LMFDB - Embedded newform 450.6.c.n.199.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(199,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("450.199");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.c (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:	$72.1727189158$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	450.199
Dual form			450.6.c.n.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -16.0000 q^{4} +233.000i q^{7} -64.0000i q^{8} +O(q^{10})$	$q+4.00000i q^{2} -16.0000 q^{4} +233.000i q^{7} -64.0000i q^{8} +498.000 q^{11} -809.000i q^{13} -932.000 q^{14} +256.000 q^{16} +1002.00i q^{17} +1705.00 q^{19} +1992.00i q^{22} +1554.00i q^{23} +3236.00 q^{26} -3728.00i q^{28} +7830.00 q^{29} +977.000 q^{31} +1024.00i q^{32} -4008.00 q^{34} -4822.00i q^{37} +6820.00i q^{38} +8148.00 q^{41} -19469.0i q^{43} -7968.00 q^{44} -6216.00 q^{46} -8418.00i q^{47} -37482.0 q^{49} +12944.0i q^{52} +17664.0i q^{53} +14912.0 q^{56} +31320.0i q^{58} +35910.0 q^{59} +3527.00 q^{61} +3908.00i q^{62} -4096.00 q^{64} +57473.0i q^{67} -16032.0i q^{68} +7548.00 q^{71} +646.000i q^{73} +19288.0 q^{74} -27280.0 q^{76} +116034. i q^{77} +22720.0 q^{79} +32592.0i q^{82} +11574.0i q^{83} +77876.0 q^{86} -31872.0i q^{88} -78960.0 q^{89} +188497. q^{91} -24864.0i q^{92} +33672.0 q^{94} +54593.0i q^{97} -149928. i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4}+O(q^{10}) $ 2 * q - 32 * q^4	$ 2 q - 32 q^{4} + 996 q^{11} - 1864 q^{14} + 512 q^{16} + 3410 q^{19} + 6472 q^{26} + 15660 q^{29} + 1954 q^{31} - 8016 q^{34} + 16296 q^{41} - 15936 q^{44} - 12432 q^{46} - 74964 q^{49} + 29824 q^{56} + 71820 q^{59} + 7054 q^{61} - 8192 q^{64} + 15096 q^{71} + 38576 q^{74} - 54560 q^{76} + 45440 q^{79} + 155752 q^{86} - 157920 q^{89} + 376994 q^{91} + 67344 q^{94}+O(q^{100}) $ 2 * q - 32 * q^4 + 996 * q^11 - 1864 * q^14 + 512 * q^16 + 3410 * q^19 + 6472 * q^26 + 15660 * q^29 + 1954 * q^31 - 8016 * q^34 + 16296 * q^41 - 15936 * q^44 - 12432 * q^46 - 74964 * q^49 + 29824 * q^56 + 71820 * q^59 + 7054 * q^61 - 8192 * q^64 + 15096 * q^71 + 38576 * q^74 - 54560 * q^76 + 45440 * q^79 + 155752 * q^86 - 157920 * q^89 + 376994 * q^91 + 67344 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	233.000i	1.79726i	0.438708	+	0.898630i	$0.355436\pi$
$7$	233.000i	1.79726i	−0.438708	+	0.898630i	$0.644564\pi$
$8$	− 64.0000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	498.000	1.24093	0.620465	−	0.784234i	$-0.286944\pi$
$11$	498.000	1.24093	0.620465	+	0.784234i	$0.286944\pi$
$12$	0	0
$13$	− 809.000i	− 1.32767i	−0.747879	−	0.663835i	$-0.768928\pi$
$13$	− 809.000i	− 1.32767i	0.747879	−	0.663835i	$-0.231072\pi$
$14$	−932.000	−1.27085
$15$	0	0
$16$	256.000	0.250000
$17$	1002.00i	0.840902i	0.907315	+	0.420451i	$0.138128\pi$
$17$	1002.00i	0.840902i	−0.907315	+	0.420451i	$0.861872\pi$
$18$	0	0
$19$	1705.00	1.08353	0.541764	−	0.840530i	$-0.317757\pi$
$19$	1705.00	1.08353	0.541764	+	0.840530i	$0.317757\pi$
$20$	0	0
$21$	0	0
$22$	1992.00i	0.877471i
$23$	1554.00i	0.612536i	0.951945	+	0.306268i	$0.0990803\pi$
$23$	1554.00i	0.612536i	−0.951945	+	0.306268i	$0.900920\pi$
$24$	0	0
$25$	0	0
$26$	3236.00	0.938804
$27$	0	0
$28$	− 3728.00i	− 0.898630i
$29$	7830.00	1.72889	0.864444	−	0.502729i	$-0.167671\pi$
$29$	7830.00	1.72889	0.864444	+	0.502729i	$0.167671\pi$
$30$	0	0
$31$	977.000	0.182596	0.0912978	−	0.995824i	$-0.470898\pi$
$31$	977.000	0.182596	0.0912978	+	0.995824i	$0.470898\pi$
$32$	1024.00i	0.176777i
$33$	0	0
$34$	−4008.00	−0.594608
$35$	0	0
$36$	0	0
$37$	− 4822.00i	− 0.579059i	−0.957169	−	0.289530i	$-0.906501\pi$
$37$	− 4822.00i	− 0.579059i	0.957169	−	0.289530i	$-0.0934989\pi$
$38$	6820.00i	0.766170i
$39$	0	0
$40$	0	0
$41$	8148.00	0.756992	0.378496	−	0.925603i	$-0.376441\pi$
$41$	8148.00	0.756992	0.378496	+	0.925603i	$0.376441\pi$
$42$	0	0
$43$	− 19469.0i	− 1.60573i	−0.596161	−	0.802865i	$-0.703308\pi$
$43$	− 19469.0i	− 1.60573i	0.596161	−	0.802865i	$-0.296692\pi$
$44$	−7968.00	−0.620465
$45$	0	0
$46$	−6216.00	−0.433128
$47$	− 8418.00i	− 0.555859i	−0.960602	−	0.277929i	$-0.910352\pi$
$47$	− 8418.00i	− 0.555859i	0.960602	−	0.277929i	$-0.0896481\pi$
$48$	0	0
$49$	−37482.0	−2.23014
$50$	0	0
$51$	0	0
$52$	12944.0i	0.663835i
$53$	17664.0i	0.863773i	0.901928	+	0.431886i	$0.142152\pi$
$53$	17664.0i	0.863773i	−0.901928	+	0.431886i	$0.857848\pi$
$54$	0	0
$55$	0	0
$56$	14912.0	0.635427
$57$	0	0
$58$	31320.0i	1.22251i
$59$	35910.0	1.34303	0.671514	−	0.740991i	$-0.265644\pi$
$59$	35910.0	1.34303	0.671514	+	0.740991i	$0.265644\pi$
$60$	0	0
$61$	3527.00	0.121361	0.0606807	−	0.998157i	$-0.480673\pi$
$61$	3527.00	0.121361	0.0606807	+	0.998157i	$0.480673\pi$
$62$	3908.00i	0.129115i
$63$	0	0
$64$	−4096.00	−0.125000
$65$	0	0
$66$	0	0
$67$	57473.0i	1.56414i	0.623188	+	0.782072i	$0.285837\pi$
$67$	57473.0i	1.56414i	−0.623188	+	0.782072i	$0.714163\pi$
$68$	− 16032.0i	− 0.420451i
$69$	0	0
$70$	0	0
$71$	7548.00	0.177699	0.0888497	−	0.996045i	$-0.471681\pi$
$71$	7548.00	0.177699	0.0888497	+	0.996045i	$0.471681\pi$
