LMFDB - Embedded newform 525.4.d.g.274.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,4,Mod(274,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	525.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$30.9760027530$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{57})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 29x^{2} + 196 $ x^4 + 29*x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.3
Root			$3.27492i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.4.d.g.274.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.27492i q^{2} -3.00000i q^{3} +2.82475 q^{4} +6.82475 q^{6} +7.00000i q^{7} +24.6254i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q+2.27492i q^{2} -3.00000i q^{3} +2.82475 q^{4} +6.82475 q^{6} +7.00000i q^{7} +24.6254i q^{8} -9.00000 q^{9} -40.7492 q^{11} -8.47425i q^{12} -53.2990i q^{13} -15.9244 q^{14} -33.4228 q^{16} +4.54983i q^{17} -20.4743i q^{18} -122.598 q^{19} +21.0000 q^{21} -92.7010i q^{22} -131.347i q^{23} +73.8762 q^{24} +121.251 q^{26} +27.0000i q^{27} +19.7733i q^{28} +216.598 q^{29} -251.794 q^{31} +120.969i q^{32} +122.248i q^{33} -10.3505 q^{34} -25.4228 q^{36} +11.8970i q^{37} -278.900i q^{38} -159.897 q^{39} -111.752 q^{41} +47.7733i q^{42} -369.196i q^{43} -115.106 q^{44} +298.804 q^{46} -262.694i q^{47} +100.268i q^{48} -49.0000 q^{49} +13.6495 q^{51} -150.556i q^{52} +567.100i q^{53} -61.4228 q^{54} -172.378 q^{56} +367.794i q^{57} +492.743i q^{58} -839.890 q^{59} -485.794 q^{61} -572.811i q^{62} -63.0000i q^{63} -542.577 q^{64} -278.103 q^{66} -333.691i q^{67} +12.8522i q^{68} -394.042 q^{69} +590.248 q^{71} -221.629i q^{72} -490.701i q^{73} -27.0647 q^{74} -346.309 q^{76} -285.244i q^{77} -363.752i q^{78} -121.691 q^{79} +81.0000 q^{81} -254.228i q^{82} -609.608i q^{83} +59.3198 q^{84} +839.890 q^{86} -649.794i q^{87} -1003.47i q^{88} -719.038 q^{89} +373.093 q^{91} -371.023i q^{92} +755.382i q^{93} +597.608 q^{94} +362.908 q^{96} -637.877i q^{97} -111.471i q^{98} +366.743 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 34 q^{4} - 18 q^{6} - 36 q^{9}+O(q^{10}) $ 4 * q - 34 * q^4 - 18 * q^6 - 36 * q^9	$ 4 q - 34 q^{4} - 18 q^{6} - 36 q^{9} - 12 q^{11} + 42 q^{14} + 274 q^{16} - 128 q^{19} + 84 q^{21} + 522 q^{24} + 636 q^{26} + 504 q^{29} + 80 q^{31} - 132 q^{34} + 306 q^{36} - 96 q^{39} - 900 q^{41} - 1608 q^{44} + 1920 q^{46} - 196 q^{49} - 36 q^{51} + 162 q^{54} - 1218 q^{56} - 1608 q^{59} - 856 q^{61} - 2578 q^{64} - 1656 q^{66} - 36 q^{69} + 1908 q^{71} - 2796 q^{74} - 3016 q^{76} + 1144 q^{79} + 324 q^{81} - 714 q^{84} + 1608 q^{86} - 732 q^{89} + 224 q^{91} + 3840 q^{94} - 1674 q^{96} + 108 q^{99}+O(q^{100}) $ 4 * q - 34 * q^4 - 18 * q^6 - 36 * q^9 - 12 * q^11 + 42 * q^14 + 274 * q^16 - 128 * q^19 + 84 * q^21 + 522 * q^24 + 636 * q^26 + 504 * q^29 + 80 * q^31 - 132 * q^34 + 306 * q^36 - 96 * q^39 - 900 * q^41 - 1608 * q^44 + 1920 * q^46 - 196 * q^49 - 36 * q^51 + 162 * q^54 - 1218 * q^56 - 1608 * q^59 - 856 * q^61 - 2578 * q^64 - 1656 * q^66 - 36 * q^69 + 1908 * q^71 - 2796 * q^74 - 3016 * q^76 + 1144 * q^79 + 324 * q^81 - 714 * q^84 + 1608 * q^86 - 732 * q^89 + 224 * q^91 + 3840 * q^94 - 1674 * q^96 + 108 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	2.27492i	0.804305i	0.915573	+	0.402152i	$0.131738\pi$
$2$	2.27492i	0.804305i	−0.915573	+	0.402152i	$0.868262\pi$
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	2.82475	0.353094
$5$	0	0
$5$	0	0
$6$	6.82475	0.464366
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	24.6254i	1.08830i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−40.7492	−1.11694	−0.558470	−	0.829525i	$-0.688611\pi$
$11$	−40.7492	−1.11694	−0.558470	+	0.829525i	$0.688611\pi$
$12$	− 8.47425i	− 0.203859i
$13$	− 53.2990i	− 1.13711i	−0.822644	−	0.568557i	$-0.807502\pi$
$13$	− 53.2990i	− 1.13711i	0.822644	−	0.568557i	$-0.192498\pi$
$14$	−15.9244	−0.303999
$15$	0	0
$16$	−33.4228	−0.522231
$17$	4.54983i	0.0649116i	0.999473	+	0.0324558i	$0.0103328\pi$
$17$	4.54983i	0.0649116i	−0.999473	+	0.0324558i	$0.989667\pi$
$18$	− 20.4743i	− 0.268102i
$19$	−122.598	−1.48031	−0.740156	−	0.672436i	$-0.765248\pi$
$19$	−122.598	−1.48031	−0.740156	+	0.672436i	$0.765248\pi$
$20$	0	0
$21$	21.0000	0.218218
$22$	− 92.7010i	− 0.898360i
$23$	− 131.347i	− 1.19077i	−0.803439	−	0.595387i	$-0.796999\pi$
$23$	− 131.347i	− 1.19077i	0.803439	−	0.595387i	$-0.203001\pi$
$24$	73.8762	0.628330
$25$	0	0
$26$	121.251	0.914586
$27$	27.0000i	0.192450i
$28$	19.7733i	0.133457i
$29$	216.598	1.38694	0.693470	−	0.720486i	$-0.256081\pi$
$29$	216.598	1.38694	0.693470	+	0.720486i	$0.256081\pi$
$30$	0	0
$31$	−251.794	−1.45882	−0.729412	−	0.684075i	$-0.760206\pi$
$31$	−251.794	−1.45882	−0.729412	+	0.684075i	$0.760206\pi$
$32$	120.969i	0.668267i
$33$	122.248i	0.644865i
$34$	−10.3505	−0.0522087
$35$	0	0
$36$	−25.4228	−0.117698
$37$	11.8970i	0.0528610i	0.999651	+	0.0264305i	$0.00841407\pi$
$37$	11.8970i	0.0528610i	−0.999651	+	0.0264305i	$0.991586\pi$
$38$	− 278.900i	− 1.19062i
$39$	−159.897	−0.656513
$40$	0	0
$41$	−111.752	−0.425678	−0.212839	−	0.977087i	$-0.568271\pi$
$41$	−111.752	−0.425678	−0.212839	+	0.977087i	$0.568271\pi$
$42$	47.7733i	0.175514i
$43$	− 369.196i	− 1.30935i	−0.755912	−	0.654673i	$-0.772806\pi$
$43$	− 369.196i	− 1.30935i	0.755912	−	0.654673i	$-0.227194\pi$
$44$	−115.106	−0.394385
$45$	0	0
$46$	298.804	0.957744
$47$	− 262.694i	− 0.815275i	−0.913144	−	0.407637i	$-0.866353\pi$
$47$	− 262.694i	− 0.815275i	0.913144	−	0.407637i	$-0.133647\pi$
$48$	100.268i	0.301510i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	13.6495	0.0374767
$52$	− 150.556i	− 0.401508i
$53$	567.100i	1.46976i	0.678199	+	0.734879i	$0.262761\pi$
$53$	567.100i	1.46976i	−0.678199	+	0.734879i	$0.737239\pi$
$54$	−61.4228	−0.154789
$55$	0	0
$56$	−172.378	−0.411339
$57$	367.794i	0.854658i
