LMFDB - Embedded newform 225.7.c.a.26.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,7,Mod(26,225)]

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([1, 0]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("225.26");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	225.c (of order $2$, degree $1$, 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:	$51.7621688145$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	225.26
Dual form			225.7.c.a.26.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.7279i q^{2} -98.0000 q^{4} -524.000 q^{7} +432.749i q^{8} +O(q^{10})$	$q-12.7279i q^{2} -98.0000 q^{4} -524.000 q^{7} +432.749i q^{8} -865.499i q^{11} -344.000 q^{13} +6669.43i q^{14} -764.000 q^{16} +7140.36i q^{17} -2320.00 q^{19} -11016.0 q^{22} -5753.02i q^{23} +4378.41i q^{26} +51352.0 q^{28} +23152.1i q^{29} -10564.0 q^{31} +37420.1i q^{32} +90882.0 q^{34} +24082.0 q^{37} +29528.8i q^{38} -108836. i q^{41} +90952.0 q^{43} +84818.9i q^{44} -73224.0 q^{46} -128959. i q^{47} +156927. q^{49} +33712.0 q^{52} +196685. i q^{53} -226761. i q^{56} +294678. q^{58} +39812.9i q^{59} +251138. q^{61} +134458. i q^{62} +427384. q^{64} +216088. q^{67} -699756. i q^{68} +53915.5i q^{71} +308176. q^{73} -306514. i q^{74} +227360. q^{76} +453521. i q^{77} -540124. q^{79} -1.38526e6 q^{82} -932346. i q^{83} -1.15763e6i q^{86} +374544. q^{88} +223413. i q^{89} +180256. q^{91} +563796. i q^{92} -1.64138e6 q^{94} +37168.0 q^{97} -1.99735e6i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 196 q^{4} - 1048 q^{7}+O(q^{10}) $ 2 * q - 196 * q^4 - 1048 * q^7	$ 2 q - 196 q^{4} - 1048 q^{7} - 688 q^{13} - 1528 q^{16} - 4640 q^{19} - 22032 q^{22} + 102704 q^{28} - 21128 q^{31} + 181764 q^{34} + 48164 q^{37} + 181904 q^{43} - 146448 q^{46} + 313854 q^{49} + 67424 q^{52} + 589356 q^{58} + 502276 q^{61} + 854768 q^{64} + 432176 q^{67} + 616352 q^{73} + 454720 q^{76} - 1080248 q^{79} - 2770524 q^{82} + 749088 q^{88} + 360512 q^{91} - 3282768 q^{94} + 74336 q^{97}+O(q^{100}) $ 2 * q - 196 * q^4 - 1048 * q^7 - 688 * q^13 - 1528 * q^16 - 4640 * q^19 - 22032 * q^22 + 102704 * q^28 - 21128 * q^31 + 181764 * q^34 + 48164 * q^37 + 181904 * q^43 - 146448 * q^46 + 313854 * q^49 + 67424 * q^52 + 589356 * q^58 + 502276 * q^61 + 854768 * q^64 + 432176 * q^67 + 616352 * q^73 + 454720 * q^76 - 1080248 * q^79 - 2770524 * q^82 + 749088 * q^88 + 360512 * q^91 - 3282768 * q^94 + 74336 * q^97

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 $

$2$	− 12.7279i	− 1.59099i	−0.605960	−	0.795495i	$-0.707211\pi$
$2$	− 12.7279i	− 1.59099i	0.605960	−	0.795495i	$-0.292789\pi$
$3$	0	0
$3$	0	0
$4$	−98.0000	−1.53125
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−524.000	−1.52770	−0.763848	−	0.645396i	$-0.776692\pi$
$7$	−524.000	−1.52770	−0.763848	+	0.645396i	$0.776692\pi$
$8$	432.749i	0.845214i
$9$	0	0
$10$	0	0
$11$	− 865.499i	− 0.650262i	−0.945669	−	0.325131i	$-0.894592\pi$
$11$	− 865.499i	− 0.650262i	0.945669	−	0.325131i	$-0.105408\pi$
$12$	0	0
$13$	−344.000	−0.156577	−0.0782886	−	0.996931i	$-0.524946\pi$
$13$	−344.000	−0.156577	−0.0782886	+	0.996931i	$0.524946\pi$
$14$	6669.43i	2.43055i
$15$	0	0
$16$	−764.000	−0.186523
$17$	7140.36i	1.45336i	0.686975	+	0.726681i	$0.258938\pi$
$17$	7140.36i	1.45336i	−0.686975	+	0.726681i	$0.741062\pi$
$18$	0	0
$19$	−2320.00	−0.338242	−0.169121	−	0.985595i	$-0.554093\pi$
$19$	−2320.00	−0.338242	−0.169121	+	0.985595i	$0.554093\pi$
$20$	0	0
$21$	0	0
$22$	−11016.0	−1.03456
$23$	− 5753.02i	− 0.472838i	−0.971651	−	0.236419i	$-0.924026\pi$
$23$	− 5753.02i	− 0.472838i	0.971651	−	0.236419i	$-0.0759738\pi$
$24$	0	0
$25$	0	0
$26$	4378.41i	0.249113i
$27$	0	0
$28$	51352.0	2.33929
$29$	23152.1i	0.949284i	0.880179	+	0.474642i	$0.157422\pi$
$29$	23152.1i	0.949284i	−0.880179	+	0.474642i	$0.842578\pi$
$30$	0	0
$31$	−10564.0	−0.354604	−0.177302	−	0.984157i	$-0.556737\pi$
$31$	−10564.0	−0.354604	−0.177302	+	0.984157i	$0.556737\pi$
$32$	37420.1i	1.14197i
$33$	0	0
$34$	90882.0	2.31228
$35$	0	0
$36$	0	0
$37$	24082.0	0.475431	0.237715	−	0.971335i	$-0.423601\pi$
$37$	24082.0	0.475431	0.237715	+	0.971335i	$0.423601\pi$
$38$	29528.8i	0.538139i
$39$	0	0
$40$	0	0
$41$	− 108836.i	− 1.57915i	−0.613655	−	0.789574i	$-0.710302\pi$
$41$	− 108836.i	− 1.57915i	0.613655	−	0.789574i	$-0.289698\pi$
$42$	0	0
$43$	90952.0	1.14395	0.571975	−	0.820271i	$-0.306178\pi$
$43$	90952.0	1.14395	0.571975	+	0.820271i	$0.306178\pi$
$44$	84818.9i	0.995714i
$45$	0	0
$46$	−73224.0	−0.752281
$47$	− 128959.i	− 1.24211i	−0.783768	−	0.621054i	$-0.786705\pi$
$47$	− 128959.i	− 1.24211i	0.783768	−	0.621054i	$-0.213295\pi$
$48$	0	0
$49$	156927.	1.33386
$50$	0	0
$51$	0	0
$52$	33712.0	0.239759
$53$	196685.i	1.32112i	0.750773	+	0.660561i	$0.229681\pi$
$53$	196685.i	1.32112i	−0.750773	+	0.660561i	$0.770319\pi$
$54$	0	0
$55$	0	0
$56$	− 226761.i	− 1.29123i
$57$	0	0
$58$	294678.	1.51030
$59$	39812.9i	0.193851i	0.995292	+	0.0969255i	$0.0309009\pi$
$59$	39812.9i	0.193851i	−0.995292	+	0.0969255i	$0.969099\pi$
$60$	0	0
$61$	251138.	1.10643	0.553214	−	0.833039i	$-0.313401\pi$
$61$	251138.	1.10643	0.553214	+	0.833039i	$0.313401\pi$
$62$	134458.i	0.564171i
$63$	0	0
$64$	427384.	1.63034
$65$	0	0
$66$	0	0