$72$	0	0
$73$	646.000i	0.0141881i	0.999975	+	0.00709407i	$0.00225813\pi$
$73$	646.000i	0.0141881i	−0.999975	+	0.00709407i	$0.997742\pi$
$74$	19288.0	0.409457
$75$	0	0
$76$	−27280.0	−0.541764
$77$	116034.i	2.23028i
$78$	0	0
$79$	22720.0	0.409582	0.204791	−	0.978806i	$-0.434349\pi$
$79$	22720.0	0.409582	0.204791	+	0.978806i	$0.434349\pi$
$80$	0	0
$81$	0	0
$82$	32592.0i	0.535274i
$83$	11574.0i	0.184412i	0.995740	+	0.0922058i	$0.0293918\pi$
$83$	11574.0i	0.184412i	−0.995740	+	0.0922058i	$0.970608\pi$
$84$	0	0
$85$	0	0
$86$	77876.0	1.13542
$87$	0	0
$88$	− 31872.0i	− 0.438735i
$89$	−78960.0	−1.05665	−0.528326	−	0.849041i	$-0.677180\pi$
$89$	−78960.0	−1.05665	−0.528326	+	0.849041i	$0.677180\pi$
$90$	0	0
$91$	188497.	2.38617
$92$	− 24864.0i	− 0.306268i
$93$	0	0
$94$	33672.0	0.393051
$95$	0	0
$96$	0	0
$97$	54593.0i	0.589125i	0.955632	+	0.294563i	$0.0951740\pi$
$97$	54593.0i	0.589125i	−0.955632	+	0.294563i	$0.904826\pi$
$98$	− 149928.i	− 1.57695i
$99$	0	0
$100$	0	0
$101$	−105552.	−1.02959	−0.514793	−	0.857314i	$-0.672131\pi$
$101$	−105552.	−1.02959	−0.514793	+	0.857314i	$0.672131\pi$
$102$	0	0
$103$	177436.i	1.64797i	0.566613	+	0.823984i	$0.308253\pi$
$103$	177436.i	1.64797i	−0.566613	+	0.823984i	$0.691747\pi$
$104$	−51776.0	−0.469402
$105$	0	0
$106$	−70656.0	−0.610779
$107$	183792.i	1.55191i	0.630787	+	0.775956i	$0.282732\pi$
$107$	183792.i	1.55191i	−0.630787	+	0.775956i	$0.717268\pi$
$108$	0	0
$109$	−169685.	−1.36797	−0.683986	−	0.729495i	$-0.739755\pi$
$109$	−169685.	−1.36797	−0.683986	+	0.729495i	$0.739755\pi$
$110$	0	0
$111$	0	0
$112$	59648.0i	0.449315i
$113$	263484.i	1.94115i	0.240808	+	0.970573i	$0.422588\pi$
$113$	263484.i	1.94115i	−0.240808	+	0.970573i	$0.577412\pi$
$114$	0	0
$115$	0	0
$116$	−125280.	−0.864444
$117$	0	0
$118$	143640.i	0.949665i
$119$	−233466.	−1.51132
$120$	0	0
$121$	86953.0	0.539910
$122$	14108.0i	0.0858155i
$123$	0	0
$124$	−15632.0	−0.0912978
$125$	0	0
$126$	0	0
$127$	− 256912.i	− 1.41343i	−0.707497	−	0.706716i	$-0.750176\pi$
$127$	− 256912.i	− 1.41343i	0.707497	−	0.706716i	$-0.249824\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	219048.	1.11522	0.557611	−	0.830103i	$-0.311718\pi$
$131$	219048.	1.11522	0.557611	+	0.830103i	$0.311718\pi$
$132$	0	0
$133$	397265.i	1.94738i
$134$	−229892.	−1.10602
$135$	0	0
$136$	64128.0	0.297304
$137$	− 228678.i	− 1.04093i	−0.853882	−	0.520467i	$-0.825758\pi$
$137$	− 228678.i	− 1.04093i	0.853882	−	0.520467i	$-0.174242\pi$
$138$	0	0
$139$	−339740.	−1.49145	−0.745727	−	0.666252i	$-0.767898\pi$
$139$	−339740.	−1.49145	−0.745727	+	0.666252i	$0.767898\pi$
$140$	0	0
$141$	0	0
$142$	30192.0i	0.125652i
$143$	− 402882.i	− 1.64755i
$144$	0	0
$145$	0	0
$146$	−2584.00	−0.0100325
$147$	0	0
$148$	77152.0i	0.289530i
$149$	306450.	1.13082	0.565411	−	0.824810i	$-0.308718\pi$
$149$	306450.	1.13082	0.565411	+	0.824810i	$0.308718\pi$
$150$	0	0
$151$	198827.	0.709632	0.354816	−	0.934936i	$-0.384544\pi$
$151$	198827.	0.709632	0.354816	+	0.934936i	$0.384544\pi$
$152$	− 109120.i	− 0.383085i
$153$	0	0
$154$	−464136.	−1.57704
$155$	0	0
$156$	0	0
$157$	361283.i	1.16976i	0.811118	+	0.584882i	$0.198859\pi$
$157$	361283.i	1.16976i	−0.811118	+	0.584882i	$0.801141\pi$
$158$	90880.0i	0.289618i
$159$	0	0
$160$	0	0
$161$	−362082.	−1.10089
$162$	0	0
$163$	− 338159.i	− 0.996901i	−0.866918	−	0.498450i	$-0.833903\pi$
$163$	− 338159.i	− 0.996901i	0.866918	−	0.498450i	$-0.166097\pi$
$164$	−130368.	−0.378496
$165$	0	0
$166$	−46296.0	−0.130399
$167$	− 430248.i	− 1.19379i	−0.802320	−	0.596895i	$-0.796401\pi$
$167$	− 430248.i	− 1.19379i	0.802320	−	0.596895i	$-0.203599\pi$
$168$	0	0
$169$	−283188.	−0.762708
$170$	0	0
$171$	0	0
$172$	311504.i	0.802865i
$173$	603354.i	1.53270i	0.642424	+	0.766350i	$0.277929\pi$
$173$	603354.i	1.53270i	−0.642424	+	0.766350i	$0.722071\pi$
$174$	0	0
$175$	0	0
$176$	127488.	0.310233
$177$	0	0
$178$	− 315840.i	− 0.747166i
$179$	−374370.	−0.873310	−0.436655	−	0.899629i	$-0.643837\pi$
$179$	−374370.	−0.873310	−0.436655	+	0.899629i	$0.643837\pi$
$180$	0	0
$181$	−232423.	−0.527330	−0.263665	−	0.964614i	$-0.584931\pi$
$181$	−232423.	−0.527330	−0.263665	+	0.964614i	$0.584931\pi$
$182$	753988.i	1.68728i
$183$	0	0
$184$	99456.0	0.216564
$185$	0	0
$186$	0	0
$187$	498996.i	1.04350i
$188$	134688.i	0.277929i
$189$	0	0
$190$	0	0
$191$	846198.	1.67837	0.839187	−	0.543843i	$-0.183031\pi$
$191$	846198.	1.67837	0.839187	+	0.543843i	$0.183031\pi$
$192$	0	0
$193$	155581.i	0.300651i	0.988637	+	0.150326i	$0.0480322\pi$
$193$	155581.i	0.300651i	−0.988637	+	0.150326i	$0.951968\pi$
$194$	−218372.	−0.416574
$195$	0	0
$196$	599712.	1.11507
$197$	103482.i	0.189976i	0.995478	+	0.0949881i	$0.0302813\pi$
$197$	103482.i	0.189976i	−0.995478	+	0.0949881i	$0.969719\pi$
$198$	0	0
$199$	140425.	0.251369	0.125685	−	0.992070i	$-0.459887\pi$
$199$	140425.	0.251369	0.125685	+	0.992070i	$0.459887\pi$
$200$	0	0
$201$	0	0
$202$	− 422208.i	− 0.728028i
$203$	1.82439e6i	3.10726i
$204$	0	0
$205$	0	0
$206$	−709744.	−1.16529
$207$	0	0
$208$	− 207104.i	− 0.331917i
