LMFDB - Embedded newform 225.10.b.e.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,10,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("225.199");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	225.b (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:	$115.883063137$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.10.b.e.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} +496.000 q^{4} -7680.00i q^{7} +4032.00i q^{8} +O(q^{10})$	$q+4.00000i q^{2} +496.000 q^{4} -7680.00i q^{7} +4032.00i q^{8} +86404.0 q^{11} +149978. i q^{13} +30720.0 q^{14} +237824. q^{16} +207622. i q^{17} -716284. q^{19} +345616. i q^{22} +1.36992e6i q^{23} -599912. q^{26} -3.80928e6i q^{28} -3.19440e6 q^{29} -2.34900e6 q^{31} +3.01568e6i q^{32} -830488. q^{34} +1.87357e7i q^{37} -2.86514e6i q^{38} +2.92826e7 q^{41} +1.51672e6i q^{43} +4.28564e7 q^{44} -5.47968e6 q^{46} -615752. i q^{47} -1.86288e7 q^{49} +7.43891e7i q^{52} +4.74743e6i q^{53} +3.09658e7 q^{56} -1.27776e7i q^{58} +6.06161e7 q^{59} -1.26746e8 q^{61} -9.39600e6i q^{62} +1.09703e8 q^{64} -1.11183e8i q^{67} +1.02981e8i q^{68} +1.75552e8 q^{71} +6.12334e7i q^{73} -7.49428e7 q^{74} -3.55277e8 q^{76} -6.63583e8i q^{77} -2.34431e8 q^{79} +1.17131e8i q^{82} +1.18910e8i q^{83} -6.06690e6 q^{86} +3.48381e8i q^{88} -3.16534e8 q^{89} +1.15183e9 q^{91} +6.79480e8i q^{92} +2.46301e6 q^{94} +2.42912e8i q^{97} -7.45152e7i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 992 q^{4}+O(q^{10}) $ 2 * q + 992 * q^4	$ 2 q + 992 q^{4} + 172808 q^{11} + 61440 q^{14} + 475648 q^{16} - 1432568 q^{19} - 1199824 q^{26} - 6388804 q^{29} - 4698000 q^{31} - 1660976 q^{34} + 58565260 q^{41} + 85712768 q^{44} - 10959360 q^{46} - 37257586 q^{49} + 61931520 q^{56} + 121232152 q^{59} - 253491364 q^{61} + 219406336 q^{64} + 351103216 q^{71} - 149885680 q^{74} - 710553728 q^{76} - 468862320 q^{79} - 12133792 q^{86} - 633068652 q^{89} + 2303662080 q^{91} + 4926016 q^{94}+O(q^{100}) $ 2 * q + 992 * q^4 + 172808 * q^11 + 61440 * q^14 + 475648 * q^16 - 1432568 * q^19 - 1199824 * q^26 - 6388804 * q^29 - 4698000 * q^31 - 1660976 * q^34 + 58565260 * q^41 + 85712768 * q^44 - 10959360 * q^46 - 37257586 * q^49 + 61931520 * q^56 + 121232152 * q^59 - 253491364 * q^61 + 219406336 * q^64 + 351103216 * q^71 - 149885680 * q^74 - 710553728 * q^76 - 468862320 * q^79 - 12133792 * q^86 - 633068652 * q^89 + 2303662080 * q^91 + 4926016 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	4.00000i	0.176777i	0.996086	+	0.0883883i	$0.0281716\pi$
$2$	4.00000i	0.176777i	−0.996086	+	0.0883883i	$0.971828\pi$
$3$	0	0
$3$	0	0
$4$	496.000	0.968750
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 7680.00i	− 1.20898i	−0.796612	−	0.604491i	$-0.793376\pi$
$7$	− 7680.00i	− 1.20898i	0.796612	−	0.604491i	$-0.206624\pi$
$8$	4032.00i	0.348029i
$9$	0	0
$10$	0	0
$11$	86404.0	1.77937	0.889686	−	0.456573i	$-0.150923\pi$
$11$	86404.0	1.77937	0.889686	+	0.456573i	$0.150923\pi$
$12$	0	0
$13$	149978.i	1.45641i	0.685361	+	0.728203i	$0.259644\pi$
$13$	149978.i	1.45641i	−0.685361	+	0.728203i	$0.740356\pi$
$14$	30720.0	0.213720
$15$	0	0
$16$	237824.	0.907227
$17$	207622.i	0.602911i	0.953480	+	0.301456i	$0.0974725\pi$
$17$	207622.i	0.602911i	−0.953480	+	0.301456i	$0.902528\pi$
$18$	0	0
$19$	−716284.	−1.26094	−0.630469	−	0.776214i	$-0.717138\pi$
$19$	−716284.	−1.26094	−0.630469	+	0.776214i	$0.717138\pi$
$20$	0	0
$21$	0	0
$22$	345616.i	0.314552i
$23$	1.36992e6i	1.02075i	0.859952	+	0.510376i	$0.170494\pi$
$23$	1.36992e6i	1.02075i	−0.859952	+	0.510376i	$0.829506\pi$
$24$	0	0
$25$	0	0
$26$	−599912.	−0.257459
$27$	0	0
$28$	− 3.80928e6i	− 1.17120i
$29$	−3.19440e6	−0.838684	−0.419342	−	0.907828i	$-0.637739\pi$
$29$	−3.19440e6	−0.838684	−0.419342	+	0.907828i	$0.637739\pi$
$30$	0	0
$31$	−2.34900e6	−0.456831	−0.228415	−	0.973564i	$-0.573354\pi$
$31$	−2.34900e6	−0.456831	−0.228415	+	0.973564i	$0.573354\pi$
$32$	3.01568e6i	0.508406i
$33$	0	0
$34$	−830488.	−0.106581
$35$	0	0
$36$	0	0
$37$	1.87357e7i	1.64347i	0.569868	+	0.821736i	$0.306994\pi$
$37$	1.87357e7i	1.64347i	−0.569868	+	0.821736i	$0.693006\pi$
$38$	− 2.86514e6i	− 0.222905i
$39$	0	0
$40$	0	0
$41$	2.92826e7	1.61839	0.809194	−	0.587541i	$-0.199904\pi$
$41$	2.92826e7	1.61839	0.809194	+	0.587541i	$0.199904\pi$
$42$	0	0
$43$	1.51672e6i	0.0676548i	0.999428	+	0.0338274i	$0.0107696\pi$
$43$	1.51672e6i	0.0676548i	−0.999428	+	0.0338274i	$0.989230\pi$
$44$	4.28564e7	1.72377
$45$	0	0
$46$	−5.47968e6	−0.180445
$47$	− 615752.i	− 0.0184063i	−0.999958	−	0.00920313i	$-0.997071\pi$
$47$	− 615752.i	− 0.0184063i	0.999958	−	0.00920313i	$-0.00292949\pi$
$48$	0	0
$49$	−1.86288e7	−0.461639
$50$	0	0
$51$	0	0
$52$	7.43891e7i	1.41089i
$53$	4.74743e6i	0.0826451i	0.999146	+	0.0413226i	$0.0131571\pi$
$53$	4.74743e6i	0.0826451i	−0.999146	+	0.0413226i	$0.986843\pi$
$54$	0	0
$55$	0	0
$56$	3.09658e7	0.420761
$57$	0	0
$58$	− 1.27776e7i	− 0.148260i
$59$	6.06161e7	0.651259	0.325630	−	0.945497i	$-0.394424\pi$
$59$	6.06161e7	0.651259	0.325630	+	0.945497i	$0.394424\pi$
$60$	0	0
$61$	−1.26746e8	−1.17206	−0.586029	−	0.810290i	$-0.699309\pi$
$61$	−1.26746e8	−1.17206	−0.586029	+	0.810290i	$0.699309\pi$
$62$	− 9.39600e6i	− 0.0807570i
$63$	0	0
$64$	1.09703e8	0.817352
$65$	0	0
$66$	0	0
