LMFDB - Embedded newform 2352.4.a.cn.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:	4.4.391168.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 40x^{2} + 382 $ x^4 - 40*x^2 + 382
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	no (minimal twist has level 1176)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.96955$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} +14.6418 q^{5} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +14.6418 q^{5} +9.00000 q^{9} +11.4490 q^{11} +20.8071 q^{13} -43.9254 q^{15} -18.8331 q^{17} +91.0178 q^{19} -28.9336 q^{23} +89.3825 q^{25} -27.0000 q^{27} +281.235 q^{29} -276.010 q^{31} -34.3469 q^{33} +250.567 q^{37} -62.4214 q^{39} -12.1422 q^{41} -65.2121 q^{43} +131.776 q^{45} +199.886 q^{47} +56.4992 q^{51} -64.1260 q^{53} +167.633 q^{55} -273.054 q^{57} -69.0791 q^{59} +656.210 q^{61} +304.654 q^{65} +419.714 q^{67} +86.8007 q^{69} -1022.46 q^{71} +103.280 q^{73} -268.148 q^{75} -260.616 q^{79} +81.0000 q^{81} +879.313 q^{83} -275.750 q^{85} -843.705 q^{87} +1480.70 q^{89} +828.029 q^{93} +1332.67 q^{95} +177.673 q^{97} +103.041 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 8 q^{5} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 + 8 * q^5 + 36 * q^9	$ 4 q - 12 q^{3} + 8 q^{5} + 36 q^{9} - 40 q^{11} + 48 q^{13} - 24 q^{15} + 72 q^{17} - 32 q^{19} - 8 q^{23} + 164 q^{25} - 108 q^{27} + 144 q^{29} - 48 q^{31} + 120 q^{33} + 48 q^{37} - 144 q^{39} + 72 q^{41} - 512 q^{43} + 72 q^{45} - 160 q^{47} - 216 q^{51} + 536 q^{53} + 336 q^{55} + 96 q^{57} - 240 q^{59} + 896 q^{61} - 136 q^{65} - 1088 q^{67} + 24 q^{69} - 1288 q^{71} + 1488 q^{73} - 492 q^{75} - 416 q^{79} + 324 q^{81} + 112 q^{83} - 1512 q^{85} - 432 q^{87} + 3160 q^{89} + 144 q^{93} + 240 q^{95} + 2384 q^{97} - 360 q^{99}+O(q^{100}) $ 4 * q - 12 * q^3 + 8 * q^5 + 36 * q^9 - 40 * q^11 + 48 * q^13 - 24 * q^15 + 72 * q^17 - 32 * q^19 - 8 * q^23 + 164 * q^25 - 108 * q^27 + 144 * q^29 - 48 * q^31 + 120 * q^33 + 48 * q^37 - 144 * q^39 + 72 * q^41 - 512 * q^43 + 72 * q^45 - 160 * q^47 - 216 * q^51 + 536 * q^53 + 336 * q^55 + 96 * q^57 - 240 * q^59 + 896 * q^61 - 136 * q^65 - 1088 * q^67 + 24 * q^69 - 1288 * q^71 + 1488 * q^73 - 492 * q^75 - 416 * q^79 + 324 * q^81 + 112 * q^83 - 1512 * q^85 - 432 * q^87 + 3160 * q^89 + 144 * q^93 + 240 * q^95 + 2384 * q^97 - 360 * 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
$5$	14.6418	1.30960	0.654802	−	0.755801i	$-0.272752\pi$
$5$	14.6418	1.30960	0.654802	+	0.755801i	$0.272752\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	11.4490	0.313817	0.156909	−	0.987613i	$-0.449847\pi$
$11$	11.4490	0.313817	0.156909	+	0.987613i	$0.449847\pi$
$12$	0	0
$13$	20.8071	0.443913	0.221956	−	0.975057i	$-0.428756\pi$
$13$	20.8071	0.443913	0.221956	+	0.975057i	$0.428756\pi$
$14$	0	0
$15$	−43.9254	−0.756100
$16$	0	0
$17$	−18.8331	−0.268688	−0.134344	−	0.990935i	$-0.542893\pi$
$17$	−18.8331	−0.268688	−0.134344	+	0.990935i	$0.542893\pi$
$18$	0	0
$19$	91.0178	1.09900	0.549498	−	0.835495i	$-0.314819\pi$
$19$	91.0178	1.09900	0.549498	+	0.835495i	$0.314819\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−28.9336	−0.262307	−0.131154	−	0.991362i	$-0.541868\pi$
$23$	−28.9336	−0.262307	−0.131154	+	0.991362i	$0.541868\pi$
$24$	0	0
$25$	89.3825	0.715060
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	281.235	1.80083	0.900414	−	0.435034i	$-0.143264\pi$
$29$	281.235	1.80083	0.900414	+	0.435034i	$0.143264\pi$
$30$	0	0
$31$	−276.010	−1.59912	−0.799561	−	0.600585i	$-0.794935\pi$
$31$	−276.010	−1.59912	−0.799561	+	0.600585i	$0.794935\pi$
$32$	0	0
$33$	−34.3469	−0.181182
$34$	0	0
$35$	0	0
$36$	0	0
$37$	250.567	1.11332	0.556662	−	0.830739i	$-0.312082\pi$
$37$	250.567	1.11332	0.556662	+	0.830739i	$0.312082\pi$
$38$	0	0
$39$	−62.4214	−0.256293
$40$	0	0
$41$	−12.1422	−0.0462512	−0.0231256	−	0.999733i	$-0.507362\pi$
$41$	−12.1422	−0.0462512	−0.0231256	+	0.999733i	$0.507362\pi$
$42$	0	0
$43$	−65.2121	−0.231273	−0.115637	−	0.993292i	$-0.536891\pi$
$43$	−65.2121	−0.231273	−0.115637	+	0.993292i	$0.536891\pi$
$44$	0	0
$45$	131.776	0.436534
$46$	0	0
$47$	199.886	0.620350	0.310175	−	0.950680i	$-0.399612\pi$
$47$	199.886	0.620350	0.310175	+	0.950680i	$0.399612\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	56.4992	0.155127
$52$	0	0
$53$	−64.1260	−0.166196	−0.0830980	−	0.996541i	$-0.526481\pi$
$53$	−64.1260	−0.166196	−0.0830980	+	0.996541i	$0.526481\pi$
$54$	0	0
$55$	167.633	0.410976
$56$	0	0
$57$	−273.054	−0.634506
$58$	0	0
$59$	−69.0791	−0.152429	−0.0762147	−	0.997091i	$-0.524283\pi$
$59$	−69.0791	−0.152429	−0.0762147	+	0.997091i	$0.524283\pi$
$60$	0	0
$61$	656.210	1.37736	0.688681	−	0.725064i	$-0.258190\pi$
