LMFDB - Embedded newform 975.2.c.i.274.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(274,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("975.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 195)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.1
Root			$0.264658 + 1.38923i$ of defining polynomial
Character	$\chi$	$=$	975.274
Dual form			975.2.c.i.274.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.77846i q^{2} -1.00000i q^{3} -5.71982 q^{4} -2.77846 q^{6} +2.71982i q^{7} +10.3354i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-2.77846i q^{2} -1.00000i q^{3} -5.71982 q^{4} -2.77846 q^{6} +2.71982i q^{7} +10.3354i q^{8} -1.00000 q^{9} -2.71982 q^{11} +5.71982i q^{12} +1.00000i q^{13} +7.55691 q^{14} +17.2767 q^{16} +2.83709i q^{17} +2.77846i q^{18} +3.55691 q^{19} +2.71982 q^{21} +7.55691i q^{22} -4.83709i q^{23} +10.3354 q^{24} +2.77846 q^{26} +1.00000i q^{27} -15.5569i q^{28} -6.00000 q^{29} +7.55691 q^{31} -27.3319i q^{32} +2.71982i q^{33} +7.88273 q^{34} +5.71982 q^{36} +4.27674i q^{37} -9.88273i q^{38} +1.00000 q^{39} +2.83709 q^{41} -7.55691i q^{42} +11.1138i q^{43} +15.5569 q^{44} -13.4396 q^{46} +11.5569i q^{47} -17.2767i q^{48} -0.397442 q^{49} +2.83709 q^{51} -5.71982i q^{52} +1.16291i q^{53} +2.77846 q^{54} -28.1104 q^{56} -3.55691i q^{57} +16.6707i q^{58} +2.11727 q^{59} +6.60256 q^{61} -20.9966i q^{62} -2.71982i q^{63} -41.3871 q^{64} +7.55691 q^{66} -1.88273i q^{67} -16.2277i q^{68} -4.83709 q^{69} -6.71982 q^{71} -10.3354i q^{72} +9.11383i q^{73} +11.8827 q^{74} -20.3449 q^{76} -7.39744i q^{77} -2.77846i q^{78} -10.2767 q^{79} +1.00000 q^{81} -7.88273i q^{82} +2.11727i q^{83} -15.5569 q^{84} +30.8793 q^{86} +6.00000i q^{87} -28.1104i q^{88} -1.16291 q^{89} -2.71982 q^{91} +27.6673i q^{92} -7.55691i q^{93} +32.1104 q^{94} -27.3319 q^{96} +10.8371i q^{97} +1.10428i q^{98} +2.71982 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 16 q^{4} - 6 q^{9}+O(q^{10}) $ 6 * q - 16 * q^4 - 6 * q^9	$ 6 q - 16 q^{4} - 6 q^{9} + 2 q^{11} + 12 q^{14} + 52 q^{16} - 12 q^{19} - 2 q^{21} + 12 q^{24} - 36 q^{29} + 12 q^{31} + 44 q^{34} + 16 q^{36} + 6 q^{39} + 2 q^{41} + 60 q^{44} - 44 q^{46} - 24 q^{49} + 2 q^{51} - 32 q^{56} + 16 q^{59} + 18 q^{61} - 60 q^{64} + 12 q^{66} - 14 q^{69} - 22 q^{71} + 68 q^{74} + 8 q^{76} - 10 q^{79} + 6 q^{81} - 60 q^{84} + 112 q^{86} - 22 q^{89} + 2 q^{91} + 56 q^{94} - 44 q^{96} - 2 q^{99}+O(q^{100}) $ 6 * q - 16 * q^4 - 6 * q^9 + 2 * q^11 + 12 * q^14 + 52 * q^16 - 12 * q^19 - 2 * q^21 + 12 * q^24 - 36 * q^29 + 12 * q^31 + 44 * q^34 + 16 * q^36 + 6 * q^39 + 2 * q^41 + 60 * q^44 - 44 * q^46 - 24 * q^49 + 2 * q^51 - 32 * q^56 + 16 * q^59 + 18 * q^61 - 60 * q^64 + 12 * q^66 - 14 * q^69 - 22 * q^71 + 68 * q^74 + 8 * q^76 - 10 * q^79 + 6 * q^81 - 60 * q^84 + 112 * q^86 - 22 * q^89 + 2 * q^91 + 56 * q^94 - 44 * q^96 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$.

$n$	$301$	$326$	$352$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 2.77846i	− 1.96467i	−0.187142	−	0.982333i	$-0.559922\pi$
$2$	− 2.77846i	− 1.96467i	0.187142	−	0.982333i	$-0.440078\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−5.71982	−2.85991
$5$	0	0
$5$	0	0
$6$	−2.77846	−1.13430
$7$	2.71982i	1.02800i	0.857791	+	0.513998i	$0.171836\pi$
$7$	2.71982i	1.02800i	−0.857791	+	0.513998i	$0.828164\pi$
$8$	10.3354i	3.65411i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−2.71982	−0.820058	−0.410029	−	0.912073i	$-0.634481\pi$
$11$	−2.71982	−0.820058	−0.410029	+	0.912073i	$0.634481\pi$
$12$	5.71982i	1.65117i
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	7.55691	2.01967
$15$	0	0
$16$	17.2767	4.31918
$17$	2.83709i	0.688095i	0.938952	+	0.344048i	$0.111798\pi$
$17$	2.83709i	0.688095i	−0.938952	+	0.344048i	$0.888202\pi$
$18$	2.77846i	0.654889i
$19$	3.55691	0.816012	0.408006	−	0.912979i	$-0.366224\pi$
$19$	3.55691	0.816012	0.408006	+	0.912979i	$0.366224\pi$
$20$	0	0
$21$	2.71982	0.593514
$22$	7.55691i	1.61114i
$23$	− 4.83709i	− 1.00860i	−0.863528	−	0.504302i	$-0.831750\pi$
$23$	− 4.83709i	− 1.00860i	0.863528	−	0.504302i	$-0.168250\pi$
$24$	10.3354	2.10970
$25$	0	0
$26$	2.77846	0.544900
$27$	1.00000i	0.192450i
$28$	− 15.5569i	− 2.93998i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	7.55691	1.35726	0.678631	−	0.734479i	$-0.262574\pi$
$31$	7.55691	1.35726	0.678631	+	0.734479i	$0.262574\pi$
$32$	− 27.3319i	− 4.83165i
$33$	2.71982i	0.473461i
$34$	7.88273	1.35188
$35$	0	0
$36$	5.71982	0.953304
$37$	4.27674i	0.703091i	0.936171	+	0.351546i	$0.114344\pi$
$37$	4.27674i	0.703091i	−0.936171	+	0.351546i	$0.885656\pi$
$38$	− 9.88273i	− 1.60319i
$39$	1.00000	0.160128
$40$	0	0
$41$	2.83709	0.443079	0.221540	−	0.975151i	$-0.428892\pi$
$41$	2.83709	0.443079	0.221540	+	0.975151i	$0.428892\pi$
$42$	− 7.55691i	− 1.16606i
$43$	11.1138i	1.69484i	0.530921	+	0.847421i	$0.321846\pi$
$43$	11.1138i	1.69484i	−0.530921	+	0.847421i	$0.678154\pi$
$44$	15.5569	2.34529
$45$	0	0
$46$	−13.4396	−1.98157
$47$	11.5569i	1.68575i	0.538110	+	0.842875i	$0.319138\pi$
$47$	11.5569i	1.68575i	−0.538110	+	0.842875i	$0.680862\pi$
$48$	− 17.2767i	− 2.49368i
$49$	−0.397442	−0.0567775
$50$	0	0
$51$	2.83709	0.397272
$52$	− 5.71982i	− 0.793197i
$53$	1.16291i	0.159738i	0.996805	+	0.0798690i	$0.0254502\pi$
$53$	1.16291i	0.159738i	−0.996805	+	0.0798690i	$0.974550\pi$
$54$	2.77846	0.378100
$55$	0	0
$56$	−28.1104	−3.75641
$57$	− 3.55691i	− 0.471125i
$58$	16.6707i	2.18898i
$59$	2.11727	0.275645	0.137822	−	0.990457i	$-0.455990\pi$
$59$	2.11727	0.275645	0.137822	+	0.990457i	$0.455990\pi$
$60$	0	0
$61$	6.60256	0.845371	0.422685	−	0.906276i	$-0.361088\pi$
