LMFDB - Embedded newform 1075.6.a.l.1.34

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:	$52$
Twist minimal:	no (minimal twist has level 215)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.34
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)

$f(q)$	$=$	$q+4.25084 q^{2} -22.3022 q^{3} -13.9304 q^{4} -94.8031 q^{6} +47.8405 q^{7} -195.243 q^{8} +254.388 q^{9} +O(q^{10})$	\(q+4.25084 q^{2} -22.3022 q^{3} -13.9304 q^{4} -94.8031 q^{6} +47.8405 q^{7} -195.243 q^{8} +254.388 q^{9} +701.161 q^{11} +310.678 q^{12} +193.920 q^{13} +203.362 q^{14} -384.174 q^{16} +2303.10 q^{17} +1081.36 q^{18} +2642.67 q^{19} -1066.95 q^{21} +2980.52 q^{22} -285.167 q^{23} +4354.34 q^{24} +824.324 q^{26} -253.988 q^{27} -666.435 q^{28} -5881.89 q^{29} -2955.63 q^{31} +4614.70 q^{32} -15637.4 q^{33} +9790.12 q^{34} -3543.72 q^{36} +4082.95 q^{37} +11233.6 q^{38} -4324.85 q^{39} -12546.4 q^{41} -4535.43 q^{42} -1849.00 q^{43} -9767.42 q^{44} -1212.20 q^{46} -2588.76 q^{47} +8567.93 q^{48} -14518.3 q^{49} -51364.3 q^{51} -2701.38 q^{52} +18038.4 q^{53} -1079.66 q^{54} -9340.50 q^{56} -58937.4 q^{57} -25003.0 q^{58} +29779.6 q^{59} +29187.8 q^{61} -12563.9 q^{62} +12170.1 q^{63} +31909.9 q^{64} -66472.3 q^{66} +19453.7 q^{67} -32083.0 q^{68} +6359.84 q^{69} +50527.8 q^{71} -49667.5 q^{72} -18020.6 q^{73} +17356.0 q^{74} -36813.3 q^{76} +33543.9 q^{77} -18384.2 q^{78} +59531.0 q^{79} -56151.9 q^{81} -53332.9 q^{82} +10977.9 q^{83} +14863.0 q^{84} -7859.80 q^{86} +131179. q^{87} -136897. q^{88} -12708.0 q^{89} +9277.24 q^{91} +3972.47 q^{92} +65917.0 q^{93} -11004.4 q^{94} -102918. q^{96} -73202.4 q^{97} -61714.9 q^{98} +178367. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 52 q + 20 q^{2} + 54 q^{3} + 826 q^{4} - 162 q^{6} + 196 q^{7} + 960 q^{8} + 4098 q^{9}+O(q^{10}) $ 52 * q + 20 * q^2 + 54 * q^3 + 826 * q^4 - 162 * q^6 + 196 * q^7 + 960 * q^8 + 4098 * q^9	\( 52 q + 20 q^{2} + 54 q^{3} + 826 q^{4} - 162 q^{6} + 196 q^{7} + 960 q^{8} + 4098 q^{9} - 664 q^{11} - 523 q^{12} + 2704 q^{13} + 150 q^{14} + 13474 q^{16} + 7266 q^{17} + 4860 q^{18} - 1970 q^{19} + 800 q^{21} + 14477 q^{22} + 9522 q^{23} + 314 q^{24} + 5514 q^{26} + 22926 q^{27} + 9408 q^{28} - 7188 q^{29} - 11556 q^{31} + 48390 q^{32} + 26136 q^{33} + 16774 q^{34} + 51872 q^{36} + 42558 q^{37} + 46208 q^{38} + 4682 q^{39} - 7746 q^{41} + 174265 q^{42} - 96148 q^{43} - 48600 q^{44} + 16182 q^{46} + 87136 q^{47} - 2912 q^{48} + 142286 q^{49} - 3710 q^{51} + 146868 q^{52} + 127034 q^{53} - 49563 q^{54} - 2849 q^{56} + 101594 q^{57} + 9480 q^{58} - 55924 q^{59} + 73702 q^{61} + 186016 q^{62} + 50120 q^{63} + 157750 q^{64} + 58211 q^{66} + 131996 q^{67} + 298560 q^{68} + 128436 q^{69} - 56284 q^{71} + 343775 q^{72} + 128620 q^{73} - 17721 q^{74} - 170410 q^{76} + 448438 q^{77} + 237616 q^{78} + 106204 q^{79} + 478568 q^{81} + 249596 q^{82} + 348616 q^{83} - 131855 q^{84} - 36980 q^{86} + 267478 q^{87} + 525216 q^{88} + 80410 q^{89} + 226376 q^{91} + 581456 q^{92} + 902902 q^{93} + 180980 q^{94} + 38543 q^{96} + 316148 q^{97} + 295095 q^{98} + 68428 q^{99}+O(q^{100}) \) 52 * q + 20 * q^2 + 54 * q^3 + 826 * q^4 - 162 * q^6 + 196 * q^7 + 960 * q^8 + 4098 * q^9 - 664 * q^11 - 523 * q^12 + 2704 * q^13 + 150 * q^14 + 13474 * q^16 + 7266 * q^17 + 4860 * q^18 - 1970 * q^19 + 800 * q^21 + 14477 * q^22 + 9522 * q^23 + 314 * q^24 + 5514 * q^26 + 22926 * q^27 + 9408 * q^28 - 7188 * q^29 - 11556 * q^31 + 48390 * q^32 + 26136 * q^33 + 16774 * q^34 + 51872 * q^36 + 42558 * q^37 + 46208 * q^38 + 4682 * q^39 - 7746 * q^41 + 174265 * q^42 - 96148 * q^43 - 48600 * q^44 + 16182 * q^46 + 87136 * q^47 - 2912 * q^48 + 142286 * q^49 - 3710 * q^51 + 146868 * q^52 + 127034 * q^53 - 49563 * q^54 - 2849 * q^56 + 101594 * q^57 + 9480 * q^58 - 55924 * q^59 + 73702 * q^61 + 186016 * q^62 + 50120 * q^63 + 157750 * q^64 + 58211 * q^66 + 131996 * q^67 + 298560 * q^68 + 128436 * q^69 - 56284 * q^71 + 343775 * q^72 + 128620 * q^73 - 17721 * q^74 - 170410 * q^76 + 448438 * q^77 + 237616 * q^78 + 106204 * q^79 + 478568 * q^81 + 249596 * q^82 + 348616 * q^83 - 131855 * q^84 - 36980 * q^86 + 267478 * q^87 + 525216 * q^88 + 80410 * q^89 + 226376 * q^91 + 581456 * q^92 + 902902 * q^93 + 180980 * q^94 + 38543 * q^96 + 316148 * q^97 + 295095 * q^98 + 68428 * 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 $

$2$	4.25084	0.751450	0.375725	−	0.926731i	$-0.377394\pi$
$2$	4.25084	0.751450	0.375725	+	0.926731i	$0.377394\pi$
$3$	−22.3022	−1.43069	−0.715344	−	0.698773i	$-0.753730\pi$
$3$	−22.3022	−1.43069	−0.715344	+	0.698773i	$0.753730\pi$
$4$	−13.9304	−0.435323
$5$	0	0
$5$	0	0
$6$	−94.8031	−1.07509
$7$	47.8405	0.369021	0.184510	−	0.982831i	$-0.440930\pi$
$7$	47.8405	0.369021	0.184510	+	0.982831i	$0.440930\pi$
$8$	−195.243	−1.07857
$9$	254.388	1.04687
$10$	0	0
$11$	701.161	1.74717	0.873587	−	0.486668i	$-0.161788\pi$
$11$	701.161	1.74717	0.873587	+	0.486668i	$0.161788\pi$
$12$	310.678	0.622812
$13$	193.920	0.318247	0.159124	−	0.987259i	$-0.449133\pi$
$13$	193.920	0.318247	0.159124	+	0.987259i	$0.449133\pi$
$14$	203.362	0.277300
$15$	0	0
$16$	−384.174	−0.375170
$17$	2303.10	1.93282	0.966409	−	0.257009i	$-0.0827370\pi$
$17$	2303.10	1.93282	0.966409	+	0.257009i	$0.0827370\pi$
$18$	1081.36	0.786667
$19$	2642.67	1.67942	0.839710	−	0.543036i	$-0.182725\pi$
$19$	2642.67	1.67942	0.839710	+	0.543036i	$0.182725\pi$
$20$	0	0
$21$	−1066.95	−0.527953
$22$	2980.52	1.31291
$23$	−285.167	−0.112403	−0.0562017	−	0.998419i	$-0.517899\pi$
$23$	−285.167	−0.112403	−0.0562017	+	0.998419i	$0.517899\pi$
$24$	4354.34	1.54310
$25$	0	0
$26$	824.324	0.239147
$27$	−253.988	−0.0670509
$28$	−666.435	−0.160643
$29$	−5881.89	−1.29874	−0.649369	−	0.760473i	$-0.724967\pi$
$29$	−5881.89	−1.29874	−0.649369	+	0.760473i	$0.724967\pi$
$30$	0	0
$31$	−2955.63	−0.552390	−0.276195	−	0.961102i	$-0.589073\pi$
$31$	−2955.63	−0.552390	−0.276195	+	0.961102i	$0.589073\pi$
