LMFDB - Embedded newform 1075.6.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,6,Mod(1,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1075.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1075.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:	$172.412606299$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 $ x^8 - 4x^7 - 173x^6 + 462x^5 + 9118x^4 - 14192x^3 - 167688x^2 + 106368*x + 681984
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.65705$ of defining polynomial
Character	$\chi$	$=$	1075.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.65705 q^{2} -7.84314 q^{3} -18.6260 q^{4} +28.6827 q^{6} +25.5214 q^{7} +185.142 q^{8} -181.485 q^{9} +O(q^{10})$	\(q-3.65705 q^{2} -7.84314 q^{3} -18.6260 q^{4} +28.6827 q^{6} +25.5214 q^{7} +185.142 q^{8} -181.485 q^{9} +512.073 q^{11} +146.086 q^{12} -862.516 q^{13} -93.3331 q^{14} -81.0406 q^{16} +1521.49 q^{17} +663.700 q^{18} -1543.11 q^{19} -200.168 q^{21} -1872.67 q^{22} +3126.31 q^{23} -1452.09 q^{24} +3154.26 q^{26} +3329.30 q^{27} -475.362 q^{28} -947.120 q^{29} +339.499 q^{31} -5628.17 q^{32} -4016.26 q^{33} -5564.17 q^{34} +3380.34 q^{36} +7448.67 q^{37} +5643.25 q^{38} +6764.84 q^{39} +5116.45 q^{41} +732.024 q^{42} +1849.00 q^{43} -9537.86 q^{44} -11433.1 q^{46} +17159.9 q^{47} +635.613 q^{48} -16155.7 q^{49} -11933.3 q^{51} +16065.2 q^{52} -18090.4 q^{53} -12175.4 q^{54} +4725.08 q^{56} +12102.9 q^{57} +3463.66 q^{58} +17031.6 q^{59} +10664.9 q^{61} -1241.56 q^{62} -4631.76 q^{63} +23175.8 q^{64} +14687.6 q^{66} +8799.60 q^{67} -28339.3 q^{68} -24520.1 q^{69} -77057.3 q^{71} -33600.5 q^{72} +7964.75 q^{73} -27240.1 q^{74} +28742.0 q^{76} +13068.8 q^{77} -24739.3 q^{78} +68997.0 q^{79} +17988.8 q^{81} -18711.1 q^{82} -40813.4 q^{83} +3728.33 q^{84} -6761.88 q^{86} +7428.39 q^{87} +94806.0 q^{88} -83692.1 q^{89} -22012.6 q^{91} -58230.7 q^{92} -2662.74 q^{93} -62754.7 q^{94} +44142.5 q^{96} -159824. q^{97} +59082.0 q^{98} -92933.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9}+O(q^{10}) $ 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9	\( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9} - 532 q^{11} + 4195 q^{12} + 2492 q^{13} - 4240 q^{14} + 1882 q^{16} + 2534 q^{17} + 3711 q^{18} - 1678 q^{19} - 2256 q^{21} - 11502 q^{22} + 2488 q^{23} + 19953 q^{24} + 4586 q^{26} + 8960 q^{27} - 18640 q^{28} - 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 12852 q^{33} + 30007 q^{34} + 67969 q^{36} + 3772 q^{37} + 6559 q^{38} + 11120 q^{39} - 10698 q^{41} - 78698 q^{42} + 14792 q^{43} - 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 118727 q^{48} + 7188 q^{49} - 80246 q^{51} + 60736 q^{52} + 62352 q^{53} + 61026 q^{54} - 144528 q^{56} + 808 q^{57} - 52951 q^{58} - 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 61768 q^{63} + 153858 q^{64} - 48516 q^{66} - 27784 q^{67} - 40507 q^{68} - 93776 q^{69} - 9504 q^{71} + 186687 q^{72} - 14260 q^{73} - 15239 q^{74} + 1279 q^{76} + 218140 q^{77} - 264170 q^{78} + 160248 q^{79} + 161076 q^{81} + 47781 q^{82} + 77176 q^{83} + 16382 q^{84} + 22188 q^{86} - 268136 q^{87} - 129544 q^{88} - 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 123860 q^{93} + 248737 q^{94} + 950817 q^{96} - 144742 q^{97} + 292244 q^{98} + 239516 q^{99}+O(q^{100}) \) 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9 - 532 * q^11 + 4195 * q^12 + 2492 * q^13 - 4240 * q^14 + 1882 * q^16 + 2534 * q^17 + 3711 * q^18 - 1678 * q^19 - 2256 * q^21 - 11502 * q^22 + 2488 * q^23 + 19953 * q^24 + 4586 * q^26 + 8960 * q^27 - 18640 * q^28 - 4360 * q^29 + 5704 * q^31 + 18294 * q^32 + 12852 * q^33 + 30007 * q^34 + 67969 * q^36 + 3772 * q^37 + 6559 * q^38 + 11120 * q^39 - 10698 * q^41 - 78698 * q^42 + 14792 * q^43 - 356 * q^44 - 19389 * q^46 + 77864 * q^47 + 118727 * q^48 + 7188 * q^49 - 80246 * q^51 + 60736 * q^52 + 62352 * q^53 + 61026 * q^54 - 144528 * q^56 + 808 * q^57 - 52951 * q^58 - 26224 * q^59 - 82540 * q^61 + 9023 * q^62 + 61768 * q^63 + 153858 * q^64 - 48516 * q^66 - 27784 * q^67 - 40507 * q^68 - 93776 * q^69 - 9504 * q^71 + 186687 * q^72 - 14260 * q^73 - 15239 * q^74 + 1279 * q^76 + 218140 * q^77 - 264170 * q^78 + 160248 * q^79 + 161076 * q^81 + 47781 * q^82 + 77176 * q^83 + 16382 * q^84 + 22188 * q^86 - 268136 * q^87 - 129544 * q^88 - 265692 * q^89 + 401148 * q^91 - 190391 * q^92 + 123860 * q^93 + 248737 * q^94 + 950817 * q^96 - 144742 * q^97 + 292244 * q^98 + 239516 * 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 $

$2$	−3.65705	−0.646481	−0.323241	−	0.946317i	$-0.604772\pi$
$2$	−3.65705	−0.646481	−0.323241	+	0.946317i	$0.604772\pi$
$3$	−7.84314	−0.503138	−0.251569	−	0.967839i	$-0.580946\pi$
$3$	−7.84314	−0.503138	−0.251569	+	0.967839i	$0.580946\pi$
$4$	−18.6260	−0.582062
$5$	0	0
$5$	0	0
$6$	28.6827	0.325269
$7$	25.5214	0.196861	0.0984305	−	0.995144i	$-0.468618\pi$
$7$	25.5214	0.196861	0.0984305	+	0.995144i	$0.468618\pi$
$8$	185.142	1.02277
$9$	−181.485	−0.746853
$10$	0	0
$11$	512.073	1.27600	0.637999	−	0.770037i	$-0.279762\pi$
$11$	512.073	1.27600	0.637999	+	0.770037i	$0.279762\pi$
$12$	146.086	0.292857
$13$	−862.516	−1.41550	−0.707749	−	0.706464i	$-0.750289\pi$
$13$	−862.516	−1.41550	−0.707749	+	0.706464i	$0.750289\pi$
$14$	−93.3331	−0.127267
$15$	0	0
$16$	−81.0406	−0.0791413
$17$	1521.49	1.27687	0.638435	−	0.769675i	$-0.279582\pi$
$17$	1521.49	1.27687	0.638435	+	0.769675i	$0.279582\pi$
$18$	663.700	0.482826
$19$	−1543.11	−0.980650	−0.490325	−	0.871540i	$-0.663122\pi$
$19$	−1543.11	−0.980650	−0.490325	+	0.871540i	$0.663122\pi$
$20$	0	0
$21$	−200.168	−0.0990481
$22$	−1872.67	−0.824908
$23$	3126.31	1.23229	0.616145	−	0.787633i	$-0.288694\pi$
$23$	3126.31	1.23229	0.616145	+	0.787633i	$0.288694\pi$
$24$	−1452.09	−0.514596
$25$	0	0
$26$	3154.26	0.915092
$27$	3329.30	0.878907
$28$	−475.362	−0.114585
$29$	−947.120	−0.209127	−0.104563	−	0.994518i	$-0.533345\pi$
$29$	−947.120	−0.209127	−0.104563	+	0.994518i	$0.533345\pi$
