LMFDB - Embedded newform 616.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [616,4,Mod(1,616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(616, 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("616.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 616 = 2^{3} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	616.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:	$36.3451765635$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	616.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.70820 q^{3} +0.763932 q^{5} -7.00000 q^{7} +5.58359 q^{9} -11.0000 q^{11} -60.6525 q^{13} +4.36068 q^{15} +36.2229 q^{17} +47.7082 q^{19} -39.9574 q^{21} -186.387 q^{23} -124.416 q^{25} -122.249 q^{27} +27.2523 q^{29} +156.079 q^{31} -62.7902 q^{33} -5.34752 q^{35} -302.721 q^{37} -346.217 q^{39} -73.2724 q^{41} -43.9675 q^{43} +4.26548 q^{45} -112.197 q^{47} +49.0000 q^{49} +206.768 q^{51} +264.249 q^{53} -8.40325 q^{55} +272.328 q^{57} -339.872 q^{59} -557.459 q^{61} -39.0851 q^{63} -46.3344 q^{65} +926.906 q^{67} -1063.93 q^{69} -78.7601 q^{71} -668.689 q^{73} -710.194 q^{75} +77.0000 q^{77} -1218.72 q^{79} -848.580 q^{81} +1006.86 q^{83} +27.6718 q^{85} +155.562 q^{87} +1146.42 q^{89} +424.567 q^{91} +890.930 q^{93} +36.4458 q^{95} +657.299 q^{97} -61.4195 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 6 q^{5} - 14 q^{7} + 38 q^{9} - 22 q^{11} - 90 q^{13} - 36 q^{15} + 144 q^{17} + 82 q^{19} + 14 q^{21} - 176 q^{23} - 222 q^{25} - 164 q^{27} + 296 q^{29} - 144 q^{31} + 22 q^{33} - 42 q^{35}+ \cdots - 418 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 6 * q^5 - 14 * q^7 + 38 * q^9 - 22 * q^11 - 90 * q^13 - 36 * q^15 + 144 * q^17 + 82 * q^19 + 14 * q^21 - 176 * q^23 - 222 * q^25 - 164 * q^27 + 296 * q^29 - 144 * q^31 + 22 * q^33 - 42 * q^35 - 516 * q^37 - 120 * q^39 + 408 * q^41 + 404 * q^43 + 174 * q^45 - 448 * q^47 + 98 * q^49 - 624 * q^51 + 448 * q^53 - 66 * q^55 + 8 * q^57 - 398 * q^59 - 1182 * q^61 - 266 * q^63 - 200 * q^65 + 700 * q^67 - 1144 * q^69 - 1204 * q^71 - 756 * q^73 + 42 * q^75 + 154 * q^77 - 1704 * q^79 - 1402 * q^81 + 122 * q^83 + 592 * q^85 - 1916 * q^87 - 140 * q^89 + 630 * q^91 + 3204 * q^93 + 216 * q^95 + 608 * q^97 - 418 * 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
$3$	5.70820	1.09854	0.549272	−	0.835644i	$-0.314905\pi$
$3$	5.70820	1.09854	0.549272	+	0.835644i	$0.314905\pi$
$4$	0	0
$5$	0.763932	0.0683282	0.0341641	−	0.999416i	$-0.489123\pi$
$5$	0.763932	0.0683282	0.0341641	+	0.999416i	$0.489123\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	5.58359	0.206800
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−60.6525	−1.29400	−0.646999	−	0.762491i	$-0.723976\pi$
$13$	−60.6525	−1.29400	−0.646999	+	0.762491i	$0.723976\pi$
$14$	0	0
$15$	4.36068	0.0750615
$16$	0	0
$17$	36.2229	0.516785	0.258393	−	0.966040i	$-0.416807\pi$
$17$	36.2229	0.516785	0.258393	+	0.966040i	$0.416807\pi$
$18$	0	0
$19$	47.7082	0.576053	0.288027	−	0.957622i	$-0.407001\pi$
$19$	47.7082	0.576053	0.288027	+	0.957622i	$0.407001\pi$
$20$	0	0
$21$	−39.9574	−0.415211
$22$	0	0
$23$	−186.387	−1.68976	−0.844878	−	0.534960i	$-0.820327\pi$
$23$	−186.387	−1.68976	−0.844878	+	0.534960i	$0.820327\pi$
$24$	0	0
$25$	−124.416	−0.995331
$26$	0	0
$27$	−122.249	−0.871366
$28$	0	0
$29$	27.2523	0.174504	0.0872522	−	0.996186i	$-0.472191\pi$
$29$	27.2523	0.174504	0.0872522	+	0.996186i	$0.472191\pi$
$30$	0	0
$31$	156.079	0.904278	0.452139	−	0.891948i	$-0.350661\pi$
$31$	156.079	0.904278	0.452139	+	0.891948i	$0.350661\pi$
$32$	0	0
$33$	−62.7902	−0.331224
$34$	0	0
$35$	−5.34752	−0.0258256
$36$	0	0
$37$	−302.721	−1.34506	−0.672528	−	0.740072i	$-0.734792\pi$
$37$	−302.721	−1.34506	−0.672528	+	0.740072i	$0.734792\pi$
$38$	0	0
$39$	−346.217	−1.42151
$40$	0	0
$41$	−73.2724	−0.279103	−0.139552	−	0.990215i	$-0.544566\pi$
$41$	−73.2724	−0.279103	−0.139552	+	0.990215i	$0.544566\pi$
$42$	0	0
$43$	−43.9675	−0.155930	−0.0779649	−	0.996956i	$-0.524842\pi$
$43$	−43.9675	−0.155930	−0.0779649	+	0.996956i	$0.524842\pi$
$44$	0	0
$45$	4.26548	0.0141302
$46$	0	0
$47$	−112.197	−0.348203	−0.174102	−	0.984728i	$-0.555702\pi$
$47$	−112.197	−0.348203	−0.174102	+	0.984728i	$0.555702\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	206.768	0.567711
$52$	0	0
$53$	264.249	0.684857	0.342429	−	0.939544i	$-0.388751\pi$
$53$	264.249	0.684857	0.342429	+	0.939544i	$0.388751\pi$
$54$	0	0
$55$	−8.40325	−0.0206017
$56$	0	0
$57$	272.328	0.632820
$58$	0	0
$59$	−339.872	−0.749959	−0.374980	−	0.927033i	$-0.622350\pi$
$59$	−339.872	−0.749959	−0.374980	+	0.927033i	$0.622350\pi$
$60$	0	0
$61$	−557.459	−1.17009	−0.585044	−	0.811002i	$-0.698923\pi$
$61$	−557.459	−1.17009	−0.585044	+	0.811002i	$0.698923\pi$
