LMFDB - Embedded newform 845.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(1,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("845.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.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:	$6.74735897080$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.08613$ of defining polynomial
Character	$\chi$	$=$	845.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.08613 q^{2} -3.08613 q^{3} +2.35194 q^{4} -1.00000 q^{5} +6.43807 q^{6} +1.35194 q^{7} -0.734191 q^{8} +6.52420 q^{9} +O(q^{10})$	\(q-2.08613 q^{2} -3.08613 q^{3} +2.35194 q^{4} -1.00000 q^{5} +6.43807 q^{6} +1.35194 q^{7} -0.734191 q^{8} +6.52420 q^{9} +2.08613 q^{10} +3.73419 q^{11} -7.25839 q^{12} -2.82032 q^{14} +3.08613 q^{15} -3.17226 q^{16} -2.70388 q^{17} -13.6103 q^{18} -0.438069 q^{19} -2.35194 q^{20} -4.17226 q^{21} -7.79001 q^{22} -5.08613 q^{23} +2.26581 q^{24} +1.00000 q^{25} -10.8761 q^{27} +3.17968 q^{28} -1.35194 q^{29} -6.43807 q^{30} +6.43807 q^{31} +8.08613 q^{32} -11.5242 q^{33} +5.64064 q^{34} -1.35194 q^{35} +15.3445 q^{36} +7.35194 q^{37} +0.913870 q^{38} +0.734191 q^{40} -6.87614 q^{41} +8.70388 q^{42} -0.209991 q^{43} +8.78259 q^{44} -6.52420 q^{45} +10.6103 q^{46} -1.35194 q^{47} +9.79001 q^{48} -5.17226 q^{49} -2.08613 q^{50} +8.34452 q^{51} -1.46838 q^{53} +22.6890 q^{54} -3.73419 q^{55} -0.992582 q^{56} +1.35194 q^{57} +2.82032 q^{58} -2.26581 q^{59} +7.25839 q^{60} +3.52420 q^{61} -13.4307 q^{62} +8.82032 q^{63} -10.5242 q^{64} +24.0410 q^{66} +11.5242 q^{67} -6.35936 q^{68} +15.6965 q^{69} +2.82032 q^{70} -0.438069 q^{71} -4.79001 q^{72} -3.69646 q^{73} -15.3371 q^{74} -3.08613 q^{75} -1.03031 q^{76} +5.04840 q^{77} +15.0484 q^{79} +3.17226 q^{80} +13.9926 q^{81} +14.3445 q^{82} +0.475800 q^{83} -9.81290 q^{84} +2.70388 q^{85} +0.438069 q^{86} +4.17226 q^{87} -2.74161 q^{88} -11.0484 q^{89} +13.6103 q^{90} -11.9623 q^{92} -19.8687 q^{93} +2.82032 q^{94} +0.438069 q^{95} -24.9549 q^{96} -3.29612 q^{97} +10.7900 q^{98} +24.3626 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} + 10 q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + q^2 - 2 * q^3 + 5 * q^4 - 3 * q^5 + 10 * q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9	$ 3 q + q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} + 10 q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9} - q^{10} + 6 q^{11} + 4 q^{14} + 2 q^{15} + 5 q^{16} - 4 q^{17} - 17 q^{18} + 8 q^{19} - 5 q^{20} + 2 q^{21} - 12 q^{22} - 8 q^{23} + 12 q^{24} + 3 q^{25} - 14 q^{27} + 22 q^{28} - 2 q^{29} - 10 q^{30} + 10 q^{31} + 17 q^{32} - 18 q^{33} - 8 q^{34} - 2 q^{35} + 17 q^{36} + 20 q^{37} + 10 q^{38} - 3 q^{40} - 2 q^{41} + 22 q^{42} - 12 q^{43} - 12 q^{44} - 3 q^{45} + 8 q^{46} - 2 q^{47} + 18 q^{48} - q^{49} + q^{50} - 4 q^{51} + 6 q^{53} + 10 q^{54} - 6 q^{55} + 24 q^{56} + 2 q^{57} - 4 q^{58} - 12 q^{59} - 6 q^{61} - 4 q^{62} + 14 q^{63} - 15 q^{64} + 12 q^{66} + 18 q^{67} - 44 q^{68} + 16 q^{69} - 4 q^{70} + 8 q^{71} - 3 q^{72} + 20 q^{73} + 10 q^{74} - 2 q^{75} - 2 q^{76} - 18 q^{77} + 12 q^{79} - 5 q^{80} + 15 q^{81} + 14 q^{82} + 18 q^{83} + 10 q^{84} + 4 q^{85} - 8 q^{86} - 2 q^{87} - 30 q^{88} + 17 q^{90} - 10 q^{92} - 14 q^{93} - 4 q^{94} - 8 q^{95} - 22 q^{96} - 14 q^{97} + 21 q^{98} + 12 q^{99}+O(q^{100}) $ 3 * q + q^2 - 2 * q^3 + 5 * q^4 - 3 * q^5 + 10 * q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9 - q^10 + 6 * q^11 + 4 * q^14 + 2 * q^15 + 5 * q^16 - 4 * q^17 - 17 * q^18 + 8 * q^19 - 5 * q^20 + 2 * q^21 - 12 * q^22 - 8 * q^23 + 12 * q^24 + 3 * q^25 - 14 * q^27 + 22 * q^28 - 2 * q^29 - 10 * q^30 + 10 * q^31 + 17 * q^32 - 18 * q^33 - 8 * q^34 - 2 * q^35 + 17 * q^36 + 20 * q^37 + 10 * q^38 - 3 * q^40 - 2 * q^41 + 22 * q^42 - 12 * q^43 - 12 * q^44 - 3 * q^45 + 8 * q^46 - 2 * q^47 + 18 * q^48 - q^49 + q^50 - 4 * q^51 + 6 * q^53 + 10 * q^54 - 6 * q^55 + 24 * q^56 + 2 * q^57 - 4 * q^58 - 12 * q^59 - 6 * q^61 - 4 * q^62 + 14 * q^63 - 15 * q^64 + 12 * q^66 + 18 * q^67 - 44 * q^68 + 16 * q^69 - 4 * q^70 + 8 * q^71 - 3 * q^72 + 20 * q^73 + 10 * q^74 - 2 * q^75 - 2 * q^76 - 18 * q^77 + 12 * q^79 - 5 * q^80 + 15 * q^81 + 14 * q^82 + 18 * q^83 + 10 * q^84 + 4 * q^85 - 8 * q^86 - 2 * q^87 - 30 * q^88 + 17 * q^90 - 10 * q^92 - 14 * q^93 - 4 * q^94 - 8 * q^95 - 22 * q^96 - 14 * q^97 + 21 * q^98 + 12 * 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 $

$2$	−2.08613	−1.47512	−0.737558	−	0.675283i	$-0.764021\pi$
$2$	−2.08613	−1.47512	−0.737558	+	0.675283i	$0.764021\pi$
$3$	−3.08613	−1.78178	−0.890889	−	0.454221i	$-0.849918\pi$
$3$	−3.08613	−1.78178	−0.890889	+	0.454221i	$0.849918\pi$
$4$	2.35194	1.17597
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	6.43807	2.62833
$7$	1.35194	0.510985	0.255492	−	0.966811i	$-0.417762\pi$
$7$	1.35194	0.510985	0.255492	+	0.966811i	$0.417762\pi$
$8$	−0.734191	−0.259576
$9$	6.52420	2.17473
$10$	2.08613	0.659692
$11$	3.73419	1.12590	0.562950	−	0.826491i	$-0.309666\pi$
$11$	3.73419	1.12590	0.562950	+	0.826491i	$0.309666\pi$
$12$	−7.25839	−2.09532
$13$	0	0
$13$	0	0
$14$	−2.82032	−0.753763
$15$	3.08613	0.796835
$16$	−3.17226	−0.793065
$17$	−2.70388	−0.655787	−0.327893	−	0.944715i	$-0.606339\pi$
$17$	−2.70388	−0.655787	−0.327893	+	0.944715i	$0.606339\pi$
$18$	−13.6103	−3.20799
$19$	−0.438069	−0.100500	−0.0502500	−	0.998737i	$-0.516002\pi$
$19$	−0.438069	−0.100500	−0.0502500	+	0.998737i	$0.516002\pi$
$20$	−2.35194	−0.525910
$21$	−4.17226	−0.910462
$22$	−7.79001	−1.66084
$23$	−5.08613	−1.06053	−0.530266	−	0.847832i	$-0.677908\pi$
$23$	−5.08613	−1.06053	−0.530266	+	0.847832i	$0.677908\pi$
$24$	2.26581	0.462506
$25$	1.00000	0.200000
$26$	0	0
$27$	−10.8761	−2.09311
$28$	3.17968	0.600903
$29$	−1.35194	−0.251049	−0.125524	−	0.992091i	$-0.540061\pi$
$29$	−1.35194	−0.251049	−0.125524	+	0.992091i	$0.540061\pi$
$30$	−6.43807	−1.17543
$31$	6.43807	1.15631	0.578156	−	0.815926i	$-0.303773\pi$
$31$	6.43807	1.15631	0.578156	+	0.815926i	$0.303773\pi$
$32$	8.08613	1.42944
$33$	−11.5242	−2.00611
$34$	5.64064	0.967362
$35$	−1.35194	−0.228519
$36$	15.3445	2.55742
$37$	7.35194	1.20865	0.604326	−	0.796737i	$-0.293443\pi$
