LMFDB - Embedded newform 608.4.a.h.1.3

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:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 28x^{3} - 8x^{2} + 73x - 12 $ x^5 - 2x^4 - 28x^3 - 8x^2 + 73x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.54663$ 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)

$f(q)$	$=$	$q+2.55330 q^{3} +7.40075 q^{5} +10.4962 q^{7} -20.4807 q^{9} -48.7928 q^{11} +4.45376 q^{13} +18.8963 q^{15} -61.5346 q^{17} -19.0000 q^{19} +26.7999 q^{21} -34.8416 q^{23} -70.2288 q^{25} -121.232 q^{27} -179.528 q^{29} -132.845 q^{31} -124.583 q^{33} +77.6798 q^{35} -99.3918 q^{37} +11.3718 q^{39} +382.310 q^{41} +129.659 q^{43} -151.572 q^{45} +488.106 q^{47} -232.830 q^{49} -157.116 q^{51} -129.199 q^{53} -361.104 q^{55} -48.5127 q^{57} +479.346 q^{59} -394.879 q^{61} -214.969 q^{63} +32.9612 q^{65} +19.8601 q^{67} -88.9610 q^{69} -221.694 q^{71} +258.902 q^{73} -179.315 q^{75} -512.139 q^{77} -469.595 q^{79} +243.436 q^{81} +373.014 q^{83} -455.402 q^{85} -458.390 q^{87} -1609.29 q^{89} +46.7476 q^{91} -339.193 q^{93} -140.614 q^{95} -1654.62 q^{97} +999.310 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{3} + 5 q^{5} - 7 q^{7} - 5 q^{9} + 13 q^{11} - 72 q^{13} - 72 q^{15} - 59 q^{17} - 95 q^{19} - 224 q^{21} - 52 q^{23} - 86 q^{25} + 54 q^{27} - 128 q^{29} + 110 q^{31} - 68 q^{33} - 45 q^{35}+ \cdots + 601 q^{99}+O(q^{100}) $ 5 * q + 6 * q^3 + 5 * q^5 - 7 * q^7 - 5 * q^9 + 13 * q^11 - 72 * q^13 - 72 * q^15 - 59 * q^17 - 95 * q^19 - 224 * q^21 - 52 * q^23 - 86 * q^25 + 54 * q^27 - 128 * q^29 + 110 * q^31 - 68 * q^33 - 45 * q^35 - 436 * q^37 - 356 * q^39 - 804 * q^41 - 143 * q^43 - 579 * q^45 - 661 * q^47 - 406 * q^49 + 570 * q^51 - 898 * q^53 - 17 * q^55 - 114 * q^57 + 196 * q^59 - 1079 * q^61 - 143 * q^63 - 1632 * q^65 + 832 * q^67 - 1644 * q^69 - 834 * q^71 - 1375 * q^73 + 1054 * q^75 - 1473 * q^77 - 154 * q^79 - 1859 * q^81 + 2224 * q^83 - 1807 * q^85 - 2004 * q^87 - 542 * q^89 + 1890 * q^91 - 2788 * q^93 - 95 * q^95 - 1528 * q^97 + 601 * 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
$3$	2.55330	0.491382	0.245691	−	0.969348i	$-0.420985\pi$
$3$	2.55330	0.491382	0.245691	+	0.969348i	$0.420985\pi$
$4$	0	0
$5$	7.40075	0.661944	0.330972	−	0.943641i	$-0.392623\pi$
$5$	7.40075	0.661944	0.330972	+	0.943641i	$0.392623\pi$
$6$	0	0
$7$	10.4962	0.566741	0.283371	−	0.959010i	$-0.408547\pi$
$7$	10.4962	0.566741	0.283371	+	0.959010i	$0.408547\pi$
$8$	0	0
$9$	−20.4807	−0.758543
$10$	0	0
$11$	−48.7928	−1.33742	−0.668709	−	0.743524i	$-0.733153\pi$
$11$	−48.7928	−1.33742	−0.668709	+	0.743524i	$0.733153\pi$
$12$	0	0
$13$	4.45376	0.0950194	0.0475097	−	0.998871i	$-0.484872\pi$
$13$	4.45376	0.0950194	0.0475097	+	0.998871i	$0.484872\pi$
$14$	0	0
$15$	18.8963	0.325267
$16$	0	0
$17$	−61.5346	−0.877901	−0.438951	−	0.898511i	$-0.644650\pi$
$17$	−61.5346	−0.877901	−0.438951	+	0.898511i	$0.644650\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	26.7999	0.278487
$22$	0	0
$23$	−34.8416	−0.315869	−0.157934	−	0.987450i	$-0.550483\pi$
$23$	−34.8416	−0.315869	−0.157934	+	0.987450i	$0.550483\pi$
$24$	0	0
$25$	−70.2288	−0.561831
$26$	0	0
$27$	−121.232	−0.864117
$28$	0	0
$29$	−179.528	−1.14957	−0.574786	−	0.818304i	$-0.694915\pi$
$29$	−179.528	−1.14957	−0.574786	+	0.818304i	$0.694915\pi$
$30$	0	0
$31$	−132.845	−0.769667	−0.384833	−	0.922986i	$-0.625741\pi$
$31$	−132.845	−0.769667	−0.384833	+	0.922986i	$0.625741\pi$
$32$	0	0
$33$	−124.583	−0.657184
$34$	0	0
$35$	77.6798	0.375151
$36$	0	0
$37$	−99.3918	−0.441619	−0.220810	−	0.975317i	$-0.570870\pi$
$37$	−99.3918	−0.441619	−0.220810	+	0.975317i	$0.570870\pi$
$38$	0	0
$39$	11.3718	0.0466908
$40$	0	0
$41$	382.310	1.45626	0.728132	−	0.685437i	$-0.240389\pi$
$41$	382.310	1.45626	0.728132	+	0.685437i	$0.240389\pi$
$42$	0	0
$43$	129.659	0.459834	0.229917	−	0.973210i	$-0.426154\pi$
$43$	129.659	0.459834	0.229917	+	0.973210i	$0.426154\pi$
$44$	0	0
$45$	−151.572	−0.502113
$46$	0	0
$47$	488.106	1.51484	0.757420	−	0.652928i	$-0.226459\pi$
$47$	488.106	1.51484	0.757420	+	0.652928i	$0.226459\pi$
$48$	0	0
$49$	−232.830	−0.678804
$50$	0	0
$51$	−157.116	−0.431385
$52$	0	0
$53$	−129.199	−0.334846	−0.167423	−	0.985885i	$-0.553545\pi$
$53$	−129.199	−0.334846	−0.167423	+	0.985885i	$0.553545\pi$
$54$	0	0
$55$	−361.104	−0.885295
$56$	0	0
$57$	−48.5127	−0.112731
$58$	0	0
$59$	479.346	1.05772	0.528861	−	0.848709i	$-0.322619\pi$
$59$	479.346	1.05772	0.528861	+	0.848709i	$0.322619\pi$
$60$	0	0
$61$	−394.879	−0.828836	−0.414418	−	0.910087i	$-0.636015\pi$
$61$	−394.879	−0.828836	−0.414418	+	0.910087i	$0.636015\pi$
$62$	0	0
$63$	−214.969	−0.429898
$64$	0	0
$65$	32.9612	0.0628975
$66$	0	0