$67$	216088.	0.718466	0.359233	−	0.933248i	$-0.383038\pi$
$67$	216088.	0.718466	0.359233	+	0.933248i	$0.383038\pi$
$68$	− 699756.i	− 2.22546i
$69$	0	0
$70$	0	0
$71$	53915.5i	0.150639i	0.997159	+	0.0753197i	$0.0239977\pi$
$71$	53915.5i	0.150639i	−0.997159	+	0.0753197i	$0.976002\pi$
$72$	0	0
$73$	308176.	0.792192	0.396096	−	0.918209i	$-0.370365\pi$
$73$	308176.	0.792192	0.396096	+	0.918209i	$0.370365\pi$
$74$	− 306514.i	− 0.756406i
$75$	0	0
$76$	227360.	0.517933
$77$	453521.i	0.993403i
$78$	0	0
$79$	−540124.	−1.09550	−0.547750	−	0.836642i	$-0.684515\pi$
$79$	−540124.	−1.09550	−0.547750	+	0.836642i	$0.684515\pi$
$80$	0	0
$81$	0	0
$82$	−1.38526e6	−2.51241
$83$	− 932346.i	− 1.63058i	−0.579051	−	0.815291i	$-0.696577\pi$
$83$	− 932346.i	− 1.63058i	0.579051	−	0.815291i	$-0.303423\pi$
$84$	0	0
$85$	0	0
$86$	− 1.15763e6i	− 1.82001i
$87$	0	0
$88$	374544.	0.549610
$89$	223413.i	0.316912i	0.987366	+	0.158456i	$0.0506516\pi$
$89$	223413.i	0.316912i	−0.987366	+	0.158456i	$0.949348\pi$
$90$	0	0
$91$	180256.	0.239202
$92$	563796.i	0.724033i
$93$	0	0
$94$	−1.64138e6	−1.97618
$95$	0	0
$96$	0	0
$97$	37168.0	0.0407243	0.0203622	−	0.999793i	$-0.493518\pi$
$97$	37168.0	0.0407243	0.0203622	+	0.999793i	$0.493518\pi$
$98$	− 1.99735e6i	− 2.12215i
$99$	0	0
$100$	0	0
$101$	559787.i	0.543323i	0.962393	+	0.271662i	$0.0875732\pi$
$101$	559787.i	0.543323i	−0.962393	+	0.271662i	$0.912427\pi$
$102$	0	0
$103$	−1.46018e6	−1.33627	−0.668136	−	0.744039i	$-0.732907\pi$
$103$	−1.46018e6	−1.33627	−0.668136	+	0.744039i	$0.732907\pi$
$104$	− 148866.i	− 0.132341i
$105$	0	0
$106$	2.50339e6	2.10189
$107$	1.29031e6i	1.05327i	0.850090	+	0.526637i	$0.176547\pi$
$107$	1.29031e6i	1.05327i	−0.850090	+	0.526637i	$0.823453\pi$
$108$	0	0
$109$	−1.43548e6	−1.10845	−0.554227	−	0.832366i	$-0.686986\pi$
$109$	−1.43548e6	−1.10845	−0.554227	+	0.832366i	$0.686986\pi$
$110$	0	0
$111$	0	0
$112$	400336.	0.284951
$113$	186426.i	0.129202i	0.997911	+	0.0646012i	$0.0205775\pi$
$113$	186426.i	0.129202i	−0.997911	+	0.0646012i	$0.979422\pi$
$114$	0	0
$115$	0	0
$116$	− 2.26890e6i	− 1.45359i
$117$	0	0
$118$	506736.	0.308415
$119$	− 3.74155e6i	− 2.22030i
$120$	0	0
$121$	1.02247e6	0.577159
$122$	− 3.19646e6i	− 1.76032i
$123$	0	0
$124$	1.03527e6	0.542987
$125$	0	0
$126$	0	0
$127$	127060.	0.0620294	0.0310147	−	0.999519i	$-0.490126\pi$
$127$	127060.	0.0620294	0.0310147	+	0.999519i	$0.490126\pi$
$128$	− 3.04482e6i	− 1.45189i
$129$	0	0
$130$	0	0
$131$	2.62348e6i	1.16698i	0.812120	+	0.583490i	$0.198313\pi$
$131$	2.62348e6i	1.16698i	−0.812120	+	0.583490i	$0.801687\pi$
$132$	0	0
$133$	1.21568e6	0.516731
$134$	− 2.75035e6i	− 1.14307i
$135$	0	0
$136$	−3.08999e6	−1.22840
$137$	− 202310.i	− 0.0786785i	−0.999226	−	0.0393393i	$-0.987475\pi$
$137$	− 202310.i	− 0.0786785i	0.999226	−	0.0393393i	$-0.0125253\pi$
$138$	0	0
$139$	2.02642e6	0.754546	0.377273	−	0.926102i	$-0.376862\pi$
$139$	2.02642e6	0.754546	0.377273	+	0.926102i	$0.376862\pi$
$140$	0	0
$141$	0	0
$142$	686232.	0.239666
$143$	297732.i	0.101816i
$144$	0	0
$145$	0	0
$146$	− 3.92244e6i	− 1.26037i
$147$	0	0
$148$	−2.36004e6	−0.728004
$149$	4.25534e6i	1.28640i	0.765699	+	0.643199i	$0.222393\pi$
$149$	4.25534e6i	1.28640i	−0.765699	+	0.643199i	$0.777607\pi$
$150$	0	0
$151$	3.74035e6	1.08638	0.543189	−	0.839610i	$-0.317217\pi$
$151$	3.74035e6	1.08638	0.543189	+	0.839610i	$0.317217\pi$
$152$	− 1.00398e6i	− 0.285886i
$153$	0	0
$154$	5.77238e6	1.58049
$155$	0	0
$156$	0	0
$157$	2.38813e6	0.617105	0.308552	−	0.951207i	$-0.400155\pi$
$157$	2.38813e6	0.617105	0.308552	+	0.951207i	$0.400155\pi$
$158$	6.87466e6i	1.74293i
$159$	0	0
$160$	0	0
$161$	3.01458e6i	0.722353i
$162$	0	0
$163$	6.74519e6	1.55751	0.778756	−	0.627327i	$-0.215851\pi$
$163$	6.74519e6	1.55751	0.778756	+	0.627327i	$0.215851\pi$
$164$	1.06660e7i	2.41807i
$165$	0	0
$166$	−1.18668e7	−2.59424
$167$	6.61699e6i	1.42073i	0.703834	+	0.710364i	$0.251470\pi$
$167$	6.61699e6i	1.42073i	−0.703834	+	0.710364i	$0.748530\pi$
$168$	0	0
$169$	−4.70847e6	−0.975484
$170$	0	0
$171$	0	0
$172$	−8.91330e6	−1.75167
$173$	5.58077e6i	1.07784i	0.842356	+	0.538922i	$0.181168\pi$
$173$	5.58077e6i	1.07784i	−0.842356	+	0.538922i	$0.818832\pi$
$174$	0	0
$175$	0	0
$176$	661241.i	0.121289i
$177$	0	0
$178$	2.84359e6	0.504204
$179$	4.87103e6i	0.849301i	0.905357	+	0.424650i	$0.139603\pi$
$179$	4.87103e6i	0.849301i	−0.905357	+	0.424650i	$0.860397\pi$
$180$	0	0
$181$	8.47546e6	1.42931	0.714657	−	0.699475i	$-0.246583\pi$
$181$	8.47546e6	1.42931	0.714657	+	0.699475i	$0.246583\pi$
$182$	− 2.29428e6i	− 0.380569i
$183$	0	0
$184$	2.48962e6	0.399649
$185$	0	0
$186$	0	0
$187$	6.17998e6	0.945066
$188$	1.26380e7i	1.90198i
$189$	0	0
$190$	0	0
$191$	97037.7i	0.0139264i	0.999976	+	0.00696322i	$0.00221648\pi$
$191$	97037.7i	0.0139264i	−0.999976	+	0.00696322i	$0.997784\pi$
$192$	0	0
$193$	−7.49473e6	−1.04252	−0.521260	−	0.853398i	$-0.674538\pi$
$193$	−7.49473e6	−1.04252	−0.521260	+	0.853398i	$0.674538\pi$
$194$	− 473071.i	− 0.0647920i
$195$	0	0
$196$	−1.53788e7	−2.04247
$197$	9.84656e6i	1.28791i	0.765063	+	0.643956i	$0.222708\pi$
$197$	9.84656e6i	1.28791i	−0.765063	+	0.643956i	$0.777292\pi$
$198$	0	0
$199$	3.54170e6	0.449420	0.224710	−	0.974426i	$-0.427856\pi$
$199$	3.54170e6	0.449420	0.224710	+	0.974426i	$0.427856\pi$