$61$	6.60256	0.845371	0.422685	+	0.906276i	$0.361088\pi$
$62$	− 20.9966i	− 2.66657i
$63$	− 2.71982i	− 0.342666i
$64$	−41.3871	−5.17339
$65$	0	0
$66$	7.55691	0.930192
$67$	− 1.88273i	− 0.230013i	−0.993365	−	0.115006i	$-0.963311\pi$
$67$	− 1.88273i	− 0.230013i	0.993365	−	0.115006i	$-0.0366888\pi$
$68$	− 16.2277i	− 1.96789i
$69$	−4.83709	−0.582317
$70$	0	0
$71$	−6.71982	−0.797496	−0.398748	−	0.917060i	$-0.630555\pi$
$71$	−6.71982	−0.797496	−0.398748	+	0.917060i	$0.630555\pi$
$72$	− 10.3354i	− 1.21804i
$73$	9.11383i	1.06669i	0.845897	+	0.533346i	$0.179066\pi$
$73$	9.11383i	1.06669i	−0.845897	+	0.533346i	$0.820934\pi$
$74$	11.8827	1.38134
$75$	0	0
$76$	−20.3449	−2.33372
$77$	− 7.39744i	− 0.843017i
$78$	− 2.77846i	− 0.314598i
$79$	−10.2767	−1.15622	−0.578112	−	0.815958i	$-0.696210\pi$
$79$	−10.2767	−1.15622	−0.578112	+	0.815958i	$0.696210\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 7.88273i	− 0.870502i
$83$	2.11727i	0.232400i	0.993226	+	0.116200i	$0.0370714\pi$
$83$	2.11727i	0.232400i	−0.993226	+	0.116200i	$0.962929\pi$
$84$	−15.5569	−1.69740
$85$	0	0
$86$	30.8793	3.32980
$87$	6.00000i	0.643268i
$88$	− 28.1104i	− 2.99658i
$89$	−1.16291	−0.123268	−0.0616341	−	0.998099i	$-0.519631\pi$
$89$	−1.16291	−0.123268	−0.0616341	+	0.998099i	$0.519631\pi$
$90$	0	0
$91$	−2.71982	−0.285115
$92$	27.6673i	2.88452i
$93$	− 7.55691i	− 0.783616i
$94$	32.1104	3.31193
$95$	0	0
$96$	−27.3319	−2.78955
$97$	10.8371i	1.10034i	0.835053	+	0.550170i	$0.185437\pi$
$97$	10.8371i	1.10034i	−0.835053	+	0.550170i	$0.814563\pi$
$98$	1.10428i	0.111549i
$99$	2.71982	0.273353
$100$	0	0
$101$	7.67418	0.763610	0.381805	−	0.924243i	$-0.375303\pi$
$101$	7.67418	0.763610	0.381805	+	0.924243i	$0.375303\pi$
$102$	− 7.88273i	− 0.780507i
$103$	3.76547i	0.371023i	0.982642	+	0.185511i	$0.0593941\pi$
$103$	3.76547i	0.371023i	−0.982642	+	0.185511i	$0.940606\pi$
$104$	−10.3354	−1.01347
$105$	0	0
$106$	3.23109	0.313832
$107$	12.6026i	1.21834i	0.793041	+	0.609168i	$0.208496\pi$
$107$	12.6026i	1.21834i	−0.793041	+	0.609168i	$0.791504\pi$
$108$	− 5.71982i	− 0.550390i
$109$	−11.4396	−1.09572	−0.547860	−	0.836570i	$-0.684557\pi$
$109$	−11.4396	−1.09572	−0.547860	+	0.836570i	$0.684557\pi$
$110$	0	0
$111$	4.27674	0.405930
$112$	46.9897i	4.44011i
$113$	− 13.1138i	− 1.23365i	−0.787102	−	0.616823i	$-0.788420\pi$
$113$	− 13.1138i	− 1.23365i	0.787102	−	0.616823i	$-0.211580\pi$
$114$	−9.88273	−0.925603
$115$	0	0
$116$	34.3189	3.18643
$117$	− 1.00000i	− 0.0924500i
$118$	− 5.88273i	− 0.541550i
$119$	−7.71639	−0.707360
$120$	0	0
$121$	−3.60256	−0.327505
$122$	− 18.3449i	− 1.66087i
$123$	− 2.83709i	− 0.255812i
$124$	−43.2242	−3.88165
$125$	0	0
$126$	−7.55691	−0.673223
$127$	13.4396i	1.19258i	0.802771	+	0.596288i	$0.203358\pi$
$127$	13.4396i	1.19258i	−0.802771	+	0.596288i	$0.796642\pi$
$128$	60.3285i	5.33234i
$129$	11.1138	0.978518
$130$	0	0
$131$	−9.43965	−0.824746	−0.412373	−	0.911015i	$-0.635300\pi$
$131$	−9.43965	−0.824746	−0.412373	+	0.911015i	$0.635300\pi$
$132$	− 15.5569i	− 1.35406i
$133$	9.67418i	0.838858i
$134$	−5.23109	−0.451898
$135$	0	0
$136$	−29.3224	−2.51437
$137$	− 1.76547i	− 0.150834i	−0.997152	−	0.0754170i	$-0.975971\pi$
$137$	− 1.76547i	− 0.150834i	0.997152	−	0.0754170i	$-0.0240288\pi$
$138$	13.4396i	1.14406i
$139$	6.27674	0.532386	0.266193	−	0.963920i	$-0.414234\pi$
$139$	6.27674	0.532386	0.266193	+	0.963920i	$0.414234\pi$
$140$	0	0
$141$	11.5569	0.973268
$142$	18.6707i	1.56681i
$143$	− 2.71982i	− 0.227443i
$144$	−17.2767	−1.43973
$145$	0	0
$146$	25.3224	2.09570
$147$	0.397442i	0.0327805i
$148$	− 24.4622i	− 2.01078i
$149$	20.8302	1.70648	0.853239	−	0.521520i	$-0.174635\pi$
$149$	20.8302	1.70648	0.853239	+	0.521520i	$0.174635\pi$
$150$	0	0
$151$	4.99656	0.406614	0.203307	−	0.979115i	$-0.434831\pi$
$151$	4.99656	0.406614	0.203307	+	0.979115i	$0.434831\pi$
$152$	36.7620i	2.98179i
$153$	− 2.83709i	− 0.229365i
$154$	−20.5535	−1.65625
$155$	0	0
$156$	−5.71982	−0.457952
$157$	− 8.87930i	− 0.708645i	−0.935123	−	0.354322i	$-0.884712\pi$
$157$	− 8.87930i	− 0.708645i	0.935123	−	0.354322i	$-0.115288\pi$
$158$	28.5535i	2.27159i
$159$	1.16291	0.0922247
$160$	0	0
$161$	13.1560	1.03684
$162$	− 2.77846i	− 0.218296i
$163$	− 13.8337i	− 1.08354i	−0.840528	−	0.541768i	$-0.817755\pi$
$163$	− 13.8337i	− 1.08354i	0.840528	−	0.541768i	$-0.182245\pi$
$164$	−16.2277	−1.26717
$165$	0	0
$166$	5.88273	0.456589
$167$	9.88273i	0.764749i	0.924007	+	0.382374i	$0.124894\pi$
$167$	9.88273i	0.764749i	−0.924007	+	0.382374i	$0.875106\pi$
$168$	28.1104i	2.16876i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−3.55691	−0.272004
$172$	− 63.5691i	− 4.84710i
$173$	13.1138i	0.997026i	0.866882	+	0.498513i	$0.166120\pi$
$173$	13.1138i	0.997026i	−0.866882	+	0.498513i	$0.833880\pi$
$174$	16.6707	1.26381
$175$	0	0
$176$	−46.9897	−3.54198
$177$	− 2.11727i	− 0.159143i
$178$	3.23109i	0.242181i
$179$	−8.55348	−0.639317	−0.319658	−	0.947533i	$-0.603568\pi$
$179$	−8.55348	−0.639317	−0.319658	+	0.947533i	$0.603568\pi$
$180$	0	0
$181$	3.72326	0.276748	0.138374	−	0.990380i	$-0.455812\pi$
$181$	3.72326	0.276748	0.138374	+	0.990380i	$0.455812\pi$
$182$	7.55691i	0.560156i
$183$	− 6.60256i	− 0.488075i
$184$	49.9931	3.68554
$185$	0	0
$186$	−20.9966	−1.53954
$187$	− 7.71639i	− 0.564278i
$188$	− 66.1035i	− 4.82109i
$189$	−2.71982	−0.197838
$190$	0	0
$191$	−4.23453	−0.306400	−0.153200	−	0.988195i	$-0.548958\pi$
$191$	−4.23453	−0.306400	−0.153200	+	0.988195i	$0.548958\pi$
$192$	41.3871i	2.98686i
$193$	− 23.3906i	− 1.68369i	−0.539719	−	0.841845i	$-0.681470\pi$
$193$	− 23.3906i	− 1.68369i	0.539719	−	0.841845i	$-0.318530\pi$