$58$	492.743i	1.11552i
$59$	−839.890	−1.85330	−0.926648	−	0.375931i	$-0.877323\pi$
$59$	−839.890	−1.85330	−0.926648	+	0.375931i	$0.877323\pi$
$60$	0	0
$61$	−485.794	−1.01966	−0.509832	−	0.860274i	$-0.670293\pi$
$61$	−485.794	−1.01966	−0.509832	+	0.860274i	$0.670293\pi$
$62$	− 572.811i	− 1.17334i
$63$	− 63.0000i	− 0.125988i
$64$	−542.577	−1.05972
$65$	0	0
$66$	−278.103	−0.518668
$67$	− 333.691i	− 0.608460i	−0.952599	−	0.304230i	$-0.901601\pi$
$67$	− 333.691i	− 0.608460i	0.952599	−	0.304230i	$-0.0983992\pi$
$68$	12.8522i	0.0229199i
$69$	−394.042	−0.687493
$70$	0	0
$71$	590.248	0.986613	0.493306	−	0.869856i	$-0.335788\pi$
$71$	590.248	0.986613	0.493306	+	0.869856i	$0.335788\pi$
$72$	− 221.629i	− 0.362767i
$73$	− 490.701i	− 0.786743i	−0.919380	−	0.393371i	$-0.871309\pi$
$73$	− 490.701i	− 0.786743i	0.919380	−	0.393371i	$-0.128691\pi$
$74$	−27.0647	−0.0425164
$75$	0	0
$76$	−346.309	−0.522689
$77$	− 285.244i	− 0.422164i
$78$	− 363.752i	− 0.528037i
$79$	−121.691	−0.173308	−0.0866539	−	0.996238i	$-0.527617\pi$
$79$	−121.691	−0.173308	−0.0866539	+	0.996238i	$0.527617\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 254.228i	− 0.342375i
$83$	− 609.608i	− 0.806183i	−0.915160	−	0.403091i	$-0.867936\pi$
$83$	− 609.608i	− 0.806183i	0.915160	−	0.403091i	$-0.132064\pi$
$84$	59.3198	0.0770514
$85$	0	0
$86$	839.890	1.05311
$87$	− 649.794i	− 0.800750i
$88$	− 1003.47i	− 1.21557i
$89$	−719.038	−0.856381	−0.428190	−	0.903689i	$-0.640849\pi$
$89$	−719.038	−0.856381	−0.428190	+	0.903689i	$0.640849\pi$
$90$	0	0
$91$	373.093	0.429789
$92$	− 371.023i	− 0.420455i
$93$	755.382i	0.842252i
$94$	597.608	0.655729
$95$	0	0
$96$	362.908	0.385824
$97$	− 637.877i	− 0.667697i	−0.942627	−	0.333849i	$-0.891653\pi$
$97$	− 637.877i	− 0.667697i	0.942627	−	0.333849i	$-0.108347\pi$
$98$	− 111.471i	− 0.114901i
$99$	366.743	0.372313
$100$	0	0
$101$	671.148	0.661205	0.330603	−	0.943770i	$-0.392748\pi$
$101$	671.148	0.661205	0.330603	+	0.943770i	$0.392748\pi$
$102$	31.0515i	0.0301427i
$103$	912.412i	0.872841i	0.899743	+	0.436420i	$0.143754\pi$
$103$	912.412i	0.872841i	−0.899743	+	0.436420i	$0.856246\pi$
$104$	1312.51	1.23752
$105$	0	0
$106$	−1290.10	−1.18213
$107$	− 116.736i	− 0.105470i	−0.998609	−	0.0527350i	$-0.983206\pi$
$107$	− 116.736i	− 0.105470i	0.998609	−	0.0527350i	$-0.0167939\pi$
$108$	76.2683i	0.0679530i
$109$	−837.176	−0.735660	−0.367830	−	0.929893i	$-0.619899\pi$
$109$	−837.176	−0.735660	−0.367830	+	0.929893i	$0.619899\pi$
$110$	0	0
$111$	35.6911	0.0305193
$112$	− 233.959i	− 0.197385i
$113$	1086.58i	0.904572i	0.891873	+	0.452286i	$0.149391\pi$
$113$	1086.58i	0.904572i	−0.891873	+	0.452286i	$0.850609\pi$
$114$	−836.701	−0.687406
$115$	0	0
$116$	611.836	0.489720
$117$	479.691i	0.379038i
$118$	− 1910.68i	− 1.49061i
$119$	−31.8488	−0.0245343
$120$	0	0
$121$	329.495	0.247554
$122$	− 1105.14i	− 0.820121i
$123$	335.257i	0.245765i
$124$	−711.256	−0.515102
$125$	0	0
$126$	143.320	0.101333
$127$	− 537.113i	− 0.375284i	−0.982237	−	0.187642i	$-0.939916\pi$
$127$	− 537.113i	− 0.375284i	0.982237	−	0.187642i	$-0.0600845\pi$
$128$	− 266.564i	− 0.184071i
$129$	−1107.59	−0.755951
$130$	0	0
$131$	1497.39	0.998683	0.499341	−	0.866405i	$-0.333575\pi$
$131$	1497.39	0.998683	0.499341	+	0.866405i	$0.333575\pi$
$132$	345.319i	0.227698i
$133$	− 858.186i	− 0.559505i
$134$	759.120	0.489388
$135$	0	0
$136$	−112.042	−0.0706433
$137$	− 1380.09i	− 0.860650i	−0.902674	−	0.430325i	$-0.858399\pi$
$137$	− 1380.09i	− 0.860650i	0.902674	−	0.430325i	$-0.141601\pi$
$138$	− 896.412i	− 0.552954i
$139$	141.980	0.0866374	0.0433187	−	0.999061i	$-0.486207\pi$
$139$	141.980	0.0866374	0.0433187	+	0.999061i	$0.486207\pi$
$140$	0	0
$141$	−788.083	−0.470699
$142$	1342.76i	0.793537i
$143$	2171.89i	1.27009i
$144$	300.805	0.174077
$145$	0	0
$146$	1116.30	0.632781
$147$	147.000i	0.0824786i
$148$	33.6061i	0.0186649i
$149$	1943.87	1.06878	0.534390	−	0.845238i	$-0.320542\pi$
$149$	1943.87	1.06878	0.534390	+	0.845238i	$0.320542\pi$
$150$	0	0
$151$	−2654.76	−1.43074	−0.715370	−	0.698746i	$-0.753742\pi$
$151$	−2654.76	−1.43074	−0.715370	+	0.698746i	$0.753742\pi$
$152$	− 3019.03i	− 1.61102i
$153$	− 40.9485i	− 0.0216372i
$154$	648.907	0.339548
$155$	0	0
$156$	−451.669	−0.231811
$157$	1665.22i	0.846489i	0.906016	+	0.423244i	$0.139109\pi$
$157$	1665.22i	0.846489i	−0.906016	+	0.423244i	$0.860891\pi$
$158$	− 276.837i	− 0.139392i
$159$	1701.30	0.848565
$160$	0	0
$161$	919.430	0.450070
$162$	184.268i	0.0893672i
$163$	33.0732i	0.0158926i	0.999968	+	0.00794629i	$0.00252941\pi$
$163$	33.0732i	0.0158926i	−0.999968	+	0.00794629i	$0.997471\pi$
$164$	−315.673	−0.150304
$165$	0	0
$166$	1386.81	0.648417
$167$	1654.48i	0.766630i	0.923618	+	0.383315i	$0.125218\pi$
$167$	1654.48i	0.766630i	−0.923618	+	0.383315i	$0.874782\pi$
$168$	517.134i	0.237486i
$169$	−643.784	−0.293029
$170$	0	0
$171$	1103.38	0.493437
$172$	− 1042.89i	− 0.462322i
$173$	− 64.1909i	− 0.0282101i	−0.999901	−	0.0141050i	$-0.995510\pi$
$173$	− 64.1909i	− 0.0282101i	0.999901	−	0.0141050i	$-0.00448992\pi$
$174$	1478.23	0.644047
$175$	0	0
$176$	1361.95	0.583300
$177$	2519.67i	1.07000i
$178$	− 1635.75i	− 0.688791i
$179$	−3914.68	−1.63462	−0.817309	−	0.576200i	$-0.804535\pi$
$179$	−3914.68	−1.63462	−0.817309	+	0.576200i	$0.804535\pi$
$180$	0	0
$181$	−2058.04	−0.845156	−0.422578	−	0.906327i	$-0.638875\pi$
$181$	−2058.04	−0.845156	−0.422578	+	0.906327i	$0.638875\pi$
$182$	848.756i	0.345681i
$183$	1457.38i	0.588704i
$184$	3234.48	1.29592
$185$	0	0
$186$	−1718.43	−0.677428
$187$	− 185.402i	− 0.0725023i
$188$	− 742.046i	− 0.287869i
$189$	−189.000	−0.0727393
$190$	0	0
$191$	428.048	0.162160	0.0810798	−	0.996708i	$-0.474163\pi$
$191$	428.048	0.162160	0.0810798	+	0.996708i	$0.474163\pi$
$192$	1627.73i	0.611830i
$193$	− 1604.93i	− 0.598576i	−0.954163	−	0.299288i	$-0.903251\pi$
$193$	− 1604.93i	− 0.598576i	0.954163	−	0.299288i	$-0.0967491\pi$
