LMFDB - Embedded newform 225.7.d.a.224.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,7,Mod(224,225)]

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, 1]))

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

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

chi := DirichletCharacter("225.224");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	225.d (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:	$51.7621688145$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			224.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	225.224
Dual form			225.7.d.a.224.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.7279 q^{2} +98.0000 q^{4} -524.000i q^{7} -432.749 q^{8} +O(q^{10})$	$q-12.7279 q^{2} +98.0000 q^{4} -524.000i q^{7} -432.749 q^{8} +865.499i q^{11} +344.000i q^{13} +6669.43i q^{14} -764.000 q^{16} +7140.36 q^{17} +2320.00 q^{19} -11016.0i q^{22} +5753.02 q^{23} -4378.41i q^{26} -51352.0i q^{28} +23152.1i q^{29} -10564.0 q^{31} +37420.1 q^{32} -90882.0 q^{34} +24082.0i q^{37} -29528.8 q^{38} +108836. i q^{41} -90952.0i q^{43} +84818.9i q^{44} -73224.0 q^{46} -128959. q^{47} -156927. q^{49} +33712.0i q^{52} -196685. q^{53} +226761. i q^{56} -294678. i q^{58} +39812.9i q^{59} +251138. q^{61} +134458. q^{62} -427384. q^{64} +216088. i q^{67} +699756. q^{68} -53915.5i q^{71} -308176. i q^{73} -306514. i q^{74} +227360. q^{76} +453521. q^{77} +540124. q^{79} -1.38526e6i q^{82} +932346. q^{83} +1.15763e6i q^{86} -374544. i q^{88} +223413. i q^{89} +180256. q^{91} +563796. q^{92} +1.64138e6 q^{94} +37168.0i q^{97} +1.99735e6 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 392 q^{4}+O(q^{10}) $ 4 * q + 392 * q^4	$ 4 q + 392 q^{4} - 3056 q^{16} + 9280 q^{19} - 42256 q^{31} - 363528 q^{34} - 292896 q^{46} - 627708 q^{49} + 1004552 q^{61} - 1709536 q^{64} + 909440 q^{76} + 2160496 q^{79} + 721024 q^{91} + 6565536 q^{94}+O(q^{100}) $ 4 * q + 392 * q^4 - 3056 * q^16 + 9280 * q^19 - 42256 * q^31 - 363528 * q^34 - 292896 * q^46 - 627708 * q^49 + 1004552 * q^61 - 1709536 * q^64 + 909440 * q^76 + 2160496 * q^79 + 721024 * q^91 + 6565536 * 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 $

$2$	−12.7279	−1.59099	−0.795495	−	0.605960i	$-0.792789\pi$
$2$	−12.7279	−1.59099	−0.795495	+	0.605960i	$0.792789\pi$
$3$	0	0
$3$	0	0
$4$	98.0000	1.53125
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 524.000i	− 1.52770i	−0.645396	−	0.763848i	$-0.723308\pi$
$7$	− 524.000i	− 1.52770i	0.645396	−	0.763848i	$-0.276692\pi$
$8$	−432.749	−0.845214
$9$	0	0
$10$	0	0
$11$	865.499i	0.650262i	0.945669	+	0.325131i	$0.105408\pi$
$11$	865.499i	0.650262i	−0.945669	+	0.325131i	$0.894592\pi$
$12$	0	0
$13$	344.000i	0.156577i	0.996931	+	0.0782886i	$0.0249456\pi$
$13$	344.000i	0.156577i	−0.996931	+	0.0782886i	$0.975054\pi$
$14$	6669.43i	2.43055i
$15$	0	0
$16$	−764.000	−0.186523
$17$	7140.36	1.45336	0.726681	−	0.686975i	$-0.241062\pi$
$17$	7140.36	1.45336	0.726681	+	0.686975i	$0.241062\pi$
$18$	0	0
$19$	2320.00	0.338242	0.169121	−	0.985595i	$-0.445907\pi$
$19$	2320.00	0.338242	0.169121	+	0.985595i	$0.445907\pi$
$20$	0	0
$21$	0	0
$22$	− 11016.0i	− 1.03456i
$23$	5753.02	0.472838	0.236419	−	0.971651i	$-0.424026\pi$
$23$	5753.02	0.472838	0.236419	+	0.971651i	$0.424026\pi$
$24$	0	0
$25$	0	0
$26$	− 4378.41i	− 0.249113i
$27$	0	0
$28$	− 51352.0i	− 2.33929i
$29$	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	$0.556737\pi$
$32$	37420.1	1.14197
$33$	0	0
$34$	−90882.0	−2.31228
$35$	0	0
$36$	0	0
$37$	24082.0i	0.475431i	0.971335	+	0.237715i	$0.0763986\pi$
$37$	24082.0i	0.475431i	−0.971335	+	0.237715i	$0.923601\pi$
$38$	−29528.8	−0.538139
$39$	0	0
$40$	0	0
$41$	108836.i	1.57915i	0.613655	+	0.789574i	$0.289698\pi$
$41$	108836.i	1.57915i	−0.613655	+	0.789574i	$0.710302\pi$
$42$	0	0
$43$	− 90952.0i	− 1.14395i	−0.820271	−	0.571975i	$-0.806178\pi$
$43$	− 90952.0i	− 1.14395i	0.820271	−	0.571975i	$-0.193822\pi$
$44$	84818.9i	0.995714i
$45$	0	0
$46$	−73224.0	−0.752281
$47$	−128959.	−1.24211	−0.621054	−	0.783768i	$-0.713295\pi$
$47$	−128959.	−1.24211	−0.621054	+	0.783768i	$0.713295\pi$
$48$	0	0
$49$	−156927.	−1.33386
$50$	0	0
$51$	0	0
$52$	33712.0i	0.239759i
$53$	−196685.	−1.32112	−0.660561	−	0.750773i	$-0.729681\pi$
$53$	−196685.	−1.32112	−0.660561	+	0.750773i	$0.729681\pi$
$54$	0	0
$55$	0	0
$56$	226761.i	1.29123i
$57$	0	0
$58$	− 294678.i	− 1.51030i
$59$	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.	0.564171
$63$	0	0
$64$	−427384.	−1.63034
$65$	0	0
$66$	0	0
$67$	216088.i	0.718466i	0.933248	+	0.359233i	$0.116962\pi$
$67$	216088.i	0.718466i	−0.933248	+	0.359233i	$0.883038\pi$
$68$	699756.	2.22546
$69$	0	0
$70$	0	0
