LMFDB - Embedded newform 693.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.54138$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.54138 q^{2} +12.6241 q^{4} -13.1655 q^{5} +7.00000 q^{7} -21.0000 q^{8} +59.7897 q^{10} +11.0000 q^{11} -54.8345 q^{13} -31.7897 q^{14} -5.62414 q^{16} +79.9104 q^{17} +21.2551 q^{19} -166.203 q^{20} -49.9552 q^{22} -119.414 q^{23} +48.3311 q^{25} +249.024 q^{26} +88.3690 q^{28} +87.4829 q^{29} +191.248 q^{31} +193.541 q^{32} -362.904 q^{34} -92.1587 q^{35} +91.6484 q^{37} -96.5277 q^{38} +276.476 q^{40} +60.4829 q^{41} -213.552 q^{43} +138.866 q^{44} +542.304 q^{46} -417.421 q^{47} +49.0000 q^{49} -219.490 q^{50} -692.238 q^{52} +414.904 q^{53} -144.821 q^{55} -147.000 q^{56} -397.293 q^{58} +358.069 q^{59} +515.297 q^{61} -868.531 q^{62} -833.952 q^{64} +721.925 q^{65} -107.572 q^{67} +1008.80 q^{68} +418.528 q^{70} +711.793 q^{71} +131.773 q^{73} -416.210 q^{74} +268.328 q^{76} +77.0000 q^{77} +10.4966 q^{79} +74.0448 q^{80} -274.676 q^{82} +618.352 q^{83} -1052.06 q^{85} +969.821 q^{86} -231.000 q^{88} -1376.44 q^{89} -383.841 q^{91} -1507.50 q^{92} +1895.67 q^{94} -279.835 q^{95} -534.752 q^{97} -222.528 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 7 q^{4} - 2 q^{5} + 14 q^{7} - 42 q^{8} + 77 q^{10} + 22 q^{11} - 134 q^{13} - 21 q^{14} + 7 q^{16} + 26 q^{17} + 152 q^{19} - 229 q^{20} - 33 q^{22} - 178 q^{23} + 48 q^{25} + 127 q^{26}+ \cdots - 147 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 + 7 * q^4 - 2 * q^5 + 14 * q^7 - 42 * q^8 + 77 * q^10 + 22 * q^11 - 134 * q^13 - 21 * q^14 + 7 * q^16 + 26 * q^17 + 152 * q^19 - 229 * q^20 - 33 * q^22 - 178 * q^23 + 48 * q^25 + 127 * q^26 + 49 * q^28 - 190 * q^29 + 346 * q^31 + 381 * q^32 - 446 * q^34 - 14 * q^35 - 206 * q^37 + 105 * q^38 + 42 * q^40 - 244 * q^41 + 242 * q^43 + 77 * q^44 + 452 * q^46 - 920 * q^47 + 98 * q^49 - 220 * q^50 - 247 * q^52 + 550 * q^53 - 22 * q^55 - 294 * q^56 - 825 * q^58 + 412 * q^59 + 252 * q^61 - 630 * q^62 - 646 * q^64 - 162 * q^65 + 16 * q^67 + 1312 * q^68 + 539 * q^70 + 572 * q^71 - 1026 * q^73 - 875 * q^74 - 467 * q^76 + 154 * q^77 - 52 * q^79 + 215 * q^80 - 744 * q^82 - 138 * q^83 - 1654 * q^85 + 1672 * q^86 - 462 * q^88 - 1512 * q^89 - 938 * q^91 - 1178 * q^92 + 1121 * q^94 + 1180 * q^95 - 1106 * q^97 - 147 * 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 $

$2$	−4.54138	−1.60562	−0.802810	−	0.596234i	$-0.796663\pi$
$2$	−4.54138	−1.60562	−0.802810	+	0.596234i	$0.796663\pi$
$3$	0	0
$3$	0	0
$4$	12.6241	1.57802
$5$	−13.1655	−1.17756	−0.588780	−	0.808293i	$-0.700392\pi$
$5$	−13.1655	−1.17756	−0.588780	+	0.808293i	$0.700392\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−21.0000	−0.928078
$9$	0	0
$10$	59.7897	1.89072
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−54.8345	−1.16987	−0.584936	−	0.811079i	$-0.698881\pi$
$13$	−54.8345	−1.16987	−0.584936	+	0.811079i	$0.698881\pi$
$14$	−31.7897	−0.606868
$15$	0	0
$16$	−5.62414	−0.0878772
$17$	79.9104	1.14007	0.570033	−	0.821622i	$-0.306930\pi$
$17$	79.9104	1.14007	0.570033	+	0.821622i	$0.306930\pi$
$18$	0	0
$19$	21.2551	0.256645	0.128323	−	0.991732i	$-0.459041\pi$
$19$	21.2551	0.256645	0.128323	+	0.991732i	$0.459041\pi$
$20$	−166.203	−1.85821
$21$	0	0
$22$	−49.9552	−0.484113
$23$	−119.414	−1.08259	−0.541294	−	0.840834i	$-0.682065\pi$
$23$	−119.414	−1.08259	−0.541294	+	0.840834i	$0.682065\pi$
$24$	0	0
$25$	48.3311	0.386648
$26$	249.024	1.87837
$27$	0	0
$28$	88.3690	0.596435
$29$	87.4829	0.560178	0.280089	−	0.959974i	$-0.409636\pi$
$29$	87.4829	0.560178	0.280089	+	0.959974i	$0.409636\pi$
$30$	0	0
$31$	191.248	1.10804	0.554019	−	0.832504i	$-0.313093\pi$
$31$	191.248	1.10804	0.554019	+	0.832504i	$0.313093\pi$
$32$	193.541	1.06918
$33$	0	0
$34$	−362.904	−1.83051
$35$	−92.1587	−0.445076
$36$	0	0
$37$	91.6484	0.407214	0.203607	−	0.979053i	$-0.434734\pi$
$37$	91.6484	0.407214	0.203607	+	0.979053i	$0.434734\pi$
$38$	−96.5277	−0.412075
$39$	0	0
$40$	276.476	1.09287
$41$	60.4829	0.230386	0.115193	−	0.993343i	$-0.463251\pi$
$41$	60.4829	0.230386	0.115193	+	0.993343i	$0.463251\pi$
$42$	0	0
$43$	−213.552	−0.757357	−0.378679	−	0.925528i	$-0.623621\pi$
$43$	−213.552	−0.757357	−0.378679	+	0.925528i	$0.623621\pi$
$44$	138.866	0.475790
$45$	0	0
$46$	542.304	1.73822
$47$	−417.421	−1.29547	−0.647735	−	0.761866i	$-0.724283\pi$
$47$	−417.421	−1.29547	−0.647735	+	0.761866i	$0.724283\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−219.490	−0.620811
$51$	0	0
$52$	−692.238	−1.84608
$53$	414.904	1.07531	0.537655	−	0.843165i	$-0.319310\pi$
$53$	414.904	1.07531	0.537655	+	0.843165i	$0.319310\pi$
$54$	0	0
$55$	−144.821	−0.355048
$56$	−147.000	−0.350780
$57$	0	0
$58$	−397.293	−0.899433
$59$	358.069	0.790112	0.395056	−	0.918657i	$-0.370725\pi$
$59$	358.069	0.790112	0.395056	+	0.918657i	$0.370725\pi$
$60$	0	0
$61$	515.297	1.08159	0.540795	−	0.841154i	$-0.318123\pi$
$61$	515.297	1.08159	0.540795	+	0.841154i	$0.318123\pi$
$62$	−868.531	−1.77909
$63$	0	0
$64$	−833.952	−1.62881
$65$	721.925	1.37760
$66$	0	0
