LMFDB - Embedded newform 324.4.e.i.217.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [324,4,Mod(109,324)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("324.109"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	324.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$19.1166188419$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.49787136.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.1
Root			$-1.09445 - 0.895644i$ of defining polynomial
Character	$\chi$	$=$	324.217
Dual form			324.4.e.i.109.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-10.5353 - 18.2477i) q^{5} +(15.7477 - 27.2759i) q^{7} +(-18.3296 + 31.7477i) q^{11} +(-28.2477 - 48.9265i) q^{13} -35.8010 q^{17} +83.4955 q^{19} +(34.7762 + 60.2341i) q^{23} +(-159.486 + 276.238i) q^{25} +(-40.8541 + 70.7614i) q^{29} +(36.4864 + 63.1962i) q^{31} -663.630 q^{35} -25.4682 q^{37} +(-199.742 - 345.964i) q^{41} +(41.7295 - 72.2777i) q^{43} +(-155.885 + 270.000i) q^{47} +(-324.482 - 562.019i) q^{49} -4.09919 q^{53} +772.432 q^{55} +(-176.097 - 305.009i) q^{59} +(-1.77046 + 3.06652i) q^{61} +(-595.198 + 1030.91i) q^{65} +(-246.216 - 426.458i) q^{67} -154.502 q^{71} +305.000 q^{73} +(577.298 + 999.909i) q^{77} +(-335.693 + 581.438i) q^{79} +(646.850 - 1120.38i) q^{83} +(377.175 + 653.286i) q^{85} -1183.91 q^{89} -1779.35 q^{91} +(-879.652 - 1523.60i) q^{95} +(-307.991 + 533.456i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{7} - 116 q^{13} + 448 q^{19} - 616 q^{25} - 368 q^{31} + 1336 q^{37} - 656 q^{43} - 1716 q^{49} + 2880 q^{55} - 1004 q^{61} - 320 q^{67} + 2440 q^{73} + 64 q^{79} - 612 q^{85} - 6976 q^{91}+ \cdots - 2024 q^{97}+O(q^{100}) $ 8 * q + 16 * q^7 - 116 * q^13 + 448 * q^19 - 616 * q^25 - 368 * q^31 + 1336 * q^37 - 656 * q^43 - 1716 * q^49 + 2880 * q^55 - 1004 * q^61 - 320 * q^67 + 2440 * q^73 + 64 * q^79 - 612 * q^85 - 6976 * q^91 - 2024 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−10.5353	−	18.2477i	−0.942309	−	1.63213i	−0.761052	−	0.648690i	$-0.775317\pi$
$5$	−10.5353	−	18.2477i	−0.942309	−	1.63213i	−0.181256	−	0.983436i	$-0.558016\pi$
$6$		0			0
$7$	15.7477	−	27.2759i	0.850297	−	1.47276i	−0.0306428	−	0.999530i	$-0.509755\pi$
$7$	15.7477	−	27.2759i	0.850297	−	1.47276i	0.880940	−	0.473228i	$-0.156911\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−18.3296	+	31.7477i	−0.502415	+	0.870209i	0.497581	+	0.867418i	$0.334222\pi$
$11$	−18.3296	+	31.7477i	−0.502415	+	0.870209i	−0.999996	+	0.00279137i	$0.999111\pi$
$12$		0			0
$13$	−28.2477	−	48.9265i	−0.602655	−	1.04383i	−0.992417	−	0.122913i	$-0.960776\pi$
$13$	−28.2477	−	48.9265i	−0.602655	−	1.04383i	0.389763	−	0.920915i	$-0.372557\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−35.8010			−0.510765			−0.255383	−	0.966840i	$-0.582201\pi$
$17$	−35.8010			−0.510765			−0.255383	+	0.966840i	$0.582201\pi$
$18$		0			0
$19$	83.4955			1.00817			0.504083	−	0.863655i	$-0.331830\pi$
$19$	83.4955			1.00817			0.504083	+	0.863655i	$0.331830\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	34.7762	+	60.2341i	0.315275	+	0.546073i	0.979496	−	0.201464i	$-0.0645698\pi$
$23$	34.7762	+	60.2341i	0.315275	+	0.546073i	−0.664221	+	0.747537i	$0.731236\pi$
$24$		0			0
$25$	−159.486	+	276.238i	−1.27589	+	2.20991i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−40.8541	+	70.7614i	−0.261601	+	0.453105i	−0.966667	−	0.256035i	$-0.917584\pi$
$29$	−40.8541	+	70.7614i	−0.261601	+	0.453105i	0.705067	+	0.709141i	$0.250917\pi$
$30$		0			0
$31$	36.4864	+	63.1962i	0.211392	+	0.366141i	0.952150	−	0.305630i	$-0.0988671\pi$
$31$	36.4864	+	63.1962i	0.211392	+	0.366141i	−0.740759	+	0.671771i	$0.765534\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−663.630			−3.20497
$36$		0			0
$37$	−25.4682			−0.113161			−0.0565803	−	0.998398i	$-0.518020\pi$
$37$	−25.4682			−0.113161			−0.0565803	+	0.998398i	$0.518020\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−199.742	−	345.964i	−0.760841	−	1.31782i	−0.942417	−	0.334439i	$-0.891453\pi$
$41$	−199.742	−	345.964i	−0.760841	−	1.31782i	0.181576	−	0.983377i	$-0.441880\pi$
$42$		0			0
$43$	41.7295	−	72.2777i	0.147993	−	0.256331i	−0.782493	−	0.622660i	$-0.786052\pi$
$43$	41.7295	−	72.2777i	0.147993	−	0.256331i	0.930486	+	0.366329i	$0.119385\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−155.885	+	270.000i	−0.483789	+	0.837948i	−0.999827	−	0.0186183i	$-0.994073\pi$
$47$	−155.885	+	270.000i	−0.483789	+	0.837948i	0.516037	+	0.856566i	$0.327407\pi$
$48$		0			0
$49$	−324.482	−	562.019i	−0.946011	−	1.63854i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.09919			−0.0106239			−0.00531196	−	0.999986i	$-0.501691\pi$
$53$	−4.09919			−0.0106239			−0.00531196	+	0.999986i	$0.501691\pi$
$54$		0			0
$55$	772.432			1.89372
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−176.097	−	305.009i	−0.388574	−	0.673031i	0.603684	−	0.797224i	$-0.293699\pi$
$59$	−176.097	−	305.009i	−0.388574	−	0.673031i	−0.992258	+	0.124193i	$0.960366\pi$
$60$		0			0
$61$	−1.77046	+	3.06652i	−0.00371613	+	0.00643652i	−0.867877	−	0.496778i	$-0.834516\pi$
$61$	−1.77046	+	3.06652i	−0.00371613	+	0.00643652i	0.864161	+	0.503215i	$0.167850\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−595.198	+	1030.91i	−1.13577	+	1.96722i
$66$		0			0
$67$	−246.216	−	426.458i	−0.448956	−	0.777615i	0.549362	−	0.835584i	$-0.314871\pi$
$67$	−246.216	−	426.458i	−0.448956	−	0.777615i	−0.998318	+	0.0579695i	$0.981537\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−154.502			−0.258253			−0.129126	−	0.991628i	$-0.541217\pi$
$71$	−154.502			−0.258253			−0.129126	+	0.991628i	$0.541217\pi$
$72$		0			0
$73$	305.000			0.489008			0.244504	−	0.969648i	$-0.421375\pi$
$73$	305.000			0.489008			0.244504	+	0.969648i	$0.421375\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	577.298	+	999.909i	0.854405	+	1.47987i
