LMFDB - Embedded newform 825.4.c.h.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{97})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 49x^{2} + 576 $ x^4 + 49*x^2 + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-5.42443i$ of defining polynomial
Character	$\chi$	$=$	825.199
Dual form			825.4.c.h.199.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.42443i q^{2} -3.00000i q^{3} -21.4244 q^{4} -16.2733 q^{6} +7.69772i q^{7} +72.8199i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q-5.42443i q^{2} -3.00000i q^{3} -21.4244 q^{4} -16.2733 q^{6} +7.69772i q^{7} +72.8199i q^{8} -9.00000 q^{9} -11.0000 q^{11} +64.2733i q^{12} +24.8489i q^{13} +41.7557 q^{14} +223.611 q^{16} +15.9420i q^{17} +48.8199i q^{18} -15.1511 q^{19} +23.0931 q^{21} +59.6687i q^{22} +17.7557i q^{23} +218.460 q^{24} +134.791 q^{26} +27.0000i q^{27} -164.919i q^{28} +128.547 q^{29} +219.395 q^{31} -630.402i q^{32} +33.0000i q^{33} +86.4763 q^{34} +192.820 q^{36} -92.0703i q^{37} +82.1863i q^{38} +74.5466 q^{39} -459.942 q^{41} -125.267i q^{42} +64.9648i q^{43} +235.669 q^{44} +96.3146 q^{46} -497.408i q^{47} -670.832i q^{48} +283.745 q^{49} +47.8260 q^{51} -532.373i q^{52} -526.919i q^{53} +146.460 q^{54} -560.547 q^{56} +45.4534i q^{57} -697.292i q^{58} +578.443 q^{59} -221.569 q^{61} -1190.09i q^{62} -69.2794i q^{63} -1630.68 q^{64} +179.006 q^{66} +860.745i q^{67} -341.548i q^{68} +53.2671 q^{69} +580.919 q^{71} -655.379i q^{72} +510.116i q^{73} -499.429 q^{74} +324.605 q^{76} -84.6749i q^{77} -404.373i q^{78} -1035.12 q^{79} +81.0000 q^{81} +2494.92i q^{82} +606.211i q^{83} -494.757 q^{84} +352.397 q^{86} -385.640i q^{87} -801.018i q^{88} +23.4411 q^{89} -191.279 q^{91} -380.406i q^{92} -658.186i q^{93} -2698.15 q^{94} -1891.20 q^{96} -719.490i q^{97} -1539.16i q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 66 q^{4} - 6 q^{6} - 36 q^{9}+O(q^{10}) $ 4 * q - 66 * q^4 - 6 * q^6 - 36 * q^9	$ 4 q - 66 q^{4} - 6 q^{6} - 36 q^{9} - 44 q^{11} + 364 q^{14} + 402 q^{16} - 100 q^{19} - 144 q^{21} + 342 q^{24} + 224 q^{26} + 396 q^{29} + 720 q^{31} + 1252 q^{34} + 594 q^{36} + 180 q^{39} - 1564 q^{41} + 726 q^{44} - 836 q^{46} - 756 q^{49} - 636 q^{51} + 54 q^{54} - 2124 q^{56} + 344 q^{59} - 1556 q^{61} - 1618 q^{64} + 66 q^{66} + 804 q^{69} + 1260 q^{71} - 4716 q^{74} + 1456 q^{76} - 1304 q^{79} + 324 q^{81} + 1212 q^{84} - 2136 q^{86} + 1512 q^{89} - 56 q^{91} - 7444 q^{94} - 5142 q^{96} + 396 q^{99}+O(q^{100}) $ 4 * q - 66 * q^4 - 6 * q^6 - 36 * q^9 - 44 * q^11 + 364 * q^14 + 402 * q^16 - 100 * q^19 - 144 * q^21 + 342 * q^24 + 224 * q^26 + 396 * q^29 + 720 * q^31 + 1252 * q^34 + 594 * q^36 + 180 * q^39 - 1564 * q^41 + 726 * q^44 - 836 * q^46 - 756 * q^49 - 636 * q^51 + 54 * q^54 - 2124 * q^56 + 344 * q^59 - 1556 * q^61 - 1618 * q^64 + 66 * q^66 + 804 * q^69 + 1260 * q^71 - 4716 * q^74 + 1456 * q^76 - 1304 * q^79 + 324 * q^81 + 1212 * q^84 - 2136 * q^86 + 1512 * q^89 - 56 * q^91 - 7444 * q^94 - 5142 * q^96 + 396 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 5.42443i	− 1.91783i	−0.283702	−	0.958913i	$-0.591563\pi$
$2$	− 5.42443i	− 1.91783i	0.283702	−	0.958913i	$-0.408437\pi$
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	−21.4244	−2.67805
$5$	0	0
$5$	0	0
$6$	−16.2733	−1.10726
$7$	7.69772i	0.415638i	0.978167	+	0.207819i	$0.0666364\pi$
$7$	7.69772i	0.415638i	−0.978167	+	0.207819i	$0.933364\pi$
$8$	72.8199i	3.21821i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	64.2733i	1.54617i
$13$	24.8489i	0.530141i	0.964229	+	0.265071i	$0.0853952\pi$
$13$	24.8489i	0.530141i	−0.964229	+	0.265071i	$0.914605\pi$
$14$	41.7557	0.797120
$15$	0	0
$16$	223.611	3.49392
$17$	15.9420i	0.227441i	0.993513	+	0.113721i	$0.0362769\pi$
$17$	15.9420i	0.227441i	−0.993513	+	0.113721i	$0.963723\pi$
$18$	48.8199i	0.639275i
$19$	−15.1511	−0.182943	−0.0914713	−	0.995808i	$-0.529157\pi$
$19$	−15.1511	−0.182943	−0.0914713	+	0.995808i	$0.529157\pi$
$20$	0	0
$21$	23.0931	0.239968
$22$	59.6687i	0.578246i
$23$	17.7557i	0.160971i	0.996756	+	0.0804853i	$0.0256470\pi$
$23$	17.7557i	0.160971i	−0.996756	+	0.0804853i	$0.974353\pi$
$24$	218.460	1.85804
$25$	0	0
$26$	134.791	1.01672
$27$	27.0000i	0.192450i
$28$	− 164.919i	− 1.11310i
$29$	128.547	0.823121	0.411560	−	0.911383i	$-0.364984\pi$
$29$	128.547	0.823121	0.411560	+	0.911383i	$0.364984\pi$
$30$	0	0
$31$	219.395	1.27112	0.635558	−	0.772053i	$-0.280770\pi$
$31$	219.395	1.27112	0.635558	+	0.772053i	$0.280770\pi$
$32$	− 630.402i	− 3.48251i
$33$	33.0000i	0.174078i
$34$	86.4763	0.436193
$35$	0	0
$36$	192.820	0.892685
$37$	− 92.0703i	− 0.409088i	−0.978857	−	0.204544i	$-0.934429\pi$
$37$	− 92.0703i	− 0.409088i	0.978857	−	0.204544i	$-0.0655712\pi$
$38$	82.1863i	0.350852i
$39$	74.5466	0.306077
$40$	0	0
$41$	−459.942	−1.75197	−0.875986	−	0.482336i	$-0.839788\pi$
$41$	−459.942	−1.75197	−0.875986	+	0.482336i	$0.839788\pi$
$42$	− 125.267i	− 0.460218i
$43$	64.9648i	0.230396i	0.993343	+	0.115198i	$0.0367503\pi$
$43$	64.9648i	0.230396i	−0.993343	+	0.115198i	$0.963250\pi$
$44$	235.669	0.807464
$45$	0	0
$46$	96.3146	0.308713
$47$	− 497.408i	− 1.54371i	−0.635799	−	0.771855i	$-0.719329\pi$
$47$	− 497.408i	− 1.54371i	0.635799	−	0.771855i	$-0.280671\pi$
$48$	− 670.832i	− 2.01721i
$49$	283.745	0.827245
$50$	0	0
$51$	47.8260	0.131313
$52$	− 532.373i	− 1.41975i
$53$	− 526.919i	− 1.36562i	−0.730596	−	0.682811i	$-0.760757\pi$
$53$	− 526.919i	− 1.36562i	0.730596	−	0.682811i	$-0.239243\pi$
$54$	146.460	0.369086
$55$	0	0
$56$	−560.547	−1.33761
$57$	45.4534i	0.105622i
$58$	− 697.292i	− 1.57860i
$59$	578.443	1.27639	0.638194	−	0.769876i	$-0.279682\pi$
$59$	578.443	1.27639	0.638194	+	0.769876i	$0.279682\pi$
$60$	0	0
