LMFDB - Embedded newform 315.4.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,4,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, 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("315.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	315.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:	$18.5856016518$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.56155 q^{2} +22.9309 q^{4} +5.00000 q^{5} -7.00000 q^{7} +83.0388 q^{8} +O(q^{10})$	\(q+5.56155 q^{2} +22.9309 q^{4} +5.00000 q^{5} -7.00000 q^{7} +83.0388 q^{8} +27.8078 q^{10} +33.6155 q^{11} -38.3542 q^{13} -38.9309 q^{14} +278.378 q^{16} -65.7235 q^{17} +33.3996 q^{19} +114.654 q^{20} +186.955 q^{22} -207.447 q^{23} +25.0000 q^{25} -213.309 q^{26} -160.516 q^{28} +189.170 q^{29} +202.108 q^{31} +883.902 q^{32} -365.525 q^{34} -35.0000 q^{35} -16.5227 q^{37} +185.754 q^{38} +415.194 q^{40} -388.617 q^{41} +41.8144 q^{43} +770.833 q^{44} -1153.73 q^{46} -368.648 q^{47} +49.0000 q^{49} +139.039 q^{50} -879.494 q^{52} -458.172 q^{53} +168.078 q^{55} -581.272 q^{56} +1052.08 q^{58} -256.216 q^{59} -123.511 q^{61} +1124.03 q^{62} +2688.85 q^{64} -191.771 q^{65} -336.277 q^{67} -1507.10 q^{68} -194.654 q^{70} +453.312 q^{71} +22.0436 q^{73} -91.8920 q^{74} +765.882 q^{76} -235.309 q^{77} +385.417 q^{79} +1391.89 q^{80} -2161.32 q^{82} -23.7501 q^{83} -328.617 q^{85} +232.553 q^{86} +2791.39 q^{88} +1482.81 q^{89} +268.479 q^{91} -4756.94 q^{92} -2050.25 q^{94} +166.998 q^{95} +51.9867 q^{97} +272.516 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 7 q^{2} + 17 q^{4} + 10 q^{5} - 14 q^{7} + 63 q^{8}+O(q^{10}) $ 2 * q + 7 * q^2 + 17 * q^4 + 10 * q^5 - 14 * q^7 + 63 * q^8	$ 2 q + 7 q^{2} + 17 q^{4} + 10 q^{5} - 14 q^{7} + 63 q^{8} + 35 q^{10} + 26 q^{11} + 14 q^{13} - 49 q^{14} + 297 q^{16} - 16 q^{17} + 174 q^{19} + 85 q^{20} + 176 q^{22} - 184 q^{23} + 50 q^{25} - 138 q^{26} - 119 q^{28} + 32 q^{29} + 330 q^{31} + 1071 q^{32} - 294 q^{34} - 70 q^{35} - 132 q^{37} + 388 q^{38} + 315 q^{40} - 200 q^{41} + 364 q^{43} + 816 q^{44} - 1120 q^{46} - 292 q^{47} + 98 q^{49} + 175 q^{50} - 1190 q^{52} - 34 q^{53} + 130 q^{55} - 441 q^{56} + 826 q^{58} - 364 q^{59} + 792 q^{61} + 1308 q^{62} + 2809 q^{64} + 70 q^{65} - 788 q^{67} - 1802 q^{68} - 245 q^{70} - 454 q^{71} + 778 q^{73} - 258 q^{74} - 68 q^{76} - 182 q^{77} + 408 q^{79} + 1485 q^{80} - 1890 q^{82} - 1136 q^{83} - 80 q^{85} + 696 q^{86} + 2944 q^{88} - 36 q^{89} - 98 q^{91} - 4896 q^{92} - 1940 q^{94} + 870 q^{95} - 498 q^{97} + 343 q^{98}+O(q^{100}) $ 2 * q + 7 * q^2 + 17 * q^4 + 10 * q^5 - 14 * q^7 + 63 * q^8 + 35 * q^10 + 26 * q^11 + 14 * q^13 - 49 * q^14 + 297 * q^16 - 16 * q^17 + 174 * q^19 + 85 * q^20 + 176 * q^22 - 184 * q^23 + 50 * q^25 - 138 * q^26 - 119 * q^28 + 32 * q^29 + 330 * q^31 + 1071 * q^32 - 294 * q^34 - 70 * q^35 - 132 * q^37 + 388 * q^38 + 315 * q^40 - 200 * q^41 + 364 * q^43 + 816 * q^44 - 1120 * q^46 - 292 * q^47 + 98 * q^49 + 175 * q^50 - 1190 * q^52 - 34 * q^53 + 130 * q^55 - 441 * q^56 + 826 * q^58 - 364 * q^59 + 792 * q^61 + 1308 * q^62 + 2809 * q^64 + 70 * q^65 - 788 * q^67 - 1802 * q^68 - 245 * q^70 - 454 * q^71 + 778 * q^73 - 258 * q^74 - 68 * q^76 - 182 * q^77 + 408 * q^79 + 1485 * q^80 - 1890 * q^82 - 1136 * q^83 - 80 * q^85 + 696 * q^86 + 2944 * q^88 - 36 * q^89 - 98 * q^91 - 4896 * q^92 - 1940 * q^94 + 870 * q^95 - 498 * q^97 + 343 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	5.56155	1.96631	0.983153	−	0.182785i	$-0.0585112\pi$
$2$	5.56155	1.96631	0.983153	+	0.182785i	$0.0585112\pi$
$3$	0	0
$3$	0	0
$4$	22.9309	2.86636
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	83.0388	3.66983
$9$	0	0
$10$	27.8078	0.879359
$11$	33.6155	0.921406	0.460703	−	0.887554i	$-0.347597\pi$
$11$	33.6155	0.921406	0.460703	+	0.887554i	$0.347597\pi$
$12$	0	0
$13$	−38.3542	−0.818272	−0.409136	−	0.912474i	$-0.634170\pi$
$13$	−38.3542	−0.818272	−0.409136	+	0.912474i	$0.634170\pi$
$14$	−38.9309	−0.743194
$15$	0	0
$16$	278.378	4.34965
$17$	−65.7235	−0.937664	−0.468832	−	0.883287i	$-0.655325\pi$
$17$	−65.7235	−0.937664	−0.468832	+	0.883287i	$0.655325\pi$
$18$	0	0
$19$	33.3996	0.403284	0.201642	−	0.979459i	$-0.435372\pi$
$19$	33.3996	0.403284	0.201642	+	0.979459i	$0.435372\pi$
$20$	114.654	1.28187
$21$	0	0
$22$	186.955	1.81177
$23$	−207.447	−1.88068	−0.940341	−	0.340234i	$-0.889494\pi$
$23$	−207.447	−1.88068	−0.940341	+	0.340234i	$0.889494\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−213.309	−1.60897
$27$	0	0
$28$	−160.516	−1.08338
$29$	189.170	1.21131	0.605656	−	0.795726i	$-0.292911\pi$
$29$	189.170	1.21131	0.605656	+	0.795726i	$0.292911\pi$
$30$	0	0
$31$	202.108	1.17096	0.585478	−	0.810688i	$-0.300907\pi$
$31$	202.108	1.17096	0.585478	+	0.810688i	$0.300907\pi$
$32$	883.902	4.88292
$33$	0	0
$34$	−365.525	−1.84373
$35$	−35.0000	−0.169031
$36$	0	0
$37$	−16.5227	−0.0734141	−0.0367070	−	0.999326i	$-0.511687\pi$
$37$	−16.5227	−0.0734141	−0.0367070	+	0.999326i	$0.511687\pi$
$38$	185.754	0.792980
$39$	0	0
$40$	415.194	1.64120
$41$	−388.617	−1.48029	−0.740144	−	0.672448i	$-0.765243\pi$
$41$	−388.617	−1.48029	−0.740144	+	0.672448i	$0.765243\pi$
$42$	0	0
$43$	41.8144	0.148294	0.0741469	−	0.997247i	$-0.476377\pi$
$43$	41.8144	0.148294	0.0741469	+	0.997247i	$0.476377\pi$
$44$	770.833	2.64108