$71$	− 53915.5i	− 0.150639i	−0.997159	−	0.0753197i	$-0.976002\pi$
$71$	− 53915.5i	− 0.150639i	0.997159	−	0.0753197i	$-0.0239977\pi$
$72$	0	0
$73$	− 308176.i	− 0.792192i	−0.918209	−	0.396096i	$-0.870365\pi$
$73$	− 308176.i	− 0.792192i	0.918209	−	0.396096i	$-0.129635\pi$
$74$	− 306514.i	− 0.756406i
$75$	0	0
$76$	227360.	0.517933
$77$	453521.	0.993403
$78$	0	0
$79$	540124.	1.09550	0.547750	−	0.836642i	$-0.315485\pi$
$79$	540124.	1.09550	0.547750	+	0.836642i	$0.315485\pi$
$80$	0	0
$81$	0	0
$82$	− 1.38526e6i	− 2.51241i
$83$	932346.	1.63058	0.815291	−	0.579051i	$-0.196577\pi$
$83$	932346.	1.63058	0.815291	+	0.579051i	$0.196577\pi$
$84$	0	0
$85$	0	0
$86$	1.15763e6i	1.82001i
$87$	0	0
$88$	− 374544.i	− 0.549610i
$89$	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.	0.724033
$93$	0	0
$94$	1.64138e6	1.97618
$95$	0	0
$96$	0	0
$97$	37168.0i	0.0407243i	0.999793	+	0.0203622i	$0.00648193\pi$
$97$	37168.0i	0.0407243i	−0.999793	+	0.0203622i	$0.993518\pi$
$98$	1.99735e6	2.12215
$99$	0	0
$100$	0	0
$101$	− 559787.i	− 0.543323i	−0.962393	−	0.271662i	$-0.912427\pi$
$101$	− 559787.i	− 0.543323i	0.962393	−	0.271662i	$-0.0875732\pi$
$102$	0	0
$103$	1.46018e6i	1.33627i	0.744039	+	0.668136i	$0.232907\pi$
$103$	1.46018e6i	1.33627i	−0.744039	+	0.668136i	$0.767093\pi$
$104$	− 148866.i	− 0.132341i
$105$	0	0
$106$	2.50339e6	2.10189
$107$	1.29031e6	1.05327	0.526637	−	0.850090i	$-0.323453\pi$
$107$	1.29031e6	1.05327	0.526637	+	0.850090i	$0.323453\pi$
$108$	0	0
$109$	1.43548e6	1.10845	0.554227	−	0.832366i	$-0.313014\pi$
$109$	1.43548e6	1.10845	0.554227	+	0.832366i	$0.313014\pi$
$110$	0	0
$111$	0	0
$112$	400336.i	0.284951i
$113$	−186426.	−0.129202	−0.0646012	−	0.997911i	$-0.520578\pi$
$113$	−186426.	−0.129202	−0.0646012	+	0.997911i	$0.520578\pi$
$114$	0	0
$115$	0	0
$116$	2.26890e6i	1.45359i
$117$	0	0
$118$	− 506736.i	− 0.308415i
$119$	− 3.74155e6i	− 2.22030i
$120$	0	0
$121$	1.02247e6	0.577159
$122$	−3.19646e6	−1.76032
$123$	0	0
$124$	−1.03527e6	−0.542987
$125$	0	0
$126$	0	0
$127$	127060.i	0.0620294i	0.999519	+	0.0310147i	$0.00987387\pi$
$127$	127060.i	0.0620294i	−0.999519	+	0.0310147i	$0.990126\pi$
$128$	3.04482e6	1.45189
$129$	0	0
$130$	0	0
$131$	− 2.62348e6i	− 1.16698i	−0.812120	−	0.583490i	$-0.801687\pi$
$131$	− 2.62348e6i	− 1.16698i	0.812120	−	0.583490i	$-0.198313\pi$
$132$	0	0
$133$	− 1.21568e6i	− 0.516731i
$134$	− 2.75035e6i	− 1.14307i
$135$	0	0
$136$	−3.08999e6	−1.22840
$137$	−202310.	−0.0786785	−0.0393393	−	0.999226i	$-0.512525\pi$
$137$	−202310.	−0.0786785	−0.0393393	+	0.999226i	$0.512525\pi$
$138$	0	0
$139$	−2.02642e6	−0.754546	−0.377273	−	0.926102i	$-0.623138\pi$
$139$	−2.02642e6	−0.754546	−0.377273	+	0.926102i	$0.623138\pi$
$140$	0	0
$141$	0	0
$142$	686232.i	0.239666i
$143$	−297732.	−0.101816
$144$	0	0
$145$	0	0
$146$	3.92244e6i	1.26037i
$147$	0	0
$148$	2.36004e6i	0.728004i
$149$	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	$0.317217\pi$
$152$	−1.00398e6	−0.285886
$153$	0	0
$154$	−5.77238e6	−1.58049
$155$	0	0
$156$	0	0
$157$	2.38813e6i	0.617105i	0.951207	+	0.308552i	$0.0998445\pi$
$157$	2.38813e6i	0.617105i	−0.951207	+	0.308552i	$0.900155\pi$
$158$	−6.87466e6	−1.74293
$159$	0	0
$160$	0	0
$161$	− 3.01458e6i	− 0.722353i
$162$	0	0
$163$	− 6.74519e6i	− 1.55751i	−0.627327	−	0.778756i	$-0.715851\pi$
$163$	− 6.74519e6i	− 1.55751i	0.627327	−	0.778756i	$-0.284149\pi$
$164$	1.06660e7i	2.41807i
$165$	0	0
$166$	−1.18668e7	−2.59424
$167$	6.61699e6	1.42073	0.710364	−	0.703834i	$-0.248530\pi$
$167$	6.61699e6	1.42073	0.710364	+	0.703834i	$0.248530\pi$
$168$	0	0
$169$	4.70847e6	0.975484
$170$	0	0
$171$	0	0
$172$	− 8.91330e6i	− 1.75167i
$173$	−5.58077e6	−1.07784	−0.538922	−	0.842356i	$-0.681168\pi$
$173$	−5.58077e6	−1.07784	−0.538922	+	0.842356i	$0.681168\pi$
$174$	0	0
$175$	0	0
$176$	− 661241.i	− 0.121289i
$177$	0	0
$178$	− 2.84359e6i	− 0.504204i
$179$	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	$0.246583\pi$
$182$	−2.29428e6	−0.380569
$183$	0	0
$184$	−2.48962e6	−0.399649
$185$	0	0
$186$	0	0
$187$	6.17998e6i	0.945066i
$188$	−1.26380e7	−1.90198
$189$	0	0
$190$	0	0
$191$	− 97037.7i	− 0.0139264i	−0.999976	−	0.00696322i	$-0.997784\pi$
$191$	− 97037.7i	− 0.0139264i	0.999976	−	0.00696322i	$-0.00221648\pi$
$192$	0	0
$193$	7.49473e6i	1.04252i	0.853398	+	0.521260i	$0.174538\pi$
$193$	7.49473e6i	1.04252i	−0.853398	+	0.521260i	$0.825462\pi$
$194$	− 473071.i	− 0.0647920i
$195$	0	0
$196$	−1.53788e7	−2.04247
$197$	9.84656e6	1.28791	0.643956	−	0.765063i	$-0.277292\pi$
$197$	9.84656e6	1.28791	0.643956	+	0.765063i	$0.277292\pi$
$198$	0	0
$199$	−3.54170e6	−0.449420	−0.224710	−	0.974426i	$-0.572144\pi$
$199$	−3.54170e6	−0.449420	−0.224710	+	0.974426i	$0.572144\pi$
$200$	0	0
$201$	0	0
$202$	7.12492e6i	0.864422i