$67$	− 1.11183e8i	− 0.674063i	−0.941493	−	0.337031i	$-0.890577\pi$
$67$	− 1.11183e8i	− 0.674063i	0.941493	−	0.337031i	$-0.109423\pi$
$68$	1.02981e8i	0.584070i
$69$	0	0
$70$	0	0
$71$	1.75552e8	0.819865	0.409932	−	0.912116i	$-0.365552\pi$
$71$	1.75552e8	0.819865	0.409932	+	0.912116i	$0.365552\pi$
$72$	0	0
$73$	6.12334e7i	0.252369i	0.992007	+	0.126184i	$0.0402731\pi$
$73$	6.12334e7i	0.252369i	−0.992007	+	0.126184i	$0.959727\pi$
$74$	−7.49428e7	−0.290528
$75$	0	0
$76$	−3.55277e8	−1.22153
$77$	− 6.63583e8i	− 2.15123i
$78$	0	0
$79$	−2.34431e8	−0.677163	−0.338582	−	0.940937i	$-0.609947\pi$
$79$	−2.34431e8	−0.677163	−0.338582	+	0.940937i	$0.609947\pi$
$80$	0	0
$81$	0	0
$82$	1.17131e8i	0.286093i
$83$	1.18910e8i	0.275023i	0.990500	+	0.137511i	$0.0439103\pi$
$83$	1.18910e8i	0.275023i	−0.990500	+	0.137511i	$0.956090\pi$
$84$	0	0
$85$	0	0
$86$	−6.06690e6	−0.0119598
$87$	0	0
$88$	3.48381e8i	0.619273i
$89$	−3.16534e8	−0.534768	−0.267384	−	0.963590i	$-0.586159\pi$
$89$	−3.16534e8	−0.534768	−0.267384	+	0.963590i	$0.586159\pi$
$90$	0	0
$91$	1.15183e9	1.76077
$92$	6.79480e8i	0.988853i
$93$	0	0
$94$	2.46301e6	0.00325380
$95$	0	0
$96$	0	0
$97$	2.42912e8i	0.278597i	0.990250	+	0.139299i	$0.0444848\pi$
$97$	2.42912e8i	0.278597i	−0.990250	+	0.139299i	$0.955515\pi$
$98$	− 7.45152e7i	− 0.0816070i
$99$	0	0
$100$	0	0
$101$	6.53803e8	0.625173	0.312587	−	0.949889i	$-0.398805\pi$
$101$	6.53803e8	0.625173	0.312587	+	0.949889i	$0.398805\pi$
$102$	0	0
$103$	− 1.40420e9i	− 1.22931i	−0.788795	−	0.614656i	$-0.789295\pi$
$103$	− 1.40420e9i	− 1.22931i	0.788795	−	0.614656i	$-0.210705\pi$
$104$	−6.04711e8	−0.506872
$105$	0	0
$106$	−1.89897e7	−0.0146097
$107$	1.83854e9i	1.35595i	0.735083	+	0.677977i	$0.237143\pi$
$107$	1.83854e9i	1.35595i	−0.735083	+	0.677977i	$0.762857\pi$
$108$	0	0
$109$	9.33452e8	0.633392	0.316696	−	0.948527i	$-0.397426\pi$
$109$	9.33452e8	0.633392	0.316696	+	0.948527i	$0.397426\pi$
$110$	0	0
$111$	0	0
$112$	− 1.82649e9i	− 1.09682i
$113$	− 9.28534e7i	− 0.0535728i	−0.999641	−	0.0267864i	$-0.991473\pi$
$113$	− 9.28534e7i	− 0.0535728i	0.999641	−	0.0267864i	$-0.00852740\pi$
$114$	0	0
$115$	0	0
$116$	−1.58442e9	−0.812476
$117$	0	0
$118$	2.42464e8i	0.115127i
$119$	1.59454e9	0.728909
$120$	0	0
$121$	5.10770e9	2.16616
$122$	− 5.06983e8i	− 0.207192i
$123$	0	0
$124$	−1.16510e9	−0.442555
$125$	0	0
$126$	0	0
$127$	1.73819e9i	0.592900i	0.955048	+	0.296450i	$0.0958028\pi$
$127$	1.73819e9i	0.592900i	−0.955048	+	0.296450i	$0.904197\pi$
$128$	1.98284e9i	0.652894i
$129$	0	0
$130$	0	0
$131$	2.49730e9	0.740882	0.370441	−	0.928856i	$-0.379207\pi$
$131$	2.49730e9	0.740882	0.370441	+	0.928856i	$0.379207\pi$
$132$	0	0
$133$	5.50106e9i	1.52445i
$134$	4.44731e8	0.119159
$135$	0	0
$136$	−8.37132e8	−0.209831
$137$	7.96226e9i	1.93105i	0.260306	+	0.965526i	$0.416177\pi$
$137$	7.96226e9i	1.93105i	−0.260306	+	0.965526i	$0.583823\pi$
$138$	0	0
$139$	2.85565e9	0.648842	0.324421	−	0.945913i	$-0.394831\pi$
$139$	2.85565e9	0.648842	0.324421	+	0.945913i	$0.394831\pi$
$140$	0	0
$141$	0	0
$142$	7.02206e8i	0.144933i
$143$	1.29587e10i	2.59149i
$144$	0	0
$145$	0	0
$146$	−2.44933e8	−0.0446129
$147$	0	0
$148$	9.29291e9i	1.59211i
$149$	−9.63383e9	−1.60126	−0.800628	−	0.599161i	$-0.795501\pi$
$149$	−9.63383e9	−1.60126	−0.800628	+	0.599161i	$0.795501\pi$
$150$	0	0
$151$	−5.38292e9	−0.842601	−0.421300	−	0.906921i	$-0.638426\pi$
$151$	−5.38292e9	−0.842601	−0.421300	+	0.906921i	$0.638426\pi$
$152$	− 2.88806e9i	− 0.438843i
$153$	0	0
$154$	2.65433e9	0.380287
$155$	0	0
$156$	0	0
$157$	5.19434e8i	0.0682310i	0.999418	+	0.0341155i	$0.0108614\pi$
$157$	5.19434e8i	0.0682310i	−0.999418	+	0.0341155i	$0.989139\pi$
$158$	− 9.37725e8i	− 0.119707i
$159$	0	0
$160$	0	0
$161$	1.05210e10	1.23407
$162$	0	0
$163$	− 9.41239e9i	− 1.04437i	−0.852831	−	0.522187i	$-0.825116\pi$
$163$	− 9.41239e9i	− 1.04437i	0.852831	−	0.522187i	$-0.174884\pi$
$164$	1.45242e10	1.56781
$165$	0	0
$166$	−4.75642e8	−0.0486176
$167$	− 9.37241e9i	− 0.932453i	−0.884665	−	0.466227i	$-0.845613\pi$
$167$	− 9.37241e9i	− 0.932453i	0.884665	−	0.466227i	$-0.154387\pi$
$168$	0	0
$169$	−1.18889e10	−1.12112
$170$	0	0
$171$	0	0
$172$	7.52295e8i	0.0655406i
$173$	1.23573e10i	1.04886i	0.851455	+	0.524428i	$0.175721\pi$
$173$	1.23573e10i	1.04886i	−0.851455	+	0.524428i	$0.824279\pi$
$174$	0	0
$175$	0	0
$176$	2.05489e10	1.61429
$177$	0	0
$178$	− 1.26614e9i	− 0.0945346i
$179$	−6.66040e8	−0.0484910	−0.0242455	−	0.999706i	$-0.507718\pi$
$179$	−6.66040e8	−0.0484910	−0.0242455	+	0.999706i	$0.507718\pi$
$180$	0	0
$181$	5.27207e9	0.365113	0.182557	−	0.983195i	$-0.441563\pi$
$181$	5.27207e9	0.365113	0.182557	+	0.983195i	$0.441563\pi$
$182$	4.60732e9i	0.311263i
$183$	0	0
$184$	−5.52352e9	−0.355251
$185$	0	0
$186$	0	0
$187$	1.79394e10i	1.07280i
$188$	− 3.05413e8i	− 0.0178311i
$189$	0	0
$190$	0	0
$191$	2.93896e10	1.59788	0.798939	−	0.601412i	$-0.205395\pi$
$191$	2.93896e10	1.59788	0.798939	+	0.601412i	$0.205395\pi$
$192$	0	0
$193$	1.48746e10i	0.771681i	0.922565	+	0.385841i	$0.126089\pi$
$193$	1.48746e10i	0.771681i	−0.922565	+	0.385841i	$0.873911\pi$
$194$	−9.71649e8	−0.0492495
$195$	0	0
$196$	−9.23988e9	−0.447213
$197$	− 4.98675e9i	− 0.235895i	−0.993020	−	0.117948i	$-0.962368\pi$
$197$	− 4.98675e9i	− 0.235895i	0.993020	−	0.117948i	$-0.0376315\pi$
$198$	0	0