$45$	0	0
$46$	−1153.73	−3.69800
$47$	−368.648	−1.14410	−0.572051	−	0.820218i	$-0.693852\pi$
$47$	−368.648	−1.14410	−0.572051	+	0.820218i	$0.693852\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	139.039	0.393261
$51$	0	0
$52$	−879.494	−2.34546
$53$	−458.172	−1.18745	−0.593725	−	0.804668i	$-0.702343\pi$
$53$	−458.172	−1.18745	−0.593725	+	0.804668i	$0.702343\pi$
$54$	0	0
$55$	168.078	0.412065
$56$	−581.272	−1.38707
$57$	0	0
$58$	1052.08	2.38181
$59$	−256.216	−0.565364	−0.282682	−	0.959214i	$-0.591224\pi$
$59$	−256.216	−0.565364	−0.282682	+	0.959214i	$0.591224\pi$
$60$	0	0
$61$	−123.511	−0.259246	−0.129623	−	0.991563i	$-0.541377\pi$
$61$	−123.511	−0.259246	−0.129623	+	0.991563i	$0.541377\pi$
$62$	1124.03	2.30246
$63$	0	0
$64$	2688.85	5.25166
$65$	−191.771	−0.365942
$66$	0	0
$67$	−336.277	−0.613175	−0.306587	−	0.951843i	$-0.599187\pi$
$67$	−336.277	−0.613175	−0.306587	+	0.951843i	$0.599187\pi$
$68$	−1507.10	−2.68768
$69$	0	0
$70$	−194.654	−0.332366
$71$	453.312	0.757722	0.378861	−	0.925454i	$-0.376316\pi$
$71$	453.312	0.757722	0.378861	+	0.925454i	$0.376316\pi$
$72$	0	0
$73$	22.0436	0.0353426	0.0176713	−	0.999844i	$-0.494375\pi$
$73$	22.0436	0.0353426	0.0176713	+	0.999844i	$0.494375\pi$
$74$	−91.8920	−0.144355
$75$	0	0
$76$	765.882	1.15596
$77$	−235.309	−0.348259
$78$	0	0
$79$	385.417	0.548896	0.274448	−	0.961602i	$-0.411505\pi$
$79$	385.417	0.548896	0.274448	+	0.961602i	$0.411505\pi$
$80$	1391.89	1.94522
$81$	0	0
$82$	−2161.32	−2.91070
$83$	−23.7501	−0.0314085	−0.0157043	−	0.999877i	$-0.504999\pi$
$83$	−23.7501	−0.0314085	−0.0157043	+	0.999877i	$0.504999\pi$
$84$	0	0
$85$	−328.617	−0.419336
$86$	232.553	0.291591
$87$	0	0
$88$	2791.39	3.38140
$89$	1482.81	1.76604	0.883020	−	0.469335i	$-0.155506\pi$
$89$	1482.81	1.76604	0.883020	+	0.469335i	$0.155506\pi$
$90$	0	0
$91$	268.479	0.309278
$92$	−4756.94	−5.39071
$93$	0	0
$94$	−2050.25	−2.24965
$95$	166.998	0.180354
$96$	0	0
$97$	51.9867	0.0544170	0.0272085	−	0.999630i	$-0.491338\pi$
$97$	51.9867	0.0544170	0.0272085	+	0.999630i	$0.491338\pi$
$98$	272.516	0.280901
$99$	0	0
$100$	573.272	0.573272
$101$	−1429.30	−1.40812	−0.704062	−	0.710138i	$-0.748632\pi$
$101$	−1429.30	−1.40812	−0.704062	+	0.710138i	$0.748632\pi$
$102$	0	0
$103$	434.212	0.415381	0.207690	−	0.978195i	$-0.433405\pi$
$103$	434.212	0.415381	0.207690	+	0.978195i	$0.433405\pi$
$104$	−3184.88	−3.00292
$105$	0	0
$106$	−2548.15	−2.33489
$107$	−666.307	−0.602003	−0.301001	−	0.953624i	$-0.597321\pi$
$107$	−666.307	−0.602003	−0.301001	+	0.953624i	$0.597321\pi$
$108$	0	0
$109$	−1199.51	−1.05406	−0.527029	−	0.849847i	$-0.676694\pi$
$109$	−1199.51	−1.05406	−0.527029	+	0.849847i	$0.676694\pi$
$110$	934.773	0.810246
$111$	0	0
$112$	−1948.64	−1.64401
$113$	81.5171	0.0678627	0.0339314	−	0.999424i	$-0.489197\pi$
$113$	81.5171	0.0678627	0.0339314	+	0.999424i	$0.489197\pi$
$114$	0	0
$115$	−1037.23	−0.841066
$116$	4337.84	3.47206
$117$	0	0
$118$	−1424.96	−1.11168
$119$	460.064	0.354404
$120$	0	0
$121$	−200.996	−0.151011
$122$	−686.915	−0.509757
$123$	0	0
$124$	4634.51	3.35638
$125$	125.000	0.0894427
$126$	0	0
$127$	−336.985	−0.235453	−0.117727	−	0.993046i	$-0.537561\pi$
$127$	−336.985	−0.235453	−0.117727	+	0.993046i	$0.537561\pi$
$128$	7882.95	5.44344
$129$	0	0
$130$	−1066.54	−0.719554
$131$	−2931.15	−1.95493	−0.977465	−	0.211097i	$-0.932297\pi$
$131$	−2931.15	−1.95493	−0.977465	+	0.211097i	$0.932297\pi$
$132$	0	0
$133$	−233.797	−0.152427
$134$	−1870.22	−1.20569
$135$	0	0
$136$	−5457.60	−3.44107
$137$	1585.07	0.988477	0.494238	−	0.869326i	$-0.335447\pi$
$137$	1585.07	0.988477	0.494238	+	0.869326i	$0.335447\pi$
$138$	0	0
$139$	−1298.85	−0.792569	−0.396284	−	0.918128i	$-0.629701\pi$
$139$	−1298.85	−0.792569	−0.396284	+	0.918128i	$0.629701\pi$
$140$	−802.580	−0.484503
$141$	0	0
$142$	2521.12	1.48991
$143$	−1289.30	−0.753960
$144$	0	0
$145$	945.852	0.541716
$146$	122.597	0.0694943
$147$	0	0
$148$	−378.881	−0.210431
$149$	2003.29	1.10145	0.550724	−	0.834687i	$-0.314352\pi$
$149$	2003.29	1.10145	0.550724	+	0.834687i	$0.314352\pi$
$150$	0	0
$151$	2740.96	1.47719	0.738596	−	0.674148i	$-0.235489\pi$
$151$	2740.96	1.47719	0.738596	+	0.674148i	$0.235489\pi$
$152$	2773.47	1.47999
$153$	0	0
$154$	−1308.68	−0.684783
$155$	1010.54	0.523668
$156$	0	0
$157$	3644.22	1.85249	0.926243	−	0.376928i	$-0.123019\pi$
$157$	3644.22	1.85249	0.926243	+	0.376928i	$0.123019\pi$
$158$	2143.52	1.07930
$159$	0	0
$160$	4419.51	2.18371
$161$	1452.13	0.710831
$162$	0	0
$163$	2774.27	1.33311	0.666557	−	0.745454i	$-0.267767\pi$
$163$	2774.27	1.33311	0.666557	+	0.745454i	$0.267767\pi$
$164$	−8911.33	−4.24304
$165$	0	0
$166$	−132.087	−0.0617588
$167$	−1154.91	−0.535149	−0.267574	−	0.963537i	$-0.586222\pi$
$167$	−1154.91	−0.535149	−0.267574	+	0.963537i	$0.586222\pi$
$168$	0	0
$169$	−725.958	−0.330432
$170$	−1827.62	−0.824543
$171$	0	0
$172$	958.841	0.425064
$173$	3387.46	1.48869	0.744346	−	0.667794i	$-0.232761\pi$
$173$	3387.46	1.48869	0.744346	+	0.667794i	$0.232761\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	9357.82	4.00780
$177$	0	0
$178$	8246.73	3.47258
$179$	−1603.32	−0.669486	−0.334743	−	0.942309i	$-0.608650\pi$
$179$	−1603.32	−0.669486	−0.334743	+	0.942309i	$0.608650\pi$
$180$	0	0
$181$	544.220	0.223489	0.111745	−	0.993737i	$-0.464356\pi$
$181$	544.220	0.223489	0.111745	+	0.993737i	$0.464356\pi$