$30$	0	0
$31$	339.499	0.0634504	0.0317252	−	0.999497i	$-0.489900\pi$
$31$	339.499	0.0634504	0.0317252	+	0.999497i	$0.489900\pi$
$32$	−5628.17	−0.971610
$33$	−4016.26	−0.642002
$34$	−5564.17	−0.825473
$35$	0	0
$36$	3380.34	0.434715
$37$	7448.67	0.894488	0.447244	−	0.894412i	$-0.352406\pi$
$37$	7448.67	0.894488	0.447244	+	0.894412i	$0.352406\pi$
$38$	5643.25	0.633972
$39$	6764.84	0.712190
$40$	0	0
$41$	5116.45	0.475345	0.237672	−	0.971345i	$-0.423616\pi$
$41$	5116.45	0.475345	0.237672	+	0.971345i	$0.423616\pi$
$42$	732.024	0.0640328
$43$	1849.00	0.152499
$43$	1849.00	0.152499
$44$	−9537.86	−0.742710
$45$	0	0
$46$	−11433.1	−0.796652
$47$	17159.9	1.13311	0.566553	−	0.824025i	$-0.308276\pi$
$47$	17159.9	1.13311	0.566553	+	0.824025i	$0.308276\pi$
$48$	635.613	0.0398189
$49$	−16155.7	−0.961246
$50$	0	0
$51$	−11933.3	−0.642442
$52$	16065.2	0.823907
$53$	−18090.4	−0.884623	−0.442312	−	0.896861i	$-0.645841\pi$
$53$	−18090.4	−0.884623	−0.442312	+	0.896861i	$0.645841\pi$
$54$	−12175.4	−0.568197
$55$	0	0
$56$	4725.08	0.201344
$57$	12102.9	0.493402
$58$	3463.66	0.135197
$59$	17031.6	0.636980	0.318490	−	0.947926i	$-0.396824\pi$
$59$	17031.6	0.636980	0.318490	+	0.947926i	$0.396824\pi$
$60$	0	0
$61$	10664.9	0.366972	0.183486	−	0.983022i	$-0.441262\pi$
$61$	10664.9	0.366972	0.183486	+	0.983022i	$0.441262\pi$
$62$	−1241.56	−0.0410195
$63$	−4631.76	−0.147026
$64$	23175.8	0.707269
$65$	0	0
$66$	14687.6	0.415042
$67$	8799.60	0.239484	0.119742	−	0.992805i	$-0.461793\pi$
$67$	8799.60	0.239484	0.119742	+	0.992805i	$0.461793\pi$
$68$	−28339.3	−0.743218
$69$	−24520.1	−0.620011
$70$	0	0
$71$	−77057.3	−1.81413	−0.907064	−	0.420992i	$-0.861682\pi$
$71$	−77057.3	−1.81413	−0.907064	+	0.420992i	$0.861682\pi$
$72$	−33600.5	−0.763861
$73$	7964.75	0.174930	0.0874652	−	0.996168i	$-0.472123\pi$
$73$	7964.75	0.174930	0.0874652	+	0.996168i	$0.472123\pi$
$74$	−27240.1	−0.578269
$75$	0	0
$76$	28742.0	0.570800
$77$	13068.8	0.251194
$78$	−24739.3	−0.460417
$79$	68997.0	1.24383	0.621917	−	0.783083i	$-0.286354\pi$
$79$	68997.0	1.24383	0.621917	+	0.783083i	$0.286354\pi$
$80$	0	0
$81$	17988.8	0.304641
$82$	−18711.1	−0.307301
$83$	−40813.4	−0.650290	−0.325145	−	0.945664i	$-0.605413\pi$
$83$	−40813.4	−0.650290	−0.325145	+	0.945664i	$0.605413\pi$
$84$	3728.33	0.0576522
$85$	0	0
$86$	−6761.88	−0.0985874
$87$	7428.39	0.105220
$88$	94806.0	1.30506
$89$	−83692.1	−1.11998	−0.559989	−	0.828500i	$-0.689195\pi$
$89$	−83692.1	−1.11998	−0.559989	+	0.828500i	$0.689195\pi$
$90$	0	0
$91$	−22012.6	−0.278656
$92$	−58230.7	−0.717269
$93$	−2662.74	−0.0319243
$94$	−62754.7	−0.732532
$95$	0	0
$96$	44142.5	0.488853
$97$	−159824.	−1.72469	−0.862346	−	0.506320i	$-0.831006\pi$
$97$	−159824.	−1.72469	−0.862346	+	0.506320i	$0.831006\pi$
$98$	59082.0	0.621427
$99$	−92933.6	−0.952982
$100$	0	0
$101$	−168968.	−1.64817	−0.824085	−	0.566466i	$-0.808310\pi$
$101$	−168968.	−1.64817	−0.824085	+	0.566466i	$0.808310\pi$
$102$	43640.5	0.415326
$103$	145084.	1.34749	0.673747	−	0.738962i	$-0.264684\pi$
$103$	145084.	1.34749	0.673747	+	0.738962i	$0.264684\pi$
$104$	−159688.	−1.44773
$105$	0	0
$106$	66157.5	0.571892
$107$	−72693.1	−0.613810	−0.306905	−	0.951740i	$-0.599293\pi$
$107$	−72693.1	−0.613810	−0.306905	+	0.951740i	$0.599293\pi$
$108$	−62011.4	−0.511579
$109$	−3254.39	−0.0262364	−0.0131182	−	0.999914i	$-0.504176\pi$
$109$	−3254.39	−0.0262364	−0.0131182	+	0.999914i	$0.504176\pi$
$110$	0	0
$111$	−58420.9	−0.450050
$112$	−2068.27	−0.0155798
$113$	−116394.	−0.857503	−0.428752	−	0.903422i	$-0.641046\pi$
$113$	−116394.	−0.857503	−0.428752	+	0.903422i	$0.641046\pi$
$114$	−44260.8	−0.318975
$115$	0	0
$116$	17641.0	0.121725
$117$	156534.	1.05717
$118$	−62285.4	−0.411795
$119$	38830.6	0.251366
$120$	0	0
$121$	101167.	0.628170
$122$	−39002.1	−0.237240
$123$	−40129.0	−0.239164
$124$	−6323.50	−0.0369321
$125$	0	0
$126$	16938.6	0.0950496
$127$	6451.01	0.0354910	0.0177455	−	0.999843i	$-0.494351\pi$
$127$	6451.01	0.0354910	0.0177455	+	0.999843i	$0.494351\pi$
$128$	95346.3	0.514374
$129$	−14502.0	−0.0767278
$130$	0	0
$131$	−48048.9	−0.244628	−0.122314	−	0.992491i	$-0.539031\pi$
$131$	−48048.9	−0.244628	−0.122314	+	0.992491i	$0.539031\pi$
$132$	74806.8	0.373685
$133$	−39382.5	−0.193052
$134$	−32180.6	−0.154822
$135$	0	0
$136$	281691.	1.30595
$137$	103084.	0.469233	0.234616	−	0.972088i	$-0.424617\pi$
$137$	103084.	0.469233	0.234616	+	0.972088i	$0.424617\pi$
$138$	89671.2	0.400826
$139$	219626.	0.964156	0.482078	−	0.876128i	$-0.339882\pi$
$139$	219626.	0.964156	0.482078	+	0.876128i	$0.339882\pi$
$140$	0	0
$141$	−134588.	−0.570109
$142$	281802.	1.17280
$143$	−441671.	−1.80617
$144$	14707.7	0.0591069
$145$	0	0
$146$	−29127.5	−0.113089
$147$	126711.	0.483639
$148$	−138739.	−0.520648
$149$	−335627.	−1.23849	−0.619243	−	0.785200i	$-0.712560\pi$
$149$	−335627.	−1.23849	−0.619243	+	0.785200i	$0.712560\pi$
$150$	0	0
$151$	−84920.3	−0.303088	−0.151544	−	0.988450i	$-0.548425\pi$
$151$	−84920.3	−0.303088	−0.151544	+	0.988450i	$0.548425\pi$
$152$	−285695.	−1.00298
$153$	−276128.	−0.953634
$154$	−47793.3	−0.162392
$155$	0	0
$156$	−126002.	−0.414539
$157$	−313767.	−1.01592	−0.507958	−	0.861382i	$-0.669599\pi$
$157$	−313767.	−1.01592	−0.507958	+	0.861382i	$0.669599\pi$
$158$	−252325.	−0.804115
$159$	141885.	0.445087
$160$	0	0
$161$	79787.9	0.242590
$162$	−65785.8	−0.196945
$163$	−239642.	−0.706469	−0.353235	−	0.935535i	$-0.614918\pi$
$163$	−239642.	−0.706469	−0.353235	+	0.935535i	$0.614918\pi$
$164$	−95298.9	−0.276680
$165$	0	0
$166$	149257.	0.420400
$167$	506675.	1.40585	0.702923	−	0.711266i	$-0.251878\pi$
$167$	506675.	1.40585	0.702923	+	0.711266i	$0.251878\pi$
$168$	−37059.4	−0.101304
$169$	372641.	1.00363
$170$	0	0
$171$	280052.	0.732401
$172$	−34439.5	−0.0887637
$173$	427174.	1.08515	0.542575	−	0.840007i	$-0.317449\pi$
$173$	427174.	1.08515	0.542575	+	0.840007i	$0.317449\pi$
$174$	−27166.0	−0.0680225
$175$	0	0
$176$	−41498.7	−0.100984
$177$	−133581.	−0.320489
