16	−0.349801	−0.174900	−	0.984586i	$-0.555960\pi$
$109$	−7.27689e16	−0.349801	−0.174900	+	0.984586i	$0.555960\pi$
$110$	0	0
$111$	−1.68951e17	−0.695848
$112$	0	0
$113$	1.81697e17	0.642957	0.321479	−	0.946917i	$-0.395820\pi$
$113$	1.81697e17	0.642957	0.321479	+	0.946917i	$0.395820\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.50848e15	0.00923718
$118$	0	0
$119$	−2.02172e16	−0.0460862
$120$	0	0
$121$	−6.79205e16	−0.134377
$122$	0	0
$123$	−1.39869e17	−0.240729
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.79786e17	0.366855	0.183428	−	0.983033i	$-0.441281\pi$
$127$	2.79786e17	0.366855	0.183428	+	0.983033i	$0.441281\pi$
$128$	0	0
$129$	4.77358e17	0.548065
$130$	0	0
$131$	8.37934e17	0.844119	0.422060	−	0.906568i	$-0.361307\pi$
$131$	8.37934e17	0.844119	0.422060	+	0.906568i	$0.361307\pi$
$132$	0	0
$133$	−5.36484e17	−0.475136
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.45426e18	1.68965	0.844823	−	0.535046i	$-0.179706\pi$
$137$	2.45426e18	1.68965	0.844823	+	0.535046i	$0.179706\pi$
$138$	0	0
$139$	−1.17108e18	−0.712786	−0.356393	−	0.934336i	$-0.615994\pi$
$139$	−1.17108e18	−0.712786	−0.356393	+	0.934336i	$0.615994\pi$
$140$	0	0
$141$	−1.04181e18	−0.561600
$142$	0	0
$143$	−2.22594e16	−0.0106453
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−7.79657e17	−0.294921
$148$	0	0
$149$	−2.93908e18	−0.991124	−0.495562	−	0.868573i	$-0.665038\pi$
$149$	−2.93908e18	−0.991124	−0.495562	+	0.868573i	$0.665038\pi$
$150$	0	0
$151$	−2.52716e18	−0.760902	−0.380451	−	0.924801i	$-0.624231\pi$
$151$	−2.52716e18	−0.760902	−0.380451	+	0.924801i	$0.624231\pi$
$152$	0	0
$153$	2.41255e17	0.0649530
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.25673e17	−0.113650	−0.0568248	−	0.998384i	$-0.518098\pi$
$157$	−5.25673e17	−0.113650	−0.0568248	+	0.998384i	$0.518098\pi$
$158$	0	0
$159$	2.50047e18	0.485452
$160$	0	0
$161$	6.13311e16	0.0107068
$162$	0	0
$163$	5.49012e18	0.862953	0.431476	−	0.902124i	$-0.357993\pi$
$163$	5.49012e18	0.862953	0.431476	+	0.902124i	$0.357993\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.54439e19	−1.97546	−0.987729	−	0.156175i	$-0.950083\pi$
$167$	−1.54439e19	−1.97546	−0.987729	+	0.156175i	$0.950083\pi$
$168$	0	0
$169$	−8.64928e18	−0.999869
$170$	0	0
$171$	6.40196e18	0.669647
$172$	0	0
$173$	−6.59357e18	−0.624782	−0.312391	−	0.949954i	$-0.601130\pi$
$173$	−6.59357e18	−0.624782	−0.312391	+	0.949954i	$0.601130\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.55466e18	0.277349
$178$	0	0
$179$	−1.38246e19	−0.980400	−0.490200	−	0.871610i	$-0.663076\pi$
$179$	−1.38246e19	−0.980400	−0.490200	+	0.871610i	$0.663076\pi$
$180$	0	0
$181$	−1.27920e19	−0.825413	−0.412707	−	0.910864i	$-0.635417\pi$
$181$	−1.27920e19	−0.825413	−0.412707	+	0.910864i	$0.635417\pi$
$182$	0	0
$183$	7.44271e18	0.437418
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.53063e18	−0.0748544
$188$	0	0
$189$	−1.01718e19	−0.454434
$190$	0	0
$191$	−1.69147e19	−0.691004	−0.345502	−	0.938418i	$-0.612291\pi$
$191$	−1.69147e19	−0.691004	−0.345502	+	0.938418i	$0.612291\pi$
$192$	0	0
$193$	−3.85104e19	−1.43993	−0.719964	−	0.694011i	$-0.755842\pi$
$193$	−3.85104e19	−1.43993	−0.719964	+	0.694011i	$0.755842\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.29078e19	1.66172	0.830858	−	0.556484i	$-0.187850\pi$
$197$	5.29078e19	1.66172	0.830858	+	0.556484i	$0.187850\pi$
$198$	0	0
$199$	−5.02252e19	−1.44767	−0.723837	−	0.689971i	$-0.757623\pi$
$199$	−5.02252e19	−1.44767	−0.723837	+	0.689971i	$0.757623\pi$
$200$	0	0
$201$	2.86060e18	0.0757340
$202$	0	0
$203$	9.87320e18	0.240293
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.31875e17	−0.0150900
$208$	0	0
$209$	−4.06169e19	−0.771727
$210$	0	0
$211$	−6.25806e19	−1.09658	−0.548288	−	0.836289i	$-0.684720\pi$
$211$	−6.25806e19	−1.09658	−0.548288	+	0.836289i	$0.684720\pi$
$212$	0	0
$213$	−3.33100e19	−0.538701
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.18813e19	0.440178
$218$	0	0
$219$	−5.71554e19	−0.729932
$220$	0	0
$221$	7.78717e16	0.000920546 0
$222$	0	0
$223$	1.15820e20	1.26821	0.634106	−	0.773246i	$-0.281368\pi$