$32$	4614.70	0.796652
$33$	−15637.4	−2.49966
$34$	9790.12	1.45242
$35$	0	0
$36$	−3543.72	−0.455725
$37$	4082.95	0.490308	0.245154	−	0.969484i	$-0.421161\pi$
$37$	4082.95	0.490308	0.245154	+	0.969484i	$0.421161\pi$
$38$	11233.6	1.26200
$39$	−4324.85	−0.455312
$40$	0	0
$41$	−12546.4	−1.16563	−0.582815	−	0.812605i	$-0.698049\pi$
$41$	−12546.4	−1.16563	−0.582815	+	0.812605i	$0.698049\pi$
$42$	−4535.43	−0.396730
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	−9767.42	−0.760586
$45$	0	0
$46$	−1212.20	−0.0844654
$47$	−2588.76	−0.170941	−0.0854707	−	0.996341i	$-0.527239\pi$
$47$	−2588.76	−0.170941	−0.0854707	+	0.996341i	$0.527239\pi$
$48$	8567.93	0.536751
$49$	−14518.3	−0.863824
$50$	0	0
$51$	−51364.3	−2.76526
$52$	−2701.38	−0.138541
$53$	18038.4	0.882080	0.441040	−	0.897487i	$-0.354610\pi$
$53$	18038.4	0.882080	0.441040	+	0.897487i	$0.354610\pi$
$54$	−1079.66	−0.0503853
$55$	0	0
$56$	−9340.50	−0.398016
$57$	−58937.4	−2.40272
$58$	−25003.0	−0.975937
$59$	29779.6	1.11375	0.556877	−	0.830595i	$-0.312000\pi$
$59$	29779.6	1.11375	0.556877	+	0.830595i	$0.312000\pi$
$60$	0	0
$61$	29187.8	1.00433	0.502166	−	0.864771i	$-0.332537\pi$
$61$	29187.8	1.00433	0.502166	+	0.864771i	$0.332537\pi$
$62$	−12563.9	−0.415093
$63$	12170.1	0.386315
$64$	31909.9	0.973814
$65$	0	0
$66$	−66472.3	−1.87837
$67$	19453.7	0.529439	0.264719	−	0.964325i	$-0.414721\pi$
$67$	19453.7	0.529439	0.264719	+	0.964325i	$0.414721\pi$
$68$	−32083.0	−0.841401
$69$	6359.84	0.160814
$70$	0	0
$71$	50527.8	1.18955	0.594777	−	0.803891i	$-0.297240\pi$
$71$	50527.8	1.18955	0.594777	+	0.803891i	$0.297240\pi$
$72$	−49667.5	−1.12912
$73$	−18020.6	−0.395788	−0.197894	−	0.980223i	$-0.563410\pi$
$73$	−18020.6	−0.395788	−0.197894	+	0.980223i	$0.563410\pi$
$74$	17356.0	0.368442
$75$	0	0
$76$	−36813.3	−0.731091
$77$	33543.9	0.644743
$78$	−18384.2	−0.342144
$79$	59531.0	1.07319	0.536593	−	0.843841i	$-0.319711\pi$
$79$	59531.0	1.07319	0.536593	+	0.843841i	$0.319711\pi$
$80$	0	0
$81$	−56151.9	−0.950937
$82$	−53332.9	−0.875912
$83$	10977.9	0.174914	0.0874571	−	0.996168i	$-0.472126\pi$
$83$	10977.9	0.174914	0.0874571	+	0.996168i	$0.472126\pi$
$84$	14863.0	0.229830
$85$	0	0
$86$	−7859.80	−0.114595
$87$	131179.	1.85809
$88$	−136897.	−1.88445
$89$	−12708.0	−0.170060	−0.0850301	−	0.996378i	$-0.527099\pi$
$89$	−12708.0	−0.170060	−0.0850301	+	0.996378i	$0.527099\pi$
$90$	0	0
$91$	9277.24	0.117440
$92$	3972.47	0.0489318
$93$	65917.0	0.790297
$94$	−11004.4	−0.128454
$95$	0	0
$96$	−102918.	−1.13976
$97$	−73202.4	−0.789944	−0.394972	−	0.918693i	$-0.629246\pi$
$97$	−73202.4	−0.789944	−0.394972	+	0.918693i	$0.629246\pi$
$98$	−61714.9	−0.649120
$99$	178367.	1.82906
$100$	0	0
$101$	−67568.7	−0.659086	−0.329543	−	0.944141i	$-0.606895\pi$
$101$	−67568.7	−0.659086	−0.329543	+	0.944141i	$0.606895\pi$
$102$	−218341.	−2.07795
$103$	19718.4	0.183138	0.0915690	−	0.995799i	$-0.470812\pi$
$103$	19718.4	0.183138	0.0915690	+	0.995799i	$0.470812\pi$
$104$	−37861.5	−0.343253
$105$	0	0
$106$	76678.3	0.662839
$107$	−21994.4	−0.185718	−0.0928588	−	0.995679i	$-0.529601\pi$
$107$	−21994.4	−0.185718	−0.0928588	+	0.995679i	$0.529601\pi$
$108$	3538.15	0.0291888
$109$	−203248.	−1.63855	−0.819275	−	0.573401i	$-0.805624\pi$
$109$	−203248.	−1.63855	−0.819275	+	0.573401i	$0.805624\pi$
$110$	0	0
$111$	−91058.7	−0.701478
$112$	−18379.1	−0.138445
$113$	144316.	1.06321	0.531605	−	0.846992i	$-0.321589\pi$
$113$	144316.	1.06321	0.531605	+	0.846992i	$0.321589\pi$
$114$	−250534.	−1.80553
$115$	0	0
$116$	81936.8	0.565371
$117$	49331.1	0.333162
$118$	126588.	0.836929
$119$	110182.	0.713250
$120$	0	0
$121$	330576.	2.05262
$122$	124073.	0.754705
$123$	279813.	1.66765
$124$	41172.9	0.240468
$125$	0	0
$126$	51733.0	0.290296
$127$	40512.0	0.222882	0.111441	−	0.993771i	$-0.464453\pi$
$127$	40512.0	0.222882	0.111441	+	0.993771i	$0.464453\pi$
$128$	−12026.4	−0.0648800
$129$	41236.8	0.218178
$130$	0	0
$131$	−310188.	−1.57923	−0.789617	−	0.613600i	$-0.789721\pi$
$131$	−310188.	−1.57923	−0.789617	+	0.613600i	$0.789721\pi$
$132$	217835.	1.08816
$133$	126427.	0.619740
$134$	82694.7	0.397847
$135$	0	0
$136$	−449664.	−2.08469
$137$	134365.	0.611624	0.305812	−	0.952092i	$-0.401072\pi$
$137$	134365.	0.611624	0.305812	+	0.952092i	$0.401072\pi$
$138$	27034.7	0.120844
$139$	−23855.7	−0.104726	−0.0523630	−	0.998628i	$-0.516675\pi$
$139$	−23855.7	−0.104726	−0.0523630	+	0.998628i	$0.516675\pi$
$140$	0	0
$141$	57735.1	0.244564
$142$	214785.	0.893890
$143$	135969.	0.556033
$144$	−97729.5	−0.392753
$145$	0	0
$146$	−76602.7	−0.297414
$147$	323790.	1.23586
$148$	−56876.9	−0.213443
$149$	238475.	0.879989	0.439994	−	0.898001i	$-0.354980\pi$
$149$	238475.	0.879989	0.439994	+	0.898001i	$0.354980\pi$
$150$	0	0
$151$	137619.	0.491176	0.245588	−	0.969374i	$-0.421019\pi$
$151$	137619.	0.491176	0.245588	+	0.969374i	$0.421019\pi$
$152$	−515962.	−1.81138
$153$	585883.	2.02340
$154$	142590.	0.484492
$155$	0	0
$156$	60246.7	0.198208
$157$	390926.	1.26574	0.632872	−	0.774256i	$-0.281876\pi$
$157$	390926.	1.26574	0.632872	+	0.774256i	$0.281876\pi$
$158$	253057.	0.806446
$159$	−402296.	−1.26198
$160$	0	0
$161$	−13642.5	−0.0414791
$162$	−238693.	−0.714582
$163$	−614192.	−1.81065	−0.905326	−	0.424718i	$-0.860373\pi$
$163$	−614192.	−1.81065	−0.905326	+	0.424718i	$0.860373\pi$
$164$	174776.	0.507426
$165$	0	0
$166$	46665.4	0.131439
$167$	458729.	1.27281	0.636407	−	0.771353i	$-0.280420\pi$
$167$	458729.	1.27281	0.636407	+	0.771353i	$0.280420\pi$
$168$	208314.	0.569436
$169$	−333688.	−0.898719
$170$	0	0
$171$	672265.	1.75813
$172$	25757.2	0.0663862
$173$	−237648.	−0.603698	−0.301849	−	0.953356i	$-0.597604\pi$
$173$	−237648.	−0.603698	−0.301849	+	0.953356i	$0.597604\pi$
$174$	557621.	1.39626
$175$	0	0
$176$	−269368.	−0.655487
$177$	−664151.	−1.59343
$178$	−54019.7	−0.127792
$179$	−801343.	−1.86933	−0.934664	−	0.355531i	$-0.884300\pi$
$179$	−801343.	−1.86933	−0.934664	+	0.355531i	$0.884300\pi$
$180$	0	0
