LMFDB - Embedded newform 1075.2.b.k.474.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,2,Mod(474,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([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("1075.474");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1075.b (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:	$8.58391821729$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 19x^{10} + 121x^{8} + 339x^{6} + 437x^{4} + 247x^{2} + 49 $ x^12 + 19x^10 + 121x^8 + 339x^6 + 437x^4 + 247*x^2 + 49
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 215)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			474.3
Root			$-0.840555i$ of defining polynomial
Character	$\chi$	$=$	1075.474
Dual form			1075.2.b.k.474.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.84056i q^{2} +0.291442i q^{3} -1.38764 q^{4} +0.536415 q^{6} -5.13977i q^{7} -1.12707i q^{8} +2.91506 q^{9} +O(q^{10})$	\(q-1.84056i q^{2} +0.291442i q^{3} -1.38764 q^{4} +0.536415 q^{6} -5.13977i q^{7} -1.12707i q^{8} +2.91506 q^{9} +5.03950 q^{11} -0.404418i q^{12} -1.68111i q^{13} -9.46004 q^{14} -4.84973 q^{16} +7.47071i q^{17} -5.36533i q^{18} +0.608281 q^{19} +1.49795 q^{21} -9.27547i q^{22} -1.48871i q^{23} +0.328477 q^{24} -3.09418 q^{26} +1.72390i q^{27} +7.13218i q^{28} +5.51410 q^{29} +1.88966 q^{31} +6.67205i q^{32} +1.46872i q^{33} +13.7503 q^{34} -4.04507 q^{36} -5.08289i q^{37} -1.11957i q^{38} +0.489946 q^{39} -6.85633 q^{41} -2.75705i q^{42} +1.00000i q^{43} -6.99303 q^{44} -2.74005 q^{46} -8.54354i q^{47} -1.41342i q^{48} -19.4173 q^{49} -2.17728 q^{51} +2.33278i q^{52} +9.18537i q^{53} +3.17293 q^{54} -5.79291 q^{56} +0.177279i q^{57} -10.1490i q^{58} -7.44326 q^{59} -8.51814 q^{61} -3.47803i q^{62} -14.9828i q^{63} +2.58081 q^{64} +2.70326 q^{66} +4.71055i q^{67} -10.3667i q^{68} +0.433871 q^{69} +3.25011 q^{71} -3.28549i q^{72} +6.48385i q^{73} -9.35534 q^{74} -0.844077 q^{76} -25.9019i q^{77} -0.901773i q^{78} +3.22785 q^{79} +8.24277 q^{81} +12.6195i q^{82} -6.51534i q^{83} -2.07862 q^{84} +1.84056 q^{86} +1.60704i q^{87} -5.67989i q^{88} +11.5677 q^{89} -8.64053 q^{91} +2.06579i q^{92} +0.550728i q^{93} -15.7249 q^{94} -1.94452 q^{96} -1.06878i q^{97} +35.7386i q^{98} +14.6904 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 14 q^{4} - 16 q^{9}+O(q^{10}) $ 12 * q - 14 * q^4 - 16 * q^9	$ 12 q - 14 q^{4} - 16 q^{9} + 8 q^{14} + 10 q^{16} - 12 q^{19} - 24 q^{21} - 64 q^{26} + 20 q^{29} - 28 q^{34} - 82 q^{36} - 16 q^{39} - 12 q^{41} - 8 q^{44} - 16 q^{46} - 40 q^{49} - 16 q^{51} + 10 q^{54} - 70 q^{56} + 40 q^{59} - 16 q^{61} - 34 q^{64} - 70 q^{66} + 84 q^{69} + 16 q^{71} - 90 q^{74} - 32 q^{76} + 32 q^{79} + 92 q^{81} - 74 q^{84} + 6 q^{86} + 48 q^{91} - 24 q^{94} - 46 q^{96} + 32 q^{99}+O(q^{100}) $ 12 * q - 14 * q^4 - 16 * q^9 + 8 * q^14 + 10 * q^16 - 12 * q^19 - 24 * q^21 - 64 * q^26 + 20 * q^29 - 28 * q^34 - 82 * q^36 - 16 * q^39 - 12 * q^41 - 8 * q^44 - 16 * q^46 - 40 * q^49 - 16 * q^51 + 10 * q^54 - 70 * q^56 + 40 * q^59 - 16 * q^61 - 34 * q^64 - 70 * q^66 + 84 * q^69 + 16 * q^71 - 90 * q^74 - 32 * q^76 + 32 * q^79 + 92 * q^81 - 74 * q^84 + 6 * q^86 + 48 * q^91 - 24 * q^94 - 46 * q^96 + 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1075\mathbb{Z}\right)^\times$.

$n$	$302$	$476$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 1.84056i	− 1.30147i	−0.759305	−	0.650735i	$-0.774461\pi$
$2$	− 1.84056i	− 1.30147i	0.759305	−	0.650735i	$-0.225539\pi$
$3$	0.291442i	0.168264i	0.996455	+	0.0841320i	$0.0268118\pi$
$3$	0.291442i	0.168264i	−0.996455	+	0.0841320i	$0.973188\pi$
$4$	−1.38764	−0.693822
$5$	0	0
$5$	0	0
$6$	0.536415	0.218991
$7$	− 5.13977i	− 1.94265i	−0.237752	−	0.971326i	$-0.576411\pi$
$7$	− 5.13977i	− 1.94265i	0.237752	−	0.971326i	$-0.423589\pi$
$8$	− 1.12707i	− 0.398481i
$9$	2.91506	0.971687
$10$	0	0
$11$	5.03950	1.51947	0.759733	−	0.650236i	$-0.225330\pi$
$11$	5.03950	1.51947	0.759733	+	0.650236i	$0.225330\pi$
$12$	− 0.404418i	− 0.116745i
$13$	− 1.68111i	− 0.466256i	−0.972446	−	0.233128i	$-0.925104\pi$
$13$	− 1.68111i	− 0.466256i	0.972446	−	0.233128i	$-0.0748962\pi$
$14$	−9.46004	−2.52830
$15$	0	0
$16$	−4.84973	−1.21243
$17$	7.47071i	1.81191i	0.423370	+	0.905957i	$0.360847\pi$
$17$	7.47071i	1.81191i	−0.423370	+	0.905957i	$0.639153\pi$
$18$	− 5.36533i	− 1.26462i
$19$	0.608281	0.139549	0.0697746	−	0.997563i	$-0.477772\pi$
$19$	0.608281	0.139549	0.0697746	+	0.997563i	$0.477772\pi$
$20$	0	0
$21$	1.49795	0.326878
$22$	− 9.27547i	− 1.97754i
$23$	− 1.48871i	− 0.310417i	−0.987882	−	0.155208i	$-0.950395\pi$
$23$	− 1.48871i	− 0.310417i	0.987882	−	0.155208i	$-0.0496049\pi$
$24$	0.328477	0.0670501
$25$	0	0
$26$	−3.09418	−0.606818
$27$	1.72390i	0.331764i
$28$	7.13218i	1.34785i
$29$	5.51410	1.02394	0.511972	−	0.859002i	$-0.328915\pi$
$29$	5.51410	1.02394	0.511972	+	0.859002i	$0.328915\pi$
$30$	0	0
$31$	1.88966	0.339394	0.169697	−	0.985496i	$-0.445721\pi$
$31$	1.88966	0.339394	0.169697	+	0.985496i	$0.445721\pi$
$32$	6.67205i	1.17946i
$33$	1.46872i	0.255671i
$34$	13.7503	2.35815
$35$	0	0
$36$	−4.04507	−0.674178
$37$	− 5.08289i	− 0.835622i	−0.908534	−	0.417811i	$-0.862797\pi$
$37$	− 5.08289i	− 0.835622i	0.908534	−	0.417811i	$-0.137203\pi$
$38$	− 1.11957i	− 0.181619i
$39$	0.489946	0.0784542
$40$	0	0
$41$	−6.85633	−1.07078	−0.535389	−	0.844605i	$-0.679835\pi$
$41$	−6.85633	−1.07078	−0.535389	+	0.844605i	$0.679835\pi$
$42$	− 2.75705i	− 0.425422i
$43$	1.00000i	0.152499i
$43$	1.00000i	0.152499i
$44$	−6.99303	−1.05424
$45$	0	0
$46$	−2.74005	−0.403998
$47$	− 8.54354i	− 1.24620i	−0.782141	−	0.623102i	$-0.785872\pi$
$47$	− 8.54354i	− 1.24620i	0.782141	−	0.623102i	$-0.214128\pi$
$48$	− 1.41342i	− 0.204009i
$49$	−19.4173	−2.77390
