LMFDB - Embedded newform 384.4.j.a.289.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(97,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("384.97");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.j (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:	$22.6567334422$
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			289.10
Character	$\chi$	$=$	384.289
Dual form			384.4.j.a.97.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.12132 + 2.12132i) q^{3} +(-0.706564 + 0.706564i) q^{5} -4.44122i q^{7} +9.00000i q^{9} +O(q^{10})$	\(q+(2.12132 + 2.12132i) q^{3} +(-0.706564 + 0.706564i) q^{5} -4.44122i q^{7} +9.00000i q^{9} +(-17.7126 + 17.7126i) q^{11} +(17.7078 + 17.7078i) q^{13} -2.99770 q^{15} -105.640 q^{17} +(40.2862 + 40.2862i) q^{19} +(9.42125 - 9.42125i) q^{21} -42.9921i q^{23} +124.002i q^{25} +(-19.0919 + 19.0919i) q^{27} +(185.646 + 185.646i) q^{29} -291.485 q^{31} -75.1483 q^{33} +(3.13800 + 3.13800i) q^{35} +(-151.040 + 151.040i) q^{37} +75.1278i q^{39} -60.3918i q^{41} +(120.532 - 120.532i) q^{43} +(-6.35907 - 6.35907i) q^{45} -500.182 q^{47} +323.276 q^{49} +(-224.097 - 224.097i) q^{51} +(-192.407 + 192.407i) q^{53} -25.0302i q^{55} +170.920i q^{57} +(-500.053 + 500.053i) q^{59} +(166.418 + 166.418i) q^{61} +39.9710 q^{63} -25.0234 q^{65} +(-575.426 - 575.426i) q^{67} +(91.2000 - 91.2000i) q^{69} -457.290i q^{71} +1082.59i q^{73} +(-263.047 + 263.047i) q^{75} +(78.6657 + 78.6657i) q^{77} +544.309 q^{79} -81.0000 q^{81} +(48.6931 + 48.6931i) q^{83} +(74.6417 - 74.6417i) q^{85} +787.630i q^{87} -43.6714i q^{89} +(78.6442 - 78.6442i) q^{91} +(-618.332 - 618.332i) q^{93} -56.9295 q^{95} +690.537 q^{97} +(-159.414 - 159.414i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 40 q^{11} - 120 q^{15} + 24 q^{19} - 400 q^{29} + 744 q^{31} - 456 q^{35} - 16 q^{37} + 1240 q^{43} - 1176 q^{49} + 744 q^{51} - 752 q^{53} - 1376 q^{59} + 912 q^{61} + 504 q^{63} + 976 q^{65} - 2256 q^{67} + 528 q^{69} + 1104 q^{75} - 1904 q^{77} - 5992 q^{79} - 1944 q^{81} + 2680 q^{83} + 240 q^{85} - 3496 q^{91} + 7728 q^{95} - 360 q^{99}+O(q^{100}) $ 24 * q - 40 * q^11 - 120 * q^15 + 24 * q^19 - 400 * q^29 + 744 * q^31 - 456 * q^35 - 16 * q^37 + 1240 * q^43 - 1176 * q^49 + 744 * q^51 - 752 * q^53 - 1376 * q^59 + 912 * q^61 + 504 * q^63 + 976 * q^65 - 2256 * q^67 + 528 * q^69 + 1104 * q^75 - 1904 * q^77 - 5992 * q^79 - 1944 * q^81 + 2680 * q^83 + 240 * q^85 - 3496 * q^91 + 7728 * q^95 - 360 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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
$3$	2.12132	+	2.12132i	0.408248	+	0.408248i
$3$	2.12132	+	2.12132i	0.408248	+	0.408248i
$4$		0			0
$5$	−0.706564	+	0.706564i	−0.0631970	+	0.0631970i	−0.737999	−	0.674802i	$-0.764229\pi$
$5$	−0.706564	+	0.706564i	−0.0631970	+	0.0631970i	0.674802	+	0.737999i	$0.264229\pi$
$6$		0			0
$7$		−	4.44122i		−	0.239803i	−0.992786	−	0.119902i	$-0.961742\pi$
$7$		−	4.44122i		−	0.239803i	0.992786	−	0.119902i	$-0.0382579\pi$
$8$		0			0
$9$			9.00000i			0.333333i
$10$		0			0
$11$	−17.7126	+	17.7126i	−0.485506	+	0.485506i	−0.906885	−	0.421379i	$-0.861546\pi$
$11$	−17.7126	+	17.7126i	−0.485506	+	0.485506i	0.421379	+	0.906885i	$0.361546\pi$
$12$		0			0
$13$	17.7078	+	17.7078i	0.377789	+	0.377789i	0.870304	−	0.492515i	$-0.163922\pi$
$13$	17.7078	+	17.7078i	0.377789	+	0.377789i	−0.492515	+	0.870304i	$0.663922\pi$
$14$		0			0
$15$	−2.99770			−0.0516001
$16$		0			0
$17$	−105.640			−1.50715			−0.753576	−	0.657361i	$-0.771673\pi$
$17$	−105.640			−1.50715			−0.753576	+	0.657361i	$0.771673\pi$
$18$		0			0
$19$	40.2862	+	40.2862i	0.486436	+	0.486436i	0.907180	−	0.420743i	$-0.138231\pi$
$19$	40.2862	+	40.2862i	0.486436	+	0.486436i	−0.420743	+	0.907180i	$0.638231\pi$
$20$		0			0
$21$	9.42125	−	9.42125i	0.0978993	−	0.0978993i
$22$		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
$27$	−19.0919	+	19.0919i	−0.136083	+	0.136083i
$28$		0			0
$29$	185.646	+	185.646i	1.18875	+	1.18875i	0.977415	+	0.211331i	$0.0677797\pi$
$29$	185.646	+	185.646i	1.18875	+	1.18875i	0.211331	+	0.977415i	$0.432220\pi$
$30$		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
$33$	−75.1483			−0.396414
$34$		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	$-0.907386\pi$
$37$	−151.040	+	151.040i	−0.671104	+	0.671104i	0.286867	+	0.957971i	$0.407386\pi$
$38$		0			0
$39$			75.1278i			0.308463i
$40$		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	$-0.652258\pi$
$43$	120.532	−	120.532i	0.427464	−	0.427464i	0.887764	+	0.460300i	$0.152258\pi$
$44$		0			0
$45$	−6.35907	−	6.35907i	−0.0210657	−	0.0210657i
$46$		0			0
$47$	−500.182			−1.55232			−0.776159	−	0.630537i	$-0.782835\pi$
$47$	−500.182			−1.55232			−0.776159	+	0.630537i	$0.782835\pi$
$48$		0			0
$49$	323.276			0.942494
$50$		0			0
$51$	−224.097	−	224.097i	−0.615292	−	0.615292i
$52$		0			0
$53$	−192.407	+	192.407i	−0.498663	+	0.498663i	−0.911022	−	0.412359i	$-0.864705\pi$
$53$	−192.407	+	192.407i	−0.498663	+	0.498663i	0.412359	+	0.911022i	$0.364705\pi$
$54$		0			0
$55$		−	25.0302i		−	0.0613650i
$56$		0			0
$57$			170.920i			0.397174i
$58$		0			0
$59$	−500.053	+	500.053i	−1.10341	+	1.10341i	−0.109417	+	0.993996i	$0.534898\pi$
$59$	−500.053	+	500.053i	−1.10341	+	1.10341i	−0.993996	+	0.109417i	$0.965102\pi$
