LMFDB - Embedded newform 25.6.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,6,Mod(1,25)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("25.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 25 = 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	25.a (trivial)

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:	yes
Analytic conductor:	$4.00959549532$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{241}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 60 $ x^2 - x - 60
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.26209$ of defining polynomial
Character	$\chi$	$=$	25.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.26209 q^{2} -25.5242 q^{3} -4.31044 q^{4} -134.310 q^{6} -131.048 q^{7} -191.069 q^{8} +408.483 q^{9} +O(q^{10})$	\(q+5.26209 q^{2} -25.5242 q^{3} -4.31044 q^{4} -134.310 q^{6} -131.048 q^{7} -191.069 q^{8} +408.483 q^{9} +290.104 q^{11} +110.020 q^{12} -68.3868 q^{13} -689.588 q^{14} -867.486 q^{16} +310.644 q^{17} +2149.48 q^{18} -2133.35 q^{19} +3344.90 q^{21} +1526.55 q^{22} -873.145 q^{23} +4876.87 q^{24} -359.857 q^{26} -4223.83 q^{27} +564.876 q^{28} -2580.97 q^{29} -9086.30 q^{31} +1549.41 q^{32} -7404.67 q^{33} +1634.64 q^{34} -1760.74 q^{36} -3990.64 q^{37} -11225.9 q^{38} +1745.52 q^{39} +16981.8 q^{41} +17601.2 q^{42} +18017.7 q^{43} -1250.48 q^{44} -4594.57 q^{46} -24864.7 q^{47} +22141.9 q^{48} +366.670 q^{49} -7928.93 q^{51} +294.777 q^{52} +7652.91 q^{53} -22226.2 q^{54} +25039.2 q^{56} +54451.9 q^{57} -13581.3 q^{58} -9233.69 q^{59} +3326.17 q^{61} -47812.9 q^{62} -53531.1 q^{63} +35912.7 q^{64} -38964.0 q^{66} +32340.7 q^{67} -1339.01 q^{68} +22286.3 q^{69} -35885.9 q^{71} -78048.4 q^{72} +26513.6 q^{73} -20999.1 q^{74} +9195.65 q^{76} -38017.7 q^{77} +9185.06 q^{78} +71705.7 q^{79} +8548.28 q^{81} +89359.9 q^{82} -39630.1 q^{83} -14418.0 q^{84} +94810.8 q^{86} +65877.1 q^{87} -55429.9 q^{88} -117441. q^{89} +8961.98 q^{91} +3763.64 q^{92} +231920. q^{93} -130840. q^{94} -39547.4 q^{96} -21878.3 q^{97} +1929.45 q^{98} +118503. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{2} - 20 q^{3} + 69 q^{4} - 191 q^{6} - 200 q^{7} - 615 q^{8} + 196 q^{9}+O(q^{10}) $ 2 * q - 5 * q^2 - 20 * q^3 + 69 * q^4 - 191 * q^6 - 200 * q^7 - 615 * q^8 + 196 * q^9	\( 2 q - 5 q^{2} - 20 q^{3} + 69 q^{4} - 191 q^{6} - 200 q^{7} - 615 q^{8} + 196 q^{9} - 196 q^{11} + 515 q^{12} + 360 q^{13} + 18 q^{14} + 1137 q^{16} - 1490 q^{17} + 4330 q^{18} - 3180 q^{19} + 2964 q^{21} + 6515 q^{22} - 1560 q^{23} + 2535 q^{24} - 4756 q^{26} - 6740 q^{27} - 4490 q^{28} - 3920 q^{29} - 1096 q^{31} - 5455 q^{32} - 10090 q^{33} + 20113 q^{34} - 17338 q^{36} - 2020 q^{37} - 485 q^{38} + 4112 q^{39} + 27754 q^{41} + 21510 q^{42} + 3000 q^{43} - 36887 q^{44} + 2454 q^{46} - 25760 q^{47} + 33215 q^{48} - 11686 q^{49} - 17876 q^{51} + 31700 q^{52} + 26980 q^{53} + 3595 q^{54} + 54270 q^{56} + 48670 q^{57} + 160 q^{58} + 11960 q^{59} - 24396 q^{61} - 129810 q^{62} - 38880 q^{63} + 43649 q^{64} - 11407 q^{66} + 40060 q^{67} - 133345 q^{68} + 18492 q^{69} - 87296 q^{71} + 12030 q^{72} + 70290 q^{73} - 41222 q^{74} - 67535 q^{76} - 4500 q^{77} - 15100 q^{78} + 65480 q^{79} + 46282 q^{81} - 21185 q^{82} - 92580 q^{83} - 42342 q^{84} + 248924 q^{86} + 58480 q^{87} + 150645 q^{88} - 72810 q^{89} - 20576 q^{91} - 46590 q^{92} + 276060 q^{93} - 121652 q^{94} - 78241 q^{96} + 126140 q^{97} + 125615 q^{98} + 221792 q^{99}+O(q^{100}) \) 2 * q - 5 * q^2 - 20 * q^3 + 69 * q^4 - 191 * q^6 - 200 * q^7 - 615 * q^8 + 196 * q^9 - 196 * q^11 + 515 * q^12 + 360 * q^13 + 18 * q^14 + 1137 * q^16 - 1490 * q^17 + 4330 * q^18 - 3180 * q^19 + 2964 * q^21 + 6515 * q^22 - 1560 * q^23 + 2535 * q^24 - 4756 * q^26 - 6740 * q^27 - 4490 * q^28 - 3920 * q^29 - 1096 * q^31 - 5455 * q^32 - 10090 * q^33 + 20113 * q^34 - 17338 * q^36 - 2020 * q^37 - 485 * q^38 + 4112 * q^39 + 27754 * q^41 + 21510 * q^42 + 3000 * q^43 - 36887 * q^44 + 2454 * q^46 - 25760 * q^47 + 33215 * q^48 - 11686 * q^49 - 17876 * q^51 + 31700 * q^52 + 26980 * q^53 + 3595 * q^54 + 54270 * q^56 + 48670 * q^57 + 160 * q^58 + 11960 * q^59 - 24396 * q^61 - 129810 * q^62 - 38880 * q^63 + 43649 * q^64 - 11407 * q^66 + 40060 * q^67 - 133345 * q^68 + 18492 * q^69 - 87296 * q^71 + 12030 * q^72 + 70290 * q^73 - 41222 * q^74 - 67535 * q^76 - 4500 * q^77 - 15100 * q^78 + 65480 * q^79 + 46282 * q^81 - 21185 * q^82 - 92580 * q^83 - 42342 * q^84 + 248924 * q^86 + 58480 * q^87 + 150645 * q^88 - 72810 * q^89 - 20576 * q^91 - 46590 * q^92 + 276060 * q^93 - 121652 * q^94 - 78241 * q^96 + 126140 * q^97 + 125615 * q^98 + 221792 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	5.26209	0.930214	0.465107	−	0.885254i	$-0.346016\pi$
$2$	5.26209	0.930214	0.465107	+	0.885254i	$0.346016\pi$
$3$	−25.5242	−1.63738	−0.818688	−	0.574238i	$-0.805298\pi$
$3$	−25.5242	−1.63738	−0.818688	+	0.574238i	$0.805298\pi$
$4$	−4.31044	−0.134701
$5$	0	0
$5$	0	0
$6$	−134.310	−1.52311
$7$	−131.048	−1.01085	−0.505425	−	0.862871i	$-0.668664\pi$
$7$	−131.048	−1.01085	−0.505425	+	0.862871i	$0.668664\pi$
$8$	−191.069	−1.05552
$9$	408.483	1.68100
$10$	0	0
$11$	290.104	0.722891	0.361445	−	0.932393i	$-0.382283\pi$
$11$	290.104	0.722891	0.361445	+	0.932393i	$0.382283\pi$
$12$	110.020	0.220557
$13$	−68.3868	−0.112231	−0.0561156	−	0.998424i	$-0.517872\pi$
$13$	−68.3868	−0.112231	−0.0561156	+	0.998424i	$0.517872\pi$
$14$	−689.588	−0.940307
$15$	0	0
$16$	−867.486	−0.847154
$17$	310.644	0.260700	0.130350	−	0.991468i	$-0.458390\pi$
$17$	310.644	0.260700	0.130350	+	0.991468i	$0.458390\pi$
$18$	2149.48	1.56369
$19$	−2133.35	−1.35574	−0.677871	−	0.735180i	$-0.737097\pi$
$19$	−2133.35	−1.35574	−0.677871	+	0.735180i	$0.737097\pi$
$20$	0	0
$21$	3344.90	1.65514
$22$	1526.55	0.672443
$23$	−873.145	−0.344165	−0.172083	−	0.985083i	$-0.555050\pi$
$23$	−873.145	−0.344165	−0.172083	+	0.985083i	$0.555050\pi$
$24$	4876.87	1.72828
$25$	0	0
$26$	−359.857	−0.104399
$27$	−4223.83	−1.11506
$28$	564.876	0.136163
$29$	−2580.97	−0.569885	−0.284943	−	0.958545i	$-0.591975\pi$