$67$	−107.572	−0.196150	−0.0980752	−	0.995179i	$-0.531269\pi$
$67$	−107.572	−0.196150	−0.0980752	+	0.995179i	$0.531269\pi$
$68$	1008.80	1.79904
$69$	0	0
$70$	418.528	0.714623
$71$	711.793	1.18978	0.594890	−	0.803807i	$-0.297196\pi$
$71$	711.793	1.18978	0.594890	+	0.803807i	$0.297196\pi$
$72$	0	0
$73$	131.773	0.211272	0.105636	−	0.994405i	$-0.466312\pi$
$73$	131.773	0.211272	0.105636	+	0.994405i	$0.466312\pi$
$74$	−416.210	−0.653831
$75$	0	0
$76$	268.328	0.404991
$77$	77.0000	0.113961
$78$	0	0
$79$	10.4966	0.0149488	0.00747441	−	0.999972i	$-0.497621\pi$
$79$	10.4966	0.0149488	0.00747441	+	0.999972i	$0.497621\pi$
$80$	74.0448	0.103481
$81$	0	0
$82$	−274.676	−0.369913
$83$	618.352	0.817747	0.408873	−	0.912591i	$-0.365922\pi$
$83$	618.352	0.817747	0.408873	+	0.912591i	$0.365922\pi$
$84$	0	0
$85$	−1052.06	−1.34250
$86$	969.821	1.21603
$87$	0	0
$88$	−231.000	−0.279826
$89$	−1376.44	−1.63935	−0.819677	−	0.572826i	$-0.805847\pi$
$89$	−1376.44	−1.63935	−0.819677	+	0.572826i	$0.805847\pi$
$90$	0	0
$91$	−383.841	−0.442170
$92$	−1507.50	−1.70834
$93$	0	0
$94$	1895.67	2.08003
$95$	−279.835	−0.302215
$96$	0	0
$97$	−534.752	−0.559751	−0.279875	−	0.960036i	$-0.590293\pi$
$97$	−534.752	−0.559751	−0.279875	+	0.960036i	$0.590293\pi$
$98$	−222.528	−0.229374
$99$	0	0
$100$	610.138	0.610138
$101$	−1620.01	−1.59601	−0.798004	−	0.602652i	$-0.794111\pi$
$101$	−1620.01	−1.59601	−0.798004	+	0.602652i	$0.794111\pi$
$102$	0	0
$103$	822.814	0.787129	0.393564	−	0.919297i	$-0.371242\pi$
$103$	822.814	0.787129	0.393564	+	0.919297i	$0.371242\pi$
$104$	1151.52	1.08573
$105$	0	0
$106$	−1884.24	−1.72654
$107$	1749.78	1.58091	0.790456	−	0.612518i	$-0.209843\pi$
$107$	1749.78	1.58091	0.790456	+	0.612518i	$0.209843\pi$
$108$	0	0
$109$	−647.607	−0.569078	−0.284539	−	0.958665i	$-0.591840\pi$
$109$	−647.607	−0.569078	−0.284539	+	0.958665i	$0.591840\pi$
$110$	657.686	0.570072
$111$	0	0
$112$	−39.3690	−0.0332145
$113$	−1151.50	−0.958622	−0.479311	−	0.877645i	$-0.659113\pi$
$113$	−1151.50	−0.958622	−0.479311	+	0.877645i	$0.659113\pi$
$114$	0	0
$115$	1572.15	1.27481
$116$	1104.40	0.883971
$117$	0	0
$118$	−1626.13	−1.26862
$119$	559.373	0.430904
$120$	0	0
$121$	121.000	0.0909091
$122$	−2340.16	−1.73662
$123$	0	0
$124$	2414.35	1.74851
$125$	1009.39	0.722259
$126$	0	0
$127$	278.146	0.194342	0.0971710	−	0.995268i	$-0.469021\pi$
$127$	278.146	0.194342	0.0971710	+	0.995268i	$0.469021\pi$
$128$	2238.96	1.54608
$129$	0	0
$130$	−3278.54	−2.21190
$131$	−2332.22	−1.55547	−0.777737	−	0.628590i	$-0.783633\pi$
$131$	−2332.22	−1.55547	−0.777737	+	0.628590i	$0.783633\pi$
$132$	0	0
$133$	148.786	0.0970028
$134$	488.528	0.314943
$135$	0	0
$136$	−1678.12	−1.05807
$137$	−2612.23	−1.62903	−0.814517	−	0.580139i	$-0.802998\pi$
$137$	−2612.23	−1.62903	−0.814517	+	0.580139i	$0.802998\pi$
$138$	0	0
$139$	−2884.68	−1.76025	−0.880126	−	0.474740i	$-0.842542\pi$
$139$	−2884.68	−1.76025	−0.880126	+	0.474740i	$0.842542\pi$
$140$	−1163.42	−0.702338
$141$	0	0
$142$	−3232.53	−1.91033
$143$	−603.179	−0.352730
$144$	0	0
$145$	−1151.76	−0.659643
$146$	−598.431	−0.339222
$147$	0	0
$148$	1156.98	0.642590
$149$	−2909.54	−1.59972	−0.799862	−	0.600184i	$-0.795094\pi$
$149$	−2909.54	−1.59972	−0.799862	+	0.600184i	$0.795094\pi$
$150$	0	0
$151$	−2327.06	−1.25413	−0.627063	−	0.778968i	$-0.715743\pi$
$151$	−2327.06	−1.25413	−0.627063	+	0.778968i	$0.715743\pi$
$152$	−446.358	−0.238187
$153$	0	0
$154$	−349.686	−0.182977
$155$	−2517.88	−1.30478
$156$	0	0
$157$	−927.572	−0.471518	−0.235759	−	0.971812i	$-0.575758\pi$
$157$	−927.572	−0.471518	−0.235759	+	0.971812i	$0.575758\pi$
$158$	−47.6689	−0.0240021
$159$	0	0
$160$	−2548.07	−1.25902
$161$	−835.897	−0.409179
$162$	0	0
$163$	−2966.31	−1.42539	−0.712696	−	0.701473i	$-0.752526\pi$
$163$	−2966.31	−1.42539	−0.712696	+	0.701473i	$0.752526\pi$
$164$	763.545	0.363554
$165$	0	0
$166$	−2808.17	−1.31299
$167$	−3856.35	−1.78691	−0.893454	−	0.449156i	$-0.851725\pi$
$167$	−3856.35	−1.78691	−0.893454	+	0.449156i	$0.851725\pi$
$168$	0	0
$169$	809.820	0.368602
$170$	4777.82	2.15554
$171$	0	0
$172$	−2695.91	−1.19512
$173$	2699.52	1.18636	0.593181	−	0.805069i	$-0.297872\pi$
$173$	2699.52	1.18636	0.593181	+	0.805069i	$0.297872\pi$
$174$	0	0
$175$	338.317	0.146139
$176$	−61.8656	−0.0264960
$177$	0	0
$178$	6250.95	2.63218
$179$	−1328.58	−0.554764	−0.277382	−	0.960760i	$-0.589467\pi$
$179$	−1328.58	−0.554764	−0.277382	+	0.960760i	$0.589467\pi$
$180$	0	0
$181$	534.944	0.219680	0.109840	−	0.993949i	$-0.464966\pi$
$181$	534.944	0.219680	0.109840	+	0.993949i	$0.464966\pi$
$182$	1743.17	0.709958
$183$	0	0
$184$	2507.69	1.00472
$185$	−1206.60	−0.479519
$186$	0	0
$187$	879.014	0.343743
$188$	−5269.58	−2.04427
$189$	0	0
$190$	1270.84	0.485243
$191$	3456.20	1.30933	0.654665	−	0.755919i	$-0.272810\pi$
$191$	3456.20	1.30933	0.654665	+	0.755919i	$0.272810\pi$
$192$	0	0
$193$	−3106.02	−1.15843	−0.579213	−	0.815176i	$-0.696640\pi$
$193$	−3106.02	−1.15843	−0.579213	+	0.815176i	$0.696640\pi$
$194$	2428.51	0.898747
$195$	0	0
$196$	618.583	0.225431
