LMFDB - Embedded newform 2400.4.a.bq.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, 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("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2400.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:	$141.604584014$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.32340.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 42x - 14 $ x^3 - 42*x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.334222$ of defining polynomial
Character	$\chi$	$=$	2400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +15.3676 q^{7} +9.00000 q^{9} -57.1135 q^{11} -5.66311 q^{13} -27.0307 q^{17} -4.68316 q^{19} +46.1028 q^{21} +145.993 q^{23} +27.0000 q^{27} +25.6832 q^{29} -144.186 q^{31} -171.340 q^{33} -9.14193 q^{37} -16.9893 q^{39} -222.832 q^{41} +234.038 q^{43} -117.042 q^{47} -106.837 q^{49} -81.0921 q^{51} +460.267 q^{53} -14.0495 q^{57} -203.759 q^{59} +356.408 q^{61} +138.308 q^{63} +20.1704 q^{67} +437.979 q^{69} -501.702 q^{71} -713.071 q^{73} -877.697 q^{77} -814.480 q^{79} +81.0000 q^{81} -924.709 q^{83} +77.0495 q^{87} -1638.27 q^{89} -87.0284 q^{91} -432.557 q^{93} +1412.86 q^{97} -514.021 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{3} - 7 q^{7} + 27 q^{9} - 21 q^{13} - 32 q^{17} + 19 q^{19} - 21 q^{21} - 60 q^{23} + 81 q^{27} + 44 q^{29} - 151 q^{31} - 330 q^{37} - 63 q^{39} + 82 q^{41} - 367 q^{43} + 246 q^{47} - 42 q^{49}+ \cdots + 351 q^{97}+O(q^{100}) $ 3 * q + 9 * q^3 - 7 * q^7 + 27 * q^9 - 21 * q^13 - 32 * q^17 + 19 * q^19 - 21 * q^21 - 60 * q^23 + 81 * q^27 + 44 * q^29 - 151 * q^31 - 330 * q^37 - 63 * q^39 + 82 * q^41 - 367 * q^43 + 246 * q^47 - 42 * q^49 - 96 * q^51 - 554 * q^53 + 57 * q^57 + 42 * q^59 - 27 * q^61 - 63 * q^63 + 837 * q^67 - 180 * q^69 - 1422 * q^71 - 498 * q^73 - 560 * q^77 - 688 * q^79 + 243 * q^81 - 702 * q^83 + 132 * q^87 - 1488 * q^89 - 735 * q^91 - 453 * q^93 + 351 * q^97

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	15.3676	0.829773	0.414886	−	0.909873i	$-0.363821\pi$
$7$	15.3676	0.829773	0.414886	+	0.909873i	$0.363821\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−57.1135	−1.56549	−0.782744	−	0.622344i	$-0.786180\pi$
$11$	−57.1135	−1.56549	−0.782744	+	0.622344i	$0.786180\pi$
$12$	0	0
$13$	−5.66311	−0.120820	−0.0604102	−	0.998174i	$-0.519241\pi$
$13$	−5.66311	−0.120820	−0.0604102	+	0.998174i	$0.519241\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−27.0307	−0.385642	−0.192821	−	0.981234i	$-0.561764\pi$
$17$	−27.0307	−0.385642	−0.192821	+	0.981234i	$0.561764\pi$
$18$	0	0
$19$	−4.68316	−0.0565469	−0.0282734	−	0.999600i	$-0.509001\pi$
$19$	−4.68316	−0.0565469	−0.0282734	+	0.999600i	$0.509001\pi$
$20$	0	0
$21$	46.1028	0.479069
$22$	0	0
$23$	145.993	1.32355	0.661774	−	0.749703i	$-0.269804\pi$
$23$	145.993	1.32355	0.661774	+	0.749703i	$0.269804\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	25.6832	0.164457	0.0822283	−	0.996614i	$-0.473796\pi$
$29$	25.6832	0.164457	0.0822283	+	0.996614i	$0.473796\pi$
$30$	0	0
$31$	−144.186	−0.835371	−0.417685	−	0.908592i	$-0.637159\pi$
$31$	−144.186	−0.835371	−0.417685	+	0.908592i	$0.637159\pi$
$32$	0	0
$33$	−171.340	−0.903835
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.14193	−0.0406196	−0.0203098	−	0.999794i	$-0.506465\pi$
$37$	−9.14193	−0.0406196	−0.0203098	+	0.999794i	$0.506465\pi$
$38$	0	0
$39$	−16.9893	−0.0697557
$40$	0	0
$41$	−222.832	−0.848794	−0.424397	−	0.905476i	$-0.639514\pi$
$41$	−222.832	−0.848794	−0.424397	+	0.905476i	$0.639514\pi$
$42$	0	0
$43$	234.038	0.830010	0.415005	−	0.909819i	$-0.363780\pi$
$43$	234.038	0.830010	0.415005	+	0.909819i	$0.363780\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−117.042	−0.363242	−0.181621	−	0.983369i	$-0.558134\pi$
$47$	−117.042	−0.363242	−0.181621	+	0.983369i	$0.558134\pi$
$48$	0	0
$49$	−106.837	−0.311477
$50$	0	0
$51$	−81.0921	−0.222650
$52$	0	0
$53$	460.267	1.19288	0.596439	−	0.802658i	$-0.296582\pi$
$53$	460.267	1.19288	0.596439	+	0.802658i	$0.296582\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−14.0495	−0.0326473
$58$	0	0
$59$	−203.759	−0.449612	−0.224806	−	0.974403i	$-0.572175\pi$
$59$	−203.759	−0.449612	−0.224806	+	0.974403i	$0.572175\pi$
$60$	0	0
$61$	356.408	0.748088	0.374044	−	0.927411i	$-0.377971\pi$
$61$	356.408	0.748088	0.374044	+	0.927411i	$0.377971\pi$
$62$	0	0
$63$	138.308	0.276591
$64$	0	0
$65$	0	0
$66$	0	0
$67$	20.1704	0.0367792	0.0183896	−	0.999831i	$-0.494146\pi$
$67$	20.1704	0.0367792	0.0183896	+	0.999831i	$0.494146\pi$
$68$	0	0
$69$	437.979	0.764151
$70$	0	0
