LMFDB - Embedded newform 207.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,4,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, 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("207.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	207.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:	$12.2133953712$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.82843 q^{2} -4.65685 q^{4} +7.31371 q^{5} -11.1716 q^{7} +23.1421 q^{8} -13.3726 q^{10} +16.2010 q^{11} -13.0294 q^{13} +20.4264 q^{14} -5.05887 q^{16} -33.8579 q^{17} -6.11775 q^{19} -34.0589 q^{20} -29.6224 q^{22} -23.0000 q^{23} -71.5097 q^{25} +23.8234 q^{26} +52.0244 q^{28} -84.6762 q^{29} -236.735 q^{31} -175.887 q^{32} +61.9066 q^{34} -81.7056 q^{35} -63.6224 q^{37} +11.1859 q^{38} +169.255 q^{40} -75.5391 q^{41} -260.912 q^{43} -75.4457 q^{44} +42.0538 q^{46} -224.902 q^{47} -218.196 q^{49} +130.750 q^{50} +60.6762 q^{52} -44.2843 q^{53} +118.489 q^{55} -258.534 q^{56} +154.824 q^{58} +466.891 q^{59} -520.593 q^{61} +432.853 q^{62} +362.068 q^{64} -95.2935 q^{65} -906.950 q^{67} +157.671 q^{68} +149.393 q^{70} +920.930 q^{71} +251.608 q^{73} +116.329 q^{74} +28.4895 q^{76} -180.991 q^{77} +1052.88 q^{79} -36.9991 q^{80} +138.118 q^{82} -143.202 q^{83} -247.627 q^{85} +477.058 q^{86} +374.926 q^{88} -1081.94 q^{89} +145.559 q^{91} +107.108 q^{92} +411.216 q^{94} -44.7434 q^{95} +1159.76 q^{97} +398.955 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 8 q^{5} - 28 q^{7} + 18 q^{8} - 72 q^{10} + 72 q^{11} - 60 q^{13} - 44 q^{14} - 78 q^{16} - 96 q^{17} - 148 q^{19} - 136 q^{20} + 184 q^{22} - 46 q^{23} + 38 q^{25} - 156 q^{26}+ \cdots + 170 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 8 * q^5 - 28 * q^7 + 18 * q^8 - 72 * q^10 + 72 * q^11 - 60 * q^13 - 44 * q^14 - 78 * q^16 - 96 * q^17 - 148 * q^19 - 136 * q^20 + 184 * q^22 - 46 * q^23 + 38 * q^25 - 156 * q^26 - 60 * q^28 + 204 * q^29 - 168 * q^31 - 414 * q^32 - 176 * q^34 + 176 * q^35 + 116 * q^37 - 532 * q^38 + 248 * q^40 - 4 * q^41 - 420 * q^43 + 296 * q^44 - 46 * q^46 + 48 * q^47 - 278 * q^49 + 550 * q^50 - 252 * q^52 - 32 * q^53 - 736 * q^55 - 172 * q^56 + 1260 * q^58 + 40 * q^59 - 764 * q^61 + 696 * q^62 + 34 * q^64 + 624 * q^65 - 988 * q^67 - 256 * q^68 + 1136 * q^70 + 224 * q^71 + 820 * q^73 + 804 * q^74 - 916 * q^76 - 1120 * q^77 + 1772 * q^79 + 1080 * q^80 + 412 * q^82 - 1480 * q^83 + 704 * q^85 - 132 * q^86 + 88 * q^88 - 744 * q^89 + 936 * q^91 - 46 * q^92 + 1456 * q^94 + 2128 * q^95 - 260 * q^97 + 170 * q^98

Coefficient data

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

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

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

$2$	−1.82843	−0.646447	−0.323223	−	0.946323i	$-0.604766\pi$
$2$	−1.82843	−0.646447	−0.323223	+	0.946323i	$0.604766\pi$
$3$	0	0
$3$	0	0
$4$	−4.65685	−0.582107
$5$	7.31371	0.654158	0.327079	−	0.944997i	$-0.393936\pi$
$5$	7.31371	0.654158	0.327079	+	0.944997i	$0.393936\pi$
$6$	0	0
$7$	−11.1716	−0.603208	−0.301604	−	0.953433i	$-0.597522\pi$
$7$	−11.1716	−0.603208	−0.301604	+	0.953433i	$0.597522\pi$
$8$	23.1421	1.02275
$9$	0	0
$10$	−13.3726	−0.422878
$11$	16.2010	0.444072	0.222036	−	0.975039i	$-0.428730\pi$
$11$	16.2010	0.444072	0.222036	+	0.975039i	$0.428730\pi$
$12$	0	0
$13$	−13.0294	−0.277978	−0.138989	−	0.990294i	$-0.544385\pi$
$13$	−13.0294	−0.277978	−0.138989	+	0.990294i	$0.544385\pi$
$14$	20.4264	0.389942
$15$	0	0
$16$	−5.05887	−0.0790449
$17$	−33.8579	−0.483043	−0.241522	−	0.970395i	$-0.577647\pi$
$17$	−33.8579	−0.483043	−0.241522	+	0.970395i	$0.577647\pi$
$18$	0	0
$19$	−6.11775	−0.0738688	−0.0369344	−	0.999318i	$-0.511759\pi$
$19$	−6.11775	−0.0738688	−0.0369344	+	0.999318i	$0.511759\pi$
$20$	−34.0589	−0.380790
$21$	0	0
$22$	−29.6224	−0.287069
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−71.5097	−0.572077
$26$	23.8234	0.179698
$27$	0	0
$28$	52.0244	0.351132
$29$	−84.6762	−0.542206	−0.271103	−	0.962550i	$-0.587388\pi$
$29$	−84.6762	−0.542206	−0.271103	+	0.962550i	$0.587388\pi$
$30$	0	0
$31$	−236.735	−1.37158	−0.685788	−	0.727801i	$-0.740542\pi$
$31$	−236.735	−1.37158	−0.685788	+	0.727801i	$0.740542\pi$
$32$	−175.887	−0.971649
$33$	0	0
$34$	61.9066	0.312262
$35$	−81.7056	−0.394593
$36$	0	0
$37$	−63.6224	−0.282688	−0.141344	−	0.989961i	$-0.545142\pi$
$37$	−63.6224	−0.282688	−0.141344	+	0.989961i	$0.545142\pi$
$38$	11.1859	0.0477523
$39$	0	0
$40$	169.255	0.669038
$41$	−75.5391	−0.287737	−0.143869	−	0.989597i	$-0.545954\pi$
$41$	−75.5391	−0.287737	−0.143869	+	0.989597i	$0.545954\pi$
$42$	0	0
$43$	−260.912	−0.925318	−0.462659	−	0.886536i	$-0.653105\pi$
$43$	−260.912	−0.925318	−0.462659	+	0.886536i	$0.653105\pi$
$44$	−75.4457	−0.258497
$45$	0	0
$46$	42.0538	0.134793
$47$	−224.902	−0.697984	−0.348992	−	0.937126i	$-0.613476\pi$
$47$	−224.902	−0.697984	−0.348992	+	0.937126i	$0.613476\pi$
$48$	0	0
$49$	−218.196	−0.636140
$50$	130.750	0.369817
$51$	0	0
$52$	60.6762	0.161813
$53$	−44.2843	−0.114772	−0.0573860	−	0.998352i	$-0.518277\pi$
$53$	−44.2843	−0.114772	−0.0573860	+	0.998352i	$0.518277\pi$
$54$	0	0
$55$	118.489	0.290493
