LMFDB - Embedded newform 225.8.b.l.199.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,8,Mod(199,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.199"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	225.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-362,0,0,0,0,0,0,-8688] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$70.2866307339$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{649})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 325x^{2} + 26244 $ x^4 + 325*x^2 + 26244
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.4
Root			$13.2377i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.8.b.l.199.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+20.2377i q^{2} -281.566 q^{4} +1369.97i q^{7} -3107.83i q^{8} +1012.43 q^{11} +3643.00i q^{13} -27725.1 q^{14} +26855.0 q^{16} -5532.37i q^{17} -23238.9 q^{19} +20489.4i q^{22} -96310.5i q^{23} -73726.1 q^{26} -385737. i q^{28} -139729. q^{29} -208208. q^{31} +145682. i q^{32} +111963. q^{34} -448306. i q^{37} -470303. i q^{38} +79718.5 q^{41} +227324. i q^{43} -285067. q^{44} +1.94911e6 q^{46} +719892. i q^{47} -1.05328e6 q^{49} -1.02575e6i q^{52} +495702. i q^{53} +4.25763e6 q^{56} -2.82779e6i q^{58} -81006.9 q^{59} +3.15701e6 q^{61} -4.21366e6i q^{62} +489161. q^{64} -1.06340e6i q^{67} +1.55773e6i q^{68} +3.00099e6 q^{71} +405758. i q^{73} +9.07270e6 q^{74} +6.54328e6 q^{76} +1.38701e6i q^{77} +4.71733e6 q^{79} +1.61332e6i q^{82} -5.14106e6i q^{83} -4.60053e6 q^{86} -3.14648e6i q^{88} -8.62652e6 q^{89} -4.99080e6 q^{91} +2.71178e7i q^{92} -1.45690e7 q^{94} +6.99663e6i q^{97} -2.13159e7i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 362 q^{4} - 8688 q^{11} - 63516 q^{14} + 66914 q^{16} - 36400 q^{19} - 408 q^{26} + 111600 q^{29} - 603552 q^{31} + 353846 q^{34} + 216972 q^{41} - 1647486 q^{44} + 4685868 q^{46} - 1645172 q^{49}+ \cdots - 17950264 q^{94}+O(q^{100}) $ 4 * q - 362 * q^4 - 8688 * q^11 - 63516 * q^14 + 66914 * q^16 - 36400 * q^19 - 408 * q^26 + 111600 * q^29 - 603552 * q^31 + 353846 * q^34 + 216972 * q^41 - 1647486 * q^44 + 4685868 * q^46 - 1645172 * q^49 + 6671700 * q^56 + 4135200 * q^59 + 1164088 * q^61 + 700958 * q^64 + 9456432 * q^71 + 16453044 * q^74 + 14100050 * q^76 + 14372400 * q^79 - 1458888 * q^86 - 11981700 * q^89 + 11634528 * q^91 - 17950264 * q^94

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 $

$2$	20.2377i	1.78878i	0.447288	+	0.894390i	$0.352390\pi$
$2$	20.2377i	1.78878i	−0.447288	+	0.894390i	$0.647610\pi$
$3$	0	0
$3$	0	0
$4$	−281.566	−2.19974
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1369.97i	1.50962i	0.655943	+	0.754811i	$0.272271\pi$
$7$	1369.97i	1.50962i	−0.655943	+	0.754811i	$0.727729\pi$
$8$	− 3107.83i	− 2.14606i
$9$	0	0
$10$	0	0
$11$	1012.43	0.229347	0.114673	−	0.993403i	$-0.463418\pi$
$11$	1012.43	0.229347	0.114673	+	0.993403i	$0.463418\pi$
$12$	0	0
$13$	3643.00i	0.459894i	0.973203	+	0.229947i	$0.0738553\pi$
$13$	3643.00i	0.459894i	−0.973203	+	0.229947i	$0.926145\pi$
$14$	−27725.1	−2.70038
$15$	0	0
$16$	26855.0	1.63910
$17$	− 5532.37i	− 0.273111i	−0.990632	−	0.136556i	$-0.956397\pi$
$17$	− 5532.37i	− 0.273111i	0.990632	−	0.136556i	$-0.0436033\pi$
$18$	0	0
$19$	−23238.9	−0.777281	−0.388640	−	0.921390i	$-0.627055\pi$
$19$	−23238.9	−0.777281	−0.388640	+	0.921390i	$0.627055\pi$
$20$	0	0
$21$	0	0
$22$	20489.4i	0.410251i
$23$	− 96310.5i	− 1.65054i	−0.564738	−	0.825270i	$-0.691023\pi$
$23$	− 96310.5i	− 1.65054i	0.564738	−	0.825270i	$-0.308977\pi$
$24$	0	0
$25$	0	0
$26$	−73726.1	−0.822649
$27$	0	0
$28$	− 385737.i	− 3.32077i
$29$	−139729.	−1.06388	−0.531940	−	0.846782i	$-0.678537\pi$
$29$	−139729.	−1.06388	−0.531940	+	0.846782i	$0.678537\pi$
$30$	0	0
$31$	−208208.	−1.25525	−0.627626	−	0.778515i	$-0.715973\pi$
$31$	−208208.	−1.25525	−0.627626	+	0.778515i	$0.715973\pi$
$32$	145682.i	0.785926i
$33$	0	0
$34$	111963.	0.488536
$35$	0	0
$36$	0	0
$37$	− 448306.i	− 1.45502i	−0.686098	−	0.727509i	$-0.740678\pi$
$37$	− 448306.i	− 1.45502i	0.686098	−	0.727509i	$-0.259322\pi$
$38$	− 470303.i	− 1.39038i
$39$	0	0
$40$	0	0
$41$	79718.5	0.180641	0.0903203	−	0.995913i	$-0.471211\pi$
$41$	79718.5	0.180641	0.0903203	+	0.995913i	$0.471211\pi$
$42$	0	0
$43$	227324.i	0.436020i	0.975947	+	0.218010i	$0.0699565\pi$
$43$	227324.i	0.436020i	−0.975947	+	0.218010i	$0.930043\pi$
$44$	−285067.	−0.504502
$45$	0	0
$46$	1.94911e6	2.95245
$47$	719892.i	1.01140i	0.862708	+	0.505702i	$0.168766\pi$
$47$	719892.i	1.01140i	−0.862708	+	0.505702i	$0.831234\pi$
$48$	0	0
$49$	−1.05328e6	−1.27896
$50$	0	0
$51$	0	0
$52$	− 1.02575e6i	− 1.01164i
$53$	495702.i	0.457357i	0.973502	+	0.228678i	$0.0734404\pi$
$53$	495702.i	0.457357i	−0.973502	+	0.228678i	$0.926560\pi$
$54$	0	0
$55$	0	0
$56$	4.25763e6	3.23974
$57$	0	0
$58$	− 2.82779e6i	− 1.90305i
$59$	−81006.9	−0.0513500	−0.0256750	−	0.999670i	$-0.508173\pi$
$59$	−81006.9	−0.0513500	−0.0256750	+	0.999670i	$0.508173\pi$
$60$	0	0
$61$	3.15701e6	1.78083	0.890414	−	0.455151i	$-0.150415\pi$
$61$	3.15701e6	1.78083	0.890414	+	0.455151i	$0.150415\pi$
$62$	− 4.21366e6i	− 2.24537i
$63$	0	0
$64$	489161.	0.233250
$65$	0	0
$66$	0	0
