LMFDB - Embedded newform 325.6.b.h.274.13

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			274.13
Root			$5.30592i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.6.b.h.274.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.30592i q^{2} -19.9083i q^{3} +13.4591 q^{4} +85.7235 q^{6} +125.783i q^{7} +195.743i q^{8} -153.341 q^{9} -709.332 q^{11} -267.947i q^{12} -169.000i q^{13} -541.610 q^{14} -412.163 q^{16} -1927.07i q^{17} -660.272i q^{18} +2166.48 q^{19} +2504.12 q^{21} -3054.33i q^{22} +305.642i q^{23} +3896.91 q^{24} +727.700 q^{26} -1784.97i q^{27} +1692.92i q^{28} -4514.28 q^{29} +4078.41 q^{31} +4489.03i q^{32} +14121.6i q^{33} +8297.79 q^{34} -2063.82 q^{36} -12350.6i q^{37} +9328.69i q^{38} -3364.50 q^{39} +14430.4 q^{41} +10782.5i q^{42} -17133.7i q^{43} -9546.95 q^{44} -1316.07 q^{46} -18349.2i q^{47} +8205.47i q^{48} +985.723 q^{49} -38364.6 q^{51} -2274.58i q^{52} +11938.3i q^{53} +7685.93 q^{54} -24621.1 q^{56} -43131.0i q^{57} -19438.1i q^{58} +39854.6 q^{59} +4490.25 q^{61} +17561.3i q^{62} -19287.6i q^{63} -32518.6 q^{64} -60806.5 q^{66} -30488.3i q^{67} -25936.5i q^{68} +6084.82 q^{69} -14561.2 q^{71} -30015.3i q^{72} -59007.3i q^{73} +53180.6 q^{74} +29158.8 q^{76} -89221.7i q^{77} -14487.3i q^{78} +43350.2 q^{79} -72797.4 q^{81} +62136.2i q^{82} +83849.9i q^{83} +33703.1 q^{84} +73776.2 q^{86} +89871.6i q^{87} -138847. i q^{88} -124592. q^{89} +21257.3 q^{91} +4113.66i q^{92} -81194.2i q^{93} +79010.1 q^{94} +89369.1 q^{96} +93842.4i q^{97} +4244.44i q^{98} +108769. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots + 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

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 $

$2$	4.30592i	0.761186i	0.924743	+	0.380593i	$0.124280\pi$
$2$	4.30592i	0.761186i	−0.924743	+	0.380593i	$0.875720\pi$
$3$	− 19.9083i	− 1.27712i	−0.769573	−	0.638559i	$-0.779531\pi$
$3$	− 19.9083i	− 1.27712i	0.769573	−	0.638559i	$-0.220469\pi$
$4$	13.4591	0.420596
$5$	0	0
$5$	0	0
$6$	85.7235	0.972125
$7$	125.783i	0.970232i	0.874450	+	0.485116i	$0.161223\pi$
$7$	125.783i	0.970232i	−0.874450	+	0.485116i	$0.838777\pi$
$8$	195.743i	1.08134i
$9$	−153.341	−0.631031
$10$	0	0
$11$	−709.332	−1.76754	−0.883768	−	0.467926i	$-0.845001\pi$
$11$	−709.332	−1.76754	−0.883768	+	0.467926i	$0.845001\pi$
$12$	− 267.947i	− 0.537151i
$13$	− 169.000i	− 0.277350i
$13$	− 169.000i	− 0.277350i
$14$	−541.610	−0.738527
$15$	0	0
$16$	−412.163	−0.402503
$17$	− 1927.07i	− 1.61724i	−0.588332	−	0.808620i	$-0.700215\pi$
$17$	− 1927.07i	− 1.61724i	0.588332	−	0.808620i	$-0.299785\pi$
$18$	− 660.272i	− 0.480332i
$19$	2166.48	1.37680	0.688400	−	0.725331i	$-0.258313\pi$
$19$	2166.48	1.37680	0.688400	+	0.725331i	$0.258313\pi$
$20$	0	0
$21$	2504.12	1.23910
$22$	− 3054.33i	− 1.34542i
$23$	305.642i	0.120474i	0.998184	+	0.0602371i	$0.0191857\pi$
$23$	305.642i	0.120474i	−0.998184	+	0.0602371i	$0.980814\pi$
$24$	3896.91	1.38100
$25$	0	0
$26$	727.700	0.211115
$27$	− 1784.97i	− 0.471217i
$28$	1692.92i	0.408076i
$29$	−4514.28	−0.996766	−0.498383	−	0.866957i	$-0.666073\pi$
$29$	−4514.28	−0.996766	−0.498383	+	0.866957i	$0.666073\pi$
$30$	0	0
$31$	4078.41	0.762231	0.381116	−	0.924527i	$-0.375540\pi$
$31$	4078.41	0.762231	0.381116	+	0.924527i	$0.375540\pi$
$32$	4489.03i	0.774958i
$33$	14121.6i	2.25735i
$34$	8297.79	1.23102
$35$	0	0
$36$	−2063.82	−0.265409
$37$	− 12350.6i	− 1.48314i	−0.670873	−	0.741572i	$-0.734080\pi$
$37$	− 12350.6i	− 1.48314i	0.670873	−	0.741572i	$-0.265920\pi$
$38$	9328.69i	1.04800i
$39$	−3364.50	−0.354209
$40$	0	0
$41$	14430.4	1.34066	0.670331	−	0.742062i	$-0.266152\pi$
$41$	14430.4	1.34066	0.670331	+	0.742062i	$0.266152\pi$
$42$	10782.5i	0.943187i
$43$	− 17133.7i	− 1.41312i	−0.707652	−	0.706561i	$-0.750246\pi$
$43$	− 17133.7i	− 1.41312i	0.707652	−	0.706561i	$-0.249754\pi$
$44$	−9546.95	−0.743418
$45$	0	0
$46$	−1316.07	−0.0917032
$47$	− 18349.2i	− 1.21164i	−0.795603	−	0.605818i	$-0.792846\pi$
$47$	− 18349.2i	− 1.21164i	0.795603	−	0.605818i	$-0.207154\pi$
$48$	8205.47i	0.514044i
$49$	985.723	0.0586496
$50$	0	0
$51$	−38364.6	−2.06541
$52$	− 2274.58i	− 0.116652i
$53$	11938.3i	0.583785i	0.956451	+	0.291893i	$0.0942850\pi$
$53$	11938.3i	0.583785i	−0.956451	+	0.291893i	$0.905715\pi$
$54$	7685.93	0.358684
$55$	0	0
$56$	−24621.1	−1.04915
$57$	− 43131.0i	− 1.75834i
$58$	− 19438.1i	− 0.758724i
$59$	39854.6	1.49056	0.745278	−	0.666754i	$-0.232317\pi$
$59$	39854.6	1.49056	0.745278	+	0.666754i	$0.232317\pi$
$60$	0	0
$61$	4490.25	0.154506	0.0772530	−	0.997012i	$-0.475385\pi$
$61$	4490.25	0.154506	0.0772530	+	0.997012i	$0.475385\pi$
$62$	17561.3i	0.580200i
$63$	− 19287.6i	− 0.612247i
$64$	−32518.6	−0.992390
$65$	0	0
$66$	−60806.5	−1.71826
$67$	− 30488.3i	− 0.829749i	−0.909879	−	0.414874i	$-0.863826\pi$
$67$	− 30488.3i	− 0.829749i	0.909879	−	0.414874i	$-0.136174\pi$
$68$	− 25936.5i	− 0.680204i
$69$	6084.82	0.153860
$70$	0	0
$71$	−14561.2	−0.342808	−0.171404	−	0.985201i	$-0.554830\pi$
$71$	−14561.2	−0.342808	−0.171404	+	0.985201i	$0.554830\pi$
$72$	− 30015.3i	− 0.682358i
$73$	− 59007.3i	− 1.29598i	−0.761649	−	0.647990i	$-0.775610\pi$
$73$	− 59007.3i	− 1.29598i	0.761649	−	0.647990i	$-0.224390\pi$
$74$	53180.6	1.12895
$75$	0	0
$76$	29158.8	0.579076
$77$	− 89221.7i	− 1.71492i
$78$	− 14487.3i	− 0.269619i
$79$	43350.2	0.781489	0.390745	−	0.920499i	$-0.372218\pi$
$79$	43350.2	0.781489	0.390745	+	0.920499i	$0.372218\pi$
$80$	0	0
$81$	−72797.4	−1.23283
$82$	62136.2i	1.02049i
$83$	83849.9i	1.33600i	0.744160	+	0.668002i	$0.232850\pi$