$56$	−258.534	−0.616930
$57$	0	0
$58$	154.824	0.350507
$59$	466.891	1.03024	0.515119	−	0.857118i	$-0.327748\pi$
$59$	466.891	1.03024	0.515119	+	0.857118i	$0.327748\pi$
$60$	0	0
$61$	−520.593	−1.09271	−0.546353	−	0.837555i	$-0.683984\pi$
$61$	−520.593	−1.09271	−0.546353	+	0.837555i	$0.683984\pi$
$62$	432.853	0.886651
$63$	0	0
$64$	362.068	0.707164
$65$	−95.2935	−0.181842
$66$	0	0
$67$	−906.950	−1.65376	−0.826878	−	0.562382i	$-0.809885\pi$
$67$	−906.950	−1.65376	−0.826878	+	0.562382i	$0.809885\pi$
$68$	157.671	0.281183
$69$	0	0
$70$	149.393	0.255084
$71$	920.930	1.53936	0.769678	−	0.638432i	$-0.220417\pi$
$71$	920.930	1.53936	0.769678	+	0.638432i	$0.220417\pi$
$72$	0	0
$73$	251.608	0.403404	0.201702	−	0.979447i	$-0.435353\pi$
$73$	251.608	0.403404	0.201702	+	0.979447i	$0.435353\pi$
$74$	116.329	0.182743
$75$	0	0
$76$	28.4895	0.0429996
$77$	−180.991	−0.267868
$78$	0	0
$79$	1052.88	1.49947	0.749734	−	0.661740i	$-0.230182\pi$
$79$	1052.88	1.49947	0.749734	+	0.661740i	$0.230182\pi$
$80$	−36.9991	−0.0517079
$81$	0	0
$82$	138.118	0.186007
$83$	−143.202	−0.189379	−0.0946894	−	0.995507i	$-0.530186\pi$
$83$	−143.202	−0.189379	−0.0946894	+	0.995507i	$0.530186\pi$
$84$	0	0
$85$	−247.627	−0.315987
$86$	477.058	0.598168
$87$	0	0
$88$	374.926	0.454173
$89$	−1081.94	−1.28859	−0.644297	−	0.764775i	$-0.722850\pi$
$89$	−1081.94	−1.28859	−0.644297	+	0.764775i	$0.722850\pi$
$90$	0	0
$91$	145.559	0.167679
$92$	107.108	0.121378
$93$	0	0
$94$	411.216	0.451210
$95$	−44.7434	−0.0483219
$96$	0	0
$97$	1159.76	1.21398	0.606990	−	0.794709i	$-0.292377\pi$
$97$	1159.76	1.21398	0.606990	+	0.794709i	$0.292377\pi$
$98$	398.955	0.411230
$99$	0	0
$100$	333.010	0.333010
$101$	−1172.72	−1.15535	−0.577676	−	0.816266i	$-0.696040\pi$
$101$	−1172.72	−1.15535	−0.577676	+	0.816266i	$0.696040\pi$
$102$	0	0
$103$	981.003	0.938458	0.469229	−	0.883077i	$-0.344532\pi$
$103$	981.003	0.938458	0.469229	+	0.883077i	$0.344532\pi$
$104$	−301.529	−0.284301
$105$	0	0
$106$	80.9706	0.0741939
$107$	899.543	0.812730	0.406365	−	0.913711i	$-0.366796\pi$
$107$	899.543	0.812730	0.406365	+	0.913711i	$0.366796\pi$
$108$	0	0
$109$	1710.34	1.50294	0.751470	−	0.659767i	$-0.229345\pi$
$109$	1710.34	1.50294	0.751470	+	0.659767i	$0.229345\pi$
$110$	−216.649	−0.187788
$111$	0	0
$112$	56.5156	0.0476805
$113$	1079.79	0.898919	0.449459	−	0.893301i	$-0.351617\pi$
$113$	1079.79	0.898919	0.449459	+	0.893301i	$0.351617\pi$
$114$	0	0
$115$	−168.215	−0.136401
$116$	394.325	0.315622
$117$	0	0
$118$	−853.677	−0.665994
$119$	378.246	0.291376
$120$	0	0
$121$	−1068.53	−0.802800
$122$	951.866	0.706376
$123$	0	0
$124$	1102.44	0.798404
$125$	−1437.21	−1.02839
$126$	0	0
$127$	−361.324	−0.252459	−0.126230	−	0.992001i	$-0.540288\pi$
$127$	−361.324	−0.252459	−0.126230	+	0.992001i	$0.540288\pi$
$128$	745.083	0.514505
$129$	0	0
$130$	174.237	0.117551
$131$	1056.56	0.704672	0.352336	−	0.935873i	$-0.385387\pi$
$131$	1056.56	0.704672	0.352336	+	0.935873i	$0.385387\pi$
$132$	0	0
$133$	68.3449	0.0445583
$134$	1658.29	1.06906
$135$	0	0
$136$	−783.543	−0.494031
$137$	−2273.44	−1.41776	−0.708879	−	0.705331i	$-0.750799\pi$
$137$	−2273.44	−1.41776	−0.708879	+	0.705331i	$0.750799\pi$
$138$	0	0
$139$	−1544.41	−0.942411	−0.471206	−	0.882023i	$-0.656181\pi$
$139$	−1544.41	−0.942411	−0.471206	+	0.882023i	$0.656181\pi$
$140$	380.491	0.229696
$141$	0	0
$142$	−1683.85	−0.995112
$143$	−211.090	−0.123442
$144$	0	0
$145$	−619.297	−0.354688
$146$	−460.047	−0.260779
$147$	0	0
$148$	296.280	0.164555
$149$	−665.076	−0.365672	−0.182836	−	0.983143i	$-0.558528\pi$
$149$	−665.076	−0.365672	−0.182836	+	0.983143i	$0.558528\pi$
$150$	0	0
$151$	−3559.98	−1.91859	−0.959294	−	0.282410i	$-0.908866\pi$
$151$	−3559.98	−1.91859	−0.959294	+	0.282410i	$0.908866\pi$
$152$	−141.578	−0.0755492
$153$	0	0
$154$	330.928	0.173162
$155$	−1731.41	−0.897228
$156$	0	0
$157$	−1391.25	−0.707221	−0.353611	−	0.935393i	$-0.615046\pi$
$157$	−1391.25	−0.707221	−0.353611	+	0.935393i	$0.615046\pi$
$158$	−1925.11	−0.969326
$159$	0	0
$160$	−1286.39	−0.635612
$161$	256.946	0.125778
$162$	0	0
$163$	975.960	0.468976	0.234488	−	0.972119i	$-0.424659\pi$
$163$	975.960	0.468976	0.234488	+	0.972119i	$0.424659\pi$
$164$	351.775	0.167494
$165$	0	0
$166$	261.834	0.122423
$167$	−2053.06	−0.951323	−0.475661	−	0.879628i	$-0.657791\pi$
$167$	−2053.06	−0.951323	−0.475661	+	0.879628i	$0.657791\pi$
$168$	0	0
$169$	−2027.23	−0.922728
$170$	452.767	0.204269
$171$	0	0
$172$	1215.03	0.538634
$173$	−1308.25	−0.574937	−0.287468	−	0.957790i	$-0.592814\pi$
$173$	−1308.25	−0.574937	−0.287468	+	0.957790i	$0.592814\pi$
$174$	0	0
$175$	798.875	0.345082
$176$	−81.9589	−0.0351016
$177$	0	0
$178$	1978.24	0.833007
$179$	829.584	0.346403	0.173201	−	0.984886i	$-0.444589\pi$
$179$	829.584	0.346403	0.173201	+	0.984886i	$0.444589\pi$
$180$	0	0
$181$	−2431.29	−0.998431	−0.499216	−	0.866478i	$-0.666378\pi$
$181$	−2431.29	−0.998431	−0.499216	+	0.866478i	$0.666378\pi$
$182$	−266.145	−0.108395
$183$	0	0
$184$	−532.269	−0.213258
$185$	−465.315	−0.184923
$186$	0	0
$187$	−548.532	−0.214506
$188$	1047.33	0.406301
$189$	0	0
$190$	81.8101	0.0312375
