LMFDB - Embedded newform 650.4.b.k.599.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [650,4,Mod(599,650)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("650.599"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(650, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	650.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-16,0,-8,0,0,-76,0,108] 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:	$38.3512415037$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 130)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.1
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.4.b.k.599.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} -7.70820i q^{3} -4.00000 q^{4} -15.4164 q^{6} +17.8885i q^{7} +8.00000i q^{8} -32.4164 q^{9} +38.1803 q^{11} +30.8328i q^{12} +13.0000i q^{13} +35.7771 q^{14} +16.0000 q^{16} +69.7771i q^{17} +64.8328i q^{18} +152.928 q^{19} +137.889 q^{21} -76.3607i q^{22} -167.125i q^{23} +61.6656 q^{24} +26.0000 q^{26} +41.7508i q^{27} -71.5542i q^{28} -187.305 q^{29} +91.7082 q^{31} -32.0000i q^{32} -294.302i q^{33} +139.554 q^{34} +129.666 q^{36} -127.974i q^{37} -305.856i q^{38} +100.207 q^{39} +239.941 q^{41} -275.777i q^{42} +203.485i q^{43} -152.721 q^{44} -334.249 q^{46} -333.495i q^{47} -123.331i q^{48} +23.0000 q^{49} +537.856 q^{51} -52.0000i q^{52} +585.712i q^{53} +83.5016 q^{54} -143.108 q^{56} -1178.80i q^{57} +374.610i q^{58} +568.895 q^{59} +278.912 q^{61} -183.416i q^{62} -579.882i q^{63} -64.0000 q^{64} -588.604 q^{66} -617.698i q^{67} -279.108i q^{68} -1288.23 q^{69} +319.656 q^{71} -259.331i q^{72} +241.574i q^{73} -255.947 q^{74} -611.712 q^{76} +682.991i q^{77} -200.413i q^{78} +60.4659 q^{79} -553.420 q^{81} -479.882i q^{82} +1204.87i q^{83} -551.554 q^{84} +406.971 q^{86} +1443.78i q^{87} +305.443i q^{88} -557.437 q^{89} -232.551 q^{91} +668.498i q^{92} -706.906i q^{93} -666.991 q^{94} -246.663 q^{96} -241.895i q^{97} -46.0000i q^{98} -1237.67 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4} - 8 q^{6} - 76 q^{9} + 108 q^{11} + 64 q^{16} + 84 q^{19} + 480 q^{21} + 32 q^{24} + 104 q^{26} - 624 q^{29} + 340 q^{31} + 272 q^{34} + 304 q^{36} + 52 q^{39} + 280 q^{41} - 432 q^{44} - 1176 q^{46}+ \cdots - 2652 q^{99}+O(q^{100}) $ 4 * q - 16 * q^4 - 8 * q^6 - 76 * q^9 + 108 * q^11 + 64 * q^16 + 84 * q^19 + 480 * q^21 + 32 * q^24 + 104 * q^26 - 624 * q^29 + 340 * q^31 + 272 * q^34 + 304 * q^36 + 52 * q^39 + 280 * q^41 - 432 * q^44 - 1176 * q^46 + 92 * q^49 + 1096 * q^51 + 656 * q^54 + 764 * q^59 + 96 * q^61 - 256 * q^64 - 816 * q^66 - 1128 * q^69 + 1860 * q^71 - 1632 * q^74 - 336 * q^76 - 1064 * q^79 - 2804 * q^81 - 1920 * q^84 + 1288 * q^86 - 584 * q^89 - 736 * q^94 - 128 * q^96 - 2652 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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
$3$	− 7.70820i	− 1.48344i	−0.670707	−	0.741722i	$-0.734009\pi$
$3$	− 7.70820i	− 1.48344i	0.670707	−	0.741722i	$-0.265991\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−15.4164	−1.04895
$7$	17.8885i	0.965891i	0.875651	+	0.482945i	$0.160433\pi$
$7$	17.8885i	0.965891i	−0.875651	+	0.482945i	$0.839567\pi$
$8$	8.00000i	0.353553i
$9$	−32.4164	−1.20061
$10$	0	0
$11$	38.1803	1.04653	0.523264	−	0.852171i	$-0.324714\pi$
$11$	38.1803	1.04653	0.523264	+	0.852171i	$0.324714\pi$
$12$	30.8328i	0.741722i
$13$	13.0000i	0.277350i
$13$	13.0000i	0.277350i
$14$	35.7771	0.682988
$15$	0	0
$16$	16.0000	0.250000
$17$	69.7771i	0.995496i	0.867322	+	0.497748i	$0.165840\pi$
$17$	69.7771i	0.995496i	−0.867322	+	0.497748i	$0.834160\pi$
$18$	64.8328i	0.848958i
$19$	152.928	1.84653	0.923266	−	0.384162i	$-0.125510\pi$
$19$	152.928	1.84653	0.923266	+	0.384162i	$0.125510\pi$
$20$	0	0
$21$	137.889	1.43285
$22$	− 76.3607i	− 0.740007i
$23$	− 167.125i	− 1.51513i	−0.652762	−	0.757563i	$-0.726390\pi$
$23$	− 167.125i	− 1.51513i	0.652762	−	0.757563i	$-0.273610\pi$
$24$	61.6656	0.524477
$25$	0	0
$26$	26.0000	0.196116
$27$	41.7508i	0.297590i
$28$	− 71.5542i	− 0.482945i
$29$	−187.305	−1.19937	−0.599684	−	0.800237i	$-0.704707\pi$
$29$	−187.305	−1.19937	−0.599684	+	0.800237i	$0.704707\pi$
$30$	0	0
$31$	91.7082	0.531332	0.265666	−	0.964065i	$-0.414408\pi$
$31$	91.7082	0.531332	0.265666	+	0.964065i	$0.414408\pi$
$32$	− 32.0000i	− 0.176777i
$33$	− 294.302i	− 1.55247i
$34$	139.554	0.703922
$35$	0	0
$36$	129.666	0.600304
$37$	− 127.974i	− 0.568615i	−0.958733	−	0.284307i	$-0.908236\pi$
$37$	− 127.974i	− 0.568615i	0.958733	−	0.284307i	$-0.0917636\pi$
$38$	− 305.856i	− 1.30569i
$39$	100.207	0.411433
$40$	0	0
$41$	239.941	0.913964	0.456982	−	0.889476i	$-0.348930\pi$
$41$	239.941	0.913964	0.456982	+	0.889476i	$0.348930\pi$
$42$	− 275.777i	− 1.01317i
$43$	203.485i	0.721656i	0.932632	+	0.360828i	$0.117506\pi$
$43$	203.485i	0.721656i	−0.932632	+	0.360828i	$0.882494\pi$
$44$	−152.721	−0.523264
$45$	0	0
$46$	−334.249	−1.07136
$47$	− 333.495i	− 1.03501i	−0.855681	−	0.517503i	$-0.826862\pi$
$47$	− 333.495i	− 1.03501i	0.855681	−	0.517503i	$-0.173138\pi$
$48$	− 123.331i	− 0.370861i
$49$	23.0000	0.0670554
$50$	0	0
$51$	537.856	1.47676
$52$	− 52.0000i	− 0.138675i
$53$	585.712i	1.51800i	0.651094	+	0.758998i	$0.274311\pi$
$53$	585.712i	1.51800i	−0.651094	+	0.758998i	$0.725689\pi$
$54$	83.5016	0.210428
$55$	0	0
$56$	−143.108	−0.341494
$57$	− 1178.80i	− 2.73923i
$58$	374.610i	0.848081i
$59$	568.895	1.25532	0.627660	−	0.778488i	$-0.284013\pi$
$59$	568.895	1.25532	0.627660	+	0.778488i	$0.284013\pi$
$60$	0	0
$61$	278.912	0.585426	0.292713	−	0.956200i	$-0.405442\pi$