$200$	0	0
$201$	0	0
$202$	7.12492e6	0.864422
$203$	− 1.21317e7i	− 1.45022i
$204$	0	0
$205$	0	0
$206$	1.85851e7i	2.12600i
$207$	0	0
$208$	262816.	0.0292053
$209$	2.00796e6i	0.219946i
$210$	0	0
$211$	−6.77298e6	−0.720996	−0.360498	−	0.932760i	$-0.617393\pi$
$211$	−6.77298e6	−0.720996	−0.360498	+	0.932760i	$0.617393\pi$
$212$	− 1.92751e7i	− 2.02297i
$213$	0	0
$214$	1.64229e7	1.67575
$215$	0	0
$216$	0	0
$217$	5.53554e6	0.541727
$218$	1.82707e7i	1.76354i
$219$	0	0
$220$	0	0
$221$	− 2.45629e6i	− 0.227563i
$222$	0	0
$223$	3.34186e6	0.301352	0.150676	−	0.988583i	$-0.451855\pi$
$223$	3.34186e6	0.301352	0.150676	+	0.988583i	$0.451855\pi$
$224$	− 1.96081e7i	− 1.74458i
$225$	0	0
$226$	2.37281e6	0.205560
$227$	− 1.62013e7i	− 1.38507i	−0.721385	−	0.692535i	$-0.756494\pi$
$227$	− 1.62013e7i	− 1.38507i	0.721385	−	0.692535i	$-0.243506\pi$
$228$	0	0
$229$	−1.66351e7	−1.38522	−0.692611	−	0.721311i	$-0.743540\pi$
$229$	−1.66351e7	−1.38522	−0.692611	+	0.721311i	$0.743540\pi$
$230$	0	0
$231$	0	0
$232$	−1.00191e7	−0.802348
$233$	8.85600e6i	0.700116i	0.936728	+	0.350058i	$0.113838\pi$
$233$	8.85600e6i	0.700116i	−0.936728	+	0.350058i	$0.886162\pi$
$234$	0	0
$235$	0	0
$236$	− 3.90167e6i	− 0.296834i
$237$	0	0
$238$	−4.76222e7	−3.53247
$239$	− 1.12995e7i	− 0.827689i	−0.910348	−	0.413845i	$-0.864186\pi$
$239$	− 1.12995e7i	− 0.827689i	0.910348	−	0.413845i	$-0.135814\pi$
$240$	0	0
$241$	1.50090e7	1.07226	0.536129	−	0.844136i	$-0.319886\pi$
$241$	1.50090e7	1.07226	0.536129	+	0.844136i	$0.319886\pi$
$242$	− 1.30140e7i	− 0.918255i
$243$	0	0
$244$	−2.46115e7	−1.69422
$245$	0	0
$246$	0	0
$247$	798080.	0.0529609
$248$	− 4.57156e6i	− 0.299716i
$249$	0	0
$250$	0	0
$251$	3.04076e7i	1.92292i	0.274950	+	0.961458i	$0.411339\pi$
$251$	3.04076e7i	1.92292i	−0.274950	+	0.961458i	$0.588661\pi$
$252$	0	0
$253$	−4.97923e6	−0.307469
$254$	− 1.61721e6i	− 0.0986882i
$255$	0	0
$256$	−1.14017e7	−0.679595
$257$	1.97422e7i	1.16305i	0.813530	+	0.581523i	$0.197543\pi$
$257$	1.97422e7i	1.16305i	−0.813530	+	0.581523i	$0.802457\pi$
$258$	0	0
$259$	−1.26190e7	−0.726314
$260$	0	0
$261$	0	0
$262$	3.33914e7	1.85666
$263$	− 3.48531e6i	− 0.191591i	−0.995401	−	0.0957954i	$-0.969461\pi$
$263$	− 3.48531e6i	− 0.191591i	0.995401	−	0.0957954i	$-0.0305394\pi$
$264$	0	0
$265$	0	0
$266$	− 1.54731e7i	− 0.822114i
$267$	0	0
$268$	−2.11766e7	−1.10015
$269$	5.65391e6i	0.290464i	0.989398	+	0.145232i	$0.0463928\pi$
$269$	5.65391e6i	0.290464i	−0.989398	+	0.145232i	$0.953607\pi$
$270$	0	0
$271$	2.91893e7	1.46662	0.733308	−	0.679897i	$-0.237976\pi$
$271$	2.91893e7	1.46662	0.733308	+	0.679897i	$0.237976\pi$
$272$	− 5.45524e6i	− 0.271086i
$273$	0	0
$274$	−2.57499e6	−0.125177
$275$	0	0
$276$	0	0
$277$	−2.29938e7	−1.08186	−0.540931	−	0.841067i	$-0.681928\pi$
$277$	−2.29938e7	−1.08186	−0.540931	+	0.841067i	$0.681928\pi$
$278$	− 2.57922e7i	− 1.20048i
$279$	0	0
$280$	0	0
$281$	− 1.25303e6i	− 0.0564730i	−0.999601	−	0.0282365i	$-0.991011\pi$
$281$	− 1.25303e6i	− 0.0564730i	0.999601	−	0.0282365i	$-0.00898916\pi$
$282$	0	0
$283$	1.45129e7	0.640317	0.320159	−	0.947364i	$-0.396264\pi$
$283$	1.45129e7	0.640317	0.320159	+	0.947364i	$0.396264\pi$
$284$	− 5.28372e6i	− 0.230666i
$285$	0	0
$286$	3.78950e6	0.161989
$287$	5.70303e7i	2.41246i
$288$	0	0
$289$	−2.68472e7	−1.11226
$290$	0	0
$291$	0	0
$292$	−3.02012e7	−1.21304
$293$	− 1.12729e7i	− 0.448160i	−0.974571	−	0.224080i	$-0.928062\pi$
$293$	− 1.12729e7i	− 0.448160i	0.974571	−	0.224080i	$-0.0719377\pi$
$294$	0	0
$295$	0	0
$296$	1.04215e7i	0.401841i
$297$	0	0
$298$	5.41616e7	2.04665
$299$	1.97904e6i	0.0740356i
$300$	0	0
$301$	−4.76588e7	−1.74761
$302$	− 4.76069e7i	− 1.72842i
$303$	0	0
$304$	1.77248e6	0.0630900
$305$	0	0
$306$	0	0
$307$	−4.51916e7	−1.56186	−0.780930	−	0.624618i	$-0.785255\pi$
$307$	−4.51916e7	−1.56186	−0.780930	+	0.624618i	$0.785255\pi$
$308$	− 4.44451e7i	− 1.52115i
$309$	0	0
$310$	0	0
$311$	− 9.74134e6i	− 0.323845i	−0.986803	−	0.161923i	$-0.948230\pi$
$311$	− 9.74134e6i	− 0.323845i	0.986803	−	0.161923i	$-0.0517695\pi$
$312$	0	0
$313$	5.30265e6	0.172926	0.0864630	−	0.996255i	$-0.472444\pi$
$313$	5.30265e6	0.172926	0.0864630	+	0.996255i	$0.472444\pi$
$314$	− 3.03959e7i	− 0.981808i
$315$	0	0
$316$	5.29322e7	1.67748
$317$	1.05462e7i	0.331068i	0.986204	+	0.165534i	$0.0529348\pi$
$317$	1.05462e7i	0.331068i	−0.986204	+	0.165534i	$0.947065\pi$
$318$	0	0
$319$	2.00381e7	0.617283
$320$	0	0
$321$	0	0
$322$	3.83694e7	1.14926
$323$	− 1.65656e7i	− 0.491587i
$324$	0	0
$325$	0	0
$326$	− 8.58523e7i	− 2.47799i
$327$	0	0
$328$	4.70989e7	1.33472
$329$	6.75747e7i	1.89756i
$330$	0	0
$331$	−3.81242e7	−1.05128	−0.525638	−	0.850708i	$-0.676174\pi$
$331$	−3.81242e7	−1.05128	−0.525638	+	0.850708i	$0.676174\pi$
$332$	9.13699e7i	2.49683i
$333$	0	0
$334$	8.42206e7	2.26037
$335$	0	0
$336$	0	0
$337$	1.22682e7	0.320548	0.160274	−	0.987073i	$-0.448762\pi$
$337$	1.22682e7	0.320548	0.160274	+	0.987073i	$0.448762\pi$
$338$	5.99291e7i	1.55198i
$339$	0	0
$340$	0	0
$341$	9.14313e6i	0.230585i
$342$	0	0
$343$	−2.05817e7	−0.510033
$344$	3.93594e7i	0.966882i
$345$	0	0
$346$	7.10317e7	1.71484
$347$	− 1.65809e7i	− 0.396843i	−0.980117	−	0.198422i	$-0.936418\pi$
$347$	− 1.65809e7i	− 0.396843i	0.980117	−	0.198422i	$-0.0635815\pi$
$348$	0	0
