LMFDB - Embedded newform 325.6.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("325.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 161x^{4} + 328x^{3} + 6584x^{2} - 10688x - 47440 $ x^6 - 2x^5 - 161x^4 + 328x^3 + 6584x^2 - 10688*x - 47440
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$9.62003$ of defining polynomial
Character	$\chi$	$=$	325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-9.62003 q^{2} -10.7175 q^{3} +60.5450 q^{4} +103.103 q^{6} -18.4289 q^{7} -274.604 q^{8} -128.135 q^{9} -80.7965 q^{11} -648.891 q^{12} +169.000 q^{13} +177.287 q^{14} +704.257 q^{16} -170.210 q^{17} +1232.67 q^{18} +722.993 q^{19} +197.512 q^{21} +777.265 q^{22} +519.385 q^{23} +2943.06 q^{24} -1625.79 q^{26} +3977.64 q^{27} -1115.78 q^{28} +2392.34 q^{29} -6731.65 q^{31} +2012.35 q^{32} +865.936 q^{33} +1637.42 q^{34} -7757.95 q^{36} -3921.11 q^{37} -6955.22 q^{38} -1811.26 q^{39} -6520.89 q^{41} -1900.07 q^{42} -20998.4 q^{43} -4891.83 q^{44} -4996.50 q^{46} -855.573 q^{47} -7547.87 q^{48} -16467.4 q^{49} +1824.22 q^{51} +10232.1 q^{52} -553.736 q^{53} -38265.0 q^{54} +5060.64 q^{56} -7748.68 q^{57} -23014.4 q^{58} -25934.8 q^{59} -697.362 q^{61} +64758.7 q^{62} +2361.39 q^{63} -41895.1 q^{64} -8330.33 q^{66} +42914.4 q^{67} -10305.3 q^{68} -5566.51 q^{69} +29581.6 q^{71} +35186.4 q^{72} -84016.4 q^{73} +37721.2 q^{74} +43773.6 q^{76} +1488.99 q^{77} +17424.3 q^{78} +37065.0 q^{79} -11493.4 q^{81} +62731.2 q^{82} -94551.6 q^{83} +11958.3 q^{84} +202005. q^{86} -25639.9 q^{87} +22187.0 q^{88} +123136. q^{89} -3114.48 q^{91} +31446.2 q^{92} +72146.4 q^{93} +8230.64 q^{94} -21567.3 q^{96} +29967.6 q^{97} +158417. q^{98} +10352.9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} - 20 q^{3} + 134 q^{4} - 52 q^{6} - 172 q^{7} + 138 q^{8} + 1034 q^{9} + 800 q^{11} + 832 q^{12} + 1014 q^{13} + 2108 q^{14} + 322 q^{16} - 4396 q^{17} - 6142 q^{18} + 5304 q^{19} + 1072 q^{21}+ \cdots + 301264 q^{99}+O(q^{100}) $ 6 * q - 2 * q^2 - 20 * q^3 + 134 * q^4 - 52 * q^6 - 172 * q^7 + 138 * q^8 + 1034 * q^9 + 800 * q^11 + 832 * q^12 + 1014 * q^13 + 2108 * q^14 + 322 * q^16 - 4396 * q^17 - 6142 * q^18 + 5304 * q^19 + 1072 * q^21 - 8008 * q^22 + 1140 * q^23 + 4756 * q^24 - 338 * q^26 - 800 * q^27 - 524 * q^28 - 3664 * q^29 + 16664 * q^31 + 9894 * q^32 + 13092 * q^33 + 1380 * q^34 + 1818 * q^36 - 4488 * q^37 + 30112 * q^38 - 3380 * q^39 + 12436 * q^41 + 68336 * q^42 - 1516 * q^43 - 35124 * q^44 - 4556 * q^46 - 212 * q^47 + 16712 * q^48 - 27842 * q^49 - 27664 * q^51 + 22646 * q^52 + 15612 * q^53 - 132912 * q^54 + 44420 * q^56 + 86796 * q^57 + 39664 * q^58 - 11896 * q^59 + 57232 * q^61 + 95488 * q^62 - 207956 * q^63 - 39914 * q^64 + 9908 * q^66 + 25428 * q^67 + 16236 * q^68 + 72404 * q^69 + 214104 * q^71 + 130806 * q^72 - 60808 * q^73 - 182104 * q^74 + 366636 * q^76 + 5192 * q^77 - 8788 * q^78 + 36024 * q^79 + 350514 * q^81 + 1124 * q^82 - 34684 * q^83 + 6024 * q^84 + 29204 * q^86 + 8640 * q^87 + 40520 * q^88 - 52028 * q^89 - 29068 * q^91 + 283232 * q^92 - 26508 * q^93 + 157572 * q^94 + 547908 * q^96 - 129316 * q^97 + 113942 * q^98 + 301264 * q^99

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 $

$2$	−9.62003	−1.70060	−0.850299	−	0.526301i	$-0.823579\pi$
$2$	−9.62003	−1.70060	−0.850299	+	0.526301i	$0.823579\pi$
$3$	−10.7175	−0.687528	−0.343764	−	0.939056i	$-0.611702\pi$
$3$	−10.7175	−0.687528	−0.343764	+	0.939056i	$0.611702\pi$
$4$	60.5450	1.89203
$5$	0	0
$5$	0	0
$6$	103.103	1.16921
$7$	−18.4289	−0.142152	−0.0710762	−	0.997471i	$-0.522643\pi$
$7$	−18.4289	−0.142152	−0.0710762	+	0.997471i	$0.522643\pi$
$8$	−274.604	−1.51699
$9$	−128.135	−0.527306
$10$	0	0
$11$	−80.7965	−0.201331	−0.100666	−	0.994920i	$-0.532097\pi$
$11$	−80.7965	−0.201331	−0.100666	+	0.994920i	$0.532097\pi$
$12$	−648.891	−1.30082
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	177.287	0.241744
$15$	0	0
$16$	704.257	0.687751
$17$	−170.210	−0.142844	−0.0714219	−	0.997446i	$-0.522754\pi$
$17$	−170.210	−0.142844	−0.0714219	+	0.997446i	$0.522754\pi$
$18$	1232.67	0.896735
$19$	722.993	0.459463	0.229731	−	0.973254i	$-0.426215\pi$
$19$	722.993	0.459463	0.229731	+	0.973254i	$0.426215\pi$
$20$	0	0
$21$	197.512	0.0977337
$22$	777.265	0.342383
$23$	519.385	0.204725	0.102362	−	0.994747i	$-0.467360\pi$
$23$	519.385	0.204725	0.102362	+	0.994747i	$0.467360\pi$
$24$	2943.06	1.04297
$25$	0	0
$26$	−1625.79	−0.471661
$27$	3977.64	1.05006
$28$	−1115.78	−0.268957
$29$	2392.34	0.528236	0.264118	−	0.964490i	$-0.414919\pi$
$29$	2392.34	0.528236	0.264118	+	0.964490i	$0.414919\pi$
$30$	0	0
$31$	−6731.65	−1.25811	−0.629053	−	0.777362i	$-0.716557\pi$
$31$	−6731.65	−1.25811	−0.629053	+	0.777362i	$0.716557\pi$
$32$	2012.35	0.347399
$33$	865.936	0.138421
$34$	1637.42	0.242920
$35$	0	0
$36$	−7757.95	−0.997679
$37$	−3921.11	−0.470875	−0.235437	−	0.971890i	$-0.575652\pi$
$37$	−3921.11	−0.470875	−0.235437	+	0.971890i	$0.575652\pi$
$38$	−6955.22	−0.781361
$39$	−1811.26	−0.190686
$40$	0	0
$41$	−6520.89	−0.605825	−0.302913	−	0.953018i	$-0.597959\pi$
$41$	−6520.89	−0.605825	−0.302913	+	0.953018i	$0.597959\pi$
$42$	−1900.07	−0.166206
$43$	−20998.4	−1.73187	−0.865933	−	0.500160i	$-0.833275\pi$
$43$	−20998.4	−1.73187	−0.865933	+	0.500160i	$0.833275\pi$
$44$	−4891.83	−0.380925
$45$	0	0
$46$	−4996.50	−0.348154
$47$	−855.573	−0.0564953	−0.0282477	−	0.999601i	$-0.508993\pi$
$47$	−855.573	−0.0564953	−0.0282477	+	0.999601i	$0.508993\pi$
$48$	−7547.87	−0.472848
$49$	−16467.4	−0.979793
$50$	0	0
$51$	1824.22	0.0982091
$52$	10232.1	0.524755
$53$	−553.736	−0.0270778	−0.0135389	−	0.999908i	$-0.504310\pi$
$53$	−553.736	−0.0270778	−0.0135389	+	0.999908i	$0.504310\pi$
