LMFDB - Embedded newform 1075.6.a.c.1.6

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:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 6 x^{12} - 251 x^{11} + 1196 x^{10} + 24182 x^{9} - 82500 x^{8} - 1103006 x^{7} + \cdots + 375746432 $ x^13 - 6x^12 - 251x^11 + 1196x^10 + 24182x^9 - 82500x^8 - 1103006x^7 + 2204256x^6 + 23945205x^5 - 13321606x^4 - 217943943x^3 - 142431932x^2 + 413097652x + 375746432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 215)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.55454$ 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)

$f(q)$	$=$	$q-0.554544 q^{2} +16.4196 q^{3} -31.6925 q^{4} -9.10541 q^{6} -14.6642 q^{7} +35.3203 q^{8} +26.6038 q^{9} +O(q^{10})$	\(q-0.554544 q^{2} +16.4196 q^{3} -31.6925 q^{4} -9.10541 q^{6} -14.6642 q^{7} +35.3203 q^{8} +26.6038 q^{9} +193.584 q^{11} -520.378 q^{12} -301.970 q^{13} +8.13197 q^{14} +994.573 q^{16} -745.431 q^{17} -14.7530 q^{18} +496.498 q^{19} -240.781 q^{21} -107.351 q^{22} +2098.96 q^{23} +579.946 q^{24} +167.456 q^{26} -3553.14 q^{27} +464.746 q^{28} +4293.63 q^{29} -4522.98 q^{31} -1681.78 q^{32} +3178.58 q^{33} +413.374 q^{34} -843.142 q^{36} +14232.5 q^{37} -275.330 q^{38} -4958.23 q^{39} -823.013 q^{41} +133.524 q^{42} -1849.00 q^{43} -6135.17 q^{44} -1163.96 q^{46} -4797.74 q^{47} +16330.5 q^{48} -16592.0 q^{49} -12239.7 q^{51} +9570.18 q^{52} +39201.4 q^{53} +1970.38 q^{54} -517.945 q^{56} +8152.30 q^{57} -2381.01 q^{58} -23854.7 q^{59} -20456.9 q^{61} +2508.19 q^{62} -390.125 q^{63} -30893.7 q^{64} -1762.66 q^{66} -39866.3 q^{67} +23624.5 q^{68} +34464.1 q^{69} -902.284 q^{71} +939.656 q^{72} -27145.3 q^{73} -7892.53 q^{74} -15735.2 q^{76} -2838.77 q^{77} +2749.56 q^{78} +98353.1 q^{79} -64806.0 q^{81} +456.397 q^{82} +80094.4 q^{83} +7630.95 q^{84} +1025.35 q^{86} +70499.8 q^{87} +6837.46 q^{88} -93722.9 q^{89} +4428.16 q^{91} -66521.1 q^{92} -74265.5 q^{93} +2660.56 q^{94} -27614.3 q^{96} -98962.9 q^{97} +9200.98 q^{98} +5150.09 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 7 q^{2} + 16 q^{3} + 123 q^{4} - 40 q^{6} + 372 q^{7} + 15 q^{8} + 107 q^{9}+O(q^{10}) $ 13 * q + 7 * q^2 + 16 * q^3 + 123 * q^4 - 40 * q^6 + 372 * q^7 + 15 * q^8 + 107 * q^9	\( 13 q + 7 q^{2} + 16 q^{3} + 123 q^{4} - 40 q^{6} + 372 q^{7} + 15 q^{8} + 107 q^{9} - 1039 q^{11} + 2668 q^{12} + 1791 q^{13} - 1389 q^{14} - 1789 q^{16} + 365 q^{17} - 293 q^{18} - 4200 q^{19} - 3452 q^{21} + 2302 q^{22} + 1315 q^{23} - 2784 q^{24} - 2753 q^{26} + 1078 q^{27} - 8067 q^{28} - 14052 q^{29} - 4479 q^{31} + 6527 q^{32} - 8626 q^{33} - 9432 q^{34} - 5709 q^{36} + 25156 q^{37} + 2391 q^{38} - 7456 q^{39} - 32227 q^{41} - 630 q^{42} - 24037 q^{43} - 52652 q^{44} - 28700 q^{46} + 28390 q^{47} + 15504 q^{48} - 48915 q^{49} - 50744 q^{51} + 32261 q^{52} + 66177 q^{53} - 81742 q^{54} - 98379 q^{56} + 126280 q^{57} + 72353 q^{58} - 103588 q^{59} - 105406 q^{61} + 120123 q^{62} + 100962 q^{63} - 148957 q^{64} - 124146 q^{66} + 139241 q^{67} + 151020 q^{68} - 166714 q^{69} - 72726 q^{71} - 74361 q^{72} + 140142 q^{73} - 153906 q^{74} - 30223 q^{76} + 74050 q^{77} + 18812 q^{78} - 41076 q^{79} - 219547 q^{81} - 16483 q^{82} - 106623 q^{83} + 59732 q^{84} - 12943 q^{86} + 41238 q^{87} - 23748 q^{88} - 180646 q^{89} - 181694 q^{91} - 193132 q^{92} - 137950 q^{93} - 125382 q^{94} + 185312 q^{96} + 119291 q^{97} + 141778 q^{98} - 14335 q^{99}+O(q^{100}) \) 13 * q + 7 * q^2 + 16 * q^3 + 123 * q^4 - 40 * q^6 + 372 * q^7 + 15 * q^8 + 107 * q^9 - 1039 * q^11 + 2668 * q^12 + 1791 * q^13 - 1389 * q^14 - 1789 * q^16 + 365 * q^17 - 293 * q^18 - 4200 * q^19 - 3452 * q^21 + 2302 * q^22 + 1315 * q^23 - 2784 * q^24 - 2753 * q^26 + 1078 * q^27 - 8067 * q^28 - 14052 * q^29 - 4479 * q^31 + 6527 * q^32 - 8626 * q^33 - 9432 * q^34 - 5709 * q^36 + 25156 * q^37 + 2391 * q^38 - 7456 * q^39 - 32227 * q^41 - 630 * q^42 - 24037 * q^43 - 52652 * q^44 - 28700 * q^46 + 28390 * q^47 + 15504 * q^48 - 48915 * q^49 - 50744 * q^51 + 32261 * q^52 + 66177 * q^53 - 81742 * q^54 - 98379 * q^56 + 126280 * q^57 + 72353 * q^58 - 103588 * q^59 - 105406 * q^61 + 120123 * q^62 + 100962 * q^63 - 148957 * q^64 - 124146 * q^66 + 139241 * q^67 + 151020 * q^68 - 166714 * q^69 - 72726 * q^71 - 74361 * q^72 + 140142 * q^73 - 153906 * q^74 - 30223 * q^76 + 74050 * q^77 + 18812 * q^78 - 41076 * q^79 - 219547 * q^81 - 16483 * q^82 - 106623 * q^83 + 59732 * q^84 - 12943 * q^86 + 41238 * q^87 - 23748 * q^88 - 180646 * q^89 - 181694 * q^91 - 193132 * q^92 - 137950 * q^93 - 125382 * q^94 + 185312 * q^96 + 119291 * q^97 + 141778 * q^98 - 14335 * 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 $

$2$	−0.554544	−0.0980305	−0.0490153	−	0.998798i	$-0.515608\pi$
$2$	−0.554544	−0.0980305	−0.0490153	+	0.998798i	$0.515608\pi$
$3$	16.4196	1.05332	0.526659	−	0.850076i	$-0.323444\pi$
$3$	16.4196	1.05332	0.526659	+	0.850076i	$0.323444\pi$
$4$	−31.6925	−0.990390
$5$	0	0
$5$	0	0
$6$	−9.10541	−0.103257
$7$	−14.6642	−0.113113	−0.0565567	−	0.998399i	$-0.518012\pi$
$7$	−14.6642	−0.113113	−0.0565567	+	0.998399i	$0.518012\pi$
$8$	35.3203	0.195119
$9$	26.6038	0.109481
$10$	0	0
$11$	193.584	0.482379	0.241190	−	0.970478i	$-0.422462\pi$
$11$	193.584	0.482379	0.241190	+	0.970478i	$0.422462\pi$
$12$	−520.378	−1.04320
$13$	−301.970	−0.495571	−0.247785	−	0.968815i	$-0.579703\pi$
$13$	−301.970	−0.495571	−0.247785	+	0.968815i	$0.579703\pi$
$14$	8.13197	0.0110886
$15$	0	0
$16$	994.573	0.971262
$17$	−745.431	−0.625583	−0.312791	−	0.949822i	$-0.601264\pi$
$17$	−745.431	−0.625583	−0.312791	+	0.949822i	$0.601264\pi$
$18$	−14.7530	−0.0107325
$19$	496.498	0.315525	0.157762	−	0.987477i	$-0.449572\pi$
$19$	496.498	0.315525	0.157762	+	0.987477i	$0.449572\pi$
$20$	0	0
$21$	−240.781	−0.119145
$22$	−107.351	−0.0472879
$23$	2098.96	0.827340	0.413670	−	0.910427i	$-0.364247\pi$
$23$	2098.96	0.827340	0.413670	+	0.910427i	$0.364247\pi$
$24$	579.946	0.205523
$25$	0	0
$26$	167.456	0.0485811
$27$	−3553.14	−0.938001
