LMFDB - Embedded newform 325.6.b.i.274.9

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:	$22$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.9
Character	$\chi$	$=$	325.274
Dual form			325.6.b.i.274.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.89241i q^{2} -3.45730i q^{3} +23.6340 q^{4} -9.99993 q^{6} +148.288i q^{7} -160.916i q^{8} +231.047 q^{9} +O(q^{10})$	\(q-2.89241i q^{2} -3.45730i q^{3} +23.6340 q^{4} -9.99993 q^{6} +148.288i q^{7} -160.916i q^{8} +231.047 q^{9} -712.658 q^{11} -81.7096i q^{12} +169.000i q^{13} +428.910 q^{14} +290.850 q^{16} +1131.92i q^{17} -668.283i q^{18} -1400.39 q^{19} +512.676 q^{21} +2061.30i q^{22} +897.571i q^{23} -556.336 q^{24} +488.818 q^{26} -1638.92i q^{27} +3504.63i q^{28} -3238.07 q^{29} -7976.53 q^{31} -5990.58i q^{32} +2463.87i q^{33} +3273.98 q^{34} +5460.56 q^{36} -4978.35i q^{37} +4050.50i q^{38} +584.284 q^{39} -15559.9 q^{41} -1482.87i q^{42} -2308.61i q^{43} -16842.9 q^{44} +2596.15 q^{46} +7602.52i q^{47} -1005.56i q^{48} -5182.35 q^{49} +3913.39 q^{51} +3994.14i q^{52} -14138.1i q^{53} -4740.44 q^{54} +23862.0 q^{56} +4841.56i q^{57} +9365.83i q^{58} -49808.7 q^{59} -2516.69 q^{61} +23071.4i q^{62} +34261.5i q^{63} -8020.03 q^{64} +7126.54 q^{66} +38549.3i q^{67} +26751.8i q^{68} +3103.17 q^{69} +68008.5 q^{71} -37179.2i q^{72} -21305.2i q^{73} -14399.4 q^{74} -33096.7 q^{76} -105679. i q^{77} -1689.99i q^{78} -3984.60 q^{79} +50478.2 q^{81} +45005.8i q^{82} -13876.6i q^{83} +12116.6 q^{84} -6677.44 q^{86} +11195.0i q^{87} +114678. i q^{88} -89289.8 q^{89} -25060.7 q^{91} +21213.1i q^{92} +27577.2i q^{93} +21989.6 q^{94} -20711.2 q^{96} +147549. i q^{97} +14989.5i q^{98} -164658. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) $ 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9	$ 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36} - 3718 q^{39} - 20020 q^{41} - 121302 q^{44} - 19716 q^{46} - 146626 q^{49} + 73212 q^{51} - 96026 q^{54} - 2524 q^{56} - 263284 q^{59} - 86730 q^{61} - 360142 q^{64} - 117214 q^{66} - 118690 q^{69} + 136840 q^{71} - 742792 q^{74} + 45034 q^{76} - 104478 q^{79} + 215014 q^{81} - 1705432 q^{84} + 404492 q^{86} - 655424 q^{89} + 70304 q^{91} - 1299948 q^{94} + 1799326 q^{96} - 853396 q^{99}+O(q^{100}) $ 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9 + 2552 * q^11 - 1156 * q^14 + 11414 * q^16 - 7040 * q^19 + 3412 * q^21 - 15446 * q^24 + 1690 * q^26 - 36850 * q^29 + 18136 * q^31 + 28910 * q^34 + 75368 * q^36 - 3718 * q^39 - 20020 * q^41 - 121302 * q^44 - 19716 * q^46 - 146626 * q^49 + 73212 * q^51 - 96026 * q^54 - 2524 * q^56 - 263284 * q^59 - 86730 * q^61 - 360142 * q^64 - 117214 * q^66 - 118690 * q^69 + 136840 * q^71 - 742792 * q^74 + 45034 * q^76 - 104478 * q^79 + 215014 * q^81 - 1705432 * q^84 + 404492 * q^86 - 655424 * q^89 + 70304 * q^91 - 1299948 * q^94 + 1799326 * q^96 - 853396 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	− 2.89241i	− 0.511311i	−0.966768	−	0.255656i	$-0.917709\pi$
$2$	− 2.89241i	− 0.511311i	0.966768	−	0.255656i	$-0.0822913\pi$
$3$	− 3.45730i	− 0.221786i	−0.993832	−	0.110893i	$-0.964629\pi$
$3$	− 3.45730i	− 0.221786i	0.993832	−	0.110893i	$-0.0353711\pi$
$4$	23.6340	0.738561
$5$	0	0
$5$	0	0
$6$	−9.99993	−0.113402
$7$	148.288i	1.14383i	0.820313	+	0.571914i	$0.193799\pi$
$7$	148.288i	1.14383i	−0.820313	+	0.571914i	$0.806201\pi$
$8$	− 160.916i	− 0.888945i
$9$	231.047	0.950811
$10$	0	0
$11$	−712.658	−1.77582	−0.887912	−	0.460014i	$-0.847844\pi$
$11$	−712.658	−1.77582	−0.887912	+	0.460014i	$0.847844\pi$
$12$	− 81.7096i	− 0.163802i
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	428.910	0.584852
$15$	0	0
$16$	290.850	0.284033
$17$	1131.92i	0.949935i	0.880003	+	0.474968i	$0.157540\pi$
$17$	1131.92i	0.949935i	−0.880003	+	0.474968i	$0.842460\pi$
$18$	− 668.283i	− 0.486160i
$19$	−1400.39	−0.889947	−0.444974	−	0.895544i	$-0.646787\pi$
$19$	−1400.39	−0.889947	−0.444974	+	0.895544i	$0.646787\pi$
$20$	0	0
$21$	512.676	0.253685
$22$	2061.30i	0.907998i
$23$	897.571i	0.353793i	0.984229	+	0.176896i	$0.0566058\pi$
$23$	897.571i	0.353793i	−0.984229	+	0.176896i	$0.943394\pi$
$24$	−556.336	−0.197156
$25$	0	0
$26$	488.818	0.141812
$27$	− 1638.92i	− 0.432662i
$28$	3504.63i	0.844787i
$29$	−3238.07	−0.714975	−0.357488	−	0.933918i	$-0.616367\pi$
$29$	−3238.07	−0.714975	−0.357488	+	0.933918i	$0.616367\pi$
$30$	0	0
$31$	−7976.53	−1.49077	−0.745383	−	0.666636i	$-0.767733\pi$
$31$	−7976.53	−1.49077	−0.745383	+	0.666636i	$0.767733\pi$
$32$	− 5990.58i	− 1.03417i
$33$	2463.87i	0.393852i
$34$	3273.98	0.485712
$35$	0	0
$36$	5460.56	0.702232
$37$	− 4978.35i	− 0.597835i	−0.954279	−	0.298917i	$-0.903374\pi$
$37$	− 4978.35i	− 0.597835i	0.954279	−	0.298917i	$-0.0966255\pi$
$38$	4050.50i	0.455040i
$39$	584.284	0.0615123
$40$	0	0
$41$	−15559.9	−1.44560	−0.722800	−	0.691057i	$-0.757145\pi$
$41$	−15559.9	−1.44560	−0.722800	+	0.691057i	$0.757145\pi$
$42$	− 1482.87i	− 0.129712i
$43$	− 2308.61i	− 0.190405i	−0.995458	−	0.0952026i	$-0.969650\pi$
$43$	− 2308.61i	− 0.190405i	0.995458	−	0.0952026i	$-0.0303499\pi$
$44$	−16842.9	−1.31155
$45$	0	0
$46$	2596.15	0.180898
$47$	7602.52i	0.502011i	0.967986	+	0.251005i	$0.0807612\pi$
$47$	7602.52i	0.502011i	−0.967986	+	0.251005i	$0.919239\pi$
$48$	− 1005.56i	− 0.0629946i
$49$	−5182.35	−0.308344
$50$	0	0
$51$	3913.39	0.210682
$52$	3994.14i	0.204840i
$53$	− 14138.1i	− 0.691357i	−0.938353	−	0.345678i	$-0.887649\pi$
$53$	− 14138.1i	− 0.691357i	0.938353	−	0.345678i	$-0.112351\pi$
$54$	−4740.44	−0.221225
$55$	0	0
$56$	23862.0	1.01680
$57$	4841.56i	0.197378i
$58$	9365.83i	0.365575i
$59$	−49808.7	−1.86284	−0.931419	−	0.363948i	$-0.881428\pi$
$59$	−49808.7	−1.86284	−0.931419	+	0.363948i	$0.881428\pi$
$60$	0	0
$61$	−2516.69	−0.0865973	−0.0432986	−	0.999062i	$-0.513787\pi$
