LMFDB - Embedded newform 2151.4.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$126.913108422$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.78487 q^{2} +14.8950 q^{4} +19.8446 q^{5} +9.72473 q^{7} -32.9917 q^{8} +O(q^{10})$	\(q-4.78487 q^{2} +14.8950 q^{4} +19.8446 q^{5} +9.72473 q^{7} -32.9917 q^{8} -94.9538 q^{10} +8.54762 q^{11} +70.2704 q^{13} -46.5316 q^{14} +38.7010 q^{16} +97.0361 q^{17} +72.5866 q^{19} +295.585 q^{20} -40.8993 q^{22} -147.733 q^{23} +268.808 q^{25} -336.235 q^{26} +144.850 q^{28} -33.0087 q^{29} +113.100 q^{31} +78.7542 q^{32} -464.305 q^{34} +192.983 q^{35} +185.104 q^{37} -347.318 q^{38} -654.706 q^{40} -347.392 q^{41} +332.997 q^{43} +127.317 q^{44} +706.884 q^{46} +161.096 q^{47} -248.430 q^{49} -1286.21 q^{50} +1046.68 q^{52} -550.417 q^{53} +169.624 q^{55} -320.835 q^{56} +157.942 q^{58} +103.680 q^{59} +280.648 q^{61} -541.170 q^{62} -686.437 q^{64} +1394.49 q^{65} -408.830 q^{67} +1445.35 q^{68} -923.400 q^{70} -530.048 q^{71} +549.032 q^{73} -885.699 q^{74} +1081.18 q^{76} +83.1232 q^{77} +934.908 q^{79} +768.005 q^{80} +1662.23 q^{82} +786.990 q^{83} +1925.64 q^{85} -1593.35 q^{86} -282.000 q^{88} +741.961 q^{89} +683.360 q^{91} -2200.48 q^{92} -770.825 q^{94} +1440.45 q^{95} -611.445 q^{97} +1188.70 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−4.78487	−1.69171	−0.845854	−	0.533415i	$-0.820909\pi$
$2$	−4.78487	−1.69171	−0.845854	+	0.533415i	$0.820909\pi$
$3$	0	0
$3$	0	0
$4$	14.8950	1.86187
$5$	19.8446	1.77495	0.887477	−	0.460852i	$-0.152456\pi$
$5$	19.8446	1.77495	0.887477	+	0.460852i	$0.152456\pi$
$6$	0	0
$7$	9.72473	0.525086	0.262543	−	0.964920i	$-0.415439\pi$
$7$	9.72473	0.525086	0.262543	+	0.964920i	$0.415439\pi$
$8$	−32.9917	−1.45804
$9$	0	0
$10$	−94.9538	−3.00270
$11$	8.54762	0.234291	0.117146	−	0.993115i	$-0.462626\pi$
$11$	8.54762	0.234291	0.117146	+	0.993115i	$0.462626\pi$
$12$	0	0
$13$	70.2704	1.49919	0.749596	−	0.661896i	$-0.230248\pi$
$13$	70.2704	1.49919	0.749596	+	0.661896i	$0.230248\pi$
$14$	−46.5316	−0.888292
$15$	0	0
$16$	38.7010	0.604703
$17$	97.0361	1.38439	0.692197	−	0.721709i	$-0.256643\pi$
$17$	97.0361	1.38439	0.692197	+	0.721709i	$0.256643\pi$
$18$	0	0
$19$	72.5866	0.876448	0.438224	−	0.898866i	$-0.355608\pi$
$19$	72.5866	0.876448	0.438224	+	0.898866i	$0.355608\pi$
$20$	295.585	3.30474
$21$	0	0
$22$	−40.8993	−0.396352
$23$	−147.733	−1.33933	−0.669663	−	0.742665i	$-0.733561\pi$
$23$	−147.733	−1.33933	−0.669663	+	0.742665i	$0.733561\pi$
$24$	0	0
$25$	268.808	2.15046
$26$	−336.235	−2.53619
$27$	0	0
$28$	144.850	0.977644
$29$	−33.0087	−0.211364	−0.105682	−	0.994400i	$-0.533703\pi$
$29$	−33.0087	−0.211364	−0.105682	+	0.994400i	$0.533703\pi$
$30$	0	0
$31$	113.100	0.655271	0.327636	−	0.944804i	$-0.393748\pi$
$31$	113.100	0.655271	0.327636	+	0.944804i	$0.393748\pi$
$32$	78.7542	0.435059
$33$	0	0
$34$	−464.305	−2.34199
$35$	192.983	0.932003
$36$	0	0
$37$	185.104	0.822458	0.411229	−	0.911532i	$-0.365100\pi$
$37$	185.104	0.822458	0.411229	+	0.911532i	$0.365100\pi$
$38$	−347.318	−1.48269
$39$	0	0
$40$	−654.706	−2.58795
$41$	−347.392	−1.32326	−0.661629	−	0.749832i	$-0.730134\pi$
$41$	−347.392	−1.32326	−0.661629	+	0.749832i	$0.730134\pi$
$42$	0	0
$43$	332.997	1.18097	0.590484	−	0.807050i	$-0.298937\pi$
$43$	332.997	1.18097	0.590484	+	0.807050i	$0.298937\pi$
$44$	127.317	0.436221
$45$	0	0
$46$	706.884	2.26575
$47$	161.096	0.499964	0.249982	−	0.968250i	$-0.419575\pi$
$47$	161.096	0.499964	0.249982	+	0.968250i	$0.419575\pi$
$48$	0	0
$49$	−248.430	−0.724285
$50$	−1286.21	−3.63795
$51$	0	0
$52$	1046.68	2.79131
$53$	−550.417	−1.42652	−0.713261	−	0.700899i	$-0.752782\pi$
$53$	−550.417	−1.42652	−0.713261	+	0.700899i	$0.752782\pi$
$54$	0	0
$55$	169.624	0.415856
$56$	−320.835	−0.765596
$57$	0	0
$58$	157.942	0.357566
$59$	103.680	0.228780	0.114390	−	0.993436i	$-0.463509\pi$
$59$	103.680	0.228780	0.114390	+	0.993436i	$0.463509\pi$
$60$	0	0
$61$	280.648	0.589070	0.294535	−	0.955641i	$-0.404835\pi$
$61$	280.648	0.589070	0.294535	+	0.955641i	$0.404835\pi$
$62$	−541.170	−1.10853
$63$	0	0
$64$	−686.437	−1.34070
$65$	1394.49	2.66100
$66$	0	0
$67$	−408.830	−0.745470	−0.372735	−	0.927938i	$-0.621580\pi$
$67$	−408.830	−0.745470	−0.372735	+	0.927938i	$0.621580\pi$
$68$	1445.35	2.57757
$69$	0	0
$70$	−923.400	−1.57668
$71$	−530.048	−0.885988	−0.442994	−	0.896525i	$-0.646084\pi$
$71$	−530.048	−0.885988	−0.442994	+	0.896525i	$0.646084\pi$
$72$	0	0
$73$	549.032	0.880265	0.440132	−	0.897933i	$-0.354931\pi$
$73$	549.032	0.880265	0.440132	+	0.897933i	$0.354931\pi$
$74$	−885.699	−1.39136
$75$	0	0
$76$	1081.18	1.63184
$77$	83.1232	0.123023
$78$	0	0
$79$	934.908	1.33146	0.665730	−	0.746192i	$-0.268120\pi$
$79$	934.908	1.33146	0.665730	+	0.746192i	$0.268120\pi$