$83$	83849.9i	1.33600i	−0.744160	+	0.668002i	$0.767150\pi$
$84$	33703.1	0.521161
$85$	0	0
$86$	73776.2	1.07565
$87$	89871.6i	1.27299i
$88$	− 138847.i	− 1.91130i
$89$	−124592.	−1.66731	−0.833656	−	0.552284i	$-0.813756\pi$
$89$	−124592.	−1.66731	−0.833656	+	0.552284i	$0.813756\pi$
$90$	0	0
$91$	21257.3	0.269094
$92$	4113.66i	0.0506709i
$93$	− 81194.2i	− 0.973459i
$94$	79010.1	0.922281
$95$	0	0
$96$	89369.1	0.989713
$97$	93842.4i	1.01267i	0.862335	+	0.506337i	$0.169001\pi$
$97$	93842.4i	1.01267i	−0.862335	+	0.506337i	$0.830999\pi$
$98$	4244.44i	0.0446432i
$99$	108769.	1.11537
$100$	0	0
$101$	−117081.	−1.14205	−0.571023	−	0.820934i	$-0.693453\pi$
$101$	−117081.	−1.14205	−0.571023	+	0.820934i	$0.693453\pi$
$102$	− 165195.i	− 1.57216i
$103$	− 116464.i	− 1.08168i	−0.841124	−	0.540842i	$-0.818106\pi$
$103$	− 116464.i	− 1.08168i	0.841124	−	0.540842i	$-0.181894\pi$
$104$	33080.6	0.299909
$105$	0	0
$106$	−51405.4	−0.444369
$107$	43117.6i	0.364079i	0.983291	+	0.182039i	$0.0582698\pi$
$107$	43117.6i	0.364079i	−0.983291	+	0.182039i	$0.941730\pi$
$108$	− 24024.0i	− 0.198192i
$109$	−32460.7	−0.261693	−0.130846	−	0.991403i	$-0.541769\pi$
$109$	−32460.7	−0.261693	−0.130846	+	0.991403i	$0.541769\pi$
$110$	0	0
$111$	−245879.	−1.89415
$112$	− 51843.0i	− 0.390522i
$113$	− 73976.7i	− 0.545003i	−0.962155	−	0.272501i	$-0.912149\pi$
$113$	− 73976.7i	− 0.545003i	0.962155	−	0.272501i	$-0.0878510\pi$
$114$	185718.	1.33842
$115$	0	0
$116$	−60757.9	−0.419236
$117$	25914.5i	0.175017i
$118$	171611.i	1.13459i
$119$	242392.	1.56910
$120$	0	0
$121$	342101.	2.12418
$122$	19334.6i	0.117608i
$123$	− 287285.i	− 1.71218i
$124$	54891.6	0.320591
$125$	0	0
$126$	83050.7	0.466034
$127$	− 250540.i	− 1.37838i	−0.724582	−	0.689189i	$-0.757967\pi$
$127$	− 250540.i	− 1.37838i	0.724582	−	0.689189i	$-0.242033\pi$
$128$	3626.46i	0.0195640i
$129$	−341103.	−1.80472
$130$	0	0
$131$	98431.4	0.501136	0.250568	−	0.968099i	$-0.419383\pi$
$131$	98431.4	0.501136	0.250568	+	0.968099i	$0.419383\pi$
$132$	190064.i	0.949433i
$133$	272506.i	1.33582i
$134$	131280.	0.631593
$135$	0	0
$136$	377210.	1.74878
$137$	− 302938.i	− 1.37896i	−0.724305	−	0.689480i	$-0.757839\pi$
$137$	− 302938.i	− 1.37896i	0.724305	−	0.689480i	$-0.242161\pi$
$138$	26200.7i	0.117116i
$139$	215720.	0.947006	0.473503	−	0.880792i	$-0.342989\pi$
$139$	215720.	0.947006	0.473503	+	0.880792i	$0.342989\pi$
$140$	0	0
$141$	−365301.	−1.54740
$142$	− 62699.3i	− 0.260941i
$143$	119877.i	0.490226i
$144$	63201.4	0.253992
$145$	0	0
$146$	254080.	0.986482
$147$	− 19624.1i	− 0.0749024i
$148$	− 166227.i	− 0.623804i
$149$	193167.	0.712798	0.356399	−	0.934334i	$-0.384004\pi$
$149$	193167.	0.712798	0.356399	+	0.934334i	$0.384004\pi$
$150$	0	0
$151$	−54010.3	−0.192768	−0.0963839	−	0.995344i	$-0.530728\pi$
$151$	−54010.3	−0.192768	−0.0963839	+	0.995344i	$0.530728\pi$
$152$	424074.i	1.48879i
$153$	295497.i	1.02053i
$154$	384181.	1.30537
$155$	0	0
$156$	−45283.1	−0.148979
$157$	386541.i	1.25155i	0.780006	+	0.625773i	$0.215216\pi$
$157$	386541.i	1.25155i	−0.780006	+	0.625773i	$0.784784\pi$
$158$	186662.i	0.594859i
$159$	237671.	0.745563
$160$	0	0
$161$	−38444.5	−0.116888
$162$	− 313460.i	− 0.938414i
$163$	− 106816.i	− 0.314896i	−0.987527	−	0.157448i	$-0.949673\pi$
$163$	− 106816.i	− 0.314896i	0.987527	−	0.157448i	$-0.0503267\pi$
$164$	194220.	0.563877
$165$	0	0
$166$	−361051.	−1.01695
$167$	− 371554.i	− 1.03093i	−0.856909	−	0.515467i	$-0.827619\pi$
$167$	− 371554.i	− 1.03093i	0.856909	−	0.515467i	$-0.172381\pi$
$168$	490164.i	1.33989i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−332209.	−0.868803
$172$	− 230603.i	− 0.594353i
$173$	− 274680.i	− 0.697769i	−0.937166	−	0.348884i	$-0.886561\pi$
$173$	− 274680.i	− 0.697769i	0.937166	−	0.348884i	$-0.113439\pi$
$174$	−386980.	−0.968981
$175$	0	0
$176$	292361.	0.711439
$177$	− 793437.i	− 1.90362i
$178$	− 536485.i	− 1.26913i
$179$	−317330.	−0.740251	−0.370125	−	0.928982i	$-0.620685\pi$
$179$	−317330.	−0.740251	−0.370125	+	0.928982i	$0.620685\pi$
$180$	0	0
$181$	−241524.	−0.547979	−0.273989	−	0.961733i	$-0.588343\pi$
$181$	−241524.	−0.547979	−0.273989	+	0.961733i	$0.588343\pi$
$182$	91532.1i	0.204831i
$183$	− 89393.2i	− 0.197322i
$184$	−59827.3	−0.130273
$185$	0	0
$186$	349616.	0.740984
$187$	1.36693e6i	2.85853i
$188$	− 246963.i	− 0.509609i
$189$	224518.	0.457190
$190$	0	0
$191$	208827.	0.414193	0.207096	−	0.978321i	$-0.433599\pi$
$191$	208827.	0.414193	0.207096	+	0.978321i	$0.433599\pi$
$192$	647391.i	1.26740i
$193$	410293.i	0.792868i	0.918063	+	0.396434i	$0.129752\pi$
$193$	410293.i	0.792868i	−0.918063	+	0.396434i	$0.870248\pi$
$194$	−404078.	−0.770834
$195$	0	0
$196$	13266.9	0.0246678
$197$	466919.i	0.857187i	0.903497	+	0.428594i	$0.140991\pi$
$197$	466919.i	0.857187i	−0.903497	+	0.428594i	$0.859009\pi$
$198$	468352.i	0.849004i
$199$	−655337.	−1.17309	−0.586546	−	0.809916i	$-0.699513\pi$
$199$	−655337.	−1.17309	−0.586546	+	0.809916i	$0.699513\pi$
$200$	0	0
$201$	−606971.	−1.05969
$202$	− 504142.i	− 0.869309i
$203$	− 567818.i	− 0.967094i
$204$	−516352.	−0.868701
$205$	0	0
$206$	501486.	0.823362
$207$	− 46867.3i	− 0.0760229i
$208$	69655.6i	0.111634i
$209$	−1.53676e6	−2.43354
$210$	0	0
$211$	−508536.	−0.786350	−0.393175	−	0.919464i	$-0.628623\pi$
$211$	−508536.	−0.786350	−0.393175	+	0.919464i	$0.628623\pi$
$212$	160679.i	0.245538i
$213$	289889.i	0.437806i
$214$	−185661.	−0.277132
$215$	0	0
$216$	349395.	0.509545
$217$	512993.i	0.739541i
$218$	− 139773.i	− 0.199197i