$67$	19.8601	0.0362135	0.0181067	−	0.999836i	$-0.494236\pi$
$67$	19.8601	0.0362135	0.0181067	+	0.999836i	$0.494236\pi$
$68$	0	0
$69$	−88.9610	−0.155212
$70$	0	0
$71$	−221.694	−0.370567	−0.185283	−	0.982685i	$-0.559320\pi$
$71$	−221.694	−0.370567	−0.185283	+	0.982685i	$0.559320\pi$
$72$	0	0
$73$	258.902	0.415099	0.207549	−	0.978225i	$-0.433451\pi$
$73$	258.902	0.415099	0.207549	+	0.978225i	$0.433451\pi$
$74$	0	0
$75$	−179.315	−0.276074
$76$	0	0
$77$	−512.139	−0.757970
$78$	0	0
$79$	−469.595	−0.668780	−0.334390	−	0.942435i	$-0.608530\pi$
$79$	−469.595	−0.668780	−0.334390	+	0.942435i	$0.608530\pi$
$80$	0	0
$81$	243.436	0.333932
$82$	0	0
$83$	373.014	0.493296	0.246648	−	0.969105i	$-0.420671\pi$
$83$	373.014	0.493296	0.246648	+	0.969105i	$0.420671\pi$
$84$	0	0
$85$	−455.402	−0.581121
$86$	0	0
$87$	−458.390	−0.564880
$88$	0	0
$89$	−1609.29	−1.91668	−0.958340	−	0.285631i	$-0.907797\pi$
$89$	−1609.29	−1.91668	−0.958340	+	0.285631i	$0.907797\pi$
$90$	0	0
$91$	46.7476	0.0538514
$92$	0	0
$93$	−339.193	−0.378201
$94$	0	0
$95$	−140.614	−0.151860
$96$	0	0
$97$	−1654.62	−1.73198	−0.865988	−	0.500065i	$-0.833309\pi$
$97$	−1654.62	−1.73198	−0.865988	+	0.500065i	$0.833309\pi$
$98$	0	0
$99$	999.310	1.01449
$100$	0	0
$101$	−660.681	−0.650893	−0.325447	−	0.945560i	$-0.605515\pi$
$101$	−660.681	−0.650893	−0.325447	+	0.945560i	$0.605515\pi$
$102$	0	0
$103$	99.1397	0.0948401	0.0474200	−	0.998875i	$-0.484900\pi$
$103$	99.1397	0.0948401	0.0474200	+	0.998875i	$0.484900\pi$
$104$	0	0
$105$	198.340	0.184342
$106$	0	0
$107$	1584.99	1.43202	0.716011	−	0.698089i	$-0.245966\pi$
$107$	1584.99	1.43202	0.716011	+	0.698089i	$0.245966\pi$
$108$	0	0
$109$	−203.005	−0.178389	−0.0891943	−	0.996014i	$-0.528429\pi$
$109$	−203.005	−0.178389	−0.0891943	+	0.996014i	$0.528429\pi$
$110$	0	0
$111$	−253.777	−0.217004
$112$	0	0
$113$	−1849.28	−1.53952	−0.769761	−	0.638332i	$-0.779625\pi$
$113$	−1849.28	−1.53952	−0.769761	+	0.638332i	$0.779625\pi$
$114$	0	0
$115$	−257.854	−0.209087
$116$	0	0
$117$	−91.2161	−0.0720763
$118$	0	0
$119$	−645.879	−0.497543
$120$	0	0
$121$	1049.74	0.788687
$122$	0	0
$123$	976.151	0.715582
$124$	0	0
$125$	−1444.84	−1.03384
$126$	0	0
$127$	1739.70	1.21554	0.607769	−	0.794114i	$-0.292065\pi$
$127$	1739.70	1.21554	0.607769	+	0.794114i	$0.292065\pi$
$128$	0	0
$129$	331.059	0.225954
$130$	0	0
$131$	1490.46	0.994060	0.497030	−	0.867733i	$-0.334424\pi$
$131$	1490.46	0.994060	0.497030	+	0.867733i	$0.334424\pi$
$132$	0	0
$133$	−199.428	−0.130019
$134$	0	0
$135$	−897.210	−0.571997
$136$	0	0
$137$	866.483	0.540355	0.270178	−	0.962810i	$-0.412918\pi$
$137$	866.483	0.540355	0.270178	+	0.962810i	$0.412918\pi$
$138$	0	0
$139$	−498.851	−0.304403	−0.152202	−	0.988349i	$-0.548636\pi$
$139$	−498.851	−0.304403	−0.152202	+	0.988349i	$0.548636\pi$
$140$	0	0
$141$	1246.28	0.744366
$142$	0	0
$143$	−217.312	−0.127081
$144$	0	0
$145$	−1328.65	−0.760952
$146$	0	0
$147$	−594.484	−0.333552
$148$	0	0
$149$	696.388	0.382888	0.191444	−	0.981504i	$-0.438683\pi$
$149$	696.388	0.382888	0.191444	+	0.981504i	$0.438683\pi$
$150$	0	0
$151$	−2834.12	−1.52740	−0.763699	−	0.645573i	$-0.776619\pi$
$151$	−2834.12	−1.52740	−0.763699	+	0.645573i	$0.776619\pi$
$152$	0	0
$153$	1260.27	0.665926
$154$	0	0
$155$	−983.153	−0.509476
$156$	0	0
$157$	2337.71	1.18834	0.594172	−	0.804338i	$-0.297480\pi$
$157$	2337.71	1.18834	0.594172	+	0.804338i	$0.297480\pi$
$158$	0	0
$159$	−329.883	−0.164537
$160$	0	0
$161$	−365.704	−0.179016
$162$	0	0
$163$	−649.785	−0.312240	−0.156120	−	0.987738i	$-0.549899\pi$
$163$	−649.785	−0.312240	−0.156120	+	0.987738i	$0.549899\pi$
$164$	0	0
$165$	−922.006	−0.435018
$166$	0	0
$167$	−3101.68	−1.43722	−0.718609	−	0.695415i	$-0.755221\pi$
$167$	−3101.68	−1.43722	−0.718609	+	0.695415i	$0.755221\pi$
$168$	0	0
$169$	−2177.16	−0.990971
$170$	0	0
$171$	389.133	0.174022
$172$	0	0
$173$	−226.436	−0.0995123	−0.0497561	−	0.998761i	$-0.515844\pi$
$173$	−226.436	−0.0995123	−0.0497561	+	0.998761i	$0.515844\pi$
$174$	0	0
$175$	−737.135	−0.318413
$176$	0	0
$177$	1223.91	0.519746
$178$	0	0
$179$	2519.69	1.05213	0.526063	−	0.850446i	$-0.323668\pi$
$179$	2519.69	1.05213	0.526063	+	0.850446i	$0.323668\pi$
$180$	0	0
$181$	−1012.54	−0.415809	−0.207904	−	0.978149i	$-0.566664\pi$
$181$	−1012.54	−0.415809	−0.207904	+	0.978149i	$0.566664\pi$
$182$	0	0
$183$	−1008.24	−0.407276
$184$	0	0
$185$	−735.575	−0.292327
$186$	0	0
$187$	3002.45	1.17412
$188$	0	0
$189$	−1272.48	−0.489731
$190$	0	0
$191$	−1941.43	−0.735480	−0.367740	−	0.929929i	$-0.619868\pi$
$191$	−1941.43	−0.735480	−0.367740	+	0.929929i	$0.619868\pi$
$192$	0	0
$193$	2069.72	0.771926	0.385963	−	0.922514i	$-0.373869\pi$
$193$	2069.72	0.771926	0.385963	+	0.922514i	$0.373869\pi$
$194$	0	0
