LMFDB - Embedded newform 325.6.b.h.274.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 378 x^{16} + 59175 x^{14} + 4956036 x^{12} + 239061247 x^{10} + 6629044458 x^{8} + \cdots + 134659641600 $ x^18 + 378x^16 + 59175x^14 + 4956036x^12 + 239061247x^10 + 6629044458x^8 + 98240243849x^6 + 625586966104x^4 + 584919029904x^2 + 134659641600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.18
Root			$9.17271i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.h.274.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.1727i q^{2} +23.0461i q^{3} -71.4841 q^{4} -234.441 q^{6} +51.0548i q^{7} -401.660i q^{8} -288.123 q^{9} +O(q^{10})$	\(q+10.1727i q^{2} +23.0461i q^{3} -71.4841 q^{4} -234.441 q^{6} +51.0548i q^{7} -401.660i q^{8} -288.123 q^{9} -512.445 q^{11} -1647.43i q^{12} +169.000i q^{13} -519.366 q^{14} +1798.48 q^{16} -978.964i q^{17} -2930.99i q^{18} -1353.56 q^{19} -1176.61 q^{21} -5212.95i q^{22} +11.3026i q^{23} +9256.71 q^{24} -1719.19 q^{26} -1039.91i q^{27} -3649.61i q^{28} +8050.51 q^{29} +759.478 q^{31} +5442.33i q^{32} -11809.9i q^{33} +9958.72 q^{34} +20596.2 q^{36} +12780.7i q^{37} -13769.3i q^{38} -3894.79 q^{39} -18129.3 q^{41} -11969.4i q^{42} +19533.0i q^{43} +36631.6 q^{44} -114.978 q^{46} -22049.0i q^{47} +41448.1i q^{48} +14200.4 q^{49} +22561.3 q^{51} -12080.8i q^{52} +13861.1i q^{53} +10578.7 q^{54} +20506.7 q^{56} -31194.2i q^{57} +81895.5i q^{58} +18164.6 q^{59} +7623.02 q^{61} +7725.95i q^{62} -14710.1i q^{63} +2188.19 q^{64} +120138. q^{66} -3764.44i q^{67} +69980.4i q^{68} -260.481 q^{69} -75797.0 q^{71} +115728. i q^{72} -86605.1i q^{73} -130015. q^{74} +96757.7 q^{76} -26162.8i q^{77} -39620.6i q^{78} -61535.6 q^{79} -46048.0 q^{81} -184425. i q^{82} -46849.0i q^{83} +84109.2 q^{84} -198704. q^{86} +185533. i q^{87} +205829. i q^{88} -1237.02 q^{89} -8628.26 q^{91} -807.955i q^{92} +17503.0i q^{93} +224298. q^{94} -125425. q^{96} -57565.4i q^{97} +144457. i q^{98} +147647. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) $ 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9	$ 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) $ 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9 - 2844 * q^11 + 684 * q^14 - 2122 * q^16 + 816 * q^19 - 7824 * q^21 + 16938 * q^24 - 1690 * q^26 + 17474 * q^29 + 1496 * q^31 + 35578 * q^34 + 1024 * q^36 + 3718 * q^39 - 57352 * q^41 + 61198 * q^44 - 24112 * q^46 + 81814 * q^49 - 62012 * q^51 + 205522 * q^54 - 46004 * q^56 + 176284 * q^59 + 56330 * q^61 + 201690 * q^64 + 85154 * q^66 + 363494 * q^69 - 141124 * q^71 + 271352 * q^74 + 92746 * q^76 + 328146 * q^79 - 139870 * q^81 + 691312 * q^84 - 589840 * q^86 + 505396 * q^89 + 4056 * q^91 + 1003212 * q^94 - 638362 * q^96 + 1515552 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	10.1727i	1.79830i	0.437642	+	0.899149i	$0.355814\pi$
$2$	10.1727i	1.79830i	−0.437642	+	0.899149i	$0.644186\pi$
$3$	23.0461i	1.47841i	0.673481	+	0.739204i	$0.264798\pi$
$3$	23.0461i	1.47841i	−0.673481	+	0.739204i	$0.735202\pi$
$4$	−71.4841	−2.23388
$5$	0	0
$5$	0	0
$6$	−234.441	−2.65862
$7$	51.0548i	0.393814i	0.980422	+	0.196907i	$0.0630898\pi$
$7$	51.0548i	0.393814i	−0.980422	+	0.196907i	$0.936910\pi$
$8$	− 401.660i	− 2.21888i
$9$	−288.123	−1.18569
$10$	0	0
$11$	−512.445	−1.27692	−0.638462	−	0.769653i	$-0.720429\pi$
$11$	−512.445	−1.27692	−0.638462	+	0.769653i	$0.720429\pi$
$12$	− 1647.43i	− 3.30258i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	−519.366	−0.708196
$15$	0	0
$16$	1798.48	1.75633
$17$	− 978.964i	− 0.821570i	−0.911732	−	0.410785i	$-0.865255\pi$
$17$	− 978.964i	− 0.821570i	0.911732	−	0.410785i	$-0.134745\pi$
$18$	− 2930.99i	− 2.13223i
$19$	−1353.56	−0.860186	−0.430093	−	0.902785i	$-0.641519\pi$
$19$	−1353.56	−0.860186	−0.430093	+	0.902785i	$0.641519\pi$
$20$	0	0
$21$	−1176.61	−0.582218
$22$	− 5212.95i	− 2.29629i
$23$	11.3026i	0.00445511i	0.999998	+	0.00222755i	$0.000709053\pi$
$23$	11.3026i	0.00445511i	−0.999998	+	0.00222755i	$0.999291\pi$
$24$	9256.71	3.28041
$25$	0	0
$26$	−1719.19	−0.498758
$27$	− 1039.91i	− 0.274528i
$28$	− 3649.61i	− 0.879733i
$29$	8050.51	1.77758	0.888788	−	0.458318i	$-0.151548\pi$
$29$	8050.51	1.77758	0.888788	+	0.458318i	$0.151548\pi$
$30$	0	0
$31$	759.478	0.141942	0.0709710	−	0.997478i	$-0.477390\pi$
$31$	759.478	0.141942	0.0709710	+	0.997478i	$0.477390\pi$
$32$	5442.33i	0.939529i
$33$	− 11809.9i	− 1.88782i
$34$	9958.72	1.47743
$35$	0	0
$36$	20596.2	2.64869
$37$	12780.7i	1.53480i	0.641170	+	0.767399i	$0.278449\pi$
$37$	12780.7i	1.53480i	−0.641170	+	0.767399i	$0.721551\pi$
$38$	− 13769.3i	− 1.54687i
$39$	−3894.79	−0.410037
$40$	0	0
$41$	−18129.3	−1.68431	−0.842156	−	0.539234i	$-0.818714\pi$
$41$	−18129.3	−1.68431	−0.842156	+	0.539234i	$0.818714\pi$
$42$	− 11969.4i	− 1.04700i
$43$	19533.0i	1.61101i	0.592589	+	0.805505i	$0.298106\pi$
$43$	19533.0i	1.61101i	−0.592589	+	0.805505i	$0.701894\pi$
$44$	36631.6	2.85249
$45$	0	0
$46$	−114.978	−0.00801161
$47$	− 22049.0i	− 1.45594i	−0.685608	−	0.727971i	$-0.740464\pi$
$47$	− 22049.0i	− 1.45594i	0.685608	−	0.727971i	$-0.259536\pi$
$48$	41448.1i	2.59658i
$49$	14200.4	0.844910
$50$	0	0
$51$	22561.3	1.21462
$52$	− 12080.8i	− 0.619566i
$53$	13861.1i	0.677811i	0.940821	+	0.338905i	$0.110057\pi$
$53$	13861.1i	0.677811i	−0.940821	+	0.338905i	$0.889943\pi$
$54$	10578.7	0.493683
$55$	0	0
$56$	20506.7	0.873827
$57$	− 31194.2i	− 1.27171i
$58$	81895.5i	3.19661i
$59$	18164.6	0.679352	0.339676	−	0.940542i	$-0.389683\pi$
$59$	18164.6	0.679352	0.339676	+	0.940542i	$0.389683\pi$
$60$	0	0
$61$	7623.02	0.262303	0.131151	−	0.991362i	$-0.458133\pi$
$61$	7623.02	0.262303	0.131151	+	0.991362i	$0.458133\pi$
$62$	7725.95i	0.255254i
$63$	− 14710.1i	− 0.466942i
$64$	2188.19	0.0667783
$65$	0	0