$199$	−1.45527e10	−0.657816	−0.328908	−	0.944362i	$-0.606681\pi$
$199$	−1.45527e10	−0.657816	−0.328908	+	0.944362i	$0.606681\pi$
$200$	0	0
$201$	0	0
$202$	2.61521e9i	0.110516i
$203$	2.45330e10i	1.01395i
$204$	0	0
$205$	0	0
$206$	5.61681e9	0.217314
$207$	0	0
$208$	3.56684e10i	1.32129i
$209$	−6.18898e10	−2.24368
$210$	0	0
$211$	5.15407e10	1.79011	0.895054	−	0.445959i	$-0.147137\pi$
$211$	5.15407e10	1.79011	0.895054	+	0.445959i	$0.147137\pi$
$212$	2.35473e9i	0.0800624i
$213$	0	0
$214$	−7.35414e9	−0.239701
$215$	0	0
$216$	0	0
$217$	1.80403e10i	0.552300i
$218$	3.73381e9i	0.111969i
$219$	0	0
$220$	0	0
$221$	−3.11387e10	−0.878083
$222$	0	0
$223$	− 4.61272e10i	− 1.24907i	−0.780998	−	0.624533i	$-0.785289\pi$
$223$	− 4.61272e10i	− 1.24907i	0.780998	−	0.624533i	$-0.214711\pi$
$224$	2.31604e10	0.614654
$225$	0	0
$226$	3.71414e8	0.00947043
$227$	3.75833e10i	0.939460i	0.882810	+	0.469730i	$0.155649\pi$
$227$	3.75833e10i	0.939460i	−0.882810	+	0.469730i	$0.844351\pi$
$228$	0	0
$229$	6.41082e10	1.54047	0.770236	−	0.637759i	$-0.220138\pi$
$229$	6.41082e10	1.54047	0.770236	+	0.637759i	$0.220138\pi$
$230$	0	0
$231$	0	0
$232$	− 1.28798e10i	− 0.291887i
$233$	6.96578e10i	1.54835i	0.632973	+	0.774174i	$0.281834\pi$
$233$	6.96578e10i	1.54835i	−0.632973	+	0.774174i	$0.718166\pi$
$234$	0	0
$235$	0	0
$236$	3.00656e10	0.630907
$237$	0	0
$238$	6.37815e9i	0.128854i
$239$	6.65825e10	1.31999	0.659993	−	0.751272i	$-0.270559\pi$
$239$	6.65825e10	1.31999	0.659993	+	0.751272i	$0.270559\pi$
$240$	0	0
$241$	−4.41659e10	−0.843354	−0.421677	−	0.906746i	$-0.638558\pi$
$241$	−4.41659e10	−0.843354	−0.421677	+	0.906746i	$0.638558\pi$
$242$	2.04308e10i	0.382927i
$243$	0	0
$244$	−6.28659e10	−1.13543
$245$	0	0
$246$	0	0
$247$	− 1.07427e11i	− 1.83644i
$248$	− 9.47117e9i	− 0.158990i
$249$	0	0
$250$	0	0
$251$	−8.36236e10	−1.32983	−0.664916	−	0.746918i	$-0.731533\pi$
$251$	−8.36236e10	−1.32983	−0.664916	+	0.746918i	$0.731533\pi$
$252$	0	0
$253$	1.18367e11i	1.81630i
$254$	−6.95277e9	−0.104811
$255$	0	0
$256$	4.82367e10	0.701936
$257$	8.65274e10i	1.23724i	0.785690	+	0.618621i	$0.212308\pi$
$257$	8.65274e10i	1.23724i	−0.785690	+	0.618621i	$0.787692\pi$
$258$	0	0
$259$	1.43890e11	1.98693
$260$	0	0
$261$	0	0
$262$	9.98919e9i	0.130971i
$263$	− 9.61535e10i	− 1.23927i	−0.784892	−	0.619633i	$-0.787282\pi$
$263$	− 9.61535e10i	− 1.23927i	0.784892	−	0.619633i	$-0.212718\pi$
$264$	0	0
$265$	0	0
$266$	−2.20042e10	−0.269488
$267$	0	0
$268$	− 5.51466e10i	− 0.652998i
$269$	−1.09505e10	−0.127511	−0.0637557	−	0.997966i	$-0.520308\pi$
$269$	−1.09505e10	−0.127511	−0.0637557	+	0.997966i	$0.520308\pi$
$270$	0	0
$271$	7.80287e10	0.878805	0.439403	−	0.898290i	$-0.355190\pi$
$271$	7.80287e10	0.878805	0.439403	+	0.898290i	$0.355190\pi$
$272$	4.93775e10i	0.546977i
$273$	0	0
$274$	−3.18491e10	−0.341365
$275$	0	0
$276$	0	0
$277$	6.56840e10i	0.670349i	0.942156	+	0.335174i	$0.108795\pi$
$277$	6.56840e10i	0.670349i	−0.942156	+	0.335174i	$0.891205\pi$
$278$	1.14226e10i	0.114700i
$279$	0	0
$280$	0	0
$281$	6.44906e10	0.617046	0.308523	−	0.951217i	$-0.400165\pi$
$281$	6.44906e10	0.617046	0.308523	+	0.951217i	$0.400165\pi$
$282$	0	0
$283$	− 9.63133e10i	− 0.892580i	−0.894888	−	0.446290i	$-0.852745\pi$
$283$	− 9.63133e10i	− 0.892580i	0.894888	−	0.446290i	$-0.147255\pi$
$284$	8.70736e10	0.794244
$285$	0	0
$286$	−5.18348e10	−0.458115
$287$	− 2.24891e11i	− 1.95660i
$288$	0	0
$289$	7.54810e10	0.636498
$290$	0	0
$291$	0	0
$292$	3.03717e10i	0.244482i
$293$	8.16308e10i	0.647068i	0.946217	+	0.323534i	$0.104871\pi$
$293$	8.16308e10i	0.647068i	−0.946217	+	0.323534i	$0.895129\pi$
$294$	0	0
$295$	0	0
$296$	−7.55424e10	−0.571976
$297$	0	0
$298$	− 3.85353e10i	− 0.283065i
$299$	−2.05458e11	−1.48663
$300$	0	0
$301$	1.16484e10	0.0817935
$302$	− 2.15317e10i	− 0.148952i
$303$	0	0
$304$	−1.70350e11	−1.14396
$305$	0	0
$306$	0	0
$307$	− 2.95582e10i	− 0.189914i	−0.995481	−	0.0949568i	$-0.969729\pi$
$307$	− 2.95582e10i	− 0.189914i	0.995481	−	0.0949568i	$-0.0302713\pi$
$308$	− 3.29137e11i	− 2.08400i
$309$	0	0
$310$	0	0
$311$	3.99071e10	0.241896	0.120948	−	0.992659i	$-0.461407\pi$
$311$	3.99071e10	0.241896	0.120948	+	0.992659i	$0.461407\pi$
$312$	0	0
$313$	− 1.85371e11i	− 1.09167i	−0.837892	−	0.545836i	$-0.816212\pi$
$313$	− 1.85371e11i	− 1.09167i	0.837892	−	0.545836i	$-0.183788\pi$
$314$	−2.07774e9	−0.0120617
$315$	0	0
$316$	−1.16278e11	−0.656002
$317$	− 2.68895e11i	− 1.49560i	−0.663924	−	0.747800i	$-0.731110\pi$
$317$	− 2.68895e11i	− 1.49560i	0.663924	−	0.747800i	$-0.268890\pi$
$318$	0	0
$319$	−2.76009e11	−1.49233
$320$	0	0
$321$	0	0
$322$	4.20839e10i	0.218155i
$323$	− 1.48716e11i	− 0.760234i
$324$	0	0
$325$	0	0
$326$	3.76496e10	0.184621
$327$	0	0
$328$	1.18068e11i	0.563246i
$329$	−4.72898e9	−0.0222528
$330$	0	0
$331$	−4.29099e11	−1.96486	−0.982430	−	0.186629i	$-0.940244\pi$
$331$	−4.29099e11	−1.96486	−0.982430	+	0.186629i	$0.940244\pi$
$332$	5.89796e10i	0.266428i
$333$	0	0
$334$	3.74896e10	0.164836
$335$	0	0
$336$	0	0
$337$	− 2.02598e10i	− 0.0855657i	−0.999084	−	0.0427828i	$-0.986378\pi$
$337$	− 2.02598e10i	− 0.0855657i	0.999084	−	0.0427828i	$-0.0136224\pi$
$338$	− 4.75556e10i	− 0.198188i
$339$	0	0
$340$	0	0
$341$	−2.02963e11	−0.812872
$342$	0	0
$343$	− 1.66847e11i	− 0.650869i
$344$	−6.11543e9	−0.0235458