$178$	306066.	0.724044
$179$	11674.9	0.0272345	0.0136172	−	0.999907i	$-0.495665\pi$
$179$	11674.9	0.0272345	0.0136172	+	0.999907i	$0.495665\pi$
$180$	0	0
$181$	71691.3	0.162656	0.0813280	−	0.996687i	$-0.474084\pi$
$181$	71691.3	0.162656	0.0813280	+	0.996687i	$0.474084\pi$
$182$	80501.3	0.180146
$183$	−83646.4	−0.184637
$184$	578811.	1.26035
$185$	0	0
$186$	9737.76	0.0206384
$187$	779114.	1.62928
$188$	−319621.	−0.659539
$189$	84968.3	0.173023
$190$	0	0
$191$	212639.	0.421755	0.210877	−	0.977513i	$-0.432368\pi$
$191$	212639.	0.421755	0.210877	+	0.977513i	$0.432368\pi$
$192$	−181771.	−0.355853
$193$	137469.	0.265651	0.132825	−	0.991139i	$-0.457595\pi$
$193$	137469.	0.265651	0.132825	+	0.991139i	$0.457595\pi$
$194$	584483.	1.11498
$195$	0	0
$196$	300915.	0.559505
$197$	28517.3	0.0523531	0.0261766	−	0.999657i	$-0.491667\pi$
$197$	28517.3	0.0523531	0.0261766	+	0.999657i	$0.491667\pi$
$198$	339863.	0.616085
$199$	−353575.	−0.632921	−0.316460	−	0.948606i	$-0.602494\pi$
$199$	−353575.	−0.632921	−0.316460	+	0.948606i	$0.602494\pi$
$200$	0	0
$201$	−69016.5	−0.120493
$202$	617926.	1.06551
$203$	−24171.8	−0.0411689
$204$	222269.	0.373941
$205$	0	0
$206$	−530580.	−0.871130
$207$	−567379.	−0.920339
$208$	69898.9	0.112024
$209$	−790187.	−1.25131
$210$	0	0
$211$	327840.	0.506939	0.253469	−	0.967343i	$-0.418428\pi$
$211$	327840.	0.506939	0.253469	+	0.967343i	$0.418428\pi$
$212$	336952.	0.514906
$213$	604371.	0.912756
$214$	265842.	0.396816
$215$	0	0
$216$	616392.	0.898923
$217$	8664.49	0.0124909
$218$	11901.5	0.0169613
$219$	−62468.7	−0.0880140
$220$	0	0
$221$	−1.31231e6	−1.80741
$222$	213648.	0.290949
$223$	−247951.	−0.333890	−0.166945	−	0.985966i	$-0.553390\pi$
$223$	−247951.	−0.333890	−0.166945	+	0.985966i	$0.553390\pi$
$224$	−143639.	−0.191272
$225$	0	0
$226$	425660.	0.554359
$227$	868910.	1.11921	0.559603	−	0.828761i	$-0.310954\pi$
$227$	868910.	1.11921	0.559603	+	0.828761i	$0.310954\pi$
$228$	−225428.	−0.287191
$229$	238721.	0.300817	0.150408	−	0.988624i	$-0.451941\pi$
$229$	238721.	0.300817	0.150408	+	0.988624i	$0.451941\pi$
$230$	0	0
$231$	−102501.	−0.126385
$232$	−175351.	−0.213889
$233$	−507726.	−0.612688	−0.306344	−	0.951921i	$-0.599106\pi$
$233$	−507726.	−0.612688	−0.306344	+	0.951921i	$0.599106\pi$
$234$	−572452.	−0.683439
$235$	0	0
$236$	−317231.	−0.370762
$237$	−541153.	−0.625819
$238$	−142005.	−0.162503
$239$	143880.	0.162932	0.0814660	−	0.996676i	$-0.474040\pi$
$239$	143880.	0.162932	0.0814660	+	0.996676i	$0.474040\pi$
$240$	0	0
$241$	398237.	0.441671	0.220835	−	0.975311i	$-0.429122\pi$
$241$	398237.	0.441671	0.220835	+	0.975311i	$0.429122\pi$
$242$	−369974.	−0.406100
$243$	−950107.	−1.03218
$244$	−198645.	−0.213600
$245$	0	0
$246$	146754.	0.154615
$247$	1.33096e6	1.38811
$248$	62855.4	0.0648953
$249$	320105.	0.327185
$250$	0	0
$251$	1.65625e6	1.65937	0.829684	−	0.558233i	$-0.188520\pi$
$251$	1.65625e6	1.65937	0.829684	+	0.558233i	$0.188520\pi$
$252$	86271.1	0.0855784
$253$	1.60090e6	1.57240
$254$	−23591.7	−0.0229443
$255$	0	0
$256$	−1.09031e6	−1.03980
$257$	−32691.4	−0.0308746	−0.0154373	−	0.999881i	$-0.504914\pi$
$257$	−32691.4	−0.0308746	−0.0154373	+	0.999881i	$0.504914\pi$
$258$	53034.4	0.0496030
$259$	190101.	0.176090
$260$	0	0
$261$	171888.	0.156187
$262$	175717.	0.158147
$263$	−1.02727e6	−0.915791	−0.457895	−	0.889006i	$-0.651397\pi$
$263$	−1.02727e6	−0.915791	−0.457895	+	0.889006i	$0.651397\pi$
$264$	−743577.	−0.656623
$265$	0	0
$266$	144024.	0.124804
$267$	656409.	0.563503
$268$	−163901.	−0.139394
$269$	−1.88917e6	−1.59180	−0.795902	−	0.605426i	$-0.793003\pi$
$269$	−1.88917e6	−1.59180	−0.795902	+	0.605426i	$0.793003\pi$
$270$	0	0
$271$	605259.	0.500631	0.250316	−	0.968164i	$-0.419466\pi$
$271$	605259.	0.500631	0.250316	+	0.968164i	$0.419466\pi$
$272$	−123303.	−0.101053
$273$	172648.	0.140202
$274$	−376982.	−0.303350
$275$	0	0
$276$	456711.	0.360885
$277$	1.16555e6	0.912710	0.456355	−	0.889798i	$-0.349155\pi$
$277$	1.16555e6	0.912710	0.456355	+	0.889798i	$0.349155\pi$
$278$	−803184.	−0.623309
$279$	−61614.0	−0.0473881
$280$	0	0
$281$	938873.	0.709318	0.354659	−	0.934996i	$-0.384597\pi$
$281$	938873.	0.709318	0.354659	+	0.934996i	$0.384597\pi$
$282$	492194.	0.368564
$283$	2.54537e6	1.88923	0.944616	−	0.328179i	$-0.106435\pi$
$283$	2.54537e6	1.88923	0.944616	+	0.328179i	$0.106435\pi$
$284$	1.43527e6	1.05594
$285$	0	0
$286$	1.61521e6	1.16766
$287$	130579.	0.0935768
$288$	1.02143e6	0.725649
$289$	895076.	0.630399
$290$	0	0
$291$	1.25352e6	0.867757
$292$	−148351.	−0.101820
$293$	1.61560e6	1.09942	0.549712	−	0.835354i	$-0.314737\pi$
$293$	1.61560e6	1.09942	0.549712	+	0.835354i	$0.314737\pi$
$294$	−463389.	−0.312663
$295$	0	0
$296$	1.37906e6	0.914858
$297$	1.70484e6	1.12148
$298$	1.22740e6	0.800657
$299$	−2.69650e6	−1.74430
$300$	0	0
$301$	47189.1	0.0300210
$302$	310558.	0.195941
$303$	1.32524e6	0.829257
$304$	125055.	0.0776099
$305$	0	0
$306$	1.00981e6	0.616506
$307$	1.49055e6	0.902613	0.451307	−	0.892369i	$-0.350958\pi$
$307$	1.49055e6	0.902613	0.451307	+	0.892369i	$0.350958\pi$
$308$	−243420.	−0.146211
$309$	−1.13792e6	−0.677975
$310$	0	0
$311$	−754959.	−0.442611	−0.221305	−	0.975205i	$-0.571032\pi$
$311$	−754959.	−0.442611	−0.221305	+	0.975205i	$0.571032\pi$
$312$	1.25245e6	0.728409
$313$	−1.31681e6	−0.759735	−0.379867	−	0.925041i	$-0.624030\pi$
$313$	−1.31681e6	−0.759735	−0.379867	+	0.925041i	$0.624030\pi$
$314$	1.14746e6	0.656770
$315$	0	0
$316$	−1.28514e6	−0.723988
$317$	2.19577e6	1.22727	0.613633	−	0.789591i	$-0.289707\pi$
$317$	2.19577e6	1.22727	0.613633	+	0.789591i	$0.289707\pi$
$318$	−518882.	−0.287740
$319$	−484994.	−0.266845
$320$	0	0
$321$	570142.	0.308831
$322$	−291788.	−0.156830
$323$	−2.34783e6	−1.25216
$324$	−335059.	−0.177320
$325$	0	0
$326$	876381.	0.456719
$327$	25524.7	0.0132005
$328$	947268.	0.486170
$329$	437946.	0.223065
$330$	0	0
$331$	727999.	0.365225	0.182613	−	0.983185i	$-0.441545\pi$
$331$	727999.	0.365225	0.182613	+	0.983185i	$0.441545\pi$
