LMFDB - Embedded newform 300.9.g.e.101.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,9,Mod(101,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("300.101");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 300 = 2^{2} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	300.g (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:	$122.213583018$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + 17581 x^{8} - 268094 x^{7} + 129938570 x^{6} - 2805075950 x^{5} + 497042336337 x^{4} + \cdots + 51\!\cdots\!56 $ x^10 - x^9 + 17581x^8 - 268094x^7 + 129938570x^6 - 2805075950x^5 + 497042336337x^4 - 12249078945417x^3 + 863119630222893x^2 - 20295663225220440x + 511594269195067956
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{17}\cdot 5^{7}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			101.3
Root			$-5.98515 + 56.3224i$ of defining polynomial
Character	$\chi$	$=$	300.101
Dual form			300.9.g.e.101.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-58.2133 - 56.3224i) q^{3} -2295.51 q^{7} +(216.575 + 6557.42i) q^{9} +17396.4i q^{11} +16463.9 q^{13} +13497.4i q^{17} +195706. q^{19} +(133629. + 129288. i) q^{21} -206229. i q^{23} +(356722. - 393927. i) q^{27} +367170. i q^{29} -933775. q^{31} +(979805. - 1.01270e6i) q^{33} -410.325 q^{37} +(-958418. - 927287. i) q^{39} -3.34234e6i q^{41} -1.94035e6 q^{43} -5.83328e6i q^{47} -495447. q^{49} +(760207. - 785730. i) q^{51} +7.57320e6i q^{53} +(-1.13927e7 - 1.10226e7i) q^{57} +2.32367e7i q^{59} +8.94563e6 q^{61} +(-497149. - 1.50526e7i) q^{63} +2.59768e7 q^{67} +(-1.16153e7 + 1.20053e7i) q^{69} -2.66873e7i q^{71} -1.77737e7 q^{73} -3.99335e7i q^{77} +3.48684e7 q^{79} +(-4.29529e7 + 2.84034e6i) q^{81} -4.67421e7i q^{83} +(2.06799e7 - 2.13742e7i) q^{87} -2.42212e7i q^{89} -3.77930e7 q^{91} +(5.43581e7 + 5.25925e7i) q^{93} -2.59451e7 q^{97} +(-1.14075e8 + 3.76761e6i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 137 q^{3} - 1048 q^{7} - 12455 q^{9} + 34472 q^{13} - 376030 q^{19} - 142540 q^{21} - 1458782 q^{27} - 410860 q^{31} - 523275 q^{33} - 110344 q^{37} + 3527870 q^{39} - 6252148 q^{43} + 13891530 q^{49}+ \cdots + 52292505 q^{99}+O(q^{100}) $ 10 * q - 137 * q^3 - 1048 * q^7 - 12455 * q^9 + 34472 * q^13 - 376030 * q^19 - 142540 * q^21 - 1458782 * q^27 - 410860 * q^31 - 523275 * q^33 - 110344 * q^37 + 3527870 * q^39 - 6252148 * q^43 + 13891530 * q^49 - 644745 * q^51 - 10355323 * q^57 - 51373540 * q^61 + 6641072 * q^63 - 82069258 * q^67 - 12562590 * q^69 - 33368314 * q^73 + 36211400 * q^79 - 75697775 * q^81 - 33748950 * q^87 + 64519480 * q^91 - 109348198 * q^93 + 248767112 * q^97 + 52292505 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$277$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−58.2133	−	56.3224i	−0.718683	−	0.695338i
$3$	−58.2133	−	56.3224i	−0.718683	−	0.695338i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2295.51			−0.956063			−0.478032	−	0.878343i	$-0.658650\pi$
$7$	−2295.51			−0.956063			−0.478032	+	0.878343i	$0.658650\pi$
$8$		0			0
$9$	216.575	+	6557.42i	0.0330094	+	0.999455i
$10$		0			0
$11$			17396.4i			1.18819i	0.804393	+	0.594097i	$0.202491\pi$
$11$			17396.4i			1.18819i	−0.804393	+	0.594097i	$0.797509\pi$
$12$		0			0
$13$	16463.9			0.576447			0.288224	−	0.957563i	$-0.406935\pi$
$13$	16463.9			0.576447			0.288224	+	0.957563i	$0.406935\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			13497.4i			0.161605i	0.996730	+	0.0808026i	$0.0257483\pi$
$17$			13497.4i			0.161605i	−0.996730	+	0.0808026i	$0.974252\pi$
$18$		0			0
$19$	195706.			1.50172			0.750860	−	0.660461i	$-0.229639\pi$
$19$	195706.			1.50172			0.750860	+	0.660461i	$0.229639\pi$
$20$		0			0
$21$	133629.	+	129288.i	0.687106	+	0.664787i
$22$		0			0
$23$		−	206229.i		−	0.736952i	−0.929637	−	0.368476i	$-0.879880\pi$
$23$		−	206229.i		−	0.736952i	0.929637	−	0.368476i	$-0.120120\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	356722.	−	393927.i	0.671236	−	0.741244i
$28$		0			0
$29$			367170.i			0.519128i	0.965726	+	0.259564i	$0.0835789\pi$
$29$			367170.i			0.519128i	−0.965726	+	0.259564i	$0.916421\pi$
$30$		0			0
$31$	−933775.			−1.01110			−0.505552	−	0.862796i	$-0.668711\pi$
$31$	−933775.			−1.01110			−0.505552	+	0.862796i	$0.668711\pi$
$32$		0			0
$33$	979805.	−	1.01270e6i	0.826197	−	0.853935i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−410.325			−0.000218938			−0.000109469	−	1.00000i	$-0.500035\pi$
$37$	−410.325			−0.000218938			−0.000109469		1.00000i	$0.500035\pi$
$38$		0			0
$39$	−958418.	−	927287.i	−0.414283	−	0.400826i
$40$		0			0
$41$		−	3.34234e6i		−	1.18281i	−0.806375	−	0.591405i	$-0.798573\pi$
$41$		−	3.34234e6i		−	1.18281i	0.806375	−	0.591405i	$-0.201427\pi$
$42$		0			0
$43$	−1.94035e6			−0.567553			−0.283776	−	0.958891i	$-0.591587\pi$
$43$	−1.94035e6			−0.567553			−0.283776	+	0.958891i	$0.591587\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	5.83328e6i		−	1.19542i	−0.801711	−	0.597712i	$-0.796077\pi$
$47$		−	5.83328e6i		−	1.19542i	0.801711	−	0.597712i	$-0.203923\pi$
$48$		0			0
$49$	−495447.			−0.0859435
$50$		0			0
$51$	760207.	−	785730.i	0.112370	−	0.116143i
$52$		0			0
$53$			7.57320e6i			0.959790i	0.877326	+	0.479895i	$0.159325\pi$
$53$			7.57320e6i			0.959790i	−0.877326	+	0.479895i	$0.840675\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.13927e7	−	1.10226e7i	−1.07926	−	1.04420i
$58$		0			0
$59$			2.32367e7i			1.91764i	0.284021	+	0.958818i	$0.408331\pi$
$59$			2.32367e7i			1.91764i	−0.284021	+	0.958818i	$0.591669\pi$
$60$		0			0
$61$	8.94563e6			0.646088			0.323044	−	0.946384i	$-0.395294\pi$
$61$	8.94563e6			0.646088			0.323044	+	0.946384i	$0.395294\pi$
$62$		0			0
$63$	−497149.	−	1.50526e7i	−0.0315591	−	0.955542i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	2.59768e7			1.28910			0.644551	−	0.764562i	$-0.277044\pi$