$209$	849090.	1.34458
$210$	0	0
$211$	−462673.	−0.715431	−0.357716	−	0.933831i	$-0.616444\pi$
$211$	−462673.	−0.715431	−0.357716	+	0.933831i	$0.616444\pi$
$212$	− 282624.i	− 0.431886i
$213$	0	0
$214$	−735168.	−1.09737
$215$	0	0
$216$	0	0
$217$	227641.i	0.328172i
$218$	− 678740.i	− 0.967302i
$219$	0	0
$220$	0	0
$221$	810618.	1.11644
$222$	0	0
$223$	735271.i	0.990114i	0.868860	+	0.495057i	$0.164853\pi$
$223$	735271.i	0.990114i	−0.868860	+	0.495057i	$0.835147\pi$
$224$	−238592.	−0.317714
$225$	0	0
$226$	−1.05394e6	−1.37260
$227$	− 967188.i	− 1.24579i	−0.782304	−	0.622897i	$-0.785956\pi$
$227$	− 967188.i	− 1.24579i	0.782304	−	0.622897i	$-0.214044\pi$
$228$	0	0
$229$	83695.0	0.105466	0.0527328	−	0.998609i	$-0.483207\pi$
$229$	83695.0	0.105466	0.0527328	+	0.998609i	$0.483207\pi$
$230$	0	0
$231$	0	0
$232$	− 501120.i	− 0.611254i
$233$	− 873876.i	− 1.05453i	−0.849700	−	0.527266i	$-0.823217\pi$
$233$	− 873876.i	− 1.05453i	0.849700	−	0.527266i	$-0.176783\pi$
$234$	0	0
$235$	0	0
$236$	−574560.	−0.671514
$237$	0	0
$238$	− 933864.i	− 1.06866i
$239$	−1.06056e6	−1.20099	−0.600497	−	0.799627i	$-0.705030\pi$
$239$	−1.06056e6	−1.20099	−0.600497	+	0.799627i	$0.705030\pi$
$240$	0	0
$241$	−756823.	−0.839367	−0.419683	−	0.907671i	$-0.637859\pi$
$241$	−756823.	−0.839367	−0.419683	+	0.907671i	$0.637859\pi$
$242$	347812.i	0.381774i
$243$	0	0
$244$	−56432.0	−0.0606807
$245$	0	0
$246$	0	0
$247$	− 1.37934e6i	− 1.43857i
$248$	− 62528.0i	− 0.0645573i
$249$	0	0
$250$	0	0
$251$	635148.	0.636342	0.318171	−	0.948033i	$-0.396931\pi$
$251$	635148.	0.636342	0.318171	+	0.948033i	$0.396931\pi$
$252$	0	0
$253$	773892.i	0.760115i
$254$	1.02765e6	0.999448
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 30708.0i	− 0.0290014i	−0.999895	−	0.0145007i	$-0.995384\pi$
$257$	− 30708.0i	− 0.0290014i	0.999895	−	0.0145007i	$-0.00461588\pi$
$258$	0	0
$259$	1.12353e6	1.04072
$260$	0	0
$261$	0	0
$262$	876192.i	0.788581i
$263$	− 189516.i	− 0.168949i	−0.996426	−	0.0844747i	$-0.973079\pi$
$263$	− 189516.i	− 0.168949i	0.996426	−	0.0844747i	$-0.0269212\pi$
$264$	0	0
$265$	0	0
$266$	−1.58906e6	−1.37701
$267$	0	0
$268$	− 919568.i	− 0.782072i
$269$	1.10997e6	0.935256	0.467628	−	0.883925i	$-0.345109\pi$
$269$	1.10997e6	0.935256	0.467628	+	0.883925i	$0.345109\pi$
$270$	0	0
$271$	211952.	0.175313	0.0876565	−	0.996151i	$-0.472062\pi$
$271$	211952.	0.175313	0.0876565	+	0.996151i	$0.472062\pi$
$272$	256512.i	0.210226i
$273$	0	0
$274$	914712.	0.736051
$275$	0	0
$276$	0	0
$277$	1.04741e6i	0.820198i	0.912041	+	0.410099i	$0.134506\pi$
$277$	1.04741e6i	0.820198i	−0.912041	+	0.410099i	$0.865494\pi$
$278$	− 1.35896e6i	− 1.05462i
$279$	0	0
$280$	0	0
$281$	−34002.0	−0.0256885	−0.0128442	−	0.999918i	$-0.504089\pi$
$281$	−34002.0	−0.0256885	−0.0128442	+	0.999918i	$0.504089\pi$
$282$	0	0
$283$	281101.i	0.208639i	0.994544	+	0.104320i	$0.0332665\pi$
$283$	281101.i	0.208639i	−0.994544	+	0.104320i	$0.966733\pi$
$284$	−120768.	−0.0888497
$285$	0	0
$286$	1.61153e6	1.16499
$287$	1.89848e6i	1.36051i
$288$	0	0
$289$	415853.	0.292884
$290$	0	0
$291$	0	0
$292$	− 10336.0i	− 0.00709407i
$293$	− 1.55851e6i	− 1.06057i	−0.847819	−	0.530285i	$-0.822085\pi$
$293$	− 1.55851e6i	− 1.06057i	0.847819	−	0.530285i	$-0.177915\pi$
$294$	0	0
$295$	0	0
$296$	−308608.	−0.204728
$297$	0	0
$298$	1.22580e6i	0.799611i
$299$	1.25719e6	0.813245
$300$	0	0
$301$	4.53628e6	2.88591
$302$	795308.i	0.501785i
$303$	0	0
$304$	436480.	0.270882
$305$	0	0
$306$	0	0
$307$	− 839917.i	− 0.508616i	−0.967123	−	0.254308i	$-0.918152\pi$
$307$	− 839917.i	− 0.508616i	0.967123	−	0.254308i	$-0.0818478\pi$
$308$	− 1.85654e6i	− 1.11514i
$309$	0	0
$310$	0	0
$311$	292698.	0.171601	0.0858003	−	0.996312i	$-0.472655\pi$
$311$	292698.	0.171601	0.0858003	+	0.996312i	$0.472655\pi$
$312$	0	0
$313$	− 127859.i	− 0.0737684i	−0.999320	−	0.0368842i	$-0.988257\pi$
$313$	− 127859.i	− 0.0737684i	0.999320	−	0.0368842i	$-0.0117433\pi$
$314$	−1.44513e6	−0.827148
$315$	0	0
$316$	−363520.	−0.204791
$317$	− 648048.i	− 0.362209i	−0.983464	−	0.181104i	$-0.942033\pi$
$317$	− 648048.i	− 0.362209i	0.983464	−	0.181104i	$-0.0579672\pi$
$318$	0	0
$319$	3.89934e6	2.14543
$320$	0	0
$321$	0	0
$322$	− 1.44833e6i	− 0.778444i
$323$	1.70841e6i	0.911141i
$324$	0	0
$325$	0	0
$326$	1.35264e6	0.704915
$327$	0	0
$328$	− 521472.i	− 0.267637i
$329$	1.96139e6	0.999022
$330$	0	0
$331$	−3.39315e6	−1.70229	−0.851144	−	0.524933i	$-0.824090\pi$
$331$	−3.39315e6	−1.70229	−0.851144	+	0.524933i	$0.824090\pi$
$332$	− 185184.i	− 0.0922058i
$333$	0	0
$334$	1.72099e6	0.844137
$335$	0	0
$336$	0	0
$337$	− 1.62085e6i	− 0.777441i	−0.921356	−	0.388720i	$-0.872917\pi$
$337$	− 1.62085e6i	− 0.777441i	0.921356	−	0.388720i	$-0.127083\pi$
$338$	− 1.13275e6i	− 0.539316i
$339$	0	0
$340$	0	0
$341$	486546.	0.226589
$342$	0	0
$343$	− 4.81728e6i	− 2.21088i
$344$	−1.24602e6	−0.567711
$345$	0	0
$346$	−2.41342e6	−1.08378
$347$	1.28638e6i	0.573517i	0.958003	+	0.286758i	$0.0925777\pi$
$347$	1.28638e6i	0.573517i	−0.958003	+	0.286758i	$0.907422\pi$
$348$	0	0
