LMFDB - Embedded newform 1150.4.b.o.599.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,4,Mod(599,1150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1150.599");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1150 = 2 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1150.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$67.8521965066$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 124x^{6} + 4272x^{4} + 28129x^{2} + 4 $ x^8 + 124x^6 + 4272x^4 + 28129*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 230)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.4
Root			$7.12571i$ of defining polynomial
Character	$\chi$	$=$	1150.599
Dual form			1150.4.b.o.599.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} +10.1257i q^{3} -4.00000 q^{4} +20.2514 q^{6} +24.6431i q^{7} +8.00000i q^{8} -75.5301 q^{9} -17.3867 q^{11} -40.5029i q^{12} +4.00699i q^{13} +49.2862 q^{14} +16.0000 q^{16} -48.2740i q^{17} +151.060i q^{18} -79.3172 q^{19} -249.529 q^{21} +34.7734i q^{22} -23.0000i q^{23} -81.0057 q^{24} +8.01398 q^{26} -491.402i q^{27} -98.5723i q^{28} +254.267 q^{29} -220.696 q^{31} -32.0000i q^{32} -176.053i q^{33} -96.5480 q^{34} +302.120 q^{36} +422.904i q^{37} +158.634i q^{38} -40.5736 q^{39} -170.251 q^{41} +499.058i q^{42} -228.920i q^{43} +69.5468 q^{44} -46.0000 q^{46} -580.087i q^{47} +162.011i q^{48} -264.282 q^{49} +488.809 q^{51} -16.0280i q^{52} -260.354i q^{53} -982.803 q^{54} -197.145 q^{56} -803.144i q^{57} -508.535i q^{58} -353.130 q^{59} -80.6108 q^{61} +441.392i q^{62} -1861.29i q^{63} -64.0000 q^{64} -352.106 q^{66} +820.011i q^{67} +193.096i q^{68} +232.891 q^{69} +614.845 q^{71} -604.241i q^{72} +511.586i q^{73} +845.808 q^{74} +317.269 q^{76} -428.462i q^{77} +81.1472i q^{78} -160.464 q^{79} +2936.48 q^{81} +340.502i q^{82} -32.5646i q^{83} +998.115 q^{84} -457.840 q^{86} +2574.64i q^{87} -139.094i q^{88} +25.0375 q^{89} -98.7445 q^{91} +92.0000i q^{92} -2234.70i q^{93} -1160.17 q^{94} +324.023 q^{96} +249.798i q^{97} +528.563i q^{98} +1313.22 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{4} + 56 q^{6} - 128 q^{9} + 42 q^{11} - 32 q^{14} + 128 q^{16} - 346 q^{19} - 240 q^{21} - 224 q^{24} + 280 q^{26} + 236 q^{29} + 34 q^{31} - 224 q^{34} + 512 q^{36} + 442 q^{39} + 278 q^{41}+ \cdots + 5490 q^{99}+O(q^{100}) $ 8 * q - 32 * q^4 + 56 * q^6 - 128 * q^9 + 42 * q^11 - 32 * q^14 + 128 * q^16 - 346 * q^19 - 240 * q^21 - 224 * q^24 + 280 * q^26 + 236 * q^29 + 34 * q^31 - 224 * q^34 + 512 * q^36 + 442 * q^39 + 278 * q^41 - 168 * q^44 - 368 * q^46 + 248 * q^49 + 878 * q^51 - 1556 * q^54 + 128 * q^56 + 906 * q^59 - 654 * q^61 - 512 * q^64 - 356 * q^66 + 644 * q^69 + 390 * q^71 + 1372 * q^74 + 1384 * q^76 + 2280 * q^79 + 2912 * q^81 + 960 * q^84 - 200 * q^86 + 4340 * q^89 + 1114 * q^91 - 1468 * q^94 + 896 * q^96 + 5490 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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
$3$	10.1257i	1.94869i	0.225050	+	0.974347i	$0.427745\pi$
$3$	10.1257i	1.94869i	−0.225050	+	0.974347i	$0.572255\pi$
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	20.2514	1.37794
$7$	24.6431i	1.33060i	0.746575	+	0.665301i	$0.231697\pi$
$7$	24.6431i	1.33060i	−0.746575	+	0.665301i	$0.768303\pi$
$8$	8.00000i	0.353553i
$9$	−75.5301	−2.79741
$10$	0	0
$11$	−17.3867	−0.476572	−0.238286	−	0.971195i	$-0.576586\pi$
$11$	−17.3867	−0.476572	−0.238286	+	0.971195i	$0.576586\pi$
$12$	− 40.5029i	− 0.974347i
$13$	4.00699i	0.0854876i	0.999086	+	0.0427438i	$0.0136099\pi$
$13$	4.00699i	0.0854876i	−0.999086	+	0.0427438i	$0.986390\pi$
$14$	49.2862	0.940877
$15$	0	0
$16$	16.0000	0.250000
$17$	− 48.2740i	− 0.688716i	−0.938838	−	0.344358i	$-0.888097\pi$
$17$	− 48.2740i	− 0.688716i	0.938838	−	0.344358i	$-0.111903\pi$
$18$	151.060i	1.97807i
$19$	−79.3172	−0.957717	−0.478859	−	0.877892i	$-0.658949\pi$
$19$	−79.3172	−0.957717	−0.478859	+	0.877892i	$0.658949\pi$
$20$	0	0
$21$	−249.529	−2.59294
$22$	34.7734i	0.336987i
$23$	− 23.0000i	− 0.208514i
$23$	− 23.0000i	− 0.208514i
$24$	−81.0057	−0.688968
$25$	0	0
$26$	8.01398	0.0604488
$27$	− 491.402i	− 3.50260i
$28$	− 98.5723i	− 0.665301i
$29$	254.267	1.62815	0.814074	−	0.580761i	$-0.197245\pi$
$29$	254.267	1.62815	0.814074	+	0.580761i	$0.197245\pi$
$30$	0	0
$31$	−220.696	−1.27865	−0.639325	−	0.768937i	$-0.720786\pi$
$31$	−220.696	−1.27865	−0.639325	+	0.768937i	$0.720786\pi$
$32$	− 32.0000i	− 0.176777i
$33$	− 176.053i	− 0.928693i
$34$	−96.5480	−0.486996
$35$	0	0
$36$	302.120	1.39870
$37$	422.904i	1.87905i	0.342476	+	0.939527i	$0.388735\pi$
$37$	422.904i	1.87905i	−0.342476	+	0.939527i	$0.611265\pi$
$38$	158.634i	0.677208i
$39$	−40.5736	−0.166589
$40$	0	0
$41$	−170.251	−0.648505	−0.324252	−	0.945971i	$-0.605113\pi$
$41$	−170.251	−0.648505	−0.324252	+	0.945971i	$0.605113\pi$
$42$	499.058i	1.83348i
$43$	− 228.920i	− 0.811860i	−0.913904	−	0.405930i	$-0.866948\pi$
$43$	− 228.920i	− 0.811860i	0.913904	−	0.405930i	$-0.133052\pi$
$44$	69.5468	0.238286
$45$	0	0
$46$	−46.0000	−0.147442
$47$	− 580.087i	− 1.80031i	−0.435572	−	0.900154i	$-0.643454\pi$
$47$	− 580.087i	− 1.80031i	0.435572	−	0.900154i	$-0.356546\pi$
$48$	162.011i	0.487174i
$49$	−264.282	−0.770500
$50$	0	0
$51$	488.809	1.34210
$52$	− 16.0280i	− 0.0427438i
$53$	− 260.354i	− 0.674762i	−0.941368	−	0.337381i	$-0.890459\pi$
$53$	− 260.354i	− 0.674762i	0.941368	−	0.337381i	$-0.109541\pi$
$54$	−982.803	−2.47671
$55$	0	0
$56$	−197.145	−0.470439
$57$	− 803.144i	− 1.86630i
$58$	− 508.535i	− 1.15127i
$59$	−353.130	−0.779215	−0.389607	−	0.920981i	$-0.627389\pi$