$71$	−501.702	−0.838607	−0.419303	−	0.907846i	$-0.637726\pi$
$71$	−501.702	−0.838607	−0.419303	+	0.907846i	$0.637726\pi$
$72$	0	0
$73$	−713.071	−1.14327	−0.571634	−	0.820508i	$-0.693690\pi$
$73$	−713.071	−1.14327	−0.571634	+	0.820508i	$0.693690\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−877.697	−1.29900
$78$	0	0
$79$	−814.480	−1.15995	−0.579976	−	0.814634i	$-0.696938\pi$
$79$	−814.480	−1.15995	−0.579976	+	0.814634i	$0.696938\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−924.709	−1.22289	−0.611446	−	0.791286i	$-0.709412\pi$
$83$	−924.709	−1.22289	−0.611446	+	0.791286i	$0.709412\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	77.0495	0.0949491
$88$	0	0
$89$	−1638.27	−1.95119	−0.975597	−	0.219570i	$-0.929535\pi$
$89$	−1638.27	−1.95119	−0.975597	+	0.219570i	$0.929535\pi$
$90$	0	0
$91$	−87.0284	−0.100253
$92$	0	0
$93$	−432.557	−0.482302
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1412.86	1.47891	0.739453	−	0.673208i	$-0.235084\pi$
$97$	1412.86	1.47891	0.739453	+	0.673208i	$0.235084\pi$
$98$	0	0
$99$	−514.021	−0.521829
$100$	0	0
$101$	−337.024	−0.332031	−0.166015	−	0.986123i	$-0.553090\pi$
$101$	−337.024	−0.332031	−0.166015	+	0.986123i	$0.553090\pi$
$102$	0	0
$103$	−2054.36	−1.96526	−0.982630	−	0.185575i	$-0.940585\pi$
$103$	−2054.36	−1.96526	−0.982630	+	0.185575i	$0.940585\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	371.354	0.335515	0.167758	−	0.985828i	$-0.446347\pi$
$107$	371.354	0.335515	0.167758	+	0.985828i	$0.446347\pi$
$108$	0	0
$109$	309.557	0.272020	0.136010	−	0.990707i	$-0.456572\pi$
$109$	309.557	0.272020	0.136010	+	0.990707i	$0.456572\pi$
$110$	0	0
$111$	−27.4258	−0.0234517
$112$	0	0
$113$	−836.695	−0.696545	−0.348273	−	0.937393i	$-0.613232\pi$
$113$	−836.695	−0.696545	−0.348273	+	0.937393i	$0.613232\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−50.9680	−0.0402734
$118$	0	0
$119$	−415.397	−0.319995
$120$	0	0
$121$	1930.95	1.45075
$122$	0	0
$123$	−668.497	−0.490051
$124$	0	0
$125$	0	0
$126$	0	0
$127$	878.510	0.613820	0.306910	−	0.951738i	$-0.400705\pi$
$127$	878.510	0.613820	0.306910	+	0.951738i	$0.400705\pi$
$128$	0	0
$129$	702.113	0.479207
$130$	0	0
$131$	207.922	0.138673	0.0693367	−	0.997593i	$-0.477912\pi$
$131$	207.922	0.138673	0.0693367	+	0.997593i	$0.477912\pi$
$132$	0	0
$133$	−71.9689	−0.0469210
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1894.02	−1.18115	−0.590574	−	0.806983i	$-0.701099\pi$
$137$	−1894.02	−1.18115	−0.590574	+	0.806983i	$0.701099\pi$
$138$	0	0
$139$	1723.74	1.05184	0.525920	−	0.850534i	$-0.323721\pi$
$139$	1723.74	1.05184	0.525920	+	0.850534i	$0.323721\pi$
$140$	0	0
$141$	−351.127	−0.209718
$142$	0	0
$143$	323.440	0.189143
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−320.510	−0.179832
$148$	0	0
$149$	−1522.07	−0.836864	−0.418432	−	0.908248i	$-0.637420\pi$
$149$	−1522.07	−0.836864	−0.418432	+	0.908248i	$0.637420\pi$
$150$	0	0
$151$	−225.226	−0.121382	−0.0606908	−	0.998157i	$-0.519330\pi$
$151$	−225.226	−0.121382	−0.0606908	+	0.998157i	$0.519330\pi$
$152$	0	0
$153$	−243.276	−0.128547
$154$	0	0
$155$	0	0
$156$	0	0
$157$	928.691	0.472087	0.236043	−	0.971743i	$-0.424149\pi$
$157$	928.691	0.472087	0.236043	+	0.971743i	$0.424149\pi$
$158$	0	0
$159$	1380.80	0.688709
$160$	0	0
$161$	2243.56	1.09824
$162$	0	0
$163$	1801.86	0.865843	0.432922	−	0.901432i	$-0.357483\pi$
$163$	1801.86	0.865843	0.432922	+	0.901432i	$0.357483\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1471.25	0.681728	0.340864	−	0.940113i	$-0.389280\pi$
$167$	1471.25	0.681728	0.340864	+	0.940113i	$0.389280\pi$
$168$	0	0
$169$	−2164.93	−0.985402
$170$	0	0
$171$	−42.1484	−0.0188490
$172$	0	0
$173$	−3633.06	−1.59663	−0.798314	−	0.602241i	$-0.794275\pi$
$173$	−3633.06	−1.59663	−0.798314	+	0.602241i	$0.794275\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−611.276	−0.259584
$178$	0	0
$179$	1329.28	0.555054	0.277527	−	0.960718i	$-0.410485\pi$
$179$	1329.28	0.555054	0.277527	+	0.960718i	$0.410485\pi$
$180$	0	0
$181$	26.3286	0.0108121	0.00540605	−	0.999985i	$-0.498279\pi$
$181$	26.3286	0.0108121	0.00540605	+	0.999985i	$0.498279\pi$
$182$	0	0
$183$	1069.22	0.431909
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1543.82	0.603717
$188$	0	0
$189$	414.925	0.159690
$190$	0	0
$191$	−2639.91	−1.00009	−0.500045	−	0.866000i	$-0.666683\pi$
$191$	−2639.91	−1.00009	−0.500045	+	0.866000i	$0.666683\pi$
$192$	0	0
$193$	−720.370	−0.268670	−0.134335	−	0.990936i	$-0.542890\pi$
$193$	−720.370	−0.268670	−0.134335	+	0.990936i	$0.542890\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1768.75	−0.639688	−0.319844	−	0.947470i	$-0.603630\pi$