$349$	4.81038e7	1.13163	0.565813	−	0.824534i	$-0.308562\pi$
$349$	4.81038e7	1.13163	0.565813	+	0.824534i	$0.308562\pi$
$350$	0	0
$351$	0	0
$352$	3.23870e7	0.742580
$353$	− 1.47570e7i	− 0.335485i	−0.985831	−	0.167742i	$-0.946352\pi$
$353$	− 1.47570e7i	− 0.335485i	0.985831	−	0.167742i	$-0.0536477\pi$
$354$	0	0
$355$	0	0
$356$	− 2.18945e7i	− 0.485272i
$357$	0	0
$358$	6.19980e7	1.35123
$359$	1.78509e7i	0.385813i	0.981217	+	0.192907i	$0.0617914\pi$
$359$	1.78509e7i	0.385813i	−0.981217	+	0.192907i	$0.938209\pi$
$360$	0	0
$361$	−4.16635e7	−0.885593
$362$	− 1.07875e8i	− 2.27403i
$363$	0	0
$364$	−1.76651e7	−0.366279
$365$	0	0
$366$	0	0
$367$	−8.67940e6	−0.175587	−0.0877934	−	0.996139i	$-0.527982\pi$
$367$	−8.67940e6	−0.175587	−0.0877934	+	0.996139i	$0.527982\pi$
$368$	4.39531e6i	0.0881954i
$369$	0	0
$370$	0	0
$371$	− 1.03063e8i	− 2.01827i
$372$	0	0
$373$	7.94052e7	1.53011	0.765055	−	0.643965i	$-0.222712\pi$
$373$	7.94052e7	1.53011	0.765055	+	0.643965i	$0.222712\pi$
$374$	− 7.86583e7i	− 1.50359i
$375$	0	0
$376$	5.58071e7	1.04985
$377$	− 7.96432e6i	− 0.148636i
$378$	0	0
$379$	−1.46346e7	−0.268821	−0.134410	−	0.990926i	$-0.542914\pi$
$379$	−1.46346e7	−0.268821	−0.134410	+	0.990926i	$0.542914\pi$
$380$	0	0
$381$	0	0
$382$	1.23509e6	0.0221568
$383$	9.16736e6i	0.163173i	0.996666	+	0.0815865i	$0.0259987\pi$
$383$	9.16736e6i	0.163173i	−0.996666	+	0.0815865i	$0.974001\pi$
$384$	0	0
$385$	0	0
$386$	9.53924e7i	1.65864i
$387$	0	0
$388$	−3.64246e6	−0.0623591
$389$	9.26668e6i	0.157426i	0.996897	+	0.0787128i	$0.0250810\pi$
$389$	9.26668e6i	0.157426i	−0.996897	+	0.0787128i	$0.974919\pi$
$390$	0	0
$391$	4.10787e7	0.687205
$392$	6.79101e7i	1.12739i
$393$	0	0
$394$	1.25326e8	2.04905
$395$	0	0
$396$	0	0
$397$	−7.25544e7	−1.15956	−0.579779	−	0.814774i	$-0.696861\pi$
$397$	−7.25544e7	−1.15956	−0.579779	+	0.814774i	$0.696861\pi$
$398$	− 4.50785e7i	− 0.715023i
$399$	0	0
$400$	0	0
$401$	− 7.37246e7i	− 1.14335i	−0.820480	−	0.571675i	$-0.806294\pi$
$401$	− 7.37246e7i	− 1.14335i	0.820480	−	0.571675i	$-0.193706\pi$
$402$	0	0
$403$	3.63402e6	0.0555228
$404$	− 5.48591e7i	− 0.831964i
$405$	0	0
$406$	−1.54411e8	−2.30728
$407$	− 2.08429e7i	− 0.309155i
$408$	0	0
$409$	3.43558e7	0.502146	0.251073	−	0.967968i	$-0.419217\pi$
$409$	3.43558e7	0.502146	0.251073	+	0.967968i	$0.419217\pi$
$410$	0	0
$411$	0	0
$412$	1.43098e8	2.04617
$413$	− 2.08620e7i	− 0.296146i
$414$	0	0
$415$	0	0
$416$	− 1.28725e7i	− 0.178806i
$417$	0	0
$418$	2.55571e7	0.349932
$419$	1.31347e8i	1.78557i	0.450479	+	0.892787i	$0.351253\pi$
$419$	1.31347e8i	1.78557i	−0.450479	+	0.892787i	$0.648747\pi$
$420$	0	0
$421$	2.36756e6	0.0317289	0.0158644	−	0.999874i	$-0.494950\pi$
$421$	2.36756e6	0.0317289	0.0158644	+	0.999874i	$0.494950\pi$
$422$	8.62060e7i	1.14710i
$423$	0	0
$424$	−8.51151e7	−1.11663
$425$	0	0
$426$	0	0
$427$	−1.31596e8	−1.69029
$428$	− 1.26450e8i	− 1.61283i
$429$	0	0
$430$	0	0
$431$	1.13049e8i	1.41200i	0.708212	+	0.705999i	$0.249502\pi$
$431$	1.13049e8i	1.41200i	−0.708212	+	0.705999i	$0.750498\pi$
$432$	0	0
$433$	−4.50927e7	−0.555447	−0.277723	−	0.960661i	$-0.589580\pi$
$433$	−4.50927e7	−0.555447	−0.277723	+	0.960661i	$0.589580\pi$
$434$	− 7.04559e7i	− 0.861882i
$435$	0	0
$436$	1.40677e8	1.69732
$437$	1.33470e7i	0.159934i
$438$	0	0
$439$	−1.61605e8	−1.91013	−0.955064	−	0.296399i	$-0.904214\pi$
$439$	−1.61605e8	−1.91013	−0.955064	+	0.296399i	$0.904214\pi$
$440$	0	0
$441$	0	0
$442$	−3.12634e7	−0.362051
$443$	− 2.54011e7i	− 0.292174i	−0.989272	−	0.146087i	$-0.953332\pi$
$443$	− 2.54011e7i	− 0.292174i	0.989272	−	0.146087i	$-0.0466679\pi$
$444$	0	0
$445$	0	0
$446$	− 4.25349e7i	− 0.479448i
$447$	0	0
$448$	−2.23949e8	−2.49067
$449$	− 9.43898e7i	− 1.04276i	−0.853323	−	0.521382i	$-0.825417\pi$
$449$	− 9.43898e7i	− 1.04276i	0.853323	−	0.521382i	$-0.174583\pi$
$450$	0	0
$451$	−9.41978e7	−1.02686
$452$	− 1.82697e7i	− 0.197841i
$453$	0	0
$454$	−2.06209e8	−2.20363
$455$	0	0
$456$	0	0
$457$	−4.68452e7	−0.490813	−0.245407	−	0.969420i	$-0.578922\pi$
$457$	−4.68452e7	−0.490813	−0.245407	+	0.969420i	$0.578922\pi$
$458$	2.11730e8i	2.20387i
$459$	0	0
$460$	0	0
$461$	2.33797e7i	0.238636i	0.992856	+	0.119318i	$0.0380708\pi$
$461$	2.33797e7i	0.238636i	−0.992856	+	0.119318i	$0.961929\pi$
$462$	0	0
$463$	4.98269e7	0.502019	0.251010	−	0.967985i	$-0.419237\pi$
$463$	4.98269e7	0.502019	0.251010	+	0.967985i	$0.419237\pi$
$464$	− 1.76882e7i	− 0.177064i
$465$	0	0
$466$	1.12718e8	1.11388
$467$	9.52369e7i	0.935092i	0.883969	+	0.467546i	$0.154862\pi$
$467$	9.52369e7i	0.935092i	−0.883969	+	0.467546i	$0.845138\pi$
$468$	0	0
$469$	−1.13230e8	−1.09760
$470$	0	0
$471$	0	0
$472$	−1.72290e7	−0.163846
$473$	− 7.87188e7i	− 0.743867i
$474$	0	0
$475$	0	0
$476$	3.66672e8i	3.39983i
$477$	0	0
$478$	−1.43820e8	−1.31685
$479$	1.97141e8i	1.79378i	0.442251	+	0.896891i	$0.354180\pi$
$479$	1.97141e8i	1.79378i	−0.442251	+	0.896891i	$0.645820\pi$
$480$	0	0
$481$	−8.28421e6	−0.0744416
$482$	− 1.91033e8i	− 1.70595i
$483$	0	0
$484$	−1.00202e8	−0.883775
$485$	0	0
$486$	0	0
$487$	−1.92602e6	−0.0166753	−0.00833765	−	0.999965i	$-0.502654\pi$
$487$	−1.92602e6	−0.0166753	−0.00833765	+	0.999965i	$0.502654\pi$
$488$	1.08680e8i	0.935167i
$489$	0	0
$490$	0	0
$491$	− 1.79722e8i	− 1.51830i	−0.650916	−	0.759150i	$-0.725615\pi$