$191$	2878.90	1.09063	0.545314	−	0.838232i	$-0.316410\pi$
$191$	2878.90	1.09063	0.545314	+	0.838232i	$0.316410\pi$
$192$	0	0
$193$	1423.90	0.531060	0.265530	−	0.964103i	$-0.414453\pi$
$193$	1423.90	0.531060	0.265530	+	0.964103i	$0.414453\pi$
$194$	−2120.54	−0.784773
$195$	0	0
$196$	1016.11	0.370301
$197$	3517.78	1.27224	0.636121	−	0.771590i	$-0.280538\pi$
$197$	3517.78	1.27224	0.636121	+	0.771590i	$0.280538\pi$
$198$	0	0
$199$	4069.01	1.44947	0.724735	−	0.689028i	$-0.241962\pi$
$199$	4069.01	1.44947	0.724735	+	0.689028i	$0.241962\pi$
$200$	−1654.89	−0.585091
$201$	0	0
$202$	2144.24	0.746873
$203$	945.966	0.327063
$204$	0	0
$205$	−552.471	−0.188226
$206$	−1793.69	−0.606663
$207$	0	0
$208$	65.9143	0.0219728
$209$	−99.1137	−0.0328031
$210$	0	0
$211$	−2631.29	−0.858510	−0.429255	−	0.903183i	$-0.641224\pi$
$211$	−2631.29	−0.858510	−0.429255	+	0.903183i	$0.641224\pi$
$212$	206.225	0.0668095
$213$	0	0
$214$	−1644.75	−0.525387
$215$	−1908.23	−0.605304
$216$	0	0
$217$	2644.70	0.827346
$218$	−3127.22	−0.971570
$219$	0	0
$220$	−551.788	−0.169098
$221$	441.149	0.134276
$222$	0	0
$223$	−2886.47	−0.866780	−0.433390	−	0.901206i	$-0.642683\pi$
$223$	−2886.47	−0.866780	−0.433390	+	0.901206i	$0.642683\pi$
$224$	1964.94	0.586107
$225$	0	0
$226$	−1974.31	−0.581103
$227$	−1819.49	−0.532000	−0.266000	−	0.963973i	$-0.585702\pi$
$227$	−1819.49	−0.532000	−0.266000	+	0.963973i	$0.585702\pi$
$228$	0	0
$229$	1038.46	0.299664	0.149832	−	0.988711i	$-0.452127\pi$
$229$	1038.46	0.299664	0.149832	+	0.988711i	$0.452127\pi$
$230$	307.569	0.0881762
$231$	0	0
$232$	−1959.59	−0.554540
$233$	1432.80	0.402858	0.201429	−	0.979503i	$-0.435441\pi$
$233$	1432.80	0.402858	0.201429	+	0.979503i	$0.435441\pi$
$234$	0	0
$235$	−1644.86	−0.456592
$236$	−2174.25	−0.599709
$237$	0	0
$238$	−691.595	−0.188359
$239$	6548.04	1.77221	0.886103	−	0.463488i	$-0.153402\pi$
$239$	6548.04	1.77221	0.886103	+	0.463488i	$0.153402\pi$
$240$	0	0
$241$	1815.36	0.485217	0.242609	−	0.970124i	$-0.421997\pi$
$241$	1815.36	0.485217	0.242609	+	0.970124i	$0.421997\pi$
$242$	1953.72	0.518968
$243$	0	0
$244$	2424.33	0.636072
$245$	−1595.82	−0.416136
$246$	0	0
$247$	79.7108	0.0205339
$248$	−5478.55	−1.40278
$249$	0	0
$250$	2627.84	0.664797
$251$	7405.10	1.86218	0.931088	−	0.364796i	$-0.118861\pi$
$251$	7405.10	1.86218	0.931088	+	0.364796i	$0.118861\pi$
$252$	0	0
$253$	−372.623	−0.0925953
$254$	660.654	0.163201
$255$	0	0
$256$	−4258.88	−1.03976
$257$	203.697	0.0494408	0.0247204	−	0.999694i	$-0.492130\pi$
$257$	203.697	0.0494408	0.0247204	+	0.999694i	$0.492130\pi$
$258$	0	0
$259$	710.762	0.170520
$260$	443.768	0.105851
$261$	0	0
$262$	−1931.84	−0.455533
$263$	−4807.59	−1.12718	−0.563591	−	0.826054i	$-0.690581\pi$
$263$	−4807.59	−1.12718	−0.563591	+	0.826054i	$0.690581\pi$
$264$	0	0
$265$	−323.882	−0.0750790
$266$	−124.964	−0.0288046
$267$	0	0
$268$	4223.54	0.962662
$269$	7183.57	1.62822	0.814108	−	0.580714i	$-0.197227\pi$
$269$	7183.57	1.62822	0.814108	+	0.580714i	$0.197227\pi$
$270$	0	0
$271$	7099.96	1.59148	0.795742	−	0.605636i	$-0.207081\pi$
$271$	7099.96	1.59148	0.795742	+	0.605636i	$0.207081\pi$
$272$	171.283	0.0381821
$273$	0	0
$274$	4156.81	0.916504
$275$	−1158.53	−0.254043
$276$	0	0
$277$	−2195.58	−0.476244	−0.238122	−	0.971235i	$-0.576532\pi$
$277$	−2195.58	−0.476244	−0.238122	+	0.971235i	$0.576532\pi$
$278$	2823.84	0.609219
$279$	0	0
$280$	−1890.84	−0.403570
$281$	646.472	0.137243	0.0686215	−	0.997643i	$-0.478140\pi$
$281$	646.472	0.137243	0.0686215	+	0.997643i	$0.478140\pi$
$282$	0	0
$283$	−3031.71	−0.636806	−0.318403	−	0.947955i	$-0.603147\pi$
$283$	−3031.71	−0.636806	−0.318403	+	0.947955i	$0.603147\pi$
$284$	−4288.64	−0.896070
$285$	0	0
$286$	385.963	0.0797988
$287$	843.891	0.173565
$288$	0	0
$289$	−3766.65	−0.766669
$290$	1132.34	0.229287
$291$	0	0
$292$	−1171.70	−0.234824
$293$	−3944.40	−0.786466	−0.393233	−	0.919439i	$-0.628643\pi$
$293$	−3944.40	−0.786466	−0.393233	+	0.919439i	$0.628643\pi$
$294$	0	0
$295$	3414.71	0.673939
$296$	−1472.36	−0.289118
$297$	0	0
$298$	1216.04	0.236388
$299$	299.677	0.0579624
$300$	0	0
$301$	2914.79	0.558159
$302$	6509.16	1.24026
$303$	0	0
$304$	30.9489	0.00583896
$305$	−3807.46	−0.714803
$306$	0	0
$307$	−3403.21	−0.632676	−0.316338	−	0.948647i	$-0.602453\pi$
$307$	−3403.21	−0.632676	−0.316338	+	0.948647i	$0.602453\pi$
$308$	842.848	0.155928
$309$	0	0
$310$	3165.76	0.580010
$311$	5024.59	0.916136	0.458068	−	0.888917i	$-0.348542\pi$
$311$	5024.59	0.916136	0.458068	+	0.888917i	$0.348542\pi$
$312$	0	0
$313$	6242.51	1.12731	0.563654	−	0.826011i	$-0.309395\pi$
$313$	6242.51	1.12731	0.563654	+	0.826011i	$0.309395\pi$
$314$	2543.80	0.457181
$315$	0	0
$316$	−4903.10	−0.872850
$317$	−5029.02	−0.891034	−0.445517	−	0.895273i	$-0.646980\pi$
$317$	−5029.02	−0.891034	−0.445517	+	0.895273i	$0.646980\pi$
$318$	0	0
$319$	−1371.84	−0.240778
$320$	2648.06	0.462597
$321$	0	0
$322$	−469.807	−0.0813085
$323$	207.134	0.0356819
$324$	0	0
$325$	931.731	0.159025
$326$	−1784.47	−0.303168
$327$	0	0
$328$	−1748.14	−0.294283
$329$	2512.50	0.421030
$330$	0	0
$331$	6574.03	1.09167	0.545833	−	0.837894i	$-0.316213\pi$
$331$	6574.03	1.09167	0.545833	+	0.837894i	$0.316213\pi$