$61$	278.912	0.585426	0.292713	+	0.956200i	$0.405442\pi$
$62$	− 183.416i	− 0.375708i
$63$	− 579.882i	− 1.15966i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−588.604	−1.09776
$67$	− 617.698i	− 1.12633i	−0.826346	−	0.563163i	$-0.809584\pi$
$67$	− 617.698i	− 1.12633i	0.826346	−	0.563163i	$-0.190416\pi$
$68$	− 279.108i	− 0.497748i
$69$	−1288.23	−2.24760
$70$	0	0
$71$	319.656	0.534312	0.267156	−	0.963653i	$-0.413916\pi$
$71$	319.656	0.534312	0.267156	+	0.963653i	$0.413916\pi$
$72$	− 259.331i	− 0.424479i
$73$	241.574i	0.387317i	0.981069	+	0.193658i	$0.0620354\pi$
$73$	241.574i	0.387317i	−0.981069	+	0.193658i	$0.937965\pi$
$74$	−255.947	−0.402071
$75$	0	0
$76$	−611.712	−0.923266
$77$	682.991i	1.01083i
$78$	− 200.413i	− 0.290927i
$79$	60.4659	0.0861133	0.0430566	−	0.999073i	$-0.486290\pi$
$79$	60.4659	0.0861133	0.0430566	+	0.999073i	$0.486290\pi$
$80$	0	0
$81$	−553.420	−0.759149
$82$	− 479.882i	− 0.646270i
$83$	1204.87i	1.59339i	0.604382	+	0.796695i	$0.293420\pi$
$83$	1204.87i	1.59339i	−0.604382	+	0.796695i	$0.706580\pi$
$84$	−551.554	−0.716423
$85$	0	0
$86$	406.971	0.510288
$87$	1443.78i	1.77920i
$88$	305.443i	0.370003i
$89$	−557.437	−0.663912	−0.331956	−	0.943295i	$-0.607709\pi$
$89$	−557.437	−0.663912	−0.331956	+	0.943295i	$0.607709\pi$
$90$	0	0
$91$	−232.551	−0.267890
$92$	668.498i	0.757563i
$93$	− 706.906i	− 0.788201i
$94$	−666.991	−0.731860
$95$	0	0
$96$	−246.663	−0.262238
$97$	− 241.895i	− 0.253203i	−0.991954	−	0.126602i	$-0.959593\pi$
$97$	− 241.895i	− 0.253203i	0.991954	−	0.126602i	$-0.0404069\pi$
$98$	− 46.0000i	− 0.0474153i
$99$	−1237.67	−1.25647
$100$	0	0
$101$	−283.102	−0.278908	−0.139454	−	0.990229i	$-0.544535\pi$
$101$	−283.102	−0.278908	−0.139454	+	0.990229i	$0.544535\pi$
$102$	− 1075.71i	− 1.04423i
$103$	− 1932.07i	− 1.84828i	−0.382059	−	0.924138i	$-0.624785\pi$
$103$	− 1932.07i	− 1.84828i	0.382059	−	0.924138i	$-0.375215\pi$
$104$	−104.000	−0.0980581
$105$	0	0
$106$	1171.42	1.07338
$107$	− 468.547i	− 0.423329i	−0.977342	−	0.211664i	$-0.932112\pi$
$107$	− 468.547i	− 0.423329i	0.977342	−	0.211664i	$-0.0678884\pi$
$108$	− 167.003i	− 0.148795i
$109$	1504.37	1.32195	0.660977	−	0.750406i	$-0.270142\pi$
$109$	1504.37	1.32195	0.660977	+	0.750406i	$0.270142\pi$
$110$	0	0
$111$	−986.447	−0.843508
$112$	286.217i	0.241473i
$113$	1311.97i	1.09221i	0.837717	+	0.546105i	$0.183890\pi$
$113$	1311.97i	1.09221i	−0.837717	+	0.546105i	$0.816110\pi$
$114$	−2357.60	−1.93693
$115$	0	0
$116$	749.220	0.599684
$117$	− 421.413i	− 0.332989i
$118$	− 1137.79i	− 0.887645i
$119$	−1248.21	−0.961540
$120$	0	0
$121$	126.738	0.0952204
$122$	− 557.823i	− 0.413959i
$123$	− 1849.52i	− 1.35581i
$124$	−366.833	−0.265666
$125$	0	0
$126$	−1159.76	−0.820000
$127$	− 1857.03i	− 1.29752i	−0.760994	−	0.648758i	$-0.775289\pi$
$127$	− 1857.03i	− 1.29752i	0.760994	−	0.648758i	$-0.224711\pi$
$128$	128.000i	0.0883883i
$129$	1568.51	1.07054
$130$	0	0
$131$	2023.27	1.34942	0.674708	−	0.738084i	$-0.264269\pi$
$131$	2023.27	1.34942	0.674708	+	0.738084i	$0.264269\pi$
$132$	1177.21i	0.776233i
$133$	2735.66i	1.78355i
$134$	−1235.40	−0.796433
$135$	0	0
$136$	−558.217	−0.351961
$137$	− 1447.75i	− 0.902842i	−0.892311	−	0.451421i	$-0.850917\pi$
$137$	− 1447.75i	− 0.902842i	0.892311	−	0.451421i	$-0.149083\pi$
$138$	2576.46i	1.58930i
$139$	−1527.02	−0.931802	−0.465901	−	0.884837i	$-0.654270\pi$
$139$	−1527.02	−0.931802	−0.465901	+	0.884837i	$0.654270\pi$
$140$	0	0
$141$	−2570.65	−1.53537
$142$	− 639.311i	− 0.377816i
$143$	496.344i	0.290255i
$144$	−518.663	−0.300152
$145$	0	0
$146$	483.149	0.273874
$147$	− 177.289i	− 0.0994730i
$148$	511.895i	0.284307i
$149$	−2307.18	−1.26854	−0.634268	−	0.773114i	$-0.718698\pi$
$149$	−2307.18	−1.26854	−0.634268	+	0.773114i	$0.718698\pi$
$150$	0	0
$151$	1093.09	0.589104	0.294552	−	0.955635i	$-0.404830\pi$
$151$	1093.09	0.589104	0.294552	+	0.955635i	$0.404830\pi$
$152$	1223.42i	0.652847i
$153$	− 2261.92i	− 1.19520i
$154$	1365.98	0.714766
$155$	0	0
$156$	−400.827	−0.205717
$157$	− 348.046i	− 0.176924i	−0.996080	−	0.0884622i	$-0.971805\pi$
$157$	− 348.046i	− 0.176924i	0.996080	−	0.0884622i	$-0.0281952\pi$
$158$	− 120.932i	− 0.0608913i
$159$	4514.79	2.25186
$160$	0	0
$161$	2989.62	1.46345
$162$	1106.84i	0.536799i
$163$	− 2388.98i	− 1.14797i	−0.818865	−	0.573986i	$-0.805396\pi$
$163$	− 2388.98i	− 1.14797i	0.818865	−	0.573986i	$-0.194604\pi$
$164$	−959.765	−0.456982
$165$	0	0
$166$	2409.73	1.12670
$167$	4264.95i	1.97624i	0.153695	+	0.988118i	$0.450883\pi$
$167$	4264.95i	1.97624i	−0.153695	+	0.988118i	$0.549117\pi$
$168$	1103.11i	0.506587i
$169$	−169.000	−0.0769231
$170$	0	0
$171$	−4957.38	−2.21696
$172$	− 813.941i	− 0.360828i
$173$	− 2871.68i	− 1.26202i	−0.775774	−	0.631011i	$-0.782640\pi$
$173$	− 2871.68i	− 1.26202i	0.775774	−	0.631011i	$-0.217360\pi$
$174$	2887.57	1.25808
$175$	0	0
$176$	610.885	0.261632
$177$	− 4385.16i	− 1.86220i
$178$	1114.87i	0.469457i
$179$	−982.845	−0.410398	−0.205199	−	0.978720i	$-0.565784\pi$
$179$	−982.845	−0.410398	−0.205199	+	0.978720i	$0.565784\pi$
$180$	0	0
$181$	3342.85	1.37277	0.686386	−	0.727237i	$-0.259196\pi$
$181$	3342.85	1.37277	0.686386	+	0.727237i	$0.259196\pi$
$182$	465.102i	0.189427i
$183$	− 2149.91i	− 0.868447i
$184$	1337.00	0.535678
$185$	0	0
$186$	−1413.81	−0.557342
$187$	2664.11i	1.04181i
$188$	1333.98i	0.517503i
$189$	−746.861	−0.287440
$190$	0	0
$191$	16.1052	0.00610123	0.00305061	−	0.999995i	$-0.499029\pi$
$191$	16.1052	0.00610123	0.00305061	+	0.999995i	$0.499029\pi$
$192$	493.325i	0.185431i
$193$	529.003i	0.197298i	0.995122	+	0.0986489i	$0.0314521\pi$