$78$		0			0
$79$	−335.693	+	581.438i	−0.478081	+	0.828061i	−0.999684	−	0.0251271i	$-0.992001\pi$
$79$	−335.693	+	581.438i	−0.478081	+	0.828061i	0.521603	+	0.853188i	$0.325334\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	646.850	−	1120.38i	0.855434	−	1.48166i	−0.0208078	−	0.999783i	$-0.506624\pi$
$83$	646.850	−	1120.38i	0.855434	−	1.48166i	0.876242	−	0.481872i	$-0.160043\pi$
$84$		0			0
$85$	377.175	+	653.286i	0.481298	+	0.833633i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1183.91			−1.41005			−0.705026	−	0.709181i	$-0.749065\pi$
$89$	−1183.91			−1.41005			−0.705026	+	0.709181i	$0.749065\pi$
$90$		0			0
$91$	−1779.35			−2.04974
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−879.652	−	1523.60i	−0.950004	−	1.64546i
$96$		0			0
$97$	−307.991	+	533.456i	−0.322389	+	0.558394i	−0.980980	−	0.194107i	$-0.937819\pi$
$97$	−307.991	+	533.456i	−0.322389	+	0.558394i	0.658591	+	0.752501i	$0.271153\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	593.744	−	1028.40i	0.584948	−	1.01316i	−0.409934	−	0.912115i	$-0.634448\pi$
$101$	593.744	−	1028.40i	0.584948	−	1.01316i	0.994882	−	0.101045i	$-0.0322185\pi$
$102$		0			0
$103$	693.459	+	1201.11i	0.663384	+	1.14901i	0.979721	+	0.200368i	$0.0642137\pi$
$103$	693.459	+	1201.11i	0.663384	+	1.14901i	−0.316337	+	0.948647i	$0.602453\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	678.360			0.612893			0.306447	−	0.951888i	$-0.400860\pi$
$107$	678.360			0.612893			0.306447	+	0.951888i	$0.400860\pi$
$108$		0			0
$109$	945.432			0.830788			0.415394	−	0.909641i	$-0.363644\pi$
$109$	945.432			0.830788			0.415394	+	0.909641i	$0.363644\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	993.300	+	1720.45i	0.826918	+	1.43226i	0.900444	+	0.434971i	$0.143241\pi$
$113$	993.300	+	1720.45i	0.826918	+	1.43226i	−0.0735260	+	0.997293i	$0.523425\pi$
$114$		0			0
$115$	732.757	−	1269.17i	0.594173	−	1.02914i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−563.784	+	976.502i	−0.434302	+	0.752234i
$120$		0			0
$121$	−6.44545	−	11.1638i	−0.00484256	−	0.00838756i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4087.13			2.92451
$126$		0			0
$127$	−1496.54			−1.04564			−0.522821	−	0.852442i	$-0.675120\pi$
$127$	−1496.54			−1.04564			−0.522821	+	0.852442i	$0.675120\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−42.6413	−	73.8568i	−0.0284396	−	0.0492588i	0.851455	−	0.524427i	$-0.175720\pi$
$131$	−42.6413	−	73.8568i	−0.0284396	−	0.0492588i	−0.879895	+	0.475168i	$0.842387\pi$
$132$		0			0
$133$	1314.86	−	2277.41i	0.857242	−	1.48479i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	449.753	−	778.995i	0.280475	−	0.485796i	−0.691027	−	0.722829i	$-0.742842\pi$
$137$	449.753	−	778.995i	0.280475	−	0.485796i	0.971502	+	0.237033i	$0.0761748\pi$
$138$		0			0
$139$	−1000.30	−	1732.58i	−0.610394	−	1.05723i	−0.991174	−	0.132567i	$-0.957678\pi$
$139$	−1000.30	−	1732.58i	−0.610394	−	1.05723i	0.380780	−	0.924666i	$-0.375655\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2071.07			1.21113
$144$		0			0
$145$	1721.65			0.986034
$146$		0			0
$147$		0			0
$148$		0			0
$149$	737.186	+	1276.84i	0.405320	+	0.702034i	0.994359	−	0.106071i	$-0.0338269\pi$
$149$	737.186	+	1276.84i	0.405320	+	0.702034i	−0.589039	+	0.808105i	$0.700494\pi$
$150$		0			0
$151$	546.486	−	946.542i	0.294519	−	0.510123i	−0.680354	−	0.732884i	$-0.738174\pi$
$151$	546.486	−	946.542i	0.294519	−	0.510123i	0.974873	+	0.222761i	$0.0715071\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	768.792	−	1331.59i	0.398393	−	0.690036i
$156$		0			0
$157$	−270.257	−	468.099i	−0.137381	−	0.237951i	0.789123	−	0.614235i	$-0.210535\pi$
$157$	−270.257	−	468.099i	−0.137381	−	0.237951i	−0.926505	+	0.376284i	$0.877202\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2190.58			1.07231
$162$		0			0
$163$	−3251.60			−1.56248			−0.781242	−	0.624228i	$-0.785414\pi$
$163$	−3251.60			−1.56248			−0.781242	+	0.624228i	$0.785414\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−854.291	−	1479.67i	−0.395850	−	0.685633i	0.597359	−	0.801974i	$-0.296217\pi$
$167$	−854.291	−	1479.67i	−0.395850	−	0.685633i	−0.993209	+	0.116341i	$0.962883\pi$
$168$		0			0
$169$	−497.368	+	861.467i	−0.226385	+	0.392111i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1314.87	−	2277.42i	0.577847	−	1.00086i	−0.417879	−	0.908503i	$-0.637227\pi$
$173$	1314.87	−	2277.42i	0.577847	−	1.00086i	0.995726	−	0.0923571i	$-0.0294401\pi$
$174$		0			0
$175$	5023.10	+	8700.26i	2.16977	+	3.75816i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4532.77			−1.89271			−0.946354	−	0.323131i	$-0.895265\pi$
$179$	−4532.77			−1.89271			−0.946354	+	0.323131i	$0.895265\pi$
$180$		0			0
$181$	−2327.65			−0.955874			−0.477937	−	0.878394i	$-0.658615\pi$
$181$	−2327.65			−0.955874			−0.477937	+	0.878394i	$0.658615\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	268.316	+	464.736i	0.106632	+	0.184692i
$186$		0			0
$187$	656.216	−	1136.60i	0.256616	−	0.444473i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	615.840	−	1066.67i	0.233302	−	0.404090i	−0.725476	−	0.688247i	$-0.758380\pi$
$191$	615.840	−	1066.67i	0.233302	−	0.404090i	0.958778	+	0.284157i	$0.0917137\pi$
$192$		0			0
$193$	−1071.99	−	1856.73i	−0.399810	−	0.692491i	0.593892	−	0.804544i	$-0.297590\pi$
$193$	−1071.99	−	1856.73i	−0.399810	−	0.692491i	−0.993702	+	0.112054i	$0.964257\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1984.55			0.717733			0.358866	−	0.933389i	$-0.383163\pi$
$197$	1984.55			0.717733			0.358866	+	0.933389i	$0.383163\pi$
$198$		0			0
$199$	−224.400			−0.0799362			−0.0399681	−	0.999201i	$-0.512726\pi$
$199$	−224.400			−0.0799362			−0.0399681	+	0.999201i	$0.512726\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1286.72	+	2228.66i	0.444876	+	0.770549i