$181$	−205412.	−0.466047	−0.233024	−	0.972471i	$-0.574862\pi$
$181$	−205412.	−0.466047	−0.233024	+	0.972471i	$0.574862\pi$
$182$	39436.1	0.0882501
$183$	−650953.	−1.43688
$184$	55676.7	0.121235
$185$	0	0
$186$	280203.	0.593868
$187$	1.61485e6	3.37697
$188$	36062.3	0.0744148
$189$	−12150.9	−0.0247431
$190$	0	0
$191$	445785.	0.884183	0.442092	−	0.896970i	$-0.354237\pi$
$191$	445785.	0.884183	0.442092	+	0.896970i	$0.354237\pi$
$192$	−711662.	−1.39322
$193$	630031.	1.21750	0.608750	−	0.793362i	$-0.291671\pi$
$193$	630031.	1.21750	0.608750	+	0.793362i	$0.291671\pi$
$194$	−311172.	−0.593603
$195$	0	0
$196$	202245.	0.376043
$197$	−170869.	−0.313687	−0.156844	−	0.987623i	$-0.550132\pi$
$197$	−170869.	−0.313687	−0.156844	+	0.987623i	$0.550132\pi$
$198$	758211.	1.37444
$199$	−412381.	−0.738186	−0.369093	−	0.929392i	$-0.620332\pi$
$199$	−412381.	−0.738186	−0.369093	+	0.929392i	$0.620332\pi$
$200$	0	0
$201$	−433861.	−0.757462
$202$	−287224.	−0.495270
$203$	−281392.	−0.479261
$204$	715522.	1.20378
$205$	0	0
$206$	83819.7	0.137619
$207$	−72543.1	−0.117671
$208$	−74499.1	−0.119397
$209$	1.85294e6	2.93424
$210$	0	0
$211$	42367.2	0.0655124	0.0327562	−	0.999463i	$-0.489572\pi$
$211$	42367.2	0.0655124	0.0327562	+	0.999463i	$0.489572\pi$
$212$	−251281.	−0.383990
$213$	−1.12688e6	−1.70188
$214$	−93494.8	−0.139557
$215$	0	0
$216$	49589.4	0.0723193
$217$	−141399.	−0.203843
$218$	−863974.	−1.23129
$219$	401899.	0.566248
$220$	0	0
$221$	446618.	0.615114
$222$	−387076.	−0.527125
$223$	986475.	1.32838	0.664192	−	0.747562i	$-0.268776\pi$
$223$	986475.	1.32838	0.664192	+	0.747562i	$0.268776\pi$
$224$	220770.	0.293981
$225$	0	0
$226$	613466.	0.798949
$227$	449657.	0.579185	0.289592	−	0.957150i	$-0.406480\pi$
$227$	449657.	0.579185	0.289592	+	0.957150i	$0.406480\pi$
$228$	821019.	1.04596
$229$	−606237.	−0.763931	−0.381965	−	0.924177i	$-0.624753\pi$
$229$	−606237.	−0.763931	−0.381965	+	0.924177i	$0.624753\pi$
$230$	0	0
$231$	−748103.	−0.922426
$232$	1.14840e6	1.40078
$233$	7380.99	0.00890687	0.00445343	−	0.999990i	$-0.498582\pi$
$233$	7380.99	0.00890687	0.00445343	+	0.999990i	$0.498582\pi$
$234$	209699.	0.250355
$235$	0	0
$236$	−414840.	−0.484843
$237$	−1.32767e6	−1.53539
$238$	468364.	0.535971
$239$	−7015.49	−0.00794444	−0.00397222	−	0.999992i	$-0.501264\pi$
$239$	−7015.49	−0.00794444	−0.00397222	+	0.999992i	$0.501264\pi$
$240$	0	0
$241$	1.06601e6	1.18228	0.591139	−	0.806570i	$-0.298679\pi$
$241$	1.06601e6	1.18228	0.591139	+	0.806570i	$0.298679\pi$
$242$	1.40523e6	1.54244
$243$	1.31403e6	1.42754
$244$	−406597.	−0.437209
$245$	0	0
$246$	1.18944e6	1.25316
$247$	512468.	0.534471
$248$	577064.	0.595793
$249$	−244832.	−0.250248
$250$	0	0
$251$	525399.	0.526386	0.263193	−	0.964743i	$-0.415224\pi$
$251$	525399.	0.526386	0.263193	+	0.964743i	$0.415224\pi$
$252$	−169533.	−0.168172
$253$	−199948.	−0.196388
$254$	172210.	0.167484
$255$	0	0
$256$	−1.07224e6	−1.02257
$257$	−1.64593e6	−1.55446	−0.777229	−	0.629218i	$-0.783375\pi$
$257$	−1.64593e6	−1.55446	−0.777229	+	0.629218i	$0.783375\pi$
$258$	175291.	0.163950
$259$	195330.	0.180934
$260$	0	0
$261$	−1.49628e6	−1.35961
$262$	−1.31856e6	−1.18671
$263$	960478.	0.856245	0.428122	−	0.903721i	$-0.359175\pi$
$263$	960478.	0.856245	0.428122	+	0.903721i	$0.359175\pi$
$264$	3.05309e6	2.69607
$265$	0	0
$266$	537420.	0.465704
$267$	283417.	0.243303
$268$	−270997.	−0.230477
$269$	−521106.	−0.439082	−0.219541	−	0.975603i	$-0.570456\pi$
$269$	−521106.	−0.439082	−0.219541	+	0.975603i	$0.570456\pi$
$270$	0	0
$271$	1.24686e6	1.03132	0.515661	−	0.856792i	$-0.327546\pi$
$271$	1.24686e6	1.03132	0.515661	+	0.856792i	$0.327546\pi$
$272$	−884792.	−0.725135
$273$	−206903.	−0.168020
$274$	571164.	0.459605
$275$	0	0
$276$	−88594.9	−0.0700061
$277$	2.13768e6	1.67395	0.836976	−	0.547239i	$-0.184321\pi$
$277$	2.13768e6	1.67395	0.836976	+	0.547239i	$0.184321\pi$
$278$	−101407.	−0.0786963
$279$	−751878.	−0.578278
$280$	0	0
$281$	1.66423e6	1.25733	0.628663	−	0.777678i	$-0.283602\pi$
$281$	1.66423e6	1.25733	0.628663	+	0.777678i	$0.283602\pi$
$282$	245423.	0.183777
$283$	206169.	0.153023	0.0765116	−	0.997069i	$-0.475622\pi$
$283$	206169.	0.153023	0.0765116	+	0.997069i	$0.475622\pi$
$284$	−703869.	−0.517841
$285$	0	0
$286$	577984.	0.417831
$287$	−600227.	−0.430141
$288$	1.17393e6	0.833988
$289$	3.88442e6	2.73579
$290$	0	0
$291$	1.63258e6	1.13016
$292$	251033.	0.172296
$293$	1.04784e6	0.713061	0.356531	−	0.934284i	$-0.383960\pi$
$293$	1.04784e6	0.713061	0.356531	+	0.934284i	$0.383960\pi$
$294$	1.37638e6	0.928688
$295$	0	0
$296$	−797165.	−0.528834
$297$	−178087.	−0.117150
$298$	1.01372e6	0.661267
$299$	−55299.6	−0.0357721
$300$	0	0
$301$	−88457.1	−0.0562751
$302$	584998.	0.369094
$303$	1.50693e6	0.942946
$304$	−1.01525e6	−0.630068
$305$	0	0
$306$	2.49049e6	1.52048
$307$	3.02895e6	1.83420	0.917098	−	0.398661i	$-0.130525\pi$
$307$	3.02895e6	1.83420	0.917098	+	0.398661i	$0.130525\pi$
$308$	−467278.	−0.280672
$309$	−439763.	−0.262013
$310$	0	0
$311$	−1.35396e6	−0.793790	−0.396895	−	0.917864i	$-0.629912\pi$
$311$	−1.35396e6	−0.793790	−0.396895	+	0.917864i	$0.629912\pi$
$312$	844395.	0.491088
$313$	−360727.	−0.208122	−0.104061	−	0.994571i	$-0.533184\pi$
$313$	−360727.	−0.208122	−0.104061	+	0.994571i	$0.533184\pi$
$314$	1.66177e6	0.951143
$315$	0	0
$316$	−829288.	−0.467183
$317$	−2.19979e6	−1.22952	−0.614758	−	0.788716i	$-0.710746\pi$
$317$	−2.19979e6	−1.22952	−0.614758	+	0.788716i	$0.710746\pi$
$318$	−1.71010e6	−0.948315
$319$	−4.12415e6	−2.26912
$320$	0	0
$321$	490524.	0.265704
$322$	−57992.1	−0.0311695
$323$	6.08634e6	3.24601
$324$	782216.	0.413965
$325$	0	0
$326$	−2.61083e6	−1.36061
$327$	4.53287e6	2.34425
$328$	2.44960e6	1.25722
$329$	−123848.	−0.0630809
$330$	0	0
$331$	−3.63323e6	−1.82273	−0.911366	−	0.411596i	$-0.864972\pi$
$331$	−3.63323e6	−1.82273	−0.911366	+	0.411596i	$0.864972\pi$
$332$	−152926.	−0.0761443
$333$	1.03865e6	0.513287
$334$	1.94998e6	0.956456
$335$	0	0
