LMFDB - Embedded newform 1176.4.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,4,Mod(1,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, 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("1176.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1176.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:	$69.3862461668$
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			1.1
Root			$13.2349$ of defining polynomial
Character	$\chi$	$=$	1176.1

$q$-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	$-0.822090\pi$
$5$	−18.9580	−1.69566	−0.847828	+	0.530271i	$0.822090\pi$
$6$	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	$-0.230011\pi$
$11$	54.7308	1.50018	0.750089	+	0.661337i	$0.230011\pi$
$12$	0	0
$13$	−62.0173	−1.32312	−0.661558	−	0.749894i	$-0.730104\pi$
$13$	−62.0173	−1.32312	−0.661558	+	0.749894i	$0.730104\pi$
$14$	0	0
$15$	56.8741	0.978988
$16$	0	0
$17$	−122.439	−1.74681	−0.873407	−	0.486991i	$-0.838095\pi$
$17$	−122.439	−1.74681	−0.873407	+	0.486991i	$0.838095\pi$
$18$	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	$-0.609556\pi$
$23$	−74.4391	−0.674853	−0.337427	+	0.941352i	$0.609556\pi$
$24$	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	$-0.650192\pi$
$41$	−238.653	−0.909056	−0.454528	+	0.890733i	$0.650192\pi$
$42$	0	0
$43$	−92.9718	−0.329722	−0.164861	−	0.986317i	$-0.552718\pi$
$43$	−92.9718	−0.329722	−0.164861	+	0.986317i	$0.552718\pi$
$44$	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	$-0.636599\pi$
$61$	−396.470	−0.832177	−0.416088	+	0.909324i	$0.636599\pi$
$62$	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	$-0.576549\pi$
$67$	−261.239	−0.476350	−0.238175	+	0.971222i	$0.576549\pi$
$68$	0	0
$69$	223.317	0.389627
$70$	0	0
$71$	−874.523	−1.46179	−0.730893	−	0.682492i	$-0.760896\pi$
$71$	−874.523	−1.46179	−0.730893	+	0.682492i	$0.760896\pi$
$72$	0	0
$73$	−152.406	−0.244354	−0.122177	−	0.992508i	$-0.538988\pi$
$73$	−152.406	−0.244354	−0.122177	+	0.992508i	$0.538988\pi$
$74$	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	$-0.366130\pi$
$79$	573.357	0.816554	0.408277	+	0.912858i	$0.366130\pi$
$80$	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	$-0.481980\pi$
$89$	95.0160	0.113165	0.0565824	+	0.998398i	$0.481980\pi$
$90$	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	$-0.181383\pi$
$97$	1608.78	1.68398	0.841992	+	0.539490i	$0.181383\pi$
$98$	0	0
$99$	492.577	0.500059
$100$	0	0
$101$	783.043	0.771442	0.385721	−	0.922615i	$-0.373953\pi$
$101$	783.043	0.771442	0.385721	+	0.922615i	$0.373953\pi$
$102$	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	$-0.395648\pi$
$107$	712.769	0.643981	0.321991	+	0.946743i	$0.395648\pi$
$108$	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	$-0.624798\pi$
$127$	−1093.73	−0.764194	−0.382097	+	0.924122i	$0.624798\pi$
$128$	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	$-0.298637\pi$
$151$	2194.13	1.18249	0.591245	+	0.806492i	$0.298637\pi$
$152$	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	$-0.482126\pi$
$157$	220.818	0.112250	0.0561248	+	0.998424i	$0.482126\pi$
$158$	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	$-0.384536\pi$
$163$	1476.86	0.709674	0.354837	+	0.934928i	$0.384536\pi$
$164$	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	$-0.652007\pi$
$173$	−2091.61	−0.919201	−0.459601	+	0.888126i	$0.652007\pi$
$174$	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	$-0.337943\pi$