$204$		0			0
$205$	−4208.70	+	7289.68i	−1.43389	+	2.48358i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1530.43	+	2650.79i	−0.506519	+	0.877316i
$210$		0			0
$211$	−1892.61	−	3278.10i	−0.617501	−	1.06954i	−0.989940	−	0.141487i	$-0.954812\pi$
$211$	−1892.61	−	3278.10i	−0.617501	−	1.06954i	0.372439	−	0.928057i	$-0.378522\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1758.54			−0.557820
$216$		0			0
$217$	2298.31			0.718983
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1011.30	+	1751.62i	0.307815	+	0.533151i
$222$		0			0
$223$	2704.16	−	4683.74i	0.812036	−	1.40649i	−0.0994007	−	0.995047i	$-0.531693\pi$
$223$	2704.16	−	4683.74i	0.812036	−	1.40649i	0.911437	−	0.411440i	$-0.134974\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2244.95	−	3888.37i	0.656400	−	1.13692i	−0.325141	−	0.945666i	$-0.605412\pi$
$227$	2244.95	−	3888.37i	0.656400	−	1.13692i	0.981541	−	0.191253i	$-0.0612550\pi$
$228$		0			0
$229$	−1558.50	−	2699.40i	−0.449731	−	0.778957i	0.548637	−	0.836061i	$-0.315147\pi$
$229$	−1558.50	−	2699.40i	−0.449731	−	0.778957i	−0.998368	+	0.0571034i	$0.981814\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1675.60			0.471124			0.235562	−	0.971859i	$-0.424307\pi$
$233$	1675.60			0.471124			0.235562	+	0.971859i	$0.424307\pi$
$234$		0			0
$235$	6569.18			1.82352
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2875.86	−	4981.13i	−0.778342	−	1.34813i	−0.932897	−	0.360144i	$-0.882728\pi$
$239$	−2875.86	−	4981.13i	−0.778342	−	1.34813i	0.154555	−	0.987984i	$-0.450606\pi$
$240$		0			0
$241$	902.705	−	1563.53i	0.241279	−	0.417908i	−0.719800	−	0.694182i	$-0.755766\pi$
$241$	902.705	−	1563.53i	0.241279	−	0.417908i	0.961079	+	0.276274i	$0.0890997\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6837.05	+	11842.1i	−1.78287	+	3.08802i
$246$		0			0
$247$	−2358.56	−	4085.14i	−0.607576	−	1.05235i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	32.6596			0.00821297			0.00410649	−	0.999992i	$-0.498693\pi$
$251$	32.6596			0.00821297			0.00410649	+	0.999992i	$0.498693\pi$
$252$		0			0
$253$	−2549.73			−0.633597
$254$		0			0
$255$		0			0
$256$		0			0
$257$	574.128	+	994.418i	0.139351	+	0.241362i	0.927251	−	0.374440i	$-0.122165\pi$
$257$	574.128	+	994.418i	0.139351	+	0.241362i	−0.787900	+	0.615803i	$0.788832\pi$
$258$		0			0
$259$	−401.066	+	694.667i	−0.0962202	+	0.166658i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2317.01	+	4013.17i	−0.543242	+	0.940923i	0.455473	+	0.890250i	$0.349470\pi$
$263$	−2317.01	+	4013.17i	−0.543242	+	0.940923i	−0.998715	+	0.0506737i	$0.983863\pi$
$264$		0			0
$265$	43.1863	+	74.8009i	0.0100110	+	0.0173396i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−8134.42			−1.84373			−0.921866	−	0.387508i	$-0.873336\pi$
$269$	−8134.42			−1.84373			−0.921866	+	0.387508i	$0.873336\pi$
$270$		0			0
$271$	6891.35			1.54472			0.772361	−	0.635184i	$-0.219076\pi$
$271$	6891.35			1.54472			0.772361	+	0.635184i	$0.219076\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5846.63	−	10126.7i	−1.28205	−	2.22058i
$276$		0			0
$277$	910.891	−	1577.71i	0.197582	−	0.342222i	−0.750162	−	0.661254i	$-0.770025\pi$
$277$	910.891	−	1577.71i	0.197582	−	0.342222i	0.947744	+	0.319032i	$0.103358\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2270.60	−	3932.80i	0.482039	−	0.834915i	−0.517749	−	0.855533i	$-0.673230\pi$
$281$	2270.60	−	3932.80i	0.482039	−	0.834915i	0.999787	+	0.0206174i	$0.00656319\pi$
$282$		0			0
$283$	1378.97	+	2388.45i	0.289652	+	0.501691i	0.973727	−	0.227720i	$-0.0731272\pi$
$283$	1378.97	+	2388.45i	0.289652	+	0.501691i	−0.684075	+	0.729412i	$0.739794\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12581.9			−2.58777
$288$		0			0
$289$	−3631.29			−0.739119
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−834.532	−	1445.45i	−0.166396	−	0.288206i	0.770754	−	0.637132i	$-0.219880\pi$
$293$	−834.532	−	1445.45i	−0.166396	−	0.288206i	−0.937150	+	0.348927i	$0.886546\pi$
$294$		0			0
$295$	−3710.48	+	6426.74i	−0.732314	+	1.26840i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1964.70	−	3402.95i	0.380004	−	0.658187i
$300$		0			0
$301$	−1314.29	−	2276.42i	−0.251676	−	0.435916i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	74.6094			0.0140069
$306$		0			0
$307$	−7500.21			−1.39433			−0.697165	−	0.716910i	$-0.745556\pi$
$307$	−7500.21			−1.39433			−0.697165	+	0.716910i	$0.745556\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4243.26	+	7349.54i	0.773676	+	1.34005i	0.935536	+	0.353232i	$0.114917\pi$
$311$	4243.26	+	7349.54i	0.773676	+	1.34005i	−0.161860	+	0.986814i	$0.551749\pi$
$312$		0			0
$313$	−4682.83	+	8110.90i	−0.845653	+	1.46471i	0.0394004	+	0.999224i	$0.487455\pi$
$313$	−4682.83	+	8110.90i	−0.845653	+	1.46471i	−0.885053	+	0.465490i	$0.845878\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	252.944	−	438.111i	0.0448162	−	0.0776239i	−0.842747	−	0.538310i	$-0.819063\pi$
$317$	252.944	−	438.111i	0.0448162	−	0.0776239i	0.887563	+	0.460686i	$0.152396\pi$
$318$		0			0
$319$	−1497.67	−	2594.05i	−0.262864	−	0.455294i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2989.22			−0.514937
$324$		0			0
$325$	18020.5			3.07569
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4909.66	+	8503.77i	0.822730	+	1.42501i
$330$		0			0
$331$	−3299.26	+	5714.48i	−0.547866	+	0.948931i	0.450555	+	0.892749i	$0.351226\pi$
$331$	−3299.26	+	5714.48i	−0.547866	+	0.948931i	−0.998421	+	0.0561825i	$0.982107\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−5187.93	+	8985.76i	−0.846110	+	1.46551i
$336$		0			0
$337$	1806.21	+	3128.45i	0.291960	+	0.505689i	0.974273	−	0.225371i	$-0.0723593\pi$
$337$	1806.21	+	3128.45i	0.291960	+	0.505689i	−0.682313	+	0.731060i	$0.739026\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2675.12			−0.424826