$67$	2.59768e7			1.28910			0.644551	+	0.764562i	$0.277044\pi$
$68$		0			0
$69$	−1.16153e7	+	1.20053e7i	−0.512431	+	0.529634i
$70$		0			0
$71$		−	2.66873e7i		−	1.05020i	−0.851041	−	0.525100i	$-0.824028\pi$
$71$		−	2.66873e7i		−	1.05020i	0.851041	−	0.525100i	$-0.175972\pi$
$72$		0			0
$73$	−1.77737e7			−0.625875			−0.312937	−	0.949774i	$-0.601313\pi$
$73$	−1.77737e7			−0.625875			−0.312937	+	0.949774i	$0.601313\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	3.99335e7i		−	1.13599i
$78$		0			0
$79$	3.48684e7			0.895209			0.447604	−	0.894232i	$-0.352277\pi$
$79$	3.48684e7			0.895209			0.447604	+	0.894232i	$0.352277\pi$
$80$		0			0
$81$	−4.29529e7	+	2.84034e6i	−0.997821	+	0.0659828i
$82$		0			0
$83$		−	4.67421e7i		−	0.984909i	−0.870338	−	0.492454i	$-0.836100\pi$
$83$		−	4.67421e7i		−	0.984909i	0.870338	−	0.492454i	$-0.163900\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.06799e7	−	2.13742e7i	0.360970	−	0.373088i
$88$		0			0
$89$		−	2.42212e7i		−	0.386043i	−0.981195	−	0.193021i	$-0.938171\pi$
$89$		−	2.42212e7i		−	0.386043i	0.981195	−	0.193021i	$-0.0618286\pi$
$90$		0			0
$91$	−3.77930e7			−0.551120
$92$		0			0
$93$	5.43581e7	+	5.25925e7i	0.726662	+	0.703059i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−2.59451e7			−0.293068			−0.146534	−	0.989206i	$-0.546812\pi$
$97$	−2.59451e7			−0.293068			−0.146534	+	0.989206i	$0.546812\pi$
$98$		0			0
$99$	−1.14075e8	+	3.76761e6i	−1.18755	+	0.0392216i
$100$		0			0
$101$			1.57296e8i			1.51158i	0.654814	+	0.755790i	$0.272747\pi$
$101$			1.57296e8i			1.51158i	−0.654814	+	0.755790i	$0.727253\pi$
$102$		0			0
$103$	−2.36343e7			−0.209987			−0.104994	−	0.994473i	$-0.533482\pi$
$103$	−2.36343e7			−0.209987			−0.104994	+	0.994473i	$0.533482\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	5.24020e7i		−	0.399773i	−0.979819	−	0.199886i	$-0.935943\pi$
$107$		−	5.24020e7i		−	0.399773i	0.979819	−	0.199886i	$-0.0640573\pi$
$108$		0			0
$109$	4.49101e7			0.318154			0.159077	−	0.987266i	$-0.449148\pi$
$109$	4.49101e7			0.318154			0.159077	+	0.987266i	$0.449148\pi$
$110$		0			0
$111$	23886.4	+	23110.5i	0.000157347	+	0.000152236i
$112$		0			0
$113$			1.45262e8i			0.890921i	0.895302	+	0.445460i	$0.146960\pi$
$113$			1.45262e8i			0.890921i	−0.895302	+	0.445460i	$0.853040\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	3.56566e6	+	1.07961e8i	0.0190282	+	0.576133i
$118$		0			0
$119$		−	3.09834e7i		−	0.154505i
$120$		0			0
$121$	−8.82745e7			−0.411807
$122$		0			0
$123$	−1.88248e8	+	1.94568e8i	−0.822453	+	0.850065i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	3.12436e8			1.20101			0.600504	−	0.799622i	$-0.294967\pi$
$127$	3.12436e8			1.20101			0.600504	+	0.799622i	$0.294967\pi$
$128$		0			0
$129$	1.12954e8	+	1.09285e8i	0.407890	+	0.394641i
$130$		0			0
$131$			4.47775e8i			1.52046i	0.649656	+	0.760229i	$0.274913\pi$
$131$			4.47775e8i			1.52046i	−0.649656	+	0.760229i	$0.725087\pi$
$132$		0			0
$133$	−4.49244e8			−1.43574
$134$		0			0
$135$		0			0
$136$		0			0
$137$			4.98335e8i			1.41462i	0.706904	+	0.707309i	$0.250091\pi$
$137$			4.98335e8i			1.41462i	−0.706904	+	0.707309i	$0.749909\pi$
$138$		0			0
$139$	−5.33850e8			−1.43008			−0.715039	−	0.699084i	$-0.753591\pi$
$139$	−5.33850e8			−1.43008			−0.715039	+	0.699084i	$0.753591\pi$
$140$		0			0
$141$	−3.28545e8	+	3.39575e8i	−0.831224	+	0.859130i
$142$		0			0
$143$			2.86412e8i			0.684932i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	2.88416e7	+	2.79048e7i	0.0617661	+	0.0597598i
$148$		0			0
$149$			2.63916e7i			0.0535452i	0.999642	+	0.0267726i	$0.00852301\pi$
$149$			2.63916e7i			0.0535452i	−0.999642	+	0.0267726i	$0.991477\pi$
$150$		0			0
$151$	−8.51062e8			−1.63702			−0.818509	−	0.574493i	$-0.805199\pi$
$151$	−8.51062e8			−1.63702			−0.818509	+	0.574493i	$0.805199\pi$
$152$		0			0
$153$	−8.85084e7	+	2.92320e6i	−0.161517	+	0.00533449i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.48170e8			0.408462			0.204231	−	0.978923i	$-0.434531\pi$
$157$	2.48170e8			0.408462			0.204231	+	0.978923i	$0.434531\pi$
$158$		0			0
$159$	4.26541e8	−	4.40861e8i	0.667379	−	0.689784i
$160$		0			0
$161$			4.73401e8i			0.704572i
$162$		0			0
$163$	3.20988e8			0.454713			0.227357	−	0.973812i	$-0.426992\pi$
$163$	3.20988e8			0.454713			0.227357	+	0.973812i	$0.426992\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1.46565e9i		−	1.88436i	−0.335108	−	0.942180i	$-0.608773\pi$
$167$		−	1.46565e9i		−	1.88436i	0.335108	−	0.942180i	$-0.391227\pi$
$168$		0			0
$169$	−5.44670e8			−0.667709
$170$		0			0
$171$	4.23849e7	+	1.28333e9i	0.0495709	+	1.50090i
$172$		0			0
$173$			1.44526e9i			1.61347i	0.590915	+	0.806734i	$0.298767\pi$
$173$			1.44526e9i			1.61347i	−0.590915	+	0.806734i	$0.701233\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	1.30875e9	−	1.35268e9i	1.33341	−	1.37817i
$178$		0			0
$179$			7.19759e8i			0.701092i	0.936546	+	0.350546i	$0.114004\pi$
$179$			7.19759e8i			0.701092i	−0.936546	+	0.350546i	$0.885996\pi$
$180$		0			0
$181$	−1.56467e9			−1.45784			−0.728918	−	0.684601i	$-0.759977\pi$
$181$	−1.56467e9			−1.45784			−0.728918	+	0.684601i	$0.759977\pi$
$182$		0			0
$183$	−5.20755e8	−	5.03839e8i	−0.464332	−	0.449250i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−2.34806e8			−0.192018
$188$		0			0
$189$	−8.18859e8	+	9.04263e8i	−0.641744	+	0.708676i
$190$		0			0
$191$			1.31338e9i			0.986864i	0.869784	+	0.493432i	$0.164258\pi$
$191$			1.31338e9i			0.986864i	−0.869784	+	0.493432i	$0.835742\pi$
$192$		0			0
$193$	−1.12320e9			−0.809523			−0.404761	−	0.914422i	$-0.632645\pi$
$193$	−1.12320e9			−0.809523			−0.404761	+	0.914422i	$0.632645\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	1.07566e9i		−	0.714185i	−0.934069	−	0.357092i	$-0.883768\pi$
