LMFDB - Embedded newform 5950.2.a.cc.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5950,2,Mod(1,5950)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5950.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5950 = 2 \cdot 5^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5950.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,7,-2,7,0,-2,-7,7,7,0,-7,-2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$47.5109892027$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 12x^{5} + 17x^{4} + 40x^{3} - 32x^{2} - 40x + 4 $ x^7 - 2x^6 - 12x^5 + 17x^4 + 40x^3 - 32x^2 - 40x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1190)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.58141$ of defining polynomial
Character	$\chi$	$=$	5950.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.58141 q^{3} +1.00000 q^{4} -1.58141 q^{6} -1.00000 q^{7} +1.00000 q^{8} -0.499141 q^{9} +2.47946 q^{11} -1.58141 q^{12} +1.48483 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -0.499141 q^{18} -4.85321 q^{19} +1.58141 q^{21} +2.47946 q^{22} +4.11626 q^{23} -1.58141 q^{24} +1.48483 q^{26} +5.53358 q^{27} -1.00000 q^{28} -3.24982 q^{29} -4.93212 q^{31} +1.00000 q^{32} -3.92105 q^{33} -1.00000 q^{34} -0.499141 q^{36} -8.92252 q^{37} -4.85321 q^{38} -2.34812 q^{39} +9.41904 q^{41} +1.58141 q^{42} -10.3912 q^{43} +2.47946 q^{44} +4.11626 q^{46} +8.26867 q^{47} -1.58141 q^{48} +1.00000 q^{49} +1.58141 q^{51} +1.48483 q^{52} -4.09224 q^{53} +5.53358 q^{54} -1.00000 q^{56} +7.67492 q^{57} -3.24982 q^{58} +9.51108 q^{59} +9.86895 q^{61} -4.93212 q^{62} +0.499141 q^{63} +1.00000 q^{64} -3.92105 q^{66} -15.8048 q^{67} -1.00000 q^{68} -6.50950 q^{69} +11.7828 q^{71} -0.499141 q^{72} -10.4393 q^{73} -8.92252 q^{74} -4.85321 q^{76} -2.47946 q^{77} -2.34812 q^{78} -15.7552 q^{79} -7.25344 q^{81} +9.41904 q^{82} -16.9990 q^{83} +1.58141 q^{84} -10.3912 q^{86} +5.13930 q^{87} +2.47946 q^{88} +15.3955 q^{89} -1.48483 q^{91} +4.11626 q^{92} +7.79970 q^{93} +8.26867 q^{94} -1.58141 q^{96} -14.0489 q^{97} +1.00000 q^{98} -1.23760 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} - 2 q^{3} + 7 q^{4} - 2 q^{6} - 7 q^{7} + 7 q^{8} + 7 q^{9} - 7 q^{11} - 2 q^{12} + q^{13} - 7 q^{14} + 7 q^{16} - 7 q^{17} + 7 q^{18} - 2 q^{19} + 2 q^{21} - 7 q^{22} - 27 q^{23} - 2 q^{24}+ \cdots - 44 q^{99}+O(q^{100}) $ 7 * q + 7 * q^2 - 2 * q^3 + 7 * q^4 - 2 * q^6 - 7 * q^7 + 7 * q^8 + 7 * q^9 - 7 * q^11 - 2 * q^12 + q^13 - 7 * q^14 + 7 * q^16 - 7 * q^17 + 7 * q^18 - 2 * q^19 + 2 * q^21 - 7 * q^22 - 27 * q^23 - 2 * q^24 + q^26 - 17 * q^27 - 7 * q^28 - 10 * q^29 - 18 * q^31 + 7 * q^32 - 8 * q^33 - 7 * q^34 + 7 * q^36 - 12 * q^37 - 2 * q^38 - 25 * q^39 + 17 * q^41 + 2 * q^42 - 7 * q^43 - 7 * q^44 - 27 * q^46 - 8 * q^47 - 2 * q^48 + 7 * q^49 + 2 * q^51 + q^52 - 20 * q^53 - 17 * q^54 - 7 * q^56 - 25 * q^57 - 10 * q^58 + q^61 - 18 * q^62 - 7 * q^63 + 7 * q^64 - 8 * q^66 - 7 * q^67 - 7 * q^68 + 3 * q^69 - 11 * q^71 + 7 * q^72 + 7 * q^73 - 12 * q^74 - 2 * q^76 + 7 * q^77 - 25 * q^78 - 26 * q^79 + 11 * q^81 + 17 * q^82 - 22 * q^83 + 2 * q^84 - 7 * q^86 - 20 * q^87 - 7 * q^88 + 21 * q^89 - q^91 - 27 * q^92 - 25 * q^93 - 8 * q^94 - 2 * q^96 + 19 * q^97 + 7 * q^98 - 44 * q^99

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.58141	−0.913028	−0.456514	−	0.889716i	$-0.650902\pi$
$3$	−1.58141	−0.913028	−0.456514	+	0.889716i	$0.650902\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.58141	−0.645608
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−0.499141	−0.166380
$10$	0	0
$11$	2.47946	0.747587	0.373793	−	0.927512i	$-0.378057\pi$
$11$	2.47946	0.747587	0.373793	+	0.927512i	$0.378057\pi$
$12$	−1.58141	−0.456514
$13$	1.48483	0.411817	0.205909	−	0.978571i	$-0.433985\pi$
$13$	1.48483	0.411817	0.205909	+	0.978571i	$0.433985\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−0.499141	−0.117649
$19$	−4.85321	−1.11340	−0.556702	−	0.830713i	$-0.687933\pi$
$19$	−4.85321	−1.11340	−0.556702	+	0.830713i	$0.687933\pi$
$20$	0	0
$21$	1.58141	0.345092
$22$	2.47946	0.528624
$23$	4.11626	0.858299	0.429150	−	0.903233i	$-0.358813\pi$
$23$	4.11626	0.858299	0.429150	+	0.903233i	$0.358813\pi$
$24$	−1.58141	−0.322804
$25$	0	0
$26$	1.48483	0.291199
$27$	5.53358	1.06494
$28$	−1.00000	−0.188982
$29$	−3.24982	−0.603477	−0.301738	−	0.953391i	$-0.597567\pi$
$29$	−3.24982	−0.603477	−0.301738	+	0.953391i	$0.597567\pi$
$30$	0	0
$31$	−4.93212	−0.885834	−0.442917	−	0.896563i	$-0.646056\pi$
$31$	−4.93212	−0.885834	−0.442917	+	0.896563i	$0.646056\pi$
$32$	1.00000	0.176777
$33$	−3.92105	−0.682568
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−0.499141	−0.0831901
$37$	−8.92252	−1.46685	−0.733427	−	0.679768i	$-0.762080\pi$
$37$	−8.92252	−1.46685	−0.733427	+	0.679768i	$0.762080\pi$
$38$	−4.85321	−0.787295
$39$	−2.34812	−0.376001
$40$	0	0
$41$	9.41904	1.47101	0.735503	−	0.677521i	$-0.236946\pi$
$41$	9.41904	1.47101	0.735503	+	0.677521i	$0.236946\pi$
$42$	1.58141	0.244017
$43$	−10.3912	−1.58465	−0.792325	−	0.610099i	$-0.791130\pi$
$43$	−10.3912	−1.58465	−0.792325	+	0.610099i	$0.791130\pi$
$44$	2.47946	0.373793
$45$	0	0
$46$	4.11626	0.606909
$47$	8.26867	1.20611	0.603055	−	0.797700i	$-0.293950\pi$
$47$	8.26867	1.20611	0.603055	+	0.797700i	$0.293950\pi$
$48$	−1.58141	−0.228257
$49$	1.00000	0.142857
$50$	0	0
$51$	1.58141	0.221442
$52$	1.48483	0.205909
$53$	−4.09224	−0.562112	−0.281056	−	0.959691i	$-0.590685\pi$
$53$	−4.09224	−0.562112	−0.281056	+	0.959691i	$0.590685\pi$