$197$	−1768.75	−0.639688	−0.319844	+	0.947470i	$0.603630\pi$
$198$	0	0
$199$	1950.50	0.694812	0.347406	−	0.937715i	$-0.387063\pi$
$199$	1950.50	0.694812	0.347406	+	0.937715i	$0.387063\pi$
$200$	0	0
$201$	60.5111	0.0212345
$202$	0	0
$203$	394.689	0.136462
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1313.94	0.441183
$208$	0	0
$209$	267.472	0.0885234
$210$	0	0
$211$	−2381.61	−0.777048	−0.388524	−	0.921439i	$-0.627015\pi$
$211$	−2381.61	−0.777048	−0.388524	+	0.921439i	$0.627015\pi$
$212$	0	0
$213$	−1505.11	−0.484170
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2215.79	−0.693168
$218$	0	0
$219$	−2139.21	−0.660067
$220$	0	0
$221$	153.078	0.0465934
$222$	0	0
$223$	−2563.32	−0.769742	−0.384871	−	0.922970i	$-0.625754\pi$
$223$	−2563.32	−0.769742	−0.384871	+	0.922970i	$0.625754\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−218.391	−0.0638550	−0.0319275	−	0.999490i	$-0.510165\pi$
$227$	−218.391	−0.0638550	−0.0319275	+	0.999490i	$0.510165\pi$
$228$	0	0
$229$	−6230.84	−1.79802	−0.899008	−	0.437933i	$-0.855711\pi$
$229$	−6230.84	−1.79802	−0.899008	+	0.437933i	$0.855711\pi$
$230$	0	0
$231$	−2633.09	−0.749977
$232$	0	0
$233$	−2873.70	−0.807992	−0.403996	−	0.914761i	$-0.632379\pi$
$233$	−2873.70	−0.807992	−0.403996	+	0.914761i	$0.632379\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2443.44	−0.669698
$238$	0	0
$239$	−2596.49	−0.702733	−0.351367	−	0.936238i	$-0.614283\pi$
$239$	−2596.49	−0.702733	−0.351367	+	0.936238i	$0.614283\pi$
$240$	0	0
$241$	4926.06	1.31666	0.658330	−	0.752729i	$-0.271263\pi$
$241$	4926.06	1.31666	0.658330	+	0.752729i	$0.271263\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	26.5213	0.00683201
$248$	0	0
$249$	−2774.13	−0.706037
$250$	0	0
$251$	4767.23	1.19883	0.599413	−	0.800440i	$-0.295401\pi$
$251$	4767.23	1.19883	0.599413	+	0.800440i	$0.295401\pi$
$252$	0	0
$253$	−8338.16	−2.07200
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4055.54	0.984348	0.492174	−	0.870497i	$-0.336202\pi$
$257$	4055.54	0.984348	0.492174	+	0.870497i	$0.336202\pi$
$258$	0	0
$259$	−140.490	−0.0337050
$260$	0	0
$261$	231.148	0.0548189
$262$	0	0
$263$	2192.57	0.514066	0.257033	−	0.966403i	$-0.417255\pi$
$263$	2192.57	0.514066	0.257033	+	0.966403i	$0.417255\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4914.81	−1.12652
$268$	0	0
$269$	5274.47	1.19550	0.597751	−	0.801682i	$-0.296061\pi$
$269$	5274.47	1.19550	0.597751	+	0.801682i	$0.296061\pi$
$270$	0	0
$271$	4338.71	0.972539	0.486269	−	0.873809i	$-0.338357\pi$
$271$	4338.71	0.972539	0.486269	+	0.873809i	$0.338357\pi$
$272$	0	0
$273$	−261.085	−0.0578813
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−6580.91	−1.42747	−0.713733	−	0.700417i	$-0.752997\pi$
$277$	−6580.91	−1.42747	−0.713733	+	0.700417i	$0.752997\pi$
$278$	0	0
$279$	−1297.67	−0.278457
$280$	0	0
$281$	3331.85	0.707336	0.353668	−	0.935371i	$-0.384934\pi$
$281$	3331.85	0.707336	0.353668	+	0.935371i	$0.384934\pi$
$282$	0	0
$283$	3393.15	0.712727	0.356364	−	0.934347i	$-0.384016\pi$
$283$	3393.15	0.712727	0.356364	+	0.934347i	$0.384016\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3424.40	−0.704306
$288$	0	0
$289$	−4182.34	−0.851280
$290$	0	0
$291$	4238.57	0.853847
$292$	0	0
$293$	−6811.67	−1.35816	−0.679082	−	0.734063i	$-0.737622\pi$
$293$	−6811.67	−1.35816	−0.679082	+	0.734063i	$0.737622\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1542.06	−0.301278
$298$	0	0
$299$	−826.774	−0.159912
$300$	0	0
$301$	3596.60	0.688720
$302$	0	0
$303$	−1011.07	−0.191698
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−4673.14	−0.868762	−0.434381	−	0.900729i	$-0.643033\pi$
$307$	−4673.14	−0.868762	−0.434381	+	0.900729i	$0.643033\pi$
$308$	0	0
$309$	−6163.07	−1.13464
$310$	0	0
$311$	−3097.51	−0.564771	−0.282385	−	0.959301i	$-0.591126\pi$
$311$	−3097.51	−0.564771	−0.282385	+	0.959301i	$0.591126\pi$
$312$	0	0
$313$	−1570.93	−0.283687	−0.141844	−	0.989889i	$-0.545303\pi$
$313$	−1570.93	−0.283687	−0.141844	+	0.989889i	$0.545303\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1734.76	−0.307362	−0.153681	−	0.988121i	$-0.549113\pi$
$317$	−1734.76	−0.307362	−0.153681	+	0.988121i	$0.549113\pi$
$318$	0	0
$319$	−1466.85	−0.257455
$320$	0	0
$321$	1114.06	0.193710
$322$	0	0
$323$	126.589	0.0218068
$324$	0	0
$325$	0	0
$326$	0	0
$327$	928.671	0.157051
$328$	0	0
$329$	−1798.66	−0.301408
$330$	0	0
$331$	7538.14	1.25176	0.625881	−	0.779918i	$-0.284739\pi$
$331$	7538.14	1.25176	0.625881	+	0.779918i	$0.284739\pi$
$332$	0	0
$333$	−82.2774	−0.0135399
