LMFDB - Embedded newform 324.4.e.i.109.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,4,Mod(109,324)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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			109.4
Root			$1.09445 - 0.895644i$ of defining polynomial
Character	$\chi$	$=$	324.109
Dual form			324.4.e.i.217.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(10.5353 - 18.2477i) q^{5} +(15.7477 + 27.2759i) q^{7} +O(q^{10})$	$q+(10.5353 - 18.2477i) q^{5} +(15.7477 + 27.2759i) q^{7} +(18.3296 + 31.7477i) q^{11} +(-28.2477 + 48.9265i) q^{13} +35.8010 q^{17} +83.4955 q^{19} +(-34.7762 + 60.2341i) q^{23} +(-159.486 - 276.238i) q^{25} +(40.8541 + 70.7614i) q^{29} +(36.4864 - 63.1962i) q^{31} +663.630 q^{35} -25.4682 q^{37} +(199.742 - 345.964i) q^{41} +(41.7295 + 72.2777i) q^{43} +(155.885 + 270.000i) q^{47} +(-324.482 + 562.019i) q^{49} +4.09919 q^{53} +772.432 q^{55} +(176.097 - 305.009i) q^{59} +(-1.77046 - 3.06652i) q^{61} +(595.198 + 1030.91i) q^{65} +(-246.216 + 426.458i) q^{67} +154.502 q^{71} +305.000 q^{73} +(-577.298 + 999.909i) q^{77} +(-335.693 - 581.438i) q^{79} +(-646.850 - 1120.38i) q^{83} +(377.175 - 653.286i) q^{85} +1183.91 q^{89} -1779.35 q^{91} +(879.652 - 1523.60i) q^{95} +(-307.991 - 533.456i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{7}+O(q^{10}) $ 8 * q + 16 * q^7	$ 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}+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

Display 100 coefficients 10 coefficients

Character values

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

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

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	0.181256	−	0.983436i	$-0.441984\pi$
$5$	10.5353	−	18.2477i	0.942309	−	1.63213i	0.761052	−	0.648690i	$-0.224683\pi$
$6$		0			0
$7$	15.7477	+	27.2759i	0.850297	+	1.47276i	0.880940	+	0.473228i	$0.156911\pi$
$7$	15.7477	+	27.2759i	0.850297	+	1.47276i	−0.0306428	+	0.999530i	$0.509755\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	18.3296	+	31.7477i	0.502415	+	0.870209i	0.999996	+	0.00279137i	$0.000888521\pi$
$11$	18.3296	+	31.7477i	0.502415	+	0.870209i	−0.497581	+	0.867418i	$0.665778\pi$
$12$		0			0
$13$	−28.2477	+	48.9265i	−0.602655	+	1.04383i	0.389763	+	0.920915i	$0.372557\pi$
$13$	−28.2477	+	48.9265i	−0.602655	+	1.04383i	−0.992417	+	0.122913i	$0.960776\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	35.8010			0.510765			0.255383	−	0.966840i	$-0.417799\pi$
$17$	35.8010			0.510765			0.255383	+	0.966840i	$0.417799\pi$
$18$		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	$-0.935430\pi$
$23$	−34.7762	+	60.2341i	−0.315275	+	0.546073i	0.664221	+	0.747537i	$0.268764\pi$
$24$		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	$-0.0824164\pi$
$29$	40.8541	+	70.7614i	0.261601	+	0.453105i	−0.705067	+	0.709141i	$0.749083\pi$
$30$		0			0
$31$	36.4864	−	63.1962i	0.211392	−	0.366141i	−0.740759	−	0.671771i	$-0.765534\pi$
$31$	36.4864	−	63.1962i	0.211392	−	0.366141i	0.952150	+	0.305630i	$0.0988671\pi$
$32$		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	−0.181576	−	0.983377i	$-0.558120\pi$
$41$	199.742	−	345.964i	0.760841	−	1.31782i	0.942417	−	0.334439i	$-0.108547\pi$
$42$		0			0
$43$	41.7295	+	72.2777i	0.147993	+	0.256331i	0.930486	−	0.366329i	$-0.119385\pi$
$43$	41.7295	+	72.2777i	0.147993	+	0.256331i	−0.782493	+	0.622660i	$0.786052\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	155.885	+	270.000i	0.483789	+	0.837948i	0.999827	−	0.0186183i	$-0.00592674\pi$
