LMFDB - Embedded newform 1502.2.a.h.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1502,2,Mod(1,1502)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1502, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1502.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1502 = 2 \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1502.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:	$11.9935303836$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 6 x^{18} - 23 x^{17} + 190 x^{16} + 128 x^{15} - 2394 x^{14} + 749 x^{13} + 15539 x^{12} + \cdots + 4222 $ x^19 - 6x^18 - 23x^17 + 190x^16 + 128x^15 - 2394x^14 + 749x^13 + 15539x^12 - 11338x^11 - 56744x^10 + 50183x^9 + 120237x^8 - 102992x^7 - 148539x^6 + 99519x^5 + 104388x^4 - 41081x^3 - 35733x^2 + 5542x + 4222
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$1.74260$ of defining polynomial
Character	$\chi$	$=$	1502.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.74260 q^{3} +1.00000 q^{4} +1.47569 q^{5} -1.74260 q^{6} +4.24779 q^{7} -1.00000 q^{8} +0.0366584 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.74260 q^{3} +1.00000 q^{4} +1.47569 q^{5} -1.74260 q^{6} +4.24779 q^{7} -1.00000 q^{8} +0.0366584 q^{9} -1.47569 q^{10} +0.362193 q^{11} +1.74260 q^{12} +6.28246 q^{13} -4.24779 q^{14} +2.57153 q^{15} +1.00000 q^{16} +7.83788 q^{17} -0.0366584 q^{18} -3.75277 q^{19} +1.47569 q^{20} +7.40221 q^{21} -0.362193 q^{22} -5.42454 q^{23} -1.74260 q^{24} -2.82235 q^{25} -6.28246 q^{26} -5.16392 q^{27} +4.24779 q^{28} -9.69191 q^{29} -2.57153 q^{30} +4.40895 q^{31} -1.00000 q^{32} +0.631158 q^{33} -7.83788 q^{34} +6.26841 q^{35} +0.0366584 q^{36} +0.581067 q^{37} +3.75277 q^{38} +10.9478 q^{39} -1.47569 q^{40} +9.56330 q^{41} -7.40221 q^{42} -5.48964 q^{43} +0.362193 q^{44} +0.0540963 q^{45} +5.42454 q^{46} +3.06302 q^{47} +1.74260 q^{48} +11.0437 q^{49} +2.82235 q^{50} +13.6583 q^{51} +6.28246 q^{52} -9.03537 q^{53} +5.16392 q^{54} +0.534483 q^{55} -4.24779 q^{56} -6.53958 q^{57} +9.69191 q^{58} -12.9457 q^{59} +2.57153 q^{60} +2.93554 q^{61} -4.40895 q^{62} +0.155717 q^{63} +1.00000 q^{64} +9.27093 q^{65} -0.631158 q^{66} -15.9238 q^{67} +7.83788 q^{68} -9.45280 q^{69} -6.26841 q^{70} -0.901407 q^{71} -0.0366584 q^{72} +1.16811 q^{73} -0.581067 q^{74} -4.91823 q^{75} -3.75277 q^{76} +1.53852 q^{77} -10.9478 q^{78} +13.8034 q^{79} +1.47569 q^{80} -9.10863 q^{81} -9.56330 q^{82} -13.2635 q^{83} +7.40221 q^{84} +11.5663 q^{85} +5.48964 q^{86} -16.8891 q^{87} -0.362193 q^{88} -12.3489 q^{89} -0.0540963 q^{90} +26.6866 q^{91} -5.42454 q^{92} +7.68304 q^{93} -3.06302 q^{94} -5.53791 q^{95} -1.74260 q^{96} +6.96468 q^{97} -11.0437 q^{98} +0.0132774 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9}+O(q^{10}) $ 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9	\( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9} - 2 q^{10} - 7 q^{11} + 6 q^{12} + 19 q^{13} - 13 q^{14} + 6 q^{15} + 19 q^{16} + 11 q^{17} - 25 q^{18} + 7 q^{19} + 2 q^{20} + 7 q^{21} + 7 q^{22} + 12 q^{23} - 6 q^{24} + 37 q^{25} - 19 q^{26} + 24 q^{27} + 13 q^{28} - 8 q^{29} - 6 q^{30} + 32 q^{31} - 19 q^{32} + 24 q^{33} - 11 q^{34} - 19 q^{35} + 25 q^{36} + 41 q^{37} - 7 q^{38} - 2 q^{39} - 2 q^{40} + 15 q^{41} - 7 q^{42} + 15 q^{43} - 7 q^{44} + 28 q^{45} - 12 q^{46} - 6 q^{47} + 6 q^{48} + 42 q^{49} - 37 q^{50} + 8 q^{51} + 19 q^{52} + 6 q^{53} - 24 q^{54} + 22 q^{55} - 13 q^{56} + 24 q^{57} + 8 q^{58} - 4 q^{59} + 6 q^{60} + 13 q^{61} - 32 q^{62} + 37 q^{63} + 19 q^{64} + 20 q^{65} - 24 q^{66} + 47 q^{67} + 11 q^{68} + 15 q^{69} + 19 q^{70} - 25 q^{72} + 64 q^{73} - 41 q^{74} - 3 q^{75} + 7 q^{76} - 4 q^{77} + 2 q^{78} + 34 q^{79} + 2 q^{80} + 27 q^{81} - 15 q^{82} + 4 q^{83} + 7 q^{84} + 21 q^{85} - 15 q^{86} + 7 q^{88} + 18 q^{89} - 28 q^{90} + 28 q^{91} + 12 q^{92} + 43 q^{93} + 6 q^{94} - 8 q^{95} - 6 q^{96} + 82 q^{97} - 42 q^{98} - 2 q^{99}+O(q^{100}) \) 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9 - 2 * q^10 - 7 * q^11 + 6 * q^12 + 19 * q^13 - 13 * q^14 + 6 * q^15 + 19 * q^16 + 11 * q^17 - 25 * q^18 + 7 * q^19 + 2 * q^20 + 7 * q^21 + 7 * q^22 + 12 * q^23 - 6 * q^24 + 37 * q^25 - 19 * q^26 + 24 * q^27 + 13 * q^28 - 8 * q^29 - 6 * q^30 + 32 * q^31 - 19 * q^32 + 24 * q^33 - 11 * q^34 - 19 * q^35 + 25 * q^36 + 41 * q^37 - 7 * q^38 - 2 * q^39 - 2 * q^40 + 15 * q^41 - 7 * q^42 + 15 * q^43 - 7 * q^44 + 28 * q^45 - 12 * q^46 - 6 * q^47 + 6 * q^48 + 42 * q^49 - 37 * q^50 + 8 * q^51 + 19 * q^52 + 6 * q^53 - 24 * q^54 + 22 * q^55 - 13 * q^56 + 24 * q^57 + 8 * q^58 - 4 * q^59 + 6 * q^60 + 13 * q^61 - 32 * q^62 + 37 * q^63 + 19 * q^64 + 20 * q^65 - 24 * q^66 + 47 * q^67 + 11 * q^68 + 15 * q^69 + 19 * q^70 - 25 * q^72 + 64 * q^73 - 41 * q^74 - 3 * q^75 + 7 * q^76 - 4 * q^77 + 2 * q^78 + 34 * q^79 + 2 * q^80 + 27 * q^81 - 15 * q^82 + 4 * q^83 + 7 * q^84 + 21 * q^85 - 15 * q^86 + 7 * q^88 + 18 * q^89 - 28 * q^90 + 28 * q^91 + 12 * q^92 + 43 * q^93 + 6 * q^94 - 8 * q^95 - 6 * q^96 + 82 * q^97 - 42 * q^98 - 2 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.74260	1.00609	0.503046	−	0.864260i	$-0.332213\pi$
$3$	1.74260	1.00609	0.503046	+	0.864260i	$0.332213\pi$
$4$	1.00000	0.500000
$5$	1.47569	0.659947	0.329973	−	0.943990i	$-0.392960\pi$
$5$	1.47569	0.659947	0.329973	+	0.943990i	$0.392960\pi$
$6$	−1.74260	−0.711414
$7$	4.24779	1.60551	0.802757	−	0.596306i	$-0.203366\pi$
$7$	4.24779	1.60551	0.802757	+	0.596306i	$0.203366\pi$
$8$	−1.00000	−0.353553
$9$	0.0366584	0.0122195
$10$	−1.47569	−0.466653
$11$	0.362193	0.109205	0.0546026	−	0.998508i	$-0.482611\pi$
$11$	0.362193	0.109205	0.0546026	+	0.998508i	$0.482611\pi$
$12$	1.74260	0.503046
$13$	6.28246	1.74244	0.871220	−	0.490893i	$-0.163329\pi$
$13$	6.28246	1.74244	0.871220	+	0.490893i	$0.163329\pi$
$14$	−4.24779	−1.13527
$15$	2.57153	0.663967
$16$	1.00000	0.250000
$17$	7.83788	1.90097	0.950483	−	0.310776i	$-0.100589\pi$
$17$	7.83788	1.90097	0.950483	+	0.310776i	$0.100589\pi$
$18$	−0.0366584	−0.00864046
$19$	−3.75277	−0.860944	−0.430472	−	0.902604i	$-0.641653\pi$
$19$	−3.75277	−0.860944	−0.430472	+	0.902604i	$0.641653\pi$
$20$	1.47569	0.329973
$21$	7.40221	1.61529
$22$	−0.362193	−0.0772198
$23$	−5.42454	−1.13109	−0.565547	−	0.824716i	$-0.691335\pi$
