LMFDB - Embedded newform 75.8.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [75,8,Mod(1,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	75.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$23.4288769113$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{31}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 31 $ x^2 - 31
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.56776$ of defining polynomial
Character	$\chi$	$=$	75.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-15.1355 q^{2} +27.0000 q^{3} +101.084 q^{4} -408.659 q^{6} -198.897 q^{7} +407.384 q^{8} +729.000 q^{9} -5268.05 q^{11} +2729.27 q^{12} -1212.88 q^{13} +3010.42 q^{14} -19104.8 q^{16} +34583.5 q^{17} -11033.8 q^{18} +18611.2 q^{19} -5370.23 q^{21} +79734.7 q^{22} +33322.1 q^{23} +10999.4 q^{24} +18357.6 q^{26} +19683.0 q^{27} -20105.4 q^{28} -178577. q^{29} -237212. q^{31} +237015. q^{32} -142237. q^{33} -523440. q^{34} +73690.4 q^{36} -482947. q^{37} -281690. q^{38} -32747.9 q^{39} +293643. q^{41} +81281.3 q^{42} -443299. q^{43} -532517. q^{44} -504348. q^{46} +48179.6 q^{47} -515829. q^{48} -783983. q^{49} +933755. q^{51} -122603. q^{52} -1.66310e6 q^{53} -297913. q^{54} -81027.7 q^{56} +502501. q^{57} +2.70285e6 q^{58} +1.75081e6 q^{59} -3.15141e6 q^{61} +3.59033e6 q^{62} -144996. q^{63} -1.14194e6 q^{64} +2.15284e6 q^{66} -2.29198e6 q^{67} +3.49585e6 q^{68} +899697. q^{69} -2.71218e6 q^{71} +296983. q^{72} +2.67756e6 q^{73} +7.30965e6 q^{74} +1.88129e6 q^{76} +1.04780e6 q^{77} +495656. q^{78} +3.44864e6 q^{79} +531441. q^{81} -4.44444e6 q^{82} -1.71969e6 q^{83} -542846. q^{84} +6.70957e6 q^{86} -4.82157e6 q^{87} -2.14612e6 q^{88} +3.52224e6 q^{89} +241239. q^{91} +3.36834e6 q^{92} -6.40472e6 q^{93} -729223. q^{94} +6.39942e6 q^{96} -1.44653e7 q^{97} +1.18660e7 q^{98} -3.84041e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 54 q^{3} + 24 q^{4} - 216 q^{6} - 86 q^{7} - 1056 q^{8} + 1458 q^{9} - 4612 q^{11} + 648 q^{12} - 9018 q^{13} + 3816 q^{14} - 19680 q^{16} + 19948 q^{17} - 5832 q^{18} - 23934 q^{19} - 2322 q^{21}+ \cdots - 3362148 q^{99}+O(q^{100}) $ 2 * q - 8 * q^2 + 54 * q^3 + 24 * q^4 - 216 * q^6 - 86 * q^7 - 1056 * q^8 + 1458 * q^9 - 4612 * q^11 + 648 * q^12 - 9018 * q^13 + 3816 * q^14 - 19680 * q^16 + 19948 * q^17 - 5832 * q^18 - 23934 * q^19 - 2322 * q^21 + 84416 * q^22 - 19812 * q^23 - 28512 * q^24 - 37336 * q^26 + 39366 * q^27 - 28808 * q^28 - 346508 * q^29 - 129178 * q^31 + 420224 * q^32 - 124524 * q^33 - 627872 * q^34 + 17496 * q^36 - 58036 * q^37 - 585272 * q^38 - 243486 * q^39 - 340928 * q^41 + 103032 * q^42 - 1087350 * q^43 - 583088 * q^44 - 883488 * q^46 + 448420 * q^47 - 531360 * q^48 - 1594780 * q^49 + 538596 * q^51 + 479048 * q^52 - 728552 * q^53 - 157464 * q^54 - 246240 * q^56 - 646218 * q^57 + 1504576 * q^58 + 907844 * q^59 - 478410 * q^61 + 4361208 * q^62 - 62694 * q^63 + 238976 * q^64 + 2279232 * q^66 - 4514702 * q^67 + 4624016 * q^68 - 534924 * q^69 - 2607296 * q^71 - 769824 * q^72 + 4070268 * q^73 + 10341616 * q^74 + 5160856 * q^76 + 1121868 * q^77 - 1008072 * q^78 + 6227360 * q^79 + 1062882 * q^81 - 8972432 * q^82 + 5769084 * q^83 - 777816 * q^84 + 2113928 * q^86 - 9355716 * q^87 - 3106176 * q^88 + 16356096 * q^89 - 639938 * q^91 + 7464144 * q^92 - 3487806 * q^93 + 2126704 * q^94 + 11346048 * q^96 - 17433442 * q^97 + 6080528 * q^98 - 3362148 * q^99

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 $

$2$	−15.1355	−1.33780	−0.668902	−	0.743350i	$-0.733236\pi$
$2$	−15.1355	−1.33780	−0.668902	+	0.743350i	$0.733236\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	101.084	0.789721
$5$	0	0
$5$	0	0
$6$	−408.659	−0.772382
$7$	−198.897	−0.219172	−0.109586	−	0.993977i	$-0.534953\pi$
$7$	−198.897	−0.219172	−0.109586	+	0.993977i	$0.534953\pi$
$8$	407.384	0.281313
$9$	729.000	0.333333
$10$	0	0
$11$	−5268.05	−1.19337	−0.596686	−	0.802475i	$-0.703516\pi$
$11$	−5268.05	−1.19337	−0.596686	+	0.802475i	$0.703516\pi$
$12$	2729.27	0.455945
$13$	−1212.88	−0.153115	−0.0765574	−	0.997065i	$-0.524393\pi$
$13$	−1212.88	−0.153115	−0.0765574	+	0.997065i	$0.524393\pi$
$14$	3010.42	0.293210
$15$	0	0
$16$	−19104.8	−1.16606
$17$	34583.5	1.70725	0.853627	−	0.520885i	$-0.174398\pi$
$17$	34583.5	1.70725	0.853627	+	0.520885i	$0.174398\pi$
$18$	−11033.8	−0.445935
$19$	18611.2	0.622495	0.311248	−	0.950329i	$-0.399253\pi$
$19$	18611.2	0.622495	0.311248	+	0.950329i	$0.399253\pi$
$20$	0	0
$21$	−5370.23	−0.126539
$22$	79734.7	1.59650
$23$	33322.1	0.571065	0.285532	−	0.958369i	$-0.407830\pi$
$23$	33322.1	0.571065	0.285532	+	0.958369i	$0.407830\pi$
$24$	10999.4	0.162416
$25$	0	0
$26$	18357.6	0.204838
$27$	19683.0	0.192450
$28$	−20105.4	−0.173085
$29$	−178577.	−1.35967	−0.679833	−	0.733367i	$-0.737948\pi$
$29$	−178577.	−1.35967	−0.679833	+	0.733367i	$0.737948\pi$
$30$	0	0
$31$	−237212.	−1.43011	−0.715057	−	0.699066i	$-0.753599\pi$
$31$	−237212.	−1.43011	−0.715057	+	0.699066i	$0.753599\pi$
$32$	237015.	1.27865
$33$	−142237.	−0.688993
$34$	−523440.	−2.28397
$35$	0	0
$36$	73690.4	0.263240
$37$	−482947.	−1.56745	−0.783724	−	0.621109i	$-0.786682\pi$
$37$	−482947.	−1.56745	−0.783724	+	0.621109i	$0.786682\pi$
$38$	−281690.	−0.832777
$39$	−32747.9	−0.0884009
$40$	0	0
$41$	293643.	0.665389	0.332694	−	0.943035i	$-0.392042\pi$
$41$	293643.	0.665389	0.332694	+	0.943035i	$0.392042\pi$
$42$	81281.3	0.169285
$43$	−443299.	−0.850271	−0.425136	−	0.905130i	$-0.639774\pi$
$43$	−443299.	−0.850271	−0.425136	+	0.905130i	$0.639774\pi$
$44$	−532517.	−0.942430
$45$	0	0
$46$	−504348.	−0.763973
$47$	48179.6	0.0676894	0.0338447	−	0.999427i	$-0.489225\pi$
$47$	48179.6	0.0676894	0.0338447	+	0.999427i	$0.489225\pi$
$48$	−515829.	−0.673226
$49$	−783983.	−0.951963
$50$	0	0
$51$	933755.	0.985683
$52$	−122603.	−0.120918
$53$	−1.66310e6	−1.53445	−0.767226	−	0.641377i	$-0.778364\pi$