$47$	155.885	+	270.000i	0.483789	+	0.837948i	−0.516037	+	0.856566i	$0.672593\pi$
$48$		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	$-0.498309\pi$
$53$	4.09919			0.0106239			0.00531196	+	0.999986i	$0.498309\pi$
$54$		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	$-0.706301\pi$
$59$	176.097	−	305.009i	0.388574	−	0.673031i	0.992258	+	0.124193i	$0.0396343\pi$
$60$		0			0
$61$	−1.77046	−	3.06652i	−0.00371613	−	0.00643652i	0.864161	−	0.503215i	$-0.167850\pi$
$61$	−1.77046	−	3.06652i	−0.00371613	−	0.00643652i	−0.867877	+	0.496778i	$0.834516\pi$
$62$		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	−0.998318	−	0.0579695i	$-0.981537\pi$
$67$	−246.216	+	426.458i	−0.448956	+	0.777615i	0.549362	+	0.835584i	$0.314871\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	154.502			0.258253			0.129126	−	0.991628i	$-0.458783\pi$
$71$	154.502			0.258253			0.129126	+	0.991628i	$0.458783\pi$
$72$		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	0.521603	−	0.853188i	$-0.325334\pi$
$79$	−335.693	−	581.438i	−0.478081	−	0.828061i	−0.999684	+	0.0251271i	$0.992001\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−646.850	−	1120.38i	−0.855434	−	1.48166i	−0.876242	−	0.481872i	$-0.839957\pi$
$83$	−646.850	−	1120.38i	−0.855434	−	1.48166i	0.0208078	−	0.999783i	$-0.493376\pi$
$84$		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	$-0.250935\pi$
$89$	1183.91			1.41005			0.705026	+	0.709181i	$0.250935\pi$
$90$		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	0.658591	−	0.752501i	$-0.271153\pi$
$97$	−307.991	−	533.456i	−0.322389	−	0.558394i	−0.980980	+	0.194107i	$0.937819\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−593.744	−	1028.40i	−0.584948	−	1.01316i	−0.994882	−	0.101045i	$-0.967781\pi$
$101$	−593.744	−	1028.40i	−0.584948	−	1.01316i	0.409934	−	0.912115i	$-0.365552\pi$
$102$		0			0
$103$	693.459	−	1201.11i	0.663384	−	1.14901i	−0.316337	−	0.948647i	$-0.602453\pi$
$103$	693.459	−	1201.11i	0.663384	−	1.14901i	0.979721	−	0.200368i	$-0.0642137\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−678.360			−0.612893			−0.306447	−	0.951888i	$-0.599140\pi$
$107$	−678.360			−0.612893			−0.306447	+	0.951888i	$0.599140\pi$
$108$		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	0.0735260	+	0.997293i	$0.476575\pi$
$113$	−993.300	+	1720.45i	−0.826918	+	1.43226i	−0.900444	+	0.434971i	$0.856759\pi$
$114$		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	$-0.824280\pi$
$131$	42.6413	−	73.8568i	0.0284396	−	0.0492588i	0.879895	+	0.475168i	$0.157613\pi$
$132$		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	$-0.257158\pi$
$137$	−449.753	−	778.995i	−0.280475	−	0.485796i	−0.971502	+	0.237033i	$0.923825\pi$
$138$		0			0
$139$	−1000.30	+	1732.58i	−0.610394	+	1.05723i	0.380780	+	0.924666i	$0.375655\pi$
$139$	−1000.30	+	1732.58i	−0.610394	+	1.05723i	−0.991174	+	0.132567i	$0.957678\pi$
$140$		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	$-0.966173\pi$
$149$	−737.186	+	1276.84i	−0.405320	+	0.702034i	0.589039	+	0.808105i	$0.299506\pi$
$150$		0			0
$151$	546.486	+	946.542i	0.294519	+	0.510123i	0.974873	−	0.222761i	$-0.0715071\pi$
$151$	546.486	+	946.542i	0.294519	+	0.510123i	−0.680354	+	0.732884i	$0.738174\pi$
$152$		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	−0.926505	−	0.376284i	$-0.877202\pi$
$157$	−270.257	+	468.099i	−0.137381	+	0.237951i	0.789123	+	0.614235i	$0.210535\pi$
$158$		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	$-0.703783\pi$