$342$		0			0
$343$	−9636.46			−1.51697
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1304.78	−	2259.95i	−0.201857	−	0.349626i	0.747270	−	0.664521i	$-0.231364\pi$
$347$	−1304.78	−	2259.95i	−0.201857	−	0.349626i	−0.949127	+	0.314894i	$0.898031\pi$
$348$		0			0
$349$	−338.764	+	586.756i	−0.0519587	+	0.0899952i	−0.890835	−	0.454327i	$-0.849880\pi$
$349$	−338.764	+	586.756i	−0.0519587	+	0.0899952i	0.838876	+	0.544322i	$0.183213\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−774.941	+	1342.24i	−0.116844	+	0.202380i	−0.918515	−	0.395385i	$-0.870611\pi$
$353$	−774.941	+	1342.24i	−0.116844	+	0.202380i	0.801671	+	0.597765i	$0.203944\pi$
$354$		0			0
$355$	1627.72	+	2819.30i	0.243354	+	0.421501i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	8381.31			1.23217			0.616084	−	0.787680i	$-0.288718\pi$
$359$	8381.31			1.23217			0.616084	+	0.787680i	$0.288718\pi$
$360$		0			0
$361$	112.491			0.0164005
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3213.28	−	5565.56i	−0.460796	−	0.798122i
$366$		0			0
$367$	2102.91	−	3642.35i	0.299104	−	0.518063i	−0.676827	−	0.736142i	$-0.736646\pi$
$367$	2102.91	−	3642.35i	0.299104	−	0.518063i	0.975931	+	0.218079i	$0.0699789\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−64.5530	+	111.809i	−0.00903348	+	0.0156465i
$372$		0			0
$373$	−4006.42	−	6939.32i	−0.556151	−	0.963282i	−0.997813	−	0.0661007i	$-0.978944\pi$
$373$	−4006.42	−	6939.32i	−0.556151	−	0.963282i	0.441662	−	0.897182i	$-0.354389\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	4616.14			0.630619
$378$		0			0
$379$	8049.26			1.09093			0.545466	−	0.838133i	$-0.316353\pi$
$379$	8049.26			1.09093			0.545466	+	0.838133i	$0.316353\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1613.73	+	2795.07i	0.215295	+	0.372902i	0.953364	−	0.301824i	$-0.0975954\pi$
$383$	1613.73	+	2795.07i	0.215295	+	0.372902i	−0.738069	+	0.674725i	$0.764262\pi$
$384$		0			0
$385$	12164.0	−	21068.7i	1.61023	−	2.78899i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4455.97	−	7717.96i	0.580788	−	1.00595i	−0.414598	−	0.910004i	$-0.636078\pi$
$389$	4455.97	−	7717.96i	0.580788	−	1.00595i	0.995386	−	0.0959494i	$-0.0305887\pi$
$390$		0			0
$391$	−1245.02	−	2156.44i	−0.161032	−	0.278915i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	14146.6			1.80200
$396$		0			0
$397$	2972.50			0.375783			0.187891	−	0.982190i	$-0.439835\pi$
$397$	2972.50			0.375783			0.187891	+	0.982190i	$0.439835\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6136.59	−	10628.9i	−0.764206	−	1.32364i	−0.940665	−	0.339336i	$-0.889798\pi$
$401$	−6136.59	−	10628.9i	−0.764206	−	1.32364i	0.176459	−	0.984308i	$-0.443536\pi$
$402$		0			0
$403$	2061.31	−	3570.30i	0.254792	−	0.441313i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	466.820	−	808.557i	0.0568536	−	0.0984734i
$408$		0			0
$409$	1124.65	+	1947.95i	0.135967	+	0.235501i	0.925966	−	0.377606i	$-0.123253\pi$
$409$	1124.65	+	1947.95i	0.135967	+	0.235501i	−0.790000	+	0.613107i	$0.789919\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−11092.5			−1.32161
$414$		0			0
$415$	−27259.1			−3.22433
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4730.46	−	8193.40i	−0.551547	−	0.955307i	−0.998163	−	0.0605814i	$-0.980705\pi$
$419$	−4730.46	−	8193.40i	−0.551547	−	0.955307i	0.446617	−	0.894725i	$-0.352629\pi$
$420$		0			0
$421$	−767.457	+	1329.27i	−0.0888446	+	0.153883i	−0.907023	−	0.421081i	$-0.861651\pi$
$421$	−767.457	+	1329.27i	−0.0888446	+	0.153883i	0.818178	+	0.574964i	$0.194984\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5709.77	−	9889.60i	0.651681	−	1.12874i
$426$		0			0
$427$	55.7613	+	96.5814i	0.00631962	+	0.0109459i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	3991.88			0.446130			0.223065	−	0.974804i	$-0.428394\pi$
$431$	3991.88			0.446130			0.223065	+	0.974804i	$0.428394\pi$
$432$		0			0
$433$	−3058.08			−0.339404			−0.169702	−	0.985495i	$-0.554281\pi$
$433$	−3058.08			−0.339404			−0.169702	+	0.985495i	$0.554281\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2903.65	+	5029.27i	0.317850	+	0.550533i
$438$		0			0
$439$	7381.89	−	12785.8i	0.802548	−	1.39005i	−0.115387	−	0.993321i	$-0.536811\pi$
$439$	7381.89	−	12785.8i	0.802548	−	1.39005i	0.917934	−	0.396733i	$-0.129856\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5782.33	+	10015.3i	−0.620150	+	1.07413i	0.369307	+	0.929307i	$0.379595\pi$
$443$	−5782.33	+	10015.3i	−0.620150	+	1.07413i	−0.989457	+	0.144824i	$0.953738\pi$
$444$		0			0
$445$	12472.9	+	21603.7i	1.32870	+	2.30138i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	17266.9			1.81486			0.907432	−	0.420198i	$-0.138039\pi$
$449$	17266.9			1.81486			0.907432	+	0.420198i	$0.138039\pi$
$450$		0			0
$451$	14644.7			1.52903
$452$		0			0
$453$		0			0
$454$		0			0
$455$	18746.0	+	32469.1i	1.93149	+	3.34544i
$456$		0			0
$457$	−6350.76	+	10999.8i	−0.650057	+	1.12593i	0.333052	+	0.942909i	$0.391922\pi$
$457$	−6350.76	+	10999.8i	−0.650057	+	1.12593i	−0.983109	+	0.183023i	$0.941412\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8249.92	−	14289.3i	0.833486	−	1.44364i	−0.0617714	−	0.998090i	$-0.519675\pi$
$461$	8249.92	−	14289.3i	0.833486	−	1.44364i	0.895257	−	0.445550i	$-0.146992\pi$
$462$		0			0
$463$	682.418	+	1181.98i	0.0684982	+	0.118642i	0.898240	−	0.439504i	$-0.144846\pi$
$463$	682.418	+	1181.98i	0.0684982	+	0.118642i	−0.829742	+	0.558147i	$0.811513\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	10333.5			1.02393			0.511966	−	0.859006i	$-0.328917\pi$
$467$	10333.5			1.02393			0.511966	+	0.859006i	$0.328917\pi$
$468$		0			0
$469$	−15509.4			−1.52698
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1529.77	+	2649.64i	0.148708	+	0.257570i