$66$	120138.	3.39486
$67$	− 3764.44i	− 0.102450i	−0.998687	−	0.0512252i	$-0.983687\pi$
$67$	− 3764.44i	− 0.102450i	0.998687	−	0.0512252i	$-0.0163126\pi$
$68$	69980.4i	1.83529i
$69$	−260.481	−0.00658647
$70$	0	0
$71$	−75797.0	−1.78446	−0.892229	−	0.451583i	$-0.850859\pi$
$71$	−75797.0	−1.78446	−0.892229	+	0.451583i	$0.850859\pi$
$72$	115728.i	2.63091i
$73$	− 86605.1i	− 1.90211i	−0.309015	−	0.951057i	$-0.599999\pi$
$73$	− 86605.1i	− 1.90211i	0.309015	−	0.951057i	$-0.400001\pi$
$74$	−130015.	−2.76003
$75$	0	0
$76$	96757.7	1.92155
$77$	− 26162.8i	− 0.502871i
$78$	− 39620.6i	− 0.737368i
$79$	−61535.6	−1.10932	−0.554662	−	0.832076i	$-0.687153\pi$
$79$	−61535.6	−1.10932	−0.554662	+	0.832076i	$0.687153\pi$
$80$	0	0
$81$	−46048.0	−0.779827
$82$	− 184425.i	− 3.02890i
$83$	− 46849.0i	− 0.746458i	−0.927739	−	0.373229i	$-0.878251\pi$
$83$	− 46849.0i	− 0.746458i	0.927739	−	0.373229i	$-0.121749\pi$
$84$	84109.2	1.30060
$85$	0	0
$86$	−198704.	−2.89708
$87$	185533.i	2.62798i
$88$	205829.i	2.83334i
$89$	−1237.02	−0.0165540	−0.00827698	−	0.999966i	$-0.502635\pi$
$89$	−1237.02	−0.0165540	−0.00827698	+	0.999966i	$0.502635\pi$
$90$	0	0
$91$	−8628.26	−0.109224
$92$	− 807.955i	− 0.00995217i
$93$	17503.0i	0.209848i
$94$	224298.	2.61822
$95$	0	0
$96$	−125425.	−1.38901
$97$	− 57565.4i	− 0.621201i	−0.950541	−	0.310600i	$-0.899470\pi$
$97$	− 57565.4i	− 0.621201i	0.950541	−	0.310600i	$-0.100530\pi$
$98$	144457.i	1.51940i
$99$	147647.	1.51404
$100$	0	0
$101$	−13535.7	−0.132031	−0.0660155	−	0.997819i	$-0.521029\pi$
$101$	−13535.7	−0.132031	−0.0660155	+	0.997819i	$0.521029\pi$
$102$	229510.i	2.18424i
$103$	− 99640.9i	− 0.925432i	−0.886507	−	0.462716i	$-0.846875\pi$
$103$	− 99640.9i	− 0.925432i	0.886507	−	0.462716i	$-0.153125\pi$
$104$	67880.6	0.615407
$105$	0	0
$106$	−141005.	−1.21891
$107$	3447.88i	0.0291134i	0.999894	+	0.0145567i	$0.00463371\pi$
$107$	3447.88i	0.0291134i	−0.999894	+	0.0145567i	$0.995366\pi$
$108$	74337.0i	0.613261i
$109$	−54235.5	−0.437238	−0.218619	−	0.975810i	$-0.570155\pi$
$109$	−54235.5	−0.437238	−0.218619	+	0.975810i	$0.570155\pi$
$110$	0	0
$111$	−294546.	−2.26906
$112$	91821.3i	0.691669i
$113$	− 103423.i	− 0.761937i	−0.924588	−	0.380969i	$-0.875591\pi$
$113$	− 103423.i	− 0.761937i	0.924588	−	0.380969i	$-0.124409\pi$
$114$	317330.	2.28691
$115$	0	0
$116$	−575483.	−3.97089
$117$	− 48692.8i	− 0.328852i
$118$	184783.i	1.22168i
$119$	49980.8	0.323546
$120$	0	0
$121$	101549.	0.630537
$122$	77546.8i	0.471698i
$123$	− 417811.i	− 2.49010i
$124$	−54290.6	−0.317081
$125$	0	0
$126$	149641.	0.839702
$127$	86597.6i	0.476427i	0.971213	+	0.238214i	$0.0765618\pi$
$127$	86597.6i	0.476427i	−0.971213	+	0.238214i	$0.923438\pi$
$128$	196415.i	1.05962i
$129$	−450160.	−2.38173
$130$	0	0
$131$	212824.	1.08354	0.541768	−	0.840528i	$-0.317755\pi$
$131$	212824.	1.08354	0.541768	+	0.840528i	$0.317755\pi$
$132$	844217.i	4.21715i
$133$	− 69105.6i	− 0.338754i
$134$	38294.6	0.184236
$135$	0	0
$136$	−393211.	−1.82297
$137$	392531.i	1.78678i	0.449277	+	0.893392i	$0.351681\pi$
$137$	392531.i	1.78678i	−0.449277	+	0.893392i	$0.648319\pi$
$138$	− 2649.79i	− 0.0118444i
$139$	−64771.0	−0.284344	−0.142172	−	0.989842i	$-0.545409\pi$
$139$	−64771.0	−0.284344	−0.142172	+	0.989842i	$0.545409\pi$
$140$	0	0
$141$	508143.	2.15248
$142$	− 771061.i	− 3.20899i
$143$	− 86603.2i	− 0.354155i
$144$	−518185.	−2.08247
$145$	0	0
$146$	881009.	3.42057
$147$	327264.i	1.24912i
$148$	− 913619.i	− 3.42855i
$149$	277263.	1.02312	0.511560	−	0.859248i	$-0.329068\pi$
$149$	277263.	1.02312	0.511560	+	0.859248i	$0.329068\pi$
$150$	0	0
$151$	−56639.1	−0.202150	−0.101075	−	0.994879i	$-0.532228\pi$
$151$	−56639.1	−0.202150	−0.101075	+	0.994879i	$0.532228\pi$
$152$	543670.i	1.90865i
$153$	282062.i	0.974128i
$154$	266146.	0.904313
$155$	0	0
$156$	278416.	0.915972
$157$	229864.i	0.744257i	0.928181	+	0.372128i	$0.121372\pi$
$157$	229864.i	0.744257i	−0.928181	+	0.372128i	$0.878628\pi$
$158$	− 625984.i	− 1.99490i
$159$	−319445.	−1.00208
$160$	0	0
$161$	−577.051	−0.00175449
$162$	− 468433.i	− 1.40236i
$163$	400206.i	1.17982i	0.807470	+	0.589908i	$0.200836\pi$
$163$	400206.i	1.17982i	−0.807470	+	0.589908i	$0.799164\pi$
$164$	1.29596e6	3.76255
$165$	0	0
$166$	476582.	1.34235
$167$	− 276559.i	− 0.767355i	−0.923467	−	0.383677i	$-0.874657\pi$
$167$	− 276559.i	− 0.767355i	0.923467	−	0.383677i	$-0.125343\pi$
$168$	472599.i	1.29187i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	389991.	1.01991
$172$	− 1.39630e6i	− 3.59880i
$173$	− 323519.i	− 0.821834i	−0.911673	−	0.410917i	$-0.865209\pi$
$173$	− 323519.i	− 0.821834i	0.911673	−	0.410917i	$-0.134791\pi$
$174$	−1.88737e6	−4.72590
$175$	0	0
$176$	−921624.	−2.24270
$177$	418623.i	1.00436i
$178$	− 12583.9i	− 0.0297690i
$179$	757066.	1.76604	0.883021	−	0.469333i	$-0.155506\pi$
$179$	757066.	1.76604	0.883021	+	0.469333i	$0.155506\pi$
$180$	0	0
$181$	640530.	1.45326	0.726629	−	0.687030i	$-0.241086\pi$
$181$	640530.	1.45326	0.726629	+	0.687030i	$0.241086\pi$
$182$	− 87772.8i	− 0.196418i
$183$	175681.i	0.387790i
$184$	4539.80	0.00988535
$185$	0	0
$186$	−178053.	−0.377370
$187$	501665.i	1.04908i
$188$	1.57615e6i	3.25239i
$189$	53092.4	0.108113
$190$	0	0
$191$	−611932.	−1.21372	−0.606862	−	0.794807i	$-0.707572\pi$
$191$	−611932.	−1.21372	−0.606862	+	0.794807i	$0.707572\pi$
$192$	50429.3i	0.0987256i
$193$	− 134676.i	− 0.260254i	−0.991497	−	0.130127i	$-0.958461\pi$
$193$	− 134676.i	− 0.260254i	0.991497	−	0.130127i	$-0.0415386\pi$
$194$	585596.	1.11710
$195$	0	0
$196$	−1.01510e6	−1.88743
$197$	435184.i	0.798927i	0.916749	+	0.399463i	$0.130804\pi$
$197$	435184.i	0.798927i	−0.916749	+	0.399463i	$0.869196\pi$
$198$	1.50197e6i	2.72269i
$199$	−179852.	−0.321946	−0.160973	−	0.986959i	$-0.551463\pi$