$167$	854.291	−	1479.67i	0.395850	−	0.685633i	0.993209	+	0.116341i	$0.0371166\pi$
$168$		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	−0.995726	−	0.0923571i	$-0.970560\pi$
$173$	−1314.87	−	2277.42i	−0.577847	−	1.00086i	0.417879	−	0.908503i	$-0.362773\pi$
$174$		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	$-0.104735\pi$
$179$	4532.77			1.89271			0.946354	+	0.323131i	$0.104735\pi$
$180$		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	$-0.241620\pi$
$191$	−615.840	−	1066.67i	−0.233302	−	0.404090i	−0.958778	+	0.284157i	$0.908286\pi$
$192$		0			0
$193$	−1071.99	+	1856.73i	−0.399810	+	0.692491i	−0.993702	−	0.112054i	$-0.964257\pi$
$193$	−1071.99	+	1856.73i	−0.399810	+	0.692491i	0.593892	+	0.804544i	$0.297590\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1984.55			−0.717733			−0.358866	−	0.933389i	$-0.616837\pi$
$197$	−1984.55			−0.717733			−0.358866	+	0.933389i	$0.616837\pi$
$198$		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	0.372439	+	0.928057i	$0.378522\pi$
$211$	−1892.61	+	3278.10i	−0.617501	+	1.06954i	−0.989940	+	0.141487i	$0.954812\pi$
$212$		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	0.911437	+	0.411440i	$0.134974\pi$
$223$	2704.16	+	4683.74i	0.812036	+	1.40649i	−0.0994007	+	0.995047i	$0.531693\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2244.95	−	3888.37i	−0.656400	−	1.13692i	−0.981541	−	0.191253i	$-0.938745\pi$
$227$	−2244.95	−	3888.37i	−0.656400	−	1.13692i	0.325141	−	0.945666i	$-0.394588\pi$
$228$		0			0
$229$	−1558.50	+	2699.40i	−0.449731	+	0.778957i	−0.998368	−	0.0571034i	$-0.981814\pi$
$229$	−1558.50	+	2699.40i	−0.449731	+	0.778957i	0.548637	+	0.836061i	$0.315147\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1675.60			−0.471124			−0.235562	−	0.971859i	$-0.575693\pi$
$233$	−1675.60			−0.471124			−0.235562	+	0.971859i	$0.575693\pi$
$234$		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	−0.154555	−	0.987984i	$-0.549394\pi$
$239$	2875.86	−	4981.13i	0.778342	−	1.34813i	0.932897	−	0.360144i	$-0.117272\pi$
$240$		0			0
$241$	902.705	+	1563.53i	0.241279	+	0.417908i	0.961079	−	0.276274i	$-0.0890997\pi$
$241$	902.705	+	1563.53i	0.241279	+	0.417908i	−0.719800	+	0.694182i	$0.755766\pi$
$242$		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	$-0.501307\pi$
$251$	−32.6596			−0.00821297			−0.00410649	+	0.999992i	$0.501307\pi$
$252$		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	$-0.877835\pi$
$257$	−574.128	+	994.418i	−0.139351	+	0.241362i	0.787900	+	0.615803i	$0.211168\pi$
$258$		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	0.998715	+	0.0506737i	$0.0161368\pi$
$263$	2317.01	+	4013.17i	0.543242	+	0.940923i	−0.455473	+	0.890250i	$0.650530\pi$
$264$		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	$-0.126664\pi$
$269$	8134.42			1.84373			0.921866	+	0.387508i	$0.126664\pi$
$270$		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	0.947744	−	0.319032i	$-0.103358\pi$
$277$	910.891	+	1577.71i	0.197582	+	0.342222i	−0.750162	+	0.661254i	$0.770025\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2270.60	−	3932.80i	−0.482039	−	0.834915i	0.517749	−	0.855533i	$-0.326770\pi$
$281$	−2270.60	−	3932.80i	−0.482039	−	0.834915i	−0.999787	+	0.0206174i	$0.993437\pi$
$282$		0			0
$283$	1378.97	−	2388.45i	0.289652	−	0.501691i	−0.684075	−	0.729412i	$-0.739794\pi$
$283$	1378.97	−	2388.45i	0.289652	−	0.501691i	0.973727	+	0.227720i	$0.0731272\pi$
$284$		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	$-0.780120\pi$
