LMFDB - Embedded newform 608.4.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [608,4,Mod(1,608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(608, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("608.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	608.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:	$35.8731612835$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 121x^{5} + 402x^{4} + 4234x^{3} - 14542x^{2} - 40996x + 141664 $ x^7 - 3x^6 - 121x^5 + 402x^4 + 4234x^3 - 14542x^2 - 40996x + 141664
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-4.00212$ of defining polynomial
Character	$\chi$	$=$	608.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.923414 q^{3} -0.670967 q^{5} -28.6780 q^{7} -26.1473 q^{9} +24.1312 q^{11} -17.5373 q^{13} +0.619580 q^{15} -61.1485 q^{17} -19.0000 q^{19} +26.4817 q^{21} +160.534 q^{23} -124.550 q^{25} +49.0770 q^{27} +287.921 q^{29} -47.5884 q^{31} -22.2831 q^{33} +19.2420 q^{35} -237.442 q^{37} +16.1942 q^{39} +197.712 q^{41} +352.718 q^{43} +17.5440 q^{45} +357.106 q^{47} +479.427 q^{49} +56.4654 q^{51} +381.501 q^{53} -16.1912 q^{55} +17.5449 q^{57} -691.149 q^{59} -827.651 q^{61} +749.852 q^{63} +11.7669 q^{65} +166.560 q^{67} -148.240 q^{69} +296.418 q^{71} -916.657 q^{73} +115.011 q^{75} -692.034 q^{77} +1229.64 q^{79} +660.659 q^{81} +547.516 q^{83} +41.0286 q^{85} -265.870 q^{87} -257.418 q^{89} +502.934 q^{91} +43.9438 q^{93} +12.7484 q^{95} +206.366 q^{97} -630.965 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{3} - 17 q^{5} + 42 q^{7} + 86 q^{9} - 33 q^{11} - 35 q^{13} + 120 q^{15} + 66 q^{17} - 133 q^{19} + 33 q^{21} + 389 q^{23} + 44 q^{25} - 39 q^{27} + 233 q^{29} + 158 q^{31} - 206 q^{33} + 123 q^{35}+ \cdots - 1591 q^{99}+O(q^{100}) $ 7 * q + 3 * q^3 - 17 * q^5 + 42 * q^7 + 86 * q^9 - 33 * q^11 - 35 * q^13 + 120 * q^15 + 66 * q^17 - 133 * q^19 + 33 * q^21 + 389 * q^23 + 44 * q^25 - 39 * q^27 + 233 * q^29 + 158 * q^31 - 206 * q^33 + 123 * q^35 - 436 * q^37 + 807 * q^39 - 94 * q^41 + 645 * q^43 + 103 * q^45 + 1451 * q^47 + 93 * q^49 + 1741 * q^51 + 3 * q^53 + 1971 * q^55 - 57 * q^57 + 297 * q^59 + 93 * q^61 + 2999 * q^63 - 788 * q^65 + 1641 * q^67 + 945 * q^69 + 2392 * q^71 + 324 * q^73 + 1909 * q^75 - 711 * q^77 + 2492 * q^79 + 143 * q^81 + 310 * q^83 + 2353 * q^85 + 4795 * q^87 - 440 * q^89 - 107 * q^91 + 900 * q^93 + 323 * q^95 - 532 * q^97 - 1591 * q^99

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
$3$	−0.923414	−0.177711	−0.0888556	−	0.996045i	$-0.528321\pi$
$3$	−0.923414	−0.177711	−0.0888556	+	0.996045i	$0.528321\pi$
$4$	0	0
$5$	−0.670967	−0.0600131	−0.0300066	−	0.999550i	$-0.509553\pi$
$5$	−0.670967	−0.0600131	−0.0300066	+	0.999550i	$0.509553\pi$
$6$	0	0
$7$	−28.6780	−1.54847	−0.774233	−	0.632901i	$-0.781864\pi$
$7$	−28.6780	−1.54847	−0.774233	+	0.632901i	$0.781864\pi$
$8$	0	0
$9$	−26.1473	−0.968419
$10$	0	0
$11$	24.1312	0.661439	0.330719	−	0.943729i	$-0.392709\pi$
$11$	24.1312	0.661439	0.330719	+	0.943729i	$0.392709\pi$
$12$	0	0
$13$	−17.5373	−0.374152	−0.187076	−	0.982345i	$-0.559901\pi$
$13$	−17.5373	−0.374152	−0.187076	+	0.982345i	$0.559901\pi$
$14$	0	0
$15$	0.619580	0.0106650
$16$	0	0
$17$	−61.1485	−0.872393	−0.436197	−	0.899851i	$-0.643675\pi$
$17$	−61.1485	−0.872393	−0.436197	+	0.899851i	$0.643675\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	26.4817	0.275180
$22$	0	0
$23$	160.534	1.45538	0.727689	−	0.685907i	$-0.240594\pi$
$23$	160.534	1.45538	0.727689	+	0.685907i	$0.240594\pi$
$24$	0	0
$25$	−124.550	−0.996398
$26$	0	0
$27$	49.0770	0.349810
$28$	0	0
$29$	287.921	1.84364	0.921819	−	0.387620i	$-0.126703\pi$
$29$	287.921	1.84364	0.921819	+	0.387620i	$0.126703\pi$
$30$	0	0
$31$	−47.5884	−0.275714	−0.137857	−	0.990452i	$-0.544021\pi$
$31$	−47.5884	−0.275714	−0.137857	+	0.990452i	$0.544021\pi$
$32$	0	0
$33$	−22.2831	−0.117545
$34$	0	0
$35$	19.2420	0.0929283
$36$	0	0
$37$	−237.442	−1.05501	−0.527504	−	0.849552i	$-0.676872\pi$
$37$	−237.442	−1.05501	−0.527504	+	0.849552i	$0.676872\pi$
$38$	0	0
$39$	16.1942	0.0664909
$40$	0	0
$41$	197.712	0.753109	0.376554	−	0.926395i	$-0.377109\pi$
$41$	197.712	0.753109	0.376554	+	0.926395i	$0.377109\pi$
$42$	0	0
$43$	352.718	1.25091	0.625454	−	0.780261i	$-0.284914\pi$
$43$	352.718	1.25091	0.625454	+	0.780261i	$0.284914\pi$
$44$	0	0
$45$	17.5440	0.0581178
$46$	0	0
$47$	357.106	1.10828	0.554141	−	0.832423i	$-0.313047\pi$
$47$	357.106	1.10828	0.554141	+	0.832423i	$0.313047\pi$
$48$	0	0
$49$	479.427	1.39775
$50$	0	0
$51$	56.4654	0.155034
$52$	0	0
$53$	381.501	0.988739	0.494369	−	0.869252i	$-0.335399\pi$
$53$	381.501	0.988739	0.494369	+	0.869252i	$0.335399\pi$
$54$	0	0
$55$	−16.1912	−0.0396950
$56$	0	0
$57$	17.5449	0.0407697
$58$	0	0