$199$	−179852.	−0.321946	−0.160973	+	0.986959i	$0.551463\pi$
$200$	0	0
$201$	86755.8	0.151464
$202$	− 137694.i	− 0.237431i
$203$	411017.i	0.700035i
$204$	−1.61277e6	−2.71330
$205$	0	0
$206$	1.01362e6	1.66420
$207$	− 3256.54i	− 0.00528238i
$208$	303944.i	0.487119i
$209$	693623.	1.09839
$210$	0	0
$211$	−1.03041e6	−1.59333	−0.796663	−	0.604424i	$-0.793403\pi$
$211$	−1.03041e6	−1.59333	−0.796663	+	0.604424i	$0.793403\pi$
$212$	− 990849.i	− 1.51415i
$213$	− 1.74683e6i	− 2.63816i
$214$	−35074.3	−0.0523546
$215$	0	0
$216$	−417690.	−0.609144
$217$	38775.0i	0.0558988i
$218$	− 551722.i	− 0.786284i
$219$	1.99591e6	2.81210
$220$	0	0
$221$	165445.	0.227862
$222$	− 2.99633e6i	− 4.08044i
$223$	− 1.04356e6i	− 1.40525i	−0.711559	−	0.702626i	$-0.752011\pi$
$223$	− 1.04356e6i	− 1.40525i	0.711559	−	0.702626i	$-0.247989\pi$
$224$	−277857.	−0.370000
$225$	0	0
$226$	1.05209e6	1.37019
$227$	− 1.04811e6i	− 1.35003i	−0.737803	−	0.675016i	$-0.764137\pi$
$227$	− 1.04811e6i	− 1.35003i	0.737803	−	0.675016i	$-0.235863\pi$
$228$	2.22989e6i	2.84084i
$229$	−718967.	−0.905984	−0.452992	−	0.891515i	$-0.649643\pi$
$229$	−718967.	−0.905984	−0.452992	+	0.891515i	$0.649643\pi$
$230$	0	0
$231$	602950.	0.743449
$232$	− 3.23357e6i	− 3.94423i
$233$	− 714611.i	− 0.862343i	−0.902270	−	0.431172i	$-0.858100\pi$
$233$	− 714611.i	− 0.862343i	0.902270	−	0.431172i	$-0.141900\pi$
$234$	495338.	0.591373
$235$	0	0
$236$	−1.29848e6	−1.51759
$237$	− 1.41816e6i	− 1.64003i
$238$	508441.i	0.581832i
$239$	−114365.	−0.129508	−0.0647541	−	0.997901i	$-0.520626\pi$
$239$	−114365.	−0.129508	−0.0647541	+	0.997901i	$0.520626\pi$
$240$	0	0
$241$	1.01606e6	1.12688	0.563440	−	0.826157i	$-0.309478\pi$
$241$	1.01606e6	1.12688	0.563440	+	0.826157i	$0.309478\pi$
$242$	1.03303e6i	1.13389i
$243$	− 1.31393e6i	− 1.42743i
$244$	−544925.	−0.585952
$245$	0	0
$246$	4.25027e6	4.47794
$247$	− 228751.i	− 0.238573i
$248$	− 305052.i	− 0.314952i
$249$	1.07969e6	1.10357
$250$	0	0
$251$	−1.13208e6	−1.13421	−0.567105	−	0.823645i	$-0.691937\pi$
$251$	−1.13208e6	−1.13421	−0.567105	+	0.823645i	$0.691937\pi$
$252$	1.05154e6i	1.04309i
$253$	− 5791.95i	− 0.00568884i
$254$	−880933.	−0.856758
$255$	0	0
$256$	−1.92805e6	−1.83873
$257$	− 357876.i	− 0.337987i	−0.985617	−	0.168993i	$-0.945948\pi$
$257$	− 357876.i	− 0.337987i	0.985617	−	0.168993i	$-0.0540516\pi$
$258$	− 4.57935e6i	− 4.28306i
$259$	−652518.	−0.604426
$260$	0	0
$261$	−2.31954e6	−2.10766
$262$	2.16500e6i	1.94852i
$263$	305594.i	0.272431i	0.990679	+	0.136215i	$0.0434939\pi$
$263$	305594.i	0.272431i	−0.990679	+	0.136215i	$0.956506\pi$
$264$	−4.74355e6	−4.18884
$265$	0	0
$266$	702991.	0.609180
$267$	− 28508.5i	− 0.0244735i
$268$	269098.i	0.228862i
$269$	667878.	0.562751	0.281375	−	0.959598i	$-0.409209\pi$
$269$	667878.	0.562751	0.281375	+	0.959598i	$0.409209\pi$
$270$	0	0
$271$	−1.05893e6	−0.875875	−0.437937	−	0.899005i	$-0.644291\pi$
$271$	−1.05893e6	−0.875875	−0.437937	+	0.899005i	$0.644291\pi$
$272$	− 1.76065e6i	− 1.44295i
$273$	− 198848.i	− 0.161478i
$274$	−3.99310e6	−3.21317
$275$	0	0
$276$	18620.2	0.0147134
$277$	− 1.91406e6i	− 1.49884i	−0.662095	−	0.749420i	$-0.730333\pi$
$277$	− 1.91406e6i	− 1.49884i	0.662095	−	0.749420i	$-0.269667\pi$
$278$	− 658896.i	− 0.511335i
$279$	−218823.	−0.168299
$280$	0	0
$281$	1.08128e6	0.816908	0.408454	−	0.912779i	$-0.366068\pi$
$281$	1.08128e6	0.816908	0.408454	+	0.912779i	$0.366068\pi$
$282$	5.16919e6i	3.87079i
$283$	− 859352.i	− 0.637830i	−0.947783	−	0.318915i	$-0.896682\pi$
$283$	− 859352.i	− 0.637830i	0.947783	−	0.318915i	$-0.103318\pi$
$284$	5.41828e6	3.98626
$285$	0	0
$286$	880989.	0.636877
$287$	− 925590.i	− 0.663306i
$288$	− 1.56806e6i	− 1.11399i
$289$	461486.	0.325023
$290$	0	0
$291$	1.32666e6	0.918389
$292$	6.19089e6i	4.24909i
$293$	2.36378e6i	1.60856i	0.594250	+	0.804280i	$0.297449\pi$
$293$	2.36378e6i	1.60856i	−0.594250	+	0.804280i	$0.702551\pi$
$294$	−3.32916e6	−2.24630
$295$	0	0
$296$	5.13351e6	3.40553
$297$	532896.i	0.350551i
$298$	2.82052e6i	1.83988i
$299$	−1910.14	−0.00123562
$300$	0	0
$301$	−997255.	−0.634439
$302$	− 576173.i	− 0.363526i
$303$	− 311944.i	− 0.195196i
$304$	−2.43435e6	−1.51077
$305$	0	0
$306$	−2.86934e6	−1.75177
$307$	− 1.61376e6i	− 0.977220i	−0.872502	−	0.488610i	$-0.837504\pi$
$307$	− 1.61376e6i	− 0.977220i	0.872502	−	0.488610i	$-0.162496\pi$
$308$	1.87022e6i	1.12335i
$309$	2.29633e6	1.36817
$310$	0	0
$311$	776229.	0.455081	0.227541	−	0.973769i	$-0.426932\pi$
$311$	776229.	0.455081	0.227541	+	0.973769i	$0.426932\pi$
$312$	1.56438e6i	0.909823i
$313$	− 656850.i	− 0.378970i	−0.981884	−	0.189485i	$-0.939318\pi$
$313$	− 656850.i	− 0.378970i	0.981884	−	0.189485i	$-0.0606819\pi$
$314$	−2.33835e6	−1.33840
$315$	0	0
$316$	4.39882e6	2.47810
$317$	1.41713e6i	0.792069i	0.918236	+	0.396034i	$0.129614\pi$
$317$	1.41713e6i	0.792069i	−0.918236	+	0.396034i	$0.870386\pi$
$318$	− 3.24962e6i	− 1.80204i
$319$	−4.12544e6	−2.26983
$320$	0	0
$321$	−79460.3	−0.0430415
$322$	− 5870.18i	− 0.00315509i
$323$	1.32508e6i	0.706703i
$324$	3.29170e6	1.74204
$325$	0	0
$326$	−4.07118e6	−2.12166
$327$	− 1.24992e6i	− 0.646416i
$328$	7.28184e6i	3.73729i
$329$	1.12571e6	0.573371
$330$	0	0
$331$	−2.12709e6	−1.06713	−0.533564	−	0.845760i	$-0.679148\pi$
$331$	−2.12709e6	−1.06713	−0.533564	+	0.845760i	$0.679148\pi$
$332$	3.34896e6i	1.66750i
$333$	− 3.68242e6i	− 1.81980i
$334$	2.81335e6	1.37993
$335$	0	0
$336$	−2.11612e6	−1.02257
$337$	483390.i	0.231859i	0.993257	+	0.115929i	$0.0369846\pi$
$337$	483390.i	0.231859i	−0.993257	+	0.115929i	$0.963015\pi$
$338$	− 290543.i	− 0.138331i
$339$	2.38349e6	1.12645
$340$	0	0
$341$	−389190.	−0.181249
$342$	3.96726e6i	1.83411i
$343$	1.58308e6i	0.726552i
$344$	7.84564e6	3.57464
$345$	0	0
$346$	3.29106e6	1.47790
$347$	1.95372e6i	0.871041i	0.900179	+	0.435521i	$0.143436\pi$