$179$	2334.54	0.974813	0.487406	+	0.873175i	$0.337943\pi$
$180$	0	0
$181$	−1758.40	−0.722105	−0.361053	−	0.932545i	$-0.617583\pi$
$181$	−1758.40	−0.722105	−0.361053	+	0.932545i	$0.617583\pi$
$182$	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	$-0.747251\pi$
$191$	−3700.68	−1.40195	−0.700973	+	0.713188i	$0.747251\pi$
$192$	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	$-0.552535\pi$
$211$	−1007.12	−0.328592	−0.164296	+	0.986411i	$0.552535\pi$
$212$	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	$-0.619245\pi$
$229$	−2536.11	−0.731837	−0.365918	+	0.930647i	$0.619245\pi$
$230$	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	$-0.380300\pi$
$239$	2713.85	0.734495	0.367248	+	0.930123i	$0.380300\pi$
$240$	0	0
$241$	4174.59	1.11580	0.557902	−	0.829907i	$-0.311606\pi$
$241$	4174.59	1.11580	0.557902	+	0.829907i	$0.311606\pi$
$242$	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	$-0.690108\pi$
$257$	−4633.91	−1.12473	−0.562365	+	0.826889i	$0.690108\pi$
$258$	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	$-0.374239\pi$
$263$	3283.23	0.769781	0.384891	+	0.922962i	$0.374239\pi$
$264$	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	$-0.471138\pi$
$269$	798.985	0.181097	0.0905483	+	0.995892i	$0.471138\pi$
$270$	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	$-0.637839\pi$
$293$	−4209.15	−0.839252	−0.419626	+	0.907697i	$0.637839\pi$
$294$	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	$-0.352352\pi$
$313$	4954.91	0.894786	0.447393	+	0.894337i	$0.352352\pi$
$314$	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	$-0.684906\pi$
$331$	−6609.49	−1.09755	−0.548777	+	0.835969i	$0.684906\pi$
$332$	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	$-0.654857\pi$
$347$	−6044.15	−0.935064	−0.467532	+	0.883976i	$0.654857\pi$
$348$	0	0
$349$	5487.93	0.841726	0.420863	−	0.907124i	$-0.361727\pi$
$349$	5487.93	0.841726	0.420863	+	0.907124i	$0.361727\pi$
$350$	0	0
$351$	1674.47	0.254634
$352$	0	0
$353$	−6880.15	−1.03738	−0.518688	−	0.854964i	$-0.673579\pi$
$353$	−6880.15	−1.03738	−0.518688	+	0.854964i	$0.673579\pi$
$354$	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	$-0.593487\pi$
$359$	−3938.34	−0.578990	−0.289495	+	0.957180i	$0.593487\pi$
$360$	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	$-0.528278\pi$
$379$	−1309.25	−0.177445	−0.0887225	+	0.996056i	$0.528278\pi$
$380$	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	$-0.759204\pi$
$397$	−11505.4	−1.45451	−0.727254	+	0.686368i	$0.759204\pi$
$398$	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	$-0.404386\pi$
$409$	4894.83	0.591769	0.295885	+	0.955224i	$0.404386\pi$
$410$	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	$-0.406276\pi$
$431$	5193.41	0.580413	0.290206	+	0.956964i	$0.406276\pi$
$432$	0	0
$433$	−6314.17	−0.700785	−0.350392	−	0.936603i	$-0.613952\pi$
$433$	−6314.17	−0.700785	−0.350392	+	0.936603i	$0.613952\pi$
$434$	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	$-0.654616\pi$
$443$	−8706.11	−0.933725	−0.466862	+	0.884330i	$0.654616\pi$
$444$	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	$-0.761967\pi$
$461$	−14514.2	−1.46637	−0.733184	+	0.680030i	$0.761967\pi$
$462$	0	0
$463$	9971.00	1.00085	0.500423	−	0.865781i	$-0.333178\pi$
$463$	9971.00	1.00085	0.500423	+	0.865781i	$0.333178\pi$