$194$	30.1104	2.16180
$195$	0	0
$196$	2.27330	0.162379
$197$	− 14.5535i	− 1.03689i	−0.855110	−	0.518446i	$-0.826511\pi$
$197$	− 14.5535i	− 1.03689i	0.855110	−	0.518446i	$-0.173489\pi$
$198$	− 7.55691i	− 0.537047i
$199$	15.1138	1.07139	0.535695	−	0.844411i	$-0.320050\pi$
$199$	15.1138	1.07139	0.535695	+	0.844411i	$0.320050\pi$
$200$	0	0
$201$	−1.88273	−0.132798
$202$	− 21.3224i	− 1.50024i
$203$	− 16.3189i	− 1.14537i
$204$	−16.2277	−1.13616
$205$	0	0
$206$	10.4622	0.728935
$207$	4.83709i	0.336201i
$208$	17.2767i	1.19793i
$209$	−9.67418	−0.669177
$210$	0	0
$211$	−18.2277	−1.25484	−0.627422	−	0.778680i	$-0.715890\pi$
$211$	−18.2277	−1.25484	−0.627422	+	0.778680i	$0.715890\pi$
$212$	− 6.65164i	− 0.456836i
$213$	6.71982i	0.460435i
$214$	35.0157	2.39362
$215$	0	0
$216$	−10.3354	−0.703233
$217$	20.5535i	1.39526i
$218$	31.7846i	2.15272i
$219$	9.11383	0.615855
$220$	0	0
$221$	−2.83709	−0.190843
$222$	− 11.8827i	− 0.797517i
$223$	10.1173i	0.677502i	0.940876	+	0.338751i	$0.110004\pi$
$223$	10.1173i	0.677502i	−0.940876	+	0.338751i	$0.889996\pi$
$224$	74.3380	4.96692
$225$	0	0
$226$	−36.4362	−2.42370
$227$	− 11.3224i	− 0.751493i	−0.926723	−	0.375746i	$-0.877386\pi$
$227$	− 11.3224i	− 0.751493i	0.926723	−	0.375746i	$-0.122614\pi$
$228$	20.3449i	1.34738i
$229$	6.23453	0.411990	0.205995	−	0.978553i	$-0.433957\pi$
$229$	6.23453	0.411990	0.205995	+	0.978553i	$0.433957\pi$
$230$	0	0
$231$	−7.39744	−0.486716
$232$	− 62.0122i	− 4.07130i
$233$	6.83709i	0.447913i	0.974599	+	0.223956i	$0.0718973\pi$
$233$	6.83709i	0.447913i	−0.974599	+	0.223956i	$0.928103\pi$
$234$	−2.77846	−0.181633
$235$	0	0
$236$	−12.1104	−0.788319
$237$	10.2767i	0.667546i
$238$	21.4396i	1.38973i
$239$	−1.28018	−0.0828077	−0.0414039	−	0.999142i	$-0.513183\pi$
$239$	−1.28018	−0.0828077	−0.0414039	+	0.999142i	$0.513183\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	10.0096i	0.643438i
$243$	− 1.00000i	− 0.0641500i
$244$	−37.7655	−2.41769
$245$	0	0
$246$	−7.88273	−0.502585
$247$	3.55691i	0.226321i
$248$	78.1035i	4.95958i
$249$	2.11727	0.134176
$250$	0	0
$251$	18.2277	1.15052	0.575260	−	0.817971i	$-0.304901\pi$
$251$	18.2277	1.15052	0.575260	+	0.817971i	$0.304901\pi$
$252$	15.5569i	0.979993i
$253$	13.1560i	0.827113i
$254$	37.3415	2.34301
$255$	0	0
$256$	84.8459	5.30287
$257$	− 1.11383i	− 0.0694787i	−0.999396	−	0.0347394i	$-0.988940\pi$
$257$	− 1.11383i	− 0.0694787i	0.999396	−	0.0347394i	$-0.0110601\pi$
$258$	− 30.8793i	− 1.92246i
$259$	−11.6320	−0.722776
$260$	0	0
$261$	6.00000	0.371391
$262$	26.2277i	1.62035i
$263$	8.00000i	0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$	8.00000i	0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$	−28.1104	−1.73007
$265$	0	0
$266$	26.8793	1.64808
$267$	1.16291i	0.0711689i
$268$	10.7689i	0.657816i
$269$	−15.6742	−0.955672	−0.477836	−	0.878449i	$-0.658579\pi$
$269$	−15.6742	−0.955672	−0.477836	+	0.878449i	$0.658579\pi$
$270$	0	0
$271$	0.443086	0.0269155	0.0134578	−	0.999909i	$-0.495716\pi$
$271$	0.443086	0.0269155	0.0134578	+	0.999909i	$0.495716\pi$
$272$	49.0157i	2.97201i
$273$	2.71982i	0.164611i
$274$	−4.90528	−0.296339
$275$	0	0
$276$	27.6673	1.66538
$277$	4.87930i	0.293168i	0.989198	+	0.146584i	$0.0468279\pi$
$277$	4.87930i	0.293168i	−0.989198	+	0.146584i	$0.953172\pi$
$278$	− 17.4396i	− 1.04596i
$279$	−7.55691	−0.452421
$280$	0	0
$281$	−9.11383	−0.543685	−0.271843	−	0.962342i	$-0.587633\pi$
$281$	−9.11383	−0.543685	−0.271843	+	0.962342i	$0.587633\pi$
$282$	− 32.1104i	− 1.91215i
$283$	33.3415i	1.98195i	0.134063	+	0.990973i	$0.457198\pi$
$283$	33.3415i	1.98195i	−0.134063	+	0.990973i	$0.542802\pi$
$284$	38.4362	2.28077
$285$	0	0
$286$	−7.55691	−0.446850
$287$	7.71639i	0.455484i
$288$	27.3319i	1.61055i
$289$	8.95092	0.526525
$290$	0	0
$291$	10.8371	0.635281
$292$	− 52.1295i	− 3.05065i
$293$	− 29.4328i	− 1.71948i	−0.510731	−	0.859740i	$-0.670625\pi$
$293$	− 29.4328i	− 1.71948i	0.510731	−	0.859740i	$-0.329375\pi$
$294$	1.10428	0.0644027
$295$	0	0
$296$	−44.2017	−2.56917
$297$	− 2.71982i	− 0.157820i
$298$	− 57.8759i	− 3.35266i
$299$	4.83709	0.279736
$300$	0	0
$301$	−30.2277	−1.74229
$302$	− 13.8827i	− 0.798862i
$303$	− 7.67418i	− 0.440870i
$304$	61.4519	3.52451
$305$	0	0
$306$	−7.88273	−0.450626
$307$	21.8337i	1.24611i	0.782177	+	0.623056i	$0.214109\pi$
$307$	21.8337i	1.24611i	−0.782177	+	0.623056i	$0.785891\pi$
$308$	42.3121i	2.41095i
$309$	3.76547	0.214210
$310$	0	0
$311$	25.1070	1.42368	0.711842	−	0.702339i	$-0.247861\pi$
$311$	25.1070	1.42368	0.711842	+	0.702339i	$0.247861\pi$
$312$	10.3354i	0.585125i
$313$	8.22766i	0.465055i	0.972590	+	0.232527i	$0.0746995\pi$
$313$	8.22766i	0.465055i	−0.972590	+	0.232527i	$0.925300\pi$
$314$	−24.6707	−1.39225
$315$	0	0
$316$	58.7811	3.30670
$317$	− 27.6742i	− 1.55434i	−0.629293	−	0.777168i	$-0.716655\pi$
$317$	− 27.6742i	− 1.55434i	0.629293	−	0.777168i	$-0.283345\pi$
$318$	− 3.23109i	− 0.181191i
$319$	16.3189	0.913685
$320$	0	0
$321$	12.6026	0.703406
$322$	− 36.5535i	− 2.03705i
$323$	10.0913i	0.561494i
$324$	−5.71982	−0.317768
$325$	0	0
$326$	−38.4362	−2.12878
$327$	11.4396i	0.632614i
$328$	29.3224i	1.61906i
$329$	−31.4328	−1.73294
$330$	0	0
$331$	−13.2311	−0.727247	−0.363623	−	0.931546i	$-0.618460\pi$
$331$	−13.2311	−0.727247	−0.363623	+	0.931546i	$0.618460\pi$
$332$	− 12.1104i	− 0.664644i
$333$	− 4.27674i	− 0.234364i
$334$	27.4588	1.50248
$335$	0	0
$336$	46.9897	2.56350
$337$	4.32582i	0.235642i	0.993035	+	0.117821i	$0.0375910\pi$
$337$	4.32582i	0.235642i	−0.993035	+	0.117821i	$0.962409\pi$
$338$	2.77846i	0.151128i
$339$	−13.1138	−0.712245
$340$	0	0
$341$	−20.5535	−1.11303