$23$	−5.42454	−1.13109	−0.565547	+	0.824716i	$0.691335\pi$
$24$	−1.74260	−0.355707
$25$	−2.82235	−0.564470
$26$	−6.28246	−1.23209
$27$	−5.16392	−0.993797
$28$	4.24779	0.802757
$29$	−9.69191	−1.79974	−0.899871	−	0.436156i	$-0.856339\pi$
$29$	−9.69191	−1.79974	−0.899871	+	0.436156i	$0.856339\pi$
$30$	−2.57153	−0.469495
$31$	4.40895	0.791871	0.395935	−	0.918278i	$-0.370420\pi$
$31$	4.40895	0.791871	0.395935	+	0.918278i	$0.370420\pi$
$32$	−1.00000	−0.176777
$33$	0.631158	0.109870
$34$	−7.83788	−1.34419
$35$	6.26841	1.05955
$36$	0.0366584	0.00610973
$37$	0.581067	0.0955269	0.0477634	−	0.998859i	$-0.484791\pi$
$37$	0.581067	0.0955269	0.0477634	+	0.998859i	$0.484791\pi$
$38$	3.75277	0.608779
$39$	10.9478	1.75305
$40$	−1.47569	−0.233326
$41$	9.56330	1.49354	0.746768	−	0.665084i	$-0.231604\pi$
$41$	9.56330	1.49354	0.746768	+	0.665084i	$0.231604\pi$
$42$	−7.40221	−1.14219
$43$	−5.48964	−0.837162	−0.418581	−	0.908179i	$-0.637472\pi$
$43$	−5.48964	−0.837162	−0.418581	+	0.908179i	$0.637472\pi$
$44$	0.362193	0.0546026
$45$	0.0540963	0.00806419
$46$	5.42454	0.799804
$47$	3.06302	0.446787	0.223393	−	0.974728i	$-0.428287\pi$
$47$	3.06302	0.446787	0.223393	+	0.974728i	$0.428287\pi$
$48$	1.74260	0.251523
$49$	11.0437	1.57768
$50$	2.82235	0.399141
$51$	13.6583	1.91255
$52$	6.28246	0.871220
$53$	−9.03537	−1.24110	−0.620552	−	0.784165i	$-0.713091\pi$
$53$	−9.03537	−1.24110	−0.620552	+	0.784165i	$0.713091\pi$
$54$	5.16392	0.702721
$55$	0.534483	0.0720697
$56$	−4.24779	−0.567635
$57$	−6.53958	−0.866188
$58$	9.69191	1.27261
$59$	−12.9457	−1.68538	−0.842692	−	0.538395i	$-0.819031\pi$
$59$	−12.9457	−1.68538	−0.842692	+	0.538395i	$0.819031\pi$
$60$	2.57153	0.331983
$61$	2.93554	0.375857	0.187929	−	0.982183i	$-0.439823\pi$
$61$	2.93554	0.375857	0.187929	+	0.982183i	$0.439823\pi$
$62$	−4.40895	−0.559937
$63$	0.155717	0.0196185
$64$	1.00000	0.125000
$65$	9.27093	1.14992
$66$	−0.631158	−0.0776901
$67$	−15.9238	−1.94540	−0.972701	−	0.232061i	$-0.925453\pi$
$67$	−15.9238	−1.94540	−0.972701	+	0.232061i	$0.925453\pi$
$68$	7.83788	0.950483
$69$	−9.45280	−1.13798
$70$	−6.26841	−0.749218
$71$	−0.901407	−0.106977	−0.0534887	−	0.998568i	$-0.517034\pi$
$71$	−0.901407	−0.106977	−0.0534887	+	0.998568i	$0.517034\pi$
$72$	−0.0366584	−0.00432023
$73$	1.16811	0.136717	0.0683587	−	0.997661i	$-0.478224\pi$
$73$	1.16811	0.136717	0.0683587	+	0.997661i	$0.478224\pi$
$74$	−0.581067	−0.0675477
$75$	−4.91823	−0.567908
$76$	−3.75277	−0.430472
$77$	1.53852	0.175331
$78$	−10.9478	−1.23960
$79$	13.8034	1.55300	0.776499	−	0.630118i	$-0.216993\pi$
$79$	13.8034	1.55300	0.776499	+	0.630118i	$0.216993\pi$
$80$	1.47569	0.164987
$81$	−9.10863	−1.01207
$82$	−9.56330	−1.05609
$83$	−13.2635	−1.45586	−0.727929	−	0.685652i	$-0.759517\pi$
$83$	−13.2635	−1.45586	−0.727929	+	0.685652i	$0.759517\pi$
$84$	7.40221	0.807647
$85$	11.5663	1.25454
$86$	5.48964	0.591963
$87$	−16.8891	−1.81070
$88$	−0.362193	−0.0386099
$89$	−12.3489	−1.30898	−0.654488	−	0.756073i	$-0.727116\pi$
$89$	−12.3489	−1.30898	−0.654488	+	0.756073i	$0.727116\pi$
$90$	−0.0540963	−0.00570225
$91$	26.6866	2.79751
$92$	−5.42454	−0.565547
$93$	7.68304	0.796694
$94$	−3.06302	−0.315926
$95$	−5.53791	−0.568177
$96$	−1.74260	−0.177853
$97$	6.96468	0.707156	0.353578	−	0.935405i	$-0.384965\pi$
$97$	6.96468	0.707156	0.353578	+	0.935405i	$0.384965\pi$
$98$	−11.0437	−1.11559
$99$	0.0132774	0.00133443
$100$	−2.82235	−0.282235
$101$	5.05830	0.503319	0.251660	−	0.967816i	$-0.419024\pi$
$101$	5.05830	0.503319	0.251660	+	0.967816i	$0.419024\pi$
$102$	−13.6583	−1.35237
$103$	11.2720	1.11067	0.555333	−	0.831628i	$-0.312591\pi$
$103$	11.2720	1.11067	0.555333	+	0.831628i	$0.312591\pi$
$104$	−6.28246	−0.616046
$105$	10.9233	1.06601
$106$	9.03537	0.877593
$107$	−7.11067	−0.687414	−0.343707	−	0.939077i	$-0.611683\pi$
$107$	−7.11067	−0.687414	−0.343707	+	0.939077i	$0.611683\pi$
$108$	−5.16392	−0.496899
$109$	1.13607	0.108816	0.0544081	−	0.998519i	$-0.482673\pi$
$109$	1.13607	0.108816	0.0544081	+	0.998519i	$0.482673\pi$
$110$	−0.534483	−0.0509609
$111$	1.01257	0.0961087
$112$	4.24779	0.401379
$113$	3.16439	0.297681	0.148841	−	0.988861i	$-0.452446\pi$
$113$	3.16439	0.297681	0.148841	+	0.988861i	$0.452446\pi$
$114$	6.53958	0.612488
$115$	−8.00491	−0.746462
$116$	−9.69191	−0.899871
$117$	0.230305	0.0212917
$118$	12.9457	1.19175
$119$	33.2937	3.05203
$120$	−2.57153	−0.234748
$121$	−10.8688	−0.988074
$122$	−2.93554	−0.265771
$123$	16.6650	1.50263
$124$	4.40895	0.395935
$125$	−11.5433	−1.03247
$126$	−0.155717	−0.0138724
$127$	14.9086	1.32293	0.661464	−	0.749977i	$-0.269936\pi$
$127$	14.9086	1.32293	0.661464	+	0.749977i	$0.269936\pi$
$128$	−1.00000	−0.0883883
$129$	−9.56625	−0.842261
$130$	−9.27093	−0.813115
$131$	21.3563	1.86591	0.932954	−	0.359995i	$-0.117221\pi$
$131$	21.3563	1.86591	0.932954	+	0.359995i	$0.117221\pi$
$132$	0.631158	0.0549352
$133$	−15.9410	−1.38226
$134$	15.9238	1.37561
$135$	−7.62033	−0.655853
$136$	−7.83788	−0.672093
$137$	0.733153	0.0626375	0.0313188	−	0.999509i	$-0.490029\pi$
$137$	0.733153	0.0626375	0.0313188	+	0.999509i	$0.490029\pi$
$138$	9.45280	0.804676
$139$	6.32332	0.536337	0.268168	−	0.963372i	$-0.413582\pi$
$139$	6.32332	0.536337	0.268168	+	0.963372i	$0.413582\pi$
$140$	6.26841	0.529777
$141$	5.33761	0.449508
$142$	0.901407	0.0756444
$143$	2.27546	0.190284
$144$	0.0366584	0.00305487
$145$	−14.3022	−1.18773
$146$	−1.16811	−0.0966738
$147$	19.2448	1.58729
$148$	0.581067	0.0477634
$149$	11.4383	0.937062	0.468531	−	0.883447i	$-0.344783\pi$
$149$	11.4383	0.937062	0.468531	+	0.883447i	$0.344783\pi$
$150$	4.91823	0.401572
$151$	−7.32718	−0.596278	−0.298139	−	0.954523i	$-0.596366\pi$
$151$	−7.32718	−0.596278	−0.298139	+	0.954523i	$0.596366\pi$
$152$	3.75277	0.304390
$153$	0.287324	0.0232288
$154$	−1.53852	−0.123977
$155$	6.50622	0.522592
$156$	10.9478	0.876527
$157$	−8.21392	−0.655542	−0.327771	−	0.944757i	$-0.606297\pi$
$157$	−8.21392	−0.655542	−0.327771	+	0.944757i	$0.606297\pi$
$158$	−13.8034	−1.09814
$159$	−15.7451	−1.24866
$160$	−1.47569	−0.116663
$161$	−23.0423	−1.81599
$162$	9.10863	0.715642
$163$	−17.4726	−1.36856	−0.684280	−	0.729219i	$-0.739883\pi$
$163$	−17.4726	−1.36856	−0.684280	+	0.729219i	$0.739883\pi$
$164$	9.56330	0.746768
$165$	0.931390	0.0725086