$54$	−38265.0	−1.78574
$55$	0	0
$56$	5060.64	0.215643
$57$	−7748.68	−0.315893
$58$	−23014.4	−0.898317
$59$	−25934.8	−0.969957	−0.484978	−	0.874526i	$-0.661173\pi$
$59$	−25934.8	−0.969957	−0.484978	+	0.874526i	$0.661173\pi$
$60$	0	0
$61$	−697.362	−0.0239957	−0.0119979	−	0.999928i	$-0.503819\pi$
$61$	−697.362	−0.0239957	−0.0119979	+	0.999928i	$0.503819\pi$
$62$	64758.7	2.13953
$63$	2361.39	0.0749578
$64$	−41895.1	−1.27854
$65$	0	0
$66$	−8330.33	−0.235398
$67$	42914.4	1.16793	0.583963	−	0.811780i	$-0.301501\pi$
$67$	42914.4	1.16793	0.583963	+	0.811780i	$0.301501\pi$
$68$	−10305.3	−0.270265
$69$	−5566.51	−0.140754
$70$	0	0
$71$	29581.6	0.696427	0.348214	−	0.937415i	$-0.386788\pi$
$71$	29581.6	0.696427	0.348214	+	0.937415i	$0.386788\pi$
$72$	35186.4	0.799916
$73$	−84016.4	−1.84526	−0.922629	−	0.385690i	$-0.873964\pi$
$73$	−84016.4	−1.84526	−0.922629	+	0.385690i	$0.873964\pi$
$74$	37721.2	0.800768
$75$	0	0
$76$	43773.6	0.869318
$77$	1488.99	0.0286197
$78$	17424.3	0.324280
$79$	37065.0	0.668184	0.334092	−	0.942540i	$-0.391570\pi$
$79$	37065.0	0.668184	0.334092	+	0.942540i	$0.391570\pi$
$80$	0	0
$81$	−11493.4	−0.194643
$82$	62731.2	1.03027
$83$	−94551.6	−1.50652	−0.753258	−	0.657725i	$-0.771519\pi$
$83$	−94551.6	−1.50652	−0.753258	+	0.657725i	$0.771519\pi$
$84$	11958.3	0.184915
$85$	0	0
$86$	202005.	2.94521
$87$	−25639.9	−0.363177
$88$	22187.0	0.305416
$89$	123136.	1.64782	0.823910	−	0.566720i	$-0.191788\pi$
$89$	123136.	1.64782	0.823910	+	0.566720i	$0.191788\pi$
$90$	0	0
$91$	−3114.48	−0.0394260
$92$	31446.2	0.387345
$93$	72146.4	0.864983
$94$	8230.64	0.0960758
$95$	0	0
$96$	−21567.3	−0.238846
$97$	29967.6	0.323387	0.161694	−	0.986841i	$-0.448304\pi$
$97$	29967.6	0.323387	0.161694	+	0.986841i	$0.448304\pi$
$98$	158417.	1.66623
$99$	10352.9	0.106163
$100$	0	0
$101$	151193.	1.47478	0.737390	−	0.675468i	$-0.236058\pi$
$101$	151193.	1.47478	0.737390	+	0.675468i	$0.236058\pi$
$102$	−17549.0	−0.167014
$103$	11776.4	0.109375	0.0546876	−	0.998504i	$-0.482584\pi$
$103$	11776.4	0.109375	0.0546876	+	0.998504i	$0.482584\pi$
$104$	−46408.0	−0.420736
$105$	0	0
$106$	5326.95	0.0460484
$107$	−146823.	−1.23975	−0.619876	−	0.784700i	$-0.712817\pi$
$107$	−146823.	−1.23975	−0.619876	+	0.784700i	$0.712817\pi$
$108$	240826.	1.98676
$109$	39831.5	0.321115	0.160558	−	0.987026i	$-0.448671\pi$
$109$	39831.5	0.321115	0.160558	+	0.987026i	$0.448671\pi$
$110$	0	0
$111$	42024.5	0.323739
$112$	−12978.7	−0.0977654
$113$	−11885.0	−0.0875594	−0.0437797	−	0.999041i	$-0.513940\pi$
$113$	−11885.0	−0.0875594	−0.0437797	+	0.999041i	$0.513940\pi$
$114$	74542.5	0.537207
$115$	0	0
$116$	144844.	0.999440
$117$	−21654.9	−0.146248
$118$	249493.	1.64951
$119$	3136.77	0.0203056
$120$	0	0
$121$	−154523.	−0.959466
$122$	6708.65	0.0408071
$123$	69887.6	0.416522
$124$	−407568.	−2.38038
$125$	0	0
$126$	−22716.7	−0.127473
$127$	245216.	1.34908	0.674541	−	0.738237i	$-0.264341\pi$
$127$	245216.	1.34908	0.674541	+	0.738237i	$0.264341\pi$
$128$	338637.	1.82688
$129$	225050.	1.19071
$130$	0	0
$131$	−43517.0	−0.221555	−0.110777	−	0.993845i	$-0.535334\pi$
$131$	−43517.0	−0.221555	−0.110777	+	0.993845i	$0.535334\pi$
$132$	52428.1	0.261896
$133$	−13324.0	−0.0653137
$134$	−412837.	−1.98617
$135$	0	0
$136$	46740.2	0.216692
$137$	323142.	1.47093	0.735466	−	0.677562i	$-0.236964\pi$
$137$	323142.	1.47093	0.735466	+	0.677562i	$0.236964\pi$
$138$	53550.0	0.239366
$139$	244416.	1.07298	0.536490	−	0.843907i	$-0.319750\pi$
$139$	244416.	1.07298	0.536490	+	0.843907i	$0.319750\pi$
$140$	0	0
$141$	9169.60	0.0388421
$142$	−284576.	−1.18434
$143$	−13654.6	−0.0558392
$144$	−90240.2	−0.362655
$145$	0	0
$146$	808240.	3.13804
$147$	176489.	0.673634
$148$	−237404.	−0.890909
$149$	−172702.	−0.637282	−0.318641	−	0.947875i	$-0.603226\pi$
$149$	−172702.	−0.637282	−0.318641	+	0.947875i	$0.603226\pi$
$150$	0	0
$151$	256350.	0.914935	0.457468	−	0.889226i	$-0.348757\pi$
$151$	256350.	0.914935	0.457468	+	0.889226i	$0.348757\pi$
$152$	−198537.	−0.696998
$153$	21809.9	0.0753224
$154$	−14324.1	−0.0486706
$155$	0	0
$156$	−109663.	−0.360784
$157$	−333694.	−1.08043	−0.540217	−	0.841525i	$-0.681658\pi$
$157$	−333694.	−1.08043	−0.540217	+	0.841525i	$0.681658\pi$
$158$	−356566.	−1.13631
$159$	5934.66	0.0186167
$160$	0	0
$161$	−9571.70	−0.0291021
$162$	110567.	0.331009
$163$	−95114.8	−0.280401	−0.140200	−	0.990123i	$-0.544775\pi$
$163$	−95114.8	−0.280401	−0.140200	+	0.990123i	$0.544775\pi$
$164$	−394807.	−1.14624
$165$	0	0
$166$	909590.	2.56198
$167$	488741.	1.35609	0.678043	−	0.735022i	$-0.262828\pi$
$167$	488741.	1.35609	0.678043	+	0.735022i	$0.262828\pi$
$168$	−54237.4	−0.148261
$169$	28561.0	0.0769231
$170$	0	0
$171$	−92641.0	−0.242277
$172$	−1.27135e6	−3.27675
$173$	219573.	0.557780	0.278890	−	0.960323i	$-0.410033\pi$
$173$	219573.	0.557780	0.278890	+	0.960323i	$0.410033\pi$
$174$	246657.	0.617618
$175$	0	0
$176$	−56901.5	−0.138466
$177$	277956.	0.666872
$178$	−1.18457e6	−2.80228
$179$	217092.	0.506421	0.253210	−	0.967411i	$-0.418514\pi$
$179$	217092.	0.506421	0.253210	+	0.967411i	$0.418514\pi$
$180$	0	0
$181$	−647747.	−1.46963	−0.734816	−	0.678266i	$-0.762732\pi$
$181$	−647747.	−1.46963	−0.734816	+	0.678266i	$0.762732\pi$
$182$	29961.4	0.0670477
$183$	7473.98	0.0164977
$184$	−142625.	−0.310564
$185$	0	0
$186$	−694051.	−1.47099
$187$	13752.3	0.0287589
$188$	−51800.7	−0.106891
$189$	−73303.5	−0.149269
$190$	0	0
$191$	−675866.	−1.34053	−0.670266	−	0.742121i	$-0.733820\pi$
$191$	−675866.	−1.34053	−0.670266	+	0.742121i	$0.733820\pi$
$192$	449010.	0.879029
$193$	69360.2	0.134035	0.0670174	−	0.997752i	$-0.478652\pi$
$193$	69360.2	0.134035	0.0670174	+	0.997752i	$0.478652\pi$