$61$	656.210	1.37736	0.688681	+	0.725064i	$0.258190\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	304.654	0.581349
$66$	0	0
$67$	419.714	0.765317	0.382658	−	0.923890i	$-0.375009\pi$
$67$	419.714	0.765317	0.382658	+	0.923890i	$0.375009\pi$
$68$	0	0
$69$	86.8007	0.151443
$70$	0	0
$71$	−1022.46	−1.70907	−0.854536	−	0.519393i	$-0.826158\pi$
$71$	−1022.46	−1.70907	−0.854536	+	0.519393i	$0.826158\pi$
$72$	0	0
$73$	103.280	0.165590	0.0827948	−	0.996567i	$-0.473615\pi$
$73$	103.280	0.165590	0.0827948	+	0.996567i	$0.473615\pi$
$74$	0	0
$75$	−268.148	−0.412840
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−260.616	−0.371159	−0.185579	−	0.982629i	$-0.559416\pi$
$79$	−260.616	−0.371159	−0.185579	+	0.982629i	$0.559416\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	879.313	1.16286	0.581428	−	0.813598i	$-0.302494\pi$
$83$	879.313	1.16286	0.581428	+	0.813598i	$0.302494\pi$
$84$	0	0
$85$	−275.750	−0.351874
$86$	0	0
$87$	−843.705	−1.03971
$88$	0	0
$89$	1480.70	1.76353	0.881763	−	0.471692i	$-0.156357\pi$
$89$	1480.70	1.76353	0.881763	+	0.471692i	$0.156357\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	828.029	0.923254
$94$	0	0
$95$	1332.67	1.43925
$96$	0	0
$97$	177.673	0.185979	0.0929895	−	0.995667i	$-0.470358\pi$
$97$	177.673	0.185979	0.0929895	+	0.995667i	$0.470358\pi$
$98$	0	0
$99$	103.041	0.104606
$100$	0	0
$101$	−872.534	−0.859608	−0.429804	−	0.902922i	$-0.641417\pi$
$101$	−872.534	−0.859608	−0.429804	+	0.902922i	$0.641417\pi$
$102$	0	0
$103$	−942.292	−0.901425	−0.450713	−	0.892669i	$-0.648830\pi$
$103$	−942.292	−0.901425	−0.450713	+	0.892669i	$0.648830\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1162.43	1.05024	0.525121	−	0.851028i	$-0.324020\pi$
$107$	1162.43	1.05024	0.525121	+	0.851028i	$0.324020\pi$
$108$	0	0
$109$	−35.2015	−0.0309330	−0.0154665	−	0.999880i	$-0.504923\pi$
$109$	−35.2015	−0.0309330	−0.0154665	+	0.999880i	$0.504923\pi$
$110$	0	0
$111$	−751.701	−0.642777
$112$	0	0
$113$	−420.632	−0.350175	−0.175087	−	0.984553i	$-0.556021\pi$
$113$	−420.632	−0.350175	−0.175087	+	0.984553i	$0.556021\pi$
$114$	0	0
$115$	−423.640	−0.343519
$116$	0	0
$117$	187.264	0.147971
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1199.92	−0.901519
$122$	0	0
$123$	36.4267	0.0267032
$124$	0	0
$125$	−521.504	−0.373158
$126$	0	0
$127$	722.609	0.504891	0.252445	−	0.967611i	$-0.418765\pi$
$127$	722.609	0.504891	0.252445	+	0.967611i	$0.418765\pi$
$128$	0	0
$129$	195.636	0.133526
$130$	0	0
$131$	−799.224	−0.533042	−0.266521	−	0.963829i	$-0.585874\pi$
$131$	−799.224	−0.533042	−0.266521	+	0.963829i	$0.585874\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−395.329	−0.252033
$136$	0	0
$137$	−2951.13	−1.84038	−0.920191	−	0.391470i	$-0.871967\pi$
$137$	−2951.13	−1.84038	−0.920191	+	0.391470i	$0.871967\pi$
$138$	0	0
$139$	102.191	0.0623579	0.0311789	−	0.999514i	$-0.490074\pi$
$139$	102.191	0.0623579	0.0311789	+	0.999514i	$0.490074\pi$
$140$	0	0
$141$	−599.659	−0.358159
$142$	0	0
$143$	238.220	0.139307
$144$	0	0
$145$	4117.79	2.35837
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1798.70	0.988963	0.494482	−	0.869188i	$-0.335358\pi$
$149$	1798.70	0.988963	0.494482	+	0.869188i	$0.335358\pi$
$150$	0	0
$151$	3196.81	1.72286	0.861432	−	0.507873i	$-0.169568\pi$
$151$	3196.81	1.72286	0.861432	+	0.507873i	$0.169568\pi$
$152$	0	0
$153$	−169.498	−0.0895626
$154$	0	0
$155$	−4041.28	−2.09422
$156$	0	0
$157$	3476.27	1.76711	0.883556	−	0.468326i	$-0.155143\pi$
$157$	3476.27	1.76711	0.883556	+	0.468326i	$0.155143\pi$
$158$	0	0
$159$	192.378	0.0959533
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3789.94	−1.82117	−0.910587	−	0.413318i	$-0.864370\pi$
$163$	−3789.94	−1.82117	−0.910587	+	0.413318i	$0.864370\pi$
$164$	0	0
$165$	−502.900	−0.237277
$166$	0	0
$167$	−2711.44	−1.25639	−0.628195	−	0.778056i	$-0.716206\pi$
$167$	−2711.44	−1.25639	−0.628195	+	0.778056i	$0.716206\pi$
$168$	0	0
$169$	−1764.06	−0.802942
$170$	0	0
$171$	819.161	0.366332
$172$	0	0
$173$	−2504.48	−1.10065	−0.550324	−	0.834951i	$-0.685496\pi$
$173$	−2504.48	−1.10065	−0.550324	+	0.834951i	$0.685496\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	207.237	0.0880052
$178$	0	0
$179$	2089.31	0.872413	0.436207	−	0.899847i	$-0.356322\pi$
$179$	2089.31	0.872413	0.436207	+	0.899847i	$0.356322\pi$
$180$	0	0
$181$	4081.29	1.67602	0.838010	−	0.545655i	$-0.183719\pi$
$181$	4081.29	1.67602	0.838010	+	0.545655i	$0.183719\pi$
$182$	0	0
$183$	−1968.63	−0.795220
$184$	0	0
$185$	3668.75	1.45801
$186$	0	0
$187$	−215.619	−0.0843189
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2056.23	−0.778970	−0.389485	−	0.921033i	$-0.627347\pi$
$191$	−2056.23	−0.778970	−0.389485	+	0.921033i	$0.627347\pi$
$192$	0	0
$193$	−1489.48	−0.555520	−0.277760	−	0.960650i	$-0.589592\pi$