$182$	1493.16	0.608134
$183$	0	0
$184$	−17226.2	−6.90179
$185$	−82.6137	−0.0328318
$186$	0	0
$187$	−2209.33	−0.863969
$188$	−8453.41	−3.27941
$189$	0	0
$190$	928.769	0.354631
$191$	−2993.44	−1.13402	−0.567010	−	0.823711i	$-0.691900\pi$
$191$	−2993.44	−1.13402	−0.567010	+	0.823711i	$0.691900\pi$
$192$	0	0
$193$	1309.32	0.488325	0.244163	−	0.969734i	$-0.421487\pi$
$193$	1309.32	0.488325	0.244163	+	0.969734i	$0.421487\pi$
$194$	289.127	0.107001
$195$	0	0
$196$	1123.61	0.409480
$197$	1141.38	0.412790	0.206395	−	0.978469i	$-0.433827\pi$
$197$	1141.38	0.412790	0.206395	+	0.978469i	$0.433827\pi$
$198$	0	0
$199$	2370.23	0.844327	0.422164	−	0.906520i	$-0.361271\pi$
$199$	2370.23	0.844327	0.422164	+	0.906520i	$0.361271\pi$
$200$	2075.97	0.733966
$201$	0	0
$202$	−7949.12	−2.76880
$203$	−1324.19	−0.457833
$204$	0	0
$205$	−1943.09	−0.662005
$206$	2414.89	0.816765
$207$	0	0
$208$	−10676.9	−3.55920
$209$	1122.75	0.371588
$210$	0	0
$211$	−687.159	−0.224199	−0.112099	−	0.993697i	$-0.535758\pi$
$211$	−687.159	−0.224199	−0.112099	+	0.993697i	$0.535758\pi$
$212$	−10506.3	−3.40366
$213$	0	0
$214$	−3705.70	−1.18372
$215$	209.072	0.0663190
$216$	0	0
$217$	−1414.76	−0.442580
$218$	−6671.15	−2.07260
$219$	0	0
$220$	3854.17	1.18113
$221$	2520.77	0.767264
$222$	0	0
$223$	990.496	0.297437	0.148719	−	0.988880i	$-0.452485\pi$
$223$	990.496	0.297437	0.148719	+	0.988880i	$0.452485\pi$
$224$	−6187.32	−1.84557
$225$	0	0
$226$	453.362	0.133439
$227$	−1479.25	−0.432517	−0.216258	−	0.976336i	$-0.569385\pi$
$227$	−1479.25	−0.432517	−0.216258	+	0.976336i	$0.569385\pi$
$228$	0	0
$229$	6704.47	1.93469	0.967345	−	0.253463i	$-0.0815696\pi$
$229$	6704.47	1.93469	0.967345	+	0.253463i	$0.0815696\pi$
$230$	−5768.64	−1.65379
$231$	0	0
$232$	15708.5	4.44531
$233$	1749.09	0.491789	0.245895	−	0.969297i	$-0.420918\pi$
$233$	1749.09	0.491789	0.245895	+	0.969297i	$0.420918\pi$
$234$	0	0
$235$	−1843.24	−0.511658
$236$	−5875.25	−1.62054
$237$	0	0
$238$	2558.67	0.696866
$239$	6320.89	1.71073	0.855365	−	0.518027i	$-0.173333\pi$
$239$	6320.89	1.71073	0.855365	+	0.518027i	$0.173333\pi$
$240$	0	0
$241$	3359.62	0.897975	0.448988	−	0.893538i	$-0.351785\pi$
$241$	3359.62	0.897975	0.448988	+	0.893538i	$0.351785\pi$
$242$	−1117.85	−0.296935
$243$	0	0
$244$	−2832.22	−0.743092
$245$	245.000	0.0638877
$246$	0	0
$247$	−1281.01	−0.329996
$248$	16782.8	4.29721
$249$	0	0
$250$	695.194	0.175872
$251$	1330.50	0.334582	0.167291	−	0.985908i	$-0.446498\pi$
$251$	1330.50	0.334582	0.167291	+	0.985908i	$0.446498\pi$
$252$	0	0
$253$	−6973.44	−1.73287
$254$	−1874.16	−0.462973
$255$	0	0
$256$	22330.6	5.45182
$257$	2476.95	0.601197	0.300599	−	0.953751i	$-0.402814\pi$
$257$	2476.95	0.601197	0.300599	+	0.953751i	$0.402814\pi$
$258$	0	0
$259$	115.659	0.0277479
$260$	−4397.47	−1.04892
$261$	0	0
$262$	−16301.8	−3.84399
$263$	5152.56	1.20806	0.604032	−	0.796960i	$-0.293560\pi$
$263$	5152.56	1.20806	0.604032	+	0.796960i	$0.293560\pi$
$264$	0	0
$265$	−2290.86	−0.531043
$266$	−1300.28	−0.299718
$267$	0	0
$268$	−7711.11	−1.75758
$269$	1150.97	0.260876	0.130438	−	0.991456i	$-0.458362\pi$
$269$	1150.97	0.260876	0.130438	+	0.991456i	$0.458362\pi$
$270$	0	0
$271$	1838.32	0.412067	0.206034	−	0.978545i	$-0.433944\pi$
$271$	1838.32	0.412067	0.206034	+	0.978545i	$0.433944\pi$
$272$	−18296.0	−4.07851
$273$	0	0
$274$	8815.43	1.94365
$275$	840.388	0.184281
$276$	0	0
$277$	568.447	0.123302	0.0616510	−	0.998098i	$-0.480363\pi$
$277$	568.447	0.123302	0.0616510	+	0.998098i	$0.480363\pi$
$278$	−7223.62	−1.55843
$279$	0	0
$280$	−2906.36	−0.620315
$281$	−6015.00	−1.27696	−0.638479	−	0.769640i	$-0.720436\pi$
$281$	−6015.00	−1.27696	−0.638479	+	0.769640i	$0.720436\pi$
$282$	0	0
$283$	3985.75	0.837202	0.418601	−	0.908170i	$-0.362521\pi$
$283$	3985.75	0.837202	0.418601	+	0.908170i	$0.362521\pi$
$284$	10394.8	2.17190
$285$	0	0
$286$	−7170.48	−1.48252
$287$	2720.32	0.559497
$288$	0	0
$289$	−593.424	−0.120787
$290$	5260.41	1.06518
$291$	0	0
$292$	505.479	0.101305
$293$	2490.01	0.496478	0.248239	−	0.968699i	$-0.420148\pi$
$293$	2490.01	0.496478	0.248239	+	0.968699i	$0.420148\pi$
$294$	0	0
$295$	−1281.08	−0.252838
$296$	−1372.03	−0.269417
$297$	0	0
$298$	11141.4	2.16578
$299$	7956.45	1.53891
$300$	0	0
$301$	−292.701	−0.0560498
$302$	15244.0	2.90461
$303$	0	0
$304$	9297.72	1.75415
$305$	−617.557	−0.115938
$306$	0	0
$307$	−141.853	−0.0263712	−0.0131856	−	0.999913i	$-0.504197\pi$
$307$	−141.853	−0.0263712	−0.0131856	+	0.999913i	$0.504197\pi$
$308$	−5395.83	−0.998234
$309$	0	0
$310$	5620.17	1.02969
$311$	−2091.92	−0.381420	−0.190710	−	0.981646i	$-0.561079\pi$
$311$	−2091.92	−0.381420	−0.190710	+	0.981646i	$0.561079\pi$
$312$	0	0
$313$	5521.44	0.997094	0.498547	−	0.866863i	$-0.333867\pi$
$313$	5521.44	0.997094	0.498547	+	0.866863i	$0.333867\pi$
$314$	20267.5	3.64255
$315$	0	0
$316$	8837.94	1.57333
$317$	−5351.63	−0.948195	−0.474097	−	0.880472i	$-0.657226\pi$
$317$	−5351.63	−0.948195	−0.474097	+	0.880472i	$0.657226\pi$
$318$	0	0
$319$	6359.06	1.11611
$320$	13444.2	2.34861
$321$	0	0
$322$	8076.09	1.39771
$323$	−2195.14	−0.378145
$324$	0	0
$325$	−958.854	−0.163654
$326$	15429.2	2.62131
$327$	0	0
$328$	−32270.3	−5.43241
$329$	2580.53	0.432430
$330$	0	0
$331$	−4383.52	−0.727916	−0.363958	−	0.931415i	$-0.618575\pi$
$331$	−4383.52	−0.727916	−0.363958	+	0.931415i	$0.618575\pi$
