LMFDB - Embedded newform 750.6.c.c.499.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [750,6,Mod(499,750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(750, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("750.499");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 750 = 2 \cdot 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	750.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$120.287864860$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.289444000000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 39x^{6} + 541x^{4} + 3084x^{2} + 5776 $ x^8 + 39x^6 + 541x^4 + 3084*x^2 + 5776
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			499.4
Root			$-3.53270i$ of defining polynomial
Character	$\chi$	$=$	750.499
Dual form			750.6.c.c.499.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +166.052i q^{7} +64.0000i q^{8} -81.0000 q^{9} +O(q^{10})$	\(q-4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +166.052i q^{7} +64.0000i q^{8} -81.0000 q^{9} +327.856 q^{11} -144.000i q^{12} -383.096i q^{13} +664.206 q^{14} +256.000 q^{16} +645.632i q^{17} +324.000i q^{18} -2344.12 q^{19} -1494.46 q^{21} -1311.42i q^{22} -4426.41i q^{23} -576.000 q^{24} -1532.38 q^{26} -729.000i q^{27} -2656.83i q^{28} -4581.94 q^{29} -4884.88 q^{31} -1024.00i q^{32} +2950.70i q^{33} +2582.53 q^{34} +1296.00 q^{36} +11001.0i q^{37} +9376.47i q^{38} +3447.86 q^{39} +15659.4 q^{41} +5977.86i q^{42} -7182.66i q^{43} -5245.69 q^{44} -17705.6 q^{46} +9868.14i q^{47} +2304.00i q^{48} -10766.1 q^{49} -5810.69 q^{51} +6129.54i q^{52} -18916.7i q^{53} -2916.00 q^{54} -10627.3 q^{56} -21097.1i q^{57} +18327.8i q^{58} +21959.2 q^{59} -39173.2 q^{61} +19539.5i q^{62} -13450.2i q^{63} -4096.00 q^{64} +11802.8 q^{66} -28560.5i q^{67} -10330.1i q^{68} +39837.6 q^{69} +1932.87 q^{71} -5184.00i q^{72} +39035.9i q^{73} +44004.2 q^{74} +37505.9 q^{76} +54441.0i q^{77} -13791.5i q^{78} +39188.7 q^{79} +6561.00 q^{81} -62637.6i q^{82} -94809.5i q^{83} +23911.4 q^{84} -28730.6 q^{86} -41237.4i q^{87} +20982.8i q^{88} +116490. q^{89} +63613.7 q^{91} +70822.5i q^{92} -43963.9i q^{93} +39472.5 q^{94} +9216.00 q^{96} -31218.6i q^{97} +43064.5i q^{98} -26556.3 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 128 q^{4} + 288 q^{6} - 648 q^{9}+O(q^{10}) $ 8 * q - 128 * q^4 + 288 * q^6 - 648 * q^9	$ 8 q - 128 q^{4} + 288 q^{6} - 648 q^{9} - 804 q^{11} + 336 q^{14} + 2048 q^{16} + 3320 q^{19} - 756 q^{21} - 4608 q^{24} - 10128 q^{26} + 11400 q^{29} - 26004 q^{31} + 17296 q^{34} + 10368 q^{36} + 22788 q^{39} + 26316 q^{41} + 12864 q^{44} - 64048 q^{46} + 16064 q^{49} - 38916 q^{51} - 23328 q^{54} - 5376 q^{56} + 21660 q^{59} - 97324 q^{61} - 32768 q^{64} - 28944 q^{66} + 144108 q^{69} - 138444 q^{71} + 223776 q^{74} - 53120 q^{76} + 346560 q^{79} + 52488 q^{81} + 12096 q^{84} - 58848 q^{86} - 123600 q^{89} - 283544 q^{91} + 190336 q^{94} + 73728 q^{96} + 65124 q^{99}+O(q^{100}) $ 8 * q - 128 * q^4 + 288 * q^6 - 648 * q^9 - 804 * q^11 + 336 * q^14 + 2048 * q^16 + 3320 * q^19 - 756 * q^21 - 4608 * q^24 - 10128 * q^26 + 11400 * q^29 - 26004 * q^31 + 17296 * q^34 + 10368 * q^36 + 22788 * q^39 + 26316 * q^41 + 12864 * q^44 - 64048 * q^46 + 16064 * q^49 - 38916 * q^51 - 23328 * q^54 - 5376 * q^56 + 21660 * q^59 - 97324 * q^61 - 32768 * q^64 - 28944 * q^66 + 144108 * q^69 - 138444 * q^71 + 223776 * q^74 - 53120 * q^76 + 346560 * q^79 + 52488 * q^81 + 12096 * q^84 - 58848 * q^86 - 123600 * q^89 - 283544 * q^91 + 190336 * q^94 + 73728 * q^96 + 65124 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/750\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 4.00000i	− 0.707107i
$2$	− 4.00000i	− 0.707107i
$3$	9.00000i	0.577350i
$3$	9.00000i	0.577350i
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	36.0000	0.408248
$7$	166.052i	1.28085i	0.768021	+	0.640424i	$0.221242\pi$
$7$	166.052i	1.28085i	−0.768021	+	0.640424i	$0.778758\pi$
$8$	64.0000i	0.353553i
$9$	−81.0000	−0.333333
$10$	0	0
$11$	327.856	0.816960	0.408480	−	0.912767i	$-0.366059\pi$
$11$	327.856	0.816960	0.408480	+	0.912767i	$0.366059\pi$
$12$	− 144.000i	− 0.288675i
$13$	− 383.096i	− 0.628709i	−0.949306	−	0.314354i	$-0.898212\pi$
$13$	− 383.096i	− 0.628709i	0.949306	−	0.314354i	$-0.101788\pi$
$14$	664.206	0.905697
$15$	0	0
$16$	256.000	0.250000
$17$	645.632i	0.541830i	0.962603	+	0.270915i	$0.0873262\pi$
$17$	645.632i	0.541830i	−0.962603	+	0.270915i	$0.912674\pi$
$18$	324.000i	0.235702i
$19$	−2344.12	−1.48969	−0.744844	−	0.667239i	$-0.767476\pi$
$19$	−2344.12	−1.48969	−0.744844	+	0.667239i	$0.767476\pi$
$20$	0	0
$21$	−1494.46	−0.739498
$22$	− 1311.42i	− 0.577678i
$23$	− 4426.41i	− 1.74474i	−0.488843	−	0.872372i	$-0.662581\pi$
$23$	− 4426.41i	− 1.74474i	0.488843	−	0.872372i	$-0.337419\pi$
$24$	−576.000	−0.204124
$25$	0	0
$26$	−1532.38	−0.444564
$27$	− 729.000i	− 0.192450i
$28$	− 2656.83i	− 0.640424i
$29$	−4581.94	−1.01171	−0.505853	−	0.862620i	$-0.668822\pi$
$29$	−4581.94	−1.01171	−0.505853	+	0.862620i	$0.668822\pi$
$30$	0	0
$31$	−4884.88	−0.912955	−0.456478	−	0.889735i	$-0.650889\pi$
$31$	−4884.88	−0.912955	−0.456478	+	0.889735i	$0.650889\pi$
$32$	− 1024.00i	− 0.176777i
$33$	2950.70i	0.471672i
$34$	2582.53	0.383131
$35$	0	0
$36$	1296.00	0.166667
$37$	11001.0i	1.32108i	0.750790	+	0.660541i	$0.229673\pi$
$37$	11001.0i	1.32108i	−0.750790	+	0.660541i	$0.770327\pi$
$38$	9376.47i	1.05337i
$39$	3447.86	0.362985
$40$	0	0
$41$	15659.4	1.45484	0.727420	−	0.686192i	$-0.240719\pi$
$41$	15659.4	1.45484	0.727420	+	0.686192i	$0.240719\pi$
$42$	5977.86i	0.522904i
$43$	− 7182.66i	− 0.592399i	−0.955126	−	0.296199i	$-0.904281\pi$
$43$	− 7182.66i	− 0.592399i	0.955126	−	0.296199i	$-0.0957192\pi$
$44$	−5245.69	−0.408480
$45$	0	0
$46$	−17705.6	−1.23372
$47$	9868.14i	0.651614i	0.945436	+	0.325807i	$0.105636\pi$
$47$	9868.14i	0.651614i	−0.945436	+	0.325807i	$0.894364\pi$
$48$	2304.00i	0.144338i
