LMFDB - Embedded newform 667.4.a.b.1.36

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [667,4,Mod(1,667)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(667, 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("667.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 667 = 23 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	667.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:	$39.3542739738$
Analytic rank:	$1$
Dimension:	$38$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.36
Character	$\chi$	$=$	667.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.71195 q^{2} -7.99074 q^{3} +14.2025 q^{4} +16.5206 q^{5} -37.6520 q^{6} -23.1601 q^{7} +29.2258 q^{8} +36.8519 q^{9} +O(q^{10})$	\(q+4.71195 q^{2} -7.99074 q^{3} +14.2025 q^{4} +16.5206 q^{5} -37.6520 q^{6} -23.1601 q^{7} +29.2258 q^{8} +36.8519 q^{9} +77.8440 q^{10} +15.9710 q^{11} -113.488 q^{12} -73.2593 q^{13} -109.129 q^{14} -132.011 q^{15} +24.0908 q^{16} -77.7780 q^{17} +173.644 q^{18} +19.9503 q^{19} +234.633 q^{20} +185.066 q^{21} +75.2547 q^{22} -23.0000 q^{23} -233.536 q^{24} +147.929 q^{25} -345.194 q^{26} -78.7241 q^{27} -328.931 q^{28} +29.0000 q^{29} -622.032 q^{30} +38.8164 q^{31} -120.292 q^{32} -127.620 q^{33} -366.486 q^{34} -382.618 q^{35} +523.389 q^{36} -254.170 q^{37} +94.0050 q^{38} +585.396 q^{39} +482.827 q^{40} -169.198 q^{41} +872.024 q^{42} +130.529 q^{43} +226.828 q^{44} +608.814 q^{45} -108.375 q^{46} -208.487 q^{47} -192.503 q^{48} +193.390 q^{49} +697.033 q^{50} +621.503 q^{51} -1040.46 q^{52} -614.465 q^{53} -370.944 q^{54} +263.850 q^{55} -676.873 q^{56} -159.418 q^{57} +136.647 q^{58} +360.086 q^{59} -1874.89 q^{60} -589.205 q^{61} +182.901 q^{62} -853.494 q^{63} -759.536 q^{64} -1210.28 q^{65} -601.341 q^{66} +475.391 q^{67} -1104.64 q^{68} +183.787 q^{69} -1802.88 q^{70} -459.883 q^{71} +1077.03 q^{72} -869.782 q^{73} -1197.64 q^{74} -1182.06 q^{75} +283.344 q^{76} -369.891 q^{77} +2758.36 q^{78} +1117.33 q^{79} +397.994 q^{80} -365.938 q^{81} -797.251 q^{82} -159.756 q^{83} +2628.40 q^{84} -1284.94 q^{85} +615.046 q^{86} -231.731 q^{87} +466.767 q^{88} +817.409 q^{89} +2868.70 q^{90} +1696.69 q^{91} -326.657 q^{92} -310.172 q^{93} -982.378 q^{94} +329.590 q^{95} +961.221 q^{96} -1346.87 q^{97} +911.245 q^{98} +588.563 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 8 q^{2} - 16 q^{3} + 152 q^{4} - 80 q^{5} - 16 q^{6} - 38 q^{7} - 138 q^{8} + 312 q^{9}+O(q^{10}) $ 38 * q - 8 * q^2 - 16 * q^3 + 152 * q^4 - 80 * q^5 - 16 * q^6 - 38 * q^7 - 138 * q^8 + 312 * q^9	\( 38 q - 8 q^{2} - 16 q^{3} + 152 q^{4} - 80 q^{5} - 16 q^{6} - 38 q^{7} - 138 q^{8} + 312 q^{9} + 30 q^{10} - 100 q^{11} - 173 q^{12} - 160 q^{13} - 304 q^{14} - 208 q^{15} + 552 q^{16} - 666 q^{17} - 315 q^{18} - 252 q^{19} - 747 q^{20} - 470 q^{21} - 323 q^{22} - 874 q^{23} - 114 q^{24} + 910 q^{25} - 119 q^{26} - 526 q^{27} - 619 q^{28} + 1102 q^{29} - 533 q^{30} + 358 q^{31} - 1272 q^{32} - 604 q^{33} - 553 q^{34} + 16 q^{35} + 1230 q^{36} - 1206 q^{37} - 2010 q^{38} - 240 q^{39} - 418 q^{40} - 1094 q^{41} - 1184 q^{42} - 936 q^{43} - 2153 q^{44} - 3654 q^{45} + 184 q^{46} - 1702 q^{47} - 1319 q^{48} + 1920 q^{49} - 1255 q^{50} - 270 q^{51} - 2202 q^{52} - 4640 q^{53} - 1529 q^{54} - 318 q^{55} - 2435 q^{56} - 1980 q^{57} - 232 q^{58} + 318 q^{59} - 3279 q^{60} - 2212 q^{61} - 541 q^{62} - 1044 q^{63} + 1186 q^{64} - 2438 q^{65} - 3503 q^{66} - 496 q^{67} - 6099 q^{68} + 368 q^{69} + 4068 q^{70} + 870 q^{71} - 3869 q^{72} - 2810 q^{73} - 2793 q^{74} - 3638 q^{75} + 1548 q^{76} - 6072 q^{77} + 2170 q^{78} - 3084 q^{79} - 6702 q^{80} + 362 q^{81} - 1183 q^{82} - 5566 q^{83} - 6518 q^{84} - 120 q^{85} - 2095 q^{86} - 464 q^{87} - 1220 q^{88} - 6506 q^{89} + 165 q^{90} + 44 q^{91} - 3496 q^{92} - 5928 q^{93} + 135 q^{94} - 2024 q^{95} - 3164 q^{96} - 7472 q^{97} - 6422 q^{98} - 944 q^{99}+O(q^{100}) \) 38 * q - 8 * q^2 - 16 * q^3 + 152 * q^4 - 80 * q^5 - 16 * q^6 - 38 * q^7 - 138 * q^8 + 312 * q^9 + 30 * q^10 - 100 * q^11 - 173 * q^12 - 160 * q^13 - 304 * q^14 - 208 * q^15 + 552 * q^16 - 666 * q^17 - 315 * q^18 - 252 * q^19 - 747 * q^20 - 470 * q^21 - 323 * q^22 - 874 * q^23 - 114 * q^24 + 910 * q^25 - 119 * q^26 - 526 * q^27 - 619 * q^28 + 1102 * q^29 - 533 * q^30 + 358 * q^31 - 1272 * q^32 - 604 * q^33 - 553 * q^34 + 16 * q^35 + 1230 * q^36 - 1206 * q^37 - 2010 * q^38 - 240 * q^39 - 418 * q^40 - 1094 * q^41 - 1184 * q^42 - 936 * q^43 - 2153 * q^44 - 3654 * q^45 + 184 * q^46 - 1702 * q^47 - 1319 * q^48 + 1920 * q^49 - 1255 * q^50 - 270 * q^51 - 2202 * q^52 - 4640 * q^53 - 1529 * q^54 - 318 * q^55 - 2435 * q^56 - 1980 * q^57 - 232 * q^58 + 318 * q^59 - 3279 * q^60 - 2212 * q^61 - 541 * q^62 - 1044 * q^63 + 1186 * q^64 - 2438 * q^65 - 3503 * q^66 - 496 * q^67 - 6099 * q^68 + 368 * q^69 + 4068 * q^70 + 870 * q^71 - 3869 * q^72 - 2810 * q^73 - 2793 * q^74 - 3638 * q^75 + 1548 * q^76 - 6072 * q^77 + 2170 * q^78 - 3084 * q^79 - 6702 * q^80 + 362 * q^81 - 1183 * q^82 - 5566 * q^83 - 6518 * q^84 - 120 * q^85 - 2095 * q^86 - 464 * q^87 - 1220 * q^88 - 6506 * q^89 + 165 * q^90 + 44 * q^91 - 3496 * q^92 - 5928 * q^93 + 135 * q^94 - 2024 * q^95 - 3164 * q^96 - 7472 * q^97 - 6422 * q^98 - 944 * q^99

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 $

$2$	4.71195	1.66593	0.832963	−	0.553328i	$-0.186642\pi$
$2$	4.71195	1.66593	0.832963	+	0.553328i	$0.186642\pi$
$3$	−7.99074	−1.53782	−0.768909	−	0.639358i	$-0.779200\pi$
$3$	−7.99074	−1.53782	−0.768909	+	0.639358i	$0.779200\pi$
$4$	14.2025	1.77531
$5$	16.5206	1.47764	0.738822	−	0.673901i	$-0.235383\pi$
$5$	16.5206	1.47764	0.738822	+	0.673901i	$0.235383\pi$
$6$	−37.6520	−2.56189
$7$	−23.1601	−1.25053	−0.625264	−	0.780413i	$-0.715009\pi$
$7$	−23.1601	−1.25053	−0.625264	+	0.780413i	$0.715009\pi$
$8$	29.2258	1.29161
$9$	36.8519	1.36489
$10$	77.8440	2.46164
$11$	15.9710	0.437768	0.218884	−	0.975751i	$-0.429758\pi$
$11$	15.9710	0.437768	0.218884	+	0.975751i	$0.429758\pi$
$12$	−113.488	−2.73011
$13$	−73.2593	−1.56296	−0.781479	−	0.623931i	$-0.785535\pi$
$13$	−73.2593	−1.56296	−0.781479	+	0.623931i	$0.785535\pi$
$14$	−109.129	−2.08329
$15$	−132.011	−2.27235
$16$	24.0908	0.376419
$17$	−77.7780	−1.10964	−0.554821	−	0.831969i	$-0.687213\pi$
