LMFDB - Embedded newform 576.4.k.b.433.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,4,Mod(145,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0]))

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

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

chi := DirichletCharacter("576.145");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	576.k (of order $4$, 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:	$33.9851001633$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			433.6
Character	$\chi$	$=$	576.433
Dual form			576.4.k.b.145.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.706564 - 0.706564i) q^{5} +4.44122i q^{7} +O(q^{10})$	$q+(-0.706564 - 0.706564i) q^{5} +4.44122i q^{7} +(-17.7126 - 17.7126i) q^{11} +(-17.7078 + 17.7078i) q^{13} +105.640 q^{17} +(-40.2862 + 40.2862i) q^{19} -42.9921i q^{23} -124.002i q^{25} +(185.646 - 185.646i) q^{29} -291.485 q^{31} +(3.13800 - 3.13800i) q^{35} +(151.040 + 151.040i) q^{37} -60.3918i q^{41} +(-120.532 - 120.532i) q^{43} +500.182 q^{47} +323.276 q^{49} +(-192.407 - 192.407i) q^{53} +25.0302i q^{55} +(-500.053 - 500.053i) q^{59} +(-166.418 + 166.418i) q^{61} +25.0234 q^{65} +(575.426 - 575.426i) q^{67} -457.290i q^{71} -1082.59i q^{73} +(78.6657 - 78.6657i) q^{77} +544.309 q^{79} +(48.6931 - 48.6931i) q^{83} +(-74.6417 - 74.6417i) q^{85} -43.6714i q^{89} +(-78.6442 - 78.6442i) q^{91} +56.9295 q^{95} +690.537 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 40 q^{11} - 24 q^{19} - 400 q^{29} + 744 q^{31} - 456 q^{35} + 16 q^{37} - 1240 q^{43} - 1176 q^{49} - 752 q^{53} - 1376 q^{59} - 912 q^{61} - 976 q^{65} + 2256 q^{67} - 1904 q^{77} - 5992 q^{79} + 2680 q^{83} - 240 q^{85} + 3496 q^{91} - 7728 q^{95}+O(q^{100}) $ 24 * q - 40 * q^11 - 24 * q^19 - 400 * q^29 + 744 * q^31 - 456 * q^35 + 16 * q^37 - 1240 * q^43 - 1176 * q^49 - 752 * q^53 - 1376 * q^59 - 912 * q^61 - 976 * q^65 + 2256 * q^67 - 1904 * q^77 - 5992 * q^79 + 2680 * q^83 - 240 * q^85 + 3496 * q^91 - 7728 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$5$	−0.706564	−	0.706564i	−0.0631970	−	0.0631970i	0.674802	−	0.737999i	$-0.264229\pi$
$5$	−0.706564	−	0.706564i	−0.0631970	−	0.0631970i	−0.737999	+	0.674802i	$0.764229\pi$
$6$		0			0
$7$			4.44122i			0.239803i	0.992786	+	0.119902i	$0.0382579\pi$
$7$			4.44122i			0.239803i	−0.992786	+	0.119902i	$0.961742\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−17.7126	−	17.7126i	−0.485506	−	0.485506i	0.421379	−	0.906885i	$-0.361546\pi$
$11$	−17.7126	−	17.7126i	−0.485506	−	0.485506i	−0.906885	+	0.421379i	$0.861546\pi$
$12$		0			0
$13$	−17.7078	+	17.7078i	−0.377789	+	0.377789i	−0.870304	−	0.492515i	$-0.836078\pi$
$13$	−17.7078	+	17.7078i	−0.377789	+	0.377789i	0.492515	+	0.870304i	$0.336078\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	105.640			1.50715			0.753576	−	0.657361i	$-0.228327\pi$
$17$	105.640			1.50715			0.753576	+	0.657361i	$0.228327\pi$
$18$		0			0
$19$	−40.2862	+	40.2862i	−0.486436	+	0.486436i	−0.907180	−	0.420743i	$-0.861769\pi$
$19$	−40.2862	+	40.2862i	−0.486436	+	0.486436i	0.420743	+	0.907180i	$0.361769\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	42.9921i		−	0.389759i	−0.980827	−	0.194880i	$-0.937568\pi$
$23$		−	42.9921i		−	0.389759i	0.980827	−	0.194880i	$-0.0624316\pi$
$24$		0			0
$25$		−	124.002i		−	0.992012i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	185.646	−	185.646i	1.18875	−	1.18875i	0.211331	−	0.977415i	$-0.432220\pi$
$29$	185.646	−	185.646i	1.18875	−	1.18875i	0.977415	−	0.211331i	$-0.0677797\pi$
$30$		0			0
$31$	−291.485			−1.68878			−0.844390	−	0.535729i	$-0.820037\pi$
$31$	−291.485			−1.68878			−0.844390	+	0.535729i	$0.820037\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.13800	−	3.13800i	0.0151548	−	0.0151548i
$36$		0			0
$37$	151.040	+	151.040i	0.671104	+	0.671104i	0.957971	−	0.286867i	$-0.0926137\pi$
$37$	151.040	+	151.040i	0.671104	+	0.671104i	−0.286867	+	0.957971i	$0.592614\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	60.3918i		−	0.230040i	−0.993363	−	0.115020i	$-0.963307\pi$
$41$		−	60.3918i		−	0.230040i	0.993363	−	0.115020i	$-0.0366931\pi$
$42$		0			0
$43$	−120.532	−	120.532i	−0.427464	−	0.427464i	0.460300	−	0.887764i	$-0.347742\pi$
$43$	−120.532	−	120.532i	−0.427464	−	0.427464i	−0.887764	+	0.460300i	$0.847742\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	500.182			1.55232			0.776159	−	0.630537i	$-0.217165\pi$
$47$	500.182			1.55232			0.776159	+	0.630537i	$0.217165\pi$
$48$		0			0
$49$	323.276			0.942494
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−192.407	−	192.407i	−0.498663	−	0.498663i	0.412359	−	0.911022i	$-0.364705\pi$
$53$	−192.407	−	192.407i	−0.498663	−	0.498663i	−0.911022	+	0.412359i	$0.864705\pi$
$54$		0			0
$55$			25.0302i			0.0613650i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−500.053	−	500.053i	−1.10341	−	1.10341i	−0.993996	−	0.109417i	$-0.965102\pi$
$59$	−500.053	−	500.053i	−1.10341	−	1.10341i	−0.109417	−	0.993996i	$-0.534898\pi$
$60$		0			0
$61$	−166.418	+	166.418i	−0.349306	+	0.349306i	−0.859851	−	0.510545i	$-0.829444\pi$
$61$	−166.418	+	166.418i	−0.349306	+	0.349306i	0.510545	+	0.859851i	$0.329444\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	25.0234			0.0477502
$66$		0			0
$67$	575.426	−	575.426i	1.04925	−	1.04925i	0.0505233	−	0.998723i	$-0.483911\pi$
$67$	575.426	−	575.426i	1.04925	−	1.04925i	0.998723	−	0.0505233i	$-0.0160889\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	457.290i		−	0.764371i	−0.924086	−	0.382186i	$-0.875172\pi$
$71$		−	457.290i		−	0.764371i	0.924086	−	0.382186i	$-0.124828\pi$
$72$		0			0
$73$		−	1082.59i		−	1.73572i	−0.496812	−	0.867858i	$-0.665496\pi$