$194$	−288289.	−0.549952
$195$	0	0
$196$	−997017.	−1.85380
$197$	1.07612e6	1.97559	0.987794	−	0.155766i	$-0.0497846\pi$
$197$	1.07612e6	1.97559	0.987794	+	0.155766i	$0.0497846\pi$
$198$	−99595.1	−0.180541
$199$	−290642.	−0.520266	−0.260133	−	0.965573i	$-0.583766\pi$
$199$	−290642.	−0.520266	−0.260133	+	0.965573i	$0.583766\pi$
$200$	0	0
$201$	−459934.	−0.802982
$202$	−1.45448e6	−2.50801
$203$	−44088.2	−0.0750901
$204$	110447.	0.185815
$205$	0	0
$206$	−113289.	−0.186003
$207$	−66551.6	−0.107953
$208$	119019.	0.190748
$209$	−58415.3	−0.0925041
$210$	0	0
$211$	1435.58	0.00221983	0.00110992	−	0.999999i	$-0.499647\pi$
$211$	1435.58	0.00221983	0.00110992	+	0.999999i	$0.499647\pi$
$212$	−33525.9	−0.0512320
$213$	−317040.	−0.478813
$214$	1.41244e6	2.10832
$215$	0	0
$216$	−1.09227e6	−1.59293
$217$	124057.	0.178843
$218$	−383181.	−0.546087
$219$	900445.	1.26867
$220$	0	0
$221$	−28765.4	−0.0396178
$222$	−404277.	−0.550550
$223$	−417548.	−0.562270	−0.281135	−	0.959668i	$-0.590711\pi$
$223$	−417548.	−0.562270	−0.281135	+	0.959668i	$0.590711\pi$
$224$	−37085.4	−0.0493836
$225$	0	0
$226$	114334.	0.148903
$227$	614640.	0.791691	0.395846	−	0.918317i	$-0.370452\pi$
$227$	614640.	0.791691	0.395846	+	0.918317i	$0.370452\pi$
$228$	−469144.	−0.597680
$229$	1.09898e6	1.38484	0.692421	−	0.721494i	$-0.256544\pi$
$229$	1.09898e6	1.38484	0.692421	+	0.721494i	$0.256544\pi$
$230$	0	0
$231$	−15958.2	−0.0196768
$232$	−656946.	−0.801327
$233$	790169.	0.953521	0.476760	−	0.879033i	$-0.341811\pi$
$233$	790169.	0.953521	0.476760	+	0.879033i	$0.341811\pi$
$234$	208321.	0.248710
$235$	0	0
$236$	−1.57022e6	−1.83519
$237$	−397244.	−0.459395
$238$	−30175.9	−0.0345316
$239$	1.62836e6	1.84397	0.921987	−	0.387221i	$-0.126565\pi$
$239$	1.62836e6	1.84397	0.921987	+	0.387221i	$0.126565\pi$
$240$	0	0
$241$	−208904.	−0.231689	−0.115844	−	0.993267i	$-0.536957\pi$
$241$	−208904.	−0.231689	−0.115844	+	0.993267i	$0.536957\pi$
$242$	1.48652e6	1.63166
$243$	−843386.	−0.916243
$244$	−42221.8	−0.0454007
$245$	0	0
$246$	−672321.	−0.708336
$247$	122186.	0.127432
$248$	1.84854e6	1.90853
$249$	1.01336e6	1.03577
$250$	0	0
$251$	1.51514e6	1.51799	0.758993	−	0.651099i	$-0.225692\pi$
$251$	1.51514e6	1.51799	0.758993	+	0.651099i	$0.225692\pi$
$252$	142971.	0.141823
$253$	−41964.5	−0.0412175
$254$	−2.35898e6	−2.29425
$255$	0	0
$256$	−1.91705e6	−1.82824
$257$	1.80241e6	1.70224	0.851120	−	0.524972i	$-0.175924\pi$
$257$	1.80241e6	1.70224	0.851120	+	0.524972i	$0.175924\pi$
$258$	−2.16499e6	−2.02491
$259$	72261.8	0.0669359
$260$	0	0
$261$	−306544.	−0.278542
$262$	418635.	0.376775
$263$	796589.	0.710141	0.355071	−	0.934839i	$-0.384457\pi$
$263$	796589.	0.710141	0.355071	+	0.934839i	$0.384457\pi$
$264$	−237789.	−0.209982
$265$	0	0
$266$	128177.	0.111072
$267$	−1.31971e6	−1.13292
$268$	2.59825e6	2.20975
$269$	−1.58930e6	−1.33913	−0.669567	−	0.742751i	$-0.733520\pi$
$269$	−1.58930e6	−1.33913	−0.669567	+	0.742751i	$0.733520\pi$
$270$	0	0
$271$	1.77761e6	1.47033	0.735163	−	0.677890i	$-0.237106\pi$
$271$	1.77761e6	1.47033	0.735163	+	0.677890i	$0.237106\pi$
$272$	−119871.	−0.0982410
$273$	33379.5	0.0271064
$274$	−3.10864e6	−2.50146
$275$	0	0
$276$	−337024.	−0.266311
$277$	−1.58991e6	−1.24501	−0.622506	−	0.782615i	$-0.713885\pi$
$277$	−1.58991e6	−1.24501	−0.622506	+	0.782615i	$0.713885\pi$
$278$	−2.35129e6	−1.82471
$279$	862562.	0.663407
$280$	0	0
$281$	−542033.	−0.409505	−0.204753	−	0.978814i	$-0.565639\pi$
$281$	−542033.	−0.409505	−0.204753	+	0.978814i	$0.565639\pi$
$282$	−88211.8	−0.0660547
$283$	−230145.	−0.170818	−0.0854092	−	0.996346i	$-0.527220\pi$
$283$	−230145.	−0.170818	−0.0854092	+	0.996346i	$0.527220\pi$
$284$	1.79102e6	1.31766
$285$	0	0
$286$	131358.	0.0949600
$287$	120173.	0.0861195
$288$	−257853.	−0.183185
$289$	−1.39089e6	−0.979596
$290$	0	0
$291$	−321178.	−0.222338
$292$	−5.08677e6	−3.49128
$293$	355633.	0.242010	0.121005	−	0.992652i	$-0.461388\pi$
$293$	355633.	0.242010	0.121005	+	0.992652i	$0.461388\pi$
$294$	−1.69783e6	−1.14558
$295$	0	0
$296$	1.07675e6	0.714310
$297$	−321380.	−0.211411
$298$	1.66140e6	1.08376
$299$	87776.1	0.0567804
$300$	0	0
$301$	386977.	0.246189
$302$	−2.46609e6	−1.55594
$303$	−1.62041e6	−1.01395
$304$	509173.	0.315996
$305$	0	0
$306$	−209811.	−0.128093
$307$	−923225.	−0.559064	−0.279532	−	0.960136i	$-0.590179\pi$
$307$	−923225.	−0.559064	−0.279532	+	0.960136i	$0.590179\pi$
$308$	90150.9	0.0541494
$309$	−126213.	−0.0751984
$310$	0	0
$311$	449003.	0.263238	0.131619	−	0.991300i	$-0.457983\pi$
$311$	449003.	0.263238	0.131619	+	0.991300i	$0.457983\pi$
$312$	497378.	0.289268
$313$	2.93310e6	1.69225	0.846127	−	0.532981i	$-0.178928\pi$
$313$	2.93310e6	1.69225	0.846127	+	0.532981i	$0.178928\pi$
$314$	3.21014e6	1.83738
$315$	0	0
$316$	2.24410e6	1.26423
$317$	1.28699e6	0.719327	0.359663	−	0.933082i	$-0.382892\pi$
$317$	1.28699e6	0.719327	0.359663	+	0.933082i	$0.382892\pi$
$318$	−57091.6	−0.0316595
$319$	−193293.	−0.106350
$320$	0	0
$321$	1.57357e6	0.852363
$322$	92080.0	0.0494910
$323$	−123060.	−0.0656314
$324$	−695871.	−0.368270
$325$	0	0
$326$	915007.	0.476849
$327$	−426894.	−0.220775
$328$	1.79066e6	0.919028
$329$	15767.3	0.00803094
$330$	0	0
$331$	−69938.9	−0.0350872	−0.0175436	−	0.999846i	$-0.505585\pi$
$331$	−69938.9	−0.0350872	−0.0175436	+	0.999846i	$0.505585\pi$
$332$	−5.72463e6	−2.85038
$333$	502433.	0.248295
$334$	−4.70170e6	−2.30616
$335$	0	0
$336$	139099.	0.0672164
$337$	1.63283e6	0.783187	0.391593	−	0.920138i	$-0.371924\pi$
$337$	1.63283e6	0.783187	0.391593	+	0.920138i	$0.371924\pi$
$338$	−274758.	−0.130815
$339$	127377.	0.0601995
$340$	0	0
$341$	543894.	0.253296
$342$	891209.	0.412016