$50$	0	0
$51$	−2.17728	−0.304880
$52$	2.33278i	0.323499i
$53$	9.18537i	1.26171i	0.775902	+	0.630854i	$0.217295\pi$
$53$	9.18537i	1.26171i	−0.775902	+	0.630854i	$0.782705\pi$
$54$	3.17293	0.431781
$55$	0	0
$56$	−5.79291	−0.774110
$57$	0.177279i	0.0234811i
$58$	− 10.1490i	− 1.33263i
$59$	−7.44326	−0.969030	−0.484515	−	0.874783i	$-0.661004\pi$
$59$	−7.44326	−0.969030	−0.484515	+	0.874783i	$0.661004\pi$
$60$	0	0
$61$	−8.51814	−1.09064	−0.545318	−	0.838229i	$-0.683591\pi$
$61$	−8.51814	−1.09064	−0.545318	+	0.838229i	$0.683591\pi$
$62$	− 3.47803i	− 0.441711i
$63$	− 14.9828i	− 1.88765i
$64$	2.58081	0.322602
$65$	0	0
$66$	2.70326	0.332748
$67$	4.71055i	0.575485i	0.957708	+	0.287743i	$0.0929047\pi$
$67$	4.71055i	0.575485i	−0.957708	+	0.287743i	$0.907095\pi$
$68$	− 10.3667i	− 1.25715i
$69$	0.433871	0.0522320
$70$	0	0
$71$	3.25011	0.385717	0.192858	−	0.981227i	$-0.438224\pi$
$71$	3.25011	0.385717	0.192858	+	0.981227i	$0.438224\pi$
$72$	− 3.28549i	− 0.387199i
$73$	6.48385i	0.758877i	0.925217	+	0.379438i	$0.123883\pi$
$73$	6.48385i	0.758877i	−0.925217	+	0.379438i	$0.876117\pi$
$74$	−9.35534	−1.08754
$75$	0	0
$76$	−0.844077	−0.0968223
$77$	− 25.9019i	− 2.95179i
$78$	− 0.901773i	− 0.102106i
$79$	3.22785	0.363162	0.181581	−	0.983376i	$-0.441879\pi$
$79$	3.22785	0.363162	0.181581	+	0.983376i	$0.441879\pi$
$80$	0	0
$81$	8.24277	0.915863
$82$	12.6195i	1.39359i
$83$	− 6.51534i	− 0.715152i	−0.933884	−	0.357576i	$-0.883603\pi$
$83$	− 6.51534i	− 0.715152i	0.933884	−	0.357576i	$-0.116397\pi$
$84$	−2.07862	−0.226796
$85$	0	0
$86$	1.84056	0.198472
$87$	1.60704i	0.172293i
$88$	− 5.67989i	− 0.605478i
$89$	11.5677	1.22617	0.613087	−	0.790016i	$-0.289928\pi$
$89$	11.5677	1.22617	0.613087	+	0.790016i	$0.289928\pi$
$90$	0	0
$91$	−8.64053	−0.905773
$92$	2.06579i	0.215374i
$93$	0.550728i	0.0571078i
$94$	−15.7249	−1.62190
$95$	0	0
$96$	−1.94452	−0.198461
$97$	− 1.06878i	− 0.108518i	−0.998527	−	0.0542591i	$-0.982720\pi$
$97$	− 1.06878i	− 0.108518i	0.998527	−	0.0542591i	$-0.0172797\pi$
$98$	35.7386i	3.61014i
$99$	14.6904	1.47644
$100$	0	0
$101$	−15.7267	−1.56487	−0.782433	−	0.622735i	$-0.786022\pi$
$101$	−15.7267	−1.56487	−0.782433	+	0.622735i	$0.786022\pi$
$102$	4.00740i	0.396792i
$103$	16.4707i	1.62290i	0.584421	+	0.811451i	$0.301322\pi$
$103$	16.4707i	1.62290i	−0.584421	+	0.811451i	$0.698678\pi$
$104$	−1.89474	−0.185794
$105$	0	0
$106$	16.9062	1.64207
$107$	− 2.13751i	− 0.206641i	−0.994648	−	0.103320i	$-0.967053\pi$
$107$	− 2.13751i	− 0.206641i	0.994648	−	0.103320i	$-0.0329467\pi$
$108$	− 2.39216i	− 0.230185i
$109$	1.91506	0.183430	0.0917148	−	0.995785i	$-0.470765\pi$
$109$	1.91506	0.183430	0.0917148	+	0.995785i	$0.470765\pi$
$110$	0	0
$111$	1.48137	0.140605
$112$	24.9265i	2.35534i
$113$	10.3309i	0.971853i	0.874000	+	0.485926i	$0.161518\pi$
$113$	10.3309i	0.971853i	−0.874000	+	0.485926i	$0.838482\pi$
$114$	0.326291	0.0305599
$115$	0	0
$116$	−7.65161	−0.710434
$117$	− 4.90054i	− 0.453055i
$118$	13.6997i	1.26116i
$119$	38.3978	3.51992
$120$	0	0
$121$	14.3965	1.30877
$122$	15.6781i	1.41943i
$123$	− 1.99822i	− 0.180174i
$124$	−2.62218	−0.235479
$125$	0	0
$126$	−27.5766	−2.45672
$127$	− 12.1166i	− 1.07517i	−0.843208	−	0.537587i	$-0.819336\pi$
$127$	− 12.1166i	− 1.07517i	0.843208	−	0.537587i	$-0.180664\pi$
$128$	8.59397i	0.759607i
$129$	−0.291442	−0.0256600
$130$	0	0
$131$	−0.00984495	−0.000860157 0	−0.000430079	−	1.00000i	$-0.500137\pi$
$131$	−0.00984495	−0.000860157 0	−0.000430079		1.00000i	$0.500137\pi$
$132$	− 2.03806i	− 0.177390i
$133$	− 3.12643i	− 0.271095i
$134$	8.67003	0.748976
$135$	0	0
$136$	8.42005	0.722013
$137$	− 7.64961i	− 0.653551i	−0.945102	−	0.326775i	$-0.894038\pi$
$137$	− 7.64961i	− 0.653551i	0.945102	−	0.326775i	$-0.105962\pi$
$138$	− 0.798564i	− 0.0679783i
$139$	−10.8055	−0.916509	−0.458255	−	0.888821i	$-0.651525\pi$
$139$	−10.8055	−0.916509	−0.458255	+	0.888821i	$0.651525\pi$
$140$	0	0
$141$	2.48995	0.209691
$142$	− 5.98200i	− 0.501999i
$143$	− 8.47195i	− 0.708460i
$144$	−14.1373	−1.17811
$145$	0	0
$146$	11.9339	0.987655
$147$	− 5.65901i	− 0.466747i
$148$	7.05324i	0.579773i
$149$	3.39172	0.277860	0.138930	−	0.990302i	$-0.455634\pi$
$149$	3.39172	0.277860	0.138930	+	0.990302i	$0.455634\pi$
$150$	0	0
$151$	0.387678	0.0315488	0.0157744	−	0.999876i	$-0.494979\pi$
$151$	0.387678	0.0315488	0.0157744	+	0.999876i	$0.494979\pi$
$152$	− 0.685578i	− 0.0556077i
$153$	21.7776i	1.76061i
$154$	−47.6738	−3.84167
$155$	0	0
$156$	−0.679871	−0.0544332
$157$	20.2046i	1.61250i	0.591573	+	0.806252i	$0.298507\pi$
$157$	20.2046i	1.61250i	−0.591573	+	0.806252i	$0.701493\pi$
$158$	− 5.94104i	− 0.472644i
$159$	−2.67700	−0.212300
$160$	0	0
$161$	−7.65161	−0.603032
$162$	− 15.1713i	− 1.19197i
$163$	10.9097i	0.854510i	0.904131	+	0.427255i	$0.140519\pi$
$163$	10.9097i	0.854510i	−0.904131	+	0.427255i	$0.859481\pi$
$164$	9.51415	0.742930
$165$	0	0
$166$	−11.9918	−0.930748
$167$	− 1.54073i	− 0.119225i	−0.998222	−	0.0596127i	$-0.981013\pi$
$167$	− 1.54073i	− 0.119225i	0.998222	−	0.0596127i	$-0.0189866\pi$
$168$	− 1.68830i	− 0.130255i
$169$	10.1739	0.782605
$170$	0	0
$171$	1.77318	0.135598
$172$	− 1.38764i	− 0.105807i
$173$	− 6.99994i	− 0.532196i	−0.963946	−	0.266098i	$-0.914266\pi$
$173$	− 6.99994i	− 0.532196i	0.963946	−	0.266098i	$-0.0857344\pi$
$174$	2.95785	0.224234
$175$	0	0
$176$	−24.4402	−1.84225
$177$	− 2.16928i	− 0.163053i
$178$	− 21.2910i	− 1.59583i
$179$	11.8268	0.883974	0.441987	−	0.897021i	$-0.354274\pi$
$179$	11.8268	0.883974	0.441987	+	0.897021i	$0.354274\pi$
$180$	0	0
$181$	−1.74546	−0.129739	−0.0648694	−	0.997894i	$-0.520663\pi$
$181$	−1.74546	−0.129739	−0.0648694	+	0.997894i	$0.520663\pi$
$182$	15.9034i	1.17884i
$183$	− 2.48254i	− 0.183515i
$184$	−1.67788	−0.123695
