LMFDB - Embedded newform 825.4.c.s.199.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 43x^{8} + 631x^{6} + 3625x^{4} + 7104x^{2} + 900 $ x^10 + 43x^8 + 631x^6 + 3625x^4 + 7104x^2 + 900
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.6
Root			$0.368634i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.s.199.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.368634i q^{2} +3.00000i q^{3} +7.86411 q^{4} -1.10590 q^{6} -26.5824i q^{7} +5.84806i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q+0.368634i q^{2} +3.00000i q^{3} +7.86411 q^{4} -1.10590 q^{6} -26.5824i q^{7} +5.84806i q^{8} -9.00000 q^{9} -11.0000 q^{11} +23.5923i q^{12} -50.4402i q^{13} +9.79918 q^{14} +60.7571 q^{16} +108.843i q^{17} -3.31771i q^{18} -19.1278 q^{19} +79.7471 q^{21} -4.05498i q^{22} -60.4434i q^{23} -17.5442 q^{24} +18.5940 q^{26} -27.0000i q^{27} -209.047i q^{28} +39.2097 q^{29} -22.4715 q^{31} +69.1816i q^{32} -33.0000i q^{33} -40.1234 q^{34} -70.7770 q^{36} -345.619i q^{37} -7.05117i q^{38} +151.321 q^{39} -96.3091 q^{41} +29.3975i q^{42} -335.088i q^{43} -86.5052 q^{44} +22.2815 q^{46} -514.880i q^{47} +182.271i q^{48} -363.623 q^{49} -326.530 q^{51} -396.667i q^{52} -131.589i q^{53} +9.95313 q^{54} +155.455 q^{56} -57.3834i q^{57} +14.4541i q^{58} +210.728 q^{59} -68.9791 q^{61} -8.28378i q^{62} +239.241i q^{63} +460.554 q^{64} +12.1649 q^{66} +202.618i q^{67} +855.955i q^{68} +181.330 q^{69} -645.234 q^{71} -52.6325i q^{72} -1021.75i q^{73} +127.407 q^{74} -150.423 q^{76} +292.406i q^{77} +55.7820i q^{78} -321.381 q^{79} +81.0000 q^{81} -35.5029i q^{82} -840.583i q^{83} +627.140 q^{84} +123.525 q^{86} +117.629i q^{87} -64.3286i q^{88} +1449.66 q^{89} -1340.82 q^{91} -475.333i q^{92} -67.4146i q^{93} +189.803 q^{94} -207.545 q^{96} +1602.70i q^{97} -134.044i q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{4} - 6 q^{6} - 90 q^{9}+O(q^{10}) $ 10 * q - 6 * q^4 - 6 * q^6 - 90 * q^9	$ 10 q - 6 q^{4} - 6 q^{6} - 90 q^{9} - 110 q^{11} - 16 q^{14} - 250 q^{16} + 114 q^{19} - 108 q^{21} - 18 q^{24} - 250 q^{26} + 214 q^{29} - 590 q^{31} - 68 q^{34} + 54 q^{36} + 186 q^{39} - 256 q^{41} + 66 q^{44} - 1154 q^{46} + 830 q^{49} - 228 q^{51} + 54 q^{54} - 264 q^{56} + 2304 q^{59} - 688 q^{61} + 1090 q^{64} + 66 q^{66} + 966 q^{69} - 1414 q^{71} + 2352 q^{74} - 3398 q^{76} + 4988 q^{79} + 810 q^{81} + 1944 q^{84} - 1598 q^{86} + 4870 q^{89} - 1608 q^{91} + 1004 q^{94} + 138 q^{96} + 990 q^{99}+O(q^{100}) $ 10 * q - 6 * q^4 - 6 * q^6 - 90 * q^9 - 110 * q^11 - 16 * q^14 - 250 * q^16 + 114 * q^19 - 108 * q^21 - 18 * q^24 - 250 * q^26 + 214 * q^29 - 590 * q^31 - 68 * q^34 + 54 * q^36 + 186 * q^39 - 256 * q^41 + 66 * q^44 - 1154 * q^46 + 830 * q^49 - 228 * q^51 + 54 * q^54 - 264 * q^56 + 2304 * q^59 - 688 * q^61 + 1090 * q^64 + 66 * q^66 + 966 * q^69 - 1414 * q^71 + 2352 * q^74 - 3398 * q^76 + 4988 * q^79 + 810 * q^81 + 1944 * q^84 - 1598 * q^86 + 4870 * q^89 - 1608 * q^91 + 1004 * q^94 + 138 * q^96 + 990 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0.368634i	0.130332i	0.997874	+	0.0651660i	$0.0207577\pi$
$2$	0.368634i	0.130332i	−0.997874	+	0.0651660i	$0.979242\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	7.86411	0.983014
$5$	0	0
$5$	0	0
$6$	−1.10590	−0.0752472
$7$	− 26.5824i	− 1.43531i	−0.696397	−	0.717657i	$-0.745215\pi$
$7$	− 26.5824i	− 1.43531i	0.696397	−	0.717657i	$-0.254785\pi$
$8$	5.84806i	0.258450i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	23.5923i	0.567543i
$13$	− 50.4402i	− 1.07612i	−0.842906	−	0.538061i	$-0.819157\pi$
$13$	− 50.4402i	− 1.07612i	0.842906	−	0.538061i	$-0.180843\pi$
$14$	9.79918	0.187067
$15$	0	0
$16$	60.7571	0.949329
$17$	108.843i	1.55284i	0.630213	+	0.776422i	$0.282968\pi$
$17$	108.843i	1.55284i	−0.630213	+	0.776422i	$0.717032\pi$
$18$	− 3.31771i	− 0.0434440i
$19$	−19.1278	−0.230959	−0.115479	−	0.993310i	$-0.536840\pi$
$19$	−19.1278	−0.230959	−0.115479	+	0.993310i	$0.536840\pi$
$20$	0	0
$21$	79.7471	0.828679
$22$	− 4.05498i	− 0.0392966i
$23$	− 60.4434i	− 0.547970i	−0.961734	−	0.273985i	$-0.911658\pi$
$23$	− 60.4434i	− 0.547970i	0.961734	−	0.273985i	$-0.0883419\pi$
$24$	−17.5442	−0.149216
$25$	0	0
$26$	18.5940	0.140253
$27$	− 27.0000i	− 0.192450i
$28$	− 209.047i	− 1.41093i
$29$	39.2097	0.251071	0.125536	−	0.992089i	$-0.459935\pi$
$29$	39.2097	0.251071	0.125536	+	0.992089i	$0.459935\pi$
$30$	0	0
$31$	−22.4715	−0.130194	−0.0650969	−	0.997879i	$-0.520736\pi$
$31$	−22.4715	−0.130194	−0.0650969	+	0.997879i	$0.520736\pi$
$32$	69.1816i	0.382178i
$33$	− 33.0000i	− 0.174078i
$34$	−40.1234	−0.202385
$35$	0	0
$36$	−70.7770	−0.327671
$37$	− 345.619i	− 1.53566i	−0.640654	−	0.767830i	$-0.721337\pi$
$37$	− 345.619i	− 1.53566i	0.640654	−	0.767830i	$-0.278663\pi$
$38$	− 7.05117i	− 0.0301013i
$39$	151.321	0.621300
$40$	0	0
$41$	−96.3091	−0.366853	−0.183426	−	0.983033i	$-0.558719\pi$
$41$	−96.3091	−0.366853	−0.183426	+	0.983033i	$0.558719\pi$
$42$	29.3975i	0.108003i
$43$	− 335.088i	− 1.18838i	−0.804324	−	0.594191i	$-0.797472\pi$
$43$	− 335.088i	− 1.18838i	0.804324	−	0.594191i	$-0.202528\pi$
$44$	−86.5052	−0.296390
$45$	0	0
$46$	22.2815	0.0714180
$47$	− 514.880i	− 1.59794i	−0.601373	−	0.798968i	$-0.705379\pi$
$47$	− 514.880i	− 1.59794i	0.601373	−	0.798968i	$-0.294621\pi$
$48$	182.271i	0.548096i
$49$	−363.623	−1.06012
$50$	0	0
$51$	−326.530	−0.896535
$52$	− 396.667i	− 1.05784i
$53$	− 131.589i	− 0.341041i	−0.985354	−	0.170521i	$-0.945455\pi$
$53$	− 131.589i	− 0.341041i	0.985354	−	0.170521i	$-0.0545450\pi$
$54$	9.95313	0.0250824
$55$	0	0
$56$	155.455	0.370957
$57$	− 57.3834i	− 0.133344i
$58$	14.4541i	0.0327226i
$59$	210.728	0.464991	0.232495	−	0.972598i	$-0.425311\pi$