$53$	−1.66310e6	−1.53445	−0.767226	+	0.641377i	$0.778364\pi$
$54$	−297913.	−0.257461
$55$	0	0
$56$	−81027.7	−0.0616560
$57$	502501.	0.359398
$58$	2.70285e6	1.81897
$59$	1.75081e6	1.10983	0.554916	−	0.831907i	$-0.312750\pi$
$59$	1.75081e6	1.10983	0.554916	+	0.831907i	$0.312750\pi$
$60$	0	0
$61$	−3.15141e6	−1.77767	−0.888835	−	0.458228i	$-0.848484\pi$
$61$	−3.15141e6	−1.77767	−0.888835	+	0.458228i	$0.848484\pi$
$62$	3.59033e6	1.91321
$63$	−144996.	−0.0730575
$64$	−1.14194e6	−0.544522
$65$	0	0
$66$	2.15284e6	0.921738
$67$	−2.29198e6	−0.931000	−0.465500	−	0.885048i	$-0.654125\pi$
$67$	−2.29198e6	−0.931000	−0.465500	+	0.885048i	$0.654125\pi$
$68$	3.49585e6	1.34825
$69$	899697.	0.329704
$70$	0	0
$71$	−2.71218e6	−0.899320	−0.449660	−	0.893200i	$-0.648455\pi$
$71$	−2.71218e6	−0.899320	−0.449660	+	0.893200i	$0.648455\pi$
$72$	296983.	0.0937709
$73$	2.67756e6	0.805582	0.402791	−	0.915292i	$-0.368040\pi$
$73$	2.67756e6	0.805582	0.402791	+	0.915292i	$0.368040\pi$
$74$	7.30965e6	2.09694
$75$	0	0
$76$	1.88129e6	0.491597
$77$	1.04780e6	0.261554
$78$	495656.	0.118263
$79$	3.44864e6	0.786960	0.393480	−	0.919333i	$-0.371271\pi$
$79$	3.44864e6	0.786960	0.393480	+	0.919333i	$0.371271\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	−4.44444e6	−0.890160
$83$	−1.71969e6	−0.330123	−0.165062	−	0.986283i	$-0.552782\pi$
$83$	−1.71969e6	−0.330123	−0.165062	+	0.986283i	$0.552782\pi$
$84$	−542846.	−0.0999307
$85$	0	0
$86$	6.70957e6	1.13750
$87$	−4.82157e6	−0.785003
$88$	−2.14612e6	−0.335711
$89$	3.52224e6	0.529607	0.264804	−	0.964302i	$-0.414693\pi$
$89$	3.52224e6	0.529607	0.264804	+	0.964302i	$0.414693\pi$
$90$	0	0
$91$	241239.	0.0335586
$92$	3.36834e6	0.450981
$93$	−6.40472e6	−0.825677
$94$	−729223.	−0.0905551
$95$	0	0
$96$	6.39942e6	0.738229
$97$	−1.44653e7	−1.60926	−0.804632	−	0.593774i	$-0.797637\pi$
$97$	−1.44653e7	−1.60926	−0.804632	+	0.593774i	$0.797637\pi$
$98$	1.18660e7	1.27354
$99$	−3.84041e6	−0.397790
$100$	0	0
$101$	−1.03861e7	−1.00307	−0.501533	−	0.865138i	$-0.667230\pi$
$101$	−1.03861e7	−1.00307	−0.501533	+	0.865138i	$0.667230\pi$
$102$	−1.41329e7	−1.31865
$103$	−5.74920e6	−0.518414	−0.259207	−	0.965822i	$-0.583461\pi$
$103$	−5.74920e6	−0.518414	−0.259207	+	0.965822i	$0.583461\pi$
$104$	−494110.	−0.0430732
$105$	0	0
$106$	2.51719e7	2.05280
$107$	−814648.	−0.0642876	−0.0321438	−	0.999483i	$-0.510233\pi$
$107$	−814648.	−0.0642876	−0.0321438	+	0.999483i	$0.510233\pi$
$108$	1.98964e6	0.151982
$109$	1.19570e6	0.0884362	0.0442181	−	0.999022i	$-0.485920\pi$
$109$	1.19570e6	0.0884362	0.0442181	+	0.999022i	$0.485920\pi$
$110$	0	0
$111$	−1.30396e7	−0.904967
$112$	3.79989e6	0.255569
$113$	2.01650e7	1.31469	0.657345	−	0.753590i	$-0.271679\pi$
$113$	2.01650e7	1.31469	0.657345	+	0.753590i	$0.271679\pi$
$114$	−7.60562e6	−0.480804
$115$	0	0
$116$	−1.80513e7	−1.07376
$117$	−884192.	−0.0510383
$118$	−2.64994e7	−1.48474
$119$	−6.87857e6	−0.374183
$120$	0	0
$121$	8.26519e6	0.424135
$122$	4.76983e7	2.37817
$123$	7.92835e6	0.384162
$124$	−2.39784e7	−1.12939
$125$	0	0
$126$	2.19459e6	0.0977366
$127$	3.18561e7	1.38000	0.690000	−	0.723809i	$-0.257611\pi$
$127$	3.18561e7	1.38000	0.690000	+	0.723809i	$0.257611\pi$
$128$	−1.30540e7	−0.550187
$129$	−1.19691e7	−0.490904
$130$	0	0
$131$	−4.60762e7	−1.79072	−0.895359	−	0.445346i	$-0.853081\pi$
$131$	−4.60762e7	−1.79072	−0.895359	+	0.445346i	$0.853081\pi$
$132$	−1.43780e7	−0.544112
$133$	−3.70171e6	−0.136434
$134$	3.46904e7	1.24550
$135$	0	0
$136$	1.40888e7	0.480272
$137$	3.10405e7	1.03135	0.515676	−	0.856784i	$-0.327541\pi$
$137$	3.10405e7	1.03135	0.515676	+	0.856784i	$0.327541\pi$
$138$	−1.36174e7	−0.441080
$139$	1.16619e7	0.368313	0.184157	−	0.982897i	$-0.441045\pi$
$139$	1.16619e7	0.368313	0.184157	+	0.982897i	$0.441045\pi$
$140$	0	0
$141$	1.30085e6	0.0390805
$142$	4.10503e7	1.20311
$143$	6.38953e6	0.182723
$144$	−1.39274e7	−0.388687
$145$	0	0
$146$	−4.05264e7	−1.07771
$147$	−2.11675e7	−0.549616
$148$	−4.88183e7	−1.23785
$149$	−2.42032e7	−0.599406	−0.299703	−	0.954033i	$-0.596888\pi$
$149$	−2.42032e7	−0.599406	−0.299703	+	0.954033i	$0.596888\pi$
$150$	0	0
$151$	5.56277e6	0.131484	0.0657418	−	0.997837i	$-0.479059\pi$
$151$	5.56277e6	0.131484	0.0657418	+	0.997837i	$0.479059\pi$
$152$	7.58190e6	0.175116
$153$	2.52114e7	0.569084
$154$	−1.58590e7	−0.349908
$155$	0	0
$156$	−3.31029e6	−0.0698120
$157$	−6.29132e7	−1.29746	−0.648729	−	0.761020i	$-0.724699\pi$
$157$	−6.29132e7	−1.29746	−0.648729	+	0.761020i	$0.724699\pi$
$158$	−5.21969e7	−1.05280
$159$	−4.49037e7	−0.885916
$160$	0	0
$161$	−6.62768e6	−0.125162
$162$	−8.04364e6	−0.148645
$163$	6.99767e7	1.26560	0.632800	−	0.774315i	$-0.281905\pi$
$163$	6.99767e7	1.26560	0.632800	+	0.774315i	$0.281905\pi$
$164$	2.96826e7	0.525471
$165$	0	0
$166$	2.60284e7	0.441640
$167$	−4.69310e7	−0.779744	−0.389872	−	0.920869i	$-0.627481\pi$
$167$	−4.69310e7	−0.779744	−0.389872	+	0.920869i	$0.627481\pi$
$168$	−2.18775e6	−0.0355971
$169$	−6.12774e7	−0.976556
$170$	0	0
$171$	1.35675e7	0.207498
$172$	−4.48106e7	−0.671477
$173$	6.98991e7	1.02638	0.513192	−	0.858274i	$-0.328463\pi$
$173$	6.98991e7	1.02638	0.513192	+	0.858274i	$0.328463\pi$
$174$	7.29771e7	1.05018
$175$	0	0
$176$	1.00645e8	1.39154
$177$	4.72719e7	0.640761
$178$	−5.33109e7	−0.708511
$179$	−7.07028e7	−0.921406	−0.460703	−	0.887554i	$-0.652403\pi$
$179$	−7.07028e7	−0.921406	−0.460703	+	0.887554i	$0.652403\pi$
$180$	0	0
$181$	−1.13942e7	−0.142826	−0.0714130	−	0.997447i	$-0.522751\pi$
$181$	−1.13942e7	−0.142826	−0.0714130	+	0.997447i	$0.522751\pi$
$182$	−3.65129e6	−0.0448948
$183$	−8.50882e7	−1.02634
$184$	1.35749e7	0.160648
$185$	0	0
$186$	9.69389e7	1.10459
$187$	−1.82188e8	−2.03739
$188$	4.87019e6	0.0534557
$189$	−3.91490e6	−0.0421798
$190$	0	0
$191$	9.68535e7	1.00577	0.502885	−	0.864353i	$-0.332272\pi$