$61$	−2516.69	−0.0865973	−0.0432986	+	0.999062i	$0.513787\pi$
$62$	23071.4i	0.762245i
$63$	34261.5i	1.08757i
$64$	−8020.03	−0.244752
$65$	0	0
$66$	7126.54	0.201381
$67$	38549.3i	1.04913i	0.851370	+	0.524566i	$0.175772\pi$
$67$	38549.3i	1.04913i	−0.851370	+	0.524566i	$0.824228\pi$
$68$	26751.8i	0.701585i
$69$	3103.17	0.0784663
$70$	0	0
$71$	68008.5	1.60110	0.800548	−	0.599269i	$-0.204542\pi$
$71$	68008.5	1.60110	0.800548	+	0.599269i	$0.204542\pi$
$72$	− 37179.2i	− 0.845219i
$73$	− 21305.2i	− 0.467927i	−0.972245	−	0.233963i	$-0.924830\pi$
$73$	− 21305.2i	− 0.467927i	0.972245	−	0.233963i	$-0.0751696\pi$
$74$	−14399.4	−0.305680
$75$	0	0
$76$	−33096.7	−0.657280
$77$	− 105679.i	− 2.03124i
$78$	− 1689.99i	− 0.0314519i
$79$	−3984.60	−0.0718318	−0.0359159	−	0.999355i	$-0.511435\pi$
$79$	−3984.60	−0.0718318	−0.0359159	+	0.999355i	$0.511435\pi$
$80$	0	0
$81$	50478.2	0.854853
$82$	45005.8i	0.739152i
$83$	− 13876.6i	− 0.221100i	−0.993871	−	0.110550i	$-0.964739\pi$
$83$	− 13876.6i	− 0.221100i	0.993871	−	0.110550i	$-0.0352612\pi$
$84$	12116.6	0.187362
$85$	0	0
$86$	−6677.44	−0.0973563
$87$	11195.0i	0.158571i
$88$	114678.i	1.57861i
$89$	−89289.8	−1.19489	−0.597443	−	0.801911i	$-0.703817\pi$
$89$	−89289.8	−1.19489	−0.597443	+	0.801911i	$0.703817\pi$
$90$	0	0
$91$	−25060.7	−0.317241
$92$	21213.1i	0.261298i
$93$	27577.2i	0.330631i
$94$	21989.6	0.256684
$95$	0	0
$96$	−20711.2	−0.229365
$97$	147549.i	1.59223i	0.605145	+	0.796116i	$0.293115\pi$
$97$	147549.i	1.59223i	−0.605145	+	0.796116i	$0.706885\pi$
$98$	14989.5i	0.157660i
$99$	−164658.	−1.68847
$100$	0	0
$101$	−10084.2	−0.0983642	−0.0491821	−	0.998790i	$-0.515661\pi$
$101$	−10084.2	−0.0983642	−0.0491821	+	0.998790i	$0.515661\pi$
$102$	− 11319.1i	− 0.107724i
$103$	175080.i	1.62608i	0.582206	+	0.813041i	$0.302190\pi$
$103$	175080.i	1.62608i	−0.582206	+	0.813041i	$0.697810\pi$
$104$	27194.9	0.246549
$105$	0	0
$106$	−40893.3	−0.353498
$107$	225117.i	1.90086i	0.310946	+	0.950428i	$0.399354\pi$
$107$	225117.i	1.90086i	−0.310946	+	0.950428i	$0.600646\pi$
$108$	− 38734.2i	− 0.319547i
$109$	−25083.5	−0.202219	−0.101109	−	0.994875i	$-0.532239\pi$
$109$	−25083.5	−0.202219	−0.101109	+	0.994875i	$0.532239\pi$
$110$	0	0
$111$	−17211.6	−0.132591
$112$	43129.6i	0.324885i
$113$	− 39258.9i	− 0.289229i	−0.989488	−	0.144615i	$-0.953806\pi$
$113$	− 39258.9i	− 0.289229i	0.989488	−	0.144615i	$-0.0461942\pi$
$114$	14003.8	0.100921
$115$	0	0
$116$	−76528.3	−0.528053
$117$	39047.0i	0.263708i
$118$	144067.i	0.952490i
$119$	−167850.	−1.08656
$120$	0	0
$121$	346831.	2.15355
$122$	7279.29i	0.0442782i
$123$	53795.4i	0.320614i
$124$	−188517.	−1.10102
$125$	0	0
$126$	99098.4	0.556084
$127$	267258.i	1.47035i	0.677875	+	0.735177i	$0.262901\pi$
$127$	267258.i	1.47035i	−0.677875	+	0.735177i	$0.737099\pi$
$128$	− 168501.i	− 0.909031i
$129$	−7981.55	−0.0422292
$130$	0	0
$131$	−157114.	−0.799903	−0.399952	−	0.916536i	$-0.630973\pi$
$131$	−157114.	−0.799903	−0.399952	+	0.916536i	$0.630973\pi$
$132$	58231.1i	0.290884i
$133$	− 207661.i	− 1.01795i
$134$	111501.	0.536433
$135$	0	0
$136$	182145.	0.844441
$137$	− 132383.i	− 0.602603i	−0.953529	−	0.301302i	$-0.902579\pi$
$137$	− 132383.i	− 0.602603i	0.953529	−	0.301302i	$-0.0974211\pi$
$138$	− 8975.65i	− 0.0401207i
$139$	285040.	1.25132	0.625661	−	0.780095i	$-0.284829\pi$
$139$	285040.	1.25132	0.625661	+	0.780095i	$0.284829\pi$
$140$	0	0
$141$	26284.2	0.111339
$142$	− 196709.i	− 0.818658i
$143$	− 120439.i	− 0.492525i
$144$	67200.1	0.270062
$145$	0	0
$146$	−61623.4	−0.239256
$147$	17916.9i	0.0683864i
$148$	− 117658.i	− 0.441537i
$149$	156686.	0.578180	0.289090	−	0.957302i	$-0.406647\pi$
$149$	156686.	0.578180	0.289090	+	0.957302i	$0.406647\pi$
$150$	0	0
$151$	422557.	1.50814	0.754072	−	0.656792i	$-0.228087\pi$
$151$	422557.	1.50814	0.754072	+	0.656792i	$0.228087\pi$
$152$	225345.i	0.791115i
$153$	261527.i	0.903209i
$154$	−305666.	−1.03859
$155$	0	0
$156$	13808.9	0.0454306
$157$	129540.i	0.419427i	0.977763	+	0.209713i	$0.0672531\pi$
$157$	129540.i	0.419427i	−0.977763	+	0.209713i	$0.932747\pi$
$158$	11525.1i	0.0367284i
$159$	−48879.7	−0.153333
$160$	0	0
$161$	−133099.	−0.404679
$162$	− 146004.i	− 0.437096i
$163$	616763.i	1.81823i	0.416542	+	0.909116i	$0.363242\pi$
$163$	616763.i	1.81823i	−0.416542	+	0.909116i	$0.636758\pi$
$164$	−367743.	−1.06766
$165$	0	0
$166$	−40136.9	−0.113051
$167$	− 512670.i	− 1.42248i	−0.702948	−	0.711241i	$-0.748133\pi$
$167$	− 512670.i	− 1.42248i	0.702948	−	0.711241i	$-0.251867\pi$
$168$	− 82498.0i	− 0.225512i
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	−323556.	−0.846172
$172$	− 54561.5i	− 0.140626i
$173$	− 462817.i	− 1.17569i	−0.808973	−	0.587846i	$-0.799976\pi$
$173$	− 462817.i	− 1.17569i	0.808973	−	0.587846i	$-0.200024\pi$
$174$	32380.5	0.0810793
$175$	0	0
$176$	−207277.	−0.504393
$177$	172204.i	0.413151i
$178$	258263.i	0.610959i
$179$	−413926.	−0.965584	−0.482792	−	0.875735i	$-0.660377\pi$
$179$	−413926.	−0.965584	−0.482792	+	0.875735i	$0.660377\pi$
$180$	0	0
$181$	−167089.	−0.379099	−0.189549	−	0.981871i	$-0.560703\pi$
$181$	−167089.	−0.379099	−0.189549	+	0.981871i	$0.560703\pi$
$182$	72485.8i	0.162209i
$183$	8700.93i	0.0192061i
$184$	144434.	0.314503
$185$	0	0
$186$	79764.7	0.169055
$187$	− 806673.i	− 1.68692i
$188$	179678.i	0.370766i
$189$	243033.	0.494892
$190$	0	0
$191$	30723.8	0.0609385	0.0304692	−	0.999536i	$-0.490300\pi$
$191$	30723.8	0.0609385	0.0304692	+	0.999536i	$0.490300\pi$
$192$	27727.6i	0.0542825i
$193$	− 1.01259e6i	− 1.95677i	−0.206784	−	0.978387i	$-0.566300\pi$
$193$	− 1.01259e6i	− 1.95677i	0.206784	−	0.978387i	$-0.433700\pi$
$194$	426772.	0.814125
$195$	0	0
$196$	−122479.	−0.227731
$197$	422275.i	0.775229i	0.921822	+	0.387614i	$0.126701\pi$
$197$	422275.i	0.775229i	−0.921822	+	0.387614i	$0.873299\pi$