$464$	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	$-0.716817\pi$
$487$	−13534.7	−1.25937	−0.629687	+	0.776849i	$0.716817\pi$
$488$	0	0
$489$	−4430.59	−0.409730
$490$	0	0
$491$	−8693.29	−0.799028	−0.399514	−	0.916727i	$-0.630821\pi$
$491$	−8693.29	−0.799028	−0.399514	+	0.916727i	$0.630821\pi$
$492$	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	$-0.442317\pi$
$499$	4017.93	0.360455	0.180228	+	0.983625i	$0.442317\pi$
$500$	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	$-0.631780\pi$
$509$	−9239.10	−0.804550	−0.402275	+	0.915519i	$0.631780\pi$
$510$	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	$-0.716763\pi$
$521$	−14973.4	−1.25911	−0.629555	+	0.776956i	$0.716763\pi$
$522$	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	$-0.575182\pi$
$547$	−5987.25	−0.468001	−0.234000	+	0.972237i	$0.575182\pi$
$548$	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	$-0.300261\pi$
$571$	16021.8	1.17424	0.587122	+	0.809498i	$0.300261\pi$
$572$	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	$-0.302552\pi$
$577$	16113.1	1.16256	0.581279	+	0.813704i	$0.302552\pi$
$578$	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	$-0.184420\pi$
$593$	24167.8	1.67361	0.836807	+	0.547499i	$0.184420\pi$
$594$	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	$-0.491082\pi$
$599$	821.331	0.0560245	0.0280122	+	0.999608i	$0.491082\pi$
$600$	0	0
$601$	14674.7	0.995996	0.497998	−	0.867178i	$-0.334069\pi$
$601$	14674.7	0.995996	0.497998	+	0.867178i	$0.334069\pi$
$602$	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	$-0.193660\pi$
$631$	26012.7	1.64113	0.820563	+	0.571556i	$0.193660\pi$
$632$	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	$-0.634794\pi$
$659$	−13903.4	−0.821851	−0.410926	+	0.911669i	$0.634794\pi$
$660$	0	0
$661$	−6306.61	−0.371102	−0.185551	−	0.982635i	$-0.559407\pi$
$661$	−6306.61	−0.371102	−0.185551	+	0.982635i	$0.559407\pi$
$662$	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	$-0.156055\pi$
$677$	31080.3	1.76442	0.882209	+	0.470858i	$0.156055\pi$
$678$	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	$-0.201843\pi$
$683$	28759.5	1.61120	0.805601	+	0.592458i	$0.201843\pi$
$684$	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	$-0.424762\pi$
$733$	9294.38	0.468344	0.234172	+	0.972195i	$0.424762\pi$
$734$	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	$-0.349552\pi$
$739$	18291.2	0.910490	0.455245	+	0.890366i	$0.349552\pi$
$740$	0	0
$741$	2326.72	0.115350
$742$	0	0
$743$	14742.4	0.727921	0.363960	−	0.931414i	$-0.381424\pi$
$743$	14742.4	0.727921	0.363960	+	0.931414i	$0.381424\pi$
$744$	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	$-0.256947\pi$
$751$	28463.4	1.38301	0.691507	+	0.722370i	$0.256947\pi$
$752$	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	$-0.891749\pi$
$761$	−39581.6	−1.88546	−0.942728	+	0.333564i	$0.891749\pi$
$762$	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	$-0.579797\pi$
$769$	−10580.3	−0.496146	−0.248073	+	0.968741i	$0.579797\pi$
$770$	0	0
$771$	13901.7	0.649363
$772$	0	0
$773$	−25922.2	−1.20615	−0.603077	−	0.797683i	$-0.706059\pi$
$773$	−25922.2	−1.20615	−0.603077	+	0.797683i	$0.706059\pi$
$774$	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	$-0.455465\pi$
$797$	6275.52	0.278909	0.139454	+	0.990228i	$0.455465\pi$
$798$	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	$-0.813897\pi$
