LMFDB - Embedded newform 950.4.b.j.799.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [950,4,Mod(799,950)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("950.799"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 950 = 2 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	950.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-24,0,-36,0,0,16,0,-136] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$56.0518145055$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.119596096.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} - 2x^{3} + 121x^{2} - 264x + 288 $ x^6 - 2x^5 + 2x^4 - 2x^3 + 121x^2 - 264*x + 288
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 190)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.5
Root			$2.25895 + 2.25895i$ of defining polynomial
Character	$\chi$	$=$	950.799
Dual form			950.4.b.j.799.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000i q^{2} +0.794314i q^{3} -4.00000 q^{4} -1.58863 q^{6} -22.9577i q^{7} -8.00000i q^{8} +26.3691 q^{9} +53.3074 q^{11} -3.17726i q^{12} +73.3488i q^{13} +45.9154 q^{14} +16.0000 q^{16} -139.402i q^{17} +52.7381i q^{18} +19.0000 q^{19} +18.2356 q^{21} +106.615i q^{22} -99.1611i q^{23} +6.35452 q^{24} -146.698 q^{26} +42.3918i q^{27} +91.8308i q^{28} -170.693 q^{29} -274.774 q^{31} +32.0000i q^{32} +42.3428i q^{33} +278.804 q^{34} -105.476 q^{36} -47.0653i q^{37} +38.0000i q^{38} -58.2620 q^{39} +62.0699 q^{41} +36.4713i q^{42} +224.192i q^{43} -213.230 q^{44} +198.322 q^{46} -376.852i q^{47} +12.7090i q^{48} -184.056 q^{49} +110.729 q^{51} -293.395i q^{52} -551.425i q^{53} -84.7836 q^{54} -183.662 q^{56} +15.0920i q^{57} -341.386i q^{58} +100.804 q^{59} +533.726 q^{61} -549.549i q^{62} -605.373i q^{63} -64.0000 q^{64} -84.6856 q^{66} +327.968i q^{67} +557.608i q^{68} +78.7651 q^{69} -344.315 q^{71} -210.953i q^{72} -543.970i q^{73} +94.1306 q^{74} -76.0000 q^{76} -1223.81i q^{77} -116.524i q^{78} -130.821 q^{79} +678.292 q^{81} +124.140i q^{82} -944.539i q^{83} -72.9425 q^{84} -448.385 q^{86} -135.584i q^{87} -426.459i q^{88} +717.584 q^{89} +1683.92 q^{91} +396.644i q^{92} -218.257i q^{93} +753.704 q^{94} -25.4181 q^{96} -190.085i q^{97} -368.111i q^{98} +1405.67 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 24 q^{4} - 36 q^{6} + 16 q^{9} - 136 q^{11} + 44 q^{14} + 96 q^{16} + 114 q^{19} - 542 q^{21} + 144 q^{24} - 484 q^{26} - 182 q^{29} - 308 q^{31} + 412 q^{34} - 64 q^{36} - 922 q^{39} + 564 q^{41}+ \cdots + 4328 q^{99}+O(q^{100}) $ 6 * q - 24 * q^4 - 36 * q^6 + 16 * q^9 - 136 * q^11 + 44 * q^14 + 96 * q^16 + 114 * q^19 - 542 * q^21 + 144 * q^24 - 484 * q^26 - 182 * q^29 - 308 * q^31 + 412 * q^34 - 64 * q^36 - 922 * q^39 + 564 * q^41 + 544 * q^44 - 812 * q^46 - 2120 * q^49 + 1938 * q^51 + 516 * q^54 - 176 * q^56 + 1218 * q^59 + 1584 * q^61 - 384 * q^64 + 1648 * q^66 - 2474 * q^69 - 624 * q^71 + 1216 * q^74 - 456 * q^76 - 3612 * q^79 + 2998 * q^81 + 2168 * q^84 - 856 * q^86 + 2020 * q^89 + 226 * q^91 + 4576 * q^94 - 576 * q^96 + 4328 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/950\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	0.794314i	0.152866i	0.997075	+	0.0764329i	$0.0243531\pi$
$3$	0.794314i	0.152866i	−0.997075	+	0.0764329i	$0.975647\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.58863	−0.108093
$7$	− 22.9577i	− 1.23960i	−0.784760	−	0.619799i	$-0.787214\pi$
$7$	− 22.9577i	− 1.23960i	0.784760	−	0.619799i	$-0.212786\pi$
$8$	− 8.00000i	− 0.353553i
$9$	26.3691	0.976632
$10$	0	0
$11$	53.3074	1.46116	0.730581	−	0.682826i	$-0.239249\pi$
$11$	53.3074	1.46116	0.730581	+	0.682826i	$0.239249\pi$
$12$	− 3.17726i	− 0.0764329i
$13$	73.3488i	1.56487i	0.622732	+	0.782435i	$0.286023\pi$
$13$	73.3488i	1.56487i	−0.622732	+	0.782435i	$0.713977\pi$
$14$	45.9154	0.876529
$15$	0	0
$16$	16.0000	0.250000
$17$	− 139.402i	− 1.98882i	−0.105592	−	0.994410i	$-0.533674\pi$
$17$	− 139.402i	− 1.98882i	0.105592	−	0.994410i	$-0.466326\pi$
$18$	52.7381i	0.690583i
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	18.2356	0.189492
$22$	106.615i	1.03320i
$23$	− 99.1611i	− 0.898979i	−0.893286	−	0.449489i	$-0.851606\pi$
$23$	− 99.1611i	− 0.898979i	0.893286	−	0.449489i	$-0.148394\pi$
$24$	6.35452	0.0540463
$25$	0	0
$26$	−146.698	−1.10653
$27$	42.3918i	0.302160i
$28$	91.8308i	0.619799i
$29$	−170.693	−1.09300	−0.546498	−	0.837461i	$-0.684039\pi$
$29$	−170.693	−1.09300	−0.546498	+	0.837461i	$0.684039\pi$
$30$	0	0
$31$	−274.774	−1.59197	−0.795983	−	0.605319i	$-0.793045\pi$
$31$	−274.774	−1.59197	−0.795983	+	0.605319i	$0.793045\pi$
$32$	32.0000i	0.176777i
$33$	42.3428i	0.223362i
$34$	278.804	1.40631
$35$	0	0
$36$	−105.476	−0.488316
$37$	− 47.0653i	− 0.209121i	−0.994519	−	0.104561i	$-0.966656\pi$
$37$	− 47.0653i	− 0.209121i	0.994519	−	0.104561i	$-0.0333437\pi$
$38$	38.0000i	0.162221i
$39$	−58.2620	−0.239215
$40$	0	0
$41$	62.0699	0.236431	0.118216	−	0.992988i	$-0.462283\pi$
$41$	62.0699	0.236431	0.118216	+	0.992988i	$0.462283\pi$
$42$	36.4713i	0.133991i
$43$	224.192i	0.795093i	0.917582	+	0.397547i	$0.130138\pi$
$43$	224.192i	0.795093i	−0.917582	+	0.397547i	$0.869862\pi$
$44$	−213.230	−0.730581
$45$	0	0
$46$	198.322	0.635674
$47$	− 376.852i	− 1.16956i	−0.811190	−	0.584782i	$-0.801180\pi$
$47$	− 376.852i	− 1.16956i	0.811190	−	0.584782i	$-0.198820\pi$
$48$	12.7090i	0.0382165i
$49$	−184.056	−0.536606
$50$	0	0
$51$	110.729	0.304023
$52$	− 293.395i	− 0.782435i
$53$	− 551.425i	− 1.42913i	−0.699567	−	0.714567i	$-0.746624\pi$
$53$	− 551.425i	− 1.42913i	0.699567	−	0.714567i	$-0.253376\pi$
$54$	−84.7836	−0.213659
$55$	0	0
$56$	−183.662	−0.438264