$185$	0	0
$186$	1.01364	0.0743240
$187$	37.6486i	2.75314i
$188$	11.8554i	0.864644i
$189$	8.86044	0.644502
$190$	0	0
$191$	21.9963	1.59159	0.795797	−	0.605564i	$-0.207052\pi$
$191$	21.9963	1.59159	0.795797	+	0.605564i	$0.207052\pi$
$192$	0.752158i	0.0542823i
$193$	15.1264i	1.08882i	0.838818	+	0.544412i	$0.183247\pi$
$193$	15.1264i	1.08882i	−0.838818	+	0.544412i	$0.816753\pi$
$194$	−1.96715	−0.141233
$195$	0	0
$196$	26.9443	1.92459
$197$	14.5354i	1.03560i	0.855501	+	0.517802i	$0.173250\pi$
$197$	14.5354i	1.03560i	−0.855501	+	0.517802i	$0.826750\pi$
$198$	− 27.0386i	− 1.92155i
$199$	7.46791	0.529386	0.264693	−	0.964333i	$-0.414729\pi$
$199$	7.46791	0.529386	0.264693	+	0.964333i	$0.414729\pi$
$200$	0	0
$201$	−1.37285	−0.0968335
$202$	28.9459i	2.03662i
$203$	− 28.3412i	− 1.98917i
$204$	3.02129	0.211532
$205$	0	0
$206$	30.3151	2.11216
$207$	− 4.33967i	− 0.301628i
$208$	8.15294i	0.565304i
$209$	3.06543	0.212040
$210$	0	0
$211$	−8.81738	−0.607014	−0.303507	−	0.952829i	$-0.598157\pi$
$211$	−8.81738	−0.607014	−0.303507	+	0.952829i	$0.598157\pi$
$212$	− 12.7460i	− 0.875401i
$213$	0.947218i	0.0649023i
$214$	−3.93421	−0.268937
$215$	0	0
$216$	1.94296	0.132202
$217$	− 9.71245i	− 0.659324i
$218$	− 3.52478i	− 0.238728i
$219$	−1.88966	−0.127692
$220$	0	0
$221$	12.5591	0.844816
$222$	− 2.72654i	− 0.182993i
$223$	− 3.76333i	− 0.252011i	−0.992030	−	0.126006i	$-0.959784\pi$
$223$	− 3.76333i	− 0.252011i	0.992030	−	0.126006i	$-0.0402157\pi$
$224$	34.2928	2.29129
$225$	0	0
$226$	19.0147	1.26484
$227$	21.9658i	1.45792i	0.684556	+	0.728960i	$0.259996\pi$
$227$	21.9658i	1.45792i	−0.684556	+	0.728960i	$0.740004\pi$
$228$	− 0.245999i	− 0.0162917i
$229$	14.2515	0.941765	0.470883	−	0.882196i	$-0.343936\pi$
$229$	14.2515	0.941765	0.470883	+	0.882196i	$0.343936\pi$
$230$	0	0
$231$	7.54889	0.496681
$232$	− 6.21481i	− 0.408022i
$233$	− 16.5427i	− 1.08375i	−0.840459	−	0.541875i	$-0.817715\pi$
$233$	− 16.5427i	− 1.08375i	0.840459	−	0.541875i	$-0.182285\pi$
$234$	−9.01972	−0.589637
$235$	0	0
$236$	10.3286	0.672335
$237$	0.940731i	0.0611071i
$238$	− 70.6732i	− 4.58106i
$239$	9.08775	0.587838	0.293919	−	0.955830i	$-0.405040\pi$
$239$	9.08775	0.587838	0.293919	+	0.955830i	$0.405040\pi$
$240$	0	0
$241$	−3.24141	−0.208797	−0.104399	−	0.994536i	$-0.533292\pi$
$241$	−3.24141	−0.208797	−0.104399	+	0.994536i	$0.533292\pi$
$242$	− 26.4976i	− 1.70333i
$243$	7.57398i	0.485871i
$244$	11.8202	0.756708
$245$	0	0
$246$	−3.67784	−0.234490
$247$	− 1.02259i	− 0.0650657i
$248$	− 2.12979i	− 0.135242i
$249$	1.89884	0.120334
$250$	0	0
$251$	8.88255	0.560661	0.280331	−	0.959903i	$-0.409556\pi$
$251$	8.88255	0.560661	0.280331	+	0.959903i	$0.409556\pi$
$252$	20.7907i	1.30969i
$253$	− 7.50233i	− 0.471667i
$254$	−22.3013	−1.39931
$255$	0	0
$256$	20.9793	1.31121
$257$	− 20.4758i	− 1.27724i	−0.769520	−	0.638622i	$-0.779505\pi$
$257$	− 20.4758i	− 1.27724i	0.769520	−	0.638622i	$-0.220495\pi$
$258$	0.536415i	0.0333957i
$259$	−26.1249	−1.62332
$260$	0	0
$261$	16.0740	0.994953
$262$	0.0181202i	0.00111947i
$263$	14.6293i	0.902083i	0.892503	+	0.451041i	$0.148947\pi$
$263$	14.6293i	0.902083i	−0.892503	+	0.451041i	$0.851053\pi$
$264$	1.65536	0.101880
$265$	0	0
$266$	−5.75436	−0.352822
$267$	3.37131i	0.206321i
$268$	− 6.53657i	− 0.399284i
$269$	12.5266	0.763762	0.381881	−	0.924211i	$-0.375276\pi$
$269$	12.5266	0.763762	0.381881	+	0.924211i	$0.375276\pi$
$270$	0	0
$271$	2.66641	0.161973	0.0809863	−	0.996715i	$-0.474193\pi$
$271$	2.66641	0.161973	0.0809863	+	0.996715i	$0.474193\pi$
$272$	− 36.2309i	− 2.19682i
$273$	− 2.51821i	− 0.152409i
$274$	−14.0795	−0.850576
$275$	0	0
$276$	−0.602059	−0.0362397
$277$	4.83731i	0.290646i	0.989384	+	0.145323i	$0.0464221\pi$
$277$	4.83731i	0.290646i	−0.989384	+	0.145323i	$0.953578\pi$
$278$	19.8881i	1.19281i
$279$	5.50849	0.329785
$280$	0	0
$281$	−27.6927	−1.65201	−0.826004	−	0.563664i	$-0.809391\pi$
$281$	−27.6927	−1.65201	−0.826004	+	0.563664i	$0.809391\pi$
$282$	− 4.58288i	− 0.272907i
$283$	− 4.82203i	− 0.286640i	−0.989676	−	0.143320i	$-0.954222\pi$
$283$	− 4.82203i	− 0.286640i	0.989676	−	0.143320i	$-0.0457778\pi$
$284$	−4.50999	−0.267619
$285$	0	0
$286$	−15.5931	−0.922039
$287$	35.2400i	2.08015i
$288$	19.4494i	1.14607i
$289$	−38.8115	−2.28303
$290$	0	0
$291$	0.311488	0.0182597
$292$	− 8.99727i	− 0.526525i
$293$	− 28.1922i	− 1.64700i	−0.567314	−	0.823502i	$-0.692017\pi$
$293$	− 28.1922i	− 1.64700i	0.567314	−	0.823502i	$-0.307983\pi$
$294$	−10.4157	−0.607457
$295$	0	0
$296$	−5.72880	−0.332979
$297$	8.68757i	0.504104i
$298$	− 6.24265i	− 0.361627i
$299$	−2.50268	−0.144734
$300$	0	0
$301$	5.13977	0.296252
$302$	− 0.713543i	− 0.0410598i
$303$	− 4.58342i	− 0.263311i
$304$	−2.95000	−0.169194
$305$	0	0
$306$	40.0828	2.29138
$307$	− 5.79727i	− 0.330868i	−0.986221	−	0.165434i	$-0.947098\pi$
$307$	− 5.79727i	− 0.330868i	0.986221	−	0.165434i	$-0.0529025\pi$
$308$	35.9426i	2.04802i
$309$	−4.80024	−0.273076
$310$	0	0
$311$	−15.0774	−0.854959	−0.427479	−	0.904025i	$-0.640598\pi$
$311$	−15.0774	−0.854959	−0.427479	+	0.904025i	$0.640598\pi$
$312$	− 0.552206i	− 0.0312625i
$313$	− 0.780361i	− 0.0441086i	−0.999757	−	0.0220543i	$-0.992979\pi$
$313$	− 0.780361i	− 0.0441086i	0.999757	−	0.0220543i	$-0.00702068\pi$
$314$	37.1877	2.09862
$315$	0	0
$316$	−4.47911	−0.251970
$317$	− 21.4989i	− 1.20750i	−0.797174	−	0.603750i	$-0.793673\pi$
$317$	− 21.4989i	− 1.20750i	0.797174	−	0.603750i	$-0.206327\pi$
$318$	4.92717i	0.276302i
$319$	27.7883	1.55585
$320$	0	0
$321$	0.622960	0.0347702
$322$	14.0832i	0.784827i
$323$	4.54429i	0.252851i
$324$	−11.4380	−0.635446
$325$	0	0
$326$	20.0798	1.11212
$327$	0.558129i	0.0308646i
$328$	7.72760i	0.426685i
$329$	−43.9119	−2.42094
$330$	0	0