$73$		−	1082.59i		−	1.73572i	0.496812	−	0.867858i	$-0.334504\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	78.6657	−	78.6657i	0.116426	−	0.116426i
$78$		0			0
$79$	544.309			0.775184			0.387592	−	0.921831i	$-0.373307\pi$
$79$	544.309			0.775184			0.387592	+	0.921831i	$0.373307\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	48.6931	−	48.6931i	0.0643947	−	0.0643947i	−0.674176	−	0.738571i	$-0.735501\pi$
$83$	48.6931	−	48.6931i	0.0643947	−	0.0643947i	0.738571	+	0.674176i	$0.235501\pi$
$84$		0			0
$85$	−74.6417	−	74.6417i	−0.0952474	−	0.0952474i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	43.6714i		−	0.0520131i	−0.999662	−	0.0260065i	$-0.991721\pi$
$89$		−	43.6714i		−	0.0520131i	0.999662	−	0.0260065i	$-0.00827907\pi$
$90$		0			0
$91$	−78.6442	−	78.6442i	−0.0905950	−	0.0905950i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	56.9295			0.0614826
$96$		0			0
$97$	690.537			0.722818			0.361409	−	0.932407i	$-0.382296\pi$
$97$	690.537			0.722818			0.361409	+	0.932407i	$0.382296\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−564.484	−	564.484i	−0.556121	−	0.556121i	0.372080	−	0.928201i	$-0.378645\pi$
$101$	−564.484	−	564.484i	−0.556121	−	0.556121i	−0.928201	+	0.372080i	$0.878645\pi$
$102$		0			0
$103$			110.408i			0.105620i	0.998605	+	0.0528098i	$0.0168177\pi$
$103$			110.408i			0.105620i	−0.998605	+	0.0528098i	$0.983182\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−254.357	−	254.357i	−0.229810	−	0.229810i	0.582803	−	0.812613i	$-0.301956\pi$
$107$	−254.357	−	254.357i	−0.229810	−	0.229810i	−0.812613	+	0.582803i	$0.801956\pi$
$108$		0			0
$109$	−205.025	+	205.025i	−0.180164	+	0.180164i	−0.791427	−	0.611263i	$-0.790662\pi$
$109$	−205.025	+	205.025i	−0.180164	+	0.180164i	0.611263	+	0.791427i	$0.290662\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	729.046			0.606928			0.303464	−	0.952843i	$-0.401857\pi$
$113$	729.046			0.606928			0.303464	+	0.952843i	$0.401857\pi$
$114$		0			0
$115$	−30.3766	+	30.3766i	−0.0246316	+	0.0246316i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			469.172i			0.361420i
$120$		0			0
$121$		−	703.525i		−	0.528569i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−175.935	+	175.935i	−0.125889	+	0.125889i
$126$		0			0
$127$	2431.21			1.69870			0.849352	−	0.527827i	$-0.176993\pi$
$127$	2431.21			1.69870			0.849352	+	0.527827i	$0.176993\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	898.181	−	898.181i	0.599042	−	0.599042i	−0.341016	−	0.940058i	$-0.610771\pi$
$131$	898.181	−	898.181i	0.599042	−	0.599042i	0.940058	+	0.341016i	$0.110771\pi$
$132$		0			0
$133$	−178.920	−	178.920i	−0.116649	−	0.116649i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	3061.79i		−	1.90939i	−0.297590	−	0.954694i	$-0.596183\pi$
$137$		−	3061.79i		−	1.90939i	0.297590	−	0.954694i	$-0.403817\pi$
$138$		0			0
$139$	−1811.69	−	1811.69i	−1.10551	−	1.10551i	−0.993733	−	0.111776i	$-0.964346\pi$
$139$	−1811.69	−	1811.69i	−1.10551	−	1.10551i	−0.111776	−	0.993733i	$-0.535654\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	627.303			0.366837
$144$		0			0
$145$	−262.342			−0.150250
$146$		0			0
$147$		0			0
$148$		0			0
$149$	869.143	+	869.143i	0.477872	+	0.477872i	0.904451	−	0.426578i	$-0.140281\pi$
$149$	869.143	+	869.143i	0.477872	+	0.477872i	−0.426578	+	0.904451i	$0.640281\pi$
$150$		0			0
$151$			1033.70i			0.557097i	0.960422	+	0.278549i	$0.0898534\pi$
$151$			1033.70i			0.557097i	−0.960422	+	0.278549i	$0.910147\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	205.952	+	205.952i	0.106726	+	0.106726i
$156$		0			0
$157$	−2314.43	+	2314.43i	−1.17651	+	1.17651i	−0.195879	+	0.980628i	$0.562756\pi$
$157$	−2314.43	+	2314.43i	−1.17651	+	1.17651i	−0.980628	+	0.195879i	$0.937244\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	190.937			0.0934656
$162$		0			0
$163$	−737.872	+	737.872i	−0.354568	+	0.354568i	−0.861806	−	0.507238i	$-0.830667\pi$
$163$	−737.872	+	737.872i	−0.354568	+	0.354568i	0.507238	+	0.861806i	$0.330667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			468.664i			0.217164i	0.994088	+	0.108582i	$0.0346309\pi$
$167$			468.664i			0.217164i	−0.994088	+	0.108582i	$0.965369\pi$
$168$		0			0
$169$			1569.87i			0.714551i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1671.65	+	1671.65i	−0.734643	+	0.734643i	−0.971536	−	0.236893i	$-0.923871\pi$
$173$	−1671.65	+	1671.65i	−0.734643	+	0.734643i	0.236893	+	0.971536i	$0.423871\pi$
$174$		0			0
$175$	550.718			0.237888
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1413.56	−	1413.56i	0.590247	−	0.590247i	−0.347451	−	0.937698i	$-0.612953\pi$
$179$	1413.56	−	1413.56i	0.590247	−	0.590247i	0.937698	+	0.347451i	$0.112953\pi$
$180$		0			0
$181$	−2515.65	−	2515.65i	−1.03308	−	1.03308i	−0.999434	−	0.0336429i	$-0.989289\pi$
$181$	−2515.65	−	2515.65i	−1.03308	−	1.03308i	−0.0336429	−	0.999434i	$-0.510711\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	213.439i		−	0.0848235i
$186$		0			0
$187$	−1871.17	−	1871.17i	−0.731730	−	0.731730i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	18.1653			0.00688165			0.00344082	−	0.999994i	$-0.498905\pi$
$191$	18.1653			0.00688165			0.00344082	+	0.999994i	$0.498905\pi$
$192$		0			0
$193$	1318.14			0.491616			0.245808	−	0.969319i	$-0.420947\pi$
$193$	1318.14			0.491616			0.245808	+	0.969319i	$0.420947\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1818.15	+	1818.15i	0.657551	+	0.657551i	0.954800	−	0.297249i	$-0.0960691\pi$
$197$	1818.15	+	1818.15i	0.657551	+	0.657551i	−0.297249	+	0.954800i	$0.596069\pi$
$198$		0			0
$199$			4114.56i			1.46570i	0.680392	+	0.732849i	$0.261810\pi$