$62$	0	0
$63$	−39.0851	−0.0781629
$64$	0	0
$65$	−46.3344	−0.0884165
$66$	0	0
$67$	926.906	1.69014	0.845071	−	0.534654i	$-0.179558\pi$
$67$	926.906	1.69014	0.845071	+	0.534654i	$0.179558\pi$
$68$	0	0
$69$	−1063.93	−1.85627
$70$	0	0
$71$	−78.7601	−0.131649	−0.0658247	−	0.997831i	$-0.520968\pi$
$71$	−78.7601	−0.131649	−0.0658247	+	0.997831i	$0.520968\pi$
$72$	0	0
$73$	−668.689	−1.07211	−0.536056	−	0.844183i	$-0.680086\pi$
$73$	−668.689	−1.07211	−0.536056	+	0.844183i	$0.680086\pi$
$74$	0	0
$75$	−710.194	−1.09342
$76$	0	0
$77$	77.0000	0.113961
$78$	0	0
$79$	−1218.72	−1.73565	−0.867824	−	0.496872i	$-0.834482\pi$
$79$	−1218.72	−1.73565	−0.867824	+	0.496872i	$0.834482\pi$
$80$	0	0
$81$	−848.580	−1.16403
$82$	0	0
$83$	1006.86	1.33153	0.665764	−	0.746162i	$-0.268106\pi$
$83$	1006.86	1.33153	0.665764	+	0.746162i	$0.268106\pi$
$84$	0	0
$85$	27.6718	0.0353110
$86$	0	0
$87$	155.562	0.191701
$88$	0	0
$89$	1146.42	1.36540	0.682699	−	0.730700i	$-0.260806\pi$
$89$	1146.42	1.36540	0.682699	+	0.730700i	$0.260806\pi$
$90$	0	0
$91$	424.567	0.489085
$92$	0	0
$93$	890.930	0.993389
$94$	0	0
$95$	36.4458	0.0393607
$96$	0	0
$97$	657.299	0.688027	0.344013	−	0.938965i	$-0.388213\pi$
$97$	657.299	0.688027	0.344013	+	0.938965i	$0.388213\pi$
$98$	0	0
$99$	−61.4195	−0.0623525
$100$	0	0
$101$	398.306	0.392405	0.196202	−	0.980563i	$-0.437139\pi$
$101$	398.306	0.392405	0.196202	+	0.980563i	$0.437139\pi$
$102$	0	0
$103$	1927.16	1.84358	0.921791	−	0.387687i	$-0.126726\pi$
$103$	1927.16	1.84358	0.921791	+	0.387687i	$0.126726\pi$
$104$	0	0
$105$	−30.5248	−0.0283706
$106$	0	0
$107$	−1783.72	−1.61158	−0.805788	−	0.592204i	$-0.798258\pi$
$107$	−1783.72	−1.61158	−0.805788	+	0.592204i	$0.798258\pi$
$108$	0	0
$109$	−172.570	−0.151644	−0.0758219	−	0.997121i	$-0.524158\pi$
$109$	−172.570	−0.151644	−0.0758219	+	0.997121i	$0.524158\pi$
$110$	0	0
$111$	−1728.00	−1.47760
$112$	0	0
$113$	50.1502	0.0417498	0.0208749	−	0.999782i	$-0.493355\pi$
$113$	50.1502	0.0417498	0.0208749	+	0.999782i	$0.493355\pi$
$114$	0	0
$115$	−142.387	−0.115458
$116$	0	0
$117$	−338.659	−0.267598
$118$	0	0
$119$	−253.560	−0.195326
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−418.254	−0.306607
$124$	0	0
$125$	−190.537	−0.136337
$126$	0	0
$127$	−2143.92	−1.49797	−0.748986	−	0.662586i	$-0.769459\pi$
$127$	−2143.92	−1.49797	−0.748986	+	0.662586i	$0.769459\pi$
$128$	0	0
$129$	−250.975	−0.171296
$130$	0	0
$131$	−1351.35	−0.901281	−0.450640	−	0.892706i	$-0.648804\pi$
$131$	−1351.35	−0.901281	−0.450640	+	0.892706i	$0.648804\pi$
$132$	0	0
$133$	−333.957	−0.217728
$134$	0	0
$135$	−93.3901	−0.0595388
$136$	0	0
$137$	1732.64	1.08051	0.540253	−	0.841503i	$-0.318329\pi$
$137$	1732.64	1.08051	0.540253	+	0.841503i	$0.318329\pi$
$138$	0	0
$139$	−2803.08	−1.71046	−0.855231	−	0.518248i	$-0.826585\pi$
$139$	−2803.08	−1.71046	−0.855231	+	0.518248i	$0.826585\pi$
$140$	0	0
$141$	−640.441	−0.382517
$142$	0	0
$143$	667.177	0.390155
$144$	0	0
$145$	20.8189	0.0119236
$146$	0	0
$147$	279.702	0.156935
$148$	0	0
$149$	−170.271	−0.0936183	−0.0468091	−	0.998904i	$-0.514905\pi$
$149$	−170.271	−0.0936183	−0.0468091	+	0.998904i	$0.514905\pi$
$150$	0	0
$151$	−1685.42	−0.908327	−0.454163	−	0.890918i	$-0.650062\pi$
$151$	−1685.42	−0.908327	−0.454163	+	0.890918i	$0.650062\pi$
$152$	0	0
$153$	202.254	0.106871
$154$	0	0
$155$	119.234	0.0617876
$156$	0	0
$157$	663.459	0.337260	0.168630	−	0.985679i	$-0.446066\pi$
$157$	663.459	0.337260	0.168630	+	0.985679i	$0.446066\pi$
$158$	0	0
$159$	1508.39	0.752346
$160$	0	0
$161$	1304.71	0.638668
$162$	0	0
$163$	−334.906	−0.160931	−0.0804657	−	0.996757i	$-0.525641\pi$
$163$	−334.906	−0.160931	−0.0804657	+	0.996757i	$0.525641\pi$
$164$	0	0
$165$	−47.9675	−0.0226319
$166$	0	0
$167$	−1724.13	−0.798903	−0.399452	−	0.916754i	$-0.630799\pi$
$167$	−1724.13	−0.798903	−0.399452	+	0.916754i	$0.630799\pi$
$168$	0	0
$169$	1481.72	0.674430
$170$	0	0
$171$	266.383	0.119128
$172$	0	0
$173$	119.044	0.0523166	0.0261583	−	0.999658i	$-0.491673\pi$
$173$	119.044	0.0523166	0.0261583	+	0.999658i	$0.491673\pi$
$174$	0	0
$175$	870.915	0.376200
$176$	0	0
$177$	−1940.06	−0.823864
$178$	0	0
$179$	−4589.64	−1.91646	−0.958229	−	0.286001i	$-0.907674\pi$
$179$	−4589.64	−1.91646	−0.958229	+	0.286001i	$0.907674\pi$
$180$	0	0
$181$	2645.91	1.08657	0.543285	−	0.839548i	$-0.317180\pi$
$181$	2645.91	1.08657	0.543285	+	0.839548i	$0.317180\pi$
$182$	0	0
$183$	−3182.09	−1.28539
$184$	0	0
$185$	−231.259	−0.0919052
$186$	0	0
$187$	−398.452	−0.155817
$188$	0	0
$189$	855.745	0.329345
$190$	0	0
$191$	794.404	0.300948	0.150474	−	0.988614i	$-0.451920\pi$
$191$	794.404	0.300948	0.150474	+	0.988614i	$0.451920\pi$
$192$	0	0