$191$	9.68535e7	1.00577	0.502885	+	0.864353i	$0.332272\pi$
$192$	−3.08325e7	−0.314380
$193$	1.75300e8	1.75522	0.877610	−	0.479375i	$-0.159137\pi$
$193$	1.75300e8	1.75522	0.877610	+	0.479375i	$0.159137\pi$
$194$	2.18940e8	2.15288
$195$	0	0
$196$	−7.92483e7	−0.751785
$197$	3.23559e7	0.301524	0.150762	−	0.988570i	$-0.451827\pi$
$197$	3.23559e7	0.301524	0.150762	+	0.988570i	$0.451827\pi$
$198$	5.81266e7	0.532166
$199$	1.87609e8	1.68760	0.843798	−	0.536661i	$-0.180315\pi$
$199$	1.87609e8	1.68760	0.843798	+	0.536661i	$0.180315\pi$
$200$	0	0
$201$	−6.18835e7	−0.537513
$202$	1.57200e8	1.34191
$203$	3.55185e7	0.298001
$204$	9.43879e7	0.778414
$205$	0	0
$206$	8.70172e7	0.693537
$207$	2.42918e7	0.190355
$208$	2.31718e7	0.178541
$209$	−9.80445e7	−0.742868
$210$	0	0
$211$	−2.34640e8	−1.71954	−0.859771	−	0.510680i	$-0.829394\pi$
$211$	−2.34640e8	−1.71954	−0.859771	+	0.510680i	$0.829394\pi$
$212$	−1.68113e8	−1.21179
$213$	−7.32289e7	−0.519223
$214$	1.23301e7	0.0860042
$215$	0	0
$216$	8.01855e6	0.0541387
$217$	4.71808e7	0.313442
$218$	−1.80976e7	−0.118310
$219$	7.22943e7	0.465103
$220$	0	0
$221$	−4.19458e7	−0.261406
$222$	1.97361e8	1.21067
$223$	1.06824e8	0.645061	0.322530	−	0.946559i	$-0.395467\pi$
$223$	1.06824e8	0.645061	0.322530	+	0.946559i	$0.395467\pi$
$224$	−4.71418e7	−0.280245
$225$	0	0
$226$	−3.05208e8	−1.75880
$227$	1.10477e8	0.626876	0.313438	−	0.949609i	$-0.398519\pi$
$227$	1.10477e8	0.626876	0.313438	+	0.949609i	$0.398519\pi$
$228$	5.07950e7	0.283824
$229$	7.64868e7	0.420884	0.210442	−	0.977606i	$-0.432510\pi$
$229$	7.64868e7	0.420884	0.210442	+	0.977606i	$0.432510\pi$
$230$	0	0
$231$	2.82906e7	0.151008
$232$	−7.27494e7	−0.382491
$233$	−1.53071e8	−0.792768	−0.396384	−	0.918085i	$-0.629735\pi$
$233$	−1.53071e8	−0.792768	−0.396384	+	0.918085i	$0.629735\pi$
$234$	1.33827e7	0.0682792
$235$	0	0
$236$	1.76979e8	0.876457
$237$	9.31132e7	0.454352
$238$	1.04111e8	0.500584
$239$	6.23039e7	0.295204	0.147602	−	0.989047i	$-0.452845\pi$
$239$	6.23039e7	0.295204	0.147602	+	0.989047i	$0.452845\pi$
$240$	0	0
$241$	2.48021e8	1.14138	0.570688	−	0.821167i	$-0.306677\pi$
$241$	2.48021e8	1.14138	0.570688	+	0.821167i	$0.306677\pi$
$242$	−1.25098e8	−0.567409
$243$	1.43489e7	0.0641500
$244$	−3.18558e8	−1.40386
$245$	0	0
$246$	−1.20000e8	−0.513934
$247$	−2.25732e7	−0.0953132
$248$	−9.66365e7	−0.402309
$249$	−4.64316e7	−0.190597
$250$	0	0
$251$	2.23677e7	0.0892819	0.0446409	−	0.999003i	$-0.485786\pi$
$251$	2.23677e7	0.0892819	0.0446409	+	0.999003i	$0.485786\pi$
$252$	−1.46568e7	−0.0576950
$253$	−1.75543e8	−0.681492
$254$	−4.82158e8	−1.84617
$255$	0	0
$256$	3.43749e8	1.28056
$257$	−2.68205e8	−0.985601	−0.492800	−	0.870142i	$-0.664027\pi$
$257$	−2.68205e8	−0.985601	−0.492800	+	0.870142i	$0.664027\pi$
$258$	1.81158e8	0.656734
$259$	9.60568e7	0.343541
$260$	0	0
$261$	−1.30182e8	−0.453222
$262$	6.97388e8	2.39563
$263$	−4.38844e8	−1.48753	−0.743763	−	0.668443i	$-0.766961\pi$
$263$	−4.38844e8	−1.48753	−0.743763	+	0.668443i	$0.766961\pi$
$264$	−5.79453e7	−0.193823
$265$	0	0
$266$	5.60274e7	0.182522
$267$	9.51005e7	0.305769
$268$	−2.31683e8	−0.735229
$269$	−4.50968e8	−1.41258	−0.706289	−	0.707923i	$-0.749632\pi$
$269$	−4.50968e8	−1.41258	−0.706289	+	0.707923i	$0.749632\pi$
$270$	0	0
$271$	−3.75385e7	−0.114574	−0.0572868	−	0.998358i	$-0.518245\pi$
$271$	−3.75385e7	−0.114574	−0.0572868	+	0.998358i	$0.518245\pi$
$272$	−6.60710e8	−1.99076
$273$	6.51346e6	0.0193750
$274$	−4.69814e8	−1.37975
$275$	0	0
$276$	9.09452e7	0.260374
$277$	−1.77406e7	−0.0501521	−0.0250760	−	0.999686i	$-0.507983\pi$
$277$	−1.77406e7	−0.0501521	−0.0250760	+	0.999686i	$0.507983\pi$
$278$	−1.76509e8	−0.492731
$279$	−1.72928e8	−0.476705
$280$	0	0
$281$	−2.17802e8	−0.585585	−0.292792	−	0.956176i	$-0.594585\pi$
$281$	−2.17802e8	−0.585585	−0.292792	+	0.956176i	$0.594585\pi$
$282$	−1.96890e7	−0.0522820
$283$	−4.34127e8	−1.13858	−0.569291	−	0.822136i	$-0.692782\pi$
$283$	−4.34127e8	−1.13858	−0.569291	+	0.822136i	$0.692782\pi$
$284$	−2.74159e8	−0.710212
$285$	0	0
$286$	−9.67089e7	−0.244447
$287$	−5.84047e7	−0.145835
$288$	1.72784e8	0.426217
$289$	7.85681e8	1.91471
$290$	0	0
$291$	−3.90564e8	−0.929109
$292$	2.70660e8	0.636185
$293$	3.93901e8	0.914850	0.457425	−	0.889248i	$-0.348772\pi$
$293$	3.93901e8	0.914850	0.457425	+	0.889248i	$0.348772\pi$
$294$	3.20382e8	0.735279
$295$	0	0
$296$	−1.96745e8	−0.440943
$297$	−1.03691e8	−0.229664
$298$	3.66328e8	0.801888
$299$	−4.04159e7	−0.0874385
$300$	0	0
$301$	8.81711e7	0.186356
$302$	−8.41955e7	−0.175899
$303$	−2.80426e8	−0.579121
$304$	−3.55562e8	−0.725868
$305$	0	0
$306$	−3.81588e8	−0.761324
$307$	1.78228e8	0.351554	0.175777	−	0.984430i	$-0.443756\pi$
$307$	1.78228e8	0.351554	0.175777	+	0.984430i	$0.443756\pi$
$308$	1.05916e8	0.206555
$309$	−1.55228e8	−0.299307
$310$	0	0
$311$	3.51118e8	0.661899	0.330949	−	0.943648i	$-0.392631\pi$
$311$	3.51118e8	0.661899	0.330949	+	0.943648i	$0.392631\pi$
$312$	−1.33410e7	−0.0248683
$313$	−4.90233e8	−0.903644	−0.451822	−	0.892108i	$-0.649226\pi$
$313$	−4.90233e8	−0.903644	−0.451822	+	0.892108i	$0.649226\pi$
$314$	9.52225e8	1.73574
$315$	0	0
$316$	3.48603e8	0.621478
$317$	9.76769e8	1.72220	0.861102	−	0.508433i	$-0.169775\pi$
$317$	9.76769e8	1.72220	0.861102	+	0.508433i	$0.169775\pi$
$318$	6.79642e8	1.18518
$319$	9.40752e8	1.62259
$320$	0	0
$321$	−2.19955e7	−0.0371164
$322$	1.00313e8	0.167442
$323$	6.43639e8	1.06276
$324$	5.37203e7	0.0877467
$325$	0	0
$326$	−1.05913e9	−1.69313
$327$	3.22840e7	0.0510587
$328$	1.19625e8	0.187182
$329$	−9.58279e6	−0.0148356
$330$	0	0
$331$	1.09787e9	1.66400	0.831998	−	0.554779i	$-0.187197\pi$
$331$	1.09787e9	1.66400	0.831998	+	0.554779i	$0.187197\pi$
$332$	−1.73833e8	−0.260705
$333$	−3.52068e8	−0.522483
$334$	7.10326e8	1.04315
$335$	0	0
$336$	1.02597e8	0.147553
$337$	3.75517e8	0.534472	0.267236	−	0.963631i	$-0.413890\pi$
