LMFDB - Embedded newform 2352.4.a.cm.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.a (trivial)

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:	yes
Analytic conductor:	$138.772492334$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 152x^{2} - 177x + 2922 $ x^4 - 2x^3 - 152x^2 - 177*x + 2922
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$13.2349$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +18.9580 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +18.9580 q^{5} +9.00000 q^{9} -54.7308 q^{11} +62.0173 q^{13} -56.8741 q^{15} +122.439 q^{17} +12.5058 q^{19} +74.4391 q^{23} +234.406 q^{25} -27.0000 q^{27} -232.572 q^{29} -10.3677 q^{31} +164.192 q^{33} -245.987 q^{37} -186.052 q^{39} +238.653 q^{41} +92.9718 q^{43} +170.622 q^{45} +485.645 q^{47} -367.317 q^{51} -378.557 q^{53} -1037.59 q^{55} -37.5173 q^{57} +182.784 q^{59} +396.470 q^{61} +1175.73 q^{65} +261.239 q^{67} -223.317 q^{69} +874.523 q^{71} +152.406 q^{73} -703.219 q^{75} -573.357 q^{79} +81.0000 q^{81} -317.754 q^{83} +2321.20 q^{85} +697.717 q^{87} -95.0160 q^{89} +31.1032 q^{93} +237.084 q^{95} -1608.78 q^{97} -492.577 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 4 q^{5} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 + 4 * q^5 + 36 * q^9	$ 4 q - 12 q^{3} + 4 q^{5} + 36 q^{9} - 14 q^{11} + 22 q^{13} - 12 q^{15} + 96 q^{17} + 26 q^{19} - 96 q^{23} + 110 q^{25} - 108 q^{27} - 76 q^{29} - 238 q^{31} + 42 q^{33} + 562 q^{37} - 66 q^{39} + 428 q^{41} + 258 q^{43} + 36 q^{45} + 80 q^{47} - 288 q^{51} - 1476 q^{55} - 78 q^{57} - 262 q^{59} - 276 q^{61} + 2196 q^{65} - 150 q^{67} + 288 q^{69} + 848 q^{71} - 218 q^{73} - 330 q^{75} - 1762 q^{79} + 324 q^{81} - 3450 q^{83} + 1452 q^{85} + 228 q^{87} - 344 q^{89} + 714 q^{93} - 2004 q^{95} - 622 q^{97} - 126 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 + 4 * q^5 + 36 * q^9 - 14 * q^11 + 22 * q^13 - 12 * q^15 + 96 * q^17 + 26 * q^19 - 96 * q^23 + 110 * q^25 - 108 * q^27 - 76 * q^29 - 238 * q^31 + 42 * q^33 + 562 * q^37 - 66 * q^39 + 428 * q^41 + 258 * q^43 + 36 * q^45 + 80 * q^47 - 288 * q^51 - 1476 * q^55 - 78 * q^57 - 262 * q^59 - 276 * q^61 + 2196 * q^65 - 150 * q^67 + 288 * q^69 + 848 * q^71 - 218 * q^73 - 330 * q^75 - 1762 * q^79 + 324 * q^81 - 3450 * q^83 + 1452 * q^85 + 228 * q^87 - 344 * q^89 + 714 * q^93 - 2004 * q^95 - 622 * q^97 - 126 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	18.9580	1.69566	0.847828	−	0.530271i	$-0.177910\pi$
$5$	18.9580	1.69566	0.847828	+	0.530271i	$0.177910\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−54.7308	−1.50018	−0.750089	−	0.661337i	$-0.769989\pi$
$11$	−54.7308	−1.50018	−0.750089	+	0.661337i	$0.769989\pi$
$12$	0	0
$13$	62.0173	1.32312	0.661558	−	0.749894i	$-0.269896\pi$
$13$	62.0173	1.32312	0.661558	+	0.749894i	$0.269896\pi$
$14$	0	0
$15$	−56.8741	−0.978988
$16$	0	0
$17$	122.439	1.74681	0.873407	−	0.486991i	$-0.161905\pi$
$17$	122.439	1.74681	0.873407	+	0.486991i	$0.161905\pi$
$18$	0	0
$19$	12.5058	0.151001	0.0755004	−	0.997146i	$-0.475945\pi$
$19$	12.5058	0.151001	0.0755004	+	0.997146i	$0.475945\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	74.4391	0.674853	0.337427	−	0.941352i	$-0.390444\pi$
$23$	74.4391	0.674853	0.337427	+	0.941352i	$0.390444\pi$
$24$	0	0
$25$	234.406	1.87525
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−232.572	−1.48923	−0.744613	−	0.667496i	$-0.767366\pi$
$29$	−232.572	−1.48923	−0.744613	+	0.667496i	$0.767366\pi$
$30$	0	0
$31$	−10.3677	−0.0600677	−0.0300339	−	0.999549i	$-0.509562\pi$
$31$	−10.3677	−0.0600677	−0.0300339	+	0.999549i	$0.509562\pi$
$32$	0	0
$33$	164.192	0.866128
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−245.987	−1.09297	−0.546486	−	0.837468i	$-0.684035\pi$
$37$	−245.987	−1.09297	−0.546486	+	0.837468i	$0.684035\pi$
$38$	0	0
$39$	−186.052	−0.763901
$40$	0	0
$41$	238.653	0.909056	0.454528	−	0.890733i	$-0.349808\pi$
$41$	238.653	0.909056	0.454528	+	0.890733i	$0.349808\pi$
$42$	0	0
$43$	92.9718	0.329722	0.164861	−	0.986317i	$-0.447282\pi$
$43$	92.9718	0.329722	0.164861	+	0.986317i	$0.447282\pi$
$44$	0	0
$45$	170.622	0.565219
$46$	0	0
$47$	485.645	1.50720	0.753601	−	0.657332i	$-0.228315\pi$
$47$	485.645	1.50720	0.753601	+	0.657332i	$0.228315\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−367.317	−1.00852
$52$	0	0
$53$	−378.557	−0.981109	−0.490555	−	0.871410i	$-0.663206\pi$
$53$	−378.557	−0.981109	−0.490555	+	0.871410i	$0.663206\pi$
$54$	0	0
$55$	−1037.59	−2.54379
$56$	0	0
$57$	−37.5173	−0.0871804
$58$	0	0
$59$	182.784	0.403329	0.201664	−	0.979455i	$-0.435365\pi$
$59$	182.784	0.403329	0.201664	+	0.979455i	$0.435365\pi$
$60$	0	0
$61$	396.470	0.832177	0.416088	−	0.909324i	$-0.363401\pi$