$57$	15.0920i	0.0350698i
$58$	− 341.386i	− 0.772864i
$59$	100.804	0.222433	0.111217	−	0.993796i	$-0.464525\pi$
$59$	100.804	0.222433	0.111217	+	0.993796i	$0.464525\pi$
$60$	0	0
$61$	533.726	1.12027	0.560136	−	0.828401i	$-0.310749\pi$
$61$	533.726	1.12027	0.560136	+	0.828401i	$0.310749\pi$
$62$	− 549.549i	− 1.12569i
$63$	− 605.373i	− 1.21063i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−84.6856	−0.157941
$67$	327.968i	0.598025i	0.954249	+	0.299012i	$0.0966571\pi$
$67$	327.968i	0.598025i	−0.954249	+	0.299012i	$0.903343\pi$
$68$	557.608i	0.994410i
$69$	78.7651	0.137423
$70$	0	0
$71$	−344.315	−0.575531	−0.287765	−	0.957701i	$-0.592912\pi$
$71$	−344.315	−0.575531	−0.287765	+	0.957701i	$0.592912\pi$
$72$	− 210.953i	− 0.345292i
$73$	− 543.970i	− 0.872149i	−0.899911	−	0.436074i	$-0.856368\pi$
$73$	− 543.970i	− 0.872149i	0.899911	−	0.436074i	$-0.143632\pi$
$74$	94.1306	0.147871
$75$	0	0
$76$	−76.0000	−0.114708
$77$	− 1223.81i	− 1.81125i
$78$	− 116.524i	− 0.169151i
$79$	−130.821	−0.186310	−0.0931548	−	0.995652i	$-0.529695\pi$
$79$	−130.821	−0.186310	−0.0931548	+	0.995652i	$0.529695\pi$
$80$	0	0
$81$	678.292	0.930442
$82$	124.140i	0.167182i
$83$	− 944.539i	− 1.24912i	−0.780979	−	0.624558i	$-0.785279\pi$
$83$	− 944.539i	− 1.24912i	0.780979	−	0.624558i	$-0.214721\pi$
$84$	−72.9425	−0.0947462
$85$	0	0
$86$	−448.385	−0.562216
$87$	− 135.584i	− 0.167082i
$88$	− 426.459i	− 0.516599i
$89$	717.584	0.854648	0.427324	−	0.904098i	$-0.359456\pi$
$89$	717.584	0.854648	0.427324	+	0.904098i	$0.359456\pi$
$90$	0	0
$91$	1683.92	1.93981
$92$	396.644i	0.449489i
$93$	− 218.257i	− 0.243357i
$94$	753.704	0.827007
$95$	0	0
$96$	−25.4181	−0.0270231
$97$	− 190.085i	− 0.198971i	−0.995039	−	0.0994856i	$-0.968280\pi$
$97$	− 190.085i	− 0.198971i	0.995039	−	0.0994856i	$-0.0317197\pi$
$98$	− 368.111i	− 0.379437i
$99$	1405.67	1.42702
$100$	0	0
$101$	−583.088	−0.574450	−0.287225	−	0.957863i	$-0.592733\pi$
$101$	−583.088	−0.574450	−0.287225	+	0.957863i	$0.592733\pi$
$102$	221.458i	0.214976i
$103$	1794.19i	1.71638i	0.513333	+	0.858189i	$0.328410\pi$
$103$	1794.19i	1.71638i	−0.513333	+	0.858189i	$0.671590\pi$
$104$	586.791	0.553265
$105$	0	0
$106$	1102.85	1.01055
$107$	258.601i	0.233644i	0.993153	+	0.116822i	$0.0372706\pi$
$107$	258.601i	0.233644i	−0.993153	+	0.116822i	$0.962729\pi$
$108$	− 169.567i	− 0.151080i
$109$	562.480	0.494273	0.247137	−	0.968981i	$-0.420510\pi$
$109$	562.480	0.494273	0.247137	+	0.968981i	$0.420510\pi$
$110$	0	0
$111$	37.3847	0.0319675
$112$	− 367.323i	− 0.309900i
$113$	706.717i	0.588340i	0.955753	+	0.294170i	$0.0950431\pi$
$113$	706.717i	0.588340i	−0.955753	+	0.294170i	$0.904957\pi$
$114$	−30.1839	−0.0247981
$115$	0	0
$116$	682.771	0.546498
$117$	1934.14i	1.52830i
$118$	201.608i	0.157284i
$119$	−3200.35	−2.46534
$120$	0	0
$121$	1510.68	1.13499
$122$	1067.45i	0.792151i
$123$	49.3030i	0.0361423i
$124$	1099.10	0.795983
$125$	0	0
$126$	1210.75	0.856046
$127$	− 2640.33i	− 1.84481i	−0.386219	−	0.922407i	$-0.626219\pi$
$127$	− 2640.33i	− 1.84481i	0.386219	−	0.922407i	$-0.373781\pi$
$128$	− 128.000i	− 0.0883883i
$129$	−178.079	−0.121543
$130$	0	0
$131$	360.905	0.240705	0.120353	−	0.992731i	$-0.461597\pi$
$131$	360.905	0.240705	0.120353	+	0.992731i	$0.461597\pi$
$132$	− 169.371i	− 0.111681i
$133$	− 436.196i	− 0.284383i
$134$	−655.936	−0.422867
$135$	0	0
$136$	−1115.22	−0.703154
$137$	49.7951i	0.0310532i	0.999879	+	0.0155266i	$0.00494247\pi$
$137$	49.7951i	0.0310532i	−0.999879	+	0.0155266i	$0.995058\pi$
$138$	157.530i	0.0971729i
$139$	−2216.11	−1.35229	−0.676144	−	0.736769i	$-0.736350\pi$
$139$	−2216.11	−1.35229	−0.676144	+	0.736769i	$0.736350\pi$
$140$	0	0
$141$	299.339	0.178787
$142$	− 688.630i	− 0.406962i
$143$	3910.03i	2.28653i
$144$	421.905	0.244158
$145$	0	0
$146$	1087.94	0.616702
$147$	− 146.198i	− 0.0820287i
$148$	188.261i	0.104561i
$149$	3381.39	1.85916	0.929579	−	0.368624i	$-0.120171\pi$
$149$	3381.39	1.85916	0.929579	+	0.368624i	$0.120171\pi$
$150$	0	0
$151$	−653.981	−0.352452	−0.176226	−	0.984350i	$-0.556389\pi$
$151$	−653.981	−0.352452	−0.176226	+	0.984350i	$0.556389\pi$
$152$	− 152.000i	− 0.0811107i
$153$	− 3675.90i	− 1.94234i
$154$	2447.63	1.28075
$155$	0	0
$156$	233.048	0.119608
$157$	− 2927.57i	− 1.48819i	−0.668076	−	0.744093i	$-0.732882\pi$
$157$	− 2927.57i	− 1.48819i	0.668076	−	0.744093i	$-0.267118\pi$
$158$	− 261.641i	− 0.131741i
$159$	438.005	0.218466
$160$	0	0
$161$	−2276.51	−1.11437
$162$	1356.58i	0.657922i
$163$	991.454i	0.476421i	0.971214	+	0.238211i	$0.0765608\pi$
$163$	991.454i	0.476421i	−0.971214	+	0.238211i	$0.923439\pi$
$164$	−248.280	−0.118216
$165$	0	0
$166$	1889.08	0.883258
$167$	− 3679.78i	− 1.70509i	−0.522653	−	0.852545i	$-0.675058\pi$
$167$	− 3679.78i	− 1.70509i	0.522653	−	0.852545i	$-0.324942\pi$
$168$	− 145.885i	− 0.0669957i
$169$	−3183.05	−1.44882
$170$	0	0
$171$	501.012	0.224055
$172$	− 896.770i	− 0.397547i
$173$	− 2017.51i	− 0.886638i	−0.896364	−	0.443319i	$-0.853801\pi$
$173$	− 2017.51i	− 0.886638i	0.896364	−	0.443319i	$-0.146199\pi$
$174$	271.168	0.118145
$175$	0	0
$176$	852.918	0.365290
$177$	80.0700i	0.0340024i
$178$	1435.17i	0.604328i
$179$	3045.55	1.27170	0.635852	−	0.771811i	$-0.280649\pi$
$179$	3045.55	1.27170	0.635852	+	0.771811i	$0.280649\pi$
$180$	0	0
$181$	2374.04	0.974924	0.487462	−	0.873144i	$-0.337923\pi$
$181$	2374.04	0.974924	0.487462	+	0.873144i	$0.337923\pi$
$182$	3367.84i	1.37165i
$183$	423.946i	0.171251i
$184$	−793.289	−0.317837
$185$	0	0
$186$	436.514	0.172080
$187$	− 7431.15i	− 2.90599i
$188$	1507.41i	0.584782i
$189$	973.218	0.374557
$190$	0	0