$332$	−544.609	−0.0900281
$333$	0	0
$334$	−6423.11	−1.05227
$335$	−1681.38	−0.274220
$336$	0	0
$337$	−7124.57	−1.15163	−0.575817	−	0.817579i	$-0.695316\pi$
$337$	−7124.57	−1.15163	−0.575817	+	0.817579i	$0.695316\pi$
$338$	−4037.46	−0.649730
$339$	0	0
$340$	−7535.48	−1.20197
$341$	6793.97	1.07893
$342$	0	0
$343$	−343.000	−0.0539949
$344$	3472.22	0.544214
$345$	0	0
$346$	18839.5	2.92722
$347$	−507.743	−0.0785506	−0.0392753	−	0.999228i	$-0.512505\pi$
$347$	−507.743	−0.0785506	−0.0392753	+	0.999228i	$0.512505\pi$
$348$	0	0
$349$	6155.14	0.944060	0.472030	−	0.881582i	$-0.343521\pi$
$349$	6155.14	0.944060	0.472030	+	0.881582i	$0.343521\pi$
$350$	−973.272	−0.148639
$351$	0	0
$352$	29712.8	4.49915
$353$	6429.56	0.969437	0.484718	−	0.874670i	$-0.338922\pi$
$353$	6429.56	0.969437	0.484718	+	0.874670i	$0.338922\pi$
$354$	0	0
$355$	2266.56	0.338864
$356$	34002.1	5.06211
$357$	0	0
$358$	−8916.97	−1.31641
$359$	−10075.4	−1.48123	−0.740614	−	0.671931i	$-0.765465\pi$
$359$	−10075.4	−1.48123	−0.740614	+	0.671931i	$0.765465\pi$
$360$	0	0
$361$	−5743.46	−0.837362
$362$	3026.71	0.439448
$363$	0	0
$364$	6156.46	0.886501
$365$	110.218	0.0158057
$366$	0	0
$367$	816.898	0.116190	0.0580950	−	0.998311i	$-0.481497\pi$
$367$	816.898	0.116190	0.0580950	+	0.998311i	$0.481497\pi$
$368$	−57748.6	−8.18031
$369$	0	0
$370$	−459.460	−0.0645573
$371$	3207.21	0.448814
$372$	0	0
$373$	−3737.85	−0.518870	−0.259435	−	0.965761i	$-0.583536\pi$
$373$	−3737.85	−0.518870	−0.259435	+	0.965761i	$0.583536\pi$
$374$	−12287.3	−1.69883
$375$	0	0
$376$	−30612.1	−4.19866
$377$	−7255.47	−0.991183
$378$	0	0
$379$	1950.47	0.264351	0.132176	−	0.991226i	$-0.457804\pi$
$379$	1950.47	0.264351	0.132176	+	0.991226i	$0.457804\pi$
$380$	3829.41	0.516960
$381$	0	0
$382$	−16648.2	−2.22983
$383$	−6762.06	−0.902155	−0.451077	−	0.892485i	$-0.648960\pi$
$383$	−6762.06	−0.902155	−0.451077	+	0.892485i	$0.648960\pi$
$384$	0	0
$385$	−1176.54	−0.155746
$386$	7281.84	0.960197
$387$	0	0
$388$	1192.10	0.155979
$389$	2551.98	0.332624	0.166312	−	0.986073i	$-0.446814\pi$
$389$	2551.98	0.332624	0.166312	+	0.986073i	$0.446814\pi$
$390$	0	0
$391$	13634.1	1.76345
$392$	4068.90	0.524262
$393$	0	0
$394$	6347.83	0.811672
$395$	1927.08	0.245474
$396$	0	0
$397$	−4097.93	−0.518058	−0.259029	−	0.965869i	$-0.583403\pi$
$397$	−4097.93	−0.518058	−0.259029	+	0.965869i	$0.583403\pi$
$398$	13182.2	1.66021
$399$	0	0
$400$	6959.45	0.869931
$401$	1046.81	0.130362	0.0651811	−	0.997873i	$-0.479238\pi$
$401$	1046.81	0.130362	0.0651811	+	0.997873i	$0.479238\pi$
$402$	0	0
$403$	−7751.68	−0.958161
$404$	−32775.1	−4.03619
$405$	0	0
$406$	−7364.57	−0.900240
$407$	−555.420	−0.0676441
$408$	0	0
$409$	6516.92	0.787876	0.393938	−	0.919137i	$-0.371113\pi$
$409$	6516.92	0.787876	0.393938	+	0.919137i	$0.371113\pi$
$410$	−10806.6	−1.30170
$411$	0	0
$412$	9956.86	1.19063
$413$	1793.51	0.213687
$414$	0	0
$415$	−118.750	−0.0140463
$416$	−33901.3	−3.99555
$417$	0	0
$418$	6244.21	0.730656
$419$	−12279.1	−1.43168	−0.715838	−	0.698267i	$-0.753955\pi$
$419$	−12279.1	−1.43168	−0.715838	+	0.698267i	$0.753955\pi$
$420$	0	0
$421$	10146.9	1.17465	0.587325	−	0.809351i	$-0.300181\pi$
$421$	10146.9	1.17465	0.587325	+	0.809351i	$0.300181\pi$
$422$	−3821.67	−0.440844
$423$	0	0
$424$	−38046.1	−4.35774
$425$	−1643.09	−0.187533
$426$	0	0
$427$	864.579	0.0979858
$428$	−15279.0	−1.72556
$429$	0	0
$430$	1162.77	0.130404
$431$	−7059.04	−0.788914	−0.394457	−	0.918914i	$-0.629067\pi$
$431$	−7059.04	−0.788914	−0.394457	+	0.918914i	$0.629067\pi$
$432$	0	0
$433$	6468.98	0.717966	0.358983	−	0.933344i	$-0.383124\pi$
$433$	6468.98	0.717966	0.358983	+	0.933344i	$0.383124\pi$
$434$	−7868.24	−0.870248
$435$	0	0
$436$	−27505.8	−3.02131
$437$	−6928.65	−0.758449
$438$	0	0
$439$	4767.13	0.518275	0.259137	−	0.965840i	$-0.416562\pi$
$439$	4767.13	0.518275	0.259137	+	0.965840i	$0.416562\pi$
$440$	13957.0	1.51221
$441$	0	0
$442$	14019.4	1.50868
$443$	−2366.55	−0.253810	−0.126905	−	0.991915i	$-0.540504\pi$
$443$	−2366.55	−0.253810	−0.126905	+	0.991915i	$0.540504\pi$
$444$	0	0
$445$	7414.05	0.789797
$446$	5508.70	0.584853
$447$	0	0
$448$	−18821.9	−1.98494
$449$	−1814.17	−0.190681	−0.0953406	−	0.995445i	$-0.530394\pi$
$449$	−1814.17	−0.190681	−0.0953406	+	0.995445i	$0.530394\pi$
$450$	0	0
$451$	−13063.6	−1.36395
$452$	1869.26	0.194519
$453$	0	0
$454$	−8226.93	−0.850460
$455$	1342.40	0.138313
$456$	0	0
$457$	8284.13	0.847955	0.423977	−	0.905673i	$-0.360634\pi$
$457$	8284.13	0.847955	0.423977	+	0.905673i	$0.360634\pi$
$458$	37287.3	3.80419
$459$	0	0
$460$	−23784.7	−2.41080
$461$	−1384.62	−0.139888	−0.0699439	−	0.997551i	$-0.522282\pi$
$461$	−1384.62	−0.139888	−0.0699439	+	0.997551i	$0.522282\pi$
$462$	0	0
$463$	−13210.3	−1.32599	−0.662994	−	0.748624i	$-0.730715\pi$
$463$	−13210.3	−1.32599	−0.662994	+	0.748624i	$0.730715\pi$
$464$	52660.9	5.26879
$465$	0	0
$466$	9727.67	0.967008
$467$	4574.24	0.453256	0.226628	−	0.973981i	$-0.427230\pi$
$467$	4574.24	0.453256	0.226628	+	0.973981i	$0.427230\pi$
$468$	0	0
$469$	2353.94	0.231758
$470$	−10251.3	−1.00608
$471$	0	0
$472$	−21275.9	−2.07479
$473$	1405.61	0.136639
$474$	0	0
$475$	834.991	0.0806568
$476$	10549.7	1.01585
$477$	0	0
$478$	35154.0	3.36382
$479$	11031.8	1.05231	0.526154	−	0.850389i	$-0.323634\pi$
