LMFDB - Embedded newform 525.8.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,8,Mod(1,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("525.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	525.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$164.002138379$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 277x^{2} + 80x + 15348 $ x^4 - 2x^3 - 277x^2 + 80*x + 15348
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-12.8099$ of defining polynomial
Character	$\chi$	$=$	525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.41438 q^{2} +27.0000 q^{3} -116.342 q^{4} +92.1883 q^{6} -343.000 q^{7} -834.277 q^{8} +729.000 q^{9} -3379.63 q^{11} -3141.23 q^{12} +14159.0 q^{13} -1171.13 q^{14} +12043.2 q^{16} +19538.0 q^{17} +2489.08 q^{18} -38098.7 q^{19} -9261.00 q^{21} -11539.4 q^{22} -103511. q^{23} -22525.5 q^{24} +48344.1 q^{26} +19683.0 q^{27} +39905.3 q^{28} +59794.5 q^{29} +68349.9 q^{31} +147908. q^{32} -91250.1 q^{33} +66710.3 q^{34} -84813.3 q^{36} -266512. q^{37} -130084. q^{38} +382292. q^{39} +655437. q^{41} -31620.6 q^{42} +775223. q^{43} +393193. q^{44} -353426. q^{46} +119729. q^{47} +325167. q^{48} +117649. q^{49} +527527. q^{51} -1.64728e6 q^{52} -224102. q^{53} +67205.3 q^{54} +286157. q^{56} -1.02867e6 q^{57} +204161. q^{58} -409602. q^{59} +1.42274e6 q^{61} +233373. q^{62} -250047. q^{63} -1.03652e6 q^{64} -311563. q^{66} -4.22037e6 q^{67} -2.27309e6 q^{68} -2.79479e6 q^{69} +2.17619e6 q^{71} -608188. q^{72} -4.43912e6 q^{73} -909973. q^{74} +4.43248e6 q^{76} +1.15921e6 q^{77} +1.30529e6 q^{78} +1.63306e6 q^{79} +531441. q^{81} +2.23791e6 q^{82} +3.49903e6 q^{83} +1.07744e6 q^{84} +2.64691e6 q^{86} +1.61445e6 q^{87} +2.81955e6 q^{88} +2.03583e6 q^{89} -4.85652e6 q^{91} +1.20427e7 q^{92} +1.84545e6 q^{93} +408799. q^{94} +3.99351e6 q^{96} -1.27049e7 q^{97} +401698. q^{98} -2.46375e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 11 q^{2} + 108 q^{3} + 141 q^{4} - 297 q^{6} - 1372 q^{7} - 2133 q^{8} + 2916 q^{9} - 2708 q^{11} + 3807 q^{12} + 2212 q^{13} + 3773 q^{14} - 9599 q^{16} + 17016 q^{17} - 8019 q^{18} + 32668 q^{19}+ \cdots - 1974132 q^{99}+O(q^{100}) $ 4 * q - 11 * q^2 + 108 * q^3 + 141 * q^4 - 297 * q^6 - 1372 * q^7 - 2133 * q^8 + 2916 * q^9 - 2708 * q^11 + 3807 * q^12 + 2212 * q^13 + 3773 * q^14 - 9599 * q^16 + 17016 * q^17 - 8019 * q^18 + 32668 * q^19 - 37044 * q^21 + 7196 * q^22 - 87696 * q^23 - 57591 * q^24 + 67354 * q^26 + 78732 * q^27 - 48363 * q^28 + 200488 * q^29 + 258852 * q^31 - 39445 * q^32 - 73116 * q^33 + 225518 * q^34 + 102789 * q^36 - 666856 * q^37 - 846556 * q^38 + 59724 * q^39 + 833000 * q^41 + 101871 * q^42 - 1051896 * q^43 + 1918236 * q^44 + 1786160 * q^46 - 529824 * q^47 - 259173 * q^48 + 470596 * q^49 + 459432 * q^51 - 211718 * q^52 - 791764 * q^53 - 216513 * q^54 + 731619 * q^56 + 882036 * q^57 - 3500242 * q^58 - 3825904 * q^59 + 212432 * q^61 - 882400 * q^62 - 1000188 * q^63 - 1515063 * q^64 + 194292 * q^66 - 5165272 * q^67 - 7423202 * q^68 - 2367792 * q^69 - 623260 * q^71 - 1554957 * q^72 + 2083228 * q^73 + 1059642 * q^74 + 6990964 * q^76 + 928844 * q^77 + 1818558 * q^78 - 1325016 * q^79 + 2125764 * q^81 - 11798982 * q^82 - 11447616 * q^83 - 1305801 * q^84 + 7525948 * q^86 + 5413176 * q^87 - 19484572 * q^88 - 5805808 * q^89 - 758716 * q^91 + 2716288 * q^92 + 6989004 * q^93 - 15552560 * q^94 - 1065015 * q^96 - 5857516 * q^97 - 1294139 * q^98 - 1974132 * q^99

Coefficient data

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

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

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

$2$	3.41438	0.301791	0.150896	−	0.988550i	$-0.451784\pi$
$2$	3.41438	0.301791	0.150896	+	0.988550i	$0.451784\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	−116.342	−0.908922
$5$	0	0
$5$	0	0
$6$	92.1883	0.174239
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	−834.277	−0.576096
$9$	729.000	0.333333
$10$	0	0
$11$	−3379.63	−0.765588	−0.382794	−	0.923834i	$-0.625038\pi$
$11$	−3379.63	−0.765588	−0.382794	+	0.923834i	$0.625038\pi$
$12$	−3141.23	−0.524766
$13$	14159.0	1.78743	0.893716	−	0.448633i	$-0.148089\pi$
$13$	14159.0	1.78743	0.893716	+	0.448633i	$0.148089\pi$
$14$	−1171.13	−0.114066
$15$	0	0
$16$	12043.2	0.735061
$17$	19538.0	0.964516	0.482258	−	0.876029i	$-0.339817\pi$
$17$	19538.0	0.964516	0.482258	+	0.876029i	$0.339817\pi$
$18$	2489.08	0.100597
$19$	−38098.7	−1.27430	−0.637152	−	0.770738i	$-0.719888\pi$
$19$	−38098.7	−1.27430	−0.637152	+	0.770738i	$0.719888\pi$
$20$	0	0
$21$	−9261.00	−0.218218
$22$	−11539.4	−0.231048
$23$	−103511.	−1.77394	−0.886969	−	0.461828i	$-0.847194\pi$
$23$	−103511.	−1.77394	−0.886969	+	0.461828i	$0.847194\pi$
$24$	−22525.5	−0.332609
$25$	0	0
$26$	48344.1	0.539432
$27$	19683.0	0.192450
$28$	39905.3	0.343540
$29$	59794.5	0.455269	0.227635	−	0.973747i	$-0.426901\pi$
$29$	59794.5	0.455269	0.227635	+	0.973747i	$0.426901\pi$
$30$	0	0
$31$	68349.9	0.412071	0.206035	−	0.978545i	$-0.433944\pi$
$31$	68349.9	0.412071	0.206035	+	0.978545i	$0.433944\pi$
$32$	147908.	0.797931
$33$	−91250.1	−0.442013
$34$	66710.3	0.291083
$35$	0	0
$36$	−84813.3	−0.302974
$37$	−266512.	−0.864989	−0.432494	−	0.901637i	$-0.642366\pi$
$37$	−266512.	−0.864989	−0.432494	+	0.901637i	$0.642366\pi$
$38$	−130084.	−0.384574
$39$	382292.	1.03197
$40$	0	0
$41$	655437.	1.48521	0.742604	−	0.669730i	$-0.233590\pi$
$41$	655437.	1.48521	0.742604	+	0.669730i	$0.233590\pi$
$42$	−31620.6	−0.0658563
$43$	775223.	1.48692	0.743459	−	0.668782i	$-0.233184\pi$
$43$	775223.	1.48692	0.743459	+	0.668782i	$0.233184\pi$
$44$	393193.	0.695860
$45$	0	0
$46$	−353426.	−0.535360
$47$	119729.	0.168211	0.0841057	−	0.996457i	$-0.473197\pi$
$47$	119729.	0.168211	0.0841057	+	0.996457i	$0.473197\pi$
$48$	325167.	0.424388