$474$		0			0
$475$	−13316.4	+	23064.7i	−1.28631	+	2.22796i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	6094.31	−	10555.7i	0.581328	−	1.00689i	−0.413994	−	0.910280i	$-0.635867\pi$
$479$	6094.31	−	10555.7i	0.581328	−	1.00689i	0.995322	−	0.0966103i	$-0.0308001\pi$
$480$		0			0
$481$	719.418	+	1246.07i	0.0681968	+	0.118120i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12979.1			1.21516
$486$		0			0
$487$	9816.46			0.913401			0.456701	−	0.889620i	$-0.349031\pi$
$487$	9816.46			0.913401			0.456701	+	0.889620i	$0.349031\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1732.01	−	2999.93i	−0.159195	−	0.275733i	0.775384	−	0.631490i	$-0.217556\pi$
$491$	−1732.01	−	2999.93i	−0.159195	−	0.275733i	−0.934578	+	0.355757i	$0.884223\pi$
$492$		0			0
$493$	1462.62	−	2533.33i	0.133616	−	0.231430i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2433.05	+	4214.16i	−0.219592	+	0.380344i
$498$		0			0
$499$	6183.45	+	10710.1i	0.554728	+	0.960817i	0.997925	+	0.0643928i	$0.0205111\pi$
$499$	6183.45	+	10710.1i	0.554728	+	0.960817i	−0.443196	+	0.896424i	$0.646156\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−6249.98			−0.554021			−0.277011	−	0.960867i	$-0.589344\pi$
$503$	−6249.98			−0.554021			−0.277011	+	0.960867i	$0.589344\pi$
$504$		0			0
$505$	−25021.2			−2.20481
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3740.71	+	6479.10i	0.325745	+	0.564207i	0.981663	−	0.190625i	$-0.0610516\pi$
$509$	3740.71	+	6479.10i	0.325745	+	0.564207i	−0.655918	+	0.754832i	$0.727718\pi$
$510$		0			0
$511$	4803.06	−	8319.14i	0.415802	−	0.720190i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	14611.6	−	25308.1i	1.25022	−	2.16545i
$516$		0			0
$517$	−5714.59	−	9897.96i	−0.486127	−	0.841996i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−17700.1			−1.48840			−0.744200	−	0.667957i	$-0.767169\pi$
$521$	−17700.1			−1.48840			−0.744200	+	0.667957i	$0.767169\pi$
$522$		0			0
$523$	21707.9			1.81495			0.907475	−	0.420107i	$-0.138007\pi$
$523$	21707.9			1.81495			0.907475	+	0.420107i	$0.138007\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1306.25	−	2262.49i	−0.107972	−	0.187012i
$528$		0			0
$529$	3664.74	−	6347.51i	0.301203	−	0.521699i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−11284.5	+	19545.4i	−0.917049	+	1.58838i
$534$		0			0
$535$	−7146.75	−	12378.5i	−0.577534	−	1.00032i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	23790.4			1.90116
$540$		0			0
$541$	−19282.0			−1.53235			−0.766173	−	0.642634i	$-0.777841\pi$
$541$	−19282.0			−1.53235			−0.766173	+	0.642634i	$0.777841\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−9960.44	−	17252.0i	−0.782859	−	1.35595i
$546$		0			0
$547$	3892.26	−	6741.60i	0.304243	−	0.526965i	−0.672849	−	0.739780i	$-0.734930\pi$
$547$	3892.26	−	6741.60i	0.304243	−	0.526965i	0.977093	+	0.212815i	$0.0682630\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−3411.13	+	5908.25i	−0.263737	+	0.456806i
$552$		0			0
$553$	10572.8	+	18312.6i	0.813023	+	1.40820i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8556.03			0.650863			0.325431	−	0.945566i	$-0.394490\pi$
$557$	8556.03			0.650863			0.325431	+	0.945566i	$0.394490\pi$
$558$		0			0
$559$	−4715.06			−0.356754
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1481.61	+	2566.23i	0.110910	+	0.192102i	0.916138	−	0.400864i	$-0.131290\pi$
$563$	1481.61	+	2566.23i	0.110910	+	0.192102i	−0.805227	+	0.592966i	$0.797957\pi$
$564$		0			0
$565$	20929.5	−	36250.9i	1.55842	−	2.69927i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−7242.04	+	12543.6i	−0.533571	+	0.924172i	0.465660	+	0.884964i	$0.345817\pi$
$569$	−7242.04	+	12543.6i	−0.533571	+	0.924172i	−0.999231	+	0.0392086i	$0.987516\pi$
$570$		0			0
$571$	−8324.90	−	14419.1i	−0.610133	−	1.05678i	−0.991217	−	0.132242i	$-0.957782\pi$
$571$	−8324.90	−	14419.1i	−0.610133	−	1.05678i	0.381084	−	0.924540i	$-0.375551\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−22185.3			−1.60903
$576$		0			0
$577$	14355.4			1.03574			0.517872	−	0.855458i	$-0.326724\pi$
$577$	14355.4			1.03574			0.517872	+	0.855458i	$0.326724\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−20372.8	−	35286.8i	−1.45475	−	2.51969i
$582$		0			0
$583$	75.1364	−	130.140i	0.00533762	−	0.00924502i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4003.11	−	6933.59i	0.281475	−	0.487530i	−0.690273	−	0.723549i	$-0.742510\pi$
$587$	4003.11	−	6933.59i	0.281475	−	0.487530i	0.971748	+	0.236019i	$0.0758429\pi$
$588$		0			0
$589$	3046.45	+	5276.60i	0.213118	+	0.369131i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−17866.7			−1.23726			−0.618632	−	0.785681i	$-0.712313\pi$
$593$	−17866.7			−1.23726			−0.618632	+	0.785681i	$0.712313\pi$
$594$		0			0
$595$	23758.6			1.63699
$596$		0			0
$597$		0			0
$598$		0			0
$599$	7906.02	+	13693.6i	0.539284	+	0.934068i	0.998943	+	0.0459720i	$0.0146385\pi$
$599$	7906.02	+	13693.6i	0.539284	+	0.934068i	−0.459658	+	0.888096i	$0.652028\pi$
$600$		0			0
$601$	−11455.2	+	19840.9i	−0.777481	+	1.34664i	0.155909	+	0.987771i	$0.450169\pi$
$601$	−11455.2	+	19840.9i	−0.777481	+	1.34664i	−0.933390	+	0.358865i	$0.883164\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−135.810	+	235.230i	−0.00912638	+	0.0158073i
$606$		0			0
$607$	−8.36137	−	14.4823i	−0.000559107	−	0.000968401i	0.865746	−	0.500484i	$-0.166845\pi$
$607$	−8.36137	−	14.4823i	−0.000559107	−	0.000968401i	−0.866305	+	0.499516i	$0.833511\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	17613.5			1.16623
$612$		0			0
$613$	162.745			0.0107230			0.00536152	−	0.999986i	$-0.498293\pi$
$613$	162.745			0.0107230			0.00536152	+	0.999986i	$0.498293\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4446.59	−	7701.72i	−0.290134	−	0.502528i	0.683707	−	0.729757i	$-0.260367\pi$