$199$			4114.56i			1.46570i	−0.680392	+	0.732849i	$0.738190\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	824.495	+	824.495i	0.285065	+	0.285065i
$204$		0			0
$205$	−42.6707	+	42.6707i	−0.0145378	+	0.0145378i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1427.15			0.472335
$210$		0			0
$211$	−1636.19	+	1636.19i	−0.533840	+	0.533840i	−0.921713	−	0.387873i	$-0.873210\pi$
$211$	−1636.19	+	1636.19i	−0.533840	+	0.533840i	0.387873	+	0.921713i	$0.373210\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			170.327i			0.0540289i
$216$		0			0
$217$		−	1294.55i		−	0.404975i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1870.66	+	1870.66i	−0.569385	+	0.569385i
$222$		0			0
$223$	3121.80			0.937450			0.468725	−	0.883344i	$-0.344714\pi$
$223$	3121.80			0.937450			0.468725	+	0.883344i	$0.344714\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3405.69	+	3405.69i	−0.995787	+	0.995787i	−0.999991	−	0.00420438i	$-0.998662\pi$
$227$	−3405.69	+	3405.69i	−0.995787	+	0.995787i	0.00420438	+	0.999991i	$0.498662\pi$
$228$		0			0
$229$	2566.84	+	2566.84i	0.740706	+	0.740706i	0.972714	−	0.232008i	$-0.0745294\pi$
$229$	2566.84	+	2566.84i	0.740706	+	0.740706i	−0.232008	+	0.972714i	$0.574529\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			1509.90i			0.424534i	0.977212	+	0.212267i	$0.0680847\pi$
$233$			1509.90i			0.424534i	−0.977212	+	0.212267i	$0.931915\pi$
$234$		0			0
$235$	−353.410	−	353.410i	−0.0981018	−	0.0981018i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2608.05			−0.705860			−0.352930	−	0.935650i	$-0.614815\pi$
$239$	−2608.05			−0.705860			−0.352930	+	0.935650i	$0.614815\pi$
$240$		0			0
$241$	2222.20			0.593962			0.296981	−	0.954883i	$-0.404020\pi$
$241$	2222.20			0.593962			0.296981	+	0.954883i	$0.404020\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−228.415	−	228.415i	−0.0595628	−	0.0595628i
$246$		0			0
$247$		−	1426.76i		−	0.367541i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2551.97	−	2551.97i	−0.641749	−	0.641749i	0.309236	−	0.950985i	$-0.399927\pi$
$251$	−2551.97	−	2551.97i	−0.641749	−	0.641749i	−0.950985	+	0.309236i	$0.899927\pi$
$252$		0			0
$253$	−761.503	+	761.503i	−0.189230	+	0.189230i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2203.05			−0.534718			−0.267359	−	0.963597i	$-0.586151\pi$
$257$	−2203.05			−0.534718			−0.267359	+	0.963597i	$0.586151\pi$
$258$		0			0
$259$	−670.802	+	670.802i	−0.160933	+	0.160933i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			7230.57i			1.69527i	0.530581	+	0.847635i	$0.321974\pi$
$263$			7230.57i			1.69527i	−0.530581	+	0.847635i	$0.678026\pi$
$264$		0			0
$265$			271.896i			0.0630280i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3750.63	+	3750.63i	−0.850112	+	0.850112i	−0.990147	−	0.140035i	$-0.955279\pi$
$269$	−3750.63	+	3750.63i	−0.850112	+	0.850112i	0.140035	+	0.990147i	$0.455279\pi$
$270$		0			0
$271$	−2751.65			−0.616792			−0.308396	−	0.951258i	$-0.599792\pi$
$271$	−2751.65			−0.616792			−0.308396	+	0.951258i	$0.599792\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2196.39	+	2196.39i	−0.481627	+	0.481627i
$276$		0			0
$277$	3621.67	+	3621.67i	0.785578	+	0.785578i	0.980766	−	0.195188i	$-0.0625318\pi$
$277$	3621.67	+	3621.67i	0.785578	+	0.785578i	−0.195188	+	0.980766i	$0.562532\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			4989.59i			1.05927i	0.848227	+	0.529633i	$0.177670\pi$
$281$			4989.59i			1.05927i	−0.848227	+	0.529633i	$0.822330\pi$
$282$		0			0
$283$	−2426.42	−	2426.42i	−0.509667	−	0.509667i	0.404757	−	0.914424i	$-0.367356\pi$
$283$	−2426.42	−	2426.42i	−0.509667	−	0.509667i	−0.914424	+	0.404757i	$0.867356\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	268.213			0.0551642
$288$		0			0
$289$	6246.91			1.27151
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3379.84	−	3379.84i	−0.673899	−	0.673899i	0.284714	−	0.958613i	$-0.408101\pi$
$293$	−3379.84	−	3379.84i	−0.673899	−	0.673899i	−0.958613	+	0.284714i	$0.908101\pi$
$294$		0			0
$295$			706.639i			0.139465i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	761.294	+	761.294i	0.147247	+	0.147247i
$300$		0			0
$301$	535.309	−	535.309i	0.102507	−	0.102507i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	235.170			0.0441502
$306$		0			0
$307$	2575.52	−	2575.52i	0.478803	−	0.478803i	−0.425946	−	0.904749i	$-0.640059\pi$
$307$	2575.52	−	2575.52i	0.478803	−	0.478803i	0.904749	+	0.425946i	$0.140059\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			2993.88i			0.545875i	0.962032	+	0.272938i	$0.0879953\pi$
$311$			2993.88i			0.545875i	−0.962032	+	0.272938i	$0.912005\pi$
$312$		0			0
$313$		−	6133.96i		−	1.10771i	−0.832614	−	0.553853i	$-0.813157\pi$
$313$		−	6133.96i		−	1.10771i	0.832614	−	0.553853i	$-0.186843\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6740.75	−	6740.75i	1.19432	−	1.19432i	0.218474	−	0.975843i	$-0.429892\pi$
$317$	6740.75	−	6740.75i	1.19432	−	1.19432i	0.975843	−	0.218474i	$-0.0701078\pi$
$318$		0			0
$319$	−6576.56			−1.15428
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4255.85	+	4255.85i	−0.733133	+	0.733133i
$324$		0			0
$325$	2195.79	+	2195.79i	0.374771	+	0.374771i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			2221.42i			0.372251i
$330$		0			0
$331$	6485.75	+	6485.75i	1.07701	+	1.07701i	0.996776	+	0.0802296i	$0.0255654\pi$
$331$	6485.75	+	6485.75i	1.07701	+	1.07701i	0.0802296	+	0.996776i	$0.474435\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−813.150			−0.132618
$336$		0			0
$337$	−9405.38			−1.52031			−0.760154	−	0.649743i	$-0.774877\pi$
$337$	−9405.38			−1.52031			−0.760154	+	0.649743i	$0.774877\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5162.96	+	5162.96i	0.819912	+	0.819912i
$342$		0			0