$59$	−691.149	−1.52508	−0.762542	−	0.646938i	$-0.776049\pi$
$59$	−691.149	−1.52508	−0.762542	+	0.646938i	$0.776049\pi$
$60$	0	0
$61$	−827.651	−1.73721	−0.868606	−	0.495504i	$-0.834983\pi$
$61$	−827.651	−1.73721	−0.868606	+	0.495504i	$0.834983\pi$
$62$	0	0
$63$	749.852	1.49956
$64$	0	0
$65$	11.7669	0.0224540
$66$	0	0
$67$	166.560	0.303710	0.151855	−	0.988403i	$-0.451475\pi$
$67$	166.560	0.303710	0.151855	+	0.988403i	$0.451475\pi$
$68$	0	0
$69$	−148.240	−0.258637
$70$	0	0
$71$	296.418	0.495470	0.247735	−	0.968828i	$-0.420314\pi$
$71$	296.418	0.495470	0.247735	+	0.968828i	$0.420314\pi$
$72$	0	0
$73$	−916.657	−1.46968	−0.734840	−	0.678241i	$-0.762743\pi$
$73$	−916.657	−1.46968	−0.734840	+	0.678241i	$0.762743\pi$
$74$	0	0
$75$	115.011	0.177071
$76$	0	0
$77$	−692.034	−1.02422
$78$	0	0
$79$	1229.64	1.75120	0.875601	−	0.483036i	$-0.160466\pi$
$79$	1229.64	1.75120	0.875601	+	0.483036i	$0.160466\pi$
$80$	0	0
$81$	660.659	0.906254
$82$	0	0
$83$	547.516	0.724069	0.362035	−	0.932165i	$-0.382082\pi$
$83$	547.516	0.724069	0.362035	+	0.932165i	$0.382082\pi$
$84$	0	0
$85$	41.0286	0.0523550
$86$	0	0
$87$	−265.870	−0.327635
$88$	0	0
$89$	−257.418	−0.306587	−0.153293	−	0.988181i	$-0.548988\pi$
$89$	−257.418	−0.306587	−0.153293	+	0.988181i	$0.548988\pi$
$90$	0	0
$91$	502.934	0.579361
$92$	0	0
$93$	43.9438	0.0489974
$94$	0	0
$95$	12.7484	0.0137679
$96$	0	0
$97$	206.366	0.216013	0.108007	−	0.994150i	$-0.465553\pi$
$97$	206.366	0.216013	0.108007	+	0.994150i	$0.465553\pi$
$98$	0	0
$99$	−630.965	−0.640550
$100$	0	0
$101$	560.665	0.552359	0.276179	−	0.961106i	$-0.410932\pi$
$101$	560.665	0.552359	0.276179	+	0.961106i	$0.410932\pi$
$102$	0	0
$103$	1325.06	1.26759	0.633796	−	0.773500i	$-0.281496\pi$
$103$	1325.06	1.26759	0.633796	+	0.773500i	$0.281496\pi$
$104$	0	0
$105$	−17.7683	−0.0165144
$106$	0	0
$107$	1278.56	1.15517	0.577583	−	0.816332i	$-0.303996\pi$
$107$	1278.56	1.15517	0.577583	+	0.816332i	$0.303996\pi$
$108$	0	0
$109$	313.965	0.275893	0.137947	−	0.990440i	$-0.455950\pi$
$109$	313.965	0.275893	0.137947	+	0.990440i	$0.455950\pi$
$110$	0	0
$111$	219.258	0.187487
$112$	0	0
$113$	2248.70	1.87203	0.936016	−	0.351958i	$-0.114484\pi$
$113$	2248.70	1.87203	0.936016	+	0.351958i	$0.114484\pi$
$114$	0	0
$115$	−107.713	−0.0873418
$116$	0	0
$117$	458.553	0.362335
$118$	0	0
$119$	1753.62	1.35087
$120$	0	0
$121$	−748.686	−0.562499
$122$	0	0
$123$	−182.570	−0.133836
$124$	0	0
$125$	167.440	0.119810
$126$	0	0
$127$	1045.71	0.730644	0.365322	−	0.930881i	$-0.380959\pi$
$127$	1045.71	0.730644	0.365322	+	0.930881i	$0.380959\pi$
$128$	0	0
$129$	−325.705	−0.222300
$130$	0	0
$131$	−1723.55	−1.14952	−0.574761	−	0.818321i	$-0.694905\pi$
$131$	−1723.55	−1.14952	−0.574761	+	0.818321i	$0.694905\pi$
$132$	0	0
$133$	544.882	0.355243
$134$	0	0
$135$	−32.9290	−0.0209932
$136$	0	0
$137$	−740.069	−0.461521	−0.230761	−	0.973011i	$-0.574121\pi$
$137$	−740.069	−0.461521	−0.230761	+	0.973011i	$0.574121\pi$
$138$	0	0
$139$	988.103	0.602948	0.301474	−	0.953474i	$-0.402521\pi$
$139$	988.103	0.602948	0.301474	+	0.953474i	$0.402521\pi$
$140$	0	0
$141$	−329.756	−0.196954
$142$	0	0
$143$	−423.196	−0.247478
$144$	0	0
$145$	−193.185	−0.110642
$146$	0	0
$147$	−442.710	−0.248395
$148$	0	0
$149$	2070.89	1.13862	0.569308	−	0.822125i	$-0.307211\pi$
$149$	2070.89	1.13862	0.569308	+	0.822125i	$0.307211\pi$
$150$	0	0
$151$	−2702.62	−1.45653	−0.728267	−	0.685294i	$-0.759674\pi$
$151$	−2702.62	−1.45653	−0.728267	+	0.685294i	$0.759674\pi$
$152$	0	0
$153$	1598.87	0.844842
$154$	0	0
$155$	31.9303	0.0165465
$156$	0	0
$157$	−190.958	−0.0970706	−0.0485353	−	0.998821i	$-0.515455\pi$
$157$	−190.958	−0.0970706	−0.0485353	+	0.998821i	$0.515455\pi$
$158$	0	0
$159$	−352.283	−0.175710
$160$	0	0
$161$	−4603.80	−2.25360
$162$	0	0
$163$	−3222.79	−1.54864	−0.774320	−	0.632795i	$-0.781908\pi$
$163$	−3222.79	−1.54864	−0.774320	+	0.632795i	$0.781908\pi$
$164$	0	0
$165$	14.9512	0.00705424
$166$	0	0
$167$	3334.86	1.54526	0.772632	−	0.634854i	$-0.218940\pi$
$167$	3334.86	1.54526	0.772632	+	0.634854i	$0.218940\pi$
$168$	0	0
$169$	−1889.44	−0.860011
$170$	0	0
$171$	496.799	0.222170
$172$	0	0
$173$	1037.10	0.455774	0.227887	−	0.973688i	$-0.426818\pi$
$173$	1037.10	0.455774	0.227887	+	0.973688i	$0.426818\pi$
$174$	0	0
$175$	3571.84	1.54289
$176$	0	0
$177$	638.217	0.271025
$178$	0	0
$179$	1524.12	0.636412	0.318206	−	0.948022i	$-0.396920\pi$
$179$	1524.12	0.636412	0.318206	+	0.948022i	$0.396920\pi$
$180$	0	0
$181$	−4005.50	−1.64490	−0.822448	−	0.568840i	$-0.807392\pi$
$181$	−4005.50	−1.64490	−0.822448	+	0.568840i	$0.807392\pi$
$182$	0	0
$183$	764.265	0.308722
$184$	0	0
$185$	159.316	0.0633143
$186$	0	0
$187$	−1475.59	−0.577035
$188$	0	0
$189$	−1407.43	−0.541669
$190$	0	0
$191$	3756.82	1.42322	0.711608	−	0.702577i	$-0.247967\pi$
