LMFDB - Embedded newform 2366.4.a.bg.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2366,4,Mod(1,2366)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2366, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2366.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2366 = 2 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2366.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:	$139.598519074$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} - 219 x^{10} + 1022 x^{9} + 17084 x^{8} - 65540 x^{7} - 566763 x^{6} + 1871300 x^{5} + \cdots + 543166 $ x^12 - 6x^11 - 219x^10 + 1022x^9 + 17084x^8 - 65540x^7 - 566763x^6 + 1871300x^5 + 6589295x^4 - 20167176x^3 + 8073768x^2 + 5594524*x + 543166
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 13 $
Twist minimal:	no (minimal twist has level 182)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$6.59050$ of defining polynomial
Character	$\chi$	$=$	2366.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.85845 q^{3} +4.00000 q^{4} +13.8421 q^{5} +9.71690 q^{6} -7.00000 q^{7} +8.00000 q^{8} -3.39547 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} +4.85845 q^{3} +4.00000 q^{4} +13.8421 q^{5} +9.71690 q^{6} -7.00000 q^{7} +8.00000 q^{8} -3.39547 q^{9} +27.6843 q^{10} +51.8429 q^{11} +19.4338 q^{12} -14.0000 q^{14} +67.2513 q^{15} +16.0000 q^{16} -105.787 q^{17} -6.79094 q^{18} -48.0058 q^{19} +55.3685 q^{20} -34.0091 q^{21} +103.686 q^{22} +22.8487 q^{23} +38.8676 q^{24} +66.6045 q^{25} -147.675 q^{27} -28.0000 q^{28} +233.181 q^{29} +134.503 q^{30} +123.298 q^{31} +32.0000 q^{32} +251.876 q^{33} -211.574 q^{34} -96.8949 q^{35} -13.5819 q^{36} +187.074 q^{37} -96.0117 q^{38} +110.737 q^{40} +354.907 q^{41} -68.0183 q^{42} +72.8028 q^{43} +207.372 q^{44} -47.0005 q^{45} +45.6974 q^{46} +470.316 q^{47} +77.7352 q^{48} +49.0000 q^{49} +133.209 q^{50} -513.961 q^{51} +159.833 q^{53} -295.350 q^{54} +717.616 q^{55} -56.0000 q^{56} -233.234 q^{57} +466.361 q^{58} +320.826 q^{59} +269.005 q^{60} +478.752 q^{61} +246.596 q^{62} +23.7683 q^{63} +64.0000 q^{64} +503.752 q^{66} -343.671 q^{67} -423.148 q^{68} +111.009 q^{69} -193.790 q^{70} -147.749 q^{71} -27.1637 q^{72} +779.080 q^{73} +374.149 q^{74} +323.595 q^{75} -192.023 q^{76} -362.900 q^{77} +962.115 q^{79} +221.474 q^{80} -625.793 q^{81} +709.815 q^{82} +1097.00 q^{83} -136.037 q^{84} -1464.32 q^{85} +145.606 q^{86} +1132.90 q^{87} +414.743 q^{88} -709.491 q^{89} -94.0010 q^{90} +91.3948 q^{92} +599.037 q^{93} +940.631 q^{94} -664.503 q^{95} +155.470 q^{96} +355.024 q^{97} +98.0000 q^{98} -176.031 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 24 q^{2} + 6 q^{3} + 48 q^{4} + 28 q^{5} + 12 q^{6} - 84 q^{7} + 96 q^{8} + 162 q^{9}+O(q^{10}) $ 12 * q + 24 * q^2 + 6 * q^3 + 48 * q^4 + 28 * q^5 + 12 * q^6 - 84 * q^7 + 96 * q^8 + 162 * q^9	\( 12 q + 24 q^{2} + 6 q^{3} + 48 q^{4} + 28 q^{5} + 12 q^{6} - 84 q^{7} + 96 q^{8} + 162 q^{9} + 56 q^{10} + 6 q^{11} + 24 q^{12} - 168 q^{14} - 138 q^{15} + 192 q^{16} - 56 q^{17} + 324 q^{18} + 158 q^{19} + 112 q^{20} - 42 q^{21} + 12 q^{22} + 414 q^{23} + 48 q^{24} + 478 q^{25} + 390 q^{27} - 336 q^{28} + 222 q^{29} - 276 q^{30} - 200 q^{31} + 384 q^{32} + 844 q^{33} - 112 q^{34} - 196 q^{35} + 648 q^{36} - 560 q^{37} + 316 q^{38} + 224 q^{40} - 66 q^{41} - 84 q^{42} + 484 q^{43} + 24 q^{44} + 542 q^{45} + 828 q^{46} + 618 q^{47} + 96 q^{48} + 588 q^{49} + 956 q^{50} + 992 q^{51} + 504 q^{53} + 780 q^{54} + 2584 q^{55} - 672 q^{56} - 1164 q^{57} + 444 q^{58} + 1460 q^{59} - 552 q^{60} - 2 q^{61} - 400 q^{62} - 1134 q^{63} + 768 q^{64} + 1688 q^{66} + 334 q^{67} - 224 q^{68} + 4660 q^{69} - 392 q^{70} - 196 q^{71} + 1296 q^{72} + 490 q^{73} - 1120 q^{74} - 338 q^{75} + 632 q^{76} - 42 q^{77} + 2942 q^{79} + 448 q^{80} + 2824 q^{81} - 132 q^{82} + 236 q^{83} - 168 q^{84} + 1352 q^{85} + 968 q^{86} + 1456 q^{87} + 48 q^{88} + 3566 q^{89} + 1084 q^{90} + 1656 q^{92} + 1884 q^{93} + 1236 q^{94} + 5754 q^{95} + 192 q^{96} + 3032 q^{97} + 1176 q^{98} + 3670 q^{99}+O(q^{100}) \) 12 * q + 24 * q^2 + 6 * q^3 + 48 * q^4 + 28 * q^5 + 12 * q^6 - 84 * q^7 + 96 * q^8 + 162 * q^9 + 56 * q^10 + 6 * q^11 + 24 * q^12 - 168 * q^14 - 138 * q^15 + 192 * q^16 - 56 * q^17 + 324 * q^18 + 158 * q^19 + 112 * q^20 - 42 * q^21 + 12 * q^22 + 414 * q^23 + 48 * q^24 + 478 * q^25 + 390 * q^27 - 336 * q^28 + 222 * q^29 - 276 * q^30 - 200 * q^31 + 384 * q^32 + 844 * q^33 - 112 * q^34 - 196 * q^35 + 648 * q^36 - 560 * q^37 + 316 * q^38 + 224 * q^40 - 66 * q^41 - 84 * q^42 + 484 * q^43 + 24 * q^44 + 542 * q^45 + 828 * q^46 + 618 * q^47 + 96 * q^48 + 588 * q^49 + 956 * q^50 + 992 * q^51 + 504 * q^53 + 780 * q^54 + 2584 * q^55 - 672 * q^56 - 1164 * q^57 + 444 * q^58 + 1460 * q^59 - 552 * q^60 - 2 * q^61 - 400 * q^62 - 1134 * q^63 + 768 * q^64 + 1688 * q^66 + 334 * q^67 - 224 * q^68 + 4660 * q^69 - 392 * q^70 - 196 * q^71 + 1296 * q^72 + 490 * q^73 - 1120 * q^74 - 338 * q^75 + 632 * q^76 - 42 * q^77 + 2942 * q^79 + 448 * q^80 + 2824 * q^81 - 132 * q^82 + 236 * q^83 - 168 * q^84 + 1352 * q^85 + 968 * q^86 + 1456 * q^87 + 48 * q^88 + 3566 * q^89 + 1084 * q^90 + 1656 * q^92 + 1884 * q^93 + 1236 * q^94 + 5754 * q^95 + 192 * q^96 + 3032 * q^97 + 1176 * q^98 + 3670 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	4.85845	0.935009	0.467505	−	0.883991i	$-0.345153\pi$
$3$	4.85845	0.935009	0.467505	+	0.883991i	$0.345153\pi$
$4$	4.00000	0.500000
$5$	13.8421	1.23808	0.619039	−	0.785360i	$-0.287522\pi$
$5$	13.8421	1.23808	0.619039	+	0.785360i	$0.287522\pi$
$6$	9.71690	0.661151
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	−3.39547	−0.125758
$10$	27.6843	0.875453
$11$	51.8429	1.42102	0.710510	−	0.703687i	$-0.248464\pi$
$11$	51.8429	1.42102	0.710510	+	0.703687i	$0.248464\pi$
$12$	19.4338	0.467505
$13$	0	0
$13$	0	0
$14$	−14.0000	−0.267261
$15$	67.2513	1.15761
$16$	16.0000	0.250000
$17$	−105.787	−1.50924	−0.754621	−	0.656161i	$-0.772179\pi$
$17$	−105.787	−1.50924	−0.754621	+	0.656161i	$0.772179\pi$
$18$	−6.79094	−0.0889244
$19$	−48.0058	−0.579647	−0.289824	−	0.957080i	$-0.593597\pi$
$19$	−48.0058	−0.579647	−0.289824	+	0.957080i	$0.593597\pi$