$349$	2.73055e6	1.20001	0.600007	−	0.799994i	$-0.295164\pi$
$349$	2.73055e6	1.20001	0.600007	+	0.799994i	$0.295164\pi$
$350$	0	0
$351$	0	0
$352$	509952.i	0.219368i
$353$	− 2.13649e6i	− 0.912564i	−0.889835	−	0.456282i	$-0.849181\pi$
$353$	− 2.13649e6i	− 0.912564i	0.889835	−	0.456282i	$-0.150819\pi$
$354$	0	0
$355$	0	0
$356$	1.26336e6	0.528326
$357$	0	0
$358$	− 1.49748e6i	− 0.617523i
$359$	3.26406e6	1.33666	0.668332	−	0.743863i	$-0.267009\pi$
$359$	3.26406e6	1.33666	0.668332	+	0.743863i	$0.267009\pi$
$360$	0	0
$361$	430926.	0.174034
$362$	− 929692.i	− 0.372879i
$363$	0	0
$364$	−3.01595e6	−1.19308
$365$	0	0
$366$	0	0
$367$	− 4.15078e6i	− 1.60866i	−0.594183	−	0.804330i	$-0.702524\pi$
$367$	− 4.15078e6i	− 1.60866i	0.594183	−	0.804330i	$-0.297476\pi$
$368$	397824.i	0.153134i
$369$	0	0
$370$	0	0
$371$	−4.11571e6	−1.55242
$372$	0	0
$373$	2.14242e6i	0.797320i	0.917099	+	0.398660i	$0.130525\pi$
$373$	2.14242e6i	0.797320i	−0.917099	+	0.398660i	$0.869475\pi$
$374$	−1.99598e6	−0.737867
$375$	0	0
$376$	−538752.	−0.196526
$377$	− 6.33447e6i	− 2.29539i
$378$	0	0
$379$	1.94699e6	0.696253	0.348126	−	0.937448i	$-0.386818\pi$
$379$	1.94699e6	0.696253	0.348126	+	0.937448i	$0.386818\pi$
$380$	0	0
$381$	0	0
$382$	3.38479e6i	1.18679i
$383$	2.65052e6i	0.923283i	0.887067	+	0.461641i	$0.152739\pi$
$383$	2.65052e6i	0.923283i	−0.887067	+	0.461641i	$0.847261\pi$
$384$	0	0
$385$	0	0
$386$	−622324.	−0.212593
$387$	0	0
$388$	− 873488.i	− 0.294563i
$389$	282540.	0.0946686	0.0473343	−	0.998879i	$-0.484927\pi$
$389$	282540.	0.0946686	0.0473343	+	0.998879i	$0.484927\pi$
$390$	0	0
$391$	−1.55711e6	−0.515083
$392$	2.39885e6i	0.788474i
$393$	0	0
$394$	−413928.	−0.134333
$395$	0	0
$396$	0	0
$397$	− 2.62066e6i	− 0.834515i	−0.908788	−	0.417257i	$-0.862991\pi$
$397$	− 2.62066e6i	− 0.834515i	0.908788	−	0.417257i	$-0.137009\pi$
$398$	561700.i	0.177745i
$399$	0	0
$400$	0	0
$401$	286248.	0.0888959	0.0444479	−	0.999012i	$-0.485847\pi$
$401$	286248.	0.0888959	0.0444479	+	0.999012i	$0.485847\pi$
$402$	0	0
$403$	− 790393.i	− 0.242427i
$404$	1.68883e6	0.514793
$405$	0	0
$406$	−7.29756e6	−2.19716
$407$	− 2.40136e6i	− 0.718572i
$408$	0	0
$409$	−4.12069e6	−1.21804	−0.609019	−	0.793155i	$-0.708437\pi$
$409$	−4.12069e6	−1.21804	−0.609019	+	0.793155i	$0.708437\pi$
$410$	0	0
$411$	0	0
$412$	− 2.83898e6i	− 0.823984i
$413$	8.36703e6i	2.41377i
$414$	0	0
$415$	0	0
$416$	828416.	0.234701
$417$	0	0
$418$	3.39636e6i	0.950765i
$419$	−2.37948e6	−0.662136	−0.331068	−	0.943607i	$-0.607409\pi$
$419$	−2.37948e6	−0.662136	−0.331068	+	0.943607i	$0.607409\pi$
$420$	0	0
$421$	−741298.	−0.203839	−0.101920	−	0.994793i	$-0.532498\pi$
$421$	−741298.	−0.203839	−0.101920	+	0.994793i	$0.532498\pi$
$422$	− 1.85069e6i	− 0.505886i
$423$	0	0
$424$	1.13050e6	0.305390
$425$	0	0
$426$	0	0
$427$	821791.i	0.218118i
$428$	− 2.94067e6i	− 0.775956i
$429$	0	0
$430$	0	0
$431$	187398.	0.0485928	0.0242964	−	0.999705i	$-0.492265\pi$
$431$	187398.	0.0485928	0.0242964	+	0.999705i	$0.492265\pi$
$432$	0	0
$433$	− 6.55110e6i	− 1.67917i	−0.543229	−	0.839585i	$-0.682798\pi$
$433$	− 6.55110e6i	− 1.67917i	0.543229	−	0.839585i	$-0.317202\pi$
$434$	−910564.	−0.232052
$435$	0	0
$436$	2.71496e6	0.683986
$437$	2.64957e6i	0.663700i
$438$	0	0
$439$	−270065.	−0.0668817	−0.0334408	−	0.999441i	$-0.510647\pi$
$439$	−270065.	−0.0668817	−0.0334408	+	0.999441i	$0.510647\pi$
$440$	0	0
$441$	0	0
$442$	3.24247e6i	0.789443i
$443$	2.77934e6i	0.672873i	0.941706	+	0.336436i	$0.109222\pi$
$443$	2.77934e6i	0.672873i	−0.941706	+	0.336436i	$0.890778\pi$
$444$	0	0
$445$	0	0
$446$	−2.94108e6	−0.700116
$447$	0	0
$448$	− 954368.i	− 0.224657i
$449$	7.86630e6	1.84143	0.920714	−	0.390238i	$-0.127607\pi$
$449$	7.86630e6	1.84143	0.920714	+	0.390238i	$0.127607\pi$
$450$	0	0
$451$	4.05770e6	0.939375
$452$	− 4.21574e6i	− 0.970573i
$453$	0	0
$454$	3.86875e6	0.880909
$455$	0	0
$456$	0	0
$457$	6.23356e6i	1.39619i	0.716004	+	0.698097i	$0.245969\pi$
$457$	6.23356e6i	1.39619i	−0.716004	+	0.698097i	$0.754031\pi$
$458$	334780.i	0.0745754i
$459$	0	0
$460$	0	0
$461$	4.68305e6	1.02630	0.513152	−	0.858298i	$-0.328478\pi$
$461$	4.68305e6	1.02630	0.513152	+	0.858298i	$0.328478\pi$
$462$	0	0
$463$	382816.i	0.0829923i	0.999139	+	0.0414961i	$0.0132124\pi$
$463$	382816.i	0.0829923i	−0.999139	+	0.0414961i	$0.986788\pi$
$464$	2.00448e6	0.432222
$465$	0	0
$466$	3.49550e6	0.745667
$467$	1.93540e6i	0.410657i	0.978693	+	0.205328i	$0.0658262\pi$
$467$	1.93540e6i	0.410657i	−0.978693	+	0.205328i	$0.934174\pi$
$468$	0	0
$469$	−1.33912e7	−2.81117
$470$	0	0
$471$	0	0
$472$	− 2.29824e6i	− 0.474832i
$473$	− 9.69556e6i	− 1.99260i
$474$	0	0
$475$	0	0
$476$	3.73546e6	0.755660
$477$	0	0
$478$	− 4.24224e6i	− 0.849230i
$479$	−5.56917e6	−1.10905	−0.554526	−	0.832167i	$-0.687100\pi$
$479$	−5.56917e6	−1.10905	−0.554526	+	0.832167i	$0.687100\pi$
$480$	0	0
$481$	−3.90100e6	−0.768799
$482$	− 3.02729e6i	− 0.593522i
$483$	0	0
$484$	−1.39125e6	−0.269955
$485$	0	0
$486$	0	0
$487$	4.22450e6i	0.807148i	0.914947	+	0.403574i	$0.132232\pi$