$61$	396.470	0.832177	0.416088	+	0.909324i	$0.363401\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1175.73	2.24355
$66$	0	0
$67$	261.239	0.476350	0.238175	−	0.971222i	$-0.423451\pi$
$67$	261.239	0.476350	0.238175	+	0.971222i	$0.423451\pi$
$68$	0	0
$69$	−223.317	−0.389627
$70$	0	0
$71$	874.523	1.46179	0.730893	−	0.682492i	$-0.239104\pi$
$71$	874.523	1.46179	0.730893	+	0.682492i	$0.239104\pi$
$72$	0	0
$73$	152.406	0.244354	0.122177	−	0.992508i	$-0.461012\pi$
$73$	152.406	0.244354	0.122177	+	0.992508i	$0.461012\pi$
$74$	0	0
$75$	−703.219	−1.08268
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−573.357	−0.816554	−0.408277	−	0.912858i	$-0.633870\pi$
$79$	−573.357	−0.816554	−0.408277	+	0.912858i	$0.633870\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−317.754	−0.420217	−0.210108	−	0.977678i	$-0.567382\pi$
$83$	−317.754	−0.420217	−0.210108	+	0.977678i	$0.567382\pi$
$84$	0	0
$85$	2321.20	2.96200
$86$	0	0
$87$	697.717	0.859806
$88$	0	0
$89$	−95.0160	−0.113165	−0.0565824	−	0.998398i	$-0.518020\pi$
$89$	−95.0160	−0.113165	−0.0565824	+	0.998398i	$0.518020\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	31.1032	0.0346801
$94$	0	0
$95$	237.084	0.256046
$96$	0	0
$97$	−1608.78	−1.68398	−0.841992	−	0.539490i	$-0.818617\pi$
$97$	−1608.78	−1.68398	−0.841992	+	0.539490i	$0.818617\pi$
$98$	0	0
$99$	−492.577	−0.500059
$100$	0	0
$101$	−783.043	−0.771442	−0.385721	−	0.922615i	$-0.626047\pi$
$101$	−783.043	−0.771442	−0.385721	+	0.922615i	$0.626047\pi$
$102$	0	0
$103$	1489.61	1.42501	0.712503	−	0.701669i	$-0.247562\pi$
$103$	1489.61	1.42501	0.712503	+	0.701669i	$0.247562\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−712.769	−0.643981	−0.321991	−	0.946743i	$-0.604352\pi$
$107$	−712.769	−0.643981	−0.321991	+	0.946743i	$0.604352\pi$
$108$	0	0
$109$	1041.85	0.915513	0.457757	−	0.889078i	$-0.348653\pi$
$109$	1041.85	0.915513	0.457757	+	0.889078i	$0.348653\pi$
$110$	0	0
$111$	737.960	0.631028
$112$	0	0
$113$	352.093	0.293116	0.146558	−	0.989202i	$-0.453181\pi$
$113$	352.093	0.293116	0.146558	+	0.989202i	$0.453181\pi$
$114$	0	0
$115$	1411.22	1.14432
$116$	0	0
$117$	558.156	0.441039
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1664.46	1.25053
$122$	0	0
$123$	−715.958	−0.524844
$124$	0	0
$125$	2074.13	1.48413
$126$	0	0
$127$	1093.73	0.764194	0.382097	−	0.924122i	$-0.375202\pi$
$127$	1093.73	0.764194	0.382097	+	0.924122i	$0.375202\pi$
$128$	0	0
$129$	−278.915	−0.190365
$130$	0	0
$131$	2166.37	1.44486	0.722430	−	0.691444i	$-0.243025\pi$
$131$	2166.37	1.44486	0.722430	+	0.691444i	$0.243025\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−511.866	−0.326329
$136$	0	0
$137$	−1964.92	−1.22536	−0.612681	−	0.790330i	$-0.709909\pi$
$137$	−1964.92	−1.22536	−0.612681	+	0.790330i	$0.709909\pi$
$138$	0	0
$139$	136.976	0.0835840	0.0417920	−	0.999126i	$-0.486693\pi$
$139$	136.976	0.0835840	0.0417920	+	0.999126i	$0.486693\pi$
$140$	0	0
$141$	−1456.93	−0.870184
$142$	0	0
$143$	−3394.26	−1.98491
$144$	0	0
$145$	−4409.11	−2.52522
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−280.105	−0.154007	−0.0770037	−	0.997031i	$-0.524535\pi$
$149$	−280.105	−0.154007	−0.0770037	+	0.997031i	$0.524535\pi$
$150$	0	0
$151$	−2194.13	−1.18249	−0.591245	−	0.806492i	$-0.701363\pi$
$151$	−2194.13	−1.18249	−0.591245	+	0.806492i	$0.701363\pi$
$152$	0	0
$153$	1101.95	0.582271
$154$	0	0
$155$	−196.552	−0.101854
$156$	0	0
$157$	−220.818	−0.112250	−0.0561248	−	0.998424i	$-0.517874\pi$
$157$	−220.818	−0.112250	−0.0561248	+	0.998424i	$0.517874\pi$
$158$	0	0
$159$	1135.67	0.566444
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1476.86	−0.709674	−0.354837	−	0.934928i	$-0.615464\pi$
$163$	−1476.86	−0.709674	−0.354837	+	0.934928i	$0.615464\pi$
$164$	0	0
$165$	3112.76	1.46866
$166$	0	0
$167$	−2197.66	−1.01832	−0.509162	−	0.860671i	$-0.670045\pi$
$167$	−2197.66	−1.01832	−0.509162	+	0.860671i	$0.670045\pi$
$168$	0	0
$169$	1649.15	0.750635
$170$	0	0
$171$	112.552	0.0503336
$172$	0	0
$173$	2091.61	0.919201	0.459601	−	0.888126i	$-0.347993\pi$
$173$	2091.61	0.919201	0.459601	+	0.888126i	$0.347993\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−548.351	−0.232862
$178$	0	0
$179$	−2334.54	−0.974813	−0.487406	−	0.873175i	$-0.662057\pi$
$179$	−2334.54	−0.974813	−0.487406	+	0.873175i	$0.662057\pi$
$180$	0	0
$181$	1758.40	0.722105	0.361053	−	0.932545i	$-0.382417\pi$
$181$	1758.40	0.722105	0.361053	+	0.932545i	$0.382417\pi$
$182$	0	0
$183$	−1189.41	−0.480457
$184$	0	0
$185$	−4663.42	−1.85330
$186$	0	0
$187$	−6701.19	−2.62053
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3700.68	1.40195	0.700973	−	0.713188i	$-0.252749\pi$
$191$	3700.68	1.40195	0.700973	+	0.713188i	$0.252749\pi$
$192$	0	0
$193$	2708.45	1.01015	0.505073	−	0.863077i	$-0.331466\pi$
$193$	2708.45	1.01015	0.505073	+	0.863077i	$0.331466\pi$