$336$	409894.	0.198072
$337$	2.15763e6	1.03491	0.517455	−	0.855710i	$-0.326879\pi$
$337$	2.15763e6	1.03491	0.517455	+	0.855710i	$0.326879\pi$
$338$	−1.41845e6	−0.675342
$339$	−3.21857e6	−1.52112
$340$	0	0
$341$	−2.07237e6	−0.965120
$342$	2.85769e6	1.32114
$343$	−1.49862e6	−0.687789
$344$	361004.	0.164481
$345$	0	0
$346$	−1.01021e6	−0.453648
$347$	−3.29321e6	−1.46824	−0.734119	−	0.679021i	$-0.762405\pi$
$347$	−3.29321e6	−1.46824	−0.734119	+	0.679021i	$0.762405\pi$
$348$	−1.82737e6	−0.808870
$349$	−3.46699e6	−1.52366	−0.761832	−	0.647774i	$-0.775700\pi$
$349$	−3.46699e6	−1.52366	−0.761832	+	0.647774i	$0.775700\pi$
$350$	0	0
$351$	−49253.5	−0.0213388
$352$	3.23565e6	1.39189
$353$	1.94445e6	0.830539	0.415270	−	0.909698i	$-0.363687\pi$
$353$	1.94445e6	0.830539	0.415270	+	0.909698i	$0.363687\pi$
$354$	−2.82320e6	−1.19738
$355$	0	0
$356$	177027.	0.0740312
$357$	−2.45729e6	−1.02044
$358$	−3.40638e6	−1.40471
$359$	924734.	0.378687	0.189344	−	0.981911i	$-0.439364\pi$
$359$	924734.	0.378687	0.189344	+	0.981911i	$0.439364\pi$
$360$	0	0
$361$	4.50761e6	1.82045
$362$	−873175.	−0.350211
$363$	−7.37257e6	−2.93665
$364$	−129235.	−0.0511243
$365$	0	0
$366$	−2.76710e6	−1.07975
$367$	−4.26910e6	−1.65452	−0.827258	−	0.561822i	$-0.810101\pi$
$367$	−4.26910e6	−1.65452	−0.827258	+	0.561822i	$0.810101\pi$
$368$	109554.	0.0421703
$369$	−3.19167e6	−1.22026
$370$	0	0
$371$	862965.	0.325506
$372$	−918247.	−0.344035
$373$	3.75519e6	1.39753	0.698763	−	0.715353i	$-0.253734\pi$
$373$	3.75519e6	1.39753	0.698763	+	0.715353i	$0.253734\pi$
$374$	6.86445e6	2.53762
$375$	0	0
$376$	505436.	0.184373
$377$	−1.14062e6	−0.413320
$378$	−51651.7	−0.0185932
$379$	−773773.	−0.276704	−0.138352	−	0.990383i	$-0.544181\pi$
$379$	−773773.	−0.276704	−0.138352	+	0.990383i	$0.544181\pi$
$380$	0	0
$381$	−903506.	−0.318874
$382$	1.89496e6	0.664419
$383$	374047.	0.130296	0.0651478	−	0.997876i	$-0.479248\pi$
$383$	374047.	0.130296	0.0651478	+	0.997876i	$0.479248\pi$
$384$	268215.	0.0928229
$385$	0	0
$386$	2.67816e6	0.914890
$387$	−470364.	−0.159646
$388$	1.01974e6	0.343881
$389$	−4.57868e6	−1.53414	−0.767072	−	0.641561i	$-0.778287\pi$
$389$	−4.57868e6	−1.53414	−0.767072	+	0.641561i	$0.778287\pi$
$390$	0	0
$391$	−656768.	−0.217255
$392$	2.83459e6	0.931697
$393$	6.91787e6	2.25939
$394$	−726335.	−0.235720
$395$	0	0
$396$	−2.48472e6	−0.796231
$397$	−3.39821e6	−1.08212	−0.541059	−	0.840985i	$-0.681976\pi$
$397$	−3.39821e6	−1.08212	−0.541059	+	0.840985i	$0.681976\pi$
$398$	−1.75297e6	−0.554710
$399$	−2.81959e6	−0.886654
$400$	0	0
$401$	1.49216e6	0.463399	0.231699	−	0.972787i	$-0.425571\pi$
$401$	1.49216e6	0.463399	0.231699	+	0.972787i	$0.425571\pi$
$402$	−1.84427e6	−0.569194
$403$	−573156.	−0.175797
$404$	941255.	0.286915
$405$	0	0
$406$	−1.19615e6	−0.360141
$407$	2.86280e6	0.856654
$408$	1.00285e7	2.98253
$409$	−4.89305e6	−1.44634	−0.723171	−	0.690669i	$-0.757316\pi$
$409$	−4.89305e6	−1.44634	−0.723171	+	0.690669i	$0.757316\pi$
$410$	0	0
$411$	−2.99664e6	−0.875043
$412$	−274684.	−0.0797242
$413$	1.42467e6	0.410998
$414$	−308369.	−0.0884240
$415$	0	0
$416$	894884.	0.253532
$417$	532034.	0.149830
$418$	7.87655e6	2.20493
$419$	−2.55494e6	−0.710960	−0.355480	−	0.934684i	$-0.615683\pi$
$419$	−2.55494e6	−0.710960	−0.355480	+	0.934684i	$0.615683\pi$
$420$	0	0
$421$	3.70919e6	1.01994	0.509969	−	0.860193i	$-0.329657\pi$
$421$	3.70919e6	1.01994	0.509969	+	0.860193i	$0.329657\pi$
$422$	180096.	0.0492293
$423$	−658551.	−0.178953
$424$	−3.52186e6	−0.951388
$425$	0	0
$426$	−4.79019e6	−1.27888
$427$	1.39636e6	0.370619
$428$	306390.	0.0808472
$429$	−3.03242e6	−0.795510
$430$	0	0
$431$	−2.16153e6	−0.560490	−0.280245	−	0.959929i	$-0.590416\pi$
$431$	−2.16153e6	−0.560490	−0.280245	+	0.959929i	$0.590416\pi$
$432$	97575.7	0.0251555
$433$	−1.91644e6	−0.491220	−0.245610	−	0.969369i	$-0.578988\pi$
$433$	−1.91644e6	−0.491220	−0.245610	+	0.969369i	$0.578988\pi$
$434$	−601063.	−0.153178
$435$	0	0
$436$	2.83131e6	0.713299
$437$	−753602.	−0.188772
$438$	1.70841e6	0.425507
$439$	565750.	0.140108	0.0700541	−	0.997543i	$-0.477683\pi$
$439$	565750.	0.140108	0.0700541	+	0.997543i	$0.477683\pi$
$440$	0	0
$441$	−3.69329e6	−0.904308
$442$	1.89850e6	0.462227
$443$	2.45240e6	0.593721	0.296860	−	0.954921i	$-0.404060\pi$
$443$	2.45240e6	0.593721	0.296860	+	0.954921i	$0.404060\pi$
$444$	1.26848e6	0.305370
$445$	0	0
$446$	4.19335e6	0.998214
$447$	−5.31852e6	−1.25899
$448$	1.52659e6	0.359357
$449$	4.47734e6	1.04810	0.524051	−	0.851687i	$-0.324420\pi$
$449$	4.47734e6	1.04810	0.524051	+	0.851687i	$0.324420\pi$
$450$	0	0
$451$	−8.79707e6	−2.03656
$452$	−2.01038e6	−0.462840
$453$	−3.06922e6	−0.702719
$454$	1.91142e6	0.435228
$455$	0	0
$456$	1.15071e7	2.59151
$457$	629542.	0.141005	0.0705025	−	0.997512i	$-0.477540\pi$
$457$	629542.	0.141005	0.0705025	+	0.997512i	$0.477540\pi$
$458$	−2.57702e6	−0.574055
$459$	−584961.	−0.129597
$460$	0	0
$461$	1.63790e6	0.358950	0.179475	−	0.983763i	$-0.442560\pi$
$461$	1.63790e6	0.358950	0.179475	+	0.983763i	$0.442560\pi$
$462$	−3.18007e6	−0.693156
$463$	8.19560e6	1.77676	0.888379	−	0.459111i	$-0.151832\pi$
$463$	8.19560e6	1.77676	0.888379	+	0.459111i	$0.151832\pi$
$464$	2.25967e6	0.487248
$465$	0	0
$466$	31375.4	0.00669306
$467$	5.59534e6	1.18723	0.593615	−	0.804750i	$-0.297700\pi$
$467$	5.59534e6	1.18723	0.593615	+	0.804750i	$0.297700\pi$
$468$	−687199.	−0.145033
$469$	930676.	0.195374
$470$	0	0
$471$	−8.71852e6	−1.81088
$472$	−5.81425e6	−1.20126
$473$	−1.29645e6	−0.266441
$474$	−5.64372e6	−1.15377
$475$	0	0
$476$	−1.53487e6	−0.310494
$477$	4.58876e6	0.923420
$478$	−29821.7	−0.00596984
$479$	−2.34468e6	−0.466923	−0.233462	−	0.972366i	$-0.575005\pi$
$479$	−2.34468e6	−0.466923	−0.233462	+	0.972366i	$0.575005\pi$
$480$	0	0
$481$	791766.	0.156039
$482$	4.53145e6	0.888422
$483$	304258.	0.0593437
$484$	−4.60504e6	−0.893552
$485$	0	0
$486$	5.58573e6	1.07273
$487$	3.17254e6	0.606156	0.303078	−	0.952966i	$-0.401986\pi$