$293$	834.532	−	1445.45i	0.166396	−	0.288206i	0.937150	+	0.348927i	$0.113454\pi$
$294$		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	0.161860	+	0.986814i	$0.448251\pi$
$311$	−4243.26	+	7349.54i	−0.773676	+	1.34005i	−0.935536	+	0.353232i	$0.885083\pi$
$312$		0			0
$313$	−4682.83	−	8110.90i	−0.845653	−	1.46471i	−0.885053	−	0.465490i	$-0.845878\pi$
$313$	−4682.83	−	8110.90i	−0.845653	−	1.46471i	0.0394004	−	0.999224i	$-0.487455\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−252.944	−	438.111i	−0.0448162	−	0.0776239i	0.842747	−	0.538310i	$-0.180937\pi$
$317$	−252.944	−	438.111i	−0.0448162	−	0.0776239i	−0.887563	+	0.460686i	$0.847604\pi$
$318$		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	−0.998421	−	0.0561825i	$-0.982107\pi$
$331$	−3299.26	−	5714.48i	−0.547866	−	0.948931i	0.450555	−	0.892749i	$-0.351226\pi$
$332$		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	−0.682313	−	0.731060i	$-0.739026\pi$
$337$	1806.21	−	3128.45i	0.291960	−	0.505689i	0.974273	+	0.225371i	$0.0723593\pi$
$338$		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	$-0.768636\pi$
$347$	1304.78	−	2259.95i	0.201857	−	0.349626i	0.949127	+	0.314894i	$0.101969\pi$
$348$		0			0
$349$	−338.764	−	586.756i	−0.0519587	−	0.0899952i	0.838876	−	0.544322i	$-0.183213\pi$
$349$	−338.764	−	586.756i	−0.0519587	−	0.0899952i	−0.890835	+	0.454327i	$0.849880\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	774.941	+	1342.24i	0.116844	+	0.202380i	0.918515	−	0.395385i	$-0.129389\pi$
$353$	774.941	+	1342.24i	0.116844	+	0.202380i	−0.801671	+	0.597765i	$0.796056\pi$
$354$		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	$-0.711282\pi$
$359$	−8381.31			−1.23217			−0.616084	+	0.787680i	$0.711282\pi$
$360$		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	0.975931	−	0.218079i	$-0.0699789\pi$
$367$	2102.91	+	3642.35i	0.299104	+	0.518063i	−0.676827	+	0.736142i	$0.736646\pi$
$368$		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	0.441662	+	0.897182i	$0.354389\pi$
$373$	−4006.42	+	6939.32i	−0.556151	+	0.963282i	−0.997813	+	0.0661007i	$0.978944\pi$
$374$		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	$-0.902405\pi$
$383$	−1613.73	+	2795.07i	−0.215295	+	0.372902i	0.738069	+	0.674725i	$0.235738\pi$
$384$		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	−0.995386	−	0.0959494i	$-0.969411\pi$
$389$	−4455.97	−	7717.96i	−0.580788	−	1.00595i	0.414598	−	0.910004i	$-0.363922\pi$
$390$		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	−0.176459	−	0.984308i	$-0.556464\pi$
$401$	6136.59	−	10628.9i	0.764206	−	1.32364i	0.940665	−	0.339336i	$-0.110202\pi$
$402$		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	−0.790000	−	0.613107i	$-0.789919\pi$
$409$	1124.65	−	1947.95i	0.135967	−	0.235501i	0.925966	+	0.377606i	$0.123253\pi$
$410$		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	−0.446617	−	0.894725i	$-0.647371\pi$
$419$	4730.46	−	8193.40i	0.551547	−	0.955307i	0.998163	−	0.0605814i	$-0.0192955\pi$
$420$		0			0
$421$	−767.457	−	1329.27i	−0.0888446	−	0.153883i	0.818178	−	0.574964i	$-0.194984\pi$
$421$	−767.457	−	1329.27i	−0.0888446	−	0.153883i	−0.907023	+	0.421081i	$0.861651\pi$
$422$		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	$-0.571606\pi$
$431$	−3991.88			−0.446130			−0.223065	+	0.974804i	$0.571606\pi$
$432$		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	0.917934	+	0.396733i	$0.129856\pi$