$345$	0	0
$346$	−4.94292e10	−0.185413
$347$	− 2.92783e10i	− 0.108409i	−0.998530	−	0.0542043i	$-0.982738\pi$
$347$	− 2.92783e10i	− 0.108409i	0.998530	−	0.0542043i	$-0.0172622\pi$
$348$	0	0
$349$	−7.05132e10	−0.254423	−0.127211	−	0.991876i	$-0.540603\pi$
$349$	−7.05132e10	−0.254423	−0.127211	+	0.991876i	$0.540603\pi$
$350$	0	0
$351$	0	0
$352$	2.60567e11i	0.904643i
$353$	− 6.57350e10i	− 0.225325i	−0.993633	−	0.112663i	$-0.964062\pi$
$353$	− 6.57350e10i	− 0.225325i	0.993633	−	0.112663i	$-0.0359380\pi$
$354$	0	0
$355$	0	0
$356$	−1.57001e11	−0.518057
$357$	0	0
$358$	− 2.66416e9i	− 0.00857209i
$359$	5.81702e11	1.84831	0.924157	−	0.382013i	$-0.124769\pi$
$359$	5.81702e11	1.84831	0.924157	+	0.382013i	$0.124769\pi$
$360$	0	0
$361$	1.90375e11	0.589967
$362$	2.10883e10i	0.0645435i
$363$	0	0
$364$	5.71308e11	1.70575
$365$	0	0
$366$	0	0
$367$	− 4.17070e11i	− 1.20008i	−0.799969	−	0.600042i	$-0.795151\pi$
$367$	− 4.17070e11i	− 1.20008i	0.799969	−	0.600042i	$-0.204849\pi$
$368$	3.25800e11i	0.926053i
$369$	0	0
$370$	0	0
$371$	3.64603e10	0.0999165
$372$	0	0
$373$	7.60417e10i	0.203405i	0.994815	+	0.101703i	$0.0324290\pi$
$373$	7.60417e10i	0.203405i	−0.994815	+	0.101703i	$0.967571\pi$
$374$	−7.17575e10	−0.189647
$375$	0	0
$376$	2.48271e9	0.00640591
$377$	− 4.79090e11i	− 1.22147i
$378$	0	0
$379$	1.79180e11	0.446080	0.223040	−	0.974809i	$-0.428402\pi$
$379$	1.79180e11	0.446080	0.223040	+	0.974809i	$0.428402\pi$
$380$	0	0
$381$	0	0
$382$	1.17558e11i	0.282467i
$383$	− 7.95018e11i	− 1.88792i	−0.330066	−	0.943958i	$-0.607071\pi$
$383$	− 7.95018e11i	− 1.88792i	0.330066	−	0.943958i	$-0.392929\pi$
$384$	0	0
$385$	0	0
$386$	−5.94985e10	−0.136415
$387$	0	0
$388$	1.20484e11i	0.269891i
$389$	−1.79533e11	−0.397532	−0.198766	−	0.980047i	$-0.563693\pi$
$389$	−1.79533e11	−0.397532	−0.198766	+	0.980047i	$0.563693\pi$
$390$	0	0
$391$	−2.84426e11	−0.615422
$392$	− 7.51113e10i	− 0.160664i
$393$	0	0
$394$	1.99470e10	0.0417008
$395$	0	0
$396$	0	0
$397$	− 3.43730e11i	− 0.694480i	−0.937776	−	0.347240i	$-0.887119\pi$
$397$	− 3.43730e11i	− 0.694480i	0.937776	−	0.347240i	$-0.112881\pi$
$398$	− 5.82108e10i	− 0.116287i
$399$	0	0
$400$	0	0
$401$	−7.72080e11	−1.49112	−0.745560	−	0.666438i	$-0.767818\pi$
$401$	−7.72080e11	−1.49112	−0.745560	+	0.666438i	$0.767818\pi$
$402$	0	0
$403$	− 3.52298e11i	− 0.665331i
$404$	3.24286e11	0.605637
$405$	0	0
$406$	−9.81320e10	−0.179244
$407$	1.61884e12i	2.92435i
$408$	0	0
$409$	−2.60632e11	−0.460546	−0.230273	−	0.973126i	$-0.573962\pi$
$409$	−2.60632e11	−0.460546	−0.230273	+	0.973126i	$0.573962\pi$
$410$	0	0
$411$	0	0
$412$	− 6.96484e11i	− 1.19090i
$413$	− 4.65531e11i	− 0.787361i
$414$	0	0
$415$	0	0
$416$	−4.52286e11	−0.740445
$417$	0	0
$418$	− 2.47559e11i	− 0.396630i
$419$	−5.60166e11	−0.887879	−0.443939	−	0.896057i	$-0.646419\pi$
$419$	−5.60166e11	−0.887879	−0.443939	+	0.896057i	$0.646419\pi$
$420$	0	0
$421$	1.68321e11	0.261137	0.130569	−	0.991439i	$-0.458320\pi$
$421$	1.68321e11	0.261137	0.130569	+	0.991439i	$0.458320\pi$
$422$	2.06163e11i	0.316449i
$423$	0	0
$424$	−1.91416e10	−0.0287629
$425$	0	0
$426$	0	0
$427$	9.73407e11i	1.41700i
$428$	9.11913e11i	1.31358i
$429$	0	0
$430$	0	0
$431$	−4.48383e11	−0.625895	−0.312948	−	0.949770i	$-0.601316\pi$
$431$	−4.48383e11	−0.625895	−0.312948	+	0.949770i	$0.601316\pi$
$432$	0	0
$433$	1.08485e12i	1.48311i	0.670891	+	0.741556i	$0.265912\pi$
$433$	1.08485e12i	1.48311i	−0.670891	+	0.741556i	$0.734088\pi$
$434$	−7.21613e10	−0.0976339
$435$	0	0
$436$	4.62992e11	0.613599
$437$	− 9.81252e11i	− 1.28711i
$438$	0	0
$439$	−4.60548e11	−0.591814	−0.295907	−	0.955217i	$-0.595622\pi$
$439$	−4.60548e11	−0.591814	−0.295907	+	0.955217i	$0.595622\pi$
$440$	0	0
$441$	0	0
$442$	− 1.24555e11i	− 0.155225i
$443$	− 1.32095e10i	− 0.0162956i	−0.999967	−	0.00814779i	$-0.997406\pi$
$443$	− 1.32095e10i	− 0.0162956i	0.999967	−	0.00814779i	$-0.00259355\pi$
$444$	0	0
$445$	0	0
$446$	1.84509e11	0.220806
$447$	0	0
$448$	− 8.42520e11i	− 0.988165i
$449$	−6.91889e11	−0.803393	−0.401696	−	0.915773i	$-0.631579\pi$
$449$	−6.91889e11	−0.803393	−0.401696	+	0.915773i	$0.631579\pi$
$450$	0	0
$451$	2.53014e12	2.87971
$452$	− 4.60553e10i	− 0.0518987i
$453$	0	0
$454$	−1.50333e11	−0.166075
$455$	0	0
$456$	0	0
$457$	− 3.73135e11i	− 0.400168i	−0.979779	−	0.200084i	$-0.935878\pi$
$457$	− 3.73135e11i	− 0.400168i	0.979779	−	0.200084i	$-0.0641216\pi$
$458$	2.56433e11i	0.272320i
$459$	0	0
$460$	0	0
$461$	1.45940e12	1.50494	0.752470	−	0.658627i	$-0.228862\pi$
$461$	1.45940e12	1.50494	0.752470	+	0.658627i	$0.228862\pi$
$462$	0	0
$463$	− 1.34213e11i	− 0.135732i	−0.997694	−	0.0678658i	$-0.978381\pi$
$463$	− 1.34213e11i	− 0.135732i	0.997694	−	0.0678658i	$-0.0216190\pi$
$464$	−7.59705e11	−0.760877
$465$	0	0
$466$	−2.78631e11	−0.273712
$467$	3.64531e10i	0.0354657i	0.999843	+	0.0177329i	$0.00564484\pi$
$467$	3.64531e10i	0.0354657i	−0.999843	+	0.0177329i	$0.994355\pi$
$468$	0	0
$469$	−8.53883e11	−0.814930
$470$	0	0
$471$	0	0
$472$	2.44404e11i	0.226657i
$473$	1.31051e11i	0.120383i
$474$	0	0
$475$	0	0
$476$	7.90890e11	0.706131
$477$	0	0
$478$	2.66330e11i	0.233343i
$479$	8.82280e11	0.765767	0.382883	−	0.923797i	$-0.374931\pi$
$479$	8.82280e11	0.765767	0.382883	+	0.923797i	$0.374931\pi$
$480$	0	0
$481$	−2.80994e12	−2.39356