$487$	3.17254e6	0.606156	0.303078	+	0.952966i	$0.401986\pi$
$488$	−5.69871e6	−1.08325
$489$	1.36978e7	2.59048
$490$	0	0
$491$	5.47203e6	1.02434	0.512170	−	0.858884i	$-0.328842\pi$
$491$	5.47203e6	1.02434	0.512170	+	0.858884i	$0.328842\pi$
$492$	−3.89789e6	−0.725968
$493$	−1.35466e7	−2.51023
$494$	2.17842e6	0.401628
$495$	0	0
$496$	1.13548e6	0.207240
$497$	2.41727e6	0.438970
$498$	−1.04074e6	−0.188048
$499$	−5.98791e6	−1.07652	−0.538262	−	0.842778i	$-0.680919\pi$
$499$	−5.98791e6	−1.07652	−0.538262	+	0.842778i	$0.680919\pi$
$500$	0	0
$501$	−1.02307e7	−1.82100
$502$	2.23339e6	0.395553
$503$	−7.88456e6	−1.38950	−0.694748	−	0.719253i	$-0.744484\pi$
$503$	−7.88456e6	−1.38950	−0.694748	+	0.719253i	$0.744484\pi$
$504$	−2.37612e6	−0.416669
$505$	0	0
$506$	−849946.	−0.147576
$507$	7.44198e6	1.28579
$508$	−564346.	−0.0970256
$509$	−8.60147e6	−1.47156	−0.735781	−	0.677220i	$-0.763185\pi$
$509$	−8.60147e6	−1.47156	−0.735781	+	0.677220i	$0.763185\pi$
$510$	0	0
$511$	−862115.	−0.146054
$512$	−4.17308e6	−0.703528
$513$	−671208.	−0.112607
$514$	−6.99659e6	−1.16810
$515$	0	0
$516$	−574443.	−0.0949779
$517$	−1.81514e6	−0.298664
$518$	830317.	0.135963
$519$	5.30008e6	0.863703
$520$	0	0
$521$	1.60380e6	0.258855	0.129427	−	0.991589i	$-0.458686\pi$
$521$	1.60380e6	0.258855	0.129427	+	0.991589i	$0.458686\pi$
$522$	−6.36047e6	−1.02167
$523$	−1.99127e6	−0.318328	−0.159164	−	0.987252i	$-0.550880\pi$
$523$	−1.99127e6	−0.318328	−0.159164	+	0.987252i	$0.550880\pi$
$524$	4.32103e6	0.687478
$525$	0	0
$526$	4.08284e6	0.643425
$527$	−6.80711e6	−1.06767
$528$	6.00750e6	0.937797
$529$	−6.35502e6	−0.987365
$530$	0	0
$531$	7.57559e6	1.16595
$532$	−1.76117e6	−0.269787
$533$	−2.43301e6	−0.370958
$534$	1.20476e6	0.182830
$535$	0	0
$536$	−3.79820e6	−0.571039
$537$	1.78717e7	2.67443
$538$	−2.21514e6	−0.329948
$539$	−1.01797e7	−1.50925
$540$	0	0
$541$	−6.28605e6	−0.923389	−0.461695	−	0.887039i	$-0.652758\pi$
$541$	−6.28605e6	−0.923389	−0.461695	+	0.887039i	$0.652758\pi$
$542$	5.30021e6	0.774987
$543$	4.58115e6	0.666768
$544$	1.06281e7	1.53978
$545$	0	0
$546$	−879512.	−0.126258
$547$	−4.73580e6	−0.676745	−0.338373	−	0.941012i	$-0.609876\pi$
$547$	−4.73580e6	−0.676745	−0.338373	+	0.941012i	$0.609876\pi$
$548$	−1.87175e6	−0.266254
$549$	7.42505e6	1.05140
$550$	0	0
$551$	−1.55439e7	−2.18113
$552$	−1.24171e6	−0.173450
$553$	2.84799e6	0.396028
$554$	9.08694e6	1.25789
$555$	0	0
$556$	332318.	0.0455897
$557$	1.01785e7	1.39010	0.695050	−	0.718961i	$-0.255382\pi$
$557$	1.01785e7	1.39010	0.695050	+	0.718961i	$0.255382\pi$
$558$	−3.19611e6	−0.434547
$559$	−358559.	−0.0485323
$560$	0	0
$561$	−3.60146e7	−4.83139
$562$	7.07438e6	0.944817
$563$	−5.30532e6	−0.705409	−0.352704	−	0.935735i	$-0.614738\pi$
$563$	−5.30532e6	−0.705409	−0.352704	+	0.935735i	$0.614738\pi$
$564$	−804270.	−0.106464
$565$	0	0
$566$	876391.	0.114989
$567$	−2.68633e6	−0.350915
$568$	−9.86517e6	−1.28302
$569$	1.60351e6	0.207630	0.103815	−	0.994597i	$-0.466895\pi$
$569$	1.60351e6	0.207630	0.103815	+	0.994597i	$0.466895\pi$
$570$	0	0
$571$	694475.	0.0891387	0.0445693	−	0.999006i	$-0.485808\pi$
$571$	694475.	0.0891387	0.0445693	+	0.999006i	$0.485808\pi$
$572$	−1.89410e6	−0.242054
$573$	−9.94199e6	−1.26499
$574$	−2.55147e6	−0.323229
$575$	0	0
$576$	8.11752e6	1.01945
$577$	−5.76221e6	−0.720526	−0.360263	−	0.932851i	$-0.617313\pi$
$577$	−5.76221e6	−0.720526	−0.360263	+	0.932851i	$0.617313\pi$
$578$	1.65121e7	2.05581
$579$	−1.40511e7	−1.74186
$580$	0	0
$581$	525189.	0.0645469
$582$	6.93982e6	0.849260
$583$	1.26478e7	1.54115
$584$	3.51839e6	0.426886
$585$	0	0
$586$	4.45421e6	0.535829
$587$	5.44914e6	0.652728	0.326364	−	0.945244i	$-0.394176\pi$
$587$	5.44914e6	0.652728	0.326364	+	0.945244i	$0.394176\pi$
$588$	−4.51051e6	−0.538000
$589$	−7.81075e6	−0.927694
$590$	0	0
$591$	3.81075e6	0.448788
$592$	−1.56856e6	−0.183949
$593$	2.94363e6	0.343753	0.171877	−	0.985118i	$-0.445017\pi$
$593$	2.94363e6	0.343753	0.171877	+	0.985118i	$0.445017\pi$
$594$	−757018.	−0.0880319
$595$	0	0
$596$	−3.32204e6	−0.383080
$597$	9.19700e6	1.05611
$598$	−235070.	−0.0268809
$599$	3.53077e6	0.402071	0.201036	−	0.979584i	$-0.435569\pi$
$599$	3.53077e6	0.402071	0.201036	+	0.979584i	$0.435569\pi$
$600$	0	0
$601$	1.50169e7	1.69588	0.847938	−	0.530096i	$-0.177844\pi$
$601$	1.50169e7	1.69588	0.847938	+	0.530096i	$0.177844\pi$
$602$	−376017.	−0.0422879
$603$	4.94880e6	0.554252
$604$	−1.91709e6	−0.213820
$605$	0	0
$606$	6.40572e6	0.708576
$607$	2.77312e6	0.305490	0.152745	−	0.988266i	$-0.451189\pi$
$607$	2.77312e6	0.305490	0.152745	+	0.988266i	$0.451189\pi$
$608$	1.21951e7	1.33791
$609$	6.27567e6	0.685673
$610$	0	0
$611$	−502013.	−0.0544016
$612$	−8.16155e6	−0.880834
$613$	7.16772e6	0.770424	0.385212	−	0.922828i	$-0.374128\pi$
$613$	7.16772e6	0.770424	0.385212	+	0.922828i	$0.374128\pi$
$614$	1.28756e7	1.37831
$615$	0	0
$616$	−6.54920e6	−0.695403
$617$	4.21461e6	0.445702	0.222851	−	0.974852i	$-0.428464\pi$
$617$	4.21461e6	0.445702	0.222851	+	0.974852i	$0.428464\pi$
$618$	−1.86936e6	−0.196890
$619$	8.04715e6	0.844142	0.422071	−	0.906563i	$-0.361303\pi$
$619$	8.04715e6	0.844142	0.422071	+	0.906563i	$0.361303\pi$
$620$	0	0
$621$	72429.0	0.00753674
$622$	−5.75548e6	−0.596493
$623$	−607957.	−0.0627557
$624$	1.66150e6	0.170820
$625$	0	0
$626$	−1.53339e6	−0.156393
$627$	−4.13246e7	−4.19798
$628$	−5.44574e6	−0.551008
$629$	9.40344e6	0.947677
$630$	0	0
$631$	−5.12480e6	−0.512394	−0.256197	−	0.966625i	$-0.582470\pi$
$631$	−5.12480e6	−0.512394	−0.256197	+	0.966625i	$0.582470\pi$
$632$	−1.16230e7	−1.15751
$633$	−944882.	−0.0937278
$634$	−9.35098e6	−0.923919
$635$	0	0
$636$	5.60412e6	0.549370
$637$	−2.81539e6	−0.274910
$638$	−1.75311e7	−1.70513
$639$	1.28537e7	1.24530
$640$	0	0
$641$	−5.81462e6	−0.558954	−0.279477	−	0.960152i	$-0.590161\pi$
$641$	−5.81462e6	−0.558954	−0.279477	+	0.960152i	$0.590161\pi$
$642$	2.08514e6	0.199663
$643$	1.60463e7	1.53055	0.765274	−	0.643704i	$-0.222603\pi$