$20$	55.3685	0.619039
$21$	−34.0091	−0.353400
$22$	103.686	1.00481
$23$	22.8487	0.207143	0.103571	−	0.994622i	$-0.466973\pi$
$23$	22.8487	0.207143	0.103571	+	0.994622i	$0.466973\pi$
$24$	38.8676	0.330576
$25$	66.6045	0.532836
$26$	0	0
$27$	−147.675	−1.05259
$28$	−28.0000	−0.188982
$29$	233.181	1.49312	0.746561	−	0.665316i	$-0.231703\pi$
$29$	233.181	1.49312	0.746561	+	0.665316i	$0.231703\pi$
$30$	134.503	0.818557
$31$	123.298	0.714354	0.357177	−	0.934037i	$-0.383739\pi$
$31$	123.298	0.714354	0.357177	+	0.934037i	$0.383739\pi$
$32$	32.0000	0.176777
$33$	251.876	1.32867
$34$	−211.574	−1.06720
$35$	−96.8949	−0.467949
$36$	−13.5819	−0.0628790
$37$	187.074	0.831212	0.415606	−	0.909545i	$-0.363570\pi$
$37$	187.074	0.831212	0.415606	+	0.909545i	$0.363570\pi$
$38$	−96.0117	−0.409872
$39$	0	0
$40$	110.737	0.437727
$41$	354.907	1.35188	0.675942	−	0.736955i	$-0.263737\pi$
$41$	354.907	1.35188	0.675942	+	0.736955i	$0.263737\pi$
$42$	−68.0183	−0.249892
$43$	72.8028	0.258193	0.129097	−	0.991632i	$-0.458792\pi$
$43$	72.8028	0.258193	0.129097	+	0.991632i	$0.458792\pi$
$44$	207.372	0.710510
$45$	−47.0005	−0.155698
$46$	45.6974	0.146472
$47$	470.316	1.45963	0.729815	−	0.683645i	$-0.239606\pi$
$47$	470.316	1.45963	0.729815	+	0.683645i	$0.239606\pi$
$48$	77.7352	0.233752
$49$	49.0000	0.142857
$50$	133.209	0.376772
$51$	−513.961	−1.41115
$52$	0	0
$53$	159.833	0.414242	0.207121	−	0.978315i	$-0.433591\pi$
$53$	159.833	0.414242	0.207121	+	0.978315i	$0.433591\pi$
$54$	−295.350	−0.744296
$55$	717.616	1.75933
$56$	−56.0000	−0.133631
$57$	−233.234	−0.541975
$58$	466.361	1.05580
$59$	320.826	0.707931	0.353966	−	0.935258i	$-0.384833\pi$
$59$	320.826	0.707931	0.353966	+	0.935258i	$0.384833\pi$
$60$	269.005	0.578807
$61$	478.752	1.00488	0.502442	−	0.864611i	$-0.332435\pi$
$61$	478.752	1.00488	0.502442	+	0.864611i	$0.332435\pi$
$62$	246.596	0.505124
$63$	23.7683	0.0475321
$64$	64.0000	0.125000
$65$	0	0
$66$	503.752	0.939509
$67$	−343.671	−0.626658	−0.313329	−	0.949645i	$-0.601444\pi$
$67$	−343.671	−0.626658	−0.313329	+	0.949645i	$0.601444\pi$
$68$	−423.148	−0.754621
$69$	111.009	0.193680
$70$	−193.790	−0.330890
$71$	−147.749	−0.246966	−0.123483	−	0.992347i	$-0.539406\pi$
$71$	−147.749	−0.246966	−0.123483	+	0.992347i	$0.539406\pi$
$72$	−27.1637	−0.0444622
$73$	779.080	1.24910	0.624551	−	0.780984i	$-0.285282\pi$
$73$	779.080	1.24910	0.624551	+	0.780984i	$0.285282\pi$
$74$	374.149	0.587755
$75$	323.595	0.498207
$76$	−192.023	−0.289824
$77$	−362.900	−0.537095
$78$	0	0
$79$	962.115	1.37021	0.685104	−	0.728445i	$-0.259757\pi$
$79$	962.115	1.37021	0.685104	+	0.728445i	$0.259757\pi$
$80$	221.474	0.309519
$81$	−625.793	−0.858427
$82$	709.815	0.955926
$83$	1097.00	1.45074	0.725369	−	0.688361i	$-0.241669\pi$
$83$	1097.00	1.45074	0.725369	+	0.688361i	$0.241669\pi$
$84$	−136.037	−0.176700
$85$	−1464.32	−1.86856
$86$	145.606	0.182570
$87$	1132.90	1.39608
$88$	414.743	0.502406
$89$	−709.491	−0.845010	−0.422505	−	0.906361i	$-0.638849\pi$
$89$	−709.491	−0.845010	−0.422505	+	0.906361i	$0.638849\pi$
$90$	−94.0010	−0.110095
$91$	0	0
$92$	91.3948	0.103571
$93$	599.037	0.667927
$94$	940.631	1.03211
$95$	−664.503	−0.717648
$96$	155.470	0.165288
$97$	355.024	0.371621	0.185811	−	0.982586i	$-0.440509\pi$
$97$	355.024	0.371621	0.185811	+	0.982586i	$0.440509\pi$
$98$	98.0000	0.101015
$99$	−176.031	−0.178705
$100$	266.418	0.266418
$101$	−476.630	−0.469569	−0.234785	−	0.972047i	$-0.575438\pi$
$101$	−476.630	−0.469569	−0.234785	+	0.972047i	$0.575438\pi$
$102$	−1027.92	−0.997837
$103$	−1564.39	−1.49655	−0.748274	−	0.663390i	$-0.769117\pi$
$103$	−1564.39	−1.49655	−0.748274	+	0.663390i	$0.769117\pi$
$104$	0	0
$105$	−470.759	−0.437537
$106$	319.667	0.292913
$107$	1641.19	1.48280	0.741399	−	0.671064i	$-0.234162\pi$
$107$	1641.19	1.48280	0.741399	+	0.671064i	$0.234162\pi$
$108$	−590.699	−0.526297
$109$	−1828.38	−1.60667	−0.803335	−	0.595527i	$-0.796943\pi$
$109$	−1828.38	−1.60667	−0.803335	+	0.595527i	$0.796943\pi$
$110$	1435.23	1.24404
$111$	908.891	0.777191
$112$	−112.000	−0.0944911
$113$	−638.758	−0.531764	−0.265882	−	0.964006i	$-0.585663\pi$
$113$	−638.758	−0.531764	−0.265882	+	0.964006i	$0.585663\pi$
$114$	−466.468	−0.383234
$115$	316.275	0.256459
$116$	932.723	0.746561
$117$	0	0
$118$	641.651	0.500583
$119$	740.509	0.570440
$120$	538.010	0.409278
$121$	1356.68	1.01930
$122$	957.504	0.710560
$123$	1724.30	1.26402
$124$	493.192	0.357177
$125$	−808.318	−0.578385
$126$	47.5366	0.0336103
$127$	−1940.17	−1.35561	−0.677805	−	0.735242i	$-0.737068\pi$
$127$	−1940.17	−1.35561	−0.677805	+	0.735242i	$0.737068\pi$
$128$	128.000	0.0883883
$129$	353.709	0.241413
$130$	0	0
$131$	698.893	0.466126	0.233063	−	0.972462i	$-0.425125\pi$
$131$	698.893	0.466126	0.233063	+	0.972462i	$0.425125\pi$
$132$	1007.50	0.664333
$133$	336.041	0.219086
$134$	−687.342	−0.443114
$135$	−2044.13	−1.30319
$136$	−846.296	−0.533598
$137$	−2273.06	−1.41752	−0.708761	−	0.705448i	$-0.750746\pi$
$137$	−2273.06	−1.41752	−0.708761	+	0.705448i	$0.750746\pi$
$138$	222.019	0.136953
$139$	−2107.46	−1.28599	−0.642993	−	0.765872i	$-0.722308\pi$
$139$	−2107.46	−1.28599	−0.642993	+	0.765872i	$0.722308\pi$
$140$	−387.580	−0.233975
$141$	2285.00	1.36477
$142$	−295.498	−0.174631
$143$	0	0
$144$	−54.3275	−0.0314395
$145$	3227.72	1.84860
$146$	1558.16	0.883249
$147$	238.064	0.133573
$148$	748.297	0.415606
$149$	−1089.78	−0.599180	−0.299590	−	0.954068i	$-0.596850\pi$
$149$	−1089.78	−0.599180	−0.299590	+	0.954068i	$0.596850\pi$
$150$	647.189	0.352285
$151$	−2968.18	−1.59965	−0.799826	−	0.600232i	$-0.795075\pi$
$151$	−2968.18	−1.59965	−0.799826	+	0.600232i	$0.795075\pi$
$152$	−384.047	−0.204936
$153$	359.196	0.189799
$154$	−725.800	−0.379783
$155$	1706.71	0.884425
$156$	0	0
$157$	3026.87	1.53866	0.769332	−	0.638849i	$-0.220589\pi$
$157$	3026.87	1.53866	0.769332	+	0.638849i	$0.220589\pi$
$158$	1924.23	0.968883
$159$	776.543	0.387320
$160$	442.948	0.218863
$161$	−159.941	−0.0782926
$162$	−1251.59	−0.606999
$163$	469.542	0.225628	0.112814	−	0.993616i	$-0.464014\pi$
$163$	469.542	0.225628	0.112814	+	0.993616i	$0.464014\pi$
$164$	1419.63	0.675942