$80$	768.005	1.07332
$81$	0	0
$82$	1662.23	2.23856
$83$	786.990	1.04076	0.520382	−	0.853934i	$-0.325790\pi$
$83$	786.990	1.04076	0.520382	+	0.853934i	$0.325790\pi$
$84$	0	0
$85$	1925.64	2.45724
$86$	−1593.35	−1.99785
$87$	0	0
$88$	−282.000	−0.341606
$89$	741.961	0.883682	0.441841	−	0.897093i	$-0.354325\pi$
$89$	741.961	0.883682	0.441841	+	0.897093i	$0.354325\pi$
$90$	0	0
$91$	683.360	0.787204
$92$	−2200.48	−2.49366
$93$	0	0
$94$	−770.825	−0.845793
$95$	1440.45	1.55565
$96$	0	0
$97$	−611.445	−0.640029	−0.320015	−	0.947413i	$-0.603688\pi$
$97$	−611.445	−0.640029	−0.320015	+	0.947413i	$0.603688\pi$
$98$	1188.70	1.22528
$99$	0	0
$100$	4003.89	4.00389
$101$	694.672	0.684380	0.342190	−	0.939631i	$-0.388831\pi$
$101$	694.672	0.684380	0.342190	+	0.939631i	$0.388831\pi$
$102$	0	0
$103$	1438.35	1.37597	0.687984	−	0.725726i	$-0.258496\pi$
$103$	1438.35	1.37597	0.687984	+	0.725726i	$0.258496\pi$
$104$	−2318.34	−2.18588
$105$	0	0
$106$	2633.68	2.41326
$107$	1029.81	0.930426	0.465213	−	0.885199i	$-0.345978\pi$
$107$	1029.81	0.930426	0.465213	+	0.885199i	$0.345978\pi$
$108$	0	0
$109$	861.356	0.756908	0.378454	−	0.925620i	$-0.376456\pi$
$109$	861.356	0.756908	0.378454	+	0.925620i	$0.376456\pi$
$110$	−811.629	−0.703507
$111$	0	0
$112$	376.357	0.317521
$113$	703.757	0.585875	0.292938	−	0.956132i	$-0.405367\pi$
$113$	703.757	0.585875	0.292938	+	0.956132i	$0.405367\pi$
$114$	0	0
$115$	−2931.70	−2.37724
$116$	−491.665	−0.393534
$117$	0	0
$118$	−496.097	−0.387029
$119$	943.649	0.726926
$120$	0	0
$121$	−1257.94	−0.945108
$122$	−1342.86	−0.996535
$123$	0	0
$124$	1684.63	1.22003
$125$	2853.80	2.04201
$126$	0	0
$127$	64.5337	0.0450901	0.0225450	−	0.999746i	$-0.492823\pi$
$127$	64.5337	0.0450901	0.0225450	+	0.999746i	$0.492823\pi$
$128$	2654.48	1.83301
$129$	0	0
$130$	−6672.44	−4.50163
$131$	−18.7940	−0.0125346	−0.00626732	−	0.999980i	$-0.501995\pi$
$131$	−18.7940	−0.0125346	−0.00626732	+	0.999980i	$0.501995\pi$
$132$	0	0
$133$	705.885	0.460210
$134$	1956.20	1.26112
$135$	0	0
$136$	−3201.38	−2.01850
$137$	−1154.78	−0.720142	−0.360071	−	0.932925i	$-0.617248\pi$
$137$	−1154.78	−0.720142	−0.360071	+	0.932925i	$0.617248\pi$
$138$	0	0
$139$	−1892.52	−1.15483	−0.577414	−	0.816452i	$-0.695938\pi$
$139$	−1892.52	−1.15483	−0.577414	+	0.816452i	$0.695938\pi$
$140$	2874.48	1.73527
$141$	0	0
$142$	2536.21	1.49883
$143$	600.644	0.351247
$144$	0	0
$145$	−655.044	−0.375162
$146$	−2627.05	−1.48915
$147$	0	0
$148$	2757.13	1.53131
$149$	−1645.18	−0.904553	−0.452276	−	0.891878i	$-0.649388\pi$
$149$	−1645.18	−0.904553	−0.452276	+	0.891878i	$0.649388\pi$
$150$	0	0
$151$	−2245.47	−1.21016	−0.605078	−	0.796166i	$-0.706858\pi$
$151$	−2245.47	−1.21016	−0.605078	+	0.796166i	$0.706858\pi$
$152$	−2394.75	−1.27790
$153$	0	0
$154$	−397.734	−0.208119
$155$	2244.43	1.16308
$156$	0	0
$157$	−2419.98	−1.23016	−0.615081	−	0.788464i	$-0.710877\pi$
$157$	−2419.98	−1.23016	−0.615081	+	0.788464i	$0.710877\pi$
$158$	−4473.42	−2.25244
$159$	0	0
$160$	1562.84	0.772210
$161$	−1436.66	−0.703261
$162$	0	0
$163$	−722.645	−0.347251	−0.173626	−	0.984812i	$-0.555548\pi$
$163$	−722.645	−0.347251	−0.173626	+	0.984812i	$0.555548\pi$
$164$	−5174.41	−2.46374
$165$	0	0
$166$	−3765.65	−1.76067
$167$	2807.10	1.30072	0.650358	−	0.759628i	$-0.274619\pi$
$167$	2807.10	1.30072	0.650358	+	0.759628i	$0.274619\pi$
$168$	0	0
$169$	2740.92	1.24758
$170$	−9213.94	−4.15692
$171$	0	0
$172$	4959.99	2.19881
$173$	−4063.67	−1.78587	−0.892933	−	0.450190i	$-0.851356\pi$
$173$	−4063.67	−1.78587	−0.892933	+	0.450190i	$0.851356\pi$
$174$	0	0
$175$	2614.08	1.12918
$176$	330.801	0.141677
$177$	0	0
$178$	−3550.19	−1.49493
$179$	−2575.22	−1.07531	−0.537656	−	0.843164i	$-0.680690\pi$
$179$	−2575.22	−1.07531	−0.537656	+	0.843164i	$0.680690\pi$
$180$	0	0
$181$	1161.25	0.476877	0.238438	−	0.971158i	$-0.423365\pi$
$181$	1161.25	0.476877	0.238438	+	0.971158i	$0.423365\pi$
$182$	−3269.79	−1.33172
$183$	0	0
$184$	4873.96	1.95279
$185$	3673.31	1.45982
$186$	0	0
$187$	829.427	0.324351
$188$	2399.53	0.930871
$189$	0	0
$190$	−6892.37	−2.63171
$191$	−3619.78	−1.37130	−0.685650	−	0.727931i	$-0.740482\pi$
$191$	−3619.78	−1.37130	−0.685650	+	0.727931i	$0.740482\pi$
$192$	0	0
$193$	3105.81	1.15835	0.579174	−	0.815204i	$-0.303375\pi$
$193$	3105.81	1.15835	0.579174	+	0.815204i	$0.303375\pi$
$194$	2925.69	1.08274
$195$	0	0
$196$	−3700.36	−1.34853
$197$	−499.680	−0.180715	−0.0903573	−	0.995909i	$-0.528801\pi$
$197$	−499.680	−0.180715	−0.0903573	+	0.995909i	$0.528801\pi$
$198$	0	0
$199$	−1884.46	−0.671286	−0.335643	−	0.941989i	$-0.608953\pi$
$199$	−1884.46	−0.671286	−0.335643	+	0.941989i	$0.608953\pi$
$200$	−8868.42	−3.13546
$201$	0	0
$202$	−3323.92	−1.15777
$203$	−321.001	−0.110984
$204$	0	0
$205$	−6893.85	−2.34872
$206$	−6882.32	−2.32774