$67$	− 1.06340e6i	− 0.431950i	−0.976399	−	0.215975i	$-0.930707\pi$
$67$	− 1.06340e6i	− 0.431950i	0.976399	−	0.215975i	$-0.0692929\pi$
$68$	1.55773e6i	0.600773i
$69$	0	0
$70$	0	0
$71$	3.00099e6	0.995087	0.497543	−	0.867439i	$-0.334235\pi$
$71$	3.00099e6	0.995087	0.497543	+	0.867439i	$0.334235\pi$
$72$	0	0
$73$	405758.i	0.122078i	0.998135	+	0.0610389i	$0.0194414\pi$
$73$	405758.i	0.122078i	−0.998135	+	0.0610389i	$0.980559\pi$
$74$	9.07270e6	2.60271
$75$	0	0
$76$	6.54328e6	1.70981
$77$	1.38701e6i	0.346227i
$78$	0	0
$79$	4.71733e6	1.07647	0.538235	−	0.842795i	$-0.319091\pi$
$79$	4.71733e6	1.07647	0.538235	+	0.842795i	$0.319091\pi$
$80$	0	0
$81$	0	0
$82$	1.61332e6i	0.323126i
$83$	− 5.14106e6i	− 0.986914i	−0.869770	−	0.493457i	$-0.835733\pi$
$83$	− 5.14106e6i	− 0.986914i	0.869770	−	0.493457i	$-0.164267\pi$
$84$	0	0
$85$	0	0
$86$	−4.60053e6	−0.779944
$87$	0	0
$88$	− 3.14648e6i	− 0.492193i
$89$	−8.62652e6	−1.29709	−0.648546	−	0.761175i	$-0.724623\pi$
$89$	−8.62652e6	−1.29709	−0.648546	+	0.761175i	$0.724623\pi$
$90$	0	0
$91$	−4.99080e6	−0.694266
$92$	2.71178e7i	3.63075i
$93$	0	0
$94$	−1.45690e7	−1.80918
$95$	0	0
$96$	0	0
$97$	6.99663e6i	0.778373i	0.921159	+	0.389187i	$0.127244\pi$
$97$	6.99663e6i	0.778373i	−0.921159	+	0.389187i	$0.872756\pi$
$98$	− 2.13159e7i	− 2.28777i
$99$	0	0
$100$	0	0
$101$	−5.34837e6	−0.516531	−0.258266	−	0.966074i	$-0.583151\pi$
$101$	−5.34837e6	−0.516531	−0.258266	+	0.966074i	$0.583151\pi$
$102$	0	0
$103$	7.09826e6i	0.640061i	0.947407	+	0.320031i	$0.103693\pi$
$103$	7.09826e6i	0.640061i	−0.947407	+	0.320031i	$0.896307\pi$
$104$	1.13218e7	0.986961
$105$	0	0
$106$	−1.00319e7	−0.818111
$107$	− 1.01014e7i	− 0.797145i	−0.917137	−	0.398573i	$-0.869506\pi$
$107$	− 1.01014e7i	− 0.797145i	0.917137	−	0.398573i	$-0.130494\pi$
$108$	0	0
$109$	1.20315e6	0.0889874	0.0444937	−	0.999010i	$-0.485833\pi$
$109$	1.20315e6	0.0889874	0.0444937	+	0.999010i	$0.485833\pi$
$110$	0	0
$111$	0	0
$112$	3.67906e7i	2.47442i
$113$	529610.i	0.0345288i	0.999851	+	0.0172644i	$0.00549570\pi$
$113$	529610.i	0.0345288i	−0.999851	+	0.0172644i	$0.994504\pi$
$114$	0	0
$115$	0	0
$116$	3.93428e7	2.34025
$117$	0	0
$118$	− 1.63940e6i	− 0.0918538i
$119$	7.57918e6	0.412295
$120$	0	0
$121$	−1.84621e7	−0.947400
$122$	6.38908e7i	3.18551i
$123$	0	0
$124$	5.86243e7	2.76122
$125$	0	0
$126$	0	0
$127$	− 1.90420e7i	− 0.824897i	−0.910981	−	0.412448i	$-0.864674\pi$
$127$	− 1.90420e7i	− 0.824897i	0.910981	−	0.412448i	$-0.135326\pi$
$128$	2.85468e7i	1.20316i
$129$	0	0
$130$	0	0
$131$	−4.51565e7	−1.75498	−0.877488	−	0.479599i	$-0.840782\pi$
$131$	−4.51565e7	−1.75498	−0.877488	+	0.479599i	$0.840782\pi$
$132$	0	0
$133$	− 3.18366e7i	− 1.17340i
$134$	2.15207e7	0.772663
$135$	0	0
$136$	−1.71937e7	−0.586114
$137$	− 3.51625e7i	− 1.16831i	−0.811642	−	0.584155i	$-0.801426\pi$
$137$	− 3.51625e7i	− 1.16831i	0.811642	−	0.584155i	$-0.198574\pi$
$138$	0	0
$139$	4.10740e7	1.29722	0.648612	−	0.761119i	$-0.275350\pi$
$139$	4.10740e7	1.29722	0.648612	+	0.761119i	$0.275350\pi$
$140$	0	0
$141$	0	0
$142$	6.07334e7i	1.77999i
$143$	3.68830e6i	0.105475i
$144$	0	0
$145$	0	0
$146$	−8.21162e6	−0.218370
$147$	0	0
$148$	1.26228e8i	3.20066i
$149$	5.46024e7	1.35226	0.676129	−	0.736783i	$-0.263656\pi$
$149$	5.46024e7	1.35226	0.676129	+	0.736783i	$0.263656\pi$
$150$	0	0
$151$	−7.68654e6	−0.181682	−0.0908409	−	0.995865i	$-0.528955\pi$
$151$	−7.68654e6	−0.181682	−0.0908409	+	0.995865i	$0.528955\pi$
$152$	7.22225e7i	1.66809i
$153$	0	0
$154$	−2.80699e7	−0.619324
$155$	0	0
$156$	0	0
$157$	4.63213e6i	0.0955283i	0.998859	+	0.0477641i	$0.0152096\pi$
$157$	4.63213e6i	0.0955283i	−0.998859	+	0.0477641i	$0.984790\pi$
$158$	9.54682e7i	1.92557i
$159$	0	0
$160$	0	0
$161$	1.31942e8	2.49169
$162$	0	0
$163$	2.15892e7i	0.390462i	0.980757	+	0.195231i	$0.0625457\pi$
$163$	2.15892e7i	0.390462i	−0.980757	+	0.195231i	$0.937454\pi$
$164$	−2.24460e7	−0.397362
$165$	0	0
$166$	1.04043e8	1.76537
$167$	3.71330e7i	0.616953i	0.951232	+	0.308476i	$0.0998191\pi$
$167$	3.71330e7i	0.616953i	−0.951232	+	0.308476i	$0.900181\pi$
$168$	0	0
$169$	4.94771e7	0.788498
$170$	0	0
$171$	0	0
$172$	− 6.40068e7i	− 0.959128i
$173$	− 6.07796e7i	− 0.892476i	−0.894914	−	0.446238i	$-0.852763\pi$
$173$	− 6.07796e7i	− 0.892476i	0.894914	−	0.446238i	$-0.147237\pi$
$174$	0	0
$175$	0	0
$176$	2.71889e7	0.375922
$177$	0	0
$178$	− 1.74581e8i	− 2.32021i
$179$	3.05526e7	0.398164	0.199082	−	0.979983i	$-0.436204\pi$
$179$	3.05526e7	0.398164	0.199082	+	0.979983i	$0.436204\pi$
$180$	0	0
$181$	−1.18125e8	−1.48070	−0.740349	−	0.672223i	$-0.765340\pi$
$181$	−1.18125e8	−1.48070	−0.740349	+	0.672223i	$0.765340\pi$
$182$	− 1.01003e8i	− 1.24189i
$183$	0	0
$184$	−2.99317e8	−3.54216
$185$	0	0
$186$	0	0
$187$	− 5.60116e6i	− 0.0626372i
$188$	− 2.02697e8i	− 2.22482i
$189$	0	0
$190$	0	0
$191$	−8.20283e7	−0.851818	−0.425909	−	0.904766i	$-0.640046\pi$
$191$	−8.20283e7	−0.851818	−0.425909	+	0.904766i	$0.640046\pi$
$192$	0	0
$193$	− 1.98988e8i	− 1.99240i	−0.0871152	−	0.996198i	$-0.527765\pi$
$193$	− 1.98988e8i	− 1.99240i	0.0871152	−	0.996198i	$-0.472235\pi$
$194$	−1.41596e8	−1.39234
$195$	0	0
$196$	2.96567e8	2.81336
$197$	− 8.48880e7i	− 0.791069i	−0.918451	−	0.395535i	$-0.870559\pi$