$487$	4.22450e6i	0.807148i	−0.914947	+	0.403574i	$0.867768\pi$
$488$	− 225728.i	− 0.0429078i
$489$	0	0
$490$	0	0
$491$	−6.96295e6	−1.30344	−0.651718	−	0.758461i	$-0.725951\pi$
$491$	−6.96295e6	−1.30344	−0.651718	+	0.758461i	$0.725951\pi$
$492$	0	0
$493$	7.84566e6i	1.45383i
$494$	5.51738e6	1.01722
$495$	0	0
$496$	250112.	0.0456489
$497$	1.75868e6i	0.319372i
$498$	0	0
$499$	9.29582e6	1.67123	0.835616	−	0.549315i	$-0.185111\pi$
$499$	9.29582e6	1.67123	0.835616	+	0.549315i	$0.185111\pi$
$500$	0	0
$501$	0	0
$502$	2.54059e6i	0.449962i
$503$	6.34136e6i	1.11754i	0.829323	+	0.558770i	$0.188726\pi$
$503$	6.34136e6i	1.11754i	−0.829323	+	0.558770i	$0.811274\pi$
$504$	0	0
$505$	0	0
$506$	−3.09557e6	−0.537482
$507$	0	0
$508$	4.11059e6i	0.706716i
$509$	−7.38309e6	−1.26312	−0.631559	−	0.775328i	$-0.717585\pi$
$509$	−7.38309e6	−1.26312	−0.631559	+	0.775328i	$0.717585\pi$
$510$	0	0
$511$	−150518.	−0.0254998
$512$	262144.i	0.0441942i
$513$	0	0
$514$	122832.	0.0205071
$515$	0	0
$516$	0	0
$517$	− 4.19216e6i	− 0.689782i
$518$	4.49410e6i	0.735900i
$519$	0	0
$520$	0	0
$521$	−4.19620e6	−0.677270	−0.338635	−	0.940918i	$-0.609965\pi$
$521$	−4.19620e6	−0.677270	−0.338635	+	0.940918i	$0.609965\pi$
$522$	0	0
$523$	3.57942e6i	0.572214i	0.958198	+	0.286107i	$0.0923613\pi$
$523$	3.57942e6i	0.572214i	−0.958198	+	0.286107i	$0.907639\pi$
$524$	−3.50477e6	−0.557611
$525$	0	0
$526$	758064.	0.119465
$527$	978954.i	0.153545i
$528$	0	0
$529$	4.02143e6	0.624800
$530$	0	0
$531$	0	0
$532$	− 6.35624e6i	− 0.973691i
$533$	− 6.59173e6i	− 1.00504i
$534$	0	0
$535$	0	0
$536$	3.67827e6	0.553009
$537$	0	0
$538$	4.43988e6i	0.661326i
$539$	−1.86660e7	−2.76745
$540$	0	0
$541$	4.95548e6	0.727934	0.363967	−	0.931412i	$-0.381422\pi$
$541$	4.95548e6	0.727934	0.363967	+	0.931412i	$0.381422\pi$
$542$	847808.i	0.123965i
$543$	0	0
$544$	−1.02605e6	−0.148652
$545$	0	0
$546$	0	0
$547$	− 5.31803e6i	− 0.759946i	−0.924998	−	0.379973i	$-0.875933\pi$
$547$	− 5.31803e6i	− 0.759946i	0.924998	−	0.379973i	$-0.124067\pi$
$548$	3.65885e6i	0.520467i
$549$	0	0
$550$	0	0
$551$	1.33502e7	1.87330
$552$	0	0
$553$	5.29376e6i	0.736125i
$554$	−4.18965e6	−0.579967
$555$	0	0
$556$	5.43584e6	0.745727
$557$	4.25794e6i	0.581516i	0.956797	+	0.290758i	$0.0939075\pi$
$557$	4.25794e6i	0.581516i	−0.956797	+	0.290758i	$0.906093\pi$
$558$	0	0
$559$	−1.57504e7	−2.13188
$560$	0	0
$561$	0	0
$562$	− 136008.i	− 0.0181645i
$563$	− 4.37617e6i	− 0.581866i	−0.956743	−	0.290933i	$-0.906034\pi$
$563$	− 4.37617e6i	− 0.581866i	0.956743	−	0.290933i	$-0.0939656\pi$
$564$	0	0
$565$	0	0
$566$	−1.12440e6	−0.147530
$567$	0	0
$568$	− 483072.i	− 0.0628262i
$569$	3.57717e6	0.463190	0.231595	−	0.972812i	$-0.425606\pi$
$569$	3.57717e6	0.463190	0.231595	+	0.972812i	$0.425606\pi$
$570$	0	0
$571$	1.94693e6	0.249896	0.124948	−	0.992163i	$-0.460124\pi$
$571$	1.94693e6	0.249896	0.124948	+	0.992163i	$0.460124\pi$
$572$	6.44611e6i	0.823773i
$573$	0	0
$574$	−7.59394e6	−0.962027
$575$	0	0
$576$	0	0
$577$	6.95576e6i	0.869772i	0.900486	+	0.434886i	$0.143211\pi$
$577$	6.95576e6i	0.869772i	−0.900486	+	0.434886i	$0.856789\pi$
$578$	1.66341e6i	0.207100i
$579$	0	0
$580$	0	0
$581$	−2.69674e6	−0.331436
$582$	0	0
$583$	8.79667e6i	1.07188i
$584$	41344.0	0.00501626
$585$	0	0
$586$	6.23402e6	0.749936
$587$	− 2.41853e6i	− 0.289705i	−0.989453	−	0.144852i	$-0.953729\pi$
$587$	− 2.41853e6i	− 0.289705i	0.989453	−	0.144852i	$-0.0462708\pi$
$588$	0	0
$589$	1.66578e6	0.197848
$590$	0	0
$591$	0	0
$592$	− 1.23443e6i	− 0.144765i
$593$	− 9.58396e6i	− 1.11920i	−0.828763	−	0.559600i	$-0.810955\pi$
$593$	− 9.58396e6i	− 1.11920i	0.828763	−	0.559600i	$-0.189045\pi$
$594$	0	0
$595$	0	0
$596$	−4.90320e6	−0.565411
$597$	0	0
$598$	5.02874e6i	0.575051i
$599$	−1.52070e6	−0.173172	−0.0865858	−	0.996244i	$-0.527596\pi$
$599$	−1.52070e6	−0.173172	−0.0865858	+	0.996244i	$0.527596\pi$
$600$	0	0
$601$	3.88283e6	0.438492	0.219246	−	0.975670i	$-0.429640\pi$
$601$	3.88283e6	0.438492	0.219246	+	0.975670i	$0.429640\pi$
$602$	1.81451e7i	2.04065i
$603$	0	0
$604$	−3.18123e6	−0.354816
$605$	0	0
$606$	0	0
$607$	− 107992.i	− 0.0118965i	−0.999982	−	0.00594826i	$-0.998107\pi$
$607$	− 107992.i	− 0.0118965i	0.999982	−	0.00594826i	$-0.00189340\pi$
$608$	1.74592e6i	0.191543i
$609$	0	0
$610$	0	0
$611$	−6.81016e6	−0.737997
$612$	0	0
$613$	− 7.49923e6i	− 0.806057i	−0.915187	−	0.403028i	$-0.867958\pi$
$613$	− 7.49923e6i	− 0.806057i	0.915187	−	0.403028i	$-0.132042\pi$
$614$	3.35967e6	0.359646
$615$	0	0
$616$	7.42618e6	0.788521
$617$	1.22695e6i	0.129752i	0.997893	+	0.0648761i	$0.0206652\pi$
$617$	1.22695e6i	0.129752i	−0.997893	+	0.0648761i	$0.979335\pi$
$618$	0	0
$619$	−9.51340e6	−0.997950	−0.498975	−	0.866616i	$-0.666290\pi$
$619$	−9.51340e6	−0.997950	−0.498975	+	0.866616i	$0.666290\pi$
$620$	0	0
$621$	0	0
$622$	1.17079e6i	0.121340i
$623$	− 1.83977e7i	− 1.89908i
$624$	0	0
$625$	0	0
$626$	511436.	0.0521621
$627$	0	0
$628$	− 5.78053e6i	− 0.584882i
$629$	4.83164e6	0.486932
$630$	0	0
$631$	−1.56449e7	−1.56423	−0.782114	−	0.623135i	$-0.785859\pi$