$29$	−2580.97	−0.569885	−0.284943	+	0.958545i	$0.591975\pi$
$30$	0	0
$31$	−9086.30	−1.69818	−0.849088	−	0.528252i	$-0.822848\pi$
$31$	−9086.30	−1.69818	−0.849088	+	0.528252i	$0.822848\pi$
$32$	1549.41	0.267480
$33$	−7404.67	−1.18364
$34$	1634.64	0.242507
$35$	0	0
$36$	−1760.74	−0.226433
$37$	−3990.64	−0.479224	−0.239612	−	0.970869i	$-0.577020\pi$
$37$	−3990.64	−0.479224	−0.239612	+	0.970869i	$0.577020\pi$
$38$	−11225.9	−1.26113
$39$	1745.52	0.183765
$40$	0	0
$41$	16981.8	1.57770	0.788851	−	0.614584i	$-0.210676\pi$
$41$	16981.8	1.57770	0.788851	+	0.614584i	$0.210676\pi$
$42$	17601.2	1.53964
$43$	18017.7	1.48603	0.743017	−	0.669273i	$-0.233394\pi$
$43$	18017.7	1.48603	0.743017	+	0.669273i	$0.233394\pi$
$44$	−1250.48	−0.0973742
$45$	0	0
$46$	−4594.57	−0.320147
$47$	−24864.7	−1.64187	−0.820933	−	0.571024i	$-0.806546\pi$
$47$	−24864.7	−1.64187	−0.820933	+	0.571024i	$0.806546\pi$
$48$	22141.9	1.38711
$49$	366.670	0.0218165
$50$	0	0
$51$	−7928.93	−0.426864
$52$	294.777	0.0151177
$53$	7652.91	0.374229	0.187114	−	0.982338i	$-0.440087\pi$
$53$	7652.91	0.374229	0.187114	+	0.982338i	$0.440087\pi$
$54$	−22226.2	−1.03724
$55$	0	0
$56$	25039.2	1.06697
$57$	54451.9	2.21986
$58$	−13581.3	−0.530116
$59$	−9233.69	−0.345339	−0.172669	−	0.984980i	$-0.555239\pi$
$59$	−9233.69	−0.345339	−0.172669	+	0.984980i	$0.555239\pi$
$60$	0	0
$61$	3326.17	0.114451	0.0572256	−	0.998361i	$-0.481775\pi$
$61$	3326.17	0.114451	0.0572256	+	0.998361i	$0.481775\pi$
$62$	−47812.9	−1.57967
$63$	−53531.1	−1.69924
$64$	35912.7	1.09597
$65$	0	0
$66$	−38964.0	−1.10104
$67$	32340.7	0.880161	0.440080	−	0.897958i	$-0.354950\pi$
$67$	32340.7	0.880161	0.440080	+	0.897958i	$0.354950\pi$
$68$	−1339.01	−0.0351165
$69$	22286.3	0.563528
$70$	0	0
$71$	−35885.9	−0.844847	−0.422424	−	0.906399i	$-0.638821\pi$
$71$	−35885.9	−0.844847	−0.422424	+	0.906399i	$0.638821\pi$
$72$	−78048.4	−1.77432
$73$	26513.6	0.582319	0.291159	−	0.956675i	$-0.405959\pi$
$73$	26513.6	0.582319	0.291159	+	0.956675i	$0.405959\pi$
$74$	−20999.1	−0.445781
$75$	0	0
$76$	9195.65	0.182620
$77$	−38017.7	−0.730733
$78$	9185.06	0.170941
$79$	71705.7	1.29266	0.646332	−	0.763056i	$-0.276302\pi$
$79$	71705.7	1.29266	0.646332	+	0.763056i	$0.276302\pi$
$80$	0	0
$81$	8548.28	0.144766
$82$	89359.9	1.46760
$83$	−39630.1	−0.631437	−0.315719	−	0.948853i	$-0.602246\pi$
$83$	−39630.1	−0.631437	−0.315719	+	0.948853i	$0.602246\pi$
$84$	−14418.0	−0.222949
$85$	0	0
$86$	94810.8	1.38233
$87$	65877.1	0.933117
$88$	−55429.9	−0.763022
$89$	−117441.	−1.57161	−0.785806	−	0.618473i	$-0.787752\pi$
$89$	−117441.	−1.57161	−0.785806	+	0.618473i	$0.787752\pi$
$90$	0	0
$91$	8961.98	0.113449
$92$	3763.64	0.0463594
$93$	231920.	2.78055
$94$	−130840.	−1.52729
$95$	0	0
$96$	−39547.4	−0.437966
$97$	−21878.3	−0.236093	−0.118047	−	0.993008i	$-0.537663\pi$
$97$	−21878.3	−0.236093	−0.118047	+	0.993008i	$0.537663\pi$
$98$	1929.45	0.0202940
$99$	118503.	1.21518
$100$	0	0
$101$	−75072.1	−0.732276	−0.366138	−	0.930561i	$-0.619320\pi$
$101$	−75072.1	−0.732276	−0.366138	+	0.930561i	$0.619320\pi$
$102$	−41722.7	−0.397075
$103$	−47928.6	−0.445145	−0.222573	−	0.974916i	$-0.571445\pi$
$103$	−47928.6	−0.445145	−0.222573	+	0.974916i	$0.571445\pi$
$104$	13066.6	0.118462
$105$	0	0
$106$	40270.3	0.348113
$107$	−92012.3	−0.776938	−0.388469	−	0.921462i	$-0.626996\pi$
$107$	−92012.3	−0.776938	−0.388469	+	0.921462i	$0.626996\pi$
$108$	18206.6	0.150199
$109$	−10647.5	−0.0858387	−0.0429194	−	0.999079i	$-0.513666\pi$
$109$	−10647.5	−0.0858387	−0.0429194	+	0.999079i	$0.513666\pi$
$110$	0	0
$111$	101858.	0.784670
$112$	113683.	0.856346
$113$	−87373.9	−0.643703	−0.321852	−	0.946790i	$-0.604305\pi$
$113$	−87373.9	−0.643703	−0.321852	+	0.946790i	$0.604305\pi$
$114$	286531.	2.06495
$115$	0	0
$116$	11125.1	0.0767642
$117$	−27934.9	−0.188661
$118$	−48588.5	−0.321239
$119$	−40709.4	−0.263528
$120$	0	0
$121$	−76890.5	−0.477429
$122$	17502.6	0.106464
$123$	−433447.	−2.58329
$124$	39165.9	0.228746
$125$	0	0
$126$	−281685.	−1.58066
$127$	−197379.	−1.08591	−0.542953	−	0.839763i	$-0.682694\pi$
$127$	−197379.	−1.08591	−0.542953	+	0.839763i	$0.682694\pi$
$128$	139395.	0.752005
$129$	−459887.	−2.43320
$130$	0	0
$131$	−118490.	−0.603258	−0.301629	−	0.953425i	$-0.597530\pi$
$131$	−118490.	−0.603258	−0.301629	+	0.953425i	$0.597530\pi$
$132$	31917.4	0.159438
$133$	279571.	1.37045
$134$	170179.	0.818738
$135$	0	0
$136$	−59354.3	−0.275173
$137$	302570.	1.37728	0.688642	−	0.725101i	$-0.258207\pi$
$137$	302570.	1.37728	0.688642	+	0.725101i	$0.258207\pi$
$138$	117272.	0.524202
$139$	157190.	0.690062	0.345031	−	0.938591i	$-0.387868\pi$
$139$	157190.	0.690062	0.345031	+	0.938591i	$0.387868\pi$
$140$	0	0
$141$	634650.	2.68835
$142$	−188835.	−0.785889
$143$	−19839.3	−0.0811309
$144$	−354354.	−1.42407
$145$	0	0
$146$	139517.	0.541681
$147$	−9358.95	−0.0357218
$148$	17201.4	0.0645520
$149$	526340.	1.94223	0.971115	−	0.238612i	$-0.0766923\pi$
$149$	526340.	1.94223	0.971115	+	0.238612i	$0.0766923\pi$
$150$	0	0
$151$	1849.08	0.00659954	0.00329977	−	0.999995i	$-0.498950\pi$
$151$	1849.08	0.00659954	0.00329977	+	0.999995i	$0.498950\pi$
$152$	407616.	1.43101
$153$	126893.	0.438237
$154$	−200052.	−0.679739
$155$	0	0
$156$	−7523.94	−0.0247533
$157$	−343342.	−1.11167	−0.555837	−	0.831292i	$-0.687602\pi$
$157$	−343342.	−1.11167	−0.555837	+	0.831292i	$0.687602\pi$
$158$	377322.	1.20246
$159$	−195334.	−0.612753
$160$	0	0
$161$	114424.	0.347899
$162$	44981.8	0.134663
$163$	267463.	0.788487	0.394243	−	0.919006i	$-0.371007\pi$
$163$	267463.	0.788487	0.394243	+	0.919006i	$0.371007\pi$
$164$	−73199.1	−0.212518
$165$	0	0
$166$	−208537.	−0.587372
$167$	−122968.	−0.341193	−0.170596	−	0.985341i	$-0.554569\pi$
$167$	−122968.	−0.341193	−0.170596	+	0.985341i	$0.554569\pi$
$168$	−639106.	−1.74703
$169$	−366616.	−0.987404
$170$	0	0
$171$	−871437.	−2.27901
$172$	−77664.3	−0.200170
$173$	−288020.	−0.731657	−0.365829	−	0.930682i	$-0.619214\pi$
$173$	−288020.	−0.731657	−0.365829	+	0.930682i	$0.619214\pi$
$174$	346651.	0.867999