$193$	−1489.48	−0.555520	−0.277760	+	0.960650i	$0.589592\pi$
$194$	0	0
$195$	−913.963	−0.335642
$196$	0	0
$197$	1591.66	0.575640	0.287820	−	0.957684i	$-0.407069\pi$
$197$	1591.66	0.575640	0.287820	+	0.957684i	$0.407069\pi$
$198$	0	0
$199$	4303.32	1.53294	0.766468	−	0.642282i	$-0.222012\pi$
$199$	4303.32	1.53294	0.766468	+	0.642282i	$0.222012\pi$
$200$	0	0
$201$	−1259.14	−0.441856
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−177.784	−0.0605707
$206$	0	0
$207$	−260.402	−0.0874358
$208$	0	0
$209$	1042.06	0.344884
$210$	0	0
$211$	−2209.01	−0.720733	−0.360367	−	0.932811i	$-0.617348\pi$
$211$	−2209.01	−0.720733	−0.360367	+	0.932811i	$0.617348\pi$
$212$	0	0
$213$	3067.39	0.986733
$214$	0	0
$215$	−954.824	−0.302876
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−309.841	−0.0956033
$220$	0	0
$221$	−391.863	−0.119274
$222$	0	0
$223$	4731.91	1.42095	0.710476	−	0.703721i	$-0.248480\pi$
$223$	4731.91	1.42095	0.710476	+	0.703721i	$0.248480\pi$
$224$	0	0
$225$	804.443	0.238353
$226$	0	0
$227$	−622.308	−0.181956	−0.0909780	−	0.995853i	$-0.528999\pi$
$227$	−622.308	−0.181956	−0.0909780	+	0.995853i	$0.528999\pi$
$228$	0	0
$229$	530.619	0.153119	0.0765595	−	0.997065i	$-0.475606\pi$
$229$	530.619	0.153119	0.0765595	+	0.997065i	$0.475606\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3189.28	0.896723	0.448362	−	0.893852i	$-0.352008\pi$
$233$	3189.28	0.896723	0.448362	+	0.893852i	$0.352008\pi$
$234$	0	0
$235$	2926.70	0.812412
$236$	0	0
$237$	781.847	0.214289
$238$	0	0
$239$	−1515.36	−0.410127	−0.205064	−	0.978749i	$-0.565740\pi$
$239$	−1515.36	−0.410127	−0.205064	+	0.978749i	$0.565740\pi$
$240$	0	0
$241$	1235.38	0.330200	0.165100	−	0.986277i	$-0.447205\pi$
$241$	1235.38	0.330200	0.165100	+	0.986277i	$0.447205\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1893.82	0.487858
$248$	0	0
$249$	−2637.94	−0.671376
$250$	0	0
$251$	−2165.31	−0.544514	−0.272257	−	0.962225i	$-0.587770\pi$
$251$	−2165.31	−0.544514	−0.272257	+	0.962225i	$0.587770\pi$
$252$	0	0
$253$	−331.259	−0.0823166
$254$	0	0
$255$	827.251	0.203155
$256$	0	0
$257$	3947.43	0.958108	0.479054	−	0.877786i	$-0.340980\pi$
$257$	3947.43	0.958108	0.479054	+	0.877786i	$0.340980\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2531.11	0.600276
$262$	0	0
$263$	997.979	0.233985	0.116992	−	0.993133i	$-0.462675\pi$
$263$	997.979	0.233985	0.116992	+	0.993133i	$0.462675\pi$
$264$	0	0
$265$	−938.921	−0.217651
$266$	0	0
$267$	−4442.10	−1.01817
$268$	0	0
$269$	6029.26	1.36658	0.683291	−	0.730146i	$-0.260548\pi$
$269$	6029.26	1.36658	0.683291	+	0.730146i	$0.260548\pi$
$270$	0	0
$271$	−3119.65	−0.699280	−0.349640	−	0.936884i	$-0.613696\pi$
$271$	−3119.65	−0.699280	−0.349640	+	0.936884i	$0.613696\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1023.34	0.224398
$276$	0	0
$277$	−2003.80	−0.434645	−0.217323	−	0.976100i	$-0.569732\pi$
$277$	−2003.80	−0.434645	−0.217323	+	0.976100i	$0.569732\pi$
$278$	0	0
$279$	−2484.09	−0.533041
$280$	0	0
$281$	−8915.35	−1.89269	−0.946344	−	0.323162i	$-0.895254\pi$
$281$	−8915.35	−1.89269	−0.946344	+	0.323162i	$0.895254\pi$
$282$	0	0
$283$	3239.09	0.680366	0.340183	−	0.940359i	$-0.389511\pi$
$283$	3239.09	0.680366	0.340183	+	0.940359i	$0.389511\pi$
$284$	0	0
$285$	−3998.00	−0.830951
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4558.32	−0.927807
$290$	0	0
$291$	−533.019	−0.107375
$292$	0	0
$293$	5997.02	1.19573	0.597867	−	0.801595i	$-0.296015\pi$
$293$	5997.02	1.19573	0.597867	+	0.801595i	$0.296015\pi$
$294$	0	0
$295$	−1011.44	−0.199622
$296$	0	0
$297$	−309.122	−0.0603942
$298$	0	0
$299$	−602.025	−0.116442
$300$	0	0
$301$	0	0
$302$	0	0
$303$	2617.60	0.496295
$304$	0	0
$305$	9608.10	1.80380
$306$	0	0
$307$	9676.32	1.79888	0.899441	−	0.437042i	$-0.143974\pi$
$307$	9676.32	1.79888	0.899441	+	0.437042i	$0.143974\pi$
$308$	0	0
$309$	2826.88	0.520438
$310$	0	0
$311$	−1458.55	−0.265938	−0.132969	−	0.991120i	$-0.542451\pi$
$311$	−1458.55	−0.265938	−0.132969	+	0.991120i	$0.542451\pi$
$312$	0	0
$313$	10115.7	1.82674	0.913371	−	0.407128i	$-0.133470\pi$
$313$	10115.7	1.82674	0.913371	+	0.407128i	$0.133470\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8516.05	1.50886	0.754431	−	0.656379i	$-0.227913\pi$
$317$	8516.05	1.50886	0.754431	+	0.656379i	$0.227913\pi$
$318$	0	0
$319$	3219.85	0.565131
$320$	0	0
$321$	−3487.28	−0.606357
$322$	0	0
$323$	−1714.15	−0.295287
$324$	0	0
$325$	1859.80	0.317424
$326$	0	0
$327$	105.605	0.0178592
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2800.56	0.465053	0.232527	−	0.972590i	$-0.425301\pi$
$331$	2800.56	0.465053	0.232527	+	0.972590i	$0.425301\pi$
$332$	0	0
$333$	2255.10	0.371108
$334$	0	0
$335$	6145.37	1.00226
$336$	0	0
$337$	−6972.73	−1.12709	−0.563544	−	0.826086i	$-0.690563\pi$