$347$	1.95372e6i	0.871041i	−0.900179	+	0.435521i	$0.856564\pi$
$348$	− 1.32626e7i	− 5.87060i
$349$	−471784.	−0.207338	−0.103669	−	0.994612i	$-0.533058\pi$
$349$	−471784.	−0.207338	−0.103669	+	0.994612i	$0.533058\pi$
$350$	0	0
$351$	175745.	0.0761403
$352$	− 2.78890e6i	− 1.19971i
$353$	− 4.45914e6i	− 1.90465i	−0.305090	−	0.952324i	$-0.598686\pi$
$353$	− 4.45914e6i	− 1.90465i	0.305090	−	0.952324i	$-0.401314\pi$
$354$	−4.25853e6	−1.80614
$355$	0	0
$356$	88427.3	0.0369795
$357$	1.15186e6i	0.478333i
$358$	7.70141e6i	3.17587i
$359$	22798.5	0.00933619	0.00466810	−	0.999989i	$-0.498514\pi$
$359$	22798.5	0.00933619	0.00466810	+	0.999989i	$0.498514\pi$
$360$	0	0
$361$	−643984.	−0.260080
$362$	6.51593e6i	2.61339i
$363$	2.34030e6i	0.932192i
$364$	616784.	0.243994
$365$	0	0
$366$	−1.78715e6	−0.697363
$367$	1.45349e6i	0.563307i	0.959516	+	0.281654i	$0.0908829\pi$
$367$	1.45349e6i	0.563307i	−0.959516	+	0.281654i	$0.909117\pi$
$368$	20327.5i	0.00782465i
$369$	5.22348e6	1.99707
$370$	0	0
$371$	−707676.	−0.266932
$372$	− 1.25119e6i	− 0.468775i
$373$	− 1.98547e6i	− 0.738910i	−0.929249	−	0.369455i	$-0.879544\pi$
$373$	− 1.98547e6i	− 0.738910i	0.929249	−	0.369455i	$-0.120456\pi$
$374$	−5.10329e6	−1.88656
$375$	0	0
$376$	−8.85620e6	−3.23056
$377$	1.36054e6i	0.493011i
$378$	540094.i	0.194419i
$379$	−1.09576e6	−0.391847	−0.195924	−	0.980619i	$-0.562770\pi$
$379$	−1.09576e6	−0.391847	−0.195924	+	0.980619i	$0.562770\pi$
$380$	0	0
$381$	−1.99574e6	−0.704354
$382$	− 6.22501e6i	− 2.18264i
$383$	− 2.86567e6i	− 0.998225i	−0.866537	−	0.499113i	$-0.833659\pi$
$383$	− 2.86567e6i	− 0.998225i	0.866537	−	0.499113i	$-0.166341\pi$
$384$	−4.52659e6	−1.56655
$385$	0	0
$386$	1.37002e6	0.468015
$387$	− 5.62791e6i	− 1.91016i
$388$	4.11501e6i	1.38769i
$389$	−1.42325e6	−0.476878	−0.238439	−	0.971158i	$-0.576636\pi$
$389$	−1.42325e6	−0.476878	−0.238439	+	0.971158i	$0.576636\pi$
$390$	0	0
$391$	11064.8	0.00366018
$392$	− 5.70374e6i	− 1.87476i
$393$	4.90477e6i	1.60191i
$394$	−4.42700e6	−1.43671
$395$	0	0
$396$	−1.05544e7	−3.38218
$397$	− 6.03873e6i	− 1.92296i	−0.274882	−	0.961478i	$-0.588639\pi$
$397$	− 6.03873e6i	− 1.92296i	0.274882	−	0.961478i	$-0.411361\pi$
$398$	− 1.82958e6i	− 0.578955i
$399$	1.59261e6	0.500816
$400$	0	0
$401$	5.69168e6	1.76758	0.883791	−	0.467881i	$-0.154983\pi$
$401$	5.69168e6	1.76758	0.883791	+	0.467881i	$0.154983\pi$
$402$	882541.i	0.272377i
$403$	128352.i	0.0393676i
$404$	967584.	0.294941
$405$	0	0
$406$	−4.18116e6	−1.25887
$407$	− 6.54942e6i	− 1.95982i
$408$	− 9.06198e6i	− 2.69509i
$409$	972884.	0.287576	0.143788	−	0.989609i	$-0.454072\pi$
$409$	972884.	0.287576	0.143788	+	0.989609i	$0.454072\pi$
$410$	0	0
$411$	−9.04630e6	−2.64160
$412$	7.12274e6i	2.06730i
$413$	927388.i	0.267539i
$414$	33127.8	0.00949930
$415$	0	0
$416$	−919755.	−0.260579
$417$	− 1.49272e6i	− 0.420376i
$418$	7.05603e6i	1.97524i
$419$	−3.58406e6	−0.997334	−0.498667	−	0.866794i	$-0.666177\pi$
$419$	−3.58406e6	−0.997334	−0.498667	+	0.866794i	$0.666177\pi$
$420$	0	0
$421$	1.21460e6	0.333986	0.166993	−	0.985958i	$-0.446594\pi$
$421$	1.21460e6	0.333986	0.166993	+	0.985958i	$0.446594\pi$
$422$	− 1.04821e7i	− 2.86528i
$423$	6.35282e6i	1.72630i
$424$	5.56746e6	1.50398
$425$	0	0
$426$	1.77700e7	4.74419
$427$	389192.i	0.103299i
$428$	− 246469.i	− 0.0650358i
$429$	1.99587e6	0.523586
$430$	0	0
$431$	3.18545e6	0.825996	0.412998	−	0.910732i	$-0.364482\pi$
$431$	3.18545e6	0.825996	0.412998	+	0.910732i	$0.364482\pi$
$432$	− 1.87026e6i	− 0.482162i
$433$	5.64608e6i	1.44720i	0.690221	+	0.723598i	$0.257513\pi$
$433$	5.64608e6i	1.44720i	−0.690221	+	0.723598i	$0.742487\pi$
$434$	−394447.	−0.100523
$435$	0	0
$436$	3.87698e6	0.976735
$437$	− 15298.7i	− 0.00383222i
$438$	2.03038e7i	5.05700i
$439$	−7.65369e6	−1.89544	−0.947719	−	0.319108i	$-0.896617\pi$
$439$	−7.65369e6	−1.89544	−0.947719	+	0.319108i	$0.896617\pi$
$440$	0	0
$441$	−4.09146e6	−1.00180
$442$	1.68302e6i	0.409765i
$443$	5.65650e6i	1.36943i	0.728813	+	0.684713i	$0.240073\pi$
$443$	5.65650e6i	1.36943i	−0.728813	+	0.684713i	$0.759927\pi$
$444$	2.10554e7	5.06880
$445$	0	0
$446$	1.06158e7	2.52706
$447$	6.38984e6i	1.51259i
$448$	111718.i	0.0262983i
$449$	1.93752e6	0.453556	0.226778	−	0.973947i	$-0.427181\pi$
$449$	1.93752e6	0.453556	0.226778	+	0.973947i	$0.427181\pi$
$450$	0	0
$451$	9.29029e6	2.15074
$452$	7.39307e6i	1.70208i
$453$	− 1.30531e6i	− 0.298860i
$454$	1.06622e7	2.42776
$455$	0	0
$456$	−1.25295e7	−2.82176
$457$	− 3.22536e6i	− 0.722417i	−0.932485	−	0.361208i	$-0.882364\pi$
$457$	− 3.22536e6i	− 0.722417i	0.932485	−	0.361208i	$-0.117636\pi$
$458$	− 7.31385e6i	− 1.62923i
$459$	−1.01803e6	−0.225544
$460$	0	0
$461$	−4.53894e6	−0.994723	−0.497362	−	0.867543i	$-0.665698\pi$
$461$	−4.53894e6	−0.994723	−0.497362	+	0.867543i	$0.665698\pi$
$462$	6.13364e6i	1.33694i
$463$	4.40536e6i	0.955056i	0.878617	+	0.477528i	$0.158467\pi$
$463$	4.40536e6i	0.955056i	−0.878617	+	0.477528i	$0.841533\pi$
$464$	1.44787e7	3.12202
$465$	0	0
$466$	7.26954e6	1.55075
$467$	7.89872e6i	1.67596i	0.545698	+	0.837982i	$0.316265\pi$
$467$	7.89872e6i	1.67596i	−0.545698	+	0.837982i	$0.683735\pi$
$468$	3.48076e6i	0.734614i
$469$	192193.	0.0403464
$470$	0	0
$471$	−5.29748e6	−1.10032
$472$	− 7.29599e6i	− 1.50740i
$473$	− 1.00096e7i	− 2.05714i
$474$	1.44265e7	2.94927
$475$	0	0
$476$	−3.57283e6	−0.722762
$477$	− 3.99370e6i	− 0.803674i
$478$	− 1.16340e6i	− 0.232895i
$479$	−1.62018e6	−0.322644	−0.161322	−	0.986902i	$-0.551576\pi$
$479$	−1.62018e6	−0.322644	−0.161322	+	0.986902i	$0.551576\pi$
$480$	0	0
$481$	−2.15994e6	−0.425676
$482$	1.03361e7i	2.02647i
$483$	− 13298.8i	− 0.00259385i
$484$	−7.25911e6	−1.40854
$485$	0	0
$486$	1.33662e7	2.56695
$487$	− 4.09572e6i	− 0.782543i	−0.920275	−	0.391272i	$-0.872035\pi$