$193$	4888.02	1.82305	0.911523	−	0.411250i	$-0.134908\pi$
$193$	4888.02	1.82305	0.911523	+	0.411250i	$0.134908\pi$
$194$	0	0
$195$	−264.486	−0.0971294
$196$	0	0
$197$	−2962.03	−1.07125	−0.535624	−	0.844457i	$-0.679923\pi$
$197$	−2962.03	−1.07125	−0.535624	+	0.844457i	$0.679923\pi$
$198$	0	0
$199$	1649.76	0.587679	0.293840	−	0.955855i	$-0.405067\pi$
$199$	1649.76	0.587679	0.293840	+	0.955855i	$0.405067\pi$
$200$	0	0
$201$	5290.97	1.85670
$202$	0	0
$203$	−190.766	−0.0659565
$204$	0	0
$205$	−55.9752	−0.0190706
$206$	0	0
$207$	−1040.71	−0.349441
$208$	0	0
$209$	−524.790	−0.173687
$210$	0	0
$211$	3304.44	1.07814	0.539069	−	0.842262i	$-0.318776\pi$
$211$	3304.44	1.07814	0.539069	+	0.842262i	$0.318776\pi$
$212$	0	0
$213$	−449.579	−0.144623
$214$	0	0
$215$	−33.5882	−0.0106544
$216$	0	0
$217$	−1092.55	−0.341785
$218$	0	0
$219$	−3817.01	−1.17776
$220$	0	0
$221$	−2197.01	−0.668719
$222$	0	0
$223$	−998.421	−0.299817	−0.149909	−	0.988700i	$-0.547898\pi$
$223$	−998.421	−0.299817	−0.149909	+	0.988700i	$0.547898\pi$
$224$	0	0
$225$	−694.690	−0.205834
$226$	0	0
$227$	3490.81	1.02067	0.510337	−	0.859974i	$-0.329521\pi$
$227$	3490.81	1.02067	0.510337	+	0.859974i	$0.329521\pi$
$228$	0	0
$229$	1228.13	0.354398	0.177199	−	0.984175i	$-0.443296\pi$
$229$	1228.13	0.354398	0.177199	+	0.984175i	$0.443296\pi$
$230$	0	0
$231$	439.532	0.125191
$232$	0	0
$233$	4361.81	1.22640	0.613201	−	0.789927i	$-0.289882\pi$
$233$	4361.81	1.22640	0.613201	+	0.789927i	$0.289882\pi$
$234$	0	0
$235$	−85.7106	−0.0237921
$236$	0	0
$237$	−6956.67	−1.90669
$238$	0	0
$239$	2804.52	0.759034	0.379517	−	0.925185i	$-0.376090\pi$
$239$	2804.52	0.759034	0.379517	+	0.925185i	$0.376090\pi$
$240$	0	0
$241$	4410.82	1.17895	0.589473	−	0.807788i	$-0.299336\pi$
$241$	4410.82	1.17895	0.589473	+	0.807788i	$0.299336\pi$
$242$	0	0
$243$	−1543.14	−0.407377
$244$	0	0
$245$	37.4327	0.00976117
$246$	0	0
$247$	−2893.62	−0.745412
$248$	0	0
$249$	5747.34	1.46274
$250$	0	0
$251$	−183.800	−0.0462204	−0.0231102	−	0.999733i	$-0.507357\pi$
$251$	−183.800	−0.0462204	−0.0231102	+	0.999733i	$0.507357\pi$
$252$	0	0
$253$	2050.26	0.509480
$254$	0	0
$255$	157.957	0.0387907
$256$	0	0
$257$	3718.53	0.902550	0.451275	−	0.892385i	$-0.350969\pi$
$257$	3718.53	0.902550	0.451275	+	0.892385i	$0.350969\pi$
$258$	0	0
$259$	2119.05	0.508384
$260$	0	0
$261$	152.166	0.0360875
$262$	0	0
$263$	−2361.30	−0.553627	−0.276813	−	0.960924i	$-0.589278\pi$
$263$	−2361.30	−0.553627	−0.276813	+	0.960924i	$0.589278\pi$
$264$	0	0
$265$	201.868	0.0467950
$266$	0	0
$267$	6544.00	1.49995
$268$	0	0
$269$	2676.42	0.606633	0.303317	−	0.952890i	$-0.401906\pi$
$269$	2676.42	0.606633	0.303317	+	0.952890i	$0.401906\pi$
$270$	0	0
$271$	−4672.14	−1.04728	−0.523639	−	0.851940i	$-0.675426\pi$
$271$	−4672.14	−1.04728	−0.523639	+	0.851940i	$0.675426\pi$
$272$	0	0
$273$	2423.52	0.537282
$274$	0	0
$275$	1368.58	0.300104
$276$	0	0
$277$	8630.05	1.87195	0.935974	−	0.352070i	$-0.114522\pi$
$277$	8630.05	1.87195	0.935974	+	0.352070i	$0.114522\pi$
$278$	0	0
$279$	871.481	0.187004
$280$	0	0
$281$	−1533.41	−0.325536	−0.162768	−	0.986664i	$-0.552042\pi$
$281$	−1533.41	−0.325536	−0.162768	+	0.986664i	$0.552042\pi$
$282$	0	0
$283$	−3075.10	−0.645921	−0.322961	−	0.946412i	$-0.604678\pi$
$283$	−3075.10	−0.645921	−0.322961	+	0.946412i	$0.604678\pi$
$284$	0	0
$285$	208.040	0.0432394
$286$	0	0
$287$	512.907	0.105491
$288$	0	0
$289$	−3600.90	−0.732933
$290$	0	0
$291$	3752.00	0.755828
$292$	0	0
$293$	−6705.90	−1.33707	−0.668537	−	0.743679i	$-0.733079\pi$
$293$	−6705.90	−1.33707	−0.668537	+	0.743679i	$0.733079\pi$
$294$	0	0
$295$	−259.639	−0.0512433
$296$	0	0
$297$	1344.74	0.262727
$298$	0	0
$299$	11304.8	2.18654
$300$	0	0
$301$	307.772	0.0589359
$302$	0	0
$303$	2273.61	0.431074
$304$	0	0
$305$	−425.861	−0.0799499
$306$	0	0
$307$	8182.05	1.52109	0.760544	−	0.649286i	$-0.224932\pi$
$307$	8182.05	1.52109	0.760544	+	0.649286i	$0.224932\pi$
$308$	0	0
$309$	11000.6	2.02526
$310$	0	0
$311$	−6563.79	−1.19678	−0.598389	−	0.801205i	$-0.704192\pi$
$311$	−6563.79	−1.19678	−0.598389	+	0.801205i	$0.704192\pi$
$312$	0	0
$313$	6609.11	1.19351	0.596755	−	0.802423i	$-0.296456\pi$
$313$	6609.11	1.19351	0.596755	+	0.802423i	$0.296456\pi$
$314$	0	0
$315$	−29.8584	−0.00534073
$316$	0	0
$317$	3777.95	0.669371	0.334686	−	0.942330i	$-0.391370\pi$
$317$	3777.95	0.669371	0.334686	+	0.942330i	$0.391370\pi$
$318$	0	0
$319$	−299.776	−0.0526151
$320$	0	0
$321$	−10181.8	−1.77039
$322$	0	0
$323$	1728.13	0.297696
$324$	0	0
$325$	7546.16	1.28796
$326$	0	0
$327$	−985.063	−0.166587
$328$	0	0
$329$	785.376	0.131608
$330$	0	0
$331$	1059.15	0.175879	0.0879397	−	0.996126i	$-0.471972\pi$
$331$	1059.15	0.175879	0.0879397	+	0.996126i	$0.471972\pi$
$332$	0	0