$223$	1.15820e20	1.26821	0.634106	+	0.773246i	$0.281368\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.47311e20	−1.38680	−0.693401	−	0.720552i	$-0.743888\pi$
$227$	−1.47311e20	−1.38680	−0.693401	+	0.720552i	$0.743888\pi$
$228$	0	0
$229$	−1.79644e20	−1.56968	−0.784841	−	0.619697i	$-0.787256\pi$
$229$	−1.79644e20	−1.56968	−0.784841	+	0.619697i	$0.787256\pi$
$230$	0	0
$231$	2.88271e19	0.233937
$232$	0	0
$233$	3.47360e19	0.261972	0.130986	−	0.991384i	$-0.458186\pi$
$233$	3.47360e19	0.261972	0.130986	+	0.991384i	$0.458186\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−5.98646e19	−0.390669
$238$	0	0
$239$	−2.75666e20	−1.67495	−0.837473	−	0.546478i	$-0.815968\pi$
$239$	−2.75666e20	−1.67495	−0.837473	+	0.546478i	$0.815968\pi$
$240$	0	0
$241$	−2.16087e20	−1.22316	−0.611579	−	0.791183i	$-0.709465\pi$
$241$	−2.16087e20	−1.22316	−0.611579	+	0.791183i	$0.709465\pi$
$242$	0	0
$243$	1.88542e20	0.994845
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.06641e18	0.00949056
$248$	0	0
$249$	−1.27865e20	−0.548350
$250$	0	0
$251$	−8.83606e19	−0.354023	−0.177012	−	0.984209i	$-0.556643\pi$
$251$	−8.83606e19	−0.354023	−0.177012	+	0.984209i	$0.556643\pi$
$252$	0	0
$253$	4.64335e18	0.0173903
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.99412e19	0.294800	0.147400	−	0.989077i	$-0.452910\pi$
$257$	8.99412e19	0.294800	0.147400	+	0.989077i	$0.452910\pi$
$258$	0	0
$259$	−2.95913e20	−0.908064
$260$	0	0
$261$	−1.17819e20	−0.338664
$262$	0	0
$263$	−4.85249e20	−1.30720	−0.653599	−	0.756841i	$-0.726742\pi$
$263$	−4.85249e20	−1.30720	−0.653599	+	0.756841i	$0.726742\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.44721e20	0.579862
$268$	0	0
$269$	−4.15720e20	−0.924499	−0.462249	−	0.886750i	$-0.652958\pi$
$269$	−4.15720e20	−0.924499	−0.462249	+	0.886750i	$0.652958\pi$
$270$	0	0
$271$	−1.31018e20	−0.273585	−0.136793	−	0.990600i	$-0.543679\pi$
$271$	−1.31018e20	−0.273585	−0.136793	+	0.990600i	$0.543679\pi$
$272$	0	0
$273$	−1.46660e18	−0.00287692
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.70408e20	0.642100	0.321050	−	0.947062i	$-0.395964\pi$
$277$	3.70408e20	0.642100	0.321050	+	0.947062i	$0.395964\pi$
$278$	0	0
$279$	−3.80445e20	−0.620378
$280$	0	0
$281$	−2.18104e20	−0.334704	−0.167352	−	0.985897i	$-0.553522\pi$
$281$	−2.18104e20	−0.334704	−0.167352	+	0.985897i	$0.553522\pi$
$282$	0	0
$283$	−1.17331e21	−1.69523	−0.847616	−	0.530611i	$-0.821963\pi$
$283$	−1.17331e21	−1.69523	−0.847616	+	0.530611i	$0.821963\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.44978e20	−0.314146
$288$	0	0
$289$	−8.21886e20	−0.993527
$290$	0	0
$291$	−1.10117e20	−0.125535
$292$	0	0
$293$	4.00436e20	0.430684	0.215342	−	0.976539i	$-0.430913\pi$
$293$	4.00436e20	0.430684	0.215342	+	0.976539i	$0.430913\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−7.70098e20	−0.738102
$298$	0	0
$299$	−2.36233e17	−0.000213863 0
$300$	0	0
$301$	8.36082e20	0.715210
$302$	0	0
$303$	2.17308e20	0.175717
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.64633e20	0.336073	0.168037	−	0.985781i	$-0.446257\pi$
$307$	4.64633e20	0.336073	0.168037	+	0.985781i	$0.446257\pi$
$308$	0	0
$309$	5.45877e20	0.373635
$310$	0	0
$311$	−1.54971e21	−1.00412	−0.502060	−	0.864833i	$-0.667424\pi$
$311$	−1.54971e21	−1.00412	−0.502060	+	0.864833i	$0.667424\pi$
$312$	0	0
$313$	2.22175e21	1.36323	0.681613	−	0.731713i	$-0.261279\pi$
$313$	2.22175e21	1.36323	0.681613	+	0.731713i	$0.261279\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.76769e21	−0.973646	−0.486823	−	0.873501i	$-0.661844\pi$
$317$	−1.76769e21	−0.973646	−0.486823	+	0.873501i	$0.661844\pi$
$318$	0	0
$319$	7.47495e20	0.390290
$320$	0	0
$321$	−9.80992e20	−0.485705
$322$	0	0
$323$	1.42093e20	0.0667348
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.62989e20	−0.153546
$328$	0	0
$329$	−1.82471e21	−0.732874
$330$	0	0
$331$	2.81224e21	1.07279	0.536394	−	0.843967i	$-0.319786\pi$
$331$	2.81224e21	1.07279	0.536394	+	0.843967i	$0.319786\pi$
$332$	0	0
$333$	3.53118e21	1.27981
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.43159e21	1.12368	0.561838	−	0.827247i	$-0.310094\pi$