$54$	5.53358	0.753025
$55$	0	0
$56$	−1.00000	−0.133631
$57$	7.67492	1.01657
$58$	−3.24982	−0.426722
$59$	9.51108	1.23824	0.619119	−	0.785297i	$-0.287490\pi$
$59$	9.51108	1.23824	0.619119	+	0.785297i	$0.287490\pi$
$60$	0	0
$61$	9.86895	1.26359	0.631795	−	0.775136i	$-0.282318\pi$
$61$	9.86895	1.26359	0.631795	+	0.775136i	$0.282318\pi$
$62$	−4.93212	−0.626379
$63$	0.499141	0.0628858
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.92105	−0.482648
$67$	−15.8048	−1.93086	−0.965431	−	0.260657i	$-0.916061\pi$
$67$	−15.8048	−1.93086	−0.965431	+	0.260657i	$0.916061\pi$
$68$	−1.00000	−0.121268
$69$	−6.50950	−0.783651
$70$	0	0
$71$	11.7828	1.39836	0.699181	−	0.714944i	$-0.253548\pi$
$71$	11.7828	1.39836	0.699181	+	0.714944i	$0.253548\pi$
$72$	−0.499141	−0.0588243
$73$	−10.4393	−1.22183	−0.610914	−	0.791697i	$-0.709198\pi$
$73$	−10.4393	−1.22183	−0.610914	+	0.791697i	$0.709198\pi$
$74$	−8.92252	−1.03722
$75$	0	0
$76$	−4.85321	−0.556702
$77$	−2.47946	−0.282561
$78$	−2.34812	−0.265873
$79$	−15.7552	−1.77260	−0.886298	−	0.463115i	$-0.846732\pi$
$79$	−15.7552	−1.77260	−0.886298	+	0.463115i	$0.846732\pi$
$80$	0	0
$81$	−7.25344	−0.805937
$82$	9.41904	1.04016
$83$	−16.9990	−1.86588	−0.932940	−	0.360032i	$-0.882766\pi$
$83$	−16.9990	−1.86588	−0.932940	+	0.360032i	$0.882766\pi$
$84$	1.58141	0.172546
$85$	0	0
$86$	−10.3912	−1.12052
$87$	5.13930	0.550991
$88$	2.47946	0.264312
$89$	15.3955	1.63192	0.815961	−	0.578107i	$-0.196208\pi$
$89$	15.3955	1.63192	0.815961	+	0.578107i	$0.196208\pi$
$90$	0	0
$91$	−1.48483	−0.155652
$92$	4.11626	0.429150
$93$	7.79970	0.808791
$94$	8.26867	0.852849
$95$	0	0
$96$	−1.58141	−0.161402
$97$	−14.0489	−1.42645	−0.713227	−	0.700933i	$-0.752767\pi$
$97$	−14.0489	−1.42645	−0.713227	+	0.700933i	$0.752767\pi$
$98$	1.00000	0.101015
$99$	−1.23760	−0.124384
$100$	0	0
$101$	8.31807	0.827679	0.413840	−	0.910350i	$-0.364187\pi$
$101$	8.31807	0.827679	0.413840	+	0.910350i	$0.364187\pi$
$102$	1.58141	0.156583
$103$	16.1282	1.58915	0.794577	−	0.607163i	$-0.207693\pi$
$103$	16.1282	1.58915	0.794577	+	0.607163i	$0.207693\pi$
$104$	1.48483	0.145599
$105$	0	0
$106$	−4.09224	−0.397473
$107$	2.19983	0.212665	0.106333	−	0.994331i	$-0.466089\pi$
$107$	2.19983	0.212665	0.106333	+	0.994331i	$0.466089\pi$
$108$	5.53358	0.532469
$109$	−4.13495	−0.396057	−0.198028	−	0.980196i	$-0.563454\pi$
$109$	−4.13495	−0.396057	−0.198028	+	0.980196i	$0.563454\pi$
$110$	0	0
$111$	14.1102	1.33928
$112$	−1.00000	−0.0944911
$113$	3.67558	0.345769	0.172885	−	0.984942i	$-0.444691\pi$
$113$	3.67558	0.345769	0.172885	+	0.984942i	$0.444691\pi$
$114$	7.67492	0.718822
$115$	0	0
$116$	−3.24982	−0.301738
$117$	−0.741138	−0.0685182
$118$	9.51108	0.875566
$119$	1.00000	0.0916698
$120$	0	0
$121$	−4.85225	−0.441114
$122$	9.86895	0.893493
$123$	−14.8954	−1.34307
$124$	−4.93212	−0.442917
$125$	0	0
$126$	0.499141	0.0444670
$127$	0.539711	0.0478916	0.0239458	−	0.999713i	$-0.492377\pi$
$127$	0.539711	0.0478916	0.0239458	+	0.999713i	$0.492377\pi$
$128$	1.00000	0.0883883
$129$	16.4328	1.44683
$130$	0	0
$131$	−8.07474	−0.705493	−0.352747	−	0.935719i	$-0.614752\pi$
$131$	−8.07474	−0.705493	−0.352747	+	0.935719i	$0.614752\pi$
$132$	−3.92105	−0.341284
$133$	4.85321	0.420827
$134$	−15.8048	−1.36533
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−10.1767	−0.869451	−0.434725	−	0.900563i	$-0.643155\pi$
$137$	−10.1767	−0.869451	−0.434725	+	0.900563i	$0.643155\pi$
$138$	−6.50950	−0.554125
$139$	−17.9150	−1.51953	−0.759767	−	0.650196i	$-0.774687\pi$
$139$	−17.9150	−1.51953	−0.759767	+	0.650196i	$0.774687\pi$
$140$	0	0
$141$	−13.0762	−1.10121
$142$	11.7828	0.988792
$143$	3.68158	0.307869
$144$	−0.499141	−0.0415950
$145$	0	0
$146$	−10.4393	−0.863963
$147$	−1.58141	−0.130433
$148$	−8.92252	−0.733427
$149$	−5.62465	−0.460789	−0.230395	−	0.973097i	$-0.574002\pi$
$149$	−5.62465	−0.460789	−0.230395	+	0.973097i	$0.574002\pi$
$150$	0	0
$151$	17.0093	1.38419	0.692097	−	0.721805i	$-0.256687\pi$
$151$	17.0093	1.38419	0.692097	+	0.721805i	$0.256687\pi$
$152$	−4.85321	−0.393647
$153$	0.499141	0.0403531
$154$	−2.47946	−0.199801
$155$	0	0
$156$	−2.34812	−0.188000
$157$	−8.76768	−0.699737	−0.349869	−	0.936799i	$-0.613774\pi$
$157$	−8.76768	−0.699737	−0.349869	+	0.936799i	$0.613774\pi$
$158$	−15.7552	−1.25341
$159$	6.47151	0.513224
$160$	0	0
$161$	−4.11626	−0.324407
$162$	−7.25344	−0.569884
$163$	−15.9342	−1.24807	−0.624033	−	0.781398i	$-0.714507\pi$
$163$	−15.9342	−1.24807	−0.624033	+	0.781398i	$0.714507\pi$
$164$	9.41904	0.735503
$165$	0	0
$166$	−16.9990	−1.31938
$167$	−5.93877	−0.459555	−0.229778	−	0.973243i	$-0.573800\pi$
$167$	−5.93877	−0.459555	−0.229778	+	0.973243i	$0.573800\pi$
$168$	1.58141	0.122008
$169$	−10.7953	−0.830407
$170$	0	0
$171$	2.42243	0.185248
$172$	−10.3912	−0.792325
$173$	−19.9065	−1.51346	−0.756731	−	0.653726i	$-0.773205\pi$
$173$	−19.9065	−1.51346	−0.756731	+	0.653726i	$0.773205\pi$
$174$	5.13930	0.389609
$175$	0	0
$176$	2.47946	0.186897
$177$	−15.0409	−1.13055
$178$	15.3955	1.15394
$179$	−13.0850	−0.978021	−0.489011	−	0.872278i	$-0.662642\pi$
$179$	−13.0850	−0.978021	−0.489011	+	0.872278i	$0.662642\pi$
$180$	0	0
$181$	5.13376	0.381589	0.190795	−	0.981630i	$-0.438894\pi$
$181$	5.13376	0.381589	0.190795	+	0.981630i	$0.438894\pi$
$182$	−1.48483	−0.110063
$183$	−15.6069	−1.15369
$184$	4.11626	0.303455
$185$	0	0
$186$	7.79970	0.571902
$187$	−2.47946	−0.181316
$188$	8.26867	0.603055
$189$	−5.53358	−0.402509
$190$	0	0