$343$	613210.	0.281432
$344$	5.76623e6	2.62722
$345$	0	0
$346$	−2.11230e6	−0.948560
$347$	1.44134e6	0.642605	0.321302	−	0.946977i	$-0.395879\pi$
$347$	1.44134e6	0.642605	0.321302	+	0.946977i	$0.395879\pi$
$348$	−1.55237e6	−0.687142
$349$	2.89043e6	1.27028	0.635140	−	0.772397i	$-0.280942\pi$
$349$	2.89043e6	1.27028	0.635140	+	0.772397i	$0.280942\pi$
$350$	0	0
$351$	672221.	0.291236
$352$	−162591.	−0.0699422
$353$	−803981.	−0.343407	−0.171704	−	0.985149i	$-0.554927\pi$
$353$	−803981.	−0.343407	−0.171704	+	0.985149i	$0.554927\pi$
$354$	−2.67394e6	−1.13408
$355$	0	0
$356$	7.45527e6	3.11773
$357$	−33618.3	−0.0139607
$358$	−2.08843e6	−0.861218
$359$	1.93534e6	0.792541	0.396270	−	0.918134i	$-0.370304\pi$
$359$	1.93534e6	0.792541	0.396270	+	0.918134i	$0.370304\pi$
$360$	0	0
$361$	−1.95338e6	−0.788894
$362$	6.23134e6	2.49925
$363$	1.65610e6	0.659659
$364$	−188566.	−0.0745952
$365$	0	0
$366$	−71899.9	−0.0280560
$367$	−3.46002e6	−1.34095	−0.670476	−	0.741931i	$-0.733910\pi$
$367$	−3.46002e6	−1.34095	−0.670476	+	0.741931i	$0.733910\pi$
$368$	365781.	0.140800
$369$	835557.	0.319455
$370$	0	0
$371$	10204.7	0.00384917
$372$	4.36810e6	1.63657
$373$	3.83742e6	1.42813	0.714064	−	0.700081i	$-0.246853\pi$
$373$	3.83742e6	1.42813	0.714064	+	0.700081i	$0.246853\pi$
$374$	−132298.	−0.0489073
$375$	0	0
$376$	234944.	0.0857026
$377$	404306.	0.146506
$378$	705182.	0.253847
$379$	4.11044e6	1.46991	0.734955	−	0.678116i	$-0.237203\pi$
$379$	4.11044e6	1.46991	0.734955	+	0.678116i	$0.237203\pi$
$380$	0	0
$381$	−2.62810e6	−0.927532
$382$	6.50185e6	2.27971
$383$	3.66515e6	1.27672	0.638358	−	0.769740i	$-0.279614\pi$
$383$	3.66515e6	1.27672	0.638358	+	0.769740i	$0.279614\pi$
$384$	−3.62934e6	−1.25603
$385$	0	0
$386$	−667248.	−0.227939
$387$	2.69063e6	0.913223
$388$	1.81439e6	0.611859
$389$	−2.07210e6	−0.694282	−0.347141	−	0.937813i	$-0.612847\pi$
$389$	−2.07210e6	−0.694282	−0.347141	+	0.937813i	$0.612847\pi$
$390$	0	0
$391$	−88404.4	−0.0292437
$392$	4.52200e6	1.48633
$393$	466393.	0.152325
$394$	−1.03523e7	−3.35968
$395$	0	0
$396$	626816.	0.200864
$397$	−5.27270e6	−1.67902	−0.839512	−	0.543341i	$-0.817159\pi$
$397$	−5.27270e6	−1.67902	−0.839512	+	0.543341i	$0.817159\pi$
$398$	2.79598e6	0.884763
$399$	142800.	0.0449050
$400$	0	0
$401$	−1.49206e6	−0.463367	−0.231683	−	0.972791i	$-0.574423\pi$
$401$	−1.49206e6	−0.463367	−0.231683	+	0.972791i	$0.574423\pi$
$402$	4.42458e6	1.36555
$403$	−1.13765e6	−0.348936
$404$	9.15395e6	2.79033
$405$	0	0
$406$	424130.	0.127698
$407$	316812.	0.0948017
$408$	−500938.	−0.148982
$409$	−1.36588e6	−0.403743	−0.201872	−	0.979412i	$-0.564702\pi$
$409$	−1.36588e6	−0.403743	−0.201872	+	0.979412i	$0.564702\pi$
$410$	0	0
$411$	−3.46327e6	−1.01131
$412$	713001.	0.206941
$413$	477949.	0.137882
$414$	640229.	0.183584
$415$	0	0
$416$	340087.	0.0963511
$417$	−2.61952e6	−0.737704
$418$	561957.	0.157312
$419$	1.06312e6	0.295835	0.147917	−	0.989000i	$-0.452743\pi$
$419$	1.06312e6	0.295835	0.147917	+	0.989000i	$0.452743\pi$
$420$	0	0
$421$	2.10790e6	0.579622	0.289811	−	0.957084i	$-0.406408\pi$
$421$	2.10790e6	0.579622	0.289811	+	0.957084i	$0.406408\pi$
$422$	−13810.3	−0.00377504
$423$	109629.	0.0297903
$424$	152058.	0.0410766
$425$	0	0
$426$	3.04994e6	0.814268
$427$	12851.6	0.00341105
$428$	−8.88940e6	−2.34565
$429$	146343.	0.0383910
$430$	0	0
$431$	−223504.	−0.0579551	−0.0289775	−	0.999580i	$-0.509225\pi$
$431$	−223504.	−0.0579551	−0.0289775	+	0.999580i	$0.509225\pi$
$432$	2.80128e6	0.722183
$433$	836386.	0.214381	0.107191	−	0.994238i	$-0.465814\pi$
$433$	836386.	0.214381	0.107191	+	0.994238i	$0.465814\pi$
$434$	−1.19343e6	−0.304140
$435$	0	0
$436$	2.41160e6	0.607560
$437$	375512.	0.0940633
$438$	−8.66231e6	−2.15749
$439$	2.41540e6	0.598173	0.299087	−	0.954226i	$-0.403318\pi$
$439$	2.41540e6	0.598173	0.299087	+	0.954226i	$0.403318\pi$
$440$	0	0
$441$	2.11005e6	0.516651
$442$	276724.	0.0673739
$443$	5.13301e6	1.24269	0.621345	−	0.783537i	$-0.286586\pi$
$443$	5.13301e6	1.24269	0.621345	+	0.783537i	$0.286586\pi$
$444$	2.54437e6	0.612525
$445$	0	0
$446$	4.01683e6	0.956194
$447$	1.85093e6	0.438149
$448$	772080.	0.181747
$449$	−3.64066e6	−0.852244	−0.426122	−	0.904666i	$-0.640121\pi$
$449$	−3.64066e6	−0.852244	−0.426122	+	0.904666i	$0.640121\pi$
$450$	0	0
$451$	526866.	0.121972
$452$	−719577.	−0.165665
$453$	−2.74743e6	−0.629043
$454$	−5.91285e6	−1.34635
$455$	0	0
$456$	2.12782e6	0.479206
$457$	−3.99375e6	−0.894521	−0.447260	−	0.894404i	$-0.647600\pi$
$457$	−3.99375e6	−0.894521	−0.447260	+	0.894404i	$0.647600\pi$
$458$	−1.05722e7	−2.35506
$459$	−677032.	−0.149995
$460$	0	0
$461$	−6.94874e6	−1.52284	−0.761419	−	0.648260i	$-0.775497\pi$
$461$	−6.94874e6	−1.52284	−0.761419	+	0.648260i	$0.775497\pi$
$462$	153519.	0.0334624
$463$	−5.09427e6	−1.10441	−0.552204	−	0.833709i	$-0.686213\pi$
$463$	−5.09427e6	−1.10441	−0.552204	+	0.833709i	$0.686213\pi$
$464$	1.68482e6	0.363295
$465$	0	0
$466$	−7.60145e6	−1.62155
$467$	−6.29319e6	−1.33530	−0.667650	−	0.744475i	$-0.732700\pi$
$467$	−6.29319e6	−1.33530	−0.667650	+	0.744475i	$0.732700\pi$
$468$	−1.31109e6	−0.276706
$469$	−790864.	−0.166024
$470$	0	0
$471$	3.57636e6	0.742829
$472$	7.12178e6	1.47141
$473$	1.69660e6	0.348679
$474$	3.82150e6	0.781246
$475$	0	0
$476$	189916.	0.0384188
$477$	70953.1	0.0142783
$478$	−1.56648e7	−3.13586
$479$	2.99562e6	0.596551	0.298276	−	0.954480i	$-0.403589\pi$
$479$	2.99562e6	0.596551	0.298276	+	0.954480i	$0.403589\pi$
$480$	0	0
$481$	−662668.	−0.130597
$482$	2.00967e6	0.394009
$483$	102585.	0.0200085
$484$	−9.35559e6	−1.81534
$485$	0	0
$486$	8.11340e6	1.55816
$487$	7.03713e6	1.34454	0.672269	−	0.740307i	$-0.265320\pi$