$482$	− 1.76663e11i	− 0.149085i
$483$	0	0
$484$	2.53342e12	2.09847
$485$	0	0
$486$	0	0
$487$	5.09840e11i	0.410727i	0.978686	+	0.205364i	$0.0658377\pi$
$487$	5.09840e11i	0.410727i	−0.978686	+	0.205364i	$0.934162\pi$
$488$	− 5.11039e11i	− 0.407910i
$489$	0	0
$490$	0	0
$491$	1.52131e12	1.18128	0.590638	−	0.806937i	$-0.298876\pi$
$491$	1.52131e12	1.18128	0.590638	+	0.806937i	$0.298876\pi$
$492$	0	0
$493$	− 6.63228e11i	− 0.505652i
$494$	4.29707e11	0.324640
$495$	0	0
$496$	−5.58649e11	−0.414449
$497$	− 1.34824e12i	− 0.991202i
$498$	0	0
$499$	−7.03413e11	−0.507876	−0.253938	−	0.967220i	$-0.581726\pi$
$499$	−7.03413e11	−0.507876	−0.253938	+	0.967220i	$0.581726\pi$
$500$	0	0
$501$	0	0
$502$	− 3.34494e11i	− 0.235083i
$503$	3.78018e8i	0 0.000263304i	1.00000		0.000131652i	$4.19061e-5\pi$
$503$	3.78018e8i	0 0.000263304i	−1.00000		0.000131652i	$0.999958\pi$
$504$	0	0
$505$	0	0
$506$	−4.73466e11	−0.321079
$507$	0	0
$508$	8.62144e11i	0.574372i
$509$	1.32057e12	0.872027	0.436013	−	0.899940i	$-0.356390\pi$
$509$	1.32057e12	0.872027	0.436013	+	0.899940i	$0.356390\pi$
$510$	0	0
$511$	4.70272e11	0.305109
$512$	1.20816e12i	0.776980i
$513$	0	0
$514$	−3.46110e11	−0.218715
$515$	0	0
$516$	0	0
$517$	− 5.32034e10i	− 0.0327516i
$518$	5.75561e11i	0.351243i
$519$	0	0
$520$	0	0
$521$	1.31853e12	0.784009	0.392005	−	0.919963i	$-0.371782\pi$
$521$	1.31853e12	0.784009	0.392005	+	0.919963i	$0.371782\pi$
$522$	0	0
$523$	1.69211e12i	0.988945i	0.869193	+	0.494472i	$0.164639\pi$
$523$	1.69211e12i	0.988945i	−0.869193	+	0.494472i	$0.835361\pi$
$524$	1.23866e12	0.717730
$525$	0	0
$526$	3.84614e11	0.219073
$527$	− 4.87704e11i	− 0.275428i
$528$	0	0
$529$	−7.55281e10	−0.0419332
$530$	0	0
$531$	0	0
$532$	2.72853e12i	1.47681i
$533$	4.39175e12i	2.35703i
$534$	0	0
$535$	0	0
$536$	4.48288e11	0.234594
$537$	0	0
$538$	− 4.38021e10i	− 0.0225411i
$539$	−1.60960e12	−0.821427
$540$	0	0
$541$	−1.86369e12	−0.935373	−0.467687	−	0.883894i	$-0.654912\pi$
$541$	−1.86369e12	−0.935373	−0.467687	+	0.883894i	$0.654912\pi$
$542$	3.12115e11i	0.155352i
$543$	0	0
$544$	−6.26122e11	−0.306523
$545$	0	0
$546$	0	0
$547$	4.37242e11i	0.208823i	0.994534	+	0.104412i	$0.0332959\pi$
$547$	4.37242e11i	0.208823i	−0.994534	+	0.104412i	$0.966704\pi$
$548$	3.94928e12i	1.87071i
$549$	0	0
$550$	0	0
$551$	2.28810e12	1.05753
$552$	0	0
$553$	1.80043e12i	0.818679i
$554$	−2.62736e11	−0.118502
$555$	0	0
$556$	1.41640e12	0.628565
$557$	7.09146e11i	0.312167i	0.987744	+	0.156084i	$0.0498869\pi$
$557$	7.09146e11i	0.312167i	−0.987744	+	0.156084i	$0.950113\pi$
$558$	0	0
$559$	−2.27475e11	−0.0985328
$560$	0	0
$561$	0	0
$562$	2.57962e11i	0.109079i
$563$	3.77472e10i	0.0158342i	0.999969	+	0.00791711i	$0.00252012\pi$
$563$	3.77472e10i	0.0158342i	−0.999969	+	0.00791711i	$0.997480\pi$
$564$	0	0
$565$	0	0
$566$	3.85253e11	0.157787
$567$	0	0
$568$	7.07824e11i	0.285337i
$569$	−3.56270e11	−0.142487	−0.0712433	−	0.997459i	$-0.522697\pi$
$569$	−3.56270e11	−0.142487	−0.0712433	+	0.997459i	$0.522697\pi$
$570$	0	0
$571$	3.87932e12	1.52719	0.763596	−	0.645694i	$-0.223432\pi$
$571$	3.87932e12	1.52719	0.763596	+	0.645694i	$0.223432\pi$
$572$	6.42751e12i	2.51050i
$573$	0	0
$574$	8.99562e11	0.345882
$575$	0	0
$576$	0	0
$577$	− 4.28876e12i	− 1.61080i	−0.592734	−	0.805399i	$-0.701951\pi$
$577$	− 4.28876e12i	− 1.61080i	0.592734	−	0.805399i	$-0.298049\pi$
$578$	3.01924e11i	0.112518i
$579$	0	0
$580$	0	0
$581$	9.13232e11	0.332498
$582$	0	0
$583$	4.10197e11i	0.147056i
$584$	−2.46893e11	−0.0878316
$585$	0	0
$586$	−3.26523e11	−0.114387
$587$	− 4.43245e12i	− 1.54089i	−0.637504	−	0.770447i	$-0.720033\pi$
$587$	− 4.43245e12i	− 1.54089i	0.637504	−	0.770447i	$-0.279967\pi$
$588$	0	0
$589$	1.68255e12	0.576036
$590$	0	0
$591$	0	0
$592$	4.45580e12i	1.49100i
$593$	− 5.10104e12i	− 1.69400i	−0.531596	−	0.846998i	$-0.678407\pi$
$593$	− 5.10104e12i	− 1.69400i	0.531596	−	0.846998i	$-0.321593\pi$
$594$	0	0
$595$	0	0
$596$	−4.77838e12	−1.55122
$597$	0	0
$598$	− 8.21831e11i	− 0.262801i
$599$	−7.04599e11	−0.223626	−0.111813	−	0.993729i	$-0.535666\pi$
$599$	−7.04599e11	−0.223626	−0.111813	+	0.993729i	$0.535666\pi$
$600$	0	0
$601$	−1.73879e12	−0.543641	−0.271821	−	0.962348i	$-0.587626\pi$
$601$	−1.73879e12	−0.543641	−0.271821	+	0.962348i	$0.587626\pi$
$602$	4.65938e10i	0.0144592i
$603$	0	0
$604$	−2.66993e12	−0.816269
$605$	0	0
$606$	0	0
$607$	− 5.78292e11i	− 0.172901i	−0.996256	−	0.0864507i	$-0.972447\pi$
$607$	− 5.78292e11i	− 0.172901i	0.996256	−	0.0864507i	$-0.0275525\pi$
$608$	− 2.16008e12i	− 0.641068i
$609$	0	0
$610$	0	0
$611$	9.23493e10	0.0268070
$612$	0	0
$613$	− 3.74595e12i	− 1.07150i	−0.844378	−	0.535748i	$-0.820030\pi$
$613$	− 3.74595e12i	− 1.07150i	0.844378	−	0.535748i	$-0.179970\pi$
$614$	1.18233e11	0.0335723
$615$	0	0
$616$	2.67557e12	0.748691
$617$	3.94875e12i	1.09692i	0.836176	+	0.548461i	$0.184786\pi$
$617$	3.94875e12i	1.09692i	−0.836176	+	0.548461i	$0.815214\pi$
$618$	0	0
$619$	−3.42253e12	−0.937000	−0.468500	−	0.883463i	$-0.655205\pi$
$619$	−3.42253e12	−0.937000	−0.468500	+	0.883463i	$0.655205\pi$
$620$	0	0
$621$	0	0
$622$	1.59628e11i	0.0427615i
$623$	2.43098e12i	0.646526i
$624$	0	0
$625$	0	0
$626$	7.41484e11	0.192982
$627$	0	0
$628$	2.57639e11i	0.0660988i
$629$	−3.88995e12	−0.990868
$630$	0	0