$198$	476258.i	0.863335i
$199$	−553506.	−0.990808	−0.495404	−	0.868663i	$-0.664980\pi$
$199$	−553506.	−0.990808	−0.495404	+	0.868663i	$0.664980\pi$
$200$	0	0
$201$	133277.	0.232683
$202$	29167.6i	0.0502947i
$203$	− 480167.i	− 0.817809i
$204$	92488.9	0.155602
$205$	0	0
$206$	506402.	0.831434
$207$	207381.i	0.336390i
$208$	49153.7i	0.0787767i
$209$	997998.	1.58039
$210$	0	0
$211$	−758171.	−1.17236	−0.586180	−	0.810181i	$-0.699369\pi$
$211$	−758171.	−1.17236	−0.586180	+	0.810181i	$0.699369\pi$
$212$	− 334140.i	− 0.510609i
$213$	− 235126.i	− 0.355100i
$214$	651132.	0.971928
$215$	0	0
$216$	−263729.	−0.384613
$217$	− 1.18282e6i	− 1.70518i
$218$	72551.8i	0.103397i
$219$	−73658.4	−0.103780
$220$	0	0
$221$	−191295.	−0.263465
$222$	49783.2i	0.0677954i
$223$	− 142765.i	− 0.192247i	−0.995369	−	0.0961236i	$-0.969356\pi$
$223$	− 142765.i	− 0.192247i	0.995369	−	0.0961236i	$-0.0306444\pi$
$224$	888332.	1.18292
$225$	0	0
$226$	−113553.	−0.147886
$227$	− 578969.i	− 0.745746i	−0.927882	−	0.372873i	$-0.878373\pi$
$227$	− 578969.i	− 0.745746i	0.927882	−	0.372873i	$-0.121627\pi$
$228$	114425.i	0.145775i
$229$	242404.	0.305457	0.152729	−	0.988268i	$-0.451194\pi$
$229$	242404.	0.305457	0.152729	+	0.988268i	$0.451194\pi$
$230$	0	0
$231$	−365363.	−0.450500
$232$	521058.i	0.635574i
$233$	− 540278.i	− 0.651970i	−0.945375	−	0.325985i	$-0.894304\pi$
$233$	− 540278.i	− 0.651970i	0.945375	−	0.325985i	$-0.105696\pi$
$234$	112940.	0.134837
$235$	0	0
$236$	−1.17718e6	−1.37582
$237$	13776.0i	0.0159313i
$238$	485493.i	0.555572i
$239$	223034.	0.252567	0.126284	−	0.991994i	$-0.459695\pi$
$239$	223034.	0.252567	0.126284	+	0.991994i	$0.459695\pi$
$240$	0	0
$241$	1.31596e6	1.45948	0.729742	−	0.683723i	$-0.239640\pi$
$241$	1.31596e6	1.45948	0.729742	+	0.683723i	$0.239640\pi$
$242$	− 1.00318e6i	− 1.10113i
$243$	− 572776.i	− 0.622256i
$244$	−59479.2	−0.0639574
$245$	0	0
$246$	155598.	0.163933
$247$	− 236666.i	− 0.246827i
$248$	1.28355e6i	1.32521i
$249$	−47975.6	−0.0490368
$250$	0	0
$251$	475886.	0.476781	0.238390	−	0.971169i	$-0.423380\pi$
$251$	475886.	0.476781	0.238390	+	0.971169i	$0.423380\pi$
$252$	809735.i	0.803233i
$253$	− 639661.i	− 0.628274i
$254$	773021.	0.751808
$255$	0	0
$256$	−744016.	−0.709549
$257$	184678.i	0.174415i	0.996190	+	0.0872074i	$0.0277943\pi$
$257$	184678.i	0.174415i	−0.996190	+	0.0872074i	$0.972206\pi$
$258$	23085.9i	0.0215923i
$259$	738230.	0.683821
$260$	0	0
$261$	−748146.	−0.679806
$262$	454439.i	0.408999i
$263$	541222.i	0.482487i	0.970465	+	0.241244i	$0.0775553\pi$
$263$	541222.i	0.482487i	−0.970465	+	0.241244i	$0.922445\pi$
$264$	396477.	0.350113
$265$	0	0
$266$	−600641.	−0.520488
$267$	308702.i	0.265009i
$268$	911073.i	0.774848i
$269$	−198419.	−0.167187	−0.0835934	−	0.996500i	$-0.526640\pi$
$269$	−198419.	−0.167187	−0.0835934	+	0.996500i	$0.526640\pi$
$270$	0	0
$271$	−258403.	−0.213734	−0.106867	−	0.994273i	$-0.534082\pi$
$271$	−258403.	−0.213734	−0.106867	+	0.994273i	$0.534082\pi$
$272$	329219.i	0.269813i
$273$	86642.3i	0.0703596i
$274$	−382907.	−0.308118
$275$	0	0
$276$	73340.2	0.0579521
$277$	− 2.19835e6i	− 1.72146i	−0.509063	−	0.860729i	$-0.670008\pi$
$277$	− 2.19835e6i	− 1.72146i	0.509063	−	0.860729i	$-0.329992\pi$
$278$	− 824454.i	− 0.639815i
$279$	−1.84295e6	−1.41744
$280$	0	0
$281$	−842.387	−0.000636423 0	−0.000318211	−	1.00000i	$-0.500101\pi$
$281$	−842.387	−0.000636423 0	−0.000318211		1.00000i	$0.500101\pi$
$282$	− 76024.7i	− 0.0569288i
$283$	− 1.93205e6i	− 1.43401i	−0.697067	−	0.717006i	$-0.745512\pi$
$283$	− 1.93205e6i	− 1.43401i	0.697067	−	0.717006i	$-0.254488\pi$
$284$	1.60731e6	1.18251
$285$	0	0
$286$	−348360.	−0.251833
$287$	− 2.30735e6i	− 1.65352i
$288$	− 1.38411e6i	− 0.983305i
$289$	138611.	0.0976230
$290$	0	0
$291$	510120.	0.353134
$292$	− 503526.i	− 0.345593i
$293$	− 1.01321e6i	− 0.689493i	−0.938696	−	0.344747i	$-0.887965\pi$
$293$	− 1.01321e6i	− 0.689493i	0.938696	−	0.344747i	$-0.112035\pi$
$294$	51823.1	0.0349667
$295$	0	0
$296$	−801098.	−0.531443
$297$	1.16799e6i	0.768332i
$298$	− 453199.i	− 0.295630i
$299$	−151689.	−0.0981245
$300$	0	0
$301$	342339.	0.217791
$302$	− 1.22221e6i	− 0.771131i
$303$	34864.0i	0.0218158i
$304$	−407303.	−0.252775
$305$	0	0
$306$	756444.	0.461821
$307$	386274.i	0.233910i	0.993137	+	0.116955i	$0.0373134\pi$
$307$	386274.i	0.233910i	−0.993137	+	0.116955i	$0.962687\pi$
$308$	− 2.49761e6i	− 1.50019i
$309$	605303.	0.360642
$310$	0	0
$311$	1.37666e6	0.807099	0.403550	−	0.914958i	$-0.367776\pi$
$311$	1.37666e6	0.807099	0.403550	+	0.914958i	$0.367776\pi$
$312$	− 94020.8i	− 0.0546811i
$313$	541373.i	0.312346i	0.987730	+	0.156173i	$0.0499157\pi$
$313$	541373.i	0.312346i	−0.987730	+	0.156173i	$0.950084\pi$
$314$	374684.	0.214458
$315$	0	0
$316$	−94171.8	−0.0530522
$317$	1.91853e6i	1.07231i	0.844120	+	0.536154i	$0.180124\pi$
$317$	1.91853e6i	1.07231i	−0.844120	+	0.536154i	$0.819876\pi$
$318$	141380.i	0.0784009i
$319$	2.30764e6	1.26967
$320$	0	0
$321$	778297.	0.421583
$322$	384977.i	0.206917i
$323$	− 1.58513e6i	− 0.845392i
$324$	1.19300e6	0.631361
$325$	0	0
$326$	1.78393e6	0.929683
$327$	86721.2i	0.0448493i
$328$	2.50385e6i	1.28506i
$329$	−1.12736e6	−0.574215
$330$	0	0
$331$	−2.40144e6	−1.20477	−0.602383	−	0.798207i	$-0.705782\pi$
$331$	−2.40144e6	−1.20477	−0.602383	+	0.798207i	$0.705782\pi$
$332$	− 327959.i	− 0.163296i
$333$	− 1.15023e6i	− 0.568428i
$334$	−1.48285e6	−0.727331
$335$	0	0
$336$	149112.	0.0720550
$337$	3.84325e6i	1.84342i	0.387882	+	0.921709i	$0.373207\pi$
$337$	3.84325e6i	1.84342i	−0.387882	+	0.921709i	$0.626793\pi$
$338$	82610.2i	0.0393316i
$339$	−135730.	−0.0641469
$340$	0	0
$341$	5.68454e6	2.64734
$342$	935856.i	0.432657i
$343$	1.72380e6i	0.791136i
$344$	−371493.	−0.169260
$345$	0	0
$346$	−1.33866e6	−0.601144