$203$	1.21317e7	1.45022
$204$	0	0
$205$	0	0
$206$	− 1.85851e7i	− 2.12600i
$207$	0	0
$208$	− 262816.i	− 0.0292053i
$209$	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	$0.617393\pi$
$212$	−1.92751e7	−2.02297
$213$	0	0
$214$	−1.64229e7	−1.67575
$215$	0	0
$216$	0	0
$217$	5.53554e6i	0.541727i
$218$	−1.82707e7	−1.76354
$219$	0	0
$220$	0	0
$221$	2.45629e6i	0.227563i
$222$	0	0
$223$	− 3.34186e6i	− 0.301352i	−0.988583	−	0.150676i	$-0.951855\pi$
$223$	− 3.34186e6i	− 0.301352i	0.988583	−	0.150676i	$-0.0481450\pi$
$224$	− 1.96081e7i	− 1.74458i
$225$	0	0
$226$	2.37281e6	0.205560
$227$	−1.62013e7	−1.38507	−0.692535	−	0.721385i	$-0.743506\pi$
$227$	−1.62013e7	−1.38507	−0.692535	+	0.721385i	$0.743506\pi$
$228$	0	0
$229$	1.66351e7	1.38522	0.692611	−	0.721311i	$-0.256460\pi$
$229$	1.66351e7	1.38522	0.692611	+	0.721311i	$0.256460\pi$
$230$	0	0
$231$	0	0
$232$	− 1.00191e7i	− 0.802348i
$233$	−8.85600e6	−0.700116	−0.350058	−	0.936728i	$-0.613838\pi$
$233$	−8.85600e6	−0.700116	−0.350058	+	0.936728i	$0.613838\pi$
$234$	0	0
$235$	0	0
$236$	3.90167e6i	0.296834i
$237$	0	0
$238$	4.76222e7i	3.53247i
$239$	− 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	$0.319886\pi$
$242$	−1.30140e7	−0.918255
$243$	0	0
$244$	2.46115e7	1.69422
$245$	0	0
$246$	0	0
$247$	798080.i	0.0529609i
$248$	4.57156e6	0.299716
$249$	0	0
$250$	0	0
$251$	− 3.04076e7i	− 1.92292i	−0.274950	−	0.961458i	$-0.588661\pi$
$251$	− 3.04076e7i	− 1.92292i	0.274950	−	0.961458i	$-0.411339\pi$
$252$	0	0
$253$	4.97923e6i	0.307469i
$254$	− 1.61721e6i	− 0.0986882i
$255$	0	0
$256$	−1.14017e7	−0.679595
$257$	1.97422e7	1.16305	0.581523	−	0.813530i	$-0.302457\pi$
$257$	1.97422e7	1.16305	0.581523	+	0.813530i	$0.302457\pi$
$258$	0	0
$259$	1.26190e7	0.726314
$260$	0	0
$261$	0	0
$262$	3.33914e7i	1.85666i
$263$	3.48531e6	0.191591	0.0957954	−	0.995401i	$-0.469461\pi$
$263$	3.48531e6	0.191591	0.0957954	+	0.995401i	$0.469461\pi$
$264$	0	0
$265$	0	0
$266$	1.54731e7i	0.822114i
$267$	0	0
$268$	2.11766e7i	1.10015i
$269$	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	$0.237976\pi$
$272$	−5.45524e6	−0.271086
$273$	0	0
$274$	2.57499e6	0.125177
$275$	0	0
$276$	0	0
$277$	− 2.29938e7i	− 1.08186i	−0.841067	−	0.540931i	$-0.818072\pi$
$277$	− 2.29938e7i	− 1.08186i	0.841067	−	0.540931i	$-0.181928\pi$
$278$	2.57922e7	1.20048
$279$	0	0
$280$	0	0
$281$	1.25303e6i	0.0564730i	0.999601	+	0.0282365i	$0.00898916\pi$
$281$	1.25303e6i	0.0564730i	−0.999601	+	0.0282365i	$0.991011\pi$
$282$	0	0
$283$	− 1.45129e7i	− 0.640317i	−0.947364	−	0.320159i	$-0.896264\pi$
$283$	− 1.45129e7i	− 0.640317i	0.947364	−	0.320159i	$-0.103736\pi$
$284$	− 5.28372e6i	− 0.230666i
$285$	0	0
$286$	3.78950e6	0.161989
$287$	5.70303e7	2.41246
$288$	0	0
$289$	2.68472e7	1.11226
$290$	0	0
$291$	0	0
$292$	− 3.02012e7i	− 1.21304i
$293$	1.12729e7	0.448160	0.224080	−	0.974571i	$-0.428062\pi$
$293$	1.12729e7	0.448160	0.224080	+	0.974571i	$0.428062\pi$
$294$	0	0
$295$	0	0
$296$	− 1.04215e7i	− 0.401841i
$297$	0	0
$298$	− 5.41616e7i	− 2.04665i
$299$	1.97904e6i	0.0740356i
$300$	0	0
$301$	−4.76588e7	−1.74761
$302$	−4.76069e7	−1.72842
$303$	0	0
$304$	−1.77248e6	−0.0630900
$305$	0	0
$306$	0	0
$307$	− 4.51916e7i	− 1.56186i	−0.624618	−	0.780930i	$-0.714745\pi$
$307$	− 4.51916e7i	− 1.56186i	0.624618	−	0.780930i	$-0.285255\pi$
$308$	4.44451e7	1.52115
$309$	0	0
$310$	0	0
$311$	9.74134e6i	0.323845i	0.986803	+	0.161923i	$0.0517695\pi$
$311$	9.74134e6i	0.323845i	−0.986803	+	0.161923i	$0.948230\pi$
$312$	0	0
$313$	− 5.30265e6i	− 0.172926i	−0.996255	−	0.0864630i	$-0.972444\pi$
$313$	− 5.30265e6i	− 0.172926i	0.996255	−	0.0864630i	$-0.0275564\pi$
$314$	− 3.03959e7i	− 0.981808i
$315$	0	0
$316$	5.29322e7	1.67748
$317$	1.05462e7	0.331068	0.165534	−	0.986204i	$-0.447065\pi$
$317$	1.05462e7	0.331068	0.165534	+	0.986204i	$0.447065\pi$
$318$	0	0
$319$	−2.00381e7	−0.617283
$320$	0	0
$321$	0	0
$322$	3.83694e7i	1.14926i
$323$	1.65656e7	0.491587
$324$	0	0
$325$	0	0
$326$	8.58523e7i	2.47799i
$327$	0	0
$328$	− 4.70989e7i	− 1.33472i
$329$	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	$0.676174\pi$
$332$	9.13699e7	2.49683
$333$	0	0
$334$	−8.42206e7	−2.26037
$335$	0	0
$336$	0	0
$337$	1.22682e7i	0.320548i	0.987073	+	0.160274i	$0.0512377\pi$
$337$	1.22682e7i	0.320548i	−0.987073	+	0.160274i	$0.948762\pi$
$338$	−5.99291e7	−1.55198
$339$	0	0
$340$	0	0
$341$	− 9.14313e6i	− 0.230585i
$342$	0	0
$343$	2.05817e7i	0.510033i
$344$	3.93594e7i	0.966882i
$345$	0	0
$346$	7.10317e7	1.71484
$347$	−1.65809e7	−0.396843	−0.198422	−	0.980117i	$-0.563582\pi$
$347$	−1.65809e7	−0.396843	−0.198422	+	0.980117i	$0.563582\pi$
$348$	0	0
$349$	−4.81038e7	−1.13163	−0.565813	−	0.824534i	$-0.691438\pi$