$333$	−1690.27	−0.278157
$334$	0	0
$335$	708.093	0.115484
$336$	0	0
$337$	−583.077	−0.0942500	−0.0471250	−	0.998889i	$-0.515006\pi$
$337$	−583.077	−0.0942500	−0.0471250	+	0.998889i	$0.515006\pi$
$338$	0	0
$339$	286.268	0.0458641
$340$	0	0
$341$	−1716.87	−0.272650
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−812.774	−0.126836
$346$	0	0
$347$	993.813	0.153748	0.0768742	−	0.997041i	$-0.475506\pi$
$347$	993.813	0.153748	0.0768742	+	0.997041i	$0.475506\pi$
$348$	0	0
$349$	−6624.74	−1.01609	−0.508043	−	0.861332i	$-0.669631\pi$
$349$	−6624.74	−1.01609	−0.508043	+	0.861332i	$0.669631\pi$
$350$	0	0
$351$	7414.72	1.12755
$352$	0	0
$353$	−1136.14	−0.171305	−0.0856524	−	0.996325i	$-0.527297\pi$
$353$	−1136.14	−0.171305	−0.0856524	+	0.996325i	$0.527297\pi$
$354$	0	0
$355$	−60.1674	−0.00899536
$356$	0	0
$357$	−1447.37	−0.214575
$358$	0	0
$359$	5362.89	0.788419	0.394210	−	0.919021i	$-0.371018\pi$
$359$	5362.89	0.788419	0.394210	+	0.919021i	$0.371018\pi$
$360$	0	0
$361$	−4582.93	−0.668163
$362$	0	0
$363$	690.693	0.0998677
$364$	0	0
$365$	−510.833	−0.0732554
$366$	0	0
$367$	11782.1	1.67581	0.837904	−	0.545818i	$-0.183781\pi$
$367$	11782.1	1.67581	0.837904	+	0.545818i	$0.183781\pi$
$368$	0	0
$369$	−409.123	−0.0577185
$370$	0	0
$371$	−1849.74	−0.258852
$372$	0	0
$373$	−5670.15	−0.787103	−0.393551	−	0.919303i	$-0.628754\pi$
$373$	−5670.15	−0.787103	−0.393551	+	0.919303i	$0.628754\pi$
$374$	0	0
$375$	−1087.63	−0.149773
$376$	0	0
$377$	−1652.92	−0.225808
$378$	0	0
$379$	4988.40	0.676087	0.338043	−	0.941131i	$-0.390235\pi$
$379$	4988.40	0.676087	0.338043	+	0.941131i	$0.390235\pi$
$380$	0	0
$381$	−12237.9	−1.64559
$382$	0	0
$383$	−6793.64	−0.906368	−0.453184	−	0.891417i	$-0.649712\pi$
$383$	−6793.64	−0.906368	−0.453184	+	0.891417i	$0.649712\pi$
$384$	0	0
$385$	58.8228	0.00778672
$386$	0	0
$387$	−245.496	−0.0322462
$388$	0	0
$389$	6302.89	0.821515	0.410757	−	0.911745i	$-0.365264\pi$
$389$	6302.89	0.821515	0.410757	+	0.911745i	$0.365264\pi$
$390$	0	0
$391$	−6751.48	−0.873240
$392$	0	0
$393$	−7713.77	−0.990097
$394$	0	0
$395$	−931.016	−0.118594
$396$	0	0
$397$	2970.47	0.375525	0.187762	−	0.982214i	$-0.439876\pi$
$397$	2970.47	0.375525	0.187762	+	0.982214i	$0.439876\pi$
$398$	0	0
$399$	−1906.30	−0.239184
$400$	0	0
$401$	11810.8	1.47084	0.735418	−	0.677614i	$-0.236986\pi$
$401$	11810.8	1.47084	0.735418	+	0.677614i	$0.236986\pi$
$402$	0	0
$403$	−9466.57	−1.17013
$404$	0	0
$405$	−648.258	−0.0795363
$406$	0	0
$407$	3329.93	0.405550
$408$	0	0
$409$	−6050.63	−0.731502	−0.365751	−	0.930713i	$-0.619188\pi$
$409$	−6050.63	−0.731502	−0.365751	+	0.930713i	$0.619188\pi$
$410$	0	0
$411$	9890.25	1.18698
$412$	0	0
$413$	2379.11	0.283458
$414$	0	0
$415$	769.170	0.0909809
$416$	0	0
$417$	−16000.6	−1.87902
$418$	0	0
$419$	−979.679	−0.114225	−0.0571127	−	0.998368i	$-0.518189\pi$
$419$	−979.679	−0.114225	−0.0571127	+	0.998368i	$0.518189\pi$
$420$	0	0
$421$	−4273.53	−0.494725	−0.247363	−	0.968923i	$-0.579564\pi$
$421$	−4273.53	−0.494725	−0.247363	+	0.968923i	$0.579564\pi$
$422$	0	0
$423$	−626.460	−0.0720083
$424$	0	0
$425$	−4506.72	−0.514372
$426$	0	0
$427$	3902.21	0.442251
$428$	0	0
$429$	3808.38	0.428603
$430$	0	0
$431$	−11923.4	−1.33255	−0.666277	−	0.745704i	$-0.732113\pi$
$431$	−11923.4	−1.33255	−0.666277	+	0.745704i	$0.732113\pi$
$432$	0	0
$433$	2942.85	0.326615	0.163308	−	0.986575i	$-0.447784\pi$
$433$	2942.85	0.326615	0.163308	+	0.986575i	$0.447784\pi$
$434$	0	0
$435$	118.839	0.0130986
$436$	0	0
$437$	−8892.19	−0.973389
$438$	0	0
$439$	9664.14	1.05067	0.525335	−	0.850895i	$-0.323940\pi$
$439$	9664.14	1.05067	0.525335	+	0.850895i	$0.323940\pi$
$440$	0	0
$441$	273.596	0.0295428
$442$	0	0
$443$	−6781.33	−0.727293	−0.363647	−	0.931537i	$-0.618468\pi$
$443$	−6781.33	−0.727293	−0.363647	+	0.931537i	$0.618468\pi$
$444$	0	0
$445$	875.788	0.0932951
$446$	0	0
$447$	−971.940	−0.102844
$448$	0	0
$449$	−8117.32	−0.853184	−0.426592	−	0.904444i	$-0.640286\pi$
$449$	−8117.32	−0.853184	−0.426592	+	0.904444i	$0.640286\pi$
$450$	0	0
$451$	805.997	0.0841528
$452$	0	0
$453$	−9620.71	−0.997837
$454$	0	0
$455$	324.341	0.0334183
$456$	0	0
$457$	12449.4	1.27431	0.637153	−	0.770737i	$-0.280112\pi$
$457$	12449.4	1.27431	0.637153	+	0.770737i	$0.280112\pi$
$458$	0	0
$459$	−4428.22	−0.450309
$460$	0	0
$461$	−12149.6	−1.22746	−0.613732	−	0.789514i	$-0.710333\pi$
$461$	−12149.6	−1.22746	−0.613732	+	0.789514i	$0.710333\pi$
$462$	0	0
$463$	−15806.0	−1.58654	−0.793269	−	0.608871i	$-0.791623\pi$
$463$	−15806.0	−1.58654	−0.793269	+	0.608871i	$0.791623\pi$
$464$	0	0
$465$	680.610	0.0678764
$466$	0	0
$467$	5659.39	0.560783	0.280391	−	0.959886i	$-0.409536\pi$
$467$	5659.39	0.560783	0.280391	+	0.959886i	$0.409536\pi$