$347$	− 1.02696e6i	− 0.457858i	−0.973443	−	0.228929i	$-0.926478\pi$
$347$	− 1.02696e6i	− 0.457858i	0.973443	−	0.228929i	$-0.0735224\pi$
$348$	264581.i	0.117115i
$349$	−1.24908e6	−0.548944	−0.274472	−	0.961595i	$-0.588503\pi$
$349$	−1.24908e6	−0.548944	−0.274472	+	0.961595i	$0.588503\pi$
$350$	0	0
$351$	276978.	0.119999
$352$	4.26924e6i	1.83651i
$353$	3.13090e6i	1.33731i	0.743573	+	0.668654i	$0.233129\pi$
$353$	3.13090e6i	1.33731i	−0.743573	+	0.668654i	$0.766871\pi$
$354$	498084.	0.211249
$355$	0	0
$356$	−2.11027e6	−0.882497
$357$	580309.i	0.240984i
$358$	1.19724e6i	0.493714i
$359$	2.63535e6	1.07920	0.539600	−	0.841921i	$-0.318575\pi$
$359$	2.63535e6	1.07920	0.539600	+	0.841921i	$0.318575\pi$
$360$	0	0
$361$	−515013.	−0.207994
$362$	483291.i	0.193837i
$363$	− 1.19910e6i	− 0.477626i
$364$	−592283.	−0.234302
$365$	0	0
$366$	25166.7	0.00982027
$367$	− 2.23239e6i	− 0.865175i	−0.901592	−	0.432588i	$-0.857601\pi$
$367$	− 2.23239e6i	− 0.865175i	0.901592	−	0.432588i	$-0.142399\pi$
$368$	261059.i	0.100489i
$369$	−3.59508e6	−1.37449
$370$	0	0
$371$	2.09651e6	0.790794
$372$	651759.i	0.244191i
$373$	3.72332e6i	1.38566i	0.721099	+	0.692832i	$0.243637\pi$
$373$	3.72332e6i	1.38566i	−0.721099	+	0.692832i	$0.756363\pi$
$374$	−2.33323e6	−0.862539
$375$	0	0
$376$	1.22337e6	0.446260
$377$	− 547233.i	− 0.198298i
$378$	− 702951.i	− 0.253044i
$379$	2.11153e6	0.755089	0.377545	−	0.925991i	$-0.376768\pi$
$379$	2.11153e6	0.755089	0.377545	+	0.925991i	$0.376768\pi$
$380$	0	0
$381$	923992.	0.326104
$382$	− 88865.9i	− 0.0311585i
$383$	414336.i	0.144330i	0.997393	+	0.0721648i	$0.0229908\pi$
$383$	414336.i	0.144330i	−0.997393	+	0.0721648i	$0.977009\pi$
$384$	−582560.	−0.201610
$385$	0	0
$386$	−2.92883e6	−1.00052
$387$	− 533397.i	− 0.181039i
$388$	3.48716e6i	1.17596i
$389$	−4.10572e6	−1.37567	−0.687837	−	0.725865i	$-0.741440\pi$
$389$	−4.10572e6	−1.37567	−0.687837	+	0.725865i	$0.741440\pi$
$390$	0	0
$391$	−1.01598e6	−0.336080
$392$	833924.i	0.274101i
$393$	543191.i	0.177407i
$394$	1.22139e6	0.396383
$395$	0	0
$396$	−3.89151e6	−1.24704
$397$	− 5.78249e6i	− 1.84136i	−0.390317	−	0.920681i	$-0.627635\pi$
$397$	− 5.78249e6i	− 1.84136i	0.390317	−	0.920681i	$-0.372365\pi$
$398$	1.60097e6i	0.506611i
$399$	−717945.	−0.225766
$400$	0	0
$401$	−3.48722e6	−1.08298	−0.541488	−	0.840708i	$-0.682139\pi$
$401$	−3.48722e6	−1.08298	−0.541488	+	0.840708i	$0.682139\pi$
$402$	− 385491.i	− 0.118973i
$403$	− 1.34803e6i	− 0.413464i
$404$	−238329.	−0.0726479
$405$	0	0
$406$	−1.38884e6	−0.418155
$407$	3.54786e6i	1.06165i
$408$	− 629729.i	− 0.187285i
$409$	3.23381e6	0.955886	0.477943	−	0.878391i	$-0.341382\pi$
$409$	3.23381e6	0.955886	0.477943	+	0.878391i	$0.341382\pi$
$410$	0	0
$411$	−457688.	−0.133649
$412$	4.13782e6i	1.20096i
$413$	− 7.38603e6i	− 2.13077i
$414$	599832.	0.172000
$415$	0	0
$416$	1.01241e6	0.286829
$417$	− 985470.i	− 0.277526i
$418$	− 2.88662e6i	− 0.808070i
$419$	−3.63720e6	−1.01212	−0.506060	−	0.862498i	$-0.668899\pi$
$419$	−3.63720e6	−1.01212	−0.506060	+	0.862498i	$0.668899\pi$
$420$	0	0
$421$	2.11596e6	0.581839	0.290920	−	0.956747i	$-0.406039\pi$
$421$	2.11596e6	0.581839	0.290920	+	0.956747i	$0.406039\pi$
$422$	2.19294e6i	0.599441i
$423$	1.75654e6i	0.477318i
$424$	−2.27505e6	−0.614578
$425$	0	0
$426$	−680081.	−0.181567
$427$	− 373194.i	− 0.0990525i
$428$	5.32041e6i	1.40390i
$429$	−416395.	−0.109235
$430$	0	0
$431$	6.14425e6	1.59322	0.796609	−	0.604495i	$-0.206625\pi$
$431$	6.14425e6	1.59322	0.796609	+	0.604495i	$0.206625\pi$
$432$	− 476681.i	− 0.122890i
$433$	6.08667e6i	1.56013i	0.625700	+	0.780064i	$0.284813\pi$
$433$	6.08667e6i	1.56013i	−0.625700	+	0.780064i	$0.715187\pi$
$434$	−3.42121e6	−0.871878
$435$	0	0
$436$	−592822.	−0.149351
$437$	− 1.25695e6i	− 0.314857i
$438$	213050.i	0.0530636i
$439$	−5.46357e6	−1.35305	−0.676527	−	0.736418i	$-0.736516\pi$
$439$	−5.46357e6	−1.35305	−0.676527	+	0.736418i	$0.736516\pi$
$440$	0	0
$441$	−1.19737e6	−0.293177
$442$	553303.i	0.134712i
$443$	4.25637e6i	1.03046i	0.857052	+	0.515229i	$0.172293\pi$
$443$	4.25637e6i	1.03046i	−0.857052	+	0.515229i	$0.827707\pi$
$444$	−406779.	−0.0979268
$445$	0	0
$446$	−412935.	−0.0982981
$447$	− 541709.i	− 0.128232i
$448$	− 1.18927e6i	− 0.279954i
$449$	−3.07736e6	−0.720382	−0.360191	−	0.932879i	$-0.617288\pi$
$449$	−3.07736e6	−0.720382	−0.360191	+	0.932879i	$0.617288\pi$
$450$	0	0
$451$	1.10889e7	2.56713
$452$	− 927843.i	− 0.213613i
$453$	− 1.46091e6i	− 0.334485i
$454$	−1.67462e6	−0.381308
$455$	0	0
$456$	779086.	0.175458
$457$	− 1.92830e6i	− 0.431901i	−0.976404	−	0.215951i	$-0.930715\pi$
$457$	− 1.92830e6i	− 0.431901i	0.976404	−	0.215951i	$-0.0692850\pi$
$458$	− 701132.i	− 0.156184i
$459$	1.85513e6	0.411001
$460$	0	0
$461$	1.53125e6	0.335578	0.167789	−	0.985823i	$-0.446337\pi$
$461$	1.53125e6	0.335578	0.167789	+	0.985823i	$0.446337\pi$
$462$	1.05678e6i	0.230346i
$463$	2.13296e6i	0.462412i	0.972905	+	0.231206i	$0.0742672\pi$
$463$	2.13296e6i	0.462412i	−0.972905	+	0.231206i	$0.925733\pi$
$464$	−941792.	−0.203077
$465$	0	0
$466$	−1.56271e6	−0.333359
$467$	− 2.47557e6i	− 0.525270i	−0.964895	−	0.262635i	$-0.915408\pi$
$467$	− 2.47557e6i	− 0.525270i	0.964895	−	0.262635i	$-0.0845916\pi$
$468$	922834.i	0.194764i
$469$	−5.71641e6	−1.20003
$470$	0	0
$471$	447860.	0.0930230
$472$	8.01503e6i	1.65596i
$473$	1.64525e6i	0.338126i
$474$	39845.7	0.00814584
$475$	0	0
$476$	−3.96697e6	−0.802493
$477$	− 3.26657e6i	− 0.657349i
$478$	− 645108.i	− 0.129140i
$479$	−6.52504e6	−1.29940	−0.649702	−	0.760189i	$-0.725106\pi$
$479$	−6.52504e6	−1.29940	−0.649702	+	0.760189i	$0.725106\pi$
$480$	0	0
$481$	841341.	0.165810
$482$	− 3.80629e6i	− 0.746250i
$483$	460163.i	0.0897520i
$484$	8.19699e6	1.59053
$485$	0	0
$486$	−1.65671e6	−0.318167
$487$	5.50061e6i	1.05097i	0.850804	+	0.525483i	$0.176115\pi$