$487$	− 4.09572e6i	− 0.782543i	0.920275	−	0.391272i	$-0.127965\pi$
$488$	− 3.06187e6i	− 0.582018i
$489$	−9.22319e6	−1.74425
$490$	0	0
$491$	−7.09337e6	−1.32785	−0.663925	−	0.747799i	$-0.731111\pi$
$491$	−7.09337e6	−1.32785	−0.663925	+	0.747799i	$0.731111\pi$
$492$	2.98668e7i	5.56258i
$493$	− 7.88116e6i	− 1.46040i
$494$	2.32702e6	0.429025
$495$	0	0
$496$	1.36591e6	0.249297
$497$	− 3.86980e6i	− 0.702745i
$498$	1.09834e7i	1.98455i
$499$	1.60514e6	0.288577	0.144289	−	0.989536i	$-0.453911\pi$
$499$	1.60514e6	0.288577	0.144289	+	0.989536i	$0.453911\pi$
$500$	0	0
$501$	6.37360e6	1.13446
$502$	− 1.15164e7i	− 2.03965i
$503$	5.32213e6i	0.937920i	0.883219	+	0.468960i	$0.155371\pi$
$503$	5.32213e6i	0.937920i	−0.883219	+	0.468960i	$0.844629\pi$
$504$	−5.90845e6	−1.03609
$505$	0	0
$506$	58919.9	0.0102302
$507$	− 658220.i	− 0.113724i
$508$	− 6.19035e6i	− 1.06428i
$509$	−2.38983e6	−0.408857	−0.204429	−	0.978881i	$-0.565534\pi$
$509$	−2.38983e6	−0.408857	−0.204429	+	0.978881i	$0.565534\pi$
$510$	0	0
$511$	4.42161e6	0.749080
$512$	− 1.33282e7i	− 2.24697i
$513$	1.40758e6i	0.236145i
$514$	3.64057e6	0.607801
$515$	0	0
$516$	3.21793e7	5.32050
$517$	1.12989e7i	1.85913i
$518$	− 6.63787e6i	− 1.08694i
$519$	7.45584e6	1.21501
$520$	0	0
$521$	−970922.	−0.156708	−0.0783538	−	0.996926i	$-0.524966\pi$
$521$	−970922.	−0.156708	−0.0783538	+	0.996926i	$0.524966\pi$
$522$	− 2.35960e7i	− 3.79020i
$523$	7.93187e6i	1.26801i	0.773330	+	0.634003i	$0.218590\pi$
$523$	7.93187e6i	1.26801i	−0.773330	+	0.634003i	$0.781410\pi$
$524$	−1.52136e7	−2.42049
$525$	0	0
$526$	−3.10872e6	−0.489912
$527$	− 743501.i	− 0.116615i
$528$	− 2.12398e7i	− 3.31563i
$529$	6.43622e6	0.999980
$530$	0	0
$531$	−5.23363e6	−0.805502
$532$	4.93995e6i	0.756734i
$533$	− 3.06386e6i	− 0.467144i
$534$	290009.	0.0440107
$535$	0	0
$536$	−1.51203e6	−0.227325
$537$	1.74474e7i	2.61093i
$538$	6.79413e6i	1.01199i
$539$	−7.27692e6	−1.07889
$540$	0	0
$541$	−6.96364e6	−1.02292	−0.511462	−	0.859306i	$-0.670896\pi$
$541$	−6.96364e6	−1.02292	−0.511462	+	0.859306i	$0.670896\pi$
$542$	− 1.07721e7i	− 1.57508i
$543$	1.47617e7i	2.14851i
$544$	5.32785e6	0.771889
$545$	0	0
$546$	2.02282e6	0.290386
$547$	− 1.01847e7i	− 1.45539i	−0.685903	−	0.727693i	$-0.740593\pi$
$547$	− 1.01847e7i	− 1.45539i	0.685903	−	0.727693i	$-0.259407\pi$
$548$	− 2.80597e7i	− 3.99146i
$549$	−2.19637e6	−0.311010
$550$	0	0
$551$	−1.08968e7	−1.52905
$552$	104625.i	0.0146146i
$553$	− 3.14169e6i	− 0.436868i
$554$	1.94711e7	2.69536
$555$	0	0
$556$	4.63009e6	0.635189
$557$	− 1.17433e7i	− 1.60381i	−0.597454	−	0.801903i	$-0.703821\pi$
$557$	− 1.17433e7i	− 1.60381i	0.597454	−	0.801903i	$-0.296179\pi$
$558$	− 2.22602e6i	− 0.302652i
$559$	−3.30108e6	−0.446814
$560$	0	0
$561$	−1.15614e7	−1.55097
$562$	1.09996e7i	1.46904i
$563$	− 1.14861e7i	− 1.52722i	−0.645677	−	0.763611i	$-0.723425\pi$
$563$	− 1.14861e7i	− 1.52722i	0.645677	−	0.763611i	$-0.276575\pi$
$564$	−3.63241e7	−4.80837
$565$	0	0
$566$	8.74194e6	1.14701
$567$	− 2.35097e6i	− 0.307107i
$568$	3.04447e7i	3.95950i
$569$	−1.51914e7	−1.96706	−0.983532	−	0.180734i	$-0.942153\pi$
$569$	−1.51914e7	−1.96706	−0.983532	+	0.180734i	$0.942153\pi$
$570$	0	0
$571$	−7.26504e6	−0.932498	−0.466249	−	0.884654i	$-0.654395\pi$
$571$	−7.26504e6	−0.932498	−0.466249	+	0.884654i	$0.654395\pi$
$572$	6.19075e6i	0.791140i
$573$	− 1.41027e7i	− 1.79438i
$574$	9.41576e6	1.19282
$575$	0	0
$576$	−630468.	−0.0791784
$577$	− 5.66789e6i	− 0.708731i	−0.935107	−	0.354366i	$-0.884697\pi$
$577$	− 5.66789e6i	− 0.708731i	0.935107	−	0.354366i	$-0.115303\pi$
$578$	4.69457e6i	0.584489i
$579$	3.10377e6	0.384762
$580$	0	0
$581$	2.39187e6	0.293966
$582$	1.34957e7i	1.65154i
$583$	− 7.10305e6i	− 0.865513i
$584$	−3.47859e7	−4.22056
$585$	0	0
$586$	−2.40460e7	−2.89267
$587$	− 1.09050e6i	− 0.130627i	−0.997865	−	0.0653133i	$-0.979195\pi$
$587$	− 1.09050e6i	− 0.130627i	0.997865	−	0.0653133i	$-0.0208047\pi$
$588$	− 2.33942e7i	− 2.79039i
$589$	−1.02800e6	−0.122096
$590$	0	0
$591$	−1.00293e7	−1.18114
$592$	2.29859e7i	2.69562i
$593$	4.95413e6i	0.578536i	0.957248	+	0.289268i	$0.0934119\pi$
$593$	4.95413e6i	0.578536i	−0.957248	+	0.289268i	$0.906588\pi$
$594$	−5.42100e6	−0.630396
$595$	0	0
$596$	−1.98199e7	−2.28553
$597$	− 4.14489e6i	− 0.475968i
$598$	− 19431.3i	− 0.00222202i
$599$	−6.54475e6	−0.745292	−0.372646	−	0.927974i	$-0.621549\pi$
$599$	−6.54475e6	−0.745292	−0.372646	+	0.927974i	$0.621549\pi$
$600$	0	0
$601$	−2.16617e6	−0.244628	−0.122314	−	0.992491i	$-0.539031\pi$
$601$	−2.16617e6	−0.244628	−0.122314	+	0.992491i	$0.539031\pi$
$602$	− 1.01448e7i	− 1.14091i
$603$	1.08462e6i	0.121475i
$604$	4.04879e6	0.451578
$605$	0	0
$606$	3.17332e6	0.351020
$607$	986604.i	0.108685i	0.998522	+	0.0543427i	$0.0173063\pi$
$607$	986604.i	0.108685i	−0.998522	+	0.0543427i	$0.982694\pi$
$608$	− 7.36651e6i	− 0.808170i
$609$	−9.47235e6	−1.03494
$610$	0	0
$611$	3.72628e6	0.403805
$612$	− 2.01630e7i	− 2.17608i
$613$	1.50041e7i	1.61272i	0.591425	+	0.806360i	$0.298565\pi$
$613$	1.50041e7i	1.61272i	−0.591425	+	0.806360i	$0.701435\pi$
$614$	1.64163e7	1.75733
$615$	0	0
$616$	−1.05085e7	−1.11581
$617$	− 1.27376e6i	− 0.134703i	−0.997729	−	0.0673514i	$-0.978545\pi$
$617$	− 1.27376e6i	− 0.134703i	0.997729	−	0.0673514i	$-0.0214548\pi$
$618$	2.33599e7i	2.46037i
$619$	2.59865e6	0.272597	0.136299	−	0.990668i	$-0.456479\pi$
$619$	2.59865e6	0.272597	0.136299	+	0.990668i	$0.456479\pi$
$620$	0	0
$621$	11753.7	0.00122305
$622$	7.89635e6i	0.818372i
$623$	− 63155.8i	− 0.00651919i
$624$	−7.00472e6	−0.720161
$625$	0	0
$626$	6.68195e6	0.681502
$627$	1.59853e7i	1.62387i
$628$	− 1.64317e7i	− 1.66258i
$629$	1.25119e7	1.26094
$630$	0	0
$631$	−1.92424e7	−1.92392	−0.961959	−	0.273194i	$-0.911920\pi$
$631$	−1.92424e7	−1.92392	−0.961959	+	0.273194i	$0.911920\pi$
