LMFDB - Embedded newform 845.4.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,4,Mod(1,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("845.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	845.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:	$49.8566139549$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 41x^{5} + 67x^{4} + 504x^{3} - 605x^{2} - 1800x + 1076 $ x^7 - 2x^6 - 41x^5 + 67x^4 + 504x^3 - 605x^2 - 1800x + 1076
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.545397$ of defining polynomial
Character	$\chi$	$=$	845.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.545397 q^{2} +7.02243 q^{3} -7.70254 q^{4} +5.00000 q^{5} -3.83001 q^{6} -22.8716 q^{7} +8.56412 q^{8} +22.3146 q^{9} +O(q^{10})$	\(q-0.545397 q^{2} +7.02243 q^{3} -7.70254 q^{4} +5.00000 q^{5} -3.83001 q^{6} -22.8716 q^{7} +8.56412 q^{8} +22.3146 q^{9} -2.72699 q^{10} -8.55617 q^{11} -54.0906 q^{12} +12.4741 q^{14} +35.1122 q^{15} +56.9495 q^{16} +32.6895 q^{17} -12.1703 q^{18} +87.1647 q^{19} -38.5127 q^{20} -160.615 q^{21} +4.66651 q^{22} -8.23980 q^{23} +60.1410 q^{24} +25.0000 q^{25} -32.9030 q^{27} +176.170 q^{28} +214.146 q^{29} -19.1501 q^{30} -279.366 q^{31} -99.5730 q^{32} -60.0851 q^{33} -17.8287 q^{34} -114.358 q^{35} -171.879 q^{36} -273.944 q^{37} -47.5394 q^{38} +42.8206 q^{40} +85.8319 q^{41} +87.5987 q^{42} -380.793 q^{43} +65.9042 q^{44} +111.573 q^{45} +4.49396 q^{46} -435.021 q^{47} +399.924 q^{48} +180.112 q^{49} -13.6349 q^{50} +229.560 q^{51} -436.042 q^{53} +17.9452 q^{54} -42.7808 q^{55} -195.875 q^{56} +612.108 q^{57} -116.795 q^{58} -749.132 q^{59} -270.453 q^{60} -70.6681 q^{61} +152.365 q^{62} -510.371 q^{63} -401.289 q^{64} +32.7702 q^{66} -206.116 q^{67} -251.792 q^{68} -57.8635 q^{69} +62.3706 q^{70} -182.480 q^{71} +191.105 q^{72} -188.587 q^{73} +149.408 q^{74} +175.561 q^{75} -671.390 q^{76} +195.694 q^{77} +923.280 q^{79} +284.747 q^{80} -833.553 q^{81} -46.8125 q^{82} -1215.55 q^{83} +1237.14 q^{84} +163.447 q^{85} +207.683 q^{86} +1503.83 q^{87} -73.2760 q^{88} +19.7604 q^{89} -60.8515 q^{90} +63.4674 q^{92} -1961.83 q^{93} +237.259 q^{94} +435.823 q^{95} -699.245 q^{96} -541.901 q^{97} -98.2326 q^{98} -190.927 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{2} - 4 q^{3} + 30 q^{4} + 35 q^{5} - 23 q^{6} - 7 q^{7} - 21 q^{8} + 87 q^{9}+O(q^{10}) $ 7 * q - 2 * q^2 - 4 * q^3 + 30 * q^4 + 35 * q^5 - 23 * q^6 - 7 * q^7 - 21 * q^8 + 87 * q^9	\( 7 q - 2 q^{2} - 4 q^{3} + 30 q^{4} + 35 q^{5} - 23 q^{6} - 7 q^{7} - 21 q^{8} + 87 q^{9} - 10 q^{10} - 87 q^{11} - 79 q^{12} + 66 q^{14} - 20 q^{15} - 134 q^{16} - 114 q^{17} - 207 q^{18} - 245 q^{19} + 150 q^{20} + 38 q^{21} + 338 q^{22} - 74 q^{23} - 334 q^{24} + 175 q^{25} - 442 q^{27} - 230 q^{28} - 88 q^{29} - 115 q^{30} - 500 q^{31} - 80 q^{32} + 194 q^{33} - 427 q^{34} - 35 q^{35} + 425 q^{36} - 633 q^{37} - 298 q^{38} - 105 q^{40} - 162 q^{41} - 1439 q^{42} - 280 q^{43} - 220 q^{44} + 435 q^{45} + 11 q^{46} - 475 q^{47} + 2281 q^{48} + 1694 q^{49} - 50 q^{50} - 430 q^{51} - 603 q^{53} - 51 q^{54} - 435 q^{55} - 1277 q^{56} - 458 q^{57} + 1213 q^{58} - 1410 q^{59} - 395 q^{60} + 412 q^{61} - 56 q^{62} - 1241 q^{63} - 1179 q^{64} + 2173 q^{66} - 1398 q^{67} - 493 q^{68} + 1080 q^{69} + 330 q^{70} + 584 q^{71} - 1545 q^{72} - 2538 q^{73} + 3840 q^{74} - 100 q^{75} - 3292 q^{76} - 2753 q^{77} + 464 q^{79} - 670 q^{80} - 473 q^{81} - 1583 q^{82} - 466 q^{83} + 3081 q^{84} - 570 q^{85} - 4929 q^{86} - 282 q^{87} + 3389 q^{88} - 443 q^{89} - 1035 q^{90} + 3091 q^{92} + 2116 q^{93} + 2017 q^{94} - 1225 q^{95} - 477 q^{96} + 1870 q^{97} - 1364 q^{98} - 5689 q^{99}+O(q^{100}) \) 7 * q - 2 * q^2 - 4 * q^3 + 30 * q^4 + 35 * q^5 - 23 * q^6 - 7 * q^7 - 21 * q^8 + 87 * q^9 - 10 * q^10 - 87 * q^11 - 79 * q^12 + 66 * q^14 - 20 * q^15 - 134 * q^16 - 114 * q^17 - 207 * q^18 - 245 * q^19 + 150 * q^20 + 38 * q^21 + 338 * q^22 - 74 * q^23 - 334 * q^24 + 175 * q^25 - 442 * q^27 - 230 * q^28 - 88 * q^29 - 115 * q^30 - 500 * q^31 - 80 * q^32 + 194 * q^33 - 427 * q^34 - 35 * q^35 + 425 * q^36 - 633 * q^37 - 298 * q^38 - 105 * q^40 - 162 * q^41 - 1439 * q^42 - 280 * q^43 - 220 * q^44 + 435 * q^45 + 11 * q^46 - 475 * q^47 + 2281 * q^48 + 1694 * q^49 - 50 * q^50 - 430 * q^51 - 603 * q^53 - 51 * q^54 - 435 * q^55 - 1277 * q^56 - 458 * q^57 + 1213 * q^58 - 1410 * q^59 - 395 * q^60 + 412 * q^61 - 56 * q^62 - 1241 * q^63 - 1179 * q^64 + 2173 * q^66 - 1398 * q^67 - 493 * q^68 + 1080 * q^69 + 330 * q^70 + 584 * q^71 - 1545 * q^72 - 2538 * q^73 + 3840 * q^74 - 100 * q^75 - 3292 * q^76 - 2753 * q^77 + 464 * q^79 - 670 * q^80 - 473 * q^81 - 1583 * q^82 - 466 * q^83 + 3081 * q^84 - 570 * q^85 - 4929 * q^86 - 282 * q^87 + 3389 * q^88 - 443 * q^89 - 1035 * q^90 + 3091 * q^92 + 2116 * q^93 + 2017 * q^94 - 1225 * q^95 - 477 * q^96 + 1870 * q^97 - 1364 * q^98 - 5689 * 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 $

$2$	−0.545397	−0.192827	−0.0964135	−	0.995341i	$-0.530737\pi$
$2$	−0.545397	−0.192827	−0.0964135	+	0.995341i	$0.530737\pi$
$3$	7.02243	1.35147	0.675734	−	0.737146i	$-0.263827\pi$
$3$	7.02243	1.35147	0.675734	+	0.737146i	$0.263827\pi$
$4$	−7.70254	−0.962818
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−3.83001	−0.260599
$7$	−22.8716	−1.23495	−0.617476	−	0.786589i	$-0.711845\pi$
$7$	−22.8716	−1.23495	−0.617476	+	0.786589i	$0.711845\pi$
$8$	8.56412	0.378484
$9$	22.3146	0.826466
$10$	−2.72699	−0.0862348
$11$	−8.55617	−0.234526	−0.117263	−	0.993101i	$-0.537412\pi$
$11$	−8.55617	−0.234526	−0.117263	+	0.993101i	$0.537412\pi$
$12$	−54.0906	−1.30122
$13$	0	0
$13$	0	0
$14$	12.4741	0.238132
$15$	35.1122	0.604395
$16$	56.9495	0.889836
$17$	32.6895	0.466374	0.233187	−	0.972432i	$-0.425085\pi$
$17$	32.6895	0.466374	0.233187	+	0.972432i	$0.425085\pi$
$18$	−12.1703	−0.159365
$19$	87.1647	1.05247	0.526236	−	0.850339i	$-0.323603\pi$
$19$	87.1647	1.05247	0.526236	+	0.850339i	$0.323603\pi$
$20$	−38.5127	−0.430585
$21$	−160.615	−1.66900
$22$	4.66651	0.0452229
$23$	−8.23980	−0.0747007	−0.0373504	−	0.999302i	$-0.511892\pi$
$23$	−8.23980	−0.0747007	−0.0373504	+	0.999302i	$0.511892\pi$