$195$	84.1598	0.0309067
$196$	0	0
$197$	4208.36	1.52200	0.760998	−	0.648754i	$-0.224710\pi$
$197$	4208.36	1.52200	0.760998	+	0.648754i	$0.224710\pi$
$198$	0	0
$199$	267.700	0.0953604	0.0476802	−	0.998863i	$-0.484817\pi$
$199$	267.700	0.0953604	0.0476802	+	0.998863i	$0.484817\pi$
$200$	0	0
$201$	50.7089	0.0177947
$202$	0	0
$203$	−1884.37	−0.651510
$204$	0	0
$205$	2829.38	0.963964
$206$	0	0
$207$	713.579	0.239600
$208$	0	0
$209$	927.064	0.306825
$210$	0	0
$211$	−1259.32	−0.410876	−0.205438	−	0.978670i	$-0.565862\pi$
$211$	−1259.32	−0.410876	−0.205438	+	0.978670i	$0.565862\pi$
$212$	0	0
$213$	−566.051	−0.182090
$214$	0	0
$215$	959.578	0.304384
$216$	0	0
$217$	−1394.37	−0.436202
$218$	0	0
$219$	661.054	0.203972
$220$	0	0
$221$	−274.060	−0.0834176
$222$	0	0
$223$	3639.90	1.09303	0.546516	−	0.837449i	$-0.315954\pi$
$223$	3639.90	1.09303	0.546516	+	0.837449i	$0.315954\pi$
$224$	0	0
$225$	1438.33	0.426173
$226$	0	0
$227$	1873.77	0.547870	0.273935	−	0.961748i	$-0.411675\pi$
$227$	1873.77	0.547870	0.273935	+	0.961748i	$0.411675\pi$
$228$	0	0
$229$	4120.43	1.18902	0.594510	−	0.804088i	$-0.297346\pi$
$229$	4120.43	1.18902	0.594510	+	0.804088i	$0.297346\pi$
$230$	0	0
$231$	−1307.64	−0.372453
$232$	0	0
$233$	562.027	0.158024	0.0790121	−	0.996874i	$-0.474823\pi$
$233$	562.027	0.158024	0.0790121	+	0.996874i	$0.474823\pi$
$234$	0	0
$235$	3612.35	1.00274
$236$	0	0
$237$	−1199.02	−0.328627
$238$	0	0
$239$	−4754.65	−1.28683	−0.643415	−	0.765517i	$-0.722483\pi$
$239$	−4754.65	−1.28683	−0.643415	+	0.765517i	$0.722483\pi$
$240$	0	0
$241$	806.451	0.215552	0.107776	−	0.994175i	$-0.465627\pi$
$241$	806.451	0.215552	0.107776	+	0.994175i	$0.465627\pi$
$242$	0	0
$243$	3894.84	1.02821
$244$	0	0
$245$	−1723.12	−0.449330
$246$	0	0
$247$	−84.6215	−0.0217989
$248$	0	0
$249$	952.415	0.242397
$250$	0	0
$251$	−2484.87	−0.624876	−0.312438	−	0.949938i	$-0.601146\pi$
$251$	−2484.87	−0.624876	−0.312438	+	0.949938i	$0.601146\pi$
$252$	0	0
$253$	1700.02	0.422448
$254$	0	0
$255$	−1162.78	−0.285553
$256$	0	0
$257$	3530.59	0.856933	0.428467	−	0.903558i	$-0.359054\pi$
$257$	3530.59	0.856933	0.428467	+	0.903558i	$0.359054\pi$
$258$	0	0
$259$	−1043.24	−0.250284
$260$	0	0
$261$	3676.86	0.872001
$262$	0	0
$263$	3025.99	0.709470	0.354735	−	0.934967i	$-0.384571\pi$
$263$	3025.99	0.709470	0.354735	+	0.934967i	$0.384571\pi$
$264$	0	0
$265$	−956.169	−0.221649
$266$	0	0
$267$	−4109.00	−0.941822
$268$	0	0
$269$	3179.99	0.720772	0.360386	−	0.932803i	$-0.382645\pi$
$269$	3179.99	0.720772	0.360386	+	0.932803i	$0.382645\pi$
$270$	0	0
$271$	−5796.15	−1.29923	−0.649614	−	0.760264i	$-0.725069\pi$
$271$	−5796.15	−1.29923	−0.649614	+	0.760264i	$0.725069\pi$
$272$	0	0
$273$	119.360	0.0264616
$274$	0	0
$275$	3426.66	0.751402
$276$	0	0
$277$	−5143.01	−1.11557	−0.557786	−	0.829984i	$-0.688349\pi$
$277$	−5143.01	−1.11557	−0.557786	+	0.829984i	$0.688349\pi$
$278$	0	0
$279$	2720.75	0.583826
$280$	0	0
$281$	2121.08	0.450296	0.225148	−	0.974325i	$-0.427713\pi$
$281$	2121.08	0.450296	0.225148	+	0.974325i	$0.427713\pi$
$282$	0	0
$283$	5446.02	1.14393	0.571965	−	0.820278i	$-0.306181\pi$
$283$	5446.02	1.14393	0.571965	+	0.820278i	$0.306181\pi$
$284$	0	0
$285$	−359.030	−0.0746215
$286$	0	0
$287$	4012.80	0.825324
$288$	0	0
$289$	−1126.50	−0.229289
$290$	0	0
$291$	−4224.75	−0.851062
$292$	0	0
$293$	6125.16	1.22128	0.610641	−	0.791908i	$-0.290912\pi$
$293$	6125.16	1.22128	0.610641	+	0.791908i	$0.290912\pi$
$294$	0	0
$295$	3547.53	0.700152
$296$	0	0
$297$	5915.27	1.15569
$298$	0	0
$299$	−155.176	−0.0300136
$300$	0	0
$301$	1360.93	0.260607
$302$	0	0
$303$	−1686.92	−0.319837
$304$	0	0
$305$	−2922.40	−0.548643
$306$	0	0
$307$	−1263.62	−0.234914	−0.117457	−	0.993078i	$-0.537474\pi$
$307$	−1263.62	−0.234914	−0.117457	+	0.993078i	$0.537474\pi$
$308$	0	0
$309$	253.133	0.0466027
$310$	0	0
$311$	−4954.26	−0.903313	−0.451657	−	0.892192i	$-0.649167\pi$
$311$	−4954.26	−0.903313	−0.451657	+	0.892192i	$0.649167\pi$
$312$	0	0
$313$	4740.07	0.855989	0.427995	−	0.903781i	$-0.359220\pi$
$313$	4740.07	0.855989	0.427995	+	0.903781i	$0.359220\pi$
$314$	0	0
$315$	−1590.93	−0.284568
$316$	0	0
$317$	−4986.86	−0.883564	−0.441782	−	0.897122i	$-0.645654\pi$
$317$	−4986.86	−0.883564	−0.441782	+	0.897122i	$0.645654\pi$
$318$	0	0
$319$	8759.71	1.53746
$320$	0	0
$321$	4046.94	0.703671
$322$	0	0
$323$	1169.16	0.201404
$324$	0	0
$325$	−312.783	−0.0533848
$326$	0	0
$327$	−518.332	−0.0876570
$328$	0	0
$329$	5123.25	0.858523
$330$	0	0
$331$	4783.52	0.794339	0.397169	−	0.917745i	$-0.369993\pi$
$331$	4783.52	0.794339	0.397169	+	0.917745i	$0.369993\pi$
$332$	0	0
$333$	2035.61	0.334987
$334$	0	0
$335$	146.980	0.0239713
$336$	0	0
$337$	5653.91	0.913912	0.456956	−	0.889489i	$-0.348940\pi$