$175$	0	0
$176$	−251662.	−0.612400
$177$	235682.	0.565450
$178$	−617986.	−1.46194
$179$	246177.	0.574268	0.287134	−	0.957890i	$-0.407298\pi$
$179$	246177.	0.574268	0.287134	+	0.957890i	$0.407298\pi$
$180$	0	0
$181$	433120.	0.982678	0.491339	−	0.870968i	$-0.336508\pi$
$181$	433120.	0.982678	0.491339	+	0.870968i	$0.336508\pi$
$182$	47158.7	0.105532
$183$	−84897.9	−0.187400
$184$	166831.	0.363272
$185$	0	0
$186$	1.22038e6	2.58651
$187$	90119.1	0.188457
$188$	107178.	0.221161
$189$	553526.	1.12715
$190$	0	0
$191$	−701011.	−1.39040	−0.695202	−	0.718814i	$-0.744685\pi$
$191$	−701011.	−1.39040	−0.695202	+	0.718814i	$0.744685\pi$
$192$	−916642.	−1.79451
$193$	−215730.	−0.416887	−0.208443	−	0.978034i	$-0.566840\pi$
$193$	−215730.	−0.416887	−0.208443	+	0.978034i	$0.566840\pi$
$194$	−115125.	−0.219618
$195$	0	0
$196$	−1580.51	−0.00293871
$197$	700484.	1.28598	0.642988	−	0.765876i	$-0.277695\pi$
$197$	700484.	1.28598	0.642988	+	0.765876i	$0.277695\pi$
$198$	623572.	1.13038
$199$	22097.5	0.0395558	0.0197779	−	0.999804i	$-0.493704\pi$
$199$	22097.5	0.0395558	0.0197779	+	0.999804i	$0.493704\pi$
$200$	0	0
$201$	−825469.	−1.44115
$202$	−395036.	−0.681174
$203$	338231.	0.576068
$204$	34177.1	0.0574990
$205$	0	0
$206$	−252205.	−0.414081
$207$	−356665.	−0.578542
$208$	59324.6	0.0950772
$209$	−618893.	−0.980054
$210$	0	0
$211$	910782.	1.40834	0.704172	−	0.710030i	$-0.251319\pi$
$211$	910782.	1.40834	0.704172	+	0.710030i	$0.251319\pi$
$212$	−32987.4	−0.0504090
$213$	915958.	1.38333
$214$	−484177.	−0.722719
$215$	0	0
$216$	807042.	1.17696
$217$	1.19074e6	1.71660
$218$	−56028.3	−0.0798484
$219$	−676737.	−0.953475
$220$	0	0
$221$	−21243.9	−0.0292587
$222$	535985.	0.729911
$223$	−132745.	−0.178754	−0.0893768	−	0.995998i	$-0.528488\pi$
$223$	−132745.	−0.178754	−0.0893768	+	0.995998i	$0.528488\pi$
$224$	−203048.	−0.270382
$225$	0	0
$226$	−459769.	−0.598782
$227$	354321.	0.456386	0.228193	−	0.973616i	$-0.426718\pi$
$227$	354321.	0.456386	0.228193	+	0.973616i	$0.426718\pi$
$228$	−234711.	−0.299018
$229$	366643.	0.462013	0.231007	−	0.972952i	$-0.425798\pi$
$229$	366643.	0.462013	0.231007	+	0.972952i	$0.425798\pi$
$230$	0	0
$231$	970370.	1.19649
$232$	493142.	0.601523
$233$	−1.02388e6	−1.23555	−0.617776	−	0.786355i	$-0.711966\pi$
$233$	−1.02388e6	−1.23555	−0.617776	+	0.786355i	$0.711966\pi$
$234$	−146996.	−0.175495
$235$	0	0
$236$	39801.2	0.0465175
$237$	−1.83023e6	−2.11658
$238$	−214216.	−0.245138
$239$	1.19966e6	1.35852	0.679258	−	0.733899i	$-0.262302\pi$
$239$	1.19966e6	1.35852	0.679258	+	0.733899i	$0.262302\pi$
$240$	0	0
$241$	−94967.5	−0.105325	−0.0526626	−	0.998612i	$-0.516771\pi$
$241$	−94967.5	−0.105325	−0.0526626	+	0.998612i	$0.516771\pi$
$242$	−404604.	−0.444112
$243$	808203.	0.878021
$244$	−14337.3	−0.0154167
$245$	0	0
$246$	−2.28084e6	−2.40302
$247$	145893.	0.152157
$248$	1.73611e6	1.79245
$249$	1.01153e6	1.03390
$250$	0	0
$251$	418053.	0.418839	0.209419	−	0.977826i	$-0.432843\pi$
$251$	418053.	0.418839	0.209419	+	0.977826i	$0.432843\pi$
$252$	230742.	0.228890
$253$	−253303.	−0.248794
$254$	−1.03863e6	−1.01012
$255$	0	0
$256$	−415700.	−0.396442
$257$	−2.04586e6	−1.93216	−0.966079	−	0.258246i	$-0.916856\pi$
$257$	−2.04586e6	−1.93216	−0.966079	+	0.258246i	$0.916856\pi$
$258$	−2.41997e6	−2.26340
$259$	522967.	0.484423
$260$	0	0
$261$	−1.05428e6	−0.957978
$262$	−623505.	−0.561160
$263$	−1.64024e6	−1.46224	−0.731119	−	0.682250i	$-0.761002\pi$
$263$	−1.64024e6	−1.46224	−0.731119	+	0.682250i	$0.761002\pi$
$264$	1.41480e6	1.24935
$265$	0	0
$266$	1.47113e6	1.27481
$267$	2.99759e6	2.57332
$268$	−139402.	−0.118559
$269$	−720582.	−0.607160	−0.303580	−	0.952806i	$-0.598182\pi$
$269$	−720582.	−0.607160	−0.303580	+	0.952806i	$0.598182\pi$
$270$	0	0
$271$	1.14186e6	0.944477	0.472238	−	0.881471i	$-0.343446\pi$
$271$	1.14186e6	0.944477	0.472238	+	0.881471i	$0.343446\pi$
$272$	−269479.	−0.220853
$273$	−228747.	−0.185759
$274$	1.59215e6	1.28117
$275$	0	0
$276$	−96063.7	−0.0759078
$277$	−377028.	−0.295239	−0.147620	−	0.989044i	$-0.547161\pi$
$277$	−377028.	−0.295239	−0.147620	+	0.989044i	$0.547161\pi$
$278$	827148.	0.641906
$279$	−3.71160e6	−2.85464
$280$	0	0
$281$	−617249.	−0.466331	−0.233166	−	0.972437i	$-0.574908\pi$
$281$	−617249.	−0.466331	−0.233166	+	0.972437i	$0.574908\pi$
$282$	3.33958e6	2.50075
$283$	1.25311e6	0.930087	0.465044	−	0.885288i	$-0.346039\pi$
$283$	1.25311e6	0.930087	0.465044	+	0.885288i	$0.346039\pi$
$284$	154684.	0.113802
$285$	0	0
$286$	−104396.	−0.0754692
$287$	−2.22544e6	−1.59482
$288$	632908.	0.449635
$289$	−1.32336e6	−0.932036
$290$	0	0
$291$	558425.	0.386574
$292$	−114285.	−0.0784390
$293$	−818972.	−0.557314	−0.278657	−	0.960391i	$-0.589889\pi$
$293$	−818972.	−0.557314	−0.278657	+	0.960391i	$0.589889\pi$
$294$	−49247.6	−0.0332290
$295$	0	0
$296$	762487.	0.505828
$297$	−1.22535e6	−0.806064
$298$	2.76965e6	1.80669
$299$	59711.6	0.0386261
$300$	0	0
$301$	−2.36119e6	−1.50216
$302$	9730.03	0.00613899
$303$	1.91615e6	1.19901
$304$	1.85065e6	1.14852
$305$	0	0
$306$	667722.	0.407654
$307$	136224.	0.0824915	0.0412458	−	0.999149i	$-0.486867\pi$
$307$	136224.	0.0824915	0.0412458	+	0.999149i	$0.486867\pi$
$308$	163873.	0.0984306
$309$	1.22334e6	0.728871
$310$	0	0
$311$	2.62886e6	1.54122	0.770612	−	0.637304i	$-0.219950\pi$
$311$	2.62886e6	1.54122	0.770612	+	0.637304i	$0.219950\pi$
$312$	−333514.	−0.193967
$313$	218161.	0.125868	0.0629341	−	0.998018i	$-0.479954\pi$
$313$	218161.	0.125868	0.0629341	+	0.998018i	$0.479954\pi$
$314$	−1.80669e6	−1.03409
$315$	0	0
$316$	−309083.	−0.174123
$317$	1.25865e6	0.703491	0.351745	−	0.936096i	$-0.385588\pi$
$317$	1.25865e6	0.703491	0.351745	+	0.936096i	$0.385588\pi$
$318$	−1.02787e6	−0.569992
$319$	−748750.	−0.411965
$320$	0	0
$321$	2.34854e6	1.27214
$322$	602110.	0.323621
$323$	−662711.	−0.353442
$324$	−36846.8	−0.0195001
$325$	0	0
$326$	1.40741e6	0.733462
$327$	271770.	0.140550
$328$	−3.24470e6	−1.66529
$329$	3.25847e6	1.65968
$330$	0	0
$331$	−3.21863e6	−1.61473	−0.807366	−	0.590051i	$-0.799108\pi$