$197$		−	1.07566e9i		−	0.714185i	0.934069	−	0.357092i	$-0.116232\pi$
$198$		0			0
$199$	−6.55984e8			−0.418293			−0.209147	−	0.977884i	$-0.567069\pi$
$199$	−6.55984e8			−0.418293			−0.209147	+	0.977884i	$0.567069\pi$
$200$		0			0
$201$	−1.51220e9	−	1.46308e9i	−0.926455	−	0.896361i
$202$		0			0
$203$		−	8.42840e8i		−	0.496319i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.35233e9	−	4.46640e7i	0.736550	−	0.0243263i
$208$		0			0
$209$			3.40457e9i			1.78434i
$210$		0			0
$211$	−1.48749e9			−0.750453			−0.375227	−	0.926933i	$-0.622435\pi$
$211$	−1.48749e9			−0.750453			−0.375227	+	0.926933i	$0.622435\pi$
$212$		0			0
$213$	−1.50309e9	+	1.55356e9i	−0.730244	+	0.754760i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	2.14349e9			0.966679
$218$		0			0
$219$	1.03467e9	+	1.00106e9i	0.449805	+	0.435195i
$220$		0			0
$221$			2.22220e8i			0.0931569i
$222$		0			0
$223$	−4.80438e9			−1.94276			−0.971378	−	0.237540i	$-0.923659\pi$
$223$	−4.80438e9			−1.94276			−0.971378	+	0.237540i	$0.923659\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	4.19807e9i		−	1.58105i	−0.612429	−	0.790525i	$-0.709808\pi$
$227$		−	4.19807e9i		−	1.58105i	0.612429	−	0.790525i	$-0.290192\pi$
$228$		0			0
$229$	−1.93560e9			−0.703841			−0.351920	−	0.936030i	$-0.614471\pi$
$229$	−1.93560e9			−0.703841			−0.351920	+	0.936030i	$0.614471\pi$
$230$		0			0
$231$	−2.24915e9	+	2.32466e9i	−0.789897	+	0.816416i
$232$		0			0
$233$			2.52861e9i			0.857942i	0.903318	+	0.428971i	$0.141124\pi$
$233$			2.52861e9i			0.857942i	−0.903318	+	0.428971i	$0.858876\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−2.02981e9	−	1.96387e9i	−0.643371	−	0.622473i
$238$		0			0
$239$		−	1.76815e9i		−	0.541910i	−0.962592	−	0.270955i	$-0.912661\pi$
$239$		−	1.76815e9i		−	0.541910i	0.962592	−	0.270955i	$-0.0873395\pi$
$240$		0			0
$241$	1.55220e9			0.460129			0.230064	−	0.973175i	$-0.426106\pi$
$241$	1.55220e9			0.460129			0.230064	+	0.973175i	$0.426106\pi$
$242$		0			0
$243$	2.66041e9	+	2.25387e9i	0.762997	+	0.646402i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	3.22208e9			0.865663
$248$		0			0
$249$	−2.63263e9	+	2.72101e9i	−0.684845	+	0.707837i
$250$		0			0
$251$			4.84318e9i			1.22021i	0.792319	+	0.610107i	$0.208873\pi$
$251$			4.84318e9i			1.22021i	−0.792319	+	0.610107i	$0.791127\pi$
$252$		0			0
$253$	3.58764e9			0.875642
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	6.97195e8i		−	0.159816i	−0.996802	−	0.0799082i	$-0.974537\pi$
$257$		−	6.97195e8i		−	0.159816i	0.996802	−	0.0799082i	$-0.0254627\pi$
$258$		0			0
$259$	941903.			0.000209318
$260$		0			0
$261$	−2.40769e9	+	7.95196e7i	−0.518845	+	0.0171361i
$262$		0			0
$263$		−	4.25701e9i		−	0.889777i	−0.895586	−	0.444889i	$-0.853243\pi$
$263$		−	4.25701e9i		−	0.889777i	0.895586	−	0.444889i	$-0.146757\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−1.36419e9	+	1.40999e9i	−0.268430	+	0.277442i
$268$		0			0
$269$		−	3.37356e9i		−	0.644288i	−0.946691	−	0.322144i	$-0.895597\pi$
$269$		−	3.37356e9i		−	0.644288i	0.946691	−	0.322144i	$-0.104403\pi$
$270$		0			0
$271$	−6.57981e9			−1.21993			−0.609966	−	0.792427i	$-0.708817\pi$
$271$	−6.57981e9			−1.21993			−0.609966	+	0.792427i	$0.708817\pi$
$272$		0			0
$273$	2.20006e9	+	2.12859e9i	0.396080	+	0.383215i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−5.05227e7			−0.00858158			−0.00429079	−	0.999991i	$-0.501366\pi$
$277$	−5.05227e7			−0.00858158			−0.00429079	+	0.999991i	$0.501366\pi$
$278$		0			0
$279$	−2.02232e8	−	6.12316e9i	−0.0333759	−	1.01055i
$280$		0			0
$281$			5.00839e9i			0.803290i	0.915795	+	0.401645i	$0.131561\pi$
$281$			5.00839e9i			0.803290i	−0.915795	+	0.401645i	$0.868439\pi$
$282$		0			0
$283$	−1.00231e10			−1.56263			−0.781314	−	0.624139i	$-0.785450\pi$
$283$	−1.00231e10			−1.56263			−0.781314	+	0.624139i	$0.785450\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			7.67236e9i			1.13084i
$288$		0			0
$289$	6.79358e9			0.973884
$290$		0			0
$291$	1.51035e9	+	1.46129e9i	0.210623	+	0.203781i
$292$		0			0
$293$			8.52756e9i			1.15706i	0.815663	+	0.578528i	$0.196373\pi$
$293$			8.52756e9i			1.15706i	−0.815663	+	0.578528i	$0.803627\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	6.85290e9	+	6.20567e9i	0.880742	+	0.797559i
$298$		0			0
$299$		−	3.39534e9i		−	0.424814i
$300$		0			0
$301$	4.45409e9			0.542616
$302$		0			0
$303$	8.85927e9	−	9.15670e9i	1.05106	−	1.08635i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.69426e9			−0.190733			−0.0953665	−	0.995442i	$-0.530402\pi$
$307$	−1.69426e9			−0.190733			−0.0953665	+	0.995442i	$0.530402\pi$
$308$		0			0
$309$	1.37583e9	+	1.33114e9i	0.150914	+	0.146012i
$310$		0			0
$311$		−	5.14916e9i		−	0.550420i	−0.961384	−	0.275210i	$-0.911253\pi$
$311$		−	5.14916e9i		−	0.550420i	0.961384	−	0.275210i	$-0.0887474\pi$
$312$		0			0
$313$	−8.32301e9			−0.867168			−0.433584	−	0.901113i	$-0.642751\pi$
$313$	−8.32301e9			−0.867168			−0.433584	+	0.901113i	$0.642751\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			1.67633e10i			1.66006i	0.557721	+	0.830029i	$0.311676\pi$
$317$			1.67633e10i			1.66006i	−0.557721	+	0.830029i	$0.688324\pi$
$318$		0			0
$319$	−6.38742e9			−0.616826
$320$		0			0
$321$	−2.95141e9	+	3.05050e9i	−0.277977	+	0.287310i
$322$		0			0
$323$			2.64152e9i			0.242686i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−2.61436e9	−	2.52944e9i	−0.228652	−	0.221225i
$328$		0			0
$329$			1.33903e10i			1.14290i
$330$		0			0
$331$	2.17827e10			1.81468			0.907341	−	0.420396i	$-0.138109\pi$
$331$	2.17827e10			1.81468			0.907341	+	0.420396i	$0.138109\pi$
$332$		0			0
$333$	−88865.9	−	2.69067e6i	−7.22700e−6	−	0.000218819i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2.33242e10			−1.80837			−0.904186	−	0.427138i	$-0.859522\pi$
$337$	−2.33242e10			−1.80837			−0.904186	+	0.427138i	$0.859522\pi$