$194$	0	0
$195$	−3527.18	−1.29531
$196$	0	0
$197$	−160.686	−0.0581138	−0.0290569	−	0.999578i	$-0.509250\pi$
$197$	−160.686	−0.0581138	−0.0290569	+	0.999578i	$0.509250\pi$
$198$	0	0
$199$	2065.99	0.735951	0.367976	−	0.929835i	$-0.380051\pi$
$199$	2065.99	0.735951	0.367976	+	0.929835i	$0.380051\pi$
$200$	0	0
$201$	−783.717	−0.275021
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4524.38	1.54145
$206$	0	0
$207$	669.952	0.224951
$208$	0	0
$209$	−684.450	−0.226528
$210$	0	0
$211$	1007.12	0.328592	0.164296	−	0.986411i	$-0.447465\pi$
$211$	1007.12	0.328592	0.164296	+	0.986411i	$0.447465\pi$
$212$	0	0
$213$	−2623.57	−0.843962
$214$	0	0
$215$	1762.56	0.559096
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−457.219	−0.141078
$220$	0	0
$221$	7593.34	2.31124
$222$	0	0
$223$	−1644.83	−0.493929	−0.246964	−	0.969025i	$-0.579433\pi$
$223$	−1644.83	−0.493929	−0.246964	+	0.969025i	$0.579433\pi$
$224$	0	0
$225$	2109.66	0.625084
$226$	0	0
$227$	−318.874	−0.0932354	−0.0466177	−	0.998913i	$-0.514844\pi$
$227$	−318.874	−0.0932354	−0.0466177	+	0.998913i	$0.514844\pi$
$228$	0	0
$229$	2536.11	0.731837	0.365918	−	0.930647i	$-0.380755\pi$
$229$	2536.11	0.731837	0.365918	+	0.930647i	$0.380755\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2332.81	0.655913	0.327957	−	0.944693i	$-0.393640\pi$
$233$	2332.81	0.655913	0.327957	+	0.944693i	$0.393640\pi$
$234$	0	0
$235$	9206.86	2.55570
$236$	0	0
$237$	1720.07	0.471438
$238$	0	0
$239$	−2713.85	−0.734495	−0.367248	−	0.930123i	$-0.619700\pi$
$239$	−2713.85	−0.734495	−0.367248	+	0.930123i	$0.619700\pi$
$240$	0	0
$241$	−4174.59	−1.11580	−0.557902	−	0.829907i	$-0.688394\pi$
$241$	−4174.59	−1.11580	−0.557902	+	0.829907i	$0.688394\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	775.573	0.199792
$248$	0	0
$249$	953.261	0.242612
$250$	0	0
$251$	−6123.58	−1.53991	−0.769954	−	0.638099i	$-0.779721\pi$
$251$	−6123.58	−1.53991	−0.769954	+	0.638099i	$0.779721\pi$
$252$	0	0
$253$	−4074.11	−1.01240
$254$	0	0
$255$	−6963.61	−1.71011
$256$	0	0
$257$	4633.91	1.12473	0.562365	−	0.826889i	$-0.309892\pi$
$257$	4633.91	1.12473	0.562365	+	0.826889i	$0.309892\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2093.15	−0.496409
$262$	0	0
$263$	−3283.23	−0.769781	−0.384891	−	0.922962i	$-0.625761\pi$
$263$	−3283.23	−0.769781	−0.384891	+	0.922962i	$0.625761\pi$
$264$	0	0
$265$	−7176.69	−1.66362
$266$	0	0
$267$	285.048	0.0653358
$268$	0	0
$269$	−798.985	−0.181097	−0.0905483	−	0.995892i	$-0.528862\pi$
$269$	−798.985	−0.181097	−0.0905483	+	0.995892i	$0.528862\pi$
$270$	0	0
$271$	6706.07	1.50319	0.751596	−	0.659624i	$-0.229284\pi$
$271$	6706.07	1.50319	0.751596	+	0.659624i	$0.229284\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−12829.3	−2.81321
$276$	0	0
$277$	1162.27	0.252108	0.126054	−	0.992023i	$-0.459769\pi$
$277$	1162.27	0.252108	0.126054	+	0.992023i	$0.459769\pi$
$278$	0	0
$279$	−93.3096	−0.0200226
$280$	0	0
$281$	2718.17	0.577055	0.288527	−	0.957472i	$-0.406834\pi$
$281$	2718.17	0.577055	0.288527	+	0.957472i	$0.406834\pi$
$282$	0	0
$283$	3009.48	0.632138	0.316069	−	0.948736i	$-0.397637\pi$
$283$	3009.48	0.632138	0.316069	+	0.948736i	$0.397637\pi$
$284$	0	0
$285$	−711.253	−0.147828
$286$	0	0
$287$	0	0
$288$	0	0
$289$	10078.3	2.05136
$290$	0	0
$291$	4826.33	0.972249
$292$	0	0
$293$	4209.15	0.839252	0.419626	−	0.907697i	$-0.362161\pi$
$293$	4209.15	0.839252	0.419626	+	0.907697i	$0.362161\pi$
$294$	0	0
$295$	3465.21	0.683907
$296$	0	0
$297$	1477.73	0.288709
$298$	0	0
$299$	4616.51	0.892909
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2349.13	0.445392
$304$	0	0
$305$	7516.28	1.41109
$306$	0	0
$307$	3114.82	0.579063	0.289531	−	0.957169i	$-0.406501\pi$
$307$	3114.82	0.579063	0.289531	+	0.957169i	$0.406501\pi$
$308$	0	0
$309$	−4468.83	−0.822727
$310$	0	0
$311$	8487.24	1.54748	0.773741	−	0.633501i	$-0.218383\pi$
$311$	8487.24	1.54748	0.773741	+	0.633501i	$0.218383\pi$
$312$	0	0
$313$	−4954.91	−0.894786	−0.447393	−	0.894337i	$-0.647648\pi$
$313$	−4954.91	−0.894786	−0.447393	+	0.894337i	$0.647648\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5608.79	−0.993757	−0.496879	−	0.867820i	$-0.665521\pi$
$317$	−5608.79	−0.993757	−0.496879	+	0.867820i	$0.665521\pi$
$318$	0	0
$319$	12728.9	2.23411
$320$	0	0
$321$	2138.31	0.371803
$322$	0	0
$323$	1531.19	0.263770
$324$	0	0
$325$	14537.3	2.48117
$326$	0	0
$327$	−3125.54	−0.528572
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6609.49	1.09755	0.548777	−	0.835969i	$-0.315094\pi$
$331$	6609.49	1.09755	0.548777	+	0.835969i	$0.315094\pi$
$332$	0	0
$333$	−2213.88	−0.364324
$334$	0	0
$335$	4952.57	0.807725
$336$	0	0
$337$	11455.5	1.85170	0.925850	−	0.377891i	$-0.123350\pi$
$337$	11455.5	1.85170	0.925850	+	0.377891i	$0.123350\pi$