$191$	−19.5123	−1.41186	−0.705932	−	0.708280i	$-0.749471\pi$
$191$	−19.5123	−1.41186	−0.705932	+	0.708280i	$0.749471\pi$
$192$	−1.58141	−0.114128
$193$	−2.08248	−0.149900	−0.0749501	−	0.997187i	$-0.523880\pi$
$193$	−2.08248	−0.149900	−0.0749501	+	0.997187i	$0.523880\pi$
$194$	−14.0489	−1.00865
$195$	0	0
$196$	1.00000	0.0714286
$197$	6.86555	0.489150	0.244575	−	0.969630i	$-0.421352\pi$
$197$	6.86555	0.489150	0.244575	+	0.969630i	$0.421352\pi$
$198$	−1.23760	−0.0879525
$199$	−7.97203	−0.565122	−0.282561	−	0.959249i	$-0.591184\pi$
$199$	−7.97203	−0.565122	−0.282561	+	0.959249i	$0.591184\pi$
$200$	0	0
$201$	24.9939	1.76293
$202$	8.31807	0.585258
$203$	3.24982	0.228093
$204$	1.58141	0.110721
$205$	0	0
$206$	16.1282	1.12370
$207$	−2.05459	−0.142804
$208$	1.48483	0.102954
$209$	−12.0334	−0.832365
$210$	0	0
$211$	9.87489	0.679815	0.339908	−	0.940459i	$-0.389604\pi$
$211$	9.87489	0.679815	0.339908	+	0.940459i	$0.389604\pi$
$212$	−4.09224	−0.281056
$213$	−18.6335	−1.27674
$214$	2.19983	0.150377
$215$	0	0
$216$	5.53358	0.376512
$217$	4.93212	0.334814
$218$	−4.13495	−0.280054
$219$	16.5088	1.11556
$220$	0	0
$221$	−1.48483	−0.0998803
$222$	14.1102	0.947013
$223$	−0.615959	−0.0412477	−0.0206238	−	0.999787i	$-0.506565\pi$
$223$	−0.615959	−0.0412477	−0.0206238	+	0.999787i	$0.506565\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	3.67558	0.244496
$227$	−13.0080	−0.863370	−0.431685	−	0.902025i	$-0.642081\pi$
$227$	−13.0080	−0.863370	−0.431685	+	0.902025i	$0.642081\pi$
$228$	7.67492	0.508284
$229$	16.7881	1.10939	0.554695	−	0.832054i	$-0.312835\pi$
$229$	16.7881	1.10939	0.554695	+	0.832054i	$0.312835\pi$
$230$	0	0
$231$	3.92105	0.257986
$232$	−3.24982	−0.213361
$233$	7.50071	0.491388	0.245694	−	0.969347i	$-0.420984\pi$
$233$	7.50071	0.491388	0.245694	+	0.969347i	$0.420984\pi$
$234$	−0.741138	−0.0484497
$235$	0	0
$236$	9.51108	0.619119
$237$	24.9154	1.61843
$238$	1.00000	0.0648204
$239$	−7.82561	−0.506196	−0.253098	−	0.967441i	$-0.581450\pi$
$239$	−7.82561	−0.506196	−0.253098	+	0.967441i	$0.581450\pi$
$240$	0	0
$241$	−22.5508	−1.45263	−0.726313	−	0.687364i	$-0.758768\pi$
$241$	−22.5508	−1.45263	−0.726313	+	0.687364i	$0.758768\pi$
$242$	−4.85225	−0.311915
$243$	−5.13007	−0.329094
$244$	9.86895	0.631795
$245$	0	0
$246$	−14.8954	−0.949694
$247$	−7.20618	−0.458519
$248$	−4.93212	−0.313190
$249$	26.8824	1.70360
$250$	0	0
$251$	28.6256	1.80683	0.903417	−	0.428764i	$-0.141051\pi$
$251$	28.6256	1.80683	0.903417	+	0.428764i	$0.141051\pi$
$252$	0.499141	0.0314429
$253$	10.2061	0.641653
$254$	0.539711	0.0338645
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.67057	0.603233	0.301617	−	0.953429i	$-0.402474\pi$
$257$	9.67057	0.603233	0.301617	+	0.953429i	$0.402474\pi$
$258$	16.4328	1.02306
$259$	8.92252	0.554419
$260$	0	0
$261$	1.62212	0.100407
$262$	−8.07474	−0.498859
$263$	−10.5324	−0.649457	−0.324728	−	0.945807i	$-0.605273\pi$
$263$	−10.5324	−0.649457	−0.324728	+	0.945807i	$0.605273\pi$
$264$	−3.92105	−0.241324
$265$	0	0
$266$	4.85321	0.297569
$267$	−24.3466	−1.48999
$268$	−15.8048	−0.965431
$269$	−9.77949	−0.596266	−0.298133	−	0.954524i	$-0.596364\pi$
$269$	−9.77949	−0.596266	−0.298133	+	0.954524i	$0.596364\pi$
$270$	0	0
$271$	−18.7381	−1.13826	−0.569130	−	0.822247i	$-0.692720\pi$
$271$	−18.7381	−1.13826	−0.569130	+	0.822247i	$0.692720\pi$
$272$	−1.00000	−0.0606339
$273$	2.34812	0.142115
$274$	−10.1767	−0.614795
$275$	0	0
$276$	−6.50950	−0.391826
$277$	−18.3954	−1.10527	−0.552637	−	0.833422i	$-0.686378\pi$
$277$	−18.3954	−1.10527	−0.552637	+	0.833422i	$0.686378\pi$
$278$	−17.9150	−1.07447
$279$	2.46182	0.147385
$280$	0	0
$281$	−15.4172	−0.919711	−0.459855	−	0.887994i	$-0.652099\pi$
$281$	−15.4172	−0.919711	−0.459855	+	0.887994i	$0.652099\pi$
$282$	−13.0762	−0.778675
$283$	3.20689	0.190630	0.0953150	−	0.995447i	$-0.469614\pi$
$283$	3.20689	0.190630	0.0953150	+	0.995447i	$0.469614\pi$
$284$	11.7828	0.699181
$285$	0	0
$286$	3.68158	0.217696
$287$	−9.41904	−0.555988
$288$	−0.499141	−0.0294121
$289$	1.00000	0.0588235
$290$	0	0
$291$	22.2171	1.30239
$292$	−10.4393	−0.610914
$293$	0.379629	0.0221782	0.0110891	−	0.999939i	$-0.496470\pi$
$293$	0.379629	0.0221782	0.0110891	+	0.999939i	$0.496470\pi$
$294$	−1.58141	−0.0922297
$295$	0	0
$296$	−8.92252	−0.518611
$297$	13.7203	0.796133
$298$	−5.62465	−0.325827
$299$	6.11194	0.353462
$300$	0	0
$301$	10.3912	0.598942
$302$	17.0093	0.978773
$303$	−13.1543	−0.755694
$304$	−4.85321	−0.278351
$305$	0	0
$306$	0.499141	0.0285340
$307$	−5.97695	−0.341123	−0.170561	−	0.985347i	$-0.554558\pi$
$307$	−5.97695	−0.341123	−0.170561	+	0.985347i	$0.554558\pi$
$308$	−2.47946	−0.141281
$309$	−25.5052	−1.45094
$310$	0	0
$311$	−30.5004	−1.72952	−0.864759	−	0.502188i	$-0.832529\pi$
$311$	−30.5004	−1.72952	−0.864759	+	0.502188i	$0.832529\pi$
$312$	−2.34812	−0.132936
$313$	20.5359	1.16076	0.580378	−	0.814347i	$-0.302905\pi$
$313$	20.5359	1.16076	0.580378	+	0.814347i	$0.302905\pi$
$314$	−8.76768	−0.494789
$315$	0	0
$316$	−15.7552	−0.886298
$317$	−9.79660	−0.550232	−0.275116	−	0.961411i	$-0.588716\pi$
$317$	−9.79660	−0.550232	−0.275116	+	0.961411i	$0.588716\pi$
$318$	6.47151	0.362904
$319$	−8.05782	−0.451151
$320$	0	0
$321$	−3.47883	−0.194169
$322$	−4.11626	−0.229390
$323$	4.85321	0.270040
$324$	−7.25344	−0.402969
$325$	0	0
$326$	−15.9342	−0.882516
$327$	6.53906	0.361611
$328$	9.41904	0.520079
$329$	−8.26867	−0.455867
$330$	0	0
$331$	−9.38887	−0.516059	−0.258030	−	0.966137i	$-0.583073\pi$