$59$	210.728	0.464991	0.232495	+	0.972598i	$0.425311\pi$
$60$	0	0
$61$	−68.9791	−0.144785	−0.0723924	−	0.997376i	$-0.523063\pi$
$61$	−68.9791	−0.144785	−0.0723924	+	0.997376i	$0.523063\pi$
$62$	− 8.28378i	− 0.0169684i
$63$	239.241i	0.478438i
$64$	460.554	0.899519
$65$	0	0
$66$	12.1649	0.0226879
$67$	202.618i	0.369459i	0.982789	+	0.184729i	$0.0591408\pi$
$67$	202.618i	0.369459i	−0.982789	+	0.184729i	$0.940859\pi$
$68$	855.955i	1.52647i
$69$	181.330	0.316371
$70$	0	0
$71$	−645.234	−1.07852	−0.539262	−	0.842138i	$-0.681297\pi$
$71$	−645.234	−1.07852	−0.539262	+	0.842138i	$0.681297\pi$
$72$	− 52.6325i	− 0.0861500i
$73$	− 1021.75i	− 1.63818i	−0.573666	−	0.819089i	$-0.694479\pi$
$73$	− 1021.75i	− 1.63818i	0.573666	−	0.819089i	$-0.305521\pi$
$74$	127.407	0.200145
$75$	0	0
$76$	−150.423	−0.227036
$77$	292.406i	0.432763i
$78$	55.7820i	0.0809752i
$79$	−321.381	−0.457699	−0.228849	−	0.973462i	$-0.573496\pi$
$79$	−321.381	−0.457699	−0.228849	+	0.973462i	$0.573496\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 35.5029i	− 0.0478126i
$83$	− 840.583i	− 1.11164i	−0.831303	−	0.555819i	$-0.812405\pi$
$83$	− 840.583i	− 1.11164i	0.831303	−	0.555819i	$-0.187595\pi$
$84$	627.140	0.814602
$85$	0	0
$86$	123.525	0.154884
$87$	117.629i	0.144956i
$88$	− 64.3286i	− 0.0779256i
$89$	1449.66	1.72655	0.863277	−	0.504731i	$-0.168408\pi$
$89$	1449.66	1.72655	0.863277	+	0.504731i	$0.168408\pi$
$90$	0	0
$91$	−1340.82	−1.54457
$92$	− 475.333i	− 0.538662i
$93$	− 67.4146i	− 0.0751674i
$94$	189.803	0.208262
$95$	0	0
$96$	−207.545	−0.220651
$97$	1602.70i	1.67762i	0.544422	+	0.838811i	$0.316749\pi$
$97$	1602.70i	1.67762i	−0.544422	+	0.838811i	$0.683251\pi$
$98$	− 134.044i	− 0.138168i
$99$	99.0000	0.100504
$100$	0	0
$101$	−970.653	−0.956273	−0.478136	−	0.878286i	$-0.658688\pi$
$101$	−970.653	−0.956273	−0.478136	+	0.878286i	$0.658688\pi$
$102$	− 120.370i	− 0.116847i
$103$	185.474i	0.177430i	0.996057	+	0.0887152i	$0.0282761\pi$
$103$	185.474i	0.177430i	−0.996057	+	0.0887152i	$0.971724\pi$
$104$	294.977	0.278124
$105$	0	0
$106$	48.5084	0.0444486
$107$	− 652.518i	− 0.589545i	−0.955567	−	0.294773i	$-0.904756\pi$
$107$	− 652.518i	− 0.589545i	0.955567	−	0.294773i	$-0.0952439\pi$
$108$	− 212.331i	− 0.189181i
$109$	915.109	0.804143	0.402071	−	0.915608i	$-0.368290\pi$
$109$	915.109	0.804143	0.402071	+	0.915608i	$0.368290\pi$
$110$	0	0
$111$	1036.86	0.886613
$112$	− 1615.07i	− 1.36258i
$113$	− 327.063i	− 0.272279i	−0.990690	−	0.136139i	$-0.956530\pi$
$113$	− 327.063i	− 0.272279i	0.990690	−	0.136139i	$-0.0434695\pi$
$114$	21.1535	0.0173790
$115$	0	0
$116$	308.349	0.246806
$117$	453.962i	0.358708i
$118$	77.6816i	0.0606031i
$119$	2893.31	2.22882
$120$	0	0
$121$	121.000	0.0909091
$122$	− 25.4281i	− 0.0188701i
$123$	− 288.927i	− 0.211802i
$124$	−176.719	−0.127982
$125$	0	0
$126$	−88.1926	−0.0623557
$127$	− 1739.39i	− 1.21532i	−0.794196	−	0.607661i	$-0.792108\pi$
$127$	− 1739.39i	− 1.21532i	0.794196	−	0.607661i	$-0.207892\pi$
$128$	723.229i	0.499414i
$129$	1005.26	0.686112
$130$	0	0
$131$	1537.93	1.02572	0.512860	−	0.858472i	$-0.328586\pi$
$131$	1537.93	1.02572	0.512860	+	0.858472i	$0.328586\pi$
$132$	− 259.516i	− 0.171121i
$133$	508.463i	0.331498i
$134$	−74.6920	−0.0481523
$135$	0	0
$136$	−636.521	−0.401333
$137$	1885.08i	1.17557i	0.809016	+	0.587787i	$0.200001\pi$
$137$	1885.08i	1.17557i	−0.809016	+	0.587787i	$0.799999\pi$
$138$	66.8445i	0.0412332i
$139$	482.323	0.294317	0.147159	−	0.989113i	$-0.452987\pi$
$139$	482.323	0.294317	0.147159	+	0.989113i	$0.452987\pi$
$140$	0	0
$141$	1544.64	0.922569
$142$	− 237.855i	− 0.140566i
$143$	554.842i	0.324463i
$144$	−546.814	−0.316443
$145$	0	0
$146$	376.653	0.213507
$147$	− 1090.87i	− 0.612063i
$148$	− 2717.98i	− 1.50957i
$149$	757.043	0.416237	0.208119	−	0.978104i	$-0.433266\pi$
$149$	757.043	0.416237	0.208119	+	0.978104i	$0.433266\pi$
$150$	0	0
$151$	−2919.01	−1.57315	−0.786576	−	0.617494i	$-0.788148\pi$
$151$	−2919.01	−1.57315	−0.786576	+	0.617494i	$0.788148\pi$
$152$	− 111.861i	− 0.0596914i
$153$	− 979.589i	− 0.517615i
$154$	−107.791	−0.0564029
$155$	0	0
$156$	1190.00	0.610746
$157$	− 1352.65i	− 0.687599i	−0.939043	−	0.343799i	$-0.888286\pi$
$157$	− 1352.65i	− 0.687599i	0.939043	−	0.343799i	$-0.111714\pi$
$158$	− 118.472i	− 0.0596527i
$159$	394.768	0.196900
$160$	0	0
$161$	−1606.73	−0.786509
$162$	29.8594i	0.0144813i
$163$	3027.63i	1.45486i	0.686181	+	0.727431i	$0.259286\pi$
$163$	3027.63i	1.45486i	−0.686181	+	0.727431i	$0.740714\pi$
$164$	−757.385	−0.360621
$165$	0	0
$166$	309.868	0.144882
$167$	− 4225.69i	− 1.95805i	−0.203749	−	0.979023i	$-0.565313\pi$
$167$	− 4225.69i	− 1.95805i	0.203749	−	0.979023i	$-0.434687\pi$
$168$	466.366i	0.214172i
$169$	−347.214	−0.158040
$170$	0	0
$171$	172.150	0.0769863
$172$	− 2635.17i	− 1.16820i
$173$	− 1515.20i	− 0.665886i	−0.942947	−	0.332943i	$-0.891958\pi$
$173$	− 1515.20i	− 0.665886i	0.942947	−	0.332943i	$-0.108042\pi$
$174$	−43.3622	−0.0188924
$175$	0	0
$176$	−668.328	−0.286234
$177$	632.184i	0.268462i
$178$	534.393i	0.225025i
$179$	−1087.42	−0.454066	−0.227033	−	0.973887i	$-0.572902\pi$
$179$	−1087.42	−0.454066	−0.227033	+	0.973887i	$0.572902\pi$
$180$	0	0
$181$	−145.994	−0.0599538	−0.0299769	−	0.999551i	$-0.509543\pi$
$181$	−145.994	−0.0599538	−0.0299769	+	0.999551i	$0.509543\pi$
$182$	− 494.273i	− 0.201307i
$183$	− 206.937i	− 0.0835915i
$184$	353.476	0.141623
$185$	0	0
$186$	24.8514	0.00979672
$187$	− 1197.28i	− 0.468200i
$188$	− 4049.08i	− 1.57079i
$189$	−717.724	−0.276226
$190$	0	0
$191$	3586.76	1.35879	0.679395	−	0.733773i	$-0.262243\pi$
$191$	3586.76	1.35879	0.679395	+	0.733773i	$0.262243\pi$
$192$	1381.66i	0.519338i