$61$	−221.569	−0.465067	−0.232533	−	0.972588i	$-0.574701\pi$
$61$	−221.569	−0.465067	−0.232533	+	0.972588i	$0.574701\pi$
$62$	− 1190.09i	− 2.43778i
$63$	− 69.2794i	− 0.138546i
$64$	−1630.68	−3.18493
$65$	0	0
$66$	179.006	0.333851
$67$	860.745i	1.56950i	0.619810	+	0.784752i	$0.287210\pi$
$67$	860.745i	1.56950i	−0.619810	+	0.784752i	$0.712790\pi$
$68$	− 341.548i	− 0.609100i
$69$	53.2671	0.0929364
$70$	0	0
$71$	580.919	0.971020	0.485510	−	0.874231i	$-0.338634\pi$
$71$	580.919	0.971020	0.485510	+	0.874231i	$0.338634\pi$
$72$	− 655.379i	− 1.07274i
$73$	510.116i	0.817871i	0.912563	+	0.408935i	$0.134100\pi$
$73$	510.116i	0.817871i	−0.912563	+	0.408935i	$0.865900\pi$
$74$	−499.429	−0.784560
$75$	0	0
$76$	324.605	0.489930
$77$	− 84.6749i	− 0.125319i
$78$	− 404.373i	− 0.587002i
$79$	−1035.12	−1.47418	−0.737088	−	0.675797i	$-0.763800\pi$
$79$	−1035.12	−1.47418	−0.737088	+	0.675797i	$0.763800\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	2494.92i	3.35998i
$83$	606.211i	0.801690i	0.916146	+	0.400845i	$0.131283\pi$
$83$	606.211i	0.801690i	−0.916146	+	0.400845i	$0.868717\pi$
$84$	−494.757	−0.642648
$85$	0	0
$86$	352.397	0.441860
$87$	− 385.640i	− 0.475229i
$88$	− 801.018i	− 0.970328i
$89$	23.4411	0.0279186	0.0139593	−	0.999903i	$-0.495556\pi$
$89$	23.4411	0.0279186	0.0139593	+	0.999903i	$0.495556\pi$
$90$	0	0
$91$	−191.279	−0.220347
$92$	− 380.406i	− 0.431088i
$93$	− 658.186i	− 0.733879i
$94$	−2698.15	−2.96057
$95$	0	0
$96$	−1891.20	−2.01063
$97$	− 719.490i	− 0.753126i	−0.926391	−	0.376563i	$-0.877106\pi$
$97$	− 719.490i	− 0.753126i	0.926391	−	0.376563i	$-0.122894\pi$
$98$	− 1539.16i	− 1.58651i
$99$	99.0000	0.100504
$100$	0	0
$101$	1871.27	1.84355	0.921774	−	0.387727i	$-0.126740\pi$
$101$	1871.27	1.84355	0.921774	+	0.387727i	$0.126740\pi$
$102$	− 259.429i	− 0.251836i
$103$	− 428.745i	− 0.410151i	−0.978746	−	0.205075i	$-0.934256\pi$
$103$	− 428.745i	− 0.410151i	0.978746	−	0.205075i	$-0.0657439\pi$
$104$	−1809.49	−1.70611
$105$	0	0
$106$	−2858.24	−2.61902
$107$	− 1148.02i	− 1.03723i	−0.855008	−	0.518616i	$-0.826448\pi$
$107$	− 1148.02i	− 1.03723i	0.855008	−	0.518616i	$-0.173552\pi$
$108$	− 578.460i	− 0.515392i
$109$	1828.32	1.60662	0.803308	−	0.595564i	$-0.203071\pi$
$109$	1828.32	1.60662	0.803308	+	0.595564i	$0.203071\pi$
$110$	0	0
$111$	−276.211	−0.236187
$112$	1721.29i	1.45220i
$113$	1126.40i	0.937722i	0.883272	+	0.468861i	$0.155335\pi$
$113$	1126.40i	0.937722i	−0.883272	+	0.468861i	$0.844665\pi$
$114$	246.559	0.202565
$115$	0	0
$116$	−2754.04	−2.20436
$117$	− 223.640i	− 0.176714i
$118$	− 3137.72i	− 2.44789i
$119$	−122.717	−0.0945332
$120$	0	0
$121$	121.000	0.0909091
$122$	1201.89i	0.891916i
$123$	1379.83i	1.01150i
$124$	−4700.42	−3.40412
$125$	0	0
$126$	−375.801	−0.265707
$127$	− 661.304i	− 0.462057i	−0.972947	−	0.231029i	$-0.925791\pi$
$127$	− 661.304i	− 0.462057i	0.972947	−	0.231029i	$-0.0742091\pi$
$128$	3802.31i	2.62562i
$129$	194.895	0.133019
$130$	0	0
$131$	622.186	0.414967	0.207483	−	0.978239i	$-0.433473\pi$
$131$	622.186	0.414967	0.207483	+	0.978239i	$0.433473\pi$
$132$	− 707.006i	− 0.466189i
$133$	− 116.629i	− 0.0760378i
$134$	4669.05	3.01003
$135$	0	0
$136$	−1160.89	−0.731955
$137$	1872.84i	1.16794i	0.811776	+	0.583969i	$0.198501\pi$
$137$	1872.84i	1.16794i	−0.811776	+	0.583969i	$0.801499\pi$
$138$	− 288.944i	− 0.178236i
$139$	954.058	0.582174	0.291087	−	0.956697i	$-0.405983\pi$
$139$	954.058	0.582174	0.291087	+	0.956697i	$0.405983\pi$
$140$	0	0
$141$	−1492.22	−0.891261
$142$	− 3151.15i	− 1.86225i
$143$	− 273.337i	− 0.159844i
$144$	−2012.50	−1.16464
$145$	0	0
$146$	2767.09	1.56853
$147$	− 851.236i	− 0.477610i
$148$	1972.55i	1.09556i
$149$	2047.01	1.12549	0.562745	−	0.826631i	$-0.309745\pi$
$149$	2047.01	1.12549	0.562745	+	0.826631i	$0.309745\pi$
$150$	0	0
$151$	475.863	0.256458	0.128229	−	0.991745i	$-0.459071\pi$
$151$	475.863	0.256458	0.128229	+	0.991745i	$0.459071\pi$
$152$	− 1103.30i	− 0.588749i
$153$	− 143.478i	− 0.0758138i
$154$	−459.313	−0.240341
$155$	0	0
$156$	−1597.12	−0.819691
$157$	647.466i	0.329130i	0.986366	+	0.164565i	$0.0526220\pi$
$157$	647.466i	0.329130i	−0.986366	+	0.164565i	$0.947378\pi$
$158$	5614.92i	2.82721i
$159$	−1580.76	−0.788442
$160$	0	0
$161$	−136.678	−0.0669054
$162$	− 439.379i	− 0.213092i
$163$	1093.23i	0.525329i	0.964887	+	0.262665i	$0.0846013\pi$
$163$	1093.23i	0.525329i	−0.964887	+	0.262665i	$0.915399\pi$
$164$	9853.99	4.69188
$165$	0	0
$166$	3288.35	1.53750
$167$	− 1123.25i	− 0.520479i	−0.965544	−	0.260240i	$-0.916198\pi$
$167$	− 1123.25i	− 0.520479i	0.965544	−	0.260240i	$-0.0838016\pi$
$168$	1681.64i	0.772270i
$169$	1579.53	0.718951
$170$	0	0
$171$	136.360	0.0609809
$172$	− 1391.83i	− 0.617014i
$173$	46.0123i	0.0202211i	0.999949	+	0.0101106i	$0.00321834\pi$
$173$	46.0123i	0.0202211i	−0.999949	+	0.0101106i	$0.996782\pi$
$174$	−2091.88	−0.911406
$175$	0	0
$176$	−2459.72	−1.05346
$177$	− 1735.33i	− 0.736923i
$178$	− 127.155i	− 0.0535430i
$179$	831.975	0.347401	0.173700	−	0.984799i	$-0.444428\pi$
$179$	831.975	0.347401	0.173700	+	0.984799i	$0.444428\pi$
$180$	0	0
$181$	−1810.63	−0.743553	−0.371776	−	0.928322i	$-0.621251\pi$
$181$	−1810.63	−0.743553	−0.371776	+	0.928322i	$0.621251\pi$
$182$	1037.58i	0.422586i
$183$	664.708i	0.268506i
$184$	−1292.97	−0.518037
$185$	0	0
$186$	−3570.28	−1.40745
$187$	− 175.362i	− 0.0685762i
$188$	10656.7i	4.13414i
$189$	−207.838	−0.0799895
$190$	0	0
$191$	458.898	0.173847	0.0869233	−	0.996215i	$-0.472296\pi$
$191$	458.898	0.173847	0.0869233	+	0.996215i	$0.472296\pi$
$192$	4892.05i	1.83882i
$193$	1778.91i	0.663465i	0.943373	+	0.331733i	$0.107633\pi$
$193$	1778.91i	0.663465i	−0.943373	+	0.331733i	$0.892367\pi$
$194$	−3902.82	−1.44436