$337$	3.43159e21	1.12368	0.561838	+	0.827247i	$0.310094\pi$
$338$	0	0
$339$	9.06350e20	0.282227
$340$	0	0
$341$	2.41372e21	0.714948
$342$	0	0
$343$	−3.39800e21	−0.957685
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.42288e20	−0.0618773	−0.0309387	−	0.999521i	$-0.509850\pi$
$347$	−2.42288e20	−0.0618773	−0.0309387	+	0.999521i	$0.509850\pi$
$348$	0	0
$349$	7.34788e21	1.78709	0.893544	−	0.448976i	$-0.148211\pi$
$349$	7.34788e21	1.78709	0.893544	+	0.448976i	$0.148211\pi$
$350$	0	0
$351$	3.91792e19	0.00907705
$352$	0	0
$353$	−2.16796e21	−0.478592	−0.239296	−	0.970947i	$-0.576917\pi$
$353$	−2.16796e21	−0.478592	−0.239296	+	0.970947i	$0.576917\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.00848e20	−0.0202296
$358$	0	0
$359$	5.57848e21	1.06712	0.533560	−	0.845762i	$-0.320854\pi$
$359$	5.57848e21	1.06712	0.533560	+	0.845762i	$0.320854\pi$
$360$	0	0
$361$	−1.70979e21	−0.311984
$362$	0	0
$363$	−3.38803e20	−0.0589850
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.15562e21	0.817748	0.408874	−	0.912591i	$-0.365922\pi$
$367$	5.15562e21	0.817748	0.408874	+	0.912591i	$0.365922\pi$
$368$	0	0
$369$	2.92336e21	0.442750
$370$	0	0
$371$	4.37952e21	0.633502
$372$	0	0
$373$	−1.16300e22	−1.60714	−0.803571	−	0.595209i	$-0.797069\pi$
$373$	−1.16300e22	−1.60714	−0.803571	+	0.595209i	$0.797069\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3.80292e19	−0.00479972
$378$	0	0
$379$	4.10676e21	0.495525	0.247763	−	0.968821i	$-0.420305\pi$
$379$	4.10676e21	0.495525	0.247763	+	0.968821i	$0.420305\pi$
$380$	0	0
$381$	1.39564e21	0.161032
$382$	0	0
$383$	7.92184e21	0.874252	0.437126	−	0.899400i	$-0.355996\pi$
$383$	7.92184e21	0.874252	0.437126	+	0.899400i	$0.355996\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−9.97712e21	−1.00800
$388$	0	0
$389$	1.41357e22	1.36693	0.683466	−	0.729983i	$-0.260472\pi$
$389$	1.41357e22	1.36693	0.683466	+	0.729983i	$0.260472\pi$
$390$	0	0
$391$	−1.62442e19	−0.00150382
$392$	0	0
$393$	4.17982e21	0.370528
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.01127e22	0.822526	0.411263	−	0.911517i	$-0.365088\pi$
$397$	1.01127e22	0.822526	0.411263	+	0.911517i	$0.365088\pi$
$398$	0	0
$399$	−2.67611e21	−0.208562
$400$	0	0
$401$	9.77567e21	0.730162	0.365081	−	0.930976i	$-0.381041\pi$
$401$	9.77567e21	0.730162	0.365081	+	0.930976i	$0.381041\pi$
$402$	0	0
$403$	−1.22799e20	−0.00879230
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.24034e22	−1.47490
$408$	0	0
$409$	1.10530e22	0.697960	0.348980	−	0.937130i	$-0.386528\pi$
$409$	1.10530e22	0.697960	0.348980	+	0.937130i	$0.386528\pi$
$410$	0	0
$411$	1.22424e22	0.741673
$412$	0	0
$413$	6.22592e21	0.361933
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−5.84161e21	−0.312879
$418$	0	0
$419$	3.43934e22	1.76871	0.884353	−	0.466819i	$-0.154600\pi$
$419$	3.43934e22	1.76871	0.884353	+	0.466819i	$0.154600\pi$
$420$	0	0
$421$	−1.69964e21	−0.0839381	−0.0419691	−	0.999119i	$-0.513363\pi$
$421$	−1.69964e21	−0.0839381	−0.0419691	+	0.999119i	$0.513363\pi$
$422$	0	0
$423$	2.17746e22	1.03290
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.30357e22	0.570820
$428$	0	0
$429$	−1.11035e20	−0.00467276
$430$	0	0
$431$	−5.25915e21	−0.212745	−0.106372	−	0.994326i	$-0.533924\pi$
$431$	−5.25915e21	−0.212745	−0.106372	+	0.994326i	$0.533924\pi$
$432$	0	0
$433$	−3.96890e22	−1.54356	−0.771778	−	0.635892i	$-0.780632\pi$
$433$	−3.96890e22	−1.54356	−0.771778	+	0.635892i	$0.780632\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.31056e20	−0.0155039
$438$	0	0
$439$	1.16302e21	0.0402383	0.0201191	−	0.999798i	$-0.493595\pi$
$439$	1.16302e21	0.0402383	0.0201191	+	0.999798i	$0.493595\pi$
$440$	0	0
$441$	1.62954e22	0.542421
$442$	0	0
$443$	1.32297e22	0.423757	0.211878	−	0.977296i	$-0.432042\pi$
$443$	1.32297e22	0.423757	0.211878	+	0.977296i	$0.432042\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.46608e22	−0.435055
$448$	0	0
$449$	−3.39299e22	−0.969368	−0.484684	−	0.874689i	$-0.661065\pi$
$449$	−3.39299e22	−0.969368	−0.484684	+	0.874689i	$0.661065\pi$
$450$	0	0
$451$	−1.85472e22	−0.510243