$487$	7.03713e6	1.34454	0.672269	+	0.740307i	$0.265320\pi$
$488$	191498.	0.0364012
$489$	1.01939e6	0.192783
$490$	0	0
$491$	−4.68700e6	−0.877388	−0.438694	−	0.898637i	$-0.644559\pi$
$491$	−4.68700e6	−0.877388	−0.438694	+	0.898637i	$0.644559\pi$
$492$	4.23135e6	0.788072
$493$	−407199.	−0.0754553
$494$	−1.17543e6	−0.216711
$495$	0	0
$496$	−4.74081e6	−0.865263
$497$	−545156.	−0.0989988
$498$	−9.74852e6	−1.76143
$499$	−7.25902e6	−1.30505	−0.652525	−	0.757768i	$-0.726290\pi$
$499$	−7.25902e6	−1.30505	−0.652525	+	0.757768i	$0.726290\pi$
$500$	0	0
$501$	−5.23808e6	−0.932347
$502$	−1.45757e7	−2.58148
$503$	−1.36235e6	−0.240087	−0.120044	−	0.992769i	$-0.538303\pi$
$503$	−1.36235e6	−0.240087	−0.120044	+	0.992769i	$0.538303\pi$
$504$	−648447.	−0.113710
$505$	0	0
$506$	403700.	0.0700943
$507$	−306102.	−0.0528867
$508$	1.48466e7	2.55251
$509$	7.22299e6	1.23573	0.617864	−	0.786285i	$-0.287998\pi$
$509$	7.22299e6	1.23573	0.617864	+	0.786285i	$0.287998\pi$
$510$	0	0
$511$	1.54833e6	0.262308
$512$	7.60574e6	1.28223
$513$	2.87581e6	0.482466
$514$	−1.73392e7	−2.89482
$515$	0	0
$516$	1.36256e7	2.25285
$517$	69127.3	0.0113743
$518$	−695161.	−0.113831
$519$	−2.35327e6	−0.383489
$520$	0	0
$521$	−4.66901e6	−0.753582	−0.376791	−	0.926298i	$-0.622973\pi$
$521$	−4.66901e6	−0.753582	−0.376791	+	0.926298i	$0.622973\pi$
$522$	2.94896e6	0.473688
$523$	−1.60688e6	−0.256880	−0.128440	−	0.991717i	$-0.540997\pi$
$523$	−1.60688e6	−0.256880	−0.128440	+	0.991717i	$0.540997\pi$
$524$	−2.63474e6	−0.419188
$525$	0	0
$526$	−7.66321e6	−1.20766
$527$	1.14579e6	0.179713
$528$	609841.	0.0951989
$529$	−6.16658e6	−0.958088
$530$	0	0
$531$	3.32316e6	0.511464
$532$	−806700.	−0.123576
$533$	−1.10203e6	−0.168026
$534$	1.26956e7	1.92664
$535$	0	0
$536$	−1.17844e7	−1.77173
$537$	−2.32668e6	−0.348178
$538$	1.52891e7	2.27733
$539$	1.33051e6	0.197263
$540$	0	0
$541$	1.09893e7	1.61428	0.807139	−	0.590362i	$-0.201015\pi$
$541$	1.09893e7	1.61428	0.807139	+	0.590362i	$0.201015\pi$
$542$	−1.71007e7	−2.50043
$543$	6.94222e6	1.01041
$544$	−342521.	−0.0496238
$545$	0	0
$546$	−321111.	−0.0460972
$547$	2.28671e6	0.326770	0.163385	−	0.986562i	$-0.447759\pi$
$547$	2.28671e6	0.326770	0.163385	+	0.986562i	$0.447759\pi$
$548$	1.95646e7	2.78305
$549$	89356.8	0.0126531
$550$	0	0
$551$	1.72965e6	0.242705
$552$	1.52858e6	0.213522
$553$	−683067.	−0.0949840
$554$	1.52950e7	2.11726
$555$	0	0
$556$	1.47981e7	2.03011
$557$	−7.22072e6	−0.986149	−0.493075	−	0.869987i	$-0.664127\pi$
$557$	−7.22072e6	−0.986149	−0.493075	+	0.869987i	$0.664127\pi$
$558$	−8.29788e6	−1.12819
$559$	−3.54872e6	−0.480333
$560$	0	0
$561$	−147391.	−0.0197725
$562$	5.21437e6	0.696404
$563$	2.89551e6	0.384994	0.192497	−	0.981298i	$-0.438341\pi$
$563$	2.89551e6	0.384994	0.192497	+	0.981298i	$0.438341\pi$
$564$	555173.	0.0734904
$565$	0	0
$566$	2.21400e6	0.290493
$567$	211812.	0.0276689
$568$	−8.12322e6	−1.05647
$569$	−1.43899e7	−1.86327	−0.931637	−	0.363391i	$-0.881619\pi$
$569$	−1.43899e7	−1.86327	−0.931637	+	0.363391i	$0.881619\pi$
$570$	0	0
$571$	−8.99971e6	−1.15515	−0.577575	−	0.816338i	$-0.696001\pi$
$571$	−8.99971e6	−1.15515	−0.577575	+	0.816338i	$0.696001\pi$
$572$	−826718.	−0.105650
$573$	7.24359e6	0.921653
$574$	−1.15607e6	−0.146455
$575$	0	0
$576$	5.36824e6	0.674180
$577$	−1.15215e7	−1.44069	−0.720344	−	0.693617i	$-0.756016\pi$
$577$	−1.15215e7	−1.44069	−0.720344	+	0.693617i	$0.756016\pi$
$578$	1.33804e7	1.66590
$579$	−743368.	−0.0921526
$580$	0	0
$581$	1.74248e6	0.214155
$582$	3.08974e6	0.378107
$583$	44739.9	0.00545160
$584$	2.30712e7	2.79923
$585$	0	0
$586$	−3.42120e6	−0.411561
$587$	1.00352e7	1.20207	0.601036	−	0.799222i	$-0.294755\pi$
$587$	1.00352e7	1.20207	0.601036	+	0.799222i	$0.294755\pi$
$588$	1.06855e7	1.27454
$589$	−4.86694e6	−0.578053
$590$	0	0
$591$	−1.15333e7	−1.35827
$592$	−2.76147e6	−0.323844
$593$	3.56031e6	0.415768	0.207884	−	0.978153i	$-0.433342\pi$
$593$	3.56031e6	0.415768	0.207884	+	0.978153i	$0.433342\pi$
$594$	3.09168e6	0.359525
$595$	0	0
$596$	−1.04562e7	−1.20576
$597$	3.11495e6	0.357697
$598$	−844409.	−0.0965606
$599$	1.00529e7	1.14479	0.572395	−	0.819978i	$-0.306015\pi$
$599$	1.00529e7	1.14479	0.572395	+	0.819978i	$0.306015\pi$
$600$	0	0
$601$	−4.56283e6	−0.515286	−0.257643	−	0.966240i	$-0.582946\pi$
$601$	−4.56283e6	−0.515286	−0.257643	+	0.966240i	$0.582946\pi$
$602$	−3.72273e6	−0.418668
$603$	−5.49885e6	−0.615855
$604$	1.55207e7	1.73109
$605$	0	0
$606$	1.55883e7	1.72432
$607$	−8.72137e6	−0.960756	−0.480378	−	0.877062i	$-0.659500\pi$
$607$	−8.72137e6	−0.960756	−0.480378	+	0.877062i	$0.659500\pi$
$608$	1.45491e6	0.159617
$609$	472515.	0.0516265
$610$	0	0
$611$	−144592.	−0.0156690
$612$	1.32048e6	0.142512
$613$	1.08948e7	1.17103	0.585514	−	0.810663i	$-0.300893\pi$
$613$	1.08948e7	1.17103	0.585514	+	0.810663i	$0.300893\pi$
$614$	8.88146e6	0.950743
$615$	0	0
$616$	−408882.	−0.0434157
$617$	857013.	0.0906306	0.0453153	−	0.998973i	$-0.485571\pi$
$617$	857013.	0.0906306	0.0453153	+	0.998973i	$0.485571\pi$
$618$	1.21418e6	0.127882
$619$	1.32215e7	1.38693	0.693465	−	0.720490i	$-0.256083\pi$
$619$	1.32215e7	1.38693	0.693465	+	0.720490i	$0.256083\pi$
$620$	0	0
$621$	2.06593e6	0.214974
$622$	−4.31942e6	−0.447661
$623$	−2.26926e6	−0.234242
$624$	−1.27559e6	−0.131144
$625$	0	0
$626$	−2.82165e7	−2.87784
$627$	626066.	0.0635991
$628$	−2.02035e7	−2.04422
$629$	667411.	0.0672615
$630$	0	0
$631$	1.67770e7	1.67742	0.838708	−	0.544581i	$-0.183311\pi$
$631$	1.67770e7	1.67742	0.838708	+	0.544581i	$0.183311\pi$
$632$	−1.01782e7	−1.01363
$633$	−15385.8	−0.00152620
$634$	−1.23809e7	−1.22329
$635$	0	0
$636$	359314.	0.0352234
$637$	−2.78299e6	−0.271746
$638$	1.85948e6	0.180859