$191$	3756.82	1.42322	0.711608	+	0.702577i	$0.247967\pi$
$192$	0	0
$193$	−774.721	−0.288941	−0.144471	−	0.989509i	$-0.546148\pi$
$193$	−774.721	−0.288941	−0.144471	+	0.989509i	$0.546148\pi$
$194$	0	0
$195$	−10.8658	−0.00399032
$196$	0	0
$197$	384.941	0.139218	0.0696089	−	0.997574i	$-0.477825\pi$
$197$	384.941	0.139218	0.0696089	+	0.997574i	$0.477825\pi$
$198$	0	0
$199$	−620.031	−0.220869	−0.110434	−	0.993883i	$-0.535224\pi$
$199$	−620.031	−0.220869	−0.110434	+	0.993883i	$0.535224\pi$
$200$	0	0
$201$	−153.804	−0.0539726
$202$	0	0
$203$	−8256.98	−2.85481
$204$	0	0
$205$	−132.658	−0.0451964
$206$	0	0
$207$	−4197.54	−1.40942
$208$	0	0
$209$	−458.492	−0.151744
$210$	0	0
$211$	−2811.84	−0.917418	−0.458709	−	0.888586i	$-0.651688\pi$
$211$	−2811.84	−0.917418	−0.458709	+	0.888586i	$0.651688\pi$
$212$	0	0
$213$	−273.716	−0.0880504
$214$	0	0
$215$	−236.662	−0.0750709
$216$	0	0
$217$	1364.74	0.426934
$218$	0	0
$219$	846.454	0.261178
$220$	0	0
$221$	1072.38	0.326407
$222$	0	0
$223$	5415.68	1.62628	0.813141	−	0.582066i	$-0.197756\pi$
$223$	5415.68	1.62628	0.813141	+	0.582066i	$0.197756\pi$
$224$	0	0
$225$	3256.64	0.964931
$226$	0	0
$227$	−3930.45	−1.14922	−0.574611	−	0.818427i	$-0.694847\pi$
$227$	−3930.45	−1.14922	−0.574611	+	0.818427i	$0.694847\pi$
$228$	0	0
$229$	2136.62	0.616558	0.308279	−	0.951296i	$-0.400247\pi$
$229$	2136.62	0.616558	0.308279	+	0.951296i	$0.400247\pi$
$230$	0	0
$231$	639.034	0.182014
$232$	0	0
$233$	2841.83	0.799031	0.399516	−	0.916726i	$-0.369178\pi$
$233$	2841.83	0.799031	0.399516	+	0.916726i	$0.369178\pi$
$234$	0	0
$235$	−239.606	−0.0665114
$236$	0	0
$237$	−1135.46	−0.311208
$238$	0	0
$239$	−4786.45	−1.29544	−0.647720	−	0.761879i	$-0.724277\pi$
$239$	−4786.45	−1.29544	−0.647720	+	0.761879i	$0.724277\pi$
$240$	0	0
$241$	3583.38	0.957783	0.478891	−	0.877874i	$-0.341039\pi$
$241$	3583.38	0.957783	0.478891	+	0.877874i	$0.341039\pi$
$242$	0	0
$243$	−1935.14	−0.510861
$244$	0	0
$245$	−321.680	−0.0838832
$246$	0	0
$247$	333.209	0.0858362
$248$	0	0
$249$	−505.584	−0.128675
$250$	0	0
$251$	−2241.78	−0.563744	−0.281872	−	0.959452i	$-0.590955\pi$
$251$	−2241.78	−0.563744	−0.281872	+	0.959452i	$0.590955\pi$
$252$	0	0
$253$	3873.88	0.962643
$254$	0	0
$255$	−37.8864	−0.00930407
$256$	0	0
$257$	−3307.64	−0.802819	−0.401410	−	0.915899i	$-0.631480\pi$
$257$	−3307.64	−0.802819	−0.401410	+	0.915899i	$0.631480\pi$
$258$	0	0
$259$	6809.37	1.63364
$260$	0	0
$261$	−7528.35	−1.78541
$262$	0	0
$263$	−5733.35	−1.34423	−0.672117	−	0.740445i	$-0.734615\pi$
$263$	−5733.35	−1.34423	−0.672117	+	0.740445i	$0.734615\pi$
$264$	0	0
$265$	−255.974	−0.0593373
$266$	0	0
$267$	237.703	0.0544839
$268$	0	0
$269$	4863.10	1.10226	0.551131	−	0.834419i	$-0.314196\pi$
$269$	4863.10	1.10226	0.551131	+	0.834419i	$0.314196\pi$
$270$	0	0
$271$	−6428.81	−1.44104	−0.720521	−	0.693433i	$-0.756097\pi$
$271$	−6428.81	−1.44104	−0.720521	+	0.693433i	$0.756097\pi$
$272$	0	0
$273$	−464.417	−0.102959
$274$	0	0
$275$	−3005.53	−0.659056
$276$	0	0
$277$	6855.48	1.48702	0.743512	−	0.668722i	$-0.233158\pi$
$277$	6855.48	1.48702	0.743512	+	0.668722i	$0.233158\pi$
$278$	0	0
$279$	1244.31	0.267007
$280$	0	0
$281$	1682.58	0.357203	0.178602	−	0.983921i	$-0.442843\pi$
$281$	1682.58	0.357203	0.178602	+	0.983921i	$0.442843\pi$
$282$	0	0
$283$	−5894.74	−1.23818	−0.619091	−	0.785319i	$-0.712499\pi$
$283$	−5894.74	−1.23818	−0.619091	+	0.785319i	$0.712499\pi$
$284$	0	0
$285$	−11.7720	−0.00244672
$286$	0	0
$287$	−5669.99	−1.16616
$288$	0	0
$289$	−1173.86	−0.238930
$290$	0	0
$291$	−190.561	−0.0383879
$292$	0	0
$293$	−6625.17	−1.32098	−0.660489	−	0.750836i	$-0.729651\pi$
$293$	−6625.17	−1.32098	−0.660489	+	0.750836i	$0.729651\pi$
$294$	0	0
$295$	463.738	0.0915251
$296$	0	0
$297$	1184.29	0.231378
$298$	0	0
$299$	−2815.34	−0.544532
$300$	0	0
$301$	−10115.3	−1.93699
$302$	0	0
$303$	−517.726	−0.0981603
$304$	0	0
$305$	555.327	0.104255
$306$	0	0
$307$	−4185.59	−0.778125	−0.389062	−	0.921211i	$-0.627201\pi$
$307$	−4185.59	−0.778125	−0.389062	+	0.921211i	$0.627201\pi$
$308$	0	0
$309$	−1223.58	−0.225265
$310$	0	0
$311$	507.256	0.0924882	0.0462441	−	0.998930i	$-0.485275\pi$
$311$	507.256	0.0924882	0.0462441	+	0.998930i	$0.485275\pi$
$312$	0	0
$313$	10307.5	1.86139	0.930696	−	0.365793i	$-0.119202\pi$
$313$	10307.5	1.86139	0.930696	+	0.365793i	$0.119202\pi$
$314$	0	0
$315$	−503.126	−0.0899935
$316$	0	0
$317$	9.58898	0.00169896	0.000849481	−	1.00000i	$-0.499730\pi$
$317$	9.58898	0.00169896	0.000849481		1.00000i	$0.499730\pi$
$318$	0	0
$319$	6947.86	1.21945
$320$	0	0
$321$	−1180.64	−0.205286
$322$	0	0
$323$	1161.82	0.200141
$324$	0	0
$325$	2184.27	0.372804
$326$	0	0
$327$	−289.919	−0.0490293
$328$	0	0
$329$	−10241.1	−1.71614
$330$	0	0
$331$	1770.81	0.294055	0.147028	−	0.989132i	$-0.453029\pi$