$452$	0	0
$453$	−1.26061e22	−0.333999
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.61346e22	1.87195	0.935976	−	0.352065i	$-0.114520\pi$
$457$	7.61346e22	1.87195	0.935976	+	0.352065i	$0.114520\pi$
$458$	0	0
$459$	2.69409e21	0.0638271
$460$	0	0
$461$	−3.05919e22	−0.698471	−0.349235	−	0.937035i	$-0.613559\pi$
$461$	−3.05919e22	−0.698471	−0.349235	+	0.937035i	$0.613559\pi$
$462$	0	0
$463$	−4.13172e22	−0.909270	−0.454635	−	0.890678i	$-0.650230\pi$
$463$	−4.13172e22	−0.909270	−0.454635	+	0.890678i	$0.650230\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.41963e22	0.494944	0.247472	−	0.968895i	$-0.420400\pi$
$467$	2.41963e22	0.494944	0.247472	+	0.968895i	$0.420400\pi$
$468$	0	0
$469$	5.01028e21	0.0988309
$470$	0	0
$471$	−2.62218e21	−0.0498867
$472$	0	0
$473$	6.32994e22	1.16166
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5.22616e22	−0.892845
$478$	0	0
$479$	−5.15334e22	−0.849644	−0.424822	−	0.905277i	$-0.639663\pi$
$479$	−5.15334e22	−0.849644	−0.424822	+	0.905277i	$0.639663\pi$
$480$	0	0
$481$	1.13979e21	0.0181380
$482$	0	0
$483$	3.05934e20	0.00469978
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.38806e22	0.628458	0.314229	−	0.949347i	$-0.398254\pi$
$487$	4.38806e22	0.628458	0.314229	+	0.949347i	$0.398254\pi$
$488$	0	0
$489$	2.73860e22	0.378794
$490$	0	0
$491$	1.04309e23	1.39357	0.696787	−	0.717278i	$-0.254612\pi$
$491$	1.04309e23	1.39357	0.696787	+	0.717278i	$0.254612\pi$
$492$	0	0
$493$	−2.61502e21	−0.0337502
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.83417e22	−0.702991
$498$	0	0
$499$	−9.97077e22	−1.16111	−0.580556	−	0.814220i	$-0.697165\pi$
$499$	−9.97077e22	−1.16111	−0.580556	+	0.814220i	$0.697165\pi$
$500$	0	0
$501$	−7.70380e22	−0.867131
$502$	0	0
$503$	7.99430e22	0.869867	0.434933	−	0.900463i	$-0.356772\pi$
$503$	7.99430e22	0.869867	0.434933	+	0.900463i	$0.356772\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−4.31447e22	−0.438894
$508$	0	0
$509$	−1.58395e23	−1.55827	−0.779133	−	0.626859i	$-0.784340\pi$
$509$	−1.58395e23	−1.55827	−0.779133	+	0.626859i	$0.784340\pi$
$510$	0	0
$511$	−1.00106e23	−0.952543
$512$	0	0
$513$	7.14906e22	0.658038
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.38148e23	−1.19035
$518$	0	0
$519$	−3.28903e22	−0.274249
$520$	0	0
$521$	−1.45303e23	−1.17261	−0.586306	−	0.810090i	$-0.699418\pi$
$521$	−1.45303e23	−1.17261	−0.586306	+	0.810090i	$0.699418\pi$
$522$	0	0
$523$	−1.64129e23	−1.28210	−0.641050	−	0.767499i	$-0.721501\pi$
$523$	−1.64129e23	−1.28210	−0.641050	+	0.767499i	$0.721501\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.44409e21	−0.0618248
$528$	0	0
$529$	−1.41001e23	−0.999651
$530$	0	0
$531$	−7.42949e22	−0.510101
$532$	0	0
$533$	9.43596e20	0.00627487
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−6.89606e22	−0.430348
$538$	0	0
$539$	−1.03385e23	−0.625106
$540$	0	0
$541$	2.16545e23	1.26873	0.634367	−	0.773032i	$-0.281261\pi$
$541$	2.16545e23	1.26873	0.634367	+	0.773032i	$0.281261\pi$
$542$	0	0
$543$	−6.38097e22	−0.362317
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.49615e23	1.86508	0.932541	−	0.361064i	$-0.117586\pi$
$547$	3.49615e23	1.86508	0.932541	+	0.361064i	$0.117586\pi$
$548$	0	0
$549$	−1.55558e23	−0.804502
$550$	0	0
$551$	−6.93922e22	−0.347954
$552$	0	0
$553$	−1.04852e23	−0.509813
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.56088e22	0.0713839	0.0356919	−	0.999363i	$-0.488636\pi$
$557$	1.56088e22	0.0713839	0.0356919	+	0.999363i	$0.488636\pi$
$558$	0	0
$559$	−3.22039e21	−0.0142859
$560$	0	0
$561$	−7.63516e21	−0.0328575
$562$	0	0
$563$	−1.54986e23	−0.647099	−0.323550	−	0.946211i	$-0.604876\pi$
$563$	−1.54986e23	−0.647099	−0.323550	+	0.946211i	$0.604876\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	6.68917e22	0.262976
$568$	0	0
$569$	2.22901e23	0.850468	0.425234	−	0.905083i	$-0.360192\pi$
$569$	2.22901e23	0.850468	0.425234	+	0.905083i	$0.360192\pi$
$570$	0	0
$571$	−4.79371e22	−0.177527	−0.0887637	−	0.996053i	$-0.528292\pi$
$571$	−4.79371e22	−0.177527	−0.0887637	+	0.996053i	$0.528292\pi$
$572$	0	0
$573$	−8.43745e22	−0.303317