$191$	−1777.35	−0.673324	−0.336662	−	0.941626i	$-0.609298\pi$
$191$	−1777.35	−0.673324	−0.336662	+	0.941626i	$0.609298\pi$
$192$	− 50.8361i	− 0.0191082i
$193$	2180.19i	0.813128i	0.913622	+	0.406564i	$0.133273\pi$
$193$	2180.19i	0.813128i	−0.913622	+	0.406564i	$0.866727\pi$
$194$	380.170	0.140694
$195$	0	0
$196$	736.223	0.268303
$197$	− 2601.13i	− 0.940724i	−0.882473	−	0.470362i	$-0.844123\pi$
$197$	− 2601.13i	− 0.940724i	0.882473	−	0.470362i	$-0.155877\pi$
$198$	2811.33i	1.00905i
$199$	−412.217	−0.146841	−0.0734204	−	0.997301i	$-0.523391\pi$
$199$	−412.217	−0.146841	−0.0734204	+	0.997301i	$0.523391\pi$
$200$	0	0
$201$	−260.510	−0.0914175
$202$	− 1166.18i	− 0.406198i
$203$	3918.71i	1.35488i
$204$	−442.916	−0.152011
$205$	0	0
$206$	−3588.38	−1.21366
$207$	− 2614.78i	− 0.877972i
$208$	1173.58i	0.391217i
$209$	1012.84	0.335214
$210$	0	0
$211$	−1123.55	−0.366579	−0.183289	−	0.983059i	$-0.558675\pi$
$211$	−1123.55	−0.366579	−0.183289	+	0.983059i	$0.558675\pi$
$212$	2205.70i	0.714567i
$213$	− 273.494i	− 0.0879790i
$214$	−517.201	−0.165211
$215$	0	0
$216$	339.135	0.106830
$217$	6308.18i	1.97340i
$218$	1124.96i	0.349504i
$219$	432.083	0.133322
$220$	0	0
$221$	10225.0	3.11224
$222$	74.7693i	0.0226045i
$223$	− 6066.46i	− 1.82170i	−0.412732	−	0.910852i	$-0.635426\pi$
$223$	− 6066.46i	− 1.82170i	0.412732	−	0.910852i	$-0.364574\pi$
$224$	734.646	0.219132
$225$	0	0
$226$	−1413.43	−0.416019
$227$	2994.36i	0.875518i	0.899092	+	0.437759i	$0.144228\pi$
$227$	2994.36i	0.875518i	−0.899092	+	0.437759i	$0.855772\pi$
$228$	− 60.3679i	− 0.0175349i
$229$	1510.63	0.435917	0.217959	−	0.975958i	$-0.430060\pi$
$229$	1510.63	0.435917	0.217959	+	0.975958i	$0.430060\pi$
$230$	0	0
$231$	972.094	0.276879
$232$	1365.54i	0.386432i
$233$	5805.95i	1.63245i	0.577736	+	0.816223i	$0.303936\pi$
$233$	5805.95i	1.63245i	−0.577736	+	0.816223i	$0.696064\pi$
$234$	−3868.28	−1.08067
$235$	0	0
$236$	−403.216	−0.111217
$237$	− 103.913i	− 0.0284804i
$238$	− 6400.69i	− 1.74326i
$239$	722.022	0.195413	0.0977066	−	0.995215i	$-0.468849\pi$
$239$	722.022	0.195413	0.0977066	+	0.995215i	$0.468849\pi$
$240$	0	0
$241$	5771.57	1.54265	0.771327	−	0.636439i	$-0.219593\pi$
$241$	5771.57	1.54265	0.771327	+	0.636439i	$0.219593\pi$
$242$	3021.35i	0.802562i
$243$	1683.36i	0.444392i
$244$	−2134.90	−0.560136
$245$	0	0
$246$	−98.6060	−0.0255565
$247$	1393.63i	0.359006i
$248$	2198.19i	0.562845i
$249$	750.261	0.190947
$250$	0	0
$251$	727.300	0.182895	0.0914477	−	0.995810i	$-0.470851\pi$
$251$	727.300	0.182895	0.0914477	+	0.995810i	$0.470851\pi$
$252$	2421.49i	0.605316i
$253$	− 5286.02i	− 1.31355i
$254$	5280.66	1.30448
$255$	0	0
$256$	256.000	0.0625000
$257$	2137.23i	0.518743i	0.965778	+	0.259371i	$0.0835154\pi$
$257$	2137.23i	0.518743i	−0.965778	+	0.259371i	$0.916485\pi$
$258$	− 356.159i	− 0.0859436i
$259$	−1080.51	−0.259227
$260$	0	0
$261$	−4501.01	−1.06745
$262$	721.809i	0.170204i
$263$	− 2997.82i	− 0.702865i	−0.936213	−	0.351433i	$-0.885695\pi$
$263$	− 2997.82i	− 0.702865i	0.936213	−	0.351433i	$-0.114305\pi$
$264$	338.743	0.0789703
$265$	0	0
$266$	872.392	0.201089
$267$	569.987i	0.130647i
$268$	− 1311.87i	− 0.299012i
$269$	−2601.91	−0.589743	−0.294872	−	0.955537i	$-0.595277\pi$
$269$	−2601.91	−0.589743	−0.294872	+	0.955537i	$0.595277\pi$
$270$	0	0
$271$	−766.382	−0.171787	−0.0858937	−	0.996304i	$-0.527375\pi$
$271$	−766.382	−0.171787	−0.0858937	+	0.996304i	$0.527375\pi$
$272$	− 2230.43i	− 0.497205i
$273$	1337.56i	0.296531i
$274$	−99.5903	−0.0219579
$275$	0	0
$276$	−315.060	−0.0687116
$277$	− 5989.96i	− 1.29928i	−0.760240	−	0.649642i	$-0.774919\pi$
$277$	− 5989.96i	− 1.29928i	0.760240	−	0.649642i	$-0.225081\pi$
$278$	− 4432.22i	− 0.956212i
$279$	−7245.54	−1.55476
$280$	0	0
$281$	−1442.46	−0.306228	−0.153114	−	0.988209i	$-0.548930\pi$
$281$	−1442.46	−0.306228	−0.153114	+	0.988209i	$0.548930\pi$
$282$	598.678i	0.126421i
$283$	3000.34i	0.630217i	0.949056	+	0.315109i	$0.102041\pi$
$283$	3000.34i	0.630217i	−0.949056	+	0.315109i	$0.897959\pi$
$284$	1377.26	0.287765
$285$	0	0
$286$	−7820.07	−1.61682
$287$	− 1424.98i	− 0.293080i
$288$	843.810i	0.172646i
$289$	−14519.9	−2.95540
$290$	0	0
$291$	150.987	0.0304159
$292$	2175.88i	0.436074i
$293$	− 750.895i	− 0.149719i	−0.997194	−	0.0748597i	$-0.976149\pi$
$293$	− 750.895i	− 0.149719i	0.997194	−	0.0748597i	$-0.0238509\pi$
$294$	292.396	0.0580030
$295$	0	0
$296$	−376.523	−0.0739356
$297$	2259.80i	0.441504i
$298$	6762.79i	1.31462i
$299$	7273.35	1.40679
$300$	0	0
$301$	5146.94	0.985597
$302$	− 1307.96i	− 0.249221i
$303$	− 463.155i	− 0.0878138i
$304$	304.000	0.0573539
$305$	0	0
$306$	7351.79	1.37344
$307$	− 339.570i	− 0.0631279i	−0.999502	−	0.0315640i	$-0.989951\pi$
$307$	− 339.570i	− 0.0631279i	0.999502	−	0.0315640i	$-0.0100488\pi$
$308$	4895.26i	0.905627i
$309$	−1425.15	−0.262376
$310$	0	0
$311$	9239.59	1.68466	0.842330	−	0.538963i	$-0.181184\pi$
$311$	9239.59	1.68466	0.842330	+	0.538963i	$0.181184\pi$
$312$	466.096i	0.0845754i
$313$	713.175i	0.128789i	0.997925	+	0.0643946i	$0.0205116\pi$
$313$	713.175i	0.128789i	−0.997925	+	0.0643946i	$0.979488\pi$
$314$	5855.13	1.05231
$315$	0	0
$316$	523.282	0.0931548
$317$	1145.46i	0.202951i	0.994838	+	0.101475i	$0.0323563\pi$
$317$	1145.46i	0.202951i	−0.994838	+	0.101475i	$0.967644\pi$
$318$	876.010i	0.154479i
$319$	−9099.19	−1.59704
$320$	0	0
$321$	−205.410	−0.0357161
$322$	− 4553.02i	− 0.787981i
$323$	− 2648.64i	− 0.456266i
$324$	−2713.17	−0.465221
$325$	0	0
$326$	−1982.91	−0.336881
$327$	446.786i	0.0755575i
$328$	− 496.559i	− 0.0835911i
$329$	−8651.66	−1.44979
$330$	0	0
$331$	−1030.08	−0.171052	−0.0855261	−	0.996336i	$-0.527257\pi$