$337$	−6972.73	−1.12709	−0.563544	+	0.826086i	$0.690563\pi$
$338$	0	0
$339$	1261.90	0.202173
$340$	0	0
$341$	−3160.02	−0.501832
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1270.92	0.198331
$346$	0	0
$347$	704.074	0.108924	0.0544621	−	0.998516i	$-0.482656\pi$
$347$	704.074	0.108924	0.0544621	+	0.998516i	$0.482656\pi$
$348$	0	0
$349$	−5350.53	−0.820652	−0.410326	−	0.911939i	$-0.634585\pi$
$349$	−5350.53	−0.820652	−0.410326	+	0.911939i	$0.634585\pi$
$350$	0	0
$351$	−561.793	−0.0854310
$352$	0	0
$353$	1230.18	0.185485	0.0927423	−	0.995690i	$-0.470437\pi$
$353$	1230.18	0.185485	0.0927423	+	0.995690i	$0.470437\pi$
$354$	0	0
$355$	−14970.7	−2.23820
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5402.57	0.794253	0.397126	−	0.917764i	$-0.370007\pi$
$359$	5402.57	0.794253	0.397126	+	0.917764i	$0.370007\pi$
$360$	0	0
$361$	1425.25	0.207792
$362$	0	0
$363$	3599.76	0.520492
$364$	0	0
$365$	1512.21	0.216857
$366$	0	0
$367$	13258.7	1.88583	0.942913	−	0.333040i	$-0.108074\pi$
$367$	13258.7	1.88583	0.942913	+	0.333040i	$0.108074\pi$
$368$	0	0
$369$	−109.280	−0.0154171
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6424.77	−0.891856	−0.445928	−	0.895069i	$-0.647126\pi$
$373$	−6424.77	−0.891856	−0.445928	+	0.895069i	$0.647126\pi$
$374$	0	0
$375$	1564.51	0.215443
$376$	0	0
$377$	5851.70	0.799410
$378$	0	0
$379$	11167.5	1.51355	0.756773	−	0.653678i	$-0.226775\pi$
$379$	11167.5	1.51355	0.756773	+	0.653678i	$0.226775\pi$
$380$	0	0
$381$	−2167.83	−0.291499
$382$	0	0
$383$	8758.23	1.16847	0.584236	−	0.811584i	$-0.301394\pi$
$383$	8758.23	1.16847	0.584236	+	0.811584i	$0.301394\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−586.909	−0.0770911
$388$	0	0
$389$	12054.8	1.57122	0.785610	−	0.618723i	$-0.212350\pi$
$389$	12054.8	1.57122	0.785610	+	0.618723i	$0.212350\pi$
$390$	0	0
$391$	544.908	0.0704788
$392$	0	0
$393$	2397.67	0.307752
$394$	0	0
$395$	−3815.88	−0.486071
$396$	0	0
$397$	−4650.10	−0.587863	−0.293932	−	0.955826i	$-0.594964\pi$
$397$	−4650.10	−0.587863	−0.293932	+	0.955826i	$0.594964\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7378.14	0.918819	0.459410	−	0.888225i	$-0.348061\pi$
$401$	7378.14	0.918819	0.459410	+	0.888225i	$0.348061\pi$
$402$	0	0
$403$	−5742.97	−0.709871
$404$	0	0
$405$	1185.99	0.145511
$406$	0	0
$407$	2868.73	0.349380
$408$	0	0
$409$	−3727.49	−0.450642	−0.225321	−	0.974285i	$-0.572343\pi$
$409$	−3727.49	−0.450642	−0.225321	+	0.974285i	$0.572343\pi$
$410$	0	0
$411$	8853.40	1.06254
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12874.7	1.52288
$416$	0	0
$417$	−306.574	−0.0360023
$418$	0	0
$419$	11146.7	1.29964	0.649822	−	0.760087i	$-0.274844\pi$
$419$	11146.7	1.29964	0.649822	+	0.760087i	$0.274844\pi$
$420$	0	0
$421$	−11777.7	−1.36344	−0.681719	−	0.731614i	$-0.738768\pi$
$421$	−11777.7	−1.36344	−0.681719	+	0.731614i	$0.738768\pi$
$422$	0	0
$423$	1798.98	0.206783
$424$	0	0
$425$	−1683.35	−0.192128
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−714.660	−0.0804292
$430$	0	0
$431$	5574.28	0.622978	0.311489	−	0.950250i	$-0.399172\pi$
$431$	5574.28	0.622978	0.311489	+	0.950250i	$0.399172\pi$
$432$	0	0
$433$	240.867	0.0267328	0.0133664	−	0.999911i	$-0.495745\pi$
$433$	240.867	0.0267328	0.0133664	+	0.999911i	$0.495745\pi$
$434$	0	0
$435$	−12353.4	−1.36161
$436$	0	0
$437$	−2633.47	−0.288275
$438$	0	0
$439$	9433.79	1.02563	0.512813	−	0.858500i	$-0.328603\pi$
$439$	9433.79	1.02563	0.512813	+	0.858500i	$0.328603\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9053.72	0.971005	0.485502	−	0.874235i	$-0.338637\pi$
$443$	9053.72	0.971005	0.485502	+	0.874235i	$0.338637\pi$
$444$	0	0
$445$	21680.1	2.30952
$446$	0	0
$447$	−5396.11	−0.570978
$448$	0	0
$449$	2726.05	0.286526	0.143263	−	0.989685i	$-0.454241\pi$
$449$	2726.05	0.286526	0.143263	+	0.989685i	$0.454241\pi$
$450$	0	0
$451$	−139.016	−0.0145144
$452$	0	0
$453$	−9590.42	−0.994696
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10628.5	1.08792	0.543960	−	0.839111i	$-0.316924\pi$
$457$	10628.5	1.08792	0.543960	+	0.839111i	$0.316924\pi$
$458$	0	0
$459$	508.493	0.0517090
$460$	0	0
$461$	11324.0	1.14406	0.572032	−	0.820232i	$-0.306155\pi$
$461$	11324.0	1.14406	0.572032	+	0.820232i	$0.306155\pi$
$462$	0	0
$463$	−11711.0	−1.17550	−0.587751	−	0.809042i	$-0.699986\pi$
$463$	−11711.0	−1.17550	−0.587751	+	0.809042i	$0.699986\pi$
$464$	0	0
$465$	12123.8	1.20910
$466$	0	0
$467$	15480.6	1.53396	0.766980	−	0.641671i	$-0.221759\pi$
$467$	15480.6	1.53396	0.766980	+	0.641671i	$0.221759\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−10428.8	−1.02024
$472$	0	0
$473$	−746.611	−0.0725776
$474$	0	0
$475$	8135.41	0.785848
$476$	0	0
$477$	−577.134	−0.0553987
$478$	0	0