$194$	1451.12	0.537032
$195$	0	0
$196$	−138.413	−0.0504420
$197$	3738.83i	1.35218i	0.736817	+	0.676092i	$0.236328\pi$
$197$	3738.83i	1.35218i	−0.736817	+	0.676092i	$0.763672\pi$
$198$	834.309i	0.299453i
$199$	349.030	0.124332	0.0621660	−	0.998066i	$-0.480199\pi$
$199$	349.030	0.124332	0.0621660	+	0.998066i	$0.480199\pi$
$200$	0	0
$201$	−1001.07	−0.351295
$202$	1526.81i	0.531810i
$203$	1516.19i	0.524214i
$204$	38.5565	0.0132328
$205$	0	0
$206$	−2075.66	−0.702030
$207$	1182.12i	0.396924i
$208$	1781.40i	0.593836i
$209$	4995.77	1.65342
$210$	0	0
$211$	2588.58	0.844574	0.422287	−	0.906462i	$-0.361227\pi$
$211$	2588.58	0.844574	0.422287	+	0.906462i	$0.361227\pi$
$212$	1601.92i	0.518962i
$213$	− 1770.74i	− 0.569621i
$214$	265.565	0.0848300
$215$	0	0
$216$	−664.886	−0.209443
$217$	− 1762.56i	− 0.551384i
$218$	− 1904.51i	− 0.591695i
$219$	−1472.10	−0.454226
$220$	0	0
$221$	242.502	0.0738119
$222$	81.1942i	0.0245468i
$223$	3236.21i	0.971804i	0.874013	+	0.485902i	$0.161509\pi$
$223$	3236.21i	0.971804i	−0.874013	+	0.485902i	$0.838491\pi$
$224$	−846.785	−0.252581
$225$	0	0
$226$	−2471.88	−0.727552
$227$	− 5631.62i	− 1.64662i	−0.567589	−	0.823312i	$-0.692124\pi$
$227$	− 5631.62i	− 1.64662i	0.567589	−	0.823312i	$-0.307876\pi$
$228$	1038.93i	0.301775i
$229$	−3770.25	−1.08797	−0.543985	−	0.839095i	$-0.683085\pi$
$229$	−3770.25	−1.08797	−0.543985	+	0.839095i	$0.683085\pi$
$230$	0	0
$231$	−855.733	−0.243736
$232$	5333.82i	1.50941i
$233$	6560.90i	1.84472i	0.386336	+	0.922358i	$0.373741\pi$
$233$	6560.90i	1.84472i	−0.386336	+	0.922358i	$0.626259\pi$
$234$	−1091.26	−0.304862
$235$	0	0
$236$	−2372.48	−0.654387
$237$	365.073i	0.100059i
$238$	− 72.4535i	− 0.0197330i
$239$	771.444	0.208789	0.104394	−	0.994536i	$-0.466710\pi$
$239$	771.444	0.208789	0.104394	+	0.994536i	$0.466710\pi$
$240$	0	0
$241$	1252.10	0.334668	0.167334	−	0.985900i	$-0.446484\pi$
$241$	1252.10	0.334668	0.167334	+	0.985900i	$0.446484\pi$
$242$	749.574i	0.199109i
$243$	− 243.000i	− 0.0641500i
$244$	−1372.25	−0.360037
$245$	0	0
$246$	−762.683	−0.197670
$247$	6534.35i	1.68328i
$248$	− 6200.53i	− 1.58764i
$249$	−1828.82	−0.465450
$250$	0	0
$251$	5166.27	1.29917	0.649586	−	0.760288i	$-0.274942\pi$
$251$	5166.27	1.29917	0.649586	+	0.760288i	$0.274942\pi$
$252$	− 177.959i	− 0.0444857i
$253$	5352.29i	1.33002i
$254$	1221.89	0.301843
$255$	0	0
$256$	−3734.21	−0.911672
$257$	2767.45i	0.671707i	0.941914	+	0.335854i	$0.109025\pi$
$257$	2767.45i	0.671707i	−0.941914	+	0.335854i	$0.890975\pi$
$258$	− 2519.67i	− 0.608015i
$259$	−83.2791	−0.0199796
$260$	0	0
$261$	−1949.38	−0.462313
$262$	3406.44i	0.803245i
$263$	4101.78i	0.961699i	0.876803	+	0.480849i	$0.159672\pi$
$263$	4101.78i	0.961699i	−0.876803	+	0.480849i	$0.840328\pi$
$264$	−3010.40	−0.701807
$265$	0	0
$266$	1952.30	0.450013
$267$	2157.11i	0.494432i
$268$	− 942.594i	− 0.214844i
$269$	6950.84	1.57546	0.787732	−	0.616018i	$-0.211255\pi$
$269$	6950.84	1.57546	0.787732	+	0.616018i	$0.211255\pi$
$270$	0	0
$271$	−7140.29	−1.60052	−0.800262	−	0.599651i	$-0.795306\pi$
$271$	−7140.29	−1.60052	−0.800262	+	0.599651i	$0.795306\pi$
$272$	− 152.068i	− 0.0338988i
$273$	− 1119.28i	− 0.248139i
$274$	3139.59	0.692225
$275$	0	0
$276$	−1113.07	−0.242750
$277$	1320.51i	0.286433i	0.989691	+	0.143217i	$0.0457446\pi$
$277$	1320.51i	0.286433i	−0.989691	+	0.143217i	$0.954255\pi$
$278$	322.993i	0.0696829i
$279$	2266.15	0.486275
$280$	0	0
$281$	−204.309	−0.0433738	−0.0216869	−	0.999765i	$-0.506904\pi$
$281$	−204.309	−0.0433738	−0.0216869	+	0.999765i	$0.506904\pi$
$282$	− 1792.82i	− 0.378585i
$283$	− 975.794i	− 0.204964i	−0.994735	−	0.102482i	$-0.967322\pi$
$283$	− 975.794i	− 0.204964i	0.994735	−	0.102482i	$-0.0326785\pi$
$284$	1667.30	0.348367
$285$	0	0
$286$	−4940.87	−1.02154
$287$	− 782.267i	− 0.160891i
$288$	− 1088.72i	− 0.222756i
$289$	4892.30	0.995786
$290$	0	0
$291$	−1913.63	−0.385495
$292$	− 1386.11i	− 0.277794i
$293$	− 607.919i	− 0.121212i	−0.998162	−	0.0606058i	$-0.980697\pi$
$293$	− 607.919i	− 0.121212i	0.998162	−	0.0606058i	$-0.0193032\pi$
$294$	−334.413	−0.0663379
$295$	0	0
$296$	−292.969	−0.0575286
$297$	− 1100.23i	− 0.214955i
$298$	4422.14i	0.859624i
$299$	−7000.67	−1.35405
$300$	0	0
$301$	2584.37	0.494886
$302$	− 6039.37i	− 1.15075i
$303$	− 2013.44i	− 0.381747i
$304$	4097.56	0.773064
$305$	0	0
$306$	93.1545	0.0174029
$307$	8037.08i	1.49414i	0.664747	+	0.747069i	$0.268539\pi$
$307$	8037.08i	1.49414i	−0.664747	+	0.747069i	$0.731461\pi$
$308$	− 805.744i	− 0.149063i
$309$	2737.24	0.503935
$310$	0	0
$311$	5311.60	0.968468	0.484234	−	0.874939i	$-0.339098\pi$
$311$	5311.60	0.968468	0.484234	+	0.874939i	$0.339098\pi$
$312$	− 3937.53i	− 0.714483i
$313$	1531.61i	0.276587i	0.990391	+	0.138293i	$0.0441617\pi$
$313$	1531.61i	0.276587i	−0.990391	+	0.138293i	$0.955838\pi$
$314$	−3788.23	−0.680835
$315$	0	0
$316$	−343.747	−0.0611939
$317$	4219.19i	0.747549i	0.927520	+	0.373775i	$0.121937\pi$
$317$	4219.19i	0.747549i	−0.927520	+	0.373775i	$0.878063\pi$
$318$	3870.31i	0.682505i
$319$	−8826.19	−1.54913
$320$	0	0
$321$	−350.208	−0.0608931
$322$	2091.63i	0.361993i
$323$	− 557.801i	− 0.0960893i
$324$	228.805	0.0392327
$325$	0	0
$326$	−75.2387	−0.0127825
$327$	2511.53i	0.424733i
$328$	− 2751.95i	− 0.463265i
$329$	1838.86	0.308145
$330$	0	0
$331$	8298.19	1.37797	0.688987	−	0.724773i	$-0.258056\pi$
$331$	8298.19	1.37797	0.688987	+	0.724773i	$0.258056\pi$
$332$	− 1721.99i	− 0.284658i
$333$	− 107.073i	− 0.0176203i
$334$	−3763.79	−0.616604
$335$	0	0
$336$	−701.878	−0.113960
$337$	− 4348.44i	− 0.702892i	−0.936208	−	0.351446i	$-0.885690\pi$
$337$	− 4348.44i	− 0.702892i	0.936208	−	0.351446i	$-0.114310\pi$
$338$	− 1464.56i	− 0.235684i
$339$	3259.73	0.522255
$340$	0	0
$341$	10260.4	1.62942
$342$	2510.10i	0.396874i
$343$	− 343.000i	− 0.0539949i
$344$	9091.60	1.42496
$345$	0	0