$49$	−10766.1	−0.640574
$50$	0	0
$51$	−5810.69	−0.312825
$52$	6129.54i	0.314354i
$53$	− 18916.7i	− 0.925028i	−0.886612	−	0.462514i	$-0.846947\pi$
$53$	− 18916.7i	− 0.925028i	0.886612	−	0.462514i	$-0.153053\pi$
$54$	−2916.00	−0.136083
$55$	0	0
$56$	−10627.3	−0.452848
$57$	− 21097.1i	− 0.860072i
$58$	18327.8i	0.715384i
$59$	21959.2	0.821270	0.410635	−	0.911800i	$-0.365307\pi$
$59$	21959.2	0.821270	0.410635	+	0.911800i	$0.365307\pi$
$60$	0	0
$61$	−39173.2	−1.34792	−0.673960	−	0.738768i	$-0.735408\pi$
$61$	−39173.2	−1.34792	−0.673960	+	0.738768i	$0.735408\pi$
$62$	19539.5i	0.645557i
$63$	− 13450.2i	− 0.426950i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	11802.8	0.333523
$67$	− 28560.5i	− 0.777283i	−0.921389	−	0.388641i	$-0.872945\pi$
$67$	− 28560.5i	− 0.777283i	0.921389	−	0.388641i	$-0.127055\pi$
$68$	− 10330.1i	− 0.270915i
$69$	39837.6	1.00733
$70$	0	0
$71$	1932.87	0.0455047	0.0227523	−	0.999741i	$-0.492757\pi$
$71$	1932.87	0.0455047	0.0227523	+	0.999741i	$0.492757\pi$
$72$	− 5184.00i	− 0.117851i
$73$	39035.9i	0.857347i	0.903460	+	0.428673i	$0.141019\pi$
$73$	39035.9i	0.857347i	−0.903460	+	0.428673i	$0.858981\pi$
$74$	44004.2	0.934145
$75$	0	0
$76$	37505.9	0.744844
$77$	54441.0i	1.04640i
$78$	− 13791.5i	− 0.256669i
$79$	39188.7	0.706469	0.353235	−	0.935535i	$-0.385082\pi$
$79$	39188.7	0.706469	0.353235	+	0.935535i	$0.385082\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	− 62637.6i	− 1.02873i
$83$	− 94809.5i	− 1.51062i	−0.655365	−	0.755312i	$-0.727485\pi$
$83$	− 94809.5i	− 1.51062i	0.655365	−	0.755312i	$-0.272515\pi$
$84$	23911.4	0.369749
$85$	0	0
$86$	−28730.6	−0.418889
$87$	− 41237.4i	− 0.584109i
$88$	20982.8i	0.288839i
$89$	116490.	1.55888	0.779439	−	0.626478i	$-0.215504\pi$
$89$	116490.	1.55888	0.779439	+	0.626478i	$0.215504\pi$
$90$	0	0
$91$	63613.7	0.805281
$92$	70822.5i	0.872372i
$93$	− 43963.9i	− 0.527095i
$94$	39472.5	0.460761
$95$	0	0
$96$	9216.00	0.102062
$97$	− 31218.6i	− 0.336887i	−0.985711	−	0.168444i	$-0.946126\pi$
$97$	− 31218.6i	− 0.336887i	0.985711	−	0.168444i	$-0.0538741\pi$
$98$	43064.5i	0.452954i
$99$	−26556.3	−0.272320
$100$	0	0
$101$	−149912.	−1.46229	−0.731144	−	0.682223i	$-0.761013\pi$
$101$	−149912.	−1.46229	−0.731144	+	0.682223i	$0.761013\pi$
$102$	23242.7i	0.221201i
$103$	− 72792.7i	− 0.676075i	−0.941133	−	0.338038i	$-0.890237\pi$
$103$	− 72792.7i	− 0.676075i	0.941133	−	0.338038i	$-0.109763\pi$
$104$	24518.1	0.222282
$105$	0	0
$106$	−75666.7	−0.654094
$107$	103204.i	0.871437i	0.900083	+	0.435718i	$0.143506\pi$
$107$	103204.i	0.871437i	−0.900083	+	0.435718i	$0.856494\pi$
$108$	11664.0i	0.0962250i
$109$	128746.	1.03793	0.518963	−	0.854797i	$-0.326318\pi$
$109$	128746.	1.03793	0.518963	+	0.854797i	$0.326318\pi$
$110$	0	0
$111$	−99009.4	−0.762727
$112$	42509.2i	0.320212i
$113$	28493.8i	0.209920i	0.994476	+	0.104960i	$0.0334715\pi$
$113$	28493.8i	0.209920i	−0.994476	+	0.104960i	$0.966529\pi$
$114$	−84388.2	−0.608162
$115$	0	0
$116$	73311.0	0.505853
$117$	31030.8i	0.209570i
$118$	− 87836.7i	− 0.580726i
$119$	−107208.	−0.694002
$120$	0	0
$121$	−53561.7	−0.332576
$122$	156693.i	0.953124i
$123$	140935.i	0.839953i
$124$	78158.0	0.456478
$125$	0	0
$126$	−53800.7	−0.301899
$127$	− 119645.i	− 0.658239i	−0.944288	−	0.329119i	$-0.893248\pi$
$127$	− 119645.i	− 0.658239i	0.944288	−	0.329119i	$-0.106752\pi$
$128$	16384.0i	0.0883883i
$129$	64643.9	0.342021
$130$	0	0
$131$	47966.6	0.244208	0.122104	−	0.992517i	$-0.461036\pi$
$131$	47966.6	0.244208	0.122104	+	0.992517i	$0.461036\pi$
$132$	− 47211.2i	− 0.235836i
$133$	− 389244.i	− 1.90806i
$134$	−114242.	−0.549622
$135$	0	0
$136$	−41320.4	−0.191566
$137$	− 395031.i	− 1.79817i	−0.437779	−	0.899083i	$-0.644235\pi$
$137$	− 395031.i	− 1.79817i	0.437779	−	0.899083i	$-0.355765\pi$
$138$	− 159351.i	− 0.712289i
$139$	−254332.	−1.11651	−0.558257	−	0.829668i	$-0.688530\pi$
$139$	−254332.	−1.11651	−0.558257	+	0.829668i	$0.688530\pi$
$140$	0	0
$141$	−88813.2	−0.376210
$142$	− 7731.47i	− 0.0321767i
$143$	− 125600.i	− 0.513630i
$144$	−20736.0	−0.0833333
$145$	0	0
$146$	156143.	0.606236
$147$	− 96895.2i	− 0.369836i
$148$	− 176017.i	− 0.660541i
$149$	99428.8	0.366899	0.183450	−	0.983029i	$-0.441274\pi$
$149$	99428.8	0.366899	0.183450	+	0.983029i	$0.441274\pi$
$150$	0	0
$151$	293386.	1.04712	0.523560	−	0.851989i	$-0.324603\pi$
$151$	293386.	1.04712	0.523560	+	0.851989i	$0.324603\pi$
$152$	− 150023.i	− 0.526684i
$153$	− 52296.2i	− 0.180610i
$154$	217764.	0.739919
$155$	0	0
$156$	−55165.8	−0.181493
$157$	− 554343.i	− 1.79486i	−0.441161	−	0.897428i	$-0.645433\pi$
$157$	− 554343.i	− 1.79486i	0.441161	−	0.897428i	$-0.354567\pi$
$158$	− 156755.i	− 0.499549i
$159$	170250.	0.534065
$160$	0	0
$161$	735012.	2.23475
$162$	− 26244.0i	− 0.0785674i
$163$	481575.i	1.41970i	0.704355	+	0.709848i	$0.251236\pi$
$163$	481575.i	1.41970i	−0.704355	+	0.709848i	$0.748764\pi$
$164$	−250550.	−0.727420
$165$	0	0
$166$	−379238.	−1.06817
$167$	457124.i	1.26836i	0.773185	+	0.634180i	$0.218662\pi$
$167$	457124.i	1.26836i	−0.773185	+	0.634180i	$0.781338\pi$
$168$	− 95645.7i	− 0.261452i
$169$	224530.	0.604726
$170$	0	0
$171$	189873.	0.496563
$172$	114923.i	0.296199i
$173$	− 428584.i	− 1.08873i	−0.838848	−	0.544365i	$-0.816771\pi$
$173$	− 428584.i	− 1.08873i	0.838848	−	0.544365i	$-0.183229\pi$
$174$	−164950.	−0.413027
$175$	0	0
$176$	83931.1	0.204240
$177$	197633.i	0.474160i
$178$	− 465959.i	− 1.10229i
$179$	475942.	1.11025	0.555126	−	0.831766i	$-0.312670\pi$
$179$	475942.	1.11025	0.555126	+	0.831766i	$0.312670\pi$
$180$	0	0
$181$	19570.4	0.0444021	0.0222011	−	0.999754i	$-0.492933\pi$
$181$	19570.4	0.0444021	0.0222011	+	0.999754i	$0.492933\pi$
$182$	− 254455.i	− 0.569419i
$183$	− 352559.i	− 0.778222i
$184$	283290.	0.616860
$185$	0	0
$186$	−175856.	−0.372712
$187$	211674.i	0.442653i
$188$	− 157890.i	− 0.325807i
$189$	121052.	0.246499
$190$	0	0
$191$	89329.1	0.177178	0.0885890	−	0.996068i	$-0.471764\pi$