$331$	−1030.08	−0.171052	−0.0855261	+	0.996336i	$0.527257\pi$
$332$	3778.15i	0.624558i
$333$	− 1241.07i	− 0.204235i
$334$	7359.57	1.20568
$335$	0	0
$336$	291.770	0.0473731
$337$	7214.68i	1.16620i	0.812401	+	0.583099i	$0.198160\pi$
$337$	7214.68i	1.16620i	−0.812401	+	0.583099i	$0.801840\pi$
$338$	− 6366.11i	− 1.02447i
$339$	−561.356	−0.0899371
$340$	0	0
$341$	−14647.5	−2.32612
$342$	1002.02i	0.158431i
$343$	− 3648.99i	− 0.574423i
$344$	1793.54	0.281108
$345$	0	0
$346$	4035.02	0.626948
$347$	− 1463.81i	− 0.226459i	−0.993569	−	0.113229i	$-0.963880\pi$
$347$	− 1463.81i	− 0.226459i	0.993569	−	0.113229i	$-0.0361195\pi$
$348$	542.335i	0.0835409i
$349$	10644.8	1.63267	0.816336	−	0.577578i	$-0.196002\pi$
$349$	10644.8	1.63267	0.816336	+	0.577578i	$0.196002\pi$
$350$	0	0
$351$	−3109.39	−0.472840
$352$	1705.84i	0.258299i
$353$	3372.46i	0.508492i	0.967140	+	0.254246i	$0.0818273\pi$
$353$	3372.46i	0.508492i	−0.967140	+	0.254246i	$0.918173\pi$
$354$	−160.140	−0.0240433
$355$	0	0
$356$	−2870.33	−0.427324
$357$	− 2542.08i	− 0.376866i
$358$	6091.10i	0.899230i
$359$	10388.6	1.52728	0.763638	−	0.645645i	$-0.223411\pi$
$359$	10388.6	1.52728	0.763638	+	0.645645i	$0.223411\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	4748.08i	0.689375i
$363$	1199.95i	0.173502i
$364$	−6735.68	−0.969906
$365$	0	0
$366$	−847.892	−0.121093
$367$	− 430.531i	− 0.0612358i	−0.999531	−	0.0306179i	$-0.990252\pi$
$367$	− 430.531i	− 0.0612358i	0.999531	−	0.0306179i	$-0.00974751\pi$
$368$	− 1586.58i	− 0.224745i
$369$	1636.72	0.230906
$370$	0	0
$371$	−12659.4	−1.77155
$372$	873.029i	0.121679i
$373$	375.132i	0.0520740i	0.999661	+	0.0260370i	$0.00828878\pi$
$373$	375.132i	0.0520740i	−0.999661	+	0.0260370i	$0.991711\pi$
$374$	14862.3	2.05484
$375$	0	0
$376$	−3014.82	−0.413504
$377$	− 12520.1i	− 1.71040i
$378$	1946.44i	0.264852i
$379$	−2575.31	−0.349036	−0.174518	−	0.984654i	$-0.555837\pi$
$379$	−2575.31	−0.349036	−0.174518	+	0.984654i	$0.555837\pi$
$380$	0	0
$381$	2097.25	0.282009
$382$	− 3554.71i	− 0.476112i
$383$	− 9007.80i	− 1.20177i	−0.799336	−	0.600884i	$-0.794815\pi$
$383$	− 9007.80i	− 1.20177i	0.799336	−	0.600884i	$-0.205185\pi$
$384$	101.672	0.0135116
$385$	0	0
$386$	−4360.39	−0.574968
$387$	5911.74i	0.776514i
$388$	760.340i	0.0994856i
$389$	−3716.86	−0.484453	−0.242227	−	0.970220i	$-0.577878\pi$
$389$	−3716.86	−0.484453	−0.242227	+	0.970220i	$0.577878\pi$
$390$	0	0
$391$	−13823.2	−1.78791
$392$	1472.45i	0.189719i
$393$	286.672i	0.0367956i
$394$	5202.26	0.665193
$395$	0	0
$396$	−5622.66	−0.713509
$397$	14192.2i	1.79418i	0.441851	+	0.897088i	$0.354322\pi$
$397$	14192.2i	1.79418i	−0.441851	+	0.897088i	$0.645678\pi$
$398$	− 824.435i	− 0.103832i
$399$	346.477	0.0434725
$400$	0	0
$401$	−9736.11	−1.21246	−0.606232	−	0.795288i	$-0.707320\pi$
$401$	−9736.11	−1.21246	−0.606232	+	0.795288i	$0.707320\pi$
$402$	− 521.019i	− 0.0646420i
$403$	− 20154.4i	− 2.49122i
$404$	2332.35	0.287225
$405$	0	0
$406$	−7837.43	−0.958042
$407$	− 2508.93i	− 0.305560i
$408$	− 885.831i	− 0.107488i
$409$	−15847.0	−1.91585	−0.957925	−	0.287020i	$-0.907335\pi$
$409$	−15847.0	−1.91585	−0.957925	+	0.287020i	$0.907335\pi$
$410$	0	0
$411$	−39.5530	−0.00474697
$412$	− 7176.77i	− 0.858189i
$413$	− 2314.22i	− 0.275728i
$414$	5229.57	0.620820
$415$	0	0
$416$	−2347.16	−0.276633
$417$	− 1760.29i	− 0.206719i
$418$	2025.68i	0.237032i
$419$	15104.2	1.76107	0.880536	−	0.473979i	$-0.157183\pi$
$419$	15104.2	1.76107	0.880536	+	0.473979i	$0.157183\pi$
$420$	0	0
$421$	1317.72	0.152546	0.0762729	−	0.997087i	$-0.475698\pi$
$421$	1317.72	0.152546	0.0762729	+	0.997087i	$0.475698\pi$
$422$	− 2247.09i	− 0.259210i
$423$	− 9937.24i	− 1.14223i
$424$	−4411.40	−0.505275
$425$	0	0
$426$	546.989	0.0622106
$427$	− 12253.1i	− 1.38869i
$428$	− 1034.40i	− 0.116822i
$429$	−3105.80	−0.349532
$430$	0	0
$431$	−16162.6	−1.80632	−0.903159	−	0.429306i	$-0.858758\pi$
$431$	−16162.6	−1.80632	−0.903159	+	0.429306i	$0.858758\pi$
$432$	678.269i	0.0755399i
$433$	− 10551.5i	− 1.17107i	−0.810647	−	0.585536i	$-0.800884\pi$
$433$	− 10551.5i	− 1.17107i	0.810647	−	0.585536i	$-0.199116\pi$
$434$	−12616.4	−1.39540
$435$	0	0
$436$	−2249.92	−0.247137
$437$	− 1884.06i	− 0.206240i
$438$	864.166i	0.0942727i
$439$	12039.2	1.30889	0.654443	−	0.756112i	$-0.272903\pi$
$439$	12039.2	1.30889	0.654443	+	0.756112i	$0.272903\pi$
$440$	0	0
$441$	−4853.38	−0.524066
$442$	20449.9i	2.20069i
$443$	− 4159.65i	− 0.446120i	−0.974805	−	0.223060i	$-0.928395\pi$
$443$	− 4159.65i	− 0.446120i	0.974805	−	0.223060i	$-0.0716045\pi$
$444$	−149.539	−0.0159838
$445$	0	0
$446$	12132.9	1.28814
$447$	2685.89i	0.284202i
$448$	1469.29i	0.154950i
$449$	−14827.6	−1.55848	−0.779242	−	0.626724i	$-0.784395\pi$
$449$	−14827.6	−1.55848	−0.779242	+	0.626724i	$0.784395\pi$
$450$	0	0
$451$	3308.78	0.345465
$452$	− 2826.87i	− 0.294170i
$453$	− 519.466i	− 0.0538778i
$454$	−5988.72	−0.619085
$455$	0	0
$456$	120.736	0.0123991
$457$	14972.2i	1.53254i	0.642518	+	0.766270i	$0.277890\pi$
$457$	14972.2i	1.53254i	−0.642518	+	0.766270i	$0.722110\pi$
$458$	3021.26i	0.308240i
$459$	5909.50	0.600941
$460$	0	0
$461$	−6764.28	−0.683392	−0.341696	−	0.939810i	$-0.611001\pi$
$461$	−6764.28	−0.683392	−0.341696	+	0.939810i	$0.611001\pi$
$462$	1944.19i	0.195783i
$463$	12092.4i	1.21378i	0.794786	+	0.606890i	$0.207583\pi$
$463$	12092.4i	1.21378i	−0.794786	+	0.606890i	$0.792417\pi$
$464$	−2731.09	−0.273249
$465$	0	0
$466$	−11611.9	−1.15431
$467$	3436.72i	0.340541i	0.985397	+	0.170270i	$0.0544641\pi$
$467$	3436.72i	0.340541i	−0.985397	+	0.170270i	$0.945536\pi$
$468$	− 7736.56i	− 0.764151i
$469$	7529.38	0.741311
$470$	0	0
$471$	2325.41	0.227493