$197$	−4886.49	−1.76725	−0.883623	−	0.468198i	$-0.844903\pi$
$197$	−4886.49	−1.76725	−0.883623	+	0.468198i	$0.844903\pi$
$198$	0	0
$199$	3319.74	1.18257	0.591283	−	0.806464i	$-0.298622\pi$
$199$	3319.74	1.18257	0.591283	+	0.806464i	$0.298622\pi$
$200$	−1014.95	−0.358840
$201$	0	0
$202$	7357.07	2.56258
$203$	612.380	0.211727
$204$	0	0
$205$	−796.289	−0.271294
$206$	−3736.71	−1.26383
$207$	0	0
$208$	308.397	0.102805
$209$	233.807	0.0773815
$210$	0	0
$211$	4262.51	1.39073	0.695364	−	0.718658i	$-0.255243\pi$
$211$	4262.51	1.39073	0.695364	+	0.718658i	$0.255243\pi$
$212$	5237.80	1.69686
$213$	0	0
$214$	−7946.42	−2.53835
$215$	2811.52	0.891834
$216$	0	0
$217$	1338.74	0.418799
$218$	2941.03	0.913723
$219$	0	0
$220$	−1828.24	−0.560272
$221$	−4381.84	−1.33373
$222$	0	0
$223$	2766.16	0.830655	0.415328	−	0.909672i	$-0.363667\pi$
$223$	2766.16	0.830655	0.415328	+	0.909672i	$0.363667\pi$
$224$	1354.79	0.404110
$225$	0	0
$226$	5229.42	1.53918
$227$	−545.724	−0.159564	−0.0797818	−	0.996812i	$-0.525422\pi$
$227$	−545.724	−0.159564	−0.0797818	+	0.996812i	$0.525422\pi$
$228$	0	0
$229$	3197.48	0.922688	0.461344	−	0.887221i	$-0.347367\pi$
$229$	3197.48	0.922688	0.461344	+	0.887221i	$0.347367\pi$
$230$	−7139.71	−2.04686
$231$	0	0
$232$	−1837.14	−0.519889
$233$	−4203.76	−1.18196	−0.590982	−	0.806684i	$-0.701260\pi$
$233$	−4203.76	−1.18196	−0.590982	+	0.806684i	$0.701260\pi$
$234$	0	0
$235$	5495.56	1.52549
$236$	4520.32	1.24681
$237$	0	0
$238$	−2540.32	−0.691869
$239$	−5720.73	−1.54830	−0.774150	−	0.633003i	$-0.781822\pi$
$239$	−5720.73	−1.54830	−0.774150	+	0.633003i	$0.781822\pi$
$240$	0	0
$241$	5191.77	1.38768	0.693841	−	0.720128i	$-0.255917\pi$
$241$	5191.77	1.38768	0.693841	+	0.720128i	$0.255917\pi$
$242$	−549.507	−0.145966
$243$	0	0
$244$	6505.18	1.70677
$245$	−645.111	−0.168223
$246$	0	0
$247$	−1165.51	−0.300243
$248$	−4016.21	−1.02835
$249$	0	0
$250$	−4584.01	−1.15967
$251$	−847.944	−0.213234	−0.106617	−	0.994300i	$-0.534002\pi$
$251$	−847.944	−0.213234	−0.106617	+	0.994300i	$0.534002\pi$
$252$	0	0
$253$	−1313.55	−0.326412
$254$	−1263.16	−0.312039
$255$	0	0
$256$	−3496.37	−0.853606
$257$	1493.87	0.362588	0.181294	−	0.983429i	$-0.441971\pi$
$257$	1493.87	0.362588	0.181294	+	0.983429i	$0.441971\pi$
$258$	0	0
$259$	641.539	0.153912
$260$	9113.68	2.17387
$261$	0	0
$262$	10591.5	2.49750
$263$	−135.848	−0.0318507	−0.0159253	−	0.999873i	$-0.505069\pi$
$263$	−135.848	−0.0318507	−0.0159253	+	0.999873i	$0.505069\pi$
$264$	0	0
$265$	−5462.42	−1.26624
$266$	−675.694	−0.155750
$267$	0	0
$268$	−1358.01	−0.309529
$269$	739.406	0.167593	0.0837963	−	0.996483i	$-0.473295\pi$
$269$	739.406	0.167593	0.0837963	+	0.996483i	$0.473295\pi$
$270$	0	0
$271$	4360.29	0.977375	0.488688	−	0.872459i	$-0.337476\pi$
$271$	4360.29	0.977375	0.488688	+	0.872459i	$0.337476\pi$
$272$	−449.428	−0.100186
$273$	0	0
$274$	11863.1	2.61561
$275$	531.642	0.116579
$276$	0	0
$277$	−5779.74	−1.25369	−0.626843	−	0.779146i	$-0.715653\pi$
$277$	−5779.74	−1.25369	−0.626843	+	0.779146i	$0.715653\pi$
$278$	13100.4	2.82630
$279$	0	0
$280$	1935.33	0.413065
$281$	4366.84	0.927059	0.463530	−	0.886081i	$-0.346583\pi$
$281$	4366.84	0.927059	0.463530	+	0.886081i	$0.346583\pi$
$282$	0	0
$283$	−8294.14	−1.74218	−0.871088	−	0.491128i	$-0.836585\pi$
$283$	−8294.14	−1.74218	−0.871088	+	0.491128i	$0.836585\pi$
$284$	8985.78	1.87749
$285$	0	0
$286$	2739.27	0.566351
$287$	423.380	0.0870778
$288$	0	0
$289$	1472.67	0.299750
$290$	5230.57	1.05914
$291$	0	0
$292$	1663.52	0.333391
$293$	−2434.82	−0.485473	−0.242736	−	0.970092i	$-0.578045\pi$
$293$	−2434.82	−0.485473	−0.242736	+	0.970092i	$0.578045\pi$
$294$	0	0
$295$	−4714.17	−0.930405
$296$	−1924.62	−0.377926
$297$	0	0
$298$	13213.3	2.56855
$299$	6547.99	1.26649
$300$	0	0
$301$	−1494.86	−0.286254
$302$	10568.1	2.01365
$303$	0	0
$304$	−119.542	−0.0225533
$305$	−6784.15	−1.27364
$306$	0	0
$307$	−5972.33	−1.11029	−0.555145	−	0.831754i	$-0.687337\pi$
$307$	−5972.33	−1.11029	−0.555145	+	0.831754i	$0.687337\pi$
$308$	972.059	0.179832
$309$	0	0
$310$	11434.7	2.09499
$311$	8278.65	1.50945	0.754725	−	0.656041i	$-0.227770\pi$
$311$	8278.65	1.50945	0.754725	+	0.656041i	$0.227770\pi$
$312$	0	0
$313$	−1386.60	−0.250400	−0.125200	−	0.992131i	$-0.539957\pi$
$313$	−1386.60	−0.250400	−0.125200	+	0.992131i	$0.539957\pi$
$314$	4212.46	0.757079
$315$	0	0
$316$	132.510	0.0235895
$317$	−11.2346	−0.00199053	−0.000995264	−	1.00000i	$-0.500317\pi$
$317$	−11.2346	−0.00199053	−0.000995264		1.00000i	$0.500317\pi$
$318$	0	0
$319$	962.312	0.168900
$320$	10979.4	1.91803
$321$	0	0
$322$	3796.13	0.656987
$323$	1698.51	0.292593
$324$	0	0
$325$	−2650.21	−0.452329
$326$	13471.1	2.28864
$327$	0	0
$328$	−1270.14	−0.213816
$329$	−2921.94	−0.489641
$330$	0	0
$331$	−4516.29	−0.749963	−0.374981	−	0.927032i	$-0.622351\pi$
$331$	−4516.29	−0.749963	−0.374981	+	0.927032i	$0.622351\pi$
$332$	7806.17	1.29042
$333$	0	0
$334$	17513.2	2.86910
$335$	1416.25	0.230979
$336$	0	0
$337$	−676.780	−0.109396	−0.0546982	−	0.998503i	$-0.517420\pi$
$337$	−676.780	−0.109396	−0.0546982	+	0.998503i	$0.517420\pi$