$337$	5653.91	0.913912	0.456956	+	0.889489i	$0.348940\pi$
$338$	0	0
$339$	−4721.77	−0.756494
$340$	0	0
$341$	6481.88	1.02937
$342$	0	0
$343$	−6044.02	−0.951448
$344$	0	0
$345$	−658.378	−0.102742
$346$	0	0
$347$	12278.2	1.89951	0.949753	−	0.312999i	$-0.101334\pi$
$347$	12278.2	1.89951	0.949753	+	0.312999i	$0.101334\pi$
$348$	0	0
$349$	−12930.5	−1.98324	−0.991621	−	0.129180i	$-0.958765\pi$
$349$	−12930.5	−1.98324	−0.991621	+	0.129180i	$0.958765\pi$
$350$	0	0
$351$	−539.940	−0.0821079
$352$	0	0
$353$	1583.46	0.238751	0.119375	−	0.992849i	$-0.461911\pi$
$353$	1583.46	0.238751	0.119375	+	0.992849i	$0.461911\pi$
$354$	0	0
$355$	−1640.70	−0.245294
$356$	0	0
$357$	−1649.12	−0.244484
$358$	0	0
$359$	−8248.96	−1.21271	−0.606355	−	0.795194i	$-0.707369\pi$
$359$	−8248.96	−1.21271	−0.606355	+	0.795194i	$0.707369\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	2680.30	0.387547
$364$	0	0
$365$	1916.07	0.274772
$366$	0	0
$367$	1330.75	0.189277	0.0946387	−	0.995512i	$-0.469830\pi$
$367$	1330.75	0.189277	0.0946387	+	0.995512i	$0.469830\pi$
$368$	0	0
$369$	−7829.97	−1.10464
$370$	0	0
$371$	−1356.10	−0.189771
$372$	0	0
$373$	−262.288	−0.0364095	−0.0182048	−	0.999834i	$-0.505795\pi$
$373$	−262.288	−0.0364095	−0.0182048	+	0.999834i	$0.505795\pi$
$374$	0	0
$375$	−3689.11	−0.508013
$376$	0	0
$377$	−799.577	−0.109232
$378$	0	0
$379$	−808.151	−0.109530	−0.0547651	−	0.998499i	$-0.517441\pi$
$379$	−808.151	−0.109530	−0.0547651	+	0.998499i	$0.517441\pi$
$380$	0	0
$381$	4441.97	0.597294
$382$	0	0
$383$	−1379.74	−0.184076	−0.0920382	−	0.995755i	$-0.529338\pi$
$383$	−1379.74	−0.184076	−0.0920382	+	0.995755i	$0.529338\pi$
$384$	0	0
$385$	−3790.22	−0.501733
$386$	0	0
$387$	−2655.51	−0.348804
$388$	0	0
$389$	15154.6	1.97524	0.987618	−	0.156880i	$-0.0501436\pi$
$389$	15154.6	1.97524	0.987618	+	0.156880i	$0.0501436\pi$
$390$	0	0
$391$	2143.96	0.277301
$392$	0	0
$393$	3805.58	0.488463
$394$	0	0
$395$	−3475.36	−0.442695
$396$	0	0
$397$	−3851.15	−0.486861	−0.243431	−	0.969918i	$-0.578273\pi$
$397$	−3851.15	−0.486861	−0.243431	+	0.969918i	$0.578273\pi$
$398$	0	0
$399$	−509.198	−0.0638892
$400$	0	0
$401$	539.933	0.0672394	0.0336197	−	0.999435i	$-0.489297\pi$
$401$	539.933	0.0672394	0.0336197	+	0.999435i	$0.489297\pi$
$402$	0	0
$403$	−591.660	−0.0731332
$404$	0	0
$405$	1801.61	0.221044
$406$	0	0
$407$	4849.61	0.590630
$408$	0	0
$409$	−8778.60	−1.06130	−0.530652	−	0.847590i	$-0.678053\pi$
$409$	−8778.60	−1.06130	−0.530652	+	0.847590i	$0.678053\pi$
$410$	0	0
$411$	2212.39	0.265521
$412$	0	0
$413$	5031.31	0.599455
$414$	0	0
$415$	2760.58	0.326534
$416$	0	0
$417$	−1273.72	−0.149578
$418$	0	0
$419$	−13534.2	−1.57802	−0.789010	−	0.614380i	$-0.789406\pi$
$419$	−13534.2	−1.57802	−0.789010	+	0.614380i	$0.789406\pi$
$420$	0	0
$421$	−4055.19	−0.469448	−0.234724	−	0.972062i	$-0.575419\pi$
$421$	−4055.19	−0.469448	−0.234724	+	0.972062i	$0.575419\pi$
$422$	0	0
$423$	−9996.73	−1.14907
$424$	0	0
$425$	4321.50	0.493232
$426$	0	0
$427$	−4144.72	−0.469736
$428$	0	0
$429$	−554.862	−0.0624452
$430$	0	0
$431$	14601.5	1.63186	0.815929	−	0.578153i	$-0.196226\pi$
$431$	14601.5	1.63186	0.815929	+	0.578153i	$0.196226\pi$
$432$	0	0
$433$	−3546.74	−0.393638	−0.196819	−	0.980440i	$-0.563061\pi$
$433$	−3546.74	−0.393638	−0.196819	+	0.980440i	$0.563061\pi$
$434$	0	0
$435$	−3392.43	−0.373918
$436$	0	0
$437$	661.990	0.0724652
$438$	0	0
$439$	4947.47	0.537881	0.268941	−	0.963157i	$-0.413326\pi$
$439$	4947.47	0.537881	0.268941	+	0.963157i	$0.413326\pi$
$440$	0	0
$441$	4768.51	0.514903
$442$	0	0
$443$	3858.07	0.413776	0.206888	−	0.978365i	$-0.433666\pi$
$443$	3858.07	0.413776	0.206888	+	0.978365i	$0.433666\pi$
$444$	0	0
$445$	−11910.0	−1.26873
$446$	0	0
$447$	1778.08	0.188144
$448$	0	0
$449$	−5045.14	−0.530278	−0.265139	−	0.964210i	$-0.585418\pi$
$449$	−5045.14	−0.530278	−0.265139	+	0.964210i	$0.585418\pi$
$450$	0	0
$451$	−18654.0	−1.94763
$452$	0	0
$453$	−7236.34	−0.750536
$454$	0	0
$455$	345.967	0.0356466
$456$	0	0
$457$	−8779.52	−0.898662	−0.449331	−	0.893365i	$-0.648338\pi$
$457$	−8779.52	−0.898662	−0.449331	+	0.893365i	$0.648338\pi$
$458$	0	0
$459$	7459.97	0.758610
$460$	0	0
$461$	−16829.5	−1.70028	−0.850138	−	0.526561i	$-0.823481\pi$
$461$	−16829.5	−1.70028	−0.850138	+	0.526561i	$0.823481\pi$
$462$	0	0
$463$	6166.41	0.618957	0.309479	−	0.950906i	$-0.399845\pi$
$463$	6166.41	0.618957	0.309479	+	0.950906i	$0.399845\pi$
$464$	0	0
$465$	−2510.28	−0.250347
$466$	0	0
$467$	−2117.84	−0.209854	−0.104927	−	0.994480i	$-0.533461\pi$
$467$	−2117.84	−0.209854	−0.104927	+	0.994480i	$0.533461\pi$
$468$	0	0
$469$	208.456	0.0205237
$470$	0	0
$471$	5968.88	0.583931
$472$	0	0
$473$	−6326.45	−0.614991
$474$	0	0