$193$	− 1015.55i	− 0.378762i	−0.981904	−	0.189381i	$-0.939352\pi$
$193$	− 1015.55i	− 0.378762i	0.981904	−	0.189381i	$-0.0606481\pi$
$194$	−590.810	−0.218648
$195$	0	0
$196$	−2859.57	−1.04212
$197$	2992.63i	1.08231i	0.840921	+	0.541157i	$0.182014\pi$
$197$	2992.63i	1.08231i	−0.840921	+	0.541157i	$0.817986\pi$
$198$	36.4948i	0.0130989i
$199$	3267.57	1.16398	0.581989	−	0.813196i	$-0.302275\pi$
$199$	3267.57	1.16398	0.581989	+	0.813196i	$0.302275\pi$
$200$	0	0
$201$	−607.854	−0.213307
$202$	− 357.816i	− 0.124633i
$203$	− 1042.29i	− 0.360366i
$204$	−2567.86	−0.881306
$205$	0	0
$206$	−68.3722	−0.0231248
$207$	543.990i	0.182657i
$208$	− 3064.60i	− 1.02159i
$209$	210.406	0.0696368
$210$	0	0
$211$	−4275.02	−1.39481	−0.697403	−	0.716679i	$-0.745661\pi$
$211$	−4275.02	−1.39481	−0.697403	+	0.716679i	$0.745661\pi$
$212$	− 1034.83i	− 0.335248i
$213$	− 1935.70i	− 0.622686i
$214$	240.541	0.0768366
$215$	0	0
$216$	157.898	0.0497387
$217$	597.347i	0.186869i
$218$	337.341i	0.104805i
$219$	3065.26	0.945803
$220$	0	0
$221$	5490.07	1.67105
$222$	382.221i	0.115554i
$223$	3771.24i	1.13247i	0.824244	+	0.566235i	$0.191601\pi$
$223$	3771.24i	1.13247i	−0.824244	+	0.566235i	$0.808399\pi$
$224$	1839.01	0.548545
$225$	0	0
$226$	120.567	0.0354866
$227$	2891.58i	0.845467i	0.906254	+	0.422733i	$0.138929\pi$
$227$	2891.58i	0.845467i	−0.906254	+	0.422733i	$0.861071\pi$
$228$	− 451.270i	− 0.131079i
$229$	−921.309	−0.265859	−0.132930	−	0.991125i	$-0.542438\pi$
$229$	−921.309	−0.265859	−0.132930	+	0.991125i	$0.542438\pi$
$230$	0	0
$231$	−877.218	−0.249856
$232$	229.301i	0.0648893i
$233$	− 2818.65i	− 0.792516i	−0.918139	−	0.396258i	$-0.870309\pi$
$233$	− 2818.65i	− 0.792516i	0.918139	−	0.396258i	$-0.129691\pi$
$234$	−167.346	−0.0467511
$235$	0	0
$236$	1657.19	0.457092
$237$	− 964.143i	− 0.264252i
$238$	1066.57i	0.290486i
$239$	−783.622	−0.212085	−0.106042	−	0.994362i	$-0.533818\pi$
$239$	−783.622	−0.212085	−0.106042	+	0.994362i	$0.533818\pi$
$240$	0	0
$241$	5891.99	1.57484	0.787420	−	0.616417i	$-0.211417\pi$
$241$	5891.99	1.57484	0.787420	+	0.616417i	$0.211417\pi$
$242$	44.6048i	0.0118484i
$243$	243.000i	0.0641500i
$244$	−542.459	−0.142325
$245$	0	0
$246$	106.509	0.0276046
$247$	964.811i	0.248540i
$248$	− 131.415i	− 0.0336486i
$249$	2521.75	0.641805
$250$	0	0
$251$	−7892.69	−1.98479	−0.992394	−	0.123101i	$-0.960716\pi$
$251$	−7892.69	−1.98479	−0.992394	+	0.123101i	$0.960716\pi$
$252$	1881.42i	0.470311i
$253$	664.877i	0.165219i
$254$	641.199	0.158395
$255$	0	0
$256$	3417.82	0.834430
$257$	3375.73i	0.819348i	0.912232	+	0.409674i	$0.134357\pi$
$257$	3375.73i	0.819348i	−0.912232	+	0.409674i	$0.865643\pi$
$258$	370.575i	0.0894224i
$259$	−9187.37	−2.20415
$260$	0	0
$261$	−352.887	−0.0836904
$262$	566.933i	0.133684i
$263$	1823.12i	0.427447i	0.976894	+	0.213724i	$0.0685592\pi$
$263$	1823.12i	0.427447i	−0.976894	+	0.213724i	$0.931441\pi$
$264$	192.986	0.0449904
$265$	0	0
$266$	−187.437	−0.0432048
$267$	4348.97i	0.996826i
$268$	1593.41i	0.363183i
$269$	7312.95	1.65754	0.828770	−	0.559590i	$-0.189041\pi$
$269$	7312.95	1.65754	0.828770	+	0.559590i	$0.189041\pi$
$270$	0	0
$271$	6168.14	1.38261	0.691306	−	0.722562i	$-0.257036\pi$
$271$	6168.14	1.38261	0.691306	+	0.722562i	$0.257036\pi$
$272$	6612.99i	1.47416i
$273$	− 4022.46i	− 0.891760i
$274$	−694.907	−0.153215
$275$	0	0
$276$	1426.00	0.310997
$277$	− 1858.13i	− 0.403048i	−0.979484	−	0.201524i	$-0.935411\pi$
$277$	− 1858.13i	− 0.403048i	0.979484	−	0.201524i	$-0.0645894\pi$
$278$	177.801i	0.0383589i
$279$	202.244	0.0433979
$280$	0	0
$281$	−7716.88	−1.63826	−0.819129	−	0.573609i	$-0.805543\pi$
$281$	−7716.88	−1.63826	−0.819129	+	0.573609i	$0.805543\pi$
$282$	569.408i	0.120240i
$283$	6582.51i	1.38265i	0.722545	+	0.691324i	$0.242972\pi$
$283$	6582.51i	1.38265i	−0.722545	+	0.691324i	$0.757028\pi$
$284$	−5074.19	−1.06020
$285$	0	0
$286$	−204.534	−0.0422879
$287$	2560.12i	0.526548i
$288$	− 622.634i	− 0.127393i
$289$	−6933.84	−1.41133
$290$	0	0
$291$	−4808.10	−0.968576
$292$	− 8035.17i	− 1.61035i
$293$	4788.12i	0.954692i	0.878715	+	0.477346i	$0.158401\pi$
$293$	4788.12i	0.954692i	−0.878715	+	0.477346i	$0.841599\pi$
$294$	402.131	0.0797714
$295$	0	0
$296$	2021.20	0.396891
$297$	297.000i	0.0580259i
$298$	279.072i	0.0542490i
$299$	−3048.78	−0.589683
$300$	0	0
$301$	−8907.43	−1.70570
$302$	− 1076.05i	− 0.205032i
$303$	− 2911.96i	− 0.552104i
$304$	−1162.15	−0.219256
$305$	0	0
$306$	361.110	0.0674617
$307$	− 6612.72i	− 1.22934i	−0.788784	−	0.614671i	$-0.789289\pi$
$307$	− 6612.72i	− 1.22934i	0.788784	−	0.614671i	$-0.210711\pi$
$308$	2299.51i	0.425412i
$309$	−556.423	−0.102439
$310$	0	0
$311$	−1915.55	−0.349264	−0.174632	−	0.984634i	$-0.555874\pi$
$311$	−1915.55	−0.349264	−0.174632	+	0.984634i	$0.555874\pi$
$312$	884.931i	0.160575i
$313$	6383.65i	1.15280i	0.817169	+	0.576398i	$0.195542\pi$
$313$	6383.65i	1.15280i	−0.817169	+	0.576398i	$0.804458\pi$
$314$	498.632	0.0896161
$315$	0	0
$316$	−2527.38	−0.449924
$317$	10128.7i	1.79459i	0.441434	+	0.897293i	$0.354470\pi$
$317$	10128.7i	1.79459i	−0.441434	+	0.897293i	$0.645530\pi$
$318$	145.525i	0.0256624i
$319$	−431.307	−0.0757008
$320$	0	0
$321$	1957.56	0.340374
$322$	− 592.295i	− 0.102507i
$323$	− 2081.93i	− 0.358643i
$324$	636.993	0.109224
$325$	0	0
$326$	−1116.09	−0.189615
$327$	2745.33i	0.464272i
$328$	− 563.221i	− 0.0948131i
$329$	−13686.7	−2.29354
$330$	0	0
$331$	3453.52	0.573483	0.286741	−	0.958008i	$-0.407428\pi$
$331$	3453.52	0.573483	0.286741	+	0.958008i	$0.407428\pi$
$332$	− 6610.44i	− 1.09276i
$333$	3110.57i	0.511886i
$334$	1557.73	0.255196
$335$	0	0
$336$	4845.20	0.786689
$337$	− 4246.90i	− 0.686480i	−0.939248	−	0.343240i	$-0.888476\pi$