$331$	−3.21863e6	−1.61473	−0.807366	+	0.590051i	$0.799108\pi$
$332$	170823.	0.0850553
$333$	−1.63011e6	−0.805576
$334$	−647067.	−0.317382
$335$	0	0
$336$	−2.90166e6	−1.40216
$337$	1.63574e6	0.784585	0.392293	−	0.919840i	$-0.371682\pi$
$337$	1.63574e6	0.784585	0.392293	+	0.919840i	$0.371682\pi$
$338$	−1.92917e6	−0.918498
$339$	2.23015e6	1.05398
$340$	0	0
$341$	−2.63597e6	−1.22760
$342$	−4.58558e6	−2.11996
$343$	2.15448e6	0.988796
$344$	−3.44262e6	−1.56853
$345$	0	0
$346$	−1.51559e6	−0.680598
$347$	−1.83815e6	−0.819514	−0.409757	−	0.912195i	$-0.634386\pi$
$347$	−1.83815e6	−0.819514	−0.409757	+	0.912195i	$0.634386\pi$
$348$	−283959.	−0.125692
$349$	−2.53806e6	−1.11542	−0.557710	−	0.830036i	$-0.688320\pi$
$349$	−2.53806e6	−1.11542	−0.557710	+	0.830036i	$0.688320\pi$
$350$	0	0
$351$	288854.	0.125144
$352$	449491.	0.193359
$353$	−1.88471e6	−0.805023	−0.402511	−	0.915415i	$-0.631863\pi$
$353$	−1.88471e6	−0.805023	−0.402511	+	0.915415i	$0.631863\pi$
$354$	1.24018e6	0.525989
$355$	0	0
$356$	506223.	0.211698
$357$	1.03907e6	0.431495
$358$	1.29540e6	0.534192
$359$	305057.	0.124924	0.0624619	−	0.998047i	$-0.480105\pi$
$359$	305057.	0.124924	0.0624619	+	0.998047i	$0.480105\pi$
$360$	0	0
$361$	2.07507e6	0.838039
$362$	2.27911e6	0.914102
$363$	1.96257e6	0.781731
$364$	−38630.0	−0.0152817
$365$	0	0
$366$	−446740.	−0.174322
$367$	−727834.	−0.282077	−0.141038	−	0.990004i	$-0.545044\pi$
$367$	−727834.	−0.282077	−0.141038	+	0.990004i	$0.545044\pi$
$368$	757441.	0.291561
$369$	6.93680e6	2.65212
$370$	0	0
$371$	−1.00290e6	−0.378289
$372$	−999677.	−0.374544
$373$	−4.77676e6	−1.77771	−0.888855	−	0.458188i	$-0.848499\pi$
$373$	−4.77676e6	−1.77771	−0.888855	+	0.458188i	$0.848499\pi$
$374$	474215.	0.175306
$375$	0	0
$376$	4.75086e6	1.73302
$377$	176504.	0.0639590
$378$	2.91270e6	1.04850
$379$	−701558.	−0.250880	−0.125440	−	0.992101i	$-0.540034\pi$
$379$	−701558.	−0.250880	−0.125440	+	0.992101i	$0.540034\pi$
$380$	0	0
$381$	5.03794e6	1.77804
$382$	−3.68878e6	−1.29337
$383$	4.01069e6	1.39708	0.698541	−	0.715570i	$-0.253833\pi$
$383$	4.01069e6	1.39708	0.698541	+	0.715570i	$0.253833\pi$
$384$	−3.55793e6	−1.23132
$385$	0	0
$386$	−1.13519e6	−0.387794
$387$	7.35994e6	2.49803
$388$	94305.0	0.0318021
$389$	−4.45952e6	−1.49422	−0.747108	−	0.664702i	$-0.768558\pi$
$389$	−4.45952e6	−1.49422	−0.747108	+	0.664702i	$0.768558\pi$
$390$	0	0
$391$	−271237.	−0.0897237
$392$	−70059.1	−0.0230276
$393$	3.02436e6	0.987761
$394$	3.68601e6	1.19623
$395$	0	0
$396$	−510799.	−0.163686
$397$	3.36993e6	1.07311	0.536555	−	0.843865i	$-0.319725\pi$
$397$	3.36993e6	1.07311	0.536555	+	0.843865i	$0.319725\pi$
$398$	116279.	0.0367953
$399$	−7.13583e6	−2.24395
$400$	0	0
$401$	−3.00679e6	−0.933775	−0.466888	−	0.884317i	$-0.654625\pi$
$401$	−3.00679e6	−0.933775	−0.466888	+	0.884317i	$0.654625\pi$
$402$	−4.34369e6	−1.34058
$403$	621383.	0.190588
$404$	323593.	0.0986384
$405$	0	0
$406$	1.77980e6	0.535867
$407$	−1.15770e6	−0.346426
$408$	1.51497e6	0.450561
$409$	−998012.	−0.295004	−0.147502	−	0.989062i	$-0.547123\pi$
$409$	−998012.	−0.295004	−0.147502	+	0.989062i	$0.547123\pi$
$410$	0	0
$411$	−7.72284e6	−2.25513
$412$	206593.	0.0599616
$413$	1.21006e6	0.349085
$414$	−1.87680e6	−0.538168
$415$	0	0
$416$	−105959.	−0.0300196
$417$	−4.01215e6	−1.12989
$418$	−3.25667e6	−0.911660
$419$	5.53743e6	1.54090	0.770448	−	0.637503i	$-0.220033\pi$
$419$	5.53743e6	1.54090	0.770448	+	0.637503i	$0.220033\pi$
$420$	0	0
$421$	1.98635e6	0.546198	0.273099	−	0.961986i	$-0.411951\pi$
$421$	1.98635e6	0.546198	0.273099	+	0.961986i	$0.411951\pi$
$422$	4.79262e6	1.31006
$423$	−1.01568e7	−2.75998
$424$	−1.46223e6	−0.395004
$425$	0	0
$426$	4.81985e6	1.28680
$427$	−435890.	−0.115693
$428$	396613.	0.104654
$429$	506382.	0.132842
$430$	0	0
$431$	116512.	0.0302118	0.0151059	−	0.999886i	$-0.495191\pi$
$431$	116512.	0.0302118	0.0151059	+	0.999886i	$0.495191\pi$
$432$	3.66411e6	0.944625
$433$	4.56166e6	1.16924	0.584619	−	0.811308i	$-0.301244\pi$
$433$	4.56166e6	1.16924	0.584619	+	0.811308i	$0.301244\pi$
$434$	6.26580e6	1.59681
$435$	0	0
$436$	45895.6	0.0115626
$437$	1.86272e6	0.466599
$438$	−3.56105e6	−0.886936
$439$	2.92172e6	0.723565	0.361782	−	0.932263i	$-0.382168\pi$
$439$	2.92172e6	0.723565	0.361782	+	0.932263i	$0.382168\pi$
$440$	0	0
$441$	149779.	0.0366736
$442$	−111787.	−0.0272168
$443$	1.59752e6	0.386756	0.193378	−	0.981124i	$-0.438056\pi$
$443$	1.59752e6	0.386756	0.193378	+	0.981124i	$0.438056\pi$
$444$	−439052.	−0.105696
$445$	0	0
$446$	−698514.	−0.166279
$447$	−1.34344e7	−3.18016
$448$	−4.70630e6	−1.10786
$449$	3.11073e6	0.728193	0.364096	−	0.931361i	$-0.381378\pi$
$449$	3.11073e6	0.728193	0.364096	+	0.931361i	$0.381378\pi$
$450$	0	0
$451$	4.92650e6	1.14051
$452$	376620.	0.0867076
$453$	−47196.3	−0.0108059
$454$	1.86447e6	0.424537
$455$	0	0
$456$	−1.04041e7	−2.34310
$457$	−6.47145e6	−1.44948	−0.724738	−	0.689025i	$-0.758039\pi$
$457$	−6.47145e6	−1.44948	−0.724738	+	0.689025i	$0.758039\pi$
$458$	1.92931e6	0.429771
$459$	−1.31211e6	−0.290695
$460$	0	0
$461$	−5.47864e6	−1.20066	−0.600330	−	0.799752i	$-0.704964\pi$
$461$	−5.47864e6	−1.20066	−0.600330	+	0.799752i	$0.704964\pi$
$462$	5.10617e6	1.11299
$463$	−2.35489e6	−0.510526	−0.255263	−	0.966872i	$-0.582162\pi$
$463$	−2.35489e6	−0.510526	−0.255263	+	0.966872i	$0.582162\pi$
$464$	2.23895e6	0.482781
$465$	0	0
$466$	−5.38776e6	−1.14933
$467$	4.56027e6	0.967606	0.483803	−	0.875177i	$-0.339255\pi$
$467$	4.56027e6	0.967606	0.483803	+	0.875177i	$0.339255\pi$
$468$	120412.	0.0254129
$469$	−4.23819e6	−0.889710
$470$	0	0
$471$	8.76351e6	1.82023
$472$	1.76427e6	0.364510
$473$	5.22702e6	1.07424
$474$	−9.63082e6	−1.96887
$475$	0	0
$476$	175475.	0.0354975
$477$	3.12609e6	0.629079
$478$	6.31274e6	1.26371
$479$	−1.88004e6	−0.374394	−0.187197	−	0.982322i	$-0.559940\pi$
$479$	−1.88004e6	−0.374394	−0.187197	+	0.982322i	$0.559940\pi$
$480$	0	0
$481$	272907.	0.0537839
$482$	−499727.	−0.0979751
$483$	−2.92058e6	−0.569642
$484$	331431.	0.0643103
$485$	0	0
$486$	4.25283e6	0.816747
$487$	−1.69396e6	−0.323654	−0.161827	−	0.986819i	$-0.551739\pi$