$342$	9.88273i	0.534397i
$343$	17.9578i	0.969630i
$344$	−114.866	−6.19314
$345$	0	0
$346$	36.4362	1.95882
$347$	− 6.27674i	− 0.336953i	−0.985706	−	0.168476i	$-0.946115\pi$
$347$	− 6.27674i	− 0.336953i	0.985706	−	0.168476i	$-0.0538847\pi$
$348$	− 34.3189i	− 1.83969i
$349$	−17.6673	−0.945709	−0.472855	−	0.881140i	$-0.656776\pi$
$349$	−17.6673	−0.945709	−0.472855	+	0.881140i	$0.656776\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	74.3380i	3.96223i
$353$	− 13.7655i	− 0.732662i	−0.930485	−	0.366331i	$-0.880614\pi$
$353$	− 13.7655i	− 0.732662i	0.930485	−	0.366331i	$-0.119386\pi$
$354$	−5.88273	−0.312664
$355$	0	0
$356$	6.65164	0.352536
$357$	7.71639i	0.408394i
$358$	23.7655i	1.25604i
$359$	−0.996562	−0.0525965	−0.0262983	−	0.999654i	$-0.508372\pi$
$359$	−0.996562	−0.0525965	−0.0262983	+	0.999654i	$0.508372\pi$
$360$	0	0
$361$	−6.34836	−0.334124
$362$	− 10.3449i	− 0.543717i
$363$	3.60256i	0.189085i
$364$	15.5569	0.815404
$365$	0	0
$366$	−18.3449	−0.958905
$367$	− 14.2277i	− 0.742678i	−0.928497	−	0.371339i	$-0.878899\pi$
$367$	− 14.2277i	− 0.742678i	0.928497	−	0.371339i	$-0.121101\pi$
$368$	− 83.5691i	− 4.35634i
$369$	−2.83709	−0.147693
$370$	0	0
$371$	−3.16291	−0.164210
$372$	43.2242i	2.24107i
$373$	15.6742i	0.811578i	0.913967	+	0.405789i	$0.133003\pi$
$373$	15.6742i	0.811578i	−0.913967	+	0.405789i	$0.866997\pi$
$374$	−21.4396	−1.10862
$375$	0	0
$376$	−119.445	−6.15991
$377$	− 6.00000i	− 0.309016i
$378$	7.55691i	0.388686i
$379$	−26.2017	−1.34589	−0.672945	−	0.739693i	$-0.734971\pi$
$379$	−26.2017	−1.34589	−0.672945	+	0.739693i	$0.734971\pi$
$380$	0	0
$381$	13.4396	0.688534
$382$	11.7655i	0.601974i
$383$	22.4362i	1.14644i	0.819403	+	0.573218i	$0.194305\pi$
$383$	22.4362i	1.14644i	−0.819403	+	0.573218i	$0.805695\pi$
$384$	60.3285	3.07863
$385$	0	0
$386$	−64.9897	−3.30789
$387$	− 11.1138i	− 0.564948i
$388$	− 61.9862i	− 3.14687i
$389$	−31.6742	−1.60594	−0.802972	−	0.596016i	$-0.796749\pi$
$389$	−31.6742	−1.60594	−0.802972	+	0.596016i	$0.796749\pi$
$390$	0	0
$391$	13.7233	0.694015
$392$	− 4.10771i	− 0.207471i
$393$	9.43965i	0.476167i
$394$	−40.4362	−2.03715
$395$	0	0
$396$	−15.5569	−0.781764
$397$	− 17.9509i	− 0.900931i	−0.892794	−	0.450465i	$-0.851258\pi$
$397$	− 17.9509i	− 0.900931i	0.892794	−	0.450465i	$-0.148742\pi$
$398$	− 41.9931i	− 2.10493i
$399$	9.67418	0.484315
$400$	0	0
$401$	13.5829	0.678297	0.339149	−	0.940733i	$-0.389861\pi$
$401$	13.5829	0.678297	0.339149	+	0.940733i	$0.389861\pi$
$402$	5.23109i	0.260903i
$403$	7.55691i	0.376437i
$404$	−43.8950	−2.18386
$405$	0	0
$406$	−45.3415	−2.25026
$407$	− 11.6320i	− 0.576576i
$408$	29.3224i	1.45167i
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	−1.76547	−0.0870841
$412$	− 21.5378i	− 1.06109i
$413$	5.75859i	0.283362i
$414$	13.4396	0.660523
$415$	0	0
$416$	27.3319	1.34006
$417$	− 6.27674i	− 0.307373i
$418$	26.8793i	1.31471i
$419$	−12.3189	−0.601820	−0.300910	−	0.953653i	$-0.597290\pi$
$419$	−12.3189	−0.601820	−0.300910	+	0.953653i	$0.597290\pi$
$420$	0	0
$421$	22.7880	1.11062	0.555310	−	0.831644i	$-0.312600\pi$
$421$	22.7880	1.11062	0.555310	+	0.831644i	$0.312600\pi$
$422$	50.6448i	2.46535i
$423$	− 11.5569i	− 0.561916i
$424$	−12.0191	−0.583699
$425$	0	0
$426$	18.6707	0.904600
$427$	17.9578i	0.869039i
$428$	− 72.0844i	− 3.48433i
$429$	−2.71982	−0.131314
$430$	0	0
$431$	−8.99656	−0.433349	−0.216675	−	0.976244i	$-0.569521\pi$
$431$	−8.99656	−0.433349	−0.216675	+	0.976244i	$0.569521\pi$
$432$	17.2767i	0.831227i
$433$	− 20.3258i	− 0.976797i	−0.872621	−	0.488398i	$-0.837581\pi$
$433$	− 20.3258i	− 0.976797i	0.872621	−	0.488398i	$-0.162419\pi$
$434$	57.1070	2.74122
$435$	0	0
$436$	65.4328	3.13366
$437$	− 17.2051i	− 0.823032i
$438$	− 25.3224i	− 1.20995i
$439$	25.3906	1.21183	0.605913	−	0.795531i	$-0.292808\pi$
$439$	25.3906	1.21183	0.605913	+	0.795531i	$0.292808\pi$
$440$	0	0
$441$	0.397442	0.0189258
$442$	7.88273i	0.374943i
$443$	10.9284i	0.519223i	0.965713	+	0.259611i	$0.0835945\pi$
$443$	10.9284i	0.519223i	−0.965713	+	0.259611i	$0.916406\pi$
$444$	−24.4622	−1.16092
$445$	0	0
$446$	28.1104	1.33107
$447$	− 20.8302i	− 0.985235i
$448$	− 112.566i	− 5.31823i
$449$	−2.83709	−0.133891	−0.0669453	−	0.997757i	$-0.521325\pi$
$449$	−2.83709	−0.133891	−0.0669453	+	0.997757i	$0.521325\pi$
$450$	0	0
$451$	−7.71639	−0.363350
$452$	75.0088i	3.52812i
$453$	− 4.99656i	− 0.234759i
$454$	−31.4588	−1.47643
$455$	0	0
$456$	36.7620	1.72154
$457$	13.7164i	0.641625i	0.947143	+	0.320813i	$0.103956\pi$
$457$	13.7164i	0.641625i	−0.947143	+	0.320813i	$0.896044\pi$
$458$	− 17.3224i	− 0.809422i
$459$	−2.83709	−0.132424
$460$	0	0
$461$	−19.6251	−0.914032	−0.457016	−	0.889458i	$-0.651082\pi$
$461$	−19.6251	−0.914032	−0.457016	+	0.889458i	$0.651082\pi$
$462$	20.5535i	0.956234i
$463$	27.0388i	1.25660i	0.777972	+	0.628299i	$0.216249\pi$
$463$	27.0388i	1.25660i	−0.777972	+	0.628299i	$0.783751\pi$
$464$	−103.660	−4.81231
$465$	0	0
$466$	18.9966	0.879999
$467$	− 28.9215i	− 1.33833i	−0.743115	−	0.669164i	$-0.766652\pi$
$467$	− 28.9215i	− 1.33833i	0.743115	−	0.669164i	$-0.233348\pi$
$468$	5.71982i	0.264399i
$469$	5.12070	0.236452
$470$	0	0
$471$	−8.87930	−0.409136
$472$	21.8827i	1.00723i
$473$	− 30.2277i	− 1.38987i
$474$	28.5535	1.31150
$475$	0	0
$476$	44.1364	2.02299
$477$	− 1.16291i	− 0.0532460i
$478$	3.55691i	0.162689i
$479$	12.1595	0.555580	0.277790	−	0.960642i	$-0.410398\pi$
$479$	12.1595	0.555580	0.277790	+	0.960642i	$0.410398\pi$
$480$	0	0
$481$	−4.27674	−0.195002
$482$	16.6707i	0.759332i
$483$	− 13.1560i	− 0.598620i
$484$	20.6060	0.936636
$485$	0	0
$486$	−2.77846	−0.126033
$487$	0.159472i	0.00722636i	0.999993	+	0.00361318i	$0.00115011\pi$