$631$	−1.56449e7	−1.56423	−0.782114	+	0.623135i	$0.785859\pi$
$632$	− 1.45408e6i	− 0.144809i
$633$	0	0
$634$	2.59219e6	0.256120
$635$	0	0
$636$	0	0
$637$	3.03229e7i	2.96089i
$638$	1.55974e7i	1.51705i
$639$	0	0
$640$	0	0
$641$	9.60395e6	0.923219	0.461609	−	0.887083i	$-0.347272\pi$
$641$	9.60395e6	0.923219	0.461609	+	0.887083i	$0.347272\pi$
$642$	0	0
$643$	8.24396e6i	0.786336i	0.919467	+	0.393168i	$0.128621\pi$
$643$	8.24396e6i	0.786336i	−0.919467	+	0.393168i	$0.871379\pi$
$644$	5.79331e6	0.550443
$645$	0	0
$646$	−6.83364e6	−0.644274
$647$	4.07353e6i	0.382570i	0.981535	+	0.191285i	$0.0612654\pi$
$647$	4.07353e6i	0.382570i	−0.981535	+	0.191285i	$0.938735\pi$
$648$	0	0
$649$	1.78832e7	1.66661
$650$	0	0
$651$	0	0
$652$	5.41054e6i	0.498450i
$653$	1.68193e7i	1.54357i	0.635886	+	0.771783i	$0.280635\pi$
$653$	1.68193e7i	1.54357i	−0.635886	+	0.771783i	$0.719365\pi$
$654$	0	0
$655$	0	0
$656$	2.08589e6	0.189248
$657$	0	0
$658$	7.84558e6i	0.706415i
$659$	2.87826e6	0.258176	0.129088	−	0.991633i	$-0.458795\pi$
$659$	2.87826e6	0.258176	0.129088	+	0.991633i	$0.458795\pi$
$660$	0	0
$661$	1.33386e7	1.18743	0.593713	−	0.804677i	$-0.297661\pi$
$661$	1.33386e7	1.18743	0.593713	+	0.804677i	$0.297661\pi$
$662$	− 1.35726e7i	− 1.20370i
$663$	0	0
$664$	740736.	0.0651993
$665$	0	0
$666$	0	0
$667$	1.21678e7i	1.05901i
$668$	6.88397e6i	0.596895i
$669$	0	0
$670$	0	0
$671$	1.75645e6	0.150601
$672$	0	0
$673$	− 6.37345e6i	− 0.542422i	−0.962520	−	0.271211i	$-0.912576\pi$
$673$	− 6.37345e6i	− 0.542422i	0.962520	−	0.271211i	$-0.0874241\pi$
$674$	6.48339e6	0.549734
$675$	0	0
$676$	4.53101e6	0.381354
$677$	2.27210e7i	1.90526i	0.304126	+	0.952632i	$0.401636\pi$
$677$	2.27210e7i	1.90526i	−0.304126	+	0.952632i	$0.598364\pi$
$678$	0	0
$679$	−1.27202e7	−1.05881
$680$	0	0
$681$	0	0
$682$	1.94618e6i	0.160222i
$683$	− 1.53612e7i	− 1.26001i	−0.776593	−	0.630003i	$-0.783054\pi$
$683$	− 1.53612e7i	− 1.26001i	0.776593	−	0.630003i	$-0.216946\pi$
$684$	0	0
$685$	0	0
$686$	1.92691e7	1.56333
$687$	0	0
$688$	− 4.98406e6i	− 0.401432i
$689$	1.42902e7	1.14680
$690$	0	0
$691$	8.05035e6	0.641386	0.320693	−	0.947183i	$-0.396084\pi$
$691$	8.05035e6	0.641386	0.320693	+	0.947183i	$0.396084\pi$
$692$	− 9.65366e6i	− 0.766350i
$693$	0	0
$694$	−5.14553e6	−0.405538
$695$	0	0
$696$	0	0
$697$	8.16430e6i	0.636556i
$698$	1.09222e7i	0.848539i
$699$	0	0
$700$	0	0
$701$	−7.27840e6	−0.559424	−0.279712	−	0.960084i	$-0.590239\pi$
$701$	−7.27840e6	−0.559424	−0.279712	+	0.960084i	$0.590239\pi$
$702$	0	0
$703$	− 8.22151e6i	− 0.627427i
$704$	−2.03981e6	−0.155116
$705$	0	0
$706$	8.54594e6	0.645280
$707$	− 2.45936e7i	− 1.85044i
$708$	0	0
$709$	−1.37498e7	−1.02726	−0.513630	−	0.858012i	$-0.671700\pi$
$709$	−1.37498e7	−1.02726	−0.513630	+	0.858012i	$0.671700\pi$
$710$	0	0
$711$	0	0
$712$	5.05344e6i	0.373583i
$713$	1.51826e6i	0.111846i
$714$	0	0
$715$	0	0
$716$	5.98992e6	0.436655
$717$	0	0
$718$	1.30562e7i	0.945164i
$719$	−9.05583e6	−0.653290	−0.326645	−	0.945147i	$-0.605918\pi$
$719$	−9.05583e6	−0.653290	−0.326645	+	0.945147i	$0.605918\pi$
$720$	0	0
$721$	−4.13426e7	−2.96183
$722$	1.72370e6i	0.123061i
$723$	0	0
$724$	3.71877e6	0.263665
$725$	0	0
$726$	0	0
$727$	− 2.20979e7i	− 1.55065i	−0.631560	−	0.775327i	$-0.717585\pi$
$727$	− 2.20979e7i	− 1.55065i	0.631560	−	0.775327i	$-0.282415\pi$
$728$	− 1.20638e7i	− 0.843638i
$729$	0	0
$730$	0	0
$731$	1.95079e7	1.35026
$732$	0	0
$733$	− 2.07337e6i	− 0.142534i	−0.997457	−	0.0712669i	$-0.977296\pi$
$733$	− 2.07337e6i	− 0.142534i	0.997457	−	0.0712669i	$-0.0227042\pi$
$734$	1.66031e7	1.13749
$735$	0	0
$736$	−1.59130e6	−0.108282
$737$	2.86216e7i	1.94100i
$738$	0	0
$739$	−1.48849e7	−1.00262	−0.501310	−	0.865268i	$-0.667148\pi$
$739$	−1.48849e7	−1.00262	−0.501310	+	0.865268i	$0.667148\pi$
$740$	0	0
$741$	0	0
$742$	− 1.64628e7i	− 1.09773i
$743$	2.58856e7i	1.72023i	0.510099	+	0.860116i	$0.329609\pi$
$743$	2.58856e7i	1.72023i	−0.510099	+	0.860116i	$0.670391\pi$
$744$	0	0
$745$	0	0
$746$	−8.56968e6	−0.563790
$747$	0	0
$748$	− 7.98394e6i	− 0.521751i
$749$	−4.28235e7	−2.78919
$750$	0	0
$751$	−7.98645e6	−0.516718	−0.258359	−	0.966049i	$-0.583182\pi$
$751$	−7.98645e6	−0.516718	−0.258359	+	0.966049i	$0.583182\pi$
$752$	− 2.15501e6i	− 0.138965i
$753$	0	0
$754$	2.53379e7	1.62309
$755$	0	0
$756$	0	0
$757$	− 1.15570e7i	− 0.733000i	−0.930418	−	0.366500i	$-0.880556\pi$
$757$	− 1.15570e7i	− 0.733000i	0.930418	−	0.366500i	$-0.119444\pi$
$758$	7.78798e6i	0.492325i
$759$	0	0
$760$	0	0
$761$	−3.16705e6	−0.198241	−0.0991205	−	0.995075i	$-0.531603\pi$
$761$	−3.16705e6	−0.198241	−0.0991205	+	0.995075i	$0.531603\pi$
$762$	0	0
$763$	− 3.95366e7i	− 2.45860i
$764$	−1.35392e7	−0.839187
$765$	0	0
$766$	−1.06021e7	−0.652860
$767$	− 2.90512e7i	− 1.78310i
$768$	0	0
$769$	−8.21560e6	−0.500983	−0.250492	−	0.968119i	$-0.580592\pi$
$769$	−8.21560e6	−0.500983	−0.250492	+	0.968119i	$0.580592\pi$
$770$	0	0
$771$	0	0
$772$	− 2.48930e6i	− 0.150326i
$773$	1.19708e7i	0.720567i	0.932843	+	0.360284i	$0.117320\pi$
$773$	1.19708e7i	0.720567i	−0.932843	+	0.360284i	$0.882680\pi$