$338$	0	0
$339$	−1056.28	−0.169231
$340$	0	0
$341$	567.434	0.0901123
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−4233.65	−0.660673
$346$	0	0
$347$	6044.15	0.935064	0.467532	−	0.883976i	$-0.345143\pi$
$347$	6044.15	0.935064	0.467532	+	0.883976i	$0.345143\pi$
$348$	0	0
$349$	−5487.93	−0.841726	−0.420863	−	0.907124i	$-0.638273\pi$
$349$	−5487.93	−0.841726	−0.420863	+	0.907124i	$0.638273\pi$
$350$	0	0
$351$	−1674.47	−0.254634
$352$	0	0
$353$	6880.15	1.03738	0.518688	−	0.854964i	$-0.326421\pi$
$353$	6880.15	1.03738	0.518688	+	0.854964i	$0.326421\pi$
$354$	0	0
$355$	16579.2	2.47869
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3938.34	0.578990	0.289495	−	0.957180i	$-0.406513\pi$
$359$	3938.34	0.578990	0.289495	+	0.957180i	$0.406513\pi$
$360$	0	0
$361$	−6702.61	−0.977199
$362$	0	0
$363$	−4993.38	−0.721996
$364$	0	0
$365$	2889.32	0.414340
$366$	0	0
$367$	−1439.63	−0.204763	−0.102381	−	0.994745i	$-0.532646\pi$
$367$	−1439.63	−0.204763	−0.102381	+	0.994745i	$0.532646\pi$
$368$	0	0
$369$	2147.87	0.303019
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2494.11	−0.346220	−0.173110	−	0.984903i	$-0.555382\pi$
$373$	−2494.11	−0.346220	−0.173110	+	0.984903i	$0.555382\pi$
$374$	0	0
$375$	−6222.39	−0.856861
$376$	0	0
$377$	−14423.5	−1.97042
$378$	0	0
$379$	1309.25	0.177445	0.0887225	−	0.996056i	$-0.471722\pi$
$379$	1309.25	0.177445	0.0887225	+	0.996056i	$0.471722\pi$
$380$	0	0
$381$	−3281.18	−0.441208
$382$	0	0
$383$	−6961.80	−0.928803	−0.464402	−	0.885625i	$-0.653731\pi$
$383$	−6961.80	−0.928803	−0.464402	+	0.885625i	$0.653731\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	836.746	0.109907
$388$	0	0
$389$	6353.45	0.828104	0.414052	−	0.910253i	$-0.364113\pi$
$389$	6353.45	0.828104	0.414052	+	0.910253i	$0.364113\pi$
$390$	0	0
$391$	9114.25	1.17884
$392$	0	0
$393$	−6499.11	−0.834191
$394$	0	0
$395$	−10869.7	−1.38459
$396$	0	0
$397$	11505.4	1.45451	0.727254	−	0.686368i	$-0.240796\pi$
$397$	11505.4	1.45451	0.727254	+	0.686368i	$0.240796\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1653.33	0.205893	0.102947	−	0.994687i	$-0.467173\pi$
$401$	1653.33	0.205893	0.102947	+	0.994687i	$0.467173\pi$
$402$	0	0
$403$	−642.979	−0.0794766
$404$	0	0
$405$	1535.60	0.188406
$406$	0	0
$407$	13463.0	1.63965
$408$	0	0
$409$	−4894.83	−0.591769	−0.295885	−	0.955224i	$-0.595614\pi$
$409$	−4894.83	−0.591769	−0.295885	+	0.955224i	$0.595614\pi$
$410$	0	0
$411$	5894.77	0.707463
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−6023.98	−0.712544
$416$	0	0
$417$	−410.929	−0.0482573
$418$	0	0
$419$	265.504	0.0309563	0.0154782	−	0.999880i	$-0.495073\pi$
$419$	265.504	0.0309563	0.0154782	+	0.999880i	$0.495073\pi$
$420$	0	0
$421$	11136.8	1.28925	0.644623	−	0.764500i	$-0.277014\pi$
$421$	11136.8	1.28925	0.644623	+	0.764500i	$0.277014\pi$
$422$	0	0
$423$	4370.80	0.502401
$424$	0	0
$425$	28700.5	3.27572
$426$	0	0
$427$	0	0
$428$	0	0
$429$	10182.8	1.14599
$430$	0	0
$431$	−5193.41	−0.580413	−0.290206	−	0.956964i	$-0.593724\pi$
$431$	−5193.41	−0.580413	−0.290206	+	0.956964i	$0.593724\pi$
$432$	0	0
$433$	6314.17	0.700785	0.350392	−	0.936603i	$-0.386048\pi$
$433$	6314.17	0.700785	0.350392	+	0.936603i	$0.386048\pi$
$434$	0	0
$435$	13227.3	1.45794
$436$	0	0
$437$	930.917	0.101903
$438$	0	0
$439$	15423.3	1.67680	0.838399	−	0.545056i	$-0.183492\pi$
$439$	15423.3	1.67680	0.838399	+	0.545056i	$0.183492\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8706.11	0.933725	0.466862	−	0.884330i	$-0.345384\pi$
$443$	8706.11	0.933725	0.466862	+	0.884330i	$0.345384\pi$
$444$	0	0
$445$	−1801.31	−0.191889
$446$	0	0
$447$	840.315	0.0889162
$448$	0	0
$449$	5495.91	0.577657	0.288829	−	0.957381i	$-0.406734\pi$
$449$	5495.91	0.577657	0.288829	+	0.957381i	$0.406734\pi$
$450$	0	0
$451$	−13061.7	−1.36375
$452$	0	0
$453$	6582.40	0.682711
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−11357.3	−1.16252	−0.581258	−	0.813719i	$-0.697439\pi$
$457$	−11357.3	−1.16252	−0.581258	+	0.813719i	$0.697439\pi$
$458$	0	0
$459$	−3305.86	−0.336175
$460$	0	0
$461$	14514.2	1.46637	0.733184	−	0.680030i	$-0.238033\pi$
$461$	14514.2	1.46637	0.733184	+	0.680030i	$0.238033\pi$
$462$	0	0
$463$	−9971.00	−1.00085	−0.500423	−	0.865781i	$-0.666822\pi$
$463$	−9971.00	−1.00085	−0.500423	+	0.865781i	$0.666822\pi$
$464$	0	0
$465$	589.655	0.0588056
$466$	0	0
$467$	397.477	0.0393856	0.0196928	−	0.999806i	$-0.493731\pi$
$467$	397.477	0.0393856	0.0196928	+	0.999806i	$0.493731\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	662.454	0.0648074
$472$	0	0
$473$	−5088.42	−0.494642
$474$	0	0
$475$	2931.43	0.283165
$476$	0	0
$477$	−3407.01	−0.327036
$478$	0	0
$479$	14060.4	1.34120	0.670602	−	0.741818i	$-0.266036\pi$
$479$	14060.4	1.34120	0.670602	+	0.741818i	$0.266036\pi$
$480$	0	0