$346$	146.029	0.0226895
$347$	− 8345.54i	− 1.29110i	−0.763718	−	0.645550i	$-0.776628\pi$
$347$	− 8345.54i	− 1.29110i	0.763718	−	0.645550i	$-0.223372\pi$
$348$	− 1835.51i	− 0.282740i
$349$	9982.54	1.53110	0.765549	−	0.643378i	$-0.222468\pi$
$349$	9982.54	1.53110	0.765549	+	0.643378i	$0.222468\pi$
$350$	0	0
$351$	1439.07	0.218838
$352$	− 4929.40i	− 0.746414i
$353$	− 8801.59i	− 1.32709i	−0.748138	−	0.663543i	$-0.769052\pi$
$353$	− 8801.59i	− 1.32709i	0.748138	−	0.663543i	$-0.230948\pi$
$354$	−5732.04	−0.860606
$355$	0	0
$356$	−2031.10	−0.302383
$357$	95.5465i	0.0141649i
$358$	− 8905.56i	− 1.31473i
$359$	524.039	0.0770409	0.0385205	−	0.999258i	$-0.487736\pi$
$359$	524.039	0.0770409	0.0385205	+	0.999258i	$0.487736\pi$
$360$	0	0
$361$	8171.27	1.19132
$362$	− 4681.88i	− 0.679763i
$363$	− 988.485i	− 0.142926i
$364$	1053.90	0.151756
$365$	0	0
$366$	−3315.42	−0.473497
$367$	− 6362.72i	− 0.904991i	−0.891767	−	0.452495i	$-0.850534\pi$
$367$	− 6362.72i	− 0.904991i	0.891767	−	0.452495i	$-0.149466\pi$
$368$	4389.99i	0.621858i
$369$	1005.77	0.141893
$370$	0	0
$371$	−3969.70	−0.555516
$372$	2133.77i	0.297394i
$373$	11265.8i	1.56387i	0.623361	+	0.781935i	$0.285767\pi$
$373$	11265.8i	1.56387i	−0.623361	+	0.781935i	$0.714233\pi$
$374$	421.774	0.0583140
$375$	0	0
$376$	6468.96	0.887263
$377$	− 11544.5i	− 1.57711i
$378$	− 429.959i	− 0.0585046i
$379$	1151.71	0.156094	0.0780470	−	0.996950i	$-0.475132\pi$
$379$	1151.71	0.156094	0.0780470	+	0.996950i	$0.475132\pi$
$380$	0	0
$381$	−1611.34	−0.216670
$382$	973.774i	0.130426i
$383$	151.554i	0.0202195i	0.999949	+	0.0101097i	$0.00321809\pi$
$383$	151.554i	0.0202195i	−0.999949	+	0.0101097i	$0.996782\pi$
$384$	−799.692	−0.106274
$385$	0	0
$386$	3651.08	0.481437
$387$	3322.76i	0.436449i
$388$	− 1801.84i	− 0.235760i
$389$	−4794.18	−0.624870	−0.312435	−	0.949939i	$-0.601145\pi$
$389$	−4794.18	−0.624870	−0.312435	+	0.949939i	$0.601145\pi$
$390$	0	0
$391$	597.608	0.0772950
$392$	− 1206.65i	− 0.155471i
$393$	− 4492.17i	− 0.576590i
$394$	−8505.52	−1.08757
$395$	0	0
$396$	1035.96	0.131462
$397$	− 4623.94i	− 0.584556i	−0.956333	−	0.292278i	$-0.905587\pi$
$397$	− 4623.94i	− 0.584556i	0.956333	−	0.292278i	$-0.0944133\pi$
$398$	794.014i	0.100001i
$399$	−2574.56	−0.323030
$400$	0	0
$401$	−3610.63	−0.449642	−0.224821	−	0.974400i	$-0.572180\pi$
$401$	−3610.63	−0.449642	−0.224821	+	0.974400i	$0.572180\pi$
$402$	− 2277.36i	− 0.282548i
$403$	13420.4i	1.65885i
$404$	1895.83	0.233467
$405$	0	0
$406$	−3449.20	−0.421628
$407$	− 484.794i	− 0.0590426i
$408$	336.125i	0.0407859i
$409$	−8959.57	−1.08318	−0.541592	−	0.840641i	$-0.682178\pi$
$409$	−8959.57	−1.08318	−0.541592	+	0.840641i	$0.682178\pi$
$410$	0	0
$411$	−4140.27	−0.496896
$412$	2577.34i	0.308195i
$413$	− 5879.23i	− 0.700480i
$414$	−2689.24	−0.319248
$415$	0	0
$416$	6447.54	0.759896
$417$	− 425.940i	− 0.0500201i
$418$	11365.0i	1.32985i
$419$	7078.28	0.825290	0.412645	−	0.910892i	$-0.364605\pi$
$419$	7078.28	0.825290	0.412645	+	0.910892i	$0.364605\pi$
$420$	0	0
$421$	11551.5	1.33725	0.668626	−	0.743599i	$-0.266883\pi$
$421$	11551.5	1.33725	0.668626	+	0.743599i	$0.266883\pi$
$422$	5888.80i	0.679295i
$423$	2364.25i	0.271758i
$424$	−13965.1	−1.59954
$425$	0	0
$426$	4028.29	0.458149
$427$	− 3400.56i	− 0.385397i
$428$	− 329.750i	− 0.0372408i
$429$	6515.67	0.733286
$430$	0	0
$431$	−4064.38	−0.454232	−0.227116	−	0.973868i	$-0.572930\pi$
$431$	−4064.38	−0.454232	−0.227116	+	0.973868i	$0.572930\pi$
$432$	− 902.415i	− 0.100503i
$433$	− 17456.3i	− 1.93740i	−0.248229	−	0.968701i	$-0.579849\pi$
$433$	− 17456.3i	− 1.93740i	0.248229	−	0.968701i	$-0.420151\pi$
$434$	4009.67	0.443480
$435$	0	0
$436$	−2364.81	−0.259757
$437$	16102.9i	1.76271i
$438$	− 3348.91i	− 0.365336i
$439$	−4595.39	−0.499604	−0.249802	−	0.968297i	$-0.580365\pi$
$439$	−4595.39	−0.499604	−0.249802	+	0.968297i	$0.580365\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	551.671i	0.0593672i
$443$	306.214i	0.0328412i	0.999865	+	0.0164206i	$0.00522708\pi$
$443$	306.214i	0.0328412i	−0.999865	+	0.0164206i	$0.994773\pi$
$444$	100.818	0.0107762
$445$	0	0
$446$	−7362.10	−0.781627
$447$	− 5831.61i	− 0.617060i
$448$	− 3798.04i	− 0.400537i
$449$	−9229.22	−0.970053	−0.485026	−	0.874500i	$-0.661190\pi$
$449$	−9229.22	−0.970053	−0.485026	+	0.874500i	$0.661190\pi$
$450$	0	0
$451$	4553.82	0.475457
$452$	3069.31i	0.319399i
$453$	7964.29i	0.826038i
$454$	12811.5	1.32439
$455$	0	0
$456$	−9057.08	−0.930124
$457$	− 10992.2i	− 1.12515i	−0.826745	−	0.562577i	$-0.809810\pi$
$457$	− 10992.2i	− 1.12515i	0.826745	−	0.562577i	$-0.190190\pi$
$458$	− 8577.01i	− 0.875059i
$459$	−122.846	−0.0124922
$460$	0	0
$461$	7387.88	0.746394	0.373197	−	0.927752i	$-0.378261\pi$
$461$	7387.88	0.746394	0.373197	+	0.927752i	$0.378261\pi$
$462$	− 1946.72i	− 0.196038i
$463$	− 10163.8i	− 1.02020i	−0.860114	−	0.510101i	$-0.829608\pi$
$463$	− 10163.8i	− 1.02020i	0.860114	−	0.510101i	$-0.170392\pi$
$464$	−7239.30	−0.724302
$465$	0	0
$466$	−14925.5	−1.48371
$467$	− 15814.6i	− 1.56705i	−0.621362	−	0.783524i	$-0.713420\pi$
$467$	− 15814.6i	− 1.56705i	0.621362	−	0.783524i	$-0.286580\pi$
$468$	1355.01i	0.133836i
$469$	2335.84	0.229976
$470$	0	0
$471$	4995.65	0.488720
$472$	− 20682.6i	− 2.01694i
$473$	15044.4i	1.46246i
$474$	−830.511	−0.0804782
$475$	0	0
$476$	−89.9651	−0.00866290
$477$	− 5103.90i	− 0.489919i
$478$	1754.97i	0.167930i
$479$	1444.85	0.137823	0.0689113	−	0.997623i	$-0.478047\pi$
$479$	1444.85	0.137823	0.0689113	+	0.997623i	$0.478047\pi$
$480$	0	0
$481$	634.099	0.0601090
$482$	2848.43i	0.269175i
$483$	− 2758.29i	− 0.259848i
$484$	930.742	0.0874100
$485$	0	0
$486$	552.805	0.0515962
$487$	− 489.402i	− 0.0455378i	−0.999741	−	0.0227689i	$-0.992752\pi$
$487$	− 489.402i	− 0.0455378i	0.999741	−	0.0227689i	$-0.00724820\pi$
$488$	− 11962.9i	− 1.10970i
$489$	99.2195	0.00917559
$490$	0	0