$338$	−3677.70	−0.591836
$339$	0	0
$340$	−13281.4	−2.11848
$341$	2103.73	0.334086
$342$	0	0
$343$	343.000	0.0539949
$344$	4484.59	0.702886
$345$	0	0
$346$	−12259.5	−1.90485
$347$	−7619.43	−1.17877	−0.589384	−	0.807853i	$-0.700629\pi$
$347$	−7619.43	−1.17877	−0.589384	+	0.807853i	$0.700629\pi$
$348$	0	0
$349$	3178.35	0.487488	0.243744	−	0.969840i	$-0.421624\pi$
$349$	3178.35	0.487488	0.243744	+	0.969840i	$0.421624\pi$
$350$	−1536.43	−0.234644
$351$	0	0
$352$	2128.96	0.322368
$353$	−1776.12	−0.267800	−0.133900	−	0.990995i	$-0.542750\pi$
$353$	−1776.12	−0.267800	−0.133900	+	0.990995i	$0.542750\pi$
$354$	0	0
$355$	−9371.13	−1.40104
$356$	−17376.4	−2.58693
$357$	0	0
$358$	6033.59	0.890740
$359$	11425.3	1.67968	0.839838	−	0.542838i	$-0.182650\pi$
$359$	11425.3	1.67968	0.839838	+	0.542838i	$0.182650\pi$
$360$	0	0
$361$	−6407.22	−0.934133
$362$	−2429.38	−0.352722
$363$	0	0
$364$	−4845.67	−0.697753
$365$	−1734.86	−0.248785
$366$	0	0
$367$	−1321.34	−0.187938	−0.0939689	−	0.995575i	$-0.529955\pi$
$367$	−1321.34	−0.187938	−0.0939689	+	0.995575i	$0.529955\pi$
$368$	671.600	0.0951348
$369$	0	0
$370$	5479.63	0.769925
$371$	2904.32	0.406429
$372$	0	0
$373$	−2011.16	−0.279180	−0.139590	−	0.990209i	$-0.544578\pi$
$373$	−2011.16	−0.279180	−0.139590	+	0.990209i	$0.544578\pi$
$374$	−3991.94	−0.551920
$375$	0	0
$376$	8765.83	1.20230
$377$	−4797.08	−0.655337
$378$	0	0
$379$	−1376.20	−0.186519	−0.0932597	−	0.995642i	$-0.529729\pi$
$379$	−1376.20	−0.186519	−0.0932597	+	0.995642i	$0.529729\pi$
$380$	−3532.68	−0.476901
$381$	0	0
$382$	−15695.9	−2.10229
$383$	−68.4429	−0.00913125	−0.00456563	−	0.999990i	$-0.501453\pi$
$383$	−68.4429	−0.00913125	−0.00456563	+	0.999990i	$0.501453\pi$
$384$	0	0
$385$	−1013.75	−0.134195
$386$	14105.6	1.85999
$387$	0	0
$388$	−6750.78	−0.883297
$389$	1796.33	0.234133	0.117066	−	0.993124i	$-0.462651\pi$
$389$	1796.33	0.234133	0.117066	+	0.993124i	$0.462651\pi$
$390$	0	0
$391$	−9542.40	−1.23422
$392$	−1029.00	−0.132583
$393$	0	0
$394$	22191.4	2.83753
$395$	−138.193	−0.0176031
$396$	0	0
$397$	4175.59	0.527876	0.263938	−	0.964540i	$-0.414978\pi$
$397$	4175.59	0.527876	0.263938	+	0.964540i	$0.414978\pi$
$398$	−15076.2	−1.89875
$399$	0	0
$400$	−271.821	−0.0339776
$401$	5735.25	0.714226	0.357113	−	0.934061i	$-0.383761\pi$
$401$	5735.25	0.714226	0.357113	+	0.934061i	$0.383761\pi$
$402$	0	0
$403$	−10487.0	−1.29626
$404$	−20451.2	−2.51853
$405$	0	0
$406$	−2781.05	−0.339954
$407$	1008.13	0.122780
$408$	0	0
$409$	4552.26	0.550355	0.275177	−	0.961394i	$-0.411263\pi$
$409$	4552.26	0.550355	0.275177	+	0.961394i	$0.411263\pi$
$410$	3616.25	0.435595
$411$	0	0
$412$	10387.3	1.24210
$413$	2506.48	0.298634
$414$	0	0
$415$	−8140.93	−0.962946
$416$	−10612.7	−1.25080
$417$	0	0
$418$	−1061.80	−0.124245
$419$	−12761.1	−1.48788	−0.743940	−	0.668247i	$-0.767045\pi$
$419$	−12761.1	−1.48788	−0.743940	+	0.668247i	$0.767045\pi$
$420$	0	0
$421$	−2838.93	−0.328649	−0.164324	−	0.986406i	$-0.552544\pi$
$421$	−2838.93	−0.328649	−0.164324	+	0.986406i	$0.552544\pi$
$422$	−19357.7	−2.23298
$423$	0	0
$424$	−8712.97	−0.997970
$425$	3862.15	0.440805
$426$	0	0
$427$	3607.08	0.408803
$428$	22089.5	2.49471
$429$	0	0
$430$	−12768.2	−1.43195
$431$	12820.2	1.43277	0.716387	−	0.697704i	$-0.245795\pi$
$431$	12820.2	1.43277	0.716387	+	0.697704i	$0.245795\pi$
$432$	0	0
$433$	−12570.9	−1.39519	−0.697595	−	0.716493i	$-0.745746\pi$
$433$	−12570.9	−1.39519	−0.697595	+	0.716493i	$0.745746\pi$
$434$	−6079.72	−0.672433
$435$	0	0
$436$	−8175.48	−0.898015
$437$	−2538.16	−0.277841
$438$	0	0
$439$	−11589.8	−1.26003	−0.630013	−	0.776585i	$-0.716950\pi$
$439$	−11589.8	−1.26003	−0.630013	+	0.776585i	$0.716950\pi$
$440$	3041.24	0.329512
$441$	0	0
$442$	19899.6	2.14147
$443$	−15298.1	−1.64071	−0.820355	−	0.571855i	$-0.806224\pi$
$443$	−15298.1	−1.64071	−0.820355	+	0.571855i	$0.806224\pi$
$444$	0	0
$445$	18121.6	1.93044
$446$	−12562.2	−1.33372
$447$	0	0
$448$	−5837.66	−0.615633
$449$	−3370.37	−0.354249	−0.177125	−	0.984188i	$-0.556680\pi$
$449$	−3370.37	−0.354249	−0.177125	+	0.984188i	$0.556680\pi$
$450$	0	0
$451$	665.312	0.0694641
$452$	−14536.7	−1.51272
$453$	0	0
$454$	2478.34	0.256199
$455$	5053.47	0.520682
$456$	0	0
$457$	−2682.12	−0.274539	−0.137269	−	0.990534i	$-0.543833\pi$
$457$	−2682.12	−0.274539	−0.137269	+	0.990534i	$0.543833\pi$
$458$	−14521.0	−1.48149
$459$	0	0
$460$	19847.0	2.01168
$461$	−1313.76	−0.132729	−0.0663645	−	0.997795i	$-0.521140\pi$
$461$	−1313.76	−0.132729	−0.0663645	+	0.997795i	$0.521140\pi$
$462$	0	0
$463$	17339.2	1.74043	0.870215	−	0.492672i	$-0.163980\pi$
$463$	17339.2	1.74043	0.870215	+	0.492672i	$0.163980\pi$
$464$	−492.016	−0.0492269
$465$	0	0
$466$	19090.9	1.89779
$467$	8810.22	0.872995	0.436497	−	0.899706i	$-0.356219\pi$
$467$	8810.22	0.872995	0.436497	+	0.899706i	$0.356219\pi$
$468$	0	0
$469$	−753.007	−0.0741379
$470$	−24957.4	−2.44936
$471$	0	0
$472$	−7519.45	−0.733285
$473$	−2349.07	−0.228352
$474$	0	0
$475$	1027.28	0.0992315
$476$	7061.60	0.679975
$477$	0	0
$478$	25980.0	2.48598