$332$	666.870	0.110239
$333$	0	0
$334$	3753.88	0.614979
$335$	−6633.17	−1.08182
$336$	0	0
$337$	6000.96	0.970010	0.485005	−	0.874511i	$-0.338818\pi$
$337$	6000.96	0.970010	0.485005	+	0.874511i	$0.338818\pi$
$338$	3706.65	0.596494
$339$	0	0
$340$	1153.16	0.183938
$341$	−3835.35	−0.609078
$342$	0	0
$343$	6269.44	0.986933
$344$	−6038.05	−0.946366
$345$	0	0
$346$	2392.03	0.371666
$347$	−3854.32	−0.596285	−0.298143	−	0.954521i	$-0.596367\pi$
$347$	−3854.32	−0.596285	−0.298143	+	0.954521i	$0.596367\pi$
$348$	0	0
$349$	4251.34	0.652061	0.326030	−	0.945359i	$-0.394289\pi$
$349$	4251.34	0.652061	0.326030	+	0.945359i	$0.394289\pi$
$350$	−1460.69	−0.223077
$351$	0	0
$352$	−2849.55	−0.431482
$353$	−11058.1	−1.66731	−0.833656	−	0.552284i	$-0.813756\pi$
$353$	−11058.1	−1.66731	−0.833656	+	0.552284i	$0.813756\pi$
$354$	0	0
$355$	6735.41	1.00698
$356$	5038.41	0.750100
$357$	0	0
$358$	−1516.83	−0.223931
$359$	−445.389	−0.0654784	−0.0327392	−	0.999464i	$-0.510423\pi$
$359$	−445.389	−0.0654784	−0.0327392	+	0.999464i	$0.510423\pi$
$360$	0	0
$361$	−6821.57	−0.994543
$362$	4445.43	0.645432
$363$	0	0
$364$	−677.848	−0.0976069
$365$	1840.19	0.263890
$366$	0	0
$367$	3992.27	0.567833	0.283917	−	0.958849i	$-0.408366\pi$
$367$	3992.27	0.567833	0.283917	+	0.958849i	$0.408366\pi$
$368$	116.354	0.0164820
$369$	0	0
$370$	850.795	0.119543
$371$	494.725	0.0692314
$372$	0	0
$373$	−6176.60	−0.857405	−0.428703	−	0.903446i	$-0.641029\pi$
$373$	−6176.60	−0.857405	−0.428703	+	0.903446i	$0.641029\pi$
$374$	1002.95	0.138667
$375$	0	0
$376$	−5204.70	−0.713862
$377$	1103.28	0.150721
$378$	0	0
$379$	−3069.15	−0.415968	−0.207984	−	0.978132i	$-0.566690\pi$
$379$	−3069.15	−0.415968	−0.207984	+	0.978132i	$0.566690\pi$
$380$	208.364	0.0281285
$381$	0	0
$382$	−5263.86	−0.705033
$383$	−1017.81	−0.135790	−0.0678952	−	0.997692i	$-0.521628\pi$
$383$	−1017.81	−0.135790	−0.0678952	+	0.997692i	$0.521628\pi$
$384$	0	0
$385$	−1323.71	−0.175228
$386$	−2603.50	−0.343302
$387$	0	0
$388$	−5400.85	−0.706666
$389$	10231.0	1.33350	0.666748	−	0.745283i	$-0.267686\pi$
$389$	10231.0	1.33350	0.666748	+	0.745283i	$0.267686\pi$
$390$	0	0
$391$	778.731	0.100722
$392$	−5049.52	−0.650610
$393$	0	0
$394$	−6432.01	−0.822436
$395$	7700.44	0.980889
$396$	0	0
$397$	−7312.11	−0.924394	−0.462197	−	0.886777i	$-0.652939\pi$
$397$	−7312.11	−0.924394	−0.462197	+	0.886777i	$0.652939\pi$
$398$	−7439.88	−0.937004
$399$	0	0
$400$	361.758	0.0452198
$401$	2859.00	0.356039	0.178020	−	0.984027i	$-0.443031\pi$
$401$	2859.00	0.356039	0.178020	+	0.984027i	$0.443031\pi$
$402$	0	0
$403$	3084.52	0.381268
$404$	5461.21	0.672538
$405$	0	0
$406$	−1729.63	−0.211429
$407$	−1030.75	−0.125534
$408$	0	0
$409$	−2100.40	−0.253932	−0.126966	−	0.991907i	$-0.540524\pi$
$409$	−2100.40	−0.253932	−0.126966	+	0.991907i	$0.540524\pi$
$410$	1010.15	0.121678
$411$	0	0
$412$	−4568.39	−0.546283
$413$	−5215.91	−0.621449
$414$	0	0
$415$	−1047.34	−0.123884
$416$	2291.71	0.270097
$417$	0	0
$418$	181.222	0.0212054
$419$	8941.83	1.04257	0.521285	−	0.853382i	$-0.325453\pi$
$419$	8941.83	1.04257	0.521285	+	0.853382i	$0.325453\pi$
$420$	0	0
$421$	−11876.0	−1.37483	−0.687413	−	0.726266i	$-0.741254\pi$
$421$	−11876.0	−1.37483	−0.687413	+	0.726266i	$0.741254\pi$
$422$	4811.13	0.554981
$423$	0	0
$424$	−1024.83	−0.117383
$425$	2421.16	0.276338
$426$	0	0
$427$	5815.84	0.659130
$428$	−4189.04	−0.473096
$429$	0	0
$430$	3489.06	0.391297
$431$	−2474.23	−0.276518	−0.138259	−	0.990396i	$-0.544151\pi$
$431$	−2474.23	−0.276518	−0.138259	+	0.990396i	$0.544151\pi$
$432$	0	0
$433$	−8210.49	−0.911249	−0.455624	−	0.890172i	$-0.650584\pi$
$433$	−8210.49	−0.911249	−0.455624	+	0.890172i	$0.650584\pi$
$434$	−4835.65	−0.534835
$435$	0	0
$436$	−7964.78	−0.874871
$437$	140.708	0.0154027
$438$	0	0
$439$	15542.0	1.68970	0.844850	−	0.535003i	$-0.179689\pi$
$439$	15542.0	1.68970	0.844850	+	0.535003i	$0.179689\pi$
$440$	2742.10	0.297101
$441$	0	0
$442$	−806.609	−0.0868019
$443$	6576.52	0.705328	0.352664	−	0.935750i	$-0.385276\pi$
$443$	6576.52	0.705328	0.352664	+	0.935750i	$0.385276\pi$
$444$	0	0
$445$	−7912.96	−0.842944
$446$	5277.69	0.560327
$447$	0	0
$448$	−4044.87	−0.426567
$449$	6648.47	0.698799	0.349400	−	0.936974i	$-0.386386\pi$
$449$	6648.47	0.698799	0.349400	+	0.936974i	$0.386386\pi$
$450$	0	0
$451$	−1223.81	−0.127776
$452$	−5028.41	−0.523267
$453$	0	0
$454$	3326.81	0.343910
$455$	1064.58	0.109688
$456$	0	0
$457$	−14901.1	−1.52526	−0.762631	−	0.646834i	$-0.776093\pi$
$457$	−14901.1	−1.52526	−0.762631	+	0.646834i	$0.776093\pi$
$458$	−1898.74	−0.193717
$459$	0	0
$460$	783.354	0.0794002
$461$	−3180.92	−0.321367	−0.160684	−	0.987006i	$-0.551370\pi$
$461$	−3180.92	−0.321367	−0.160684	+	0.987006i	$0.551370\pi$
$462$	0	0
$463$	10841.3	1.08821	0.544104	−	0.839018i	$-0.316870\pi$
$463$	10841.3	1.08821	0.544104	+	0.839018i	$0.316870\pi$
$464$	428.366	0.0428586
$465$	0	0
$466$	−2619.77	−0.260426
$467$	−16873.2	−1.67195	−0.835973	−	0.548770i	$-0.815096\pi$
$467$	−16873.2	−1.67195	−0.835973	+	0.548770i	$0.815096\pi$
$468$	0	0
$469$	10132.1	0.997559
$470$	3007.52	0.295162
$471$	0	0
$472$	10804.9	1.05367
$473$	−4227.03	−0.410907