$166$	13.2635	1.02945
$167$	−0.875011	−0.0677104	−0.0338552	−	0.999427i	$-0.510778\pi$
$167$	−0.875011	−0.0677104	−0.0338552	+	0.999427i	$0.510778\pi$
$168$	−7.40221	−0.571093
$169$	26.4693	2.03610
$170$	−11.5663	−0.887091
$171$	−0.137570	−0.0105203
$172$	−5.48964	−0.418581
$173$	−5.85534	−0.445173	−0.222587	−	0.974913i	$-0.571450\pi$
$173$	−5.85534	−0.445173	−0.222587	+	0.974913i	$0.571450\pi$
$174$	16.8891	1.28036
$175$	−11.9888	−0.906265
$176$	0.362193	0.0273013
$177$	−22.5592	−1.69565
$178$	12.3489	0.925585
$179$	16.2185	1.21223	0.606113	−	0.795379i	$-0.292728\pi$
$179$	16.2185	1.21223	0.606113	+	0.795379i	$0.292728\pi$
$180$	0.0540963	0.00403210
$181$	6.80339	0.505692	0.252846	−	0.967507i	$-0.418633\pi$
$181$	6.80339	0.505692	0.252846	+	0.967507i	$0.418633\pi$
$182$	−26.6866	−1.97814
$183$	5.11547	0.378147
$184$	5.42454	0.399902
$185$	0.857473	0.0630427
$186$	−7.68304	−0.563348
$187$	2.83883	0.207595
$188$	3.06302	0.223393
$189$	−21.9353	−1.59556
$190$	5.53791	0.401762
$191$	16.3505	1.18308	0.591540	−	0.806276i	$-0.298520\pi$
$191$	16.3505	1.18308	0.591540	+	0.806276i	$0.298520\pi$
$192$	1.74260	0.125761
$193$	11.6518	0.838718	0.419359	−	0.907820i	$-0.362255\pi$
$193$	11.6518	0.838718	0.419359	+	0.907820i	$0.362255\pi$
$194$	−6.96468	−0.500035
$195$	16.1555	1.15692
$196$	11.0437	0.788838
$197$	9.44520	0.672942	0.336471	−	0.941694i	$-0.390767\pi$
$197$	9.44520	0.672942	0.336471	+	0.941694i	$0.390767\pi$
$198$	−0.0132774	−0.000943584 0
$199$	−12.5325	−0.888403	−0.444201	−	0.895927i	$-0.646513\pi$
$199$	−12.5325	−0.888403	−0.444201	+	0.895927i	$0.646513\pi$
$200$	2.82235	0.199570
$201$	−27.7488	−1.95725
$202$	−5.05830	−0.355900
$203$	−41.1692	−2.88951
$204$	13.6583	0.956273
$205$	14.1124	0.985655
$206$	−11.2720	−0.785360
$207$	−0.198855	−0.0138214
$208$	6.28246	0.435610
$209$	−1.35923	−0.0940196
$210$	−10.9233	−0.753781
$211$	22.2106	1.52904	0.764521	−	0.644598i	$-0.222975\pi$
$211$	22.2106	1.52904	0.764521	+	0.644598i	$0.222975\pi$
$212$	−9.03537	−0.620552
$213$	−1.57079	−0.107629
$214$	7.11067	0.486075
$215$	−8.10098	−0.552482
$216$	5.16392	0.351360
$217$	18.7283	1.27136
$218$	−1.13607	−0.0769447
$219$	2.03556	0.137550
$220$	0.534483	0.0360348
$221$	49.2412	3.31232
$222$	−1.01257	−0.0679591
$223$	22.4147	1.50100	0.750501	−	0.660869i	$-0.229812\pi$
$223$	22.4147	1.50100	0.750501	+	0.660869i	$0.229812\pi$
$224$	−4.24779	−0.283818
$225$	−0.103463	−0.00689752
$226$	−3.16439	−0.210492
$227$	−11.1098	−0.737383	−0.368691	−	0.929552i	$-0.620194\pi$
$227$	−11.1098	−0.737383	−0.368691	+	0.929552i	$0.620194\pi$
$228$	−6.53958	−0.433094
$229$	−7.08664	−0.468298	−0.234149	−	0.972201i	$-0.575230\pi$
$229$	−7.08664	−0.468298	−0.234149	+	0.972201i	$0.575230\pi$
$230$	8.00491	0.527828
$231$	2.68103	0.176399
$232$	9.69191	0.636305
$233$	−1.30048	−0.0851970	−0.0425985	−	0.999092i	$-0.513564\pi$
$233$	−1.30048	−0.0851970	−0.0425985	+	0.999092i	$0.513564\pi$
$234$	−0.230305	−0.0150555
$235$	4.52005	0.294856
$236$	−12.9457	−0.842692
$237$	24.0537	1.56246
$238$	−33.2937	−2.15811
$239$	−9.54584	−0.617469	−0.308734	−	0.951148i	$-0.599905\pi$
$239$	−9.54584	−0.617469	−0.308734	+	0.951148i	$0.599905\pi$
$240$	2.57153	0.165992
$241$	−9.64517	−0.621300	−0.310650	−	0.950524i	$-0.600547\pi$
$241$	−9.64517	−0.621300	−0.310650	+	0.950524i	$0.600547\pi$
$242$	10.8688	0.698674
$243$	−0.380944	−0.0244376
$244$	2.93554	0.187929
$245$	16.2971	1.04118
$246$	−16.6650	−1.06252
$247$	−23.5766	−1.50014
$248$	−4.40895	−0.279969
$249$	−23.1130	−1.46473
$250$	11.5433	0.730065
$251$	−10.6277	−0.670815	−0.335407	−	0.942073i	$-0.608874\pi$
$251$	−10.6277	−0.670815	−0.335407	+	0.942073i	$0.608874\pi$
$252$	0.155717	0.00980926
$253$	−1.96473	−0.123521
$254$	−14.9086	−0.935451
$255$	20.1554	1.26218
$256$	1.00000	0.0625000
$257$	−2.86819	−0.178913	−0.0894563	−	0.995991i	$-0.528513\pi$
$257$	−2.86819	−0.178913	−0.0894563	+	0.995991i	$0.528513\pi$
$258$	9.56625	0.595569
$259$	2.46825	0.153370
$260$	9.27093	0.574959
$261$	−0.355290	−0.0219919
$262$	−21.3563	−1.31940
$263$	−7.54183	−0.465049	−0.232525	−	0.972591i	$-0.574699\pi$
$263$	−7.54183	−0.465049	−0.232525	+	0.972591i	$0.574699\pi$
$264$	−0.631158	−0.0388451
$265$	−13.3334	−0.819063
$266$	15.9410	0.977404
$267$	−21.5191	−1.31695
$268$	−15.9238	−0.972701
$269$	−29.8812	−1.82189	−0.910944	−	0.412531i	$-0.864645\pi$
$269$	−29.8812	−1.82189	−0.910944	+	0.412531i	$0.864645\pi$
$270$	7.62033	0.463758
$271$	−7.76244	−0.471535	−0.235767	−	0.971810i	$-0.575760\pi$
$271$	−7.76244	−0.471535	−0.235767	+	0.971810i	$0.575760\pi$
$272$	7.83788	0.475242
$273$	46.5040	2.81455
$274$	−0.733153	−0.0442914
$275$	−1.02224	−0.0616431
$276$	−9.45280	−0.568992
$277$	1.53731	0.0923681	0.0461840	−	0.998933i	$-0.485294\pi$
$277$	1.53731	0.0923681	0.0461840	+	0.998933i	$0.485294\pi$
$278$	−6.32332	−0.379247
$279$	0.161625	0.00967623
$280$	−6.26841	−0.374609
$281$	−28.1054	−1.67663	−0.838313	−	0.545189i	$-0.816458\pi$
$281$	−28.1054	−1.67663	−0.838313	+	0.545189i	$0.816458\pi$
$282$	−5.33761	−0.317850
$283$	−26.9429	−1.60159	−0.800795	−	0.598938i	$-0.795590\pi$
$283$	−26.9429	−1.60159	−0.800795	+	0.598938i	$0.795590\pi$
$284$	−0.901407	−0.0534887
$285$	−9.65036	−0.571638
$286$	−2.27546	−0.134551
$287$	40.6229	2.39789
$288$	−0.0366584	−0.00216012
$289$	44.4324	2.61367
$290$	14.3022	0.839855
$291$	12.1367	0.711463
$292$	1.16811	0.0683587
$293$	23.6055	1.37905	0.689523	−	0.724264i	$-0.257820\pi$
$293$	23.6055	1.37905	0.689523	+	0.724264i	$0.257820\pi$
$294$	−19.2448	−1.12238
$295$	−19.1038	−1.11226
$296$	−0.581067	−0.0337738
$297$	−1.87034	−0.108528
$298$	−11.4383	−0.662603
$299$	−34.0794	−1.97086
$300$	−4.91823	−0.283954
$301$	−23.3188	−1.34408
$302$	7.32718	0.421632
$303$	8.81459	0.506385
$304$	−3.75277	−0.215236
$305$	4.33194	0.248046
$306$	−0.287324	−0.0164252
$307$	21.7758	1.24281	0.621405	−	0.783490i	$-0.286562\pi$
$307$	21.7758	1.24281	0.621405	+	0.783490i	$0.286562\pi$
$308$	1.53852	0.0876653
$309$	19.6427	1.11743
$310$	−6.50622	−0.369529
$311$	−20.4976	−1.16231	−0.581155	−	0.813793i	$-0.697399\pi$
$311$	−20.4976	−1.16231	−0.581155	+	0.813793i	$0.697399\pi$
$312$	−10.9478	−0.619798
$313$	14.5233	0.820908	0.410454	−	0.911881i	$-0.365370\pi$
$313$	14.5233	0.820908	0.410454	+	0.911881i	$0.365370\pi$