$207$	0	0
$208$	2719.53	0.906566
$209$	620.442	0.205344
$210$	0	0
$211$	−4769.84	−1.55625	−0.778127	−	0.628107i	$-0.783830\pi$
$211$	−4769.84	−1.55625	−0.778127	+	0.628107i	$0.783830\pi$
$212$	−8198.46	−2.65600
$213$	0	0
$214$	−4927.51	−1.57401
$215$	6608.19	2.09616
$216$	0	0
$217$	1099.87	0.344074
$218$	−4121.48	−1.28047
$219$	0	0
$220$	2526.55	0.774272
$221$	6818.76	2.07547
$222$	0	0
$223$	−6102.53	−1.83254	−0.916268	−	0.400565i	$-0.868814\pi$
$223$	−6102.53	−1.83254	−0.916268	+	0.400565i	$0.868814\pi$
$224$	765.863	0.228444
$225$	0	0
$226$	−3367.39	−0.991130
$227$	−6734.80	−1.96918	−0.984591	−	0.174875i	$-0.944048\pi$
$227$	−6734.80	−1.96918	−0.984591	+	0.174875i	$0.944048\pi$
$228$	0	0
$229$	4481.93	1.29334	0.646669	−	0.762771i	$-0.276161\pi$
$229$	4481.93	1.29334	0.646669	+	0.762771i	$0.276161\pi$
$230$	14027.8	4.02160
$231$	0	0
$232$	1089.01	0.308178
$233$	−3307.94	−0.930087	−0.465044	−	0.885288i	$-0.653961\pi$
$233$	−3307.94	−0.930087	−0.465044	+	0.885288i	$0.653961\pi$
$234$	0	0
$235$	3196.89	0.887413
$236$	1544.32	0.425960
$237$	0	0
$238$	−4515.24	−1.22975
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	5155.80	1.37807	0.689033	−	0.724730i	$-0.258035\pi$
$241$	5155.80	1.37807	0.689033	+	0.724730i	$0.258035\pi$
$242$	6019.07	1.59885
$243$	0	0
$244$	4180.25	1.09678
$245$	−4929.98	−1.28557
$246$	0	0
$247$	5100.69	1.31396
$248$	−3731.37	−0.955412
$249$	0	0
$250$	−13655.1	−3.45449
$251$	2292.76	0.576564	0.288282	−	0.957546i	$-0.406916\pi$
$251$	2292.76	0.576564	0.288282	+	0.957546i	$0.406916\pi$
$252$	0	0
$253$	−1262.77	−0.313792
$254$	−308.786	−0.0762792
$255$	0	0
$256$	−7209.84	−1.76022
$257$	7868.90	1.90992	0.954958	−	0.296741i	$-0.0958999\pi$
$257$	7868.90	1.90992	0.954958	+	0.296741i	$0.0958999\pi$
$258$	0	0
$259$	1800.09	0.431861
$260$	20770.9	4.95444
$261$	0	0
$262$	89.9268	0.0212050
$263$	−4584.99	−1.07499	−0.537495	−	0.843267i	$-0.680629\pi$
$263$	−4584.99	−1.07499	−0.537495	+	0.843267i	$0.680629\pi$
$264$	0	0
$265$	−10922.8	−2.53201
$266$	−3377.57	−0.778541
$267$	0	0
$268$	−6089.52	−1.38797
$269$	3099.36	0.702497	0.351248	−	0.936282i	$-0.385757\pi$
$269$	3099.36	0.702497	0.351248	+	0.936282i	$0.385757\pi$
$270$	0	0
$271$	74.6626	0.0167359	0.00836795	−	0.999965i	$-0.497336\pi$
$271$	74.6626	0.0167359	0.00836795	+	0.999965i	$0.497336\pi$
$272$	3755.39	0.837147
$273$	0	0
$274$	5525.47	1.21827
$275$	2297.66	0.503834
$276$	0	0
$277$	4505.76	0.977346	0.488673	−	0.872467i	$-0.337481\pi$
$277$	4505.76	0.977346	0.488673	+	0.872467i	$0.337481\pi$
$278$	9055.44	1.95363
$279$	0	0
$280$	−6366.84	−1.35890
$281$	1282.60	0.272290	0.136145	−	0.990689i	$-0.456529\pi$
$281$	1282.60	0.272290	0.136145	+	0.990689i	$0.456529\pi$
$282$	0	0
$283$	7160.55	1.50406	0.752032	−	0.659126i	$-0.229074\pi$
$283$	7160.55	1.50406	0.752032	+	0.659126i	$0.229074\pi$
$284$	−7895.07	−1.64960
$285$	0	0
$286$	−2874.00	−0.594208
$287$	−3378.29	−0.694824
$288$	0	0
$289$	4503.00	0.916547
$290$	3134.30	0.634664
$291$	0	0
$292$	8177.83	1.63894
$293$	4723.96	0.941900	0.470950	−	0.882160i	$-0.343911\pi$
$293$	4723.96	0.941900	0.470950	+	0.882160i	$0.343911\pi$
$294$	0	0
$295$	2057.49	0.406074
$296$	−6106.90	−1.19918
$297$	0	0
$298$	7871.97	1.53024
$299$	−10381.3	−2.00791
$300$	0	0
$301$	3238.31	0.620109
$302$	10744.3	2.04723
$303$	0	0
$304$	2809.17	0.529991
$305$	5569.34	1.04557
$306$	0	0
$307$	−932.353	−0.173330	−0.0866648	−	0.996238i	$-0.527621\pi$
$307$	−932.353	−0.173330	−0.0866648	+	0.996238i	$0.527621\pi$
$308$	1238.12	0.229053
$309$	0	0
$310$	−10739.3	−1.96758
$311$	−8737.59	−1.59313	−0.796565	−	0.604553i	$-0.793352\pi$
$311$	−8737.59	−1.59313	−0.796565	+	0.604553i	$0.793352\pi$
$312$	0	0
$313$	−9685.91	−1.74914	−0.874569	−	0.484902i	$-0.838855\pi$
$313$	−9685.91	−1.74914	−0.874569	+	0.484902i	$0.838855\pi$
$314$	11579.3	2.08107
$315$	0	0
$316$	13925.5	2.47901
$317$	−7268.72	−1.28786	−0.643931	−	0.765084i	$-0.722698\pi$
$317$	−7268.72	−1.28786	−0.643931	+	0.765084i	$0.722698\pi$
$318$	0	0
$319$	−282.146	−0.0495208
$320$	−13622.1	−2.37967
$321$	0	0
$322$	6874.25	1.18971
$323$	7043.52	1.21335
$324$	0	0
$325$	18889.2	3.22395
$326$	3457.77	0.587448
$327$	0	0
$328$	11461.1	1.92936
$329$	1566.62	0.262524
$330$	0	0
$331$	−2629.34	−0.436622	−0.218311	−	0.975879i	$-0.570055\pi$
$331$	−2629.34	−0.436622	−0.218311	+	0.975879i	$0.570055\pi$
$332$	11722.2	1.93777
$333$	0	0
$334$	−13431.6	−2.20043
$335$	−8113.06	−1.32318
$336$	0	0
$337$	−822.253	−0.132911	−0.0664555	−	0.997789i	$-0.521169\pi$
$337$	−822.253	−0.132911	−0.0664555	+	0.997789i	$0.521169\pi$
$338$	−13115.0	−2.11053
$339$	0	0
$340$	28682.4	4.57507
$341$	966.738	0.153524
$342$	0	0
$343$	−5751.49	−0.905398
$344$	−10986.1	−1.72190
$345$	0	0
$346$	19444.1	3.02116