$191$	89329.1	0.177178	0.0885890	+	0.996068i	$0.471764\pi$
$192$	− 36864.0i	− 0.0721688i
$193$	− 899778.i	− 1.73877i	−0.494135	−	0.869385i	$-0.664515\pi$
$193$	− 899778.i	− 1.73877i	0.494135	−	0.869385i	$-0.335485\pi$
$194$	−124875.	−0.238215
$195$	0	0
$196$	172258.	0.320287
$197$	490816.i	0.901058i	0.892762	+	0.450529i	$0.148765\pi$
$197$	490816.i	0.901058i	−0.892762	+	0.450529i	$0.851235\pi$
$198$	106225.i	0.192559i
$199$	−48467.7	−0.0867600	−0.0433800	−	0.999059i	$-0.513813\pi$
$199$	−48467.7	−0.0867600	−0.0433800	+	0.999059i	$0.513813\pi$
$200$	0	0
$201$	257045.	0.448764
$202$	599648.i	1.03399i
$203$	− 760838.i	− 1.29584i
$204$	92971.0	0.156413
$205$	0	0
$206$	−291171.	−0.478057
$207$	358539.i	0.581581i
$208$	− 98072.6i	− 0.157177i
$209$	−768532.	−1.21702
$210$	0	0
$211$	−395706.	−0.611880	−0.305940	−	0.952051i	$-0.598971\pi$
$211$	−395706.	−0.611880	−0.305940	+	0.952051i	$0.598971\pi$
$212$	302667.i	0.462514i
$213$	17395.8i	0.0262721i
$214$	412815.	0.616199
$215$	0	0
$216$	46656.0	0.0680414
$217$	− 811142.i	− 1.16936i
$218$	− 514982.i	− 0.733924i
$219$	−351323.	−0.494989
$220$	0	0
$221$	247339.	0.340653
$222$	396038.i	0.539329i
$223$	− 279453.i	− 0.376311i	−0.982139	−	0.188155i	$-0.939749\pi$
$223$	− 279453.i	− 0.376311i	0.982139	−	0.188155i	$-0.0602509\pi$
$224$	170037.	0.226424
$225$	0	0
$226$	113975.	0.148436
$227$	− 1.53323e6i	− 1.97489i	−0.157953	−	0.987447i	$-0.550490\pi$
$227$	− 1.53323e6i	− 1.97489i	0.157953	−	0.987447i	$-0.449510\pi$
$228$	337553.i	0.430036i
$229$	−576172.	−0.726044	−0.363022	−	0.931781i	$-0.618255\pi$
$229$	−576172.	−0.726044	−0.363022	+	0.931781i	$0.618255\pi$
$230$	0	0
$231$	−489969.	−0.604141
$232$	− 293244.i	− 0.357692i
$233$	− 424153.i	− 0.511838i	−0.966698	−	0.255919i	$-0.917622\pi$
$233$	− 424153.i	− 0.511838i	0.966698	−	0.255919i	$-0.0823780\pi$
$234$	124123.	0.148188
$235$	0	0
$236$	−351347.	−0.410635
$237$	352698.i	0.407880i
$238$	428833.i	0.490733i
$239$	920319.	1.04218	0.521091	−	0.853501i	$-0.325525\pi$
$239$	920319.	1.04218	0.521091	+	0.853501i	$0.325525\pi$
$240$	0	0
$241$	1.09033e6	1.20925	0.604627	−	0.796509i	$-0.293322\pi$
$241$	1.09033e6	1.20925	0.604627	+	0.796509i	$0.293322\pi$
$242$	214247.i	0.235167i
$243$	59049.0i	0.0641500i
$244$	626771.	0.673960
$245$	0	0
$246$	563738.	0.593936
$247$	898022.i	0.936579i
$248$	− 312632.i	− 0.322778i
$249$	853285.	0.872160
$250$	0	0
$251$	1.07657e6	1.07859	0.539296	−	0.842116i	$-0.318691\pi$
$251$	1.07657e6	1.07859	0.539296	+	0.842116i	$0.318691\pi$
$252$	215203.i	0.213475i
$253$	− 1.45122e6i	− 1.42539i
$254$	−478578.	−0.465445
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 594491.i	− 0.561452i	−0.959788	−	0.280726i	$-0.909425\pi$
$257$	− 594491.i	− 0.561452i	0.959788	−	0.280726i	$-0.0905753\pi$
$258$	− 258576.i	− 0.241846i
$259$	−1.82674e6	−1.69211
$260$	0	0
$261$	371137.	0.337235
$262$	− 191866.i	− 0.172681i
$263$	− 379861.i	− 0.338638i	−0.985561	−	0.169319i	$-0.945843\pi$
$263$	− 379861.i	− 0.338638i	0.985561	−	0.169319i	$-0.0541568\pi$
$264$	−188845.	−0.166761
$265$	0	0
$266$	−1.55698e6	−1.34921
$267$	1.04841e6i	0.900019i
$268$	456968.i	0.388641i
$269$	−1.98366e6	−1.67143	−0.835714	−	0.549166i	$-0.814946\pi$
$269$	−1.98366e6	−1.67143	−0.835714	+	0.549166i	$0.814946\pi$
$270$	0	0
$271$	−1.92474e6	−1.59202	−0.796011	−	0.605283i	$-0.793060\pi$
$271$	−1.92474e6	−1.59202	−0.796011	+	0.605283i	$0.793060\pi$
$272$	165282.i	0.135457i
$273$	572523.i	0.464929i
$274$	−1.58012e6	−1.27150
$275$	0	0
$276$	−637402.	−0.503664
$277$	− 1.30578e6i	− 1.02251i	−0.859428	−	0.511257i	$-0.829180\pi$
$277$	− 1.30578e6i	− 1.02251i	0.859428	−	0.511257i	$-0.170820\pi$
$278$	1.01733e6i	0.789495i
$279$	395675.	0.304318
$280$	0	0
$281$	139928.	0.105715	0.0528577	−	0.998602i	$-0.483167\pi$
$281$	139928.	0.105715	0.0528577	+	0.998602i	$0.483167\pi$
$282$	355253.i	0.266020i
$283$	− 1.52204e6i	− 1.12969i	−0.825196	−	0.564847i	$-0.808935\pi$
$283$	− 1.52204e6i	− 1.12969i	0.825196	−	0.564847i	$-0.191065\pi$
$284$	−30925.9	−0.0227523
$285$	0	0
$286$	−502401.	−0.363191
$287$	2.60027e6i	1.86343i
$288$	82944.0i	0.0589256i
$289$	1.00302e6	0.706421
$290$	0	0
$291$	280968.	0.194502
$292$	− 624574.i	− 0.428673i
$293$	− 284972.i	− 0.193925i	−0.995288	−	0.0969624i	$-0.969087\pi$
$293$	− 284972.i	− 0.193925i	0.995288	−	0.0969624i	$-0.0309127\pi$
$294$	−387581.	−0.261513
$295$	0	0
$296$	−704067.	−0.467073
$297$	− 239007.i	− 0.157224i
$298$	− 397715.i	− 0.259437i
$299$	−1.69574e6	−1.09694
$300$	0	0
$301$	1.19269e6	0.758773
$302$	− 1.17354e6i	− 0.740426i
$303$	− 1.34921e6i	− 0.844253i
$304$	−600094.	−0.372422
$305$	0	0
$306$	−209185.	−0.127710
$307$	1.40636e6i	0.851629i	0.904810	+	0.425815i	$0.140012\pi$
$307$	1.40636e6i	0.851629i	−0.904810	+	0.425815i	$0.859988\pi$
$308$	− 871055.i	− 0.523201i
$309$	655135.	0.390332
$310$	0	0
$311$	75432.3	0.0442239	0.0221119	−	0.999756i	$-0.492961\pi$
$311$	75432.3	0.0442239	0.0221119	+	0.999756i	$0.492961\pi$
$312$	220663.i	0.128335i
$313$	− 3.18952e6i	− 1.84020i	−0.391688	−	0.920098i	$-0.628109\pi$
$313$	− 3.18952e6i	− 1.84020i	0.391688	−	0.920098i	$-0.371891\pi$
$314$	−2.21737e6	−1.26915
$315$	0	0
$316$	−627019.	−0.353235
$317$	− 577345.i	− 0.322691i	−0.986898	−	0.161346i	$-0.948417\pi$
$317$	− 577345.i	− 0.322691i	0.986898	−	0.161346i	$-0.0515834\pi$
$318$	− 681000.i	− 0.377641i
$319$	−1.50221e6	−0.826524
$320$	0	0
$321$	−928833.	−0.503124
$322$	− 2.94005e6i	− 1.58021i
$323$	− 1.51344e6i	− 0.807157i
$324$	−104976.	−0.0555556
$325$	0	0
$326$	1.92630e6	1.00388
$327$	1.15871e6i	0.599246i
$328$	1.00220e6i	0.514364i
$329$	−1.63862e6	−0.834619
$330$	0	0
$331$	539380.	0.270598	0.135299	−	0.990805i	$-0.456800\pi$
$331$	539380.	0.270598	0.135299	+	0.990805i	$0.456800\pi$
$332$	1.51695e6i	0.755312i
$333$	− 891084.i	− 0.440360i
$334$	1.82850e6	0.896866
$335$	0	0
$336$	−382583.	−0.184875