$468$	0	0
$469$	−6488.34	−0.638814
$470$	0	0
$471$	3787.16	0.370495
$472$	0	0
$473$	483.642	0.0470146
$474$	0	0
$475$	−5935.68	−0.573364
$476$	0	0
$477$	1475.46	0.141628
$478$	0	0
$479$	−16251.4	−1.55020	−0.775101	−	0.631837i	$-0.782301\pi$
$479$	−16251.4	−1.55020	−0.775101	+	0.631837i	$0.782301\pi$
$480$	0	0
$481$	18360.8	1.74050
$482$	0	0
$483$	7447.54	0.701605
$484$	0	0
$485$	502.132	0.0470116
$486$	0	0
$487$	−18908.5	−1.75939	−0.879696	−	0.475536i	$-0.842254\pi$
$487$	−18908.5	−1.75939	−0.879696	+	0.475536i	$0.842254\pi$
$488$	0	0
$489$	−1911.71	−0.176790
$490$	0	0
$491$	−46.7115	−0.00429340	−0.00214670	−	0.999998i	$-0.500683\pi$
$491$	−46.7115	−0.00429340	−0.00214670	+	0.999998i	$0.500683\pi$
$492$	0	0
$493$	987.159	0.0901813
$494$	0	0
$495$	−46.9203	−0.00426043
$496$	0	0
$497$	551.321	0.0497588
$498$	0	0
$499$	15520.5	1.39237	0.696183	−	0.717864i	$-0.254880\pi$
$499$	15520.5	1.39237	0.696183	+	0.717864i	$0.254880\pi$
$500$	0	0
$501$	−9841.66	−0.877631
$502$	0	0
$503$	−12124.3	−1.07474	−0.537369	−	0.843347i	$-0.680582\pi$
$503$	−12124.3	−1.07474	−0.537369	+	0.843347i	$0.680582\pi$
$504$	0	0
$505$	304.278	0.0268123
$506$	0	0
$507$	8457.98	0.740891
$508$	0	0
$509$	−21982.8	−1.91429	−0.957143	−	0.289616i	$-0.906473\pi$
$509$	−21982.8	−1.91429	−0.957143	+	0.289616i	$0.906473\pi$
$510$	0	0
$511$	4680.82	0.405220
$512$	0	0
$513$	−5832.29	−0.501953
$514$	0	0
$515$	1472.22	0.125969
$516$	0	0
$517$	1234.16	0.104987
$518$	0	0
$519$	679.529	0.0574721
$520$	0	0
$521$	−20822.5	−1.75096	−0.875479	−	0.483257i	$-0.839454\pi$
$521$	−20822.5	−1.75096	−0.875479	+	0.483257i	$0.839454\pi$
$522$	0	0
$523$	−9380.10	−0.784251	−0.392125	−	0.919912i	$-0.628260\pi$
$523$	−9380.10	−0.784251	−0.392125	+	0.919912i	$0.628260\pi$
$524$	0	0
$525$	4971.36	0.413272
$526$	0	0
$527$	5653.63	0.467317
$528$	0	0
$529$	22573.1	1.85527
$530$	0	0
$531$	−1897.71	−0.155091
$532$	0	0
$533$	4444.15	0.361159
$534$	0	0
$535$	−1362.64	−0.110116
$536$	0	0
$537$	−26198.6	−2.10531
$538$	0	0
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−1561.29	−0.124076	−0.0620382	−	0.998074i	$-0.519760\pi$
$541$	−1561.29	−0.124076	−0.0620382	+	0.998074i	$0.519760\pi$
$542$	0	0
$543$	15103.4	1.19365
$544$	0	0
$545$	−131.832	−0.0103615
$546$	0	0
$547$	13736.3	1.07372	0.536859	−	0.843672i	$-0.319611\pi$
$547$	13736.3	1.07372	0.536859	+	0.843672i	$0.319611\pi$
$548$	0	0
$549$	−3112.62	−0.241974
$550$	0	0
$551$	1300.16	0.100524
$552$	0	0
$553$	8531.01	0.656013
$554$	0	0
$555$	−1320.07	−0.100962
$556$	0	0
$557$	15896.8	1.20928	0.604639	−	0.796499i	$-0.293317\pi$
$557$	15896.8	1.20928	0.604639	+	0.796499i	$0.293317\pi$
$558$	0	0
$559$	2666.74	0.201773
$560$	0	0
$561$	−2274.45	−0.171171
$562$	0	0
$563$	−3750.20	−0.280732	−0.140366	−	0.990100i	$-0.544828\pi$
$563$	−3750.20	−0.280732	−0.140366	+	0.990100i	$0.544828\pi$
$564$	0	0
$565$	38.3113	0.00285269
$566$	0	0
$567$	5940.06	0.439963
$568$	0	0
$569$	6212.45	0.457714	0.228857	−	0.973460i	$-0.426501\pi$
$569$	6212.45	0.457714	0.228857	+	0.973460i	$0.426501\pi$
$570$	0	0
$571$	−9826.17	−0.720162	−0.360081	−	0.932921i	$-0.617251\pi$
$571$	−9826.17	−0.720162	−0.360081	+	0.932921i	$0.617251\pi$
$572$	0	0
$573$	4534.62	0.330605
$574$	0	0
$575$	23189.6	1.68187
$576$	0	0
$577$	−16223.1	−1.17050	−0.585248	−	0.810854i	$-0.699003\pi$
$577$	−16223.1	−1.17050	−0.585248	+	0.810854i	$0.699003\pi$
$578$	0	0
$579$	27901.8	2.00270
$580$	0	0
$581$	−7048.00	−0.503271
$582$	0	0
$583$	−2906.74	−0.206492
$584$	0	0
$585$	−258.712	−0.0182845
$586$	0	0
$587$	−6898.95	−0.485094	−0.242547	−	0.970140i	$-0.577983\pi$
$587$	−6898.95	−0.485094	−0.242547	+	0.970140i	$0.577983\pi$
$588$	0	0
$589$	7446.25	0.520912
$590$	0	0
$591$	−16907.9	−1.17681
$592$	0	0
$593$	−17914.3	−1.24056	−0.620282	−	0.784379i	$-0.712982\pi$
$593$	−17914.3	−1.24056	−0.620282	+	0.784379i	$0.712982\pi$
$594$	0	0
$595$	−193.703	−0.0133463
$596$	0	0
$597$	9417.15	0.645592
$598$	0	0
$599$	−26256.9	−1.79103	−0.895517	−	0.445027i	$-0.853194\pi$
$599$	−26256.9	−1.79103	−0.895517	+	0.445027i	$0.853194\pi$
$600$	0	0
$601$	−25416.8	−1.72508	−0.862538	−	0.505991i	$-0.831127\pi$
$601$	−25416.8	−1.72508	−0.862538	+	0.505991i	$0.831127\pi$
$602$	0	0
$603$	5175.46	0.349521
$604$	0	0
$605$	92.4358	0.00621165
$606$	0	0
$607$	−11099.0	−0.742166	−0.371083	−	0.928600i	$-0.621013\pi$
$607$	−11099.0	−0.742166	−0.371083	+	0.928600i	$0.621013\pi$
$608$	0	0
$609$	−1088.93	−0.0724561
$610$	0	0
$611$	6805.00	0.450574
$612$	0	0
$613$	−8870.82	−0.584484	−0.292242	−	0.956344i	$-0.594401\pi$
$613$	−8870.82	−0.584484	−0.292242	+	0.956344i	$0.594401\pi$
$614$	0	0
$615$	−319.518	−0.0209499
$616$	0	0
$617$	−5428.99	−0.354235	−0.177117	−	0.984190i	$-0.556677\pi$
