LMFDB - Embedded newform 1224.4.c.e.577.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,4,Mod(577,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1224.577");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1224 = 2^{3} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1224.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$72.2183378470$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152 $ x^8 + 95x^6 + 756x^4 + 1780*x^2 + 1152
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.4
Root			$1.60125i$ of defining polynomial
Character	$\chi$	$=$	1224.577
Dual form			1224.4.c.e.577.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.89575i q^{5} -4.46235i q^{7} +O(q^{10})$	$q-4.89575i q^{5} -4.46235i q^{7} -60.3865i q^{11} -56.1938 q^{13} +(-25.6861 - 65.2168i) q^{17} -134.845 q^{19} -39.1633i q^{23} +101.032 q^{25} +113.700i q^{29} +306.516i q^{31} -21.8465 q^{35} -61.9621i q^{37} +317.272i q^{41} +122.660 q^{43} -303.233 q^{47} +323.087 q^{49} -133.743 q^{53} -295.637 q^{55} +130.660 q^{59} -772.618i q^{61} +275.111i q^{65} +378.907 q^{67} +465.116i q^{71} +664.801i q^{73} -269.466 q^{77} -925.763i q^{79} +723.561 q^{83} +(-319.285 + 125.752i) q^{85} -889.439 q^{89} +250.757i q^{91} +660.168i q^{95} +1506.43i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 44 q^{13} - 28 q^{17} + 48 q^{19} - 520 q^{25} + 1064 q^{35} + 8 q^{43} - 312 q^{47} - 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 1660 q^{77} - 2472 q^{83} - 2160 q^{85} - 68 q^{89}+O(q^{100}) $ 8 * q + 44 * q^13 - 28 * q^17 + 48 * q^19 - 520 * q^25 + 1064 * q^35 + 8 * q^43 - 312 * q^47 - 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 1660 * q^77 - 2472 * q^83 - 2160 * q^85 - 68 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$.

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	4.89575i		−	0.437889i	−0.975737	−	0.218944i	$-0.929739\pi$
$5$		−	4.89575i		−	0.437889i	0.975737	−	0.218944i	$-0.0702614\pi$
$6$		0			0
$7$		−	4.46235i		−	0.240944i	−0.992717	−	0.120472i	$-0.961559\pi$
$7$		−	4.46235i		−	0.240944i	0.992717	−	0.120472i	$-0.0384408\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	60.3865i		−	1.65520i	−0.561318	−	0.827600i	$-0.689706\pi$
$11$		−	60.3865i		−	1.65520i	0.561318	−	0.827600i	$-0.310294\pi$
$12$		0			0
$13$	−56.1938			−1.19887			−0.599437	−	0.800422i	$-0.704609\pi$
$13$	−56.1938			−1.19887			−0.599437	+	0.800422i	$0.704609\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−25.6861	−	65.2168i	−0.366458	−	0.930435i
$17$	−25.6861	−	65.2168i	−0.366458	−	0.930435i
$18$		0			0
$19$	−134.845			−1.62819			−0.814095	−	0.580731i	$-0.802767\pi$
$19$	−134.845			−1.62819			−0.814095	+	0.580731i	$0.802767\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	39.1633i		−	0.355049i	−0.984116	−	0.177524i	$-0.943191\pi$
$23$		−	39.1633i		−	0.355049i	0.984116	−	0.177524i	$-0.0568088\pi$
$24$		0			0
$25$	101.032			0.808253
$26$		0			0
$27$		0			0
$28$		0			0
$29$			113.700i			0.728053i	0.931389	+	0.364027i	$0.118598\pi$
$29$			113.700i			0.728053i	−0.931389	+	0.364027i	$0.881402\pi$
$30$		0			0
$31$			306.516i			1.77587i	0.459969	+	0.887935i	$0.347861\pi$
$31$			306.516i			1.77587i	−0.459969	+	0.887935i	$0.652139\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−21.8465			−0.105507
$36$		0			0
$37$		−	61.9621i		−	0.275311i	−0.990480	−	0.137656i	$-0.956043\pi$
$37$		−	61.9621i		−	0.275311i	0.990480	−	0.137656i	$-0.0439567\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			317.272i			1.20853i	0.796785	+	0.604263i	$0.206532\pi$
$41$			317.272i			1.20853i	−0.796785	+	0.604263i	$0.793468\pi$
$42$		0			0
$43$	122.660			0.435009			0.217505	−	0.976059i	$-0.430208\pi$
$43$	122.660			0.435009			0.217505	+	0.976059i	$0.430208\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−303.233			−0.941086			−0.470543	−	0.882377i	$-0.655942\pi$
$47$	−303.233			−0.941086			−0.470543	+	0.882377i	$0.655942\pi$
$48$		0			0
$49$	323.087			0.941946
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−133.743			−0.346623			−0.173311	−	0.984867i	$-0.555447\pi$
$53$	−133.743			−0.346623			−0.173311	+	0.984867i	$0.555447\pi$
$54$		0			0
$55$	−295.637			−0.724794
$56$		0			0
$57$		0			0
$58$		0			0
$59$	130.660			0.288312			0.144156	−	0.989555i	$-0.453953\pi$
$59$	130.660			0.288312			0.144156	+	0.989555i	$0.453953\pi$
$60$		0			0
$61$		−	772.618i		−	1.62170i	−0.585256	−	0.810849i	$-0.699006\pi$
$61$		−	772.618i		−	1.62170i	0.585256	−	0.810849i	$-0.300994\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			275.111i			0.524974i
$66$		0			0
$67$	378.907			0.690909			0.345455	−	0.938435i	$-0.387725\pi$
$67$	378.907			0.690909			0.345455	+	0.938435i	$0.387725\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			465.116i			0.777452i	0.921353	+	0.388726i	$0.127085\pi$
$71$			465.116i			0.777452i	−0.921353	+	0.388726i	$0.872915\pi$
$72$		0			0
$73$			664.801i			1.06588i	0.846154	+	0.532939i	$0.178913\pi$
