LMFDB - Embedded newform 315.4.a.k.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-1.82843 q^{2} -4.65685 q^{4} -5.00000 q^{5} -7.00000 q^{7} +23.1421 q^{8} +O(q^{10})$	\(q-1.82843 q^{2} -4.65685 q^{4} -5.00000 q^{5} -7.00000 q^{7} +23.1421 q^{8} +9.14214 q^{10} +64.5685 q^{11} -32.3431 q^{13} +12.7990 q^{14} -5.05887 q^{16} +56.3431 q^{17} -2.74517 q^{19} +23.2843 q^{20} -118.059 q^{22} -88.1665 q^{23} +25.0000 q^{25} +59.1371 q^{26} +32.5980 q^{28} -246.735 q^{29} -110.912 q^{31} -175.887 q^{32} -103.019 q^{34} +35.0000 q^{35} +120.676 q^{37} +5.01934 q^{38} -115.711 q^{40} +176.274 q^{41} -443.362 q^{43} -300.686 q^{44} +161.206 q^{46} +345.206 q^{47} +49.0000 q^{49} -45.7107 q^{50} +150.617 q^{52} -260.981 q^{53} -322.843 q^{55} -161.995 q^{56} +451.137 q^{58} -628.999 q^{59} -115.206 q^{61} +202.794 q^{62} +362.068 q^{64} +161.716 q^{65} -951.480 q^{67} -262.382 q^{68} -63.9949 q^{70} -356.264 q^{71} -656.754 q^{73} -220.648 q^{74} +12.7838 q^{76} -451.980 q^{77} +440.195 q^{79} +25.2944 q^{80} -322.304 q^{82} +54.4121 q^{83} -281.716 q^{85} +810.656 q^{86} +1494.25 q^{88} +1018.78 q^{89} +226.402 q^{91} +410.579 q^{92} -631.184 q^{94} +13.7258 q^{95} -724.108 q^{97} -89.5929 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 10 q^{5} - 14 q^{7} + 18 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 10 * q^5 - 14 * q^7 + 18 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 10 q^{5} - 14 q^{7} + 18 q^{8} - 10 q^{10} + 16 q^{11} - 76 q^{13} - 14 q^{14} - 78 q^{16} + 124 q^{17} - 96 q^{19} - 10 q^{20} - 304 q^{22} + 16 q^{23} + 50 q^{25} - 108 q^{26} - 14 q^{28} - 188 q^{29} - 120 q^{31} - 414 q^{32} + 156 q^{34} + 70 q^{35} - 132 q^{37} - 352 q^{38} - 90 q^{40} - 100 q^{41} - 536 q^{43} - 624 q^{44} + 560 q^{46} + 928 q^{47} + 98 q^{49} + 50 q^{50} - 140 q^{52} - 884 q^{53} - 80 q^{55} - 126 q^{56} + 676 q^{58} - 104 q^{59} - 468 q^{61} + 168 q^{62} + 34 q^{64} + 380 q^{65} - 1688 q^{67} + 188 q^{68} + 70 q^{70} + 136 q^{71} + 508 q^{73} - 1188 q^{74} - 608 q^{76} - 112 q^{77} - 432 q^{79} + 390 q^{80} - 1380 q^{82} + 584 q^{83} - 620 q^{85} + 456 q^{86} + 1744 q^{88} + 1404 q^{89} + 532 q^{91} + 1104 q^{92} + 1600 q^{94} + 480 q^{95} - 1188 q^{97} + 98 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 10 * q^5 - 14 * q^7 + 18 * q^8 - 10 * q^10 + 16 * q^11 - 76 * q^13 - 14 * q^14 - 78 * q^16 + 124 * q^17 - 96 * q^19 - 10 * q^20 - 304 * q^22 + 16 * q^23 + 50 * q^25 - 108 * q^26 - 14 * q^28 - 188 * q^29 - 120 * q^31 - 414 * q^32 + 156 * q^34 + 70 * q^35 - 132 * q^37 - 352 * q^38 - 90 * q^40 - 100 * q^41 - 536 * q^43 - 624 * q^44 + 560 * q^46 + 928 * q^47 + 98 * q^49 + 50 * q^50 - 140 * q^52 - 884 * q^53 - 80 * q^55 - 126 * q^56 + 676 * q^58 - 104 * q^59 - 468 * q^61 + 168 * q^62 + 34 * q^64 + 380 * q^65 - 1688 * q^67 + 188 * q^68 + 70 * q^70 + 136 * q^71 + 508 * q^73 - 1188 * q^74 - 608 * q^76 - 112 * q^77 - 432 * q^79 + 390 * q^80 - 1380 * q^82 + 584 * q^83 - 620 * q^85 + 456 * q^86 + 1744 * q^88 + 1404 * q^89 + 532 * q^91 + 1104 * q^92 + 1600 * q^94 + 480 * q^95 - 1188 * q^97 + 98 * 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 $

$2$	−1.82843	−0.646447	−0.323223	−	0.946323i	$-0.604766\pi$
$2$	−1.82843	−0.646447	−0.323223	+	0.946323i	$0.604766\pi$
$3$	0	0
$3$	0	0
$4$	−4.65685	−0.582107
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	23.1421	1.02275
$9$	0	0
$10$	9.14214	0.289100
$11$	64.5685	1.76983	0.884916	−	0.465751i	$-0.154216\pi$
$11$	64.5685	1.76983	0.884916	+	0.465751i	$0.154216\pi$
$12$	0	0
$13$	−32.3431	−0.690029	−0.345014	−	0.938597i	$-0.612126\pi$
$13$	−32.3431	−0.690029	−0.345014	+	0.938597i	$0.612126\pi$
$14$	12.7990	0.244334
$15$	0	0
$16$	−5.05887	−0.0790449
$17$	56.3431	0.803836	0.401918	−	0.915676i	$-0.368344\pi$
$17$	56.3431	0.803836	0.401918	+	0.915676i	$0.368344\pi$
$18$	0	0
$19$	−2.74517	−0.0331465	−0.0165733	−	0.999863i	$-0.505276\pi$
$19$	−2.74517	−0.0331465	−0.0165733	+	0.999863i	$0.505276\pi$
$20$	23.2843	0.260326
$21$	0	0
$22$	−118.059	−1.14410
$23$	−88.1665	−0.799304	−0.399652	−	0.916667i	$-0.630869\pi$
$23$	−88.1665	−0.799304	−0.399652	+	0.916667i	$0.630869\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	59.1371	0.446067
$27$	0	0
$28$	32.5980	0.220016
$29$	−246.735	−1.57992	−0.789958	−	0.613161i	$-0.789898\pi$
$29$	−246.735	−1.57992	−0.789958	+	0.613161i	$0.789898\pi$
$30$	0	0
$31$	−110.912	−0.642591	−0.321296	−	0.946979i	$-0.604118\pi$
$31$	−110.912	−0.642591	−0.321296	+	0.946979i	$0.604118\pi$
$32$	−175.887	−0.971649
$33$	0	0
$34$	−103.019	−0.519637
$35$	35.0000	0.169031
$36$	0	0
$37$	120.676	0.536190	0.268095	−	0.963392i	$-0.413606\pi$
$37$	120.676	0.536190	0.268095	+	0.963392i	$0.413606\pi$
$38$	5.01934	0.0214275
$39$	0	0
$40$	−115.711	−0.457387
$41$	176.274	0.671449	0.335724	−	0.941960i	$-0.391019\pi$
$41$	176.274	0.671449	0.335724	+	0.941960i	$0.391019\pi$
$42$	0	0
$43$	−443.362	−1.57238	−0.786188	−	0.617988i	$-0.787948\pi$
$43$	−443.362	−1.57238	−0.786188	+	0.617988i	$0.787948\pi$
$44$	−300.686	−1.03023
$45$	0	0
$46$	161.206	0.516707
$47$	345.206	1.07135	0.535675	−	0.844424i	$-0.320057\pi$
$47$	345.206	1.07135	0.535675	+	0.844424i	$0.320057\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−45.7107	−0.129289
$51$	0	0
$52$	150.617	0.401670