$481$	−15255.4	−1.44613
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−30499.2	−2.85546
$486$	0	0
$487$	13534.7	1.25937	0.629687	−	0.776849i	$-0.283183\pi$
$487$	13534.7	1.25937	0.629687	+	0.776849i	$0.283183\pi$
$488$	0	0
$489$	4430.59	0.409730
$490$	0	0
$491$	8693.29	0.799028	0.399514	−	0.916727i	$-0.369179\pi$
$491$	8693.29	0.799028	0.399514	+	0.916727i	$0.369179\pi$
$492$	0	0
$493$	−28475.9	−2.60140
$494$	0	0
$495$	−9338.29	−0.847929
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4017.93	−0.360455	−0.180228	−	0.983625i	$-0.557683\pi$
$499$	−4017.93	−0.360455	−0.180228	+	0.983625i	$0.557683\pi$
$500$	0	0
$501$	6592.98	0.587929
$502$	0	0
$503$	−52.2455	−0.00463124	−0.00231562	−	0.999997i	$-0.500737\pi$
$503$	−52.2455	−0.00463124	−0.00231562	+	0.999997i	$0.500737\pi$
$504$	0	0
$505$	−14844.9	−1.30810
$506$	0	0
$507$	−4947.44	−0.433379
$508$	0	0
$509$	9239.10	0.804550	0.402275	−	0.915519i	$-0.368220\pi$
$509$	9239.10	0.804550	0.402275	+	0.915519i	$0.368220\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−337.655	−0.0290601
$514$	0	0
$515$	28240.0	2.41632
$516$	0	0
$517$	−26579.7	−2.26107
$518$	0	0
$519$	−6274.82	−0.530701
$520$	0	0
$521$	14973.4	1.25911	0.629555	−	0.776956i	$-0.283237\pi$
$521$	14973.4	1.25911	0.629555	+	0.776956i	$0.283237\pi$
$522$	0	0
$523$	−13603.5	−1.13736	−0.568681	−	0.822558i	$-0.692546\pi$
$523$	−13603.5	−1.13736	−0.568681	+	0.822558i	$0.692546\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1269.42	−0.104927
$528$	0	0
$529$	−6625.82	−0.544573
$530$	0	0
$531$	1645.05	0.134443
$532$	0	0
$533$	14800.6	1.20279
$534$	0	0
$535$	−13512.7	−1.09197
$536$	0	0
$537$	7003.61	0.562808
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−16886.4	−1.34196	−0.670981	−	0.741474i	$-0.734127\pi$
$541$	−16886.4	−1.34196	−0.670981	+	0.741474i	$0.734127\pi$
$542$	0	0
$543$	−5275.21	−0.416908
$544$	0	0
$545$	19751.4	1.55240
$546$	0	0
$547$	5987.25	0.468001	0.234000	−	0.972237i	$-0.424818\pi$
$547$	5987.25	0.468001	0.234000	+	0.972237i	$0.424818\pi$
$548$	0	0
$549$	3568.23	0.277392
$550$	0	0
$551$	−2908.49	−0.224875
$552$	0	0
$553$	0	0
$554$	0	0
$555$	13990.3	1.07001
$556$	0	0
$557$	2238.71	0.170300	0.0851499	−	0.996368i	$-0.472863\pi$
$557$	2238.71	0.170300	0.0851499	+	0.996368i	$0.472863\pi$
$558$	0	0
$559$	5765.86	0.436261
$560$	0	0
$561$	20103.6	1.51297
$562$	0	0
$563$	−8452.84	−0.632762	−0.316381	−	0.948632i	$-0.602468\pi$
$563$	−8452.84	−0.632762	−0.316381	+	0.948632i	$0.602468\pi$
$564$	0	0
$565$	6674.98	0.497024
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14953.6	1.10173	0.550866	−	0.834594i	$-0.314298\pi$
$569$	14953.6	1.10173	0.550866	+	0.834594i	$0.314298\pi$
$570$	0	0
$571$	−16021.8	−1.17424	−0.587122	−	0.809498i	$-0.699739\pi$
$571$	−16021.8	−1.17424	−0.587122	+	0.809498i	$0.699739\pi$
$572$	0	0
$573$	−11102.0	−0.809414
$574$	0	0
$575$	17449.0	1.26552
$576$	0	0
$577$	−16113.1	−1.16256	−0.581279	−	0.813704i	$-0.697448\pi$
$577$	−16113.1	−1.16256	−0.581279	+	0.813704i	$0.697448\pi$
$578$	0	0
$579$	−8125.34	−0.583208
$580$	0	0
$581$	0	0
$582$	0	0
$583$	20718.7	1.47184
$584$	0	0
$585$	10581.5	0.747850
$586$	0	0
$587$	9552.04	0.671644	0.335822	−	0.941925i	$-0.390986\pi$
$587$	9552.04	0.671644	0.335822	+	0.941925i	$0.390986\pi$
$588$	0	0
$589$	−129.656	−0.00907028
$590$	0	0
$591$	482.058	0.0335520
$592$	0	0
$593$	−24167.8	−1.67361	−0.836807	−	0.547499i	$-0.815580\pi$
$593$	−24167.8	−1.67361	−0.836807	+	0.547499i	$0.815580\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6197.98	−0.424902
$598$	0	0
$599$	−821.331	−0.0560245	−0.0280122	−	0.999608i	$-0.508918\pi$
$599$	−821.331	−0.0560245	−0.0280122	+	0.999608i	$0.508918\pi$
$600$	0	0
$601$	−14674.7	−0.995996	−0.497998	−	0.867178i	$-0.665931\pi$
$601$	−14674.7	−0.995996	−0.497998	+	0.867178i	$0.665931\pi$
$602$	0	0
$603$	2351.15	0.158783
$604$	0	0
$605$	31554.9	2.12048
$606$	0	0
$607$	−5083.12	−0.339897	−0.169949	−	0.985453i	$-0.554360\pi$
$607$	−5083.12	−0.339897	−0.169949	+	0.985453i	$0.554360\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30118.4	1.99420
$612$	0	0
$613$	−22202.0	−1.46286	−0.731428	−	0.681919i	$-0.761146\pi$
$613$	−22202.0	−1.46286	−0.731428	+	0.681919i	$0.761146\pi$
$614$	0	0
$615$	−13573.1	−0.889954
$616$	0	0
$617$	−14990.6	−0.978121	−0.489060	−	0.872250i	$-0.662660\pi$
$617$	−14990.6	−0.978121	−0.489060	+	0.872250i	$0.662660\pi$
$618$	0	0
$619$	−3073.02	−0.199540	−0.0997700	−	0.995011i	$-0.531811\pi$
$619$	−3073.02	−0.199540	−0.0997700	+	0.995011i	$0.531811\pi$
$620$	0	0
$621$	−2009.86	−0.129876
$622$	0	0
$623$	0	0
$624$	0	0
$625$	10020.6	0.641317
$626$	0	0
$627$	2053.35	0.130786
$628$	0	0
$629$	−30118.4	−1.90922
$630$	0	0
$631$	−26012.7	−1.64113	−0.820563	−	0.571556i	$-0.806340\pi$