$28$	464.746	0.112026
$29$	4293.63	0.948047	0.474023	−	0.880512i	$-0.342801\pi$
$29$	4293.63	0.948047	0.474023	+	0.880512i	$0.342801\pi$
$30$	0	0
$31$	−4522.98	−0.845318	−0.422659	−	0.906289i	$-0.638903\pi$
$31$	−4522.98	−0.845318	−0.422659	+	0.906289i	$0.638903\pi$
$32$	−1681.78	−0.290332
$33$	3178.58	0.508099
$34$	413.374	0.0613262
$35$	0	0
$36$	−843.142	−0.108429
$37$	14232.5	1.70913	0.854566	−	0.519344i	$-0.173824\pi$
$37$	14232.5	1.70913	0.854566	+	0.519344i	$0.173824\pi$
$38$	−275.330	−0.0309310
$39$	−4958.23	−0.521994
$40$	0	0
$41$	−823.013	−0.0764623	−0.0382311	−	0.999269i	$-0.512172\pi$
$41$	−823.013	−0.0764623	−0.0382311	+	0.999269i	$0.512172\pi$
$42$	133.524	0.0116798
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	−6135.17	−0.477743
$45$	0	0
$46$	−1163.96	−0.0811045
$47$	−4797.74	−0.316805	−0.158402	−	0.987375i	$-0.550634\pi$
$47$	−4797.74	−0.316805	−0.158402	+	0.987375i	$0.550634\pi$
$48$	16330.5	1.02305
$49$	−16592.0	−0.987205
$50$	0	0
$51$	−12239.7	−0.658938
$52$	9570.18	0.490808
$53$	39201.4	1.91695	0.958476	−	0.285172i	$-0.0920506\pi$
$53$	39201.4	1.91695	0.958476	+	0.285172i	$0.0920506\pi$
$54$	1970.38	0.0919527
$55$	0	0
$56$	−517.945	−0.0220706
$57$	8152.30	0.332348
$58$	−2381.01	−0.0929375
$59$	−23854.7	−0.892164	−0.446082	−	0.894992i	$-0.647181\pi$
$59$	−23854.7	−0.892164	−0.446082	+	0.894992i	$0.647181\pi$
$60$	0	0
$61$	−20456.9	−0.703908	−0.351954	−	0.936017i	$-0.614483\pi$
$61$	−20456.9	−0.703908	−0.351954	+	0.936017i	$0.614483\pi$
$62$	2508.19	0.0828670
$63$	−390.125	−0.0123838
$64$	−30893.7	−0.942801
$65$	0	0
$66$	−1762.66	−0.0498092
$67$	−39866.3	−1.08497	−0.542487	−	0.840064i	$-0.682517\pi$
$67$	−39866.3	−1.08497	−0.542487	+	0.840064i	$0.682517\pi$
$68$	23624.5	0.619571
$69$	34464.1	0.871452
$70$	0	0
$71$	−902.284	−0.0212421	−0.0106211	−	0.999944i	$-0.503381\pi$
$71$	−902.284	−0.0212421	−0.0106211	+	0.999944i	$0.503381\pi$
$72$	939.656	0.0213618
$73$	−27145.3	−0.596194	−0.298097	−	0.954536i	$-0.596352\pi$
$73$	−27145.3	−0.596194	−0.298097	+	0.954536i	$0.596352\pi$
$74$	−7892.53	−0.167547
$75$	0	0
$76$	−15735.2	−0.312492
$77$	−2838.77	−0.0545636
$78$	2749.56	0.0511714
$79$	98353.1	1.77305	0.886523	−	0.462684i	$-0.153113\pi$
$79$	98353.1	1.77305	0.886523	+	0.462684i	$0.153113\pi$
$80$	0	0
$81$	−64806.0	−1.09749
$82$	456.397	0.00749564
$83$	80094.4	1.27617	0.638083	−	0.769968i	$-0.279728\pi$
$83$	80094.4	1.27617	0.638083	+	0.769968i	$0.279728\pi$
$84$	7630.95	0.118000
$85$	0	0
$86$	1025.35	0.0149495
$87$	70499.8	0.998596
$88$	6837.46	0.0941213
$89$	−93722.9	−1.25421	−0.627106	−	0.778934i	$-0.715761\pi$
$89$	−93722.9	−1.25421	−0.627106	+	0.778934i	$0.715761\pi$
$90$	0	0
$91$	4428.16	0.0560557
$92$	−66521.1	−0.819389
$93$	−74265.5	−0.890389
$94$	2660.56	0.0310565
$95$	0	0
$96$	−27614.3	−0.305813
$97$	−98962.9	−1.06793	−0.533966	−	0.845506i	$-0.679299\pi$
$97$	−98962.9	−1.06793	−0.533966	+	0.845506i	$0.679299\pi$
$98$	9200.98	0.0967763
$99$	5150.09	0.0528113
$100$	0	0
$101$	7310.91	0.0713129	0.0356564	−	0.999364i	$-0.488648\pi$
$101$	7310.91	0.0713129	0.0356564	+	0.999364i	$0.488648\pi$
$102$	6787.45	0.0645961
$103$	136241.	1.26536	0.632681	−	0.774413i	$-0.281955\pi$
$103$	136241.	1.26536	0.632681	+	0.774413i	$0.281955\pi$
$104$	−10665.7	−0.0966953
$105$	0	0
$106$	−21738.9	−0.187920
$107$	−126567.	−1.06871	−0.534356	−	0.845259i	$-0.679446\pi$
$107$	−126567.	−1.06871	−0.534356	+	0.845259i	$0.679446\pi$
$108$	112608.	0.928987
$109$	36606.3	0.295114	0.147557	−	0.989054i	$-0.452859\pi$
$109$	36606.3	0.295114	0.147557	+	0.989054i	$0.452859\pi$
$110$	0	0
$111$	233691.	1.80026
$112$	−14584.6	−0.109863
$113$	44380.8	0.326963	0.163481	−	0.986546i	$-0.447728\pi$
$113$	44380.8	0.326963	0.163481	+	0.986546i	$0.447728\pi$
$114$	−4520.81	−0.0325803
$115$	0	0
$116$	−136076.	−0.938936
$117$	−8033.57	−0.0542555
$118$	13228.5	0.0874593
$119$	10931.2	0.0707618
$120$	0	0
$121$	−123576.	−0.767310
$122$	11344.3	0.0690045
$123$	−13513.6	−0.0805391
$124$	143344.	0.837194
$125$	0	0
$126$	216.342	0.00121399
$127$	54795.0	0.301461	0.150731	−	0.988575i	$-0.451837\pi$
$127$	54795.0	0.301461	0.150731	+	0.988575i	$0.451837\pi$
$128$	70949.0	0.382756
$129$	−30359.9	−0.160630
$130$	0	0
$131$	−186554.	−0.949786	−0.474893	−	0.880044i	$-0.657513\pi$
$131$	−186554.	−0.949786	−0.474893	+	0.880044i	$0.657513\pi$
$132$	−100737.	−0.503216
$133$	−7280.76	−0.0356901
$134$	22107.7	0.106361
$135$	0	0
$136$	−26328.8	−0.122063
$137$	−143733.	−0.654269	−0.327134	−	0.944978i	$-0.606083\pi$
$137$	−143733.	−0.654269	−0.327134	+	0.944978i	$0.606083\pi$
$138$	−19111.9	−0.0854289
$139$	−310110.	−1.36138	−0.680688	−	0.732573i	$-0.738319\pi$
$139$	−310110.	−1.36138	−0.680688	+	0.732573i	$0.738319\pi$
$140$	0	0
$141$	−78777.0	−0.333697
$142$	500.357	0.00208237
$143$	−58456.7	−0.239053
$144$	26459.5	0.106335
$145$	0	0
$146$	15053.3	0.0584452
$147$	−272434.	−1.03984
$148$	−451062.	−1.69271
$149$	−330792.	−1.22065	−0.610323	−	0.792153i	$-0.708960\pi$
$149$	−330792.	−1.22065	−0.610323	+	0.792153i	$0.708960\pi$
$150$	0	0
$151$	228755.	0.816447	0.408223	−	0.912882i	$-0.366148\pi$
$151$	228755.	0.816447	0.408223	+	0.912882i	$0.366148\pi$
$152$	17536.5	0.0615648
$153$	−19831.3	−0.0684893
$154$	1574.22	0.00534889
$155$	0	0
$156$	157139.	0.516978
$157$	−490780.	−1.58905	−0.794525	−	0.607231i	$-0.792280\pi$
$157$	−490780.	−1.58905	−0.794525	+	0.607231i	$0.792280\pi$
$158$	−54541.1	−0.173813
$159$	643671.	2.01916
$160$	0	0
$161$	−30779.6	−0.0935832
$162$	35937.8	0.107588
$163$	281808.	0.830777	0.415389	−	0.909644i	$-0.363646\pi$
$163$	281808.	0.830777	0.415389	+	0.909644i	$0.363646\pi$
$164$	26083.3	0.0757275
$165$	0	0
$166$	−44415.9	−0.125103
$167$	−87652.5	−0.243205	−0.121603	−	0.992579i	$-0.538803\pi$
$167$	−87652.5	−0.243205	−0.121603	+	0.992579i	$0.538803\pi$
$168$	−8504.46	−0.0232474
$169$	−280107.	−0.754410
$170$	0	0
$171$	13208.7	0.0345439
$172$	58599.4	0.151033
$173$	−169083.	−0.429522	−0.214761	−	0.976667i	$-0.568897\pi$