$491$	−3941.30	−0.362257	−0.181129	−	0.983459i	$-0.557975\pi$
$491$	−3941.30	−0.362257	−0.181129	+	0.983459i	$0.557975\pi$
$492$	947.019i	0.0867783i
$493$	985.485i	0.0900284i
$494$	−14865.1	−1.35387
$495$	0	0
$496$	8415.65	0.761843
$497$	4131.73i	0.372905i
$498$	− 4160.42i	− 0.374363i
$499$	−11.0894	−0.000994850 0	−0.000497425	−	1.00000i	$-0.500158\pi$
$499$	−11.0894	−0.000994850 0	−0.000497425		1.00000i	$0.500158\pi$
$500$	0	0
$501$	4963.43	0.442614
$502$	11752.8i	1.04493i
$503$	− 7088.41i	− 0.628343i	−0.949366	−	0.314172i	$-0.898273\pi$
$503$	− 7088.41i	− 0.628343i	0.949366	−	0.314172i	$-0.101727\pi$
$504$	1551.40	0.137113
$505$	0	0
$506$	−12176.0	−1.06974
$507$	1931.35i	0.169180i
$508$	− 1517.21i	− 0.132511i
$509$	−17588.4	−1.53162	−0.765810	−	0.643067i	$-0.777662\pi$
$509$	−17588.4	−1.53162	−0.765810	+	0.643067i	$0.777662\pi$
$510$	0	0
$511$	3434.91	0.297361
$512$	− 10627.5i	− 0.917333i
$513$	− 3310.15i	− 0.284886i
$514$	−6295.72	−0.540257
$515$	0	0
$516$	−3128.66	−0.266922
$517$	10704.6i	0.910613i
$518$	− 189.453i	− 0.0160697i
$519$	−192.573	−0.0162871
$520$	0	0
$521$	−11646.6	−0.979360	−0.489680	−	0.871902i	$-0.662886\pi$
$521$	−11646.6	−0.979360	−0.489680	+	0.871902i	$0.662886\pi$
$522$	− 4434.68i	− 0.371841i
$523$	− 8965.82i	− 0.749614i	−0.927103	−	0.374807i	$-0.877709\pi$
$523$	− 8965.82i	− 0.749614i	0.927103	−	0.374807i	$-0.122291\pi$
$524$	4229.75	0.352629
$525$	0	0
$526$	−9331.22	−0.773499
$527$	− 1145.62i	− 0.0946946i
$528$	− 4085.85i	− 0.336769i
$529$	−5085.08	−0.417941
$530$	0	0
$531$	7559.01	0.617765
$532$	− 2424.16i	− 0.197558i
$533$	5956.30i	0.484045i
$534$	−4907.26	−0.397674
$535$	0	0
$536$	8217.28	0.662187
$537$	11744.0i	0.943747i
$538$	15812.6i	1.26715i
$539$	1996.71	0.159563
$540$	0	0
$541$	−195.272	−0.0155183	−0.00775914	−	0.999970i	$-0.502470\pi$
$541$	−195.272	−0.0155183	−0.00775914	+	0.999970i	$0.502470\pi$
$542$	− 16243.6i	− 1.28731i
$543$	6174.13i	0.487951i
$544$	−550.390	−0.0433783
$545$	0	0
$546$	2546.27	0.199579
$547$	− 1399.26i	− 0.109375i	−0.998504	−	0.0546874i	$-0.982584\pi$
$547$	− 1399.26i	− 0.109375i	0.998504	−	0.0546874i	$-0.0174162\pi$
$548$	− 3898.41i	− 0.303890i
$549$	4372.15	0.339888
$550$	0	0
$551$	−26554.5	−2.05310
$552$	− 9703.44i	− 0.748199i
$553$	− 851.837i	− 0.0655042i
$554$	−3004.06	−0.230380
$555$	0	0
$556$	401.059	0.0305911
$557$	43.0467i	0.00327459i	0.999999	+	0.00163730i	$0.000521167\pi$
$557$	43.0467i	0.00327459i	−0.999999	+	0.00163730i	$0.999479\pi$
$558$	5155.30i	0.391113i
$559$	−19677.8	−1.48888
$560$	0	0
$561$	−556.206	−0.0418592
$562$	− 464.786i	− 0.0348858i
$563$	19232.9i	1.43973i	0.694114	+	0.719865i	$0.255797\pi$
$563$	19232.9i	1.43973i	−0.694114	+	0.719865i	$0.744203\pi$
$564$	−2226.14	−0.166201
$565$	0	0
$566$	2219.85	0.164854
$567$	567.000i	0.0419961i
$568$	14535.1i	1.07373i
$569$	−5163.98	−0.380466	−0.190233	−	0.981739i	$-0.560924\pi$
$569$	−5163.98	−0.380466	−0.190233	+	0.981739i	$0.560924\pi$
$570$	0	0
$571$	−10231.9	−0.749899	−0.374950	−	0.927045i	$-0.622340\pi$
$571$	−10231.9	−0.749899	−0.374950	+	0.927045i	$0.622340\pi$
$572$	6135.05i	0.448460i
$573$	− 1284.14i	− 0.0936229i
$574$	1779.59	0.129406
$575$	0	0
$576$	4883.20	0.353240
$577$	16563.7i	1.19507i	0.801842	+	0.597537i	$0.203854\pi$
$577$	16563.7i	1.19507i	−0.801842	+	0.597537i	$0.796146\pi$
$578$	11129.6i	0.800916i
$579$	−4814.78	−0.345588
$580$	0	0
$581$	4267.26	0.304708
$582$	− 4353.35i	− 0.310055i
$583$	− 23108.8i	− 1.64163i
$584$	12083.7	0.856212
$585$	0	0
$586$	1382.96	0.0974910
$587$	16020.6i	1.12648i	0.826294	+	0.563239i	$0.190445\pi$
$587$	16020.6i	1.12648i	−0.826294	+	0.563239i	$0.809555\pi$
$588$	415.238i	0.0291227i
$589$	30869.4	2.15951
$590$	0	0
$591$	11216.5	0.780684
$592$	− 397.631i	− 0.0276057i
$593$	6771.14i	0.468900i	0.972128	+	0.234450i	$0.0753289\pi$
$593$	6771.14i	0.468900i	−0.972128	+	0.234450i	$0.924671\pi$
$594$	2502.93	0.172889
$595$	0	0
$596$	5490.95	0.377379
$597$	− 1047.09i	− 0.0717831i
$598$	− 15926.0i	− 1.08906i
$599$	−11070.2	−0.755120	−0.377560	−	0.925985i	$-0.623237\pi$
$599$	−11070.2	−0.755120	−0.377560	+	0.925985i	$0.623237\pi$
$600$	0	0
$601$	−24187.7	−1.64166	−0.820830	−	0.571173i	$-0.806489\pi$
$601$	−24187.7	−1.64166	−0.820830	+	0.571173i	$0.806489\pi$
$602$	5879.23i	0.398039i
$603$	3003.22i	0.202820i
$604$	−7499.05	−0.505185
$605$	0	0
$606$	4580.42	0.307041
$607$	− 10074.1i	− 0.673631i	−0.941571	−	0.336816i	$-0.890650\pi$
$607$	− 10074.1i	− 0.673631i	0.941571	−	0.336816i	$-0.109350\pi$
$608$	− 14830.6i	− 0.989243i
$609$	4548.56	0.302655
$610$	0	0
$611$	−14001.3	−0.927060
$612$	− 115.669i	− 0.00763996i
$613$	11114.6i	0.732323i	0.930551	+	0.366161i	$0.119328\pi$
$613$	11114.6i	0.732323i	−0.930551	+	0.366161i	$0.880672\pi$
$614$	−18283.7	−1.20174
$615$	0	0
$616$	7024.26	0.459441
$617$	20496.4i	1.33737i	0.743548	+	0.668683i	$0.233142\pi$
$617$	20496.4i	1.33737i	−0.743548	+	0.668683i	$0.766858\pi$
$618$	6226.98i	0.405317i
$619$	16714.4	1.08532	0.542658	−	0.839954i	$-0.317418\pi$
$619$	16714.4	1.08532	0.542658	+	0.839954i	$0.317418\pi$
$620$	0	0
$621$	3546.37	0.229164
$622$	12083.5i	0.778943i
$623$	− 5033.27i	− 0.323682i
$624$	5344.20	0.342851
$625$	0	0
$626$	−3484.28	−0.222460
$627$	− 14987.3i	− 0.954602i
$628$	4703.82i	0.298890i
$629$	−54.1295	−0.00343129
$630$	0	0
$631$	9168.53	0.578437	0.289218	−	0.957263i	$-0.406605\pi$
$631$	9168.53	0.578437	0.289218	+	0.957263i	$0.406605\pi$
$632$	− 2996.69i	− 0.188611i
$633$	− 7765.73i	− 0.487615i
$634$	−9598.30	−0.601257
$635$	0	0
$636$	4805.75	0.299623
$637$	2611.65i	0.162445i
$638$	− 20078.9i	− 1.24597i
$639$	−5312.23	−0.328871
$640$	0	0
$641$	−4273.37	−0.263319	−0.131660	−	0.991295i	$-0.542031\pi$
$641$	−4273.37	−0.263319	−0.131660	+	0.991295i	$0.542031\pi$
$642$	− 796.694i	− 0.0489766i
$643$	− 2955.75i	− 0.181281i	−0.995884	−	0.0906404i	$-0.971109\pi$