$331$	17.6324	0.969163	0.484581	−	0.874746i	$-0.338972\pi$
$331$	17.6324	0.969163	0.484581	+	0.874746i	$0.338972\pi$
$332$	9.04098i	0.496188i
$333$	− 14.8169i	− 0.811963i
$334$	−2.83580	−0.155168
$335$	0	0
$336$	−7.26463	−0.396318
$337$	− 26.9345i	− 1.46722i	−0.679573	−	0.733608i	$-0.737835\pi$
$337$	− 26.9345i	− 1.46722i	0.679573	−	0.733608i	$-0.262165\pi$
$338$	− 18.7256i	− 1.01854i
$339$	−3.01087	−0.163528
$340$	0	0
$341$	9.52296	0.515697
$342$	− 3.26363i	− 0.176477i
$343$	63.8219i	3.44606i
$344$	1.12707	0.0607678
$345$	0	0
$346$	−12.8838	−0.692636
$347$	− 11.4381i	− 0.614028i	−0.951705	−	0.307014i	$-0.900670\pi$
$347$	− 11.4381i	− 0.614028i	0.951705	−	0.307014i	$-0.0993298\pi$
$348$	− 2.23000i	− 0.119541i
$349$	−9.05483	−0.484694	−0.242347	−	0.970190i	$-0.577917\pi$
$349$	−9.05483	−0.484694	−0.242347	+	0.970190i	$0.577917\pi$
$350$	0	0
$351$	2.89806	0.154687
$352$	33.6238i	1.79215i
$353$	9.57877i	0.509827i	0.966964	+	0.254913i	$0.0820470\pi$
$353$	9.57877i	0.509827i	−0.966964	+	0.254913i	$0.917953\pi$
$354$	−3.99268	−0.212208
$355$	0	0
$356$	−16.0518	−0.850746
$357$	11.1907i	0.592276i
$358$	− 21.7678i	− 1.15047i
$359$	−1.30335	−0.0687879	−0.0343940	−	0.999408i	$-0.510950\pi$
$359$	−1.30335	−0.0687879	−0.0343940	+	0.999408i	$0.510950\pi$
$360$	0	0
$361$	−18.6300	−0.980526
$362$	3.21261i	0.168851i
$363$	4.19575i	0.220220i
$364$	11.9900	0.628446
$365$	0	0
$366$	−4.56926	−0.238839
$367$	17.4149i	0.909053i	0.890733	+	0.454526i	$0.150191\pi$
$367$	17.4149i	0.909053i	−0.890733	+	0.454526i	$0.849809\pi$
$368$	7.21983i	0.376359i
$369$	−19.9866	−1.04046
$370$	0	0
$371$	47.2107	2.45106
$372$	− 0.764214i	− 0.0396226i
$373$	13.5500i	0.701591i	0.936452	+	0.350796i	$0.114089\pi$
$373$	13.5500i	0.701591i	−0.936452	+	0.350796i	$0.885911\pi$
$374$	69.2944	3.58313
$375$	0	0
$376$	−9.62921	−0.496589
$377$	− 9.26982i	− 0.477420i
$378$	− 16.3081i	− 0.838800i
$379$	−8.25854	−0.424213	−0.212106	−	0.977247i	$-0.568032\pi$
$379$	−8.25854	−0.424213	−0.212106	+	0.977247i	$0.568032\pi$
$380$	0	0
$381$	3.53128	0.180913
$382$	− 40.4853i	− 2.07141i
$383$	− 17.7331i	− 0.906118i	−0.891481	−	0.453059i	$-0.850333\pi$
$383$	− 17.7331i	− 0.906118i	0.891481	−	0.453059i	$-0.149667\pi$
$384$	−2.50464	−0.127815
$385$	0	0
$386$	27.8410	1.41707
$387$	2.91506i	0.148181i
$388$	1.48309i	0.0752923i
$389$	25.0836	1.27179	0.635894	−	0.771776i	$-0.280632\pi$
$389$	25.0836	1.27179	0.635894	+	0.771776i	$0.280632\pi$
$390$	0	0
$391$	11.1217	0.562448
$392$	21.8847i	1.10535i
$393$	− 0.00286923i	0 0.000144734i
$394$	26.7532	1.34781
$395$	0	0
$396$	−20.3851	−1.02439
$397$	1.18706i	0.0595770i	0.999556	+	0.0297885i	$0.00948338\pi$
$397$	1.18706i	0.0595770i	−0.999556	+	0.0297885i	$0.990517\pi$
$398$	− 13.7451i	− 0.688980i
$399$	0.911171	0.0456156
$400$	0	0
$401$	34.8646	1.74105	0.870527	−	0.492121i	$-0.163778\pi$
$401$	34.8646	1.74105	0.870527	+	0.492121i	$0.163778\pi$
$402$	2.52681i	0.126026i
$403$	− 3.17674i	− 0.158244i
$404$	21.8231	1.08574
$405$	0	0
$406$	−52.1636	−2.58884
$407$	− 25.6152i	− 1.26970i
$408$	2.45396i	0.121489i
$409$	−27.2193	−1.34591	−0.672954	−	0.739684i	$-0.734975\pi$
$409$	−27.2193	−1.34591	−0.672954	+	0.739684i	$0.734975\pi$
$410$	0	0
$411$	2.22942	0.109969
$412$	− 22.8554i	− 1.12600i
$413$	38.2567i	1.88249i
$414$	−7.98740	−0.392560
$415$	0	0
$416$	11.2165	0.549932
$417$	− 3.14917i	− 0.154216i
$418$	− 5.64209i	− 0.275964i
$419$	−1.70123	−0.0831104	−0.0415552	−	0.999136i	$-0.513231\pi$
$419$	−1.70123	−0.0831104	−0.0415552	+	0.999136i	$0.513231\pi$
$420$	0	0
$421$	−12.3991	−0.604293	−0.302147	−	0.953261i	$-0.597703\pi$
$421$	−12.3991	−0.604293	−0.302147	+	0.953261i	$0.597703\pi$
$422$	16.2289i	0.790009i
$423$	− 24.9049i	− 1.21092i
$424$	10.3526	0.502767
$425$	0	0
$426$	1.74341	0.0844683
$427$	43.7813i	2.11873i
$428$	2.96610i	0.143372i
$429$	2.46908	0.119208
$430$	0	0
$431$	−15.2271	−0.733465	−0.366733	−	0.930326i	$-0.619524\pi$
$431$	−15.2271	−0.733465	−0.366733	+	0.930326i	$0.619524\pi$
$432$	− 8.36044i	− 0.402242i
$433$	5.77913i	0.277727i	0.990312	+	0.138864i	$0.0443450\pi$
$433$	5.77913i	0.277727i	−0.990312	+	0.138864i	$0.955655\pi$
$434$	−17.8763	−0.858090
$435$	0	0
$436$	−2.65742	−0.127268
$437$	− 0.905551i	− 0.0433184i
$438$	3.47803i	0.166187i
$439$	−7.89384	−0.376753	−0.188376	−	0.982097i	$-0.560322\pi$
$439$	−7.89384	−0.376753	−0.188376	+	0.982097i	$0.560322\pi$
$440$	0	0
$441$	−56.6025	−2.69536
$442$	− 23.1157i	− 1.09950i
$443$	28.7110i	1.36410i	0.731306	+	0.682050i	$0.238911\pi$
$443$	28.7110i	1.36410i	−0.731306	+	0.682050i	$0.761089\pi$
$444$	−2.05561	−0.0975549
$445$	0	0
$446$	−6.92662	−0.327985
$447$	0.988489i	0.0467539i
$448$	− 13.2648i	− 0.626703i
$449$	9.97648	0.470819	0.235410	−	0.971896i	$-0.424357\pi$
$449$	9.97648	0.470819	0.235410	+	0.971896i	$0.424357\pi$
$450$	0	0
$451$	−34.5525	−1.62701
$452$	− 14.3357i	− 0.674293i
$453$	0.112986i	0.00530853i
$454$	40.4292	1.89744
$455$	0	0
$456$	0.199806	0.00935678
$457$	12.5755i	0.588255i	0.955766	+	0.294128i	$0.0950291\pi$
$457$	12.5755i	0.588255i	−0.955766	+	0.294128i	$0.904971\pi$
$458$	− 26.2307i	− 1.22568i
$459$	−12.8787	−0.601128
$460$	0	0
$461$	24.2571	1.12977	0.564883	−	0.825171i	$-0.308921\pi$
$461$	24.2571	1.12977	0.564883	+	0.825171i	$0.308921\pi$
$462$	− 13.8942i	− 0.646414i
$463$	− 26.4560i	− 1.22951i	−0.788716	−	0.614757i	$-0.789254\pi$
$463$	− 26.4560i	− 1.22951i	0.788716	−	0.614757i	$-0.210746\pi$
$464$	−26.7419	−1.24146
$465$	0	0
$466$	−30.4478	−1.41047
$467$	− 4.66803i	− 0.216011i	−0.994150	−	0.108005i	$-0.965554\pi$
$467$	− 4.66803i	− 0.216011i	0.994150	−	0.108005i	$-0.0344464\pi$
$468$	6.80021i	0.314340i
$469$	24.2112	1.11797
$470$	0	0
$471$	−5.88847	−0.271326
$472$	8.38912i	0.386140i
$473$	5.03950i	0.231716i