$37$	7.35194	1.20865	0.604326	+	0.796737i	$0.293443\pi$
$38$	0.913870	0.148249
$39$	0	0
$40$	0.734191	0.116086
$41$	−6.87614	−1.07387	−0.536936	−	0.843623i	$-0.680418\pi$
$41$	−6.87614	−1.07387	−0.536936	+	0.843623i	$0.680418\pi$
$42$	8.70388	1.34304
$43$	−0.209991	−0.0320234	−0.0160117	−	0.999872i	$-0.505097\pi$
$43$	−0.209991	−0.0320234	−0.0160117	+	0.999872i	$0.505097\pi$
$44$	8.78259	1.32403
$45$	−6.52420	−0.972570
$46$	10.6103	1.56441
$47$	−1.35194	−0.197201	−0.0986003	−	0.995127i	$-0.531437\pi$
$47$	−1.35194	−0.197201	−0.0986003	+	0.995127i	$0.531437\pi$
$48$	9.79001	1.41307
$49$	−5.17226	−0.738894
$50$	−2.08613	−0.295023
$51$	8.34452	1.16847
$52$	0	0
$53$	−1.46838	−0.201698	−0.100849	−	0.994902i	$-0.532156\pi$
$53$	−1.46838	−0.201698	−0.100849	+	0.994902i	$0.532156\pi$
$54$	22.6890	3.08759
$55$	−3.73419	−0.503518
$56$	−0.992582	−0.132639
$57$	1.35194	0.179069
$58$	2.82032	0.370326
$59$	−2.26581	−0.294983	−0.147492	−	0.989063i	$-0.547120\pi$
$59$	−2.26581	−0.294983	−0.147492	+	0.989063i	$0.547120\pi$
$60$	7.25839	0.937054
$61$	3.52420	0.451228	0.225614	−	0.974217i	$-0.427561\pi$
$61$	3.52420	0.451228	0.225614	+	0.974217i	$0.427561\pi$
$62$	−13.4307	−1.70569
$63$	8.82032	1.11126
$64$	−10.5242	−1.31552
$65$	0	0
$66$	24.0410	2.95924
$67$	11.5242	1.40791	0.703953	−	0.710247i	$-0.251417\pi$
$67$	11.5242	1.40791	0.703953	+	0.710247i	$0.251417\pi$
$68$	−6.35936	−0.771185
$69$	15.6965	1.88963
$70$	2.82032	0.337093
$71$	−0.438069	−0.0519893	−0.0259946	−	0.999662i	$-0.508275\pi$
$71$	−0.438069	−0.0519893	−0.0259946	+	0.999662i	$0.508275\pi$
$72$	−4.79001	−0.564508
$73$	−3.69646	−0.432638	−0.216319	−	0.976323i	$-0.569405\pi$
$73$	−3.69646	−0.432638	−0.216319	+	0.976323i	$0.569405\pi$
$74$	−15.3371	−1.78290
$75$	−3.08613	−0.356356
$76$	−1.03031	−0.118185
$77$	5.04840	0.575318
$78$	0	0
$79$	15.0484	1.69308	0.846539	−	0.532327i	$-0.178682\pi$
$79$	15.0484	1.69308	0.846539	+	0.532327i	$0.178682\pi$
$80$	3.17226	0.354669
$81$	13.9926	1.55473
$82$	14.3445	1.58409
$83$	0.475800	0.0522259	0.0261129	−	0.999659i	$-0.491687\pi$
$83$	0.475800	0.0522259	0.0261129	+	0.999659i	$0.491687\pi$
$84$	−9.81290	−1.07068
$85$	2.70388	0.293277
$86$	0.438069	0.0472382
$87$	4.17226	0.447313
$88$	−2.74161	−0.292257
$89$	−11.0484	−1.17113	−0.585564	−	0.810626i	$-0.699127\pi$
$89$	−11.0484	−1.17113	−0.585564	+	0.810626i	$0.699127\pi$
$90$	13.6103	1.43465
$91$	0	0
$92$	−11.9623	−1.24715
$93$	−19.8687	−2.06029
$94$	2.82032	0.290894
$95$	0.438069	0.0449450
$96$	−24.9549	−2.54694
$97$	−3.29612	−0.334670	−0.167335	−	0.985900i	$-0.553516\pi$
$97$	−3.29612	−0.334670	−0.167335	+	0.985900i	$0.553516\pi$
$98$	10.7900	1.08996
$99$	24.3626	2.44853
$100$	2.35194	0.235194
$101$	16.1723	1.60920	0.804600	−	0.593817i	$-0.202380\pi$
$101$	16.1723	1.60920	0.804600	+	0.593817i	$0.202380\pi$
$102$	−17.4078	−1.72362
$103$	10.5545	1.03997	0.519983	−	0.854176i	$-0.325938\pi$
$103$	10.5545	1.03997	0.519983	+	0.854176i	$0.325938\pi$
$104$	0	0
$105$	4.17226	0.407171
$106$	3.06324	0.297528
$107$	13.4307	1.29839	0.649195	−	0.760622i	$-0.275106\pi$
$107$	13.4307	1.29839	0.649195	+	0.760622i	$0.275106\pi$
$108$	−25.5800	−2.46144
$109$	11.6406	1.11497	0.557486	−	0.830187i	$-0.311766\pi$
$109$	11.6406	1.11497	0.557486	+	0.830187i	$0.311766\pi$
$110$	7.79001	0.742748
$111$	−22.6890	−2.15355
$112$	−4.28870	−0.405244
$113$	−13.7523	−1.29371	−0.646853	−	0.762615i	$-0.723915\pi$
$113$	−13.7523	−1.29371	−0.646853	+	0.762615i	$0.723915\pi$
$114$	−2.82032	−0.264147
$115$	5.08613	0.474284
$116$	−3.17968	−0.295226
$117$	0	0
$118$	4.72677	0.435135
$119$	−3.65548	−0.335097
$120$	−2.26581	−0.206839
$121$	2.94418	0.267653
$122$	−7.35194	−0.665613
$123$	21.2207	1.91340
$124$	15.1419	1.35979
$125$	−1.00000	−0.0894427
$126$	−18.4003	−1.63923
$127$	−3.96227	−0.351595	−0.175797	−	0.984426i	$-0.556250\pi$
$127$	−3.96227	−0.351595	−0.175797	+	0.984426i	$0.556250\pi$
$128$	5.78259	0.511114
$129$	0.648061	0.0570586
$130$	0	0
$131$	−11.0484	−0.965303	−0.482652	−	0.875812i	$-0.660326\pi$
$131$	−11.0484	−0.965303	−0.482652	+	0.875812i	$0.660326\pi$
$132$	−27.1042	−2.35912
$133$	−0.592243	−0.0513540
$134$	−24.0410	−2.07682
$135$	10.8761	0.936069
$136$	1.98516	0.170226
$137$	12.5168	1.06938	0.534690	−	0.845048i	$-0.320428\pi$
$137$	12.5168	1.06938	0.534690	+	0.845048i	$0.320428\pi$
$138$	−32.7449	−2.78743
$139$	1.64064	0.139157	0.0695787	−	0.997576i	$-0.477834\pi$
$139$	1.64064	0.139157	0.0695787	+	0.997576i	$0.477834\pi$
$140$	−3.17968	−0.268732
$141$	4.17226	0.351368
$142$	0.913870	0.0766903
$143$	0	0
$144$	−20.6965	−1.72470
$145$	1.35194	0.112272
$146$	7.71130	0.638191
$147$	15.9623	1.31655
$148$	17.2913	1.42134
$149$	−3.29612	−0.270029	−0.135014	−	0.990844i	$-0.543108\pi$
$149$	−3.29612	−0.270029	−0.135014	+	0.990844i	$0.543108\pi$
$150$	6.43807	0.525666
$151$	−9.65873	−0.786016	−0.393008	−	0.919535i	$-0.628566\pi$
$151$	−9.65873	−0.786016	−0.393008	+	0.919535i	$0.628566\pi$
$152$	0.321627	0.0260874
$153$	−17.6406	−1.42616
$154$	−10.5316	−0.848662
$155$	−6.43807	−0.517118
$156$	0	0
$157$	11.5800	0.924186	0.462093	−	0.886831i	$-0.347099\pi$
$157$	11.5800	0.924186	0.462093	+	0.886831i	$0.347099\pi$
$158$	−31.3929	−2.49749
$159$	4.53162	0.359381
$160$	−8.08613	−0.639265
$161$	−6.87614	−0.541916
$162$	−29.1903	−2.29341
$163$	18.9926	1.48761	0.743807	−	0.668395i	$-0.233018\pi$
$163$	18.9926	1.48761	0.743807	+	0.668395i	$0.233018\pi$
$164$	−16.1723	−1.26284
$165$	11.5242	0.897158
$166$	−0.992582	−0.0770393
$167$	16.9320	1.31023	0.655117	−	0.755527i	$-0.272619\pi$
$167$	16.9320	1.31023	0.655117	+	0.755527i	$0.272619\pi$
$168$	3.06324	0.236334
$169$	0	0
$170$	−5.64064	−0.432618
$171$	−2.85805	−0.218561
$172$	−0.493887	−0.0376585
$173$	−10.7645	−0.818410	−0.409205	−	0.912442i	$-0.634194\pi$
$173$	−10.7645	−0.818410	−0.409205	+	0.912442i	$0.634194\pi$
$174$	−8.70388	−0.659839
$175$	1.35194	0.102197
$176$	−11.8458	−0.892913
$177$	6.99258	0.525595
$178$	23.0484	1.72755
$179$	18.5168	1.38401	0.692005	−	0.721893i	$-0.256728\pi$
$179$	18.5168	1.38401	0.692005	+	0.721893i	$0.256728\pi$
$180$	−15.3445	−1.14371
$181$	−12.2887	−0.913412	−0.456706	−	0.889618i	$-0.650971\pi$