$197$	− 8.48880e7i	− 0.791069i	0.918451	−	0.395535i	$-0.129441\pi$
$198$	0	0
$199$	4.12974e7	0.371481	0.185741	−	0.982599i	$-0.440532\pi$
$199$	4.12974e7	0.371481	0.185741	+	0.982599i	$0.440532\pi$
$200$	0	0
$201$	0	0
$202$	− 1.08239e8i	− 0.923961i
$203$	− 1.91424e8i	− 1.60605i
$204$	0	0
$205$	0	0
$206$	−1.43653e8	−1.14493
$207$	0	0
$208$	9.78328e7i	0.753812i
$209$	−2.35279e7	−0.178267
$210$	0	0
$211$	−1.55077e7	−0.113647	−0.0568236	−	0.998384i	$-0.518097\pi$
$211$	−1.55077e7	−0.113647	−0.0568236	+	0.998384i	$0.518097\pi$
$212$	− 1.39573e8i	− 1.00606i
$213$	0	0
$214$	2.04429e8	1.42592
$215$	0	0
$216$	0	0
$217$	− 2.85238e8i	− 1.89496i
$218$	2.43491e7i	0.159179i
$219$	0	0
$220$	0	0
$221$	2.01544e7	0.125602
$222$	0	0
$223$	4.16316e6i	0.0251395i	0.999921	+	0.0125697i	$0.00400118\pi$
$223$	4.16316e6i	0.0251395i	−0.999921	+	0.0125697i	$0.995999\pi$
$224$	−1.99580e8	−1.18645
$225$	0	0
$226$	−1.07181e7	−0.0617645
$227$	6.41270e7i	0.363873i	0.983310	+	0.181937i	$0.0582366\pi$
$227$	6.41270e7i	0.363873i	−0.983310	+	0.181937i	$0.941763\pi$
$228$	0	0
$229$	−5.41678e7	−0.298069	−0.149035	−	0.988832i	$-0.547617\pi$
$229$	−5.41678e7	−0.298069	−0.149035	+	0.988832i	$0.547617\pi$
$230$	0	0
$231$	0	0
$232$	4.34253e8i	2.28315i
$233$	2.74855e8i	1.42350i	0.702431	+	0.711752i	$0.252098\pi$
$233$	2.74855e8i	1.42350i	−0.702431	+	0.711752i	$0.747902\pi$
$234$	0	0
$235$	0	0
$236$	2.28088e7	0.112956
$237$	0	0
$238$	1.53385e8i	0.737505i
$239$	−1.86000e8	−0.881291	−0.440646	−	0.897681i	$-0.645250\pi$
$239$	−1.86000e8	−0.881291	−0.440646	+	0.897681i	$0.645250\pi$
$240$	0	0
$241$	2.37791e8	1.09430	0.547150	−	0.837034i	$-0.315713\pi$
$241$	2.37791e8	1.09430	0.547150	+	0.837034i	$0.315713\pi$
$242$	− 3.73632e8i	− 1.69469i
$243$	0	0
$244$	−8.88908e8	−3.91735
$245$	0	0
$246$	0	0
$247$	− 8.46593e7i	− 0.357467i
$248$	6.47075e8i	2.69385i
$249$	0	0
$250$	0	0
$251$	−3.28304e8	−1.31044	−0.655221	−	0.755438i	$-0.727424\pi$
$251$	−3.28304e8	−1.31044	−0.655221	+	0.755438i	$0.727424\pi$
$252$	0	0
$253$	− 9.75081e7i	− 0.378546i
$254$	3.85367e8	1.47556
$255$	0	0
$256$	−5.15111e8	−1.91894
$257$	− 1.01567e8i	− 0.373238i	−0.982432	−	0.186619i	$-0.940247\pi$
$257$	− 1.01567e8i	− 0.373238i	0.982432	−	0.186619i	$-0.0597530\pi$
$258$	0	0
$259$	6.14166e8	2.19653
$260$	0	0
$261$	0	0
$262$	− 9.13866e8i	− 3.13927i
$263$	− 1.98884e8i	− 0.674147i	−0.941478	−	0.337073i	$-0.890563\pi$
$263$	− 1.98884e8i	− 0.674147i	0.941478	−	0.337073i	$-0.109437\pi$
$264$	0	0
$265$	0	0
$266$	6.44301e8	2.09895
$267$	0	0
$268$	2.99416e8i	0.950175i
$269$	2.44943e8	0.767242	0.383621	−	0.923491i	$-0.374677\pi$
$269$	2.44943e8	0.767242	0.383621	+	0.923491i	$0.374677\pi$
$270$	0	0
$271$	−3.91023e8	−1.19347	−0.596733	−	0.802440i	$-0.703535\pi$
$271$	−3.91023e8	−1.19347	−0.596733	+	0.802440i	$0.703535\pi$
$272$	− 1.48572e8i	− 0.447657i
$273$	0	0
$274$	7.11610e8	2.08985
$275$	0	0
$276$	0	0
$277$	5.08902e8i	1.43865i	0.694674	+	0.719325i	$0.255549\pi$
$277$	5.08902e8i	1.43865i	−0.694674	+	0.719325i	$0.744451\pi$
$278$	8.31245e8i	2.32045i
$279$	0	0
$280$	0	0
$281$	1.55563e8	0.418248	0.209124	−	0.977889i	$-0.432939\pi$
$281$	1.55563e8	0.418248	0.209124	+	0.977889i	$0.432939\pi$
$282$	0	0
$283$	1.14570e8i	0.300481i	0.988649	+	0.150241i	$0.0480049\pi$
$283$	1.14570e8i	0.300481i	−0.988649	+	0.150241i	$0.951995\pi$
$284$	−8.44978e8	−2.18893
$285$	0	0
$286$	−7.46429e7	−0.188672
$287$	1.09212e8i	0.272699i
$288$	0	0
$289$	3.79732e8	0.925410
$290$	0	0
$291$	0	0
$292$	− 1.14248e8i	− 0.268539i
$293$	− 8.42345e8i	− 1.95638i	−0.207713	−	0.978190i	$-0.566602\pi$
$293$	− 8.42345e8i	− 1.95638i	0.207713	−	0.978190i	$-0.433398\pi$
$294$	0	0
$295$	0	0
$296$	−1.39326e9	−3.12256
$297$	0	0
$298$	1.10503e9i	2.41889i
$299$	3.50859e8	0.759073
$300$	0	0
$301$	−3.11427e8	−0.658225
$302$	− 1.55558e8i	− 0.324989i
$303$	0	0
$304$	−6.24080e8	−1.27404
$305$	0	0
$306$	0	0
$307$	− 2.38090e7i	− 0.0469631i	−0.999724	−	0.0234816i	$-0.992525\pi$
$307$	− 2.38090e7i	− 0.0469631i	0.999724	−	0.0234816i	$-0.00747510\pi$
$308$	− 3.90534e8i	− 0.761607i
$309$	0	0
$310$	0	0
$311$	−7.31704e8	−1.37935	−0.689674	−	0.724120i	$-0.742246\pi$
$311$	−7.31704e8	−1.37935	−0.689674	+	0.724120i	$0.742246\pi$
$312$	0	0
$313$	− 9.21577e8i	− 1.69874i	−0.527799	−	0.849369i	$-0.676983\pi$
$313$	− 9.21577e8i	− 1.69874i	0.527799	−	0.849369i	$-0.323017\pi$
$314$	−9.37438e7	−0.170879
$315$	0	0
$316$	−1.32824e9	−2.36795
$317$	− 7.85353e7i	− 0.138471i	−0.997600	−	0.0692353i	$-0.977944\pi$
$317$	− 7.85353e7i	− 0.138471i	0.997600	−	0.0692353i	$-0.0220559\pi$
$318$	0	0
$319$	−1.41466e8	−0.243997
$320$	0	0
$321$	0	0
$322$	2.67022e9i	4.45709i
$323$	1.28566e8i	0.212284i
$324$	0	0
$325$	0	0
$326$	−4.36916e8	−0.698451
$327$	0	0
$328$	− 2.47752e8i	− 0.387666i
$329$	−9.86230e8	−1.52684
$330$	0	0
$331$	3.78878e8	0.574250	0.287125	−	0.957893i	$-0.407300\pi$
$331$	3.78878e8	0.574250	0.287125	+	0.957893i	$0.407300\pi$
$332$	1.44755e9i	2.17095i
$333$	0	0
$334$	−7.51487e8	−1.10359
$335$	0	0
$336$	0	0
$337$	− 1.08031e9i	− 1.53760i	−0.639489	−	0.768800i	$-0.720854\pi$
$337$	− 1.08031e9i	− 1.53760i	0.639489	−	0.768800i	$-0.279146\pi$