$631$	−26012.7	−1.64113	−0.820563	+	0.571556i	$0.806340\pi$
$632$	0	0
$633$	−3021.36	−0.189713
$634$	0	0
$635$	20734.9	1.29581
$636$	0	0
$637$	0	0
$638$	0	0
$639$	7870.70	0.487262
$640$	0	0
$641$	8032.23	0.494936	0.247468	−	0.968896i	$-0.420401\pi$
$641$	8032.23	0.494936	0.247468	+	0.968896i	$0.420401\pi$
$642$	0	0
$643$	24887.7	1.52640	0.763200	−	0.646162i	$-0.223627\pi$
$643$	24887.7	1.52640	0.763200	+	0.646162i	$0.223627\pi$
$644$	0	0
$645$	−5287.68	−0.322794
$646$	0	0
$647$	21084.3	1.28116	0.640580	−	0.767891i	$-0.278694\pi$
$647$	21084.3	1.28116	0.640580	+	0.767891i	$0.278694\pi$
$648$	0	0
$649$	−10003.9	−0.605065
$650$	0	0
$651$	0	0
$652$	0	0
$653$	13166.3	0.789033	0.394516	−	0.918889i	$-0.370912\pi$
$653$	13166.3	0.789033	0.394516	+	0.918889i	$0.370912\pi$
$654$	0	0
$655$	41070.1	2.44999
$656$	0	0
$657$	1371.66	0.0814513
$658$	0	0
$659$	13903.4	0.821851	0.410926	−	0.911669i	$-0.365206\pi$
$659$	13903.4	0.821851	0.410926	+	0.911669i	$0.365206\pi$
$660$	0	0
$661$	6306.61	0.371102	0.185551	−	0.982635i	$-0.440593\pi$
$661$	6306.61	0.371102	0.185551	+	0.982635i	$0.440593\pi$
$662$	0	0
$663$	−22780.0	−1.33439
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−17312.5	−1.00501
$668$	0	0
$669$	4934.50	0.285170
$670$	0	0
$671$	−21699.1	−1.24841
$672$	0	0
$673$	−24407.6	−1.39798	−0.698992	−	0.715129i	$-0.746368\pi$
$673$	−24407.6	−1.39798	−0.698992	+	0.715129i	$0.746368\pi$
$674$	0	0
$675$	−6328.97	−0.360892
$676$	0	0
$677$	−31080.3	−1.76442	−0.882209	−	0.470858i	$-0.843945\pi$
$677$	−31080.3	−1.76442	−0.882209	+	0.470858i	$0.843945\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	956.623	0.0538295
$682$	0	0
$683$	−28759.5	−1.61120	−0.805601	−	0.592458i	$-0.798157\pi$
$683$	−28759.5	−1.61120	−0.805601	+	0.592458i	$0.798157\pi$
$684$	0	0
$685$	−37251.0	−2.07779
$686$	0	0
$687$	−7608.32	−0.422526
$688$	0	0
$689$	−23477.1	−1.29812
$690$	0	0
$691$	−24895.4	−1.37057	−0.685287	−	0.728273i	$-0.740323\pi$
$691$	−24895.4	−1.37057	−0.685287	+	0.728273i	$0.740323\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2596.80	0.141730
$696$	0	0
$697$	29220.4	1.58795
$698$	0	0
$699$	−6998.44	−0.378692
$700$	0	0
$701$	−1702.74	−0.0917427	−0.0458714	−	0.998947i	$-0.514606\pi$
$701$	−1702.74	−0.0917427	−0.0458714	+	0.998947i	$0.514606\pi$
$702$	0	0
$703$	−3076.25	−0.165040
$704$	0	0
$705$	−27620.6	−1.47553
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6523.21	0.345535	0.172767	−	0.984963i	$-0.444729\pi$
$709$	6523.21	0.345535	0.172767	+	0.984963i	$0.444729\pi$
$710$	0	0
$711$	−5160.22	−0.272185
$712$	0	0
$713$	−771.765	−0.0405369
$714$	0	0
$715$	−64348.4	−3.36572
$716$	0	0
$717$	8141.55	0.424061
$718$	0	0
$719$	25254.4	1.30992	0.654959	−	0.755664i	$-0.272686\pi$
$719$	25254.4	1.30992	0.654959	+	0.755664i	$0.272686\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	12523.8	0.644210
$724$	0	0
$725$	−54516.4	−2.79268
$726$	0	0
$727$	27964.9	1.42663	0.713316	−	0.700843i	$-0.247192\pi$
$727$	27964.9	1.42663	0.713316	+	0.700843i	$0.247192\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	11383.4	0.575964
$732$	0	0
$733$	−9294.38	−0.468344	−0.234172	−	0.972195i	$-0.575238\pi$
$733$	−9294.38	−0.468344	−0.234172	+	0.972195i	$0.575238\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14297.8	−0.714609
$738$	0	0
$739$	−18291.2	−0.910490	−0.455245	−	0.890366i	$-0.650448\pi$
$739$	−18291.2	−0.910490	−0.455245	+	0.890366i	$0.650448\pi$
$740$	0	0
$741$	−2326.72	−0.115350
$742$	0	0
$743$	−14742.4	−0.727921	−0.363960	−	0.931414i	$-0.618576\pi$
$743$	−14742.4	−0.727921	−0.363960	+	0.931414i	$0.618576\pi$
$744$	0	0
$745$	−5310.24	−0.261144
$746$	0	0
$747$	−2859.78	−0.140072
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−28463.4	−1.38301	−0.691507	−	0.722370i	$-0.743053\pi$
$751$	−28463.4	−1.38301	−0.691507	+	0.722370i	$0.743053\pi$
$752$	0	0
$753$	18370.7	0.889067
$754$	0	0
$755$	−41596.4	−2.00510
$756$	0	0
$757$	−20336.7	−0.976422	−0.488211	−	0.872726i	$-0.662350\pi$
$757$	−20336.7	−0.976422	−0.488211	+	0.872726i	$0.662350\pi$
$758$	0	0
$759$	12222.3	0.584509
$760$	0	0
$761$	39581.6	1.88546	0.942728	−	0.333564i	$-0.108251\pi$
$761$	39581.6	1.88546	0.942728	+	0.333564i	$0.108251\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	20890.8	0.987332
$766$	0	0
$767$	11335.7	0.533650
$768$	0	0
$769$	10580.3	0.496146	0.248073	−	0.968741i	$-0.420203\pi$
$769$	10580.3	0.496146	0.248073	+	0.968741i	$0.420203\pi$
$770$	0	0
$771$	−13901.7	−0.649363
$772$	0	0
$773$	25922.2	1.20615	0.603077	−	0.797683i	$-0.293941\pi$
$773$	25922.2	1.20615	0.603077	+	0.797683i	$0.293941\pi$
$774$	0	0
$775$	−2430.26	−0.112642
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2984.53	0.137268
$780$	0	0
$781$	−47863.3	−2.19294
$782$	0	0
$783$	6279.45	0.286602