$219$	−1.17473e6	−1.65512
$220$	0	0
$221$	−325674.	−0.448542
$222$	− 1.05874e6i	− 1.44180i
$223$	− 97604.2i	− 0.131434i	−0.997838	−	0.0657168i	$-0.979067\pi$
$223$	− 97604.2i	− 0.131434i	0.997838	−	0.0657168i	$-0.0209334\pi$
$224$	−564643.	−0.751889
$225$	0	0
$226$	318538.	0.414849
$227$	− 363749.i	− 0.468530i	−0.972173	−	0.234265i	$-0.924732\pi$
$227$	− 363749.i	− 0.468530i	0.972173	−	0.234265i	$-0.0752683\pi$
$228$	− 580502.i	− 0.739549i
$229$	−549995.	−0.693059	−0.346530	−	0.938039i	$-0.612640\pi$
$229$	−549995.	−0.693059	−0.346530	+	0.938039i	$0.612640\pi$
$230$	0	0
$231$	−1.77625e6	−2.19016
$232$	− 883638.i	− 1.07784i
$233$	444960.i	0.536947i	0.963287	+	0.268473i	$0.0865191\pi$
$233$	444960.i	0.536947i	−0.963287	+	0.268473i	$0.913481\pi$
$234$	−111586.	−0.133220
$235$	0	0
$236$	536406.	0.626922
$237$	− 863028.i	− 0.998054i
$238$	1.04372e6i	1.19438i
$239$	−1.06065e6	−1.20110	−0.600549	−	0.799588i	$-0.705051\pi$
$239$	−1.06065e6	−1.20110	−0.600549	+	0.799588i	$0.705051\pi$
$240$	0	0
$241$	642027.	0.712051	0.356025	−	0.934476i	$-0.384132\pi$
$241$	642027.	0.712051	0.356025	+	0.934476i	$0.384132\pi$
$242$	1.47306e6i	1.61690i
$243$	1.01553e6i	1.10325i
$244$	60434.5	0.0649846
$245$	0	0
$246$	1.23703e6	1.30329
$247$	− 366135.i	− 0.381856i
$248$	798320.i	0.824229i
$249$	1.66931e6	1.70623
$250$	0	0
$251$	386423.	0.387149	0.193575	−	0.981086i	$-0.437992\pi$
$251$	386423.	0.387149	0.193575	+	0.981086i	$0.437992\pi$
$252$	− 259593.i	− 0.257508i
$253$	− 216802.i	− 0.212942i
$254$	1.07881e6	1.04920
$255$	0	0
$256$	−1.05621e6	−1.00728
$257$	432043.i	0.408032i	0.978968	+	0.204016i	$0.0653994\pi$
$257$	432043.i	0.408032i	−0.978968	+	0.204016i	$0.934601\pi$
$258$	− 1.46876e6i	− 1.37373i
$259$	1.55349e6	1.43899
$260$	0	0
$261$	692222.	0.628990
$262$	423837.i	0.381457i
$263$	− 120603.i	− 0.107515i	−0.998554	−	0.0537575i	$-0.982880\pi$
$263$	− 120603.i	− 0.107515i	0.998554	−	0.0537575i	$-0.0171198\pi$
$264$	−2.76421e6	−2.44096
$265$	0	0
$266$	−1.17339e6	−1.01680
$267$	2.48043e6i	2.12935i
$268$	− 410344.i	− 0.348989i
$269$	2.20375e6	1.85687	0.928436	−	0.371494i	$-0.121154\pi$
$269$	2.20375e6	1.85687	0.928436	+	0.371494i	$0.121154\pi$
$270$	0	0
$271$	599069.	0.495511	0.247756	−	0.968823i	$-0.420307\pi$
$271$	599069.	0.495511	0.247756	+	0.968823i	$0.420307\pi$
$272$	794266.i	0.650944i
$273$	− 423196.i	− 0.343665i
$274$	1.30442e6	1.04965
$275$	0	0
$276$	81896.0	0.0647127
$277$	67945.4i	0.0532060i	0.999646	+	0.0266030i	$0.00846900\pi$
$277$	67945.4i	0.0532060i	−0.999646	+	0.0266030i	$0.991531\pi$
$278$	928871.i	0.720847i
$279$	−625386.	−0.480991
$280$	0	0
$281$	79333.1	0.0599361	0.0299681	−	0.999551i	$-0.490459\pi$
$281$	79333.1	0.0599361	0.0299681	+	0.999551i	$0.490459\pi$
$282$	− 1.57296e6i	− 1.17786i
$283$	810998.i	0.601941i	0.953633	+	0.300970i	$0.0973105\pi$
$283$	810998.i	0.601941i	−0.953633	+	0.300970i	$0.902689\pi$
$284$	−195980.	−0.144184
$285$	0	0
$286$	−516181.	−0.373153
$287$	1.81510e6i	1.30075i
$288$	− 688351.i	− 0.489022i
$289$	−2.29373e6	−1.61546
$290$	0	0
$291$	1.86824e6	1.29331
$292$	− 794183.i	− 0.545084i
$293$	2.31923e6i	1.57825i	0.614236	+	0.789123i	$0.289464\pi$
$293$	2.31923e6i	1.57825i	−0.614236	+	0.789123i	$0.710536\pi$
$294$	84499.7	0.0570147
$295$	0	0
$296$	2.41754e6	1.60378
$297$	1.26614e6i	0.832893i
$298$	831760.i	0.542572i
$299$	51653.5	0.0334135
$300$	0	0
$301$	2.15512e6	1.37106
$302$	− 232564.i	− 0.146732i
$303$	2.33089e6i	1.45853i
$304$	−892944.	−0.554167
$305$	0	0
$306$	−1.27239e6	−0.776812
$307$	− 804892.i	− 0.487407i	−0.969850	−	0.243703i	$-0.921638\pi$
$307$	− 804892.i	− 0.487407i	0.969850	−	0.243703i	$-0.0783624\pi$
$308$	− 1.20084e6i	− 0.721288i
$309$	−2.31861e6	−1.38144
$310$	0	0
$311$	−2.49761e6	−1.46428	−0.732139	−	0.681156i	$-0.761478\pi$
$311$	−2.49761e6	−1.46428	−0.732139	+	0.681156i	$0.761478\pi$
$312$	− 658578.i	− 0.383019i
$313$	− 146716.i	− 0.0846480i	−0.999104	−	0.0423240i	$-0.986524\pi$
$313$	− 146716.i	− 0.0846480i	0.999104	−	0.0423240i	$-0.0134762\pi$
$314$	−1.66441e6	−0.952659
$315$	0	0
$316$	583453.	0.328691
$317$	2.37920e6i	1.32979i	0.746936	+	0.664896i	$0.231524\pi$
$317$	2.37920e6i	1.32979i	−0.746936	+	0.664896i	$0.768476\pi$
$318$	1.02339e6i	0.567512i
$319$	3.20212e6	1.76182
$320$	0	0
$321$	858399.	0.464972
$322$	− 165539.i	− 0.0889734i
$323$	− 4.17495e6i	− 2.22662i
$324$	−979785.	−0.518523
$325$	0	0
$326$	459941.	0.239695
$327$	646237.i	0.334212i
$328$	2.82466e6i	1.44971i
$329$	2.30801e6	1.17557
$330$	0	0
$331$	2.55669e6	1.28265	0.641324	−	0.767270i	$-0.278385\pi$
$331$	2.55669e6	1.28265	0.641324	+	0.767270i	$0.278385\pi$
$332$	1.12854e6i	0.561917i
$333$	1.89385e6i	0.935910i
$334$	1.59988e6	0.784732
$335$	0	0
$336$	−1.03211e6	−0.498742
$337$	2.38453e6i	1.14374i	0.820344	+	0.571871i	$0.193782\pi$
$337$	2.38453e6i	1.14374i	−0.820344	+	0.571871i	$0.806218\pi$
$338$	− 122981.i	− 0.0585528i
$339$	−1.47275e6	−0.696033
$340$	0	0
$341$	−2.89295e6	−1.34727
$342$	− 1.43047e6i	− 0.661321i
$343$	2.23802e6i	1.02714i
$344$	3.35380e6	1.52806
$345$	0	0
$346$	1.18275e6	0.531132
$347$	− 427794.i	− 0.190727i	−0.995443	−	0.0953633i	$-0.969599\pi$
$347$	− 427794.i	− 0.190727i	0.995443	−	0.0953633i	$-0.0304013\pi$
$348$	1.20959e6i	0.535413i
$349$	3.91198e6	1.71922	0.859612	−	0.510947i	$-0.170705\pi$
$349$	3.91198e6	1.71922	0.859612	+	0.510947i	$0.170705\pi$
$350$	0	0
$351$	−301660.	−0.130692
$352$	− 3.18422e6i	− 1.36977i
$353$	− 1.85968e6i	− 0.794330i	−0.917747	−	0.397165i	$-0.869994\pi$