$343$			2959.08i			0.465817i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1557.96	−	1557.96i	−0.241025	−	0.241025i	0.576249	−	0.817274i	$-0.304516\pi$
$347$	−1557.96	−	1557.96i	−0.241025	−	0.241025i	−0.817274	+	0.576249i	$0.804516\pi$
$348$		0			0
$349$	−3041.63	+	3041.63i	−0.466518	+	0.466518i	−0.900784	−	0.434266i	$-0.857008\pi$
$349$	−3041.63	+	3041.63i	−0.466518	+	0.466518i	0.434266	+	0.900784i	$0.357008\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1629.66			−0.245717			−0.122859	−	0.992424i	$-0.539206\pi$
$353$	−1629.66			−0.245717			−0.122859	+	0.992424i	$0.539206\pi$
$354$		0			0
$355$	−323.104	+	323.104i	−0.0483059	+	0.0483059i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	9276.15i		−	1.36372i	−0.731482	−	0.681861i	$-0.761171\pi$
$359$		−	9276.15i		−	1.36372i	0.731482	−	0.681861i	$-0.238829\pi$
$360$		0			0
$361$			3613.04i			0.526759i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−764.917	+	764.917i	−0.109692	+	0.109692i
$366$		0			0
$367$	−1016.63			−0.144599			−0.0722994	−	0.997383i	$-0.523034\pi$
$367$	−1016.63			−0.144599			−0.0722994	+	0.997383i	$0.523034\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	854.522	−	854.522i	0.119581	−	0.119581i
$372$		0			0
$373$	−8897.07	−	8897.07i	−1.23505	−	1.23505i	−0.962001	−	0.273047i	$-0.911969\pi$
$373$	−8897.07	−	8897.07i	−1.23505	−	1.23505i	−0.273047	−	0.962001i	$-0.588031\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			6574.76i			0.898190i
$378$		0			0
$379$	5154.79	+	5154.79i	0.698638	+	0.698638i	0.964117	−	0.265478i	$-0.0855299\pi$
$379$	5154.79	+	5154.79i	0.698638	+	0.698638i	−0.265478	+	0.964117i	$0.585530\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10573.7			1.41068			0.705340	−	0.708870i	$-0.250795\pi$
$383$	10573.7			1.41068			0.705340	+	0.708870i	$0.250795\pi$
$384$		0			0
$385$	−111.165			−0.0147155
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7789.01	−	7789.01i	−1.01522	−	1.01522i	−0.999882	−	0.0153326i	$-0.995119\pi$
$389$	−7789.01	−	7789.01i	−1.01522	−	1.01522i	−0.0153326	−	0.999882i	$-0.504881\pi$
$390$		0			0
$391$		−	4541.70i		−	0.587426i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−384.589	−	384.589i	−0.0489893	−	0.0489893i
$396$		0			0
$397$	−1219.69	+	1219.69i	−0.154193	+	0.154193i	−0.779988	−	0.625795i	$-0.784775\pi$
$397$	−1219.69	+	1219.69i	−0.154193	+	0.154193i	0.625795	+	0.779988i	$0.284775\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5714.97			0.711701			0.355850	−	0.934543i	$-0.384191\pi$
$401$	5714.97			0.711701			0.355850	+	0.934543i	$0.384191\pi$
$402$		0			0
$403$	5161.55	−	5161.55i	0.638003	−	0.638003i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	5350.64i		−	0.651649i
$408$		0			0
$409$		−	1504.54i		−	0.181894i	−0.995856	−	0.0909469i	$-0.971011\pi$
$409$		−	1504.54i		−	0.181894i	0.995856	−	0.0909469i	$-0.0289894\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2220.85	−	2220.85i	0.264602	−	0.264602i
$414$		0			0
$415$	−68.8095			−0.00813910
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9630.38	−	9630.38i	1.12285	−	1.12285i	0.131541	−	0.991311i	$-0.458007\pi$
$419$	9630.38	−	9630.38i	1.12285	−	1.12285i	0.991311	−	0.131541i	$-0.0419927\pi$
$420$		0			0
$421$	2831.19	+	2831.19i	0.327752	+	0.327752i	0.851731	−	0.523979i	$-0.175553\pi$
$421$	2831.19	+	2831.19i	0.327752	+	0.327752i	−0.523979	+	0.851731i	$0.675553\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	13099.6i		−	1.49511i
$426$		0			0
$427$	−739.100	−	739.100i	−0.0837647	−	0.0837647i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5555.36			0.620864			0.310432	−	0.950596i	$-0.399526\pi$
$431$	5555.36			0.620864			0.310432	+	0.950596i	$0.399526\pi$
$432$		0			0
$433$	−14582.8			−1.61848			−0.809241	−	0.587476i	$-0.800122\pi$
$433$	−14582.8			−1.61848			−0.809241	+	0.587476i	$0.800122\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1731.99	+	1731.99i	0.189593	+	0.189593i
$438$		0			0
$439$		−	4862.86i		−	0.528682i	−0.964429	−	0.264341i	$-0.914846\pi$
$439$		−	4862.86i		−	0.528682i	0.964429	−	0.264341i	$-0.0851545\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	9325.47	+	9325.47i	1.00015	+	1.00015i	1.00000		0.000149687i	$4.76467e-5\pi$
$443$	9325.47	+	9325.47i	1.00015	+	1.00015i	0.000149687		1.00000i	$0.499952\pi$
$444$		0			0
$445$	−30.8566	+	30.8566i	−0.00328707	+	0.00328707i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−11461.1			−1.20464			−0.602318	−	0.798256i	$-0.705756\pi$
$449$	−11461.1			−1.20464			−0.602318	+	0.798256i	$0.705756\pi$
$450$		0			0
$451$	−1069.70	+	1069.70i	−0.111685	+	0.111685i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			111.134i			0.0114507i
$456$		0			0
$457$			13856.7i			1.41835i	0.705031	+	0.709177i	$0.250933\pi$
$457$			13856.7i			1.41835i	−0.705031	+	0.709177i	$0.749067\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3166.73	−	3166.73i	0.319934	−	0.319934i	−0.528808	−	0.848741i	$-0.677361\pi$
$461$	3166.73	−	3166.73i	0.319934	−	0.319934i	0.848741	+	0.528808i	$0.177361\pi$
$462$		0			0
$463$	−1566.96			−0.157285			−0.0786426	−	0.996903i	$-0.525059\pi$
$463$	−1566.96			−0.157285			−0.0786426	+	0.996903i	$0.525059\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	5309.28	−	5309.28i	0.526090	−	0.526090i	−0.393314	−	0.919404i	$-0.628671\pi$
$467$	5309.28	−	5309.28i	0.526090	−	0.526090i	0.919404	+	0.393314i	$0.128671\pi$
$468$		0			0
$469$	2555.59	+	2555.59i	0.251613	+	0.251613i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			4269.88i			0.415072i
$474$		0			0
$475$	4995.55	+	4995.55i	0.482551	+	0.482551i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4034.12			−0.384809			−0.192405	−	0.981316i	$-0.561629\pi$