$181$	−12.2887	−0.913412	−0.456706	+	0.889618i	$0.650971\pi$
$182$	0	0
$183$	−10.8761	−0.803987
$184$	3.73419	0.275288
$185$	−7.35194	−0.540525
$186$	41.4487	3.03917
$187$	−10.0968	−0.738351
$188$	−3.17968	−0.231902
$189$	−14.7039	−1.06955
$190$	−0.913870	−0.0662991
$191$	5.40776	0.391292	0.195646	−	0.980675i	$-0.437320\pi$
$191$	5.40776	0.391292	0.195646	+	0.980675i	$0.437320\pi$
$192$	32.4791	2.34397
$193$	−19.1090	−1.37550	−0.687749	−	0.725949i	$-0.741401\pi$
$193$	−19.1090	−1.37550	−0.687749	+	0.725949i	$0.741401\pi$
$194$	6.87614	0.493678
$195$	0	0
$196$	−12.1648	−0.868917
$197$	12.2839	0.875191	0.437596	−	0.899172i	$-0.355830\pi$
$197$	12.2839	0.875191	0.437596	+	0.899172i	$0.355830\pi$
$198$	−50.8236	−3.61187
$199$	0.764504	0.0541942	0.0270971	−	0.999633i	$-0.491374\pi$
$199$	0.764504	0.0541942	0.0270971	+	0.999633i	$0.491374\pi$
$200$	−0.734191	−0.0519151
$201$	−35.5652	−2.50857
$202$	−33.7374	−2.37376
$203$	−1.82774	−0.128282
$204$	19.6258	1.37408
$205$	6.87614	0.480250
$206$	−22.0181	−1.53407
$207$	−33.1829	−2.30637
$208$	0	0
$209$	−1.63583	−0.113153
$210$	−8.70388	−0.600625
$211$	−7.92454	−0.545548	−0.272774	−	0.962078i	$-0.587941\pi$
$211$	−7.92454	−0.545548	−0.272774	+	0.962078i	$0.587941\pi$
$212$	−3.45355	−0.237190
$213$	1.35194	0.0926333
$214$	−28.0181	−1.91528
$215$	0.209991	0.0143213
$216$	7.98516	0.543322
$217$	8.70388	0.590858
$218$	−24.2839	−1.64471
$219$	11.4078	0.770865
$220$	−8.78259	−0.592122
$221$	0	0
$222$	47.3323	3.17674
$223$	4.28870	0.287193	0.143596	−	0.989636i	$-0.454133\pi$
$223$	4.28870	0.287193	0.143596	+	0.989636i	$0.454133\pi$
$224$	10.9320	0.730422
$225$	6.52420	0.434947
$226$	28.6890	1.90837
$227$	25.5094	1.69312	0.846558	−	0.532297i	$-0.178671\pi$
$227$	25.5094	1.69312	0.846558	+	0.532297i	$0.178671\pi$
$228$	3.17968	0.210579
$229$	25.1090	1.65925	0.829626	−	0.558320i	$-0.188554\pi$
$229$	25.1090	1.65925	0.829626	+	0.558320i	$0.188554\pi$
$230$	−10.6103	−0.699624
$231$	−15.5800	−1.02509
$232$	0.992582	0.0651662
$233$	15.2961	1.00208	0.501041	−	0.865423i	$-0.332951\pi$
$233$	15.2961	1.00208	0.501041	+	0.865423i	$0.332951\pi$
$234$	0	0
$235$	1.35194	0.0881908
$236$	−5.32905	−0.346891
$237$	−46.4413	−3.01669
$238$	7.62581	0.494308
$239$	13.6736	0.884469	0.442235	−	0.896899i	$-0.354186\pi$
$239$	13.6736	0.884469	0.442235	+	0.896899i	$0.354186\pi$
$240$	−9.79001	−0.631942
$241$	12.5168	0.806277	0.403138	−	0.915139i	$-0.367919\pi$
$241$	12.5168	0.806277	0.403138	+	0.915139i	$0.367919\pi$
$242$	−6.14195	−0.394819
$243$	−10.5545	−0.677072
$244$	8.28870	0.530630
$245$	5.17226	0.330444
$246$	−44.2691	−2.82249
$247$	0	0
$248$	−4.72677	−0.300150
$249$	−1.46838	−0.0930549
$250$	2.08613	0.131938
$251$	−14.6284	−0.923337	−0.461669	−	0.887052i	$-0.652749\pi$
$251$	−14.6284	−0.923337	−0.461669	+	0.887052i	$0.652749\pi$
$252$	20.7449	1.30680
$253$	−18.9926	−1.19405
$254$	8.26581	0.518643
$255$	−8.34452	−0.522554
$256$	8.98516	0.561573
$257$	28.0968	1.75263	0.876315	−	0.481738i	$-0.159994\pi$
$257$	28.0968	1.75263	0.876315	+	0.481738i	$0.159994\pi$
$258$	−1.35194	−0.0841681
$259$	9.93937	0.617603
$260$	0	0
$261$	−8.82032	−0.545964
$262$	23.0484	1.42393
$263$	15.7752	0.972739	0.486369	−	0.873753i	$-0.338321\pi$
$263$	15.7752	0.972739	0.486369	+	0.873753i	$0.338321\pi$
$264$	8.46096	0.520736
$265$	1.46838	0.0902020
$266$	1.23550	0.0757531
$267$	34.0968	2.08669
$268$	27.1042	1.65565
$269$	−4.17226	−0.254387	−0.127194	−	0.991878i	$-0.540597\pi$
$269$	−4.17226	−0.254387	−0.127194	+	0.991878i	$0.540597\pi$
$270$	−22.6890	−1.38081
$271$	6.07871	0.369255	0.184628	−	0.982809i	$-0.440892\pi$
$271$	6.07871	0.369255	0.184628	+	0.982809i	$0.440892\pi$
$272$	8.57741	0.520082
$273$	0	0
$274$	−26.1116	−1.57746
$275$	3.73419	0.225180
$276$	36.9171	2.22215
$277$	−26.1574	−1.57165	−0.785824	−	0.618451i	$-0.787761\pi$
$277$	−26.1574	−1.57165	−0.785824	+	0.618451i	$0.787761\pi$
$278$	−3.42259	−0.205274
$279$	42.0032	2.51467
$280$	0.992582	0.0593181
$281$	6.64325	0.396303	0.198152	−	0.980171i	$-0.436506\pi$
$281$	6.64325	0.396303	0.198152	+	0.980171i	$0.436506\pi$
$282$	−8.70388	−0.518308
$283$	17.5423	1.04278	0.521390	−	0.853318i	$-0.325414\pi$
$283$	17.5423	1.04278	0.521390	+	0.853318i	$0.325414\pi$
$284$	−1.03031	−0.0611378
$285$	−1.35194	−0.0800820
$286$	0	0
$287$	−9.29612	−0.548733
$288$	52.7555	3.10865
$289$	−9.68904	−0.569944
$290$	−2.82032	−0.165615
$291$	10.1723	0.596308
$292$	−8.69385	−0.508769
$293$	−13.9442	−0.814628	−0.407314	−	0.913288i	$-0.633535\pi$
$293$	−13.9442	−0.814628	−0.407314	+	0.913288i	$0.633535\pi$
$294$	−33.2994	−1.94206
$295$	2.26581	0.131921
$296$	−5.39773	−0.313737
$297$	−40.6136	−2.35664
$298$	6.87614	0.398324
$299$	0	0
$300$	−7.25839	−0.419063
$301$	−0.283896	−0.0163635
$302$	20.1494	1.15947
$303$	−49.9097	−2.86724
$304$	1.38967	0.0797031
$305$	−3.52420	−0.201795
$306$	36.8007	2.10375
$307$	28.2132	1.61021	0.805107	−	0.593129i	$-0.202108\pi$
$307$	28.2132	1.61021	0.805107	+	0.593129i	$0.202108\pi$
$308$	11.8735	0.676557
$309$	−32.5726	−1.85299
$310$	13.4307	0.762810
$311$	7.23550	0.410287	0.205144	−	0.978732i	$-0.434234\pi$
$311$	7.23550	0.410287	0.205144	+	0.978732i	$0.434234\pi$
$312$	0	0
$313$	−21.7523	−1.22951	−0.614756	−	0.788718i	$-0.710745\pi$
$313$	−21.7523	−1.22951	−0.614756	+	0.788718i	$0.710745\pi$
$314$	−24.1574	−1.36328
$315$	−8.82032	−0.496969
$316$	35.3929	1.99101
$317$	14.7449	0.828154	0.414077	−	0.910242i	$-0.364104\pi$
$317$	14.7449	0.828154	0.414077	+	0.910242i	$0.364104\pi$
$318$	−9.45355	−0.530128
$319$	−5.04840	−0.282656
$320$	10.5242	0.588321
$321$	−41.4487	−2.31344
$322$	14.3445	0.799389
$323$	1.18449	0.0659066
$324$	32.9097	1.82832
$325$	0	0
$326$	−39.6210	−2.19440
$327$	−35.9245	−1.98663
$328$	5.04840	0.278751
$329$	−1.82774	−0.100767
$330$	−24.0410	−1.32341
$331$	34.0181	1.86980	0.934902	−	0.354907i	$-0.115488\pi$
$331$	34.0181	1.86980	0.934902	+	0.354907i	$0.115488\pi$
$332$	1.11905	0.0614160
$333$	47.9655	2.62849
$334$	−35.3223	−1.93275
$335$	−11.5242	−0.629634
$336$	13.2355	0.722056
$337$	15.3929	0.838506	0.419253	−	0.907869i	$-0.362292\pi$
$337$	15.3929	0.838506	0.419253	+	0.907869i	$0.362292\pi$