$314$	8.21392	0.463538
$315$	0.229790	0.0129472
$316$	13.8034	0.776499
$317$	13.4088	0.753111	0.376555	−	0.926394i	$-0.377108\pi$
$317$	13.4088	0.753111	0.376555	+	0.926394i	$0.377108\pi$
$318$	15.7451	0.882939
$319$	−3.51034	−0.196541
$320$	1.47569	0.0824934
$321$	−12.3911	−0.691601
$322$	23.0423	1.28410
$323$	−29.4138	−1.63663
$324$	−9.10863	−0.506035
$325$	−17.7313	−0.983555
$326$	17.4726	0.967718
$327$	1.97972	0.109479
$328$	−9.56330	−0.528045
$329$	13.0111	0.717323
$330$	−0.931390	−0.0512714
$331$	−23.4244	−1.28752	−0.643760	−	0.765228i	$-0.722626\pi$
$331$	−23.4244	−1.28752	−0.643760	+	0.765228i	$0.722626\pi$
$332$	−13.2635	−0.727929
$333$	0.0213010	0.00116729
$334$	0.875011	0.0478785
$335$	−23.4985	−1.28386
$336$	7.40221	0.403823
$337$	6.80168	0.370511	0.185256	−	0.982690i	$-0.440689\pi$
$337$	6.80168	0.370511	0.185256	+	0.982690i	$0.440689\pi$
$338$	−26.4693	−1.43974
$339$	5.51427	0.299494
$340$	11.5663	0.627268
$341$	1.59689	0.0864764
$342$	0.137570	0.00743896
$343$	17.1769	0.927468
$344$	5.48964	0.295981
$345$	−13.9494	−0.751009
$346$	5.85534	0.314785
$347$	9.56433	0.513440	0.256720	−	0.966486i	$-0.417358\pi$
$347$	9.56433	0.513440	0.256720	+	0.966486i	$0.417358\pi$
$348$	−16.8891	−0.905352
$349$	17.5639	0.940175	0.470088	−	0.882620i	$-0.344222\pi$
$349$	17.5639	0.940175	0.470088	+	0.882620i	$0.344222\pi$
$350$	11.9888	0.640826
$351$	−32.4421	−1.73163
$352$	−0.362193	−0.0193049
$353$	−26.4332	−1.40690	−0.703448	−	0.710747i	$-0.748357\pi$
$353$	−26.4332	−1.40690	−0.703448	+	0.710747i	$0.748357\pi$
$354$	22.5592	1.19901
$355$	−1.33019	−0.0705993
$356$	−12.3489	−0.654488
$357$	58.0176	3.07062
$358$	−16.2185	−0.857173
$359$	−10.5379	−0.556170	−0.278085	−	0.960556i	$-0.589700\pi$
$359$	−10.5379	−0.556170	−0.278085	+	0.960556i	$0.589700\pi$
$360$	−0.0540963	−0.00285112
$361$	−4.91673	−0.258775
$362$	−6.80339	−0.357578
$363$	−18.9400	−0.994093
$364$	26.6866	1.39876
$365$	1.72377	0.0902262
$366$	−5.11547	−0.267390
$367$	−27.4026	−1.43041	−0.715203	−	0.698917i	$-0.753666\pi$
$367$	−27.4026	−1.43041	−0.715203	+	0.698917i	$0.753666\pi$
$368$	−5.42454	−0.282774
$369$	0.350575	0.0182502
$370$	−0.857473	−0.0445779
$371$	−38.3804	−1.99261
$372$	7.68304	0.398347
$373$	7.89964	0.409028	0.204514	−	0.978864i	$-0.434439\pi$
$373$	7.89964	0.409028	0.204514	+	0.978864i	$0.434439\pi$
$374$	−2.83883	−0.146792
$375$	−20.1154	−1.03876
$376$	−3.06302	−0.157963
$377$	−60.8890	−3.13594
$378$	21.9353	1.12823
$379$	−16.5781	−0.851558	−0.425779	−	0.904827i	$-0.640000\pi$
$379$	−16.5781	−0.851558	−0.425779	+	0.904827i	$0.640000\pi$
$380$	−5.53791	−0.284089
$381$	25.9798	1.33099
$382$	−16.3505	−0.836564
$383$	17.6097	0.899815	0.449908	−	0.893075i	$-0.351457\pi$
$383$	17.6097	0.899815	0.449908	+	0.893075i	$0.351457\pi$
$384$	−1.74260	−0.0889267
$385$	2.27037	0.115709
$386$	−11.6518	−0.593063
$387$	−0.201241	−0.0102297
$388$	6.96468	0.353578
$389$	4.17943	0.211905	0.105953	−	0.994371i	$-0.466211\pi$
$389$	4.17943	0.211905	0.105953	+	0.994371i	$0.466211\pi$
$390$	−16.1555	−0.818068
$391$	−42.5169	−2.15017
$392$	−11.0437	−0.557793
$393$	37.2155	1.87727
$394$	−9.44520	−0.475842
$395$	20.3694	1.02490
$396$	0.0132774	0.000667215 0
$397$	4.76007	0.238901	0.119451	−	0.992840i	$-0.461887\pi$
$397$	4.76007	0.238901	0.119451	+	0.992840i	$0.461887\pi$
$398$	12.5325	0.628196
$399$	−27.7788	−1.39068
$400$	−2.82235	−0.141118
$401$	38.5043	1.92282	0.961408	−	0.275128i	$-0.0887203\pi$
$401$	38.5043	1.92282	0.961408	+	0.275128i	$0.0887203\pi$
$402$	27.7488	1.38399
$403$	27.6990	1.37979
$404$	5.05830	0.251660
$405$	−13.4415	−0.667912
$406$	41.1692	2.04319
$407$	0.210458	0.0104320
$408$	−13.6583	−0.676187
$409$	12.7106	0.628498	0.314249	−	0.949341i	$-0.398247\pi$
$409$	12.7106	0.628498	0.314249	+	0.949341i	$0.398247\pi$
$410$	−14.1124	−0.696963
$411$	1.27759	0.0630190
$412$	11.2720	0.555333
$413$	−54.9906	−2.70591
$414$	0.198855	0.00977318
$415$	−19.5728	−0.960789
$416$	−6.28246	−0.308023
$417$	11.0190	0.539604
$418$	1.35923	0.0664819
$419$	6.70254	0.327440	0.163720	−	0.986507i	$-0.447651\pi$
$419$	6.70254	0.327440	0.163720	+	0.986507i	$0.447651\pi$
$420$	10.9233	0.533004
$421$	−27.0260	−1.31716	−0.658582	−	0.752509i	$-0.728843\pi$
$421$	−27.0260	−1.31716	−0.658582	+	0.752509i	$0.728843\pi$
$422$	−22.2106	−1.08120
$423$	0.112285	0.00545949
$424$	9.03537	0.438797
$425$	−22.1213	−1.07304
$426$	1.57079	0.0761052
$427$	12.4696	0.603444
$428$	−7.11067	−0.343707
$429$	3.96522	0.191443
$430$	8.10098	0.390664
$431$	−15.7038	−0.756423	−0.378212	−	0.925719i	$-0.623461\pi$
$431$	−15.7038	−0.756423	−0.378212	+	0.925719i	$0.623461\pi$
$432$	−5.16392	−0.248449
$433$	9.93323	0.477361	0.238680	−	0.971098i	$-0.423285\pi$
$433$	9.93323	0.477361	0.238680	+	0.971098i	$0.423285\pi$
$434$	−18.7283	−0.898987
$435$	−24.9230	−1.19497
$436$	1.13607	0.0544081
$437$	20.3570	0.973809
$438$	−2.03556	−0.0972626
$439$	19.8807	0.948852	0.474426	−	0.880295i	$-0.342656\pi$
$439$	19.8807	0.948852	0.474426	+	0.880295i	$0.342656\pi$
$440$	−0.534483	−0.0254805
$441$	0.404845	0.0192784
$442$	−49.2412	−2.34216
$443$	−19.0577	−0.905460	−0.452730	−	0.891648i	$-0.649550\pi$
$443$	−19.0577	−0.905460	−0.452730	+	0.891648i	$0.649550\pi$
$444$	1.01257	0.0480544
$445$	−18.2230	−0.863854
$446$	−22.4147	−1.06137
$447$	19.9324	0.942770
$448$	4.24779	0.200689
$449$	18.8105	0.887723	0.443862	−	0.896095i	$-0.353608\pi$
$449$	18.8105	0.887723	0.443862	+	0.896095i	$0.353608\pi$
$450$	0.103463	0.00487728
$451$	3.46376	0.163102
$452$	3.16439	0.148841
$453$	−12.7684	−0.599910
$454$	11.1098	0.521408
$455$	39.3810	1.84621
$456$	6.53958	0.306244
$457$	−9.52248	−0.445443	−0.222722	−	0.974882i	$-0.571494\pi$
$457$	−9.52248	−0.445443	−0.222722	+	0.974882i	$0.571494\pi$
$458$	7.08664	0.331137
$459$	−40.4742	−1.88918
$460$	−8.00491	−0.373231
$461$	−16.2426	−0.756495	−0.378248	−	0.925704i	$-0.623473\pi$
$461$	−16.2426	−0.756495	−0.378248	+	0.925704i	$0.623473\pi$
$462$	−2.68103	−0.124733
$463$	−13.3656	−0.621154	−0.310577	−	0.950548i	$-0.600522\pi$
$463$	−13.3656	−0.621154	−0.310577	+	0.950548i	$0.600522\pi$
$464$	−9.69191	−0.449935
$465$	11.3378	0.525776
$466$	1.30048	0.0602433
$467$	15.2497	0.705674	0.352837	−	0.935685i	$-0.385217\pi$