$337$	− 4246.90i	− 0.686480i	0.939248	−	0.343240i	$-0.111524\pi$
$338$	− 127.995i	− 0.0205977i
$339$	981.190	0.157200
$340$	0	0
$341$	247.187	0.0392549
$342$	63.4605i	0.0100338i
$343$	548.198i	0.0862971i
$344$	1959.61	0.307137
$345$	0	0
$346$	558.554	0.0867863
$347$	10332.8i	1.59854i	0.600975	+	0.799268i	$0.294779\pi$
$347$	10332.8i	1.59854i	−0.600975	+	0.799268i	$0.705221\pi$
$348$	925.048i	0.142494i
$349$	3899.90	0.598156	0.299078	−	0.954229i	$-0.403321\pi$
$349$	3899.90	0.598156	0.299078	+	0.954229i	$0.403321\pi$
$350$	0	0
$351$	−1361.89	−0.207100
$352$	− 760.998i	− 0.115231i
$353$	7564.03i	1.14049i	0.821475	+	0.570245i	$0.193152\pi$
$353$	7564.03i	1.14049i	−0.821475	+	0.570245i	$0.806848\pi$
$354$	−233.045	−0.0349892
$355$	0	0
$356$	11400.3	1.69723
$357$	8679.93i	1.28681i
$358$	− 400.861i	− 0.0591793i
$359$	11198.7	1.64637	0.823185	−	0.567773i	$-0.192195\pi$
$359$	11198.7	1.64637	0.823185	+	0.567773i	$0.192195\pi$
$360$	0	0
$361$	−6493.13	−0.946658
$362$	− 53.8183i	− 0.00781389i
$363$	363.000i	0.0524864i
$364$	−10544.4	−1.51834
$365$	0	0
$366$	76.2842	0.0108946
$367$	− 5939.65i	− 0.844816i	−0.906406	−	0.422408i	$-0.861185\pi$
$367$	− 5939.65i	− 0.844816i	0.906406	−	0.422408i	$-0.138815\pi$
$368$	− 3672.36i	− 0.520204i
$369$	866.782	0.122284
$370$	0	0
$371$	−3497.96	−0.489501
$372$	− 530.156i	− 0.0738906i
$373$	20.1820i	0.00280157i	0.999999	+	0.00140078i	$0.000445884\pi$
$373$	20.1820i	0.00280157i	−0.999999	+	0.00140078i	$0.999554\pi$
$374$	441.357	0.0610214
$375$	0	0
$376$	3011.05	0.412987
$377$	− 1977.75i	− 0.270183i
$378$	− 264.578i	− 0.0360011i
$379$	−8241.81	−1.11703	−0.558514	−	0.829495i	$-0.688628\pi$
$379$	−8241.81	−1.11703	−0.558514	+	0.829495i	$0.688628\pi$
$380$	0	0
$381$	5218.17	0.701667
$382$	1322.20i	0.177094i
$383$	9971.48i	1.33034i	0.746694	+	0.665168i	$0.231640\pi$
$383$	9971.48i	1.33034i	−0.746694	+	0.665168i	$0.768360\pi$
$384$	−2169.69	−0.288337
$385$	0	0
$386$	374.367	0.0493647
$387$	3015.79i	0.396127i
$388$	12603.8i	1.64913i
$389$	−6897.31	−0.898991	−0.449496	−	0.893283i	$-0.648396\pi$
$389$	−6897.31	−0.898991	−0.449496	+	0.893283i	$0.648396\pi$
$390$	0	0
$391$	6578.85	0.850913
$392$	− 2126.49i	− 0.273989i
$393$	4613.78i	0.592200i
$394$	−1103.19	−0.141060
$395$	0	0
$396$	778.547	0.0987966
$397$	− 10587.7i	− 1.33849i	−0.743043	−	0.669243i	$-0.766618\pi$
$397$	− 10587.7i	− 1.33849i	0.743043	−	0.669243i	$-0.233382\pi$
$398$	1204.54i	0.151704i
$399$	−1525.39	−0.191391
$400$	0	0
$401$	5700.87	0.709944	0.354972	−	0.934877i	$-0.384490\pi$
$401$	5700.87	0.709944	0.354972	+	0.934877i	$0.384490\pi$
$402$	− 224.076i	− 0.0278007i
$403$	1133.47i	0.140105i
$404$	−7633.32	−0.940029
$405$	0	0
$406$	384.223	0.0469672
$407$	3801.81i	0.463019i
$408$	− 1909.56i	− 0.231710i
$409$	−2389.92	−0.288934	−0.144467	−	0.989510i	$-0.546147\pi$
$409$	−2389.92	−0.288934	−0.144467	+	0.989510i	$0.546147\pi$
$410$	0	0
$411$	−5655.25	−0.678718
$412$	1458.59i	0.174416i
$413$	− 5601.65i	− 0.667407i
$414$	−200.534	−0.0238060
$415$	0	0
$416$	3489.53	0.411270
$417$	1446.97i	0.169924i
$418$	77.5629i	0.00907589i
$419$	−2338.30	−0.272633	−0.136316	−	0.990665i	$-0.543526\pi$
$419$	−2338.30	−0.272633	−0.136316	+	0.990665i	$0.543526\pi$
$420$	0	0
$421$	−9848.22	−1.14008	−0.570039	−	0.821618i	$-0.693072\pi$
$421$	−9848.22	−1.14008	−0.570039	+	0.821618i	$0.693072\pi$
$422$	− 1575.92i	− 0.181788i
$423$	4633.92i	0.532646i
$424$	769.542	0.0881422
$425$	0	0
$426$	713.566	0.0811559
$427$	1833.63i	0.207811i
$428$	− 5131.48i	− 0.579531i
$429$	−1664.53	−0.187329
$430$	0	0
$431$	13734.9	1.53501	0.767504	−	0.641045i	$-0.221499\pi$
$431$	13734.9	1.53501	0.767504	+	0.641045i	$0.221499\pi$
$432$	− 1640.44i	− 0.182699i
$433$	300.831i	0.0333881i	0.999861	+	0.0166940i	$0.00531412\pi$
$433$	300.831i	0.0333881i	−0.999861	+	0.0166940i	$0.994686\pi$
$434$	−220.203	−0.0243550
$435$	0	0
$436$	7196.52	0.790483
$437$	1156.15i	0.126559i
$438$	1129.96i	0.123268i
$439$	−12002.9	−1.30494	−0.652468	−	0.757816i	$-0.726266\pi$
$439$	−12002.9	−1.30494	−0.652468	+	0.757816i	$0.726266\pi$
$440$	0	0
$441$	3272.60	0.353375
$442$	2023.83i	0.217791i
$443$	− 9448.03i	− 1.01329i	−0.862153	−	0.506647i	$-0.830885\pi$
$443$	− 9448.03i	− 1.01329i	0.862153	−	0.506647i	$-0.169115\pi$
$444$	8153.95	0.871553
$445$	0	0
$446$	−1390.21	−0.147597
$447$	2271.13i	0.240315i
$448$	− 12242.6i	− 1.29109i
$449$	5412.07	0.568845	0.284422	−	0.958699i	$-0.408198\pi$
$449$	5412.07	0.568845	0.284422	+	0.958699i	$0.408198\pi$
$450$	0	0
$451$	1059.40	0.110610
$452$	− 2572.06i	− 0.267654i
$453$	− 8757.04i	− 0.908259i
$454$	−1065.94	−0.110191
$455$	0	0
$456$	335.582	0.0344628
$457$	10009.7i	1.02458i	0.858812	+	0.512291i	$0.171203\pi$
$457$	10009.7i	1.02458i	−0.858812	+	0.512291i	$0.828797\pi$
$458$	− 339.626i	− 0.0346500i
$459$	2938.77	0.298845
$460$	0	0
$461$	9058.72	0.915199	0.457599	−	0.889159i	$-0.348709\pi$
$461$	9058.72	0.915199	0.457599	+	0.889159i	$0.348709\pi$
$462$	− 323.373i	− 0.0325642i
$463$	16017.2i	1.60774i	0.594804	+	0.803871i	$0.297230\pi$
$463$	16017.2i	1.60774i	−0.594804	+	0.803871i	$0.702770\pi$
$464$	2382.27	0.238349
$465$	0	0
$466$	1039.05	0.103290
$467$	− 10318.0i	− 1.02240i	−0.859461	−	0.511201i	$-0.829201\pi$
$467$	− 10318.0i	− 1.02240i	0.859461	−	0.511201i	$-0.170799\pi$
$468$	3570.00i	0.352614i
$469$	5386.07	0.530289
$470$	0	0
$471$	4057.94	0.396985
$472$	1232.35i	0.120177i
$473$	3685.97i	0.358311i
$474$	355.416	0.0344405
$475$	0	0
$476$	22753.3	2.19096
$477$	1184.30i	0.113680i
$478$	− 288.870i	− 0.0276414i
$479$	9810.96	0.935854	0.467927	−	0.883767i	$-0.345001\pi$
$479$	9810.96	0.935854	0.467927	+	0.883767i	$0.345001\pi$
$480$	0	0
$481$	−17433.1	−1.65256