$475$	1334.35	0.128893
$476$	0	0
$477$	2646.08	0.253995
$478$	0	0
$479$	3832.94	0.365619	0.182809	−	0.983148i	$-0.441481\pi$
$479$	3832.94	0.365619	0.182809	+	0.983148i	$0.441481\pi$
$480$	0	0
$481$	−442.668	−0.0419624
$482$	0	0
$483$	−933.752	−0.0879652
$484$	0	0
$485$	−12245.5	−1.14647
$486$	0	0
$487$	−3080.97	−0.286678	−0.143339	−	0.989674i	$-0.545784\pi$
$487$	−3080.97	−0.286678	−0.143339	+	0.989674i	$0.545784\pi$
$488$	0	0
$489$	−1659.10	−0.153429
$490$	0	0
$491$	9947.19	0.914278	0.457139	−	0.889395i	$-0.348874\pi$
$491$	9947.19	0.914278	0.457139	+	0.889395i	$0.348874\pi$
$492$	0	0
$493$	11047.2	1.00921
$494$	0	0
$495$	7395.65	0.671535
$496$	0	0
$497$	−2326.94	−0.210015
$498$	0	0
$499$	−13670.7	−1.22642	−0.613211	−	0.789919i	$-0.710122\pi$
$499$	−13670.7	−1.22642	−0.613211	+	0.789919i	$0.710122\pi$
$500$	0	0
$501$	−7919.51	−0.706223
$502$	0	0
$503$	−2851.63	−0.252779	−0.126390	−	0.991981i	$-0.540339\pi$
$503$	−2851.63	−0.252779	−0.126390	+	0.991981i	$0.540339\pi$
$504$	0	0
$505$	−4889.54	−0.430855
$506$	0	0
$507$	−5558.95	−0.486946
$508$	0	0
$509$	2228.47	0.194057	0.0970286	−	0.995282i	$-0.469066\pi$
$509$	2228.47	0.194057	0.0970286	+	0.995282i	$0.469066\pi$
$510$	0	0
$511$	2717.49	0.235254
$512$	0	0
$513$	2303.41	0.198242
$514$	0	0
$515$	733.709	0.0627788
$516$	0	0
$517$	−23816.1	−2.02597
$518$	0	0
$519$	−578.159	−0.0488986
$520$	0	0
$521$	−12531.3	−1.05375	−0.526877	−	0.849942i	$-0.676637\pi$
$521$	−12531.3	−1.05375	−0.526877	+	0.849942i	$0.676637\pi$
$522$	0	0
$523$	−11402.8	−0.953362	−0.476681	−	0.879076i	$-0.658160\pi$
$523$	−11402.8	−0.953362	−0.476681	+	0.879076i	$0.658160\pi$
$524$	0	0
$525$	−1882.13	−0.156462
$526$	0	0
$527$	8174.56	0.675691
$528$	0	0
$529$	−10953.1	−0.900227
$530$	0	0
$531$	−9817.34	−0.802328
$532$	0	0
$533$	1702.72	0.138373
$534$	0	0
$535$	11730.1	0.947918
$536$	0	0
$537$	6433.52	0.516996
$538$	0	0
$539$	11360.4	0.907845
$540$	0	0
$541$	−1217.02	−0.0967171	−0.0483585	−	0.998830i	$-0.515399\pi$
$541$	−1217.02	−0.0967171	−0.0483585	+	0.998830i	$0.515399\pi$
$542$	0	0
$543$	−2585.31	−0.204321
$544$	0	0
$545$	−1502.39	−0.118083
$546$	0	0
$547$	−13940.4	−1.08967	−0.544833	−	0.838545i	$-0.683407\pi$
$547$	−13940.4	−1.08967	−0.544833	+	0.838545i	$0.683407\pi$
$548$	0	0
$549$	8087.38	0.628708
$550$	0	0
$551$	3411.04	0.263730
$552$	0	0
$553$	−4928.97	−0.379025
$554$	0	0
$555$	−1878.14	−0.143644
$556$	0	0
$557$	10002.4	0.760889	0.380445	−	0.924804i	$-0.375771\pi$
$557$	10002.4	0.760889	0.380445	+	0.924804i	$0.375771\pi$
$558$	0	0
$559$	577.473	0.0436932
$560$	0	0
$561$	7666.14	0.576942
$562$	0	0
$563$	−8869.58	−0.663958	−0.331979	−	0.943287i	$-0.607716\pi$
$563$	−8869.58	−0.663958	−0.331979	+	0.943287i	$0.607716\pi$
$564$	0	0
$565$	−13686.1	−1.01908
$566$	0	0
$567$	2555.15	0.189253
$568$	0	0
$569$	8649.42	0.637263	0.318631	−	0.947879i	$-0.396777\pi$
$569$	8649.42	0.637263	0.318631	+	0.947879i	$0.396777\pi$
$570$	0	0
$571$	6544.85	0.479673	0.239837	−	0.970813i	$-0.422906\pi$
$571$	6544.85	0.479673	0.239837	+	0.970813i	$0.422906\pi$
$572$	0	0
$573$	−4957.04	−0.361402
$574$	0	0
$575$	2446.88	0.177465
$576$	0	0
$577$	4202.01	0.303175	0.151587	−	0.988444i	$-0.451561\pi$
$577$	4202.01	0.303175	0.151587	+	0.988444i	$0.451561\pi$
$578$	0	0
$579$	5284.61	0.379311
$580$	0	0
$581$	3915.23	0.279571
$582$	0	0
$583$	6303.98	0.447829
$584$	0	0
$585$	−675.068	−0.0477105
$586$	0	0
$587$	−21992.1	−1.54635	−0.773177	−	0.634190i	$-0.781334\pi$
$587$	−21992.1	−1.54635	−0.773177	+	0.634190i	$0.781334\pi$
$588$	0	0
$589$	2524.05	0.176574
$590$	0	0
$591$	10745.2	0.747882
$592$	0	0
$593$	−1596.92	−0.110586	−0.0552932	−	0.998470i	$-0.517609\pi$
$593$	−1596.92	−0.110586	−0.0552932	+	0.998470i	$0.517609\pi$
$594$	0	0
$595$	−4779.99	−0.329345
$596$	0	0
$597$	683.517	0.0468584
$598$	0	0
$599$	−9965.77	−0.679783	−0.339892	−	0.940465i	$-0.610390\pi$
$599$	−9965.77	−0.679783	−0.339892	+	0.940465i	$0.610390\pi$
$600$	0	0
$601$	−12790.5	−0.868110	−0.434055	−	0.900886i	$-0.642918\pi$
$601$	−12790.5	−0.868110	−0.434055	+	0.900886i	$0.642918\pi$
$602$	0	0
$603$	−406.749	−0.0274695
$604$	0	0
$605$	7768.88	0.522066
$606$	0	0
$607$	25391.9	1.69790	0.848949	−	0.528474i	$-0.177236\pi$
$607$	25391.9	1.69790	0.848949	+	0.528474i	$0.177236\pi$
$608$	0	0
$609$	−4811.35	−0.320141
$610$	0	0
$611$	2173.91	0.143939
$612$	0	0
$613$	−9439.11	−0.621929	−0.310964	−	0.950422i	$-0.600652\pi$
$613$	−9439.11	−0.621929	−0.310964	+	0.950422i	$0.600652\pi$
$614$	0	0
$615$	7224.26	0.473675
$616$	0	0
$617$	3247.93	0.211923	0.105962	−	0.994370i	$-0.466208\pi$
$617$	3247.93	0.211923	0.105962	+	0.994370i	$0.466208\pi$
$618$	0	0
$619$	−2781.32	−0.180599	−0.0902995	−	0.995915i	$-0.528782\pi$