$467$	15.2497	0.705674	0.352837	+	0.935685i	$0.385217\pi$
$468$	0.230305	0.0106458
$469$	−67.6410	−3.12337
$470$	−4.52005	−0.208494
$471$	−14.3136	−0.659535
$472$	12.9457	0.595873
$473$	−1.98831	−0.0914225
$474$	−24.0537	−1.10482
$475$	10.5916	0.485977
$476$	33.2937	1.52601
$477$	−0.331222	−0.0151656
$478$	9.54584	0.436616
$479$	−12.9035	−0.589578	−0.294789	−	0.955562i	$-0.595249\pi$
$479$	−12.9035	−0.589578	−0.294789	+	0.955562i	$0.595249\pi$
$480$	−2.57153	−0.117374
$481$	3.65053	0.166450
$482$	9.64517	0.439325
$483$	−40.1535	−1.82705
$484$	−10.8688	−0.494037
$485$	10.2777	0.466685
$486$	0.380944	0.0172800
$487$	4.16837	0.188887	0.0944435	−	0.995530i	$-0.469893\pi$
$487$	4.16837	0.188887	0.0944435	+	0.995530i	$0.469893\pi$
$488$	−2.93554	−0.132886
$489$	−30.4478	−1.37690
$490$	−16.2971	−0.736227
$491$	−13.1177	−0.591995	−0.295998	−	0.955189i	$-0.595652\pi$
$491$	−13.1177	−0.591995	−0.295998	+	0.955189i	$0.595652\pi$
$492$	16.6650	0.751317
$493$	−75.9640	−3.42125
$494$	23.5766	1.06076
$495$	0.0195933	0.000880652 0
$496$	4.40895	0.197968
$497$	−3.82899	−0.171754
$498$	23.1130	1.03572
$499$	−7.66161	−0.342981	−0.171490	−	0.985186i	$-0.554858\pi$
$499$	−7.66161	−0.342981	−0.171490	+	0.985186i	$0.554858\pi$
$500$	−11.5433	−0.516234
$501$	−1.52479	−0.0681228
$502$	10.6277	0.474338
$503$	−23.8495	−1.06339	−0.531697	−	0.846934i	$-0.678446\pi$
$503$	−23.8495	−1.06339	−0.531697	+	0.846934i	$0.678446\pi$
$504$	−0.155717	−0.00693619
$505$	7.46446	0.332164
$506$	1.96473	0.0873428
$507$	46.1254	2.04850
$508$	14.9086	0.661464
$509$	17.8495	0.791164	0.395582	−	0.918431i	$-0.370543\pi$
$509$	17.8495	0.791164	0.395582	+	0.918431i	$0.370543\pi$
$510$	−20.1554	−0.892495
$511$	4.96190	0.219502
$512$	−1.00000	−0.0441942
$513$	19.3790	0.855604
$514$	2.86819	0.126510
$515$	16.6340	0.732981
$516$	−9.56625	−0.421131
$517$	1.10940	0.0487915
$518$	−2.46825	−0.108449
$519$	−10.2035	−0.447885
$520$	−9.27093	−0.406557
$521$	−6.43146	−0.281768	−0.140884	−	0.990026i	$-0.544994\pi$
$521$	−6.43146	−0.281768	−0.140884	+	0.990026i	$0.544994\pi$
$522$	0.355290	0.0155506
$523$	12.9757	0.567387	0.283693	−	0.958915i	$-0.408440\pi$
$523$	12.9757	0.567387	0.283693	+	0.958915i	$0.408440\pi$
$524$	21.3563	0.932954
$525$	−20.8916	−0.911785
$526$	7.54183	0.328840
$527$	34.5568	1.50532
$528$	0.631158	0.0274676
$529$	6.42560	0.279374
$530$	13.3334	0.579165
$531$	−0.474568	−0.0205945
$532$	−15.9410	−0.691129
$533$	60.0810	2.60240
$534$	21.5191	0.931223
$535$	−10.4931	−0.453657
$536$	15.9238	0.687804
$537$	28.2623	1.21961
$538$	29.8812	1.28827
$539$	3.99996	0.172291
$540$	−7.62033	−0.327927
$541$	13.0646	0.561690	0.280845	−	0.959753i	$-0.409385\pi$
$541$	13.0646	0.561690	0.280845	+	0.959753i	$0.409385\pi$
$542$	7.76244	0.333425
$543$	11.8556	0.508772
$544$	−7.83788	−0.336047
$545$	1.67649	0.0718129
$546$	−46.5040	−1.99019
$547$	−16.2738	−0.695817	−0.347909	−	0.937528i	$-0.613108\pi$
$547$	−16.2738	−0.695817	−0.347909	+	0.937528i	$0.613108\pi$
$548$	0.733153	0.0313188
$549$	0.107612	0.00459277
$550$	1.02224	0.0435883
$551$	36.3715	1.54948
$552$	9.45280	0.402338
$553$	58.6338	2.49336
$554$	−1.53731	−0.0653141
$555$	1.49423	0.0634267
$556$	6.32332	0.268168
$557$	−19.5958	−0.830302	−0.415151	−	0.909752i	$-0.636271\pi$
$557$	−19.5958	−0.830302	−0.415151	+	0.909752i	$0.636271\pi$
$558$	−0.161625	−0.00684213
$559$	−34.4884	−1.45870
$560$	6.26841	0.264889
$561$	4.94694	0.208860
$562$	28.1054	1.18555
$563$	−37.8066	−1.59336	−0.796679	−	0.604402i	$-0.793412\pi$
$563$	−37.8066	−1.59336	−0.796679	+	0.604402i	$0.793412\pi$
$564$	5.33761	0.224754
$565$	4.66965	0.196454
$566$	26.9429	1.13250
$567$	−38.6916	−1.62489
$568$	0.901407	0.0378222
$569$	25.8067	1.08187	0.540936	−	0.841064i	$-0.318070\pi$
$569$	25.8067	1.08187	0.540936	+	0.841064i	$0.318070\pi$
$570$	9.65036	0.404209
$571$	−32.2680	−1.35037	−0.675186	−	0.737647i	$-0.735937\pi$
$571$	−32.2680	−1.35037	−0.675186	+	0.737647i	$0.735937\pi$
$572$	2.27546	0.0951418
$573$	28.4924	1.19029
$574$	−40.6229	−1.69557
$575$	15.3099	0.638469
$576$	0.0366584	0.00152743
$577$	43.7299	1.82050	0.910249	−	0.414060i	$-0.135890\pi$
$577$	43.7299	1.82050	0.910249	+	0.414060i	$0.135890\pi$
$578$	−44.4324	−1.84815
$579$	20.3045	0.843827
$580$	−14.3022	−0.593867
$581$	−56.3406	−2.33740
$582$	−12.1367	−0.503081
$583$	−3.27255	−0.135535
$584$	−1.16811	−0.0483369
$585$	0.339857	0.0140514
$586$	−23.6055	−0.975133
$587$	21.3086	0.879498	0.439749	−	0.898121i	$-0.355067\pi$
$587$	21.3086	0.879498	0.439749	+	0.898121i	$0.355067\pi$
$588$	19.2448	0.793643
$589$	−16.5458	−0.681756
$590$	19.1038	0.786490
$591$	16.4592	0.677041
$592$	0.581067	0.0238817
$593$	−5.69728	−0.233959	−0.116980	−	0.993134i	$-0.537321\pi$
$593$	−5.69728	−0.233959	−0.116980	+	0.993134i	$0.537321\pi$
$594$	1.87034	0.0767408
$595$	49.1310	2.01418
$596$	11.4383	0.468531
$597$	−21.8391	−0.893814
$598$	34.0794	1.39361
$599$	5.89794	0.240983	0.120492	−	0.992714i	$-0.461553\pi$
$599$	5.89794	0.240983	0.120492	+	0.992714i	$0.461553\pi$
$600$	4.91823	0.200786
$601$	21.7309	0.886420	0.443210	−	0.896418i	$-0.353840\pi$
$601$	21.7309	0.886420	0.443210	+	0.896418i	$0.353840\pi$
$602$	23.3188	0.950405
$603$	−0.583741	−0.0237718
$604$	−7.32718	−0.298139
$605$	−16.0390	−0.652076
$606$	−8.81459	−0.358068
$607$	−20.1769	−0.818957	−0.409478	−	0.912320i	$-0.634289\pi$
$607$	−20.1769	−0.818957	−0.409478	+	0.912320i	$0.634289\pi$
$608$	3.75277	0.152195
$609$	−71.7415	−2.90711
$610$	−4.33194	−0.175395
$611$	19.2433	0.778499
$612$	0.287324	0.0116144
$613$	30.0077	1.21200	0.606000	−	0.795464i	$-0.292773\pi$
$613$	30.0077	1.21200	0.606000	+	0.795464i	$0.292773\pi$
$614$	−21.7758	−0.878799
$615$	24.5923	0.991658
$616$	−1.53852	−0.0619887
$617$	−11.6915	−0.470680	−0.235340	−	0.971913i	$-0.575620\pi$
$617$	−11.6915	−0.470680	−0.235340	+	0.971913i	$0.575620\pi$
$618$	−19.6427	−0.790144
$619$	−12.7734	−0.513405	−0.256703	−	0.966490i	$-0.582636\pi$
$619$	−12.7734	−0.513405	−0.256703	+	0.966490i	$0.582636\pi$
$620$	6.50622	0.261296
$621$	28.0119	1.12408
$622$	20.4976	0.821878
$623$	−52.4553	−2.10158
$624$	10.9478	0.438263
$625$	−2.92258	−0.116903
$626$	−14.5233	−0.580469
$627$	−2.36859	−0.0945923
$628$	−8.21392	−0.327771
$629$	4.55434	0.181593