$487$	5.50061e6i	1.05097i	−0.850804	+	0.525483i	$0.823885\pi$
$488$	404976.i	0.0769803i
$489$	2.13233e6	0.403258
$490$	0	0
$491$	8.54562e6	1.59970	0.799852	−	0.600197i	$-0.204911\pi$
$491$	8.54562e6	1.59970	0.799852	+	0.600197i	$0.204911\pi$
$492$	1.27140e6i	0.236793i
$493$	− 3.66524e6i	− 0.679180i
$494$	−684534.	−0.126205
$495$	0	0
$496$	−2.31997e6	−0.423427
$497$	1.00848e7i	1.83138i
$498$	138765.i	0.0250730i
$499$	−2.98189e6	−0.536094	−0.268047	−	0.963406i	$-0.586378\pi$
$499$	−2.98189e6	−0.536094	−0.268047	+	0.963406i	$0.586378\pi$
$500$	0	0
$501$	−1.77245e6	−0.315486
$502$	− 1.37646e6i	− 0.243783i
$503$	− 4.24939e6i	− 0.748871i	−0.927253	−	0.374436i	$-0.877836\pi$
$503$	− 4.24939e6i	− 0.748871i	0.927253	−	0.374436i	$-0.122164\pi$
$504$	5.51324e6	0.966786
$505$	0	0
$506$	−1.85016e6	−0.321243
$507$	98743.9i	0.0170604i
$508$	6.31637e6i	1.08595i
$509$	1.22747e6	0.209999	0.105000	−	0.994472i	$-0.466516\pi$
$509$	1.22747e6	0.209999	0.105000	+	0.994472i	$0.466516\pi$
$510$	0	0
$511$	3.15930e6	0.535228
$512$	− 3.24004e6i	− 0.546230i
$513$	2.29513e6i	0.385047i
$514$	534166.	0.0891802
$515$	0	0
$516$	−188635.	−0.0311888
$517$	− 5.41800e6i	− 0.891483i
$518$	− 2.13527e6i	− 0.349645i
$519$	−1.60010e6	−0.260752
$520$	0	0
$521$	−7.91981e6	−1.27826	−0.639131	−	0.769097i	$-0.720706\pi$
$521$	−7.91981e6	−1.27826	−0.639131	+	0.769097i	$0.720706\pi$
$522$	2.16395e6i	0.347592i
$523$	− 4.76548e6i	− 0.761821i	−0.924612	−	0.380910i	$-0.875611\pi$
$523$	− 4.76548e6i	− 0.761821i	0.924612	−	0.380910i	$-0.124389\pi$
$524$	−3.71323e6	−0.590777
$525$	0	0
$526$	1.56544e6	0.246701
$527$	− 9.02880e6i	− 1.41613i
$528$	716618.i	0.111867i
$529$	5.63071e6	0.874831
$530$	0	0
$531$	−1.15082e7	−1.77121
$532$	− 4.90784e6i	− 0.751816i
$533$	− 2.62963e6i	− 0.400938i
$534$	892892.	0.135502
$535$	0	0
$536$	6.20322e6	0.932621
$537$	1.43107e6i	0.214153i
$538$	573909.i	0.0854845i
$539$	3.69324e6	0.547565
$540$	0	0
$541$	−5.85407e6	−0.859934	−0.429967	−	0.902845i	$-0.641475\pi$
$541$	−5.85407e6	−0.859934	−0.429967	+	0.902845i	$0.641475\pi$
$542$	747408.i	0.109285i
$543$	577678.i	0.0840787i
$544$	6.78087e6	0.982399
$545$	0	0
$546$	250605.	0.0359756
$547$	587042.i	0.0838882i	0.999120	+	0.0419441i	$0.0133551\pi$
$547$	587042.i	0.0838882i	−0.999120	+	0.0419441i	$0.986645\pi$
$548$	− 3.12874e6i	− 0.445059i
$549$	−581473.	−0.0823377
$550$	0	0
$551$	4.53455e6	0.636290
$552$	− 499351.i	− 0.0697522i
$553$	− 590868.i	− 0.0821633i
$554$	−6.35852e6	−0.880201
$555$	0	0
$556$	6.73663e6	0.924178
$557$	− 3.23702e6i	− 0.442086i	−0.975264	−	0.221043i	$-0.929054\pi$
$557$	− 3.23702e6i	− 0.442086i	0.975264	−	0.221043i	$-0.0709462\pi$
$558$	5.33058e6i	0.724751i
$559$	390155.	0.0528089
$560$	0	0
$561$	−2.78891e6	−0.374134
$562$	2436.53i	0 0.000325410i
$563$	− 1.09630e7i	− 1.45767i	−0.684689	−	0.728835i	$-0.740062\pi$
$563$	− 1.09630e7i	− 1.45767i	0.684689	−	0.728835i	$-0.259938\pi$
$564$	621200.	0.0822306
$565$	0	0
$566$	−5.58829e6	−0.733226
$567$	7.48531e6i	0.977805i
$568$	− 1.09437e7i	− 1.42329i
$569$	−4.35413e6	−0.563795	−0.281897	−	0.959445i	$-0.590964\pi$
$569$	−4.35413e6	−0.563795	−0.281897	+	0.959445i	$0.590964\pi$
$570$	0	0
$571$	4.59961e6	0.590379	0.295189	−	0.955439i	$-0.404617\pi$
$571$	4.59961e6	0.590379	0.295189	+	0.955439i	$0.404617\pi$
$572$	− 2.84646e6i	− 0.363760i
$573$	− 106221.i	− 0.0135153i
$574$	−6.67382e6	−0.845463
$575$	0	0
$576$	−1.85300e6	−0.232713
$577$	− 7.57061e6i	− 0.946654i	−0.880887	−	0.473327i	$-0.843053\pi$
$577$	− 7.57061e6i	− 0.946654i	0.880887	−	0.473327i	$-0.156947\pi$
$578$	− 400919.i	− 0.0499157i
$579$	−3.50083e6	−0.433985
$580$	0	0
$581$	2.05773e6	0.252900
$582$	− 1.47548e6i	− 0.180561i
$583$	1.00757e7i	1.22773i
$584$	−3.42835e6	−0.415962
$585$	0	0
$586$	−2.93062e6	−0.352546
$587$	1.48215e7i	1.77541i	0.460416	+	0.887703i	$0.347700\pi$
$587$	1.48215e7i	1.77541i	−0.460416	+	0.887703i	$0.652300\pi$
$588$	423448.i	0.0505076i
$589$	1.11702e7	1.32670
$590$	0	0
$591$	1.45993e6	0.171935
$592$	− 1.44795e6i	− 0.169805i
$593$	− 1.00021e6i	− 0.116803i	−0.998293	−	0.0584017i	$-0.981400\pi$
$593$	− 1.00021e6i	− 0.116803i	0.998293	−	0.0584017i	$-0.0186004\pi$
$594$	3.37831e6	0.392856
$595$	0	0
$596$	3.70310e6	0.427021
$597$	1.91363e6i	0.219747i
$598$	438749.i	0.0501721i
$599$	8.35745e6	0.951715	0.475857	−	0.879522i	$-0.342138\pi$
$599$	8.35745e6	0.951715	0.475857	+	0.879522i	$0.342138\pi$
$600$	0	0
$601$	−6.04104e6	−0.682222	−0.341111	−	0.940023i	$-0.610803\pi$
$601$	−6.04104e6	−0.682222	−0.341111	+	0.940023i	$0.610803\pi$
$602$	− 990185.i	− 0.111359i
$603$	8.90671e6i	0.997526i
$604$	9.98669e6	1.11386
$605$	0	0
$606$	100841.	0.0111547
$607$	− 137224.i	− 0.0151168i	−0.999971	−	0.00755838i	$-0.997594\pi$
$607$	− 137224.i	− 0.0151168i	0.999971	−	0.00755838i	$-0.00240593\pi$
$608$	8.38914e6i	0.920361i
$609$	−1.66008e6	−0.181378
$610$	0	0
$611$	−1.28483e6	−0.139233
$612$	6.18092e6i	0.667075i
$613$	− 1.09895e7i	− 1.18121i	−0.806960	−	0.590606i	$-0.798889\pi$
$613$	− 1.09895e7i	− 1.18121i	0.806960	−	0.590606i	$-0.201111\pi$
$614$	1.11726e6	0.119601
$615$	0	0
$616$	−1.70054e7	−1.80566
$617$	3.42268e6i	0.361954i	0.983487	+	0.180977i	$0.0579259\pi$
$617$	3.42268e6i	0.361954i	−0.983487	+	0.180977i	$0.942074\pi$
$618$	− 1.75078e6i	− 0.184400i
$619$	1.22543e7	1.28547	0.642736	−	0.766088i	$-0.277799\pi$
$619$	1.22543e7	1.28547	0.642736	+	0.766088i	$0.277799\pi$
$620$	0	0
$621$	1.47105e6	0.153073
$622$	− 3.98188e6i	− 0.412679i
$623$	− 1.32406e7i	− 1.36675i
$624$	169939.	0.0174715
$625$	0	0
$626$	1.56587e6	0.159706
$627$	− 3.45038e6i	− 0.350508i
$628$	3.06155e6i	0.309772i
$629$	5.63510e6	0.567904
$630$	0	0
$631$	7.99788e6	0.799653	0.399827	−	0.916591i	$-0.369070\pi$
$631$	7.99788e6	0.799653	0.399827	+	0.916591i	$0.369070\pi$
$632$	641187.i	0.0638546i
$633$	2.62122e6i	0.260013i