$60$		0			0
$61$	166.418	+	166.418i	0.349306	+	0.349306i	0.859851	−	0.510545i	$-0.170556\pi$
$61$	166.418	+	166.418i	0.349306	+	0.349306i	−0.510545	+	0.859851i	$0.670556\pi$
$62$		0			0
$63$	39.9710			0.0799344
$64$		0			0
$65$	−25.0234			−0.0477502
$66$		0			0
$67$	−575.426	−	575.426i	−1.04925	−	1.04925i	−0.998723	−	0.0505233i	$-0.983911\pi$
$67$	−575.426	−	575.426i	−1.04925	−	1.04925i	−0.0505233	−	0.998723i	$-0.516089\pi$
$68$		0			0
$69$	91.2000	−	91.2000i	0.159119	−	0.159119i
$70$		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	$0.334504\pi$
$73$			1082.59i			1.73572i	−0.496812	+	0.867858i	$0.665496\pi$
$74$		0			0
$75$	−263.047	+	263.047i	−0.404987	+	0.404987i
$76$		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
$81$	−81.0000			−0.111111
$82$		0			0
$83$	48.6931	+	48.6931i	0.0643947	+	0.0643947i	0.738571	−	0.674176i	$-0.235501\pi$
$83$	48.6931	+	48.6931i	0.0643947	+	0.0643947i	−0.674176	+	0.738571i	$0.735501\pi$
$84$		0			0
$85$	74.6417	−	74.6417i	0.0952474	−	0.0952474i
$86$		0			0
$87$			787.630i			0.970607i
$88$		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
$93$	−618.332	−	618.332i	−0.689442	−	0.689442i
$94$		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
$99$	−159.414	−	159.414i	−0.161835	−	0.161835i
$100$		0			0
$101$	−564.484	+	564.484i	−0.556121	+	0.556121i	−0.928201	−	0.372080i	$-0.878645\pi$
$101$	−564.484	+	564.484i	−0.556121	+	0.556121i	0.372080	+	0.928201i	$0.378645\pi$
$102$		0			0
$103$		−	110.408i		−	0.105620i	−0.998605	−	0.0528098i	$-0.983182\pi$
$103$		−	110.408i		−	0.105620i	0.998605	−	0.0528098i	$-0.0168177\pi$
$104$		0			0
$105$			13.3134i			0.0123739i
$106$		0			0
$107$	−254.357	+	254.357i	−0.229810	+	0.229810i	−0.812613	−	0.582803i	$-0.801956\pi$
$107$	−254.357	+	254.357i	−0.229810	+	0.229810i	0.582803	+	0.812613i	$0.301956\pi$
$108$		0			0
$109$	205.025	+	205.025i	0.180164	+	0.180164i	0.791427	−	0.611263i	$-0.209338\pi$
$109$	205.025	+	205.025i	0.180164	+	0.180164i	−0.611263	+	0.791427i	$0.709338\pi$
$110$		0			0
$111$	−640.809			−0.547954
$112$		0			0
$113$	−729.046			−0.606928			−0.303464	−	0.952843i	$-0.598143\pi$
$113$	−729.046			−0.606928			−0.303464	+	0.952843i	$0.598143\pi$
$114$		0			0
$115$	30.3766	+	30.3766i	0.0246316	+	0.0246316i
$116$		0			0
$117$	−159.370	+	159.370i	−0.125930	+	0.125930i
$118$		0			0
$119$			469.172i			0.361420i
$120$		0			0
$121$			703.525i			0.528569i
$122$		0			0
$123$	128.110	−	128.110i	0.0939132	−	0.0939132i
$124$		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
$129$	511.374			0.349023
$130$		0			0
$131$	898.181	+	898.181i	0.599042	+	0.599042i	0.940058	−	0.341016i	$-0.110771\pi$
$131$	898.181	+	898.181i	0.599042	+	0.599042i	−0.341016	+	0.940058i	$0.610771\pi$
$132$		0			0
$133$	178.920	−	178.920i	0.116649	−	0.116649i
$134$		0			0
$135$		−	26.9793i		−	0.0172000i
$136$		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	0.111776	−	0.993733i	$-0.464346\pi$
$139$	1811.69	−	1811.69i	1.10551	−	1.10551i	0.993733	−	0.111776i	$-0.0356538\pi$
$140$		0			0
$141$	−1061.05	−	1061.05i	−0.633731	−	0.633731i
$142$		0			0
$143$	−627.303			−0.366837
$144$		0			0
$145$	−262.342			−0.150250
$146$		0			0
$147$	685.771	+	685.771i	0.384772	+	0.384772i
$148$		0			0
$149$	869.143	−	869.143i	0.477872	−	0.477872i	−0.426578	−	0.904451i	$-0.640281\pi$
$149$	869.143	−	869.143i	0.477872	−	0.477872i	0.904451	+	0.426578i	$0.140281\pi$
$150$		0			0
$151$		−	1033.70i		−	0.557097i	−0.960422	−	0.278549i	$-0.910147\pi$
$151$		−	1033.70i		−	0.557097i	0.960422	−	0.278549i	$-0.0898534\pi$
$152$		0			0
$153$		−	950.764i		−	0.502384i
$154$		0			0
$155$	205.952	−	205.952i	0.106726	−	0.106726i
$156$		0			0
$157$	2314.43	+	2314.43i	1.17651	+	1.17651i	0.980628	+	0.195879i	$0.0627561\pi$
$157$	2314.43	+	2314.43i	1.17651	+	1.17651i	0.195879	+	0.980628i	$0.437244\pi$
$158$		0			0
$159$	−816.314			−0.407157
$160$		0			0
$161$	−190.937			−0.0934656
$162$		0			0
$163$	737.872	+	737.872i	0.354568	+	0.354568i	0.861806	−	0.507238i	$-0.169333\pi$
$163$	737.872	+	737.872i	0.354568	+	0.354568i	−0.507238	+	0.861806i	$0.669333\pi$
$164$		0			0
$165$	53.0971	−	53.0971i	0.0250521	−	0.0250521i
$166$		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
$171$	−362.576	+	362.576i	−0.162145	+	0.162145i
$172$		0			0
$173$	−1671.65	−	1671.65i	−0.734643	−	0.734643i	0.236893	−	0.971536i	$-0.423871\pi$
$173$	−1671.65	−	1671.65i	−0.734643	−	0.734643i	−0.971536	+	0.236893i	$0.923871\pi$
$174$		0			0
$175$	550.718			0.237888
$176$		0			0
$177$	−2121.55			−0.900933
$178$		0			0
$179$	1413.56	+	1413.56i	0.590247	+	0.590247i	0.937698	−	0.347451i	$-0.112953\pi$
$179$	1413.56	+	1413.56i	0.590247	+	0.590247i	−0.347451	+	0.937698i	$0.612953\pi$
$180$		0			0
$181$	2515.65	−	2515.65i	1.03308	−	1.03308i	0.0336429	−	0.999434i	$-0.489289\pi$
$181$	2515.65	−	2515.65i	1.03308	−	1.03308i	0.999434	−	0.0336429i	$-0.0107109\pi$
$182$		0			0
$183$			706.053i			0.285207i
$184$		0			0
$185$		−	213.439i		−	0.0848235i
$186$		0			0
$187$	1871.17	−	1871.17i	0.731730	−	0.731730i
$188$		0			0
$189$	84.7912	+	84.7912i	0.0326331	+	0.0326331i
$190$		0			0
$191$	−18.1653			−0.00688165			−0.00344082	−	0.999994i	$-0.501095\pi$