$331$	−9.38887	−0.516059	−0.258030	+	0.966137i	$0.583073\pi$
$332$	−16.9990	−0.932940
$333$	4.45359	0.244055
$334$	−5.93877	−0.324955
$335$	0	0
$336$	1.58141	0.0862730
$337$	3.89003	0.211904	0.105952	−	0.994371i	$-0.466211\pi$
$337$	3.89003	0.211904	0.105952	+	0.994371i	$0.466211\pi$
$338$	−10.7953	−0.587186
$339$	−5.81260	−0.315697
$340$	0	0
$341$	−12.2290	−0.662238
$342$	2.42243	0.130990
$343$	−1.00000	−0.0539949
$344$	−10.3912	−0.560259
$345$	0	0
$346$	−19.9065	−1.07018
$347$	5.56440	0.298713	0.149356	−	0.988783i	$-0.452280\pi$
$347$	5.56440	0.298713	0.149356	+	0.988783i	$0.452280\pi$
$348$	5.13930	0.275495
$349$	16.9111	0.905231	0.452615	−	0.891706i	$-0.350491\pi$
$349$	16.9111	0.905231	0.452615	+	0.891706i	$0.350491\pi$
$350$	0	0
$351$	8.21641	0.438560
$352$	2.47946	0.132156
$353$	1.79189	0.0953729	0.0476865	−	0.998862i	$-0.484815\pi$
$353$	1.79189	0.0953729	0.0476865	+	0.998862i	$0.484815\pi$
$354$	−15.0409	−0.799416
$355$	0	0
$356$	15.3955	0.815961
$357$	−1.58141	−0.0836971
$358$	−13.0850	−0.691565
$359$	6.81835	0.359859	0.179929	−	0.983680i	$-0.442413\pi$
$359$	6.81835	0.359859	0.179929	+	0.983680i	$0.442413\pi$
$360$	0	0
$361$	4.55366	0.239667
$362$	5.13376	0.269824
$363$	7.67341	0.402749
$364$	−1.48483	−0.0778261
$365$	0	0
$366$	−15.6069	−0.815784
$367$	−25.1760	−1.31418	−0.657089	−	0.753813i	$-0.728212\pi$
$367$	−25.1760	−1.31418	−0.657089	+	0.753813i	$0.728212\pi$
$368$	4.11626	0.214575
$369$	−4.70142	−0.244746
$370$	0	0
$371$	4.09224	0.212458
$372$	7.79970	0.404396
$373$	30.7407	1.59169	0.795846	−	0.605499i	$-0.207026\pi$
$373$	30.7407	1.59169	0.795846	+	0.605499i	$0.207026\pi$
$374$	−2.47946	−0.128210
$375$	0	0
$376$	8.26867	0.426424
$377$	−4.82542	−0.248522
$378$	−5.53358	−0.284617
$379$	−13.1580	−0.675881	−0.337940	−	0.941168i	$-0.609730\pi$
$379$	−13.1580	−0.675881	−0.337940	+	0.941168i	$0.609730\pi$
$380$	0	0
$381$	−0.853504	−0.0437264
$382$	−19.5123	−0.998338
$383$	−25.8537	−1.32106	−0.660532	−	0.750798i	$-0.729669\pi$
$383$	−25.8537	−1.32106	−0.660532	+	0.750798i	$0.729669\pi$
$384$	−1.58141	−0.0807010
$385$	0	0
$386$	−2.08248	−0.105995
$387$	5.18669	0.263654
$388$	−14.0489	−0.713227
$389$	28.0199	1.42067	0.710334	−	0.703865i	$-0.248544\pi$
$389$	28.0199	1.42067	0.710334	+	0.703865i	$0.248544\pi$
$390$	0	0
$391$	−4.11626	−0.208168
$392$	1.00000	0.0505076
$393$	12.7695	0.644135
$394$	6.86555	0.345881
$395$	0	0
$396$	−1.23760	−0.0621918
$397$	22.4810	1.12829	0.564144	−	0.825677i	$-0.309206\pi$
$397$	22.4810	1.12829	0.564144	+	0.825677i	$0.309206\pi$
$398$	−7.97203	−0.399602
$399$	−7.67492	−0.384227
$400$	0	0
$401$	−24.3231	−1.21464	−0.607318	−	0.794459i	$-0.707754\pi$
$401$	−24.3231	−1.21464	−0.607318	+	0.794459i	$0.707754\pi$
$402$	24.9939	1.24658
$403$	−7.32334	−0.364802
$404$	8.31807	0.413840
$405$	0	0
$406$	3.24982	0.161286
$407$	−22.1231	−1.09660
$408$	1.58141	0.0782915
$409$	28.8624	1.42716	0.713578	−	0.700576i	$-0.247073\pi$
$409$	28.8624	1.42716	0.713578	+	0.700576i	$0.247073\pi$
$410$	0	0
$411$	16.0935	0.793833
$412$	16.1282	0.794577
$413$	−9.51108	−0.468010
$414$	−2.05459	−0.100978
$415$	0	0
$416$	1.48483	0.0727997
$417$	28.3310	1.38738
$418$	−12.0334	−0.588571
$419$	16.3304	0.797790	0.398895	−	0.916997i	$-0.369394\pi$
$419$	16.3304	0.797790	0.398895	+	0.916997i	$0.369394\pi$
$420$	0	0
$421$	−26.6939	−1.30098	−0.650491	−	0.759514i	$-0.725437\pi$
$421$	−26.6939	−1.30098	−0.650491	+	0.759514i	$0.725437\pi$
$422$	9.87489	0.480702
$423$	−4.12723	−0.200673
$424$	−4.09224	−0.198737
$425$	0	0
$426$	−18.6335	−0.902794
$427$	−9.86895	−0.477592
$428$	2.19983	0.106333
$429$	−5.82209	−0.281093
$430$	0	0
$431$	−6.02989	−0.290450	−0.145225	−	0.989399i	$-0.546391\pi$
$431$	−6.02989	−0.290450	−0.145225	+	0.989399i	$0.546391\pi$
$432$	5.53358	0.266234
$433$	−16.9902	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$433$	−16.9902	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$434$	4.93212	0.236749
$435$	0	0
$436$	−4.13495	−0.198028
$437$	−19.9771	−0.955633
$438$	16.5088	0.788822
$439$	−20.3664	−0.972035	−0.486018	−	0.873949i	$-0.661551\pi$
$439$	−20.3664	−0.972035	−0.486018	+	0.873949i	$0.661551\pi$
$440$	0	0
$441$	−0.499141	−0.0237686
$442$	−1.48483	−0.0706261
$443$	−12.5581	−0.596654	−0.298327	−	0.954464i	$-0.596429\pi$
$443$	−12.5581	−0.596654	−0.298327	+	0.954464i	$0.596429\pi$
$444$	14.1102	0.669639
$445$	0	0
$446$	−0.615959	−0.0291665
$447$	8.89488	0.420714
$448$	−1.00000	−0.0472456
$449$	25.7519	1.21531	0.607654	−	0.794202i	$-0.292111\pi$
$449$	25.7519	1.21531	0.607654	+	0.794202i	$0.292111\pi$
$450$	0	0
$451$	23.3542	1.09971
$452$	3.67558	0.172885
$453$	−26.8986	−1.26381
$454$	−13.0080	−0.610495
$455$	0	0
$456$	7.67492	0.359411
$457$	28.4616	1.33138	0.665689	−	0.746229i	$-0.268138\pi$
$457$	28.4616	1.33138	0.665689	+	0.746229i	$0.268138\pi$
$458$	16.7881	0.784457
$459$	−5.53358	−0.258285
$460$	0	0
$461$	−3.91476	−0.182329	−0.0911643	−	0.995836i	$-0.529059\pi$
$461$	−3.91476	−0.182329	−0.0911643	+	0.995836i	$0.529059\pi$
$462$	3.92105	0.182424
$463$	24.4445	1.13603	0.568016	−	0.823017i	$-0.307711\pi$
$463$	24.4445	1.13603	0.568016	+	0.823017i	$0.307711\pi$
$464$	−3.24982	−0.150869
$465$	0	0
$466$	7.50071	0.347464
$467$	−15.8270	−0.732385	−0.366192	−	0.930539i	$-0.619339\pi$
$467$	−15.8270	−0.732385	−0.366192	+	0.930539i	$0.619339\pi$
$468$	−0.741138	−0.0342591
$469$	15.8048	0.729798
$470$	0	0
$471$	13.8653	0.638879
$472$	9.51108	0.437783
$473$	−25.7647	−1.18466
$474$	24.9154	1.14440