$479$	−4034.12			−0.384809			−0.192405	+	0.981316i	$0.561629\pi$
$480$		0			0
$481$	−5349.17			−0.507071
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−487.908	−	487.908i	−0.0456799	−	0.0456799i
$486$		0			0
$487$		−	16499.3i		−	1.53522i	−0.640915	−	0.767612i	$-0.721445\pi$
$487$		−	16499.3i		−	1.53522i	0.640915	−	0.767612i	$-0.278555\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1060.25	+	1060.25i	0.0974506	+	0.0974506i	0.754151	−	0.656701i	$-0.228049\pi$
$491$	1060.25	+	1060.25i	0.0974506	+	0.0974506i	−0.656701	+	0.754151i	$0.728049\pi$
$492$		0			0
$493$	19611.7	−	19611.7i	1.79162	−	1.79162i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2030.93			0.183299
$498$		0			0
$499$	−4275.19	+	4275.19i	−0.383534	+	0.383534i	−0.872374	−	0.488839i	$-0.837420\pi$
$499$	−4275.19	+	4275.19i	−0.383534	+	0.383534i	0.488839	+	0.872374i	$0.337420\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	13976.3i		−	1.23891i	−0.785031	−	0.619456i	$-0.787353\pi$
$503$		−	13976.3i		−	1.23891i	0.785031	−	0.619456i	$-0.212647\pi$
$504$		0			0
$505$			797.687i			0.0702903i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9315.16	−	9315.16i	0.811174	−	0.811174i	−0.173636	−	0.984810i	$-0.555552\pi$
$509$	9315.16	−	9315.16i	0.811174	−	0.811174i	0.984810	+	0.173636i	$0.0555517\pi$
$510$		0			0
$511$	4808.01			0.416230
$512$		0			0
$513$		0			0
$514$		0			0
$515$	78.0102	−	78.0102i	0.00667484	−	0.00667484i
$516$		0			0
$517$	−8859.53	−	8859.53i	−0.753659	−	0.753659i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	10404.0i		−	0.874870i	−0.899250	−	0.437435i	$-0.855887\pi$
$521$		−	10404.0i		−	0.874870i	0.899250	−	0.437435i	$-0.144113\pi$
$522$		0			0
$523$	−2520.38	−	2520.38i	−0.210724	−	0.210724i	0.593851	−	0.804575i	$-0.297607\pi$
$523$	−2520.38	−	2520.38i	−0.210724	−	0.210724i	−0.804575	+	0.593851i	$0.797607\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−30792.6			−2.54525
$528$		0			0
$529$	10318.7			0.848088
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1069.41	+	1069.41i	0.0869064	+	0.0869064i
$534$		0			0
$535$			359.439i			0.0290465i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5726.06	−	5726.06i	−0.457586	−	0.457586i
$540$		0			0
$541$	13635.3	−	13635.3i	1.08360	−	1.08360i	0.0874292	−	0.996171i	$-0.472135\pi$
$541$	13635.3	−	13635.3i	1.08360	−	1.08360i	0.996171	−	0.0874292i	$-0.0278651\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	289.726			0.0227716
$546$		0			0
$547$	1729.49	−	1729.49i	0.135188	−	0.135188i	−0.636275	−	0.771463i	$-0.719526\pi$
$547$	1729.49	−	1729.49i	0.135188	−	0.135188i	0.771463	+	0.636275i	$0.219526\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			14958.0i			1.15650i
$552$		0			0
$553$			2417.39i			0.185892i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3758.29	+	3758.29i	−0.285896	+	0.285896i	−0.835455	−	0.549559i	$-0.814796\pi$
$557$	−3758.29	+	3758.29i	−0.285896	+	0.285896i	0.549559	+	0.835455i	$0.314796\pi$
$558$		0			0
$559$	4268.71			0.322982
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1676.08	−	1676.08i	0.125468	−	0.125468i	−0.641585	−	0.767052i	$-0.721723\pi$
$563$	1676.08	−	1676.08i	0.125468	−	0.125468i	0.767052	+	0.641585i	$0.221723\pi$
$564$		0			0
$565$	−515.118	−	515.118i	−0.0383560	−	0.0383560i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	11314.4i		−	0.833611i	−0.908996	−	0.416805i	$-0.863150\pi$
$569$		−	11314.4i		−	0.833611i	0.908996	−	0.416805i	$-0.136850\pi$
$570$		0			0
$571$	−4555.47	−	4555.47i	−0.333871	−	0.333871i	0.520184	−	0.854055i	$-0.325864\pi$
$571$	−4555.47	−	4555.47i	−0.333871	−	0.333871i	−0.854055	+	0.520184i	$0.825864\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−5331.08			−0.386646
$576$		0			0
$577$	15216.2			1.09785			0.548925	−	0.835872i	$-0.315037\pi$
$577$	15216.2			1.09785			0.548925	+	0.835872i	$0.315037\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	216.257	+	216.257i	0.0154421	+	0.0154421i
$582$		0			0
$583$			6816.07i			0.484207i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3111.52	+	3111.52i	0.218784	+	0.218784i	0.807986	−	0.589202i	$-0.200558\pi$
$587$	3111.52	+	3111.52i	0.218784	+	0.218784i	−0.589202	+	0.807986i	$0.700558\pi$
$588$		0			0
$589$	11742.8	−	11742.8i	0.821484	−	0.821484i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	7275.10			0.503799			0.251899	−	0.967753i	$-0.418945\pi$
$593$	7275.10			0.503799			0.251899	+	0.967753i	$0.418945\pi$
$594$		0			0
$595$	331.500	−	331.500i	0.0228406	−	0.0228406i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	18975.0i		−	1.29432i	−0.762355	−	0.647159i	$-0.775957\pi$
$599$		−	18975.0i		−	1.29432i	0.762355	−	0.647159i	$-0.224043\pi$
$600$		0			0
$601$			2493.99i			0.169271i	0.996412	+	0.0846357i	$0.0269726\pi$
$601$			2493.99i			0.169271i	−0.996412	+	0.0846357i	$0.973027\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−497.085	+	497.085i	−0.0334039	+	0.0334039i
$606$		0			0
$607$	−13075.7			−0.874341			−0.437171	−	0.899379i	$-0.644019\pi$
$607$	−13075.7			−0.874341			−0.437171	+	0.899379i	$0.644019\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8857.11	+	8857.11i	−0.586449	+	0.586449i
$612$		0			0
$613$	10449.8	+	10449.8i	0.688523	+	0.688523i	0.961906	−	0.273382i	$-0.0881423\pi$
$613$	10449.8	+	10449.8i	0.688523	+	0.688523i	−0.273382	+	0.961906i	$0.588142\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			6927.08i			0.451983i	0.974129	+	0.225992i	$0.0725622\pi$
$617$			6927.08i			0.451983i	−0.974129	+	0.225992i	$0.927438\pi$
$618$		0			0
$619$	16635.4	+	16635.4i	1.08018	+	1.08018i	0.996492	+	0.0836922i	$0.0266713\pi$