$191$	−18.1653			−0.00688165			−0.00344082	+	0.999994i	$0.501095\pi$
$192$		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
$195$	−53.0826	−	53.0826i	−0.0194940	−	0.0194940i
$196$		0			0
$197$	1818.15	−	1818.15i	0.657551	−	0.657551i	−0.297249	−	0.954800i	$-0.596069\pi$
$197$	1818.15	−	1818.15i	0.657551	−	0.657551i	0.954800	+	0.297249i	$0.0960691\pi$
$198$		0			0
$199$		−	4114.56i		−	1.46570i	−0.680392	−	0.732849i	$-0.738190\pi$
$199$		−	4114.56i		−	1.46570i	0.680392	−	0.732849i	$-0.261810\pi$
$200$		0			0
$201$		−	2441.33i		−	0.856706i
$202$		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
$207$	386.929			0.129920
$208$		0			0
$209$	−1427.15			−0.472335
$210$		0			0
$211$	1636.19	+	1636.19i	0.533840	+	0.533840i	0.921713	−	0.387873i	$-0.126790\pi$
$211$	1636.19	+	1636.19i	0.533840	+	0.533840i	−0.387873	+	0.921713i	$0.626790\pi$
$212$		0			0
$213$	970.059	−	970.059i	0.312053	−	0.312053i
$214$		0			0
$215$			170.327i			0.0540289i
$216$		0			0
$217$			1294.55i			0.404975i
$218$		0			0
$219$	−2296.51	+	2296.51i	−0.708603	+	0.708603i
$220$		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
$225$	−1116.01			−0.330671
$226$		0			0
$227$	−3405.69	−	3405.69i	−0.995787	−	0.995787i	0.00420438	−	0.999991i	$-0.498662\pi$
$227$	−3405.69	−	3405.69i	−0.995787	−	0.995787i	−0.999991	+	0.00420438i	$0.998662\pi$
$228$		0			0
$229$	−2566.84	+	2566.84i	−0.740706	+	0.740706i	−0.972714	−	0.232008i	$-0.925471\pi$
$229$	−2566.84	+	2566.84i	−0.740706	+	0.740706i	0.232008	+	0.972714i	$0.425471\pi$
$230$		0			0
$231$			333.750i			0.0950613i
$232$		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
$237$	1154.65	+	1154.65i	0.316468	+	0.316468i
$238$		0			0
$239$	2608.05			0.705860			0.352930	−	0.935650i	$-0.385185\pi$
$239$	2608.05			0.705860			0.352930	+	0.935650i	$0.385185\pi$
$240$		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
$243$	−171.827	−	171.827i	−0.0453609	−	0.0453609i
$244$		0			0
$245$	−228.415	+	228.415i	−0.0595628	+	0.0595628i
$246$		0			0
$247$			1426.76i			0.367541i
$248$		0			0
$249$			206.587i			0.0525781i
$250$		0			0
$251$	−2551.97	+	2551.97i	−0.641749	+	0.641749i	−0.950985	−	0.309236i	$-0.899927\pi$
$251$	−2551.97	+	2551.97i	−0.641749	+	0.641749i	0.309236	+	0.950985i	$0.399927\pi$
$252$		0			0
$253$	761.503	+	761.503i	0.189230	+	0.189230i
$254$		0			0
$255$	316.678			0.0777692
$256$		0			0
$257$	2203.05			0.534718			0.267359	−	0.963597i	$-0.413849\pi$
$257$	2203.05			0.534718			0.267359	+	0.963597i	$0.413849\pi$
$258$		0			0
$259$	670.802	+	670.802i	0.160933	+	0.160933i
$260$		0			0
$261$	−1670.81	+	1670.81i	−0.396248	+	0.396248i
$262$		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
$267$	92.6411	−	92.6411i	0.0212342	−	0.0212342i
$268$		0			0
$269$	−3750.63	−	3750.63i	−0.850112	−	0.850112i	0.140035	−	0.990147i	$-0.455279\pi$
$269$	−3750.63	−	3750.63i	−0.850112	−	0.850112i	−0.990147	+	0.140035i	$0.955279\pi$
$270$		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
$273$	333.659			0.0739705
$274$		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	$-0.937468\pi$
$277$	−3621.67	+	3621.67i	−0.785578	+	0.785578i	0.195188	+	0.980766i	$0.437468\pi$
$278$		0			0
$279$		−	2623.36i		−	0.562927i
$280$		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	$-0.632644\pi$
$283$	2426.42	−	2426.42i	0.509667	−	0.509667i	0.914424	+	0.404757i	$0.132644\pi$
$284$		0			0
$285$	−120.766	−	120.766i	−0.0251002	−	0.0251002i
$286$		0			0
$287$	−268.213			−0.0551642
$288$		0			0
$289$	6246.91			1.27151
$290$		0			0
$291$	1464.85	+	1464.85i	0.295089	+	0.295089i
$292$		0			0
$293$	−3379.84	+	3379.84i	−0.673899	+	0.673899i	−0.958613	−	0.284714i	$-0.908101\pi$
$293$	−3379.84	+	3379.84i	−0.673899	+	0.673899i	0.284714	+	0.958613i	$0.408101\pi$
$294$		0			0
$295$		−	706.639i		−	0.139465i
$296$		0			0
$297$		−	676.335i		−	0.132138i
$298$		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
$303$	−2394.90			−0.454071
$304$		0			0
$305$	−235.170			−0.0441502
$306$		0			0
$307$	−2575.52	−	2575.52i	−0.478803	−	0.478803i	0.425946	−	0.904749i	$-0.359941\pi$
$307$	−2575.52	−	2575.52i	−0.478803	−	0.478803i	−0.904749	+	0.425946i	$0.859941\pi$
$308$		0			0
$309$	234.211	−	234.211i	0.0431190	−	0.0431190i
$310$		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	$0.186843\pi$
$313$			6133.96i			1.10771i	−0.832614	+	0.553853i	$0.813157\pi$
$314$		0			0
$315$	−28.2420	+	28.2420i	−0.00505161	+	0.00505161i
$316$		0			0
$317$	6740.75	+	6740.75i	1.19432	+	1.19432i	0.975843	+	0.218474i	$0.0701078\pi$
$317$	6740.75	+	6740.75i	1.19432	+	1.19432i	0.218474	+	0.975843i	$0.429892\pi$
$318$		0			0
$319$	−6576.56			−1.15428
$320$		0			0
$321$	−1079.15			−0.187639
$322$		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
$327$			869.848i			0.147103i
$328$		0			0
$329$			2221.42i			0.372251i
$330$		0			0
$331$	−6485.75	+	6485.75i	−1.07701	+	1.07701i	−0.0802296	+	0.996776i	$0.525565\pi$
$331$	−6485.75	+	6485.75i	−1.07701	+	1.07701i	−0.996776	+	0.0802296i	$0.974435\pi$
$332$		0			0
$333$	−1359.36	−	1359.36i	−0.223701	−	0.223701i
$334$		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
$339$	−1546.54	−	1546.54i	−0.247777	−	0.247777i
$340$		0			0
$341$	5162.96	−	5162.96i	0.819912	−	0.819912i
$342$		0			0