$643$	1.60463e7	1.53055	0.765274	+	0.643704i	$0.222603\pi$
$644$	190045.	0.0180568
$645$	0	0
$646$	2.58721e7	2.43921
$647$	9.81116e6	0.921424	0.460712	−	0.887550i	$-0.347594\pi$
$647$	9.81116e6	0.921424	0.460712	+	0.887550i	$0.347594\pi$
$648$	1.09632e7	1.02566
$649$	2.08803e7	1.94592
$650$	0	0
$651$	3.15350e6	0.291636
$652$	8.55591e6	0.788219
$653$	1.79332e7	1.64579	0.822897	−	0.568190i	$-0.192356\pi$
$653$	1.79332e7	1.64579	0.822897	+	0.568190i	$0.192356\pi$
$654$	1.92685e7	1.76159
$655$	0	0
$656$	4.82001e6	0.437309
$657$	−4.58423e6	−0.414337
$658$	−526456.	−0.0474021
$659$	1.29854e7	1.16477	0.582387	−	0.812912i	$-0.302119\pi$
$659$	1.29854e7	1.16477	0.582387	+	0.812912i	$0.302119\pi$
$660$	0	0
$661$	1.41202e7	1.25700	0.628501	−	0.777809i	$-0.283669\pi$
$661$	1.41202e7	1.25700	0.628501	+	0.777809i	$0.283669\pi$
$662$	−1.54443e7	−1.36969
$663$	−9.96057e6	−0.880036
$664$	−2.14336e6	−0.188658
$665$	0	0
$666$	4.41516e6	0.385710
$667$	1.67732e6	0.145982
$668$	−6.39025e6	−0.554086
$669$	−2.20006e7	−1.90050
$670$	0	0
$671$	2.04654e7	1.75474
$672$	−4.92365e6	−0.420595
$673$	−2.17468e7	−1.85079	−0.925395	−	0.379005i	$-0.876266\pi$
$673$	−2.17468e7	−1.85079	−0.925395	+	0.379005i	$0.876266\pi$
$674$	9.17175e6	0.777683
$675$	0	0
$676$	4.64839e6	0.391233
$677$	1.04409e7	0.875523	0.437761	−	0.899091i	$-0.355771\pi$
$677$	1.04409e7	0.875523	0.437761	+	0.899091i	$0.355771\pi$
$678$	−1.36816e7	−1.14305
$679$	−3.50204e6	−0.291505
$680$	0	0
$681$	−1.00284e7	−0.828632
$682$	−8.80932e6	−0.725239
$683$	−632493.	−0.0518805	−0.0259402	−	0.999663i	$-0.508258\pi$
$683$	−632493.	−0.0518805	−0.0259402	+	0.999663i	$0.508258\pi$
$684$	−9.36489e6	−0.765354
$685$	0	0
$686$	−6.37038e6	−0.516839
$687$	1.35204e7	1.09295
$688$	710338.	0.0572129
$689$	3.49801e6	0.280720
$690$	0	0
$691$	−2.97090e6	−0.236697	−0.118349	−	0.992972i	$-0.537760\pi$
$691$	−2.97090e6	−0.236697	−0.118349	+	0.992972i	$0.537760\pi$
$692$	3.31052e6	0.262804
$693$	8.53318e6	0.674960
$694$	−1.39989e7	−1.10331
$695$	0	0
$696$	−2.56117e7	−2.00408
$697$	−2.88957e7	−2.25295
$698$	−1.47376e7	−1.14496
$699$	−164612.	−0.0127429
$700$	0	0
$701$	−1.97783e7	−1.52017	−0.760086	−	0.649822i	$-0.774843\pi$
$701$	−1.97783e7	−1.52017	−0.760086	+	0.649822i	$0.774843\pi$
$702$	−209369.	−0.0160350
$703$	1.07899e7	0.823433
$704$	2.23740e7	1.70142
$705$	0	0
$706$	8.26555e6	0.624108
$707$	−3.23252e6	−0.243216
$708$	9.25186e6	0.693659
$709$	1.42591e7	1.06531	0.532657	−	0.846331i	$-0.321193\pi$
$709$	1.42591e7	1.06531	0.532657	+	0.846331i	$0.321193\pi$
$710$	0	0
$711$	1.51440e7	1.12348
$712$	2.48114e6	0.183422
$713$	842846.	0.0620904
$714$	−1.04456e7	−0.766807
$715$	0	0
$716$	1.11630e7	0.813763
$717$	156461.	0.0113660
$718$	3.93090e6	0.284564
$719$	1.65663e7	1.19509	0.597547	−	0.801834i	$-0.296142\pi$
$719$	1.65663e7	1.19509	0.597547	+	0.801834i	$0.296142\pi$
$720$	0	0
$721$	943337.	0.0675816
$722$	1.91611e7	1.36798
$723$	−2.37744e7	−1.69147
$724$	2.86147e6	0.202881
$725$	0	0
$726$	−3.13396e7	−2.20675
$727$	−2.20916e7	−1.55022	−0.775108	−	0.631829i	$-0.782304\pi$
$727$	−2.20916e7	−1.55022	−0.775108	+	0.631829i	$0.782304\pi$
$728$	−1.81131e6	−0.126667
$729$	−1.56609e7	−1.09143
$730$	0	0
$731$	−4.25844e6	−0.294752
$732$	9.06800e6	0.625510
$733$	−1.55837e7	−1.07130	−0.535649	−	0.844441i	$-0.679933\pi$
$733$	−1.55837e7	−1.07130	−0.535649	+	0.844441i	$0.679933\pi$
$734$	−1.81473e7	−1.24329
$735$	0	0
$736$	−1.31596e6	−0.0895463
$737$	1.36402e7	0.925022
$738$	−1.35673e7	−0.916962
$739$	2.66982e7	1.79834	0.899169	−	0.437602i	$-0.144172\pi$
$739$	2.66982e7	1.79834	0.899169	+	0.437602i	$0.144172\pi$
$740$	0	0
$741$	−1.14292e7	−0.764661
$742$	3.66833e6	0.244601
$743$	1.24089e7	0.824637	0.412318	−	0.911040i	$-0.364719\pi$
$743$	1.24089e7	0.824637	0.412318	+	0.911040i	$0.364719\pi$
$744$	−1.28698e7	−0.852393
$745$	0	0
$746$	1.59627e7	1.05017
$747$	2.79266e6	0.183112
$748$	−2.24954e7	−1.47007
$749$	−1.05222e6	−0.0685336
$750$	0	0
$751$	2.66187e7	1.72221	0.861106	−	0.508426i	$-0.169772\pi$
$751$	2.66187e7	1.72221	0.861106	+	0.508426i	$0.169772\pi$
$752$	994534.	0.0641321
$753$	−1.17175e7	−0.753094
$754$	−4.84858e6	−0.310589
$755$	0	0
$756$	169267.	0.0107713
$757$	2.34018e7	1.48426	0.742131	−	0.670255i	$-0.233815\pi$
$757$	2.34018e7	1.48426	0.742131	+	0.670255i	$0.233815\pi$
$758$	−3.28919e6	−0.207929
$759$	4.45928e6	0.280970
$760$	0	0
$761$	2.40376e6	0.150463	0.0752315	−	0.997166i	$-0.476030\pi$
$761$	2.40376e6	0.150463	0.0752315	+	0.997166i	$0.476030\pi$
$762$	−3.84066e6	−0.239618
$763$	−9.72347e6	−0.604658
$764$	−6.20994e6	−0.384906
$765$	0	0
$766$	1.59002e6	0.0979106
$767$	5.77487e6	0.354449
$768$	2.39133e7	1.46297
$769$	2.33330e7	1.42284	0.711419	−	0.702768i	$-0.248053\pi$
$769$	2.33330e7	1.42284	0.711419	+	0.702768i	$0.248053\pi$
$770$	0	0
$771$	3.67079e7	2.22394
$772$	−8.77656e6	−0.530006
$773$	3.21124e7	1.93296	0.966481	−	0.256738i	$-0.0826476\pi$
$773$	3.21124e7	1.93296	0.966481	+	0.256738i	$0.0826476\pi$
$774$	−1.99944e6	−0.119966
$775$	0	0
$776$	1.42922e7	0.852012
$777$	−4.35629e6	−0.258860
$778$	−1.94632e7	−1.15283
$779$	−3.31561e7	−1.95758
$780$	0	0
$781$	3.54281e7	2.07836
$782$	−2.79182e6	−0.163256
$783$	1.49393e6	0.0870815
$784$	5.57755e6	0.324081
$785$	0	0
$786$	2.94068e7	1.69782
$787$	1.21395e7	0.698656	0.349328	−	0.937000i	$-0.386410\pi$
$787$	1.21395e7	0.698656	0.349328	+	0.937000i	$0.386410\pi$
$788$	2.38026e6	0.136555
$789$	−2.14208e7	−1.22502
$790$	0	0
$791$	6.90416e6	0.392347
$792$	−3.48249e7	−1.97277
$793$	5.66011e6	0.319626
$794$	−1.44453e7	−0.813157
$795$	0	0
$796$	5.74461e6	0.321350
$797$	1.44554e6	0.0806089	0.0403045	−	0.999187i	$-0.487167\pi$
$797$	1.44554e6	0.0806089	0.0403045	+	0.999187i	$0.487167\pi$
$798$	−1.19856e7	−0.666276
$799$	−5.96218e6	−0.330398
$800$	0	0
$801$	−3.23277e6	−0.178030
$802$	6.34294e6	0.348221
$803$	−1.26353e7	−0.691510
$804$	6.04384e6	0.329741
$805$	0	0
$806$	−2.43640e6	−0.132102
$807$	1.16218e7	0.628189