$24$	60.1410	0.511509
$25$	25.0000	0.200000
$26$	0	0
$27$	−32.9030	−0.234526
$28$	176.170	1.18903
$29$	214.146	1.37124	0.685620	−	0.727959i	$-0.259531\pi$
$29$	214.146	1.37124	0.685620	+	0.727959i	$0.259531\pi$
$30$	−19.1501	−0.116544
$31$	−279.366	−1.61857	−0.809283	−	0.587419i	$-0.800144\pi$
$31$	−279.366	−1.61857	−0.809283	+	0.587419i	$0.800144\pi$
$32$	−99.5730	−0.550069
$33$	−60.0851	−0.316954
$34$	−17.8287	−0.0899295
$35$	−114.358	−0.552288
$36$	−171.879	−0.795736
$37$	−273.944	−1.21719	−0.608596	−	0.793480i	$-0.708267\pi$
$37$	−273.944	−1.21719	−0.608596	+	0.793480i	$0.708267\pi$
$38$	−47.5394	−0.202945
$39$	0	0
$40$	42.8206	0.169263
$41$	85.8319	0.326944	0.163472	−	0.986548i	$-0.447731\pi$
$41$	85.8319	0.326944	0.163472	+	0.986548i	$0.447731\pi$
$42$	87.5987	0.321828
$43$	−380.793	−1.35047	−0.675237	−	0.737601i	$-0.735958\pi$
$43$	−380.793	−1.35047	−0.675237	+	0.737601i	$0.735958\pi$
$44$	65.9042	0.225805
$45$	111.573	0.369607
$46$	4.49396	0.0144043
$47$	−435.021	−1.35009	−0.675046	−	0.737776i	$-0.735876\pi$
$47$	−435.021	−1.35009	−0.675046	+	0.737776i	$0.735876\pi$
$48$	399.924	1.20258
$49$	180.112	0.525108
$50$	−13.6349	−0.0385654
$51$	229.560	0.630290
$52$	0	0
$53$	−436.042	−1.13009	−0.565046	−	0.825059i	$-0.691142\pi$
$53$	−436.042	−1.13009	−0.565046	+	0.825059i	$0.691142\pi$
$54$	17.9452	0.0452229
$55$	−42.7808	−0.104883
$56$	−195.875	−0.467410
$57$	612.108	1.42238
$58$	−116.795	−0.264412
$59$	−749.132	−1.65303	−0.826514	−	0.562916i	$-0.809680\pi$
$59$	−749.132	−1.65303	−0.826514	+	0.562916i	$0.809680\pi$
$60$	−270.453	−0.581922
$61$	−70.6681	−0.148330	−0.0741649	−	0.997246i	$-0.523629\pi$
$61$	−70.6681	−0.148330	−0.0741649	+	0.997246i	$0.523629\pi$
$62$	152.365	0.312103
$63$	−510.371	−1.02065
$64$	−401.289	−0.783768
$65$	0	0
$66$	32.7702	0.0611172
$67$	−206.116	−0.375837	−0.187918	−	0.982185i	$-0.560174\pi$
$67$	−206.116	−0.375837	−0.187918	+	0.982185i	$0.560174\pi$
$68$	−251.792	−0.449033
$69$	−57.8635	−0.100956
$70$	62.3706	0.106496
$71$	−182.480	−0.305020	−0.152510	−	0.988302i	$-0.548736\pi$
$71$	−182.480	−0.305020	−0.152510	+	0.988302i	$0.548736\pi$
$72$	191.105	0.312804
$73$	−188.587	−0.302363	−0.151181	−	0.988506i	$-0.548308\pi$
$73$	−188.587	−0.302363	−0.151181	+	0.988506i	$0.548308\pi$
$74$	149.408	0.234707
$75$	175.561	0.270294
$76$	−671.390	−1.01334
$77$	195.694	0.289628
$78$	0	0
$79$	923.280	1.31490	0.657450	−	0.753498i	$-0.271635\pi$
$79$	923.280	1.31490	0.657450	+	0.753498i	$0.271635\pi$
$80$	284.747	0.397947
$81$	−833.553	−1.14342
$82$	−46.8125	−0.0630436
$83$	−1215.55	−1.60752	−0.803759	−	0.594955i	$-0.797170\pi$
$83$	−1215.55	−1.60752	−0.803759	+	0.594955i	$0.797170\pi$
$84$	1237.14	1.60694
$85$	163.447	0.208569
$86$	207.683	0.260408
$87$	1503.83	1.85319
$88$	−73.2760	−0.0887642
$89$	19.7604	0.0235348	0.0117674	−	0.999931i	$-0.496254\pi$
$89$	19.7604	0.0235348	0.0117674	+	0.999931i	$0.496254\pi$
$90$	−60.8515	−0.0712702
$91$	0	0
$92$	63.4674	0.0719232
$93$	−1961.83	−2.18744
$94$	237.259	0.260334
$95$	435.823	0.470679
$96$	−699.245	−0.743400
$97$	−541.901	−0.567234	−0.283617	−	0.958938i	$-0.591534\pi$
$97$	−541.901	−0.567234	−0.283617	+	0.958938i	$0.591534\pi$
$98$	−98.2326	−0.101255
$99$	−190.927	−0.193827
$100$	−192.564	−0.192564
$101$	1246.04	1.22758	0.613792	−	0.789468i	$-0.289643\pi$
$101$	1246.04	1.22758	0.613792	+	0.789468i	$0.289643\pi$
$102$	−125.201	−0.121537
$103$	439.445	0.420387	0.210193	−	0.977660i	$-0.432591\pi$
$103$	439.445	0.420387	0.210193	+	0.977660i	$0.432591\pi$
$104$	0	0
$105$	−803.073	−0.746399
$106$	237.816	0.217912
$107$	−1218.08	−1.10053	−0.550263	−	0.834992i	$-0.685472\pi$
$107$	−1218.08	−1.10053	−0.550263	+	0.834992i	$0.685472\pi$
$108$	253.437	0.225805
$109$	−503.852	−0.442755	−0.221377	−	0.975188i	$-0.571055\pi$
$109$	−503.852	−0.442755	−0.221377	+	0.975188i	$0.571055\pi$
$110$	23.3325	0.0202243
$111$	−1923.75	−1.64500
$112$	−1302.53	−1.09891
$113$	1311.41	1.09174	0.545870	−	0.837870i	$-0.316199\pi$
$113$	1311.41	1.09174	0.545870	+	0.837870i	$0.316199\pi$
$114$	−333.842	−0.274273
$115$	−41.1990	−0.0334072
$116$	−1649.47	−1.32025
$117$	0	0
$118$	408.574	0.318748
$119$	−747.662	−0.575950
$120$	300.705	0.228754
$121$	−1257.79	−0.944998
$122$	38.5421	0.0286020
$123$	602.749	0.441854
$124$	2151.82	1.55838
$125$	125.000	0.0894427
$126$	278.355	0.196808
$127$	2438.11	1.70352	0.851760	−	0.523933i	$-0.175536\pi$
$127$	2438.11	1.70352	0.851760	+	0.523933i	$0.175536\pi$
$128$	1015.45	0.701200
$129$	−2674.09	−1.82512
$130$	0	0
$131$	−1854.96	−1.23716	−0.618582	−	0.785720i	$-0.712293\pi$
$131$	−1854.96	−1.23716	−0.618582	+	0.785720i	$0.712293\pi$
$132$	462.808	0.305169
$133$	−1993.60	−1.29975
$134$	112.415	0.0724715
$135$	−164.515	−0.104883
$136$	279.956	0.176515
$137$	2867.68	1.78834	0.894168	−	0.447732i	$-0.147768\pi$
$137$	2867.68	1.78834	0.894168	+	0.447732i	$0.147768\pi$
$138$	31.5586	0.0194670
$139$	2004.87	1.22339	0.611693	−	0.791095i	$-0.290489\pi$
$139$	2004.87	1.22339	0.611693	+	0.791095i	$0.290489\pi$
$140$	880.849	0.531752
$141$	−3054.91	−1.82461
$142$	99.5242	0.0588161
$143$	0	0
$144$	1270.80	0.735419
$145$	1070.73	0.613237
$146$	102.855	0.0583037
$147$	1264.83	0.709667
$148$	2110.06	1.17193
$149$	34.9820	0.0192338	0.00961690	−	0.999954i	$-0.496939\pi$
$149$	34.9820	0.0192338	0.00961690	+	0.999954i	$0.496939\pi$
$150$	−95.7504	−0.0521199
$151$	2486.15	1.33987	0.669934	−	0.742421i	$-0.266322\pi$
$151$	2486.15	1.33987	0.669934	+	0.742421i	$0.266322\pi$
$152$	746.489	0.398344
$153$	729.452	0.385442
$154$	−106.731	−0.0558481
$155$	−1396.83	−0.723844
$156$	0	0
$157$	3322.18	1.68878	0.844391	−	0.535728i	$-0.179963\pi$
$157$	3322.18	1.68878	0.844391	+	0.535728i	$0.179963\pi$
$158$	−503.554	−0.253548
$159$	−3062.07	−1.52728
$160$	−497.865	−0.245998
$161$	188.458	0.0922519
$162$	454.617	0.220482
$163$	−625.412	−0.300528	−0.150264	−	0.988646i	$-0.548012\pi$
$163$	−625.412	−0.300528	−0.150264	+	0.988646i	$0.548012\pi$
$164$	−661.124	−0.314787
$165$	−300.426	−0.141746
$166$	662.958	0.309973
$167$	502.790	0.232976	0.116488	−	0.993192i	$-0.462836\pi$
$167$	502.790	0.232976	0.116488	+	0.993192i	$0.462836\pi$