$479$	11031.8	1.05231	0.526154	+	0.850389i	$0.323634\pi$
$480$	0	0
$481$	633.716	0.0600726
$482$	18684.7	1.76569
$483$	0	0
$484$	−4609.02	−0.432853
$485$	259.934	0.0243360
$486$	0	0
$487$	−5194.06	−0.483296	−0.241648	−	0.970364i	$-0.577688\pi$
$487$	−5194.06	−0.483296	−0.241648	+	0.970364i	$0.577688\pi$
$488$	−10256.2	−0.951389
$489$	0	0
$490$	1362.58	0.125623
$491$	−11954.7	−1.09880	−0.549398	−	0.835561i	$-0.685143\pi$
$491$	−11954.7	−1.09880	−0.549398	+	0.835561i	$0.685143\pi$
$492$	0	0
$493$	−12432.9	−1.13580
$494$	−7124.43	−0.648873
$495$	0	0
$496$	56262.4	5.09326
$497$	−3173.19	−0.286392
$498$	0	0
$499$	2566.05	0.230205	0.115102	−	0.993354i	$-0.463280\pi$
$499$	2566.05	0.230205	0.115102	+	0.993354i	$0.463280\pi$
$500$	2866.36	0.256375
$501$	0	0
$502$	7399.62	0.657891
$503$	−21103.5	−1.87069	−0.935347	−	0.353731i	$-0.884913\pi$
$503$	−21103.5	−1.87069	−0.935347	+	0.353731i	$0.884913\pi$
$504$	0	0
$505$	−7146.50	−0.629732
$506$	−38783.1	−3.40735
$507$	0	0
$508$	−7727.36	−0.674894
$509$	781.732	0.0680740	0.0340370	−	0.999421i	$-0.489164\pi$
$509$	781.732	0.0680740	0.0340370	+	0.999421i	$0.489164\pi$
$510$	0	0
$511$	−154.305	−0.0133582
$512$	61129.5	5.27650
$513$	0	0
$514$	13775.7	1.18214
$515$	2171.06	0.185764
$516$	0	0
$517$	−12392.3	−1.05418
$518$	643.244	0.0545609
$519$	0	0
$520$	−15924.4	−1.34295
$521$	−14013.0	−1.17835	−0.589176	−	0.808005i	$-0.700547\pi$
$521$	−14013.0	−1.17835	−0.589176	+	0.808005i	$0.700547\pi$
$522$	0	0
$523$	−10310.7	−0.862052	−0.431026	−	0.902339i	$-0.641848\pi$
$523$	−10310.7	−0.862052	−0.431026	+	0.902339i	$0.641848\pi$
$524$	−67213.8	−5.60353
$525$	0	0
$526$	28656.3	2.37542
$527$	−13283.2	−1.09796
$528$	0	0
$529$	30867.2	2.53696
$530$	−12740.7	−1.04419
$531$	0	0
$532$	−5361.18	−0.436911
$533$	14905.1	1.21128
$534$	0	0
$535$	−3331.53	−0.269224
$536$	−27924.0	−2.25025
$537$	0	0
$538$	6401.16	0.512962
$539$	1647.16	0.131629
$540$	0	0
$541$	−17562.9	−1.39572	−0.697862	−	0.716232i	$-0.745865\pi$
$541$	−17562.9	−1.39572	−0.697862	+	0.716232i	$0.745865\pi$
$542$	10223.9	0.810250
$543$	0	0
$544$	−58093.1	−4.57853
$545$	−5997.56	−0.471389
$546$	0	0
$547$	−19889.6	−1.55469	−0.777347	−	0.629072i	$-0.783435\pi$
$547$	−19889.6	−1.55469	−0.777347	+	0.629072i	$0.783435\pi$
$548$	36346.9	2.83333
$549$	0	0
$550$	4673.86	0.362353
$551$	6318.22	0.488503
$552$	0	0
$553$	−2697.92	−0.207463
$554$	3161.45	0.242450
$555$	0	0
$556$	−29783.8	−2.27179
$557$	−5579.54	−0.424439	−0.212219	−	0.977222i	$-0.568069\pi$
$557$	−5579.54	−0.424439	−0.212219	+	0.977222i	$0.568069\pi$
$558$	0	0
$559$	−1603.76	−0.121345
$560$	−9743.22	−0.735226
$561$	0	0
$562$	−33452.8	−2.51089
$563$	24463.2	1.83126	0.915630	−	0.402022i	$-0.131693\pi$
$563$	24463.2	1.83126	0.915630	+	0.402022i	$0.131693\pi$
$564$	0	0
$565$	407.586	0.0303491
$566$	22167.0	1.64620
$567$	0	0
$568$	37642.5	2.78071
$569$	8582.14	0.632306	0.316153	−	0.948708i	$-0.397609\pi$
$569$	8582.14	0.632306	0.316153	+	0.948708i	$0.397609\pi$
$570$	0	0
$571$	17580.8	1.28850	0.644248	−	0.764816i	$-0.277170\pi$
$571$	17580.8	1.28850	0.644248	+	0.764816i	$0.277170\pi$
$572$	−29564.7	−2.16112
$573$	0	0
$574$	15129.2	1.10014
$575$	−5186.17	−0.376136
$576$	0	0
$577$	−8692.57	−0.627169	−0.313585	−	0.949560i	$-0.601530\pi$
$577$	−8692.57	−0.627169	−0.313585	+	0.949560i	$0.601530\pi$
$578$	−3300.36	−0.237503
$579$	0	0
$580$	21689.2	1.55275
$581$	166.250	0.0118713
$582$	0	0
$583$	−15401.7	−1.09412
$584$	1830.47	0.129701
$585$	0	0
$586$	13848.3	0.976227
$587$	−3584.61	−0.252049	−0.126024	−	0.992027i	$-0.540222\pi$
$587$	−3584.61	−0.252049	−0.126024	+	0.992027i	$0.540222\pi$
$588$	0	0
$589$	6750.33	0.472228
$590$	−7124.79	−0.497158
$591$	0	0
$592$	−4599.56	−0.319326
$593$	21853.6	1.51335	0.756676	−	0.653790i	$-0.226822\pi$
$593$	21853.6	1.51335	0.756676	+	0.653790i	$0.226822\pi$
$594$	0	0
$595$	2300.32	0.158494
$596$	45937.1	3.15714
$597$	0	0
$598$	44250.2	3.02596
$599$	−9090.48	−0.620078	−0.310039	−	0.950724i	$-0.600342\pi$
$599$	−9090.48	−0.620078	−0.310039	+	0.950724i	$0.600342\pi$
$600$	0	0
$601$	−19546.1	−1.32663	−0.663314	−	0.748341i	$-0.730851\pi$
$601$	−19546.1	−1.32663	−0.663314	+	0.748341i	$0.730851\pi$
$602$	−1627.87	−0.110211
$603$	0	0
$604$	62852.6	4.23416
$605$	−1004.98	−0.0675344
$606$	0	0
$607$	−15726.0	−1.05157	−0.525783	−	0.850619i	$-0.676227\pi$
$607$	−15726.0	−1.05157	−0.525783	+	0.850619i	$0.676227\pi$
$608$	29522.0	1.96920
$609$	0	0
$610$	−3434.57	−0.227970
$611$	14139.2	0.936186
$612$	0	0
$613$	−13572.5	−0.894269	−0.447135	−	0.894467i	$-0.647556\pi$
$613$	−13572.5	−0.894269	−0.447135	+	0.894467i	$0.647556\pi$
$614$	−788.921	−0.0518538
$615$	0	0
$616$	−19539.8	−1.27805
$617$	−17378.5	−1.13393	−0.566964	−	0.823743i	$-0.691882\pi$
$617$	−17378.5	−1.13393	−0.566964	+	0.823743i	$0.691882\pi$
$618$	0	0
$619$	−25113.3	−1.63068	−0.815338	−	0.578985i	$-0.803449\pi$
$619$	−25113.3	−1.63068	−0.815338	+	0.578985i	$0.803449\pi$
$620$	23172.6	1.50102
$621$	0	0
$622$	−11634.3	−0.749989
$623$	−10379.7	−0.667501
$624$	0	0
$625$	625.000	0.0400000
$626$	30707.8	1.96059
$627$	0	0
$628$	83565.1	5.30989
$629$	1085.93	0.0688377
$630$	0	0
$631$	−10814.4	−0.682276	−0.341138	−	0.940013i	$-0.610812\pi$
$631$	−10814.4	−0.682276	−0.341138	+	0.940013i	$0.610812\pi$
$632$	32004.5	2.01436
$633$	0	0
$634$	−29763.4	−1.86444
$635$	−1684.92	−0.105298