$331$	1770.81	0.294055	0.147028	+	0.989132i	$0.453029\pi$
$332$	0	0
$333$	6208.48	1.02169
$334$	0	0
$335$	−111.756	−0.0182266
$336$	0	0
$337$	1494.60	0.241590	0.120795	−	0.992677i	$-0.461456\pi$
$337$	1494.60	0.241590	0.120795	+	0.992677i	$0.461456\pi$
$338$	0	0
$339$	−2076.48	−0.332681
$340$	0	0
$341$	−1148.36	−0.182368
$342$	0	0
$343$	−3912.46	−0.615898
$344$	0	0
$345$	99.4638	0.0155216
$346$	0	0
$347$	12738.1	1.97065	0.985326	−	0.170685i	$-0.0545979\pi$
$347$	12738.1	1.97065	0.985326	+	0.170685i	$0.0545979\pi$
$348$	0	0
$349$	2634.06	0.404006	0.202003	−	0.979385i	$-0.435255\pi$
$349$	2634.06	0.404006	0.202003	+	0.979385i	$0.435255\pi$
$350$	0	0
$351$	−860.677	−0.130882
$352$	0	0
$353$	−8694.21	−1.31090	−0.655448	−	0.755241i	$-0.727520\pi$
$353$	−8694.21	−1.31090	−0.655448	+	0.755241i	$0.727520\pi$
$354$	0	0
$355$	−198.887	−0.0297347
$356$	0	0
$357$	−1619.31	−0.240065
$358$	0	0
$359$	−4297.38	−0.631774	−0.315887	−	0.948797i	$-0.602302\pi$
$359$	−4297.38	−0.631774	−0.315887	+	0.948797i	$0.602302\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	691.347	0.0999623
$364$	0	0
$365$	615.047	0.0882000
$366$	0	0
$367$	7786.34	1.10748	0.553738	−	0.832691i	$-0.313201\pi$
$367$	7786.34	1.10748	0.553738	+	0.832691i	$0.313201\pi$
$368$	0	0
$369$	−5169.64	−0.729324
$370$	0	0
$371$	−10940.7	−1.53103
$372$	0	0
$373$	3268.99	0.453785	0.226893	−	0.973920i	$-0.427143\pi$
$373$	3268.99	0.453785	0.226893	+	0.973920i	$0.427143\pi$
$374$	0	0
$375$	−154.616	−0.0212916
$376$	0	0
$377$	−5049.35	−0.689800
$378$	0	0
$379$	2284.10	0.309569	0.154784	−	0.987948i	$-0.450532\pi$
$379$	2284.10	0.309569	0.154784	+	0.987948i	$0.450532\pi$
$380$	0	0
$381$	−965.624	−0.129844
$382$	0	0
$383$	9724.98	1.29745	0.648725	−	0.761023i	$-0.275303\pi$
$383$	9724.98	1.29745	0.648725	+	0.761023i	$0.275303\pi$
$384$	0	0
$385$	464.332	0.0614663
$386$	0	0
$387$	−9222.64	−1.21140
$388$	0	0
$389$	−4135.23	−0.538984	−0.269492	−	0.963003i	$-0.586856\pi$
$389$	−4135.23	−0.538984	−0.269492	+	0.963003i	$0.586856\pi$
$390$	0	0
$391$	−9816.43	−1.26966
$392$	0	0
$393$	1591.55	0.204283
$394$	0	0
$395$	−825.045	−0.105095
$396$	0	0
$397$	1908.00	0.241208	0.120604	−	0.992701i	$-0.461517\pi$
$397$	1908.00	0.241208	0.120604	+	0.992701i	$0.461517\pi$
$398$	0	0
$399$	−503.152	−0.0631305
$400$	0	0
$401$	7253.91	0.903349	0.451675	−	0.892183i	$-0.350827\pi$
$401$	7253.91	0.903349	0.451675	+	0.892183i	$0.350827\pi$
$402$	0	0
$403$	834.572	0.103159
$404$	0	0
$405$	−443.280	−0.0543871
$406$	0	0
$407$	−5729.77	−0.697823
$408$	0	0
$409$	−3288.67	−0.397589	−0.198795	−	0.980041i	$-0.563703\pi$
$409$	−3288.67	−0.397589	−0.198795	+	0.980041i	$0.563703\pi$
$410$	0	0
$411$	683.390	0.0820174
$412$	0	0
$413$	19820.8	2.36154
$414$	0	0
$415$	−367.365	−0.0434536
$416$	0	0
$417$	−912.429	−0.107151
$418$	0	0
$419$	−2946.70	−0.343570	−0.171785	−	0.985134i	$-0.554953\pi$
$419$	−2946.70	−0.343570	−0.171785	+	0.985134i	$0.554953\pi$
$420$	0	0
$421$	−3705.67	−0.428987	−0.214493	−	0.976725i	$-0.568810\pi$
$421$	−3705.67	−0.428987	−0.214493	+	0.976725i	$0.568810\pi$
$422$	0	0
$423$	−9337.35	−1.07328
$424$	0	0
$425$	7616.03	0.869251
$426$	0	0
$427$	23735.4	2.69001
$428$	0	0
$429$	390.785	0.0439796
$430$	0	0
$431$	7988.10	0.892746	0.446373	−	0.894847i	$-0.352716\pi$
$431$	7988.10	0.892746	0.446373	+	0.894847i	$0.352716\pi$
$432$	0	0
$433$	8406.05	0.932954	0.466477	−	0.884533i	$-0.345523\pi$
$433$	8406.05	0.932954	0.466477	+	0.884533i	$0.345523\pi$
$434$	0	0
$435$	178.390	0.0196624
$436$	0	0
$437$	−3050.15	−0.333887
$438$	0	0
$439$	984.085	0.106988	0.0534941	−	0.998568i	$-0.482964\pi$
$439$	984.085	0.106988	0.0534941	+	0.998568i	$0.482964\pi$
$440$	0	0
$441$	−12535.7	−1.35360
$442$	0	0
$443$	7540.20	0.808681	0.404341	−	0.914608i	$-0.367501\pi$
$443$	7540.20	0.808681	0.404341	+	0.914608i	$0.367501\pi$
$444$	0	0
$445$	172.719	0.0183992
$446$	0	0
$447$	−1912.29	−0.202345
$448$	0	0
$449$	−2931.72	−0.308143	−0.154072	−	0.988060i	$-0.549239\pi$
$449$	−2931.72	−0.308143	−0.154072	+	0.988060i	$0.549239\pi$
$450$	0	0
$451$	4771.03	0.498135
$452$	0	0
$453$	2495.64	0.258842
$454$	0	0
$455$	−337.452	−0.0347692
$456$	0	0
$457$	6833.35	0.699455	0.349727	−	0.936852i	$-0.386274\pi$
$457$	6833.35	0.699455	0.349727	+	0.936852i	$0.386274\pi$
$458$	0	0
$459$	−3000.98	−0.305172
$460$	0	0
$461$	4006.80	0.404805	0.202402	−	0.979302i	$-0.435125\pi$
$461$	4006.80	0.404805	0.202402	+	0.979302i	$0.435125\pi$
$462$	0	0
$463$	−1786.36	−0.179307	−0.0896536	−	0.995973i	$-0.528576\pi$
$463$	−1786.36	−0.179307	−0.0896536	+	0.995973i	$0.528576\pi$
$464$	0	0
$465$	−29.4848	−0.00294049
$466$	0	0
$467$	10194.6	1.01017	0.505085	−	0.863070i	$-0.331461\pi$
$467$	10194.6	1.01017	0.505085	+	0.863070i	$0.331461\pi$
$468$	0	0