$195$	0	0
$196$	−6079.08	−2.21541
$197$	− 5304.53i	− 1.91844i	−0.282666	−	0.959218i	$-0.591219\pi$
$197$	− 5304.53i	− 1.91844i	0.282666	−	0.959218i	$-0.408781\pi$
$198$	− 537.018i	− 0.192749i
$199$	5138.40	1.83041	0.915205	−	0.402989i	$-0.132029\pi$
$199$	5138.40	1.83041	0.915205	+	0.402989i	$0.132029\pi$
$200$	0	0
$201$	2582.24	0.906153
$202$	− 10150.6i	− 3.53560i
$203$	989.515i	0.342120i
$204$	−1024.65	−0.351664
$205$	0	0
$206$	−2325.70	−0.786597
$207$	− 159.801i	− 0.0536568i
$208$	5556.47i	1.85227i
$209$	166.663	0.0551593
$210$	0	0
$211$	−4262.36	−1.39068	−0.695339	−	0.718682i	$-0.744746\pi$
$211$	−4262.36	−1.39068	−0.695339	+	0.718682i	$0.744746\pi$
$212$	11288.9i	3.65721i
$213$	− 1742.76i	− 0.560619i
$214$	−6227.38	−1.98923
$215$	0	0
$216$	−1966.14	−0.619345
$217$	1688.84i	0.528323i
$218$	− 9917.58i	− 3.08121i
$219$	1530.35	0.472198
$220$	0	0
$221$	−396.141	−0.120576
$222$	1498.29i	0.452966i
$223$	− 1377.80i	− 0.413740i	−0.978368	−	0.206870i	$-0.933672\pi$
$223$	− 1377.80i	− 0.413740i	0.978368	−	0.206870i	$-0.0663277\pi$
$224$	4852.65	1.44746
$225$	0	0
$226$	6110.06	1.79839
$227$	1227.28i	0.358843i	0.983772	+	0.179422i	$0.0574226\pi$
$227$	1227.28i	0.358843i	−0.983772	+	0.179422i	$0.942577\pi$
$228$	− 973.814i	− 0.282861i
$229$	−3890.28	−1.12261	−0.561304	−	0.827610i	$-0.689700\pi$
$229$	−3890.28	−1.12261	−0.561304	+	0.827610i	$0.689700\pi$
$230$	0	0
$231$	−254.025	−0.0723532
$232$	9360.74i	2.64898i
$233$	− 3218.14i	− 0.904837i	−0.891806	−	0.452419i	$-0.850561\pi$
$233$	− 3218.14i	− 0.904837i	0.891806	−	0.452419i	$-0.149439\pi$
$234$	−1213.12	−0.338906
$235$	0	0
$236$	−12392.8	−3.41823
$237$	3105.35i	0.851115i
$238$	665.670i	0.181298i
$239$	428.098	0.115864	0.0579318	−	0.998321i	$-0.481549\pi$
$239$	428.098	0.115864	0.0579318	+	0.998321i	$0.481549\pi$
$240$	0	0
$241$	1231.16	0.329070	0.164535	−	0.986371i	$-0.447388\pi$
$241$	1231.16	0.329070	0.164535	+	0.986371i	$0.447388\pi$
$242$	− 656.356i	− 0.174348i
$243$	− 243.000i	− 0.0641500i
$244$	4747.00	1.24547
$245$	0	0
$246$	7484.77	1.93988
$247$	− 376.489i	− 0.0969854i
$248$	15976.3i	4.09072i
$249$	1818.63	0.462856
$250$	0	0
$251$	2838.22	0.713732	0.356866	−	0.934156i	$-0.383845\pi$
$251$	2838.22	0.713732	0.356866	+	0.934156i	$0.383845\pi$
$252$	1484.27i	0.371033i
$253$	− 195.313i	− 0.0485344i
$254$	−3587.20	−0.886145
$255$	0	0
$256$	7579.90	1.85056
$257$	− 342.007i	− 0.0830110i	−0.999138	−	0.0415055i	$-0.986785\pi$
$257$	− 342.007i	− 0.0830110i	0.999138	−	0.0415055i	$-0.0132154\pi$
$258$	− 1057.19i	− 0.255108i
$259$	708.731	0.170032
$260$	0	0
$261$	−1156.92	−0.274374
$262$	− 3375.01i	− 0.795834i
$263$	5895.00i	1.38213i	0.722791	+	0.691067i	$0.242859\pi$
$263$	5895.00i	1.38213i	−0.722791	+	0.691067i	$0.757141\pi$
$264$	−2403.06	−0.560219
$265$	0	0
$266$	−632.647	−0.145827
$267$	− 70.3234i	− 0.0161188i
$268$	− 18441.0i	− 4.20322i
$269$	2496.18	0.565779	0.282890	−	0.959152i	$-0.408707\pi$
$269$	2496.18	0.565779	0.282890	+	0.959152i	$0.408707\pi$
$270$	0	0
$271$	−2249.68	−0.504274	−0.252137	−	0.967692i	$-0.581133\pi$
$271$	−2249.68	−0.504274	−0.252137	+	0.967692i	$0.581133\pi$
$272$	3564.80i	0.794662i
$273$	573.838i	0.127217i
$274$	10159.1	2.23990
$275$	0	0
$276$	−1141.22	−0.248889
$277$	− 4082.59i	− 0.885556i	−0.896631	−	0.442778i	$-0.853993\pi$
$277$	− 4082.59i	− 0.885556i	0.896631	−	0.442778i	$-0.146007\pi$
$278$	− 5175.22i	− 1.11651i
$279$	−1974.56	−0.423705
$280$	0	0
$281$	−1033.79	−0.219468	−0.109734	−	0.993961i	$-0.535000\pi$
$281$	−1033.79	−0.219468	−0.109734	+	0.993961i	$0.535000\pi$
$282$	8094.46i	1.70928i
$283$	− 7809.14i	− 1.64030i	−0.572148	−	0.820150i	$-0.693890\pi$
$283$	− 7809.14i	− 1.64030i	0.572148	−	0.820150i	$-0.306110\pi$
$284$	−12445.9	−2.60044
$285$	0	0
$286$	−1482.70	−0.306552
$287$	− 3540.50i	− 0.728186i
$288$	5673.61i	1.16084i
$289$	4658.85	0.948270
$290$	0	0
$291$	−2158.47	−0.434817
$292$	− 10928.9i	− 2.19030i
$293$	− 1949.19i	− 0.388645i	−0.980938	−	0.194323i	$-0.937749\pi$
$293$	− 1949.19i	− 0.388645i	0.980938	−	0.194323i	$-0.0622509\pi$
$294$	−4617.47	−0.915973
$295$	0	0
$296$	6704.55	1.31653
$297$	− 297.000i	− 0.0580259i
$298$	− 11103.9i	− 2.15849i
$299$	−441.209	−0.0853371
$300$	0	0
$301$	−500.081	−0.0957614
$302$	− 2581.28i	− 0.491842i
$303$	− 5613.81i	− 1.06437i
$304$	−3387.96	−0.639187
$305$	0	0
$306$	−778.286	−0.145398
$307$	− 2364.09i	− 0.439497i	−0.975557	−	0.219748i	$-0.929476\pi$
$307$	− 2364.09i	− 0.439497i	0.975557	−	0.219748i	$-0.0705236\pi$
$308$	1814.11i	0.335612i
$309$	−1286.24	−0.236801
$310$	0	0
$311$	−1989.17	−0.362686	−0.181343	−	0.983420i	$-0.558044\pi$
$311$	−1989.17	−0.362686	−0.181343	+	0.983420i	$0.558044\pi$
$312$	5428.47i	0.985021i
$313$	− 3878.67i	− 0.700433i	−0.936669	−	0.350216i	$-0.886108\pi$
$313$	− 3878.67i	− 0.700433i	0.936669	−	0.350216i	$-0.113892\pi$
$314$	3512.13	0.631214
$315$	0	0
$316$	22176.8	3.94792
$317$	− 2913.73i	− 0.516251i	−0.966111	−	0.258126i	$-0.916895\pi$
$317$	− 2913.73i	− 0.516251i	0.966111	−	0.258126i	$-0.0831048\pi$
$318$	8574.71i	1.51209i
$319$	−1414.01	−0.248180
$320$	0	0
$321$	−3444.07	−0.598846
$322$	741.402i	0.128313i
$323$	− 241.540i	− 0.0416087i
$324$	−1735.38	−0.297562
$325$	0	0
$326$	5930.17	1.00749
$327$	− 5484.95i	− 0.927580i
$328$	− 33492.9i	− 5.63822i
$329$	3828.90	0.641624
$330$	0	0
$331$	8104.46	1.34580	0.672902	−	0.739731i	$-0.265047\pi$
$331$	8104.46	1.34580	0.672902	+	0.739731i	$0.265047\pi$
$332$	− 12987.7i	− 2.14697i
$333$	828.633i	0.136363i
$334$	−6093.02	−0.998189
$335$	0	0
$336$	5163.88	0.838430
$337$	− 5919.19i	− 0.956792i	−0.878144	−	0.478396i	$-0.841218\pi$
$337$	− 5919.19i	− 0.956792i	0.878144	−	0.478396i	$-0.158782\pi$
$338$	− 8568.07i	− 1.37882i