$574$	0	0
$575$	0	0
$576$	0	0
$577$	8.37199e22	0.283684	0.141842	−	0.989889i	$-0.454698\pi$
$577$	8.37199e22	0.283684	0.141842	+	0.989889i	$0.454698\pi$
$578$	0	0
$579$	−1.92099e23	−0.632059
$580$	0	0
$581$	−2.23953e23	−0.715583
$582$	0	0
$583$	3.31571e23	1.02895
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.07053e23	0.899063	0.449531	−	0.893265i	$-0.351591\pi$
$587$	3.07053e23	0.899063	0.449531	+	0.893265i	$0.351591\pi$
$588$	0	0
$589$	−2.24073e23	−0.637396
$590$	0	0
$591$	2.63917e23	0.729413
$592$	0	0
$593$	1.81111e23	0.486385	0.243193	−	0.969978i	$-0.421805\pi$
$593$	1.81111e23	0.486385	0.243193	+	0.969978i	$0.421805\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.50535e23	−0.635458
$598$	0	0
$599$	−1.26579e23	−0.312057	−0.156028	−	0.987753i	$-0.549869\pi$
$599$	−1.26579e23	−0.312057	−0.156028	+	0.987753i	$0.549869\pi$
$600$	0	0
$601$	−2.83226e23	−0.678734	−0.339367	−	0.940654i	$-0.610213\pi$
$601$	−2.83226e23	−0.678734	−0.339367	+	0.940654i	$0.610213\pi$
$602$	0	0
$603$	−5.97885e22	−0.139290
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7.56311e23	−1.66570	−0.832849	−	0.553500i	$-0.813292\pi$
$607$	−7.56311e23	−1.66570	−0.832849	+	0.553500i	$0.813292\pi$
$608$	0	0
$609$	4.92499e22	0.105477
$610$	0	0
$611$	7.02836e21	0.0146387
$612$	0	0
$613$	5.21867e23	1.05717	0.528586	−	0.848880i	$-0.322722\pi$
$613$	5.21867e23	1.05717	0.528586	+	0.848880i	$0.322722\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.91259e21	0.0190004	0.00950021	−	0.999955i	$-0.496976\pi$
$617$	9.91259e21	0.0190004	0.00950021	+	0.999955i	$0.496976\pi$
$618$	0	0
$619$	7.36588e23	1.37358	0.686790	−	0.726856i	$-0.259019\pi$
$619$	7.36588e23	1.37358	0.686790	+	0.726856i	$0.259019\pi$
$620$	0	0
$621$	−8.17284e21	−0.0148284
$622$	0	0
$623$	4.28623e23	0.756705
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.02607e23	−0.338751
$628$	0	0
$629$	7.83757e22	0.127541
$630$	0	0
$631$	6.98066e23	1.10572	0.552862	−	0.833273i	$-0.313536\pi$
$631$	6.98066e23	1.10572	0.552862	+	0.833273i	$0.313536\pi$
$632$	0	0
$633$	−3.12167e23	−0.481344
$634$	0	0
$635$	0	0
$636$	0	0
$637$	5.25978e21	0.00768745
$638$	0	0
$639$	6.96202e23	0.990782
$640$	0	0
$641$	6.23825e23	0.864509	0.432254	−	0.901752i	$-0.357718\pi$
$641$	6.23825e23	0.864509	0.432254	+	0.901752i	$0.357718\pi$
$642$	0	0
$643$	3.46371e23	0.467465	0.233732	−	0.972301i	$-0.424906\pi$
$643$	3.46371e23	0.467465	0.233732	+	0.972301i	$0.424906\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.47826e23	−0.317293	−0.158647	−	0.987335i	$-0.550713\pi$
$647$	−2.47826e23	−0.317293	−0.158647	+	0.987335i	$0.550713\pi$
$648$	0	0
$649$	4.71361e23	0.587860
$650$	0	0
$651$	1.59032e23	0.193217
$652$	0	0
$653$	−1.24507e24	−1.47378	−0.736891	−	0.676011i	$-0.763707\pi$
$653$	−1.24507e24	−1.47378	−0.736891	+	0.676011i	$0.763707\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.19459e24	1.34249
$658$	0	0
$659$	1.18538e24	1.29817	0.649085	−	0.760716i	$-0.275152\pi$
$659$	1.18538e24	1.29817	0.649085	+	0.760716i	$0.275152\pi$
$660$	0	0
$661$	−7.63244e23	−0.814613	−0.407306	−	0.913292i	$-0.633532\pi$
$661$	−7.63244e23	−0.814613	−0.407306	+	0.913292i	$0.633532\pi$
$662$	0	0
$663$	3.88443e20	0.000404075 0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.93295e21	0.00784089
$668$	0	0
$669$	5.77738e23	0.556684
$670$	0	0
$671$	9.86930e23	0.927139
$672$	0	0
$673$	1.42179e23	0.130229	0.0651144	−	0.997878i	$-0.479259\pi$
$673$	1.42179e23	0.130229	0.0651144	+	0.997878i	$0.479259\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.54090e24	1.34205	0.671027	−	0.741433i	$-0.265853\pi$
$677$	1.54090e24	1.34205	0.671027	+	0.741433i	$0.265853\pi$
$678$	0	0
$679$	−1.92868e23	−0.163820
$680$	0	0
$681$	−7.34821e23	−0.608739
$682$	0	0
$683$	−2.32059e24	−1.87509	−0.937545	−	0.347865i	$-0.886907\pi$
$683$	−2.32059e24	−1.87509	−0.937545	+	0.347865i	$0.886907\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−8.96109e23	−0.689015
$688$	0	0
$689$	−1.68689e22	−0.0126538
$690$	0	0
$691$	3.27287e23	0.239533	0.119766	−	0.992802i	$-0.461785\pi$