$332$	760190.	0.378509
$333$	−1.35182e6	−0.668050
$334$	−1.85293e6	−0.908853
$335$	0	0
$336$	16221.7	0.00783879
$337$	−626937.	−0.300711	−0.150355	−	0.988632i	$-0.548042\pi$
$337$	−626937.	−0.300711	−0.150355	+	0.988632i	$0.548042\pi$
$338$	−1.36277e6	−0.648829
$339$	912897.	0.431442
$340$	0	0
$341$	173848.	0.0809625
$342$	−1.02417e6	−0.473484
$343$	−841254.	−0.386093
$344$	342327.	0.155971
$345$	0	0
$346$	−1.56220e6	−0.701529
$347$	3.30393e6	1.47302	0.736508	−	0.676428i	$-0.236473\pi$
$347$	3.30393e6	1.47302	0.736508	+	0.676428i	$0.236473\pi$
$348$	−138361.	−0.0612444
$349$	−2.93486e6	−1.28980	−0.644902	−	0.764265i	$-0.723102\pi$
$349$	−2.93486e6	−1.28980	−0.644902	+	0.764265i	$0.723102\pi$
$350$	0	0
$351$	−2.87157e6	−1.24409
$352$	−2.88203e6	−1.23977
$353$	−633691.	−0.270670	−0.135335	−	0.990800i	$-0.543211\pi$
$353$	−633691.	−0.270670	−0.135335	+	0.990800i	$0.543211\pi$
$354$	488513.	0.207190
$355$	0	0
$356$	1.55885e6	0.651897
$357$	−304554.	−0.126472
$358$	−42695.6	−0.0176066
$359$	3.08062e6	1.26154	0.630771	−	0.775969i	$-0.282739\pi$
$359$	3.08062e6	1.26154	0.630771	+	0.775969i	$0.282739\pi$
$360$	0	0
$361$	−94896.2	−0.0383249
$362$	−262179.	−0.105154
$363$	−793470.	−0.316056
$364$	410007.	0.162195
$365$	0	0
$366$	305899.	0.119364
$367$	1.32881e6	0.514988	0.257494	−	0.966280i	$-0.417103\pi$
$367$	1.32881e6	0.514988	0.257494	+	0.966280i	$0.417103\pi$
$368$	−253358.	−0.0975250
$369$	−928559.	−0.355012
$370$	0	0
$371$	−461692.	−0.174148
$372$	49596.1	0.0185819
$373$	1.91319e6	0.712011	0.356006	−	0.934484i	$-0.384138\pi$
$373$	1.91319e6	0.712011	0.356006	+	0.934484i	$0.384138\pi$
$374$	−2.84926e6	−1.05330
$375$	0	0
$376$	3.17702e6	1.15891
$377$	816906.	0.296019
$378$	−310733.	−0.111856
$379$	703691.	0.251642	0.125821	−	0.992053i	$-0.459843\pi$
$379$	703691.	0.251642	0.125821	+	0.992053i	$0.459843\pi$
$380$	0	0
$381$	−50596.2	−0.0178569
$382$	−777632.	−0.272656
$383$	−1.73748e6	−0.605234	−0.302617	−	0.953112i	$-0.597860\pi$
$383$	−1.73748e6	−0.605234	−0.302617	+	0.953112i	$0.597860\pi$
$384$	−747815.	−0.258801
$385$	0	0
$386$	−502731.	−0.171738
$387$	−335566.	−0.113894
$388$	2.97687e6	1.00388
$389$	3.93096e6	1.31712	0.658559	−	0.752529i	$-0.271166\pi$
$389$	3.93096e6	1.31712	0.658559	+	0.752529i	$0.271166\pi$
$390$	0	0
$391$	4.75666e6	1.57347
$392$	−2.99109e6	−0.983136
$393$	376854.	0.123081
$394$	−104289.	−0.0338453
$395$	0	0
$396$	1.73098e6	0.554695
$397$	−3.99784e6	−1.27306	−0.636530	−	0.771252i	$-0.719631\pi$
$397$	−3.99784e6	−1.27306	−0.636530	+	0.771252i	$0.719631\pi$
$398$	1.29304e6	0.409171
$399$	308882.	0.0971316
$400$	0	0
$401$	4.39985e6	1.36640	0.683198	−	0.730233i	$-0.260588\pi$
$401$	4.39985e6	1.36640	0.683198	+	0.730233i	$0.260588\pi$
$402$	252397.	0.0778966
$403$	−292823.	−0.0898138
$404$	3.14721e6	0.959338
$405$	0	0
$406$	88397.6	0.0266149
$407$	3.81426e6	1.14136
$408$	−2.20934e6	−0.657072
$409$	−1.64907e6	−0.487451	−0.243726	−	0.969844i	$-0.578370\pi$
$409$	−1.64907e6	−0.487451	−0.243726	+	0.969844i	$0.578370\pi$
$410$	0	0
$411$	−808499.	−0.236089
$412$	−2.70234e6	−0.784326
$413$	434671.	0.125396
$414$	2.07493e6	0.594982
$415$	0	0
$416$	4.85439e6	1.37531
$417$	−1.72256e6	−0.485103
$418$	2.88975e6	0.808947
$419$	−5.93011e6	−1.65017	−0.825083	−	0.565012i	$-0.808871\pi$
$419$	−5.93011e6	−1.65017	−0.825083	+	0.565012i	$0.808871\pi$
$420$	0	0
$421$	4.85544e6	1.33513	0.667565	−	0.744552i	$-0.267337\pi$
$421$	4.85544e6	1.33513	0.667565	+	0.744552i	$0.267337\pi$
$422$	−1.19893e6	−0.327726
$423$	−3.11427e6	−0.846264
$424$	−3.34929e6	−0.904769
$425$	0	0
$426$	−2.21022e6	−0.590080
$427$	272184.	0.0722424
$428$	1.35398e6	0.357275
$429$	3.46409e6	0.908752
$430$	0	0
$431$	2.59168e6	0.672029	0.336015	−	0.941857i	$-0.390921\pi$
$431$	2.59168e6	0.672029	0.336015	+	0.941857i	$0.390921\pi$
$432$	−269808.	−0.0695578
$433$	5.08691e6	1.30387	0.651934	−	0.758275i	$-0.273958\pi$
$433$	5.08691e6	1.30387	0.651934	+	0.758275i	$0.273958\pi$
$434$	−31686.5	−0.00807513
$435$	0	0
$436$	60616.3	0.0152712
$437$	−4.82426e6	−1.20845
$438$	228451.	0.0568994
$439$	−5.71552e6	−1.41545	−0.707725	−	0.706488i	$-0.750278\pi$
$439$	−5.71552e6	−1.41545	−0.707725	+	0.706488i	$0.750278\pi$
$440$	0	0
$441$	2.93201e6	0.717909
$442$	4.79918e6	1.16845
$443$	4.81790e6	1.16640	0.583201	−	0.812328i	$-0.301800\pi$
$443$	4.81790e6	1.16640	0.583201	+	0.812328i	$0.301800\pi$
$444$	1.08815e6	0.261957
$445$	0	0
$446$	906769.	0.215854
$447$	2.63237e6	0.623128
$448$	591479.	0.139234
$449$	−2.36680e6	−0.554046	−0.277023	−	0.960863i	$-0.589348\pi$
$449$	−2.36680e6	−0.554046	−0.277023	+	0.960863i	$0.589348\pi$
$450$	0	0
$451$	2.61999e6	0.606539
$452$	2.16796e6	0.499120
$453$	666042.	0.152495
$454$	−3.17765e6	−0.723546
$455$	0	0
$456$	2.24074e6	0.504638
$457$	−1.78378e6	−0.399531	−0.199766	−	0.979844i	$-0.564018\pi$
$457$	−1.78378e6	−0.399531	−0.199766	+	0.979844i	$0.564018\pi$
$458$	−873015.	−0.194472
$459$	5.06549e6	1.12225
$460$	0	0
$461$	−4.26575e6	−0.934852	−0.467426	−	0.884032i	$-0.654819\pi$
$461$	−4.26575e6	−0.934852	−0.467426	+	0.884032i	$0.654819\pi$
$462$	374850.	0.0817056
$463$	2.33897e6	0.507074	0.253537	−	0.967326i	$-0.418406\pi$
$463$	2.33897e6	0.507074	0.253537	+	0.967326i	$0.418406\pi$
$464$	76755.2	0.0165506
$465$	0	0
$466$	1.85678e6	0.396091
$467$	6.84641e6	1.45268	0.726342	−	0.687334i	$-0.241219\pi$
$467$	6.84641e6	1.45268	0.726342	+	0.687334i	$0.241219\pi$
$468$	−2.91560e6	−0.615337
$469$	224578.	0.0471450
$470$	0	0
$471$	2.46091e6	0.511145
$472$	3.15326e6	0.651486
$473$	946822.	0.194588
$474$	1.97902e6	0.404580
$475$	0	0
$476$	−723258.	−0.146311
$477$	3.28314e6	0.660683
$478$	−526177.	−0.105332
$479$	9.12451e6	1.81707	0.908533	−	0.417814i	$-0.137203\pi$
$479$	9.12451e6	1.81707	0.908533	+	0.417814i	$0.137203\pi$
$480$	0	0
$481$	−6.42460e6	−1.26614
$482$	−1.45637e6	−0.285532
$483$	−625788.	−0.122056
$484$	−1.88434e6	−0.365634
$485$	0	0
$486$	3.47459e6	0.667287