$474$	0	0
$475$	437.478	0.0422587
$476$	−1761.43	−0.169612
$477$	0	0
$478$	−11972.6	−1.14564
$479$	−5673.54	−0.541192	−0.270596	−	0.962693i	$-0.587221\pi$
$479$	−5673.54	−0.541192	−0.270596	+	0.962693i	$0.587221\pi$
$480$	0	0
$481$	828.964	0.0785811
$482$	−3319.25	−0.313667
$483$	0	0
$484$	4975.98	0.467316
$485$	8482.17	0.794135
$486$	0	0
$487$	16387.7	1.52484	0.762421	−	0.647082i	$-0.224011\pi$
$487$	16387.7	1.52484	0.762421	+	0.647082i	$0.224011\pi$
$488$	−12047.6	−1.11756
$489$	0	0
$490$	2917.84	0.269010
$491$	−16060.1	−1.47614	−0.738068	−	0.674726i	$-0.764262\pi$
$491$	−16060.1	−1.47614	−0.738068	+	0.674726i	$0.764262\pi$
$492$	0	0
$493$	2866.95	0.261909
$494$	−145.745	−0.0132741
$495$	0	0
$496$	1197.61	0.108416
$497$	−10288.2	−0.928552
$498$	0	0
$499$	−6254.22	−0.561077	−0.280539	−	0.959843i	$-0.590513\pi$
$499$	−6254.22	−0.561077	−0.280539	+	0.959843i	$0.590513\pi$
$500$	6692.90	0.598631
$501$	0	0
$502$	−13539.7	−1.20380
$503$	−14085.9	−1.24862	−0.624312	−	0.781175i	$-0.714621\pi$
$503$	−14085.9	−1.24862	−0.624312	+	0.781175i	$0.714621\pi$
$504$	0	0
$505$	−8576.97	−0.755782
$506$	681.314	0.0598579
$507$	0	0
$508$	1682.63	0.146958
$509$	−6902.54	−0.601080	−0.300540	−	0.953769i	$-0.597167\pi$
$509$	−6902.54	−0.601080	−0.300540	+	0.953769i	$0.597167\pi$
$510$	0	0
$511$	−2810.86	−0.243337
$512$	1826.38	0.157647
$513$	0	0
$514$	−372.446	−0.0319608
$515$	7174.77	0.613899
$516$	0	0
$517$	−3643.63	−0.309955
$518$	−1299.58	−0.110232
$519$	0	0
$520$	−2205.30	−0.185978
$521$	13831.4	1.16308	0.581538	−	0.813519i	$-0.302451\pi$
$521$	13831.4	1.16308	0.581538	+	0.813519i	$0.302451\pi$
$522$	0	0
$523$	4141.92	0.346297	0.173149	−	0.984896i	$-0.444606\pi$
$523$	4141.92	0.346297	0.173149	+	0.984896i	$0.444606\pi$
$524$	−4920.25	−0.410195
$525$	0	0
$526$	8790.34	0.728663
$527$	8015.34	0.662531
$528$	0	0
$529$	529.000	0.0434783
$530$	592.195	0.0485345
$531$	0	0
$532$	−318.272	−0.0259377
$533$	984.232	0.0799847
$534$	0	0
$535$	6579.00	0.531654
$536$	−20988.8	−1.69137
$537$	0	0
$538$	−13134.6	−1.05255
$539$	−3534.99	−0.282492
$540$	0	0
$541$	−4441.37	−0.352956	−0.176478	−	0.984305i	$-0.556470\pi$
$541$	−4441.37	−0.352956	−0.176478	+	0.984305i	$0.556470\pi$
$542$	−12981.8	−1.02881
$543$	0	0
$544$	5955.17	0.469349
$545$	12508.9	0.983160
$546$	0	0
$547$	−2858.01	−0.223399	−0.111700	−	0.993742i	$-0.535629\pi$
$547$	−2858.01	−0.223399	−0.111700	+	0.993742i	$0.535629\pi$
$548$	10587.1	0.825286
$549$	0	0
$550$	2118.29	0.164225
$551$	518.028	0.0400521
$552$	0	0
$553$	−11762.3	−0.904491
$554$	4014.46	0.307866
$555$	0	0
$556$	7192.09	0.548584
$557$	−24614.7	−1.87246	−0.936230	−	0.351388i	$-0.885710\pi$
$557$	−24614.7	−1.87246	−0.936230	+	0.351388i	$0.885710\pi$
$558$	0	0
$559$	3399.53	0.257218
$560$	413.339	0.0311906
$561$	0	0
$562$	−1182.03	−0.0887202
$563$	21799.4	1.63186	0.815928	−	0.578154i	$-0.196227\pi$
$563$	21799.4	1.63186	0.815928	+	0.578154i	$0.196227\pi$
$564$	0	0
$565$	7897.25	0.588035
$566$	5543.25	0.411661
$567$	0	0
$568$	21312.3	1.57437
$569$	18323.3	1.35000	0.675002	−	0.737816i	$-0.264143\pi$
$569$	18323.3	1.35000	0.675002	+	0.737816i	$0.264143\pi$
$570$	0	0
$571$	8424.79	0.617454	0.308727	−	0.951151i	$-0.400097\pi$
$571$	8424.79	0.617454	0.308727	+	0.951151i	$0.400097\pi$
$572$	983.016	0.0718565
$573$	0	0
$574$	−1542.99	−0.112201
$575$	1644.72	0.119286
$576$	0	0
$577$	−9565.96	−0.690184	−0.345092	−	0.938569i	$-0.612152\pi$
$577$	−9565.96	−0.690184	−0.345092	+	0.938569i	$0.612152\pi$
$578$	6887.04	0.495611
$579$	0	0
$580$	2883.98	0.206467
$581$	1599.79	0.114235
$582$	0	0
$583$	−717.450	−0.0509670
$584$	5822.75	0.412581
$585$	0	0
$586$	7212.05	0.508408
$587$	27041.1	1.90137	0.950686	−	0.310156i	$-0.100381\pi$
$587$	27041.1	1.90137	0.950686	+	0.310156i	$0.100381\pi$
$588$	0	0
$589$	1448.29	0.101317
$590$	−6243.55	−0.435666
$591$	0	0
$592$	321.858	0.0223450
$593$	25057.0	1.73519	0.867595	−	0.497272i	$-0.165665\pi$
$593$	25057.0	1.73519	0.867595	+	0.497272i	$0.165665\pi$
$594$	0	0
$595$	2766.38	0.190606
$596$	3097.16	0.212860
$597$	0	0
$598$	−547.938	−0.0374696
$599$	4596.80	0.313556	0.156778	−	0.987634i	$-0.449889\pi$
$599$	4596.80	0.313556	0.156778	+	0.987634i	$0.449889\pi$
$600$	0	0
$601$	7811.09	0.530152	0.265076	−	0.964228i	$-0.414603\pi$
$601$	7811.09	0.530152	0.265076	+	0.964228i	$0.414603\pi$
$602$	−5329.49	−0.360820
$603$	0	0
$604$	16578.3	1.11682
$605$	−7814.90	−0.525158
$606$	0	0
$607$	15143.2	1.01259	0.506296	−	0.862360i	$-0.331014\pi$
$607$	15143.2	1.01259	0.506296	+	0.862360i	$0.331014\pi$
$608$	1076.03	0.0717746
$609$	0	0
$610$	6961.67	0.462082
$611$	2930.34	0.194024
$612$	0	0
$613$	−20487.4	−1.34988	−0.674941	−	0.737872i	$-0.735831\pi$
$613$	−20487.4	−1.34988	−0.674941	+	0.737872i	$0.735831\pi$
$614$	6222.52	0.408991
$615$	0	0
$616$	−4188.51	−0.273961
$617$	−25547.8	−1.66696	−0.833480	−	0.552549i	$-0.813655\pi$
$617$	−25547.8	−1.66696	−0.833480	+	0.552549i	$0.813655\pi$
$618$	0	0
$619$	15251.1	0.990294	0.495147	−	0.868809i	$-0.335114\pi$
$619$	15251.1	0.990294	0.495147	+	0.868809i	$0.335114\pi$
$620$	8062.93	0.522282
$621$	0	0
$622$	−9187.10	−0.592233