$73$			664.801i			1.06588i	−0.846154	+	0.532939i	$0.821087\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−269.466			−0.398811
$78$		0			0
$79$		−	925.763i		−	1.31844i	−0.751952	−	0.659218i	$-0.770887\pi$
$79$		−	925.763i		−	1.31844i	0.751952	−	0.659218i	$-0.229113\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	723.561			0.956882			0.478441	−	0.878120i	$-0.341202\pi$
$83$	723.561			0.956882			0.478441	+	0.878120i	$0.341202\pi$
$84$		0			0
$85$	−319.285	+	125.752i	−0.407427	+	0.160468i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−889.439			−1.05933			−0.529665	−	0.848207i	$-0.677682\pi$
$89$	−889.439			−1.05933			−0.529665	+	0.848207i	$0.677682\pi$
$90$		0			0
$91$			250.757i			0.288862i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			660.168i			0.712967i
$96$		0			0
$97$			1506.43i			1.57686i	0.615127	+	0.788428i	$0.289105\pi$
$97$			1506.43i			1.57686i	−0.615127	+	0.788428i	$0.710895\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−606.815			−0.597825			−0.298912	−	0.954281i	$-0.596624\pi$
$101$	−606.815			−0.597825			−0.298912	+	0.954281i	$0.596624\pi$
$102$		0			0
$103$	582.633			0.557364			0.278682	−	0.960383i	$-0.410102\pi$
$103$	582.633			0.557364			0.278682	+	0.960383i	$0.410102\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	14.1057i		−	0.0127444i	−0.999980	−	0.00637221i	$-0.997972\pi$
$107$		−	14.1057i		−	0.0127444i	0.999980	−	0.00637221i	$-0.00202835\pi$
$108$		0			0
$109$			1599.48i			1.40553i	0.711422	+	0.702765i	$0.248051\pi$
$109$			1599.48i			1.40553i	−0.711422	+	0.702765i	$0.751949\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			620.655i			0.516693i	0.966052	+	0.258347i	$0.0831777\pi$
$113$			620.655i			0.516693i	−0.966052	+	0.258347i	$0.916822\pi$
$114$		0			0
$115$	−191.734			−0.155472
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−291.020	+	114.620i	−0.224183	+	0.0882960i
$120$		0			0
$121$	−2315.52			−1.73969
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1106.59i		−	0.791814i
$126$		0			0
$127$	−970.396			−0.678021			−0.339011	−	0.940783i	$-0.610092\pi$
$127$	−970.396			−0.678021			−0.339011	+	0.940783i	$0.610092\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	705.717i		−	0.470678i	−0.971913	−	0.235339i	$-0.924380\pi$
$131$		−	705.717i		−	0.470678i	0.971913	−	0.235339i	$-0.0756200\pi$
$132$		0			0
$133$			601.727i			0.392303i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2470.01			1.54035			0.770173	−	0.637835i	$-0.220170\pi$
$137$	2470.01			1.54035			0.770173	+	0.637835i	$0.220170\pi$
$138$		0			0
$139$			435.484i			0.265736i	0.991134	+	0.132868i	$0.0424186\pi$
$139$			435.484i			0.265736i	−0.991134	+	0.132868i	$0.957581\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			3393.35i			1.98438i
$144$		0			0
$145$	556.646			0.318806
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2428.83			−1.33542			−0.667709	−	0.744422i	$-0.732725\pi$
$149$	−2428.83			−1.33542			−0.667709	+	0.744422i	$0.732725\pi$
$150$		0			0
$151$	2608.09			1.40558			0.702792	−	0.711396i	$-0.251937\pi$
$151$	2608.09			1.40558			0.702792	+	0.711396i	$0.251937\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1500.63			0.777634
$156$		0			0
$157$	164.006			0.0833703			0.0416851	−	0.999131i	$-0.486727\pi$
$157$	164.006			0.0833703			0.0416851	+	0.999131i	$0.486727\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−174.760			−0.0855469
$162$		0			0
$163$			289.416i			0.139072i	0.997579	+	0.0695362i	$0.0221519\pi$
$163$			289.416i			0.139072i	−0.997579	+	0.0695362i	$0.977848\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			2147.52i			0.995090i	0.867438	+	0.497545i	$0.165765\pi$
$167$			2147.52i			0.995090i	−0.867438	+	0.497545i	$0.834235\pi$
$168$		0			0
$169$	960.747			0.437299
$170$		0			0
$171$		0			0
$172$		0			0
$173$			4383.99i			1.92664i	0.268357	+	0.963320i	$0.413519\pi$
$173$			4383.99i			1.92664i	−0.268357	+	0.963320i	$0.586481\pi$
$174$		0			0
$175$		−	450.839i		−	0.194744i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3793.10			−1.58385			−0.791926	−	0.610617i	$-0.790921\pi$
$179$	−3793.10			−1.58385			−0.791926	+	0.610617i	$0.790921\pi$
$180$		0			0
$181$			1726.75i			0.709108i	0.935036	+	0.354554i	$0.115367\pi$
$181$			1726.75i			0.709108i	−0.935036	+	0.354554i	$0.884633\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−303.351			−0.120556
$186$		0			0
$187$	−3938.21	+	1551.09i	−1.54006	+	0.606561i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1286.01			−0.487184			−0.243592	−	0.969878i	$-0.578326\pi$
$191$	−1286.01			−0.487184			−0.243592	+	0.969878i	$0.578326\pi$
$192$		0			0
$193$		−	1682.07i		−	0.627349i	−0.949531	−	0.313674i	$-0.898440\pi$
$193$		−	1682.07i		−	0.627349i	0.949531	−	0.313674i	$-0.101560\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	415.578i		−	0.150298i	−0.997172	−	0.0751489i	$-0.976057\pi$
$197$		−	415.578i		−	0.150298i	0.997172	−	0.0751489i	$-0.0239432\pi$
$198$		0			0
$199$		−	2761.31i		−	0.983637i	−0.870698	−	0.491819i	$-0.836332\pi$