$634$	5.54917e6	0.548283
$635$	0	0
$636$	−1.15522e6	−0.113246
$637$	− 875816.i	− 0.0855194i
$638$	− 6.67463e6i	− 0.649196i
$639$	1.57132e7	1.52234
$640$	0	0
$641$	8.18872e6	0.787175	0.393587	−	0.919287i	$-0.371234\pi$
$641$	8.18872e6	0.787175	0.393587	+	0.919287i	$0.371234\pi$
$642$	− 2.25116e6i	− 0.215560i
$643$	1.64743e7i	1.57137i	0.618626	+	0.785686i	$0.287690\pi$
$643$	1.64743e7i	1.57137i	−0.618626	+	0.785686i	$0.712310\pi$
$644$	−3.14566e6	−0.298880
$645$	0	0
$646$	−4.58485e6	−0.432258
$647$	− 3.65606e6i	− 0.343362i	−0.985153	−	0.171681i	$-0.945080\pi$
$647$	− 3.65606e6i	− 0.343362i	0.985153	−	0.171681i	$-0.0549199\pi$
$648$	− 8.12277e6i	− 0.759917i
$649$	3.54966e7	3.30807
$650$	0	0
$651$	−4.08938e6	−0.378185
$652$	1.45766e7i	1.34288i
$653$	− 1.22553e7i	− 1.12471i	−0.826896	−	0.562355i	$-0.809896\pi$
$653$	− 1.22553e7i	− 1.12471i	0.826896	−	0.562355i	$-0.190104\pi$
$654$	250833.	0.0229319
$655$	0	0
$656$	−4.52561e6	−0.410599
$657$	− 4.92250e6i	− 0.444910i
$658$	3.26080e6i	0.293602i
$659$	−708926.	−0.0635898	−0.0317949	−	0.999494i	$-0.510122\pi$
$659$	−708926.	−0.0635898	−0.0317949	+	0.999494i	$0.510122\pi$
$660$	0	0
$661$	1.69284e7	1.50700	0.753500	−	0.657448i	$-0.228364\pi$
$661$	1.69284e7	1.50700	0.753500	+	0.657448i	$0.228364\pi$
$662$	6.94597e6i	0.616010i
$663$	661363.i	0.0584327i
$664$	−2.23297e6	−0.196546
$665$	0	0
$666$	−3.32695e6	−0.290644
$667$	− 2.90639e6i	− 0.252953i
$668$	− 1.21164e7i	− 1.05059i
$669$	−493582.	−0.0426377
$670$	0	0
$671$	1.79354e6	0.153781
$672$	− 3.07123e6i	− 0.262355i
$673$	1.00330e7i	0.853873i	0.904281	+	0.426937i	$0.140407\pi$
$673$	1.00330e7i	0.853873i	−0.904281	+	0.426937i	$0.859593\pi$
$674$	1.11163e7	0.942560
$675$	0	0
$676$	−675009.	−0.0568124
$677$	− 1.54623e7i	− 1.29659i	−0.761388	−	0.648296i	$-0.775482\pi$
$677$	− 1.54623e7i	− 1.29659i	0.761388	−	0.648296i	$-0.224518\pi$
$678$	392586.i	0.0327990i
$679$	−2.18797e7	−1.82124
$680$	0	0
$681$	−2.00167e6	−0.165396
$682$	− 1.64420e7i	− 1.35361i
$683$	− 9.19066e6i	− 0.753868i	−0.926240	−	0.376934i	$-0.876978\pi$
$683$	− 9.19066e6i	− 0.753868i	0.926240	−	0.376934i	$-0.123022\pi$
$684$	−7.64689e6	−0.624949
$685$	0	0
$686$	4.98593e6	0.404516
$687$	− 838062.i	− 0.0677461i
$688$	− 671459.i	− 0.0540814i
$689$	2.38934e6	0.191748
$690$	0	0
$691$	−1.38192e7	−1.10100	−0.550502	−	0.834834i	$-0.685564\pi$
$691$	−1.38192e7	−1.10100	−0.550502	+	0.834834i	$0.685564\pi$
$692$	− 1.09382e7i	− 0.868320i
$693$	− 2.44168e7i	− 1.93132i
$694$	−2.97040e6	−0.234108
$695$	0	0
$696$	1.80145e6	0.140961
$697$	− 1.76126e7i	− 1.37323i
$698$	3.61286e6i	0.280681i
$699$	−1.86790e6	−0.144598
$700$	0	0
$701$	1.58946e7	1.22167	0.610834	−	0.791758i	$-0.290834\pi$
$701$	1.58946e7	1.22167	0.610834	+	0.791758i	$0.290834\pi$
$702$	− 801134.i	− 0.0613568i
$703$	6.97162e6i	0.532041i
$704$	5.71554e6	0.434636
$705$	0	0
$706$	9.05584e6	0.683781
$707$	− 1.49536e6i	− 0.112512i
$708$	4.06985e6i	0.305137i
$709$	−1.20549e7	−0.900635	−0.450317	−	0.892869i	$-0.648689\pi$
$709$	−1.20549e7	−0.900635	−0.450317	+	0.892869i	$0.648689\pi$
$710$	0	0
$711$	−920630.	−0.0682985
$712$	1.43682e7i	1.06219i
$713$	− 7.15950e6i	− 0.527423i
$714$	1.67849e6	0.123218
$715$	0	0
$716$	−9.78271e6	−0.713143
$717$	− 771097.i	− 0.0560159i
$718$	− 7.62252e6i	− 0.551807i
$719$	−1.21480e7	−0.876357	−0.438178	−	0.898888i	$-0.644376\pi$
$719$	−1.21480e7	−0.876357	−0.438178	+	0.898888i	$0.644376\pi$
$720$	0	0
$721$	−2.59622e7	−1.85996
$722$	1.48963e6i	0.106350i
$723$	− 4.54966e6i	− 0.323693i
$724$	−3.94898e6	−0.279988
$725$	0	0
$726$	−3.46829e6	−0.244216
$727$	1.37550e7i	0.965215i	0.875837	+	0.482607i	$0.160310\pi$
$727$	1.37550e7i	0.965215i	−0.875837	+	0.482607i	$0.839690\pi$
$728$	4.03267e6i	0.282010i
$729$	1.02859e7	0.716845
$730$	0	0
$731$	2.61316e6	0.180873
$732$	205637.i	0.0141848i
$733$	5.10914e6i	0.351227i	0.984459	+	0.175614i	$0.0561909\pi$
$733$	5.10914e6i	0.351227i	−0.984459	+	0.175614i	$0.943809\pi$
$734$	−6.45698e6	−0.442374
$735$	0	0
$736$	5.37697e6	0.365884
$737$	− 2.74725e7i	− 1.86307i
$738$	1.03985e7i	0.702794i
$739$	−4.07246e6	−0.274312	−0.137156	−	0.990549i	$-0.543796\pi$
$739$	−4.07246e6	−0.274312	−0.137156	+	0.990549i	$0.543796\pi$
$740$	0	0
$741$	−818224.	−0.0547427
$742$	− 6.06398e6i	− 0.404342i
$743$	1.86637e7i	1.24029i	0.784485	+	0.620147i	$0.212927\pi$
$743$	1.86637e7i	1.24029i	−0.784485	+	0.620147i	$0.787073\pi$
$744$	4.43763e6	0.293913
$745$	0	0
$746$	1.07694e7	0.708505
$747$	− 3.20615e6i	− 0.210224i
$748$	− 1.90649e7i	− 1.24589i
$749$	−3.33822e7	−2.17425
$750$	0	0
$751$	−2.20898e7	−1.42919	−0.714597	−	0.699536i	$-0.753390\pi$
$751$	−2.20898e7	−1.42919	−0.714597	+	0.699536i	$0.753390\pi$
$752$	2.21119e6i	0.142588i
$753$	− 1.64528e6i	− 0.105743i
$754$	−1.58282e6	−0.101392
$755$	0	0
$756$	5.74382e6	0.365508
$757$	2.27912e7i	1.44553i	0.691094	+	0.722765i	$0.257129\pi$
$757$	2.27912e7i	1.44553i	−0.691094	+	0.722765i	$0.742871\pi$
$758$	− 6.10740e6i	− 0.386085i
$759$	−2.21150e6	−0.139342
$760$	0	0
$761$	2.05216e7	1.28455	0.642274	−	0.766475i	$-0.277991\pi$
$761$	2.05216e7	1.28455	0.642274	+	0.766475i	$0.277991\pi$
$762$	− 2.67257e6i	− 0.166740i
$763$	− 3.71958e6i	− 0.231304i
$764$	726125.	0.0450068
$765$	0	0
$766$	1.19843e6	0.0737973
$767$	− 8.41767e6i	− 0.516658i
$768$	2.57229e6i	0.157368i
$769$	2.11516e7	1.28982	0.644908	−	0.764260i	$-0.276896\pi$
$769$	2.11516e7	1.28982	0.644908	+	0.764260i	$0.276896\pi$
$770$	0	0
$771$	638488.	0.0386827
$772$	− 2.39315e7i	− 1.44520i
$773$	− 6.75134e6i	− 0.406389i	−0.979138	−	0.203194i	$-0.934868\pi$
$773$	− 6.75134e6i	− 0.406389i	0.979138	−	0.203194i	$-0.0651323\pi$
$774$	−1.54280e6	−0.0925675
$775$	0	0
$776$	2.37430e7	1.41541
$777$	− 2.55228e6i	− 0.151662i
$778$	1.18754e7i	0.703398i
$779$	2.17900e7	1.28651
$780$	0	0