$487$	0.159472i	0.00722636i	−0.999993	+	0.00361318i	$0.998850\pi$
$488$	68.2399i	3.08907i
$489$	−13.8337	−0.625579
$490$	0	0
$491$	−42.2277	−1.90571	−0.952854	−	0.303430i	$-0.901868\pi$
$491$	−42.2277	−1.90571	−0.952854	+	0.303430i	$0.901868\pi$
$492$	16.2277i	0.731599i
$493$	− 17.0225i	− 0.766657i
$494$	9.88273	0.444645
$495$	0	0
$496$	130.559	5.86226
$497$	− 18.2767i	− 0.819824i
$498$	− 5.88273i	− 0.263612i
$499$	−7.79145	−0.348793	−0.174397	−	0.984676i	$-0.555797\pi$
$499$	−7.79145	−0.348793	−0.174397	+	0.984676i	$0.555797\pi$
$500$	0	0
$501$	9.88273	0.441528
$502$	− 50.6448i	− 2.26039i
$503$	27.3484i	1.21940i	0.792631	+	0.609702i	$0.208711\pi$
$503$	27.3484i	1.21940i	−0.792631	+	0.609702i	$0.791289\pi$
$504$	28.1104	1.25214
$505$	0	0
$506$	36.5535	1.62500
$507$	1.00000i	0.0444116i
$508$	− 76.8724i	− 3.41066i
$509$	33.4819	1.48406	0.742029	−	0.670368i	$-0.233864\pi$
$509$	33.4819	1.48406	0.742029	+	0.670368i	$0.233864\pi$
$510$	0	0
$511$	−24.7880	−1.09656
$512$	− 115.084i	− 5.08603i
$513$	3.55691i	0.157042i
$514$	−3.09472	−0.136502
$515$	0	0
$516$	−63.5691	−2.79848
$517$	− 31.4328i	− 1.38241i
$518$	32.3189i	1.42001i
$519$	13.1138	0.575633
$520$	0	0
$521$	−17.3484	−0.760045	−0.380023	−	0.924977i	$-0.624084\pi$
$521$	−17.3484	−0.760045	−0.380023	+	0.924977i	$0.624084\pi$
$522$	− 16.6707i	− 0.729659i
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	53.9931	2.35870
$525$	0	0
$526$	22.2277	0.969172
$527$	21.4396i	0.933926i
$528$	46.9897i	2.04496i
$529$	−0.397442	−0.0172801
$530$	0	0
$531$	−2.11727	−0.0918815
$532$	− 55.3346i	− 2.39906i
$533$	2.83709i	0.122888i
$534$	3.23109	0.139823
$535$	0	0
$536$	19.4588	0.840490
$537$	8.55348i	0.369110i
$538$	43.5500i	1.87758i
$539$	1.08097	0.0465608
$540$	0	0
$541$	−32.6448	−1.40351	−0.701754	−	0.712419i	$-0.747599\pi$
$541$	−32.6448	−1.40351	−0.701754	+	0.712419i	$0.747599\pi$
$542$	− 1.23109i	− 0.0528800i
$543$	− 3.72326i	− 0.159780i
$544$	77.5432	3.32464
$545$	0	0
$546$	7.55691	0.323406
$547$	− 34.2277i	− 1.46347i	−0.681590	−	0.731734i	$-0.738711\pi$
$547$	− 34.2277i	− 1.46347i	0.681590	−	0.731734i	$-0.261289\pi$
$548$	10.0982i	0.431372i
$549$	−6.60256	−0.281790
$550$	0	0
$551$	−21.3415	−0.909178
$552$	− 49.9931i	− 2.12785i
$553$	− 27.9509i	− 1.18859i
$554$	13.5569	0.575978
$555$	0	0
$556$	−35.9018	−1.52258
$557$	− 6.65164i	− 0.281839i	−0.990021	−	0.140919i	$-0.954994\pi$
$557$	− 6.65164i	− 0.281839i	0.990021	−	0.140919i	$-0.0450059\pi$
$558$	20.9966i	0.888855i
$559$	−11.1138	−0.470065
$560$	0	0
$561$	−7.71639	−0.325786
$562$	25.3224i	1.06816i
$563$	40.2699i	1.69717i	0.529057	+	0.848586i	$0.322546\pi$
$563$	40.2699i	1.69717i	−0.529057	+	0.848586i	$0.677454\pi$
$564$	−66.1035	−2.78346
$565$	0	0
$566$	92.6379	3.89386
$567$	2.71982i	0.114222i
$568$	− 69.4519i	− 2.91414i
$569$	13.4328	0.563131	0.281566	−	0.959542i	$-0.409146\pi$
$569$	13.4328	0.563131	0.281566	+	0.959542i	$0.409146\pi$
$570$	0	0
$571$	−35.7164	−1.49468	−0.747342	−	0.664439i	$-0.768670\pi$
$571$	−35.7164	−1.49468	−0.747342	+	0.664439i	$0.768670\pi$
$572$	15.5569i	0.650467i
$573$	4.23453i	0.176900i
$574$	21.4396	0.894874
$575$	0	0
$576$	41.3871	1.72446
$577$	13.7164i	0.571021i	0.958376	+	0.285510i	$0.0921631\pi$
$577$	13.7164i	0.571021i	−0.958376	+	0.285510i	$0.907837\pi$
$578$	− 24.8697i	− 1.03444i
$579$	−23.3906	−0.972079
$580$	0	0
$581$	−5.75859	−0.238907
$582$	− 30.1104i	− 1.24812i
$583$	− 3.16291i	− 0.130994i
$584$	−94.1948	−3.89781
$585$	0	0
$586$	−81.7777	−3.37821
$587$	− 30.6707i	− 1.26592i	−0.774186	−	0.632959i	$-0.781840\pi$
$587$	− 30.6707i	− 1.26592i	0.774186	−	0.632959i	$-0.218160\pi$
$588$	− 2.27330i	− 0.0937493i
$589$	26.8793	1.10754
$590$	0	0
$591$	−14.5535	−0.598650
$592$	73.8881i	3.03678i
$593$	− 45.6673i	− 1.87533i	−0.347538	−	0.937666i	$-0.612982\pi$
$593$	− 45.6673i	− 1.87533i	0.347538	−	0.937666i	$-0.387018\pi$
$594$	−7.55691	−0.310064
$595$	0	0
$596$	−119.145	−4.88038
$597$	− 15.1138i	− 0.618568i
$598$	− 13.4396i	− 0.549588i
$599$	40.2208	1.64338	0.821688	−	0.569937i	$-0.193032\pi$
$599$	40.2208	1.64338	0.821688	+	0.569937i	$0.193032\pi$
$600$	0	0
$601$	17.3974	0.709656	0.354828	−	0.934932i	$-0.384539\pi$
$601$	17.3974	0.709656	0.354828	+	0.934932i	$0.384539\pi$
$602$	83.9862i	3.42302i
$603$	1.88273i	0.0766708i
$604$	−28.5795	−1.16288
$605$	0	0
$606$	−21.3224	−0.866163
$607$	14.2277i	0.577483i	0.957407	+	0.288741i	$0.0932368\pi$
$607$	14.2277i	0.577483i	−0.957407	+	0.288741i	$0.906763\pi$
$608$	− 97.2173i	− 3.94268i
$609$	−16.3189	−0.661277
$610$	0	0
$611$	−11.5569	−0.467543
$612$	16.2277i	0.655964i
$613$	− 40.8302i	− 1.64912i	−0.565777	−	0.824558i	$-0.691424\pi$
$613$	− 40.8302i	− 1.64912i	0.565777	−	0.824558i	$-0.308576\pi$
$614$	60.6639	2.44819
$615$	0	0
$616$	76.4553	3.08047
$617$	− 5.11383i	− 0.205875i	−0.994688	−	0.102937i	$-0.967176\pi$
$617$	− 5.11383i	− 0.205875i	0.994688	−	0.102937i	$-0.0328242\pi$
$618$	− 10.4622i	− 0.420851i
$619$	−11.5569	−0.464512	−0.232256	−	0.972655i	$-0.574611\pi$
$619$	−11.5569	−0.464512	−0.232256	+	0.972655i	$0.574611\pi$
$620$	0	0
$621$	4.83709	0.194106
$622$	− 69.7586i	− 2.79706i
$623$	− 3.16291i	− 0.126719i
$624$	17.2767	0.691623
$625$	0	0
$626$	22.8602	0.913677
$627$	9.67418i	0.386350i
$628$	50.7880i	2.02666i
$629$	−12.1335	−0.483794
$630$	0	0
$631$	35.2242	1.40225	0.701127	−	0.713036i	$-0.252681\pi$
$631$	35.2242	1.40225	0.701127	+	0.713036i	$0.252681\pi$
$632$	− 106.214i	− 4.22496i
$633$	18.2277i	0.724484i
$634$	−76.8915	−3.05375
$635$	0	0
$636$	−6.65164	−0.263755
$637$	− 0.397442i	− 0.0157472i
$638$	− 45.3415i	− 1.79509i