$173$	−169083.	−0.429522	−0.214761	+	0.976667i	$0.568897\pi$
$174$	−39095.3	−0.0978929
$175$	0	0
$176$	192534.	0.468517
$177$	−391686.	−0.939733
$178$	51973.5	0.122951
$179$	−168630.	−0.393371	−0.196686	−	0.980467i	$-0.563018\pi$
$179$	−168630.	−0.393371	−0.196686	+	0.980467i	$0.563018\pi$
$180$	0	0
$181$	−335049.	−0.760172	−0.380086	−	0.924951i	$-0.624106\pi$
$181$	−335049.	−0.760172	−0.380086	+	0.924951i	$0.624106\pi$
$182$	−2455.61	−0.00549517
$183$	−335895.	−0.741440
$184$	74135.8	0.161430
$185$	0	0
$186$	41183.5	0.0872853
$187$	−144304.	−0.301768
$188$	152052.	0.313760
$189$	52104.1	0.106100
$190$	0	0
$191$	70043.7	0.138927	0.0694633	−	0.997585i	$-0.477871\pi$
$191$	70043.7	0.138927	0.0694633	+	0.997585i	$0.477871\pi$
$192$	−507263.	−0.993070
$193$	−821092.	−1.58671	−0.793357	−	0.608757i	$-0.791668\pi$
$193$	−821092.	−1.58671	−0.793357	+	0.608757i	$0.791668\pi$
$194$	54879.4	0.104690
$195$	0	0
$196$	525840.	0.977718
$197$	949590.	1.74329	0.871647	−	0.490134i	$-0.163052\pi$
$197$	949590.	1.74329	0.871647	+	0.490134i	$0.163052\pi$
$198$	−2855.95	−0.00517712
$199$	144439.	0.258555	0.129278	−	0.991608i	$-0.458734\pi$
$199$	144439.	0.258555	0.129278	+	0.991608i	$0.458734\pi$
$200$	0	0
$201$	−654590.	−1.14282
$202$	−4054.22	−0.00699084
$203$	−62962.8	−0.107237
$204$	387906.	0.652606
$205$	0	0
$206$	−75551.6	−0.124044
$207$	55840.3	0.0905778
$208$	−300331.	−0.481329
$209$	96114.2	0.152202
$210$	0	0
$211$	−942390.	−1.45722	−0.728609	−	0.684930i	$-0.759833\pi$
$211$	−942390.	−1.45722	−0.728609	+	0.684930i	$0.759833\pi$
$212$	−1.24239e6	−1.89853
$213$	−14815.2	−0.0223747
$214$	70187.0	0.104766
$215$	0	0
$216$	−125498.	−0.183022
$217$	66326.0	0.0956168
$218$	−20299.8	−0.0289302
$219$	−445716.	−0.627982
$220$	0	0
$221$	225098.	0.310021
$222$	−129592.	−0.176480
$223$	307727.	0.414384	0.207192	−	0.978300i	$-0.433567\pi$
$223$	307727.	0.414384	0.207192	+	0.978300i	$0.433567\pi$
$224$	24662.1	0.0328405
$225$	0	0
$226$	−24611.1	−0.0320524
$227$	−253558.	−0.326597	−0.163299	−	0.986577i	$-0.552213\pi$
$227$	−253558.	−0.326597	−0.163299	+	0.986577i	$0.552213\pi$
$228$	−258367.	−0.329154
$229$	−342004.	−0.430966	−0.215483	−	0.976508i	$-0.569133\pi$
$229$	−342004.	−0.430966	−0.215483	+	0.976508i	$0.569133\pi$
$230$	0	0
$231$	−46611.4	−0.0574728
$232$	151652.	0.184982
$233$	520020.	0.627524	0.313762	−	0.949502i	$-0.398411\pi$
$233$	520020.	0.627524	0.313762	+	0.949502i	$0.398411\pi$
$234$	4454.97	0.00531870
$235$	0	0
$236$	756016.	0.883590
$237$	1.61492e6	1.86758
$238$	−6061.82	−0.00693682
$239$	−343268.	−0.388722	−0.194361	−	0.980930i	$-0.562263\pi$
$239$	−343268.	−0.388722	−0.194361	+	0.980930i	$0.562263\pi$
$240$	0	0
$241$	−808674.	−0.896873	−0.448437	−	0.893815i	$-0.648019\pi$
$241$	−808674.	−0.896873	−0.448437	+	0.893815i	$0.648019\pi$
$242$	68528.4	0.0752198
$243$	−200676.	−0.218011
$244$	648331.	0.697144
$245$	0	0
$246$	7493.87	0.00789530
$247$	−149927.	−0.156365
$248$	−159753.	−0.164938
$249$	1.31512e6	1.34421
$250$	0	0
$251$	−362336.	−0.363018	−0.181509	−	0.983389i	$-0.558098\pi$
$251$	−362336.	−0.363018	−0.181509	+	0.983389i	$0.558098\pi$
$252$	12364.0	0.0122647
$253$	406325.	0.399091
$254$	−30386.3	−0.0295524
$255$	0	0
$256$	949254.	0.905279
$257$	1.38706e6	1.30997	0.654986	−	0.755641i	$-0.272674\pi$
$257$	1.38706e6	1.30997	0.654986	+	0.755641i	$0.272674\pi$
$258$	16835.9	0.0157466
$259$	−208708.	−0.193326
$260$	0	0
$261$	114227.	0.103793
$262$	103452.	0.0931080
$263$	−1.34751e6	−1.20128	−0.600639	−	0.799521i	$-0.705087\pi$
$263$	−1.34751e6	−1.20128	−0.600639	+	0.799521i	$0.705087\pi$
$264$	112268.	0.0991398
$265$	0	0
$266$	4037.50	0.00349872
$267$	−1.53889e6	−1.32109
$268$	1.26346e6	1.07455
$269$	578581.	0.487510	0.243755	−	0.969837i	$-0.421621\pi$
$269$	578581.	0.487510	0.243755	+	0.969837i	$0.421621\pi$
$270$	0	0
$271$	−1.11862e6	−0.925249	−0.462624	−	0.886554i	$-0.653092\pi$
$271$	−1.11862e6	−0.925249	−0.462624	+	0.886554i	$0.653092\pi$
$272$	−741385.	−0.607605
$273$	72708.7	0.0590445
$274$	79706.6	0.0641383
$275$	0	0
$276$	−1.09225e6	−0.863078
$277$	1.00925e6	0.790310	0.395155	−	0.918614i	$-0.370691\pi$
$277$	1.00925e6	0.790310	0.395155	+	0.918614i	$0.370691\pi$
$278$	171970.	0.133456
$279$	−120329.	−0.0925461
$280$	0	0
$281$	−1.77997e6	−1.34477	−0.672383	−	0.740203i	$-0.734729\pi$
$281$	−1.77997e6	−1.34477	−0.672383	+	0.740203i	$0.734729\pi$
$282$	43685.3	0.0327124
$283$	−759595.	−0.563788	−0.281894	−	0.959445i	$-0.590963\pi$
$283$	−759595.	−0.563788	−0.281894	+	0.959445i	$0.590963\pi$
$284$	28595.6	0.0210380
$285$	0	0
$286$	32416.8	0.0234345
$287$	12068.9	0.00864891
$288$	−44741.9	−0.0317858
$289$	−864190.	−0.608646
$290$	0	0
$291$	−1.62493e6	−1.12487
$292$	860302.	0.590465
$293$	1.47742e6	1.00539	0.502697	−	0.864463i	$-0.332341\pi$
$293$	1.47742e6	1.00539	0.502697	+	0.864463i	$0.332341\pi$
$294$	151077.	0.101936
$295$	0	0
$296$	502695.	0.333484
$297$	−687833.	−0.452472
$298$	183439.	0.119661
$299$	−633822.	−0.410005
$300$	0	0
$301$	27114.2	0.0172496
$302$	−126855.	−0.0800367
$303$	120042.	0.0751152
$304$	493803.	0.306457
$305$	0	0
$306$	10997.3	0.00671405
$307$	313383.	0.189771	0.0948854	−	0.995488i	$-0.469752\pi$
$307$	313383.	0.189771	0.0948854	+	0.995488i	$0.469752\pi$
$308$	89967.5	0.0540392
$309$	2.23702e6	1.33283
$310$	0	0
$311$	−1.81292e6	−1.06286	−0.531432	−	0.847101i	$-0.678346\pi$
$311$	−1.81292e6	−1.06286	−0.531432	+	0.847101i	$0.678346\pi$
$312$	−175126.	−0.101851
$313$	951089.	0.548732	0.274366	−	0.961625i	$-0.411532\pi$
$313$	951089.	0.548732	0.274366	+	0.961625i	$0.411532\pi$
$314$	272159.	0.155775
$315$	0	0
$316$	−3.11705e6	−1.75601
$317$	334904.	0.187185	0.0935926	−	0.995611i	$-0.470165\pi$
$317$	334904.	0.187185	0.0935926	+	0.995611i	$0.470165\pi$
$318$	−356944.	−0.197940
$319$	831180.	0.457318
$320$	0	0
$321$	−2.07818e6	−1.12570
$322$	17068.6	0.00917401
$323$	−370105.	−0.197387
$324$	2.05386e6	1.08695
$325$	0	0
$326$	−156275.	−0.0814415