$49$	117649.	0.142857
$50$	0	0
$51$	527527.	0.556864
$52$	−1.64728e6	−1.62464
$53$	−224102.	−0.206767	−0.103383	−	0.994642i	$-0.532967\pi$
$53$	−224102.	−0.206767	−0.103383	+	0.994642i	$0.532967\pi$
$54$	67205.3	0.0580798
$55$	0	0
$56$	286157.	0.217744
$57$	−1.02867e6	−0.735720
$58$	204161.	0.137396
$59$	−409602.	−0.259645	−0.129823	−	0.991537i	$-0.541441\pi$
$59$	−409602.	−0.259645	−0.129823	+	0.991537i	$0.541441\pi$
$60$	0	0
$61$	1.42274e6	0.802548	0.401274	−	0.915958i	$-0.368568\pi$
$61$	1.42274e6	0.802548	0.401274	+	0.915958i	$0.368568\pi$
$62$	233373.	0.124359
$63$	−250047.	−0.125988
$64$	−1.03652e6	−0.494252
$65$	0	0
$66$	−311563.	−0.133396
$67$	−4.22037e6	−1.71431	−0.857153	−	0.515061i	$-0.827769\pi$
$67$	−4.22037e6	−1.71431	−0.857153	+	0.515061i	$0.827769\pi$
$68$	−2.27309e6	−0.876670
$69$	−2.79479e6	−1.02418
$70$	0	0
$71$	2.17619e6	0.721595	0.360798	−	0.932644i	$-0.382505\pi$
$71$	2.17619e6	0.721595	0.360798	+	0.932644i	$0.382505\pi$
$72$	−608188.	−0.192032
$73$	−4.43912e6	−1.33557	−0.667785	−	0.744354i	$-0.732758\pi$
$73$	−4.43912e6	−1.33557	−0.667785	+	0.744354i	$0.732758\pi$
$74$	−909973.	−0.261046
$75$	0	0
$76$	4.43248e6	1.15824
$77$	1.15921e6	0.289365
$78$	1.30529e6	0.311441
$79$	1.63306e6	0.372656	0.186328	−	0.982488i	$-0.440341\pi$
$79$	1.63306e6	0.372656	0.186328	+	0.982488i	$0.440341\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	2.23791e6	0.448223
$83$	3.49903e6	0.671698	0.335849	−	0.941916i	$-0.390977\pi$
$83$	3.49903e6	0.671698	0.335849	+	0.941916i	$0.390977\pi$
$84$	1.07744e6	0.198343
$85$	0	0
$86$	2.64691e6	0.448739
$87$	1.61445e6	0.262850
$88$	2.81955e6	0.441053
$89$	2.03583e6	0.306110	0.153055	−	0.988218i	$-0.451089\pi$
$89$	2.03583e6	0.306110	0.153055	+	0.988218i	$0.451089\pi$
$90$	0	0
$91$	−4.85652e6	−0.675586
$92$	1.20427e7	1.61237
$93$	1.84545e6	0.237909
$94$	408799.	0.0507648
$95$	0	0
$96$	3.99351e6	0.460686
$97$	−1.27049e7	−1.41342	−0.706709	−	0.707505i	$-0.749821\pi$
$97$	−1.27049e7	−1.41342	−0.706709	+	0.707505i	$0.749821\pi$
$98$	401698.	0.0431131
$99$	−2.46375e6	−0.255196
$100$	0	0
$101$	−1.86609e7	−1.80222	−0.901110	−	0.433590i	$-0.857247\pi$
$101$	−1.86609e7	−1.80222	−0.901110	+	0.433590i	$0.857247\pi$
$102$	1.80118e6	0.168057
$103$	−5.30772e6	−0.478606	−0.239303	−	0.970945i	$-0.576919\pi$
$103$	−5.30772e6	−0.478606	−0.239303	+	0.970945i	$0.576919\pi$
$104$	−1.18125e7	−1.02973
$105$	0	0
$106$	−765171.	−0.0624005
$107$	−1.74818e7	−1.37957	−0.689784	−	0.724015i	$-0.742295\pi$
$107$	−1.74818e7	−1.37957	−0.689784	+	0.724015i	$0.742295\pi$
$108$	−2.28996e6	−0.174922
$109$	467681.	0.0345905	0.0172952	−	0.999850i	$-0.494494\pi$
$109$	467681.	0.0345905	0.0172952	+	0.999850i	$0.494494\pi$
$110$	0	0
$111$	−7.19582e6	−0.499401
$112$	−4.13083e6	−0.277827
$113$	1.45523e7	0.948763	0.474382	−	0.880319i	$-0.342672\pi$
$113$	1.45523e7	0.948763	0.474382	+	0.880319i	$0.342672\pi$
$114$	−3.51226e6	−0.222034
$115$	0	0
$116$	−6.95661e6	−0.413804
$117$	1.03219e7	0.595811
$118$	−1.39854e6	−0.0783587
$119$	−6.70154e6	−0.364553
$120$	0	0
$121$	−8.06524e6	−0.413874
$122$	4.85777e6	0.242202
$123$	1.76968e7	0.857486
$124$	−7.95196e6	−0.374540
$125$	0	0
$126$	−853756.	−0.0380222
$127$	−2.04101e6	−0.0884161	−0.0442081	−	0.999022i	$-0.514076\pi$
$127$	−2.04101e6	−0.0884161	−0.0442081	+	0.999022i	$0.514076\pi$
$128$	−2.24713e7	−0.947093
$129$	2.09310e7	0.858472
$130$	0	0
$131$	−9.14330e6	−0.355348	−0.177674	−	0.984089i	$-0.556857\pi$
$131$	−9.14330e6	−0.355348	−0.177674	+	0.984089i	$0.556857\pi$
$132$	1.06162e7	0.401755
$133$	1.30679e7	0.481642
$134$	−1.44099e7	−0.517363
$135$	0	0
$136$	−1.63001e7	−0.555654
$137$	−3.62725e7	−1.20519	−0.602596	−	0.798047i	$-0.705867\pi$
$137$	−3.62725e7	−1.20519	−0.602596	+	0.798047i	$0.705867\pi$
$138$	−9.54249e6	−0.309090
$139$	−2.28823e7	−0.722682	−0.361341	−	0.932434i	$-0.617681\pi$
$139$	−2.28823e7	−0.722682	−0.361341	+	0.932434i	$0.617681\pi$
$140$	0	0
$141$	3.23267e6	0.0971169
$142$	7.43036e6	0.217771
$143$	−4.78521e7	−1.36844
$144$	8.77952e6	0.245020
$145$	0	0
$146$	−1.51568e7	−0.403064
$147$	3.17652e6	0.0824786
$148$	3.10065e7	0.786207
$149$	−7.18870e7	−1.78032	−0.890161	−	0.455646i	$-0.849408\pi$
$149$	−7.18870e7	−1.78032	−0.890161	+	0.455646i	$0.849408\pi$
$150$	0	0
$151$	6.39974e7	1.51267	0.756333	−	0.654187i	$-0.226989\pi$
$151$	6.39974e7	1.51267	0.756333	+	0.654187i	$0.226989\pi$
$152$	3.17849e7	0.734122
$153$	1.42432e7	0.321505
$154$	3.95800e6	0.0873280
$155$	0	0
$156$	−4.44766e7	−0.937984
$157$	−6.00497e6	−0.123840	−0.0619202	−	0.998081i	$-0.519722\pi$
$157$	−6.00497e6	−0.123840	−0.0619202	+	0.998081i	$0.519722\pi$
$158$	5.57589e6	0.112464
$159$	−6.05077e6	−0.119377
$160$	0	0
$161$	3.55042e7	0.670486
$162$	1.81454e6	0.0335324
$163$	−9.83028e7	−1.77791	−0.888954	−	0.457997i	$-0.848567\pi$
$163$	−9.83028e7	−1.77791	−0.888954	+	0.457997i	$0.848567\pi$
$164$	−7.62549e7	−1.34994
$165$	0	0
$166$	1.19470e7	0.202713
$167$	−5.80791e7	−0.964966	−0.482483	−	0.875905i	$-0.660265\pi$
$167$	−5.80791e7	−0.964966	−0.482483	+	0.875905i	$0.660265\pi$
$168$	7.72624e6	0.125715
$169$	1.37728e8	2.19491
$170$	0	0
$171$	−2.77740e7	−0.424768
$172$	−9.01909e7	−1.35149
$173$	2.31746e7	0.340291	0.170146	−	0.985419i	$-0.445576\pi$
$173$	2.31746e7	0.340291	0.170146	+	0.985419i	$0.445576\pi$
$174$	5.51235e6	0.0793258
$175$	0	0
$176$	−4.07017e7	−0.562754
$177$	−1.10593e7	−0.149906
$178$	6.95111e6	0.0923814
$179$	3.77902e7	0.492486	0.246243	−	0.969208i	$-0.420804\pi$
$179$	3.77902e7	0.492486	0.246243	+	0.969208i	$0.420804\pi$
$180$	0	0
$181$	−9.09325e6	−0.113984	−0.0569920	−	0.998375i	$-0.518151\pi$
$181$	−9.09325e6	−0.113984	−0.0569920	+	0.998375i	$0.518151\pi$
$182$	−1.65820e7	−0.203886
$183$	3.84140e7	0.463351
$184$	8.63567e7	1.02196
$185$	0	0
$186$	6.30106e6	0.0717990
$187$	−6.60314e7	−0.738422