$338$	0	0
$339$	42.4413	2.30510
$340$	6.35936	0.344885
$341$	24.0410	1.30189
$342$	5.96227	0.322403
$343$	−16.4562	−0.888549
$344$	0.154174	0.00831249
$345$	−15.6965	−0.845069
$346$	22.4562	1.20725
$347$	−2.89903	−0.155628	−0.0778141	−	0.996968i	$-0.524794\pi$
$347$	−2.89903	−0.155628	−0.0778141	+	0.996968i	$0.524794\pi$
$348$	9.81290	0.526027
$349$	−18.2839	−0.978714	−0.489357	−	0.872083i	$-0.662769\pi$
$349$	−18.2839	−0.978714	−0.489357	+	0.872083i	$0.662769\pi$
$350$	−2.82032	−0.150753
$351$	0	0
$352$	30.1952	1.60941
$353$	21.1042	1.12326	0.561632	−	0.827387i	$-0.310174\pi$
$353$	21.1042	1.12326	0.561632	+	0.827387i	$0.310174\pi$
$354$	−14.5874	−0.775313
$355$	0.438069	0.0232503
$356$	−25.9852	−1.37721
$357$	11.2813	0.597069
$358$	−38.6284	−2.04158
$359$	−34.5349	−1.82268	−0.911340	−	0.411654i	$-0.864951\pi$
$359$	−34.5349	−1.82268	−0.911340	+	0.411654i	$0.864951\pi$
$360$	4.79001	0.252456
$361$	−18.8081	−0.989900
$362$	25.6358	1.34739
$363$	−9.08613	−0.476898
$364$	0	0
$365$	3.69646	0.193482
$366$	22.6890	1.18598
$367$	1.60291	0.0836713	0.0418356	−	0.999125i	$-0.486679\pi$
$367$	1.60291	0.0836713	0.0418356	+	0.999125i	$0.486679\pi$
$368$	16.1345	0.841070
$369$	−44.8613	−2.33539
$370$	15.3371	0.797338
$371$	−1.98516	−0.103065
$372$	−46.7300	−2.42284
$373$	19.5800	1.01381	0.506907	−	0.862000i	$-0.330789\pi$
$373$	19.5800	1.01381	0.506907	+	0.862000i	$0.330789\pi$
$374$	21.0632	1.08915
$375$	3.08613	0.159367
$376$	0.992582	0.0511885
$377$	0	0
$378$	30.6742	1.57771
$379$	−6.36261	−0.326825	−0.163413	−	0.986558i	$-0.552250\pi$
$379$	−6.36261	−0.326825	−0.163413	+	0.986558i	$0.552250\pi$
$380$	1.03031	0.0528539
$381$	12.2281	0.626463
$382$	−11.2813	−0.577201
$383$	−15.9804	−0.816558	−0.408279	−	0.912857i	$-0.633871\pi$
$383$	−15.9804	−0.816558	−0.408279	+	0.912857i	$0.633871\pi$
$384$	−17.8458	−0.910691
$385$	−5.04840	−0.257290
$386$	39.8639	2.02902
$387$	−1.37003	−0.0696423
$388$	−7.75228	−0.393562
$389$	−21.5046	−1.09032	−0.545162	−	0.838331i	$-0.683532\pi$
$389$	−21.5046	−1.09032	−0.545162	+	0.838331i	$0.683532\pi$
$390$	0	0
$391$	13.7523	0.695483
$392$	3.79743	0.191799
$393$	34.0968	1.71996
$394$	−25.6258	−1.29101
$395$	−15.0484	−0.757167
$396$	57.2994	2.87940
$397$	−15.4636	−0.776095	−0.388047	−	0.921639i	$-0.626850\pi$
$397$	−15.4636	−0.776095	−0.388047	+	0.921639i	$0.626850\pi$
$398$	−1.59485	−0.0799428
$399$	1.82774	0.0915014
$400$	−3.17226	−0.158613
$401$	25.7523	1.28601	0.643004	−	0.765863i	$-0.277688\pi$
$401$	25.7523	1.28601	0.643004	+	0.765863i	$0.277688\pi$
$402$	74.1936	3.70044
$403$	0	0
$404$	38.0362	1.89237
$405$	−13.9926	−0.695297
$406$	3.81290	0.189231
$407$	27.4535	1.36082
$408$	−6.12647	−0.303306
$409$	−10.5316	−0.520755	−0.260377	−	0.965507i	$-0.583847\pi$
$409$	−10.5316	−0.520755	−0.260377	+	0.965507i	$0.583847\pi$
$410$	−14.3445	−0.708425
$411$	−38.6284	−1.90540
$412$	24.8236	1.22297
$413$	−3.06324	−0.150732
$414$	69.2239	3.40217
$415$	−0.475800	−0.0233561
$416$	0	0
$417$	−5.06324	−0.247948
$418$	3.41256	0.166914
$419$	7.46838	0.364854	0.182427	−	0.983219i	$-0.441605\pi$
$419$	7.46838	0.364854	0.182427	+	0.983219i	$0.441605\pi$
$420$	9.81290	0.478821
$421$	35.4897	1.72966	0.864832	−	0.502062i	$-0.167425\pi$
$421$	35.4897	1.72966	0.864832	+	0.502062i	$0.167425\pi$
$422$	16.5316	0.804747
$423$	−8.82032	−0.428859
$424$	1.07807	0.0523558
$425$	−2.70388	−0.131157
$426$	−2.82032	−0.136645
$427$	4.76450	0.230570
$428$	31.5881	1.52687
$429$	0	0
$430$	−0.438069	−0.0211256
$431$	−0.154174	−0.00742629	−0.00371315	−	0.999993i	$-0.501182\pi$
$431$	−0.154174	−0.00742629	−0.00371315	+	0.999993i	$0.501182\pi$
$432$	34.5019	1.65998
$433$	5.65548	0.271785	0.135892	−	0.990724i	$-0.456610\pi$
$433$	5.65548	0.271785	0.135892	+	0.990724i	$0.456610\pi$
$434$	−18.1574	−0.871584
$435$	−4.17226	−0.200045
$436$	27.3781	1.31117
$437$	2.22808	0.106583
$438$	−23.7981	−1.13712
$439$	26.9729	1.28735	0.643674	−	0.765300i	$-0.277409\pi$
$439$	26.9729	1.28735	0.643674	+	0.765300i	$0.277409\pi$
$440$	2.74161	0.130701
$441$	−33.7449	−1.60690
$442$	0	0
$443$	29.3700	1.39541	0.697706	−	0.716384i	$-0.254204\pi$
$443$	29.3700	1.39541	0.697706	+	0.716384i	$0.254204\pi$
$444$	−53.3632	−2.53251
$445$	11.0484	0.523744
$446$	−8.94679	−0.423643
$447$	10.1723	0.481131
$448$	−14.2281	−0.672214
$449$	31.6768	1.49492	0.747461	−	0.664306i	$-0.231273\pi$
$449$	31.6768	1.49492	0.747461	+	0.664306i	$0.231273\pi$
$450$	−13.6103	−0.641597
$451$	−25.6768	−1.20907
$452$	−32.3445	−1.52136
$453$	29.8081	1.40051
$454$	−53.2159	−2.49754
$455$	0	0
$456$	−0.992582	−0.0464819
$457$	21.1696	0.990274	0.495137	−	0.868815i	$-0.335118\pi$
$457$	21.1696	0.990274	0.495137	+	0.868815i	$0.335118\pi$
$458$	−52.3807	−2.44759
$459$	29.4078	1.37264
$460$	11.9623	0.557744
$461$	−35.2058	−1.63970	−0.819849	−	0.572579i	$-0.805943\pi$
$461$	−35.2058	−1.63970	−0.819849	+	0.572579i	$0.805943\pi$
$462$	32.5019	1.51213
$463$	−1.35194	−0.0628299	−0.0314150	−	0.999506i	$-0.510001\pi$
$463$	−1.35194	−0.0628299	−0.0314150	+	0.999506i	$0.510001\pi$
$464$	4.28870	0.199098
$465$	19.8687	0.921390
$466$	−31.9097	−1.47819
$467$	−31.6635	−1.46521	−0.732607	−	0.680652i	$-0.761697\pi$
$467$	−31.6635	−1.46521	−0.732607	+	0.680652i	$0.761697\pi$
$468$	0	0
$469$	15.5800	0.719418
$470$	−2.82032	−0.130092
$471$	−35.7374	−1.64669
$472$	1.66354	0.0765705
$473$	−0.784148	−0.0360552
$474$	96.8826	4.44997
$475$	−0.438069	−0.0201000
$476$	−8.59746	−0.394064
$477$	−9.58002	−0.438639
$478$	−28.5248	−1.30470
$479$	13.3142	0.608342	0.304171	−	0.952617i	$-0.401621\pi$
$479$	13.3142	0.608342	0.304171	+	0.952617i	$0.401621\pi$
$480$	24.9549	1.13903
$481$	0	0
$482$	−26.1116	−1.18935
$483$	21.2207	0.965573
$484$	6.92454	0.314752
$485$	3.29612	0.149669
$486$	22.0181	0.998761
$487$	8.89578	0.403106	0.201553	−	0.979478i	$-0.435401\pi$
$487$	8.89578	0.403106	0.201553	+	0.979478i	$0.435401\pi$
$488$	−2.58744	−0.117128
$489$	−58.6136	−2.65060
$490$	−10.7900	−0.487443
$491$	−9.52901	−0.430038	−0.215019	−	0.976610i	$-0.568981\pi$
$491$	−9.52901	−0.430038	−0.215019	+	0.976610i	$0.568981\pi$
$492$	49.9097	2.25010
$493$	3.65548	0.164635
$494$	0	0