$199$		−	2761.31i		−	0.983637i	0.870698	−	0.491819i	$-0.163668\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	507.369			0.175420
$204$		0			0
$205$	1553.28			0.529200
$206$		0			0
$207$		0			0
$208$		0			0
$209$			8142.83i			2.69498i
$210$		0			0
$211$		−	2713.80i		−	0.885431i	−0.896662	−	0.442715i	$-0.854015\pi$
$211$		−	2713.80i		−	0.885431i	0.896662	−	0.442715i	$-0.145985\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	600.510i		−	0.190486i
$216$		0			0
$217$	1367.78			0.427886
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1443.40	+	3664.78i	0.439337	+	1.11547i
$222$		0			0
$223$	−6395.76			−1.92059			−0.960296	−	0.278984i	$-0.910002\pi$
$223$	−6395.76			−1.92059			−0.960296	+	0.278984i	$0.910002\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	3844.30i		−	1.12403i	−0.827127	−	0.562016i	$-0.810026\pi$
$227$		−	3844.30i		−	1.12403i	0.827127	−	0.562016i	$-0.189974\pi$
$228$		0			0
$229$	−4849.35			−1.39936			−0.699682	−	0.714455i	$-0.746675\pi$
$229$	−4849.35			−1.39936			−0.699682	+	0.714455i	$0.746675\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	2719.86i		−	0.764738i	−0.924010	−	0.382369i	$-0.875108\pi$
$233$		−	2719.86i		−	0.764738i	0.924010	−	0.382369i	$-0.124892\pi$
$234$		0			0
$235$			1484.55i			0.412091i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2524.92			−0.683362			−0.341681	−	0.939816i	$-0.610996\pi$
$239$	−2524.92			−0.683362			−0.341681	+	0.939816i	$0.610996\pi$
$240$		0			0
$241$			1943.29i			0.519413i	0.965688	+	0.259706i	$0.0836258\pi$
$241$			1943.29i			0.519413i	−0.965688	+	0.259706i	$0.916374\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	1581.75i		−	0.412468i
$246$		0			0
$247$	7577.47			1.95200
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2332.58			−0.586578			−0.293289	−	0.956024i	$-0.594750\pi$
$251$	−2332.58			−0.586578			−0.293289	+	0.956024i	$0.594750\pi$
$252$		0			0
$253$	−2364.93			−0.587676
$254$		0			0
$255$		0			0
$256$		0			0
$257$	807.810			0.196069			0.0980346	−	0.995183i	$-0.468744\pi$
$257$	807.810			0.196069			0.0980346	+	0.995183i	$0.468744\pi$
$258$		0			0
$259$	−276.497			−0.0663347
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6420.82			1.50542			0.752708	−	0.658354i	$-0.228747\pi$
$263$	6420.82			1.50542			0.752708	+	0.658354i	$0.228747\pi$
$264$		0			0
$265$			654.772i			0.151782i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			6335.06i			1.43589i	0.696097	+	0.717947i	$0.254918\pi$
$269$			6335.06i			1.43589i	−0.696097	+	0.717947i	$0.745082\pi$
$270$		0			0
$271$	−1429.35			−0.320394			−0.160197	−	0.987085i	$-0.551213\pi$
$271$	−1429.35			−0.320394			−0.160197	+	0.987085i	$0.551213\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	6100.94i		−	1.33782i
$276$		0			0
$277$			7878.89i			1.70901i	0.519441	+	0.854506i	$0.326140\pi$
$277$			7878.89i			1.70901i	−0.519441	+	0.854506i	$0.673860\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2967.99			−0.630090			−0.315045	−	0.949077i	$-0.602020\pi$
$281$	−2967.99			−0.630090			−0.315045	+	0.949077i	$0.602020\pi$
$282$		0			0
$283$			401.204i			0.0842725i	0.999112	+	0.0421363i	$0.0134164\pi$
$283$			401.204i			0.0842725i	−0.999112	+	0.0421363i	$0.986584\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1415.78			0.291188
$288$		0			0
$289$	−3593.45	+	3350.32i	−0.731417	+	0.681930i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−7390.62			−1.47360			−0.736800	−	0.676111i	$-0.763664\pi$
$293$	−7390.62			−1.47360			−0.736800	+	0.676111i	$0.763664\pi$
$294$		0			0
$295$		−	639.676i		−	0.126249i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			2200.74i			0.425659i
$300$		0			0
$301$		−	547.350i		−	0.104813i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3782.54			−0.710123
$306$		0			0
$307$	−50.2688			−0.00934524			−0.00467262	−	0.999989i	$-0.501487\pi$
$307$	−50.2688			−0.00934524			−0.00467262	+	0.999989i	$0.501487\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	215.028i		−	0.0392062i	−0.999808	−	0.0196031i	$-0.993760\pi$
$311$		−	215.028i		−	0.0392062i	0.999808	−	0.0196031i	$-0.00624026\pi$
$312$		0			0
$313$			4639.52i			0.837831i	0.908025	+	0.418915i	$0.137590\pi$
$313$			4639.52i			0.837831i	−0.908025	+	0.418915i	$0.862410\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	3965.63i		−	0.702624i	−0.936259	−	0.351312i	$-0.885736\pi$
$317$		−	3965.63i		−	0.702624i	0.936259	−	0.351312i	$-0.114264\pi$
$318$		0			0
$319$	6865.94			1.20507
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3463.64	+	8794.17i	0.596663	+	1.51492i
$324$		0			0
$325$	−5677.36			−0.968994
$326$		0			0
$327$		0			0
$328$		0			0
$329$			1353.13i			0.226749i
$330$		0			0
$331$	7497.94			1.24509			0.622544	−	0.782585i	$-0.286099\pi$
$331$	7497.94			1.24509			0.622544	+	0.782585i	$0.286099\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	1855.03i		−	0.302541i
$336$		0			0
$337$		−	798.500i		−	0.129071i	−0.997915	−	0.0645357i	$-0.979443\pi$
$337$		−	798.500i		−	0.129071i	0.997915	−	0.0645357i	$-0.0205567\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	18509.4			2.93942