$59$	−353.130	−0.779215	−0.389607	+	0.920981i	$0.627389\pi$
$60$	0	0
$61$	−80.6108	−0.169199	−0.0845996	−	0.996415i	$-0.526961\pi$
$61$	−80.6108	−0.169199	−0.0845996	+	0.996415i	$0.526961\pi$
$62$	441.392i	0.904142i
$63$	− 1861.29i	− 3.72224i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−352.106	−0.656685
$67$	820.011i	1.49523i	0.664133	+	0.747614i	$0.268801\pi$
$67$	820.011i	1.49523i	−0.664133	+	0.747614i	$0.731199\pi$
$68$	193.096i	0.344358i
$69$	232.891	0.406331
$70$	0	0
$71$	614.845	1.02773	0.513864	−	0.857872i	$-0.328214\pi$
$71$	614.845	1.02773	0.513864	+	0.857872i	$0.328214\pi$
$72$	− 604.241i	− 0.989034i
$73$	511.586i	0.820228i	0.912034	+	0.410114i	$0.134511\pi$
$73$	511.586i	0.820228i	−0.912034	+	0.410114i	$0.865489\pi$
$74$	845.808	1.32869
$75$	0	0
$76$	317.269	0.478859
$77$	− 428.462i	− 0.634127i
$78$	81.1472i	0.117796i
$79$	−160.464	−0.228526	−0.114263	−	0.993451i	$-0.536451\pi$
$79$	−160.464	−0.228526	−0.114263	+	0.993451i	$0.536451\pi$
$80$	0	0
$81$	2936.48	4.02809
$82$	340.502i	0.458562i
$83$	− 32.5646i	− 0.0430654i	−0.999768	−	0.0215327i	$-0.993145\pi$
$83$	− 32.5646i	− 0.0430654i	0.999768	−	0.0215327i	$-0.00685460\pi$
$84$	998.115	1.29647
$85$	0	0
$86$	−457.840	−0.574072
$87$	2574.64i	3.17276i
$88$	− 139.094i	− 0.168494i
$89$	25.0375	0.0298199	0.0149100	−	0.999889i	$-0.495254\pi$
$89$	25.0375	0.0298199	0.0149100	+	0.999889i	$0.495254\pi$
$90$	0	0
$91$	−98.7445	−0.113750
$92$	92.0000i	0.104257i
$93$	− 2234.70i	− 2.49170i
$94$	−1160.17	−1.27301
$95$	0	0
$96$	324.023	0.344484
$97$	249.798i	0.261476i	0.991417	+	0.130738i	$0.0417347\pi$
$97$	249.798i	0.261476i	−0.991417	+	0.130738i	$0.958265\pi$
$98$	528.563i	0.544826i
$99$	1313.22	1.33317
$100$	0	0
$101$	620.493	0.611300	0.305650	−	0.952144i	$-0.401126\pi$
$101$	620.493	0.611300	0.305650	+	0.952144i	$0.401126\pi$
$102$	− 977.618i	− 0.949006i
$103$	1473.24i	1.40935i	0.709532	+	0.704673i	$0.248906\pi$
$103$	1473.24i	1.40935i	−0.709532	+	0.704673i	$0.751094\pi$
$104$	−32.0559	−0.0302244
$105$	0	0
$106$	−520.708	−0.477129
$107$	940.141i	0.849410i	0.905332	+	0.424705i	$0.139622\pi$
$107$	940.141i	0.849410i	−0.905332	+	0.424705i	$0.860378\pi$
$108$	1965.61i	1.75130i
$109$	636.264	0.559111	0.279555	−	0.960130i	$-0.409813\pi$
$109$	636.264	0.559111	0.279555	+	0.960130i	$0.409813\pi$
$110$	0	0
$111$	−4282.20	−3.66170
$112$	394.289i	0.332650i
$113$	832.451i	0.693013i	0.938048	+	0.346506i	$0.112632\pi$
$113$	832.451i	0.693013i	−0.938048	+	0.346506i	$0.887368\pi$
$114$	−1606.29	−1.31967
$115$	0	0
$116$	−1017.07	−0.814074
$117$	− 302.648i	− 0.239144i
$118$	706.261i	0.550988i
$119$	1189.62	0.916406
$120$	0	0
$121$	−1028.70	−0.772879
$122$	161.222i	0.119642i
$123$	− 1723.91i	− 1.26374i
$124$	882.783	0.639325
$125$	0	0
$126$	−3722.59	−2.63202
$127$	− 1614.60i	− 1.12813i	−0.825729	−	0.564067i	$-0.809236\pi$
$127$	− 1614.60i	− 1.12813i	0.825729	−	0.564067i	$-0.190764\pi$
$128$	128.000i	0.0883883i
$129$	2317.98	1.58207
$130$	0	0
$131$	−1974.56	−1.31693	−0.658464	−	0.752612i	$-0.728794\pi$
$131$	−1974.56	−1.31693	−0.658464	+	0.752612i	$0.728794\pi$
$132$	704.211i	0.464346i
$133$	− 1954.62i	− 1.27434i
$134$	1640.02	1.05729
$135$	0	0
$136$	386.192	0.243498
$137$	− 753.805i	− 0.470087i	−0.971985	−	0.235044i	$-0.924477\pi$
$137$	− 753.805i	− 0.470087i	0.971985	−	0.235044i	$-0.0755233\pi$
$138$	− 465.783i	− 0.287319i
$139$	−1014.10	−0.618810	−0.309405	−	0.950930i	$-0.600130\pi$
$139$	−1014.10	−0.618810	−0.309405	+	0.950930i	$0.600130\pi$
$140$	0	0
$141$	5873.80	3.50825
$142$	− 1229.69i	− 0.726714i
$143$	− 69.6683i	− 0.0407410i
$144$	−1208.48	−0.699352
$145$	0	0
$146$	1023.17	0.579989
$147$	− 2676.04i	− 1.50147i
$148$	− 1691.62i	− 0.939527i
$149$	2771.80	1.52399	0.761995	−	0.647583i	$-0.224220\pi$
$149$	2771.80	1.52399	0.761995	+	0.647583i	$0.224220\pi$
$150$	0	0
$151$	3108.62	1.67534	0.837668	−	0.546180i	$-0.183918\pi$
$151$	3108.62	1.67534	0.837668	+	0.546180i	$0.183918\pi$
$152$	− 634.538i	− 0.338604i
$153$	3646.14i	1.92662i
$154$	−856.924	−0.448396
$155$	0	0
$156$	162.294	0.0832946
$157$	− 3712.87i	− 1.88739i	−0.330822	−	0.943693i	$-0.607326\pi$
$157$	− 3712.87i	− 1.88739i	0.330822	−	0.943693i	$-0.392674\pi$
$158$	320.928i	0.161593i
$159$	2636.27	1.31490
$160$	0	0
$161$	566.791	0.277450
$162$	− 5872.96i	− 2.84829i
$163$	915.791i	0.440063i	0.975493	+	0.220032i	$0.0706161\pi$
$163$	915.791i	0.440063i	−0.975493	+	0.220032i	$0.929384\pi$
$164$	681.003	0.324252
$165$	0	0
$166$	−65.1292	−0.0304518
$167$	− 1432.32i	− 0.663688i	−0.943334	−	0.331844i	$-0.892329\pi$
$167$	− 1432.32i	− 0.663688i	0.943334	−	0.331844i	$-0.107671\pi$
$168$	− 1996.23i	− 0.916741i
$169$	2180.94	0.992692
$170$	0	0
$171$	5990.84	2.67913
$172$	915.680i	0.405930i
$173$	− 3479.54i	− 1.52916i	−0.644529	−	0.764580i	$-0.722946\pi$
$173$	− 3479.54i	− 1.52916i	0.644529	−	0.764580i	$-0.277054\pi$
$174$	5149.28	2.24348
$175$	0	0
$176$	−278.187	−0.119143
$177$	− 3575.70i	− 1.51845i
$178$	− 50.0751i	− 0.0210859i
$179$	−3642.71	−1.52105	−0.760527	−	0.649307i	$-0.775059\pi$
$179$	−3642.71	−1.52105	−0.760527	+	0.649307i	$0.775059\pi$
$180$	0	0
$181$	2409.15	0.989342	0.494671	−	0.869080i	$-0.335289\pi$
$181$	2409.15	0.989342	0.494671	+	0.869080i	$0.335289\pi$
$182$	197.489i	0.0804333i
$183$	− 816.241i	− 0.329717i
$184$	184.000	0.0737210
$185$	0	0
$186$	−4469.41	−1.76190
$187$	839.326i	0.328223i
$188$	2320.35i	0.900154i
$189$	12109.6	4.66057
$190$	0	0
$191$	−189.608	−0.0718299	−0.0359150	−	0.999355i	$-0.511435\pi$
$191$	−189.608	−0.0718299	−0.0359150	+	0.999355i	$0.511435\pi$
$192$	− 648.046i	− 0.243587i