$165$	3486.50	1.64499
$166$	2194.00	1.02583
$167$	−1376.82	−0.637974	−0.318987	−	0.947759i	$-0.603343\pi$
$167$	−1376.82	−0.637974	−0.318987	+	0.947759i	$0.603343\pi$
$168$	−272.073	−0.124946
$169$	0	0
$170$	−2928.63	−1.32127
$171$	163.002	0.0728953
$172$	291.211	0.129097
$173$	346.341	0.152207	0.0761035	−	0.997100i	$-0.475752\pi$
$173$	346.341	0.152207	0.0761035	+	0.997100i	$0.475752\pi$
$174$	2265.79	0.987180
$175$	−466.232	−0.201393
$176$	829.486	0.355255
$177$	1558.71	0.661922
$178$	−1418.98	−0.597512
$179$	1807.66	0.754811	0.377406	−	0.926048i	$-0.376816\pi$
$179$	1807.66	0.754811	0.377406	+	0.926048i	$0.376816\pi$
$180$	−188.002	−0.0778491
$181$	2138.29	0.878110	0.439055	−	0.898460i	$-0.355313\pi$
$181$	2138.29	0.878110	0.439055	+	0.898460i	$0.355313\pi$
$182$	0	0
$183$	2325.99	0.939575
$184$	182.790	0.0732360
$185$	2589.51	1.02910
$186$	1198.07	0.472296
$187$	−5484.30	−2.14466
$188$	1881.26	0.729815
$189$	1033.72	0.397843
$190$	−1329.01	−0.507454
$191$	−2405.05	−0.911115	−0.455558	−	0.890206i	$-0.650560\pi$
$191$	−2405.05	−0.911115	−0.455558	+	0.890206i	$0.650560\pi$
$192$	310.941	0.116876
$193$	1341.73	0.500412	0.250206	−	0.968193i	$-0.419502\pi$
$193$	1341.73	0.500412	0.250206	+	0.968193i	$0.419502\pi$
$194$	710.049	0.262776
$195$	0	0
$196$	196.000	0.0714286
$197$	4329.75	1.56590	0.782949	−	0.622086i	$-0.213715\pi$
$197$	4329.75	1.56590	0.782949	+	0.622086i	$0.213715\pi$
$198$	−352.062	−0.126363
$199$	1853.27	0.660176	0.330088	−	0.943950i	$-0.392921\pi$
$199$	1853.27	0.660176	0.330088	+	0.943950i	$0.392921\pi$
$200$	532.836	0.188386
$201$	−1669.71	−0.585931
$202$	−953.260	−0.332035
$203$	−1632.26	−0.564347
$204$	−2055.84	−0.705577
$205$	4912.67	1.67374
$206$	−3128.79	−1.05822
$207$	−77.5820	−0.0260499
$208$	0	0
$209$	−2488.76	−0.823690
$210$	−941.518	−0.309385
$211$	−1358.15	−0.443121	−0.221561	−	0.975147i	$-0.571115\pi$
$211$	−1358.15	−0.443121	−0.221561	+	0.975147i	$0.571115\pi$
$212$	639.334	0.207121
$213$	−717.831	−0.230915
$214$	3282.37	1.04850
$215$	1007.75	0.319663
$216$	−1181.40	−0.372148
$217$	−863.086	−0.270000
$218$	−3656.76	−1.13609
$219$	3785.12	1.16792
$220$	2870.46	0.879666
$221$	0	0
$222$	1817.78	0.549557
$223$	−6094.90	−1.83025	−0.915123	−	0.403175i	$-0.867907\pi$
$223$	−6094.90	−1.83025	−0.915123	+	0.403175i	$0.867907\pi$
$224$	−224.000	−0.0668153
$225$	−226.154	−0.0670085
$226$	−1277.52	−0.376014
$227$	5436.03	1.58943	0.794717	−	0.606980i	$-0.207619\pi$
$227$	5436.03	1.58943	0.794717	+	0.606980i	$0.207619\pi$
$228$	−932.936	−0.270988
$229$	−3794.17	−1.09487	−0.547437	−	0.836847i	$-0.684396\pi$
$229$	−3794.17	−1.09487	−0.547437	+	0.836847i	$0.684396\pi$
$230$	632.549	0.181344
$231$	−1763.13	−0.502189
$232$	1865.45	0.527899
$233$	1070.66	0.301035	0.150518	−	0.988607i	$-0.451906\pi$
$233$	1070.66	0.301035	0.150518	+	0.988607i	$0.451906\pi$
$234$	0	0
$235$	6510.17	1.80713
$236$	1283.30	0.353966
$237$	4674.39	1.28116
$238$	1481.02	0.403362
$239$	−5205.17	−1.40876	−0.704382	−	0.709821i	$-0.748776\pi$
$239$	−5205.17	−1.40876	−0.704382	+	0.709821i	$0.748776\pi$
$240$	1076.02	0.289403
$241$	−3077.22	−0.822494	−0.411247	−	0.911524i	$-0.634907\pi$
$241$	−3077.22	−0.822494	−0.411247	+	0.911524i	$0.634907\pi$
$242$	2713.37	0.720752
$243$	946.836	0.249957
$244$	1915.01	0.502442
$245$	678.264	0.176868
$246$	3448.60	0.893799
$247$	0	0
$248$	986.384	0.252562
$249$	5329.71	1.35645
$250$	−1616.64	−0.408980
$251$	1338.28	0.336540	0.168270	−	0.985741i	$-0.446182\pi$
$251$	1338.28	0.336540	0.168270	+	0.985741i	$0.446182\pi$
$252$	95.0731	0.0237660
$253$	1184.54	0.294354
$254$	−3880.34	−0.958561
$255$	−7114.31	−1.74712
$256$	256.000	0.0625000
$257$	−5716.43	−1.38747	−0.693737	−	0.720228i	$-0.744037\pi$
$257$	−5716.43	−1.38747	−0.693737	+	0.720228i	$0.744037\pi$
$258$	707.417	0.170705
$259$	−1309.52	−0.314169
$260$	0	0
$261$	−791.758	−0.187772
$262$	1397.79	0.329601
$263$	−5068.87	−1.18844	−0.594220	−	0.804303i	$-0.702539\pi$
$263$	−5068.87	−1.18844	−0.594220	+	0.804303i	$0.702539\pi$
$264$	2015.01	0.469754
$265$	2212.43	0.512863
$266$	672.082	0.154917
$267$	−3447.03	−0.790092
$268$	−1374.68	−0.313329
$269$	−6243.48	−1.41514	−0.707568	−	0.706645i	$-0.750208\pi$
$269$	−6243.48	−1.41514	−0.707568	+	0.706645i	$0.750208\pi$
$270$	−4088.27	−0.921497
$271$	−4648.16	−1.04190	−0.520951	−	0.853587i	$-0.674423\pi$
$271$	−4648.16	−1.04190	−0.520951	+	0.853587i	$0.674423\pi$
$272$	−1692.59	−0.377310
$273$	0	0
$274$	−4546.12	−1.00234
$275$	3452.97	0.757171
$276$	444.037	0.0968402
$277$	−2699.24	−0.585493	−0.292746	−	0.956190i	$-0.594569\pi$
$277$	−2699.24	−0.585493	−0.292746	+	0.956190i	$0.594569\pi$
$278$	−4214.91	−0.909330
$279$	−418.654	−0.0898358
$280$	−775.159	−0.165445
$281$	3742.77	0.794573	0.397286	−	0.917695i	$-0.369952\pi$
$281$	3742.77	0.794573	0.397286	+	0.917695i	$0.369952\pi$
$282$	4570.01	0.965036
$283$	−5419.64	−1.13839	−0.569194	−	0.822203i	$-0.692745\pi$
$283$	−5419.64	−1.13839	−0.569194	+	0.822203i	$0.692745\pi$
$284$	−590.996	−0.123483
$285$	−3228.45	−0.671007
$286$	0	0
$287$	−2484.35	−0.510964
$288$	−108.655	−0.0222311
$289$	6277.88	1.27781
$290$	6455.43	1.30716
$291$	1724.87	0.347469
$292$	3116.32	0.624551
$293$	−2507.16	−0.499897	−0.249948	−	0.968259i	$-0.580414\pi$
$293$	−2507.16	−0.499897	−0.249948	+	0.968259i	$0.580414\pi$
$294$	476.128	0.0944502
$295$	4440.91	0.876474
$296$	1496.59	0.293878
$297$	−7655.89	−1.49576
$298$	−2179.55	−0.423684
$299$	0	0
$300$	1294.38	0.249103
$301$	−509.619	−0.0975879
$302$	−5936.37	−1.13112
$303$	−2315.68	−0.439051
$304$	−768.093	−0.144912
$305$	6626.95	1.24412
$306$	718.393	0.134208
$307$	−3558.50	−0.661545	−0.330773	−	0.943710i	$-0.607309\pi$
$307$	−3558.50	−0.661545	−0.330773	+	0.943710i	$0.607309\pi$
$308$	−1451.60	−0.268547
$309$	−7600.53	−1.39929
$310$	3413.41	0.625383
$311$	−790.495	−0.144131	−0.0720657	−	0.997400i	$-0.522959\pi$
$311$	−790.495	−0.144131	−0.0720657	+	0.997400i	$0.522959\pi$
$312$	0	0
$313$	6589.31	1.18994	0.594968	−	0.803749i	$-0.297165\pi$
$313$	6589.31	1.18994	0.594968	+	0.803749i	$0.297165\pi$
$314$	6053.73	1.08800
$315$	329.004	0.0588484
$316$	3848.46	0.685104