$475$	0	0
$476$	1.00000	0.0458349
$477$	2.04260	0.0935243
$478$	−7.82561	−0.357935
$479$	34.7707	1.58871	0.794357	−	0.607452i	$-0.207808\pi$
$479$	34.7707	1.58871	0.794357	+	0.607452i	$0.207808\pi$
$480$	0	0
$481$	−13.2484	−0.604076
$482$	−22.5508	−1.02716
$483$	6.50950	0.296192
$484$	−4.85225	−0.220557
$485$	0	0
$486$	−5.13007	−0.232705
$487$	−30.7487	−1.39336	−0.696679	−	0.717383i	$-0.745340\pi$
$487$	−30.7487	−1.39336	−0.696679	+	0.717383i	$0.745340\pi$
$488$	9.86895	0.446747
$489$	25.1986	1.13952
$490$	0	0
$491$	33.6199	1.51724	0.758621	−	0.651532i	$-0.225873\pi$
$491$	33.6199	1.51724	0.758621	+	0.651532i	$0.225873\pi$
$492$	−14.8954	−0.671535
$493$	3.24982	0.146365
$494$	−7.20618	−0.324222
$495$	0	0
$496$	−4.93212	−0.221459
$497$	−11.7828	−0.528531
$498$	26.8824	1.20463
$499$	−31.2932	−1.40087	−0.700437	−	0.713714i	$-0.747012\pi$
$499$	−31.2932	−1.40087	−0.700437	+	0.713714i	$0.747012\pi$
$500$	0	0
$501$	9.39163	0.419587
$502$	28.6256	1.27762
$503$	−18.5131	−0.825459	−0.412730	−	0.910854i	$-0.635425\pi$
$503$	−18.5131	−0.825459	−0.412730	+	0.910854i	$0.635425\pi$
$504$	0.499141	0.0222335
$505$	0	0
$506$	10.2061	0.453717
$507$	17.0718	0.758184
$508$	0.539711	0.0239458
$509$	−16.4622	−0.729674	−0.364837	−	0.931071i	$-0.618875\pi$
$509$	−16.4622	−0.729674	−0.364837	+	0.931071i	$0.618875\pi$
$510$	0	0
$511$	10.4393	0.461807
$512$	1.00000	0.0441942
$513$	−26.8556	−1.18570
$514$	9.67057	0.426550
$515$	0	0
$516$	16.4328	0.723415
$517$	20.5019	0.901672
$518$	8.92252	0.392033
$519$	31.4803	1.38183
$520$	0	0
$521$	−30.6959	−1.34481	−0.672406	−	0.740183i	$-0.734739\pi$
$521$	−30.6959	−1.34481	−0.672406	+	0.740183i	$0.734739\pi$
$522$	1.62212	0.0709982
$523$	18.5329	0.810389	0.405194	−	0.914231i	$-0.367204\pi$
$523$	18.5329	0.810389	0.405194	+	0.914231i	$0.367204\pi$
$524$	−8.07474	−0.352747
$525$	0	0
$526$	−10.5324	−0.459235
$527$	4.93212	0.214846
$528$	−3.92105	−0.170642
$529$	−6.05641	−0.263322
$530$	0	0
$531$	−4.74737	−0.206018
$532$	4.85321	0.210413
$533$	13.9857	0.605786
$534$	−24.3466	−1.05358
$535$	0	0
$536$	−15.8048	−0.682663
$537$	20.6928	0.892961
$538$	−9.77949	−0.421624
$539$	2.47946	0.106798
$540$	0	0
$541$	25.0160	1.07552	0.537761	−	0.843097i	$-0.319270\pi$
$541$	25.0160	1.07552	0.537761	+	0.843097i	$0.319270\pi$
$542$	−18.7381	−0.804872
$543$	−8.11858	−0.348402
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	2.34812	0.100490
$547$	12.0619	0.515730	0.257865	−	0.966181i	$-0.416981\pi$
$547$	12.0619	0.515730	0.257865	+	0.966181i	$0.416981\pi$
$548$	−10.1767	−0.434725
$549$	−4.92599	−0.210236
$550$	0	0
$551$	15.7721	0.671913
$552$	−6.50950	−0.277063
$553$	15.7552	0.669978
$554$	−18.3954	−0.781547
$555$	0	0
$556$	−17.9150	−0.759767
$557$	17.5136	0.742073	0.371037	−	0.928618i	$-0.379002\pi$
$557$	17.5136	0.742073	0.371037	+	0.928618i	$0.379002\pi$
$558$	2.46182	0.104217
$559$	−15.4292	−0.652586
$560$	0	0
$561$	3.92105	0.165547
$562$	−15.4172	−0.650334
$563$	−17.5043	−0.737720	−0.368860	−	0.929485i	$-0.620252\pi$
$563$	−17.5043	−0.737720	−0.368860	+	0.929485i	$0.620252\pi$
$564$	−13.0762	−0.550606
$565$	0	0
$566$	3.20689	0.134796
$567$	7.25344	0.304616
$568$	11.7828	0.494396
$569$	35.7406	1.49832	0.749162	−	0.662387i	$-0.230457\pi$
$569$	35.7406	1.49832	0.749162	+	0.662387i	$0.230457\pi$
$570$	0	0
$571$	−27.4858	−1.15024	−0.575122	−	0.818067i	$-0.695046\pi$
$571$	−27.4858	−1.15024	−0.575122	+	0.818067i	$0.695046\pi$
$572$	3.68158	0.153935
$573$	30.8570	1.28907
$574$	−9.41904	−0.393143
$575$	0	0
$576$	−0.499141	−0.0207975
$577$	18.4175	0.766730	0.383365	−	0.923597i	$-0.374765\pi$
$577$	18.4175	0.766730	0.383365	+	0.923597i	$0.374765\pi$
$578$	1.00000	0.0415945
$579$	3.29326	0.136863
$580$	0	0
$581$	16.9990	0.705236
$582$	22.2171	0.920930
$583$	−10.1466	−0.420228
$584$	−10.4393	−0.431981
$585$	0	0
$586$	0.379629	0.0156823
$587$	39.8719	1.64569	0.822845	−	0.568265i	$-0.192385\pi$
$587$	39.8719	1.64569	0.822845	+	0.568265i	$0.192385\pi$
$588$	−1.58141	−0.0652163
$589$	23.9366	0.986291
$590$	0	0
$591$	−10.8572	−0.446608
$592$	−8.92252	−0.366713
$593$	15.8586	0.651234	0.325617	−	0.945502i	$-0.394428\pi$
$593$	15.8586	0.651234	0.325617	+	0.945502i	$0.394428\pi$
$594$	13.7203	0.562951
$595$	0	0
$596$	−5.62465	−0.230395
$597$	12.6071	0.515972
$598$	6.11194	0.249936
$599$	6.37001	0.260272	0.130136	−	0.991496i	$-0.458459\pi$
$599$	6.37001	0.260272	0.130136	+	0.991496i	$0.458459\pi$
$600$	0	0
$601$	4.47924	0.182712	0.0913559	−	0.995818i	$-0.470880\pi$
$601$	4.47924	0.182712	0.0913559	+	0.995818i	$0.470880\pi$
$602$	10.3912	0.423516
$603$	7.88881	0.321257
$604$	17.0093	0.692097
$605$	0	0
$606$	−13.1543	−0.534356
$607$	−5.46432	−0.221790	−0.110895	−	0.993832i	$-0.535372\pi$
$607$	−5.46432	−0.221790	−0.110895	+	0.993832i	$0.535372\pi$
$608$	−4.85321	−0.196824
$609$	−5.13930	−0.208255
$610$	0	0
$611$	12.2776	0.496697
$612$	0.499141	0.0201766
$613$	29.5706	1.19435	0.597173	−	0.802112i	$-0.296290\pi$
$613$	29.5706	1.19435	0.597173	+	0.802112i	$0.296290\pi$
$614$	−5.97695	−0.241210
$615$	0	0
$616$	−2.47946	−0.0999005
$617$	3.08987	0.124394	0.0621968	−	0.998064i	$-0.480189\pi$
$617$	3.08987	0.124394	0.0621968	+	0.998064i	$0.480189\pi$
$618$	−25.5052	−1.02597
$619$	−33.1577	−1.33272	−0.666361	−	0.745629i	$-0.732149\pi$
$619$	−33.1577	−1.33272	−0.666361	+	0.745629i	$0.732149\pi$
$620$	0	0
$621$	22.7776	0.914035
$622$	−30.5004	−1.22295
$623$	−15.3955	−0.616808
$624$	−2.34812	−0.0940001
$625$	0	0