$636$	0	0
$637$	−1879.35	−0.116896
$638$	35366.3	2.19461
$639$	0	0
$640$	39414.7	2.43438
$641$	16359.0	1.00802	0.504010	−	0.863698i	$-0.331857\pi$
$641$	16359.0	1.00802	0.504010	+	0.863698i	$0.331857\pi$
$642$	0	0
$643$	8819.47	0.540911	0.270456	−	0.962732i	$-0.412826\pi$
$643$	8819.47	0.540911	0.270456	+	0.962732i	$0.412826\pi$
$644$	33298.6	2.03750
$645$	0	0
$646$	−12208.4	−0.743549
$647$	13828.8	0.840290	0.420145	−	0.907457i	$-0.361979\pi$
$647$	13828.8	0.840290	0.420145	+	0.907457i	$0.361979\pi$
$648$	0	0
$649$	−8612.83	−0.520930
$650$	−5332.72	−0.321794
$651$	0	0
$652$	63616.4	3.82118
$653$	−23988.7	−1.43760	−0.718798	−	0.695219i	$-0.755307\pi$
$653$	−23988.7	−1.43760	−0.718798	+	0.695219i	$0.755307\pi$
$654$	0	0
$655$	−14655.8	−0.874271
$656$	−108182.	−6.43874
$657$	0	0
$658$	14351.8	0.850289
$659$	−3109.28	−0.183794	−0.0918972	−	0.995769i	$-0.529293\pi$
$659$	−3109.28	−0.183794	−0.0918972	+	0.995769i	$0.529293\pi$
$660$	0	0
$661$	22695.0	1.33545	0.667726	−	0.744407i	$-0.267268\pi$
$661$	22695.0	1.33545	0.667726	+	0.744407i	$0.267268\pi$
$662$	−24379.2	−1.43131
$663$	0	0
$664$	−1972.18	−0.115264
$665$	−1168.99	−0.0681675
$666$	0	0
$667$	−39242.8	−2.27809
$668$	−26483.2	−1.53393
$669$	0	0
$670$	−9351.10	−0.539201
$671$	−4151.90	−0.238871
$672$	0	0
$673$	22073.4	1.26429	0.632145	−	0.774850i	$-0.282175\pi$
$673$	22073.4	1.26429	0.632145	+	0.774850i	$0.282175\pi$
$674$	−39623.7	−2.26446
$675$	0	0
$676$	−16646.9	−0.947136
$677$	2489.50	0.141328	0.0706642	−	0.997500i	$-0.477488\pi$
$677$	2489.50	0.141328	0.0706642	+	0.997500i	$0.477488\pi$
$678$	0	0
$679$	−363.907	−0.0205677
$680$	−27288.0	−1.53889
$681$	0	0
$682$	37785.0	2.12150
$683$	−7970.98	−0.446561	−0.223280	−	0.974754i	$-0.571677\pi$
$683$	−7970.98	−0.446561	−0.223280	+	0.974754i	$0.571677\pi$
$684$	0	0
$685$	7925.33	0.442060
$686$	−1907.61	−0.106171
$687$	0	0
$688$	11640.2	0.645027
$689$	17572.8	0.971656
$690$	0	0
$691$	−23892.7	−1.31537	−0.657687	−	0.753292i	$-0.728465\pi$
$691$	−23892.7	−1.31537	−0.657687	+	0.753292i	$0.728465\pi$
$692$	77677.4	4.26712
$693$	0	0
$694$	−2823.84	−0.154454
$695$	−6494.25	−0.354447
$696$	0	0
$697$	25541.3	1.38801
$698$	34232.1	1.85631
$699$	0	0
$700$	−4012.90	−0.216676
$701$	−12197.0	−0.657170	−0.328585	−	0.944474i	$-0.606572\pi$
$701$	−12197.0	−0.657170	−0.328585	+	0.944474i	$0.606572\pi$
$702$	0	0
$703$	−551.853	−0.0296067
$704$	90387.0	4.83891
$705$	0	0
$706$	35758.4	1.90621
$707$	10005.1	0.532221
$708$	0	0
$709$	−8982.28	−0.475792	−0.237896	−	0.971291i	$-0.576458\pi$
$709$	−8982.28	−0.475792	−0.237896	+	0.971291i	$0.576458\pi$
$710$	12605.6	0.666310
$711$	0	0
$712$	123131.	6.48107
$713$	−41926.7	−2.20220
$714$	0	0
$715$	−6446.48	−0.337181
$716$	−36765.6	−1.91899
$717$	0	0
$718$	−56035.0	−2.91255
$719$	6501.61	0.337231	0.168616	−	0.985682i	$-0.446070\pi$
$719$	6501.61	0.337231	0.168616	+	0.985682i	$0.446070\pi$
$720$	0	0
$721$	−3039.49	−0.156999
$722$	−31942.6	−1.64651
$723$	0	0
$724$	12479.4	0.640600
$725$	4729.26	0.242263
$726$	0	0
$727$	−24228.7	−1.23603	−0.618013	−	0.786168i	$-0.712062\pi$
$727$	−24228.7	−1.23603	−0.618013	+	0.786168i	$0.712062\pi$
$728$	22294.2	1.13500
$729$	0	0
$730$	612.983	0.0310788
$731$	−2748.19	−0.139050
$732$	0	0
$733$	36719.1	1.85027	0.925136	−	0.379636i	$-0.123951\pi$
$733$	36719.1	1.85027	0.925136	+	0.379636i	$0.123951\pi$
$734$	4543.22	0.228465
$735$	0	0
$736$	−183363.	−9.18321
$737$	−11304.1	−0.564983
$738$	0	0
$739$	23304.5	1.16004	0.580021	−	0.814602i	$-0.303044\pi$
$739$	23304.5	1.16004	0.580021	+	0.814602i	$0.303044\pi$
$740$	−1894.40	−0.0941076
$741$	0	0
$742$	17837.0	0.882505
$743$	−6875.35	−0.339478	−0.169739	−	0.985489i	$-0.554292\pi$
$743$	−6875.35	−0.339478	−0.169739	+	0.985489i	$0.554292\pi$
$744$	0	0
$745$	10016.4	0.492582
$746$	−20788.3	−1.02026
$747$	0	0
$748$	−50661.8	−2.47644
$749$	4664.15	0.227536
$750$	0	0
$751$	1182.65	0.0574640	0.0287320	−	0.999587i	$-0.490853\pi$
$751$	1182.65	0.0574640	0.0287320	+	0.999587i	$0.490853\pi$
$752$	−102623.	−4.97645
$753$	0	0
$754$	−40351.7	−1.94897
$755$	13704.8	0.660620
$756$	0	0
$757$	25226.8	1.21121	0.605604	−	0.795766i	$-0.292932\pi$
$757$	25226.8	1.21121	0.605604	+	0.795766i	$0.292932\pi$
$758$	10847.7	0.519795
$759$	0	0
$760$	13867.3	0.661870
$761$	10909.2	0.519655	0.259827	−	0.965655i	$-0.416334\pi$
$761$	10909.2	0.519655	0.259827	+	0.965655i	$0.416334\pi$
$762$	0	0
$763$	8396.58	0.398397
$764$	−68642.2	−3.25051
$765$	0	0
$766$	−37607.6	−1.77391
$767$	9826.95	0.462621
$768$	0	0
$769$	−12771.6	−0.598903	−0.299451	−	0.954112i	$-0.596804\pi$
$769$	−12771.6	−0.598903	−0.299451	+	0.954112i	$0.596804\pi$
$770$	−6543.41	−0.306244
$771$	0	0
$772$	30023.8	1.39972
$773$	−2199.06	−0.102322	−0.0511610	−	0.998690i	$-0.516292\pi$
$773$	−2199.06	−0.102322	−0.0511610	+	0.998690i	$0.516292\pi$
$774$	0	0
$775$	5052.70	0.234191
$776$	4316.92	0.199701
$777$	0	0
$778$	14193.0	0.654041
$779$	−12979.7	−0.596977
$780$	0	0
$781$	15238.3	0.698170
$782$	75827.0	3.46748
$783$	0	0
$784$	13640.5	0.621379
$785$	18221.1	0.828457
$786$	0	0
$787$	−19587.7	−0.887201	−0.443601	−	0.896225i	$-0.646299\pi$
$787$	−19587.7	−0.887201	−0.443601	+	0.896225i	$0.646299\pi$
$788$	26172.8	1.18321
$789$	0	0
$790$	10717.6	0.482676
$791$	−570.620	−0.0256497
$792$	0	0
$793$	4737.17	0.212134