$339$	3379.19	0.541394
$340$	0	0
$341$	−2413.35	−0.383256
$342$	− 739.677i	− 0.116951i
$343$	4824.51i	0.759472i
$344$	−4730.73	−0.741465
$345$	0	0
$346$	249.590	0.0387805
$347$	8540.59i	1.32128i	0.750705	+	0.660638i	$0.229714\pi$
$347$	8540.59i	1.32128i	−0.750705	+	0.660638i	$0.770286\pi$
$348$	8262.11i	1.27269i
$349$	−937.337	−0.143767	−0.0718833	−	0.997413i	$-0.522901\pi$
$349$	−937.337	−0.143767	−0.0718833	+	0.997413i	$0.522901\pi$
$350$	0	0
$351$	−670.919	−0.102026
$352$	6934.42i	1.05002i
$353$	− 211.118i	− 0.0318319i	−0.999873	−	0.0159160i	$-0.994934\pi$
$353$	− 211.118i	− 0.0318319i	0.999873	−	0.0159160i	$-0.00506642\pi$
$354$	−9413.17	−1.41329
$355$	0	0
$356$	−502.213	−0.0747675
$357$	368.151i	0.0545788i
$358$	− 4512.99i	− 0.666254i
$359$	1376.31	0.202337	0.101169	−	0.994869i	$-0.467742\pi$
$359$	1376.31	0.202337	0.101169	+	0.994869i	$0.467742\pi$
$360$	0	0
$361$	−6629.44	−0.966532
$362$	9821.63i	1.42600i
$363$	− 363.000i	− 0.0524864i
$364$	4098.05	0.590100
$365$	0	0
$366$	3605.66	0.514948
$367$	− 1030.45i	− 0.146564i	−0.997311	−	0.0732821i	$-0.976653\pi$
$367$	− 1030.45i	− 0.146564i	0.997311	−	0.0732821i	$-0.0233473\pi$
$368$	3970.37i	0.562418i
$369$	4139.48	0.583991
$370$	0	0
$371$	4056.07	0.567603
$372$	14101.3i	1.96537i
$373$	9365.39i	1.30006i	0.759909	+	0.650029i	$0.225243\pi$
$373$	9365.39i	1.30006i	−0.759909	+	0.650029i	$0.774757\pi$
$374$	−951.239	−0.131517
$375$	0	0
$376$	36221.2	4.96799
$377$	3194.24i	0.436370i
$378$	1127.40i	0.153406i
$379$	7120.23	0.965017	0.482509	−	0.875891i	$-0.339726\pi$
$379$	7120.23	0.965017	0.482509	+	0.875891i	$0.339726\pi$
$380$	0	0
$381$	−1983.91	−0.266769
$382$	− 2489.26i	− 0.333407i
$383$	− 1163.56i	− 0.155235i	−0.996983	−	0.0776176i	$-0.975269\pi$
$383$	− 1163.56i	− 0.155235i	0.996983	−	0.0776176i	$-0.0247313\pi$
$384$	11406.9	1.51590
$385$	0	0
$386$	9649.57	1.27241
$387$	− 584.684i	− 0.0767988i
$388$	15414.7i	2.01691i
$389$	10958.9	1.42838	0.714188	−	0.699954i	$-0.246796\pi$
$389$	10958.9	1.42838	0.714188	+	0.699954i	$0.246796\pi$
$390$	0	0
$391$	−283.062	−0.0366114
$392$	20662.3i	2.66225i
$393$	− 1866.56i	− 0.239581i
$394$	−28774.0	−3.67923
$395$	0	0
$396$	−2121.02	−0.269155
$397$	2172.09i	0.274595i	0.990530	+	0.137298i	$0.0438417\pi$
$397$	2172.09i	0.274595i	−0.990530	+	0.137298i	$0.956158\pi$
$398$	− 27872.9i	− 3.51041i
$399$	−349.888	−0.0439005
$400$	0	0
$401$	7830.71	0.975180	0.487590	−	0.873073i	$-0.337876\pi$
$401$	7830.71	0.975180	0.487590	+	0.873073i	$0.337876\pi$
$402$	− 14007.2i	− 1.73784i
$403$	5451.73i	0.673870i
$404$	−40090.9	−4.93712
$405$	0	0
$406$	5367.55	0.656126
$407$	1012.77i	0.123345i
$408$	3482.68i	0.422594i
$409$	10731.2	1.29736	0.648682	−	0.761060i	$-0.275321\pi$
$409$	10731.2	1.29736	0.648682	+	0.761060i	$0.275321\pi$
$410$	0	0
$411$	5618.52	0.674309
$412$	9185.62i	1.09841i
$413$	4452.69i	0.530515i
$414$	−866.831	−0.102904
$415$	0	0
$416$	15664.8	1.84622
$417$	− 2862.17i	− 0.336118i
$418$	− 904.049i	− 0.105786i
$419$	7315.88	0.852994	0.426497	−	0.904489i	$-0.359748\pi$
$419$	7315.88	0.852994	0.426497	+	0.904489i	$0.359748\pi$
$420$	0	0
$421$	−12495.7	−1.44657	−0.723284	−	0.690551i	$-0.757368\pi$
$421$	−12495.7	−1.44657	−0.723284	+	0.690551i	$0.757368\pi$
$422$	23120.9i	2.66708i
$423$	4476.67i	0.514570i
$424$	38370.2	4.39486
$425$	0	0
$426$	−9453.46	−1.07517
$427$	− 1705.58i	− 0.193299i
$428$	24595.8i	2.77776i
$429$	−820.012	−0.0922857
$430$	0	0
$431$	6075.01	0.678939	0.339470	−	0.940617i	$-0.389752\pi$
$431$	6075.01	0.678939	0.339470	+	0.940617i	$0.389752\pi$
$432$	6037.49i	0.672405i
$433$	5641.79i	0.626160i	0.949727	+	0.313080i	$0.101361\pi$
$433$	5641.79i	0.626160i	−0.949727	+	0.313080i	$0.898639\pi$
$434$	9161.01	1.01323
$435$	0	0
$436$	−39170.7	−4.30260
$437$	− 269.019i	− 0.0294484i
$438$	− 8301.26i	− 0.905593i
$439$	−10897.0	−1.18470	−0.592351	−	0.805680i	$-0.701800\pi$
$439$	−10897.0	−1.18470	−0.592351	+	0.805680i	$0.701800\pi$
$440$	0	0
$441$	−2553.71	−0.275748
$442$	2148.84i	0.231244i
$443$	− 7720.83i	− 0.828054i	−0.910265	−	0.414027i	$-0.864122\pi$
$443$	− 7720.83i	− 0.828054i	0.910265	−	0.414027i	$-0.135878\pi$
$444$	5917.66	0.632522
$445$	0	0
$446$	−7473.76	−0.793481
$447$	− 6141.04i	− 0.649802i
$448$	− 12552.5i	− 1.32378i
$449$	7473.86	0.785553	0.392776	−	0.919634i	$-0.371515\pi$
$449$	7473.86	0.785553	0.392776	+	0.919634i	$0.371515\pi$
$450$	0	0
$451$	5059.36	0.528240
$452$	− 24132.4i	− 2.51127i
$453$	− 1427.59i	− 0.148066i
$454$	6657.29	0.688198
$455$	0	0
$456$	−3309.91	−0.339914
$457$	− 11140.5i	− 1.14033i	−0.821529	−	0.570167i	$-0.806879\pi$
$457$	− 11140.5i	− 1.14033i	0.821529	−	0.570167i	$-0.193121\pi$
$458$	21102.6i	2.15296i
$459$	−430.434	−0.0437711
$460$	0	0
$461$	14328.8	1.44763	0.723817	−	0.689992i	$-0.242386\pi$
$461$	14328.8	1.44763	0.723817	+	0.689992i	$0.242386\pi$
$462$	1377.94i	0.138761i
$463$	11760.7i	1.18049i	0.807223	+	0.590246i	$0.200969\pi$
$463$	11760.7i	1.18049i	−0.807223	+	0.590246i	$0.799031\pi$
$464$	28744.4	2.87592
$465$	0	0
$466$	−17456.5	−1.73532
$467$	− 11854.9i	− 1.17469i	−0.809338	−	0.587343i	$-0.800174\pi$
$467$	− 11854.9i	− 1.17469i	0.809338	−	0.587343i	$-0.199826\pi$
$468$	4791.35i	0.473249i
$469$	−6625.77	−0.652345
$470$	0	0
$471$	1942.40	0.190023
$472$	42122.1i	4.10769i
$473$	− 714.613i	− 0.0694671i
$474$	16844.8	1.63229
$475$	0	0
$476$	2629.14	0.253165
$477$	4742.27i	0.455207i
$478$	− 2322.19i	− 0.222206i
$479$	1324.68	0.126359	0.0631796	−	0.998002i	$-0.479876\pi$
$479$	1324.68	0.126359	0.0631796	+	0.998002i	$0.479876\pi$
$480$	0	0
$481$	2287.84	0.216874
$482$	− 6678.34i	− 0.631100i
$483$	410.035i	0.0386278i
$484$	−2592.36	−0.243459
$485$	0	0
$486$	−1318.14	−0.123029