$487$	−2.68151e6	−0.512338	−0.256169	−	0.966632i	$-0.582460\pi$
$487$	−2.68151e6	−0.512338	−0.256169	+	0.966632i	$0.582460\pi$
$488$	1.97452e6	0.375329
$489$	1.87954e6	0.355451
$490$	0	0
$491$	4.83155e6	0.904446	0.452223	−	0.891905i	$-0.350631\pi$
$491$	4.83155e6	0.904446	0.452223	+	0.891905i	$0.350631\pi$
$492$	747442.	0.139208
$493$	−1.44103e6	−0.267028
$494$	−4.86739e6	−0.897385
$495$	0	0
$496$	−27513.2	−0.00502154
$497$	−1.96661e6	−0.357131
$498$	−1.17064e6	−0.211519
$499$	8.26537e6	1.48597	0.742987	−	0.669306i	$-0.233409\pi$
$499$	8.26537e6	1.48597	0.742987	+	0.669306i	$0.233409\pi$
$500$	0	0
$501$	−3.97392e6	−0.707334
$502$	−6.05701e6	−1.07275
$503$	1.06078e7	1.86941	0.934703	−	0.355429i	$-0.115665\pi$
$503$	1.06078e7	1.86941	0.934703	+	0.355429i	$0.115665\pi$
$504$	−857532.	−0.150374
$505$	0	0
$506$	−5.85457e6	−1.01653
$507$	−2.92268e6	−0.504965
$508$	−120157.	−0.0206580
$509$	−1.89756e6	−0.324639	−0.162320	−	0.986738i	$-0.551898\pi$
$509$	−1.89756e6	−0.324639	−0.162320	+	0.986738i	$0.551898\pi$
$510$	0	0
$511$	203272.	0.0344370
$512$	936238.	0.157838
$513$	−5.13749e6	−0.861901
$514$	119554.	0.0199598
$515$	0	0
$516$	270113.	0.0446603
$517$	8.78713e6	1.44584
$518$	−695207.	−0.113839
$519$	−3.35039e6	−0.545980
$520$	0	0
$521$	−1.76155e6	−0.284316	−0.142158	−	0.989844i	$-0.545404\pi$
$521$	−1.76155e6	−0.284316	−0.142158	+	0.989844i	$0.545404\pi$
$522$	−628604.	−0.100972
$523$	7.12167e6	1.13849	0.569243	−	0.822170i	$-0.307236\pi$
$523$	7.12167e6	1.13849	0.569243	+	0.822170i	$0.307236\pi$
$524$	894959.	0.142388
$525$	0	0
$526$	3.75679e6	0.592041
$527$	516544.	0.0810179
$528$	325480.	0.0508089
$529$	3.33749e6	0.518538
$530$	0	0
$531$	−3.09099e6	−0.475730
$532$	733538.	0.112368
$533$	−4.41302e6	−0.672849
$534$	−2.40052e6	−0.364294
$535$	0	0
$536$	1.62917e6	0.244937
$537$	−91567.6	−0.0137027
$538$	6.90877e6	1.02907
$539$	−8.27287e6	−1.22655
$540$	0	0
$541$	6.26443e6	0.920214	0.460107	−	0.887864i	$-0.347811\pi$
$541$	6.26443e6	0.920214	0.460107	+	0.887864i	$0.347811\pi$
$542$	−2.21346e6	−0.323649
$543$	−562285.	−0.0818383
$544$	−8.56320e6	−1.24062
$545$	0	0
$546$	−631383.	−0.0906382
$547$	3.19734e6	0.456899	0.228449	−	0.973556i	$-0.426634\pi$
$547$	3.19734e6	0.456899	0.228449	+	0.973556i	$0.426634\pi$
$548$	−1.92003e6	−0.273123
$549$	−1.93552e6	−0.274074
$550$	0	0
$551$	1.46151e6	0.205080
$552$	−4.53969e6	−0.634131
$553$	1.76090e6	0.244862
$554$	−4.26248e6	−0.590049
$555$	0	0
$556$	−4.09076e6	−0.561199
$557$	−1.13299e7	−1.54735	−0.773677	−	0.633581i	$-0.781584\pi$
$557$	−1.13299e7	−1.54735	−0.773677	+	0.633581i	$0.781584\pi$
$558$	225326.	0.0306355
$559$	−1.59479e6	−0.215861
$560$	0	0
$561$	−6.11070e6	−0.819754
$562$	−3.43350e6	−0.458561
$563$	4.20790e6	0.559492	0.279746	−	0.960074i	$-0.409750\pi$
$563$	4.20790e6	0.559492	0.279746	+	0.960074i	$0.409750\pi$
$564$	2.50683e6	0.331839
$565$	0	0
$566$	−9.30855e6	−1.22135
$567$	459099.	0.0599720
$568$	−1.42665e7	−1.85544
$569$	−3.28836e6	−0.425794	−0.212897	−	0.977075i	$-0.568290\pi$
$569$	−3.28836e6	−0.425794	−0.212897	+	0.977075i	$0.568290\pi$
$570$	0	0
$571$	−2.54135e6	−0.326192	−0.163096	−	0.986610i	$-0.552148\pi$
$571$	−2.54135e6	−0.326192	−0.163096	+	0.986610i	$0.552148\pi$
$572$	8.22656e6	1.05130
$573$	−1.66776e6	−0.212201
$574$	−477534.	−0.0604956
$575$	0	0
$576$	−4.20606e6	−0.528225
$577$	1.69533e6	0.211990	0.105995	−	0.994367i	$-0.466197\pi$
$577$	1.69533e6	0.211990	0.105995	+	0.994367i	$0.466197\pi$
$578$	−3.27334e6	−0.407541
$579$	−1.07819e6	−0.133659
$580$	0	0
$581$	−1.04161e6	−0.128017
$582$	−4.58418e6	−0.560988
$583$	−9.26360e6	−1.12878
$584$	1.47461e6	0.178914
$585$	0	0
$586$	−5.90834e6	−0.710757
$587$	−6.20781e6	−0.743607	−0.371803	−	0.928311i	$-0.621260\pi$
$587$	−6.20781e6	−0.743607	−0.371803	+	0.928311i	$0.621260\pi$
$588$	−2.36012e6	−0.281508
$589$	−523886.	−0.0622226
$590$	0	0
$591$	−223665.	−0.0263408
$592$	−603645.	−0.0707909
$593$	−3.01403e6	−0.351974	−0.175987	−	0.984393i	$-0.556312\pi$
$593$	−3.01403e6	−0.351974	−0.175987	+	0.984393i	$0.556312\pi$
$594$	−6.23469e6	−0.725018
$595$	0	0
$596$	6.25138e6	0.720876
$597$	2.77314e6	0.318446
$598$	9.86122e6	1.12766
$599$	1.21463e7	1.38317	0.691586	−	0.722294i	$-0.256912\pi$
$599$	1.21463e7	1.38317	0.691586	+	0.722294i	$0.256912\pi$
$600$	0	0
$601$	1.68363e6	0.190134	0.0950671	−	0.995471i	$-0.469693\pi$
$601$	1.68363e6	0.190134	0.0950671	+	0.995471i	$0.469693\pi$
$602$	−172573.	−0.0194080
$603$	−1.59700e6	−0.178859
$604$	1.58173e6	0.176416
$605$	0	0
$606$	−4.84648e6	−0.536099
$607$	5.11718e6	0.563714	0.281857	−	0.959456i	$-0.409050\pi$
$607$	5.11718e6	0.563714	0.281857	+	0.959456i	$0.409050\pi$
$608$	8.68491e6	0.952810
$609$	189583.	0.0207136
$610$	0	0
$611$	−1.48007e7	−1.60391
$612$	5.14316e6	0.555075
$613$	−4.10152e6	−0.440853	−0.220426	−	0.975404i	$-0.570745\pi$
$613$	−4.10152e6	−0.440853	−0.220426	+	0.975404i	$0.570745\pi$
$614$	−5.45103e6	−0.583522
$615$	0	0
$616$	2.41958e6	0.256915
$617$	−1.52311e7	−1.61071	−0.805355	−	0.592792i	$-0.798025\pi$
$617$	−1.52311e7	−1.61071	−0.805355	+	0.592792i	$0.798025\pi$
$618$	4.16141e6	0.438298
$619$	5.27961e6	0.553829	0.276914	−	0.960895i	$-0.410688\pi$
$619$	5.27961e6	0.553829	0.276914	+	0.960895i	$0.410688\pi$
$620$	0	0
$621$	1.04084e7	1.08307
$622$	2.76092e6	0.286140
$623$	−2.13594e6	−0.220480
$624$	−548227.	−0.0563636
$625$	0	0
$626$	4.81564e6	0.491154
$627$	6.19754e6	0.629580
$628$	5.84421e6	0.591326
$629$	1.13331e7	1.14215
$630$	0	0
$631$	−1.84888e7	−1.84857	−0.924283	−	0.381709i	$-0.875336\pi$
$631$	−1.84888e7	−1.84857	−0.924283	+	0.381709i	$0.875336\pi$
$632$	1.27742e7	1.27216
$633$	−2.57129e6	−0.255060
$634$	−8.03004e6	−0.793404
$635$	0	0
$636$	−2.64276e6	−0.259068
$637$	1.39345e7	1.36064
$638$	1.77365e6	0.172511
$639$	1.39848e7	1.35489
$640$	0	0
$641$	−3.87212e6	−0.372223	−0.186112	−	0.982529i	$-0.559589\pi$
$641$	−3.87212e6	−0.372223	−0.186112	+	0.982529i	$0.559589\pi$