$808$	1.31923e7	0.710872
$809$	−7.38343e6	−0.396631	−0.198316	−	0.980138i	$-0.563547\pi$
$809$	−7.38343e6	−0.396631	−0.198316	+	0.980138i	$0.563547\pi$
$810$	0	0
$811$	3.43516e7	1.83398	0.916991	−	0.398907i	$-0.130610\pi$
$811$	3.43516e7	1.83398	0.916991	+	0.398907i	$0.130610\pi$
$812$	3.91989e6	0.208634
$813$	−2.78077e7	−1.47550
$814$	1.21693e7	0.643732
$815$	0	0
$816$	1.97328e7	1.03744
$817$	−4.88630e6	−0.256109
$818$	−2.07996e7	−1.08685
$819$	2.36002e6	0.122944
$820$	0	0
$821$	−1.04157e7	−0.539302	−0.269651	−	0.962958i	$-0.586908\pi$
$821$	−1.04157e7	−0.539302	−0.269651	+	0.962958i	$0.586908\pi$
$822$	−1.27382e7	−0.657551
$823$	−2.03481e7	−1.04718	−0.523592	−	0.851969i	$-0.675409\pi$
$823$	−2.03481e7	−1.04718	−0.523592	+	0.851969i	$0.675409\pi$
$824$	−3.84987e6	−0.197528
$825$	0	0
$826$	6.05605e6	0.308844
$827$	−2.68396e7	−1.36462	−0.682311	−	0.731062i	$-0.739025\pi$
$827$	−2.68396e7	−1.36462	−0.682311	+	0.731062i	$0.739025\pi$
$828$	1.01055e6	0.0512250
$829$	2.76988e7	1.39983	0.699914	−	0.714227i	$-0.253222\pi$
$829$	2.76988e7	1.39983	0.699914	+	0.714227i	$0.253222\pi$
$830$	0	0
$831$	−4.76750e7	−2.39490
$832$	6.18798e6	0.309914
$833$	−3.34371e7	−1.66961
$834$	2.26159e6	0.112590
$835$	0	0
$836$	−2.58121e7	−1.27734
$837$	750695.	0.0370382
$838$	−1.08606e7	−0.534250
$839$	−1.79904e7	−0.882340	−0.441170	−	0.897424i	$-0.645436\pi$
$839$	−1.79904e7	−0.882340	−0.441170	+	0.897424i	$0.645436\pi$
$840$	0	0
$841$	1.40854e7	0.686722
$842$	1.57672e7	0.766433
$843$	−3.71160e7	−1.79884
$844$	−590190.	−0.0285191
$845$	0	0
$846$	−2.79939e6	−0.134474
$847$	1.58149e7	0.757457
$848$	−6.92988e6	−0.330930
$849$	−4.59802e6	−0.218928
$850$	0	0
$851$	−1.16432e6	−0.0551123
$852$	1.56978e7	0.740868
$853$	3.75285e7	1.76599	0.882995	−	0.469383i	$-0.155524\pi$
$853$	3.75285e7	1.76599	0.882995	+	0.469383i	$0.155524\pi$
$854$	5.93570e6	0.278502
$855$	0	0
$856$	4.29425e6	0.200310
$857$	−2.53395e7	−1.17854	−0.589272	−	0.807935i	$-0.700585\pi$
$857$	−2.53395e7	−1.17854	−0.589272	+	0.807935i	$0.700585\pi$
$858$	−1.28903e7	−0.597786
$859$	3.29625e7	1.52418	0.762092	−	0.647469i	$-0.224173\pi$
$859$	3.29625e7	1.52418	0.762092	+	0.647469i	$0.224173\pi$
$860$	0	0
$861$	1.33864e7	0.615397
$862$	−9.18831e6	−0.421180
$863$	−1.34000e7	−0.612460	−0.306230	−	0.951958i	$-0.599068\pi$
$863$	−1.34000e7	−0.612460	−0.306230	+	0.951958i	$0.599068\pi$
$864$	−1.17208e6	−0.0534162
$865$	0	0
$866$	−8.14648e6	−0.369127
$867$	−8.66312e7	−3.91405
$868$	1.96973e6	0.0887377
$869$	4.17408e7	1.87504
$870$	0	0
$871$	3.77247e6	0.168493
$872$	3.96826e7	1.76730
$873$	−1.86219e7	−0.826965
$874$	−3.20344e6	−0.141853
$875$	0	0
$876$	−5.59860e6	−0.246501
$877$	−1.34154e7	−0.588987	−0.294493	−	0.955653i	$-0.595151\pi$
$877$	−1.34154e7	−0.588987	−0.294493	+	0.955653i	$0.595151\pi$
$878$	2.40491e6	0.105284
$879$	−2.33692e7	−1.02017
$880$	0	0
$881$	−3.21619e7	−1.39605	−0.698026	−	0.716072i	$-0.745938\pi$
$881$	−3.21619e7	−1.39605	−0.698026	+	0.716072i	$0.745938\pi$
$882$	−1.56996e7	−0.679542
$883$	−8.73621e6	−0.377069	−0.188535	−	0.982067i	$-0.560374\pi$
$883$	−8.73621e6	−0.377069	−0.188535	+	0.982067i	$0.560374\pi$
$884$	−6.22155e6	−0.267774
$885$	0	0
$886$	1.04248e7	0.446151
$887$	2.96975e7	1.26739	0.633695	−	0.773583i	$-0.281537\pi$
$887$	2.96975e7	1.26739	0.633695	+	0.773583i	$0.281537\pi$
$888$	1.77785e7	0.756595
$889$	1.93811e6	0.0822479
$890$	0	0
$891$	−3.93715e7	−1.66145
$892$	−1.37419e7	−0.578277
$893$	−6.84124e6	−0.287082
$894$	−2.26082e7	−0.946067
$895$	0	0
$896$	−575349.	−0.0239420
$897$	1.23330e6	0.0511786
$898$	1.90324e7	0.787596
$899$	1.73847e7	0.717410
$900$	0	0
$901$	4.15443e7	1.70490
$902$	−3.73949e7	−1.53037
$903$	1.97279e6	0.0805121
$904$	−2.81767e7	−1.14675
$905$	0	0
$906$	−1.30467e7	−0.528058
$907$	1.49256e7	0.602440	0.301220	−	0.953555i	$-0.402606\pi$
$907$	1.49256e7	0.602440	0.301220	+	0.953555i	$0.402606\pi$
$908$	−6.26389e6	−0.252133
$909$	−1.71887e7	−0.689975
$910$	0	0
$911$	1.90413e7	0.760154	0.380077	−	0.924955i	$-0.375897\pi$
$911$	1.90413e7	0.760154	0.380077	+	0.924955i	$0.375897\pi$
$912$	2.26422e7	0.901430
$913$	7.69729e6	0.305605
$914$	2.67608e6	0.105958
$915$	0	0
$916$	8.44510e6	0.332557
$917$	−1.48395e7	−0.582770
$918$	−2.48658e6	−0.0973857
$919$	347656.	0.0135788	0.00678939	−	0.999977i	$-0.497839\pi$
$919$	347656.	0.0135788	0.00678939	+	0.999977i	$0.497839\pi$
$920$	0	0
$921$	−6.75522e7	−2.62416
$922$	6.96243e6	0.269733
$923$	9.79836e6	0.378572
$924$	1.04213e7	0.401554
$925$	0	0
$926$	3.48382e7	1.33514
$927$	5.01613e6	0.191721
$928$	−2.71432e7	−1.03464
$929$	1.96862e6	0.0748380	0.0374190	−	0.999300i	$-0.488086\pi$
$929$	1.96862e6	0.0748380	0.0374190	+	0.999300i	$0.488086\pi$
$930$	0	0
$931$	−3.83671e7	−1.45072
$932$	−102820.	−0.00387737
$933$	3.01964e7	1.13567
$934$	2.37849e7	0.892143
$935$	0	0
$936$	−9.63153e6	−0.359340
$937$	−1.07301e7	−0.399261	−0.199630	−	0.979871i	$-0.563974\pi$
$937$	−1.07301e7	−0.399261	−0.199630	+	0.979871i	$0.563974\pi$
$938$	3.95616e6	0.146814
$939$	8.04500e6	0.297757
$940$	0	0
$941$	−4.10531e7	−1.51138	−0.755688	−	0.654932i	$-0.772697\pi$
$941$	−4.10531e7	−1.51138	−0.755688	+	0.654932i	$0.772697\pi$
$942$	−3.70611e7	−1.36079
$943$	3.57782e6	0.131021
$944$	−1.14406e7	−0.417847
$945$	0	0
$946$	−5.51099e6	−0.200217
$947$	−3.62736e7	−1.31436	−0.657182	−	0.753732i	$-0.728252\pi$
$947$	−3.62736e7	−1.31436	−0.657182	+	0.753732i	$0.728252\pi$
$948$	1.84949e7	0.668393
$949$	−3.49456e6	−0.125958
$950$	0	0
$951$	4.90603e7	1.75905
$952$	−2.15121e7	−0.769292
$953$	−4.31225e7	−1.53806	−0.769028	−	0.639215i	$-0.779259\pi$
$953$	−4.31225e7	−1.53806	−0.769028	+	0.639215i	$0.779259\pi$
$954$	1.95061e7	0.693903
$955$	0	0
$956$	97728.2	0.00345840
$957$	9.19777e7	3.24640
$958$	−9.96688e6	−0.350869
$959$	6.42809e6	0.225702
$960$	0	0
$961$	−1.98934e7	−0.694866
$962$	3.36567e6	0.117256
$963$	−5.59513e6	−0.194421
$964$	−1.48499e7	−0.514673
$965$	0	0
$966$	1.29335e6	0.0445938