$479$	−4747.39	−0.452847	−0.226423	−	0.974029i	$-0.572703\pi$
$479$	−4747.39	−0.452847	−0.226423	+	0.974029i	$0.572703\pi$
$480$	0	0
$481$	5213.58	0.494218
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2601.45	0.243559
$486$	0	0
$487$	7970.85	0.741671	0.370835	−	0.928699i	$-0.379071\pi$
$487$	7970.85	0.741671	0.370835	+	0.928699i	$0.379071\pi$
$488$	0	0
$489$	11369.8	1.05145
$490$	0	0
$491$	−13019.1	−1.19662	−0.598311	−	0.801264i	$-0.704161\pi$
$491$	−13019.1	−1.19662	−0.598311	+	0.801264i	$0.704161\pi$
$492$	0	0
$493$	−5296.52	−0.483860
$494$	0	0
$495$	1508.70	0.136992
$496$	0	0
$497$	0	0
$498$	0	0
$499$	15490.8	1.38971	0.694854	−	0.719151i	$-0.255469\pi$
$499$	15490.8	1.38971	0.694854	+	0.719151i	$0.255469\pi$
$500$	0	0
$501$	8134.31	0.725377
$502$	0	0
$503$	−8646.86	−0.766490	−0.383245	−	0.923647i	$-0.625194\pi$
$503$	−8646.86	−0.766490	−0.383245	+	0.923647i	$0.625194\pi$
$504$	0	0
$505$	−12775.5	−1.12574
$506$	0	0
$507$	5292.19	0.463579
$508$	0	0
$509$	−10252.0	−0.892753	−0.446376	−	0.894845i	$-0.647286\pi$
$509$	−10252.0	−0.892753	−0.446376	+	0.894845i	$0.647286\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2457.48	−0.211502
$514$	0	0
$515$	−13796.9	−1.18051
$516$	0	0
$517$	2288.49	0.194676
$518$	0	0
$519$	7513.45	0.635460
$520$	0	0
$521$	−10025.3	−0.843023	−0.421512	−	0.906823i	$-0.638500\pi$
$521$	−10025.3	−0.843023	−0.421512	+	0.906823i	$0.638500\pi$
$522$	0	0
$523$	186.782	0.0156164	0.00780822	−	0.999970i	$-0.497515\pi$
$523$	186.782	0.0156164	0.00780822	+	0.999970i	$0.497515\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5198.11	0.429665
$528$	0	0
$529$	−11329.8	−0.931195
$530$	0	0
$531$	−621.712	−0.0508098
$532$	0	0
$533$	−252.646	−0.0205315
$534$	0	0
$535$	17020.0	1.37540
$536$	0	0
$537$	−6267.92	−0.503688
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−9729.13	−0.773176	−0.386588	−	0.922253i	$-0.626346\pi$
$541$	−9729.13	−0.773176	−0.386588	+	0.922253i	$0.626346\pi$
$542$	0	0
$543$	−12243.9	−0.967651
$544$	0	0
$545$	−515.414	−0.0405099
$546$	0	0
$547$	19387.1	1.51541	0.757707	−	0.652595i	$-0.226320\pi$
$547$	19387.1	1.51541	0.757707	+	0.652595i	$0.226320\pi$
$548$	0	0
$549$	5905.89	0.459121
$550$	0	0
$551$	25597.4	1.97910
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−11006.3	−0.841783
$556$	0	0
$557$	13182.0	1.00276	0.501381	−	0.865227i	$-0.332826\pi$
$557$	13182.0	1.00276	0.501381	+	0.865227i	$0.332826\pi$
$558$	0	0
$559$	−1356.88	−0.102665
$560$	0	0
$561$	646.857	0.0486815
$562$	0	0
$563$	14211.0	1.06381	0.531903	−	0.846805i	$-0.321477\pi$
$563$	14211.0	1.06381	0.531903	+	0.846805i	$0.321477\pi$
$564$	0	0
$565$	−6158.81	−0.458590
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−855.367	−0.0630208	−0.0315104	−	0.999503i	$-0.510032\pi$
$569$	−855.367	−0.0630208	−0.0315104	+	0.999503i	$0.510032\pi$
$570$	0	0
$571$	−16827.3	−1.23327	−0.616637	−	0.787247i	$-0.711505\pi$
$571$	−16827.3	−1.23327	−0.616637	+	0.787247i	$0.711505\pi$
$572$	0	0
$573$	6168.68	0.449738
$574$	0	0
$575$	−2586.16	−0.187566
$576$	0	0
$577$	3961.45	0.285818	0.142909	−	0.989736i	$-0.454354\pi$
$577$	3961.45	0.285818	0.142909	+	0.989736i	$0.454354\pi$
$578$	0	0
$579$	4468.45	0.320730
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−734.176	−0.0521552
$584$	0	0
$585$	2741.89	0.193783
$586$	0	0
$587$	130.006	0.00914124	0.00457062	−	0.999990i	$-0.498545\pi$
$587$	130.006	0.00914124	0.00457062	+	0.999990i	$0.498545\pi$
$588$	0	0
$589$	−25121.8	−1.75743
$590$	0	0
$591$	−4774.98	−0.332346
$592$	0	0
$593$	−11730.9	−0.812361	−0.406180	−	0.913793i	$-0.633140\pi$
$593$	−11730.9	−0.812361	−0.406180	+	0.913793i	$0.633140\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−12910.0	−0.885041
$598$	0	0
$599$	2349.76	0.160281	0.0801407	−	0.996784i	$-0.474463\pi$
$599$	2349.76	0.160281	0.0801407	+	0.996784i	$0.474463\pi$
$600$	0	0
$601$	−21900.5	−1.48642	−0.743210	−	0.669059i	$-0.766698\pi$
$601$	−21900.5	−1.48642	−0.743210	+	0.669059i	$0.766698\pi$
$602$	0	0
$603$	3777.42	0.255106
$604$	0	0
$605$	−17569.0	−1.18063
$606$	0	0
$607$	−23463.6	−1.56896	−0.784480	−	0.620154i	$-0.787070\pi$
$607$	−23463.6	−1.56896	−0.784480	+	0.620154i	$0.787070\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4159.07	0.275381
$612$	0	0
$613$	−9012.69	−0.593832	−0.296916	−	0.954904i	$-0.595958\pi$
$613$	−9012.69	−0.593832	−0.296916	+	0.954904i	$0.595958\pi$
$614$	0	0
$615$	533.353	0.0349705
$616$	0	0
$617$	18605.8	1.21401	0.607004	−	0.794699i	$-0.292371\pi$
$617$	18605.8	1.21401	0.607004	+	0.794699i	$0.292371\pi$
$618$	0	0
$619$	7917.28	0.514091	0.257045	−	0.966399i	$-0.417251\pi$
$619$	7917.28	0.514091	0.257045	+	0.966399i	$0.417251\pi$
$620$	0	0
$621$	781.207	0.0504811