$168$	−1375.52	−0.631690
$169$	0	0
$170$	−89.1437	−0.0402177
$171$	1945.04	0.869832
$172$	2933.07	1.30026
$173$	−2244.74	−0.986498	−0.493249	−	0.869888i	$-0.664191\pi$
$173$	−2244.74	−0.986498	−0.493249	+	0.869888i	$0.664191\pi$
$174$	−820.183	−0.357345
$175$	−571.791	−0.246991
$176$	−487.269	−0.208689
$177$	−5260.73	−2.23402
$178$	−10.7773	−0.00453815
$179$	−3100.82	−1.29478	−0.647392	−	0.762157i	$-0.724140\pi$
$179$	−3100.82	−1.29478	−0.647392	+	0.762157i	$0.724140\pi$
$180$	−859.395	−0.355864
$181$	74.4030	0.0305543	0.0152772	−	0.999883i	$-0.495137\pi$
$181$	74.4030	0.0305543	0.0152772	+	0.999883i	$0.495137\pi$
$182$	0	0
$183$	−496.262	−0.200463
$184$	−70.5666	−0.0282730
$185$	−1369.72	−0.544345
$186$	1069.97	0.421797
$187$	−279.696	−0.109377
$188$	3350.77	1.29989
$189$	752.547	0.289628
$190$	−237.697	−0.0907597
$191$	−4867.31	−1.84391	−0.921953	−	0.387302i	$-0.873407\pi$
$191$	−4867.31	−1.84391	−0.921953	+	0.387302i	$0.873407\pi$
$192$	−2818.03	−1.05924
$193$	−247.127	−0.0921688	−0.0460844	−	0.998938i	$-0.514674\pi$
$193$	−247.127	−0.0921688	−0.0460844	+	0.998938i	$0.514674\pi$
$194$	295.551	0.109378
$195$	0	0
$196$	−1387.32	−0.505583
$197$	−3513.77	−1.27079	−0.635395	−	0.772187i	$-0.719163\pi$
$197$	−3513.77	−1.27079	−0.635395	+	0.772187i	$0.719163\pi$
$198$	104.131	0.0373752
$199$	−3360.99	−1.19726	−0.598629	−	0.801026i	$-0.704288\pi$
$199$	−3360.99	−1.19726	−0.598629	+	0.801026i	$0.704288\pi$
$200$	214.103	0.0756968
$201$	−1447.44	−0.507932
$202$	−679.589	−0.236711
$203$	−4897.88	−1.69342
$204$	−1768.19	−0.606854
$205$	429.160	0.146214
$206$	−239.672	−0.0810619
$207$	−183.868	−0.0617376
$208$	0	0
$209$	−745.796	−0.246831
$210$	437.994	0.143926
$211$	1588.94	0.518423	0.259211	−	0.965821i	$-0.416537\pi$
$211$	1588.94	0.518423	0.259211	+	0.965821i	$0.416537\pi$
$212$	3358.63	1.08807
$213$	−1281.46	−0.412225
$214$	664.337	0.212211
$215$	−1903.96	−0.603950
$216$	−281.786	−0.0887643
$217$	6389.55	1.99885
$218$	274.799	0.0853750
$219$	−1324.34	−0.408633
$220$	329.521	0.100983
$221$	0	0
$222$	1049.21	0.317199
$223$	2850.19	0.855887	0.427944	−	0.903805i	$-0.359238\pi$
$223$	2850.19	0.855887	0.427944	+	0.903805i	$0.359238\pi$
$224$	2277.40	0.679309
$225$	557.865	0.165293
$226$	−715.237	−0.210517
$227$	−3762.83	−1.10021	−0.550106	−	0.835095i	$-0.685413\pi$
$227$	−3762.83	−1.10021	−0.550106	+	0.835095i	$0.685413\pi$
$228$	−4714.79	−1.36949
$229$	2209.34	0.637543	0.318772	−	0.947832i	$-0.396730\pi$
$229$	2209.34	0.637543	0.318772	+	0.947832i	$0.396730\pi$
$230$	22.4698	0.00644181
$231$	1374.25	0.391423
$232$	1833.97	0.518993
$233$	−1399.97	−0.393627	−0.196813	−	0.980441i	$-0.563059\pi$
$233$	−1399.97	−0.393627	−0.196813	+	0.980441i	$0.563059\pi$
$234$	0	0
$235$	−2175.10	−0.603779
$236$	5770.22	1.59157
$237$	6483.68	1.77705
$238$	407.772	0.111059
$239$	158.031	0.0427706	0.0213853	−	0.999771i	$-0.493192\pi$
$239$	158.031	0.0427706	0.0213853	+	0.999771i	$0.493192\pi$
$240$	1999.62	0.537812
$241$	−2669.64	−0.713556	−0.356778	−	0.934189i	$-0.616125\pi$
$241$	−2669.64	−0.713556	−0.356778	+	0.934189i	$0.616125\pi$
$242$	685.996	0.182221
$243$	−4965.19	−1.31077
$244$	544.324	0.142815
$245$	900.560	0.234835
$246$	−328.738	−0.0852014
$247$	0	0
$248$	−2392.52	−0.612601
$249$	−8536.12	−2.17251
$250$	−68.1746	−0.0172470
$251$	−6826.55	−1.71669	−0.858343	−	0.513076i	$-0.828506\pi$
$251$	−6826.55	−1.71669	−0.858343	+	0.513076i	$0.828506\pi$
$252$	3931.16	0.982696
$253$	70.5011	0.0175192
$254$	−1329.74	−0.328484
$255$	1147.80	0.281874
$256$	2656.49	0.648557
$257$	2752.19	0.668005	0.334002	−	0.942572i	$-0.391601\pi$
$257$	2752.19	0.668005	0.334002	+	0.942572i	$0.391601\pi$
$258$	1458.44	0.351933
$259$	6265.54	1.50317
$260$	0	0
$261$	4778.58	1.13328
$262$	1011.69	0.238559
$263$	2734.17	0.641050	0.320525	−	0.947240i	$-0.396141\pi$
$263$	2734.17	0.641050	0.320525	+	0.947240i	$0.396141\pi$
$264$	−514.576	−0.119962
$265$	−2180.21	−0.505393
$266$	1087.30	0.250627
$267$	138.766	0.0318066
$268$	1587.62	0.361862
$269$	5715.83	1.29554	0.647770	−	0.761836i	$-0.275702\pi$
$269$	5715.83	1.29554	0.647770	+	0.761836i	$0.275702\pi$
$270$	89.7261	0.0202243
$271$	1314.54	0.294659	0.147330	−	0.989087i	$-0.452932\pi$
$271$	1314.54	0.294659	0.147330	+	0.989087i	$0.452932\pi$
$272$	1861.65	0.414996
$273$	0	0
$274$	−1564.02	−0.344839
$275$	−213.904	−0.0469051
$276$	445.696	0.0972019
$277$	−2033.87	−0.441168	−0.220584	−	0.975368i	$-0.570796\pi$
$277$	−2033.87	−0.441168	−0.220584	+	0.975368i	$0.570796\pi$
$278$	−1093.45	−0.235902
$279$	−6233.92	−1.33769
$280$	−979.377	−0.209032
$281$	3958.11	0.840289	0.420144	−	0.907457i	$-0.361979\pi$
$281$	3958.11	0.840289	0.420144	+	0.907457i	$0.361979\pi$
$282$	1666.14	0.351833
$283$	−6095.65	−1.28039	−0.640193	−	0.768214i	$-0.721145\pi$
$283$	−6095.65	−1.28039	−0.640193	+	0.768214i	$0.721145\pi$
$284$	1405.56	0.293679
$285$	3060.54	0.636108
$286$	0	0
$287$	−1963.12	−0.403760
$288$	−2221.93	−0.454613
$289$	−3844.40	−0.782495
$290$	−583.974	−0.118249
$291$	−3805.46	−0.766599
$292$	1452.60	0.291120
$293$	3584.79	0.714762	0.357381	−	0.933959i	$-0.383670\pi$
$293$	3584.79	0.714762	0.357381	+	0.933959i	$0.383670\pi$
$294$	−689.832	−0.136843
$295$	−3745.66	−0.739257
$296$	−2346.09	−0.460688
$297$	281.524	0.0550023
$298$	−19.0791	−0.00370880
$299$	0	0
$300$	−1352.26	−0.260243
$301$	8709.35	1.66777
$302$	−1355.94	−0.258363
$303$	8750.26	1.65904
$304$	4963.98	0.936526
$305$	−353.340	−0.0663351
$306$	−397.841	−0.0743237
$307$	2943.87	0.547283	0.273641	−	0.961832i	$-0.411772\pi$
$307$	2943.87	0.547283	0.273641	+	0.961832i	$0.411772\pi$
$308$	−1507.34	−0.278859
$309$	3085.98	0.568139
$310$	761.826	0.139577
$311$	8067.29	1.47091	0.735457	−	0.677571i	$-0.236967\pi$
$311$	8067.29	1.47091	0.735457	+	0.677571i	$0.236967\pi$
$312$	0	0
$313$	−6091.78	−1.10009	−0.550044	−	0.835135i	$-0.685389\pi$
$313$	−6091.78	−1.10009	−0.550044	+	0.835135i	$0.685389\pi$
$314$	−1811.91	−0.325643
$315$	−2551.86	−0.456447
$316$	−7111.61	−1.26601
$317$	−2247.21	−0.398157	−0.199079	−	0.979984i	$-0.563795\pi$
$317$	−2247.21	−0.398157	−0.199079	+	0.979984i	$0.563795\pi$