$472$	− 806.431i	− 0.0786419i
$473$	11951.1i	1.16176i
$474$	207.825	0.0201387
$475$	0	0
$476$	12801.4	1.23267
$477$	− 14540.6i	− 1.39574i
$478$	1444.04i	0.138178i
$479$	−13749.1	−1.31151	−0.655756	−	0.754973i	$-0.727650\pi$
$479$	−13749.1	−1.31151	−0.655756	+	0.754973i	$0.727650\pi$
$480$	0	0
$481$	3452.19	0.327248
$482$	11543.1i	1.09082i
$483$	− 1808.26i	− 0.170350i
$484$	−6042.71	−0.567497
$485$	0	0
$486$	−3366.71	−0.314233
$487$	6265.57i	0.582999i	0.956571	+	0.291499i	$0.0941541\pi$
$487$	6265.57i	0.582999i	−0.956571	+	0.291499i	$0.905846\pi$
$488$	− 4269.80i	− 0.396076i
$489$	−787.526	−0.0728285
$490$	0	0
$491$	−184.155	−0.0169263	−0.00846314	−	0.999964i	$-0.502694\pi$
$491$	−184.155	−0.0169263	−0.00846314	+	0.999964i	$0.502694\pi$
$492$	− 197.212i	− 0.0180711i
$493$	23794.9i	2.17377i
$494$	−2787.26	−0.253855
$495$	0	0
$496$	−4396.39	−0.397991
$497$	7904.68i	0.713427i
$498$	1500.52i	0.135020i
$499$	−670.487	−0.0601506	−0.0300753	−	0.999548i	$-0.509575\pi$
$499$	−670.487	−0.0601506	−0.0300753	+	0.999548i	$0.509575\pi$
$500$	0	0
$501$	2922.91	0.260650
$502$	1454.60i	0.129327i
$503$	8875.83i	0.786786i	0.919370	+	0.393393i	$0.128699\pi$
$503$	8875.83i	0.786786i	−0.919370	+	0.393393i	$0.871301\pi$
$504$	−4842.98	−0.428023
$505$	0	0
$506$	10572.0	0.928823
$507$	− 2528.34i	− 0.221475i
$508$	10561.3i	0.922407i
$509$	−12882.1	−1.12179	−0.560893	−	0.827888i	$-0.689542\pi$
$509$	−12882.1	−1.12179	−0.560893	+	0.827888i	$0.689542\pi$
$510$	0	0
$511$	−12488.3	−1.08111
$512$	512.000i	0.0441942i
$513$	805.445i	0.0693202i
$514$	−4274.47	−0.366807
$515$	0	0
$516$	712.317	0.0607713
$517$	− 20089.0i	− 1.70892i
$518$	− 2161.02i	− 0.183301i
$519$	1602.54	0.135537
$520$	0	0
$521$	11785.0	0.990996	0.495498	−	0.868609i	$-0.334986\pi$
$521$	11785.0	0.990996	0.495498	+	0.868609i	$0.334986\pi$
$522$	− 9002.02i	− 0.754804i
$523$	− 51.4550i	− 0.00430204i	−0.999998	−	0.00215102i	$-0.999315\pi$
$523$	− 51.4550i	− 0.00430204i	0.999998	−	0.00215102i	$-0.000684692\pi$
$524$	−1443.62	−0.120353
$525$	0	0
$526$	5995.64	0.497001
$527$	38304.1i	3.16613i
$528$	677.485i	0.0558404i
$529$	2334.08	0.191837
$530$	0	0
$531$	2658.10	0.217235
$532$	1744.78i	0.142192i
$533$	4552.75i	0.369984i
$534$	−1139.97	−0.0923811
$535$	0	0
$536$	2623.74	0.211434
$537$	2419.12i	0.194400i
$538$	− 5203.81i	− 0.417012i
$539$	−9811.53	−0.784067
$540$	0	0
$541$	4414.29	0.350804	0.175402	−	0.984497i	$-0.443877\pi$
$541$	4414.29	0.350804	0.175402	+	0.984497i	$0.443877\pi$
$542$	− 1532.76i	− 0.121472i
$543$	1885.74i	0.149033i
$544$	4460.86	0.351577
$545$	0	0
$546$	−2675.12	−0.209679
$547$	11425.2i	0.893067i	0.894767	+	0.446533i	$0.147342\pi$
$547$	11425.2i	0.893067i	−0.894767	+	0.446533i	$0.852658\pi$
$548$	− 199.181i	− 0.0155266i
$549$	14073.8	1.09409
$550$	0	0
$551$	−3243.16	−0.250750
$552$	− 630.121i	− 0.0485864i
$553$	3003.34i	0.230949i
$554$	11979.9	0.918732
$555$	0	0
$556$	8864.44	0.676144
$557$	19442.2i	1.47898i	0.673166	+	0.739491i	$0.264934\pi$
$557$	19442.2i	1.47898i	−0.673166	+	0.739491i	$0.735066\pi$
$558$	− 14491.1i	− 1.09938i
$559$	−16444.3	−1.24422
$560$	0	0
$561$	5902.67	0.444226
$562$	− 2884.93i	− 0.216536i
$563$	9459.96i	0.708153i	0.935217	+	0.354076i	$0.115205\pi$
$563$	9459.96i	0.708153i	−0.935217	+	0.354076i	$0.884795\pi$
$564$	−1197.36	−0.0893933
$565$	0	0
$566$	−6000.67	−0.445631
$567$	− 15572.0i	− 1.15338i
$568$	2754.52i	0.203481i
$569$	19606.0	1.44451	0.722256	−	0.691626i	$-0.243105\pi$
$569$	19606.0	1.44451	0.722256	+	0.691626i	$0.243105\pi$
$570$	0	0
$571$	5156.19	0.377898	0.188949	−	0.981987i	$-0.439492\pi$
$571$	5156.19	0.377898	0.188949	+	0.981987i	$0.439492\pi$
$572$	− 15640.1i	− 1.14326i
$573$	− 1411.78i	− 0.102928i
$574$	2849.96	0.207239
$575$	0	0
$576$	−1687.62	−0.122079
$577$	− 17093.9i	− 1.23332i	−0.787228	−	0.616662i	$-0.788485\pi$
$577$	− 17093.9i	− 1.23332i	0.787228	−	0.616662i	$-0.211515\pi$
$578$	− 29039.8i	− 2.08978i
$579$	−1731.76	−0.124300
$580$	0	0
$581$	−21684.4	−1.54840
$582$	301.975i	0.0215073i
$583$	− 29395.0i	− 2.08820i
$584$	−4351.76	−0.308351
$585$	0	0
$586$	1501.79	0.105868
$587$	− 10003.2i	− 0.703366i	−0.936119	−	0.351683i	$-0.885610\pi$
$587$	− 10003.2i	− 0.703366i	0.936119	−	0.351683i	$-0.114390\pi$
$588$	584.792i	0.0410143i
$589$	−5220.71	−0.365222
$590$	0	0
$591$	2066.11	0.143805
$592$	− 753.045i	− 0.0522803i
$593$	5615.44i	0.388868i	0.980916	+	0.194434i	$0.0622870\pi$
$593$	5615.44i	0.388868i	−0.980916	+	0.194434i	$0.937713\pi$
$594$	−4519.59	−0.312191
$595$	0	0
$596$	−13525.6	−0.929579
$597$	− 327.430i	− 0.0224470i
$598$	14546.7i	0.994747i
$599$	2294.71	0.156526	0.0782631	−	0.996933i	$-0.475063\pi$
$599$	2294.71	0.156526	0.0782631	+	0.996933i	$0.475063\pi$
$600$	0	0
$601$	−11049.3	−0.749935	−0.374967	−	0.927038i	$-0.622346\pi$
$601$	−11049.3	−0.749935	−0.374967	+	0.927038i	$0.622346\pi$
$602$	10293.9i	0.696922i
$603$	8648.20i	0.584050i
$604$	2615.92	0.176226
$605$	0	0
$606$	926.311	0.0620937
$607$	3081.49i	0.206053i	0.994679	+	0.103026i	$0.0328526\pi$
$607$	3081.49i	0.206053i	−0.994679	+	0.103026i	$0.967147\pi$
$608$	608.000i	0.0405554i
$609$	−3112.69	−0.207114
$610$	0	0
$611$	27641.7	1.83022
$612$	14703.6i	0.971172i
$613$	8913.61i	0.587304i	0.955912	+	0.293652i	$0.0948707\pi$
$613$	8913.61i	0.587304i	−0.955912	+	0.293652i	$0.905129\pi$
$614$	679.140	0.0446382
$615$	0	0
$616$	−9790.52	−0.640375
$617$	17711.0i	1.15562i	0.816170	+	0.577811i	$0.196093\pi$
$617$	17711.0i	1.15562i	−0.816170	+	0.577811i	$0.803907\pi$
$618$	− 2850.31i	− 0.185528i
$619$	9489.11	0.616154	0.308077	−	0.951361i	$-0.400315\pi$