$193$	− 1855.45i	− 0.692012i	−0.938232	−	0.346006i	$-0.887538\pi$
$193$	− 1855.45i	− 0.692012i	0.938232	−	0.346006i	$-0.112462\pi$
$194$	499.597	0.184891
$195$	0	0
$196$	1057.13	0.385250
$197$	− 2429.85i	− 0.878779i	−0.898297	−	0.439390i	$-0.855195\pi$
$197$	− 2429.85i	− 0.878779i	0.898297	−	0.439390i	$-0.144805\pi$
$198$	− 2626.44i	− 0.942691i
$199$	−4333.49	−1.54368	−0.771842	−	0.635815i	$-0.780664\pi$
$199$	−4333.49	−1.54368	−0.771842	+	0.635815i	$0.780664\pi$
$200$	0	0
$201$	−8303.20	−2.91374
$202$	− 1240.99i	− 0.432255i
$203$	6265.94i	2.16642i
$204$	−1955.24	−0.671048
$205$	0	0
$206$	2946.48	0.996558
$207$	1737.19i	0.583300i
$208$	64.1118i	0.0213719i
$209$	1379.07	0.456421
$210$	0	0
$211$	−816.788	−0.266493	−0.133247	−	0.991083i	$-0.542540\pi$
$211$	−816.788	−0.266493	−0.133247	+	0.991083i	$0.542540\pi$
$212$	1041.42i	0.337381i
$213$	6225.75i	2.00273i
$214$	1880.28	0.600624
$215$	0	0
$216$	3931.21	1.23836
$217$	− 5438.63i	− 1.70137i
$218$	− 1272.53i	− 0.395351i
$219$	−5180.18	−1.59837
$220$	0	0
$221$	193.433	0.0588767
$222$	8564.41i	2.58921i
$223$	− 4513.80i	− 1.35546i	−0.735313	−	0.677728i	$-0.762965\pi$
$223$	− 4513.80i	− 1.35546i	0.735313	−	0.677728i	$-0.237035\pi$
$224$	788.579	0.235219
$225$	0	0
$226$	1664.90	0.490034
$227$	− 2792.85i	− 0.816599i	−0.912848	−	0.408300i	$-0.866122\pi$
$227$	− 2792.85i	− 0.816599i	0.912848	−	0.408300i	$-0.133878\pi$
$228$	3212.57i	0.933149i
$229$	−1404.50	−0.405292	−0.202646	−	0.979252i	$-0.564954\pi$
$229$	−1404.50	−0.405292	−0.202646	+	0.979252i	$0.564954\pi$
$230$	0	0
$231$	4338.48	1.23572
$232$	2034.14i	0.575637i
$233$	1073.79i	0.301916i	0.988540	+	0.150958i	$0.0482359\pi$
$233$	1073.79i	0.301916i	−0.988540	+	0.150958i	$0.951764\pi$
$234$	−605.296	−0.169100
$235$	0	0
$236$	1412.52	0.389607
$237$	− 1624.81i	− 0.445328i
$238$	− 2379.24i	− 0.647997i
$239$	2573.18	0.696423	0.348212	−	0.937416i	$-0.386789\pi$
$239$	2573.18	0.696423	0.348212	+	0.937416i	$0.386789\pi$
$240$	0	0
$241$	−696.127	−0.186064	−0.0930321	−	0.995663i	$-0.529656\pi$
$241$	−696.127	−0.186064	−0.0930321	+	0.995663i	$0.529656\pi$
$242$	2057.40i	0.546508i
$243$	16466.1i	4.34692i
$244$	322.443	0.0845996
$245$	0	0
$246$	−3447.82	−0.893598
$247$	− 317.823i	− 0.0818729i
$248$	− 1765.57i	− 0.452071i
$249$	329.740	0.0839213
$250$	0	0
$251$	6467.81	1.62647	0.813236	−	0.581934i	$-0.197704\pi$
$251$	6467.81	1.62647	0.813236	+	0.581934i	$0.197704\pi$
$252$	7445.18i	1.86112i
$253$	399.894i	0.0993721i
$254$	−3229.21	−0.797711
$255$	0	0
$256$	256.000	0.0625000
$257$	− 5332.72i	− 1.29434i	−0.762344	−	0.647172i	$-0.775952\pi$
$257$	− 5332.72i	− 1.29434i	0.762344	−	0.647172i	$-0.224048\pi$
$258$	− 4635.96i	− 1.11869i
$259$	−10421.7	−2.50027
$260$	0	0
$261$	−19204.8	−4.55460
$262$	3949.11i	0.931209i
$263$	− 6872.85i	− 1.61140i	−0.592325	−	0.805699i	$-0.701790\pi$
$263$	− 6872.85i	− 1.61140i	0.592325	−	0.805699i	$-0.298210\pi$
$264$	1408.42	0.328342
$265$	0	0
$266$	−3909.24	−0.901094
$267$	253.523i	0.0581099i
$268$	− 3280.04i	− 0.747614i
$269$	1926.13	0.436572	0.218286	−	0.975885i	$-0.429953\pi$
$269$	1926.13	0.436572	0.218286	+	0.975885i	$0.429953\pi$
$270$	0	0
$271$	−3653.06	−0.818847	−0.409423	−	0.912344i	$-0.634270\pi$
$271$	−3653.06	−0.818847	−0.409423	+	0.912344i	$0.634270\pi$
$272$	− 772.384i	− 0.172179i
$273$	− 999.859i	− 0.221664i
$274$	−1507.61	−0.332402
$275$	0	0
$276$	−931.566	−0.203165
$277$	1047.37i	0.227185i	0.993527	+	0.113592i	$0.0362358\pi$
$277$	1047.37i	0.227185i	−0.993527	+	0.113592i	$0.963764\pi$
$278$	2028.19i	0.437565i
$279$	16669.2	3.57691
$280$	0	0
$281$	−2758.90	−0.585701	−0.292851	−	0.956158i	$-0.594604\pi$
$281$	−2758.90	−0.585701	−0.292851	+	0.956158i	$0.594604\pi$
$282$	− 11747.6i	− 2.48071i
$283$	3355.58i	0.704837i	0.935843	+	0.352418i	$0.114641\pi$
$283$	3355.58i	0.704837i	−0.935843	+	0.352418i	$0.885359\pi$
$284$	−2459.38	−0.513864
$285$	0	0
$286$	−139.337	−0.0288082
$287$	− 4195.50i	− 0.862902i
$288$	2416.96i	0.494517i
$289$	2582.62	0.525670
$290$	0	0
$291$	−2529.39	−0.509537
$292$	− 2046.35i	− 0.410114i
$293$	419.625i	0.0836681i	0.999125	+	0.0418341i	$0.0133201\pi$
$293$	419.625i	0.0836681i	−0.999125	+	0.0418341i	$0.986680\pi$
$294$	−5352.08	−1.06170
$295$	0	0
$296$	−3383.23	−0.664346
$297$	8543.85i	1.66924i
$298$	− 5543.59i	− 1.07762i
$299$	92.1607	0.0178254
$300$	0	0
$301$	5641.30	1.08026
$302$	− 6217.24i	− 1.18464i
$303$	6282.93i	1.19124i
$304$	−1269.08	−0.239429
$305$	0	0
$306$	7292.28	1.36233
$307$	4133.41i	0.768425i	0.923245	+	0.384212i	$0.125527\pi$
$307$	4133.41i	0.768425i	−0.923245	+	0.384212i	$0.874473\pi$
$308$	1713.85i	0.317064i
$309$	−14917.6	−2.74639
$310$	0	0
$311$	−6991.03	−1.27468	−0.637339	−	0.770584i	$-0.719965\pi$
$311$	−6991.03	−1.27468	−0.637339	+	0.770584i	$0.719965\pi$
$312$	− 324.589i	− 0.0588982i
$313$	9380.53i	1.69399i	0.531600	+	0.846996i	$0.321591\pi$
$313$	9380.53i	1.69399i	−0.531600	+	0.846996i	$0.678409\pi$
$314$	−7425.75	−1.33458
$315$	0	0
$316$	641.855	0.114263
$317$	7995.35i	1.41660i	0.705910	+	0.708302i	$0.250538\pi$
$317$	7995.35i	1.41660i	−0.705910	+	0.708302i	$0.749462\pi$
$318$	− 5272.54i	− 0.929778i
$319$	−4420.87	−0.775929
$320$	0	0
$321$	−9519.60	−1.65524
$322$	− 1133.58i	− 0.196186i
$323$	3828.96i	0.659595i
$324$	−11745.9	−2.01405
$325$	0	0
$326$	1831.58	0.311172
$327$	6442.63i	1.08954i
$328$	− 1362.01i	− 0.229281i
$329$	14295.1	2.39549
$330$	0	0
$331$	−4798.86	−0.796886	−0.398443	−	0.917193i	$-0.630449\pi$
$331$	−4798.86	−0.796886	−0.398443	+	0.917193i	$0.630449\pi$
$332$	130.258i	0.0215327i
$333$	− 31942.0i	− 5.25648i
$334$	−2864.63	−0.469298
$335$	0	0
$336$	−3992.46	−0.648234