$619$	16635.4	+	16635.4i	1.08018	+	1.08018i	0.0836922	+	0.996492i	$0.473329\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	193.954			0.0124729
$624$		0			0
$625$	−15251.6			−0.976101
$626$		0			0
$627$		0			0
$628$		0			0
$629$	15955.9	+	15955.9i	1.01146	+	1.01146i
$630$		0			0
$631$		−	22231.3i		−	1.40256i	−0.712888	−	0.701278i	$-0.752613\pi$
$631$		−	22231.3i		−	1.40256i	0.712888	−	0.701278i	$-0.247387\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1717.81	−	1717.81i	−0.107353	−	0.107353i
$636$		0			0
$637$	−5724.50	+	5724.50i	−0.356064	+	0.356064i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8447.90			−0.520549			−0.260274	−	0.965535i	$-0.583813\pi$
$641$	−8447.90			−0.520549			−0.260274	+	0.965535i	$0.583813\pi$
$642$		0			0
$643$	−5326.38	+	5326.38i	−0.326675	+	0.326675i	−0.851321	−	0.524646i	$-0.824198\pi$
$643$	−5326.38	+	5326.38i	−0.326675	+	0.326675i	0.524646	+	0.851321i	$0.324198\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	7105.36i		−	0.431747i	−0.976421	−	0.215874i	$-0.930740\pi$
$647$		−	7105.36i		−	0.431747i	0.976421	−	0.215874i	$-0.0692599\pi$
$648$		0			0
$649$			17714.5i			1.07143i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1518.80	+	1518.80i	−0.0910189	+	0.0910189i	−0.751150	−	0.660131i	$-0.770501\pi$
$653$	−1518.80	+	1518.80i	−0.0910189	+	0.0910189i	0.660131	+	0.751150i	$0.270501\pi$
$654$		0			0
$655$	−1269.24			−0.0757152
$656$		0			0
$657$		0			0
$658$		0			0
$659$	5425.29	−	5425.29i	0.320697	−	0.320697i	−0.528337	−	0.849035i	$-0.677184\pi$
$659$	5425.29	−	5425.29i	0.320697	−	0.320697i	0.849035	+	0.528337i	$0.177184\pi$
$660$		0			0
$661$	−18128.4	−	18128.4i	−1.06674	−	1.06674i	−0.997608	−	0.0691300i	$-0.977978\pi$
$661$	−18128.4	−	18128.4i	−1.06674	−	1.06674i	−0.0691300	−	0.997608i	$-0.522022\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			252.837i			0.0147437i
$666$		0			0
$667$	−7981.31	−	7981.31i	−0.463325	−	0.463325i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5895.41			0.339180
$672$		0			0
$673$	1072.11			0.0614071			0.0307036	−	0.999529i	$-0.490225\pi$
$673$	1072.11			0.0614071			0.0307036	+	0.999529i	$0.490225\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4806.53	−	4806.53i	−0.272865	−	0.272865i	0.557387	−	0.830253i	$-0.311804\pi$
$677$	−4806.53	−	4806.53i	−0.272865	−	0.272865i	−0.830253	+	0.557387i	$0.811804\pi$
$678$		0			0
$679$			3066.82i			0.173334i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	21855.0	+	21855.0i	1.22439	+	1.22439i	0.966056	+	0.258334i	$0.0831737\pi$
$683$	21855.0	+	21855.0i	1.22439	+	1.22439i	0.258334	+	0.966056i	$0.416826\pi$
$684$		0			0
$685$	−2163.35	+	2163.35i	−0.120667	+	0.120667i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6814.21			0.376779
$690$		0			0
$691$	585.815	−	585.815i	0.0322510	−	0.0322510i	−0.690797	−	0.723048i	$-0.742740\pi$
$691$	585.815	−	585.815i	0.0322510	−	0.0322510i	0.723048	+	0.690797i	$0.242740\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			2560.15i			0.139730i
$696$		0			0
$697$		−	6379.82i		−	0.346704i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	7273.96	−	7273.96i	0.391917	−	0.391917i	−0.483453	−	0.875370i	$-0.660618\pi$
$701$	7273.96	−	7273.96i	0.391917	−	0.391917i	0.875370	+	0.483453i	$0.160618\pi$
$702$		0			0
$703$	−12169.7			−0.652899
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2507.00	−	2507.00i	0.133360	−	0.133360i
$708$		0			0
$709$	17600.7	+	17600.7i	0.932309	+	0.932309i	0.997850	−	0.0655405i	$-0.0208772\pi$
$709$	17600.7	+	17600.7i	0.932309	+	0.932309i	−0.0655405	+	0.997850i	$0.520877\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			12531.5i			0.658218i
$714$		0			0
$715$	−443.230	−	443.230i	−0.0231830	−	0.0231830i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	14431.8			0.748562			0.374281	−	0.927315i	$-0.377890\pi$
$719$	14431.8			0.748562			0.374281	+	0.927315i	$0.377890\pi$
$720$		0			0
$721$	−490.346			−0.0253279
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−23020.4	−	23020.4i	−1.17925	−	1.17925i
$726$		0			0
$727$			5802.18i			0.295998i	0.988987	+	0.147999i	$0.0472833\pi$
$727$			5802.18i			0.295998i	−0.988987	+	0.147999i	$0.952717\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−12733.1	−	12733.1i	−0.644253	−	0.644253i
$732$		0			0
$733$	−16743.4	+	16743.4i	−0.843699	+	0.843699i	−0.989338	−	0.145639i	$-0.953476\pi$
$733$	−16743.4	+	16743.4i	−0.843699	+	0.843699i	0.145639	+	0.989338i	$0.453476\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−20384.6			−1.01883
$738$		0			0
$739$	−5259.84	+	5259.84i	−0.261822	+	0.261822i	−0.825794	−	0.563972i	$-0.809273\pi$
$739$	−5259.84	+	5259.84i	−0.261822	+	0.261822i	0.563972	+	0.825794i	$0.309273\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			29413.6i			1.45233i	0.687522	+	0.726163i	$0.258698\pi$
$743$			29413.6i			1.45233i	−0.687522	+	0.726163i	$0.741302\pi$
$744$		0			0
$745$		−	1228.21i		−	0.0604002i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1129.66	−	1129.66i	0.0551091	−	0.0551091i
$750$		0			0
$751$	−37261.5			−1.81051			−0.905254	−	0.424870i	$-0.860320\pi$
$751$	−37261.5			−1.81051			−0.905254	+	0.424870i	$0.860320\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	730.378	−	730.378i	0.0352069	−	0.0352069i
$756$		0			0
$757$	4651.56	+	4651.56i	0.223334	+	0.223334i	0.809901	−	0.586567i	$-0.199521\pi$
$757$	4651.56	+	4651.56i	0.223334	+	0.223334i	−0.586567	+	0.809901i	$0.699521\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			25144.7i			1.19776i	0.800839	+	0.598879i	$0.204387\pi$
$761$			25144.7i			1.19776i	−0.800839	+	0.598879i	$0.795613\pi$
$762$		0			0
$763$	−910.561	−	910.561i	−0.0432038	−	0.0432038i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	17709.7			0.833715