$623$	12086.9	0.777291
$624$	0	0
$625$	−1572.66	−0.100650
$626$	−11414.0	−0.728745
$627$	0	0
$628$	6478.84	0.411678
$629$	2154.12	0.136551
$630$	0	0
$631$	−11888.3	−0.750024	−0.375012	−	0.927020i	$-0.622361\pi$
$631$	−11888.3	−0.750024	−0.375012	+	0.927020i	$0.622361\pi$
$632$	24365.8	1.53358
$633$	0	0
$634$	9195.19	0.576006
$635$	−2642.62	−0.165148
$636$	0	0
$637$	2842.97	0.176833
$638$	2508.31	0.155650
$639$	0	0
$640$	5449.32	0.336568
$641$	−11163.5	−0.687881	−0.343941	−	0.938991i	$-0.611762\pi$
$641$	−11163.5	−0.687881	−0.343941	+	0.938991i	$0.611762\pi$
$642$	0	0
$643$	−22456.6	−1.37730	−0.688648	−	0.725096i	$-0.741795\pi$
$643$	−22456.6	−1.37730	−0.688648	+	0.725096i	$0.741795\pi$
$644$	−1196.56	−0.0732160
$645$	0	0
$646$	−378.729	−0.0230664
$647$	−27307.0	−1.65927	−0.829635	−	0.558306i	$-0.811451\pi$
$647$	−27307.0	−1.65927	−0.829635	+	0.558306i	$0.811451\pi$
$648$	0	0
$649$	7564.11	0.457500
$650$	−1703.60	−0.102801
$651$	0	0
$652$	−4544.90	−0.272994
$653$	21965.4	1.31634	0.658171	−	0.752868i	$-0.271330\pi$
$653$	21965.4	1.31634	0.658171	+	0.752868i	$0.271330\pi$
$654$	0	0
$655$	7727.37	0.460967
$656$	382.143	0.0227442
$657$	0	0
$658$	−4593.93	−0.272173
$659$	27620.9	1.63271	0.816356	−	0.577549i	$-0.195991\pi$
$659$	27620.9	1.63271	0.816356	+	0.577549i	$0.195991\pi$
$660$	0	0
$661$	4999.00	0.294158	0.147079	−	0.989125i	$-0.453013\pi$
$661$	4999.00	0.294158	0.147079	+	0.989125i	$0.453013\pi$
$662$	−12020.1	−0.705703
$663$	0	0
$664$	−3314.00	−0.193687
$665$	499.855	0.0291482
$666$	0	0
$667$	1947.55	0.113058
$668$	9560.82	0.553771
$669$	0	0
$670$	12128.3	0.699337
$671$	−8434.13	−0.485240
$672$	0	0
$673$	−18499.9	−1.05961	−0.529807	−	0.848118i	$-0.677736\pi$
$673$	−18499.9	−1.05961	−0.529807	+	0.848118i	$0.677736\pi$
$674$	−10972.3	−0.627059
$675$	0	0
$676$	9440.53	0.537126
$677$	1611.99	0.0915121	0.0457560	−	0.998953i	$-0.485430\pi$
$677$	1611.99	0.0915121	0.0457560	+	0.998953i	$0.485430\pi$
$678$	0	0
$679$	−12956.4	−0.732283
$680$	−5730.61	−0.323175
$681$	0	0
$682$	7012.65	0.393737
$683$	−2745.18	−0.153794	−0.0768972	−	0.997039i	$-0.524501\pi$
$683$	−2745.18	−0.153794	−0.0768972	+	0.997039i	$0.524501\pi$
$684$	0	0
$685$	−16627.2	−0.927437
$686$	−11463.2	−0.638000
$687$	0	0
$688$	1319.92	0.0731417
$689$	576.999	0.0319041
$690$	0	0
$691$	6774.46	0.372956	0.186478	−	0.982459i	$-0.440293\pi$
$691$	6774.46	0.372956	0.186478	+	0.982459i	$0.440293\pi$
$692$	6092.31	0.334675
$693$	0	0
$694$	7047.35	0.385467
$695$	−11295.4	−0.616486
$696$	0	0
$697$	2557.59	0.138990
$698$	−7773.27	−0.421522
$699$	0	0
$700$	−3720.25	−0.200874
$701$	−19717.2	−1.06235	−0.531176	−	0.847261i	$-0.678250\pi$
$701$	−19717.2	−1.06235	−0.531176	+	0.847261i	$0.678250\pi$
$702$	0	0
$703$	389.226	0.0208818
$704$	5865.87	0.314032
$705$	0	0
$706$	20218.9	1.07783
$707$	13101.2	0.696918
$708$	0	0
$709$	25897.2	1.37178	0.685889	−	0.727706i	$-0.259414\pi$
$709$	25897.2	1.37178	0.685889	+	0.727706i	$0.259414\pi$
$710$	−12315.2	−0.650960
$711$	0	0
$712$	−25038.3	−1.31791
$713$	5444.91	0.285993
$714$	0	0
$715$	−1543.85	−0.0807507
$716$	−3863.25	−0.201643
$717$	0	0
$718$	814.362	0.0423283
$719$	−14013.5	−0.726865	−0.363433	−	0.931620i	$-0.618395\pi$
$719$	−14013.5	−0.726865	−0.363433	+	0.931620i	$0.618395\pi$
$720$	0	0
$721$	−10959.4	−0.566085
$722$	12472.7	0.642919
$723$	0	0
$724$	11322.1	0.581194
$725$	6055.17	0.310184
$726$	0	0
$727$	−6116.58	−0.312038	−0.156019	−	0.987754i	$-0.549866\pi$
$727$	−6116.58	−0.312038	−0.156019	+	0.987754i	$0.549866\pi$
$728$	3368.55	0.171493
$729$	0	0
$730$	−3364.65	−0.170591
$731$	8833.91	0.446969
$732$	0	0
$733$	−25067.8	−1.26317	−0.631583	−	0.775308i	$-0.717594\pi$
$733$	−25067.8	−1.26317	−0.631583	+	0.775308i	$0.717594\pi$
$734$	−7299.58	−0.367074
$735$	0	0
$736$	4045.41	0.202603
$737$	−14693.5	−0.734386
$738$	0	0
$739$	14908.2	0.742095	0.371047	−	0.928614i	$-0.378999\pi$
$739$	14908.2	0.742095	0.371047	+	0.928614i	$0.378999\pi$
$740$	2166.91	0.107645
$741$	0	0
$742$	−904.569	−0.0447544
$743$	−17125.0	−0.845566	−0.422783	−	0.906231i	$-0.638947\pi$
$743$	−17125.0	−0.845566	−0.422783	+	0.906231i	$0.638947\pi$
$744$	0	0
$745$	−4864.18	−0.239208
$746$	11293.5	0.554267
$747$	0	0
$748$	2554.43	0.124865
$749$	−10049.3	−0.490246
$750$	0	0
$751$	−16547.1	−0.804009	−0.402004	−	0.915638i	$-0.631686\pi$
$751$	−16547.1	−0.804009	−0.402004	+	0.915638i	$0.631686\pi$
$752$	1137.75	0.0551721
$753$	0	0
$754$	−2017.27	−0.0974333
$755$	−26036.6	−1.25506
$756$	0	0
$757$	33624.0	1.61438	0.807189	−	0.590293i	$-0.200988\pi$
$757$	33624.0	1.61438	0.807189	+	0.590293i	$0.200988\pi$
$758$	5611.72	0.268901
$759$	0	0
$760$	−1035.46	−0.0494211
$761$	−1430.59	−0.0681459	−0.0340729	−	0.999419i	$-0.510848\pi$
$761$	−1430.59	−0.0681459	−0.0340729	+	0.999419i	$0.510848\pi$
$762$	0	0
$763$	−19107.1	−0.906586
$764$	−13406.6	−0.634862
$765$	0	0
$766$	1860.99	0.0877813
$767$	−6083.33	−0.286384
$768$	0	0
$769$	−32902.7	−1.54291	−0.771457	−	0.636282i	$-0.780471\pi$
$769$	−32902.7	−1.54291	−0.771457	+	0.636282i	$0.780471\pi$
$770$	2420.31	0.113275
$771$	0	0
$772$	−6630.90	−0.309134