$318$	1670.05	0.294502
$319$	−1832.27	−0.321591
$320$	−2006.45	−0.350512
$321$	−8553.89	−1.48733
$322$	−102.784	−0.0177886
$323$	2849.37	0.490845
$324$	6420.48	1.10091
$325$	0	0
$326$	341.098	0.0579499
$327$	−3538.27	−0.598369
$328$	735.075	0.123743
$329$	9949.64	1.66730
$330$	163.851	0.0273325
$331$	−9367.84	−1.55560	−0.777799	−	0.628513i	$-0.783664\pi$
$331$	−9367.84	−1.55560	−0.777799	+	0.628513i	$0.783664\pi$
$332$	9362.83	1.54775
$333$	−6112.94	−1.00597
$334$	−274.220	−0.0449241
$335$	−1030.58	−0.168079
$336$	−9146.92	−1.48514
$337$	2123.63	0.343269	0.171635	−	0.985161i	$-0.445095\pi$
$337$	2123.63	0.343269	0.171635	+	0.985161i	$0.445095\pi$
$338$	0	0
$339$	9209.26	1.47545
$340$	−1258.96	−0.200814
$341$	2390.30	0.379595
$342$	−1060.82	−0.167727
$343$	3725.51	0.586469
$344$	−3261.15	−0.511133
$345$	−289.317	−0.0451487
$346$	1224.27	0.190224
$347$	−7948.85	−1.22973	−0.614865	−	0.788632i	$-0.710790\pi$
$347$	−7948.85	−1.22973	−0.614865	+	0.788632i	$0.710790\pi$
$348$	−11583.3	−1.78428
$349$	2404.11	0.368737	0.184369	−	0.982857i	$-0.440976\pi$
$349$	2404.11	0.368737	0.184369	+	0.982857i	$0.440976\pi$
$350$	311.853	0.0476264
$351$	0	0
$352$	851.964	0.129005
$353$	−2111.79	−0.318412	−0.159206	−	0.987245i	$-0.550893\pi$
$353$	−2111.79	−0.318412	−0.159206	+	0.987245i	$0.550893\pi$
$354$	2869.19	0.430778
$355$	−912.401	−0.136409
$356$	−152.205	−0.0226598
$357$	−5250.41	−0.778378
$358$	1691.18	0.249669
$359$	9615.51	1.41361	0.706807	−	0.707407i	$-0.250135\pi$
$359$	9615.51	1.41361	0.706807	+	0.707407i	$0.250135\pi$
$360$	955.524	0.139890
$361$	738.682	0.107695
$362$	−40.5792	−0.00589170
$363$	−8832.76	−1.27713
$364$	0	0
$365$	−942.936	−0.135221
$366$	270.660	0.0386547
$367$	−3736.11	−0.531399	−0.265699	−	0.964056i	$-0.585603\pi$
$367$	−3736.11	−0.531399	−0.265699	+	0.964056i	$0.585603\pi$
$368$	−469.252	−0.0664714
$369$	1915.30	0.270208
$370$	747.041	0.104964
$371$	9972.99	1.39561
$372$	15111.0	2.10611
$373$	485.602	0.0674089	0.0337044	−	0.999432i	$-0.489270\pi$
$373$	485.602	0.0674089	0.0337044	+	0.999432i	$0.489270\pi$
$374$	152.546	0.0210908
$375$	877.804	0.120879
$376$	−3725.57	−0.510988
$377$	0	0
$378$	−410.437	−0.0558481
$379$	3381.69	0.458326	0.229163	−	0.973388i	$-0.426401\pi$
$379$	3381.69	0.458326	0.229163	+	0.973388i	$0.426401\pi$
$380$	−3356.95	−0.453178
$381$	17121.4	2.30225
$382$	2654.62	0.355555
$383$	9759.00	1.30199	0.650994	−	0.759083i	$-0.274352\pi$
$383$	9759.00	1.30199	0.650994	+	0.759083i	$0.274352\pi$
$384$	7130.90	0.947650
$385$	978.468	0.129526
$386$	134.782	0.0177726
$387$	−8497.23	−1.11612
$388$	4174.02	0.546143
$389$	−6561.59	−0.855233	−0.427617	−	0.903960i	$-0.640647\pi$
$389$	−6561.59	−0.855233	−0.427617	+	0.903960i	$0.640647\pi$
$390$	0	0
$391$	−269.355	−0.0348385
$392$	1542.50	0.198745
$393$	−13026.3	−1.67199
$394$	1916.40	0.245043
$395$	4616.40	0.588041
$396$	1470.63	0.186620
$397$	−1813.89	−0.229311	−0.114655	−	0.993405i	$-0.536576\pi$
$397$	−1813.89	−0.229311	−0.114655	+	0.993405i	$0.536576\pi$
$398$	1833.08	0.230864
$399$	−13999.9	−1.75657
$400$	1423.74	0.177967
$401$	9045.10	1.12641	0.563206	−	0.826317i	$-0.309568\pi$
$401$	9045.10	1.12641	0.563206	+	0.826317i	$0.309568\pi$
$402$	789.427	0.0979429
$403$	0	0
$404$	−9597.71	−1.18194
$405$	−4167.77	−0.511353
$406$	2671.29	0.326536
$407$	2343.91	0.285463
$408$	1965.98	0.238555
$409$	10109.9	1.22225	0.611125	−	0.791534i	$-0.290717\pi$
$409$	10109.9	1.22225	0.611125	+	0.791534i	$0.290717\pi$
$410$	−234.062	−0.0281939
$411$	20138.1	2.41688
$412$	−3384.85	−0.404756
$413$	17133.9	2.04141
$414$	100.281	0.0119047
$415$	−6077.75	−0.718904
$416$	0	0
$417$	14079.0	1.65337
$418$	406.755	0.0475957
$419$	2519.34	0.293742	0.146871	−	0.989156i	$-0.453080\pi$
$419$	2519.34	0.293742	0.146871	+	0.989156i	$0.453080\pi$
$420$	6185.70	0.718646
$421$	1683.18	0.194853	0.0974264	−	0.995243i	$-0.468939\pi$
$421$	1683.18	0.194853	0.0974264	+	0.995243i	$0.468939\pi$
$422$	−866.603	−0.0999659
$423$	−9707.31	−1.11581
$424$	−3734.31	−0.427722
$425$	817.237	0.0932748
$426$	698.902	0.0794880
$427$	1616.29	0.183180
$428$	9382.31	1.05961
$429$	0	0
$430$	1038.42	0.116458
$431$	−4048.81	−0.452492	−0.226246	−	0.974070i	$-0.572645\pi$
$431$	−4048.81	−0.452492	−0.226246	+	0.974070i	$0.572645\pi$
$432$	−1873.81	−0.208689
$433$	1697.78	0.188430	0.0942151	−	0.995552i	$-0.469966\pi$
$433$	1697.78	0.188430	0.0942151	+	0.995552i	$0.469966\pi$
$434$	−3484.84	−0.385432
$435$	7519.14	0.828771
$436$	3880.94	0.426292
$437$	−718.220	−0.0786204
$438$	722.292	0.0787956
$439$	6527.57	0.709667	0.354833	−	0.934930i	$-0.384538\pi$
$439$	6527.57	0.709667	0.354833	+	0.934930i	$0.384538\pi$
$440$	−366.380	−0.0396966
$441$	4019.13	0.433984
$442$	0	0
$443$	−992.738	−0.106470	−0.0532352	−	0.998582i	$-0.516953\pi$
$443$	−992.738	−0.106470	−0.0532352	+	0.998582i	$0.516953\pi$
$444$	14817.8	1.58383
$445$	98.8021	0.0105251
$446$	−1554.48	−0.165038
$447$	245.659	0.0259939
$448$	9178.14	0.967916
$449$	8797.56	0.924682	0.462341	−	0.886702i	$-0.347010\pi$
$449$	8797.56	0.924682	0.462341	+	0.886702i	$0.347010\pi$
$450$	−304.258	−0.0318730
$451$	−734.392	−0.0766767
$452$	−10101.2	−1.05115
$453$	17458.8	1.81079
$454$	2052.24	0.212151
$455$	0	0
$456$	5242.17	0.538349
$457$	6601.60	0.675733	0.337866	−	0.941194i	$-0.390295\pi$
$457$	6601.60	0.675733	0.337866	+	0.941194i	$0.390295\pi$
$458$	−1204.97	−0.122936
$459$	−1075.58	−0.109377
$460$	317.337	0.0321650
$461$	−16958.7	−1.71334	−0.856668	−	0.515869i	$-0.827469\pi$
$461$	−16958.7	−1.71334	−0.856668	+	0.515869i	$0.827469\pi$
$462$	−749.509	−0.0754769
$463$	−5002.16	−0.502095	−0.251048	−	0.967975i	$-0.580775\pi$
$463$	−5002.16	−0.502095	−0.251048	+	0.967975i	$0.580775\pi$
$464$	12195.5	1.22018
$465$	−9809.13	−0.978253
$466$	763.539	0.0759019
$467$	896.692	0.0888522	0.0444261	−	0.999013i	$-0.485854\pi$
$467$	896.692	0.0888522	0.0444261	+	0.999013i	$0.485854\pi$
$468$	0	0
$469$	4714.21	0.464141
$470$	1186.30	0.116425
$471$	23329.8	2.28233
$472$	−6415.66	−0.625645
$473$	3258.13	0.316720
$474$	−3536.18	−0.342662
$475$	2179.12	0.210494
$476$	5758.90	0.554535