$474$	1.73147	0.0795289
$475$	0	0
$476$	−53.2824	−2.44220
$477$	26.7759i	1.22599i
$478$	− 16.7265i	− 0.765052i
$479$	20.2163	0.923707	0.461853	−	0.886956i	$-0.347185\pi$
$479$	20.2163	0.923707	0.461853	+	0.886956i	$0.347185\pi$
$480$	0	0
$481$	−8.54490	−0.389614
$482$	5.96599i	0.271743i
$483$	− 2.23000i	− 0.101469i
$484$	−19.9772	−0.908057
$485$	0	0
$486$	13.9403	0.632346
$487$	23.6802i	1.07305i	0.843883	+	0.536527i	$0.180264\pi$
$487$	23.6802i	1.07305i	−0.843883	+	0.536527i	$0.819736\pi$
$488$	9.60059i	0.434598i
$489$	−3.17953	−0.143783
$490$	0	0
$491$	0.723299	0.0326420	0.0163210	−	0.999867i	$-0.494805\pi$
$491$	0.723299	0.0326420	0.0163210	+	0.999867i	$0.494805\pi$
$492$	2.77282i	0.125008i
$493$	41.1943i	1.85530i
$494$	−1.88213	−0.0846810
$495$	0	0
$496$	−9.16437	−0.411492
$497$	− 16.7048i	− 0.749314i
$498$	− 3.49493i	− 0.156611i
$499$	0.367937	0.0164711	0.00823556	−	0.999966i	$-0.497379\pi$
$499$	0.367937	0.0164711	0.00823556	+	0.999966i	$0.497379\pi$
$500$	0	0
$501$	0.449034	0.0200614
$502$	− 16.3488i	− 0.729683i
$503$	− 9.38700i	− 0.418546i	−0.977857	−	0.209273i	$-0.932890\pi$
$503$	− 9.38700i	− 0.418546i	0.977857	−	0.209273i	$-0.0671097\pi$
$504$	−16.8867	−0.752193
$505$	0	0
$506$	−13.8085	−0.613861
$507$	2.96509i	0.131684i
$508$	16.8135i	0.745979i
$509$	13.6908	0.606836	0.303418	−	0.952858i	$-0.401872\pi$
$509$	13.6908	0.606836	0.303418	+	0.952858i	$0.401872\pi$
$510$	0	0
$511$	33.3255	1.47423
$512$	− 21.4256i	− 0.946888i
$513$	1.04861i	0.0462974i
$514$	−37.6868	−1.66229
$515$	0	0
$516$	0.404418	0.0178035
$517$	− 43.0551i	− 1.89356i
$518$	48.0843i	2.11270i
$519$	2.04008	0.0895494
$520$	0	0
$521$	−7.85102	−0.343960	−0.171980	−	0.985100i	$-0.555016\pi$
$521$	−7.85102	−0.343960	−0.171980	+	0.985100i	$0.555016\pi$
$522$	− 29.5850i	− 1.29490i
$523$	− 6.26463i	− 0.273933i	−0.990576	−	0.136967i	$-0.956265\pi$
$523$	− 6.26463i	− 0.273933i	0.990576	−	0.136967i	$-0.0437353\pi$
$524$	0.0136613	0.000596796 0
$525$	0	0
$526$	26.9261	1.17403
$527$	14.1171i	0.614952i
$528$	− 7.12290i	− 0.309984i
$529$	20.7838	0.903641
$530$	0	0
$531$	−21.6976	−0.941594
$532$	4.33837i	0.188092i
$533$	11.5263i	0.499257i
$534$	6.20509	0.268520
$535$	0	0
$536$	5.30914	0.229320
$537$	3.44682i	0.148741i
$538$	− 23.0560i	− 0.994013i
$539$	−97.8533	−4.21484
$540$	0	0
$541$	−37.2677	−1.60226	−0.801131	−	0.598488i	$-0.795768\pi$
$541$	−37.2677	−1.60226	−0.801131	+	0.598488i	$0.795768\pi$
$542$	− 4.90767i	− 0.210802i
$543$	− 0.508699i	− 0.0218304i
$544$	−49.8450	−2.13709
$545$	0	0
$546$	−4.63491	−0.198356
$547$	41.7401i	1.78468i	0.451366	+	0.892339i	$0.350937\pi$
$547$	41.7401i	1.78468i	−0.451366	+	0.892339i	$0.649063\pi$
$548$	10.6149i	0.453448i
$549$	−24.8309	−1.05976
$550$	0	0
$551$	3.35412	0.142890
$552$	− 0.489006i	− 0.0208135i
$553$	− 16.5904i	− 0.705497i
$554$	8.90334	0.378267
$555$	0	0
$556$	14.9942	0.635894
$557$	13.6095i	0.576652i	0.957532	+	0.288326i	$0.0930986\pi$
$557$	13.6095i	0.576652i	−0.957532	+	0.288326i	$0.906901\pi$
$558$	− 10.1387i	− 0.429205i
$559$	1.68111	0.0711034
$560$	0	0
$561$	−10.9724	−0.463255
$562$	50.9700i	2.15004i
$563$	41.4317i	1.74614i	0.487598	+	0.873068i	$0.337873\pi$
$563$	41.4317i	1.74614i	−0.487598	+	0.873068i	$0.662127\pi$
$564$	−3.45516	−0.145488
$565$	0	0
$566$	−8.87522	−0.373053
$567$	− 42.3660i	− 1.77920i
$568$	− 3.66312i	− 0.153701i
$569$	−1.50042	−0.0629007	−0.0314504	−	0.999505i	$-0.510013\pi$
$569$	−1.50042	−0.0629007	−0.0314504	+	0.999505i	$0.510013\pi$
$570$	0	0
$571$	42.5501	1.78066	0.890332	−	0.455311i	$-0.150472\pi$
$571$	42.5501	1.78066	0.890332	+	0.455311i	$0.150472\pi$
$572$	11.7561i	0.491545i
$573$	6.41063i	0.267808i
$574$	64.8611	2.70725
$575$	0	0
$576$	7.52323	0.313468
$577$	− 14.1072i	− 0.587290i	−0.955915	−	0.293645i	$-0.905132\pi$
$577$	− 14.1072i	− 0.587290i	0.955915	−	0.293645i	$-0.0948684\pi$
$578$	71.4348i	2.97129i
$579$	−4.40847	−0.183210
$580$	0	0
$581$	−33.4874	−1.38929
$582$	− 0.573310i	− 0.0237645i
$583$	46.2896i	1.91712i
$584$	7.30778	0.302398
$585$	0	0
$586$	−51.8892	−2.14352
$587$	− 41.2458i	− 1.70240i	−0.524845	−	0.851198i	$-0.675877\pi$
$587$	− 41.2458i	− 1.70240i	0.524845	−	0.851198i	$-0.324123\pi$
$588$	7.85269i	0.323839i
$589$	1.14945	0.0473621
$590$	0	0
$591$	−4.23622	−0.174255
$592$	24.6506i	1.01314i
$593$	6.83172i	0.280545i	0.990113	+	0.140273i	$0.0447979\pi$
$593$	6.83172i	0.280545i	−0.990113	+	0.140273i	$0.955202\pi$
$594$	15.9900	0.656076
$595$	0	0
$596$	−4.70650	−0.192786
$597$	2.17646i	0.0890767i
$598$	4.60632i	0.188367i
$599$	0.441871	0.0180543	0.00902717	−	0.999959i	$-0.497127\pi$
$599$	0.441871	0.0180543	0.00902717	+	0.999959i	$0.497127\pi$
$600$	0	0
$601$	−22.3310	−0.910899	−0.455450	−	0.890262i	$-0.650521\pi$
$601$	−22.3310	−0.910899	−0.455450	+	0.890262i	$0.650521\pi$
$602$	− 9.46004i	− 0.385562i
$603$	13.7315i	0.559192i
$604$	−0.537959	−0.0218893
$605$	0	0
$606$	−8.43604	−0.342691
$607$	− 21.6504i	− 0.878763i	−0.898301	−	0.439381i	$-0.855198\pi$
$607$	− 21.6504i	− 0.878763i	0.898301	−	0.439381i	$-0.144802\pi$
$608$	4.05848i	0.164593i
$609$	8.25983	0.334705
$610$	0	0
$611$	−14.3626	−0.581050
$612$	− 30.2195i	− 1.22155i
$613$	− 25.4719i	− 1.02880i	−0.857551	−	0.514399i	$-0.828015\pi$
$613$	− 25.4719i	− 1.02880i	0.857551	−	0.514399i	$-0.171985\pi$
$614$	−10.6702	−0.430614
$615$	0	0
$616$	−29.1933	−1.17623
$617$	− 33.9269i	− 1.36585i	−0.730490	−	0.682923i	$-0.760708\pi$
$617$	− 33.9269i	− 1.36585i	0.730490	−	0.682923i	$-0.239292\pi$
$618$	8.83510i	0.355400i
$619$	36.1265	1.45205	0.726024	−	0.687670i	$-0.241366\pi$
$619$	36.1265	1.45205	0.726024	+	0.687670i	$0.241366\pi$
$620$	0	0
$621$	2.56638	0.102985
$622$	27.7507i	1.11270i
$623$	− 59.4553i	− 2.38203i