$193$	529.003i	0.197298i	−0.995122	+	0.0986489i	$0.968548\pi$
$194$	−483.790	−0.179042
$195$	0	0
$196$	−92.0000	−0.0335277
$197$	− 4716.72i	− 1.70585i	−0.522034	−	0.852924i	$-0.674827\pi$
$197$	− 4716.72i	− 1.70585i	0.522034	−	0.852924i	$-0.325173\pi$
$198$	2475.34i	0.888458i
$199$	4072.65	1.45077	0.725383	−	0.688346i	$-0.241663\pi$
$199$	4072.65	1.45077	0.725383	+	0.688346i	$0.241663\pi$
$200$	0	0
$201$	−4761.34	−1.67084
$202$	566.204i	0.197218i
$203$	− 3350.61i	− 1.15846i
$204$	−2151.42	−0.738381
$205$	0	0
$206$	−3864.14	−1.30693
$207$	5417.58i	1.81907i
$208$	208.000i	0.0693375i
$209$	5838.84	1.93245
$210$	0	0
$211$	−4689.58	−1.53006	−0.765032	−	0.643992i	$-0.777277\pi$
$211$	−4689.58	−1.53006	−0.765032	+	0.643992i	$0.777277\pi$
$212$	− 2342.85i	− 0.758998i
$213$	− 2463.97i	− 0.792622i
$214$	−937.094	−0.299339
$215$	0	0
$216$	−334.006	−0.105214
$217$	1640.53i	0.513208i
$218$	− 3008.75i	− 0.934762i
$219$	1862.10	0.574563
$220$	0	0
$221$	−907.102	−0.276101
$222$	1972.89i	0.596451i
$223$	− 2158.45i	− 0.648164i	−0.946029	−	0.324082i	$-0.894945\pi$
$223$	− 2158.45i	− 0.648164i	0.946029	−	0.324082i	$-0.105055\pi$
$224$	572.433	0.170747
$225$	0	0
$226$	2623.94	0.772309
$227$	4362.74i	1.27562i	0.770195	+	0.637808i	$0.220159\pi$
$227$	4362.74i	1.27562i	−0.770195	+	0.637808i	$0.779841\pi$
$228$	4715.20i	1.36961i
$229$	1071.83	0.309296	0.154648	−	0.987970i	$-0.450576\pi$
$229$	1071.83	0.309296	0.154648	+	0.987970i	$0.450576\pi$
$230$	0	0
$231$	5264.63	1.49951
$232$	− 1498.44i	− 0.424040i
$233$	− 1472.88i	− 0.414128i	−0.978327	−	0.207064i	$-0.933609\pi$
$233$	− 1472.88i	− 0.414128i	0.978327	−	0.207064i	$-0.0663908\pi$
$234$	−842.827	−0.235459
$235$	0	0
$236$	−2275.58	−0.627660
$237$	− 466.084i	− 0.127744i
$238$	2496.42i	0.679912i
$239$	2380.83	0.644363	0.322182	−	0.946678i	$-0.395584\pi$
$239$	2380.83	0.644363	0.322182	+	0.946678i	$0.395584\pi$
$240$	0	0
$241$	4376.77	1.16984	0.584922	−	0.811090i	$-0.301125\pi$
$241$	4376.77	1.16984	0.584922	+	0.811090i	$0.301125\pi$
$242$	− 253.477i	− 0.0673310i
$243$	5393.14i	1.42375i
$244$	−1115.65	−0.292713
$245$	0	0
$246$	−3699.03	−0.958706
$247$	1988.06i	0.512136i
$248$	733.666i	0.187854i
$249$	9287.36	2.36370
$250$	0	0
$251$	1087.80	0.273551	0.136776	−	0.990602i	$-0.456326\pi$
$251$	1087.80	0.273551	0.136776	+	0.990602i	$0.456326\pi$
$252$	2319.53i	0.579828i
$253$	− 6380.87i	− 1.58562i
$254$	−3714.06	−0.917483
$255$	0	0
$256$	256.000	0.0625000
$257$	− 5112.49i	− 1.24089i	−0.784251	−	0.620444i	$-0.786953\pi$
$257$	− 5112.49i	− 1.24089i	0.784251	−	0.620444i	$-0.213047\pi$
$258$	− 3137.01i	− 0.756984i
$259$	2289.26	0.549220
$260$	0	0
$261$	6071.75	1.43997
$262$	− 4046.53i	− 0.954182i
$263$	3790.88i	0.888805i	0.895827	+	0.444402i	$0.146584\pi$
$263$	3790.88i	0.888805i	−0.895827	+	0.444402i	$0.853416\pi$
$264$	2354.41	0.548880
$265$	0	0
$266$	5471.32	1.26116
$267$	4296.83i	0.984876i
$268$	2470.79i	0.563163i
$269$	−3523.30	−0.798585	−0.399293	−	0.916824i	$-0.630744\pi$
$269$	−3523.30	−0.798585	−0.399293	+	0.916824i	$0.630744\pi$
$270$	0	0
$271$	4254.59	0.953682	0.476841	−	0.878990i	$-0.341782\pi$
$271$	4254.59	0.953682	0.476841	+	0.878990i	$0.341782\pi$
$272$	1116.43i	0.248874i
$273$	1792.55i	0.397400i
$274$	−2895.49	−0.638405
$275$	0	0
$276$	5152.92	1.12380
$277$	− 1312.77i	− 0.284754i	−0.989813	−	0.142377i	$-0.954526\pi$
$277$	− 1312.77i	− 0.284754i	0.989813	−	0.142377i	$-0.0454745\pi$
$278$	3054.05i	0.658883i
$279$	−2972.85	−0.637921
$280$	0	0
$281$	3112.38	0.660743	0.330372	−	0.943851i	$-0.392826\pi$
$281$	3112.38	0.660743	0.330372	+	0.943851i	$0.392826\pi$
$282$	5141.30i	1.08567i
$283$	3983.53i	0.836736i	0.908278	+	0.418368i	$0.137398\pi$
$283$	3983.53i	0.836736i	−0.908278	+	0.418368i	$0.862602\pi$
$284$	−1278.62	−0.267156
$285$	0	0
$286$	992.689	0.205241
$287$	4292.20i	0.882789i
$288$	1037.33i	0.212239i
$289$	44.1580	0.00898800
$290$	0	0
$291$	−1864.57	−0.375613
$292$	− 966.297i	− 0.193658i
$293$	− 6710.42i	− 1.33798i	−0.743273	−	0.668988i	$-0.766728\pi$
$293$	− 6710.42i	− 1.33798i	0.743273	−	0.668988i	$-0.233272\pi$
$294$	−354.577	−0.0703380
$295$	0	0
$296$	1023.79	0.201036
$297$	1594.06i	0.311437i
$298$	4614.37i	0.896990i
$299$	2172.62	0.420220
$300$	0	0
$301$	−3640.06	−0.697041
$302$	− 2186.19i	− 0.416559i
$303$	2182.21i	0.413745i
$304$	2446.85	0.461633
$305$	0	0
$306$	−4523.85	−0.845134
$307$	3883.81i	0.722023i	0.932561	+	0.361011i	$0.117568\pi$
$307$	3883.81i	0.722023i	−0.932561	+	0.361011i	$0.882432\pi$
$308$	− 2731.96i	− 0.505416i
$309$	−14892.8	−2.74181
$310$	0	0
$311$	−9188.01	−1.67525	−0.837627	−	0.546242i	$-0.816058\pi$
$311$	−9188.01	−1.67525	−0.837627	+	0.546242i	$0.816058\pi$
$312$	801.653i	0.145464i
$313$	7475.45i	1.34996i	0.737836	+	0.674980i	$0.235848\pi$
$313$	7475.45i	1.34996i	−0.737836	+	0.674980i	$0.764152\pi$
$314$	−696.093	−0.125104
$315$	0	0
$316$	−241.864	−0.0430566
$317$	4668.43i	0.827145i	0.910471	+	0.413572i	$0.135719\pi$
$317$	4668.43i	0.827145i	−0.910471	+	0.413572i	$0.864281\pi$
$318$	− 9029.58i	− 1.59231i
$319$	−7151.37	−1.25517
$320$	0	0
$321$	−3611.66	−0.627985
$322$	− 5979.23i	− 1.03481i
$323$	10670.9i	1.83821i
$324$	2213.68	0.379574
$325$	0	0
$326$	−4777.96	−0.811738
$327$	− 11596.0i	− 1.96104i
$328$	1919.53i	0.323135i
$329$	5965.75	0.999703
$330$	0	0
$331$	1544.67	0.256503	0.128252	−	0.991742i	$-0.459063\pi$
$331$	1544.67	0.256503	0.128252	+	0.991742i	$0.459063\pi$
$332$	− 4819.47i	− 0.796695i
$333$	4148.45i	0.682683i
$334$	8529.89	1.39741
$335$	0	0
$336$	2206.22	0.358211
$337$	3788.99i	0.612461i	0.951957	+	0.306231i	$0.0990679\pi$