$491$	− 1.79722e8i	− 1.51830i	0.650916	−	0.759150i	$-0.274385\pi$
$492$	0	0
$493$	−1.65314e8	−1.37965
$494$	− 1.01579e7i	− 0.0842603i
$495$	0	0
$496$	8.07090e6	0.0661419
$497$	− 2.82517e7i	− 0.230131i
$498$	0	0
$499$	1.54018e8	1.23956	0.619782	−	0.784774i	$-0.287221\pi$
$499$	1.54018e8	1.23956	0.619782	+	0.784774i	$0.287221\pi$
$500$	0	0
$501$	0	0
$502$	3.87025e8	3.05934
$503$	− 2.25142e7i	− 0.176910i	−0.996080	−	0.0884551i	$-0.971807\pi$
$503$	− 2.25142e7i	− 0.176910i	0.996080	−	0.0884551i	$-0.0281930\pi$
$504$	0	0
$505$	0	0
$506$	6.33753e7i	0.489180i
$507$	0	0
$508$	−1.24519e7	−0.0949825
$509$	− 2.16126e8i	− 1.63891i	−0.573147	−	0.819453i	$-0.694278\pi$
$509$	− 2.16126e8i	− 1.63891i	0.573147	−	0.819453i	$-0.305722\pi$
$510$	0	0
$511$	−1.61484e8	−1.21023
$512$	− 4.97487e7i	− 0.370656i
$513$	0	0
$514$	2.51278e8	1.85040
$515$	0	0
$516$	0	0
$517$	−1.11614e8	−0.807695
$518$	1.60613e8i	1.15556i
$519$	0	0
$520$	0	0
$521$	8.35166e7i	0.590554i	0.955412	+	0.295277i	$0.0954120\pi$
$521$	8.35166e7i	0.590554i	−0.955412	+	0.295277i	$0.904588\pi$
$522$	0	0
$523$	−2.08856e8	−1.45997	−0.729983	−	0.683466i	$-0.760472\pi$
$523$	−2.08856e8	−1.45997	−0.729983	+	0.683466i	$0.760472\pi$
$524$	− 2.57101e8i	− 1.78694i
$525$	0	0
$526$	−4.43608e7	−0.304819
$527$	− 7.54308e7i	− 0.515367i
$528$	0	0
$529$	1.14939e8	0.776424
$530$	0	0
$531$	0	0
$532$	−1.19137e8	−0.791244
$533$	3.74397e7i	0.247258i
$534$	0	0
$535$	0	0
$536$	9.35119e7i	0.607257i
$537$	0	0
$538$	7.19625e7	0.462125
$539$	− 1.35820e8i	− 0.867357i
$540$	0	0
$541$	−1.21245e8	−0.765727	−0.382863	−	0.923805i	$-0.625062\pi$
$541$	−1.21245e8	−0.765727	−0.382863	+	0.923805i	$0.625062\pi$
$542$	− 3.71519e8i	− 2.33337i
$543$	0	0
$544$	−2.67193e8	−1.65970
$545$	0	0
$546$	0	0
$547$	1.33857e8	0.817861	0.408931	−	0.912565i	$-0.365902\pi$
$547$	1.33857e8	0.817861	0.408931	+	0.912565i	$0.365902\pi$
$548$	1.98264e7i	0.120477i
$549$	0	0
$550$	0	0
$551$	− 5.37128e7i	− 0.321087i
$552$	0	0
$553$	2.83025e8	1.67359
$554$	2.92664e8i	1.72123i
$555$	0	0
$556$	−1.98590e8	−1.15540
$557$	8.20694e7i	0.474915i	0.971398	+	0.237457i	$0.0763140\pi$
$557$	8.20694e7i	0.474915i	−0.971398	+	0.237457i	$0.923686\pi$
$558$	0	0
$559$	−3.12875e7	−0.179116
$560$	0	0
$561$	0	0
$562$	−1.59484e7	−0.0898480
$563$	2.21977e8i	1.24389i	0.783059	+	0.621947i	$0.213658\pi$
$563$	2.21977e8i	1.24389i	−0.783059	+	0.621947i	$0.786342\pi$
$564$	0	0
$565$	0	0
$566$	− 1.84719e8i	− 1.01874i
$567$	0	0
$568$	−2.33319e7	−0.127322
$569$	2.88087e8i	1.56382i	0.623391	+	0.781911i	$0.285755\pi$
$569$	2.88087e8i	1.56382i	−0.623391	+	0.781911i	$0.714245\pi$
$570$	0	0
$571$	1.83227e8	0.984197	0.492098	−	0.870540i	$-0.336230\pi$
$571$	1.83227e8	0.984197	0.492098	+	0.870540i	$0.336230\pi$
$572$	− 2.91777e7i	− 0.155906i
$573$	0	0
$574$	7.25877e8	3.83820
$575$	0	0
$576$	0	0
$577$	2.07783e8	1.08164	0.540820	−	0.841139i	$-0.318114\pi$
$577$	2.07783e8	1.08164	0.540820	+	0.841139i	$0.318114\pi$
$578$	3.41709e8i	1.76959i
$579$	0	0
$580$	0	0
$581$	4.88549e8i	2.49104i
$582$	0	0
$583$	1.70230e8	0.859075
$584$	1.33363e8i	0.669571i
$585$	0	0
$586$	−1.43481e8	−0.713018
$587$	3.28908e8i	1.62615i	0.582161	+	0.813073i	$0.302207\pi$
$587$	3.28908e8i	1.62615i	−0.582161	+	0.813073i	$0.697793\pi$
$588$	0	0
$589$	2.45085e7	0.119942
$590$	0	0
$591$	0	0
$592$	−1.83986e7	−0.0886790
$593$	1.97249e7i	0.0945914i	0.998881	+	0.0472957i	$0.0150603\pi$
$593$	1.97249e7i	0.0945914i	−0.998881	+	0.0472957i	$0.984940\pi$
$594$	0	0
$595$	0	0
$596$	− 4.17023e8i	− 1.96980i
$597$	0	0
$598$	2.51891e7	0.117790
$599$	9.38486e7i	0.436664i	0.975875	+	0.218332i	$0.0700616\pi$
$599$	9.38486e7i	0.436664i	−0.975875	+	0.218332i	$0.929938\pi$
$600$	0	0
$601$	1.53106e8	0.705293	0.352646	−	0.935757i	$-0.385282\pi$
$601$	1.53106e8	0.705293	0.352646	+	0.935757i	$0.385282\pi$
$602$	6.06598e8i	2.78043i
$603$	0	0
$604$	−3.66554e8	−1.66352
$605$	0	0
$606$	0	0
$607$	−1.08279e8	−0.484147	−0.242074	−	0.970258i	$-0.577828\pi$
$607$	−1.08279e8	−0.484147	−0.242074	+	0.970258i	$0.577828\pi$
$608$	− 8.68146e7i	− 0.386262i
$609$	0	0
$610$	0	0
$611$	4.43620e7i	0.194486i
$612$	0	0
$613$	2.60288e8	1.12999	0.564993	−	0.825096i	$-0.308879\pi$
$613$	2.60288e8	1.12999	0.564993	+	0.825096i	$0.308879\pi$
$614$	5.75195e8i	2.48491i
$615$	0	0
$616$	−1.96261e8	−0.839638
$617$	− 1.65376e8i	− 0.704073i	−0.935986	−	0.352036i	$-0.885489\pi$
$617$	− 1.65376e8i	− 0.704073i	0.935986	−	0.352036i	$-0.114511\pi$
$618$	0	0
$619$	−1.36836e8	−0.576935	−0.288468	−	0.957490i	$-0.593146\pi$
$619$	−1.36836e8	−0.576935	−0.288468	+	0.957490i	$0.593146\pi$
$620$	0	0
$621$	0	0
$622$	−1.23987e8	−0.515235
$623$	− 1.17069e8i	− 0.484146i
$624$	0	0
$625$	0	0
$626$	− 6.74918e7i	− 0.275124i
$627$	0	0
$628$	−2.34037e8	−0.944942
$629$	1.71954e8i	0.690973i
$630$	0	0
$631$	2.66941e8	1.06249	0.531247	−	0.847217i	$-0.321723\pi$
$631$	2.66941e8	1.06249	0.531247	+	0.847217i	$0.321723\pi$
$632$	− 2.33738e8i	− 0.925931i
$633$	0	0
$634$	1.34231e8	0.526727
$635$	0	0
$636$	0	0
$637$	−5.39829e7	−0.208852
$638$	− 2.55043e8i	− 0.982092i
$639$	0	0
$640$	0	0
$641$	− 3.69866e8i	− 1.40433i	−0.712013	−	0.702167i	$-0.752216\pi$
$641$	− 3.69866e8i	− 1.40433i	0.712013	−	0.702167i	$-0.247784\pi$
$642$	0	0