$337$	3.75517e8	0.534472	0.267236	+	0.963631i	$0.413890\pi$
$338$	9.27466e8	1.30644
$339$	5.44455e8	0.759036
$340$	0	0
$341$	1.24964e9	1.70666
$342$	−2.05352e8	−0.277592
$343$	3.19733e8	0.427817
$344$	−1.80593e8	−0.239192
$345$	0	0
$346$	−1.05796e9	−1.37310
$347$	−3.19461e8	−0.410454	−0.205227	−	0.978714i	$-0.565793\pi$
$347$	−3.19461e8	−0.410454	−0.205227	+	0.978714i	$0.565793\pi$
$348$	−4.87385e8	−0.619933
$349$	−4.55816e8	−0.573985	−0.286992	−	0.957933i	$-0.592655\pi$
$349$	−4.55816e8	−0.573985	−0.286992	+	0.957933i	$0.592655\pi$
$350$	0	0
$351$	−2.38732e7	−0.0294670
$352$	−1.24861e9	−1.52590
$353$	−4.26817e8	−0.516452	−0.258226	−	0.966085i	$-0.583138\pi$
$353$	−4.26817e8	−0.516452	−0.258226	+	0.966085i	$0.583138\pi$
$354$	−7.15485e8	−0.857214
$355$	0	0
$356$	3.56043e8	0.418242
$357$	−1.85721e8	−0.216035
$358$	1.07012e9	1.23266
$359$	4.26515e7	0.0486524	0.0243262	−	0.999704i	$-0.492256\pi$
$359$	4.26515e7	0.0486524	0.0243262	+	0.999704i	$0.492256\pi$
$360$	0	0
$361$	−5.47496e8	−0.612500
$362$	1.72457e8	0.191073
$363$	2.23160e8	0.244874
$364$	2.43855e7	0.0265019
$365$	0	0
$366$	1.28785e9	1.37304
$367$	−1.37406e9	−1.45102	−0.725509	−	0.688212i	$-0.758396\pi$
$367$	−1.37406e9	−1.45102	−0.725509	+	0.688212i	$0.758396\pi$
$368$	−6.36611e8	−0.665897
$369$	2.14065e8	0.221796
$370$	0	0
$371$	3.30787e8	0.336310
$372$	−6.47417e8	−0.652054
$373$	1.55411e8	0.155060	0.0775302	−	0.996990i	$-0.475297\pi$
$373$	1.55411e8	0.155060	0.0775302	+	0.996990i	$0.475297\pi$
$374$	2.75751e9	2.72563
$375$	0	0
$376$	1.96276e7	0.0190419
$377$	2.16593e8	0.208185
$378$	5.92540e7	0.0564283
$379$	−1.37612e9	−1.29844	−0.649218	−	0.760603i	$-0.724904\pi$
$379$	−1.37612e9	−1.29844	−0.649218	+	0.760603i	$0.724904\pi$
$380$	0	0
$381$	8.60114e8	0.796743
$382$	−1.46593e9	−1.34552
$383$	1.45533e9	1.32362	0.661812	−	0.749669i	$-0.269788\pi$
$383$	1.45533e9	1.32362	0.661812	+	0.749669i	$0.269788\pi$
$384$	−3.52459e8	−0.317650
$385$	0	0
$386$	−2.65326e9	−2.34814
$387$	−3.23165e8	−0.283424
$388$	−1.46222e9	−1.27087
$389$	−8.43958e8	−0.726938	−0.363469	−	0.931606i	$-0.618408\pi$
$389$	−8.43958e8	−0.726938	−0.363469	+	0.931606i	$0.618408\pi$
$390$	0	0
$391$	1.15240e9	0.974952
$392$	−3.19382e8	−0.267799
$393$	−1.24406e9	−1.03387
$394$	−4.89724e8	−0.403380
$395$	0	0
$396$	−3.88205e8	−0.314143
$397$	1.62020e8	0.129958	0.0649788	−	0.997887i	$-0.479302\pi$
$397$	1.62020e8	0.129958	0.0649788	+	0.997887i	$0.479302\pi$
$398$	−2.83957e9	−2.25767
$399$	−9.99462e7	−0.0787701
$400$	0	0
$401$	9.76942e8	0.756595	0.378298	−	0.925684i	$-0.376510\pi$
$401$	9.76942e8	0.756595	0.378298	+	0.925684i	$0.376510\pi$
$402$	9.36640e8	0.719087
$403$	2.87710e8	0.218972
$404$	−1.04988e9	−0.792142
$405$	0	0
$406$	−5.37591e8	−0.398667
$407$	2.54419e9	1.87055
$408$	3.80397e8	0.277285
$409$	−2.23718e9	−1.61685	−0.808426	−	0.588598i	$-0.799680\pi$
$409$	−2.23718e9	−1.61685	−0.808426	+	0.588598i	$0.799680\pi$
$410$	0	0
$411$	8.38094e8	0.595451
$412$	−5.81153e8	−0.409403
$413$	−3.48232e8	−0.243244
$414$	−3.67670e8	−0.254658
$415$	0	0
$416$	−2.87472e8	−0.195780
$417$	3.14871e8	0.212646
$418$	1.48396e9	0.993812
$419$	2.75074e9	1.82684	0.913421	−	0.407016i	$-0.133431\pi$
$419$	2.75074e9	1.82684	0.913421	+	0.407016i	$0.133431\pi$
$420$	0	0
$421$	−2.91092e9	−1.90126	−0.950632	−	0.310320i	$-0.899564\pi$
$421$	−2.91092e9	−1.90126	−0.950632	+	0.310320i	$0.899564\pi$
$422$	3.55140e9	2.30041
$423$	3.51229e7	0.0225631
$424$	−6.77522e8	−0.431661
$425$	0	0
$426$	1.10836e9	0.694619
$427$	6.26808e8	0.389616
$428$	−8.23481e7	−0.0507692
$429$	1.72517e8	0.105495
$430$	0	0
$431$	3.53062e8	0.212412	0.106206	−	0.994344i	$-0.466130\pi$
$431$	3.53062e8	0.212412	0.106206	+	0.994344i	$0.466130\pi$
$432$	−3.76039e8	−0.224409
$433$	−3.53908e8	−0.209499	−0.104750	−	0.994499i	$-0.533404\pi$
$433$	−3.53908e8	−0.209499	−0.104750	+	0.994499i	$0.533404\pi$
$434$	−7.14107e8	−0.419324
$435$	0	0
$436$	1.20867e8	0.0698399
$437$	6.20163e8	0.355485
$438$	−1.09421e9	−0.622217
$439$	1.69079e9	0.953812	0.476906	−	0.878954i	$-0.341758\pi$
$439$	1.69079e9	0.953812	0.476906	+	0.878954i	$0.341758\pi$
$440$	0	0
$441$	−5.71523e8	−0.317321
$442$	6.34872e8	0.349710
$443$	2.84211e9	1.55320	0.776601	−	0.629993i	$-0.216942\pi$
$443$	2.84211e9	1.55320	0.776601	+	0.629993i	$0.216942\pi$
$444$	−1.31809e9	−0.714671
$445$	0	0
$446$	−1.61683e9	−0.862965
$447$	−6.53487e8	−0.346067
$448$	2.27130e8	0.119344
$449$	−2.35632e9	−1.22849	−0.614245	−	0.789115i	$-0.710539\pi$
$449$	−2.35632e9	−1.22849	−0.614245	+	0.789115i	$0.710539\pi$
$450$	0	0
$451$	−1.54692e9	−0.794056
$452$	2.03836e9	1.03824
$453$	1.50195e8	0.0759121
$454$	−1.67213e9	−0.838638
$455$	0	0
$456$	2.04711e8	0.101103
$457$	2.00995e9	0.985094	0.492547	−	0.870286i	$-0.336066\pi$
$457$	2.00995e9	0.985094	0.492547	+	0.870286i	$0.336066\pi$
$458$	−1.15767e9	−0.563061
$459$	6.80707e8	0.328561
$460$	0	0
$461$	−2.79024e9	−1.32644	−0.663222	−	0.748423i	$-0.730811\pi$
$461$	−2.79024e9	−1.32644	−0.663222	+	0.748423i	$0.730811\pi$
$462$	−4.28194e8	−0.202020
$463$	−8.25871e8	−0.386704	−0.193352	−	0.981129i	$-0.561936\pi$
$463$	−8.25871e8	−0.386704	−0.193352	+	0.981129i	$0.561936\pi$
$464$	3.41167e9	1.58545
$465$	0	0
$466$	2.31681e9	1.06057
$467$	−1.39293e8	−0.0632878	−0.0316439	−	0.999499i	$-0.510074\pi$
$467$	−1.39293e8	−0.0632878	−0.0316439	+	0.999499i	$0.510074\pi$
$468$	−8.93779e7	−0.0403060
$469$	4.55869e8	0.204049
$470$	0	0
$471$	−1.69866e9	−0.749087
$472$	7.13253e8	0.312210
$473$	2.33532e9	1.01469
$474$	−1.40932e9	−0.607833
$475$	0	0
$476$	−6.95315e8	−0.295500
$477$	−1.21240e9	−0.511484
$478$	−9.43002e8	−0.394925
$479$	−4.59217e9	−1.90917	−0.954583	−	0.297946i	$-0.903699\pi$
$479$	−4.59217e9	−1.90917	−0.954583	+	0.297946i	$0.903699\pi$
$480$	0	0
$481$	5.85758e8	0.240000