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3806.78	0.615337	0.307668	−	0.951494i	$-0.400451\pi$
$337$	3806.78	0.615337	0.307668	+	0.951494i	$0.400451\pi$
$338$	0	0
$339$	−2510.08	−0.402151
$340$	0	0
$341$	8234.94	1.30776
$342$	0	0
$343$	−6912.91	−1.08823
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9470.86	−1.46519	−0.732597	−	0.680662i	$-0.761692\pi$
$347$	−9470.86	−1.46519	−0.732597	+	0.680662i	$0.761692\pi$
$348$	0	0
$349$	872.978	0.133895	0.0669476	−	0.997756i	$-0.478674\pi$
$349$	872.978	0.133895	0.0669476	+	0.997756i	$0.478674\pi$
$350$	0	0
$351$	−152.904	−0.0232519
$352$	0	0
$353$	−9301.55	−1.40247	−0.701234	−	0.712931i	$-0.747367\pi$
$353$	−9301.55	−1.40247	−0.701234	+	0.712931i	$0.747367\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1246.19	−0.184749
$358$	0	0
$359$	−8712.81	−1.28090	−0.640452	−	0.767998i	$-0.721253\pi$
$359$	−8712.81	−1.28090	−0.640452	+	0.767998i	$0.721253\pi$
$360$	0	0
$361$	−6837.07	−0.996802
$362$	0	0
$363$	5792.85	0.837591
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4504.04	0.640624	0.320312	−	0.947312i	$-0.396212\pi$
$367$	4504.04	0.640624	0.320312	+	0.947312i	$0.396212\pi$
$368$	0	0
$369$	−2005.49	−0.282931
$370$	0	0
$371$	7073.20	0.989818
$372$	0	0
$373$	−6677.28	−0.926907	−0.463454	−	0.886121i	$-0.653390\pi$
$373$	−6677.28	−0.926907	−0.463454	+	0.886121i	$0.653390\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−145.447	−0.0198697
$378$	0	0
$379$	6455.83	0.874970	0.437485	−	0.899226i	$-0.355869\pi$
$379$	6455.83	0.874970	0.437485	+	0.899226i	$0.355869\pi$
$380$	0	0
$381$	2635.53	0.354389
$382$	0	0
$383$	−27.2888	−0.00364072	−0.00182036	−	0.999998i	$-0.500579\pi$
$383$	−27.2888	−0.00364072	−0.00182036	+	0.999998i	$0.500579\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2106.34	0.276670
$388$	0	0
$389$	−10957.5	−1.42820	−0.714098	−	0.700046i	$-0.753163\pi$
$389$	−10957.5	−1.42820	−0.714098	+	0.700046i	$0.753163\pi$
$390$	0	0
$391$	−3946.29	−0.510416
$392$	0	0
$393$	623.765	0.0800631
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−9751.46	−1.23278	−0.616388	−	0.787443i	$-0.711405\pi$
$397$	−9751.46	−1.23278	−0.616388	+	0.787443i	$0.711405\pi$
$398$	0	0
$399$	−215.907	−0.0270899
$400$	0	0
$401$	7247.97	0.902609	0.451305	−	0.892370i	$-0.350959\pi$
$401$	7247.97	0.902609	0.451305	+	0.892370i	$0.350959\pi$
$402$	0	0
$403$	816.539	0.100930
$404$	0	0
$405$	0	0
$406$	0	0
$407$	522.127	0.0635894
$408$	0	0
$409$	−5904.08	−0.713785	−0.356892	−	0.934146i	$-0.616164\pi$
$409$	−5904.08	−0.713785	−0.356892	+	0.934146i	$0.616164\pi$
$410$	0	0
$411$	−5682.07	−0.681936
$412$	0	0
$413$	−3131.28	−0.373076
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5171.23	0.607281
$418$	0	0
$419$	8541.55	0.995900	0.497950	−	0.867206i	$-0.334086\pi$
$419$	8541.55	0.995900	0.497950	+	0.867206i	$0.334086\pi$
$420$	0	0
$421$	13152.6	1.52261	0.761304	−	0.648395i	$-0.224559\pi$
$421$	13152.6	1.52261	0.761304	+	0.648395i	$0.224559\pi$
$422$	0	0
$423$	−1053.38	−0.121081
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5477.14	0.620743
$428$	0	0
$429$	970.320	0.109202
$430$	0	0
$431$	−10476.7	−1.17087	−0.585437	−	0.810718i	$-0.699077\pi$
$431$	−10476.7	−1.17087	−0.585437	+	0.810718i	$0.699077\pi$
$432$	0	0
$433$	−10752.1	−1.19333	−0.596665	−	0.802491i	$-0.703508\pi$
$433$	−10752.1	−1.19333	−0.596665	+	0.802491i	$0.703508\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−683.708	−0.0748425
$438$	0	0
$439$	16182.5	1.75934	0.879670	−	0.475584i	$-0.157763\pi$
$439$	16182.5	1.75934	0.879670	+	0.475584i	$0.157763\pi$
$440$	0	0
$441$	−961.531	−0.103826
$442$	0	0
$443$	11147.7	1.19558	0.597790	−	0.801653i	$-0.296046\pi$
$443$	11147.7	1.19558	0.597790	+	0.801653i	$0.296046\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−4566.21	−0.483164
$448$	0	0
$449$	−11000.0	−1.15618	−0.578089	−	0.815973i	$-0.696201\pi$
$449$	−11000.0	−1.15618	−0.578089	+	0.815973i	$0.696201\pi$
$450$	0	0
$451$	12726.7	1.32878
$452$	0	0
$453$	−675.678	−0.0700797
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−14825.8	−1.51755	−0.758774	−	0.651353i	$-0.774202\pi$
$457$	−14825.8	−1.51755	−0.758774	+	0.651353i	$0.774202\pi$
$458$	0	0
$459$	−729.829	−0.0742168
$460$	0	0
$461$	5857.48	0.591779	0.295890	−	0.955222i	$-0.404384\pi$
$461$	5857.48	0.591779	0.295890	+	0.955222i	$0.404384\pi$
$462$	0	0
$463$	4134.37	0.414990	0.207495	−	0.978236i	$-0.433469\pi$
$463$	4134.37	0.414990	0.207495	+	0.978236i	$0.433469\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5734.51	0.568226	0.284113	−	0.958791i	$-0.408301\pi$
$467$	5734.51	0.568226	0.284113	+	0.958791i	$0.408301\pi$