$619$	−2781.32	−0.180599	−0.0902995	+	0.995915i	$0.528782\pi$
$620$	0	0
$621$	4223.93	0.272947
$622$	0	0
$623$	−16891.4	−1.08626
$624$	0	0
$625$	−1914.31	−0.122516
$626$	0	0
$627$	2367.07	0.150768
$628$	0	0
$629$	6116.03	0.387698
$630$	0	0
$631$	−877.112	−0.0553364	−0.0276682	−	0.999617i	$-0.508808\pi$
$631$	−877.112	−0.0553364	−0.0276682	+	0.999617i	$0.508808\pi$
$632$	0	0
$633$	−3215.41	−0.201897
$634$	0	0
$635$	12875.1	0.804618
$636$	0	0
$637$	−1036.97	−0.0644996
$638$	0	0
$639$	4540.44	0.281091
$640$	0	0
$641$	18518.8	1.14111	0.570554	−	0.821260i	$-0.306728\pi$
$641$	18518.8	1.14111	0.570554	+	0.821260i	$0.306728\pi$
$642$	0	0
$643$	13493.0	0.827548	0.413774	−	0.910380i	$-0.364210\pi$
$643$	13493.0	0.827548	0.413774	+	0.910380i	$0.364210\pi$
$644$	0	0
$645$	2450.09	0.149569
$646$	0	0
$647$	−9832.93	−0.597485	−0.298742	−	0.954334i	$-0.596567\pi$
$647$	−9832.93	−0.597485	−0.298742	+	0.954334i	$0.596567\pi$
$648$	0	0
$649$	−23388.7	−1.41462
$650$	0	0
$651$	−3560.23	−0.214342
$652$	0	0
$653$	4752.63	0.284816	0.142408	−	0.989808i	$-0.454515\pi$
$653$	4752.63	0.284816	0.142408	+	0.989808i	$0.454515\pi$
$654$	0	0
$655$	11030.5	0.658012
$656$	0	0
$657$	−5302.49	−0.314870
$658$	0	0
$659$	−2061.33	−0.121848	−0.0609242	−	0.998142i	$-0.519405\pi$
$659$	−2061.33	−0.121848	−0.0609242	+	0.998142i	$0.519405\pi$
$660$	0	0
$661$	−2408.74	−0.141739	−0.0708693	−	0.997486i	$-0.522577\pi$
$661$	−2408.74	−0.141739	−0.0708693	+	0.997486i	$0.522577\pi$
$662$	0	0
$663$	−699.758	−0.0409900
$664$	0	0
$665$	−1475.92	−0.0860655
$666$	0	0
$667$	6255.06	0.363114
$668$	0	0
$669$	9293.76	0.537096
$670$	0	0
$671$	19267.2	1.10850
$672$	0	0
$673$	−22916.0	−1.31255	−0.656275	−	0.754522i	$-0.727869\pi$
$673$	−22916.0	−1.31255	−0.656275	+	0.754522i	$0.727869\pi$
$674$	0	0
$675$	8514.00	0.485487
$676$	0	0
$677$	−11902.9	−0.675724	−0.337862	−	0.941196i	$-0.609704\pi$
$677$	−11902.9	−0.675724	−0.337862	+	0.941196i	$0.609704\pi$
$678$	0	0
$679$	−17367.3	−0.981582
$680$	0	0
$681$	4784.29	0.269214
$682$	0	0
$683$	25810.3	1.44598	0.722989	−	0.690860i	$-0.242768\pi$
$683$	25810.3	1.44598	0.722989	+	0.690860i	$0.242768\pi$
$684$	0	0
$685$	6412.63	0.357685
$686$	0	0
$687$	10520.7	0.584264
$688$	0	0
$689$	−575.421	−0.0318168
$690$	0	0
$691$	−1966.02	−0.108236	−0.0541180	−	0.998535i	$-0.517235\pi$
$691$	−1966.02	−0.108236	−0.0541180	+	0.998535i	$0.517235\pi$
$692$	0	0
$693$	10489.0	0.574953
$694$	0	0
$695$	−3691.88	−0.201498
$696$	0	0
$697$	−23525.3	−1.27846
$698$	0	0
$699$	1435.02	0.0776503
$700$	0	0
$701$	29611.2	1.59543	0.797717	−	0.603032i	$-0.206041\pi$
$701$	29611.2	1.59543	0.797717	+	0.603032i	$0.206041\pi$
$702$	0	0
$703$	1888.44	0.101314
$704$	0	0
$705$	9223.40	0.492728
$706$	0	0
$707$	−6934.64	−0.368888
$708$	0	0
$709$	−27618.7	−1.46297	−0.731483	−	0.681860i	$-0.761171\pi$
$709$	−27618.7	−1.46297	−0.731483	+	0.681860i	$0.761171\pi$
$710$	0	0
$711$	9617.63	0.507299
$712$	0	0
$713$	4628.53	0.243113
$714$	0	0
$715$	−1608.27	−0.0841202
$716$	0	0
$717$	−12140.0	−0.632326
$718$	0	0
$719$	−32303.4	−1.67554	−0.837770	−	0.546023i	$-0.816141\pi$
$719$	−32303.4	−1.67554	−0.837770	+	0.546023i	$0.816141\pi$
$720$	0	0
$721$	1040.59	0.0537498
$722$	0	0
$723$	2059.11	0.105919
$724$	0	0
$725$	12608.1	0.645865
$726$	0	0
$727$	−30854.5	−1.57405	−0.787023	−	0.616924i	$-0.788379\pi$
$727$	−30854.5	−1.57405	−0.787023	+	0.616924i	$0.788379\pi$
$728$	0	0
$729$	3371.90	0.171310
$730$	0	0
$731$	−7978.54	−0.403689
$732$	0	0
$733$	−22692.4	−1.14347	−0.571735	−	0.820438i	$-0.693729\pi$
$733$	−22692.4	−1.14347	−0.571735	+	0.820438i	$0.693729\pi$
$734$	0	0
$735$	−4399.63	−0.220793
$736$	0	0
$737$	−969.033	−0.0484325
$738$	0	0
$739$	−34965.3	−1.74048	−0.870242	−	0.492624i	$-0.836038\pi$
$739$	−34965.3	−1.74048	−0.870242	+	0.492624i	$0.836038\pi$
$740$	0	0
$741$	−216.064	−0.0107116
$742$	0	0
$743$	−18699.8	−0.923325	−0.461663	−	0.887056i	$-0.652747\pi$
$743$	−18699.8	−0.923325	−0.461663	+	0.887056i	$0.652747\pi$
$744$	0	0
$745$	5153.79	0.253450
$746$	0	0
$747$	−7639.57	−0.374187
$748$	0	0
$749$	16636.3	0.811586
$750$	0	0
$751$	20650.7	1.00340	0.501700	−	0.865041i	$-0.332708\pi$
$751$	20650.7	1.00340	0.501700	+	0.865041i	$0.332708\pi$
$752$	0	0
$753$	−6344.62	−0.307053
$754$	0	0
$755$	−20974.6	−1.01105
$756$	0	0
$757$	33118.8	1.59012	0.795062	−	0.606528i	$-0.207438\pi$
$757$	33118.8	1.59012	0.795062	+	0.606528i	$0.207438\pi$
$758$	0	0
$759$	4340.66	0.207584
$760$	0	0
$761$	16513.6	0.786618	0.393309	−	0.919406i	$-0.371330\pi$
$761$	16513.6	0.786618	0.393309	+	0.919406i	$0.371330\pi$
$762$	0	0
$763$	−2130.78	−0.101100
$764$	0	0
$765$	9326.94	0.440806
$766$	0	0
$767$	2134.90	0.100504
$768$	0	0