$477$	−9730.09	−0.933983
$478$	−86.1897	−0.00824733
$479$	14629.8	1.39551	0.697757	−	0.716335i	$-0.254182\pi$
$479$	14629.8	1.39551	0.697757	+	0.716335i	$0.254182\pi$
$480$	−3496.23	−0.332459
$481$	0	0
$482$	1456.02	0.137593
$483$	1323.43	0.124675
$484$	9688.20	0.909861
$485$	−2709.51	−0.253675
$486$	2708.00	0.252752
$487$	5871.09	0.546292	0.273146	−	0.961973i	$-0.411936\pi$
$487$	5871.09	0.546292	0.273146	+	0.961973i	$0.411936\pi$
$488$	−605.210	−0.0561405
$489$	−4391.92	−0.406154
$490$	−491.163	−0.0452826
$491$	8389.73	0.771126	0.385563	−	0.922681i	$-0.374007\pi$
$491$	8389.73	0.771126	0.385563	+	0.922681i	$0.374007\pi$
$492$	−4642.70	−0.425425
$493$	7000.33	0.639511
$494$	0	0
$495$	−954.636	−0.0866823
$496$	−15909.7	−1.44026
$497$	4173.62	0.376685
$498$	4655.58	0.418918
$499$	−12745.1	−1.14339	−0.571693	−	0.820468i	$-0.693713\pi$
$499$	−12745.1	−1.14339	−0.571693	+	0.820468i	$0.693713\pi$
$500$	−962.818	−0.0861170
$501$	3530.81	0.314860
$502$	3723.18	0.331023
$503$	−15038.8	−1.33310	−0.666549	−	0.745461i	$-0.732229\pi$
$503$	−15038.8	−1.33310	−0.666549	+	0.745461i	$0.732229\pi$
$504$	−4370.88	−0.386299
$505$	6230.22	0.548992
$506$	−38.4511	−0.00337818
$507$	0	0
$508$	−18779.6	−1.64018
$509$	18597.7	1.61950	0.809751	−	0.586773i	$-0.199602\pi$
$509$	18597.7	1.61950	0.809751	+	0.586773i	$0.199602\pi$
$510$	−626.006	−0.0543529
$511$	4313.30	0.373404
$512$	−9572.41	−0.826259
$513$	−2867.98	−0.246831
$514$	−1501.04	−0.128809
$515$	2197.23	0.188003
$516$	20597.3	1.75726
$517$	3722.11	0.316631
$518$	−3417.21	−0.289852
$519$	−15763.5	−1.33322
$520$	0	0
$521$	22308.4	1.87591	0.937953	−	0.346761i	$-0.112719\pi$
$521$	22308.4	1.87591	0.937953	+	0.346761i	$0.112719\pi$
$522$	−2606.23	−0.218528
$523$	−4538.60	−0.379463	−0.189731	−	0.981836i	$-0.560762\pi$
$523$	−4538.60	−0.379463	−0.189731	+	0.981836i	$0.560762\pi$
$524$	14287.9	1.19116
$525$	−4015.37	−0.333800
$526$	−1491.21	−0.123612
$527$	−9132.31	−0.754857
$528$	−3421.82	−0.282037
$529$	−12099.1	−0.994420
$530$	1189.08	0.0974534
$531$	−16716.6	−1.36617
$532$	15355.8	1.25142
$533$	0	0
$534$	−75.6827	−0.00613317
$535$	−6090.40	−0.492170
$536$	−1765.20	−0.142248
$537$	−21775.3	−1.74986
$538$	−3117.40	−0.249815
$539$	−1541.07	−0.123151
$540$	1267.19	0.100983
$541$	−23005.3	−1.82824	−0.914118	−	0.405448i	$-0.867116\pi$
$541$	−23005.3	−1.82824	−0.914118	+	0.405448i	$0.867116\pi$
$542$	−716.947	−0.0568182
$543$	522.490	0.0412932
$544$	−3254.99	−0.256538
$545$	−2519.26	−0.198006
$546$	0	0
$547$	−1258.68	−0.0983863	−0.0491932	−	0.998789i	$-0.515665\pi$
$547$	−1258.68	−0.0983863	−0.0491932	+	0.998789i	$0.515665\pi$
$548$	−22088.4	−1.72184
$549$	−1576.93	−0.122590
$550$	116.663	0.00904457
$551$	18666.0	1.44319
$552$	−495.550	−0.0382101
$553$	−21116.9	−1.62384
$554$	1109.27	0.0850690
$555$	−9618.76	−0.735664
$556$	−15442.6	−1.17790
$557$	−17062.2	−1.29793	−0.648967	−	0.760816i	$-0.724799\pi$
$557$	−17062.2	−1.29793	−0.648967	+	0.760816i	$0.724799\pi$
$558$	3399.96	0.257943
$559$	0	0
$560$	−6512.64	−0.491445
$561$	−1964.15	−0.147819
$562$	−2158.74	−0.162030
$563$	13505.2	1.01097	0.505485	−	0.862835i	$-0.331314\pi$
$563$	13505.2	1.01097	0.505485	+	0.862835i	$0.331314\pi$
$564$	23530.5	1.75676
$565$	6557.03	0.488241
$566$	3324.55	0.246893
$567$	19064.7	1.41207
$568$	−1562.78	−0.115445
$569$	4804.67	0.353993	0.176997	−	0.984211i	$-0.443362\pi$
$569$	4804.67	0.353993	0.176997	+	0.984211i	$0.443362\pi$
$570$	−1669.21	−0.122659
$571$	18967.1	1.39010	0.695051	−	0.718961i	$-0.255382\pi$
$571$	18967.1	1.39010	0.695051	+	0.718961i	$0.255382\pi$
$572$	0	0
$573$	−34180.4	−2.49198
$574$	1070.68	0.0778558
$575$	−205.995	−0.0149401
$576$	−8954.60	−0.647757
$577$	−6023.19	−0.434573	−0.217286	−	0.976108i	$-0.569721\pi$
$577$	−6023.19	−0.434573	−0.217286	+	0.976108i	$0.569721\pi$
$578$	2096.72	0.150886
$579$	−1735.43	−0.124563
$580$	−8247.35	−0.590436
$581$	27801.6	1.98521
$582$	2075.49	0.147821
$583$	3730.84	0.265036
$584$	−1615.08	−0.114439
$585$	0	0
$586$	−1955.13	−0.137825
$587$	14239.7	1.00125	0.500625	−	0.865664i	$-0.333104\pi$
$587$	14239.7	1.00125	0.500625	+	0.865664i	$0.333104\pi$
$588$	−9742.37	−0.683280
$589$	−24350.8	−1.70349
$590$	2042.87	0.142549
$591$	−24675.2	−1.71743
$592$	−15601.0	−1.08310
$593$	−27625.9	−1.91309	−0.956544	−	0.291587i	$-0.905817\pi$
$593$	−27625.9	−1.91309	−0.956544	+	0.291587i	$0.905817\pi$
$594$	−153.542	−0.0106059
$595$	−3738.31	−0.257573
$596$	−269.450	−0.0185186
$597$	−23602.4	−1.61806
$598$	0	0
$599$	4314.14	0.294276	0.147138	−	0.989116i	$-0.452994\pi$
$599$	4314.14	0.294276	0.147138	+	0.989116i	$0.452994\pi$
$600$	1503.52	0.102302
$601$	11304.8	0.767276	0.383638	−	0.923483i	$-0.374671\pi$
$601$	11304.8	0.767276	0.383638	+	0.923483i	$0.374671\pi$
$602$	−4750.06	−0.321591
$603$	−4599.39	−0.310616
$604$	−19149.7	−1.29005
$605$	−6288.96	−0.422616
$606$	−4772.37	−0.319908
$607$	−2544.23	−0.170127	−0.0850636	−	0.996376i	$-0.527109\pi$
$607$	−2544.23	−0.170127	−0.0850636	+	0.996376i	$0.527109\pi$
$608$	−8679.25	−0.578931
$609$	−34395.0	−2.28860
$610$	192.711	0.0127912
$611$	0	0
$612$	−5618.63	−0.371111
$613$	1239.83	0.0816902	0.0408451	−	0.999165i	$-0.486995\pi$
$613$	1239.83	0.0816902	0.0408451	+	0.999165i	$0.486995\pi$
$614$	−1605.58	−0.105531
$615$	3013.75	0.197603
$616$	1675.94	0.109620
$617$	−9336.66	−0.609206	−0.304603	−	0.952479i	$-0.598524\pi$
$617$	−9336.66	−0.609206	−0.304603	+	0.952479i	$0.598524\pi$
$618$	−1683.08	−0.109553
$619$	−10049.7	−0.652553	−0.326276	−	0.945274i	$-0.605794\pi$
$619$	−10049.7	−0.652553	−0.326276	+	0.945274i	$0.605794\pi$
$620$	10759.1	0.696930
$621$	271.114	0.0175192
$622$	−4399.88	−0.283632
$623$	−451.953	−0.0290644
$624$	0	0
$625$	625.000	0.0400000
$626$	3322.44	0.212127
$627$	−5237.30	−0.333585
$628$	−25589.2	−1.62599
$629$	−8955.07	−0.567666
$630$	1391.77	0.0880153
$631$	14261.1	0.899725	0.449862	−	0.893098i	$-0.351473\pi$
$631$	14261.1	0.899725	0.449862	+	0.893098i	$0.351473\pi$
$632$	7907.08	0.497669
$633$	11158.2	0.700632
$634$	1225.62	0.0767754
$635$	12190.5	0.761837
$636$	23585.8	1.47050
$637$	0	0
$638$	999.315	0.0620114
$639$	−4071.97	−0.252089