$317$	−9740.30	−1.72577	−0.862886	−	0.505398i	$-0.831346\pi$
$317$	−9740.30	−1.72577	−0.862886	+	0.505398i	$0.831346\pi$
$318$	1553.09	0.273876
$319$	12088.8	2.12176
$320$	885.896	0.154760
$321$	7973.62	1.38643
$322$	−319.882	−0.0553612
$323$	5078.39	0.874828
$324$	−2503.17	−0.429213
$325$	0	0
$326$	939.084	0.159543
$327$	−8883.09	−1.50225
$328$	2839.26	0.477963
$329$	−3292.21	−0.551688
$330$	6973.00	1.16318
$331$	−1309.24	−0.217410	−0.108705	−	0.994074i	$-0.534670\pi$
$331$	−1309.24	−0.217410	−0.108705	+	0.994074i	$0.534670\pi$
$332$	4387.99	0.725369
$333$	−635.205	−0.104532
$334$	−2753.64	−0.451116
$335$	−4757.14	−0.775851
$336$	−544.146	−0.0883501
$337$	−7265.34	−1.17439	−0.587194	−	0.809446i	$-0.699767\pi$
$337$	−7265.34	−1.17439	−0.587194	+	0.809446i	$0.699767\pi$
$338$	0	0
$339$	−3103.38	−0.497204
$340$	−5857.27	−0.934279
$341$	6392.12	1.01511
$342$	326.005	0.0515448
$343$	−343.000	−0.0539949
$344$	582.422	0.0912852
$345$	1536.60	0.239791
$346$	692.682	0.107627
$347$	2244.17	0.347186	0.173593	−	0.984817i	$-0.444462\pi$
$347$	2244.17	0.347186	0.173593	+	0.984817i	$0.444462\pi$
$348$	4531.59	0.698042
$349$	−7509.56	−1.15180	−0.575899	−	0.817521i	$-0.695348\pi$
$349$	−7509.56	−1.15180	−0.575899	+	0.817521i	$0.695348\pi$
$350$	−932.463	−0.142406
$351$	0	0
$352$	1658.97	0.251203
$353$	8419.25	1.26944	0.634719	−	0.772743i	$-0.281116\pi$
$353$	8419.25	1.26944	0.634719	+	0.772743i	$0.281116\pi$
$354$	3117.43	0.468049
$355$	−2045.16	−0.305763
$356$	−2837.96	−0.422505
$357$	3597.72	0.533366
$358$	3615.33	0.533732
$359$	−3581.38	−0.526513	−0.263256	−	0.964726i	$-0.584797\pi$
$359$	−3581.38	−0.526513	−0.263256	+	0.964726i	$0.584797\pi$
$360$	−376.004	−0.0550476
$361$	−4554.44	−0.664009
$362$	4276.58	0.620918
$363$	6591.38	0.953052
$364$	0	0
$365$	10784.1	1.54649
$366$	4651.98	0.664380
$367$	−7202.07	−1.02437	−0.512187	−	0.858874i	$-0.671164\pi$
$367$	−7202.07	−1.02437	−0.512187	+	0.858874i	$0.671164\pi$
$368$	365.579	0.0517857
$369$	−1205.08	−0.170010
$370$	5179.01	0.727687
$371$	−1118.83	−0.156569
$372$	2396.15	0.333964
$373$	4731.06	0.656743	0.328372	−	0.944549i	$-0.393500\pi$
$373$	4731.06	0.656743	0.328372	+	0.944549i	$0.393500\pi$
$374$	−10968.6	−1.51651
$375$	−3927.17	−0.540795
$376$	3762.52	0.516057
$377$	0	0
$378$	2067.45	0.281318
$379$	−6574.68	−0.891079	−0.445539	−	0.895262i	$-0.646988\pi$
$379$	−6574.68	−0.891079	−0.445539	+	0.895262i	$0.646988\pi$
$380$	−2658.01	−0.358824
$381$	−9426.23	−1.26751
$382$	−4810.09	−0.644256
$383$	5046.06	0.673215	0.336608	−	0.941645i	$-0.390720\pi$
$383$	5046.06	0.673215	0.336608	+	0.941645i	$0.390720\pi$
$384$	621.882	0.0826439
$385$	−5023.31	−0.664965
$386$	2683.45	0.353845
$387$	−247.199	−0.0324699
$388$	1420.10	0.185811
$389$	8101.69	1.05597	0.527985	−	0.849254i	$-0.322948\pi$
$389$	8101.69	1.05597	0.527985	+	0.849254i	$0.322948\pi$
$390$	0	0
$391$	−2417.10	−0.312629
$392$	392.000	0.0505076
$393$	3395.54	0.435832
$394$	8659.50	1.10726
$395$	13317.7	1.69642
$396$	−704.123	−0.0893523
$397$	−1924.21	−0.243258	−0.121629	−	0.992576i	$-0.538812\pi$
$397$	−1924.21	−0.243258	−0.121629	+	0.992576i	$0.538812\pi$
$398$	3706.55	0.466815
$399$	1632.64	0.204847
$400$	1065.67	0.133209
$401$	12535.6	1.56109	0.780544	−	0.625101i	$-0.214942\pi$
$401$	12535.6	1.56109	0.780544	+	0.625101i	$0.214942\pi$
$402$	−3339.42	−0.414316
$403$	0	0
$404$	−1906.52	−0.234785
$405$	−8662.31	−1.06280
$406$	−3264.53	−0.399054
$407$	9698.47	1.18117
$408$	−4111.69	−0.498919
$409$	5310.06	0.641970	0.320985	−	0.947084i	$-0.395986\pi$
$409$	5310.06	0.641970	0.320985	+	0.947084i	$0.395986\pi$
$410$	9825.35	1.18351
$411$	−11043.5	−1.32540
$412$	−6257.58	−0.748274
$413$	−2245.78	−0.267573
$414$	−155.164	−0.0184200
$415$	15184.8	1.79613
$416$	0	0
$417$	−10239.0	−1.20241
$418$	−4977.52	−0.582437
$419$	12040.8	1.40389	0.701947	−	0.712229i	$-0.252314\pi$
$419$	12040.8	1.40389	0.701947	+	0.712229i	$0.252314\pi$
$420$	−1883.04	−0.218768
$421$	4855.80	0.562131	0.281066	−	0.959689i	$-0.409312\pi$
$421$	4855.80	0.562131	0.281066	+	0.959689i	$0.409312\pi$
$422$	−2716.29	−0.313334
$423$	−1596.94	−0.183560
$424$	1278.67	0.146457
$425$	−7045.89	−0.804179
$426$	−1435.66	−0.163282
$427$	−3351.26	−0.379810
$428$	6564.75	0.741399
$429$	0	0
$430$	2015.49	0.226036
$431$	−12615.6	−1.40991	−0.704954	−	0.709253i	$-0.749032\pi$
$431$	−12615.6	−1.40991	−0.704954	+	0.709253i	$0.749032\pi$
$432$	−2362.80	−0.263148
$433$	10733.9	1.19131	0.595655	−	0.803240i	$-0.296892\pi$
$433$	10733.9	1.19131	0.595655	+	0.803240i	$0.296892\pi$
$434$	−1726.17	−0.190919
$435$	15681.7	1.72846
$436$	−7313.52	−0.803335
$437$	−1096.87	−0.120070
$438$	7570.25	0.825846
$439$	−1434.13	−0.155916	−0.0779580	−	0.996957i	$-0.524840\pi$
$439$	−1434.13	−0.155916	−0.0779580	+	0.996957i	$0.524840\pi$
$440$	5740.93	0.622018
$441$	−166.378	−0.0179654
$442$	0	0
$443$	1155.06	0.123879	0.0619395	−	0.998080i	$-0.480271\pi$
$443$	1155.06	0.123879	0.0619395	+	0.998080i	$0.480271\pi$
$444$	3635.57	0.388595
$445$	−9820.87	−1.04619
$446$	−12189.8	−1.29418
$447$	−5294.62	−0.560239
$448$	−448.000	−0.0472456
$449$	4988.94	0.524371	0.262185	−	0.965018i	$-0.415557\pi$
$449$	4988.94	0.524371	0.262185	+	0.965018i	$0.415557\pi$
$450$	−452.307	−0.0473821
$451$	18399.4	1.92105
$452$	−2555.03	−0.265882
$453$	−14420.8	−1.49569
$454$	10872.1	1.12390
$455$	0	0
$456$	−1865.87	−0.191617
$457$	−15322.2	−1.56836	−0.784181	−	0.620532i	$-0.786917\pi$
$457$	−15322.2	−1.56836	−0.784181	+	0.620532i	$0.786917\pi$
$458$	−7588.35	−0.774193
$459$	15622.1	1.58862
$460$	1265.10	0.128229
$461$	11903.8	1.20264	0.601319	−	0.799009i	$-0.294642\pi$
$461$	11903.8	1.20264	0.601319	+	0.799009i	$0.294642\pi$
$462$	−3526.26	−0.355101
$463$	15221.5	1.52787	0.763936	−	0.645292i	$-0.223264\pi$
$463$	15221.5	1.52787	0.763936	+	0.645292i	$0.223264\pi$
$464$	3730.89	0.373281
$465$	8291.95	0.826946
$466$	2141.32	0.212864
$467$	8807.37	0.872712	0.436356	−	0.899774i	$-0.356269\pi$
$467$	8807.37	0.872712	0.436356	+	0.899774i	$0.356269\pi$
$468$	0	0
$469$	2405.70	0.236854
$470$	13020.3	1.27784
$471$	14705.9	1.43867
$472$	2566.60	0.250291
$473$	3774.30	0.366898
$474$	9348.78	0.905915