$479$	708.948	0.0676256	0.0338128	−	0.999428i	$-0.489235\pi$
$479$	708.948	0.0676256	0.0338128	+	0.999428i	$0.489235\pi$
$480$	0	0
$481$	−5025.49	−0.476388
$482$	−23577.8	−2.22809
$483$	0	0
$484$	1527.52	0.143456
$485$	7040.29	0.659140
$486$	0	0
$487$	12253.6	1.14017	0.570085	−	0.821586i	$-0.306910\pi$
$487$	12253.6	1.14017	0.570085	+	0.821586i	$0.306910\pi$
$488$	−10821.2	−1.00380
$489$	0	0
$490$	2929.69	0.270102
$491$	5429.37	0.499031	0.249515	−	0.968371i	$-0.419729\pi$
$491$	5429.37	0.499031	0.249515	+	0.968371i	$0.419729\pi$
$492$	0	0
$493$	6990.79	0.638640
$494$	5293.04	0.482076
$495$	0	0
$496$	−1075.61	−0.0973714
$497$	4982.55	0.449694
$498$	0	0
$499$	−14531.6	−1.30366	−0.651829	−	0.758366i	$-0.725998\pi$
$499$	−14531.6	−1.30366	−0.651829	+	0.758366i	$0.725998\pi$
$500$	12742.6	1.13974
$501$	0	0
$502$	3850.83	0.342373
$503$	6349.27	0.562823	0.281412	−	0.959587i	$-0.409197\pi$
$503$	6349.27	0.562823	0.281412	+	0.959587i	$0.409197\pi$
$504$	0	0
$505$	21328.2	1.87940
$506$	5965.34	0.524094
$507$	0	0
$508$	3511.35	0.306675
$509$	2082.16	0.181317	0.0906585	−	0.995882i	$-0.471103\pi$
$509$	2082.16	0.181317	0.0906585	+	0.995882i	$0.471103\pi$
$510$	0	0
$511$	922.410	0.0798532
$512$	−2033.36	−0.175513
$513$	0	0
$514$	−6784.24	−0.582179
$515$	−10832.8	−0.926892
$516$	0	0
$517$	−4591.63	−0.390599
$518$	−2913.47	−0.247125
$519$	0	0
$520$	−15160.4	−1.27852
$521$	17339.3	1.45806	0.729030	−	0.684481i	$-0.239971\pi$
$521$	17339.3	1.45806	0.729030	+	0.684481i	$0.239971\pi$
$522$	0	0
$523$	−11310.6	−0.945660	−0.472830	−	0.881154i	$-0.656768\pi$
$523$	−11310.6	−0.945660	−0.472830	+	0.881154i	$0.656768\pi$
$524$	−29442.3	−2.45457
$525$	0	0
$526$	616.936	0.0511401
$527$	15282.7	1.26324
$528$	0	0
$529$	2092.66	0.171995
$530$	24806.9	2.03310
$531$	0	0
$532$	1878.30	0.153072
$533$	−3316.55	−0.269523
$534$	0	0
$535$	−23036.8	−1.86162
$536$	2259.02	0.182043
$537$	0	0
$538$	−3357.93	−0.269090
$539$	539.000	0.0430730
$540$	0	0
$541$	−1483.01	−0.117855	−0.0589275	−	0.998262i	$-0.518768\pi$
$541$	−1483.01	−0.117855	−0.0589275	+	0.998262i	$0.518768\pi$
$542$	−19801.7	−1.56929
$543$	0	0
$544$	15466.0	1.21893
$545$	8526.08	0.670123
$546$	0	0
$547$	3512.31	0.274544	0.137272	−	0.990533i	$-0.456167\pi$
$547$	3512.31	0.274544	0.137272	+	0.990533i	$0.456167\pi$
$548$	−32977.2	−2.57065
$549$	0	0
$550$	−2414.39	−0.187181
$551$	1859.46	0.143767
$552$	0	0
$553$	73.4760	0.00565012
$554$	26248.0	2.01294
$555$	0	0
$556$	−36416.6	−2.77771
$557$	−14069.1	−1.07025	−0.535123	−	0.844774i	$-0.679735\pi$
$557$	−14069.1	−1.07025	−0.535123	+	0.844774i	$0.679735\pi$
$558$	0	0
$559$	11710.0	0.886012
$560$	518.314	0.0391121
$561$	0	0
$562$	−19831.5	−1.48851
$563$	−12982.7	−0.971860	−0.485930	−	0.873998i	$-0.661519\pi$
$563$	−12982.7	−0.971860	−0.485930	+	0.873998i	$0.661519\pi$
$564$	0	0
$565$	15160.1	1.12884
$566$	37666.9	2.79727
$567$	0	0
$568$	−14947.7	−1.10421
$569$	1668.21	0.122909	0.0614545	−	0.998110i	$-0.480426\pi$
$569$	1668.21	0.122909	0.0614545	+	0.998110i	$0.480426\pi$
$570$	0	0
$571$	−22037.1	−1.61510	−0.807552	−	0.589797i	$-0.799208\pi$
$571$	−22037.1	−1.61510	−0.807552	+	0.589797i	$0.799208\pi$
$572$	−7614.62	−0.556614
$573$	0	0
$574$	−1922.73	−0.139814
$575$	−5771.39	−0.418581
$576$	0	0
$577$	22415.8	1.61730	0.808649	−	0.588292i	$-0.200199\pi$
$577$	22415.8	1.61730	0.808649	+	0.588292i	$0.200199\pi$
$578$	−6687.96	−0.481284
$579$	0	0
$580$	−14540.0	−1.04093
$581$	4328.47	0.309079
$582$	0	0
$583$	4563.94	0.324218
$584$	−2767.23	−0.196077
$585$	0	0
$586$	11057.4	0.779485
$587$	24252.2	1.70527	0.852636	−	0.522506i	$-0.175003\pi$
$587$	24252.2	1.70527	0.852636	+	0.522506i	$0.175003\pi$
$588$	0	0
$589$	4065.01	0.284373
$590$	21408.8	1.49388
$591$	0	0
$592$	−515.444	−0.0357848
$593$	−18666.9	−1.29268	−0.646340	−	0.763050i	$-0.723701\pi$
$593$	−18666.9	−1.29268	−0.646340	+	0.763050i	$0.723701\pi$
$594$	0	0
$595$	−7364.44	−0.507416
$596$	−36730.5	−2.52439
$597$	0	0
$598$	−29736.9	−2.03350
$599$	11400.1	0.777620	0.388810	−	0.921318i	$-0.372886\pi$
$599$	11400.1	0.777620	0.388810	+	0.921318i	$0.372886\pi$
$600$	0	0
$601$	−9722.11	−0.659856	−0.329928	−	0.944006i	$-0.607024\pi$
$601$	−9722.11	−0.659856	−0.329928	+	0.944006i	$0.607024\pi$
$602$	6788.75	0.459616
$603$	0	0
$604$	−29377.1	−1.97903
$605$	−1593.03	−0.107051
$606$	0	0
$607$	−20616.5	−1.37858	−0.689291	−	0.724484i	$-0.742078\pi$
$607$	−20616.5	−1.37858	−0.689291	+	0.724484i	$0.742078\pi$
$608$	4113.75	0.274399
$609$	0	0
$610$	30809.4	2.04498
$611$	22889.0	1.51553
$612$	0	0
$613$	−363.351	−0.0239406	−0.0119703	−	0.999928i	$-0.503810\pi$
$613$	−363.351	−0.0239406	−0.0119703	+	0.999928i	$0.503810\pi$
$614$	27122.6	1.78270
$615$	0	0
$616$	−1617.00	−0.105764
$617$	19599.7	1.27886	0.639429	−	0.768850i	$-0.279171\pi$
$617$	19599.7	1.27886	0.639429	+	0.768850i	$0.279171\pi$
$618$	0	0
$619$	10178.6	0.660926	0.330463	−	0.943819i	$-0.392795\pi$
$619$	10178.6	0.660926	0.330463	+	0.943819i	$0.392795\pi$
$620$	−31786.1	−2.05897
$621$	0	0
$622$	−37596.5	−2.42361
$623$	−9635.09	−0.619618