$482$	2171.99i	0.205252i
$483$	− 4820.19i	− 0.454091i
$484$	951.557	0.0893649
$485$	0	0
$486$	−89.5782	−0.00836080
$487$	8081.87i	0.752001i	0.926619	+	0.376001i	$0.122701\pi$
$487$	8081.87i	0.752001i	−0.926619	+	0.376001i	$0.877299\pi$
$488$	− 403.394i	− 0.0374196i
$489$	−9082.90	−0.839965
$490$	0	0
$491$	−5645.54	−0.518900	−0.259450	−	0.965757i	$-0.583541\pi$
$491$	−5645.54	−0.518900	−0.259450	+	0.965757i	$0.583541\pi$
$492$	− 2272.16i	− 0.208205i
$493$	4267.71i	0.389874i
$494$	−355.662	−0.0323927
$495$	0	0
$496$	−1365.31	−0.123597
$497$	17151.8i	1.54802i
$498$	929.604i	0.0836477i
$499$	−4176.04	−0.374640	−0.187320	−	0.982299i	$-0.559980\pi$
$499$	−4176.04	−0.374640	−0.187320	+	0.982299i	$0.559980\pi$
$500$	0	0
$501$	12677.1	1.13048
$502$	− 2909.52i	− 0.258681i
$503$	− 9131.76i	− 0.809473i	−0.914433	−	0.404737i	$-0.867363\pi$
$503$	− 9131.76i	− 0.809473i	0.914433	−	0.404737i	$-0.132637\pi$
$504$	−1399.10	−0.123652
$505$	0	0
$506$	−245.097	−0.0215333
$507$	− 1041.64i	− 0.0912444i
$508$	− 13678.8i	− 1.19468i
$509$	−21062.6	−1.83415	−0.917076	−	0.398712i	$-0.869457\pi$
$509$	−21062.6	−1.83415	−0.917076	+	0.398712i	$0.869457\pi$
$510$	0	0
$511$	−27160.6	−2.35130
$512$	7045.76i	0.608167i
$513$	516.451i	0.0444481i
$514$	−1244.41	−0.106787
$515$	0	0
$516$	7905.50	0.674458
$517$	5663.68i	0.481796i
$518$	− 3386.78i	− 0.287271i
$519$	4545.59	0.384450
$520$	0	0
$521$	−6858.67	−0.576744	−0.288372	−	0.957518i	$-0.593114\pi$
$521$	−6858.67	−0.576744	−0.288372	+	0.957518i	$0.593114\pi$
$522$	− 130.086i	− 0.0109075i
$523$	− 5752.51i	− 0.480956i	−0.970655	−	0.240478i	$-0.922696\pi$
$523$	− 5752.51i	− 0.480956i	0.970655	−	0.240478i	$-0.0773041\pi$
$524$	12094.4	1.00830
$525$	0	0
$526$	−672.067	−0.0557101
$527$	− 2445.87i	− 0.202171i
$528$	− 2004.98i	− 0.165257i
$529$	8513.60	0.699729
$530$	0	0
$531$	−1896.55	−0.154997
$532$	3998.61i	0.325868i
$533$	4857.85i	0.394778i
$534$	−1603.18	−0.129918
$535$	0	0
$536$	−1184.92	−0.0954866
$537$	− 3262.27i	− 0.262155i
$538$	2695.80i	0.216030i
$539$	3999.85	0.319639
$540$	0	0
$541$	4957.66	0.393986	0.196993	−	0.980405i	$-0.436882\pi$
$541$	4957.66	0.393986	0.196993	+	0.980405i	$0.436882\pi$
$542$	2273.79i	0.180199i
$543$	− 437.981i	− 0.0346143i
$544$	−7529.95	−0.593463
$545$	0	0
$546$	1482.82	0.116225
$547$	− 19515.0i	− 1.52542i	−0.646742	−	0.762709i	$-0.723869\pi$
$547$	− 19515.0i	− 1.52542i	0.646742	−	0.762709i	$-0.276131\pi$
$548$	14824.5i	1.15560i
$549$	620.812	0.0482616
$550$	0	0
$551$	−749.996	−0.0579871
$552$	1060.43i	0.0817660i
$553$	8543.07i	0.656941i
$554$	684.971	0.0525301
$555$	0	0
$556$	3793.04	0.289318
$557$	− 227.163i	− 0.0172804i	−0.999963	−	0.00864022i	$-0.997250\pi$
$557$	− 227.163i	− 0.0172804i	0.999963	−	0.00864022i	$-0.00275030\pi$
$558$	74.5541i	0.00565614i
$559$	−16901.9	−1.27884
$560$	0	0
$561$	3591.83	0.270316
$562$	− 2844.71i	− 0.213517i
$563$	18313.6i	1.37092i	0.728110	+	0.685460i	$0.240399\pi$
$563$	18313.6i	1.37092i	−0.728110	+	0.685460i	$0.759601\pi$
$564$	12147.2	0.906898
$565$	0	0
$566$	−2426.54	−0.180203
$567$	− 2153.17i	− 0.159479i
$568$	− 3773.36i	− 0.278744i
$569$	5703.35	0.420206	0.210103	−	0.977679i	$-0.432620\pi$
$569$	5703.35	0.420206	0.210103	+	0.977679i	$0.432620\pi$
$570$	0	0
$571$	−14797.1	−1.08448	−0.542241	−	0.840223i	$-0.682424\pi$
$571$	−14797.1	−1.08448	−0.542241	+	0.840223i	$0.682424\pi$
$572$	4363.34i	0.318952i
$573$	10760.3i	0.784497i
$574$	−943.750	−0.0686261
$575$	0	0
$576$	−4144.98	−0.299840
$577$	2045.44i	0.147578i	0.997274	+	0.0737891i	$0.0235092\pi$
$577$	2045.44i	0.147578i	−0.997274	+	0.0737891i	$0.976491\pi$
$578$	− 2556.05i	− 0.183941i
$579$	3046.65	0.218678
$580$	0	0
$581$	−22344.7	−1.59555
$582$	− 1772.43i	− 0.126236i
$583$	1447.48i	0.102828i
$584$	5975.27	0.423387
$585$	0	0
$586$	−1765.06	−0.124427
$587$	− 10482.6i	− 0.737075i	−0.929613	−	0.368537i	$-0.879859\pi$
$587$	− 10482.6i	− 0.737075i	0.929613	−	0.368537i	$-0.120141\pi$
$588$	− 8578.70i	− 0.601666i
$589$	429.831	0.0300694
$590$	0	0
$591$	−8977.89	−0.624875
$592$	− 20998.8i	− 1.45785i
$593$	938.019i	0.0649575i	0.999472	+	0.0324788i	$0.0103401\pi$
$593$	938.019i	0.0649575i	−0.999472	+	0.0324788i	$0.989660\pi$
$594$	−109.484	−0.00756263
$595$	0	0
$596$	5953.47	0.409167
$597$	9802.70i	0.672023i
$598$	− 1123.88i	− 0.0768546i
$599$	−2885.83	−0.196848	−0.0984239	−	0.995145i	$-0.531380\pi$
$599$	−2885.83	−0.196848	−0.0984239	+	0.995145i	$0.531380\pi$
$600$	0	0
$601$	−25636.7	−1.74000	−0.870001	−	0.493049i	$-0.835882\pi$
$601$	−25636.7	−1.74000	−0.870001	+	0.493049i	$0.835882\pi$
$602$	− 3283.59i	− 0.222307i
$603$	− 1823.56i	− 0.123153i
$604$	−22955.4	−1.54643
$605$	0	0
$606$	1073.45	0.0719568
$607$	− 9924.91i	− 0.663657i	−0.943340	−	0.331828i	$-0.892335\pi$
$607$	− 9924.91i	− 0.663657i	0.943340	−	0.331828i	$-0.107665\pi$
$608$	− 1323.29i	− 0.0882674i
$609$	3126.86	0.208057
$610$	0	0
$611$	−25970.7	−1.71958
$612$	− 7703.59i	− 0.508822i
$613$	9514.05i	0.626866i	0.949610	+	0.313433i	$0.101479\pi$
$613$	9514.05i	0.626866i	−0.949610	+	0.313433i	$0.898521\pi$
$614$	2437.68	0.160222
$615$	0	0
$616$	−1710.01	−0.111848
$617$	− 25714.5i	− 1.67784i	−0.544255	−	0.838920i	$-0.683188\pi$
$617$	− 25714.5i	− 1.67784i	0.544255	−	0.838920i	$-0.316812\pi$
$618$	− 205.117i	− 0.0133511i
$619$	17933.5	1.16447	0.582235	−	0.813020i	$-0.302178\pi$
$619$	17933.5	1.16447	0.582235	+	0.813020i	$0.302178\pi$
$620$	0	0
$621$	−1631.97	−0.105457
$622$	− 706.139i	− 0.0455203i
$623$	− 38535.3i	− 2.47815i
$624$	9193.80	0.589818
$625$	0	0
$626$	−2353.23	−0.150246
$627$	631.218i	0.0402048i
$628$	− 10637.4i	− 0.675919i
$629$	37618.3	2.38464
$630$	0	0
$631$	−3836.00	−0.242011	−0.121005	−	0.992652i	$-0.538612\pi$