$769$	16599.5	0.778406	0.389203	−	0.921152i	$-0.372750\pi$
$769$	16599.5	0.778406	0.389203	+	0.921152i	$0.372750\pi$
$770$	0	0
$771$	9014.64	0.421082
$772$	0	0
$773$	−22940.9	−1.06743	−0.533717	−	0.845663i	$-0.679205\pi$
$773$	−22940.9	−1.06743	−0.533717	+	0.845663i	$0.679205\pi$
$774$	0	0
$775$	9329.55	0.432422
$776$	0	0
$777$	−2663.69	−0.122985
$778$	0	0
$779$	−7263.89	−0.334090
$780$	0	0
$781$	10817.1	0.495603
$782$	0	0
$783$	21764.6	0.993365
$784$	0	0
$785$	17300.8	0.786616
$786$	0	0
$787$	15086.1	0.683304	0.341652	−	0.939826i	$-0.389014\pi$
$787$	15086.1	0.683304	0.341652	+	0.939826i	$0.389014\pi$
$788$	0	0
$789$	7726.25	0.348621
$790$	0	0
$791$	−19410.4	−0.872511
$792$	0	0
$793$	−1758.70	−0.0787555
$794$	0	0
$795$	−2441.38	−0.108914
$796$	0	0
$797$	−36309.4	−1.61373	−0.806867	−	0.590733i	$-0.798839\pi$
$797$	−36309.4	−1.61373	−0.806867	+	0.590733i	$0.798839\pi$
$798$	0	0
$799$	−30035.4	−1.32988
$800$	0	0
$801$	32959.4	1.45388
$802$	0	0
$803$	−12632.6	−0.555161
$804$	0	0
$805$	−2706.49	−0.118498
$806$	0	0
$807$	8119.47	0.354175
$808$	0	0
$809$	32700.4	1.42112	0.710560	−	0.703637i	$-0.248442\pi$
$809$	32700.4	1.42112	0.710560	+	0.703637i	$0.248442\pi$
$810$	0	0
$811$	6484.51	0.280767	0.140383	−	0.990097i	$-0.455166\pi$
$811$	6484.51	0.280767	0.140383	+	0.990097i	$0.455166\pi$
$812$	0	0
$813$	−14799.3	−0.638418
$814$	0	0
$815$	−4808.90	−0.206685
$816$	0	0
$817$	−2463.53	−0.105493
$818$	0	0
$819$	−957.422	−0.0408486
$820$	0	0
$821$	−8864.19	−0.376812	−0.188406	−	0.982091i	$-0.560332\pi$
$821$	−8864.19	−0.376812	−0.188406	+	0.982091i	$0.560332\pi$
$822$	0	0
$823$	−24083.4	−1.02004	−0.510022	−	0.860161i	$-0.670363\pi$
$823$	−24083.4	−1.02004	−0.510022	+	0.860161i	$0.670363\pi$
$824$	0	0
$825$	8749.29	0.369226
$826$	0	0
$827$	35052.0	1.47385	0.736927	−	0.675972i	$-0.236276\pi$
$827$	35052.0	1.47385	0.736927	+	0.675972i	$0.236276\pi$
$828$	0	0
$829$	1914.79	0.0802214	0.0401107	−	0.999195i	$-0.487229\pi$
$829$	1914.79	0.0802214	0.0401107	+	0.999195i	$0.487229\pi$
$830$	0	0
$831$	−13131.6	−0.548173
$832$	0	0
$833$	14327.1	0.595923
$834$	0	0
$835$	−22954.8	−0.951357
$836$	0	0
$837$	16105.1	0.665082
$838$	0	0
$839$	19785.9	0.814164	0.407082	−	0.913392i	$-0.366546\pi$
$839$	19785.9	0.814164	0.407082	+	0.913392i	$0.366546\pi$
$840$	0	0
$841$	7841.48	0.321517
$842$	0	0
$843$	5415.75	0.221267
$844$	0	0
$845$	−16112.7	−0.655967
$846$	0	0
$847$	11018.3	0.446981
$848$	0	0
$849$	13905.3	0.562107
$850$	0	0
$851$	3462.97	0.139494
$852$	0	0
$853$	−287.455	−0.0115384	−0.00576920	−	0.999983i	$-0.501836\pi$
$853$	−287.455	−0.0115384	−0.00576920	+	0.999983i	$0.501836\pi$
$854$	0	0
$855$	2879.88	0.115193
$856$	0	0
$857$	2352.88	0.0937839	0.0468919	−	0.998900i	$-0.485068\pi$
$857$	2352.88	0.0937839	0.0468919	+	0.998900i	$0.485068\pi$
$858$	0	0
$859$	29976.3	1.19066	0.595331	−	0.803480i	$-0.297021\pi$
$859$	29976.3	1.19066	0.595331	+	0.803480i	$0.297021\pi$
$860$	0	0
$861$	10245.9	0.405550
$862$	0	0
$863$	28858.3	1.13829	0.569147	−	0.822236i	$-0.307274\pi$
$863$	28858.3	1.13829	0.569147	+	0.822236i	$0.307274\pi$
$864$	0	0
$865$	−1675.80	−0.0658715
$866$	0	0
$867$	−2876.28	−0.112669
$868$	0	0
$869$	22912.9	0.894438
$870$	0	0
$871$	88.4524	0.00344098
$872$	0	0
$873$	33887.8	1.31378
$874$	0	0
$875$	−15165.3	−0.585922
$876$	0	0
$877$	45124.0	1.73743	0.868716	−	0.495310i	$-0.164946\pi$
$877$	45124.0	1.73743	0.868716	+	0.495310i	$0.164946\pi$
$878$	0	0
$879$	15639.4	0.600116
$880$	0	0
$881$	46543.5	1.77990	0.889949	−	0.456060i	$-0.150740\pi$
$881$	46543.5	1.77990	0.889949	+	0.456060i	$0.150740\pi$
$882$	0	0
$883$	−10533.3	−0.401442	−0.200721	−	0.979648i	$-0.564329\pi$
$883$	−10533.3	−0.401442	−0.200721	+	0.979648i	$0.564329\pi$
$884$	0	0
$885$	9057.89	0.344042
$886$	0	0
$887$	−28070.3	−1.06258	−0.531289	−	0.847191i	$-0.678292\pi$
$887$	−28070.3	−1.06258	−0.531289	+	0.847191i	$0.678292\pi$
$888$	0	0
$889$	18260.2	0.688896
$890$	0	0
$891$	−11877.9	−0.446606
$892$	0	0
$893$	−9274.01	−0.347528
$894$	0	0
$895$	18647.6	0.696448
$896$	0	0
$897$	−396.211	−0.0147482
$898$	0	0
$899$	23849.5	0.884788
$900$	0	0
$901$	7950.19	0.293962
$902$	0	0
$903$	3474.86	0.128058
$904$	0	0
$905$	−7493.55	−0.275242
$906$	0	0
$907$	26589.1	0.973404	0.486702	−	0.873568i	$-0.338200\pi$
$907$	26589.1	0.973404	0.486702	+	0.873568i	$0.338200\pi$
$908$	0	0
$909$	13531.2	0.493731
$910$	0	0
$911$	32296.8	1.17458	0.587290	−	0.809377i	$-0.300195\pi$
$911$	32296.8	1.17458	0.587290	+	0.809377i	$0.300195\pi$
$912$	0	0
$913$	−18200.4	−0.659743
$914$	0	0
$915$	−7461.76	−0.269593
$916$	0	0
$917$	15644.1	0.563375
$918$	0	0
$919$	−35127.6	−1.26089	−0.630443	−	0.776236i	$-0.717127\pi$
$919$	−35127.6	−1.26089	−0.630443	+	0.776236i	$0.717127\pi$