$337$	− 6895.51i	− 1.11461i	−0.830309	−	0.557303i	$-0.811836\pi$
$337$	− 6895.51i	− 1.11461i	0.830309	−	0.557303i	$-0.188164\pi$
$338$	− 4361.89i	− 0.701939i
$339$	−8429.16	−1.35047
$340$	0	0
$341$	3837.17	0.609368
$342$	− 11981.7i	− 1.89443i
$343$	1939.86i	0.305372i
$344$	1831.36	0.287036
$345$	0	0
$346$	−6959.08	−1.08128
$347$	8744.17i	1.35277i	0.736548	+	0.676386i	$0.236455\pi$
$347$	8744.17i	1.35277i	−0.736548	+	0.676386i	$0.763545\pi$
$348$	− 10298.6i	− 1.58638i
$349$	6387.08	0.979635	0.489818	−	0.871825i	$-0.337063\pi$
$349$	6387.08	0.979635	0.489818	+	0.871825i	$0.337063\pi$
$350$	0	0
$351$	1969.04	0.299429
$352$	556.375i	0.0842468i
$353$	− 589.384i	− 0.0888661i	−0.999012	−	0.0444331i	$-0.985852\pi$
$353$	− 589.384i	− 0.0888661i	0.999012	−	0.0444331i	$-0.0141481\pi$
$354$	−7151.39	−1.07371
$355$	0	0
$356$	−100.150	−0.0149100
$357$	12045.8i	1.78580i
$358$	7285.41i	1.07555i
$359$	−7214.13	−1.06058	−0.530289	−	0.847817i	$-0.677917\pi$
$359$	−7214.13	−1.06058	−0.530289	+	0.847817i	$0.677917\pi$
$360$	0	0
$361$	−567.774	−0.0827780
$362$	− 4818.30i	− 0.699570i
$363$	− 10416.3i	− 1.50611i
$364$	394.978	0.0568749
$365$	0	0
$366$	−1632.48	−0.233145
$367$	− 12356.4i	− 1.75750i	−0.477286	−	0.878748i	$-0.658379\pi$
$367$	− 12356.4i	− 1.75750i	0.477286	−	0.878748i	$-0.341621\pi$
$368$	− 368.000i	− 0.0521286i
$369$	12859.1	1.81413
$370$	0	0
$371$	6415.92	0.897839
$372$	8938.81i	1.24585i
$373$	7145.98i	0.991969i	0.868331	+	0.495985i	$0.165193\pi$
$373$	7145.98i	0.991969i	−0.868331	+	0.495985i	$0.834807\pi$
$374$	1678.65	0.232088
$375$	0	0
$376$	4640.70	0.636505
$377$	1018.85i	0.139186i
$378$	− 24219.3i	− 3.29552i
$379$	−2170.69	−0.294197	−0.147099	−	0.989122i	$-0.546993\pi$
$379$	−2170.69	−0.294197	−0.147099	+	0.989122i	$0.546993\pi$
$380$	0	0
$381$	16349.0	2.19839
$382$	379.215i	0.0507914i
$383$	− 7967.21i	− 1.06294i	−0.847078	−	0.531469i	$-0.821640\pi$
$383$	− 7967.21i	− 1.06294i	0.847078	−	0.531469i	$-0.178360\pi$
$384$	−1296.09	−0.172242
$385$	0	0
$386$	−3710.90	−0.489326
$387$	17290.3i	2.27111i
$388$	− 999.193i	− 0.130738i
$389$	568.951	0.0741567	0.0370783	−	0.999312i	$-0.488195\pi$
$389$	568.951	0.0741567	0.0370783	+	0.999312i	$0.488195\pi$
$390$	0	0
$391$	−1110.30	−0.143607
$392$	− 2114.25i	− 0.272413i
$393$	− 19993.8i	− 2.56629i
$394$	−4859.70	−0.621391
$395$	0	0
$396$	−5252.88	−0.666583
$397$	− 8564.88i	− 1.08277i	−0.840775	−	0.541384i	$-0.817900\pi$
$397$	− 8564.88i	− 1.08277i	0.840775	−	0.541384i	$-0.182100\pi$
$398$	8666.98i	1.09155i
$399$	19791.9	2.48330
$400$	0	0
$401$	−12455.6	−1.55113	−0.775563	−	0.631270i	$-0.782534\pi$
$401$	−12455.6	−1.55113	−0.775563	+	0.631270i	$0.782534\pi$
$402$	16606.4i	2.06033i
$403$	− 884.325i	− 0.109309i
$404$	−2481.97	−0.305650
$405$	0	0
$406$	12531.9	1.53189
$407$	− 7352.91i	− 0.895504i
$408$	3910.47i	0.474503i
$409$	−11838.9	−1.43129	−0.715645	−	0.698465i	$-0.753867\pi$
$409$	−11838.9	−1.43129	−0.715645	+	0.698465i	$0.753867\pi$
$410$	0	0
$411$	7632.82	0.916056
$412$	− 5892.96i	− 0.704673i
$413$	− 8702.22i	− 1.03682i
$414$	3474.38	0.412456
$415$	0	0
$416$	128.224	0.0151122
$417$	− 10268.5i	− 1.20587i
$418$	− 2758.13i	− 0.322738i
$419$	1531.77	0.178596	0.0892981	−	0.996005i	$-0.471538\pi$
$419$	1531.77	0.178596	0.0892981	+	0.996005i	$0.471538\pi$
$420$	0	0
$421$	−9985.06	−1.15592	−0.577959	−	0.816066i	$-0.696151\pi$
$421$	−9985.06	−1.15592	−0.577959	+	0.816066i	$0.696151\pi$
$422$	1633.58i	0.188439i
$423$	43814.0i	5.03620i
$424$	2082.83	0.238564
$425$	0	0
$426$	12451.5	1.41614
$427$	− 1986.50i	− 0.225137i
$428$	− 3760.56i	− 0.424705i
$429$	705.441	0.0793917
$430$	0	0
$431$	−13786.7	−1.54079	−0.770396	−	0.637566i	$-0.779941\pi$
$431$	−13786.7	−1.54079	−0.770396	+	0.637566i	$0.779941\pi$
$432$	− 7862.42i	− 0.875651i
$433$	− 2621.92i	− 0.290996i	−0.989359	−	0.145498i	$-0.953522\pi$
$433$	− 2621.92i	− 0.290996i	0.989359	−	0.145498i	$-0.0464784\pi$
$434$	−10877.3	−1.20305
$435$	0	0
$436$	−2545.06	−0.279555
$437$	1824.30i	0.199698i
$438$	10360.4i	1.13022i
$439$	−12062.7	−1.31143	−0.655717	−	0.755007i	$-0.727633\pi$
$439$	−12062.7	−1.31143	−0.655717	+	0.755007i	$0.727633\pi$
$440$	0	0
$441$	19961.2	2.15541
$442$	− 386.867i	− 0.0416321i
$443$	3659.26i	0.392453i	0.980559	+	0.196227i	$0.0628689\pi$
$443$	3659.26i	0.392453i	−0.980559	+	0.196227i	$0.937131\pi$
$444$	17128.8	1.83085
$445$	0	0
$446$	−9027.60	−0.958452
$447$	28066.4i	2.96979i
$448$	− 1577.16i	− 0.166325i
$449$	10529.6	1.10674	0.553368	−	0.832937i	$-0.313342\pi$
$449$	10529.6	1.10674	0.553368	+	0.832937i	$0.313342\pi$
$450$	0	0
$451$	2960.10	0.309059
$452$	− 3329.81i	− 0.346506i
$453$	31477.0i	3.26472i
$454$	−5585.70	−0.577423
$455$	0	0
$456$	6425.15	0.659836
$457$	− 6443.23i	− 0.659522i	−0.944064	−	0.329761i	$-0.893032\pi$
$457$	− 6443.23i	− 0.659522i	0.944064	−	0.329761i	$-0.106968\pi$
$458$	2809.00i	0.286585i
$459$	−23721.9	−2.41230
$460$	0	0
$461$	−3263.86	−0.329747	−0.164873	−	0.986315i	$-0.552722\pi$
$461$	−3263.86	−0.329747	−0.164873	+	0.986315i	$0.552722\pi$
$462$	− 8676.97i	− 0.873786i
$463$	9518.12i	0.955388i	0.878526	+	0.477694i	$0.158527\pi$
$463$	9518.12i	0.955388i	−0.878526	+	0.477694i	$0.841473\pi$
$464$	4068.28	0.407037
$465$	0	0
$466$	2147.59	0.213487
$467$	− 19092.6i	− 1.89187i	−0.324360	−	0.945934i	$-0.605149\pi$
$467$	− 19092.6i	− 1.89187i	0.324360	−	0.945934i	$-0.394851\pi$
$468$	1210.59i	0.119572i
$469$	−20207.6	−1.98955
$470$	0	0
$471$	37595.5	3.67794
$472$	− 2825.04i	− 0.275494i
$473$	3980.17i	0.386910i
$474$	−3249.62	−0.314895
$475$	0	0
$476$	−4758.48	−0.458203
$477$	19664.6i	1.88758i