$639$	−3.79045e6	−0.367230
$640$	0	0
$641$	−4.32082e6	−0.415357	−0.207679	−	0.978197i	$-0.566591\pi$
$641$	−4.32082e6	−0.415357	−0.207679	+	0.978197i	$0.566591\pi$
$642$	−1.51378e7	−1.44953
$643$	1.77399e7	1.69209	0.846045	−	0.533111i	$-0.178977\pi$
$643$	1.77399e7	1.69209	0.846045	+	0.533111i	$0.178977\pi$
$644$	−579519.	−0.0550621
$645$	0	0
$646$	1.18384e6	0.111613
$647$	1.50885e7	1.41705	0.708525	−	0.705686i	$-0.249361\pi$
$647$	1.50885e7	1.41705	0.708525	+	0.705686i	$0.249361\pi$
$648$	3.15614e6	0.295270
$649$	2.09544e6	0.195282
$650$	0	0
$651$	−1.32958e6	−0.122959
$652$	−5.75873e6	−0.530527
$653$	−9.81391e6	−0.900657	−0.450328	−	0.892863i	$-0.648693\pi$
$653$	−9.81391e6	−0.900657	−0.450328	+	0.892863i	$0.648693\pi$
$654$	4.10673e6	0.375450
$655$	0	0
$656$	−4.59238e6	−0.416657
$657$	1.07655e7	0.973015
$658$	−151682.	−0.0136574
$659$	−8.40142e6	−0.753597	−0.376798	−	0.926295i	$-0.622975\pi$
$659$	−8.40142e6	−0.753597	−0.376798	+	0.926295i	$0.622975\pi$
$660$	0	0
$661$	3.18929e6	0.283916	0.141958	−	0.989873i	$-0.454660\pi$
$661$	3.18929e6	0.283916	0.141958	+	0.989873i	$0.454660\pi$
$662$	672814.	0.0596692
$663$	308293.	0.0272383
$664$	2.59642e7	2.28536
$665$	0	0
$666$	−4.83342e6	−0.422250
$667$	1.24255e6	0.108143
$668$	2.95908e7	2.56576
$669$	4.47507e6	0.386576
$670$	0	0
$671$	56344.5	0.00483109
$672$	397462.	0.0339526
$673$	−1.51187e7	−1.28669	−0.643347	−	0.765575i	$-0.722455\pi$
$673$	−1.51187e7	−1.28669	−0.643347	+	0.765575i	$0.722455\pi$
$674$	−1.57078e7	−1.33189
$675$	0	0
$676$	1.72923e6	0.145541
$677$	−7.55195e6	−0.633268	−0.316634	−	0.948548i	$-0.602553\pi$
$677$	−7.55195e6	−0.633268	−0.316634	+	0.948548i	$0.602553\pi$
$678$	−1.22537e6	−0.102375
$679$	−552270.	−0.0459703
$680$	0	0
$681$	−6.58740e6	−0.544310
$682$	−5.23228e6	−0.430754
$683$	−2.23148e7	−1.83038	−0.915191	−	0.403021i	$-0.867960\pi$
$683$	−2.23148e7	−1.83038	−0.915191	+	0.403021i	$0.867960\pi$
$684$	−5.60895e6	−0.458396
$685$	0	0
$686$	−5.89910e6	−0.478603
$687$	−1.17783e7	−0.952117
$688$	−1.47882e7	−1.19109
$689$	−93581.3	−0.00751002
$690$	0	0
$691$	1.32161e7	1.05295	0.526474	−	0.850191i	$-0.323514\pi$
$691$	1.32161e7	1.05295	0.526474	+	0.850191i	$0.323514\pi$
$692$	1.32940e7	1.05534
$693$	−190792.	−0.0150913
$694$	−1.38658e7	−1.09281
$695$	0	0
$696$	7.04082e6	0.550934
$697$	1.10992e6	0.0865384
$698$	−2.78061e7	−2.16023
$699$	−8.46863e6	−0.655572
$700$	0	0
$701$	−9.22726e6	−0.709215	−0.354607	−	0.935015i	$-0.615385\pi$
$701$	−9.22726e6	−0.709215	−0.354607	+	0.935015i	$0.615385\pi$
$702$	−6.46679e6	−0.495274
$703$	−2.83494e6	−0.216349
$704$	3.38498e6	0.257409
$705$	0	0
$706$	7.73433e6	0.583997
$707$	−2.78631e6	−0.209643
$708$	1.68288e7	1.26174
$709$	1.54395e7	1.15350	0.576751	−	0.816920i	$-0.304320\pi$
$709$	1.54395e7	1.15350	0.576751	+	0.816920i	$0.304320\pi$
$710$	0	0
$711$	−4.74934e6	−0.352338
$712$	−3.38136e7	−2.49972
$713$	−3.49632e6	−0.257565
$714$	323410.	0.0237415
$715$	0	0
$716$	1.31438e7	0.958164
$717$	−1.74519e7	−1.26778
$718$	−1.86180e7	−1.34779
$719$	−6.33241e6	−0.456822	−0.228411	−	0.973565i	$-0.573353\pi$
$719$	−6.33241e6	−0.456822	−0.228411	+	0.973565i	$0.573353\pi$
$720$	0	0
$721$	−217026.	−0.0155479
$722$	1.87916e7	1.34159
$723$	2.23893e6	0.159292
$724$	−3.92178e7	−2.78059
$725$	0	0
$726$	−1.59317e7	−1.12181
$727$	−1.26906e6	−0.0890522	−0.0445261	−	0.999008i	$-0.514178\pi$
$727$	−1.26906e6	−0.0890522	−0.0445261	+	0.999008i	$0.514178\pi$
$728$	855249.	0.0598087
$729$	1.18319e7	0.824585
$730$	0	0
$731$	3.57412e6	0.247386
$732$	452512.	0.0312142
$733$	−3.39046e6	−0.233077	−0.116538	−	0.993186i	$-0.537180\pi$
$733$	−3.39046e6	−0.233077	−0.116538	+	0.993186i	$0.537180\pi$
$734$	3.32855e7	2.28042
$735$	0	0
$736$	1.04518e6	0.0711211
$737$	−3.46733e6	−0.235140
$738$	−8.03808e6	−0.543265
$739$	−1.16232e7	−0.782917	−0.391458	−	0.920196i	$-0.628029\pi$
$739$	−1.16232e7	−0.782917	−0.391458	+	0.920196i	$0.628029\pi$
$740$	0	0
$741$	−1.30953e6	−0.0876130
$742$	−98169.9	−0.00654589
$743$	−1.56391e7	−1.03929	−0.519647	−	0.854381i	$-0.673937\pi$
$743$	−1.56391e7	−1.03929	−0.519647	+	0.854381i	$0.673937\pi$
$744$	−1.98117e7	−1.31217
$745$	0	0
$746$	−3.69161e7	−2.42867
$747$	1.21154e7	0.794395
$748$	832635.	0.0544128
$749$	2.70579e6	0.176234
$750$	0	0
$751$	2.41507e7	1.56253	0.781267	−	0.624196i	$-0.214574\pi$
$751$	2.41507e7	1.56253	0.781267	+	0.624196i	$0.214574\pi$
$752$	−602543.	−0.0388547
$753$	−1.62385e7	−1.04366
$754$	−3.88944e6	−0.249148
$755$	0	0
$756$	−4.43816e6	−0.282422
$757$	1.27880e7	0.811078	0.405539	−	0.914078i	$-0.367084\pi$
$757$	1.27880e7	0.811078	0.405539	+	0.914078i	$0.367084\pi$
$758$	−3.95426e7	−2.49972
$759$	449755.	0.0283381
$760$	0	0
$761$	2.35049e7	1.47129	0.735644	−	0.677369i	$-0.236880\pi$
$761$	2.35049e7	1.47129	0.735644	+	0.677369i	$0.236880\pi$
$762$	2.52824e7	1.57736
$763$	−734051.	−0.0456473
$764$	−4.09203e7	−2.53633
$765$	0	0
$766$	−3.52588e7	−2.17118
$767$	−4.38298e6	−0.269018
$768$	2.05460e7	1.25697
$769$	1.61889e7	0.987190	0.493595	−	0.869692i	$-0.335683\pi$
$769$	1.61889e7	0.987190	0.493595	+	0.869692i	$0.335683\pi$
$770$	0	0
$771$	−1.93173e7	−1.17034
$772$	4.19942e6	0.253598
$773$	1.34926e7	0.812172	0.406086	−	0.913835i	$-0.366893\pi$
$773$	1.34926e7	0.812172	0.406086	+	0.913835i	$0.366893\pi$
$774$	−2.58840e7	−1.55303
$775$	0	0
$776$	−8.22922e6	−0.490574
$777$	−774465.	−0.0460203
$778$	1.99336e7	1.18069
$779$	−4.71456e6	−0.278354
$780$	0	0
$781$	−2.39009e6	−0.140212
$782$	850453.	0.0497317
$783$	9.51588e6	0.554683
$784$	−1.15973e7	−0.673853
$785$	0	0
$786$	−4.48672e6	−0.259043
$787$	2.77228e7	1.59551	0.797757	−	0.602979i	$-0.206020\pi$