$17$	−77.7780	−1.10964	−0.554821	+	0.831969i	$0.687213\pi$
$18$	173.644	2.27380
$19$	19.9503	0.240891	0.120445	−	0.992720i	$-0.461568\pi$
$19$	19.9503	0.240891	0.120445	+	0.992720i	$0.461568\pi$
$20$	234.633	2.62328
$21$	185.066	1.92308
$22$	75.2547	0.729289
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	−233.536	−1.98626
$25$	147.929	1.18343
$26$	−345.194	−2.60377
$27$	−78.7241	−0.561128
$28$	−328.931	−2.22008
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	−622.032	−3.78556
$31$	38.8164	0.224891	0.112446	−	0.993658i	$-0.464132\pi$
$31$	38.8164	0.224891	0.112446	+	0.993658i	$0.464132\pi$
$32$	−120.292	−0.664525
$33$	−127.620	−0.673207
$34$	−366.486	−1.84858
$35$	−382.618	−1.84783
$36$	523.389	2.42310
$37$	−254.170	−1.12933	−0.564665	−	0.825320i	$-0.690995\pi$
$37$	−254.170	−1.12933	−0.564665	+	0.825320i	$0.690995\pi$
$38$	94.0050	0.401306
$39$	585.396	2.40355
$40$	482.827	1.90854
$41$	−169.198	−0.644493	−0.322247	−	0.946656i	$-0.604438\pi$
$41$	−169.198	−0.644493	−0.322247	+	0.946656i	$0.604438\pi$
$42$	872.024	3.20372
$43$	130.529	0.462918	0.231459	−	0.972845i	$-0.425650\pi$
$43$	130.529	0.462918	0.231459	+	0.972845i	$0.425650\pi$
$44$	226.828	0.777174
$45$	608.814	2.01681
$46$	−108.375	−0.347370
$47$	−208.487	−0.647040	−0.323520	−	0.946221i	$-0.604866\pi$
$47$	−208.487	−0.647040	−0.323520	+	0.946221i	$0.604866\pi$
$48$	−192.503	−0.578864
$49$	193.390	0.563820
$50$	697.033	1.97151
$51$	621.503	1.70643
$52$	−1040.46	−2.77474
$53$	−614.465	−1.59251	−0.796257	−	0.604958i	$-0.793190\pi$
$53$	−614.465	−1.59251	−0.796257	+	0.604958i	$0.793190\pi$
$54$	−370.944	−0.934798
$55$	263.850	0.646865
$56$	−676.873	−1.61520
$57$	−159.418	−0.370446
$58$	136.647	0.309355
$59$	360.086	0.794564	0.397282	−	0.917697i	$-0.369954\pi$
$59$	360.086	0.794564	0.397282	+	0.917697i	$0.369954\pi$
$60$	−1874.89	−4.03412
$61$	−589.205	−1.23672	−0.618360	−	0.785895i	$-0.712203\pi$
$61$	−589.205	−1.23672	−0.618360	+	0.785895i	$0.712203\pi$
$62$	182.901	0.374652
$63$	−853.494	−1.70683
$64$	−759.536	−1.48347
$65$	−1210.28	−2.30949
$66$	−601.341	−1.12151
$67$	475.391	0.866840	0.433420	−	0.901192i	$-0.357307\pi$
$67$	475.391	0.866840	0.433420	+	0.901192i	$0.357307\pi$
$68$	−1104.64	−1.96996
$69$	183.787	0.320657
$70$	−1802.88	−3.07836
$71$	−459.883	−0.768705	−0.384352	−	0.923186i	$-0.625575\pi$
$71$	−459.883	−0.768705	−0.384352	+	0.923186i	$0.625575\pi$
$72$	1077.03	1.76290
$73$	−869.782	−1.39453	−0.697263	−	0.716816i	$-0.745599\pi$
$73$	−869.782	−1.39453	−0.697263	+	0.716816i	$0.745599\pi$
$74$	−1197.64	−1.88138
$75$	−1182.06	−1.81990
$76$	283.344	0.427656
$77$	−369.891	−0.547441
$78$	2758.36	4.00413
$79$	1117.33	1.59125	0.795627	−	0.605787i	$-0.207141\pi$
$79$	1117.33	1.59125	0.795627	+	0.605787i	$0.207141\pi$
$80$	397.994	0.556213
$81$	−365.938	−0.501973
$82$	−797.251	−1.07368
$83$	−159.756	−0.211271	−0.105635	−	0.994405i	$-0.533688\pi$
$83$	−159.756	−0.211271	−0.105635	+	0.994405i	$0.533688\pi$
$84$	2628.40	3.41407
$85$	−1284.94	−1.63966
$86$	615.046	0.771187
$87$	−231.731	−0.285566
$88$	466.767	0.565426
$89$	817.409	0.973541	0.486771	−	0.873530i	$-0.338175\pi$
$89$	817.409	0.973541	0.486771	+	0.873530i	$0.338175\pi$
$90$	2868.70	3.35986
$91$	1696.69	1.95452
$92$	−326.657	−0.370178
$93$	−310.172	−0.345842
$94$	−982.378	−1.07792
$95$	329.590	0.355950
$96$	961.221	1.02192
$97$	−1346.87	−1.40983	−0.704917	−	0.709290i	$-0.749016\pi$
$97$	−1346.87	−1.40983	−0.704917	+	0.709290i	$0.749016\pi$
$98$	911.245	0.939283
$99$	588.563	0.597503
$100$	2100.96	2.10096
$101$	−280.785	−0.276625	−0.138313	−	0.990389i	$-0.544168\pi$
$101$	−280.785	−0.276625	−0.138313	+	0.990389i	$0.544168\pi$
$102$	2928.49	2.84279
$103$	−620.840	−0.593915	−0.296957	−	0.954891i	$-0.595972\pi$
$103$	−620.840	−0.593915	−0.296957	+	0.954891i	$0.595972\pi$
$104$	−2141.06	−2.01874
$105$	3057.40	2.84163
$106$	−2895.33	−2.65301
$107$	−119.980	−0.108401	−0.0542006	−	0.998530i	$-0.517261\pi$
$107$	−119.980	−0.108401	−0.0542006	+	0.998530i	$0.517261\pi$
$108$	−1118.08	−0.996177
$109$	329.717	0.289736	0.144868	−	0.989451i	$-0.453724\pi$
$109$	329.717	0.289736	0.144868	+	0.989451i	$0.453724\pi$
$110$	1243.25	1.07763
$111$	2031.00	1.73671
$112$	−557.946	−0.470722
$113$	2228.59	1.85530	0.927648	−	0.373456i	$-0.121827\pi$
$113$	2228.59	1.85530	0.927648	+	0.373456i	$0.121827\pi$
$114$	−751.169	−0.617136
$115$	−379.973	−0.308110
$116$	411.872	0.329667
$117$	−2699.74	−2.13326
$118$	1696.71	1.32369
$119$	1801.35	1.38764
$120$	−3858.14	−2.93499
$121$	−1075.93	−0.808359
$122$	−2776.30	−2.06028
$123$	1352.01	0.991113
$124$	551.290	0.399252
$125$	378.794	0.271043
$126$	−4021.62	−2.84345
$127$	2261.32	1.57999	0.789997	−	0.613110i	$-0.210082\pi$
$127$	2261.32	1.57999	0.789997	+	0.613110i	$0.210082\pi$
$128$	−2616.56	−1.80683
$129$	−1043.02	−0.711884
$130$	−5702.80	−3.84745
$131$	89.3192	0.0595714	0.0297857	−	0.999556i	$-0.490518\pi$
$131$	89.3192	0.0595714	0.0297857	+	0.999556i	$0.490518\pi$
$132$	−1812.53	−1.19515
$133$	−462.052	−0.301240
$134$	2240.02	1.44409
$135$	−1300.57	−0.829147
$136$	−2273.13	−1.43323
$137$	−391.947	−0.244425	−0.122213	−	0.992504i	$-0.538999\pi$
$137$	−391.947	−0.244425	−0.122213	+	0.992504i	$0.538999\pi$
$138$	865.996	0.534192
$139$	1155.51	0.705102	0.352551	−	0.935793i	$-0.385314\pi$
$139$	1155.51	0.705102	0.352551	+	0.935793i	$0.385314\pi$
$140$	−5434.12	−3.28048
$141$	1665.96	0.995030
$142$	−2166.95	−1.28061
$143$	−1170.03	−0.684213
$144$	887.793	0.513769
$145$	479.096	0.274391
$146$	−4098.37	−2.32318
$147$	−1545.33	−0.867053
$148$	−3609.84	−2.00491
$149$	2523.41	1.38742	0.693711	−	0.720253i	$-0.255974\pi$
$149$	2523.41	1.38742	0.693711	+	0.720253i	$0.255974\pi$
$150$	−5569.81	−3.03182
$151$	−2310.13	−1.24500	−0.622502	−	0.782618i	$-0.713884\pi$
$151$	−2310.13	−1.24500	−0.622502	+	0.782618i	$0.713884\pi$
$152$	583.065	0.311137
$153$	−2866.27	−1.51454
$154$	−1742.91	−0.911996
$155$	641.268	0.332309
$156$	8314.08	4.26704
$157$	1346.00	0.684220	0.342110	−	0.939660i	$-0.388859\pi$
$157$	1346.00	0.684220	0.342110	+	0.939660i	$0.388859\pi$
$158$	5264.79	2.65091
$159$	4910.03	2.44900
$160$	−1987.29	−0.981931
$161$	532.682	0.260753
$162$	−1724.28	−0.836250
$163$	2085.63	1.00220	0.501101	−	0.865389i	$-0.332929\pi$
$163$	2085.63	1.00220	0.501101	+	0.865389i	$0.332929\pi$