$691$	3.27287e23	0.239533	0.119766	+	0.992802i	$0.461785\pi$
$692$	0	0
$693$	−6.02507e23	−0.430258
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.48849e22	0.0441230
$698$	0	0
$699$	1.73272e23	0.114993
$700$	0	0
$701$	5.06478e22	0.0328063	0.0164032	−	0.999865i	$-0.494778\pi$
$701$	5.06478e22	0.0328063	0.0164032	+	0.999865i	$0.494778\pi$
$702$	0	0
$703$	2.07978e24	1.31491
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.80610e23	0.229306
$708$	0	0
$709$	2.97489e24	1.74975	0.874877	−	0.484344i	$-0.160942\pi$
$709$	2.97489e24	1.74975	0.874877	+	0.484344i	$0.160942\pi$
$710$	0	0
$711$	1.25121e24	0.718520
$712$	0	0
$713$	2.56161e22	0.0143632
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.37509e24	−0.735220
$718$	0	0
$719$	2.64437e24	1.38079	0.690393	−	0.723435i	$-0.257438\pi$
$719$	2.64437e24	1.38079	0.690393	+	0.723435i	$0.257438\pi$
$720$	0	0
$721$	9.56091e23	0.487584
$722$	0	0
$723$	−1.07789e24	−0.536907
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−3.89057e24	−1.84914	−0.924572	−	0.381007i	$-0.875577\pi$
$727$	−3.89057e24	−1.84914	−0.924572	+	0.381007i	$0.875577\pi$
$728$	0	0
$729$	−4.82447e22	−0.0224009
$730$	0	0
$731$	−2.21445e23	−0.100454
$732$	0	0
$733$	2.92056e24	1.29444	0.647220	−	0.762304i	$-0.275932\pi$
$733$	2.92056e24	1.29444	0.647220	+	0.762304i	$0.275932\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.79326e23	0.160523
$738$	0	0
$739$	−2.80453e24	−1.15980	−0.579898	−	0.814689i	$-0.696908\pi$
$739$	−2.80453e24	−1.15980	−0.579898	+	0.814689i	$0.696908\pi$
$740$	0	0
$741$	1.03077e22	0.00416590
$742$	0	0
$743$	3.22775e24	1.27496	0.637478	−	0.770468i	$-0.279978\pi$
$743$	3.22775e24	1.27496	0.637478	+	0.770468i	$0.279978\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.67247e24	1.00853
$748$	0	0
$749$	−1.71819e24	−0.633833
$750$	0	0
$751$	9.11135e23	0.328582	0.164291	−	0.986412i	$-0.447466\pi$
$751$	9.11135e23	0.328582	0.164291	+	0.986412i	$0.447466\pi$
$752$	0	0
$753$	−4.40764e23	−0.155399
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−7.82526e23	−0.263745	−0.131872	−	0.991267i	$-0.542099\pi$
$757$	−7.82526e23	−0.263745	−0.131872	+	0.991267i	$0.542099\pi$
$758$	0	0
$759$	2.31621e22	0.00763349
$760$	0	0
$761$	−2.33932e24	−0.753912	−0.376956	−	0.926231i	$-0.623029\pi$
$761$	−2.33932e24	−0.753912	−0.376956	+	0.926231i	$0.623029\pi$
$762$	0	0
$763$	−6.35767e23	−0.200373
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.39807e22	−0.00722939
$768$	0	0
$769$	−3.17224e24	−0.935388	−0.467694	−	0.883891i	$-0.654915\pi$
$769$	−3.17224e24	−0.935388	−0.467694	+	0.883891i	$0.654915\pi$
$770$	0	0
$771$	4.48648e23	0.129403
$772$	0	0
$773$	1.00492e24	0.283534	0.141767	−	0.989900i	$-0.454722\pi$
$773$	1.00492e24	0.283534	0.141767	+	0.989900i	$0.454722\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.47609e24	−0.398596
$778$	0	0
$779$	1.72179e24	0.454895
$780$	0	0
$781$	−4.41702e24	−1.14182
$782$	0	0
$783$	−1.31568e24	−0.332794
$784$	0	0
$785$	0	0
$786$	0	0
$787$	8.15000e24	1.97412	0.987058	−	0.160365i	$-0.0512671\pi$
$787$	8.15000e24	1.97412	0.987058	+	0.160365i	$0.0512671\pi$
$788$	0	0
$789$	−2.42054e24	−0.573796
$790$	0	0
$791$	1.58745e24	0.368299
$792$	0	0
$793$	−5.02106e22	−0.0114018
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.83723e24	0.834876	0.417438	−	0.908705i	$-0.362928\pi$
$797$	3.83723e24	0.834876	0.417438	+	0.908705i	$0.362928\pi$
$798$	0	0
$799$	4.83294e23	0.102935
$800$	0	0
$801$	−5.11483e24	−1.06649
$802$	0	0
$803$	−7.57901e24	−1.54714
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.07371e24	−0.405810
$808$	0	0
$809$	−8.12096e24	−1.55613	−0.778064	−	0.628185i	$-0.783798\pi$
$809$	−8.12096e24	−1.55613	−0.778064	+	0.628185i	$0.783798\pi$
$810$	0	0
$811$	6.16668e24	1.15711	0.578554	−	0.815644i	$-0.303617\pi$
$811$	6.16668e24	1.15711	0.578554	+	0.815644i	$0.303617\pi$
$812$	0	0
$813$	−6.53550e23	−0.120091
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−5.87627e24	−1.03565
$818$	0	0
$819$	3.06528e22	0.00529124
$820$	0	0
$821$	−3.44858e24	−0.583074	−0.291537	−	0.956560i	$-0.594167\pi$