$338$		0			0
$339$	8.18152e9	−	8.45619e9i	0.619491	−	0.640289i
$340$		0			0
$341$		−	1.62443e10i		−	1.20139i
$342$		0			0
$343$	1.43704e10			1.03823
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	1.42872e10i		−	0.985437i	−0.870189	−	0.492718i	$-0.836003\pi$
$347$		−	1.42872e10i		−	0.985437i	0.870189	−	0.492718i	$-0.163997\pi$
$348$		0			0
$349$	−1.25048e10			−0.842897			−0.421449	−	0.906852i	$-0.638478\pi$
$349$	−1.25048e10			−0.842897			−0.421449	+	0.906852i	$0.638478\pi$
$350$		0			0
$351$	5.87304e9	−	6.48558e9i	0.386932	−	0.427288i
$352$		0			0
$353$		−	3.48520e9i		−	0.224454i	−0.993683	−	0.112227i	$-0.964202\pi$
$353$		−	3.48520e9i		−	0.224454i	0.993683	−	0.112227i	$-0.0357985\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−1.74506e9	+	1.80365e9i	−0.107433	+	0.111040i
$358$		0			0
$359$		−	1.70468e10i		−	1.02628i	−0.858305	−	0.513139i	$-0.828482\pi$
$359$		−	1.70468e10i		−	1.02628i	0.858305	−	0.513139i	$-0.171518\pi$
$360$		0			0
$361$	2.13172e10			1.25517
$362$		0			0
$363$	5.13875e9	+	4.97183e9i	0.295959	+	0.286345i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	5.41628e9			0.298563			0.149282	−	0.988795i	$-0.452304\pi$
$367$	5.41628e9			0.298563			0.149282	+	0.988795i	$0.452304\pi$
$368$		0			0
$369$	2.19171e10	−	7.23865e8i	1.18217	−	0.0390438i
$370$		0			0
$371$		−	1.73843e10i		−	0.917620i
$372$		0			0
$373$	1.31731e10			0.680539			0.340269	−	0.940328i	$-0.389482\pi$
$373$	1.31731e10			0.680539			0.340269	+	0.940328i	$0.389482\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			6.04505e9i			0.299250i
$378$		0			0
$379$	7.75719e8			0.0375965			0.0187983	−	0.999823i	$-0.494016\pi$
$379$	7.75719e8			0.0375965			0.0187983	+	0.999823i	$0.494016\pi$
$380$		0			0
$381$	−1.81879e10	−	1.75971e10i	−0.863143	−	0.835106i
$382$		0			0
$383$		−	3.66582e10i		−	1.70363i	−0.523841	−	0.851816i	$-0.675502\pi$
$383$		−	3.66582e10i		−	1.70363i	0.523841	−	0.851816i	$-0.324498\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4.20230e8	−	1.27237e10i	−0.0187346	−	0.567243i
$388$		0			0
$389$		−	1.07813e10i		−	0.470839i	−0.971894	−	0.235419i	$-0.924354\pi$
$389$		−	1.07813e10i		−	0.470839i	0.971894	−	0.235419i	$-0.0756463\pi$
$390$		0			0
$391$	2.78356e9			0.119095
$392$		0			0
$393$	2.52197e10	−	2.60664e10i	1.05723	−	1.09273i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−3.48948e10			−1.40475			−0.702375	−	0.711807i	$-0.747877\pi$
$397$	−3.48948e10			−1.40475			−0.702375	+	0.711807i	$0.747877\pi$
$398$		0			0
$399$	2.61520e10	+	2.53025e10i	1.03184	+	0.998325i
$400$		0			0
$401$		−	3.53994e10i		−	1.36905i	−0.728991	−	0.684524i	$-0.760010\pi$
$401$		−	3.53994e10i		−	1.36905i	0.728991	−	0.684524i	$-0.239990\pi$
$402$		0			0
$403$	−1.53736e10			−0.582848
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	7.13816e6i		−	0.000260141i
$408$		0			0
$409$	−2.49105e10			−0.890205			−0.445102	−	0.895480i	$-0.646833\pi$
$409$	−2.49105e10			−0.890205			−0.445102	+	0.895480i	$0.646833\pi$
$410$		0			0
$411$	2.80674e10	−	2.90097e10i	0.983638	−	1.01666i
$412$		0			0
$413$		−	5.33400e10i		−	1.83338i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	3.10772e10	+	3.00677e10i	1.02777	+	0.994388i
$418$		0			0
$419$			2.06624e10i			0.670385i	0.942150	+	0.335192i	$0.108801\pi$
$419$			2.06624e10i			0.670385i	−0.942150	+	0.335192i	$0.891199\pi$
$420$		0			0
$421$	4.40984e10			1.40377			0.701883	−	0.712292i	$-0.252343\pi$
$421$	4.40984e10			1.40377			0.701883	+	0.712292i	$0.252343\pi$
$422$		0			0
$423$	3.82513e10	−	1.26334e9i	1.19477	−	0.0394602i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.05348e10			−0.617701
$428$		0			0
$429$	1.61314e10	−	1.66730e10i	0.476259	−	0.492248i
$430$		0			0
$431$			6.54249e9i			0.189598i	0.995496	+	0.0947990i	$0.0302208\pi$
$431$			6.54249e9i			0.189598i	−0.995496	+	0.0947990i	$0.969779\pi$
$432$		0			0
$433$	−1.78927e10			−0.509008			−0.254504	−	0.967072i	$-0.581912\pi$
$433$	−1.78927e10			−0.509008			−0.254504	+	0.967072i	$0.581912\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	4.03603e10i		−	1.10670i
$438$		0			0
$439$	4.08114e10			1.09881			0.549406	−	0.835556i	$-0.314854\pi$
$439$	4.08114e10			1.09881			0.549406	+	0.835556i	$0.314854\pi$
$440$		0			0
$441$	−1.07301e8	−	3.24886e9i	−0.00283694	−	0.0858967i
$442$		0			0
$443$			4.66125e10i			1.21028i	0.796117	+	0.605142i	$0.206884\pi$
$443$			4.66125e10i			1.21028i	−0.796117	+	0.605142i	$0.793116\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	1.48644e9	−	1.53634e9i	0.0372320	−	0.0384820i
$448$		0			0
$449$		−	1.18770e9i		−	0.0292229i	−0.999893	−	0.0146114i	$-0.995349\pi$
$449$		−	1.18770e9i		−	0.0292229i	0.999893	−	0.0146114i	$-0.00465113\pi$
$450$		0			0
$451$	5.81445e10			1.40541
$452$		0			0
$453$	4.95431e10	+	4.79339e10i	1.17650	+	1.13828i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3.89499e10			−0.892979			−0.446489	−	0.894789i	$-0.647326\pi$
$457$	−3.89499e10			−0.892979			−0.446489	+	0.894789i	$0.647326\pi$
$458$		0			0
$459$	5.31700e9	+	4.81483e9i	0.119789	+	0.108475i
$460$		0			0
$461$			4.68231e8i			0.0103671i	0.999987	+	0.00518354i	$0.00164998\pi$
$461$			4.68231e8i			0.0103671i	−0.999987	+	0.00518354i	$0.998350\pi$
$462$		0			0
$463$	−6.51315e10			−1.41732			−0.708658	−	0.705552i	$-0.750699\pi$
$463$	−6.51315e10			−1.41732			−0.708658	+	0.705552i	$0.750699\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			9.36135e10i			1.96821i	0.177594	+	0.984104i	$0.443169\pi$
$467$			9.36135e10i			1.96821i	−0.177594	+	0.984104i	$0.556831\pi$
$468$		0			0
$469$	−5.96300e10			−1.23246
$470$		0			0
$471$	−1.44468e10	−	1.39776e10i	−0.293554	−	0.284019i
$472$		0			0
$473$		−	3.37550e10i		−	0.674363i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−4.96607e10	+	1.64016e9i	−0.959267	+	0.0316821i