$327$	601062.	0.310849
$328$	−29069.1	−0.0149192
$329$	70355.1	0.0358349
$330$	0	0
$331$	−1.03811e6	−0.520804	−0.260402	−	0.965500i	$-0.583855\pi$
$331$	−1.03811e6	−0.520804	−0.260402	+	0.965500i	$0.583855\pi$
$332$	−2.53839e6	−1.26390
$333$	378638.	0.187117
$334$	48607.2	0.0238416
$335$	0	0
$336$	−239474.	−0.115721
$337$	468908.	0.224912	0.112456	−	0.993657i	$-0.464128\pi$
$337$	468908.	0.224912	0.112456	+	0.993657i	$0.464128\pi$
$338$	155332.	0.0739552
$339$	728715.	0.344396
$340$	0	0
$341$	−875577.	−0.407764
$342$	−7324.84	−0.00338636
$343$	489770.	0.224780
$344$	−65307.3	−0.0297554
$345$	0	0
$346$	93764.2	0.0421063
$347$	−3.42593e6	−1.52741	−0.763704	−	0.645566i	$-0.776621\pi$
$347$	−3.42593e6	−1.52741	−0.763704	+	0.645566i	$0.776621\pi$
$348$	−2.23431e6	−0.988999
$349$	−3.70653e6	−1.62894	−0.814468	−	0.580209i	$-0.802971\pi$
$349$	−3.70653e6	−1.62894	−0.814468	+	0.580209i	$0.802971\pi$
$350$	0	0
$351$	1.07294e6	0.464846
$352$	−325567.	−0.140050
$353$	−354914.	−0.151595	−0.0757977	−	0.997123i	$-0.524150\pi$
$353$	−354914.	−0.151595	−0.0757977	+	0.997123i	$0.524150\pi$
$354$	217207.	0.0921225
$355$	0	0
$356$	2.97031e6	1.24216
$357$	179486.	0.0745348
$358$	93512.9	0.0385624
$359$	2.78446e6	1.14026	0.570132	−	0.821553i	$-0.306892\pi$
$359$	2.78446e6	1.14026	0.570132	+	0.821553i	$0.306892\pi$
$360$	0	0
$361$	−2.22959e6	−0.900444
$362$	185800.	0.0745201
$363$	−2.02907e6	−0.808223
$364$	−140339.	−0.0555170
$365$	0	0
$366$	186269.	0.0726838
$367$	−3.42809e6	−1.32858	−0.664289	−	0.747476i	$-0.731266\pi$
$367$	−3.42809e6	−1.32858	−0.664289	+	0.747476i	$0.731266\pi$
$368$	2.08756e6	0.803564
$369$	−21895.3	−0.00837115
$370$	0	0
$371$	−574858.	−0.216833
$372$	2.35366e6	0.881833
$373$	−461278.	−0.171669	−0.0858343	−	0.996309i	$-0.527356\pi$
$373$	−461278.	−0.171669	−0.0858343	+	0.996309i	$0.527356\pi$
$374$	80022.8	0.0295825
$375$	0	0
$376$	−169458.	−0.0618146
$377$	−1.29655e6	−0.469824
$378$	−28894.0	−0.0104011
$379$	3.69849e6	1.32259	0.661297	−	0.750124i	$-0.270006\pi$
$379$	3.69849e6	1.32259	0.661297	+	0.750124i	$0.270006\pi$
$380$	0	0
$381$	899713.	0.317535
$382$	−38842.3	−0.0136191
$383$	−3.53033e6	−1.22975	−0.614877	−	0.788623i	$-0.710794\pi$
$383$	−3.53033e6	−1.22975	−0.614877	+	0.788623i	$0.710794\pi$
$384$	1.16496e6	0.403164
$385$	0	0
$386$	455332.	0.155546
$387$	−49190.5	−0.0166957
$388$	3.13638e6	1.05767
$389$	4.44092e6	1.48799	0.743993	−	0.668188i	$-0.232930\pi$
$389$	4.44092e6	1.48799	0.743993	+	0.668188i	$0.232930\pi$
$390$	0	0
$391$	−1.56463e6	−0.517569
$392$	−586033.	−0.192623
$393$	−3.06314e6	−1.00043
$394$	−526590.	−0.170896
$395$	0	0
$396$	−163219.	−0.0523038
$397$	−416292.	−0.132563	−0.0662815	−	0.997801i	$-0.521114\pi$
$397$	−416292.	−0.132563	−0.0662815	+	0.997801i	$0.521114\pi$
$398$	−80098.1	−0.0253463
$399$	−119547.	−0.0375930
$400$	0	0
$401$	−88679.5	−0.0275399	−0.0137700	−	0.999905i	$-0.504383\pi$
$401$	−88679.5	−0.0275399	−0.0137700	+	0.999905i	$0.504383\pi$
$402$	362999.	0.112032
$403$	1.36580e6	0.418915
$404$	−231701.	−0.0706275
$405$	0	0
$406$	34915.7	0.0105125
$407$	2.75518e6	0.824449
$408$	−432309.	−0.128571
$409$	−3.13874e6	−0.927785	−0.463892	−	0.885892i	$-0.653548\pi$
$409$	−3.13874e6	−0.927785	−0.463892	+	0.885892i	$0.653548\pi$
$410$	0	0
$411$	−2.36005e6	−0.689154
$412$	−4.31781e6	−1.25320
$413$	349811.	0.100916
$414$	−30965.9	−0.00887939
$415$	0	0
$416$	507849.	0.143880
$417$	−5.09188e6	−1.43396
$418$	−53299.6	−0.0149205
$419$	3.64758e6	1.01501	0.507505	−	0.861649i	$-0.330568\pi$
$419$	3.64758e6	1.01501	0.507505	+	0.861649i	$0.330568\pi$
$420$	0	0
$421$	433225.	0.119126	0.0595632	−	0.998225i	$-0.481029\pi$
$421$	433225.	0.119126	0.0595632	+	0.998225i	$0.481029\pi$
$422$	522597.	0.142852
$423$	−127638.	−0.0346841
$424$	1.38460e6	0.374034
$425$	0	0
$426$	8215.67	0.00219340
$427$	299985.	0.0796215
$428$	4.01122e6	1.05844
$429$	−959836.	−0.251799
$430$	0	0
$431$	55256.5	0.0143281	0.00716407	−	0.999974i	$-0.497720\pi$
$431$	55256.5	0.0143281	0.00716407	+	0.999974i	$0.497720\pi$
$432$	−3.53386e6	−0.911045
$433$	2.01236e6	0.515805	0.257903	−	0.966171i	$-0.416969\pi$
$433$	2.01236e6	0.515805	0.257903	+	0.966171i	$0.416969\pi$
$434$	−36780.7	−0.00937336
$435$	0	0
$436$	−1.16015e6	−0.292278
$437$	1.04213e6	0.261046
$438$	247169.	0.0615615
$439$	−4.23994e6	−1.05002	−0.525011	−	0.851095i	$-0.675939\pi$
$439$	−4.23994e6	−1.05002	−0.525011	+	0.851095i	$0.675939\pi$
$440$	0	0
$441$	−441410.	−0.108080
$442$	−124827.	−0.0303915
$443$	−244501.	−0.0591931	−0.0295966	−	0.999562i	$-0.509422\pi$
$443$	−244501.	−0.0591931	−0.0295966	+	0.999562i	$0.509422\pi$
$444$	−7.40626e6	−1.78296
$445$	0	0
$446$	−170648.	−0.0406223
$447$	−5.43148e6	−1.28573
$448$	453032.	0.106643
$449$	−704348.	−0.164881	−0.0824407	−	0.996596i	$-0.526271\pi$
$449$	−704348.	−0.164881	−0.0824407	+	0.996596i	$0.526271\pi$
$450$	0	0
$451$	−159322.	−0.0368838
$452$	−1.40654e6	−0.323821
$453$	3.75607e6	0.859979
$454$	140609.	0.0320165
$455$	0	0
$456$	287942.	0.0648474
$457$	−3.39124e6	−0.759571	−0.379785	−	0.925075i	$-0.624002\pi$
$457$	−3.39124e6	−0.759571	−0.379785	+	0.925075i	$0.624002\pi$
$458$	189657.	0.0422478
$459$	2.64862e6	0.586797
$460$	0	0
$461$	4.75642e6	1.04239	0.521193	−	0.853439i	$-0.325487\pi$
$461$	4.75642e6	1.04239	0.521193	+	0.853439i	$0.325487\pi$
$462$	25848.1	0.00563409
$463$	−1.59218e6	−0.345176	−0.172588	−	0.984994i	$-0.555213\pi$
$463$	−1.59218e6	−0.345176	−0.172588	+	0.984994i	$0.555213\pi$
$464$	4.27033e6	0.920802
$465$	0	0
$466$	−288374.	−0.0615165
$467$	−1.03259e6	−0.219097	−0.109549	−	0.993981i	$-0.534941\pi$
$467$	−1.03259e6	−0.219097	−0.109549	+	0.993981i	$0.534941\pi$
$468$	254604.	0.0537341
$469$	584609.	0.122725
$470$	0	0
$471$	−8.05842e6	−1.67378
$472$	−842556.	−0.174078
$473$	−357937.	−0.0735621
$474$	−895545.	−0.183080
$475$	0	0
$476$	−346436.	−0.0700818
$477$	1.04291e6	0.209870
$478$	190358.	0.0381066
$479$	−4.92602e6	−0.980974	−0.490487	−	0.871448i	$-0.663181\pi$