$337$	3788.99i	0.612461i	−0.951957	+	0.306231i	$0.900932\pi$
$338$	338.000i	0.0543928i
$339$	10112.9	1.62023
$340$	0	0
$341$	3501.45	0.556053
$342$	9914.75i	1.56763i
$343$	6547.21i	1.03066i
$344$	−1627.88	−0.255144
$345$	0	0
$346$	−5743.36	−0.892385
$347$	4545.59i	0.703227i	0.936145	+	0.351614i	$0.114367\pi$
$347$	4545.59i	0.703227i	−0.936145	+	0.351614i	$0.885633\pi$
$348$	− 5775.14i	− 0.889598i
$349$	−1545.04	−0.236975	−0.118487	−	0.992956i	$-0.537805\pi$
$349$	−1545.04	−0.236975	−0.118487	+	0.992956i	$0.537805\pi$
$350$	0	0
$351$	−542.760	−0.0825367
$352$	− 1221.77i	− 0.185002i
$353$	4271.28i	0.644015i	0.946737	+	0.322008i	$0.104358\pi$
$353$	4271.28i	0.644015i	−0.946737	+	0.322008i	$0.895642\pi$
$354$	−8770.32	−1.31677
$355$	0	0
$356$	2229.75	0.331956
$357$	9621.46i	1.42639i
$358$	1965.69i	0.290195i
$359$	−1106.68	−0.162697	−0.0813484	−	0.996686i	$-0.525923\pi$
$359$	−1106.68	−0.162697	−0.0813484	+	0.996686i	$0.525923\pi$
$360$	0	0
$361$	16528.0	2.40968
$362$	− 6685.69i	− 0.970697i
$363$	− 976.925i	− 0.141254i
$364$	930.204	0.133945
$365$	0	0
$366$	−4299.82	−0.614085
$367$	83.1275i	0.0118235i	0.999983	+	0.00591175i	$0.00188178\pi$
$367$	83.1275i	0.0118235i	−0.999983	+	0.00591175i	$0.998118\pi$
$368$	− 2673.99i	− 0.378781i
$369$	−7778.03	−1.09731
$370$	0	0
$371$	−10477.5	−1.46622
$372$	2827.62i	0.394100i
$373$	− 2509.17i	− 0.348310i	−0.984718	−	0.174155i	$-0.944281\pi$
$373$	− 2509.17i	− 0.348310i	0.984718	−	0.174155i	$-0.0557194\pi$
$374$	5328.23	0.736674
$375$	0	0
$376$	2667.96	0.365930
$377$	− 2434.96i	− 0.332645i
$378$	1493.72i	0.203251i
$379$	−5959.76	−0.807738	−0.403869	−	0.914817i	$-0.632335\pi$
$379$	−5959.76	−0.807738	−0.403869	+	0.914817i	$0.632335\pi$
$380$	0	0
$381$	−14314.4	−1.92479
$382$	− 32.2105i	− 0.00431422i
$383$	2191.58i	0.292388i	0.989256	+	0.146194i	$0.0467023\pi$
$383$	2191.58i	0.292388i	−0.989256	+	0.146194i	$0.953298\pi$
$384$	986.650	0.131119
$385$	0	0
$386$	1058.01	0.139511
$387$	− 6596.26i	− 0.866426i
$388$	967.579i	0.126602i
$389$	4157.52	0.541889	0.270944	−	0.962595i	$-0.412664\pi$
$389$	4157.52	0.541889	0.270944	+	0.962595i	$0.412664\pi$
$390$	0	0
$391$	11661.5	1.50830
$392$	184.000i	0.0237077i
$393$	− 15595.7i	− 2.00178i
$394$	−9433.44	−1.20622
$395$	0	0
$396$	4950.68	0.628235
$397$	− 13622.1i	− 1.72210i	−0.508524	−	0.861048i	$-0.669809\pi$
$397$	− 13622.1i	− 1.72210i	0.508524	−	0.861048i	$-0.330191\pi$
$398$	− 8145.29i	− 1.02585i
$399$	21087.0	2.64579
$400$	0	0
$401$	−1679.41	−0.209142	−0.104571	−	0.994517i	$-0.533347\pi$
$401$	−1679.41	−0.209142	−0.104571	+	0.994517i	$0.533347\pi$
$402$	9522.69i	1.18146i
$403$	1192.21i	0.147365i
$404$	1132.41	0.139454
$405$	0	0
$406$	−6701.23	−0.819153
$407$	− 4886.08i	− 0.595071i
$408$	4302.85i	0.522115i
$409$	15460.6	1.86914	0.934569	−	0.355781i	$-0.115785\pi$
$409$	15460.6	1.86914	0.934569	+	0.355781i	$0.115785\pi$
$410$	0	0
$411$	−11159.5	−1.33932
$412$	7728.28i	0.924138i
$413$	10176.7i	1.21250i
$414$	10835.2	1.28628
$415$	0	0
$416$	416.000	0.0490290
$417$	11770.6i	1.38228i
$418$	− 11677.7i	− 1.36645i
$419$	13864.6	1.61654	0.808271	−	0.588811i	$-0.200404\pi$
$419$	13864.6	1.61654	0.808271	+	0.588811i	$0.200404\pi$
$420$	0	0
$421$	−3626.36	−0.419806	−0.209903	−	0.977722i	$-0.567315\pi$
$421$	−3626.36	−0.419806	−0.209903	+	0.977722i	$0.567315\pi$
$422$	9379.15i	1.08192i
$423$	10810.7i	1.24264i
$424$	−4685.70	−0.536692
$425$	0	0
$426$	−4927.94	−0.560468
$427$	4989.33i	0.565458i
$428$	1874.19i	0.211664i
$429$	3825.92	0.430577
$430$	0	0
$431$	13189.5	1.47405	0.737023	−	0.675868i	$-0.236231\pi$
$431$	13189.5	1.47405	0.737023	+	0.675868i	$0.236231\pi$
$432$	668.012i	0.0743976i
$433$	3061.68i	0.339804i	0.985461	+	0.169902i	$0.0543451\pi$
$433$	3061.68i	0.339804i	−0.985461	+	0.169902i	$0.945655\pi$
$434$	3281.05	0.362893
$435$	0	0
$436$	−6017.50	−0.660977
$437$	− 25558.0i	− 2.79773i
$438$	− 3724.21i	− 0.406277i
$439$	−12751.4	−1.38632	−0.693158	−	0.720786i	$-0.743781\pi$
$439$	−12751.4	−1.38632	−0.693158	+	0.720786i	$0.743781\pi$
$440$	0	0
$441$	−745.577	−0.0805072
$442$	1814.20i	0.195233i
$443$	6375.24i	0.683740i	0.939747	+	0.341870i	$0.111060\pi$
$443$	6375.24i	0.683740i	−0.939747	+	0.341870i	$0.888940\pi$
$444$	3945.79	0.421754
$445$	0	0
$446$	−4316.90	−0.458321
$447$	17784.2i	1.88180i
$448$	− 1144.87i	− 0.120736i
$449$	−14821.9	−1.55789	−0.778943	−	0.627095i	$-0.784244\pi$
$449$	−14821.9	−1.55789	−0.778943	+	0.627095i	$0.784244\pi$
$450$	0	0
$451$	9161.04	0.956488
$452$	− 5247.88i	− 0.546105i
$453$	− 8425.79i	− 0.873903i
$454$	8725.47	0.901997
$455$	0	0
$456$	9430.40	0.968463
$457$	9791.53i	1.00225i	0.865375	+	0.501125i	$0.167080\pi$
$457$	9791.53i	1.00225i	−0.865375	+	0.501125i	$0.832920\pi$
$458$	− 2143.67i	− 0.218705i
$459$	−2913.25	−0.296250
$460$	0	0
$461$	7532.58	0.761013	0.380507	−	0.924778i	$-0.375750\pi$
$461$	7532.58	0.761013	0.380507	+	0.924778i	$0.375750\pi$
$462$	− 10529.3i	− 1.06032i
$463$	− 2045.90i	− 0.205359i	−0.994715	−	0.102679i	$-0.967258\pi$
$463$	− 2045.90i	− 0.205359i	0.994715	−	0.102679i	$-0.0327415\pi$
$464$	−2996.88	−0.299842
$465$	0	0
$466$	−2945.77	−0.292832
$467$	17289.0i	1.71315i	0.516023	+	0.856574i	$0.327412\pi$
$467$	17289.0i	1.71315i	−0.516023	+	0.856574i	$0.672588\pi$
$468$	1685.65i	0.166494i
$469$	11049.7	1.08791
$470$	0	0
$471$	−2682.81	−0.262458
$472$	4551.16i	0.443823i
$473$	7769.14i	0.755233i
$474$	−932.167	−0.0903288
$475$	0	0
$476$	4992.84	0.480770
$477$	− 18986.7i	− 1.82252i
$478$	− 4761.65i	− 0.455634i
$479$	−14488.7	−1.38206	−0.691028	−	0.722828i	$-0.742842\pi$