$353$	− 1.85968e6i	− 0.794330i	0.917747	−	0.397165i	$-0.130006\pi$
$354$	3.41648e6	1.44901
$355$	0	0
$356$	−1.67690e6	−0.701264
$357$	− 4.82560e6i	− 2.00392i
$358$	− 1.36640e6i	− 0.563468i
$359$	−693152.	−0.283852	−0.141926	−	0.989877i	$-0.545330\pi$
$359$	−693152.	−0.283852	−0.141926	+	0.989877i	$0.545330\pi$
$360$	0	0
$361$	2.21754e6	0.895578
$362$	− 1.03998e6i	− 0.417114i
$363$	− 6.81066e6i	− 2.71283i
$364$	286103.	0.113180
$365$	0	0
$366$	384920.	0.150199
$367$	2.25998e6i	0.875870i	0.899007	+	0.437935i	$0.144290\pi$
$367$	2.25998e6i	0.875870i	−0.899007	+	0.437935i	$0.855710\pi$
$368$	− 125975.i	− 0.0484912i
$369$	−2.21277e6	−0.846000
$370$	0	0
$371$	−1.50163e6	−0.566407
$372$	− 1.09280e6i	− 0.409433i
$373$	− 802038.i	− 0.298485i	−0.988801	−	0.149243i	$-0.952316\pi$
$373$	− 802038.i	− 0.298485i	0.988801	−	0.149243i	$-0.0476836\pi$
$374$	−5.88589e6	−2.17587
$375$	0	0
$376$	3.59173e6	1.31019
$377$	762913.i	0.276453i
$378$	966756.i	0.348007i
$379$	−1.44664e6	−0.517325	−0.258662	−	0.965968i	$-0.583282\pi$
$379$	−1.44664e6	−0.517325	−0.258662	+	0.965968i	$0.583282\pi$
$380$	0	0
$381$	−4.98783e6	−1.76035
$382$	899191.i	0.315278i
$383$	− 3.55257e6i	− 1.23750i	−0.785587	−	0.618751i	$-0.787639\pi$
$383$	− 3.55257e6i	− 1.23750i	0.785587	−	0.618751i	$-0.212361\pi$
$384$	72196.7	0.0249856
$385$	0	0
$386$	−1.76669e6	−0.603520
$387$	2.62729e6i	0.891724i
$388$	1.26303e6i	0.425927i
$389$	5.57943e6	1.86946	0.934730	−	0.355359i	$-0.115642\pi$
$389$	5.57943e6	1.86946	0.934730	+	0.355359i	$0.115642\pi$
$390$	0	0
$391$	588993.	0.194836
$392$	192948.i	0.0634200i
$393$	− 1.95960e6i	− 0.640009i
$394$	−2.01051e6	−0.652479
$395$	0	0
$396$	1.46393e6	0.469120
$397$	− 3.43746e6i	− 1.09462i	−0.836931	−	0.547308i	$-0.815653\pi$
$397$	− 3.43746e6i	− 1.09462i	0.836931	−	0.547308i	$-0.184347\pi$
$398$	− 2.82183e6i	− 0.892942i
$399$	5.42513e6	1.70599
$400$	0	0
$401$	−4.49645e6	−1.39640	−0.698199	−	0.715904i	$-0.746015\pi$
$401$	−4.49645e6	−1.39640	−0.698199	+	0.715904i	$0.746015\pi$
$402$	− 2.61357e6i	− 0.806619i
$403$	− 689251.i	− 0.211405i
$404$	−1.57580e6	−0.480340
$405$	0	0
$406$	2.44498e6	0.736139
$407$	8.76068e6i	2.62151i
$408$	− 7.50961e6i	− 2.23340i
$409$	4.00809e6	1.18476	0.592379	−	0.805659i	$-0.298189\pi$
$409$	4.00809e6	1.18476	0.592379	+	0.805659i	$0.298189\pi$
$410$	0	0
$411$	−6.03097e6	−1.76109
$412$	− 1.56750e6i	− 0.454952i
$413$	5.01302e6i	1.44619i
$414$	201807.	0.0578676
$415$	0	0
$416$	758647.	0.214935
$417$	− 4.29461e6i	− 1.20944i
$418$	− 6.61714e6i	− 1.85238i
$419$	30427.3	0.00846697	0.00423349	−	0.999991i	$-0.498652\pi$
$419$	30427.3	0.00846697	0.00423349	+	0.999991i	$0.498652\pi$
$420$	0	0
$421$	1.28719e6	0.353946	0.176973	−	0.984216i	$-0.443369\pi$
$421$	1.28719e6	0.353946	0.176973	+	0.984216i	$0.443369\pi$
$422$	− 2.18972e6i	− 0.598559i
$423$	2.81367e6i	0.764580i
$424$	−2.33684e6	−0.631269
$425$	0	0
$426$	−1.24824e6	−0.333252
$427$	564795.i	0.149907i
$428$	580323.i	0.153130i
$429$	2.38655e6	0.626077
$430$	0	0
$431$	−3.27469e6	−0.849137	−0.424568	−	0.905396i	$-0.639574\pi$
$431$	−3.27469e6	−0.849137	−0.424568	+	0.905396i	$0.639574\pi$
$432$	735699.i	0.189666i
$433$	2.56449e6i	0.657327i	0.944447	+	0.328664i	$0.106598\pi$
$433$	2.56449e6i	0.657327i	−0.944447	+	0.328664i	$0.893402\pi$
$434$	−2.20891e6	−0.562928
$435$	0	0
$436$	−436890.	−0.110067
$437$	662168.i	0.165869i
$438$	− 5.05831e6i	− 1.25985i
$439$	−973053.	−0.240977	−0.120488	−	0.992715i	$-0.538446\pi$
$439$	−973053.	−0.240977	−0.120488	+	0.992715i	$0.538446\pi$
$440$	0	0
$441$	−151151.	−0.0370097
$442$	− 1.40233e6i	− 0.341424i
$443$	− 1.98241e6i	− 0.479936i	−0.970781	−	0.239968i	$-0.922863\pi$
$443$	− 1.98241e6i	− 0.479936i	0.970781	−	0.239968i	$-0.0771369\pi$
$444$	−3.30931e6	−0.796672
$445$	0	0
$446$	420276.	0.100045
$447$	− 3.84562e6i	− 0.910327i
$448$	− 4.09028e6i	− 0.962849i
$449$	−1.98450e6	−0.464552	−0.232276	−	0.972650i	$-0.574617\pi$
$449$	−1.98450e6	−0.464552	−0.232276	+	0.972650i	$0.574617\pi$
$450$	0	0
$451$	−1.02360e7	−2.36967
$452$	− 995657.i	− 0.229226i
$453$	1.07525e6i	0.246187i
$454$	1.56627e6	0.356639
$455$	0	0
$456$	8.44258e6	1.90136
$457$	− 6.95085e6i	− 1.55685i	−0.627736	−	0.778427i	$-0.716018\pi$
$457$	− 6.95085e6i	− 1.55685i	0.627736	−	0.778427i	$-0.283982\pi$
$458$	− 2.36824e6i	− 0.527547i
$459$	−3.43975e6	−0.762071
$460$	0	0
$461$	−2.63571e6	−0.577625	−0.288812	−	0.957386i	$-0.593260\pi$
$461$	−2.63571e6	−0.577625	−0.288812	+	0.957386i	$0.593260\pi$
$462$	− 7.64840e6i	− 1.66712i
$463$	6.75578e6i	1.46461i	0.680975	+	0.732306i	$0.261556\pi$
$463$	6.75578e6i	1.46461i	−0.680975	+	0.732306i	$0.738444\pi$
$464$	1.86062e6	0.401202
$465$	0	0
$466$	−1.91596e6	−0.408716
$467$	− 7.34426e6i	− 1.55832i	−0.626826	−	0.779159i	$-0.715646\pi$
$467$	− 7.34426e6i	− 1.55832i	0.626826	−	0.779159i	$-0.284354\pi$
$468$	348786.i	0.0736112i
$469$	3.83490e6	0.805049
$470$	0	0
$471$	7.69538e6	1.59837
$472$	7.80126e6i	1.61179i
$473$	1.21535e7i	2.49774i
$474$	3.71613e6	0.759705
$475$	0	0
$476$	3.26236e6	0.659956
$477$	− 1.83063e6i	− 0.368387i
$478$	− 4.56708e6i	− 0.914259i
$479$	6.14073e6	1.22287	0.611436	−	0.791294i	$-0.290592\pi$
$479$	6.14073e6	1.22287	0.611436	+	0.791294i	$0.290592\pi$
$480$	0	0
$481$	−2.08725e6	−0.411350
$482$	2.76452e6i	0.542003i
$483$	765365.i	0.149280i
$484$	4.60437e6	0.893422
$485$	0	0
$486$	−4.37277e6	−0.839781
$487$	1.73635e6i	0.331752i	0.986147	+	0.165876i	$0.0530452\pi$
$487$	1.73635e6i	0.331752i	−0.986147	+	0.165876i	$0.946955\pi$
$488$	878934.i	0.167073i
$489$	−2.12653e6	−0.402160