$338$	1.00130e9i	1.41045i
$339$	0	0
$340$	0	0
$341$	−2.10797e8	−0.287888
$342$	0	0
$343$	− 3.14726e8i	− 0.421117i
$344$	7.06485e8	0.935726
$345$	0	0
$346$	1.23004e9	1.59644
$347$	8.77465e8i	1.12740i	0.825981	+	0.563698i	$0.190622\pi$
$347$	8.77465e8i	1.12740i	−0.825981	+	0.563698i	$0.809378\pi$
$348$	0	0
$349$	−3.31907e8	−0.417953	−0.208976	−	0.977921i	$-0.567013\pi$
$349$	−3.31907e8	−0.417953	−0.208976	+	0.977921i	$0.567013\pi$
$350$	0	0
$351$	0	0
$352$	1.47494e8i	0.180250i
$353$	− 9.88344e8i	− 1.19590i	−0.801532	−	0.597952i	$-0.795981\pi$
$353$	− 9.88344e8i	− 1.19590i	0.801532	−	0.597952i	$-0.204019\pi$
$354$	0	0
$355$	0	0
$356$	2.42894e9	2.85326
$357$	0	0
$358$	6.18315e8i	0.712228i
$359$	1.41167e9	1.61028	0.805140	−	0.593084i	$-0.202090\pi$
$359$	1.41167e9	1.61028	0.805140	+	0.593084i	$0.202090\pi$
$360$	0	0
$361$	−3.53826e8	−0.395835
$362$	− 2.39058e9i	− 2.64864i
$363$	0	0
$364$	1.40524e9	1.52720
$365$	0	0
$366$	0	0
$367$	− 7.57010e8i	− 0.799412i	−0.916643	−	0.399706i	$-0.869112\pi$
$367$	− 7.57010e8i	− 0.799412i	0.916643	−	0.399706i	$-0.130888\pi$
$368$	− 2.58642e9i	− 2.70540i
$369$	0	0
$370$	0	0
$371$	−6.79097e8	−0.690435
$372$	0	0
$373$	2.96259e8i	0.295591i	0.989018	+	0.147796i	$0.0472178\pi$
$373$	2.96259e8i	0.295591i	−0.989018	+	0.147796i	$0.952782\pi$
$374$	1.13355e8	0.112044
$375$	0	0
$376$	2.23730e9	2.17054
$377$	− 5.09032e8i	− 0.489272i
$378$	0	0
$379$	−1.32923e9	−1.25419	−0.627093	−	0.778944i	$-0.715756\pi$
$379$	−1.32923e9	−1.25419	−0.627093	+	0.778944i	$0.715756\pi$
$380$	0	0
$381$	0	0
$382$	− 1.66007e9i	− 1.52372i
$383$	8.35565e8i	0.759949i	0.924997	+	0.379974i	$0.124067\pi$
$383$	8.35565e8i	0.759949i	−0.924997	+	0.379974i	$0.875933\pi$
$384$	0	0
$385$	0	0
$386$	4.02706e9	3.56396
$387$	0	0
$388$	− 1.97001e9i	− 1.71221i
$389$	1.65397e8	0.142464	0.0712320	−	0.997460i	$-0.477307\pi$
$389$	1.65397e8	0.142464	0.0712320	+	0.997460i	$0.477307\pi$
$390$	0	0
$391$	−5.32825e8	−0.450781
$392$	3.27340e9i	2.74472i
$393$	0	0
$394$	1.71794e9	1.41505
$395$	0	0
$396$	0	0
$397$	− 2.01752e8i	− 0.161827i	−0.996721	−	0.0809134i	$-0.974216\pi$
$397$	− 2.01752e8i	− 0.161827i	0.996721	−	0.0809134i	$-0.0257837\pi$
$398$	8.35765e8i	0.664498i
$399$	0	0
$400$	0	0
$401$	2.44937e8	0.189692	0.0948462	−	0.995492i	$-0.469764\pi$
$401$	2.44937e8	0.189692	0.0948462	+	0.995492i	$0.469764\pi$
$402$	0	0
$403$	− 7.58502e8i	− 0.577283i
$404$	1.50592e9	1.13623
$405$	0	0
$406$	3.87399e9	2.87288
$407$	− 4.53881e8i	− 0.333704i
$408$	0	0
$409$	−1.72583e8	−0.124728	−0.0623642	−	0.998053i	$-0.519864\pi$
$409$	−1.72583e8	−0.124728	−0.0623642	+	0.998053i	$0.519864\pi$
$410$	0	0
$411$	0	0
$412$	− 1.99863e9i	− 1.40797i
$413$	− 1.10977e8i	− 0.0775190i
$414$	0	0
$415$	0	0
$416$	−5.30721e8	−0.361443
$417$	0	0
$418$	− 4.76151e8i	− 0.318880i
$419$	−2.64174e9	−1.75445	−0.877225	−	0.480079i	$-0.840608\pi$
$419$	−2.64174e9	−1.75445	−0.877225	+	0.480079i	$0.840608\pi$
$420$	0	0
$421$	1.11997e9	0.731509	0.365754	−	0.930711i	$-0.380811\pi$
$421$	1.11997e9	0.731509	0.365754	+	0.930711i	$0.380811\pi$
$422$	− 3.13841e8i	− 0.203290i
$423$	0	0
$424$	1.54056e9	0.981516
$425$	0	0
$426$	0	0
$427$	4.32501e9i	2.68838i
$428$	2.84421e9i	1.75351i
$429$	0	0
$430$	0	0
$431$	2.56747e9	1.54466	0.772332	−	0.635219i	$-0.219090\pi$
$431$	2.56747e9	1.54466	0.772332	+	0.635219i	$0.219090\pi$
$432$	0	0
$433$	9.65650e8i	0.571626i	0.958285	+	0.285813i	$0.0922637\pi$
$433$	9.65650e8i	0.571626i	−0.958285	+	0.285813i	$0.907736\pi$
$434$	5.77258e9	3.38966
$435$	0	0
$436$	−3.38767e8	−0.195749
$437$	2.23815e9i	1.28293i
$438$	0	0
$439$	−2.67168e9	−1.50716	−0.753580	−	0.657356i	$-0.771675\pi$
$439$	−2.67168e9	−1.50716	−0.753580	+	0.657356i	$0.771675\pi$
$440$	0	0
$441$	0	0
$442$	4.07880e8i	0.224675i
$443$	− 9.25736e8i	− 0.505911i	−0.967478	−	0.252955i	$-0.918597\pi$
$443$	− 9.25736e8i	− 0.505911i	0.967478	−	0.252955i	$-0.0814026\pi$
$444$	0	0
$445$	0	0
$446$	−8.42530e7	−0.0449690
$447$	0	0
$448$	6.70136e8i	0.352119i
$449$	−3.43969e9	−1.79332	−0.896658	−	0.442724i	$-0.854012\pi$
$449$	−3.43969e9	−1.79332	−0.896658	+	0.442724i	$0.854012\pi$
$450$	0	0
$451$	8.07098e7	0.0414294
$452$	− 1.49120e8i	− 0.0759542i
$453$	0	0
$454$	−1.29778e9	−0.650889
$455$	0	0
$456$	0	0
$457$	4.40856e8i	0.216068i	0.994147	+	0.108034i	$0.0344555\pi$
$457$	4.40856e8i	0.216068i	−0.994147	+	0.108034i	$0.965544\pi$
$458$	− 1.09623e9i	− 0.533180i
$459$	0	0
$460$	0	0
$461$	−2.97064e9	−1.41220	−0.706100	−	0.708112i	$-0.749547\pi$
$461$	−2.97064e9	−1.41220	−0.706100	+	0.708112i	$0.749547\pi$
$462$	0	0
$463$	− 1.59975e9i	− 0.749062i	−0.927214	−	0.374531i	$-0.877804\pi$
$463$	− 1.59975e9i	− 0.749062i	0.927214	−	0.374531i	$-0.122196\pi$
$464$	−3.75241e9	−1.74380
$465$	0	0
$466$	−5.56245e9	−2.54634
$467$	− 1.62694e9i	− 0.739201i	−0.929191	−	0.369600i	$-0.879495\pi$
$467$	− 1.62694e9i	− 0.739201i	0.929191	−	0.369600i	$-0.120505\pi$
$468$	0	0
$469$	1.45682e9	0.652080
$470$	0	0
$471$	0	0
$472$	2.51756e8i	0.110200i
$473$	2.30151e8i	0.0999998i
$474$	0	0
$475$	0	0
$476$	−2.13404e9	−0.906939
$477$	0	0
$478$	− 3.76421e9i	− 1.57644i
$479$	−3.93755e8	−0.163701	−0.0818507	−	0.996645i	$-0.526083\pi$