$475$	−3197.41	−0.308857
$476$	2962.04	0.285220
$477$	−542.709	−0.0520942
$478$	−10410.3	−0.996147
$479$	10143.7	0.967590	0.483795	−	0.875181i	$-0.339258\pi$
$479$	10143.7	0.967590	0.483795	+	0.875181i	$0.339258\pi$
$480$	2152.04	0.204639
$481$	0	0
$482$	−6154.44	−0.581591
$483$	−777.065	−0.0732043
$484$	5426.74	0.509648
$485$	4914.29	0.460096
$486$	1893.67	0.176746
$487$	−9386.36	−0.873381	−0.436691	−	0.899612i	$-0.643850\pi$
$487$	−9386.36	−0.873381	−0.436691	+	0.899612i	$0.643850\pi$
$488$	3830.02	0.355280
$489$	2281.25	0.210964
$490$	1356.53	0.125065
$491$	11886.9	1.09256	0.546281	−	0.837602i	$-0.316043\pi$
$491$	11886.9	1.09256	0.546281	+	0.837602i	$0.316043\pi$
$492$	6897.20	0.632012
$493$	−24667.5	−2.25348
$494$	0	0
$495$	−2436.64	−0.221250
$496$	1972.77	0.178588
$497$	1034.24	0.0933443
$498$	10659.4	0.959157
$499$	−5665.99	−0.508306	−0.254153	−	0.967164i	$-0.581797\pi$
$499$	−5665.99	−0.508306	−0.254153	+	0.967164i	$0.581797\pi$
$500$	−3233.27	−0.289193
$501$	−6689.22	−0.596511
$502$	2676.56	0.237970
$503$	6305.31	0.558926	0.279463	−	0.960156i	$-0.409844\pi$
$503$	6305.31	0.558926	0.279463	+	0.960156i	$0.409844\pi$
$504$	190.146	0.0168051
$505$	−6597.58	−0.581363
$506$	2369.08	0.208140
$507$	0	0
$508$	−7760.69	−0.677805
$509$	6832.62	0.594992	0.297496	−	0.954723i	$-0.403849\pi$
$509$	6832.62	0.594992	0.297496	+	0.954723i	$0.403849\pi$
$510$	−14228.6	−1.23540
$511$	−5453.56	−0.472116
$512$	512.000	0.0441942
$513$	7089.25	0.610133
$514$	−11432.9	−0.981092
$515$	−21654.5	−1.85284
$516$	1414.83	0.120707
$517$	24382.5	2.07416
$518$	−2619.04	−0.222151
$519$	1682.68	0.142315
$520$	0	0
$521$	17145.8	1.44178	0.720892	−	0.693047i	$-0.243732\pi$
$521$	17145.8	1.44178	0.720892	+	0.693047i	$0.243732\pi$
$522$	−1583.52	−0.132775
$523$	−9121.38	−0.762620	−0.381310	−	0.924447i	$-0.624527\pi$
$523$	−9121.38	−0.762620	−0.381310	+	0.924447i	$0.624527\pi$
$524$	2795.57	0.233063
$525$	−2265.16	−0.188304
$526$	−10137.7	−0.840354
$527$	−13043.3	−1.07813
$528$	4030.02	0.332167
$529$	−11644.9	−0.957092
$530$	4424.87	0.362649
$531$	−1089.35	−0.0890280
$532$	1344.16	0.109543
$533$	0	0
$534$	−6894.05	−0.558680
$535$	22717.5	1.83582
$536$	−2749.37	−0.221557
$537$	8782.45	0.705755
$538$	−12487.0	−1.00065
$539$	2540.30	0.203003
$540$	−8176.54	−0.651597
$541$	−14595.4	−1.15990	−0.579950	−	0.814652i	$-0.696928\pi$
$541$	−14595.4	−1.15990	−0.579950	+	0.814652i	$0.696928\pi$
$542$	−9296.31	−0.736736
$543$	10388.8	0.821041
$544$	−3385.18	−0.266799
$545$	−25308.7	−1.98918
$546$	0	0
$547$	6798.55	0.531417	0.265708	−	0.964053i	$-0.414394\pi$
$547$	6798.55	0.531417	0.265708	+	0.964053i	$0.414394\pi$
$548$	−9092.24	−0.708761
$549$	−1625.59	−0.126372
$550$	6905.94	0.535401
$551$	−11194.0	−0.865484
$552$	888.074	0.0684764
$553$	−6734.81	−0.517890
$554$	−5398.47	−0.414006
$555$	12581.0	0.962222
$556$	−8429.82	−0.642993
$557$	1050.99	0.0799498	0.0399749	−	0.999201i	$-0.487272\pi$
$557$	1050.99	0.0799498	0.0399749	+	0.999201i	$0.487272\pi$
$558$	−837.309	−0.0635235
$559$	0	0
$560$	−1550.32	−0.116987
$561$	−26645.2	−2.00528
$562$	7485.54	0.561848
$563$	−8634.05	−0.646326	−0.323163	−	0.946343i	$-0.604746\pi$
$563$	−8634.05	−0.646326	−0.323163	+	0.946343i	$0.604746\pi$
$564$	9140.02	0.682383
$565$	−8841.78	−0.658365
$566$	−10839.3	−0.804962
$567$	4380.55	0.324455
$568$	−1181.99	−0.0873156
$569$	−17056.1	−1.25664	−0.628321	−	0.777954i	$-0.716258\pi$
$569$	−17056.1	−1.25664	−0.628321	+	0.777954i	$0.716258\pi$
$570$	−6456.91	−0.474474
$571$	−10954.2	−0.802833	−0.401416	−	0.915896i	$-0.631482\pi$
$571$	−10954.2	−0.802833	−0.401416	+	0.915896i	$0.631482\pi$
$572$	0	0
$573$	−11684.8	−0.851901
$574$	−4968.70	−0.361306
$575$	1521.83	0.110373
$576$	−217.310	−0.0157198
$577$	−5216.96	−0.376404	−0.188202	−	0.982130i	$-0.560266\pi$
$577$	−5216.96	−0.376404	−0.188202	+	0.982130i	$0.560266\pi$
$578$	12555.8	0.903549
$579$	6518.71	0.467890
$580$	12910.9	0.924301
$581$	−7678.99	−0.548327
$582$	3449.74	0.245698
$583$	8286.22	0.588645
$584$	6232.64	0.441624
$585$	0	0
$586$	−5014.31	−0.353480
$587$	−1205.16	−0.0847399	−0.0423699	−	0.999102i	$-0.513491\pi$
$587$	−1205.16	−0.0847399	−0.0423699	+	0.999102i	$0.513491\pi$
$588$	952.256	0.0667864
$589$	−5919.02	−0.414073
$590$	8881.82	0.619760
$591$	21035.9	1.46413
$592$	2993.19	0.207803
$593$	−3957.51	−0.274057	−0.137028	−	0.990567i	$-0.543755\pi$
$593$	−3957.51	−0.274057	−0.137028	+	0.990567i	$0.543755\pi$
$594$	−15311.8	−1.05766
$595$	10250.2	0.706249
$596$	−4359.10	−0.299590
$597$	9004.04	0.617271
$598$	0	0
$599$	18807.0	1.28286	0.641432	−	0.767180i	$-0.278341\pi$
$599$	18807.0	1.28286	0.641432	+	0.767180i	$0.278341\pi$
$600$	2588.76	0.176143
$601$	1149.80	0.0780388	0.0390194	−	0.999238i	$-0.487577\pi$
$601$	1149.80	0.0780388	0.0390194	+	0.999238i	$0.487577\pi$
$602$	−1019.24	−0.0690051
$603$	1166.92	0.0788073
$604$	−11872.7	−0.799826
$605$	18779.4	1.26197
$606$	−4631.37	−0.310456
$607$	10560.1	0.706128	0.353064	−	0.935599i	$-0.385140\pi$
$607$	10560.1	0.706128	0.353064	+	0.935599i	$0.385140\pi$
$608$	−1536.19	−0.102468
$609$	−7930.28	−0.527670
$610$	13253.9	0.879729
$611$	0	0
$612$	1436.79	0.0948997
$613$	28888.4	1.90341	0.951707	−	0.307008i	$-0.0993279\pi$
$613$	28888.4	1.90341	0.951707	+	0.307008i	$0.0993279\pi$
$614$	−7117.00	−0.467783
$615$	23868.0	1.56496
$616$	−2903.20	−0.189892
$617$	−27432.2	−1.78992	−0.894958	−	0.446151i	$-0.852795\pi$
$617$	−27432.2	−1.78992	−0.894958	+	0.446151i	$0.852795\pi$
$618$	−15201.1	−0.989444
$619$	−3690.47	−0.239633	−0.119816	−	0.992796i	$-0.538231\pi$
$619$	−3690.47	−0.239633	−0.119816	+	0.992796i	$0.538231\pi$
$620$	6826.82	0.442213
$621$	−3374.18	−0.218037
$622$	−1580.99	−0.101916
$623$	4966.44	0.319384
$624$	0	0
$625$	−19514.4	−1.24892
$626$	13178.6	0.841412
$627$	−12091.5	−0.770157
$628$	12107.5	0.769332
$629$	−19790.0	−1.25450
$630$	658.007	0.0416121
$631$	7976.61	0.503239	0.251619	−	0.967826i	$-0.419037\pi$
$631$	7976.61	0.503239	0.251619	+	0.967826i	$0.419037\pi$
$632$	7696.92	0.484442
$633$	−6598.48	−0.414322
$634$	−19480.6	−1.22031
$635$	−26856.1	−1.67835
$636$	3106.17	0.193660
$637$	0	0
$638$	24177.5	1.50031