$478$		0			0
$479$		−	8.18125e10i		−	1.55410i	−0.629442	−	0.777048i	$-0.716716\pi$
$479$		−	8.18125e10i		−	1.55410i	0.629442	−	0.777048i	$-0.283284\pi$
$480$		0			0
$481$	−6.75555e6			−0.000126206
$482$		0			0
$483$	2.66631e10	−	2.75582e10i	0.489916	−	0.506364i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−1.53575e10			−0.273026			−0.136513	−	0.990638i	$-0.543590\pi$
$487$	−1.53575e10			−0.273026			−0.136513	+	0.990638i	$0.543590\pi$
$488$		0			0
$489$	−1.86857e10	−	1.80788e10i	−0.326795	−	0.316180i
$490$		0			0
$491$		−	7.76212e10i		−	1.33553i	−0.744371	−	0.667766i	$-0.767251\pi$
$491$		−	7.76212e10i		−	1.33553i	0.744371	−	0.667766i	$-0.232749\pi$
$492$		0			0
$493$	−4.95584e9			−0.0838938
$494$		0			0
$495$		0			0
$496$		0			0
$497$			6.12610e10i			1.00406i
$498$		0			0
$499$	−3.55417e10			−0.573239			−0.286620	−	0.958044i	$-0.592532\pi$
$499$	−3.55417e10			−0.573239			−0.286620	+	0.958044i	$0.592532\pi$
$500$		0			0
$501$	−8.25488e10	+	8.53202e10i	−1.31027	+	1.35426i
$502$		0			0
$503$			1.19756e11i			1.87079i	0.353603	+	0.935396i	$0.384956\pi$
$503$			1.19756e11i			1.87079i	−0.353603	+	0.935396i	$0.615044\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	3.17071e10	+	3.06771e10i	0.479871	+	0.464283i
$508$		0			0
$509$			4.51523e10i			0.672680i	0.941741	+	0.336340i	$0.109189\pi$
$509$			4.51523e10i			0.672680i	−0.941741	+	0.336340i	$0.890811\pi$
$510$		0			0
$511$	4.07997e10			0.598376
$512$		0			0
$513$	6.98126e10	−	7.70938e10i	1.00801	−	1.11314i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	1.01478e11			1.42040
$518$		0			0
$519$	8.14003e10	−	8.41331e10i	1.12191	−	1.15957i
$520$		0			0
$521$			1.25898e11i			1.70871i	0.519687	+	0.854357i	$0.326049\pi$
$521$			1.25898e11i			1.70871i	−0.519687	+	0.854357i	$0.673951\pi$
$522$		0			0
$523$	−1.27416e11			−1.70301			−0.851507	−	0.524343i	$-0.824311\pi$
$523$	−1.27416e11			−1.70301			−0.851507	+	0.524343i	$0.824311\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	1.26036e10i		−	0.163400i
$528$		0			0
$529$	3.57805e10			0.456902
$530$		0			0
$531$	−1.52373e11	+	5.03248e9i	−1.91659	+	0.0633000i
$532$		0			0
$533$		−	5.50279e10i		−	0.681827i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	4.05385e10	−	4.18995e10i	0.487496	−	0.503862i
$538$		0			0
$539$		−	8.61898e9i		−	0.102118i
$540$		0			0
$541$	1.05813e11			1.23524			0.617620	−	0.786476i	$-0.288097\pi$
$541$	1.05813e11			1.23524			0.617620	+	0.786476i	$0.288097\pi$
$542$		0			0
$543$	9.10847e10	+	8.81260e10i	1.04772	+	1.01369i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.17218e10			−0.130932			−0.0654661	−	0.997855i	$-0.520853\pi$
$547$	−1.17218e10			−0.130932			−0.0654661	+	0.997855i	$0.520853\pi$
$548$		0			0
$549$	1.93740e9	+	5.86603e10i	0.0213270	+	0.645736i
$550$		0			0
$551$			7.18572e10i			0.779586i
$552$		0			0
$553$	−8.00408e10			−0.855876
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	1.90665e10i		−	0.198085i	−0.995083	−	0.0990423i	$-0.968422\pi$
$557$		−	1.90665e10i		−	0.198085i	0.995083	−	0.0990423i	$-0.0315779\pi$
$558$		0			0
$559$	−3.19457e10			−0.327164
$560$		0			0
$561$	1.36688e10	+	1.32248e10i	0.138000	+	0.133518i
$562$		0			0
$563$			4.94309e10i			0.492000i	0.969270	+	0.246000i	$0.0791163\pi$
$563$			4.94309e10i			0.492000i	−0.969270	+	0.246000i	$0.920884\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	9.85987e10	−	6.52003e9i	0.953980	−	0.0630837i
$568$		0			0
$569$			7.53303e10i			0.718655i	0.933211	+	0.359328i	$0.116994\pi$
$569$			7.53303e10i			0.718655i	−0.933211	+	0.359328i	$0.883006\pi$
$570$		0			0
$571$	−7.43397e10			−0.699321			−0.349661	−	0.936876i	$-0.613703\pi$
$571$	−7.43397e10			−0.699321			−0.349661	+	0.936876i	$0.613703\pi$
$572$		0			0
$573$	7.39728e10	−	7.64562e10i	0.686204	−	0.709242i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.46246e11			−1.31941			−0.659706	−	0.751524i	$-0.729319\pi$
$577$	−1.46246e11			−1.31941			−0.659706	+	0.751524i	$0.729319\pi$
$578$		0			0
$579$	6.53854e10	+	6.32615e10i	0.581790	+	0.562892i
$580$		0			0
$581$			1.07297e11i			0.941635i
$582$		0			0
$583$	−1.31746e11			−1.14042
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	1.88952e11i		−	1.59147i	−0.605646	−	0.795735i	$-0.707085\pi$
$587$		−	1.88952e11i		−	1.59147i	0.605646	−	0.795735i	$-0.292915\pi$
$588$		0			0
$589$	−1.82745e11			−1.51840
$590$		0			0
$591$	−6.05838e10	+	6.26178e10i	−0.496600	+	0.513272i
$592$		0			0
$593$			5.67120e10i			0.458624i	0.973353	+	0.229312i	$0.0736476\pi$
$593$			5.67120e10i			0.458624i	−0.973353	+	0.229312i	$0.926352\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	3.81870e10	+	3.69466e10i	0.300620	+	0.290855i
$598$		0			0
$599$			4.00737e10i			0.311281i	0.987814	+	0.155640i	$0.0497441\pi$
$599$			4.00737e10i			0.311281i	−0.987814	+	0.155640i	$0.950256\pi$
$600$		0			0
$601$	−2.27069e11			−1.74044			−0.870220	−	0.492663i	$-0.836024\pi$
$601$	−2.27069e11			−1.74044			−0.870220	+	0.492663i	$0.836024\pi$
$602$		0			0
$603$	5.62592e9	+	1.70341e11i	0.0425524	+	1.28840i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	6.87812e10			0.506658			0.253329	−	0.967380i	$-0.418474\pi$
$607$	6.87812e10			0.506658			0.253329	+	0.967380i	$0.418474\pi$
$608$		0			0
$609$	−4.74708e10	+	4.90645e10i	−0.345110	+	0.356696i
$610$		0			0
$611$		−	9.60387e10i		−	0.689098i
$612$		0			0
$613$	−2.22523e11			−1.57592			−0.787959	−	0.615728i	$-0.788862\pi$
$613$	−2.22523e11			−1.57592			−0.787959	+	0.615728i	$0.788862\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.41332e11i		−	0.975215i	−0.873063	−	0.487608i	$-0.837870\pi$
$617$		−	1.41332e11i		−	0.975215i	0.873063	−	0.487608i	$-0.162130\pi$
$618$		0			0
$619$	−1.28936e11			−0.878236			−0.439118	−	0.898429i	$-0.644709\pi$