$632$	2.47164e7i	2.46146i
$633$	− 2.37470e7i	− 2.35559i
$634$	−1.44161e7	−1.42438
$635$	0	0
$636$	2.28352e7	2.23853
$637$	2.39987e6i	0.234336i
$638$	− 4.19669e7i	− 4.08184i
$639$	2.18389e7	2.11582
$640$	0	0
$641$	1.83373e7	1.76275	0.881375	−	0.472417i	$-0.156618\pi$
$641$	1.83373e7	1.76275	0.881375	+	0.472417i	$0.156618\pi$
$642$	− 808326.i	− 0.0774015i
$643$	− 1.43215e6i	− 0.136603i	−0.997665	−	0.0683015i	$-0.978242\pi$
$643$	− 1.43215e6i	− 0.136603i	0.997665	−	0.0683015i	$-0.0217580\pi$
$644$	41250.0	0.00391931
$645$	0	0
$646$	−1.34797e7	−1.27086
$647$	− 1.04396e7i	− 0.980440i	−0.871599	−	0.490220i	$-0.836916\pi$
$647$	− 1.04396e7i	− 0.980440i	0.871599	−	0.490220i	$-0.163084\pi$
$648$	1.84957e7i	1.73034i
$649$	−9.30834e6	−0.867482
$650$	0	0
$651$	−893612.	−0.0826412
$652$	− 2.86084e7i	− 2.63557i
$653$	− 1.32094e7i	− 1.21227i	−0.795360	−	0.606137i	$-0.792718\pi$
$653$	− 1.32094e7i	− 1.21227i	0.795360	−	0.606137i	$-0.207282\pi$
$654$	1.27151e7	1.16245
$655$	0	0
$656$	−3.26053e7	−2.95821
$657$	2.49529e7i	2.25532i
$658$	1.14515e7i	1.03109i
$659$	−3.20415e6	−0.287408	−0.143704	−	0.989621i	$-0.545901\pi$
$659$	−3.20415e6	−0.287408	−0.143704	+	0.989621i	$0.545901\pi$
$660$	0	0
$661$	−2.22430e7	−1.98011	−0.990057	−	0.140667i	$-0.955075\pi$
$661$	−2.22430e7	−1.98011	−0.990057	+	0.140667i	$0.955075\pi$
$662$	− 2.16383e7i	− 1.91901i
$663$	3.81286e6i	0.336874i
$664$	−1.88174e7	−1.65630
$665$	0	0
$666$	3.74602e7	3.27254
$667$	90991.6i	0.00791930i
$668$	1.97696e7i	1.71418i
$669$	2.40500e7	2.07754
$670$	0	0
$671$	−3.90638e6	−0.334941
$672$	− 6.40353e6i	− 0.547011i
$673$	1.08386e7i	0.922434i	0.887287	+	0.461217i	$0.152587\pi$
$673$	1.08386e7i	0.922434i	−0.887287	+	0.461217i	$0.847413\pi$
$674$	−4.91739e6	−0.416951
$675$	0	0
$676$	2.04166e6	0.171837
$677$	− 1.21301e7i	− 1.01717i	−0.861012	−	0.508585i	$-0.830169\pi$
$677$	− 1.21301e7i	− 1.01717i	0.861012	−	0.508585i	$-0.169831\pi$
$678$	2.42465e7i	2.02570i
$679$	2.93899e6	0.244638
$680$	0	0
$681$	2.41550e7	1.99590
$682$	− 3.95912e6i	− 0.325940i
$683$	1.86818e7i	1.53238i	0.642613	+	0.766191i	$0.277851\pi$
$683$	1.86818e7i	1.53238i	−0.642613	+	0.766191i	$0.722149\pi$
$684$	−2.78781e7	−2.27837
$685$	0	0
$686$	−1.61042e7	−1.30656
$687$	− 1.65694e7i	− 1.33941i
$688$	3.51298e7i	2.82947i
$689$	−2.34253e6	−0.187991
$690$	0	0
$691$	−1.25804e7	−1.00230	−0.501151	−	0.865360i	$-0.667090\pi$
$691$	−1.25804e7	−1.00230	−0.501151	+	0.865360i	$0.667090\pi$
$692$	2.31264e7i	1.83588i
$693$	7.53810e6i	0.596250i
$694$	−1.98746e7	−1.56639
$695$	0	0
$696$	7.45212e7	5.83118
$697$	1.77480e7i	1.38378i
$698$	− 4.79933e6i	− 0.372856i
$699$	1.64690e7	1.27490
$700$	0	0
$701$	8.22912e6	0.632497	0.316249	−	0.948676i	$-0.397577\pi$
$701$	8.22912e6	0.632497	0.316249	+	0.948676i	$0.397577\pi$
$702$	1.78780e6i	0.136923i
$703$	− 1.72994e7i	− 1.32021i
$704$	−1.12133e6	−0.0852709
$705$	0	0
$706$	4.53616e7	3.42512
$707$	− 691060.i	− 0.0519957i
$708$	− 2.99248e7i	− 2.24362i
$709$	−2.08389e7	−1.55690	−0.778449	−	0.627708i	$-0.783993\pi$
$709$	−2.08389e7	−1.55690	−0.778449	+	0.627708i	$0.783993\pi$
$710$	0	0
$711$	1.77298e7	1.31532
$712$	496862.i	0.0367313i
$713$	8584.06i	0 0.000632367i
$714$	−1.17176e7	−0.860186
$715$	0	0
$716$	−5.41182e7	−3.94512
$717$	− 2.63566e6i	− 0.191466i
$718$	231922.i	0.0167893i
$719$	−1.98244e7	−1.43014	−0.715068	−	0.699055i	$-0.753604\pi$
$719$	−1.98244e7	−1.43014	−0.715068	+	0.699055i	$0.753604\pi$
$720$	0	0
$721$	5.08715e6	0.364448
$722$	− 6.55107e6i	− 0.467702i
$723$	2.34163e7i	1.66599i
$724$	−4.57877e7	−3.24640
$725$	0	0
$726$	−2.38072e7	−1.67636
$727$	9.64612e6i	0.676888i	0.940987	+	0.338444i	$0.109901\pi$
$727$	9.64612e6i	0.676888i	−0.940987	+	0.338444i	$0.890099\pi$
$728$	3.46563e6i	0.242356i
$729$	1.90912e7	1.33050
$730$	0	0
$731$	1.91221e7	1.32356
$732$	− 1.25584e7i	− 0.866276i
$733$	− 1.74331e7i	− 1.19844i	−0.800585	−	0.599219i	$-0.795478\pi$
$733$	− 1.74331e7i	− 1.19844i	0.800585	−	0.599219i	$-0.204522\pi$
$734$	−1.47859e7	−1.01299
$735$	0	0
$736$	−61512.5	−0.00418570
$737$	1.92907e6i	0.130821i
$738$	5.31370e7i	3.59133i
$739$	3.56942e6	0.240429	0.120214	−	0.992748i	$-0.461642\pi$
$739$	3.56942e6	0.240429	0.120214	+	0.992748i	$0.461642\pi$
$740$	0	0
$741$	5.27182e6	0.352708
$742$	− 7.19899e6i	− 0.480023i
$743$	− 9.04400e6i	− 0.601019i	−0.953779	−	0.300510i	$-0.902843\pi$
$743$	− 9.04400e6i	− 0.601019i	0.953779	−	0.300510i	$-0.0971567\pi$
$744$	7.03026e6	0.465628
$745$	0	0
$746$	2.01976e7	1.32878
$747$	1.34983e7i	0.885069i
$748$	− 3.58611e7i	− 2.34352i
$749$	−176031.	−0.0114653
$750$	0	0
$751$	−2.27403e7	−1.47128	−0.735642	−	0.677371i	$-0.763119\pi$
$751$	−2.27403e7	−1.47128	−0.735642	+	0.677371i	$0.763119\pi$
$752$	− 3.96547e7i	− 2.55712i
$753$	− 2.60901e7i	− 1.67683i
$754$	−1.38403e7	−0.886581
$755$	0	0
$756$	−3.79526e6	−0.241511
$757$	1.51736e7i	0.962385i	0.876615	+	0.481192i	$0.159796\pi$
$757$	1.51736e7i	0.962385i	−0.876615	+	0.481192i	$0.840204\pi$
$758$	− 1.11468e7i	− 0.704659i
$759$	133482.	0.00841043
$760$	0	0
$761$	1.74364e7	1.09143	0.545714	−	0.837971i	$-0.316258\pi$
$761$	1.74364e7	1.09143	0.545714	+	0.837971i	$0.316258\pi$
$762$	− 2.03021e7i	− 1.26664i
$763$	− 2.76898e6i	− 0.172190i
$764$	4.37434e7	2.71131
$765$	0	0
$766$	2.91516e7	1.79511
$767$	3.06981e6i	0.188418i
$768$	− 4.44340e7i	− 2.71839i
$769$	−1.64867e7	−1.00535	−0.502675	−	0.864476i	$-0.667651\pi$
$769$	−1.64867e7	−1.00535	−0.502675	+	0.864476i	$0.667651\pi$
$770$	0	0
$771$	8.24764e6	0.499682
$772$	9.62722e6i	0.581377i
$773$	1.70607e7i	1.02695i	0.858105	+	0.513474i	$0.171642\pi$
$773$	1.70607e7i	1.02695i	−0.858105	+	0.513474i	$0.828358\pi$
$774$	5.72512e7	3.43504
$775$	0	0
$776$	−2.31217e7	−1.37837
$777$	− 1.50380e7i	− 0.893588i