$624$	−2.37611	−0.0951204
$625$	0	0
$626$	−1.43630	−0.0574060
$627$	0.893394i	0.0356787i
$628$	− 28.0368i	− 1.11879i
$629$	37.9728	1.51407
$630$	0	0
$631$	−7.67874	−0.305686	−0.152843	−	0.988251i	$-0.548843\pi$
$631$	−7.67874	−0.305686	−0.152843	+	0.988251i	$0.548843\pi$
$632$	− 3.63803i	− 0.144713i
$633$	− 2.56975i	− 0.102139i
$634$	−39.5699	−1.57152
$635$	0	0
$636$	3.71473	0.147298
$637$	32.6426i	1.29335i
$638$	− 51.1459i	− 2.02489i
$639$	9.47427	0.374796
$640$	0	0
$641$	9.35930	0.369670	0.184835	−	0.982770i	$-0.440825\pi$
$641$	9.35930	0.369670	0.184835	+	0.982770i	$0.440825\pi$
$642$	− 1.14659i	− 0.0452524i
$643$	5.80473i	0.228916i	0.993428	+	0.114458i	$0.0365131\pi$
$643$	5.80473i	0.228916i	−0.993428	+	0.114458i	$0.963487\pi$
$644$	10.6177	0.418397
$645$	0	0
$646$	8.36402	0.329078
$647$	26.8056i	1.05384i	0.849915	+	0.526919i	$0.176653\pi$
$647$	26.8056i	1.05384i	−0.849915	+	0.526919i	$0.823347\pi$
$648$	− 9.29022i	− 0.364954i
$649$	−37.5103	−1.47241
$650$	0	0
$651$	2.83061	0.110941
$652$	− 15.1387i	− 0.592878i
$653$	− 7.24374i	− 0.283469i	−0.989905	−	0.141735i	$-0.954732\pi$
$653$	− 7.24374i	− 0.283469i	0.989905	−	0.141735i	$-0.0452680\pi$
$654$	1.02727	0.0401694
$655$	0	0
$656$	33.2514	1.29825
$657$	18.9008i	0.737391i
$658$	80.8222i	3.15078i
$659$	−30.2151	−1.17701	−0.588507	−	0.808492i	$-0.700284\pi$
$659$	−30.2151	−1.17701	−0.588507	+	0.808492i	$0.700284\pi$
$660$	0	0
$661$	−1.97273	−0.0767304	−0.0383652	−	0.999264i	$-0.512215\pi$
$661$	−1.97273	−0.0767304	−0.0383652	+	0.999264i	$0.512215\pi$
$662$	− 32.4534i	− 1.26134i
$663$	3.66025i	0.142152i
$664$	−7.34328	−0.284975
$665$	0	0
$666$	−27.2714	−1.05674
$667$	− 8.20888i	− 0.317849i
$668$	2.13799i	0.0827212i
$669$	1.09679	0.0424044
$670$	0	0
$671$	−42.9272	−1.65718
$672$	9.99437i	0.385541i
$673$	− 51.0499i	− 1.96783i	−0.178638	−	0.983915i	$-0.557169\pi$
$673$	− 51.0499i	− 1.96783i	0.178638	−	0.983915i	$-0.442831\pi$
$674$	−49.5744	−1.90954
$675$	0	0
$676$	−14.1177	−0.542989
$677$	− 18.6165i	− 0.715490i	−0.933819	−	0.357745i	$-0.883546\pi$
$677$	− 18.6165i	− 0.715490i	0.933819	−	0.357745i	$-0.116454\pi$
$678$	5.54167i	0.212827i
$679$	−5.49329	−0.210813
$680$	0	0
$681$	−6.40175	−0.245316
$682$	− 17.5275i	− 0.671164i
$683$	5.53670i	0.211856i	0.994374	+	0.105928i	$0.0337813\pi$
$683$	5.53670i	0.211856i	−0.994374	+	0.105928i	$0.966219\pi$
$684$	−2.46054	−0.0940810
$685$	0	0
$686$	117.468	4.48494
$687$	4.15348i	0.158465i
$688$	− 4.84973i	− 0.184894i
$689$	15.4416	0.588279
$690$	0	0
$691$	−18.0622	−0.687120	−0.343560	−	0.939131i	$-0.611633\pi$
$691$	−18.0622	−0.687120	−0.343560	+	0.939131i	$0.611633\pi$
$692$	9.71343i	0.369249i
$693$	− 75.5055i	− 2.86822i
$694$	−21.0524	−0.799139
$695$	0	0
$696$	1.81126	0.0686555
$697$	− 51.2217i	− 1.94016i
$698$	16.6659i	0.630815i
$699$	4.82124	0.182356
$700$	0	0
$701$	−25.7231	−0.971547	−0.485774	−	0.874085i	$-0.661462\pi$
$701$	−25.7231	−0.971547	−0.485774	+	0.874085i	$0.661462\pi$
$702$	− 5.33404i	− 0.201320i
$703$	− 3.09182i	− 0.116610i
$704$	13.0060	0.490182
$705$	0	0
$706$	17.6303	0.663524
$707$	80.8317i	3.03999i
$708$	3.01019i	0.113130i
$709$	−22.8874	−0.859556	−0.429778	−	0.902935i	$-0.641408\pi$
$709$	−22.8874	−0.859556	−0.429778	+	0.902935i	$0.641408\pi$
$710$	0	0
$711$	9.40938	0.352879
$712$	− 13.0377i	− 0.488607i
$713$	− 2.81316i	− 0.105354i
$714$	20.5971	0.770828
$715$	0	0
$716$	−16.4113	−0.613321
$717$	2.64855i	0.0989119i
$718$	2.39888i	0.0895254i
$719$	−33.5392	−1.25080	−0.625400	−	0.780304i	$-0.715064\pi$
$719$	−33.5392	−1.25080	−0.625400	+	0.780304i	$0.715064\pi$
$720$	0	0
$721$	84.6554	3.15273
$722$	34.2895i	1.27612i
$723$	− 0.944682i	− 0.0351331i
$724$	2.42207	0.0900157
$725$	0	0
$726$	7.72251	0.286609
$727$	− 47.3713i	− 1.75690i	−0.477831	−	0.878452i	$-0.658577\pi$
$727$	− 47.3713i	− 1.75690i	0.477831	−	0.878452i	$-0.341423\pi$
$728$	9.73852i	0.360934i
$729$	22.5209	0.834109
$730$	0	0
$731$	−7.47071	−0.276314
$732$	3.44489i	0.127327i
$733$	− 28.8063i	− 1.06399i	−0.846749	−	0.531993i	$-0.821443\pi$
$733$	− 28.8063i	− 1.06399i	0.846749	−	0.531993i	$-0.178557\pi$
$734$	32.0532	1.18310
$735$	0	0
$736$	9.93272	0.366125
$737$	23.7388i	0.874430i
$738$	36.7865i	1.35413i
$739$	44.3061	1.62982	0.814912	−	0.579584i	$-0.196785\pi$
$739$	44.3061	1.62982	0.814912	+	0.579584i	$0.196785\pi$
$740$	0	0
$741$	0.298025	0.0109482
$742$	− 86.8939i	− 3.18998i
$743$	− 8.81533i	− 0.323403i	−0.986840	−	0.161702i	$-0.948302\pi$
$743$	− 8.81533i	− 0.323403i	0.986840	−	0.161702i	$-0.0516982\pi$
$744$	0.620711	0.0227564
$745$	0	0
$746$	24.9395	0.913099
$747$	− 18.9926i	− 0.694904i
$748$	− 52.2429i	− 1.91019i
$749$	−10.9863	−0.401431
$750$	0	0
$751$	4.39589	0.160408	0.0802041	−	0.996778i	$-0.474443\pi$
$751$	4.39589	0.160408	0.0802041	+	0.996778i	$0.474443\pi$
$752$	41.4339i	1.51094i
$753$	2.58875i	0.0943392i
$754$	−17.0616	−0.621347
$755$	0	0
$756$	−12.2951	−0.447170
$757$	− 3.60759i	− 0.131120i	−0.997849	−	0.0655600i	$-0.979117\pi$
$757$	− 3.60759i	− 0.131120i	0.997849	−	0.0655600i	$-0.0208834\pi$
$758$	15.2003i	0.552100i
$759$	2.18649	0.0793647
$760$	0	0
$761$	11.6606	0.422696	0.211348	−	0.977411i	$-0.432215\pi$
$761$	11.6606	0.422696	0.211348	+	0.977411i	$0.432215\pi$
$762$	− 6.49952i	− 0.235453i
$763$	− 9.84298i	− 0.356340i
$764$	−30.5230	−1.10428
$765$	0	0
$766$	−32.6387	−1.17928
$767$	12.5130i	0.451816i
$768$	6.11425i	0.220629i
$769$	1.46086	0.0526800	0.0263400	−	0.999653i	$-0.491615\pi$
$769$	1.46086	0.0526800	0.0263400	+	0.999653i	$0.491615\pi$
$770$	0	0
$771$	5.96750	0.214914
$772$	− 20.9901i	− 0.755450i
$773$	11.5351i	0.414887i	0.978247	+	0.207444i	$0.0665143\pi$
$773$	11.5351i	0.414887i	−0.978247	+	0.207444i	$0.933486\pi$
$774$	5.36533	0.192853