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−18808.6	−1.20375
$626$	0	0
$627$	−3126.18	−0.199119
$628$	0	0
$629$	−4718.95	−0.299136
$630$	0	0
$631$	10620.3	0.670025	0.335013	−	0.942214i	$-0.391259\pi$
$631$	10620.3	0.670025	0.335013	+	0.942214i	$0.391259\pi$
$632$	0	0
$633$	6627.04	0.416116
$634$	0	0
$635$	10580.3	0.661207
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−9202.17	−0.569690
$640$	0	0
$641$	−6334.25	−0.390309	−0.195154	−	0.980773i	$-0.562521\pi$
$641$	−6334.25	−0.390309	−0.195154	+	0.980773i	$0.562521\pi$
$642$	0	0
$643$	−25268.3	−1.54974	−0.774872	−	0.632118i	$-0.782186\pi$
$643$	−25268.3	−1.54974	−0.774872	+	0.632118i	$0.782186\pi$
$644$	0	0
$645$	2864.47	0.174866
$646$	0	0
$647$	2288.39	0.139051	0.0695256	−	0.997580i	$-0.477851\pi$
$647$	2288.39	0.139051	0.0695256	+	0.997580i	$0.477851\pi$
$648$	0	0
$649$	−790.884	−0.0478350
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12721.2	−0.762357	−0.381179	−	0.924501i	$-0.624482\pi$
$653$	−12721.2	−0.762357	−0.381179	+	0.924501i	$0.624482\pi$
$654$	0	0
$655$	−11702.1	−0.698073
$656$	0	0
$657$	929.523	0.0551966
$658$	0	0
$659$	−15931.0	−0.941707	−0.470853	−	0.882211i	$-0.656054\pi$
$659$	−15931.0	−0.941707	−0.470853	+	0.882211i	$0.656054\pi$
$660$	0	0
$661$	−3790.79	−0.223063	−0.111531	−	0.993761i	$-0.535576\pi$
$661$	−3790.79	−0.223063	−0.111531	+	0.993761i	$0.535576\pi$
$662$	0	0
$663$	1175.59	0.0688628
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8137.13	−0.472370
$668$	0	0
$669$	−14195.7	−0.820387
$670$	0	0
$671$	7512.92	0.432240
$672$	0	0
$673$	−27903.0	−1.59819	−0.799096	−	0.601203i	$-0.794688\pi$
$673$	−27903.0	−1.59819	−0.799096	+	0.601203i	$0.794688\pi$
$674$	0	0
$675$	−2413.33	−0.137613
$676$	0	0
$677$	12845.9	0.729257	0.364629	−	0.931153i	$-0.381196\pi$
$677$	12845.9	0.729257	0.364629	+	0.931153i	$0.381196\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	1866.92	0.105052
$682$	0	0
$683$	27871.5	1.56146	0.780728	−	0.624870i	$-0.214848\pi$
$683$	27871.5	1.56146	0.780728	+	0.624870i	$0.214848\pi$
$684$	0	0
$685$	−43209.9	−2.41017
$686$	0	0
$687$	−1591.86	−0.0884033
$688$	0	0
$689$	−1334.28	−0.0737765
$690$	0	0
$691$	−12700.3	−0.699190	−0.349595	−	0.936901i	$-0.613681\pi$
$691$	−12700.3	−0.699190	−0.349595	+	0.936901i	$0.613681\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1496.26	0.0816641
$696$	0	0
$697$	228.676	0.0124271
$698$	0	0
$699$	−9567.83	−0.517723
$700$	0	0
$701$	−18491.7	−0.996320	−0.498160	−	0.867085i	$-0.665991\pi$
$701$	−18491.7	−0.996320	−0.498160	+	0.867085i	$0.665991\pi$
$702$	0	0
$703$	22806.1	1.22354
$704$	0	0
$705$	−8780.10	−0.469046
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−10589.8	−0.560944	−0.280472	−	0.959862i	$-0.590491\pi$
$709$	−10589.8	−0.560944	−0.280472	+	0.959862i	$0.590491\pi$
$710$	0	0
$711$	−2345.54	−0.123720
$712$	0	0
$713$	7985.95	0.419462
$714$	0	0
$715$	3487.97	0.182437
$716$	0	0
$717$	4546.07	0.236787
$718$	0	0
$719$	−4494.41	−0.233120	−0.116560	−	0.993184i	$-0.537187\pi$
$719$	−4494.41	−0.233120	−0.116560	+	0.993184i	$0.537187\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−3706.15	−0.190641
$724$	0	0
$725$	25137.5	1.28770
$726$	0	0
$727$	3251.31	0.165866	0.0829328	−	0.996555i	$-0.473571\pi$
$727$	3251.31	0.165866	0.0829328	+	0.996555i	$0.473571\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	1228.15	0.0621403
$732$	0	0
$733$	17052.9	0.859293	0.429647	−	0.902997i	$-0.358638\pi$
$733$	17052.9	0.859293	0.429647	+	0.902997i	$0.358638\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4805.29	0.240170
$738$	0	0
$739$	−33680.4	−1.67653	−0.838265	−	0.545264i	$-0.816429\pi$
$739$	−33680.4	−1.67653	−0.838265	+	0.545264i	$0.816429\pi$
$740$	0	0
$741$	−5681.47	−0.281665
$742$	0	0
$743$	18689.5	0.922813	0.461407	−	0.887189i	$-0.347345\pi$
$743$	18689.5	0.922813	0.461407	+	0.887189i	$0.347345\pi$
$744$	0	0
$745$	26336.3	1.29515
$746$	0	0
$747$	7913.81	0.387619
$748$	0	0
$749$	0	0
$750$	0	0
$751$	25028.2	1.21610	0.608052	−	0.793897i	$-0.291951\pi$
$751$	25028.2	1.21610	0.608052	+	0.793897i	$0.291951\pi$
$752$	0	0
$753$	6495.93	0.314375
$754$	0	0
$755$	46807.0	2.25627
$756$	0	0
$757$	−18630.2	−0.894488	−0.447244	−	0.894412i	$-0.647594\pi$
$757$	−18630.2	−0.894488	−0.447244	+	0.894412i	$0.647594\pi$
$758$	0	0
$759$	993.778	0.0475255
$760$	0	0
$761$	28611.1	1.36288	0.681440	−	0.731874i	$-0.261354\pi$
$761$	28611.1	1.36288	0.681440	+	0.731874i	$0.261354\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2481.75	−0.117291
$766$	0	0
$767$	−1437.34	−0.0676654
$768$	0	0
$769$	15072.5	0.706798	0.353399	−	0.935473i	$-0.385026\pi$
$769$	15072.5	0.706798	0.353399	+	0.935473i	$0.385026\pi$
$770$	0	0
$771$	−11842.3	−0.553164
$772$	0	0