$624$	0	0
$625$	−19330.5	−1.23715
$626$	6297.08	0.402048
$627$	0	0
$628$	−11709.8	−0.744064
$629$	7323.66	0.464250
$630$	0	0
$631$	−13503.9	−0.851949	−0.425975	−	0.904735i	$-0.640069\pi$
$631$	−13503.9	−0.851949	−0.425975	+	0.904735i	$0.640069\pi$
$632$	−220.428	−0.0138737
$633$	0	0
$634$	51.0205	0.00319603
$635$	−3661.93	−0.228849
$636$	0	0
$637$	−2686.89	−0.167125
$638$	−4370.22	−0.271189
$639$	0	0
$640$	−29477.1	−1.82060
$641$	−29389.5	−1.81095	−0.905473	−	0.424404i	$-0.860483\pi$
$641$	−29389.5	−1.81095	−0.905473	+	0.424404i	$0.860483\pi$
$642$	0	0
$643$	−11632.9	−0.713461	−0.356730	−	0.934207i	$-0.616108\pi$
$643$	−11632.9	−0.713461	−0.356730	+	0.934207i	$0.616108\pi$
$644$	−10552.5	−0.645692
$645$	0	0
$646$	−7713.56	−0.469793
$647$	7522.70	0.457106	0.228553	−	0.973531i	$-0.426601\pi$
$647$	7522.70	0.457106	0.228553	+	0.973531i	$0.426601\pi$
$648$	0	0
$649$	3938.76	0.238228
$650$	12035.6	0.726270
$651$	0	0
$652$	−37447.1	−2.24929
$653$	−31890.8	−1.91115	−0.955577	−	0.294743i	$-0.904766\pi$
$653$	−31890.8	−1.91115	−0.955577	+	0.294743i	$0.904766\pi$
$654$	0	0
$655$	30704.9	1.83166
$656$	−340.164	−0.0202457
$657$	0	0
$658$	13269.7	0.786178
$659$	−20234.0	−1.19606	−0.598029	−	0.801474i	$-0.704049\pi$
$659$	−20234.0	−1.19606	−0.598029	+	0.801474i	$0.704049\pi$
$660$	0	0
$661$	16493.0	0.970505	0.485252	−	0.874374i	$-0.338728\pi$
$661$	16493.0	0.970505	0.485252	+	0.874374i	$0.338728\pi$
$662$	20510.2	1.20416
$663$	0	0
$664$	−12985.4	−0.758932
$665$	−1958.85	−0.114227
$666$	0	0
$667$	−10446.7	−0.606441
$668$	−48683.1	−2.81977
$669$	0	0
$670$	−6431.72	−0.370864
$671$	5668.26	0.326112
$672$	0	0
$673$	7338.87	0.420346	0.210173	−	0.977664i	$-0.432597\pi$
$673$	7338.87	0.420346	0.210173	+	0.977664i	$0.432597\pi$
$674$	3073.52	0.175649
$675$	0	0
$676$	10223.3	0.581661
$677$	−10296.8	−0.584549	−0.292274	−	0.956334i	$-0.594412\pi$
$677$	−10296.8	−0.584549	−0.292274	+	0.956334i	$0.594412\pi$
$678$	0	0
$679$	−3743.26	−0.211566
$680$	22093.3	1.24594
$681$	0	0
$682$	−9553.85	−0.536416
$683$	−19884.5	−1.11400	−0.556998	−	0.830514i	$-0.688047\pi$
$683$	−19884.5	−1.11400	−0.556998	+	0.830514i	$0.688047\pi$
$684$	0	0
$685$	34391.4	1.91829
$686$	−1557.69	−0.0866954
$687$	0	0
$688$	1201.05	0.0665545
$689$	−22751.0	−1.25797
$690$	0	0
$691$	26756.6	1.47304	0.736519	−	0.676417i	$-0.236468\pi$
$691$	26756.6	1.47304	0.736519	+	0.676417i	$0.236468\pi$
$692$	34079.1	1.87210
$693$	0	0
$694$	34602.7	1.89265
$695$	37978.3	2.07280
$696$	0	0
$697$	4833.21	0.262656
$698$	−14434.1	−0.782720
$699$	0	0
$700$	4270.97	0.230611
$701$	7556.29	0.407128	0.203564	−	0.979062i	$-0.434747\pi$
$701$	7556.29	0.407128	0.203564	+	0.979062i	$0.434747\pi$
$702$	0	0
$703$	1948.00	0.104510
$704$	−9173.47	−0.491105
$705$	0	0
$706$	8066.03	0.429985
$707$	−11340.1	−0.603234
$708$	0	0
$709$	18371.1	0.973117	0.486558	−	0.873648i	$-0.338252\pi$
$709$	18371.1	0.973117	0.486558	+	0.873648i	$0.338252\pi$
$710$	42557.9	2.24953
$711$	0	0
$712$	28905.3	1.52145
$713$	−22837.7	−1.19955
$714$	0	0
$715$	7941.17	0.415361
$716$	−16772.2	−0.875427
$717$	0	0
$718$	−51886.6	−2.69692
$719$	−24627.6	−1.27740	−0.638702	−	0.769454i	$-0.720528\pi$
$719$	−24627.6	−1.27740	−0.638702	+	0.769454i	$0.720528\pi$
$720$	0	0
$721$	5759.70	0.297507
$722$	29097.6	1.49986
$723$	0	0
$724$	6753.20	0.346659
$725$	4228.14	0.216592
$726$	0	0
$727$	19040.6	0.971360	0.485680	−	0.874137i	$-0.338572\pi$
$727$	19040.6	0.971360	0.485680	+	0.874137i	$0.338572\pi$
$728$	8060.67	0.410368
$729$	0	0
$730$	7878.65	0.399455
$731$	−17065.0	−0.863437
$732$	0	0
$733$	−35623.2	−1.79505	−0.897526	−	0.440961i	$-0.854638\pi$
$733$	−35623.2	−1.79505	−0.897526	+	0.440961i	$0.854638\pi$
$734$	6000.69	0.301757
$735$	0	0
$736$	−23111.5	−1.15748
$737$	−1183.30	−0.0591415
$738$	0	0
$739$	7248.54	0.360814	0.180407	−	0.983592i	$-0.442258\pi$
$739$	7248.54	0.360814	0.180407	+	0.983592i	$0.442258\pi$
$740$	−15232.3	−0.756689
$741$	0	0
$742$	−13189.6	−0.652570
$743$	29033.2	1.43354	0.716772	−	0.697307i	$-0.245619\pi$
$743$	29033.2	1.43354	0.716772	+	0.697307i	$0.245619\pi$
$744$	0	0
$745$	38305.6	1.88377
$746$	9133.46	0.448257
$747$	0	0
$748$	11096.8	0.542432
$749$	12248.5	0.597529
$750$	0	0
$751$	−32062.0	−1.55787	−0.778933	−	0.627107i	$-0.784239\pi$
$751$	−32062.0	−1.55787	−0.778933	+	0.627107i	$0.784239\pi$
$752$	2347.63	0.113842
$753$	0	0
$754$	21785.4	1.05222
$755$	30636.9	1.47681
$756$	0	0
$757$	16132.9	0.774584	0.387292	−	0.921957i	$-0.373411\pi$
$757$	16132.9	0.774584	0.387292	+	0.921957i	$0.373411\pi$
$758$	6249.87	0.299479
$759$	0	0
$760$	5876.54	0.280479
$761$	37232.9	1.77358	0.886789	−	0.462175i	$-0.152931\pi$
$761$	37232.9	1.77358	0.886789	+	0.462175i	$0.152931\pi$
$762$	0	0
$763$	−4533.25	−0.215091
$764$	43631.6	2.06615
$765$	0	0
$766$	310.825	0.0146613
$767$	−19634.5	−0.924331
$768$	0	0
$769$	36009.7	1.68861	0.844305	−	0.535862i	$-0.180013\pi$
$769$	36009.7	1.68861	0.844305	+	0.535862i	$0.180013\pi$
$770$	4603.80	0.215467
$771$	0	0
$772$	−39210.9	−1.82802