$479$	−14488.7	−1.38206	−0.691028	+	0.722828i	$0.742842\pi$
$480$	0	0
$481$	1663.66	0.157705
$482$	− 8753.54i	− 0.827205i
$483$	− 23044.6i	− 2.17094i
$484$	−506.953	−0.0476102
$485$	0	0
$486$	10786.3	1.00674
$487$	− 3451.25i	− 0.321132i	−0.987025	−	0.160566i	$-0.948668\pi$
$487$	− 3451.25i	− 0.321132i	0.987025	−	0.160566i	$-0.0513319\pi$
$488$	2231.29i	0.206979i
$489$	−18414.7	−1.70295
$490$	0	0
$491$	−20328.9	−1.86849	−0.934246	−	0.356630i	$-0.883926\pi$
$491$	−20328.9	−1.86849	−0.934246	+	0.356630i	$0.883926\pi$
$492$	7398.06i	0.677907i
$493$	− 13069.6i	− 1.19397i
$494$	3976.13	0.362135
$495$	0	0
$496$	1467.33	0.132833
$497$	5718.17i	0.516087i
$498$	− 18574.7i	− 1.67139i
$499$	4756.36	0.426701	0.213351	−	0.976976i	$-0.431562\pi$
$499$	4756.36	0.426701	0.213351	+	0.976976i	$0.431562\pi$
$500$	0	0
$501$	32875.1	2.93164
$502$	− 2175.60i	− 0.193430i
$503$	− 18875.2i	− 1.67317i	−0.547840	−	0.836583i	$-0.684550\pi$
$503$	− 18875.2i	− 1.67317i	0.547840	−	0.836583i	$-0.315450\pi$
$504$	4639.06	0.410000
$505$	0	0
$506$	−12761.7	−1.12120
$507$	1302.69i	0.114111i
$508$	7428.11i	0.648758i
$509$	−13108.3	−1.14148	−0.570740	−	0.821131i	$-0.693343\pi$
$509$	−13108.3	−1.14148	−0.570740	+	0.821131i	$0.693343\pi$
$510$	0	0
$511$	−4321.41	−0.374106
$512$	− 512.000i	− 0.0441942i
$513$	6384.86i	0.549510i
$514$	−10225.0	−0.877440
$515$	0	0
$516$	−6274.02	−0.535268
$517$	− 12733.0i	− 1.08316i
$518$	− 4578.53i	− 0.388357i
$519$	−22135.5	−1.87214
$520$	0	0
$521$	−1065.58	−0.0896047	−0.0448023	−	0.998996i	$-0.514266\pi$
$521$	−1065.58	−0.0896047	−0.0448023	+	0.998996i	$0.514266\pi$
$522$	− 12143.5i	− 1.01821i
$523$	− 11752.4i	− 0.982591i	−0.870993	−	0.491296i	$-0.836524\pi$
$523$	− 11752.4i	− 0.982591i	0.870993	−	0.491296i	$-0.163476\pi$
$524$	−8093.06	−0.674708
$525$	0	0
$526$	7581.76	0.628480
$527$	6399.13i	0.528938i
$528$	− 4708.83i	− 0.388116i
$529$	−15763.6	−1.29561
$530$	0	0
$531$	−18441.5	−1.50715
$532$	− 10942.6i	− 0.891774i
$533$	3119.24i	0.253488i
$534$	8593.67	0.696413
$535$	0	0
$536$	4941.59	0.398216
$537$	7575.97i	0.608803i
$538$	7046.60i	0.564685i
$539$	878.148	0.0701753
$540$	0	0
$541$	−11910.5	−0.946532	−0.473266	−	0.880920i	$-0.656925\pi$
$541$	−11910.5	−0.946532	−0.473266	+	0.880920i	$0.656925\pi$
$542$	− 8509.17i	− 0.674355i
$543$	− 25767.3i	− 2.03643i
$544$	2232.87	0.175980
$545$	0	0
$546$	3585.10	0.281004
$547$	− 22283.2i	− 1.74180i	−0.491464	−	0.870898i	$-0.663538\pi$
$547$	− 22283.2i	− 1.74180i	0.491464	−	0.870898i	$-0.336462\pi$
$548$	5790.98i	0.451421i
$549$	−9041.32	−0.702867
$550$	0	0
$551$	−28644.2	−2.21467
$552$	− 10305.8i	− 0.794648i
$553$	1081.65i	0.0831760i
$554$	−2625.54	−0.201351
$555$	0	0
$556$	6108.09	0.465901
$557$	8012.19i	0.609493i	0.952434	+	0.304746i	$0.0985717\pi$
$557$	8012.19i	0.609493i	−0.952434	+	0.304746i	$0.901428\pi$
$558$	5945.70i	0.451078i
$559$	−2645.31	−0.200151
$560$	0	0
$561$	20535.5	1.54547
$562$	− 6224.76i	− 0.467216i
$563$	1086.28i	0.0813167i	0.999173	+	0.0406583i	$0.0129455\pi$
$563$	1086.28i	0.0813167i	−0.999173	+	0.0406583i	$0.987054\pi$
$564$	10282.6	0.767687
$565$	0	0
$566$	7967.06	0.591661
$567$	− 9899.87i	− 0.733255i
$568$	2557.24i	0.188908i
$569$	−1084.26	−0.0798850	−0.0399425	−	0.999202i	$-0.512717\pi$
$569$	−1084.26	−0.0798850	−0.0399425	+	0.999202i	$0.512717\pi$
$570$	0	0
$571$	14671.2	1.07525	0.537627	−	0.843183i	$-0.319321\pi$
$571$	14671.2	1.07525	0.537627	+	0.843183i	$0.319321\pi$
$572$	− 1985.38i	− 0.145127i
$573$	− 124.143i	− 0.00905083i
$574$	8584.40	0.624226
$575$	0	0
$576$	2074.65	0.150076
$577$	25311.7i	1.82624i	0.407696	+	0.913118i	$0.366332\pi$
$577$	25311.7i	1.82624i	−0.407696	+	0.913118i	$0.633668\pi$
$578$	− 88.3161i	− 0.00635548i
$579$	4077.66	0.292680
$580$	0	0
$581$	−21553.3	−1.53904
$582$	3729.15i	0.265598i
$583$	22362.7i	1.58862i
$584$	−1932.59	−0.136937
$585$	0	0
$586$	−13420.8	−0.946092
$587$	− 11336.2i	− 0.797093i	−0.917148	−	0.398546i	$-0.869515\pi$
$587$	− 11336.2i	− 0.797093i	0.917148	−	0.398546i	$-0.130485\pi$
$588$	709.155i	0.0497365i
$589$	14024.8	0.981120
$590$	0	0
$591$	−36357.4	−2.53053
$592$	− 2047.58i	− 0.142154i
$593$	9575.02i	0.663067i	0.943443	+	0.331534i	$0.107566\pi$
$593$	9575.02i	0.663067i	−0.943443	+	0.331534i	$0.892434\pi$
$594$	3188.12	0.220219
$595$	0	0
$596$	9228.73	0.634268
$597$	− 31392.8i	− 2.15213i
$598$	− 4345.24i	− 0.297141i
$599$	−15495.6	−1.05698	−0.528491	−	0.848939i	$-0.677242\pi$
$599$	−15495.6	−1.05698	−0.528491	+	0.848939i	$0.677242\pi$
$600$	0	0
$601$	11100.9	0.753433	0.376717	−	0.926329i	$-0.377053\pi$
$601$	11100.9	0.753433	0.376717	+	0.926329i	$0.377053\pi$
$602$	7280.11i	0.492882i
$603$	20023.6i	1.35228i
$604$	−4372.37	−0.294552
$605$	0	0
$606$	4364.42	0.292562
$607$	25928.7i	1.73379i	0.498487	+	0.866897i	$0.333889\pi$
$607$	25928.7i	1.73379i	−0.498487	+	0.866897i	$0.666111\pi$
$608$	− 4893.70i	− 0.326424i
$609$	−25827.2	−1.71851
$610$	0	0
$611$	4335.44	0.287059
$612$	9047.69i	0.597600i
$613$	14208.7i	0.936188i	0.883679	+	0.468094i	$0.155059\pi$
$613$	14208.7i	0.936188i	−0.883679	+	0.468094i	$0.844941\pi$
$614$	7767.63	0.510547
$615$	0	0
$616$	−5463.93	−0.357383
$617$	− 21425.7i	− 1.39800i	−0.715121	−	0.699000i	$-0.753629\pi$
$617$	− 21425.7i	− 1.39800i	0.715121	−	0.699000i	$-0.246371\pi$
$618$	29785.6i	1.93876i
$619$	−22816.0	−1.48151	−0.740755	−	0.671776i	$-0.765532\pi$
$619$	−22816.0	−1.48151	−0.740755	+	0.671776i	$0.765532\pi$
$620$	0	0
$621$	6977.58	0.450887
$622$	18376.0i	1.18458i
$623$	− 9971.73i	− 0.641266i
$624$	1603.31	0.102858