$778$	− 1.44783e7i	− 0.857569i
$779$	2.45391e7	1.44882
$780$	0	0
$781$	3.88418e7	2.27862
$782$	112559.i	0.00658210i
$783$	− 8.37180e6i	− 0.487994i
$784$	2.55392e7	1.48394
$785$	0	0
$786$	−4.98949e7	−2.88071
$787$	− 2.38884e6i	− 0.137483i	−0.997634	−	0.0687417i	$-0.978102\pi$
$787$	− 2.38884e6i	− 0.137483i	0.997634	−	0.0687417i	$-0.0218984\pi$
$788$	− 3.11087e7i	− 1.78471i
$789$	−7.04276e6	−0.402764
$790$	0	0
$791$	5.28022e6	0.300062
$792$	− 5.93040e7i	− 3.35947i
$793$	1.28829e6i	0.0727496i
$794$	6.14303e7	3.45805
$795$	0	0
$796$	1.28566e7	0.719188
$797$	− 7.46487e6i	− 0.416272i	−0.978100	−	0.208136i	$-0.933260\pi$
$797$	− 7.46487e6i	− 0.416272i	0.978100	−	0.208136i	$-0.0667396\pi$
$798$	1.62012e7i	0.900617i
$799$	−2.15852e7	−1.19616
$800$	0	0
$801$	356414.	0.0196279
$802$	5.78999e7i	3.17864i
$803$	4.43804e7i	2.42886i
$804$	−6.20166e6	−0.338351
$805$	0	0
$806$	−1.30569e6	−0.0707947
$807$	1.53920e7i	0.831976i
$808$	5.43674e6i	0.292961i
$809$	1.91578e7	1.02914	0.514569	−	0.857449i	$-0.327952\pi$
$809$	1.91578e7	1.02914	0.514569	+	0.857449i	$0.327952\pi$
$810$	0	0
$811$	2.86526e6	0.152972	0.0764859	−	0.997071i	$-0.475630\pi$
$811$	2.86526e6	0.152972	0.0764859	+	0.997071i	$0.475630\pi$
$812$	− 2.93812e7i	− 1.56379i
$813$	− 2.44041e7i	− 1.29490i
$814$	6.66253e7	3.52435
$815$	0	0
$816$	4.05762e7	2.13327
$817$	− 2.64391e7i	− 1.38577i
$818$	9.89687e6i	0.517147i
$819$	2.48600e6	0.129507
$820$	0	0
$821$	−1.42195e7	−0.736254	−0.368127	−	0.929775i	$-0.620001\pi$
$821$	−1.42195e7	−0.736254	−0.368127	+	0.929775i	$0.620001\pi$
$822$	− 9.20255e7i	− 4.75038i
$823$	8.97879e6i	0.462081i	0.972944	+	0.231041i	$0.0742130\pi$
$823$	8.97879e6i	0.462081i	−0.972944	+	0.231041i	$0.925787\pi$
$824$	−4.00218e7	−2.05342
$825$	0	0
$826$	−9.43406e6	−0.481114
$827$	− 1.46294e7i	− 0.743813i	−0.928270	−	0.371907i	$-0.878704\pi$
$827$	− 1.46294e7i	− 0.743813i	0.928270	−	0.371907i	$-0.121296\pi$
$828$	232790.i	0.0118002i
$829$	−6.70195e6	−0.338700	−0.169350	−	0.985556i	$-0.554167\pi$
$829$	−6.70195e6	−0.338700	−0.169350	+	0.985556i	$0.554167\pi$
$830$	0	0
$831$	4.41115e7	2.21590
$832$	369804.i	0.0185210i
$833$	− 1.39017e7i	− 0.694153i
$834$	1.51850e7	0.755961
$835$	0	0
$836$	−4.95830e7	−2.45368
$837$	− 789788.i	− 0.0389670i
$838$	− 3.64597e7i	− 1.79350i
$839$	1.00902e7	0.494876	0.247438	−	0.968904i	$-0.420411\pi$
$839$	1.00902e7	0.494876	0.247438	+	0.968904i	$0.420411\pi$
$840$	0	0
$841$	4.42995e7	2.15978
$842$	1.23558e7i	0.600607i
$843$	2.49193e7i	1.20772i
$844$	7.36580e7	3.55930
$845$	0	0
$846$	−6.46254e7	−3.10440
$847$	5.18455e6i	0.248315i
$848$	2.49290e7i	1.19046i
$849$	1.98047e7	0.942973
$850$	0	0
$851$	−144455.	−0.00683769
$852$	1.24870e8i	5.89332i
$853$	8.26689e6i	0.389018i	0.980901	+	0.194509i	$0.0623113\pi$
$853$	8.26689e6i	0.389018i	−0.980901	+	0.194509i	$0.937689\pi$
$854$	−3.95914e6	−0.185762
$855$	0	0
$856$	1.38488e6	0.0645992
$857$	1.34533e6i	0.0625714i	0.999510	+	0.0312857i	$0.00996017\pi$
$857$	1.34533e6i	0.0625714i	−0.999510	+	0.0312857i	$0.990040\pi$
$858$	2.03034e7i	0.941564i
$859$	2.57272e7	1.18962	0.594812	−	0.803865i	$-0.297226\pi$
$859$	2.57272e7	1.18962	0.594812	+	0.803865i	$0.297226\pi$
$860$	0	0
$861$	2.13313e7	0.980637
$862$	3.24047e7i	1.48539i
$863$	− 9.60150e6i	− 0.438846i	−0.975630	−	0.219423i	$-0.929583\pi$
$863$	− 9.60150e6i	− 0.438846i	0.975630	−	0.219423i	$-0.0704175\pi$
$864$	5.65954e6	0.257927
$865$	0	0
$866$	−5.74360e7	−2.60249
$867$	1.06355e7i	0.480517i
$868$	− 2.77179e6i	− 0.124871i
$869$	3.15336e7	1.41652
$870$	0	0
$871$	636191.	0.0284146
$872$	2.17843e7i	0.970178i
$873$	1.65859e7i	0.736552i
$874$	155629.	0.00689148
$875$	0	0
$876$	−1.42676e8	−6.28189
$877$	3.88143e6i	0.170409i	0.996363	+	0.0852046i	$0.0271544\pi$
$877$	3.88143e6i	0.170409i	−0.996363	+	0.0852046i	$0.972846\pi$
$878$	− 7.78588e7i	− 3.40856i
$879$	−5.44759e7	−2.37811
$880$	0	0
$881$	−1.47028e7	−0.638204	−0.319102	−	0.947720i	$-0.603381\pi$
$881$	−1.47028e7	−0.638204	−0.319102	+	0.947720i	$0.603381\pi$
$882$	− 4.16213e7i	− 1.80154i
$883$	− 4.37837e7i	− 1.88978i	−0.327392	−	0.944889i	$-0.606170\pi$
$883$	− 4.37837e7i	− 1.88978i	0.327392	−	0.944889i	$-0.393830\pi$
$884$	−1.18267e7	−0.509017
$885$	0	0
$886$	−5.75420e7	−2.46264
$887$	− 9.31739e6i	− 0.397636i	−0.980036	−	0.198818i	$-0.936290\pi$
$887$	− 9.31739e6i	− 0.397636i	0.980036	−	0.198818i	$-0.0637102\pi$
$888$	1.18307e8i	5.03477i
$889$	−4.42122e6	−0.187624
$890$	0	0
$891$	2.35971e7	0.995781
$892$	7.45978e7i	3.13916i
$893$	2.98445e7i	1.25238i
$894$	−6.50020e7	−2.72009
$895$	0	0
$896$	−1.00279e7	−0.417292
$897$	− 44021.2i	− 0.00182676i
$898$	1.97098e7i	0.815628i
$899$	6.11418e6	0.252313
$900$	0	0
$901$	1.35695e7	0.556869
$902$	9.45074e7i	3.86767i
$903$	− 2.29828e7i	− 0.937960i
$904$	−4.15407e7	−1.69065
$905$	0	0
$906$	1.32785e7	0.537440
$907$	− 1.32431e7i	− 0.534529i	−0.963623	−	0.267265i	$-0.913880\pi$
$907$	− 1.32431e7i	− 0.534529i	0.963623	−	0.267265i	$-0.0861198\pi$
$908$	7.49235e7i	3.01581i
$909$	3.89993e6	0.156548
$910$	0	0
$911$	2.78672e7	1.11249	0.556246	−	0.831018i	$-0.312241\pi$
$911$	2.78672e7	1.11249	0.556246	+	0.831018i	$0.312241\pi$
$912$	− 5.61023e7i	− 2.23354i
$913$	2.40075e7i	0.953171i
$914$	3.28107e7	1.29912
$915$	0	0
$916$	5.13947e7	2.02386
$917$	1.08657e7i	0.426712i
$918$	− 1.03562e7i	− 0.405595i
$919$	1.81646e7	0.709474	0.354737	−	0.934966i	$-0.384570\pi$
$919$	1.81646e7	0.709474	0.354737	+	0.934966i	$0.384570\pi$
$920$	0	0
$921$	3.71908e7	1.44473
$922$	− 4.61733e7i	− 1.78881i
$923$	− 1.28097e7i	− 0.494920i
$924$	−4.31013e7	−1.66077
$925$	0	0
$926$	−4.48145e7	−1.71748
$927$	2.87088e7i	1.09728i
$928$	4.38136e7i	1.67009i
$929$	−3.97985e7	−1.51296	−0.756480	−	0.654017i	$-0.773083\pi$
$929$	−3.97985e7	−1.51296	−0.756480	+	0.654017i	$0.773083\pi$