$478$	− 5146.36i	− 0.492446i
$479$	6324.81	0.603315	0.301658	−	0.953416i	$-0.402460\pi$
$479$	6324.81	0.603315	0.301658	+	0.953416i	$0.402460\pi$
$480$	0	0
$481$	−1694.57	−0.160636
$482$	1392.25i	0.131567i
$483$	5739.16i	0.540664i
$484$	4114.81	0.386440
$485$	0	0
$486$	32932.2	3.07373
$487$	7873.07i	0.732573i	0.930502	+	0.366286i	$0.119371\pi$
$487$	7873.07i	0.732573i	−0.930502	+	0.366286i	$0.880629\pi$
$488$	− 644.886i	− 0.0598209i
$489$	−9273.04	−0.857549
$490$	0	0
$491$	−3556.82	−0.326918	−0.163459	−	0.986550i	$-0.552265\pi$
$491$	−3556.82	−0.326918	−0.163459	+	0.986550i	$0.552265\pi$
$492$	6895.64i	0.631869i
$493$	− 12274.5i	− 1.12133i
$494$	−635.646	−0.0578929
$495$	0	0
$496$	−3531.13	−0.319662
$497$	15151.7i	1.36750i
$498$	− 659.479i	− 0.0593413i
$499$	1933.37	0.173446	0.0867231	−	0.996232i	$-0.472360\pi$
$499$	1933.37	0.173446	0.0867231	+	0.996232i	$0.472360\pi$
$500$	0	0
$501$	14503.2	1.29332
$502$	− 12935.6i	− 1.15009i
$503$	2114.36i	0.187425i	0.995599	+	0.0937123i	$0.0298734\pi$
$503$	2114.36i	0.187425i	−0.995599	+	0.0937123i	$0.970127\pi$
$504$	14890.4	1.31601
$505$	0	0
$506$	799.789	0.0702667
$507$	22083.6i	1.93445i
$508$	6458.42i	0.564067i
$509$	316.452	0.0275570	0.0137785	−	0.999905i	$-0.495614\pi$
$509$	316.452	0.0275570	0.0137785	+	0.999905i	$0.495614\pi$
$510$	0	0
$511$	−12607.1	−1.09140
$512$	− 512.000i	− 0.0441942i
$513$	38976.6i	3.35450i
$514$	−10665.4	−0.915239
$515$	0	0
$516$	−9271.92	−0.791034
$517$	10085.8i	0.857976i
$518$	20843.3i	1.76796i
$519$	35232.8	2.97987
$520$	0	0
$521$	−309.041	−0.0259872	−0.0129936	−	0.999916i	$-0.504136\pi$
$521$	−309.041	−0.0259872	−0.0129936	+	0.999916i	$0.504136\pi$
$522$	38409.7i	3.22059i
$523$	− 5892.46i	− 0.492656i	−0.969186	−	0.246328i	$-0.920776\pi$
$523$	− 5892.46i	− 0.492656i	0.969186	−	0.246328i	$-0.0792241\pi$
$524$	7898.22	0.658464
$525$	0	0
$526$	−13745.7	−1.13943
$527$	10653.9i	0.880626i
$528$	− 2816.84i	− 0.232173i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	26672.0	2.17978
$532$	7818.49i	0.637170i
$533$	− 682.193i	− 0.0554391i
$534$	507.046	0.0410899
$535$	0	0
$536$	−6560.09	−0.528643
$537$	− 36885.0i	− 2.96407i
$538$	− 3852.25i	− 0.308703i
$539$	4594.99	0.367199
$540$	0	0
$541$	1560.55	0.124017	0.0620085	−	0.998076i	$-0.480249\pi$
$541$	1560.55	0.124017	0.0620085	+	0.998076i	$0.480249\pi$
$542$	7306.12i	0.579012i
$543$	24394.4i	1.92792i
$544$	−1544.77	−0.121749
$545$	0	0
$546$	−1999.72	−0.156740
$547$	− 17756.3i	− 1.38795i	−0.720001	−	0.693973i	$-0.755859\pi$
$547$	− 17756.3i	− 1.38795i	0.720001	−	0.693973i	$-0.244141\pi$
$548$	3015.22i	0.235044i
$549$	6088.54	0.473319
$550$	0	0
$551$	−20167.8	−1.55930
$552$	1863.13i	0.143660i
$553$	− 3954.32i	− 0.304078i
$554$	2094.73	0.160644
$555$	0	0
$556$	4056.39	0.309405
$557$	− 1212.77i	− 0.0922559i	−0.998936	−	0.0461280i	$-0.985312\pi$
$557$	− 1212.77i	− 0.0922559i	0.998936	−	0.0461280i	$-0.0146882\pi$
$558$	− 33338.3i	− 2.52926i
$559$	917.280	0.0694040
$560$	0	0
$561$	−8498.78	−0.639605
$562$	5517.80i	0.414153i
$563$	12558.2i	0.940078i	0.882646	+	0.470039i	$0.155760\pi$
$563$	12558.2i	0.940078i	−0.882646	+	0.470039i	$0.844240\pi$
$564$	−23495.2	−1.75412
$565$	0	0
$566$	6711.17	0.498395
$567$	72363.9i	5.35978i
$568$	4918.76i	0.363357i
$569$	−9776.72	−0.720319	−0.360159	−	0.932891i	$-0.617278\pi$
$569$	−9776.72	−0.720319	−0.360159	+	0.932891i	$0.617278\pi$
$570$	0	0
$571$	−18733.9	−1.37301	−0.686505	−	0.727125i	$-0.740856\pi$
$571$	−18733.9	−1.37301	−0.686505	+	0.727125i	$0.740856\pi$
$572$	278.673i	0.0203705i
$573$	− 1919.91i	− 0.139975i
$574$	−8391.01	−0.610164
$575$	0	0
$576$	4833.92	0.349676
$577$	− 5113.58i	− 0.368944i	−0.982838	−	0.184472i	$-0.940942\pi$
$577$	− 5113.58i	− 0.368944i	0.982838	−	0.184472i	$-0.0590576\pi$
$578$	− 5165.24i	− 0.371705i
$579$	18787.8	1.34852
$580$	0	0
$581$	802.492	0.0573029
$582$	5058.77i	0.360297i
$583$	4526.70i	0.321572i
$584$	−4092.69	−0.289995
$585$	0	0
$586$	839.250	0.0591623
$587$	5379.95i	0.378287i	0.981949	+	0.189143i	$0.0605711\pi$
$587$	5379.95i	0.378287i	−0.981949	+	0.189143i	$0.939429\pi$
$588$	10704.2i	0.750735i
$589$	17505.0	1.22458
$590$	0	0
$591$	24603.9	1.71247
$592$	6766.46i	0.469763i
$593$	− 15060.3i	− 1.04292i	−0.853275	−	0.521461i	$-0.825387\pi$
$593$	− 15060.3i	− 1.04292i	0.853275	−	0.521461i	$-0.174613\pi$
$594$	17087.7	1.18033
$595$	0	0
$596$	−11087.2	−0.761995
$597$	− 43879.7i	− 3.00817i
$598$	− 184.321i	− 0.0126045i
$599$	3772.04	0.257298	0.128649	−	0.991690i	$-0.458936\pi$
$599$	3772.04	0.257298	0.128649	+	0.991690i	$0.458936\pi$
$600$	0	0
$601$	−14663.9	−0.995261	−0.497631	−	0.867389i	$-0.665796\pi$
$601$	−14663.9	−0.995261	−0.497631	+	0.867389i	$0.665796\pi$
$602$	− 11282.6i	− 0.763861i
$603$	− 61935.5i	− 4.18277i
$604$	−12434.5	−0.837668
$605$	0	0
$606$	12565.9	0.842332
$607$	24514.5i	1.63923i	0.572915	+	0.819614i	$0.305812\pi$
$607$	24514.5i	1.63923i	−0.572915	+	0.819614i	$0.694188\pi$
$608$	2538.15i	0.169302i
$609$	−63447.1	−4.22168
$610$	0	0
$611$	2324.40	0.153904
$612$	− 14584.6i	− 0.963310i
$613$	14451.2i	0.952167i	0.879400	+	0.476084i	$0.157944\pi$
$613$	14451.2i	0.952167i	−0.879400	+	0.476084i	$0.842056\pi$
$614$	8266.83	0.543358
$615$	0	0
$616$	3427.70	0.224198
$617$	− 3292.19i	− 0.214811i	−0.994215	−	0.107406i	$-0.965746\pi$
$617$	− 3292.19i	− 0.214811i	0.994215	−	0.107406i	$-0.0342544\pi$
$618$	29835.2i	1.94199i
$619$	12595.5	0.817858	0.408929	−	0.912566i	$-0.365902\pi$
$619$	12595.5	0.817858	0.408929	+	0.912566i	$0.365902\pi$
$620$	0	0
$621$	−11302.2	−0.730343
$622$	13982.1i	0.901333i
$623$	617.002i	0.0396785i