$479$	−4.92602e6	−0.980974	−0.490487	+	0.871448i	$0.663181\pi$
$480$	0	0
$481$	−4.29777e6	−0.846995
$482$	448446.	0.0879210
$483$	−505389.	−0.0985730
$484$	3.91643e6	0.759937
$485$	0	0
$486$	111284.	0.0213718
$487$	−6.40568e6	−1.22389	−0.611946	−	0.790900i	$-0.709613\pi$
$487$	−6.40568e6	−1.22389	−0.611946	+	0.790900i	$0.709613\pi$
$488$	−722546.	−0.137346
$489$	4.62718e6	0.875073
$490$	0	0
$491$	7.62840e6	1.42800	0.714002	−	0.700143i	$-0.246881\pi$
$491$	7.62840e6	1.42800	0.714002	+	0.700143i	$0.246881\pi$
$492$	428278.	0.0797652
$493$	−3.20060e6	−0.593082
$494$	83141.5	0.0153285
$495$	0	0
$496$	−4.49843e6	−0.821025
$497$	13231.3	0.00240277
$498$	−729292.	−0.131774
$499$	−8.75840e6	−1.57461	−0.787306	−	0.616563i	$-0.788525\pi$
$499$	−8.75840e6	−1.57461	−0.787306	+	0.616563i	$0.788525\pi$
$500$	0	0
$501$	−1.43922e6	−0.256173
$502$	200932.	0.0355868
$503$	−5.44967e6	−0.960395	−0.480198	−	0.877160i	$-0.659435\pi$
$503$	−5.44967e6	−0.960395	−0.480198	+	0.877160i	$0.659435\pi$
$504$	−13779.3	−0.00241631
$505$	0	0
$506$	−225325.	−0.0391231
$507$	−4.59925e6	−0.794634
$508$	−1.73659e6	−0.298564
$509$	−9.38817e6	−1.60615	−0.803076	−	0.595877i	$-0.796805\pi$
$509$	−9.38817e6	−1.60615	−0.803076	+	0.595877i	$0.796805\pi$
$510$	0	0
$511$	398065.	0.0674375
$512$	−2.79677e6	−0.471501
$513$	−1.76413e6	−0.295962
$514$	−769186.	−0.128417
$515$	0	0
$516$	962180.	0.159086
$517$	−928767.	−0.152820
$518$	115738.	0.0189518
$519$	−2.77628e6	−0.452424
$520$	0	0
$521$	−1.95674e6	−0.315819	−0.157910	−	0.987454i	$-0.550476\pi$
$521$	−1.95674e6	−0.315819	−0.157910	+	0.987454i	$0.550476\pi$
$522$	−63344.0	−0.0101749
$523$	−8.31436e6	−1.32915	−0.664576	−	0.747221i	$-0.731388\pi$
$523$	−8.31436e6	−1.32915	−0.664576	+	0.747221i	$0.731388\pi$
$524$	5.91235e6	0.940659
$525$	0	0
$526$	747255.	0.117762
$527$	3.37156e6	0.528816
$528$	3.16133e6	0.493498
$529$	−2.03073e6	−0.315509
$530$	0	0
$531$	−634628.	−0.0976748
$532$	230745.	0.0353471
$533$	248525.	0.0378925
$534$	853386.	0.129507
$535$	0	0
$536$	−1.40809e6	−0.211699
$537$	−2.76884e6	−0.414345
$538$	−320849.	−0.0477909
$539$	−3.21194e6	−0.476207
$540$	0	0
$541$	1.91201e6	0.280864	0.140432	−	0.990090i	$-0.455151\pi$
$541$	1.91201e6	0.280864	0.140432	+	0.990090i	$0.455151\pi$
$542$	620323.	0.0907026
$543$	−5.50138e6	−0.800704
$544$	1.25365e6	0.181627
$545$	0	0
$546$	−40320.2	−0.00578817
$547$	1.19168e6	0.170291	0.0851455	−	0.996369i	$-0.472865\pi$
$547$	1.19168e6	0.170291	0.0851455	+	0.996369i	$0.472865\pi$
$548$	4.55527e6	0.647981
$549$	−544233.	−0.0770645
$550$	0	0
$551$	2.13178e6	0.299132
$552$	1.21728e6	0.170037
$553$	−1.44227e6	−0.200555
$554$	−559672.	−0.0774745
$555$	0	0
$556$	9.82814e6	1.34829
$557$	4.86430e6	0.664328	0.332164	−	0.943222i	$-0.392221\pi$
$557$	4.86430e6	0.664328	0.332164	+	0.943222i	$0.392221\pi$
$558$	66727.5	0.00907234
$559$	558343.	0.0755738
$560$	0	0
$561$	−2.36941e6	−0.317858
$562$	987072.	0.131828
$563$	−9.38674e6	−1.24808	−0.624042	−	0.781391i	$-0.714511\pi$
$563$	−9.38674e6	−1.24808	−0.624042	+	0.781391i	$0.714511\pi$
$564$	2.49664e6	0.330490
$565$	0	0
$566$	421229.	0.0552685
$567$	950330.	0.124141
$568$	−31869.0	−0.00414474
$569$	872934.	0.113032	0.0565159	−	0.998402i	$-0.482001\pi$
$569$	872934.	0.113032	0.0565159	+	0.998402i	$0.482001\pi$
$570$	0	0
$571$	−1.20552e7	−1.54733	−0.773665	−	0.633595i	$-0.781578\pi$
$571$	−1.20552e7	−1.54733	−0.773665	+	0.633595i	$0.781578\pi$
$572$	1.85264e6	0.236756
$573$	1.15009e6	0.146334
$574$	−6692.72	−0.000847857 0
$575$	0	0
$576$	−821891.	−0.103219
$577$	5.43685e6	0.679842	0.339921	−	0.940454i	$-0.389600\pi$
$577$	5.43685e6	0.679842	0.339921	+	0.940454i	$0.389600\pi$
$578$	479232.	0.0596659
$579$	−1.34820e7	−1.67132
$580$	0	0
$581$	−1.17452e6	−0.144351
$582$	901098.	0.110272
$583$	7.58877e6	0.924698
$584$	−958781.	−0.116329
$585$	0	0
$586$	−819297.	−0.0985593
$587$	−3.17650e6	−0.380499	−0.190250	−	0.981736i	$-0.560930\pi$
$587$	−3.17650e6	−0.380499	−0.190250	+	0.981736i	$0.560930\pi$
$588$	8.63410e6	1.02985
$589$	−2.24565e6	−0.266719
$590$	0	0
$591$	1.55919e7	1.83624
$592$	1.41552e7	1.66001
$593$	9.86552e6	1.15208	0.576041	−	0.817421i	$-0.304597\pi$
$593$	9.86552e6	1.15208	0.576041	+	0.817421i	$0.304597\pi$
$594$	381434.	0.0443561
$595$	0	0
$596$	1.04836e7	1.20892
$597$	2.37164e6	0.272341
$598$	351483.	0.0401930
$599$	−2.46934e6	−0.281199	−0.140599	−	0.990067i	$-0.544903\pi$
$599$	−2.46934e6	−0.281199	−0.140599	+	0.990067i	$0.544903\pi$
$600$	0	0
$601$	−3.46379e6	−0.391169	−0.195585	−	0.980687i	$-0.562660\pi$
$601$	−3.46379e6	−0.391169	−0.195585	+	0.980687i	$0.562660\pi$
$602$	−15036.0	−0.00169099
$603$	−1.06060e6	−0.118784
$604$	−7.24981e6	−0.808601
$605$	0	0
$606$	−66568.8	−0.00736358
$607$	5.37970e6	0.592634	0.296317	−	0.955090i	$-0.404241\pi$
$607$	5.37970e6	0.592634	0.296317	+	0.955090i	$0.404241\pi$
$608$	−835002.	−0.0916070
$609$	−1.03383e6	−0.112955
$610$	0	0
$611$	1.44877e6	0.156999
$612$	628504.	0.0678312
$613$	−1.06429e6	−0.114395	−0.0571977	−	0.998363i	$-0.518217\pi$
$613$	−1.06429e6	−0.114395	−0.0571977	+	0.998363i	$0.518217\pi$
$614$	−173785.	−0.0186033
$615$	0	0
$616$	−100266.	−0.0106464
$617$	8.18415e6	0.865487	0.432744	−	0.901517i	$-0.357546\pi$
$617$	8.18415e6	0.865487	0.432744	+	0.901517i	$0.357546\pi$
$618$	−1.24053e6	−0.130658
$619$	1.11551e7	1.17017	0.585083	−	0.810973i	$-0.301062\pi$
$619$	1.11551e7	1.17017	0.585083	+	0.810973i	$0.301062\pi$
$620$	0	0
$621$	−7.45789e6	−0.776045
$622$	1.00534e6	0.104193
$623$	1.37437e6	0.141868
$624$	−4.93132e6	−0.506993
$625$	0	0
$626$	−527421.	−0.0537925
$627$	1.57816e6	0.160318
$628$	1.55540e7	1.57378
$629$	−1.06093e7	−1.06920
$630$	0	0
$631$	4.15356e6	0.415286	0.207643	−	0.978205i	$-0.433421\pi$
$631$	4.15356e6	0.415286	0.207643	+	0.978205i	$0.433421\pi$
$632$	3.47386e6	0.345955
$633$	−1.54737e7	−1.53492
$634$	−185719.	−0.0183499
$635$	0	0
$636$	−2.03995e7	−1.99976
$637$	5.01028e6	0.489230
$638$	−460926.	−0.0448311
$639$	−24004.2	−0.00232560