$773$	−32904.3	−1.53103	−0.765515	−	0.643418i	$-0.777516\pi$
$773$	−32904.3	−1.53103	−0.765515	+	0.643418i	$0.777516\pi$
$774$	0	0
$775$	−24670.4	−1.14347
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1105.16	−0.0508299
$780$	0	0
$781$	−11706.1	−0.536336
$782$	0	0
$783$	−7593.34	−0.346570
$784$	0	0
$785$	50898.9	2.31422
$786$	0	0
$787$	−31033.3	−1.40562	−0.702808	−	0.711380i	$-0.748071\pi$
$787$	−31033.3	−1.40562	−0.702808	+	0.711380i	$0.748071\pi$
$788$	0	0
$789$	−2993.94	−0.135091
$790$	0	0
$791$	0	0
$792$	0	0
$793$	13653.9	0.611428
$794$	0	0
$795$	2816.76	0.125661
$796$	0	0
$797$	5906.64	0.262514	0.131257	−	0.991348i	$-0.458099\pi$
$797$	5906.64	0.262514	0.131257	+	0.991348i	$0.458099\pi$
$798$	0	0
$799$	−3764.48	−0.166680
$800$	0	0
$801$	13326.3	0.587842
$802$	0	0
$803$	1182.45	0.0519649
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−18087.8	−0.788997
$808$	0	0
$809$	−9652.69	−0.419494	−0.209747	−	0.977756i	$-0.567264\pi$
$809$	−9652.69	−0.419494	−0.209747	+	0.977756i	$0.567264\pi$
$810$	0	0
$811$	−29469.2	−1.27596	−0.637980	−	0.770053i	$-0.720230\pi$
$811$	−29469.2	−1.27596	−0.637980	+	0.770053i	$0.720230\pi$
$812$	0	0
$813$	9358.94	0.403730
$814$	0	0
$815$	−55491.6	−2.38501
$816$	0	0
$817$	−5935.47	−0.254169
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−667.350	−0.0283687	−0.0141843	−	0.999899i	$-0.504515\pi$
$821$	−667.350	−0.0283687	−0.0141843	+	0.999899i	$0.504515\pi$
$822$	0	0
$823$	18030.5	0.763673	0.381836	−	0.924230i	$-0.375292\pi$
$823$	18030.5	0.763673	0.381836	+	0.924230i	$0.375292\pi$
$824$	0	0
$825$	−3070.01	−0.129556
$826$	0	0
$827$	2362.88	0.0993535	0.0496767	−	0.998765i	$-0.484181\pi$
$827$	2362.88	0.0993535	0.0496767	+	0.998765i	$0.484181\pi$
$828$	0	0
$829$	−26128.9	−1.09468	−0.547342	−	0.836909i	$-0.684360\pi$
$829$	−26128.9	−1.09468	−0.547342	+	0.836909i	$0.684360\pi$
$830$	0	0
$831$	6011.40	0.250942
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−39700.3	−1.64537
$836$	0	0
$837$	7452.26	0.307751
$838$	0	0
$839$	−39017.9	−1.60554	−0.802769	−	0.596290i	$-0.796641\pi$
$839$	−39017.9	−1.60554	−0.802769	+	0.596290i	$0.796641\pi$
$840$	0	0
$841$	54704.1	2.24298
$842$	0	0
$843$	26746.0	1.09274
$844$	0	0
$845$	−25829.1	−1.05153
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−9717.26	−0.392810
$850$	0	0
$851$	−7249.80	−0.292033
$852$	0	0
$853$	−24487.4	−0.982922	−0.491461	−	0.870900i	$-0.663537\pi$
$853$	−24487.4	−0.982922	−0.491461	+	0.870900i	$0.663537\pi$
$854$	0	0
$855$	11994.0	0.479750
$856$	0	0
$857$	2351.90	0.0937450	0.0468725	−	0.998901i	$-0.485075\pi$
$857$	2351.90	0.0937450	0.0468725	+	0.998901i	$0.485075\pi$
$858$	0	0
$859$	−5413.01	−0.215005	−0.107503	−	0.994205i	$-0.534285\pi$
$859$	−5413.01	−0.215005	−0.107503	+	0.994205i	$0.534285\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32772.7	−1.29270	−0.646348	−	0.763043i	$-0.723705\pi$
$863$	−32772.7	−1.29270	−0.646348	+	0.763043i	$0.723705\pi$
$864$	0	0
$865$	−36670.1	−1.44141
$866$	0	0
$867$	13674.9	0.535670
$868$	0	0
$869$	−2983.78	−0.116476
$870$	0	0
$871$	8733.05	0.339734
$872$	0	0
$873$	1599.06	0.0619930
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−40592.0	−1.56294	−0.781468	−	0.623946i	$-0.785529\pi$
$877$	−40592.0	−1.56294	−0.781468	+	0.623946i	$0.785529\pi$
$878$	0	0
$879$	−17991.1	−0.690357
$880$	0	0
$881$	21588.5	0.825578	0.412789	−	0.910827i	$-0.364555\pi$
$881$	21588.5	0.825578	0.412789	+	0.910827i	$0.364555\pi$
$882$	0	0
$883$	−11223.7	−0.427757	−0.213878	−	0.976860i	$-0.568610\pi$
$883$	−11223.7	−0.427757	−0.213878	+	0.976860i	$0.568610\pi$
$884$	0	0
$885$	3034.33	0.115252
$886$	0	0
$887$	36557.9	1.38387	0.691936	−	0.721958i	$-0.256758\pi$
$887$	36557.9	1.38387	0.691936	+	0.721958i	$0.256758\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	927.365	0.0348686
$892$	0	0
$893$	18193.2	0.681762
$894$	0	0
$895$	30591.2	1.14252
$896$	0	0
$897$	1806.08	0.0672276
$898$	0	0
$899$	−77623.5	−2.87974
$900$	0	0
$901$	1207.69	0.0446548
$902$	0	0
$903$	0	0
$904$	0	0
$905$	59757.4	2.19492
$906$	0	0
$907$	19252.0	0.704798	0.352399	−	0.935850i	$-0.385366\pi$
$907$	19252.0	0.704798	0.352399	+	0.935850i	$0.385366\pi$
$908$	0	0
$909$	−7852.81	−0.286536
$910$	0	0
$911$	−4910.11	−0.178572	−0.0892861	−	0.996006i	$-0.528459\pi$
$911$	−4910.11	−0.178572	−0.0892861	+	0.996006i	$0.528459\pi$
$912$	0	0
$913$	10067.2	0.364925
$914$	0	0
$915$	−28824.3	−1.04142
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1908.48	0.0685039	0.0342520	−	0.999413i	$-0.489095\pi$
$919$	1908.48	0.0685039	0.0342520	+	0.999413i	$0.489095\pi$
$920$	0	0
$921$	−29029.0	−1.03859
$922$	0	0
$923$	−21274.5	−0.758678
$924$	0	0
$925$	22396.3	0.796093
$926$	0	0
$927$	−8480.63	−0.300475
$928$	0	0