$439$	7381.89	+	12785.8i	0.802548	+	1.39005i	−0.115387	+	0.993321i	$0.536811\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	5782.33	+	10015.3i	0.620150	+	1.07413i	0.989457	+	0.144824i	$0.0462617\pi$
$443$	5782.33	+	10015.3i	0.620150	+	1.07413i	−0.369307	+	0.929307i	$0.620405\pi$
$444$		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	$-0.861961\pi$
$449$	−17266.9			−1.81486			−0.907432	+	0.420198i	$0.861961\pi$
$450$		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	−0.983109	−	0.183023i	$-0.941412\pi$
$457$	−6350.76	−	10999.8i	−0.650057	−	1.12593i	0.333052	−	0.942909i	$-0.391922\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8249.92	−	14289.3i	−0.833486	−	1.44364i	−0.895257	−	0.445550i	$-0.853008\pi$
$461$	−8249.92	−	14289.3i	−0.833486	−	1.44364i	0.0617714	−	0.998090i	$-0.480325\pi$
$462$		0			0
$463$	682.418	−	1181.98i	0.0684982	−	0.118642i	−0.829742	−	0.558147i	$-0.811513\pi$
$463$	682.418	−	1181.98i	0.0684982	−	0.118642i	0.898240	+	0.439504i	$0.144846\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10333.5			−1.02393			−0.511966	−	0.859006i	$-0.671083\pi$
$467$	−10333.5			−1.02393			−0.511966	+	0.859006i	$0.671083\pi$
$468$		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	−0.995322	−	0.0966103i	$-0.969200\pi$
$479$	−6094.31	−	10555.7i	−0.581328	−	1.00689i	0.413994	−	0.910280i	$-0.364133\pi$
$480$		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	$-0.782444\pi$
$491$	1732.01	−	2999.93i	0.159195	−	0.275733i	0.934578	+	0.355757i	$0.115777\pi$
$492$		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	−0.443196	−	0.896424i	$-0.646156\pi$
$499$	6183.45	−	10710.1i	0.554728	−	0.960817i	0.997925	−	0.0643928i	$-0.0205111\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6249.98			0.554021			0.277011	−	0.960867i	$-0.410656\pi$
$503$	6249.98			0.554021			0.277011	+	0.960867i	$0.410656\pi$
$504$		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	$-0.938948\pi$
$509$	−3740.71	+	6479.10i	−0.325745	+	0.564207i	0.655918	+	0.754832i	$0.272282\pi$
$510$		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	$-0.232831\pi$
$521$	17700.1			1.48840			0.744200	+	0.667957i	$0.232831\pi$
$522$		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	0.977093	−	0.212815i	$-0.0682630\pi$
$547$	3892.26	+	6741.60i	0.304243	+	0.526965i	−0.672849	+	0.739780i	$0.734930\pi$
$548$		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	$-0.605510\pi$
$557$	−8556.03			−0.650863			−0.325431	+	0.945566i	$0.605510\pi$
$558$		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	$-0.868710\pi$
$563$	−1481.61	+	2566.23i	−0.110910	+	0.192102i	0.805227	+	0.592966i	$0.202043\pi$
$564$		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	0.999231	+	0.0392086i	$0.0124837\pi$
$569$	7242.04	+	12543.6i	0.533571	+	0.924172i	−0.465660	+	0.884964i	$0.654183\pi$
$570$		0			0
$571$	−8324.90	+	14419.1i	−0.610133	+	1.05678i	0.381084	+	0.924540i	$0.375551\pi$
$571$	−8324.90	+	14419.1i	−0.610133	+	1.05678i	−0.991217	+	0.132242i	$0.957782\pi$
$572$		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	$-0.257490\pi$
$587$	−4003.11	−	6933.59i	−0.281475	−	0.487530i	−0.971748	+	0.236019i	$0.924157\pi$
$588$		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	$-0.287687\pi$
$593$	17866.7			1.23726			0.618632	+	0.785681i	$0.287687\pi$
$594$		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	0.459658	+	0.888096i	$0.347972\pi$
$599$	−7906.02	+	13693.6i	−0.539284	+	0.934068i	−0.998943	+	0.0459720i	$0.985361\pi$
$600$		0			0
$601$	−11455.2	−	19840.9i	−0.777481	−	1.34664i	−0.933390	−	0.358865i	$-0.883164\pi$