$337$	− 785793.i	− 0.376906i	−0.982082	−	0.188453i	$-0.939653\pi$
$337$	− 785793.i	− 0.376906i	0.982082	−	0.188453i	$-0.0603474\pi$
$338$	− 898122.i	− 0.427606i
$339$	−256444.	−0.121197
$340$	0	0
$341$	−1.60153e6	−0.745848
$342$	− 759494.i	− 0.351123i
$343$	1.00310e6i	0.460370i
$344$	459690.	0.209444
$345$	0	0
$346$	−1.71433e6	−0.769848
$347$	− 1.17245e6i	− 0.522722i	−0.965241	−	0.261361i	$-0.915829\pi$
$347$	− 1.17245e6i	− 0.522722i	0.965241	−	0.261361i	$-0.0841714\pi$
$348$	659799.i	0.292054i
$349$	−177432.	−0.0779773	−0.0389887	−	0.999240i	$-0.512414\pi$
$349$	−177432.	−0.0779773	−0.0389887	+	0.999240i	$0.512414\pi$
$350$	0	0
$351$	−279277.	−0.120995
$352$	− 335724.i	− 0.144420i
$353$	1.02032e6i	0.435811i	0.975970	+	0.217905i	$0.0699224\pi$
$353$	1.02032e6i	0.435811i	−0.975970	+	0.217905i	$0.930078\pi$
$354$	790530.	0.335282
$355$	0	0
$356$	−1.86383e6	−0.779439
$357$	− 964874.i	− 0.400682i
$358$	− 1.90377e6i	− 0.785066i
$359$	2.43774e6	0.998276	0.499138	−	0.866523i	$-0.333650\pi$
$359$	2.43774e6	0.998276	0.499138	+	0.866523i	$0.333650\pi$
$360$	0	0
$361$	3.01878e6	1.21917
$362$	− 78281.7i	− 0.0313970i
$363$	− 482055.i	− 0.192013i
$364$	−1.01782e6	−0.402640
$365$	0	0
$366$	−1.41023e6	−0.550286
$367$	− 1.73291e6i	− 0.671599i	−0.941933	−	0.335800i	$-0.890993\pi$
$367$	− 1.73291e6i	− 0.671599i	0.941933	−	0.335800i	$-0.109007\pi$
$368$	− 1.13316e6i	− 0.436186i
$369$	−1.26841e6	−0.484947
$370$	0	0
$371$	3.14114e6	1.18482
$372$	703422.i	0.263547i
$373$	4.41738e6i	1.64397i	0.569512	+	0.821983i	$0.307132\pi$
$373$	4.41738e6i	1.64397i	−0.569512	+	0.821983i	$0.692868\pi$
$374$	846696.	0.313003
$375$	0	0
$376$	−631561.	−0.230380
$377$	1.75532e6i	0.636068i
$378$	− 484206.i	− 0.174301i
$379$	3.25898e6	1.16542	0.582711	−	0.812679i	$-0.301992\pi$
$379$	3.25898e6	1.16542	0.582711	+	0.812679i	$0.301992\pi$
$380$	0	0
$381$	1.07680e6	0.380034
$382$	− 357317.i	− 0.125284i
$383$	− 1.67933e6i	− 0.584976i	−0.956269	−	0.292488i	$-0.905517\pi$
$383$	− 1.67933e6i	− 0.584976i	0.956269	−	0.292488i	$-0.0944832\pi$
$384$	−147456.	−0.0510310
$385$	0	0
$386$	−3.59911e6	−1.22950
$387$	581795.i	0.197466i
$388$	499498.i	0.168444i
$389$	−3.77143e6	−1.26366	−0.631832	−	0.775105i	$-0.717697\pi$
$389$	−3.77143e6	−1.26366	−0.631832	+	0.775105i	$0.717697\pi$
$390$	0	0
$391$	2.85783e6	0.945354
$392$	− 689032.i	− 0.226477i
$393$	431699.i	0.140994i
$394$	1.96326e6	0.637145
$395$	0	0
$396$	424901.	0.136160
$397$	− 2.53116e6i	− 0.806016i	−0.915197	−	0.403008i	$-0.867965\pi$
$397$	− 2.53116e6i	− 0.806016i	0.915197	−	0.403008i	$-0.132035\pi$
$398$	193871.i	0.0613486i
$399$	3.50320e6	1.10162
$400$	0	0
$401$	−3.73150e6	−1.15884	−0.579419	−	0.815030i	$-0.696720\pi$
$401$	−3.73150e6	−1.15884	−0.579419	+	0.815030i	$0.696720\pi$
$402$	− 1.02818e6i	− 0.317324i
$403$	1.87138e6i	0.573983i
$404$	2.39859e6	0.731144
$405$	0	0
$406$	−3.04335e6	−0.916299
$407$	3.60675e6i	1.07927i
$408$	− 371884.i	− 0.110600i
$409$	6.23800e6	1.84390	0.921950	−	0.387309i	$-0.126595\pi$
$409$	6.23800e6	1.84390	0.921950	+	0.387309i	$0.126595\pi$
$410$	0	0
$411$	3.55528e6	1.03817
$412$	1.16468e6i	0.338038i
$413$	3.64636e6i	1.05192i
$414$	1.43416e6	0.411240
$415$	0	0
$416$	−392290.	−0.111141
$417$	− 2.28899e6i	− 0.644620i
$418$	3.07413e6i	0.860560i
$419$	−6.57122e6	−1.82857	−0.914284	−	0.405074i	$-0.867246\pi$
$419$	−6.57122e6	−1.82857	−0.914284	+	0.405074i	$0.867246\pi$
$420$	0	0
$421$	−597410.	−0.164273	−0.0821367	−	0.996621i	$-0.526174\pi$
$421$	−597410.	−0.164273	−0.0821367	+	0.996621i	$0.526174\pi$
$422$	1.58282e6i	0.432664i
$423$	− 799319.i	− 0.217205i
$424$	1.21067e6	0.327047
$425$	0	0
$426$	69583.2	0.0185772
$427$	− 6.50477e6i	− 1.72648i
$428$	− 1.65126e6i	− 0.435718i
$429$	1.13040e6	0.296544
$430$	0	0
$431$	2.87546e6	0.745615	0.372807	−	0.927909i	$-0.378395\pi$
$431$	2.87546e6	0.745615	0.372807	+	0.927909i	$0.378395\pi$
$432$	− 186624.i	− 0.0481125i
$433$	− 2.26751e6i	− 0.581205i	−0.956844	−	0.290603i	$-0.906144\pi$
$433$	− 2.26751e6i	− 0.581205i	0.956844	−	0.290603i	$-0.0938558\pi$
$434$	−3.24457e6	−0.826861
$435$	0	0
$436$	−2.05993e6	−0.518963
$437$	1.03760e7i	2.59912i
$438$	1.40529e6i	0.350010i
$439$	7.25933e6	1.79778	0.898888	−	0.438179i	$-0.144376\pi$
$439$	7.25933e6	1.79778	0.898888	+	0.438179i	$0.144376\pi$
$440$	0	0
$441$	872056.	0.213525
$442$	− 989356.i	− 0.240878i
$443$	7.18267e6i	1.73891i	0.494014	+	0.869454i	$0.335529\pi$
$443$	7.18267e6i	1.73891i	−0.494014	+	0.869454i	$0.664471\pi$
$444$	1.58415e6	0.381363
$445$	0	0
$446$	−1.11781e6	−0.266092
$447$	894860.i	0.211829i
$448$	− 680147.i	− 0.160106i
$449$	9964.15	0.00233251	0.00116626	−	0.999999i	$-0.499629\pi$
$449$	9964.15	0.00233251	0.00116626	+	0.999999i	$0.499629\pi$
$450$	0	0
$451$	5.13402e6	1.18855
$452$	− 455901.i	− 0.104960i
$453$	2.64047e6i	0.604555i
$454$	−6.13293e6	−1.39646
$455$	0	0
$456$	1.35021e6	0.304081
$457$	− 1.31032e6i	− 0.293485i	−0.989175	−	0.146742i	$-0.953121\pi$
$457$	− 1.31032e6i	− 0.293485i	0.989175	−	0.146742i	$-0.0468788\pi$
$458$	2.30469e6i	0.513391i
$459$	470666.	0.104275
$460$	0	0
$461$	1.71722e6	0.376334	0.188167	−	0.982137i	$-0.439745\pi$
$461$	1.71722e6	0.376334	0.188167	+	0.982137i	$0.439745\pi$
$462$	1.95987e6i	0.427192i
$463$	3.94866e6i	0.856046i	0.903768	+	0.428023i	$0.140790\pi$
$463$	3.94866e6i	0.856046i	−0.903768	+	0.428023i	$0.859210\pi$
$464$	−1.17298e6	−0.252926
$465$	0	0
$466$	−1.69661e6	−0.361924
$467$	− 952927.i	− 0.202194i	−0.994877	−	0.101097i	$-0.967765\pi$
$467$	− 952927.i	− 0.202194i	0.994877	−	0.101097i	$-0.0322352\pi$
$468$	− 496493.i	− 0.104785i
$469$	4.74252e6	0.995582
$470$	0	0
$471$	4.98909e6	1.03626
$472$	1.40539e6i	0.290363i
$473$	− 2.35487e6i	− 0.483966i
$474$	1.41079e6	0.288415
$475$	0	0
$476$	1.71533e6	0.347001
$477$	1.53225e6i	0.308343i
$478$	− 3.68127e6i	− 0.736934i
$479$	−3.34857e6	−0.666838	−0.333419	−	0.942779i	$-0.608202\pi$
$479$	−3.34857e6	−0.666838	−0.333419	+	0.942779i	$0.608202\pi$