$342$		0			0
$343$		−	2972.32i		−	0.467901i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	4014.43i		−	0.621054i	−0.950564	−	0.310527i	$-0.899494\pi$
$347$		−	4014.43i		−	0.621054i	0.950564	−	0.310527i	$-0.100506\pi$
$348$		0			0
$349$	4747.53			0.728165			0.364083	−	0.931367i	$-0.381383\pi$
$349$	4747.53			0.728165			0.364083	+	0.931367i	$0.381383\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9653.02			−1.45546			−0.727731	−	0.685862i	$-0.759425\pi$
$353$	−9653.02			−1.45546			−0.727731	+	0.685862i	$0.759425\pi$
$354$		0			0
$355$	2277.09			0.340438
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6641.04			0.976325			0.488162	−	0.872753i	$-0.337667\pi$
$359$	6641.04			0.976325			0.488162	+	0.872753i	$0.337667\pi$
$360$		0			0
$361$	11324.2			1.65100
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3254.70			0.466736
$366$		0			0
$367$			3305.96i			0.470217i	0.971969	+	0.235108i	$0.0755445\pi$
$367$			3305.96i			0.470217i	−0.971969	+	0.235108i	$0.924455\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			596.808i			0.0835168i
$372$		0			0
$373$	238.635			0.0331261			0.0165631	−	0.999863i	$-0.494728\pi$
$373$	238.635			0.0331261			0.0165631	+	0.999863i	$0.494728\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	6389.23i		−	0.872844i
$378$		0			0
$379$		−	11081.6i		−	1.50191i	−0.660353	−	0.750955i	$-0.729593\pi$
$379$		−	11081.6i		−	1.50191i	0.660353	−	0.750955i	$-0.270407\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−7956.45			−1.06150			−0.530752	−	0.847527i	$-0.678090\pi$
$383$	−7956.45			−1.06150			−0.530752	+	0.847527i	$0.678090\pi$
$384$		0			0
$385$			1319.24i			0.174635i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5128.06			−0.668388			−0.334194	−	0.942504i	$-0.608464\pi$
$389$	−5128.06			−0.668388			−0.334194	+	0.942504i	$0.608464\pi$
$390$		0			0
$391$	−2554.11	+	1005.95i	−0.330349	+	0.130110i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4532.30			−0.577329
$396$		0			0
$397$		−	2933.96i		−	0.370910i	−0.982653	−	0.185455i	$-0.940624\pi$
$397$		−	2933.96i		−	0.370910i	0.982653	−	0.185455i	$-0.0593759\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	5278.07i		−	0.657293i	−0.944453	−	0.328646i	$-0.893408\pi$
$401$		−	5278.07i		−	0.657293i	0.944453	−	0.328646i	$-0.106592\pi$
$402$		0			0
$403$		−	17224.3i		−	2.12904i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3741.67			−0.455695
$408$		0			0
$409$	−6880.00			−0.831770			−0.415885	−	0.909417i	$-0.636528\pi$
$409$	−6880.00			−0.831770			−0.415885	+	0.909417i	$0.636528\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	583.049i		−	0.0694672i
$414$		0			0
$415$		−	3542.37i		−	0.419008i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			3048.43i			0.355431i	0.984082	+	0.177715i	$0.0568707\pi$
$419$			3048.43i			0.355431i	−0.984082	+	0.177715i	$0.943129\pi$
$420$		0			0
$421$	3607.83			0.417660			0.208830	−	0.977952i	$-0.433034\pi$
$421$	3607.83			0.417660			0.208830	+	0.977952i	$0.433034\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2595.11	−	6588.96i	−0.296191	−	0.752027i
$426$		0			0
$427$	−3447.69			−0.390739
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	8866.93i		−	0.990962i	−0.868619	−	0.495481i	$-0.834992\pi$
$431$		−	8866.93i		−	0.990962i	0.868619	−	0.495481i	$-0.165008\pi$
$432$		0			0
$433$	−11353.2			−1.26005			−0.630024	−	0.776575i	$-0.716955\pi$
$433$	−11353.2			−1.26005			−0.630024	+	0.776575i	$0.716955\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			5280.99i			0.578087i
$438$		0			0
$439$		−	2786.53i		−	0.302947i	−0.988461	−	0.151474i	$-0.951598\pi$
$439$		−	2786.53i		−	0.302947i	0.988461	−	0.151474i	$-0.0484019\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−15244.4			−1.63495			−0.817474	−	0.575965i	$-0.804626\pi$
$443$	−15244.4			−1.63495			−0.817474	+	0.575965i	$0.804626\pi$
$444$		0			0
$445$			4354.47i			0.463869i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			17473.1i			1.83654i	0.395958	+	0.918269i	$0.370413\pi$
$449$			17473.1i			1.83654i	−0.395958	+	0.918269i	$0.629587\pi$
$450$		0			0
$451$	19158.9			2.00035
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1227.64			0.126489
$456$		0			0
$457$	−16422.8			−1.68102			−0.840508	−	0.541800i	$-0.817743\pi$
$457$	−16422.8			−1.68102			−0.840508	+	0.541800i	$0.817743\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	12802.9			1.29347			0.646737	−	0.762713i	$-0.276133\pi$
$461$	12802.9			1.29347			0.646737	+	0.762713i	$0.276133\pi$
$462$		0			0
$463$	−2291.00			−0.229961			−0.114980	−	0.993368i	$-0.536680\pi$
$463$	−2291.00			−0.229961			−0.114980	+	0.993368i	$0.536680\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	14464.0			1.43322			0.716610	−	0.697474i	$-0.245693\pi$
$467$	14464.0			1.43322			0.716610	+	0.697474i	$0.245693\pi$
$468$		0			0
$469$		−	1690.82i		−	0.166471i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	7406.97i		−	0.720028i