$164$	−2403.03	−1.14418
$165$	−2108.36	−0.994760
$166$	−752.761	−0.351961
$167$	−763.206	−0.353644	−0.176822	−	0.984243i	$-0.556582\pi$
$167$	−763.206	−0.353644	−0.176822	+	0.984243i	$0.556582\pi$
$168$	5408.72	2.48388
$169$	3169.92	1.44284
$170$	−6054.55	−2.73155
$171$	735.208	0.328788
$172$	1853.84	0.821824
$173$	3580.98	1.57374	0.786870	−	0.617119i	$-0.211700\pi$
$173$	3580.98	1.57374	0.786870	+	0.617119i	$0.211700\pi$
$174$	−1091.91	−0.475732
$175$	−3426.04	−1.47991
$176$	384.755	0.164784
$177$	−2877.36	−1.22189
$178$	3851.59	1.62185
$179$	145.375	0.0607029	0.0303515	−	0.999539i	$-0.490337\pi$
$179$	145.375	0.0607029	0.0303515	+	0.999539i	$0.490337\pi$
$180$	8646.68	3.58047
$181$	1228.72	0.504585	0.252292	−	0.967651i	$-0.418816\pi$
$181$	1228.72	0.504585	0.252292	+	0.967651i	$0.418816\pi$
$182$	7994.73	3.25609
$183$	4708.18	1.90185
$184$	−672.194	−0.269320
$185$	−4199.02	−1.66875
$186$	−1461.51	−0.576148
$187$	−1242.19	−0.485766
$188$	−2961.03	−1.14870
$189$	1823.26	0.701706
$190$	1553.01	0.592987
$191$	3461.16	1.31121	0.655604	−	0.755105i	$-0.272414\pi$
$191$	3461.16	1.31121	0.655604	+	0.755105i	$0.272414\pi$
$192$	6069.26	2.28131
$193$	−3295.10	−1.22895	−0.614473	−	0.788938i	$-0.710631\pi$
$193$	−3295.10	−1.22895	−0.614473	+	0.788938i	$0.710631\pi$
$194$	−6346.39	−2.34868
$195$	9671.06	3.55158
$196$	2746.62	1.00096
$197$	4120.56	1.49024	0.745121	−	0.666929i	$-0.232392\pi$
$197$	4120.56	1.49024	0.745121	+	0.666929i	$0.232392\pi$
$198$	2773.28	0.995396
$199$	−2342.28	−0.834370	−0.417185	−	0.908822i	$-0.636983\pi$
$199$	−2342.28	−0.834370	−0.417185	+	0.908822i	$0.636983\pi$
$200$	4323.34	1.52853
$201$	−3798.73	−1.33304
$202$	−1323.05	−0.460837
$203$	−671.643	−0.232217
$204$	8826.90	3.02944
$205$	−2795.24	−0.952331
$206$	−2925.37	−0.989418
$207$	−847.594	−0.284598
$208$	−1764.88	−0.588327
$209$	318.627	0.105454
$210$	14406.3	4.73395
$211$	−5079.90	−1.65742	−0.828708	−	0.559682i	$-0.810923\pi$
$211$	−5079.90	−1.65742	−0.828708	+	0.559682i	$0.810923\pi$
$212$	−8726.94	−2.82721
$213$	3674.80	1.18213
$214$	−565.341	−0.180589
$215$	2156.41	0.684028
$216$	−2300.78	−0.724760
$217$	−898.992	−0.281233
$218$	1553.61	0.482678
$219$	6950.20	2.14453
$220$	3747.33	1.14839
$221$	5697.96	1.73433
$222$	9569.99	2.89322
$223$	−3591.50	−1.07850	−0.539249	−	0.842147i	$-0.681292\pi$
$223$	−3591.50	−1.07850	−0.539249	+	0.842147i	$0.681292\pi$
$224$	2785.97	0.831007
$225$	5451.45	1.61525
$226$	10501.0	3.09079
$227$	−2793.63	−0.816827	−0.408413	−	0.912797i	$-0.633918\pi$
$227$	−2793.63	−0.816827	−0.408413	+	0.912797i	$0.633918\pi$
$228$	−2264.13	−0.657657
$229$	−1979.63	−0.571256	−0.285628	−	0.958341i	$-0.592202\pi$
$229$	−1979.63	−0.571256	−0.285628	+	0.958341i	$0.592202\pi$
$230$	−1790.41	−0.513288
$231$	2955.70	0.841865
$232$	847.549	0.239846
$233$	4512.89	1.26888	0.634440	−	0.772972i	$-0.281231\pi$
$233$	4512.89	1.26888	0.634440	+	0.772972i	$0.281231\pi$
$234$	−12721.1	−3.55385
$235$	−3444.31	−0.956094
$236$	5114.12	1.41060
$237$	−8928.27	−2.44706
$238$	8487.85	2.31171
$239$	238.046	0.0644265	0.0322132	−	0.999481i	$-0.489744\pi$
$239$	238.046	0.0644265	0.0322132	+	0.999481i	$0.489744\pi$
$240$	−3180.26	−0.855355
$241$	945.994	0.252850	0.126425	−	0.991976i	$-0.459650\pi$
$241$	945.994	0.252850	0.126425	+	0.991976i	$0.459650\pi$
$242$	−5069.71	−1.34667
$243$	5049.67	1.33307
$244$	−8368.17	−2.19556
$245$	3194.91	0.833125
$246$	6370.62	1.65112
$247$	−1461.55	−0.376502
$248$	1134.44	0.290472
$249$	1276.57	0.324896
$250$	1784.86	0.451538
$251$	4042.83	1.01666	0.508329	−	0.861163i	$-0.330264\pi$
$251$	4042.83	1.01666	0.508329	+	0.861163i	$0.330264\pi$
$252$	−12121.7	−3.03015
$253$	−367.334	−0.0912809
$254$	10655.2	2.63216
$255$	10267.6	2.52149
$256$	−6252.83	−1.52657
$257$	3575.71	0.867887	0.433943	−	0.900940i	$-0.357122\pi$
$257$	3575.71	0.867887	0.433943	+	0.900940i	$0.357122\pi$
$258$	−4914.67	−1.18595
$259$	5886.60	1.41226
$260$	−17189.0	−4.10007
$261$	1068.71	0.253453
$262$	420.868	0.0992416
$263$	−7211.14	−1.69071	−0.845357	−	0.534202i	$-0.820612\pi$
$263$	−7211.14	−1.69071	−0.845357	+	0.534202i	$0.820612\pi$
$264$	−3729.81	−0.869523
$265$	−10151.3	−2.35317
$266$	−2177.17	−0.501844
$267$	−6531.70	−1.49713
$268$	6751.74	1.53891
$269$	−5775.81	−1.30914	−0.654568	−	0.756003i	$-0.727149\pi$
$269$	−5775.81	−1.30914	−0.654568	+	0.756003i	$0.727149\pi$
$270$	−6128.20	−1.38130
$271$	6709.45	1.50395	0.751975	−	0.659192i	$-0.229102\pi$
$271$	6709.45	1.50395	0.751975	+	0.659192i	$0.229102\pi$
$272$	−1873.73	−0.417691
$273$	−13557.8	−3.00570
$274$	−1846.83	−0.407195
$275$	2362.57	0.518067
$276$	2610.23	0.569267
$277$	−3613.19	−0.783739	−0.391870	−	0.920021i	$-0.628172\pi$
$277$	−3613.19	−0.783739	−0.391870	+	0.920021i	$0.628172\pi$
$278$	5444.72	1.17465
$279$	1430.46	0.306951
$280$	−11182.3	−2.38668
$281$	8437.66	1.79128	0.895639	−	0.444783i	$-0.146719\pi$
$281$	8437.66	1.79128	0.895639	+	0.444783i	$0.146719\pi$
$282$	7849.93	1.65765
$283$	−502.740	−0.105600	−0.0528000	−	0.998605i	$-0.516815\pi$
$283$	−502.740	−0.105600	−0.0528000	+	0.998605i	$0.516815\pi$
$284$	−6531.48	−1.36469
$285$	−2633.67	−0.547387
$286$	−5513.10	−1.13985
$287$	3918.63	0.805957
$288$	−4432.99	−0.907001
$289$	1136.41	0.231307
$290$	2257.48	0.457116
$291$	10762.5	2.16807
$292$	−12353.1	−2.47572
$293$	−1968.39	−0.392473	−0.196236	−	0.980557i	$-0.562872\pi$
$293$	−1968.39	−0.392473	−0.196236	+	0.980557i	$0.562872\pi$
$294$	−7281.53	−1.44445
$295$	5948.83	1.17408
$296$	−7428.32	−1.45866
$297$	−1257.30	−0.245644
$298$	11890.2	2.31134
$299$	1684.96	0.325899
$300$	−16788.2	−3.23089
$301$	−3023.06	−0.578892
$302$	−10885.2	−2.07409
$303$	2243.68	0.425400
$304$	480.620	0.0906758
$305$	−9733.99	−1.82743
$306$	−13505.7	−2.52311
$307$	1415.84	0.263212	0.131606	−	0.991302i	$-0.457987\pi$
$307$	1415.84	0.263212	0.131606	+	0.991302i	$0.457987\pi$
$308$	−5253.37	−0.971878
$309$	4960.97	0.913333
$310$	3021.63	0.553603
$311$	−8043.73	−1.46662	−0.733309	−	0.679896i	$-0.762025\pi$
$311$	−8043.73	−1.46662	−0.733309	+	0.679896i	$0.762025\pi$
$312$	17108.7	3.10445
$313$	−3059.84	−0.552563	−0.276282	−	0.961077i	$-0.589102\pi$
$313$	−3059.84	−0.552563	−0.276282	+	0.961077i	$0.589102\pi$
$314$	6342.29	1.13986
$315$	−14100.2	−2.52208