$625$	0	0
$626$	14950.9	0.954565
$627$	− 45007.0i	− 2.86668i
$628$	1392.19i	0.0884622i
$629$	8929.63	0.566054
$630$	0	0
$631$	27919.5	1.76142	0.880710	−	0.473656i	$-0.157066\pi$
$631$	27919.5	1.76142	0.880710	+	0.473656i	$0.157066\pi$
$632$	483.727i	0.0304456i
$633$	36148.2i	2.26977i
$634$	9336.85	0.584880
$635$	0	0
$636$	−18059.2	−1.12593
$637$	299.000i	0.0185978i
$638$	14302.7i	0.887540i
$639$	−10362.1	−0.641499
$640$	0	0
$641$	−27914.7	−1.72007	−0.860033	−	0.510238i	$-0.829557\pi$
$641$	−27914.7	−1.72007	−0.860033	+	0.510238i	$0.829557\pi$
$642$	7223.32i	0.444052i
$643$	25491.8i	1.56345i	0.623622	+	0.781726i	$0.285661\pi$
$643$	25491.8i	1.56345i	−0.623622	+	0.781726i	$0.714339\pi$
$644$	−11958.5	−0.731723
$645$	0	0
$646$	21341.7	1.29981
$647$	− 30447.3i	− 1.85009i	−0.379857	−	0.925045i	$-0.624027\pi$
$647$	− 30447.3i	− 1.85009i	0.379857	−	0.925045i	$-0.375973\pi$
$648$	− 4427.36i	− 0.268400i
$649$	21720.6	1.31373
$650$	0	0
$651$	12645.5	0.761316
$652$	9555.92i	0.573986i
$653$	− 14357.5i	− 0.860418i	−0.902729	−	0.430209i	$-0.858440\pi$
$653$	− 14357.5i	− 0.860418i	0.902729	−	0.430209i	$-0.141560\pi$
$654$	−23192.1	−1.38667
$655$	0	0
$656$	3839.06	0.228491
$657$	− 7830.97i	− 0.465016i
$658$	− 11931.5i	− 0.706897i
$659$	7815.23	0.461970	0.230985	−	0.972957i	$-0.425805\pi$
$659$	7815.23	0.461970	0.230985	+	0.972957i	$0.425805\pi$
$660$	0	0
$661$	−16755.3	−0.985938	−0.492969	−	0.870047i	$-0.664088\pi$
$661$	−16755.3	−0.985938	−0.492969	+	0.870047i	$0.664088\pi$
$662$	− 3089.34i	− 0.181375i
$663$	6992.13i	0.409580i
$664$	−9638.93	−0.563348
$665$	0	0
$666$	8296.89	0.482730
$667$	31303.3i	1.81719i
$668$	− 17059.8i	− 0.988118i
$669$	−16637.8	−0.961516
$670$	0	0
$671$	10648.9	0.612665
$672$	− 4412.43i	− 0.253294i
$673$	15704.8i	0.899517i	0.893150	+	0.449758i	$0.148490\pi$
$673$	15704.8i	0.899517i	−0.893150	+	0.449758i	$0.851510\pi$
$674$	7577.98	0.433076
$675$	0	0
$676$	676.000	0.0384615
$677$	12623.1i	0.716608i	0.933605	+	0.358304i	$0.116645\pi$
$677$	12623.1i	0.716608i	−0.933605	+	0.358304i	$0.883355\pi$
$678$	− 20225.8i	− 1.14568i
$679$	4327.14	0.244566
$680$	0	0
$681$	33628.9	1.89231
$682$	− 7002.90i	− 0.393189i
$683$	− 6822.45i	− 0.382216i	−0.981569	−	0.191108i	$-0.938792\pi$
$683$	− 6822.45i	− 0.382216i	0.981569	−	0.191108i	$-0.0612081\pi$
$684$	19829.5	1.10848
$685$	0	0
$686$	13094.4	0.728786
$687$	− 8261.91i	− 0.458823i
$688$	3255.76i	0.180414i
$689$	−7614.26	−0.421016
$690$	0	0
$691$	−26024.5	−1.43273	−0.716365	−	0.697725i	$-0.754196\pi$
$691$	−26024.5	−1.43273	−0.716365	+	0.697725i	$0.754196\pi$
$692$	11486.7i	0.631011i
$693$	− 22140.1i	− 1.21361i
$694$	9091.17	0.497257
$695$	0	0
$696$	−11550.3	−0.629040
$697$	16742.4i	0.909847i
$698$	3090.09i	0.167567i
$699$	−11353.3	−0.614335
$700$	0	0
$701$	−3921.15	−0.211269	−0.105635	−	0.994405i	$-0.533687\pi$
$701$	−3921.15	−0.211269	−0.105635	+	0.994405i	$0.533687\pi$
$702$	1085.52i	0.0583623i
$703$	− 19570.8i	− 1.04996i
$704$	−2443.54	−0.130816
$705$	0	0
$706$	8542.56	0.455387
$707$	− 5064.29i	− 0.269395i
$708$	17540.6i	0.931099i
$709$	28145.7	1.49088	0.745440	−	0.666572i	$-0.232239\pi$
$709$	28145.7	1.49088	0.745440	+	0.666572i	$0.232239\pi$
$710$	0	0
$711$	−1960.09	−0.103388
$712$	− 4459.49i	− 0.234728i
$713$	− 15326.7i	− 0.805034i
$714$	19242.9	1.00861
$715$	0	0
$716$	3931.38	0.205199
$717$	− 18351.9i	− 0.955877i
$718$	2213.35i	0.115044i
$719$	−24216.6	−1.25609	−0.628043	−	0.778179i	$-0.716144\pi$
$719$	−24216.6	−1.25609	−0.628043	+	0.778179i	$0.716144\pi$
$720$	0	0
$721$	34561.9	1.78523
$722$	− 33056.0i	− 1.70390i
$723$	− 33737.0i	− 1.73540i
$724$	−13371.4	−0.686386
$725$	0	0
$726$	−1953.85	−0.0998818
$727$	− 12683.1i	− 0.647027i	−0.946224	−	0.323513i	$-0.895136\pi$
$727$	− 12683.1i	− 0.647027i	0.946224	−	0.323513i	$-0.104864\pi$
$728$	− 1860.41i	− 0.0947134i
$729$	26629.1	1.35290
$730$	0	0
$731$	−14198.6	−0.718406
$732$	8599.63i	0.434224i
$733$	19816.4i	0.998546i	0.866445	+	0.499273i	$0.166400\pi$
$733$	19816.4i	0.998546i	−0.866445	+	0.499273i	$0.833600\pi$
$734$	166.255	0.00836047
$735$	0	0
$736$	−5347.99	−0.267839
$737$	− 23583.9i	− 1.17873i
$738$	15556.1i	0.775917i
$739$	−19586.7	−0.974976	−0.487488	−	0.873130i	$-0.662087\pi$
$739$	−19586.7	−0.974976	−0.487488	+	0.873130i	$0.662087\pi$
$740$	0	0
$741$	15324.4	0.759725
$742$	20955.1i	1.03677i
$743$	− 12246.4i	− 0.604680i	−0.953200	−	0.302340i	$-0.902232\pi$
$743$	− 12246.4i	− 0.604680i	0.953200	−	0.302340i	$-0.0977678\pi$
$744$	5655.24	0.278671
$745$	0	0
$746$	−5018.33	−0.246293
$747$	− 39057.5i	− 1.91304i
$748$	− 10656.5i	− 0.520907i
$749$	8381.63	0.408889
$750$	0	0
$751$	−6566.74	−0.319073	−0.159537	−	0.987192i	$-0.551000\pi$
$751$	−6566.74	−0.319073	−0.159537	+	0.987192i	$0.551000\pi$
$752$	− 5335.93i	− 0.258752i
$753$	− 8384.99i	− 0.405798i
$754$	−4869.93	−0.235215
$755$	0	0
$756$	2987.44	0.143720
$757$	7608.85i	0.365321i	0.983176	+	0.182661i	$0.0584710\pi$
$757$	7608.85i	0.365321i	−0.983176	+	0.182661i	$0.941529\pi$
$758$	11919.5i	0.571157i
$759$	−49185.1	−2.35218
$760$	0	0
$761$	5763.27	0.274532	0.137266	−	0.990534i	$-0.456169\pi$
$761$	5763.27	0.274532	0.137266	+	0.990534i	$0.456169\pi$
$762$	28628.7i	1.36104i
$763$	26911.1i	1.27686i
$764$	−64.4210	−0.00305061
$765$	0	0
$766$	4383.16	0.206749
$767$	7395.64i	0.348163i
$768$	− 1973.30i	− 0.0927153i
$769$	11841.4	0.555282	0.277641	−	0.960685i	$-0.410447\pi$
$769$	11841.4	0.555282	0.277641	+	0.960685i	$0.410447\pi$
$770$	0	0
$771$	−39408.1	−1.84079
$772$	− 2116.01i	− 0.0986489i
$773$	5247.85i	0.244181i	0.992519	+	0.122091i	$0.0389598\pi$