$930$	0	0
$931$	−1.92210e7	−0.726780
$932$	5.10833e7i	1.92637i
$933$	1.78891e7i	0.672796i
$934$	−8.03515e7	−3.01388
$935$	0	0
$936$	−1.95580e7	−0.729683
$937$	3.23323e7i	1.20306i	0.798850	+	0.601531i	$0.205442\pi$
$937$	3.23323e7i	1.20306i	−0.798850	+	0.601531i	$0.794558\pi$
$938$	1.95512e6i	0.0725550i
$939$	1.51378e7	0.560273
$940$	0	0
$941$	−1.24433e7	−0.458102	−0.229051	−	0.973414i	$-0.573562\pi$
$941$	−1.24433e7	−0.458102	−0.229051	+	0.973414i	$0.573562\pi$
$942$	− 5.38898e7i	− 1.97870i
$943$	− 204908.i	− 0.00750379i
$944$	3.26687e7	1.19317
$945$	0	0
$946$	1.01825e8	3.69935
$947$	3.32452e7i	1.20463i	0.798258	+	0.602316i	$0.205755\pi$
$947$	3.32452e7i	1.20463i	−0.798258	+	0.602316i	$0.794245\pi$
$948$	1.01376e8i	3.66364i
$949$	1.46363e7	0.527552
$950$	0	0
$951$	−3.26594e7	−1.17100
$952$	− 2.00753e7i	− 0.717910i
$953$	− 4.55359e7i	− 1.62413i	−0.583566	−	0.812066i	$-0.698343\pi$
$953$	− 4.55359e7i	− 1.62413i	0.583566	−	0.812066i	$-0.301657\pi$
$954$	4.06268e7	1.44525
$955$	0	0
$956$	8.17526e6	0.289306
$957$	− 9.50754e7i	− 3.35574i
$958$	− 1.64816e7i	− 0.580210i
$959$	−2.00406e7	−0.703661
$960$	0	0
$961$	−2.80523e7	−0.979852
$962$	− 2.19725e7i	− 0.765493i
$963$	− 993414.i	− 0.0345195i
$964$	−7.26323e7	−2.51731
$965$	0	0
$966$	135285.	0.00466451
$967$	1.52863e7i	0.525697i	0.964837	+	0.262849i	$0.0846620\pi$
$967$	1.52863e7i	0.525697i	−0.964837	+	0.262849i	$0.915338\pi$
$968$	− 4.07881e7i	− 1.39909i
$969$	−3.05380e7	−1.04480
$970$	0	0
$971$	−1.92581e7	−0.655490	−0.327745	−	0.944766i	$-0.606289\pi$
$971$	−1.92581e7	−0.655490	−0.327745	+	0.944766i	$0.606289\pi$
$972$	9.39248e7i	3.18871i
$973$	− 3.30687e6i	− 0.111979i
$974$	4.16646e7	1.40725
$975$	0	0
$976$	1.37099e7	0.460690
$977$	795487.i	0.0266622i	0.999911	+	0.0133311i	$0.00424355\pi$
$977$	795487.i	0.0266622i	−0.999911	+	0.0133311i	$0.995756\pi$
$978$	− 9.38248e7i	− 3.13668i
$979$	633905.	0.0211382
$980$	0	0
$981$	1.56265e7	0.518429
$982$	− 7.21588e7i	− 2.38787i
$983$	− 7.51037e6i	− 0.247901i	−0.992288	−	0.123950i	$-0.960444\pi$
$983$	− 7.51037e6i	− 0.247901i	0.992288	−	0.123950i	$-0.0395563\pi$
$984$	−1.67818e8	−5.52524
$985$	0	0
$986$	8.01728e7	2.62624
$987$	2.59432e7i	0.847676i
$988$	1.63521e7i	0.532942i
$989$	−220774.	−0.00717723
$990$	0	0
$991$	5.90076e7	1.90864	0.954319	−	0.298788i	$-0.0965824\pi$
$991$	5.90076e7	1.90864	0.954319	+	0.298788i	$0.0965824\pi$
$992$	4.13333e6i	0.133359i
$993$	− 4.90212e7i	− 1.57765i
$994$	3.93664e7	1.26375
$995$	0	0
$996$	−7.71805e7	−2.46524
$997$	4.32622e7i	1.37839i	0.724577	+	0.689193i	$0.242035\pi$
$997$	4.32622e7i	1.37839i	−0.724577	+	0.689193i	$0.757965\pi$
$998$	1.63287e7i	0.518948i
$999$	1.32908e7	0.421345

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.h.274.18		18
5.2	odd	4		325.6.a.h.1.1	✓	9
5.3	odd	4		325.6.a.i.1.9	yes	9
5.4	even	2	inner	325.6.b.h.274.1		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.1	✓	9	5.2	odd	4
325.6.a.i.1.9	yes	9	5.3	odd	4
325.6.b.h.274.1		18	5.4	even	2	inner
325.6.b.h.274.18		18	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+10.1727i q^{2} +23.0461i q^{3} -71.4841 q^{4} -234.441 q^{6} +51.0548i q^{7} -401.660i q^{8} -288.123 q^{9} +O(q^{10})\)	\(q+10.1727i q^{2} +23.0461i q^{3} -71.4841 q^{4} -234.441 q^{6} +51.0548i q^{7} -401.660i q^{8} -288.123 q^{9} -512.445 q^{11} -1647.43i q^{12} +169.000i q^{13} -519.366 q^{14} +1798.48 q^{16} -978.964i q^{17} -2930.99i q^{18} -1353.56 q^{19} -1176.61 q^{21} -5212.95i q^{22} +11.3026i q^{23} +9256.71 q^{24} -1719.19 q^{26} -1039.91i q^{27} -3649.61i q^{28} +8050.51 q^{29} +759.478 q^{31} +5442.33i q^{32} -11809.9i q^{33} +9958.72 q^{34} +20596.2 q^{36} +12780.7i q^{37} -13769.3i q^{38} -3894.79 q^{39} -18129.3 q^{41} -11969.4i q^{42} +19533.0i q^{43} +36631.6 q^{44} -114.978 q^{46} -22049.0i q^{47} +41448.1i q^{48} +14200.4 q^{49} +22561.3 q^{51} -12080.8i q^{52} +13861.1i q^{53} +10578.7 q^{54} +20506.7 q^{56} -31194.2i q^{57} +81895.5i q^{58} +18164.6 q^{59} +7623.02 q^{61} +7725.95i q^{62} -14710.1i q^{63} +2188.19 q^{64} +120138. q^{66} -3764.44i q^{67} +69980.4i q^{68} -260.481 q^{69} -75797.0 q^{71} +115728. i q^{72} -86605.1i q^{73} -130015. q^{74} +96757.7 q^{76} -26162.8i q^{77} -39620.6i q^{78} -61535.6 q^{79} -46048.0 q^{81} -184425. i q^{82} -46849.0i q^{83} +84109.2 q^{84} -198704. q^{86} +185533. i q^{87} +205829. i q^{88} -1237.02 q^{89} -8628.26 q^{91} -807.955i q^{92} +17503.0i q^{93} +224298. q^{94} -125425. q^{96} -57565.4i q^{97} +144457. i q^{98} +147647. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9}+O(q^{10}) \) 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9	\( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39} - 57352 q^{41} + 61198 q^{44} - 24112 q^{46} + 81814 q^{49} - 62012 q^{51} + 205522 q^{54} - 46004 q^{56} + 176284 q^{59} + 56330 q^{61} + 201690 q^{64} + 85154 q^{66} + 363494 q^{69} - 141124 q^{71} + 271352 q^{74} + 92746 q^{76} + 328146 q^{79} - 139870 q^{81} + 691312 q^{84} - 589840 q^{86} + 505396 q^{89} + 4056 q^{91} + 1003212 q^{94} - 638362 q^{96} + 1515552 q^{99}+O(q^{100}) \) 18 * q - 182 * q^4 - 166 * q^6 - 1124 * q^9 - 2844 * q^11 + 684 * q^14 - 2122 * q^16 + 816 * q^19 - 7824 * q^21 + 16938 * q^24 - 1690 * q^26 + 17474 * q^29 + 1496 * q^31 + 35578 * q^34 + 1024 * q^36 + 3718 * q^39 - 57352 * q^41 + 61198 * q^44 - 24112 * q^46 + 81814 * q^49 - 62012 * q^51 + 205522 * q^54 - 46004 * q^56 + 176284 * q^59 + 56330 * q^61 + 201690 * q^64 + 85154 * q^66 + 363494 * q^69 - 141124 * q^71 + 271352 * q^74 + 92746 * q^76 + 328146 * q^79 - 139870 * q^81 + 691312 * q^84 - 589840 * q^86 + 505396 * q^89 + 4056 * q^91 + 1003212 * q^94 - 638362 * q^96 + 1515552 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more