$316$	15868.8	2.82497
$317$	−4012.33	−0.710899	−0.355449	−	0.934696i	$-0.615672\pi$
$317$	−4012.33	−0.710899	−0.355449	+	0.934696i	$0.615672\pi$
$318$	23135.8	4.07985
$319$	463.160	0.0812914
$320$	−12548.0	−2.19204
$321$	958.731	0.166701
$322$	2509.97	0.434395
$323$	−1551.70	−0.267302
$324$	−5197.23	−0.891158
$325$	−10837.1	−1.84965
$326$	9827.37	1.66959
$327$	−2634.68	−0.445561
$328$	−4944.94	−0.832435
$329$	4828.57	0.809142
$330$	−9934.48	−1.65720
$331$	−3327.75	−0.552597	−0.276299	−	0.961072i	$-0.589108\pi$
$331$	−3327.75	−0.552597	−0.276299	+	0.961072i	$0.589108\pi$
$332$	−2268.93	−0.375071
$333$	−9366.64	−1.54141
$334$	−3596.19	−0.589146
$335$	7853.72	1.28088
$336$	4458.40	0.723886
$337$	5314.40	0.859032	0.429516	−	0.903059i	$-0.358684\pi$
$337$	5314.40	0.859032	0.429516	+	0.903059i	$0.358684\pi$
$338$	14936.5	2.40366
$339$	−17808.1	−2.85311
$340$	−18249.3	−2.91090
$341$	619.938	0.0984502
$342$	3464.26	0.547737
$343$	3464.98	0.545455
$344$	3814.82	0.597910
$345$	3036.26	0.473817
$346$	16873.4	2.62173
$347$	2011.50	0.311190	0.155595	−	0.987821i	$-0.450270\pi$
$347$	2011.50	0.311190	0.155595	+	0.987821i	$0.450270\pi$
$348$	−3291.16	−0.506968
$349$	5959.49	0.914052	0.457026	−	0.889453i	$-0.348915\pi$
$349$	5959.49	0.914052	0.457026	+	0.889453i	$0.348915\pi$
$350$	−16143.3	−2.46542
$351$	5767.27	0.877020
$352$	−1921.19	−0.290908
$353$	5532.37	0.834159	0.417080	−	0.908870i	$-0.363054\pi$
$353$	5532.37	0.834159	0.417080	+	0.908870i	$0.363054\pi$
$354$	−13558.0	−2.03559
$355$	−7597.52	−1.13587
$356$	11609.2	1.72834
$357$	−14394.1	−2.13394
$358$	684.999	0.101127
$359$	−12063.0	−1.77342	−0.886711	−	0.462324i	$-0.847016\pi$
$359$	−12063.0	−1.77342	−0.886711	+	0.462324i	$0.847016\pi$
$360$	17793.1	2.60494
$361$	−6460.98	−0.941972
$362$	5789.66	0.840601
$363$	8597.45	1.24311
$364$	24097.2	3.46989
$365$	−14369.3	−2.06061
$366$	22184.7	3.16834
$367$	−411.382	−0.0585122	−0.0292561	−	0.999572i	$-0.509314\pi$
$367$	−411.382	−0.0585122	−0.0292561	+	0.999572i	$0.509314\pi$
$368$	−554.089	−0.0784888
$369$	−6235.25	−0.879660
$370$	−19785.6	−2.78001
$371$	14231.1	1.99148
$372$	−4405.21	−0.613977
$373$	9037.17	1.25450	0.627248	−	0.778820i	$-0.284181\pi$
$373$	9037.17	1.25450	0.627248	+	0.778820i	$0.284181\pi$
$374$	−5853.16	−0.809250
$375$	−3026.84	−0.416815
$376$	−6093.19	−0.835725
$377$	−2124.52	−0.290234
$378$	8591.10	1.16899
$379$	7861.39	1.06547	0.532734	−	0.846283i	$-0.321164\pi$
$379$	7861.39	1.06547	0.532734	+	0.846283i	$0.321164\pi$
$380$	4681.01	0.631922
$381$	−18069.6	−2.42975
$382$	16308.8	2.18438
$383$	−13701.7	−1.82800	−0.914002	−	0.405709i	$-0.867025\pi$
$383$	−13701.7	−1.82800	−0.914002	+	0.405709i	$0.867025\pi$
$384$	20908.3	2.77857
$385$	−6110.80	−0.808922
$386$	−15526.4	−2.04733
$387$	4810.24	0.631830
$388$	−19128.9	−2.50289
$389$	−11047.2	−1.43989	−0.719944	−	0.694032i	$-0.755833\pi$
$389$	−11047.2	−1.43989	−0.719944	+	0.694032i	$0.755833\pi$
$390$	45569.6	5.91668
$391$	1788.89	0.231377
$392$	5651.99	0.728236
$393$	−713.727	−0.0916100
$394$	19415.9	2.48263
$395$	18458.9	2.35131
$396$	8359.06	1.06075
$397$	−8265.00	−1.04486	−0.522429	−	0.852683i	$-0.674974\pi$
$397$	−8265.00	−1.04486	−0.522429	+	0.852683i	$0.674974\pi$
$398$	−11036.7	−1.39000
$399$	3692.13	0.463253
$400$	3563.72	0.445465
$401$	7477.09	0.931142	0.465571	−	0.885010i	$-0.345849\pi$
$401$	7477.09	0.931142	0.465571	+	0.885010i	$0.345849\pi$
$402$	−17899.4	−2.22075
$403$	−2843.66	−0.351496
$404$	−3987.85	−0.491096
$405$	−6045.50	−0.741736
$406$	−3164.75	−0.386857
$407$	−4059.35	−0.494385
$408$	18164.0	2.20404
$409$	−6621.39	−0.800506	−0.400253	−	0.916405i	$-0.631078\pi$
$409$	−6621.39	−0.800506	−0.400253	+	0.916405i	$0.631078\pi$
$410$	−13171.0	−1.58651
$411$	3131.94	0.375882
$412$	−8817.48	−1.05438
$413$	−8339.64	−0.993624
$414$	−3993.82	−0.474120
$415$	−2639.25	−0.312182
$416$	8812.50	1.03863
$417$	−9233.39	−1.08432
$418$	1501.36	0.175679
$419$	11200.6	1.30593	0.652963	−	0.757389i	$-0.273526\pi$
$419$	11200.6	1.30593	0.652963	+	0.757389i	$0.273526\pi$
$420$	43422.7	5.04478
$421$	−5729.06	−0.663224	−0.331612	−	0.943416i	$-0.607593\pi$
$421$	−5729.06	−0.663224	−0.331612	+	0.943416i	$0.607593\pi$
$422$	−23936.2	−2.76113
$423$	−7683.13	−0.883136
$424$	−17958.3	−2.05691
$425$	−11505.6	−1.31318
$426$	17315.5	1.96934
$427$	13646.0	1.54655
$428$	−1704.02	−0.192446
$429$	9349.37	1.05220
$430$	10160.9	1.13954
$431$	16670.6	1.86310	0.931548	−	0.363619i	$-0.118459\pi$
$431$	16670.6	1.86310	0.931548	+	0.363619i	$0.118459\pi$
$432$	−1896.53	−0.211219
$433$	−8775.67	−0.973976	−0.486988	−	0.873409i	$-0.661905\pi$
$433$	−8775.67	−0.973976	−0.486988	+	0.873409i	$0.661905\pi$
$434$	−4236.01	−0.468513
$435$	−3828.33	−0.421964
$436$	4682.81	0.514371
$437$	−458.858	−0.0502291
$438$	32749.0	3.57262
$439$	−14378.0	−1.56316	−0.781580	−	0.623806i	$-0.785586\pi$
$439$	−14378.0	−1.56316	−0.781580	+	0.623806i	$0.785586\pi$
$440$	7711.24	0.835498
$441$	7126.80	0.769550
$442$	26848.5	2.88926
$443$	1930.30	0.207023	0.103512	−	0.994628i	$-0.466992\pi$
$443$	1930.30	0.207023	0.103512	+	0.994628i	$0.466992\pi$
$444$	28845.3	3.08319
$445$	13504.0	1.43855
$446$	−16923.0	−1.79670
$447$	−20163.9	−2.13360
$448$	17590.9	1.85512
$449$	−10784.8	−1.13355	−0.566776	−	0.823872i	$-0.691809\pi$
$449$	−10784.8	−1.13355	−0.566776	+	0.823872i	$0.691809\pi$
$450$	25687.0	2.69088
$451$	−2702.26	−0.282138
$452$	31651.6	3.29373
$453$	18459.6	1.91459
$454$	−13163.4	−1.36077
$455$	28030.3	2.88809
$456$	−4659.12	−0.478472
$457$	−5612.69	−0.574509	−0.287255	−	0.957854i	$-0.592743\pi$
$457$	−5612.69	−0.574509	−0.287255	+	0.957854i	$0.592743\pi$
$458$	−9327.92	−0.951671
$459$	6123.00	0.622652
$460$	−5396.56	−0.546991
$461$	14070.4	1.42153	0.710765	−	0.703430i	$-0.248349\pi$
$461$	14070.4	1.42153	0.710765	+	0.703430i	$0.248349\pi$
$462$	13927.1	1.40248
$463$	−18184.8	−1.82531	−0.912656	−	0.408728i	$-0.865972\pi$
$463$	−18184.8	−1.82531	−0.912656	+	0.408728i	$0.865972\pi$
$464$	698.634	0.0698993
$465$	−5124.21	−0.511031
$466$	21264.5	2.11386
$467$	−1122.60	−0.111237	−0.0556186	−	0.998452i	$-0.517713\pi$
$467$	−1122.60	−0.111237	−0.0556186	+	0.998452i	$0.517713\pi$
$468$	−38343.1	−3.78720
$469$	−11010.1	−1.08401
$470$	−16229.4	−1.59278
$471$	−10755.5	−1.05221