$601$	−11455.2	−	19840.9i	−0.777481	−	1.34664i	0.155909	−	0.987771i	$-0.450169\pi$
$602$		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	−0.866305	−	0.499516i	$-0.833511\pi$
$607$	−8.36137	+	14.4823i	−0.000559107	+	0.000968401i	0.865746	+	0.500484i	$0.166845\pi$
$608$		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	$-0.739633\pi$
$617$	4446.59	−	7701.72i	0.290134	−	0.502528i	0.973841	+	0.227229i	$0.0729666\pi$
$618$		0			0
$619$	−10778.8	−	18669.5i	−0.699898	−	1.21226i	−0.968501	−	0.249009i	$-0.919895\pi$
$619$	−10778.8	−	18669.5i	−0.699898	−	1.21226i	0.268603	−	0.963251i	$-0.413438\pi$
$620$		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	$-0.293213\pi$
$641$	−6283.27	−	10882.9i	−0.387167	−	0.670593i	−0.992067	+	0.125708i	$0.959880\pi$
$642$		0			0
$643$	−438.114	+	758.835i	−0.0268702	+	0.0465405i	−0.879148	−	0.476549i	$-0.841887\pi$
$643$	−438.114	+	758.835i	−0.0268702	+	0.0465405i	0.852278	+	0.523090i	$0.175221\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	11125.4			0.676022			0.338011	−	0.941142i	$-0.390246\pi$
$647$	11125.4			0.676022			0.338011	+	0.941142i	$0.390246\pi$
$648$		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	0.370574	+	0.928803i	$0.379161\pi$
$653$	−10330.4	+	17892.8i	−0.619080	+	1.07228i	−0.989654	+	0.143475i	$0.954172\pi$
$654$		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	$-0.201108\pi$
$659$	−1826.87	−	3164.24i	−0.107989	−	0.187043i	−0.914956	+	0.403555i	$0.867775\pi$
$660$		0			0
$661$	6670.38	−	11553.4i	0.392508	−	0.679844i	−0.600272	−	0.799796i	$-0.704941\pi$
$661$	6670.38	−	11553.4i	0.392508	−	0.679844i	0.992780	+	0.119953i	$0.0382742\pi$
$662$		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	0.920127	−	0.391621i	$-0.128086\pi$
$673$	2110.98	+	3656.32i	0.120910	+	0.209422i	−0.799217	+	0.601043i	$0.794752\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−9341.03	−	16179.1i	−0.530288	−	0.918485i	−0.999376	−	0.0353338i	$-0.988751\pi$
$677$	−9341.03	−	16179.1i	−0.530288	−	0.918485i	0.469088	−	0.883152i	$-0.344583\pi$
$678$		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	$-0.324778\pi$
$683$	18674.1			1.04619			0.523093	+	0.852275i	$0.324778\pi$
$684$		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	−0.669748	−	0.742589i	$-0.733598\pi$
$691$	−17764.2	−	30768.4i	−0.977974	−	1.69390i	−0.308227	−	0.951313i	$-0.599735\pi$
$692$		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	$-0.379470\pi$
$701$	13722.2			0.739346			0.369673	+	0.929162i	$0.379470\pi$
$702$		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	0.824270	−	0.566197i	$-0.191586\pi$
$709$	−1476.42	−	2557.23i	−0.0782059	−	0.135457i	−0.902476	+	0.430740i	$0.858252\pi$
$710$		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	$-0.432618\pi$
$719$	8101.57			0.420219			0.210109	+	0.977678i	$0.432618\pi$
$720$		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	0.819393	−	0.573232i	$-0.194311\pi$
$727$	−1700.22	−	2944.87i	−0.0867368	−	0.150233i	−0.906130	+	0.422999i	$0.860977\pi$
$728$		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	−0.485532	−	0.874219i	$-0.661374\pi$
$733$	10207.0	−	17679.0i	0.514330	−	0.890845i	0.999862	−	0.0166265i	$-0.00529264\pi$
$734$		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	$-0.833625\pi$
$743$	−18.5803	+	32.1820i	−0.000917422	+	0.00158902i	0.865566	+	0.500794i	$0.166959\pi$
$744$		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	−0.962766	−	0.270335i	$-0.912865\pi$