$643$	− 2955.75i	− 0.181281i	0.995884	−	0.0906404i	$-0.0288914\pi$
$644$	2597.16	0.158917
$645$	0	0
$646$	1268.95	0.0772851
$647$	22701.2i	1.37941i	0.724091	+	0.689704i	$0.242259\pi$
$647$	22701.2i	1.37941i	−0.724091	+	0.689704i	$0.757741\pi$
$648$	1994.66i	0.120922i
$649$	34224.8	2.07002
$650$	0	0
$651$	−5287.67	−0.318341
$652$	93.4235i	0.00561158i
$653$	− 1537.81i	− 0.0921582i	−0.998938	−	0.0460791i	$-0.985327\pi$
$653$	− 1537.81i	− 0.0921582i	0.998938	−	0.0460791i	$-0.0146726\pi$
$654$	−5713.52	−0.341615
$655$	0	0
$656$	3735.08	0.222302
$657$	4416.31i	0.262248i
$658$	4183.26i	0.247842i
$659$	−12338.1	−0.729323	−0.364661	−	0.931140i	$-0.618815\pi$
$659$	−12338.1	−0.729323	−0.364661	+	0.931140i	$0.618815\pi$
$660$	0	0
$661$	1845.10	0.108572	0.0542859	−	0.998525i	$-0.482712\pi$
$661$	1845.10	0.108572	0.0542859	+	0.998525i	$0.482712\pi$
$662$	18877.7i	1.10831i
$663$	− 727.505i	− 0.0426153i
$664$	15011.8	0.877369
$665$	0	0
$666$	243.583	0.0141721
$667$	− 28449.5i	− 1.65153i
$668$	4673.48i	0.270692i
$669$	9708.62	0.561072
$670$	0	0
$671$	19795.7	1.13890
$672$	2540.36i	0.145828i
$673$	− 23955.4i	− 1.37208i	−0.727563	−	0.686041i	$-0.759347\pi$
$673$	− 23955.4i	− 1.37208i	0.727563	−	0.686041i	$-0.240653\pi$
$674$	9892.34	0.565339
$675$	0	0
$676$	−1818.53	−0.103467
$677$	− 3678.26i	− 0.208814i	−0.994535	−	0.104407i	$-0.966706\pi$
$677$	− 3678.26i	− 0.208814i	0.994535	−	0.104407i	$-0.0332944\pi$
$678$	7415.63i	0.420052i
$679$	4465.14	0.252366
$680$	0	0
$681$	−16894.9	−0.950679
$682$	23341.6i	1.31055i
$683$	− 4390.87i	− 0.245991i	−0.992407	−	0.122996i	$-0.960750\pi$
$683$	− 4390.87i	− 0.245991i	0.992407	−	0.122996i	$-0.0392501\pi$
$684$	3116.78	0.174230
$685$	0	0
$686$	780.297	0.0434284
$687$	11310.7i	0.628140i
$688$	12339.6i	0.683781i
$689$	30225.8	1.67128
$690$	0	0
$691$	10371.7	0.570994	0.285497	−	0.958380i	$-0.407841\pi$
$691$	10371.7	0.570994	0.285497	+	0.958380i	$0.407841\pi$
$692$	− 181.323i	− 0.00996081i
$693$	2567.20i	0.140721i
$694$	18985.4	1.03844
$695$	0	0
$696$	16001.4	0.871456
$697$	− 508.455i	− 0.0276314i
$698$	22709.4i	1.23147i
$699$	19682.7	1.06505
$700$	0	0
$701$	109.675	0.00590922	0.00295461	−	0.999996i	$-0.499060\pi$
$701$	109.675	0.00590922	0.00295461	+	0.999996i	$0.499060\pi$
$702$	3273.77i	0.176012i
$703$	− 1458.55i	− 0.0782508i
$704$	22109.6	1.18364
$705$	0	0
$706$	20022.9	1.06738
$707$	4698.03i	0.249912i
$708$	7117.45i	0.377811i
$709$	−26918.8	−1.42589	−0.712944	−	0.701221i	$-0.752639\pi$
$709$	−26918.8	−1.42589	−0.712944	+	0.701221i	$0.752639\pi$
$710$	0	0
$711$	1095.22	0.0577693
$712$	− 17706.6i	− 0.931999i
$713$	33072.4i	1.73713i
$714$	−217.360	−0.0113929
$715$	0	0
$716$	−11058.0	−0.577174
$717$	− 2314.33i	− 0.120544i
$718$	1192.14i	0.0619644i
$719$	−15170.8	−0.786889	−0.393445	−	0.919348i	$-0.628717\pi$
$719$	−15170.8	−0.786889	−0.393445	+	0.919348i	$0.628717\pi$
$720$	0	0
$721$	−6386.88	−0.329903
$722$	18589.0i	0.958185i
$723$	− 3756.31i	− 0.193221i
$724$	−5813.46	−0.298419
$725$	0	0
$726$	2248.72	0.114956
$727$	− 33286.9i	− 1.69813i	−0.528288	−	0.849066i	$-0.677166\pi$
$727$	− 33286.9i	− 1.69813i	0.528288	−	0.849066i	$-0.322834\pi$
$728$	9187.57i	0.467739i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	1679.78	0.0849917
$732$	4116.74i	0.207868i
$733$	− 20544.0i	− 1.03521i	−0.855619	−	0.517607i	$-0.826823\pi$
$733$	− 20544.0i	− 1.03521i	0.855619	−	0.517607i	$-0.173177\pi$
$734$	14474.7	0.727888
$735$	0	0
$736$	15889.0	0.795755
$737$	13597.6i	0.679614i
$738$	2288.05i	0.114125i
$739$	−34357.2	−1.71022	−0.855109	−	0.518449i	$-0.826510\pi$
$739$	−34357.2	−1.71022	−0.855109	+	0.518449i	$0.826510\pi$
$740$	0	0
$741$	19603.1	0.971844
$742$	− 9030.73i	− 0.446804i
$743$	− 8166.99i	− 0.403254i	−0.979462	−	0.201627i	$-0.935377\pi$
$743$	− 8166.99i	− 0.403254i	0.979462	−	0.201627i	$-0.0646229\pi$
$744$	−18601.6	−0.916623
$745$	0	0
$746$	−25628.9	−1.25783
$747$	5486.47i	0.268728i
$748$	− 523.715i	− 0.0256001i
$749$	817.151	0.0398639
$750$	0	0
$751$	17080.1	0.829909	0.414954	−	0.909842i	$-0.363798\pi$
$751$	17080.1	0.829909	0.414954	+	0.909842i	$0.363798\pi$
$752$	8779.97i	0.425761i
$753$	− 15498.8i	− 0.750077i
$754$	26262.7	1.26848
$755$	0	0
$756$	−533.878	−0.0256838
$757$	− 16324.0i	− 0.783758i	−0.920017	−	0.391879i	$-0.871825\pi$
$757$	− 16324.0i	− 0.783758i	0.920017	−	0.391879i	$-0.128175\pi$
$758$	2620.06i	0.125547i
$759$	16056.9	0.767888
$760$	0	0
$761$	−32366.2	−1.54175	−0.770875	−	0.636986i	$-0.780181\pi$
$761$	−32366.2	−1.54175	−0.770875	+	0.636986i	$0.780181\pi$
$762$	− 3665.66i	− 0.174269i
$763$	− 5860.23i	− 0.278053i
$764$	1209.13	0.0572576
$765$	0	0
$766$	−344.774	−0.0162626
$767$	44765.3i	2.10741i
$768$	11202.6i	0.526354i
$769$	7948.44	0.372728	0.186364	−	0.982481i	$-0.440330\pi$
$769$	7948.44	0.372728	0.186364	+	0.982481i	$0.440330\pi$
$770$	0	0
$771$	8302.35	0.387810
$772$	− 4533.52i	− 0.211354i
$773$	− 17819.3i	− 0.829127i	−0.910020	−	0.414564i	$-0.863934\pi$
$773$	− 17819.3i	− 0.829127i	0.910020	−	0.414564i	$-0.136066\pi$
$774$	−7559.01	−0.351038
$775$	0	0
$776$	15708.0	0.726655
$777$	249.837i	0.0115352i
$778$	− 10906.4i	− 0.502586i
$779$	13700.6	0.630136
$780$	0	0
$781$	−24052.1	−1.10199
$782$	1359.51i	0.0621687i
$783$	5848.15i	0.266917i
$784$	1637.72	0.0746044
$785$	0	0
$786$	10219.3	0.463754
$787$	2912.38i	0.131912i	0.997823	+	0.0659562i	$0.0210098\pi$
$787$	2912.38i	0.131912i	−0.997823	+	0.0659562i	$0.978990\pi$
$788$	10561.3i	0.477448i
$789$	12305.3	0.555237
$790$	0	0
$791$	−7606.05	−0.341896
$792$	9031.19i	0.405188i
$793$	25892.3i	1.15948i
$794$	10519.1	0.470162
$795$	0	0
$796$	985.923	0.0439009
$797$	− 33789.1i	− 1.50172i	−0.660460	−	0.750861i	$-0.729639\pi$
$797$	− 33789.1i	− 1.50172i	0.660460	−	0.750861i	$-0.270361\pi$
$798$	− 5856.91i	− 0.259815i
$799$	1195.22	0.0529208
$800$	0	0
$801$	6471.34	0.285460