$479$	−3.93755e8	−0.163701	−0.0818507	+	0.996645i	$0.526083\pi$
$480$	0	0
$481$	1.63318e9	0.669154
$482$	4.81236e9i	1.95746i
$483$	0	0
$484$	5.19831e9	2.08403
$485$	0	0
$486$	0	0
$487$	2.90720e9i	1.14058i	0.821445	+	0.570288i	$0.193168\pi$
$487$	2.90720e9i	1.14058i	−0.821445	+	0.570288i	$0.806832\pi$
$488$	− 9.81146e9i	− 3.82177i
$489$	0	0
$490$	0	0
$491$	−1.49340e9	−0.569364	−0.284682	−	0.958622i	$-0.591888\pi$
$491$	−1.49340e9	−0.569364	−0.284682	+	0.958622i	$0.591888\pi$
$492$	0	0
$493$	7.73030e8i	0.290558i
$494$	1.71331e9	0.639429
$495$	0	0
$496$	−5.59142e9	−2.05748
$497$	4.11127e9i	1.50220i
$498$	0	0
$499$	−2.32305e9	−0.836965	−0.418482	−	0.908225i	$-0.637438\pi$
$499$	−2.32305e9	−0.836965	−0.418482	+	0.908225i	$0.637438\pi$
$500$	0	0
$501$	0	0
$502$	− 6.64412e9i	− 2.34409i
$503$	− 4.58290e9i	− 1.60566i	−0.596210	−	0.802828i	$-0.703328\pi$
$503$	− 4.58290e9i	− 1.60566i	0.596210	−	0.802828i	$-0.296672\pi$
$504$	0	0
$505$	0	0
$506$	1.97334e9	0.677136
$507$	0	0
$508$	5.36158e9i	1.81455i
$509$	−3.08836e9	−1.03804	−0.519022	−	0.854761i	$-0.673704\pi$
$509$	−3.08836e9	−1.03804	−0.519022	+	0.854761i	$0.673704\pi$
$510$	0	0
$511$	−5.55876e8	−0.184291
$512$	− 6.77068e9i	− 2.22940i
$513$	0	0
$514$	2.05548e9	0.667641
$515$	0	0
$516$	0	0
$517$	7.28843e8i	0.231962i
$518$	1.24293e10i	3.92910i
$519$	0	0
$520$	0	0
$521$	−3.95517e9	−1.22527	−0.612637	−	0.790364i	$-0.709891\pi$
$521$	−3.95517e9	−1.22527	−0.612637	+	0.790364i	$0.709891\pi$
$522$	0	0
$523$	2.97750e9i	0.910113i	0.890463	+	0.455057i	$0.150381\pi$
$523$	2.97750e9i	0.910113i	−0.890463	+	0.455057i	$0.849619\pi$
$524$	1.27145e10	3.86048
$525$	0	0
$526$	4.02496e9	1.20590
$527$	1.15188e9i	0.342824i
$528$	0	0
$529$	−5.87088e9	−1.72428
$530$	0	0
$531$	0	0
$532$	8.96410e9i	2.58117i
$533$	2.90415e8i	0.0830756i
$534$	0	0
$535$	0	0
$536$	−3.30485e9	−0.926991
$537$	0	0
$538$	4.95710e9i	1.37243i
$539$	−1.06637e9	−0.293324
$540$	0	0
$541$	−4.75653e9	−1.29152	−0.645758	−	0.763542i	$-0.723459\pi$
$541$	−4.75653e9	−1.29152	−0.645758	+	0.763542i	$0.723459\pi$
$542$	− 7.91342e9i	− 2.13485i
$543$	0	0
$544$	8.05968e8	0.214645
$545$	0	0
$546$	0	0
$547$	− 4.58098e9i	− 1.19675i	−0.801217	−	0.598374i	$-0.795814\pi$
$547$	− 4.58098e9i	− 1.19675i	0.801217	−	0.598374i	$-0.204186\pi$
$548$	9.90057e9i	2.56997i
$549$	0	0
$550$	0	0
$551$	3.24714e9	0.826933
$552$	0	0
$553$	6.46260e9i	1.62506i
$554$	−1.02990e10	−2.57343
$555$	0	0
$556$	−1.15650e10	−2.85355
$557$	1.98969e9i	0.487856i	0.969793	+	0.243928i	$0.0784361\pi$
$557$	1.98969e9i	0.487856i	−0.969793	+	0.243928i	$0.921564\pi$
$558$	0	0
$559$	−8.28143e8	−0.200523
$560$	0	0
$561$	0	0
$562$	3.14824e9i	0.748154i
$563$	1.51071e9i	0.356781i	0.983960	+	0.178390i	$0.0570890\pi$
$563$	1.51071e9i	0.356781i	−0.983960	+	0.178390i	$0.942911\pi$
$564$	0	0
$565$	0	0
$566$	−2.31863e9	−0.537495
$567$	0	0
$568$	− 9.32658e9i	− 2.13552i
$569$	1.75619e8	0.0399650	0.0199825	−	0.999800i	$-0.493639\pi$
$569$	1.75619e8	0.0399650	0.0199825	+	0.999800i	$0.493639\pi$
$570$	0	0
$571$	−2.21767e8	−0.0498506	−0.0249253	−	0.999689i	$-0.507935\pi$
$571$	−2.21767e8	−0.0498506	−0.0249253	+	0.999689i	$0.507935\pi$
$572$	− 1.03850e9i	− 0.232018i
$573$	0	0
$574$	−2.21020e9	−0.487798
$575$	0	0
$576$	0	0
$577$	7.51519e9i	1.62864i	0.580418	+	0.814319i	$0.302889\pi$
$577$	7.51519e9i	1.62864i	−0.580418	+	0.814319i	$0.697111\pi$
$578$	7.68491e9i	1.65536i
$579$	0	0
$580$	0	0
$581$	7.04310e9	1.48987
$582$	0	0
$583$	5.01866e8i	0.104893i
$584$	1.26103e9	0.261987
$585$	0	0
$586$	1.70472e10	3.49953
$587$	− 5.57215e9i	− 1.13708i	−0.822657	−	0.568538i	$-0.807509\pi$
$587$	− 5.57215e9i	− 1.13708i	0.822657	−	0.568538i	$-0.192491\pi$
$588$	0	0
$589$	4.83852e9	0.975683
$590$	0	0
$591$	0	0
$592$	− 1.20393e10i	− 2.38492i
$593$	− 2.27962e9i	− 0.448922i	−0.974483	−	0.224461i	$-0.927938\pi$
$593$	− 2.27962e9i	− 0.448922i	0.974483	−	0.224461i	$-0.0720621\pi$
$594$	0	0
$595$	0	0
$596$	−1.53742e10	−2.97461
$597$	0	0
$598$	7.10060e9i	1.35782i
$599$	3.90822e9	0.742993	0.371497	−	0.928434i	$-0.378845\pi$
$599$	3.90822e9	0.742993	0.371497	+	0.928434i	$0.378845\pi$
$600$	0	0
$601$	−2.15505e9	−0.404946	−0.202473	−	0.979288i	$-0.564898\pi$
$601$	−2.15505e9	−0.404946	−0.202473	+	0.979288i	$0.564898\pi$
$602$	− 6.30259e9i	− 1.17742i
$603$	0	0
$604$	2.16427e9	0.399652
$605$	0	0
$606$	0	0
$607$	− 5.06857e9i	− 0.919867i	−0.887953	−	0.459933i	$-0.847873\pi$
$607$	− 5.06857e9i	− 0.919867i	0.887953	−	0.459933i	$-0.152127\pi$
$608$	− 3.38549e9i	− 0.610885i
$609$	0	0
$610$	0	0
$611$	−2.62257e9	−0.465138
$612$	0	0
$613$	− 6.56079e9i	− 1.15039i	−0.818017	−	0.575194i	$-0.804927\pi$
$613$	− 6.56079e9i	− 1.15039i	0.818017	−	0.575194i	$-0.195073\pi$
$614$	4.81841e8	0.0840068
$615$	0	0
$616$	4.31058e9	0.743024
$617$	1.11814e8i	0.0191645i	0.999954	+	0.00958223i	$0.00305017\pi$
$617$	1.11814e8i	0.0191645i	−0.999954	+	0.00958223i	$0.996950\pi$
$618$	0	0
$619$	−1.48440e9	−0.251556	−0.125778	−	0.992058i	$-0.540143\pi$
$619$	−1.48440e9	−0.251556	−0.125778	+	0.992058i	$0.540143\pi$
$620$	0	0
$621$	0	0
$622$	− 1.48080e10i	− 2.46735i
$623$	− 1.18181e10i	− 1.95812i
$624$	0	0
$625$	0	0
$626$	1.86506e10	3.03867