$472$	10523.8	1.02627
$473$	2084.68	0.202651
$474$	−42069.6	−4.07662
$475$	2951.23	0.285077
$476$	25583.6	2.46349
$477$	−22644.2	−2.17360
$478$	1121.66	0.107330
$479$	1340.10	0.127831	0.0639153	−	0.997955i	$-0.479641\pi$
$479$	1340.10	0.127831	0.0639153	+	0.997955i	$0.479641\pi$
$480$	15879.9	1.51003
$481$	18620.3	1.76510
$482$	4457.48	0.421229
$483$	−4256.53	−0.400991
$484$	−15280.8	−1.43509
$485$	−22251.0	−2.08323
$486$	23793.8	2.22080
$487$	−2983.91	−0.277646	−0.138823	−	0.990317i	$-0.544332\pi$
$487$	−2983.91	−0.277646	−0.138823	+	0.990317i	$0.544332\pi$
$488$	−17220.0	−1.59736
$489$	−16665.7	−1.54120
$490$	15054.3	1.38792
$491$	15734.9	1.44625	0.723123	−	0.690719i	$-0.242706\pi$
$491$	15734.9	1.44625	0.723123	+	0.690719i	$0.242706\pi$
$492$	19202.0	1.75954
$493$	−2255.56	−0.206056
$494$	−6886.74	−0.627225
$495$	9723.39	0.882896
$496$	935.119	0.0846534
$497$	10650.9	0.961287
$498$	6015.11	0.541252
$499$	−3470.08	−0.311307	−0.155653	−	0.987812i	$-0.549748\pi$
$499$	−3470.08	−0.311307	−0.155653	+	0.987812i	$0.549748\pi$
$500$	5379.82	0.481186
$501$	6098.58	0.543841
$502$	19049.6	1.69368
$503$	−2675.94	−0.237205	−0.118603	−	0.992942i	$-0.537841\pi$
$503$	−2675.94	−0.237205	−0.118603	+	0.992942i	$0.537841\pi$
$504$	−24944.1	−2.20456
$505$	−4638.72	−0.408753
$506$	−1730.86	−0.152067
$507$	−25330.0	−2.21883
$508$	32116.3	2.80498
$509$	−15871.3	−1.38209	−0.691046	−	0.722811i	$-0.742850\pi$
$509$	−15871.3	−1.38209	−0.691046	+	0.722811i	$0.742850\pi$
$510$	48380.3	4.20062
$511$	20144.2	1.74389
$512$	−8530.53	−0.736327
$513$	−1570.57	−0.135170
$514$	16848.6	1.44584
$515$	−10256.6	−0.877594
$516$	−14813.5	−1.26382
$517$	−3329.74	−0.283253
$518$	27737.4	2.35272
$519$	−28614.7	−2.42013
$520$	−35371.5	−2.98297
$521$	−402.598	−0.0338544	−0.0169272	−	0.999857i	$-0.505388\pi$
$521$	−402.598	−0.0338544	−0.0169272	+	0.999857i	$0.505388\pi$
$522$	5035.69	0.422234
$523$	9723.58	0.812969	0.406484	−	0.913658i	$-0.366755\pi$
$523$	9723.58	0.812969	0.406484	+	0.913658i	$0.366755\pi$
$524$	1268.56	0.105758
$525$	27376.6	2.27583
$526$	−33978.5	−2.81661
$527$	−3019.06	−0.249549
$528$	−3074.48	−0.253408
$529$	529.000	0.0434783
$530$	−47832.5	−3.92021
$531$	13269.9	1.08449
$532$	−6562.28	−0.534795
$533$	12395.3	1.00732
$534$	−30777.1	−2.49411
$535$	−1982.14	−0.160178
$536$	13893.7	1.11962
$537$	−1161.65	−0.0933501
$538$	−27215.3	−2.18092
$539$	3088.64	0.246822
$540$	−18471.3	−1.47199
$541$	4756.06	0.377965	0.188983	−	0.981980i	$-0.439481\pi$
$541$	4756.06	0.377965	0.188983	+	0.981980i	$0.439481\pi$
$542$	31614.6	2.50547
$543$	−9818.36	−0.775960
$544$	9356.06	0.737386
$545$	5447.11	0.428126
$546$	−63883.8	−5.00728
$547$	−13487.1	−1.05424	−0.527120	−	0.849791i	$-0.676728\pi$
$547$	−13487.1	−1.05424	−0.527120	+	0.849791i	$0.676728\pi$
$548$	−5566.62	−0.433931
$549$	−21713.3	−1.68798
$550$	11132.3	0.863062
$551$	578.560	0.0447322
$552$	5371.33	0.414165
$553$	−25877.4	−1.98991
$554$	−17025.2	−1.30565
$555$	33553.3	2.56623
$556$	16411.1	1.25178
$557$	3741.93	0.284651	0.142326	−	0.989820i	$-0.454542\pi$
$557$	3741.93	0.284651	0.142326	+	0.989820i	$0.454542\pi$
$558$	6740.25	0.511358
$559$	−9562.45	−0.723522
$560$	−9217.57	−0.695560
$561$	9926.05	0.747020
$562$	39757.9	2.98414
$563$	−19496.6	−1.45947	−0.729737	−	0.683728i	$-0.760357\pi$
$563$	−19496.6	−1.45947	−0.729737	+	0.683728i	$0.760357\pi$
$564$	23660.8	1.76649
$565$	36817.6	2.74146
$566$	−2368.89	−0.175922
$567$	8475.16	0.627731
$568$	−13440.5	−0.992868
$569$	−19600.8	−1.44413	−0.722063	−	0.691827i	$-0.756806\pi$
$569$	−19600.8	−1.44413	−0.722063	+	0.691827i	$0.756806\pi$
$570$	−12409.7	−0.911906
$571$	−18326.0	−1.34311	−0.671557	−	0.740952i	$-0.734374\pi$
$571$	−18326.0	−1.34311	−0.671557	+	0.740952i	$0.734374\pi$
$572$	−16617.3	−1.21469
$573$	−27657.2	−2.01640
$574$	18464.4	1.34266
$575$	−3402.36	−0.246762
$576$	−27990.4	−2.02477
$577$	12738.6	0.919086	0.459543	−	0.888155i	$-0.348013\pi$
$577$	12738.6	0.919086	0.459543	+	0.888155i	$0.348013\pi$
$578$	5354.72	0.385341
$579$	26330.3	1.88990
$580$	6804.36	0.487130
$581$	3699.96	0.264200
$582$	50712.3	3.61184
$583$	−9813.64	−0.697152
$584$	−25420.1	−1.80119
$585$	−44601.3	−3.15220
$586$	−9274.95	−0.653831
$587$	6732.59	0.473396	0.236698	−	0.971583i	$-0.423935\pi$
$587$	6732.59	0.473396	0.236698	+	0.971583i	$0.423935\pi$
$588$	−21947.5	−1.53929
$589$	774.400	0.0541742
$590$	28030.6	1.95593
$591$	−32926.3	−2.29172
$592$	−6123.16	−0.425102
$593$	−3288.37	−0.227719	−0.113859	−	0.993497i	$-0.536321\pi$
$593$	−3288.37	−0.227719	−0.113859	+	0.993497i	$0.536321\pi$
$594$	−5924.36	−0.409225
$595$	29759.2	2.05044
$596$	35838.7	2.46311
$597$	18716.5	1.28311
$598$	7939.46	0.542924
$599$	−8081.70	−0.551268	−0.275634	−	0.961263i	$-0.588888\pi$
$599$	−8081.70	−0.551268	−0.275634	+	0.961263i	$0.588888\pi$
$600$	−34546.7	−2.35060
$601$	−27562.8	−1.87073	−0.935367	−	0.353679i	$-0.884930\pi$
$601$	−27562.8	−1.87073	−0.935367	+	0.353679i	$0.884930\pi$
$602$	−14244.5	−0.964391
$603$	17519.1	1.18314
$604$	−32809.6	−2.21027
$605$	−17774.9	−1.19447
$606$	10572.1	0.708684
$607$	8256.18	0.552072	0.276036	−	0.961147i	$-0.410979\pi$
$607$	8256.18	0.552072	0.276036	+	0.961147i	$0.410979\pi$
$608$	−2399.86	−0.160078
$609$	5366.92	0.357108
$610$	−45866.1	−3.04437
$611$	15273.6	1.01130
$612$	−40708.1	−2.68877
$613$	9714.13	0.640049	0.320025	−	0.947409i	$-0.396309\pi$
$613$	9714.13	0.640049	0.320025	+	0.947409i	$0.396309\pi$
$614$	6671.35	0.438492
$615$	22336.0	1.46451
$616$	−10810.4	−0.707081
$617$	5225.56	0.340961	0.170481	−	0.985361i	$-0.445468\pi$
$617$	5225.56	0.340961	0.170481	+	0.985361i	$0.445468\pi$
$618$	23375.9	1.52155
$619$	−17451.8	−1.13320	−0.566598	−	0.823994i	$-0.691741\pi$
$619$	−17451.8	−1.13320	−0.566598	+	0.823994i	$0.691741\pi$
$620$	9107.61	0.589952
$621$	1810.65	0.117003
$622$	−37901.7	−2.44328
$623$	−18931.3	−1.21744
$624$	14102.7	0.904741
$625$	−12233.2	−0.782925
$626$	−14417.8	−0.920530
$627$	−2546.07	−0.162169
$628$	19116.5	1.21470
$629$	19768.8	1.25315
$630$	−66439.4	−4.20160
$631$	−25334.3	−1.59832	−0.799161	−	0.601117i	$-0.794723\pi$
$631$	−25334.3	−1.59832	−0.799161	+	0.601117i	$0.794723\pi$
$632$	32654.8	2.05528
$633$	40592.1	2.54880
$634$	−18905.9	−1.18431
$635$	37358.2	2.33467
$636$	69734.7	4.34773