$188$	−1.39295e7	−0.152891
$189$	−6.75127e6	−0.0727393
$190$	0	0
$191$	−5.34900e7	−0.555464	−0.277732	−	0.960659i	$-0.589583\pi$
$191$	−5.34900e7	−0.555464	−0.277732	+	0.960659i	$0.589583\pi$
$192$	−2.79861e7	−0.285357
$193$	8.24940e7	0.825984	0.412992	−	0.910735i	$-0.364484\pi$
$193$	8.24940e7	0.825984	0.412992	+	0.910735i	$0.364484\pi$
$194$	−4.33794e7	−0.426557
$195$	0	0
$196$	−1.36875e7	−0.129846
$197$	−7.55328e7	−0.703888	−0.351944	−	0.936021i	$-0.614479\pi$
$197$	−7.55328e7	−0.703888	−0.351944	+	0.936021i	$0.614479\pi$
$198$	−8.41219e6	−0.0770160
$199$	−1.05546e6	−0.00949412	−0.00474706	−	0.999989i	$-0.501511\pi$
$199$	−1.05546e6	−0.00949412	−0.00474706	+	0.999989i	$0.501511\pi$
$200$	0	0
$201$	−1.13950e8	−0.989756
$202$	−6.37154e7	−0.543895
$203$	−2.05095e7	−0.172076
$204$	−6.13735e7	−0.506146
$205$	0	0
$206$	−1.81226e7	−0.144439
$207$	−7.54594e7	−0.591313
$208$	1.70520e8	1.31387
$209$	1.28760e8	0.975593
$210$	0	0
$211$	1.14474e7	0.0838916	0.0419458	−	0.999120i	$-0.486644\pi$
$211$	1.14474e7	0.0838916	0.0419458	+	0.999120i	$0.486644\pi$
$212$	2.60725e7	0.187935
$213$	5.87573e7	0.416613
$214$	−5.96896e7	−0.416342
$215$	0	0
$216$	−1.64211e7	−0.110870
$217$	−2.34440e7	−0.155748
$218$	1.59684e6	0.0104391
$219$	−1.19856e8	−0.771092
$220$	0	0
$221$	2.76638e8	1.72401
$222$	−2.45693e7	−0.150715
$223$	−2.81598e8	−1.70044	−0.850222	−	0.526425i	$-0.823532\pi$
$223$	−2.81598e8	−1.70044	−0.850222	+	0.526425i	$0.823532\pi$
$224$	−5.07323e7	−0.301590
$225$	0	0
$226$	4.96872e7	0.286329
$227$	4.00301e7	0.227141	0.113571	−	0.993530i	$-0.463771\pi$
$227$	4.00301e7	0.227141	0.113571	+	0.993530i	$0.463771\pi$
$228$	1.19677e8	0.668712
$229$	−1.24592e8	−0.685591	−0.342796	−	0.939410i	$-0.611374\pi$
$229$	−1.24592e8	−0.685591	−0.342796	+	0.939410i	$0.611374\pi$
$230$	0	0
$231$	3.12988e7	0.167065
$232$	−4.98852e7	−0.262279
$233$	3.21485e8	1.66500	0.832500	−	0.554025i	$-0.186909\pi$
$233$	3.21485e8	1.66500	0.832500	+	0.554025i	$0.186909\pi$
$234$	3.52428e7	0.179811
$235$	0	0
$236$	4.76540e7	0.235997
$237$	4.40927e7	0.215153
$238$	−2.28816e7	−0.110019
$239$	−2.95076e8	−1.39811	−0.699054	−	0.715068i	$-0.746395\pi$
$239$	−2.95076e8	−1.39811	−0.699054	+	0.715068i	$0.746395\pi$
$240$	0	0
$241$	1.00887e8	0.464273	0.232137	−	0.972683i	$-0.425428\pi$
$241$	1.00887e8	0.464273	0.232137	+	0.972683i	$0.425428\pi$
$242$	−2.75378e7	−0.124904
$243$	1.43489e7	0.0641500
$244$	−1.65524e8	−0.729453
$245$	0	0
$246$	6.04236e7	0.258782
$247$	−5.39439e8	−2.27773
$248$	−5.70227e7	−0.237393
$249$	9.44737e7	0.387805
$250$	0	0
$251$	−9.30364e7	−0.371360	−0.185680	−	0.982610i	$-0.559449\pi$
$251$	−9.30364e7	−0.371360	−0.185680	+	0.982610i	$0.559449\pi$
$252$	2.90910e7	0.114513
$253$	3.49829e8	1.35811
$254$	−6.96877e6	−0.0266832
$255$	0	0
$256$	5.59493e7	0.208428
$257$	−3.64584e8	−1.33978	−0.669888	−	0.742462i	$-0.733658\pi$
$257$	−3.64584e8	−1.33978	−0.669888	+	0.742462i	$0.733658\pi$
$258$	7.14664e7	0.259080
$259$	9.14136e7	0.326935
$260$	0	0
$261$	4.35902e7	0.151756
$262$	−3.12187e7	−0.107241
$263$	−1.11633e8	−0.378396	−0.189198	−	0.981939i	$-0.560589\pi$
$263$	−1.11633e8	−0.378396	−0.189198	+	0.981939i	$0.560589\pi$
$264$	7.61279e7	0.254642
$265$	0	0
$266$	4.46187e7	0.145355
$267$	5.49675e7	0.176733
$268$	4.91006e8	1.55817
$269$	−1.53396e8	−0.480487	−0.240244	−	0.970713i	$-0.577227\pi$
$269$	−1.53396e8	−0.480487	−0.240244	+	0.970713i	$0.577227\pi$
$270$	0	0
$271$	−4.17128e8	−1.27314	−0.636571	−	0.771218i	$-0.719648\pi$
$271$	−4.17128e8	−1.27314	−0.636571	+	0.771218i	$0.719648\pi$
$272$	2.35301e8	0.708978
$273$	−1.31126e8	−0.390050
$274$	−1.23848e8	−0.363717
$275$	0	0
$276$	3.25152e8	0.930903
$277$	4.05683e8	1.14685	0.573426	−	0.819257i	$-0.305614\pi$
$277$	4.05683e8	1.14685	0.573426	+	0.819257i	$0.305614\pi$
$278$	−7.81288e7	−0.218099
$279$	4.98271e7	0.137357
$280$	0	0
$281$	−1.50774e8	−0.405371	−0.202686	−	0.979244i	$-0.564967\pi$
$281$	−1.50774e8	−0.405371	−0.202686	+	0.979244i	$0.564967\pi$
$282$	1.10376e7	0.0293091
$283$	3.69597e7	0.0969338	0.0484669	−	0.998825i	$-0.484566\pi$
$283$	3.69597e7	0.0969338	0.0484669	+	0.998825i	$0.484566\pi$
$284$	−2.53183e8	−0.655874
$285$	0	0
$286$	−1.63385e8	−0.412983
$287$	−2.24815e8	−0.561356
$288$	1.07825e8	0.265977
$289$	−2.86042e7	−0.0697088
$290$	0	0
$291$	−3.43032e8	−0.816037
$292$	5.16456e8	1.21393
$293$	6.00426e8	1.39451	0.697257	−	0.716822i	$-0.254404\pi$
$293$	6.00426e8	1.39451	0.697257	+	0.716822i	$0.254404\pi$
$294$	1.08459e7	0.0248913
$295$	0	0
$296$	2.22345e8	0.498317
$297$	−6.65214e7	−0.147338
$298$	−2.45450e8	−0.537286
$299$	−1.46561e9	−3.17079
$300$	0	0
$301$	−2.65901e8	−0.562002
$302$	2.18511e8	0.456510
$303$	−5.03844e8	−1.04051
$304$	−4.58832e8	−0.936691
$305$	0	0
$306$	4.86318e7	0.0970276
$307$	−9.08890e8	−1.79278	−0.896390	−	0.443267i	$-0.853819\pi$
$307$	−9.08890e8	−1.79278	−0.896390	+	0.443267i	$0.853819\pi$
$308$	−1.34865e8	−0.263010
$309$	−1.43309e8	−0.276323
$310$	0	0
$311$	6.45862e8	1.21753	0.608763	−	0.793352i	$-0.291666\pi$
$311$	6.45862e8	1.21753	0.608763	+	0.793352i	$0.291666\pi$
$312$	−3.18937e8	−0.594517
$313$	8.96603e8	1.65270	0.826352	−	0.563154i	$-0.190412\pi$
$313$	8.96603e8	1.65270	0.826352	+	0.563154i	$0.190412\pi$
$314$	−2.05033e7	−0.0373740
$315$	0	0
$316$	−1.89994e8	−0.338715
$317$	−8.47295e8	−1.49392	−0.746959	−	0.664870i	$-0.768487\pi$
$317$	−8.47295e8	−1.49392	−0.746959	+	0.664870i	$0.768487\pi$
$318$	−2.06596e7	−0.0360269
$319$	−2.02084e8	−0.348549
$320$	0	0
$321$	−4.72009e8	−0.796494
$322$	1.21225e8	0.202347
$323$	−7.44374e8	−1.22909
$324$	−6.18289e7	−0.100991
$325$	0	0
$326$	−3.35643e8	−0.536557
$327$	1.26274e7	0.0199708
$328$	−5.46816e8	−0.855623
$329$	−4.10669e7	−0.0635779
$330$	0	0
$331$	7.42401e8	1.12523	0.562614	−	0.826720i	$-0.309796\pi$