$617$	−4446.59	−	7701.72i	−0.290134	−	0.502528i	−0.973841	+	0.227229i	$0.927033\pi$
$618$		0			0
$619$	−10778.8	+	18669.5i	−0.699898	+	1.21226i	0.268603	+	0.963251i	$0.413438\pi$
$619$	−10778.8	+	18669.5i	−0.699898	+	1.21226i	−0.968501	+	0.249009i	$0.919895\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−18644.0	+	32292.3i	−1.19896	+	2.07667i
$624$		0			0
$625$	−23123.5	−	40051.1i	−1.47990	−	2.56327i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	911.785			0.0577985
$630$		0			0
$631$	3085.90			0.194687			0.0973437	−	0.995251i	$-0.468965\pi$
$631$	3085.90			0.194687			0.0973437	+	0.995251i	$0.468965\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	15766.6	+	27308.5i	0.985317	+	1.70662i
$636$		0			0
$637$	−18331.7	+	31751.5i	−1.14024	+	1.97495i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6283.27	−	10882.9i	0.387167	−	0.670593i	−0.604900	−	0.796301i	$-0.706787\pi$
$641$	6283.27	−	10882.9i	0.387167	−	0.670593i	0.992067	+	0.125708i	$0.0401203\pi$
$642$		0			0
$643$	−438.114	−	758.835i	−0.0268702	−	0.0465405i	0.852278	−	0.523090i	$-0.175221\pi$
$643$	−438.114	−	758.835i	−0.0268702	−	0.0465405i	−0.879148	+	0.476549i	$0.841887\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−11125.4			−0.676022			−0.338011	−	0.941142i	$-0.609754\pi$
$647$	−11125.4			−0.676022			−0.338011	+	0.941142i	$0.609754\pi$
$648$		0			0
$649$	12911.1			0.780903
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10330.4	+	17892.8i	0.619080	+	1.07228i	0.989654	+	0.143475i	$0.0458278\pi$
$653$	10330.4	+	17892.8i	0.619080	+	1.07228i	−0.370574	+	0.928803i	$0.620839\pi$
$654$		0			0
$655$	−898.479	+	1556.21i	−0.0535977	+	0.0928340i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1826.87	−	3164.24i	0.107989	−	0.187043i	−0.806966	−	0.590597i	$-0.798892\pi$
$659$	1826.87	−	3164.24i	0.107989	−	0.187043i	0.914956	+	0.403555i	$0.132225\pi$
$660$		0			0
$661$	6670.38	+	11553.4i	0.392508	+	0.679844i	0.992780	−	0.119953i	$-0.0382742\pi$
$661$	6670.38	+	11553.4i	0.392508	+	0.679844i	−0.600272	+	0.799796i	$0.704941\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−55410.1			−3.23114
$666$		0			0
$667$	−5683.00			−0.329905
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−64.9034	−	112.416i	−0.00373408	−	0.00646761i
$672$		0			0
$673$	2110.98	−	3656.32i	0.120910	−	0.209422i	−0.799217	−	0.601043i	$-0.794752\pi$
$673$	2110.98	−	3656.32i	0.120910	−	0.209422i	0.920127	+	0.391621i	$0.128086\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9341.03	−	16179.1i	0.530288	−	0.918485i	−0.469088	−	0.883152i	$-0.655417\pi$
$677$	9341.03	−	16179.1i	0.530288	−	0.918485i	0.999376	−	0.0353338i	$-0.0112494\pi$
$678$		0			0
$679$	9700.31	+	16801.4i	0.548253	+	0.949602i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−18674.1			−1.04619			−0.523093	−	0.852275i	$-0.675222\pi$
$683$	−18674.1			−1.04619			−0.523093	+	0.852275i	$0.675222\pi$
$684$		0			0
$685$	−18953.2			−1.05717
$686$		0			0
$687$		0			0
$688$		0			0
$689$	115.793	+	200.559i	0.00640255	+	0.0110895i
$690$		0			0
$691$	−17764.2	+	30768.4i	−0.977974	+	1.69390i	−0.308227	+	0.951313i	$0.599735\pi$
$691$	−17764.2	+	30768.4i	−0.977974	+	1.69390i	−0.669748	+	0.742589i	$0.733598\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−21077.1	+	36506.6i	−1.15036	+	1.99248i
$696$		0			0
$697$	7150.96	+	12385.8i	0.388611	+	0.673094i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−13722.2			−0.739346			−0.369673	−	0.929162i	$-0.620530\pi$
$701$	−13722.2			−0.739346			−0.369673	+	0.929162i	$0.620530\pi$
$702$		0			0
$703$	−2126.48			−0.114085
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−18700.2	−	32389.8i	−0.994760	−	1.72297i
$708$		0			0
$709$	−1476.42	+	2557.23i	−0.0782059	+	0.135457i	−0.902476	−	0.430740i	$-0.858252\pi$
$709$	−1476.42	+	2557.23i	−0.0782059	+	0.135457i	0.824270	+	0.566197i	$0.191586\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2537.71	+	4395.45i	−0.133293	+	0.230871i
$714$		0			0
$715$	−21819.4	−	37792.4i	−1.14126	−	1.97672i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8101.57			−0.420219			−0.210109	−	0.977678i	$-0.567382\pi$
$719$	−8101.57			−0.420219			−0.210109	+	0.977678i	$0.567382\pi$
$720$		0			0
$721$	43681.6			2.25629
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−13031.3	−	22570.9i	−0.667547	−	1.15623i
$726$		0			0
$727$	−1700.22	+	2944.87i	−0.0867368	+	0.150233i	−0.906130	−	0.422999i	$-0.860977\pi$
$727$	−1700.22	+	2944.87i	−0.0867368	+	0.150233i	0.819393	+	0.573232i	$0.194311\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1493.96	+	2587.61i	−0.0755896	+	0.130925i
$732$		0			0
$733$	10207.0	+	17679.0i	0.514330	+	0.890845i	0.999862	+	0.0166265i	$0.00529264\pi$
$733$	10207.0	+	17679.0i	0.514330	+	0.890845i	−0.485532	+	0.874219i	$0.661374\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	18052.1			0.902250
$738$		0			0
$739$	−9505.82			−0.473176			−0.236588	−	0.971610i	$-0.576029\pi$
$739$	−9505.82			−0.473176			−0.236588	+	0.971610i	$0.576029\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	18.5803	+	32.1820i	0.000917422	+	0.00158902i	0.866484	−	0.499205i	$-0.166375\pi$
$743$	18.5803	+	32.1820i	0.000917422	+	0.00158902i	−0.865566	+	0.500794i	$0.833041\pi$
$744$		0			0
$745$	15533.0	−	26903.9i	0.763872	−	1.32307i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	10682.6	−	18502.9i	0.521141	−	0.902643i
$750$		0			0
$751$	−5088.90	−	8814.24i	−0.247266	−	0.428277i	0.715500	−	0.698612i	$-0.246199\pi$
$751$	−5088.90	−	8814.24i	−0.247266	−	0.428277i	−0.962766	+	0.270335i	$0.912865\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−23029.7			−1.11011
$756$		0			0
$757$	17129.3			0.822425			0.411213	−	0.911539i	$-0.365105\pi$