$474$		0			0
$475$	−13623.6			−1.31599
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	6988.15i		−	0.666590i	−0.942823	−	0.333295i	$-0.891839\pi$
$479$		−	6988.15i		−	0.666590i	0.942823	−	0.333295i	$-0.108161\pi$
$480$		0			0
$481$			3481.89i			0.330063i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7375.11			0.690488
$486$		0			0
$487$			8494.76i			0.790420i	0.918591	+	0.395210i	$0.129328\pi$
$487$			8494.76i			0.790420i	−0.918591	+	0.395210i	$0.870672\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5195.82			0.477565			0.238782	−	0.971073i	$-0.423252\pi$
$491$	5195.82			0.477565			0.238782	+	0.971073i	$0.423252\pi$
$492$		0			0
$493$	7415.14	−	2920.50i	0.677406	−	0.266801i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2075.51			0.187323
$498$		0			0
$499$		−	8790.68i		−	0.788627i	−0.918976	−	0.394313i	$-0.870982\pi$
$499$		−	8790.68i		−	0.788627i	0.918976	−	0.394313i	$-0.129018\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	7764.74i		−	0.688295i	−0.938916	−	0.344148i	$-0.888168\pi$
$503$		−	7764.74i		−	0.688295i	0.938916	−	0.344148i	$-0.111832\pi$
$504$		0			0
$505$			2970.81i			0.261781i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9919.29			−0.863782			−0.431891	−	0.901926i	$-0.642153\pi$
$509$	−9919.29			−0.863782			−0.431891	+	0.901926i	$0.642153\pi$
$510$		0			0
$511$	2966.58			0.256817
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	2852.42i		−	0.244064i
$516$		0			0
$517$			18311.2i			1.55769i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	4230.44i		−	0.355737i	−0.984054	−	0.177868i	$-0.943080\pi$
$521$		−	4230.44i		−	0.355737i	0.984054	−	0.177868i	$-0.0569201\pi$
$522$		0			0
$523$	10566.8			0.883465			0.441732	−	0.897147i	$-0.354364\pi$
$523$	10566.8			0.883465			0.441732	+	0.897147i	$0.354364\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	19990.0	−	7873.20i	1.65233	−	0.650782i
$528$		0			0
$529$	10633.2			0.873941
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	17828.7i		−	1.44887i
$534$		0			0
$535$	−69.0581			−0.00558064
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	19510.1i		−	1.55911i
$540$		0			0
$541$		−	1980.86i		−	0.157419i	−0.996898	−	0.0787094i	$-0.974920\pi$
$541$		−	1980.86i		−	0.157419i	0.996898	−	0.0787094i	$-0.0250799\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7830.67			0.615466
$546$		0			0
$547$			3940.22i			0.307992i	0.988071	+	0.153996i	$0.0492143\pi$
$547$			3940.22i			0.307992i	−0.988071	+	0.153996i	$0.950786\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	15331.9i		−	1.18541i
$552$		0			0
$553$	−4131.08			−0.317670
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3709.12			−0.282155			−0.141077	−	0.989999i	$-0.545057\pi$
$557$	−3709.12			−0.282155			−0.141077	+	0.989999i	$0.545057\pi$
$558$		0			0
$559$	−6892.71			−0.521522
$560$		0			0
$561$		0			0
$562$		0			0
$563$	3393.34			0.254018			0.127009	−	0.991902i	$-0.459462\pi$
$563$	3393.34			0.254018			0.127009	+	0.991902i	$0.459462\pi$
$564$		0			0
$565$	3038.57			0.226254
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2927.65			0.215700			0.107850	−	0.994167i	$-0.465603\pi$
$569$	2927.65			0.215700			0.107850	+	0.994167i	$0.465603\pi$
$570$		0			0
$571$		−	19781.8i		−	1.44981i	−0.688846	−	0.724907i	$-0.741883\pi$
$571$		−	19781.8i		−	1.44981i	0.688846	−	0.724907i	$-0.258117\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	3956.74i		−	0.286969i
$576$		0			0
$577$	17520.1			1.26407			0.632037	−	0.774938i	$-0.282219\pi$
$577$	17520.1			1.26407			0.632037	+	0.774938i	$0.282219\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	3228.79i		−	0.230555i
$582$		0			0
$583$			8076.26i			0.573730i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−7846.90			−0.551748			−0.275874	−	0.961194i	$-0.588967\pi$
$587$	−7846.90			−0.551748			−0.275874	+	0.961194i	$0.588967\pi$
$588$		0			0
$589$		−	41332.3i		−	2.89145i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−10572.9			−0.732173			−0.366086	−	0.930581i	$-0.619303\pi$
$593$	−10572.9			−0.732173			−0.366086	+	0.930581i	$0.619303\pi$
$594$		0			0
$595$	561.152	+	1424.76i	0.0386638	+	0.0981672i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	27156.5			1.85240			0.926198	−	0.377038i	$-0.123057\pi$
$599$	27156.5			1.85240			0.926198	+	0.377038i	$0.123057\pi$
$600$		0			0
$601$			11180.6i			0.758849i	0.925223	+	0.379424i	$0.123878\pi$
$601$			11180.6i			0.758849i	−0.925223	+	0.379424i	$0.876122\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			11336.2i			0.761790i
$606$		0			0
$607$			19444.2i			1.30019i	0.759851	+	0.650097i	$0.225272\pi$
$607$			19444.2i			1.30019i	−0.759851	+	0.650097i	$0.774728\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	17039.8			1.12824
$612$		0			0
$613$	−16630.9			−1.09579			−0.547893	−	0.836548i	$-0.684570\pi$
$613$	−16630.9			−1.09579			−0.547893	+	0.836548i	$0.684570\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	8035.03i		−	0.524276i	−0.965030	−	0.262138i	$-0.915572\pi$