$642$	−2.08504e6	−0.199653
$643$	−3.24879e6	−0.309880	−0.154940	−	0.987924i	$-0.549518\pi$
$643$	−3.24879e6	−0.309880	−0.154940	+	0.987924i	$0.549518\pi$
$644$	−1.48613e6	−0.141202
$645$	0	0
$646$	8.58615e6	0.809500
$647$	−1.85942e7	−1.74629	−0.873147	−	0.487456i	$-0.837925\pi$
$647$	−1.85942e7	−1.74629	−0.873147	+	0.487456i	$0.837925\pi$
$648$	3.33047e6	0.311579
$649$	8.72142e6	0.812785
$650$	0	0
$651$	−67956.8	−0.00628464
$652$	4.46356e6	0.411209
$653$	−9.77909e6	−0.897461	−0.448730	−	0.893667i	$-0.648124\pi$
$653$	−9.77909e6	−0.897461	−0.448730	+	0.893667i	$0.648124\pi$
$654$	−93345.0	−0.00853388
$655$	0	0
$656$	−414640.	−0.0376194
$657$	−1.44548e6	−0.130647
$658$	−1.60159e6	−0.144207
$659$	1.92292e7	1.72484	0.862418	−	0.506196i	$-0.168949\pi$
$659$	1.92292e7	1.72484	0.862418	+	0.506196i	$0.168949\pi$
$660$	0	0
$661$	−1.39614e7	−1.24287	−0.621434	−	0.783467i	$-0.713449\pi$
$661$	−1.39614e7	−1.24287	−0.621434	+	0.783467i	$0.713449\pi$
$662$	−2.66233e6	−0.236111
$663$	1.02926e7	0.909374
$664$	−7.55626e6	−0.665099
$665$	0	0
$666$	4.94368e6	0.431882
$667$	−2.96099e6	−0.257705
$668$	−9.43732e6	−0.818290
$669$	1.94471e6	0.167993
$670$	0	0
$671$	5.46121e6	0.468255
$672$	1.12658e6	0.0962362
$673$	−1.13333e7	−0.964537	−0.482269	−	0.876023i	$-0.660187\pi$
$673$	−1.13333e7	−0.964537	−0.482269	+	0.876023i	$0.660187\pi$
$674$	2.29274e6	0.194404
$675$	0	0
$676$	−6.94082e6	−0.584176
$677$	9.84009e6	0.825140	0.412570	−	0.910926i	$-0.364631\pi$
$677$	9.84009e6	0.825140	0.412570	+	0.910926i	$0.364631\pi$
$678$	−3.33851e6	−0.278919
$679$	−4.07892e6	−0.339524
$680$	0	0
$681$	−6.81498e6	−0.563115
$682$	−635771.	−0.0523407
$683$	−4282.68	−0.000351288 0	−0.000175644	−	1.00000i	$-0.500056\pi$
$683$	−4282.68	−0.000351288 0	−0.000175644		1.00000i	$0.500056\pi$
$684$	−5.21625e6	−0.426303
$685$	0	0
$686$	3.07651e6	0.249602
$687$	−1.87232e6	−0.151352
$688$	−149844.	−0.0120689
$689$	1.56033e7	1.25218
$690$	0	0
$691$	5.24701e6	0.418039	0.209019	−	0.977911i	$-0.432973\pi$
$691$	5.24701e6	0.418039	0.209019	+	0.977911i	$0.432973\pi$
$692$	−7.95655e6	−0.631625
$693$	−2.37180e6	−0.187605
$694$	−1.20826e7	−0.952277
$695$	0	0
$696$	1.37531e6	0.107616
$697$	7.78462e6	0.606954
$698$	1.07329e7	0.833834
$699$	3.98217e6	0.308267
$700$	0	0
$701$	−3.73379e6	−0.286982	−0.143491	−	0.989652i	$-0.545833\pi$
$701$	−3.73379e6	−0.286982	−0.143491	+	0.989652i	$0.545833\pi$
$702$	1.05015e7	0.804281
$703$	−1.14942e7	−0.877180
$704$	1.18677e7	0.902473
$705$	0	0
$706$	2.31744e6	0.174983
$707$	−4.31231e6	−0.324461
$708$	2.48808e6	0.186544
$709$	−1.21242e7	−0.905810	−0.452905	−	0.891559i	$-0.649612\pi$
$709$	−1.21242e7	−0.905810	−0.452905	+	0.891559i	$0.649612\pi$
$710$	0	0
$711$	−1.25219e7	−0.928960
$712$	−1.54949e7	−1.14548
$713$	1.06138e6	0.0781892
$714$	1.11377e6	0.0817615
$715$	0	0
$716$	−217456.	−0.0158522
$717$	−1.12847e6	−0.0819772
$718$	−1.12660e7	−0.815563
$719$	1.59176e7	1.14830	0.574149	−	0.818751i	$-0.305333\pi$
$719$	1.59176e7	1.14830	0.574149	+	0.818751i	$0.305333\pi$
$720$	0	0
$721$	3.70275e6	0.265269
$722$	347040.	0.0247763
$723$	−3.12342e6	−0.222221
$724$	−1.33532e6	−0.0946759
$725$	0	0
$726$	2.90176e6	0.204324
$727$	633198.	0.0444328	0.0222164	−	0.999753i	$-0.492928\pi$
$727$	633198.	0.0444328	0.0222164	+	0.999753i	$0.492928\pi$
$728$	−4.07546e6	−0.285002
$729$	3.08055e6	0.214689
$730$	0	0
$731$	2.81324e6	0.194721
$732$	1.55800e6	0.107470
$733$	2.64288e7	1.81684	0.908421	−	0.418057i	$-0.137289\pi$
$733$	2.64288e7	1.81684	0.908421	+	0.418057i	$0.137289\pi$
$734$	−4.85952e6	−0.332930
$735$	0	0
$736$	−1.75954e7	−1.19730
$737$	4.50603e6	0.305581
$738$	3.39579e6	0.229509
$739$	2.15407e6	0.145094	0.0725470	−	0.997365i	$-0.476887\pi$
$739$	2.15407e6	0.145094	0.0725470	+	0.997365i	$0.476887\pi$
$740$	0	0
$741$	−1.04389e7	−0.698409
$742$	1.68843e6	0.112583
$743$	−1.76951e7	−1.17593	−0.587964	−	0.808887i	$-0.700070\pi$
$743$	−1.76951e7	−1.17593	−0.587964	+	0.808887i	$0.700070\pi$
$744$	−492984.	−0.0326513
$745$	0	0
$746$	−6.99664e6	−0.460302
$747$	7.40702e6	0.485671
$748$	−1.45118e7	−0.948345
$749$	−1.85523e6	−0.120835
$750$	0	0
$751$	2.58345e7	1.67148	0.835739	−	0.549127i	$-0.185040\pi$
$751$	2.58345e7	1.67148	0.835739	+	0.549127i	$0.185040\pi$
$752$	−1.39065e6	−0.0896755
$753$	−1.29902e7	−0.834891
$754$	−2.98747e6	−0.191370
$755$	0	0
$756$	−1.58262e6	−0.100710
$757$	2.47558e7	1.57013	0.785067	−	0.619411i	$-0.212629\pi$
$757$	2.47558e7	1.57013	0.785067	+	0.619411i	$0.212629\pi$
$758$	−2.57343e6	−0.162682
$759$	−1.25561e7	−0.791133
$760$	0	0
$761$	−1.12818e7	−0.706183	−0.353092	−	0.935589i	$-0.614870\pi$
$761$	−1.12818e7	−0.706183	−0.353092	+	0.935589i	$0.614870\pi$
$762$	185033.	0.0115441
$763$	−83056.7	−0.00516492
$764$	−3.96062e6	−0.245488
$765$	0	0
$766$	6.35406e6	0.391272
$767$	−1.46900e7	−0.901643
$768$	8.55146e6	0.523163
$769$	−1.45315e7	−0.886124	−0.443062	−	0.896491i	$-0.646108\pi$
$769$	−1.45315e7	−0.886124	−0.443062	+	0.896491i	$0.646108\pi$
$770$	0	0
$771$	256403.	0.0155341
$772$	−2.56050e6	−0.154625
$773$	5.49667e6	0.330865	0.165433	−	0.986221i	$-0.447098\pi$
$773$	5.49667e6	0.330865	0.165433	+	0.986221i	$0.447098\pi$
$774$	1.22718e6	0.0736303
$775$	0	0
$776$	−2.95900e7	−1.76397
$777$	−1.49099e6	−0.0885974
$778$	−1.43757e7	−0.851492
$779$	−7.89526e6	−0.466147
$780$	0	0
$781$	−3.94590e7	−2.31482
$782$	−1.73953e7	−1.01722
$783$	−3.15324e6	−0.183803
$784$	1.30926e6	0.0760742
$785$	0	0
$786$	−1.37817e6	−0.0795697
$787$	−1.62860e7	−0.937297	−0.468649	−	0.883385i	$-0.655259\pi$
$787$	−1.62860e7	−0.937297	−0.468649	+	0.883385i	$0.655259\pi$
$788$	−531163.	−0.0304728
$789$	8.05704e6	0.460769
$790$	0	0
$791$	−2.97055e6	−0.168809
$792$	−1.72059e7	−0.974685
$793$	−9.19866e6	−0.519447
$794$	1.46203e7	0.823009
$795$	0	0
$796$	6.58569e6	0.368399
$797$	−1.65610e7	−0.923506	−0.461753	−	0.887009i	$-0.652779\pi$
$797$	−1.65610e7	−0.923506	−0.461753	+	0.887009i	$0.652779\pi$
$798$	−1.12960e6	−0.0627937