$53$	−260.981	−0.676386	−0.338193	−	0.941077i	$-0.609816\pi$
$53$	−260.981	−0.676386	−0.338193	+	0.941077i	$0.609816\pi$
$54$	0	0
$55$	−322.843	−0.791493
$56$	−161.995	−0.386562
$57$	0	0
$58$	451.137	1.02133
$59$	−628.999	−1.38794	−0.693972	−	0.720002i	$-0.744141\pi$
$59$	−628.999	−1.38794	−0.693972	+	0.720002i	$0.744141\pi$
$60$	0	0
$61$	−115.206	−0.241814	−0.120907	−	0.992664i	$-0.538580\pi$
$61$	−115.206	−0.241814	−0.120907	+	0.992664i	$0.538580\pi$
$62$	202.794	0.415401
$63$	0	0
$64$	362.068	0.707164
$65$	161.716	0.308590
$66$	0	0
$67$	−951.480	−1.73495	−0.867476	−	0.497479i	$-0.834259\pi$
$67$	−951.480	−1.73495	−0.867476	+	0.497479i	$0.834259\pi$
$68$	−262.382	−0.467919
$69$	0	0
$70$	−63.9949	−0.109269
$71$	−356.264	−0.595504	−0.297752	−	0.954643i	$-0.596237\pi$
$71$	−356.264	−0.595504	−0.297752	+	0.954643i	$0.596237\pi$
$72$	0	0
$73$	−656.754	−1.05298	−0.526488	−	0.850183i	$-0.676491\pi$
$73$	−656.754	−1.05298	−0.526488	+	0.850183i	$0.676491\pi$
$74$	−220.648	−0.346618
$75$	0	0
$76$	12.7838	0.0192948
$77$	−451.980	−0.668933
$78$	0	0
$79$	440.195	0.626909	0.313455	−	0.949603i	$-0.398514\pi$
$79$	440.195	0.626909	0.313455	+	0.949603i	$0.398514\pi$
$80$	25.2944	0.0353500
$81$	0	0
$82$	−322.304	−0.434056
$83$	54.4121	0.0719579	0.0359790	−	0.999353i	$-0.488545\pi$
$83$	54.4121	0.0719579	0.0359790	+	0.999353i	$0.488545\pi$
$84$	0	0
$85$	−281.716	−0.359487
$86$	810.656	1.01646
$87$	0	0
$88$	1494.25	1.81009
$89$	1018.78	1.21338	0.606690	−	0.794938i	$-0.292497\pi$
$89$	1018.78	1.21338	0.606690	+	0.794938i	$0.292497\pi$
$90$	0	0
$91$	226.402	0.260806
$92$	410.579	0.465280
$93$	0	0
$94$	−631.184	−0.692571
$95$	13.7258	0.0148236
$96$	0	0
$97$	−724.108	−0.757959	−0.378979	−	0.925405i	$-0.623725\pi$
$97$	−724.108	−0.757959	−0.378979	+	0.925405i	$0.623725\pi$
$98$	−89.5929	−0.0923495
$99$	0	0
$100$	−116.421	−0.116421
$101$	−268.725	−0.264744	−0.132372	−	0.991200i	$-0.542259\pi$
$101$	−268.725	−0.264744	−0.132372	+	0.991200i	$0.542259\pi$
$102$	0	0
$103$	−1840.63	−1.76080	−0.880399	−	0.474233i	$-0.842725\pi$
$103$	−1840.63	−1.76080	−0.880399	+	0.474233i	$0.842725\pi$
$104$	−748.489	−0.705725
$105$	0	0
$106$	477.184	0.437247
$107$	243.087	0.219627	0.109813	−	0.993952i	$-0.464975\pi$
$107$	243.087	0.219627	0.109813	+	0.993952i	$0.464975\pi$
$108$	0	0
$109$	−405.176	−0.356044	−0.178022	−	0.984027i	$-0.556970\pi$
$109$	−405.176	−0.356044	−0.178022	+	0.984027i	$0.556970\pi$
$110$	590.294	0.511658
$111$	0	0
$112$	35.4121	0.0298762
$113$	28.1766	0.0234569	0.0117285	−	0.999931i	$-0.496267\pi$
$113$	28.1766	0.0234569	0.0117285	+	0.999931i	$0.496267\pi$
$114$	0	0
$115$	440.833	0.357460
$116$	1149.01	0.919680
$117$	0	0
$118$	1150.08	0.897232
$119$	−394.402	−0.303822
$120$	0	0
$121$	2838.10	2.13230
$122$	210.646	0.156320
$123$	0	0
$124$	516.500	0.374057
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2740.90	−1.91508	−0.957541	−	0.288298i	$-0.906911\pi$
$127$	−2740.90	−1.91508	−0.957541	+	0.288298i	$0.906911\pi$
$128$	745.083	0.514505
$129$	0	0
$130$	−295.685	−0.199487
$131$	1832.04	1.22188	0.610938	−	0.791678i	$-0.290792\pi$
$131$	1832.04	1.22188	0.610938	+	0.791678i	$0.290792\pi$
$132$	0	0
$133$	19.2162	0.0125282
$134$	1739.71	1.12155
$135$	0	0
$136$	1303.90	0.822122
$137$	−382.747	−0.238688	−0.119344	−	0.992853i	$-0.538079\pi$
$137$	−382.747	−0.238688	−0.119344	+	0.992853i	$0.538079\pi$
$138$	0	0
$139$	3053.60	1.86333	0.931667	−	0.363314i	$-0.118355\pi$
$139$	3053.60	1.86333	0.931667	+	0.363314i	$0.118355\pi$
$140$	−162.990	−0.0983940
$141$	0	0
$142$	651.403	0.384961
$143$	−2088.35	−1.22123
$144$	0	0
$145$	1233.68	0.706560
$146$	1200.83	0.680692
$147$	0	0
$148$	−561.971	−0.312120
$149$	−3560.60	−1.95769	−0.978843	−	0.204611i	$-0.934407\pi$
$149$	−3560.60	−1.95769	−0.978843	+	0.204611i	$0.934407\pi$
$150$	0	0
$151$	3261.80	1.75789	0.878945	−	0.476923i	$-0.158248\pi$
$151$	3261.80	1.75789	0.878945	+	0.476923i	$0.158248\pi$
$152$	−63.5290	−0.0339005
$153$	0	0
$154$	826.412	0.432430
$155$	554.558	0.287376
$156$	0	0
$157$	2878.46	1.46322	0.731611	−	0.681723i	$-0.238769\pi$
$157$	2878.46	1.46322	0.731611	+	0.681723i	$0.238769\pi$
$158$	−804.865	−0.405263
$159$	0	0
$160$	879.437	0.434535
$161$	617.166	0.302108
$162$	0	0
$163$	−927.537	−0.445708	−0.222854	−	0.974852i	$-0.571537\pi$
$163$	−927.537	−0.445708	−0.222854	+	0.974852i	$0.571537\pi$
$164$	−820.883	−0.390855
$165$	0	0
$166$	−99.4886	−0.0465169
$167$	−1094.52	−0.507164	−0.253582	−	0.967314i	$-0.581609\pi$
$167$	−1094.52	−0.507164	−0.253582	+	0.967314i	$0.581609\pi$
$168$	0	0
$169$	−1150.92	−0.523860
$170$	515.097	0.232389
$171$	0	0
$172$	2064.67	0.915290
$173$	1713.25	0.752926	0.376463	−	0.926432i	$-0.377140\pi$
$173$	1713.25	0.752926	0.376463	+	0.926432i	$0.377140\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	−326.644	−0.139896
$177$	0	0
$178$	−1862.77	−0.784386
$179$	4065.58	1.69763	0.848816	−	0.528689i	$-0.177316\pi$
$179$	4065.58	1.69763	0.848816	+	0.528689i	$0.177316\pi$
$180$	0	0
$181$	−2791.40	−1.14631	−0.573157	−	0.819445i	$-0.694282\pi$
$181$	−2791.40	−1.14631	−0.573157	+	0.819445i	$0.694282\pi$
$182$	−413.960	−0.168597
$183$	0	0
$184$	−2040.36	−0.817486
$185$	−603.381	−0.239792
$186$	0	0
$187$	3637.99	1.42266
$188$	−1607.57	−0.623640
$189$	0	0
$190$	−25.0967	−0.00958266
$191$	634.185	0.240251	0.120126	−	0.992759i	$-0.461670\pi$