$468$	0	0
$469$	309.970	0.0305183
$470$	0	0
$471$	2786.07	0.272559
$472$	0	0
$473$	−13366.7	−1.29937
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4142.40	0.397626
$478$	0	0
$479$	1050.39	0.100195	0.0500976	−	0.998744i	$-0.484047\pi$
$479$	1050.39	0.100195	0.0500976	+	0.998744i	$0.484047\pi$
$480$	0	0
$481$	51.7718	0.00490767
$482$	0	0
$483$	6730.68	0.634072
$484$	0	0
$485$	0	0
$486$	0	0
$487$	15368.0	1.42996	0.714978	−	0.699147i	$-0.246437\pi$
$487$	15368.0	1.42996	0.714978	+	0.699147i	$0.246437\pi$
$488$	0	0
$489$	5405.58	0.499895
$490$	0	0
$491$	4469.40	0.410797	0.205398	−	0.978678i	$-0.434151\pi$
$491$	4469.40	0.410797	0.205398	+	0.978678i	$0.434151\pi$
$492$	0	0
$493$	−694.234	−0.0634214
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7709.96	−0.695853
$498$	0	0
$499$	17597.2	1.57867	0.789336	−	0.613961i	$-0.210425\pi$
$499$	17597.2	1.57867	0.789336	+	0.613961i	$0.210425\pi$
$500$	0	0
$501$	4413.75	0.393596
$502$	0	0
$503$	1120.29	0.0993069	0.0496535	−	0.998767i	$-0.484188\pi$
$503$	1120.29	0.0993069	0.0496535	+	0.998767i	$0.484188\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−6494.79	−0.568922
$508$	0	0
$509$	−6857.11	−0.597124	−0.298562	−	0.954390i	$-0.596507\pi$
$509$	−6857.11	−0.597124	−0.298562	+	0.954390i	$0.596507\pi$
$510$	0	0
$511$	−10958.2	−0.948653
$512$	0	0
$513$	−126.445	−0.0108824
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6684.70	0.568651
$518$	0	0
$519$	−10899.2	−0.921814
$520$	0	0
$521$	1950.10	0.163984	0.0819919	−	0.996633i	$-0.473872\pi$
$521$	1950.10	0.163984	0.0819919	+	0.996633i	$0.473872\pi$
$522$	0	0
$523$	−9869.11	−0.825136	−0.412568	−	0.910927i	$-0.635368\pi$
$523$	−9869.11	−0.825136	−0.412568	+	0.910927i	$0.635368\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3897.44	0.322154
$528$	0	0
$529$	9146.92	0.751781
$530$	0	0
$531$	−1833.83	−0.149871
$532$	0	0
$533$	1261.92	0.102552
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3987.83	0.320461
$538$	0	0
$539$	6101.82	0.487614
$540$	0	0
$541$	−20927.4	−1.66311	−0.831554	−	0.555444i	$-0.812548\pi$
$541$	−20927.4	−1.66311	−0.831554	+	0.555444i	$0.812548\pi$
$542$	0	0
$543$	78.9858	0.00624237
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−13965.8	−1.09165	−0.545826	−	0.837899i	$-0.683784\pi$
$547$	−13965.8	−1.09165	−0.545826	+	0.837899i	$0.683784\pi$
$548$	0	0
$549$	3207.67	0.249363
$550$	0	0
$551$	−120.278	−0.00929951
$552$	0	0
$553$	−12516.6	−0.962496
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−311.693	−0.0237107	−0.0118554	−	0.999930i	$-0.503774\pi$
$557$	−311.693	−0.0237107	−0.0118554	+	0.999930i	$0.503774\pi$
$558$	0	0
$559$	−1325.38	−0.100282
$560$	0	0
$561$	4631.45	0.348556
$562$	0	0
$563$	5476.71	0.409975	0.204988	−	0.978765i	$-0.434285\pi$
$563$	5476.71	0.409975	0.204988	+	0.978765i	$0.434285\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1244.78	0.0921970
$568$	0	0
$569$	−13928.0	−1.02617	−0.513087	−	0.858337i	$-0.671498\pi$
$569$	−13928.0	−1.02617	−0.513087	+	0.858337i	$0.671498\pi$
$570$	0	0
$571$	26597.3	1.94932	0.974662	−	0.223681i	$-0.0718074\pi$
$571$	26597.3	1.94932	0.974662	+	0.223681i	$0.0718074\pi$
$572$	0	0
$573$	−7919.72	−0.577402
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3556.02	−0.256567	−0.128283	−	0.991738i	$-0.540947\pi$
$577$	−3556.02	−0.256567	−0.128283	+	0.991738i	$0.540947\pi$
$578$	0	0
$579$	−2161.11	−0.155117
$580$	0	0
$581$	−14210.6	−1.01472
$582$	0	0
$583$	−26287.5	−1.86744
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14010.8	−0.985156	−0.492578	−	0.870268i	$-0.663945\pi$
$587$	−14010.8	−0.985156	−0.492578	+	0.870268i	$0.663945\pi$
$588$	0	0
$589$	675.244	0.0472376
$590$	0	0
$591$	−5306.26	−0.369324
$592$	0	0
$593$	15250.5	1.05609	0.528046	−	0.849216i	$-0.322925\pi$
$593$	15250.5	1.05609	0.528046	+	0.849216i	$0.322925\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5851.51	0.401150
$598$	0	0
$599$	−22833.8	−1.55754	−0.778768	−	0.627311i	$-0.784155\pi$
$599$	−22833.8	−1.55754	−0.778768	+	0.627311i	$0.784155\pi$
$600$	0	0
$601$	27732.7	1.88226	0.941131	−	0.338043i	$-0.109765\pi$
$601$	27732.7	1.88226	0.941131	+	0.338043i	$0.109765\pi$
$602$	0	0
$603$	181.533	0.0122597
$604$	0	0
$605$	0	0
$606$	0	0
$607$	26899.7	1.79873	0.899363	−	0.437203i	$-0.144031\pi$
$607$	26899.7	1.79873	0.899363	+	0.437203i	$0.144031\pi$
$608$	0	0
$609$	1184.07	0.0787862
$610$	0	0
$611$	662.824	0.0438870
$612$	0	0
$613$	−18811.6	−1.23947	−0.619733	−	0.784813i	$-0.712759\pi$
$613$	−18811.6	−1.23947	−0.619733	+	0.784813i	$0.712759\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−19764.1	−1.28959	−0.644793	−	0.764357i	$-0.723056\pi$