$349$	−4.81038e7	−1.13163	−0.565813	+	0.824534i	$0.691438\pi$
$350$	0	0
$351$	0	0
$352$	3.23870e7i	0.742580i
$353$	1.47570e7	0.335485	0.167742	−	0.985831i	$-0.446352\pi$
$353$	1.47570e7	0.335485	0.167742	+	0.985831i	$0.446352\pi$
$354$	0	0
$355$	0	0
$356$	2.18945e7i	0.485272i
$357$	0	0
$358$	− 6.19980e7i	− 1.35123i
$359$	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
$362$	−1.07875e8	−2.27403
$363$	0	0
$364$	1.76651e7	0.366279
$365$	0	0
$366$	0	0
$367$	− 8.67940e6i	− 0.175587i	−0.996139	−	0.0877934i	$-0.972018\pi$
$367$	− 8.67940e6i	− 0.175587i	0.996139	−	0.0877934i	$-0.0279815\pi$
$368$	−4.39531e6	−0.0881954
$369$	0	0
$370$	0	0
$371$	1.03063e8i	2.01827i
$372$	0	0
$373$	− 7.94052e7i	− 1.53011i	−0.643965	−	0.765055i	$-0.722712\pi$
$373$	− 7.94052e7i	− 1.53011i	0.643965	−	0.765055i	$-0.277288\pi$
$374$	− 7.86583e7i	− 1.50359i
$375$	0	0
$376$	5.58071e7	1.04985
$377$	−7.96432e6	−0.148636
$378$	0	0
$379$	1.46346e7	0.268821	0.134410	−	0.990926i	$-0.457086\pi$
$379$	1.46346e7	0.268821	0.134410	+	0.990926i	$0.457086\pi$
$380$	0	0
$381$	0	0
$382$	1.23509e6i	0.0221568i
$383$	−9.16736e6	−0.163173	−0.0815865	−	0.996666i	$-0.525999\pi$
$383$	−9.16736e6	−0.163173	−0.0815865	+	0.996666i	$0.525999\pi$
$384$	0	0
$385$	0	0
$386$	− 9.53924e7i	− 1.65864i
$387$	0	0
$388$	3.64246e6i	0.0623591i
$389$	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
$392$	6.79101e7	1.12739
$393$	0	0
$394$	−1.25326e8	−2.04905
$395$	0	0
$396$	0	0
$397$	− 7.25544e7i	− 1.15956i	−0.814774	−	0.579779i	$-0.803139\pi$
$397$	− 7.25544e7i	− 1.15956i	0.814774	−	0.579779i	$-0.196861\pi$
$398$	4.50785e7	0.715023
$399$	0	0
$400$	0	0
$401$	7.37246e7i	1.14335i	0.820480	+	0.571675i	$0.193706\pi$
$401$	7.37246e7i	1.14335i	−0.820480	+	0.571675i	$0.806294\pi$
$402$	0	0
$403$	− 3.63402e6i	− 0.0555228i
$404$	− 5.48591e7i	− 0.831964i
$405$	0	0
$406$	−1.54411e8	−2.30728
$407$	−2.08429e7	−0.309155
$408$	0	0
$409$	−3.43558e7	−0.502146	−0.251073	−	0.967968i	$-0.580783\pi$
$409$	−3.43558e7	−0.502146	−0.251073	+	0.967968i	$0.580783\pi$
$410$	0	0
$411$	0	0
$412$	1.43098e8i	2.04617i
$413$	2.08620e7	0.296146
$414$	0	0
$415$	0	0
$416$	1.28725e7i	0.178806i
$417$	0	0
$418$	− 2.55571e7i	− 0.349932i
$419$	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	$0.494950\pi$
$422$	8.62060e7	1.14710
$423$	0	0
$424$	8.51151e7	1.11663
$425$	0	0
$426$	0	0
$427$	− 1.31596e8i	− 1.69029i
$428$	1.26450e8	1.61283
$429$	0	0
$430$	0	0
$431$	− 1.13049e8i	− 1.41200i	−0.708212	−	0.705999i	$-0.750498\pi$
$431$	− 1.13049e8i	− 1.41200i	0.708212	−	0.705999i	$-0.249502\pi$
$432$	0	0
$433$	4.50927e7i	0.555447i	0.960661	+	0.277723i	$0.0895799\pi$
$433$	4.50927e7i	0.555447i	−0.960661	+	0.277723i	$0.910420\pi$
$434$	− 7.04559e7i	− 0.861882i
$435$	0	0
$436$	1.40677e8	1.69732
$437$	1.33470e7	0.159934
$438$	0	0
$439$	1.61605e8	1.91013	0.955064	−	0.296399i	$-0.0957858\pi$
$439$	1.61605e8	1.91013	0.955064	+	0.296399i	$0.0957858\pi$
$440$	0	0
$441$	0	0
$442$	− 3.12634e7i	− 0.362051i
$443$	2.54011e7	0.292174	0.146087	−	0.989272i	$-0.453332\pi$
$443$	2.54011e7	0.292174	0.146087	+	0.989272i	$0.453332\pi$
$444$	0	0
$445$	0	0
$446$	4.25349e7i	0.479448i
$447$	0	0
$448$	2.23949e8i	2.49067i
$449$	− 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
$452$	−1.82697e7	−0.197841
$453$	0	0
$454$	2.06209e8	2.20363
$455$	0	0
$456$	0	0
$457$	− 4.68452e7i	− 0.490813i	−0.969420	−	0.245407i	$-0.921078\pi$
$457$	− 4.68452e7i	− 0.490813i	0.969420	−	0.245407i	$-0.0789215\pi$
$458$	−2.11730e8	−2.20387
$459$	0	0
$460$	0	0
$461$	− 2.33797e7i	− 0.238636i	−0.992856	−	0.119318i	$-0.961929\pi$
$461$	− 2.33797e7i	− 0.238636i	0.992856	−	0.119318i	$-0.0380708\pi$
$462$	0	0
$463$	− 4.98269e7i	− 0.502019i	−0.967985	−	0.251010i	$-0.919237\pi$
$463$	− 4.98269e7i	− 0.502019i	0.967985	−	0.251010i	$-0.0807626\pi$
$464$	− 1.76882e7i	− 0.177064i
$465$	0	0
$466$	1.12718e8	1.11388
$467$	9.52369e7	0.935092	0.467546	−	0.883969i	$-0.345138\pi$
$467$	9.52369e7	0.935092	0.467546	+	0.883969i	$0.345138\pi$
$468$	0	0
$469$	1.13230e8	1.09760
$470$	0	0
$471$	0	0
$472$	− 1.72290e7i	− 0.163846i
$473$	7.87188e7	0.743867
$474$	0	0
$475$	0	0
$476$	− 3.66672e8i	− 3.39983i
$477$	0	0
$478$	1.43820e8i	1.31685i
$479$	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
$482$	−1.91033e8	−1.70595
$483$	0	0
$484$	1.00202e8	0.883775
$485$	0	0
$486$	0	0
$487$	− 1.92602e6i	− 0.0166753i	−0.999965	−	0.00833765i	$-0.997346\pi$
$487$	− 1.92602e6i	− 0.0166753i	0.999965	−	0.00833765i	$-0.00265399\pi$
$488$	−1.08680e8	−0.935167
$489$	0	0
$490$	0	0
$491$	1.79722e8i	1.51830i	0.650916	+	0.759150i	$0.274385\pi$
$491$	1.79722e8i	1.51830i	−0.650916	+	0.759150i	$0.725615\pi$
$492$	0	0
$493$	1.65314e8i	1.37965i
$494$	− 1.01579e7i	− 0.0842603i