$773$	5247.85i	0.244181i	−0.992519	+	0.122091i	$0.961040\pi$
$774$	−13192.5	−0.612656
$775$	0	0
$776$	1935.16	0.0895208
$777$	− 17646.1i	− 0.814737i
$778$	− 8315.05i	− 0.383173i
$779$	36693.7	1.68766
$780$	0	0
$781$	12204.6	0.559172
$782$	− 23322.9i	− 1.06653i
$783$	− 7820.13i	− 0.356920i
$784$	368.000	0.0167638
$785$	0	0
$786$	−31191.5	−1.41548
$787$	33333.8i	1.50981i	0.655835	+	0.754905i	$0.272317\pi$
$787$	33333.8i	1.50981i	−0.655835	+	0.754905i	$0.727683\pi$
$788$	18866.9i	0.852924i
$789$	29220.9	1.31849
$790$	0	0
$791$	−23469.2	−1.05495
$792$	− 9901.36i	− 0.444229i
$793$	3625.85i	0.162368i
$794$	−27244.1	−1.21771
$795$	0	0
$796$	−16290.6	−0.725383
$797$	6999.18i	0.311071i	0.987830	+	0.155535i	$0.0497103\pi$
$797$	6999.18i	0.311071i	−0.987830	+	0.155535i	$0.950290\pi$
$798$	− 42174.0i	− 1.87086i
$799$	23270.3	1.03034
$800$	0	0
$801$	18070.1	0.797098
$802$	3358.82i	0.147886i
$803$	9223.39i	0.405338i
$804$	19045.4	0.835421
$805$	0	0
$806$	2384.41	0.104203
$807$	27158.3i	1.18466i
$808$	− 2264.82i	− 0.0986089i
$809$	−10977.3	−0.477058	−0.238529	−	0.971135i	$-0.576665\pi$
$809$	−10977.3	−0.477058	−0.238529	+	0.971135i	$0.576665\pi$
$810$	0	0
$811$	31470.2	1.36260	0.681299	−	0.732005i	$-0.261415\pi$
$811$	31470.2	1.36260	0.681299	+	0.732005i	$0.261415\pi$
$812$	13402.5i	0.579229i
$813$	− 32795.2i	− 1.41473i
$814$	−9772.16	−0.420779
$815$	0	0
$816$	8605.70	0.369191
$817$	31118.6i	1.33256i
$818$	− 30921.2i	− 1.32168i
$819$	7538.47	0.321631
$820$	0	0
$821$	970.042	0.0412359	0.0206180	−	0.999787i	$-0.493437\pi$
$821$	970.042	0.0412359	0.0206180	+	0.999787i	$0.493437\pi$
$822$	22319.0i	0.947039i
$823$	2108.31i	0.0892964i	0.999003	+	0.0446482i	$0.0142167\pi$
$823$	2108.31i	0.0892964i	−0.999003	+	0.0446482i	$0.985783\pi$
$824$	15456.6	0.653464
$825$	0	0
$826$	20353.4	0.857368
$827$	35137.2i	1.47744i	0.674013	+	0.738719i	$0.264569\pi$
$827$	35137.2i	1.47744i	−0.674013	+	0.738719i	$0.735431\pi$
$828$	− 21670.3i	− 0.909536i
$829$	20626.9	0.864175	0.432088	−	0.901832i	$-0.357777\pi$
$829$	20626.9	0.864175	0.432088	+	0.901832i	$0.357777\pi$
$830$	0	0
$831$	−10119.1	−0.422416
$832$	− 832.000i	− 0.0346688i
$833$	1604.87i	0.0667534i
$834$	23541.2	0.977417
$835$	0	0
$836$	−23355.4	−0.966223
$837$	3828.89i	0.158119i
$838$	− 27729.2i	− 1.14307i
$839$	36738.4	1.51174	0.755870	−	0.654722i	$-0.227214\pi$
$839$	36738.4	1.51174	0.755870	+	0.654722i	$0.227214\pi$
$840$	0	0
$841$	10694.1	0.438482
$842$	7252.73i	0.296847i
$843$	− 23990.8i	− 0.980176i
$844$	18758.3	0.765032
$845$	0	0
$846$	21621.4	0.878677
$847$	2267.16i	0.0919725i
$848$	9371.39i	0.379499i
$849$	30705.8	1.24125
$850$	0	0
$851$	−21387.6	−0.861523
$852$	9855.88i	0.396311i
$853$	14073.1i	0.564893i	0.959283	+	0.282446i	$0.0911459\pi$
$853$	14073.1i	0.564893i	−0.959283	+	0.282446i	$0.908854\pi$
$854$	9978.65	0.399839
$855$	0	0
$856$	3748.38	0.149669
$857$	48358.8i	1.92754i	0.266726	+	0.963772i	$0.414058\pi$
$857$	48358.8i	1.92754i	−0.266726	+	0.963772i	$0.585942\pi$
$858$	− 7651.85i	− 0.304464i
$859$	−25856.3	−1.02702	−0.513508	−	0.858085i	$-0.671654\pi$
$859$	−25856.3	−1.02702	−0.513508	+	0.858085i	$0.671654\pi$
$860$	0	0
$861$	33085.1	1.30957
$862$	− 26378.9i	− 1.04231i
$863$	15097.4i	0.595507i	0.954643	+	0.297754i	$0.0962374\pi$
$863$	15097.4i	0.595507i	−0.954643	+	0.297754i	$0.903763\pi$
$864$	1336.02	0.0526070
$865$	0	0
$866$	6123.36	0.240277
$867$	− 340.379i	− 0.0133332i
$868$	− 6562.10i	− 0.256604i
$869$	2308.61	0.0901199
$870$	0	0
$871$	8030.08	0.312387
$872$	12035.0i	0.467381i
$873$	7841.36i	0.303997i
$874$	−51116.1	−1.97829
$875$	0	0
$876$	−7448.42	−0.287282
$877$	8455.97i	0.325585i	0.986660	+	0.162792i	$0.0520501\pi$
$877$	8455.97i	0.325585i	−0.986660	+	0.162792i	$0.947950\pi$
$878$	25502.9i	0.980273i
$879$	−51725.3	−1.98481
$880$	0	0
$881$	−5047.67	−0.193031	−0.0965154	−	0.995331i	$-0.530770\pi$
$881$	−5047.67	−0.193031	−0.0965154	+	0.995331i	$0.530770\pi$
$882$	1491.15i	0.0569272i
$883$	25885.9i	0.986558i	0.869871	+	0.493279i	$0.164202\pi$
$883$	25885.9i	0.986558i	−0.869871	+	0.493279i	$0.835798\pi$
$884$	3628.41	0.138050
$885$	0	0
$886$	12750.5	0.483477
$887$	12787.6i	0.484065i	0.970268	+	0.242032i	$0.0778141\pi$
$887$	12787.6i	0.484065i	−0.970268	+	0.242032i	$0.922186\pi$
$888$	− 7891.58i	− 0.298225i
$889$	33219.5	1.25326
$890$	0	0
$891$	−21129.7	−0.794470
$892$	8633.81i	0.324082i
$893$	− 51000.8i	− 1.91117i
$894$	35568.5	1.33063
$895$	0	0
$896$	−2289.73	−0.0853735
$897$	− 16747.0i	− 0.623373i
$898$	29643.9i	1.10159i
$899$	−17177.4	−0.637262
$900$	0	0
$901$	−40869.3	−1.51116
$902$	− 18322.1i	− 0.676339i
$903$	28058.3i	1.03402i
$904$	−10495.8	−0.386154
$905$	0	0
$906$	−16851.6	−0.617943
$907$	− 4778.50i	− 0.174937i	−0.996167	−	0.0874683i	$-0.972122\pi$
$907$	− 4778.50i	− 0.174937i	0.996167	−	0.0874683i	$-0.0278776\pi$
$908$	− 17450.9i	− 0.637808i
$909$	9177.15	0.334859
$910$	0	0
$911$	−581.524	−0.0211490	−0.0105745	−	0.999944i	$-0.503366\pi$
$911$	−581.524	−0.0211490	−0.0105745	+	0.999944i	$0.503366\pi$
$912$	− 18860.8i	− 0.684807i
$913$	46002.2i	1.66753i
$914$	19583.1	0.708698
$915$	0	0
$916$	−4287.33	−0.154648
$917$	36193.3i	1.30339i
$918$	5826.50i	0.209480i
$919$	−11885.1	−0.426610	−0.213305	−	0.976986i	$-0.568423\pi$
$919$	−11885.1	−0.426610	−0.213305	+	0.976986i	$0.568423\pi$
$920$	0	0
$921$	29937.2	1.07108
$922$	− 15065.2i	− 0.538117i
$923$	4155.52i	0.148191i
$924$	−21058.5	−0.749756
$925$	0	0
$926$	−4091.80	−0.145210
$927$	62630.7i	2.21905i
$928$	5993.76i	0.212020i