$627$	0	0
$628$	− 1.30425e9i	− 0.210137i
$629$	−2.48019e9	−0.397382
$630$	0	0
$631$	6.22722e9	0.986714	0.493357	−	0.869827i	$-0.335770\pi$
$631$	6.22722e9	0.986714	0.493357	+	0.869827i	$0.335770\pi$
$632$	− 1.46607e10i	− 2.31017i
$633$	0	0
$634$	1.58938e9	0.247693
$635$	0	0
$636$	0	0
$637$	− 3.83708e9i	− 0.588184i
$638$	− 2.86295e9i	− 0.436458i
$639$	0	0
$640$	0	0
$641$	−6.72145e9	−1.00800	−0.503999	−	0.863704i	$-0.668139\pi$
$641$	−6.72145e9	−1.00800	−0.503999	+	0.863704i	$0.668139\pi$
$642$	0	0
$643$	− 6.13000e9i	− 0.909332i	−0.890662	−	0.454666i	$-0.849759\pi$
$643$	− 6.13000e9i	− 0.909332i	0.890662	−	0.454666i	$-0.150241\pi$
$644$	−3.71505e10	−5.48106
$645$	0	0
$646$	−2.60189e9	−0.379730
$647$	1.10514e10i	1.60418i	0.597204	+	0.802089i	$0.296278\pi$
$647$	1.10514e10i	1.60418i	−0.597204	+	0.802089i	$0.703722\pi$
$648$	0	0
$649$	−8.20142e7	−0.0117769
$650$	0	0
$651$	0	0
$652$	− 6.07878e9i	− 0.858914i
$653$	1.90379e9i	0.267561i	0.991011	+	0.133781i	$0.0427117\pi$
$653$	1.90379e9i	0.267561i	−0.991011	+	0.133781i	$0.957288\pi$
$654$	0	0
$655$	0	0
$656$	2.14084e9	0.296088
$657$	0	0
$658$	− 1.99591e10i	− 2.73117i
$659$	1.16276e9	0.158268	0.0791339	−	0.996864i	$-0.474785\pi$
$659$	1.16276e9	0.158268	0.0791339	+	0.996864i	$0.474785\pi$
$660$	0	0
$661$	−2.20847e9	−0.297431	−0.148716	−	0.988880i	$-0.547514\pi$
$661$	−2.20847e9	−0.297431	−0.148716	+	0.988880i	$0.547514\pi$
$662$	7.66763e9i	1.02721i
$663$	0	0
$664$	−1.59775e10	−2.11798
$665$	0	0
$666$	0	0
$667$	1.34573e10i	1.75598i
$668$	− 1.04554e10i	− 1.35713i
$669$	0	0
$670$	0	0
$671$	3.19627e9	0.408427
$672$	0	0
$673$	− 9.22930e9i	− 1.16712i	−0.812070	−	0.583561i	$-0.801659\pi$
$673$	− 9.22930e9i	− 1.16712i	0.812070	−	0.583561i	$-0.198341\pi$
$674$	2.18630e10	2.75043
$675$	0	0
$676$	−1.39311e10	−1.73449
$677$	− 5.75119e9i	− 0.712357i	−0.934418	−	0.356178i	$-0.884080\pi$
$677$	− 5.75119e9i	− 0.712357i	0.934418	−	0.356178i	$-0.115920\pi$
$678$	0	0
$679$	−9.58518e9	−1.17505
$680$	0	0
$681$	0	0
$682$	− 4.26605e9i	− 0.514969i
$683$	− 2.96482e9i	− 0.356063i	−0.984025	−	0.178031i	$-0.943027\pi$
$683$	− 2.96482e9i	− 0.356063i	0.984025	−	0.178031i	$-0.0569729\pi$
$684$	0	0
$685$	0	0
$686$	6.36934e9	0.753287
$687$	0	0
$688$	6.10479e9i	0.714680i
$689$	−1.80584e9	−0.210336
$690$	0	0
$691$	1.35266e10	1.55961	0.779805	−	0.626023i	$-0.215318\pi$
$691$	1.35266e10	1.55961	0.779805	+	0.626023i	$0.215318\pi$
$692$	1.71135e10i	1.96321i
$693$	0	0
$694$	−1.77579e10	−2.01666
$695$	0	0
$696$	0	0
$697$	− 4.41032e8i	− 0.0493350i
$698$	− 6.71704e9i	− 0.747626i
$699$	0	0
$700$	0	0
$701$	−8.89669e9	−0.975473	−0.487737	−	0.872991i	$-0.662177\pi$
$701$	−8.89669e9	−0.975473	−0.487737	+	0.872991i	$0.662177\pi$
$702$	0	0
$703$	1.04181e10i	1.13096i
$704$	4.95244e8	0.0534952
$705$	0	0
$706$	2.00018e10	2.13921
$707$	− 7.32710e9i	− 0.779766i
$708$	0	0
$709$	4.02759e9	0.424408	0.212204	−	0.977225i	$-0.431936\pi$
$709$	4.02759e9	0.424408	0.212204	+	0.977225i	$0.431936\pi$
$710$	0	0
$711$	0	0
$712$	2.68098e10i	2.78364i
$713$	2.00526e10i	2.07184i
$714$	0	0
$715$	0	0
$716$	−8.60256e9	−0.875855
$717$	0	0
$718$	2.85689e10i	2.88044i
$719$	−9.22714e9	−0.925798	−0.462899	−	0.886411i	$-0.653191\pi$
$719$	−9.22714e9	−0.925798	−0.462899	+	0.886411i	$0.653191\pi$
$720$	0	0
$721$	−9.72440e9	−0.966250
$722$	− 7.16063e9i	− 0.708062i
$723$	0	0
$724$	3.32600e10	3.25714
$725$	0	0
$726$	0	0
$727$	− 9.12103e9i	− 0.880386i	−0.897903	−	0.440193i	$-0.854910\pi$
$727$	− 9.12103e9i	− 0.880386i	0.897903	−	0.440193i	$-0.145090\pi$
$728$	1.55106e10i	1.48994i
$729$	0	0
$730$	0	0
$731$	1.25764e9	0.119082
$732$	0	0
$733$	− 1.72029e10i	− 1.61338i	−0.590974	−	0.806691i	$-0.701256\pi$
$733$	− 1.72029e10i	− 1.61338i	0.590974	−	0.806691i	$-0.298744\pi$
$734$	1.53202e10	1.42997
$735$	0	0
$736$	1.40307e10	1.29720
$737$	− 1.07662e9i	− 0.0990663i
$738$	0	0
$739$	1.73905e10	1.58510	0.792552	−	0.609805i	$-0.208752\pi$
$739$	1.73905e10	1.58510	0.792552	+	0.609805i	$0.208752\pi$
$740$	0	0
$741$	0	0
$742$	− 1.37434e10i	− 1.23504i
$743$	4.62464e9i	0.413635i	0.978380	+	0.206817i	$0.0663106\pi$
$743$	4.62464e9i	0.413635i	−0.978380	+	0.206817i	$0.933689\pi$
$744$	0	0
$745$	0	0
$746$	−5.99562e9	−0.528748
$747$	0	0
$748$	1.57710e9i	0.137785i
$749$	1.38386e10	1.20339
$750$	0	0
$751$	−1.36007e10	−1.17171	−0.585856	−	0.810415i	$-0.699242\pi$
$751$	−1.36007e10	−1.17171	−0.585856	+	0.810415i	$0.699242\pi$
$752$	1.93327e10i	1.65779i
$753$	0	0
$754$	1.03017e10	0.875200
$755$	0	0
$756$	0	0
$757$	− 1.78432e10i	− 1.49499i	−0.664269	−	0.747493i	$-0.731257\pi$
$757$	− 1.78432e10i	− 1.49499i	0.664269	−	0.747493i	$-0.268743\pi$
$758$	− 2.69006e10i	− 2.24346i
$759$	0	0
$760$	0	0
$761$	1.41211e10	1.16151	0.580754	−	0.814079i	$-0.302758\pi$
$761$	1.41211e10	1.16151	0.580754	+	0.814079i	$0.302758\pi$
$762$	0	0
$763$	1.64829e9i	0.134337i
$764$	2.30964e10	1.87377
$765$	0	0
$766$	−1.69099e10	−1.35938
$767$	− 2.95108e8i	− 0.0236155i
$768$	0	0
$769$	6.98040e9	0.553526	0.276763	−	0.960938i	$-0.410738\pi$
$769$	6.98040e9	0.553526	0.276763	+	0.960938i	$0.410738\pi$
$770$	0	0
$771$	0	0
$772$	5.60282e10i	4.38274i
$773$	− 5.57792e9i	− 0.434354i	−0.976132	−	0.217177i	$-0.930315\pi$