$619$	9489.11	0.616154	0.308077	+	0.951361i	$0.400315\pi$
$620$	0	0
$621$	4203.62	0.271635
$622$	18479.2i	1.19123i
$623$	− 16474.1i	− 1.05942i
$624$	−932.193	−0.0598038
$625$	0	0
$626$	−1426.35	−0.0910677
$627$	804.514i	0.0512427i
$628$	11710.3i	0.744093i
$629$	−6560.99	−0.415905
$630$	0	0
$631$	−8443.93	−0.532722	−0.266361	−	0.963873i	$-0.585821\pi$
$631$	−8443.93	−0.532722	−0.266361	+	0.963873i	$0.585821\pi$
$632$	1046.56i	0.0658704i
$633$	− 892.449i	− 0.0560374i
$634$	−2290.92	−0.143508
$635$	0	0
$636$	−1752.02	−0.109233
$637$	− 13500.3i	− 0.839718i
$638$	− 18198.4i	− 1.12928i
$639$	−9079.27	−0.562082
$640$	0	0
$641$	2397.17	0.147711	0.0738554	−	0.997269i	$-0.476470\pi$
$641$	2397.17	0.147711	0.0738554	+	0.997269i	$0.476470\pi$
$642$	− 410.820i	− 0.0252551i
$643$	19474.3i	1.19439i	0.802097	+	0.597193i	$0.203718\pi$
$643$	19474.3i	1.19439i	−0.802097	+	0.597193i	$0.796282\pi$
$644$	9106.04	0.557187
$645$	0	0
$646$	5297.27	0.322629
$647$	− 27920.1i	− 1.69652i	−0.529577	−	0.848262i	$-0.677649\pi$
$647$	− 27920.1i	− 1.69652i	0.529577	−	0.848262i	$-0.322351\pi$
$648$	− 5426.34i	− 0.328961i
$649$	5373.59	0.325011
$650$	0	0
$651$	−5010.68	−0.301665
$652$	− 3965.81i	− 0.238211i
$653$	465.094i	0.0278722i	0.999903	+	0.0139361i	$0.00443614\pi$
$653$	465.094i	0.0278722i	−0.999903	+	0.0139361i	$0.995564\pi$
$654$	−893.572	−0.0534272
$655$	0	0
$656$	993.118	0.0591079
$657$	− 14344.0i	− 0.851768i
$658$	− 17303.3i	− 1.02516i
$659$	−8363.60	−0.494385	−0.247192	−	0.968966i	$-0.579508\pi$
$659$	−8363.60	−0.494385	−0.247192	+	0.968966i	$0.579508\pi$
$660$	0	0
$661$	27046.1	1.59149	0.795743	−	0.605635i	$-0.207081\pi$
$661$	27046.1	1.59149	0.795743	+	0.605635i	$0.207081\pi$
$662$	− 2060.16i	− 0.120952i
$663$	8121.84i	0.475756i
$664$	−7556.31	−0.441629
$665$	0	0
$666$	2482.14	0.144416
$667$	16926.1i	0.982580i
$668$	14719.1i	0.852545i
$669$	4818.68	0.278477
$670$	0	0
$671$	28451.5	1.63690
$672$	583.540i	0.0334978i
$673$	3938.39i	0.225578i	0.993619	+	0.112789i	$0.0359784\pi$
$673$	3938.39i	0.225578i	−0.993619	+	0.112789i	$0.964022\pi$
$674$	−14429.4	−0.824626
$675$	0	0
$676$	12732.2	0.724409
$677$	− 7796.77i	− 0.442621i	−0.975203	−	0.221310i	$-0.928967\pi$
$677$	− 7796.77i	− 0.442621i	0.975203	−	0.221310i	$-0.0710334\pi$
$678$	− 1122.71i	− 0.0635951i
$679$	−4363.91	−0.246645
$680$	0	0
$681$	−2378.46	−0.133837
$682$	− 29295.0i	− 1.64481i
$683$	− 21022.7i	− 1.17776i	−0.808221	−	0.588880i	$-0.799569\pi$
$683$	− 21022.7i	− 1.17776i	0.808221	−	0.588880i	$-0.200431\pi$
$684$	−2004.05	−0.112027
$685$	0	0
$686$	7297.99	0.406179
$687$	1199.91i	0.0666369i
$688$	3587.08i	0.198773i
$689$	40446.4	2.23641
$690$	0	0
$691$	−4955.79	−0.272832	−0.136416	−	0.990652i	$-0.543558\pi$
$691$	−4955.79	−0.272832	−0.136416	+	0.990652i	$0.543558\pi$
$692$	8070.03i	0.443319i
$693$	− 32270.8i	− 1.76893i
$694$	2927.61	0.160131
$695$	0	0
$696$	−1084.67	−0.0590723
$697$	− 8652.66i	− 0.470219i
$698$	21289.6i	1.15447i
$699$	−4611.75	−0.249545
$700$	0	0
$701$	−34916.0	−1.88126	−0.940628	−	0.339440i	$-0.889762\pi$
$701$	−34916.0	−1.88126	−0.940628	+	0.339440i	$0.889762\pi$
$702$	− 6218.78i	− 0.334349i
$703$	− 894.241i	− 0.0479757i
$704$	−3411.67	−0.182645
$705$	0	0
$706$	−6744.91	−0.359558
$707$	13386.4i	0.712088i
$708$	− 320.280i	− 0.0170012i
$709$	21741.3	1.15164	0.575820	−	0.817576i	$-0.304683\pi$
$709$	21741.3	1.15164	0.575820	+	0.817576i	$0.304683\pi$
$710$	0	0
$711$	−3449.62	−0.181956
$712$	− 5740.67i	− 0.302164i
$713$	27246.9i	1.43114i
$714$	5084.16	0.266485
$715$	0	0
$716$	−12182.2	−0.635852
$717$	573.513i	0.0298720i
$718$	20777.3i	1.07995i
$719$	22241.1	1.15362	0.576810	−	0.816879i	$-0.304297\pi$
$719$	22241.1	1.15362	0.576810	+	0.816879i	$0.304297\pi$
$720$	0	0
$721$	41190.5	2.12762
$722$	722.000i	0.0372161i
$723$	4584.44i	0.235819i
$724$	−9496.17	−0.487462
$725$	0	0
$726$	−2399.90	−0.122684
$727$	2988.44i	0.152456i	0.997090	+	0.0762278i	$0.0242876\pi$
$727$	2988.44i	0.152456i	−0.997090	+	0.0762278i	$0.975712\pi$
$728$	− 13471.4i	− 0.685827i
$729$	16976.8	0.862510
$730$	0	0
$731$	31252.8	1.58130
$732$	− 1695.78i	− 0.0856256i
$733$	12240.9i	0.616816i	0.951254	+	0.308408i	$0.0997962\pi$
$733$	12240.9i	0.616816i	−0.951254	+	0.308408i	$0.900204\pi$
$734$	861.062	0.0433003
$735$	0	0
$736$	3173.15	0.158919
$737$	17483.1i	0.873811i
$738$	3273.45i	0.163276i
$739$	−19051.6	−0.948340	−0.474170	−	0.880433i	$-0.657252\pi$
$739$	−19051.6	−0.948340	−0.474170	+	0.880433i	$0.657252\pi$
$740$	0	0
$741$	−1106.98	−0.0548797
$742$	− 25318.9i	− 1.25268i
$743$	21009.1i	1.03735i	0.854973	+	0.518673i	$0.173574\pi$
$743$	21009.1i	1.03735i	−0.854973	+	0.518673i	$0.826426\pi$
$744$	−1746.06	−0.0860398
$745$	0	0
$746$	−750.264	−0.0368219
$747$	− 24906.6i	− 1.21993i
$748$	29724.6i	1.45299i
$749$	5936.87	0.289624
$750$	0	0
$751$	−17060.4	−0.828951	−0.414475	−	0.910061i	$-0.636035\pi$
$751$	−17060.4	−0.828951	−0.414475	+	0.910061i	$0.636035\pi$
$752$	− 6029.63i	− 0.292391i
$753$	577.705i	0.0279585i
$754$	25040.2	1.20943
$755$	0	0
$756$	−3892.87	−0.187278
$757$	21029.9i	1.00970i	0.863207	+	0.504850i	$0.168452\pi$
$757$	21029.9i	1.00970i	−0.863207	+	0.504850i	$0.831548\pi$
$758$	− 5150.61i	− 0.246806i
$759$	4198.76	0.200798
$760$	0	0
$761$	25781.2	1.22808	0.614040	−	0.789275i	$-0.289543\pi$
$761$	25781.2	1.22808	0.614040	+	0.789275i	$0.289543\pi$
$762$	4194.50i	0.199411i
$763$	− 12913.2i	− 0.612701i
$764$	7109.42	0.336662
$765$	0	0
$766$	18015.6	0.849778
$767$	7393.85i	0.348079i
$768$	203.344i	0.00955412i
$769$	26905.2	1.26167	0.630835	−	0.775917i	$-0.282712\pi$
$769$	26905.2	1.26167	0.630835	+	0.775917i	$0.282712\pi$