$794$	−22790.9	−1.01866
$795$	0	0
$796$	54351.4	2.42014
$797$	−21699.3	−0.964401	−0.482200	−	0.876061i	$-0.660162\pi$
$797$	−21699.3	−0.964401	−0.482200	+	0.876061i	$0.660162\pi$
$798$	0	0
$799$	24228.8	1.07278
$800$	22097.6	0.976583
$801$	0	0
$802$	5821.89	0.256332
$803$	741.007	0.0325649
$804$	0	0
$805$	7260.64	0.317893
$806$	−43111.4	−1.88404
$807$	0	0
$808$	−118687.	−5.16758
$809$	15649.4	0.680103	0.340051	−	0.940407i	$-0.389556\pi$
$809$	15649.4	0.680103	0.340051	+	0.940407i	$0.389556\pi$
$810$	0	0
$811$	−33267.3	−1.44041	−0.720206	−	0.693760i	$-0.755953\pi$
$811$	−33267.3	−1.44041	−0.720206	+	0.693760i	$0.755953\pi$
$812$	−30364.9	−1.31231
$813$	0	0
$814$	−3089.00	−0.133009
$815$	13871.3	0.596186
$816$	0	0
$817$	1396.59	0.0598046
$818$	36244.2	1.54920
$819$	0	0
$820$	−44556.7	−1.89754
$821$	−5158.58	−0.219288	−0.109644	−	0.993971i	$-0.534971\pi$
$821$	−5158.58	−0.219288	−0.109644	+	0.993971i	$0.534971\pi$
$822$	0	0
$823$	26333.6	1.11535	0.557674	−	0.830060i	$-0.311694\pi$
$823$	26333.6	1.11535	0.557674	+	0.830060i	$0.311694\pi$
$824$	36056.5	1.52438
$825$	0	0
$826$	9974.71	0.420175
$827$	−19572.7	−0.822988	−0.411494	−	0.911413i	$-0.634993\pi$
$827$	−19572.7	−0.822988	−0.411494	+	0.911413i	$0.634993\pi$
$828$	0	0
$829$	9642.26	0.403968	0.201984	−	0.979389i	$-0.435261\pi$
$829$	9642.26	0.403968	0.201984	+	0.979389i	$0.435261\pi$
$830$	−660.436	−0.0276194
$831$	0	0
$832$	−103128.	−4.29728
$833$	−3220.45	−0.133952
$834$	0	0
$835$	−5774.57	−0.239326
$836$	25745.5	1.06511
$837$	0	0
$838$	−68290.7	−2.81511
$839$	31081.1	1.27895	0.639475	−	0.768812i	$-0.279152\pi$
$839$	31081.1	1.27895	0.639475	+	0.768812i	$0.279152\pi$
$840$	0	0
$841$	11396.5	0.467278
$842$	56432.3	2.30972
$843$	0	0
$844$	−15757.1	−0.642634
$845$	−3629.79	−0.147774
$846$	0	0
$847$	1406.97	0.0570770
$848$	−127545.	−5.16499
$849$	0	0
$850$	−9138.11	−0.368747
$851$	3427.59	0.138068
$852$	0	0
$853$	25780.9	1.03484	0.517421	−	0.855731i	$-0.326892\pi$
$853$	25780.9	1.03484	0.517421	+	0.855731i	$0.326892\pi$
$854$	4808.40	0.192670
$855$	0	0
$856$	−55329.3	−2.20925
$857$	−14452.6	−0.576069	−0.288035	−	0.957620i	$-0.593002\pi$
$857$	−14452.6	−0.576069	−0.288035	+	0.957620i	$0.593002\pi$
$858$	0	0
$859$	889.366	0.0353257	0.0176628	−	0.999844i	$-0.494377\pi$
$859$	889.366	0.0353257	0.0176628	+	0.999844i	$0.494377\pi$
$860$	4794.20	0.190094
$861$	0	0
$862$	−39259.2	−1.55125
$863$	−41460.3	−1.63537	−0.817685	−	0.575665i	$-0.804743\pi$
$863$	−41460.3	−1.63537	−0.817685	+	0.575665i	$0.804743\pi$
$864$	0	0
$865$	16937.3	0.665763
$866$	35977.6	1.41174
$867$	0	0
$868$	−32441.6	−1.26859
$869$	12956.0	0.505756
$870$	0	0
$871$	12897.6	0.501744
$872$	−99606.0	−3.86822
$873$	0	0
$874$	−38534.1	−1.49134
$875$	−875.000	−0.0338062
$876$	0	0
$877$	21173.0	0.815236	0.407618	−	0.913153i	$-0.366360\pi$
$877$	21173.0	0.815236	0.407618	+	0.913153i	$0.366360\pi$
$878$	26512.6	1.01909
$879$	0	0
$880$	46789.1	1.79234
$881$	−9883.39	−0.377957	−0.188978	−	0.981981i	$-0.560518\pi$
$881$	−9883.39	−0.377957	−0.188978	+	0.981981i	$0.560518\pi$
$882$	0	0
$883$	−45273.9	−1.72547	−0.862734	−	0.505658i	$-0.831250\pi$
$883$	−45273.9	−1.72547	−0.862734	+	0.505658i	$0.831250\pi$
$884$	57803.4	2.19925
$885$	0	0
$886$	−13161.7	−0.499069
$887$	−644.388	−0.0243928	−0.0121964	−	0.999926i	$-0.503882\pi$
$887$	−644.388	−0.0243928	−0.0121964	+	0.999926i	$0.503882\pi$
$888$	0	0
$889$	2358.89	0.0889930
$890$	41233.6	1.55298
$891$	0	0
$892$	22712.9	0.852562
$893$	−12312.7	−0.461398
$894$	0	0
$895$	−8016.62	−0.299403
$896$	−55180.6	−2.05743
$897$	0	0
$898$	−10089.6	−0.374937
$899$	38232.8	1.41839
$900$	0	0
$901$	30112.7	1.11343
$902$	−72653.8	−2.68194
$903$	0	0
$904$	6769.09	0.249045
$905$	2721.10	0.0999474
$906$	0	0
$907$	−15065.2	−0.551522	−0.275761	−	0.961226i	$-0.588930\pi$
$907$	−15065.2	−0.551522	−0.275761	+	0.961226i	$0.588930\pi$
$908$	−33920.5	−1.23975
$909$	0	0
$910$	7465.80	0.271966
$911$	−28789.9	−1.04704	−0.523520	−	0.852014i	$-0.675381\pi$
$911$	−28789.9	−1.04704	−0.523520	+	0.852014i	$0.675381\pi$
$912$	0	0
$913$	−798.371	−0.0289400
$914$	46072.6	1.66734
$915$	0	0
$916$	153739.	5.54552
$917$	20518.1	0.738894
$918$	0	0
$919$	−24163.8	−0.867345	−0.433673	−	0.901070i	$-0.642783\pi$
$919$	−24163.8	−0.867345	−0.433673	+	0.901070i	$0.642783\pi$
$920$	−86130.8	−3.08657
$921$	0	0
$922$	−7700.65	−0.275062
$923$	−17386.4	−0.620023
$924$	0	0
$925$	−413.068	−0.0146828
$926$	−73469.5	−2.60730
$927$	0	0
$928$	167208.	5.91474
$929$	35115.4	1.24015	0.620075	−	0.784542i	$-0.287102\pi$
$929$	35115.4	1.24015	0.620075	+	0.784542i	$0.287102\pi$
$930$	0	0
$931$	1636.58	0.0576120
$932$	40108.2	1.40964
$933$	0	0
$934$	25439.9	0.891239
$935$	−11046.6	−0.386379
$936$	0	0
$937$	−15512.6	−0.540849	−0.270424	−	0.962741i	$-0.587164\pi$
$937$	−15512.6	−0.540849	−0.270424	+	0.962741i	$0.587164\pi$
$938$	13091.5	0.455708
$939$	0	0
$940$	−42267.1	−1.46660
$941$	53283.8	1.84591	0.922956	−	0.384905i	$-0.125766\pi$
$941$	53283.8	1.84591	0.922956	+	0.384905i	$0.125766\pi$
$942$	0	0
$943$	80617.5	2.78395
$944$	−71324.8	−2.45914
$945$	0	0
$946$	7817.39	0.268674
$947$	55509.7	1.90478	0.952388	−	0.304890i	$-0.0986196\pi$
$947$	55509.7	1.90478	0.952388	+	0.304890i	$0.0986196\pi$
$948$	0	0
$949$	−845.464	−0.0289198
$950$	4643.84	0.158596
$951$	0	0
$952$	38203.2	1.30060