$495$	−24.3626	−1.09502
$496$	−20.4232	−0.917030
$497$	−0.592243	−0.0265657
$498$	3.06324	0.137267
$499$	−25.4716	−1.14027	−0.570133	−	0.821552i	$-0.693109\pi$
$499$	−25.4716	−1.14027	−0.570133	+	0.821552i	$0.693109\pi$
$500$	−2.35194	−0.105182
$501$	−52.2542	−2.33455
$502$	30.5168	1.36203
$503$	−32.6661	−1.45651	−0.728256	−	0.685305i	$-0.759669\pi$
$503$	−32.6661	−1.45651	−0.728256	+	0.685305i	$0.759669\pi$
$504$	−6.47580	−0.288455
$505$	−16.1723	−0.719656
$506$	39.6210	1.76137
$507$	0	0
$508$	−9.31902	−0.413464
$509$	8.41998	0.373209	0.186605	−	0.982435i	$-0.440252\pi$
$509$	8.41998	0.373209	0.186605	+	0.982435i	$0.440252\pi$
$510$	17.4078	0.770828
$511$	−4.99739	−0.221071
$512$	−30.3094	−1.33950
$513$	4.76450	0.210358
$514$	−58.6136	−2.58533
$515$	−10.5545	−0.465087
$516$	1.52420	0.0670991
$517$	−5.04840	−0.222028
$518$	−20.7348	−0.911036
$519$	33.2207	1.45823
$520$	0	0
$521$	27.3371	1.19766	0.598830	−	0.800876i	$-0.295632\pi$
$521$	27.3371	1.19766	0.598830	+	0.800876i	$0.295632\pi$
$522$	18.4003	0.805361
$523$	−7.77517	−0.339985	−0.169992	−	0.985445i	$-0.554374\pi$
$523$	−7.77517	−0.339985	−0.169992	+	0.985445i	$0.554374\pi$
$524$	−25.9852	−1.13517
$525$	−4.17226	−0.182092
$526$	−32.9091	−1.43490
$527$	−17.4078	−0.758294
$528$	36.5578	1.59097
$529$	2.86872	0.124727
$530$	−3.06324	−0.133058
$531$	−14.7826	−0.641510
$532$	−1.39292	−0.0603907
$533$	0	0
$534$	−71.1304	−3.07811
$535$	−13.4307	−0.580658
$536$	−8.46096	−0.365458
$537$	−57.1452	−2.46600
$538$	8.70388	0.375251
$539$	−19.3142	−0.831922
$540$	25.5800	1.10079
$541$	2.77934	0.119493	0.0597466	−	0.998214i	$-0.480971\pi$
$541$	2.77934	0.119493	0.0597466	+	0.998214i	$0.480971\pi$
$542$	−12.6810	−0.544695
$543$	37.9245	1.62750
$544$	−21.8639	−0.937408
$545$	−11.6406	−0.498630
$546$	0	0
$547$	−38.4184	−1.64265	−0.821327	−	0.570458i	$-0.806766\pi$
$547$	−38.4184	−1.64265	−0.821327	+	0.570458i	$0.806766\pi$
$548$	29.4387	1.25756
$549$	22.9926	0.981299
$550$	−7.79001	−0.332167
$551$	0.592243	0.0252304
$552$	−11.5242	−0.490503
$553$	20.3445	0.865137
$554$	54.5678	2.31836
$555$	22.6890	0.963096
$556$	3.85869	0.163645
$557$	−7.11905	−0.301644	−0.150822	−	0.988561i	$-0.548192\pi$
$557$	−7.11905	−0.301644	−0.150822	+	0.988561i	$0.548192\pi$
$558$	−87.6242	−3.70943
$559$	0	0
$560$	4.28870	0.181231
$561$	31.1600	1.31558
$562$	−13.8587	−0.584594
$563$	32.8236	1.38335	0.691674	−	0.722210i	$-0.256873\pi$
$563$	32.8236	1.38335	0.691674	+	0.722210i	$0.256873\pi$
$564$	9.81290	0.413198
$565$	13.7523	0.578563
$566$	−36.5955	−1.53822
$567$	18.9171	0.794444
$568$	0.321627	0.0134952
$569$	15.4126	0.646128	0.323064	−	0.946377i	$-0.395287\pi$
$569$	15.4126	0.646128	0.323064	+	0.946377i	$0.395287\pi$
$570$	2.82032	0.118130
$571$	31.5142	1.31883	0.659413	−	0.751780i	$-0.270805\pi$
$571$	31.5142	1.31883	0.659413	+	0.751780i	$0.270805\pi$
$572$	0	0
$573$	−16.6890	−0.697195
$574$	19.3929	0.809445
$575$	−5.08613	−0.212106
$576$	−68.6620	−2.86092
$577$	−44.3855	−1.84779	−0.923896	−	0.382643i	$-0.875014\pi$
$577$	−44.3855	−1.84779	−0.923896	+	0.382643i	$0.875014\pi$
$578$	20.2126	0.840733
$579$	58.9729	2.45083
$580$	3.17968	0.132029
$581$	0.643253	0.0266866
$582$	−21.2207	−0.879625
$583$	−5.48322	−0.227092
$584$	2.71391	0.112302
$585$	0	0
$586$	29.0894	1.20167
$587$	−13.3519	−0.551094	−0.275547	−	0.961288i	$-0.588859\pi$
$587$	−13.3519	−0.551094	−0.275547	+	0.961288i	$0.588859\pi$
$588$	37.5423	1.54822
$589$	−2.82032	−0.116209
$590$	−4.72677	−0.194598
$591$	−37.9097	−1.55940
$592$	−23.3223	−0.958539
$593$	−37.3929	−1.53554	−0.767772	−	0.640724i	$-0.778634\pi$
$593$	−37.3929	−1.53554	−0.767772	+	0.640724i	$0.778634\pi$
$594$	84.7252	3.47632
$595$	3.65548	0.149860
$596$	−7.75228	−0.317546
$597$	−2.35936	−0.0965621
$598$	0	0
$599$	13.8277	0.564986	0.282493	−	0.959269i	$-0.408839\pi$
$599$	13.8277	0.564986	0.282493	+	0.959269i	$0.408839\pi$
$600$	2.26581	0.0925013
$601$	6.34452	0.258798	0.129399	−	0.991593i	$-0.458695\pi$
$601$	6.34452	0.258798	0.129399	+	0.991593i	$0.458695\pi$
$602$	0.592243	0.0241380
$603$	75.1862	3.06182
$604$	−22.7167	−0.924331
$605$	−2.94418	−0.119698
$606$	104.118	4.22951
$607$	−16.2462	−0.659411	−0.329706	−	0.944084i	$-0.606950\pi$
$607$	−16.2462	−0.659411	−0.329706	+	0.944084i	$0.606950\pi$
$608$	−3.54229	−0.143659
$609$	5.64064	0.228570
$610$	7.35194	0.297671
$611$	0	0
$612$	−41.4897	−1.67712
$613$	22.6890	0.916402	0.458201	−	0.888849i	$-0.348494\pi$
$613$	22.6890	0.916402	0.458201	+	0.888849i	$0.348494\pi$
$614$	−58.8565	−2.37525
$615$	−21.2207	−0.855700
$616$	−3.70649	−0.149339
$617$	−9.01223	−0.362819	−0.181409	−	0.983408i	$-0.558066\pi$
$617$	−9.01223	−0.362819	−0.181409	+	0.983408i	$0.558066\pi$
$618$	67.9507	2.73338
$619$	−3.45030	−0.138679	−0.0693395	−	0.997593i	$-0.522089\pi$
$619$	−3.45030	−0.138679	−0.0693395	+	0.997593i	$0.522089\pi$
$620$	−15.1419	−0.608115
$621$	55.3175	2.21981
$622$	−15.0942	−0.605222
$623$	−14.9368	−0.598429
$624$	0	0
$625$	1.00000	0.0400000
$626$	45.3781	1.81367
$627$	5.04840	0.201614
$628$	27.2355	1.08681
$629$	−19.8787	−0.792618
$630$	18.4003	0.733087
$631$	24.9549	0.993437	0.496718	−	0.867912i	$-0.334538\pi$
$631$	24.9549	0.993437	0.496718	+	0.867912i	$0.334538\pi$
$632$	−11.0484	−0.439482
$633$	24.4562	0.972045
$634$	−30.7597	−1.22162
$635$	3.96227	0.157238
$636$	10.6581	0.422621
$637$	0	0
$638$	10.5316	0.416951
$639$	−2.85805	−0.113063
$640$	−5.78259	−0.228577
$641$	41.1304	1.62455	0.812276	−	0.583274i	$-0.198228\pi$
$641$	41.1304	1.62455	0.812276	+	0.583274i	$0.198228\pi$
$642$	86.4675	3.41260
$643$	−7.22547	−0.284945	−0.142472	−	0.989799i	$-0.545505\pi$
$643$	−7.22547	−0.284945	−0.142472	+	0.989799i	$0.545505\pi$
$644$	−16.1723	−0.637276
$645$	−0.648061	−0.0255174
$646$	−2.47099	−0.0972199
$647$	−8.45771	−0.332507	−0.166254	−	0.986083i	$-0.553167\pi$
$647$	−8.45771	−0.332507	−0.166254	+	0.986083i	$0.553167\pi$
$648$	−10.2732	−0.403570
$649$	−8.46096	−0.332122
$650$	0	0
$651$	−26.8613	−1.05278
$652$	44.6694	1.74939
$653$	−16.3807	−0.641026	−0.320513	−	0.947244i	$-0.603855\pi$
$653$	−16.3807	−0.641026	−0.320513	+	0.947244i	$0.603855\pi$
$654$	74.9433	2.93051
$655$	11.0484	0.431697