$640$	5077.23	0.313586
$641$	−22478.6	−1.38511	−0.692553	−	0.721367i	$-0.743514\pi$
$641$	−22478.6	−1.38511	−0.692553	+	0.721367i	$0.743514\pi$
$642$	4665.26	0.286796
$643$	−1146.30	−0.0703044	−0.0351522	−	0.999382i	$-0.511192\pi$
$643$	−1146.30	−0.0703044	−0.0351522	+	0.999382i	$0.511192\pi$
$644$	−1451.60	−0.0888217
$645$	−13370.5	−0.816219
$646$	−1554.04	−0.0946482
$647$	11834.4	0.719099	0.359550	−	0.933126i	$-0.382930\pi$
$647$	11834.4	0.719099	0.359550	+	0.933126i	$0.382930\pi$
$648$	−7138.65	−0.432766
$649$	6409.70	0.387677
$650$	0	0
$651$	44870.2	2.70138
$652$	4817.26	0.289354
$653$	−26538.5	−1.59040	−0.795201	−	0.606347i	$-0.792634\pi$
$653$	−26538.5	−1.59040	−0.795201	+	0.606347i	$0.792634\pi$
$654$	1929.76	0.115382
$655$	−9274.80	−0.553277
$656$	4888.08	0.290926
$657$	−4208.25	−0.249892
$658$	−5426.51	−0.321500
$659$	−6579.32	−0.388913	−0.194457	−	0.980911i	$-0.562294\pi$
$659$	−6579.32	−0.388913	−0.194457	+	0.980911i	$0.562294\pi$
$660$	2314.04	0.136476
$661$	3475.94	0.204536	0.102268	−	0.994757i	$-0.467390\pi$
$661$	3475.94	0.204536	0.102268	+	0.994757i	$0.467390\pi$
$662$	5109.19	0.299961
$663$	0	0
$664$	−10410.1	−0.608420
$665$	−9968.00	−0.581267
$666$	3333.98	0.193978
$667$	−1764.52	−0.102433
$668$	−3872.76	−0.224314
$669$	20015.3	1.15670
$670$	562.075	0.0324102
$671$	604.648	0.0347871
$672$	15992.9	0.918064
$673$	8273.19	0.473860	0.236930	−	0.971527i	$-0.423859\pi$
$673$	8273.19	0.473860	0.236930	+	0.971527i	$0.423859\pi$
$674$	−1158.22	−0.0661916
$675$	−822.576	−0.0469051
$676$	0	0
$677$	−24565.9	−1.39460	−0.697299	−	0.716781i	$-0.745615\pi$
$677$	−24565.9	−1.39460	−0.697299	+	0.716781i	$0.745615\pi$
$678$	−5022.71	−0.284507
$679$	12394.2	0.700507
$680$	1399.78	0.0789400
$681$	−26424.3	−1.48690
$682$	−1303.66	−0.0731961
$683$	648.410	0.0363261	0.0181631	−	0.999835i	$-0.494218\pi$
$683$	648.410	0.0363261	0.0181631	+	0.999835i	$0.494218\pi$
$684$	−14981.8	−0.837489
$685$	14338.4	0.799768
$686$	−2031.88	−0.113087
$687$	15515.0	0.861620
$688$	−21685.9	−1.20170
$689$	0	0
$690$	157.793	0.00870590
$691$	−12078.5	−0.664959	−0.332480	−	0.943110i	$-0.607885\pi$
$691$	−12078.5	−0.664959	−0.332480	+	0.943110i	$0.607885\pi$
$692$	17290.2	0.949818
$693$	4366.82	0.239368
$694$	4335.28	0.237125
$695$	10024.3	0.547115
$696$	12879.0	0.701402
$697$	2805.80	0.152478
$698$	−1311.20	−0.0711024
$699$	−9831.19	−0.531974
$700$	4404.24	0.237807
$701$	−30979.5	−1.66916	−0.834579	−	0.550888i	$-0.814289\pi$
$701$	−30979.5	−1.66916	−0.834579	+	0.550888i	$0.814289\pi$
$702$	0	0
$703$	−23878.2	−1.28106
$704$	3433.50	0.183814
$705$	−15274.5	−0.815989
$706$	1151.76	0.0613984
$707$	−28499.1	−1.51601
$708$	40521.0	2.15095
$709$	−31683.2	−1.67826	−0.839131	−	0.543929i	$-0.816936\pi$
$709$	−31683.2	−1.67826	−0.839131	+	0.543929i	$0.816936\pi$
$710$	497.621	0.0263033
$711$	20602.6	1.08672
$712$	169.231	0.00890756
$713$	2301.92	0.120908
$714$	2863.56	0.150092
$715$	0	0
$716$	23884.2	1.24664
$717$	1109.76	0.0578031
$718$	−5244.27	−0.272583
$719$	8.77282	0.000455036 0	0.000227518	−	1.00000i	$-0.499928\pi$
$719$	8.77282	0.000455036 0	0.000227518		1.00000i	$0.499928\pi$
$720$	6354.02	0.328889
$721$	−10050.8	−0.519158
$722$	−402.875	−0.0207666
$723$	−18747.4	−0.964348
$724$	−573.092	−0.0294182
$725$	5353.66	0.274248
$726$	4817.36	0.246266
$727$	−19474.1	−0.993470	−0.496735	−	0.867902i	$-0.665468\pi$
$727$	−19474.1	−0.993470	−0.496735	+	0.867902i	$0.665468\pi$
$728$	0	0
$729$	−12361.8	−0.628044
$730$	514.275	0.0260742
$731$	−12447.9	−0.629826
$732$	3822.48	0.193009
$733$	−9433.64	−0.475361	−0.237680	−	0.971343i	$-0.576387\pi$
$733$	−9433.64	−0.475361	−0.237680	+	0.971343i	$0.576387\pi$
$734$	2037.66	0.102468
$735$	6324.13	0.317373
$736$	820.462	0.0410905
$737$	1763.56	0.0881434
$738$	−1044.60	−0.0521034
$739$	−7030.90	−0.349981	−0.174990	−	0.984570i	$-0.555989\pi$
$739$	−7030.90	−0.349981	−0.174990	+	0.984570i	$0.555989\pi$
$740$	10550.3	0.524105
$741$	0	0
$742$	−5439.24	−0.269111
$743$	4778.52	0.235945	0.117972	−	0.993017i	$-0.462361\pi$
$743$	4778.52	0.235945	0.117972	+	0.993017i	$0.462361\pi$
$744$	−16801.3	−0.827911
$745$	174.910	0.00860162
$746$	−264.846	−0.0129983
$747$	−27124.5	−1.32856
$748$	2154.37	0.105310
$749$	27859.5	1.35910
$750$	−478.752	−0.0233087
$751$	−2726.58	−0.132482	−0.0662412	−	0.997804i	$-0.521101\pi$
$751$	−2726.58	−0.132482	−0.0662412	+	0.997804i	$0.521101\pi$
$752$	−24774.2	−1.20136
$753$	−47939.0	−2.32005
$754$	0	0
$755$	12430.8	0.599207
$756$	−5796.52	−0.278859
$757$	25316.0	1.21549	0.607744	−	0.794133i	$-0.292075\pi$
$757$	25316.0	1.21549	0.607744	+	0.794133i	$0.292075\pi$
$758$	−1844.36	−0.0883776
$759$	495.089	0.0236767
$760$	3732.44	0.178145
$761$	27388.0	1.30462	0.652310	−	0.757952i	$-0.273800\pi$
$761$	27388.0	1.30462	0.652310	+	0.757952i	$0.273800\pi$
$762$	−9337.98	−0.443936
$763$	11523.9	0.546781
$764$	37490.6	1.77535
$765$	3647.26	0.172375
$766$	−5322.53	−0.251058
$767$	0	0
$768$	18655.0	0.876505
$769$	−10783.1	−0.505653	−0.252827	−	0.967512i	$-0.581360\pi$
$769$	−10783.1	−0.505653	−0.252827	+	0.967512i	$0.581360\pi$
$770$	−533.654	−0.0249760
$771$	19327.1	0.902787
$772$	1903.50	0.0887417
$773$	24613.9	1.14528	0.572639	−	0.819808i	$-0.305920\pi$
$773$	24613.9	1.14528	0.572639	+	0.819808i	$0.305920\pi$
$774$	4634.36	0.215218
$775$	−6984.14	−0.323713
$776$	−4640.90	−0.214689
$777$	43999.4	2.03149
$778$	3578.67	0.164912
$779$	7481.51	0.344099
$780$	0	0
$781$	1561.33	0.0715350
$782$	146.905	0.00671780
$783$	−7046.06	−0.321591
$784$	10257.3	0.467260
$785$	16610.9	0.755246
$786$	7104.52	0.322405
$787$	−2725.32	−0.123440	−0.0617200	−	0.998094i	$-0.519659\pi$
$787$	−2725.32	−0.123440	−0.0617200	+	0.998094i	$0.519659\pi$
$788$	27065.0	1.22354
$789$	19200.5	0.866359
$790$	−2517.77	−0.113390
$791$	−29994.0	−1.34825
$792$	−1635.12	−0.0733606
$793$	0	0
$794$	989.288	0.0442173
$795$	−15310.4	−0.683022
$796$	25888.2	1.15274
$797$	16687.3	0.741648	0.370824	−	0.928703i	$-0.379075\pi$
$797$	16687.3	0.741648	0.370824	+	0.928703i	$0.379075\pi$
$798$	7635.52	0.338715
$799$	−14220.6	−0.629648
$800$	−2489.33	−0.110014
$801$	440.946	0.0194507