$191$	634.185	0.240251	0.120126	+	0.992759i	$0.461670\pi$
$192$	0	0
$193$	−254.999	−0.0951049	−0.0475524	−	0.998869i	$-0.515142\pi$
$193$	−254.999	−0.0951049	−0.0475524	+	0.998869i	$0.515142\pi$
$194$	1323.98	0.489980
$195$	0	0
$196$	−228.186	−0.0831581
$197$	−4172.37	−1.50898	−0.754490	−	0.656311i	$-0.772116\pi$
$197$	−4172.37	−1.50898	−0.754490	+	0.656311i	$0.772116\pi$
$198$	0	0
$199$	−4626.48	−1.64805	−0.824026	−	0.566552i	$-0.808277\pi$
$199$	−4626.48	−1.64805	−0.824026	+	0.566552i	$0.808277\pi$
$200$	578.553	0.204550
$201$	0	0
$202$	491.344	0.171143
$203$	1727.15	0.597152
$204$	0	0
$205$	−881.371	−0.300281
$206$	3365.45	1.13826
$207$	0	0
$208$	163.620	0.0545433
$209$	−177.251	−0.0586638
$210$	0	0
$211$	−1562.64	−0.509843	−0.254921	−	0.966962i	$-0.582050\pi$
$211$	−1562.64	−0.509843	−0.254921	+	0.966962i	$0.582050\pi$
$212$	1215.35	0.393729
$213$	0	0
$214$	−444.466	−0.141977
$215$	2216.81	0.703188
$216$	0	0
$217$	776.382	0.242877
$218$	740.834	0.230163
$219$	0	0
$220$	1503.43	0.460733
$221$	−1822.31	−0.554670
$222$	0	0
$223$	−1236.39	−0.371278	−0.185639	−	0.982618i	$-0.559436\pi$
$223$	−1236.39	−0.371278	−0.185639	+	0.982618i	$0.559436\pi$
$224$	1231.21	0.367249
$225$	0	0
$226$	−51.5189	−0.0151637
$227$	−4181.82	−1.22272	−0.611359	−	0.791353i	$-0.709377\pi$
$227$	−4181.82	−1.22272	−0.611359	+	0.791353i	$0.709377\pi$
$228$	0	0
$229$	−484.774	−0.139890	−0.0699449	−	0.997551i	$-0.522282\pi$
$229$	−484.774	−0.139890	−0.0699449	+	0.997551i	$0.522282\pi$
$230$	−806.030	−0.231079
$231$	0	0
$232$	−5709.98	−1.61585
$233$	−2080.54	−0.584982	−0.292491	−	0.956268i	$-0.594484\pi$
$233$	−2080.54	−0.584982	−0.292491	+	0.956268i	$0.594484\pi$
$234$	0	0
$235$	−1726.03	−0.479123
$236$	2929.16	0.807932
$237$	0	0
$238$	721.135	0.196404
$239$	−6814.10	−1.84422	−0.922108	−	0.386933i	$-0.873534\pi$
$239$	−6814.10	−1.84422	−0.922108	+	0.386933i	$0.873534\pi$
$240$	0	0
$241$	−3921.84	−1.04825	−0.524125	−	0.851642i	$-0.675607\pi$
$241$	−3921.84	−1.04825	−0.524125	+	0.851642i	$0.675607\pi$
$242$	−5189.25	−1.37842
$243$	0	0
$244$	536.498	0.140761
$245$	−245.000	−0.0638877
$246$	0	0
$247$	88.7873	0.0228721
$248$	−2566.73	−0.657209
$249$	0	0
$250$	228.553	0.0578199
$251$	5219.10	1.31246	0.656228	−	0.754562i	$-0.272151\pi$
$251$	5219.10	1.31246	0.656228	+	0.754562i	$0.272151\pi$
$252$	0	0
$253$	−5692.78	−1.41463
$254$	5011.53	1.23800
$255$	0	0
$256$	−4258.88	−1.03976
$257$	−6975.71	−1.69312	−0.846562	−	0.532289i	$-0.821332\pi$
$257$	−6975.71	−1.69312	−0.846562	+	0.532289i	$0.821332\pi$
$258$	0	0
$259$	−844.733	−0.202661
$260$	−753.087	−0.179632
$261$	0	0
$262$	−3349.75	−0.789878
$263$	3607.36	0.845776	0.422888	−	0.906182i	$-0.361016\pi$
$263$	3607.36	0.845776	0.422888	+	0.906182i	$0.361016\pi$
$264$	0	0
$265$	1304.90	0.302489
$266$	−35.1354	−0.00809882
$267$	0	0
$268$	4430.90	1.00993
$269$	−5.88572	−0.00133405	−0.000667023	−	1.00000i	$-0.500212\pi$
$269$	−5.88572	−0.00133405	−0.000667023		1.00000i	$0.500212\pi$
$270$	0	0
$271$	6916.32	1.55032	0.775160	−	0.631765i	$-0.217669\pi$
$271$	6916.32	1.55032	0.775160	+	0.631765i	$0.217669\pi$
$272$	−285.033	−0.0635392
$273$	0	0
$274$	699.825	0.154299
$275$	1614.21	0.353966
$276$	0	0
$277$	−2119.46	−0.459733	−0.229867	−	0.973222i	$-0.573829\pi$
$277$	−2119.46	−0.459733	−0.229867	+	0.973222i	$0.573829\pi$
$278$	−5583.29	−1.20455
$279$	0	0
$280$	809.975	0.172876
$281$	239.917	0.0509334	0.0254667	−	0.999676i	$-0.491893\pi$
$281$	239.917	0.0509334	0.0254667	+	0.999676i	$0.491893\pi$
$282$	0	0
$283$	−4542.12	−0.954067	−0.477034	−	0.878885i	$-0.658288\pi$
$283$	−4542.12	−0.954067	−0.477034	+	0.878885i	$0.658288\pi$
$284$	1659.07	0.346647
$285$	0	0
$286$	3818.40	0.789463
$287$	−1233.92	−0.253784
$288$	0	0
$289$	−1738.45	−0.353847
$290$	−2255.69	−0.456753
$291$	0	0
$292$	3058.41	0.612944
$293$	2171.70	0.433010	0.216505	−	0.976281i	$-0.430534\pi$
$293$	2171.70	0.433010	0.216505	+	0.976281i	$0.430534\pi$
$294$	0	0
$295$	3145.00	0.620708
$296$	2792.70	0.548387
$297$	0	0
$298$	6510.29	1.26554
$299$	2851.58	0.551543
$300$	0	0
$301$	3103.54	0.594302
$302$	−5963.96	−1.13638
$303$	0	0
$304$	13.8875	0.00262007
$305$	576.030	0.108142
$306$	0	0
$307$	−3508.64	−0.652276	−0.326138	−	0.945322i	$-0.605747\pi$
$307$	−3508.64	−0.652276	−0.326138	+	0.945322i	$0.605747\pi$
$308$	2104.80	0.389391
$309$	0	0
$310$	−1013.97	−0.185773
$311$	3133.25	0.571287	0.285643	−	0.958336i	$-0.407793\pi$
$311$	3133.25	0.571287	0.285643	+	0.958336i	$0.407793\pi$
$312$	0	0
$313$	6389.59	1.15387	0.576935	−	0.816790i	$-0.304249\pi$
$313$	6389.59	1.15387	0.576935	+	0.816790i	$0.304249\pi$
$314$	−5263.05	−0.945895
$315$	0	0
$316$	−2049.92	−0.364928
$317$	−1634.44	−0.289587	−0.144794	−	0.989462i	$-0.546252\pi$
$317$	−1634.44	−0.289587	−0.144794	+	0.989462i	$0.546252\pi$
$318$	0	0
$319$	−15931.3	−2.79618
$320$	−1810.34	−0.316253
$321$	0	0
$322$	−1128.44	−0.195297
$323$	−154.671	−0.0266444
$324$	0	0
$325$	−808.579	−0.138006
$326$	1695.93	0.288126
$327$	0	0
$328$	4079.36	0.686723
$329$	−2416.44	−0.404932
$330$	0	0
$331$	4386.17	0.728355	0.364177	−	0.931330i	$-0.381350\pi$
$331$	4386.17	0.728355	0.364177	+	0.931330i	$0.381350\pi$
$332$	−253.389	−0.0418872
$333$	0	0
$334$	2001.25	0.327854
$335$	4757.40	0.775894
$336$	0	0
$337$	1713.98	0.277051	0.138526	−	0.990359i	$-0.455764\pi$