$773$	−15196.9	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$773$	−15196.9	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$774$	0	0
$775$	16928.8	0.784648
$776$	26839.4	1.24160
$777$	0	0
$778$	−18706.6	−0.862034
$779$	462.129	0.0212548
$780$	0	0
$781$	14920.0	0.683585
$782$	−1423.85	−0.0651111
$783$	0	0
$784$	1103.83	0.0502836
$785$	−10175.2	−0.462634
$786$	0	0
$787$	−33540.5	−1.51917	−0.759586	−	0.650406i	$-0.774599\pi$
$787$	−33540.5	−1.51917	−0.759586	+	0.650406i	$0.774599\pi$
$788$	−16381.8	−0.740580
$789$	0	0
$790$	−14079.7	−0.634092
$791$	−12062.9	−0.542235
$792$	0	0
$793$	6783.03	0.303749
$794$	13369.7	0.597571
$795$	0	0
$796$	−18948.8	−0.843746
$797$	−10451.6	−0.464508	−0.232254	−	0.972655i	$-0.574610\pi$
$797$	−10451.6	−0.464508	−0.232254	+	0.972655i	$0.574610\pi$
$798$	0	0
$799$	7614.69	0.337157
$800$	12577.6	0.555859
$801$	0	0
$802$	−5227.47	−0.230160
$803$	4076.31	0.179140
$804$	0	0
$805$	1879.23	0.0822784
$806$	−5639.83	−0.246470
$807$	0	0
$808$	−27139.4	−1.18163
$809$	−44402.8	−1.92969	−0.964845	−	0.262820i	$-0.915348\pi$
$809$	−44402.8	−1.92969	−0.964845	+	0.262820i	$0.915348\pi$
$810$	0	0
$811$	−17243.5	−0.746611	−0.373305	−	0.927708i	$-0.621776\pi$
$811$	−17243.5	−0.746611	−0.373305	+	0.927708i	$0.621776\pi$
$812$	−4405.23	−0.190386
$813$	0	0
$814$	1884.65	0.0811508
$815$	7137.88	0.306784
$816$	0	0
$817$	1596.19	0.0683521
$818$	3840.43	0.164154
$819$	0	0
$820$	2572.78	0.109567
$821$	22572.4	0.959541	0.479771	−	0.877394i	$-0.340720\pi$
$821$	22572.4	0.959541	0.479771	+	0.877394i	$0.340720\pi$
$822$	0	0
$823$	−25704.3	−1.08870	−0.544348	−	0.838859i	$-0.683223\pi$
$823$	−25704.3	−1.08870	−0.544348	+	0.838859i	$0.683223\pi$
$824$	22702.5	0.959805
$825$	0	0
$826$	9536.92	0.401733
$827$	6681.58	0.280945	0.140472	−	0.990085i	$-0.455138\pi$
$827$	6681.58	0.280945	0.140472	+	0.990085i	$0.455138\pi$
$828$	0	0
$829$	20180.8	0.845487	0.422743	−	0.906249i	$-0.361067\pi$
$829$	20180.8	0.845487	0.422743	+	0.906249i	$0.361067\pi$
$830$	1914.98	0.0800842
$831$	0	0
$832$	−4717.54	−0.196576
$833$	7387.65	0.307283
$834$	0	0
$835$	−15015.5	−0.622315
$836$	461.558	0.0190949
$837$	0	0
$838$	−16349.5	−0.673966
$839$	−15142.7	−0.623104	−0.311552	−	0.950229i	$-0.600849\pi$
$839$	−15142.7	−0.623104	−0.311552	+	0.950229i	$0.600849\pi$
$840$	0	0
$841$	−17218.9	−0.706013
$842$	21714.5	0.888752
$843$	0	0
$844$	12253.6	0.499745
$845$	−14826.6	−0.603610
$846$	0	0
$847$	11937.1	0.484256
$848$	224.029	0.00907214
$849$	0	0
$850$	−4426.92	−0.178638
$851$	1463.31	0.0589445
$852$	0	0
$853$	−29073.6	−1.16701	−0.583507	−	0.812108i	$-0.698320\pi$
$853$	−29073.6	−1.16701	−0.583507	+	0.812108i	$0.698320\pi$
$854$	−10633.8	−0.426092
$855$	0	0
$856$	20817.4	0.831218
$857$	24579.3	0.979713	0.489857	−	0.871803i	$-0.337049\pi$
$857$	24579.3	0.979713	0.489857	+	0.871803i	$0.337049\pi$
$858$	0	0
$859$	20359.2	0.808670	0.404335	−	0.914611i	$-0.367503\pi$
$859$	20359.2	0.808670	0.404335	+	0.914611i	$0.367503\pi$
$860$	8886.36	0.352352
$861$	0	0
$862$	4523.95	0.178754
$863$	31440.0	1.24013	0.620064	−	0.784551i	$-0.287107\pi$
$863$	31440.0	1.24013	0.620064	+	0.784551i	$0.287107\pi$
$864$	0	0
$865$	−9568.13	−0.376100
$866$	15012.3	0.589074
$867$	0	0
$868$	−12316.0	−0.481604
$869$	17057.7	0.665871
$870$	0	0
$871$	11817.1	0.459708
$872$	39580.8	1.53713
$873$	0	0
$874$	−257.275	−0.00995703
$875$	16055.9	0.620331
$876$	0	0
$877$	33232.1	1.27955	0.639777	−	0.768561i	$-0.279027\pi$
$877$	33232.1	1.27955	0.639777	+	0.768561i	$0.279027\pi$
$878$	−28417.4	−1.09230
$879$	0	0
$880$	−599.423	−0.0229620
$881$	−46530.5	−1.77940	−0.889700	−	0.456545i	$-0.849087\pi$
$881$	−46530.5	−1.77940	−0.889700	+	0.456545i	$0.849087\pi$
$882$	0	0
$883$	−9010.26	−0.343397	−0.171698	−	0.985150i	$-0.554925\pi$
$883$	−9010.26	−0.343397	−0.171698	+	0.985150i	$0.554925\pi$
$884$	−2054.37	−0.0781627
$885$	0	0
$886$	−12024.7	−0.455957
$887$	−16773.0	−0.634930	−0.317465	−	0.948270i	$-0.602832\pi$
$887$	−16773.0	−0.634930	−0.317465	+	0.948270i	$0.602832\pi$
$888$	0	0
$889$	4036.56	0.152285
$890$	14468.3	0.544918
$891$	0	0
$892$	13441.9	0.504559
$893$	1375.89	0.0515593
$894$	0	0
$895$	6067.34	0.226602
$896$	−8323.75	−0.310354
$897$	0	0
$898$	−12156.3	−0.451736
$899$	20045.8	0.743677
$900$	0	0
$901$	1499.37	0.0554398
$902$	2237.65	0.0826003
$903$	0	0
$904$	24988.6	0.919367
$905$	−17781.7	−0.653132
$906$	0	0
$907$	−42971.0	−1.57313	−0.786565	−	0.617507i	$-0.788143\pi$
$907$	−42971.0	−1.57313	−0.786565	+	0.617507i	$0.788143\pi$
$908$	8473.12	0.309681
$909$	0	0
$910$	−1946.50	−0.0709077
$911$	4801.23	0.174612	0.0873061	−	0.996182i	$-0.472174\pi$
$911$	4801.23	0.174612	0.0873061	+	0.996182i	$0.472174\pi$
$912$	0	0
$913$	−2320.02	−0.0840978
$914$	27245.6	0.986001
$915$	0	0
$916$	−4835.94	−0.174437
$917$	−11803.4	−0.425064
$918$	0	0
$919$	31232.7	1.12108	0.560539	−	0.828128i	$-0.310594\pi$
$919$	31232.7	1.12108	0.560539	+	0.828128i	$0.310594\pi$
$920$	−3892.86	−0.139504
$921$	0	0
$922$	5816.09	0.207747
$923$	−11999.2	−0.427907
$924$	0	0
$925$	4549.61	0.161719
$926$	−19822.6	−0.703468
$927$	0	0
$928$	14893.5	0.526834
$929$	23023.4	0.813104	0.406552	−	0.913628i	$-0.366731\pi$