$624$	−649.178	−0.0416473
$625$	0	0
$626$	18761.1	1.19783
$627$	13964.0i	0.889425i
$628$	14851.5i	0.943693i
$629$	20415.3	1.29413
$630$	0	0
$631$	19889.3	1.25480	0.627401	−	0.778697i	$-0.284119\pi$
$631$	19889.3	1.25480	0.627401	+	0.778697i	$0.284119\pi$
$632$	− 1283.71i	− 0.0807963i
$633$	− 8270.56i	− 0.519313i
$634$	15990.7	1.00169
$635$	0	0
$636$	−10545.1	−0.657452
$637$	− 1058.97i	− 0.0658682i
$638$	8841.75i	0.548665i
$639$	−46439.3	−2.87498
$640$	0	0
$641$	−19276.0	−1.18776	−0.593880	−	0.804553i	$-0.702405\pi$
$641$	−19276.0	−1.18776	−0.593880	+	0.804553i	$0.702405\pi$
$642$	19039.2i	1.17043i
$643$	− 10219.2i	− 0.626758i	−0.949628	−	0.313379i	$-0.898539\pi$
$643$	− 10219.2i	− 0.626758i	0.949628	−	0.313379i	$-0.101461\pi$
$644$	−2267.16	−0.138725
$645$	0	0
$646$	7657.93	0.466404
$647$	− 20818.4i	− 1.26500i	−0.774560	−	0.632500i	$-0.782029\pi$
$647$	− 20818.4i	− 1.26500i	0.774560	−	0.632500i	$-0.217971\pi$
$648$	23491.8i	1.42415i
$649$	6139.77	0.371352
$650$	0	0
$651$	55070.0	3.31546
$652$	− 3663.17i	− 0.220032i
$653$	− 15135.3i	− 0.907031i	−0.891248	−	0.453516i	$-0.850170\pi$
$653$	− 15135.3i	− 0.907031i	0.891248	−	0.453516i	$-0.149830\pi$
$654$	12885.3	0.770418
$655$	0	0
$656$	−2724.01	−0.162126
$657$	− 38640.2i	− 2.29451i
$658$	− 28590.3i	− 1.69387i
$659$	13207.9	0.780737	0.390369	−	0.920659i	$-0.372348\pi$
$659$	13207.9	0.780737	0.390369	+	0.920659i	$0.372348\pi$
$660$	0	0
$661$	6671.34	0.392564	0.196282	−	0.980547i	$-0.437113\pi$
$661$	6671.34	0.392564	0.196282	+	0.980547i	$0.437113\pi$
$662$	9597.72i	0.563484i
$663$	1958.65i	0.114733i
$664$	260.517	0.0152259
$665$	0	0
$666$	−63883.9	−3.71689
$667$	− 5848.15i	− 0.339492i
$668$	5729.26i	0.331844i
$669$	45705.5	2.64137
$670$	0	0
$671$	1401.56	0.0806355
$672$	7984.92i	0.458371i
$673$	− 9534.41i	− 0.546099i	−0.962000	−	0.273049i	$-0.911968\pi$
$673$	− 9534.41i	− 0.546099i	0.962000	−	0.273049i	$-0.0880322\pi$
$674$	−13791.0	−0.788146
$675$	0	0
$676$	−8723.78	−0.496346
$677$	− 17748.4i	− 1.00757i	−0.863828	−	0.503787i	$-0.831940\pi$
$677$	− 17748.4i	− 1.00757i	0.863828	−	0.503787i	$-0.168060\pi$
$678$	16858.3i	0.954927i
$679$	−6155.80	−0.347920
$680$	0	0
$681$	28279.6	1.59130
$682$	− 7674.35i	− 0.430888i
$683$	9583.57i	0.536904i	0.963293	+	0.268452i	$0.0865120\pi$
$683$	9583.57i	0.536904i	−0.963293	+	0.268452i	$0.913488\pi$
$684$	−23963.3	−1.33956
$685$	0	0
$686$	3879.73	0.215931
$687$	− 14221.5i	− 0.789790i
$688$	− 3662.72i	− 0.202965i
$689$	1043.24	0.0576837
$690$	0	0
$691$	9410.74	0.518092	0.259046	−	0.965865i	$-0.416592\pi$
$691$	9410.74	0.518092	0.259046	+	0.965865i	$0.416592\pi$
$692$	13918.2i	0.764580i
$693$	32361.8i	1.77391i
$694$	17488.3	0.956554
$695$	0	0
$696$	−20597.1	−1.12174
$697$	8218.69i	0.446636i
$698$	− 12774.2i	− 0.692707i
$699$	−10872.9	−0.588343
$700$	0	0
$701$	−19517.9	−1.05161	−0.525806	−	0.850605i	$-0.676236\pi$
$701$	−19517.9	−1.05161	−0.525806	+	0.850605i	$0.676236\pi$
$702$	− 3938.08i	− 0.211728i
$703$	− 33543.6i	− 1.79960i
$704$	1112.75	0.0595715
$705$	0	0
$706$	−1178.77	−0.0628378
$707$	15290.9i	0.813397i
$708$	14302.8i	0.759225i
$709$	28430.7	1.50598	0.752990	−	0.658033i	$-0.228611\pi$
$709$	28430.7	1.50598	0.752990	+	0.658033i	$0.228611\pi$
$710$	0	0
$711$	12119.8	0.639282
$712$	200.300i	0.0105429i
$713$	5076.00i	0.266617i
$714$	24091.5	1.26275
$715$	0	0
$716$	14570.8	0.760527
$717$	26055.3i	1.35712i
$718$	14428.3i	0.749942i
$719$	−18716.0	−0.970779	−0.485390	−	0.874298i	$-0.661322\pi$
$719$	−18716.0	−0.970779	−0.485390	+	0.874298i	$0.661322\pi$
$720$	0	0
$721$	−36305.2	−1.87528
$722$	1135.55i	0.0585329i
$723$	− 7048.78i	− 0.362582i
$724$	−9636.61	−0.494671
$725$	0	0
$726$	−20832.7	−1.06498
$727$	− 18419.5i	− 0.939670i	−0.882754	−	0.469835i	$-0.844313\pi$
$727$	− 18419.5i	− 0.939670i	0.882754	−	0.469835i	$-0.155687\pi$
$728$	− 789.956i	− 0.0402167i
$729$	−87446.1	−4.44272
$730$	0	0
$731$	−11050.9	−0.559141
$732$	3264.97i	0.164859i
$733$	− 21548.4i	− 1.08582i	−0.839790	−	0.542912i	$-0.817322\pi$
$733$	− 21548.4i	− 1.08582i	0.839790	−	0.542912i	$-0.182678\pi$
$734$	−24712.9	−1.24274
$735$	0	0
$736$	−736.000	−0.0368605
$737$	− 14257.3i	− 0.712584i
$738$	− 25718.1i	− 1.28279i
$739$	−12066.4	−0.600636	−0.300318	−	0.953839i	$-0.597093\pi$
$739$	−12066.4	−0.600636	−0.300318	+	0.953839i	$0.597093\pi$
$740$	0	0
$741$	3218.19	0.159545
$742$	− 12831.8i	− 0.634868i
$743$	− 21951.0i	− 1.08385i	−0.840426	−	0.541926i	$-0.817695\pi$
$743$	− 21951.0i	− 1.08385i	0.840426	−	0.541926i	$-0.182305\pi$
$744$	17877.6	0.880948
$745$	0	0
$746$	14292.0	0.701428
$747$	2459.61i	0.120472i
$748$	− 3357.31i	− 0.164111i
$749$	−23168.0	−1.13023
$750$	0	0
$751$	−3112.43	−0.151230	−0.0756152	−	0.997137i	$-0.524092\pi$
$751$	−3112.43	−0.151230	−0.0756152	+	0.997137i	$0.524092\pi$
$752$	− 9281.40i	− 0.450077i
$753$	65491.2i	3.16950i
$754$	2037.69	0.0984197
$755$	0	0
$756$	−48438.6	−2.33028
$757$	7684.64i	0.368960i	0.982836	+	0.184480i	$0.0590602\pi$
$757$	7684.64i	0.368960i	−0.982836	+	0.184480i	$0.940940\pi$
$758$	4341.38i	0.208029i
$759$	−4049.21	−0.193646
$760$	0	0
$761$	−36484.6	−1.73793	−0.868965	−	0.494874i	$-0.835214\pi$
$761$	−36484.6	−1.73793	−0.868965	+	0.494874i	$0.835214\pi$
$762$	− 32698.0i	− 1.55450i
$763$	15679.5i	0.743954i
$764$	758.430	0.0359150
$765$	0	0
$766$	−15934.4	−0.751611
$767$	− 1414.99i	− 0.0666132i
$768$	2592.18i	0.121793i
$769$	−2004.39	−0.0939925	−0.0469962	−	0.998895i	$-0.514965\pi$
$769$	−2004.39	−0.0939925	−0.0469962	+	0.998895i	$0.514965\pi$
$770$	0	0
$771$	53997.6	2.52228
$772$	7421.81i	0.346006i
$773$	7716.17i	0.359031i	0.983755	+	0.179516i	$0.0574531\pi$