$343$		−	2959.08i		−	0.465817i
$344$		0			0
$345$			128.877i			0.0201116i
$346$		0			0
$347$	−1557.96	+	1557.96i	−0.241025	+	0.241025i	−0.817274	−	0.576249i	$-0.804516\pi$
$347$	−1557.96	+	1557.96i	−0.241025	+	0.241025i	0.576249	+	0.817274i	$0.304516\pi$
$348$		0			0
$349$	3041.63	+	3041.63i	0.466518	+	0.466518i	0.900784	−	0.434266i	$-0.142992\pi$
$349$	3041.63	+	3041.63i	0.466518	+	0.466518i	−0.434266	+	0.900784i	$0.642992\pi$
$350$		0			0
$351$	−676.150			−0.102821
$352$		0			0
$353$	1629.66			0.245717			0.122859	−	0.992424i	$-0.460794\pi$
$353$	1629.66			0.245717			0.122859	+	0.992424i	$0.460794\pi$
$354$		0			0
$355$	323.104	+	323.104i	0.0483059	+	0.0483059i
$356$		0			0
$357$	−995.265	+	995.265i	−0.147549	+	0.147549i
$358$		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
$363$	−1492.40	+	1492.40i	−0.215787	+	0.215787i
$364$		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
$369$	543.527			0.0766798
$370$		0			0
$371$	854.522	+	854.522i	0.119581	+	0.119581i
$372$		0			0
$373$	8897.07	−	8897.07i	1.23505	−	1.23505i	0.273047	−	0.962001i	$-0.411969\pi$
$373$	8897.07	−	8897.07i	1.23505	−	1.23505i	0.962001	−	0.273047i	$-0.0880313\pi$
$374$		0			0
$375$		−	746.431i		−	0.102788i
$376$		0			0
$377$			6574.76i			0.898190i
$378$		0			0
$379$	−5154.79	+	5154.79i	−0.698638	+	0.698638i	−0.964117	−	0.265478i	$-0.914470\pi$
$379$	−5154.79	+	5154.79i	−0.698638	+	0.698638i	0.265478	+	0.964117i	$0.414470\pi$
$380$		0			0
$381$	5157.39	+	5157.39i	0.693493	+	0.693493i
$382$		0			0
$383$	−10573.7			−1.41068			−0.705340	−	0.708870i	$-0.749205\pi$
$383$	−10573.7			−1.41068			−0.705340	+	0.708870i	$0.749205\pi$
$384$		0			0
$385$	−111.165			−0.0147155
$386$		0			0
$387$	1084.79	+	1084.79i	0.142488	+	0.142488i
$388$		0			0
$389$	−7789.01	+	7789.01i	−1.01522	+	1.01522i	−0.0153326	+	0.999882i	$0.504881\pi$
$389$	−7789.01	+	7789.01i	−1.01522	+	1.01522i	−0.999882	+	0.0153326i	$0.995119\pi$
$390$		0			0
$391$			4541.70i			0.587426i
$392$		0			0
$393$			3810.66i			0.489115i
$394$		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	$-0.215225\pi$
$397$	1219.69	+	1219.69i	0.154193	+	0.154193i	−0.625795	+	0.779988i	$0.715225\pi$
$398$		0			0
$399$	759.093			0.0952435
$400$		0			0
$401$	−5714.97			−0.711701			−0.355850	−	0.934543i	$-0.615809\pi$
$401$	−5714.97			−0.711701			−0.355850	+	0.934543i	$0.615809\pi$
$402$		0			0
$403$	−5161.55	−	5161.55i	−0.638003	−	0.638003i
$404$		0			0
$405$	57.2316	−	57.2316i	0.00702189	−	0.00702189i
$406$		0			0
$407$		−	5350.64i		−	0.651649i
$408$		0			0
$409$			1504.54i			0.181894i	0.995856	+	0.0909469i	$0.0289894\pi$
$409$			1504.54i			0.181894i	−0.995856	+	0.0909469i	$0.971011\pi$
$410$		0			0
$411$	6495.03	−	6495.03i	0.779504	−	0.779504i
$412$		0			0
$413$	2220.85	+	2220.85i	0.264602	+	0.264602i
$414$		0			0
$415$	−68.8095			−0.00813910
$416$		0			0
$417$	7686.36			0.902644
$418$		0			0
$419$	9630.38	+	9630.38i	1.12285	+	1.12285i	0.991311	+	0.131541i	$0.0419927\pi$
$419$	9630.38	+	9630.38i	1.12285	+	1.12285i	0.131541	+	0.991311i	$0.458007\pi$
$420$		0			0
$421$	−2831.19	+	2831.19i	−0.327752	+	0.327752i	−0.851731	−	0.523979i	$-0.824447\pi$
$421$	−2831.19	+	2831.19i	−0.327752	+	0.327752i	0.523979	+	0.851731i	$0.324447\pi$
$422$		0			0
$423$		−	4501.63i		−	0.517440i
$424$		0			0
$425$		−	13099.6i		−	1.49511i
$426$		0			0
$427$	739.100	−	739.100i	0.0837647	−	0.0837647i
$428$		0			0
$429$	−1330.71	−	1330.71i	−0.149761	−	0.149761i
$430$		0			0
$431$	−5555.36			−0.620864			−0.310432	−	0.950596i	$-0.600474\pi$
$431$	−5555.36			−0.620864			−0.310432	+	0.950596i	$0.600474\pi$
$432$		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
$435$	−556.510	−	556.510i	−0.0613394	−	0.0613394i
$436$		0			0
$437$	1731.99	−	1731.99i	0.189593	−	0.189593i
$438$		0			0
$439$			4862.86i			0.528682i	0.964429	+	0.264341i	$0.0851545\pi$
$439$			4862.86i			0.528682i	−0.964429	+	0.264341i	$0.914846\pi$
$440$		0			0
$441$			2909.48i			0.314165i
$442$		0			0
$443$	9325.47	−	9325.47i	1.00015	−	1.00015i	0.000149687	−	1.00000i	$-0.499952\pi$
$443$	9325.47	−	9325.47i	1.00015	−	1.00015i	1.00000		0.000149687i	$-4.76467e-5\pi$
$444$		0			0
$445$	30.8566	+	30.8566i	0.00328707	+	0.00328707i
$446$		0			0
$447$	3687.46			0.390181
$448$		0			0
$449$	11461.1			1.20464			0.602318	−	0.798256i	$-0.294244\pi$
$449$	11461.1			1.20464			0.602318	+	0.798256i	$0.294244\pi$
$450$		0			0
$451$	1069.70	+	1069.70i	0.111685	+	0.111685i
$452$		0			0
$453$	2192.82	−	2192.82i	0.227434	−	0.227434i
$454$		0			0
$455$			111.134i			0.0114507i
$456$		0			0
$457$		−	13856.7i		−	1.41835i	−0.705031	−	0.709177i	$-0.749067\pi$
$457$		−	13856.7i		−	1.41835i	0.705031	−	0.709177i	$-0.250933\pi$
$458$		0			0
$459$	2016.88	−	2016.88i	0.205097	−	0.205097i
$460$		0			0
$461$	3166.73	+	3166.73i	0.319934	+	0.319934i	0.848741	−	0.528808i	$-0.177361\pi$
$461$	3166.73	+	3166.73i	0.319934	+	0.319934i	−0.528808	+	0.848741i	$0.677361\pi$
$462$		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
$465$	873.782			0.0871412
$466$		0			0
$467$	5309.28	+	5309.28i	0.526090	+	0.526090i	0.919404	−	0.393314i	$-0.128671\pi$
$467$	5309.28	+	5309.28i	0.526090	+	0.526090i	−0.393314	+	0.919404i	$0.628671\pi$
$468$		0			0
$469$	−2555.59	+	2555.59i	−0.251613	+	0.251613i