$337$	1713.98	0.277051	0.138526	+	0.990359i	$0.455764\pi$
$338$	2104.38	0.338648
$339$	0	0
$340$	1311.91	0.209260
$341$	−7161.41	−1.13728
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−10260.4	−1.60814
$345$	0	0
$346$	−3132.56	−0.486727
$347$	1744.83	0.269935	0.134967	−	0.990850i	$-0.456907\pi$
$347$	1744.83	0.269935	0.134967	+	0.990850i	$0.456907\pi$
$348$	0	0
$349$	7046.78	1.08082	0.540409	−	0.841403i	$-0.318270\pi$
$349$	7046.78	1.08082	0.540409	+	0.841403i	$0.318270\pi$
$350$	319.975	0.0488668
$351$	0	0
$352$	−11356.8	−1.71966
$353$	12668.5	1.91013	0.955064	−	0.296400i	$-0.0957863\pi$
$353$	12668.5	1.91013	0.955064	+	0.296400i	$0.0957863\pi$
$354$	0	0
$355$	1781.32	0.266317
$356$	−4744.33	−0.706317
$357$	0	0
$358$	−7433.62	−1.09743
$359$	−37.7844	−0.00555483	−0.00277742	−	0.999996i	$-0.500884\pi$
$359$	−37.7844	−0.00555483	−0.00277742	+	0.999996i	$0.500884\pi$
$360$	0	0
$361$	−6851.46	−0.998901
$362$	5103.87	0.741031
$363$	0	0
$364$	−1054.32	−0.151817
$365$	3283.77	0.470905
$366$	0	0
$367$	−759.829	−0.108073	−0.0540364	−	0.998539i	$-0.517209\pi$
$367$	−759.829	−0.108073	−0.0540364	+	0.998539i	$0.517209\pi$
$368$	446.023	0.0631809
$369$	0	0
$370$	1103.24	0.155012
$371$	1826.86	0.255650
$372$	0	0
$373$	719.320	0.0998525	0.0499263	−	0.998753i	$-0.484101\pi$
$373$	719.320	0.0998525	0.0499263	+	0.998753i	$0.484101\pi$
$374$	−6651.81	−0.919671
$375$	0	0
$376$	7988.81	1.09572
$377$	7980.19	1.09019
$378$	0	0
$379$	572.559	0.0775999	0.0388000	−	0.999247i	$-0.487646\pi$
$379$	572.559	0.0775999	0.0388000	+	0.999247i	$0.487646\pi$
$380$	−63.9192	−0.00862891
$381$	0	0
$382$	−1159.56	−0.155310
$383$	−4513.18	−0.602122	−0.301061	−	0.953605i	$-0.597341\pi$
$383$	−4513.18	−0.602122	−0.301061	+	0.953605i	$0.597341\pi$
$384$	0	0
$385$	2259.90	0.299156
$386$	466.247	0.0614802
$387$	0	0
$388$	3372.06	0.441213
$389$	6902.13	0.899619	0.449810	−	0.893124i	$-0.351492\pi$
$389$	6902.13	0.899619	0.449810	+	0.893124i	$0.351492\pi$
$390$	0	0
$391$	−4967.58	−0.642510
$392$	1133.96	0.146107
$393$	0	0
$394$	7628.88	0.975475
$395$	−2200.98	−0.280362
$396$	0	0
$397$	4124.58	0.521427	0.260714	−	0.965416i	$-0.416042\pi$
$397$	4124.58	0.521427	0.260714	+	0.965416i	$0.416042\pi$
$398$	8459.18	1.06538
$399$	0	0
$400$	−126.472	−0.0158090
$401$	1002.50	0.124844	0.0624219	−	0.998050i	$-0.480118\pi$
$401$	1002.50	0.124844	0.0624219	+	0.998050i	$0.480118\pi$
$402$	0	0
$403$	3587.23	0.443406
$404$	1251.41	0.154109
$405$	0	0
$406$	−3157.96	−0.386027
$407$	7791.89	0.948967
$408$	0	0
$409$	10335.0	1.24947	0.624736	−	0.780836i	$-0.285206\pi$
$409$	10335.0	1.24947	0.624736	+	0.780836i	$0.285206\pi$
$410$	1611.52	0.194116
$411$	0	0
$412$	8571.53	1.02497
$413$	4402.99	0.524594
$414$	0	0
$415$	−272.061	−0.0321806
$416$	5688.75	0.670466
$417$	0	0
$418$	324.091	0.0379230
$419$	−3183.21	−0.371145	−0.185573	−	0.982631i	$-0.559414\pi$
$419$	−3183.21	−0.371145	−0.185573	+	0.982631i	$0.559414\pi$
$420$	0	0
$421$	−6944.34	−0.803911	−0.401956	−	0.915659i	$-0.631669\pi$
$421$	−6944.34	−0.803911	−0.401956	+	0.915659i	$0.631669\pi$
$422$	2857.18	0.329586
$423$	0	0
$424$	−6039.65	−0.691772
$425$	1408.58	0.160767
$426$	0	0
$427$	806.442	0.0913969
$428$	−1132.02	−0.127846
$429$	0	0
$430$	−4053.28	−0.454573
$431$	−3868.41	−0.432331	−0.216166	−	0.976357i	$-0.569355\pi$
$431$	−3868.41	−0.432331	−0.216166	+	0.976357i	$0.569355\pi$
$432$	0	0
$433$	−6132.96	−0.680673	−0.340336	−	0.940304i	$-0.610541\pi$
$433$	−6132.96	−0.680673	−0.340336	+	0.940304i	$0.610541\pi$
$434$	−1419.56	−0.157007
$435$	0	0
$436$	1886.84	0.207256
$437$	242.032	0.0264942
$438$	0	0
$439$	−4090.14	−0.444673	−0.222337	−	0.974970i	$-0.571368\pi$
$439$	−4090.14	−0.444673	−0.222337	+	0.974970i	$0.571368\pi$
$440$	−7471.27	−0.809497
$441$	0	0
$442$	3331.97	0.358565
$443$	−12434.5	−1.33359	−0.666795	−	0.745241i	$-0.732334\pi$
$443$	−12434.5	−1.33359	−0.666795	+	0.745241i	$0.732334\pi$
$444$	0	0
$445$	−5093.92	−0.542640
$446$	2260.66	0.240012
$447$	0	0
$448$	−2534.48	−0.267283
$449$	−883.046	−0.0928141	−0.0464071	−	0.998923i	$-0.514777\pi$
$449$	−883.046	−0.0928141	−0.0464071	+	0.998923i	$0.514777\pi$
$450$	0	0
$451$	11381.8	1.18835
$452$	−131.214	−0.0136544
$453$	0	0
$454$	7646.15	0.790422
$455$	−1132.01	−0.116636
$456$	0	0
$457$	−9068.44	−0.928235	−0.464118	−	0.885774i	$-0.653629\pi$
$457$	−9068.44	−0.928235	−0.464118	+	0.885774i	$0.653629\pi$
$458$	886.373	0.0904312
$459$	0	0
$460$	−2052.89	−0.208080
$461$	−12508.9	−1.26377	−0.631885	−	0.775063i	$-0.717718\pi$
$461$	−12508.9	−1.26377	−0.631885	+	0.775063i	$0.717718\pi$
$462$	0	0
$463$	12688.7	1.27363	0.636817	−	0.771015i	$-0.280251\pi$
$463$	12688.7	1.27363	0.636817	+	0.771015i	$0.280251\pi$
$464$	1248.20	0.124884
$465$	0	0
$466$	3804.12	0.378160
$467$	10136.5	1.00442	0.502208	−	0.864747i	$-0.332521\pi$
$467$	10136.5	1.00442	0.502208	+	0.864747i	$0.332521\pi$
$468$	0	0
$469$	6660.36	0.655750
$470$	3155.92	0.309727
$471$	0	0
$472$	−14556.4	−1.41952
$473$	−28627.3	−2.78284
$474$	0	0
$475$	−68.6292	−0.00662931
$476$	1836.67	0.176857
$477$	0	0
$478$	12459.1	1.19219
$479$	11361.1	1.08372	0.541861	−	0.840468i	$-0.317720\pi$
$479$	11361.1	1.08372	0.541861	+	0.840468i	$0.317720\pi$
$480$	0	0
$481$	−3903.05	−0.369987
$482$	7170.80	0.677637
$483$	0	0
$484$	−13216.6	−1.24123