$643$	9.29168e7	0.349511	0.174756	−	0.984612i	$-0.444086\pi$
$643$	9.29168e7	0.349511	0.174756	+	0.984612i	$0.444086\pi$
$644$	− 2.95429e8i	− 1.10610i
$645$	0	0
$646$	−2.10846e8	−0.782111
$647$	− 9.21336e7i	− 0.340177i	−0.985429	−	0.170089i	$-0.945595\pi$
$647$	− 9.21336e7i	− 0.340177i	0.985429	−	0.170089i	$-0.0544054\pi$
$648$	0	0
$649$	3.44580e7	0.126054
$650$	0	0
$651$	0	0
$652$	−6.61029e8	−2.38494
$653$	2.20689e8i	0.792576i	0.918126	+	0.396288i	$0.129702\pi$
$653$	2.20689e8i	0.792576i	−0.918126	+	0.396288i	$0.870298\pi$
$654$	0	0
$655$	0	0
$656$	8.31511e7i	0.294548i
$657$	0	0
$658$	8.60085e8	3.01900
$659$	− 5.01619e8i	− 1.75274i	−0.481639	−	0.876370i	$-0.659958\pi$
$659$	− 5.01619e8i	− 1.75274i	0.481639	−	0.876370i	$-0.340042\pi$
$660$	0	0
$661$	−2.78166e8	−0.963163	−0.481582	−	0.876401i	$-0.659937\pi$
$661$	−2.78166e8	−0.963163	−0.481582	+	0.876401i	$0.659937\pi$
$662$	4.85242e8i	1.67257i
$663$	0	0
$664$	4.03472e8	1.37819
$665$	0	0
$666$	0	0
$667$	1.33194e8	0.448858
$668$	− 6.48465e8i	− 2.17549i
$669$	0	0
$670$	0	0
$671$	− 2.17360e8i	− 0.719468i
$672$	0	0
$673$	−5.34850e8	−1.75464	−0.877318	−	0.479910i	$-0.840669\pi$
$673$	−5.34850e8	−1.75464	−0.877318	+	0.479910i	$0.840669\pi$
$674$	− 1.56149e8i	− 0.509988i
$675$	0	0
$676$	4.61430e8	1.49371
$677$	− 1.41358e6i	− 0.00455568i	−0.999997	−	0.00227784i	$-0.999275\pi$
$677$	− 1.41358e6i	− 0.00455568i	0.999997	−	0.00227784i	$-0.000725059\pi$
$678$	0	0
$679$	−1.94760e7	−0.0622144
$680$	0	0
$681$	0	0
$682$	1.16373e8	0.366859
$683$	− 2.41616e8i	− 0.758340i	−0.925327	−	0.379170i	$-0.876209\pi$
$683$	− 2.41616e8i	− 0.758340i	0.925327	−	0.379170i	$-0.123791\pi$
$684$	0	0
$685$	0	0
$686$	2.61962e8i	0.811458i
$687$	0	0
$688$	−6.94873e7	−0.213373
$689$	− 6.76595e7i	− 0.206857i
$690$	0	0
$691$	5.30543e8	1.60800	0.804001	−	0.594628i	$-0.202701\pi$
$691$	5.30543e8	1.60800	0.804001	+	0.594628i	$0.202701\pi$
$692$	− 5.46916e8i	− 1.65045i
$693$	0	0
$694$	−2.11040e8	−0.631374
$695$	0	0
$696$	0	0
$697$	7.77132e8	2.29507
$698$	− 6.12261e8i	− 1.80041i
$699$	0	0
$700$	0	0
$701$	− 5.49256e7i	− 0.159449i	−0.996817	−	0.0797244i	$-0.974596\pi$
$701$	− 5.49256e7i	− 0.159449i	0.996817	−	0.0797244i	$-0.0254040\pi$
$702$	0	0
$703$	−5.58702e7	−0.160811
$704$	− 3.69900e8i	− 1.06015i
$705$	0	0
$706$	−1.87826e8	−0.533753
$707$	− 2.93328e8i	− 0.830034i
$708$	0	0
$709$	3.16706e7	0.0888622	0.0444311	−	0.999012i	$-0.485852\pi$
$709$	3.16706e7	0.0888622	0.0444311	+	0.999012i	$0.485852\pi$
$710$	0	0
$711$	0	0
$712$	−9.66819e7	−0.267858
$713$	6.07749e7i	0.167670i
$714$	0	0
$715$	0	0
$716$	− 4.77361e8i	− 1.30049i
$717$	0	0
$718$	2.27205e8	0.613825
$719$	− 4.37723e8i	− 1.17764i	−0.808264	−	0.588820i	$-0.799593\pi$
$719$	− 4.37723e8i	− 1.17764i	0.808264	−	0.588820i	$-0.200407\pi$
$720$	0	0
$721$	7.65134e8	2.04142
$722$	5.30290e8i	1.40897i
$723$	0	0
$724$	−8.30595e8	−2.18864
$725$	0	0
$726$	0	0
$727$	4.64180e8	1.20805	0.604023	−	0.796967i	$-0.293564\pi$
$727$	4.64180e8	1.20805	0.604023	+	0.796967i	$0.293564\pi$
$728$	7.80057e7i	0.202177i
$729$	0	0
$730$	0	0
$731$	6.49430e8i	1.66257i
$732$	0	0
$733$	−7.04886e8	−1.78981	−0.894905	−	0.446256i	$-0.852757\pi$
$733$	−7.04886e8	−1.78981	−0.894905	+	0.446256i	$0.852757\pi$
$734$	1.10471e8i	0.279357i
$735$	0	0
$736$	2.15279e8	0.539967
$737$	− 1.87024e8i	− 0.467191i
$738$	0	0
$739$	−2.83900e8	−0.703448	−0.351724	−	0.936104i	$-0.614404\pi$
$739$	−2.83900e8	−0.703448	−0.351724	+	0.936104i	$0.614404\pi$
$740$	0	0
$741$	0	0
$742$	−1.31177e9	−3.21105
$743$	− 5.01018e8i	− 1.22148i	−0.791830	−	0.610741i	$-0.790872\pi$
$743$	− 5.01018e8i	− 1.22148i	0.791830	−	0.610741i	$-0.209128\pi$
$744$	0	0
$745$	0	0
$746$	− 1.01066e9i	− 2.43439i
$747$	0	0
$748$	−6.05638e8	−1.44713
$749$	− 6.76120e8i	− 1.60908i
$750$	0	0
$751$	−1.48249e8	−0.350004	−0.175002	−	0.984568i	$-0.555993\pi$
$751$	−1.48249e8	−0.350004	−0.175002	+	0.984568i	$0.555993\pi$
$752$	9.85249e7i	0.231682i
$753$	0	0
$754$	−1.01369e8	−0.236479
$755$	0	0
$756$	0	0
$757$	−2.14422e8	−0.494291	−0.247145	−	0.968978i	$-0.579493\pi$
$757$	−2.14422e8	−0.494291	−0.247145	+	0.968978i	$0.579493\pi$
$758$	1.86268e8i	0.427691i
$759$	0	0
$760$	0	0
$761$	2.69760e7i	0.0612101i	0.999532	+	0.0306051i	$0.00974341\pi$
$761$	2.69760e7i	0.0612101i	−0.999532	+	0.0306051i	$0.990257\pi$
$762$	0	0
$763$	7.52192e8	1.69338
$764$	− 9.50969e6i	− 0.0213249i
$765$	0	0
$766$	1.16681e8	0.259607
$767$	− 1.36957e7i	− 0.0303526i
$768$	0	0
$769$	−4.90064e8	−1.07764	−0.538821	−	0.842421i	$-0.681130\pi$
$769$	−4.90064e8	−1.07764	−0.538821	+	0.842421i	$0.681130\pi$
$770$	0	0
$771$	0	0
$772$	7.34484e8	1.59636
$773$	4.17032e8i	0.902882i	0.892301	+	0.451441i	$0.149090\pi$
$773$	4.17032e8i	0.902882i	−0.892301	+	0.451441i	$0.850910\pi$
$774$	0	0
$775$	0	0
$776$	1.60844e7i	0.0344208i
$777$	0	0
$778$	1.17946e8	0.250463
$779$	2.52501e8i	0.534134i
$780$	0	0
$781$	4.66638e7	0.0979550
$782$	− 5.22846e8i	− 1.09334i
$783$	0	0
$784$	−1.19892e8	−0.248796
$785$	0	0
$786$	0	0
$787$	3.46111e8	0.710054	0.355027	−	0.934856i	$-0.384472\pi$
$787$	3.46111e8	0.710054	0.355027	+	0.934856i	$0.384472\pi$
$788$	− 9.64963e8i	− 1.97211i
$789$	0	0
$790$	0	0
$791$	− 9.76872e7i	− 0.197382i
$792$	0	0
$793$	−8.63915e7	−0.173241
$794$	9.23467e8i	1.84484i