$331$	7.42401e8	1.12523	0.562614	+	0.826720i	$0.309796\pi$
$332$	−4.07084e8	−0.610521
$333$	−1.94287e8	−0.288330
$334$	−1.98304e8	−0.291219
$335$	0	0
$336$	−1.11532e8	−0.160403
$337$	7.24512e8	1.03120	0.515598	−	0.856831i	$-0.327570\pi$
$337$	7.24512e8	1.03120	0.515598	+	0.856831i	$0.327570\pi$
$338$	4.70254e8	0.662406
$339$	3.92913e8	0.547769
$340$	0	0
$341$	−2.30998e8	−0.315477
$342$	−9.48310e7	−0.128191
$343$	−4.03536e7	−0.0539949
$344$	−6.46750e8	−0.856608
$345$	0	0
$346$	7.91269e7	0.102697
$347$	−5.09846e8	−0.655068	−0.327534	−	0.944839i	$-0.606218\pi$
$347$	−5.09846e8	−0.655068	−0.327534	+	0.944839i	$0.606218\pi$
$348$	−1.87829e8	−0.238910
$349$	−8.76052e8	−1.10317	−0.551583	−	0.834120i	$-0.685976\pi$
$349$	−8.76052e8	−1.10317	−0.551583	+	0.834120i	$0.685976\pi$
$350$	0	0
$351$	2.78691e8	0.343991
$352$	−4.99874e8	−0.610887
$353$	−1.26670e9	−1.53272	−0.766358	−	0.642413i	$-0.777933\pi$
$353$	−1.26670e9	−1.53272	−0.766358	+	0.642413i	$0.777933\pi$
$354$	−3.77605e7	−0.0452404
$355$	0	0
$356$	−2.36853e8	−0.278230
$357$	−1.80942e8	−0.210475
$358$	1.29030e8	0.148628
$359$	1.66558e9	1.89992	0.949961	−	0.312369i	$-0.101122\pi$
$359$	1.66558e9	1.89992	0.949961	+	0.312369i	$0.101122\pi$
$360$	0	0
$361$	5.57643e8	0.623851
$362$	−3.10478e7	−0.0343994
$363$	−2.17761e8	−0.238950
$364$	5.65018e8	0.614055
$365$	0	0
$366$	1.31160e8	0.139835
$367$	−1.67575e9	−1.76961	−0.884807	−	0.465957i	$-0.845710\pi$
$367$	−1.67575e9	−1.76961	−0.884807	+	0.465957i	$0.845710\pi$
$368$	−1.24661e9	−1.30395
$369$	4.77814e8	0.495070
$370$	0	0
$371$	7.68672e7	0.0781506
$372$	−2.14703e8	−0.216241
$373$	−1.47694e9	−1.47360	−0.736802	−	0.676109i	$-0.763665\pi$
$373$	−1.47694e9	−1.47360	−0.736802	+	0.676109i	$0.763665\pi$
$374$	−2.25456e8	−0.222850
$375$	0	0
$376$	−9.98868e7	−0.0969060
$377$	8.46628e8	0.813763
$378$	−2.30514e7	−0.0219521
$379$	−1.64279e9	−1.55005	−0.775024	−	0.631932i	$-0.782262\pi$
$379$	−1.64279e9	−1.55005	−0.775024	+	0.631932i	$0.782262\pi$
$380$	0	0
$381$	−5.51072e7	−0.0510471
$382$	−1.82635e8	−0.167634
$383$	1.89784e9	1.72609	0.863044	−	0.505129i	$-0.168555\pi$
$383$	1.89784e9	1.72609	0.863044	+	0.505129i	$0.168555\pi$
$384$	−6.06724e8	−0.546804
$385$	0	0
$386$	2.81666e8	0.249275
$387$	5.65137e8	0.495639
$388$	1.47811e9	1.28469
$389$	1.05432e9	0.908131	0.454065	−	0.890968i	$-0.349973\pi$
$389$	1.05432e9	0.908131	0.454065	+	0.890968i	$0.349973\pi$
$390$	0	0
$391$	−2.02240e9	−1.71099
$392$	−9.81518e7	−0.0822995
$393$	−2.46869e8	−0.205160
$394$	−2.57898e8	−0.212428
$395$	0	0
$396$	2.86638e8	0.231953
$397$	−2.96848e8	−0.238104	−0.119052	−	0.992888i	$-0.537986\pi$
$397$	−2.96848e8	−0.238104	−0.119052	+	0.992888i	$0.537986\pi$
$398$	−3.60373e6	−0.00286525
$399$	3.52832e8	0.278076
$400$	0	0
$401$	−7.79941e8	−0.604028	−0.302014	−	0.953304i	$-0.597659\pi$
$401$	−7.79941e8	−0.604028	−0.302014	+	0.953304i	$0.597659\pi$
$402$	−3.89069e8	−0.298700
$403$	9.67763e8	0.736549
$404$	2.17105e9	1.63808
$405$	0	0
$406$	−7.00273e7	−0.0519310
$407$	9.00713e8	0.662225
$408$	−4.40103e8	−0.320807
$409$	8.75152e8	0.632488	0.316244	−	0.948678i	$-0.397578\pi$
$409$	8.75152e8	0.632488	0.316244	+	0.948678i	$0.397578\pi$
$410$	0	0
$411$	−9.79359e8	−0.695818
$412$	6.17511e8	0.435015
$413$	1.40494e8	0.0981367
$414$	−2.57647e8	−0.178453
$415$	0	0
$416$	2.09422e9	1.42625
$417$	−6.17821e8	−0.417241
$418$	4.39635e8	0.294426
$419$	1.07058e9	0.711000	0.355500	−	0.934676i	$-0.384311\pi$
$419$	1.07058e9	0.711000	0.355500	+	0.934676i	$0.384311\pi$
$420$	0	0
$421$	2.54184e9	1.66020	0.830100	−	0.557614i	$-0.188283\pi$
$421$	2.54184e9	1.66020	0.830100	+	0.557614i	$0.188283\pi$
$422$	3.90858e7	0.0253178
$423$	8.72822e7	0.0560705
$424$	1.86963e8	0.119118
$425$	0	0
$426$	2.00620e8	0.125730
$427$	−4.87999e8	−0.303335
$428$	2.03387e9	1.25392
$429$	−1.29201e9	−0.790068
$430$	0	0
$431$	8.91747e8	0.536502	0.268251	−	0.963349i	$-0.413554\pi$
$431$	8.91747e8	0.536502	0.268251	+	0.963349i	$0.413554\pi$
$432$	2.37047e8	0.141463
$433$	1.03966e9	0.615440	0.307720	−	0.951477i	$-0.400434\pi$
$433$	1.03966e9	0.615440	0.307720	+	0.951477i	$0.400434\pi$
$434$	−8.00468e7	−0.0470035
$435$	0	0
$436$	−5.44109e7	−0.0314400
$437$	3.94363e9	2.26054
$438$	−4.09235e8	−0.232709
$439$	−1.55931e9	−0.879641	−0.439821	−	0.898086i	$-0.644958\pi$
$439$	−1.55931e9	−0.879641	−0.439821	+	0.898086i	$0.644958\pi$
$440$	0	0
$441$	8.57661e7	0.0476190
$442$	9.44548e8	0.520291
$443$	2.79970e8	0.153002	0.0765012	−	0.997069i	$-0.475625\pi$
$443$	2.79970e8	0.153002	0.0765012	+	0.997069i	$0.475625\pi$
$444$	8.37176e8	0.453917
$445$	0	0
$446$	−9.61482e8	−0.513179
$447$	−1.94095e9	−1.02787
$448$	3.55527e8	0.186810
$449$	6.18452e8	0.322436	0.161218	−	0.986919i	$-0.448458\pi$
$449$	6.18452e8	0.322436	0.161218	+	0.986919i	$0.448458\pi$
$450$	0	0
$451$	−2.21514e9	−1.13706
$452$	−1.69305e9	−0.862352
$453$	1.72793e9	0.873338
$454$	1.36678e8	0.0685494
$455$	0	0
$456$	8.58192e8	0.423846
$457$	−9.07329e8	−0.444691	−0.222345	−	0.974968i	$-0.571371\pi$
$457$	−9.07329e8	−0.444691	−0.222345	+	0.974968i	$0.571371\pi$
$458$	−4.25404e8	−0.206906
$459$	3.84567e8	0.185621
$460$	0	0
$461$	−2.41729e9	−1.14915	−0.574573	−	0.818453i	$-0.694832\pi$
$461$	−2.41729e9	−1.14915	−0.574573	+	0.818453i	$0.694832\pi$
$462$	1.06866e8	0.0504188
$463$	3.14336e8	0.147184	0.0735921	−	0.997288i	$-0.476554\pi$
$463$	3.14336e8	0.147184	0.0735921	+	0.997288i	$0.476554\pi$
$464$	7.20120e8	0.334651
$465$	0	0
$466$	1.09767e9	0.502483
$467$	−4.74332e7	−0.0215513	−0.0107757	−	0.999942i	$-0.503430\pi$
$467$	−4.74332e7	−0.0215513	−0.0107757	+	0.999942i	$0.503430\pi$
$468$	−1.20087e9	−0.541545
$469$	1.44759e9	0.647947
$470$	0	0
$471$	−1.62134e8	−0.0714993
$472$	3.41722e8	0.149581
$473$	−2.61997e9	−1.13837
$474$	1.50549e8	0.0649313
$475$	0	0
$476$	7.79671e8	0.331350
$477$	−1.63371e8	−0.0689223