$347$	−2068.50	−0.320009	−0.160004	−	0.987116i	$-0.551151\pi$
$347$	−2068.50	−0.320009	−0.160004	+	0.987116i	$0.551151\pi$
$348$	0	0
$349$	4309.43	0.660970	0.330485	−	0.943811i	$-0.392788\pi$
$349$	4309.43	0.660970	0.330485	+	0.943811i	$0.392788\pi$
$350$	−12508.0	−1.91024
$351$	0	0
$352$	673.161	0.101931
$353$	−1240.22	−0.186998	−0.0934989	−	0.995619i	$-0.529805\pi$
$353$	−1240.22	−0.186998	−0.0934989	+	0.995619i	$0.529805\pi$
$354$	0	0
$355$	−10518.6	−1.57259
$356$	11051.5	1.64531
$357$	0	0
$358$	12322.1	1.81911
$359$	−7485.52	−1.10048	−0.550238	−	0.835008i	$-0.685463\pi$
$359$	−7485.52	−1.10048	−0.550238	+	0.835008i	$0.685463\pi$
$360$	0	0
$361$	−1590.18	−0.231839
$362$	−5556.41	−0.806736
$363$	0	0
$364$	10178.6	1.46568
$365$	10895.3	1.56243
$366$	0	0
$367$	−8439.72	−1.20041	−0.600204	−	0.799847i	$-0.704914\pi$
$367$	−8439.72	−1.20041	−0.600204	+	0.799847i	$0.704914\pi$
$368$	−5717.42	−0.809894
$369$	0	0
$370$	−17576.3	−2.46960
$371$	−5352.66	−0.749046
$372$	0	0
$373$	12520.7	1.73807	0.869033	−	0.494755i	$-0.164742\pi$
$373$	12520.7	1.73807	0.869033	+	0.494755i	$0.164742\pi$
$374$	−3968.70	−0.548708
$375$	0	0
$376$	−5314.84	−0.728968
$377$	−2319.53	−0.316875
$378$	0	0
$379$	1526.01	0.206824	0.103412	−	0.994639i	$-0.467024\pi$
$379$	1526.01	0.206824	0.103412	+	0.994639i	$0.467024\pi$
$380$	21455.5	2.89643
$381$	0	0
$382$	17320.2	2.31984
$383$	10304.4	1.37475	0.687375	−	0.726302i	$-0.258763\pi$
$383$	10304.4	1.37475	0.687375	+	0.726302i	$0.258763\pi$
$384$	0	0
$385$	1649.55	0.218360
$386$	−14860.9	−1.95959
$387$	0	0
$388$	−9107.47	−1.19165
$389$	10412.4	1.35714	0.678571	−	0.734534i	$-0.262599\pi$
$389$	10412.4	1.35714	0.678571	+	0.734534i	$0.262599\pi$
$390$	0	0
$391$	−14335.4	−1.85415
$392$	8196.11	1.05604
$393$	0	0
$394$	2390.91	0.305716
$395$	18552.9	2.36328
$396$	0	0
$397$	−432.529	−0.0546801	−0.0273400	−	0.999626i	$-0.508704\pi$
$397$	−432.529	−0.0546801	−0.0273400	+	0.999626i	$0.508704\pi$
$398$	9016.90	1.13562
$399$	0	0
$400$	10403.1	1.30039
$401$	−7984.92	−0.994384	−0.497192	−	0.867641i	$-0.665636\pi$
$401$	−7984.92	−0.994384	−0.497192	+	0.867641i	$0.665636\pi$
$402$	0	0
$403$	7947.60	0.982377
$404$	10347.1	1.27423
$405$	0	0
$406$	1535.95	0.187753
$407$	1582.20	0.192695
$408$	0	0
$409$	2505.56	0.302914	0.151457	−	0.988464i	$-0.451604\pi$
$409$	2505.56	0.302914	0.151457	+	0.988464i	$0.451604\pi$
$410$	32986.2	3.97335
$411$	0	0
$412$	21424.2	2.56188
$413$	1008.26	0.120129
$414$	0	0
$415$	15617.5	1.84731
$416$	5534.08	0.652237
$417$	0	0
$418$	−2968.74	−0.347382
$419$	2380.41	0.277543	0.138772	−	0.990324i	$-0.455685\pi$
$419$	2380.41	0.277543	0.138772	+	0.990324i	$0.455685\pi$
$420$	0	0
$421$	2540.60	0.294112	0.147056	−	0.989128i	$-0.453020\pi$
$421$	2540.60	0.294112	0.147056	+	0.989128i	$0.453020\pi$
$422$	22823.1	2.63273
$423$	0	0
$424$	18159.2	2.07993
$425$	26084.0	2.97709
$426$	0	0
$427$	2729.23	0.309313
$428$	15339.0	1.73234
$429$	0	0
$430$	−31619.3	−3.54609
$431$	−6366.35	−0.711500	−0.355750	−	0.934581i	$-0.615774\pi$
$431$	−6366.35	−0.711500	−0.355750	+	0.934581i	$0.615774\pi$
$432$	0	0
$433$	7620.55	0.845774	0.422887	−	0.906182i	$-0.361017\pi$
$433$	7620.55	0.845774	0.422887	+	0.906182i	$0.361017\pi$
$434$	−5262.73	−0.582072
$435$	0	0
$436$	12829.9	1.40927
$437$	−10723.4	−1.17385
$438$	0	0
$439$	−222.146	−0.0241513	−0.0120757	−	0.999927i	$-0.503844\pi$
$439$	−222.146	−0.0241513	−0.0120757	+	0.999927i	$0.503844\pi$
$440$	−5596.18	−0.606335
$441$	0	0
$442$	−32626.9	−3.51109
$443$	−3699.08	−0.396724	−0.198362	−	0.980129i	$-0.563562\pi$
$443$	−3699.08	−0.396724	−0.198362	+	0.980129i	$0.563562\pi$
$444$	0	0
$445$	14723.9	1.56850
$446$	29199.8	3.10012
$447$	0	0
$448$	−6675.41	−0.703981
$449$	14076.6	1.47954	0.739772	−	0.672858i	$-0.234933\pi$
$449$	14076.6	1.47954	0.739772	+	0.672858i	$0.234933\pi$
$450$	0	0
$451$	−2969.38	−0.310028
$452$	10482.5	1.09083
$453$	0	0
$454$	32225.1	3.33128
$455$	13561.0	1.39725
$456$	0	0
$457$	−13989.7	−1.43197	−0.715987	−	0.698114i	$-0.754023\pi$
$457$	−13989.7	−1.43197	−0.715987	+	0.698114i	$0.754023\pi$
$458$	−21445.5	−2.18795
$459$	0	0
$460$	−43667.7	−4.42612
$461$	2517.77	0.254369	0.127185	−	0.991879i	$-0.459406\pi$
$461$	2517.77	0.254369	0.127185	+	0.991879i	$0.459406\pi$
$462$	0	0
$463$	−19554.7	−1.96281	−0.981406	−	0.191942i	$-0.938522\pi$
$463$	−19554.7	−1.96281	−0.981406	+	0.191942i	$0.938522\pi$
$464$	−1277.47	−0.127813
$465$	0	0
$466$	15828.1	1.57344
$467$	−5374.11	−0.532514	−0.266257	−	0.963902i	$-0.585787\pi$
$467$	−5374.11	−0.532514	−0.266257	+	0.963902i	$0.585787\pi$
$468$	0	0
$469$	−3975.76	−0.391436
$470$	−15296.7	−1.50124
$471$	0	0
$472$	−3420.59	−0.333571
$473$	2846.33	0.276690
$474$	0	0
$475$	19511.8	1.88477
$476$	14055.7	1.35344
$477$	0	0
$478$	−1143.58	−0.109427