$768$		0			0
$769$	−29241.9			−1.37125			−0.685625	−	0.727955i	$-0.740471\pi$
$769$	−29241.9			−1.37125			−0.685625	+	0.727955i	$0.740471\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6698.95	+	6698.95i	0.311700	+	0.311700i	0.845568	−	0.533868i	$-0.179262\pi$
$773$	6698.95	+	6698.95i	0.311700	+	0.311700i	−0.533868	+	0.845568i	$0.679262\pi$
$774$		0			0
$775$			36144.5i			1.67529i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2432.96	+	2432.96i	0.111900	+	0.111900i
$780$		0			0
$781$	−8099.81	+	8099.81i	−0.371106	+	0.371106i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3270.58			0.148703
$786$		0			0
$787$	4317.53	−	4317.53i	0.195557	−	0.195557i	−0.602535	−	0.798092i	$-0.705843\pi$
$787$	4317.53	−	4317.53i	0.195557	−	0.195557i	0.798092	+	0.602535i	$0.205843\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			3237.85i			0.145543i
$792$		0			0
$793$		−	5893.80i		−	0.263928i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−12492.2	+	12492.2i	−0.555201	+	0.555201i	−0.927937	−	0.372736i	$-0.878420\pi$
$797$	−12492.2	+	12492.2i	−0.555201	+	0.555201i	0.372736	+	0.927937i	$0.378420\pi$
$798$		0			0
$799$	52839.4			2.33958
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−19175.5	+	19175.5i	−0.842700	+	0.842700i
$804$		0			0
$805$	−134.909	−	134.909i	−0.00590674	−	0.00590674i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	4276.80i		−	0.185864i	−0.995672	−	0.0929321i	$-0.970376\pi$
$809$		−	4276.80i		−	0.185864i	0.995672	−	0.0929321i	$-0.0296239\pi$
$810$		0			0
$811$	15331.3	+	15331.3i	0.663817	+	0.663817i	0.956277	−	0.292461i	$-0.0944741\pi$
$811$	15331.3	+	15331.3i	0.663817	+	0.663817i	−0.292461	+	0.956277i	$0.594474\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1042.71			0.0448152
$816$		0			0
$817$	9711.55			0.415868
$818$		0			0
$819$		0			0
$820$		0			0
$821$	12527.1	+	12527.1i	0.532521	+	0.532521i	0.921322	−	0.388801i	$-0.127110\pi$
$821$	12527.1	+	12527.1i	0.532521	+	0.532521i	−0.388801	+	0.921322i	$0.627110\pi$
$822$		0			0
$823$		−	15241.1i		−	0.645532i	−0.946479	−	0.322766i	$-0.895387\pi$
$823$		−	15241.1i		−	0.645532i	0.946479	−	0.322766i	$-0.104613\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−25529.5	−	25529.5i	−1.07345	−	1.07345i	−0.997079	−	0.0763756i	$-0.975665\pi$
$827$	−25529.5	−	25529.5i	−1.07345	−	1.07345i	−0.0763756	−	0.997079i	$-0.524335\pi$
$828$		0			0
$829$	−23542.4	+	23542.4i	−0.986322	+	0.986322i	−0.999908	−	0.0135854i	$-0.995676\pi$
$829$	−23542.4	+	23542.4i	−0.986322	+	0.986322i	0.0135854	+	0.999908i	$0.495676\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	34151.0			1.42048
$834$		0			0
$835$	331.141	−	331.141i	0.0137241	−	0.0137241i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			3385.26i			0.139299i	0.997572	+	0.0696497i	$0.0221882\pi$
$839$			3385.26i			0.139299i	−0.997572	+	0.0696497i	$0.977812\pi$
$840$		0			0
$841$		−	44540.0i		−	1.82623i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1109.21	−	1109.21i	0.0451575	−	0.0451575i
$846$		0			0
$847$	3124.51			0.126753
$848$		0			0
$849$		0			0
$850$		0			0
$851$	6493.53	−	6493.53i	0.261569	−	0.261569i
$852$		0			0
$853$	4371.94	+	4371.94i	0.175489	+	0.175489i	0.789386	−	0.613897i	$-0.210399\pi$
$853$	4371.94	+	4371.94i	0.175489	+	0.175489i	−0.613897	+	0.789386i	$0.710399\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			32205.4i			1.28368i	0.766838	+	0.641840i	$0.221829\pi$
$857$			32205.4i			1.28368i	−0.766838	+	0.641840i	$0.778171\pi$
$858$		0			0
$859$	−2875.17	−	2875.17i	−0.114202	−	0.114202i	0.647696	−	0.761898i	$-0.275732\pi$
$859$	−2875.17	−	2875.17i	−0.114202	−	0.114202i	−0.761898	+	0.647696i	$0.775732\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	18181.9			0.717173			0.358586	−	0.933497i	$-0.383259\pi$
$863$	18181.9			0.717173			0.358586	+	0.933497i	$0.383259\pi$
$864$		0			0
$865$	2362.26			0.0928544
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−9641.15	−	9641.15i	−0.376356	−	0.376356i
$870$		0			0
$871$			20379.0i			0.792787i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−781.368	−	781.368i	−0.0301886	−	0.0301886i
$876$		0			0
$877$	−26665.8	+	26665.8i	−1.02673	+	1.02673i	−0.0270950	+	0.999633i	$0.508626\pi$
$877$	−26665.8	+	26665.8i	−1.02673	+	1.02673i	−0.999633	+	0.0270950i	$0.991374\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−26033.2			−0.995552			−0.497776	−	0.867305i	$-0.665850\pi$
$881$	−26033.2			−0.995552			−0.497776	+	0.867305i	$0.665850\pi$
$882$		0			0
$883$	27917.9	−	27917.9i	1.06400	−	1.06400i	0.0661953	−	0.997807i	$-0.478914\pi$
$883$	27917.9	−	27917.9i	1.06400	−	1.06400i	0.997807	−	0.0661953i	$-0.0210860\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			28947.2i			1.09577i	0.836553	+	0.547887i	$0.184567\pi$
$887$			28947.2i			1.09577i	−0.836553	+	0.547887i	$0.815433\pi$
$888$		0			0
$889$			10797.6i			0.407355i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−20150.4	+	20150.4i	−0.755104	+	0.755104i
$894$		0			0
$895$	−1997.54			−0.0746036
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−54113.0	+	54113.0i	−2.00753	+	2.00753i
$900$		0			0
$901$	−20326.0	−	20326.0i	−0.751561	−	0.751561i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			3554.94i			0.130575i
$906$		0			0
$907$	−3191.29	−	3191.29i	−0.116830	−	0.116830i	0.646275	−	0.763105i	$-0.276326\pi$
$907$	−3191.29	−	3191.29i	−0.116830	−	0.116830i	−0.763105	+	0.646275i	$0.776326\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−40016.6			−1.45534			−0.727668	−	0.685930i	$-0.759396\pi$
$911$	−40016.6			−1.45534			−0.727668	+	0.685930i	$0.759396\pi$
$912$		0			0