$770$	0	0
$771$	−1697.63	−0.0792981
$772$	− 8720.77i	− 0.406564i
$773$	10683.7i	0.497111i	0.968618	+	0.248556i	$0.0799559\pi$
$773$	10683.7i	0.497111i	−0.968618	+	0.248556i	$0.920044\pi$
$774$	−11823.5	−0.549078
$775$	0	0
$776$	−1520.68	−0.0703470
$777$	− 858.266i	− 0.0396269i
$778$	− 7433.72i	− 0.342560i
$779$	1179.33	0.0542411
$780$	0	0
$781$	−18354.5	−0.840943
$782$	− 27646.5i	− 1.26424i
$783$	− 7235.98i	− 0.330259i
$784$	−2944.89	−0.134151
$785$	0	0
$786$	−573.343	−0.0260184
$787$	6017.52i	0.272556i	0.990671	+	0.136278i	$0.0435140\pi$
$787$	6017.52i	0.272556i	−0.990671	+	0.136278i	$0.956486\pi$
$788$	10404.5i	0.470362i
$789$	2381.21	0.107444
$790$	0	0
$791$	16224.6	0.729305
$792$	− 11245.3i	− 0.504527i
$793$	39148.1i	1.75308i
$794$	−28384.5	−1.26867
$795$	0	0
$796$	1648.87	0.0734204
$797$	36703.4i	1.63124i	0.578587	+	0.815621i	$0.303604\pi$
$797$	36703.4i	1.63124i	−0.578587	+	0.815621i	$0.696396\pi$
$798$	692.954i	0.0307397i
$799$	−52533.9	−2.32605
$800$	0	0
$801$	18922.0	0.834677
$802$	− 19472.2i	− 0.857341i
$803$	− 28997.6i	− 1.27435i
$804$	1042.04	0.0457088
$805$	0	0
$806$	40308.8	1.76156
$807$	− 2066.73i	− 0.0901516i
$808$	4664.71i	0.203099i
$809$	−3237.98	−0.140718	−0.0703592	−	0.997522i	$-0.522415\pi$
$809$	−3237.98	−0.140718	−0.0703592	+	0.997522i	$0.522415\pi$
$810$	0	0
$811$	−25947.0	−1.12346	−0.561728	−	0.827322i	$-0.689863\pi$
$811$	−25947.0	−1.12346	−0.561728	+	0.827322i	$0.689863\pi$
$812$	− 15674.9i	− 0.677438i
$813$	− 608.748i	− 0.0262604i
$814$	5017.86	0.216064
$815$	0	0
$816$	1771.66	0.0760056
$817$	4259.66i	0.182407i
$818$	− 31693.9i	− 1.35471i
$819$	44403.4	1.89448
$820$	0	0
$821$	11190.8	0.475713	0.237856	−	0.971300i	$-0.423555\pi$
$821$	11190.8	0.475713	0.237856	+	0.971300i	$0.423555\pi$
$822$	− 79.1060i	− 0.00335662i
$823$	− 14476.1i	− 0.613129i	−0.951850	−	0.306564i	$-0.900821\pi$
$823$	− 14476.1i	− 0.613129i	0.951850	−	0.306564i	$-0.0991795\pi$
$824$	14353.5	0.606832
$825$	0	0
$826$	4628.45	0.194969
$827$	− 21055.8i	− 0.885349i	−0.896682	−	0.442674i	$-0.854030\pi$
$827$	− 21055.8i	− 0.885349i	0.896682	−	0.442674i	$-0.145970\pi$
$828$	10459.1i	0.438986i
$829$	−11156.5	−0.467407	−0.233704	−	0.972308i	$-0.575085\pi$
$829$	−11156.5	−0.467407	−0.233704	+	0.972308i	$0.575085\pi$
$830$	0	0
$831$	4757.91	0.198616
$832$	− 4694.33i	− 0.195609i
$833$	25657.7i	1.06721i
$834$	3520.58	0.146172
$835$	0	0
$836$	−4051.36	−0.167607
$837$	− 11648.2i	− 0.481028i
$838$	30208.4i	1.24527i
$839$	39171.2	1.61185	0.805923	−	0.592021i	$-0.201670\pi$
$839$	39171.2	1.61185	0.805923	+	0.592021i	$0.201670\pi$
$840$	0	0
$841$	4747.05	0.194639
$842$	2635.44i	0.107866i
$843$	− 1145.77i	− 0.0468118i
$844$	4494.18	0.183289
$845$	0	0
$846$	19874.5	0.807682
$847$	− 34681.7i	− 1.40694i
$848$	− 8822.80i	− 0.357283i
$849$	−2383.21	−0.0963388
$850$	0	0
$851$	−4667.05	−0.187996
$852$	1093.98i	0.0439895i
$853$	30337.0i	1.21773i	0.793275	+	0.608863i	$0.208374\pi$
$853$	30337.0i	1.21773i	−0.793275	+	0.608863i	$0.791626\pi$
$854$	24506.2	0.981950
$855$	0	0
$856$	2068.80	0.0826055
$857$	22146.0i	0.882724i	0.897329	+	0.441362i	$0.145504\pi$
$857$	22146.0i	0.882724i	−0.897329	+	0.441362i	$0.854496\pi$
$858$	− 6211.59i	− 0.247157i
$859$	−35168.0	−1.39687	−0.698437	−	0.715671i	$-0.746121\pi$
$859$	−35168.0	−1.39687	−0.698437	+	0.715671i	$0.746121\pi$
$860$	0	0
$861$	1131.88	0.0448020
$862$	− 32325.1i	− 1.27726i
$863$	− 3982.75i	− 0.157097i	−0.996910	−	0.0785483i	$-0.974972\pi$
$863$	− 3982.75i	− 0.157097i	0.996910	−	0.0785483i	$-0.0250285\pi$
$864$	−1356.54	−0.0534148
$865$	0	0
$866$	21103.0	0.828073
$867$	− 11533.4i	− 0.451780i
$868$	− 25232.7i	− 0.986699i
$869$	−6973.70	−0.272229
$870$	0	0
$871$	−24056.1	−0.935831
$872$	− 4499.84i	− 0.174752i
$873$	− 5012.36i	− 0.194322i
$874$	3768.12	0.145834
$875$	0	0
$876$	−1728.33	−0.0666609
$877$	22118.0i	0.851621i	0.904812	+	0.425810i	$0.140011\pi$
$877$	22118.0i	0.851621i	−0.904812	+	0.425810i	$0.859989\pi$
$878$	24078.4i	0.925522i
$879$	596.447	0.0228870
$880$	0	0
$881$	44513.2	1.70225	0.851127	−	0.524959i	$-0.175919\pi$
$881$	44513.2	1.70225	0.851127	+	0.524959i	$0.175919\pi$
$882$	− 9706.75i	− 0.370571i
$883$	14433.7i	0.550093i	0.961431	+	0.275047i	$0.0886933\pi$
$883$	14433.7i	0.550093i	−0.961431	+	0.275047i	$0.911307\pi$
$884$	−40899.9	−1.55612
$885$	0	0
$886$	8319.30	0.315454
$887$	19559.0i	0.740392i	0.928954	+	0.370196i	$0.120709\pi$
$887$	19559.0i	0.740392i	−0.928954	+	0.370196i	$0.879291\pi$
$888$	− 299.077i	− 0.0113022i
$889$	−60615.9	−2.28683
$890$	0	0
$891$	36158.0	1.35953
$892$	24265.8i	0.910852i
$893$	− 7160.19i	− 0.268317i
$894$	−5371.78	−0.200961
$895$	0	0
$896$	−2938.58	−0.109566
$897$	5777.33i	0.215049i
$898$	− 29655.3i	− 1.10201i
$899$	46902.0	1.74001
$900$	0	0
$901$	−76869.7	−2.84229
$902$	6617.57i	0.244280i
$903$	4088.29i	0.150664i
$904$	5653.74	0.208009
$905$	0	0
$906$	1038.93	0.0380974
$907$	− 23674.9i	− 0.866715i	−0.901222	−	0.433358i	$-0.857329\pi$
$907$	− 23674.9i	− 0.866715i	0.901222	−	0.433358i	$-0.142671\pi$
$908$	− 11977.4i	− 0.437759i
$909$	−15375.5	−0.561026
$910$	0	0
$911$	−6452.53	−0.234667	−0.117334	−	0.993093i	$-0.537435\pi$
$911$	−6452.53	−0.234667	−0.117334	+	0.993093i	$0.537435\pi$
$912$	241.472i	0.00876746i
$913$	− 50350.9i	− 1.82516i
$914$	−29944.4	−1.08367
$915$	0	0
$916$	−6042.51	−0.217959
$917$	− 8285.54i	− 0.298378i
$918$	11819.0i	0.424929i
$919$	5928.94	0.212816	0.106408	−	0.994323i	$-0.466065\pi$
$919$	5928.94	0.212816	0.106408	+	0.994323i	$0.466065\pi$
$920$	0	0
$921$	269.725	0.00965010
$922$	− 13528.6i	− 0.483231i
$923$	− 25255.1i	− 0.900631i