$478$	−1.00750e9	−0.421937
$479$	−2.08794e9	−0.868048	−0.434024	−	0.900901i	$-0.642907\pi$
$479$	−2.08794e9	−0.868048	−0.434024	+	0.900901i	$0.642907\pi$
$480$	0	0
$481$	−3.77353e9	−1.54611
$482$	3.44465e8	0.140114
$483$	9.58614e8	0.387105
$484$	9.38326e8	0.376179
$485$	0	0
$486$	4.89926e7	0.0193599
$487$	2.19273e9	0.860267	0.430134	−	0.902765i	$-0.358466\pi$
$487$	2.19273e9	0.860267	0.430134	+	0.902765i	$0.358466\pi$
$488$	−1.18696e9	−0.462345
$489$	−2.65417e9	−1.02648
$490$	0	0
$491$	2.19027e9	0.835050	0.417525	−	0.908666i	$-0.362898\pi$
$491$	2.19027e9	0.835050	0.417525	+	0.908666i	$0.362898\pi$
$492$	−2.05888e9	−0.779387
$493$	1.16827e9	0.439115
$494$	−1.84185e9	−0.687400
$495$	0	0
$496$	8.23154e8	0.302897
$497$	−7.46435e8	−0.272737
$498$	3.22569e8	0.117036
$499$	6.56737e8	0.236614	0.118307	−	0.992977i	$-0.462253\pi$
$499$	6.56737e8	0.236614	0.118307	+	0.992977i	$0.462253\pi$
$500$	0	0
$501$	−1.56813e9	−0.557123
$502$	−3.17662e8	−0.112073
$503$	2.58040e9	0.904065	0.452033	−	0.892001i	$-0.350699\pi$
$503$	2.58040e9	0.904065	0.452033	+	0.892001i	$0.350699\pi$
$504$	2.08608e8	0.0725813
$505$	0	0
$506$	1.19445e9	0.409865
$507$	3.71864e9	1.26723
$508$	2.37455e8	0.0803633
$509$	−2.07943e9	−0.698928	−0.349464	−	0.936950i	$-0.613636\pi$
$509$	−2.07943e9	−0.698928	−0.349464	+	0.936950i	$0.613636\pi$
$510$	0	0
$511$	1.52262e9	0.504798
$512$	3.06735e9	1.00999
$513$	−7.49898e8	−0.245240
$514$	−1.24483e9	−0.404333
$515$	0	0
$516$	−2.43516e9	−0.780284
$517$	−4.04639e8	−0.128781
$518$	3.12121e8	0.0986662
$519$	6.25714e8	0.196467
$520$	0	0
$521$	−2.82646e9	−0.875609	−0.437804	−	0.899070i	$-0.644244\pi$
$521$	−2.82646e9	−0.875609	−0.437804	+	0.899070i	$0.644244\pi$
$522$	1.48834e8	0.0457988
$523$	−4.81610e9	−1.47211	−0.736055	−	0.676922i	$-0.763313\pi$
$523$	−4.81610e9	−1.47211	−0.736055	+	0.676922i	$0.763313\pi$
$524$	1.06375e9	0.322983
$525$	0	0
$526$	−3.81157e8	−0.114197
$527$	1.33542e9	0.397449
$528$	−1.09895e9	−0.324906
$529$	7.30968e9	2.14686
$530$	0	0
$531$	−2.98600e8	−0.0865484
$532$	−1.52034e9	−0.437775
$533$	9.28030e9	2.65471
$534$	1.87680e8	0.0533364
$535$	0	0
$536$	3.52096e9	0.987606
$537$	1.02034e9	0.284337
$538$	−5.23754e8	−0.145007
$539$	−3.97611e8	−0.109370
$540$	0	0
$541$	1.56108e9	0.423871	0.211936	−	0.977284i	$-0.432023\pi$
$541$	1.56108e9	0.423871	0.211936	+	0.977284i	$0.432023\pi$
$542$	−1.42423e9	−0.384223
$543$	−2.45518e8	−0.0658087
$544$	2.88982e9	0.769618
$545$	0	0
$546$	−4.47714e8	−0.117714
$547$	3.82511e8	0.0999282	0.0499641	−	0.998751i	$-0.484089\pi$
$547$	3.82511e8	0.0999282	0.0499641	+	0.998751i	$0.484089\pi$
$548$	4.22002e9	1.09543
$549$	1.03718e9	0.267516
$550$	0	0
$551$	−2.27810e9	−0.580152
$552$	2.33163e9	0.590029
$553$	−5.60140e8	−0.140851
$554$	1.38516e9	0.346110
$555$	0	0
$556$	2.66217e9	0.656861
$557$	7.35207e9	1.80267	0.901335	−	0.433122i	$-0.142588\pi$
$557$	7.35207e9	1.80267	0.901335	+	0.433122i	$0.142588\pi$
$558$	1.70129e8	0.0414532
$559$	1.09763e10	2.65776
$560$	0	0
$561$	−1.78285e9	−0.426328
$562$	−5.14798e8	−0.122338
$563$	−7.91584e9	−1.86947	−0.934733	−	0.355352i	$-0.884361\pi$
$563$	−7.91584e9	−1.86947	−0.934733	+	0.355352i	$0.884361\pi$
$564$	−3.76096e8	−0.0882717
$565$	0	0
$566$	1.26194e8	0.0292538
$567$	−1.82284e8	−0.0419961
$568$	−1.81555e9	−0.415708
$569$	4.49464e9	1.02283	0.511414	−	0.859335i	$-0.329122\pi$
$569$	4.49464e9	1.02283	0.511414	+	0.859335i	$0.329122\pi$
$570$	0	0
$571$	−2.15052e9	−0.483412	−0.241706	−	0.970350i	$-0.577707\pi$
$571$	−2.15052e9	−0.483412	−0.241706	+	0.970350i	$0.577707\pi$
$572$	5.56721e9	1.24380
$573$	−1.44423e9	−0.320697
$574$	−7.67604e8	−0.169412
$575$	0	0
$576$	−7.55624e8	−0.164751
$577$	1.73438e9	0.375862	0.187931	−	0.982182i	$-0.439822\pi$
$577$	1.73438e9	0.375862	0.187931	+	0.982182i	$0.439822\pi$
$578$	−9.76656e7	−0.0210375
$579$	2.22734e9	0.476882
$580$	0	0
$581$	−1.20017e9	−0.253878
$582$	−1.17124e9	−0.246273
$583$	7.57385e8	0.158298
$584$	3.70345e9	0.769418
$585$	0	0
$586$	2.05008e9	0.420852
$587$	2.98775e9	0.609692	0.304846	−	0.952402i	$-0.401395\pi$
$587$	2.98775e9	0.609692	0.304846	+	0.952402i	$0.401395\pi$
$588$	−3.69563e8	−0.0749666
$589$	−2.60405e9	−0.525104
$590$	0	0
$591$	−2.03939e9	−0.406390
$592$	−3.20967e9	−0.635819
$593$	−5.56485e9	−1.09588	−0.547938	−	0.836519i	$-0.684587\pi$
$593$	−5.56485e9	−1.09588	−0.547938	+	0.836519i	$0.684587\pi$
$594$	−2.27129e8	−0.0444652
$595$	0	0
$596$	8.36348e9	1.61817
$597$	−2.84973e7	−0.00548143
$598$	−5.00414e9	−0.956919
$599$	−5.39399e9	−1.02545	−0.512727	−	0.858552i	$-0.671365\pi$
$599$	−5.39399e9	−1.02545	−0.512727	+	0.858552i	$0.671365\pi$
$600$	0	0
$601$	−5.88616e8	−0.110604	−0.0553021	−	0.998470i	$-0.517612\pi$
$601$	−5.88616e8	−0.110604	−0.0553021	+	0.998470i	$0.517612\pi$
$602$	−9.07888e8	−0.169607
$603$	−3.07665e9	−0.571436
$604$	−7.44558e9	−1.37489
$605$	0	0
$606$	−1.72032e9	−0.314018
$607$	−5.38304e9	−0.976938	−0.488469	−	0.872581i	$-0.662444\pi$
$607$	−5.38304e9	−0.976938	−0.488469	+	0.872581i	$0.662444\pi$
$608$	−5.63509e9	−1.01681
$609$	−5.53757e8	−0.0993479
$610$	0	0
$611$	1.69523e9	0.300667
$612$	−1.65708e9	−0.292223
$613$	−8.87502e9	−1.55617	−0.778086	−	0.628158i	$-0.783809\pi$
$613$	−8.87502e9	−1.55617	−0.778086	+	0.628158i	$0.783809\pi$
$614$	−3.10330e9	−0.541046
$615$	0	0
$616$	−9.67106e8	−0.166702
$617$	−5.94996e6	−0.00101980	−0.000509901	−	1.00000i	$-0.500162\pi$
$617$	−5.94996e6	−0.00101980	−0.000509901		1.00000i	$0.500162\pi$
$618$	−4.89310e8	−0.0833920
$619$	8.21425e9	1.39204	0.696019	−	0.718024i	$-0.254953\pi$
$619$	8.21425e9	1.39204	0.696019	+	0.718024i	$0.254953\pi$
$620$	0	0
$621$	−2.03740e9	−0.341395
$622$	2.20522e9	0.367439
$623$	−6.98291e8	−0.115699
$624$	4.60403e9	0.758564
$625$	0	0
$626$	3.06134e9	0.498772
$627$	3.47652e9	0.563259
$628$	6.98631e8	0.112561