$469$	−4776.61	−0.470284
$470$	0	0
$471$	176.333	0.0172505
$472$	0	0
$473$	8511.51	0.827399
$474$	0	0
$475$	2366.45	0.228589
$476$	0	0
$477$	−9975.21	−0.957513
$478$	0	0
$479$	153.262	0.0146195	0.00730973	−	0.999973i	$-0.497673\pi$
$479$	153.262	0.0146195	0.00730973	+	0.999973i	$0.497673\pi$
$480$	0	0
$481$	4164.10	0.394733
$482$	0	0
$483$	4251.21	0.400491
$484$	0	0
$485$	−138.465	−0.0129636
$486$	0	0
$487$	−3387.94	−0.315241	−0.157620	−	0.987500i	$-0.550382\pi$
$487$	−3387.94	−0.315241	−0.157620	+	0.987500i	$0.550382\pi$
$488$	0	0
$489$	2975.97	0.275210
$490$	0	0
$491$	8555.18	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$491$	8555.18	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$492$	0	0
$493$	−17605.9	−1.60838
$494$	0	0
$495$	423.357	0.0384414
$496$	0	0
$497$	−8500.67	−0.767218
$498$	0	0
$499$	12148.2	1.08984	0.544919	−	0.838489i	$-0.316560\pi$
$499$	12148.2	1.08984	0.544919	+	0.838489i	$0.316560\pi$
$500$	0	0
$501$	−3079.46	−0.274611
$502$	0	0
$503$	−4758.81	−0.421839	−0.210919	−	0.977503i	$-0.567646\pi$
$503$	−4758.81	−0.421839	−0.210919	+	0.977503i	$0.567646\pi$
$504$	0	0
$505$	−376.188	−0.0331488
$506$	0	0
$507$	1744.74	0.152833
$508$	0	0
$509$	−11419.1	−0.994386	−0.497193	−	0.867640i	$-0.665636\pi$
$509$	−11419.1	−0.994386	−0.497193	+	0.867640i	$0.665636\pi$
$510$	0	0
$511$	26287.9	2.27575
$512$	0	0
$513$	−932.462	−0.0802519
$514$	0	0
$515$	−889.071	−0.0760722
$516$	0	0
$517$	8617.38	0.733060
$518$	0	0
$519$	−957.669	−0.0809962
$520$	0	0
$521$	8619.66	0.724826	0.362413	−	0.932018i	$-0.381953\pi$
$521$	8619.66	0.724826	0.362413	+	0.932018i	$0.381953\pi$
$522$	0	0
$523$	20411.3	1.70654	0.853271	−	0.521467i	$-0.174615\pi$
$523$	20411.3	1.70654	0.853271	+	0.521467i	$0.174615\pi$
$524$	0	0
$525$	−3298.29	−0.274189
$526$	0	0
$527$	2909.96	0.240531
$528$	0	0
$529$	13604.2	1.11813
$530$	0	0
$531$	18071.7	1.47692
$532$	0	0
$533$	−3467.34	−0.281777
$534$	0	0
$535$	−857.869	−0.0693250
$536$	0	0
$537$	−1407.39	−0.113097
$538$	0	0
$539$	11569.1	0.924524
$540$	0	0
$541$	−4553.14	−0.361838	−0.180919	−	0.983498i	$-0.557907\pi$
$541$	−4553.14	−0.361838	−0.180919	+	0.983498i	$0.557907\pi$
$542$	0	0
$543$	3698.73	0.292316
$544$	0	0
$545$	−210.660	−0.0165572
$546$	0	0
$547$	−16216.8	−1.26761	−0.633804	−	0.773494i	$-0.718507\pi$
$547$	−16216.8	−1.26761	−0.633804	+	0.773494i	$0.718507\pi$
$548$	0	0
$549$	21640.9	1.68235
$550$	0	0
$551$	−5470.49	−0.422960
$552$	0	0
$553$	−35263.5	−2.71168
$554$	0	0
$555$	−147.115	−0.0112517
$556$	0	0
$557$	175.385	0.0133417	0.00667084	−	0.999978i	$-0.497877\pi$
$557$	175.385	0.0133417	0.00667084	+	0.999978i	$0.497877\pi$
$558$	0	0
$559$	−6185.73	−0.468029
$560$	0	0
$561$	1362.58	0.102545
$562$	0	0
$563$	−6323.70	−0.473378	−0.236689	−	0.971585i	$-0.576062\pi$
$563$	−6323.70	−0.473378	−0.236689	+	0.971585i	$0.576062\pi$
$564$	0	0
$565$	−1508.80	−0.112346
$566$	0	0
$567$	−18946.4	−1.40330
$568$	0	0
$569$	12606.5	0.928811	0.464406	−	0.885623i	$-0.346268\pi$
$569$	12606.5	0.928811	0.464406	+	0.885623i	$0.346268\pi$
$570$	0	0
$571$	8389.93	0.614899	0.307450	−	0.951564i	$-0.400524\pi$
$571$	8389.93	0.614899	0.307450	+	0.951564i	$0.400524\pi$
$572$	0	0
$573$	−3469.10	−0.252921
$574$	0	0
$575$	−19994.5	−1.45014
$576$	0	0
$577$	−22630.3	−1.63278	−0.816389	−	0.577502i	$-0.804027\pi$
$577$	−22630.3	−1.63278	−0.816389	+	0.577502i	$0.804027\pi$
$578$	0	0
$579$	715.388	0.0513480
$580$	0	0
$581$	−15701.7	−1.12120
$582$	0	0
$583$	9206.06	0.653990
$584$	0	0
$585$	−307.674	−0.0217449
$586$	0	0
$587$	5085.58	0.357589	0.178794	−	0.983886i	$-0.442780\pi$
$587$	5085.58	0.357589	0.178794	+	0.983886i	$0.442780\pi$
$588$	0	0
$589$	904.180	0.0632531
$590$	0	0
$591$	−355.460	−0.0247405
$592$	0	0
$593$	11917.6	0.825293	0.412647	−	0.910891i	$-0.364604\pi$
$593$	11917.6	0.825293	0.412647	+	0.910891i	$0.364604\pi$
$594$	0	0
$595$	−1176.62	−0.0810700
$596$	0	0
$597$	572.545	0.0392508
$598$	0	0
$599$	3541.30	0.241559	0.120779	−	0.992679i	$-0.461461\pi$
$599$	3541.30	0.241559	0.120779	+	0.992679i	$0.461461\pi$
$600$	0	0
$601$	8131.45	0.551895	0.275947	−	0.961173i	$-0.411008\pi$
$601$	8131.45	0.551895	0.275947	+	0.961173i	$0.411008\pi$
$602$	0	0
$603$	−4355.10	−0.294118
$604$	0	0
$605$	502.344	0.0337573
$606$	0	0
$607$	18658.0	1.24762	0.623809	−	0.781577i	$-0.285584\pi$
$607$	18658.0	1.24762	0.623809	+	0.781577i	$0.285584\pi$
$608$	0	0
$609$	7624.62	0.507332
$610$	0	0
$611$	−6262.67	−0.414665
$612$	0	0
$613$	23985.1	1.58034	0.790172	−	0.612885i	$-0.209991\pi$
$613$	23985.1	1.58034	0.790172	+	0.612885i	$0.209991\pi$
$614$	0	0
$615$	122.499	0.00803190
$616$	0	0
$617$	−12011.0	−0.783706	−0.391853	−	0.920028i	$-0.628166\pi$
$617$	−12011.0	−0.783706	−0.391853	+	0.920028i	$0.628166\pi$
$618$	0	0