$637$	−14167.6	−0.881227
$638$	2182.39	0.135426
$639$	−16947.6	−1.04919
$640$	−43227.1	−2.66984
$641$	24550.7	1.51278	0.756391	−	0.654120i	$-0.226961\pi$
$641$	24550.7	1.51278	0.756391	+	0.654120i	$0.226961\pi$
$642$	4517.50	0.277712
$643$	−12940.7	−0.793675	−0.396837	−	0.917889i	$-0.629892\pi$
$643$	−12940.7	−0.793675	−0.396837	+	0.917889i	$0.629892\pi$
$644$	7565.42	0.462918
$645$	−17231.3	−1.05191
$646$	−7311.52	−0.445306
$647$	−14398.1	−0.874879	−0.437440	−	0.899248i	$-0.644115\pi$
$647$	−14398.1	−0.874879	−0.437440	+	0.899248i	$0.644115\pi$
$648$	−10694.8	−0.648354
$649$	5750.95	0.347834
$650$	−51064.1	−3.08138
$651$	7183.61	0.432485
$652$	29621.1	1.77922
$653$	14063.3	0.842788	0.421394	−	0.906878i	$-0.361541\pi$
$653$	14063.3	0.842788	0.421394	+	0.906878i	$0.361541\pi$
$654$	−12414.5	−0.742272
$655$	1475.60	0.0880253
$656$	−4076.11	−0.242599
$657$	−32053.1	−1.90337
$658$	22752.0	1.34797
$659$	−434.446	−0.0256807	−0.0128404	−	0.999918i	$-0.504087\pi$
$659$	−434.446	−0.0256807	−0.0128404	+	0.999918i	$0.504087\pi$
$660$	−29943.9	−1.76601
$661$	−19410.3	−1.14217	−0.571084	−	0.820891i	$-0.693477\pi$
$661$	−19410.3	−1.14217	−0.571084	+	0.820891i	$0.693477\pi$
$662$	−15680.2	−0.920586
$663$	−45530.9	−2.66708
$664$	−4668.99	−0.272880
$665$	−7633.35	−0.445126
$666$	−44135.2	−2.56787
$667$	−667.000	−0.0387202
$668$	−10839.4	−0.627829
$669$	28698.8	1.65853
$670$	37006.4	2.13385
$671$	−9410.20	−0.541396
$672$	−22262.0	−1.27794
$673$	27922.2	1.59929	0.799646	−	0.600472i	$-0.205021\pi$
$673$	27922.2	1.59929	0.799646	+	0.600472i	$0.205021\pi$
$674$	25041.2	1.43108
$675$	−11645.5	−0.664055
$676$	45020.7	2.56149
$677$	10620.6	0.602928	0.301464	−	0.953478i	$-0.402525\pi$
$677$	10620.6	0.602928	0.301464	+	0.953478i	$0.402525\pi$
$678$	−83910.9	−4.75307
$679$	31193.6	1.76304
$680$	−37553.3	−2.11780
$681$	22323.2	1.25613
$682$	2921.12	0.164011
$683$	28802.8	1.61363	0.806815	−	0.590805i	$-0.201190\pi$
$683$	28802.8	1.61363	0.806815	+	0.590805i	$0.201190\pi$
$684$	10441.8	0.583701
$685$	−6475.18	−0.361173
$686$	16326.8	0.908689
$687$	15818.7	0.878488
$688$	3144.55	0.174251
$689$	45015.3	2.48903
$690$	14306.7	0.789344
$691$	28658.3	1.57773	0.788867	−	0.614565i	$-0.210668\pi$
$691$	28658.3	1.57773	0.788867	+	0.614565i	$0.210668\pi$
$692$	50858.8	2.79388
$693$	−13631.2	−0.747194
$694$	9478.09	0.518420
$695$	19089.7	1.04189
$696$	−6772.55	−0.368840
$697$	13159.8	0.715157
$698$	28080.8	1.52274
$699$	−36061.3	−1.95131
$700$	−48658.3	−2.62730
$701$	12820.9	0.690783	0.345391	−	0.938459i	$-0.387746\pi$
$701$	12820.9	0.690783	0.345391	+	0.938459i	$0.387746\pi$
$702$	27175.1	1.46105
$703$	−5070.77	−0.272045
$704$	−12130.6	−0.649415
$705$	27522.6	1.47030
$706$	26068.3	1.38965
$707$	6503.01	0.345928
$708$	−40865.6	−2.16924
$709$	−26681.6	−1.41333	−0.706664	−	0.707550i	$-0.749801\pi$
$709$	−26681.6	−1.41333	−0.706664	+	0.707550i	$0.749801\pi$
$710$	−35799.1	−1.89228
$711$	41175.6	2.17188
$712$	23889.5	1.25744
$713$	−892.777	−0.0468931
$714$	−67824.2	−3.55498
$715$	−19329.5	−1.01102
$716$	2064.68	0.107767
$717$	−1902.17	−0.0990762
$718$	−56840.1	−2.95439
$719$	−27347.7	−1.41849	−0.709247	−	0.704960i	$-0.750965\pi$
$719$	−27347.7	−1.41849	−0.709247	+	0.704960i	$0.750965\pi$
$720$	14666.8	0.759167
$721$	14378.7	0.742707
$722$	−30443.8	−1.56926
$723$	−7559.19	−0.388837
$724$	17450.8	0.895795
$725$	4289.93	0.219757
$726$	40510.8	2.07093
$727$	−26203.4	−1.33677	−0.668384	−	0.743817i	$-0.733014\pi$
$727$	−26203.4	−1.33677	−0.668384	+	0.743817i	$0.733014\pi$
$728$	49587.2	2.52449
$729$	−30470.2	−1.54805
$730$	−67707.4	−3.43283
$731$	−10152.3	−0.513674
$732$	66867.9	3.37638
$733$	−27875.3	−1.40464	−0.702318	−	0.711863i	$-0.747852\pi$
$733$	−27875.3	−1.40464	−0.702318	+	0.711863i	$0.747852\pi$
$734$	−1938.41	−0.0974770
$735$	−25529.7	−1.28119
$736$	2766.71	0.138563
$737$	7592.49	0.379475
$738$	−29380.2	−1.46545
$739$	4843.40	0.241093	0.120546	−	0.992708i	$-0.461535\pi$
$739$	4843.40	0.241093	0.120546	+	0.992708i	$0.461535\pi$
$740$	−59636.6	−2.96255
$741$	11678.8	0.578992
$742$	67056.1	3.31767
$743$	13142.5	0.648928	0.324464	−	0.945898i	$-0.394816\pi$
$743$	13142.5	0.648928	0.324464	+	0.945898i	$0.394816\pi$
$744$	−9065.03	−0.446694
$745$	41688.1	2.05011
$746$	42582.7	2.08990
$747$	−5887.30	−0.288360
$748$	−17642.3	−0.862386
$749$	2778.76	0.135559
$750$	−14262.3	−0.694383
$751$	−14952.8	−0.726547	−0.363273	−	0.931683i	$-0.618341\pi$
$751$	−14952.8	−0.726547	−0.363273	+	0.931683i	$0.618341\pi$
$752$	−5022.61	−0.243558
$753$	−32305.2	−1.56343
$754$	−10010.6	−0.483509
$755$	−38164.6	−1.83967
$756$	25894.8	1.24575
$757$	−23030.7	−1.10577	−0.552884	−	0.833258i	$-0.686473\pi$
$757$	−23030.7	−1.10577	−0.552884	+	0.833258i	$0.686473\pi$
$758$	37042.5	1.77499
$759$	2935.27	0.140373
$760$	9632.56	0.459750
$761$	13010.1	0.619729	0.309865	−	0.950781i	$-0.399716\pi$
$761$	13010.1	0.619729	0.309865	+	0.950781i	$0.399716\pi$
$762$	−85143.0	−4.04778
$763$	−7636.28	−0.362323
$764$	49157.1	2.32780
$765$	−47352.3	−2.23794
$766$	−64561.9	−3.04532
$767$	−26379.7	−1.24187
$768$	49964.7	2.34759
$769$	−27132.8	−1.27235	−0.636173	−	0.771546i	$-0.719484\pi$
$769$	−27132.8	−1.27235	−0.636173	+	0.771546i	$0.719484\pi$
$770$	−28793.8	−1.34760
$771$	−28572.6	−1.33465
$772$	−46798.7	−2.18176
$773$	14482.9	0.673887	0.336944	−	0.941525i	$-0.390607\pi$
$773$	14482.9	0.673887	0.336944	+	0.941525i	$0.390607\pi$
$774$	22665.6	1.05258
$775$	5742.06	0.266143
$776$	−39363.4	−1.82096
$777$	−47038.2	−2.17180
$778$	−52054.0	−2.39875
$779$	−3375.55	−0.155252
$780$	137353.	6.30517
$781$	−7344.80	−0.336514
$782$	8429.18	0.385456
$783$	−2283.00	−0.104199
$784$	4658.93	0.212233
$785$	22236.7	1.01103
$786$	−3363.05	−0.152616
$787$	−24281.4	−1.09979	−0.549896	−	0.835233i	$-0.685333\pi$
$787$	−24281.4	−1.09979	−0.549896	+	0.835233i	$0.685333\pi$
$788$	58522.2	2.64564
$789$	57622.3	2.60001
$790$	86977.2	3.91710
$791$	−51614.4	−2.32010
$792$	17201.2	0.771742
$793$	43164.7	1.93294
$794$	−38944.3	−1.74066
$795$	81116.4	3.61875
$796$	−33266.2	−1.48127
$797$	−33254.3	−1.47795	−0.738975	−	0.673733i	$-0.764690\pi$
$797$	−33254.3	−1.47795	−0.738975	+	0.673733i	$0.764690\pi$
$798$	17397.2	0.771745
$799$	16215.7	0.717983
$800$	−17794.6	−0.786419
$801$	30123.1	1.32877