$775$	0	0
$776$	−1.20460	−0.0432425
$777$	− 7.61389i	− 0.273147i
$778$	− 46.1677i	− 1.65519i
$779$	−4.17057	−0.149426
$780$	0	0
$781$	16.3789	0.586083
$782$	− 20.4701i	− 0.732009i
$783$	9.50575i	0.339708i
$784$	94.1686	3.36316
$785$	0	0
$786$	−0.00528098	−0.000188366 0
$787$	1.47008i	0.0524028i	0.999657	+	0.0262014i	$0.00834111\pi$
$787$	1.47008i	0.0524028i	−0.999657	+	0.0262014i	$0.991659\pi$
$788$	− 20.1700i	− 0.718525i
$789$	−4.26360	−0.151788
$790$	0	0
$791$	53.0987	1.88797
$792$	− 16.5572i	− 0.588335i
$793$	14.3199i	0.508516i
$794$	2.18486	0.0775377
$795$	0	0
$796$	−10.3628	−0.367300
$797$	43.1788i	1.52947i	0.644345	+	0.764735i	$0.277130\pi$
$797$	43.1788i	1.52947i	−0.644345	+	0.764735i	$0.722870\pi$
$798$	− 1.67706i	− 0.0593673i
$799$	63.8263	2.25801
$800$	0	0
$801$	33.7206	1.19146
$802$	− 64.1702i	− 2.26593i
$803$	32.6753i	1.15309i
$804$	1.90503	0.0671852
$805$	0	0
$806$	−5.84696	−0.205950
$807$	3.65079i	0.128514i
$808$	17.7252i	0.623569i
$809$	21.8926	0.769704	0.384852	−	0.922978i	$-0.374252\pi$
$809$	21.8926	0.769704	0.384852	+	0.922978i	$0.374252\pi$
$810$	0	0
$811$	41.6105	1.46114	0.730572	−	0.682836i	$-0.239254\pi$
$811$	41.6105	1.46114	0.730572	+	0.682836i	$0.239254\pi$
$812$	39.3276i	1.38013i
$813$	0.777102i	0.0272542i
$814$	−47.1462	−1.65247
$815$	0	0
$816$	10.5592	0.369647
$817$	0.608281i	0.0212810i
$818$	50.0987i	1.75166i
$819$	−25.1877	−0.880128
$820$	0	0
$821$	−14.9071	−0.520262	−0.260131	−	0.965573i	$-0.583766\pi$
$821$	−14.9071	−0.520262	−0.260131	+	0.965573i	$0.583766\pi$
$822$	− 4.10337i	− 0.143121i
$823$	− 26.2455i	− 0.914860i	−0.889246	−	0.457430i	$-0.848770\pi$
$823$	− 26.2455i	− 0.914860i	0.889246	−	0.457430i	$-0.151230\pi$
$824$	18.5637	0.646696
$825$	0	0
$826$	70.4136	2.45000
$827$	6.14518i	0.213689i	0.994276	+	0.106844i	$0.0340747\pi$
$827$	6.14518i	0.213689i	−0.994276	+	0.106844i	$0.965925\pi$
$828$	6.02192i	0.209276i
$829$	−23.4428	−0.814200	−0.407100	−	0.913384i	$-0.633460\pi$
$829$	−23.4428	−0.814200	−0.407100	+	0.913384i	$0.633460\pi$
$830$	0	0
$831$	−1.40979	−0.0489052
$832$	− 4.33864i	− 0.150415i
$833$	− 145.061i	− 5.02606i
$834$	−5.79622	−0.200707
$835$	0	0
$836$	−4.25372	−0.147118
$837$	3.25759i	0.112599i
$838$	3.13120i	0.108166i
$839$	−37.5665	−1.29694	−0.648470	−	0.761240i	$-0.724591\pi$
$839$	−37.5665	−1.29694	−0.648470	+	0.761240i	$0.724591\pi$
$840$	0	0
$841$	1.40533	0.0484598
$842$	22.8212i	0.786469i
$843$	− 8.07082i	− 0.277974i
$844$	12.2354	0.421159
$845$	0	0
$846$	−45.8389	−1.57598
$847$	− 73.9948i	− 2.54249i
$848$	− 44.5466i	− 1.52974i
$849$	1.40534	0.0482312
$850$	0	0
$851$	−7.56693	−0.259391
$852$	− 1.31440i	− 0.0450307i
$853$	22.1842i	0.759571i	0.925075	+	0.379786i	$0.124002\pi$
$853$	22.1842i	0.759571i	−0.925075	+	0.379786i	$0.875998\pi$
$854$	80.5820	2.75746
$855$	0	0
$856$	−2.40913	−0.0823425
$857$	24.9840i	0.853437i	0.904385	+	0.426718i	$0.140330\pi$
$857$	24.9840i	0.853437i	−0.904385	+	0.426718i	$0.859670\pi$
$858$	− 4.54448i	− 0.155146i
$859$	18.7825	0.640850	0.320425	−	0.947274i	$-0.396174\pi$
$859$	18.7825	0.640850	0.320425	+	0.947274i	$0.396174\pi$
$860$	0	0
$861$	−10.2704	−0.350015
$862$	28.0264i	0.954583i
$863$	35.7173i	1.21583i	0.794001	+	0.607916i	$0.207994\pi$
$863$	35.7173i	1.21583i	−0.794001	+	0.607916i	$0.792006\pi$
$864$	−11.5019	−0.391304
$865$	0	0
$866$	10.6368	0.361454
$867$	− 11.3113i	− 0.384152i
$868$	13.4774i	0.457454i
$869$	16.2667	0.551811
$870$	0	0
$871$	7.91895	0.268324
$872$	− 2.15842i	− 0.0730932i
$873$	− 3.11556i	− 0.105446i
$874$	−1.66672	−0.0563776
$875$	0	0
$876$	2.62218	0.0885953
$877$	42.4124i	1.43216i	0.698016	+	0.716082i	$0.254066\pi$
$877$	42.4124i	1.43216i	−0.698016	+	0.716082i	$0.745934\pi$
$878$	14.5291i	0.490332i
$879$	8.21638	0.277132
$880$	0	0
$881$	−26.5019	−0.892873	−0.446437	−	0.894815i	$-0.647307\pi$
$881$	−26.5019	−0.892873	−0.446437	+	0.894815i	$0.647307\pi$
$882$	104.180i	3.50793i
$883$	− 9.84464i	− 0.331299i	−0.986185	−	0.165649i	$-0.947028\pi$
$883$	− 9.84464i	− 0.331299i	0.986185	−	0.165649i	$-0.0529720\pi$
$884$	−17.4276	−0.586152
$885$	0	0
$886$	52.8441	1.77533
$887$	− 48.1046i	− 1.61519i	−0.589735	−	0.807597i	$-0.700768\pi$
$887$	− 48.1046i	− 1.61519i	0.589735	−	0.807597i	$-0.299232\pi$
$888$	− 1.66961i	− 0.0560285i
$889$	−62.2765	−2.08869
$890$	0	0
$891$	41.5394	1.39162
$892$	5.22216i	0.174851i
$893$	− 5.19687i	− 0.173907i
$894$	1.81937	0.0608488
$895$	0	0
$896$	44.1711	1.47565
$897$	− 0.729386i	− 0.0243535i
$898$	− 18.3623i	− 0.612757i
$899$	10.4198	0.347520
$900$	0	0
$901$	−68.6212	−2.28610
$902$	63.5957i	2.11751i
$903$	1.49795i	0.0498485i
$904$	11.6437	0.387265
$905$	0	0
$906$	0.207956	0.00690889
$907$	44.8522i	1.48929i	0.667459	+	0.744646i	$0.267382\pi$
$907$	44.8522i	1.48929i	−0.667459	+	0.744646i	$0.732618\pi$
$908$	− 30.4807i	− 1.01154i
$909$	−45.8443	−1.52056
$910$	0	0
$911$	−49.9719	−1.65564	−0.827822	−	0.560991i	$-0.810420\pi$
$911$	−49.9719	−1.65564	−0.827822	+	0.560991i	$0.810420\pi$
$912$	− 0.859753i	− 0.0284693i
$913$	− 32.8340i	− 1.08665i
$914$	23.1458	0.765596
$915$	0	0
$916$	−19.7760	−0.653417
$917$	0.0506008i	0.00167099i
$918$	23.7040i	0.782350i
$919$	−12.3429	−0.407155	−0.203577	−	0.979059i	$-0.565257\pi$
$919$	−12.3429	−0.407155	−0.203577	+	0.979059i	$0.565257\pi$
$920$	0	0
$921$	1.68957	0.0556732
$922$	− 44.6466i	− 1.47036i
$923$	− 5.46379i	− 0.179843i
$924$	−10.4752	−0.344608
$925$	0	0
$926$	−48.6937	−1.60018
$927$	48.0130i	1.57695i
$928$	36.7904i	1.20770i
$929$	−20.2913	−0.665735	−0.332867	−	0.942974i	$-0.608016\pi$
$929$	−20.2913	−0.665735	−0.332867	+	0.942974i	$0.608016\pi$
$930$	0	0
$931$	−11.8112	−0.387095
$932$	22.9554i	0.751930i
$933$	− 4.39417i	− 0.143859i
$934$	−8.59177	−0.281131
$935$	0	0