$626$	20.5359	0.820778
$627$	19.0297	0.759973
$628$	−8.76768	−0.349869
$629$	8.92252	0.355764
$630$	0	0
$631$	−9.39738	−0.374104	−0.187052	−	0.982350i	$-0.559893\pi$
$631$	−9.39738	−0.374104	−0.187052	+	0.982350i	$0.559893\pi$
$632$	−15.7552	−0.626707
$633$	−15.6162	−0.620690
$634$	−9.79660	−0.389073
$635$	0	0
$636$	6.47151	0.256612
$637$	1.48483	0.0588310
$638$	−8.05782	−0.319012
$639$	−5.88128	−0.232660
$640$	0	0
$641$	41.7840	1.65037	0.825185	−	0.564862i	$-0.191071\pi$
$641$	41.7840	1.65037	0.825185	+	0.564862i	$0.191071\pi$
$642$	−3.47883	−0.137298
$643$	−0.121038	−0.00477329	−0.00238664	−	0.999997i	$-0.500760\pi$
$643$	−0.121038	−0.00477329	−0.00238664	+	0.999997i	$0.500760\pi$
$644$	−4.11626	−0.162203
$645$	0	0
$646$	4.85321	0.190947
$647$	15.0998	0.593633	0.296817	−	0.954934i	$-0.404075\pi$
$647$	15.0998	0.593633	0.296817	+	0.954934i	$0.404075\pi$
$648$	−7.25344	−0.284942
$649$	23.5824	0.925690
$650$	0	0
$651$	−7.79970	−0.305694
$652$	−15.9342	−0.624033
$653$	−30.3759	−1.18870	−0.594350	−	0.804206i	$-0.702591\pi$
$653$	−30.3759	−1.18870	−0.594350	+	0.804206i	$0.702591\pi$
$654$	6.53906	0.255698
$655$	0	0
$656$	9.41904	0.367752
$657$	5.21068	0.203288
$658$	−8.26867	−0.322346
$659$	−26.6257	−1.03719	−0.518596	−	0.855020i	$-0.673545\pi$
$659$	−26.6257	−1.03719	−0.518596	+	0.855020i	$0.673545\pi$
$660$	0	0
$661$	−37.2936	−1.45055	−0.725276	−	0.688458i	$-0.758288\pi$
$661$	−37.2936	−1.45055	−0.725276	+	0.688458i	$0.758288\pi$
$662$	−9.38887	−0.364909
$663$	2.34812	0.0911935
$664$	−16.9990	−0.659688
$665$	0	0
$666$	4.45359	0.172573
$667$	−13.3771	−0.517964
$668$	−5.93877	−0.229778
$669$	0.974084	0.0376603
$670$	0	0
$671$	24.4697	0.944643
$672$	1.58141	0.0610042
$673$	−20.4641	−0.788835	−0.394417	−	0.918931i	$-0.629054\pi$
$673$	−20.4641	−0.788835	−0.394417	+	0.918931i	$0.629054\pi$
$674$	3.89003	0.149838
$675$	0	0
$676$	−10.7953	−0.415203
$677$	−35.0668	−1.34773	−0.673864	−	0.738856i	$-0.735366\pi$
$677$	−35.0668	−1.34773	−0.673864	+	0.738856i	$0.735366\pi$
$678$	−5.81260	−0.223231
$679$	14.0489	0.539149
$680$	0	0
$681$	20.5709	0.788280
$682$	−12.2290	−0.468273
$683$	−9.05059	−0.346311	−0.173156	−	0.984894i	$-0.555396\pi$
$683$	−9.05059	−0.346311	−0.173156	+	0.984894i	$0.555396\pi$
$684$	2.42243	0.0926241
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	−26.5489	−1.01290
$688$	−10.3912	−0.396163
$689$	−6.07627	−0.231487
$690$	0	0
$691$	−38.3558	−1.45912	−0.729562	−	0.683914i	$-0.760276\pi$
$691$	−38.3558	−1.45912	−0.729562	+	0.683914i	$0.760276\pi$
$692$	−19.9065	−0.756731
$693$	1.23760	0.0470126
$694$	5.56440	0.211222
$695$	0	0
$696$	5.13930	0.194805
$697$	−9.41904	−0.356772
$698$	16.9111	0.640095
$699$	−11.8617	−0.448651
$700$	0	0
$701$	−38.0045	−1.43541	−0.717706	−	0.696347i	$-0.754808\pi$
$701$	−38.0045	−1.43541	−0.717706	+	0.696347i	$0.754808\pi$
$702$	8.21641	0.310108
$703$	43.3029	1.63320
$704$	2.47946	0.0934483
$705$	0	0
$706$	1.79189	0.0674389
$707$	−8.31807	−0.312833
$708$	−15.0409	−0.565273
$709$	31.3781	1.17843	0.589214	−	0.807977i	$-0.299438\pi$
$709$	31.3781	1.17843	0.589214	+	0.807977i	$0.299438\pi$
$710$	0	0
$711$	7.86405	0.294925
$712$	15.3955	0.576971
$713$	−20.3019	−0.760311
$714$	−1.58141	−0.0591828
$715$	0	0
$716$	−13.0850	−0.489011
$717$	12.3755	0.462171
$718$	6.81835	0.254459
$719$	41.6679	1.55395	0.776975	−	0.629531i	$-0.216753\pi$
$719$	41.6679	1.55395	0.776975	+	0.629531i	$0.216753\pi$
$720$	0	0
$721$	−16.1282	−0.600644
$722$	4.55366	0.169470
$723$	35.6621	1.32629
$724$	5.13376	0.190795
$725$	0	0
$726$	7.67341	0.284787
$727$	29.0321	1.07674	0.538371	−	0.842708i	$-0.319040\pi$
$727$	29.0321	1.07674	0.538371	+	0.842708i	$0.319040\pi$
$728$	−1.48483	−0.0550314
$729$	29.8731	1.10641
$730$	0	0
$731$	10.3912	0.384334
$732$	−15.6069	−0.576846
$733$	−14.3941	−0.531659	−0.265829	−	0.964020i	$-0.585646\pi$
$733$	−14.3941	−0.531659	−0.265829	+	0.964020i	$0.585646\pi$
$734$	−25.1760	−0.929264
$735$	0	0
$736$	4.11626	0.151727
$737$	−39.1874	−1.44349
$738$	−4.70142	−0.173062
$739$	−23.5565	−0.866539	−0.433270	−	0.901264i	$-0.642640\pi$
$739$	−23.5565	−0.866539	−0.433270	+	0.901264i	$0.642640\pi$
$740$	0	0
$741$	11.3959	0.418640
$742$	4.09224	0.150231
$743$	10.5929	0.388614	0.194307	−	0.980941i	$-0.437754\pi$
$743$	10.5929	0.388614	0.194307	+	0.980941i	$0.437754\pi$
$744$	7.79970	0.285951
$745$	0	0
$746$	30.7407	1.12550
$747$	8.48488	0.310446
$748$	−2.47946	−0.0906582
$749$	−2.19983	−0.0803799
$750$	0	0
$751$	−11.2811	−0.411653	−0.205827	−	0.978588i	$-0.565988\pi$
$751$	−11.2811	−0.411653	−0.205827	+	0.978588i	$0.565988\pi$
$752$	8.26867	0.301528
$753$	−45.2689	−1.64969
$754$	−4.82542	−0.175732
$755$	0	0
$756$	−5.53358	−0.201254
$757$	−13.9885	−0.508420	−0.254210	−	0.967149i	$-0.581815\pi$
$757$	−13.9885	−0.508420	−0.254210	+	0.967149i	$0.581815\pi$
$758$	−13.1580	−0.477920
$759$	−16.1401	−0.585847
$760$	0	0
$761$	23.9575	0.868460	0.434230	−	0.900802i	$-0.357021\pi$
$761$	23.9575	0.868460	0.434230	+	0.900802i	$0.357021\pi$
$762$	−0.853504	−0.0309192
$763$	4.13495	0.149695
$764$	−19.5123	−0.705932
$765$	0	0
$766$	−25.8537	−0.934133
$767$	14.1223	0.509927
$768$	−1.58141	−0.0570642
$769$	14.8024	0.533789	0.266895	−	0.963726i	$-0.414002\pi$
$769$	14.8024	0.533789	0.266895	+	0.963726i	$0.414002\pi$
$770$	0	0
$771$	−15.2931	−0.550769
$772$	−2.08248	−0.0749501
$773$	−13.8809	−0.499262	−0.249631	−	0.968341i	$-0.580309\pi$
$773$	−13.8809	−0.499262	−0.249631	+	0.968341i	$0.580309\pi$