$495$	0	0
$496$	8.07090e6	0.0661419
$497$	−2.82517e7	−0.230131
$498$	0	0
$499$	−1.54018e8	−1.23956	−0.619782	−	0.784774i	$-0.712779\pi$
$499$	−1.54018e8	−1.23956	−0.619782	+	0.784774i	$0.712779\pi$
$500$	0	0
$501$	0	0
$502$	3.87025e8i	3.05934i
$503$	2.25142e7	0.176910	0.0884551	−	0.996080i	$-0.471807\pi$
$503$	2.25142e7	0.176910	0.0884551	+	0.996080i	$0.471807\pi$
$504$	0	0
$505$	0	0
$506$	− 6.33753e7i	− 0.489180i
$507$	0	0
$508$	1.24519e7i	0.0949825i
$509$	− 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
$512$	−4.97487e7	−0.370656
$513$	0	0
$514$	−2.51278e8	−1.85040
$515$	0	0
$516$	0	0
$517$	− 1.11614e8i	− 0.807695i
$518$	−1.60613e8	−1.15556
$519$	0	0
$520$	0	0
$521$	− 8.35166e7i	− 0.590554i	−0.955412	−	0.295277i	$-0.904588\pi$
$521$	− 8.35166e7i	− 0.590554i	0.955412	−	0.295277i	$-0.0954120\pi$
$522$	0	0
$523$	2.08856e8i	1.45997i	0.683466	+	0.729983i	$0.260472\pi$
$523$	2.08856e8i	1.45997i	−0.683466	+	0.729983i	$0.739528\pi$
$524$	− 2.57101e8i	− 1.78694i
$525$	0	0
$526$	−4.43608e7	−0.304819
$527$	−7.54308e7	−0.515367
$528$	0	0
$529$	−1.14939e8	−0.776424
$530$	0	0
$531$	0	0
$532$	− 1.19137e8i	− 0.791244i
$533$	−3.74397e7	−0.247258
$534$	0	0
$535$	0	0
$536$	− 9.35119e7i	− 0.607257i
$537$	0	0
$538$	− 7.19625e7i	− 0.462125i
$539$	− 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	$0.625062\pi$
$542$	−3.71519e8	−2.33337
$543$	0	0
$544$	2.67193e8	1.65970
$545$	0	0
$546$	0	0
$547$	1.33857e8i	0.817861i	0.912565	+	0.408931i	$0.134098\pi$
$547$	1.33857e8i	0.817861i	−0.912565	+	0.408931i	$0.865902\pi$
$548$	−1.98264e7	−0.120477
$549$	0	0
$550$	0	0
$551$	5.37128e7i	0.321087i
$552$	0	0
$553$	− 2.83025e8i	− 1.67359i
$554$	2.92664e8i	1.72123i
$555$	0	0
$556$	−1.98590e8	−1.15540
$557$	8.20694e7	0.474915	0.237457	−	0.971398i	$-0.423686\pi$
$557$	8.20694e7	0.474915	0.237457	+	0.971398i	$0.423686\pi$
$558$	0	0
$559$	3.12875e7	0.179116
$560$	0	0
$561$	0	0
$562$	− 1.59484e7i	− 0.0898480i
$563$	−2.21977e8	−1.24389	−0.621947	−	0.783059i	$-0.713658\pi$
$563$	−2.21977e8	−1.24389	−0.621947	+	0.783059i	$0.713658\pi$
$564$	0	0
$565$	0	0
$566$	1.84719e8i	1.01874i
$567$	0	0
$568$	2.33319e7i	0.127322i
$569$	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	$0.336230\pi$
$572$	−2.91777e7	−0.155906
$573$	0	0
$574$	−7.25877e8	−3.83820
$575$	0	0
$576$	0	0
$577$	2.07783e8i	1.08164i	0.841139	+	0.540820i	$0.181886\pi$
$577$	2.07783e8i	1.08164i	−0.841139	+	0.540820i	$0.818114\pi$
$578$	−3.41709e8	−1.76959
$579$	0	0
$580$	0	0
$581$	− 4.88549e8i	− 2.49104i
$582$	0	0
$583$	− 1.70230e8i	− 0.859075i
$584$	1.33363e8i	0.669571i
$585$	0	0
$586$	−1.43481e8	−0.713018
$587$	3.28908e8	1.62615	0.813073	−	0.582161i	$-0.197793\pi$
$587$	3.28908e8	1.62615	0.813073	+	0.582161i	$0.197793\pi$
$588$	0	0
$589$	−2.45085e7	−0.119942
$590$	0	0
$591$	0	0
$592$	− 1.83986e7i	− 0.0886790i
$593$	−1.97249e7	−0.0945914	−0.0472957	−	0.998881i	$-0.515060\pi$
$593$	−1.97249e7	−0.0945914	−0.0472957	+	0.998881i	$0.515060\pi$
$594$	0	0
$595$	0	0
$596$	4.17023e8i	1.96980i
$597$	0	0
$598$	− 2.51891e7i	− 0.117790i
$599$	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	$0.385282\pi$
$602$	6.06598e8	2.78043
$603$	0	0
$604$	3.66554e8	1.66352
$605$	0	0
$606$	0	0
$607$	− 1.08279e8i	− 0.484147i	−0.970258	−	0.242074i	$-0.922172\pi$
$607$	− 1.08279e8i	− 0.484147i	0.970258	−	0.242074i	$-0.0778276\pi$
$608$	8.68146e7	0.386262
$609$	0	0
$610$	0	0
$611$	− 4.43620e7i	− 0.194486i
$612$	0	0
$613$	− 2.60288e8i	− 1.12999i	−0.825096	−	0.564993i	$-0.808879\pi$
$613$	− 2.60288e8i	− 1.12999i	0.825096	−	0.564993i	$-0.191121\pi$
$614$	5.75195e8i	2.48491i
$615$	0	0
$616$	−1.96261e8	−0.839638
$617$	−1.65376e8	−0.704073	−0.352036	−	0.935986i	$-0.614511\pi$
$617$	−1.65376e8	−0.704073	−0.352036	+	0.935986i	$0.614511\pi$
$618$	0	0
$619$	1.36836e8	0.576935	0.288468	−	0.957490i	$-0.406854\pi$
$619$	1.36836e8	0.576935	0.288468	+	0.957490i	$0.406854\pi$
$620$	0	0
$621$	0	0
$622$	− 1.23987e8i	− 0.515235i
$623$	1.17069e8	0.484146
$624$	0	0
$625$	0	0
$626$	6.74918e7i	0.275124i
$627$	0	0
$628$	2.34037e8i	0.944942i
$629$	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	$0.321723\pi$
$632$	−2.33738e8	−0.925931
$633$	0	0
$634$	−1.34231e8	−0.526727
$635$	0	0
$636$	0	0
$637$	− 5.39829e7i	− 0.208852i
$638$	2.55043e8	0.982092
$639$	0	0
$640$	0	0
$641$	3.69866e8i	1.40433i	0.712013	+	0.702167i	$0.247784\pi$
$641$	3.69866e8i	1.40433i	−0.712013	+	0.702167i	$0.752216\pi$
$642$	0	0
$643$	− 9.29168e7i	− 0.349511i	−0.984612	−	0.174756i	$-0.944086\pi$
$643$	− 9.29168e7i	− 0.349511i	0.984612	−	0.174756i	$-0.0559136\pi$
$644$	− 2.95429e8i	− 1.10610i
$645$	0	0
$646$	−2.10846e8	−0.782111
$647$	−9.21336e7	−0.340177	−0.170089	−	0.985429i	$-0.554405\pi$