$821$	−3.44858e24	−0.583074	−0.291537	+	0.956560i	$0.594167\pi$
$822$	0	0
$823$	−5.80386e23	−0.0961210	−0.0480605	−	0.998844i	$-0.515304\pi$
$823$	−5.80386e23	−0.0961210	−0.0480605	+	0.998844i	$0.515304\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.09602e24	0.492048	0.246024	−	0.969264i	$-0.420876\pi$
$827$	3.09602e24	0.492048	0.246024	+	0.969264i	$0.420876\pi$
$828$	0	0
$829$	−3.40942e24	−0.530844	−0.265422	−	0.964132i	$-0.585511\pi$
$829$	−3.40942e24	−0.530844	−0.265422	+	0.964132i	$0.585511\pi$
$830$	0	0
$831$	1.84768e24	0.281851
$832$	0	0
$833$	3.61680e23	0.0540558
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4.24843e24	−0.609624
$838$	0	0
$839$	−9.38827e24	−1.32011	−0.660053	−	0.751219i	$-0.729466\pi$
$839$	−9.38827e24	−1.32011	−0.660053	+	0.751219i	$0.729466\pi$
$840$	0	0
$841$	−5.98009e24	−0.824027
$842$	0	0
$843$	−1.08796e24	−0.146919
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5.93407e23	−0.0769739
$848$	0	0
$849$	−5.85276e24	−0.744125
$850$	0	0
$851$	−2.37761e23	−0.0296306
$852$	0	0
$853$	−9.91412e23	−0.121112	−0.0605560	−	0.998165i	$-0.519287\pi$
$853$	−9.91412e23	−0.121112	−0.0605560	+	0.998165i	$0.519287\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.06301e24	−0.829188	−0.414594	−	0.910007i	$-0.636076\pi$
$857$	−7.06301e24	−0.829188	−0.414594	+	0.910007i	$0.636076\pi$
$858$	0	0
$859$	−1.24591e24	−0.143398	−0.0716992	−	0.997426i	$-0.522842\pi$
$859$	−1.24591e24	−0.143398	−0.0716992	+	0.997426i	$0.522842\pi$
$860$	0	0
$861$	−1.22201e24	−0.137895
$862$	0	0
$863$	1.46337e25	1.61906	0.809528	−	0.587082i	$-0.199723\pi$
$863$	1.46337e25	1.61906	0.809528	+	0.587082i	$0.199723\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−4.09976e24	−0.436110
$868$	0	0
$869$	−7.93826e24	−0.828051
$870$	0	0
$871$	−1.92984e22	−0.00197409
$872$	0	0
$873$	2.30153e24	0.230885
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5.04985e24	0.487284	0.243642	−	0.969865i	$-0.421658\pi$
$877$	5.04985e24	0.487284	0.243642	+	0.969865i	$0.421658\pi$
$878$	0	0
$879$	1.99747e24	0.189049
$880$	0	0
$881$	−1.61016e25	−1.49477	−0.747384	−	0.664393i	$-0.768690\pi$
$881$	−1.61016e25	−1.49477	−0.747384	+	0.664393i	$0.768690\pi$
$882$	0	0
$883$	−1.06552e25	−0.970273	−0.485137	−	0.874438i	$-0.661230\pi$
$883$	−1.06552e25	−0.970273	−0.485137	+	0.874438i	$0.661230\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−5.73815e24	−0.502830	−0.251415	−	0.967879i	$-0.580896\pi$
$887$	−5.73815e24	−0.502830	−0.251415	+	0.967879i	$0.580896\pi$
$888$	0	0
$889$	2.44443e24	0.210142
$890$	0	0
$891$	5.06433e24	0.427132
$892$	0	0
$893$	1.28247e25	1.06123
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.17839e21	−9.38753e−5 0
$898$	0	0
$899$	4.12373e24	0.322354
$900$	0	0
$901$	−1.15996e24	−0.0889780
$902$	0	0
$903$	4.17058e24	0.313943
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.60345e25	1.16250	0.581250	−	0.813725i	$-0.302564\pi$
$907$	1.60345e25	1.16250	0.581250	+	0.813725i	$0.302564\pi$
$908$	0	0
$909$	−4.54188e24	−0.323179
$910$	0	0
$911$	2.07808e25	1.45130	0.725648	−	0.688066i	$-0.241540\pi$
$911$	2.07808e25	1.45130	0.725648	+	0.688066i	$0.241540\pi$
$912$	0	0
$913$	−1.69554e25	−1.16227
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.32085e24	0.483529
$918$	0	0
$919$	−1.13639e25	−0.736796	−0.368398	−	0.929668i	$-0.620094\pi$
$919$	−1.13639e25	−0.736796	−0.368398	+	0.929668i	$0.620094\pi$
$920$	0	0
$921$	2.31770e24	0.147520
$922$	0	0
$923$	2.24718e23	0.0140418
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.14092e25	−0.687191
$928$	0	0
$929$	1.86032e25	1.10015	0.550077	−	0.835114i	$-0.314598\pi$
$929$	1.86032e25	1.10015	0.550077	+	0.835114i	$0.314598\pi$
$930$	0	0
$931$	9.59757e24	0.557300
$932$	0	0
$933$	−7.73030e24	−0.440760
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.86174e25	1.02360	0.511802	−	0.859103i	$-0.328978\pi$
$937$	1.86174e25	1.02360	0.511802	+	0.859103i	$0.328978\pi$
$938$	0	0
$939$	1.10826e25	0.598390
$940$	0	0
$941$	−2.30780e25	−1.22373	−0.611865	−	0.790962i	$-0.709581\pi$
$941$	−2.30780e25	−1.22373	−0.611865	+	0.790962i	$0.709581\pi$