$639$	6.71982	0.265832
$640$	0	0
$641$	−21.9018	−0.865071	−0.432535	−	0.901617i	$-0.642381\pi$
$641$	−21.9018	−0.865071	−0.432535	+	0.901617i	$0.642381\pi$
$642$	− 35.0157i	− 1.38196i
$643$	7.50783i	0.296080i	0.988981	+	0.148040i	$0.0472964\pi$
$643$	7.50783i	0.296080i	−0.988981	+	0.148040i	$0.952704\pi$
$644$	−75.2502	−2.96527
$645$	0	0
$646$	28.0382	1.10315
$647$	14.0422i	0.552056i	0.961150	+	0.276028i	$0.0890183\pi$
$647$	14.0422i	0.552056i	−0.961150	+	0.276028i	$0.910982\pi$
$648$	10.3354i	0.406012i
$649$	−5.75859	−0.226044
$650$	0	0
$651$	20.5535	0.805554
$652$	79.1261i	3.09882i
$653$	7.99312i	0.312795i	0.987694	+	0.156398i	$0.0499881\pi$
$653$	7.99312i	0.312795i	−0.987694	+	0.156398i	$0.950012\pi$
$654$	31.7846	1.24288
$655$	0	0
$656$	49.0157	1.91374
$657$	− 9.11383i	− 0.355564i
$658$	87.3346i	3.40466i
$659$	25.3415	0.987164	0.493582	−	0.869699i	$-0.335687\pi$
$659$	25.3415	0.987164	0.493582	+	0.869699i	$0.335687\pi$
$660$	0	0
$661$	27.4396	1.06728	0.533639	−	0.845712i	$-0.320824\pi$
$661$	27.4396	1.06728	0.533639	+	0.845712i	$0.320824\pi$
$662$	36.7620i	1.42880i
$663$	2.83709i	0.110183i
$664$	−21.8827	−0.849215
$665$	0	0
$666$	−11.8827	−0.460447
$667$	29.0225i	1.12376i
$668$	− 56.5275i	− 2.18711i
$669$	10.1173	0.391156
$670$	0	0
$671$	−17.9578	−0.693253
$672$	− 74.3380i	− 2.86765i
$673$	27.1070i	1.04490i	0.852671	+	0.522448i	$0.174981\pi$
$673$	27.1070i	1.04490i	−0.852671	+	0.522448i	$0.825019\pi$
$674$	12.0191	0.462959
$675$	0	0
$676$	5.71982	0.219993
$677$	36.5957i	1.40649i	0.710949	+	0.703243i	$0.248265\pi$
$677$	36.5957i	1.40649i	−0.710949	+	0.703243i	$0.751735\pi$
$678$	36.4362i	1.39932i
$679$	−29.4750	−1.13115
$680$	0	0
$681$	−11.3224	−0.433875
$682$	57.1070i	2.18674i
$683$	− 13.4656i	− 0.515248i	−0.966245	−	0.257624i	$-0.917060\pi$
$683$	− 13.4656i	− 0.515248i	0.966245	−	0.257624i	$-0.0829396\pi$
$684$	20.3449	0.777908
$685$	0	0
$686$	49.8950	1.90500
$687$	− 6.23453i	− 0.237862i
$688$	192.011i	7.32034i
$689$	−1.16291	−0.0443033
$690$	0	0
$691$	29.5500	1.12414	0.562068	−	0.827091i	$-0.310006\pi$
$691$	29.5500	1.12414	0.562068	+	0.827091i	$0.310006\pi$
$692$	− 75.0088i	− 2.85141i
$693$	7.39744i	0.281006i
$694$	−17.4396	−0.662000
$695$	0	0
$696$	−62.0122	−2.35057
$697$	8.04908i	0.304881i
$698$	49.0878i	1.85800i
$699$	6.83709	0.258603
$700$	0	0
$701$	43.6604	1.64903	0.824516	−	0.565839i	$-0.191448\pi$
$701$	43.6604	1.64903	0.824516	+	0.565839i	$0.191448\pi$
$702$	2.77846i	0.104866i
$703$	15.2120i	0.573731i
$704$	112.566	4.24248
$705$	0	0
$706$	−38.2468	−1.43944
$707$	20.8724i	0.784988i
$708$	12.1104i	0.455136i
$709$	26.7880	1.00604	0.503022	−	0.864273i	$-0.332221\pi$
$709$	26.7880	1.00604	0.503022	+	0.864273i	$0.332221\pi$
$710$	0	0
$711$	10.2767	0.385408
$712$	− 12.0191i	− 0.450435i
$713$	− 36.5535i	− 1.36894i
$714$	21.4396	0.802359
$715$	0	0
$716$	48.9244	1.82839
$717$	1.28018i	0.0478091i
$718$	2.76891i	0.103335i
$719$	34.8793	1.30078	0.650389	−	0.759601i	$-0.274606\pi$
$719$	34.8793	1.30078	0.650389	+	0.759601i	$0.274606\pi$
$720$	0	0
$721$	−10.2414	−0.381410
$722$	17.6386i	0.656443i
$723$	6.00000i	0.223142i
$724$	−21.2964	−0.791475
$725$	0	0
$726$	10.0096	0.371489
$727$	− 37.4396i	− 1.38856i	−0.719705	−	0.694280i	$-0.755723\pi$
$727$	− 37.4396i	− 1.38856i	0.719705	−	0.694280i	$-0.244277\pi$
$728$	− 28.1104i	− 1.04184i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−31.5309	−1.16621
$732$	37.7655i	1.39585i
$733$	− 47.1560i	− 1.74175i	−0.491506	−	0.870874i	$-0.663554\pi$
$733$	− 47.1560i	− 1.74175i	0.491506	−	0.870874i	$-0.336446\pi$
$734$	−39.5309	−1.45911
$735$	0	0
$736$	−132.207	−4.87322
$737$	5.12070i	0.188624i
$738$	7.88273i	0.290167i
$739$	31.7914	1.16947	0.584734	−	0.811225i	$-0.301199\pi$
$739$	31.7914	1.16947	0.584734	+	0.811225i	$0.301199\pi$
$740$	0	0
$741$	3.55691	0.130667
$742$	8.78801i	0.322618i
$743$	16.6776i	0.611842i	0.952057	+	0.305921i	$0.0989644\pi$
$743$	16.6776i	0.611842i	−0.952057	+	0.305921i	$0.901036\pi$
$744$	78.1035	2.86341
$745$	0	0
$746$	43.5500	1.59448
$747$	− 2.11727i	− 0.0774667i
$748$	44.1364i	1.61379i
$749$	−34.2767	−1.25244
$750$	0	0
$751$	16.1855	0.590616	0.295308	−	0.955402i	$-0.404578\pi$
$751$	16.1855	0.590616	0.295308	+	0.955402i	$0.404578\pi$
$752$	199.666i	7.28106i
$753$	− 18.2277i	− 0.664253i
$754$	−16.6707	−0.607113
$755$	0	0
$756$	15.5569	0.565800
$757$	− 12.3258i	− 0.447990i	−0.974590	−	0.223995i	$-0.928090\pi$
$757$	− 12.3258i	− 0.447990i	0.974590	−	0.223995i	$-0.0719099\pi$
$758$	72.8002i	2.64422i
$759$	13.1560	0.477534
$760$	0	0
$761$	−0.00687569	−0.000249244 0	−0.000124622	−	1.00000i	$-0.500040\pi$
$761$	−0.00687569	−0.000249244 0	−0.000124622		1.00000i	$0.500040\pi$
$762$	− 37.3415i	− 1.35274i
$763$	− 31.1138i	− 1.12640i
$764$	24.2208	0.876277
$765$	0	0
$766$	62.3380	2.25237
$767$	2.11727i	0.0764501i
$768$	− 84.8459i	− 3.06161i
$769$	20.3258	0.732968	0.366484	−	0.930424i	$-0.380561\pi$
$769$	20.3258	0.732968	0.366484	+	0.930424i	$0.380561\pi$
$770$	0	0
$771$	−1.11383	−0.0401136
$772$	133.790i	4.81520i
$773$	9.90184i	0.356144i	0.984017	+	0.178072i	$0.0569861\pi$
$773$	9.90184i	0.356144i	−0.984017	+	0.178072i	$0.943014\pi$
$774$	−30.8793	−1.10993
$775$	0	0
$776$	−112.005	−4.02076
$777$	11.6320i	0.417295i
$778$	88.0054i	3.15514i
$779$	10.0913	0.361558
$780$	0	0
$781$	18.2767	0.653993
$782$	− 38.1295i	− 1.36351i
$783$	− 6.00000i	− 0.214423i
$784$	−6.86651	−0.245232
$785$	0	0
$786$	26.2277	0.935510
$787$	− 36.3449i	− 1.29556i	−0.761829	−	0.647778i	$-0.775698\pi$
$787$	− 36.3449i	− 1.29556i	0.761829	−	0.647778i	$-0.224302\pi$
$788$	83.2433i	2.96542i
$789$	8.00000	0.284808