$647$	−9.21336e7	−0.340177	−0.170089	+	0.985429i	$0.554405\pi$
$648$	0	0
$649$	−3.44580e7	−0.126054
$650$	0	0
$651$	0	0
$652$	− 6.61029e8i	− 2.38494i
$653$	−2.20689e8	−0.792576	−0.396288	−	0.918126i	$-0.629702\pi$
$653$	−2.20689e8	−0.792576	−0.396288	+	0.918126i	$0.629702\pi$
$654$	0	0
$655$	0	0
$656$	− 8.31511e7i	− 0.294548i
$657$	0	0
$658$	− 8.60085e8i	− 3.01900i
$659$	− 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	$0.659937\pi$
$662$	4.85242e8	1.67257
$663$	0	0
$664$	−4.03472e8	−1.37819
$665$	0	0
$666$	0	0
$667$	1.33194e8i	0.448858i
$668$	6.48465e8	2.17549
$669$	0	0
$670$	0	0
$671$	2.17360e8i	0.719468i
$672$	0	0
$673$	5.34850e8i	1.75464i	0.479910	+	0.877318i	$0.340669\pi$
$673$	5.34850e8i	1.75464i	−0.479910	+	0.877318i	$0.659331\pi$
$674$	− 1.56149e8i	− 0.509988i
$675$	0	0
$676$	4.61430e8	1.49371
$677$	−1.41358e6	−0.00455568	−0.00227784	−	0.999997i	$-0.500725\pi$
$677$	−1.41358e6	−0.00455568	−0.00227784	+	0.999997i	$0.500725\pi$
$678$	0	0
$679$	1.94760e7	0.0622144
$680$	0	0
$681$	0	0
$682$	1.16373e8i	0.366859i
$683$	2.41616e8	0.758340	0.379170	−	0.925327i	$-0.376209\pi$
$683$	2.41616e8	0.758340	0.379170	+	0.925327i	$0.376209\pi$
$684$	0	0
$685$	0	0
$686$	− 2.61962e8i	− 0.811458i
$687$	0	0
$688$	6.94873e7i	0.213373i
$689$	− 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	$0.202701\pi$
$692$	−5.46916e8	−1.65045
$693$	0	0
$694$	2.11040e8	0.631374
$695$	0	0
$696$	0	0
$697$	7.77132e8i	2.29507i
$698$	6.12261e8	1.80041
$699$	0	0
$700$	0	0
$701$	5.49256e7i	0.159449i	0.996817	+	0.0797244i	$0.0254040\pi$
$701$	5.49256e7i	0.159449i	−0.996817	+	0.0797244i	$0.974596\pi$
$702$	0	0
$703$	5.58702e7i	0.160811i
$704$	− 3.69900e8i	− 1.06015i
$705$	0	0
$706$	−1.87826e8	−0.533753
$707$	−2.93328e8	−0.830034
$708$	0	0
$709$	−3.16706e7	−0.0888622	−0.0444311	−	0.999012i	$-0.514148\pi$
$709$	−3.16706e7	−0.0888622	−0.0444311	+	0.999012i	$0.514148\pi$
$710$	0	0
$711$	0	0
$712$	− 9.66819e7i	− 0.267858i
$713$	−6.07749e7	−0.167670
$714$	0	0
$715$	0	0
$716$	4.77361e8i	1.30049i
$717$	0	0
$718$	− 2.27205e8i	− 0.613825i
$719$	− 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
$722$	5.30290e8	1.40897
$723$	0	0
$724$	8.30595e8	2.18864
$725$	0	0
$726$	0	0
$727$	4.64180e8i	1.20805i	0.796967	+	0.604023i	$0.206436\pi$
$727$	4.64180e8i	1.20805i	−0.796967	+	0.604023i	$0.793564\pi$
$728$	−7.80057e7	−0.202177
$729$	0	0
$730$	0	0
$731$	− 6.49430e8i	− 1.66257i
$732$	0	0
$733$	7.04886e8i	1.78981i	0.446256	+	0.894905i	$0.352757\pi$
$733$	7.04886e8i	1.78981i	−0.446256	+	0.894905i	$0.647243\pi$
$734$	1.10471e8i	0.279357i
$735$	0	0
$736$	2.15279e8	0.539967
$737$	−1.87024e8	−0.467191
$738$	0	0
$739$	2.83900e8	0.703448	0.351724	−	0.936104i	$-0.385596\pi$
$739$	2.83900e8	0.703448	0.351724	+	0.936104i	$0.385596\pi$
$740$	0	0
$741$	0	0
$742$	− 1.31177e9i	− 3.21105i
$743$	5.01018e8	1.22148	0.610741	−	0.791830i	$-0.290872\pi$
$743$	5.01018e8	1.22148	0.610741	+	0.791830i	$0.290872\pi$
$744$	0	0
$745$	0	0
$746$	1.01066e9i	2.43439i
$747$	0	0
$748$	6.05638e8i	1.44713i
$749$	− 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	$0.555993\pi$
$752$	9.85249e7	0.231682
$753$	0	0
$754$	1.01369e8	0.236479
$755$	0	0
$756$	0	0
$757$	− 2.14422e8i	− 0.494291i	−0.968978	−	0.247145i	$-0.920507\pi$
$757$	− 2.14422e8i	− 0.494291i	0.968978	−	0.247145i	$-0.0794925\pi$
$758$	−1.86268e8	−0.427691
$759$	0	0
$760$	0	0
$761$	− 2.69760e7i	− 0.0612101i	−0.999532	−	0.0306051i	$-0.990257\pi$
$761$	− 2.69760e7i	− 0.0612101i	0.999532	−	0.0306051i	$-0.00974341\pi$
$762$	0	0
$763$	− 7.52192e8i	− 1.69338i
$764$	− 9.50969e6i	− 0.0213249i
$765$	0	0
$766$	1.16681e8	0.259607
$767$	−1.36957e7	−0.0303526
$768$	0	0
$769$	4.90064e8	1.07764	0.538821	−	0.842421i	$-0.318870\pi$
$769$	4.90064e8	1.07764	0.538821	+	0.842421i	$0.318870\pi$
$770$	0	0
$771$	0	0
$772$	7.34484e8i	1.59636i
$773$	−4.17032e8	−0.902882	−0.451441	−	0.892301i	$-0.649090\pi$
$773$	−4.17032e8	−0.902882	−0.451441	+	0.892301i	$0.649090\pi$
$774$	0	0
$775$	0	0
$776$	− 1.60844e7i	− 0.0344208i
$777$	0	0
$778$	− 1.17946e8i	− 0.250463i
$779$	2.52501e8i	0.534134i
$780$	0	0
$781$	4.66638e7	0.0979550
$782$	−5.22846e8	−1.09334
$783$	0	0
$784$	1.19892e8	0.248796
$785$	0	0
$786$	0	0
$787$	3.46111e8i	0.710054i	0.934856	+	0.355027i	$0.115528\pi$
$787$	3.46111e8i	0.710054i	−0.934856	+	0.355027i	$0.884472\pi$
$788$	9.64963e8	1.97211
$789$	0	0
$790$	0	0
$791$	9.76872e7i	0.197382i
$792$	0	0
$793$	8.63915e7i	0.173241i
$794$	9.23467e8i	1.84484i
$795$	0	0
$796$	−3.47087e8	−0.688175
$797$	7.25306e8	1.43267	0.716335	−	0.697756i	$-0.245818\pi$
$797$	7.25306e8	1.43267	0.716335	+	0.697756i	$0.245818\pi$
$798$	0	0
$799$	−9.20816e8	−1.80523
$800$	0	0
$801$	0	0