$631$	5.84755e12	1.46839	0.734196	−	0.678938i	$-0.237560\pi$
$631$	5.84755e12	1.46839	0.734196	+	0.678938i	$0.237560\pi$
$632$	− 9.45226e11i	− 0.235673i
$633$	0	0
$634$	1.07558e12	0.264387
$635$	0	0
$636$	0	0
$637$	− 2.79391e12i	− 0.672334i
$638$	− 1.10404e12i	− 0.263809i
$639$	0	0
$640$	0	0
$641$	−2.66671e12	−0.623899	−0.311950	−	0.950099i	$-0.600982\pi$
$641$	−2.66671e12	−0.623899	−0.311950	+	0.950099i	$0.600982\pi$
$642$	0	0
$643$	− 9.68716e10i	− 0.0223484i	−0.999938	−	0.0111742i	$-0.996443\pi$
$643$	− 9.68716e10i	− 0.0223484i	0.999938	−	0.0111742i	$-0.00355694\pi$
$644$	5.21841e12	1.19551
$645$	0	0
$646$	5.94865e11	0.134392
$647$	− 4.47368e10i	− 0.0100368i	−0.999987	−	0.00501840i	$-0.998403\pi$
$647$	− 4.47368e10i	− 0.0100368i	0.999987	−	0.00501840i	$-0.00159741\pi$
$648$	0	0
$649$	5.23747e12	1.15883
$650$	0	0
$651$	0	0
$652$	− 4.66855e12i	− 1.01174i
$653$	4.95385e12i	1.06619i	0.846056	+	0.533094i	$0.178971\pi$
$653$	4.95385e12i	1.06619i	−0.846056	+	0.533094i	$0.821029\pi$
$654$	0	0
$655$	0	0
$656$	6.96411e12	1.46824
$657$	0	0
$658$	− 1.89159e10i	− 0.00393378i
$659$	−5.85077e12	−1.20845	−0.604225	−	0.796814i	$-0.706517\pi$
$659$	−5.85077e12	−1.20845	−0.604225	+	0.796814i	$0.706517\pi$
$660$	0	0
$661$	−8.81007e11	−0.179503	−0.0897517	−	0.995964i	$-0.528607\pi$
$661$	−8.81007e11	−0.179503	−0.0897517	+	0.995964i	$0.528607\pi$
$662$	− 1.71640e12i	− 0.347342i
$663$	0	0
$664$	−4.79447e11	−0.0957159
$665$	0	0
$666$	0	0
$667$	− 4.37608e12i	− 0.856088i
$668$	− 4.64872e12i	− 0.903314i
$669$	0	0
$670$	0	0
$671$	−1.09513e13	−2.08553
$672$	0	0
$673$	8.66521e12i	1.62821i	0.580715	+	0.814107i	$0.302773\pi$
$673$	8.66521e12i	1.62821i	−0.580715	+	0.814107i	$0.697227\pi$
$674$	8.10390e10	0.0151260
$675$	0	0
$676$	−5.89689e12	−1.08608
$677$	− 8.98549e12i	− 1.64397i	−0.569512	−	0.821983i	$-0.692868\pi$
$677$	− 8.98549e12i	− 1.64397i	0.569512	−	0.821983i	$-0.307132\pi$
$678$	0	0
$679$	1.86557e12	0.336819
$680$	0	0
$681$	0	0
$682$	− 8.11852e11i	− 0.143697i
$683$	− 3.86477e12i	− 0.679564i	−0.940504	−	0.339782i	$-0.889647\pi$
$683$	− 3.86477e12i	− 0.679564i	0.940504	−	0.339782i	$-0.110353\pi$
$684$	0	0
$685$	0	0
$686$	6.67386e11	0.115059
$687$	0	0
$688$	3.60713e11i	0.0613782i
$689$	−7.12010e11	−0.120365
$690$	0	0
$691$	−7.08564e12	−1.18230	−0.591150	−	0.806561i	$-0.701326\pi$
$691$	−7.08564e12	−1.18230	−0.591150	+	0.806561i	$0.701326\pi$
$692$	6.12922e12i	1.01608i
$693$	0	0
$694$	1.17113e11	0.0191641
$695$	0	0
$696$	0	0
$697$	6.07972e12i	0.975744i
$698$	− 2.82053e11i	− 0.0449760i
$699$	0	0
$700$	0	0
$701$	−4.69380e12	−0.734165	−0.367083	−	0.930188i	$-0.619643\pi$
$701$	−4.69380e12	−0.734165	−0.367083	+	0.930188i	$0.619643\pi$
$702$	0	0
$703$	− 1.34201e13i	− 2.07232i
$704$	9.47879e12	1.45437
$705$	0	0
$706$	2.62940e11	0.0398323
$707$	− 5.02120e12i	− 0.755824i
$708$	0	0
$709$	−1.06645e13	−1.58501	−0.792503	−	0.609868i	$-0.791223\pi$
$709$	−1.06645e13	−1.58501	−0.792503	+	0.609868i	$0.791223\pi$
$710$	0	0
$711$	0	0
$712$	− 1.27627e12i	− 0.186115i
$713$	− 3.21794e12i	− 0.466311i
$714$	0	0
$715$	0	0
$716$	−3.30356e11	−0.0469757
$717$	0	0
$718$	2.32681e12i	0.326739i
$719$	−8.15663e12	−1.13823	−0.569116	−	0.822257i	$-0.692715\pi$
$719$	−8.15663e12	−1.13823	−0.569116	+	0.822257i	$0.692715\pi$
$720$	0	0
$721$	−1.07843e13	−1.48622
$722$	7.61500e11i	0.104292i
$723$	0	0
$724$	2.61495e12	0.353703
$725$	0	0
$726$	0	0
$727$	6.64771e12i	0.882606i	0.897358	+	0.441303i	$0.145484\pi$
$727$	6.64771e12i	0.882606i	−0.897358	+	0.441303i	$0.854516\pi$
$728$	4.64418e12i	0.612799i
$729$	0	0
$730$	0	0
$731$	−3.14905e11	−0.0407898
$732$	0	0
$733$	− 7.07821e12i	− 0.905640i	−0.891602	−	0.452820i	$-0.850418\pi$
$733$	− 7.07821e12i	− 0.905640i	0.891602	−	0.452820i	$-0.149582\pi$
$734$	1.66828e12	0.212147
$735$	0	0
$736$	−4.13124e12	−0.518956
$737$	− 9.60663e12i	− 1.19941i
$738$	0	0
$739$	2.61052e12	0.321979	0.160989	−	0.986956i	$-0.448532\pi$
$739$	2.61052e12	0.321979	0.160989	+	0.986956i	$0.448532\pi$
$740$	0	0
$741$	0	0
$742$	1.45841e11i	0.0176629i
$743$	− 1.41841e13i	− 1.70747i	−0.520709	−	0.853734i	$-0.674333\pi$
$743$	− 1.41841e13i	− 1.70747i	0.520709	−	0.853734i	$-0.325667\pi$
$744$	0	0
$745$	0	0
$746$	−3.04167e11	−0.0359573
$747$	0	0
$748$	8.89793e12i	1.03928i
$749$	1.41199e13	1.63932
$750$	0	0
$751$	8.44355e11	0.0968603	0.0484301	−	0.998827i	$-0.484578\pi$
$751$	8.44355e11	0.0968603	0.0484301	+	0.998827i	$0.484578\pi$
$752$	− 1.46441e11i	− 0.0166986i
$753$	0	0
$754$	1.91636e12	0.215927
$755$	0	0
$756$	0	0
$757$	9.05305e12i	1.00199i	0.865450	+	0.500995i	$0.167032\pi$
$757$	9.05305e12i	1.00199i	−0.865450	+	0.500995i	$0.832968\pi$
$758$	7.16719e11i	0.0788565i
$759$	0	0
$760$	0	0
$761$	6.97701e12	0.754116	0.377058	−	0.926190i	$-0.376936\pi$
$761$	6.97701e12	0.754116	0.377058	+	0.926190i	$0.376936\pi$
$762$	0	0
$763$	− 7.16891e12i	− 0.765760i
$764$	1.45772e13	1.54794
$765$	0	0
$766$	3.18007e12	0.333739
$767$	9.09108e12i	0.948498i
$768$	0	0
$769$	1.07233e13	1.10576	0.552879	−	0.833261i	$-0.313529\pi$
$769$	1.07233e13	1.10576	0.552879	+	0.833261i	$0.313529\pi$
$770$	0	0
$771$	0	0
$772$	7.37781e12i	0.747566i
$773$	− 1.37568e13i	− 1.38583i	−0.721019	−	0.692916i	$-0.756326\pi$
$773$	− 1.37568e13i	− 1.38583i	0.721019	−	0.692916i	$-0.243674\pi$
$774$	0	0
$775$	0	0