$487$	−1.69396e6	−0.323654	−0.161827	+	0.986819i	$0.551739\pi$
$488$	−635528.	−0.120805
$489$	−6.82677e6	−1.29105
$490$	0	0
$491$	1.48645e6	0.278258	0.139129	−	0.990274i	$-0.455570\pi$
$491$	1.48645e6	0.278258	0.139129	+	0.990274i	$0.455570\pi$
$492$	1.86835e6	0.347972
$493$	−801762.	−0.148569
$494$	767700.	0.141538
$495$	0	0
$496$	7.88224e6	1.43862
$497$	4.70279e6	0.854013
$498$	5.32274e6	0.961749
$499$	7.09934e6	1.27634	0.638170	−	0.769896i	$-0.279692\pi$
$499$	7.09934e6	1.27634	0.638170	+	0.769896i	$0.279692\pi$
$500$	0	0
$501$	3.13865e6	0.558661
$502$	2.19983e6	0.389610
$503$	−9.24224e6	−1.62876	−0.814381	−	0.580331i	$-0.802923\pi$
$503$	−9.24224e6	−1.62876	−0.814381	+	0.580331i	$0.802923\pi$
$504$	1.02281e7	1.79357
$505$	0	0
$506$	−1.33290e6	−0.231431
$507$	9.35758e6	1.61675
$508$	850790.	0.146273
$509$	−8.12506e6	−1.39006	−0.695028	−	0.718983i	$-0.744608\pi$
$509$	−8.12506e6	−1.39006	−0.695028	+	0.718983i	$0.744608\pi$
$510$	0	0
$511$	−3.47456e6	−0.588637
$512$	−6.64807e6	−1.12078
$513$	9.01089e6	1.51173
$514$	−1.07655e7	−1.79732
$515$	0	0
$516$	1.98232e6	0.327754
$517$	−7.21335e6	−1.18689
$518$	2.75190e6	0.450617
$519$	7.35148e6	1.19800
$520$	0	0
$521$	5.06245e6	0.817084	0.408542	−	0.912740i	$-0.366037\pi$
$521$	5.06245e6	0.817084	0.408542	+	0.912740i	$0.366037\pi$
$522$	−5.54773e6	−0.891125
$523$	4.76222e6	0.761299	0.380649	−	0.924719i	$-0.375700\pi$
$523$	4.76222e6	0.761299	0.380649	+	0.924719i	$0.375700\pi$
$524$	510744.	0.0812596
$525$	0	0
$526$	−8.63109e6	−1.36019
$527$	−2.82260e6	−0.442714
$528$	6.42345e6	1.00273
$529$	−5.67396e6	−0.881550
$530$	0	0
$531$	−3.77181e6	−0.580515
$532$	−1.20508e6	−0.184601
$533$	−1.16133e6	−0.177067
$534$	1.57736e7	2.39374
$535$	0	0
$536$	−6.17929e6	−0.929023
$537$	−6.28346e6	−0.940292
$538$	−3.79177e6	−0.564789
$539$	106373.	0.0157709
$540$	0	0
$541$	−2.89920e6	−0.425877	−0.212939	−	0.977066i	$-0.568303\pi$
$541$	−2.89920e6	−0.425877	−0.212939	+	0.977066i	$0.568303\pi$
$542$	6.00859e6	0.878566
$543$	−1.10550e7	−1.60901
$544$	481315.	0.0697320
$545$	0	0
$546$	−1.20369e6	−0.172795
$547$	5.74434e6	0.820866	0.410433	−	0.911891i	$-0.365378\pi$
$547$	5.74434e6	0.820866	0.410433	+	0.911891i	$0.365378\pi$
$548$	−1.30421e6	−0.185522
$549$	1.35869e6	0.192393
$550$	0	0
$551$	5.50610e6	0.772618
$552$	−4.25822e6	−0.594812
$553$	−9.39691e6	−1.30669
$554$	−1.98395e6	−0.274636
$555$	0	0
$556$	−677558.	−0.0929522
$557$	7.29174e6	0.995848	0.497924	−	0.867221i	$-0.334096\pi$
$557$	7.29174e6	0.995848	0.497924	+	0.867221i	$0.334096\pi$
$558$	−1.95308e7	−2.65542
$559$	−1.23217e6	−0.166779
$560$	0	0
$561$	−2.30022e6	−0.308576
$562$	−3.24802e6	−0.433788
$563$	−6.65348e6	−0.884663	−0.442331	−	0.896852i	$-0.645849\pi$
$563$	−6.65348e6	−0.884663	−0.442331	+	0.896852i	$0.645849\pi$
$564$	−2.73562e6	−0.362124
$565$	0	0
$566$	6.59398e6	0.865181
$567$	−1.12024e6	−0.146336
$568$	6.85667e6	0.891749
$569$	5.78715e6	0.749349	0.374675	−	0.927156i	$-0.377754\pi$
$569$	5.78715e6	0.749349	0.374675	+	0.927156i	$0.377754\pi$
$570$	0	0
$571$	−1.22059e7	−1.56667	−0.783336	−	0.621599i	$-0.786483\pi$
$571$	−1.22059e7	−1.56667	−0.783336	+	0.621599i	$0.786483\pi$
$572$	85516.1	0.0109284
$573$	1.78927e7	2.27662
$574$	−1.17105e7	−1.48352
$575$	0	0
$576$	1.46697e7	1.84232
$577$	1.02981e7	1.28771	0.643853	−	0.765149i	$-0.277335\pi$
$577$	1.02981e7	1.28771	0.643853	+	0.765149i	$0.277335\pi$
$578$	−6.96362e6	−0.866993
$579$	5.50634e6	0.682601
$580$	0	0
$581$	5.19346e6	0.638288
$582$	2.93848e6	0.359597
$583$	2.22014e6	0.270526
$584$	−5.06591e6	−0.614647
$585$	0	0
$586$	−4.30950e6	−0.518422
$587$	−1.30519e7	−1.56343	−0.781715	−	0.623636i	$-0.785655\pi$
$587$	−1.30519e7	−1.56343	−0.781715	+	0.623636i	$0.785655\pi$
$588$	40341.1	0.00481177
$589$	1.93842e7	2.30229
$590$	0	0
$591$	−1.78793e7	−2.10563
$592$	3.46183e6	0.405977
$593$	6.43920e6	0.751961	0.375980	−	0.926628i	$-0.377306\pi$
$593$	6.43920e6	0.751961	0.375980	+	0.926628i	$0.377306\pi$
$594$	−6.44791e6	−0.749812
$595$	0	0
$596$	−2.26876e6	−0.261621
$597$	−564020.	−0.0647677
$598$	314208.	0.0359305
$599$	−1.00760e7	−1.14741	−0.573707	−	0.819061i	$-0.694495\pi$
$599$	−1.00760e7	−1.14741	−0.573707	+	0.819061i	$0.694495\pi$
$600$	0	0
$601$	1.57050e6	0.177358	0.0886791	−	0.996060i	$-0.471735\pi$
$601$	1.57050e6	0.177358	0.0886791	+	0.996060i	$0.471735\pi$
$602$	−1.24248e7	−1.39733
$603$	1.32106e7	1.47955
$604$	−7970.35	−0.000888966 0
$605$	0	0
$606$	1.00830e7	1.11534
$607$	−7.31039e6	−0.805321	−0.402660	−	0.915349i	$-0.631914\pi$
$607$	−7.31039e6	−0.805321	−0.402660	+	0.915349i	$0.631914\pi$
$608$	−3.30543e6	−0.362634
$609$	−8.63308e6	−0.943241
$610$	0	0
$611$	1.70041e6	0.184269
$612$	−546964.	−0.0590310
$613$	1.31997e7	1.41878	0.709389	−	0.704817i	$-0.248971\pi$
$613$	1.31997e7	1.41878	0.709389	+	0.704817i	$0.248971\pi$
$614$	716825.	0.0767348
$615$	0	0
$616$	7.26399e6	0.771300
$617$	1.02423e7	1.08314	0.541570	−	0.840655i	$-0.317830\pi$
$617$	1.02423e7	1.08314	0.541570	+	0.840655i	$0.317830\pi$
$618$	6.43732e6	0.678006
$619$	1.05614e7	1.10788	0.553942	−	0.832555i	$-0.313123\pi$
$619$	1.05614e7	1.10788	0.553942	+	0.832555i	$0.313123\pi$
$620$	0	0
$621$	3.68802e6	0.383764
$622$	1.38333e7	1.43367
$623$	1.53905e7	1.58866
$624$	−1.51421e6	−0.155677
$625$	0	0
$626$	1.14798e6	0.117084
$627$	1.57967e7	1.60472
$628$	1.47995e6	0.149744
$629$	−1.23967e6	−0.124933
$630$	0	0
$631$	1.90535e7	1.90503	0.952513	−	0.304497i	$-0.0984883\pi$
$631$	1.90535e7	1.90503	0.952513	+	0.304497i	$0.0984883\pi$
$632$	−1.37007e7	−1.36443
$633$	−2.32470e7	−2.30599
$634$	6.62315e6	0.654397
$635$	0	0
$636$	841976.	0.0825385
$637$	−25075.4	−0.00244849
$638$	−3.93999e6	−0.383216
$639$	−1.46588e7	−1.42019
$640$	0	0
$641$	8.56937e6	0.823766	0.411883	−	0.911237i	$-0.364871\pi$
$641$	8.56937e6	0.823766	0.411883	+	0.911237i	$0.364871\pi$
$642$	1.23582e7	1.18336
$643$	−1.79513e7	−1.71226	−0.856130	−	0.516761i	$-0.827138\pi$
$643$	−1.79513e7	−1.71226	−0.856130	+	0.516761i	$0.827138\pi$
$644$	−493218.	−0.0468624