$656$	21.8129	0.851651
$657$	−24.1164	−0.940872
$658$	3.81290	0.148642
$659$	−0.308348	−0.0120115	−0.00600576	−	0.999982i	$-0.501912\pi$
$659$	−0.308348	−0.0120115	−0.00600576	+	0.999982i	$0.501912\pi$
$660$	27.1042	1.05503
$661$	32.7858	1.27522	0.637611	−	0.770359i	$-0.279923\pi$
$661$	32.7858	1.27522	0.637611	+	0.770359i	$0.279923\pi$
$662$	−70.9662	−2.75818
$663$	0	0
$664$	−0.349328	−0.0135566
$665$	0.592243	0.0229662
$666$	−100.062	−3.87734
$667$	6.87614	0.266245
$668$	39.8229	1.54080
$669$	−13.2355	−0.511714
$670$	24.0410	0.928784
$671$	13.1600	0.508037
$672$	−33.7374	−1.30145
$673$	−21.7523	−0.838489	−0.419244	−	0.907873i	$-0.637705\pi$
$673$	−21.7523	−0.838489	−0.419244	+	0.907873i	$0.637705\pi$
$674$	−32.1116	−1.23689
$675$	−10.8761	−0.418623
$676$	0	0
$677$	−14.4200	−0.554205	−0.277102	−	0.960840i	$-0.589374\pi$
$677$	−14.4200	−0.554205	−0.277102	+	0.960840i	$0.589374\pi$
$678$	−88.5381	−3.40029
$679$	−4.45616	−0.171012
$680$	−1.98516	−0.0761275
$681$	−78.7252	−3.01676
$682$	−50.1526	−1.92044
$683$	39.1797	1.49917	0.749584	−	0.661909i	$-0.230253\pi$
$683$	39.1797	1.49917	0.749584	+	0.661909i	$0.230253\pi$
$684$	−6.72197	−0.257021
$685$	−12.5168	−0.478242
$686$	34.3297	1.31071
$687$	−77.4897	−2.95642
$688$	0.666147	0.0253966
$689$	0	0
$690$	32.7449	1.24658
$691$	5.56193	0.211586	0.105793	−	0.994388i	$-0.466262\pi$
$691$	5.56193	0.211586	0.105793	+	0.994388i	$0.466262\pi$
$692$	−25.3175	−0.962425
$693$	32.9368	1.25116
$694$	6.04776	0.229570
$695$	−1.64064	−0.0622331
$696$	−3.06324	−0.116112
$697$	18.5922	0.704231
$698$	38.1426	1.44372
$699$	−47.2058	−1.78549
$700$	3.17968	0.120181
$701$	−9.58002	−0.361832	−0.180916	−	0.983499i	$-0.557906\pi$
$701$	−9.58002	−0.361832	−0.180916	+	0.983499i	$0.557906\pi$
$702$	0	0
$703$	−3.22066	−0.121469
$704$	−39.2994	−1.48115
$705$	−4.17226	−0.157136
$706$	−44.0261	−1.65695
$707$	21.8639	0.822277
$708$	16.4461	0.618083
$709$	−12.1574	−0.456582	−0.228291	−	0.973593i	$-0.573314\pi$
$709$	−12.1574	−0.456582	−0.228291	+	0.973593i	$0.573314\pi$
$710$	−0.913870	−0.0342969
$711$	98.1788	3.68199
$712$	8.11164	0.303996
$713$	−32.7449	−1.22630
$714$	−23.5342	−0.880746
$715$	0	0
$716$	43.5503	1.62755
$717$	−42.1984	−1.57593
$718$	72.0442	2.68867
$719$	21.1452	0.788583	0.394291	−	0.918985i	$-0.370990\pi$
$719$	21.1452	0.788583	0.394291	+	0.918985i	$0.370990\pi$
$720$	20.6965	0.771312
$721$	14.2691	0.531408
$722$	39.2361	1.46022
$723$	−38.6284	−1.43661
$724$	−28.9023	−1.07414
$725$	−1.35194	−0.0502098
$726$	18.9549	0.703480
$727$	−9.72938	−0.360843	−0.180421	−	0.983589i	$-0.557746\pi$
$727$	−9.72938	−0.360843	−0.180421	+	0.983589i	$0.557746\pi$
$728$	0	0
$729$	−9.40515	−0.348339
$730$	−7.71130	−0.285408
$731$	0.567791	0.0210005
$732$	−25.5800	−0.945465
$733$	4.40515	0.162708	0.0813539	−	0.996685i	$-0.474076\pi$
$733$	4.40515	0.162708	0.0813539	+	0.996685i	$0.474076\pi$
$734$	−3.34388	−0.123425
$735$	−15.9623	−0.588777
$736$	−41.1271	−1.51597
$737$	43.0336	1.58516
$738$	93.5865	3.44497
$739$	12.4891	0.459418	0.229709	−	0.973259i	$-0.426223\pi$
$739$	12.4891	0.459418	0.229709	+	0.973259i	$0.426223\pi$
$740$	−17.2913	−0.635641
$741$	0	0
$742$	4.14131	0.152032
$743$	−23.7571	−0.871563	−0.435781	−	0.900053i	$-0.643528\pi$
$743$	−23.7571	−0.871563	−0.435781	+	0.900053i	$0.643528\pi$
$744$	14.5874	0.534801
$745$	3.29612	0.120761
$746$	−40.8465	−1.49550
$747$	3.10422	0.113577
$748$	−23.7471	−0.868278
$749$	18.1574	0.663458
$750$	−6.43807	−0.235085
$751$	−32.1574	−1.17344	−0.586721	−	0.809789i	$-0.699581\pi$
$751$	−32.1574	−1.17344	−0.586721	+	0.809789i	$0.699581\pi$
$752$	4.28870	0.156393
$753$	45.1452	1.64518
$754$	0	0
$755$	9.65873	0.351517
$756$	−34.5826	−1.25776
$757$	−36.1723	−1.31470	−0.657352	−	0.753584i	$-0.728323\pi$
$757$	−36.1723	−1.31470	−0.657352	+	0.753584i	$0.728323\pi$
$758$	13.2732	0.482105
$759$	58.6136	2.12754
$760$	−0.321627	−0.0116666
$761$	−33.8639	−1.22757	−0.613783	−	0.789475i	$-0.710353\pi$
$761$	−33.8639	−1.22757	−0.613783	+	0.789475i	$0.710353\pi$
$762$	−25.5094	−0.924107
$763$	15.7374	0.569734
$764$	12.7187	0.460147
$765$	17.6406	0.637799
$766$	33.3371	1.20452
$767$	0	0
$768$	−27.7294	−1.00060
$769$	21.8639	0.788433	0.394216	−	0.919018i	$-0.371016\pi$
$769$	21.8639	0.788433	0.394216	+	0.919018i	$0.371016\pi$
$770$	10.5316	0.379533
$771$	−86.7104	−3.12280
$772$	−44.9433	−1.61754
$773$	22.1378	0.796241	0.398120	−	0.917333i	$-0.369663\pi$
$773$	22.1378	0.796241	0.398120	+	0.917333i	$0.369663\pi$
$774$	2.85805	0.102731
$775$	6.43807	0.231262
$776$	2.41998	0.0868723
$777$	−30.6742	−1.10043
$778$	44.8613	1.60836
$779$	3.01223	0.107924
$780$	0	0
$781$	−1.63583	−0.0585348
$782$	−28.6890	−1.02592
$783$	14.7039	0.525474
$784$	16.4078	0.585991
$785$	−11.5800	−0.413309
$786$	−71.1304	−2.53714
$787$	−15.2616	−0.544019	−0.272009	−	0.962295i	$-0.587688\pi$
$787$	−15.2616	−0.544019	−0.272009	+	0.962295i	$0.587688\pi$
$788$	28.8910	1.02920
$789$	−48.6842	−1.73320
$790$	31.3929	1.11691
$791$	−18.5922	−0.661064
$792$	−17.8868	−0.635580
$793$	0	0
$794$	32.2590	1.14483
$795$	−4.53162	−0.160720
$796$	1.79807	0.0637308
$797$	9.22066	0.326613	0.163306	−	0.986575i	$-0.447784\pi$
$797$	9.22066	0.326613	0.163306	+	0.986575i	$0.447784\pi$
$798$	−3.81290	−0.134975
$799$	3.65548	0.129322
$800$	8.08613	0.285888
$801$	−72.0820	−2.54689
$802$	−53.7226	−1.89701
$803$	−13.8033	−0.487107
$804$	−83.6471	−2.95001
$805$	6.87614	0.242352
$806$	0	0
$807$	12.8761	0.453262
$808$	−11.8735	−0.417709
$809$	0.167453	0.00588733	0.00294366	−	0.999996i	$-0.499063\pi$
$809$	0.167453	0.00588733	0.00294366	+	0.999996i	$0.499063\pi$
$810$	29.1903	1.02564
$811$	36.0032	1.26425	0.632123	−	0.774868i	$-0.282184\pi$
$811$	36.0032	1.26425	0.632123	+	0.774868i	$0.282184\pi$
$812$	−4.29873	−0.150856
$813$	−18.7597	−0.657931
$814$	−57.2717	−2.00737
$815$	−18.9926	−0.665281
$816$	−26.4710	−0.926670
$817$	0.0919908	0.00321835
$818$	21.9703	0.768174
$819$	0	0
$820$	16.1723	0.564760
$821$	5.04840	0.176190	0.0880952	−	0.996112i	$-0.471922\pi$
$821$	5.04840	0.176190	0.0880952	+	0.996112i	$0.471922\pi$
$822$	80.5839	2.81069
$823$	−11.3094	−0.394221	−0.197110	−	0.980381i	$-0.563156\pi$