$487$	− 18636.4i	− 1.73408i	−0.498239	−	0.867040i	$-0.666020\pi$
$487$	− 18636.4i	− 1.73408i	0.498239	−	0.867040i	$-0.333980\pi$
$488$	− 16134.7i	− 1.49668i
$489$	3279.70	0.303299
$490$	0	0
$491$	124.552	0.0114480	0.00572398	−	0.999984i	$-0.498178\pi$
$491$	124.552	0.0114480	0.00572398	+	0.999984i	$0.498178\pi$
$492$	− 29562.0i	− 2.70886i
$493$	2049.29i	0.187212i
$494$	−2042.24	−0.186001
$495$	0	0
$496$	49059.2	4.44117
$497$	4471.75i	0.403592i
$498$	− 9865.04i	− 0.887677i
$499$	−10230.2	−0.917768	−0.458884	−	0.888496i	$-0.651751\pi$
$499$	−10230.2	−0.917768	−0.458884	+	0.888496i	$0.651751\pi$
$500$	0	0
$501$	−3369.76	−0.300499
$502$	− 15395.7i	− 1.36881i
$503$	5150.81i	0.456587i	0.973592	+	0.228294i	$0.0733146\pi$
$503$	5150.81i	0.456587i	−0.973592	+	0.228294i	$0.926685\pi$
$504$	5044.92	0.445870
$505$	0	0
$506$	−1059.46	−0.0930806
$507$	− 4738.60i	− 0.415086i
$508$	14168.1i	1.23741i
$509$	22.7715	0.00198296	0.000991481	−	1.00000i	$-0.499684\pi$
$509$	22.7715	0.00198296	0.000991481		1.00000i	$0.499684\pi$
$510$	0	0
$511$	−3926.73	−0.339938
$512$	− 10698.1i	− 0.923429i
$513$	− 409.081i	− 0.0352073i
$514$	−1855.19	−0.159201
$515$	0	0
$516$	−4175.50	−0.356233
$517$	5471.49i	0.465446i
$518$	− 3844.46i	− 0.326093i
$519$	138.037	0.0116747
$520$	0	0
$521$	21521.7	1.80976	0.904879	−	0.425669i	$-0.139961\pi$
$521$	21521.7	1.80976	0.904879	+	0.425669i	$0.139961\pi$
$522$	6275.63i	0.526201i
$523$	2923.36i	0.244416i	0.992504	+	0.122208i	$0.0389975\pi$
$523$	2923.36i	0.244416i	−0.992504	+	0.122208i	$0.961002\pi$
$524$	−13330.0	−1.11130
$525$	0	0
$526$	31977.0	2.65069
$527$	3497.60i	0.289104i
$528$	7379.15i	0.608213i
$529$	11851.7	0.974088
$530$	0	0
$531$	−5205.99	−0.425462
$532$	2498.71i	0.203633i
$533$	− 11429.0i	− 0.928792i
$534$	−381.464	−0.0309130
$535$	0	0
$536$	−62679.3	−5.05100
$537$	− 2495.93i	− 0.200572i
$538$	− 13540.3i	− 1.08507i
$539$	−3121.20	−0.249424
$540$	0	0
$541$	21272.8	1.69056	0.845278	−	0.534327i	$-0.179435\pi$
$541$	21272.8	1.69056	0.845278	+	0.534327i	$0.179435\pi$
$542$	12203.2i	0.967108i
$543$	5431.89i	0.429290i
$544$	10049.9	0.792067
$545$	0	0
$546$	3112.75	0.243980
$547$	18730.5i	1.46409i	0.681256	+	0.732046i	$0.261434\pi$
$547$	18730.5i	1.46409i	−0.681256	+	0.732046i	$0.738566\pi$
$548$	− 40124.5i	− 3.12780i
$549$	1994.12	0.155022
$550$	0	0
$551$	−1947.63	−0.150584
$552$	3878.91i	0.299089i
$553$	− 7968.04i	− 0.612723i
$554$	−22145.7	−1.69834
$555$	0	0
$556$	−20440.1	−1.55909
$557$	18885.0i	1.43659i	0.695736	+	0.718297i	$0.255078\pi$
$557$	18885.0i	1.43659i	−0.695736	+	0.718297i	$0.744922\pi$
$558$	10710.9i	0.812593i
$559$	−1614.30	−0.122143
$560$	0	0
$561$	−526.086	−0.0395925
$562$	5607.71i	0.420902i
$563$	10285.1i	0.769922i	0.922933	+	0.384961i	$0.125785\pi$
$563$	10285.1i	0.769922i	−0.922933	+	0.384961i	$0.874215\pi$
$564$	31970.0	2.38685
$565$	0	0
$566$	−42360.1	−3.14581
$567$	623.515i	0.0461820i
$568$	42302.5i	3.12495i
$569$	−18008.8	−1.32683	−0.663415	−	0.748251i	$-0.730894\pi$
$569$	−18008.8	−1.32683	−0.663415	+	0.748251i	$0.730894\pi$
$570$	0	0
$571$	−7010.79	−0.513822	−0.256911	−	0.966435i	$-0.582705\pi$
$571$	−7010.79	−0.513822	−0.256911	+	0.966435i	$0.582705\pi$
$572$	5856.10i	0.428070i
$573$	− 1376.69i	− 0.100370i
$574$	−19205.2	−1.39653
$575$	0	0
$576$	14676.1	1.06164
$577$	16398.9i	1.18318i	0.806240	+	0.591589i	$0.201499\pi$
$577$	16398.9i	1.18318i	−0.806240	+	0.591589i	$0.798501\pi$
$578$	− 25271.6i	− 1.81862i
$579$	5336.73	0.383052
$580$	0	0
$581$	−4666.44	−0.333213
$582$	11708.5i	0.833903i
$583$	5796.11i	0.411750i
$584$	−37146.6	−2.63208
$585$	0	0
$586$	−10573.3	−0.745354
$587$	− 12823.5i	− 0.901671i	−0.892607	−	0.450836i	$-0.851126\pi$
$587$	− 12823.5i	− 0.901671i	0.892607	−	0.450836i	$-0.148874\pi$
$588$	18237.2i	1.27907i
$589$	−3324.09	−0.232541
$590$	0	0
$591$	−15913.6	−1.10761
$592$	− 20587.9i	− 1.42932i
$593$	16899.5i	1.17029i	0.810929	+	0.585144i	$0.198962\pi$
$593$	16899.5i	1.17029i	−0.810929	+	0.585144i	$0.801038\pi$
$594$	−1611.06	−0.111284
$595$	0	0
$596$	−43856.1	−3.01412
$597$	− 15415.2i	− 1.05679i
$598$	2393.31i	0.163662i
$599$	15074.9	1.02829	0.514143	−	0.857704i	$-0.328110\pi$
$599$	15074.9	1.02829	0.514143	+	0.857704i	$0.328110\pi$
$600$	0	0
$601$	−11418.8	−0.775014	−0.387507	−	0.921867i	$-0.626664\pi$
$601$	−11418.8	−0.775014	−0.387507	+	0.921867i	$0.626664\pi$
$602$	2712.65i	0.183654i
$603$	− 7746.71i	− 0.523168i
$604$	−10195.1	−0.686809
$605$	0	0
$606$	−30451.7	−2.04128
$607$	17952.8i	1.20046i	0.799826	+	0.600232i	$0.204925\pi$
$607$	17952.8i	1.20046i	−0.799826	+	0.600232i	$0.795075\pi$
$608$	9551.30i	0.637100i
$609$	2968.54	0.197523
$610$	0	0
$611$	12360.0	0.818384
$612$	3073.94i	0.203033i
$613$	12528.9i	0.825507i	0.910843	+	0.412753i	$0.135433\pi$
$613$	12528.9i	0.825507i	−0.910843	+	0.412753i	$0.864567\pi$
$614$	−12823.8	−0.842878
$615$	0	0
$616$	6166.01	0.403305
$617$	8586.10i	0.560232i	0.959966	+	0.280116i	$0.0903729\pi$
$617$	8586.10i	0.560232i	−0.959966	+	0.280116i	$0.909627\pi$
$618$	6977.09i	0.454142i
$619$	−18415.4	−1.19576	−0.597882	−	0.801584i	$-0.703991\pi$
$619$	−18415.4	−1.19576	−0.597882	+	0.801584i	$0.703991\pi$
$620$	0	0
$621$	−479.404	−0.0309788
$622$	10790.1i	0.695569i
$623$	180.443i	0.0116040i
$624$	16669.4	1.06941
$625$	0	0
$626$	−21039.6	−1.34331
$627$	− 499.988i	− 0.0318462i
$628$	− 13871.6i	− 0.881427i
$629$	1467.79	0.0930436
$630$	0	0
$631$	2374.38	0.149798	0.0748989	−	0.997191i	$-0.476137\pi$
$631$	2374.38	0.149798	0.0748989	+	0.997191i	$0.476137\pi$
$632$	− 75377.1i	− 4.74421i
$633$	12787.1i	0.802909i
$634$	−15805.3	−0.990080
$635$	0	0
$636$	33866.8	2.11149
$637$	7050.74i	0.438557i
$638$	7670.21i	0.475966i
$639$	−5228.27	−0.323673