$920$	0	0
$921$	−3226.40	−0.115433
$922$	0	0
$923$	−987.373	−0.0352110
$924$	0	0
$925$	6980.17	0.248115
$926$	0	0
$927$	−2030.45	−0.0719403
$928$	0	0
$929$	−14258.8	−0.503570	−0.251785	−	0.967783i	$-0.581018\pi$
$929$	−14258.8	−0.503570	−0.251785	+	0.967783i	$0.581018\pi$
$930$	0	0
$931$	4423.77	0.155728
$932$	0	0
$933$	−12649.7	−0.443872
$934$	0	0
$935$	22220.4	0.777202
$936$	0	0
$937$	−40641.2	−1.41696	−0.708479	−	0.705732i	$-0.750618\pi$
$937$	−40641.2	−1.41696	−0.708479	+	0.705732i	$0.750618\pi$
$938$	0	0
$939$	12102.8	0.420618
$940$	0	0
$941$	−22364.8	−0.774782	−0.387391	−	0.921915i	$-0.626624\pi$
$941$	−22364.8	−0.774782	−0.387391	+	0.921915i	$0.626624\pi$
$942$	0	0
$943$	−13320.3	−0.459988
$944$	0	0
$945$	−9417.30	−0.324174
$946$	0	0
$947$	27719.2	0.951166	0.475583	−	0.879671i	$-0.342237\pi$
$947$	27719.2	0.951166	0.475583	+	0.879671i	$0.342237\pi$
$948$	0	0
$949$	1153.09	0.0394424
$950$	0	0
$951$	−12732.9	−0.434168
$952$	0	0
$953$	−709.901	−0.0241300	−0.0120650	−	0.999927i	$-0.503841\pi$
$953$	−709.901	−0.0241300	−0.0120650	+	0.999927i	$0.503841\pi$
$954$	0	0
$955$	−14368.0	−0.486846
$956$	0	0
$957$	22366.1	0.755480
$958$	0	0
$959$	9094.78	0.306242
$960$	0	0
$961$	−12143.2	−0.407613
$962$	0	0
$963$	−32461.6	−1.08625
$964$	0	0
$965$	15317.5	0.510972
$966$	0	0
$967$	−2175.29	−0.0723400	−0.0361700	−	0.999346i	$-0.511516\pi$
$967$	−2175.29	−0.0723400	−0.0361700	+	0.999346i	$0.511516\pi$
$968$	0	0
$969$	2985.20	0.0989666
$970$	0	0
$971$	13964.7	0.461533	0.230766	−	0.973009i	$-0.425877\pi$
$971$	13964.7	0.461533	0.230766	+	0.973009i	$0.425877\pi$
$972$	0	0
$973$	−5236.04	−0.172518
$974$	0	0
$975$	−798.627	−0.0262323
$976$	0	0
$977$	−8452.26	−0.276778	−0.138389	−	0.990378i	$-0.544192\pi$
$977$	−8452.26	−0.276778	−0.138389	+	0.990378i	$0.544192\pi$
$978$	0	0
$979$	78521.9	2.56340
$980$	0	0
$981$	4157.68	0.135316
$982$	0	0
$983$	−55328.2	−1.79521	−0.897607	−	0.440798i	$-0.854696\pi$
$983$	−55328.2	−1.79521	−0.897607	+	0.440798i	$0.854696\pi$
$984$	0	0
$985$	31145.0	1.00748
$986$	0	0
$987$	13081.2	0.421863
$988$	0	0
$989$	−4517.54	−0.145247
$990$	0	0
$991$	−33520.4	−1.07448	−0.537240	−	0.843429i	$-0.680533\pi$
$991$	−33520.4	−1.07448	−0.537240	+	0.843429i	$0.680533\pi$
$992$	0	0
$993$	12213.8	0.390324
$994$	0	0
$995$	1981.18	0.0631232
$996$	0	0
$997$	−36267.7	−1.15206	−0.576032	−	0.817427i	$-0.695400\pi$
$997$	−36267.7	−1.15206	−0.576032	+	0.817427i	$0.695400\pi$
$998$	0	0
$999$	12049.5	0.381611

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.4.a.h.1.3	yes	5
4.3	odd	2		608.4.a.e.1.3	✓	5
8.3	odd	2		1216.4.a.bd.1.3		5
8.5	even	2		1216.4.a.y.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
608.4.a.e.1.3	✓	5	4.3	odd	2
608.4.a.h.1.3	yes	5	1.1	even	1	trivial
1216.4.a.y.1.3		5	8.5	even	2
1216.4.a.bd.1.3		5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	608.a (trivial)

\(f(q)\)	\(=\)	\(q+2.55330 q^{3} +7.40075 q^{5} +10.4962 q^{7} -20.4807 q^{9} -48.7928 q^{11} +4.45376 q^{13} +18.8963 q^{15} -61.5346 q^{17} -19.0000 q^{19} +26.7999 q^{21} -34.8416 q^{23} -70.2288 q^{25} -121.232 q^{27} -179.528 q^{29} -132.845 q^{31} -124.583 q^{33} +77.6798 q^{35} -99.3918 q^{37} +11.3718 q^{39} +382.310 q^{41} +129.659 q^{43} -151.572 q^{45} +488.106 q^{47} -232.830 q^{49} -157.116 q^{51} -129.199 q^{53} -361.104 q^{55} -48.5127 q^{57} +479.346 q^{59} -394.879 q^{61} -214.969 q^{63} +32.9612 q^{65} +19.8601 q^{67} -88.9610 q^{69} -221.694 q^{71} +258.902 q^{73} -179.315 q^{75} -512.139 q^{77} -469.595 q^{79} +243.436 q^{81} +373.014 q^{83} -455.402 q^{85} -458.390 q^{87} -1609.29 q^{89} +46.7476 q^{91} -339.193 q^{93} -140.614 q^{95} -1654.62 q^{97} +999.310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{3} + 5 q^{5} - 7 q^{7} - 5 q^{9} + 13 q^{11} - 72 q^{13} - 72 q^{15} - 59 q^{17} - 95 q^{19} - 224 q^{21} - 52 q^{23} - 86 q^{25} + 54 q^{27} - 128 q^{29} + 110 q^{31} - 68 q^{33} - 45 q^{35}+ \cdots + 601 q^{99}+O(q^{100}) \) 5 * q + 6 * q^3 + 5 * q^5 - 7 * q^7 - 5 * q^9 + 13 * q^11 - 72 * q^13 - 72 * q^15 - 59 * q^17 - 95 * q^19 - 224 * q^21 - 52 * q^23 - 86 * q^25 + 54 * q^27 - 128 * q^29 + 110 * q^31 - 68 * q^33 - 45 * q^35 - 436 * q^37 - 356 * q^39 - 804 * q^41 - 143 * q^43 - 579 * q^45 - 661 * q^47 - 406 * q^49 + 570 * q^51 - 898 * q^53 - 17 * q^55 - 114 * q^57 + 196 * q^59 - 1079 * q^61 - 143 * q^63 - 1632 * q^65 + 832 * q^67 - 1644 * q^69 - 834 * q^71 - 1375 * q^73 + 1054 * q^75 - 1473 * q^77 - 154 * q^79 - 1859 * q^81 + 2224 * q^83 - 1807 * q^85 - 2004 * q^87 - 542 * q^89 + 1890 * q^91 - 2788 * q^93 - 95 * q^95 - 1528 * q^97 + 601 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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