$480$	0	0
$481$	4.21446e6	0.830575
$482$	− 4.36134e6i	− 0.855071i
$483$	6.61510e6i	1.29024i
$484$	856986.	0.166288
$485$	0	0
$486$	236196.	0.0453609
$487$	336542.i	0.0643010i	0.999483	+	0.0321505i	$0.0102356\pi$
$487$	336542.i	0.0643010i	−0.999483	+	0.0321505i	$0.989764\pi$
$488$	− 2.50708e6i	− 0.476562i
$489$	−4.33418e6	−0.819662
$490$	0	0
$491$	−2.06089e6	−0.385790	−0.192895	−	0.981219i	$-0.561788\pi$
$491$	−2.06089e6	−0.385790	−0.192895	+	0.981219i	$0.561788\pi$
$492$	− 2.25495e6i	− 0.419976i
$493$	− 2.95825e6i	− 0.548172i
$494$	3.59209e6	0.662262
$495$	0	0
$496$	−1.25053e6	−0.228239
$497$	320956.i	0.0582846i
$498$	− 3.41314e6i	− 0.616710i
$499$	6.12856e6	1.10181	0.550905	−	0.834568i	$-0.314283\pi$
$499$	6.12856e6	1.10181	0.550905	+	0.834568i	$0.314283\pi$
$500$	0	0
$501$	−4.11412e6	−0.732288
$502$	− 4.30627e6i	− 0.762679i
$503$	6.54877e6i	1.15409i	0.816712	+	0.577045i	$0.195794\pi$
$503$	6.54877e6i	1.15409i	−0.816712	+	0.577045i	$0.804206\pi$
$504$	860811.	0.150949
$505$	0	0
$506$	−5.80489e6	−1.00790
$507$	2.02077e6i	0.349138i
$508$	1.91431e6i	0.329119i
$509$	1.18339e6	0.202458	0.101229	−	0.994863i	$-0.467723\pi$
$509$	1.18339e6	0.202458	0.101229	+	0.994863i	$0.467723\pi$
$510$	0	0
$511$	−6.48197e6	−1.09813
$512$	− 262144.i	− 0.0441942i
$513$	1.70886e6i	0.286691i
$514$	−2.37797e6	−0.397007
$515$	0	0
$516$	−1.03430e6	−0.171011
$517$	3.23532e6i	0.532343i
$518$	7.30696e6i	1.19650i
$519$	3.85725e6	0.628579
$520$	0	0
$521$	1.17195e6	0.189154	0.0945770	−	0.995518i	$-0.469850\pi$
$521$	1.17195e6	0.189154	0.0945770	+	0.995518i	$0.469850\pi$
$522$	− 1.48455e6i	− 0.238461i
$523$	6.59266e6i	1.05392i	0.849891	+	0.526959i	$0.176668\pi$
$523$	6.59266e6i	1.05392i	−0.849891	+	0.526959i	$0.823332\pi$
$524$	−767465.	−0.122104
$525$	0	0
$526$	−1.51944e6	−0.239453
$527$	− 3.15383e6i	− 0.494666i
$528$	755379.i	0.117918i
$529$	−1.31567e7	−2.04413
$530$	0	0
$531$	−1.77869e6	−0.273757
$532$	6.22791e6i	0.954032i
$533$	− 5.99905e6i	− 0.914671i
$534$	4.19363e6	0.636409
$535$	0	0
$536$	1.82787e6	0.274811
$537$	4.28348e6i	0.641004i
$538$	7.93466e6i	1.18188i
$539$	−3.52974e6	−0.523324
$540$	0	0
$541$	9.48222e6	1.39289	0.696445	−	0.717610i	$-0.254764\pi$
$541$	9.48222e6	1.39289	0.696445	+	0.717610i	$0.254764\pi$
$542$	7.69896e6i	1.12573i
$543$	176134.i	0.0256356i
$544$	661127.	0.0957828
$545$	0	0
$546$	2.29009e6	0.328754
$547$	− 3.83291e6i	− 0.547723i	−0.961769	−	0.273861i	$-0.911699\pi$
$547$	− 3.83291e6i	− 0.547723i	0.961769	−	0.273861i	$-0.0883009\pi$
$548$	6.32049e6i	0.899083i
$549$	3.17303e6	0.449307
$550$	0	0
$551$	1.07406e7	1.50713
$552$	2.54961e6i	0.356144i
$553$	6.50735e6i	0.904880i
$554$	−5.22310e6	−0.723026
$555$	0	0
$556$	4.06931e6	0.558257
$557$	− 3.47806e6i	− 0.475006i	−0.971387	−	0.237503i	$-0.923671\pi$
$557$	− 3.47806e6i	− 0.475006i	0.971387	−	0.237503i	$-0.0763289\pi$
$558$	− 1.58270e6i	− 0.215186i
$559$	−2.75165e6	−0.372446
$560$	0	0
$561$	−1.90507e6	−0.255566
$562$	− 559711.i	− 0.0747521i
$563$	8.40012e6i	1.11690i	0.829538	+	0.558450i	$0.188604\pi$
$563$	8.40012e6i	1.11690i	−0.829538	+	0.558450i	$0.811396\pi$
$564$	1.42101e6	0.188105
$565$	0	0
$566$	−6.08817e6	−0.798815
$567$	1.08946e6i	0.142317i
$568$	123703.i	0.0160883i
$569$	−1.00693e7	−1.30382	−0.651909	−	0.758297i	$-0.726032\pi$
$569$	−1.00693e7	−1.30382	−0.651909	+	0.758297i	$0.726032\pi$
$570$	0	0
$571$	380625.	0.0488548	0.0244274	−	0.999702i	$-0.492224\pi$
$571$	380625.	0.0488548	0.0244274	+	0.999702i	$0.492224\pi$
$572$	2.00960e6i	0.256815i
$573$	803962.i	0.102294i
$574$	1.04011e7	1.31764
$575$	0	0
$576$	331776.	0.0416667
$577$	2.54734e6i	0.318528i	0.987236	+	0.159264i	$0.0509121\pi$
$577$	2.54734e6i	0.318528i	−0.987236	+	0.159264i	$0.949088\pi$
$578$	− 4.01207e6i	− 0.499515i
$579$	8.09801e6	1.00388
$580$	0	0
$581$	1.57433e7	1.93488
$582$	− 1.12387e6i	− 0.137534i
$583$	− 6.20194e6i	− 0.755711i
$584$	−2.49829e6	−0.303118
$585$	0	0
$586$	−1.13989e6	−0.137126
$587$	− 2.18467e6i	− 0.261692i	−0.991403	−	0.130846i	$-0.958231\pi$
$587$	− 2.18467e6i	− 0.261692i	0.991403	−	0.130846i	$-0.0417693\pi$
$588$	1.55032e6i	0.184918i
$589$	1.14507e7	1.36002
$590$	0	0
$591$	−4.41734e6	−0.520226
$592$	2.81627e6i	0.330270i
$593$	− 1.09317e7i	− 1.27658i	−0.769795	−	0.638292i	$-0.779641\pi$
$593$	− 1.09317e7i	− 1.27658i	0.769795	−	0.638292i	$-0.220359\pi$
$594$	−956027.	−0.111174
$595$	0	0
$596$	−1.59086e6	−0.183450
$597$	− 436209.i	− 0.0500909i
$598$	6.78295e6i	0.775650i
$599$	−318854.	−0.0363099	−0.0181549	−	0.999835i	$-0.505779\pi$
$599$	−318854.	−0.0363099	−0.0181549	+	0.999835i	$0.505779\pi$
$600$	0	0
$601$	1.35924e7	1.53500	0.767502	−	0.641046i	$-0.221499\pi$
$601$	1.35924e7	1.53500	0.767502	+	0.641046i	$0.221499\pi$
$602$	− 4.77077e6i	− 0.536534i
$603$	2.31340e6i	0.259094i
$604$	−4.69417e6	−0.523560
$605$	0	0
$606$	−5.39683e6	−0.596977
$607$	3.67049e6i	0.404345i	0.979350	+	0.202173i	$0.0648002\pi$
$607$	3.67049e6i	0.404345i	−0.979350	+	0.202173i	$0.935200\pi$
$608$	2.40038e6i	0.263342i
$609$	6.84754e6	0.748155
$610$	0	0
$611$	3.78044e6	0.409675
$612$	836739.i	0.0903049i
$613$	7.33774e6i	0.788699i	0.918961	+	0.394350i	$0.129030\pi$
$613$	7.33774e6i	0.788699i	−0.918961	+	0.394350i	$0.870970\pi$
$614$	5.62544e6	0.602193
$615$	0	0
$616$	−3.48422e6	−0.369959
$617$	− 7.19762e6i	− 0.761160i	−0.924748	−	0.380580i	$-0.875724\pi$
$617$	− 7.19762e6i	− 0.761160i	0.924748	−	0.380580i	$-0.124276\pi$
$618$	− 2.62054e6i	− 0.276007i
$619$	1.22698e7	1.28709	0.643546	−	0.765407i	$-0.277462\pi$
$619$	1.22698e7	1.28709	0.643546	+	0.765407i	$0.277462\pi$
$620$	0	0
$621$	−3.22685e6	−0.335776
$622$	− 301729.i	− 0.0312710i
$623$	1.93433e7i	1.99669i
$624$	882653.	0.0907463
$625$	0	0
$626$	−1.27581e7	−1.30121
$627$	− 6.91679e6i	− 0.702644i
$628$	8.86949e6i	0.897428i
$629$	−7.10262e6	−0.715801
$630$	0	0
$631$	−4.05544e6	−0.405475	−0.202738	−	0.979233i	$-0.564984\pi$