$629$	−5.20711e9	−0.834295
$630$	0	0
$631$	2.53051e9	0.400964	0.200482	−	0.979697i	$-0.435749\pi$
$631$	2.53051e9	0.400964	0.200482	+	0.979697i	$0.435749\pi$
$632$	−1.36243e9	−0.214686
$633$	3.09080e8	0.0484348
$634$	−2.89299e9	−0.450852
$635$	0	0
$636$	7.03958e8	0.108504
$637$	1.66579e9	0.255347
$638$	−6.89990e8	−0.105189
$639$	1.58645e9	0.240532
$640$	0	0
$641$	6.96698e9	1.04482	0.522410	−	0.852694i	$-0.325033\pi$
$641$	6.96698e9	1.04482	0.522410	+	0.852694i	$0.325033\pi$
$642$	−1.61162e9	−0.240375
$643$	−1.21823e10	−1.80714	−0.903572	−	0.428437i	$-0.859064\pi$
$643$	−1.21823e10	−1.80714	−0.903572	+	0.428437i	$0.859064\pi$
$644$	−4.13063e9	−0.609419
$645$	0	0
$646$	−2.54158e9	−0.370928
$647$	9.95454e9	1.44496	0.722481	−	0.691391i	$-0.243002\pi$
$647$	9.95454e9	1.44496	0.722481	+	0.691391i	$0.243002\pi$
$648$	−4.43369e8	−0.0640107
$649$	1.38431e9	0.198781
$650$	0	0
$651$	−6.32988e8	−0.0899213
$652$	1.14367e10	1.61598
$653$	−2.38576e9	−0.335297	−0.167649	−	0.985847i	$-0.553617\pi$
$653$	−2.38576e9	−0.335297	−0.167649	+	0.985847i	$0.553617\pi$
$654$	4.31147e7	0.00602702
$655$	0	0
$656$	7.89358e9	1.09172
$657$	−3.23612e9	−0.445190
$658$	−1.40218e8	−0.0191873
$659$	1.98573e8	0.0270284	0.0135142	−	0.999909i	$-0.495698\pi$
$659$	1.98573e8	0.0270284	0.0135142	+	0.999909i	$0.495698\pi$
$660$	0	0
$661$	1.68735e9	0.227247	0.113624	−	0.993524i	$-0.463754\pi$
$661$	1.68735e9	0.227247	0.113624	+	0.993524i	$0.463754\pi$
$662$	2.53484e9	0.339584
$663$	7.46923e9	0.995356
$664$	−2.91916e9	−0.386963
$665$	0	0
$666$	−6.63370e8	−0.0870154
$667$	−6.18938e9	−0.807620
$668$	6.75704e9	0.877079
$669$	−7.60314e9	−0.981752
$670$	0	0
$671$	−4.80834e9	−0.614421
$672$	−1.36977e9	−0.174123
$673$	5.02352e9	0.635266	0.317633	−	0.948214i	$-0.397112\pi$
$673$	5.02352e9	0.635266	0.317633	+	0.948214i	$0.397112\pi$
$674$	2.47376e9	0.311206
$675$	0	0
$676$	−1.60235e10	−1.99501
$677$	1.03907e10	1.28702	0.643510	−	0.765438i	$-0.277477\pi$
$677$	1.03907e10	1.28702	0.643510	+	0.765438i	$0.277477\pi$
$678$	1.34155e9	0.165312
$679$	4.35778e9	0.534222
$680$	0	0
$681$	1.08081e9	0.131140
$682$	−7.88714e8	−0.0952082
$683$	−3.61608e9	−0.434276	−0.217138	−	0.976141i	$-0.569672\pi$
$683$	−3.61608e9	−0.434276	−0.217138	+	0.976141i	$0.569672\pi$
$684$	3.23128e9	0.386081
$685$	0	0
$686$	−1.37783e8	−0.0162952
$687$	−3.36398e9	−0.395826
$688$	9.33619e9	1.09297
$689$	−3.17306e9	−0.369582
$690$	0	0
$691$	1.65856e10	1.91231	0.956153	−	0.292869i	$-0.0946100\pi$
$691$	1.65856e10	1.91231	0.956153	+	0.292869i	$0.0946100\pi$
$692$	−2.69618e9	−0.309298
$693$	8.45068e8	0.0964551
$694$	−1.74081e9	−0.197694
$695$	0	0
$696$	−1.34690e9	−0.151427
$697$	1.28059e10	1.43251
$698$	−2.99117e9	−0.332926
$699$	8.68008e9	0.961289
$700$	0	0
$701$	1.42971e10	1.56760	0.783798	−	0.621015i	$-0.213280\pi$
$701$	1.42971e10	1.56760	0.783798	+	0.621015i	$0.213280\pi$
$702$	9.51556e8	0.103814
$703$	1.01538e10	1.10226
$704$	3.50306e9	0.378394
$705$	0	0
$706$	−4.32499e9	−0.462561
$707$	6.40069e9	0.681175
$708$	1.28666e9	0.136253
$709$	4.66391e9	0.491460	0.245730	−	0.969338i	$-0.420972\pi$
$709$	4.66391e9	0.491460	0.245730	+	0.969338i	$0.420972\pi$
$710$	0	0
$711$	1.19050e9	0.124219
$712$	−1.69845e9	−0.176349
$713$	−7.07496e9	−0.730989
$714$	−6.17804e8	−0.0635195
$715$	0	0
$716$	−4.39659e9	−0.447631
$717$	−7.96705e9	−0.807199
$718$	5.68694e9	0.573380
$719$	4.82173e9	0.483784	0.241892	−	0.970303i	$-0.422232\pi$
$719$	4.82173e9	0.483784	0.241892	+	0.970303i	$0.422232\pi$
$720$	0	0
$721$	1.82055e9	0.180896
$722$	1.90400e9	0.188273
$723$	2.72394e9	0.268048
$724$	1.05793e9	0.103603
$725$	0	0
$726$	−7.43521e8	−0.0721132
$727$	−2.17621e9	−0.210054	−0.105027	−	0.994469i	$-0.533493\pi$
$727$	−2.17621e9	−0.210054	−0.105027	+	0.994469i	$0.533493\pi$
$728$	4.05168e9	0.389203
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	1.51463e10	1.43416
$732$	−4.46916e9	−0.421150
$733$	−4.65228e9	−0.436317	−0.218158	−	0.975913i	$-0.570005\pi$
$733$	−4.65228e9	−0.436317	−0.218158	+	0.975913i	$0.570005\pi$
$734$	−5.72166e9	−0.534055
$735$	0	0
$736$	−1.53100e10	−1.41548
$737$	1.42633e10	1.31245
$738$	1.63144e9	0.149408
$739$	4.05338e8	0.0369456	0.0184728	−	0.999829i	$-0.494120\pi$
$739$	4.05338e8	0.0369456	0.0184728	+	0.999829i	$0.494120\pi$
$740$	0	0
$741$	−1.45648e10	−1.31505
$742$	2.62454e8	0.0235852
$743$	−1.08140e10	−0.967223	−0.483611	−	0.875283i	$-0.660675\pi$
$743$	−1.08140e10	−0.967223	−0.483611	+	0.875283i	$0.660675\pi$
$744$	−1.53961e9	−0.137059
$745$	0	0
$746$	−5.04282e9	−0.444721
$747$	2.55079e9	0.223899
$748$	7.68222e9	0.671168
$749$	5.99626e9	0.521428
$750$	0	0
$751$	−1.42070e10	−1.22394	−0.611972	−	0.790880i	$-0.709623\pi$
$751$	−1.42070e10	−1.22394	−0.611972	+	0.790880i	$0.709623\pi$
$752$	1.44192e9	0.123646
$753$	−2.51198e9	−0.214405
$754$	2.89071e9	0.245587
$755$	0	0
$756$	7.85456e8	0.0661143
$757$	−3.78427e9	−0.317064	−0.158532	−	0.987354i	$-0.550676\pi$
$757$	−3.78427e9	−0.317064	−0.158532	+	0.987354i	$0.550676\pi$
$758$	−5.60911e9	−0.467791
$759$	9.44538e9	0.784103
$760$	0	0
$761$	1.01511e10	0.834966	0.417483	−	0.908685i	$-0.362912\pi$
$761$	1.01511e10	0.834966	0.417483	+	0.908685i	$0.362912\pi$
$762$	−1.88157e8	−0.0154056
$763$	−1.60414e8	−0.0130740
$764$	6.22314e9	0.504874
$765$	0	0
$766$	6.47993e9	0.520919
$767$	−5.79954e9	−0.464098
$768$	1.51063e9	0.120336
$769$	−1.31076e10	−1.03939	−0.519696	−	0.854351i	$-0.673955\pi$
$769$	−1.31076e10	−1.03939	−0.519696	+	0.854351i	$0.673955\pi$
$770$	0	0
$771$	−9.84378e9	−0.773520
$772$	−9.59751e9	−0.750755
$773$	1.40370e10	1.09307	0.546533	−	0.837438i	$-0.315947\pi$
$773$	1.40370e10	1.09307	0.546533	+	0.837438i	$0.315947\pi$
$774$	1.92959e9	0.149580
$775$	0	0
$776$	1.05994e10	0.814265
$777$	2.46817e9	0.188756
$778$	3.59985e9	0.274066
$779$	−2.49713e10	−1.89261