$479$	−7240.49	−0.690660	−0.345330	−	0.938481i	$-0.612233\pi$
$479$	−7240.49	−0.690660	−0.345330	+	0.938481i	$0.612233\pi$
$480$	0	0
$481$	13007.3	1.23302
$482$	−24669.8	−2.33129
$483$	0	0
$484$	−18737.0	−1.75967
$485$	−12133.9	−1.13602
$486$	0	0
$487$	−12572.3	−1.16983	−0.584913	−	0.811096i	$-0.698871\pi$
$487$	−12572.3	−1.16983	−0.584913	+	0.811096i	$0.698871\pi$
$488$	−9259.05	−0.858888
$489$	0	0
$490$	23589.3	2.17481
$491$	−6112.04	−0.561777	−0.280888	−	0.959740i	$-0.590629\pi$
$491$	−6112.04	−0.561777	−0.280888	+	0.959740i	$0.590629\pi$
$492$	0	0
$493$	−3203.03	−0.292611
$494$	−24406.1	−2.22284
$495$	0	0
$496$	4377.09	0.396244
$497$	−5154.57	−0.465220
$498$	0	0
$499$	−8599.20	−0.771449	−0.385724	−	0.922614i	$-0.626048\pi$
$499$	−8599.20	−0.771449	−0.385724	+	0.922614i	$0.626048\pi$
$500$	42507.4	3.80198
$501$	0	0
$502$	−10970.6	−0.975378
$503$	1731.25	0.153465	0.0767324	−	0.997052i	$-0.475551\pi$
$503$	1731.25	0.153465	0.0767324	+	0.997052i	$0.475551\pi$
$504$	0	0
$505$	13785.5	1.21474
$506$	6042.17	0.530845
$507$	0	0
$508$	961.229	0.0839521
$509$	−9134.65	−0.795454	−0.397727	−	0.917504i	$-0.630201\pi$
$509$	−9134.65	−0.795454	−0.397727	+	0.917504i	$0.630201\pi$
$510$	0	0
$511$	5339.19	0.462215
$512$	13262.4	1.14476
$513$	0	0
$514$	−37651.7	−3.23102
$515$	28543.4	2.44228
$516$	0	0
$517$	1376.99	0.117137
$518$	−8613.18	−0.730582
$519$	0	0
$520$	−46006.4	−3.87984
$521$	4233.05	0.355956	0.177978	−	0.984034i	$-0.443044\pi$
$521$	4233.05	0.355956	0.177978	+	0.984034i	$0.443044\pi$
$522$	0	0
$523$	9909.37	0.828502	0.414251	−	0.910163i	$-0.364044\pi$
$523$	9909.37	0.828502	0.414251	+	0.910163i	$0.364044\pi$
$524$	−279.937	−0.0233379
$525$	0	0
$526$	21938.6	1.81857
$527$	10974.8	0.907154
$528$	0	0
$529$	9658.08	0.793793
$530$	52264.2	4.28342
$531$	0	0
$532$	10514.2	0.856854
$533$	−24411.4	−1.98382
$534$	0	0
$535$	20436.2	1.65146
$536$	13488.0	1.08693
$537$	0	0
$538$	−14830.1	−1.18842
$539$	−2123.48	−0.169694
$540$	0	0
$541$	−20780.5	−1.65143	−0.825714	−	0.564089i	$-0.809227\pi$
$541$	−20780.5	−1.65143	−0.825714	+	0.564089i	$0.809227\pi$
$542$	−357.251	−0.0283123
$543$	0	0
$544$	7641.99	0.602294
$545$	17093.3	1.34348
$546$	0	0
$547$	−17988.3	−1.40608	−0.703040	−	0.711150i	$-0.748175\pi$
$547$	−17988.3	−1.40608	−0.703040	+	0.711150i	$0.748175\pi$
$548$	−17200.4	−1.34081
$549$	0	0
$550$	−10994.0	−0.852340
$551$	−2395.99	−0.185250
$552$	0	0
$553$	9091.73	0.699131
$554$	−21559.5	−1.65338
$555$	0	0
$556$	−28189.0	−2.15014
$557$	−5673.16	−0.431561	−0.215781	−	0.976442i	$-0.569230\pi$
$557$	−5673.16	−0.431561	−0.215781	+	0.976442i	$0.569230\pi$
$558$	0	0
$559$	23399.8	1.77050
$560$	7468.64	0.563585
$561$	0	0
$562$	−6137.07	−0.460635
$563$	10444.6	0.781862	0.390931	−	0.920420i	$-0.372153\pi$
$563$	10444.6	0.781862	0.390931	+	0.920420i	$0.372153\pi$
$564$	0	0
$565$	13965.8	1.03990
$566$	−34262.3	−2.54444
$567$	0	0
$568$	17487.2	1.29181
$569$	18182.8	1.33965	0.669827	−	0.742517i	$-0.266368\pi$
$569$	18182.8	1.33965	0.669827	+	0.742517i	$0.266368\pi$
$570$	0	0
$571$	−3278.87	−0.240309	−0.120154	−	0.992755i	$-0.538339\pi$
$571$	−3278.87	−0.240309	−0.120154	+	0.992755i	$0.538339\pi$
$572$	8946.59	0.653979
$573$	0	0
$574$	16164.7	1.17544
$575$	−39711.8	−2.88017
$576$	0	0
$577$	11157.3	0.805000	0.402500	−	0.915420i	$-0.368141\pi$
$577$	11157.3	0.805000	0.402500	+	0.915420i	$0.368141\pi$
$578$	−21546.3	−1.55053
$579$	0	0
$580$	−9756.88	−0.698504
$581$	7653.27	0.546490
$582$	0	0
$583$	−4704.76	−0.334221
$584$	−18113.5	−1.28346
$585$	0	0
$586$	−22603.5	−1.59342
$587$	16447.0	1.15646	0.578230	−	0.815874i	$-0.303744\pi$
$587$	16447.0	1.15646	0.578230	+	0.815874i	$0.303744\pi$
$588$	0	0
$589$	8209.56	0.574311
$590$	−9844.84	−0.686959
$591$	0	0
$592$	7163.71	0.497343
$593$	19783.1	1.36997	0.684987	−	0.728556i	$-0.259808\pi$
$593$	19783.1	1.36997	0.684987	+	0.728556i	$0.259808\pi$
$594$	0	0
$595$	18726.3	1.29026
$596$	−24504.9	−1.68416
$597$	0	0
$598$	49673.0	3.39679
$599$	−8064.17	−0.550072	−0.275036	−	0.961434i	$-0.588690\pi$
$599$	−8064.17	−0.550072	−0.275036	+	0.961434i	$0.588690\pi$
$600$	0	0
$601$	12131.3	0.823368	0.411684	−	0.911327i	$-0.364941\pi$
$601$	12131.3	0.823368	0.411684	+	0.911327i	$0.364941\pi$
$602$	−15494.9	−1.04904
$603$	0	0
$604$	−33446.2	−2.25316
$605$	−24963.3	−1.67752
$606$	0	0
$607$	−374.686	−0.0250544	−0.0125272	−	0.999922i	$-0.503988\pi$
$607$	−374.686	−0.0250544	−0.0125272	+	0.999922i	$0.503988\pi$
$608$	5716.50	0.381307
$609$	0	0
$610$	−26648.6	−1.76880
$611$	11320.3	0.749542
$612$	0	0
$613$	−10459.3	−0.689144	−0.344572	−	0.938760i	$-0.611976\pi$
$613$	−10459.3	−0.689144	−0.344572	+	0.938760i	$0.611976\pi$
$614$	4461.19	0.293223
$615$	0	0
$616$	−2742.38	−0.179372
$617$	−8628.61	−0.563006	−0.281503	−	0.959560i	$-0.590833\pi$