$485$	3620.54	0.338969
$486$	0	0
$487$	7929.53	0.737826	0.368913	−	0.929464i	$-0.379730\pi$
$487$	7929.53	0.737826	0.368913	+	0.929464i	$0.379730\pi$
$488$	−2666.11	−0.247314
$489$	0	0
$490$	447.965	0.0413000
$491$	−8111.51	−0.745555	−0.372777	−	0.927921i	$-0.621595\pi$
$491$	−8111.51	−0.745555	−0.372777	+	0.927921i	$0.621595\pi$
$492$	0	0
$493$	−13901.8	−1.26999
$494$	−162.341	−0.0147856
$495$	0	0
$496$	561.088	0.0507936
$497$	2493.85	0.225079
$498$	0	0
$499$	16816.6	1.50865	0.754324	−	0.656502i	$-0.227965\pi$
$499$	16816.6	1.50865	0.754324	+	0.656502i	$0.227965\pi$
$500$	582.107	0.0520652
$501$	0	0
$502$	−9542.74	−0.848433
$503$	17764.6	1.57472	0.787362	−	0.616491i	$-0.211446\pi$
$503$	17764.6	1.57472	0.787362	+	0.616491i	$0.211446\pi$
$504$	0	0
$505$	1343.62	0.118397
$506$	10408.8	0.914485
$507$	0	0
$508$	12764.0	1.11478
$509$	−13908.8	−1.21120	−0.605598	−	0.795771i	$-0.707066\pi$
$509$	−13908.8	−1.21120	−0.605598	+	0.795771i	$0.707066\pi$
$510$	0	0
$511$	4597.27	0.397987
$512$	1826.38	0.157647
$513$	0	0
$514$	12754.6	1.09451
$515$	9203.13	0.787453
$516$	0	0
$517$	22289.5	1.89611
$518$	1544.53	0.131009
$519$	0	0
$520$	3742.45	0.315610
$521$	8639.68	0.726510	0.363255	−	0.931690i	$-0.381665\pi$
$521$	8639.68	0.726510	0.363255	+	0.931690i	$0.381665\pi$
$522$	0	0
$523$	23242.2	1.94323	0.971617	−	0.236561i	$-0.0760203\pi$
$523$	23242.2	1.94323	0.971617	+	0.236561i	$0.0760203\pi$
$524$	−8531.53	−0.711263
$525$	0	0
$526$	−6595.79	−0.546749
$527$	−6249.11	−0.516538
$528$	0	0
$529$	−4393.66	−0.361113
$530$	−2385.92	−0.195543
$531$	0	0
$532$	−89.4869	−0.00729276
$533$	−5701.26	−0.463319
$534$	0	0
$535$	−1215.43	−0.0982201
$536$	−22019.3	−1.77442
$537$	0	0
$538$	10.7616	0.000862390 0
$539$	3163.86	0.252833
$540$	0	0
$541$	11395.2	0.905577	0.452789	−	0.891618i	$-0.350429\pi$
$541$	11395.2	0.905577	0.452789	+	0.891618i	$0.350429\pi$
$542$	−12646.0	−1.00220
$543$	0	0
$544$	−9910.04	−0.781047
$545$	2025.88	0.159228
$546$	0	0
$547$	−7870.21	−0.615184	−0.307592	−	0.951518i	$-0.599523\pi$
$547$	−7870.21	−0.615184	−0.307592	+	0.951518i	$0.599523\pi$
$548$	1782.40	0.138942
$549$	0	0
$550$	−2951.47	−0.228820
$551$	677.329	0.0523687
$552$	0	0
$553$	−3081.37	−0.236949
$554$	3875.28	0.297193
$555$	0	0
$556$	−14220.2	−1.08466
$557$	−17769.8	−1.35176	−0.675880	−	0.737012i	$-0.736236\pi$
$557$	−17769.8	−1.35176	−0.675880	+	0.737012i	$0.736236\pi$
$558$	0	0
$559$	14339.7	1.08498
$560$	−177.061	−0.0133610
$561$	0	0
$562$	−438.672	−0.0329257
$563$	−15192.8	−1.13730	−0.568651	−	0.822579i	$-0.692534\pi$
$563$	−15192.8	−1.13730	−0.568651	+	0.822579i	$0.692534\pi$
$564$	0	0
$565$	−140.883	−0.0104903
$566$	8304.93	0.616753
$567$	0	0
$568$	−8244.71	−0.609050
$569$	23300.0	1.71667	0.858335	−	0.513090i	$-0.171499\pi$
$569$	23300.0	1.71667	0.858335	+	0.513090i	$0.171499\pi$
$570$	0	0
$571$	10638.2	0.779673	0.389837	−	0.920884i	$-0.372531\pi$
$571$	10638.2	0.779673	0.389837	+	0.920884i	$0.372531\pi$
$572$	9725.14	0.710889
$573$	0	0
$574$	2256.13	0.164058
$575$	−2204.16	−0.159861
$576$	0	0
$577$	−897.258	−0.0647372	−0.0323686	−	0.999476i	$-0.510305\pi$
$577$	−897.258	−0.0647372	−0.0323686	+	0.999476i	$0.510305\pi$
$578$	3178.63	0.228743
$579$	0	0
$580$	−5745.05	−0.411293
$581$	−380.885	−0.0271975
$582$	0	0
$583$	−16851.1	−1.19709
$584$	−15198.7	−1.07693
$585$	0	0
$586$	−3970.79	−0.279918
$587$	14712.9	1.03452	0.517261	−	0.855828i	$-0.326952\pi$
$587$	14712.9	1.03452	0.517261	+	0.855828i	$0.326952\pi$
$588$	0	0
$589$	304.471	0.0212997
$590$	−5750.40	−0.401254
$591$	0	0
$592$	−610.486	−0.0423831
$593$	7216.29	0.499726	0.249863	−	0.968281i	$-0.419614\pi$
$593$	7216.29	0.499726	0.249863	+	0.968281i	$0.419614\pi$
$594$	0	0
$595$	1972.01	0.135873
$596$	16581.2	1.13958
$597$	0	0
$598$	−5213.91	−0.356543
$599$	20885.5	1.42464	0.712320	−	0.701855i	$-0.247645\pi$
$599$	20885.5	1.42464	0.712320	+	0.701855i	$0.247645\pi$
$600$	0	0
$601$	−11047.7	−0.749823	−0.374911	−	0.927061i	$-0.622327\pi$
$601$	−11047.7	−0.749823	−0.374911	+	0.927061i	$0.622327\pi$
$602$	−5674.59	−0.384185
$603$	0	0
$604$	−15189.7	−1.02328
$605$	−14190.5	−0.953595
$606$	0	0
$607$	−9434.94	−0.630894	−0.315447	−	0.948943i	$-0.602154\pi$
$607$	−9434.94	−0.630894	−0.315447	+	0.948943i	$0.602154\pi$
$608$	482.840	0.0322068
$609$	0	0
$610$	−1053.23	−0.0699082
$611$	−11165.0	−0.739263
$612$	0	0
$613$	−17662.6	−1.16376	−0.581881	−	0.813274i	$-0.697683\pi$
$613$	−17662.6	−1.16376	−0.581881	+	0.813274i	$0.697683\pi$
$614$	6415.30	0.421662
$615$	0	0
$616$	−10459.8	−0.684150
$617$	10817.4	0.705820	0.352910	−	0.935657i	$-0.385192\pi$
$617$	10817.4	0.705820	0.352910	+	0.935657i	$0.385192\pi$
$618$	0	0
$619$	29073.9	1.88785	0.943926	−	0.330158i	$-0.107102\pi$
$619$	29073.9	1.88785	0.943926	+	0.330158i	$0.107102\pi$
$620$	−2582.50	−0.167283
$621$	0	0
$622$	−5728.92	−0.369306
$623$	−7131.49	−0.458615
$624$	0	0
$625$	625.000	0.0400000
$626$	−11682.9	−0.745915
$627$	0	0
$628$	−13404.5	−0.851751
$629$	6799.28	0.431009
$630$	0	0
$631$	−2203.17	−0.138996	−0.0694981	−	0.997582i	$-0.522140\pi$
$631$	−2203.17	−0.138996	−0.0694981	+	0.997582i	$0.522140\pi$
$632$	10187.1	0.641170
$633$	0	0
$634$	2988.45	0.187203
$635$	13704.5	0.856451
$636$	0	0
$637$	−1584.81	−0.0985755
$638$	29129.3	1.80758
$639$	0	0
$640$	−3725.42	−0.230094