$787$	2.77228e7	1.59551	0.797757	+	0.602979i	$0.206020\pi$
$788$	6.51539e7	3.73787
$789$	−8.53743e6	−0.488242
$790$	0	0
$791$	219027.	0.0124468
$792$	−2.84294e6	−0.161048
$793$	−117854.	−0.00665522
$794$	5.07236e7	2.85534
$795$	0	0
$796$	−1.75969e7	−0.984360
$797$	1.70875e7	0.952870	0.476435	−	0.879210i	$-0.341929\pi$
$797$	1.70875e7	0.952870	0.476435	+	0.879210i	$0.341929\pi$
$798$	−1.37374e6	−0.0763653
$799$	145627.	0.00807001
$800$	0	0
$801$	−1.57781e7	−0.868906
$802$	1.43537e7	0.788001
$803$	6.78823e6	0.371508
$804$	−2.78467e7	−1.51927
$805$	0	0
$806$	1.09442e7	0.593399
$807$	1.70333e7	0.920692
$808$	−4.15180e7	−2.23722
$809$	3.28199e7	1.76305	0.881527	−	0.472133i	$-0.156516\pi$
$809$	3.28199e7	1.76305	0.881527	+	0.472133i	$0.156516\pi$
$810$	0	0
$811$	−3.04825e6	−0.162742	−0.0813708	−	0.996684i	$-0.525930\pi$
$811$	−3.04825e6	−0.162742	−0.0813708	+	0.996684i	$0.525930\pi$
$812$	−2.66932e6	−0.142073
$813$	−1.90515e7	−1.01089
$814$	−3.04775e6	−0.161220
$815$	0	0
$816$	1.28472e6	0.0675434
$817$	−1.51817e7	−0.795728
$818$	1.31398e7	0.686605
$819$	399075.	0.0207896
$820$	0	0
$821$	4.88552e6	0.252961	0.126480	−	0.991969i	$-0.459632\pi$
$821$	4.88552e6	0.252961	0.126480	+	0.991969i	$0.459632\pi$
$822$	3.33168e7	1.71982
$823$	−7.45432e6	−0.383626	−0.191813	−	0.981431i	$-0.561437\pi$
$823$	−7.45432e6	−0.383626	−0.191813	+	0.981431i	$0.561437\pi$
$824$	−3.23384e6	−0.165921
$825$	0	0
$826$	−4.59789e6	−0.234481
$827$	1.47591e7	0.750405	0.375202	−	0.926943i	$-0.377573\pi$
$827$	1.47591e7	0.750405	0.375202	+	0.926943i	$0.377573\pi$
$828$	−4.02937e6	−0.204250
$829$	6.19808e6	0.313236	0.156618	−	0.987659i	$-0.449941\pi$
$829$	6.19808e6	0.313236	0.156618	+	0.987659i	$0.449941\pi$
$830$	0	0
$831$	1.70399e7	0.855980
$832$	−7.08027e6	−0.354602
$833$	2.80290e6	0.139957
$834$	2.51999e7	1.25454
$835$	0	0
$836$	−3.53676e6	−0.175021
$837$	−2.67761e7	−1.32109
$838$	−1.02273e7	−0.503096
$839$	2.44941e7	1.20131	0.600657	−	0.799507i	$-0.294906\pi$
$839$	2.44941e7	1.20131	0.600657	+	0.799507i	$0.294906\pi$
$840$	0	0
$841$	−1.47878e7	−0.720966
$842$	−2.02781e7	−0.985703
$843$	5.80923e6	0.281546
$844$	86917.0	0.00419999
$845$	0	0
$846$	−1.05464e6	−0.0506613
$847$	2.84769e6	0.136390
$848$	−389972.	−0.0186228
$849$	2.46657e6	0.117442
$850$	0	0
$851$	−2.03657e6	−0.0963996
$852$	−1.91952e7	−0.905929
$853$	−4.89553e6	−0.230371	−0.115185	−	0.993344i	$-0.536746\pi$
$853$	−4.89553e6	−0.230371	−0.115185	+	0.993344i	$0.536746\pi$
$854$	−123633.	−0.00580082
$855$	0	0
$856$	4.03181e7	1.88069
$857$	−2.35363e7	−1.09468	−0.547338	−	0.836912i	$-0.684359\pi$
$857$	−2.35363e7	−1.09468	−0.547338	+	0.836912i	$0.684359\pi$
$858$	−1.40783e6	−0.0652876
$859$	3.00121e7	1.38776	0.693878	−	0.720093i	$-0.255901\pi$
$859$	3.00121e7	1.38776	0.693878	+	0.720093i	$0.255901\pi$
$860$	0	0
$861$	−1.28795e6	−0.0592095
$862$	2.15011e6	0.0985582
$863$	2.04116e6	0.0932932	0.0466466	−	0.998911i	$-0.485147\pi$
$863$	2.04116e6	0.0932932	0.0466466	+	0.998911i	$0.485147\pi$
$864$	8.00440e6	0.364791
$865$	0	0
$866$	−8.04606e6	−0.364576
$867$	1.49068e7	0.673499
$868$	7.51102e6	0.338376
$869$	−2.99472e6	−0.134526
$870$	0	0
$871$	7.25253e6	0.323925
$872$	−1.09379e7	−0.487127
$873$	−3.83991e6	−0.170524
$874$	−3.61244e6	−0.159964
$875$	0	0
$876$	5.45174e7	2.40035
$877$	1.16699e7	0.512352	0.256176	−	0.966630i	$-0.417537\pi$
$877$	1.16699e7	0.512352	0.256176	+	0.966630i	$0.417537\pi$
$878$	−2.32362e7	−1.01725
$879$	−3.81149e6	−0.166388
$880$	0	0
$881$	1.23442e7	0.535826	0.267913	−	0.963443i	$-0.413666\pi$
$881$	1.23442e7	0.535826	0.267913	+	0.963443i	$0.413666\pi$
$882$	−2.02988e7	−0.878614
$883$	1.15726e7	0.499492	0.249746	−	0.968311i	$-0.419653\pi$
$883$	1.15726e7	0.499492	0.249746	+	0.968311i	$0.419653\pi$
$884$	−1.74160e6	−0.0749580
$885$	0	0
$886$	−4.93797e7	−2.11332
$887$	2.94130e7	1.25525	0.627625	−	0.778516i	$-0.284027\pi$
$887$	2.94130e7	1.25525	0.627625	+	0.778516i	$0.284027\pi$
$888$	−1.15401e7	−0.491108
$889$	−4.51905e6	−0.191775
$890$	0	0
$891$	928630.	0.0391876
$892$	−2.52805e7	−1.06383
$893$	−618573.	−0.0259575
$894$	−1.78060e7	−0.745115
$895$	0	0
$896$	−6.24070e6	−0.259695
$897$	−940740.	−0.0390381
$898$	3.50232e7	1.44932
$899$	−1.61044e7	−0.664578
$900$	0	0
$901$	94251.1	0.00386789
$902$	−5.06846e6	−0.207424
$903$	−4.14742e6	−0.169262
$904$	3.26366e6	0.132826
$905$	0	0
$906$	2.64303e7	1.06975
$907$	−1.15733e7	−0.467130	−0.233565	−	0.972341i	$-0.575039\pi$
$907$	−1.15733e7	−0.467130	−0.233565	+	0.972341i	$0.575039\pi$
$908$	3.72133e7	1.49790
$909$	−1.93731e7	−0.777660
$910$	0	0
$911$	2.25526e7	0.900328	0.450164	−	0.892946i	$-0.351366\pi$
$911$	2.25526e7	0.900328	0.450164	+	0.892946i	$0.351366\pi$
$912$	−5.45706e6	−0.217256
$913$	7.63944e6	0.303309
$914$	3.84200e7	1.52122
$915$	0	0
$916$	6.65376e7	2.62016
$917$	801971.	0.0314945
$918$	6.51307e6	0.255082
$919$	−3.93029e7	−1.53510	−0.767548	−	0.640991i	$-0.778524\pi$
$919$	−3.93029e7	−1.53510	−0.767548	+	0.640991i	$0.778524\pi$
$920$	0	0
$921$	9.89466e6	0.384372
$922$	6.68471e7	2.58973
$923$	4.99929e6	0.193154
$924$	−966192.	−0.0372292
$925$	0	0
$926$	4.90071e7	1.87815
$927$	−1.50897e6	−0.0576742
$928$	4.81423e6	0.183509
$929$	−8.90117e6	−0.338383	−0.169191	−	0.985583i	$-0.554116\pi$
$929$	−8.90117e6	−0.338383	−0.169191	+	0.985583i	$0.554116\pi$
$930$	0	0
$931$	−1.19058e7	−0.450178
$932$	4.78408e7	1.80409
$933$	−4.81218e6	−0.180983
$934$	6.05407e7	2.27081
$935$	0	0
$936$	5.94651e6	0.221857
$937$	9.66249e6	0.359534	0.179767	−	0.983709i	$-0.442466\pi$
$937$	9.66249e6	0.359534	0.179767	+	0.983709i	$0.442466\pi$
$938$	7.60814e6	0.282339
$939$	−3.14355e7	−1.16347
$940$	0	0