$639$	501.677	0.0310580
$640$	1771.79	0.109432
$641$	3729.99	0.229837	0.114919	−	0.993375i	$-0.463339\pi$
$641$	3729.99	0.229837	0.114919	+	0.993375i	$0.463339\pi$
$642$	15947.2	0.980354
$643$	−21830.9	−1.33892	−0.669462	−	0.742846i	$-0.733475\pi$
$643$	−21830.9	−1.33892	−0.669462	+	0.742846i	$0.733475\pi$
$644$	−639.764	−0.0391463
$645$	4896.08	0.298888
$646$	10156.8	0.618597
$647$	6708.66	0.407642	0.203821	−	0.979008i	$-0.434664\pi$
$647$	6708.66	0.407642	0.203821	+	0.979008i	$0.434664\pi$
$648$	−5006.35	−0.303500
$649$	16632.5	1.00598
$650$	0	0
$651$	−4193.26	−0.252453
$652$	1878.17	0.112814
$653$	24968.6	1.49632	0.748159	−	0.663519i	$-0.230938\pi$
$653$	24968.6	1.49632	0.748159	+	0.663519i	$0.230938\pi$
$654$	−17766.2	−1.06225
$655$	9674.17	0.577101
$656$	5678.52	0.337971
$657$	−2645.34	−0.157085
$658$	−6584.42	−0.390102
$659$	25676.0	1.51775	0.758874	−	0.651237i	$-0.225750\pi$
$659$	25676.0	1.51775	0.758874	+	0.651237i	$0.225750\pi$
$660$	13946.0	0.822496
$661$	−10584.5	−0.622829	−0.311414	−	0.950274i	$-0.600803\pi$
$661$	−10584.5	−0.622829	−0.311414	+	0.950274i	$0.600803\pi$
$662$	−2618.49	−0.153732
$663$	0	0
$664$	8775.99	0.512913
$665$	4651.52	0.271245
$666$	−1270.41	−0.0739150
$667$	5327.88	0.309290
$668$	−5507.29	−0.318987
$669$	−29611.8	−1.71130
$670$	−9514.27	−0.548610
$671$	24819.9	1.42796
$672$	−1088.29	−0.0624729
$673$	−5615.04	−0.321611	−0.160805	−	0.986986i	$-0.551409\pi$
$673$	−5615.04	−0.321611	−0.160805	+	0.986986i	$0.551409\pi$
$674$	−14530.7	−0.830417
$675$	−9835.81	−0.560860
$676$	0	0
$677$	22934.3	1.30197	0.650986	−	0.759090i	$-0.274356\pi$
$677$	22934.3	1.30197	0.650986	+	0.759090i	$0.274356\pi$
$678$	−6206.75	−0.351577
$679$	−2485.17	−0.140460
$680$	−11714.5	−0.660635
$681$	26410.7	1.48614
$682$	12784.2	0.717792
$683$	9747.92	0.546111	0.273055	−	0.961998i	$-0.411966\pi$
$683$	9747.92	0.546111	0.273055	+	0.961998i	$0.411966\pi$
$684$	652.009	0.0364476
$685$	−31464.0	−1.75500
$686$	−686.000	−0.0381802
$687$	−18433.8	−1.02372
$688$	1164.84	0.0645484
$689$	0	0
$690$	3073.21	0.169558
$691$	−13732.4	−0.756014	−0.378007	−	0.925803i	$-0.623390\pi$
$691$	−13732.4	−0.756014	−0.378007	+	0.925803i	$0.623390\pi$
$692$	1385.36	0.0761035
$693$	1232.22	0.0675440
$694$	4488.35	0.245498
$695$	−29171.7	−1.59215
$696$	9063.17	0.493590
$697$	−37544.6	−2.04032
$698$	−15019.1	−0.814445
$699$	5201.74	0.281471
$700$	−1864.93	−0.100697
$701$	−19528.0	−1.05216	−0.526078	−	0.850436i	$-0.676338\pi$
$701$	−19528.0	−1.05216	−0.526078	+	0.850436i	$0.676338\pi$
$702$	0	0
$703$	−8980.66	−0.481809
$704$	3317.94	0.177627
$705$	31629.3	1.68969
$706$	16838.5	0.897628
$707$	3336.41	0.177480
$708$	6234.86	0.330961
$709$	−1305.38	−0.0691460	−0.0345730	−	0.999402i	$-0.511007\pi$
$709$	−1305.38	−0.0691460	−0.0345730	+	0.999402i	$0.511007\pi$
$710$	−4090.32	−0.216207
$711$	−3266.83	−0.172315
$712$	−5675.93	−0.298756
$713$	2817.20	0.147973
$714$	7195.45	0.377147
$715$	0	0
$716$	7230.66	0.377406
$717$	−25289.1	−1.31721
$718$	−7162.76	−0.372301
$719$	20578.6	1.06739	0.533693	−	0.845678i	$-0.320804\pi$
$719$	20578.6	1.06739	0.533693	+	0.845678i	$0.320804\pi$
$720$	−752.008	−0.0389246
$721$	10950.8	0.565642
$722$	−9108.88	−0.469525
$723$	−14950.5	−0.769039
$724$	8553.17	0.439055
$725$	15530.9	0.795590
$726$	13182.8	0.673909
$727$	−19534.4	−0.996548	−0.498274	−	0.867020i	$-0.666033\pi$
$727$	−19534.4	−0.996548	−0.498274	+	0.867020i	$0.666033\pi$
$728$	0	0
$729$	21496.6	1.09214
$730$	21568.3	1.09353
$731$	−7701.58	−0.389676
$732$	9303.97	0.469788
$733$	−23047.0	−1.16134	−0.580669	−	0.814140i	$-0.697209\pi$
$733$	−23047.0	−1.16134	−0.580669	+	0.814140i	$0.697209\pi$
$734$	−14404.1	−0.724341
$735$	3295.31	0.165373
$736$	731.158	0.0366180
$737$	−17816.9	−0.890493
$738$	−2410.15	−0.120215
$739$	−7582.68	−0.377447	−0.188724	−	0.982030i	$-0.560435\pi$
$739$	−7582.68	−0.377447	−0.188724	+	0.982030i	$0.560435\pi$
$740$	10358.0	0.514552
$741$	0	0
$742$	−2237.67	−0.110711
$743$	2781.28	0.137329	0.0686645	−	0.997640i	$-0.478126\pi$
$743$	2781.28	0.137329	0.0686645	+	0.997640i	$0.478126\pi$
$744$	4792.30	0.236148
$745$	−15084.8	−0.741832
$746$	9462.13	0.464388
$747$	−3724.82	−0.182442
$748$	−21937.2	−1.07233
$749$	−11488.3	−0.560445
$750$	−7854.34	−0.382400
$751$	26274.8	1.27667	0.638337	−	0.769757i	$-0.279623\pi$
$751$	26274.8	1.27667	0.638337	+	0.769757i	$0.279623\pi$
$752$	7525.05	0.364907
$753$	6501.97	0.314668
$754$	0	0
$755$	−41086.0	−1.98049
$756$	4134.90	0.198922
$757$	−37501.6	−1.80055	−0.900276	−	0.435320i	$-0.856635\pi$
$757$	−37501.6	−1.80055	−0.900276	+	0.435320i	$0.856635\pi$
$758$	−13149.4	−0.630088
$759$	5755.04	0.275224
$760$	−5316.02	−0.253727
$761$	7853.66	0.374107	0.187053	−	0.982350i	$-0.440106\pi$
$761$	7853.66	0.374107	0.187053	+	0.982350i	$0.440106\pi$
$762$	−18852.5	−0.896263
$763$	12798.7	0.607264
$764$	−9620.18	−0.455558
$765$	4972.04	0.234986
$766$	10092.1	0.476035
$767$	0	0
$768$	1243.76	0.0584381
$769$	36073.7	1.69161	0.845806	−	0.533490i	$-0.179120\pi$
$769$	36073.7	1.69161	0.845806	+	0.533490i	$0.179120\pi$
$770$	−10046.6	−0.470201
$771$	−27773.0	−1.29730
$772$	5366.91	0.250206
$773$	−14832.4	−0.690149	−0.345075	−	0.938575i	$-0.612146\pi$
$773$	−14832.4	−0.690149	−0.345075	+	0.938575i	$0.612146\pi$
$774$	−494.399	−0.0229597
$775$	8212.20	0.380634
$776$	2840.20	0.131388
$777$	−6362.24	−0.293750
$778$	16203.4	0.746683
$779$	−17037.6	−0.783615
$780$	0	0
$781$	−7659.73	−0.350943
$782$	−4834.19	−0.221062
$783$	−34434.9	−1.57165
$784$	784.000	0.0357143
$785$	41898.3	1.90499
$786$	6791.07	0.308180
$787$	13747.7	0.622684	0.311342	−	0.950298i	$-0.399222\pi$
$787$	13747.7	0.622684	0.311342	+	0.950298i	$0.399222\pi$
$788$	17319.0	0.782949
$789$	−24626.8	−1.11120
$790$	26635.4	1.19955
$791$	4471.31	0.200988
$792$	−1408.25	−0.0631816
$793$	0	0
$794$	−3848.43	−0.172010
$795$	10749.0	0.479532
$796$	7413.10	0.330088
$797$	281.955	0.0125312	0.00626560	−	0.999980i	$-0.498006\pi$
$797$	281.955	0.0125312	0.00626560	+	0.999980i	$0.498006\pi$
$798$	3265.28	0.144849
$799$	−49753.3	−2.20293
$800$	2131.34	0.0941930
$801$	2409.05	0.106267
$802$	25071.1	1.10386