$757$	17129.3			0.822425			0.411213	+	0.911539i	$0.365105\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−11468.9	−	19864.7i	−0.546317	−	0.946248i	−0.998523	−	0.0543348i	$-0.982696\pi$
$761$	−11468.9	−	19864.7i	−0.546317	−	0.946248i	0.452206	−	0.891913i	$-0.350637\pi$
$762$		0			0
$763$	14888.4	−	25787.5i	0.706417	−	1.22355i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9948.68	+	17231.6i	−0.468352	+	0.811210i
$768$		0			0
$769$	−2635.08	−	4564.09i	−0.123567	−	0.214025i	0.797605	−	0.603181i	$-0.206100\pi$
$769$	−2635.08	−	4564.09i	−0.123567	−	0.214025i	−0.921172	+	0.389156i	$0.872767\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	31.1847			0.00145102			0.000725509	−	1.00000i	$-0.499769\pi$
$773$	31.1847			0.00145102			0.000725509		1.00000i	$0.499769\pi$
$774$		0			0
$775$	−23276.3			−1.07885
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−16677.6	−	28886.4i	−0.767055	−	1.32858i
$780$		0			0
$781$	2831.95	−	4905.07i	0.129750	−	0.224734i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−5694.49	+	9863.15i	−0.258911	+	0.448447i
$786$		0			0
$787$	−11531.6	−	19973.3i	−0.522307	−	0.904663i	−0.999663	−	0.0259526i	$-0.991738\pi$
$787$	−11531.6	−	19973.3i	−0.522307	−	0.904663i	0.477356	−	0.878710i	$-0.341595\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	62568.8			2.81251
$792$		0			0
$793$	200.045			0.00895816
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17022.0	+	29483.1i	0.756527	+	1.31034i	0.944612	+	0.328190i	$0.106439\pi$
$797$	17022.0	+	29483.1i	0.756527	+	1.31034i	−0.188085	+	0.982153i	$0.560228\pi$
$798$		0			0
$799$	5580.82	−	9666.26i	0.247103	−	0.427995i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−5590.52	+	9683.06i	−0.245685	+	0.425539i
$804$		0			0
$805$	−23078.5	−	39973.1i	−1.01045	−	1.75015i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−16017.6			−0.696103			−0.348052	−	0.937475i	$-0.613157\pi$
$809$	−16017.6			−0.696103			−0.348052	+	0.937475i	$0.613157\pi$
$810$		0			0
$811$	15085.2			0.653160			0.326580	−	0.945170i	$-0.394104\pi$
$811$	15085.2			0.653160			0.326580	+	0.945170i	$0.394104\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	34256.7	+	59334.3i	1.47234	+	2.55017i
$816$		0			0
$817$	3484.23	−	6034.86i	0.149202	−	0.258425i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5347.65	−	9262.39i	0.227325	−	0.393739i	−0.729689	−	0.683779i	$-0.760335\pi$
$821$	5347.65	−	9262.39i	0.227325	−	0.393739i	0.957015	+	0.290040i	$0.0936686\pi$
$822$		0			0
$823$	14691.9	+	25447.1i	0.622268	+	1.07780i	0.989062	+	0.147498i	$0.0471219\pi$
$823$	14691.9	+	25447.1i	0.622268	+	1.07780i	−0.366794	+	0.930302i	$0.619545\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	5402.42			0.227159			0.113580	−	0.993529i	$-0.463768\pi$
$827$	5402.42			0.227159			0.113580	+	0.993529i	$0.463768\pi$
$828$		0			0
$829$	25212.9			1.05631			0.528155	−	0.849148i	$-0.322884\pi$
$829$	25212.9			1.05631			0.528155	+	0.849148i	$0.322884\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	11616.8	+	20120.8i	0.483190	+	0.836909i
$834$		0			0
$835$	−18000.5	+	31177.7i	−0.746026	+	1.29216i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−15158.4	+	26255.1i	−0.623748	+	1.08036i	0.365033	+	0.930994i	$0.381058\pi$
$839$	−15158.4	+	26255.1i	−0.623748	+	1.08036i	−0.988782	+	0.149369i	$0.952276\pi$
$840$		0			0
$841$	8856.39	+	15339.7i	0.363130	+	0.628960i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	20959.8			0.853299
$846$		0			0
$847$	−406.005			−0.0164705
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−885.686	−	1534.05i	−0.0356767	−	0.0617939i
$852$		0			0
$853$	12664.5	−	21935.5i	0.508350	−	0.880488i	−0.491603	−	0.870819i	$-0.663589\pi$
$853$	12664.5	−	21935.5i	0.508350	−	0.880488i	0.999953	−	0.00966889i	$-0.00307775\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1005.46	+	1741.51i	−0.0400770	+	0.0694154i	−0.885368	−	0.464891i	$-0.846094\pi$
$857$	−1005.46	+	1741.51i	−0.0400770	+	0.0694154i	0.845291	+	0.534306i	$0.179427\pi$
$858$		0			0
$859$	−3482.25	−	6031.43i	−0.138315	−	0.239569i	0.788544	−	0.614979i	$-0.210835\pi$
$859$	−3482.25	−	6031.43i	−0.138315	−	0.239569i	−0.926859	+	0.375410i	$0.877502\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	14772.4			0.582686			0.291343	−	0.956619i	$-0.405898\pi$
$863$	14772.4			0.582686			0.291343	+	0.956619i	$0.405898\pi$
$864$		0			0
$865$	−55410.2			−2.17804
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−12306.2	−	21315.0i	−0.480391	−	0.832062i
$870$		0			0
$871$	−13910.1	+	24093.0i	−0.541131	+	0.937266i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	64363.1	−	111480.i	2.48671	−	4.30710i
$876$		0			0
$877$	12561.5	+	21757.1i	0.483661	+	0.837726i	0.999824	−	0.0187648i	$-0.00597336\pi$
$877$	12561.5	+	21757.1i	0.483661	+	0.837726i	−0.516163	+	0.856491i	$0.672640\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	20881.3			0.798533			0.399267	−	0.916835i	$-0.369265\pi$
$881$	20881.3			0.798533			0.399267	+	0.916835i	$0.369265\pi$
$882$		0			0
$883$	−38366.4			−1.46221			−0.731106	−	0.682264i	$-0.760995\pi$
$883$	−38366.4			−1.46221			−0.731106	+	0.682264i	$0.760995\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−20493.3	−	35495.4i	−0.775758	−	1.34365i	−0.934367	−	0.356311i	$-0.884034\pi$
$887$	−20493.3	−	35495.4i	−0.775758	−	1.34365i	0.158609	−	0.987341i	$-0.449299\pi$
$888$		0			0
$889$	−23567.1	+	40819.4i	−0.889107	+	1.53998i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−13015.7	+	22543.8i	−0.487740	+	0.844791i
$894$		0			0
$895$	47754.2	+	82712.7i	1.78352	+	3.08914i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5962.47			−0.221201
$900$		0			0
$901$	146.755			0.00542632
$902$		0			0
$903$		0			0
$904$		0			0
$905$	24522.6	+	42474.4i	0.900728	+	1.56011i
$906$		0			0