$802$	− 9.38361e8i	− 1.81906i
$803$	2.66726e8	0.515132
$804$	0	0
$805$	0	0
$806$	4.62535e7i	0.0883363i
$807$	0	0
$808$	2.42247e8i	0.459224i
$809$	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	$0.515962\pi$
$812$	1.18891e9	2.22065
$813$	0	0
$814$	2.65287e8	0.491862
$815$	0	0
$816$	0	0
$817$	− 2.11009e8i	− 0.386931i
$818$	4.37278e8	0.798909
$819$	0	0
$820$	0	0
$821$	− 1.07674e9i	− 1.94573i	−0.231377	−	0.972864i	$-0.574323\pi$
$821$	− 1.07674e9i	− 1.94573i	0.231377	−	0.972864i	$-0.425677\pi$
$822$	0	0
$823$	8.88441e7i	0.159378i	0.996820	+	0.0796891i	$0.0253928\pi$
$823$	8.88441e7i	0.159378i	−0.996820	+	0.0796891i	$0.974607\pi$
$824$	− 6.31892e8i	− 1.12943i
$825$	0	0
$826$	−2.65530e8	−0.471165
$827$	−9.82047e7	−0.173626	−0.0868132	−	0.996225i	$-0.527668\pi$
$827$	−9.82047e7	−0.173626	−0.0868132	+	0.996225i	$0.527668\pi$
$828$	0	0
$829$	−2.25045e8	−0.395008	−0.197504	−	0.980302i	$-0.563283\pi$
$829$	−2.25045e8	−0.395008	−0.197504	+	0.980302i	$0.563283\pi$
$830$	0	0
$831$	0	0
$832$	− 1.47020e8i	− 0.255274i
$833$	−1.12052e9	−1.93858
$834$	0	0
$835$	0	0
$836$	1.96780e8i	0.336792i
$837$	0	0
$838$	− 1.67177e9i	− 2.84083i
$839$	− 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
$842$	−3.01341e7	−0.0504803
$843$	0	0
$844$	−6.63752e8	−1.10402
$845$	0	0
$846$	0	0
$847$	− 5.35776e8i	− 0.881724i
$848$	1.50267e8	0.246420
$849$	0	0
$850$	0	0
$851$	1.38544e8i	0.224802i
$852$	0	0
$853$	− 7.97906e7i	− 0.128560i	−0.997932	−	0.0642798i	$-0.979525\pi$
$853$	− 7.97906e7i	− 0.128560i	0.997932	−	0.0642798i	$-0.0204750\pi$
$854$	1.67495e9i	2.68923i
$855$	0	0
$856$	−5.58379e8	−0.890241
$857$	−1.06107e9	−1.68579	−0.842894	−	0.538080i	$-0.819150\pi$
$857$	−1.06107e9	−1.68579	−0.842894	+	0.538080i	$0.819150\pi$
$858$	0	0
$859$	−3.34286e8	−0.527399	−0.263699	−	0.964605i	$-0.584943\pi$
$859$	−3.34286e8	−0.527399	−0.263699	+	0.964605i	$0.584943\pi$
$860$	0	0
$861$	0	0
$862$	1.43888e9i	2.24648i
$863$	−2.65292e7	−0.0412754	−0.0206377	−	0.999787i	$-0.506570\pi$
$863$	−2.65292e7	−0.0412754	−0.0206377	+	0.999787i	$0.506570\pi$
$864$	0	0
$865$	0	0
$866$	− 5.73936e8i	− 0.883711i
$867$	0	0
$868$	5.42483e8i	0.829519i
$869$	4.67477e8i	0.712362i
$870$	0	0
$871$	−7.43343e7	−0.112495
$872$	−6.21203e8	−0.936880
$873$	0	0
$874$	−1.69880e8	−0.254453
$875$	0	0
$876$	0	0
$877$	1.30791e8i	0.193901i	0.995289	+	0.0969504i	$0.0309088\pi$
$877$	1.30791e8i	0.193901i	−0.995289	+	0.0969504i	$0.969091\pi$
$878$	−2.05690e9	−3.03900
$879$	0	0
$880$	0	0
$881$	9.18953e8i	1.34390i	0.740598	+	0.671948i	$0.234542\pi$
$881$	9.18953e8i	1.34390i	−0.740598	+	0.671948i	$0.765458\pi$
$882$	0	0
$883$	− 1.09112e9i	− 1.58486i	−0.609961	−	0.792432i	$-0.708815\pi$
$883$	− 1.09112e9i	− 1.58486i	0.609961	−	0.792432i	$-0.291185\pi$
$884$	2.40716e8i	0.348456i
$885$	0	0
$886$	−3.23303e8	−0.464846
$887$	5.36427e8	0.768670	0.384335	−	0.923194i	$-0.374431\pi$
$887$	5.36427e8	0.768670	0.384335	+	0.923194i	$0.374431\pi$
$888$	0	0
$889$	6.65794e7	0.0947621
$890$	0	0
$891$	0	0
$892$	− 3.27502e8i	− 0.461445i
$893$	−2.99186e8	−0.420133
$894$	0	0
$895$	0	0
$896$	− 1.59549e9i	− 2.21804i
$897$	0	0
$898$	1.20139e9i	1.65903i
$899$	− 2.44579e8i	− 0.336620i
$900$	0	0
$901$	−1.40440e9	−1.92007
$902$	1.19894e9	1.63372
$903$	0	0
$904$	8.06757e7	0.109204
$905$	0	0
$906$	0	0
$907$	4.60985e8i	0.617824i	0.951091	+	0.308912i	$0.0999649\pi$
$907$	4.60985e8i	0.617824i	−0.951091	+	0.308912i	$0.900035\pi$
$908$	−1.58772e9	−2.12089
$909$	0	0
$910$	0	0
$911$	8.68400e8i	1.14859i	0.818649	+	0.574295i	$0.194724\pi$
$911$	8.68400e8i	1.14859i	−0.818649	+	0.574295i	$0.805276\pi$
$912$	0	0
$913$	8.06944e8i	1.06031i
$914$	5.96242e8i	0.780879i
$915$	0	0
$916$	1.63024e9	2.12112
$917$	−1.37470e9	−1.78279
$918$	0	0
$919$	−1.28061e9	−1.64995	−0.824976	−	0.565168i	$-0.808811\pi$
$919$	−1.28061e9	−1.64995	−0.824976	+	0.565168i	$0.808811\pi$
$920$	0	0
$921$	0	0
$922$	2.97575e8i	0.379668i
$923$	1.85469e7	0.0235867
$924$	0	0
$925$	0	0
$926$	6.34192e8i	0.798708i
$927$	0	0
$928$	8.66353e8i	1.08405i
$929$	− 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
$932$	−8.67888e8	−1.07205
$933$	0	0
$934$	−1.21217e9	−1.48772
$935$	0	0
$936$	0	0
$937$	1.33188e9i	1.61900i	0.587123	+	0.809498i	$0.300261\pi$
$937$	1.33188e9i	1.61900i	−0.587123	+	0.809498i	$0.699739\pi$
$938$	−1.44118e9	−1.74627
$939$	0	0
$940$	0	0
$941$	− 8.24869e8i	− 0.989957i	−0.868905	−	0.494978i	$-0.835176\pi$
$941$	− 8.24869e8i	− 0.989957i	0.868905	−	0.494978i	$-0.164824\pi$
$942$	0	0
$943$	6.26138e8i	0.746681i
$944$	− 3.04171e7i	− 0.0361578i
$945$	0	0
$946$	−1.00193e9	−1.18349
$947$	−7.33660e8	−0.863863	−0.431931	−	0.901906i	$-0.642168\pi$
$947$	−7.33660e8	−0.863863	−0.431931	+	0.901906i	$0.642168\pi$