$640$	0	0
$641$	1.62261e7	1.55980	0.779901	−	0.625903i	$-0.215269\pi$
$641$	1.62261e7	1.55980	0.779901	+	0.625903i	$0.215269\pi$
$642$	1.15244e6	0.110353
$643$	2.01525e7	1.92221	0.961105	−	0.276184i	$-0.0890699\pi$
$643$	2.01525e7	1.92221	0.961105	+	0.276184i	$0.0890699\pi$
$644$	975481.	0.0926839
$645$	0	0
$646$	205239.	0.0193499
$647$	−4.01029e6	−0.376630	−0.188315	−	0.982109i	$-0.560303\pi$
$647$	−4.01029e6	−0.376630	−0.188315	+	0.982109i	$0.560303\pi$
$648$	−2.28897e6	−0.214142
$649$	−4.61790e6	−0.430361
$650$	0	0
$651$	1.08905e6	0.100715
$652$	−8.93120e6	−0.822793
$653$	−1.51215e7	−1.38775	−0.693876	−	0.720095i	$-0.744098\pi$
$653$	−1.51215e7	−1.38775	−0.693876	+	0.720095i	$0.744098\pi$
$654$	−333315.	−0.0304727
$655$	0	0
$656$	−818546.	−0.0742649
$657$	−722169.	−0.0652718
$658$	−39015.0	−0.00351291
$659$	1.35872e7	1.21875	0.609376	−	0.792881i	$-0.291420\pi$
$659$	1.35872e7	1.21875	0.609376	+	0.792881i	$0.291420\pi$
$660$	0	0
$661$	−7.90334e6	−0.703570	−0.351785	−	0.936081i	$-0.614425\pi$
$661$	−7.90334e6	−0.703570	−0.351785	+	0.936081i	$0.614425\pi$
$662$	575680.	0.0510547
$663$	3.69602e6	0.326551
$664$	2.82896e6	0.249004
$665$	0	0
$666$	−209972.	−0.0183432
$667$	9.01214e6	0.784357
$668$	2.77793e6	0.240868
$669$	5.05276e6	0.436479
$670$	0	0
$671$	−3.96014e6	−0.339551
$672$	404942.	0.0345915
$673$	6.38860e6	0.543711	0.271856	−	0.962338i	$-0.412363\pi$
$673$	6.38860e6	0.543711	0.271856	+	0.962338i	$0.412363\pi$
$674$	−260030.	−0.0220482
$675$	0	0
$676$	8.87729e6	0.747160
$677$	1.36085e7	1.14114	0.570568	−	0.821251i	$-0.306723\pi$
$677$	1.36085e7	1.14114	0.570568	+	0.821251i	$0.306723\pi$
$678$	−404105.	−0.0337614
$679$	1.45122e6	0.120797
$680$	0	0
$681$	−4.16332e6	−0.344011
$682$	485547.	0.0399733
$683$	7.83412e6	0.642597	0.321298	−	0.946978i	$-0.395881\pi$
$683$	7.83412e6	0.642597	0.321298	+	0.946978i	$0.395881\pi$
$684$	−418618.	−0.0342119
$685$	0	0
$686$	−271599.	−0.0220353
$687$	−5.61558e6	−0.453945
$688$	−1.83896e6	−0.148116
$689$	−1.18376e7	−0.949986
$690$	0	0
$691$	1.86342e7	1.48462	0.742309	−	0.670058i	$-0.233731\pi$
$691$	1.86342e7	1.48462	0.742309	+	0.670058i	$0.233731\pi$
$692$	5.35867e6	0.425395
$693$	−75522.1	−0.00597366
$694$	1.89983e6	0.149733
$695$	0	0
$696$	2.49007e6	0.194845
$697$	613499.	0.0478335
$698$	2.05544e6	0.159685
$699$	8.53853e6	0.660983
$700$	0	0
$701$	1.64231e7	1.26230	0.631148	−	0.775663i	$-0.282584\pi$
$701$	1.64231e7	1.26230	0.631148	+	0.775663i	$0.282584\pi$
$702$	−594994.	−0.0455691
$703$	7.06638e6	0.539273
$704$	−5.98054e6	−0.454788
$705$	0	0
$706$	196815.	0.0148610
$707$	−107209.	−0.00806644
$708$	1.24135e7	0.930702
$709$	1.99747e7	1.49233	0.746163	−	0.665763i	$-0.231894\pi$
$709$	1.99747e7	1.49233	0.746163	+	0.665763i	$0.231894\pi$
$710$	0	0
$711$	2.61657e6	0.194115
$712$	−3.31032e6	−0.244721
$713$	−9.49353e6	−0.699365
$714$	−99532.7	−0.00730668
$715$	0	0
$716$	5.34430e6	0.389591
$717$	−5.63633e6	−0.409448
$718$	−1.54411e6	−0.111781
$719$	2.26023e7	1.63053	0.815267	−	0.579085i	$-0.196590\pi$
$719$	2.26023e7	1.63053	0.815267	+	0.579085i	$0.196590\pi$
$720$	0	0
$721$	−1.99787e6	−0.143129
$722$	1.23641e6	0.0882710
$723$	−1.32781e7	−0.944693
$724$	1.06185e7	0.752867
$725$	0	0
$726$	1.12521e6	0.0792305
$727$	−1.10845e7	−0.777821	−0.388910	−	0.921276i	$-0.627148\pi$
$727$	−1.10845e7	−0.777821	−0.388910	+	0.921276i	$0.627148\pi$
$728$	156404.	0.0109375
$729$	1.24528e7	0.867859
$730$	0	0
$731$	1.37830e6	0.0954005
$732$	1.06454e7	0.734315
$733$	−1.15876e7	−0.796591	−0.398295	−	0.917257i	$-0.630398\pi$
$733$	−1.15876e7	−0.796591	−0.398295	+	0.917257i	$0.630398\pi$
$734$	1.90103e6	0.130241
$735$	0	0
$736$	−3.52999e6	−0.240203
$737$	−7.71750e6	−0.523369
$738$	12141.9	0.000820629 0
$739$	1.27754e7	0.860524	0.430262	−	0.902704i	$-0.358421\pi$
$739$	1.27754e7	0.860524	0.430262	+	0.902704i	$0.358421\pi$
$740$	0	0
$741$	−2.46175e6	−0.164702
$742$	318784.	0.0212563
$743$	8.82224e6	0.586282	0.293141	−	0.956069i	$-0.405299\pi$
$743$	8.82224e6	0.586282	0.293141	+	0.956069i	$0.405299\pi$
$744$	−2.62308e6	−0.173732
$745$	0	0
$746$	255799.	0.0168288
$747$	2.13082e6	0.139716
$748$	4.57334e6	0.298868
$749$	1.85601e6	0.120886
$750$	0	0
$751$	−6.70112e6	−0.433558	−0.216779	−	0.976221i	$-0.569555\pi$
$751$	−6.70112e6	−0.433558	−0.216779	+	0.976221i	$0.569555\pi$
$752$	−4.77170e6	−0.307701
$753$	−5.94943e6	−0.382373
$754$	718994.	0.0460571
$755$	0	0
$756$	−1.65131e6	−0.105081
$757$	2.83394e7	1.79742	0.898712	−	0.438540i	$-0.144504\pi$
$757$	2.83394e7	1.79742	0.898712	+	0.438540i	$0.144504\pi$
$758$	−2.05098e6	−0.129655
$759$	6.67170e6	0.420370
$760$	0	0
$761$	3.06943e7	1.92130	0.960652	−	0.277756i	$-0.0895905\pi$
$761$	3.06943e7	1.92130	0.960652	+	0.277756i	$0.0895905\pi$
$762$	−498931.	−0.0311281
$763$	−536803.	−0.0333814
$764$	−2.21986e6	−0.137592
$765$	0	0
$766$	1.95772e6	0.120553
$767$	7.20342e6	0.442130
$768$	1.55864e7	0.953548
$769$	4.46977e6	0.272565	0.136282	−	0.990670i	$-0.456485\pi$
$769$	4.46977e6	0.272565	0.136282	+	0.990670i	$0.456485\pi$
$770$	0	0
$771$	2.27750e7	1.37982
$772$	2.60224e7	1.57146
$773$	−1.66252e7	−1.00073	−0.500365	−	0.865814i	$-0.666801\pi$
$773$	−1.66252e7	−1.00073	−0.500365	+	0.865814i	$0.666801\pi$
$774$	27278.3	0.00163669
$775$	0	0
$776$	−3.49540e6	−0.208374
$777$	−3.42690e6	−0.203634
$778$	−2.46269e6	−0.145868
$779$	−408624.	−0.0241257
$780$	0	0
$781$	−174668.	−0.0102467
$782$	867655.	0.0507376
$783$	−1.52559e7	−0.889269
$784$	−1.65019e7	−0.958835
$785$	0	0
$786$	1.69865e6	0.0980725
$787$	1.05009e7	0.604351	0.302176	−	0.953252i	$-0.402287\pi$
$787$	1.05009e7	0.604351	0.302176	+	0.953252i	$0.402287\pi$
$788$	−3.00949e7	−1.72654
$789$	−2.21256e7	−1.26533
$790$	0	0
$791$	−650809.	−0.0369839
$792$	181903.	0.0103045
$793$	6.17739e6	0.348836
$794$	230853.	0.0129952
$795$	0	0
$796$	−4.57765e6	−0.256070
$797$	1.10547e7	0.616452	0.308226	−	0.951313i	$-0.400265\pi$
$797$	1.10547e7	0.616452	0.308226	+	0.951313i	$0.400265\pi$
$798$	66294.3	0.00368526