$630$	−0.229790	−0.00915504
$631$	−15.6640	−0.623575	−0.311787	−	0.950152i	$-0.600928\pi$
$631$	−15.6640	−0.623575	−0.311787	+	0.950152i	$0.600928\pi$
$632$	−13.8034	−0.549068
$633$	38.7043	1.53836
$634$	−13.4088	−0.532530
$635$	22.0005	0.873062
$636$	−15.7451	−0.624332
$637$	69.3818	2.74901
$638$	3.51034	0.138976
$639$	−0.0330441	−0.00130721
$640$	−1.47569	−0.0583316
$641$	15.0769	0.595502	0.297751	−	0.954644i	$-0.403764\pi$
$641$	15.0769	0.595502	0.297751	+	0.954644i	$0.403764\pi$
$642$	12.3911	0.489036
$643$	11.8482	0.467246	0.233623	−	0.972327i	$-0.424942\pi$
$643$	11.8482	0.467246	0.233623	+	0.972327i	$0.424942\pi$
$644$	−23.0423	−0.907994
$645$	−14.1168	−0.555848
$646$	29.4138	1.15727
$647$	34.1534	1.34271	0.671354	−	0.741137i	$-0.265713\pi$
$647$	34.1534	1.34271	0.671354	+	0.741137i	$0.265713\pi$
$648$	9.10863	0.357821
$649$	−4.68883	−0.184053
$650$	17.7313	0.695479
$651$	32.6359	1.27910
$652$	−17.4726	−0.684280
$653$	−42.2800	−1.65455	−0.827273	−	0.561800i	$-0.810109\pi$
$653$	−42.2800	−1.65455	−0.827273	+	0.561800i	$0.810109\pi$
$654$	−1.97972	−0.0774133
$655$	31.5152	1.23140
$656$	9.56330	0.373384
$657$	0.0428212	0.00167061
$658$	−13.0111	−0.507224
$659$	21.9221	0.853962	0.426981	−	0.904261i	$-0.359577\pi$
$659$	21.9221	0.853962	0.426981	+	0.904261i	$0.359577\pi$
$660$	0.931390	0.0362543
$661$	38.7532	1.50733	0.753663	−	0.657261i	$-0.228285\pi$
$661$	38.7532	1.50733	0.753663	+	0.657261i	$0.228285\pi$
$662$	23.4244	0.910414
$663$	85.8077	3.33250
$664$	13.2635	0.514724
$665$	−23.5239	−0.912217
$666$	−0.0213010	−0.000825396 0
$667$	52.5741	2.03568
$668$	−0.875011	−0.0338552
$669$	39.0600	1.51014
$670$	23.4985	0.907828
$671$	1.06323	0.0410456
$672$	−7.40221	−0.285546
$673$	−34.6645	−1.33622	−0.668110	−	0.744063i	$-0.732896\pi$
$673$	−34.6645	−1.33622	−0.668110	+	0.744063i	$0.732896\pi$
$674$	−6.80168	−0.261991
$675$	14.5744	0.560969
$676$	26.4693	1.01805
$677$	−36.7141	−1.41104	−0.705518	−	0.708692i	$-0.749286\pi$
$677$	−36.7141	−1.41104	−0.705518	+	0.708692i	$0.749286\pi$
$678$	−5.51427	−0.211774
$679$	29.5845	1.13535
$680$	−11.5663	−0.443546
$681$	−19.3599	−0.741874
$682$	−1.59689	−0.0611481
$683$	1.58567	0.0606740	0.0303370	−	0.999540i	$-0.490342\pi$
$683$	1.58567	0.0606740	0.0303370	+	0.999540i	$0.490342\pi$
$684$	−0.137570	−0.00526014
$685$	1.08190	0.0413374
$686$	−17.1769	−0.655819
$687$	−12.3492	−0.471151
$688$	−5.48964	−0.209290
$689$	−56.7644	−2.16255
$690$	13.9494	0.531043
$691$	1.74983	0.0665667	0.0332834	−	0.999446i	$-0.489404\pi$
$691$	1.74983	0.0665667	0.0332834	+	0.999446i	$0.489404\pi$
$692$	−5.85534	−0.222587
$693$	0.0563996	0.00214245
$694$	−9.56433	−0.363057
$695$	9.33123	0.353954
$696$	16.8891	0.640181
$697$	74.9560	2.83916
$698$	−17.5639	−0.664804
$699$	−2.26621	−0.0857159
$700$	−11.9888	−0.453132
$701$	−22.6849	−0.856796	−0.428398	−	0.903590i	$-0.640922\pi$
$701$	−22.6849	−0.856796	−0.428398	+	0.903590i	$0.640922\pi$
$702$	32.4421	1.22445
$703$	−2.18061	−0.0822433
$704$	0.362193	0.0136507
$705$	7.87664	0.296652
$706$	26.4332	0.994825
$707$	21.4866	0.808086
$708$	−22.5592	−0.847825
$709$	45.9032	1.72393	0.861966	−	0.506966i	$-0.169233\pi$
$709$	45.9032	1.72393	0.861966	+	0.506966i	$0.169233\pi$
$710$	1.33019	0.0499213
$711$	0.506009	0.0189768
$712$	12.3489	0.462793
$713$	−23.9165	−0.895680
$714$	−58.0176	−2.17126
$715$	3.35787	0.125577
$716$	16.2185	0.606113
$717$	−16.6346	−0.621230
$718$	10.5379	0.393272
$719$	−38.3778	−1.43125	−0.715625	−	0.698484i	$-0.753858\pi$
$719$	−38.3778	−1.43125	−0.715625	+	0.698484i	$0.753858\pi$
$720$	0.0540963	0.00201605
$721$	47.8813	1.78319
$722$	4.91673	0.182982
$723$	−16.8077	−0.625084
$724$	6.80339	0.252846
$725$	27.3540	1.01590
$726$	18.9400	0.702930
$727$	−30.9765	−1.14885	−0.574427	−	0.818556i	$-0.694775\pi$
$727$	−30.9765	−1.14885	−0.574427	+	0.818556i	$0.694775\pi$
$728$	−26.6866	−0.989070
$729$	26.6621	0.987484
$730$	−1.72377	−0.0637995
$731$	−43.0271	−1.59142
$732$	5.11547	0.189073
$733$	−5.60573	−0.207052	−0.103526	−	0.994627i	$-0.533013\pi$
$733$	−5.60573	−0.207052	−0.103526	+	0.994627i	$0.533013\pi$
$734$	27.4026	1.01145
$735$	28.3993	1.04752
$736$	5.42454	0.199951
$737$	−5.76749	−0.212448
$738$	−0.350575	−0.0129048
$739$	−7.55673	−0.277979	−0.138989	−	0.990294i	$-0.544385\pi$
$739$	−7.55673	−0.277979	−0.138989	+	0.990294i	$0.544385\pi$
$740$	0.857473	0.0315213
$741$	−41.0846	−1.50928
$742$	38.3804	1.40899
$743$	4.74341	0.174019	0.0870094	−	0.996207i	$-0.472269\pi$
$743$	4.74341	0.174019	0.0870094	+	0.996207i	$0.472269\pi$
$744$	−7.68304	−0.281674
$745$	16.8793	0.618411
$746$	−7.89964	−0.289226
$747$	−0.486219	−0.0177898
$748$	2.83883	0.103798
$749$	−30.2046	−1.10365
$750$	20.1154	0.734511
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	3.06302	0.111697
$753$	−18.5198	−0.674901
$754$	60.8890	2.21745
$755$	−10.8126	−0.393511
$756$	−21.9353	−0.797778
$757$	5.76728	0.209615	0.104808	−	0.994493i	$-0.466577\pi$
$757$	5.76728	0.209615	0.104808	+	0.994493i	$0.466577\pi$
$758$	16.5781	0.602142
$759$	−3.42374	−0.124274
$760$	5.53791	0.200881
$761$	−0.0886674	−0.00321419	−0.00160710	−	0.999999i	$-0.500512\pi$
$761$	−0.0886674	−0.00321419	−0.00160710	+	0.999999i	$0.500512\pi$
$762$	−25.9798	−0.941149
$763$	4.82581	0.174706
$764$	16.3505	0.591540
$765$	0.424000	0.0153298
$766$	−17.6097	−0.636265
$767$	−81.3307	−2.93668
$768$	1.74260	0.0628807
$769$	−26.4060	−0.952226	−0.476113	−	0.879384i	$-0.657955\pi$
$769$	−26.4060	−0.952226	−0.476113	+	0.879384i	$0.657955\pi$
$770$	−2.27037	−0.0818185
$771$	−4.99811	−0.180002
$772$	11.6518	0.419359
$773$	33.3308	1.19882	0.599412	−	0.800441i	$-0.295401\pi$
$773$	33.3308	1.19882	0.599412	+	0.800441i	$0.295401\pi$
$774$	0.201241	0.00723347
$775$	−12.4436	−0.446987
$776$	−6.96468	−0.250017
$777$	4.30118	0.154304
$778$	−4.17943	−0.149840
$779$	−35.8888	−1.28585
$780$	16.1555	0.578461
$781$	−0.326483	−0.0116825
$782$	42.5169	1.52040
$783$	50.0483	1.78858
$784$	11.0437	0.394419
$785$	−12.1212	−0.432623
$786$	−37.2155	−1.32743
$787$	−13.1181	−0.467611	−0.233806	−	0.972283i	$-0.575118\pi$
$787$	−13.1181	−0.467611	−0.233806	+	0.972283i	$0.575118\pi$
$788$	9.44520	0.336471
$789$	−13.1424	−0.467882
$790$	−20.3694	−0.724711
$791$	13.4417	0.477931
$792$	−0.0132774	−0.000471792 0