$802$	35231.7	1.55121
$803$	−13891.3	−0.610478
$804$	−53951.4	−2.36657
$805$	8800.21	0.385300
$806$	−13399.2	−0.585566
$807$	46153.0	2.01321
$808$	−8206.18	−0.357293
$809$	36404.3	1.58209	0.791044	−	0.611760i	$-0.209538\pi$
$809$	36404.3	1.58209	0.791044	+	0.611760i	$0.209538\pi$
$810$	−28486.1	−1.23568
$811$	5502.37	0.238242	0.119121	−	0.992880i	$-0.461992\pi$
$811$	5502.37	0.238242	0.119121	+	0.992880i	$0.461992\pi$
$812$	−9539.00	−0.412258
$813$	−53613.5	−2.31280
$814$	−19127.5	−0.823609
$815$	34455.7	1.48090
$816$	14972.5	0.642332
$817$	2604.10	0.111513
$818$	−31199.7	−1.33358
$819$	62526.3	2.66770
$820$	−39699.3	−1.69068
$821$	6663.70	0.283270	0.141635	−	0.989919i	$-0.454764\pi$
$821$	6663.70	0.283270	0.141635	+	0.989919i	$0.454764\pi$
$822$	14757.6	0.626191
$823$	34513.9	1.46182	0.730911	−	0.682473i	$-0.239096\pi$
$823$	34513.9	1.46182	0.730911	+	0.682473i	$0.239096\pi$
$824$	−18144.6	−0.767107
$825$	−18878.7	−0.796693
$826$	−39296.0	−1.65530
$827$	−11855.1	−0.498480	−0.249240	−	0.968442i	$-0.580181\pi$
$827$	−11855.1	−0.498480	−0.249240	+	0.968442i	$0.580181\pi$
$828$	−12037.9	−0.505251
$829$	−16843.3	−0.705661	−0.352831	−	0.935687i	$-0.614781\pi$
$829$	−16843.3	−0.705661	−0.352831	+	0.935687i	$0.614781\pi$
$830$	−12436.0	−0.520073
$831$	28872.1	1.20525
$832$	55643.1	2.31860
$833$	−15041.5	−0.625639
$834$	−43507.3	−1.80640
$835$	−12608.6	−0.522560
$836$	4525.30	0.187214
$837$	−3055.79	−0.126193
$838$	52776.5	2.17558
$839$	4207.14	0.173119	0.0865593	−	0.996247i	$-0.472413\pi$
$839$	4207.14	0.173119	0.0865593	+	0.996247i	$0.472413\pi$
$840$	89355.0	3.67029
$841$	841.000	0.0344828
$842$	−26995.1	−1.10488
$843$	−67423.2	−2.75466
$844$	−72147.2	−2.94243
$845$	52368.8	2.13200
$846$	−36202.5	−1.47124
$847$	24918.6	1.01088
$848$	−14803.0	−0.599453
$849$	4017.27	0.162394
$850$	−54213.8	−2.18767
$851$	5845.90	0.235482
$852$	52191.4	2.09865
$853$	27486.2	1.10329	0.551646	−	0.834078i	$-0.314000\pi$
$853$	27486.2	1.10329	0.551646	+	0.834078i	$0.314000\pi$
$854$	64299.5	2.57644
$855$	12146.0	0.485831
$856$	−3506.53	−0.140012
$857$	−28271.9	−1.12689	−0.563447	−	0.826152i	$-0.690525\pi$
$857$	−28271.9	−1.12689	−0.563447	+	0.826152i	$0.690525\pi$
$858$	44053.8	1.75288
$859$	25239.7	1.00252	0.501261	−	0.865296i	$-0.332870\pi$
$859$	25239.7	1.00252	0.501261	+	0.865296i	$0.332870\pi$
$860$	30626.4	1.21436
$861$	−31312.8	−1.23942
$862$	78551.0	3.10378
$863$	33283.8	1.31286	0.656428	−	0.754389i	$-0.272067\pi$
$863$	33283.8	1.31286	0.656428	+	0.754389i	$0.272067\pi$
$864$	9469.87	0.372884
$865$	59159.8	2.32543
$866$	−41350.5	−1.62257
$867$	−9080.78	−0.355709
$868$	−12767.9	−0.499276
$869$	17844.9	0.696600
$870$	−18038.9	−0.702961
$871$	−34826.8	−1.35483
$872$	9636.26	0.374226
$873$	−49634.7	−1.92426
$874$	−2162.12	−0.0836781
$875$	−8772.91	−0.338947
$876$	98710.2	3.80720
$877$	33793.3	1.30116	0.650582	−	0.759436i	$-0.274525\pi$
$877$	33793.3	1.30116	0.650582	+	0.759436i	$0.274525\pi$
$878$	−67748.7	−2.60411
$879$	15728.9	0.603552
$880$	6356.37	0.243492
$881$	−17394.5	−0.665194	−0.332597	−	0.943069i	$-0.607925\pi$
$881$	−17394.5	−0.665194	−0.332597	+	0.943069i	$0.607925\pi$
$882$	33581.1	1.28201
$883$	44136.9	1.68213	0.841067	−	0.540931i	$-0.181928\pi$
$883$	44136.9	1.68213	0.841067	+	0.540931i	$0.181928\pi$
$884$	80925.2	3.07897
$885$	−47535.5	−1.80552
$886$	9095.47	0.344885
$887$	14666.1	0.555176	0.277588	−	0.960700i	$-0.410465\pi$
$887$	14666.1	0.555176	0.277588	+	0.960700i	$0.410465\pi$
$888$	59357.8	2.24315
$889$	−52372.3	−1.97583
$890$	63630.4	2.39651
$891$	−5844.41	−0.219747
$892$	−51008.3	−1.91467
$893$	−4159.38	−0.155866
$894$	−95011.4	−3.55443
$895$	2401.67	0.0896973
$896$	60599.9	2.25949
$897$	−13464.1	−0.501174
$898$	−50817.3	−1.88841
$899$	1125.68	0.0417613
$900$	77424.2	2.86756
$901$	47791.8	1.76712
$902$	−12732.9	−0.470022
$903$	24156.5	0.890231
$904$	65132.5	2.39632
$905$	20299.1	0.745596
$906$	86980.9	3.18957
$907$	22669.4	0.829905	0.414953	−	0.909843i	$-0.363798\pi$
$907$	22669.4	0.829905	0.414953	+	0.909843i	$0.363798\pi$
$908$	−39676.5	−1.45012
$909$	−10347.5	−0.377562
$910$	132077.	4.81134
$911$	15752.2	0.572879	0.286439	−	0.958098i	$-0.407528\pi$
$911$	15752.2	0.572879	0.286439	+	0.958098i	$0.407528\pi$
$912$	−3840.51	−0.139443
$913$	−2551.46	−0.0924874
$914$	−26446.7	−0.957090
$915$	77781.7	2.81026
$916$	−28115.7	−1.01416
$917$	−2068.64	−0.0744957
$918$	28851.3	1.03729
$919$	4459.98	0.160088	0.0800441	−	0.996791i	$-0.474494\pi$
$919$	4459.98	0.160088	0.0800441	+	0.996791i	$0.474494\pi$
$920$	−11105.0	−0.397958
$921$	−11313.6	−0.404772
$922$	66299.2	2.36816
$923$	33690.7	1.20145
$924$	41978.3	1.49457
$925$	−37599.0	−1.33648
$926$	−85685.9	−3.04084
$927$	−22879.2	−0.810626
$928$	−3488.47	−0.123399
$929$	13051.4	0.460928	0.230464	−	0.973081i	$-0.425976\pi$
$929$	13051.4	0.460928	0.230464	+	0.973081i	$0.425976\pi$
$930$	−24145.0	−0.851340
$931$	3858.20	0.135819
$932$	64094.2	2.25266
$933$	64275.4	2.25539
$934$	−5289.64	−0.185313
$935$	−20521.7	−0.717789
$936$	−78902.3	−2.75534
$937$	22802.3	0.795003	0.397501	−	0.917602i	$-0.369877\pi$
$937$	22802.3	0.795003	0.397501	+	0.917602i	$0.369877\pi$
$938$	−51879.1	−1.80588
$939$	24450.4	0.849742
$940$	−48917.8	−1.69737
$941$	−32817.0	−1.13688	−0.568439	−	0.822725i	$-0.692452\pi$
$941$	−32817.0	−1.13688	−0.568439	+	0.822725i	$0.692452\pi$
$942$	−50679.5	−1.75290
$943$	3891.54	0.134386
$944$	8674.78	0.299089
$945$	30121.2	1.03687
$946$	9822.92	0.337601
$947$	−20120.7	−0.690429	−0.345214	−	0.938524i	$-0.612194\pi$
$947$	−20120.7	−0.690429	−0.345214	+	0.938524i	$0.612194\pi$
$948$	−126804.	−4.34429
$949$	63719.6	2.17959
$950$	13906.0	0.474917
$951$	32061.5	1.09323
$952$	52645.8	1.79229
$953$	−44350.1	−1.50749	−0.753747	−	0.657165i	$-0.771756\pi$
$953$	−44350.1	−1.50749	−0.753747	+	0.657165i	$0.771756\pi$
$954$	−106698.	−3.62106
$955$	57180.3	1.93750
$956$	3380.85	0.114377
$957$	−3700.99	−0.125011
$958$	6314.50	0.212956
$959$	9077.53	0.305661
$960$	100267.	3.37096
$961$	−28284.3	−0.949424
$962$	87737.9	2.94052
$963$	−4421.50	−0.147955
$964$	13435.5	0.448887
$965$	−54436.9	−1.81594
$966$	−20056.5	−0.668021
$967$	−14064.3	−0.467712	−0.233856	−	0.972271i	$-0.575134\pi$
$967$	−14064.3	−0.467712	−0.233856	+	0.972271i	$0.575134\pi$