$799$	2.61087e7	1.44683
$800$	0	0
$801$	1.51889e7	0.836458
$802$	−1.60905e7	−0.883350
$803$	4.07853e6	0.223211
$804$	1.28550e6	0.0701346
$805$	0	0
$806$	1.07087e6	0.0580629
$807$	1.48170e7	0.800896
$808$	−3.12831e7	−1.68571
$809$	−1.93881e7	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$809$	−1.93881e7	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$810$	0	0
$811$	1.99327e7	1.06418	0.532089	−	0.846688i	$-0.321407\pi$
$811$	1.99327e7	1.06418	0.532089	+	0.846688i	$0.321407\pi$
$812$	450224.	0.0239629
$813$	−4.74713e6	−0.251886
$814$	−1.39489e7	−0.737870
$815$	0	0
$816$	967079.	0.0508436
$817$	−2.85322e6	−0.149548
$818$	6.03074e6	0.315128
$819$	3.99497e6	0.208115
$820$	0	0
$821$	−2.97106e7	−1.53834	−0.769172	−	0.639041i	$-0.779331\pi$
$821$	−2.97106e7	−1.53834	−0.769172	+	0.639041i	$0.779331\pi$
$822$	2.95672e6	0.152627
$823$	3.70742e6	0.190797	0.0953987	−	0.995439i	$-0.469587\pi$
$823$	3.70742e6	0.190797	0.0953987	+	0.995439i	$0.469587\pi$
$824$	2.68611e7	1.37818
$825$	0	0
$826$	−1.58961e6	−0.0810664
$827$	5.42093e6	0.275620	0.137810	−	0.990459i	$-0.455994\pi$
$827$	5.42093e6	0.275620	0.137810	+	0.990459i	$0.455994\pi$
$828$	1.05680e7	0.535694
$829$	−3.24596e7	−1.64043	−0.820214	−	0.572056i	$-0.806146\pi$
$829$	−3.24596e7	−1.64043	−0.820214	+	0.572056i	$0.806146\pi$
$830$	0	0
$831$	−9.14159e6	−0.459218
$832$	−1.99895e7	−1.00114
$833$	−2.45807e7	−1.22739
$834$	6.29948e6	0.313610
$835$	0	0
$836$	1.47180e7	0.728339
$837$	1.13029e6	0.0557670
$838$	2.16867e7	1.06680
$839$	−6.99699e6	−0.343168	−0.171584	−	0.985170i	$-0.554888\pi$
$839$	−6.99699e6	−0.343168	−0.171584	+	0.985170i	$0.554888\pi$
$840$	0	0
$841$	−1.96141e7	−0.956266
$842$	−1.77566e7	−0.863136
$843$	−7.36371e6	−0.356885
$844$	−6.10634e6	−0.295070
$845$	0	0
$846$	1.13891e7	0.547094
$847$	2.58194e6	0.123662
$848$	1.46606e6	0.0700102
$849$	−1.99637e7	−0.950543
$850$	0	0
$851$	2.32869e7	1.10227
$852$	−1.12570e7	−0.531281
$853$	5.68399e6	0.267474	0.133737	−	0.991017i	$-0.457302\pi$
$853$	5.68399e6	0.267474	0.133737	+	0.991017i	$0.457302\pi$
$854$	−995389.	−0.0467033
$855$	0	0
$856$	−1.34585e7	−0.627788
$857$	8.71203e6	0.405198	0.202599	−	0.979262i	$-0.435061\pi$
$857$	8.71203e6	0.405198	0.202599	+	0.979262i	$0.435061\pi$
$858$	−1.26683e7	−0.587491
$859$	1.41568e7	0.654609	0.327304	−	0.944919i	$-0.393860\pi$
$859$	1.41568e7	0.654609	0.327304	+	0.944919i	$0.393860\pi$
$860$	0	0
$861$	−1.02415e6	−0.0470820
$862$	−9.47790e6	−0.434454
$863$	−9.82195e6	−0.448922	−0.224461	−	0.974483i	$-0.572062\pi$
$863$	−9.82195e6	−0.448922	−0.224461	+	0.974483i	$0.572062\pi$
$864$	−1.87378e7	−0.853955
$865$	0	0
$866$	−1.86031e7	−0.842927
$867$	−7.02021e6	−0.317177
$868$	−161385.	−0.00727048
$869$	3.53315e7	1.58713
$870$	0	0
$871$	−7.58980e6	−0.338988
$872$	−602524.	−0.0268339
$873$	2.90056e7	1.28809
$874$	1.76426e7	0.781237
$875$	0	0
$876$	1.16354e6	0.0512297
$877$	−2.13034e7	−0.935298	−0.467649	−	0.883914i	$-0.654899\pi$
$877$	−2.13034e7	−0.935298	−0.467649	+	0.883914i	$0.654899\pi$
$878$	2.09019e7	0.915061
$879$	−1.26714e7	−0.553162
$880$	0	0
$881$	3.10088e7	1.34600	0.673000	−	0.739642i	$-0.265005\pi$
$881$	3.10088e7	1.34600	0.673000	+	0.739642i	$0.265005\pi$
$882$	−1.07225e7	−0.464114
$883$	2.21207e7	0.954766	0.477383	−	0.878695i	$-0.341585\pi$
$883$	2.21207e7	0.954766	0.477383	+	0.878695i	$0.341585\pi$
$884$	2.44431e7	1.05202
$885$	0	0
$886$	−1.76193e7	−0.754057
$887$	−4.29991e7	−1.83506	−0.917530	−	0.397666i	$-0.869820\pi$
$887$	−4.29991e7	−1.83506	−0.917530	+	0.397666i	$0.869820\pi$
$888$	−1.08162e7	−0.460299
$889$	164639.	0.00698680
$890$	0	0
$891$	9.21156e6	0.388722
$892$	4.61833e6	0.194345
$893$	−2.64797e7	−1.11118
$894$	−9.62669e6	−0.402841
$895$	0	0
$896$	2.43337e6	0.101260
$897$	2.11490e7	0.877624
$898$	8.65551e6	0.358181
$899$	−321546.	−0.0132692
$900$	0	0
$901$	−2.75244e7	−1.12955
$902$	−9.58144e6	−0.392116
$903$	−370111.	−0.0151047
$904$	−2.15494e7	−0.877031
$905$	0	0
$906$	−2.43575e6	−0.0985852
$907$	−3.34186e7	−1.34887	−0.674435	−	0.738334i	$-0.735613\pi$
$907$	−3.34186e7	−1.34887	−0.674435	+	0.738334i	$0.735613\pi$
$908$	−1.61843e7	−0.651448
$909$	3.06653e7	1.23094
$910$	0	0
$911$	−3.77821e7	−1.50831	−0.754155	−	0.656697i	$-0.771953\pi$
$911$	−3.77821e7	−1.50831	−0.754155	+	0.656697i	$0.771953\pi$
$912$	−980824.	−0.0390485
$913$	−2.08994e7	−0.829769
$914$	6.52337e6	0.258289
$915$	0	0
$916$	−4.44642e6	−0.175094
$917$	−1.22628e6	−0.0481576
$918$	−1.85248e7	−0.725514
$919$	3.71228e7	1.44995	0.724973	−	0.688777i	$-0.241852\pi$
$919$	3.71228e7	1.44995	0.724973	+	0.688777i	$0.241852\pi$
$920$	0	0
$921$	−1.16906e7	−0.454139
$922$	1.56000e7	0.604364
$923$	6.64632e7	2.56789
$924$	1.90917e6	0.0735641
$925$	0	0
$926$	−8.55371e6	−0.327814
$927$	−2.63306e7	−1.00638
$928$	5.33055e6	0.203190
$929$	4.73143e6	0.179868	0.0899339	−	0.995948i	$-0.471334\pi$
$929$	4.73143e6	0.179868	0.0899339	+	0.995948i	$0.471334\pi$
$930$	0	0
$931$	2.49300e7	0.942646
$932$	9.45690e6	0.356623
$933$	5.92124e6	0.222694
$934$	−2.50377e7	−0.939132
$935$	0	0
$936$	2.89810e7	1.08124
$937$	1.74328e7	0.648660	0.324330	−	0.945944i	$-0.394861\pi$
$937$	1.74328e7	0.648660	0.324330	+	0.945944i	$0.394861\pi$
$938$	−821293.	−0.0304783
$939$	1.03279e7	0.382251
$940$	0	0
$941$	3.82735e7	1.40904	0.704521	−	0.709683i	$-0.251162\pi$
$941$	3.82735e7	1.40904	0.704521	+	0.709683i	$0.251162\pi$
$942$	−8.99968e6	−0.330446
$943$	1.59956e7	0.585762
$944$	−1.38025e6	−0.0504114
$945$	0	0
$946$	−3.46258e6	−0.125797
$947$	1.65041e7	0.598021	0.299011	−	0.954250i	$-0.403343\pi$
$947$	1.65041e7	0.598021	0.299011	+	0.954250i	$0.403343\pi$
$948$	1.00795e7	0.364266
$949$	−6.86973e6	−0.247613
$950$	0	0
$951$	−1.72217e7	−0.617484
$952$	7.18916e6	0.257090
$953$	2.15575e7	0.768894	0.384447	−	0.923147i	$-0.374392\pi$
$953$	2.15575e7	0.768894	0.384447	+	0.923147i	$0.374392\pi$
$954$	−1.20066e7	−0.427119
$955$	0	0
$956$	−2.67991e6	−0.0948365
$957$	3.80388e6	0.134260