$793$	18.4424	0.654909
$794$	−4.76007	−0.168929
$795$	−23.2348	−0.824052
$796$	−12.5325	−0.444201
$797$	−28.7268	−1.01756	−0.508778	−	0.860898i	$-0.669903\pi$
$797$	−28.7268	−1.01756	−0.508778	+	0.860898i	$0.669903\pi$
$798$	27.7788	0.983358
$799$	24.0076	0.849327
$800$	2.82235	0.0997852
$801$	−0.452689	−0.0159950
$802$	−38.5043	−1.35964
$803$	0.423082	0.0149303
$804$	−27.7488	−0.978626
$805$	−34.0032	−1.19846
$806$	−27.6990	−0.975657
$807$	−52.0710	−1.83298
$808$	−5.05830	−0.177950
$809$	40.4465	1.42202	0.711012	−	0.703180i	$-0.248237\pi$
$809$	40.4465	1.42202	0.711012	+	0.703180i	$0.248237\pi$
$810$	13.4415	0.472285
$811$	43.3101	1.52082	0.760412	−	0.649441i	$-0.224997\pi$
$811$	43.3101	1.52082	0.760412	+	0.649441i	$0.224997\pi$
$812$	−41.1692	−1.44476
$813$	−13.5268	−0.474407
$814$	−0.210458	−0.00737656
$815$	−25.7841	−0.903177
$816$	13.6583	0.478136
$817$	20.6013	0.720750
$818$	−12.7106	−0.444415
$819$	0.978286	0.0341841
$820$	14.1124	0.492827
$821$	33.5386	1.17051	0.585253	−	0.810851i	$-0.300995\pi$
$821$	33.5386	1.17051	0.585253	+	0.810851i	$0.300995\pi$
$822$	−1.27759	−0.0445612
$823$	−11.6999	−0.407834	−0.203917	−	0.978988i	$-0.565367\pi$
$823$	−11.6999	−0.407834	−0.203917	+	0.978988i	$0.565367\pi$
$824$	−11.2720	−0.392680
$825$	−1.78135	−0.0620186
$826$	54.9906	1.91337
$827$	21.2105	0.737560	0.368780	−	0.929517i	$-0.379775\pi$
$827$	21.2105	0.737560	0.368780	+	0.929517i	$0.379775\pi$
$828$	−0.198855	−0.00691068
$829$	−12.9959	−0.451368	−0.225684	−	0.974201i	$-0.572462\pi$
$829$	−12.9959	−0.451368	−0.225684	+	0.974201i	$0.572462\pi$
$830$	19.5728	0.679381
$831$	2.67892	0.0929307
$832$	6.28246	0.217805
$833$	86.5595	2.99911
$834$	−11.0190	−0.381558
$835$	−1.29124	−0.0446852
$836$	−1.35923	−0.0470098
$837$	−22.7675	−0.786959
$838$	−6.70254	−0.231535
$839$	−45.8238	−1.58201	−0.791006	−	0.611809i	$-0.790442\pi$
$839$	−45.8238	−1.58201	−0.791006	+	0.611809i	$0.790442\pi$
$840$	−10.9233	−0.376891
$841$	64.9331	2.23907
$842$	27.0260	0.931376
$843$	−48.9764	−1.68684
$844$	22.2106	0.764521
$845$	39.0603	1.34372
$846$	−0.112285	−0.00386044
$847$	−46.1685	−1.58637
$848$	−9.03537	−0.310276
$849$	−46.9508	−1.61135
$850$	22.1213	0.758753
$851$	−3.15202	−0.108050
$852$	−1.57079	−0.0538145
$853$	14.0142	0.479837	0.239918	−	0.970793i	$-0.422879\pi$
$853$	14.0142	0.479837	0.239918	+	0.970793i	$0.422879\pi$
$854$	−12.4696	−0.426700
$855$	−0.203011	−0.00694282
$856$	7.11067	0.243038
$857$	−5.89890	−0.201503	−0.100751	−	0.994912i	$-0.532125\pi$
$857$	−5.89890	−0.201503	−0.100751	+	0.994912i	$0.532125\pi$
$858$	−3.96522	−0.135370
$859$	28.4297	0.970009	0.485004	−	0.874512i	$-0.338818\pi$
$859$	28.4297	0.970009	0.485004	+	0.874512i	$0.338818\pi$
$860$	−8.10098	−0.276241
$861$	70.7895	2.41250
$862$	15.7038	0.534872
$863$	−41.4908	−1.41236	−0.706181	−	0.708031i	$-0.749584\pi$
$863$	−41.4908	−1.41236	−0.706181	+	0.708031i	$0.749584\pi$
$864$	5.16392	0.175680
$865$	−8.64064	−0.293791
$866$	−9.93323	−0.337545
$867$	77.4280	2.62959
$868$	18.7283	0.635680
$869$	4.99948	0.169596
$870$	24.9230	0.844970
$871$	−100.041	−3.38975
$872$	−1.13607	−0.0384723
$873$	0.255314	0.00864107
$874$	−20.3570	−0.688587
$875$	−49.0337	−1.65764
$876$	2.03556	0.0687751
$877$	−4.34492	−0.146717	−0.0733587	−	0.997306i	$-0.523372\pi$
$877$	−4.34492	−0.146717	−0.0733587	+	0.997306i	$0.523372\pi$
$878$	−19.8807	−0.670940
$879$	41.1349	1.38745
$880$	0.534483	0.0180174
$881$	43.0038	1.44883	0.724417	−	0.689362i	$-0.242109\pi$
$881$	43.0038	1.44883	0.724417	+	0.689362i	$0.242109\pi$
$882$	−0.404845	−0.0136319
$883$	10.2344	0.344416	0.172208	−	0.985061i	$-0.444910\pi$
$883$	10.2344	0.344416	0.172208	+	0.985061i	$0.444910\pi$
$884$	49.2412	1.65616
$885$	−33.2902	−1.11904
$886$	19.0577	0.640257
$887$	44.2307	1.48512	0.742561	−	0.669779i	$-0.233611\pi$
$887$	44.2307	1.48512	0.742561	+	0.669779i	$0.233611\pi$
$888$	−1.01257	−0.0339796
$889$	63.3288	2.12398
$890$	18.2230	0.610837
$891$	−3.29908	−0.110523
$892$	22.4147	0.750501
$893$	−11.4948	−0.384658
$894$	−19.9324	−0.666639
$895$	23.9334	0.800004
$896$	−4.24779	−0.141909
$897$	−59.3868	−1.98287
$898$	−18.8105	−0.627715
$899$	−42.7311	−1.42516
$900$	−0.103463	−0.00344876
$901$	−70.8182	−2.35930
$902$	−3.46376	−0.115331
$903$	−40.6354	−1.35226
$904$	−3.16439	−0.105246
$905$	10.0397	0.333730
$906$	12.7684	0.424200
$907$	10.4047	0.345484	0.172742	−	0.984967i	$-0.444737\pi$
$907$	10.4047	0.345484	0.172742	+	0.984967i	$0.444737\pi$
$908$	−11.1098	−0.368691
$909$	0.185429	0.00615029
$910$	−39.3810	−1.30547
$911$	20.7399	0.687142	0.343571	−	0.939127i	$-0.388363\pi$
$911$	20.7399	0.687142	0.343571	+	0.939127i	$0.388363\pi$
$912$	−6.53958	−0.216547
$913$	−4.80395	−0.158987
$914$	9.52248	0.314976
$915$	7.54883	0.249557
$916$	−7.08664	−0.234149
$917$	90.7171	2.99574
$918$	40.4742	1.33585
$919$	34.8031	1.14805	0.574024	−	0.818839i	$-0.305382\pi$
$919$	34.8031	1.14805	0.574024	+	0.818839i	$0.305382\pi$
$920$	8.00491	0.263914
$921$	37.9465	1.25038
$922$	16.2426	0.534923
$923$	−5.66305	−0.186402
$924$	2.68103	0.0881993
$925$	−1.63998	−0.0539221
$926$	13.3656	0.439222
$927$	0.413215	0.0135717
$928$	9.69191	0.318152
$929$	−20.8073	−0.682665	−0.341332	−	0.939943i	$-0.610878\pi$
$929$	−20.8073	−0.682665	−0.341332	+	0.939943i	$0.610878\pi$
$930$	−11.3378	−0.371780
$931$	−41.4446	−1.35829
$932$	−1.30048	−0.0425985
$933$	−35.7191	−1.16939
$934$	−15.2497	−0.498987
$935$	4.18921	0.137002
$936$	−0.230305	−0.00752774
$937$	−57.6185	−1.88232	−0.941158	−	0.337968i	$-0.890260\pi$
$937$	−57.6185	−1.88232	−0.941158	+	0.337968i	$0.890260\pi$
$938$	67.6410	2.20856
$939$	25.3084	0.825908
$940$	4.52005	0.147428
$941$	45.1570	1.47208	0.736039	−	0.676939i	$-0.236694\pi$
$941$	45.1570	1.47208	0.736039	+	0.676939i	$0.236694\pi$
$942$	14.3136	0.466362
$943$	−51.8765	−1.68933
$944$	−12.9457	−0.421346
$945$	−32.3696	−1.05298
$946$	1.98831	0.0646454
$947$	37.9315	1.23261	0.616303	−	0.787509i	$-0.288629\pi$
$947$	37.9315	1.23261	0.616303	+	0.787509i	$0.288629\pi$
$948$	24.0537	0.781229
$949$	7.33862	0.238222
$950$	−10.5916	−0.343638
$951$	23.3661	0.757698
$952$	−33.2937	−1.07906
$953$	18.0070	0.583302	0.291651	−	0.956525i	$-0.405795\pi$
$953$	18.0070	0.583302	0.291651	+	0.956525i	$0.405795\pi$