$470$		0			0
$471$			9819.30i			0.960614i
$472$		0			0
$473$			4269.88i			0.415072i
$474$		0			0
$475$	−4995.55	+	4995.55i	−0.482551	+	0.482551i
$476$		0			0
$477$	−1731.66	−	1731.66i	−0.166221	−	0.166221i
$478$		0			0
$479$	4034.12			0.384809			0.192405	−	0.981316i	$-0.438371\pi$
$479$	4034.12			0.384809			0.192405	+	0.981316i	$0.438371\pi$
$480$		0			0
$481$	−5349.17			−0.507071
$482$		0			0
$483$	−405.039	−	405.039i	−0.0381572	−	0.0381572i
$484$		0			0
$485$	−487.908	+	487.908i	−0.0456799	+	0.0456799i
$486$		0			0
$487$			16499.3i			1.53522i	0.640915	+	0.767612i	$0.278555\pi$
$487$			16499.3i			1.53522i	−0.640915	+	0.767612i	$0.721445\pi$
$488$		0			0
$489$			3130.53i			0.289504i
$490$		0			0
$491$	1060.25	−	1060.25i	0.0974506	−	0.0974506i	−0.656701	−	0.754151i	$-0.728049\pi$
$491$	1060.25	−	1060.25i	0.0974506	−	0.0974506i	0.754151	+	0.656701i	$0.228049\pi$
$492$		0			0
$493$	−19611.7	−	19611.7i	−1.79162	−	1.79162i
$494$		0			0
$495$	225.272			0.0204550
$496$		0			0
$497$	−2030.93			−0.183299
$498$		0			0
$499$	4275.19	+	4275.19i	0.383534	+	0.383534i	0.872374	−	0.488839i	$-0.162580\pi$
$499$	4275.19	+	4275.19i	0.383534	+	0.383534i	−0.488839	+	0.872374i	$0.662580\pi$
$500$		0			0
$501$	−994.187	+	994.187i	−0.0886567	+	0.0886567i
$502$		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
$507$	3330.19	−	3330.19i	0.291714	−	0.291714i
$508$		0			0
$509$	9315.16	+	9315.16i	0.811174	+	0.811174i	0.984810	−	0.173636i	$-0.0555517\pi$
$509$	9315.16	+	9315.16i	0.811174	+	0.811174i	−0.173636	+	0.984810i	$0.555552\pi$
$510$		0			0
$511$	4808.01			0.416230
$512$		0			0
$513$	−1538.28			−0.132391
$514$		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
$519$		−	7092.21i		−	0.599834i
$520$		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	$-0.702393\pi$
$523$	2520.38	−	2520.38i	0.210724	−	0.210724i	0.804575	+	0.593851i	$0.202393\pi$
$524$		0			0
$525$	1168.25	+	1168.25i	0.0971173	+	0.0971173i
$526$		0			0
$527$	30792.6			2.54525
$528$		0			0
$529$	10318.7			0.848088
$530$		0			0
$531$	−4500.48	−	4500.48i	−0.367804	−	0.367804i
$532$		0			0
$533$	1069.41	−	1069.41i	0.0869064	−	0.0869064i
$534$		0			0
$535$		−	359.439i		−	0.0290465i
$536$		0			0
$537$			5997.21i			0.481935i
$538$		0			0
$539$	−5726.06	+	5726.06i	−0.457586	+	0.457586i
$540$		0			0
$541$	−13635.3	−	13635.3i	−1.08360	−	1.08360i	−0.996171	−	0.0874292i	$-0.972135\pi$
$541$	−13635.3	−	13635.3i	−1.08360	−	1.08360i	−0.0874292	−	0.996171i	$-0.527865\pi$
$542$		0			0
$543$	10673.0			0.843504
$544$		0			0
$545$	−289.726			−0.0227716
$546$		0			0
$547$	−1729.49	−	1729.49i	−0.135188	−	0.135188i	0.636275	−	0.771463i	$-0.280474\pi$
$547$	−1729.49	−	1729.49i	−0.135188	−	0.135188i	−0.771463	+	0.636275i	$0.780474\pi$
$548$		0			0
$549$	−1497.76	+	1497.76i	−0.116435	+	0.116435i
$550$		0			0
$551$			14958.0i			1.15650i
$552$		0			0
$553$		−	2417.39i		−	0.185892i
$554$		0			0
$555$	452.772	−	452.772i	0.0346290	−	0.0346290i
$556$		0			0
$557$	−3758.29	−	3758.29i	−0.285896	−	0.285896i	0.549559	−	0.835455i	$-0.314796\pi$
$557$	−3758.29	−	3758.29i	−0.285896	−	0.285896i	−0.835455	+	0.549559i	$0.814796\pi$
$558$		0			0
$559$	4268.71			0.322982
$560$		0			0
$561$	7938.71			0.597455
$562$		0			0
$563$	1676.08	+	1676.08i	0.125468	+	0.125468i	0.767052	−	0.641585i	$-0.221723\pi$
$563$	1676.08	+	1676.08i	0.125468	+	0.125468i	−0.641585	+	0.767052i	$0.721723\pi$
$564$		0			0
$565$	515.118	−	515.118i	0.0383560	−	0.0383560i
$566$		0			0
$567$			359.739i			0.0266448i
$568$		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	$-0.674136\pi$
$571$	4555.47	−	4555.47i	0.333871	−	0.333871i	0.854055	+	0.520184i	$0.174136\pi$
$572$		0			0
$573$	−38.5344	−	38.5344i	−0.00280942	−	0.00280942i
$574$		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
$579$	2796.20	+	2796.20i	0.200701	+	0.200701i
$580$		0			0
$581$	216.257	−	216.257i	0.0154421	−	0.0154421i
$582$		0			0
$583$		−	6816.07i		−	0.484207i
$584$		0			0
$585$		−	225.210i		−	0.0159167i
$586$		0			0
$587$	3111.52	−	3111.52i	0.218784	−	0.218784i	−0.589202	−	0.807986i	$-0.700558\pi$
$587$	3111.52	−	3111.52i	0.218784	−	0.218784i	0.807986	+	0.589202i	$0.200558\pi$
$588$		0			0
$589$	−11742.8	−	11742.8i	−0.821484	−	0.821484i
$590$		0			0
$591$	7713.74			0.536888
$592$		0			0
$593$	−7275.10			−0.503799			−0.251899	−	0.967753i	$-0.581055\pi$
$593$	−7275.10			−0.503799			−0.251899	+	0.967753i	$0.581055\pi$
$594$		0			0
$595$	−331.500	−	331.500i	−0.0228406	−	0.0228406i
$596$		0			0
$597$	8728.31	−	8728.31i	0.598368	−	0.598368i
$598$		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	$-0.973027\pi$
$601$		−	2493.99i		−	0.169271i	0.996412	−	0.0846357i	$-0.0269726\pi$
$602$		0			0
$603$	5178.84	−	5178.84i	0.349749	−	0.349749i
$604$		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
$609$	3498.04			0.232755
$610$		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	$-0.911858\pi$
$613$	−10449.8	+	10449.8i	−0.688523	+	0.688523i	0.273382	+	0.961906i	$0.411858\pi$
$614$		0			0
$615$			181.036i			0.0118701i
$616$		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	−0.0836922	+	0.996492i	$0.526671\pi$
$619$	−16635.4	+	16635.4i	−1.08018	+	1.08018i	−0.996492	+	0.0836922i	$0.973329\pi$
$620$		0			0