$790$	0	0
$791$	35.6673	1.26818
$792$	28.1104i	0.998859i
$793$	6.60256i	0.234464i
$794$	−49.8759	−1.77003
$795$	0	0
$796$	−86.4484	−3.06408
$797$	− 18.8371i	− 0.667244i	−0.942707	−	0.333622i	$-0.891729\pi$
$797$	− 18.8371i	− 0.667244i	0.942707	−	0.333622i	$-0.108271\pi$
$798$	− 26.8793i	− 0.951517i
$799$	−32.7880	−1.15996
$800$	0	0
$801$	1.16291	0.0410894
$802$	− 37.7395i	− 1.33263i
$803$	− 24.7880i	− 0.874750i
$804$	10.7689	0.379790
$805$	0	0
$806$	20.9966	0.739572
$807$	15.6742i	0.551757i
$808$	79.3155i	2.79031i
$809$	−32.2277	−1.13306	−0.566532	−	0.824040i	$-0.691715\pi$
$809$	−32.2277	−1.13306	−0.566532	+	0.824040i	$0.691715\pi$
$810$	0	0
$811$	−23.0034	−0.807760	−0.403880	−	0.914812i	$-0.632339\pi$
$811$	−23.0034	−0.807760	−0.403880	+	0.914812i	$0.632339\pi$
$812$	93.3415i	3.27564i
$813$	− 0.443086i	− 0.0155397i
$814$	−32.3189	−1.13278
$815$	0	0
$816$	49.0157	1.71589
$817$	39.5309i	1.38301i
$818$	− 38.8984i	− 1.36005i
$819$	2.71982	0.0950383
$820$	0	0
$821$	49.9372	1.74282	0.871410	−	0.490556i	$-0.163206\pi$
$821$	49.9372	1.74282	0.871410	+	0.490556i	$0.163206\pi$
$822$	4.90528i	0.171091i
$823$	28.2345i	0.984194i	0.870540	+	0.492097i	$0.163769\pi$
$823$	28.2345i	0.984194i	−0.870540	+	0.492097i	$0.836231\pi$
$824$	−38.9175	−1.35576
$825$	0	0
$826$	16.0000	0.556711
$827$	9.55004i	0.332087i	0.986118	+	0.166044i	$0.0530993\pi$
$827$	9.55004i	0.332087i	−0.986118	+	0.166044i	$0.946901\pi$
$828$	− 27.6673i	− 0.961505i
$829$	37.9862	1.31932	0.659658	−	0.751565i	$-0.270701\pi$
$829$	37.9862	1.31932	0.659658	+	0.751565i	$0.270701\pi$
$830$	0	0
$831$	4.87930	0.169261
$832$	− 41.3871i	− 1.43484i
$833$	− 1.12758i	− 0.0390683i
$834$	−17.4396	−0.603886
$835$	0	0
$836$	55.3346	1.91379
$837$	7.55691i	0.261205i
$838$	34.2277i	1.18237i
$839$	4.72670	0.163184	0.0815919	−	0.996666i	$-0.474000\pi$
$839$	4.72670	0.163184	0.0815919	+	0.996666i	$0.474000\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 63.3155i	− 2.18200i
$843$	9.11383i	0.313897i
$844$	104.259	3.58874
$845$	0	0
$846$	−32.1104	−1.10398
$847$	− 9.79832i	− 0.336674i
$848$	20.0913i	0.689938i
$849$	33.3415	1.14428
$850$	0	0
$851$	20.6870	0.709140
$852$	− 38.4362i	− 1.31680i
$853$	− 48.3611i	− 1.65585i	−0.560836	−	0.827927i	$-0.689520\pi$
$853$	− 48.3611i	− 1.65585i	0.560836	−	0.827927i	$-0.310480\pi$
$854$	49.8950	1.70737
$855$	0	0
$856$	−130.252	−4.45193
$857$	− 6.83709i	− 0.233551i	−0.993158	−	0.116775i	$-0.962744\pi$
$857$	− 6.83709i	− 0.233551i	0.993158	−	0.116775i	$-0.0372557\pi$
$858$	7.55691i	0.257989i
$859$	−12.6026	−0.429994	−0.214997	−	0.976615i	$-0.568974\pi$
$859$	−12.6026	−0.429994	−0.214997	+	0.976615i	$0.568974\pi$
$860$	0	0
$861$	7.71639	0.262974
$862$	24.9966i	0.851386i
$863$	8.20855i	0.279422i	0.990192	+	0.139711i	$0.0446174\pi$
$863$	8.20855i	0.279422i	−0.990192	+	0.139711i	$0.955383\pi$
$864$	27.3319	0.929851
$865$	0	0
$866$	−56.4744	−1.91908
$867$	− 8.95092i	− 0.303989i
$868$	− 117.562i	− 3.99032i
$869$	27.9509	0.948170
$870$	0	0
$871$	1.88273	0.0637940
$872$	− 118.233i	− 4.00387i
$873$	− 10.8371i	− 0.366780i
$874$	−47.8037	−1.61698
$875$	0	0
$876$	−52.1295	−1.76129
$877$	− 13.5309i	− 0.456907i	−0.973555	−	0.228454i	$-0.926633\pi$
$877$	− 13.5309i	− 0.456907i	0.973555	−	0.228454i	$-0.0733669\pi$
$878$	− 70.5466i	− 2.38083i
$879$	−29.4328	−0.992743
$880$	0	0
$881$	−9.34836	−0.314954	−0.157477	−	0.987523i	$-0.550336\pi$
$881$	−9.34836	−0.314954	−0.157477	+	0.987523i	$0.550336\pi$
$882$	− 1.10428i	− 0.0371829i
$883$	− 55.1001i	− 1.85427i	−0.374733	−	0.927133i	$-0.622266\pi$
$883$	− 55.1001i	− 1.85427i	0.374733	−	0.927133i	$-0.377734\pi$
$884$	16.2277	0.545795
$885$	0	0
$886$	30.3640	1.02010
$887$	− 0.133492i	− 0.00448223i	−0.999997	−	0.00224112i	$-0.999287\pi$
$887$	− 0.133492i	− 0.00448223i	0.999997	−	0.00224112i	$-0.000713370\pi$
$888$	44.2017i	1.48331i
$889$	−36.5535	−1.22596
$890$	0	0
$891$	−2.71982	−0.0911175
$892$	− 57.8690i	− 1.93760i
$893$	41.1070i	1.37559i
$894$	−57.8759	−1.93566
$895$	0	0
$896$	−164.083	−5.48162
$897$	− 4.83709i	− 0.161506i
$898$	7.88273i	0.263050i
$899$	−45.3415	−1.51222
$900$	0	0
$901$	−3.29928	−0.109915
$902$	21.4396i	0.713862i
$903$	30.2277i	1.00591i
$904$	135.536	4.50787
$905$	0	0
$906$	−13.8827	−0.461223
$907$	58.5466i	1.94401i	0.234964	+	0.972004i	$0.424503\pi$
$907$	58.5466i	1.94401i	−0.234964	+	0.972004i	$0.575497\pi$
$908$	64.7620i	2.14920i
$909$	−7.67418	−0.254537
$910$	0	0
$911$	50.4622	1.67189	0.835943	−	0.548816i	$-0.184921\pi$
$911$	50.4622	1.67189	0.835943	+	0.548816i	$0.184921\pi$
$912$	− 61.4519i	− 2.03487i
$913$	− 5.75859i	− 0.190582i
$914$	38.1104	1.26058
$915$	0	0
$916$	−35.6604	−1.17825
$917$	− 25.6742i	− 0.847836i
$918$	7.88273i	0.260169i
$919$	−56.9735	−1.87938	−0.939691	−	0.342026i	$-0.888887\pi$
$919$	−56.9735	−1.87938	−0.939691	+	0.342026i	$0.888887\pi$
$920$	0	0
$921$	21.8337	0.719443
$922$	54.5275i	1.79577i
$923$	− 6.71982i	− 0.221186i
$924$	42.3121	1.39196
$925$	0	0
$926$	75.1261	2.46880
$927$	− 3.76547i	− 0.123674i
$928$	163.992i	5.38329i
$929$	36.5957	1.20067	0.600333	−	0.799750i	$-0.295035\pi$
$929$	36.5957	1.20067	0.600333	+	0.799750i	$0.295035\pi$
$930$	0	0
$931$	−1.41367	−0.0463311
$932$	− 39.1070i	− 1.28099i
$933$	− 25.1070i	− 0.821965i
$934$	−80.3572	−2.62937
$935$	0	0
$936$	10.3354	0.337822
$937$	47.1070i	1.53892i	0.638697	+	0.769459i	$0.279474\pi$
$937$	47.1070i	1.53892i	−0.638697	+	0.769459i	$0.720526\pi$
$938$	− 14.2277i	− 0.464549i
$939$	8.22766	0.268499
$940$	0	0
$941$	−36.3611	−1.18534	−0.592670	−	0.805446i	$-0.701926\pi$
$941$	−36.3611	−1.18534	−0.592670	+	0.805446i	$0.701926\pi$