$640$	0	0
$641$	11086.0	0.683104	0.341552	−	0.939863i	$-0.389047\pi$
$641$	11086.0	0.683104	0.341552	+	0.939863i	$0.389047\pi$
$642$	18682.1i	1.14848i
$643$	19934.1i	1.22259i	0.791403	+	0.611294i	$0.209351\pi$
$643$	19934.1i	1.22259i	−0.791403	+	0.611294i	$0.790649\pi$
$644$	2928.26	0.179176
$645$	0	0
$646$	−1310.21	−0.0797983
$647$	30634.8i	1.86148i	0.365684	+	0.930739i	$0.380835\pi$
$647$	30634.8i	1.86148i	−0.365684	+	0.930739i	$0.619165\pi$
$648$	5898.41i	0.357579i
$649$	−6362.87	−0.384845
$650$	0	0
$651$	5066.53	0.305028
$652$	− 23421.9i	− 1.40686i
$653$	− 9818.07i	− 0.588378i	−0.955747	−	0.294189i	$-0.904951\pi$
$653$	− 9818.07i	− 0.588378i	0.955747	−	0.294189i	$-0.0950494\pi$
$654$	−29752.7	−1.77894
$655$	0	0
$656$	−102848.	−6.12125
$657$	− 4591.04i	− 0.272624i
$658$	− 20769.6i	− 1.23052i
$659$	−16478.5	−0.974070	−0.487035	−	0.873383i	$-0.661922\pi$
$659$	−16478.5	−0.974070	−0.487035	+	0.873383i	$0.661922\pi$
$660$	0	0
$661$	2958.12	0.174066	0.0870328	−	0.996205i	$-0.472262\pi$
$661$	2958.12	0.174066	0.0870328	+	0.996205i	$0.472262\pi$
$662$	− 43962.1i	− 2.58102i
$663$	1188.42i	0.0696146i
$664$	−44144.2	−2.58001
$665$	0	0
$666$	4494.86	0.261520
$667$	2282.44i	0.132498i
$668$	24065.1i	1.39387i
$669$	−4133.39	−0.238873
$670$	0	0
$671$	2437.26	0.140223
$672$	− 14558.0i	− 0.835692i
$673$	− 29960.3i	− 1.71602i	−0.513630	−	0.858012i	$-0.671700\pi$
$673$	− 29960.3i	− 1.71602i	0.513630	−	0.858012i	$-0.328300\pi$
$674$	−32108.2	−1.83496
$675$	0	0
$676$	−33840.6	−1.92539
$677$	4514.73i	0.256300i	0.991755	+	0.128150i	$0.0409039\pi$
$677$	4514.73i	0.256300i	−0.991755	+	0.128150i	$0.959096\pi$
$678$	− 18330.2i	− 1.03830i
$679$	5538.43	0.313027
$680$	0	0
$681$	3681.84	0.207178
$682$	13091.0i	0.735018i
$683$	− 13555.7i	− 0.759438i	−0.925102	−	0.379719i	$-0.876021\pi$
$683$	− 13555.7i	− 0.759438i	0.925102	−	0.379719i	$-0.123979\pi$
$684$	−2921.44	−0.163310
$685$	0	0
$686$	26170.2	1.45653
$687$	11670.8i	0.648137i
$688$	14526.8i	0.804986i
$689$	13093.3	0.723972
$690$	0	0
$691$	−11471.3	−0.631535	−0.315768	−	0.948837i	$-0.602262\pi$
$691$	−11471.3	−0.631535	−0.315768	+	0.948837i	$0.602262\pi$
$692$	− 985.787i	− 0.0541532i
$693$	762.074i	0.0417731i
$694$	46327.8	2.53398
$695$	0	0
$696$	28082.2	1.52939
$697$	− 7332.40i	− 0.398471i
$698$	5084.52i	0.275719i
$699$	−9654.41	−0.522408
$700$	0	0
$701$	22229.0	1.19769	0.598843	−	0.800866i	$-0.295627\pi$
$701$	22229.0	1.19769	0.598843	+	0.800866i	$0.295627\pi$
$702$	3639.35i	0.195667i
$703$	1394.97i	0.0748397i
$704$	17937.5	0.960292
$705$	0	0
$706$	−1145.19	−0.0610480
$707$	14404.5i	0.766248i
$708$	37178.4i	1.97352i
$709$	15081.2	0.798851	0.399426	−	0.916766i	$-0.369210\pi$
$709$	15081.2	0.798851	0.399426	+	0.916766i	$0.369210\pi$
$710$	0	0
$711$	9316.06	0.491392
$712$	1706.98i	0.0898480i
$713$	3895.52i	0.204612i
$714$	1997.01	0.104673
$715$	0	0
$716$	−17824.6	−0.930358
$717$	− 1284.30i	− 0.0668938i
$718$	− 7465.71i	− 0.388047i
$719$	7399.80	0.383819	0.191910	−	0.981413i	$-0.438532\pi$
$719$	7399.80	0.383819	0.191910	+	0.981413i	$0.438532\pi$
$720$	0	0
$721$	3300.36	0.170474
$722$	35960.9i	1.85364i
$723$	− 3693.48i	− 0.189989i
$724$	38791.7	1.99127
$725$	0	0
$726$	−1969.07	−0.100660
$727$	− 1705.77i	− 0.0870202i	−0.999053	−	0.0435101i	$-0.986146\pi$
$727$	− 1705.77i	− 0.0870202i	0.999053	−	0.0435101i	$-0.0138541\pi$
$728$	− 13928.9i	− 0.709122i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−1035.67	−0.0524017
$732$	− 14241.0i	− 0.719074i
$733$	37122.6i	1.87061i	0.353847	+	0.935303i	$0.384873\pi$
$733$	37122.6i	1.87061i	−0.353847	+	0.935303i	$0.615127\pi$
$734$	−5589.60	−0.281084
$735$	0	0
$736$	11193.2	0.560581
$737$	− 9468.20i	− 0.473223i
$738$	− 22454.3i	− 1.11999i
$739$	−34256.3	−1.70520	−0.852598	−	0.522568i	$-0.824974\pi$
$739$	−34256.3	−1.70520	−0.852598	+	0.522568i	$0.824974\pi$
$740$	0	0
$741$	−1129.47	−0.0559945
$742$	− 22001.9i	− 1.08856i
$743$	1567.88i	0.0774160i	0.999251	+	0.0387080i	$0.0123242\pi$
$743$	1567.88i	0.0774160i	−0.999251	+	0.0387080i	$0.987676\pi$
$744$	47929.0	2.36178
$745$	0	0
$746$	50801.9	2.49328
$747$	− 5455.90i	− 0.267230i
$748$	3757.03i	0.183651i
$749$	8837.17	0.431112
$750$	0	0
$751$	−955.613	−0.0464325	−0.0232163	−	0.999730i	$-0.507391\pi$
$751$	−955.613	−0.0464325	−0.0232163	+	0.999730i	$0.507391\pi$
$752$	− 111226.i	− 5.39360i
$753$	− 8514.65i	− 0.412073i
$754$	17326.9	0.836881
$755$	0	0
$756$	4452.82	0.214216
$757$	14015.4i	0.672918i	0.941698	+	0.336459i	$0.109229\pi$
$757$	14015.4i	0.672918i	−0.941698	+	0.336459i	$0.890771\pi$
$758$	− 38623.2i	− 1.85073i
$759$	−585.938	−0.0280214
$760$	0	0
$761$	−36271.0	−1.72776	−0.863879	−	0.503699i	$-0.831972\pi$
$761$	−36271.0	−1.72776	−0.863879	+	0.503699i	$0.831972\pi$
$762$	10761.6i	0.511616i
$763$	14073.9i	0.667770i
$764$	−9831.63	−0.465571
$765$	0	0
$766$	−6311.64	−0.297714
$767$	14373.6i	0.676665i
$768$	− 22739.7i	− 1.06842i
$769$	18163.6	0.851749	0.425874	−	0.904782i	$-0.359967\pi$
$769$	18163.6	0.851749	0.425874	+	0.904782i	$0.359967\pi$
$770$	0	0
$771$	−1026.02	−0.0479264
$772$	− 38112.1i	− 1.77680i
$773$	− 8345.65i	− 0.388321i	−0.980970	−	0.194160i	$-0.937802\pi$
$773$	− 8345.65i	− 0.388321i	0.980970	−	0.194160i	$-0.0621982\pi$
$774$	−3171.57	−0.147287
$775$	0	0
$776$	52393.2	2.42372
$777$	− 2126.19i	− 0.0981683i
$778$	− 59445.7i	− 2.73937i
$779$	6968.65	0.320510
$780$	0	0
$781$	−6390.11	−0.292774
$782$	1535.45i	0.0702142i
$783$	3470.76i	0.158410i
$784$	63448.5	2.89033
$785$	0	0
$786$	−10125.0	−0.459475
$787$	22996.2i	1.04158i	0.853684	+	0.520791i	$0.174363\pi$
$787$	22996.2i	1.04158i	−0.853684	+	0.520791i	$0.825637\pi$
$788$	113647.i	5.13768i
$789$	17685.0	0.797975
$790$	0	0