$641$	−22466.5	−1.38436	−0.692180	−	0.721725i	$-0.743349\pi$
$641$	−22466.5	−1.38436	−0.692180	+	0.721725i	$0.743349\pi$
$642$	0	0
$643$	−12347.0	−0.757257	−0.378629	−	0.925549i	$-0.623604\pi$
$643$	−12347.0	−0.757257	−0.378629	+	0.925549i	$0.623604\pi$
$644$	−2874.05	−0.175859
$645$	0	0
$646$	282.805	0.0172242
$647$	24114.0	1.46525	0.732626	−	0.680631i	$-0.238294\pi$
$647$	24114.0	1.46525	0.732626	+	0.680631i	$0.238294\pi$
$648$	0	0
$649$	−40613.6	−2.45643
$650$	1478.43	0.0892134
$651$	0	0
$652$	4319.41	0.259449
$653$	−7843.33	−0.470035	−0.235018	−	0.971991i	$-0.575515\pi$
$653$	−7843.33	−0.470035	−0.235018	+	0.971991i	$0.575515\pi$
$654$	0	0
$655$	−9160.18	−0.546440
$656$	−891.749	−0.0530746
$657$	0	0
$658$	4418.29	0.261767
$659$	−21242.8	−1.25569	−0.627846	−	0.778338i	$-0.716063\pi$
$659$	−21242.8	−1.25569	−0.627846	+	0.778338i	$0.716063\pi$
$660$	0	0
$661$	−22221.7	−1.30760	−0.653801	−	0.756667i	$-0.726827\pi$
$661$	−22221.7	−1.30760	−0.653801	+	0.756667i	$0.726827\pi$
$662$	−8019.79	−0.470843
$663$	0	0
$664$	1259.21	0.0735948
$665$	−96.0808	−0.00560279
$666$	0	0
$667$	21753.8	1.26283
$668$	5097.01	0.295223
$669$	0	0
$670$	−8698.56	−0.501574
$671$	−7438.69	−0.427969
$672$	0	0
$673$	−3787.85	−0.216955	−0.108478	−	0.994099i	$-0.534598\pi$
$673$	−3787.85	−0.216955	−0.108478	+	0.994099i	$0.534598\pi$
$674$	−3133.88	−0.179099
$675$	0	0
$676$	5359.67	0.304943
$677$	11296.8	0.641314	0.320657	−	0.947195i	$-0.396096\pi$
$677$	11296.8	0.641314	0.320657	+	0.947195i	$0.396096\pi$
$678$	0	0
$679$	5068.75	0.286481
$680$	−6519.50	−0.367664
$681$	0	0
$682$	13094.1	0.735190
$683$	−4807.14	−0.269312	−0.134656	−	0.990892i	$-0.542993\pi$
$683$	−4807.14	−0.269312	−0.134656	+	0.990892i	$0.542993\pi$
$684$	0	0
$685$	1913.73	0.106745
$686$	627.151	0.0349048
$687$	0	0
$688$	2242.92	0.124288
$689$	8440.94	0.466726
$690$	0	0
$691$	5393.47	0.296928	0.148464	−	0.988918i	$-0.452567\pi$
$691$	5393.47	0.296928	0.148464	+	0.988918i	$0.452567\pi$
$692$	−7978.37	−0.438283
$693$	0	0
$694$	−3190.29	−0.174498
$695$	−15268.0	−0.833308
$696$	0	0
$697$	9931.84	0.539735
$698$	−12884.5	−0.698691
$699$	0	0
$700$	814.949	0.0440031
$701$	−2404.77	−0.129568	−0.0647838	−	0.997899i	$-0.520636\pi$
$701$	−2404.77	−0.129568	−0.0647838	+	0.997899i	$0.520636\pi$
$702$	0	0
$703$	−331.276	−0.0177729
$704$	23378.2	1.25156
$705$	0	0
$706$	−23163.4	−1.23480
$707$	1881.07	0.100064
$708$	0	0
$709$	−21617.3	−1.14507	−0.572535	−	0.819881i	$-0.694040\pi$
$709$	−21617.3	−1.14507	−0.572535	+	0.819881i	$0.694040\pi$
$710$	−3257.01	−0.172160
$711$	0	0
$712$	23576.8	1.24098
$713$	9778.70	0.513626
$714$	0	0
$715$	10441.7	0.546153
$716$	−18932.8	−0.988203
$717$	0	0
$718$	69.0860	0.00359090
$719$	18228.6	0.945498	0.472749	−	0.881197i	$-0.343262\pi$
$719$	18228.6	0.945498	0.472749	+	0.881197i	$0.343262\pi$
$720$	0	0
$721$	12884.4	0.665519
$722$	12527.4	0.645736
$723$	0	0
$724$	12999.1	0.667278
$725$	−6168.38	−0.315983
$726$	0	0
$727$	20196.5	1.03033	0.515164	−	0.857092i	$-0.327731\pi$
$727$	20196.5	1.03033	0.515164	+	0.857092i	$0.327731\pi$
$728$	5239.43	0.266739
$729$	0	0
$730$	−6004.13	−0.304415
$731$	−24980.4	−1.26393
$732$	0	0
$733$	−15264.9	−0.769196	−0.384598	−	0.923084i	$-0.625660\pi$
$733$	−15264.9	−0.769196	−0.384598	+	0.923084i	$0.625660\pi$
$734$	1389.29	0.0698633
$735$	0	0
$736$	15507.4	0.776643
$737$	−61435.7	−3.07057
$738$	0	0
$739$	13906.0	0.692207	0.346103	−	0.938196i	$-0.387505\pi$
$739$	13906.0	0.692207	0.346103	+	0.938196i	$0.387505\pi$
$740$	2809.86	0.139584
$741$	0	0
$742$	−3340.29	−0.165264
$743$	−4592.87	−0.226778	−0.113389	−	0.993551i	$-0.536171\pi$
$743$	−4592.87	−0.226778	−0.113389	+	0.993551i	$0.536171\pi$
$744$	0	0
$745$	17803.0	0.875504
$746$	−1315.22	−0.0645493
$747$	0	0
$748$	−16941.6	−0.828137
$749$	−1701.61	−0.0830111
$750$	0	0
$751$	−8390.80	−0.407702	−0.203851	−	0.979002i	$-0.565346\pi$
$751$	−8390.80	−0.407702	−0.203851	+	0.979002i	$0.565346\pi$
$752$	−1746.35	−0.0846848
$753$	0	0
$754$	−14591.2	−0.704748
$755$	−16309.0	−0.786153
$756$	0	0
$757$	−1368.89	−0.0657240	−0.0328620	−	0.999460i	$-0.510462\pi$
$757$	−1368.89	−0.0657240	−0.0328620	+	0.999460i	$0.510462\pi$
$758$	−1046.88	−0.0501642
$759$	0	0
$760$	317.645	0.0151608
$761$	1623.77	0.0773478	0.0386739	−	0.999252i	$-0.487687\pi$
$761$	1623.77	0.0773478	0.0386739	+	0.999252i	$0.487687\pi$
$762$	0	0
$763$	2836.23	0.134572
$764$	−2953.31	−0.139852
$765$	0	0
$766$	8252.03	0.389240
$767$	20343.8	0.957722
$768$	0	0
$769$	−26842.5	−1.25873	−0.629366	−	0.777109i	$-0.716685\pi$
$769$	−26842.5	−1.25873	−0.629366	+	0.777109i	$0.716685\pi$
$770$	−4132.06	−0.193388
$771$	0	0
$772$	1187.49	0.0553612
$773$	20961.4	0.975330	0.487665	−	0.873031i	$-0.337849\pi$
$773$	20961.4	0.975330	0.487665	+	0.873031i	$0.337849\pi$
$774$	0	0
$775$	−2772.79	−0.128518
$776$	−16757.4	−0.775200
$777$	0	0
$778$	−12620.0	−0.581556
$779$	−483.902	−0.0222562
$780$	0	0
$781$	−23003.5	−1.05394
$782$	9082.86	0.415348
$783$	0	0
$784$	−247.885	−0.0112921
$785$	−14392.3	−0.654373
$786$	0	0
$787$	35333.2	1.60037	0.800187	−	0.599751i	$-0.204734\pi$
$787$	35333.2	1.60037	0.800187	+	0.599751i	$0.204734\pi$
$788$	19430.1	0.878388
$789$	0	0
$790$	4024.32	0.181239
$791$	−197.236	−0.00886589
$792$	0	0
$793$	3726.13	0.166858
$794$	−7541.49	−0.337075
$795$	0	0
$796$	21544.8	0.959342