$781$	−4.84668e7	−2.84326
$782$	2.93863e6i	0.171842i
$783$	5.30694e6i	0.309343i
$784$	−1.50729e6	−0.0875801
$785$	0	0
$786$	1.57113e6	0.0907103
$787$	1.78318e7i	1.02626i	0.858310	+	0.513132i	$0.171515\pi$
$787$	1.78318e7i	1.02626i	−0.858310	+	0.513132i	$0.828485\pi$
$788$	9.98003e6i	0.572554i
$789$	1.87117e6	0.107009
$790$	0	0
$791$	5.82163e6	0.330829
$792$	2.64961e7i	1.50096i
$793$	− 425320.i	− 0.0240178i
$794$	−1.67254e7	−0.941508
$795$	0	0
$796$	−1.30815e7	−0.731772
$797$	− 1.22612e7i	− 0.683732i	−0.939749	−	0.341866i	$-0.888941\pi$
$797$	− 1.22612e7i	− 0.683732i	0.939749	−	0.341866i	$-0.111059\pi$
$798$	2.07659e6i	0.115437i
$799$	−8.60546e6	−0.476878
$800$	0	0
$801$	−2.06301e7	−1.13611
$802$	1.00865e7i	0.553738i
$803$	1.51833e7i	0.830956i
$804$	3.14985e6	0.171850
$805$	0	0
$806$	−3.89907e6	−0.211409
$807$	685993.i	0.0370797i
$808$	1.62271e6i	0.0874404i
$809$	1.86542e7	1.00208	0.501042	−	0.865423i	$-0.332950\pi$
$809$	1.86542e7	1.00208	0.501042	+	0.865423i	$0.332950\pi$
$810$	0	0
$811$	2.65131e6	0.141550	0.0707748	−	0.997492i	$-0.477453\pi$
$811$	2.65131e6	0.141550	0.0707748	+	0.997492i	$0.477453\pi$
$812$	− 1.13482e7i	− 0.604002i
$813$	893377.i	0.0474033i
$814$	1.02619e7	0.542833
$815$	0	0
$816$	1.13821e6	0.0598408
$817$	3.23295e6i	0.169451i
$818$	− 9.35351e6i	− 0.488755i
$819$	−5.79020e6	−0.301636
$820$	0	0
$821$	2.04948e7	1.06117	0.530587	−	0.847630i	$-0.321972\pi$
$821$	2.04948e7	1.06117	0.530587	+	0.847630i	$0.321972\pi$
$822$	1.32382e6i	0.0683361i
$823$	1.26841e7i	0.652768i	0.945237	+	0.326384i	$0.105830\pi$
$823$	1.26841e7i	0.652768i	−0.945237	+	0.326384i	$0.894170\pi$
$824$	2.81732e7	1.44550
$825$	0	0
$826$	−2.13635e7	−1.08949
$827$	− 9.90644e6i	− 0.503679i	−0.967769	−	0.251839i	$-0.918965\pi$
$827$	− 9.90644e6i	− 0.503679i	0.967769	−	0.251839i	$-0.0810355\pi$
$828$	4.90124e6i	0.248445i
$829$	−8.05866e6	−0.407265	−0.203632	−	0.979047i	$-0.565275\pi$
$829$	−8.05866e6	−0.407265	−0.203632	+	0.979047i	$0.565275\pi$
$830$	0	0
$831$	−7.60034e6	−0.381795
$832$	− 1.35538e6i	− 0.0678819i
$833$	− 5.86601e6i	− 0.292907i
$834$	−2.85039e6	−0.141902
$835$	0	0
$836$	2.35866e7	1.16721
$837$	1.30729e7i	0.644998i
$838$	1.05203e7i	0.517508i
$839$	−2.15392e7	−1.05639	−0.528196	−	0.849122i	$-0.677131\pi$
$839$	−2.15392e7	−1.05639	−0.528196	+	0.849122i	$0.677131\pi$
$840$	0	0
$841$	−1.00261e7	−0.488811
$842$	− 6.12024e6i	− 0.297501i
$843$	2912.38i	0 0.000141150i
$844$	−1.79186e7	−0.865860
$845$	0	0
$846$	5.08064e6	0.244058
$847$	5.14309e7i	2.46329i
$848$	− 4.11207e6i	− 0.196368i
$849$	−6.67968e6	−0.318044
$850$	0	0
$851$	4.46842e6	0.211510
$852$	− 5.55695e6i	− 0.262263i
$853$	1.55629e7i	0.732351i	0.930546	+	0.366175i	$0.119333\pi$
$853$	1.55629e7i	0.732351i	−0.930546	+	0.366175i	$0.880667\pi$
$854$	−1.07943e6	−0.0506466
$855$	0	0
$856$	3.62250e7	1.68976
$857$	3.66500e7i	1.70460i	0.523055	+	0.852299i	$0.324792\pi$
$857$	3.66500e7i	1.70460i	−0.523055	+	0.852299i	$0.675208\pi$
$858$	1.20438e6i	0.0558531i
$859$	−3.16288e7	−1.46251	−0.731257	−	0.682102i	$-0.761066\pi$
$859$	−3.16288e7	−1.46251	−0.731257	+	0.682102i	$0.761066\pi$
$860$	0	0
$861$	−7.97721e6	−0.366727
$862$	− 1.77717e7i	− 0.814630i
$863$	1.63149e7i	0.745691i	0.927894	+	0.372845i	$0.121618\pi$
$863$	1.63149e7i	0.745691i	−0.927894	+	0.372845i	$0.878382\pi$
$864$	−9.81810e6	−0.447448
$865$	0	0
$866$	1.76052e7	0.797710
$867$	− 479219.i	− 0.0216514i
$868$	− 2.79548e7i	− 1.25938i
$869$	2.83966e6	0.127561
$870$	0	0
$871$	−6.51484e6	−0.290977
$872$	4.03634e6i	0.179762i
$873$	3.40907e7i	1.51391i
$874$	−3.63561e6	−0.160990
$875$	0	0
$876$	−1.74084e6	−0.0766475
$877$	2.64601e7i	1.16170i	0.814012	+	0.580848i	$0.197279\pi$
$877$	2.64601e7i	1.16170i	−0.814012	+	0.580848i	$0.802721\pi$
$878$	1.58029e7i	0.691831i
$879$	−3.50297e6	−0.152920
$880$	0	0
$881$	1.51155e7	0.656119	0.328059	−	0.944657i	$-0.393605\pi$
$881$	1.51155e7	0.656119	0.328059	+	0.944657i	$0.393605\pi$
$882$	3.46328e6i	0.149905i
$883$	− 2.08156e7i	− 0.898437i	−0.893422	−	0.449218i	$-0.851703\pi$
$883$	− 2.08156e7i	− 0.898437i	0.893422	−	0.449218i	$-0.148297\pi$
$884$	−4.52105e6	−0.194585
$885$	0	0
$886$	1.23112e7	0.526885
$887$	1.01116e7i	0.431529i	0.976445	+	0.215764i	$0.0692243\pi$
$887$	1.01116e7i	0.431529i	−0.976445	+	0.215764i	$0.930776\pi$
$888$	2.76964e6i	0.117866i
$889$	−3.96312e7	−1.68183
$890$	0	0
$891$	−3.59737e7	−1.51807
$892$	− 3.37410e6i	− 0.141986i
$893$	− 1.06465e7i	− 0.446763i
$894$	−1.56684e6	−0.0655665
$895$	0	0
$896$	2.49867e7	1.03978
$897$	524436.i	0.0217626i
$898$	8.90100e6i	0.368339i
$899$	2.58285e7	1.06586
$900$	0	0
$901$	1.60032e7	0.656744
$902$	− 3.20737e7i	− 1.31260i
$903$	− 1.18357e6i	− 0.0483030i
$904$	−6.31740e6	−0.257109
$905$	0	0
$906$	−4.22554e6	−0.171026
$907$	− 3.04365e6i	− 0.122850i	−0.998112	−	0.0614251i	$-0.980435\pi$
$907$	− 3.04365e6i	− 0.122850i	0.998112	−	0.0614251i	$-0.0195645\pi$
$908$	− 1.36833e7i	− 0.550779i
$909$	−2.32992e6	−0.0935257
$910$	0	0
$911$	−2.90597e7	−1.16010	−0.580050	−	0.814581i	$-0.696967\pi$
$911$	−2.90597e7	−1.16010	−0.580050	+	0.814581i	$0.696967\pi$
$912$	1.40817e6i	0.0560618i
$913$	9.88928e6i	0.392634i
$914$	−5.57744e6	−0.220836
$915$	0	0
$916$	5.72896e6	0.225599
$917$	− 2.32982e7i	− 0.914952i
$918$	− 5.36581e6i	− 0.210149i
$919$	−3.38583e7	−1.32244	−0.661220	−	0.750192i	$-0.729961\pi$
$919$	−3.38583e7	−1.32244	−0.661220	+	0.750192i	$0.729961\pi$
$920$	0	0
$921$	1.33546e6	0.0518780
$922$	− 4.42900e6i	− 0.171585i
$923$	1.14934e7i	0.444064i
$924$	−8.63497e6	−0.332722
$925$	0	0
$926$	6.16939e6	0.236437
$927$	4.04516e7i	1.54610i
$928$	1.93979e7i	0.739409i
$929$	−3.78136e7	−1.43750	−0.718751	−	0.695267i	$-0.755286\pi$
$929$	−3.78136e7	−1.43750	−0.718751	+	0.695267i	$0.755286\pi$
$930$	0	0
$931$	7.25729e6	0.274410
$932$	− 1.27689e7i	− 0.481519i