$617$		−	8035.03i		−	0.524276i	0.965030	−	0.262138i	$-0.0844275\pi$
$618$		0			0
$619$		−	16161.6i		−	1.04942i	−0.851282	−	0.524709i	$-0.824174\pi$
$619$		−	16161.6i		−	1.04942i	0.851282	−	0.524709i	$-0.175826\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			3968.99i			0.255239i
$624$		0			0
$625$	7211.35			0.461527
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4040.97	+	1591.56i	−0.256159	+	0.100890i
$630$		0			0
$631$	−10043.9			−0.633665			−0.316833	−	0.948481i	$-0.602619\pi$
$631$	−10043.9			−0.633665			−0.316833	+	0.948481i	$0.602619\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			4750.81i			0.296898i
$636$		0			0
$637$	−18155.5			−1.12927
$638$		0			0
$639$		0			0
$640$		0			0
$641$			15316.1i			0.943757i	0.881664	+	0.471879i	$0.156424\pi$
$641$			15316.1i			0.943757i	−0.881664	+	0.471879i	$0.843576\pi$
$642$		0			0
$643$		−	24684.6i		−	1.51394i	−0.653449	−	0.756970i	$-0.726679\pi$
$643$		−	24684.6i		−	1.51394i	0.653449	−	0.756970i	$-0.273321\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7774.11			−0.472383			−0.236192	−	0.971707i	$-0.575899\pi$
$647$	−7774.11			−0.472383			−0.236192	+	0.971707i	$0.575899\pi$
$648$		0			0
$649$		−	7890.07i		−	0.477214i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			752.623i			0.0451032i	0.999746	+	0.0225516i	$0.00717901\pi$
$653$			752.623i			0.0451032i	−0.999746	+	0.0225516i	$0.992821\pi$
$654$		0			0
$655$	−3455.01			−0.206105
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4887.68			0.288918			0.144459	−	0.989511i	$-0.453856\pi$
$659$	4887.68			0.288918			0.144459	+	0.989511i	$0.453856\pi$
$660$		0			0
$661$	−7122.91			−0.419136			−0.209568	−	0.977794i	$-0.567206\pi$
$661$	−7122.91			−0.419136			−0.209568	+	0.977794i	$0.567206\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2945.90			0.171785
$666$		0			0
$667$	4452.87			0.258494
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−46655.6			−2.68423
$672$		0			0
$673$		−	32249.6i		−	1.84715i	−0.383421	−	0.923574i	$-0.625254\pi$
$673$		−	32249.6i		−	1.84715i	0.383421	−	0.923574i	$-0.374746\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	17447.9i		−	0.990511i	−0.868748	−	0.495255i	$-0.835075\pi$
$677$		−	17447.9i		−	0.990511i	0.868748	−	0.495255i	$-0.164925\pi$
$678$		0			0
$679$	6722.23			0.379934
$680$		0			0
$681$		0			0
$682$		0			0
$683$			13935.3i			0.780701i	0.920666	+	0.390351i	$0.127646\pi$
$683$			13935.3i			0.780701i	−0.920666	+	0.390351i	$0.872354\pi$
$684$		0			0
$685$		−	12092.6i		−	0.674500i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7515.53			0.415557
$690$		0			0
$691$			22358.6i			1.23092i	0.788170	+	0.615458i	$0.211029\pi$
$691$			22358.6i			1.23092i	−0.788170	+	0.615458i	$0.788971\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2132.02			0.116363
$696$		0			0
$697$	20691.5	−	8149.47i	1.12445	−	0.442874i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−30456.5			−1.64098			−0.820491	−	0.571660i	$-0.806300\pi$
$701$	−30456.5			−1.64098			−0.820491	+	0.571660i	$0.806300\pi$
$702$		0			0
$703$			8355.30i			0.448259i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			2707.82i			0.144042i
$708$		0			0
$709$			36945.7i			1.95702i	0.206210	+	0.978508i	$0.433887\pi$
$709$			36945.7i			1.95702i	−0.206210	+	0.978508i	$0.566113\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	12004.2			0.630520
$714$		0			0
$715$	16613.0			0.868937
$716$		0			0
$717$		0			0
$718$		0			0
$719$			12442.4i			0.645375i	0.946506	+	0.322687i	$0.104586\pi$
$719$			12442.4i			0.645375i	−0.946506	+	0.322687i	$0.895414\pi$
$720$		0			0
$721$		−	2599.91i		−	0.134294i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			11487.3i			0.588451i
$726$		0			0
$727$	−29943.3			−1.52756			−0.763780	−	0.645477i	$-0.776659\pi$
$727$	−29943.3			−1.52756			−0.763780	+	0.645477i	$0.776659\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3150.64	−	7999.46i	−0.159413	−	0.404748i
$732$		0			0
$733$	19615.5			0.988423			0.494212	−	0.869342i	$-0.335457\pi$
$733$	19615.5			0.988423			0.494212	+	0.869342i	$0.335457\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	22880.9i		−	1.14359i
$738$		0			0
$739$	−3459.26			−0.172194			−0.0860968	−	0.996287i	$-0.527439\pi$
$739$	−3459.26			−0.172194			−0.0860968	+	0.996287i	$0.527439\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			37392.9i			1.84632i	0.384421	+	0.923158i	$0.374401\pi$
$743$			37392.9i			1.84632i	−0.384421	+	0.923158i	$0.625599\pi$
$744$		0			0
$745$			11890.9i			0.584765i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−62.9447			−0.00307069
$750$		0			0
$751$		−	34379.7i		−	1.67048i	−0.549884	−	0.835241i	$-0.685328\pi$
$751$		−	34379.7i		−	1.67048i	0.549884	−	0.835241i	$-0.314672\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	12768.5i		−	0.615489i
$756$		0			0
$757$	−20120.9			−0.966057			−0.483028	−	0.875605i	$-0.660463\pi$
$757$	−20120.9			−0.966057			−0.483028	+	0.875605i	$0.660463\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	21739.7			1.03556			0.517781	−	0.855513i	$-0.326758\pi$