$621$	820.800	+	820.800i	0.0530395	+	0.0530395i
$622$		0			0
$623$	−193.954			−0.0124729
$624$		0			0
$625$	−15251.6			−0.976101
$626$		0			0
$627$	−3027.44	−	3027.44i	−0.192830	−	0.192830i
$628$		0			0
$629$	15955.9	−	15955.9i	1.01146	−	1.01146i
$630$		0			0
$631$			22231.3i			1.40256i	0.712888	+	0.701278i	$0.247387\pi$
$631$			22231.3i			1.40256i	−0.712888	+	0.701278i	$0.752613\pi$
$632$		0			0
$633$			6941.78i			0.435878i
$634$		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
$639$	4115.61			0.254790
$640$		0			0
$641$	8447.90			0.520549			0.260274	−	0.965535i	$-0.416187\pi$
$641$	8447.90			0.520549			0.260274	+	0.965535i	$0.416187\pi$
$642$		0			0
$643$	5326.38	+	5326.38i	0.326675	+	0.326675i	0.851321	−	0.524646i	$-0.175802\pi$
$643$	5326.38	+	5326.38i	0.326675	+	0.326675i	−0.524646	+	0.851321i	$0.675802\pi$
$644$		0			0
$645$	−361.318	+	361.318i	−0.0220572	+	0.0220572i
$646$		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
$651$	−2746.15	+	2746.15i	−0.165330	+	0.165330i
$652$		0			0
$653$	−1518.80	−	1518.80i	−0.0910189	−	0.0910189i	0.660131	−	0.751150i	$-0.270501\pi$
$653$	−1518.80	−	1518.80i	−0.0910189	−	0.0910189i	−0.751150	+	0.660131i	$0.770501\pi$
$654$		0			0
$655$	−1269.24			−0.0757152
$656$		0			0
$657$	−9743.29			−0.578572
$658$		0			0
$659$	5425.29	+	5425.29i	0.320697	+	0.320697i	0.849035	−	0.528337i	$-0.177184\pi$
$659$	5425.29	+	5425.29i	0.320697	+	0.320697i	−0.528337	+	0.849035i	$0.677184\pi$
$660$		0			0
$661$	18128.4	−	18128.4i	1.06674	−	1.06674i	0.0691300	−	0.997608i	$-0.477978\pi$
$661$	18128.4	−	18128.4i	1.06674	−	1.06674i	0.997608	−	0.0691300i	$-0.0220223\pi$
$662$		0			0
$663$		−	7936.53i		−	0.464901i
$664$		0			0
$665$			252.837i			0.0147437i
$666$		0			0
$667$	7981.31	−	7981.31i	0.463325	−	0.463325i
$668$		0			0
$669$	6622.34	+	6622.34i	0.382712	+	0.382712i
$670$		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
$675$	−2367.42	−	2367.42i	−0.134996	−	0.134996i
$676$		0			0
$677$	−4806.53	+	4806.53i	−0.272865	+	0.272865i	−0.830253	−	0.557387i	$-0.811804\pi$
$677$	−4806.53	+	4806.53i	−0.272865	+	0.272865i	0.557387	+	0.830253i	$0.311804\pi$
$678$		0			0
$679$		−	3066.82i		−	0.173334i
$680$		0			0
$681$		−	14449.1i		−	0.813057i
$682$		0			0
$683$	21855.0	−	21855.0i	1.22439	−	1.22439i	0.258334	−	0.966056i	$-0.416826\pi$
$683$	21855.0	−	21855.0i	1.22439	−	1.22439i	0.966056	−	0.258334i	$-0.0831737\pi$
$684$		0			0
$685$	2163.35	+	2163.35i	0.120667	+	0.120667i
$686$		0			0
$687$	−10890.2			−0.604784
$688$		0			0
$689$	−6814.21			−0.376779
$690$		0			0
$691$	−585.815	−	585.815i	−0.0322510	−	0.0322510i	0.690797	−	0.723048i	$-0.257260\pi$
$691$	−585.815	−	585.815i	−0.0322510	−	0.0322510i	−0.723048	+	0.690797i	$0.757260\pi$
$692$		0			0
$693$	−707.991	+	707.991i	−0.0388086	+	0.0388086i
$694$		0			0
$695$			2560.15i			0.139730i
$696$		0			0
$697$			6379.82i			0.346704i
$698$		0			0
$699$	−3202.97	+	3202.97i	−0.173315	+	0.173315i
$700$		0			0
$701$	7273.96	+	7273.96i	0.391917	+	0.391917i	0.875370	−	0.483453i	$-0.160618\pi$
$701$	7273.96	+	7273.96i	0.391917	+	0.391917i	−0.483453	+	0.875370i	$0.660618\pi$
$702$		0			0
$703$	−12169.7			−0.652899
$704$		0			0
$705$	1499.39			0.0800998
$706$		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	$-0.979123\pi$
$709$	−17600.7	+	17600.7i	−0.932309	+	0.932309i	0.0655405	+	0.997850i	$0.479123\pi$
$710$		0			0
$711$			4898.78i			0.258395i
$712$		0			0
$713$			12531.5i			0.658218i
$714$		0			0
$715$	443.230	−	443.230i	0.0231830	−	0.0231830i
$716$		0			0
$717$	5532.50	+	5532.50i	0.288166	+	0.288166i
$718$		0			0
$719$	−14431.8			−0.748562			−0.374281	−	0.927315i	$-0.622110\pi$
$719$	−14431.8			−0.748562			−0.374281	+	0.927315i	$0.622110\pi$
$720$		0			0
$721$	−490.346			−0.0253279
$722$		0			0
$723$	4714.01	+	4714.01i	0.242484	+	0.242484i
$724$		0			0
$725$	−23020.4	+	23020.4i	−1.17925	+	1.17925i
$726$		0			0
$727$		−	5802.18i		−	0.295998i	−0.988987	−	0.147999i	$-0.952717\pi$
$727$		−	5802.18i		−	0.295998i	0.988987	−	0.147999i	$-0.0472833\pi$
$728$		0			0
$729$		−	729.000i		−	0.0370370i
$730$		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	$-0.0465237\pi$
$733$	16743.4	+	16743.4i	0.843699	+	0.843699i	−0.145639	+	0.989338i	$0.546524\pi$
$734$		0			0
$735$	−969.082			−0.0486328
$736$		0			0
$737$	20384.6			1.01883
$738$		0			0
$739$	5259.84	+	5259.84i	0.261822	+	0.261822i	0.825794	−	0.563972i	$-0.190727\pi$
$739$	5259.84	+	5259.84i	0.261822	+	0.261822i	−0.563972	+	0.825794i	$0.690727\pi$
$740$		0			0
$741$	−3026.61	+	3026.61i	−0.150048	+	0.150048i
$742$		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
$747$	−438.238	+	438.238i	−0.0214649	+	0.0214649i
$748$		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
$753$	−10827.1			−0.523986
$754$		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	$-0.800479\pi$
$757$	−4651.56	+	4651.56i	−0.223334	+	0.223334i	0.586567	+	0.809901i	$0.300479\pi$
$758$		0			0
$759$			3230.78i			0.154506i
$760$		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
$765$	671.775	+	671.775i	0.0317491	+	0.0317491i
$766$		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
$771$	4673.38	+	4673.38i	0.218298	+	0.218298i
$772$		0			0