$803$	40389.8	1.77500
$804$	−6678.83	−0.292965
$805$	−2213.92	−0.0969323
$806$	0	0
$807$	−30333.6	−1.32317
$808$	−3813.04	−0.166018
$809$	3977.34	0.172850	0.0864252	−	0.996258i	$-0.472456\pi$
$809$	3977.34	0.172850	0.0864252	+	0.996258i	$0.472456\pi$
$810$	−17324.6	−0.751512
$811$	8748.05	0.378774	0.189387	−	0.981903i	$-0.439350\pi$
$811$	8748.05	0.378774	0.189387	+	0.981903i	$0.439350\pi$
$812$	−6529.06	−0.282174
$813$	−22582.8	−0.974188
$814$	19396.9	0.835212
$815$	6499.46	0.279345
$816$	−8223.37	−0.352789
$817$	−3494.96	−0.149661
$818$	10620.1	0.453941
$819$	0	0
$820$	19650.7	0.836868
$821$	10646.7	0.452586	0.226293	−	0.974059i	$-0.427339\pi$
$821$	10646.7	0.452586	0.226293	+	0.974059i	$0.427339\pi$
$822$	−22087.1	−0.937197
$823$	−4003.17	−0.169552	−0.0847762	−	0.996400i	$-0.527018\pi$
$823$	−4003.17	−0.169552	−0.0847762	+	0.996400i	$0.527018\pi$
$824$	−12515.2	−0.529109
$825$	16776.1	0.707961
$826$	−4491.56	−0.189203
$827$	−36320.0	−1.52717	−0.763585	−	0.645708i	$-0.776563\pi$
$827$	−36320.0	−1.52717	−0.763585	+	0.645708i	$0.776563\pi$
$828$	−310.328	−0.0130249
$829$	−12103.1	−0.507065	−0.253532	−	0.967327i	$-0.581592\pi$
$829$	−12103.1	−0.507065	−0.253532	+	0.967327i	$0.581592\pi$
$830$	30369.6	1.27005
$831$	−13114.1	−0.547441
$832$	0	0
$833$	−5183.56	−0.215606
$834$	−20477.9	−0.850231
$835$	−19058.1	−0.789861
$836$	−9955.04	−0.411845
$837$	−18208.0	−0.751925
$838$	24081.6	0.992703
$839$	34851.0	1.43408	0.717039	−	0.697033i	$-0.245497\pi$
$839$	34851.0	1.43408	0.717039	+	0.697033i	$0.245497\pi$
$840$	−3766.07	−0.154693
$841$	29984.2	1.22942
$842$	9711.60	0.397487
$843$	18184.1	0.742933
$844$	−5432.58	−0.221561
$845$	0	0
$846$	−3193.88	−0.129797
$847$	−9496.79	−0.385258
$848$	2557.33	0.103560
$849$	−26331.0	−1.06440
$850$	−14091.8	−0.568640
$851$	4274.41	0.172180
$852$	−2871.32	−0.115458
$853$	14775.3	0.593081	0.296541	−	0.955020i	$-0.404167\pi$
$853$	14775.3	0.593081	0.296541	+	0.955020i	$0.404167\pi$
$854$	−6702.53	−0.268566
$855$	2256.30	0.0902500
$856$	13129.5	0.524249
$857$	−42512.0	−1.69450	−0.847248	−	0.531198i	$-0.821742\pi$
$857$	−42512.0	−1.69450	−0.847248	+	0.531198i	$0.821742\pi$
$858$	0	0
$859$	−9574.43	−0.380297	−0.190149	−	0.981755i	$-0.560897\pi$
$859$	−9574.43	−0.380297	−0.190149	+	0.981755i	$0.560897\pi$
$860$	4030.98	0.159832
$861$	−12070.1	−0.477756
$862$	−25231.1	−0.996955
$863$	−48476.2	−1.91211	−0.956053	−	0.293194i	$-0.905282\pi$
$863$	−48476.2	−1.91211	−0.956053	+	0.293194i	$0.905282\pi$
$864$	−4725.60	−0.186074
$865$	4794.10	0.188444
$866$	21467.8	0.842384
$867$	30500.8	1.19476
$868$	−3452.34	−0.135000
$869$	49878.8	1.94709
$870$	31363.4	1.22221
$871$	0	0
$872$	−14627.0	−0.568044
$873$	−1205.47	−0.0467344
$874$	−2193.74	−0.0849021
$875$	5658.22	0.218609
$876$	15140.5	0.583961
$877$	39895.5	1.53612	0.768058	−	0.640380i	$-0.221223\pi$
$877$	39895.5	1.53612	0.768058	+	0.640380i	$0.221223\pi$
$878$	−2868.25	−0.110249
$879$	−12180.9	−0.467408
$880$	11481.9	0.439833
$881$	−34536.7	−1.32074	−0.660369	−	0.750942i	$-0.729600\pi$
$881$	−34536.7	−1.32074	−0.660369	+	0.750942i	$0.729600\pi$
$882$	−332.756	−0.0127035
$883$	3167.28	0.120711	0.0603553	−	0.998177i	$-0.480777\pi$
$883$	3167.28	0.120711	0.0603553	+	0.998177i	$0.480777\pi$
$884$	0	0
$885$	21575.9	0.819511
$886$	2310.11	0.0875957
$887$	21597.6	0.817561	0.408781	−	0.912633i	$-0.365954\pi$
$887$	21597.6	0.817561	0.408781	+	0.912633i	$0.365954\pi$
$888$	7271.13	0.274778
$889$	13581.2	0.512372
$890$	−19641.7	−0.739767
$891$	−32442.9	−1.21984
$892$	−24379.6	−0.915123
$893$	−22577.9	−0.846070
$894$	−10589.2	−0.396149
$895$	25021.9	0.934515
$896$	−896.000	−0.0334077
$897$	0	0
$898$	9977.87	0.370786
$899$	28750.7	1.06662
$900$	−904.614	−0.0335042
$901$	−16908.3	−0.625191
$902$	36798.8	1.35839
$903$	−2475.96	−0.0912456
$904$	−5110.07	−0.188007
$905$	29598.5	1.08717
$906$	−28841.5	−1.05761
$907$	−15400.3	−0.563790	−0.281895	−	0.959445i	$-0.590963\pi$
$907$	−15400.3	−0.563790	−0.281895	+	0.959445i	$0.590963\pi$
$908$	21744.1	0.794717
$909$	1618.38	0.0590521
$910$	0	0
$911$	49710.6	1.80789	0.903945	−	0.427650i	$-0.140658\pi$
$911$	49710.6	1.80789	0.903945	+	0.427650i	$0.140658\pi$
$912$	−3731.74	−0.135494
$913$	56871.5	2.06153
$914$	−30644.4	−1.10900
$915$	32196.7	1.16327
$916$	−15176.7	−0.547437
$917$	−4892.25	−0.176179
$918$	31244.2	1.12332
$919$	−16580.4	−0.595142	−0.297571	−	0.954700i	$-0.596177\pi$
$919$	−16580.4	−0.595142	−0.297571	+	0.954700i	$0.596177\pi$
$920$	2530.20	0.0906719
$921$	−17288.8	−0.618551
$922$	23807.6	0.850393
$923$	0	0
$924$	−7052.53	−0.251094
$925$	12460.0	0.442900
$926$	30443.1	1.08037
$927$	5311.85	0.188203
$928$	7461.78	0.263949
$929$	21186.5	0.748232	0.374116	−	0.927382i	$-0.377946\pi$
$929$	21186.5	0.748232	0.374116	+	0.927382i	$0.377946\pi$
$930$	16583.9	0.584739
$931$	−2352.29	−0.0828067
$932$	4282.64	0.150518
$933$	−3840.58	−0.134764
$934$	17614.7	0.617101
$935$	−75914.4	−2.65526
$936$	0	0
$937$	−37667.5	−1.31328	−0.656640	−	0.754204i	$-0.728023\pi$
$937$	−37667.5	−1.31328	−0.656640	+	0.754204i	$0.728023\pi$
$938$	4811.39	0.167481
$939$	32013.8	1.11260
$940$	26040.7	0.903567
$941$	3401.39	0.117834	0.0589172	−	0.998263i	$-0.481235\pi$
$941$	3401.39	0.117834	0.0589172	+	0.998263i	$0.481235\pi$
$942$	29411.8	1.01729
$943$	8109.17	0.280033
$944$	5133.21	0.176983
$945$	14308.9	0.492561
$946$	7548.61	0.259436
$947$	−8950.65	−0.307135	−0.153568	−	0.988138i	$-0.549076\pi$
$947$	−8950.65	−0.307135	−0.153568	+	0.988138i	$0.549076\pi$
$948$	18697.6	0.640578
$949$	0	0
$950$	−6394.81	−0.218395
$951$	−47322.8	−1.61361
$952$	5924.07	0.201681
$953$	−26947.3	−0.915960	−0.457980	−	0.888963i	$-0.651427\pi$
$953$	−26947.3	−0.915960	−0.457980	+	0.888963i	$0.651427\pi$
$954$	−1085.42	−0.0368362
$955$	−33291.0	−1.12803
$956$	−20820.7	−0.704382
$957$	58732.6	1.98386
$958$	20287.3	0.684190
$959$	15911.4	0.535773
$960$	4304.08	0.144702
$961$	−14588.6	−0.489699
$962$	0	0
$963$	−5572.60	−0.186474
$964$	−12308.9	−0.411247
$965$	18572.3	0.619549
$966$	−1554.13	−0.0517633
$967$	7549.32	0.251054	0.125527	−	0.992090i	$-0.459938\pi$
$967$	7549.32	0.251054	0.125527	+	0.992090i	$0.459938\pi$