$933$	− 4.75954e6i	− 0.179003i
$934$	−7.16037e6	−0.268577
$935$	0	0
$936$	6.28329e6	0.234422
$937$	2.00563e7i	0.746281i	0.927775	+	0.373140i	$0.121719\pi$
$937$	2.00563e7i	0.746281i	−0.927775	+	0.373140i	$0.878281\pi$
$938$	1.65342e7i	0.613587i
$939$	1.87169e6	0.0692739
$940$	0	0
$941$	−2.54644e7	−0.937473	−0.468737	−	0.883338i	$-0.655291\pi$
$941$	−2.54644e7	−0.937473	−0.468737	+	0.883338i	$0.655291\pi$
$942$	− 1.29540e6i	− 0.0475637i
$943$	− 1.39662e7i	− 0.511443i
$944$	−1.44869e7	−0.529108
$945$	0	0
$946$	4.75874e6	0.172888
$947$	− 1.42435e7i	− 0.516110i	−0.966130	−	0.258055i	$-0.916918\pi$
$947$	− 1.42435e7i	− 0.516110i	0.966130	−	0.258055i	$-0.0830815\pi$
$948$	325580.i	0.0117662i
$949$	3.60058e6	0.129780
$950$	0	0
$951$	6.63292e6	0.237823
$952$	2.70099e7i	0.965896i
$953$	− 3.95389e7i	− 1.41024i	−0.709089	−	0.705119i	$-0.750894\pi$
$953$	− 3.95389e7i	− 1.41024i	0.709089	−	0.705119i	$-0.249106\pi$
$954$	−9.44827e6	−0.336110
$955$	0	0
$956$	5.27119e6	0.186536
$957$	− 7.97819e6i	− 0.281595i
$958$	1.88731e7i	0.664399i
$959$	1.96308e7	0.689275
$960$	0	0
$961$	3.49958e7	1.22238
$962$	− 2.43351e6i	− 0.0847803i
$963$	5.20127e7i	1.80735i
$964$	3.11013e7	1.07792
$965$	0	0
$966$	1.33098e6	0.0458912
$967$	− 4.28774e7i	− 1.47456i	−0.675588	−	0.737279i	$-0.736110\pi$
$967$	− 4.28774e7i	− 1.47456i	0.675588	−	0.737279i	$-0.263890\pi$
$968$	− 5.58108e7i	− 1.91439i
$969$	−5.48027e6	−0.187496
$970$	0	0
$971$	1.56895e7	0.534023	0.267011	−	0.963693i	$-0.413964\pi$
$971$	1.56895e7	0.534023	0.267011	+	0.963693i	$0.413964\pi$
$972$	− 1.35370e7i	− 0.459574i
$973$	4.22681e7i	1.43130i
$974$	1.59100e7	0.537370
$975$	0	0
$976$	−731978.	−0.0245965
$977$	2.22874e7i	0.747004i	0.927629	+	0.373502i	$0.121843\pi$
$977$	2.22874e7i	0.747004i	−0.927629	+	0.373502i	$0.878157\pi$
$978$	− 6.16759e6i	− 0.206190i
$979$	6.36331e7	2.12191
$980$	0	0
$981$	−5.79547e6	−0.192272
$982$	− 2.47174e7i	− 0.817946i
$983$	2.46890e6i	0.0814929i	0.999170	+	0.0407465i	$0.0129736\pi$
$983$	2.46890e6i	0.0814929i	−0.999170	+	0.0407465i	$0.987026\pi$
$984$	8.65656e6	0.285008
$985$	0	0
$986$	−1.06014e7	−0.347272
$987$	3.89763e6i	0.127353i
$988$	− 5.59334e6i	− 0.182297i
$989$	2.07214e6	0.0673640
$990$	0	0
$991$	1.73304e7	0.560563	0.280281	−	0.959918i	$-0.409572\pi$
$991$	1.73304e7	0.560563	0.280281	+	0.959918i	$0.409572\pi$
$992$	4.77840e7i	1.54171i
$993$	8.30251e6i	0.267200i
$994$	2.91695e7	0.936405
$995$	0	0
$996$	−1.13385e6	−0.0362166
$997$	5.90647e7i	1.88187i	0.338586	+	0.940936i	$0.390052\pi$
$997$	5.90647e7i	1.88187i	−0.338586	+	0.940936i	$0.609948\pi$
$998$	8.62487e6i	0.274111i
$999$	−8.15913e6	−0.258661

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.b.i.274.9		22
5.2	odd	4		325.6.a.k.1.7	yes	11
5.3	odd	4		325.6.a.j.1.5	✓	11
5.4	even	2	inner	325.6.b.i.274.14		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.j.1.5	✓	11	5.3	odd	4
325.6.a.k.1.7	yes	11	5.2	odd	4
325.6.b.i.274.9		22	1.1	even	1	trivial
325.6.b.i.274.14		22	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-2.89241i q^{2} -3.45730i q^{3} +23.6340 q^{4} -9.99993 q^{6} +148.288i q^{7} -160.916i q^{8} +231.047 q^{9} +O(q^{10})\)	\(q-2.89241i q^{2} -3.45730i q^{3} +23.6340 q^{4} -9.99993 q^{6} +148.288i q^{7} -160.916i q^{8} +231.047 q^{9} -712.658 q^{11} -81.7096i q^{12} +169.000i q^{13} +428.910 q^{14} +290.850 q^{16} +1131.92i q^{17} -668.283i q^{18} -1400.39 q^{19} +512.676 q^{21} +2061.30i q^{22} +897.571i q^{23} -556.336 q^{24} +488.818 q^{26} -1638.92i q^{27} +3504.63i q^{28} -3238.07 q^{29} -7976.53 q^{31} -5990.58i q^{32} +2463.87i q^{33} +3273.98 q^{34} +5460.56 q^{36} -4978.35i q^{37} +4050.50i q^{38} +584.284 q^{39} -15559.9 q^{41} -1482.87i q^{42} -2308.61i q^{43} -16842.9 q^{44} +2596.15 q^{46} +7602.52i q^{47} -1005.56i q^{48} -5182.35 q^{49} +3913.39 q^{51} +3994.14i q^{52} -14138.1i q^{53} -4740.44 q^{54} +23862.0 q^{56} +4841.56i q^{57} +9365.83i q^{58} -49808.7 q^{59} -2516.69 q^{61} +23071.4i q^{62} +34261.5i q^{63} -8020.03 q^{64} +7126.54 q^{66} +38549.3i q^{67} +26751.8i q^{68} +3103.17 q^{69} +68008.5 q^{71} -37179.2i q^{72} -21305.2i q^{73} -14399.4 q^{74} -33096.7 q^{76} -105679. i q^{77} -1689.99i q^{78} -3984.60 q^{79} +50478.2 q^{81} +45005.8i q^{82} -13876.6i q^{83} +12116.6 q^{84} -6677.44 q^{86} +11195.0i q^{87} +114678. i q^{88} -89289.8 q^{89} -25060.7 q^{91} +21213.1i q^{92} +27577.2i q^{93} +21989.6 q^{94} -20711.2 q^{96} +147549. i q^{97} +14989.5i q^{98} -164658. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9}+O(q^{10}) \) 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9	\( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36} - 3718 q^{39} - 20020 q^{41} - 121302 q^{44} - 19716 q^{46} - 146626 q^{49} + 73212 q^{51} - 96026 q^{54} - 2524 q^{56} - 263284 q^{59} - 86730 q^{61} - 360142 q^{64} - 117214 q^{66} - 118690 q^{69} + 136840 q^{71} - 742792 q^{74} + 45034 q^{76} - 104478 q^{79} + 215014 q^{81} - 1705432 q^{84} + 404492 q^{86} - 655424 q^{89} + 70304 q^{91} - 1299948 q^{94} + 1799326 q^{96} - 853396 q^{99}+O(q^{100}) \) 22 * q - 374 * q^4 + 702 * q^6 - 2744 * q^9 + 2552 * q^11 - 1156 * q^14 + 11414 * q^16 - 7040 * q^19 + 3412 * q^21 - 15446 * q^24 + 1690 * q^26 - 36850 * q^29 + 18136 * q^31 + 28910 * q^34 + 75368 * q^36 - 3718 * q^39 - 20020 * q^41 - 121302 * q^44 - 19716 * q^46 - 146626 * q^49 + 73212 * q^51 - 96026 * q^54 - 2524 * q^56 - 263284 * q^59 - 86730 * q^61 - 360142 * q^64 - 117214 * q^66 - 118690 * q^69 + 136840 * q^71 - 742792 * q^74 + 45034 * q^76 - 104478 * q^79 + 215014 * q^81 - 1705432 * q^84 + 404492 * q^86 - 655424 * q^89 + 70304 * q^91 - 1299948 * q^94 + 1799326 * q^96 - 853396 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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