$823$	−11.3094	−0.394221	−0.197110	+	0.980381i	$0.563156\pi$
$824$	−7.74903	−0.269950
$825$	−11.5242	−0.401221
$826$	6.39031	0.222347
$827$	−43.8687	−1.52546	−0.762732	−	0.646714i	$-0.776143\pi$
$827$	−43.8687	−1.52546	−0.762732	+	0.646714i	$0.776143\pi$
$828$	−78.0442	−2.71222
$829$	−44.4461	−1.54368	−0.771839	−	0.635818i	$-0.780663\pi$
$829$	−44.4461	−1.54368	−0.771839	+	0.635818i	$0.780663\pi$
$830$	0.992582	0.0344530
$831$	80.7252	2.80033
$832$	0	0
$833$	13.9852	0.484557
$834$	10.5626	0.365752
$835$	−16.9320	−0.585955
$836$	−3.84738	−0.133065
$837$	−70.0213	−2.42029
$838$	−15.5800	−0.538203
$839$	−1.44068	−0.0497378	−0.0248689	−	0.999691i	$-0.507917\pi$
$839$	−1.44068	−0.0497378	−0.0248689	+	0.999691i	$0.507917\pi$
$840$	−3.06324	−0.105692
$841$	−27.1723	−0.936974
$842$	−74.0362	−2.55146
$843$	−20.5019	−0.706124
$844$	−18.6380	−0.641548
$845$	0	0
$846$	18.4003	0.632617
$847$	3.98036	0.136767
$848$	4.65809	0.159959
$849$	−54.1378	−1.85800
$850$	5.64064	0.193472
$851$	−37.3929	−1.28181
$852$	3.17968	0.108934
$853$	0.992582	0.0339853	0.0169927	−	0.999856i	$-0.494591\pi$
$853$	0.992582	0.0339853	0.0169927	+	0.999856i	$0.494591\pi$
$854$	−9.93937	−0.340118
$855$	2.85805	0.0977433
$856$	−9.86066	−0.337031
$857$	−10.6071	−0.362331	−0.181165	−	0.983453i	$-0.557987\pi$
$857$	−10.6071	−0.362331	−0.181165	+	0.983453i	$0.557987\pi$
$858$	0	0
$859$	−40.9123	−1.39591	−0.697955	−	0.716142i	$-0.745906\pi$
$859$	−40.9123	−1.39591	−0.697955	+	0.716142i	$0.745906\pi$
$860$	0.493887	0.0168414
$861$	28.6890	0.977720
$862$	0.321627	0.0109546
$863$	−48.2494	−1.64243	−0.821215	−	0.570619i	$-0.806703\pi$
$863$	−48.2494	−1.64243	−0.821215	+	0.570619i	$0.806703\pi$
$864$	−87.9459	−2.99198
$865$	10.7645	0.366004
$866$	−11.7981	−0.400915
$867$	29.9016	1.01551
$868$	20.4710	0.694831
$869$	56.1936	1.90624
$870$	8.70388	0.295089
$871$	0	0
$872$	−8.54645	−0.289419
$873$	−21.5046	−0.727819
$874$	−4.64806	−0.157223
$875$	−1.35194	−0.0457039
$876$	26.8304	0.906514
$877$	−4.81551	−0.162608	−0.0813042	−	0.996689i	$-0.525909\pi$
$877$	−4.81551	−0.162608	−0.0813042	+	0.996689i	$0.525909\pi$
$878$	−56.2691	−1.89899
$879$	43.0336	1.45149
$880$	11.8458	0.399323
$881$	21.6210	0.728430	0.364215	−	0.931315i	$-0.381337\pi$
$881$	21.6210	0.728430	0.364215	+	0.931315i	$0.381337\pi$
$882$	70.3962	2.37036
$883$	−21.1616	−0.712144	−0.356072	−	0.934458i	$-0.615884\pi$
$883$	−21.1616	−0.712144	−0.356072	+	0.934458i	$0.615884\pi$
$884$	0	0
$885$	−6.99258	−0.235053
$886$	−61.2697	−2.05840
$887$	−9.10097	−0.305581	−0.152790	−	0.988259i	$-0.548826\pi$
$887$	−9.10097	−0.305581	−0.152790	+	0.988259i	$0.548826\pi$
$888$	16.6581	0.559009
$889$	−5.35675	−0.179660
$890$	−23.0484	−0.772584
$891$	52.2510	1.75047
$892$	10.0868	0.337730
$893$	0.592243	0.0198187
$894$	−21.2207	−0.709725
$895$	−18.5168	−0.618948
$896$	7.81771	0.261171
$897$	0	0
$898$	−66.0820	−2.20518
$899$	−8.70388	−0.290291
$900$	15.3445	0.511484
$901$	3.97033	0.132271
$902$	53.5652	1.78353
$903$	0.876139	0.0291561
$904$	10.0968	0.335815
$905$	12.2887	0.408490
$906$	−62.1836	−2.06591
$907$	43.2436	1.43588	0.717939	−	0.696106i	$-0.245086\pi$
$907$	43.2436	1.43588	0.717939	+	0.696106i	$0.245086\pi$
$908$	59.9965	1.99105
$909$	105.511	3.49958
$910$	0	0
$911$	23.2058	0.768843	0.384422	−	0.923158i	$-0.374401\pi$
$911$	23.2058	0.768843	0.384422	+	0.923158i	$0.374401\pi$
$912$	−4.28870	−0.142013
$913$	1.77673	0.0588012
$914$	−44.1626	−1.46077
$915$	10.8761	0.359554
$916$	59.0549	1.95123
$917$	−14.9368	−0.493255
$918$	−61.3484	−2.02480
$919$	0.419983	0.0138540	0.00692698	−	0.999976i	$-0.497795\pi$
$919$	0.419983	0.0138540	0.00692698	+	0.999976i	$0.497795\pi$
$920$	−3.73419	−0.123113
$921$	−87.0697	−2.86905
$922$	73.4439	2.41875
$923$	0	0
$924$	−36.6433	−1.20547
$925$	7.35194	0.241730
$926$	2.82032	0.0926815
$927$	68.8597	2.26165
$928$	−10.9320	−0.358859
$929$	39.2207	1.28679	0.643394	−	0.765535i	$-0.277526\pi$
$929$	39.2207	1.28679	0.643394	+	0.765535i	$0.277526\pi$
$930$	−41.4487	−1.35916
$931$	2.26581	0.0742589
$932$	35.9755	1.17842
$933$	−22.3297	−0.731041
$934$	66.0543	2.16136
$935$	10.0968	0.330201
$936$	0	0
$937$	−22.5774	−0.737572	−0.368786	−	0.929514i	$-0.620226\pi$
$937$	−22.5774	−0.737572	−0.368786	+	0.929514i	$0.620226\pi$
$938$	−32.5019	−1.06123
$939$	67.1304	2.19072
$940$	3.17968	0.103710
$941$	18.0510	0.588446	0.294223	−	0.955737i	$-0.404939\pi$
$941$	18.0510	0.588446	0.294223	+	0.955737i	$0.404939\pi$
$942$	74.5530	2.42907
$943$	34.9729	1.13888
$944$	7.18774	0.233941
$945$	14.7039	0.478317
$946$	1.63583	0.0531856
$947$	−24.5578	−0.798020	−0.399010	−	0.916947i	$-0.630646\pi$
$947$	−24.5578	−0.798020	−0.399010	+	0.916947i	$0.630646\pi$
$948$	−109.227	−3.54753
$949$	0	0
$950$	0.913870	0.0296499
$951$	−45.5046	−1.47559
$952$	2.68382	0.0869831
$953$	34.9219	1.13123	0.565616	−	0.824669i	$-0.308638\pi$
$953$	34.9219	1.13123	0.565616	+	0.824669i	$0.308638\pi$
$954$	19.9852	0.647044
$955$	−5.40776	−0.174991
$956$	32.1594	1.04011
$957$	15.5800	0.503630
$958$	−27.7752	−0.897375
$959$	16.9219	0.546438
$960$	−32.4791	−1.04826
$961$	10.4487	0.337056
$962$	0	0
$963$	87.6242	2.82365
$964$	29.4387	0.948157
$965$	19.1090	0.615141
$966$	−44.2691	−1.42433
$967$	5.16484	0.166090	0.0830451	−	0.996546i	$-0.473535\pi$
$967$	5.16484	0.166090	0.0830451	+	0.996546i	$0.473535\pi$
$968$	−2.16159	−0.0694762
$969$	−3.65548	−0.117431
$970$	−6.87614	−0.220780
$971$	14.2935	0.458701	0.229350	−	0.973344i	$-0.426340\pi$
$971$	14.2935	0.458701	0.229350	+	0.973344i	$0.426340\pi$
$972$	−24.8236	−0.796216
$973$	2.21805	0.0711074
$974$	−18.5578	−0.594629
$975$	0	0
$976$	−11.1797	−0.357853
$977$	0.424790	0.0135902	0.00679512	−	0.999977i	$-0.497837\pi$
$977$	0.424790	0.0135902	0.00679512	+	0.999977i	$0.497837\pi$
$978$	122.276	3.90994
$979$	−41.2568	−1.31857
$980$	12.1648	0.388592
$981$	75.9459	2.42477
$982$	19.8787	0.634356
$983$	17.8081	0.567990	0.283995	−	0.958826i	$-0.408340\pi$
$983$	17.8081	0.567990	0.283995	+	0.958826i	$0.408340\pi$
$984$	−15.5800	−0.496673
$985$	−12.2839	−0.391397
$986$	−7.62581	−0.242855
$987$	5.64064	0.179544
$988$	0	0
$989$	1.06804	0.0339618