$953$	28080.6	0.954480	0.477240	−	0.878773i	$-0.341637\pi$
$953$	28080.6	0.954480	0.477240	+	0.878773i	$0.341637\pi$
$954$	0	0
$955$	−14967.2	−0.507149
$956$	144943.	4.90356
$957$	0	0
$958$	61353.9	2.06916
$959$	−11095.5	−0.373609
$960$	0	0
$961$	11056.6	0.371140
$962$	3524.44	0.118121
$963$	0	0
$964$	77039.0	2.57392
$965$	6546.59	0.218386
$966$	0	0
$967$	56609.3	1.88256	0.941278	−	0.337634i	$-0.109626\pi$
$967$	56609.3	1.88256	0.941278	+	0.337634i	$0.109626\pi$
$968$	−16690.5	−0.554187
$969$	0	0
$970$	1445.63	0.0478521
$971$	6782.17	0.224151	0.112075	−	0.993700i	$-0.464250\pi$
$971$	6782.17	0.224151	0.112075	+	0.993700i	$0.464250\pi$
$972$	0	0
$973$	9091.95	0.299563
$974$	−28887.0	−0.950309
$975$	0	0
$976$	−34382.8	−1.12763
$977$	45655.0	1.49502	0.747509	−	0.664252i	$-0.231250\pi$
$977$	45655.0	1.49502	0.747509	+	0.664252i	$0.231250\pi$
$978$	0	0
$979$	49845.5	1.62724
$980$	5618.06	0.183125
$981$	0	0
$982$	−66486.8	−2.16057
$983$	10102.3	0.327785	0.163893	−	0.986478i	$-0.447595\pi$
$983$	10102.3	0.327785	0.163893	+	0.986478i	$0.447595\pi$
$984$	0	0
$985$	5706.88	0.184606
$986$	−69146.4	−2.23334
$987$	0	0
$988$	−29374.8	−0.945887
$989$	−8674.27	−0.278894
$990$	0	0
$991$	25416.2	0.814705	0.407353	−	0.913271i	$-0.366452\pi$
$991$	25416.2	0.814705	0.407353	+	0.913271i	$0.366452\pi$
$992$	178644.	5.71768
$993$	0	0
$994$	−17647.8	−0.563135
$995$	11851.1	0.377595
$996$	0	0
$997$	48152.5	1.52959	0.764797	−	0.644271i	$-0.222839\pi$
$997$	48152.5	1.52959	0.764797	+	0.644271i	$0.222839\pi$
$998$	14271.2	0.452653
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.4.a.m.1.2		2
3.2	odd	2		105.4.a.c.1.1	✓	2
5.4	even	2		1575.4.a.m.1.1		2
7.6	odd	2		2205.4.a.bh.1.2		2
12.11	even	2		1680.4.a.bk.1.2		2
15.2	even	4		525.4.d.i.274.1		4
15.8	even	4		525.4.d.i.274.4		4
15.14	odd	2		525.4.a.p.1.2		2
21.20	even	2		735.4.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.c.1.1	✓	2	3.2	odd	2
315.4.a.m.1.2		2	1.1	even	1	trivial
525.4.a.p.1.2		2	15.14	odd	2
525.4.d.i.274.1		4	15.2	even	4
525.4.d.i.274.4		4	15.8	even	4
735.4.a.k.1.1		2	21.20	even	2
1575.4.a.m.1.1		2	5.4	even	2
1680.4.a.bk.1.2		2	12.11	even	2
2205.4.a.bh.1.2		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q+5.56155 q^{2} +22.9309 q^{4} +5.00000 q^{5} -7.00000 q^{7} +83.0388 q^{8} +O(q^{10})\)	\(q+5.56155 q^{2} +22.9309 q^{4} +5.00000 q^{5} -7.00000 q^{7} +83.0388 q^{8} +27.8078 q^{10} +33.6155 q^{11} -38.3542 q^{13} -38.9309 q^{14} +278.378 q^{16} -65.7235 q^{17} +33.3996 q^{19} +114.654 q^{20} +186.955 q^{22} -207.447 q^{23} +25.0000 q^{25} -213.309 q^{26} -160.516 q^{28} +189.170 q^{29} +202.108 q^{31} +883.902 q^{32} -365.525 q^{34} -35.0000 q^{35} -16.5227 q^{37} +185.754 q^{38} +415.194 q^{40} -388.617 q^{41} +41.8144 q^{43} +770.833 q^{44} -1153.73 q^{46} -368.648 q^{47} +49.0000 q^{49} +139.039 q^{50} -879.494 q^{52} -458.172 q^{53} +168.078 q^{55} -581.272 q^{56} +1052.08 q^{58} -256.216 q^{59} -123.511 q^{61} +1124.03 q^{62} +2688.85 q^{64} -191.771 q^{65} -336.277 q^{67} -1507.10 q^{68} -194.654 q^{70} +453.312 q^{71} +22.0436 q^{73} -91.8920 q^{74} +765.882 q^{76} -235.309 q^{77} +385.417 q^{79} +1391.89 q^{80} -2161.32 q^{82} -23.7501 q^{83} -328.617 q^{85} +232.553 q^{86} +2791.39 q^{88} +1482.81 q^{89} +268.479 q^{91} -4756.94 q^{92} -2050.25 q^{94} +166.998 q^{95} +51.9867 q^{97} +272.516 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 7 q^{2} + 17 q^{4} + 10 q^{5} - 14 q^{7} + 63 q^{8}+O(q^{10}) \) 2 * q + 7 * q^2 + 17 * q^4 + 10 * q^5 - 14 * q^7 + 63 * q^8	\( 2 q + 7 q^{2} + 17 q^{4} + 10 q^{5} - 14 q^{7} + 63 q^{8} + 35 q^{10} + 26 q^{11} + 14 q^{13} - 49 q^{14} + 297 q^{16} - 16 q^{17} + 174 q^{19} + 85 q^{20} + 176 q^{22} - 184 q^{23} + 50 q^{25} - 138 q^{26} - 119 q^{28} + 32 q^{29} + 330 q^{31} + 1071 q^{32} - 294 q^{34} - 70 q^{35} - 132 q^{37} + 388 q^{38} + 315 q^{40} - 200 q^{41} + 364 q^{43} + 816 q^{44} - 1120 q^{46} - 292 q^{47} + 98 q^{49} + 175 q^{50} - 1190 q^{52} - 34 q^{53} + 130 q^{55} - 441 q^{56} + 826 q^{58} - 364 q^{59} + 792 q^{61} + 1308 q^{62} + 2809 q^{64} + 70 q^{65} - 788 q^{67} - 1802 q^{68} - 245 q^{70} - 454 q^{71} + 778 q^{73} - 258 q^{74} - 68 q^{76} - 182 q^{77} + 408 q^{79} + 1485 q^{80} - 1890 q^{82} - 1136 q^{83} - 80 q^{85} + 696 q^{86} + 2944 q^{88} - 36 q^{89} - 98 q^{91} - 4896 q^{92} - 1940 q^{94} + 870 q^{95} - 498 q^{97} + 343 q^{98}+O(q^{100}) \) 2 * q + 7 * q^2 + 17 * q^4 + 10 * q^5 - 14 * q^7 + 63 * q^8 + 35 * q^10 + 26 * q^11 + 14 * q^13 - 49 * q^14 + 297 * q^16 - 16 * q^17 + 174 * q^19 + 85 * q^20 + 176 * q^22 - 184 * q^23 + 50 * q^25 - 138 * q^26 - 119 * q^28 + 32 * q^29 + 330 * q^31 + 1071 * q^32 - 294 * q^34 - 70 * q^35 - 132 * q^37 + 388 * q^38 + 315 * q^40 - 200 * q^41 + 364 * q^43 + 816 * q^44 - 1120 * q^46 - 292 * q^47 + 98 * q^49 + 175 * q^50 - 1190 * q^52 - 34 * q^53 + 130 * q^55 - 441 * q^56 + 826 * q^58 - 364 * q^59 + 792 * q^61 + 1308 * q^62 + 2809 * q^64 + 70 * q^65 - 788 * q^67 - 1802 * q^68 - 245 * q^70 - 454 * q^71 + 778 * q^73 - 258 * q^74 - 68 * q^76 - 182 * q^77 + 408 * q^79 + 1485 * q^80 - 1890 * q^82 - 1136 * q^83 - 80 * q^85 + 696 * q^86 + 2944 * q^88 - 36 * q^89 - 98 * q^91 - 4896 * q^92 - 1940 * q^94 + 870 * q^95 - 498 * q^97 + 343 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more