$773$	− 5.57792e9i	− 0.434354i	0.976132	−	0.217177i	$-0.0696849\pi$
$774$	0	0
$775$	0	0
$776$	2.17443e10	1.67044
$777$	0	0
$778$	3.34727e9i	0.254837i
$779$	−1.85257e9	−0.140408
$780$	0	0
$781$	3.03831e9	0.228220
$782$	− 1.07832e10i	− 0.806349i
$783$	0	0
$784$	−2.82857e10	−2.09634
$785$	0	0
$786$	0	0
$787$	1.26349e10i	0.923978i	0.886886	+	0.461989i	$0.152864\pi$
$787$	1.26349e10i	0.923978i	−0.886886	+	0.461989i	$0.847136\pi$
$788$	2.39016e10i	1.74014i
$789$	0	0
$790$	0	0
$791$	−7.25550e8	−0.0521254
$792$	0	0
$793$	1.15010e10i	0.818992i
$794$	4.08300e9	0.289473
$795$	0	0
$796$	−1.16279e10	−0.817160
$797$	− 9.24597e9i	− 0.646916i	−0.946242	−	0.323458i	$-0.895154\pi$
$797$	− 9.24597e9i	− 0.646916i	0.946242	−	0.323458i	$-0.104846\pi$
$798$	0	0
$799$	3.98271e9	0.276226
$800$	0	0
$801$	0	0
$802$	4.95698e9i	0.339318i
$803$	4.10803e8i	0.0279982i
$804$	0	0
$805$	0	0
$806$	1.53504e10	1.03263
$807$	0	0
$808$	1.66218e10i	1.10851i
$809$	3.77594e9	0.250729	0.125365	−	0.992111i	$-0.459990\pi$
$809$	3.77594e9	0.250729	0.125365	+	0.992111i	$0.459990\pi$
$810$	0	0
$811$	−8.24089e9	−0.542502	−0.271251	−	0.962509i	$-0.587437\pi$
$811$	−8.24089e9	−0.542502	−0.271251	+	0.962509i	$0.587437\pi$
$812$	5.38985e10i	3.53290i
$813$	0	0
$814$	9.18552e9	0.596923
$815$	0	0
$816$	0	0
$817$	− 5.28276e9i	− 0.338910i
$818$	− 3.49268e9i	− 0.223112i
$819$	0	0
$820$	0	0
$821$	−8.07649e9	−0.509356	−0.254678	−	0.967026i	$-0.581969\pi$
$821$	−8.07649e9	−0.509356	−0.254678	+	0.967026i	$0.581969\pi$
$822$	0	0
$823$	1.76555e10i	1.10403i	0.833834	+	0.552016i	$0.186141\pi$
$823$	1.76555e10i	1.10403i	−0.833834	+	0.552016i	$0.813859\pi$
$824$	2.20602e10	1.37361
$825$	0	0
$826$	2.24593e9	0.138664
$827$	− 2.60329e9i	− 0.160049i	−0.996793	−	0.0800246i	$-0.974500\pi$
$827$	− 2.60329e9i	− 0.160049i	0.996793	−	0.0800246i	$-0.0254999\pi$
$828$	0	0
$829$	−9.16053e9	−0.558444	−0.279222	−	0.960227i	$-0.590077\pi$
$829$	−9.16053e9	−0.558444	−0.279222	+	0.960227i	$0.590077\pi$
$830$	0	0
$831$	0	0
$832$	1.78202e9i	0.107270i
$833$	5.82711e9i	0.349297i
$834$	0	0
$835$	0	0
$836$	6.62465e9	0.392140
$837$	0	0
$838$	− 5.34629e10i	− 3.13833i
$839$	2.51395e10	1.46957	0.734784	−	0.678301i	$-0.237283\pi$
$839$	2.51395e10	1.46957	0.734784	+	0.678301i	$0.237283\pi$
$840$	0	0
$841$	2.27422e9	0.131840
$842$	2.26657e10i	1.30851i
$843$	0	0
$844$	4.36644e9	0.249994
$845$	0	0
$846$	0	0
$847$	− 2.52926e10i	− 1.43022i
$848$	1.33121e10i	0.749653i
$849$	0	0
$850$	0	0
$851$	−4.31766e10	−2.40157
$852$	0	0
$853$	− 1.37385e10i	− 0.757908i	−0.925415	−	0.378954i	$-0.876284\pi$
$853$	− 1.37385e10i	− 0.757908i	0.925415	−	0.378954i	$-0.123716\pi$
$854$	−8.75285e10	−4.80891
$855$	0	0
$856$	−3.13934e10	−1.71072
$857$	2.41390e10i	1.31004i	0.755610	+	0.655022i	$0.227341\pi$
$857$	2.41390e10i	1.31004i	−0.755610	+	0.655022i	$0.772659\pi$
$858$	0	0
$859$	2.62094e10	1.41085	0.705426	−	0.708783i	$-0.250756\pi$
$859$	2.62094e10	1.41085	0.705426	+	0.708783i	$0.250756\pi$
$860$	0	0
$861$	0	0
$862$	5.19597e10i	2.76307i
$863$	− 1.61957e10i	− 0.857750i	−0.903364	−	0.428875i	$-0.858910\pi$
$863$	− 1.61957e10i	− 0.857750i	0.903364	−	0.428875i	$-0.141090\pi$
$864$	0	0
$865$	0	0
$866$	−1.95426e10	−1.02251
$867$	0	0
$868$	8.03135e10i	4.16840i
$869$	4.77599e9	0.246885
$870$	0	0
$871$	3.87395e9	0.198651
$872$	− 3.73920e9i	− 0.190973i
$873$	0	0
$874$	−4.52951e10	−2.29488
$875$	0	0
$876$	0	0
$877$	− 1.25717e10i	− 0.629356i	−0.949198	−	0.314678i	$-0.898103\pi$
$877$	− 1.25717e10i	− 0.629356i	0.949198	−	0.314678i	$-0.101897\pi$
$878$	− 5.40688e10i	− 2.69598i
$879$	0	0
$880$	0	0
$881$	3.40786e10	1.67906	0.839530	−	0.543313i	$-0.182830\pi$
$881$	3.40786e10	1.67906	0.839530	+	0.543313i	$0.182830\pi$
$882$	0	0
$883$	− 2.78949e10i	− 1.36352i	−0.731574	−	0.681762i	$-0.761214\pi$
$883$	− 2.78949e10i	− 1.36352i	0.731574	−	0.681762i	$-0.238786\pi$
$884$	−5.67480e9	−0.276292
$885$	0	0
$886$	1.87348e10	0.904963
$887$	3.29903e10i	1.58728i	0.608387	+	0.793641i	$0.291817\pi$
$887$	3.29903e10i	1.58728i	−0.608387	+	0.793641i	$0.708183\pi$
$888$	0	0
$889$	2.60870e10	1.24528
$890$	0	0
$891$	0	0
$892$	− 1.17221e9i	− 0.0553002i
$893$	− 1.67295e10i	− 0.786144i
$894$	0	0
$895$	0	0
$896$	−3.91083e10	−1.81631
$897$	0	0
$898$	− 6.96115e10i	− 3.20785i
$899$	2.90926e10	1.33544
$900$	0	0
$901$	2.74241e9	0.124909
$902$	1.63338e9i	0.0741080i
$903$	0	0
$904$	1.64594e9	0.0741010
$905$	0	0
$906$	0	0
$907$	1.66008e10i	0.738759i	0.929279	+	0.369380i	$0.120430\pi$
$907$	1.66008e10i	0.738759i	−0.929279	+	0.369380i	$0.879570\pi$
$908$	− 1.80560e10i	− 0.800425i
$909$	0	0
$910$	0	0
$911$	2.80189e9	0.122782	0.0613912	−	0.998114i	$-0.480446\pi$
$911$	2.80189e9	0.122782	0.0613912	+	0.998114i	$0.480446\pi$
$912$	0	0
$913$	− 5.20499e9i	− 0.226346i
$914$	−8.92193e9	−0.386498
$915$	0	0
$916$	1.52518e10	0.655673
$917$	− 6.18631e10i	− 2.64935i
$918$	0	0
$919$	7.97528e9	0.338955	0.169477	−	0.985534i	$-0.445792\pi$
$919$	7.97528e9	0.338955	0.169477	+	0.985534i	$0.445792\pi$
$920$	0	0
$921$	0	0
$922$	− 6.01190e10i	− 2.52612i
$923$	1.09326e10i	0.457635i
$924$	0	0
$925$	0	0
$926$	3.23753e10	1.33991
$927$	0	0
$928$	− 2.03560e10i	− 0.836131i