$619$	21339.0	1.38560	0.692800	−	0.721129i	$-0.256377\pi$
$619$	21339.0	1.38560	0.692800	+	0.721129i	$0.256377\pi$
$620$	0	0
$621$	7878.53	0.509106
$622$	0	0
$623$	7382.22	0.474739
$624$	0	0
$625$	15456.4	0.989208
$626$	0	0
$627$	423.378	0.0269667
$628$	0	0
$629$	14519.3	0.920382
$630$	0	0
$631$	−595.619	−0.0375772	−0.0187886	−	0.999823i	$-0.505981\pi$
$631$	−595.619	−0.0375772	−0.0187886	+	0.999823i	$0.505981\pi$
$632$	0	0
$633$	2596.50	0.163035
$634$	0	0
$635$	−701.637	−0.0438482
$636$	0	0
$637$	−8407.86	−0.522969
$638$	0	0
$639$	−7750.53	−0.479822
$640$	0	0
$641$	7385.91	0.455111	0.227555	−	0.973765i	$-0.426927\pi$
$641$	7385.91	0.455111	0.227555	+	0.973765i	$0.426927\pi$
$642$	0	0
$643$	29627.4	1.81710	0.908548	−	0.417781i	$-0.137192\pi$
$643$	29627.4	1.81710	0.908548	+	0.417781i	$0.137192\pi$
$644$	0	0
$645$	218.537	0.0133409
$646$	0	0
$647$	8698.54	0.528555	0.264277	−	0.964447i	$-0.414867\pi$
$647$	8698.54	0.528555	0.264277	+	0.964447i	$0.414867\pi$
$648$	0	0
$649$	−16678.3	−1.00875
$650$	0	0
$651$	−1260.22	−0.0758709
$652$	0	0
$653$	−3594.96	−0.215439	−0.107719	−	0.994181i	$-0.534355\pi$
$653$	−3594.96	−0.215439	−0.107719	+	0.994181i	$0.534355\pi$
$654$	0	0
$655$	1156.45	0.0689864
$656$	0	0
$657$	23968.1	1.42327
$658$	0	0
$659$	−25031.1	−1.47962	−0.739811	−	0.672814i	$-0.765085\pi$
$659$	−25031.1	−1.47962	−0.739811	+	0.672814i	$0.765085\pi$
$660$	0	0
$661$	22574.8	1.32838	0.664191	−	0.747563i	$-0.268776\pi$
$661$	22574.8	1.32838	0.664191	+	0.747563i	$0.268776\pi$
$662$	0	0
$663$	−990.250	−0.0580062
$664$	0	0
$665$	−365.598	−0.0213192
$666$	0	0
$667$	46221.1	2.68319
$668$	0	0
$669$	−5000.92	−0.289009
$670$	0	0
$671$	−19972.2	−1.14906
$672$	0	0
$673$	−19302.8	−1.10560	−0.552799	−	0.833315i	$-0.686440\pi$
$673$	−19302.8	−1.10560	−0.552799	+	0.833315i	$0.686440\pi$
$674$	0	0
$675$	−6112.53	−0.348550
$676$	0	0
$677$	23517.4	1.33508	0.667538	−	0.744576i	$-0.267348\pi$
$677$	23517.4	1.33508	0.667538	+	0.744576i	$0.267348\pi$
$678$	0	0
$679$	−5918.16	−0.334489
$680$	0	0
$681$	3629.44	0.204230
$682$	0	0
$683$	−14773.7	−0.827671	−0.413835	−	0.910352i	$-0.635811\pi$
$683$	−14773.7	−0.827671	−0.413835	+	0.910352i	$0.635811\pi$
$684$	0	0
$685$	496.562	0.0276973
$686$	0	0
$687$	−1972.98	−0.109569
$688$	0	0
$689$	−6690.49	−0.369938
$690$	0	0
$691$	4165.18	0.229306	0.114653	−	0.993406i	$-0.463424\pi$
$691$	4165.18	0.229306	0.114653	+	0.993406i	$0.463424\pi$
$692$	0	0
$693$	18094.8	0.991869
$694$	0	0
$695$	−662.985	−0.0361848
$696$	0	0
$697$	−12089.8	−0.657007
$698$	0	0
$699$	−2624.18	−0.141997
$700$	0	0
$701$	19303.8	1.04008	0.520039	−	0.854143i	$-0.325917\pi$
$701$	19303.8	1.04008	0.520039	+	0.854143i	$0.325917\pi$
$702$	0	0
$703$	4511.41	0.242035
$704$	0	0
$705$	221.256	0.0118198
$706$	0	0
$707$	−16078.7	−0.855309
$708$	0	0
$709$	−7913.04	−0.419155	−0.209577	−	0.977792i	$-0.567209\pi$
$709$	−7913.04	−0.419155	−0.209577	+	0.977792i	$0.567209\pi$
$710$	0	0
$711$	−32151.7	−1.69590
$712$	0	0
$713$	−7639.57	−0.401268
$714$	0	0
$715$	283.950	0.0148519
$716$	0	0
$717$	4419.88	0.230214
$718$	0	0
$719$	35512.6	1.84200	0.921000	−	0.389563i	$-0.127374\pi$
$719$	35512.6	1.84200	0.921000	+	0.389563i	$0.127374\pi$
$720$	0	0
$721$	−38000.1	−1.96282
$722$	0	0
$723$	−3308.94	−0.170209
$724$	0	0
$725$	−35860.4	−1.83700
$726$	0	0
$727$	−2510.77	−0.128087	−0.0640435	−	0.997947i	$-0.520400\pi$
$727$	−2510.77	−0.128087	−0.0640435	+	0.997947i	$0.520400\pi$
$728$	0	0
$729$	−16050.9	−0.815468
$730$	0	0
$731$	−21568.2	−1.09128
$732$	0	0
$733$	−5261.38	−0.265121	−0.132560	−	0.991175i	$-0.542320\pi$
$733$	−5261.38	−0.265121	−0.132560	+	0.991175i	$0.542320\pi$
$734$	0	0
$735$	297.044	0.0149070
$736$	0	0
$737$	4019.29	0.200885
$738$	0	0
$739$	10267.1	0.511071	0.255535	−	0.966800i	$-0.417748\pi$
$739$	10267.1	0.511071	0.255535	+	0.966800i	$0.417748\pi$
$740$	0	0
$741$	−307.689	−0.0152541
$742$	0	0
$743$	19722.5	0.973819	0.486910	−	0.873452i	$-0.338124\pi$
$743$	19722.5	0.973819	0.486910	+	0.873452i	$0.338124\pi$
$744$	0	0
$745$	−1389.50	−0.0683318
$746$	0	0
$747$	−14316.1	−0.701202
$748$	0	0
$749$	−36666.4	−1.78873
$750$	0	0
$751$	−27422.3	−1.33243	−0.666214	−	0.745760i	$-0.732086\pi$
$751$	−27422.3	−1.33243	−0.666214	+	0.745760i	$0.732086\pi$
$752$	0	0
$753$	2070.09	0.100184
$754$	0	0
$755$	1813.37	0.0874111
$756$	0	0
$757$	−6549.67	−0.314467	−0.157234	−	0.987561i	$-0.550258\pi$
$757$	−6549.67	−0.314467	−0.157234	+	0.987561i	$0.550258\pi$
$758$	0	0
$759$	−3577.20	−0.171072
$760$	0	0
$761$	24815.5	1.18208	0.591038	−	0.806644i	$-0.298718\pi$
$761$	24815.5	1.18208	0.591038	+	0.806644i	$0.298718\pi$
$762$	0	0
$763$	−9003.87	−0.427211
$764$	0	0
$765$	−1072.79	−0.0507016
$766$	0	0
$767$	12120.9	0.570613
$768$	0	0
$769$	1759.56	0.0825116	0.0412558	−	0.999149i	$-0.486864\pi$