$774$	0	0
$775$	0	0
$776$	3.49395e6	0.208287
$777$	0	0
$778$	1.13016e6i	0.0669408i
$779$	1.38923e7	0.820223
$780$	0	0
$781$	3.75890e6	0.220513
$782$	− 6.22843e6i	− 0.364218i
$783$	0	0
$784$	−9.59539e6	−0.557536
$785$	0	0
$786$	0	0
$787$	− 1.71154e7i	− 0.985032i	−0.870304	−	0.492516i	$-0.836077\pi$
$787$	− 1.71154e7i	− 0.985032i	0.870304	−	0.492516i	$-0.163923\pi$
$788$	− 1.65571e6i	− 0.0949881i
$789$	0	0
$790$	0	0
$791$	−6.13918e7	−3.48874
$792$	0	0
$793$	− 2.85334e6i	− 0.161128i
$794$	1.04826e7	0.590091
$795$	0	0
$796$	−2.24680e6	−0.125685
$797$	2.80753e7i	1.56559i	0.622277	+	0.782797i	$0.286208\pi$
$797$	2.80753e7i	1.56559i	−0.622277	+	0.782797i	$0.713792\pi$
$798$	0	0
$799$	8.43484e6	0.467423
$800$	0	0
$801$	0	0
$802$	1.14499e6i	0.0628589i
$803$	321708.i	0.0176065i
$804$	0	0
$805$	0	0
$806$	3.16157e6	0.171422
$807$	0	0
$808$	6.75533e6i	0.364014i
$809$	9.90816e6	0.532257	0.266129	−	0.963938i	$-0.414255\pi$
$809$	9.90816e6	0.532257	0.266129	+	0.963938i	$0.414255\pi$
$810$	0	0
$811$	5.72573e6	0.305688	0.152844	−	0.988250i	$-0.451157\pi$
$811$	5.72573e6	0.305688	0.152844	+	0.988250i	$0.451157\pi$
$812$	− 2.91902e7i	− 1.55363i
$813$	0	0
$814$	9.60542e6	0.508107
$815$	0	0
$816$	0	0
$817$	− 3.31946e7i	− 1.73985i
$818$	− 1.64827e7i	− 0.861284i
$819$	0	0
$820$	0	0
$821$	5.96570e6	0.308890	0.154445	−	0.988001i	$-0.450641\pi$
$821$	5.96570e6	0.308890	0.154445	+	0.988001i	$0.450641\pi$
$822$	0	0
$823$	1.97778e7i	1.01784i	0.860815	+	0.508918i	$0.169954\pi$
$823$	1.97778e7i	1.01784i	−0.860815	+	0.508918i	$0.830046\pi$
$824$	1.13559e7	0.582645
$825$	0	0
$826$	−3.34681e7	−1.70679
$827$	− 2.04045e7i	− 1.03744i	−0.854945	−	0.518719i	$-0.826409\pi$
$827$	− 2.04045e7i	− 1.03744i	0.854945	−	0.518719i	$-0.173591\pi$
$828$	0	0
$829$	2.77183e7	1.40081	0.700406	−	0.713745i	$-0.253002\pi$
$829$	2.77183e7	1.40081	0.700406	+	0.713745i	$0.253002\pi$
$830$	0	0
$831$	0	0
$832$	3.31366e6i	0.165959i
$833$	− 3.75570e7i	− 1.87533i
$834$	0	0
$835$	0	0
$836$	−1.35854e7	−0.672292
$837$	0	0
$838$	− 9.51792e6i	− 0.468201i
$839$	−7.79841e6	−0.382473	−0.191237	−	0.981544i	$-0.561250\pi$
$839$	−7.79841e6	−0.382473	−0.191237	+	0.981544i	$0.561250\pi$
$840$	0	0
$841$	4.07978e7	1.98905
$842$	− 2.96519e6i	− 0.144136i
$843$	0	0
$844$	7.40277e6	0.357716
$845$	0	0
$846$	0	0
$847$	2.02600e7i	0.970358i
$848$	4.52198e6i	0.215943i
$849$	0	0
$850$	0	0
$851$	7.49339e6	0.354694
$852$	0	0
$853$	− 6.22554e6i	− 0.292957i	−0.989214	−	0.146479i	$-0.953206\pi$
$853$	− 6.22554e6i	− 0.292957i	0.989214	−	0.146479i	$-0.0467940\pi$
$854$	−3.28716e6	−0.154233
$855$	0	0
$856$	1.17627e7	0.548684
$857$	− 3.39757e7i	− 1.58022i	−0.612968	−	0.790108i	$-0.710024\pi$
$857$	− 3.39757e7i	− 1.58022i	0.612968	−	0.790108i	$-0.289976\pi$
$858$	0	0
$859$	1.47282e7	0.681033	0.340516	−	0.940239i	$-0.389398\pi$
$859$	1.47282e7	0.681033	0.340516	+	0.940239i	$0.389398\pi$
$860$	0	0
$861$	0	0
$862$	749592.i	0.0343603i
$863$	− 1.88594e7i	− 0.861988i	−0.902355	−	0.430994i	$-0.858163\pi$
$863$	− 1.88594e7i	− 0.861988i	0.902355	−	0.430994i	$-0.141837\pi$
$864$	0	0
$865$	0	0
$866$	2.62044e7	1.18735
$867$	0	0
$868$	− 3.64226e6i	− 0.164086i
$869$	1.13146e7	0.508263
$870$	0	0
$871$	4.64957e7	2.07667
$872$	1.08598e7i	0.483651i
$873$	0	0
$874$	−1.05983e7	−0.469307
$875$	0	0
$876$	0	0
$877$	− 1.82112e7i	− 0.799540i	−0.916615	−	0.399770i	$-0.869090\pi$
$877$	− 1.82112e7i	− 0.799540i	0.916615	−	0.399770i	$-0.130910\pi$
$878$	− 1.08026e6i	− 0.0472925i
$879$	0	0
$880$	0	0
$881$	7.65425e6	0.332248	0.166124	−	0.986105i	$-0.446875\pi$
$881$	7.65425e6	0.332248	0.166124	+	0.986105i	$0.446875\pi$
$882$	0	0
$883$	597451.i	0.0257870i	0.999917	+	0.0128935i	$0.00410424\pi$
$883$	597451.i	0.0257870i	−0.999917	+	0.0128935i	$0.995896\pi$
$884$	−1.29699e7	−0.558220
$885$	0	0
$886$	−1.11174e7	−0.475793
$887$	− 2.06888e6i	− 0.0882929i	−0.999025	−	0.0441465i	$-0.985943\pi$
$887$	− 2.06888e6i	− 0.0882929i	0.999025	−	0.0441465i	$-0.0140568\pi$
$888$	0	0
$889$	5.98605e7	2.54031
$890$	0	0
$891$	0	0
$892$	− 1.17643e7i	− 0.495057i
$893$	− 1.43527e7i	− 0.602289i
$894$	0	0
$895$	0	0
$896$	3.81747e6	0.158857
$897$	0	0
$898$	3.14652e7i	1.30209i
$899$	7.64991e6	0.315687
$900$	0	0
$901$	−1.76993e7	−0.726348
$902$	1.62308e7i	0.664238i
$903$	0	0
$904$	1.68630e7	0.686299
$905$	0	0
$906$	0	0
$907$	7.83331e6i	0.316175i	0.987425	+	0.158087i	$0.0505327\pi$
$907$	7.83331e6i	0.316175i	−0.987425	+	0.158087i	$0.949467\pi$
$908$	1.54750e7i	0.622897i
$909$	0	0
$910$	0	0
$911$	−4.08133e7	−1.62932	−0.814659	−	0.579940i	$-0.803076\pi$
$911$	−4.08133e7	−1.62932	−0.814659	+	0.579940i	$0.803076\pi$
$912$	0	0
$913$	5.76385e6i	0.228842i
$914$	−2.49342e7	−0.987258
$915$	0	0
$916$	−1.33912e6	−0.0527328
$917$	5.10382e7i	2.00434i
$918$	0	0
$919$	−6.21579e6	−0.242777	−0.121389	−	0.992605i	$-0.538735\pi$
$919$	−6.21579e6	−0.242777	−0.121389	+	0.992605i	$0.538735\pi$
$920$	0	0
$921$	0	0
$922$	1.87322e7i	0.725707i
$923$	− 6.10633e6i	− 0.235926i
$924$	0	0
$925$	0	0
$926$	−1.53126e6	−0.0586844
$927$	0	0