$645$	0	0
$646$	−3.48724e6	−0.328777
$647$	−1.05470e7	−0.990534	−0.495267	−	0.868741i	$-0.664930\pi$
$647$	−1.05470e7	−0.990534	−0.495267	+	0.868741i	$0.664930\pi$
$648$	−1.63331e6	−0.152803
$649$	−2.67873e6	−0.249642
$650$	0	0
$651$	−3.03928e7	−2.81072
$652$	−1.15288e6	−0.106210
$653$	−1.00324e7	−0.920712	−0.460356	−	0.887734i	$-0.652278\pi$
$653$	−1.00324e7	−0.920712	−0.460356	+	0.887734i	$0.652278\pi$
$654$	1.43008e6	0.130742
$655$	0	0
$656$	−1.47315e7	−1.33656
$657$	1.08304e7	0.978879
$658$	1.71464e7	1.54386
$659$	−8.99161e6	−0.806536	−0.403268	−	0.915082i	$-0.632126\pi$
$659$	−8.99161e6	−0.806536	−0.403268	+	0.915082i	$0.632126\pi$
$660$	0	0
$661$	2.39297e6	0.213027	0.106513	−	0.994311i	$-0.466031\pi$
$661$	2.39297e6	0.213027	0.106513	+	0.994311i	$0.466031\pi$
$662$	−1.69367e7	−1.50205
$663$	542234.	0.0479074
$664$	7.57208e6	0.666492
$665$	0	0
$666$	−8.57779e6	−0.749359
$667$	2.25356e6	0.196135
$668$	530044.	0.0459590
$669$	3.38820e6	0.292687
$670$	0	0
$671$	964938.	0.0827357
$672$	5.18262e6	0.442717
$673$	−1.53612e7	−1.30733	−0.653666	−	0.756783i	$-0.726770\pi$
$673$	−1.53612e7	−1.30733	−0.653666	+	0.756783i	$0.726770\pi$
$674$	8.60742e6	0.729833
$675$	0	0
$676$	1.58028e6	0.133004
$677$	−1.16026e7	−0.972934	−0.486467	−	0.873699i	$-0.661715\pi$
$677$	−1.16026e7	−0.972934	−0.486467	+	0.873699i	$0.661715\pi$
$678$	1.17352e7	0.980432
$679$	2.86711e6	0.238655
$680$	0	0
$681$	−9.04375e6	−0.747275
$682$	−1.38707e7	−1.14193
$683$	1.20315e7	0.986890	0.493445	−	0.869777i	$-0.335737\pi$
$683$	1.20315e7	0.986890	0.493445	+	0.869777i	$0.335737\pi$
$684$	3.75627e6	0.306985
$685$	0	0
$686$	1.13371e7	0.919792
$687$	−9.35825e6	−0.756490
$688$	−1.56301e7	−1.25890
$689$	−523358.	−0.0420001
$690$	0	0
$691$	5.18616e6	0.413191	0.206595	−	0.978426i	$-0.433762\pi$
$691$	5.18616e6	0.413191	0.206595	+	0.978426i	$0.433762\pi$
$692$	1.24149e6	0.0985550
$693$	−1.55296e7	−1.22836
$694$	−9.67248e6	−0.762323
$695$	0	0
$696$	−1.25870e7	−0.984919
$697$	5.27530e6	0.411306
$698$	−1.33555e7	−1.03758
$699$	2.61338e7	2.02306
$700$	0	0
$701$	6.00859e6	0.461825	0.230913	−	0.972974i	$-0.425829\pi$
$701$	6.00859e6	0.461825	0.230913	+	0.972974i	$0.425829\pi$
$702$	1.51998e6	0.116411
$703$	8.51342e6	0.649704
$704$	1.04184e7	0.792265
$705$	0	0
$706$	−9.91752e6	−0.748844
$707$	9.83807e6	0.740221
$708$	−1.01589e6	−0.0761667
$709$	5.90083e6	0.440857	0.220429	−	0.975403i	$-0.429254\pi$
$709$	5.90083e6	0.440857	0.220429	+	0.975403i	$0.429254\pi$
$710$	0	0
$711$	2.92906e7	2.17297
$712$	2.24393e7	1.65886
$713$	7.93365e6	0.584453
$714$	5.46769e6	0.401383
$715$	0	0
$716$	−1.06113e6	−0.0773545
$717$	−3.06204e7	−2.22440
$718$	1.60524e6	0.116206
$719$	−1.36592e7	−0.985382	−0.492691	−	0.870204i	$-0.663987\pi$
$719$	−1.36592e7	−0.985382	−0.492691	+	0.870204i	$0.663987\pi$
$720$	0	0
$721$	6.28097e6	0.449975
$722$	1.09192e7	0.779556
$723$	2.42397e6	0.172457
$724$	−1.86693e6	−0.132368
$725$	0	0
$726$	1.03272e7	0.727178
$727$	1.11594e7	0.783079	0.391539	−	0.920161i	$-0.371943\pi$
$727$	1.11594e7	0.783079	0.391539	+	0.920161i	$0.371943\pi$
$728$	−1.71235e6	−0.119747
$729$	−2.27059e7	−1.58242
$730$	0	0
$731$	5.59710e6	0.387409
$732$	365947.	0.0252430
$733$	1.52510e7	1.04843	0.524215	−	0.851586i	$-0.324359\pi$
$733$	1.52510e7	1.04843	0.524215	+	0.851586i	$0.324359\pi$
$734$	−3.82993e6	−0.262392
$735$	0	0
$736$	−1.35286e6	−0.0920573
$737$	9.38217e6	0.636260
$738$	3.65020e7	2.46704
$739$	−1.11820e7	−0.753196	−0.376598	−	0.926377i	$-0.622906\pi$
$739$	−1.11820e7	−0.753196	−0.376598	+	0.926377i	$0.622906\pi$
$740$	0	0
$741$	−3.72379e6	−0.249138
$742$	−5.27735e6	−0.351890
$743$	−7.71450e6	−0.512667	−0.256334	−	0.966588i	$-0.582515\pi$
$743$	−7.71450e6	−0.512667	−0.256334	+	0.966588i	$0.582515\pi$
$744$	−4.43127e7	−2.93492
$745$	0	0
$746$	−2.51357e7	−1.65365
$747$	−1.61883e7	−1.06145
$748$	−388453.	−0.0253854
$749$	1.20581e7	0.785367
$750$	0	0
$751$	−2.23973e7	−1.44909	−0.724545	−	0.689228i	$-0.757950\pi$
$751$	−2.23973e7	−1.44909	−0.724545	+	0.689228i	$0.757950\pi$
$752$	2.15698e7	1.39091
$753$	−1.06704e7	−0.685796
$754$	928780.	0.0594955
$755$	0	0
$756$	−2.38594e6	−0.151829
$757$	−2.57267e7	−1.63171	−0.815857	−	0.578254i	$-0.803734\pi$
$757$	−2.57267e7	−1.63171	−0.815857	+	0.578254i	$0.803734\pi$
$758$	−3.69166e6	−0.233372
$759$	6.46535e6	0.407369
$760$	0	0
$761$	−1.48340e7	−0.928533	−0.464267	−	0.885696i	$-0.653682\pi$
$761$	−1.48340e7	−0.928533	−0.464267	+	0.885696i	$0.653682\pi$
$762$	2.65101e7	1.65395
$763$	1.39534e6	0.0867700
$764$	3.02166e6	0.187289
$765$	0	0
$766$	2.11046e7	1.29959
$767$	631463.	0.0387578
$768$	1.06104e7	0.649125
$769$	−5.57112e6	−0.339724	−0.169862	−	0.985468i	$-0.554332\pi$
$769$	−5.57112e6	−0.339724	−0.169862	+	0.985468i	$0.554332\pi$
$770$	0	0
$771$	5.22188e7	3.16367
$772$	929892.	0.0561551
$773$	1.58230e7	0.952447	0.476224	−	0.879324i	$-0.342005\pi$
$773$	1.58230e7	0.952447	0.476224	+	0.879324i	$0.342005\pi$
$774$	3.87287e7	2.32370
$775$	0	0
$776$	4.18026e6	0.249200
$777$	−1.33483e7	−0.793183
$778$	−2.34664e7	−1.38994
$779$	−3.62281e7	−2.13896
$780$	0	0
$781$	−1.04107e7	−0.610732
$782$	−1.42727e6	−0.0834623
$783$	1.09016e7	0.635454
$784$	−318081.	−0.0184819
$785$	0	0
$786$	1.59144e7	0.918830
$787$	1.31529e7	0.756981	0.378491	−	0.925605i	$-0.376443\pi$
$787$	1.31529e7	0.756981	0.378491	+	0.925605i	$0.376443\pi$
$788$	−3.01939e6	−0.173222
$789$	4.18658e7	2.39423
$790$	0	0
$791$	1.14502e7	0.650687
$792$	−2.26422e7	−1.28264
$793$	−227466.	−0.0128450
$794$	1.77328e7	0.998222
$795$	0	0
$796$	−95249.7	−0.00532821
$797$	2.58443e7	1.44118	0.720590	−	0.693361i	$-0.243871\pi$
$797$	2.58443e7	1.44118	0.720590	+	0.693361i	$0.243871\pi$
$798$	−3.75494e7	−2.08735
$799$	−7.72406e6	−0.428034
$800$	0	0
$801$	−4.79728e7	−2.64188
$802$	−1.58220e7	−0.868611
$803$	7.69170e6	0.420953
$804$	3.55813e6	0.194125
$805$	0	0
$806$	3.26977e6	0.177288
$807$	1.83923e7	0.994149
$808$	1.43439e7	0.772929
$809$	1.78857e7	0.960804	0.480402	−	0.877048i	$-0.340491\pi$
$809$	1.78857e7	0.960804	0.480402	+	0.877048i	$0.340491\pi$