$631$	−4.05544e6	−0.405475	−0.202738	+	0.979233i	$0.564984\pi$
$632$	2.50808e6i	0.249775i
$633$	− 3.56135e6i	− 0.353269i
$634$	−2.30938e6	−0.228177
$635$	0	0
$636$	−2.72400e6	−0.267033
$637$	4.12446e6i	0.402734i
$638$	6.00886e6i	0.584440i
$639$	−156562.	−0.0151682
$640$	0	0
$641$	−8.56387e6	−0.823237	−0.411619	−	0.911356i	$-0.635036\pi$
$641$	−8.56387e6	−0.823237	−0.411619	+	0.911356i	$0.635036\pi$
$642$	3.71533e6i	0.355763i
$643$	5.53738e6i	0.528174i	0.964499	+	0.264087i	$0.0850706\pi$
$643$	5.53738e6i	0.528174i	−0.964499	+	0.264087i	$0.914929\pi$
$644$	−1.17602e7	−1.11738
$645$	0	0
$646$	−6.05375e6	−0.570746
$647$	− 1.48104e7i	− 1.39093i	−0.718558	−	0.695467i	$-0.755198\pi$
$647$	− 1.48104e7i	− 1.39093i	0.718558	−	0.695467i	$-0.244802\pi$
$648$	419904.i	0.0392837i
$649$	7.19944e6	0.670945
$650$	0	0
$651$	7.30027e6	0.675129
$652$	− 7.70520e6i	− 0.709848i
$653$	1.01023e7i	0.927119i	0.886066	+	0.463559i	$0.153428\pi$
$653$	1.01023e7i	0.927119i	−0.886066	+	0.463559i	$0.846572\pi$
$654$	4.63484e6	0.423731
$655$	0	0
$656$	4.00881e6	0.363710
$657$	− 3.16190e6i	− 0.285782i
$658$	6.55448e6i	0.590165i
$659$	−9.57370e6	−0.858749	−0.429374	−	0.903127i	$-0.641266\pi$
$659$	−9.57370e6	−0.858749	−0.429374	+	0.903127i	$0.641266\pi$
$660$	0	0
$661$	−2.03192e7	−1.80886	−0.904428	−	0.426627i	$-0.859702\pi$
$661$	−2.03192e7	−1.80886	−0.904428	+	0.426627i	$0.859702\pi$
$662$	− 2.15752e6i	− 0.191342i
$663$	2.22605e6i	0.196676i
$664$	6.06781e6	0.534087
$665$	0	0
$666$	−3.56434e6	−0.311382
$667$	2.02815e7i	1.76517i
$668$	− 7.31398e6i	− 0.634180i
$669$	2.51508e6	0.217263
$670$	0	0
$671$	−1.28431e7	−1.10120
$672$	1.53033e6i	0.130726i
$673$	− 1.62949e7i	− 1.38680i	−0.720552	−	0.693401i	$-0.756111\pi$
$673$	− 1.62949e7i	− 1.38680i	0.720552	−	0.693401i	$-0.243889\pi$
$674$	−3.14317e6	−0.266513
$675$	0	0
$676$	−3.59249e6	−0.302363
$677$	5.36393e6i	0.449791i	0.974383	+	0.224896i	$0.0722041\pi$
$677$	5.36393e6i	0.449791i	−0.974383	+	0.224896i	$0.927796\pi$
$678$	1.02578e6i	0.0856995i
$679$	5.18390e6	0.431502
$680$	0	0
$681$	1.37991e7	1.14021
$682$	6.40614e6i	0.527394i
$683$	− 3.53133e6i	− 0.289658i	−0.989457	−	0.144829i	$-0.953737\pi$
$683$	− 3.53133e6i	− 0.289658i	0.989457	−	0.144829i	$-0.0462633\pi$
$684$	−3.03798e6	−0.248281
$685$	0	0
$686$	4.01239e6	0.325531
$687$	− 5.18554e6i	− 0.419182i
$688$	− 1.83876e6i	− 0.148100i
$689$	−7.24690e6	−0.581573
$690$	0	0
$691$	−6.48600e6	−0.516752	−0.258376	−	0.966044i	$-0.583187\pi$
$691$	−6.48600e6	−0.516752	−0.258376	+	0.966044i	$0.583187\pi$
$692$	6.85734e6i	0.544365i
$693$	− 4.40972e6i	− 0.348801i
$694$	−4.68981e6	−0.369621
$695$	0	0
$696$	2.63920e6	0.206514
$697$	1.01102e7i	0.788276i
$698$	709728.i	0.0551383i
$699$	3.81737e6	0.295510
$700$	0	0
$701$	785811.	0.0603981	0.0301990	−	0.999544i	$-0.490386\pi$
$701$	785811.	0.0603981	0.0301990	+	0.999544i	$0.490386\pi$
$702$	1.11711e6i	0.0855564i
$703$	− 2.57877e7i	− 1.96800i
$704$	−1.34290e6	−0.102120
$705$	0	0
$706$	4.08126e6	0.308165
$707$	− 2.48931e7i	− 1.87297i
$708$	− 3.16212e6i	− 0.237080i
$709$	3.91649e6	0.292605	0.146302	−	0.989240i	$-0.453263\pi$
$709$	3.91649e6	0.292605	0.146302	+	0.989240i	$0.453263\pi$
$710$	0	0
$711$	−3.17429e6	−0.235490
$712$	7.45534e6i	0.551147i
$713$	2.16224e7i	1.59287i
$714$	−3.85950e6	−0.283325
$715$	0	0
$716$	−7.61507e6	−0.555126
$717$	8.28287e6i	0.601704i
$718$	− 9.75094e6i	− 0.705888i
$719$	−2.25686e7	−1.62811	−0.814054	−	0.580790i	$-0.802744\pi$
$719$	−2.25686e7	−1.62811	−0.814054	+	0.580790i	$0.802744\pi$
$720$	0	0
$721$	1.20874e7	0.865950
$722$	− 1.20751e7i	− 0.862083i
$723$	9.81301e6i	0.698163i
$724$	−313127.	−0.0222011
$725$	0	0
$726$	−1.92822e6	−0.135773
$727$	− 1.95991e7i	− 1.37531i	−0.726039	−	0.687654i	$-0.758641\pi$
$727$	− 1.95991e7i	− 1.37531i	0.726039	−	0.687654i	$-0.241359\pi$
$728$	4.07128e6i	0.284710i
$729$	−531441.	−0.0370370
$730$	0	0
$731$	4.63735e6	0.320979
$732$	5.64094e6i	0.389111i
$733$	9.13629e6i	0.628073i	0.949411	+	0.314036i	$0.101681\pi$
$733$	9.13629e6i	0.628073i	−0.949411	+	0.314036i	$0.898319\pi$
$734$	−6.93163e6	−0.474892
$735$	0	0
$736$	−4.53264e6	−0.308430
$737$	− 9.36372e6i	− 0.635009i
$738$	5.07365e6i	0.342909i
$739$	−1.34336e7	−0.904861	−0.452431	−	0.891800i	$-0.649443\pi$
$739$	−1.34336e7	−0.904861	−0.452431	+	0.891800i	$0.649443\pi$
$740$	0	0
$741$	−8.08220e6	−0.540734
$742$	− 1.25646e7i	− 0.837795i
$743$	− 1.05557e7i	− 0.701482i	−0.936473	−	0.350741i	$-0.885930\pi$
$743$	− 1.05557e7i	− 0.701482i	0.936473	−	0.350741i	$-0.114070\pi$
$744$	2.81369e6	0.186356
$745$	0	0
$746$	1.76695e7	1.16246
$747$	7.67957e6i	0.503542i
$748$	− 3.38679e6i	− 0.221327i
$749$	−1.71371e7	−1.11618
$750$	0	0
$751$	4.67998e6	0.302792	0.151396	−	0.988473i	$-0.451623\pi$
$751$	4.67998e6	0.302792	0.151396	+	0.988473i	$0.451623\pi$
$752$	2.52624e6i	0.162904i
$753$	9.68911e6i	0.622725i
$754$	7.02129e6	0.449768
$755$	0	0
$756$	−1.93683e6	−0.123250
$757$	− 1.98855e7i	− 1.26123i	−0.776094	−	0.630617i	$-0.782802\pi$
$757$	− 1.98855e7i	− 1.26123i	0.776094	−	0.630617i	$-0.217198\pi$
$758$	− 1.30359e7i	− 0.824078i
$759$	1.30610e7	0.822947
$760$	0	0
$761$	−2.05039e7	−1.28343	−0.641717	−	0.766941i	$-0.721778\pi$
$761$	−2.05039e7	−1.28343	−0.641717	+	0.766941i	$0.721778\pi$
$762$	− 4.30720e6i	− 0.268725i
$763$	2.13784e7i	1.32943i
$764$	−1.42927e6	−0.0885890
$765$	0	0
$766$	−6.71731e6	−0.413641
$767$	− 8.41247e6i	− 0.516339i
$768$	589824.i	0.0360844i
$769$	−1.27733e7	−0.778909	−0.389454	−	0.921046i	$-0.627336\pi$
$769$	−1.27733e7	−0.778909	−0.389454	+	0.921046i	$0.627336\pi$
$770$	0	0
$771$	5.35042e6	0.324155
$772$	1.43965e7i	0.869385i
$773$	604803.i	0.0364054i	0.999834	+	0.0182027i	$0.00579441\pi$
$773$	604803.i	0.0364054i	−0.999834	+	0.0182027i	$0.994206\pi$
$774$	2.32718e6	0.139630
$775$	0	0
$776$	1.99799e6	0.119108
$777$	− 1.64407e7i	− 0.976938i
$778$	1.50857e7i	0.893546i
$779$	−3.67075e7	−2.16726
$780$	0	0