$799$	3.57638e6	0.198188
$800$	0	0
$801$	−2.49339e6	−0.137312
$802$	49176.8	0.00269975
$803$	−5.25491e6	−0.287592
$804$	2.07456e7	1.13184
$805$	0	0
$806$	−757399.	−0.0410664
$807$	9.50008e6	0.513504
$808$	258224.	0.0139145
$809$	−1.76974e7	−0.950687	−0.475344	−	0.879800i	$-0.657676\pi$
$809$	−1.76974e7	−0.950687	−0.475344	+	0.879800i	$0.657676\pi$
$810$	0	0
$811$	2.54637e7	1.35947	0.679735	−	0.733458i	$-0.262095\pi$
$811$	2.54637e7	1.35947	0.679735	+	0.733458i	$0.262095\pi$
$812$	1.99545e6	0.106206
$813$	−1.83673e7	−0.974582
$814$	−1.52787e6	−0.0808212
$815$	0	0
$816$	−1.21733e7	−0.640002
$817$	−918024.	−0.0481171
$818$	1.74057e6	0.0909513
$819$	117806.	0.00613702
$820$	0	0
$821$	2.65810e6	0.137630	0.0688152	−	0.997629i	$-0.478078\pi$
$821$	2.65810e6	0.137630	0.0688152	+	0.997629i	$0.478078\pi$
$822$	1.30875e6	0.0675581
$823$	2.30681e7	1.18717	0.593584	−	0.804772i	$-0.297712\pi$
$823$	2.30681e7	1.18717	0.593584	+	0.804772i	$0.297712\pi$
$824$	4.81207e6	0.246896
$825$	0	0
$826$	−193986.	−0.00989282
$827$	4.42757e6	0.225113	0.112557	−	0.993645i	$-0.464096\pi$
$827$	4.42757e6	0.225113	0.112557	+	0.993645i	$0.464096\pi$
$828$	−1.76972e6	−0.0897074
$829$	−1.47703e6	−0.0746454	−0.0373227	−	0.999303i	$-0.511883\pi$
$829$	−1.47703e6	−0.0746454	−0.0373227	+	0.999303i	$0.511883\pi$
$830$	0	0
$831$	1.65714e7	0.832449
$832$	9.32898e6	0.467225
$833$	1.23682e7	0.617579
$834$	2.82367e6	0.140572
$835$	0	0
$836$	−3.04610e6	−0.150740
$837$	1.60708e7	0.792909
$838$	−2.02275e6	−0.0995020
$839$	−2.23061e7	−1.09400	−0.547001	−	0.837132i	$-0.684231\pi$
$839$	−2.23061e7	−1.09400	−0.547001	+	0.837132i	$0.684231\pi$
$840$	0	0
$841$	−2.07588e6	−0.101207
$842$	−240242.	−0.0116780
$843$	−2.92264e7	−1.41647
$844$	2.98667e7	1.44321
$845$	0	0
$846$	70781.1	0.00340010
$847$	1.81215e6	0.0867931
$848$	3.89886e7	1.86186
$849$	−1.24723e7	−0.593849
$850$	0	0
$851$	2.98733e7	1.41403
$852$	469529.	0.0221597
$853$	2.37698e7	1.11854	0.559271	−	0.828985i	$-0.311081\pi$
$853$	2.37698e7	1.11854	0.559271	+	0.828985i	$0.311081\pi$
$854$	−166355.	−0.00780534
$855$	0	0
$856$	−4.47039e6	−0.208526
$857$	−1.39366e7	−0.648192	−0.324096	−	0.946024i	$-0.605060\pi$
$857$	−1.39366e7	−0.648192	−0.324096	+	0.946024i	$0.605060\pi$
$858$	532272.	0.0246840
$859$	−3.86225e7	−1.78590	−0.892950	−	0.450156i	$-0.851368\pi$
$859$	−3.86225e7	−1.78590	−0.892950	+	0.450156i	$0.851368\pi$
$860$	0	0
$861$	198166.	0.00911006
$862$	−30642.2	−0.00140460
$863$	2.87245e7	1.31288	0.656441	−	0.754377i	$-0.272061\pi$
$863$	2.87245e7	1.31288	0.656441	+	0.754377i	$0.272061\pi$
$864$	5.97562e6	0.272332
$865$	0	0
$866$	−1.11594e6	−0.0505647
$867$	−1.41897e7	−0.641098
$868$	−2.10203e6	−0.0946979
$869$	1.90396e7	0.855281
$870$	0	0
$871$	1.20384e7	0.537681
$872$	1.29295e6	0.0575824
$873$	−2.63279e6	−0.116918
$874$	−577906.	−0.0255905
$875$	0	0
$876$	1.41258e7	0.621948
$877$	−3.26310e6	−0.143262	−0.0716310	−	0.997431i	$-0.522820\pi$
$877$	−3.26310e6	−0.143262	−0.0716310	+	0.997431i	$0.522820\pi$
$878$	2.35124e6	0.102934
$879$	2.42587e7	1.05900
$880$	0	0
$881$	4.29011e7	1.86221	0.931104	−	0.364754i	$-0.118847\pi$
$881$	4.29011e7	1.86221	0.931104	+	0.364754i	$0.118847\pi$
$882$	244781.	0.0105951
$883$	941800.	0.0406497	0.0203248	−	0.999793i	$-0.493530\pi$
$883$	941800.	0.0406497	0.0203248	+	0.999793i	$0.493530\pi$
$884$	−7.13391e6	−0.307041
$885$	0	0
$886$	135587.	0.00580273
$887$	8.53166e6	0.364103	0.182052	−	0.983289i	$-0.441726\pi$
$887$	8.53166e6	0.364103	0.182052	+	0.983289i	$0.441726\pi$
$888$	8.25405e6	0.351265
$889$	−803526.	−0.0340993
$890$	0	0
$891$	−1.25454e7	−0.529409
$892$	−9.75263e6	−0.410402
$893$	−2.38207e6	−0.0999597
$894$	3.01200e6	0.126041
$895$	0	0
$896$	−1.04041e6	−0.0432948
$897$	−1.04071e7	−0.431866
$898$	390592.	0.0161634
$899$	−1.94200e7	−0.801401
$900$	0	0
$901$	−2.92219e7	−1.19921
$902$	88351.4	0.00361574
$903$	445204.	0.0181694
$904$	1.56754e6	0.0637967
$905$	0	0
$906$	−2.08291e6	−0.0843042
$907$	−7.62903e6	−0.307929	−0.153965	−	0.988076i	$-0.549204\pi$
$907$	−7.62903e6	−0.307929	−0.153965	+	0.988076i	$0.549204\pi$
$908$	8.03588e6	0.323459
$909$	194498.	0.00780739
$910$	0	0
$911$	1.17046e6	0.0467262	0.0233631	−	0.999727i	$-0.492563\pi$
$911$	1.17046e6	0.0467262	0.0233631	+	0.999727i	$0.492563\pi$
$912$	8.10806e6	0.322797
$913$	1.55050e7	0.615596
$914$	1.88059e6	0.0744611
$915$	0	0
$916$	1.08390e7	0.426824
$917$	2.73567e6	0.107434
$918$	−1.46878e6	−0.0575240
$919$	−4.41575e7	−1.72471	−0.862353	−	0.506307i	$-0.831010\pi$
$919$	−4.41575e7	−1.72471	−0.862353	+	0.506307i	$0.831010\pi$
$920$	0	0
$921$	5.14563e6	0.199889
$922$	−2.63765e6	−0.102186
$923$	272463.	0.0105270
$924$	1.47723e6	0.0569205
$925$	0	0
$926$	882936.	0.0338378
$927$	3.62453e6	0.138533
$928$	−7.22096e6	−0.275249
$929$	−3.89558e7	−1.48092	−0.740462	−	0.672098i	$-0.765393\pi$
$929$	−3.89558e7	−1.48092	−0.740462	+	0.672098i	$0.765393\pi$
$930$	0	0
$931$	−8.23787e6	−0.311488
$932$	−1.64807e7	−0.621493
$933$	−2.97675e7	−1.11953
$934$	572619.	0.0214782
$935$	0	0
$936$	−283748.	−0.0105863
$937$	2.39651e6	0.0891723	0.0445862	−	0.999006i	$-0.485803\pi$
$937$	2.39651e6	0.0891723	0.0445862	+	0.999006i	$0.485803\pi$
$938$	−324192.	−0.0120308
$939$	1.56165e7	0.577990
$940$	0	0
$941$	3.59229e7	1.32251	0.661253	−	0.750163i	$-0.270025\pi$
$941$	3.59229e7	1.32251	0.661253	+	0.750163i	$0.270025\pi$
$942$	4.46875e6	0.164081
$943$	−1.72747e6	−0.0632602
$944$	−2.37253e7	−0.866525
$945$	0	0
$946$	198492.	0.00721133
$947$	8.20191e6	0.297194	0.148597	−	0.988898i	$-0.452524\pi$
$947$	8.20191e6	0.297194	0.148597	+	0.988898i	$0.452524\pi$
$948$	−5.11808e7	−1.84964
$949$	8.19707e6	0.295456
$950$	0	0
$951$	5.49899e6	0.197166
$952$	386092.	0.0138070
$953$	3.66003e7	1.30543	0.652714	−	0.757604i	$-0.273630\pi$
$953$	3.66003e7	1.30543	0.652714	+	0.757604i	$0.273630\pi$
$954$	−578338.	−0.0205736
$955$	0	0
$956$	1.08790e7	0.384986
$957$	1.36477e7	0.481702
$958$	2.73170e6	0.0961654
$959$	2.10774e6	0.0740066