$797$	8137.04	0.361642	0.180821	−	0.983516i	$-0.442125\pi$
$797$	8137.04	0.361642	0.180821	+	0.983516i	$0.442125\pi$
$798$	0	0
$799$	19450.0	0.861191
$800$	−4397.18	−0.194330
$801$	0	0
$802$	−1832.99	−0.0807049
$803$	−42405.6	−1.86359
$804$	0	0
$805$	−3085.83	−0.135107
$806$	−6558.99	−0.286639
$807$	0	0
$808$	−6218.87	−0.270766
$809$	36281.0	1.57673	0.788364	−	0.615209i	$-0.210928\pi$
$809$	36281.0	1.57673	0.788364	+	0.615209i	$0.210928\pi$
$810$	0	0
$811$	−34237.2	−1.48240	−0.741202	−	0.671282i	$-0.765744\pi$
$811$	−34237.2	−1.48240	−0.741202	+	0.671282i	$0.765744\pi$
$812$	−8043.06	−0.347606
$813$	0	0
$814$	−14246.9	−0.613456
$815$	4637.69	0.199326
$816$	0	0
$817$	1217.10	0.0521188
$818$	−18896.8	−0.807717
$819$	0	0
$820$	4104.42	0.174796
$821$	23247.8	0.988250	0.494125	−	0.869391i	$-0.335489\pi$
$821$	23247.8	0.988250	0.494125	+	0.869391i	$0.335489\pi$
$822$	0	0
$823$	42934.0	1.81845	0.909225	−	0.416306i	$-0.136675\pi$
$823$	42934.0	1.81845	0.909225	+	0.416306i	$0.136675\pi$
$824$	−42596.0	−1.80085
$825$	0	0
$826$	−8050.55	−0.339122
$827$	781.391	0.0328557	0.0164278	−	0.999865i	$-0.494771\pi$
$827$	781.391	0.0328557	0.0164278	+	0.999865i	$0.494771\pi$
$828$	0	0
$829$	−33493.3	−1.40322	−0.701611	−	0.712561i	$-0.747535\pi$
$829$	−33493.3	−1.40322	−0.701611	+	0.712561i	$0.747535\pi$
$830$	497.443	0.0208030
$831$	0	0
$832$	−11710.4	−0.487964
$833$	2760.81	0.114834
$834$	0	0
$835$	5472.59	0.226811
$836$	825.434	0.0341486
$837$	0	0
$838$	5820.27	0.239926
$839$	−15155.7	−0.623639	−0.311819	−	0.950141i	$-0.600938\pi$
$839$	−15155.7	−0.623639	−0.311819	+	0.950141i	$0.600938\pi$
$840$	0	0
$841$	36489.2	1.49613
$842$	12697.2	0.519686
$843$	0	0
$844$	7277.01	0.296783
$845$	5754.60	0.234277
$846$	0	0
$847$	−19866.7	−0.805935
$848$	1320.27	0.0534649
$849$	0	0
$850$	−2575.48	−0.103927
$851$	−10639.6	−0.428579
$852$	0	0
$853$	−2917.48	−0.117107	−0.0585537	−	0.998284i	$-0.518649\pi$
$853$	−2917.48	−0.117107	−0.0585537	+	0.998284i	$0.518649\pi$
$854$	−1474.52	−0.0590832
$855$	0	0
$856$	5625.54	0.224623
$857$	31560.7	1.25799	0.628993	−	0.777411i	$-0.283468\pi$
$857$	31560.7	1.25799	0.628993	+	0.777411i	$0.283468\pi$
$858$	0	0
$859$	−1404.81	−0.0557991	−0.0278995	−	0.999611i	$-0.508882\pi$
$859$	−1404.81	−0.0557991	−0.0278995	+	0.999611i	$0.508882\pi$
$860$	−10323.4	−0.409330
$861$	0	0
$862$	7073.11	0.279479
$863$	9808.24	0.386879	0.193439	−	0.981112i	$-0.438036\pi$
$863$	9808.24	0.386879	0.193439	+	0.981112i	$0.438036\pi$
$864$	0	0
$865$	−8566.27	−0.336719
$866$	11213.7	0.440018
$867$	0	0
$868$	−3615.50	−0.141380
$869$	28422.8	1.10952
$870$	0	0
$871$	30773.9	1.19717
$872$	−9376.63	−0.364143
$873$	0	0
$874$	−442.537	−0.0171271
$875$	875.000	0.0338062
$876$	0	0
$877$	−7196.05	−0.277073	−0.138537	−	0.990357i	$-0.544240\pi$
$877$	−7196.05	−0.277073	−0.138537	+	0.990357i	$0.544240\pi$
$878$	7478.52	0.287458
$879$	0	0
$880$	1633.22	0.0625635
$881$	3183.27	0.121733	0.0608667	−	0.998146i	$-0.480614\pi$
$881$	3183.27	0.121733	0.0608667	+	0.998146i	$0.480614\pi$
$882$	0	0
$883$	25392.5	0.967751	0.483876	−	0.875137i	$-0.339229\pi$
$883$	25392.5	0.967751	0.483876	+	0.875137i	$0.339229\pi$
$884$	8486.25	0.322877
$885$	0	0
$886$	22735.6	0.862095
$887$	30634.2	1.15964	0.579818	−	0.814746i	$-0.303124\pi$
$887$	30634.2	1.15964	0.579818	+	0.814746i	$0.303124\pi$
$888$	0	0
$889$	19186.3	0.723833
$890$	9313.86	0.350788
$891$	0	0
$892$	5757.71	0.216124
$893$	−947.648	−0.0355116
$894$	0	0
$895$	−20327.9	−0.759204
$896$	−5215.58	−0.194465
$897$	0	0
$898$	1614.59	0.0599994
$899$	27365.8	1.01524
$900$	0	0
$901$	−14704.5	−0.543704
$902$	−20810.7	−0.768206
$903$	0	0
$904$	652.067	0.0239905
$905$	13957.0	0.512648
$906$	0	0
$907$	−28089.9	−1.02834	−0.514172	−	0.857687i	$-0.671901\pi$
$907$	−28089.9	−1.02834	−0.514172	+	0.857687i	$0.671901\pi$
$908$	19474.1	0.711753
$909$	0	0
$910$	2069.80	0.0753990
$911$	36102.7	1.31299	0.656495	−	0.754330i	$-0.272038\pi$
$911$	36102.7	1.31299	0.656495	+	0.754330i	$0.272038\pi$
$912$	0	0
$913$	3513.31	0.127353
$914$	16581.0	0.600055
$915$	0	0
$916$	2257.52	0.0814308
$917$	−12824.3	−0.461826
$918$	0	0
$919$	−14533.8	−0.521682	−0.260841	−	0.965382i	$-0.584000\pi$
$919$	−14533.8	−0.521682	−0.260841	+	0.965382i	$0.584000\pi$
$920$	10201.8	0.365591
$921$	0	0
$922$	22871.6	0.816959
$923$	11522.7	0.410915
$924$	0	0
$925$	3016.90	0.107238
$926$	−23200.3	−0.823336
$927$	0	0
$928$	43397.6	1.53512
$929$	−16539.6	−0.584118	−0.292059	−	0.956400i	$-0.594340\pi$
$929$	−16539.6	−0.584118	−0.292059	+	0.956400i	$0.594340\pi$
$930$	0	0
$931$	−134.513	−0.00473522
$932$	9688.79	0.340522
$933$	0	0
$934$	−18533.9	−0.649301
$935$	−18190.0	−0.636231
$936$	0	0
$937$	30212.3	1.05335	0.526677	−	0.850065i	$-0.323438\pi$
$937$	30212.3	1.05335	0.526677	+	0.850065i	$0.323438\pi$
$938$	−12178.0	−0.423908
$939$	0	0
$940$	8037.87	0.278900
$941$	26414.4	0.915074	0.457537	−	0.889191i	$-0.348732\pi$
$941$	26414.4	0.915074	0.457537	+	0.889191i	$0.348732\pi$
$942$	0	0
$943$	−15541.5	−0.536692
$944$	3182.03	0.109710
$945$	0	0
$946$	52342.9	1.79896
$947$	−10187.3	−0.349570	−0.174785	−	0.984607i	$-0.555923\pi$
$947$	−10187.3	−0.349570	−0.174785	+	0.984607i	$0.555923\pi$
$948$	0	0
$949$	21241.5	0.726583
$950$	125.483	0.00428549
$951$	0	0
$952$	−9127.31	−0.310733
$953$	−2211.39	−0.0751669	−0.0375834	−	0.999293i	$-0.511966\pi$