$751$	−5088.90	+	8814.24i	−0.247266	+	0.428277i	0.715500	+	0.698612i	$0.246199\pi$
$752$		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	−0.452206	−	0.891913i	$-0.649363\pi$
$761$	11468.9	−	19864.7i	0.546317	−	0.946248i	0.998523	−	0.0543348i	$-0.0173038\pi$
$762$		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	−0.921172	−	0.389156i	$-0.872767\pi$
$769$	−2635.08	+	4564.09i	−0.123567	+	0.214025i	0.797605	+	0.603181i	$0.206100\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−31.1847			−0.00145102			−0.000725509	−	1.00000i	$-0.500231\pi$
$773$	−31.1847			−0.00145102			−0.000725509		1.00000i	$0.500231\pi$
$774$		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	0.477356	+	0.878710i	$0.341595\pi$
$787$	−11531.6	+	19973.3i	−0.522307	+	0.904663i	−0.999663	+	0.0259526i	$0.991738\pi$
$788$		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	0.188085	+	0.982153i	$0.439772\pi$
$797$	−17022.0	+	29483.1i	−0.756527	+	1.31034i	−0.944612	+	0.328190i	$0.893561\pi$
$798$		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	$-0.386843\pi$
$809$	16017.6			0.696103			0.348052	+	0.937475i	$0.386843\pi$
$810$		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	$-0.239665\pi$
$821$	−5347.65	−	9262.39i	−0.227325	−	0.393739i	−0.957015	+	0.290040i	$0.906331\pi$
$822$		0			0
$823$	14691.9	−	25447.1i	0.622268	−	1.07780i	−0.366794	−	0.930302i	$-0.619545\pi$
$823$	14691.9	−	25447.1i	0.622268	−	1.07780i	0.989062	−	0.147498i	$-0.0471219\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−5402.42			−0.227159			−0.113580	−	0.993529i	$-0.536232\pi$
$827$	−5402.42			−0.227159			−0.113580	+	0.993529i	$0.536232\pi$
$828$		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	0.988782	+	0.149369i	$0.0477243\pi$
$839$	15158.4	+	26255.1i	0.623748	+	1.08036i	−0.365033	+	0.930994i	$0.618942\pi$
$840$		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	0.999953	+	0.00966889i	$0.00307775\pi$
$853$	12664.5	+	21935.5i	0.508350	+	0.880488i	−0.491603	+	0.870819i	$0.663589\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1005.46	+	1741.51i	0.0400770	+	0.0694154i	0.885368	−	0.464891i	$-0.153906\pi$
$857$	1005.46	+	1741.51i	0.0400770	+	0.0694154i	−0.845291	+	0.534306i	$0.820573\pi$
$858$		0			0
$859$	−3482.25	+	6031.43i	−0.138315	+	0.239569i	−0.926859	−	0.375410i	$-0.877502\pi$
$859$	−3482.25	+	6031.43i	−0.138315	+	0.239569i	0.788544	+	0.614979i	$0.210835\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−14772.4			−0.582686			−0.291343	−	0.956619i	$-0.594102\pi$
$863$	−14772.4			−0.582686			−0.291343	+	0.956619i	$0.594102\pi$
$864$		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	−0.516163	−	0.856491i	$-0.672640\pi$
$877$	12561.5	−	21757.1i	0.483661	−	0.837726i	0.999824	+	0.0187648i	$0.00597336\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−20881.3			−0.798533			−0.399267	−	0.916835i	$-0.630735\pi$
$881$	−20881.3			−0.798533			−0.399267	+	0.916835i	$0.630735\pi$
$882$		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	−0.158609	−	0.987341i	$-0.550701\pi$
$887$	20493.3	−	35495.4i	0.775758	−	1.34365i	0.934367	−	0.356311i	$-0.115966\pi$
$888$		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	0.989739	+	0.142890i	$0.0456396\pi$
$907$	16897.9	+	29268.0i	0.618616	+	1.07147i	−0.371123	+	0.928584i	$0.621027\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−26068.8	−	45152.6i	−0.948078	−	1.64212i	−0.749467	−	0.662042i	$-0.769690\pi$
$911$	−26068.8	−	45152.6i	−0.948078	−	1.64212i	−0.198612	−	0.980078i	$-0.563643\pi$