$631$	−3836.00	−0.242011	−0.121005	+	0.992652i	$0.538612\pi$
$632$	− 1879.45i	− 0.118292i
$633$	− 12825.0i	− 0.805292i
$634$	−3733.78	−0.233892
$635$	0	0
$636$	3104.50	0.193556
$637$	18341.2i	1.14082i
$638$	− 158.995i	− 0.00986623i
$639$	5807.10	0.359508
$640$	0	0
$641$	20430.4	1.25890	0.629449	−	0.777042i	$-0.283281\pi$
$641$	20430.4	1.25890	0.629449	+	0.777042i	$0.283281\pi$
$642$	721.622i	0.0443616i
$643$	− 14252.9i	− 0.874152i	−0.899424	−	0.437076i	$-0.856014\pi$
$643$	− 14252.9i	− 0.874152i	0.899424	−	0.437076i	$-0.143986\pi$
$644$	−12635.5	−0.773149
$645$	0	0
$646$	767.472	0.0467427
$647$	18729.4i	1.13807i	0.822315	+	0.569033i	$0.192682\pi$
$647$	18729.4i	1.13807i	−0.822315	+	0.569033i	$0.807318\pi$
$648$	473.693i	0.0287167i
$649$	−2318.01	−0.140200
$650$	0	0
$651$	−1792.04	−0.107889
$652$	23809.6i	1.43015i
$653$	4415.88i	0.264635i	0.991207	+	0.132318i	$0.0422419\pi$
$653$	4415.88i	0.264635i	−0.991207	+	0.132318i	$0.957758\pi$
$654$	−1012.02	−0.0605095
$655$	0	0
$656$	−5851.46	−0.348264
$657$	9195.77i	0.546060i
$658$	− 5045.40i	− 0.298922i
$659$	−7391.89	−0.436946	−0.218473	−	0.975843i	$-0.570107\pi$
$659$	−7391.89	−0.436946	−0.218473	+	0.975843i	$0.570107\pi$
$660$	0	0
$661$	−640.842	−0.0377093	−0.0188547	−	0.999822i	$-0.506002\pi$
$661$	−640.842	−0.0377093	−0.0188547	+	0.999822i	$0.506002\pi$
$662$	1273.09i	0.0747431i
$663$	16470.2i	0.964782i
$664$	4915.78	0.287303
$665$	0	0
$666$	−1146.66	−0.0667152
$667$	− 2369.97i	− 0.137579i
$668$	− 33231.3i	− 1.92479i
$669$	−11313.7	−0.653832
$670$	0	0
$671$	758.770	0.0436542
$672$	5517.03i	0.316703i
$673$	− 19129.0i	− 1.09564i	−0.836595	−	0.547822i	$-0.815457\pi$
$673$	− 19129.0i	− 1.09564i	0.836595	−	0.547822i	$-0.184543\pi$
$674$	1565.56	0.0894702
$675$	0	0
$676$	−2730.53	−0.155355
$677$	4043.27i	0.229536i	0.993392	+	0.114768i	$0.0366124\pi$
$677$	4043.27i	0.229536i	−0.993392	+	0.114768i	$0.963388\pi$
$678$	361.700i	0.0204882i
$679$	42603.5	2.40791
$680$	0	0
$681$	−8674.74	−0.488130
$682$	91.1216i	0.00511617i
$683$	26633.3i	1.49209i	0.665897	+	0.746044i	$0.268049\pi$
$683$	26633.3i	1.49209i	−0.665897	+	0.746044i	$0.731951\pi$
$684$	1353.81	0.0756786
$685$	0	0
$686$	−202.085	−0.0112473
$687$	− 2763.93i	− 0.153494i
$688$	− 20359.0i	− 1.12817i
$689$	−6637.40	−0.367002
$690$	0	0
$691$	30662.4	1.68806	0.844032	−	0.536293i	$-0.180176\pi$
$691$	30662.4	1.68806	0.844032	+	0.536293i	$0.180176\pi$
$692$	− 11915.7i	− 0.654575i
$693$	− 2631.66i	− 0.144254i
$694$	−3809.01	−0.208340
$695$	0	0
$696$	−687.902	−0.0374639
$697$	− 10482.6i	− 0.569665i
$698$	1437.64i	0.0779589i
$699$	8455.96	0.457559
$700$	0	0
$701$	18443.3	0.993717	0.496858	−	0.867832i	$-0.334487\pi$
$701$	18443.3	0.993717	0.496858	+	0.867832i	$0.334487\pi$
$702$	− 502.038i	− 0.0269917i
$703$	6610.93i	0.354674i
$704$	−5066.09	−0.271215
$705$	0	0
$706$	−2788.36	−0.148642
$707$	25802.3i	1.37255i
$708$	4971.56i	0.263902i
$709$	21693.3	1.14910	0.574548	−	0.818471i	$-0.305178\pi$
$709$	21693.3	1.14910	0.574548	+	0.818471i	$0.305178\pi$
$710$	0	0
$711$	2892.43	0.152566
$712$	8477.67i	0.446228i
$713$	1358.26i	0.0713423i
$714$	−3199.72	−0.167712
$715$	0	0
$716$	−8551.61	−0.446353
$717$	− 2350.87i	− 0.122447i
$718$	4128.24i	0.214575i
$719$	28555.1	1.48112	0.740560	−	0.671990i	$-0.234560\pi$
$719$	28555.1	1.48112	0.740560	+	0.671990i	$0.234560\pi$
$720$	0	0
$721$	4930.35	0.254668
$722$	− 2393.59i	− 0.123380i
$723$	17676.0i	0.909234i
$724$	−1148.11	−0.0589354
$725$	0	0
$726$	−133.814	−0.00684065
$727$	− 21629.1i	− 1.10341i	−0.834040	−	0.551704i	$-0.813978\pi$
$727$	− 21629.1i	− 1.10341i	0.834040	−	0.551704i	$-0.186022\pi$
$728$	− 7841.19i	− 0.399195i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	36472.0	1.84537
$732$	− 1627.38i	− 0.0821716i
$733$	13081.1i	0.659157i	0.944128	+	0.329579i	$0.106907\pi$
$733$	13081.1i	0.659157i	−0.944128	+	0.329579i	$0.893093\pi$
$734$	2189.56	0.110106
$735$	0	0
$736$	4181.57	0.209422
$737$	− 2228.80i	− 0.111396i
$738$	319.526i	0.0159375i
$739$	24296.5	1.20942	0.604710	−	0.796446i	$-0.293289\pi$
$739$	24296.5	1.20942	0.604710	+	0.796446i	$0.293289\pi$
$740$	0	0
$741$	−2894.43	−0.143495
$742$	− 1289.47i	− 0.0637977i
$743$	7572.81i	0.373916i	0.982368	+	0.186958i	$0.0598628\pi$
$743$	7572.81i	0.373916i	−0.982368	+	0.186958i	$0.940137\pi$
$744$	394.245	0.0194270
$745$	0	0
$746$	−7.43978	−0.000365134 0
$747$	7565.25i	0.370546i
$748$	− 9415.50i	− 0.460247i
$749$	−17345.5	−0.846182
$750$	0	0
$751$	38506.1	1.87098	0.935492	−	0.353347i	$-0.114957\pi$
$751$	38506.1	1.87098	0.935492	+	0.353347i	$0.114957\pi$
$752$	− 31282.6i	− 1.51697i
$753$	− 23678.1i	− 1.14592i
$754$	729.065	0.0352135
$755$	0	0
$756$	−5644.26	−0.271534
$757$	− 22513.0i	− 1.08091i	−0.841373	−	0.540455i	$-0.818252\pi$
$757$	− 22513.0i	− 1.08091i	0.841373	−	0.540455i	$-0.181748\pi$
$758$	− 3038.21i	− 0.145584i
$759$	−1994.63	−0.0953894
$760$	0	0
$761$	−29062.0	−1.38436	−0.692179	−	0.721726i	$-0.743349\pi$
$761$	−29062.0	−1.38436	−0.692179	+	0.721726i	$0.743349\pi$
$762$	1923.60i	0.0914496i
$763$	− 24325.8i	− 1.15420i
$764$	28206.7	1.33571
$765$	0	0
$766$	−3675.83	−0.173385
$767$	− 10629.2i	− 0.500387i
$768$	10253.5i	0.481758i
$769$	17769.2	0.833257	0.416628	−	0.909077i	$-0.363212\pi$
$769$	17769.2	0.833257	0.416628	+	0.909077i	$0.363212\pi$
$770$	0	0
$771$	−10127.2	−0.473051
$772$	− 7986.41i	− 0.372328i
$773$	− 15074.0i	− 0.701391i	−0.936490	−	0.350695i	$-0.885945\pi$
$773$	− 15074.0i	− 0.701391i	0.936490	−	0.350695i	$-0.114055\pi$
$774$	−1111.72	−0.0516280
$775$	0	0
$776$	−9372.67	−0.433582
$777$	− 27562.1i	− 1.27257i
$778$	− 2542.59i	− 0.117167i
$779$	1842.18	0.0847279
$780$	0	0