$953$	−2211.39	−0.0751669	−0.0375834	+	0.999293i	$0.511966\pi$
$954$	0	0
$955$	−3170.92	−0.107444
$956$	31732.3	1.07353
$957$	0	0
$958$	−20773.0	−0.700569
$959$	2679.23	0.0902156
$960$	0	0
$961$	−17489.6	−0.587077
$962$	7136.44	0.239177
$963$	0	0
$964$	18263.4	0.610193
$965$	1275.00	0.0425322
$966$	0	0
$967$	7955.89	0.264575	0.132287	−	0.991211i	$-0.457768\pi$
$967$	7955.89	0.264575	0.132287	+	0.991211i	$0.457768\pi$
$968$	65679.6	2.18081
$969$	0	0
$970$	−6619.89	−0.219126
$971$	−53071.2	−1.75400	−0.877001	−	0.480488i	$-0.840459\pi$
$971$	−53071.2	−1.75400	−0.877001	+	0.480488i	$0.840459\pi$
$972$	0	0
$973$	−21375.2	−0.704274
$974$	−14498.6	−0.476965
$975$	0	0
$976$	582.813	0.0191141
$977$	22448.2	0.735089	0.367545	−	0.930006i	$-0.380199\pi$
$977$	22448.2	0.735089	0.367545	+	0.930006i	$0.380199\pi$
$978$	0	0
$979$	65781.4	2.14748
$980$	1140.93	0.0371894
$981$	0	0
$982$	14831.3	0.481961
$983$	−21712.8	−0.704509	−0.352254	−	0.935904i	$-0.614585\pi$
$983$	−21712.8	−0.704509	−0.352254	+	0.935904i	$0.614585\pi$
$984$	0	0
$985$	20861.9	0.674837
$986$	25418.5	0.820983
$987$	0	0
$988$	−413.470	−0.0133140
$989$	39089.7	1.25681
$990$	0	0
$991$	−37849.2	−1.21324	−0.606620	−	0.794992i	$-0.707475\pi$
$991$	−37849.2	−1.21324	−0.606620	+	0.794992i	$0.707475\pi$
$992$	19508.0	0.624373
$993$	0	0
$994$	−4559.82	−0.145502
$995$	23132.4	0.737031
$996$	0	0
$997$	−39573.1	−1.25707	−0.628533	−	0.777783i	$-0.716344\pi$
$997$	−39573.1	−1.25707	−0.628533	+	0.777783i	$0.716344\pi$
$998$	−30748.0	−0.975261
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.4.a.k.1.1		2
3.2	odd	2		105.4.a.e.1.2	✓	2
5.4	even	2		1575.4.a.q.1.2		2
7.6	odd	2		2205.4.a.bb.1.1		2
12.11	even	2		1680.4.a.bo.1.2		2
15.2	even	4		525.4.d.l.274.3		4
15.8	even	4		525.4.d.l.274.2		4
15.14	odd	2		525.4.a.l.1.1		2
21.20	even	2		735.4.a.o.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.e.1.2	✓	2	3.2	odd	2
315.4.a.k.1.1		2	1.1	even	1	trivial
525.4.a.l.1.1		2	15.14	odd	2
525.4.d.l.274.2		4	15.8	even	4
525.4.d.l.274.3		4	15.2	even	4
735.4.a.o.1.2		2	21.20	even	2
1575.4.a.q.1.2		2	5.4	even	2
1680.4.a.bo.1.2		2	12.11	even	2
2205.4.a.bb.1.1		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)

\(f(q)\)	\(=\)	\(q-1.82843 q^{2} -4.65685 q^{4} -5.00000 q^{5} -7.00000 q^{7} +23.1421 q^{8} +O(q^{10})\)	\(q-1.82843 q^{2} -4.65685 q^{4} -5.00000 q^{5} -7.00000 q^{7} +23.1421 q^{8} +9.14214 q^{10} +64.5685 q^{11} -32.3431 q^{13} +12.7990 q^{14} -5.05887 q^{16} +56.3431 q^{17} -2.74517 q^{19} +23.2843 q^{20} -118.059 q^{22} -88.1665 q^{23} +25.0000 q^{25} +59.1371 q^{26} +32.5980 q^{28} -246.735 q^{29} -110.912 q^{31} -175.887 q^{32} -103.019 q^{34} +35.0000 q^{35} +120.676 q^{37} +5.01934 q^{38} -115.711 q^{40} +176.274 q^{41} -443.362 q^{43} -300.686 q^{44} +161.206 q^{46} +345.206 q^{47} +49.0000 q^{49} -45.7107 q^{50} +150.617 q^{52} -260.981 q^{53} -322.843 q^{55} -161.995 q^{56} +451.137 q^{58} -628.999 q^{59} -115.206 q^{61} +202.794 q^{62} +362.068 q^{64} +161.716 q^{65} -951.480 q^{67} -262.382 q^{68} -63.9949 q^{70} -356.264 q^{71} -656.754 q^{73} -220.648 q^{74} +12.7838 q^{76} -451.980 q^{77} +440.195 q^{79} +25.2944 q^{80} -322.304 q^{82} +54.4121 q^{83} -281.716 q^{85} +810.656 q^{86} +1494.25 q^{88} +1018.78 q^{89} +226.402 q^{91} +410.579 q^{92} -631.184 q^{94} +13.7258 q^{95} -724.108 q^{97} -89.5929 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 10 q^{5} - 14 q^{7} + 18 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 10 * q^5 - 14 * q^7 + 18 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 10 q^{5} - 14 q^{7} + 18 q^{8} - 10 q^{10} + 16 q^{11} - 76 q^{13} - 14 q^{14} - 78 q^{16} + 124 q^{17} - 96 q^{19} - 10 q^{20} - 304 q^{22} + 16 q^{23} + 50 q^{25} - 108 q^{26} - 14 q^{28} - 188 q^{29} - 120 q^{31} - 414 q^{32} + 156 q^{34} + 70 q^{35} - 132 q^{37} - 352 q^{38} - 90 q^{40} - 100 q^{41} - 536 q^{43} - 624 q^{44} + 560 q^{46} + 928 q^{47} + 98 q^{49} + 50 q^{50} - 140 q^{52} - 884 q^{53} - 80 q^{55} - 126 q^{56} + 676 q^{58} - 104 q^{59} - 468 q^{61} + 168 q^{62} + 34 q^{64} + 380 q^{65} - 1688 q^{67} + 188 q^{68} + 70 q^{70} + 136 q^{71} + 508 q^{73} - 1188 q^{74} - 608 q^{76} - 112 q^{77} - 432 q^{79} + 390 q^{80} - 1380 q^{82} + 584 q^{83} - 620 q^{85} + 456 q^{86} + 1744 q^{88} + 1404 q^{89} + 532 q^{91} + 1104 q^{92} + 1600 q^{94} + 480 q^{95} - 1188 q^{97} + 98 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 10 * q^5 - 14 * q^7 + 18 * q^8 - 10 * q^10 + 16 * q^11 - 76 * q^13 - 14 * q^14 - 78 * q^16 + 124 * q^17 - 96 * q^19 - 10 * q^20 - 304 * q^22 + 16 * q^23 + 50 * q^25 - 108 * q^26 - 14 * q^28 - 188 * q^29 - 120 * q^31 - 414 * q^32 + 156 * q^34 + 70 * q^35 - 132 * q^37 - 352 * q^38 - 90 * q^40 - 100 * q^41 - 536 * q^43 - 624 * q^44 + 560 * q^46 + 928 * q^47 + 98 * q^49 + 50 * q^50 - 140 * q^52 - 884 * q^53 - 80 * q^55 - 126 * q^56 + 676 * q^58 - 104 * q^59 - 468 * q^61 + 168 * q^62 + 34 * q^64 + 380 * q^65 - 1688 * q^67 + 188 * q^68 + 70 * q^70 + 136 * q^71 + 508 * q^73 - 1188 * q^74 - 608 * q^76 - 112 * q^77 - 432 * q^79 + 390 * q^80 - 1380 * q^82 + 584 * q^83 - 620 * q^85 + 456 * q^86 + 1744 * q^88 + 1404 * q^89 + 532 * q^91 + 1104 * q^92 + 1600 * q^94 + 480 * q^95 - 1188 * q^97 + 98 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more