$482$	−3.75393e9	−1.52694
$483$	−1.78947e8	−0.0722621
$484$	8.35480e8	0.334948
$485$	0	0
$486$	−2.17178e8	−0.0858202
$487$	−3.28997e9	−1.29075	−0.645373	−	0.763867i	$-0.723298\pi$
$487$	−3.28997e9	−1.29075	−0.645373	+	0.763867i	$0.723298\pi$
$488$	−1.28384e9	−0.500081
$489$	1.88937e9	0.730695
$490$	0	0
$491$	4.19357e9	1.59882	0.799409	−	0.600787i	$-0.205146\pi$
$491$	4.19357e9	1.59882	0.799409	+	0.600787i	$0.205146\pi$
$492$	8.01431e8	0.303381
$493$	−6.17581e9	−2.32129
$494$	3.41657e8	0.127510
$495$	0	0
$496$	4.53188e9	1.66760
$497$	5.39446e8	0.197106
$498$	7.02766e8	0.254981
$499$	4.22173e8	0.152103	0.0760517	−	0.997104i	$-0.475769\pi$
$499$	4.22173e8	0.152103	0.0760517	+	0.997104i	$0.475769\pi$
$500$	0	0
$501$	−1.26714e9	−0.450186
$502$	−3.38547e8	−0.119442
$503$	2.35612e9	0.825487	0.412743	−	0.910847i	$-0.364571\pi$
$503$	2.35612e9	0.825487	0.412743	+	0.910847i	$0.364571\pi$
$504$	−5.90692e7	−0.0205520
$505$	0	0
$506$	2.65693e9	0.911703
$507$	−1.65449e9	−0.563815
$508$	3.22015e9	1.08981
$509$	3.48158e9	1.17021	0.585105	−	0.810957i	$-0.301053\pi$
$509$	3.48158e9	1.17021	0.585105	+	0.810957i	$0.301053\pi$
$510$	0	0
$511$	−5.32561e8	−0.176561
$512$	−3.53190e9	−1.16296
$513$	3.66323e8	0.119799
$514$	4.05943e9	1.31854
$515$	0	0
$516$	−1.20989e9	−0.387677
$517$	−2.53812e8	−0.0807785
$518$	−1.45387e9	−0.459591
$519$	1.88728e9	0.592584
$520$	0	0
$521$	−1.50714e9	−0.466898	−0.233449	−	0.972369i	$-0.575001\pi$
$521$	−1.50714e9	−0.466898	−0.233449	+	0.972369i	$0.575001\pi$
$522$	1.97038e9	0.606322
$523$	−2.71312e9	−0.829302	−0.414651	−	0.909981i	$-0.636096\pi$
$523$	−2.71312e9	−0.829302	−0.414651	+	0.909981i	$0.636096\pi$
$524$	−4.65758e9	−1.41417
$525$	0	0
$526$	6.64213e9	1.99002
$527$	−8.20362e9	−2.44157
$528$	2.71741e9	0.803409
$529$	−2.29446e9	−0.673885
$530$	0	0
$531$	1.27634e9	0.369944
$532$	−3.74185e8	−0.107745
$533$	−3.56154e8	−0.101881
$534$	−1.43940e9	−0.409059
$535$	0	0
$536$	−9.33718e8	−0.261902
$537$	−1.90898e9	−0.531974
$538$	6.82564e9	1.88975
$539$	4.13006e9	1.13605
$540$	0	0
$541$	−5.65472e8	−0.153540	−0.0767699	−	0.997049i	$-0.524461\pi$
$541$	−5.65472e8	−0.153540	−0.0767699	+	0.997049i	$0.524461\pi$
$542$	5.68165e8	0.153277
$543$	−3.07642e8	−0.0824607
$544$	8.19683e9	2.18298
$545$	0	0
$546$	−9.85847e7	−0.0259200
$547$	6.12811e9	1.60092	0.800462	−	0.599383i	$-0.204587\pi$
$547$	6.12811e9	1.60092	0.800462	+	0.599383i	$0.204587\pi$
$548$	3.13771e9	0.814480
$549$	−2.29738e9	−0.592556
$550$	0	0
$551$	−3.32352e9	−0.846385
$552$	3.66523e8	0.0927500
$553$	−6.85925e8	−0.172480
$554$	2.68513e8	0.0670936
$555$	0	0
$556$	1.17883e9	0.290864
$557$	−1.99978e9	−0.490329	−0.245165	−	0.969481i	$-0.578842\pi$
$557$	−1.99978e9	−0.490329	−0.245165	+	0.969481i	$0.578842\pi$
$558$	2.61735e9	0.637738
$559$	5.37670e8	0.130189
$560$	0	0
$561$	−4.91907e9	−1.17629
$562$	3.29655e9	0.783398
$563$	3.80780e9	0.899279	0.449639	−	0.893210i	$-0.351552\pi$
$563$	3.80780e9	0.899279	0.449639	+	0.893210i	$0.351552\pi$
$564$	1.31495e8	0.0308626
$565$	0	0
$566$	6.57074e9	1.52320
$567$	−1.05702e8	−0.0243525
$568$	−1.10490e9	−0.252990
$569$	−5.48649e9	−1.24854	−0.624269	−	0.781210i	$-0.714603\pi$
$569$	−5.48649e9	−1.24854	−0.624269	+	0.781210i	$0.714603\pi$
$570$	0	0
$571$	−5.09068e9	−1.14433	−0.572163	−	0.820140i	$-0.693895\pi$
$571$	−5.09068e9	−1.14433	−0.572163	+	0.820140i	$0.693895\pi$
$572$	6.45881e8	0.144300
$573$	2.61505e9	0.580682
$574$	8.83987e8	0.195099
$575$	0	0
$576$	−8.32478e8	−0.181507
$577$	−3.65448e8	−0.0791972	−0.0395986	−	0.999216i	$-0.512608\pi$
$577$	−3.65448e8	−0.0791972	−0.0395986	+	0.999216i	$0.512608\pi$
$578$	−1.18917e10	−2.56151
$579$	4.73310e9	1.01338
$580$	0	0
$581$	3.42041e8	0.0723539
$582$	5.91139e9	1.24297
$583$	8.76130e9	1.83117
$584$	1.09080e9	0.226621
$585$	0	0
$586$	−5.96189e9	−1.22389
$587$	−1.71450e9	−0.349867	−0.174934	−	0.984580i	$-0.555971\pi$
$587$	−1.71450e9	−0.349867	−0.174934	+	0.984580i	$0.555971\pi$
$588$	−2.13970e9	−0.434043
$589$	−4.41479e9	−0.890239
$590$	0	0
$591$	8.73609e8	0.174085
$592$	9.22658e9	1.82774
$593$	−4.03168e9	−0.793952	−0.396976	−	0.917829i	$-0.629940\pi$
$593$	−4.03168e9	−0.793952	−0.396976	+	0.917829i	$0.629940\pi$
$594$	1.56942e9	0.307246
$595$	0	0
$596$	−2.44656e9	−0.473363
$597$	5.06545e9	0.974334
$598$	6.11715e8	0.116976
$599$	6.61745e9	1.25805	0.629024	−	0.777386i	$-0.283455\pi$
$599$	6.61745e9	1.25805	0.629024	+	0.777386i	$0.283455\pi$
$600$	0	0
$601$	−3.15832e9	−0.593465	−0.296732	−	0.954961i	$-0.595897\pi$
$601$	−3.15832e9	−0.593465	−0.296732	+	0.954961i	$0.595897\pi$
$602$	−1.33452e9	−0.249308
$603$	−1.67086e9	−0.310333
$604$	5.62308e8	0.103835
$605$	0	0
$606$	4.24440e9	0.774750
$607$	−6.46297e9	−1.17293	−0.586464	−	0.809975i	$-0.699481\pi$
$607$	−6.46297e9	−1.17293	−0.586464	+	0.809975i	$0.699481\pi$
$608$	4.41113e9	0.795953
$609$	9.58998e8	0.172051
$610$	0	0
$611$	−5.84362e7	−0.0103642
$612$	2.54847e9	0.449418
$613$	−7.34328e9	−1.28759	−0.643796	−	0.765197i	$-0.722642\pi$
$613$	−7.34328e9	−1.28759	−0.643796	+	0.765197i	$0.722642\pi$
$614$	−2.69758e9	−0.470311
$615$	0	0
$616$	4.26858e8	0.0735785
$617$	2.52089e9	0.432071	0.216036	−	0.976385i	$-0.430687\pi$
$617$	2.52089e9	0.432071	0.216036	+	0.976385i	$0.430687\pi$
$618$	2.34946e9	0.400414
$619$	5.86398e9	0.993746	0.496873	−	0.867823i	$-0.334482\pi$
$619$	5.86398e9	0.993746	0.496873	+	0.867823i	$0.334482\pi$
$620$	0	0
$621$	6.55879e8	0.109901
$622$	−5.31436e9	−0.885491
$623$	−7.00564e8	−0.116075
$624$	6.25640e8	0.103081
$625$	0	0
$626$	7.41993e9	1.20890
$627$	−2.64720e9	−0.428895
$628$	−6.35953e9	−1.02463
$629$	−1.67020e10	−2.67603
$630$	0	0
$631$	−9.56879e8	−0.151619	−0.0758096	−	0.997122i	$-0.524154\pi$
$631$	−9.56879e8	−0.151619	−0.0758096	+	0.997122i	$0.524154\pi$
$632$	1.40492e9	0.221382
$633$	−6.33527e9	−0.992778
$634$	−1.47839e10	−2.30397