$769$	1759.56	0.0825116	0.0412558	+	0.999149i	$0.486864\pi$
$770$	0	0
$771$	3054.32	0.142670
$772$	0	0
$773$	−20707.5	−0.963517	−0.481758	−	0.876304i	$-0.660002\pi$
$773$	−20707.5	−0.963517	−0.481758	+	0.876304i	$0.660002\pi$
$774$	0	0
$775$	5927.13	0.274721
$776$	0	0
$777$	−6287.87	−0.290317
$778$	0	0
$779$	−3756.53	−0.172775
$780$	0	0
$781$	7152.91	0.327723
$782$	0	0
$783$	14130.3	0.644923
$784$	0	0
$785$	128.126	0.00582551
$786$	0	0
$787$	−16060.1	−0.727422	−0.363711	−	0.931512i	$-0.618490\pi$
$787$	−16060.1	−0.727422	−0.363711	+	0.931512i	$0.618490\pi$
$788$	0	0
$789$	5294.26	0.238885
$790$	0	0
$791$	−64488.1	−2.89878
$792$	0	0
$793$	14514.8	0.649980
$794$	0	0
$795$	236.370	0.0105449
$796$	0	0
$797$	−6369.68	−0.283094	−0.141547	−	0.989932i	$-0.545208\pi$
$797$	−6369.68	−0.283094	−0.141547	+	0.989932i	$0.545208\pi$
$798$	0	0
$799$	−21836.5	−0.966857
$800$	0	0
$801$	6730.78	0.296904
$802$	0	0
$803$	−22120.0	−0.972103
$804$	0	0
$805$	3089.00	0.135246
$806$	0	0
$807$	−4490.66	−0.195884
$808$	0	0
$809$	18129.0	0.787863	0.393932	−	0.919140i	$-0.371115\pi$
$809$	18129.0	0.787863	0.393932	+	0.919140i	$0.371115\pi$
$810$	0	0
$811$	29297.4	1.26852	0.634260	−	0.773120i	$-0.281305\pi$
$811$	29297.4	1.26852	0.634260	+	0.773120i	$0.281305\pi$
$812$	0	0
$813$	5936.45	0.256089
$814$	0	0
$815$	2162.38	0.0929386
$816$	0	0
$817$	−6701.65	−0.286978
$818$	0	0
$819$	−13150.4	−0.561064
$820$	0	0
$821$	−42698.9	−1.81510	−0.907552	−	0.419939i	$-0.862051\pi$
$821$	−42698.9	−1.81510	−0.907552	+	0.419939i	$0.862051\pi$
$822$	0	0
$823$	−27961.5	−1.18430	−0.592149	−	0.805829i	$-0.701720\pi$
$823$	−27961.5	−1.18430	−0.592149	+	0.805829i	$0.701720\pi$
$824$	0	0
$825$	2775.35	0.117122
$826$	0	0
$827$	22115.7	0.929912	0.464956	−	0.885334i	$-0.346070\pi$
$827$	22115.7	0.929912	0.464956	+	0.885334i	$0.346070\pi$
$828$	0	0
$829$	−7144.36	−0.299317	−0.149659	−	0.988738i	$-0.547817\pi$
$829$	−7144.36	−0.299317	−0.149659	+	0.988738i	$0.547817\pi$
$830$	0	0
$831$	−6330.45	−0.264261
$832$	0	0
$833$	−29316.3	−1.21939
$834$	0	0
$835$	−2237.58	−0.0927361
$836$	0	0
$837$	−2335.50	−0.0964475
$838$	0	0
$839$	−9019.32	−0.371134	−0.185567	−	0.982632i	$-0.559412\pi$
$839$	−9019.32	−0.371134	−0.185567	+	0.982632i	$0.559412\pi$
$840$	0	0
$841$	58509.2	2.39900
$842$	0	0
$843$	−1553.72	−0.0634790
$844$	0	0
$845$	1267.75	0.0516119
$846$	0	0
$847$	21470.8	0.871011
$848$	0	0
$849$	5443.28	0.220039
$850$	0	0
$851$	−38117.6	−1.53544
$852$	0	0
$853$	−37503.6	−1.50539	−0.752695	−	0.658369i	$-0.771246\pi$
$853$	−37503.6	−1.50539	−0.752695	+	0.658369i	$0.771246\pi$
$854$	0	0
$855$	−333.336	−0.0133331
$856$	0	0
$857$	19556.1	0.779490	0.389745	−	0.920923i	$-0.372563\pi$
$857$	19556.1	0.779490	0.389745	+	0.920923i	$0.372563\pi$
$858$	0	0
$859$	−38405.1	−1.52545	−0.762727	−	0.646720i	$-0.776140\pi$
$859$	−38405.1	−1.52545	−0.762727	+	0.646720i	$0.776140\pi$
$860$	0	0
$861$	5235.75	0.207240
$862$	0	0
$863$	22495.9	0.887334	0.443667	−	0.896192i	$-0.353677\pi$
$863$	22495.9	0.887334	0.443667	+	0.896192i	$0.353677\pi$
$864$	0	0
$865$	−695.857	−0.0273524
$866$	0	0
$867$	1083.96	0.0424604
$868$	0	0
$869$	29672.6	1.15831
$870$	0	0
$871$	−2921.01	−0.113633
$872$	0	0
$873$	−5395.91	−0.209191
$874$	0	0
$875$	−4801.83	−0.185522
$876$	0	0
$877$	44597.3	1.71716	0.858578	−	0.512683i	$-0.171349\pi$
$877$	44597.3	1.71716	0.858578	+	0.512683i	$0.171349\pi$
$878$	0	0
$879$	6117.78	0.234753
$880$	0	0
$881$	36144.0	1.38220	0.691102	−	0.722758i	$-0.257126\pi$
$881$	36144.0	1.38220	0.691102	+	0.722758i	$0.257126\pi$
$882$	0	0
$883$	−14216.5	−0.541815	−0.270908	−	0.962605i	$-0.587324\pi$
$883$	−14216.5	−0.541815	−0.270908	+	0.962605i	$0.587324\pi$
$884$	0	0
$885$	−428.223	−0.0162650
$886$	0	0
$887$	−22404.0	−0.848088	−0.424044	−	0.905642i	$-0.639390\pi$
$887$	−22404.0	−0.848088	−0.424044	+	0.905642i	$0.639390\pi$
$888$	0	0
$889$	−29988.9	−1.13138
$890$	0	0
$891$	15942.5	0.599431
$892$	0	0
$893$	−6785.01	−0.254257
$894$	0	0
$895$	−1022.63	−0.0381930
$896$	0	0
$897$	2599.72	0.0967694
$898$	0	0
$899$	−13701.7	−0.508317
$900$	0	0
$901$	−23328.2	−0.862569
$902$	0	0
$903$	9340.57	0.344224
$904$	0	0
$905$	2687.56	0.0987153
$906$	0	0
$907$	−16211.5	−0.593489	−0.296745	−	0.954957i	$-0.595901\pi$
$907$	−16211.5	−0.593489	−0.296745	+	0.954957i	$0.595901\pi$
$908$	0	0
$909$	−14659.9	−0.534915
$910$	0	0
$911$	−8529.11	−0.310189	−0.155094	−	0.987900i	$-0.549568\pi$
$911$	−8529.11	−0.310189	−0.155094	+	0.987900i	$0.549568\pi$
$912$	0	0
$913$	13212.2	0.478927
$914$	0	0
$915$	−512.796	−0.0185274
$916$	0	0
$917$	49428.0	1.78000
$918$	0	0
$919$	18155.1	0.651666	0.325833	−	0.945427i	$-0.394355\pi$
$919$	18155.1	0.651666	0.325833	+	0.945427i	$0.394355\pi$
$920$	0	0
$921$	3865.03	0.138281