$784$	0	0
$785$	−4186.27	−0.190337
$786$	0	0
$787$	−17220.0	−0.779958	−0.389979	−	0.920824i	$-0.627518\pi$
$787$	−17220.0	−0.779958	−0.389979	+	0.920824i	$0.627518\pi$
$788$	0	0
$789$	9849.68	0.444433
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24588.0	1.10107
$794$	0	0
$795$	21530.1	0.960494
$796$	0	0
$797$	−6275.52	−0.278909	−0.139454	−	0.990228i	$-0.544535\pi$
$797$	−6275.52	−0.278909	−0.139454	+	0.990228i	$0.544535\pi$
$798$	0	0
$799$	59461.9	2.63280
$800$	0	0
$801$	−855.144	−0.0377216
$802$	0	0
$803$	−8341.33	−0.366574
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2396.96	0.104556
$808$	0	0
$809$	−4604.03	−0.200085	−0.100043	−	0.994983i	$-0.531898\pi$
$809$	−4604.03	−0.200085	−0.100043	+	0.994983i	$0.531898\pi$
$810$	0	0
$811$	−5104.36	−0.221009	−0.110505	−	0.993876i	$-0.535247\pi$
$811$	−5104.36	−0.221009	−0.110505	+	0.993876i	$0.535247\pi$
$812$	0	0
$813$	−20118.2	−0.867868
$814$	0	0
$815$	−27998.4	−1.20336
$816$	0	0
$817$	1162.68	0.0497884
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−45635.4	−1.93993	−0.969967	−	0.243237i	$-0.921791\pi$
$821$	−45635.4	−1.93993	−0.969967	+	0.243237i	$0.921791\pi$
$822$	0	0
$823$	39377.1	1.66780	0.833901	−	0.551915i	$-0.186103\pi$
$823$	39377.1	1.66780	0.833901	+	0.551915i	$0.186103\pi$
$824$	0	0
$825$	38487.8	1.62421
$826$	0	0
$827$	30916.3	1.29996	0.649978	−	0.759953i	$-0.274778\pi$
$827$	30916.3	1.29996	0.649978	+	0.759953i	$0.274778\pi$
$828$	0	0
$829$	−2543.33	−0.106554	−0.0532771	−	0.998580i	$-0.516967\pi$
$829$	−2543.33	−0.106554	−0.0532771	+	0.998580i	$0.516967\pi$
$830$	0	0
$831$	−3486.81	−0.145555
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−41663.3	−1.72673
$836$	0	0
$837$	279.929	0.0115600
$838$	0	0
$839$	7930.82	0.326343	0.163172	−	0.986598i	$-0.447828\pi$
$839$	7930.82	0.326343	0.163172	+	0.986598i	$0.447828\pi$
$840$	0	0
$841$	29700.8	1.21780
$842$	0	0
$843$	−8154.51	−0.333163
$844$	0	0
$845$	31264.5	1.27282
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−9028.45	−0.364965
$850$	0	0
$851$	−18311.0	−0.737595
$852$	0	0
$853$	41983.2	1.68520	0.842601	−	0.538538i	$-0.181023\pi$
$853$	41983.2	1.68520	0.842601	+	0.538538i	$0.181023\pi$
$854$	0	0
$855$	2133.76	0.0853485
$856$	0	0
$857$	−11854.3	−0.472504	−0.236252	−	0.971692i	$-0.575919\pi$
$857$	−11854.3	−0.472504	−0.236252	+	0.971692i	$0.575919\pi$
$858$	0	0
$859$	8113.88	0.322284	0.161142	−	0.986931i	$-0.448482\pi$
$859$	8113.88	0.322284	0.161142	+	0.986931i	$0.448482\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−45817.0	−1.80722	−0.903609	−	0.428358i	$-0.859092\pi$
$863$	−45817.0	−1.80722	−0.903609	+	0.428358i	$0.859092\pi$
$864$	0	0
$865$	39652.7	1.55865
$866$	0	0
$867$	−30235.0	−1.18435
$868$	0	0
$869$	31380.3	1.22498
$870$	0	0
$871$	16201.3	0.630266
$872$	0	0
$873$	−14479.0	−0.561328
$874$	0	0
$875$	0	0
$876$	0	0
$877$	31813.0	1.22491	0.612456	−	0.790504i	$-0.290182\pi$
$877$	31813.0	1.22491	0.612456	+	0.790504i	$0.290182\pi$
$878$	0	0
$879$	−12627.4	−0.484543
$880$	0	0
$881$	43551.6	1.66548	0.832742	−	0.553661i	$-0.186770\pi$
$881$	43551.6	1.66548	0.832742	+	0.553661i	$0.186770\pi$
$882$	0	0
$883$	−40645.1	−1.54906	−0.774528	−	0.632540i	$-0.782012\pi$
$883$	−40645.1	−1.54906	−0.774528	+	0.632540i	$0.782012\pi$
$884$	0	0
$885$	−10395.6	−0.394854
$886$	0	0
$887$	−34028.2	−1.28811	−0.644056	−	0.764978i	$-0.722750\pi$
$887$	−34028.2	−1.28811	−0.644056	+	0.764978i	$0.722750\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4433.20	−0.166686
$892$	0	0
$893$	6073.35	0.227589
$894$	0	0
$895$	−44258.2	−1.65295
$896$	0	0
$897$	−13849.5	−0.515521
$898$	0	0
$899$	2411.25	0.0894545
$900$	0	0
$901$	−46350.2	−1.71382
$902$	0	0
$903$	0	0
$904$	0	0
$905$	33335.8	1.22444
$906$	0	0
$907$	−50804.7	−1.85992	−0.929958	−	0.367666i	$-0.880157\pi$
$907$	−50804.7	−1.85992	−0.929958	+	0.367666i	$0.880157\pi$
$908$	0	0
$909$	−7047.38	−0.257147
$910$	0	0
$911$	−28738.3	−1.04516	−0.522582	−	0.852589i	$-0.675031\pi$
$911$	−28738.3	−1.04516	−0.522582	+	0.852589i	$0.675031\pi$
$912$	0	0
$913$	17390.9	0.630400
$914$	0	0
$915$	−22548.9	−0.814691
$916$	0	0
$917$	0	0
$918$	0	0
$919$	54742.2	1.96494	0.982470	−	0.186420i	$-0.0596886\pi$
$919$	54742.2	1.96494	0.982470	+	0.186420i	$0.0596886\pi$
$920$	0	0
$921$	−9344.46	−0.334322
$922$	0	0
$923$	54235.5	1.93411
$924$	0	0
$925$	−57660.9	−2.04960
$926$	0	0
$927$	13406.5	0.475002
$928$	0	0
$929$	43775.8	1.54600	0.773002	−	0.634403i	$-0.218754\pi$
$929$	43775.8	1.54600	0.773002	+	0.634403i	$0.218754\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−25461.7	−0.893440
$934$	0	0
$935$	−127041.	−4.44352
$936$	0	0
$937$	35090.9	1.22345	0.611724	−	0.791071i	$-0.290476\pi$
$937$	35090.9	1.22345	0.611724	+	0.791071i	$0.290476\pi$
$938$	0	0