$617$	−19764.1	−1.28959	−0.644793	+	0.764357i	$0.723056\pi$
$618$	0	0
$619$	−4748.71	−0.308347	−0.154174	−	0.988044i	$-0.549272\pi$
$619$	−4748.71	−0.308347	−0.154174	+	0.988044i	$0.549272\pi$
$620$	0	0
$621$	3941.81	0.254717
$622$	0	0
$623$	−25176.3	−1.61905
$624$	0	0
$625$	0	0
$626$	0	0
$627$	802.415	0.0511090
$628$	0	0
$629$	247.113	0.0156646
$630$	0	0
$631$	−29801.4	−1.88015	−0.940076	−	0.340965i	$-0.889246\pi$
$631$	−29801.4	−1.88015	−0.940076	+	0.340965i	$0.889246\pi$
$632$	0	0
$633$	−7144.84	−0.448629
$634$	0	0
$635$	0	0
$636$	0	0
$637$	605.028	0.0376328
$638$	0	0
$639$	−4515.32	−0.279536
$640$	0	0
$641$	5350.51	0.329692	0.164846	−	0.986319i	$-0.447287\pi$
$641$	5350.51	0.329692	0.164846	+	0.986319i	$0.447287\pi$
$642$	0	0
$643$	−18428.5	−1.13025	−0.565125	−	0.825005i	$-0.691172\pi$
$643$	−18428.5	−1.13025	−0.565125	+	0.825005i	$0.691172\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1269.82	0.0771588	0.0385794	−	0.999256i	$-0.487717\pi$
$647$	1269.82	0.0771588	0.0385794	+	0.999256i	$0.487717\pi$
$648$	0	0
$649$	11637.4	0.703863
$650$	0	0
$651$	−6647.36	−0.400201
$652$	0	0
$653$	29619.2	1.77502	0.887510	−	0.460788i	$-0.152433\pi$
$653$	29619.2	1.77502	0.887510	+	0.460788i	$0.152433\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−6417.64	−0.381090
$658$	0	0
$659$	−30435.2	−1.79907	−0.899534	−	0.436850i	$-0.856094\pi$
$659$	−30435.2	−1.79907	−0.899534	+	0.436850i	$0.856094\pi$
$660$	0	0
$661$	10059.8	0.591953	0.295976	−	0.955195i	$-0.404355\pi$
$661$	10059.8	0.591953	0.295976	+	0.955195i	$0.404355\pi$
$662$	0	0
$663$	459.234	0.0269007
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3749.56	0.217666
$668$	0	0
$669$	−7689.96	−0.444411
$670$	0	0
$671$	−20355.7	−1.17112
$672$	0	0
$673$	17171.0	0.983499	0.491750	−	0.870737i	$-0.336358\pi$
$673$	17171.0	0.983499	0.491750	+	0.870737i	$0.336358\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12992.5	0.737579	0.368790	−	0.929513i	$-0.379772\pi$
$677$	12992.5	0.737579	0.368790	+	0.929513i	$0.379772\pi$
$678$	0	0
$679$	21712.2	1.22716
$680$	0	0
$681$	−655.172	−0.0368667
$682$	0	0
$683$	18595.6	1.04179	0.520893	−	0.853622i	$-0.325599\pi$
$683$	18595.6	1.04179	0.520893	+	0.853622i	$0.325599\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−18692.5	−1.03808
$688$	0	0
$689$	−2606.54	−0.144124
$690$	0	0
$691$	21617.6	1.19012	0.595059	−	0.803682i	$-0.297129\pi$
$691$	21617.6	1.19012	0.595059	+	0.803682i	$0.297129\pi$
$692$	0	0
$693$	−7899.28	−0.433000
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6023.31	0.327330
$698$	0	0
$699$	−8621.09	−0.466495
$700$	0	0
$701$	−26307.6	−1.41744	−0.708719	−	0.705491i	$-0.750727\pi$
$701$	−26307.6	−1.41744	−0.708719	+	0.705491i	$0.750727\pi$
$702$	0	0
$703$	42.8131	0.00229691
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5179.25	−0.275510
$708$	0	0
$709$	−4112.11	−0.217819	−0.108909	−	0.994052i	$-0.534736\pi$
$709$	−4112.11	−0.217819	−0.108909	+	0.994052i	$0.534736\pi$
$710$	0	0
$711$	−7330.32	−0.386650
$712$	0	0
$713$	−21050.1	−1.10565
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7789.48	−0.405723
$718$	0	0
$719$	−29343.0	−1.52199	−0.760993	−	0.648760i	$-0.775288\pi$
$719$	−29343.0	−1.52199	−0.760993	+	0.648760i	$0.775288\pi$
$720$	0	0
$721$	−31570.5	−1.63072
$722$	0	0
$723$	14778.2	0.760174
$724$	0	0
$725$	0	0
$726$	0	0
$727$	10792.5	0.550578	0.275289	−	0.961362i	$-0.411226\pi$
$727$	10792.5	0.550578	0.275289	+	0.961362i	$0.411226\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−6326.21	−0.320087
$732$	0	0
$733$	12900.9	0.650077	0.325038	−	0.945701i	$-0.394623\pi$
$733$	12900.9	0.650077	0.325038	+	0.945701i	$0.394623\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1152.00	−0.0575773
$738$	0	0
$739$	32561.0	1.62080	0.810402	−	0.585874i	$-0.199249\pi$
$739$	32561.0	1.62080	0.810402	+	0.585874i	$0.199249\pi$
$740$	0	0
$741$	79.5638	0.00394446
$742$	0	0
$743$	−4417.40	−0.218114	−0.109057	−	0.994035i	$-0.534783\pi$
$743$	−4417.40	−0.218114	−0.109057	+	0.994035i	$0.534783\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−8322.38	−0.407631
$748$	0	0
$749$	5706.82	0.278402
$750$	0	0
$751$	−3091.88	−0.150232	−0.0751161	−	0.997175i	$-0.523933\pi$
$751$	−3091.88	−0.150232	−0.0751161	+	0.997175i	$0.523933\pi$
$752$	0	0
$753$	14301.7	0.692142
$754$	0	0
$755$	0	0
$756$	0	0
$757$	23570.0	1.13166	0.565831	−	0.824521i	$-0.308556\pi$
$757$	23570.0	1.13166	0.565831	+	0.824521i	$0.308556\pi$
$758$	0	0
$759$	−25014.5	−1.19627
$760$	0	0