$929$	23023.4	0.813104	0.406552	+	0.913628i	$0.366731\pi$
$930$	0	0
$931$	1334.87	0.0469909
$932$	−6672.35	−0.234507
$933$	0	0
$934$	30851.4	1.08082
$935$	−4011.80	−0.140321
$936$	0	0
$937$	−11599.2	−0.404405	−0.202203	−	0.979344i	$-0.564810\pi$
$937$	−11599.2	−0.404405	−0.202203	+	0.979344i	$0.564810\pi$
$938$	−18525.7	−0.644869
$939$	0	0
$940$	7659.89	0.265785
$941$	20423.2	0.707522	0.353761	−	0.935336i	$-0.384903\pi$
$941$	20423.2	0.707522	0.353761	+	0.935336i	$0.384903\pi$
$942$	0	0
$943$	1737.40	0.0599974
$944$	−2361.95	−0.0814351
$945$	0	0
$946$	7728.82	0.265630
$947$	−22021.1	−0.755638	−0.377819	−	0.925880i	$-0.623326\pi$
$947$	−22021.1	−0.755638	−0.377819	+	0.925880i	$0.623326\pi$
$948$	0	0
$949$	−3278.31	−0.112138
$950$	−799.897	−0.0273180
$951$	0	0
$952$	8753.41	0.298004
$953$	−46234.5	−1.57154	−0.785772	−	0.618517i	$-0.787734\pi$
$953$	−46234.5	−1.57154	−0.785772	+	0.618517i	$0.787734\pi$
$954$	0	0
$955$	21055.4	0.713443
$956$	−30493.3	−1.03161
$957$	0	0
$958$	10373.7	0.349852
$959$	25397.9	0.855203
$960$	0	0
$961$	26252.5	0.881222
$962$	−1515.70	−0.0507985
$963$	0	0
$964$	−8453.85	−0.282448
$965$	10414.0	0.347397
$966$	0	0
$967$	32443.4	1.07891	0.539456	−	0.842014i	$-0.318630\pi$
$967$	32443.4	1.07891	0.539456	+	0.842014i	$0.318630\pi$
$968$	−24728.0	−0.821062
$969$	0	0
$970$	−15509.0	−0.513366
$971$	−10183.9	−0.336579	−0.168290	−	0.985738i	$-0.553824\pi$
$971$	−10183.9	−0.336579	−0.168290	+	0.985738i	$0.553824\pi$
$972$	0	0
$973$	17253.5	0.568470
$974$	−29963.7	−0.985729
$975$	0	0
$976$	2633.61	0.0863729
$977$	26704.7	0.874472	0.437236	−	0.899347i	$-0.355957\pi$
$977$	26704.7	0.874472	0.437236	+	0.899347i	$0.355957\pi$
$978$	0	0
$979$	−17528.4	−0.572228
$980$	7431.51	0.242236
$981$	0	0
$982$	29364.8	0.954243
$983$	12350.6	0.400735	0.200368	−	0.979721i	$-0.435786\pi$
$983$	12350.6	0.400735	0.200368	+	0.979721i	$0.435786\pi$
$984$	0	0
$985$	25728.0	0.832247
$986$	−5242.02	−0.169310
$987$	0	0
$988$	−371.202	−0.0119529
$989$	6000.97	0.192942
$990$	0	0
$991$	18050.6	0.578602	0.289301	−	0.957238i	$-0.406577\pi$
$991$	18050.6	0.578602	0.289301	+	0.957238i	$0.406577\pi$
$992$	41638.7	1.33269
$993$	0	0
$994$	18811.3	0.600260
$995$	29759.5	0.948182
$996$	0	0
$997$	14290.5	0.453946	0.226973	−	0.973901i	$-0.427117\pi$
$997$	14290.5	0.453946	0.226973	+	0.973901i	$0.427117\pi$
$998$	11435.4	0.362706
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.4.a.b.1.1		2
3.2	odd	2		69.4.a.b.1.2	✓	2
12.11	even	2		1104.4.a.q.1.1		2
15.14	odd	2		1725.4.a.m.1.1		2
69.68	even	2		1587.4.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.a.b.1.2	✓	2	3.2	odd	2
207.4.a.b.1.1		2	1.1	even	1	trivial
1104.4.a.q.1.1		2	12.11	even	2
1587.4.a.c.1.2		2	69.68	even	2
1725.4.a.m.1.1		2	15.14	odd	2

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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-1.82843 q^{2} -4.65685 q^{4} +7.31371 q^{5} -11.1716 q^{7} +23.1421 q^{8} -13.3726 q^{10} +16.2010 q^{11} -13.0294 q^{13} +20.4264 q^{14} -5.05887 q^{16} -33.8579 q^{17} -6.11775 q^{19} -34.0589 q^{20} -29.6224 q^{22} -23.0000 q^{23} -71.5097 q^{25} +23.8234 q^{26} +52.0244 q^{28} -84.6762 q^{29} -236.735 q^{31} -175.887 q^{32} +61.9066 q^{34} -81.7056 q^{35} -63.6224 q^{37} +11.1859 q^{38} +169.255 q^{40} -75.5391 q^{41} -260.912 q^{43} -75.4457 q^{44} +42.0538 q^{46} -224.902 q^{47} -218.196 q^{49} +130.750 q^{50} +60.6762 q^{52} -44.2843 q^{53} +118.489 q^{55} -258.534 q^{56} +154.824 q^{58} +466.891 q^{59} -520.593 q^{61} +432.853 q^{62} +362.068 q^{64} -95.2935 q^{65} -906.950 q^{67} +157.671 q^{68} +149.393 q^{70} +920.930 q^{71} +251.608 q^{73} +116.329 q^{74} +28.4895 q^{76} -180.991 q^{77} +1052.88 q^{79} -36.9991 q^{80} +138.118 q^{82} -143.202 q^{83} -247.627 q^{85} +477.058 q^{86} +374.926 q^{88} -1081.94 q^{89} +145.559 q^{91} +107.108 q^{92} +411.216 q^{94} -44.7434 q^{95} +1159.76 q^{97} +398.955 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 8 q^{5} - 28 q^{7} + 18 q^{8} - 72 q^{10} + 72 q^{11} - 60 q^{13} - 44 q^{14} - 78 q^{16} - 96 q^{17} - 148 q^{19} - 136 q^{20} + 184 q^{22} - 46 q^{23} + 38 q^{25} - 156 q^{26}+ \cdots + 170 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 8 * q^5 - 28 * q^7 + 18 * q^8 - 72 * q^10 + 72 * q^11 - 60 * q^13 - 44 * q^14 - 78 * q^16 - 96 * q^17 - 148 * q^19 - 136 * q^20 + 184 * q^22 - 46 * q^23 + 38 * q^25 - 156 * q^26 - 60 * q^28 + 204 * q^29 - 168 * q^31 - 414 * q^32 - 176 * q^34 + 176 * q^35 + 116 * q^37 - 532 * q^38 + 248 * q^40 - 4 * q^41 - 420 * q^43 + 296 * q^44 - 46 * q^46 + 48 * q^47 - 278 * q^49 + 550 * q^50 - 252 * q^52 - 32 * q^53 - 736 * q^55 - 172 * q^56 + 1260 * q^58 + 40 * q^59 - 764 * q^61 + 696 * q^62 + 34 * q^64 + 624 * q^65 - 988 * q^67 - 256 * q^68 + 1136 * q^70 + 224 * q^71 + 820 * q^73 + 804 * q^74 - 916 * q^76 - 1120 * q^77 + 1772 * q^79 + 1080 * q^80 + 412 * q^82 - 1480 * q^83 + 704 * q^85 - 132 * q^86 + 88 * q^88 - 744 * q^89 + 936 * q^91 - 46 * q^92 + 1456 * q^94 + 2128 * q^95 - 260 * q^97 + 170 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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