$941$	3.06781e7	1.12942	0.564709	−	0.825290i	$-0.308988\pi$
$941$	3.06781e7	1.12942	0.564709	+	0.825290i	$0.308988\pi$
$942$	−3.44047e7	−1.26325
$943$	−3.38686e6	−0.124027
$944$	−1.82647e7	−0.667088
$945$	0	0
$946$	−1.63213e7	−0.592962
$947$	9.20730e6	0.333624	0.166812	−	0.985989i	$-0.446653\pi$
$947$	9.20730e6	0.333624	0.166812	+	0.985989i	$0.446653\pi$
$948$	−2.40511e7	−0.869190
$949$	−1.41988e7	−0.511782
$950$	0	0
$951$	−1.37933e7	−0.494557
$952$	−861370.	−0.0308033
$953$	8.44885e6	0.301346	0.150673	−	0.988584i	$-0.451856\pi$
$953$	8.44885e6	0.301346	0.150673	+	0.988584i	$0.451856\pi$
$954$	−682571.	−0.0242816
$955$	0	0
$956$	9.85888e7	3.48886
$957$	2.07162e6	0.0731189
$958$	−2.88179e7	−1.01449
$959$	−5.95515e6	−0.209096
$960$	0	0
$961$	1.66860e7	0.582831
$962$	6.37489e6	0.222093
$963$	1.88132e7	0.653728
$964$	−1.26481e7	−0.438362
$965$	0	0
$966$	−986867.	−0.0340264
$967$	7.96585e6	0.273947	0.136973	−	0.990575i	$-0.456263\pi$
$967$	7.96585e6	0.273947	0.136973	+	0.990575i	$0.456263\pi$
$968$	4.24326e7	1.45550
$969$	1.31890e6	0.0451234
$970$	0	0
$971$	−1.54790e7	−0.526858	−0.263429	−	0.964679i	$-0.584853\pi$
$971$	−1.54790e7	−0.526858	−0.263429	+	0.964679i	$0.584853\pi$
$972$	−5.10628e7	−1.73356
$973$	−4.50431e6	−0.152527
$974$	−6.76974e7	−2.28652
$975$	0	0
$976$	−491122.	−0.0165031
$977$	4.35353e7	1.45917	0.729584	−	0.683891i	$-0.239714\pi$
$977$	4.35353e7	1.45917	0.729584	+	0.683891i	$0.239714\pi$
$978$	−9.80658e6	−0.327847
$979$	−9.94896e6	−0.331758
$980$	0	0
$981$	−5.10383e6	−0.169326
$982$	4.50891e7	1.49208
$983$	−1.41883e7	−0.468323	−0.234161	−	0.972198i	$-0.575234\pi$
$983$	−1.41883e7	−0.468323	−0.234161	+	0.972198i	$0.575234\pi$
$984$	−1.91914e7	−0.631857
$985$	0	0
$986$	3.91727e6	0.128319
$987$	−168986.	−0.00552149
$988$	7.39774e6	0.241105
$989$	−1.09062e7	−0.354556
$990$	0	0
$991$	1.43414e7	0.463881	0.231940	−	0.972730i	$-0.425493\pi$
$991$	1.43414e7	0.463881	0.231940	+	0.972730i	$0.425493\pi$
$992$	−1.35464e7	−0.437065
$993$	749570.	0.0241234
$994$	5.24442e6	0.168357
$995$	0	0
$996$	6.13537e7	1.95971
$997$	129946.	0.00414024	0.00207012	−	0.999998i	$-0.499341\pi$
$997$	129946.	0.00414024	0.00207012	+	0.999998i	$0.499341\pi$
$998$	6.98320e7	2.21936
$999$	−1.55968e7	−0.494449

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.f.1.1		6
5.2	odd	4		325.6.b.f.274.2		12
5.3	odd	4		325.6.b.f.274.11		12
5.4	even	2		65.6.a.e.1.6	✓	6
15.14	odd	2		585.6.a.k.1.1		6
20.19	odd	2		1040.6.a.r.1.3		6
65.64	even	2		845.6.a.g.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.e.1.6	✓	6	5.4	even	2
325.6.a.f.1.1		6	1.1	even	1	trivial
325.6.b.f.274.2		12	5.2	odd	4
325.6.b.f.274.11		12	5.3	odd	4
585.6.a.k.1.1		6	15.14	odd	2
845.6.a.g.1.1		6	65.64	even	2
1040.6.a.r.1.3		6	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

\(f(q)\)	\(=\)	\(q-9.62003 q^{2} -10.7175 q^{3} +60.5450 q^{4} +103.103 q^{6} -18.4289 q^{7} -274.604 q^{8} -128.135 q^{9} -80.7965 q^{11} -648.891 q^{12} +169.000 q^{13} +177.287 q^{14} +704.257 q^{16} -170.210 q^{17} +1232.67 q^{18} +722.993 q^{19} +197.512 q^{21} +777.265 q^{22} +519.385 q^{23} +2943.06 q^{24} -1625.79 q^{26} +3977.64 q^{27} -1115.78 q^{28} +2392.34 q^{29} -6731.65 q^{31} +2012.35 q^{32} +865.936 q^{33} +1637.42 q^{34} -7757.95 q^{36} -3921.11 q^{37} -6955.22 q^{38} -1811.26 q^{39} -6520.89 q^{41} -1900.07 q^{42} -20998.4 q^{43} -4891.83 q^{44} -4996.50 q^{46} -855.573 q^{47} -7547.87 q^{48} -16467.4 q^{49} +1824.22 q^{51} +10232.1 q^{52} -553.736 q^{53} -38265.0 q^{54} +5060.64 q^{56} -7748.68 q^{57} -23014.4 q^{58} -25934.8 q^{59} -697.362 q^{61} +64758.7 q^{62} +2361.39 q^{63} -41895.1 q^{64} -8330.33 q^{66} +42914.4 q^{67} -10305.3 q^{68} -5566.51 q^{69} +29581.6 q^{71} +35186.4 q^{72} -84016.4 q^{73} +37721.2 q^{74} +43773.6 q^{76} +1488.99 q^{77} +17424.3 q^{78} +37065.0 q^{79} -11493.4 q^{81} +62731.2 q^{82} -94551.6 q^{83} +11958.3 q^{84} +202005. q^{86} -25639.9 q^{87} +22187.0 q^{88} +123136. q^{89} -3114.48 q^{91} +31446.2 q^{92} +72146.4 q^{93} +8230.64 q^{94} -21567.3 q^{96} +29967.6 q^{97} +158417. q^{98} +10352.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} - 20 q^{3} + 134 q^{4} - 52 q^{6} - 172 q^{7} + 138 q^{8} + 1034 q^{9} + 800 q^{11} + 832 q^{12} + 1014 q^{13} + 2108 q^{14} + 322 q^{16} - 4396 q^{17} - 6142 q^{18} + 5304 q^{19} + 1072 q^{21}+ \cdots + 301264 q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 - 20 * q^3 + 134 * q^4 - 52 * q^6 - 172 * q^7 + 138 * q^8 + 1034 * q^9 + 800 * q^11 + 832 * q^12 + 1014 * q^13 + 2108 * q^14 + 322 * q^16 - 4396 * q^17 - 6142 * q^18 + 5304 * q^19 + 1072 * q^21 - 8008 * q^22 + 1140 * q^23 + 4756 * q^24 - 338 * q^26 - 800 * q^27 - 524 * q^28 - 3664 * q^29 + 16664 * q^31 + 9894 * q^32 + 13092 * q^33 + 1380 * q^34 + 1818 * q^36 - 4488 * q^37 + 30112 * q^38 - 3380 * q^39 + 12436 * q^41 + 68336 * q^42 - 1516 * q^43 - 35124 * q^44 - 4556 * q^46 - 212 * q^47 + 16712 * q^48 - 27842 * q^49 - 27664 * q^51 + 22646 * q^52 + 15612 * q^53 - 132912 * q^54 + 44420 * q^56 + 86796 * q^57 + 39664 * q^58 - 11896 * q^59 + 57232 * q^61 + 95488 * q^62 - 207956 * q^63 - 39914 * q^64 + 9908 * q^66 + 25428 * q^67 + 16236 * q^68 + 72404 * q^69 + 214104 * q^71 + 130806 * q^72 - 60808 * q^73 - 182104 * q^74 + 366636 * q^76 + 5192 * q^77 - 8788 * q^78 + 36024 * q^79 + 350514 * q^81 + 1124 * q^82 - 34684 * q^83 + 6024 * q^84 + 29204 * q^86 + 8640 * q^87 + 40520 * q^88 - 52028 * q^89 - 29068 * q^91 + 283232 * q^92 - 26508 * q^93 + 157572 * q^94 + 547908 * q^96 - 129316 * q^97 + 113942 * q^98 + 301264 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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