$773$	7716.17i	0.359031i	−0.983755	+	0.179516i	$0.942547\pi$
$774$	34580.7	1.60591
$775$	0	0
$776$	−1998.39	−0.0924457
$777$	− 105527.i	− 4.87226i
$778$	− 1137.90i	− 0.0524367i
$779$	13503.8	0.621084
$780$	0	0
$781$	−10690.1	−0.489786
$782$	2220.61i	0.101546i
$783$	− 124947.i	− 5.70275i
$784$	−4228.51	−0.192625
$785$	0	0
$786$	−39987.6	−1.81464
$787$	57.0149i	0.00258241i	0.999999	+	0.00129121i	$0.000411004\pi$
$787$	57.0149i	0.00258241i	−0.999999	+	0.00129121i	$0.999589\pi$
$788$	9719.39i	0.439390i
$789$	69592.5	3.14012
$790$	0	0
$791$	−20514.2	−0.922124
$792$	10505.8i	0.471346i
$793$	− 323.006i	− 0.0144644i
$794$	−17129.8	−0.765633
$795$	0	0
$796$	17334.0	0.771842
$797$	15184.7i	0.674870i	0.941349	+	0.337435i	$0.109559\pi$
$797$	15184.7i	0.674870i	−0.941349	+	0.337435i	$0.890441\pi$
$798$	− 39583.9i	− 1.75596i
$799$	−28003.2	−1.23990
$800$	0	0
$801$	−1891.09	−0.0834186
$802$	24911.1i	1.09681i
$803$	− 8894.80i	− 0.390898i
$804$	33212.8	1.45687
$805$	0	0
$806$	−1768.65	−0.0772929
$807$	19503.4i	0.850746i
$808$	4963.94i	0.216127i
$809$	30286.4	1.31621	0.658105	−	0.752926i	$-0.271358\pi$
$809$	30286.4	1.31621	0.658105	+	0.752926i	$0.271358\pi$
$810$	0	0
$811$	−43936.2	−1.90235	−0.951176	−	0.308650i	$-0.900123\pi$
$811$	−43936.2	−1.90235	−0.951176	+	0.308650i	$0.900123\pi$
$812$	− 25063.7i	− 1.08321i
$813$	− 36989.8i	− 1.59568i
$814$	−14705.8	−0.633217
$815$	0	0
$816$	7820.94	0.335524
$817$	18157.3i	0.777532i
$818$	23677.9i	1.01207i
$819$	7458.18	0.318205
$820$	0	0
$821$	−5245.69	−0.222991	−0.111496	−	0.993765i	$-0.535564\pi$
$821$	−5245.69	−0.222991	−0.111496	+	0.993765i	$0.535564\pi$
$822$	− 15265.6i	− 0.647750i
$823$	10678.0i	0.452260i	0.974097	+	0.226130i	$0.0726074\pi$
$823$	10678.0i	0.452260i	−0.974097	+	0.226130i	$0.927393\pi$
$824$	−11785.9	−0.498279
$825$	0	0
$826$	−17404.4	−0.733145
$827$	− 3393.69i	− 0.142697i	−0.997451	−	0.0713484i	$-0.977270\pi$
$827$	− 3393.69i	− 0.142697i	0.997451	−	0.0713484i	$-0.0227302\pi$
$828$	− 6948.77i	− 0.291650i
$829$	−9601.74	−0.402270	−0.201135	−	0.979564i	$-0.564463\pi$
$829$	−9601.74	−0.402270	−0.201135	+	0.979564i	$0.564463\pi$
$830$	0	0
$831$	−10605.3	−0.442713
$832$	− 256.447i	− 0.0106859i
$833$	12757.9i	0.530656i
$834$	−20536.9	−0.852680
$835$	0	0
$836$	−5516.26	−0.228210
$837$	108450.i	4.47860i
$838$	− 3063.54i	− 0.126287i
$839$	−11992.8	−0.493489	−0.246744	−	0.969081i	$-0.579361\pi$
$839$	−11992.8	−0.493489	−0.246744	+	0.969081i	$0.579361\pi$
$840$	0	0
$841$	40263.0	1.65087
$842$	19970.1i	0.817358i
$843$	− 27935.8i	− 1.14135i
$844$	3267.15	0.133247
$845$	0	0
$846$	87628.1	3.56113
$847$	− 25350.4i	− 1.02839i
$848$	− 4165.66i	− 0.168690i
$849$	−33977.7	−1.37351
$850$	0	0
$851$	9726.79	0.391810
$852$	− 24903.0i	− 1.00136i
$853$	− 15441.5i	− 0.619821i	−0.950766	−	0.309911i	$-0.899701\pi$
$853$	− 15441.5i	− 0.619821i	0.950766	−	0.309911i	$-0.100299\pi$
$854$	−3973.00	−0.159196
$855$	0	0
$856$	−7521.13	−0.300312
$857$	44572.4i	1.77662i	0.459244	+	0.888310i	$0.348120\pi$
$857$	44572.4i	1.77662i	−0.459244	+	0.888310i	$0.651880\pi$
$858$	− 1410.88i	− 0.0561384i
$859$	−2519.56	−0.100077	−0.0500386	−	0.998747i	$-0.515934\pi$
$859$	−2519.56	−0.100077	−0.0500386	+	0.998747i	$0.515934\pi$
$860$	0	0
$861$	42482.5	1.68153
$862$	27573.4i	1.08950i
$863$	28980.1i	1.14310i	0.820568	+	0.571548i	$0.193657\pi$
$863$	28980.1i	1.14310i	−0.820568	+	0.571548i	$0.806343\pi$
$864$	−15724.8	−0.619178
$865$	0	0
$866$	−5243.83	−0.205765
$867$	26150.9i	1.02437i
$868$	21754.5i	0.850687i
$869$	2789.94	0.108909
$870$	0	0
$871$	−3285.77	−0.127823
$872$	5090.11i	0.197675i
$873$	− 18867.3i	− 0.731455i
$874$	3648.59	0.141208
$875$	0	0
$876$	20720.7	0.799187
$877$	7955.28i	0.306307i	0.988202	+	0.153153i	$0.0489428\pi$
$877$	7955.28i	0.306307i	−0.988202	+	0.153153i	$0.951057\pi$
$878$	24125.3i	0.927323i
$879$	−4249.00	−0.163044
$880$	0	0
$881$	29722.0	1.13662	0.568309	−	0.822815i	$-0.307598\pi$
$881$	29722.0	1.13662	0.568309	+	0.822815i	$0.307598\pi$
$882$	− 39922.4i	− 1.52410i
$883$	− 19379.7i	− 0.738596i	−0.929311	−	0.369298i	$-0.879598\pi$
$883$	− 19379.7i	− 0.738596i	0.929311	−	0.369298i	$-0.120402\pi$
$884$	−773.734	−0.0294383
$885$	0	0
$886$	7318.53	0.277507
$887$	− 22901.7i	− 0.866925i	−0.901172	−	0.433463i	$-0.857292\pi$
$887$	− 22901.7i	− 0.866925i	0.901172	−	0.433463i	$-0.142708\pi$
$888$	− 34257.6i	− 1.29461i
$889$	39788.8	1.50110
$890$	0	0
$891$	−51055.7	−1.91967
$892$	18055.2i	0.677728i
$893$	46010.9i	1.72419i
$894$	56132.9	2.09996
$895$	0	0
$896$	−3154.31	−0.117610
$897$	933.193i	0.0347362i
$898$	− 21059.3i	− 0.782581i
$899$	−56115.8	−2.08183
$900$	0	0
$901$	−12568.3	−0.464719
$902$	− 5920.20i	− 0.218538i
$903$	57122.2i	2.10510i
$904$	−6659.61	−0.245017
$905$	0	0
$906$	62953.9	2.30850
$907$	− 42542.2i	− 1.55743i	−0.627376	−	0.778716i	$-0.715871\pi$
$907$	− 42542.2i	− 1.55743i	0.627376	−	0.778716i	$-0.284129\pi$
$908$	11171.4i	0.408300i
$909$	−46865.9	−1.71006
$910$	0	0
$911$	220.864	0.00803243	0.00401622	−	0.999992i	$-0.498722\pi$
$911$	220.864	0.00803243	0.00401622	+	0.999992i	$0.498722\pi$
$912$	− 12850.3i	− 0.466574i
$913$	566.191i	0.0205237i
$914$	−12886.5	−0.466352
$915$	0	0
$916$	5617.99	0.202646
$917$	− 48659.1i	− 1.75231i
$918$	47443.9i	1.70575i
$919$	−27835.4	−0.999135	−0.499567	−	0.866275i	$-0.666508\pi$
$919$	−27835.4	−0.999135	−0.499567	+	0.866275i	$0.666508\pi$
$920$	0	0
$921$	−41853.8	−1.49743
$922$	6527.72i	0.233166i
$923$	2463.68i	0.0878580i
$924$	−17353.9	−0.617860
$925$	0	0
$926$	19036.2	0.675562
$927$	− 111274.i	− 3.94252i
$928$	− 8136.56i	− 0.287819i