$774$	5.18669	0.186432
$775$	0	0
$776$	−14.0489	−0.504327
$777$	−14.1102	−0.506200
$778$	28.0199	1.00456
$779$	−45.7126	−1.63782
$780$	0	0
$781$	29.2151	1.04540
$782$	−4.11626	−0.147197
$783$	−17.9831	−0.642665
$784$	1.00000	0.0357143
$785$	0	0
$786$	12.7695	0.455472
$787$	−12.2264	−0.435823	−0.217912	−	0.975968i	$-0.569924\pi$
$787$	−12.2264	−0.435823	−0.217912	+	0.975968i	$0.569924\pi$
$788$	6.86555	0.244575
$789$	16.6561	0.592972
$790$	0	0
$791$	−3.67558	−0.130688
$792$	−1.23760	−0.0439763
$793$	14.6537	0.520368
$794$	22.4810	0.797820
$795$	0	0
$796$	−7.97203	−0.282561
$797$	32.2049	1.14076	0.570378	−	0.821382i	$-0.306797\pi$
$797$	32.2049	1.14076	0.570378	+	0.821382i	$0.306797\pi$
$798$	−7.67492	−0.271689
$799$	−8.26867	−0.292525
$800$	0	0
$801$	−7.68453	−0.271519
$802$	−24.3231	−0.858877
$803$	−25.8839	−0.913422
$804$	24.9939	0.881466
$805$	0	0
$806$	−7.32334	−0.257954
$807$	15.4654	0.544408
$808$	8.31807	0.292629
$809$	5.71907	0.201072	0.100536	−	0.994933i	$-0.467944\pi$
$809$	5.71907	0.201072	0.100536	+	0.994933i	$0.467944\pi$
$810$	0	0
$811$	53.4561	1.87710	0.938549	−	0.345146i	$-0.112171\pi$
$811$	53.4561	1.87710	0.938549	+	0.345146i	$0.112171\pi$
$812$	3.24982	0.114046
$813$	29.6327	1.03926
$814$	−22.1231	−0.775414
$815$	0	0
$816$	1.58141	0.0553604
$817$	50.4309	1.76435
$818$	28.8624	1.00915
$819$	0.741138	0.0258975
$820$	0	0
$821$	46.8617	1.63548	0.817742	−	0.575585i	$-0.195226\pi$
$821$	46.8617	1.63548	0.817742	+	0.575585i	$0.195226\pi$
$822$	16.0935	0.561325
$823$	−42.3243	−1.47533	−0.737666	−	0.675166i	$-0.764072\pi$
$823$	−42.3243	−1.47533	−0.737666	+	0.675166i	$0.764072\pi$
$824$	16.1282	0.561851
$825$	0	0
$826$	−9.51108	−0.330933
$827$	50.2887	1.74871	0.874354	−	0.485289i	$-0.161285\pi$
$827$	50.2887	1.74871	0.874354	+	0.485289i	$0.161285\pi$
$828$	−2.05459	−0.0714020
$829$	27.5620	0.957269	0.478634	−	0.878014i	$-0.341132\pi$
$829$	27.5620	0.957269	0.478634	+	0.878014i	$0.341132\pi$
$830$	0	0
$831$	29.0907	1.00915
$832$	1.48483	0.0514771
$833$	−1.00000	−0.0346479
$834$	28.3310	0.981023
$835$	0	0
$836$	−12.0334	−0.416183
$837$	−27.2922	−0.943358
$838$	16.3304	0.564123
$839$	14.6191	0.504707	0.252353	−	0.967635i	$-0.418795\pi$
$839$	14.6191	0.504707	0.252353	+	0.967635i	$0.418795\pi$
$840$	0	0
$841$	−18.4387	−0.635816
$842$	−26.6939	−0.919934
$843$	24.3809	0.839722
$844$	9.87489	0.339908
$845$	0	0
$846$	−4.12723	−0.141897
$847$	4.85225	0.166725
$848$	−4.09224	−0.140528
$849$	−5.07142	−0.174051
$850$	0	0
$851$	−36.7274	−1.25900
$852$	−18.6335	−0.638372
$853$	26.4546	0.905788	0.452894	−	0.891564i	$-0.350392\pi$
$853$	26.4546	0.905788	0.452894	+	0.891564i	$0.350392\pi$
$854$	−9.86895	−0.337709
$855$	0	0
$856$	2.19983	0.0751885
$857$	−0.294081	−0.0100456	−0.00502280	−	0.999987i	$-0.501599\pi$
$857$	−0.294081	−0.0100456	−0.00502280	+	0.999987i	$0.501599\pi$
$858$	−5.82209	−0.198763
$859$	16.6946	0.569612	0.284806	−	0.958585i	$-0.408071\pi$
$859$	16.6946	0.569612	0.284806	+	0.958585i	$0.408071\pi$
$860$	0	0
$861$	14.8954	0.507633
$862$	−6.02989	−0.205379
$863$	−7.15399	−0.243525	−0.121762	−	0.992559i	$-0.538855\pi$
$863$	−7.15399	−0.243525	−0.121762	+	0.992559i	$0.538855\pi$
$864$	5.53358	0.188256
$865$	0	0
$866$	−16.9902	−0.577350
$867$	−1.58141	−0.0537075
$868$	4.93212	0.167407
$869$	−39.0644	−1.32517
$870$	0	0
$871$	−23.4674	−0.795163
$872$	−4.13495	−0.140027
$873$	7.01239	0.237334
$874$	−19.9771	−0.675735
$875$	0	0
$876$	16.5088	0.557781
$877$	19.9502	0.673670	0.336835	−	0.941564i	$-0.390644\pi$
$877$	19.9502	0.673670	0.336835	+	0.941564i	$0.390644\pi$
$878$	−20.3664	−0.687333
$879$	−0.600349	−0.0202493
$880$	0	0
$881$	57.6910	1.94366	0.971829	−	0.235685i	$-0.0757334\pi$
$881$	57.6910	1.94366	0.971829	+	0.235685i	$0.0757334\pi$
$882$	−0.499141	−0.0168069
$883$	−1.74844	−0.0588396	−0.0294198	−	0.999567i	$-0.509366\pi$
$883$	−1.74844	−0.0588396	−0.0294198	+	0.999567i	$0.509366\pi$
$884$	−1.48483	−0.0499402
$885$	0	0
$886$	−12.5581	−0.421898
$887$	5.63604	0.189240	0.0946199	−	0.995513i	$-0.469836\pi$
$887$	5.63604	0.189240	0.0946199	+	0.995513i	$0.469836\pi$
$888$	14.1102	0.473506
$889$	−0.539711	−0.0181013
$890$	0	0
$891$	−17.9846	−0.602508
$892$	−0.615959	−0.0206238
$893$	−40.1296	−1.34289
$894$	8.89488	0.297489
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	−9.66548	−0.322721
$898$	25.7519	0.859353
$899$	16.0285	0.534580
$900$	0	0
$901$	4.09224	0.136332
$902$	23.3542	0.777609
$903$	−16.4328	−0.546850
$904$	3.67558	0.122248
$905$	0	0
$906$	−26.8986	−0.893647
$907$	−38.5509	−1.28006	−0.640031	−	0.768349i	$-0.721079\pi$
$907$	−38.5509	−1.28006	−0.640031	+	0.768349i	$0.721079\pi$
$908$	−13.0080	−0.431685
$909$	−4.15189	−0.137709
$910$	0	0
$911$	2.35799	0.0781237	0.0390618	−	0.999237i	$-0.487563\pi$
$911$	2.35799	0.0781237	0.0390618	+	0.999237i	$0.487563\pi$
$912$	7.67492	0.254142
$913$	−42.1484	−1.39491
$914$	28.4616	0.941426
$915$	0	0
$916$	16.7881	0.554695
$917$	8.07474	0.266651
$918$	−5.53358	−0.182635
$919$	47.1055	1.55387	0.776934	−	0.629582i	$-0.216774\pi$
$919$	47.1055	1.55387	0.776934	+	0.629582i	$0.216774\pi$
$920$	0	0
$921$	9.45202	0.311455
$922$	−3.91476	−0.128926
$923$	17.4955	0.575870
$924$	3.92105	0.128993
$925$	0	0
$926$	24.4445	0.803296
$927$	−8.05022	−0.264404
$928$	−3.24982	−0.106681
$929$	−1.58032	−0.0518487	−0.0259243	−	0.999664i	$-0.508253\pi$
$929$	−1.58032	−0.0518487	−0.0259243	+	0.999664i	$0.508253\pi$
$930$	0	0
$931$	−4.85321	−0.159058