$635$	0	0
$636$	−4.53906e9	−0.699626
$637$	9.50880e8	0.145760
$638$	−1.42388e10	−2.17070
$639$	−1.97718e9	−0.299773
$640$	0	0
$641$	6.36740e9	0.954902	0.477451	−	0.878658i	$-0.341561\pi$
$641$	6.36740e9	0.954902	0.477451	+	0.878658i	$0.341561\pi$
$642$	3.32914e8	0.0496545
$643$	5.80000e9	0.860378	0.430189	−	0.902739i	$-0.358447\pi$
$643$	5.80000e9	0.860378	0.430189	+	0.902739i	$0.358447\pi$
$644$	−6.69954e8	−0.0988427
$645$	0	0
$646$	−9.74182e9	−1.42176
$647$	2.28138e9	0.331156	0.165578	−	0.986197i	$-0.447051\pi$
$647$	2.28138e9	0.331156	0.165578	+	0.986197i	$0.447051\pi$
$648$	2.16501e8	0.0312570
$649$	−9.22336e9	−1.32444
$650$	0	0
$651$	1.27388e9	0.180966
$652$	7.07354e9	0.999471
$653$	7.78616e9	1.09428	0.547138	−	0.837042i	$-0.315717\pi$
$653$	7.78616e9	1.09428	0.547138	+	0.837042i	$0.315717\pi$
$654$	−4.88635e8	−0.0683065
$655$	0	0
$656$	−5.60997e9	−0.775885
$657$	1.95194e9	0.268527
$658$	1.45041e8	0.0198472
$659$	−5.97501e9	−0.813279	−0.406640	−	0.913589i	$-0.633300\pi$
$659$	−5.97501e9	−0.813279	−0.406640	+	0.913589i	$0.633300\pi$
$660$	0	0
$661$	−5.31993e9	−0.716475	−0.358237	−	0.933631i	$-0.616622\pi$
$661$	−5.31993e9	−0.716475	−0.358237	+	0.933631i	$0.616622\pi$
$662$	−1.66168e10	−2.22610
$663$	−1.13254e9	−0.150923
$664$	−7.00574e8	−0.0928679
$665$	0	0
$666$	5.32874e9	0.698980
$667$	−5.95056e9	−0.776457
$668$	−4.74399e9	−0.615780
$669$	2.88424e9	0.372426
$670$	0	0
$671$	1.66018e10	2.12142
$672$	−1.27283e9	−0.161799
$673$	5.24622e9	0.663427	0.331714	−	0.943380i	$-0.392373\pi$
$673$	5.24622e9	0.663427	0.331714	+	0.943380i	$0.392373\pi$
$674$	−5.68365e9	−0.715019
$675$	0	0
$676$	−6.19418e9	−0.771206
$677$	6.58821e9	0.816032	0.408016	−	0.912975i	$-0.366221\pi$
$677$	6.58821e9	0.816032	0.408016	+	0.912975i	$0.366221\pi$
$678$	−8.24061e9	−1.01544
$679$	2.87712e9	0.352706
$680$	0	0
$681$	2.98288e9	0.361927
$682$	−1.89140e10	−2.28317
$683$	5.44287e9	0.653666	0.326833	−	0.945082i	$-0.394019\pi$
$683$	5.44287e9	0.653666	0.326833	+	0.945082i	$0.394019\pi$
$684$	1.37146e9	0.163866
$685$	0	0
$686$	−4.83932e9	−0.572335
$687$	2.06514e9	0.242998
$688$	8.46913e9	0.991469
$689$	2.01715e9	0.234947
$690$	0	0
$691$	2.06828e9	0.238472	0.119236	−	0.992866i	$-0.461956\pi$
$691$	2.06828e9	0.238472	0.119236	+	0.992866i	$0.461956\pi$
$692$	7.06570e9	0.810557
$693$	7.63847e8	0.0871847
$694$	4.83521e9	0.549107
$695$	0	0
$696$	−1.96423e9	−0.220831
$697$	1.01552e10	1.13599
$698$	6.89902e9	0.767880
$699$	−4.13291e9	−0.457705
$700$	0	0
$701$	4.56061e9	0.500045	0.250023	−	0.968240i	$-0.419562\pi$
$701$	4.56061e9	0.500045	0.250023	+	0.968240i	$0.419562\pi$
$702$	3.61333e8	0.0394210
$703$	−8.98820e9	−0.975729
$704$	6.01582e9	0.649816
$705$	0	0
$706$	6.46010e9	0.690912
$707$	2.06578e9	0.219845
$708$	4.77844e9	0.506023
$709$	−1.07854e9	−0.113652	−0.0568259	−	0.998384i	$-0.518098\pi$
$709$	−1.07854e9	−0.113652	−0.0568259	+	0.998384i	$0.518098\pi$
$710$	0	0
$711$	2.51406e9	0.262320
$712$	1.43491e9	0.148985
$713$	−7.90441e9	−0.816687
$714$	2.81099e9	0.289012
$715$	0	0
$716$	−7.14694e9	−0.727653
$717$	1.68221e9	0.170436
$718$	−6.45553e8	−0.0650873
$719$	1.63222e10	1.63767	0.818835	−	0.574029i	$-0.194620\pi$
$719$	1.63222e10	1.63767	0.818835	+	0.574029i	$0.194620\pi$
$720$	0	0
$721$	1.14350e9	0.113622
$722$	8.28665e9	0.819405
$723$	6.69656e9	0.658973
$724$	−1.15177e9	−0.112793
$725$	0	0
$726$	−3.37765e9	−0.327594
$727$	−1.01395e8	−0.00978689	−0.00489344	−	0.999988i	$-0.501558\pi$
$727$	−1.01395e8	−0.00978689	−0.00489344	+	0.999988i	$0.501558\pi$
$728$	9.82772e7	0.00944045
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−1.53309e10	−1.45163
$732$	−8.60107e9	−0.810520
$733$	1.10496e10	1.03630	0.518148	−	0.855291i	$-0.326622\pi$
$733$	1.10496e10	1.03630	0.518148	+	0.855291i	$0.326622\pi$
$734$	2.07971e10	1.94118
$735$	0	0
$736$	7.89786e9	0.730192
$737$	1.20743e10	1.11103
$738$	−3.23999e9	−0.296720
$739$	2.04328e10	1.86239	0.931197	−	0.364515i	$-0.118765\pi$
$739$	2.04328e10	1.86239	0.931197	+	0.364515i	$0.118765\pi$
$740$	0	0
$741$	−6.09476e8	−0.0550291
$742$	−5.00663e9	−0.449916
$743$	6.12561e9	0.547884	0.273942	−	0.961746i	$-0.411672\pi$
$743$	6.12561e9	0.547884	0.273942	+	0.961746i	$0.411672\pi$
$744$	−2.60918e9	−0.232273
$745$	0	0
$746$	−2.35223e9	−0.207440
$747$	−1.25365e9	−0.110041
$748$	−1.84163e10	−1.60897
$749$	1.62031e8	0.0140901
$750$	0	0
$751$	2.22759e10	1.91909	0.959544	−	0.281558i	$-0.0908510\pi$
$751$	2.22759e10	1.91909	0.959544	+	0.281558i	$0.0908510\pi$
$752$	−9.20459e8	−0.0789300
$753$	6.03928e8	0.0515469
$754$	−3.27825e9	−0.278511
$755$	0	0
$756$	−3.95734e8	−0.0333102
$757$	3.89337e8	0.0326205	0.0163102	−	0.999867i	$-0.494808\pi$
$757$	3.89337e8	0.0326205	0.0163102	+	0.999867i	$0.494808\pi$
$758$	2.08284e10	1.73705
$759$	−4.73965e9	−0.393460
$760$	0	0
$761$	−1.53303e10	−1.26097	−0.630483	−	0.776203i	$-0.717143\pi$
$761$	−1.53303e10	−1.26097	−0.630483	+	0.776203i	$0.717143\pi$
$762$	−1.30183e10	−1.06589
$763$	−2.37822e8	−0.0193828
$764$	9.79036e9	0.794277
$765$	0	0
$766$	−2.20271e10	−1.77075
$767$	−2.12353e9	−0.169932
$768$	9.28122e9	0.739334
$769$	6.57649e8	0.0521497	0.0260749	−	0.999660i	$-0.491699\pi$
$769$	6.57649e8	0.0521497	0.0260749	+	0.999660i	$0.491699\pi$
$770$	0	0
$771$	−7.24154e9	−0.569037
$772$	1.77201e10	1.38613
$773$	−2.59174e9	−0.201819	−0.100910	−	0.994896i	$-0.532175\pi$
$773$	−2.59174e9	−0.201819	−0.100910	+	0.994896i	$0.532175\pi$
$774$	4.89128e9	0.379166
$775$	0	0
$776$	−5.89295e9	−0.452706
$777$	2.59353e9	0.198344
$778$	1.27738e10	0.972501
$779$	5.46503e9	0.414201
$780$	0	0
$781$	1.42879e10	1.07322
$782$	−1.74421e10	−1.30429
$783$	−3.51493e9	−0.261668
$784$	1.49778e10	1.11005
$785$	0	0
$786$	1.88295e10	1.38312
$787$	−4.52650e9	−0.331017	−0.165509	−	0.986208i	$-0.552927\pi$
$787$	−4.52650e9	−0.331017	−0.165509	+	0.986208i	$0.552927\pi$