$942$	0	0
$943$	−1.96836e23	−0.0102507
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.45502e24	−0.0730959	−0.0365479	−	0.999332i	$-0.511636\pi$
$947$	−1.45502e24	−0.0730959	−0.0365479	+	0.999332i	$0.511636\pi$
$948$	0	0
$949$	3.85586e23	0.0190265
$950$	0	0
$951$	−8.81764e24	−0.427384
$952$	0	0
$953$	−1.25990e25	−0.599854	−0.299927	−	0.953962i	$-0.596962\pi$
$953$	−1.25990e25	−0.599854	−0.299927	+	0.953962i	$0.596962\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.72868e24	0.171318
$958$	0	0
$959$	2.14424e25	0.967864
$960$	0	0
$961$	−9.23427e24	−0.409500
$962$	0	0
$963$	2.05034e25	0.893311
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.96037e24	0.0824543	0.0412271	−	0.999150i	$-0.486873\pi$
$967$	1.96037e24	0.0824543	0.0412271	+	0.999150i	$0.486873\pi$
$968$	0	0
$969$	7.08795e23	0.0292933
$970$	0	0
$971$	4.74466e25	1.92682	0.963411	−	0.268029i	$-0.0863724\pi$
$971$	4.74466e25	1.92682	0.963411	+	0.268029i	$0.0863724\pi$
$972$	0	0
$973$	−1.02314e25	−0.408298
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.12147e25	−0.432199	−0.216099	−	0.976371i	$-0.569333\pi$
$977$	−1.12147e25	−0.432199	−0.216099	+	0.976371i	$0.569333\pi$
$978$	0	0
$979$	3.24508e25	1.22906
$980$	0	0
$981$	7.58672e24	0.282402
$982$	0	0
$983$	1.85575e25	0.678912	0.339456	−	0.940622i	$-0.389757\pi$
$983$	1.85575e25	0.678912	0.339456	+	0.940622i	$0.389757\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−9.10210e24	−0.321696
$988$	0	0
$989$	6.71778e23	0.0233377
$990$	0	0
$991$	4.41082e24	0.150624	0.0753119	−	0.997160i	$-0.476005\pi$
$991$	4.41082e24	0.150624	0.0753119	+	0.997160i	$0.476005\pi$
$992$	0	0
$993$	1.40281e25	0.470902
$994$	0	0
$995$	0	0
$996$	0	0
$997$	6.58711e24	0.213691	0.106845	−	0.994276i	$-0.465925\pi$
$997$	6.58711e24	0.213691	0.106845	+	0.994276i	$0.465925\pi$
$998$	0	0
$999$	3.94327e25	1.25762

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	100.18.a.f.1.5		8
5.2	odd	4		20.18.c.a.9.4	✓	8
5.3	odd	4		20.18.c.a.9.5	yes	8
5.4	even	2	inner	100.18.a.f.1.4		8
20.3	even	4		80.18.c.c.49.4		8
20.7	even	4		80.18.c.c.49.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
20.18.c.a.9.4	✓	8	5.2	odd	4
20.18.c.a.9.5	yes	8	5.3	odd	4
80.18.c.c.49.4		8	20.3	even	4
80.18.c.c.49.5		8	20.7	even	4
100.18.a.f.1.4		8	5.4	even	2	inner
100.18.a.f.1.5		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 100 = 2^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	100.a (trivial)

\(f(q)\)	\(=\)	\(q+4988.24 q^{3} +8.73679e6 q^{7} -1.04258e8 q^{9} +6.61458e8 q^{11} -3.36520e7 q^{13} -2.31403e9 q^{17} -6.14052e10 q^{19} +4.35812e10 q^{21} +7.01987e9 q^{23} -1.16424e12 q^{27} +1.13007e12 q^{29} +3.64909e12 q^{31} +3.29951e12 q^{33} -3.38698e13 q^{37} -1.67864e11 q^{39} -2.80398e13 q^{41} +9.56968e13 q^{43} -2.08854e14 q^{47} -1.56299e14 q^{49} -1.15429e13 q^{51} +5.01273e14 q^{53} -3.06304e14 q^{57} +7.12609e14 q^{59} +1.49205e15 q^{61} -9.10877e14 q^{63} +5.73469e14 q^{67} +3.50168e13 q^{69} -6.67771e15 q^{71} -1.14580e16 q^{73} +5.77902e15 q^{77} -1.20012e16 q^{79} +7.65632e15 q^{81} -2.56333e16 q^{83} +5.63707e15 q^{87} +4.90595e16 q^{89} -2.94011e14 q^{91} +1.82025e16 q^{93} -2.20754e16 q^{97} -6.89620e16 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 376848248 q^{9} - 396200640 q^{11} - 11821646592 q^{19} - 420670059472 q^{21} - 1543712861232 q^{29} - 13722543013312 q^{31} + 79298226023808 q^{39} + 29824965017568 q^{41} + 535615080275736 q^{49}+ \cdots + 23\!\cdots\!60 q^{99}+O(q^{100}) \) 8 * q + 376848248 * q^9 - 396200640 * q^11 - 11821646592 * q^19 - 420670059472 * q^21 - 1543712861232 * q^29 - 13722543013312 * q^31 + 79298226023808 * q^39 + 29824965017568 * q^41 + 535615080275736 * q^49 - 764079320419264 * q^51 - 94539545151744 * q^59 - 3380827226969024 * q^61 - 5559752175554416 * q^69 + 20155606060098624 * q^71 + 3634057920126144 * q^79 + 15949137882627512 * q^81 - 103487949043655568 * q^89 - 69783870139121088 * q^91 + 239420571619653760 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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