$773$	6698.95	−	6698.95i	0.311700	−	0.311700i	−0.533868	−	0.845568i	$-0.679262\pi$
$773$	6698.95	−	6698.95i	0.311700	−	0.311700i	0.845568	+	0.533868i	$0.179262\pi$
$774$		0			0
$775$		−	36144.5i		−	1.67529i
$776$		0			0
$777$			2845.97i			0.131401i
$778$		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
$783$	−7088.67			−0.323536
$784$		0			0
$785$	−3270.58			−0.148703
$786$		0			0
$787$	−4317.53	−	4317.53i	−0.195557	−	0.195557i	0.602535	−	0.798092i	$-0.294157\pi$
$787$	−4317.53	−	4317.53i	−0.195557	−	0.195557i	−0.798092	+	0.602535i	$0.794157\pi$
$788$		0			0
$789$	−15338.3	+	15338.3i	−0.692091	+	0.692091i
$790$		0			0
$791$			3237.85i			0.145543i
$792$		0			0
$793$			5893.80i			0.263928i
$794$		0			0
$795$	576.778	−	576.778i	0.0257311	−	0.0257311i
$796$		0			0
$797$	−12492.2	−	12492.2i	−0.555201	−	0.555201i	0.372736	−	0.927937i	$-0.378420\pi$
$797$	−12492.2	−	12492.2i	−0.555201	−	0.555201i	−0.927937	+	0.372736i	$0.878420\pi$
$798$		0			0
$799$	52839.4			2.33958
$800$		0			0
$801$	393.043			0.0173377
$802$		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
$807$		−	15912.6i		−	0.694113i
$808$		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	$-0.905526\pi$
$811$	−15331.3	+	15331.3i	−0.663817	+	0.663817i	0.292461	+	0.956277i	$0.405526\pi$
$812$		0			0
$813$	−5837.13	−	5837.13i	−0.251804	−	0.251804i
$814$		0			0
$815$	−1042.71			−0.0448152
$816$		0			0
$817$	9711.55			0.415868
$818$		0			0
$819$	707.797	+	707.797i	0.0301983	+	0.0301983i
$820$		0			0
$821$	12527.1	−	12527.1i	0.532521	−	0.532521i	−0.388801	−	0.921322i	$-0.627110\pi$
$821$	12527.1	−	12527.1i	0.532521	−	0.532521i	0.921322	+	0.388801i	$0.127110\pi$
$822$		0			0
$823$			15241.1i			0.645532i	0.946479	+	0.322766i	$0.104613\pi$
$823$			15241.1i			0.645532i	−0.946479	+	0.322766i	$0.895387\pi$
$824$		0			0
$825$		−	9318.51i		−	0.393247i
$826$		0			0
$827$	−25529.5	+	25529.5i	−1.07345	+	1.07345i	−0.0763756	+	0.997079i	$0.524335\pi$
$827$	−25529.5	+	25529.5i	−1.07345	+	1.07345i	−0.997079	+	0.0763756i	$0.975665\pi$
$828$		0			0
$829$	23542.4	+	23542.4i	0.986322	+	0.986322i	0.999908	−	0.0135854i	$-0.00432449\pi$
$829$	23542.4	+	23542.4i	0.986322	+	0.986322i	−0.0135854	+	0.999908i	$0.504324\pi$
$830$		0			0
$831$	−15365.4			−0.641421
$832$		0			0
$833$	−34151.0			−1.42048
$834$		0			0
$835$	−331.141	−	331.141i	−0.0137241	−	0.0137241i
$836$		0			0
$837$	5564.99	−	5564.99i	0.229814	−	0.229814i
$838$		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
$843$	−10584.5	+	10584.5i	−0.432444	+	0.432444i
$844$		0			0
$845$	1109.21	+	1109.21i	0.0451575	+	0.0451575i
$846$		0			0
$847$	3124.51			0.126753
$848$		0			0
$849$	10294.4			0.416141
$850$		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	$-0.789601\pi$
$853$	−4371.94	+	4371.94i	−0.175489	+	0.175489i	0.613897	+	0.789386i	$0.289601\pi$
$854$		0			0
$855$		−	512.366i		−	0.0204942i
$856$		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	$-0.724268\pi$
$859$	2875.17	−	2875.17i	0.114202	−	0.114202i	0.761898	+	0.647696i	$0.224268\pi$
$860$		0			0
$861$	−568.966	−	568.966i	−0.0225207	−	0.0225207i
$862$		0			0
$863$	−18181.9			−0.717173			−0.358586	−	0.933497i	$-0.616741\pi$
$863$	−18181.9			−0.717173			−0.358586	+	0.933497i	$0.616741\pi$
$864$		0			0
$865$	2362.26			0.0928544
$866$		0			0
$867$	13251.7	+	13251.7i	0.519090	+	0.519090i
$868$		0			0
$869$	−9641.15	+	9641.15i	−0.376356	+	0.376356i
$870$		0			0
$871$		−	20379.0i		−	0.792787i
$872$		0			0
$873$			6214.83i			0.240939i
$874$		0			0
$875$	−781.368	+	781.368i	−0.0301886	+	0.0301886i
$876$		0			0
$877$	26665.8	+	26665.8i	1.02673	+	1.02673i	0.999633	+	0.0270950i	$0.00862565\pi$
$877$	26665.8	+	26665.8i	1.02673	+	1.02673i	0.0270950	+	0.999633i	$0.491374\pi$
$878$		0			0
$879$	−14339.4			−0.550236
$880$		0			0
$881$	26033.2			0.995552			0.497776	−	0.867305i	$-0.334150\pi$
$881$	26033.2			0.995552			0.497776	+	0.867305i	$0.334150\pi$
$882$		0			0
$883$	−27917.9	−	27917.9i	−1.06400	−	1.06400i	−0.997807	−	0.0661953i	$-0.978914\pi$
$883$	−27917.9	−	27917.9i	−1.06400	−	1.06400i	−0.0661953	−	0.997807i	$-0.521086\pi$
$884$		0			0
$885$	1499.01	−	1499.01i	0.0569362	−	0.0569362i
$886$		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
$891$	1434.72	−	1434.72i	0.0539451	−	0.0539451i
$892$		0			0
$893$	−20150.4	−	20150.4i	−0.755104	−	0.755104i
$894$		0			0
$895$	−1997.54			−0.0746036
$896$		0			0
$897$	3229.90			0.120227
$898$		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
$903$		−	2271.12i		−	0.0836968i
$904$		0			0
$905$			3554.94i			0.130575i
$906$		0			0
$907$	3191.29	−	3191.29i	0.116830	−	0.116830i	−0.646275	−	0.763105i	$-0.723674\pi$
$907$	3191.29	−	3191.29i	0.116830	−	0.116830i	0.763105	+	0.646275i	$0.223674\pi$
$908$		0			0
$909$	−5080.35	−	5080.35i	−0.185374	−	0.185374i
$910$		0			0
$911$	40016.6			1.45534			0.727668	−	0.685930i	$-0.240604\pi$
$911$	40016.6			1.45534			0.727668	+	0.685930i	$0.240604\pi$
$912$		0			0
$913$	−1724.97			−0.0625280
$914$		0			0
$915$	−498.871	−	498.871i	−0.0180242	−	0.0180242i
$916$		0			0
$917$	3989.02	−	3989.02i	0.143652	−	0.143652i
$918$		0			0
$919$		−	15342.1i		−	0.550694i	−0.961345	−	0.275347i	$-0.911207\pi$
$919$		−	15342.1i		−	0.550694i	0.961345	−	0.275347i	$-0.0887928\pi$
$920$		0			0