$788$	3.27067e9	0.238120
$789$	−1.18488e10	−0.858824
$790$	0	0
$791$	−4.01076e9	−0.288144
$792$	−1.56452e9	−0.111904
$793$	3.82230e9	0.272188
$794$	−2.45226e9	−0.173858
$795$	0	0
$796$	1.89643e10	1.33273
$797$	−2.39388e9	−0.167494	−0.0837468	−	0.996487i	$-0.526689\pi$
$797$	−2.39388e9	−0.167494	−0.0837468	+	0.996487i	$0.526689\pi$
$798$	1.51274e9	0.105379
$799$	1.66622e9	0.115563
$800$	0	0
$801$	2.56771e9	0.176536
$802$	−1.47865e10	−1.01218
$803$	−1.41055e10	−0.961359
$804$	−6.25545e9	−0.424485
$805$	0	0
$806$	−4.35465e9	−0.292941
$807$	−1.21761e10	−0.815552
$808$	−4.23115e9	−0.282175
$809$	−5.76341e9	−0.382701	−0.191350	−	0.981522i	$-0.561287\pi$
$809$	−5.76341e9	−0.382701	−0.191350	+	0.981522i	$0.561287\pi$
$810$	0	0
$811$	2.15723e10	1.42012	0.710059	−	0.704142i	$-0.248668\pi$
$811$	2.15723e10	1.42012	0.710059	+	0.704142i	$0.248668\pi$
$812$	3.59036e9	0.235338
$813$	−1.01354e9	−0.0661490
$814$	−3.85076e10	−2.50243
$815$	0	0
$816$	−1.78392e10	−1.14937
$817$	−8.25032e9	−0.529290
$818$	3.38610e10	2.16303
$819$	1.75864e8	0.0111862
$820$	0	0
$821$	1.56091e10	0.984414	0.492207	−	0.870478i	$-0.336190\pi$
$821$	1.56091e10	0.984414	0.492207	+	0.870478i	$0.336190\pi$
$822$	−1.26850e10	−0.796597
$823$	−9.41194e9	−0.588545	−0.294273	−	0.955722i	$-0.595077\pi$
$823$	−9.41194e9	−0.588545	−0.294273	+	0.955722i	$0.595077\pi$
$824$	−2.34213e9	−0.145837
$825$	0	0
$826$	5.27067e9	0.325414
$827$	8.53219e9	0.524555	0.262278	−	0.964992i	$-0.415526\pi$
$827$	8.53219e9	0.524555	0.262278	+	0.964992i	$0.415526\pi$
$828$	2.45552e9	0.150327
$829$	2.01245e10	1.22683	0.613413	−	0.789762i	$-0.289796\pi$
$829$	2.01245e10	1.22683	0.613413	+	0.789762i	$0.289796\pi$
$830$	0	0
$831$	−4.78996e8	−0.0289553
$832$	1.38505e9	0.0833744
$833$	−2.71129e10	−1.62524
$834$	−4.76574e9	−0.284478
$835$	0	0
$836$	−9.91076e9	−0.586658
$837$	−4.66904e9	−0.275226
$838$	−4.16340e10	−2.44396
$839$	−6.92191e9	−0.404631	−0.202316	−	0.979320i	$-0.564847\pi$
$839$	−6.92191e9	−0.404631	−0.202316	+	0.979320i	$0.564847\pi$
$840$	0	0
$841$	1.46398e10	0.848690
$842$	4.40583e10	2.54352
$843$	−5.88066e9	−0.338088
$844$	−2.37184e10	−1.35796
$845$	0	0
$846$	−5.31604e8	−0.0301850
$847$	−1.64392e9	−0.0929587
$848$	3.17732e10	1.78927
$849$	−1.17214e10	−0.657360
$850$	0	0
$851$	−1.60928e10	−0.895114
$852$	−7.40229e9	−0.410041
$853$	−2.25584e10	−1.24447	−0.622237	−	0.782829i	$-0.713776\pi$
$853$	−2.25584e10	−1.24447	−0.622237	+	0.782829i	$0.713776\pi$
$854$	−9.48707e9	−0.521230
$855$	0	0
$856$	−3.31875e8	−0.0180849
$857$	−1.21039e10	−0.656889	−0.328444	−	0.944523i	$-0.606524\pi$
$857$	−1.21039e10	−0.656889	−0.328444	+	0.944523i	$0.606524\pi$
$858$	−2.61114e9	−0.141132
$859$	4.57575e9	0.246312	0.123156	−	0.992387i	$-0.460698\pi$
$859$	4.57575e9	0.246312	0.123156	+	0.992387i	$0.460698\pi$
$860$	0	0
$861$	−1.57693e9	−0.0841978
$862$	−5.34377e9	−0.284166
$863$	−1.10973e10	−0.587730	−0.293865	−	0.955847i	$-0.594942\pi$
$863$	−1.10973e10	−0.587730	−0.293865	+	0.955847i	$0.594942\pi$
$864$	4.66517e9	0.246076
$865$	0	0
$866$	5.35658e9	0.280269
$867$	2.12134e10	1.10546
$868$	4.76924e9	0.247531
$869$	−1.81676e10	−0.939135
$870$	0	0
$871$	2.77991e9	0.142550
$872$	4.87110e8	0.0248782
$873$	−1.05452e10	−0.536421
$874$	−9.38650e9	−0.475569
$875$	0	0
$876$	7.30781e9	0.367302
$877$	1.63401e10	0.818008	0.409004	−	0.912533i	$-0.365876\pi$
$877$	1.63401e10	0.818008	0.409004	+	0.912533i	$0.365876\pi$
$878$	−2.55909e10	−1.27601
$879$	1.06353e10	0.528189
$880$	0	0
$881$	−6.59279e9	−0.324828	−0.162414	−	0.986723i	$-0.551928\pi$
$881$	−6.59279e9	−0.324828	−0.162414	+	0.986723i	$0.551928\pi$
$882$	8.65031e9	0.424514
$883$	1.56099e10	0.763025	0.381512	−	0.924364i	$-0.375403\pi$
$883$	1.56099e10	0.763025	0.381512	+	0.924364i	$0.375403\pi$
$884$	−4.24006e9	−0.206438
$885$	0	0
$886$	−4.30169e10	−2.07788
$887$	4.43368e9	0.213320	0.106660	−	0.994296i	$-0.465984\pi$
$887$	4.43368e9	0.213320	0.106660	+	0.994296i	$0.465984\pi$
$888$	−5.31211e9	−0.254579
$889$	−6.33609e9	−0.302458
$890$	0	0
$891$	−2.79966e9	−0.132597
$892$	1.07982e10	0.509418
$893$	8.96678e8	0.0421363
$894$	9.89086e9	0.462970
$895$	0	0
$896$	2.59641e9	0.120586
$897$	−1.09123e9	−0.0504826
$898$	3.56641e10	1.64348
$899$	4.23605e10	1.94448
$900$	0	0
$901$	−5.75159e10	−2.61970
$902$	2.34135e10	1.06229
$903$	2.38062e9	0.107593
$904$	8.21490e9	0.369839
$905$	0	0
$906$	−2.27328e9	−0.101556
$907$	−3.50279e10	−1.55879	−0.779397	−	0.626530i	$-0.784475\pi$
$907$	−3.50279e10	−1.55879	−0.779397	+	0.626530i	$0.784475\pi$
$908$	1.11675e10	0.495057
$909$	−7.57150e9	−0.334355
$910$	0	0
$911$	1.11229e10	0.487421	0.243711	−	0.969848i	$-0.421635\pi$
$911$	1.11229e10	0.487421	0.243711	+	0.969848i	$0.421635\pi$
$912$	−9.60017e9	−0.419080
$913$	9.05940e9	0.393960
$914$	−3.04216e10	−1.31786
$915$	0	0
$916$	7.73161e9	0.332381
$917$	9.16444e9	0.392476
$918$	−1.03029e10	−0.439550
$919$	−1.12769e10	−0.479274	−0.239637	−	0.970863i	$-0.577028\pi$
$919$	−1.12769e10	−0.479274	−0.239637	+	0.970863i	$0.577028\pi$
$920$	0	0
$921$	4.81216e9	0.202970
$922$	4.22318e10	1.77452
$923$	3.28956e9	0.137699
$924$	2.85974e9	0.119254
$925$	0	0
$926$	1.25000e10	0.517334
$927$	−4.19117e9	−0.172805
$928$	−4.23255e10	−1.73854
$929$	2.38263e10	0.974994	0.487497	−	0.873125i	$-0.337910\pi$
$929$	2.38263e10	0.974994	0.487497	+	0.873125i	$0.337910\pi$
$930$	0	0
$931$	−1.45908e10	−0.592593
$932$	−1.54730e10	−0.626065
$933$	9.48018e9	0.382147
$934$	2.10827e9	0.0846667
$935$	0	0
$936$	−3.60206e8	−0.0143577
$937$	2.68304e10	1.06546	0.532731	−	0.846285i	$-0.321166\pi$
$937$	2.68304e10	1.06546	0.532731	+	0.846285i	$0.321166\pi$
$938$	−6.89982e9	−0.272978
$939$	−1.32363e10	−0.521719
$940$	0	0
$941$	−2.71249e10	−1.06122	−0.530610	−	0.847616i	$-0.678037\pi$
$941$	−2.71249e10	−1.06122	−0.530610	+	0.847616i	$0.678037\pi$