$773$	−20156.0	−0.937856	−0.468928	−	0.883236i	$-0.655360\pi$
$773$	−20156.0	−0.937856	−0.468928	+	0.883236i	$0.655360\pi$
$774$	0	0
$775$	9243.23	0.428421
$776$	11229.8	0.519492
$777$	0	0
$778$	−8157.83	−0.375929
$779$	1285.57	0.0591276
$780$	0	0
$781$	7829.73	0.358732
$782$	43335.7	1.98169
$783$	0	0
$784$	−275.583	−0.0125539
$785$	12212.0	0.555241
$786$	0	0
$787$	−19711.9	−0.892823	−0.446411	−	0.894828i	$-0.647298\pi$
$787$	−19711.9	−0.892823	−0.446411	+	0.894828i	$0.647298\pi$
$788$	−61687.7	−2.78875
$789$	0	0
$790$	627.587	0.0282640
$791$	−8060.52	−0.362325
$792$	0	0
$793$	−28256.0	−1.26532
$794$	−18963.0	−0.847569
$795$	0	0
$796$	41908.9	1.86611
$797$	8274.02	0.367730	0.183865	−	0.982952i	$-0.441139\pi$
$797$	8274.02	0.367730	0.183865	+	0.982952i	$0.441139\pi$
$798$	0	0
$799$	−33356.2	−1.47692
$800$	9354.06	0.413395
$801$	0	0
$802$	−26045.9	−1.14678
$803$	1449.50	0.0637009
$804$	0	0
$805$	11005.0	0.481833
$806$	47625.5	2.08131
$807$	0	0
$808$	34020.2	1.48122
$809$	−15039.4	−0.653593	−0.326797	−	0.945095i	$-0.605969\pi$
$809$	−15039.4	−0.653593	−0.326797	+	0.945095i	$0.605969\pi$
$810$	0	0
$811$	−31217.1	−1.35164	−0.675820	−	0.737067i	$-0.736210\pi$
$811$	−31217.1	−1.35164	−0.675820	+	0.737067i	$0.736210\pi$
$812$	7730.77	0.334110
$813$	0	0
$814$	−4578.31	−0.197137
$815$	39053.0	1.67849
$816$	0	0
$817$	−4539.08	−0.194372
$818$	−20673.6	−0.883661
$819$	0	0
$820$	−10052.5	−0.428106
$821$	−6151.17	−0.261483	−0.130741	−	0.991417i	$-0.541736\pi$
$821$	−6151.17	−0.261483	−0.130741	+	0.991417i	$0.541736\pi$
$822$	0	0
$823$	−34084.1	−1.44362	−0.721808	−	0.692093i	$-0.756689\pi$
$823$	−34084.1	−1.44362	−0.721808	+	0.692093i	$0.756689\pi$
$824$	−17279.1	−0.730517
$825$	0	0
$826$	−11382.9	−0.479494
$827$	−19451.9	−0.817907	−0.408953	−	0.912555i	$-0.634106\pi$
$827$	−19451.9	−0.817907	−0.408953	+	0.912555i	$0.634106\pi$
$828$	0	0
$829$	26492.7	1.10993	0.554963	−	0.831875i	$-0.312732\pi$
$829$	26492.7	1.10993	0.554963	+	0.831875i	$0.312732\pi$
$830$	36971.1	1.54613
$831$	0	0
$832$	45729.3	1.90550
$833$	3915.61	0.162867
$834$	0	0
$835$	50770.9	2.10419
$836$	2951.61	0.122109
$837$	0	0
$838$	57953.1	2.38897
$839$	8978.20	0.369442	0.184721	−	0.982791i	$-0.440862\pi$
$839$	8978.20	0.369442	0.184721	+	0.982791i	$0.440862\pi$
$840$	0	0
$841$	−16735.7	−0.686201
$842$	12892.7	0.527685
$843$	0	0
$844$	53810.6	2.19459
$845$	−10661.7	−0.434052
$846$	0	0
$847$	847.000	0.0343604
$848$	−2333.48	−0.0944952
$849$	0	0
$850$	−17539.5	−0.707765
$851$	−10944.1	−0.440844
$852$	0	0
$853$	−27980.8	−1.12315	−0.561574	−	0.827427i	$-0.689804\pi$
$853$	−27980.8	−1.12315	−0.561574	+	0.827427i	$0.689804\pi$
$854$	−16381.1	−0.656382
$855$	0	0
$856$	−36745.4	−1.46721
$857$	4175.08	0.166416	0.0832078	−	0.996532i	$-0.473483\pi$
$857$	4175.08	0.166416	0.0832078	+	0.996532i	$0.473483\pi$
$858$	0	0
$859$	38793.4	1.54088	0.770439	−	0.637514i	$-0.220037\pi$
$859$	38793.4	1.54088	0.770439	+	0.637514i	$0.220037\pi$
$860$	35493.1	1.40733
$861$	0	0
$862$	−58221.2	−2.30049
$863$	19429.6	0.766385	0.383193	−	0.923668i	$-0.374825\pi$
$863$	19429.6	0.766385	0.383193	+	0.923668i	$0.374825\pi$
$864$	0	0
$865$	−35540.6	−1.39701
$866$	57089.1	2.24015
$867$	0	0
$868$	16900.4	0.660873
$869$	115.462	0.00450724
$870$	0	0
$871$	5898.68	0.229471
$872$	13599.7	0.528148
$873$	0	0
$874$	11526.7	0.446107
$875$	7065.71	0.272988
$876$	0	0
$877$	−17547.9	−0.675656	−0.337828	−	0.941208i	$-0.609692\pi$
$877$	−17547.9	−0.675656	−0.337828	+	0.941208i	$0.609692\pi$
$878$	52633.8	2.02312
$879$	0	0
$880$	814.493	0.0312006
$881$	44948.7	1.71891	0.859455	−	0.511212i	$-0.170803\pi$
$881$	44948.7	1.71891	0.859455	+	0.511212i	$0.170803\pi$
$882$	0	0
$883$	−37819.3	−1.44136	−0.720680	−	0.693268i	$-0.756170\pi$
$883$	−37819.3	−1.44136	−0.720680	+	0.693268i	$0.756170\pi$
$884$	−55317.0	−2.10465
$885$	0	0
$886$	69474.4	2.63436
$887$	7203.82	0.272695	0.136348	−	0.990661i	$-0.456464\pi$
$887$	7203.82	0.272695	0.136348	+	0.990661i	$0.456464\pi$
$888$	0	0
$889$	1947.02	0.0734543
$890$	−82297.0	−3.09955
$891$	0	0
$892$	34920.5	1.31079
$893$	−8872.33	−0.332476
$894$	0	0
$895$	17491.5	0.653268
$896$	15672.7	0.584363
$897$	0	0
$898$	15306.2	0.568790
$899$	16731.0	0.620699
$900$	0	0
$901$	33155.1	1.22592
$902$	−3021.43	−0.111533
$903$	0	0
$904$	24181.6	0.889676
$905$	−7042.81	−0.258686
$906$	0	0
$907$	9420.96	0.344893	0.172446	−	0.985019i	$-0.444833\pi$
$907$	9420.96	0.344893	0.172446	+	0.985019i	$0.444833\pi$
$908$	−6889.30	−0.251794
$909$	0	0
$910$	−22949.7	−0.836018
$911$	−40045.0	−1.45637	−0.728184	−	0.685382i	$-0.759635\pi$
$911$	−40045.0	−1.45637	−0.728184	+	0.685382i	$0.759635\pi$
$912$	0	0
$913$	6801.87	0.246560
$914$	12180.5	0.440805
$915$	0	0
$916$	40365.5	1.45602
$917$	−16325.6	−0.587914
$918$	0	0
$919$	−8881.69	−0.318803	−0.159402	−	0.987214i	$-0.550956\pi$
$919$	−8881.69	−0.318803	−0.159402	+	0.987214i	$0.550956\pi$
$920$	−33015.1	−1.18312
$921$	0	0
$922$	5966.30	0.213112
$923$	−39030.8	−1.39189
$924$	0	0
$925$	4429.46	0.157448
$926$	−78743.7	−2.79447
$927$	0	0
$928$	16931.6	0.598928