$960$	0	0
$961$	−8.17184e6	−0.285438
$962$	2.38331e6	0.0830314
$963$	−3.36717e6	−0.117004
$964$	2.56289e7	0.888254
$965$	0	0
$966$	280261.	0.00966316
$967$	−1.88614e7	−0.648647	−0.324323	−	0.945946i	$-0.605137\pi$
$967$	−1.88614e7	−0.648647	−0.324323	+	0.945946i	$0.605137\pi$
$968$	−4.36475e6	−0.149717
$969$	−6.07698e6	−0.207911
$970$	0	0
$971$	−1.29233e7	−0.439870	−0.219935	−	0.975515i	$-0.570585\pi$
$971$	−1.29233e7	−0.439870	−0.219935	+	0.975515i	$0.570585\pi$
$972$	6.35991e6	0.215916
$973$	4.54752e6	0.153990
$974$	3.55224e6	0.119979
$975$	0	0
$976$	−2.03459e7	−0.683680
$977$	−2.24389e7	−0.752080	−0.376040	−	0.926603i	$-0.622715\pi$
$977$	−2.24389e7	−0.752080	−0.376040	+	0.926603i	$0.622715\pi$
$978$	−2.56598e6	−0.0857839
$979$	−1.81433e7	−0.605006
$980$	0	0
$981$	973869.	0.0323093
$982$	−4.23029e6	−0.139988
$983$	−4.09662e7	−1.35220	−0.676102	−	0.736808i	$-0.736332\pi$
$983$	−4.09662e7	−1.35220	−0.676102	+	0.736808i	$0.736332\pi$
$984$	−477303.	−0.0157147
$985$	0	0
$986$	1.77488e6	0.0581401
$987$	1.15520e6	0.0377456
$988$	4.75157e6	0.154862
$989$	−3.88097e6	−0.126168
$990$	0	0
$991$	5.27270e6	0.170549	0.0852745	−	0.996357i	$-0.472823\pi$
$991$	5.27270e6	0.170549	0.0852745	+	0.996357i	$0.472823\pi$
$992$	7.60667e6	0.245423
$993$	−1.70454e7	−0.548573
$994$	−7337.35	−0.000235545 0
$995$	0	0
$996$	−4.16794e7	−1.33129
$997$	−3.26542e7	−1.04040	−0.520202	−	0.854043i	$-0.674143\pi$
$997$	−3.26542e7	−1.04040	−0.520202	+	0.854043i	$0.674143\pi$
$998$	4.85692e6	0.154360
$999$	−5.05699e7	−1.60317

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.6.a.c.1.6		13
5.4	even	2		215.6.a.a.1.8	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.6.a.a.1.8	✓	13	5.4	even	2
1075.6.a.c.1.6		13	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)

\(f(q)\)	\(=\)	\(q-0.554544 q^{2} +16.4196 q^{3} -31.6925 q^{4} -9.10541 q^{6} -14.6642 q^{7} +35.3203 q^{8} +26.6038 q^{9} +O(q^{10})\)	\(q-0.554544 q^{2} +16.4196 q^{3} -31.6925 q^{4} -9.10541 q^{6} -14.6642 q^{7} +35.3203 q^{8} +26.6038 q^{9} +193.584 q^{11} -520.378 q^{12} -301.970 q^{13} +8.13197 q^{14} +994.573 q^{16} -745.431 q^{17} -14.7530 q^{18} +496.498 q^{19} -240.781 q^{21} -107.351 q^{22} +2098.96 q^{23} +579.946 q^{24} +167.456 q^{26} -3553.14 q^{27} +464.746 q^{28} +4293.63 q^{29} -4522.98 q^{31} -1681.78 q^{32} +3178.58 q^{33} +413.374 q^{34} -843.142 q^{36} +14232.5 q^{37} -275.330 q^{38} -4958.23 q^{39} -823.013 q^{41} +133.524 q^{42} -1849.00 q^{43} -6135.17 q^{44} -1163.96 q^{46} -4797.74 q^{47} +16330.5 q^{48} -16592.0 q^{49} -12239.7 q^{51} +9570.18 q^{52} +39201.4 q^{53} +1970.38 q^{54} -517.945 q^{56} +8152.30 q^{57} -2381.01 q^{58} -23854.7 q^{59} -20456.9 q^{61} +2508.19 q^{62} -390.125 q^{63} -30893.7 q^{64} -1762.66 q^{66} -39866.3 q^{67} +23624.5 q^{68} +34464.1 q^{69} -902.284 q^{71} +939.656 q^{72} -27145.3 q^{73} -7892.53 q^{74} -15735.2 q^{76} -2838.77 q^{77} +2749.56 q^{78} +98353.1 q^{79} -64806.0 q^{81} +456.397 q^{82} +80094.4 q^{83} +7630.95 q^{84} +1025.35 q^{86} +70499.8 q^{87} +6837.46 q^{88} -93722.9 q^{89} +4428.16 q^{91} -66521.1 q^{92} -74265.5 q^{93} +2660.56 q^{94} -27614.3 q^{96} -98962.9 q^{97} +9200.98 q^{98} +5150.09 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 7 q^{2} + 16 q^{3} + 123 q^{4} - 40 q^{6} + 372 q^{7} + 15 q^{8} + 107 q^{9}+O(q^{10}) \) 13 * q + 7 * q^2 + 16 * q^3 + 123 * q^4 - 40 * q^6 + 372 * q^7 + 15 * q^8 + 107 * q^9	\( 13 q + 7 q^{2} + 16 q^{3} + 123 q^{4} - 40 q^{6} + 372 q^{7} + 15 q^{8} + 107 q^{9} - 1039 q^{11} + 2668 q^{12} + 1791 q^{13} - 1389 q^{14} - 1789 q^{16} + 365 q^{17} - 293 q^{18} - 4200 q^{19} - 3452 q^{21} + 2302 q^{22} + 1315 q^{23} - 2784 q^{24} - 2753 q^{26} + 1078 q^{27} - 8067 q^{28} - 14052 q^{29} - 4479 q^{31} + 6527 q^{32} - 8626 q^{33} - 9432 q^{34} - 5709 q^{36} + 25156 q^{37} + 2391 q^{38} - 7456 q^{39} - 32227 q^{41} - 630 q^{42} - 24037 q^{43} - 52652 q^{44} - 28700 q^{46} + 28390 q^{47} + 15504 q^{48} - 48915 q^{49} - 50744 q^{51} + 32261 q^{52} + 66177 q^{53} - 81742 q^{54} - 98379 q^{56} + 126280 q^{57} + 72353 q^{58} - 103588 q^{59} - 105406 q^{61} + 120123 q^{62} + 100962 q^{63} - 148957 q^{64} - 124146 q^{66} + 139241 q^{67} + 151020 q^{68} - 166714 q^{69} - 72726 q^{71} - 74361 q^{72} + 140142 q^{73} - 153906 q^{74} - 30223 q^{76} + 74050 q^{77} + 18812 q^{78} - 41076 q^{79} - 219547 q^{81} - 16483 q^{82} - 106623 q^{83} + 59732 q^{84} - 12943 q^{86} + 41238 q^{87} - 23748 q^{88} - 180646 q^{89} - 181694 q^{91} - 193132 q^{92} - 137950 q^{93} - 125382 q^{94} + 185312 q^{96} + 119291 q^{97} + 141778 q^{98} - 14335 q^{99}+O(q^{100}) \) 13 * q + 7 * q^2 + 16 * q^3 + 123 * q^4 - 40 * q^6 + 372 * q^7 + 15 * q^8 + 107 * q^9 - 1039 * q^11 + 2668 * q^12 + 1791 * q^13 - 1389 * q^14 - 1789 * q^16 + 365 * q^17 - 293 * q^18 - 4200 * q^19 - 3452 * q^21 + 2302 * q^22 + 1315 * q^23 - 2784 * q^24 - 2753 * q^26 + 1078 * q^27 - 8067 * q^28 - 14052 * q^29 - 4479 * q^31 + 6527 * q^32 - 8626 * q^33 - 9432 * q^34 - 5709 * q^36 + 25156 * q^37 + 2391 * q^38 - 7456 * q^39 - 32227 * q^41 - 630 * q^42 - 24037 * q^43 - 52652 * q^44 - 28700 * q^46 + 28390 * q^47 + 15504 * q^48 - 48915 * q^49 - 50744 * q^51 + 32261 * q^52 + 66177 * q^53 - 81742 * q^54 - 98379 * q^56 + 126280 * q^57 + 72353 * q^58 - 103588 * q^59 - 105406 * q^61 + 120123 * q^62 + 100962 * q^63 - 148957 * q^64 - 124146 * q^66 + 139241 * q^67 + 151020 * q^68 - 166714 * q^69 - 72726 * q^71 - 74361 * q^72 + 140142 * q^73 - 153906 * q^74 - 30223 * q^76 + 74050 * q^77 + 18812 * q^78 - 41076 * q^79 - 219547 * q^81 - 16483 * q^82 - 106623 * q^83 + 59732 * q^84 - 12943 * q^86 + 41238 * q^87 - 23748 * q^88 - 180646 * q^89 - 181694 * q^91 - 193132 * q^92 - 137950 * q^93 - 125382 * q^94 + 185312 * q^96 + 119291 * q^97 + 141778 * q^98 - 14335 * q^99

Introduction

L-functions

Modular forms

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Database

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