$810$	0	0
$811$	−1.41608e7	−0.756026	−0.378013	−	0.925800i	$-0.623393\pi$
$811$	−1.41608e7	−0.756026	−0.378013	+	0.925800i	$0.623393\pi$
$812$	−1.45793e6	−0.0775970
$813$	−2.91451e7	−1.54646
$814$	−6.09193e6	−0.322251
$815$	0	0
$816$	6.87824e6	0.361619
$817$	−3.84380e7	−2.01468
$818$	−5.25162e6	−0.274417
$819$	3.66082e6	0.190708
$820$	0	0
$821$	−3.46248e7	−1.79279	−0.896394	−	0.443258i	$-0.853823\pi$
$821$	−3.46248e7	−1.79279	−0.896394	+	0.443258i	$0.853823\pi$
$822$	−4.06382e7	−2.09776
$823$	−2.13360e7	−1.09803	−0.549015	−	0.835813i	$-0.684997\pi$
$823$	−2.13360e7	−1.09803	−0.549015	+	0.835813i	$0.684997\pi$
$824$	9.15766e6	0.469858
$825$	0	0
$826$	6.36744e6	0.324724
$827$	1.59813e6	0.0812548	0.0406274	−	0.999174i	$-0.487064\pi$
$827$	1.59813e6	0.0812548	0.0406274	+	0.999174i	$0.487064\pi$
$828$	1.53738e6	0.0779303
$829$	−2.53923e7	−1.28327	−0.641633	−	0.767012i	$-0.721743\pi$
$829$	−2.53923e7	−1.28327	−0.641633	+	0.767012i	$0.721743\pi$
$830$	0	0
$831$	9.62333e6	0.483418
$832$	−2.45595e6	−0.123002
$833$	113904.	0.00568755
$834$	−2.11123e7	−1.05104
$835$	0	0
$836$	2.66770e6	0.132014
$837$	3.83790e7	1.89356
$838$	2.91384e7	1.43336
$839$	1.98528e7	0.973681	0.486841	−	0.873491i	$-0.338149\pi$
$839$	1.98528e7	0.973681	0.486841	+	0.873491i	$0.338149\pi$
$840$	0	0
$841$	−1.38498e7	−0.675231
$842$	1.04523e7	0.508081
$843$	1.57548e7	0.763560
$844$	−3.92587e6	−0.189705
$845$	0	0
$846$	−5.34460e7	−2.56737
$847$	1.00764e7	0.482609
$848$	−6.63879e6	−0.317029
$849$	−3.19846e7	−1.52290
$850$	0	0
$851$	3.48441e6	0.164932
$852$	−3.94818e6	−0.186337
$853$	−1.59794e7	−0.751948	−0.375974	−	0.926630i	$-0.622692\pi$
$853$	−1.59794e7	−0.751948	−0.375974	+	0.926630i	$0.622692\pi$
$854$	−2.29369e6	−0.107619
$855$	0	0
$856$	1.75807e7	0.820070
$857$	7.00157e6	0.325644	0.162822	−	0.986655i	$-0.447940\pi$
$857$	7.00157e6	0.325644	0.162822	+	0.986655i	$0.447940\pi$
$858$	2.66463e6	0.123571
$859$	−7.28414e6	−0.336818	−0.168409	−	0.985717i	$-0.553863\pi$
$859$	−7.28414e6	−0.336818	−0.168409	+	0.985717i	$0.553863\pi$
$860$	0	0
$861$	5.68026e7	2.61132
$862$	613094.	0.0281034
$863$	−1.76361e7	−0.806075	−0.403037	−	0.915184i	$-0.632046\pi$
$863$	−1.76361e7	−0.806075	−0.403037	+	0.915184i	$0.632046\pi$
$864$	−6.54444e6	−0.298255
$865$	0	0
$866$	2.40038e7	1.08764
$867$	3.37776e7	1.52609
$868$	−5.13263e6	−0.231228
$869$	2.08021e7	0.934455
$870$	0	0
$871$	−2.21167e6	−0.0987816
$872$	2.03441e6	0.0906041
$873$	−8.93692e6	−0.396874
$874$	9.80180e6	0.434037
$875$	0	0
$876$	2.91703e6	0.128434
$877$	−2.69004e7	−1.18102	−0.590512	−	0.807029i	$-0.701074\pi$
$877$	−2.69004e7	−1.18102	−0.590512	+	0.807029i	$0.701074\pi$
$878$	1.53743e7	0.673070
$879$	2.09036e7	0.912533
$880$	0	0
$881$	2.51911e7	1.09347	0.546735	−	0.837306i	$-0.315870\pi$
$881$	2.51911e7	1.09347	0.546735	+	0.837306i	$0.315870\pi$
$882$	788148.	0.0341143
$883$	3.22126e7	1.39035	0.695175	−	0.718840i	$-0.255327\pi$
$883$	3.22126e7	1.39035	0.695175	+	0.718840i	$0.255327\pi$
$884$	91570.7	0.00394117
$885$	0	0
$886$	8.40629e6	0.359766
$887$	−8.96139e6	−0.382443	−0.191221	−	0.981547i	$-0.561245\pi$
$887$	−8.96139e6	−0.382443	−0.191221	+	0.981547i	$0.561245\pi$
$888$	−1.94618e7	−0.828231
$889$	2.58662e7	1.09769
$890$	0	0
$891$	2.47989e6	0.104650
$892$	572187.	0.0240783
$893$	5.30449e7	2.22595
$894$	−7.06930e7	−2.95823
$895$	0	0
$896$	−1.82674e7	−0.760164
$897$	−1.52409e6	−0.0632454
$898$	1.63689e7	0.677376
$899$	2.34514e7	0.967765
$900$	0	0
$901$	2.37733e6	0.0975613
$902$	2.59237e7	1.06092
$903$	6.02675e7	2.45960
$904$	1.66944e7	0.679439
$905$	0	0
$906$	−248351.	−0.0100518
$907$	5.81689e6	0.234786	0.117393	−	0.993086i	$-0.462546\pi$
$907$	5.81689e6	0.234786	0.117393	+	0.993086i	$0.462546\pi$
$908$	−1.52728e6	−0.0614757
$909$	−3.06657e7	−1.23096
$910$	0	0
$911$	−1.96435e7	−0.784192	−0.392096	−	0.919924i	$-0.628250\pi$
$911$	−1.96435e7	−0.784192	−0.392096	+	0.919924i	$0.628250\pi$
$912$	−4.72363e7	−1.88057
$913$	−1.14969e7	−0.456460
$914$	−3.40533e7	−1.34832
$915$	0	0
$916$	−1.58039e6	−0.0622337
$917$	1.55279e7	0.609803
$918$	−6.90442e6	−0.270409
$919$	−89962.4	−0.00351376	−0.00175688	−	0.999998i	$-0.500559\pi$
$919$	−89962.4	−0.00351376	−0.00175688	+	0.999998i	$0.500559\pi$
$920$	0	0
$921$	−3.47702e6	−0.135070
$922$	−2.88291e7	−1.11687
$923$	2.45412e6	0.0948183
$924$	−4.18272e6	−0.161168
$925$	0	0
$926$	−1.23916e7	−0.474899
$927$	−1.95781e7	−0.748290
$928$	−3.99898e6	−0.152433
$929$	3.65192e7	1.38830	0.694149	−	0.719832i	$-0.255781\pi$
$929$	3.65192e7	1.38830	0.694149	+	0.719832i	$0.255781\pi$
$930$	0	0
$931$	−782234.	−0.0295776
$932$	4.41338e6	0.166430
$933$	−6.70994e7	−2.52357
$934$	2.39966e7	0.900081
$935$	0	0
$936$	5.33748e6	0.199135
$937$	3.58659e7	1.33454	0.667272	−	0.744814i	$-0.267462\pi$
$937$	3.58659e7	1.33454	0.667272	+	0.744814i	$0.267462\pi$
$938$	−2.23017e7	−0.827621
$939$	−5.56838e6	−0.206094
$940$	0	0
$941$	−3.19693e7	−1.17695	−0.588476	−	0.808515i	$-0.700272\pi$
$941$	−3.19693e7	−1.17695	−0.588476	+	0.808515i	$0.700272\pi$
$942$	4.61143e7	1.69320
$943$	−1.48276e7	−0.542990
$944$	8.01010e6	0.292555
$945$	0	0
$946$	2.75050e7	0.999273
$947$	−4.71846e7	−1.70972	−0.854861	−	0.518858i	$-0.826357\pi$
$947$	−4.71846e7	−1.70972	−0.854861	+	0.518858i	$0.826357\pi$
$948$	7.88908e6	0.285106
$949$	−1.81318e6	−0.0653544
$950$	0	0
$951$	−3.21261e7	−1.15188
$952$	7.77829e6	0.278158
$953$	−1.65226e6	−0.0589315	−0.0294657	−	0.999566i	$-0.509381\pi$
$953$	−1.65226e6	−0.0589315	−0.0294657	+	0.999566i	$0.509381\pi$
$954$	1.64497e7	0.585178
$955$	0	0
$956$	−5.17108e6	−0.182994
$957$	1.91112e7	0.674541
$958$	−9.89295e6	−0.348267
$959$	−3.96512e7	−1.39223
$960$	0	0
$961$	5.39316e7	1.88380
$962$	1.43606e6	0.0500306
$963$	−3.75855e7	−1.30603
$964$	409351.	0.0141874
$965$	0	0
$966$	−1.53684e7	−0.529889
$967$	3.23040e7	1.11094	0.555470	−	0.831537i	$-0.312538\pi$
$967$	3.23040e7	1.11094	0.555470	+	0.831537i	$0.312538\pi$
$968$	1.46914e7	0.503934
$969$	1.69151e7	0.578717
$970$	0	0
$971$	1.15927e7	0.394582	0.197291	−	0.980345i	$-0.436786\pi$