$954$	0.331222	0.0107237
$955$	24.1282	0.780770
$956$	−9.54584	−0.308734
$957$	−6.11712	−0.197738
$958$	12.9035	0.416894
$959$	3.11428	0.100565
$960$	2.57153	0.0829958
$961$	−11.5612	−0.372941
$962$	−3.65053	−0.117698
$963$	−0.260666	−0.00839983
$964$	−9.64517	−0.310650
$965$	17.1945	0.553509
$966$	40.1535	1.29192
$967$	−13.7092	−0.440857	−0.220429	−	0.975403i	$-0.570746\pi$
$967$	−13.7092	−0.440857	−0.220429	+	0.975403i	$0.570746\pi$
$968$	10.8688	0.349337
$969$	−51.2565	−1.64659
$970$	−10.2777	−0.329996
$971$	30.5049	0.978950	0.489475	−	0.872017i	$-0.337188\pi$
$971$	30.5049	0.978950	0.489475	+	0.872017i	$0.337188\pi$
$972$	−0.380944	−0.0122188
$973$	26.8601	0.861097
$974$	−4.16837	−0.133563
$975$	−30.8986	−0.989546
$976$	2.93554	0.0939643
$977$	50.1327	1.60389	0.801944	−	0.597399i	$-0.203799\pi$
$977$	50.1327	1.60389	0.801944	+	0.597399i	$0.203799\pi$
$978$	30.4478	0.973613
$979$	−4.47266	−0.142947
$980$	16.2971	0.520591
$981$	0.0416466	0.00132967
$982$	13.1177	0.418604
$983$	−6.59057	−0.210207	−0.105103	−	0.994461i	$-0.533517\pi$
$983$	−6.59057	−0.210207	−0.105103	+	0.994461i	$0.533517\pi$
$984$	−16.6650	−0.531261
$985$	13.9381	0.444106
$986$	75.9640	2.41919
$987$	22.6731	0.721692
$988$	−23.5766	−0.750072
$989$	29.7787	0.946909
$990$	−0.0195933	−0.000622715 0
$991$	−32.9630	−1.04710	−0.523551	−	0.851994i	$-0.675393\pi$
$991$	−32.9630	−1.04710	−0.523551	+	0.851994i	$0.675393\pi$
$992$	−4.40895	−0.139984
$993$	−40.8193	−1.29536
$994$	3.82899	0.121448
$995$	−18.4940	−0.586299
$996$	−23.1130	−0.732363
$997$	−28.4156	−0.899932	−0.449966	−	0.893046i	$-0.648564\pi$
$997$	−28.4156	−0.899932	−0.449966	+	0.893046i	$0.648564\pi$
$998$	7.66161	0.242524
$999$	−3.00059	−0.0949343

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1502.2.a.h.1.13	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1502.2.a.h.1.13	✓	19	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:	\( N \)	\(=\)	\( 1502 = 2 \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1502.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.74260 q^{3} +1.00000 q^{4} +1.47569 q^{5} -1.74260 q^{6} +4.24779 q^{7} -1.00000 q^{8} +0.0366584 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.74260 q^{3} +1.00000 q^{4} +1.47569 q^{5} -1.74260 q^{6} +4.24779 q^{7} -1.00000 q^{8} +0.0366584 q^{9} -1.47569 q^{10} +0.362193 q^{11} +1.74260 q^{12} +6.28246 q^{13} -4.24779 q^{14} +2.57153 q^{15} +1.00000 q^{16} +7.83788 q^{17} -0.0366584 q^{18} -3.75277 q^{19} +1.47569 q^{20} +7.40221 q^{21} -0.362193 q^{22} -5.42454 q^{23} -1.74260 q^{24} -2.82235 q^{25} -6.28246 q^{26} -5.16392 q^{27} +4.24779 q^{28} -9.69191 q^{29} -2.57153 q^{30} +4.40895 q^{31} -1.00000 q^{32} +0.631158 q^{33} -7.83788 q^{34} +6.26841 q^{35} +0.0366584 q^{36} +0.581067 q^{37} +3.75277 q^{38} +10.9478 q^{39} -1.47569 q^{40} +9.56330 q^{41} -7.40221 q^{42} -5.48964 q^{43} +0.362193 q^{44} +0.0540963 q^{45} +5.42454 q^{46} +3.06302 q^{47} +1.74260 q^{48} +11.0437 q^{49} +2.82235 q^{50} +13.6583 q^{51} +6.28246 q^{52} -9.03537 q^{53} +5.16392 q^{54} +0.534483 q^{55} -4.24779 q^{56} -6.53958 q^{57} +9.69191 q^{58} -12.9457 q^{59} +2.57153 q^{60} +2.93554 q^{61} -4.40895 q^{62} +0.155717 q^{63} +1.00000 q^{64} +9.27093 q^{65} -0.631158 q^{66} -15.9238 q^{67} +7.83788 q^{68} -9.45280 q^{69} -6.26841 q^{70} -0.901407 q^{71} -0.0366584 q^{72} +1.16811 q^{73} -0.581067 q^{74} -4.91823 q^{75} -3.75277 q^{76} +1.53852 q^{77} -10.9478 q^{78} +13.8034 q^{79} +1.47569 q^{80} -9.10863 q^{81} -9.56330 q^{82} -13.2635 q^{83} +7.40221 q^{84} +11.5663 q^{85} +5.48964 q^{86} -16.8891 q^{87} -0.362193 q^{88} -12.3489 q^{89} -0.0540963 q^{90} +26.6866 q^{91} -5.42454 q^{92} +7.68304 q^{93} -3.06302 q^{94} -5.53791 q^{95} -1.74260 q^{96} +6.96468 q^{97} -11.0437 q^{98} +0.0132774 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9}+O(q^{10}) \) 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9	\( 19 q - 19 q^{2} + 6 q^{3} + 19 q^{4} + 2 q^{5} - 6 q^{6} + 13 q^{7} - 19 q^{8} + 25 q^{9} - 2 q^{10} - 7 q^{11} + 6 q^{12} + 19 q^{13} - 13 q^{14} + 6 q^{15} + 19 q^{16} + 11 q^{17} - 25 q^{18} + 7 q^{19} + 2 q^{20} + 7 q^{21} + 7 q^{22} + 12 q^{23} - 6 q^{24} + 37 q^{25} - 19 q^{26} + 24 q^{27} + 13 q^{28} - 8 q^{29} - 6 q^{30} + 32 q^{31} - 19 q^{32} + 24 q^{33} - 11 q^{34} - 19 q^{35} + 25 q^{36} + 41 q^{37} - 7 q^{38} - 2 q^{39} - 2 q^{40} + 15 q^{41} - 7 q^{42} + 15 q^{43} - 7 q^{44} + 28 q^{45} - 12 q^{46} - 6 q^{47} + 6 q^{48} + 42 q^{49} - 37 q^{50} + 8 q^{51} + 19 q^{52} + 6 q^{53} - 24 q^{54} + 22 q^{55} - 13 q^{56} + 24 q^{57} + 8 q^{58} - 4 q^{59} + 6 q^{60} + 13 q^{61} - 32 q^{62} + 37 q^{63} + 19 q^{64} + 20 q^{65} - 24 q^{66} + 47 q^{67} + 11 q^{68} + 15 q^{69} + 19 q^{70} - 25 q^{72} + 64 q^{73} - 41 q^{74} - 3 q^{75} + 7 q^{76} - 4 q^{77} + 2 q^{78} + 34 q^{79} + 2 q^{80} + 27 q^{81} - 15 q^{82} + 4 q^{83} + 7 q^{84} + 21 q^{85} - 15 q^{86} + 7 q^{88} + 18 q^{89} - 28 q^{90} + 28 q^{91} + 12 q^{92} + 43 q^{93} + 6 q^{94} - 8 q^{95} - 6 q^{96} + 82 q^{97} - 42 q^{98} - 2 q^{99}+O(q^{100}) \) 19 * q - 19 * q^2 + 6 * q^3 + 19 * q^4 + 2 * q^5 - 6 * q^6 + 13 * q^7 - 19 * q^8 + 25 * q^9 - 2 * q^10 - 7 * q^11 + 6 * q^12 + 19 * q^13 - 13 * q^14 + 6 * q^15 + 19 * q^16 + 11 * q^17 - 25 * q^18 + 7 * q^19 + 2 * q^20 + 7 * q^21 + 7 * q^22 + 12 * q^23 - 6 * q^24 + 37 * q^25 - 19 * q^26 + 24 * q^27 + 13 * q^28 - 8 * q^29 - 6 * q^30 + 32 * q^31 - 19 * q^32 + 24 * q^33 - 11 * q^34 - 19 * q^35 + 25 * q^36 + 41 * q^37 - 7 * q^38 - 2 * q^39 - 2 * q^40 + 15 * q^41 - 7 * q^42 + 15 * q^43 - 7 * q^44 + 28 * q^45 - 12 * q^46 - 6 * q^47 + 6 * q^48 + 42 * q^49 - 37 * q^50 + 8 * q^51 + 19 * q^52 + 6 * q^53 - 24 * q^54 + 22 * q^55 - 13 * q^56 + 24 * q^57 + 8 * q^58 - 4 * q^59 + 6 * q^60 + 13 * q^61 - 32 * q^62 + 37 * q^63 + 19 * q^64 + 20 * q^65 - 24 * q^66 + 47 * q^67 + 11 * q^68 + 15 * q^69 + 19 * q^70 - 25 * q^72 + 64 * q^73 - 41 * q^74 - 3 * q^75 + 7 * q^76 - 4 * q^77 + 2 * q^78 + 34 * q^79 + 2 * q^80 + 27 * q^81 - 15 * q^82 + 4 * q^83 + 7 * q^84 + 21 * q^85 - 15 * q^86 + 7 * q^88 + 18 * q^89 - 28 * q^90 + 28 * q^91 + 12 * q^92 + 43 * q^93 + 6 * q^94 - 8 * q^95 - 6 * q^96 + 82 * q^97 - 42 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more