$929$	30463.0	1.07584	0.537921	−	0.842995i	$-0.319210\pi$
$929$	30463.0	1.07584	0.537921	+	0.842995i	$0.319210\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	4375.65	0.153540
$934$	0	0
$935$	−3157.05	−0.110424
$936$	0	0
$937$	468.146	0.0163219	0.00816097	−	0.999967i	$-0.497402\pi$
$937$	468.146	0.0163219	0.00816097	+	0.999967i	$0.497402\pi$
$938$	0	0
$939$	−30347.0	−1.05467
$940$	0	0
$941$	−8443.67	−0.292514	−0.146257	−	0.989247i	$-0.546723\pi$
$941$	−8443.67	−0.292514	−0.146257	+	0.989247i	$0.546723\pi$
$942$	0	0
$943$	351.319	0.0121320
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7269.71	0.249455	0.124727	−	0.992191i	$-0.460194\pi$
$947$	7269.71	0.249455	0.124727	+	0.992191i	$0.460194\pi$
$948$	0	0
$949$	2148.97	0.0735074
$950$	0	0
$951$	−25548.2	−0.871142
$952$	0	0
$953$	25806.9	0.877195	0.438598	−	0.898684i	$-0.355475\pi$
$953$	25806.9	0.877195	0.438598	+	0.898684i	$0.355475\pi$
$954$	0	0
$955$	−30106.9	−1.02014
$956$	0	0
$957$	−9659.54	−0.326279
$958$	0	0
$959$	0	0
$960$	0	0
$961$	46390.3	1.55719
$962$	0	0
$963$	10461.8	0.350081
$964$	0	0
$965$	−21808.7	−0.727511
$966$	0	0
$967$	−7155.61	−0.237962	−0.118981	−	0.992897i	$-0.537963\pi$
$967$	−7155.61	−0.237962	−0.118981	+	0.992897i	$0.537963\pi$
$968$	0	0
$969$	5142.44	0.170484
$970$	0	0
$971$	−29970.2	−0.990515	−0.495257	−	0.868746i	$-0.664926\pi$
$971$	−29970.2	−0.990515	−0.495257	+	0.868746i	$0.664926\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−5579.39	−0.183265
$976$	0	0
$977$	−46223.3	−1.51363	−0.756813	−	0.653631i	$-0.773245\pi$
$977$	−46223.3	−1.51363	−0.756813	+	0.653631i	$0.773245\pi$
$978$	0	0
$979$	16952.5	0.553425
$980$	0	0
$981$	−316.814	−0.0103110
$982$	0	0
$983$	−41135.8	−1.33472	−0.667360	−	0.744736i	$-0.732576\pi$
$983$	−41135.8	−1.33472	−0.667360	+	0.744736i	$0.732576\pi$
$984$	0	0
$985$	23304.8	0.753860
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1886.82	0.0606647
$990$	0	0
$991$	4228.11	0.135530	0.0677651	−	0.997701i	$-0.478413\pi$
$991$	4228.11	0.135530	0.0677651	+	0.997701i	$0.478413\pi$
$992$	0	0
$993$	−8401.67	−0.268499
$994$	0	0
$995$	63008.4	2.00754
$996$	0	0
$997$	−38767.3	−1.23147	−0.615734	−	0.787954i	$-0.711140\pi$
$997$	−38767.3	−1.23147	−0.615734	+	0.787954i	$0.711140\pi$
$998$	0	0
$999$	−6765.31	−0.214259

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.cn.1.4		4
4.3	odd	2		1176.4.a.be.1.4	yes	4
7.6	odd	2		2352.4.a.co.1.1		4
28.27	even	2		1176.4.a.z.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.4.a.z.1.1	✓	4	28.27	even	2
1176.4.a.be.1.4	yes	4	4.3	odd	2
2352.4.a.cn.1.4		4	1.1	even	1	trivial
2352.4.a.co.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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +14.6418 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +14.6418 q^{5} +9.00000 q^{9} +11.4490 q^{11} +20.8071 q^{13} -43.9254 q^{15} -18.8331 q^{17} +91.0178 q^{19} -28.9336 q^{23} +89.3825 q^{25} -27.0000 q^{27} +281.235 q^{29} -276.010 q^{31} -34.3469 q^{33} +250.567 q^{37} -62.4214 q^{39} -12.1422 q^{41} -65.2121 q^{43} +131.776 q^{45} +199.886 q^{47} +56.4992 q^{51} -64.1260 q^{53} +167.633 q^{55} -273.054 q^{57} -69.0791 q^{59} +656.210 q^{61} +304.654 q^{65} +419.714 q^{67} +86.8007 q^{69} -1022.46 q^{71} +103.280 q^{73} -268.148 q^{75} -260.616 q^{79} +81.0000 q^{81} +879.313 q^{83} -275.750 q^{85} -843.705 q^{87} +1480.70 q^{89} +828.029 q^{93} +1332.67 q^{95} +177.673 q^{97} +103.041 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 8 q^{5} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 + 8 * q^5 + 36 * q^9	\( 4 q - 12 q^{3} + 8 q^{5} + 36 q^{9} - 40 q^{11} + 48 q^{13} - 24 q^{15} + 72 q^{17} - 32 q^{19} - 8 q^{23} + 164 q^{25} - 108 q^{27} + 144 q^{29} - 48 q^{31} + 120 q^{33} + 48 q^{37} - 144 q^{39} + 72 q^{41} - 512 q^{43} + 72 q^{45} - 160 q^{47} - 216 q^{51} + 536 q^{53} + 336 q^{55} + 96 q^{57} - 240 q^{59} + 896 q^{61} - 136 q^{65} - 1088 q^{67} + 24 q^{69} - 1288 q^{71} + 1488 q^{73} - 492 q^{75} - 416 q^{79} + 324 q^{81} + 112 q^{83} - 1512 q^{85} - 432 q^{87} + 3160 q^{89} + 144 q^{93} + 240 q^{95} + 2384 q^{97} - 360 q^{99}+O(q^{100}) \) 4 * q - 12 * q^3 + 8 * q^5 + 36 * q^9 - 40 * q^11 + 48 * q^13 - 24 * q^15 + 72 * q^17 - 32 * q^19 - 8 * q^23 + 164 * q^25 - 108 * q^27 + 144 * q^29 - 48 * q^31 + 120 * q^33 + 48 * q^37 - 144 * q^39 + 72 * q^41 - 512 * q^43 + 72 * q^45 - 160 * q^47 - 216 * q^51 + 536 * q^53 + 336 * q^55 + 96 * q^57 - 240 * q^59 + 896 * q^61 - 136 * q^65 - 1088 * q^67 + 24 * q^69 - 1288 * q^71 + 1488 * q^73 - 492 * q^75 - 416 * q^79 + 324 * q^81 + 112 * q^83 - 1512 * q^85 - 432 * q^87 + 3160 * q^89 + 144 * q^93 + 240 * q^95 + 2384 * q^97 - 360 * q^99

Introduction

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