$619$	−1.28936e11			−0.878236			−0.439118	+	0.898429i	$0.644709\pi$
$620$		0			0
$621$	−8.12393e10	−	7.35666e10i	−0.546261	−	0.494668i
$622$		0			0
$623$			5.55999e10i			0.369081i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	1.91753e11	−	1.98191e11i	1.24072	−	1.28237i
$628$		0			0
$629$		−	5.53833e6i		−	3.53815e-5i
$630$		0			0
$631$	1.64177e11			1.03561			0.517804	−	0.855499i	$-0.326750\pi$
$631$	1.64177e11			1.03561			0.517804	+	0.855499i	$0.326750\pi$
$632$		0			0
$633$	8.65916e10	+	8.37789e10i	0.539338	+	0.521819i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−8.15700e9			−0.0495419
$638$		0			0
$639$	1.75000e11	−	5.77980e9i	1.04963	−	0.0346664i
$640$		0			0
$641$			3.26564e10i			0.193436i	0.995312	+	0.0967178i	$0.0308344\pi$
$641$			3.26564e10i			0.193436i	−0.995312	+	0.0967178i	$0.969166\pi$
$642$		0			0
$643$	−2.64540e11			−1.54756			−0.773781	−	0.633454i	$-0.781637\pi$
$643$	−2.64540e11			−1.54756			−0.773781	+	0.633454i	$0.781637\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	1.87714e11i		−	1.07122i	−0.844466	−	0.535610i	$-0.820082\pi$
$647$		−	1.87714e11i		−	1.07122i	0.844466	−	0.535610i	$-0.179918\pi$
$648$		0			0
$649$	−4.04234e11			−2.27853
$650$		0			0
$651$	−1.24779e11	−	1.20726e11i	−0.694735	−	0.672169i
$652$		0			0
$653$		−	1.62529e11i		−	0.893879i	−0.894564	−	0.446939i	$-0.852514\pi$
$653$		−	1.62529e11i		−	0.893879i	0.894564	−	0.446939i	$-0.147486\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−3.84934e9	−	1.16550e11i	−0.0206597	−	0.625533i
$658$		0			0
$659$			1.36472e11i			0.723608i	0.932254	+	0.361804i	$0.117839\pi$
$659$			1.36472e11i			0.723608i	−0.932254	+	0.361804i	$0.882161\pi$
$660$		0			0
$661$	−2.89793e11			−1.51804			−0.759019	−	0.651069i	$-0.774321\pi$
$661$	−2.89793e11			−1.51804			−0.759019	+	0.651069i	$0.774321\pi$
$662$		0			0
$663$	1.25160e10	−	1.29362e10i	0.0647755	−	0.0669502i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	7.57211e10			0.382572
$668$		0			0
$669$	2.79679e11	+	2.70594e11i	1.39622	+	1.35087i
$670$		0			0
$671$			1.55621e11i			0.767678i
$672$		0			0
$673$	1.95738e10			0.0954145			0.0477072	−	0.998861i	$-0.484809\pi$
$673$	1.95738e10			0.0954145			0.0477072	+	0.998861i	$0.484809\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			1.54461e11i			0.735297i	0.929965	+	0.367649i	$0.119837\pi$
$677$			1.54461e11i			0.735297i	−0.929965	+	0.367649i	$0.880163\pi$
$678$		0			0
$679$	5.95571e10			0.280191
$680$		0			0
$681$	−2.36445e11	+	2.44383e11i	−1.09937	+	1.13627i
$682$		0			0
$683$		−	3.32703e10i		−	0.152888i	−0.997074	−	0.0764440i	$-0.975643\pi$
$683$		−	3.32703e10i		−	0.152888i	0.997074	−	0.0764440i	$-0.0243566\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.12678e11	+	1.09018e11i	0.505838	+	0.489407i
$688$		0			0
$689$			1.24685e11i			0.553268i
$690$		0			0
$691$	3.41253e11			1.49680			0.748401	−	0.663246i	$-0.230822\pi$
$691$	3.41253e11			1.49680			0.748401	+	0.663246i	$0.230822\pi$
$692$		0			0
$693$	2.61861e11	−	8.64858e9i	1.13537	−	0.0374983i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	4.51130e10			0.191148
$698$		0			0
$699$	1.42417e11	−	1.47199e11i	0.596560	−	0.616588i
$700$		0			0
$701$			3.40014e11i			1.40807i	0.710165	+	0.704036i	$0.248620\pi$
$701$			3.40014e11i			1.40807i	−0.710165	+	0.704036i	$0.751380\pi$
$702$		0			0
$703$	−8.03029e7			−0.000328783
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	3.61073e11i		−	1.44517i
$708$		0			0
$709$	−2.31712e11			−0.916987			−0.458493	−	0.888698i	$-0.651611\pi$
$709$	−2.31712e11			−0.916987			−0.458493	+	0.888698i	$0.651611\pi$
$710$		0			0
$711$	7.55162e9	+	2.28647e11i	0.0295503	+	0.894721i
$712$		0			0
$713$			1.92572e11i			0.745134i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−9.95864e10	+	1.02930e11i	−0.376811	+	0.389461i
$718$		0			0
$719$		−	1.94401e11i		−	0.727417i	−0.931513	−	0.363709i	$-0.881510\pi$
$719$		−	1.94401e11i		−	0.727417i	0.931513	−	0.363709i	$-0.118490\pi$
$720$		0			0
$721$	5.42526e10			0.200761
$722$		0			0
$723$	−9.03586e10	−	8.74236e10i	−0.330687	−	0.319945i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−5.06699e10			−0.181390			−0.0906948	−	0.995879i	$-0.528909\pi$
$727$	−5.06699e10			−0.181390			−0.0906948	+	0.995879i	$0.528909\pi$
$728$		0			0
$729$	−2.79278e10	−	2.81045e11i	−0.0988843	−	0.995099i
$730$		0			0
$731$		−	2.61897e10i		−	0.0917195i
$732$		0			0
$733$	−3.08608e11			−1.06903			−0.534517	−	0.845158i	$-0.679506\pi$
$733$	−3.08608e11			−1.06903			−0.534517	+	0.845158i	$0.679506\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			4.51902e11i			1.53170i
$738$		0			0
$739$	5.24612e11			1.75898			0.879489	−	0.475920i	$-0.157885\pi$
$739$	5.24612e11			1.75898			0.879489	+	0.475920i	$0.157885\pi$
$740$		0			0
$741$	−1.87568e11	−	1.81475e11i	−0.622137	−	0.601928i
$742$		0			0
$743$			4.07793e11i			1.33809i	0.743223	+	0.669043i	$0.233296\pi$
$743$			4.07793e11i			1.33809i	−0.743223	+	0.669043i	$0.766704\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	3.06508e11	−	1.01232e10i	0.984372	−	0.0325112i
$748$		0			0
$749$			1.20289e11i			0.382208i
$750$		0			0
$751$	−2.27808e11			−0.716160			−0.358080	−	0.933691i	$-0.616568\pi$
$751$	−2.27808e11			−0.716160			−0.358080	+	0.933691i	$0.616568\pi$
$752$		0			0
$753$	2.72780e11	−	2.81938e11i	0.848461	−	0.876946i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−3.82063e11			−1.16346			−0.581730	−	0.813382i	$-0.697624\pi$
$757$	−3.82063e11			−1.16346			−0.581730	+	0.813382i	$0.697624\pi$
$758$		0			0
$759$	−2.08848e11	−	2.02064e11i	−0.629309	−	0.608867i
$760$		0			0
$761$		−	3.24968e11i		−	0.968950i	−0.874805	−	0.484475i	$-0.839011\pi$
$761$		−	3.24968e11i		−	0.968950i	0.874805	−	0.484475i	$-0.160989\pi$
$762$		0			0
$763$	−1.03091e11			−0.304175
$764$		0			0
$765$		0			0