$929$	−15219.6	−0.537502	−0.268751	−	0.963210i	$-0.586611\pi$
$929$	−15219.6	−0.537502	−0.268751	+	0.963210i	$0.586611\pi$
$930$	0	0
$931$	3517.34	0.123820
$932$	5891.53i	0.207064i
$933$	70823.1i	2.48515i
$934$	34578.0	1.21138
$935$	0	0
$936$	3371.31	0.117729
$937$	− 48722.0i	− 1.69870i	−0.527832	−	0.849349i	$-0.676995\pi$
$937$	− 48722.0i	− 1.69870i	0.527832	−	0.849349i	$-0.323005\pi$
$938$	− 22099.4i	− 0.769267i
$939$	57622.3	2.00259
$940$	0	0
$941$	−27151.1	−0.940597	−0.470299	−	0.882507i	$-0.655854\pi$
$941$	−27151.1	−0.940597	−0.470299	+	0.882507i	$0.655854\pi$
$942$	5365.63i	0.185585i
$943$	− 40100.1i	− 1.38477i
$944$	9102.33	0.313830
$945$	0	0
$946$	15538.3	0.534031
$947$	26266.5i	0.901317i	0.892696	+	0.450659i	$0.148811\pi$
$947$	26266.5i	0.901317i	−0.892696	+	0.450659i	$0.851189\pi$
$948$	1864.33i	0.0638721i
$949$	−3140.47	−0.107422
$950$	0	0
$951$	35985.2	1.22702
$952$	− 9985.68i	− 0.339956i
$953$	27675.9i	0.940725i	0.882473	+	0.470363i	$0.155877\pi$
$953$	27675.9i	0.940725i	−0.882473	+	0.470363i	$0.844123\pi$
$954$	−37973.4	−1.28871
$955$	0	0
$956$	−9523.30	−0.322182
$957$	55124.2i	1.86198i
$958$	28977.4i	0.977261i
$959$	25898.1	0.872046
$960$	0	0
$961$	−21380.6	−0.717687
$962$	− 3327.32i	− 0.111515i
$963$	15188.6i	0.508252i
$964$	−17507.1	−0.584922
$965$	0	0
$966$	−46089.1	−1.53509
$967$	− 1676.34i	− 0.0557471i	−0.999611	−	0.0278736i	$-0.991126\pi$
$967$	− 1676.34i	− 0.0557471i	0.999611	−	0.0278736i	$-0.00887358\pi$
$968$	1013.91i	0.0336655i
$969$	82253.3	2.72689
$970$	0	0
$971$	12176.4	0.402431	0.201215	−	0.979547i	$-0.435511\pi$
$971$	12176.4	0.402431	0.201215	+	0.979547i	$0.435511\pi$
$972$	− 21572.6i	− 0.711873i
$973$	− 27316.2i	− 0.900018i
$974$	−6902.51	−0.227075
$975$	0	0
$976$	4462.59	0.146357
$977$	− 4710.79i	− 0.154259i	−0.997021	−	0.0771297i	$-0.975424\pi$
$977$	− 4710.79i	− 0.154259i	0.997021	−	0.0771297i	$-0.0245756\pi$
$978$	36829.5i	1.20417i
$979$	−21283.1	−0.694802
$980$	0	0
$981$	−48766.4	−1.58715
$982$	40657.8i	1.32122i
$983$	− 4834.94i	− 0.156878i	−0.996919	−	0.0784388i	$-0.975006\pi$
$983$	− 4834.94i	− 0.156878i	0.996919	−	0.0784388i	$-0.0249935\pi$
$984$	14796.1	0.479353
$985$	0	0
$986$	−26139.2	−0.844261
$987$	− 45985.2i	− 1.48300i
$988$	− 7952.26i	− 0.256068i
$989$	34007.4	1.09340
$990$	0	0
$991$	28637.9	0.917973	0.458987	−	0.888443i	$-0.348213\pi$
$991$	28637.9	0.917973	0.458987	+	0.888443i	$0.348213\pi$
$992$	− 2934.66i	− 0.0939270i
$993$	− 11906.6i	− 0.380509i
$994$	11436.3	0.364928
$995$	0	0
$996$	−37149.4	−1.18185
$997$	− 27133.9i	− 0.861924i	−0.902370	−	0.430962i	$-0.858174\pi$
$997$	− 27133.9i	− 0.861924i	0.902370	−	0.430962i	$-0.141826\pi$
$998$	− 9512.72i	− 0.301723i
$999$	5343.00	0.169214

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.4.b.k.599.1		4
5.2	odd	4		650.4.a.s.1.1		2
5.3	odd	4		130.4.a.e.1.2	✓	2
5.4	even	2	inner	650.4.b.k.599.4		4
15.8	even	4		1170.4.a.z.1.2		2
20.3	even	4		1040.4.a.j.1.1		2
65.38	odd	4		1690.4.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.4.a.e.1.2	✓	2	5.3	odd	4
650.4.a.s.1.1		2	5.2	odd	4
650.4.b.k.599.1		4	1.1	even	1	trivial
650.4.b.k.599.4		4	5.4	even	2	inner
1040.4.a.j.1.1		2	20.3	even	4
1170.4.a.z.1.2		2	15.8	even	4
1690.4.a.t.1.2		2	65.38	odd	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:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	650.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -7.70820i q^{3} -4.00000 q^{4} -15.4164 q^{6} +17.8885i q^{7} +8.00000i q^{8} -32.4164 q^{9} +38.1803 q^{11} +30.8328i q^{12} +13.0000i q^{13} +35.7771 q^{14} +16.0000 q^{16} +69.7771i q^{17} +64.8328i q^{18} +152.928 q^{19} +137.889 q^{21} -76.3607i q^{22} -167.125i q^{23} +61.6656 q^{24} +26.0000 q^{26} +41.7508i q^{27} -71.5542i q^{28} -187.305 q^{29} +91.7082 q^{31} -32.0000i q^{32} -294.302i q^{33} +139.554 q^{34} +129.666 q^{36} -127.974i q^{37} -305.856i q^{38} +100.207 q^{39} +239.941 q^{41} -275.777i q^{42} +203.485i q^{43} -152.721 q^{44} -334.249 q^{46} -333.495i q^{47} -123.331i q^{48} +23.0000 q^{49} +537.856 q^{51} -52.0000i q^{52} +585.712i q^{53} +83.5016 q^{54} -143.108 q^{56} -1178.80i q^{57} +374.610i q^{58} +568.895 q^{59} +278.912 q^{61} -183.416i q^{62} -579.882i q^{63} -64.0000 q^{64} -588.604 q^{66} -617.698i q^{67} -279.108i q^{68} -1288.23 q^{69} +319.656 q^{71} -259.331i q^{72} +241.574i q^{73} -255.947 q^{74} -611.712 q^{76} +682.991i q^{77} -200.413i q^{78} +60.4659 q^{79} -553.420 q^{81} -479.882i q^{82} +1204.87i q^{83} -551.554 q^{84} +406.971 q^{86} +1443.78i q^{87} +305.443i q^{88} -557.437 q^{89} -232.551 q^{91} +668.498i q^{92} -706.906i q^{93} -666.991 q^{94} -246.663 q^{96} -241.895i q^{97} -46.0000i q^{98} -1237.67 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} - 8 q^{6} - 76 q^{9} + 108 q^{11} + 64 q^{16} + 84 q^{19} + 480 q^{21} + 32 q^{24} + 104 q^{26} - 624 q^{29} + 340 q^{31} + 272 q^{34} + 304 q^{36} + 52 q^{39} + 280 q^{41} - 432 q^{44} - 1176 q^{46}+ \cdots - 2652 q^{99}+O(q^{100}) \) 4 * q - 16 * q^4 - 8 * q^6 - 76 * q^9 + 108 * q^11 + 64 * q^16 + 84 * q^19 + 480 * q^21 + 32 * q^24 + 104 * q^26 - 624 * q^29 + 340 * q^31 + 272 * q^34 + 304 * q^36 + 52 * q^39 + 280 * q^41 - 432 * q^44 - 1176 * q^46 + 92 * q^49 + 1096 * q^51 + 656 * q^54 + 764 * q^59 + 96 * q^61 - 256 * q^64 - 816 * q^66 - 1128 * q^69 + 1860 * q^71 - 1632 * q^74 - 336 * q^76 - 1064 * q^79 - 2804 * q^81 - 1920 * q^84 + 1288 * q^86 - 584 * q^89 - 736 * q^94 - 128 * q^96 - 2652 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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