$948$	0	0
$949$	1.06013e8	0.124039
$950$	0	0
$951$	0	0
$952$	1.61915e9i	1.87662i
$953$	6.87744e8	0.794600	0.397300	−	0.917689i	$-0.369947\pi$
$953$	6.87744e8	0.794600	0.397300	+	0.917689i	$0.369947\pi$
$954$	0	0
$955$	0	0
$956$	− 1.10736e9i	− 1.26740i
$957$	0	0
$958$	− 2.50919e9i	− 2.85389i
$959$	1.06011e8i	0.120197i
$960$	0	0
$961$	−7.75906e8	−0.874256
$962$	1.05441e8	0.118436
$963$	0	0
$964$	1.47088e9	1.64190
$965$	0	0
$966$	0	0
$967$	− 1.43001e9i	− 1.58147i	−0.612162	−	0.790733i	$-0.709700\pi$
$967$	− 1.43001e9i	− 1.58147i	0.612162	−	0.790733i	$-0.290300\pi$
$968$	−4.42475e8	−0.487823
$969$	0	0
$970$	0	0
$971$	1.05628e9i	1.15377i	0.816824	+	0.576887i	$0.195733\pi$
$971$	1.05628e9i	1.15377i	−0.816824	+	0.576887i	$0.804267\pi$
$972$	0	0
$973$	1.06185e9i	1.15272i
$974$	2.45142e7i	0.0265303i
$975$	0	0
$976$	−1.91869e8	−0.206375
$977$	−8.72371e8	−0.935443	−0.467722	−	0.883876i	$-0.654925\pi$
$977$	−8.72371e8	−0.935443	−0.467722	+	0.883876i	$0.654925\pi$
$978$	0	0
$979$	−1.93364e8	−0.206076
$980$	0	0
$981$	0	0
$982$	− 2.28749e9i	− 2.41560i
$983$	−7.90860e8	−0.832605	−0.416302	−	0.909226i	$-0.636674\pi$
$983$	−7.90860e8	−0.832605	−0.416302	+	0.909226i	$0.636674\pi$
$984$	0	0
$985$	0	0
$986$	− 2.10411e9i	− 2.19501i
$987$	0	0
$988$	7.82118e7i	0.0810964i
$989$	− 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	$0.477283\pi$
$992$	−3.95306e8	−0.404947
$993$	0	0
$994$	3.59586e8	0.366137
$995$	0	0
$996$	0	0
$997$	3.29036e8i	0.332015i	0.986124	+	0.166008i	$0.0530876\pi$
$997$	3.29036e8i	0.332015i	−0.986124	+	0.166008i	$0.946912\pi$
$998$	1.96032e9	1.97213
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.7.d.a.224.1		4
3.2	odd	2	inner	225.7.d.a.224.3		4
5.2	odd	4		9.7.b.a.8.1	✓	2
5.3	odd	4		225.7.c.a.26.2		2
5.4	even	2	inner	225.7.d.a.224.4		4
15.2	even	4		9.7.b.a.8.2	yes	2
15.8	even	4		225.7.c.a.26.1		2
15.14	odd	2	inner	225.7.d.a.224.2		4
20.7	even	4		144.7.e.a.17.1		2
35.27	even	4		441.7.b.a.197.1		2
40.27	even	4		576.7.e.a.449.2		2
40.37	odd	4		576.7.e.l.449.2		2
45.2	even	12		81.7.d.d.53.2		4
45.7	odd	12		81.7.d.d.53.1		4
45.22	odd	12		81.7.d.d.26.2		4
45.32	even	12		81.7.d.d.26.1		4
60.47	odd	4		144.7.e.a.17.2		2
105.62	odd	4		441.7.b.a.197.2		2
120.77	even	4		576.7.e.l.449.1		2
120.107	odd	4		576.7.e.a.449.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9.7.b.a.8.1	✓	2	5.2	odd	4
9.7.b.a.8.2	yes	2	15.2	even	4
81.7.d.d.26.1		4	45.32	even	12
81.7.d.d.26.2		4	45.22	odd	12
81.7.d.d.53.1		4	45.7	odd	12
81.7.d.d.53.2		4	45.2	even	12
144.7.e.a.17.1		2	20.7	even	4
144.7.e.a.17.2		2	60.47	odd	4
225.7.c.a.26.1		2	15.8	even	4
225.7.c.a.26.2		2	5.3	odd	4
225.7.d.a.224.1		4	1.1	even	1	trivial
225.7.d.a.224.2		4	15.14	odd	2	inner
225.7.d.a.224.3		4	3.2	odd	2	inner
225.7.d.a.224.4		4	5.4	even	2	inner
441.7.b.a.197.1		2	35.27	even	4
441.7.b.a.197.2		2	105.62	odd	4
576.7.e.a.449.1		2	120.107	odd	4
576.7.e.a.449.2		2	40.27	even	4
576.7.e.l.449.1		2	120.77	even	4
576.7.e.l.449.2		2	40.37	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:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	225.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-12.7279 q^{2} +98.0000 q^{4} -524.000i q^{7} -432.749 q^{8} +O(q^{10})\)	\(q-12.7279 q^{2} +98.0000 q^{4} -524.000i q^{7} -432.749 q^{8} +865.499i q^{11} +344.000i q^{13} +6669.43i q^{14} -764.000 q^{16} +7140.36 q^{17} +2320.00 q^{19} -11016.0i q^{22} +5753.02 q^{23} -4378.41i q^{26} -51352.0i q^{28} +23152.1i q^{29} -10564.0 q^{31} +37420.1 q^{32} -90882.0 q^{34} +24082.0i q^{37} -29528.8 q^{38} +108836. i q^{41} -90952.0i q^{43} +84818.9i q^{44} -73224.0 q^{46} -128959. q^{47} -156927. q^{49} +33712.0i q^{52} -196685. q^{53} +226761. i q^{56} -294678. i q^{58} +39812.9i q^{59} +251138. q^{61} +134458. q^{62} -427384. q^{64} +216088. i q^{67} +699756. q^{68} -53915.5i q^{71} -308176. i q^{73} -306514. i q^{74} +227360. q^{76} +453521. q^{77} +540124. q^{79} -1.38526e6i q^{82} +932346. q^{83} +1.15763e6i q^{86} -374544. i q^{88} +223413. i q^{89} +180256. q^{91} +563796. q^{92} +1.64138e6 q^{94} +37168.0i q^{97} +1.99735e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 392 q^{4}+O(q^{10}) \) 4 * q + 392 * q^4	\( 4 q + 392 q^{4} - 3056 q^{16} + 9280 q^{19} - 42256 q^{31} - 363528 q^{34} - 292896 q^{46} - 627708 q^{49} + 1004552 q^{61} - 1709536 q^{64} + 909440 q^{76} + 2160496 q^{79} + 721024 q^{91} + 6565536 q^{94}+O(q^{100}) \) 4 * q + 392 * q^4 - 3056 * q^16 + 9280 * q^19 - 42256 * q^31 - 363528 * q^34 - 292896 * q^46 - 627708 * q^49 + 1004552 * q^61 - 1709536 * q^64 + 909440 * q^76 + 2160496 * q^79 + 721024 * q^91 + 6565536 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more