$781$	633701.	0.0371755
$782$	− 1.14313e7i	− 0.668466i
$783$	3.34023e6i	0.194703i
$784$	−2.75613e6	−0.160144
$785$	0	0
$786$	1.72680e6	0.0996976
$787$	1.29619e7i	0.745989i	0.927834	+	0.372994i	$0.121669\pi$
$787$	1.29619e7i	0.745989i	−0.927834	+	0.372994i	$0.878331\pi$
$788$	− 7.85305e6i	− 0.450529i
$789$	3.41875e6	0.195513
$790$	0	0
$791$	−4.73144e6	−0.268876
$792$	− 1.69960e6i	− 0.0962797i
$793$	1.50071e7i	0.847449i
$794$	−1.01246e7	−0.569939
$795$	0	0
$796$	775483.	0.0433800
$797$	1.22236e7i	0.681635i	0.940130	+	0.340817i	$0.110704\pi$
$797$	1.22236e7i	0.681635i	−0.940130	+	0.340817i	$0.889296\pi$
$798$	− 1.40128e7i	− 0.778964i
$799$	−6.37118e6	−0.353064
$800$	0	0
$801$	−9.43566e6	−0.519626
$802$	1.49260e7i	0.819422i
$803$	1.27981e7i	0.700418i
$804$	−4.11271e6	−0.224382
$805$	0	0
$806$	7.48551e6	0.405867
$807$	− 1.78530e7i	− 0.964999i
$808$	− 9.59437e6i	− 0.516997i
$809$	−1.45698e7	−0.782675	−0.391338	−	0.920247i	$-0.627988\pi$
$809$	−1.45698e7	−0.782675	−0.391338	+	0.920247i	$0.627988\pi$
$810$	0	0
$811$	−3.26176e7	−1.74140	−0.870702	−	0.491811i	$-0.836335\pi$
$811$	−3.26176e7	−1.74140	−0.870702	+	0.491811i	$0.836335\pi$
$812$	1.21734e7i	0.647921i
$813$	− 1.73227e7i	− 0.919154i
$814$	1.44270e7	0.763160
$815$	0	0
$816$	−1.48754e6	−0.0782064
$817$	1.68370e7i	0.882489i
$818$	− 2.49520e7i	− 1.30383i
$819$	−5.15271e6	−0.268427
$820$	0	0
$821$	3.57082e7	1.84889	0.924443	−	0.381321i	$-0.124531\pi$
$821$	3.57082e7	1.84889	0.924443	+	0.381321i	$0.124531\pi$
$822$	− 1.42211e7i	− 0.734098i
$823$	8.53267e6i	0.439122i	0.975599	+	0.219561i	$0.0704625\pi$
$823$	8.53267e6i	0.439122i	−0.975599	+	0.219561i	$0.929538\pi$
$824$	4.65874e6	0.239029
$825$	0	0
$826$	1.45854e7	0.743822
$827$	3.35441e7i	1.70550i	0.522316	+	0.852752i	$0.325068\pi$
$827$	3.35441e7i	1.70550i	−0.522316	+	0.852752i	$0.674932\pi$
$828$	− 5.73662e6i	− 0.290791i
$829$	−2.52589e7	−1.27652	−0.638261	−	0.769820i	$-0.720346\pi$
$829$	−2.52589e7	−1.27652	−0.638261	+	0.769820i	$0.720346\pi$
$830$	0	0
$831$	1.17520e7	0.590348
$832$	1.56916e6i	0.0785886i
$833$	− 6.95096e6i	− 0.347082i
$834$	−9.15596e6	−0.455815
$835$	0	0
$836$	1.22965e7	0.608508
$837$	3.56108e6i	0.175698i
$838$	2.62849e7i	1.29299i
$839$	560713.	0.0275002	0.0137501	−	0.999905i	$-0.495623\pi$
$839$	560713.	0.0275002	0.0137501	+	0.999905i	$0.495623\pi$
$840$	0	0
$841$	483010.	0.0235487
$842$	2.38964e6i	0.116159i
$843$	1.25935e6i	0.0610348i
$844$	6.33129e6	0.305940
$845$	0	0
$846$	−3.19728e6	−0.153587
$847$	− 8.89400e6i	− 0.425979i
$848$	− 4.84267e6i	− 0.231257i
$849$	1.36984e7	0.652229
$850$	0	0
$851$	4.86951e7	2.30495
$852$	− 278333.i	− 0.0131361i
$853$	− 3.84041e7i	− 1.80719i	−0.428385	−	0.903596i	$-0.640917\pi$
$853$	− 3.84041e7i	− 1.80719i	0.428385	−	0.903596i	$-0.359083\pi$
$854$	−2.60191e7	−1.22081
$855$	0	0
$856$	−6.60504e6	−0.308099
$857$	3.60191e7i	1.67525i	0.546243	+	0.837627i	$0.316058\pi$
$857$	3.60191e7i	1.67525i	−0.546243	+	0.837627i	$0.683942\pi$
$858$	− 4.52161e6i	− 0.209689i
$859$	4.21545e7	1.94922	0.974611	−	0.223906i	$-0.0718807\pi$
$859$	4.21545e7	1.94922	0.974611	+	0.223906i	$0.0718807\pi$
$860$	0	0
$861$	−2.34024e7	−1.07585
$862$	− 1.15018e7i	− 0.527229i
$863$	− 2.85022e7i	− 1.30272i	−0.758768	−	0.651360i	$-0.774199\pi$
$863$	− 2.85022e7i	− 1.30272i	0.758768	−	0.651360i	$-0.225801\pi$
$864$	−746496.	−0.0340207
$865$	0	0
$866$	−9.07004e6	−0.410974
$867$	9.02715e6i	0.407852i
$868$	1.29783e7i	0.584679i
$869$	1.28482e7	0.577157
$870$	0	0
$871$	−1.09414e7	−0.488684
$872$	8.23972e6i	0.366962i
$873$	2.52871e6i	0.112296i
$874$	4.15040e7	1.83786
$875$	0	0
$876$	5.62116e6	0.247495
$877$	2.34178e7i	1.02813i	0.857752	+	0.514063i	$0.171860\pi$
$877$	2.34178e7i	1.02813i	−0.857752	+	0.514063i	$0.828140\pi$
$878$	− 2.90373e7i	− 1.27122i
$879$	2.56475e6	0.111963
$880$	0	0
$881$	−3.30045e7	−1.43263	−0.716313	−	0.697779i	$-0.754172\pi$
$881$	−3.30045e7	−1.43263	−0.716313	+	0.697779i	$0.754172\pi$
$882$	− 3.48823e6i	− 0.150985i
$883$	− 2.93497e7i	− 1.26678i	−0.773831	−	0.633392i	$-0.781662\pi$
$883$	− 2.93497e7i	− 1.26678i	0.773831	−	0.633392i	$-0.218338\pi$
$884$	−3.95743e6	−0.170326
$885$	0	0
$886$	2.87307e7	1.22959
$887$	7.83982e6i	0.334578i	0.985908	+	0.167289i	$0.0535012\pi$
$887$	7.83982e6i	0.334578i	−0.985908	+	0.167289i	$0.946499\pi$
$888$	− 6.33660e6i	− 0.269665i
$889$	1.98672e7	0.843105
$890$	0	0
$891$	2.15106e6	0.0907734
$892$	4.47125e6i	0.188155i
$893$	− 2.31321e7i	− 0.970702i
$894$	3.57944e6	0.149786
$895$	0	0
$896$	−2.72059e6	−0.113212
$897$	− 1.52616e7i	− 0.633316i
$898$	− 39856.6i	− 0.00164934i
$899$	2.23822e7	0.923642
$900$	0	0
$901$	1.22132e7	0.501208
$902$	− 2.05361e7i	− 0.840430i
$903$	1.07342e7i	0.438078i
$904$	−1.82360e6	−0.0742180
$905$	0	0
$906$	1.05619e7	0.427485
$907$	3.24989e7i	1.31175i	0.754871	+	0.655874i	$0.227700\pi$
$907$	3.24989e7i	1.31175i	−0.754871	+	0.655874i	$0.772300\pi$
$908$	2.45317e7i	0.987447i
$909$	1.21429e7	0.487430
$910$	0	0
$911$	−1.11021e7	−0.443210	−0.221605	−	0.975136i	$-0.571130\pi$
$911$	−1.11021e7	−0.443210	−0.221605	+	0.975136i	$0.571130\pi$
$912$	− 5.40084e6i	− 0.215018i
$913$	− 3.10838e7i	− 1.23412i
$914$	−5.24126e6	−0.207525
$915$	0	0
$916$	9.21874e6	0.363022
$917$	7.96492e6i	0.312794i
$918$	− 1.88266e6i	− 0.0737337i
$919$	−3.09377e7	−1.20837	−0.604184	−	0.796845i	$-0.706501\pi$
$919$	−3.09377e7	−1.20837	−0.604184	+	0.796845i	$0.706501\pi$
$920$	0	0
$921$	−1.26572e7	−0.491688
$922$	− 6.86887e6i	− 0.266108i
$923$	− 740474.i	− 0.0286092i
$924$	7.83950e6	0.302070
$925$	0	0
$926$	1.57946e7	0.605316
$927$	5.89621e6i	0.225358i
$928$	4.69190e6i	0.178846i
$929$	−3.81135e7	−1.44890	−0.724452	−	0.689325i	$-0.757907\pi$
$929$	−3.81135e7	−1.44890	−0.724452	+	0.689325i	$0.757907\pi$
$930$	0	0
$931$	2.52371e7	0.954255
$932$	6.78644e6i	0.255919i
$933$	678891.i	0.0255327i
$934$	−3.81171e6	−0.142972
$935$	0	0
$936$	−1.98597e6	−0.0740940