$761$	21739.7			1.03556			0.517781	+	0.855513i	$0.326758\pi$
$762$		0			0
$763$	7137.46			0.338654
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7342.26			−0.345650
$768$		0			0
$769$	26884.9			1.26072			0.630360	−	0.776303i	$-0.282907\pi$
$769$	26884.9			1.26072			0.630360	+	0.776303i	$0.282907\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	21927.0			1.02026			0.510128	−	0.860099i	$-0.329598\pi$
$773$	21927.0			1.02026			0.510128	+	0.860099i	$0.329598\pi$
$774$		0			0
$775$			30967.9i			1.43535i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	42782.6i		−	1.96771i
$780$		0			0
$781$	28086.7			1.28684
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	802.934i		−	0.0365069i
$786$		0			0
$787$		−	28401.4i		−	1.28641i	−0.765695	−	0.643203i	$-0.777605\pi$
$787$		−	28401.4i		−	1.28641i	0.765695	−	0.643203i	$-0.222395\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	2769.58			0.124494
$792$		0			0
$793$			43416.3i			1.94421i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	19462.7			0.864999			0.432499	−	0.901634i	$-0.357632\pi$
$797$	19462.7			0.864999			0.432499	+	0.901634i	$0.357632\pi$
$798$		0			0
$799$	7788.86	+	19775.9i	0.344869	+	0.875619i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	40145.0			1.76424
$804$		0			0
$805$			855.583i			0.0374601i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	11073.0i		−	0.481220i	−0.970622	−	0.240610i	$-0.922653\pi$
$809$		−	11073.0i		−	0.481220i	0.970622	−	0.240610i	$-0.0773474\pi$
$810$		0			0
$811$		−	8512.57i		−	0.368578i	−0.982872	−	0.184289i	$-0.941002\pi$
$811$		−	8512.57i		−	0.368578i	0.982872	−	0.184289i	$-0.0589982\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1416.91			0.0608983
$816$		0			0
$817$	−16540.1			−0.708278
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	26291.2i		−	1.11762i	−0.829294	−	0.558812i	$-0.811257\pi$
$821$		−	26291.2i		−	1.11762i	0.829294	−	0.558812i	$-0.188743\pi$
$822$		0			0
$823$		−	4379.37i		−	0.185486i	−0.995690	−	0.0927432i	$-0.970436\pi$
$823$		−	4379.37i		−	0.185486i	0.995690	−	0.0927432i	$-0.0295636\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			2994.98i			0.125932i	0.998016	+	0.0629659i	$0.0200559\pi$
$827$			2994.98i			0.125932i	−0.998016	+	0.0629659i	$0.979944\pi$
$828$		0			0
$829$	−33497.1			−1.40338			−0.701690	−	0.712482i	$-0.747571\pi$
$829$	−33497.1			−1.40338			−0.701690	+	0.712482i	$0.747571\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−8298.84	−	21070.7i	−0.345184	−	0.876419i
$834$		0			0
$835$	10513.7			0.435739
$836$		0			0
$837$		0			0
$838$		0			0
$839$			25588.9i			1.05295i	0.850189	+	0.526477i	$0.176487\pi$
$839$			25588.9i			1.05295i	−0.850189	+	0.526477i	$0.823513\pi$
$840$		0			0
$841$	11461.3			0.469938
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	4703.57i		−	0.191489i
$846$		0			0
$847$			10332.7i			0.419168i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2426.64			−0.0977488
$852$		0			0
$853$			31609.1i			1.26879i	0.773010	+	0.634394i	$0.218750\pi$
$853$			31609.1i			1.26879i	−0.773010	+	0.634394i	$0.781250\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			7519.34i			0.299715i	0.988708	+	0.149858i	$0.0478815\pi$
$857$			7519.34i			0.299715i	−0.988708	+	0.149858i	$0.952119\pi$
$858$		0			0
$859$	−3923.30			−0.155834			−0.0779169	−	0.996960i	$-0.524827\pi$
$859$	−3923.30			−0.155834			−0.0779169	+	0.996960i	$0.524827\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−32254.8			−1.27227			−0.636133	−	0.771580i	$-0.719467\pi$
$863$	−32254.8			−1.27227			−0.636133	+	0.771580i	$0.719467\pi$
$864$		0			0
$865$	21462.9			0.843654
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−55903.5			−2.18228
$870$		0			0
$871$	−21292.3			−0.828313
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4938.01			−0.190783
$876$		0			0
$877$		−	9587.97i		−	0.369171i	−0.982816	−	0.184585i	$-0.940906\pi$
$877$		−	9587.97i		−	0.369171i	0.982816	−	0.184585i	$-0.0590942\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			10243.5i			0.391726i	0.980631	+	0.195863i	$0.0627508\pi$
$881$			10243.5i			0.391726i	−0.980631	+	0.195863i	$0.937249\pi$
$882$		0			0
$883$	−17802.2			−0.678474			−0.339237	−	0.940701i	$-0.610169\pi$
$883$	−17802.2			−0.678474			−0.339237	+	0.940701i	$0.610169\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	5409.67i		−	0.204779i	−0.994744	−	0.102389i	$-0.967351\pi$
$887$		−	5409.67i		−	0.204779i	0.994744	−	0.102389i	$-0.0326488\pi$
$888$		0			0
$889$			4330.25i			0.163365i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	40889.5			1.53227
$894$		0			0
$895$			18570.1i			0.693551i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−34850.9			−1.29293
$900$		0			0
$901$	3435.33	+	8722.28i	0.127023	+	0.322510i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8453.74			0.310510
$906$		0			0
$907$			27596.6i			1.01029i	0.863036	+	0.505143i	$0.168560\pi$