$942$	24.6707i	0.803816i
$943$	− 13.7233i	− 0.446891i
$944$	36.5795	1.19056
$945$	0	0
$946$	−83.9862	−2.73063
$947$	− 44.5795i	− 1.44864i	−0.689465	−	0.724319i	$-0.742154\pi$
$947$	− 44.5795i	− 1.44864i	0.689465	−	0.724319i	$-0.257846\pi$
$948$	− 58.7811i	− 1.90912i
$949$	−9.11383	−0.295847
$950$	0	0
$951$	−27.6742	−0.897397
$952$	− 79.7517i	− 2.58477i
$953$	19.8596i	0.643317i	0.946856	+	0.321658i	$0.104240\pi$
$953$	19.8596i	0.643317i	−0.946856	+	0.321658i	$0.895760\pi$
$954$	−3.23109	−0.104611
$955$	0	0
$956$	7.32238	0.236823
$957$	− 16.3189i	− 0.527517i
$958$	− 33.7846i	− 1.09153i
$959$	4.80176	0.155057
$960$	0	0
$961$	26.1070	0.842160
$962$	11.8827i	0.383115i
$963$	− 12.6026i	− 0.406112i
$964$	34.3189	1.10534
$965$	0	0
$966$	−36.5535	−1.17609
$967$	− 47.4068i	− 1.52450i	−0.647283	−	0.762250i	$-0.724095\pi$
$967$	− 47.4068i	− 1.52450i	0.647283	−	0.762250i	$-0.275905\pi$
$968$	− 37.2338i	− 1.19674i
$969$	10.0913	0.324179
$970$	0	0
$971$	−10.6448	−0.341607	−0.170803	−	0.985305i	$-0.554636\pi$
$971$	−10.6448	−0.341607	−0.170803	+	0.985305i	$0.554636\pi$
$972$	5.71982i	0.183463i
$973$	17.0716i	0.547291i
$974$	0.443086	0.0141974
$975$	0	0
$976$	114.071	3.65131
$977$	1.21199i	0.0387750i	0.999812	+	0.0193875i	$0.00617162\pi$
$977$	1.21199i	0.0387750i	−0.999812	+	0.0193875i	$0.993828\pi$
$978$	38.4362i	1.22905i
$979$	3.16291	0.101087
$980$	0	0
$981$	11.4396	0.365240
$982$	117.328i	3.74408i
$983$	51.8759i	1.65458i	0.561773	+	0.827291i	$0.310119\pi$
$983$	51.8759i	1.65458i	−0.561773	+	0.827291i	$0.689881\pi$
$984$	29.3224	0.934763
$985$	0	0
$986$	−47.2964	−1.50622
$987$	31.4328i	1.00052i
$988$	− 20.3449i	− 0.647258i
$989$	53.7586	1.70942
$990$	0	0
$991$	−21.6251	−0.686944	−0.343472	−	0.939163i	$-0.611603\pi$
$991$	−21.6251	−0.686944	−0.343472	+	0.939163i	$0.611603\pi$
$992$	− 206.545i	− 6.55781i
$993$	13.2311i	0.419876i
$994$	−50.7811	−1.61068
$995$	0	0
$996$	−12.1104	−0.383732
$997$	− 23.2051i	− 0.734913i	−0.930041	−	0.367457i	$-0.880229\pi$
$997$	− 23.2051i	− 0.734913i	0.930041	−	0.367457i	$-0.119771\pi$
$998$	21.6482i	0.685262i
$999$	−4.27674	−0.135310

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.c.i.274.1		6
3.2	odd	2		2925.2.c.w.2224.6		6
5.2	odd	4		195.2.a.e.1.3	✓	3
5.3	odd	4		975.2.a.o.1.1		3
5.4	even	2	inner	975.2.c.i.274.6		6
15.2	even	4		585.2.a.n.1.1		3
15.8	even	4		2925.2.a.bh.1.3		3
15.14	odd	2		2925.2.c.w.2224.1		6
20.7	even	4		3120.2.a.bj.1.3		3
35.27	even	4		9555.2.a.bq.1.3		3
60.47	odd	4		9360.2.a.dd.1.3		3
65.12	odd	4		2535.2.a.bc.1.1		3
195.77	even	4		7605.2.a.bx.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.a.e.1.3	✓	3	5.2	odd	4
585.2.a.n.1.1		3	15.2	even	4
975.2.a.o.1.1		3	5.3	odd	4
975.2.c.i.274.1		6	1.1	even	1	trivial
975.2.c.i.274.6		6	5.4	even	2	inner
2535.2.a.bc.1.1		3	65.12	odd	4
2925.2.a.bh.1.3		3	15.8	even	4
2925.2.c.w.2224.1		6	15.14	odd	2
2925.2.c.w.2224.6		6	3.2	odd	2
3120.2.a.bj.1.3		3	20.7	even	4
7605.2.a.bx.1.3		3	195.77	even	4
9360.2.a.dd.1.3		3	60.47	odd	4
9555.2.a.bq.1.3		3	35.27	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.77846i q^{2} -1.00000i q^{3} -5.71982 q^{4} -2.77846 q^{6} +2.71982i q^{7} +10.3354i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-2.77846i q^{2} -1.00000i q^{3} -5.71982 q^{4} -2.77846 q^{6} +2.71982i q^{7} +10.3354i q^{8} -1.00000 q^{9} -2.71982 q^{11} +5.71982i q^{12} +1.00000i q^{13} +7.55691 q^{14} +17.2767 q^{16} +2.83709i q^{17} +2.77846i q^{18} +3.55691 q^{19} +2.71982 q^{21} +7.55691i q^{22} -4.83709i q^{23} +10.3354 q^{24} +2.77846 q^{26} +1.00000i q^{27} -15.5569i q^{28} -6.00000 q^{29} +7.55691 q^{31} -27.3319i q^{32} +2.71982i q^{33} +7.88273 q^{34} +5.71982 q^{36} +4.27674i q^{37} -9.88273i q^{38} +1.00000 q^{39} +2.83709 q^{41} -7.55691i q^{42} +11.1138i q^{43} +15.5569 q^{44} -13.4396 q^{46} +11.5569i q^{47} -17.2767i q^{48} -0.397442 q^{49} +2.83709 q^{51} -5.71982i q^{52} +1.16291i q^{53} +2.77846 q^{54} -28.1104 q^{56} -3.55691i q^{57} +16.6707i q^{58} +2.11727 q^{59} +6.60256 q^{61} -20.9966i q^{62} -2.71982i q^{63} -41.3871 q^{64} +7.55691 q^{66} -1.88273i q^{67} -16.2277i q^{68} -4.83709 q^{69} -6.71982 q^{71} -10.3354i q^{72} +9.11383i q^{73} +11.8827 q^{74} -20.3449 q^{76} -7.39744i q^{77} -2.77846i q^{78} -10.2767 q^{79} +1.00000 q^{81} -7.88273i q^{82} +2.11727i q^{83} -15.5569 q^{84} +30.8793 q^{86} +6.00000i q^{87} -28.1104i q^{88} -1.16291 q^{89} -2.71982 q^{91} +27.6673i q^{92} -7.55691i q^{93} +32.1104 q^{94} -27.3319 q^{96} +10.8371i q^{97} +1.10428i q^{98} +2.71982 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 16 q^{4} - 6 q^{9}+O(q^{10}) \) 6 * q - 16 * q^4 - 6 * q^9	\( 6 q - 16 q^{4} - 6 q^{9} + 2 q^{11} + 12 q^{14} + 52 q^{16} - 12 q^{19} - 2 q^{21} + 12 q^{24} - 36 q^{29} + 12 q^{31} + 44 q^{34} + 16 q^{36} + 6 q^{39} + 2 q^{41} + 60 q^{44} - 44 q^{46} - 24 q^{49} + 2 q^{51} - 32 q^{56} + 16 q^{59} + 18 q^{61} - 60 q^{64} + 12 q^{66} - 14 q^{69} - 22 q^{71} + 68 q^{74} + 8 q^{76} - 10 q^{79} + 6 q^{81} - 60 q^{84} + 112 q^{86} - 22 q^{89} + 2 q^{91} + 56 q^{94} - 44 q^{96} - 2 q^{99}+O(q^{100}) \) 6 * q - 16 * q^4 - 6 * q^9 + 2 * q^11 + 12 * q^14 + 52 * q^16 - 12 * q^19 - 2 * q^21 + 12 * q^24 - 36 * q^29 + 12 * q^31 + 44 * q^34 + 16 * q^36 + 6 * q^39 + 2 * q^41 + 60 * q^44 - 44 * q^46 - 24 * q^49 + 2 * q^51 - 32 * q^56 + 16 * q^59 + 18 * q^61 - 60 * q^64 + 12 * q^66 - 14 * q^69 - 22 * q^71 + 68 * q^74 + 8 * q^76 - 10 * q^79 + 6 * q^81 - 60 * q^84 + 112 * q^86 - 22 * q^89 + 2 * q^91 + 56 * q^94 - 44 * q^96 - 2 * q^99

Introduction

L-functions

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