$932$	7.50071	0.245694
$933$	48.2336	1.57910
$934$	−15.8270	−0.517874
$935$	0	0
$936$	−0.741138	−0.0242248
$937$	−32.8668	−1.07371	−0.536856	−	0.843674i	$-0.680388\pi$
$937$	−32.8668	−1.07371	−0.536856	+	0.843674i	$0.680388\pi$
$938$	15.8048	0.516045
$939$	−32.4756	−1.05980
$940$	0	0
$941$	7.67130	0.250077	0.125039	−	0.992152i	$-0.460095\pi$
$941$	7.67130	0.250077	0.125039	+	0.992152i	$0.460095\pi$
$942$	13.8653	0.451756
$943$	38.7712	1.26256
$944$	9.51108	0.309559
$945$	0	0
$946$	−25.7647	−0.837684
$947$	−43.5425	−1.41494	−0.707470	−	0.706744i	$-0.750163\pi$
$947$	−43.5425	−1.41494	−0.707470	+	0.706744i	$0.750163\pi$
$948$	24.9154	0.809215
$949$	−15.5006	−0.503170
$950$	0	0
$951$	15.4924	0.502377
$952$	1.00000	0.0324102
$953$	40.6687	1.31739	0.658694	−	0.752411i	$-0.271109\pi$
$953$	40.6687	1.31739	0.658694	+	0.752411i	$0.271109\pi$
$954$	2.04260	0.0661317
$955$	0	0
$956$	−7.82561	−0.253098
$957$	12.7427	0.411914
$958$	34.7707	1.12339
$959$	10.1767	0.328622
$960$	0	0
$961$	−6.67423	−0.215298
$962$	−13.2484	−0.427146
$963$	−1.09802	−0.0353833
$964$	−22.5508	−0.726313
$965$	0	0
$966$	6.50950	0.209440
$967$	−29.1570	−0.937625	−0.468813	−	0.883298i	$-0.655318\pi$
$967$	−29.1570	−0.937625	−0.468813	+	0.883298i	$0.655318\pi$
$968$	−4.85225	−0.155957
$969$	−7.67492	−0.246554
$970$	0	0
$971$	36.8404	1.18227	0.591133	−	0.806574i	$-0.298681\pi$
$971$	36.8404	1.18227	0.591133	+	0.806574i	$0.298681\pi$
$972$	−5.13007	−0.164547
$973$	17.9150	0.574330
$974$	−30.7487	−0.985252
$975$	0	0
$976$	9.86895	0.315897
$977$	−11.2487	−0.359876	−0.179938	−	0.983678i	$-0.557590\pi$
$977$	−11.2487	−0.359876	−0.179938	+	0.983678i	$0.557590\pi$
$978$	25.1986	0.805762
$979$	38.1726	1.22000
$980$	0	0
$981$	2.06392	0.0658960
$982$	33.6199	1.07285
$983$	34.7209	1.10743	0.553713	−	0.832708i	$-0.313211\pi$
$983$	34.7209	1.10743	0.553713	+	0.832708i	$0.313211\pi$
$984$	−14.8954	−0.474847
$985$	0	0
$986$	3.24982	0.103495
$987$	13.0762	0.416219
$988$	−7.20618	−0.229259
$989$	−42.7731	−1.36010
$990$	0	0
$991$	9.11416	0.289521	0.144760	−	0.989467i	$-0.453759\pi$
$991$	9.11416	0.289521	0.144760	+	0.989467i	$0.453759\pi$
$992$	−4.93212	−0.156595
$993$	14.8477	0.471176
$994$	−11.7828	−0.373728
$995$	0	0
$996$	26.8824	0.851800
$997$	23.6110	0.747768	0.373884	−	0.927476i	$-0.378026\pi$
$997$	23.6110	0.747768	0.373884	+	0.927476i	$0.378026\pi$
$998$	−31.2932	−0.990568
$999$	−49.3735	−1.56211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5950.2.a.cc.1.3		7
5.2	odd	4		1190.2.e.g.239.12	yes	14
5.3	odd	4		1190.2.e.g.239.3	✓	14
5.4	even	2		5950.2.a.cb.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1190.2.e.g.239.3	✓	14	5.3	odd	4
1190.2.e.g.239.12	yes	14	5.2	odd	4
5950.2.a.cb.1.5		7	5.4	even	2
5950.2.a.cc.1.3		7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5950 = 2 \cdot 5^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5950.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.58141 q^{3} +1.00000 q^{4} -1.58141 q^{6} -1.00000 q^{7} +1.00000 q^{8} -0.499141 q^{9} +2.47946 q^{11} -1.58141 q^{12} +1.48483 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -0.499141 q^{18} -4.85321 q^{19} +1.58141 q^{21} +2.47946 q^{22} +4.11626 q^{23} -1.58141 q^{24} +1.48483 q^{26} +5.53358 q^{27} -1.00000 q^{28} -3.24982 q^{29} -4.93212 q^{31} +1.00000 q^{32} -3.92105 q^{33} -1.00000 q^{34} -0.499141 q^{36} -8.92252 q^{37} -4.85321 q^{38} -2.34812 q^{39} +9.41904 q^{41} +1.58141 q^{42} -10.3912 q^{43} +2.47946 q^{44} +4.11626 q^{46} +8.26867 q^{47} -1.58141 q^{48} +1.00000 q^{49} +1.58141 q^{51} +1.48483 q^{52} -4.09224 q^{53} +5.53358 q^{54} -1.00000 q^{56} +7.67492 q^{57} -3.24982 q^{58} +9.51108 q^{59} +9.86895 q^{61} -4.93212 q^{62} +0.499141 q^{63} +1.00000 q^{64} -3.92105 q^{66} -15.8048 q^{67} -1.00000 q^{68} -6.50950 q^{69} +11.7828 q^{71} -0.499141 q^{72} -10.4393 q^{73} -8.92252 q^{74} -4.85321 q^{76} -2.47946 q^{77} -2.34812 q^{78} -15.7552 q^{79} -7.25344 q^{81} +9.41904 q^{82} -16.9990 q^{83} +1.58141 q^{84} -10.3912 q^{86} +5.13930 q^{87} +2.47946 q^{88} +15.3955 q^{89} -1.48483 q^{91} +4.11626 q^{92} +7.79970 q^{93} +8.26867 q^{94} -1.58141 q^{96} -14.0489 q^{97} +1.00000 q^{98} -1.23760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} - 2 q^{3} + 7 q^{4} - 2 q^{6} - 7 q^{7} + 7 q^{8} + 7 q^{9} - 7 q^{11} - 2 q^{12} + q^{13} - 7 q^{14} + 7 q^{16} - 7 q^{17} + 7 q^{18} - 2 q^{19} + 2 q^{21} - 7 q^{22} - 27 q^{23} - 2 q^{24}+ \cdots - 44 q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 - 2 * q^3 + 7 * q^4 - 2 * q^6 - 7 * q^7 + 7 * q^8 + 7 * q^9 - 7 * q^11 - 2 * q^12 + q^13 - 7 * q^14 + 7 * q^16 - 7 * q^17 + 7 * q^18 - 2 * q^19 + 2 * q^21 - 7 * q^22 - 27 * q^23 - 2 * q^24 + q^26 - 17 * q^27 - 7 * q^28 - 10 * q^29 - 18 * q^31 + 7 * q^32 - 8 * q^33 - 7 * q^34 + 7 * q^36 - 12 * q^37 - 2 * q^38 - 25 * q^39 + 17 * q^41 + 2 * q^42 - 7 * q^43 - 7 * q^44 - 27 * q^46 - 8 * q^47 - 2 * q^48 + 7 * q^49 + 2 * q^51 + q^52 - 20 * q^53 - 17 * q^54 - 7 * q^56 - 25 * q^57 - 10 * q^58 + q^61 - 18 * q^62 - 7 * q^63 + 7 * q^64 - 8 * q^66 - 7 * q^67 - 7 * q^68 + 3 * q^69 - 11 * q^71 + 7 * q^72 + 7 * q^73 - 12 * q^74 - 2 * q^76 + 7 * q^77 - 25 * q^78 - 26 * q^79 + 11 * q^81 + 17 * q^82 - 22 * q^83 + 2 * q^84 - 7 * q^86 - 20 * q^87 - 7 * q^88 + 21 * q^89 - q^91 - 27 * q^92 - 25 * q^93 - 8 * q^94 - 2 * q^96 + 19 * q^97 + 7 * q^98 - 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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