$907$			27596.6i			1.01029i	−0.863036	+	0.505143i	$0.831440\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			47306.9i			1.72047i	0.509898	+	0.860235i	$0.329683\pi$
$911$			47306.9i			1.72047i	−0.509898	+	0.860235i	$0.670317\pi$
$912$		0			0
$913$		−	43693.3i		−	1.58383i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−3149.16			−0.113407
$918$		0			0
$919$	40635.5			1.45859			0.729293	−	0.684202i	$-0.239849\pi$
$919$	40635.5			1.45859			0.729293	+	0.684202i	$0.239849\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	26136.6i		−	0.932067i
$924$		0			0
$925$		−	6260.14i		−	0.222521i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			34143.2i			1.20582i	0.797811	+	0.602908i	$0.205991\pi$
$929$			34143.2i			1.20582i	−0.797811	+	0.602908i	$0.794009\pi$
$930$		0			0
$931$	−43566.8			−1.53367
$932$		0			0
$933$		0			0
$934$		0			0
$935$	7593.75	+	19280.5i	0.265606	+	0.674373i
$936$		0			0
$937$	4097.67			0.142865			0.0714327	−	0.997445i	$-0.477243\pi$
$937$	4097.67			0.142865			0.0714327	+	0.997445i	$0.477243\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			32801.4i			1.13634i	0.822912	+	0.568169i	$0.192348\pi$
$941$			32801.4i			1.13634i	−0.822912	+	0.568169i	$0.807652\pi$
$942$		0			0
$943$	12425.4			0.429086
$944$		0			0
$945$		0			0
$946$		0			0
$947$			32528.3i			1.11618i	0.829779	+	0.558092i	$0.188466\pi$
$947$			32528.3i			1.11618i	−0.829779	+	0.558092i	$0.811534\pi$
$948$		0			0
$949$		−	37357.7i		−	1.27785i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−14335.6			−0.487276			−0.243638	−	0.969866i	$-0.578341\pi$
$953$	−14335.6			−0.487276			−0.243638	+	0.969866i	$0.578341\pi$
$954$		0			0
$955$			6295.96i			0.213332i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	11022.1i		−	0.371138i
$960$		0			0
$961$	−64161.3			−2.15371
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8235.01			−0.274709
$966$		0			0
$967$	−688.081			−0.0228823			−0.0114412	−	0.999935i	$-0.503642\pi$
$967$	−688.081			−0.0228823			−0.0114412	+	0.999935i	$0.503642\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−49985.2			−1.65201			−0.826005	−	0.563663i	$-0.809391\pi$
$971$	−49985.2			−1.65201			−0.826005	+	0.563663i	$0.809391\pi$
$972$		0			0
$973$	1943.28			0.0640276
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−38752.4			−1.26899			−0.634493	−	0.772929i	$-0.718791\pi$
$977$	−38752.4			−1.26899			−0.634493	+	0.772929i	$0.718791\pi$
$978$		0			0
$979$			53710.1i			1.75340i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			21516.4i			0.698135i	0.937098	+	0.349067i	$0.113502\pi$
$983$			21516.4i			0.698135i	−0.937098	+	0.349067i	$0.886498\pi$
$984$		0			0
$985$	−2034.56			−0.0658138
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	4803.75i		−	0.154449i
$990$		0			0
$991$		−	6941.00i		−	0.222491i	−0.993793	−	0.111245i	$-0.964516\pi$
$991$		−	6941.00i		−	0.222491i	0.993793	−	0.111245i	$-0.0354839\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−13518.7			−0.430724
$996$		0			0
$997$		−	44135.5i		−	1.40199i	−0.713165	−	0.700996i	$-0.752739\pi$
$997$		−	44135.5i		−	1.40199i	0.713165	−	0.700996i	$-0.247261\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1224.4.c.e.577.4		8
3.2	odd	2		136.4.b.b.33.6	yes	8
12.11	even	2		272.4.b.f.33.3		8
17.16	even	2	inner	1224.4.c.e.577.5		8
51.38	odd	4		2312.4.a.k.1.6		8
51.47	odd	4		2312.4.a.k.1.3		8
51.50	odd	2		136.4.b.b.33.3	✓	8
204.203	even	2		272.4.b.f.33.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.b.b.33.3	✓	8	51.50	odd	2
136.4.b.b.33.6	yes	8	3.2	odd	2
272.4.b.f.33.3		8	12.11	even	2
272.4.b.f.33.6		8	204.203	even	2
1224.4.c.e.577.4		8	1.1	even	1	trivial
1224.4.c.e.577.5		8	17.16	even	2	inner
2312.4.a.k.1.3		8	51.47	odd	4
2312.4.a.k.1.6		8	51.38	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1224.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.89575i q^{5} -4.46235i q^{7} +O(q^{10})\)	\(q-4.89575i q^{5} -4.46235i q^{7} -60.3865i q^{11} -56.1938 q^{13} +(-25.6861 - 65.2168i) q^{17} -134.845 q^{19} -39.1633i q^{23} +101.032 q^{25} +113.700i q^{29} +306.516i q^{31} -21.8465 q^{35} -61.9621i q^{37} +317.272i q^{41} +122.660 q^{43} -303.233 q^{47} +323.087 q^{49} -133.743 q^{53} -295.637 q^{55} +130.660 q^{59} -772.618i q^{61} +275.111i q^{65} +378.907 q^{67} +465.116i q^{71} +664.801i q^{73} -269.466 q^{77} -925.763i q^{79} +723.561 q^{83} +(-319.285 + 125.752i) q^{85} -889.439 q^{89} +250.757i q^{91} +660.168i q^{95} +1506.43i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 44 q^{13} - 28 q^{17} + 48 q^{19} - 520 q^{25} + 1064 q^{35} + 8 q^{43} - 312 q^{47} - 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 1660 q^{77} - 2472 q^{83} - 2160 q^{85} - 68 q^{89}+O(q^{100}) \) 8 * q + 44 * q^13 - 28 * q^17 + 48 * q^19 - 520 * q^25 + 1064 * q^35 + 8 * q^43 - 312 * q^47 - 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 1660 * q^77 - 2472 * q^83 - 2160 * q^85 - 68 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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