$968$	10853.5	0.360376
$969$	24673.1	0.817972
$970$	9828.59	0.325337
$971$	−46922.7	−1.55080	−0.775398	−	0.631473i	$-0.782451\pi$
$971$	−46922.7	−1.55080	−0.775398	+	0.631473i	$0.782451\pi$
$972$	3787.35	0.124979
$973$	14752.2	0.486057
$974$	−18772.7	−0.617574
$975$	0	0
$976$	7660.03	0.251221
$977$	−2568.52	−0.0841086	−0.0420543	−	0.999115i	$-0.513390\pi$
$977$	−2568.52	−0.0841086	−0.0420543	+	0.999115i	$0.513390\pi$
$978$	4562.49	0.149174
$979$	−36782.1	−1.20078
$980$	2713.06	0.0884341
$981$	6208.21	0.202052
$982$	23773.8	0.772559
$983$	49875.4	1.61829	0.809144	−	0.587610i	$-0.199931\pi$
$983$	49875.4	1.61829	0.809144	+	0.587610i	$0.199931\pi$
$984$	13794.4	0.446900
$985$	59932.9	1.93870
$986$	−49335.0	−1.59345
$987$	−15995.0	−0.515833
$988$	0	0
$989$	1663.45	0.0534829
$990$	−4873.28	−0.156448
$991$	−26464.3	−0.848299	−0.424150	−	0.905592i	$-0.639427\pi$
$991$	−26464.3	−0.848299	−0.424150	+	0.905592i	$0.639427\pi$
$992$	3945.53	0.126281
$993$	−6360.90	−0.203280
$994$	2068.49	0.0660044
$995$	25653.3	0.817350
$996$	21318.8	0.678226
$997$	−22873.6	−0.726592	−0.363296	−	0.931674i	$-0.618349\pi$
$997$	−22873.6	−0.726592	−0.363296	+	0.931674i	$0.618349\pi$
$998$	−11332.0	−0.359426
$999$	−27626.2	−0.874928

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2366.4.a.bg.1.9		12
13.6	odd	12		182.4.m.b.127.2	yes	24
13.11	odd	12		182.4.m.b.43.2	✓	24
13.12	even	2		2366.4.a.bd.1.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.4.m.b.43.2	✓	24	13.11	odd	12
182.4.m.b.127.2	yes	24	13.6	odd	12
2366.4.a.bd.1.9		12	13.12	even	2
2366.4.a.bg.1.9		12	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:	\( N \)	\(=\)	\( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2366.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.85845 q^{3} +4.00000 q^{4} +13.8421 q^{5} +9.71690 q^{6} -7.00000 q^{7} +8.00000 q^{8} -3.39547 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} +4.85845 q^{3} +4.00000 q^{4} +13.8421 q^{5} +9.71690 q^{6} -7.00000 q^{7} +8.00000 q^{8} -3.39547 q^{9} +27.6843 q^{10} +51.8429 q^{11} +19.4338 q^{12} -14.0000 q^{14} +67.2513 q^{15} +16.0000 q^{16} -105.787 q^{17} -6.79094 q^{18} -48.0058 q^{19} +55.3685 q^{20} -34.0091 q^{21} +103.686 q^{22} +22.8487 q^{23} +38.8676 q^{24} +66.6045 q^{25} -147.675 q^{27} -28.0000 q^{28} +233.181 q^{29} +134.503 q^{30} +123.298 q^{31} +32.0000 q^{32} +251.876 q^{33} -211.574 q^{34} -96.8949 q^{35} -13.5819 q^{36} +187.074 q^{37} -96.0117 q^{38} +110.737 q^{40} +354.907 q^{41} -68.0183 q^{42} +72.8028 q^{43} +207.372 q^{44} -47.0005 q^{45} +45.6974 q^{46} +470.316 q^{47} +77.7352 q^{48} +49.0000 q^{49} +133.209 q^{50} -513.961 q^{51} +159.833 q^{53} -295.350 q^{54} +717.616 q^{55} -56.0000 q^{56} -233.234 q^{57} +466.361 q^{58} +320.826 q^{59} +269.005 q^{60} +478.752 q^{61} +246.596 q^{62} +23.7683 q^{63} +64.0000 q^{64} +503.752 q^{66} -343.671 q^{67} -423.148 q^{68} +111.009 q^{69} -193.790 q^{70} -147.749 q^{71} -27.1637 q^{72} +779.080 q^{73} +374.149 q^{74} +323.595 q^{75} -192.023 q^{76} -362.900 q^{77} +962.115 q^{79} +221.474 q^{80} -625.793 q^{81} +709.815 q^{82} +1097.00 q^{83} -136.037 q^{84} -1464.32 q^{85} +145.606 q^{86} +1132.90 q^{87} +414.743 q^{88} -709.491 q^{89} -94.0010 q^{90} +91.3948 q^{92} +599.037 q^{93} +940.631 q^{94} -664.503 q^{95} +155.470 q^{96} +355.024 q^{97} +98.0000 q^{98} -176.031 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 24 q^{2} + 6 q^{3} + 48 q^{4} + 28 q^{5} + 12 q^{6} - 84 q^{7} + 96 q^{8} + 162 q^{9}+O(q^{10}) \) 12 * q + 24 * q^2 + 6 * q^3 + 48 * q^4 + 28 * q^5 + 12 * q^6 - 84 * q^7 + 96 * q^8 + 162 * q^9	\( 12 q + 24 q^{2} + 6 q^{3} + 48 q^{4} + 28 q^{5} + 12 q^{6} - 84 q^{7} + 96 q^{8} + 162 q^{9} + 56 q^{10} + 6 q^{11} + 24 q^{12} - 168 q^{14} - 138 q^{15} + 192 q^{16} - 56 q^{17} + 324 q^{18} + 158 q^{19} + 112 q^{20} - 42 q^{21} + 12 q^{22} + 414 q^{23} + 48 q^{24} + 478 q^{25} + 390 q^{27} - 336 q^{28} + 222 q^{29} - 276 q^{30} - 200 q^{31} + 384 q^{32} + 844 q^{33} - 112 q^{34} - 196 q^{35} + 648 q^{36} - 560 q^{37} + 316 q^{38} + 224 q^{40} - 66 q^{41} - 84 q^{42} + 484 q^{43} + 24 q^{44} + 542 q^{45} + 828 q^{46} + 618 q^{47} + 96 q^{48} + 588 q^{49} + 956 q^{50} + 992 q^{51} + 504 q^{53} + 780 q^{54} + 2584 q^{55} - 672 q^{56} - 1164 q^{57} + 444 q^{58} + 1460 q^{59} - 552 q^{60} - 2 q^{61} - 400 q^{62} - 1134 q^{63} + 768 q^{64} + 1688 q^{66} + 334 q^{67} - 224 q^{68} + 4660 q^{69} - 392 q^{70} - 196 q^{71} + 1296 q^{72} + 490 q^{73} - 1120 q^{74} - 338 q^{75} + 632 q^{76} - 42 q^{77} + 2942 q^{79} + 448 q^{80} + 2824 q^{81} - 132 q^{82} + 236 q^{83} - 168 q^{84} + 1352 q^{85} + 968 q^{86} + 1456 q^{87} + 48 q^{88} + 3566 q^{89} + 1084 q^{90} + 1656 q^{92} + 1884 q^{93} + 1236 q^{94} + 5754 q^{95} + 192 q^{96} + 3032 q^{97} + 1176 q^{98} + 3670 q^{99}+O(q^{100}) \) 12 * q + 24 * q^2 + 6 * q^3 + 48 * q^4 + 28 * q^5 + 12 * q^6 - 84 * q^7 + 96 * q^8 + 162 * q^9 + 56 * q^10 + 6 * q^11 + 24 * q^12 - 168 * q^14 - 138 * q^15 + 192 * q^16 - 56 * q^17 + 324 * q^18 + 158 * q^19 + 112 * q^20 - 42 * q^21 + 12 * q^22 + 414 * q^23 + 48 * q^24 + 478 * q^25 + 390 * q^27 - 336 * q^28 + 222 * q^29 - 276 * q^30 - 200 * q^31 + 384 * q^32 + 844 * q^33 - 112 * q^34 - 196 * q^35 + 648 * q^36 - 560 * q^37 + 316 * q^38 + 224 * q^40 - 66 * q^41 - 84 * q^42 + 484 * q^43 + 24 * q^44 + 542 * q^45 + 828 * q^46 + 618 * q^47 + 96 * q^48 + 588 * q^49 + 956 * q^50 + 992 * q^51 + 504 * q^53 + 780 * q^54 + 2584 * q^55 - 672 * q^56 - 1164 * q^57 + 444 * q^58 + 1460 * q^59 - 552 * q^60 - 2 * q^61 - 400 * q^62 - 1134 * q^63 + 768 * q^64 + 1688 * q^66 + 334 * q^67 - 224 * q^68 + 4660 * q^69 - 392 * q^70 - 196 * q^71 + 1296 * q^72 + 490 * q^73 - 1120 * q^74 - 338 * q^75 + 632 * q^76 - 42 * q^77 + 2942 * q^79 + 448 * q^80 + 2824 * q^81 - 132 * q^82 + 236 * q^83 - 168 * q^84 + 1352 * q^85 + 968 * q^86 + 1456 * q^87 + 48 * q^88 + 3566 * q^89 + 1084 * q^90 + 1656 * q^92 + 1884 * q^93 + 1236 * q^94 + 5754 * q^95 + 192 * q^96 + 3032 * q^97 + 1176 * q^98 + 3670 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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