$776$	−9.79422e11	−0.0969599
$777$	0	0
$778$	− 7.18134e11i	− 0.0702744i
$779$	−2.09747e13	−2.04069
$780$	0	0
$781$	1.51684e13	1.45884
$782$	− 1.13770e12i	− 0.108792i
$783$	0	0
$784$	−4.43037e12	−0.418811
$785$	0	0
$786$	0	0
$787$	− 1.27539e13i	− 1.18510i	−0.805533	−	0.592551i	$-0.798121\pi$
$787$	− 1.27539e13i	− 1.18510i	0.805533	−	0.592551i	$-0.201879\pi$
$788$	− 2.47343e12i	− 0.228524i
$789$	0	0
$790$	0	0
$791$	−7.13114e11	−0.0647686
$792$	0	0
$793$	− 1.90091e13i	− 1.70699i
$794$	1.37492e12	0.122768
$795$	0	0
$796$	−7.21814e12	−0.637260
$797$	− 1.27845e13i	− 1.12233i	−0.827704	−	0.561164i	$-0.810354\pi$
$797$	− 1.27845e13i	− 1.12233i	0.827704	−	0.561164i	$-0.189646\pi$
$798$	0	0
$799$	1.27844e11	0.0110973
$800$	0	0
$801$	0	0
$802$	− 3.08832e12i	− 0.263595i
$803$	5.29081e12i	0.449057i
$804$	0	0
$805$	0	0
$806$	1.40919e12	0.117615
$807$	0	0
$808$	2.63613e12i	0.217579i
$809$	6.03746e12	0.495548	0.247774	−	0.968818i	$-0.420301\pi$
$809$	6.03746e12	0.495548	0.247774	+	0.968818i	$0.420301\pi$
$810$	0	0
$811$	−2.50043e12	−0.202965	−0.101482	−	0.994837i	$-0.532359\pi$
$811$	−2.50043e12	−0.202965	−0.101482	+	0.994837i	$0.532359\pi$
$812$	1.21684e13i	0.982269i
$813$	0	0
$814$	−6.47536e12	−0.516957
$815$	0	0
$816$	0	0
$817$	− 1.08641e12i	− 0.0853085i
$818$	− 1.04253e12i	− 0.0814138i
$819$	0	0
$820$	0	0
$821$	4.21082e12	0.323461	0.161731	−	0.986835i	$-0.448292\pi$
$821$	4.21082e12	0.323461	0.161731	+	0.986835i	$0.448292\pi$
$822$	0	0
$823$	− 2.08206e11i	− 0.0158196i	−0.999969	−	0.00790978i	$-0.997482\pi$
$823$	− 2.08206e11i	− 0.0158196i	0.999969	−	0.00790978i	$-0.00251779\pi$
$824$	5.66174e12	0.427836
$825$	0	0
$826$	1.86213e12	0.139187
$827$	9.26106e12i	0.688472i	0.938883	+	0.344236i	$0.111862\pi$
$827$	9.26106e12i	0.688472i	−0.938883	+	0.344236i	$0.888138\pi$
$828$	0	0
$829$	−2.42762e13	−1.78519	−0.892597	−	0.450856i	$-0.851119\pi$
$829$	−2.42762e13	−1.78519	−0.892597	+	0.450856i	$0.851119\pi$
$830$	0	0
$831$	0	0
$832$	1.64531e13i	1.19040i
$833$	− 3.86775e12i	− 0.278327i
$834$	0	0
$835$	0	0
$836$	−3.06973e13	−2.17356
$837$	0	0
$838$	− 2.24066e12i	− 0.156956i
$839$	−1.25546e13	−0.874728	−0.437364	−	0.899285i	$-0.644088\pi$
$839$	−1.25546e13	−0.874728	−0.437364	+	0.899285i	$0.644088\pi$
$840$	0	0
$841$	−4.30294e12	−0.296608
$842$	6.73284e11i	0.0461630i
$843$	0	0
$844$	2.55642e13	1.73417
$845$	0	0
$846$	0	0
$847$	− 3.92272e13i	− 2.61886i
$848$	1.12905e12i	0.0749778i
$849$	0	0
$850$	0	0
$851$	−2.56664e13	−1.67758
$852$	0	0
$853$	1.36320e13i	0.881632i	0.897597	+	0.440816i	$0.145311\pi$
$853$	1.36320e13i	0.881632i	−0.897597	+	0.440816i	$0.854689\pi$
$854$	−3.89363e12	−0.250492
$855$	0	0
$856$	−7.41297e12	−0.471911
$857$	− 1.73987e13i	− 1.10180i	−0.834570	−	0.550901i	$-0.814284\pi$
$857$	− 1.73987e13i	− 1.10180i	0.834570	−	0.550901i	$-0.185716\pi$
$858$	0	0
$859$	6.43355e10	0.00403163	0.00201582	−	0.999998i	$-0.499358\pi$
$859$	6.43355e10	0.00403163	0.00201582	+	0.999998i	$0.499358\pi$
$860$	0	0
$861$	0	0
$862$	− 1.79353e12i	− 0.110644i
$863$	3.32120e12i	0.203820i	0.994794	+	0.101910i	$0.0324953\pi$
$863$	3.32120e12i	0.203820i	−0.994794	+	0.101910i	$0.967505\pi$
$864$	0	0
$865$	0	0
$866$	−4.33940e12	−0.262180
$867$	0	0
$868$	8.94800e12i	0.535041i
$869$	−2.02558e13	−1.20493
$870$	0	0
$871$	1.66750e13	0.981709
$872$	3.76368e12i	0.220439i
$873$	0	0
$874$	3.92501e12	0.227530
$875$	0	0
$876$	0	0
$877$	1.95832e12i	0.111786i	0.998437	+	0.0558928i	$0.0178005\pi$
$877$	1.95832e12i	0.111786i	−0.998437	+	0.0558928i	$0.982200\pi$
$878$	− 1.84219e12i	− 0.104619i
$879$	0	0
$880$	0	0
$881$	3.02253e13	1.69036	0.845179	−	0.534483i	$-0.179494\pi$
$881$	3.02253e13	1.69036	0.845179	+	0.534483i	$0.179494\pi$
$882$	0	0
$883$	9.30097e12i	0.514879i	0.966294	+	0.257439i	$0.0828788\pi$
$883$	9.30097e12i	0.514879i	−0.966294	+	0.257439i	$0.917121\pi$
$884$	−1.54448e13	−0.850643
$885$	0	0
$886$	5.28380e10	0.00288068
$887$	− 9.27171e12i	− 0.502925i	−0.967867	−	0.251463i	$-0.919088\pi$
$887$	− 9.27171e12i	− 0.502925i	0.967867	−	0.251463i	$-0.0809115\pi$
$888$	0	0
$889$	1.33493e13	0.716805
$890$	0	0
$891$	0	0
$892$	− 2.28791e13i	− 1.21003i
$893$	4.41053e11i	0.0232092i
$894$	0	0
$895$	0	0
$896$	1.52282e13	0.789338
$897$	0	0
$898$	− 2.76756e12i	− 0.142021i
$899$	7.50365e12	0.383137
$900$	0	0
$901$	−9.85671e11	−0.0498276
$902$	1.01205e13i	0.509066i
$903$	0	0
$904$	3.74385e11	0.0186449
$905$	0	0
$906$	0	0
$907$	1.32868e12i	0.0651908i	0.999469	+	0.0325954i	$0.0103773\pi$
$907$	1.32868e12i	0.0651908i	−0.999469	+	0.0325954i	$0.989623\pi$
$908$	1.86413e13i	0.910102i
$909$	0	0
$910$	0	0
$911$	−2.71297e12	−0.130501	−0.0652503	−	0.997869i	$-0.520785\pi$
$911$	−2.71297e12	−0.130501	−0.0652503	+	0.997869i	$0.520785\pi$
$912$	0	0
$913$	1.02743e13i	0.489368i
$914$	1.49254e12	0.0707404
$915$	0	0
$916$	3.17977e13	1.49233
$917$	− 1.91792e13i	− 0.895714i
$918$	0	0
$919$	−1.32139e12	−0.0611100	−0.0305550	−	0.999533i	$-0.509727\pi$
$919$	−1.32139e12	−0.0611100	−0.0305550	+	0.999533i	$0.509727\pi$
$920$	0	0
$921$	0	0
$922$	5.83758e12i	0.266038i
$923$	2.63289e13i	1.19406i
$924$	0	0
$925$	0	0
$926$	5.36853e11	0.0239942
$927$	0	0
$928$	− 9.63329e12i	− 0.426392i
$929$	1.16124e12	0.0511507	0.0255754	−	0.999673i	$-0.491858\pi$
$929$	1.16124e12	0.0511507	0.0255754	+	0.999673i	$0.491858\pi$