$490$	0	0
$491$	−1.66007e6	−0.310759	−0.155380	−	0.987855i	$-0.549660\pi$
$491$	−1.66007e6	−0.310759	−0.155380	+	0.987855i	$0.549660\pi$
$492$	− 3.86659e6i	− 0.720138i
$493$	8.69931e6i	1.61201i
$494$	1.57655e6	0.290663
$495$	0	0
$496$	−1.68097e6	−0.306801
$497$	− 1.83155e6i	− 0.332603i
$498$	7.18791e6i	1.29876i
$499$	−7.82789e6	−1.40732	−0.703660	−	0.710536i	$-0.748452\pi$
$499$	−7.82789e6	−1.40732	−0.703660	+	0.710536i	$0.748452\pi$
$500$	0	0
$501$	−7.39701e6	−1.31662
$502$	1.66390e6i	0.294692i
$503$	4.93710e6i	0.870066i	0.900415	+	0.435033i	$0.143263\pi$
$503$	4.93710e6i	0.870066i	−0.900415	+	0.435033i	$0.856737\pi$
$504$	3.77541e6	0.662045
$505$	0	0
$506$	933531.	0.162089
$507$	568601.i	0.0982399i
$508$	− 3.37204e6i	− 0.579740i
$509$	8.26567e6	1.41411	0.707056	−	0.707158i	$-0.250023\pi$
$509$	8.26567e6	1.41411	0.707056	+	0.707158i	$0.250023\pi$
$510$	0	0
$511$	7.42209e6	1.25740
$512$	− 4.43192e6i	− 0.747165i
$513$	− 3.86710e6i	− 0.648772i
$514$	−1.86034e6	−0.310588
$515$	0	0
$516$	−4.59092e6	−0.759059
$517$	1.30157e7i	2.14161i
$518$	6.68920e6i	1.09534i
$519$	−5.46841e6	−0.891133
$520$	0	0
$521$	7.91723e6	1.27785	0.638924	−	0.769270i	$-0.279380\pi$
$521$	7.91723e6	1.27785	0.638924	+	0.769270i	$0.279380\pi$
$522$	2.98065e6i	0.478779i
$523$	− 6.02805e6i	− 0.963657i	−0.876265	−	0.481829i	$-0.839973\pi$
$523$	− 6.02805e6i	− 0.963657i	0.876265	−	0.481829i	$-0.160027\pi$
$524$	1.32479e6	0.210776
$525$	0	0
$526$	519307.	0.0818389
$527$	− 7.85937e6i	− 1.23271i
$528$	− 5.82041e6i	− 0.908592i
$529$	6.34293e6	0.985486
$530$	0	0
$531$	−6.11132e6	−0.940587
$532$	3.66767e6i	0.561838i
$533$	− 2.43874e6i	− 0.371833i
$534$	−1.06805e7	−1.62083
$535$	0	0
$536$	5.96788e6	0.897239
$537$	6.31750e6i	0.945388i
$538$	9.48917e6i	1.41342i
$539$	−699205.	−0.103665
$540$	0	0
$541$	9.65945e6	1.41892	0.709462	−	0.704744i	$-0.248938\pi$
$541$	9.65945e6	1.41892	0.709462	+	0.704744i	$0.248938\pi$
$542$	2.57954e6i	0.377176i
$543$	4.80833e6i	0.699834i
$544$	8.65067e6	1.25329
$545$	0	0
$546$	1.82225e6	0.261593
$547$	− 1.23221e7i	− 1.76083i	−0.474203	−	0.880415i	$-0.657264\pi$
$547$	− 1.23221e7i	− 1.76083i	0.474203	−	0.880415i	$-0.342736\pi$
$548$	− 4.07726e6i	− 0.579985i
$549$	−688537.	−0.0974981
$550$	0	0
$551$	−9.78009e6	−1.37235
$552$	1.19106e6i	0.166374i
$553$	5.45270e6i	0.758226i
$554$	−292567.	−0.0404997
$555$	0	0
$556$	2.90338e6	0.398307
$557$	6.45087e6i	0.881009i	0.897750	+	0.440504i	$0.145200\pi$
$557$	6.45087e6i	0.881009i	−0.897750	+	0.440504i	$0.854800\pi$
$558$	− 2.69286e6i	− 0.366124i
$559$	−2.89559e6	−0.391929
$560$	0	0
$561$	2.72133e7	3.65068
$562$	341602.i	0.0456226i
$563$	− 5.38104e6i	− 0.715476i	−0.933822	−	0.357738i	$-0.883548\pi$
$563$	− 5.38104e6i	− 0.715476i	0.933822	−	0.357738i	$-0.116452\pi$
$564$	−4.91661e6	−0.650831
$565$	0	0
$566$	−3.49209e6	−0.458189
$567$	− 9.15665e6i	− 1.19613i
$568$	− 2.85025e6i	− 0.370691i
$569$	1.19810e6	0.155136	0.0775680	−	0.996987i	$-0.475285\pi$
$569$	1.19810e6	0.155136	0.0775680	+	0.996987i	$0.475285\pi$
$570$	0	0
$571$	−5.65494e6	−0.725834	−0.362917	−	0.931821i	$-0.618219\pi$
$571$	−5.65494e6	−0.725834	−0.362917	+	0.931821i	$0.618219\pi$
$572$	1.61343e6i	0.206187i
$573$	− 4.15739e6i	− 0.528973i
$574$	−7.81566e6	−0.990116
$575$	0	0
$576$	4.98643e6	0.626229
$577$	5.03350e6i	0.629405i	0.949190	+	0.314703i	$0.101905\pi$
$577$	5.03350e6i	0.629405i	−0.949190	+	0.314703i	$0.898095\pi$
$578$	− 9.87661e6i	− 1.22967i
$579$	8.16823e6	1.01259
$580$	0	0
$581$	−1.05469e7	−1.29623
$582$	8.04451e6i	0.984446i
$583$	− 8.46823e6i	− 1.03186i
$584$	1.15503e7	1.40139
$585$	0	0
$586$	−9.98641e6	−1.20134
$587$	− 7.95091e6i	− 0.952405i	−0.879336	−	0.476203i	$-0.842013\pi$
$587$	− 7.95091e6i	− 0.952405i	0.879336	−	0.476203i	$-0.157987\pi$
$588$	− 264122.i	− 0.0315036i
$589$	8.83580e6	1.04944
$590$	0	0
$591$	9.29556e6	1.09473
$592$	5.09046e6i	0.596971i
$593$	− 3.79733e6i	− 0.443447i	−0.975110	−	0.221724i	$-0.928832\pi$
$593$	− 3.79733e6i	− 0.443447i	0.975110	−	0.221724i	$-0.0711683\pi$
$594$	−5.45188e6	−0.633986
$595$	0	0
$596$	2.59984e6	0.299800
$597$	1.30467e7i	1.49818i
$598$	222416.i	0.0254339i
$599$	−1.28647e7	−1.46498	−0.732489	−	0.680779i	$-0.761641\pi$
$599$	−1.28647e7	−1.46498	−0.732489	+	0.680779i	$0.761641\pi$
$600$	0	0
$601$	5.01991e6	0.566904	0.283452	−	0.958986i	$-0.408520\pi$
$601$	5.01991e6	0.566904	0.283452	+	0.958986i	$0.408520\pi$
$602$	9.27977e6i	1.04363i
$603$	4.67510e6i	0.523597i
$604$	−726929.	−0.0810773
$605$	0	0
$606$	−1.00366e7	−1.11021
$607$	− 1.26908e7i	− 1.39803i	−0.715105	−	0.699017i	$-0.753621\pi$
$607$	− 1.26908e7i	− 1.39803i	0.715105	−	0.699017i	$-0.246379\pi$
$608$	9.72541e6i	1.06696i
$609$	−1.13043e7	−1.23509
$610$	0	0
$611$	−3.10101e6	−0.336047
$612$	3.97712e6i	0.429230i
$613$	1.60182e7i	1.72172i	0.508840	+	0.860861i	$0.330075\pi$
$613$	1.60182e7i	1.72172i	−0.508840	+	0.860861i	$0.669925\pi$
$614$	3.46580e6	0.371007
$615$	0	0
$616$	1.74645e7	1.85441
$617$	− 4.75362e6i	− 0.502703i	−0.967896	−	0.251352i	$-0.919125\pi$
$617$	− 4.75362e6i	− 0.502703i	0.967896	−	0.251352i	$-0.0808751\pi$
$618$	− 9.98374e6i	− 1.05153i
$619$	1.08346e7	1.13655	0.568274	−	0.822839i	$-0.307611\pi$
$619$	1.08346e7	1.13655	0.568274	+	0.822839i	$0.307611\pi$
$620$	0	0
$621$	545562.	0.0567695
$622$	− 1.07545e7i	− 1.11459i
$623$	− 1.56716e7i	− 1.61768i
$624$	1.38673e6	0.142570
$625$	0	0
$626$	631747.	0.0644329
$627$	3.05942e7i	3.10792i
$628$	5.20248e6i	0.526395i
$629$	−2.38004e7	−2.39860
$630$	0	0