$617$	−5428.99	−0.354235	−0.177117	+	0.984190i	$0.556677\pi$
$618$	0	0
$619$	20949.9	1.36033	0.680167	−	0.733058i	$-0.261907\pi$
$619$	20949.9	1.36033	0.680167	+	0.733058i	$0.261907\pi$
$620$	0	0
$621$	22785.7	1.47239
$622$	0	0
$623$	−8024.95	−0.516072
$624$	0	0
$625$	15406.5	0.986016
$626$	0	0
$627$	−2995.61	−0.190802
$628$	0	0
$629$	−10965.4	−0.695105
$630$	0	0
$631$	21743.8	1.37180	0.685900	−	0.727696i	$-0.259409\pi$
$631$	21743.8	1.37180	0.685900	+	0.727696i	$0.259409\pi$
$632$	0	0
$633$	18862.4	1.18438
$634$	0	0
$635$	−1637.81	−0.102354
$636$	0	0
$637$	−2971.97	−0.184857
$638$	0	0
$639$	−439.764	−0.0272250
$640$	0	0
$641$	−16087.6	−0.991299	−0.495650	−	0.868523i	$-0.665070\pi$
$641$	−16087.6	−0.991299	−0.495650	+	0.868523i	$0.665070\pi$
$642$	0	0
$643$	−17284.9	−1.06011	−0.530055	−	0.847963i	$-0.677829\pi$
$643$	−17284.9	−1.06011	−0.530055	+	0.847963i	$0.677829\pi$
$644$	0	0
$645$	−191.728	−0.0117043
$646$	0	0
$647$	−18965.9	−1.15244	−0.576220	−	0.817295i	$-0.695473\pi$
$647$	−18965.9	−1.15244	−0.576220	+	0.817295i	$0.695473\pi$
$648$	0	0
$649$	3738.60	0.226121
$650$	0	0
$651$	−6236.51	−0.375466
$652$	0	0
$653$	24866.7	1.49021	0.745105	−	0.666947i	$-0.232399\pi$
$653$	24866.7	1.49021	0.745105	+	0.666947i	$0.232399\pi$
$654$	0	0
$655$	−1032.34	−0.0615829
$656$	0	0
$657$	−3733.69	−0.221712
$658$	0	0
$659$	−4941.55	−0.292102	−0.146051	−	0.989277i	$-0.546656\pi$
$659$	−4941.55	−0.292102	−0.146051	+	0.989277i	$0.546656\pi$
$660$	0	0
$661$	−13734.2	−0.808166	−0.404083	−	0.914722i	$-0.632409\pi$
$661$	−13734.2	−0.808166	−0.404083	+	0.914722i	$0.632409\pi$
$662$	0	0
$663$	−12541.0	−0.734617
$664$	0	0
$665$	−255.121	−0.0148769
$666$	0	0
$667$	−5079.48	−0.294870
$668$	0	0
$669$	−5699.19	−0.329362
$670$	0	0
$671$	6132.05	0.352795
$672$	0	0
$673$	10123.4	0.579833	0.289916	−	0.957052i	$-0.406373\pi$
$673$	10123.4	0.579833	0.289916	+	0.957052i	$0.406373\pi$
$674$	0	0
$675$	15209.8	0.867298
$676$	0	0
$677$	20357.5	1.15569	0.577845	−	0.816147i	$-0.303894\pi$
$677$	20357.5	1.15569	0.577845	+	0.816147i	$0.303894\pi$
$678$	0	0
$679$	−4601.09	−0.260050
$680$	0	0
$681$	19926.2	1.12126
$682$	0	0
$683$	−32651.4	−1.82924	−0.914620	−	0.404314i	$-0.867510\pi$
$683$	−32651.4	−1.82924	−0.914620	+	0.404314i	$0.867510\pi$
$684$	0	0
$685$	1323.62	0.0738289
$686$	0	0
$687$	7010.42	0.389322
$688$	0	0
$689$	−16027.4	−0.886203
$690$	0	0
$691$	24765.5	1.36342	0.681711	−	0.731621i	$-0.261236\pi$
$691$	24765.5	1.36342	0.681711	+	0.731621i	$0.261236\pi$
$692$	0	0
$693$	429.937	0.0235670
$694$	0	0
$695$	−2141.36	−0.116873
$696$	0	0
$697$	−2654.14	−0.144236
$698$	0	0
$699$	24898.1	1.34726
$700$	0	0
$701$	−7358.42	−0.396467	−0.198234	−	0.980155i	$-0.563520\pi$
$701$	−7358.42	−0.396467	−0.198234	+	0.980155i	$0.563520\pi$
$702$	0	0
$703$	−14442.3	−0.774824
$704$	0	0
$705$	−489.253	−0.0261367
$706$	0	0
$707$	−2788.14	−0.148315
$708$	0	0
$709$	27601.6	1.46206	0.731030	−	0.682345i	$-0.239040\pi$
$709$	27601.6	1.46206	0.731030	+	0.682345i	$0.239040\pi$
$710$	0	0
$711$	−6804.81	−0.358931
$712$	0	0
$713$	−29091.1	−1.52801
$714$	0	0
$715$	509.678	0.0266586
$716$	0	0
$717$	16008.8	0.833832
$718$	0	0
$719$	−16126.4	−0.836459	−0.418229	−	0.908341i	$-0.637349\pi$
$719$	−16126.4	−0.836459	−0.418229	+	0.908341i	$0.637349\pi$
$720$	0	0
$721$	−13490.1	−0.696809
$722$	0	0
$723$	25177.9	1.29512
$724$	0	0
$725$	−3390.64	−0.173690
$726$	0	0
$727$	−5873.33	−0.299628	−0.149814	−	0.988714i	$-0.547868\pi$
$727$	−5873.33	−0.299628	−0.149814	+	0.988714i	$0.547868\pi$
$728$	0	0
$729$	14103.1	0.716512
$730$	0	0
$731$	−1592.63	−0.0805821
$732$	0	0
$733$	−20612.5	−1.03866	−0.519332	−	0.854573i	$-0.673819\pi$
$733$	−20612.5	−1.03866	−0.519332	+	0.854573i	$0.673819\pi$
$734$	0	0
$735$	213.673	0.0107231
$736$	0	0
$737$	−10196.0	−0.509597
$738$	0	0
$739$	5091.41	0.253438	0.126719	−	0.991939i	$-0.459555\pi$
$739$	5091.41	0.253438	0.126719	+	0.991939i	$0.459555\pi$
$740$	0	0
$741$	−16517.4	−0.818868
$742$	0	0
$743$	20493.0	1.01186	0.505931	−	0.862574i	$-0.331149\pi$
$743$	20493.0	1.01186	0.505931	+	0.862574i	$0.331149\pi$
$744$	0	0
$745$	−130.075	−0.00639677
$746$	0	0
$747$	5621.88	0.275360
$748$	0	0
$749$	12486.0	0.609119
$750$	0	0
$751$	2488.12	0.120896	0.0604479	−	0.998171i	$-0.480747\pi$
$751$	2488.12	0.120896	0.0604479	+	0.998171i	$0.480747\pi$
$752$	0	0
$753$	−1049.17	−0.0507752
$754$	0	0
$755$	−1287.54	−0.0620643
$756$	0	0
$757$	7386.41	0.354642	0.177321	−	0.984153i	$-0.443257\pi$
$757$	7386.41	0.354642	0.177321	+	0.984153i	$0.443257\pi$
$758$	0	0
$759$	11703.3	0.559687
$760$	0	0
$761$	−22577.9	−1.07549	−0.537746	−	0.843107i	$-0.680724\pi$
$761$	−22577.9	−1.07549	−0.537746	+	0.843107i	$0.680724\pi$
$762$	0	0