$937$	− 2.23247e7i	− 0.830685i	−0.909665	−	0.415343i	$-0.863662\pi$
$937$	− 2.23247e7i	− 0.830685i	0.909665	−	0.415343i	$-0.136338\pi$
$938$	− 1.89701e7i	− 0.703982i
$939$	2.87057e7	1.06244
$940$	0	0
$941$	−1.26981e7	−0.467482	−0.233741	−	0.972299i	$-0.575097\pi$
$941$	−1.26981e7	−0.467482	−0.233741	+	0.972299i	$0.575097\pi$
$942$	− 1.99564e7i	− 0.732747i
$943$	− 6.93148e7i	− 2.53832i
$944$	5.62155e6	0.205318
$945$	0	0
$946$	−9.41950e6	−0.342216
$947$	− 2.92392e7i	− 1.05948i	−0.848161	−	0.529738i	$-0.822290\pi$
$947$	− 2.92392e7i	− 1.05948i	0.848161	−	0.529738i	$-0.177710\pi$
$948$	− 5.64317e6i	− 0.203940i
$949$	1.49545e7	0.539021
$950$	0	0
$951$	5.19610e6	0.186306
$952$	− 6.86133e6i	− 0.245367i
$953$	1.39219e7i	0.496555i	0.968689	+	0.248277i	$0.0798644\pi$
$953$	1.39219e7i	0.496555i	−0.968689	+	0.248277i	$0.920136\pi$
$954$	6.12900e6	0.218031
$955$	0	0
$956$	−1.47251e7	−0.521091
$957$	− 1.35199e7i	− 0.477194i
$958$	1.33943e7i	0.471526i
$959$	6.55955e7	2.30318
$960$	0	0
$961$	−4.76713e6	−0.166513
$962$	− 1.68578e7i	− 0.587305i
$963$	− 8.35950e6i	− 0.290479i
$964$	−1.74454e7	−0.604627
$965$	0	0
$966$	2.64604e7	0.912334
$967$	1.49004e7i	0.512427i	0.966620	+	0.256214i	$0.0824751\pi$
$967$	1.49004e7i	0.512427i	−0.966620	+	0.256214i	$0.917525\pi$
$968$	− 3.42795e6i	− 0.117583i
$969$	1.36209e7	0.466012
$970$	0	0
$971$	−1.51017e7	−0.514017	−0.257008	−	0.966409i	$-0.582737\pi$
$971$	−1.51017e7	−0.514017	−0.257008	+	0.966409i	$0.582737\pi$
$972$	− 944784.i	− 0.0320750i
$973$	− 4.22323e7i	− 1.43009i
$974$	1.34617e6	0.0454676
$975$	0	0
$976$	−1.00283e7	−0.336980
$977$	− 3.55923e7i	− 1.19294i	−0.802634	−	0.596472i	$-0.796569\pi$
$977$	− 3.55923e7i	− 1.19294i	0.802634	−	0.596472i	$-0.203431\pi$
$978$	1.73367e7i	0.579588i
$979$	3.81918e7	1.27354
$980$	0	0
$981$	−1.04284e7	−0.345975
$982$	8.24355e6i	0.272794i
$983$	5.43235e6i	0.179310i	0.995973	+	0.0896549i	$0.0285764\pi$
$983$	5.43235e6i	0.179310i	−0.995973	+	0.0896549i	$0.971424\pi$
$984$	−9.01981e6	−0.296968
$985$	0	0
$986$	−1.18330e7	−0.387616
$987$	− 1.47476e7i	− 0.481868i
$988$	− 1.43684e7i	− 0.468290i
$989$	−3.17933e7	−1.03358
$990$	0	0
$991$	−4.93387e7	−1.59589	−0.797947	−	0.602728i	$-0.794080\pi$
$991$	−4.93387e7	−1.59589	−0.797947	+	0.602728i	$0.794080\pi$
$992$	5.00211e6i	0.161389i
$993$	4.85442e6i	0.156230i
$994$	1.28382e6	0.0412135
$995$	0	0
$996$	−1.36526e7	−0.436080
$997$	− 2.42081e7i	− 0.771300i	−0.922645	−	0.385650i	$-0.873977\pi$
$997$	− 2.42081e7i	− 0.771300i	0.922645	−	0.385650i	$-0.126023\pi$
$998$	− 2.45142e7i	− 0.779098i
$999$	8.01976e6	0.254242

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	750.6.c.c.499.4		8
5.2	odd	4		750.6.a.h.1.1	yes	4
5.3	odd	4		750.6.a.a.1.4	✓	4
5.4	even	2	inner	750.6.c.c.499.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.6.a.a.1.4	✓	4	5.3	odd	4
750.6.a.h.1.1	yes	4	5.2	odd	4
750.6.c.c.499.4		8	1.1	even	1	trivial
750.6.c.c.499.5		8	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:	\( N \)	\(=\)	\( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	750.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +166.052i q^{7} +64.0000i q^{8} -81.0000 q^{9} +O(q^{10})\)	\(q-4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} +166.052i q^{7} +64.0000i q^{8} -81.0000 q^{9} +327.856 q^{11} -144.000i q^{12} -383.096i q^{13} +664.206 q^{14} +256.000 q^{16} +645.632i q^{17} +324.000i q^{18} -2344.12 q^{19} -1494.46 q^{21} -1311.42i q^{22} -4426.41i q^{23} -576.000 q^{24} -1532.38 q^{26} -729.000i q^{27} -2656.83i q^{28} -4581.94 q^{29} -4884.88 q^{31} -1024.00i q^{32} +2950.70i q^{33} +2582.53 q^{34} +1296.00 q^{36} +11001.0i q^{37} +9376.47i q^{38} +3447.86 q^{39} +15659.4 q^{41} +5977.86i q^{42} -7182.66i q^{43} -5245.69 q^{44} -17705.6 q^{46} +9868.14i q^{47} +2304.00i q^{48} -10766.1 q^{49} -5810.69 q^{51} +6129.54i q^{52} -18916.7i q^{53} -2916.00 q^{54} -10627.3 q^{56} -21097.1i q^{57} +18327.8i q^{58} +21959.2 q^{59} -39173.2 q^{61} +19539.5i q^{62} -13450.2i q^{63} -4096.00 q^{64} +11802.8 q^{66} -28560.5i q^{67} -10330.1i q^{68} +39837.6 q^{69} +1932.87 q^{71} -5184.00i q^{72} +39035.9i q^{73} +44004.2 q^{74} +37505.9 q^{76} +54441.0i q^{77} -13791.5i q^{78} +39188.7 q^{79} +6561.00 q^{81} -62637.6i q^{82} -94809.5i q^{83} +23911.4 q^{84} -28730.6 q^{86} -41237.4i q^{87} +20982.8i q^{88} +116490. q^{89} +63613.7 q^{91} +70822.5i q^{92} -43963.9i q^{93} +39472.5 q^{94} +9216.00 q^{96} -31218.6i q^{97} +43064.5i q^{98} -26556.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 128 q^{4} + 288 q^{6} - 648 q^{9}+O(q^{10}) \) 8 * q - 128 * q^4 + 288 * q^6 - 648 * q^9	\( 8 q - 128 q^{4} + 288 q^{6} - 648 q^{9} - 804 q^{11} + 336 q^{14} + 2048 q^{16} + 3320 q^{19} - 756 q^{21} - 4608 q^{24} - 10128 q^{26} + 11400 q^{29} - 26004 q^{31} + 17296 q^{34} + 10368 q^{36} + 22788 q^{39} + 26316 q^{41} + 12864 q^{44} - 64048 q^{46} + 16064 q^{49} - 38916 q^{51} - 23328 q^{54} - 5376 q^{56} + 21660 q^{59} - 97324 q^{61} - 32768 q^{64} - 28944 q^{66} + 144108 q^{69} - 138444 q^{71} + 223776 q^{74} - 53120 q^{76} + 346560 q^{79} + 52488 q^{81} + 12096 q^{84} - 58848 q^{86} - 123600 q^{89} - 283544 q^{91} + 190336 q^{94} + 73728 q^{96} + 65124 q^{99}+O(q^{100}) \) 8 * q - 128 * q^4 + 288 * q^6 - 648 * q^9 - 804 * q^11 + 336 * q^14 + 2048 * q^16 + 3320 * q^19 - 756 * q^21 - 4608 * q^24 - 10128 * q^26 + 11400 * q^29 - 26004 * q^31 + 17296 * q^34 + 10368 * q^36 + 22788 * q^39 + 26316 * q^41 + 12864 * q^44 - 64048 * q^46 + 16064 * q^49 - 38916 * q^51 - 23328 * q^54 - 5376 * q^56 + 21660 * q^59 - 97324 * q^61 - 32768 * q^64 - 28944 * q^66 + 144108 * q^69 - 138444 * q^71 + 223776 * q^74 - 53120 * q^76 + 346560 * q^79 + 52488 * q^81 + 12096 * q^84 - 58848 * q^86 - 123600 * q^89 - 283544 * q^91 + 190336 * q^94 + 73728 * q^96 + 65124 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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