$617$	−8628.61	−0.563006	−0.281503	+	0.959560i	$0.590833\pi$
$618$	0	0
$619$	−66.5127	−0.00431885	−0.00215943	−	0.999998i	$-0.500687\pi$
$619$	−66.5127	−0.00431885	−0.00215943	+	0.999998i	$0.500687\pi$
$620$	33430.7	2.16550
$621$	0	0
$622$	41808.3	2.69511
$623$	7215.37	0.464009
$624$	0	0
$625$	23031.6	1.47402
$626$	46345.8	2.95903
$627$	0	0
$628$	−36045.6	−2.29041
$629$	17961.8	1.13861
$630$	0	0
$631$	18876.9	1.19093	0.595466	−	0.803381i	$-0.296968\pi$
$631$	18876.9	1.19093	0.595466	+	0.803381i	$0.296968\pi$
$632$	−30844.2	−1.94132
$633$	0	0
$634$	34779.9	2.17869
$635$	1280.64	0.0800328
$636$	0	0
$637$	−17457.2	−1.08584
$638$	1350.03	0.0837747
$639$	0	0
$640$	52677.0	3.25350
$641$	18816.8	1.15947	0.579734	−	0.814806i	$-0.303157\pi$
$641$	18816.8	1.15947	0.579734	+	0.814806i	$0.303157\pi$
$642$	0	0
$643$	12561.3	0.770403	0.385202	−	0.922832i	$-0.374132\pi$
$643$	12561.3	0.770403	0.385202	+	0.922832i	$0.374132\pi$
$644$	−21399.1	−1.30938
$645$	0	0
$646$	−33702.3	−2.05263
$647$	12355.1	0.750740	0.375370	−	0.926875i	$-0.377516\pi$
$647$	12355.1	0.750740	0.375370	+	0.926875i	$0.377516\pi$
$648$	0	0
$649$	886.220	0.0536012
$650$	−90382.4	−5.45399
$651$	0	0
$652$	−10763.8	−0.646538
$653$	16862.8	1.01056	0.505278	−	0.862957i	$-0.331390\pi$
$653$	16862.8	1.01056	0.505278	+	0.862957i	$0.331390\pi$
$654$	0	0
$655$	−372.959	−0.0222484
$656$	−13444.4	−0.800178
$657$	0	0
$658$	−7496.07	−0.444114
$659$	101.409	0.00599441	0.00299721	−	0.999996i	$-0.499046\pi$
$659$	101.409	0.00599441	0.00299721	+	0.999996i	$0.499046\pi$
$660$	0	0
$661$	780.657	0.0459365	0.0229682	−	0.999736i	$-0.492688\pi$
$661$	780.657	0.0459365	0.0229682	+	0.999736i	$0.492688\pi$
$662$	12581.1	0.738636
$663$	0	0
$664$	−25964.1	−1.51748
$665$	14008.0	0.816852
$666$	0	0
$667$	4876.48	0.283086
$668$	41811.7	2.42177
$669$	0	0
$670$	38820.0	2.23843
$671$	2398.87	0.138014
$672$	0	0
$673$	−8497.67	−0.486718	−0.243359	−	0.969936i	$-0.578249\pi$
$673$	−8497.67	−0.486718	−0.243359	+	0.969936i	$0.578249\pi$
$674$	3934.38	0.224846
$675$	0	0
$676$	40826.0	2.32283
$677$	3736.10	0.212098	0.106049	−	0.994361i	$-0.466180\pi$
$677$	3736.10	0.212098	0.106049	+	0.994361i	$0.466180\pi$
$678$	0	0
$679$	−5946.13	−0.336070
$680$	−63530.1	−3.58275
$681$	0	0
$682$	−4625.72	−0.259718
$683$	−29098.8	−1.63021	−0.815107	−	0.579311i	$-0.803322\pi$
$683$	−29098.8	−1.63021	−0.815107	+	0.579311i	$0.803322\pi$
$684$	0	0
$685$	−22916.1	−1.27822
$686$	27520.2	1.53167
$687$	0	0
$688$	12887.3	0.714134
$689$	−38678.0	−2.13863
$690$	0	0
$691$	5688.32	0.313160	0.156580	−	0.987665i	$-0.449953\pi$
$691$	5688.32	0.313160	0.156580	+	0.987665i	$0.449953\pi$
$692$	−60528.3	−3.32506
$693$	0	0
$694$	9897.51	0.541361
$695$	−37556.2	−2.04977
$696$	0	0
$697$	−33709.6	−1.83191
$698$	−20620.1	−1.11817
$699$	0	0
$700$	38936.7	2.10239
$701$	−21585.3	−1.16300	−0.581502	−	0.813545i	$-0.697535\pi$
$701$	−21585.3	−1.16300	−0.581502	+	0.813545i	$0.697535\pi$
$702$	0	0
$703$	13436.1	0.720841
$704$	−5867.40	−0.314113
$705$	0	0
$706$	5934.29	0.316346
$707$	6755.49	0.359358
$708$	0	0
$709$	28328.5	1.50057	0.750283	−	0.661117i	$-0.229917\pi$
$709$	28328.5	1.50057	0.750283	+	0.661117i	$0.229917\pi$
$710$	50330.1	2.66036
$711$	0	0
$712$	−24478.5	−1.28844
$713$	−16708.7	−0.877621
$714$	0	0
$715$	11919.5	0.623448
$716$	−38357.9	−2.00210
$717$	0	0
$718$	35817.3	1.86168
$719$	22840.1	1.18469	0.592344	−	0.805685i	$-0.298202\pi$
$719$	22840.1	1.18469	0.592344	+	0.805685i	$0.298202\pi$
$720$	0	0
$721$	13987.6	0.722502
$722$	7608.83	0.392204
$723$	0	0
$724$	17296.7	0.887885
$725$	−8872.99	−0.454531
$726$	0	0
$727$	−25402.2	−1.29590	−0.647948	−	0.761685i	$-0.724372\pi$
$727$	−25402.2	−1.29590	−0.647948	+	0.761685i	$0.724372\pi$
$728$	−22545.2	−1.14778
$729$	0	0
$730$	−52132.7	−2.64317
$731$	32312.7	1.63492
$732$	0	0
$733$	−27288.3	−1.37505	−0.687527	−	0.726159i	$-0.741304\pi$
$733$	−27288.3	−1.37505	−0.687527	+	0.726159i	$0.741304\pi$
$734$	40383.0	2.03074
$735$	0	0
$736$	−11634.6	−0.582686
$737$	−3494.52	−0.174657
$738$	0	0
$739$	36966.9	1.84012	0.920061	−	0.391775i	$-0.128139\pi$
$739$	36966.9	1.84012	0.920061	+	0.391775i	$0.128139\pi$
$740$	54714.0	2.71801
$741$	0	0
$742$	25611.8	1.26717
$743$	1086.94	0.0536686	0.0268343	−	0.999640i	$-0.491457\pi$
$743$	1086.94	0.0536686	0.0268343	+	0.999640i	$0.491457\pi$
$744$	0	0
$745$	−32647.9	−1.60554
$746$	−59910.0	−2.94030
$747$	0	0
$748$	12354.3	0.603902
$749$	10014.6	0.488553
$750$	0	0
$751$	28459.7	1.38284	0.691418	−	0.722455i	$-0.256986\pi$
$751$	28459.7	1.38284	0.691418	+	0.722455i	$0.256986\pi$
$752$	6234.59	0.302330
$753$	0	0
$754$	11098.7	0.536061
$755$	−44560.4	−2.14797
$756$	0	0
$757$	23083.0	1.10828	0.554139	−	0.832424i	$-0.313048\pi$