$922$	0	0
$923$	−5198.37	−0.185381
$924$	0	0
$925$	29573.4	1.05121
$926$	0	0
$927$	−34646.8	−1.22756
$928$	0	0
$929$	26472.1	0.934898	0.467449	−	0.884020i	$-0.345173\pi$
$929$	26472.1	0.934898	0.467449	+	0.884020i	$0.345173\pi$
$930$	0	0
$931$	−9109.12	−0.320665
$932$	0	0
$933$	−468.407	−0.0164362
$934$	0	0
$935$	990.069	0.0346296
$936$	0	0
$937$	−3891.08	−0.135663	−0.0678313	−	0.997697i	$-0.521608\pi$
$937$	−3891.08	−0.135663	−0.0678313	+	0.997697i	$0.521608\pi$
$938$	0	0
$939$	−9518.12	−0.330790
$940$	0	0
$941$	−5700.50	−0.197483	−0.0987413	−	0.995113i	$-0.531482\pi$
$941$	−5700.50	−0.197483	−0.0987413	+	0.995113i	$0.531482\pi$
$942$	0	0
$943$	31739.6	1.09606
$944$	0	0
$945$	944.338	0.0325072
$946$	0	0
$947$	16631.9	0.570712	0.285356	−	0.958422i	$-0.407888\pi$
$947$	16631.9	0.570712	0.285356	+	0.958422i	$0.407888\pi$
$948$	0	0
$949$	16075.7	0.549883
$950$	0	0
$951$	−8.85460	−0.000301924 0
$952$	0	0
$953$	−12213.3	−0.415140	−0.207570	−	0.978220i	$-0.566556\pi$
$953$	−12213.3	−0.415140	−0.207570	+	0.978220i	$0.566556\pi$
$954$	0	0
$955$	−2520.70	−0.0854116
$956$	0	0
$957$	−6415.75	−0.216710
$958$	0	0
$959$	21223.7	0.714650
$960$	0	0
$961$	−27526.3	−0.923982
$962$	0	0
$963$	−33430.8	−1.11868
$964$	0	0
$965$	519.812	0.0173402
$966$	0	0
$967$	34551.6	1.14902	0.574512	−	0.818496i	$-0.305192\pi$
$967$	34551.6	1.14902	0.574512	+	0.818496i	$0.305192\pi$
$968$	0	0
$969$	−1072.84	−0.0355672
$970$	0	0
$971$	−55055.9	−1.81959	−0.909797	−	0.415053i	$-0.863763\pi$
$971$	−55055.9	−1.81959	−0.909797	+	0.415053i	$0.863763\pi$
$972$	0	0
$973$	−28336.8	−0.933645
$974$	0	0
$975$	−2016.98	−0.0662514
$976$	0	0
$977$	−23388.2	−0.765871	−0.382935	−	0.923775i	$-0.625087\pi$
$977$	−23388.2	−0.765871	−0.382935	+	0.923775i	$0.625087\pi$
$978$	0	0
$979$	−6211.79	−0.202788
$980$	0	0
$981$	−8209.33	−0.267180
$982$	0	0
$983$	−9127.57	−0.296159	−0.148079	−	0.988975i	$-0.547309\pi$
$983$	−9127.57	−0.296159	−0.148079	+	0.988975i	$0.547309\pi$
$984$	0	0
$985$	−258.282	−0.00835489
$986$	0	0
$987$	9456.75	0.304976
$988$	0	0
$989$	56623.4	1.82054
$990$	0	0
$991$	39058.6	1.25201	0.626003	−	0.779820i	$-0.284690\pi$
$991$	39058.6	1.25201	0.626003	+	0.779820i	$0.284690\pi$
$992$	0	0
$993$	−1635.19	−0.0522569
$994$	0	0
$995$	416.020	0.0132550
$996$	0	0
$997$	−9990.46	−0.317353	−0.158677	−	0.987331i	$-0.550723\pi$
$997$	−9990.46	−0.317353	−0.158677	+	0.987331i	$0.550723\pi$
$998$	0	0
$999$	−11653.0	−0.369052

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.4.a.k.1.4	yes	7
4.3	odd	2		608.4.a.j.1.4	✓	7
8.3	odd	2		1216.4.a.bg.1.4		7
8.5	even	2		1216.4.a.bf.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.j.1.4	✓	7	4.3	odd	2
608.4.a.k.1.4	yes	7	1.1	even	1	trivial
1216.4.a.bf.1.4		7	8.5	even	2
1216.4.a.bg.1.4		7	8.3	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:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	608.a (trivial)

\(f(q)\)	\(=\)	\(q-0.923414 q^{3} -0.670967 q^{5} -28.6780 q^{7} -26.1473 q^{9} +24.1312 q^{11} -17.5373 q^{13} +0.619580 q^{15} -61.1485 q^{17} -19.0000 q^{19} +26.4817 q^{21} +160.534 q^{23} -124.550 q^{25} +49.0770 q^{27} +287.921 q^{29} -47.5884 q^{31} -22.2831 q^{33} +19.2420 q^{35} -237.442 q^{37} +16.1942 q^{39} +197.712 q^{41} +352.718 q^{43} +17.5440 q^{45} +357.106 q^{47} +479.427 q^{49} +56.4654 q^{51} +381.501 q^{53} -16.1912 q^{55} +17.5449 q^{57} -691.149 q^{59} -827.651 q^{61} +749.852 q^{63} +11.7669 q^{65} +166.560 q^{67} -148.240 q^{69} +296.418 q^{71} -916.657 q^{73} +115.011 q^{75} -692.034 q^{77} +1229.64 q^{79} +660.659 q^{81} +547.516 q^{83} +41.0286 q^{85} -265.870 q^{87} -257.418 q^{89} +502.934 q^{91} +43.9438 q^{93} +12.7484 q^{95} +206.366 q^{97} -630.965 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{3} - 17 q^{5} + 42 q^{7} + 86 q^{9} - 33 q^{11} - 35 q^{13} + 120 q^{15} + 66 q^{17} - 133 q^{19} + 33 q^{21} + 389 q^{23} + 44 q^{25} - 39 q^{27} + 233 q^{29} + 158 q^{31} - 206 q^{33} + 123 q^{35}+ \cdots - 1591 q^{99}+O(q^{100}) \) 7 * q + 3 * q^3 - 17 * q^5 + 42 * q^7 + 86 * q^9 - 33 * q^11 - 35 * q^13 + 120 * q^15 + 66 * q^17 - 133 * q^19 + 33 * q^21 + 389 * q^23 + 44 * q^25 - 39 * q^27 + 233 * q^29 + 158 * q^31 - 206 * q^33 + 123 * q^35 - 436 * q^37 + 807 * q^39 - 94 * q^41 + 645 * q^43 + 103 * q^45 + 1451 * q^47 + 93 * q^49 + 1741 * q^51 + 3 * q^53 + 1971 * q^55 - 57 * q^57 + 297 * q^59 + 93 * q^61 + 2999 * q^63 - 788 * q^65 + 1641 * q^67 + 945 * q^69 + 2392 * q^71 + 324 * q^73 + 1909 * q^75 - 711 * q^77 + 2492 * q^79 + 143 * q^81 + 310 * q^83 + 2353 * q^85 + 4795 * q^87 - 440 * q^89 - 107 * q^91 + 900 * q^93 + 323 * q^95 - 532 * q^97 - 1591 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

Properties

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