$780$	0	0
$781$	−7.35474e9	−0.552445
$782$	−6.90524e9	−0.516363
$783$	1.17694e9	0.0876166
$784$	1.41687e9	0.105009
$785$	0	0
$786$	−8.42905e8	−0.0619155
$787$	3.87288e9	0.283219	0.141610	−	0.989923i	$-0.454772\pi$
$787$	3.87288e9	0.283219	0.141610	+	0.989923i	$0.454772\pi$
$788$	8.78764e9	0.639780
$789$	−3.01409e9	−0.218467
$790$	0	0
$791$	−4.99145e9	−0.358599
$792$	2.05545e9	0.147018
$793$	2.01445e10	1.43450
$794$	−1.01355e9	−0.0718578
$795$	0	0
$796$	1.22794e8	0.00862942
$797$	8.56688e9	0.599403	0.299701	−	0.954033i	$-0.403113\pi$
$797$	8.56688e9	0.599403	0.299701	+	0.954033i	$0.403113\pi$
$798$	1.20470e9	0.0839210
$799$	2.33926e9	0.162243
$800$	0	0
$801$	1.48412e9	0.102037
$802$	−2.66302e9	−0.182290
$803$	1.50026e10	1.02250
$804$	1.32572e10	0.899611
$805$	0	0
$806$	3.30431e9	0.222284
$807$	−4.14170e9	−0.277410
$808$	1.55684e10	1.03825
$809$	−1.49503e10	−0.992731	−0.496365	−	0.868114i	$-0.665332\pi$
$809$	−1.49503e10	−0.992731	−0.496365	+	0.868114i	$0.665332\pi$
$810$	0	0
$811$	−8.61940e9	−0.567419	−0.283710	−	0.958910i	$-0.591565\pi$
$811$	−8.61940e9	−0.567419	−0.283710	+	0.958910i	$0.591565\pi$
$812$	2.38612e9	0.156403
$813$	−1.12625e10	−0.735049
$814$	3.07538e9	0.199854
$815$	0	0
$816$	6.35313e9	0.409329
$817$	−2.95350e10	−1.89478
$818$	2.98810e9	0.190879
$819$	−3.54041e9	−0.225195
$820$	0	0
$821$	2.53165e10	1.59663	0.798313	−	0.602243i	$-0.205726\pi$
$821$	2.53165e10	1.59663	0.798313	+	0.602243i	$0.205726\pi$
$822$	−3.34390e9	−0.209992
$823$	−1.51790e9	−0.0949171	−0.0474586	−	0.998873i	$-0.515112\pi$
$823$	−1.51790e9	−0.0949171	−0.0474586	+	0.998873i	$0.515112\pi$
$824$	4.42811e9	0.275723
$825$	0	0
$826$	4.79699e8	0.0296168
$827$	−1.68322e10	−1.03484	−0.517418	−	0.855733i	$-0.673107\pi$
$827$	−1.68322e10	−1.03484	−0.517418	+	0.855733i	$0.673107\pi$
$828$	8.77910e9	0.537457
$829$	−1.54259e10	−0.940396	−0.470198	−	0.882561i	$-0.655818\pi$
$829$	−1.54259e10	−0.940396	−0.470198	+	0.882561i	$0.655818\pi$
$830$	0	0
$831$	1.09534e10	0.662135
$832$	−1.46761e10	−0.883442
$833$	2.29863e9	0.137788
$834$	−2.10948e9	−0.125920
$835$	0	0
$836$	−1.49802e10	−0.886737
$837$	1.34533e9	0.0793031
$838$	3.65536e9	0.214574
$839$	−1.06044e9	−0.0619894	−0.0309947	−	0.999520i	$-0.509868\pi$
$839$	−1.06044e9	−0.0619894	−0.0309947	+	0.999520i	$0.509868\pi$
$840$	0	0
$841$	−1.36745e10	−0.792730
$842$	8.67880e9	0.501034
$843$	−4.07089e9	−0.234041
$844$	−1.33181e9	−0.0762509
$845$	0	0
$846$	2.98015e8	0.0169216
$847$	2.76638e9	0.156430
$848$	−2.69892e9	−0.151986
$849$	9.97911e8	0.0559648
$850$	0	0
$851$	2.75869e10	1.53444
$852$	−6.83594e9	−0.378669
$853$	−3.03756e10	−1.67573	−0.837864	−	0.545878i	$-0.816196\pi$
$853$	−3.03756e10	−1.67573	−0.837864	+	0.545878i	$0.816196\pi$
$854$	−1.66622e9	−0.0915438
$855$	0	0
$856$	1.45847e10	0.794764
$857$	1.27509e10	0.692001	0.346001	−	0.938234i	$-0.387539\pi$
$857$	1.27509e10	0.692001	0.346001	+	0.938234i	$0.387539\pi$
$858$	−4.41140e9	−0.238436
$859$	1.08205e10	0.582467	0.291233	−	0.956652i	$-0.405934\pi$
$859$	1.08205e10	0.582467	0.291233	+	0.956652i	$0.405934\pi$
$860$	0	0
$861$	−6.07000e9	−0.324099
$862$	3.04476e9	0.161912
$863$	−6.05794e9	−0.320839	−0.160419	−	0.987049i	$-0.551285\pi$
$863$	−6.05794e9	−0.320839	−0.160419	+	0.987049i	$0.551285\pi$
$864$	2.91127e9	0.153562
$865$	0	0
$866$	3.54981e9	0.185734
$867$	−7.72313e8	−0.0402464
$868$	2.72752e9	0.141563
$869$	−5.51915e9	−0.285301
$870$	0	0
$871$	−5.97560e10	−3.06421
$872$	−3.90175e8	−0.0199274
$873$	−9.26188e9	−0.471139
$874$	1.34651e10	0.682211
$875$	0	0
$876$	1.39443e10	0.700863
$877$	−5.64172e9	−0.282431	−0.141216	−	0.989979i	$-0.545101\pi$
$877$	−5.64172e9	−0.282431	−0.141216	+	0.989979i	$0.545101\pi$
$878$	−5.32407e9	−0.265468
$879$	1.62115e10	0.805123
$880$	0	0
$881$	−1.41128e9	−0.0695339	−0.0347669	−	0.999395i	$-0.511069\pi$
$881$	−1.41128e9	−0.0695339	−0.0347669	+	0.999395i	$0.511069\pi$
$882$	2.92838e8	0.0143710
$883$	−1.83461e10	−0.896773	−0.448386	−	0.893840i	$-0.648001\pi$
$883$	−1.83461e10	−0.896773	−0.448386	+	0.893840i	$0.648001\pi$
$884$	−3.21846e10	−1.56699
$885$	0	0
$886$	9.55923e8	0.0461748
$887$	−6.77821e9	−0.326124	−0.163062	−	0.986616i	$-0.552137\pi$
$887$	−6.77821e9	−0.326124	−0.163062	+	0.986616i	$0.552137\pi$
$888$	6.00330e9	0.287703
$889$	7.00065e8	0.0334182
$890$	0	0
$891$	−1.79608e9	−0.0850654
$892$	3.27617e10	1.54557
$893$	−4.56151e9	−0.214353
$894$	−6.62714e9	−0.310202
$895$	0	0
$896$	7.70764e9	0.357967
$897$	−3.95714e10	−1.83066
$898$	2.11163e9	0.0973085
$899$	4.08695e9	0.187603
$900$	0	0
$901$	−4.37852e9	−0.199430
$902$	−7.56332e9	−0.343155
$903$	−7.17934e9	−0.324472
$904$	−1.21407e10	−0.546579
$905$	0	0
$906$	5.89981e9	0.263566
$907$	3.32813e9	0.148107	0.0740535	−	0.997254i	$-0.476406\pi$
$907$	3.32813e9	0.148107	0.0740535	+	0.997254i	$0.476406\pi$
$908$	−4.65718e9	−0.206454
$909$	−1.36038e10	−0.600740
$910$	0	0
$911$	7.48533e9	0.328017	0.164009	−	0.986459i	$-0.447558\pi$
$911$	7.48533e9	0.328017	0.164009	+	0.986459i	$0.447558\pi$
$912$	−1.23885e10	−0.540799
$913$	−1.18254e10	−0.514244
$914$	−3.09797e9	−0.134204
$915$	0	0
$916$	1.44953e10	0.623149
$917$	3.13615e9	0.134309
$918$	1.31306e9	0.0560189
$919$	3.84326e10	1.63341	0.816705	−	0.577056i	$-0.195799\pi$
$919$	3.84326e10	1.63341	0.816705	+	0.577056i	$0.195799\pi$
$920$	0	0
$921$	−2.45400e10	−1.03506
$922$	−8.25354e9	−0.346802
$923$	3.08127e10	1.28980
$924$	−3.64136e9	−0.151849
$925$	0	0
$926$	1.07326e9	0.0444189
$927$	−3.86933e9	−0.159535
$928$	8.84406e9	0.363274
$929$	2.74203e10	1.12206	0.561032	−	0.827794i	$-0.310404\pi$
$929$	2.74203e10	1.12206	0.561032	+	0.827794i	$0.310404\pi$
$930$	0	0
$931$	−4.48228e9	−0.182043
$932$	−3.74022e10	−1.51336
$933$	1.74383e10	0.702939
$934$	−1.61955e8	−0.00650400
$935$	0	0
$936$	−8.61130e9	−0.343244