$929$	−39037.4	−1.37866	−0.689331	−	0.724447i	$-0.742095\pi$
$929$	−39037.4	−1.37866	−0.689331	+	0.724447i	$0.742095\pi$
$930$	0	0
$931$	1041.50	0.0366636
$932$	−53068.9	−1.86516
$933$	0	0
$934$	−40010.6	−1.40170
$935$	−11572.7	−0.404778
$936$	0	0
$937$	17549.7	0.611870	0.305935	−	0.952052i	$-0.401031\pi$
$937$	17549.7	0.611870	0.305935	+	0.952052i	$0.401031\pi$
$938$	3419.69	0.119037
$939$	0	0
$940$	69376.8	2.40726
$941$	28176.8	0.976128	0.488064	−	0.872808i	$-0.337703\pi$
$941$	28176.8	0.976128	0.488064	+	0.872808i	$0.337703\pi$
$942$	0	0
$943$	−7222.49	−0.249413
$944$	−2013.83	−0.0694329
$945$	0	0
$946$	10668.0	0.366646
$947$	11092.6	0.380636	0.190318	−	0.981722i	$-0.439048\pi$
$947$	11092.6	0.380636	0.190318	+	0.981722i	$0.439048\pi$
$948$	0	0
$949$	−7225.69	−0.247161
$950$	−4665.28	−0.159328
$951$	0	0
$952$	−11746.8	−0.399913
$953$	−21883.0	−0.743820	−0.371910	−	0.928269i	$-0.621297\pi$
$953$	−21883.0	−0.743820	−0.371910	+	0.928269i	$0.621297\pi$
$954$	0	0
$955$	−45502.7	−1.54181
$956$	−72219.4	−2.44324
$957$	0	0
$958$	−3219.60	−0.108581
$959$	−18285.6	−0.615717
$960$	0	0
$961$	6784.91	0.227750
$962$	22822.7	0.764899
$963$	0	0
$964$	65541.7	2.18979
$965$	40892.4	1.36412
$966$	0	0
$967$	40235.4	1.33804	0.669019	−	0.743245i	$-0.266714\pi$
$967$	40235.4	1.33804	0.669019	+	0.743245i	$0.266714\pi$
$968$	−2541.00	−0.0843707
$969$	0	0
$970$	−31972.6	−1.05833
$971$	−44470.2	−1.46974	−0.734870	−	0.678208i	$-0.762757\pi$
$971$	−44470.2	−1.46974	−0.734870	+	0.678208i	$0.762757\pi$
$972$	0	0
$973$	−20192.7	−0.665313
$974$	−55648.2	−1.83068
$975$	0	0
$976$	−2898.10	−0.0950472
$977$	55989.4	1.83343	0.916715	−	0.399542i	$-0.130831\pi$
$977$	55989.4	1.83343	0.916715	+	0.399542i	$0.130831\pi$
$978$	0	0
$979$	−15140.9	−0.494284
$980$	−8143.97	−0.265459
$981$	0	0
$982$	−24656.8	−0.801254
$983$	5987.22	0.194265	0.0971326	−	0.995271i	$-0.469033\pi$
$983$	5987.22	0.194265	0.0971326	+	0.995271i	$0.469033\pi$
$984$	0	0
$985$	64333.1	2.08104
$986$	−31747.8	−1.02541
$987$	0	0
$988$	−14713.6	−0.473788
$989$	25501.1	0.819905
$990$	0	0
$991$	−1965.53	−0.0630043	−0.0315022	−	0.999504i	$-0.510029\pi$
$991$	−1965.53	−0.0630043	−0.0315022	+	0.999504i	$0.510029\pi$
$992$	37014.5	1.18469
$993$	0	0
$994$	−22627.7	−0.722039
$995$	−43706.2	−1.39254
$996$	0	0
$997$	−10802.9	−0.343161	−0.171581	−	0.985170i	$-0.554887\pi$
$997$	−10802.9	−0.343161	−0.171581	+	0.985170i	$0.554887\pi$
$998$	65993.7	2.09318
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.h.1.1		2
3.2	odd	2		231.4.a.i.1.2	✓	2
21.20	even	2		1617.4.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.i.1.2	✓	2	3.2	odd	2
693.4.a.h.1.1		2	1.1	even	1	trivial
1617.4.a.l.1.2		2	21.20	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-4.54138 q^{2} +12.6241 q^{4} -13.1655 q^{5} +7.00000 q^{7} -21.0000 q^{8} +59.7897 q^{10} +11.0000 q^{11} -54.8345 q^{13} -31.7897 q^{14} -5.62414 q^{16} +79.9104 q^{17} +21.2551 q^{19} -166.203 q^{20} -49.9552 q^{22} -119.414 q^{23} +48.3311 q^{25} +249.024 q^{26} +88.3690 q^{28} +87.4829 q^{29} +191.248 q^{31} +193.541 q^{32} -362.904 q^{34} -92.1587 q^{35} +91.6484 q^{37} -96.5277 q^{38} +276.476 q^{40} +60.4829 q^{41} -213.552 q^{43} +138.866 q^{44} +542.304 q^{46} -417.421 q^{47} +49.0000 q^{49} -219.490 q^{50} -692.238 q^{52} +414.904 q^{53} -144.821 q^{55} -147.000 q^{56} -397.293 q^{58} +358.069 q^{59} +515.297 q^{61} -868.531 q^{62} -833.952 q^{64} +721.925 q^{65} -107.572 q^{67} +1008.80 q^{68} +418.528 q^{70} +711.793 q^{71} +131.773 q^{73} -416.210 q^{74} +268.328 q^{76} +77.0000 q^{77} +10.4966 q^{79} +74.0448 q^{80} -274.676 q^{82} +618.352 q^{83} -1052.06 q^{85} +969.821 q^{86} -231.000 q^{88} -1376.44 q^{89} -383.841 q^{91} -1507.50 q^{92} +1895.67 q^{94} -279.835 q^{95} -534.752 q^{97} -222.528 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 7 q^{4} - 2 q^{5} + 14 q^{7} - 42 q^{8} + 77 q^{10} + 22 q^{11} - 134 q^{13} - 21 q^{14} + 7 q^{16} + 26 q^{17} + 152 q^{19} - 229 q^{20} - 33 q^{22} - 178 q^{23} + 48 q^{25} + 127 q^{26}+ \cdots - 147 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 + 7 * q^4 - 2 * q^5 + 14 * q^7 - 42 * q^8 + 77 * q^10 + 22 * q^11 - 134 * q^13 - 21 * q^14 + 7 * q^16 + 26 * q^17 + 152 * q^19 - 229 * q^20 - 33 * q^22 - 178 * q^23 + 48 * q^25 + 127 * q^26 + 49 * q^28 - 190 * q^29 + 346 * q^31 + 381 * q^32 - 446 * q^34 - 14 * q^35 - 206 * q^37 + 105 * q^38 + 42 * q^40 - 244 * q^41 + 242 * q^43 + 77 * q^44 + 452 * q^46 - 920 * q^47 + 98 * q^49 - 220 * q^50 - 247 * q^52 + 550 * q^53 - 22 * q^55 - 294 * q^56 - 825 * q^58 + 412 * q^59 + 252 * q^61 - 630 * q^62 - 646 * q^64 - 162 * q^65 + 16 * q^67 + 1312 * q^68 + 539 * q^70 + 572 * q^71 - 1026 * q^73 - 875 * q^74 - 467 * q^76 + 154 * q^77 - 52 * q^79 + 215 * q^80 - 744 * q^82 - 138 * q^83 - 1654 * q^85 + 1672 * q^86 - 462 * q^88 - 1512 * q^89 - 938 * q^91 - 1178 * q^92 + 1121 * q^94 + 1180 * q^95 - 1106 * q^97 - 147 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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