$802$	− 8213.90i	− 0.361649i
$803$	19995.7i	0.878744i
$804$	−2827.78	−0.124040
$805$	0	0
$806$	−30530.2	−1.33422
$807$	− 20852.5i	− 0.909595i
$808$	16527.3i	0.719589i
$809$	−1252.13	−0.0544159	−0.0272079	−	0.999630i	$-0.508662\pi$
$809$	−1252.13	−0.0544159	−0.0272079	+	0.999630i	$0.508662\pi$
$810$	0	0
$811$	−31913.1	−1.38178	−0.690889	−	0.722961i	$-0.742781\pi$
$811$	−31913.1	−1.38178	−0.690889	+	0.722961i	$0.742781\pi$
$812$	4282.85i	0.185097i
$813$	21420.9i	0.924063i
$814$	1102.87	0.0474882
$815$	0	0
$816$	−456.204	−0.0195715
$817$	45262.7i	1.93824i
$818$	− 20382.3i	− 0.871210i
$819$	−3357.84	−0.143263
$820$	0	0
$821$	30742.4	1.30684	0.653421	−	0.756995i	$-0.273333\pi$
$821$	30742.4	1.30684	0.653421	+	0.756995i	$0.273333\pi$
$822$	− 9418.77i	− 0.399656i
$823$	13822.6i	0.585449i	0.956197	+	0.292724i	$0.0945619\pi$
$823$	13822.6i	0.585449i	−0.956197	+	0.292724i	$0.905438\pi$
$824$	−22468.5	−0.949913
$825$	0	0
$826$	13374.8	0.563399
$827$	− 42107.1i	− 1.77051i	−0.465110	−	0.885253i	$-0.653985\pi$
$827$	− 42107.1i	− 1.77051i	0.465110	−	0.885253i	$-0.346015\pi$
$828$	3339.21i	0.140152i
$829$	38763.8	1.62403	0.812015	−	0.583636i	$-0.198371\pi$
$829$	38763.8	1.62403	0.812015	+	0.583636i	$0.198371\pi$
$830$	0	0
$831$	3961.54	0.165372
$832$	28918.8i	1.20502i
$833$	− 222.942i	− 0.00927308i
$834$	968.979	0.0402314
$835$	0	0
$836$	14111.8	0.583812
$837$	− 6798.44i	− 0.280751i
$838$	16102.5i	0.663784i
$839$	−16896.3	−0.695262	−0.347631	−	0.937631i	$-0.613014\pi$
$839$	−16896.3	−0.695262	−0.347631	+	0.937631i	$0.613014\pi$
$840$	0	0
$841$	22525.7	0.923601
$842$	26278.6i	1.07556i
$843$	612.927i	0.0250419i
$844$	7312.09	0.298214
$845$	0	0
$846$	−5378.47	−0.218576
$847$	2306.47i	0.0935668i
$848$	− 18954.0i	− 0.767552i
$849$	−2927.38	−0.118336
$850$	0	0
$851$	1562.64	0.0629455
$852$	− 5001.91i	− 0.201130i
$853$	− 46429.3i	− 1.86367i	−0.362887	−	0.931833i	$-0.618209\pi$
$853$	− 46429.3i	− 1.86367i	0.362887	−	0.931833i	$-0.381791\pi$
$854$	7735.99	0.309977
$855$	0	0
$856$	2874.67	0.114783
$857$	21206.4i	0.845272i	0.906300	+	0.422636i	$0.138895\pi$
$857$	21206.4i	0.845272i	−0.906300	+	0.422636i	$0.861105\pi$
$858$	14822.6i	0.589785i
$859$	−13876.2	−0.551163	−0.275581	−	0.961278i	$-0.588870\pi$
$859$	−13876.2	−0.551163	−0.275581	+	0.961278i	$0.588870\pi$
$860$	0	0
$861$	−2346.80	−0.0928906
$862$	− 9246.12i	− 0.365341i
$863$	− 14337.1i	− 0.565515i	−0.959191	−	0.282757i	$-0.908751\pi$
$863$	− 14337.1i	− 0.565515i	0.959191	−	0.282757i	$-0.0912491\pi$
$864$	−3266.17	−0.128608
$865$	0	0
$866$	39711.6	1.55826
$867$	− 14676.9i	− 0.574918i
$868$	− 4978.79i	− 0.194690i
$869$	4958.81	0.193574
$870$	0	0
$871$	−17785.4	−0.691889
$872$	− 20615.8i	− 0.800619i
$873$	5740.89i	0.222566i
$874$	−36632.8	−1.41776
$875$	0	0
$876$	−4158.33	−0.160384
$877$	− 24369.3i	− 0.938304i	−0.883118	−	0.469152i	$-0.844560\pi$
$877$	− 24369.3i	− 0.938304i	0.883118	−	0.469152i	$-0.155440\pi$
$878$	− 10454.1i	− 0.401834i
$879$	−1823.76	−0.0699815
$880$	0	0
$881$	−26127.0	−0.999140	−0.499570	−	0.866273i	$-0.666509\pi$
$881$	−26127.0	−0.999140	−0.499570	+	0.866273i	$0.666509\pi$
$882$	1003.24i	0.0383002i
$883$	15713.1i	0.598855i	0.954119	+	0.299428i	$0.0967957\pi$
$883$	15713.1i	0.598855i	−0.954119	+	0.299428i	$0.903204\pi$
$884$	685.007	0.0260625
$885$	0	0
$886$	−696.611	−0.0264143
$887$	13139.5i	0.497385i	0.968583	+	0.248692i	$0.0800008\pi$
$887$	13139.5i	0.497385i	−0.968583	+	0.248692i	$0.919999\pi$
$888$	878.907i	0.0332142i
$889$	3759.79	0.141844
$890$	0	0
$891$	−3300.68	−0.124104
$892$	9141.48i	0.343138i
$893$	32205.8i	1.20686i
$894$	13266.4	0.496304
$895$	0	0
$896$	1865.95	0.0695725
$897$	21002.0i	0.781758i
$898$	− 20995.7i	− 0.780218i
$899$	−54538.1	−2.02330
$900$	0	0
$901$	−2580.21	−0.0954043
$902$	10359.6i	0.382412i
$903$	− 7753.12i	− 0.285723i
$904$	−26757.4	−0.984446
$905$	0	0
$906$	−18118.1	−0.664386
$907$	− 3799.71i	− 0.139104i	−0.997578	−	0.0695519i	$-0.977843\pi$
$907$	− 3799.71i	− 0.139104i	0.997578	−	0.0695519i	$-0.0221570\pi$
$908$	− 15907.9i	− 0.581413i
$909$	−6040.33	−0.220402
$910$	0	0
$911$	−51528.4	−1.87400	−0.936998	−	0.349334i	$-0.886408\pi$
$911$	−51528.4	−1.87400	−0.936998	+	0.349334i	$0.886408\pi$
$912$	− 12292.7i	− 0.446329i
$913$	24841.0i	0.900458i
$914$	25006.4	0.904966
$915$	0	0
$916$	−10650.0	−0.384156
$917$	10481.7i	0.377467i
$918$	− 279.463i	− 0.0100476i
$919$	16984.7	0.609657	0.304828	−	0.952407i	$-0.401401\pi$
$919$	16984.7	0.609657	0.304828	+	0.952407i	$0.401401\pi$
$920$	0	0
$921$	24111.2	0.862641
$922$	16806.8i	0.600329i
$923$	− 31459.6i	− 1.12189i
$924$	−2417.23	−0.0860618
$925$	0	0
$926$	23121.9	0.820553
$927$	− 8211.71i	− 0.290947i
$928$	26201.7i	0.926846i
$929$	5451.85	0.192540	0.0962699	−	0.995355i	$-0.469309\pi$
$929$	5451.85	0.192540	0.0962699	+	0.995355i	$0.469309\pi$
$930$	0	0
$931$	6007.30	0.211473
$932$	18532.9i	0.651358i
$933$	− 15934.8i	− 0.559145i
$934$	35976.8	1.26038
$935$	0	0
$936$	−11812.6	−0.412507
$937$	42429.4i	1.47930i	0.672989	+	0.739652i	$0.265010\pi$
$937$	42429.4i	1.47930i	−0.672989	+	0.739652i	$0.734990\pi$
$938$	5313.84i	0.184971i
$939$	4594.82	0.159687
$940$	0	0
$941$	−32977.9	−1.14245	−0.571226	−	0.820793i	$-0.693532\pi$
$941$	−32977.9	−1.14245	−0.571226	+	0.820793i	$0.693532\pi$
$942$	11364.7i	0.393080i
$943$	14678.4i	0.506886i
$944$	28071.5	0.967848
$945$	0	0
$946$	−34224.8	−1.17626
$947$	− 23753.4i	− 0.815082i	−0.913187	−	0.407541i	$-0.866386\pi$
$947$	− 23753.4i	− 0.815082i	0.913187	−	0.407541i	$-0.133614\pi$
$948$	1031.24i	0.0353303i
$949$	−26153.9	−0.894616
$950$	0	0
$951$	12657.6	0.431598
$952$	− 784.291i	− 0.0267006i
$953$	28074.3i	0.954267i	0.878831	+	0.477134i	$0.158324\pi$
$953$	28074.3i	0.954267i	−0.878831	+	0.477134i	$0.841676\pi$
$954$	11610.9	0.394044
$955$	0	0
$956$	2179.14	0.0737221
$957$	26478.6i	0.894389i
$958$	3286.92i	0.110851i