$631$	−1.16256e7	−1.16236	−0.581182	−	0.813774i	$-0.697409\pi$
$631$	−1.16256e7	−1.16236	−0.581182	+	0.813774i	$0.697409\pi$
$632$	8.48549e6i	0.845054i
$633$	1.01241e7i	1.00426i
$634$	−1.02447e7	−1.01222
$635$	0	0
$636$	3.19884e6	0.313581
$637$	− 166587.i	− 0.0162665i
$638$	1.37881e7i	1.34107i
$639$	2.23282e6	0.216323
$640$	0	0
$641$	−1.49693e7	−1.43899	−0.719495	−	0.694498i	$-0.755627\pi$
$641$	−1.49693e7	−1.43899	−0.719495	+	0.694498i	$0.755627\pi$
$642$	3.69619e6i	0.353930i
$643$	1.67965e6i	0.160210i	0.996786	+	0.0801051i	$0.0255256\pi$
$643$	1.67965e6i	0.160210i	−0.996786	+	0.0801051i	$0.974474\pi$
$644$	−517427.	−0.0491625
$645$	0	0
$646$	1.79770e7	1.69487
$647$	6.56503e6i	0.616561i	0.951295	+	0.308281i	$0.0997536\pi$
$647$	6.56503e6i	0.616561i	−0.951295	+	0.308281i	$0.900246\pi$
$648$	− 1.42496e7i	− 1.33311i
$649$	−2.82702e7	−2.63461
$650$	0	0
$651$	1.02128e7	0.944482
$652$	− 1.43764e6i	− 0.132444i
$653$	1.78321e7i	1.63652i	0.574850	+	0.818259i	$0.305060\pi$
$653$	1.78321e7i	1.63652i	−0.574850	+	0.818259i	$0.694940\pi$
$654$	−2.78264e6	−0.254398
$655$	0	0
$656$	−5.94769e6	−0.539621
$657$	9.04820e6i	0.817804i
$658$	9.93810e6i	0.894826i
$659$	1.18216e6	0.106038	0.0530190	−	0.998594i	$-0.483116\pi$
$659$	1.18216e6	0.106038	0.0530190	+	0.998594i	$0.483116\pi$
$660$	0	0
$661$	1.20695e7	1.07445	0.537225	−	0.843439i	$-0.319472\pi$
$661$	1.20695e7	1.07445	0.537225	+	0.843439i	$0.319472\pi$
$662$	1.10089e7i	0.976334i
$663$	6.48362e6i	0.572841i
$664$	−1.64130e7	−1.44467
$665$	0	0
$666$	−8.15475e6	−0.712402
$667$	− 1.37975e6i	− 0.120084i
$668$	− 5.00077e6i	− 0.433606i
$669$	−1.94313e6	−0.167856
$670$	0	0
$671$	−3.18508e6	−0.273095
$672$	1.12411e7i	0.960251i
$673$	− 1.89473e7i	− 1.61254i	−0.591548	−	0.806270i	$-0.701483\pi$
$673$	− 1.89473e7i	− 1.61254i	0.591548	−	0.806270i	$-0.298517\pi$
$674$	−1.02676e7	−0.870600
$675$	0	0
$676$	−384404.	−0.0323535
$677$	817954.i	0.0685894i	0.999412	+	0.0342947i	$0.0109185\pi$
$677$	817954.i	0.0685894i	−0.999412	+	0.0342947i	$0.989082\pi$
$678$	− 6.34154e6i	− 0.529811i
$679$	−1.18038e7	−0.982530
$680$	0	0
$681$	−7.24163e6	−0.598368
$682$	− 1.24568e7i	− 1.02552i
$683$	3.54928e6i	0.291131i	0.989349	+	0.145565i	$0.0465001\pi$
$683$	3.54928e6i	0.291131i	−0.989349	+	0.145565i	$0.953500\pi$
$684$	−4.47123e6	−0.365415
$685$	0	0
$686$	−9.63671e6	−0.781841
$687$	1.09495e7i	0.885118i
$688$	7.06188e6i	0.568786i
$689$	2.01757e6	0.161913
$690$	0	0
$691$	2.39285e6	0.190643	0.0953214	−	0.995447i	$-0.469612\pi$
$691$	2.39285e6	0.190643	0.0953214	+	0.995447i	$0.469612\pi$
$692$	− 3.69693e6i	− 0.293479i
$693$	1.36813e7i	1.08217i
$694$	1.84205e6	0.145178
$695$	0	0
$696$	−1.75917e7	−1.37653
$697$	− 2.78084e7i	− 2.16817i
$698$	1.68446e7i	1.30865i
$699$	8.85840e6	0.685744
$700$	0	0
$701$	1.83475e7	1.41020	0.705101	−	0.709106i	$-0.250901\pi$
$701$	1.83475e7	1.41020	0.705101	+	0.709106i	$0.250901\pi$
$702$	− 1.29892e6i	− 0.0994810i
$703$	− 2.67573e7i	− 2.04199i
$704$	2.30665e7	1.75409
$705$	0	0
$706$	8.00762e6	0.604633
$707$	− 1.47268e7i	− 1.10805i
$708$	− 1.06789e7i	− 0.800653i
$709$	3.07694e6	0.229881	0.114941	−	0.993372i	$-0.463332\pi$
$709$	3.07694e6	0.229881	0.114941	+	0.993372i	$0.463332\pi$
$710$	0	0
$711$	−6.64733e6	−0.493144
$712$	− 2.43881e7i	− 1.80293i
$713$	1.24653e6i	0.0918291i
$714$	2.07787e7	1.52536
$715$	0	0
$716$	−4.27097e6	−0.311346
$717$	2.11158e7i	1.53394i
$718$	− 2.98466e6i	− 0.216064i
$719$	−1.80795e7	−1.30426	−0.652129	−	0.758108i	$-0.726124\pi$
$719$	−1.80795e7	−1.30426	−0.652129	+	0.758108i	$0.726124\pi$
$720$	0	0
$721$	1.46492e7	1.04948
$722$	9.54855e6i	0.681702i
$723$	− 1.27817e7i	− 0.909373i
$724$	−3.25069e6	−0.230478
$725$	0	0
$726$	2.93261e7	2.06497
$727$	2.35372e7i	1.65165i	0.563925	+	0.825826i	$0.309291\pi$
$727$	2.35372e7i	1.65165i	−0.563925	+	0.825826i	$0.690709\pi$
$728$	4.16096e6i	0.290981i
$729$	2.52763e6	0.176155
$730$	0	0
$731$	−3.30177e7	−2.28536
$732$	− 1.20315e6i	− 0.0829930i
$733$	4.96555e6i	0.341356i	0.985327	+	0.170678i	$0.0545957\pi$
$733$	4.96555e6i	0.341356i	−0.985327	+	0.170678i	$0.945404\pi$
$734$	−9.73130e6	−0.666700
$735$	0	0
$736$	−1.37204e6	−0.0933624
$737$	2.16264e7i	1.46661i
$738$	− 9.52800e6i	− 0.643963i
$739$	2.40104e7	1.61729	0.808646	−	0.588296i	$-0.200201\pi$
$739$	2.40104e7	1.61729	0.808646	+	0.588296i	$0.200201\pi$
$740$	0	0
$741$	−7.28913e6	−0.487675
$742$	− 6.46591e6i	− 0.431141i
$743$	− 1.15136e7i	− 0.765140i	−0.923927	−	0.382570i	$-0.875039\pi$
$743$	− 1.15136e7i	− 0.765140i	0.923927	−	0.382570i	$-0.124961\pi$
$744$	1.58932e7	1.05264
$745$	0	0
$746$	3.45351e6	0.227203
$747$	− 1.28576e7i	− 0.843060i
$748$	1.83976e7i	1.20228i
$749$	−5.42345e6	−0.353241
$750$	0	0
$751$	−2.24902e7	−1.45510	−0.727552	−	0.686052i	$-0.759342\pi$
$751$	−2.24902e7	−1.45510	−0.727552	+	0.686052i	$0.759342\pi$
$752$	7.56286e6i	0.487688i
$753$	− 7.69302e6i	− 0.494435i
$754$	−3.28504e6	−0.210432
$755$	0	0
$756$	3.02180e6	0.192292
$757$	− 788552.i	− 0.0500139i	−0.999687	−	0.0250069i	$-0.992039\pi$
$757$	− 788552.i	− 0.0500139i	0.999687	−	0.0250069i	$-0.00796079\pi$
$758$	− 6.22913e6i	− 0.393780i
$759$	−4.31616e6	−0.271952
$760$	0	0
$761$	−5.67255e6	−0.355072	−0.177536	−	0.984114i	$-0.556813\pi$
$761$	−5.67255e6	−0.355072	−0.177536	+	0.984114i	$0.556813\pi$
$762$	− 2.14772e7i	− 1.33995i
$763$	− 4.08299e6i	− 0.253903i
$764$	2.81061e6	0.174208
$765$	0	0
$766$	1.52971e7	0.941970
$767$	− 6.73543e6i	− 0.413406i
$768$	2.10274e7i	1.28642i
$769$	2.85131e6	0.173872	0.0869358	−	0.996214i	$-0.472293\pi$