$766$		0			0
$767$			3.82567e11i			1.10542i
$768$		0			0
$769$	−5.24436e11			−1.49964			−0.749821	−	0.661641i	$-0.769860\pi$
$769$	−5.24436e11			−1.49964			−0.749821	+	0.661641i	$0.769860\pi$
$770$		0			0
$771$	−3.92677e10	+	4.05860e10i	−0.111127	+	0.114857i
$772$		0			0
$773$			3.98964e10i			0.111742i	0.998438	+	0.0558710i	$0.0177935\pi$
$773$			3.98964e10i			0.111742i	−0.998438	+	0.0558710i	$0.982206\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−5.48313e7	−	5.30503e7i	−0.000150433	−	0.000145547i
$778$		0			0
$779$		−	6.54115e11i		−	1.77625i
$780$		0			0
$781$	4.64262e11			1.24784
$782$		0			0
$783$	1.44638e11	+	1.30978e11i	0.384801	+	0.348458i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	1.80418e10			0.0470307			0.0235154	−	0.999723i	$-0.492514\pi$
$787$	1.80418e10			0.0470307			0.0235154	+	0.999723i	$0.492514\pi$
$788$		0			0
$789$	−2.39765e11	+	2.47814e11i	−0.618696	+	0.639468i
$790$		0			0
$791$		−	3.33451e11i		−	0.851776i
$792$		0			0
$793$	1.47280e11			0.372436
$794$		0			0
$795$		0			0
$796$		0			0
$797$			6.64874e11i			1.64781i	0.566730	+	0.823903i	$0.308208\pi$
$797$			6.64874e11i			1.64781i	−0.566730	+	0.823903i	$0.691792\pi$
$798$		0			0
$799$	7.87343e10			0.193187
$800$		0			0
$801$	1.58829e11	−	5.24569e9i	0.385832	−	0.0127430i
$802$		0			0
$803$		−	3.09198e11i		−	0.743661i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−1.90007e11	+	1.96386e11i	−0.447998	+	0.463038i
$808$		0			0
$809$			5.10440e11i			1.19165i	0.803113	+	0.595827i	$0.203176\pi$
$809$			5.10440e11i			1.19165i	−0.803113	+	0.595827i	$0.796824\pi$
$810$		0			0
$811$	5.30337e11			1.22594			0.612969	−	0.790107i	$-0.289975\pi$
$811$	5.30337e11			1.22594			0.612969	+	0.790107i	$0.289975\pi$
$812$		0			0
$813$	3.83032e11	+	3.70590e11i	0.876744	+	0.848266i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−3.79738e11			−0.852306
$818$		0			0
$819$	−8.18501e9	−	2.47825e11i	−0.0181921	−	0.550820i
$820$		0			0
$821$			1.20378e11i			0.264956i	0.991186	+	0.132478i	$0.0422934\pi$
$821$			1.20378e11i			0.264956i	−0.991186	+	0.132478i	$0.957707\pi$
$822$		0			0
$823$	−8.09679e11			−1.76487			−0.882436	−	0.470432i	$-0.844098\pi$
$823$	−8.09679e11			−1.76487			−0.882436	+	0.470432i	$0.844098\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	5.02238e11i		−	1.07371i	−0.843674	−	0.536856i	$-0.819612\pi$
$827$		−	5.02238e11i		−	1.07371i	0.843674	−	0.536856i	$-0.180388\pi$
$828$		0			0
$829$	5.45793e11			1.15561			0.577803	−	0.816176i	$-0.303910\pi$
$829$	5.45793e11			1.15561			0.577803	+	0.816176i	$0.303910\pi$
$830$		0			0
$831$	2.94109e9	+	2.84556e9i	0.00616743	+	0.00596710i
$832$		0			0
$833$		−	6.68726e9i		−	0.0138889i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−3.33099e11	+	3.67840e11i	−0.678689	+	0.749474i
$838$		0			0
$839$		−	7.36835e11i		−	1.48704i	−0.668715	−	0.743519i	$-0.733155\pi$
$839$		−	7.36835e11i		−	1.48704i	0.668715	−	0.743519i	$-0.266845\pi$
$840$		0			0
$841$	3.65433e11			0.730506
$842$		0			0
$843$	2.82084e11	−	2.91555e11i	0.558558	−	0.577311i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	2.02635e11			0.393714
$848$		0			0
$849$	5.83476e11	+	5.64524e11i	1.12303	+	1.08655i
$850$		0			0
$851$			8.46210e7i			0.000161347i
$852$		0			0
$853$	4.95665e11			0.936250			0.468125	−	0.883662i	$-0.344930\pi$
$853$	4.95665e11			0.936250			0.468125	+	0.883662i	$0.344930\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1.67085e11i		−	0.309753i	−0.987934	−	0.154876i	$-0.950502\pi$
$857$		−	1.67085e11i		−	0.309753i	0.987934	−	0.154876i	$-0.0494979\pi$
$858$		0			0
$859$	6.89277e11			1.26596			0.632982	−	0.774167i	$-0.281831\pi$
$859$	6.89277e11			1.26596			0.632982	+	0.774167i	$0.281831\pi$
$860$		0			0
$861$	4.32126e11	−	4.46633e11i	0.786317	−	0.812716i
$862$		0			0
$863$		−	6.64209e11i		−	1.19746i	−0.800950	−	0.598731i	$-0.795672\pi$
$863$		−	6.64209e11i		−	1.19746i	0.800950	−	0.598731i	$-0.204328\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−3.95476e11	−	3.82631e11i	−0.699913	−	0.677179i
$868$		0			0
$869$			6.06584e11i			1.06368i
$870$		0			0
$871$	4.27680e11			0.743099
$872$		0			0
$873$	−5.61904e9	−	1.70133e11i	−0.00967398	−	0.292908i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	3.97348e11			0.671695			0.335848	−	0.941916i	$-0.390977\pi$
$877$	3.97348e11			0.671695			0.335848	+	0.941916i	$0.390977\pi$
$878$		0			0
$879$	4.80292e11	−	4.96417e11i	0.804545	−	0.831556i
$880$		0			0
$881$		−	7.33038e11i		−	1.21681i	−0.793627	−	0.608405i	$-0.791810\pi$
$881$		−	7.33038e11i		−	1.21681i	0.793627	−	0.608405i	$-0.208190\pi$
$882$		0			0
$883$	1.10665e12			1.82040			0.910199	−	0.414170i	$-0.135928\pi$
$883$	1.10665e12			1.82040			0.910199	+	0.414170i	$0.135928\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			7.21287e11i			1.16524i	0.812746	+	0.582618i	$0.197972\pi$
$887$			7.21287e11i			1.16524i	−0.812746	+	0.582618i	$0.802028\pi$
$888$		0			0
$889$	−7.17198e11			−1.14824
$890$		0			0
$891$	−4.94116e10	−	7.47224e11i	−0.0784004	−	1.18561i
$892$		0			0
$893$		−	1.14161e12i		−	1.79519i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1.91234e11	+	1.97654e11i	−0.295389	+	0.305306i
$898$		0			0
$899$		−	3.42854e11i		−	0.524892i
$900$		0			0
$901$	−1.02219e11			−0.155107
$902$		0			0
$903$	−2.59287e11	−	2.50865e11i	−0.389969	−	0.377302i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−2.73405e11			−0.403996			−0.201998	−	0.979386i	$-0.564743\pi$
$907$	−2.73405e11			−0.403996			−0.201998	+	0.979386i	$0.564743\pi$
$908$		0			0
$909$	−1.03145e12	+	3.40662e10i	−1.51076	+	0.0498963i
$910$		0			0
$911$		−	5.29336e11i		−	0.768525i	−0.923224	−	0.384263i	$-0.874456\pi$
$911$		−	5.29336e11i		−	0.768525i	0.923224	−	0.384263i	$-0.125544\pi$
$912$		0			0
$913$	8.13143e11			1.17026