$967$	1.33992e7	0.460801	0.230400	−	0.973096i	$-0.425996\pi$
$967$	1.33992e7	0.460801	0.230400	+	0.973096i	$0.425996\pi$
$968$	−6.45425e7	−2.21390
$969$	−1.35739e8	−4.64403
$970$	0	0
$971$	−2.39219e7	−0.814229	−0.407115	−	0.913377i	$-0.633465\pi$
$971$	−2.39219e7	−0.814229	−0.407115	+	0.913377i	$0.633465\pi$
$972$	−1.83049e7	−0.621444
$973$	−1.14127e6	−0.0386460
$974$	1.34859e7	0.455495
$975$	0	0
$976$	−1.12132e7	−0.376795
$977$	−7.03883e6	−0.235919	−0.117960	−	0.993018i	$-0.537635\pi$
$977$	−7.03883e6	−0.235919	−0.117960	+	0.993018i	$0.537635\pi$
$978$	5.82273e7	1.94661
$979$	−8.91036e6	−0.297125
$980$	0	0
$981$	−5.17039e7	−1.71534
$982$	2.32607e7	0.769740
$983$	−2.90533e7	−0.958984	−0.479492	−	0.877546i	$-0.659179\pi$
$983$	−2.90533e7	−0.958984	−0.479492	+	0.877546i	$0.659179\pi$
$984$	−5.46314e7	−1.79868
$985$	0	0
$986$	−5.75844e7	−1.88631
$987$	2.76207e6	0.0902490
$988$	−7.13885e6	−0.232668
$989$	527273.	0.0171413
$990$	0	0
$991$	−4.78303e7	−1.54710	−0.773552	−	0.633733i	$-0.781522\pi$
$991$	−4.78303e7	−1.54710	−0.773552	+	0.633733i	$0.781522\pi$
$992$	−1.36393e7	−0.440062
$993$	8.10291e7	2.60776
$994$	1.02754e7	0.329864
$995$	0	0
$996$	3.41060e6	0.108939
$997$	3.83079e7	1.22053	0.610267	−	0.792196i	$-0.291062\pi$
$997$	3.83079e7	1.22053	0.610267	+	0.792196i	$0.291062\pi$
$998$	−2.54536e7	−0.808954
$999$	−1.03702e6	−0.0328756

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.l.1.34		52
5.2	odd	4		215.6.b.a.44.69	yes	104
5.3	odd	4		215.6.b.a.44.36	✓	104
5.4	even	2		1075.6.a.k.1.19		52

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.6.b.a.44.36	✓	104	5.3	odd	4
215.6.b.a.44.69	yes	104	5.2	odd	4
1075.6.a.k.1.19		52	5.4	even	2
1075.6.a.l.1.34		52	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)

\(f(q)\)	\(=\)	\(q+4.25084 q^{2} -22.3022 q^{3} -13.9304 q^{4} -94.8031 q^{6} +47.8405 q^{7} -195.243 q^{8} +254.388 q^{9} +O(q^{10})\)	\(q+4.25084 q^{2} -22.3022 q^{3} -13.9304 q^{4} -94.8031 q^{6} +47.8405 q^{7} -195.243 q^{8} +254.388 q^{9} +701.161 q^{11} +310.678 q^{12} +193.920 q^{13} +203.362 q^{14} -384.174 q^{16} +2303.10 q^{17} +1081.36 q^{18} +2642.67 q^{19} -1066.95 q^{21} +2980.52 q^{22} -285.167 q^{23} +4354.34 q^{24} +824.324 q^{26} -253.988 q^{27} -666.435 q^{28} -5881.89 q^{29} -2955.63 q^{31} +4614.70 q^{32} -15637.4 q^{33} +9790.12 q^{34} -3543.72 q^{36} +4082.95 q^{37} +11233.6 q^{38} -4324.85 q^{39} -12546.4 q^{41} -4535.43 q^{42} -1849.00 q^{43} -9767.42 q^{44} -1212.20 q^{46} -2588.76 q^{47} +8567.93 q^{48} -14518.3 q^{49} -51364.3 q^{51} -2701.38 q^{52} +18038.4 q^{53} -1079.66 q^{54} -9340.50 q^{56} -58937.4 q^{57} -25003.0 q^{58} +29779.6 q^{59} +29187.8 q^{61} -12563.9 q^{62} +12170.1 q^{63} +31909.9 q^{64} -66472.3 q^{66} +19453.7 q^{67} -32083.0 q^{68} +6359.84 q^{69} +50527.8 q^{71} -49667.5 q^{72} -18020.6 q^{73} +17356.0 q^{74} -36813.3 q^{76} +33543.9 q^{77} -18384.2 q^{78} +59531.0 q^{79} -56151.9 q^{81} -53332.9 q^{82} +10977.9 q^{83} +14863.0 q^{84} -7859.80 q^{86} +131179. q^{87} -136897. q^{88} -12708.0 q^{89} +9277.24 q^{91} +3972.47 q^{92} +65917.0 q^{93} -11004.4 q^{94} -102918. q^{96} -73202.4 q^{97} -61714.9 q^{98} +178367. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 52 q + 20 q^{2} + 54 q^{3} + 826 q^{4} - 162 q^{6} + 196 q^{7} + 960 q^{8} + 4098 q^{9}+O(q^{10}) \) 52 * q + 20 * q^2 + 54 * q^3 + 826 * q^4 - 162 * q^6 + 196 * q^7 + 960 * q^8 + 4098 * q^9	\( 52 q + 20 q^{2} + 54 q^{3} + 826 q^{4} - 162 q^{6} + 196 q^{7} + 960 q^{8} + 4098 q^{9} - 664 q^{11} - 523 q^{12} + 2704 q^{13} + 150 q^{14} + 13474 q^{16} + 7266 q^{17} + 4860 q^{18} - 1970 q^{19} + 800 q^{21} + 14477 q^{22} + 9522 q^{23} + 314 q^{24} + 5514 q^{26} + 22926 q^{27} + 9408 q^{28} - 7188 q^{29} - 11556 q^{31} + 48390 q^{32} + 26136 q^{33} + 16774 q^{34} + 51872 q^{36} + 42558 q^{37} + 46208 q^{38} + 4682 q^{39} - 7746 q^{41} + 174265 q^{42} - 96148 q^{43} - 48600 q^{44} + 16182 q^{46} + 87136 q^{47} - 2912 q^{48} + 142286 q^{49} - 3710 q^{51} + 146868 q^{52} + 127034 q^{53} - 49563 q^{54} - 2849 q^{56} + 101594 q^{57} + 9480 q^{58} - 55924 q^{59} + 73702 q^{61} + 186016 q^{62} + 50120 q^{63} + 157750 q^{64} + 58211 q^{66} + 131996 q^{67} + 298560 q^{68} + 128436 q^{69} - 56284 q^{71} + 343775 q^{72} + 128620 q^{73} - 17721 q^{74} - 170410 q^{76} + 448438 q^{77} + 237616 q^{78} + 106204 q^{79} + 478568 q^{81} + 249596 q^{82} + 348616 q^{83} - 131855 q^{84} - 36980 q^{86} + 267478 q^{87} + 525216 q^{88} + 80410 q^{89} + 226376 q^{91} + 581456 q^{92} + 902902 q^{93} + 180980 q^{94} + 38543 q^{96} + 316148 q^{97} + 295095 q^{98} + 68428 q^{99}+O(q^{100}) \) 52 * q + 20 * q^2 + 54 * q^3 + 826 * q^4 - 162 * q^6 + 196 * q^7 + 960 * q^8 + 4098 * q^9 - 664 * q^11 - 523 * q^12 + 2704 * q^13 + 150 * q^14 + 13474 * q^16 + 7266 * q^17 + 4860 * q^18 - 1970 * q^19 + 800 * q^21 + 14477 * q^22 + 9522 * q^23 + 314 * q^24 + 5514 * q^26 + 22926 * q^27 + 9408 * q^28 - 7188 * q^29 - 11556 * q^31 + 48390 * q^32 + 26136 * q^33 + 16774 * q^34 + 51872 * q^36 + 42558 * q^37 + 46208 * q^38 + 4682 * q^39 - 7746 * q^41 + 174265 * q^42 - 96148 * q^43 - 48600 * q^44 + 16182 * q^46 + 87136 * q^47 - 2912 * q^48 + 142286 * q^49 - 3710 * q^51 + 146868 * q^52 + 127034 * q^53 - 49563 * q^54 - 2849 * q^56 + 101594 * q^57 + 9480 * q^58 - 55924 * q^59 + 73702 * q^61 + 186016 * q^62 + 50120 * q^63 + 157750 * q^64 + 58211 * q^66 + 131996 * q^67 + 298560 * q^68 + 128436 * q^69 - 56284 * q^71 + 343775 * q^72 + 128620 * q^73 - 17721 * q^74 - 170410 * q^76 + 448438 * q^77 + 237616 * q^78 + 106204 * q^79 + 478568 * q^81 + 249596 * q^82 + 348616 * q^83 - 131855 * q^84 - 36980 * q^86 + 267478 * q^87 + 525216 * q^88 + 80410 * q^89 + 226376 * q^91 + 581456 * q^92 + 902902 * q^93 + 180980 * q^94 + 38543 * q^96 + 316148 * q^97 + 295095 * q^98 + 68428 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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