$924$	−3888.37	−0.138440
$925$	0	0
$926$	−24184.8	−0.858273
$927$	47311.2i	1.67627i
$928$	− 5462.17i	− 0.193216i
$929$	−39057.4	−1.37937	−0.689684	−	0.724111i	$-0.742250\pi$
$929$	−39057.4	−1.37937	−0.689684	+	0.724111i	$0.742250\pi$
$930$	0	0
$931$	−3497.06	−0.123106
$932$	− 23223.8i	− 0.816223i
$933$	7339.14i	0.257527i
$934$	−6873.44	−0.240799
$935$	0	0
$936$	15473.1	0.540336
$937$	11163.3i	0.389208i	0.980882	+	0.194604i	$0.0623421\pi$
$937$	11163.3i	0.389208i	−0.980882	+	0.194604i	$0.937658\pi$
$938$	15058.8i	0.524186i
$939$	−566.485	−0.0196875
$940$	0	0
$941$	7071.91	0.244992	0.122496	−	0.992469i	$-0.460910\pi$
$941$	7071.91	0.244992	0.122496	+	0.992469i	$0.460910\pi$
$942$	4650.82i	0.160862i
$943$	− 6154.92i	− 0.212547i
$944$	1612.86	0.0556083
$945$	0	0
$946$	−23902.2	−0.821488
$947$	11977.8i	0.411010i	0.978656	+	0.205505i	$0.0658837\pi$
$947$	11977.8i	0.411010i	−0.978656	+	0.205505i	$0.934116\pi$
$948$	415.651i	0.0142402i
$949$	39899.6	1.36480
$950$	0	0
$951$	−909.854	−0.0310242
$952$	25602.8i	0.871629i
$953$	− 10362.5i	− 0.352228i	−0.984370	−	0.176114i	$-0.943647\pi$
$953$	− 10362.5i	− 0.352228i	0.984370	−	0.176114i	$-0.0563527\pi$
$954$	29081.1	0.986935
$955$	0	0
$956$	−2888.09	−0.0977066
$957$	− 7227.62i	− 0.244133i
$958$	− 27498.3i	− 0.927378i
$959$	1143.18	0.0384935
$960$	0	0
$961$	45709.9	1.53435
$962$	6904.37i	0.231399i
$963$	6819.05i	0.228184i
$964$	−23086.3	−0.771327
$965$	0	0
$966$	3616.53	0.120455
$967$	− 47698.3i	− 1.58622i	−0.609079	−	0.793110i	$-0.708461\pi$
$967$	− 47698.3i	− 1.58622i	0.609079	−	0.793110i	$-0.291539\pi$
$968$	− 12085.4i	− 0.401281i
$969$	2103.85	0.0697476
$970$	0	0
$971$	1174.76	0.0388259	0.0194130	−	0.999812i	$-0.493820\pi$
$971$	1174.76	0.0388259	0.0194130	+	0.999812i	$0.493820\pi$
$972$	− 6733.43i	− 0.222196i
$973$	50876.8i	1.67630i
$974$	−12531.1	−0.412242
$975$	0	0
$976$	8539.61	0.280068
$977$	58282.8i	1.90853i	0.298964	+	0.954264i	$0.403359\pi$
$977$	58282.8i	1.90853i	−0.298964	+	0.954264i	$0.596641\pi$
$978$	− 1575.05i	− 0.0514976i
$979$	38252.5	1.24878
$980$	0	0
$981$	14832.1	0.482723
$982$	− 368.310i	− 0.0119687i
$983$	− 49969.4i	− 1.62134i	−0.585504	−	0.810669i	$-0.699103\pi$
$983$	− 49969.4i	− 1.62134i	0.585504	−	0.810669i	$-0.300897\pi$
$984$	394.424	0.0127782
$985$	0	0
$986$	−47589.8	−1.53709
$987$	− 6872.14i	− 0.221624i
$988$	− 5574.51i	− 0.179503i
$989$	22231.2	0.714772
$990$	0	0
$991$	−3449.31	−0.110566	−0.0552830	−	0.998471i	$-0.517606\pi$
$991$	−3449.31	−0.110566	−0.0552830	+	0.998471i	$0.517606\pi$
$992$	− 8792.78i	− 0.281422i
$993$	− 818.207i	− 0.0261481i
$994$	−15809.4	−0.504469
$995$	0	0
$996$	−3001.04	−0.0954736
$997$	− 49274.3i	− 1.56523i	−0.622508	−	0.782614i	$-0.713886\pi$
$997$	− 49274.3i	− 1.56523i	0.622508	−	0.782614i	$-0.286114\pi$
$998$	− 1340.97i	− 0.0425329i
$999$	1995.18	0.0631880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.4.b.j.799.5		6
5.2	odd	4		950.4.a.m.1.2		3
5.3	odd	4		190.4.a.h.1.2	✓	3
5.4	even	2	inner	950.4.b.j.799.2		6
15.8	even	4		1710.4.a.x.1.1		3
20.3	even	4		1520.4.a.p.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.4.a.h.1.2	✓	3	5.3	odd	4
950.4.a.m.1.2		3	5.2	odd	4
950.4.b.j.799.2		6	5.4	even	2	inner
950.4.b.j.799.5		6	1.1	even	1	trivial
1520.4.a.p.1.2		3	20.3	even	4
1710.4.a.x.1.1		3	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	950.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} +0.794314i q^{3} -4.00000 q^{4} -1.58863 q^{6} -22.9577i q^{7} -8.00000i q^{8} +26.3691 q^{9} +53.3074 q^{11} -3.17726i q^{12} +73.3488i q^{13} +45.9154 q^{14} +16.0000 q^{16} -139.402i q^{17} +52.7381i q^{18} +19.0000 q^{19} +18.2356 q^{21} +106.615i q^{22} -99.1611i q^{23} +6.35452 q^{24} -146.698 q^{26} +42.3918i q^{27} +91.8308i q^{28} -170.693 q^{29} -274.774 q^{31} +32.0000i q^{32} +42.3428i q^{33} +278.804 q^{34} -105.476 q^{36} -47.0653i q^{37} +38.0000i q^{38} -58.2620 q^{39} +62.0699 q^{41} +36.4713i q^{42} +224.192i q^{43} -213.230 q^{44} +198.322 q^{46} -376.852i q^{47} +12.7090i q^{48} -184.056 q^{49} +110.729 q^{51} -293.395i q^{52} -551.425i q^{53} -84.7836 q^{54} -183.662 q^{56} +15.0920i q^{57} -341.386i q^{58} +100.804 q^{59} +533.726 q^{61} -549.549i q^{62} -605.373i q^{63} -64.0000 q^{64} -84.6856 q^{66} +327.968i q^{67} +557.608i q^{68} +78.7651 q^{69} -344.315 q^{71} -210.953i q^{72} -543.970i q^{73} +94.1306 q^{74} -76.0000 q^{76} -1223.81i q^{77} -116.524i q^{78} -130.821 q^{79} +678.292 q^{81} +124.140i q^{82} -944.539i q^{83} -72.9425 q^{84} -448.385 q^{86} -135.584i q^{87} -426.459i q^{88} +717.584 q^{89} +1683.92 q^{91} +396.644i q^{92} -218.257i q^{93} +753.704 q^{94} -25.4181 q^{96} -190.085i q^{97} -368.111i q^{98} +1405.67 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 24 q^{4} - 36 q^{6} + 16 q^{9} - 136 q^{11} + 44 q^{14} + 96 q^{16} + 114 q^{19} - 542 q^{21} + 144 q^{24} - 484 q^{26} - 182 q^{29} - 308 q^{31} + 412 q^{34} - 64 q^{36} - 922 q^{39} + 564 q^{41}+ \cdots + 4328 q^{99}+O(q^{100}) \) 6 * q - 24 * q^4 - 36 * q^6 + 16 * q^9 - 136 * q^11 + 44 * q^14 + 96 * q^16 + 114 * q^19 - 542 * q^21 + 144 * q^24 - 484 * q^26 - 182 * q^29 - 308 * q^31 + 412 * q^34 - 64 * q^36 - 922 * q^39 + 564 * q^41 + 544 * q^44 - 812 * q^46 - 2120 * q^49 + 1938 * q^51 + 516 * q^54 - 176 * q^56 + 1218 * q^59 + 1584 * q^61 - 384 * q^64 + 1648 * q^66 - 2474 * q^69 - 624 * q^71 + 1216 * q^74 - 456 * q^76 - 3612 * q^79 + 2998 * q^81 + 2168 * q^84 - 856 * q^86 + 2020 * q^89 + 226 * q^91 + 4576 * q^94 - 576 * q^96 + 4328 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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