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	1.02787e12i		−	1.45365i
$918$		0			0
$919$	3.37368e11			0.472978			0.236489	−	0.971634i	$-0.424003\pi$
$919$	3.37368e11			0.472978			0.236489	+	0.971634i	$0.424003\pi$
$920$		0			0
$921$	9.86283e10	+	9.54246e10i	0.137076	+	0.132624i
$922$		0			0
$923$		−	4.39378e11i		−	0.605384i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−5.11858e9	−	1.54980e11i	−0.00693156	−	0.209873i
$928$		0			0
$929$			3.33858e11i			0.448229i	0.974563	+	0.224114i	$0.0719489\pi$
$929$			3.33858e11i			0.448229i	−0.974563	+	0.224114i	$0.928051\pi$
$930$		0			0
$931$	−9.69619e10			−0.129063
$932$		0			0
$933$	−2.90013e11	+	2.99749e11i	−0.382728	+	0.395577i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−5.13902e11			−0.666688			−0.333344	−	0.942805i	$-0.608177\pi$
$937$	−5.13902e11			−0.666688			−0.333344	+	0.942805i	$0.608177\pi$
$938$		0			0
$939$	4.84510e11	+	4.68772e11i	0.623218	+	0.602975i
$940$		0			0
$941$		−	6.37307e11i		−	0.812812i	−0.913693	−	0.406406i	$-0.866782\pi$
$941$		−	6.37307e11i		−	0.812812i	0.913693	−	0.406406i	$-0.133218\pi$
$942$		0			0
$943$	−6.89288e11			−0.871674
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	1.13296e12i		−	1.40869i	−0.709857	−	0.704346i	$-0.751240\pi$
$947$		−	1.13296e12i		−	1.40869i	0.709857	−	0.704346i	$-0.248760\pi$
$948$		0			0
$949$	−2.92625e11			−0.360784
$950$		0			0
$951$	9.44150e11	−	9.75848e11i	1.15430	−	1.19305i
$952$		0			0
$953$			1.04326e11i			0.126480i	0.997998	+	0.0632400i	$0.0201434\pi$
$953$			1.04326e11i			0.126480i	−0.997998	+	0.0632400i	$0.979857\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	3.71832e11	+	3.59755e11i	0.443302	+	0.428902i
$958$		0			0
$959$		−	1.14393e12i		−	1.35246i
$960$		0			0
$961$	1.90453e10			0.0223303
$962$		0			0
$963$	3.43622e11	−	1.13489e10i	0.399555	−	0.0131962i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−4.71576e11			−0.539319			−0.269659	−	0.962956i	$-0.586911\pi$
$967$	−4.71576e11			−0.539319			−0.269659	+	0.962956i	$0.586911\pi$
$968$		0			0
$969$	1.48777e11	−	1.53772e11i	0.168749	−	0.174414i
$970$		0			0
$971$		−	4.60220e11i		−	0.517713i	−0.965916	−	0.258856i	$-0.916654\pi$
$971$		−	4.60220e11i		−	0.517713i	0.965916	−	0.258856i	$-0.0833456\pi$
$972$		0			0
$973$	1.22546e12			1.36725
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1.41831e12i			1.55665i	0.627860	+	0.778327i	$0.283931\pi$
$977$			1.41831e12i			1.55665i	−0.627860	+	0.778327i	$0.716069\pi$
$978$		0			0
$979$	4.21360e11			0.458694
$980$		0			0
$981$	9.72638e9	+	2.94494e11i	0.0105021	+	0.317981i
$982$		0			0
$983$			1.48643e12i			1.59196i	0.605326	+	0.795978i	$0.293043\pi$
$983$			1.48643e12i			1.59196i	−0.605326	+	0.795978i	$0.706957\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	7.54177e11	−	7.79496e11i	0.794702	−	0.821382i
$988$		0			0
$989$			4.00157e11i			0.418259i
$990$		0			0
$991$	5.45711e11			0.565807			0.282903	−	0.959148i	$-0.408702\pi$
$991$	5.45711e11			0.565807			0.282903	+	0.959148i	$0.408702\pi$
$992$		0			0
$993$	−1.26804e12	−	1.22686e12i	−1.30418	−	1.26182i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.51985e12			−1.53823			−0.769114	−	0.639112i	$-0.779302\pi$
$997$	−1.51985e12			−1.53823			−0.769114	+	0.639112i	$0.779302\pi$
$998$		0			0
$999$	−1.46372e8	+	1.61638e8i	−0.000146959	+	0.000162286i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	300.9.g.e.101.3	✓	10
3.2	odd	2	inner	300.9.g.e.101.4	yes	10
5.2	odd	4		300.9.b.e.149.8		20
5.3	odd	4		300.9.b.e.149.13		20
5.4	even	2		300.9.g.g.101.8	yes	10
15.2	even	4		300.9.b.e.149.14		20
15.8	even	4		300.9.b.e.149.7		20
15.14	odd	2		300.9.g.g.101.7	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.9.b.e.149.7		20	15.8	even	4
300.9.b.e.149.8		20	5.2	odd	4
300.9.b.e.149.13		20	5.3	odd	4
300.9.b.e.149.14		20	15.2	even	4
300.9.g.e.101.3	✓	10	1.1	even	1	trivial
300.9.g.e.101.4	yes	10	3.2	odd	2	inner
300.9.g.g.101.7	yes	10	15.14	odd	2
300.9.g.g.101.8	yes	10	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	300.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-58.2133 - 56.3224i) q^{3} -2295.51 q^{7} +(216.575 + 6557.42i) q^{9} +17396.4i q^{11} +16463.9 q^{13} +13497.4i q^{17} +195706. q^{19} +(133629. + 129288. i) q^{21} -206229. i q^{23} +(356722. - 393927. i) q^{27} +367170. i q^{29} -933775. q^{31} +(979805. - 1.01270e6i) q^{33} -410.325 q^{37} +(-958418. - 927287. i) q^{39} -3.34234e6i q^{41} -1.94035e6 q^{43} -5.83328e6i q^{47} -495447. q^{49} +(760207. - 785730. i) q^{51} +7.57320e6i q^{53} +(-1.13927e7 - 1.10226e7i) q^{57} +2.32367e7i q^{59} +8.94563e6 q^{61} +(-497149. - 1.50526e7i) q^{63} +2.59768e7 q^{67} +(-1.16153e7 + 1.20053e7i) q^{69} -2.66873e7i q^{71} -1.77737e7 q^{73} -3.99335e7i q^{77} +3.48684e7 q^{79} +(-4.29529e7 + 2.84034e6i) q^{81} -4.67421e7i q^{83} +(2.06799e7 - 2.13742e7i) q^{87} -2.42212e7i q^{89} -3.77930e7 q^{91} +(5.43581e7 + 5.25925e7i) q^{93} -2.59451e7 q^{97} +(-1.14075e8 + 3.76761e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 137 q^{3} - 1048 q^{7} - 12455 q^{9} + 34472 q^{13} - 376030 q^{19} - 142540 q^{21} - 1458782 q^{27} - 410860 q^{31} - 523275 q^{33} - 110344 q^{37} + 3527870 q^{39} - 6252148 q^{43} + 13891530 q^{49}+ \cdots + 52292505 q^{99}+O(q^{100}) \) 10 * q - 137 * q^3 - 1048 * q^7 - 12455 * q^9 + 34472 * q^13 - 376030 * q^19 - 142540 * q^21 - 1458782 * q^27 - 410860 * q^31 - 523275 * q^33 - 110344 * q^37 + 3527870 * q^39 - 6252148 * q^43 + 13891530 * q^49 - 644745 * q^51 - 10355323 * q^57 - 51373540 * q^61 + 6641072 * q^63 - 82069258 * q^67 - 12562590 * q^69 - 33368314 * q^73 + 36211400 * q^79 - 75697775 * q^81 - 33748950 * q^87 + 64519480 * q^91 - 109348198 * q^93 + 248767112 * q^97 + 52292505 * q^99

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