$769$	2.85131e6	0.173872	0.0869358	+	0.996214i	$0.472293\pi$
$770$	0	0
$771$	8.60124e6	0.521105
$772$	5.52216e6i	0.333477i
$773$	1.94976e7i	1.17363i	0.809720	+	0.586817i	$0.199619\pi$
$773$	1.94976e7i	1.17363i	−0.809720	+	0.586817i	$0.800381\pi$
$774$	−1.13129e7	−0.678768
$775$	0	0
$776$	−1.83690e7	−1.09504
$777$	− 3.09274e7i	− 1.83777i
$778$	2.40246e7i	1.42301i
$779$	3.12632e7	1.84582
$780$	0	0
$781$	1.03287e7	0.605925
$782$	2.53616e6i	0.148306i
$783$	8.05784e6i	0.469693i
$784$	−406279.	−0.0236066
$785$	0	0
$786$	8.43788e6	0.487166
$787$	− 8.39703e6i	− 0.483269i	−0.970367	−	0.241634i	$-0.922317\pi$
$787$	− 8.39703e6i	− 0.483269i	0.970367	−	0.241634i	$-0.0776835\pi$
$788$	6.28429e6i	0.360529i
$789$	−2.40100e6	−0.137309
$790$	0	0
$791$	9.30499e6	0.528779
$792$	2.12909e7i	1.20609i
$793$	− 758851.i	− 0.0428523i
$794$	1.48014e7	0.833206
$795$	0	0
$796$	−8.82023e6	−0.493398
$797$	− 1.20402e6i	− 0.0671411i	−0.999436	−	0.0335705i	$-0.989312\pi$
$797$	− 1.20402e6i	− 0.0671411i	0.999436	−	0.0335705i	$-0.0106878\pi$
$798$	2.33602e7i	1.29858i
$799$	−3.53601e7	−1.95951
$800$	0	0
$801$	1.91051e7	1.05213
$802$	− 1.93614e7i	− 1.06292i
$803$	4.18558e7i	2.29069i
$804$	−8.16926e6	−0.445700
$805$	0	0
$806$	2.96786e6	0.160918
$807$	− 4.38729e7i	− 2.37144i
$808$	− 2.29178e7i	− 1.23494i
$809$	−8.10437e6	−0.435360	−0.217680	−	0.976020i	$-0.569849\pi$
$809$	−8.10437e6	−0.435360	−0.217680	+	0.976020i	$0.569849\pi$
$810$	0	0
$811$	−4.53486e6	−0.242109	−0.121055	−	0.992646i	$-0.538628\pi$
$811$	−4.53486e6	−0.242109	−0.121055	+	0.992646i	$0.538628\pi$
$812$	− 7.64230e6i	− 0.406756i
$813$	− 1.19264e7i	− 0.632827i
$814$	−3.77228e7	−1.99546
$815$	0	0
$816$	1.58125e7	0.831333
$817$	− 3.71198e7i	− 1.94559i
$818$	1.72585e7i	0.901821i
$819$	−3.25960e6	−0.169807
$820$	0	0
$821$	1.59739e6	0.0827092	0.0413546	−	0.999145i	$-0.486833\pi$
$821$	1.59739e6	0.0827092	0.0413546	+	0.999145i	$0.486833\pi$
$822$	− 2.59689e7i	− 1.34052i
$823$	− 2.00928e7i	− 1.03405i	−0.855971	−	0.517024i	$-0.827040\pi$
$823$	− 2.00928e7i	− 1.03405i	0.855971	−	0.517024i	$-0.172960\pi$
$824$	2.27971e7	1.16967
$825$	0	0
$826$	−2.15856e7	−1.10082
$827$	1.57246e7i	0.799498i	0.916625	+	0.399749i	$0.130903\pi$
$827$	1.57246e7i	0.799498i	−0.916625	+	0.399749i	$0.869097\pi$
$828$	− 630791.i	− 0.0319749i
$829$	2.66412e7	1.34638	0.673190	−	0.739470i	$-0.264924\pi$
$829$	2.66412e7	1.34638	0.673190	+	0.739470i	$0.264924\pi$
$830$	0	0
$831$	1.35268e6	0.0679503
$832$	5.49565e6i	0.275240i
$833$	− 1.89955e6i	− 0.0948504i
$834$	1.84922e7	0.920607
$835$	0	0
$836$	−2.06833e7	−1.02354
$837$	− 7.27983e6i	− 0.359176i
$838$	131017.i	0.00644494i
$839$	2.06873e7	1.01461	0.507306	−	0.861766i	$-0.330641\pi$
$839$	2.06873e7	1.01461	0.507306	+	0.861766i	$0.330641\pi$
$840$	0	0
$841$	−132454.	−0.00645768
$842$	5.54253e6i	0.269419i
$843$	− 1.57939e6i	− 0.0765455i
$844$	−6.84442e6	−0.330735
$845$	0	0
$846$	−1.21155e7	−0.581988
$847$	4.30304e7i	2.06095i
$848$	− 4.92054e6i	− 0.234976i
$849$	1.61456e7	0.768750
$850$	0	0
$851$	3.77486e6	0.178681
$852$	3.90163e6i	0.184140i
$853$	706242.i	0.0332339i	0.999862	+	0.0166169i	$0.00528958\pi$
$853$	706242.i	0.0332339i	−0.999862	+	0.0166169i	$0.994710\pi$
$854$	−2.43196e6	−0.114107
$855$	0	0
$856$	−8.43997e6	−0.393692
$857$	1.54355e7i	0.717908i	0.933355	+	0.358954i	$0.116867\pi$
$857$	1.54355e7i	0.717908i	−0.933355	+	0.358954i	$0.883133\pi$
$858$	1.02763e7i	0.476561i
$859$	−2.09391e7	−0.968222	−0.484111	−	0.875007i	$-0.660857\pi$
$859$	−2.09391e7	−0.968222	−0.484111	+	0.875007i	$0.660857\pi$
$860$	0	0
$861$	3.61355e7	1.66122
$862$	− 1.41006e7i	− 0.646351i
$863$	− 1.50017e7i	− 0.685665i	−0.939397	−	0.342833i	$-0.888614\pi$
$863$	− 1.50017e7i	− 0.685665i	0.939397	−	0.342833i	$-0.111386\pi$
$864$	8.01278e6	0.365173
$865$	0	0
$866$	−1.10425e7	−0.500348
$867$	4.56642e7i	2.06314i
$868$	6.90441e6i	0.311048i
$869$	−3.07497e7	−1.38131
$870$	0	0
$871$	−5.15253e6	−0.230131
$872$	− 6.35395e6i	− 0.282978i
$873$	− 1.43898e7i	− 0.639029i
$874$	−2.85124e6	−0.126257
$875$	0	0
$876$	−1.58108e7	−0.696136
$877$	− 2.64018e7i	− 1.15914i	−0.814923	−	0.579569i	$-0.803221\pi$
$877$	− 2.64018e7i	− 1.15914i	0.814923	−	0.579569i	$-0.196779\pi$
$878$	− 4.18989e6i	− 0.183428i
$879$	4.61719e7	2.01561
$880$	0	0
$881$	−3.15138e7	−1.36792	−0.683960	−	0.729520i	$-0.739744\pi$
$881$	−3.15138e7	−1.36792	−0.683960	+	0.729520i	$0.739744\pi$
$882$	− 650845.i	− 0.0281713i
$883$	− 1.04657e7i	− 0.451718i	−0.974160	−	0.225859i	$-0.927481\pi$
$883$	− 1.04657e7i	− 0.451718i	0.974160	−	0.225859i	$-0.0725189\pi$
$884$	−4.38327e6	−0.188655
$885$	0	0
$886$	8.53607e6	0.365320
$887$	− 2.29824e7i	− 0.980815i	−0.871493	−	0.490408i	$-0.836848\pi$
$887$	− 2.29824e7i	− 0.980815i	0.871493	−	0.490408i	$-0.163152\pi$
$888$	− 4.81292e7i	− 2.04822i
$889$	3.15136e7	1.33735
$890$	0	0
$891$	5.16376e7	2.17907
$892$	− 1.31366e6i	− 0.0552804i
$893$	− 3.97532e7i	− 1.66818i
$894$	1.65589e7	0.692928
$895$	0	0
$896$	−456146.	−0.0189816
$897$	− 1.02833e6i	− 0.0426730i
$898$	− 8.54508e6i	− 0.353610i
$899$	−1.84111e7	−0.759766
$900$	0	0
$901$	2.30059e7	0.944121
$902$	− 4.40752e7i	− 1.80376i
$903$	− 4.29048e7i	− 1.75100i
$904$	1.44804e7	0.589332
$905$	0	0
$906$	−4.62996e6	−0.187394
$907$	3.45508e7i	1.39457i	0.716795	+	0.697284i	$0.245608\pi$
$907$	3.45508e7i	1.39457i	−0.716795	+	0.697284i	$0.754392\pi$
$908$	− 4.89572e6i	− 0.197062i
$909$	1.79533e7	0.720666
$910$	0	0
$911$	−2.24713e7	−0.897081	−0.448540	−	0.893763i	$-0.648056\pi$
$911$	−2.24713e7	−0.897081	−0.448540	+	0.893763i	$0.648056\pi$