$939$	14864.7	0.516605
$940$	0	0
$941$	−31784.8	−1.10112	−0.550560	−	0.834795i	$-0.685586\pi$
$941$	−31784.8	−1.10112	−0.550560	+	0.834795i	$0.685586\pi$
$942$	0	0
$943$	17765.1	0.613479
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−56500.9	−1.93879	−0.969394	−	0.245510i	$-0.921045\pi$
$947$	−56500.9	−1.93879	−0.969394	+	0.245510i	$0.921045\pi$
$948$	0	0
$949$	9451.84	0.323308
$950$	0	0
$951$	16826.4	0.573746
$952$	0	0
$953$	36669.1	1.24641	0.623204	−	0.782059i	$-0.285831\pi$
$953$	36669.1	1.24641	0.623204	+	0.782059i	$0.285831\pi$
$954$	0	0
$955$	70157.5	2.37722
$956$	0	0
$957$	−38186.6	−1.28986
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29683.5	−0.996392
$962$	0	0
$963$	−6414.92	−0.214660
$964$	0	0
$965$	51346.8	1.71286
$966$	0	0
$967$	12012.4	0.399474	0.199737	−	0.979850i	$-0.435991\pi$
$967$	12012.4	0.399474	0.199737	+	0.979850i	$0.435991\pi$
$968$	0	0
$969$	−4593.58	−0.152288
$970$	0	0
$971$	37296.2	1.23264	0.616320	−	0.787496i	$-0.288623\pi$
$971$	37296.2	1.23264	0.616320	+	0.787496i	$0.288623\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−43611.8	−1.43251
$976$	0	0
$977$	9788.45	0.320532	0.160266	−	0.987074i	$-0.448765\pi$
$977$	9788.45	0.320532	0.160266	+	0.987074i	$0.448765\pi$
$978$	0	0
$979$	5200.30	0.169767
$980$	0	0
$981$	9376.63	0.305171
$982$	0	0
$983$	4620.40	0.149917	0.0749583	−	0.997187i	$-0.476118\pi$
$983$	4620.40	0.149917	0.0749583	+	0.997187i	$0.476118\pi$
$984$	0	0
$985$	−3046.29	−0.0985410
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6920.73	0.222514
$990$	0	0
$991$	−40365.3	−1.29389	−0.646946	−	0.762536i	$-0.723954\pi$
$991$	−40365.3	−1.29389	−0.646946	+	0.762536i	$0.723954\pi$
$992$	0	0
$993$	−19828.5	−0.633673
$994$	0	0
$995$	39167.1	1.24792
$996$	0	0
$997$	−2994.35	−0.0951175	−0.0475587	−	0.998868i	$-0.515144\pi$
$997$	−2994.35	−0.0951175	−0.0475587	+	0.998868i	$0.515144\pi$
$998$	0	0
$999$	6641.64	0.210343

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.cm.1.4		4
4.3	odd	2		1176.4.a.bd.1.4		4
7.2	even	3		336.4.q.m.193.1		8
7.4	even	3		336.4.q.m.289.1		8
7.6	odd	2		2352.4.a.cp.1.1		4
28.11	odd	6		168.4.q.f.121.1	yes	8
28.23	odd	6		168.4.q.f.25.1	✓	8
28.27	even	2		1176.4.a.ba.1.1		4
84.11	even	6		504.4.s.j.289.4		8
84.23	even	6		504.4.s.j.361.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.f.25.1	✓	8	28.23	odd	6
168.4.q.f.121.1	yes	8	28.11	odd	6
336.4.q.m.193.1		8	7.2	even	3
336.4.q.m.289.1		8	7.4	even	3
504.4.s.j.289.4		8	84.11	even	6
504.4.s.j.361.4		8	84.23	even	6
1176.4.a.ba.1.1		4	28.27	even	2
1176.4.a.bd.1.4		4	4.3	odd	2
2352.4.a.cm.1.4		4	1.1	even	1	trivial
2352.4.a.cp.1.1		4	7.6	odd	2

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:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +18.9580 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +18.9580 q^{5} +9.00000 q^{9} -54.7308 q^{11} +62.0173 q^{13} -56.8741 q^{15} +122.439 q^{17} +12.5058 q^{19} +74.4391 q^{23} +234.406 q^{25} -27.0000 q^{27} -232.572 q^{29} -10.3677 q^{31} +164.192 q^{33} -245.987 q^{37} -186.052 q^{39} +238.653 q^{41} +92.9718 q^{43} +170.622 q^{45} +485.645 q^{47} -367.317 q^{51} -378.557 q^{53} -1037.59 q^{55} -37.5173 q^{57} +182.784 q^{59} +396.470 q^{61} +1175.73 q^{65} +261.239 q^{67} -223.317 q^{69} +874.523 q^{71} +152.406 q^{73} -703.219 q^{75} -573.357 q^{79} +81.0000 q^{81} -317.754 q^{83} +2321.20 q^{85} +697.717 q^{87} -95.0160 q^{89} +31.1032 q^{93} +237.084 q^{95} -1608.78 q^{97} -492.577 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 4 q^{5} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 + 4 * q^5 + 36 * q^9	\( 4 q - 12 q^{3} + 4 q^{5} + 36 q^{9} - 14 q^{11} + 22 q^{13} - 12 q^{15} + 96 q^{17} + 26 q^{19} - 96 q^{23} + 110 q^{25} - 108 q^{27} - 76 q^{29} - 238 q^{31} + 42 q^{33} + 562 q^{37} - 66 q^{39} + 428 q^{41} + 258 q^{43} + 36 q^{45} + 80 q^{47} - 288 q^{51} - 1476 q^{55} - 78 q^{57} - 262 q^{59} - 276 q^{61} + 2196 q^{65} - 150 q^{67} + 288 q^{69} + 848 q^{71} - 218 q^{73} - 330 q^{75} - 1762 q^{79} + 324 q^{81} - 3450 q^{83} + 1452 q^{85} + 228 q^{87} - 344 q^{89} + 714 q^{93} - 2004 q^{95} - 622 q^{97} - 126 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 + 4 * q^5 + 36 * q^9 - 14 * q^11 + 22 * q^13 - 12 * q^15 + 96 * q^17 + 26 * q^19 - 96 * q^23 + 110 * q^25 - 108 * q^27 - 76 * q^29 - 238 * q^31 + 42 * q^33 + 562 * q^37 - 66 * q^39 + 428 * q^41 + 258 * q^43 + 36 * q^45 + 80 * q^47 - 288 * q^51 - 1476 * q^55 - 78 * q^57 - 262 * q^59 - 276 * q^61 + 2196 * q^65 - 150 * q^67 + 288 * q^69 + 848 * q^71 - 218 * q^73 - 330 * q^75 - 1762 * q^79 + 324 * q^81 - 3450 * q^83 + 1452 * q^85 + 228 * q^87 - 344 * q^89 + 714 * q^93 - 2004 * q^95 - 622 * q^97 - 126 * q^99

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