$757$	23083.0	1.10828	0.554139	+	0.832424i	$0.313048\pi$
$758$	−7301.79	−0.349885
$759$	0	0
$760$	−47522.9	−2.26821
$761$	36.9783	0.00176145	0.000880724	−	1.00000i	$-0.499720\pi$
$761$	36.9783	0.00176145	0.000880724		1.00000i	$0.499720\pi$
$762$	0	0
$763$	8376.45	0.397442
$764$	−53916.7	−2.55319
$765$	0	0
$766$	−49305.2	−2.32568
$767$	7285.65	0.342985
$768$	0	0
$769$	−9659.87	−0.452983	−0.226492	−	0.974013i	$-0.572726\pi$
$769$	−9659.87	−0.452983	−0.226492	+	0.974013i	$0.572726\pi$
$770$	−7892.87	−0.369402
$771$	0	0
$772$	46261.1	2.15670
$773$	−35662.2	−1.65935	−0.829676	−	0.558245i	$-0.811475\pi$
$773$	−35662.2	−1.65935	−0.829676	+	0.558245i	$0.811475\pi$
$774$	0	0
$775$	30402.2	1.40913
$776$	20172.6	0.933188
$777$	0	0
$778$	−49821.9	−2.29589
$779$	−25216.0	−1.15977
$780$	0	0
$781$	−4530.65	−0.207579
$782$	68593.2	3.13669
$783$	0	0
$784$	−9614.48	−0.437977
$785$	−48023.5	−2.18348
$786$	0	0
$787$	39166.5	1.77400	0.886999	−	0.461771i	$-0.152786\pi$
$787$	39166.5	1.77400	0.886999	+	0.461771i	$0.152786\pi$
$788$	−7442.74	−0.336468
$789$	0	0
$790$	−88773.1	−3.99798
$791$	6843.85	0.307635
$792$	0	0
$793$	19721.2	0.883129
$794$	2069.59	0.0925027
$795$	0	0
$796$	−28069.0	−1.24985
$797$	2563.21	0.113919	0.0569595	−	0.998376i	$-0.481859\pi$
$797$	2563.21	0.113919	0.0569595	+	0.998376i	$0.481859\pi$
$798$	0	0
$799$	15632.2	0.692148
$800$	21169.7	0.935578
$801$	0	0
$802$	38206.8	1.68221
$803$	4692.92	0.206238
$804$	0	0
$805$	−28510.0	−1.24826
$806$	−38028.2	−1.66189
$807$	0	0
$808$	−22918.4	−0.997854
$809$	13667.1	0.593954	0.296977	−	0.954885i	$-0.404022\pi$
$809$	13667.1	0.593954	0.296977	+	0.954885i	$0.404022\pi$
$810$	0	0
$811$	7391.23	0.320026	0.160013	−	0.987115i	$-0.448846\pi$
$811$	7391.23	0.320026	0.160013	+	0.987115i	$0.448846\pi$
$812$	−4781.30	−0.206639
$813$	0	0
$814$	−7570.62	−0.325983
$815$	−14340.6	−0.616355
$816$	0	0
$817$	24171.1	1.03506
$818$	−11988.8	−0.512442
$819$	0	0
$820$	−102684.	−4.37302
$821$	40299.0	1.71309	0.856544	−	0.516074i	$-0.172607\pi$
$821$	40299.0	1.71309	0.856544	+	0.516074i	$0.172607\pi$
$822$	0	0
$823$	41727.0	1.76733	0.883664	−	0.468121i	$-0.155069\pi$
$823$	41727.0	1.76733	0.883664	+	0.468121i	$0.155069\pi$
$824$	−47453.6	−2.00622
$825$	0	0
$826$	−4824.41	−0.203224
$827$	34063.5	1.43229	0.716145	−	0.697952i	$-0.245905\pi$
$827$	34063.5	1.43229	0.716145	+	0.697952i	$0.245905\pi$
$828$	0	0
$829$	−36976.2	−1.54914	−0.774571	−	0.632488i	$-0.782034\pi$
$829$	−36976.2	−1.54914	−0.774571	+	0.632488i	$0.782034\pi$
$830$	−74727.7	−3.12511
$831$	0	0
$832$	−48236.1	−2.00996
$833$	−24106.6	−1.00270
$834$	0	0
$835$	55705.7	2.30871
$836$	9241.49	0.382325
$837$	0	0
$838$	−11390.0	−0.469522
$839$	−9709.26	−0.399524	−0.199762	−	0.979844i	$-0.564017\pi$
$839$	−9709.26	−0.399524	−0.199762	+	0.979844i	$0.564017\pi$
$840$	0	0
$841$	−23299.4	−0.955325
$842$	−12156.4	−0.497552
$843$	0	0
$844$	−71046.8	−2.89755
$845$	54392.5	2.21439
$846$	0	0
$847$	−12233.1	−0.496263
$848$	−21301.7	−0.862622
$849$	0	0
$850$	−124809.	−5.03636
$851$	−27346.0	−1.10154
$852$	0	0
$853$	30934.7	1.24172	0.620858	−	0.783923i	$-0.286784\pi$
$853$	30934.7	1.24172	0.620858	+	0.783923i	$0.286784\pi$
$854$	−13059.0	−0.523266
$855$	0	0
$856$	−33975.2	−1.35660
$857$	−17180.4	−0.684799	−0.342400	−	0.939554i	$-0.611240\pi$
$857$	−17180.4	−0.684799	−0.342400	+	0.939554i	$0.611240\pi$
$858$	0	0
$859$	−18577.7	−0.737910	−0.368955	−	0.929447i	$-0.620284\pi$
$859$	−18577.7	−0.737910	−0.368955	+	0.929447i	$0.620284\pi$
$860$	98429.0	3.90279
$861$	0	0
$862$	30462.2	1.20365
$863$	−35850.9	−1.41411	−0.707056	−	0.707158i	$-0.749977\pi$
$863$	−35850.9	−1.41411	−0.707056	+	0.707158i	$0.749977\pi$
$864$	0	0
$865$	−80641.8	−3.16983
$866$	−36463.4	−1.43080
$867$	0	0
$868$	16382.5	0.640622
$869$	7991.24	0.311950
$870$	0	0
$871$	−28728.6	−1.11760
$872$	−28417.6	−1.10360
$873$	0	0
$874$	51310.3	1.98581
$875$	27752.4	1.07223
$876$	0	0
$877$	−34877.5	−1.34291	−0.671454	−	0.741046i	$-0.734330\pi$
$877$	−34877.5	−1.34291	−0.671454	+	0.741046i	$0.734330\pi$
$878$	1062.94	0.0408570
$879$	0	0
$880$	6564.61	0.251469
$881$	40799.7	1.56025	0.780124	−	0.625625i	$-0.215156\pi$
$881$	40799.7	1.56025	0.780124	+	0.625625i	$0.215156\pi$
$882$	0	0
$883$	−20747.1	−0.790710	−0.395355	−	0.918528i	$-0.629378\pi$
$883$	−20747.1	−0.790710	−0.395355	+	0.918528i	$0.629378\pi$
$884$	101565.	3.86427
$885$	0	0
$886$	17699.6	0.671141
$887$	−35205.1	−1.33266	−0.666332	−	0.745655i	$-0.732137\pi$
$887$	−35205.1	−1.33266	−0.666332	+	0.745655i	$0.732137\pi$
$888$	0	0
$889$	627.573	0.0236762
$890$	−70452.0	−2.65344
$891$	0	0
$892$	−90897.2	−3.41195
$893$	11693.4	0.438193
$894$	0	0
$895$	−51104.1	−1.90863
$896$	25814.1	0.962486
$897$	0	0
$898$	−67354.7	−2.50296
$899$	−3733.29	−0.138501
$900$	0	0