$907$	16897.9	−	29268.0i	0.618616	−	1.07147i	−0.371123	−	0.928584i	$-0.621027\pi$
$907$	16897.9	−	29268.0i	0.618616	−	1.07147i	0.989739	−	0.142890i	$-0.0456396\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	26068.8	−	45152.6i	0.948078	−	1.64212i	0.198612	−	0.980078i	$-0.436357\pi$
$911$	26068.8	−	45152.6i	0.948078	−	1.64212i	0.749467	−	0.662042i	$-0.230310\pi$
$912$		0			0
$913$	23713.0	+	41072.0i	0.859567	+	1.48881i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2686.01			−0.0967284
$918$		0			0
$919$	28749.5			1.03194			0.515972	−	0.856605i	$-0.327431\pi$
$919$	28749.5			1.03194			0.515972	+	0.856605i	$0.327431\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4364.32	+	7559.22i	0.155637	+	0.269572i
$924$		0			0
$925$	4061.83	−	7035.29i	0.144381	−	0.250075i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4030.63	+	6981.26i	−0.142347	+	0.246553i	−0.928380	−	0.371632i	$-0.878798\pi$
$929$	−4030.63	+	6981.26i	−0.142347	+	0.246553i	0.786033	+	0.618185i	$0.212132\pi$
$930$		0			0
$931$	−27092.8	−	46926.0i	−0.953737	−	1.65192i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−27653.8			−0.967247
$936$		0			0
$937$	−38362.0			−1.33749			−0.668746	−	0.743491i	$-0.733169\pi$
$937$	−38362.0			−1.33749			−0.668746	+	0.743491i	$0.733169\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12891.3	+	22328.4i	0.446594	+	0.773523i	0.998162	−	0.0606066i	$-0.0193035\pi$
$941$	12891.3	+	22328.4i	0.446594	+	0.773523i	−0.551568	+	0.834130i	$0.685970\pi$
$942$		0			0
$943$	13892.5	−	24062.6i	0.479749	−	0.830949i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	24029.5	−	41620.3i	0.824554	−	1.42817i	−0.0777053	−	0.996976i	$-0.524759\pi$
$947$	24029.5	−	41620.3i	0.824554	−	1.42817i	0.902260	−	0.431193i	$-0.141907\pi$
$948$		0			0
$949$	−8615.56	−	14922.6i	−0.294703	−	0.510440i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	49570.9			1.68495			0.842476	−	0.538733i	$-0.181097\pi$
$953$	49570.9			1.68495			0.842476	+	0.538733i	$0.181097\pi$
$954$		0			0
$955$	−25952.3			−0.879368
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−14165.2	−	24534.8i	−0.476974	−	0.826142i
$960$		0			0
$961$	12233.0	−	21188.2i	0.410627	−	0.711227i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−22587.5	+	39122.6i	−0.753488	+	1.30508i
$966$		0			0
$967$	21589.6	+	37394.3i	0.717968	+	1.24356i	0.961804	+	0.273741i	$0.0882610\pi$
$967$	21589.6	+	37394.3i	0.717968	+	1.24356i	−0.243836	+	0.969817i	$0.578406\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	23494.7			0.776498			0.388249	−	0.921554i	$-0.373080\pi$
$971$	23494.7			0.776498			0.388249	+	0.921554i	$0.373080\pi$
$972$		0			0
$973$	−63010.1			−2.07606
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−10096.4	−	17487.4i	−0.330616	−	0.572643i	0.652017	−	0.758204i	$-0.273923\pi$
$977$	−10096.4	−	17487.4i	−0.330616	−	0.572643i	−0.982633	+	0.185561i	$0.940590\pi$
$978$		0			0
$979$	21700.6	−	37586.6i	0.708432	−	1.22704i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−24610.9	+	42627.4i	−0.798542	+	1.38312i	0.122023	+	0.992527i	$0.461062\pi$
$983$	−24610.9	+	42627.4i	−0.798542	+	1.38312i	−0.920566	+	0.390588i	$0.872272\pi$
$984$		0			0
$985$	−20907.9	−	36213.5i	−0.676326	−	1.17143i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5804.77			0.186634
$990$		0			0
$991$	38703.5			1.24062			0.620312	−	0.784355i	$-0.287006\pi$
$991$	38703.5			1.24062			0.620312	+	0.784355i	$0.287006\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2364.13	+	4094.79i	0.0753245	+	0.130466i
$996$		0			0
$997$	−4171.39	+	7225.06i	−0.132507	+	0.229508i	−0.924642	−	0.380837i	$-0.875636\pi$
$997$	−4171.39	+	7225.06i	−0.132507	+	0.229508i	0.792136	+	0.610345i	$0.208969\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.4.e.i.217.1		8
3.2	odd	2	inner	324.4.e.i.217.4		8
9.2	odd	6		324.4.a.e.1.1	✓	4
9.4	even	3	inner	324.4.e.i.109.1		8
9.5	odd	6	inner	324.4.e.i.109.4		8
9.7	even	3		324.4.a.e.1.4	yes	4
36.7	odd	6		1296.4.a.z.1.4		4
36.11	even	6		1296.4.a.z.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.4.a.e.1.1	✓	4	9.2	odd	6
324.4.a.e.1.4	yes	4	9.7	even	3
324.4.e.i.109.1		8	9.4	even	3	inner
324.4.e.i.109.4		8	9.5	odd	6	inner
324.4.e.i.217.1		8	1.1	even	1	trivial
324.4.e.i.217.4		8	3.2	odd	2	inner
1296.4.a.z.1.1		4	36.11	even	6
1296.4.a.z.1.4		4	36.7	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-10.5353 - 18.2477i) q^{5} +(15.7477 - 27.2759i) q^{7} +(-18.3296 + 31.7477i) q^{11} +(-28.2477 - 48.9265i) q^{13} -35.8010 q^{17} +83.4955 q^{19} +(34.7762 + 60.2341i) q^{23} +(-159.486 + 276.238i) q^{25} +(-40.8541 + 70.7614i) q^{29} +(36.4864 + 63.1962i) q^{31} -663.630 q^{35} -25.4682 q^{37} +(-199.742 - 345.964i) q^{41} +(41.7295 - 72.2777i) q^{43} +(-155.885 + 270.000i) q^{47} +(-324.482 - 562.019i) q^{49} -4.09919 q^{53} +772.432 q^{55} +(-176.097 - 305.009i) q^{59} +(-1.77046 + 3.06652i) q^{61} +(-595.198 + 1030.91i) q^{65} +(-246.216 - 426.458i) q^{67} -154.502 q^{71} +305.000 q^{73} +(577.298 + 999.909i) q^{77} +(-335.693 + 581.438i) q^{79} +(646.850 - 1120.38i) q^{83} +(377.175 + 653.286i) q^{85} -1183.91 q^{89} -1779.35 q^{91} +(-879.652 - 1523.60i) q^{95} +(-307.991 + 533.456i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{7} - 116 q^{13} + 448 q^{19} - 616 q^{25} - 368 q^{31} + 1336 q^{37} - 656 q^{43} - 1716 q^{49} + 2880 q^{55} - 1004 q^{61} - 320 q^{67} + 2440 q^{73} + 64 q^{79} - 612 q^{85} - 6976 q^{91}+ \cdots - 2024 q^{97}+O(q^{100}) \) 8 * q + 16 * q^7 - 116 * q^13 + 448 * q^19 - 616 * q^25 - 368 * q^31 + 1336 * q^37 - 656 * q^43 - 1716 * q^49 + 2880 * q^55 - 1004 * q^61 - 320 * q^67 + 2440 * q^73 + 64 * q^79 - 612 * q^85 - 6976 * q^91 - 2024 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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