$761$	6901.48	0.328750	0.164375	−	0.986398i	$-0.447439\pi$
$761$	6901.48	0.328750	0.164375	+	0.986398i	$0.447439\pi$
$762$	0	0
$763$	4757.15	0.225715
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1153.91	0.0543223
$768$	0	0
$769$	88.6468	0.00415694	0.00207847	−	0.999998i	$-0.499338\pi$
$769$	88.6468	0.00415694	0.00207847	+	0.999998i	$0.499338\pi$
$770$	0	0
$771$	12166.6	0.568313
$772$	0	0
$773$	13345.9	0.620980	0.310490	−	0.950577i	$-0.399507\pi$
$773$	13345.9	0.620980	0.310490	+	0.950577i	$0.399507\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−421.469	−0.0194596
$778$	0	0
$779$	1043.56	0.0479966
$780$	0	0
$781$	28653.9	1.31283
$782$	0	0
$783$	693.445	0.0316497
$784$	0	0
$785$	0	0
$786$	0	0
$787$	2185.46	0.0989877	0.0494939	−	0.998774i	$-0.484239\pi$
$787$	2185.46	0.0989877	0.0494939	+	0.998774i	$0.484239\pi$
$788$	0	0
$789$	6577.70	0.296796
$790$	0	0
$791$	−12858.0	−0.577974
$792$	0	0
$793$	−2018.38	−0.0903842
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20791.9	0.924073	0.462036	−	0.886861i	$-0.347119\pi$
$797$	20791.9	0.924073	0.462036	+	0.886861i	$0.347119\pi$
$798$	0	0
$799$	3163.74	0.140081
$800$	0	0
$801$	−14744.4	−0.650398
$802$	0	0
$803$	40726.0	1.78977
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15823.4	0.690223
$808$	0	0
$809$	29004.3	1.26049	0.630245	−	0.776396i	$-0.282955\pi$
$809$	29004.3	1.26049	0.630245	+	0.776396i	$0.282955\pi$
$810$	0	0
$811$	−1088.27	−0.0471199	−0.0235600	−	0.999722i	$-0.507500\pi$
$811$	−1088.27	−0.0471199	−0.0235600	+	0.999722i	$0.507500\pi$
$812$	0	0
$813$	13016.1	0.561496
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1096.04	−0.0469345
$818$	0	0
$819$	−783.256	−0.0334178
$820$	0	0
$821$	23237.5	0.987815	0.493907	−	0.869515i	$-0.335568\pi$
$821$	23237.5	0.987815	0.493907	+	0.869515i	$0.335568\pi$
$822$	0	0
$823$	−10035.1	−0.425032	−0.212516	−	0.977158i	$-0.568166\pi$
$823$	−10035.1	−0.425032	−0.212516	+	0.977158i	$0.568166\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−33397.7	−1.40429	−0.702147	−	0.712032i	$-0.747775\pi$
$827$	−33397.7	−1.40429	−0.702147	+	0.712032i	$0.747775\pi$
$828$	0	0
$829$	−28989.3	−1.21453	−0.607263	−	0.794501i	$-0.707732\pi$
$829$	−28989.3	−1.21453	−0.607263	+	0.794501i	$0.707732\pi$
$830$	0	0
$831$	−19742.7	−0.824148
$832$	0	0
$833$	2887.87	0.120119
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3893.01	−0.160767
$838$	0	0
$839$	37189.9	1.53032	0.765160	−	0.643840i	$-0.222660\pi$
$839$	37189.9	1.53032	0.765160	+	0.643840i	$0.222660\pi$
$840$	0	0
$841$	−23729.4	−0.972954
$842$	0	0
$843$	9995.55	0.408381
$844$	0	0
$845$	0	0
$846$	0	0
$847$	29674.1	1.20379
$848$	0	0
$849$	10179.4	0.411493
$850$	0	0
$851$	−1334.66	−0.0537620
$852$	0	0
$853$	36074.0	1.44801	0.724004	−	0.689796i	$-0.242300\pi$
$853$	36074.0	1.44801	0.724004	+	0.689796i	$0.242300\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2666.63	0.106290	0.0531449	−	0.998587i	$-0.483075\pi$
$857$	2666.63	0.106290	0.0531449	+	0.998587i	$0.483075\pi$
$858$	0	0
$859$	12818.1	0.509134	0.254567	−	0.967055i	$-0.418067\pi$
$859$	12818.1	0.509134	0.254567	+	0.967055i	$0.418067\pi$
$860$	0	0
$861$	−10273.2	−0.406631
$862$	0	0
$863$	2882.93	0.113715	0.0568575	−	0.998382i	$-0.481892\pi$
$863$	2882.93	0.113715	0.0568575	+	0.998382i	$0.481892\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−12547.0	−0.491487
$868$	0	0
$869$	46517.8	1.81589
$870$	0	0
$871$	−114.227	−0.00444367
$872$	0	0
$873$	12715.7	0.492969
$874$	0	0
$875$	0	0
$876$	0	0
$877$	989.168	0.0380865	0.0190432	−	0.999819i	$-0.493938\pi$
$877$	989.168	0.0380865	0.0190432	+	0.999819i	$0.493938\pi$
$878$	0	0
$879$	−20435.0	−0.784136
$880$	0	0
$881$	−39678.2	−1.51736	−0.758678	−	0.651466i	$-0.774154\pi$
$881$	−39678.2	−1.51736	−0.758678	+	0.651466i	$0.774154\pi$
$882$	0	0
$883$	−15619.3	−0.595278	−0.297639	−	0.954679i	$-0.596199\pi$
$883$	−15619.3	−0.595278	−0.297639	+	0.954679i	$0.596199\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30078.6	1.13860	0.569301	−	0.822129i	$-0.307214\pi$
$887$	30078.6	1.13860	0.569301	+	0.822129i	$0.307214\pi$
$888$	0	0
$889$	13500.6	0.509331
$890$	0	0
$891$	−4626.19	−0.173943
$892$	0	0
$893$	548.128	0.0205402
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2480.32	−0.0923250
$898$	0	0
$899$	−3703.14	−0.137382
$900$	0	0
$901$	−12441.3	−0.460024
$902$	0	0
$903$	10789.8	0.397632
$904$	0	0
$905$	0	0
$906$	0	0
$907$	52235.2	1.91229	0.956143	−	0.292901i	$-0.0946207\pi$
$907$	52235.2	1.91229	0.956143	+	0.292901i	$0.0946207\pi$
$908$	0	0
$909$	−3033.21	−0.110677
$910$	0	0