$958$	−3.33688e7	−1.17470
$959$	2.63084e6	0.0923736
$960$	0	0
$961$	−2.85139e7	−0.995974
$962$	2.34951e7	0.818539
$963$	1.31927e7	0.458425
$964$	−7.41755e6	−0.257080
$965$	0	0
$966$	2.28854e6	0.0789069
$967$	3.38528e7	1.16420	0.582101	−	0.813116i	$-0.302231\pi$
$967$	3.38528e7	1.16420	0.582101	+	0.813116i	$0.302231\pi$
$968$	1.87303e7	0.642476
$969$	1.84144e7	0.630011
$970$	0	0
$971$	1.12966e7	0.384502	0.192251	−	0.981346i	$-0.438421\pi$
$971$	1.12966e7	0.384502	0.192251	+	0.981346i	$0.438421\pi$
$972$	1.76967e7	0.600795
$973$	5.60517e6	0.189805
$974$	9.80641e6	0.331217
$975$	0	0
$976$	−864291.	−0.0290426
$977$	2.28369e7	0.765423	0.382711	−	0.923868i	$-0.374990\pi$
$977$	2.28369e7	0.765423	0.382711	+	0.923868i	$0.374990\pi$
$978$	−6.87358e6	−0.229793
$979$	−4.28564e7	−1.42909
$980$	0	0
$981$	590624.	0.0195947
$982$	−1.76692e7	−0.584707
$983$	−1.60143e7	−0.528598	−0.264299	−	0.964441i	$-0.585140\pi$
$983$	−1.60143e7	−0.528598	−0.264299	+	0.964441i	$0.585140\pi$
$984$	−7.42955e6	−0.244610
$985$	0	0
$986$	5.26993e6	0.172629
$987$	−3.43487e6	−0.112232
$988$	−2.47905e7	−0.807965
$989$	5.78055e6	0.187922
$990$	0	0
$991$	1.70581e7	0.551755	0.275878	−	0.961193i	$-0.411032\pi$
$991$	1.70581e7	0.551755	0.275878	+	0.961193i	$0.411032\pi$
$992$	−1.91076e6	−0.0616490
$993$	−5.70980e6	−0.183759
$994$	7.19200e6	0.230879
$995$	0	0
$996$	−5.96227e6	−0.190442
$997$	2.65006e7	0.844341	0.422171	−	0.906516i	$-0.361268\pi$
$997$	2.65006e7	0.844341	0.422171	+	0.906516i	$0.361268\pi$
$998$	−3.02269e7	−0.960653
$999$	2.47988e7	0.786172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.a.1.3		8
5.4	even	2		43.6.a.a.1.6	✓	8
15.14	odd	2		387.6.a.c.1.3		8
20.19	odd	2		688.6.a.e.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.a.1.6	✓	8	5.4	even	2
387.6.a.c.1.3		8	15.14	odd	2
688.6.a.e.1.3		8	20.19	odd	2
1075.6.a.a.1.3		8	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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)

\(f(q)\)	\(=\)	\(q-3.65705 q^{2} -7.84314 q^{3} -18.6260 q^{4} +28.6827 q^{6} +25.5214 q^{7} +185.142 q^{8} -181.485 q^{9} +O(q^{10})\)	\(q-3.65705 q^{2} -7.84314 q^{3} -18.6260 q^{4} +28.6827 q^{6} +25.5214 q^{7} +185.142 q^{8} -181.485 q^{9} +512.073 q^{11} +146.086 q^{12} -862.516 q^{13} -93.3331 q^{14} -81.0406 q^{16} +1521.49 q^{17} +663.700 q^{18} -1543.11 q^{19} -200.168 q^{21} -1872.67 q^{22} +3126.31 q^{23} -1452.09 q^{24} +3154.26 q^{26} +3329.30 q^{27} -475.362 q^{28} -947.120 q^{29} +339.499 q^{31} -5628.17 q^{32} -4016.26 q^{33} -5564.17 q^{34} +3380.34 q^{36} +7448.67 q^{37} +5643.25 q^{38} +6764.84 q^{39} +5116.45 q^{41} +732.024 q^{42} +1849.00 q^{43} -9537.86 q^{44} -11433.1 q^{46} +17159.9 q^{47} +635.613 q^{48} -16155.7 q^{49} -11933.3 q^{51} +16065.2 q^{52} -18090.4 q^{53} -12175.4 q^{54} +4725.08 q^{56} +12102.9 q^{57} +3463.66 q^{58} +17031.6 q^{59} +10664.9 q^{61} -1241.56 q^{62} -4631.76 q^{63} +23175.8 q^{64} +14687.6 q^{66} +8799.60 q^{67} -28339.3 q^{68} -24520.1 q^{69} -77057.3 q^{71} -33600.5 q^{72} +7964.75 q^{73} -27240.1 q^{74} +28742.0 q^{76} +13068.8 q^{77} -24739.3 q^{78} +68997.0 q^{79} +17988.8 q^{81} -18711.1 q^{82} -40813.4 q^{83} +3728.33 q^{84} -6761.88 q^{86} +7428.39 q^{87} +94806.0 q^{88} -83692.1 q^{89} -22012.6 q^{91} -58230.7 q^{92} -2662.74 q^{93} -62754.7 q^{94} +44142.5 q^{96} -159824. q^{97} +59082.0 q^{98} -92933.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9}+O(q^{10}) \) 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9	\( 8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9} - 532 q^{11} + 4195 q^{12} + 2492 q^{13} - 4240 q^{14} + 1882 q^{16} + 2534 q^{17} + 3711 q^{18} - 1678 q^{19} - 2256 q^{21} - 11502 q^{22} + 2488 q^{23} + 19953 q^{24} + 4586 q^{26} + 8960 q^{27} - 18640 q^{28} - 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 12852 q^{33} + 30007 q^{34} + 67969 q^{36} + 3772 q^{37} + 6559 q^{38} + 11120 q^{39} - 10698 q^{41} - 78698 q^{42} + 14792 q^{43} - 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 118727 q^{48} + 7188 q^{49} - 80246 q^{51} + 60736 q^{52} + 62352 q^{53} + 61026 q^{54} - 144528 q^{56} + 808 q^{57} - 52951 q^{58} - 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 61768 q^{63} + 153858 q^{64} - 48516 q^{66} - 27784 q^{67} - 40507 q^{68} - 93776 q^{69} - 9504 q^{71} + 186687 q^{72} - 14260 q^{73} - 15239 q^{74} + 1279 q^{76} + 218140 q^{77} - 264170 q^{78} + 160248 q^{79} + 161076 q^{81} + 47781 q^{82} + 77176 q^{83} + 16382 q^{84} + 22188 q^{86} - 268136 q^{87} - 129544 q^{88} - 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 123860 q^{93} + 248737 q^{94} + 950817 q^{96} - 144742 q^{97} + 292244 q^{98} + 239516 q^{99}+O(q^{100}) \) 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9 - 532 * q^11 + 4195 * q^12 + 2492 * q^13 - 4240 * q^14 + 1882 * q^16 + 2534 * q^17 + 3711 * q^18 - 1678 * q^19 - 2256 * q^21 - 11502 * q^22 + 2488 * q^23 + 19953 * q^24 + 4586 * q^26 + 8960 * q^27 - 18640 * q^28 - 4360 * q^29 + 5704 * q^31 + 18294 * q^32 + 12852 * q^33 + 30007 * q^34 + 67969 * q^36 + 3772 * q^37 + 6559 * q^38 + 11120 * q^39 - 10698 * q^41 - 78698 * q^42 + 14792 * q^43 - 356 * q^44 - 19389 * q^46 + 77864 * q^47 + 118727 * q^48 + 7188 * q^49 - 80246 * q^51 + 60736 * q^52 + 62352 * q^53 + 61026 * q^54 - 144528 * q^56 + 808 * q^57 - 52951 * q^58 - 26224 * q^59 - 82540 * q^61 + 9023 * q^62 + 61768 * q^63 + 153858 * q^64 - 48516 * q^66 - 27784 * q^67 - 40507 * q^68 - 93776 * q^69 - 9504 * q^71 + 186687 * q^72 - 14260 * q^73 - 15239 * q^74 + 1279 * q^76 + 218140 * q^77 - 264170 * q^78 + 160248 * q^79 + 161076 * q^81 + 47781 * q^82 + 77176 * q^83 + 16382 * q^84 + 22188 * q^86 - 268136 * q^87 - 129544 * q^88 - 265692 * q^89 + 401148 * q^91 - 190391 * q^92 + 123860 * q^93 + 248737 * q^94 + 950817 * q^96 - 144742 * q^97 + 292244 * q^98 + 239516 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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