$930$	0	0
$931$	1.33435e13	0.582098
$932$	3.45503e13i	1.49996i
$933$	0	0
$934$	−1.45813e11	−0.00626952
$935$	0	0
$936$	0	0
$937$	3.40914e13i	1.44483i	0.691460	+	0.722415i	$0.256968\pi$
$937$	3.40914e13i	1.44483i	−0.691460	+	0.722415i	$0.743032\pi$
$938$	− 3.41553e12i	− 0.144061i
$939$	0	0
$940$	0	0
$941$	1.60215e12	0.0666114	0.0333057	−	0.999445i	$-0.489397\pi$
$941$	1.60215e12	0.0666114	0.0333057	+	0.999445i	$0.489397\pi$
$942$	0	0
$943$	4.01149e13i	1.65197i
$944$	1.44160e13	0.590840
$945$	0	0
$946$	−5.24204e11	−0.0212809
$947$	3.38850e13i	1.36909i	0.728969	+	0.684546i	$0.240000\pi$
$947$	3.38850e13i	1.36909i	−0.728969	+	0.684546i	$0.760000\pi$
$948$	0	0
$949$	−9.18366e12	−0.367551
$950$	0	0
$951$	0	0
$952$	6.42917e12i	0.253682i
$953$	2.15757e13i	0.847320i	0.905821	+	0.423660i	$0.139255\pi$
$953$	2.15757e13i	0.847320i	−0.905821	+	0.423660i	$0.860745\pi$
$954$	0	0
$955$	0	0
$956$	3.30249e13	1.27874
$957$	0	0
$958$	3.52912e12i	0.135370i
$959$	6.11502e13	2.33461
$960$	0	0
$961$	−2.09218e13	−0.791306
$962$	− 1.12398e13i	− 0.423126i
$963$	0	0
$964$	−2.19063e13	−0.816999
$965$	0	0
$966$	0	0
$967$	3.06249e13i	1.12630i	0.826353	+	0.563152i	$0.190411\pi$
$967$	3.06249e13i	1.12630i	−0.826353	+	0.563152i	$0.809589\pi$
$968$	2.05943e13i	0.753888i
$969$	0	0
$970$	0	0
$971$	−1.92365e12	−0.0694447	−0.0347224	−	0.999397i	$-0.511055\pi$
$971$	−1.92365e12	−0.0694447	−0.0347224	+	0.999397i	$0.511055\pi$
$972$	0	0
$973$	− 2.19314e13i	− 0.784438i
$974$	−2.03936e12	−0.0726070
$975$	0	0
$976$	−3.01432e13	−1.06332
$977$	1.83893e13i	0.645714i	0.946448	+	0.322857i	$0.104643\pi$
$977$	1.83893e13i	0.645714i	−0.946448	+	0.322857i	$0.895357\pi$
$978$	0	0
$979$	−2.73498e13	−0.951552
$980$	0	0
$981$	0	0
$982$	6.08524e12i	0.208822i
$983$	− 4.63273e13i	− 1.58251i	−0.611488	−	0.791254i	$-0.709429\pi$
$983$	− 4.63273e13i	− 1.58251i	0.611488	−	0.791254i	$-0.290571\pi$
$984$	0	0
$985$	0	0
$986$	2.65291e12	0.0893875
$987$	0	0
$988$	− 5.32837e13i	− 1.77905i
$989$	−2.07779e12	−0.0690587
$990$	0	0
$991$	2.76252e12	0.0909857	0.0454929	−	0.998965i	$-0.485514\pi$
$991$	2.76252e12	0.0909857	0.0454929	+	0.998965i	$0.485514\pi$
$992$	− 7.08383e12i	− 0.232255i
$993$	0	0
$994$	5.39295e12	0.175221
$995$	0	0
$996$	0	0
$997$	− 1.74502e13i	− 0.559337i	−0.960097	−	0.279668i	$-0.909776\pi$
$997$	− 1.74502e13i	− 0.559337i	0.960097	−	0.279668i	$-0.0902245\pi$
$998$	− 2.81365e12i	− 0.0897807i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.10.b.e.199.2		2
3.2	odd	2		75.10.b.d.49.1		2
5.2	odd	4		225.10.a.c.1.1		1
5.3	odd	4		45.10.a.b.1.1		1
5.4	even	2	inner	225.10.b.e.199.1		2
15.2	even	4		75.10.a.c.1.1		1
15.8	even	4		15.10.a.a.1.1	✓	1
15.14	odd	2		75.10.b.d.49.2		2
60.23	odd	4		240.10.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.10.a.a.1.1	✓	1	15.8	even	4
45.10.a.b.1.1		1	5.3	odd	4
75.10.a.c.1.1		1	15.2	even	4
75.10.b.d.49.1		2	3.2	odd	2
75.10.b.d.49.2		2	15.14	odd	2
225.10.a.c.1.1		1	5.2	odd	4
225.10.b.e.199.1		2	5.4	even	2	inner
225.10.b.e.199.2		2	1.1	even	1	trivial
240.10.a.c.1.1		1	60.23	odd	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} +496.000 q^{4} -7680.00i q^{7} +4032.00i q^{8} +O(q^{10})\)	\(q+4.00000i q^{2} +496.000 q^{4} -7680.00i q^{7} +4032.00i q^{8} +86404.0 q^{11} +149978. i q^{13} +30720.0 q^{14} +237824. q^{16} +207622. i q^{17} -716284. q^{19} +345616. i q^{22} +1.36992e6i q^{23} -599912. q^{26} -3.80928e6i q^{28} -3.19440e6 q^{29} -2.34900e6 q^{31} +3.01568e6i q^{32} -830488. q^{34} +1.87357e7i q^{37} -2.86514e6i q^{38} +2.92826e7 q^{41} +1.51672e6i q^{43} +4.28564e7 q^{44} -5.47968e6 q^{46} -615752. i q^{47} -1.86288e7 q^{49} +7.43891e7i q^{52} +4.74743e6i q^{53} +3.09658e7 q^{56} -1.27776e7i q^{58} +6.06161e7 q^{59} -1.26746e8 q^{61} -9.39600e6i q^{62} +1.09703e8 q^{64} -1.11183e8i q^{67} +1.02981e8i q^{68} +1.75552e8 q^{71} +6.12334e7i q^{73} -7.49428e7 q^{74} -3.55277e8 q^{76} -6.63583e8i q^{77} -2.34431e8 q^{79} +1.17131e8i q^{82} +1.18910e8i q^{83} -6.06690e6 q^{86} +3.48381e8i q^{88} -3.16534e8 q^{89} +1.15183e9 q^{91} +6.79480e8i q^{92} +2.46301e6 q^{94} +2.42912e8i q^{97} -7.45152e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 992 q^{4}+O(q^{10}) \) 2 * q + 992 * q^4	\( 2 q + 992 q^{4} + 172808 q^{11} + 61440 q^{14} + 475648 q^{16} - 1432568 q^{19} - 1199824 q^{26} - 6388804 q^{29} - 4698000 q^{31} - 1660976 q^{34} + 58565260 q^{41} + 85712768 q^{44} - 10959360 q^{46} - 37257586 q^{49} + 61931520 q^{56} + 121232152 q^{59} - 253491364 q^{61} + 219406336 q^{64} + 351103216 q^{71} - 149885680 q^{74} - 710553728 q^{76} - 468862320 q^{79} - 12133792 q^{86} - 633068652 q^{89} + 2303662080 q^{91} + 4926016 q^{94}+O(q^{100}) \) 2 * q + 992 * q^4 + 172808 * q^11 + 61440 * q^14 + 475648 * q^16 - 1432568 * q^19 - 1199824 * q^26 - 6388804 * q^29 - 4698000 * q^31 - 1660976 * q^34 + 58565260 * q^41 + 85712768 * q^44 - 10959360 * q^46 - 37257586 * q^49 + 61931520 * q^56 + 121232152 * q^59 - 253491364 * q^61 + 219406336 * q^64 + 351103216 * q^71 - 149885680 * q^74 - 710553728 * q^76 - 468862320 * q^79 - 12133792 * q^86 - 633068652 * q^89 + 2303662080 * q^91 + 4926016 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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