$911$	−48299.7	−1.75658	−0.878288	−	0.478132i	$-0.841314\pi$
$911$	−48299.7	−1.75658	−0.878288	+	0.478132i	$0.841314\pi$
$912$	0	0
$913$	52813.4	1.91442
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3195.26	0.115067
$918$	0	0
$919$	23957.4	0.859935	0.429968	−	0.902844i	$-0.358525\pi$
$919$	23957.4	0.859935	0.429968	+	0.902844i	$0.358525\pi$
$920$	0	0
$921$	−14019.4	−0.501580
$922$	0	0
$923$	2841.19	0.101321
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−18489.2	−0.655087
$928$	0	0
$929$	9838.50	0.347460	0.173730	−	0.984793i	$-0.444418\pi$
$929$	9838.50	0.347460	0.173730	+	0.984793i	$0.444418\pi$
$930$	0	0
$931$	500.333	0.0176131
$932$	0	0
$933$	−9292.53	−0.326070
$934$	0	0
$935$	0	0
$936$	0	0
$937$	55492.0	1.93473	0.967367	−	0.253380i	$-0.0815422\pi$
$937$	55492.0	1.93473	0.967367	+	0.253380i	$0.0815422\pi$
$938$	0	0
$939$	−4712.79	−0.163787
$940$	0	0
$941$	9714.31	0.336533	0.168266	−	0.985742i	$-0.446183\pi$
$941$	9714.31	0.336533	0.168266	+	0.985742i	$0.446183\pi$
$942$	0	0
$943$	−32531.9	−1.12342
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−10533.5	−0.361451	−0.180725	−	0.983534i	$-0.557845\pi$
$947$	−10533.5	−0.361451	−0.180725	+	0.983534i	$0.557845\pi$
$948$	0	0
$949$	4038.20	0.138130
$950$	0	0
$951$	−5204.28	−0.177456
$952$	0	0
$953$	−24457.5	−0.831328	−0.415664	−	0.909518i	$-0.636451\pi$
$953$	−24457.5	−0.831328	−0.415664	+	0.909518i	$0.636451\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4400.56	−0.148642
$958$	0	0
$959$	−29106.6	−0.980085
$960$	0	0
$961$	−9001.52	−0.302156
$962$	0	0
$963$	3342.19	0.111838
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−12612.1	−0.419419	−0.209710	−	0.977764i	$-0.567252\pi$
$967$	−12612.1	−0.419419	−0.209710	+	0.977764i	$0.567252\pi$
$968$	0	0
$969$	379.767	0.0125902
$970$	0	0
$971$	47076.0	1.55586	0.777930	−	0.628350i	$-0.216270\pi$
$971$	47076.0	1.55586	0.777930	+	0.628350i	$0.216270\pi$
$972$	0	0
$973$	26489.8	0.872789
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12409.7	0.406367	0.203183	−	0.979141i	$-0.434871\pi$
$977$	12409.7	0.406367	0.203183	+	0.979141i	$0.434871\pi$
$978$	0	0
$979$	93567.3	3.05457
$980$	0	0
$981$	2786.01	0.0906733
$982$	0	0
$983$	−18983.3	−0.615943	−0.307972	−	0.951396i	$-0.599650\pi$
$983$	−18983.3	−0.615943	−0.307972	+	0.951396i	$0.599650\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5395.98	−0.174018
$988$	0	0
$989$	34167.9	1.09856
$990$	0	0
$991$	45928.0	1.47220	0.736101	−	0.676872i	$-0.236665\pi$
$991$	45928.0	1.47220	0.736101	+	0.676872i	$0.236665\pi$
$992$	0	0
$993$	22614.4	0.722706
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−34448.3	−1.09427	−0.547136	−	0.837043i	$-0.684282\pi$
$997$	−34448.3	−1.09427	−0.547136	+	0.837043i	$0.684282\pi$
$998$	0	0
$999$	−246.832	−0.00781724

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.bq.1.3	yes	3
4.3	odd	2		2400.4.a.bk.1.1	✓	3
5.4	even	2		2400.4.a.bl.1.1	yes	3
20.19	odd	2		2400.4.a.br.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.4.a.bk.1.1	✓	3	4.3	odd	2
2400.4.a.bl.1.1	yes	3	5.4	even	2
2400.4.a.bq.1.3	yes	3	1.1	even	1	trivial
2400.4.a.br.1.3	yes	3	20.19	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +15.3676 q^{7} +9.00000 q^{9} -57.1135 q^{11} -5.66311 q^{13} -27.0307 q^{17} -4.68316 q^{19} +46.1028 q^{21} +145.993 q^{23} +27.0000 q^{27} +25.6832 q^{29} -144.186 q^{31} -171.340 q^{33} -9.14193 q^{37} -16.9893 q^{39} -222.832 q^{41} +234.038 q^{43} -117.042 q^{47} -106.837 q^{49} -81.0921 q^{51} +460.267 q^{53} -14.0495 q^{57} -203.759 q^{59} +356.408 q^{61} +138.308 q^{63} +20.1704 q^{67} +437.979 q^{69} -501.702 q^{71} -713.071 q^{73} -877.697 q^{77} -814.480 q^{79} +81.0000 q^{81} -924.709 q^{83} +77.0495 q^{87} -1638.27 q^{89} -87.0284 q^{91} -432.557 q^{93} +1412.86 q^{97} -514.021 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} - 7 q^{7} + 27 q^{9} - 21 q^{13} - 32 q^{17} + 19 q^{19} - 21 q^{21} - 60 q^{23} + 81 q^{27} + 44 q^{29} - 151 q^{31} - 330 q^{37} - 63 q^{39} + 82 q^{41} - 367 q^{43} + 246 q^{47} - 42 q^{49}+ \cdots + 351 q^{97}+O(q^{100}) \) 3 * q + 9 * q^3 - 7 * q^7 + 27 * q^9 - 21 * q^13 - 32 * q^17 + 19 * q^19 - 21 * q^21 - 60 * q^23 + 81 * q^27 + 44 * q^29 - 151 * q^31 - 330 * q^37 - 63 * q^39 + 82 * q^41 - 367 * q^43 + 246 * q^47 - 42 * q^49 - 96 * q^51 - 554 * q^53 + 57 * q^57 + 42 * q^59 - 27 * q^61 - 63 * q^63 + 837 * q^67 - 180 * q^69 - 1422 * q^71 - 498 * q^73 - 560 * q^77 - 688 * q^79 + 243 * q^81 - 702 * q^83 + 132 * q^87 - 1488 * q^89 - 735 * q^91 - 453 * q^93 + 351 * q^97

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