$901$	−53410.3	−1.97487
$902$	14208.1	0.524476
$903$	0	0
$904$	−23218.1	−0.854230
$905$	23044.4	0.846434
$906$	0	0
$907$	33086.7	1.21127	0.605636	−	0.795741i	$-0.292919\pi$
$907$	33086.7	1.21127	0.605636	+	0.795741i	$0.292919\pi$
$908$	−100315.	−3.66637
$909$	0	0
$910$	−64887.6	−2.36374
$911$	19349.1	0.703693	0.351846	−	0.936058i	$-0.385554\pi$
$911$	19349.1	0.703693	0.351846	+	0.936058i	$0.385554\pi$
$912$	0	0
$913$	6726.89	0.243842
$914$	66939.1	2.42248
$915$	0	0
$916$	66758.4	2.40803
$917$	−182.766	−0.00658177
$918$	0	0
$919$	−7490.62	−0.268871	−0.134436	−	0.990922i	$-0.542922\pi$
$919$	−7490.62	−0.268871	−0.134436	+	0.990922i	$0.542922\pi$
$920$	96721.8	3.46611
$921$	0	0
$922$	−12047.2	−0.430318
$923$	−37246.7	−1.32827
$924$	0	0
$925$	49757.4	1.76866
$926$	93566.6	3.32051
$927$	0	0
$928$	−2599.57	−0.0919560
$929$	14979.3	0.529013	0.264507	−	0.964384i	$-0.414791\pi$
$929$	14979.3	0.529013	0.264507	+	0.964384i	$0.414791\pi$
$930$	0	0
$931$	−18032.7	−0.634798
$932$	−49271.7	−1.73171
$933$	0	0
$934$	25714.4	0.900858
$935$	16459.6	0.575709
$936$	0	0
$937$	35932.6	1.25279	0.626397	−	0.779504i	$-0.284529\pi$
$937$	35932.6	1.25279	0.626397	+	0.779504i	$0.284529\pi$
$938$	19023.5	0.662195
$939$	0	0
$940$	47617.7	1.65225
$941$	40959.5	1.41896	0.709480	−	0.704726i	$-0.248930\pi$
$941$	40959.5	1.41896	0.709480	+	0.704726i	$0.248930\pi$
$942$	0	0
$943$	51321.3	1.77227
$944$	4012.53	0.138344
$945$	0	0
$946$	−13619.3	−0.468079
$947$	−25559.8	−0.877068	−0.438534	−	0.898715i	$-0.644502\pi$
$947$	−25559.8	−0.877068	−0.438534	+	0.898715i	$0.644502\pi$
$948$	0	0
$949$	38580.7	1.31969
$950$	−93361.6	−3.18847
$951$	0	0
$952$	−31132.6	−1.05989
$953$	49571.0	1.68496	0.842478	−	0.538731i	$-0.181096\pi$
$953$	49571.0	1.68496	0.842478	+	0.538731i	$0.181096\pi$
$954$	0	0
$955$	−71833.1	−2.43400
$956$	3559.90	0.120435
$957$	0	0
$958$	34644.8	1.16840
$959$	−11229.9	−0.378136
$960$	0	0
$961$	−16999.3	−0.570620
$962$	−62238.4	−2.08591
$963$	0	0
$964$	76795.6	2.56579
$965$	61633.5	2.05601
$966$	0	0
$967$	−17045.0	−0.566837	−0.283419	−	0.958996i	$-0.591469\pi$
$967$	−17045.0	−0.566837	−0.283419	+	0.958996i	$0.591469\pi$
$968$	41501.5	1.37800
$969$	0	0
$970$	58059.0	1.92182
$971$	55060.0	1.81973	0.909866	−	0.414902i	$-0.136184\pi$
$971$	55060.0	1.81973	0.909866	+	0.414902i	$0.136184\pi$
$972$	0	0
$973$	−18404.2	−0.606384
$974$	60156.8	1.97900
$975$	0	0
$976$	10861.4	0.356213
$977$	4980.17	0.163080	0.0815402	−	0.996670i	$-0.474016\pi$
$977$	4980.17	0.163080	0.0815402	+	0.996670i	$0.474016\pi$
$978$	0	0
$979$	6342.00	0.207039
$980$	−73432.1	−2.39357
$981$	0	0
$982$	29245.3	0.950362
$983$	−44793.9	−1.45341	−0.726706	−	0.686949i	$-0.758950\pi$
$983$	−44793.9	−1.45341	−0.726706	+	0.686949i	$0.758950\pi$
$984$	0	0
$985$	−9915.95	−0.320760
$986$	15326.1	0.495013
$987$	0	0
$988$	75974.7	2.44644
$989$	−49194.7	−1.58170
$990$	0	0
$991$	−14339.8	−0.459657	−0.229829	−	0.973231i	$-0.573817\pi$
$991$	−14339.8	−0.459657	−0.229829	+	0.973231i	$0.573817\pi$
$992$	8907.12	0.285082
$993$	0	0
$994$	24664.0	0.787016
$995$	−37396.3	−1.19150
$996$	0	0
$997$	−34566.7	−1.09803	−0.549017	−	0.835811i	$-0.684998\pi$
$997$	−34566.7	−1.09803	−0.549017	+	0.835811i	$0.684998\pi$
$998$	41146.1	1.30507
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.e.1.5		32
3.2	odd	2		717.4.a.c.1.28	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.c.1.28	✓	32	3.2	odd	2
2151.4.a.e.1.5		32	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-4.78487 q^{2} +14.8950 q^{4} +19.8446 q^{5} +9.72473 q^{7} -32.9917 q^{8} +O(q^{10})\)	\(q-4.78487 q^{2} +14.8950 q^{4} +19.8446 q^{5} +9.72473 q^{7} -32.9917 q^{8} -94.9538 q^{10} +8.54762 q^{11} +70.2704 q^{13} -46.5316 q^{14} +38.7010 q^{16} +97.0361 q^{17} +72.5866 q^{19} +295.585 q^{20} -40.8993 q^{22} -147.733 q^{23} +268.808 q^{25} -336.235 q^{26} +144.850 q^{28} -33.0087 q^{29} +113.100 q^{31} +78.7542 q^{32} -464.305 q^{34} +192.983 q^{35} +185.104 q^{37} -347.318 q^{38} -654.706 q^{40} -347.392 q^{41} +332.997 q^{43} +127.317 q^{44} +706.884 q^{46} +161.096 q^{47} -248.430 q^{49} -1286.21 q^{50} +1046.68 q^{52} -550.417 q^{53} +169.624 q^{55} -320.835 q^{56} +157.942 q^{58} +103.680 q^{59} +280.648 q^{61} -541.170 q^{62} -686.437 q^{64} +1394.49 q^{65} -408.830 q^{67} +1445.35 q^{68} -923.400 q^{70} -530.048 q^{71} +549.032 q^{73} -885.699 q^{74} +1081.18 q^{76} +83.1232 q^{77} +934.908 q^{79} +768.005 q^{80} +1662.23 q^{82} +786.990 q^{83} +1925.64 q^{85} -1593.35 q^{86} -282.000 q^{88} +741.961 q^{89} +683.360 q^{91} -2200.48 q^{92} -770.825 q^{94} +1440.45 q^{95} -611.445 q^{97} +1188.70 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * q^98

Introduction

L-functions

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