$942$	2.57101e10	1.00213
$943$	9.78479e9	0.379980
$944$	−3.34488e10	−1.29413
$945$	0	0
$946$	−3.53464e10	−1.35746
$947$	1.61557e10	0.618158	0.309079	−	0.951036i	$-0.399979\pi$
$947$	1.61557e10	0.618158	0.309079	+	0.951036i	$0.399979\pi$
$948$	9.41228e9	0.358811
$949$	−3.24757e9	−0.123347
$950$	0	0
$951$	2.63728e10	0.994314
$952$	−2.80222e9	−0.105262
$953$	−2.62529e10	−0.982543	−0.491271	−	0.871007i	$-0.663468\pi$
$953$	−2.62529e10	−0.982543	−0.491271	+	0.871007i	$0.663468\pi$
$954$	1.83503e10	0.684266
$955$	0	0
$956$	6.29794e9	0.233129
$957$	2.54003e10	0.936800
$958$	6.95049e10	2.55409
$959$	−6.17388e9	−0.226044
$960$	0	0
$961$	2.87569e10	1.04523
$962$	−8.86576e9	−0.321073
$963$	−5.93879e8	−0.0214292
$964$	2.50710e10	0.901368
$965$	0	0
$966$	2.70846e9	0.0966726
$967$	−1.93702e10	−0.688878	−0.344439	−	0.938809i	$-0.611931\pi$
$967$	−1.93702e10	−0.688878	−0.344439	+	0.938809i	$0.611931\pi$
$968$	3.36711e9	0.119315
$969$	1.73783e10	0.613583
$970$	0	0
$971$	1.24120e10	0.435086	0.217543	−	0.976051i	$-0.430196\pi$
$971$	1.24120e10	0.435086	0.217543	+	0.976051i	$0.430196\pi$
$972$	1.45045e9	0.0506606
$973$	−2.31952e9	−0.0807241
$974$	4.97955e10	1.72677
$975$	0	0
$976$	6.02070e10	2.07287
$977$	−2.77251e10	−0.951134	−0.475567	−	0.879679i	$-0.657757\pi$
$977$	−2.77251e10	−0.951134	−0.475567	+	0.879679i	$0.657757\pi$
$978$	−2.85966e10	−0.977526
$979$	−1.85553e10	−0.632018
$980$	0	0
$981$	8.71667e8	0.0294787
$982$	−6.34720e10	−2.13891
$983$	−4.14298e8	−0.0139115	−0.00695577	−	0.999976i	$-0.502214\pi$
$983$	−4.14298e8	−0.0139115	−0.00695577	+	0.999976i	$0.502214\pi$
$984$	3.22989e9	0.108070
$985$	0	0
$986$	9.34742e10	3.10544
$987$	−2.58735e8	−0.00856536
$988$	−2.28179e9	−0.0752708
$989$	−1.47717e10	−0.485560
$990$	0	0
$991$	−4.38575e10	−1.43148	−0.715741	−	0.698366i	$-0.753911\pi$
$991$	−4.38575e10	−1.43148	−0.715741	+	0.698366i	$0.753911\pi$
$992$	−5.62229e10	−1.82862
$993$	2.96424e10	0.960708
$994$	−8.16480e9	−0.263690
$995$	0	0
$996$	−4.69350e9	−0.150518
$997$	−9.42882e9	−0.301317	−0.150659	−	0.988586i	$-0.548139\pi$
$997$	−9.42882e9	−0.301317	−0.150659	+	0.988586i	$0.548139\pi$
$998$	−6.38982e9	−0.203485
$999$	−9.50584e9	−0.301656

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.8.a.d.1.1	✓	2
3.2	odd	2		225.8.a.u.1.2		2
5.2	odd	4		75.8.b.e.49.1		4
5.3	odd	4		75.8.b.e.49.4		4
5.4	even	2		75.8.a.f.1.2	yes	2
15.2	even	4		225.8.b.o.199.4		4
15.8	even	4		225.8.b.o.199.1		4
15.14	odd	2		225.8.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.8.a.d.1.1	✓	2	1.1	even	1	trivial
75.8.a.f.1.2	yes	2	5.4	even	2
75.8.b.e.49.1		4	5.2	odd	4
75.8.b.e.49.4		4	5.3	odd	4
225.8.a.l.1.1		2	15.14	odd	2
225.8.a.u.1.2		2	3.2	odd	2
225.8.b.o.199.1		4	15.8	even	4
225.8.b.o.199.4		4	15.2	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

\(f(q)\)	\(=\)	\(q-15.1355 q^{2} +27.0000 q^{3} +101.084 q^{4} -408.659 q^{6} -198.897 q^{7} +407.384 q^{8} +729.000 q^{9} -5268.05 q^{11} +2729.27 q^{12} -1212.88 q^{13} +3010.42 q^{14} -19104.8 q^{16} +34583.5 q^{17} -11033.8 q^{18} +18611.2 q^{19} -5370.23 q^{21} +79734.7 q^{22} +33322.1 q^{23} +10999.4 q^{24} +18357.6 q^{26} +19683.0 q^{27} -20105.4 q^{28} -178577. q^{29} -237212. q^{31} +237015. q^{32} -142237. q^{33} -523440. q^{34} +73690.4 q^{36} -482947. q^{37} -281690. q^{38} -32747.9 q^{39} +293643. q^{41} +81281.3 q^{42} -443299. q^{43} -532517. q^{44} -504348. q^{46} +48179.6 q^{47} -515829. q^{48} -783983. q^{49} +933755. q^{51} -122603. q^{52} -1.66310e6 q^{53} -297913. q^{54} -81027.7 q^{56} +502501. q^{57} +2.70285e6 q^{58} +1.75081e6 q^{59} -3.15141e6 q^{61} +3.59033e6 q^{62} -144996. q^{63} -1.14194e6 q^{64} +2.15284e6 q^{66} -2.29198e6 q^{67} +3.49585e6 q^{68} +899697. q^{69} -2.71218e6 q^{71} +296983. q^{72} +2.67756e6 q^{73} +7.30965e6 q^{74} +1.88129e6 q^{76} +1.04780e6 q^{77} +495656. q^{78} +3.44864e6 q^{79} +531441. q^{81} -4.44444e6 q^{82} -1.71969e6 q^{83} -542846. q^{84} +6.70957e6 q^{86} -4.82157e6 q^{87} -2.14612e6 q^{88} +3.52224e6 q^{89} +241239. q^{91} +3.36834e6 q^{92} -6.40472e6 q^{93} -729223. q^{94} +6.39942e6 q^{96} -1.44653e7 q^{97} +1.18660e7 q^{98} -3.84041e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 54 q^{3} + 24 q^{4} - 216 q^{6} - 86 q^{7} - 1056 q^{8} + 1458 q^{9} - 4612 q^{11} + 648 q^{12} - 9018 q^{13} + 3816 q^{14} - 19680 q^{16} + 19948 q^{17} - 5832 q^{18} - 23934 q^{19} - 2322 q^{21}+ \cdots - 3362148 q^{99}+O(q^{100}) \) 2 * q - 8 * q^2 + 54 * q^3 + 24 * q^4 - 216 * q^6 - 86 * q^7 - 1056 * q^8 + 1458 * q^9 - 4612 * q^11 + 648 * q^12 - 9018 * q^13 + 3816 * q^14 - 19680 * q^16 + 19948 * q^17 - 5832 * q^18 - 23934 * q^19 - 2322 * q^21 + 84416 * q^22 - 19812 * q^23 - 28512 * q^24 - 37336 * q^26 + 39366 * q^27 - 28808 * q^28 - 346508 * q^29 - 129178 * q^31 + 420224 * q^32 - 124524 * q^33 - 627872 * q^34 + 17496 * q^36 - 58036 * q^37 - 585272 * q^38 - 243486 * q^39 - 340928 * q^41 + 103032 * q^42 - 1087350 * q^43 - 583088 * q^44 - 883488 * q^46 + 448420 * q^47 - 531360 * q^48 - 1594780 * q^49 + 538596 * q^51 + 479048 * q^52 - 728552 * q^53 - 157464 * q^54 - 246240 * q^56 - 646218 * q^57 + 1504576 * q^58 + 907844 * q^59 - 478410 * q^61 + 4361208 * q^62 - 62694 * q^63 + 238976 * q^64 + 2279232 * q^66 - 4514702 * q^67 + 4624016 * q^68 - 534924 * q^69 - 2607296 * q^71 - 769824 * q^72 + 4070268 * q^73 + 10341616 * q^74 + 5160856 * q^76 + 1121868 * q^77 - 1008072 * q^78 + 6227360 * q^79 + 1062882 * q^81 - 8972432 * q^82 + 5769084 * q^83 - 777816 * q^84 + 2113928 * q^86 - 9355716 * q^87 - 3106176 * q^88 + 16356096 * q^89 - 639938 * q^91 + 7464144 * q^92 - 3487806 * q^93 + 2126704 * q^94 + 11346048 * q^96 - 17433442 * q^97 + 6080528 * q^98 - 3362148 * q^99

Introduction

L-functions

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