$968$	−31444.8	−1.04409
$969$	12399.2	0.411063
$970$	−104846.	−3.47051
$971$	−42934.6	−1.41899	−0.709493	−	0.704712i	$-0.751076\pi$
$971$	−42934.6	−1.41899	−0.709493	+	0.704712i	$0.751076\pi$
$972$	71717.8	2.36662
$973$	−26761.8	−0.881750
$974$	−14060.0	−0.462538
$975$	86596.8	2.84443
$976$	−14194.4	−0.465525
$977$	−27806.2	−0.910541	−0.455270	−	0.890353i	$-0.650457\pi$
$977$	−27806.2	−0.910541	−0.455270	+	0.890353i	$0.650457\pi$
$978$	−78527.9	−2.56753
$979$	13054.9	0.426185
$980$	45375.7	1.47906
$981$	12150.7	0.395456
$982$	74142.2	2.40934
$983$	−15012.9	−0.487120	−0.243560	−	0.969886i	$-0.578315\pi$
$983$	−15012.9	−0.487120	−0.243560	+	0.969886i	$0.578315\pi$
$984$	39513.7	1.28013
$985$	68073.9	2.20205
$986$	−10628.1	−0.343273
$987$	−38583.8	−1.24431
$988$	−20757.6	−0.668408
$989$	−3002.17	−0.0965251
$990$	45816.1	1.47084
$991$	48419.8	1.55208	0.776038	−	0.630687i	$-0.217227\pi$
$991$	48419.8	1.55208	0.776038	+	0.630687i	$0.217227\pi$
$992$	−4669.30	−0.149446
$993$	26591.2	0.849794
$994$	50186.7	1.60143
$995$	−38695.7	−1.23290
$996$	18130.4	0.576791
$997$	−5423.05	−0.172266	−0.0861332	−	0.996284i	$-0.527451\pi$
$997$	−5423.05	−0.172266	−0.0861332	+	0.996284i	$0.527451\pi$
$998$	−16350.8	−0.518614
$999$	20009.3	0.633699

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.4.a.b.1.36	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.4.a.b.1.36	✓	38	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:	\( N \)	\(=\)	\( 667 = 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	667.a (trivial)

\(f(q)\)	\(=\)	\(q+4.71195 q^{2} -7.99074 q^{3} +14.2025 q^{4} +16.5206 q^{5} -37.6520 q^{6} -23.1601 q^{7} +29.2258 q^{8} +36.8519 q^{9} +O(q^{10})\)	\(q+4.71195 q^{2} -7.99074 q^{3} +14.2025 q^{4} +16.5206 q^{5} -37.6520 q^{6} -23.1601 q^{7} +29.2258 q^{8} +36.8519 q^{9} +77.8440 q^{10} +15.9710 q^{11} -113.488 q^{12} -73.2593 q^{13} -109.129 q^{14} -132.011 q^{15} +24.0908 q^{16} -77.7780 q^{17} +173.644 q^{18} +19.9503 q^{19} +234.633 q^{20} +185.066 q^{21} +75.2547 q^{22} -23.0000 q^{23} -233.536 q^{24} +147.929 q^{25} -345.194 q^{26} -78.7241 q^{27} -328.931 q^{28} +29.0000 q^{29} -622.032 q^{30} +38.8164 q^{31} -120.292 q^{32} -127.620 q^{33} -366.486 q^{34} -382.618 q^{35} +523.389 q^{36} -254.170 q^{37} +94.0050 q^{38} +585.396 q^{39} +482.827 q^{40} -169.198 q^{41} +872.024 q^{42} +130.529 q^{43} +226.828 q^{44} +608.814 q^{45} -108.375 q^{46} -208.487 q^{47} -192.503 q^{48} +193.390 q^{49} +697.033 q^{50} +621.503 q^{51} -1040.46 q^{52} -614.465 q^{53} -370.944 q^{54} +263.850 q^{55} -676.873 q^{56} -159.418 q^{57} +136.647 q^{58} +360.086 q^{59} -1874.89 q^{60} -589.205 q^{61} +182.901 q^{62} -853.494 q^{63} -759.536 q^{64} -1210.28 q^{65} -601.341 q^{66} +475.391 q^{67} -1104.64 q^{68} +183.787 q^{69} -1802.88 q^{70} -459.883 q^{71} +1077.03 q^{72} -869.782 q^{73} -1197.64 q^{74} -1182.06 q^{75} +283.344 q^{76} -369.891 q^{77} +2758.36 q^{78} +1117.33 q^{79} +397.994 q^{80} -365.938 q^{81} -797.251 q^{82} -159.756 q^{83} +2628.40 q^{84} -1284.94 q^{85} +615.046 q^{86} -231.731 q^{87} +466.767 q^{88} +817.409 q^{89} +2868.70 q^{90} +1696.69 q^{91} -326.657 q^{92} -310.172 q^{93} -982.378 q^{94} +329.590 q^{95} +961.221 q^{96} -1346.87 q^{97} +911.245 q^{98} +588.563 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 8 q^{2} - 16 q^{3} + 152 q^{4} - 80 q^{5} - 16 q^{6} - 38 q^{7} - 138 q^{8} + 312 q^{9}+O(q^{10}) \) 38 * q - 8 * q^2 - 16 * q^3 + 152 * q^4 - 80 * q^5 - 16 * q^6 - 38 * q^7 - 138 * q^8 + 312 * q^9	\( 38 q - 8 q^{2} - 16 q^{3} + 152 q^{4} - 80 q^{5} - 16 q^{6} - 38 q^{7} - 138 q^{8} + 312 q^{9} + 30 q^{10} - 100 q^{11} - 173 q^{12} - 160 q^{13} - 304 q^{14} - 208 q^{15} + 552 q^{16} - 666 q^{17} - 315 q^{18} - 252 q^{19} - 747 q^{20} - 470 q^{21} - 323 q^{22} - 874 q^{23} - 114 q^{24} + 910 q^{25} - 119 q^{26} - 526 q^{27} - 619 q^{28} + 1102 q^{29} - 533 q^{30} + 358 q^{31} - 1272 q^{32} - 604 q^{33} - 553 q^{34} + 16 q^{35} + 1230 q^{36} - 1206 q^{37} - 2010 q^{38} - 240 q^{39} - 418 q^{40} - 1094 q^{41} - 1184 q^{42} - 936 q^{43} - 2153 q^{44} - 3654 q^{45} + 184 q^{46} - 1702 q^{47} - 1319 q^{48} + 1920 q^{49} - 1255 q^{50} - 270 q^{51} - 2202 q^{52} - 4640 q^{53} - 1529 q^{54} - 318 q^{55} - 2435 q^{56} - 1980 q^{57} - 232 q^{58} + 318 q^{59} - 3279 q^{60} - 2212 q^{61} - 541 q^{62} - 1044 q^{63} + 1186 q^{64} - 2438 q^{65} - 3503 q^{66} - 496 q^{67} - 6099 q^{68} + 368 q^{69} + 4068 q^{70} + 870 q^{71} - 3869 q^{72} - 2810 q^{73} - 2793 q^{74} - 3638 q^{75} + 1548 q^{76} - 6072 q^{77} + 2170 q^{78} - 3084 q^{79} - 6702 q^{80} + 362 q^{81} - 1183 q^{82} - 5566 q^{83} - 6518 q^{84} - 120 q^{85} - 2095 q^{86} - 464 q^{87} - 1220 q^{88} - 6506 q^{89} + 165 q^{90} + 44 q^{91} - 3496 q^{92} - 5928 q^{93} + 135 q^{94} - 2024 q^{95} - 3164 q^{96} - 7472 q^{97} - 6422 q^{98} - 944 q^{99}+O(q^{100}) \) 38 * q - 8 * q^2 - 16 * q^3 + 152 * q^4 - 80 * q^5 - 16 * q^6 - 38 * q^7 - 138 * q^8 + 312 * q^9 + 30 * q^10 - 100 * q^11 - 173 * q^12 - 160 * q^13 - 304 * q^14 - 208 * q^15 + 552 * q^16 - 666 * q^17 - 315 * q^18 - 252 * q^19 - 747 * q^20 - 470 * q^21 - 323 * q^22 - 874 * q^23 - 114 * q^24 + 910 * q^25 - 119 * q^26 - 526 * q^27 - 619 * q^28 + 1102 * q^29 - 533 * q^30 + 358 * q^31 - 1272 * q^32 - 604 * q^33 - 553 * q^34 + 16 * q^35 + 1230 * q^36 - 1206 * q^37 - 2010 * q^38 - 240 * q^39 - 418 * q^40 - 1094 * q^41 - 1184 * q^42 - 936 * q^43 - 2153 * q^44 - 3654 * q^45 + 184 * q^46 - 1702 * q^47 - 1319 * q^48 + 1920 * q^49 - 1255 * q^50 - 270 * q^51 - 2202 * q^52 - 4640 * q^53 - 1529 * q^54 - 318 * q^55 - 2435 * q^56 - 1980 * q^57 - 232 * q^58 + 318 * q^59 - 3279 * q^60 - 2212 * q^61 - 541 * q^62 - 1044 * q^63 + 1186 * q^64 - 2438 * q^65 - 3503 * q^66 - 496 * q^67 - 6099 * q^68 + 368 * q^69 + 4068 * q^70 + 870 * q^71 - 3869 * q^72 - 2810 * q^73 - 2793 * q^74 - 3638 * q^75 + 1548 * q^76 - 6072 * q^77 + 2170 * q^78 - 3084 * q^79 - 6702 * q^80 + 362 * q^81 - 1183 * q^82 - 5566 * q^83 - 6518 * q^84 - 120 * q^85 - 2095 * q^86 - 464 * q^87 - 1220 * q^88 - 6506 * q^89 + 165 * q^90 + 44 * q^91 - 3496 * q^92 - 5928 * q^93 + 135 * q^94 - 2024 * q^95 - 3164 * q^96 - 7472 * q^97 - 6422 * q^98 - 944 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more