$781$	7097.57	0.325187
$782$	2425.19i	0.110901i
$783$	− 1058.66i	− 0.0483187i
$784$	−22092.6	−1.00641
$785$	0	0
$786$	−1700.80	−0.0771826
$787$	− 15445.3i	− 0.699573i	−0.936829	−	0.349787i	$-0.886254\pi$
$787$	− 15445.3i	− 0.699573i	0.936829	−	0.349787i	$-0.113746\pi$
$788$	23534.4i	1.06393i
$789$	−5469.37	−0.246787
$790$	0	0
$791$	−8694.12	−0.390806
$792$	578.958i	0.0259752i
$793$	3479.32i	0.155806i
$794$	3902.97	0.174448
$795$	0	0
$796$	25696.5	1.14421
$797$	− 14902.8i	− 0.662337i	−0.943572	−	0.331169i	$-0.892557\pi$
$797$	− 14902.8i	− 0.662337i	0.943572	−	0.331169i	$-0.107443\pi$
$798$	− 562.310i	− 0.0249443i
$799$	56041.2	2.48135
$800$	0	0
$801$	−13046.9	−0.575518
$802$	2101.54i	0.0925284i
$803$	11239.3i	0.493930i
$804$	−4780.23	−0.209684
$805$	0	0
$806$	−417.836	−0.0182601
$807$	21938.8i	0.956981i
$808$	− 5676.43i	− 0.247149i
$809$	−19642.4	−0.853633	−0.426816	−	0.904338i	$-0.640365\pi$
$809$	−19642.4	−0.853633	−0.426816	+	0.904338i	$0.640365\pi$
$810$	0	0
$811$	16729.7	0.724365	0.362182	−	0.932107i	$-0.382032\pi$
$811$	16729.7	0.724365	0.362182	+	0.932107i	$0.382032\pi$
$812$	− 8196.66i	− 0.354244i
$813$	18504.4i	0.798252i
$814$	−1401.48	−0.0603461
$815$	0	0
$816$	−19839.0	−0.851107
$817$	6409.50i	0.274467i
$818$	− 881.007i	− 0.0376573i
$819$	12067.4	0.514858
$820$	0	0
$821$	−10838.2	−0.460726	−0.230363	−	0.973105i	$-0.573991\pi$
$821$	−10838.2	−0.460726	−0.230363	+	0.973105i	$0.573991\pi$
$822$	− 2084.72i	− 0.0884586i
$823$	− 12375.6i	− 0.524164i	−0.965046	−	0.262082i	$-0.915591\pi$
$823$	− 12375.6i	− 0.524164i	0.965046	−	0.262082i	$-0.0844090\pi$
$824$	−1084.66	−0.0458569
$825$	0	0
$826$	2064.96	0.0869845
$827$	− 30336.4i	− 1.27558i	−0.770212	−	0.637788i	$-0.779850\pi$
$827$	− 30336.4i	− 1.27558i	0.770212	−	0.637788i	$-0.220150\pi$
$828$	4278.00i	0.179554i
$829$	26159.0	1.09595	0.547973	−	0.836496i	$-0.315400\pi$
$829$	26159.0	1.09595	0.547973	+	0.836496i	$0.315400\pi$
$830$	0	0
$831$	5574.40	0.232700
$832$	− 23230.4i	− 0.967993i
$833$	− 39577.9i	− 1.64621i
$834$	−533.402	−0.0221465
$835$	0	0
$836$	1654.65	0.0684539
$837$	606.732i	0.0250558i
$838$	− 861.976i	− 0.0355328i
$839$	−22058.7	−0.907690	−0.453845	−	0.891081i	$-0.649948\pi$
$839$	−22058.7	−0.907690	−0.453845	+	0.891081i	$0.649948\pi$
$840$	0	0
$841$	−22851.6	−0.936963
$842$	− 3630.39i	− 0.148589i
$843$	− 23150.6i	− 0.945849i
$844$	−33619.2	−1.37111
$845$	0	0
$846$	−1708.22	−0.0694207
$847$	− 3216.47i	− 0.130483i
$848$	− 7994.99i	− 0.323761i
$849$	−19747.5	−0.798272
$850$	0	0
$851$	−20890.4	−0.841496
$852$	− 15222.6i	− 0.612109i
$853$	41626.5i	1.67088i	0.549578	+	0.835442i	$0.314788\pi$
$853$	41626.5i	1.67088i	−0.549578	+	0.835442i	$0.685212\pi$
$854$	−675.938	−0.0270845
$855$	0	0
$856$	3815.96	0.152368
$857$	44478.1i	1.77286i	0.462860	+	0.886431i	$0.346823\pi$
$857$	44478.1i	1.77286i	−0.462860	+	0.886431i	$0.653177\pi$
$858$	− 613.602i	− 0.0244149i
$859$	5250.15	0.208537	0.104268	−	0.994549i	$-0.466750\pi$
$859$	5250.15	0.208537	0.104268	+	0.994549i	$0.466750\pi$
$860$	0	0
$861$	−7680.37	−0.304003
$862$	5063.17i	0.200061i
$863$	8110.49i	0.319912i	0.987124	+	0.159956i	$0.0511353\pi$
$863$	8110.49i	0.319912i	−0.987124	+	0.159956i	$0.948865\pi$
$864$	1867.90	0.0735502
$865$	0	0
$866$	−110.897	−0.00435153
$867$	− 20801.5i	− 0.814829i
$868$	4697.60i	0.183695i
$869$	3535.19	0.138001
$870$	0	0
$871$	10220.1	0.397583
$872$	5351.61i	0.207831i
$873$	− 14424.3i	− 0.559208i
$874$	−426.196	−0.0164946
$875$	0	0
$876$	24105.5	0.929737
$877$	− 11508.7i	− 0.443126i	−0.975146	−	0.221563i	$-0.928884\pi$
$877$	− 11508.7i	− 0.443126i	0.975146	−	0.221563i	$-0.0711159\pi$
$878$	− 4424.68i	− 0.170075i
$879$	−14364.4	−0.551192
$880$	0	0
$881$	−15689.1	−0.599978	−0.299989	−	0.953943i	$-0.596983\pi$
$881$	−15689.1	−0.599978	−0.299989	+	0.953943i	$0.596983\pi$
$882$	1206.39i	0.0460560i
$883$	12353.4i	0.470809i	0.971897	+	0.235405i	$0.0756415\pi$
$883$	12353.4i	0.470809i	−0.971897	+	0.235405i	$0.924358\pi$
$884$	43174.5	1.64267
$885$	0	0
$886$	3482.87	0.132065
$887$	− 5146.08i	− 0.194801i	−0.995245	−	0.0974005i	$-0.968947\pi$
$887$	− 5146.08i	− 0.194801i	0.995245	−	0.0974005i	$-0.0310528\pi$
$888$	6063.60i	0.229145i
$889$	−46237.1	−1.74437
$890$	0	0
$891$	−891.000	−0.0335013
$892$	29657.4i	1.11323i
$893$	9848.53i	0.369058i
$894$	−837.216	−0.0313207
$895$	0	0
$896$	19225.1	0.716816
$897$	− 9146.33i	− 0.340454i
$898$	1995.07i	0.0741386i
$899$	−881.103	−0.0326879
$900$	0	0
$901$	14322.6	0.529584
$902$	390.531i	0.0144160i
$903$	− 26722.3i	− 0.984786i
$904$	1912.68	0.0703705
$905$	0	0
$906$	3228.15	0.118375
$907$	19902.0i	0.728594i	0.931283	+	0.364297i	$0.118691\pi$
$907$	19902.0i	0.728594i	−0.931283	+	0.364297i	$0.881309\pi$
$908$	22739.7i	0.831105i
$909$	8735.88	0.318758
$910$	0	0
$911$	−6980.14	−0.253856	−0.126928	−	0.991912i	$-0.540512\pi$
$911$	−6980.14	−0.253856	−0.126928	+	0.991912i	$0.540512\pi$
$912$	− 3486.45i	− 0.126588i
$913$	9246.42i	0.335172i
$914$	−3689.92	−0.133536
$915$	0	0
$916$	−7245.27	−0.261343
$917$	− 40881.8i	− 1.47223i
$918$	1083.33i	0.0389491i
$919$	13632.2	0.489320	0.244660	−	0.969609i	$-0.421324\pi$
$919$	13632.2	0.489320	0.244660	+	0.969609i	$0.421324\pi$
$920$	0	0
$921$	19838.2	0.709760
$922$	3339.36i	0.119280i
$923$	32545.7i	1.16062i
$924$	−6898.54	−0.245612
$925$	0	0
$926$	−5904.50	−0.209540
$927$	− 1669.27i	− 0.0591435i
$928$	2712.59i	0.0959538i
$929$	−48008.6	−1.69549	−0.847746	−	0.530403i	$-0.822041\pi$
$929$	−48008.6	−1.69549	−0.847746	+	0.530403i	$0.822041\pi$
$930$	0	0
$931$	6955.30	0.244845
$932$	− 22166.2i	− 0.779054i
$933$	− 5746.66i	− 0.201648i
$934$	3803.58	0.133252
$935$	0	0
$936$	−2654.79	−0.0927080