$802$	−4933.17	−0.217202
$803$	1613.58	0.0709118
$804$	11148.9	0.489045
$805$	942.289	0.0412563
$806$	0	0
$807$	40139.0	1.75088
$808$	10671.3	0.464621
$809$	−21008.4	−0.913000	−0.456500	−	0.889723i	$-0.650897\pi$
$809$	−21008.4	−0.913000	−0.456500	+	0.889723i	$0.650897\pi$
$810$	2273.09	0.0986026
$811$	6817.34	0.295178	0.147589	−	0.989049i	$-0.452849\pi$
$811$	6817.34	0.295178	0.147589	+	0.989049i	$0.452849\pi$
$812$	37726.1	1.63045
$813$	9231.28	0.398223
$814$	−1278.36	−0.0550449
$815$	−3127.06	−0.134400
$816$	13073.3	0.560854
$817$	−33191.7	−1.42133
$818$	−5513.88	−0.235683
$819$	0	0
$820$	−3305.62	−0.140777
$821$	29880.2	1.27019	0.635095	−	0.772434i	$-0.280961\pi$
$821$	29880.2	1.27019	0.635095	+	0.772434i	$0.280961\pi$
$822$	−10983.2	−0.466039
$823$	33068.2	1.40059	0.700294	−	0.713855i	$-0.253052\pi$
$823$	33068.2	1.40059	0.700294	+	0.713855i	$0.253052\pi$
$824$	3763.46	0.159110
$825$	−1502.13	−0.0633908
$826$	−9344.77	−0.393639
$827$	−5187.68	−0.218130	−0.109065	−	0.994035i	$-0.534786\pi$
$827$	−5187.68	−0.218130	−0.109065	+	0.994035i	$0.534786\pi$
$828$	1416.25	0.0594421
$829$	−873.675	−0.0366031	−0.0183016	−	0.999833i	$-0.505826\pi$
$829$	−873.675	−0.0366031	−0.0183016	+	0.999833i	$0.505826\pi$
$830$	3314.79	0.138624
$831$	−14282.7	−0.596224
$832$	0	0
$833$	5887.77	0.244897
$834$	−7678.67	−0.318814
$835$	2513.95	0.104190
$836$	5744.52	0.237654
$837$	9191.97	0.379595
$838$	−1374.04	−0.0566413
$839$	−10638.0	−0.437741	−0.218871	−	0.975754i	$-0.570237\pi$
$839$	−10638.0	−0.437741	−0.218871	+	0.975754i	$0.570237\pi$
$840$	−6877.61	−0.282500
$841$	21469.6	0.880300
$842$	−918.000	−0.0375729
$843$	27795.6	1.13562
$844$	−12238.9	−0.499146
$845$	0	0
$846$	5294.34	0.215157
$847$	28767.8	1.16703
$848$	−24832.3	−1.00560
$849$	−42806.3	−1.73040
$850$	−445.718	−0.0179859
$851$	2257.24	0.0909251
$852$	9870.46	0.396897
$853$	−36699.3	−1.47311	−0.736554	−	0.676379i	$-0.763548\pi$
$853$	−36699.3	−1.47311	−0.736554	+	0.676379i	$0.763548\pi$
$854$	−881.522	−0.0353221
$855$	9725.22	0.389000
$856$	−10431.8	−0.416532
$857$	−9154.44	−0.364889	−0.182444	−	0.983216i	$-0.558401\pi$
$857$	−9154.44	−0.364889	−0.182444	+	0.983216i	$0.558401\pi$
$858$	0	0
$859$	22184.1	0.881156	0.440578	−	0.897714i	$-0.354774\pi$
$859$	22184.1	0.881156	0.440578	+	0.897714i	$0.354774\pi$
$860$	14665.4	0.581494
$861$	−13785.9	−0.545669
$862$	2208.21	0.0872527
$863$	44982.7	1.77431	0.887155	−	0.461472i	$-0.152679\pi$
$863$	44982.7	1.77431	0.887155	+	0.461472i	$0.152679\pi$
$864$	3276.26	0.129005
$865$	−11223.7	−0.441176
$866$	−925.966	−0.0363344
$867$	−26997.0	−1.05752
$868$	−49215.8	−1.92453
$869$	−7899.74	−0.308378
$870$	−4100.92	−0.159809
$871$	0	0
$872$	−4315.05	−0.167576
$873$	−12092.3	−0.468800
$874$	391.715	0.0151601
$875$	−2858.96	−0.110458
$876$	10200.8	0.393440
$877$	−16617.0	−0.639812	−0.319906	−	0.947449i	$-0.603651\pi$
$877$	−16617.0	−0.639812	−0.319906	+	0.947449i	$0.603651\pi$
$878$	−3560.12	−0.136843
$879$	25173.9	0.965979
$880$	−2436.35	−0.0933287
$881$	25734.2	0.984116	0.492058	−	0.870563i	$-0.336245\pi$
$881$	25734.2	0.984116	0.492058	+	0.870563i	$0.336245\pi$
$882$	−2192.02	−0.0836838
$883$	18857.6	0.718697	0.359349	−	0.933203i	$-0.382999\pi$
$883$	18857.6	0.718697	0.359349	+	0.933203i	$0.382999\pi$
$884$	0	0
$885$	−26303.7	−0.999082
$886$	541.436	0.0205304
$887$	−36070.6	−1.36543	−0.682713	−	0.730687i	$-0.739200\pi$
$887$	−36070.6	−1.36543	−0.682713	+	0.730687i	$0.739200\pi$
$888$	−16475.2	−0.622605
$889$	−55763.5	−2.10377
$890$	−53.8864	−0.00202952
$891$	7132.02	0.268161
$892$	−21953.7	−0.824063
$893$	−37918.5	−1.42093
$894$	−133.982	−0.00501232
$895$	−15504.1	−0.579045
$896$	−23224.9	−0.865949
$897$	0	0
$898$	−4798.16	−0.178304
$899$	−59825.1	−2.21944
$900$	−4296.98	−0.159147
$901$	−14254.0	−0.527046
$902$	400.535	0.0147853
$903$	61160.9	2.25394
$904$	11231.0	0.413207
$905$	372.015	0.0136643
$906$	−9521.99	−0.349169
$907$	−29257.6	−1.07110	−0.535548	−	0.844505i	$-0.679895\pi$
$907$	−29257.6	−1.07110	−0.535548	+	0.844505i	$0.679895\pi$
$908$	28983.4	1.05930
$909$	27804.9	1.01456
$910$	0	0
$911$	10407.8	0.378512	0.189256	−	0.981928i	$-0.439392\pi$
$911$	10407.8	0.378512	0.189256	+	0.981928i	$0.439392\pi$
$912$	34859.3	1.26569
$913$	10400.5	0.377004
$914$	−3600.49	−0.130300
$915$	−2481.31	−0.0896498
$916$	−17017.6	−0.613838
$917$	42426.0	1.52784
$918$	586.620	0.0210908
$919$	13585.8	0.487655	0.243827	−	0.969819i	$-0.421597\pi$
$919$	13585.8	0.487655	0.243827	+	0.969819i	$0.421597\pi$
$920$	−352.833	−0.0126441
$921$	20673.2	0.739635
$922$	9249.25	0.330377
$923$	0	0
$924$	−10585.2	−0.376869
$925$	−6848.59	−0.243438
$926$	2728.16	0.0968175
$927$	9806.04	0.347435
$928$	−21323.2	−0.754276
$929$	16146.3	0.570230	0.285115	−	0.958493i	$-0.407968\pi$
$929$	16146.3	0.570230	0.285115	+	0.958493i	$0.407968\pi$
$930$	5349.87	0.188633
$931$	15699.4	0.552661
$932$	10783.3	0.378991
$933$	56652.0	1.98789
$934$	−489.053	−0.0171331
$935$	−1398.48	−0.0489147
$936$	0	0
$937$	4077.89	0.142176	0.0710880	−	0.997470i	$-0.477353\pi$
$937$	4077.89	0.142176	0.0710880	+	0.997470i	$0.477353\pi$
$938$	−2571.12	−0.0894988
$939$	−42779.1	−1.48674
$940$	16753.8	0.581330
$941$	13410.0	0.464562	0.232281	−	0.972649i	$-0.425381\pi$
$941$	13410.0	0.464562	0.232281	+	0.972649i	$0.425381\pi$
$942$	−12724.0	−0.440096
$943$	−707.238	−0.0244229
$944$	−42662.7	−1.47092
$945$	3762.73	0.129526
$946$	−1776.97	−0.0610722
$947$	17163.3	0.588947	0.294474	−	0.955660i	$-0.404856\pi$
$947$	17163.3	0.588947	0.294474	+	0.955660i	$0.404856\pi$
$948$	−49940.8	−1.71097
$949$	0	0
$950$	−1188.48	−0.0405890
$951$	−15780.9	−0.538097
$952$	−6403.06	−0.217988
$953$	32616.5	1.10866	0.554329	−	0.832298i	$-0.312975\pi$
$953$	32616.5	1.10866	0.554329	+	0.832298i	$0.312975\pi$
$954$	5306.76	0.180097
$955$	−24336.5	−0.824620
$956$	−1217.24	−0.0411803
$957$	−12867.0	−0.434620
$958$	−7979.03	−0.269093
$959$	−65588.4	−2.20851
$960$	−14090.1	−0.473705
$961$	48254.1	1.61975
$962$	0	0
$963$	−27180.9	−0.909547
$964$	20563.0	0.687024
$965$	−1235.63	−0.0412191
$966$	−721.796	−0.0240408
$967$	48167.6	1.60183	0.800913	−	0.598781i	$-0.204348\pi$