$929$	2172.71	0.0767323	0.0383661	−	0.999264i	$-0.487785\pi$
$929$	2172.71	0.0767323	0.0383661	+	0.999264i	$0.487785\pi$
$930$	0	0
$931$	20962.1	0.737921
$932$	− 4295.17i	− 0.150958i
$933$	− 70789.1i	− 2.48396i
$934$	−38185.3	−1.33775
$935$	0	0
$936$	2421.18	0.0845501
$937$	− 54906.5i	− 1.91432i	−0.289560	−	0.957160i	$-0.593509\pi$
$937$	− 54906.5i	− 1.91432i	0.289560	−	0.957160i	$-0.406491\pi$
$938$	40415.2i	1.40683i
$939$	−94984.6	−3.30107
$940$	0	0
$941$	5980.06	0.207167	0.103584	−	0.994621i	$-0.466969\pi$
$941$	5980.06	0.207167	0.103584	+	0.994621i	$0.466969\pi$
$942$	− 75191.0i	− 2.60070i
$943$	3915.77i	0.135223i
$944$	−5650.09	−0.194804
$945$	0	0
$946$	7960.33	0.273586
$947$	20268.6i	0.695504i	0.937587	+	0.347752i	$0.113055\pi$
$947$	20268.6i	0.695504i	−0.937587	+	0.347752i	$0.886945\pi$
$948$	6499.24i	0.222664i
$949$	−2049.92	−0.0701193
$950$	0	0
$951$	−80958.6	−2.76053
$952$	9516.97i	0.323999i
$953$	− 21797.9i	− 0.740925i	−0.928847	−	0.370463i	$-0.879199\pi$
$953$	− 21797.9i	− 0.740925i	0.928847	−	0.370463i	$-0.120801\pi$
$954$	39329.1	1.33472
$955$	0	0
$956$	−10292.7	−0.348212
$957$	− 44764.5i	− 1.51205i
$958$	− 12649.6i	− 0.426608i
$959$	18576.1	0.625499
$960$	0	0
$961$	18915.6	0.634945
$962$	3389.14i	0.113587i
$963$	− 71008.9i	− 2.37615i
$964$	2784.51	0.0930321
$965$	0	0
$966$	11478.3	0.382308
$967$	− 48950.6i	− 1.62786i	−0.580960	−	0.813932i	$-0.697323\pi$
$967$	− 48950.6i	− 1.62786i	0.580960	−	0.813932i	$-0.302677\pi$
$968$	− 8229.62i	− 0.273254i
$969$	−38771.0	−1.28535
$970$	0	0
$971$	−26426.2	−0.873386	−0.436693	−	0.899611i	$-0.643850\pi$
$971$	−26426.2	−0.873386	−0.436693	+	0.899611i	$0.643850\pi$
$972$	− 65864.4i	− 2.17346i
$973$	− 24990.5i	− 0.823389i
$974$	15746.1	0.518007
$975$	0	0
$976$	−1289.77	−0.0422998
$977$	− 5770.09i	− 0.188947i	−0.995527	−	0.0944736i	$-0.969883\pi$
$977$	− 5770.09i	− 0.188947i	0.995527	−	0.0944736i	$-0.0301168\pi$
$978$	18546.1i	0.606379i
$979$	−435.320	−0.0142113
$980$	0	0
$981$	−48057.1	−1.56406
$982$	7113.64i	0.231166i
$983$	− 484.532i	− 0.0157214i	−0.999969	−	0.00786071i	$-0.997498\pi$
$983$	− 484.532i	− 0.0157214i	0.999969	−	0.00786071i	$-0.00250217\pi$
$984$	13791.3	0.446799
$985$	0	0
$986$	−24549.0	−0.792901
$987$	144749.i	4.66808i
$988$	1271.29i	0.0409365i
$989$	−5265.16	−0.169285
$990$	0	0
$991$	−26511.5	−0.849813	−0.424907	−	0.905237i	$-0.639693\pi$
$991$	−26511.5	−0.849813	−0.424907	+	0.905237i	$0.639693\pi$
$992$	7062.27i	0.226035i
$993$	− 48591.9i	− 1.55289i
$994$	30303.4	0.966966
$995$	0	0
$996$	−1318.96	−0.0419606
$997$	4628.20i	0.147018i	0.997295	+	0.0735088i	$0.0234197\pi$
$997$	4628.20i	0.147018i	−0.997295	+	0.0735088i	$0.976580\pi$
$998$	− 3866.74i	− 0.122645i
$999$	207816.	6.58158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1150.4.b.o.599.4		8
5.2	odd	4		230.4.a.j.1.4	✓	4
5.3	odd	4		1150.4.a.n.1.1		4
5.4	even	2	inner	1150.4.b.o.599.5		8
15.2	even	4		2070.4.a.bg.1.1		4
20.7	even	4		1840.4.a.k.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.j.1.4	✓	4	5.2	odd	4
1150.4.a.n.1.1		4	5.3	odd	4
1150.4.b.o.599.4		8	1.1	even	1	trivial
1150.4.b.o.599.5		8	5.4	even	2	inner
1840.4.a.k.1.1		4	20.7	even	4
2070.4.a.bg.1.1		4	15.2	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:	\( N \)	\(=\)	\( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1150.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} +10.1257i q^{3} -4.00000 q^{4} +20.2514 q^{6} +24.6431i q^{7} +8.00000i q^{8} -75.5301 q^{9} -17.3867 q^{11} -40.5029i q^{12} +4.00699i q^{13} +49.2862 q^{14} +16.0000 q^{16} -48.2740i q^{17} +151.060i q^{18} -79.3172 q^{19} -249.529 q^{21} +34.7734i q^{22} -23.0000i q^{23} -81.0057 q^{24} +8.01398 q^{26} -491.402i q^{27} -98.5723i q^{28} +254.267 q^{29} -220.696 q^{31} -32.0000i q^{32} -176.053i q^{33} -96.5480 q^{34} +302.120 q^{36} +422.904i q^{37} +158.634i q^{38} -40.5736 q^{39} -170.251 q^{41} +499.058i q^{42} -228.920i q^{43} +69.5468 q^{44} -46.0000 q^{46} -580.087i q^{47} +162.011i q^{48} -264.282 q^{49} +488.809 q^{51} -16.0280i q^{52} -260.354i q^{53} -982.803 q^{54} -197.145 q^{56} -803.144i q^{57} -508.535i q^{58} -353.130 q^{59} -80.6108 q^{61} +441.392i q^{62} -1861.29i q^{63} -64.0000 q^{64} -352.106 q^{66} +820.011i q^{67} +193.096i q^{68} +232.891 q^{69} +614.845 q^{71} -604.241i q^{72} +511.586i q^{73} +845.808 q^{74} +317.269 q^{76} -428.462i q^{77} +81.1472i q^{78} -160.464 q^{79} +2936.48 q^{81} +340.502i q^{82} -32.5646i q^{83} +998.115 q^{84} -457.840 q^{86} +2574.64i q^{87} -139.094i q^{88} +25.0375 q^{89} -98.7445 q^{91} +92.0000i q^{92} -2234.70i q^{93} -1160.17 q^{94} +324.023 q^{96} +249.798i q^{97} +528.563i q^{98} +1313.22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{4} + 56 q^{6} - 128 q^{9} + 42 q^{11} - 32 q^{14} + 128 q^{16} - 346 q^{19} - 240 q^{21} - 224 q^{24} + 280 q^{26} + 236 q^{29} + 34 q^{31} - 224 q^{34} + 512 q^{36} + 442 q^{39} + 278 q^{41}+ \cdots + 5490 q^{99}+O(q^{100}) \) 8 * q - 32 * q^4 + 56 * q^6 - 128 * q^9 + 42 * q^11 - 32 * q^14 + 128 * q^16 - 346 * q^19 - 240 * q^21 - 224 * q^24 + 280 * q^26 + 236 * q^29 + 34 * q^31 - 224 * q^34 + 512 * q^36 + 442 * q^39 + 278 * q^41 - 168 * q^44 - 368 * q^46 + 248 * q^49 + 878 * q^51 - 1556 * q^54 + 128 * q^56 + 906 * q^59 - 654 * q^61 - 512 * q^64 - 356 * q^66 + 644 * q^69 + 390 * q^71 + 1372 * q^74 + 1384 * q^76 + 2280 * q^79 + 2912 * q^81 + 960 * q^84 - 200 * q^86 + 4340 * q^89 + 1114 * q^91 - 1468 * q^94 + 896 * q^96 + 5490 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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