$990$	50.8236	1.61528
$991$	−34.7497	−1.10386	−0.551930	−	0.833891i	$-0.686108\pi$
$991$	−34.7497	−1.10386	−0.551930	+	0.833891i	$0.686108\pi$
$992$	52.0591	1.65288
$993$	−104.984	−3.33157
$994$	1.23550	0.0391876
$995$	−0.764504	−0.0242364
$996$	−3.45355	−0.109430
$997$	48.0213	1.52085	0.760425	−	0.649425i	$-0.224990\pi$
$997$	48.0213	1.52085	0.760425	+	0.649425i	$0.224990\pi$
$998$	53.1371	1.68203
$999$	−79.9607	−2.52984

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.a.k.1.1		3
3.2	odd	2		7605.2.a.bs.1.3		3
5.4	even	2		4225.2.a.bc.1.3		3
13.2	odd	12		845.2.m.h.316.2		12
13.3	even	3		845.2.e.i.191.3		6
13.4	even	6		845.2.e.k.146.1		6
13.5	odd	4		65.2.c.a.51.5	yes	6
13.6	odd	12		845.2.m.h.361.5		12
13.7	odd	12		845.2.m.h.361.2		12
13.8	odd	4		65.2.c.a.51.2	✓	6
13.9	even	3		845.2.e.i.146.3		6
13.10	even	6		845.2.e.k.191.1		6
13.11	odd	12		845.2.m.h.316.5		12
13.12	even	2		845.2.a.i.1.3		3
39.5	even	4		585.2.b.g.181.2		6
39.8	even	4		585.2.b.g.181.5		6
39.38	odd	2		7605.2.a.cc.1.1		3
52.31	even	4		1040.2.k.d.961.6		6
52.47	even	4		1040.2.k.d.961.5		6
65.8	even	4		325.2.d.f.324.1		6
65.18	even	4		325.2.d.e.324.5		6
65.34	odd	4		325.2.c.g.51.5		6
65.44	odd	4		325.2.c.g.51.2		6
65.47	even	4		325.2.d.e.324.6		6
65.57	even	4		325.2.d.f.324.2		6
65.64	even	2		4225.2.a.be.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.c.a.51.2	✓	6	13.8	odd	4
65.2.c.a.51.5	yes	6	13.5	odd	4
325.2.c.g.51.2		6	65.44	odd	4
325.2.c.g.51.5		6	65.34	odd	4
325.2.d.e.324.5		6	65.18	even	4
325.2.d.e.324.6		6	65.47	even	4
325.2.d.f.324.1		6	65.8	even	4
325.2.d.f.324.2		6	65.57	even	4
585.2.b.g.181.2		6	39.5	even	4
585.2.b.g.181.5		6	39.8	even	4
845.2.a.i.1.3		3	13.12	even	2
845.2.a.k.1.1		3	1.1	even	1	trivial
845.2.e.i.146.3		6	13.9	even	3
845.2.e.i.191.3		6	13.3	even	3
845.2.e.k.146.1		6	13.4	even	6
845.2.e.k.191.1		6	13.10	even	6
845.2.m.h.316.2		12	13.2	odd	12
845.2.m.h.316.5		12	13.11	odd	12
845.2.m.h.361.2		12	13.7	odd	12
845.2.m.h.361.5		12	13.6	odd	12
1040.2.k.d.961.5		6	52.47	even	4
1040.2.k.d.961.6		6	52.31	even	4
4225.2.a.bc.1.3		3	5.4	even	2
4225.2.a.be.1.1		3	65.64	even	2
7605.2.a.bs.1.3		3	3.2	odd	2
7605.2.a.cc.1.1		3	39.38	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)

\(f(q)\)	\(=\)	\(q-2.08613 q^{2} -3.08613 q^{3} +2.35194 q^{4} -1.00000 q^{5} +6.43807 q^{6} +1.35194 q^{7} -0.734191 q^{8} +6.52420 q^{9} +O(q^{10})\)	\(q-2.08613 q^{2} -3.08613 q^{3} +2.35194 q^{4} -1.00000 q^{5} +6.43807 q^{6} +1.35194 q^{7} -0.734191 q^{8} +6.52420 q^{9} +2.08613 q^{10} +3.73419 q^{11} -7.25839 q^{12} -2.82032 q^{14} +3.08613 q^{15} -3.17226 q^{16} -2.70388 q^{17} -13.6103 q^{18} -0.438069 q^{19} -2.35194 q^{20} -4.17226 q^{21} -7.79001 q^{22} -5.08613 q^{23} +2.26581 q^{24} +1.00000 q^{25} -10.8761 q^{27} +3.17968 q^{28} -1.35194 q^{29} -6.43807 q^{30} +6.43807 q^{31} +8.08613 q^{32} -11.5242 q^{33} +5.64064 q^{34} -1.35194 q^{35} +15.3445 q^{36} +7.35194 q^{37} +0.913870 q^{38} +0.734191 q^{40} -6.87614 q^{41} +8.70388 q^{42} -0.209991 q^{43} +8.78259 q^{44} -6.52420 q^{45} +10.6103 q^{46} -1.35194 q^{47} +9.79001 q^{48} -5.17226 q^{49} -2.08613 q^{50} +8.34452 q^{51} -1.46838 q^{53} +22.6890 q^{54} -3.73419 q^{55} -0.992582 q^{56} +1.35194 q^{57} +2.82032 q^{58} -2.26581 q^{59} +7.25839 q^{60} +3.52420 q^{61} -13.4307 q^{62} +8.82032 q^{63} -10.5242 q^{64} +24.0410 q^{66} +11.5242 q^{67} -6.35936 q^{68} +15.6965 q^{69} +2.82032 q^{70} -0.438069 q^{71} -4.79001 q^{72} -3.69646 q^{73} -15.3371 q^{74} -3.08613 q^{75} -1.03031 q^{76} +5.04840 q^{77} +15.0484 q^{79} +3.17226 q^{80} +13.9926 q^{81} +14.3445 q^{82} +0.475800 q^{83} -9.81290 q^{84} +2.70388 q^{85} +0.438069 q^{86} +4.17226 q^{87} -2.74161 q^{88} -11.0484 q^{89} +13.6103 q^{90} -11.9623 q^{92} -19.8687 q^{93} +2.82032 q^{94} +0.438069 q^{95} -24.9549 q^{96} -3.29612 q^{97} +10.7900 q^{98} +24.3626 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} + 10 q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + q^2 - 2 * q^3 + 5 * q^4 - 3 * q^5 + 10 * q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + q^{2} - 2 q^{3} + 5 q^{4} - 3 q^{5} + 10 q^{6} + 2 q^{7} + 3 q^{8} + 3 q^{9} - q^{10} + 6 q^{11} + 4 q^{14} + 2 q^{15} + 5 q^{16} - 4 q^{17} - 17 q^{18} + 8 q^{19} - 5 q^{20} + 2 q^{21} - 12 q^{22} - 8 q^{23} + 12 q^{24} + 3 q^{25} - 14 q^{27} + 22 q^{28} - 2 q^{29} - 10 q^{30} + 10 q^{31} + 17 q^{32} - 18 q^{33} - 8 q^{34} - 2 q^{35} + 17 q^{36} + 20 q^{37} + 10 q^{38} - 3 q^{40} - 2 q^{41} + 22 q^{42} - 12 q^{43} - 12 q^{44} - 3 q^{45} + 8 q^{46} - 2 q^{47} + 18 q^{48} - q^{49} + q^{50} - 4 q^{51} + 6 q^{53} + 10 q^{54} - 6 q^{55} + 24 q^{56} + 2 q^{57} - 4 q^{58} - 12 q^{59} - 6 q^{61} - 4 q^{62} + 14 q^{63} - 15 q^{64} + 12 q^{66} + 18 q^{67} - 44 q^{68} + 16 q^{69} - 4 q^{70} + 8 q^{71} - 3 q^{72} + 20 q^{73} + 10 q^{74} - 2 q^{75} - 2 q^{76} - 18 q^{77} + 12 q^{79} - 5 q^{80} + 15 q^{81} + 14 q^{82} + 18 q^{83} + 10 q^{84} + 4 q^{85} - 8 q^{86} - 2 q^{87} - 30 q^{88} + 17 q^{90} - 10 q^{92} - 14 q^{93} - 4 q^{94} - 8 q^{95} - 22 q^{96} - 14 q^{97} + 21 q^{98} + 12 q^{99}+O(q^{100}) \) 3 * q + q^2 - 2 * q^3 + 5 * q^4 - 3 * q^5 + 10 * q^6 + 2 * q^7 + 3 * q^8 + 3 * q^9 - q^10 + 6 * q^11 + 4 * q^14 + 2 * q^15 + 5 * q^16 - 4 * q^17 - 17 * q^18 + 8 * q^19 - 5 * q^20 + 2 * q^21 - 12 * q^22 - 8 * q^23 + 12 * q^24 + 3 * q^25 - 14 * q^27 + 22 * q^28 - 2 * q^29 - 10 * q^30 + 10 * q^31 + 17 * q^32 - 18 * q^33 - 8 * q^34 - 2 * q^35 + 17 * q^36 + 20 * q^37 + 10 * q^38 - 3 * q^40 - 2 * q^41 + 22 * q^42 - 12 * q^43 - 12 * q^44 - 3 * q^45 + 8 * q^46 - 2 * q^47 + 18 * q^48 - q^49 + q^50 - 4 * q^51 + 6 * q^53 + 10 * q^54 - 6 * q^55 + 24 * q^56 + 2 * q^57 - 4 * q^58 - 12 * q^59 - 6 * q^61 - 4 * q^62 + 14 * q^63 - 15 * q^64 + 12 * q^66 + 18 * q^67 - 44 * q^68 + 16 * q^69 - 4 * q^70 + 8 * q^71 - 3 * q^72 + 20 * q^73 + 10 * q^74 - 2 * q^75 - 2 * q^76 - 18 * q^77 + 12 * q^79 - 5 * q^80 + 15 * q^81 + 14 * q^82 + 18 * q^83 + 10 * q^84 + 4 * q^85 - 8 * q^86 - 2 * q^87 - 30 * q^88 + 17 * q^90 - 10 * q^92 - 14 * q^93 - 4 * q^94 - 8 * q^95 - 22 * q^96 - 14 * q^97 + 21 * q^98 + 12 * q^99

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