$795$	0	0
$796$	−3.47087e8	−0.688175
$797$	7.25306e8i	1.43267i	0.697756	+	0.716335i	$0.254182\pi$
$797$	7.25306e8i	1.43267i	−0.697756	+	0.716335i	$0.745818\pi$
$798$	0	0
$799$	9.20816e8	1.80523
$800$	0	0
$801$	0	0
$802$	−9.38361e8	−1.81906
$803$	− 2.66726e8i	− 0.515132i
$804$	0	0
$805$	0	0
$806$	− 4.62535e7i	− 0.0883363i
$807$	0	0
$808$	−2.42247e8	−0.459224
$809$	4.93161e8i	0.931415i	0.884939	+	0.465707i	$0.154200\pi$
$809$	4.93161e8i	0.931415i	−0.884939	+	0.465707i	$0.845800\pi$
$810$	0	0
$811$	−5.34731e7	−0.100247	−0.0501237	−	0.998743i	$-0.515962\pi$
$811$	−5.34731e7	−0.100247	−0.0501237	+	0.998743i	$0.515962\pi$
$812$	1.18891e9i	2.22065i
$813$	0	0
$814$	−2.65287e8	−0.491862
$815$	0	0
$816$	0	0
$817$	−2.11009e8	−0.386931
$818$	− 4.37278e8i	− 0.798909i
$819$	0	0
$820$	0	0
$821$	1.07674e9i	1.94573i	0.231377	+	0.972864i	$0.425677\pi$
$821$	1.07674e9i	1.94573i	−0.231377	+	0.972864i	$0.574323\pi$
$822$	0	0
$823$	−8.88441e7	−0.159378	−0.0796891	−	0.996820i	$-0.525393\pi$
$823$	−8.88441e7	−0.159378	−0.0796891	+	0.996820i	$0.525393\pi$
$824$	− 6.31892e8i	− 1.12943i
$825$	0	0
$826$	−2.65530e8	−0.471165
$827$	− 9.82047e7i	− 0.173626i	−0.996225	−	0.0868132i	$-0.972332\pi$
$827$	− 9.82047e7i	− 0.173626i	0.996225	−	0.0868132i	$-0.0276683\pi$
$828$	0	0
$829$	2.25045e8	0.395008	0.197504	−	0.980302i	$-0.436717\pi$
$829$	2.25045e8	0.395008	0.197504	+	0.980302i	$0.436717\pi$
$830$	0	0
$831$	0	0
$832$	−1.47020e8	−0.255274
$833$	1.12052e9i	1.93858i
$834$	0	0
$835$	0	0
$836$	− 1.96780e8i	− 0.336792i
$837$	0	0
$838$	1.67177e9	2.84083
$839$	− 7.84433e8i	− 1.32822i	−0.747635	−	0.664110i	$-0.768811\pi$
$839$	− 7.84433e8i	− 1.32822i	0.747635	−	0.664110i	$-0.231189\pi$
$840$	0	0
$841$	5.88040e7	0.0988597
$842$	− 3.01341e7i	− 0.0504803i
$843$	0	0
$844$	6.63752e8	1.10402
$845$	0	0
$846$	0	0
$847$	−5.35776e8	−0.881724
$848$	− 1.50267e8i	− 0.246420i
$849$	0	0
$850$	0	0
$851$	− 1.38544e8i	− 0.224802i
$852$	0	0
$853$	7.97906e7	0.128560	0.0642798	−	0.997932i	$-0.479525\pi$
$853$	7.97906e7	0.128560	0.0642798	+	0.997932i	$0.479525\pi$
$854$	1.67495e9i	2.68923i
$855$	0	0
$856$	−5.58379e8	−0.890241
$857$	− 1.06107e9i	− 1.68579i	−0.538080	−	0.842894i	$-0.680850\pi$
$857$	− 1.06107e9i	− 1.68579i	0.538080	−	0.842894i	$-0.319150\pi$
$858$	0	0
$859$	3.34286e8	0.527399	0.263699	−	0.964605i	$-0.415057\pi$
$859$	3.34286e8	0.527399	0.263699	+	0.964605i	$0.415057\pi$
$860$	0	0
$861$	0	0
$862$	1.43888e9	2.24648
$863$	2.65292e7i	0.0412754i	0.999787	+	0.0206377i	$0.00656965\pi$
$863$	2.65292e7i	0.0412754i	−0.999787	+	0.0206377i	$0.993430\pi$
$864$	0	0
$865$	0	0
$866$	5.73936e8i	0.883711i
$867$	0	0
$868$	−5.42483e8	−0.829519
$869$	4.67477e8i	0.712362i
$870$	0	0
$871$	−7.43343e7	−0.112495
$872$	− 6.21203e8i	− 0.936880i
$873$	0	0
$874$	1.69880e8	0.254453
$875$	0	0
$876$	0	0
$877$	1.30791e8	0.193901	0.0969504	−	0.995289i	$-0.469091\pi$
$877$	1.30791e8	0.193901	0.0969504	+	0.995289i	$0.469091\pi$
$878$	2.05690e9i	3.03900i
$879$	0	0
$880$	0	0
$881$	− 9.18953e8i	− 1.34390i	−0.740598	−	0.671948i	$-0.765458\pi$
$881$	− 9.18953e8i	− 1.34390i	0.740598	−	0.671948i	$-0.234542\pi$
$882$	0	0
$883$	1.09112e9	1.58486	0.792432	−	0.609961i	$-0.208815\pi$
$883$	1.09112e9	1.58486	0.792432	+	0.609961i	$0.208815\pi$
$884$	2.40716e8i	0.348456i
$885$	0	0
$886$	−3.23303e8	−0.464846
$887$	5.36427e8i	0.768670i	0.923194	+	0.384335i	$0.125569\pi$
$887$	5.36427e8i	0.768670i	−0.923194	+	0.384335i	$0.874431\pi$
$888$	0	0
$889$	−6.65794e7	−0.0947621
$890$	0	0
$891$	0	0
$892$	−3.27502e8	−0.461445
$893$	2.99186e8i	0.420133i
$894$	0	0
$895$	0	0
$896$	1.59549e9i	2.21804i
$897$	0	0
$898$	−1.20139e9	−1.65903
$899$	− 2.44579e8i	− 0.336620i
$900$	0	0
$901$	−1.40440e9	−1.92007
$902$	1.19894e9i	1.63372i
$903$	0	0
$904$	−8.06757e7	−0.109204
$905$	0	0
$906$	0	0
$907$	4.60985e8	0.617824	0.308912	−	0.951091i	$-0.400035\pi$
$907$	4.60985e8	0.617824	0.308912	+	0.951091i	$0.400035\pi$
$908$	1.58772e9i	2.12089i
$909$	0	0
$910$	0	0
$911$	− 8.68400e8i	− 1.14859i	−0.818649	−	0.574295i	$-0.805276\pi$
$911$	− 8.68400e8i	− 1.14859i	0.818649	−	0.574295i	$-0.194724\pi$
$912$	0	0
$913$	−8.06944e8	−1.06031
$914$	5.96242e8i	0.780879i
$915$	0	0
$916$	1.63024e9	2.12112
$917$	− 1.37470e9i	− 1.78279i
$918$	0	0
$919$	1.28061e9	1.64995	0.824976	−	0.565168i	$-0.191189\pi$
$919$	1.28061e9	1.64995	0.824976	+	0.565168i	$0.191189\pi$
$920$	0	0
$921$	0	0
$922$	2.97575e8	0.379668
$923$	− 1.85469e7i	− 0.0235867i
$924$	0	0
$925$	0	0
$926$	− 6.34192e8i	− 0.798708i
$927$	0	0
$928$	−8.66353e8	−1.08405
$929$	− 3.23960e8i	− 0.404058i	−0.979380	−	0.202029i	$-0.935246\pi$
$929$	− 3.23960e8i	− 0.404058i	0.979380	−	0.202029i	$-0.0647536\pi$
$930$	0	0
$931$	−3.64071e8	−0.451166
$932$	− 8.67888e8i	− 1.07205i
$933$	0	0
$934$	1.21217e9	1.48772
$935$	0	0
$936$	0	0
$937$	1.33188e9	1.61900	0.809498	−	0.587123i	$-0.199739\pi$
$937$	1.33188e9	1.61900	0.809498	+	0.587123i	$0.199739\pi$
$938$	1.44118e9i	1.74627i
$939$	0	0
$940$	0	0
$941$	8.24869e8i	0.989957i	0.868905	+	0.494978i	$0.164824\pi$
$941$	8.24869e8i	0.989957i	−0.868905	+	0.494978i	$0.835176\pi$
$942$	0	0
$943$	−6.26138e8	−0.746681
$944$	− 3.04171e7i	− 0.0361578i
$945$	0	0
$946$	−1.00193e9	−1.18349
$947$	− 7.33660e8i	− 0.863863i	−0.901906	−	0.431931i	$-0.857832\pi$