$763$	1207.99	0.0573160
$764$	0	0
$765$	154.508	0.00730230
$766$	0	0
$767$	20614.1	0.970446
$768$	0	0
$769$	−23886.7	−1.12012	−0.560062	−	0.828451i	$-0.689223\pi$
$769$	−23886.7	−1.12012	−0.560062	+	0.828451i	$0.689223\pi$
$770$	0	0
$771$	21226.1	0.991492
$772$	0	0
$773$	5087.32	0.236712	0.118356	−	0.992971i	$-0.462238\pi$
$773$	5087.32	0.236712	0.118356	+	0.992971i	$0.462238\pi$
$774$	0	0
$775$	−19418.8	−0.900056
$776$	0	0
$777$	12096.0	0.558482
$778$	0	0
$779$	−3495.70	−0.160778
$780$	0	0
$781$	866.361	0.0396938
$782$	0	0
$783$	−3331.58	−0.152057
$784$	0	0
$785$	506.838	0.0230443
$786$	0	0
$787$	3776.28	0.171042	0.0855209	−	0.996336i	$-0.472745\pi$
$787$	3776.28	0.171042	0.0855209	+	0.996336i	$0.472745\pi$
$788$	0	0
$789$	−13478.8	−0.608184
$790$	0	0
$791$	−351.051	−0.0157800
$792$	0	0
$793$	33811.3	1.51409
$794$	0	0
$795$	1152.31	0.0514064
$796$	0	0
$797$	1500.95	0.0667083	0.0333542	−	0.999444i	$-0.489381\pi$
$797$	1500.95	0.0667083	0.0333542	+	0.999444i	$0.489381\pi$
$798$	0	0
$799$	−4064.09	−0.179946
$800$	0	0
$801$	6401.15	0.282364
$802$	0	0
$803$	7355.58	0.323254
$804$	0	0
$805$	996.709	0.0436390
$806$	0	0
$807$	15277.6	0.666413
$808$	0	0
$809$	3940.35	0.171243	0.0856213	−	0.996328i	$-0.472712\pi$
$809$	3940.35	0.171243	0.0856213	+	0.996328i	$0.472712\pi$
$810$	0	0
$811$	17138.0	0.742044	0.371022	−	0.928624i	$-0.379008\pi$
$811$	17138.0	0.742044	0.371022	+	0.928624i	$0.379008\pi$
$812$	0	0
$813$	−26669.5	−1.15048
$814$	0	0
$815$	−255.845	−0.0109961
$816$	0	0
$817$	−2097.61	−0.0898238
$818$	0	0
$819$	2370.61	0.101143
$820$	0	0
$821$	−6162.19	−0.261951	−0.130976	−	0.991386i	$-0.541811\pi$
$821$	−6162.19	−0.261951	−0.130976	+	0.991386i	$0.541811\pi$
$822$	0	0
$823$	−44961.7	−1.90433	−0.952166	−	0.305581i	$-0.901149\pi$
$823$	−44961.7	−1.90433	−0.952166	+	0.305581i	$0.901149\pi$
$824$	0	0
$825$	7812.14	0.329677
$826$	0	0
$827$	−23796.0	−1.00057	−0.500283	−	0.865862i	$-0.666771\pi$
$827$	−23796.0	−1.00057	−0.500283	+	0.865862i	$0.666771\pi$
$828$	0	0
$829$	5578.25	0.233704	0.116852	−	0.993149i	$-0.462720\pi$
$829$	5578.25	0.233704	0.116852	+	0.993149i	$0.462720\pi$
$830$	0	0
$831$	49262.1	2.05642
$832$	0	0
$833$	1774.92	0.0738264
$834$	0	0
$835$	−1317.11	−0.0545876
$836$	0	0
$837$	−19080.5	−0.787956
$838$	0	0
$839$	1184.55	0.0487427	0.0243713	−	0.999703i	$-0.492242\pi$
$839$	1184.55	0.0487427	0.0243713	+	0.999703i	$0.492242\pi$
$840$	0	0
$841$	−23646.3	−0.969548
$842$	0	0
$843$	−8753.03	−0.357616
$844$	0	0
$845$	1131.94	0.0460826
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	−17553.3	−0.709573
$850$	0	0
$851$	56423.3	2.27282
$852$	0	0
$853$	−3979.23	−0.159726	−0.0798630	−	0.996806i	$-0.525448\pi$
$853$	−3979.23	−0.159726	−0.0798630	+	0.996806i	$0.525448\pi$
$854$	0	0
$855$	203.499	0.00813977
$856$	0	0
$857$	−23815.6	−0.949271	−0.474636	−	0.880182i	$-0.657420\pi$
$857$	−23815.6	−0.949271	−0.474636	+	0.880182i	$0.657420\pi$
$858$	0	0
$859$	41886.5	1.66374	0.831869	−	0.554973i	$-0.187271\pi$
$859$	41886.5	1.66374	0.831869	+	0.554973i	$0.187271\pi$
$860$	0	0
$861$	2927.78	0.115887
$862$	0	0
$863$	−8287.00	−0.326875	−0.163437	−	0.986554i	$-0.552258\pi$
$863$	−8287.00	−0.326875	−0.163437	+	0.986554i	$0.552258\pi$
$864$	0	0
$865$	90.9417	0.00357469
$866$	0	0
$867$	−20554.7	−0.805160
$868$	0	0
$869$	13405.9	0.523317
$870$	0	0
$871$	−56219.1	−2.18704
$872$	0	0
$873$	3670.09	0.142284
$874$	0	0
$875$	1333.76	0.0515307
$876$	0	0
$877$	32181.9	1.23912	0.619559	−	0.784950i	$-0.287311\pi$
$877$	32181.9	1.23912	0.619559	+	0.784950i	$0.287311\pi$
$878$	0	0
$879$	−38278.6	−1.46884
$880$	0	0
$881$	26654.6	1.01932	0.509658	−	0.860377i	$-0.329772\pi$
$881$	26654.6	1.01932	0.509658	+	0.860377i	$0.329772\pi$
$882$	0	0
$883$	8951.70	0.341165	0.170582	−	0.985343i	$-0.445435\pi$
$883$	8951.70	0.341165	0.170582	+	0.985343i	$0.445435\pi$
$884$	0	0
$885$	−1482.07	−0.0562931
$886$	0	0
$887$	6233.31	0.235957	0.117979	−	0.993016i	$-0.462359\pi$
$887$	6233.31	0.235957	0.117979	+	0.993016i	$0.462359\pi$
$888$	0	0
$889$	15007.5	0.566180
$890$	0	0
$891$	9334.39	0.350969
$892$	0	0
$893$	−5352.70	−0.200584
$894$	0	0
$895$	−3506.18	−0.130948
$896$	0	0
$897$	64530.3	2.40201
$898$	0	0
$899$	4253.51	0.157801
$900$	0	0
$901$	9571.88	0.353924
$902$	0	0
$903$	1756.83	0.0647437
$904$	0	0
$905$	2021.30	0.0742434
$906$	0	0
$907$	15545.4	0.569104	0.284552	−	0.958661i	$-0.408155\pi$
$907$	15545.4	0.569104	0.284552	+	0.958661i	$0.408155\pi$
$908$	0	0
$909$	2223.98	0.0811492
$910$	0	0
$911$	−12274.7	−0.446409	−0.223205	−	0.974772i	$-0.571652\pi$
$911$	−12274.7	−0.446409	−0.223205	+	0.974772i	$0.571652\pi$
$912$	0	0
$913$	−11075.4	−0.401471
$914$	0	0
$915$	−2430.90	−0.0878285
$916$	0	0