$791$	−8670.69	−0.389752
$792$	7209.17i	0.323443i
$793$	− 5505.75i	− 0.246551i
$794$	11782.4	0.526626
$795$	0	0
$796$	−110087.	−4.90194
$797$	2743.82i	0.121946i	0.998139	+	0.0609730i	$0.0194204\pi$
$797$	2743.82i	0.121946i	−0.998139	+	0.0609730i	$0.980580\pi$
$798$	1897.94i	0.0841934i
$799$	7929.68	0.351104
$800$	0	0
$801$	−210.970	−0.00930619
$802$	− 42477.1i	− 1.87022i
$803$	− 5611.28i	− 0.246597i
$804$	−55322.9	−2.42673
$805$	0	0
$806$	29572.5	1.29237
$807$	− 7488.53i	− 0.326653i
$808$	136266.i	5.93293i
$809$	−41241.7	−1.79231	−0.896156	−	0.443738i	$-0.853652\pi$
$809$	−41241.7	−1.79231	−0.896156	+	0.443738i	$0.853652\pi$
$810$	0	0
$811$	12832.9	0.555641	0.277820	−	0.960633i	$-0.410388\pi$
$811$	12832.9	0.555641	0.277820	+	0.960633i	$0.410388\pi$
$812$	− 21199.8i	− 0.916215i
$813$	6749.03i	0.291142i
$814$	5493.72	0.236554
$815$	0	0
$816$	10694.4	0.458798
$817$	− 984.292i	− 0.0421493i
$818$	− 58210.4i	− 2.48812i
$819$	1721.51	0.0734488
$820$	0	0
$821$	16368.5	0.695817	0.347908	−	0.937529i	$-0.386892\pi$
$821$	16368.5	0.695817	0.347908	+	0.937529i	$0.386892\pi$
$822$	− 30477.3i	− 1.29321i
$823$	3869.53i	0.163892i	0.996637	+	0.0819461i	$0.0261135\pi$
$823$	3869.53i	0.163892i	−0.996637	+	0.0819461i	$0.973886\pi$
$824$	31221.2	1.31995
$825$	0	0
$826$	24153.3	1.01743
$827$	− 7388.69i	− 0.310677i	−0.987861	−	0.155339i	$-0.950353\pi$
$827$	− 7388.69i	− 0.310677i	0.987861	−	0.155339i	$-0.0496469\pi$
$828$	3423.65i	0.143696i
$829$	−23990.1	−1.00508	−0.502539	−	0.864554i	$-0.667601\pi$
$829$	−23990.1	−1.00508	−0.502539	+	0.864554i	$0.667601\pi$
$830$	0	0
$831$	−12247.8	−0.511276
$832$	− 40520.6i	− 1.68846i
$833$	4523.47i	0.188150i
$834$	−15525.7	−0.644616
$835$	0	0
$836$	−3570.65	−0.147720
$837$	5923.68i	0.244626i
$838$	− 39684.5i	− 1.63589i
$839$	18228.3	0.750074	0.375037	−	0.927010i	$-0.377630\pi$
$839$	18228.3	0.750074	0.375037	+	0.927010i	$0.377630\pi$
$840$	0	0
$841$	−7864.78	−0.322472
$842$	67782.3i	2.77427i
$843$	3101.36i	0.126710i
$844$	91318.7	3.72431
$845$	0	0
$846$	24283.4	0.986855
$847$	931.424i	0.0377852i
$848$	− 117825.i	− 4.77137i
$849$	−23427.4	−0.947028
$850$	0	0
$851$	1634.77	0.0658511
$852$	37337.6i	1.50137i
$853$	− 21737.3i	− 0.872534i	−0.899817	−	0.436267i	$-0.856300\pi$
$853$	− 21737.3i	− 0.872534i	0.899817	−	0.436267i	$-0.143700\pi$
$854$	−9251.79	−0.370714
$855$	0	0
$856$	83599.0	3.33803
$857$	18712.2i	0.745852i	0.927861	+	0.372926i	$0.121645\pi$
$857$	18712.2i	0.745852i	−0.927861	+	0.372926i	$0.878355\pi$
$858$	4448.10i	0.176988i
$859$	−30527.6	−1.21256	−0.606279	−	0.795252i	$-0.707339\pi$
$859$	−30527.6	−1.21256	−0.606279	+	0.795252i	$0.707339\pi$
$860$	0	0
$861$	−10621.5	−0.420418
$862$	− 32953.4i	− 1.30209i
$863$	10906.4i	0.430196i	0.976592	+	0.215098i	$0.0690071\pi$
$863$	10906.4i	0.430196i	−0.976592	+	0.215098i	$0.930993\pi$
$864$	17020.8	0.670209
$865$	0	0
$866$	30603.5	1.20087
$867$	− 13976.6i	− 0.547484i
$868$	− 36182.5i	− 1.41488i
$869$	11386.3	0.444481
$870$	0	0
$871$	−21388.5	−0.832058
$872$	133138.i	5.17043i
$873$	6475.41i	0.251042i
$874$	−1459.28	−0.0564768
$875$	0	0
$876$	−32786.8	−1.26457
$877$	− 21770.9i	− 0.838256i	−0.907927	−	0.419128i	$-0.862336\pi$
$877$	− 21770.9i	− 0.838256i	0.907927	−	0.419128i	$-0.137664\pi$
$878$	59109.8i	2.27205i
$879$	−5847.58	−0.224384
$880$	0	0
$881$	47206.9	1.80527	0.902634	−	0.430409i	$-0.141631\pi$
$881$	47206.9	1.80527	0.902634	+	0.430409i	$0.141631\pi$
$882$	13852.4i	0.528837i
$883$	− 6059.68i	− 0.230945i	−0.993311	−	0.115473i	$-0.963162\pi$
$883$	− 6059.68i	− 0.230945i	0.993311	−	0.115473i	$-0.0368382\pi$
$884$	8487.09	0.322909
$885$	0	0
$886$	−41881.1	−1.58806
$887$	37130.2i	1.40553i	0.711420	+	0.702767i	$0.248052\pi$
$887$	37130.2i	1.40553i	−0.711420	+	0.702767i	$0.751948\pi$
$888$	− 20113.6i	− 0.760101i
$889$	5090.53	0.192048
$890$	0	0
$891$	−891.000	−0.0335013
$892$	29518.5i	1.10802i
$893$	7536.30i	0.282410i
$894$	−33311.6	−1.24621
$895$	0	0
$896$	−29269.1	−1.09131
$897$	1323.63i	0.0492694i
$898$	− 40541.4i	− 1.50655i
$899$	28202.5	1.04628
$900$	0	0
$901$	8400.15	0.310599
$902$	− 27444.1i	− 1.01307i
$903$	1500.24i	0.0552879i
$904$	−82024.1	−3.01779
$905$	0	0
$906$	−7743.85	−0.283965
$907$	− 1182.94i	− 0.0433064i	−0.999766	−	0.0216532i	$-0.993107\pi$
$907$	− 1182.94i	− 0.0433064i	0.999766	−	0.0216532i	$-0.00689297\pi$
$908$	− 26293.8i	− 0.961001i
$909$	−16841.4	−0.614516
$910$	0	0
$911$	37676.4	1.37022	0.685112	−	0.728438i	$-0.259753\pi$
$911$	37676.4	1.37022	0.685112	+	0.728438i	$0.259753\pi$
$912$	10163.9i	0.369035i
$913$	− 6668.32i	− 0.241719i
$914$	−60431.1	−2.18696
$915$	0	0
$916$	83347.1	3.00640
$917$	4789.41i	0.172476i
$918$	2334.86i	0.0839454i
$919$	−8697.82	−0.312203	−0.156101	−	0.987741i	$-0.549893\pi$
$919$	−8697.82	−0.312203	−0.156101	+	0.987741i	$0.549893\pi$
$920$	0	0
$921$	−7092.26	−0.253744
$922$	− 77725.7i	− 2.77631i
$923$	14435.2i	0.514778i
$924$	5442.33	0.193766
$925$	0	0
$926$	63795.3	2.26398
$927$	3858.71i	0.136717i
$928$	− 81036.0i	− 2.86653i
$929$	17247.5	0.609119	0.304559	−	0.952493i	$-0.401491\pi$
$929$	17247.5	0.609119	0.304559	+	0.952493i	$0.401491\pi$
$930$	0	0
$931$	−4299.06	−0.151338
$932$	68946.7i	2.42320i
$933$	5967.51i	0.209397i
$934$	−64306.0	−2.25284
$935$	0	0
$936$	16285.4	0.568702
$937$	41812.4i	1.45779i	0.684624	+	0.728896i	$0.259966\pi$
$937$	41812.4i	1.45779i	−0.684624	+	0.728896i	$0.740034\pi$
$938$	35941.0i	1.25108i
$939$	−11636.0	−0.404395
$940$	0	0
$941$	37655.9	1.30451	0.652257	−	0.757998i	$-0.273822\pi$
$941$	37655.9	1.30451	0.652257	+	0.757998i	$0.273822\pi$
$942$	− 10536.4i	− 0.364431i
$943$	− 8166.60i	− 0.282016i
$944$	129346.	4.45959
$945$	0	0
$946$	−3876.37	−0.133226