$928$	8.01792e6i	0.305627i
$929$	−4.64847e6	−0.176714	−0.0883570	−	0.996089i	$-0.528162\pi$
$929$	−4.64847e6	−0.176714	−0.0883570	+	0.996089i	$0.528162\pi$
$930$	0	0
$931$	−6.39068e7	−2.41642
$932$	1.39820e7i	0.527266i
$933$	0	0
$934$	−7.74161e6	−0.290378
$935$	0	0
$936$	0	0
$937$	− 1.33603e7i	− 0.497127i	−0.968616	−	0.248563i	$-0.920042\pi$
$937$	− 1.33603e7i	− 0.497127i	0.968616	−	0.248563i	$-0.0799584\pi$
$938$	− 5.35648e7i	− 1.98780i
$939$	0	0
$940$	0	0
$941$	−3.32569e7	−1.22436	−0.612178	−	0.790720i	$-0.709706\pi$
$941$	−3.32569e7	−1.22436	−0.612178	+	0.790720i	$0.709706\pi$
$942$	0	0
$943$	1.26620e7i	0.463685i
$944$	9.19296e6	0.335757
$945$	0	0
$946$	3.87822e7	1.40898
$947$	− 4.21530e7i	− 1.52740i	−0.645569	−	0.763702i	$-0.723380\pi$
$947$	− 4.21530e7i	− 1.52740i	0.645569	−	0.763702i	$-0.276620\pi$
$948$	0	0
$949$	522614.	0.0188372
$950$	0	0
$951$	0	0
$952$	1.49418e7i	0.534332i
$953$	2.24691e7i	0.801406i	0.916208	+	0.400703i	$0.131234\pi$
$953$	2.24691e7i	0.801406i	−0.916208	+	0.400703i	$0.868766\pi$
$954$	0	0
$955$	0	0
$956$	1.69690e7	0.600497
$957$	0	0
$958$	− 2.22767e7i	− 0.784218i
$959$	5.32820e7	1.87083
$960$	0	0
$961$	−2.76746e7	−0.966659
$962$	− 1.56040e7i	− 0.543623i
$963$	0	0
$964$	1.21092e7	0.419683
$965$	0	0
$966$	0	0
$967$	− 1.75000e7i	− 0.601826i	−0.953652	−	0.300913i	$-0.902709\pi$
$967$	− 1.75000e7i	− 0.601826i	0.953652	−	0.300913i	$-0.0972913\pi$
$968$	− 5.56499e6i	− 0.190887i
$969$	0	0
$970$	0	0
$971$	−5.42920e7	−1.84794	−0.923970	−	0.382465i	$-0.875075\pi$
$971$	−5.42920e7	−1.84794	−0.923970	+	0.382465i	$0.875075\pi$
$972$	0	0
$973$	− 7.91594e7i	− 2.68053i
$974$	−1.68980e7	−0.570740
$975$	0	0
$976$	902912.	0.0303404
$977$	− 2.55925e7i	− 0.857782i	−0.903356	−	0.428891i	$-0.858904\pi$
$977$	− 2.55925e7i	− 0.857782i	0.903356	−	0.428891i	$-0.141096\pi$
$978$	0	0
$979$	−3.93221e7	−1.31123
$980$	0	0
$981$	0	0
$982$	− 2.78518e7i	− 0.921668i
$983$	− 4.82488e6i	− 0.159258i	−0.996825	−	0.0796292i	$-0.974626\pi$
$983$	− 4.82488e6i	− 0.159258i	0.996825	−	0.0796292i	$-0.0253736\pi$
$984$	0	0
$985$	0	0
$986$	−3.13826e7	−1.02801
$987$	0	0
$988$	2.20695e7i	0.719284i
$989$	3.02548e7	0.983567
$990$	0	0
$991$	1.18448e6	0.0383127	0.0191563	−	0.999817i	$-0.493902\pi$
$991$	1.18448e6	0.0383127	0.0191563	+	0.999817i	$0.493902\pi$
$992$	1.00045e6i	0.0322786i
$993$	0	0
$994$	−7.03474e6	−0.225830
$995$	0	0
$996$	0	0
$997$	− 2.58178e7i	− 0.822585i	−0.911503	−	0.411293i	$-0.865077\pi$
$997$	− 2.58178e7i	− 0.822585i	0.911503	−	0.411293i	$-0.134923\pi$
$998$	3.71833e7i	1.18174i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.c.n.199.2		2
3.2	odd	2		150.6.c.e.49.1		2
5.2	odd	4		450.6.a.a.1.1		1
5.3	odd	4		450.6.a.x.1.1		1
5.4	even	2	inner	450.6.c.n.199.1		2
15.2	even	4		150.6.a.l.1.1	yes	1
15.8	even	4		150.6.a.c.1.1	✓	1
15.14	odd	2		150.6.c.e.49.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.6.a.c.1.1	✓	1	15.8	even	4
150.6.a.l.1.1	yes	1	15.2	even	4
150.6.c.e.49.1		2	3.2	odd	2
150.6.c.e.49.2		2	15.14	odd	2
450.6.a.a.1.1		1	5.2	odd	4
450.6.a.x.1.1		1	5.3	odd	4
450.6.c.n.199.1		2	5.4	even	2	inner
450.6.c.n.199.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -16.0000 q^{4} +233.000i q^{7} -64.0000i q^{8} +O(q^{10})\)	\(q+4.00000i q^{2} -16.0000 q^{4} +233.000i q^{7} -64.0000i q^{8} +498.000 q^{11} -809.000i q^{13} -932.000 q^{14} +256.000 q^{16} +1002.00i q^{17} +1705.00 q^{19} +1992.00i q^{22} +1554.00i q^{23} +3236.00 q^{26} -3728.00i q^{28} +7830.00 q^{29} +977.000 q^{31} +1024.00i q^{32} -4008.00 q^{34} -4822.00i q^{37} +6820.00i q^{38} +8148.00 q^{41} -19469.0i q^{43} -7968.00 q^{44} -6216.00 q^{46} -8418.00i q^{47} -37482.0 q^{49} +12944.0i q^{52} +17664.0i q^{53} +14912.0 q^{56} +31320.0i q^{58} +35910.0 q^{59} +3527.00 q^{61} +3908.00i q^{62} -4096.00 q^{64} +57473.0i q^{67} -16032.0i q^{68} +7548.00 q^{71} +646.000i q^{73} +19288.0 q^{74} -27280.0 q^{76} +116034. i q^{77} +22720.0 q^{79} +32592.0i q^{82} +11574.0i q^{83} +77876.0 q^{86} -31872.0i q^{88} -78960.0 q^{89} +188497. q^{91} -24864.0i q^{92} +33672.0 q^{94} +54593.0i q^{97} -149928. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4}+O(q^{10}) \) 2 * q - 32 * q^4	\( 2 q - 32 q^{4} + 996 q^{11} - 1864 q^{14} + 512 q^{16} + 3410 q^{19} + 6472 q^{26} + 15660 q^{29} + 1954 q^{31} - 8016 q^{34} + 16296 q^{41} - 15936 q^{44} - 12432 q^{46} - 74964 q^{49} + 29824 q^{56} + 71820 q^{59} + 7054 q^{61} - 8192 q^{64} + 15096 q^{71} + 38576 q^{74} - 54560 q^{76} + 45440 q^{79} + 155752 q^{86} - 157920 q^{89} + 376994 q^{91} + 67344 q^{94}+O(q^{100}) \) 2 * q - 32 * q^4 + 996 * q^11 - 1864 * q^14 + 512 * q^16 + 3410 * q^19 + 6472 * q^26 + 15660 * q^29 + 1954 * q^31 - 8016 * q^34 + 16296 * q^41 - 15936 * q^44 - 12432 * q^46 - 74964 * q^49 + 29824 * q^56 + 71820 * q^59 + 7054 * q^61 - 8192 * q^64 + 15096 * q^71 + 38576 * q^74 - 54560 * q^76 + 45440 * q^79 + 155752 * q^86 - 157920 * q^89 + 376994 * q^91 + 67344 * q^94

Introduction

L-functions

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