$917$	9459.43	0.340652
$918$	0	0
$919$	12058.6	0.432836	0.216418	−	0.976301i	$-0.430563\pi$
$919$	12058.6	0.432836	0.216418	+	0.976301i	$0.430563\pi$
$920$	0	0
$921$	46704.8	1.67098
$922$	0	0
$923$	4776.99	0.170354
$924$	0	0
$925$	37663.5	1.33878
$926$	0	0
$927$	10760.5	0.381252
$928$	0	0
$929$	−29126.1	−1.02863	−0.514315	−	0.857601i	$-0.671954\pi$
$929$	−29126.1	−1.02863	−0.514315	+	0.857601i	$0.671954\pi$
$930$	0	0
$931$	2337.70	0.0822933
$932$	0	0
$933$	−37467.4	−1.31471
$934$	0	0
$935$	−304.390	−0.0106467
$936$	0	0
$937$	−21291.5	−0.742331	−0.371165	−	0.928567i	$-0.621042\pi$
$937$	−21291.5	−0.742331	−0.371165	+	0.928567i	$0.621042\pi$
$938$	0	0
$939$	37726.1	1.31112
$940$	0	0
$941$	33899.2	1.17437	0.587184	−	0.809453i	$-0.300236\pi$
$941$	33899.2	1.17437	0.587184	+	0.809453i	$0.300236\pi$
$942$	0	0
$943$	13657.0	0.471616
$944$	0	0
$945$	653.731	0.0225036
$946$	0	0
$947$	3661.41	0.125639	0.0628194	−	0.998025i	$-0.479991\pi$
$947$	3661.41	0.125639	0.0628194	+	0.998025i	$0.479991\pi$
$948$	0	0
$949$	40557.6	1.38731
$950$	0	0
$951$	21565.3	0.735334
$952$	0	0
$953$	−6827.41	−0.232069	−0.116034	−	0.993245i	$-0.537018\pi$
$953$	−6827.41	−0.232069	−0.116034	+	0.993245i	$0.537018\pi$
$954$	0	0
$955$	606.871	0.0205632
$956$	0	0
$957$	−1711.18	−0.0578000
$958$	0	0
$959$	−12128.5	−0.408393
$960$	0	0
$961$	−5430.37	−0.182282
$962$	0	0
$963$	−9959.56	−0.333274
$964$	0	0
$965$	3734.12	0.124565
$966$	0	0
$967$	18184.0	0.604714	0.302357	−	0.953195i	$-0.402226\pi$
$967$	18184.0	0.604714	0.302357	+	0.953195i	$0.402226\pi$
$968$	0	0
$969$	9864.52	0.327032
$970$	0	0
$971$	−25014.6	−0.826731	−0.413366	−	0.910565i	$-0.635647\pi$
$971$	−25014.6	−0.826731	−0.413366	+	0.910565i	$0.635647\pi$
$972$	0	0
$973$	19621.6	0.646494
$974$	0	0
$975$	43075.0	1.41488
$976$	0	0
$977$	−35342.1	−1.15731	−0.578656	−	0.815572i	$-0.696423\pi$
$977$	−35342.1	−1.15731	−0.578656	+	0.815572i	$0.696423\pi$
$978$	0	0
$979$	−12610.6	−0.411683
$980$	0	0
$981$	−963.559	−0.0313599
$982$	0	0
$983$	−10482.2	−0.340114	−0.170057	−	0.985434i	$-0.554395\pi$
$983$	−10482.2	−0.340114	−0.170057	+	0.985434i	$0.554395\pi$
$984$	0	0
$985$	−2262.79	−0.0731964
$986$	0	0
$987$	4483.09	0.144578
$988$	0	0
$989$	8194.97	0.263483
$990$	0	0
$991$	−12086.9	−0.387440	−0.193720	−	0.981057i	$-0.562055\pi$
$991$	−12086.9	−0.387440	−0.193720	+	0.981057i	$0.562055\pi$
$992$	0	0
$993$	6045.84	0.193211
$994$	0	0
$995$	1260.30	0.0401550
$996$	0	0
$997$	25242.6	0.801847	0.400924	−	0.916111i	$-0.368689\pi$
$997$	25242.6	0.801847	0.400924	+	0.916111i	$0.368689\pi$
$998$	0	0
$999$	37007.5	1.17204

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	616.4.a.d.1.2	✓	2
4.3	odd	2		1232.4.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.d.1.2	✓	2	1.1	even	1	trivial
1232.4.a.l.1.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	616.a (trivial)

\(f(q)\)	\(=\)	\(q+5.70820 q^{3} +0.763932 q^{5} -7.00000 q^{7} +5.58359 q^{9} -11.0000 q^{11} -60.6525 q^{13} +4.36068 q^{15} +36.2229 q^{17} +47.7082 q^{19} -39.9574 q^{21} -186.387 q^{23} -124.416 q^{25} -122.249 q^{27} +27.2523 q^{29} +156.079 q^{31} -62.7902 q^{33} -5.34752 q^{35} -302.721 q^{37} -346.217 q^{39} -73.2724 q^{41} -43.9675 q^{43} +4.26548 q^{45} -112.197 q^{47} +49.0000 q^{49} +206.768 q^{51} +264.249 q^{53} -8.40325 q^{55} +272.328 q^{57} -339.872 q^{59} -557.459 q^{61} -39.0851 q^{63} -46.3344 q^{65} +926.906 q^{67} -1063.93 q^{69} -78.7601 q^{71} -668.689 q^{73} -710.194 q^{75} +77.0000 q^{77} -1218.72 q^{79} -848.580 q^{81} +1006.86 q^{83} +27.6718 q^{85} +155.562 q^{87} +1146.42 q^{89} +424.567 q^{91} +890.930 q^{93} +36.4458 q^{95} +657.299 q^{97} -61.4195 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 6 q^{5} - 14 q^{7} + 38 q^{9} - 22 q^{11} - 90 q^{13} - 36 q^{15} + 144 q^{17} + 82 q^{19} + 14 q^{21} - 176 q^{23} - 222 q^{25} - 164 q^{27} + 296 q^{29} - 144 q^{31} + 22 q^{33} - 42 q^{35}+ \cdots - 418 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 6 * q^5 - 14 * q^7 + 38 * q^9 - 22 * q^11 - 90 * q^13 - 36 * q^15 + 144 * q^17 + 82 * q^19 + 14 * q^21 - 176 * q^23 - 222 * q^25 - 164 * q^27 + 296 * q^29 - 144 * q^31 + 22 * q^33 - 42 * q^35 - 516 * q^37 - 120 * q^39 + 408 * q^41 + 404 * q^43 + 174 * q^45 - 448 * q^47 + 98 * q^49 - 624 * q^51 + 448 * q^53 - 66 * q^55 + 8 * q^57 - 398 * q^59 - 1182 * q^61 - 266 * q^63 - 200 * q^65 + 700 * q^67 - 1144 * q^69 - 1204 * q^71 - 756 * q^73 + 42 * q^75 + 154 * q^77 - 1704 * q^79 - 1402 * q^81 + 122 * q^83 + 592 * q^85 - 1916 * q^87 - 140 * q^89 + 630 * q^91 + 3204 * q^93 + 216 * q^95 + 608 * q^97 - 418 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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