$959$	9660.63	0.325295
$960$	0	0
$961$	33609.2	1.12817
$962$	1442.52i	0.0483460i
$963$	1050.62i	0.0351567i
$964$	3536.88	0.118169
$965$	0	0
$966$	6274.88	0.208997
$967$	− 11150.3i	− 0.370806i	−0.982663	−	0.185403i	$-0.940641\pi$
$967$	− 11150.3i	− 0.370806i	0.982663	−	0.185403i	$-0.0593590\pi$
$968$	8113.95i	0.269414i
$969$	−1673.40	−0.0554772
$970$	0	0
$971$	6059.04	0.200251	0.100126	−	0.994975i	$-0.468076\pi$
$971$	6059.04	0.200251	0.100126	+	0.994975i	$0.468076\pi$
$972$	− 686.415i	− 0.0226510i
$973$	993.861i	0.0327459i
$974$	1113.35	0.0366263
$975$	0	0
$976$	16236.6	0.532500
$977$	5700.49i	0.186668i	0.995635	+	0.0933341i	$0.0297525\pi$
$977$	5700.49i	0.186668i	−0.995635	+	0.0933341i	$0.970248\pi$
$978$	225.716i	0.00737997i
$979$	29300.2	0.956526
$980$	0	0
$981$	7534.59	0.245220
$982$	− 8966.12i	− 0.291365i
$983$	− 197.480i	− 0.00640757i	−0.999995	−	0.00320378i	$-0.998980\pi$
$983$	− 197.480i	− 0.00640757i	0.999995	−	0.00320378i	$-0.00101980\pi$
$984$	−8255.85	−0.267466
$985$	0	0
$986$	−2241.90	−0.0724103
$987$	− 5516.58i	− 0.177908i
$988$	18457.9i	0.594357i
$989$	−48492.9	−1.55913
$990$	0	0
$991$	20620.8	0.660990	0.330495	−	0.943808i	$-0.392784\pi$
$991$	20620.8	0.660990	0.330495	+	0.943808i	$0.392784\pi$
$992$	− 30459.3i	− 0.974884i
$993$	− 24894.6i	− 0.795574i
$994$	−9399.35	−0.299929
$995$	0	0
$996$	−5165.97	−0.164348
$997$	19326.8i	0.613928i	0.951721	+	0.306964i	$0.0993132\pi$
$997$	19326.8i	0.613928i	−0.951721	+	0.306964i	$0.900687\pi$
$998$	− 25.2275i	0 0.000800162i
$999$	−321.220	−0.0101731

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.4.d.g.274.3		4
5.2	odd	4		525.4.a.n.1.1		2
5.3	odd	4		21.4.a.c.1.2	✓	2
5.4	even	2	inner	525.4.d.g.274.2		4
15.2	even	4		1575.4.a.p.1.2		2
15.8	even	4		63.4.a.e.1.1		2
20.3	even	4		336.4.a.m.1.1		2
35.3	even	12		147.4.e.m.79.1		4
35.13	even	4		147.4.a.i.1.2		2
35.18	odd	12		147.4.e.l.79.1		4
35.23	odd	12		147.4.e.l.67.1		4
35.33	even	12		147.4.e.m.67.1		4
40.3	even	4		1344.4.a.bo.1.2		2
40.13	odd	4		1344.4.a.bg.1.2		2
60.23	odd	4		1008.4.a.ba.1.2		2
105.23	even	12		441.4.e.q.361.2		4
105.38	odd	12		441.4.e.p.226.2		4
105.53	even	12		441.4.e.q.226.2		4
105.68	odd	12		441.4.e.p.361.2		4
105.83	odd	4		441.4.a.r.1.1		2
140.83	odd	4		2352.4.a.bz.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.4.a.c.1.2	✓	2	5.3	odd	4
63.4.a.e.1.1		2	15.8	even	4
147.4.a.i.1.2		2	35.13	even	4
147.4.e.l.67.1		4	35.23	odd	12
147.4.e.l.79.1		4	35.18	odd	12
147.4.e.m.67.1		4	35.33	even	12
147.4.e.m.79.1		4	35.3	even	12
336.4.a.m.1.1		2	20.3	even	4
441.4.a.r.1.1		2	105.83	odd	4
441.4.e.p.226.2		4	105.38	odd	12
441.4.e.p.361.2		4	105.68	odd	12
441.4.e.q.226.2		4	105.53	even	12
441.4.e.q.361.2		4	105.23	even	12
525.4.a.n.1.1		2	5.2	odd	4
525.4.d.g.274.2		4	5.4	even	2	inner
525.4.d.g.274.3		4	1.1	even	1	trivial
1008.4.a.ba.1.2		2	60.23	odd	4
1344.4.a.bg.1.2		2	40.13	odd	4
1344.4.a.bo.1.2		2	40.3	even	4
1575.4.a.p.1.2		2	15.2	even	4
2352.4.a.bz.1.2		2	140.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.27492i q^{2} -3.00000i q^{3} +2.82475 q^{4} +6.82475 q^{6} +7.00000i q^{7} +24.6254i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q+2.27492i q^{2} -3.00000i q^{3} +2.82475 q^{4} +6.82475 q^{6} +7.00000i q^{7} +24.6254i q^{8} -9.00000 q^{9} -40.7492 q^{11} -8.47425i q^{12} -53.2990i q^{13} -15.9244 q^{14} -33.4228 q^{16} +4.54983i q^{17} -20.4743i q^{18} -122.598 q^{19} +21.0000 q^{21} -92.7010i q^{22} -131.347i q^{23} +73.8762 q^{24} +121.251 q^{26} +27.0000i q^{27} +19.7733i q^{28} +216.598 q^{29} -251.794 q^{31} +120.969i q^{32} +122.248i q^{33} -10.3505 q^{34} -25.4228 q^{36} +11.8970i q^{37} -278.900i q^{38} -159.897 q^{39} -111.752 q^{41} +47.7733i q^{42} -369.196i q^{43} -115.106 q^{44} +298.804 q^{46} -262.694i q^{47} +100.268i q^{48} -49.0000 q^{49} +13.6495 q^{51} -150.556i q^{52} +567.100i q^{53} -61.4228 q^{54} -172.378 q^{56} +367.794i q^{57} +492.743i q^{58} -839.890 q^{59} -485.794 q^{61} -572.811i q^{62} -63.0000i q^{63} -542.577 q^{64} -278.103 q^{66} -333.691i q^{67} +12.8522i q^{68} -394.042 q^{69} +590.248 q^{71} -221.629i q^{72} -490.701i q^{73} -27.0647 q^{74} -346.309 q^{76} -285.244i q^{77} -363.752i q^{78} -121.691 q^{79} +81.0000 q^{81} -254.228i q^{82} -609.608i q^{83} +59.3198 q^{84} +839.890 q^{86} -649.794i q^{87} -1003.47i q^{88} -719.038 q^{89} +373.093 q^{91} -371.023i q^{92} +755.382i q^{93} +597.608 q^{94} +362.908 q^{96} -637.877i q^{97} -111.471i q^{98} +366.743 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 34 q^{4} - 18 q^{6} - 36 q^{9}+O(q^{10}) \) 4 * q - 34 * q^4 - 18 * q^6 - 36 * q^9	\( 4 q - 34 q^{4} - 18 q^{6} - 36 q^{9} - 12 q^{11} + 42 q^{14} + 274 q^{16} - 128 q^{19} + 84 q^{21} + 522 q^{24} + 636 q^{26} + 504 q^{29} + 80 q^{31} - 132 q^{34} + 306 q^{36} - 96 q^{39} - 900 q^{41} - 1608 q^{44} + 1920 q^{46} - 196 q^{49} - 36 q^{51} + 162 q^{54} - 1218 q^{56} - 1608 q^{59} - 856 q^{61} - 2578 q^{64} - 1656 q^{66} - 36 q^{69} + 1908 q^{71} - 2796 q^{74} - 3016 q^{76} + 1144 q^{79} + 324 q^{81} - 714 q^{84} + 1608 q^{86} - 732 q^{89} + 224 q^{91} + 3840 q^{94} - 1674 q^{96} + 108 q^{99}+O(q^{100}) \) 4 * q - 34 * q^4 - 18 * q^6 - 36 * q^9 - 12 * q^11 + 42 * q^14 + 274 * q^16 - 128 * q^19 + 84 * q^21 + 522 * q^24 + 636 * q^26 + 504 * q^29 + 80 * q^31 - 132 * q^34 + 306 * q^36 - 96 * q^39 - 900 * q^41 - 1608 * q^44 + 1920 * q^46 - 196 * q^49 - 36 * q^51 + 162 * q^54 - 1218 * q^56 - 1608 * q^59 - 856 * q^61 - 2578 * q^64 - 1656 * q^66 - 36 * q^69 + 1908 * q^71 - 2796 * q^74 - 3016 * q^76 + 1144 * q^79 + 324 * q^81 - 714 * q^84 + 1608 * q^86 - 732 * q^89 + 224 * q^91 + 3840 * q^94 - 1674 * q^96 + 108 * q^99

Introduction

L-functions

Modular forms

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Database

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