$947$	21244.4i	0.728986i	0.931206	+	0.364493i	$0.118758\pi$
$947$	21244.4i	0.728986i	−0.931206	+	0.364493i	$0.881242\pi$
$948$	− 66530.4i	− 2.27933i
$949$	−12675.8	−0.433587
$950$	0	0
$951$	−8741.20	−0.298058
$952$	− 8936.24i	− 0.304228i
$953$	1324.27i	0.0450130i	0.999747	+	0.0225065i	$0.00716465\pi$
$953$	1324.27i	0.0450130i	−0.999747	+	0.0225065i	$0.992835\pi$
$954$	25724.1	0.873007
$955$	0	0
$956$	−9171.77	−0.310289
$957$	4242.04i	0.143287i
$958$	− 7185.62i	− 0.242335i
$959$	−14416.6	−0.485439
$960$	0	0
$961$	18343.4	0.615735
$962$	− 12410.2i	− 0.415927i
$963$	10332.2i	0.345744i
$964$	−26376.9	−0.881268
$965$	0	0
$966$	2224.21	0.0740815
$967$	− 52267.1i	− 1.73815i	−0.494676	−	0.869077i	$-0.664713\pi$
$967$	− 52267.1i	− 1.73815i	0.494676	−	0.869077i	$-0.335287\pi$
$968$	8811.20i	0.292565i
$969$	−724.619	−0.0240228
$970$	0	0
$971$	−52489.8	−1.73479	−0.867394	−	0.497622i	$-0.834207\pi$
$971$	−52489.8	−1.73479	−0.867394	+	0.497622i	$0.834207\pi$
$972$	5206.14i	0.171797i
$973$	7344.07i	0.241973i
$974$	−101092.	−3.32566
$975$	0	0
$976$	−49545.3	−1.62490
$977$	− 8324.11i	− 0.272581i	−0.990669	−	0.136291i	$-0.956482\pi$
$977$	− 8324.11i	− 0.272581i	0.990669	−	0.136291i	$-0.0435181\pi$
$978$	− 17790.5i	− 0.581675i
$979$	−257.852	−0.00841777
$980$	0	0
$981$	−16454.9	−0.535539
$982$	− 675.623i	− 0.0219552i
$983$	− 44407.1i	− 1.44086i	−0.693527	−	0.720431i	$-0.743944\pi$
$983$	− 44407.1i	− 1.44086i	0.693527	−	0.720431i	$-0.256056\pi$
$984$	−100479.	−3.25523
$985$	0	0
$986$	11116.2	0.359039
$987$	− 11486.7i	− 0.370442i
$988$	8066.05i	0.259732i
$989$	−1153.50	−0.0370870
$990$	0	0
$991$	−45124.7	−1.44645	−0.723226	−	0.690612i	$-0.757341\pi$
$991$	−45124.7	−1.44645	−0.723226	+	0.690612i	$0.757341\pi$
$992$	− 138307.i	− 4.42667i
$993$	− 24313.4i	− 0.777001i
$994$	24256.7	0.774020
$995$	0	0
$996$	−38963.2	−1.23955
$997$	− 5480.61i	− 0.174095i	−0.996204	−	0.0870474i	$-0.972257\pi$
$997$	− 5480.61i	− 0.174095i	0.996204	−	0.0870474i	$-0.0277432\pi$
$998$	55492.9i	1.76012i
$999$	2485.90	0.0787291

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.c.h.199.1		4
5.2	odd	4		33.4.a.c.1.2	✓	2
5.3	odd	4		825.4.a.l.1.1		2
5.4	even	2	inner	825.4.c.h.199.4		4
15.2	even	4		99.4.a.f.1.1		2
15.8	even	4		2475.4.a.p.1.2		2
20.7	even	4		528.4.a.p.1.1		2
35.27	even	4		1617.4.a.k.1.2		2
40.27	even	4		2112.4.a.bg.1.2		2
40.37	odd	4		2112.4.a.bn.1.2		2
55.32	even	4		363.4.a.i.1.1		2
60.47	odd	4		1584.4.a.bj.1.2		2
165.32	odd	4		1089.4.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.c.1.2	✓	2	5.2	odd	4
99.4.a.f.1.1		2	15.2	even	4
363.4.a.i.1.1		2	55.32	even	4
528.4.a.p.1.1		2	20.7	even	4
825.4.a.l.1.1		2	5.3	odd	4
825.4.c.h.199.1		4	1.1	even	1	trivial
825.4.c.h.199.4		4	5.4	even	2	inner
1089.4.a.u.1.2		2	165.32	odd	4
1584.4.a.bj.1.2		2	60.47	odd	4
1617.4.a.k.1.2		2	35.27	even	4
2112.4.a.bg.1.2		2	40.27	even	4
2112.4.a.bn.1.2		2	40.37	odd	4
2475.4.a.p.1.2		2	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.42443i q^{2} -3.00000i q^{3} -21.4244 q^{4} -16.2733 q^{6} +7.69772i q^{7} +72.8199i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q-5.42443i q^{2} -3.00000i q^{3} -21.4244 q^{4} -16.2733 q^{6} +7.69772i q^{7} +72.8199i q^{8} -9.00000 q^{9} -11.0000 q^{11} +64.2733i q^{12} +24.8489i q^{13} +41.7557 q^{14} +223.611 q^{16} +15.9420i q^{17} +48.8199i q^{18} -15.1511 q^{19} +23.0931 q^{21} +59.6687i q^{22} +17.7557i q^{23} +218.460 q^{24} +134.791 q^{26} +27.0000i q^{27} -164.919i q^{28} +128.547 q^{29} +219.395 q^{31} -630.402i q^{32} +33.0000i q^{33} +86.4763 q^{34} +192.820 q^{36} -92.0703i q^{37} +82.1863i q^{38} +74.5466 q^{39} -459.942 q^{41} -125.267i q^{42} +64.9648i q^{43} +235.669 q^{44} +96.3146 q^{46} -497.408i q^{47} -670.832i q^{48} +283.745 q^{49} +47.8260 q^{51} -532.373i q^{52} -526.919i q^{53} +146.460 q^{54} -560.547 q^{56} +45.4534i q^{57} -697.292i q^{58} +578.443 q^{59} -221.569 q^{61} -1190.09i q^{62} -69.2794i q^{63} -1630.68 q^{64} +179.006 q^{66} +860.745i q^{67} -341.548i q^{68} +53.2671 q^{69} +580.919 q^{71} -655.379i q^{72} +510.116i q^{73} -499.429 q^{74} +324.605 q^{76} -84.6749i q^{77} -404.373i q^{78} -1035.12 q^{79} +81.0000 q^{81} +2494.92i q^{82} +606.211i q^{83} -494.757 q^{84} +352.397 q^{86} -385.640i q^{87} -801.018i q^{88} +23.4411 q^{89} -191.279 q^{91} -380.406i q^{92} -658.186i q^{93} -2698.15 q^{94} -1891.20 q^{96} -719.490i q^{97} -1539.16i q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 66 q^{4} - 6 q^{6} - 36 q^{9}+O(q^{10}) \) 4 * q - 66 * q^4 - 6 * q^6 - 36 * q^9	\( 4 q - 66 q^{4} - 6 q^{6} - 36 q^{9} - 44 q^{11} + 364 q^{14} + 402 q^{16} - 100 q^{19} - 144 q^{21} + 342 q^{24} + 224 q^{26} + 396 q^{29} + 720 q^{31} + 1252 q^{34} + 594 q^{36} + 180 q^{39} - 1564 q^{41} + 726 q^{44} - 836 q^{46} - 756 q^{49} - 636 q^{51} + 54 q^{54} - 2124 q^{56} + 344 q^{59} - 1556 q^{61} - 1618 q^{64} + 66 q^{66} + 804 q^{69} + 1260 q^{71} - 4716 q^{74} + 1456 q^{76} - 1304 q^{79} + 324 q^{81} + 1212 q^{84} - 2136 q^{86} + 1512 q^{89} - 56 q^{91} - 7444 q^{94} - 5142 q^{96} + 396 q^{99}+O(q^{100}) \) 4 * q - 66 * q^4 - 6 * q^6 - 36 * q^9 - 44 * q^11 + 364 * q^14 + 402 * q^16 - 100 * q^19 - 144 * q^21 + 342 * q^24 + 224 * q^26 + 396 * q^29 + 720 * q^31 + 1252 * q^34 + 594 * q^36 + 180 * q^39 - 1564 * q^41 + 726 * q^44 - 836 * q^46 - 756 * q^49 - 636 * q^51 + 54 * q^54 - 2124 * q^56 + 344 * q^59 - 1556 * q^61 - 1618 * q^64 + 66 * q^66 + 804 * q^69 + 1260 * q^71 - 4716 * q^74 + 1456 * q^76 - 1304 * q^79 + 324 * q^81 + 1212 * q^84 - 2136 * q^86 + 1512 * q^89 - 56 * q^91 - 7444 * q^94 - 5142 * q^96 + 396 * q^99

Introduction

L-functions

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