$913$	−1724.97			−0.0625280
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3989.02	+	3989.02i	0.143652	+	0.143652i
$918$		0			0
$919$			15342.1i			0.550694i	0.961345	+	0.275347i	$0.0887928\pi$
$919$			15342.1i			0.550694i	−0.961345	+	0.275347i	$0.911207\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	8097.60	+	8097.60i	0.288771	+	0.288771i
$924$		0			0
$925$	18729.2	−	18729.2i	0.665743	−	0.665743i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	104.186			0.00367946			0.00183973	−	0.999998i	$-0.499414\pi$
$929$	104.186			0.00367946			0.00183973	+	0.999998i	$0.499414\pi$
$930$		0			0
$931$	−13023.5	+	13023.5i	−0.458464	+	0.458464i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			2644.20i			0.0924863i
$936$		0			0
$937$		−	3137.81i		−	0.109400i	−0.998503	−	0.0547001i	$-0.982580\pi$
$937$		−	3137.81i		−	0.109400i	0.998503	−	0.0547001i	$-0.0174203\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7984.14	+	7984.14i	−0.276595	+	0.276595i	−0.831748	−	0.555153i	$-0.812660\pi$
$941$	−7984.14	+	7984.14i	−0.276595	+	0.276595i	0.555153	+	0.831748i	$0.312660\pi$
$942$		0			0
$943$	−2596.37			−0.0896601
$944$		0			0
$945$		0			0
$946$		0			0
$947$	20247.7	−	20247.7i	0.694787	−	0.694787i	−0.268495	−	0.963281i	$-0.586526\pi$
$947$	20247.7	−	20247.7i	0.694787	−	0.694787i	0.963281	+	0.268495i	$0.0865261\pi$
$948$		0			0
$949$	19170.2	+	19170.2i	0.655734	+	0.655734i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	34966.0i		−	1.18852i	−0.804272	−	0.594261i	$-0.797445\pi$
$953$		−	34966.0i		−	1.18852i	0.804272	−	0.594261i	$-0.202555\pi$
$954$		0			0
$955$	−12.8349	−	12.8349i	−0.000434899	−	0.000434899i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	13598.1			0.457877
$960$		0			0
$961$	55172.3			1.85198
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−931.351	−	931.351i	−0.0310686	−	0.0310686i
$966$		0			0
$967$		−	15161.3i		−	0.504194i	−0.967702	−	0.252097i	$-0.918880\pi$
$967$		−	15161.3i		−	0.504194i	0.967702	−	0.252097i	$-0.0811201\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	13515.9	+	13515.9i	0.446701	+	0.446701i	0.894256	−	0.447555i	$-0.147705\pi$
$971$	13515.9	+	13515.9i	0.446701	+	0.446701i	−0.447555	+	0.894256i	$0.647705\pi$
$972$		0			0
$973$	8046.12	−	8046.12i	0.265105	−	0.265105i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−22582.6			−0.739490			−0.369745	−	0.929133i	$-0.620555\pi$
$977$	−22582.6			−0.739490			−0.369745	+	0.929133i	$0.620555\pi$
$978$		0			0
$979$	−773.536	+	773.536i	−0.0252526	+	0.0252526i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	39898.2i		−	1.29456i	−0.762252	−	0.647281i	$-0.775906\pi$
$983$		−	39898.2i		−	1.29456i	0.762252	−	0.647281i	$-0.224094\pi$
$984$		0			0
$985$		−	2569.27i		−	0.0831105i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−5181.92	+	5181.92i	−0.166608	+	0.166608i
$990$		0			0
$991$	36767.4			1.17856			0.589281	−	0.807928i	$-0.299411\pi$
$991$	36767.4			1.17856			0.589281	+	0.807928i	$0.299411\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2907.20	−	2907.20i	0.0926276	−	0.0926276i
$996$		0			0
$997$	6130.71	+	6130.71i	0.194746	+	0.194746i	0.797743	−	0.602997i	$-0.206027\pi$
$997$	6130.71	+	6130.71i	0.194746	+	0.194746i	−0.602997	+	0.797743i	$0.706027\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.4.k.b.433.6		24
3.2	odd	2		192.4.j.a.49.4		24
4.3	odd	2		144.4.k.b.37.2		24
12.11	even	2		48.4.j.a.37.11	yes	24
16.3	odd	4		144.4.k.b.109.2		24
16.13	even	4	inner	576.4.k.b.145.6		24
24.5	odd	2		384.4.j.a.97.10		24
24.11	even	2		384.4.j.b.97.3		24
48.5	odd	4		384.4.j.a.289.10		24
48.11	even	4		384.4.j.b.289.3		24
48.29	odd	4		192.4.j.a.145.4		24
48.35	even	4		48.4.j.a.13.11	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.4.j.a.13.11	✓	24	48.35	even	4
48.4.j.a.37.11	yes	24	12.11	even	2
144.4.k.b.37.2		24	4.3	odd	2
144.4.k.b.109.2		24	16.3	odd	4
192.4.j.a.49.4		24	3.2	odd	2
192.4.j.a.145.4		24	48.29	odd	4
384.4.j.a.97.10		24	24.5	odd	2
384.4.j.a.289.10		24	48.5	odd	4
384.4.j.b.97.3		24	24.11	even	2
384.4.j.b.289.3		24	48.11	even	4
576.4.k.b.145.6		24	16.13	even	4	inner
576.4.k.b.433.6		24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	576.k (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.706564 - 0.706564i) q^{5} +4.44122i q^{7} +O(q^{10})\)	\(q+(-0.706564 - 0.706564i) q^{5} +4.44122i q^{7} +(-17.7126 - 17.7126i) q^{11} +(-17.7078 + 17.7078i) q^{13} +105.640 q^{17} +(-40.2862 + 40.2862i) q^{19} -42.9921i q^{23} -124.002i q^{25} +(185.646 - 185.646i) q^{29} -291.485 q^{31} +(3.13800 - 3.13800i) q^{35} +(151.040 + 151.040i) q^{37} -60.3918i q^{41} +(-120.532 - 120.532i) q^{43} +500.182 q^{47} +323.276 q^{49} +(-192.407 - 192.407i) q^{53} +25.0302i q^{55} +(-500.053 - 500.053i) q^{59} +(-166.418 + 166.418i) q^{61} +25.0234 q^{65} +(575.426 - 575.426i) q^{67} -457.290i q^{71} -1082.59i q^{73} +(78.6657 - 78.6657i) q^{77} +544.309 q^{79} +(48.6931 - 48.6931i) q^{83} +(-74.6417 - 74.6417i) q^{85} -43.6714i q^{89} +(-78.6442 - 78.6442i) q^{91} +56.9295 q^{95} +690.537 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 40 q^{11} - 24 q^{19} - 400 q^{29} + 744 q^{31} - 456 q^{35} + 16 q^{37} - 1240 q^{43} - 1176 q^{49} - 752 q^{53} - 1376 q^{59} - 912 q^{61} - 976 q^{65} + 2256 q^{67} - 1904 q^{77} - 5992 q^{79} + 2680 q^{83} - 240 q^{85} + 3496 q^{91} - 7728 q^{95}+O(q^{100}) \) 24 * q - 40 * q^11 - 24 * q^19 - 400 * q^29 + 744 * q^31 - 456 * q^35 + 16 * q^37 - 1240 * q^43 - 1176 * q^49 - 752 * q^53 - 1376 * q^59 - 912 * q^61 - 976 * q^65 + 2256 * q^67 - 1904 * q^77 - 5992 * q^79 + 2680 * q^83 - 240 * q^85 + 3496 * q^91 - 7728 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more