$937$	2.33959e9	0.0929077	0.0464539	−	0.998920i	$-0.485208\pi$
$937$	2.33959e9	0.0929077	0.0464539	+	0.998920i	$0.485208\pi$
$938$	4.94261e9	0.195545
$939$	2.42083e10	0.954189
$940$	0	0
$941$	−4.04198e10	−1.58136	−0.790680	−	0.612230i	$-0.790273\pi$
$941$	−4.04198e10	−1.58136	−0.790680	+	0.612230i	$0.790273\pi$
$942$	−5.53588e8	−0.0215779
$943$	−6.78449e10	−2.63467
$944$	−4.93294e9	−0.190855
$945$	0	0
$946$	−8.94557e9	−0.343549
$947$	−2.68103e10	−1.02583	−0.512917	−	0.858438i	$-0.671435\pi$
$947$	−2.68103e10	−1.02583	−0.512917	+	0.858438i	$0.671435\pi$
$948$	−5.12983e9	−0.195557
$949$	−6.28533e10	−2.38724
$950$	0	0
$951$	−2.28770e10	−0.862514
$952$	5.59094e9	0.210018
$953$	6.72263e9	0.251602	0.125801	−	0.992056i	$-0.459850\pi$
$953$	6.72263e9	0.251602	0.125801	+	0.992056i	$0.459850\pi$
$954$	−5.57810e8	−0.0208002
$955$	0	0
$956$	3.43297e10	1.27077
$957$	−5.45626e9	−0.201235
$958$	−7.12902e9	−0.261970
$959$	1.24415e10	0.455520
$960$	0	0
$961$	−2.28409e10	−0.830198
$962$	−1.28843e10	−0.466602
$963$	−1.27442e10	−0.459856
$964$	−1.17373e10	−0.421988
$965$	0	0
$966$	3.27307e9	0.116825
$967$	−2.39023e10	−0.850055	−0.425027	−	0.905180i	$-0.639736\pi$
$967$	−2.39023e10	−0.850055	−0.425027	+	0.905180i	$0.639736\pi$
$968$	6.72864e9	0.238431
$969$	−2.00981e10	−0.709614
$970$	0	0
$971$	1.65771e10	0.581088	0.290544	−	0.956862i	$-0.406164\pi$
$971$	1.65771e10	0.581088	0.290544	+	0.956862i	$0.406164\pi$
$972$	−1.66938e9	−0.0583074
$973$	7.84862e9	0.273148
$974$	7.48681e9	0.259621
$975$	0	0
$976$	1.71344e10	0.589921
$977$	2.03177e10	0.697018	0.348509	−	0.937305i	$-0.386688\pi$
$977$	2.03177e10	0.697018	0.348509	+	0.937305i	$0.386688\pi$
$978$	−9.06236e9	−0.309781
$979$	−6.88038e9	−0.234354
$980$	0	0
$981$	3.40939e8	0.0115302
$982$	7.47841e9	0.252011
$983$	−4.78793e10	−1.60772	−0.803861	−	0.594818i	$-0.797224\pi$
$983$	−4.78793e10	−1.60772	−0.803861	+	0.594818i	$0.797224\pi$
$984$	−1.47640e10	−0.493994
$985$	0	0
$986$	3.98891e9	0.132521
$987$	−1.10881e9	−0.0367067
$988$	6.27594e10	2.07028
$989$	−8.02440e10	−2.63770
$990$	0	0
$991$	−1.99787e10	−0.652092	−0.326046	−	0.945354i	$-0.605716\pi$
$991$	−1.99787e10	−0.652092	−0.326046	+	0.945354i	$0.605716\pi$
$992$	1.01095e10	0.328804
$993$	2.00448e10	0.649650
$994$	−2.54861e9	−0.0823098
$995$	0	0
$996$	−1.09913e10	−0.352484
$997$	−4.49634e10	−1.43690	−0.718450	−	0.695579i	$-0.755148\pi$
$997$	−4.49634e10	−1.43690	−0.718450	+	0.695579i	$0.755148\pi$
$998$	2.24235e9	0.0714080
$999$	−5.24575e9	−0.166467

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.8.a.i.1.3		4
5.4	even	2		105.8.a.h.1.2	✓	4
15.14	odd	2		315.8.a.g.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.8.a.h.1.2	✓	4	5.4	even	2
315.8.a.g.1.3		4	15.14	odd	2
525.8.a.i.1.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

\(f(q)\)	\(=\)	\(q+3.41438 q^{2} +27.0000 q^{3} -116.342 q^{4} +92.1883 q^{6} -343.000 q^{7} -834.277 q^{8} +729.000 q^{9} -3379.63 q^{11} -3141.23 q^{12} +14159.0 q^{13} -1171.13 q^{14} +12043.2 q^{16} +19538.0 q^{17} +2489.08 q^{18} -38098.7 q^{19} -9261.00 q^{21} -11539.4 q^{22} -103511. q^{23} -22525.5 q^{24} +48344.1 q^{26} +19683.0 q^{27} +39905.3 q^{28} +59794.5 q^{29} +68349.9 q^{31} +147908. q^{32} -91250.1 q^{33} +66710.3 q^{34} -84813.3 q^{36} -266512. q^{37} -130084. q^{38} +382292. q^{39} +655437. q^{41} -31620.6 q^{42} +775223. q^{43} +393193. q^{44} -353426. q^{46} +119729. q^{47} +325167. q^{48} +117649. q^{49} +527527. q^{51} -1.64728e6 q^{52} -224102. q^{53} +67205.3 q^{54} +286157. q^{56} -1.02867e6 q^{57} +204161. q^{58} -409602. q^{59} +1.42274e6 q^{61} +233373. q^{62} -250047. q^{63} -1.03652e6 q^{64} -311563. q^{66} -4.22037e6 q^{67} -2.27309e6 q^{68} -2.79479e6 q^{69} +2.17619e6 q^{71} -608188. q^{72} -4.43912e6 q^{73} -909973. q^{74} +4.43248e6 q^{76} +1.15921e6 q^{77} +1.30529e6 q^{78} +1.63306e6 q^{79} +531441. q^{81} +2.23791e6 q^{82} +3.49903e6 q^{83} +1.07744e6 q^{84} +2.64691e6 q^{86} +1.61445e6 q^{87} +2.81955e6 q^{88} +2.03583e6 q^{89} -4.85652e6 q^{91} +1.20427e7 q^{92} +1.84545e6 q^{93} +408799. q^{94} +3.99351e6 q^{96} -1.27049e7 q^{97} +401698. q^{98} -2.46375e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 11 q^{2} + 108 q^{3} + 141 q^{4} - 297 q^{6} - 1372 q^{7} - 2133 q^{8} + 2916 q^{9} - 2708 q^{11} + 3807 q^{12} + 2212 q^{13} + 3773 q^{14} - 9599 q^{16} + 17016 q^{17} - 8019 q^{18} + 32668 q^{19}+ \cdots - 1974132 q^{99}+O(q^{100}) \) 4 * q - 11 * q^2 + 108 * q^3 + 141 * q^4 - 297 * q^6 - 1372 * q^7 - 2133 * q^8 + 2916 * q^9 - 2708 * q^11 + 3807 * q^12 + 2212 * q^13 + 3773 * q^14 - 9599 * q^16 + 17016 * q^17 - 8019 * q^18 + 32668 * q^19 - 37044 * q^21 + 7196 * q^22 - 87696 * q^23 - 57591 * q^24 + 67354 * q^26 + 78732 * q^27 - 48363 * q^28 + 200488 * q^29 + 258852 * q^31 - 39445 * q^32 - 73116 * q^33 + 225518 * q^34 + 102789 * q^36 - 666856 * q^37 - 846556 * q^38 + 59724 * q^39 + 833000 * q^41 + 101871 * q^42 - 1051896 * q^43 + 1918236 * q^44 + 1786160 * q^46 - 529824 * q^47 - 259173 * q^48 + 470596 * q^49 + 459432 * q^51 - 211718 * q^52 - 791764 * q^53 - 216513 * q^54 + 731619 * q^56 + 882036 * q^57 - 3500242 * q^58 - 3825904 * q^59 + 212432 * q^61 - 882400 * q^62 - 1000188 * q^63 - 1515063 * q^64 + 194292 * q^66 - 5165272 * q^67 - 7423202 * q^68 - 2367792 * q^69 - 623260 * q^71 - 1554957 * q^72 + 2083228 * q^73 + 1059642 * q^74 + 6990964 * q^76 + 928844 * q^77 + 1818558 * q^78 - 1325016 * q^79 + 2125764 * q^81 - 11798982 * q^82 - 11447616 * q^83 - 1305801 * q^84 + 7525948 * q^86 + 5413176 * q^87 - 19484572 * q^88 - 5805808 * q^89 - 758716 * q^91 + 2716288 * q^92 + 6989004 * q^93 - 15552560 * q^94 - 1065015 * q^96 - 5857516 * q^97 - 1294139 * q^98 - 1974132 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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