$921$		−	10927.0i		−	0.390941i
$922$		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
$927$	993.671			0.0352065
$928$		0			0
$929$	−104.186			−0.00367946			−0.00183973	−	0.999998i	$-0.500586\pi$
$929$	−104.186			−0.00367946			−0.00183973	+	0.999998i	$0.500586\pi$
$930$		0			0
$931$	13023.5	+	13023.5i	0.458464	+	0.458464i
$932$		0			0
$933$	−6350.97	+	6350.97i	−0.222853	+	0.222853i
$934$		0			0
$935$			2644.20i			0.0924863i
$936$		0			0
$937$			3137.81i			0.109400i	0.998503	+	0.0547001i	$0.0174203\pi$
$937$			3137.81i			0.109400i	−0.998503	+	0.0547001i	$0.982580\pi$
$938$		0			0
$939$	−13012.1	+	13012.1i	−0.452219	+	0.452219i
$940$		0			0
$941$	−7984.14	−	7984.14i	−0.276595	−	0.276595i	0.555153	−	0.831748i	$-0.312660\pi$
$941$	−7984.14	−	7984.14i	−0.276595	−	0.276595i	−0.831748	+	0.555153i	$0.812660\pi$
$942$		0			0
$943$	−2596.37			−0.0896601
$944$		0			0
$945$	−119.821			−0.00412462
$946$		0			0
$947$	20247.7	+	20247.7i	0.694787	+	0.694787i	0.963281	−	0.268495i	$-0.0865261\pi$
$947$	20247.7	+	20247.7i	0.694787	+	0.694787i	−0.268495	+	0.963281i	$0.586526\pi$
$948$		0			0
$949$	−19170.2	+	19170.2i	−0.655734	+	0.655734i
$950$		0			0
$951$			28598.6i			0.975155i
$952$		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
$957$	−13951.0	−	13951.0i	−0.471235	−	0.471235i
$958$		0			0
$959$	−13598.1			−0.457877
$960$		0			0
$961$	55172.3			1.85198
$962$		0			0
$963$	−2289.21	−	2289.21i	−0.0766032	−	0.0766032i
$964$		0			0
$965$	−931.351	+	931.351i	−0.0310686	+	0.0310686i
$966$		0			0
$967$			15161.3i			0.504194i	0.967702	+	0.252097i	$0.0811201\pi$
$967$			15161.3i			0.504194i	−0.967702	+	0.252097i	$0.918880\pi$
$968$		0			0
$969$		−	18056.1i		−	0.598601i
$970$		0			0
$971$	13515.9	−	13515.9i	0.446701	−	0.446701i	−0.447555	−	0.894256i	$-0.647705\pi$
$971$	13515.9	−	13515.9i	0.446701	−	0.446701i	0.894256	+	0.447555i	$0.147705\pi$
$972$		0			0
$973$	−8046.12	−	8046.12i	−0.265105	−	0.265105i
$974$		0			0
$975$	−9315.96			−0.305999
$976$		0			0
$977$	22582.6			0.739490			0.369745	−	0.929133i	$-0.379445\pi$
$977$	22582.6			0.739490			0.369745	+	0.929133i	$0.379445\pi$
$978$		0			0
$979$	773.536	+	773.536i	0.0252526	+	0.0252526i
$980$		0			0
$981$	−1845.23	+	1845.23i	−0.0600546	+	0.0600546i
$982$		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
$987$	−4712.33	+	4712.33i	−0.151971	+	0.151971i
$988$		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
$993$	−27516.7			−0.879372
$994$		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	$-0.793973\pi$
$997$	−6130.71	+	6130.71i	−0.194746	+	0.194746i	0.602997	+	0.797743i	$0.293973\pi$
$998$		0			0
$999$		−	5767.28i		−	0.182651i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.4.j.a.13.11	✓	24	8.3	odd	2
48.4.j.a.37.11	yes	24	16.3	odd	4
144.4.k.b.37.2		24	48.35	even	4
144.4.k.b.109.2		24	24.11	even	2
192.4.j.a.49.4		24	16.13	even	4
192.4.j.a.145.4		24	8.5	even	2
384.4.j.a.97.10		24	16.5	even	4	inner
384.4.j.a.289.10		24	1.1	even	1	trivial
384.4.j.b.97.3		24	16.11	odd	4
384.4.j.b.289.3		24	4.3	odd	2
576.4.k.b.145.6		24	24.5	odd	2
576.4.k.b.433.6		24	48.29	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.12132 + 2.12132i) q^{3} +(-0.706564 + 0.706564i) q^{5} -4.44122i q^{7} +9.00000i q^{9} +O(q^{10})\)	\(q+(2.12132 + 2.12132i) q^{3} +(-0.706564 + 0.706564i) q^{5} -4.44122i q^{7} +9.00000i q^{9} +(-17.7126 + 17.7126i) q^{11} +(17.7078 + 17.7078i) q^{13} -2.99770 q^{15} -105.640 q^{17} +(40.2862 + 40.2862i) q^{19} +(9.42125 - 9.42125i) q^{21} -42.9921i q^{23} +124.002i q^{25} +(-19.0919 + 19.0919i) q^{27} +(185.646 + 185.646i) q^{29} -291.485 q^{31} -75.1483 q^{33} +(3.13800 + 3.13800i) q^{35} +(-151.040 + 151.040i) q^{37} +75.1278i q^{39} -60.3918i q^{41} +(120.532 - 120.532i) q^{43} +(-6.35907 - 6.35907i) q^{45} -500.182 q^{47} +323.276 q^{49} +(-224.097 - 224.097i) q^{51} +(-192.407 + 192.407i) q^{53} -25.0302i q^{55} +170.920i q^{57} +(-500.053 + 500.053i) q^{59} +(166.418 + 166.418i) q^{61} +39.9710 q^{63} -25.0234 q^{65} +(-575.426 - 575.426i) q^{67} +(91.2000 - 91.2000i) q^{69} -457.290i q^{71} +1082.59i q^{73} +(-263.047 + 263.047i) q^{75} +(78.6657 + 78.6657i) q^{77} +544.309 q^{79} -81.0000 q^{81} +(48.6931 + 48.6931i) q^{83} +(74.6417 - 74.6417i) q^{85} +787.630i q^{87} -43.6714i q^{89} +(78.6442 - 78.6442i) q^{91} +(-618.332 - 618.332i) q^{93} -56.9295 q^{95} +690.537 q^{97} +(-159.414 - 159.414i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 40 q^{11} - 120 q^{15} + 24 q^{19} - 400 q^{29} + 744 q^{31} - 456 q^{35} - 16 q^{37} + 1240 q^{43} - 1176 q^{49} + 744 q^{51} - 752 q^{53} - 1376 q^{59} + 912 q^{61} + 504 q^{63} + 976 q^{65} - 2256 q^{67} + 528 q^{69} + 1104 q^{75} - 1904 q^{77} - 5992 q^{79} - 1944 q^{81} + 2680 q^{83} + 240 q^{85} - 3496 q^{91} + 7728 q^{95} - 360 q^{99}+O(q^{100}) \) 24 * q - 40 * q^11 - 120 * q^15 + 24 * q^19 - 400 * q^29 + 744 * q^31 - 456 * q^35 - 16 * q^37 + 1240 * q^43 - 1176 * q^49 + 744 * q^51 - 752 * q^53 - 1376 * q^59 + 912 * q^61 + 504 * q^63 + 976 * q^65 - 2256 * q^67 + 528 * q^69 + 1104 * q^75 - 1904 * q^77 - 5992 * q^79 - 1944 * q^81 + 2680 * q^83 + 240 * q^85 - 3496 * q^91 + 7728 * q^95 - 360 * q^99

Introduction

L-functions

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