$912$	1.77770e7i	0.707736i
$913$	− 5.94775e7i	− 2.36143i
$914$	2.99298e7	1.18506
$915$	0	0
$916$	−7.40242e6	−0.291498
$917$	1.23810e7i	0.486218i
$918$	− 1.48113e7i	− 0.580078i
$919$	−1.60273e7	−0.625995	−0.312997	−	0.949754i	$-0.601333\pi$
$919$	−1.60273e7	−0.625995	−0.312997	+	0.949754i	$0.601333\pi$
$920$	0	0
$921$	−1.60240e7	−0.622476
$922$	− 1.13492e7i	− 0.439680i
$923$	2.46084e6i	0.0950779i
$924$	−2.39067e7	−0.921170
$925$	0	0
$926$	−2.90898e7	−1.11484
$927$	1.78587e7i	0.682576i
$928$	− 2.02647e7i	− 0.772451i
$929$	2.25541e6	0.0857404	0.0428702	−	0.999081i	$-0.486350\pi$
$929$	2.25541e6	0.0857404	0.0428702	+	0.999081i	$0.486350\pi$
$930$	0	0
$931$	2.13555e6	0.0807487
$932$	5.98875e6i	0.225838i
$933$	4.97231e7i	1.87006i
$934$	3.16238e7	1.18617
$935$	0	0
$936$	−5.07259e6	−0.189252
$937$	− 1.45299e7i	− 0.540645i	−0.962770	−	0.270323i	$-0.912870\pi$
$937$	− 1.45299e7i	− 0.540645i	0.962770	−	0.270323i	$-0.0871304\pi$
$938$	1.65128e7i	0.612792i
$939$	−2.92087e6	−0.108106
$940$	0	0
$941$	−2.81974e7	−1.03809	−0.519046	−	0.854746i	$-0.673713\pi$
$941$	−2.81974e7	−1.03809	−0.519046	+	0.854746i	$0.673713\pi$
$942$	3.31357e7i	1.21666i
$943$	4.41055e6i	0.161515i
$944$	−1.64266e7	−0.599954
$945$	0	0
$946$	−5.23319e7	−1.90125
$947$	1.11561e7i	0.404238i	0.979361	+	0.202119i	$0.0647828\pi$
$947$	1.11561e7i	0.404238i	−0.979361	+	0.202119i	$0.935217\pi$
$948$	− 1.16155e7i	− 0.419777i
$949$	−9.97223e6	−0.359440
$950$	0	0
$951$	4.73659e7	1.69830
$952$	4.74465e7i	1.69672i
$953$	− 2.78329e7i	− 0.992719i	−0.868117	−	0.496359i	$-0.834670\pi$
$953$	− 2.78329e7i	− 0.992719i	0.868117	−	0.496359i	$-0.165330\pi$
$954$	7.88253e6	0.280411
$955$	0	0
$956$	−1.42754e7	−0.505177
$957$	− 6.37488e7i	− 2.25005i
$958$	2.64415e7i	0.930833i
$959$	3.81043e7	1.33791
$960$	0	0
$961$	−1.19957e7	−0.419004
$962$	− 8.98753e6i	− 0.313114i
$963$	− 6.61168e6i	− 0.229745i
$964$	8.64109e6	0.299486
$965$	0	0
$966$	−3.29560e6	−0.113630
$967$	− 2.30836e7i	− 0.793847i	−0.917852	−	0.396924i	$-0.870078\pi$
$967$	− 2.30836e7i	− 0.793847i	0.917852	−	0.396924i	$-0.129922\pi$
$968$	6.69640e7i	2.29696i
$969$	−8.31162e7	−2.84365
$970$	0	0
$971$	1.79497e7	0.610955	0.305477	−	0.952199i	$-0.401184\pi$
$971$	1.79497e7	0.610955	0.305477	+	0.952199i	$0.401184\pi$
$972$	1.36680e7i	0.464024i
$973$	2.71338e7i	0.918815i
$974$	−7.47657e6	−0.252525
$975$	0	0
$976$	−1.85072e6	−0.0621892
$977$	5.63392e7i	1.88832i	0.329494	+	0.944158i	$0.393122\pi$
$977$	5.63392e7i	1.88832i	−0.329494	+	0.944158i	$0.606878\pi$
$978$	− 9.15665e6i	− 0.306118i
$979$	8.83775e7	2.94703
$980$	0	0
$981$	4.97754e6	0.165136
$982$	− 7.14815e6i	− 0.236545i
$983$	− 4.50208e6i	− 0.148604i	−0.997236	−	0.0743018i	$-0.976327\pi$
$983$	− 4.50208e6i	− 0.148604i	0.997236	−	0.0743018i	$-0.0236728\pi$
$984$	5.62341e7	1.85145
$985$	0	0
$986$	−3.74585e7	−1.22704
$987$	− 4.59486e7i	− 1.50134i
$988$	− 4.92784e6i	− 0.160607i
$989$	5.23678e6	0.170245
$990$	0	0
$991$	4.18623e7	1.35406	0.677032	−	0.735954i	$-0.263266\pi$
$991$	4.18623e7	1.35406	0.677032	+	0.735954i	$0.263266\pi$
$992$	1.83081e7i	0.590697i
$993$	− 5.08993e7i	− 1.63809i
$994$	7.88649e6	0.253173
$995$	0	0
$996$	2.24674e7	0.717635
$997$	5.44586e7i	1.73512i	0.497337	+	0.867558i	$0.334311\pi$
$997$	5.44586e7i	1.73512i	−0.497337	+	0.867558i	$0.665689\pi$
$998$	− 3.37062e7i	− 1.07123i
$999$	−2.20454e7	−0.698883

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.h.274.13		18
5.2	odd	4		325.6.a.i.1.3	yes	9
5.3	odd	4		325.6.a.h.1.7	✓	9
5.4	even	2	inner	325.6.b.h.274.6		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.7	✓	9	5.3	odd	4
325.6.a.i.1.3	yes	9	5.2	odd	4
325.6.b.h.274.6		18	5.4	even	2	inner
325.6.b.h.274.13		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)

\(f(q)\)	\(=\)	\(q+4.30592i q^{2} -19.9083i q^{3} +13.4591 q^{4} +85.7235 q^{6} +125.783i q^{7} +195.743i q^{8} -153.341 q^{9} -709.332 q^{11} -267.947i q^{12} -169.000i q^{13} -541.610 q^{14} -412.163 q^{16} -1927.07i q^{17} -660.272i q^{18} +2166.48 q^{19} +2504.12 q^{21} -3054.33i q^{22} +305.642i q^{23} +3896.91 q^{24} +727.700 q^{26} -1784.97i q^{27} +1692.92i q^{28} -4514.28 q^{29} +4078.41 q^{31} +4489.03i q^{32} +14121.6i q^{33} +8297.79 q^{34} -2063.82 q^{36} -12350.6i q^{37} +9328.69i q^{38} -3364.50 q^{39} +14430.4 q^{41} +10782.5i q^{42} -17133.7i q^{43} -9546.95 q^{44} -1316.07 q^{46} -18349.2i q^{47} +8205.47i q^{48} +985.723 q^{49} -38364.6 q^{51} -2274.58i q^{52} +11938.3i q^{53} +7685.93 q^{54} -24621.1 q^{56} -43131.0i q^{57} -19438.1i q^{58} +39854.6 q^{59} +4490.25 q^{61} +17561.3i q^{62} -19287.6i q^{63} -32518.6 q^{64} -60806.5 q^{66} -30488.3i q^{67} -25936.5i q^{68} +6084.82 q^{69} -14561.2 q^{71} -30015.3i q^{72} -59007.3i q^{73} +53180.6 q^{74} +29158.8 q^{76} -89221.7i q^{77} -14487.3i q^{78} +43350.2 q^{79} -72797.4 q^{81} +62136.2i q^{82} +83849.9i q^{83} +33703.1 q^{84} +73776.2 q^{86} +89871.6i q^{87} -138847. i q^{88} -124592. q^{89} +21257.3 q^{91} +4113.66i q^{92} -81194.2i q^{93} +79010.1 q^{94} +89369.1 q^{96} +93842.4i q^{97} +4244.44i q^{98} +108769. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 182 q^{4} - 166 q^{6} - 1124 q^{9} - 2844 q^{11} + 684 q^{14} - 2122 q^{16} + 816 q^{19} - 7824 q^{21} + 16938 q^{24} - 1690 q^{26} + 17474 q^{29} + 1496 q^{31} + 35578 q^{34} + 1024 q^{36} + 3718 q^{39}+ \cdots + 1515552 q^{99}+O(q^{100}) \) 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

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