$936$	−5.52328	−0.180534
$937$	5.50728i	0.179915i	0.995946	+	0.0899576i	$0.0286731\pi$
$937$	5.50728i	0.179915i	−0.995946	+	0.0899576i	$0.971327\pi$
$938$	− 44.5620i	− 1.45500i
$939$	0.227430	0.00742190
$940$	0	0
$941$	29.3989	0.958378	0.479189	−	0.877712i	$-0.340931\pi$
$941$	29.3989	0.958378	0.479189	+	0.877712i	$0.340931\pi$
$942$	10.8381i	0.353123i
$943$	10.2071i	0.332388i
$944$	36.0978	1.17488
$945$	0	0
$946$	9.27547	0.301572
$947$	10.8670i	0.353130i	0.984289	+	0.176565i	$0.0564986\pi$
$947$	10.8670i	0.353130i	−0.984289	+	0.176565i	$0.943501\pi$
$948$	− 1.30540i	− 0.0423974i
$949$	10.9001	0.353831
$950$	0	0
$951$	6.26568	0.203179
$952$	− 43.2772i	− 1.40262i
$953$	51.5990i	1.67146i	0.549144	+	0.835728i	$0.314954\pi$
$953$	51.5990i	1.67146i	−0.549144	+	0.835728i	$0.685046\pi$
$954$	49.2826	1.59558
$955$	0	0
$956$	−12.6106	−0.407855
$957$	8.09868i	0.261793i
$958$	− 37.2092i	− 1.20218i
$959$	−39.3173	−1.26962
$960$	0	0
$961$	−27.4292	−0.884812
$962$	15.7274i	0.507070i
$963$	− 6.23097i	− 0.200790i
$964$	4.49792	0.144868
$965$	0	0
$966$	−4.10444	−0.132058
$967$	49.7367i	1.59942i	0.600383	+	0.799712i	$0.295015\pi$
$967$	49.7367i	1.59942i	−0.600383	+	0.799712i	$0.704985\pi$
$968$	− 16.2260i	− 0.521522i
$969$	−1.32440	−0.0425457
$970$	0	0
$971$	23.7725	0.762896	0.381448	−	0.924390i	$-0.375426\pi$
$971$	23.7725	0.762896	0.381448	+	0.924390i	$0.375426\pi$
$972$	− 10.5100i	− 0.337108i
$973$	55.5377i	1.78046i
$974$	43.5848	1.39655
$975$	0	0
$976$	41.3107	1.32232
$977$	− 40.5334i	− 1.29678i	−0.761309	−	0.648389i	$-0.775443\pi$
$977$	− 40.5334i	− 1.29678i	0.761309	−	0.648389i	$-0.224557\pi$
$978$	5.85210i	0.187130i
$979$	58.2954	1.86313
$980$	0	0
$981$	5.58252	0.178236
$982$	− 1.33127i	− 0.0424826i
$983$	60.6697i	1.93506i	0.252749	+	0.967532i	$0.418665\pi$
$983$	60.6697i	1.93506i	−0.252749	+	0.967532i	$0.581335\pi$
$984$	−2.25215	−0.0717958
$985$	0	0
$986$	75.8203	2.41461
$987$	− 12.7978i	− 0.407357i
$988$	1.41899i	0.0451440i
$989$	1.48871	0.0473381
$990$	0	0
$991$	−35.5820	−1.13030	−0.565150	−	0.824988i	$-0.691182\pi$
$991$	−35.5820	−1.13030	−0.565150	+	0.824988i	$0.691182\pi$
$992$	12.6079i	0.400302i
$993$	5.13881i	0.163075i
$994$	−30.7461	−0.975209
$995$	0	0
$996$	−2.63492	−0.0834906
$997$	− 0.728331i	− 0.0230665i	−0.999933	−	0.0115332i	$-0.996329\pi$
$997$	− 0.728331i	− 0.0230665i	0.999933	−	0.0115332i	$-0.00367122\pi$
$998$	− 0.677208i	− 0.0214366i
$999$	8.76238	0.277229

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.2.b.k.474.3		12
5.2	odd	4		215.2.a.d.1.5	✓	6
5.3	odd	4		1075.2.a.p.1.2		6
5.4	even	2	inner	1075.2.b.k.474.10		12
15.2	even	4		1935.2.a.z.1.2		6
15.8	even	4		9675.2.a.cl.1.5		6
20.7	even	4		3440.2.a.x.1.4		6
215.42	even	4		9245.2.a.n.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.2.a.d.1.5	✓	6	5.2	odd	4
1075.2.a.p.1.2		6	5.3	odd	4
1075.2.b.k.474.3		12	1.1	even	1	trivial
1075.2.b.k.474.10		12	5.4	even	2	inner
1935.2.a.z.1.2		6	15.2	even	4
3440.2.a.x.1.4		6	20.7	even	4
9245.2.a.n.1.2		6	215.42	even	4
9675.2.a.cl.1.5		6	15.8	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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1075.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.84056i q^{2} +0.291442i q^{3} -1.38764 q^{4} +0.536415 q^{6} -5.13977i q^{7} -1.12707i q^{8} +2.91506 q^{9} +O(q^{10})\)	\(q-1.84056i q^{2} +0.291442i q^{3} -1.38764 q^{4} +0.536415 q^{6} -5.13977i q^{7} -1.12707i q^{8} +2.91506 q^{9} +5.03950 q^{11} -0.404418i q^{12} -1.68111i q^{13} -9.46004 q^{14} -4.84973 q^{16} +7.47071i q^{17} -5.36533i q^{18} +0.608281 q^{19} +1.49795 q^{21} -9.27547i q^{22} -1.48871i q^{23} +0.328477 q^{24} -3.09418 q^{26} +1.72390i q^{27} +7.13218i q^{28} +5.51410 q^{29} +1.88966 q^{31} +6.67205i q^{32} +1.46872i q^{33} +13.7503 q^{34} -4.04507 q^{36} -5.08289i q^{37} -1.11957i q^{38} +0.489946 q^{39} -6.85633 q^{41} -2.75705i q^{42} +1.00000i q^{43} -6.99303 q^{44} -2.74005 q^{46} -8.54354i q^{47} -1.41342i q^{48} -19.4173 q^{49} -2.17728 q^{51} +2.33278i q^{52} +9.18537i q^{53} +3.17293 q^{54} -5.79291 q^{56} +0.177279i q^{57} -10.1490i q^{58} -7.44326 q^{59} -8.51814 q^{61} -3.47803i q^{62} -14.9828i q^{63} +2.58081 q^{64} +2.70326 q^{66} +4.71055i q^{67} -10.3667i q^{68} +0.433871 q^{69} +3.25011 q^{71} -3.28549i q^{72} +6.48385i q^{73} -9.35534 q^{74} -0.844077 q^{76} -25.9019i q^{77} -0.901773i q^{78} +3.22785 q^{79} +8.24277 q^{81} +12.6195i q^{82} -6.51534i q^{83} -2.07862 q^{84} +1.84056 q^{86} +1.60704i q^{87} -5.67989i q^{88} +11.5677 q^{89} -8.64053 q^{91} +2.06579i q^{92} +0.550728i q^{93} -15.7249 q^{94} -1.94452 q^{96} -1.06878i q^{97} +35.7386i q^{98} +14.6904 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 14 q^{4} - 16 q^{9}+O(q^{10}) \) 12 * q - 14 * q^4 - 16 * q^9	\( 12 q - 14 q^{4} - 16 q^{9} + 8 q^{14} + 10 q^{16} - 12 q^{19} - 24 q^{21} - 64 q^{26} + 20 q^{29} - 28 q^{34} - 82 q^{36} - 16 q^{39} - 12 q^{41} - 8 q^{44} - 16 q^{46} - 40 q^{49} - 16 q^{51} + 10 q^{54} - 70 q^{56} + 40 q^{59} - 16 q^{61} - 34 q^{64} - 70 q^{66} + 84 q^{69} + 16 q^{71} - 90 q^{74} - 32 q^{76} + 32 q^{79} + 92 q^{81} - 74 q^{84} + 6 q^{86} + 48 q^{91} - 24 q^{94} - 46 q^{96} + 32 q^{99}+O(q^{100}) \) 12 * q - 14 * q^4 - 16 * q^9 + 8 * q^14 + 10 * q^16 - 12 * q^19 - 24 * q^21 - 64 * q^26 + 20 * q^29 - 28 * q^34 - 82 * q^36 - 16 * q^39 - 12 * q^41 - 8 * q^44 - 16 * q^46 - 40 * q^49 - 16 * q^51 + 10 * q^54 - 70 * q^56 + 40 * q^59 - 16 * q^61 - 34 * q^64 - 70 * q^66 + 84 * q^69 + 16 * q^71 - 90 * q^74 - 32 * q^76 + 32 * q^79 + 92 * q^81 - 74 * q^84 + 6 * q^86 + 48 * q^91 - 24 * q^94 - 46 * q^96 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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