$937$	19708.7i	0.687144i	0.939126	+	0.343572i	$0.111637\pi$
$937$	19708.7i	0.687144i	−0.939126	+	0.343572i	$0.888363\pi$
$938$	1985.49i	0.0691136i
$939$	−19150.9	−0.665567
$940$	0	0
$941$	−40078.3	−1.38843	−0.694216	−	0.719767i	$-0.744249\pi$
$941$	−40078.3	−1.38843	−0.694216	+	0.719767i	$0.744249\pi$
$942$	1495.90i	0.0517399i
$943$	5821.25i	0.201024i
$944$	12803.2	0.441429
$945$	0	0
$946$	−1358.77	−0.0466993
$947$	34849.6i	1.19584i	0.801556	+	0.597919i	$0.204006\pi$
$947$	34849.6i	1.19584i	−0.801556	+	0.597919i	$0.795994\pi$
$948$	− 7582.13i	− 0.259764i
$949$	−51537.4	−1.76288
$950$	0	0
$951$	−30386.1	−1.03611
$952$	16920.2i	0.576038i
$953$	− 40133.0i	− 1.36415i	−0.731282	−	0.682076i	$-0.761077\pi$
$953$	− 40133.0i	− 1.36415i	0.731282	−	0.682076i	$-0.238923\pi$
$954$	−436.575	−0.0148162
$955$	0	0
$956$	−6162.49	−0.208482
$957$	− 1293.92i	− 0.0437059i
$958$	3616.66i	0.121972i
$959$	50110.0	1.68732
$960$	0	0
$961$	−29286.0	−0.983050
$962$	− 6426.43i	− 0.215381i
$963$	5872.67i	0.196515i
$964$	46335.2	1.54809
$965$	0	0
$966$	1776.89	0.0591826
$967$	36825.4i	1.22464i	0.790610	+	0.612320i	$0.209763\pi$
$967$	36825.4i	1.22464i	−0.790610	+	0.612320i	$0.790237\pi$
$968$	707.615i	0.0234955i
$969$	6245.80	0.207063
$970$	0	0
$971$	72.3675	0.00239174	0.00119587	−	0.999999i	$-0.499619\pi$
$971$	72.3675	0.00239174	0.00119587	+	0.999999i	$0.499619\pi$
$972$	1910.98i	0.0630604i
$973$	− 12821.3i	− 0.422437i
$974$	−2979.26	−0.0980098
$975$	0	0
$976$	−4190.97	−0.137448
$977$	25134.8i	0.823065i	0.911395	+	0.411533i	$0.135006\pi$
$977$	25134.8i	0.823065i	−0.911395	+	0.411533i	$0.864994\pi$
$978$	− 3348.27i	− 0.109474i
$979$	−15946.2	−0.520576
$980$	0	0
$981$	−8235.98	−0.268048
$982$	− 2081.14i	− 0.0676292i
$983$	− 43144.8i	− 1.39990i	−0.714190	−	0.699952i	$-0.753205\pi$
$983$	− 43144.8i	− 1.39990i	0.714190	−	0.699952i	$-0.246795\pi$
$984$	1689.66	0.0547403
$985$	0	0
$986$	−1573.23	−0.0508131
$987$	− 41060.2i	− 1.32418i
$988$	7587.38i	0.244318i
$989$	−20253.8	−0.651198
$990$	0	0
$991$	28318.0	0.907721	0.453860	−	0.891073i	$-0.350046\pi$
$991$	28318.0	0.907721	0.453860	+	0.891073i	$0.350046\pi$
$992$	− 1554.62i	− 0.0497572i
$993$	10360.6i	0.331100i
$994$	−6322.76	−0.201756
$995$	0	0
$996$	19831.3	0.630903
$997$	17969.5i	0.570811i	0.958407	+	0.285405i	$0.0921283\pi$
$997$	17969.5i	0.570811i	−0.958407	+	0.285405i	$0.907872\pi$
$998$	− 1539.43i	− 0.0488275i
$999$	−9331.71	−0.295538

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.s.199.6		10
5.2	odd	4		825.4.a.x.1.3	✓	5
5.3	odd	4		825.4.a.y.1.3	yes	5
5.4	even	2	inner	825.4.c.s.199.5		10
15.2	even	4		2475.4.a.bj.1.3		5
15.8	even	4		2475.4.a.bi.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.x.1.3	✓	5	5.2	odd	4
825.4.a.y.1.3	yes	5	5.3	odd	4
825.4.c.s.199.5		10	5.4	even	2	inner
825.4.c.s.199.6		10	1.1	even	1	trivial
2475.4.a.bi.1.3		5	15.8	even	4
2475.4.a.bj.1.3		5	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.368634i q^{2} +3.00000i q^{3} +7.86411 q^{4} -1.10590 q^{6} -26.5824i q^{7} +5.84806i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q+0.368634i q^{2} +3.00000i q^{3} +7.86411 q^{4} -1.10590 q^{6} -26.5824i q^{7} +5.84806i q^{8} -9.00000 q^{9} -11.0000 q^{11} +23.5923i q^{12} -50.4402i q^{13} +9.79918 q^{14} +60.7571 q^{16} +108.843i q^{17} -3.31771i q^{18} -19.1278 q^{19} +79.7471 q^{21} -4.05498i q^{22} -60.4434i q^{23} -17.5442 q^{24} +18.5940 q^{26} -27.0000i q^{27} -209.047i q^{28} +39.2097 q^{29} -22.4715 q^{31} +69.1816i q^{32} -33.0000i q^{33} -40.1234 q^{34} -70.7770 q^{36} -345.619i q^{37} -7.05117i q^{38} +151.321 q^{39} -96.3091 q^{41} +29.3975i q^{42} -335.088i q^{43} -86.5052 q^{44} +22.2815 q^{46} -514.880i q^{47} +182.271i q^{48} -363.623 q^{49} -326.530 q^{51} -396.667i q^{52} -131.589i q^{53} +9.95313 q^{54} +155.455 q^{56} -57.3834i q^{57} +14.4541i q^{58} +210.728 q^{59} -68.9791 q^{61} -8.28378i q^{62} +239.241i q^{63} +460.554 q^{64} +12.1649 q^{66} +202.618i q^{67} +855.955i q^{68} +181.330 q^{69} -645.234 q^{71} -52.6325i q^{72} -1021.75i q^{73} +127.407 q^{74} -150.423 q^{76} +292.406i q^{77} +55.7820i q^{78} -321.381 q^{79} +81.0000 q^{81} -35.5029i q^{82} -840.583i q^{83} +627.140 q^{84} +123.525 q^{86} +117.629i q^{87} -64.3286i q^{88} +1449.66 q^{89} -1340.82 q^{91} -475.333i q^{92} -67.4146i q^{93} +189.803 q^{94} -207.545 q^{96} +1602.70i q^{97} -134.044i q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{4} - 6 q^{6} - 90 q^{9}+O(q^{10}) \) 10 * q - 6 * q^4 - 6 * q^6 - 90 * q^9	\( 10 q - 6 q^{4} - 6 q^{6} - 90 q^{9} - 110 q^{11} - 16 q^{14} - 250 q^{16} + 114 q^{19} - 108 q^{21} - 18 q^{24} - 250 q^{26} + 214 q^{29} - 590 q^{31} - 68 q^{34} + 54 q^{36} + 186 q^{39} - 256 q^{41} + 66 q^{44} - 1154 q^{46} + 830 q^{49} - 228 q^{51} + 54 q^{54} - 264 q^{56} + 2304 q^{59} - 688 q^{61} + 1090 q^{64} + 66 q^{66} + 966 q^{69} - 1414 q^{71} + 2352 q^{74} - 3398 q^{76} + 4988 q^{79} + 810 q^{81} + 1944 q^{84} - 1598 q^{86} + 4870 q^{89} - 1608 q^{91} + 1004 q^{94} + 138 q^{96} + 990 q^{99}+O(q^{100}) \) 10 * q - 6 * q^4 - 6 * q^6 - 90 * q^9 - 110 * q^11 - 16 * q^14 - 250 * q^16 + 114 * q^19 - 108 * q^21 - 18 * q^24 - 250 * q^26 + 214 * q^29 - 590 * q^31 - 68 * q^34 + 54 * q^36 + 186 * q^39 - 256 * q^41 + 66 * q^44 - 1154 * q^46 + 830 * q^49 - 228 * q^51 + 54 * q^54 - 264 * q^56 + 2304 * q^59 - 688 * q^61 + 1090 * q^64 + 66 * q^66 + 966 * q^69 - 1414 * q^71 + 2352 * q^74 - 3398 * q^76 + 4988 * q^79 + 810 * q^81 + 1944 * q^84 - 1598 * q^86 + 4870 * q^89 - 1608 * q^91 + 1004 * q^94 + 138 * q^96 + 990 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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