$947$	− 7.33660e8i	− 0.863863i	0.901906	−	0.431931i	$-0.142168\pi$
$948$	0	0
$949$	−1.06013e8	−0.124039
$950$	0	0
$951$	0	0
$952$	1.61915e9	1.87662
$953$	− 6.87744e8i	− 0.794600i	−0.917689	−	0.397300i	$-0.869947\pi$
$953$	− 6.87744e8i	− 0.794600i	0.917689	−	0.397300i	$-0.130053\pi$
$954$	0	0
$955$	0	0
$956$	1.10736e9i	1.26740i
$957$	0	0
$958$	2.50919e9	2.85389
$959$	1.06011e8i	0.120197i
$960$	0	0
$961$	−7.75906e8	−0.874256
$962$	1.05441e8i	0.118436i
$963$	0	0
$964$	−1.47088e9	−1.64190
$965$	0	0
$966$	0	0
$967$	−1.43001e9	−1.58147	−0.790733	−	0.612162i	$-0.790300\pi$
$967$	−1.43001e9	−1.58147	−0.790733	+	0.612162i	$0.790300\pi$
$968$	4.42475e8i	0.487823i
$969$	0	0
$970$	0	0
$971$	− 1.05628e9i	− 1.15377i	−0.816824	−	0.576887i	$-0.804267\pi$
$971$	− 1.05628e9i	− 1.15377i	0.816824	−	0.576887i	$-0.195733\pi$
$972$	0	0
$973$	−1.06185e9	−1.15272
$974$	2.45142e7i	0.0265303i
$975$	0	0
$976$	−1.91869e8	−0.206375
$977$	− 8.72371e8i	− 0.935443i	−0.883876	−	0.467722i	$-0.845075\pi$
$977$	− 8.72371e8i	− 0.935443i	0.883876	−	0.467722i	$-0.154925\pi$
$978$	0	0
$979$	1.93364e8	0.206076
$980$	0	0
$981$	0	0
$982$	−2.28749e9	−2.41560
$983$	7.90860e8i	0.832605i	0.909226	+	0.416302i	$0.136674\pi$
$983$	7.90860e8i	0.832605i	−0.909226	+	0.416302i	$0.863326\pi$
$984$	0	0
$985$	0	0
$986$	2.10411e9i	2.19501i
$987$	0	0
$988$	−7.82118e7	−0.0810964
$989$	− 5.23249e8i	− 0.540903i
$990$	0	0
$991$	1.38796e8	0.142612	0.0713062	−	0.997454i	$-0.477283\pi$
$991$	1.38796e8	0.142612	0.0713062	+	0.997454i	$0.477283\pi$
$992$	− 3.95306e8i	− 0.404947i
$993$	0	0
$994$	−3.59586e8	−0.366137
$995$	0	0
$996$	0	0
$997$	3.29036e8	0.332015	0.166008	−	0.986124i	$-0.446912\pi$
$997$	3.29036e8	0.332015	0.166008	+	0.986124i	$0.446912\pi$
$998$	− 1.96032e9i	− 1.97213i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.7.c.a.26.1		2
3.2	odd	2	inner	225.7.c.a.26.2		2
5.2	odd	4		225.7.d.a.224.3		4
5.3	odd	4		225.7.d.a.224.2		4
5.4	even	2		9.7.b.a.8.2	yes	2
15.2	even	4		225.7.d.a.224.1		4
15.8	even	4		225.7.d.a.224.4		4
15.14	odd	2		9.7.b.a.8.1	✓	2
20.19	odd	2		144.7.e.a.17.2		2
35.34	odd	2		441.7.b.a.197.2		2
40.19	odd	2		576.7.e.a.449.1		2
40.29	even	2		576.7.e.l.449.1		2
45.4	even	6		81.7.d.d.26.1		4
45.14	odd	6		81.7.d.d.26.2		4
45.29	odd	6		81.7.d.d.53.1		4
45.34	even	6		81.7.d.d.53.2		4
60.59	even	2		144.7.e.a.17.1		2
105.104	even	2		441.7.b.a.197.1		2
120.29	odd	2		576.7.e.l.449.2		2
120.59	even	2		576.7.e.a.449.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.7.b.a.8.1	✓	2	15.14	odd	2
9.7.b.a.8.2	yes	2	5.4	even	2
81.7.d.d.26.1		4	45.4	even	6
81.7.d.d.26.2		4	45.14	odd	6
81.7.d.d.53.1		4	45.29	odd	6
81.7.d.d.53.2		4	45.34	even	6
144.7.e.a.17.1		2	60.59	even	2
144.7.e.a.17.2		2	20.19	odd	2
225.7.c.a.26.1		2	1.1	even	1	trivial
225.7.c.a.26.2		2	3.2	odd	2	inner
225.7.d.a.224.1		4	15.2	even	4
225.7.d.a.224.2		4	5.3	odd	4
225.7.d.a.224.3		4	5.2	odd	4
225.7.d.a.224.4		4	15.8	even	4
441.7.b.a.197.1		2	105.104	even	2
441.7.b.a.197.2		2	35.34	odd	2
576.7.e.a.449.1		2	40.19	odd	2
576.7.e.a.449.2		2	120.59	even	2
576.7.e.l.449.1		2	40.29	even	2
576.7.e.l.449.2		2	120.29	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	225.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-12.7279i q^{2} -98.0000 q^{4} -524.000 q^{7} +432.749i q^{8} +O(q^{10})\)	\(q-12.7279i q^{2} -98.0000 q^{4} -524.000 q^{7} +432.749i q^{8} -865.499i q^{11} -344.000 q^{13} +6669.43i q^{14} -764.000 q^{16} +7140.36i q^{17} -2320.00 q^{19} -11016.0 q^{22} -5753.02i q^{23} +4378.41i q^{26} +51352.0 q^{28} +23152.1i q^{29} -10564.0 q^{31} +37420.1i q^{32} +90882.0 q^{34} +24082.0 q^{37} +29528.8i q^{38} -108836. i q^{41} +90952.0 q^{43} +84818.9i q^{44} -73224.0 q^{46} -128959. i q^{47} +156927. q^{49} +33712.0 q^{52} +196685. i q^{53} -226761. i q^{56} +294678. q^{58} +39812.9i q^{59} +251138. q^{61} +134458. i q^{62} +427384. q^{64} +216088. q^{67} -699756. i q^{68} +53915.5i q^{71} +308176. q^{73} -306514. i q^{74} +227360. q^{76} +453521. i q^{77} -540124. q^{79} -1.38526e6 q^{82} -932346. i q^{83} -1.15763e6i q^{86} +374544. q^{88} +223413. i q^{89} +180256. q^{91} +563796. i q^{92} -1.64138e6 q^{94} +37168.0 q^{97} -1.99735e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 196 q^{4} - 1048 q^{7}+O(q^{10}) \) 2 * q - 196 * q^4 - 1048 * q^7	\( 2 q - 196 q^{4} - 1048 q^{7} - 688 q^{13} - 1528 q^{16} - 4640 q^{19} - 22032 q^{22} + 102704 q^{28} - 21128 q^{31} + 181764 q^{34} + 48164 q^{37} + 181904 q^{43} - 146448 q^{46} + 313854 q^{49} + 67424 q^{52} + 589356 q^{58} + 502276 q^{61} + 854768 q^{64} + 432176 q^{67} + 616352 q^{73} + 454720 q^{76} - 1080248 q^{79} - 2770524 q^{82} + 749088 q^{88} + 360512 q^{91} - 3282768 q^{94} + 74336 q^{97}+O(q^{100}) \) 2 * q - 196 * q^4 - 1048 * q^7 - 688 * q^13 - 1528 * q^16 - 4640 * q^19 - 22032 * q^22 + 102704 * q^28 - 21128 * q^31 + 181764 * q^34 + 48164 * q^37 + 181904 * q^43 - 146448 * q^46 + 313854 * q^49 + 67424 * q^52 + 589356 * q^58 + 502276 * q^61 + 854768 * q^64 + 432176 * q^67 + 616352 * q^73 + 454720 * q^76 - 1080248 * q^79 - 2770524 * q^82 + 749088 * q^88 + 360512 * q^91 - 3282768 * q^94 + 74336 * q^97

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