$967$	48167.6	1.60183	0.800913	+	0.598781i	$0.204348\pi$
$968$	−10771.9	−0.357667
$969$	20009.5	0.663362
$970$	1477.76	0.0489153
$971$	19561.8	0.646516	0.323258	−	0.946311i	$-0.395222\pi$
$971$	19561.8	0.646516	0.323258	+	0.946311i	$0.395222\pi$
$972$	38244.6	1.26203
$973$	−45854.6	−1.51082
$974$	−3202.07	−0.105340
$975$	0	0
$976$	−4024.51	−0.131989
$977$	−24956.7	−0.817233	−0.408616	−	0.912706i	$-0.633989\pi$
$977$	−24956.7	−0.817233	−0.408616	+	0.912706i	$0.633989\pi$
$978$	2395.34	0.0783174
$979$	−169.073	−0.00551952
$980$	−6936.60	−0.226104
$981$	−11243.2	−0.365922
$982$	−4575.73	−0.148694
$983$	−5283.69	−0.171438	−0.0857190	−	0.996319i	$-0.527319\pi$
$983$	−5283.69	−0.171438	−0.0857190	+	0.996319i	$0.527319\pi$
$984$	5162.02	0.167235
$985$	−17568.9	−0.568315
$986$	−3817.96	−0.123315
$987$	69870.7	2.25330
$988$	0	0
$989$	3137.66	0.100881
$990$	520.656	0.0167147
$991$	50914.9	1.63205	0.816026	−	0.578015i	$-0.196173\pi$
$991$	50914.9	1.63205	0.816026	+	0.578015i	$0.196173\pi$
$992$	27817.3	0.890322
$993$	−65785.0	−2.10234
$994$	−2276.28	−0.0726351
$995$	−16805.0	−0.535430
$996$	65749.8	2.09173
$997$	35570.6	1.12992	0.564962	−	0.825117i	$-0.308891\pi$
$997$	35570.6	1.12992	0.564962	+	0.825117i	$0.308891\pi$
$998$	6951.14	0.220476
$999$	9013.58	0.285463

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.4.a.h.1.4		7
13.4	even	6		65.4.e.a.16.4	✓	14
13.10	even	6		65.4.e.a.61.4	yes	14
13.12	even	2		845.4.a.k.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.4.e.a.16.4	✓	14	13.4	even	6
65.4.e.a.61.4	yes	14	13.10	even	6
845.4.a.h.1.4		7	1.1	even	1	trivial
845.4.a.k.1.4		7	13.12	even	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:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)

\(f(q)\)	\(=\)	\(q-0.545397 q^{2} +7.02243 q^{3} -7.70254 q^{4} +5.00000 q^{5} -3.83001 q^{6} -22.8716 q^{7} +8.56412 q^{8} +22.3146 q^{9} +O(q^{10})\)	\(q-0.545397 q^{2} +7.02243 q^{3} -7.70254 q^{4} +5.00000 q^{5} -3.83001 q^{6} -22.8716 q^{7} +8.56412 q^{8} +22.3146 q^{9} -2.72699 q^{10} -8.55617 q^{11} -54.0906 q^{12} +12.4741 q^{14} +35.1122 q^{15} +56.9495 q^{16} +32.6895 q^{17} -12.1703 q^{18} +87.1647 q^{19} -38.5127 q^{20} -160.615 q^{21} +4.66651 q^{22} -8.23980 q^{23} +60.1410 q^{24} +25.0000 q^{25} -32.9030 q^{27} +176.170 q^{28} +214.146 q^{29} -19.1501 q^{30} -279.366 q^{31} -99.5730 q^{32} -60.0851 q^{33} -17.8287 q^{34} -114.358 q^{35} -171.879 q^{36} -273.944 q^{37} -47.5394 q^{38} +42.8206 q^{40} +85.8319 q^{41} +87.5987 q^{42} -380.793 q^{43} +65.9042 q^{44} +111.573 q^{45} +4.49396 q^{46} -435.021 q^{47} +399.924 q^{48} +180.112 q^{49} -13.6349 q^{50} +229.560 q^{51} -436.042 q^{53} +17.9452 q^{54} -42.7808 q^{55} -195.875 q^{56} +612.108 q^{57} -116.795 q^{58} -749.132 q^{59} -270.453 q^{60} -70.6681 q^{61} +152.365 q^{62} -510.371 q^{63} -401.289 q^{64} +32.7702 q^{66} -206.116 q^{67} -251.792 q^{68} -57.8635 q^{69} +62.3706 q^{70} -182.480 q^{71} +191.105 q^{72} -188.587 q^{73} +149.408 q^{74} +175.561 q^{75} -671.390 q^{76} +195.694 q^{77} +923.280 q^{79} +284.747 q^{80} -833.553 q^{81} -46.8125 q^{82} -1215.55 q^{83} +1237.14 q^{84} +163.447 q^{85} +207.683 q^{86} +1503.83 q^{87} -73.2760 q^{88} +19.7604 q^{89} -60.8515 q^{90} +63.4674 q^{92} -1961.83 q^{93} +237.259 q^{94} +435.823 q^{95} -699.245 q^{96} -541.901 q^{97} -98.2326 q^{98} -190.927 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} - 4 q^{3} + 30 q^{4} + 35 q^{5} - 23 q^{6} - 7 q^{7} - 21 q^{8} + 87 q^{9}+O(q^{10}) \) 7 * q - 2 * q^2 - 4 * q^3 + 30 * q^4 + 35 * q^5 - 23 * q^6 - 7 * q^7 - 21 * q^8 + 87 * q^9	\( 7 q - 2 q^{2} - 4 q^{3} + 30 q^{4} + 35 q^{5} - 23 q^{6} - 7 q^{7} - 21 q^{8} + 87 q^{9} - 10 q^{10} - 87 q^{11} - 79 q^{12} + 66 q^{14} - 20 q^{15} - 134 q^{16} - 114 q^{17} - 207 q^{18} - 245 q^{19} + 150 q^{20} + 38 q^{21} + 338 q^{22} - 74 q^{23} - 334 q^{24} + 175 q^{25} - 442 q^{27} - 230 q^{28} - 88 q^{29} - 115 q^{30} - 500 q^{31} - 80 q^{32} + 194 q^{33} - 427 q^{34} - 35 q^{35} + 425 q^{36} - 633 q^{37} - 298 q^{38} - 105 q^{40} - 162 q^{41} - 1439 q^{42} - 280 q^{43} - 220 q^{44} + 435 q^{45} + 11 q^{46} - 475 q^{47} + 2281 q^{48} + 1694 q^{49} - 50 q^{50} - 430 q^{51} - 603 q^{53} - 51 q^{54} - 435 q^{55} - 1277 q^{56} - 458 q^{57} + 1213 q^{58} - 1410 q^{59} - 395 q^{60} + 412 q^{61} - 56 q^{62} - 1241 q^{63} - 1179 q^{64} + 2173 q^{66} - 1398 q^{67} - 493 q^{68} + 1080 q^{69} + 330 q^{70} + 584 q^{71} - 1545 q^{72} - 2538 q^{73} + 3840 q^{74} - 100 q^{75} - 3292 q^{76} - 2753 q^{77} + 464 q^{79} - 670 q^{80} - 473 q^{81} - 1583 q^{82} - 466 q^{83} + 3081 q^{84} - 570 q^{85} - 4929 q^{86} - 282 q^{87} + 3389 q^{88} - 443 q^{89} - 1035 q^{90} + 3091 q^{92} + 2116 q^{93} + 2017 q^{94} - 1225 q^{95} - 477 q^{96} + 1870 q^{97} - 1364 q^{98} - 5689 q^{99}+O(q^{100}) \) 7 * q - 2 * q^2 - 4 * q^3 + 30 * q^4 + 35 * q^5 - 23 * q^6 - 7 * q^7 - 21 * q^8 + 87 * q^9 - 10 * q^10 - 87 * q^11 - 79 * q^12 + 66 * q^14 - 20 * q^15 - 134 * q^16 - 114 * q^17 - 207 * q^18 - 245 * q^19 + 150 * q^20 + 38 * q^21 + 338 * q^22 - 74 * q^23 - 334 * q^24 + 175 * q^25 - 442 * q^27 - 230 * q^28 - 88 * q^29 - 115 * q^30 - 500 * q^31 - 80 * q^32 + 194 * q^33 - 427 * q^34 - 35 * q^35 + 425 * q^36 - 633 * q^37 - 298 * q^38 - 105 * q^40 - 162 * q^41 - 1439 * q^42 - 280 * q^43 - 220 * q^44 + 435 * q^45 + 11 * q^46 - 475 * q^47 + 2281 * q^48 + 1694 * q^49 - 50 * q^50 - 430 * q^51 - 603 * q^53 - 51 * q^54 - 435 * q^55 - 1277 * q^56 - 458 * q^57 + 1213 * q^58 - 1410 * q^59 - 395 * q^60 + 412 * q^61 - 56 * q^62 - 1241 * q^63 - 1179 * q^64 + 2173 * q^66 - 1398 * q^67 - 493 * q^68 + 1080 * q^69 + 330 * q^70 + 584 * q^71 - 1545 * q^72 - 2538 * q^73 + 3840 * q^74 - 100 * q^75 - 3292 * q^76 - 2753 * q^77 + 464 * q^79 - 670 * q^80 - 473 * q^81 - 1583 * q^82 - 466 * q^83 + 3081 * q^84 - 570 * q^85 - 4929 * q^86 - 282 * q^87 + 3389 * q^88 - 443 * q^89 - 1035 * q^90 + 3091 * q^92 + 2116 * q^93 + 2017 * q^94 - 1225 * q^95 - 477 * q^96 + 1870 * q^97 - 1364 * q^98 - 5689 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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