$971$	1.15927e7	0.394582	0.197291	+	0.980345i	$0.436786\pi$
$972$	−3.48371e6	−0.118270
$973$	−2.05995e7	−0.697549
$974$	−8.91377e6	−0.301068
$975$	0	0
$976$	−2.88541e6	−0.0969579
$977$	−2.58947e7	−0.867909	−0.433954	−	0.900935i	$-0.642882\pi$
$977$	−2.58947e7	−0.867909	−0.433954	+	0.900935i	$0.642882\pi$
$978$	−3.59231e7	−1.20095
$979$	−3.40702e7	−1.13610
$980$	0	0
$981$	−4.34935e6	−0.144295
$982$	7.82184e6	0.258839
$983$	−3.46040e7	−1.14220	−0.571101	−	0.820880i	$-0.693483\pi$
$983$	−3.46040e7	−1.14220	−0.571101	+	0.820880i	$0.693483\pi$
$984$	8.28182e7	2.72670
$985$	0	0
$986$	−4.21894e6	−0.138201
$987$	−8.31698e7	−2.71752
$988$	−628861.	−0.0204957
$989$	−1.57321e7	−0.511441
$990$	0	0
$991$	−3.71464e7	−1.20152	−0.600762	−	0.799428i	$-0.705136\pi$
$991$	−3.71464e7	−1.20152	−0.600762	+	0.799428i	$0.705136\pi$
$992$	−1.40784e7	−0.454228
$993$	8.21528e7	2.64393
$994$	2.47465e7	0.794415
$995$	0	0
$996$	−4.36012e6	−0.139268
$997$	2.47350e7	0.788086	0.394043	−	0.919092i	$-0.371076\pi$
$997$	2.47350e7	0.788086	0.394043	+	0.919092i	$0.371076\pi$
$998$	3.73573e7	1.18727
$999$	1.68558e7	0.534362

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.a.b.1.2	✓	2
3.2	odd	2		225.6.a.s.1.1		2
4.3	odd	2		400.6.a.w.1.2		2
5.2	odd	4		25.6.b.b.24.3		4
5.3	odd	4		25.6.b.b.24.2		4
5.4	even	2		25.6.a.d.1.1	yes	2
15.2	even	4		225.6.b.i.199.2		4
15.8	even	4		225.6.b.i.199.3		4
15.14	odd	2		225.6.a.l.1.2		2
20.3	even	4		400.6.c.n.49.4		4
20.7	even	4		400.6.c.n.49.1		4
20.19	odd	2		400.6.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.6.a.b.1.2	✓	2	1.1	even	1	trivial
25.6.a.d.1.1	yes	2	5.4	even	2
25.6.b.b.24.2		4	5.3	odd	4
25.6.b.b.24.3		4	5.2	odd	4
225.6.a.l.1.2		2	15.14	odd	2
225.6.a.s.1.1		2	3.2	odd	2
225.6.b.i.199.2		4	15.2	even	4
225.6.b.i.199.3		4	15.8	even	4
400.6.a.o.1.1		2	20.19	odd	2
400.6.a.w.1.2		2	4.3	odd	2
400.6.c.n.49.1		4	20.7	even	4
400.6.c.n.49.4		4	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	25.a (trivial)

\(f(q)\)	\(=\)	\(q+5.26209 q^{2} -25.5242 q^{3} -4.31044 q^{4} -134.310 q^{6} -131.048 q^{7} -191.069 q^{8} +408.483 q^{9} +O(q^{10})\)	\(q+5.26209 q^{2} -25.5242 q^{3} -4.31044 q^{4} -134.310 q^{6} -131.048 q^{7} -191.069 q^{8} +408.483 q^{9} +290.104 q^{11} +110.020 q^{12} -68.3868 q^{13} -689.588 q^{14} -867.486 q^{16} +310.644 q^{17} +2149.48 q^{18} -2133.35 q^{19} +3344.90 q^{21} +1526.55 q^{22} -873.145 q^{23} +4876.87 q^{24} -359.857 q^{26} -4223.83 q^{27} +564.876 q^{28} -2580.97 q^{29} -9086.30 q^{31} +1549.41 q^{32} -7404.67 q^{33} +1634.64 q^{34} -1760.74 q^{36} -3990.64 q^{37} -11225.9 q^{38} +1745.52 q^{39} +16981.8 q^{41} +17601.2 q^{42} +18017.7 q^{43} -1250.48 q^{44} -4594.57 q^{46} -24864.7 q^{47} +22141.9 q^{48} +366.670 q^{49} -7928.93 q^{51} +294.777 q^{52} +7652.91 q^{53} -22226.2 q^{54} +25039.2 q^{56} +54451.9 q^{57} -13581.3 q^{58} -9233.69 q^{59} +3326.17 q^{61} -47812.9 q^{62} -53531.1 q^{63} +35912.7 q^{64} -38964.0 q^{66} +32340.7 q^{67} -1339.01 q^{68} +22286.3 q^{69} -35885.9 q^{71} -78048.4 q^{72} +26513.6 q^{73} -20999.1 q^{74} +9195.65 q^{76} -38017.7 q^{77} +9185.06 q^{78} +71705.7 q^{79} +8548.28 q^{81} +89359.9 q^{82} -39630.1 q^{83} -14418.0 q^{84} +94810.8 q^{86} +65877.1 q^{87} -55429.9 q^{88} -117441. q^{89} +8961.98 q^{91} +3763.64 q^{92} +231920. q^{93} -130840. q^{94} -39547.4 q^{96} -21878.3 q^{97} +1929.45 q^{98} +118503. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{2} - 20 q^{3} + 69 q^{4} - 191 q^{6} - 200 q^{7} - 615 q^{8} + 196 q^{9}+O(q^{10}) \) 2 * q - 5 * q^2 - 20 * q^3 + 69 * q^4 - 191 * q^6 - 200 * q^7 - 615 * q^8 + 196 * q^9	\( 2 q - 5 q^{2} - 20 q^{3} + 69 q^{4} - 191 q^{6} - 200 q^{7} - 615 q^{8} + 196 q^{9} - 196 q^{11} + 515 q^{12} + 360 q^{13} + 18 q^{14} + 1137 q^{16} - 1490 q^{17} + 4330 q^{18} - 3180 q^{19} + 2964 q^{21} + 6515 q^{22} - 1560 q^{23} + 2535 q^{24} - 4756 q^{26} - 6740 q^{27} - 4490 q^{28} - 3920 q^{29} - 1096 q^{31} - 5455 q^{32} - 10090 q^{33} + 20113 q^{34} - 17338 q^{36} - 2020 q^{37} - 485 q^{38} + 4112 q^{39} + 27754 q^{41} + 21510 q^{42} + 3000 q^{43} - 36887 q^{44} + 2454 q^{46} - 25760 q^{47} + 33215 q^{48} - 11686 q^{49} - 17876 q^{51} + 31700 q^{52} + 26980 q^{53} + 3595 q^{54} + 54270 q^{56} + 48670 q^{57} + 160 q^{58} + 11960 q^{59} - 24396 q^{61} - 129810 q^{62} - 38880 q^{63} + 43649 q^{64} - 11407 q^{66} + 40060 q^{67} - 133345 q^{68} + 18492 q^{69} - 87296 q^{71} + 12030 q^{72} + 70290 q^{73} - 41222 q^{74} - 67535 q^{76} - 4500 q^{77} - 15100 q^{78} + 65480 q^{79} + 46282 q^{81} - 21185 q^{82} - 92580 q^{83} - 42342 q^{84} + 248924 q^{86} + 58480 q^{87} + 150645 q^{88} - 72810 q^{89} - 20576 q^{91} - 46590 q^{92} + 276060 q^{93} - 121652 q^{94} - 78241 q^{96} + 126140 q^{97} + 125615 q^{98} + 221792 q^{99}+O(q^{100}) \) 2 * q - 5 * q^2 - 20 * q^3 + 69 * q^4 - 191 * q^6 - 200 * q^7 - 615 * q^8 + 196 * q^9 - 196 * q^11 + 515 * q^12 + 360 * q^13 + 18 * q^14 + 1137 * q^16 - 1490 * q^17 + 4330 * q^18 - 3180 * q^19 + 2964 * q^21 + 6515 * q^22 - 1560 * q^23 + 2535 * q^24 - 4756 * q^26 - 6740 * q^27 - 4490 * q^28 - 3920 * q^29 - 1096 * q^31 - 5455 * q^32 - 10090 * q^33 + 20113 * q^34 - 17338 * q^36 - 2020 * q^37 - 485 * q^38 + 4112 * q^39 + 27754 * q^41 + 21510 * q^42 + 3000 * q^43 - 36887 * q^44 + 2454 * q^46 - 25760 * q^47 + 33215 * q^48 - 11686 * q^49 - 17876 * q^51 + 31700 * q^52 + 26980 * q^53 + 3595 * q^54 + 54270 * q^56 + 48670 * q^57 + 160 * q^58 + 11960 * q^59 - 24396 * q^61 - 129810 * q^62 - 38880 * q^63 + 43649 * q^64 - 11407 * q^66 + 40060 * q^67 - 133345 * q^68 + 18492 * q^69 - 87296 * q^71 + 12030 * q^72 + 70290 * q^73 - 41222 * q^74 - 67535 * q^76 - 4500 * q^77 - 15100 * q^78 + 65480 * q^79 + 46282 * q^81 - 21185 * q^82 - 92580 * q^83 - 42342 * q^84 + 248924 * q^86 + 58480 * q^87 + 150645 * q^88 - 72810 * q^89 - 20576 * q^91 - 46590 * q^92 + 276060 * q^93 - 121652 * q^94 - 78241 * q^96 + 126140 * q^97 + 125615 * q^98 + 221792 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more