$912$		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	$-0.121202\pi$
$929$	4030.63	+	6981.26i	0.142347	+	0.246553i	−0.786033	+	0.618185i	$0.787868\pi$
$930$		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	$-0.980696\pi$
$941$	−12891.3	+	22328.4i	−0.446594	+	0.773523i	0.551568	+	0.834130i	$0.314030\pi$
$942$		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	−0.902260	−	0.431193i	$-0.858093\pi$
$947$	−24029.5	−	41620.3i	−0.824554	−	1.42817i	0.0777053	−	0.996976i	$-0.475241\pi$
$948$		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	$-0.818903\pi$
$953$	−49570.9			−1.68495			−0.842476	+	0.538733i	$0.818903\pi$
$954$		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	−0.243836	−	0.969817i	$-0.578406\pi$
$967$	21589.6	−	37394.3i	0.717968	−	1.24356i	0.961804	−	0.273741i	$-0.0882610\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−23494.7			−0.776498			−0.388249	−	0.921554i	$-0.626920\pi$
$971$	−23494.7			−0.776498			−0.388249	+	0.921554i	$0.626920\pi$
$972$		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	$-0.726077\pi$
$977$	10096.4	−	17487.4i	0.330616	−	0.572643i	0.982633	+	0.185561i	$0.0594103\pi$
$978$		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	0.920566	+	0.390588i	$0.127728\pi$
$983$	24610.9	+	42627.4i	0.798542	+	1.38312i	−0.122023	+	0.992527i	$0.538938\pi$
$984$		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	0.792136	−	0.610345i	$-0.208969\pi$
$997$	−4171.39	−	7225.06i	−0.132507	−	0.229508i	−0.924642	+	0.380837i	$0.875636\pi$
$998$		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.109.4		8
3.2	odd	2	inner	324.4.e.i.109.1		8
9.2	odd	6	inner	324.4.e.i.217.1		8
9.4	even	3		324.4.a.e.1.1	✓	4
9.5	odd	6		324.4.a.e.1.4	yes	4
9.7	even	3	inner	324.4.e.i.217.4		8
36.23	even	6		1296.4.a.z.1.4		4
36.31	odd	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	9.4	even	3
324.4.a.e.1.4	yes	4	9.5	odd	6
324.4.e.i.109.1		8	3.2	odd	2	inner
324.4.e.i.109.4		8	1.1	even	1	trivial
324.4.e.i.217.1		8	9.2	odd	6	inner
324.4.e.i.217.4		8	9.7	even	3	inner
1296.4.a.z.1.1		4	36.31	odd	6
1296.4.a.z.1.4		4	36.23	even	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(10.5353 - 18.2477i) q^{5} +(15.7477 + 27.2759i) q^{7} +O(q^{10})\)	\(q+(10.5353 - 18.2477i) q^{5} +(15.7477 + 27.2759i) q^{7} +(18.3296 + 31.7477i) q^{11} +(-28.2477 + 48.9265i) q^{13} +35.8010 q^{17} +83.4955 q^{19} +(-34.7762 + 60.2341i) q^{23} +(-159.486 - 276.238i) q^{25} +(40.8541 + 70.7614i) q^{29} +(36.4864 - 63.1962i) q^{31} +663.630 q^{35} -25.4682 q^{37} +(199.742 - 345.964i) q^{41} +(41.7295 + 72.2777i) q^{43} +(155.885 + 270.000i) q^{47} +(-324.482 + 562.019i) q^{49} +4.09919 q^{53} +772.432 q^{55} +(176.097 - 305.009i) q^{59} +(-1.77046 - 3.06652i) q^{61} +(595.198 + 1030.91i) q^{65} +(-246.216 + 426.458i) q^{67} +154.502 q^{71} +305.000 q^{73} +(-577.298 + 999.909i) q^{77} +(-335.693 - 581.438i) q^{79} +(-646.850 - 1120.38i) q^{83} +(377.175 - 653.286i) q^{85} +1183.91 q^{89} -1779.35 q^{91} +(879.652 - 1523.60i) q^{95} +(-307.991 - 533.456i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{7}+O(q^{10}) \) 8 * q + 16 * q^7	\( 8 q + 16 q^{7} - 116 q^{13} + 448 q^{19} - 616 q^{25} - 368 q^{31} + 1336 q^{37} - 656 q^{43} - 1716 q^{49} + 2880 q^{55} - 1004 q^{61} - 320 q^{67} + 2440 q^{73} + 64 q^{79} - 612 q^{85} - 6976 q^{91} - 2024 q^{97}+O(q^{100}) \) 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

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