$823$	−39377.1	−1.66780	−0.833901	+	0.551915i	$0.813897\pi$
$824$	0	0
$825$	−38487.8	−1.62421
$826$	0	0
$827$	−30916.3	−1.29996	−0.649978	−	0.759953i	$-0.725222\pi$
$827$	−30916.3	−1.29996	−0.649978	+	0.759953i	$0.725222\pi$
$828$	0	0
$829$	2543.33	0.106554	0.0532771	−	0.998580i	$-0.483033\pi$
$829$	2543.33	0.106554	0.0532771	+	0.998580i	$0.483033\pi$
$830$	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	$-0.818977\pi$
$853$	−41983.2	−1.68520	−0.842601	+	0.538538i	$0.818977\pi$
$854$	0	0
$855$	−2133.76	−0.0853485
$856$	0	0
$857$	11854.3	0.472504	0.236252	−	0.971692i	$-0.424081\pi$
$857$	11854.3	0.472504	0.236252	+	0.971692i	$0.424081\pi$
$858$	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	$-0.140908\pi$
$863$	45817.0	1.80722	0.903609	+	0.428358i	$0.140908\pi$
$864$	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	$-0.813230\pi$
$881$	−43551.6	−1.66548	−0.832742	+	0.553661i	$0.813230\pi$
$882$	0	0
$883$	40645.1	1.54906	0.774528	−	0.632540i	$-0.217988\pi$
$883$	40645.1	1.54906	0.774528	+	0.632540i	$0.217988\pi$
$884$	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	$-0.119843\pi$
$907$	50804.7	1.85992	0.929958	+	0.367666i	$0.119843\pi$
$908$	0	0
$909$	7047.38	0.257147
$910$	0	0
$911$	28738.3	1.04516	0.522582	−	0.852589i	$-0.324969\pi$
$911$	28738.3	1.04516	0.522582	+	0.852589i	$0.324969\pi$
$912$	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	$-0.940311\pi$
$919$	−54742.2	−1.96494	−0.982470	+	0.186420i	$0.940311\pi$
$920$	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	$-0.781246\pi$
$929$	−43775.8	−1.54600	−0.773002	+	0.634403i	$0.781246\pi$
$930$	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	$-0.709524\pi$
$937$	−35090.9	−1.22345	−0.611724	+	0.791071i	$0.709524\pi$
$938$	0	0
$939$	−14864.7	−0.516605
$940$	0	0
$941$	31784.8	1.10112	0.550560	−	0.834795i	$-0.314414\pi$
$941$	31784.8	1.10112	0.550560	+	0.834795i	$0.314414\pi$
$942$	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	$-0.0789554\pi$
$947$	56500.9	1.93879	0.969394	+	0.245510i	$0.0789554\pi$
$948$	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	$-0.564009\pi$
$967$	−12012.4	−0.399474	−0.199737	+	0.979850i	$0.564009\pi$
$968$	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	$-0.276046\pi$
$991$	40365.3	1.29389	0.646946	+	0.762536i	$0.276046\pi$
$992$	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	$-0.484856\pi$
$997$	2994.35	0.0951175	0.0475587	+	0.998868i	$0.484856\pi$
$998$	0	0
$999$	6641.64	0.210343

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.4.a.ba.1.1		4
4.3	odd	2		2352.4.a.cp.1.1		4
7.3	odd	6		168.4.q.f.121.1	yes	8
7.5	odd	6		168.4.q.f.25.1	✓	8
7.6	odd	2		1176.4.a.bd.1.4		4
21.5	even	6		504.4.s.j.361.4		8
21.17	even	6		504.4.s.j.289.4		8
28.3	even	6		336.4.q.m.289.1		8
28.19	even	6		336.4.q.m.193.1		8
28.27	even	2		2352.4.a.cm.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.q.f.25.1	✓	8	7.5	odd	6
168.4.q.f.121.1	yes	8	7.3	odd	6
336.4.q.m.193.1		8	28.19	even	6
336.4.q.m.289.1		8	28.3	even	6
504.4.s.j.289.4		8	21.17	even	6
504.4.s.j.361.4		8	21.5	even	6
1176.4.a.ba.1.1		4	1.1	even	1	trivial
1176.4.a.bd.1.4		4	7.6	odd	2
2352.4.a.cm.1.4		4	28.27	even	2
2352.4.a.cp.1.1		4	4.3	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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1176.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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