$929$	−4.40021e10	−1.80061	−0.900303	−	0.435264i	$-0.856655\pi$
$929$	−4.40021e10	−1.80061	−0.900303	+	0.435264i	$0.856655\pi$
$930$	0	0
$931$	2.44769e10	0.994107
$932$	− 7.73900e10i	− 3.13133i
$933$	0	0
$934$	3.29256e10	1.32227
$935$	0	0
$936$	0	0
$937$	− 1.85516e10i	− 0.736702i	−0.929687	−	0.368351i	$-0.879922\pi$
$937$	− 1.85516e10i	− 0.736702i	0.929687	−	0.368351i	$-0.120078\pi$
$938$	2.94828e10i	1.16643i
$939$	0	0
$940$	0	0
$941$	3.98382e8	0.0155860	0.00779302	−	0.999970i	$-0.497519\pi$
$941$	3.98382e8	0.0155860	0.00779302	+	0.999970i	$0.497519\pi$
$942$	0	0
$943$	− 7.67772e9i	− 0.298155i
$944$	−2.17544e9	−0.0841677
$945$	0	0
$946$	−4.65774e9	−0.178878
$947$	− 1.22967e10i	− 0.470504i	−0.971934	−	0.235252i	$-0.924409\pi$
$947$	− 1.22967e10i	− 0.470504i	0.971934	−	0.235252i	$-0.0755915\pi$
$948$	0	0
$949$	−1.47818e9	−0.0561429
$950$	0	0
$951$	0	0
$952$	− 2.35548e10i	− 0.884810i
$953$	2.00329e10i	0.749754i	0.927075	+	0.374877i	$0.122315\pi$
$953$	2.00329e10i	0.749754i	−0.927075	+	0.374877i	$0.877685\pi$
$954$	0	0
$955$	0	0
$956$	5.23712e10	1.93861
$957$	0	0
$958$	− 7.96872e9i	− 0.292826i
$959$	4.81716e10	1.76370
$960$	0	0
$961$	1.58379e10	0.575659
$962$	3.30519e10i	1.19697i
$963$	0	0
$964$	−6.69540e10	−2.40717
$965$	0	0
$966$	0	0
$967$	− 2.01344e10i	− 0.716055i	−0.933711	−	0.358027i	$-0.883449\pi$
$967$	− 2.01344e10i	− 0.716055i	0.933711	−	0.358027i	$-0.116551\pi$
$968$	5.73772e10i	2.03318i
$969$	0	0
$970$	0	0
$971$	5.02424e10	1.76118	0.880589	−	0.473881i	$-0.157147\pi$
$971$	5.02424e10	1.76118	0.880589	+	0.473881i	$0.157147\pi$
$972$	0	0
$973$	5.62702e10i	1.95832i
$974$	−5.88352e10	−2.04024
$975$	0	0
$976$	8.47816e10	2.91895
$977$	− 3.03613e10i	− 1.04157i	−0.853687	−	0.520786i	$-0.825639\pi$
$977$	− 3.03613e10i	− 1.04157i	0.853687	−	0.520786i	$-0.174361\pi$
$978$	0	0
$979$	−8.73379e9	−0.297484
$980$	0	0
$981$	0	0
$982$	− 3.02230e10i	− 1.01847i
$983$	3.64309e10i	1.22330i	0.791129	+	0.611650i	$0.209494\pi$
$983$	3.64309e10i	1.22330i	−0.791129	+	0.611650i	$0.790506\pi$
$984$	0	0
$985$	0	0
$986$	−1.56444e10	−0.519744
$987$	0	0
$988$	2.38372e10i	0.786332i
$989$	2.18937e10	0.719668
$990$	0	0
$991$	−2.79867e10	−0.913470	−0.456735	−	0.889603i	$-0.650981\pi$
$991$	−2.79867e10	−0.913470	−0.456735	+	0.889603i	$0.650981\pi$
$992$	− 3.03322e10i	− 0.986536i
$993$	0	0
$994$	−8.32029e10	−2.68711
$995$	0	0
$996$	0	0
$997$	3.06088e10i	0.978169i	0.872236	+	0.489084i	$0.162669\pi$
$997$	3.06088e10i	0.978169i	−0.872236	+	0.489084i	$0.837331\pi$
$998$	− 4.70133e10i	− 1.49715i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.8.b.l.199.4		4
3.2	odd	2		25.8.b.b.24.1		4
5.2	odd	4		225.8.a.k.1.1		2
5.3	odd	4		225.8.a.v.1.2		2
5.4	even	2	inner	225.8.b.l.199.1		4
12.11	even	2		400.8.c.s.49.1		4
15.2	even	4		25.8.a.e.1.2	yes	2
15.8	even	4		25.8.a.c.1.1	✓	2
15.14	odd	2		25.8.b.b.24.4		4
60.23	odd	4		400.8.a.bd.1.2		2
60.47	odd	4		400.8.a.v.1.1		2
60.59	even	2		400.8.c.s.49.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.a.c.1.1	✓	2	15.8	even	4
25.8.a.e.1.2	yes	2	15.2	even	4
25.8.b.b.24.1		4	3.2	odd	2
25.8.b.b.24.4		4	15.14	odd	2
225.8.a.k.1.1		2	5.2	odd	4
225.8.a.v.1.2		2	5.3	odd	4
225.8.b.l.199.1		4	5.4	even	2	inner
225.8.b.l.199.4		4	1.1	even	1	trivial
400.8.a.v.1.1		2	60.47	odd	4
400.8.a.bd.1.2		2	60.23	odd	4
400.8.c.s.49.1		4	12.11	even	2
400.8.c.s.49.4		4	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+20.2377i q^{2} -281.566 q^{4} +1369.97i q^{7} -3107.83i q^{8} +1012.43 q^{11} +3643.00i q^{13} -27725.1 q^{14} +26855.0 q^{16} -5532.37i q^{17} -23238.9 q^{19} +20489.4i q^{22} -96310.5i q^{23} -73726.1 q^{26} -385737. i q^{28} -139729. q^{29} -208208. q^{31} +145682. i q^{32} +111963. q^{34} -448306. i q^{37} -470303. i q^{38} +79718.5 q^{41} +227324. i q^{43} -285067. q^{44} +1.94911e6 q^{46} +719892. i q^{47} -1.05328e6 q^{49} -1.02575e6i q^{52} +495702. i q^{53} +4.25763e6 q^{56} -2.82779e6i q^{58} -81006.9 q^{59} +3.15701e6 q^{61} -4.21366e6i q^{62} +489161. q^{64} -1.06340e6i q^{67} +1.55773e6i q^{68} +3.00099e6 q^{71} +405758. i q^{73} +9.07270e6 q^{74} +6.54328e6 q^{76} +1.38701e6i q^{77} +4.71733e6 q^{79} +1.61332e6i q^{82} -5.14106e6i q^{83} -4.60053e6 q^{86} -3.14648e6i q^{88} -8.62652e6 q^{89} -4.99080e6 q^{91} +2.71178e7i q^{92} -1.45690e7 q^{94} +6.99663e6i q^{97} -2.13159e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 362 q^{4} - 8688 q^{11} - 63516 q^{14} + 66914 q^{16} - 36400 q^{19} - 408 q^{26} + 111600 q^{29} - 603552 q^{31} + 353846 q^{34} + 216972 q^{41} - 1647486 q^{44} + 4685868 q^{46} - 1645172 q^{49}+ \cdots - 17950264 q^{94}+O(q^{100}) \) 4 * q - 362 * q^4 - 8688 * q^11 - 63516 * q^14 + 66914 * q^16 - 36400 * q^19 - 408 * q^26 + 111600 * q^29 - 603552 * q^31 + 353846 * q^34 + 216972 * q^41 - 1647486 * q^44 + 4685868 * q^46 - 1645172 * q^49 + 6671700 * q^56 + 4135200 * q^59 + 1164088 * q^61 + 700958 * q^64 + 9456432 * q^71 + 16453044 * q^74 + 14100050 * q^76 + 14372400 * q^79 - 1458888 * q^86 - 11981700 * q^89 + 11634528 * q^91 - 17950264 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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